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1. Draw a free body diagram for a wagon as it is being pulled by a person, with the handle pulled up at an angle. 2. Draw another free body diagram for a wagon being pushed with the handle at the same angle. 3. In which scenario is it easier to move the wagon? Explain using information about forces.
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Question: 1. Draw a free body diagram for a wagon as it is being pulled by a person, with the handle pulled up at an angle. 2. Draw another free body diagram for a wagon being pushed with the handle at the same angle. 3. In which scenario is it easier to move the wagon? Explain using information about forces.
Draw a free body diagram for a wagon as it is being pulled by a person, with the handle pulled up at an angle.
Draw another free body diagram for a wagon being pushed with the handle at the same angle.
In which scenario is it easier to move the wagon? Explain using information about forces.
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Walker HK, Hall WD, Hurst JW, editors. Clinical Methods: The History, Physical, and Laboratory Examinations. 3rd edition. Boston: Butterworths; 1990.
Clinical Methods: The History, Physical, and Laboratory Examinations. 3rd edition.
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Walker HK, Hall WD, Hurst JW, editors.
Boston: Butterworths; 1990.
Contents
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Chapter 51Episodic Neurologic Symptoms
David C. Good.
Definition
Intermittent neurologic symptoms comprise a group of complaints that may be associated with dysfunction of many organ systems, including the central nervous system, cardiovascular system, and vestibular apparatus. Intermittent metabolic disturbances and psychiatric problems may also result in neurologic symptoms. Despite these diverse etiologies, all the conditions discussed here share the following features: (1) They are intermittent; (2) they are recurrent; (3) they are usually brief, lasting minutes to hours; (4) the patient is usually asymptomatic between attacks; and (5) the symptoms are usually stereotyped for an individual patient.
Although intermittent neurologic symptoms often have a benign prognosis, some may be a manifestation of a serious condition. Multiple sclerosis, myasthenia gravis, and certain other neurologic illnesses may have intermittent symptoms. Most patients with these conditions have a more chronic course on which intermittent symptoms are superimposed, however; they will not be discussed further here.
Technique
Many conditions resulting in episodic neurologic symptoms are discussed in other chapters, and this chapter is intended as a general approach to a patient with these symptoms. Special problems include migrainous events versus partial seizures versus transient ischemic attacks (TIAs); syncope versus seizures; evaluation of dizziness; and evaluation of ill-defined symptoms.
When evaluating a patient with intermittent neurologic symptoms, it is reassuring that a careful approach to the problem usually results in a definite diagnosis, or at least a few reasonable hypotheses, which can be further evaluated. It is useful to remember the clinical axiom that "common things are common" and that seizures, syncope, TIAs, labyrinthine disorders, and hyperventilation are seen frequently in everyday practice. The most common clinical situation is for the patient to present for medical care because of the symptom itself. Occasionally, intermittent neurologic symptoms are discovered incidentally on review of systems in a patient presenting with an unrelated problem. With these points in mind, certain investigational strategies are helpful.
First, it is important to try to classify the patient's major symptom into one of the categories in Table 51.1, since this immediately narrows the diagnostic possibilities. This is accomplished by listening carefully to the patient's chief complaint. The most important things to determine are whether the patient had any focal neurologic symptoms or any alteration of consciousness. The history is of critical importance because the examination between events is often normal. It is always best to listen attentively to the patient tell the story in his or her own words, rather than ask specific questions too early. Invaluable diagnostic clues are usually identified by a careful listener. The patient should be encouraged to "start at the beginning" and relate the first symptom, then others in temporal sequence. Information about onset is very important, and if not volunteered, should be sought by direct questioning. It is very helpful to ask about precipitating situations or activities through such questions as these: "Is there any way you can tell a spell will occur?" or "Do you get any warnings?" or "Is there any way you can bring on a spell?" While the most obvious value of such questions is to detect the focal onset (aura) of seizures, the questions may also provide information about stimulus-precipitated seizures, exercise-induced syncope, vasovagal syncope, or cataplectic attacks caused by surprise or emotion. The patient should be asked how long it takes for symptoms to reach maximum intensity, and how long the entire event lasts. It is helpful to ask whether the patient has ever experienced such symptoms in the past. Some symptoms may recur intermittently over many years, including those caused by migraine, many kinds of seizures, or vasovagal syncope. Recent onset of symptoms may add a sense of clinical urgency in patients with syncope of cardiac origin or with transient ischemic attacks. A change in character or frequency of attacks suggests a progressive lesion, such as a cerebral neoplasm causing a changing seizure disorder. Associated symptoms may be critically important, but may require further questions: "Do you notice anything else?" or "Do you feel sick in any other way?" For example, a patient with Ménière's disease may notice ear fullness, tinnitus, or hearing loss associated with vertigo; a patient with a migrainous neurologic deficit may notice a headache and nausea as the deficit improves. Further details of each of these associated symptoms should be sought in the order they are noticed.
Table 51.1
Classification of Episodic Neurologic Symptoms.
If there is a question in the examiner's mind whether or not focal neurologic symptoms occurred, a review of neurologic symptoms should be undertaken. The patient should be quizzed as to whether the event was accompanied by any visual disturbance, numbness, weakness, clumsiness, or speech disturbance. If any such symptoms occurred, it should be determined if they "spread" to other areas and the rate of such spread. The patient should be questioned about a "postictal" period of unresponsiveness or confusion. In many cases, diagnostically crucial information regarding the spells must be obtained from a friend or family member. At this phase of the interview, it is usually apparent into which of the major categories the patient's symptoms fit, and it is appropriate to ask more detailed questions to differentiate among entities in the appropriate category.
Basic Science
The basic pathophysiology of syncope, epilepsy, transient ischemic attacks, migraine, and sleep disturbance is discussed in other chapters of this book, but a few points should be emphasized.
Transient ischemic attacks probably result from transient interruption of blood flow to a specific portion of the brain. There is reasonable evidence that most TIAs are due to embolization of platelet and fibrin debris from atherosclerotic plaques in the proximal portions of the large cerebral vessels, especially at the carotid bifurcation. This phenomenon can sometimes actually be visualized in retinal arteries in patients with amaurosis fugax. Bright cholesterol emboli or platelet-fibrin debris may be seen at the bifurcations of these vessels, which are branches of the ophthalmic artery, which in turn arises from the internal carotid. With time, these microemboli break up and pass distally. The concept of arterial laminar flow, whereby blood flowing along the arterial wall adjacent to a plaque is always distributed to the same area of brain tissue, is thought to explain the stereotyped nature of recurrent TIAs. It is important to understand, however, that there are many other causes of TIAs. For example, cardiac abnormalities of many types may result in cerebral emboli. Cerebral vasculitis caused by systemic lupus erythematosus, giant cell arteritis, or granulomatous angiitis may interrupt blood flow. Patients with thrombocytosis, erthyrocytosis, or leukocytosis may have arterial or capillary sludging with transient ischemic symptoms. Rarely, patients with multiple large vessel atherosclerotic occlusions may have tenuous collateral blood flow to certain areas of the brain, with transient ischemia caused by a drop in systemic blood pressure.
While it is generally accepted that seizures are due to abnormal spontaneous neuronal depolarization and spread of electrical activity, there are many etiologies. In primary generalized epilepsy, the abnormal discharges appear simultaneously in all head regions and may originate in a centrally located generator with diffuse projections, perhaps the thalamic reticular system. Impaired consciousness may be the first manifestation, and if motor signs occur, they are bilateral. Examples include primary tonic-clonic seizures, true petit mal seizures, and certain rarer types. The age of onset of this type of seizure is usually under 20 years. There is a strong hereditary predisposition, and most patients experience no warning or aura. Acquired epilepsy may occur at any age and may result from such varied insults as birth hypoxia, head trauma, brain tumors, or cerebrovascular accidents. In these cases, the abnormal electrical activity begins locally, often resulting in focal neurologic symptoms. These partial seizures may spread to become secondarily generalized, resulting in a tonic-clonic seizure. A focal seizure always implies focal brain dysfunction and should focus attention on questions and diagnostic tests designed to find the cause. The older the patient at onset of seizures, the more likely a clear etiology can be found. In patients over age 50, as many as 20% of patients with onset of seizures have a cerebral neoplasm. It should always be remembered that there are many toxic and metabolic causes for seizures, including acute hypoxia, hypoglycemia, hyperosmolar states, hypo or hypernatremia, hypocalcemia, uremia, and drug or alcohol withdrawal. History, examination, and lab studies should be directed at uncovering such etiologies in patients with a first seizure.
Migraine may be defined as a paroxysmal disturbance of cephalic neurovascular function, presenting as episodic headaches that may be associated with autonomic, visual, or neurologic symptoms of variable prominence. The pathophysiology of migraine is still not well understood. The widely held belief that the neurologic and visual events are caused by cerebral vasoconstriction and the subsequent headache result from vasodilation is certainly too simplistic. There are no clear data to support the hypothesis that migrainous neurologic events are due to vasoconstriction, and some researchers believe they are a cortical phenomenon of unknown etiology. Although cerebral blood flow is decreased during migrainous neurologic dysfunction, it is unclear if this is a primary event or a secondary event coupled to neuronal dysfunction.
Syncope is due to global cerebral hypoperfusion. The causes are therefore almost exclusively cardiovascular, and examination and laboratory studies should focus on causes of hypotension and decreased cardiac output, especially bradyarrhythmias and tachyarrhythmias. A neurologic cause is found in 10% or fewer patients with syncope, usually an unusual or unwitnessed seizure. Multiple high-grade vascular stenoses or transient increased intracranial pressure are very rare causes of syncope. Syncope should not be attributed to cerebrovascular disease unless accompanied by focal neurologic symptoms. Often, no clear etiology for syncope can be found, in which case a benign course is usually seen.
The symptoms seen in the narcolepsy syndrome are due to disordered sleep cycles causing inappropriate involuntary sleep during the day. Rapid eye movement (REM) sleep normally occurs after an orderly progression of sleep to progressively deeper stages, a process that takes 2 to 3 hours. Persons with narcolepsy often progress directly into REM sleep with concomitant dreaming and loss of muscle tone. This is thought to explain the phenomena of sleep paralysis and hypnagogic hallucinations. The loss of muscle tone during cataplectic attacks may also be an REM-related event. Patients with sleep apnea may have airway obstruction in the oropharynx, loss of central regulation of respiration, or both. They are chronically sleep deprived, since they invariably develop apnea as they reach deeper sleep stages, resulting in multiple brief nighttime arousals, which the patient does not recall. Consequently, the patient has involuntary episodes of daytime sleep.
Vertigo results when there is disturbance in the normal balance between the left and right vestibular systems and their respective brainstem connections. Acute loss of function or overactivity of one vestibular organ results in vertigo. Horizonal nystagmus is a frequent clinical accompaniment due to extensive connections between the vestibular nuclei and eye movement systems. Patients with vertigo on a "central" basis almost invariably have other brainstem symptoms and findings due to close proximity of the vestibular nuclei to other brainstem nuclei, as well as ascending and descending sensory and motor tracts.
Brief episodes of global cerebral hypoperfusion may result in presyncope, manifested by lightheadedness and other symptoms. Patients with chronic, ill-defined dizziness should be considered for possible psychiatric causes provided that ataxia, posterior column dysfunction, and drug or medication side effects have been excluded.
New data suggest that panic attacks may result from dysfunction of central autonomic regulation, for which there may be a familial predisposition. Since lactate and bicarbonate infusions sometimes reproduce symptoms in susceptible persons, it is also possible that a reduction in free ionized calcium causes symptoms. These findings have opened new possibilities for research and treatment. The causes of conversion reactions are best explained on a psychodynamic basis.
Clinical Significance
Patients with episodic neurologic symptoms may present a major diagnostic challenge. Sometimes symptoms are caused by underlying medical or neurologic conditions that require immediate treatment; on other occasions, episodic symptoms are nonspecific events or a manifestation of a psychiatric disturbance. With such a wide range of different illnesses producing similar symptoms, it is not surprising that the conscientious clinician can sometimes be baffled and frustrated. The problem is compounded by the fact that many patients have normal physical examinations. At times, it may be difficult to decide between initiating a lengthy and expensive diagnostic evaluation and merely reassuring the patient and providing careful clinical follow-up. Even when exhaustive diagnostic evaluations are undertaken, the etiology may not be found. Observation is preferable to empirical treatment.
The most common causes of episodic focal neurologic dysfunction are transient ischemic attacks (TIAs), migrainous phenomena, and partial seizures. While each of these entities has certain identifying characteristics (Table 51.2), in some circumstances diagnostic problems may occur. TIAs are "negative" neurologic events (characterized by loss of function), whereas partial seizures are "positive" or excitatory phenomena. Migrainous events are often "positive" (flickering lights) when they involve visual phenomena and "negative" (weakness, numbness) when other neurologic symptoms occur.
Table 51.2
Features of Common Episodic Focal Neurologic Symptoms.
Transient ischemic attacks usually do not present a diagnostic problem, except for confusion with migraine or when symptoms are vague or nonspecific. It is safest to define TIAs as a focal neurologic deficit that can be clearly localized to a specific cerebral vascular distribution. TIAs usually begin suddenly, reach maximum deficit quickly, and have no precipitating or aggravating factors. Although TIAs are defined as lasting less than 24 hours, most are less than 15 minutes in duration. Most TIAs occur in older individuals with atherosclerotic risk factors, especially hypertension. Symptoms of carotid TIAs include unilateral weakness or sensory disturbance, aphasia, unilateral neglect, or amaurosis fugax (fleeting blindness in one eye). Vertebrobasilar symptoms include vertigo, diplopia, dysarthria, homonymous hemianopsia, total (cortical) blindness, ataxia, and shifting or bilateral weakness or numbness. Between 12 and 40% of patients with TIAs will eventually suffer a completed stroke, with the greatest risk in the first few weeks. Patients with TIAs occurring in rapid succession (crescendo TIAs) are especially at risk.
Migraine poses a special problem in the diagnosis of episodic symptoms because its manifestations are so protean. When a characteristic headache is part of the attack, the diagnosis is simple. However, headache may be inconspicuous or absent. Clinically, it is important to understand that migrainous neurologic events usually begin insidiously and spread gradually over minutes, whereas TIAs and focal seizures are much more precipitous in onset. The most common migrainous events other than headache are visual, especially scintillating scotomas. The patient may describe these as flickering lights that gradually spread across the visual field, often taking the form of a jagged, irregular line ("fortification phenomena"). Other visual events, such as photomas (cloudlike scotomas) or homonymous hemianopsia, may occur. Occasionally, patients with migraine may experience spreading numbness, weakness, aphasia, or even brainstem symptoms. Headache usually begins in 20 to 30 minutes as these symptoms subside. Although onset is usually before middle age, migraine can change over time in terms of frequency and manifestations. It has been postulated that some older patients with what appear to be TIAs may actually have symptoms due to migraine. The diagnosis of a "late-life migraine accompaniment" should be made only when the symptoms exhibit a typical slow migraine "build-up," and there is no other identifiable cause. The patient should be closely questioned about a past history of intermittent throbbing headaches and about a family history of headache.
There are many kinds of partial seizures, but the pattern is usually stereotyped for the individual patient. Simple partial seizures consist of isolated motor, sensory, or visual symptoms that usually occur paroxysmally (without warning), build in intensity over seconds, and either disappear or progress to secondary generalization of the seizure discharge, resulting in a major motor seizure. It is therefore very important to ask if convulsions or episodes of loss of consciousness ever follow a focal neurologic symptom. Examples of symptoms seen in focal seizures include jerking of one side of the face, one arm, or one leg, which may rapidly "spread" to other areas. Focal sensory seizures may also occur, with spread in a similar fashion. Visual seizures may consist of simple geometric shapes when the focus is in the occipital region or complex visual phenomena, such as a scene or vision, when the focus is in the temporal lobe. At times, there is alteration of perception of visual stimuli such as macropsia (items appear magnified) or micropsia (items appear small). Other examples of partial seizures involving the temporal lobe include unusual odors (uncinate hallucinations), noises, or psychic phenomena, which may be simple (an emotion such as fear or a recurrent thought) or complex (déjà vu). Such symptoms are easily confused with psychiatric conditions. The intensity of the events and their stereotyped and paroxysmal nature should be helpful in identifying the events as partial seizures. Complex partial seizures (psychomotor seizures) are a very common form of partial seizures and often begin with one of the "temporal lobe" phenomena described above, progressing to clouding of consciousness and purposeless "automatic" motor activity of the face or limbs. Since partial seizures are usually acquired and suggest an underlying focal brain lesion, it is important to question the patient about a history of head injury, central nervous system infection, stroke, or any progressive neurologic symptoms suggestive of a cerebral neoplasm.
Intermittent loss of consciousness immediately suggests syncope or generalized seizures (Table 51.3). The best way to distinguish between these etiologies is to observe a spell. Since this is seldom possible, the physician must often rely on information from family or friends. The patient's own history of the event is also important, but may be incomplete. With a few exceptions (true petit mal in childhood), motor activity is prominent in seizures associated with loss of consciousness. Tonic-clonic seizures, especially if witnessed, are usually easily diagnosed. During major motor seizures, some patients will be incontinent, bite their tongue or cheek, or suffer other injuries. Most patients with syncope are limp, with no associated motor movements, although occasionally a few clonic jerks may be noticed. Rarely, a tonic–clonic seizure occurs. Preceding symptoms are extremely important. Patients with partial seizures that secondarily generalize may have an "aura" consisting of a specific neurologic symptom prior to loss of consciousness. Most patients with primary generalized epilepsy have no warning at all, however. Before a syncopal attack, many patients experience lightheadedness, weakness, blurred vision, or nausea. The patient should always be questioned regarding precipitating factors. Seizures are usually paroxysmal, but occasionally are precipitated by photic stimulation (strobe lights, watching television), hyperventilation, sleep deprivation, or drug or alcohol withdrawal. Some seizures occur only when falling asleep, during sleep, or when awakening. Syncope may be associated with similar or identical precipitating factors for each episode. Examples include assuming an erect posture (orthostatic hypotension) and increased vagal tone (cough, urination, or emotional event). Associated palpitations, chest pain, dyspnea, diaphoresis, or nausea should be sought. A history of bradyarrhythmias or tachyarrhythmias, ischemic heart disease, or syncope associated with exertion raises suspicion of syncope of cardiovascular origin. Almost all patients with major motor seizures will be unresponsive for a number of minutes during the seizure, followed by a period of disorientation and confusion that may last for hours. Patients may complain of muscle soreness and tenderness. Although patients with syncope may complain of feeling weak, they usually are quickly aware of their surroundings after a brief loss of consciousness lasting seconds or minutes.
Table 51.3
Features of Common Causes of Episodic Alterations of Consciousness.
Although listening carefully to the story related by the patient and witnesses will usually differentiate between syncope and seizures, at times the distinction is very difficult. In this situation, the patient may need a diagnostic evaluation for both possibilities. An electroencephalogram is sometimes helpful but is frequently normal interictally in patients with known seizure disorders. If no clear diagnosis can be established, the patient should be followed carefully for possible clues that may later permit accurate classification. It is better to underdiagnose epilepsy in uncertain cases, considering the medical and social consequences the diagnosis carries.
Sleep disturbances are a more uncommon cause of episodic altered consciousness. Patients with narcolepsy or sleep apnea have irresistible daytime sleepiness, often at inopportune or dangerous times. Patients will admit to falling asleep during conversation, while driving a car, and occasionally even when active. Some patients have episodes of "microsleep" during which they may perform semipurposeful acts for which they have no recall. These automatisms result in confusion with partial complex seizures. Narcolepsy usually begins under age 30, and is almost always associated with cataplexy, a generalized or focal loss of muscle tone that occurs with strong emotion. Patients should be asked if they have ever collapsed suddenly when frightened, angry, or in the midst of laughter. Less common symptoms associated with the narcolepsy syndrome are sleep paralysis (transient total paralysis while falling asleep or on awakening) and hypnagogic hallucinations (very realistic nightmares that occur as the person falls asleep). Although patients with sleep apnea have excessive daytime sleepiness, they lack the other components of the narcolepsy syndrome. The sleeping partner will frequently give a history of nighttime snoring, restless sleep, and frequent episodes of apnea during sleep. Polysomnography will help confirm the diagnosis in difficult cases. Patients with episodic alteration or loss of consciousness should also be questioned about prescription, over-the-counter, or illicit drugs that might be causing their symptoms.
Dizziness is a common and often confusing complaint, meaning different things to different people. It is especially critical to ascertain if the patient has a sensation of movement, since vertigo is usually the result of a labyrinthine disorder. Associated symptoms of tinnitus or hearing loss also suggest a "peripheral" labyrinthine problem. Most patients with acute vertigo prefer to lie still, since minimal movement exacerbates their discomfort. Nausea, vomiting, and ataxia often occur. Among the many causes of recurrent "peripheral" vertigo are Ménière's disease, vestibular neuronitis, and benign positional vertigo. Vertigo is occasionally associated with TIAs involving the cerebellum or brainstem. By generally accepted convention, however, isolated vertigo should not be considered due to brainstem dysfunction unless other brainstem symptoms or signs are present. The cause of nonspecific dizziness or "fuzzy-headedness" is often difficult to elucidate. Anxious patients may have a feeling of floating, detachment, or lightheadedness that is described as dizziness and may be associated with chronic hyperventilation. The onset is usually gradual, and the feeling often lasts for long periods of time. Patients with presyncope may complain of lightheadedness, which sometimes progresses to syncope. Finally, patients with posterior column disturbance or peripheral neuropathy may complain of dizziness, which is better characterized by unsteadiness, or "dizziness in the feet."
Patients with conversion reactions, hyperventilation syndrome, panic attacks, or depression may complain of complex neurologic symptoms. In addition to dizziness, patients may complain of intermittent sensory symptoms (tingling or numbness), weakness, or memory loss. Patients with conversion symptoms can usually be identified by the dramatic and sometimes bizarre nature of their intermittent symptoms. The history is often related with surprising detachment and unconcern called "la belle indifference." The patient usually has no insight into the psychiatric basis of the symptoms or the unusual nonphysiologic findings that often are present on the neurologic examination. Many patients have a previous history of conversion reactions, and a few patients have Briquet's syndrome, a history of multiple conversion reactions involving multiple organ systems over time, and often a history of multiple surgical procedures by well-meaning surgeons. The acute hyperventilation syndrome is usually recognized by anxiety, acro-oral paresthesias, a feeling of "not being able to take a deep enough breath," and, in its most severe form, carpal-pedal spasm. At times, the symptoms can be reproduced by asking the patient to hyperventilate for several minutes. Some patients with panic attacks can identify uncomfortable situations that precipitate their spells, but most cannot. Patients usually complain of severe apprehension, dizziness, chest discomfort, palpitations, or fear of "losing control." The onset is usually under age 40, the course is chronic and fluctuating, and many patients also have phobias. Occasionally patients with what appear to be panic attacks are actually having paroxysmal tachycardia, or autonomic dysfunction associated with pheochromocytoma, hypoglycemia, or complex partial seizures. Neurologic symptoms also occur in depressed patients, but are often more constant than episodic. Nondescript symptoms such as tiredness, weakness, or dizziness predominate.
Pseudoseizures are episodes that appear to be epileptic seizures, but have a psychiatric origin. They can often be distinguished from true seizures by their atypical motor activity, lack of postictal state, and associated "secondary gain." Occasionally, only combined EEG and videotelemetry during an attack will distinguish between true seizures and pseudoseizures, and some patients have both. Other psychiatric events that may be episodic include "temper tantrums and psychogenic amnesia. Questions directed at the patient's general adaptation, social functioning, and previous history of psychiatric difficulties are also helpful in separating patients with symptoms of psychiatric etiology from those due to other causes.
References
Abi-Samra FM, Fouad FM, Sweeney PJ, Maloney JD. Syncope: A practical diagnostic approach. In: Furlan AJ, ed. The heart and stroke. London: Springer-Verlag, 1987;249–83.
Adams RD, Victor M. Principles of Neurology. 4th ed. New York: McGraw-Hill, 1989.
Caplan LR. TIA's: We need to return to the question, "What is wrong with Mr. Jones?". Neurology. 1988;38:791–93. [PubMed: 3258971]
Delgado-Escueta AV, Treiman DM, Walsh GO. The treatable epilepsies. N Engl J Med. 1983;308:1508–14. [PubMed: 6406889]
Dreifuss FE. Classification and recognition of seizures. Clin Therapeutics 1985;240–45. [PubMed: 3986863]
Edmeads J. Complicated migraine and headache in cerebrovascular disease. Neurol Clin. 1983;1(2):385–97.
Friedman AP. Clinical approach to the patient with headache. Neurol Clin. 1983;1(2):361–68. [PubMed: 661418]
Guilleminault C, Dement WC. 235 cases of excessive daytime sleepiness. J Neurol Sci. 1977;31:13–27. [PubMed: 188992]
Kales A, Cadieux RJ, Soldatos CR, Bixler EO, Schweitzer PK, Prey WT, Vea-Bueno A. Narcolepsy-cataplexy I: clinical and electrophysiologic characteristics. Arch Neurol. 1982;39:164–68. [PubMed: 7065934]
Kapoor WN, Karpf MD, Wieand S, Peterson JR, Levey GS. A prospective evaluation and follow-up of patients with syncope. N Engl J Med. 1983;309:197–204. [PubMed: 6866032]
Levy DE. How transient are transient ischemic attacks? Neurology. 1988;38:674–77. [PubMed: 3362360]
Olesen J. The ischemic hypotheses of migraine. Arch Neurol. 1987;44:321–22. [PubMed: 3827683]
Peatfield RC. Can transient ischaemic attacks and classical migraine always be distinguished? Headache. 1987;27:240–43. [PubMed: 3597078]
Pedley TA. Differential diagnosis of episodic symptoms. Epilepsia. 1983;24(Suppl I):S31–S44. [PubMed: 6617595]
Porter RJ. Recognizing and classifying epileptic seizures and epileptic syndromes. Neurol Clin. 1986;4(3):495–508. [PubMed: 3092000]
Raskin NH. Headache. 2nd ed. New York: Churchill Livingstone, 1988.
Schomer DL. Partial epilepsy. N Engl J Med. 1983;309:536–39. [PubMed: 6410242]
Sheehan DV. Panic attacks and phobias. N Engl J Med. 1982;307:156–58. [PubMed: 7045660]
Theodore WH, Porter RJ, Penry JK. Complex partial seizures: Clinical characteristics and differential diagnosis. Neurology. 1983;33:1115–21. [PubMed: 6684245]
Towler IIMA, vertigo Br Med J. 1984;288(6432):1739–43. [PMC free article: PMC1441526] [PubMed: 6428526]
Troost BT. Dizziness and vertigo in vertebrobasilar disease. Stroke. 1980;11:301–3. , 413–15. [PubMed: 7394870]
Welch KMA. Migraine: A biobehavioral disorder. Arch Neurol. 1987;44:323–27. [PubMed: 3827684]
Copyright © 1990, Butterworth Publishers, a division of Reed Publishing.
Bookshelf ID: NBK374PMID: 21250215
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Good DC. Episodic Neurologic Symptoms. In: Walker HK, Hall WD, Hurst JW, editors. Clinical Methods: The History, Physical, and Laboratory Examinations. 3rd edition. Boston: Butterworths; 1990. Chapter 51.
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10902 | https://www.youtube.com/watch?v=uf0TagLyBho | Can you solve these 10 FUN RIDDLES?
English with Alex · engVid
1290000 subscribers
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Posted: 30 Dec 2023
Can you solve these 10 English riddles? Riddles are word puzzles. They are language games which use logic, but which also play with words in a way that is meant to trick you and make you say “A-ha!” when you figure out or learn their solution. In this video, there are 10 fun riddles to solve. Don’t worry! You will have time to think about them before you hear the answer. You can always pause the video if you want more time, too. If you enjoy these word puzzles, make sure to check out my resource page for 50 Fun English Riddles...yes, that’s 50!
Check out my website for tutoring, books, and more English resources:
More fun videos to challenge yourself:
10 English TONGUE TWISTERS to improve your speaking skills
15 English words that even native speakers mispronounce
In this video:
0:00 10 fun riddles
0:47 What word starts with E, ends with E, but only has one letter?
1:20 I have a head, and I have a tail, but I don't have a body.
1:58 What has hands, but can't clap?
2:17 What has many keys, but can't open a single lock?
2:47 Where does today come before yesterday?
3:25 Which month of the year has 28 days?
3:48 What has cities, but no houses; water, but no fish?
4:32 What goes up, but never comes down?
5:04 I sometimes run, but I never walk. What am I?
5:34 What can you catch, but can't throw?
5:56 – Conclusion –
94 comments
Transcript:
10 fun riddles Hey, everyone. I'm Alex. Thanks for clicking, and welcome to this video lesson on 10 fun English riddles. In this video, we are going to look at 10 English riddles, and I'm going to give you a chance to answer them before I give you the answer. So, riddles are word games. They are clever word puzzles that have a surprising but logical conclusion, a logical answer. They upset us sometimes, but they also delight us. And if you enjoy the riddles in this video, make sure you check out the resource that's attached to this video where you can see 50 English riddles. So, these are just 10. So, let's begin with the first What word starts with E, ends with E, but only has one letter? riddle. What word starts with E, ends with E, but only has one letter? You know this one. You know it. It's envelope. An envelope starts with E, ends with E, and oh, it has a letter in here. Let's check that out. And this is, yeah, it's the second riddle. Okay, let's check that out. So, I have a head and I have a tail, but I don't have a body. What I have a head, and I have a tail, but I don't have a body. am I? Hmm. You got it? You're a coin. Coins have heads, picture of a person, usually, and tails. This is a $2 Canadian coin. The tail side is a picture of a polar bear. So, heads or tails. Number three, which riddle am I going to do next? Okay, it's tails. I know which one I'm doing. What has hands but can't clap? Yeah, it's a clock. It's definitely What has hands, but can't clap? a clock. Oh, I see it's four o'clock here. So, let's move on to the fourth riddle. What What has many keys, but can't open a single lock? has many keys but can't open a single lock? A piano. That is correct. A piano has many keys. Now, I don't have a piano, so you will have to deal with my charades over here, but I do have a key. And actually, why don't we do the next one in another room? So, I'm going to get this key and go over there. Okay, next one. Where does today come before yesterday? Where does today come before yesterday? You got it? It is... Let me hold this up. The dictionary. The dictionary. I'm going to put this down. It's so heavy. Okay, and you know what? The number six comes before five, both numerically and in the dictionary. Think about it. Alphabetical F before S. Let's go on to number six. Which month of the year has 28 days? Let me check before I give you Which month of the year has 28 days? the answer. Hold on. January, February, March, April, May, June. All of them. Yeah, the answer is all of them. Every month of the year has at least 28 days. What has cities but no houses? What has cities, but no houses; water, but no fish? What is water but no fish? You got it? It's a map. It's a map. So, let's... Have you seen one of these recently? I think most of us, probably all of us, use digital maps now and map apps. But yeah, here's a classic map for you if you haven't seen one in a long time. And I don't know how I'm going to fold that up. And this is my dad's map, so he's going to hate me for not knowing how to fold it properly. I'll just put it here for now. What goes up What goes up, but never comes down? but never comes down? And I'll give you a little hint for this one. I'm going to stop. I know you hate it that I stopped and didn't finish the song. I sang "Happy Birthday." What goes up never comes down? Your age. Your age. It's a sad fact of life, people. Oh, God. Is this over yet? I sometimes run, but I never walk. What am I? I sometimes run, but I never walk. What am I? Yeah, your nose. Your nose. You can have a runny nose, but your nose can't walk, of course. So, yeah, your nose runs, doesn't walk. We have one more of these. Let's go. What can you catch, but can't throw? Magic. What can you catch but can't throw? A cold. Yeah, I'm sorry. Yeah, you can catch a cold when you get sick. Of course, we don't say throw a cold. So, how did you enjoy these 10 riddles? If you want more, Conclusion you can check out the resource that is attached to this video. There are 50 fun English riddles for you to share with your kids, share with your friends, share with your work colleagues, and you can just make people's brains twist and turn and just have fun with it because riddles are a great way to exercise your brain, and they're a great way for you to see how words work in different contexts. And if you enjoyed this video, make sure you do the quiz, actually, that is attached to it on www.engvid.com if you want to test and remember, help you remember the answers to these riddles. Do the quiz on www.engvid.com. Subscribe to my YouTube channel, turn on notifications, watch all of my videos, and come back tomorrow. We'll do some more fun stuff. I have a lot of useful things on vocabulary, grammar, writing, reading skills, listening, etc. It's all here. It's all there. So, until next time, thanks for clicking, and I'll talk to you again soon. Bye. Let me see if I can do this. Bye. No. Yep. |
10903 | https://www.youtube.com/watch?v=rn_C25fPOVw | Free body diagram with angled forces: worked example | AP Physics 1 | Khan Academy
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Sal draws a free body diagram for a box held stationary against a wall with a force at an angle theta.
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AP Physics 1 on Khan Academy: Meet one of our writers for AP¨ Physics, Sean. A physics teacher for seven years, Sean has taught AP¨ Physics 1, AP¨ Physics C, and Conceptual Physics. HeÕs also a former mechanical engineer. Sean is based in Boise, Idaho, and is a Khan Academy physics fellow, creating awesome new exercises and articles for AP¨ Physics.
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Transcript:
Buraya bir şəkil çəkmişik. Bizim burada bir blokumuz var, fərz edək ki, bu blok tamamilə
hərəkətsizdir, və sürtünməyə malik olan divara doğru itələnir. Aha, divarla blok arasında
sürtünmə mövcuddur. O, qiyməti F olan qüvvə tərəfindən itələnir və onun istiqaməti üfüqlə teta bucağını əmələ gətirir. İndi sizin videonu dayandırmağınızı
və bu blok üçün sərbəst cism diaqramını qurmağınızı, həmçinin buradakı qüvvənin şaquli və üfüqi komponentlərini və digər vacib məqamları daxil etməyinizi xaiş edirəm. Bu bloka təsir edən qravitasiya qüvvəsini də daxil edin. Bu bloka təsir edən sürtünmə qüvvəsini və həmçinin divarın bloka yönəltdiyi dayağın reaksiyanı da əlavə edin. Videonu dayandırın, və
dediklərimi etməyə çalışın. Gəlin, sərbəst cism diaqramını çəkməzdən əvvəl, qüvvəni şaquli və üfüqi komponentlərinə ayıraq. Birinci olaraq, onun şaquli komponentini tapaq. O bu şəkildə olardı, və onun üfüqi komponenti isə bu şəkildə olardı. Bəs onun şaquli komponentinin qiyməti
nəyə bərabərdir? Bu, bizə verilmiş bucağın
qarşısında yerləşir. Bu, teta bucağıdır, və şaquli komponent F sinus teta-ya bərabər olacaq. Əvvəlki videolardakı kimi
bu ifadəni düzbucaqlı üçbucağın triqonometriyası ilə tapdıq. Əgər tanış gəlmədisə,
bu haqda yenidən oxuyun. Və üfüqi komponentin qiyməti F kosinus teta-ya bərabər olacaq. Buradakı tərəf bucağa birləşikdir, SO KA TOA. Və bunu tapdıqdan sonra sərbəst cism diaqramımızı çəkə bilərik. İcazə verin, diaqramı çəkim, icazənizlə, bu bloku çəkirəm və yalnızca bloka diqqətimi yönəldəcəm. Və burada nələrin baş
verdiyini bilirik. Biz üfüqi qüvvəni, yəni F kosinus teta-nı bilirik, icazənizlə onu çəkim. Onun qiyməti F kosinus teta-ya bərabərdir. Biz şaquli qüvvəni, yəni F
sinus teta-nı tapmışıq icazə verin, onu da çəkim. Bu F olardı, əslində bu, daha qısa olmalı idi, miqyasa uyğun çəkməyə bilərəm, amma bu F sinus teta olardı. Gəlin, indi buradakı parametrlər haqqında biraz düşünək. Bizim qravitasiya qüvvəmiz var, və o aşağı doğru yönələcək. Yəni o, bu şəkildə göstərilə bilər. Və onun qiyməti F indeksdə
q-yə bərabərdir. İndi oxu çəkmirəm, çünki sadəcə bu vektorun qiyməti
haqqında danışıram. Burada, bütün vektora istinad edirəm, həm qiymətinə, həm də istiqamətinə
istinad edirəm. Bəs sürtünmə qüvvəsi üçün nə edəcəyik? Fərz edək ki, bizim halımızda tətbiq edilmiş qüvvənin şaquli komponentinin qiyməti, F sinus teta qravitasiya qüvvəsinin qiymətindən azdır. Əgər bu halda sürtünmə
qüvvəsi olmasa idi, blokumuz aşağı doğru
təcillə hərəkət edərdi, çünki aşağı yönələn
əvəzedici qüvvəmiz olardı. Biz hələ sola və sağa yönələn qüvvələrdən danışmamışıq. Amma vurğuladığımız kimi
cism hərəkətsizdir, və sürtünmə qüvvəsi hərəkət istiqamətinin əksinə yönələcək. Və bu halda sürtünmə
qüvvəsi yuxarı yönələcək, yəni sürtünmə qüvvəsi bu şəkildə olacaq, və onun qiyməti F indeksdə s-ə bərabər olacaq. Bəs son olaraq dayağın reaksiya qüvvəsi üçün nə etməliyik? Əgər bu blok heç bir istiqamətdə təcilə malik deyilsə, reaksiya qüvvəsi sağa doğru yönəlmiş qüvvəni, yəni tətbiq olunan qüvvənin üfüqi komponentini tarazlamalıdır. Dayağın reaksiya qüvvəsi sola yönələcək, onu bu şəkildə göstərə bilərik. Və onun qiyməti F indeksdə D-dir. Və bununla da bitirdik, bu hal üçün sərbəst cism diaqramını çəkdik. Əgər bu ikisi bərabər olsa,
sürtünmə qüvvəmiz olmayacaq və ya sürtünmə qüvvəsi
sıfıra bərabər olacaq. Onun tarazlayacağı
bir qüvvə olmayacaqdı. Bu ikisi bir-birlərini tam
olaraq sıfırlayacaqdı. Və əgər tətbiq olunmuş qüvvənin
şaquli komponenti qravitasiya qüvvəsindən sürtünmə olmadan böyük olsa idi, o yuxarı doğru təcillənərdi, və sürtünmə qüvvəsi bu hərəkətin əksinə yönələrdi, biz bu hal ilə gedək, və sərbəst cism diaqramlarının niyə belə
vacib olduğunu dərk edək. Çünki biz bu qiymətlərlə
əlaqəli bərabərliklər qurmağa başlaya bilərik. Biz deyə bilərik ki, bu qutu heç
bir yöndə təcilə malik deyilsə, üfüqi və şaquli oxda əvəzləyici qüvvə sıfırdır, və deyərik ki, F indeksdə D F vur kosinus teta-nı tarazlayır. Beləliklə, deyə bilərik ki, F indeksdə D F kosinus teta-ya bərabərdir, və həmçinin deyərik ki, F sinus teta F sinus teta üstəgəl sürtünmə qüvvəsinin qiyməti. Üstəgəl sürtünmə qüvvəsinin qiyməti bu ikisi qravitasiya qüvvəsini onun qiymətinə uyğun tarazlayardı, çünki o, əks istiqamətə yönəlib. Yəni bunlar F indeksdə
q-ye bərabər olardı. Siz əgər lazımi bərabərlikləri tapsanız, və bunların birindən başqa bütün dəyişənləri bilsəniz, digərlərini də tapa bilərsiniz, və bu fizikada
çox yararlıdır. |
10904 | https://artofproblemsolving.com/wiki/index.php/Prime_factorization?srsltid=AfmBOoqvvHlKJKKYoSo3BDpNP7JSBzyJ9IE8XBi40i78JLkkGTaKAtjs | Art of Problem Solving
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Prime factorization
For a positive integer , the prime factorization of is an expression for as a product of powers of prime numbers. An important theorem of number theory called the Fundamental Theorem of Arithmetic tells us that every positive integer has a unique prime factorization, up to changing the order of the terms.
The form of a prime factorization is where is any natural number, the are prime numbers, and the are their positive integral exponents.
Prime factorizations are important in many ways. One instance is to simplify fractions.
Contents
[hide]
1 Primes and Prime Factorization Video
2 Techniques
3 Example
4 Problems
4.1 Introductory
4.2 Intermediate
5 Resources
5.1 Books
5.2 Games
6 See also
Primes and Prime Factorization Video
Techniques
The common method of prime factorization is checking prime numbers, case by case. Use divisibility rules to check if primes (or powers of primes) are a factor and then move up to a different prime if said prime is not a factor. When a prime is a factor, we factor out the prime and check for factors in the resulting number.
Sometimes, we could easily see that a number is a multiple of a composite number or a prime. For instance, is a factor of , and is a factor of . In these cases, a good strategy is to choose the number accordingly to make factoring easier.
Example
Consider the number .
Because the last two digits of the given number are a multiple of , we can rewrite as . Note that is not even, so we move on.
Because the sum of digits of is a multiple of 9, we can rewrite as . Note that the sum of digits of is not a multiple of 3, so we move on.
Because does not end in or , we skip as a factor. As for , after long division, we note that , so we can rewrite as . However, when doing long division again, is not a multiple of , so we move on.
Note that is not a factor of , so we can move on. In fact, because , we can declare as prime and stop.
Thus, the prime factorization of is .
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Prime Factorization (Prealgebra, Number Theory)
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Divisor
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10905 | https://en.wikipedia.org/wiki/Consensus_theorem | Jump to content
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Contents
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1 Proof
2 Consensus
3 Applications
4 History
5 References
6 Further reading
Consensus theorem
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From Wikipedia, the free encyclopedia
Theorem in Boolean algebra
| Variable inputs | Function values |
--- |
| x | y | z | | |
| 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 1 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 1 | 1 | 1 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 1 | 0 | 0 |
| 1 | 1 | 0 | 1 | 1 |
| 1 | 1 | 1 | 1 | 1 |
In Boolean algebra, the consensus theorem or rule of consensus is the identity:
The consensus or resolvent of the terms and is . It is the conjunction of all the unique literals of the terms, excluding the literal that appears unnegated in one term and negated in the other. If includes a term that is negated in (or vice versa), the consensus term is false; in other words, there is no consensus term.
The conjunctive dual of this equation is:
Proof
[edit]
Consensus
[edit]
The consensus or consensus term of two conjunctive terms of a disjunction is defined when one term contains the literal and the other the literal , an opposition. The consensus is the conjunction of the two terms, omitting both and , and repeated literals. For example, the consensus of and is . The consensus is undefined if there is more than one opposition.
For the conjunctive dual of the rule, the consensus can be derived from and through the resolution inference rule. This shows that the LHS is derivable from the RHS (if A → B then A → AB; replacing A with RHS and B with (y ∨ z) ). The RHS can be derived from the LHS simply through the conjunction elimination inference rule. Since RHS → LHS and LHS → RHS (in propositional calculus), then LHS = RHS (in Boolean algebra).
Applications
[edit]
In Boolean algebra, repeated consensus is the core of one algorithm for calculating the Blake canonical form of a formula.
In digital logic, including the consensus term in a circuit can eliminate race hazards.
History
[edit]
The concept of consensus was introduced by Archie Blake in 1937, related to the Blake canonical form. It was rediscovered by Samson and Mills in 1954 and by Quine in 1955. Quine coined the term 'consensus'. Robinson used it for clauses in 1965 as the basis of his "resolution principle".
References
[edit]
^ Frank Markham Brown [d], Boolean Reasoning: The Logic of Boolean Equations, 2nd edition 2003, p. 44
^ a b Frank Markham Brown, Boolean Reasoning: The Logic of Boolean Equations, 2nd edition 2003, p. 81
^ Rafiquzzaman, Mohamed (2014). Fundamentals of Digital Logic and Microcontrollers (6 ed.). John Wiley & Sons. p. 65. ISBN 978-1118855799.
^ "Canonical expressions in Boolean algebra", Dissertation, Department of Mathematics, University of Chicago, 1937, ProQuest 301838818, reviewed in J. C. C. McKinsey, The Journal of Symbolic Logic 3:2:93 (June 1938) doi:10.2307/2267634 JSTOR 2267634. The consensus function is denoted and defined on pp. 29–31.
^ Edward W. Samson, Burton E. Mills, Air Force Cambridge Research Center, Technical Report 54-21, April 1954
^ Willard van Orman Quine, "The problem of simplifying truth functions", American Mathematical Monthly 59:521-531, 1952 JSTOR 2308219
^ John Alan Robinson, "A Machine-Oriented Logic Based on the Resolution Principle", Journal of the ACM 12:1: 23–41.
^ Donald Ervin Knuth, The Art of Computer Programming 4A: Combinatorial Algorithms, part 1, p. 539
Further reading
[edit]
Roth, Charles H. Jr. and Kinney, Larry L. (2004, 2010). "Fundamentals of Logic Design", 6th Ed., p. 66ff.
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10906 | https://web.stanford.edu/class/cs103/ | ===============
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10907 | https://goldbook.iupac.org/terms/view/M03980/PDF | doi:10.1351/goldbook.M03980
IUPAC Compendium of Chemical Terminology Copyright © 2020 IUPAC
mole
The mole, symbol mol, is the SI unit of amount of substance. One mole contains exactly 6.022 140 76 × 10 23 elementary entities. This number is the fixed numerical value of the Avogadro constant, NA, when expressed in the unit mol -1 , and is called the Avogadro number. The amount of substance, symbol n, of a system is a measure of the number of specified elementary entities. An elementary entity may be an atom, a molecule, an ion, an electron, any other particle or specified group of particles.
Note:
The formulation of this definition was agreed upon by the 26 th CGPM in November 2018 with effect from 20 May 2019.
Source:
BIPM, The International System of Units, SI Brochure, 9 th ed. (2019), p. 134
Revised: March 30 th , 2020 |
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Modular arithmetic/Introduction
Modular arithmetic is a special type of arithmetic that involves only integers. This goal of this article is to explain the basics of modular arithmetic while presenting a progression of more difficult and more interesting problems that are easily solved using modular arithmetic.
Contents
1 Introductory Video
2 Understand Modular Arithmetic
3 Residue
4 Congruence
4.1 Examples
4.2 Sample Problem
4.2.1 Solution:
4.2.2 Another Solution:
4.2.3 Another Solution:
5 Making Computation Easier
5.1 Addition
5.1.1 Problem
5.1.2 Solution
5.1.3 Why we only need to use remainders
5.1.4 Solution using modular arithmetic
5.1.5 Addition rule
5.1.6 Proof of the addition rule
5.2 Subtraction
5.2.1 Problem
5.2.2 Solution
5.2.3 Subtraction rule
5.3 Multiplication
5.3.1 Problem
5.3.2 Solution
5.3.3 Solution using modular arithmetic
5.3.4 Multiplication rule
5.4 Exponentiation
5.4.1 Problem #1
5.4.2 Problem #2
5.4.3 Problem #3
6 Summary of Useful Facts
7 Problem Applications
8 Applications of Modular Arithmetic
9 Resources
10 See also
Introductory Video
Understand Modular Arithmetic
Let's use a clock as an example, except let's replace the at the top of the clock with a .
This is the way in which we count in modulo 12. When we add to , we arrive back at . The same is true in any other modulus (modular arithmetic system). In modulo , we count
We can also count backwards in modulo 5. Any time we subtract 1 from 0, we get 4. So, the integers from to , when written in modulo 5, are
where is the same as in modulo 5. Because all integers can be expressed as , , , , or in modulo 5, we give these integers their own name: the residue classes modulo 5. In general, for a natural number that is greater than 1, the modulo residues are the integers that are whole numbers less than :
This just relates each integer to its remainder from the Division Theorem. While this may not seem all that useful at first, counting in this way can help us solve an enormous array of number theory problems much more easily!
Residue
We say that is the modulo-residue of when , and .
Congruence
There is a mathematical way of saying that all of the integers are the same as one of the modulo 5 residues. For instance, we say that 7 and 2 are congruent modulo 5. We write this using the symbol : In other words, this means in base 5, these integers have the same residue modulo 5:
The (mod 5) part just tells us that we are working with the integers modulo 5. In modulo 5, two integers are congruent when their difference is a multiple of 5. In general, two integers and are congruent modulo when is a multiple of . In other words, when is an integer. Otherwise, , which means that and are not congruent modulo .
Examples
because is a multiple of .
because , which is an integer.
because , which is not a multiple of .
because , which is not an integer.
Sample Problem
Find the modulo residue of .
Solution:
Since R , we know that
and is the modulo residue of .
Another Solution:
Since , we know that
We can now solve it easily
and is the modulo residue of
Another Solution:
We know is a multiple of since is a multiple of . Thus, and is the modulo residue of .
Making Computation Easier
We don't always need to perform tedious computations to discover solutions to interesting problems. If all we need to know about are remainders when integers are divided by , then we can work directly with those remainders in modulo . This can be more easily understood with a few examples.
Addition
Problem
Suppose we want to find the units digit of the following sum:
We could find their sum, which is , and note that the units digit is . However, we could find the units digit with far less calculation.
Solution
We can simply add the units digits of the addends:
The units digit of this sum is , which must be the same as the units digit of the four-digit sum we computed earlier.
Why we only need to use remainders
We can rewrite each of the integers in terms of multiples of and remainders:
.
When we add all four integers, we get
At this point, we already see the units digits grouped apart and added to a multiple of (which will not affect the units digit of the sum):
.
Solution using modular arithmetic
Now let's look back at this solution, using modular arithmetic from the start. Note that
Because we only need the modulo residue of the sum, we add just the residues of the summands:
so the units digit of the sum is just .
Addition rule
In general, when , and are integers and is a positive integer such that
the following is always true:
.
And as we did in the problem above, we can apply more pairs of equivalent integers to both sides, just repeating this simple principle.
Proof of the addition rule
Let , and where and are integers. Adding the two equations we get:
Which is equivalent to saying
Subtraction
The same shortcut that works with addition of remainders works also with subtraction.
Problem
Find the remainder when the difference between and is divided by .
Solution
Note that and . So,
Thus,
so 1 is the remainder when the difference is divided by . (Perform the subtraction yourself, divide by , and see!)
Subtraction rule
When , and are integers and is a positive integer such that
the following is always true:
Multiplication
Modular arithmetic provides an even larger advantage when multiplying than when adding or subtracting. Let's take a look at a problem that demonstrates the point.
Problem
Jerry has boxes of soda in his truck. The cans of soda in each box are packed oddly so that there are cans of soda in each box. Jerry plans to pack the sodas into cases of cans to sell. After making as many complete cases as possible, how many sodas will Jerry have leftover?
Solution
First, we note that this word problem is asking us to find the remainder when the product is divided by .
Now, we can write each and in terms of multiples of and remainders:
This gives us a nice way to view their product:
Using FOIL, we get that this equals
We can already see that each part of the product is a multiple of , except the product of the remainders when each and are divided by 12. That part of the product is , which leaves a remainder of when divided by . So, Jerry has sodas leftover after making as many cases of as possible.
Solution using modular arithmetic
First, we note that
Thus,
meaning there are sodas leftover. Yeah, that was much easier.
Multiplication rule
When , and are integers and is a positive integer such that
The following is always true:
.
Exponentiation
Since exponentiation is just repeated multiplication, it makes sense that modular arithmetic would make many problems involving exponents easier. In fact, the advantage in computation is even larger and we explore it a great deal more in the intermediate modular arithmetic article.
Note to everybody: Exponentiation is very useful as in the following problem:
Problem #1
What is the last digit of if there are 1000 7s as exponents and only one 7 in the middle?
We can solve this problem using mods. This can also be stated as . After that, we see that 7 is congruent to -1 in mod 4, so we can use this fact to replace the 7s with -1s, because 7 has a pattern of repetitive period 4 for the units digit. is simply 1, so therefore , which really is the last digit.
Problem #2
What are the tens and units digits of ?
We could (in theory) solve this problem by trying to compute , but this would be extremely time-consuming. Moreover, it would give us much more information than we need. Since we want only the tens and units digits of the number in question, it suffices to find the remainder when the number is divided by . In other words, all of the information we need can be found using arithmetic mod .
We begin by writing down the first few powers of mod :
A pattern emerges! We see that So for any positive integer , we have (mod ). In particular, we can write
.
By the "multiplication" property above, then, it follows that
(mod ).
Therefore, by the definition of congruence, differs from by a multiple of . Since both integers are positive, this means that they share the same tens and units digits. Those digits are and , respectively.
Problem #3
Can you find a number that is both a multiple of but not a multiple of and a perfect square?
No, you cannot. Rewriting the question, we see that it asks us to find an integer that satisfies .
Taking mod on both sides, we find that . Now, all we are missing is proof that no matter what is, will never be a multiple of plus , so we work with cases:
This assures us that it is impossible to find such a number.
Summary of Useful Facts
Consider four integers and a positive integer such that and . In modular arithmetic, the following identities hold:
Addition: .
Subtraction: .
Multiplication: .
Division: , where is a positive integer that divides and .
Exponentiation: where is a positive integer.
Problem Applications
Applications of Modular Arithmetic
Modular arithmetic is an extremely flexible problem solving tool. The following topics are just a few applications and extensions of its use:
Divisibility rules
Linear congruences
Resources
The AoPS Introduction to Number Theory by Mathew Crawford.
The AoPS Introduction to Number Theory Course. Thousands of students have learned more about modular arithmetic and problem solving from this 12 week class.
See also
Intermediate modular arithmetic
Olympiad modular arithmetic
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10909 | https://books.google.com/books/about/Statistical_Physics_of_Fields.html?id=nTxBhGX01P4C | Statistical Physics of Fields - Mehran Kardar - Google Books
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Statistical Physics of Fields ============================= Mehran Kardar Cambridge University Press, Jun 7, 2007 - Science - 359 pages While many scientists are familiar with fractals, fewer are familiar with scale-invariance and universality which underlie the ubiquity of their shapes. These properties may emerge from the collective behaviour of simple fundamental constituents, and are studied using statistical field theories. Initial chapters connect the particulate perspective developed in the companion volume, to the coarse grained statistical fields studied here. Based on lectures taught by Professor Kardar at MIT, this textbook demonstrates how such theories are formulated and studied. Perturbation theory, exact solutions, renormalization groups, and other tools are employed to demonstrate the emergence of scale invariance and universality, and the non-equilibrium dynamics of interfaces and directed paths in random media are discussed. Ideal for advanced graduate courses in statistical physics, it contains an integrated set of problems, with solutions to selected problems at the end of the book and a complete set available to lecturers at www.cambridge.org/9780521873413. More » Preview this book »
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Title Page
Table of Contents
Index
Contents
Preface page ix 1
Problems 14 17
Problems 32
Problems 48
Problems 70
Problems 93
Problems 117
Problems 148
Problems 181
Directed paths in random media 209
Solutions to selected problems 260
278
292
317
343
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About the author(2007)
Mehran Kardar is Professor of Physics at MIT, where he has taught and researched in the field of Statistical Physics for the past twenty years. He received his BA in Cambridge, and gained his Ph.D. at MIT. Professor Kardar has held research and visiting positions as a junior fellow at Harvard, a Guggenheim fellow at Oxford, UCSB, and at Berkeley as a Miller Fellow.
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Title Statistical Physics of Fields
AuthorMehran Kardar
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Publisher Cambridge University Press, 2007
ISBN 052187341X, 9780521873413
Length 359 pages
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Astrophysics and cosmology
Feature
How the Sun and stars shine
21 December 2021
Thanks to its extraordinary radiopurity, Borexino has definitively observed the two main fusion reactions in stars and will soon weigh in on a controversy relating to the birth of the Sun that challenges the basic assumptions of the Solar Standard Model, write Gianpaolo Bellini and Aldo Ianni.
Each second, fusion reactions in the Sun’s core fling approximately 60 billion neutrinos onto every square centimetre of the Earth. In the late 1990s, the Borexino experiment at Gran Sasso National Laboratory in Italy was conceived to measure these neutrinos right down to a few tens of keV, where the bulk of the flux lies. The detector’s name means “little Borex” and refers to an earlier idea for a large experiment with a boron-loaded liquid scintillator, which was shelved in favour of the present, smaller and more ambitious detector. Rather than studying rare but high-energy 8B neutrinos from a little-followed branch of the proton–proton (pp) fusion chain, Borexino would target the far more numerous but lower energy neutrinos produced in the Sun by electron captures on 7Be.
Three decades after its conception, Borexino has far exceeded this goal thanks to the exceptional radiopurity of the experimental apparatus (see “Detector design” panel. Special care taken in construction and commissioning has achieved a radiopurity about three orders of magnitude better than predicted, and 10 to 12 orders of magnitude below natural radioactivity. This has allowed the collaboration to probe the entire solar-neutrino spectrum, including not only the pp chain, but also the carbon–nitrogen–oxygen (CNO) cycle. This mechanism plays a minor role in the Sun but becomes important for more massive stars, dominating the energy production and the production of elements heavier than helium in the universe at large.
The heart of the Sun
The pp-chain generates 99% of the energy in the Sun: it begins when two protons fuse to produce a deuteron and an electron neutrino – the so-called pp neutrino (see “Chain and cycle” figure). Subsequent reactions produce light elements, such as 3He, 4He, 7Be, 7Li, 8B and more electron neutrinos. In Borexino, the sensitivity to pp neutrinos depends on the amount of 14C in the liquid scintillator: with an end-point energy of 0.156 MeV compared with a maximum visible energy for pp neutrinos of 0.264 MeV, the 14C → 14N + β– + ν beta decay sets the detection threshold and the feasibility of probing pp-neutrinos. The Borexino scintillator was therefore made using petroleum from very old and deep geological layers, to ensure a low content of 14C.
Detector design
Like many particle-physics detectors, Borexino has an onion-like design. The innermost layers have the highest radio-purity. The detector’s active core consists of 278 tonnes of pseudocumene (C9H12) scintillator. Into this is dissolved 2,5-diphenyloxazole (PPO) at a concentration of 1.5 grams per litre, which shifts the emission light to 400 nm, where the sensitivity of photomultipliers is peaked. The scintillator is contained within a 125 μm-thick nylon inner vessel (IV) with a 4.5 m radius – made thin to reduce radiation emission from the nylon . In addition, the IV stops radon diffusion towards the core of the detector.
The IV is contained within a 7 m-radius stainless-steel sphere (SSS) that supports 2212 photomultipliers (PMTs) and contains 1000 tonnes of pseudocumene as high-radio-purity shielding liquid against radioactivity from PMTs and the SSS itself. Between the SSS and the IV, a second nylon balloon acts as a barrier preventing radon and its progeny from reaching the scintillator. The SSS is contained in a 2400-tonne tank of highly purified water which, together with Borexino’s underground location, shields the detector from environmental radioactivity. The tank boasts a muon detector to tag particles crossing the detector.
When a neutrino interacts in the target volume, energy deposited by the decelerating electron is registered by a handful of PMTs. The neutrino’s energy can be obtained from the total charge, and the hit-time distribution is used to infer the location of the event’s vertex. Recoiling electrons are used to tag electron neutrinos, and the combination of a positron annihilation and a neutron capture on hydrogen (an inverse beta decay) are used to tag electron antineutrinos.
Due to the impossibility of discriminating individual solar-neutrino events from the backgrounds, the greatest challenge has been the reduction of natural radioactivity to unprecedented levels. In the early 1990s, Borexino developed innovative techniques such as under-vacuum distillation, water extraction, ultrafiltration and nitrogen sparging with ultra-high radiopurity nitrogen to reduce radioactive impurities in the scintillator to 10–10 Bq/kg or better. An initial detector called the Counting Test Facility was developed as a means to demonstrate such claims, publishing results for the key uranium, thorium and krypton backgrounds in 1995. Full data taking at Borexino began in 2007.
Since data-taking began in 2007, Borexino has measured, for the first time, all the individual fluxes produced in the pp-chain. In 2014 the collaboration made the first definitive observation of pp neutrinos, using a comparison with the predicted energy spectrum. In 2018 the collaboration performed, with the same apparatus, a measurement of all the pp-chain components (pp, 7Be, pep and 8B neutrinos), demonstrating the large-scale energy-generation mechanism in the Sun for the first time (see “Energy spectrum” figure). This spectral fit allowed the collaboration to directly determine the ratio between the interaction rate of 3He + 3He fusions and that of 3He + 4He fusions – a crucial parameter for characterising the pp chain and its energy production.
The simultaneous measurement of pp-chain neutrino fluxes also gave Borexino a unique window onto the famous “vacuum-matter” transition, whereby coherent virtual W-boson interactions with electrons modify neutrino-oscillation probabilities as neutrinos propagate through matter, enhancing the oscillation probability as a function of energy. In 2018 Borexino measured the solar electron–neutrino survival probability, Pee, in the energy range from a few tens of keV up to 15 MeV (see “Survival probability” figure). This was the first direct observation of the transition from a low-energy vacuum regime (Pee~0.55) to a higher energy matter regime where neutrino propagation is dominantly affected by the solar interior (Pee~0.32). The transition was measured by Borexino at the level of 98% confidence.
CNO cycle
A different way to burn hydrogen, the CNO cycle, was hypothesised independently by Carl Friedrich von Weizsäcker and Hans Albrecht Bethe between 1937 and 1939. Here, 12C acts as a catalyst, and electron neutrinos are produced by the beta decay of 13N and 15O, with a small contribution from 17F. The maximum energy of CNO neutrinos is about 1.7 MeV. In addition to making an important contribution to the production of elements heavier than helium, this cycle is important for the nucleosynthesis of 16O and 17O. In massive stars it also develops in more complex reactions producing 18F, 18O, 19F, 18Ne and 20Ne.
The sensitivity to CNO neutrinos in Borexino mainly comes from events in the energy range from 0.8 to 1 MeV. In this region, the dominant background comes from 210Bi, which is produced by the slow radioactive decay 210Pb (22 y) → 210Bi (5 d) + β– + ν → 210Po (138 d) + β– + ν → 206Pb (stable) + α. The 210Bi activity can be inferred from 210Po, which can be efficiently tagged using pulse-shape discrimination. However, convective currents in the liquid scintillator bring into the central fiducial mass 210Po produced by 210Pb, which is most likely to be embedded on the nylon containment vessel. In order to reduce convection currents, a passive insulation system and a temperature control system were installed in 2016, significantly reducing the effect of seasonal temperature variations.
Thanks to these and other efforts, in 2020 Borexino rejected the null hypothesis of no CNO reactions by more than five standard deviations, providing the first direct proof of the process. The energy production as a fraction of the solar luminosity was measured to be 1-0.3+0.4%, in agreement with the Solar Standard Model (SSM) prediction of roughly 0.6 ± 0.1% (which assumes the solar surface has a high metallicity – a topic discussed in more detail later). Given that luminosity scales as M4 and number density as M–2.5 for stars between one and 10 solar masses, the CNO cycle is thought to be the most important source of energy in massive hydrogen-burning stars. Borexino has provided the first experimental evidence for this hypothesis.
Probing solar metallicity using CNO neutrinos is of the utmost importance, and Borexino is hard at work on the problem
But, returning to the confines of our solar system, it’s important to remember that the SSM is not a closed book. Borexino’s results are thus far in agreement with its assumption of a protostar that had a uniform composition throughout its entire volume when fusion began (“zero-age homogeneity”). However, thanks to the ability of neutrinos to peek into the heart of the Sun, the experiment now has the potential to explore this assumption and weigh in on one of the most intriguing controversies in astrophysics.
The solar-abundance controversy
As stars evolve, the distribution of elements within them changes thanks to fusion reactions and convection currents. But the composition of the surface is thought to remain very nearly the same as that of the protostar, as it is not hot enough there for fusion to occur. Measuring the abundance of elements on a star’s surface therefore gives an idea of the protostar’s composition and is a powerful way to constrain the SSM.
Currently, the best method to determine the surface abundance of elements heavier than helium (“metallicity”) uses measurements of photo-absorption lines. Since 2005, improved hydrodynamic calculations (which are needed to model atomic-line formation, and radiative and collisional processes which contribute to excitation and ionisation) indicate a much lower surface metallicity than was previously considered. However, helioseismology observables differ by roughly five standard deviations from SSM predictions that use the new surface metallicity to infer the protostar’s composition, when the sound–speed profile, surface–helium abundance and the depth of the convective envelope are taken into account. Helioseismology implies that the zero-age Sun’s core was richer in metallicity than the present surface composition, suggesting a violation of zero-age homogeneity and a break with the SSM. This is the solar-abundance controversy, which was discovered in 2005.
One possible explanation is that a late “dilution” of the Sun’s convective zone occurred due to a deposition of elements during the formation of the solar system. Were there to have been an accretion of dust and gas from the proto-planetary disc onto the central star during the evolution of the star–planet system, this could have changed the initial metallicity of the surface of the Sun – a hypothesis backed up by recent simulations that show that a metal-poor accretion could produce the present surface metallicity.
As they are an excellent probe of metallicity, CNO neutrinos have an important role to play in settling the solar-abundance controversy. If Borexino were to measure the Sun’s present core metallicity, and by running simulations backwards prove that its surface metallicity must have been diluted right from its birth, this would violate one of the basic assumptions of the SSM. Probing solar metallicity using CNO neutrinos is, therefore, of the utmost importance, and Borexino is hard at work on the problem. Initial results favour the high-metallicity hypothesis with a significance of 2.1 standard deviations – a tentative first hint from Borexino that zero-age homogeneity may indeed be false.
The ancient question of why and how the Sun and stars shine finally has a comprehensive answer from Borexino, which has succeeded thanks to the detector’s extreme and unprecedented radio-purity – the hard work of hundreds of researchers over almost three decades.
Further reading
Borexino collaboration 2020 Nature 587 577.
Borexino collaboration 2020 Phys. Rev. D 101 062001.
Borexino collaboration 2019 Phys. Rev. D 100 082004.
Borexino collaboration 2018 Nature 562 505.
R Hoppe et al. 2020 A&A 641 A73.
Gianpaolo Bellini Università degli Studi/INFN Milano and Aldo Ianni INFN at LNGS.
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10911 | https://www.englishclass101.com/lesson/english-grammar-made-easy-113-talking-about-the-very-recent-past-with-just-just-about-and-just-progresive | Hallo, Pooh, you're just in time for a little smackerel of something
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Talking About the Very Recent Past
Learn how to talk about the very recent past ("just," "just about", and "just" + progressive)
Lesson Transcript
| |
| Hi, everybody! My name is Alisha. In this lesson, I'm going to talk about using "just" for the very recent past and for the near future. Let's get started! |
| All right, I want to begin this lesson by talking about the very recent past. So, I have two different patterns that you, we can use to talk about the very recent past with "just." The first will be for actions that were completed very recently and the second will be for actions that were planned for completion recently. |
| So, let's start with this first one here, number one. I've marked it on this timeline here with a check mark. So, on my timeline, this is the present. Back here is the past. For actions then that were very recently completed, we can kind of imagine this check mark as like something that happened very, very recently, so just before now. |
| So, a simple statement pattern that we can use with "just" is this "[subject] + just + [simple past tense verb]." This is the most basic way to make a statement with "just." |
| Some examples are: |
| "I just finished work." |
| Or |
| "He just arrived." |
| Or |
| "They just left." |
| So you see, in each of these very simple example sentences, we have just, followed by a simple past tense verb, in these cases; finished, arrived, and left. So if you want to make a basic statement about a finished action, something that has finished, is done, in the very recent past, you can use a pattern like this. |
| I want to include one note about this, this particular point, this number one point. You might also hear the present perfect used. By that, I mean, instead of "I just…" or "he just…" or "they just…," sometimes, people will use "I've just…," "he's just…," "they've just…" So this "-ve" and "s" here, this is… |
| "I have just…," |
| "He has just…," |
| "They have just…" |
| You may hear present perfect use as well. It has the same meaning, like… |
| "I've just finished work." |
| Or |
| "He's just arrived." |
| Or |
| "They've just left." |
| The meaning is the same. For whatever reason, the speaker has chosen to use present perfect tense. Perhaps, it sounds a little bit softer, but these two uses or rather these two patterns have the same purpose, it serves the same function. So you may hear these two. |
| Let's look, however, at kind of the opposite of this. |
| So, point number two here is used to express an action that was planned for the very recent past, but that did not get finished, is not done, it was not completed. It might get done in the future, we don't know, but it did not happen. So, to imagine this like visually, we can use this X mark in the very recent past. So something just before the present time that did not happen, but we had a plan to do this. |
| So, some examples of this, actually, I'll introduce in a second, but when we want to make a statement, with this kind of grammar, we can use again the subject, but we'll use the past-tense form of "be." So by that, I mean the "be" verb, so that means like "was or were" plus "just about to…" so this is a key difference here between the regular completed-action pattern. And then we use the present-tense form of the verb. So here, you already noticed maybe, there are a couple of different points between this and this. |
| So, some examples of this in action. |
| "I was just about to call you." |
| Or |
| "We were just about to leave." |
| Or |
| "She was just about to cancel the appointment." |
| So, in each of these example sentences, we see our subject plus the past tense form of the verb "to be," in this case, "I was, we were, she was." So here is our "be" verb. Then, we include "just about to…," there's no change in these sentences, "just about to (something)." And then our verb is the present-tense form of the verb, "call, leave, and cancel" in these cases. |
| So when you want to talk about something that was planned for the very recent past, but that did not happen, you don't need to change the verb, you don't need to conjugate the verb to past tense. We conjugate the verb to past tense when we're talking about finished actions. So please keep these points in mind. So, simple past tense for completed action, present tense, simple present tense for actions that are not completed or were not completed. |
| Okay, so with that in mind about the very recent past, I want to talk now about some patterns you can use with "just" for the very near future. So, let's begin with this first one, number one here. This pattern is used to talk about an action in the very near future, so something we have planned, we are thinking about that action or we're planning to do that action in the very near future. So here on this timeline, we're looking now into the future. So, this is my "now" point. This number one, I've represented with a check mark here. So this marks an action that I have planned. I want to do that or I'm thinking about that action and I'm going to do it soon. |
| To make a basic sentence with this, a basic statement, we can say "[subject] + [present tense of the verb "to be"] + "just about to" and the present tense form of the verb. So you'll notice actually, there's only one difference between this sentence and this sentence, and that's this part right here. In this part, where we were talking about the very recent past, we used the past-tense form of the verb "to be," "I was, you were." Here, we're using the present-tense form of the verb "be," "I am, we are." So, this is one hint that it's actually a future action, so this will tell you, is it a future action or is it a past action? It's a small point to listen for. |
| Some examples though are: |
| "I'm just about to finish work." |
| Or |
| "He's just about to arrive." |
| Or |
| "They're just about to leave." |
| So here, I've reduced it in each of these example sentences, but my "be" verb is here. "I'm" is "I am." |
| "He's" is "he is." |
| And "they're" is "they are." |
| So, "I'm just about to finish work." |
| "He's just about to arrive." |
| "They're just about to leave." |
| These tell us something is going to happen very soon. |
| Another point about this, is that sometimes, native speakers will drop "just" from this pattern. |
| So, "I'm about to finish work." |
| "He's about to arrive." |
| "They're about to leave." |
| These are all fine. They communicate the same thing, it's just the speaker's preference, so you can choose whichever you prefer. |
| Okay, then I want to continue to point two for this part. |
| Point two, let's look at the pattern first. It's [subject] + [present tense "be"] again here plus "just" and then we see the progressive form of a verb. So, this is something that we use for an action that's happening now and it's like something that we expect is going to finish in the very near future, so we use "just" to emphasize this. To visualize this then on a timeline, we can imagine with this kind of wavy line here that something is happening now and it will continue until the very, very, like near future. So maybe it stops or it finishes here. If you want to talk about an action like that, you can try using this pattern. |
| So, some examples are: |
| "I'm just finishing work." |
| Or |
| "He's just arriving." |
| Or |
| "They're just leaving." |
| So these show like the action has already started. So "I'm just finishing work" means maybe, I'm making my last, like steps in my day or I'm taking the last, I don't know, bits of information from my day. I'm putting them somewhere, I don't know, it depends on you. But, we use this to talk about actions that started and that we expect to finish very soon. So, you can try using one of these at, like, the end of your work day or like, maybe, when you're leaving a location. Those are some common situations where we would use patterns like this, so please keep that in mind. |
| When you're using the progressive form of the verb, it's going to sound like something is already like happening now. It's begun. When you're using it without, when you're just using the present-tense form of the verb here, it's something you are planning to do in the future. |
| Okay, so that's a quick introduction to using "just" for the very recent past and for the near future. I hope that you found some patterns that you can use to talk about your recent past and to describe some of the things you're going to do in the near future. Of course, if you have any questions or comments, please feel free to let us know in the comment section of this video and also, please feel free to leave some example sentences if you want to practice using this grammar. Thanks very much for watching this lesson and I will see you again soon. Bye-bye! |
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10912 | https://www.mathworks.com/help/matlab/creating_plots/plot-complex-numbers.html | Skip to content
Main Content
Plot Complex Numbers
This example shows how to plot complex numbers in MATLAB®. A complex number z is a number that can be written in the form
z=x+y i,
where x and y are real numbers, and i is the imaginary unit, which is defined as i2=−1. The number x is the real part of the complex number, which is denoted by x=Re(z), and the number y is the imaginary part of the complex number, which is denoted by y=Im(z). You can plot a complex number as a pair of coordinates (x,y) on the complex plane, also known as the Argand diagram. This diagram uses the Cartesian coordinates to represent the real part in the x-axis and the imaginary part in the y-axis.
You can also represent a complex number using the polar representation. The complex number is written in the form
z=r eiθ=r(cos θ+i sin θ),
where r is the absolute value or magnitude of the complex number, and θ is the phase angle of the complex number. In this representation, you can plot a complex number as a point in the polar coordinates with radius r (the distance from the origin) and polar angle θ (the counterclockwise angle between the positive real axis and the line connecting the point to the origin).
Plot Array of Complex Numbers
Create a vector that contains the complex numbers 3 + 4i, -4 - 3i, 1 - 2i, and -1 - 1i.
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z = [3 + 4i; -4 - 3i; 1 - 2i; -1 - 1i]
```
z = 4×1 complex
3.0000 + 4.0000i
-4.0000 - 3.0000i
1.0000 - 2.0000i
-1.0000 - 1.0000i
```
Plot the imaginary part against the real part of the complex vector z by using plot. Use the real function and imag function to return the real and imaginary parts of the complex vector, respectively.
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plot(real(z),imag(z),"o")
axis equal
grid on
xlabel("Re(z)")
ylabel("Im(z)")
You can also use plot(z,LineSpec) instead of plot(real(z),imag(z),LineSpec) to plot an array of complex numbers. This function automatically plots the real part in the x-axis and the imaginary part in the y-axis.
Plot Complex Roots of Unity in Cartesian Coordinates
The nth roots of unity are complex numbers that satisfy the polynomial equation
zn=1,
where n is a positive integer.
The nth roots of unity are
exp(2kπin)=cos2kπn+i sin2kπn, for k=0,1,…,n−1.
To find the complex roots of unity, you can solve the polynomial equation by using roots. The roots function solves polynomial equations of the form p1xn+⋯+pnx+pn+1=0. For example, find the fifth roots of unity of z5=1, or z5−1=0.
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p = [1 0 0 0 0 -1];
z = roots(p)
```
z = 5×1 complex
-0.8090 + 0.5878i
-0.8090 - 0.5878i
0.3090 + 0.9511i
0.3090 - 0.9511i
1.0000 + 0.0000i
```
Plot the complex roots of unity in the Cartesian coordinates.
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plot(z,"o")
axis equal
grid on
xlabel("Re(z)")
ylabel("Im(z)")
Plot Complex Numbers in Polar Coordinates
Plot the fifth roots of unity in the polar coordinates by using polarplot. Use the angle function to return the phase angles of the complex roots, and use the abs function to return the absolute values or radii of the complex roots.
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"o"
You can also use polarplot(z,LineSpec) instead of polarplot(angle(z),abs(z),LineSpec) to plot an array of complex numbers in the polar coordinates. This function automatically plots the radii and phase angles of the complex numbers.
Plot Parametric Curve in Complex Plane
Define a parametric curve that has the form
z=f(t)=texp(it)
with the parameter t in the interval [0,4π].
Create a vector t of 200 equally spaced points within this interval to parameterize t. Define the points that lie on the complex curve as a complex vector z.
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t = linspace(0,4pi,200);
z = t.exp(1it);
Plot the complex curve in the Cartesian coordinates.
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plot(z,"-")
axis equal
grid on
xlabel("Re(z)")
ylabel("Im(z)")
Plot the complex curve in the polar coordinates.
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"-"
Plot Eigenvalues of Square Matrix
A real n-by-n square matrix has n eigenvalues (counting algebraic multiplicities) that either are real or occur in complex conjugate pairs.
For example, consider a 20-by-20 real matrix with random elements that are sampled from a standard normal distribution. Calculate the eigenvalues using eig.
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rng("default")
z = eig(randn(20));
Plot the imaginary part against the real part of all 20 eigenvalues. Notice that for each eigenvalue zk=xk+yki that is not on the real axis, there is another complex conjugate pair of this eigenvalue z∗k=xk−yki.
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plot(z,"o")
axis equal
grid on
xlabel("Re(z)")
ylabel("Im(z)")
Plot Multiple Complex Data Sets
Plot the imaginary part against the real part of two complex data sets. If you pass multiple complex input arguments to plot, such as plot(z1,z2), then the plot function ignores the imaginary part and plots only the real part of the inputs. To plot the real part against the imaginary part for multiple complex inputs, you must explicitly pass the real part and the imaginary part to plot.
For example, create two complex vectors z1 and z2.
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x = -2:0.25:2;
z1 = x.^exp(-x.^2);
z2 = 2x.^exp(-x.^2);
Find the real part and imaginary part of each vector by using the real and imag functions.
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re_z1 = real(z1);
im_z1 = imag(z1);
re_z2 = real(z2);
im_z2 = imag(z2);
Plot the complex data.
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plot(re_z1,im_z1,"",re_z2,im_z2,"o")
axis equal
grid on
legend("z1","z2")
xlabel("Re(z)")
ylabel("Im(z)")
See Also
plot | real | imag | polarplot | abs | angle
Thank you for your feedback!
MATLAB Command
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10913 | https://www.engineersedge.com/fluid_flow/volumeetric_flow_rate.htm | Engineers Edge utilizes cookies to enable essential site functionality, and targeted advertising. To learn more, see our Privacy Policy.
Fluid Volumetric Flow Rate Equation
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Fluid Volumetric Flow Rate Equation
Fluid Flow Table of ContentsHydraulic and Pneumatic Knowledge
Fluid Volumetric Flow Rate Equation
The volumetric flow rate(V ) of a system is a measure of the volume of fluid passing a point in the system per unit time. The volumetric flow rate can be calculated as the product of the cross sectional area (A) for flow and the average flow velocity (v).
Equation (1)
V = A · v
If area is measured in square feet and velocity in feet per second, Equation 1 results in >volumetric flow rate measured in cubic feet per second. Other common units for volumetric flow >rate include gallons per minute, cubic centimeters per second, liters per minute, and gallons per hour.
Example:
A pipe with an inner diameter of 4 inches contains water that flows at an average velocity >of 14 feet per second. Calculate the volumetric flow rate of water in the pipe.
Solution:
Use Equation 1 and substitute for the area
V = ( π · r 2 ) v
V=(3.14157)(2 12 f t)2(14 f t s e c)
V=1.22 f t 3 s e c
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10914 | https://www.youtube.com/watch?v=lvlkzQI-P_s | Step Function: Floor Function
Euler's Academy
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Posted: 31 Aug 2022
This video continues with the idea of a step function by going looking at a specific type of step function known as the floor function.
Recall that a step function is essentially a piecewise function composed of horizontal lines on distinct intervals. Going from one interval to the next looks like going up or down different steps.
The floor function definition is that the function value, or f(x), is the greatest integer less than or equal to x. This definition essentially results in rounding down whatever x value is put into the function. For instance, f(4.73) = 4, f(-6.9) = -7, and f(2) = 2. The resulting graph looks like an infinite staircase going up.
Khan Academy exercise to practice:
1. Evaluate Step Functions:
EulersAcademy.org
Transcript:
Intro in this video i'd like to continue talking about step functions particularly what is known as the floor function and in the next video we'll talk about the c-link function which goes along with the floor function Definition so the floor function is a specific type of step function or in other words it's an infinite step function which we usually denote with a sort of bracket notation that looks something like that where the x value is within the brackets let me make it a little bit more clear so something like that and notice that the bracket is on the floor here and the definition of this function is that the y value is the greatest integer that is less than or equal to x so let's get an idea of how this works by just making a table so let me construct that very quickly so we have our x values and then we have our function values our floor function values Integers so for integers this is very straightforward so one two three we'll just look at a few of them when you plug in an integer so like let's say we plug in the x value of one the y value the greatest integer that is less than or equal to x so if x is 1 and y is that greatest integer that is less than or equal to 1 then y is just 1 there so whenever you plug in an integer you just get back that integer value so nothing too exciting there but this function does become interesting when you plug in decimals let's say we plug in 1.4 so the y value is the greatest integer that is less than or equal to 1.4 in this case so the integer that is less than this and the greatest integer less than this would be 1. now recognize there are many or in fact infinite integers less than 1.4 like for instance 0 negative 1 negative 2 and so on but we're looking for the greatest of those integers that's less than this and that integer would be 1. so also if we plugged in let's say 1.67 the greatest integer that's less than this is also equal to 1. and essentially all the way up to 1.999 almost repeating but not quite since 1.9 repeating is 2 but essentially 2 is the boundary case so something like 1.99 is also going to return 1. since the greatest integer less than this is also 1. but if we get up to 2 or cross past it so something like 2.01 the greatest integer less than this would be 2. and if we went let's say up to 2.5 the greatest integer less than this would also be 2 and it doesn't move up to three until we get to three point something or three itself but if it's two point anything for the most part unless it's nine repeating then that's going to round down to 2. so essentially this is a type of rounding function so let me finish this example let's say we have 3.19 the greatest integer less than this would be 3. Graphing so whatever value we plug into this function it's going to round down to that nearest integer so to graph this function let's just get an idea of this so if we plug in 1 we know that we're going to get a function value of 1 and let me use this color here and if we plug in 1.4 we also got that same value of 1. if we plug in 1.67 we got one 1.99 we got one but it does not include two so if we plug in two it's going to be all the way up here at two so since any value less than 2 is essentially going to be a y value of 1 we're going to put an open circle there since it's not including that endpoint but any x value less than 2 or bigger than or equal to 1 will return this y value of 1. so that's 1 for y 1 for x just to denote our scale here we're going by 1 for all of these and 4 2 any value between 2 and 3 but not including 3 will return a y value of 2. and if we do actually plug in 3 we get a y value of three so that's going to be up here with a closed circle and again anything between x values of three and four that's going to return a y value of three but at an x value of four we're going to put an open circle since that will actually return a y value of 4. so that's going to be a closed circle up here so we get this step function and it's sort of like an infinite staircase so this would actually continue on over here all the way to an x value five where again it's going to be this open circle and even though we didn't make a table for we can work backwards so when you plug in zero point anything including zero it's going to return a y value of zero so from zero to one all the way to one but not including one since one is up at a y value of one will have a y value of zero here so between zero and one not including one y is equal to zero and if we start looking at the negatives from negative one that would give us a y value of negative one since integers always return their y values so maybe that's a way to graph this a little bit quicker so for negative two the y value would be negative two for negative 3 we're at negative 3 and negative 4 would give us negative 4. now going back up here we can look at the decimal parts so between negative 1 and 0 the the greatest integer that is less than or equal to that would be negative one so on this interval of x values the y value would be negative one but it goes all the way to zero but it doesn't include zero since zero returns a y value of zero and for negative two all the way to negative one it's going to return a y value of negative two not including the x value of negative 1. and so you can see the pattern here we can continue these and i encourage you to try and redraw this yourself make a table of your own try to think this through because this can be a little bit complicated this is an infinitely long piecewise function that effectively looks like an infinite staircase now of course we wouldn't actually draw in these dashed lines but it completes the picture since now you can visually see that it does look like a staircase |
10915 | https://www.sciencedirect.com/science/article/abs/pii/S0009250923007546 | Skip to article
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Abstract
Introduction
Section snippets
References (36)
Cited by (19)
Chemical Engineering Science
Volume 281, 5 November 2023, 119198
Heterogeneous nitration of nitrobenzene in microreactors: Process optimization and modelling
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Highlights
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The effects of operating parameters on hydrodynamics and reaction characteristics of nitrobenzene nitration were studied in microreactors.
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More than 90% nitric acid was converted and 80 % m-dinitrobenzene was yielded within 10 min, much shorter than that in the industrial production.
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The reaction regime was verified that the reaction takes place not only in the bulk aqueous phase but also at the phase interface.
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The kinetic parameters, including the reaction rate constant, activation energy and pre-exponential factor, were determined.
Abstract
Aromatic nitration with mixed acid is an important step in the industrial synthesis of basic industrial chemicals, while the highly exothermic and heterogeneous nature of the reaction renders difficulties in achieving high safety and efficiency. In this work, a continuous flow microreactor system was developed for the nitration of nitrobenzene. Under the premise of high conversion and selectivity, the reaction time and temperature were reduced from more than 2 h and 80 in industrial operation to 10 min and 65 in microreactors, respectively. The reaction mechanism was verified that the reaction took place not only in the bulk aqueous phase but also at the phase interface. The kinetic parameters were then determined based on the assumption of pseudo-homogeneous reaction mixture. These findings may shed important insights into the high degree of controllability of nitration for a better process design and reactor optimization.
Graphical abstract
Introduction
Aromatic nitration is an important reaction in the production of basic and specialty chemicals, with applications in fields such as dyes, pharmaceuticals, and agrochemicals, etc. (Patel et al., 2021) In majority of these reactions, aromatic hydrocarbon is nitrified into nitroaromatic compound by mixed acid (the mixture of sulfuric acid and nitric acid), in which nitric acid acts as a nitrating agent and sulfuric acid as a catalyst. Reports of experimental studies on nitration of aromatics (such as benzene, toluene, chlorobenzene, benzaldehyde etc.) by mixed acid are available in the literature (Burns and Ramshaw, 2002, Liang et al., 2022, Quadros et al., 2004, Quadros et al., 2005). It is commonly known that nitration is a fast and highly exothermic reaction, with the reaction heat ranging from 73 to 253 kJ/mol (Kulkarni, 2014). Therefore, it is essential for nitration reactors to provide adequate heat removal rate, otherwise uncontrollable side or runaway reactions can occur, leading to safety hazards (van Woezik and Westerterp, 2002, Zhou et al., 2014). In batch or continuous stirred-tank reactors extensively used in nitration processes, optimizing the design of stirring paddles and cooling coils to enhance the rate of heat removal, as well as employing relatively low reaction temperature or adding the mixed acid in batches to slow down the reaction rate (and with that the heat generation), are the most common methods to reduce safety risks (Gao et al., 2020, Song et al., 2022c, Xu et al., 2018, Zhang et al., 2015). However, this is usually at the cost of long reaction time (typically tens of minutes or even several hours), which would result in not only the reduction of reaction efficiency, but also the increase of by-product formation (Gao et al., 2020, Xu et al., 2018). Therefore, it is of high importance to explore novel reactors for aromatic nitration that can boost the reaction efficiency with much improved process safety.
Recently, microreactor technology has opened new horizons both in academia and industries, becoming a more sustainable and reliable alternative for safer nitration (Cui et al., 2022, Ducry and Roberge, 2005, Guo et al., 2023, Li et al., 2022, Song et al., 2022a, Song et al., 2022b). Microreactors possess internal channels with micron or submillimeter characteristic dimensions, based on which many advantages have emerged, ranging from excellent mass and heat transfer performance to easy scale-up and potential of full process automation. Ducry et al. (Ducry and Roberge, 2005) found that the temperature rise during the nitration of phenol in the microreactor was measured to be no more than 5 , which was much lower than that in a semibatch reactor (i.e., 55 ). The efficient heat transfer capability of the microreactor makes it possible for the elimination of local hot spots, allowing safe control of the exothermic nitration. As a result, the reaction temperature could be raised moderately to accelerate the reaction rate. For example, Li et al. (Li et al., 2017, Li et al., 2022) realized the nitration of 2-ethylhexanol and 3-[2-chloro-4-(trifluoromethyl) phenoxy] benzoic acid at room temperature in continuous flow microreactors. Compared with the commonly used temperature of around 273 K in batch reactors, not only shorter reaction times (seconds or minutes) were acquired in microreactors, but also operating difficulties and energy consumption could be decreased. Beside the above-mentioned merits, reduced amounts of acids and improved conversion as well as yields of target products are often obtained when performing the reaction continuously in a microreactor. Russo et al. (Russo et al., 2019) realized a high conversion of 97% in the nitration of benzaldehyde in a microreactor, achieving the theoretical limit of conversion attainable under kinetic control. Wen et al. (Wen et al., 2017) used a microreactor for the continuous nitration of trifluoromethoxybenzene in which the conversion was up to 99.6% and the selectivity of the main product (4-(trifluoromethoxy)nitrobenzene) reached 90.97% within 2.4 min. Furthermore, microreactors have shown advantages in the characterization of nitration kinetics, mainly due to the high degree of controllability over the temperature and concentrations in the multiphase microenvironment (Cui et al., 2022, Li et al., 2017, Li et al., 2022, Song et al., 2022a, Song et al., 2022b).
Among various nitroaromatic compounds, m-dinitrobenzene (m-DNB) is an important basic industrial chemical, which mainly used to synthesize m-phenylenediamine and m-nitroaniline (Duan et al., 2022, Zhao et al., 2007). m-DNB is generally prepared by nitration of nitrobenzene (NB), as expressed by Eq. (1). In this process, other two isomers, i.e., o-dinitrobenzene (o-DNB) and p-dinitrobenzene (p-DNB), are also produced.
In the industrial production, nitration of nitrobenzene is generally carried out in tank reactors at 5090 °C with the reaction time of 26 h, the molar ratios of 0.92.2:1 and 1.011.12:1 for H2SO4/HNO3 and HNO3/NB, respectively (Gao et al., 2020, Xu et al., 2018, Zhang et al., 2020). Under such conditions, the products, o-DNB, m-DNB, p-DNB, are yielded in a ratio of around 12:85:3. As discussed above, the fundamental drawbacks of macroscale reactors (e.g., low space time yield, long reaction time and high safety risk) still hinder the process improvement towards economically viable and environmentally sustainable manufacturing.
Motivated by the compelling advantages of microreactors, a continuous flow microreactor system was developed to perform the safe and efficient nitration of nitrobenzene in this work. The reaction performance was studied in relation to reaction temperature, sulfuric acid strength, molar ratio of substrates and catalysts, flow rate and the residence time. According to experimental results, a kinetic model was developed, based on which the kinetic parameters, such as the reaction rate constant, pre-exponential factor and activation energy, were obtained. The developed kinetic model was further validated by comparing the predicted data with the experimental results, and proved applicable even though the reaction conditions were changed.
Section snippets
Chemicals
Nitrobenzene (C6H5NO2, 99%), fuming nitric acid (HNO3, 98%) and sulfuric acid (H2SO4, 98%) were purchased from Sinopharm Chemical Reagent Beijing Co., Ltd. Methanol (CH3OH, HPLC) was procured from Merck Drugs Biotechnology.
Experimental setup and procedures
Fig. 1 shows the continuous flow experimental setup for nitrobenzene nitration. It contained two high-pressure syringe pumps (Harvard 11 ELITE, USA, 1.26 pL/min-88.4 mL/min, ±0.5%) equipped with two glass syringes (10 mL), two feeding microchannels (fluorinated ethylene
Reaction regime
On the basis of the results collected during the previous investigations, nitronium ion (NO2+) is the real nitrating species (Guo et al., 2023, Li et al., 2022, Rahaman et al., 2010, Song et al., 2022a, Song et al., 2022b), hence a simplified reaction scheme for the nitration of nitrobenzene has been proposed as shown in Eqs. (9), (10) (Rahaman et al., 2010). Firstly, the reactive species NO2+ is released through the protonation of nitric acid in the presence of sulfuric acid. Then, NO2+
Conclusions
To develop a safe and efficient process for the synthesis of m-dinitrobenzene, nitration of nitrobenzene with mixed acid was investigated in a continuous flow microreactor. Various operating parameters were firstly examined to quantify their effects on the reaction characteristics. Furthermore, a pseudo-homogeneous kinetic model was developed, based on which the kinetic parameters of the reaction were determined. Finally, the model was validated by comparison with the experimental data.
The
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgements
We gratefully acknowledge the financial supports from Youth Innovation Team Plan of Shandong Province (Nos. 2022KJ270, 2019KJC012), National Natural Science Foundation of China (Nos. 21808194, 21978250), and Natural Science Foundation of Shandong Province (Nos. ZR2020QE211, 2019JZZY010410).
References (36)
X. Duan et al.
### Hydrogenation kinetics of m-dinitrobenzene in a continuous micro-packed bed reactor
### Chem. Eng. Sci.
(2022)
S. Guo et al.
### Nitration of o-xylene in the microreactor: reaction kinetics and process intensification
### Chem. Eng. J.
(2023)
A.A. Kulkarni
### Continuous flow nitration in miniaturized devices
### Beilstein J. Org. Chem.
(2014)
L. Li et al.
### Experimental and kinetic study of the nitration of 2-ethylhexanol in capillary microreactors
### Chem. Eng. Process.
(2017)
S. Li et al.
### Continuous flow nitration of 3-[2-chloro-4-(trifluoromethyl) phenoxy] benzoic acid and its chemical kinetics within droplet-based microreactors
### Chem. Eng. Sci.
(2022)
P.A. Quadros et al.
### Benzene nitration: validation of heterogeneous reaction models
### Chem. Eng. Sci.
(2004)
P.A. Quadros et al.
### Continuous adiabatic industrial benzene nitration with mixed acid at a pilot plant scale
### Chem. Eng. J.
(2005)
S.V. Ravikumar Bandaru et al.
### Mass transfer of chlorobenzene in concentrated sulfuric acid
### Int. J. Heat Mass Transf.
(2011)
D. Russo et al.
### Intensification of nitrobenzaldehydes synthesis from benzyl alcohol in a microreactor
### Org. Process Res. Dev.
(2017)
D. Russo et al.
### Heterogeneous benzaldehyde nitration in batch and continuous flow microreactor
### Chem. Eng. J.
(2019)
J. Song et al.
### Determination of nitration kinetics of p-nitrotoluene with a homogeneously continuous microflow
### Chem. Eng. Sci.
(2022)
- B.A.A. van Woezik et al.
### Runaway behavior and thermally safe operation of multiple liquid-liquid reactions in the semi-batch reactor: The nitric acid oxidation of 2-octanol
### Chem. Eng. Process.
(2002)
- Z. Wen et al.
### Process development and scale-up of the continuous flow nitration of trifluoromethoxybenzene
### Org. Process Res. Dev.
(2017)
- C.Q. Yao et al.
### Multiphase processes with ionic liquids in microreactors: hydrodynamics, mass transfer and applications
### Chem. Eng. Sci.
(2018)
- J.M. Zaldivar et al.
### Aromatic nitrations by mixed acid. Slow liquid-liquid reaction regime
### Chem. Eng. Process.
(1995)
- L. Zhang et al.
### Continuous-flow process for the preparation of m-dinitrobenzene from nitrobenzene
### Org. Prep. Proced. Int.
(2020)
- S. Zhao et al.
### Selective hydrogenation of m-dinitrobenzene to m-nitroaniline catalyzed by PVP-Ru/Al2O3
### Catal. Commun.
(2007)
- J.R. Burns et al.
### A microreactor for the nitration of benzene and toluene
### Chem. Eng. Commun.
(2002)
Cited by (19)
Biogenic synthesis of silver nanoparticles/reduced graphene oxide (AgNPs/rGO) mediated Nephelium lappaceum leaf extract as an effective solid acid catalyst for liquid-phase benzene nitration
2025, Results in Chemistry
Researchers are increasingly required to adopt green chemistry principles in developing nanomaterials and their applications to support a sustainable chemical industry and contribute to environmental balance. This study presents the green synthesis of AgNPs/rGO nanocomposites using Nephelium lappaceum as both a bio-reductant and a capping agent, alongside their characterization and application as a solid acid catalyst in benzene nitration reactions. Graphene oxide (GO) was synthesized from graphite using a modified Hummers method, and the AgNPs/rGO nanocomposites were prepared via an ex-situ approach. In this process, Nephelium lappaceum leaf extract served to reduce both silver ions and GO to rGO. The results revealed that the silver nanoparticles were spherical, with an average diameter of 6.76 nm, and were uniformly deposited on the reduced graphene oxide (rGO) surface. Catalyst performance tests demonstrated that the synthesized AgNPs E5/rGO catalyst achieved excellent yields and selectivity in converting benzene into nitrobenzene. Furthermore, the catalyst exhibited remarkable reusability, with only a slight decrease in yield after five uses, while maintaining consistent selectivity toward nitrobenzene as an essential and promising chemical intermediate for many derived chemicals.
### Design and optimization of novel vortex microreactors for ultrasound-assisted synthesis of high-performance Fe3O4 nanoparticles
2024, Chemical Engineering Journal
Microreactors excel in nanomaterial preparation but are limited by microchannel clogging for sustained long-term use. This study reports an innovative design of an ultrasound-assisted vortex microreactor for the continuous synthesis of high-performance nano-Fe3O4 particles. Combining visual experiments with computational fluid dynamics (CFD) simulations, four vortex microreactors were designed, and their mixing and heat transfer processes were investigated. Through comprehensive analysis, the microreactor-4 was identified as the optimal configuration, with an optimal flow rate of 1 mL/min and a temperature of 70 °C. By coupling the microreactor with ultrasound, a continuous preparation method for nano-Fe3O4 was realized. Scanning electron microscopy (SEM), transmission electron microscopy (TEM), and X-ray diffraction (XRD) analyses revealed that the synthesized nano-Fe3O4 particles exhibit a spherical crystal morphology with an average particle size of approximately 6.68 nm, which is 24.4 % and 20.5 % smaller than those prepared by the beaker method and by a stirred-field coupled microreactor reported in the literature, respectively. Vibrating sample magnetometry (VSM) measurements indicated a saturation magnetization of 45.75 emu/g for the nano-Fe3O4, representing a 32.3 % increase over the beaker method and demonstrating excellent superparamagnetic properties. This study provides a novel and effective pathway for the continuous preparation of nanoscale magnetic materials.
### Characteristics of numbering-up and flow distribution of multi-channel microreactor with 2-D constructal inlet distributors
2024, Chemical Engineering Science
Citation Excerpt :
Microchannel reactor often claimed to have better mass and heat transfer characteristics (Guo et al., 2021; Musci et al., 2020). It is safer in operation with highly endothermic and exothermic reactions (Abolhasani et al., 2014; Jin et al., 2023; Kuijpers et al., 2017; Su et al., 2016; Yue, 2022). It also helps reduce the lag-time between lab and industrial production (Pereira et al., 2021; Sen et al., 2022; Xie et al., 2021).
In this work, the flow distributions of 16-, 8-, 4-, 2-channels microreactors with 2-D constructal inlet distributors were measured. The results showed an increase in channel numbers caused a decreasing uniformity of the flow distribution. Both asymmetrical velocity profiles before a bifurcation point and asymmetrical flow resistance after a bifurcation point played an important role on the flow distribution in microchannels. The uniformity of flow distribution showed a trend of firstly increasing and then remaining unchanged with the increase of Re (4 ¼ 680). The pressure drops of microreactors were also investigated and the empirical method is in the acceptable range for estimating the energy consumption. Extending the channel length after bifurcation point from 124.46 mm to 160.95 mm and incorporating a bypass pressure equalization chamber at bifurcation point could improve the uniformity of flow distribution.
### 3D-printed devices for continuous-flow lithium recovery of brines
2024, Desalination
Adsorption processes were widely used in the recovery of lithium from salt lakes. Fixed-bed continuous flow reactors were mainly used in adsorption due to their simple structure and stable reaction. 3D printing, due to its ability to greatly reduce the packing and fixed-bed reactor design cycle and cost, was gradually becoming as one of the key technologies for intelligent manufacturing of fixed beds. This work utilizes the advantages of digital preparation by 3D printing to address the low elution efficiency and adsorption capacity of titanium-based lithium-ion sieves (Ti-LIS) in fixed-bed lithium recovery. 3D printed fixed bed packing with high adsorption capacity were prepared using an in-situ growth strategy, acidic gel and reactor design were utilized to achieve efficient elution of Ti-LIS. The dynamic adsorption showed that the adsorption capacity of 3D printed fixed bed packing could reach 16 mg·g1 and the elution efficiency can be increased from 78 % to 95 % with the same amount of acid.
### Highly selective oxidation of glucose to formic acid and exploring reaction pathways using continuous flow microreactors
2024, Fuel
Citation Excerpt :
Additionally, circulation eddies triggered by inter-phase velocity disparities intensify phase mixing and boundary layer renewal, resulting in a more robust driving force and inter-phase mass transfer rate . These characteristics make microchannel reactors capable of circumventing the challenges associated with wide residence time distribution and low selectivity due to back-mixing . Moreover, by increasing parallel channel numbers and optimizing distributor design, the scalability of these reactors is enhanced, facilitating potential industrial-scale applications.
The process of converting glucose into formic acid (FA) has been explored as a promising route for using biomass as a renewable feedstock for green chemical manufacturing. However, the most commonly used batch-tank reactor systems currently suffer from several drawbacks, including extended processing time, limited product selectivity, and high energy consumption, while the underlying reaction paths have been poorly understood. In this work, a continuous flow microchannel reactor was employed for the glucose conversion catalyzed by H5PV2Mo10O40 with O2 as an oxidant. The influence of various parameters on the conversion rate of glucose and the FA yield was characterized. Experimental results indicated that with a residence time of less than 3 min, the glucose was completely converted, and the highest FA yield reached 82.40 %. Extensive examination on the apparent by-products revealed that most of them acted as intermediates to produce FA through various pathways, including glyoxal, glyceraldehyde and glycolaldehyde as key intermediates for the oxidation of glucose to FA. Furthermore, the density functional theory (DFT) method was used to determine the bond energies of different CC bond cleavage modes of substrate glucose and fructose produced by the isomerization of glucose. Experimentally, the conversion of three main intermediates and some other possible intermediates and their FA yield were measured with different residence time. It was also found that the CO2 was produced through the decarboxylation of α-hydroxy and α-carbonyl carboxylic acid compounds, while the aldehyde groups in the compounds were more likely converted to FA by the α-carbon bond cleavage. Finally, plausible reaction pathways were proposed for the process of glucose-to-FA catalyzed by HPA-2, providing useful guidance for the identification of side reaction pathways and further improvement of FA yield.
### Continuous-Flow Microreactor System for Enhanced Selectivity and Safety in Nitrobenzene Nitration
2025, ACS Omega
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10916 | https://www.webassign.net/sdsucalclab1/assets/lab_04.html | Lab 4: Simplifying Algebraic Expressions
The following material should be read prior to attending lab. You are responsible for preparing for lab so that you don't slow down your group.
This section focuses on how to simplify algebraic expressions. In calculus, you will often need to simplify algebraic expressions when differentiating so that you can find a second derivative, find where the derivative equals 0, or to solve an application problem. This might be a good time to go back and review Lab 1 on factoring expressions. Two really important things to remember when simplifying expressions are:
When factoring out a common factor, ALWAYS factor out the smallest power—this can be tricky when dealing with negative exponents and fractional exponents.
Some expressions may be easier to deal with in calculus if you eliminate all negative exponents.
In general, when you learn the product rule, quotient rule, and chain rule, your calculus instructor may not make you simplify your answers. However, later on in the course, you will NEED to simplify your answers so you can think of simplifying your answers early in the course as practice for when you need it. The following examples will illustrate types of problems that you may need to simplify in your calculus course.
Example 4.1:
Simplify the expression $x^{3/2}e^{x} + \frac{3}{2}x^{1/2}e^{x}$ and write your answer using only positive exponents.
| | | | | |
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| Take out the common factors of $e^{x}$ and $x^{1/2}$. Remember that you always factor out the smallest power of x and 1/2 is smaller than 3/2. | $x^{3/2}e^{x} + \frac{3}{2}x^{1/2}e^{x}$ | $=$ | $e^{x}x^{1/2}\left(x + \frac{3}{2}\right)$ | When you factor out $x^{1/2}$, what are you left with in the first term? You need to find the power for x that you need so that $x^{1/2}x^{\Box{}} = x^{3/2}$. Using the laws of exponents and keeping the base and adding exponents will help you see that the number needed in the box is 1. |
| $ = $ | $ \sqrt{x}e^{x}\left(\frac{2x \,+\, 3}{2}\right) $ |
Example 4.2:
Simplify the expression $\frac{(1 \,-\, t^2)(3t^2) \,-\, t^3(-2t)}{(1 \,-\, t^2)^2}$ and write your answer using only positive exponents.
| | | | | | |
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| Often when we look at this denominator, we might be tempted to expand it by multiplying it out—DO NOT DO THIS! Often with problems such as these, you want the denominator in factored form because something may reduce later on. In this example, there is no reduction, but we still should leave it in factored form. | $\frac{(1 \,-\, t^2)(3t^2) \,-\, t^3(-2t)}{(1 \,-\, t^2)^2}$ | $=$ | $\frac{t^2[3(1 \,-\, t^2) \,+\, 2t^2]}{(1 \,-\, t^2)^2}$ | | There is a common factor of $t^2$ in both terms in the numerator so we factor that out. Then we simplify the portion in brackets by using the distributive property and combining like terms. |
| | $ = $ | $ \frac{t^2[3 \,-\, 3t^2 \,+\, 2t^2]}{(1 \,-\, t^2)^2} $ | |
$=$ | $\frac{t^2[3 \,-\, t^2]}{(1 \,-\, t^2)^2}$ |
Eventually, you will recognize the previous examples as derivatives that were found using the product and quotient rules. Another important derivative rule is the chain rule and the next two examples focus on it, combined with the product and quotient rules.
Example 4.3:
Simplify the expression $(2t - 5)^4[-3(8t^2 - 5)^{-4}(16t)] + (8t^2 - 5)^{-3}[4(2t - 5)^3(2)]$ and write your answer using only positive exponents.
| | |
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| $\small{(2t - 5)^4[-3(8t^2 - 5)^{-4}(16t)] + (8t^2 - 5)^{-3}[4(2t - 5)^3(2)]}$ | |
| $ = $ | $\small{8(2t - 5)^3(8t^2 - 5)^{-4}[-6t(2t - 5) + (8t^2 - 5)]}$ | We first factor out the common factors of 8, $(2t - 5)^3$, and $(8t^2 - 5)^{-4}$. This leaves us with $-6t(2t - 5)$ in the first term and $(8t^2 - 5)$ in the second term. |
| $ = $ | $\small{8(2t - 5)^3(8t^2 - 5)^{-4}[-12t^2 + 30t + 8t^2 - 5]}$ |
| $ = $ | $\small{8(2t - 5)^3(8t^2 - 5)^{-4}[-4t^2 + 30t - 5]}$ |
| $ = $ | $\frac{8(2t \,-\, 5)^3(-4t^2 \,+\, 30t \,-\, 5)}{(8t^2 \,-\, 5)^4}$ | Then we use algebra to simplify the expression in brackets and write the final answer with positive exponents only. |
Example 4.4:
Simplify the expression $\frac{\sin^2(x^2 \,+\, 1)(2x) \,-\, (x^2 \,+\, 1)[2 \, \sin(x^2 \,+\, 1)\cos(x^2 \,+\, 1)(2x)]}{\sin^4(x^2 \,+\, 1)}$ and write your answer using only positive exponents.
| | |
--- |
| $\frac{\sin^2(x^2 \,+\, 1)(2x) \,-\, (x^2 \,+\, 1)[2 \, \sin(x^2 \,+\, 1)\cos(x^2 \,+\, 1)(2x)]}{\sin^4(x^2 \,+\, 1)}$ | |
| $ = $ | $\frac{2x \, \sin(x^2 \,+\, 1)[\sin(x^2 \,+\, 1) \,-\, 2(x^2 \,+\, 1)\cos(x^2 \,+\, 1)]}{\sin^4(x^2 \,+\, 1)}$ | Here we factor out the common factors from each term in the numerator. What are they? |
| $ = $ | $\frac{2x[\sin(x^2 \,+\, 1) \,-\, 2(x^2 \,+\, 1)\cos(x^2 \,+\, 1)]}{\sin^3(x^2 \,+\, 1)}$ | There isn't much simplification in this problem, but we can reduce the power in the denominator by dividing out one of our common factors. | |
10917 | https://brainly.com/question/28729451 | [FREE] What is the answer to this? 1.93 × 9 - brainly.com
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What is the answer to this?
1.93 × 9
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The result of multiplying 1.93 by 9 is 17.57. You multiply the two numbers as usual, ensuring the decimal is placed correctly. Therefore, 1.93×9=17.57.
Explanation
To solve the question 1.93×9, we will start by multiplying the two numbers step-by-step.
Understanding the Numbers: We have 1.93 (a decimal number) and 9 (a whole number).
Performing the Multiplication: You multiply them as follows:
1.93×9=17.57
To do this, you can treat it like a standard multiplication. First, multiply 9 by the whole number part of 1.93 (which is 1), and then multiply 9 by the decimal part (which is 0.93).
Calculation:
9×1=9
9×0.93=8.37
Now, add these two results together:
9+8.37=17.37
Final Answer:
The final answer is 17.57. Therefore, 1.93×9=17.57.
Remember: When multiplying decimals, keep track of the decimal place, and if you need to round it, count the digits in the multiplied numbers after the decimal point to decide where to place the decimal in the result.
Examples & Evidence
For example, if you want to calculate 2.5×4, you would do: 2.5×4=10.0. Similarly, for 3.2×5, it would be 3.2×5=16.0.
Multiplying decimals follows the same rules as whole numbers; ensure that the placement of the decimal in the final answer reflects the digits in the original numbers being multiplied.
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10918 | https://ceridap.eu/ensuring-the-rule-of-law-under-martial-law-a-comparative-study-of-constitutional-mechanisms/ | La Rivista
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Ensuring the Rule of Law under Martial Law: A Comparative Study of Constitutional Mechanisms
Post author Stella Kelbia, Roman Lutskyi, Vitalii Skomorovskyi, Olexandr Polishchuk, Maksym Lysak
Ago 4, 2025
DOI: 10.13130/2723-9195/2025-3-37
Tags: Anticorruzione, Diritti e Libertà fondamentali, Diritto amministrativo comparato, Pubblica amministrazione
Ensuring the Rule of Law under Martial Law: A Comparative Study of Constitutional Mechanisms
Post author Stella Kelbia, Roman Lutskyi, Vitalii Skomorovskyi, Olexandr Polishchuk, Maksym Lysak |
4 Agosto 2025
DOI: 10.13130/2723-9195/2025-3-37
Tags: Anticorruzione, Diritti e Libertà fondamentali, Diritto amministrativo comparato, Pubblica amministrazione
Scopo dello studio è di esaminare le peculiarità inerenti alla garanzia dello stato di diritto in regime di legge marziale e l’esperienza e la prassi internazionale di diversi Paesi (tra cui gli Stati Uniti, il Regno Unito, la Francia, Israele e l’Ucraina). Esso pone in evidenza la necessità di trovare un equilibrio tra sicurezza nazionale e protezione dei diritti umani. Lo studio ha rilevato che le misure restrittive, come i coprifuoco, le restrizioni alla libertà di movimento e di riunione, la censura e i tribunali militari, sono d’uso comune in molti paesi durante la legge marziale. Tuttavia, la portata e la durata di tali restrizioni variano notevolmente. Lo stato di diritto richiede una disciplina normativa chiara, una magistratura indipendente, meccanismi di governance efficaci e la trasparenza degli organi statali. Lo studio ha sottolineato il ruolo della cooperazione internazionale e dello scambio di esperienze nel rafforzamento dello stato di diritto durante i conflitti militari.
The study explores the rule of law under martial law in various states (including the United States, the United Kingdom, France, Israel, and Ukraine) and highlights the need for a balance between national security and protection of human rights. The study found that restrictive measures, such as curfews, restrictions on freedom of movement and assembly, censorship and military courts, were common in many countries during martial law. However, the extent and duration of these restrictions vary greatly. An effective rule of law requires clear legal regulation, an independent judiciary, effective governance mechanisms, and transparency of state bodies. Successful cases demonstrate the importance of international cooperation and exchange of experience in strengthening the rule of law during military conflicts.
Summary: 1. Introduction.- 2. Materials and Methods.- 3. Results.- 3.1. Interrelationships and factors affecting the rule of law under martial law.- 3.2. Chronological analysis and comparative historical analysis of sources.- 4. Discussion.- 5. Conclusions.
1. Introduction
Ensuring the rule of law under martial law is a pressing issue of our time, especially given the growing number of armed conflicts and crises in the world. Martial law, as an emergency legal regime, provides the state with the expanded powers necessary to effectively respond to threats to national security. However, these powers carry the risk of restricting human rights and freedoms, which may lead to a violation of the fundamental principles of the rule of law. Therefore, finding an optimal balance between security and human rights under martial law is an important task for legal scholarship and practice.
The issue of ensuring the rule of law under martial law has been actively studied in the international scientific discourse. Dyzenhaus examined the constitutional aspects of the legal regime during emergencies, focusing on the mechanisms of control over the actions of the authorities and the protection of human rights. He pointed out the importance of preserving the independence of the judiciary and the role of civil society in ensuring the rule of law during martial law. Neff studied the law of armed conflict, analysing permissible legal restrictions and their impact on the legal status of individuals. He noted that international humanitarian law establishes the limits of permissible restrictions on human rights during war, aimed at protecting civilians and preventing unnecessary suffering.
In the Ukrainian context, the issue of ensuring the rule of law under martial law has become particularly relevant after the outbreak of full-scale aggression in 2022. Kozyubra studied the issue of ensuring human rights and freedoms under martial law, emphasizing the need for strict compliance with international human rights standards. The author highlighted that even under martial law, the state is obliged to ensure fundamental human rights, such as the right to life, freedom from torture and the right to a fair trial. The study shows that Ukraine has a positive track record in fulfilling its international human rights obligations, even under martial law.
In a broader context, Hirschl noted the importance of comparative analysis of constitutional law in different countries to understand the peculiarities of ensuring the rule of law in crisis conditions and situations. His research described that different countries have different approaches to balancing security and human rights during martial law, and this experience may be useful for Ukraine. Sikkink examined the role of the international community in ensuring human rights during armed conflicts and crises. She emphasized the importance of international pressure and cooperation to ensure respect for human rights during wartime. Cane studied the historical development of UK constitutional law, including periods of war and emergency. His work demonstrated that even in such difficult conditions, the British legal system remained committed to the fundamental principles of the rule of law, such as the independence of the judiciary and respect for human rights.
The gap in research is the lack of a comprehensive approach to the problem of ensuring the rule of law under martial law. Existing research works have mainly focused on certain aspects of this problem. However, there is a lack of studies that would combine these different aspects into a single conceptual framework and consider them in interconnection. In particular, insufficient attention has been paid to how the theoretical foundations of the rule of law, such as the principle of legality, independence of the judiciary, and protection of human rights, can be implemented in practice under martial law, when the state has expanded powers and faces the need to take swift and decisive action. In addition, existing studies have not always considered the specifics of different countries and their historical experience in ensuring the rule of law during wartime. For instance, studies conducted in countries with developed democracies and strong legal institutions may not be fully applicable to countries with less developed legal cultures or those experiencing protracted armed conflicts.
There has also been insufficient attention paid to the interaction of different legal institutions and mechanisms that ensure the rule of law under martial law. For example, how the judiciary, parliament, civil society and international organizations can cooperate to ensure respect for human rights and limit the excessive use of state powers. Thus, there is a need for a comprehensive study that would take into account both the theoretical foundations of the rule of law and the practical experience of different countries in ensuring this principle during martial law, as well as analyse the interaction of various legal institutions and mechanisms in this process. The research was conducted to determine the details of ensuring the rule of law under martial law and to study international experience and practice.
2. Materials and Methods
The article uses a systematic method to study the peculiarities of ensuring the rule of law under martial law. The author defines the boundaries of the rule of law system, identifies its main elements (legislation, judiciary, law enforcement agencies, civil society) and defines their functions under martial law. The author analyses the interrelationships between the elements of the system, identifies direct and reverse relationships, and determines the factors affecting their effectiveness during wartime (restriction of rights and freedoms). Models have been developed that reflect the functioning of the rule of law system in different martial law conditions (e.g., in a full-scale war, local conflicts, occupation). This made it possible to assess the impact of various factors on the rule of law and to predict possible scenarios.
Comparative legal methods were used to analyse the legal systems of different countries that have introduced martial law. The legal norms of the United States, the United Kingdom, France, Israel and Ukraine were studied. This made it possible to identify common features and differences in the approach to the introduction of martial law, to assess the effectiveness of various legal mechanisms and their impact on human rights. Through thematic categorisation of institutional tasks within six functional domains (legislative, executive, judiciary, law enforcement, civil society, and international cooperation) comparison tables (Tables 1-4) were created. Each domain’s content was taken from government decrees, court rulings, and national legal frameworks. These categories were chosen in light of frequent cross-jurisdictional references in scholarly commentary and legal writings. This methodology made it possible to compare institutional and legal practices under martial law in the five case study countries in a methodical manner.
Historical methods were used, namely chronological analysis and comparative historical analysis of sources. To study the evolution of legal regulation of martial law and its impact on the rule of law. An analysis was made of historical data on the introduction of martial law in different countries, as well as its impact on the legal system and society. This made it possible to understand how the approach to the legal regulation of martial law has changed over time and what lessons can be learnt from experience.
Analytical methods were used to analyse the information received, identify important trends and formulate results. International and national legal norms, judicial decisions of international and national courts, scientific publications and reports of international organizations such as Amnesty International, Human Rights Watch and Freedom House were analysed. This made it possible to assess the degree of human rights observance under martial law and identify the factors that affect the effectiveness of legal mechanisms. The analysis used included reports and studies, policy, and case law. The main sources of research were international treaties, domestic laws, Conseil constitutionnel, court rulings and reports of international organizations. The main sources include the judgments of the European Court of Human Rights, the Supreme Court of Ukraine and domestic legislation on the state of emergency in the country selected for comparative analysis.
In order to maintain methodological clarity, the study’s empirical component is based on a purposeful selection of court rulings, human rights monitoring reports, and normative legal acts from 2000 to 2024. Targeted searches in publicly available legal databases (such as HUDOC, HeinOnline, Legifrance, the official websites of the Knesset and the Constitutional Court of Ukraine), as well as repositories of global human rights organisations like Amnesty International, Human Rights Watch, and Freedom House, were used to find the source materials. The inclusion criteria included the suitability, relevance, and accessibility of the information related to the introduction of martial law. The sample was representative and included documents from different countries and international organizations. Violations were classified according to their category and evaluated based on whether or not there were adequate legal remedies. Fisher’s exact test (significance threshold: p<0.05) was used for statistical testing to assess the association between the frequency of violations and the availability of legal remedies. The R software environment (v.4.3.1) was used for data processing, and established protocols for categorical data analysis were followed to guarantee analytical robustness. The analysis of statistical data helped to identify crucial trends and problems in ensuring the rule of law under martial law.
3. Results
3.1. Interrelationships and factors affecting the rule of law under martial law
The rule of law under martial law is a complex mechanism whose effectiveness depends on the interaction of its components and the influence of various factors. These relationships fall into three main categories: horizontal (between different branches of government and civil society), vertical (hierarchy and control) and external (international cooperation). In wartime, vertical ties become particularly important as they ensure prompt and efficient decision-making. The parliament adopts laws that regulate this specific regime, the executive branch (government, president) is responsible for their implementation, and the judiciary controls the legality of the actions of other branches of government. Law enforcement agencies and military forces ensure security and law and order, while civil society plays a key role in monitoring human rights and providing quality legal aid. The state can also cooperate with the United Nations (UN), the Organization for Security and Cooperation in Europe (OSCE) and other international organizations in the field of human rights protection, share experiences with other states on best practices for ensuring the rule of law under martial law and ensure compliance with international humanitarian law during hostilities.
The effectiveness of the rule of law system under martial law depends not only on the interaction of its elements, but also on several internal and external factors. Internal factors include the legal framework, institutional capacity, resources and public opinion. External factors include international support, information warfare and the geopolitical situation. The clarity and integrity of the legal framework and its compliance with international standards are crucial. Legislation that complies with international standards, such as the International Covenant on Civil and Political Rights and the European Convention for the Protection of Human Rights and Fundamental Freedoms, prevents ambiguous interpretation and abuse of power, and provides a predictable and stable legal environment even under martial law.
The judiciary plays a key role in ensuring the rule of law during martial law, controlling the legality of restrictions and protecting human rights in the United States, the United Kingdom, France, Ukraine and Israel. In the United States, the Supreme Court limits executive power and ensures that measures are proportionate. In the UK, the courts review government actions for compliance with the Human Rights Act. The French Council of State controls the legality of restrictions during a state of emergency. In Israel, the Supreme Court restricts the actions of the military in the occupied territories. In Ukraine, the courts are functioning, and the Constitutional Court controls the legality of government decisions. All these examples show that the judiciary is an essential element of the system of checks and balances that guarantees human rights and prevents the state from abusing its power during martial law. The independence of the courts is a key to their effectiveness.
Restrictive measures during martial law are measures necessary to ensure state security and public order. They include curfews, restrictions on freedom of movement, bans on public events, censorship, and the establishment of military courts. The specific measures and their duration vary from country to country and from case to case. The legality of these measures is determined by the principle of proportionality. According to this principle, the restriction of human rights and freedoms must be appropriate and necessary to achieve a legitimate aim, such as national security. Restrictive measures must be clearly defined in law, applied only to the extent necessary, and must not violate fundamental human rights.
International humanitarian law (IHL) plays a pivotal role in establishing the permissible limits of human rights restrictions during conflicts, including martial law. The main purpose of IHL is to protect those not taking part in hostilities and to limit the methods and means of warfare. IHL establishes the rules for the treatment of prisoners of war and guarantees the right to humane treatment, medical care and communication, as well as other fundamental rights. The Geneva Conventions relative to the Treatment of Prisoners of War define in detail the conditions of detention, rights, and obligations of prisoners of war. For example, the Geneva Conventions prohibit the use of weapons “which by their nature are likely to cause unnecessary injury or suffering”. The International Covenant on Civil and Political Rights also allows for the restriction of certain rights in times of public emergency, but these restrictions must be proportionate to the threat and non-discriminatory. International courts, such as the International Criminal Court, play an important role in ensuring compliance with IHL. The Rome Statute of the International Criminal Court provides for jurisdiction over war crimes, including violations of IHL. The International Criminal Court investigates and prosecutes war crimes, including human rights violations, during conflict. The United States Constitution authorizes the President to declare martial law, but this power is not absolute and is subject to strict judicial review. The US Supreme Court has repeatedly emphasized the need to respect human rights even under martial law. In subsequent decisions, the Supreme Court has repeatedly pointed out that restrictive measures during martial law must be necessary and proportionate to the threat and must not violate fundamental human rights.
In the United Kingdom, the Armed Forces Act 2006 clearly defines the rights and responsibilities of the armed forces and establishes the legal framework within which they operate. In particular, it guarantees the right to a fair trial, protection from discrimination and access to medical care. The Constitution of France provides that certain rights and freedoms may be restricted during a state of emergency (l’état d’urgence), which is introduced in case of a serious threat to the institutions of the Republic, the independence and territorial integrity of the state or the fulfilment of France’s international obligations. In addition, all restrictive measures are subject to judicial review. This means that anyone who believes that their rights have been violated during the state of emergency can go to court to protect their rights. The Conseil d’État, the highest administrative court in France, has the right to review the legality and legitimacy of restrictive measures and to cancel them if they do not comply with the Constitution or the law. The Basic Laws of Israel do not contain specific provisions on martial law, but give the Knesset (parliament) the power to pass laws necessary to ensure the security of the state. The Defence Regulations (Times of Emergency), adopted before Israel’s independence, gives the government broad powers in the event of war or a threat of war. The Defence Regulations give the government the right to restrict freedom of movement, impose censorship, detain persons without trial and take other measures necessary to ensure security. While the Defence Regulations give the government broad powers, the Supreme Court plays a major role in overseeing the exercise of these powers and ensuring that human rights are respected.
After the start of Russia’s large-scale invasion in 2022, Ukraine was forced to declare martial law in accordance with the Decree of the President of Ukraine No. 11/2022 “On the Report on the Results of the Review of Public Security and Civil Defence”. This measure was necessary to ensure the security and defence of the country, but at the same time led to certain restrictions on the rights and freedoms of citizens. Ukraine has ratified important international human rights treaties, such as the European Convention for the Protection of Human Rights and Fundamental Freedoms and the International Covenant on Civil and Political Rights, and continues to fulfil its obligations under these treaties. Restrictions have been imposed on freedom of movement, freedom of assembly, freedom of expression and access to information. In addition, the martial law affects the judicial system, restricting access to justice and the ability to appeal against government decisions. There are also restrictions on freedom of expression, including on the dissemination of information that could harm national security. Thus, martial law creates a complex situation in which the state must strike a balance between ensuring national security and protecting human rights. Although states are obliged to comply with international human rights standards, martial law inevitably leads to certain restrictions. It is critical that these restrictions are proportionate to the threat, clearly defined in law and subject to effective judicial review. A study of the legislation and case law of the United States, the United Kingdom, France, Israel, and Ukraine confirms that no country is immune to this challenge.
A systematic analysis of the rule of law under martial law allows us to consider the legal system as a complex organism consisting of interconnected elements: legislative, executive and judicial branches of government, law enforcement agencies, military forces and civil society. The functioning of this organism changes under different martial law scenarios, requiring adaptation and flexibility to ensure the rule of law and the protection of human rights (Table 1-3).
Table 1: The “Full-scale war” model.
| | |
--- |
| Legislature | Adopts laws aimed at mobilizing resources by restricting certain rights and freedoms of citizens in the interests of national security. |
| Executive power | It carries out operational management of military operations and ensures the functioning of the state in times of war. |
| Judiciary | It plays a pivotal role in monitoring the legality of the actions of other branches of government and protecting human rights, although its functioning can be complicated. |
| Law enforcement agencies and military formations | They are strengthening their role in ensuring security and law and order. |
| Civil society | It plays a major part in monitoring human rights, providing assistance to victims and supporting the military, despite possible restrictions on freedom of speech and assembly. |
| International community | It can provide military, financial and humanitarian assistance, facilitate a diplomatic settlement of the conflict, monitor compliance with international humanitarian law and assist in the investigation of war crimes. |
Source: created by the authors based on United States Constitution, Anti-terrorism, Crime and Security Act 2001, Title VIII: Judiciary (Articles 64 to 66), Constitution of 4 October 1958, Constitution for Israel, Decree of the President of Ukraine No. 11/2022 “On the Report on the Results of the Review of Public Security and Civil Defence”, European Convention for the Protection of Human Rights and Fundamental Freedoms.
A situation where the state’s constitutional architecture is maintained but is prone to increased centralisation and quick legal adaptation is reflected in the full-scale war model. One of the most important findings is the change in institutional balance: the legislative now plays a reactive rather than an initiating role, while the executive branch takes on an extended operational role that is frequently justified by urgency and national security imperatives. Despite being legally retained, the judiciary has severe restrictions on institutional independence, access, and procedural protections. This results in a structurally imbalanced arrangement where checks and balances are undermined via tangible limitations, like restricted court operations, lowered requirements for admissible evidence, or postponed legal remedies, rather than by official suspension. Additionally, the paradigm implies that civil society, while active, functions under conditions of limited autonomy and is frequently mobilised to support rather than to challenge official authority. When wartime demands are not backed by concurrent investments in legal continuity and rights-based oversight systems, this pattern highlights the dangers of normative disintegration.
The local conflict model, on the other hand, shows a more unique and flexible constitutional response (Table 2). In this case, context-sensitive governance is made possible by the decentralised structure, in which the executive collaborates with non-state actors and regional institutions rather than monopolising power.
Table 2: Local conflicts model.
| | |
--- |
| Legislature | Adopts special laws to resolve conflict situations. |
| Executive power | Focuses on conflict resolution and security in the region. |
| Judiciary | He actively protects human rights and ensures fair trials. |
| Law enforcement agencies and military formations | They are involved in ensuring security in the conflict zone, conducting anti-terrorist operations and protecting civilians. |
| Civil society | More actively involved in peaceful settlement and protection of the rights of the local population. |
| International community | The international community can play a mediating role in conflict resolution, provide humanitarian aid, monitor human rights and provide expert support in restoring law and order. |
Source: created by the authors based on United States Constitution, Anti-terrorism, Crime and Security Act 2001, Title VIII: Judiciary (Articles 64 to 66), Constitution of 4 October 1958, Constitution for Israel, Decree of the President of Ukraine No. 11/2022 “On the Report on the Results of the Review of Public Security and Civil Defence”, European Convention for the Protection of Human Rights and Fundamental Freedoms.
According to Table 2, the judiciary maintains its adjudicative validity and functional accessibility, especially while handling human rights complaints and overseeing emergency procedures. This implies a situation when martial law is imposed proportionately and selectively. Notably, civil society is not only present in this model but also has a significant impact on efforts for accountability, rights monitoring, and conflict mediation. The model shows that the risk of arbitrary governance is much decreased when legal institutions are integrated into a responsive and networked structure. Furthermore, the public’s opinion of state legitimacy is influenced by the ability to maintain procedural justice in conflict areas. This model supports the importance of distinct legal frameworks that take into account the temporal and spatial subtleties of conflict dynamics.
The model that deviates from constitutional normativity the greatest is the occupation model (Table 3). The state’s capacity to function as a sovereign legal system is essentially suspended when legislative and judicial institutions are either abolished or appropriated. If executive authority does exist, it is either severely limited by the occupying military or is only symbolic. According to this concept, any horizontal accountability mechanisms are undermined when civil society collapses or is suppressed, which results in systematic repression, arbitrary rule, and widespread legal discontinuity. This arrangement highlights the importance of external players as the only sources of legal intervention and normative reference, including international institutions, human rights courts, and transnational investigative bodies. This approach shows how urgently international legal frameworks, such as occupation law and international humanitarian law, are needed to record transgressions and reinstate minimum rights. It also implies that previous documenting and acknowledgement of institutional collapse during occupation are crucial for post-conflict legal reconstruction. Thus, in the lack of institutional and territorial sovereignty, the paradigm emphasises the limitations of domestic constitutionalism.
Table 3: The “Occupation” model.
| | |
--- |
| Legislature | Significantly weakened or replaced by the occupation administration. |
| Executive power | Faces obstacles in the administration of justice. |
| Judiciary | Faces obstacles in the administration of justice. |
| Law enforcement agencies and military formations | They can continue to fight in the form of guerrilla or underground activities, or be disbanded and replaced by the occupying forces. |
| Civil society | He is subjected to repression. |
| International community | It plays a key role in protecting human rights and upholding the rule of law. |
Source: created by the authors based on United States Constitution, Anti-terrorism, Crime and Security Act 2001, Title VIII: Judiciary (Articles 64 to 66), Constitution of 4 October 1958, Constitution for Israel, Decree of the President of Ukraine No. 11/2022 “On the Report on the Results of the Review of Public Security and Civil Defence”, European Convention for the Protection of Human Rights and Fundamental Freedoms.
The functioning of the rule of law system under martial law shows that it depends on many factors, including the type of conflict, its intensity, the political situation in the country and the international situation. A systemic approach allows taking into account all these factors and developing the most effective strategies to ensure the rule of law and human rights protection in each case.
International cooperation and the exchange of experience play a pivotal role in strengthening the rule of law in times of armed conflict. International organizations such as the United Nations (UN), the Organization for Security and Cooperation in Europe (OSCE) and the Council of Europe have developed standards and guidelines for the protection of human rights in times of war. The Office of the United Nations High Commissioner for Human Rights (OHCHR) monitored the human rights situation in the conflict zones, provided expert assistance to states and investigated violations. The OSCE monitored the human rights situation in the conflict zones, supported war crimes investigations and facilitated dialogue between the parties to the conflict. The Council of Europe developed theEuropean Convention for the Protection of Human Rights and Fundamental Freedoms. The European Court of Human Rights (ECHR) has considered and ruled on numerous cases of human rights violations during military conflicts. States have exchanged information on war crimes suspects, provided legal assistance in investigations, and transferred suspects for prosecution. For example, cooperation between Ukraine and the Netherlands led to the establishment of a Joint Investigation Team (JIT) to investigate the downing of flight MH17 in eastern Ukraine in 2014. International cooperation and exchange of experience are important tools for strengthening the rule of law during armed conflicts. This helps to protect human rights, ensure justice and bring perpetrators of war crimes to justice. Table 4 shows the results of a comparative analysis of the legal systems of each country (the United States, the United Kingdom, France, Israel, and Ukraine). Elements and functions of the rule of law.
Table 4: Elements and functions of the rule of law
| | | | | |
--- ---
| USA | Ukraine | Israel | France | United Kingdom |
| Legislature |
| The US Congress enacts laws governing martial law (the Insurrection Act, the Martial Law Act). Congress has the power to declare war, allocate funds for defence, and oversee the executive branch during martial law. | The Verkhovna Rada of Ukraine adopts laws regulating the legal regime of martial law. It also approves presidential decrees on the introduction and cancellation of martial law, and exercises control over the activities of the executive branch. | The Knesset (Israel’s parliament) adopts laws that regulate the activities of the military and law enforcement agencies in the security sector (the Basic Defence Law). The Knesset also approves government decisions to declare a state of war and can amend legislation. | The French parliament (National Assembly and Senate) adopts laws that regulate the state of emergency (l’état d’urgence) and the scope of the president’s powers during this period. Parliament also exercises control over the government’s actions during the state of emergency. | The UK Parliament makes laws governing the government’s emergency powers (Emergency Powers Acts) in times of war or other emergencies. Parliament also exercises control over the government’s actions during the emergency powers. |
| Executive power |
| The US judicial system continues to function under martial law, but some cases may be transferred to military courts. The U.S. Supreme Court has the power to review decisions of military courts and other authorities regarding the legality of their actions during martial law. | The Ukrainian judicial system continues to function during martial law, but some cases may be transferred to military courts. The Constitutional Court of Ukraine has the right to review the decisions of other authorities regarding the legality of their actions during martial law. | The Israeli judicial system continues to function during martial law, but some cases may be transferred to military courts. The Supreme Court of Israel has the right to review decisions of military courts and other authorities regarding the legality of their actions during martial law. | The French judicial system continues to function during a state of emergency, but some powers may be transferred to administrative authorities. The Conseil d’État (Council of State) oversees the legality of the government’s actions during a state of emergency. | The UK judicial system continues to function during the emergency powers, but the government may impose restrictions on certain court procedures. The UK courts have the power to review the government’s decisions on the legality of its actions during the emergency powers. |
| Law enforcement agencies |
| The FBI, state police, and other law enforcement agencies continue to operate under martial law, but may be given additional powers and tasks. | The SSU, the National Police and other law enforcement agencies continue to operate under martial law, but may be given additional powers and tasks. | The Israeli Police, Shin Bet (security service) and other law enforcement agencies continue to operate under martial law, but may receive additional powers and tasks. | The police, gendarmerie, and other law enforcement agencies continue to operate during the state of emergency, but may be given additional powers and tasks. | The police, intelligence services (MI5, MI6) and other law enforcement agencies continue to operate under emergency powers, but may be given additional powers and tasks. |
| Military formations |
| The United States Army plays a key role in ensuring national security and defence during martial law. They can be called upon to perform a variety of tasks. | The Armed Forces of Ukraine play a key role in ensuring national security and defence during martial law. They conduct combat operations and protect the country’s territorial integrity. | The IDF (Israeli Defence Forces) plays a key role in ensuring national security and defence during martial law. It conducts combat operations, protects the integrity of the country and exercises control over the occupied territories. | The Forces armées françaises (French armed forces) may be called upon to maintain law and order during a state of emergency, especially in the event of terrorist threats. | The British Armed Forces may be involved in maintaining law and order during emergency powers, especially in the event of terrorist attacks. |
Source: created by the authors based on ArtII.S2.C1.1.14 Martial Law Generally, Anti-terrorism, Crime and Security Act 2001, Title VIII: Judiciary (Articles 64 to 66), Constitution of 4 October 1958 (2024), Constitution for Israel (1958), Decree of the President of Ukraine No. 11/2022 “On the Report on the Results of the Review of Public Security and Civil Defence”.
Table 4 compares the roles of the main elements of the rule of law (legislative, executive and judicial branches, law enforcement agencies, the army and civil society) under martial law. The table shows which institutions are responsible for making laws, providing security, administering justice, maintaining law and order, and protecting the rights of citizens under martial law in each country.
3.2. Chronological analysis and comparative historical analysis of sources
These four conventions and their additional protocols form the basis of international humanitarian law (IHL). During armed conflict, states parties to the Geneva Conventions are obliged to protect medical personnel and facilities from attack. The International Covenant on Civil and Political Rights is one of the fundamental international human rights treaties adopted by the UN General Assembly in 1966. The fundamental rights guaranteed by the International Covenant on Civil and Political Rights include the right to life; freedom from torture and cruel, inhuman or degrading treatment or punishment; liberty and security of person; the right to a fair trial; freedom of thought, conscience, and religion; freedom of expression; freedom of peaceful assembly and freedom of association. However, such restrictions must be prescribed by law, be necessary to protect national security, public order, public health or morals, or the rights and freedoms of others, be proportionate to the aims pursued and not exceed what is strictly necessary.
Most countries have constitutional provisions and specific laws governing the introduction and operation of martial law. During martial law, the state may restrict freedom of movement, but such restrictions must be aimed at achieving a legitimate aim (e.g., protection of national security) and must not be excessive. For example, the Constitution of Ukraine (Articles 64, 85) and the Law of Ukraine No. 389-VIII “On the Legal Regime of Martial Law” define the conditions and procedure for the introduction of martial law. With the beginning of Russia’s full-scale invasion of Ukraine on 24 February 2022, martial law was introduced in the country. This allowed the government to introduce a number of restrictions aimed at ensuring national security and defence.
In its case law, the ECtHR has repeatedly emphasized that even under martial law, states are obliged to respect the fundamental human rights enshrined in the European Convention for the Protection of Human Rights and Fundamental Freedoms. Some rights, such as the right to life and the prohibition of torture, are absolute and cannot be restricted even during martial law. Other rights, such as freedom of movement, freedom of expression and freedom of assembly, may be restricted during martial law, but only if such restrictions are absolutely necessary for the achievement of a legitimate aim and are proportionate to that aim. The ECtHR has also repeatedly pointed out the importance of ensuring effective remedies for victims of human rights violations during martial law. This includes access to court, the right to a fair trial and the right to an effective investigation of complaints. In the case of Ukraine and the Netherlands v. Russia, the ECtHR found that Russia had exercised control over parts of eastern Ukraine since May 2014 and was responsible for numerous human rights violations in those territories. The ECHR case law plays an important role in shaping international standards for the protection of human rights under martial law; it is binding on Council of Europe member states and influences the development of national legislation and practice. In addition, the ECHR case law is an essential source for other international and national courts considering cases related to martial law. Thus, the ECHR is a crucial mechanism for protecting human rights under martial law and helps to ensure compliance with international standards and prevent states from abusing their powers.
Courts in different countries also considered cases related to martial law, and their decisions contributed to the development of national legislation and its application. The courts assessed the legality and proportionality of restrictions on rights and freedoms, protected the rights of detainees and prisoners of war, and controlled the actions of the authorities. The reports of Amnesty International, Human Rights Watch and Freedom House are a critical source of information on the human rights situation in the world, especially in countries with martial law or other emergency regimes. These organizations conduct independent research, document human rights violations, analyse the causes and consequences of such violations, and offer recommendations to governments and the international community to address them.
Figure 1: The number of cases of human rights violations during martial law in the presence/absence of effective legal protection mechanisms.
Source: created by the authors, based on Amnesty International and Human Rights Watch.
The test results showed that there is a statistically significant relationship between the type of human rights violation and the availability of effective remedies (p<0.05). This means that the availability of effective legal remedies has a significant impact on the likelihood of human rights violations under martial law. Countries with effective remedies, such as independent courts and active human rights organizations, are less likely to experience human rights violations than countries with no or ineffective remedies.
The results of the study show that the right to life and the right not to be subjected to torture are the most frequently violated rights, especially in countries where legal mechanisms are ineffective. This indicates that these rights are the most vulnerable under martial law and require special protection. At the same time, the research findings show that human rights violations occur even where effective legal mechanisms are in place. This indicates that even in the presence of an appropriate legal framework and institutions, martial law creates special conditions that complicate the protection of human rights. Therefore, this analysis highlights the importance of strengthening legal mechanisms for the protection of human rights in all countries, especially those under the threat of armed conflict. This includes ensuring the independence of the judiciary, establishing effective mechanisms for challenging government decisions, supporting the work of human rights organizations and raising legal awareness.
4. Discussion
The study confirms the complexity and multifaceted nature of the problem of ensuring the rule of law during martial law, which echoes the findings of many researchers in this area. For example, Dyzenhaus emphasized that martial law should not lead to a complete abandonment of legal principles, but rather aims to find a balance between security and human rights. This study expands on this assertion by showing that this balance is achieved not only by legal measures, but also by a comprehensive approach that includes the effective functioning of the judiciary, public oversight and international cooperation. Similarly, Hart, in his concept of law, pointed out the importance of primary and secondary rules for the structure and legitimacy of the legal system. In the context of martial law, this study has demonstrated that secondary rules, such as judicial oversight and parliamentary control, are critical to ensuring that the rule of law is upheld and preventing abuses of power. The study also found that restrictions on rights and freedoms during martial law should be clearly defined by law, necessary and proportionate to the threat, which is consistent with the position by Bandurka on the need to preserve the rule of law even during emergencies.
A historical analysis of the legal regulation of martial law in different countries has revealed a variety of approaches and decisions that have developed under the influence of historical, political and cultural factors. This was confirmed by the study by Hirschl, which emphasized the importance of taking into account the national context when analysing legal systems. However, like the study by Dyzenhaus, the present study revealed some general trends, such as the increasing role of international law and judicial review, which indicates the universality of some legal principles. These findings also echo those of Raymond, who found that international norms and standards increasingly influence national human rights legislation, even during times of armed conflict. The study by Miles also highlighted the influence of international factors on domestic policies, including the decision to impose martial law and restrictions on human rights. It is important to note that Cane’s study emphasized that the historical development of constitutionalism in the UK demonstrates the importance of a balance between the executive and other branches of government, especially in times of crisis, which was confirmed by the need for judicial and parliamentary control identified in this study.
A comparative analysis of the legal frameworks of different countries has shown that there is no single model of martial law regulation. This is consistent with the findings of the study by Neff, which emphasized that international humanitarian law establishes only a general framework, and that specific human rights protection mechanisms depend on national legislation. However, this study went further, highlighting some general principles that should be taken into account when developing and applying legal norms during martial law, such as the principle of legality, necessity, and proportionality of restrictions, the principle of non-discrimination and the principle of access to justice. These principles are also emphasized in the studies of other authors, such as Dworkin, who highlighted the importance of respecting human rights even in emergency situations. In addition, the study by Finnis emphasized the importance of natural law, which is the basis for any legal system, including the legal norms applied during martial law. Raz’sstudy also pointed out the importance of moral constraints on the exercise of state power, even in times of emergency, which is consistent with the findings of this study on the need to balance security and human rights. These findings are further supported by Chemerinsky, who analysed US constitutional law and emphasized the importance of judicial review of government actions during emergencies such as martial law.
The statistical analysis of the data showed that there is a correlation between the level of development of the legal system, the independence of the judiciary and the effectiveness of human rights protection during martial law. The results coincide with the study of Doronin, where he pointed out that national security is the main function of the state. This also confirms the findings of the study by Gur and Jackson, which showed that adherence to the rule of law is an important factor in ensuring public support and legitimacy of the legal system. In addition, the study by Kravchuk emphasized the importance of the independence of the judiciary to ensure effective human rights protection, especially during martial law, when there is an increased risk of abuse of power. This is also consistent with the arguments of Hart, who put an emphasis on the importance of an independent court to interpret and apply legal norms, especially in complex situations such as martial law. Levy’s study also emphasized the importance of institutional mechanisms, such as independent courts and parliamentary oversight, to ensure the rule of law and the protection of human rights during crises.
The study also identified certain challenges in ensuring the rule of law during martial law, such as the lack of a clear legal framework, insufficient judicial oversight, restrictions on civil society and lack of transparency in the activities of state bodies. This is in line with the findings of Sikkink, which highlighted the importance of international pressure and cooperation to ensure human rights during crises. In addition, a study by Reiter showed that martial law can be used by authoritarian regimes to consolidate power and restrict political freedoms. These conclusions are confirmed by Articles ArtII.S2.C1.1.14 Martial Law Generally and ArtII.S2.C1.1.13 President as Commander of Armed Forces, which analysed the historical experience of the United States and showed that martial law can lead to significant restrictions on civil liberties and abuse of power. The study by Neocleous also emphasized the potential risks of martial law for human rights and democratic values, as well as the study by Kostiuk.
Given these conclusions, it is important to point out that ensuring the rule of law during martial law is not only a legal, but also a political and moral necessity. As noted by Hayek in his work, the rule of law is the foundation of freedom and justice in society. During martial law, when there is an increased risk of abuse of power and human rights violations, the rule of law becomes even more relevant. This requires the state not only to comply with formal legal procedures, but also to adhere to fundamental principles of justice and human rights. As noted by Brooke-Holland and Mills, even during martial law, the state must ensure the protection of the rights of military personnel and veterans, which is an essential aspect of the rule of law and social justice.
5. Conclusions
The study provided a better understanding of the complex and multifaceted nature of ensuring the rule of law under martial law. It confirmed that this process requires a comprehensive approach that takes into account not only legal norms but also political, social and economic factors. The analysis of international legal norms, national legal norms, court practice and reports of international organizations showed that even under martial law, states are obliged to respect fundamental human rights and ensure the functioning of the legal system. This means that restrictions on rights and freedoms under martial law must be legitimate, necessary, and proportionate to the threat.
The study found that clear legal norms, an independent judiciary, effective administration and transparency of state institutions are key elements for ensuring the rule of law under martial law. Clear legal norms guarantee predictability and stability of the legal environment, an independent judiciary protects human rights and checks the legality of government actions, effective public administration ensures the implementation of legal norms and the maintenance of law and order, and transparency of state institutions increases public trust and prevents abuse of power.
An important outcome of this study was the identification of the role of international humanitarian law and international human rights law in establishing the limits of permissible restrictions on human rights during war. These norms help to ensure minimum human rights protection in armed conflict and prevent unnecessary suffering of the civilian population. The importance of international cooperation and exchange of experience in strengthening the rule of law during armed conflicts was also highlighted. International organizations and other states can provide support to countries at war in the form of expert assistance, financial support, monitoring of the human rights situation and assistance in the investigation of war crimes.
Statistical analysis of the data using the Fisher’s exact test confirms the hypothesis that the existence of effective legal protection mechanisms significantly affects the likelihood of human rights violations under martial law. In other words, strengthening the judiciary, supporting the work of human rights organizations and raising legal awareness are crucial steps to ensure that human rights are respected in wartime. However, the study also revealed certain difficulties in ensuring the rule of law under martial law. In particular, the absence of a clear legal framework and inadequate judicial oversight can lead to abuse of power and human rights violations. This underscores the need for further research in this area to identify and address these challenges.
As this study examined the experience of only five countries (the United States, the United Kingdom, France, Israel, and Ukraine), it cannot be concluded that the patterns and trends identified can be generalized. Future research could also examine the experience of countries such as India, South Africa, Colombia, Turkey, and Japan. As these countries have different legal systems and historical backgrounds, it may be possible to gain a broader picture of how to ensure the rule of law under martial law and identify the most successful practices. In addition, a historical study of martial law regulation will help to understand the history of martial law and identify the reasons for the emergence of modern legal norms.
D. Dyzenhaus, Legality in a time of emergency, in The Constitution of Law: Legality in a Time of Emergency, Cambridge University Press, Cambridge, 2006. ↑
S. Neff, The law of armed conflict, in R. Lesaffer, J.E. Nijman (ed.), The Cambridge Companion to Hugo Grotius, Cambridge University Press, Cambridge, 2021. ↑
M.I. Kozyubra, Philosophy of law, jurisprudence and science: Commonalities and differences, in Philosophy of Law and General Theory of Law, vol. 2, 2019, pp. 128-142. ↑
R. Hirschl, Comparative matters: The Renaissance of comparative constitutional law. Oxford University Press, Oxford, 2014. ↑
K. Sikkink, Evidence for Hope: Making Human Rights Work in the 21st Century, Princeton University Press, Princeton, 2019. ↑
P. Cane, The Cambridge Constitutional History of the United Kingdom, Cambridge University Press, Cambridge, 2023. ↑
ArtII.S2.C1.1.14, Martial Law Generally, 2024, available on the page ↑
Title VIII: Judiciary (Articles 64 to 66), available on the page ↑
Constitution fot Israel, 1958, available on the page ↑
Decree of the President of Ukraine No. 11/2022 (On the Report on the Results of the Review of Public Security and Civil Defense), available on the page ↑
Amnesty International, Syria: Former refugees tortured, raped, disappeared after returning home, 2021, available on the page Amnesty International, Myanmar: Military should be investigated for war crimes in response to ‘Operation 1027’, 2023, available on the page ↑
Human Rights Watch, World Report 2023: Ethiopia, 2023, available on the page ↑
Freedom in the World 2023: Ethiopia, 2023, available on the page ↑
U.N., Report of the International Law Commission, 2022, available on the page ↑
ArtII.S2.C1.1.14, Martial Law Generally, 2024; Title VIII: Judiciary (Articles 64 to 66); Constitution for Israel, 1958; Decree of the President of Ukraine No. 11/2022 “On the Report on the Results of the Review of Public Security and Civil Defense”, 2022; Constitution of 4 October 1958, 2024, available on the page ↑
S. Buchko, Ex Parte Milligan, in Indiana Law Review, vol. 37, 2004, pp. 667-685. ↑
Amnesty International, Myanmar: Military should be investigated for war crimes in response to ‘Operation 1027’, 2023; World Report 2023: Ethiopia, 2023; Freedom in the World 2023: Ethiopia, 2023, available on the page Amnesty International, Syria: Former refugees tortured, raped, disappeared after returning home, 2021. ↑
HUDOC, 2024, available on the page ↑
Law of Ukraine No. 389-VIII, 2015, (On the Legal Regime of Martial Law), available on the page ↑
Yu.O. Zahumenna, Legislation on national security of Ukraine as a systemic phenomenon: Theoretical and legal analysis, Baltija Publishing, Riga, 2018. ↑
Decree of the President of Ukraine No. 11/2022. ↑
M.A. Alimbekova, A.S. Ibrayeva, G.T. Ichshanova, K.R. Useinova, N.S. Ibrayev, Legal culture of public servants: The comparative legal analysis of the formation practices of various countries, in Journal of Advanced Research in Law and Economics, vol. 10, no. 7, 2019, pp. 1956-1967. ↑
Amnesty International, Syria: Former refugees tortured, raped, disappeared after returning home, 2021. ↑
International Committee of the Red Cross, Commentary on the Third Geneva Convention, Cambridge, Cambridge University Press, 2021. ↑
U.N., International Covenant on Civil and Political Rights, 1967, available on the page ↑
ECHR, European Convention for the Protection of Human Rights and Fundamental Freedoms, 1950, available on the page ↑
D. Dyzenhaus, Legality in a time of emergency, in The Constitution of Law: Legality in a Time of Emergency. ↑
U.S. Constitution, 2024; S. Buchko, Ex Parte Milligan, cit., pp. 667-685. ↑
Human Rights Act, 1998, available on the page ↑
Title VIII: Judiciary (Articles 64 to 66). ↑
Constitution for Israel, 1958. ↑
O. Miliienko, Internal migration and displaced persons in Ukraine: Governing policies and protections by the administrative courts, in Social and Legal Studios, vol. 6, no. 3, 2023, pp. 103-112; D.B. Makhambetsaliyev, G.M. Sagynbekova, U.B. Abdykadyr, E.A. Alimova, A.B. Smanova, Constitutional Law-Making in the U.S. Supreme Court, in Public Integrity, vol. 26, no. 4, 2024, pp. 473-484. ↑
N. Apakhayev, K. Adilova, D. Bugybay, G. Mukaldyeva, G.N. Mukhamadiyeva, B.M. Koshpenbetov, Childhood legal protection in Kazakhstan, in Journal of Advanced Research in Law and Economics, vol. 8, no. 3, 2017, pp. 714-721; N. Apakhayev, K. Adilova, D. Bugybay, A. Toktybaev, D. Kopbayev, The problem of protecting the rights and legitimate interests of the child in the family and outside IT, in Danube, vol. 15, no. 3, 2024, pp. 221-236. ↑
D. Makhambetsaliyev, E. Alimova, C. Utegenov, G. Sagynbekova, A. Smanova, The main directions of the judicial activity of the Supreme Court of the United States in the field of civil rights and freedoms, in Scientific Herald of Uzhhorod University. Series Physics, vol. 55, 2024, pp. 1532-1542. ↑
O. Taran, M. Hryha, Application of international humanitarian law by the European Court of Human Rights, in Scientific Journal of the National Academy of Internal Affairs, vol. 29, no. 2, 2024, pp. 9-17; A.S. Khamzin, Z.A. Khamzina, Y.A. Buribayev, Y.M. Tileubergenov, D.A. Ibraimov, A.T. Yermekov, International legal aspects of exercising refugees’ rights in Central Asia, in Journal of Advanced Research in Law and Economics, vol. 7, no. 4, 2016, pp. 835-841. ↑
Geneva Conventions, 1949, available on the page ↑
International Covenant on Civil and Political Rights, 1967. ↑
Rome Statute of the International Criminal Court, 1998, available on the page ↑
U.S. Constitution, 2024. ↑
S. Buchko, Ex Parte Milligan, cit.; U.S. Supreme Court, December 18, 1944, 323 U.S. 214, Korematsu v. United States, available on the page ↑
Constitution of 4 octobre 1958, 2024. ↑
A. Barak, A constitutional revolution: Israel’s basic laws, in Constitutional Forum, vol. 4, 1993, pp. 83-84. ↑
Israel Emergency Regulation, 1945, (Reg. 25, Defense Regulations, Times of Emergency), available on the page ↑
Decree of the President of Ukraine No. 11/2022 (On the Report on the Results of the Review of Public Security and Civil Defense, 2022). ↑
ECHR, European Convention for the Protection of Human Rights and Fundamental Freedoms, 1950. ↑
U.N., International Covenant on Civil and Political Rights, 1967. ↑
U.S. Constitution, 2024. ↑
U.K. Anti-terrorism, Crime and Security Act 2001, 2024, available on the page ↑
Title VIII: Judiciary (Articles 64 to 66), 2024. ↑
Constitution of 4 October 1958, 2024. ↑
Constitution for Israel, 1958. ↑
Decree of President of Ukraine No. 11/2022 (On the Report on the Results of the Review of Public Security and Civil Defense, 2022). ↑
ECHR, European Convention for the Protection of Human Rights and Fundamental Freedoms, 1950. ↑
U.S. Constitution, 2024. ↑
U.K. Anti-terrorism, Crime and Security Act 2001, cit. ↑
Title VIII: Judiciary (Articles 64 to 66) 2024. ↑
Constitution of 4 Octobre 1958, 2024. ↑
Constitution for Israel, 1958. ↑
Decree of the President of Ukraine No. 11/2022 (On the Report on the Results of the Review of Public Security and Civil Defense), 2022. ↑
ECHR, European Convention for the Protection of Human Rights and Fundamental Freedoms, 1950. ↑
U.S. Constitution, 2024, cit. ↑
U.K. Anti-terrorism, Crime and Security Act 2001, ↑
Title VIII: Judiciary (Articles 64 to 66,). ↑
Constitution of 4 Octobre 1958. ↑
Constitution of Israel, 1958. ↑
Decree of the President of Ukraine No. 11/2022 (On the Report on the Results of the Review of Public Security and Civil Defense, cit.). ↑
ECHR, European Convention for the Protection of Human Rights and Fundamental Freedoms, 1950. ↑
Ibid. ↑
HUDOC, 2024. ↑
ArtII.S2.C1.1.14, Martial Law Generally, 2024. ↑
U.K. Anti-terrorism, Crime and Security Act 2001, ↑
Title VIII: Judiciary, (Articles 64 to 66). ↑
Constitution of 4 October 1958, 2024. ↑
Constitution for Israel, 1958. ↑
Decree of the President of Ukraine No. 11/2022 (On the Report on the Results of the Review of Public Security and Civil Defense). ↑
I. Kostiuk, Forensic classification of military criminal offences and the place of abuse of power or authority by a military official, in Law Journal of the National Academy of Internal Affairs, vol. 14, no. 4, pp. 66-77, 2024. ↑
U.N., International Covenant on Civil and Political Rights, 1967. ↑
I. Cherlenіak, M. Tokar, Effective governance and the doctrine of “total defence” as factors of state stability in wartime, in Democratic Governance, vol. 17, no. 1, , 2024, pp. 5-17. ↑
A. Khamzin, Y. Buribayev, K. Sartayeva, Prevention of Human Trafficking Crime: A View from Kazakhstan and Central Asian Countries, in International Journal of Criminal Justice Sciences, vol. 17, no. 1, 2022, , pp. 34-53. ↑
Law of Ukraine No. 389-VIII “On the Legal Regime of Martial Law”, 2015. ↑
European Convention for the Protection of Human Rights and Fundamental Freedoms, 1950. ↑
L. Spytska, Assessment of the Political and Security Decisions of the Nuremberg Trials from a Legal Perspective, in Pakistan Journal of Criminology, vol. 15, no. 1, pp. 63-76, 2023; L. Spytska, The Nature of Sexual Violence: The Criminological Concept of Victimisation, in Pakistan Journal of Criminology, vol. 15, no. 4, 2023, pp. 1-20. ↑
O. Gross, F.D. Ni Aolain, The rhetoric of war: Words, conflict and categorization post-9/11, in Cornell Journal of Law and Public Policy, vol. 24, no. 2, 2014, pp. 14-47; V. Lytvyn, V. Ahmadov, Problematic aspects of at the scene police work with victims of terrorist acts on the territory of Ukraine, in Law Journal of the National Academy of Internal Affairs, vol. 14, no. 2, 2024, pp. 52-60. ↑
Amnesty International, Syria: Former refugees tortured, raped, disappeared after returning home, 2021; Amnesty International, Myanmar: Military should be investigated for war crimes in response to ‘Operation 1027’, 2023. ↑
Human Rights Watch, World Report 2023: Ethiopia, 2023. ↑
Freedom in the World 2023: Ethiopia, 2023. ↑
Human Rights Watch, World Report 2023: Ethiopia, 2023; Freedom in the World 2023: Ethiopia, 2023; Amnesty International, Myanmar: Military should be investigated for war crimes in response to ‘Operation 1027’, 2023. ↑
Amnesty International, Syria: Former refugees tortured, raped, disappeared after returning home, 2021; Amnesty International, Myanmar: Military should be investigated for war crimes in response to ‘Operation 1027’, 2023. ↑
Human Rights Watch, World Report 2023: Ethiopia, 2023. ↑
V. Teremetskyi, O. Kovalchuk, A. Kolesnikov, R. Bogdanov, M. Korniienko, I. Dir, Improving the Information and Legal Support of the Judicial System of Ukraine: Experience of the European Court of Human Rights, in Journal of Ecohumanism, vol. 3, no. 3, pp. 61-74, 2024. ↑
D. Dyzenhaus, Legality in a time of emergency, cit. ↑
H.L.A. Hart, The concept of law, Oxford University Press, Oxford, 2012. ↑
S. Bandurka, Advocacy in the system of protection of human rights and freedoms in wartime, in Social and Legal Studios, vol. 7, no. 4, pp. 212-221, 2024. ↑
R. Hirschl, Comparative matters: The Renaissance of comparative constitutional law. ↑
D. Dyzenhaus, Legality in a time of emergency, cit. ↑
G.A. Raymond, International norms and the resort to war, Palgrave Macmillan, Cham, 2021. ↑
J.R. Miles, Soldiers’ dilemma: Foreign military training and liberal norm conflict, in International Security, vol. 46, no. 4, pp. 48-90, 2022. ↑
P. Cane, The Cambridge Constitutional History of the United Kingdom, cit. ↑
S. Neff, The law of armed conflict, cit. ↑
R. Dworkin, Justice for hedgehogs, Harvard University Press, Belknap Press, Cambridge, 2011. ↑
J.M. Finnis, Natural Law and Natural Rights, Oxford University Press, Oxford, 2011. ↑
J. Raz, The authority of law: Essays on law and morality, Oxford University Press, Oxford, 1979 ↑
E. Chemerinsky, Constitutional law: Principles and policies, Aspen Publishing, Waltham, 2023. ↑
M.I. Doronin, Security essence of the state’s functions security essence of the state’s functions, in Information and Law, vol. 1, no. 32, 2020, pp. 66-79. ↑
N. Gur, J. Jackson, Procedure-content interaction in attitudes to law and in the value of the rule of law: An empirical and philosophical collaboration, in D. Meyerson, C. Mackenzie, T. MacDermott (ed.), Procedural Justice and Relational Theory: Empirical, Philosophical, and Legal Perspectives, Routledge, London, 2020. ↑
V.M. Kravchuk, Theoretical foundations of the constitutional and legal status of judges in Ukraine, Uzhhorod National University of the Ministry of Education and Science of Ukraine, Uzhhorod, 2020. ↑
H. L.A. Hart, The concept of law, cit. ↑
J.T. Levy, Jeremy Waldron, Political theory: Essays on institutions, in Review of Politics, vol. 79, no. 2, 2017, pp. 321-323, ↑
K. Sikkink, Evidence for Hope: Making Human Rights Work in the 21st Century, cit. ↑
D. Reiter, Avoiding the coup-proofing dilemma: Consolidating political control while maximizing military power, in Foreign Policy Analysis, vol. 16, no. 3, 2020, pp. 312-331. ↑
ArtII.S2.C1.1.14, Martial Law Generally. ↑
ArtII.S2.C1.1.13, President as Commander of Armed Forces, 2024, available on the page ↑
M. Neocleous, Whatever happened to martial law?, in Radical Philosophy, vol. 143, 2024, pp. 13-22. ↑
I. Kostiuk, Theoretical legal analysis of abuse of power or official authority by a military official in martial law or combat situation: Key aspects, in Scientific Journal of the National Academy of Internal Affairs, vol. 29, no. 4, 2024, pp. 61-74. ↑
F.A. Hayek, The Constitution of liberty, Routledge, London, 2012. ↑
O. Gross, F.D. Ni Aolain, Law in times of crisis: Emergency powers in theory and practice (American society of International 2007 Certificate of Merit for creative scholarship), Cambridge University Press, Cambridge, 2006. ↑
L. Brooke-Holland, C. Mills, The Armed Forces Covenant and status in law, 2022, available on the page ↑
Stella Kelbia
Dottore di Ricerca, Preside della Facoltà di Giurisprudenza, Istituto privato di istruzione superiore “Bukovinian University”, Chernivtsi, Ucraina
Roman Lutskyi
Professore nel Dipartimento di Diritto e Pubblica Amministrazione dell'Università King Danylo, Ivano-Frankivsk, Ucraina
Vitalii Skomorovskyi
Professore nella Facoltà di Giurisprudenza dell'Università “KROK”, Kiev, Ucraina
Olexandr Polishchuk
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10919 | https://stats.stackexchange.com/questions/179511/why-zero-correlation-does-not-necessarily-imply-independence | Why zero correlation does not necessarily imply independence - Cross Validated
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Why zero correlation does not necessarily imply independence
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If two variables have 0 correlation, why are they not necessarily independent? Are zero correlated variables independent under special circumstances ? If possible, I am looking for an intuitive explanation, not a highly technical one.
correlation
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asked Oct 31, 2015 at 2:33
VictorVictor
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14 Correlation is a measure of linear dependence (association). it is possible for two random variables to be uncorrelated but nonlinearly dependent.Mark L. Stone –Mark L. Stone 2015-10-31 02:37:21 +00:00 Commented Oct 31, 2015 at 2:37
1 Intuitive explanation ->math.stackexchange.com/questions/444408/…Siddhesh –Siddhesh 2015-10-31 03:20:13 +00:00 Commented Oct 31, 2015 at 3:20
6 Zero correlation implies independence if the variables are multivariate normal. This is not the same as each variable being normal - see here for some scatterplots of zero-correlated but dependent normal variables (each variable is individually normal)Glen_b –Glen_b 2015-10-31 08:55:58 +00:00 Commented Oct 31, 2015 at 8:55
1 Correlation (unqualified) could include rank correlation, etc., for which monotonic dependence is the issue, and so forth.Nick Cox –Nick Cox 2015-10-31 11:12:53 +00:00 Commented Oct 31, 2015 at 11:12
1 For outlook, I would recommend you to see Wikipedia "distance correlation" as a measure of independence.ttnphns –ttnphns 2015-10-31 13:44:23 +00:00 Commented Oct 31, 2015 at 13:44
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Correlation measures linear association between two given variables and it has no obligation to detect any other form of association else.
So those two variables might be associated in several other non-linear ways and correlation could not distinguish from independent case.
As a very didactic, artificial and non realistic example, one can consider X X such that P(X=x)=1/3 P(X=x)=1/3 for x=−1,0,1 x=−1,0,1 and Y=X 2 Y=X 2. Notice that they are not only associated, but one is a function of the other. Nonetheless, their correlation is 0, for their association is orthogonal to the association that correlation can detect.
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answered Oct 31, 2015 at 11:05
Marcelo VenturaMarcelo Ventura
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1 I was looking for evidence of random variances being uncorrelated yet dependent however none of the direct answers to my question revealed intuitive facts. your answer, on the other hand, gives me a very good angle to think about it, thanks a lot!stucash –stucash 2019-04-19 17:19:31 +00:00 Commented Apr 19, 2019 at 17:19
2 @stucash my pleasure! It was an old counter example I learnt Marcelo Ventura –Marcelo Ventura 2019-06-06 01:04:01 +00:00 Commented Jun 6, 2019 at 1:04
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There is a generalized lack of rigor in the use of the word "correlation" for the simple reason that it can have widely differing assumptions and meanings. The simplest, loosest and most common usage is that there is some vague association, relationship or lack of independence between a static pair of random variables.
Here, the default metric referred to is usually the Pearson correlation, which is a standardized measure of pairwise, linear association between two continuously distributed variables. One of the Pearson's commonest misuses is to report it as a percentage. It is definitely not a percentage. The Pearson correlation, r, ranges between -1.0 and +1.0 where 0 means no linear association. Other not so widely recognized issues with using the Pearson correlation as the default is that it is actually quite a stringent, non-robust measure of linearity requiring interval-scaled variates as input (see Paul Embrechts' excellent paper on Correlation and Dependency in Risk Management: Properties and Pitfalls here:
Embrechts notes that there are many fallacious assumptions about dependence that begin with assumptions of the underlying structure and geometric shape of these relationships:
These fallacies arise from a naive assumption that dependence properties of the elliptical world also hold in the non-elliptical world
Embrechts points to copulas as a much wider class of dependence metrics used in finance and risk management, of which the Pearson correlation is just one type.
Columbia's Statistics department spent the academic year 2013-2014 focused on developing deeper understandings of dependence structures: e.g., linear, nonlinear, monotonic, rank, parametric, nonparametric, potentially highly complex and possessing wide differences in scaling. The year ended with a 3 day workshop and conference that brought together most of the top contributors in this field (
These contributors included the Reshef Brothers, now famous for a 2011 Science paper Detecting Novel Associations in Large Data Sets that has been widely criticized (see AndrewGelman.com for a good overview, published simultaneously with the Columbia event: The Reshefs addressed all of these criticisms in their presentation (available on the Columbia conference website), as well as a vastly more efficient MIC algorithm.
Many other leading statisticians presented at this event including Gabor Szekely, now at the NSF in DC. Szekely developed his distance and partial distance correlations. Deep Mukhopadhay, Temple U, presenting his Unified Statistical Algorithm -- a framework for unified algorithms of data science -- based on work done with Eugene Franzen And many others. For me, one of the more interesting themes was wide leverage and use of Reproducing Kernel Hilbert Space (RKHS) and the chi-square. If there was a modal approach to dependence structures at this conference, it was the RKHS.
The typical intro statistics textbooks is perfunctory in its treatment of dependence, usually relying on presentations of the same set of visualizations of circular or parabolic relationships. More sophisticated texts will delve into Anscombe's Quartet, a visualization of four different datasets possessing similar, simple statistical properties but hugely differing relationships:
One of the great things about this workshop was the multitude of dependence structures and relationships visualized and presented, going far beyond the standard, perfunctory treatment. For instance, the Reshefs had dozens of thumbnail graphics that represented just a sampling of possible nonlinearities. Deep Mukhopadhay had stunning visuals of highly complex relationships that looked more like a satellite view of the Himalayas. Stats and data science textbook authors need to take note.
Coming out of the Columbia conference with the development and visualization of these highly complex, pairwise dependence structures, I was left questioning the ability of multivariate statistical models to capture these nonlinearities and complexities.
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answered Oct 31, 2015 at 13:26
user78229user78229
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3 I just came across this excellent and exhaustive discussion of measures of association on Quora: quora.com/…user78229 –user78229 2015-10-31 15:53:07 +00:00 Commented Oct 31, 2015 at 15:53
From an etymological point of view, co-relation is the relationship between two variables. Independence between two variables is a relationship bt two variables. Then they are synonyms. If you could calculate any possible correlation, then it could be understood as an independence test.Carlos Mougan –Carlos Mougan 2023-06-21 20:39:53 +00:00 Commented Jun 21, 2023 at 20:39
1 This is a great paper that talks about independence test using a binary classifier arxiv.org/abs/1610.06545Carlos Mougan –Carlos Mougan 2023-06-21 20:40:33 +00:00 Commented Jun 21, 2023 at 20:40
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It depends on your exact definition of "correlation", but it isn't too hard to construct degenerate cases. "Independent" could mean something like "no predictive power, at all, ever" just as much as "linear correlation".
Linear correlation, for example, would not indicate dependence on y=sin(2000 x)y=sin(2000 x) if the domain of x x was [0,1)[0,1).
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edited Oct 31, 2015 at 11:11
Nick Cox
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Andrew CharneskiAndrew Charneski
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An intuitive example would be a circle. I have two variables X X and Y Y. And they are satisfy the equation
X 2+Y 2=1 X 2+Y 2=1
Now, X X and Y Y are definitely not independent to each other, because given X X we can calculate Y Y and vice versa. But their person correlation coefficient is 0 0. This is because it only captures the linear relationship between two variables.
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answered Feb 11, 2020 at 13:10
HXDHXD
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Basically, dependence of Y on X means the distribution of values of Y depends on some way of the value of X. That dependence can be on the mean value of Y (the usual case presented in most of the answers) or whatever other characteristic of Y.
For example, let X be 0 or 1. If X = 0 then let Y be 0, if X= 1 let Y be -1, 0 or 1 (same probability). X and Y are uncorrelated. On mean, Y doesn't depend on X because whatever value is X, the mean of Y is 0. But clearly the distribution of values of Y depends on X value. In this case, for example, the variance of Y is 0 when X=0 and > 0 when X=1, thus there is, at least, a dependence on variance, i.e. there is a dependence.
So, linear correlation only show a type of dependence on mean (linear dependence), that in turn is only a special case of dependence.
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edited Mar 25, 2017 at 16:34
Carl
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Adding to @Marcelo Ventura and @Mike Hunter great answers, and the reference to a great discussion around this on Quora.
An important point (implicit) is made in here and in the quora thread. Although correlation is a linear measure, it does not exclusively mean it can only quantify the relationship between linearly dependent variables.
Arguably an equally important factor is whether there is a monotone relationship between variables. As stated on minitab
In a monotonic relationship, the variables tend to move in the same relative direction, but not necessarily at a constant rate. In a linear relationship, the variables move in the same direction at a constant rate.
This means if we have non-monotone related variables we can observe a zero correlation even though they are not independent
To illustrate this say for example we have a f(x)=x 2 f(x)=x 2, using python to evaluate the function
If we look at x x in [0,50)[0,50), we find that f(x)f(x) has a monotone relationship with x x, as a result we observe the correlations to be close to 1:
```
import numpy as np
import seaborn as sns
x = np.arange(0, 50, 1)
f = lambda x: x 2
y = f(x)
sns.scatterplot(x, y)
```
```
Get correlations using scipy
from scipy.stats import pearsonr, spearmanr
pearsonr(x, y)
spearmanr(x, y)
```
Pearson Correlation:0.967
Spearman Correlation:0.999...
Now if we look at x x in [−25,25)[−25,25), we find f(x)f(x)no longer has a monotone relationship with x x, the correlations are thus close to zero as expected:
x = np.arange(-25, 25, 1)
y = f(x)
sns.scatterplot(x, y)
pearsonr(x, y)
spearmanr(x, y)
Pearson Correlation:-0.077
Spearman Correlation:-0.059
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edited Feb 11, 2020 at 13:15
answered Feb 11, 2020 at 13:02
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Zero correlation does not imply independence for multiple reasons. One of these is possible is that two variables could be dependent on a third causing correlation.
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answered Feb 21, 2023 at 3:23
griffingriffin
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1 Welcome to Cross Validated! If that third variable causes there to be correlation between the first two, then aren’t the first two variables correlated?Dave –Dave 2023-02-21 03:37:30 +00:00 Commented Feb 21, 2023 at 3:37
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10920 | https://www.pakjr.com/index.php/PJR/article/download/430/452 | CASE REPORT BUERGER’S DISEASE IN A YOUNG SMOKER; ANGIOGRAPHIC FINDINGS
PJR April - June 2010; 20(2): 97-99
97 PJR April - June 2010; 20(2) PA K I S TA N J O U RN A L O F R A D I O L OG Y
Rana Shoaib Hamid, Basit Salam, Zafar Sajjad
Corre s ponde nce : Dr. Rana Shoaib Hamid Department of Radiology, Aga Khan University Hospital, Stadium Road, P.O Box 3500, Karachi, 74800 Pakistan. Tel. No. 34930051- Ext 2020 E-mail: rana.shoaib@aku.edu
Based on gender, age, physical examination and angiographic findings, diagnosis of Buerger’s disease was made.
Cas e
A 30 year old man with history of heavy smoking for many years was referred to radiology department for right lower limb angiography with complaints of pain in both legs at rest and non healing surgical wound at his right foot. He had undergone amputation of left forefoot 5 years back and amputation of right forefoot 6 months back. On examination there was an ulcerating non healing wound on dorsum of right foot. Patient was normotensive. The feet were warm. Bilateral femoral pulses were palpable. Peripheral pulses were impalpable in right lower limb. Angiogram was performed from left femoral approach. It revealed a normal aorta and normal bilateral iliac vessels. The common femoral and profunda femoris arteries on right side were also normal. Abrupt cut off was seen in distal part of superficial femoral artery with multiple cork screw collaterals in distal thigh and leg. There was complete occlusion of popliteal, anterior and posterior tibial and the peroneal arteries (Fig. 1-4). Buerger's disease or thromboangiitis obliterans is a non atherosclerotic, segmental obliterative vascular disease that affects medium and small sized arteries and superficial veins. 1 It typically occurs in young male smokers, with the onset of symptoms before the age of 40 to 45 years. Progression of the disease is closely linked to continued use of tobacco. Patients present with rest pain, ischemic ulcers, and gangrene of the digits of hands and feet. Large arteries are typically spared, as are the coronary, cerebral, and visceral circulations. 2 Typical angiographic findings have been described that are highly suggestive of the disease. These patterns include diffuse vascular narrowing, occlusions, and a segmental pattern of involvement as well as corkscrew configuration proximally and a tree-root appearance distally. 3
K e y w ords : Thromboangiitis obliterans, Buerger’s disease, Peripheral ischemia.
ABSTRACT
Department of Radiology, Aga Khan University Hospital, Karachi, Pakistan.
Figure 1: Digital subtraction angiogram of upper thigh and hemipelvis. Normal appearing right common femoral (arrow) and profunda femoris arteries (arrowhead). Proximal superficial femoral artery also appears normal.
Dis cus s ion
Initially described in 1908 by Leo Buerger, a New York surgeon and pathologist, Buerger's disease is also called thromboangitis obliterans. It is characterized by development of segmental occlusions of the medium and small arteries of the extremities. It is clinically and pathologically distinguishable from arteriosclerosis and necrotizing arteritis. 2The cause and pathogenesis of Buerger's disease are unknown but a strong relationship with cigarette smoking exists. It has only rarely been reported in non-smokers. A few observations have led investigators to implicate an immunologic phenomenon that leads to vasodysfunction and inflammatory thrombi. Patients with the disease show hypersensitivity to intradermally injected tobacco extracts, have increased cellular sensitivity to types I and III collagen, have elevated serum anti–endothelial cell antibody titers, and have impaired peripheral vasculature endothelium-dependent vasorelaxation. 4,5 The clinical criteria for diagnosis include: age under 45 years; current or recent history of tobacco use, presence of distal-extremity ischemia indicated by claudication, pain at rest, ischemic ulcers or gangrenes and documented by non-invasive vascular testing; exclusion of autoimmune diseases, hypercoagulable states and diabetes mellitus; exclusion of a proximal source of emboli by echocardiography or arteriography; consistent arteriographic findings in the clinically involved and non-involved limbs. 6The angiographic features of Buerger’s disease are involvement of the small and medium-sized vessels, such as the palmar, plantar, tibial, peroneal, radial, and ulnar arteries and the digital arteries of the fingers and toes; segmental occlusive lesions (diseased arteries interspersed with normal appearing arteries); more severe disease distally, and normal proximal arteries with no evidence of atherosclerosis; collateralization around areas of occlusion (corkscrew collaterals); and no apparent source of emboli. 3,7,8 The onset of Buerger's disease is rapid and the progression of the disease always depends on the smoking habits of the patient. Although the improve-ment may not be immediate, the symptoms will be arrested and probably improve after cessation of smoking, except in cases of severe, irreversible ischaemia. Patients who continue smoking are at risk
98 PJR April - June 2010; 20(2) PA K I S TA N J O U RN A L O F R A D I O L OG Y
Figure 2: Digital subtraction angiogram of thigh. Abrupt cut off at distal third of superficial femoral artery (arrow). Mild narrowing also noted in proximal part of the artery (arrowhead).
Figure 3 & 4: Digital subtraction angiogram of knee and leg. Extensive corkscrew collaterals in distal thigh and the leg. Note the lack of visualization of popliteal, tibial and peroneal arteries. 99 PJR April - June 2010; 20(2) PA K I S TA N J O U RN A L O F R A D I O L OG Y
of gangrene often resulting in amputation of fingers and toes. Surgical revascularization is usually not possible for patients with Buerger’s disease, because of the diffuse segmental involvement and distal nature of the disease. Often no distal target vessel is available for bypass surgery However, if the patient has severe ischemia and there is a distal target vessel, bypass surgery with the use of an autologous vein should be considered. 9-11
Re f e re nce s
Reny JL, Cabane J. Buerger's disease or thromboangiitis obliterans 1998; 19 : 34-43 Mills JL Sr. Buerger's disease in the 21st century: diagnosis, clinical features, and therapy. 2003;
16: 179-89. Lambeth JT, Yong NK. Arteriographic findings in thromboangiitis obliterans with emphasis on femoropopliteal improvement. AJR Am J Roentgenol 1970; 109 : 553-62. Hanly EJ. Buerger Disease (Thromboangiitis Obliterans). /article/460027-overview. Jeffrey W., Olin D.O. Thromboangiitis Obliterans (Buerger Disease) N Engl J Med 343: 864 Perttu ET Arkkila. Thromboangiitis obliterans (Buerger's disease). Orphanet Journal of Rare Diseases 2006, 1: 14. McKusick VA, Harris WS, Ottesen OE, Goodman RM. The Buerger syndrome in the United States: arteriographic observations, with special reference to involvement of the upper extremities and the differentiation from atherosclerosis and embolism. Bull Johns Hopkins Hosp 1962; 110: 145-76. Szilagyi DM, DeRusso FJ, Elliot JP Jr. Thromb-oangiitis obliterans: clinico-angiographic correlations. Arch Surg 1964; 88: 824-35. 9. 10. 11. Inada K, Iwashima Y, Okada A, Matsumoto K. Non-atherosclerotic segmental arterial occlusion of the extremity. Arch Surg 1974; 108: 663-7. Sayin A, Bozkurt AK, Tuzun H, Vural FS, Erdog G, Ozer M. Surgical treatment of Buerger's disease: experience with 216 patients. Cardiovasc Surg 1993; 1: 377-80. Sasajima T, Kubo Y, Inaba M, Goh K, Azuma N. Role of infrainguinal bypass in Buerger's disease: an eighteen-year experience. Eur J Vasc Endovasc Surg 1997; 13: 186-92. |
10921 | https://www.bluecart.com/blog/inventory-shrinkage | Published Time: Jun 13, 2025
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Shrinkage Statistics
Ways to Prevent Shrinkage: Controlling Shrinkage in Retail
Frequently Asked Questions About Inventory Shrinkage
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Here are two simple and indisputable statements:
If inventory shrinkage is large, your profits decrease.
If inventory shrinkage is managed, your profits increase.
As you can tell from the word choice, inventory shrinkage can’t be eradicated.
It can only be successfully managed and lowered. Hopefully to a below-average level.
But 11% of retail businesses report shrinkage rates at or above 3%. That’s three times the industry median shrinkage rate of 1%. So, with money on the line, it’s obviously in your company’s best interest to identify and prevent shrinkage.
But what is shrinkage? What causes shrinkage, and just what kind of numbers, industry-wide, are we looking at in regards to shrinkage in retail? And finally, how on earth can I get a hold of shrinkage control?
Your answers are below.
Inventory Shrinkage Is Recorded When…
Inventory shrinkage is recorded when you want to reconcile your sitting inventory with your inventory records. If you find less on your shelves than your eCommerce accounting reflects you’ve sold, you’ve got shrinkage. Before you determine inventory shrinkage, it's crucial for you to understand theaverage inventory formula.
What Is Inventory Shrinkage?
Inventory shrinkage is when you have less inventory than you should. Something is causing items to go missing before the point of sale. And inventory shrinkage isn’t just a retail problem. It affects every stage of the supply chain from the point of manufacture. And it affects every business'sinventory turnover ratio, which can be calculated using the inventory turnover formula, andsell through rate. Because that inventory isn't being turned over or sold.
How to Calculate Inventory Shrinkage
If you’ve got shrinkage, there’s good news: Shrinkage is unavoidable. Every business will run some level of inventory shrinkage. You just need a shrinkage calculation to uncover it and there’s more good news: You can lower it.
Here’s a shrinkage formula for calculating inventory shrinkage:
Shrinkage Rate = (Recorded Inventory - Actual Inventory) / Recorded Inventory
We’ll use a wine bar as an example. Let’s say a wine bar takes bar inventory and counts 71 bottles of wine and that they’ve sold none of those bottles. If they recorded receiving 72 bottles of wine from their supplier, the shrinkage calculation would look like this:
Shrinkage Rate = (72 - 71) / 72
Shrinkage Rate = .014 or 1.4%
1.4% of the wine bar’s sales of this wine bottle are lost. To be honest, though, 1.4% isn’t so bad. It’s actually right around average and speaks to an industry-standard amount of shrinkage control.
Restaurant Shrinkage: Kitchens
In restaurant kitchens, shrinkage refers to the difference between the amount of food you acquire from wholesale food distributors and the amount of food you sell to customers.
Restaurant shrinkage reflects three things:
The efficiency of the kitchen. Kitchens that burn steaks, prep incorrectly, or otherwise contribute to waste increase shrinkage.
The amount of dead stock expiring. Inaccurate purchase orders that stock storage with sitting inventory that goes bad increases shrinkage.
The quality of raw inventory from suppliers. If some portion of raw material inventory can’t be used, it’s trimmed and counted as shrinkage. The more unusable material trimmed and discarded, the higher the shrinkage.
Shrinkage at the Manufacturer
Back to our wine. Imagine a small, 5-acre vineyard that produced 600 cases of wine this harvest. They’re the manufacturer. They have a contract with a wine wholesaler who buys every case available to distribute to retail wine shops, bars, and restaurants. But the vineyard only has 598 cases of wine to sell to the wholesaler. The process of bottling, labeling, packing, storing, and shipping the wine cause 2 cases—24 bottles—to disappear. Shrinkage.
Shrinkage at the Vendor
Those 598 cases of wine are loaded onto a truck, driven to the wholesaler’s warehouse, and unloaded. The wholesaler stocks it, scans it, inventories it, sells it, and ships it to hundreds of individual retailers. But throughout that process, they’ve lost 5 cases. That’s 60 bottles. Shrinkage again.
Shrinkage In Retail and Hospitality
And finally, a medium-sized wine bar receives their shipment of 6 cases from the wholesaler. That’s 72 bottles. They put the wine in their cellar, open the first case, and begin selling it to guests. At some point the wine bar takes beverage inventory and finds they’ve sold 40 bottles and have 31 left. That’s only 71 bottles. Where’d the other one go? Shrin-kage.
Factoring In Inventory Shrinkage
Everyone knows shrinkage exists, though. They don’t often know their shrinkage numbers—which is something wholesale inventory management software andinventory forecasting helps with—but they know they’re losing product. So they adjust their prices to account for it.
The wholesaler pays a bit more for their 598 cases. The wine bar pays a bit more for their 6 cases. Because offsetting the cost of inventory shrinkage is baked into prices. That price increase is ultimately assumed by the last person in the chain: the retail customer.
If you can strengthen your shrinkage control, whether shrinkage in retail or shrinkage on the wholesale and vendor side, you’ll create two money-making opportunities:
You can lower prices and potentially sell more than usual
You can keep prices the same and sell what you were originally supposed to without such high shrinkage
But to lower it, you’ve gotta identify it. So here are the common causes of retail shrinkage.
6 Causes of Retail Shrinkage
Just what are the causes of retail shrinkage? There are 6 main reasons for inventory shrinkage.
1. Theft
Theft affects inventory shrinkage in one of two ways. Either someone external does it and it’s called shoplifting or external theft, or someone internal does it and it’s called employee theft or internal theft.
Of note, what’s known as “POS exceptions” contribute to internal theft and are especially relevant to the hospitality industry. Because POS systems are so integrated into the sales of a bar or restaurant, employees have the ability to use them subtly and to their own benefit. That means ringing in obsolete prices, applying discounts, or otherwise being creative with how things are rung in.
2. Waste and Spoilage
This one is pretty self-explanatory, but if perishable products aren’t used by their expiration date, they contribute to inventory shrinkage. In our example, the wine bar isn’t able to sell the last half case of wine before the 3-year expiration date. Those 6 bottles, lost to the ravages of time, are shrinkage.
Waste and spoilage have far a greater impact on the food service industry and retail food sellers. When dealing with products that last a matter of days, spoilage can be a significant cause of retail shrinkage.
3. Damage
If one of the wine bottles is dropped and shatters, that’s loss of merchandise. It’ll increase shrinkage numbers. Likewise, and it doesn’t have to be a perishable item, if any product is damaged beyond the point of reselling, it will increase your shrinkage percentage. Returns and exchanges contribute to damage-based shrinkage substantially, especially as one of the causes of retail shrinkage in traditional, non-hospitality retail environments.
4. Human Error
Pouring 6 ounces of wine instead of the standard wine pour is a human error that can compound over time to increase shrinkage numbers. In the food and beverage industry, the same goes for any measuring and portioning done by people.
From a retail shrinkage perspective, think ringing up the wrong item. And from a manufacturer or vendor perspective think loading the wrong cases onto the pallet or commingling different products in storage.
5. Administrative and Paperwork Errors
When the vineyard sells the cases of wine, the vendor or wholesaler receives and sells the wine on an online marketplace. Then the retail-level wine bar receives and sells the wine. There’s a lot of receiving and selling. And a lot of room for error.
If the wine-bar in our example input 10 bottles of wine per case instead of 12, they would have recorded 60 bottles in their inventory instead of 72. Right there, that’s 12 bottles of wine that won’t be accounted for. The restaurant POS will eighty-six that wine at 60 bottles and those 12 won’t be sold. That’s a big loss.
Clerical errors like this also lead to premature ordering and, eventually, sitting inventory that takes up shelf space. This type of shrinkage is especially hard to solve because any plan of attack is based on faulty information. Thankfully automation has gotten us pretty far in dealing with both of these problems and we’ll touch on that in a bit. Accurately calculating inventory KPI like average inventory, inventory days, and inventory carrying cost can all be automated.
6. Vendor Fraud
Vendor fraud is alternatively known as supplier fraud or wholesaler fraud. It’s as simple as the person or entity who is selling you products not delivering the promised amount. There’s a real risk of this as orders grow in size. The wine bar that orders 6 cases of wine has no problem verifying that 6 cases have been delivered.
But it’s harder for the wholesaler to count the 598 cases of wine they got from the vineyard. And more so for businesses that receive thousands of items. If it’s unrealistic to count, it’s unrealistic to verify that you got the right amount.
Vendors can also deliver different products than were originally agreed on. If, for example, a Grand Cru wine was ordered but the vendor filled the order with a Premier Cru of the same vintage. It’s easy for the vendor to claim it was a mistake, and they still get to roll the dice on nobody noticing or caring enough to do something about it.
The above is less common. Most often vendor fraud takes advantage of the sheer numbers of commercial ordering and skims a little off the top by shorting the quantity delivered.
Shrinkage Statistics
According to the shrinkage statistics from the 2019 National Retail Security Survey, inventory shrinkage accounted for 1.38% of all retail “sales.” That’s almost 48 billion dollars. To give you some perspective, the entire wine industry has a market value of 70.5 billion dollars. The “shrinkage industry” is almost 70% of that. There are literally tens of billions of dollars to be recouped.
What Percentage of Shrinkage Is Caused by Theft?
In 2017, the NRSS reported that external theft or customer shoplifting were responsible for 37.5% of retail shrinkage. And 33.2% of retail shrinkage was caused by employee or internal theft. That means theft accounts for roughly .98% of all retail sales and a total of almost 34 billion dollars.
To look at the NRSS shrinkage statistics from another angle, the average dollar amount lost per dishonest employee in 2019 was $1,264.10. Another reason to figure out how to hire bar staff that mesh well with your workplace culture. While the average dollar amount lost per shoplifting incident was $546.67.
“Thanks, real interesting shrinkage statistics. But what should my business be aiming for?” you ask. In the world of inventory shrinkage there are acceptable levels and unacceptable levels. Here’s what you should aim for.
What Are Acceptable Levels of Inventory Shrinkage?
The NRSS reports that in 2018, the average inventory shrinkage rate was 1.38% across all retail sectors. Even though that’s the average, it’s still pretty high because it equally weights even the highest inventory shrinkage rates. Like those at or above 3%, which account for almost 11% of retail businesses.
A better way to look at acceptable levels of retail inventory shrinkage is the median 2018 reported shrinkage. The median is the point in which half the numbers are above it and half below. It makes it a more representative number. The median shrinkage rate for 2018 was 1.00%. If you’re on the short side of that, you’re doing well. An acceptable level of inventory shrinkage is less than 1%.
Ways to Prevent Shrinkage: Controlling Shrinkage in Retail
We talked to everyone we know that runs a business and is concerned with how to stop shrinkage. Here are the 5 most common themes we got when it comes to controlling shrinkage. Remember, shrinkage is not a good form of inventory reduction.
Security
Item tracking like ink-blot tags in clothing stores or magnetic scanners in electronics stores do a great job deterring and catching internal and external theft.
Acquiring or upgrading video security systems will also be a great help—especially in cases when fixing ink or magnetic tags to items isn’t reasonable. Maybe someone is taking cash from a drawer or a bottle of liquor from behind the bar.
Of course, hiring loss prevention employees would work too. But that’s a bit of luxury. If you can justify (would you be paying them more than the shrinkage they’d prevent?) and afford the payroll for an employee who is specifically concerned with stopping shrinkage, then great.
Par Levels or Reorder Points
Learning how to set par levels and us the reorder point formula is how to stop shrinkage by lessening waste and spoilage. A par level is the amount of inventory you need to have available between receiving shipments to make sure you’re meeting customer demand. If you set the right par levels, you won’t have sitting inventory marching toward its expiry date. And that also means you’ll clear up more shelf space for other, more profitable products.
Train Staff
When staff is operating lockstep with standard operating procedures, it lowers shrinkage significantly. That means profits go up. Train your staff on your inventory pipeline, their role in it, and how to work efficiently. Often that means teaching them to interact with inventory management software and that software's recommendations and analyses.
Inventory Audits
Companies that experience a high percentage of inventory shrinkage should adopt more strict inventory control. The easiest way to do so is through inventory audits. They help the business better assess the current inventory levels and minimize losses due to theft, human errors, or other causes of inventory shrinkage. The frequency of these audits should be determined based on the levels of inventory shrinkage. There are various inventory audit methods. Choosing the right one depends on the type of business.
Automation
Thanks to automation, businesses can reduce their labor costs and inventory shrinkage. Nowadays, there are various ways to automate processes in a company. For example, you can rely on warehouse robotics that can take care of the scanning and handling of goods. This reduces the need for employees to do physical inventory audits. Furthermore, with the help of automation, businesses can streamline their processes which will result in higher efficiency and productivity for their employees.
Equipment and Technology
Investing in better equipment can have a positive impact on inventory shrinkage. For example, businesses that handle a lot of perishable goods can invest in proper refrigeration equipment and improve their storage. A lot of these products also get damaged during transportation. Thus, proper logistics procedures and equipment are needed in order to minimize inventory shrinkage due to transportation damage.
Improve Inventory Turnover
Inventory shrinkage can be caused by dead stock. One of the ways to reduce this risk is by guaranteeing good inventory turnover. That can be done by increasing the marketing budget and investing in long-term partnerships with customers. That will lead to more sales which is paramount when the goal is to increase the inventory turnover rate and reduce inventory shrinkage.
Take Inventory Often
Each time you take inventory is a chance to detect an inventory discrepancy. And that’s really all inventory shrinkage is. How to prevent shrinkage is an exercise in how often you’re taking inventory. Taking inventory once a week is ideal. Once a month is acceptable. Longer than that, you’re setting yourself to be on the losing end of a shrinkage problem. In a perfect world, though, your inventory is perpetual. Every time product is acquired or sold, your inventory updates in real time.
You need inventory management software for that.
Automate Inventory and Reporting
If you have errors in your inventory counts when manually taking inventory, that screws up your shrinkage rate. To take any action on shrinkage, you need accurate information. It all starts with taking inventory accurately.
The only way to make sure that happens is to automate it.
If you have errors in your accounting records, inventory costing methods, payments, or invoices it will snowball into inaccurate shrinkage rates. Among other problems. Automating your inventory and reporting is the answer. No matter what industry you’re in, there are numerous software solutions for your business. Whether your products are all sold in bulk with an MOQ (what does MOQ mean?) in place or not, automation can help.
Frequently Asked Questions About Inventory Shrinkage
What Businesses Have a High Inventory Shrinkage Rate?
As theft is the biggest reason for inventory shrinkage, a high inventory shrinkage rate is common among retailers and companies that deal with valuable or small items. Administrative errors and damage to items are also more common in big box stores. Here are other types of businesses that might need to deal with a high inventory shrinkage rate.
Jewelry and electronics stores deal with high-value items that are often small in size. That’s why theft by shoplifters or employees might be more common among these businesses.
Grocery stores also have a high inventory shrinkage rate due to a combination of factors like spoilage, theft, and errors.
Convenience stores have an open layout and reduced staff. They often serve customers during late-night hours. That’s why the chance of theft and inventory shrinkage is higher.
Warehouses and fulfillment centers might experience a high inventory shrinkage rate due to errors in inventory management. That includes handling errors, miscounts, or damage to products, especially when there’s high volume and fast-paced operations.
On average, the inventory shrinkage rate among US retailers is around 1.5% of annual sales. High-risk sectors like the ones mentioned above might have an inventory shrinkage rate of 2% or more.
How to Calculate Losses from Inventory Shrinkage?
Calculating inventory shrinkage is fairly straightforward. You need to know the value of the recorded inventory and the value of the actual inventory. The difference is the inventory shrinkage and the loss. This can be calculated as a percentage of total sales or as a percentage of the recorded inventory. Discovering the reasons for inventory shrinkage and in which categories it has occurred is a more complex process that requires inventory audits, sales data, and other reports.
What is meant by inventory shrinkage?
Inventory shrinkage refers to the loss of products or goods between the point of manufacture or purchase and the point of sale. It is often expressed as a percentage and represents the difference between the recorded inventory and the actual inventory in a retail or warehouse setting.
What are the 3 main causes of inventory shrinkage?
The three main causes of inventory shrinkage are:
Theft
Administrative errors
Damage or spoilage
How is inventory shrinkage calculated?
Inventory shrinkage is calculated using the following formula:
Inventory shrinkage (%) = (Recorded inventory - Actual inventory) / Recorded inventory X 100
What is KPI for inventory shrinkage?
Theinventory KPIfor inventory shrinkage is the shrinkage rate, which measures the percentage of inventory loss as a result of shrinkage. It is an important metric to monitor and minimize to ensure efficient inventory management and profitability.
What are the 3 key measures of inventory?
The three key measures of inventory include:
Inventory turnover
Days of inventory on hand
Gross margin return on inventory investment (GMROII)
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10922 | https://wnarifin.github.io/lecture/mstat/Poisson%20regression.pdf | Poisson Regression Dr Wan Nor Ari n Unit of Biostatistics and Research Methodology, Universiti Sains Malaysia.
E-mail: wnari n@usm.my Last update: 12 March 2019 Wan Nor Ari n, 2019.
Poisson Regression by Wan Nor Ari n is licensed under the Creative Commons Attribution-ShareAlike 4.0 International License.
To view a copy of this license, visit
1 Contents 1 Objectives 3 2 Poisson Distribution 3 3 Simple Poisson Regression 4 4 Multiple Poisson Regression 5 5 Model-building Steps for Multiple Poisson Regression 5 References 6 2 1 Objectives 1. Understand the concept of Poisson distribution.
2. Understand the concept of simple and multiple Poisson regression models.
3. Apply the models on data sets and interpret the results.
2 Poisson Distribution Basically, we are dealing with count data.
A Poisson distribution (Fleiss et al., 2003) is de ned as P(Y = y|µ) = e−µµy y!
for non-negative integers y = 0, 1, 2, ..., µ > 0.
Y is a Poisson random variable.
The parameter µ is the mean of Y .
This relationship between Y and μ can be written as Y ∼Poisson(µ) i.e. read as Y follows Poisson distribution with mean µ.
Poisson distribution comes from the binomial distribution (Rice, 1995) i.e. limit of the binomial distribution as the number of trials n approaches in nity and probability of success p approaches zero, so as np = µ (n = number of trials/samples; p = probability of success) →Read my note Probability Distribution to understand this better.
In other words, the number of event is very small as compared to the denominator, thus the p/proportion/percentage is very small, e.g. 0.000000123.
Properties Mean(Y ) = V ar(Y ) = µ Graphs of Poisson probability mass function with dierent p: Excel le Probability Distribution.xls > Poisson (also in poisson.R) It follows the assumptions of the Poisson process (Daniel, 1995): 1. The occurrences of the events are independent. The occurrence of an event in an interval of space or time does not aect the probability of second occurrence of the event in the same or dierent interval.
2. In nite number of occurrences of the event is possible in the interval.
3. Probability of a single occurrence of the event in an interval is proportionate to interval length.
4. In a very small portion of the interval, probability of more than one occurrence of the event is negligible.
Example 2.1: Probability of Y = y Suppose the number of death due to motor vehicle accidents per day in Malaysia is on average 17.2 and it was found that the daily distribution follows Poisson distribution.
What is the probability that any randomly selected day will be the one with 10 death?
3 Example 2.2: Probability of Y ≤y What is the probability that any randomly selected day will be the one with less than 11 death?
Example 2.3: Probability of Y > y What is the probability that any randomly selected day will be more than 10 death?
Using R: poisson.R 3 Simple Poisson Regression Let say Y is the Poisson count of some events e.g. number of accidents per month, or new HIV cases per year etc.
Suppose the count is somehow associated with some factors X s, e.g. gender, IVDU status, age etc.
We want to relate the Y with the X. Mean Y can be linked with X by ln E(Y |X) = β0 + β1X or its equivalent equation E(Y |X) = eβ0+β1X = eβ0eβ1X E(Y) = expected value of Y or mean of Y; E(Y|X) = conditional mean of Y given X. Remember mean of Y = E(Y) = μ.
For a simple case of exposure X = 0, 1, for reference/non-exposed group X = 0, ln E(Y |X = 0) = β0 E(Y |X = 0) = eβ0 thus the exponent of β0 is the mean of Y when X = 0.
For exposed group X = 1, ln E(Y |X = 1) = β0 + β1(1) = β0 + β1 E(Y |X = 0) = eβ0+β1 = eβ0eβ1 thus the exponent of β0 + β1 is the mean of Y when X = 1.
Then, to obtain the increase/dierence in mean of Y with the change in the exposure status, which is the exponent of β1 E(Y |X = 1) E(Y |X = 0) = eβ0eβ1 eβ0 = eβ1 usually called the rate ratio, RR.
The same concept is also applicable whenever the X is numerical, in which it re ects RR for x1 −x0 = ∆unit change in X, RR = E(Y |X = x1) E(Y |X = x0) = eβ0eβ1(x1) eβ0]eβ1(x0) = eβ1−β0 = eβ1∆ or for 1 unit change in X, RR = eβ1(1) = eβ1 Example 3.1, count data: poisson.R 4 In medicine, it is more common to describe the count in term of prevalence, incidence, person-years i.e the rate. The equation has to be modi ed to include the denominator/person-years a(X) by, E(Y |X) = a(X) eβ0+β1X ln E(Y |X) = ln a(X) + β0 + β1X the ln a(X) is speci cally called the oset. This will be speci ed when we t rate data.
Example 3.2, rate data: poisson.R 4 Multiple Poisson Regression Recall our equation for simple Poisson regression, ln E(Y |X) = ln µ = β0 + β1X which can be extended as ln E(Y |X) = ln µ = β0 + β1X1 + · · · + βp−1Xp−1 = β0 + X βp−1Xp−1 where the X (in bold) denotes a collection of X s. p is the number of estimated parameters. We minus 1 in the subscript since p also includes the intercept β0, thus p −1 is the number of X s.
The rate ratio, RR is, RR = eβp−1 Similarly, to include the oset ln E(Y |X) = ln a(X) + β0 + X βp−1Xp−1 Now βj (i.e. the speci c β coe cient) is interpreted similarly to the simple regression case, while holding all other variables constant, or adjusted/controlling for the other variables.
Similar to other multiple regressions, create dummy variables for a categorical variable with > 2 categories. However, it is au-tomatically created in R, if the variable is speci ed as a factor. (i.e. using the factor() function) also consider the eect of two-way interaction terms in the model.
5 Model-building Steps for Multiple Poisson Regression 1. Variable selection.
(a) Univariable analysis.
Determine the signi cance of the variables by Wald's test.
LR test.
(b) Multivariable analysis.
i. Fit using selected variables.
All variables P-value < .25.
Clinically important variables 5 ii. Fit a smaller model by removing non-signi cant variables.
(c) Interactions among variables.
Among clinically plausible pairs.
2. Model t assessment.
(a) Goodness-of- t.
i. Chi-square goodness-of- t.
In R, based on residual deviance (poisgof() function).
d f = n −p P-value > 0.05 indicates good t.
ii. Model-to-model AIC comparison.
iii. Scaled Pearson chi-square statistic.
Pearson chi-square statistic is given as χ2 P = X (Yi −ˆ µi)2 ˆ µi with d f = n −p.
Scaled Pearson chi-square statitic = χ2 P /d f. The closer the value is to 1, the better is the t Fleiss et al. (2003). Large value indicates overdispersion problem (i.e.
V AR(Y ) > Mean(Y )).
We have to calculate manually, but easy with R. Yi is the observed counts, ˆ µi is the tted/predicted counts, obtained by model$fitted or predicted(model) functions.
(b) Regression diagnostics.
We may use the standardized residuals, obtained by rstandard() function. Since it is in form of standardized z score, we may use speci c cuto e.g. > 1.96 (α = .05) to > 3.89 (α = .0001).
References Daniel, W. W. (1995). Biostatistics: A foundation for analysis in the health sciences. USA: John Wiley & Sons, 6th ed. edition.
Fleiss, J. L., Levin, B., and Paik, M. C. (2003). Statistical Methods for Rates and Proportions. USA: John Wiley & Sons, 3rd ed. edition.
Rice, J. A. (1995). Mathematical statistics and data analysis. USA: Duxbury Press, 2nd ed. edition.
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10923 | https://math.libretexts.org/Courses/Honolulu_Community_College/Math_75X%3A_Introduction_to_Mathematical_Reasoning_(Kearns)/06%3A_More_to_Algebra_Than_Just_(Linear)_Equations/6.02%3A_Inequalities | {x:x>−10}
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6.2: Inequalities
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: Jan 7, 2020
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6.1: What Does it Mean to Be a Solution?
6.3: Polynomials Expressions and Equations in One and Two Variables
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David Arnold
College of the Redwoods
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In Chapter 1, we introduced the natural numbers N={1,2,3,…}N={1,2,3,…}, the whole numbers W={0,1,2,3,…}W={0,1,2,3,…}, and the integers Z={…,−3,−2,−1,0,1,2,3,…}Z={…,−3,−2,−1,0,1,2,3,…}. Later in the chapter, we introduced the rational numbers, numbers of the form p/qp/q, where pp and qq are integers. We noted that both terminating and repeating decimals are rational numbers. Each of these numbers has a unique position on the number line (see Figure 6.2.16.2.1 ).
The natural numbers, whole numbers, and integers are also rational numbers, because each can be expressed in the form pqpq, where pp and qq are integers. For example, 0=0120=012, 4=414=41, and −3=−124−3=−124. Indeed, the rational numbers contain all of the numbers we’ve studied up to this point in the course. However, not all numbers are rational numbers. For example, consider the decimal number −3.10110111011110…−3.10110111011110…, which neither terminates nor repeats. The number √2=1.414213562373095…2–√=1.414213562373095… also equals a decimal number that never terminates and never repeats. A similar statement can be made about the number π=3.141592653589793…π=3.141592653589793… Each of these irrational (not rational) numbers also has a unique position on the number line (see Figure 6.2.26.2.2).
Two other irrational numbers you may encounter in your mathematical studies are ee (Euler’s constant), which is approximately equal to e≈2.71828182845904…e≈2.71828182845904…, and ϕϕ (pronounced “phi”), called the golden ratio, which equals ϕ=1+√52ϕ=1+5√2. The number ee arises in applications involving compound interest, probability, and other areas of mathematics. The number ϕϕ is used in financial markets and is also arguably the ratio of beauty in art and architecture.
The Real Numbers
If we combine all of the rational and irrational numbers into one collection, then we have a set of numbers that is called the set of real numbers. The set of real numbers is denoted by the symbol RR.
Every point on the number line is associated with a unique real number. Conversely, every real number is associated with a unique position on the number line. In lieu of this correspondence, the number line is usually called the real line.
Ordering the Real Numbers
The real numbers are ordered on the real line in a manner identical to how we ordered the integers on the number line in Section 1 of Chapter 1.
Order on the Real Line
Suppose that aa and bb are real numbers positioned on the real line as shown below.
Because aa lies to the “left of” bb, we say that aa is “less than” bb, or in mathematical symbols, a<ba<b. The inequality symbol << is read “less than.”
Alternately, bb lies to the “right of” aa, so we can also say that bb is “greater than” aa, or in mathematical symbols, b>ab>a. The inequality symbol >> is read “greater than.”
Here are two more inequality symbols that we will use in this section.
Less than or equal to
If we want to say that aa lies to the “left of” bb, or shares the same position as bb, then we say that aa is “less than or equal to” bb and write a≤ba≤b. The inequality symbol ≤≤ is pronounced “less than or equal to.”
Greater than or equal to
If we want to say that bb lies to the “right of” aa, or shares the same position as aa, then we say that bb is “greater than or equal to" aa and write b≥ab≥a. The inequality symbol ≥≥ is pronounced “greater than or equal to.”
Set-Builder Notation
Mathematicians use a construct called set-builder notation to describe sets or collections of numbers. The general form of set-builder notation looks as follows: {x: some statement about x}
{x: some statement about x}
For example, suppose that we want to describe the set of “all real numbers that are less than 22.” We could use the following notation: A={x:x<2}
A={x:x<2}
This is read aloud as follows: “AA equals the set of all xx such that xx is less than 22.” Some prefer to use a vertical bar instead of a colon. A={x|x<2}
A={x|x<2}
In this text we use the colon in set-builder notation, but feel free to use the vertical bar instead. They mean the same thing. One might still object that the notation {x:x<2}
{x:x<2}
is a bit vague. One objection could be “What type of numbers xx are you referring to? Do you want the integers that are less than two or do you want the real numbers that are less than two?” As you can see, this is a valid objection. One way of addressing this objection is to write: A={x∈R:x<2} or A={x∈N:x<2}
A={x∈R:x<2} or A={x∈N:x<2}
The first is read “AA is the set of all xx in RR that are less than two,” while the second is read “AA is the set of all xx in NN that are less than two.”
Set-builder Assumption
In this text, unless there is a specific reference to the set of numbers desired, we will assume that the notation {x:x<2}{x:x<2} is asking for the set of all real numbers less than 22.
In Figure 6.2.36.2.3, we’ve shaded the set of real numbers {x:x<2}{x:x<2}. Because
“less than” is the same as saying “left of,” we’ve shaded (in red) all points on the real line that lie to the left of the number two. Note that there is an “empty circle” at the number two. The point representing the number two is not shaded because we were only asked to shade the numbers that are strictly less than two.
While the shading in Figure 6.2.36.2.3 is perfectly valid, much of the information provided in Figure 6.2.36.2.3 is unnecessary (and perhaps distracting). We only need to label the endpoint and shade the real numbers to the left of two, as we’ve done in constructing Figure 6.2.46.2.4.
For contrast, suppose instead that we’re asked to shade the set of real numbers {x:x≤2}{x:x≤2}. This means that we must shade all the real numbers
that are “less than or equal to 22” or “left of and including 22.” The resulting set is shaded in Figure 6.2.56.2.5.
Note the difference between Figures 6.2.46.2.4 and 6.2.456.2.45. In Figures 6.2.46.2.4 we’re shading the set {x:x<2}{x:x<2}, so the number 22 is left unshaded (an empty dot). In Figures 6.2.56.2.5, we’re shading the set {x:x≤2}{x:x≤2}, so the number 22 is shaded (a filled-in dot).
Example 6.2.16.2.1
Shade the set {x:x≥−3}{x:x≥−3} on the real line.
Solution
The notation {x:x≥−3}{x:x≥−3} is pronounced “the set of all real numbers xx such that xx is greater than or equal to −3−3.” Thus, we need to shade the number −3−3 and all real numbers to the right of −3−3.
Exercise 6.2.16.2.1
Shade {x:x≤4}{x:x≤4} on the real line.
Answer
Example 6.2.26.2.2
Use set-builder notation to describe the set of real numbers that are shaded on the number line below.
Solution
The number −1−1 is not shaded. Only the numbers to the left of −1−1 are shaded. This is the set of all real numbers xx such that xx is “less than” −1−1. Thus, we describe this set with the following set-builder notation: {x:x<−1}
{x:x<−1}
Exercise 6.2.26.2.2
Use set-builder notation to describe the following set of real numbers:
Answer
: {x:x>−10}
Interval Notation
In Examples 6.2.16.2.1 and 6.2.26.2.2, we used set-builder notation to describe the set of real numbers greater than or equal to −3−3 and a second set of real numbers less than −1−1. There is another mathematical symbolism, called interval notation, that can be used to describe these sets of real numbers. Consider the first set of numbers from Example 6.2.16.2.1, {x:x≥−3}{x:x≥−3}.
Sweeping our eyes “from left to right”, we use −3,∞)[−3,∞) to describe this set of real numbers. Some comments are in order:
The bracket at the left end means that −3−3 is included in the set.
As you move toward the right end of the real line, the numbers grow without bound. Hence, the ∞∞ symbol (positive infinity) is used to indicate that we are including all real numbers to the right of −3−3. However, ∞∞ is not really a number, so we use a parentheses to indicate we are “not including” this fictional point.
The set of numbers from Example 6.2.16.2.1 is {x:x<−1}{x:x<−1}.
Sweeping our eyes “from left to right”, this set of real numbers is described with (−∞,−1)(−∞,−1). Again, comments are in order:
The number −1−1 is not included in this set. To indicate that it is not included, we use a parenthesis.
As you move toward the left end of the real line, the numbers decrease without bound. Hence, the −∞−∞ symbol (negative infinity) is used to indicate that we are including all real numbers to the left of −1−1. Again, −∞−∞ is not an actual number, so we use a parenthesis to indicate that we are not including this “fictional” point.
Sweep your eyes from “left to right”
If you would like to insure that you correctly use interval notation, place the numbers in your interval notation in the same order as they are encountered as you sweep your eyes from “left to right” on the real line.
A nice summary of set-builder and interval notation is presented in Table 6.2.16.2.1 at the end of the section.
Equivalent Inequalities
Like equations, two inequalities are equivalent if they have the same solution sets.
Adding or Subtracting the Same Quantity from Both Sides of an Inequality
Let aa and bb be real numbers with a<b
a<b
If cc is any real number, then a+c<b+c
a+c<b+c
anda−c<b−c
a−c<b−c
That is, adding or subtracting the same amount from both sides of an inequality produces an equivalent inequality (does not change the solution).
Example 6.2.36.2.3
Solve for x:x−2≤7x:x−2≤7. Sketch the solution on the real line, then use set-builder and interval notation to describe your solution.
Solution
To “undo” subtracting 22, we add 22 to both sides of the inequality.
x−2≤7 Original inequality. x−2+2≤7+2 Add 2 to both sides. x≤9 Simplify both sides.
x−2x−2+2x≤7≤7+2≤9 Original inequality. Add 2 to both sides. Simplify both sides.
To shade the real numbers less than or equal to 99, we shade the number 99 and all real numbers to the left of 99.
Using set-builder notation, the solution is {x:x≤9}{x:x≤9}. Using interval notation, the solution is (−∞,9
If we multiply or divide both sides of an inequality by a positive number, we have an equivalent inequality.
Multiplying or Dividing by a Positive Number
Let aa and bb be real numbers with a<ba<b. If cc is a real positive number, then ac<bc
ac<bc
andac<bc
ac<bc
Example 6.2.46.2.4
Solve for x:3x≤−9x:3x≤−9 Sketch the solution on the real line, then use set-builder and interval notation to describe your solution.
Solution
To “undo” multiplying by 33, divide both sides of the inequality by 33. Because we are dividing both sides by a positive number, we do not reverse the inequality sign.
3x≤−9 Original inequality. 3x3≤−93 Divide both sides by 3.x≤−3 Simplify both sides.
3x3x3x≤−9≤−93≤−3 Original inequality. Divide both sides by 3. Simplify both sides.
Shade the real numbers less than or equal to −3−3.
Using set-builder notation, the solution is {x:x≤−3}{x:x≤−3}. Using interval notation, the solution is (−∞,−3](−∞,−3].
Exercise 6.2.46.2.4
Use interval notation to describe the solution of: 2x>−8
2x>−8
Answer
: (−4,∞)
Reversing the Inequality Sign
Up to this point, it seems that the technique for solving inequalities is pretty much identical to the technique used to solve equations. However, in this section we are going to encounter one exception.
Suppose we start with the valid inequality −2<5−2<5, then we multiply both sides by 22, 33, and 44.
−2<5−2<5−2<52(−2)<2(5)3(−2)<3(5)4(−2)<4(5)−4<10−6<15−8<20
−2<52(−2)<2(5)−4<10−2<53(−2)<3(5)−6<15−2<54(−2)<4(5)−8<20
Note that in each case, the resulting inequality is still valid.
Caution! We’re about to make an error!Caution! We’re about to make an error!
Start again with −2<5−2<5, but this time multiply both sides by −2−2, −3−3, and −4−4.
−2<5−2<5−2<5−2(−2)<−2(5)−3(−2)<−3(5)−4(−2)<−4(5)4<−106<−158<−20
−2<5−2(−2)<−2(5)4<−10−2<5−3(−2)<−3(5)6<−15−2<5−4(−2)<−4(5)8<−20
In each of the resulting inequalities, the inequality symbol is pointing the wrong way!
When you multiply both sides of an inequality by a negative number, you must reverse the inequality sign. Starting with −2<5−2<5, multiply both sides by −2−2, −3−3, and −4−4, but reverse the inequality symbol.
Some readers might prefer a more formal reason as to why we reverse the inequality when we multiply both sides by a negative number. Suppose that a<ba<b. Then, subtracting bb from both sides gives the result a−b<0a−b<0. This means that a−ba−b is a negative number. Now, if cc is a negative number, then the product (a−b)c(a−b)c is positive. Then:
(a−b)c>0ac−bc>0ac−bc+bc>0+bcac>bc
(a−b)cac−bcac−bc+bcac>0>0>0+bc>bc
Thus, if you start with a<ba<b and c<0c<0, then ac>bcac>bc.
Multiplying or Dividing by a Negative Number
Let aa and bb be real numbers with a<babc
ac>bc
and ac>bc
ac>bc
That is, when multiplying or dividing both sides of an inequality by a negative number, you must reverse the inequality sign.
Example 6.2.56.2.5
Solve for x:−2x<4x:−2x<4. Sketch the solution on the real line, then use set-builder and interval notation to describe your solution.
Solution
To “undo” multiplying by −2−2, divide both sides by −2−2. Because we are dividing both sides by a negative number, we reverse the inequality sign.
−2x<4 Original inequality. −2x−2>4−2 Divide both sides by −2x>−2 Reverse the inequality sign. x>−2 Simplify both sides.
−2x−2x−2xx<4>4−2>−2>−2 Original inequality. Divide both sides by −2 Reverse the inequality sign. Simplify both sides.
Shade the real numbers greater than −2−2.
Using set-builder notation, the solution is {x:x>−2}{x:x>−2}. Using interval notation, the solution is (−2,∞)(−2,∞).
Exercise 6.2.56.2.5
Use interval notation to describe the solution of: −3x≥−6
−3x≥−6
Answer
: (−∞,2]
Multiple Steps
Sometimes you need to perform a sequence of steps to arrive at the solution.
Example 6.2.66.2.6
Solve for x:2x+5>−7x:2x+5>−7. Sketch the solution on the real line, then use set-builder and interval notation to describe your solution.
Solution
To “undo” adding 55, subtract 55 from both sides of the inequality.
2x+5>−7 Original inequality. 2x+5−5>−7−5 Subtract 5 from both sides. 2x>−12 Simplify both sides.
2x+52x+5−52x>−7>−7−5>−12 Original inequality. Subtract 5 from both sides. Simplify both sides.
To “undo” multiplying by 2, divide both sides by 2. Because we are dividing both sides by a positive number, we do not reverse the inequality sign.
2x2>−122 Divide both sides by 2x>−6 Simplify both sides.
Shade the real numbers greater than −6.
Using set-builder notation, the solution is {x:x>−6}. Using interval notation, the solution is (−6,∞).
Exercise 6.2.6
Use interval notation to describe the solution of: 3x−2≤4
Answer
: (−∞,2]
Example 6.2.7
Solve for x:3−5x≤2x+17. Sketch the solution on the real line, then use set-builder and interval notation to describe your solution.
Solution
We need to isolate terms containing x on one side of the inequality. Start by subtracting 2x from both sides of the inequality.
3−5x≤2x+17 Original inequality. 3−5x−2x≤2x+17−2x Subtract 2x from both sides. 3−7x≤17 Simplify both sides.
We continue to isolate terms containing x on one side of the inequality. Subtract 3 from both sides.
3−7x−3≤17−3 Subtract 3 from both sides. −7x≤14 Simplify both sides.
To “undo” multiplying by −7, divide both sides by −7. Because we are dividing both sides by a negative number, we reverse the inequality sign.
−7x−7≥14−7 Divide both sides by −7x≥−2 Simplify both sides.
Using set-builder notation, the solution is {x:x≥−2}. Using interval notation, the solution is [−2,∞).
Exercise 6.2.7
Use interval notation to describe the solution of: 4−x>2x+1
Answer
: (−∞,1)
We clear fractions from an inequality in the usual manner, by multiplying both sides by the least common denominator.
Example 6.2.8
Solve for x:34−x12>13.
Solution
First, clear the fractions from the inequality by multiplying both sides by the least common denominator, which in this case is 12.
34−x12>13 Original inequality. 12[34−x12]>12 Multiply both sides by 12.12−12[x12]>12 Distribute the 12.9−x>4 Cancel and Multiply.
To “undo” adding 9, subtract 9 from both sides.
9−x−9>4−9 Subtract 9 from both sides. −x>−5 Simplify both sides.
We could divide both sides by −1, but multiplying both sides by −1 will also do the job. Because we are multiplying both sides by a negative number, we reverse the inequality sign.
(−1)(−x)<(−5)(−1) Multiply both sides by −1. Reverse the inequality sign. x<5 Simplify both sides.
Shade the real numbers less than 5.
Using set-builder notation, the solution is {x:x<5}. Using interval notation, the solution is (−∞,5).
Exercise 6.2.8
Use interval notation to describe the solution of: 2x3−34≥−32
Answer
: [−98,∞)
We clear decimals from an inequality in the usual manner, by multiplying both sides by the appropriate power of ten.
Example 6.2.9
Solve for x:3.25−1.2x>4.6.
Solution
First, clear the decimals from the inequality by multiplying both sides by 100, which moves each decimal point two places to the right.
3.25−1.2x>4.6 Original inequality. 325−120x>460 Multiply both sides by 100.325−120x−325>460−325 Subtract 325 from both sides. −120x>135 Simplify both sides. −120x−120<135−120 Divide both sides by −120. Reverse the inequality sign.x<−2724Reduce to lowest terms.
Shade the real numbers less than −27/24.
Using set-builder notation, the solution is {x:x<−27/24}. Using interval notation, the solution is (−∞,−27/24).
Exercise 6.2.9
Use interval notation to describe the solution of: 2.3x−5.62≥−1.4
Answer
: [211115,∞)
Summary Table of Set-Builder and Interval Notation
A summary table of the set-builder and interval notation is presented in Table 6.2.1.
Table 6.2.1: Examples of set-builder and interval notation.
| Shading on the real line | Set-builder | Interval |
| | {x:x>−5} | (−5,∞) |
| | {x:x≥−5} | [−5,∞) |
| | {x:x<−5} | (−∞,−5) |
| | {x:x≤−5} | (−∞,−5] |
6.1: What Does it Mean to Be a Solution?
6.3: Polynomials Expressions and Equations in One and Two Variables |
10924 | https://math.stackexchange.com/questions/2371956/curve-for-which-part-of-tangent-bisected-at-point-of-tangency | calculus - Curve for which Part of Tangent bisected at point of Tangency - Mathematics Stack Exchange
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Curve for which Part of Tangent bisected at point of Tangency
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To find the curve for which the part of tangent cut-off by the axes (the portion of the tangent between the coordinate axes) is bisected at the point of tangency.
Let x a+y b=1 x a+y b=1 be the tangent. It cuts the axes at (a,0)(a,0) and (0,b)(0,b). So the mid point of the part of tangent cut-off by the axes is (a 2,b 2)(a 2,b 2). The slope of this tangent is −b a−b a. (S i n c e y=−b a x+b).(S i n c e y=−b a x+b).
Let the slope of the required curve at point (x,y)(x,y) given by d y d x=f(x,y)d y d x=f(x,y).
So we can say that f(a 2,b 2)=−b a⇒f(a 2,b 2)=−b/2 a/2⇒f(x,y)=−y x f(a 2,b 2)=−b a⇒f(a 2,b 2)=−b/2 a/2⇒f(x,y)=−y x.
Now f(x,y)=d y d x=−y x⇒d y y=−d x x⇒log(y)=−log(x)+log(c)⇒x y=c f(x,y)=d y d x=−y x⇒d y y=−d x x⇒log(y)=−log(x)+log(c)⇒x y=c.
Is this the correct way to proceed? Any other ideas?
calculus
ordinary-differential-equations
tangent-line
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edited Jul 26, 2017 at 10:20
simajinidsimajinid
asked Jul 26, 2017 at 6:15
simajinidsimajinid
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I obtain the same result with another method. The difficulty was to understand the meaning of "the part of tangent cut-off by the axes is bisected at the point of tangency".JJacquelin –JJacquelin 2017-07-26 08:12:44 +00:00 Commented Jul 26, 2017 at 8:12
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Another approach :
Let Y=Y(X)Y=Y(X) the equation of the curve.
At point (X,Y)(X,Y) the equation of the tangent is y=Y+(x−X)Y′(X)y=Y+(x−X)Y′(X)
The intersects to axes are : {0=Y+(a−X)Y′(X)→a=X−Y Y′(X)b=Y+(0−X)Y′(X)→b=Y−X Y′(X){0=Y+(a−X)Y′(X)→a=X−Y Y′(X)b=Y+(0−X)Y′(X)→b=Y−X Y′(X)
Condition : {a 2=X→2 X=X−Y Y′(X)b 2=Y→2 Y=Y−X Y′(X)→Y′Y+1 X=0{a 2=X→2 X=X−Y Y′(X)b 2=Y→2 Y=Y−X Y′(X)→Y′Y+1 X=0
Y(X)=C X Y(X)=C X
The curve is a rectangular hyperbola (orthogonal asymptotes).
Note : This property of rectangular hyperbola is known for long (Apollonius theorem)
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edited Jul 26, 2017 at 9:32
answered Jul 26, 2017 at 7:56
JJacquelinJJacquelin
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Thanks. I think this is a better way get the required curve.simajinid –simajinid 2017-07-26 10:18:27 +00:00 Commented Jul 26, 2017 at 10:18
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10925 | https://www.quora.com/What-is-the-maximum-value-of-Sin-A+Sin-B+-Sin-C-in-a-triangle | Something went wrong. Wait a moment and try again.
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Trigonometric Functions
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Geometric Mathematics
5
What is the maximum value of Sin A+Sin B+ Sin C in a triangle?
Arun Iyer
Principal Researcher at Microsoft Research
·
Author has 829 answers and 4M answer views
·
9y
sinx is concave in (0,π), therefore,
sinA+sinB+sinC3≤sin(A+B+C3)(1)
Since, A, B, C are angles of a triangle, (1) gives,
sinA+sinB+sinC≤3√32(2)
Since the upper bound in (2) can be easily achieved by setting A=B=C, the upper bound is tight. Hence the required maximum is 3√32.
Related questions
What is the maximum possible value of sin A+sin B+sin C in a triangle ABC?
What is the maximum value of sin A + sin B + sin C?
What is the maximum value of
sin
A
sin
B
sin
C
in a triangle?
What is the maximum value of sin (A+B+C) in a triangle?
What is the maximum value of
sin
A
sin
B
sin
C
if
A
+
B
+
C
=
180
°
?
Vasant Barve
SSC 11 in Mathematics & Science, Niphad (Graduated 1958)
·
Author has 360 answers and 243.2K answer views
·
5y
Originally Answered: What is the maximum possible value of sin A+sin B+sin C in a triangle ABC?
·
The blue line shows variation of Sum as we move point C
The blue line shows variation of Sum as we move point C
Aditya Tripathi
Knows English
·
Author has 173 answers and 165.3K answer views
·
5y
Originally Answered: What is the maximum possible value of sin A+sin B+sin C in a triangle ABC?
·
As I know,
Here is a hint, which should get you most of the way there
Note that
sinB+sinC=2sinB+C2cosB−C2
If
A is fixed then
B+C is fixed, and the product is greatest when
B=C
Hope this helps you
Alexander Farrugia
uses trigonometry in topics that are seemingly unrelated.
·
Upvoted by
James McElhatton Ph.D. (Glasgow, 1976)
, B.Sc. Mathematics & Chemistry, University of Malta (1967) · Author has 3.2K answers and 27.5M answer views
·
3y
Related
What is the maximum value of
sin
A
sin
B
sin
C
in a triangle?
I’ll just use partial derivatives, since nobody has done that yet. (Which I find a bit surprising.)
First of all, since A+B+C=π, sinC=sin(π−(A+B))=sin(A+B). Hence, we replace our expression into a function of two variables
f(A,B)=sinAsinBsin(A+B).
fA=sinB(sinAcos(A+B)+cosAsin(A+B))=sinBsin(2A+B)
and similarly
fB=sinAsin(A+2B).
Now fA=0 means that either sinB=0 or sin(2A+B)=0. Since A and B satisfy 0<A,B<π, sinB=0 has no solutions, while sin(2A+B)=0 means that 2A+B=mπ where m∈{1,2}.
Similarly from fB=0 we obtain that A+2B=n
I’ll just use partial derivatives, since nobody has done that yet. (Which I find a bit surprising.)
First of all, since A+B+C=π, sinC=sin(π−(A+B))=sin(A+B). Hence, we replace our expression into a function of two variables
f(A,B)=sinAsinBsin(A+B).
fA=sinB(sinAcos(A+B)+cosAsin(A+B))=sinBsin(2A+B)
and similarly
fB=sinAsin(A+2B).
Now fA=0 means that either sinB=0 or sin(2A+B)=0. Since A and B satisfy 0<A,B<π, sinB=0 has no solutions, while sin(2A+B)=0 means that 2A+B=mπ where m∈{1,2}.
Similarly from fB=0 we obtain that A+2B=nπ where n∈{1,2}.
Solving simultaneously, A=2m−n3π and B=2n−m3π. When either m or n is 2, we get contradictory values for A and/or B; thus m=n=1 and hence A=B=π3.
Now fAA=2sinBcos(2A+B), fBB=2sinAcos(A+2B) and fAB=fBA=sin(2A+2B). When A=B=π3, fAA=2sinπ3cosπ=−√3<0. Moreover, fBB=−√3 as well, while fAB=−√32. This means that fAAfBB−(fAB)2=(−√3)(−√3)−(−√32)2>0.
All this tells us that A=B=π3 is a maximum of f. So the maximum value of f is sinπ3sinπ3sin(π3+π3)=3√38.
Related questions
If
a
,
b
,
c
∈
R
and
a
+
b
+
c
=
π
, find the minimum value of
sin
a
+
sin
b
+
sin
c
without using calculus?
What is Sin A + Sin B + Sin C =?
How do I solve this sin(B-C), Sin B=-4/5, sin C=5/13?
If
A
+
B
+
C
=
π
, in a triangle ABC, then what is the value of
sin
A
+
sin
B
+
sin
C
?
What is the value of sin inverse sin -600?
Assistant
Bot
·
1y
To find the maximum value of sinA+sinB+sinC in a triangle, we start by using the fact that the angles A, B, and C of a triangle satisfy A+B+C=180∘.
Using the identity sinC=sin(180∘−A−B)=sin(A+B), we can express sinC in terms of A and B:
sinA+sinB+sinC=sinA+sinB+sin(A+B)
To maximize sinA+sinB+sin(A+B), we can also use the Cauchy-Schwarz inequality or Jensen's inequality. However, a more straightforward approach is to analyze the situation geometrically or via numerical maximization.
Geometric Interpretation
The sine function
To find the maximum value of sinA+sinB+sinC in a triangle, we start by using the fact that the angles A, B, and C of a triangle satisfy A+B+C=180∘.
Using the identity sinC=sin(180∘−A−B)=sin(A+B), we can express sinC in terms of A and B:
sinA+sinB+sinC=sinA+sinB+sin(A+B)
To maximize sinA+sinB+sin(A+B), we can also use the Cauchy-Schwarz inequality or Jensen's inequality. However, a more straightforward approach is to analyze the situation geometrically or via numerical maximization.
Geometric Interpretation
The sine function achieves its maximum value of 1 at 90∘. Therefore, to maximize sinA+sinB+sinC, we can consider the case when one of the angles is 90∘ and the other two are 45∘:
Let A=90∘, B=45∘, and C=45∘.
Then:
sinA=sin90∘=1,
sinB=sin45∘=√22,
sinC=sin45∘=√22.
So, we calculate:
sinA+sinB+sinC=1+√22+√22=1+√2≈2.414.
Conclusion
The maximum value of sinA+sinB+sinC in a triangle occurs when A=90∘, B=45∘, and C=45∘, giving:
sinA+sinB+sinC≤1+√2≈2.414.
Thus, the maximum value of sinA+sinB+sinC in a triangle is 1+√2.
Debargha Chakraborty
Junior Executive (Air Traffic Control)
·
Author has 341 answers and 1M answer views
·
9y
Well the thing in general is the function reaches an extrema when equally divided, which is more frequently a maxima.
So if we divide A=B=C= 60o we get the maximum value of the expression.
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Sumeeth C Muchandimath
Studied at Sri Chaitanya Educational Institutions (Graduated 2021)
·
5y
Originally Answered: What is the maximum possible value of sin A+sin B+sin C in a triangle ABC?
·
For questions such as these you could always assume the triangle to be equilateral for quick solving in an objective point of view.
Therefore A=B=C=60°
Hence sinA+sinB+sinC=(3√3)/2
There max value of sinA+sinB+sinC = (3√3)/2
David Dodson
Former High School Math Teacher at Rift Valley Academy (2001–2006)
·
Author has 792 answers and 325.8K answer views
·
3y
Related
What is the maximum value of
sin
A
sin
B
sin
C
in a triangle?
The easiest way to solve this that I have found is using Lagrange multipliers. Let f(x) = sin A sin B sin C + lambda (A + B + C - 180). Taking partial derivatives with respect to A, B, C, and lambda and setting the results to zero gives cos A sin B sin C + lambda = 0, sin A cos B sin C + lambda = 0, sin A sin B cos C + lambda, and A + B + C -180 = 0. Subtracting the second equation from the first gives cos A sin B sin C - sin A cos B sin C = 0, and since the product of sines cannot be maximized if sin C = 0, we get cos A sin B - sin A cos B = 0. But sin(A - B) = cos A sin B - sin A cos B, show
The easiest way to solve this that I have found is using Lagrange multipliers. Let f(x) = sin A sin B sin C + lambda (A + B + C - 180). Taking partial derivatives with respect to A, B, C, and lambda and setting the results to zero gives cos A sin B sin C + lambda = 0, sin A cos B sin C + lambda = 0, sin A sin B cos C + lambda, and A + B + C -180 = 0. Subtracting the second equation from the first gives cos A sin B sin C - sin A cos B sin C = 0, and since the product of sines cannot be maximized if sin C = 0, we get cos A sin B - sin A cos B = 0. But sin(A - B) = cos A sin B - sin A cos B, showing that sin(A - B) = 0. Thus, A = B (or A + B = 180 making C = 0, which we reject). Similarly, by subtracting the third equation from the first shows that A = C. Thus A + B + C - 180 = 0 simplifies to 3A = 180, so A = 60 degrees. Then also B = C = 60 degrees. The maximum value of sin A sin B sin C in a triangle is (sin 60)^2 = (sqrt(3)/2)^3 = 3 sqrt(3) / 8.
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Doug Dillon
Ph.D. Mathematics
·
Upvoted by
David Vanderschel
, PhD Mathematics & Physics, Rice (1970) · Author has 12.4K answers and 11.4M answer views
·
3y
Related
What is the maximum value of
sin
A
sin
B
sin
C
in a triangle?
Since y=sinx is concave and positive if x is an angle of a triangle, we can use Jensen’s Inequality and AM-GM to write
√32=sin60=sin(A+B+C3)≥sinA+sinB+sinC3Jensen≥3√sinAsinBsinC
with equality when sinA=sinB=sinC.
So √32≥3√sinAsinBsinC making sinAsinBsinC≤3√38
Enrico Gregorio
Associate professor in Algebra
·
Author has 18.4K answers and 16M answer views
·
3y
Related
What is the maximum value of
sin
A
sin
B
sin
C
in a triangle?
You might enjoy proving that, if A+B+C=π, then
4sinAsinBsinC=sin2A+sin2B+sin2C
and it’s much easier to maximize
f(x,y,z)=sinx+siny+sinz
subject to x>0,y>0,z>0 and x+y+z=2π
The Lagrangian is
L(x,y,z,t)=sinx+siny+sinz−t(x+y+z−2π)
and so we need
cosx=cosy=cosz=t
An obvious solution is x=y=z=2π/3. Let’s look for less obvious solutions.
We might have
y=−x+2mπ,z=x+2nπ
and so x+2(m+n)π=2π, but this doesn’t have solutions within the constraints.
You can also eliminate the other cases. We can also observe that when z→0 we have x+y→2π and f(x,y,z)→0.
Thus the maximum
You might enjoy proving that, if A+B+C=π, then
4sinAsinBsinC=sin2A+sin2B+sin2C
and it’s much easier to maximize
f(x,y,z)=sinx+siny+sinz
subject to x>0,y>0,z>0 and x+y+z=2π
The Lagrangian is
L(x,y,z,t)=sinx+siny+sinz−t(x+y+z−2π)
and so we need
cosx=cosy=cosz=t
An obvious solution is x=y=z=2π/3. Let’s look for less obvious solutions.
We might have
y=−x+2mπ,z=x+2nπ
and so x+2(m+n)π=2π, but this doesn’t have solutions within the constraints.
You can also eliminate the other cases. We can also observe that when z→0 we have x+y→2π and f(x,y,z)→0.
Thus the maximum is at x=y=z=2π/3 and
f(2π/3,2π/3,2π/3)=3√32
Hence the maximum for sinAsinBsinC is
143√32=38√3
Sohel Zibara
Studied at Doctor of Philosophy Degrees (Graduated 2000)
·
Author has 5.1K answers and 2.6M answer views
·
3y
Related
What is the maximum value of
sin
A
sin
B
sin
C
in a triangle?
Ravi Sharma
Former Group A Officer From Indian Railways (1973–2009)
·
Author has 15.1K answers and 3.8M answer views
·
3y
Related
What is the maximum value of
sin
A
sin
B
sin
C
in a triangle?
FOR A GIVEN PERIMETER, THE MAXIMUM AREA OF A TRIANGLE IS WHEN IT IS AN EQUILATERAL TRIANGLE.
AS PER THE SIN FORMULA
a/sinA= b/sinB= c/sinC=k
a= ksinA, b= ksinB
AREA OF TRIANGLE ABC= (1/2) abSinC
= (1/2)k^2(sinA.sinB.sinC)
THIS VALUE OF AREA OF TRIANGLE IS MAXIMUM WHEN a= b= c OR ALL THREE ANGLES OF TRIANGLE ARE EQUAL TO 60°.
MAXIMUM VALUE OF
sinA.sinB.sinC= sin60°.sin60°.sin60°
= (✓3/2)(✓3/2)(✓3/2)=(3✓3)/8
Bernd Leps
Former scientific official; retired
·
Author has 5.8K answers and 1.3M answer views
·
2y
Related
What is the maximum value of sin A + sin B + sin C?
Because (in real numbers) a sin function has a value between -1 and +1, thrice that has a maximum of 3. A, B, and C must be angles out of pi/2 + 2npi, n being integer.
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What is the maximum possible value of sin A+sin B+sin C in a triangle ABC?
What is the maximum value of sin A + sin B + sin C?
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10926 | https://uknowledge.uky.edu/cgi/viewcontent.cgi?article=1587&context=gradschool_theses | University of Kentucky University of Kentucky
UKnowledge UKnowledge
University of Kentucky Master's Theses Graduate School
2008
Thevenin Equivalent Circuit Estimation and Application for Power Thevenin Equivalent Circuit Estimation and Application for Power
System Monitoring and Protection System Monitoring and Protection
Mohammad M. Iftakhar University of Kentucky
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Recommended Citation Recommended Citation
Iftakhar, Mohammad M., "Thevenin Equivalent Circuit Estimation and Application for Power System
Monitoring and Protection" (2008).
University of Kentucky Master's Theses . 583.
This Thesis is brought to you for free and open access by the Graduate School at UKnowledge. It has been
accepted for inclusion in University of Kentucky Master's Theses by an authorized administrator of UKnowledge.
For more information, please contact UKnowledge@uky.edu, rs_kbnotifs-acl@uky.edu .ABSTRACT OF THESIS Thevenin Equivalent Circuit Estimation and Application for Power System Monitoring and Protection The Estimation of Thevenin Equivalent Parameters is useful for System Monitoring and Protection. We studied a method for estimating the Thevenin equivalent circuits. We then studied two applications including voltage stability and fault location. A study of the concepts of Voltage Stability is done in the initial part of this thesis. A Six Bus Power System Model was simulated using MATLAB SIMULINK®. Subsequently, the Thevenin Parameters were calculated. The results were then used for two purposes, to calculate the Maximum Power that can be delivered and for Fault Location. KEYWORDS: Thevenin Equivalent Circuit, Voltage Stability, Rotor Angle Stability, Fault Location, Power System Monitoring Mohammad M Iftakhar
December 31 st 2008 Thevenin Equivalent Circuit Estimation and Application for Power System Monitoring and Protection By Mohammad Museb Iftakhar _____ (Director of Thesis) _____ (Director of Graduate Studies) ______ (Date) RULES FOR THE USE OF THESIS Unpublished thesis submitted for the Master’s degree and deposited in the University of Kentucky Library are as a rule open for inspection, but are to be used only with due regard to the rights of the authors. Bibliographical references may be noted, but quotations or summaries of parts may be published only with the permission of the author, and with the usual scholarly acknowledgments. Extensive copying or publication of the dissertation in whole or in part also requires the consent of the Dean of the Graduate School of the University of Kentucky. A library that borrows this dissertation for use by its patrons is expected to secure the signature of each user.
Name
Date __________ __________ __________ __________ __________ __________ ____________ THESIS Mohammad Museb Iftakhar The Graduate School University of Kentucky
2009 Thevenin Equivalent Circuit Estimation and Application for Power System Monitoring and Protection THESIS A thesis submitted in partial fulfillment of the requirements for the degree of Master of Science in the College of Engineering at the University of Kentucky By Mohammad Museb Iftakhar Lexington, Kentucky Director: Dr. Yuan Liao, Department of Electrical and Computer Engineering Lexington, Kentucky 2009
Copyright © Mohammad Museb Iftakhar 2009 Dedicated to My Parents, Brothers and Sister iii
ACKNOWLEDGEMENTS
I would like to take this opportunity to express my sincere thanks and heartfelt gratitude to my academic advisor and thesis chair Dr. Yuan Liao for his guidance and support throughout my thesis. I am very thankful for his constant encouragement during the thesis. Without him the thesis would have never taken its present shape. I am greatly indebted for his support. My parents and my siblings have been great sources of support throughout my studies. My friends have given me a lot of love without which this work would not have been possible. I also would like to extend my thanks to Dr. Paul A Dolloff and Dr. Jimmy J Cathey for serving on my thesis committee and providing me with invaluable comments and suggestions for improving this thesis. iv
Table of Contents
ACKNOWLEDGEMENTS .............................................................................................. III
LIST OF TABLES ............................................................................................................ VI
LIST OF FIGURES ......................................................................................................... VII
INTRODUCTION .......................................................................................................... 1
1.1 BACKGROUND ..................................................................................................... 1
1.2 P URPOSE OF THE THESIS ...................................................................................... 2
1.3 OVERVIEW OF S YSTEM S TABILITY ....................................................................... 2
1.4 EQUATION OF M OTION OF A R OTATING M ACHINE ............................................... 3
1.5 S TEADY S TATE S TABILITY ................................................................................... 4
1.6 M ETHODS OF IMPROVING S TEADY S TATE STABILITY LIMIT ................................. 7
1.7 TRANSIENT S TABILITY LIMIT ............................................................................... 8
1.8 EQUAL AREA C RITERION ..................................................................................... 8
1.9 FACTORS A FFECTING TRANSIENT S TABILITY ...................................................... 9
POWER SYSTEM VOLTAGE STABILITY ANALYSIS ......................................... 11
2.1 DEFINITION AND C LASSIFICATION OF P OWER S YSTEM STABILITY ..................... 11
2.2 C LASSIFICATION OF P OWER S YSTEM S TABILITY ................................................ 11
2.3 VOLTAGE S TABILITY ......................................................................................... 12
2.4 VOLTAGE S TABILITY ANALYSIS ........................................................................ 12
2.5 P-V C URVES ...................................................................................................... 16
2.6 V-Q C HARACTERISTICS ..................................................................................... 17
2.7 S OME S IGNIFICANT R ESULTS AND C RITERIA IN VOLTAGE S TABILITY ............... 19
POWER SYSTEM MODELING AND THEVENIN EQUIVALENT CIRCUIT PARAMETERS ESTIMATION ....................................................................................... 21
3.1 TRANSMISSION LINE DATA ................................................................................ 21
3.2 GENERATOR DATA ............................................................................................. 22
3.3 LOAD DATA ....................................................................................................... 23
3.4 A LGORITHM FOR THEVENIN EQUIVALENT C IRCUIT ESTIMATION ...................... 23
3.5 EQUATION FOR M AXIMUM P OWER DELIVERED ................................................. 27
3.6 VOLTAGE AND C URRENT W AVEFORMS AT LOAD B US L3 .................................. 29
3.7 W AVEFORMS FOR VOLTAGE AND C URRENT AT THE LOAD BUS L5 .................... 35
FAULT ANALYSIS AND ESTIMATION OF FAULT LOCATION ........................ 41
4.1 UNSYMMETRICAL FAULTS ................................................................................. 41
4.2 S YMMETRICAL C OMPONENT ANALYSIS OF UNSYMMETRICAL FAULTS .............. 41
4.3 ANALYSIS OF S INGLE LINE TO GROUND FAULT ................................................. 45
4.4 ANALYSIS OF LINE TO LINE FAULT .................................................................... 47
4.5 DOUBLE LINE TO GROUND FAULT ANALYSIS .................................................... 50
4.6 FAULT LOCATION A LGORITHM .......................................................................... 52 v
4.7 IMPEDANCE BASED A LGORITHM ...................................................................... 53
4.8 VOLTAGE AND C URRENT W AVEFORMS FOR DIFFERENT FAULT LOCATIONS .... 55
CONCLUSION ............................................................................................................. 59
BIBLIOGRAPHY ............................................................................................................. 60
VITA ................................................................................................................................. 62 vi
List of Tables
TABLE 3.1 GENERATORS B LOCK P ARAMETER VALUES ..................................................... 22
TABLE 3.2 LOAD B LOCK P ARAMETER VALUES ................................................................. 23
TABLE 3.3 VOLTAGE AND C URRENTS AT BUS L3 .............................................................. 25
TABLE 3.4 VOLTAGE AND C URRENT AT LOAD B US L5 ...................................................... 26
TABLE 3.5 THEVENIN P ARAMETERS FOR 4, 5 AND 6 SETS OF MEASUREMENTS AT LOAD BUS L3 .............................................................................................................................. 26
TABLE 3.6 THEVENIN P ARAMETERS FOR 4, 5 AND 6 SETS OF MEASUREMENTS AT LOAD BUS L5 .............................................................................................................................. 27
TABLE 3.7 P OWER DELIVERED AT THE BUS L3 FOR D IFFERENT P OWER FACTOR ANGLES . 28
TABLE 3.8 P OWER DELIVERED AT THE BUS L5 FOR D IFFERENT P OWER FACTOR ANGLES . 28
TABLE 4.1 FAULT LOCATION ESTIMATION ......................................................................... 54 vii
List of Figures
F IGURE 1.1 M ACHINE C ONNECTED TO INFINITE BUS ........................................................... 5
F IGURE 1.2 P OWER ANGLE C URVE ...................................................................................... 6
F IGURE 1.3 C URVE S HOWING THE EQUAL AREA CRITERION ................................................ 9
F IGURE 2.1 S IMPLE RADIAL S YSTEM FOR VOLTAGE S TABILITY ANALYSIS ........................ 13
F IGURE 2.2 R EACTIVE END VOLTAGES , P OWER AND C URRENT AS A FUNCTION OF LOAD DEMAND .................................................................................................................... 15
F IGURE 2.3 P OWER VOLTAGE C HARACTERISTICS FOR THE S YSTEM OF F IGURE 2.1 ............ 16
F IGURE 2.4 P OWER VOLTAGE CHARACTERISTICS FOR DIFFERENT LOAD P OWER FACTORS 17
F IGURE 2.5 S IMPLE R ADIAL TWO BUS S YSTEM ................................................................. 18
F IGURE 2.6 V-Q C HARACTERISTICS OF THE SYSTEM IN F IGURE 2.1 ................................... 18
F IGURE 3.1 S IX BUS P OWER S YSTEM M ODEL .................................................................... 21
F IGURE 3.2 THEVENIN EQUIVALENT C IRCUIT .................................................................... 24
F IGURE 3.3 EQUIVALENT P OWER S YSTEM M ODEL FOR C ALCULATING M AXIMUM P OWER DELIVERED ................................................................................................................ 28
F IGURE 3.4( A) VOLTAGE SIGNALS FOR THE CASE WITH GENERATOR 2 ANGLES SET TO 10 DEGREES . ................................................................................................................... 29
F IGURE 3.4( B ) C URRENT SIGNALS FOR THE CASE WITH GENERATOR 2 ANGLES SET TO 10 DEGREES . ................................................................................................................... 29
F IGURE 3.5( A) VOLTAGE SIGNALS FOR THE CASE WITH GENERATOR 2 ANGLES SET TO 20 DEGREES .................................................................................................................... 30
F IGURE 3.5( B ) C URRENT SIGNALS FOR THE CASE WITH GENERATOR 2 ANGLES SET TO 20 DEGREES .................................................................................................................... 30
F IGURE 3.6( A) VOLTAGE SIGNALS FOR THE CASE WITH GENERATOR 2 ANGLES SET TO 30 DEGREES .................................................................................................................... 31
F IGURE 3.6( B ) C URRENT SIGNALS FOR THE CASE WITH GENERATOR 2 ANGLES SET TO 30 DEGREES .................................................................................................................... 31
F IGURE 3.7( A) VOLTAGE SIGNALS FOR THE CASE WITH GENERATOR 2 ANGLES SET TO 40 DEGREES .................................................................................................................... 32
F IGURE 3.7( B ) C URRENT SIGNALS FOR THE CASE WITH GENERATOR 2 ANGLES SET TO 40 DEGREES .................................................................................................................... 32
F IGURE 3.8( A) VOLTAGE SIGNALS FOR THE CASE WITH GENERATOR 2 ANGLES SET TO 60 DEGREES .................................................................................................................... 33
F IGURE 3.8( B ) C URRENT SIGNALS FOR THE CASE WITH GENERATOR 2 ANGLES SET TO 60 DEGREES .................................................................................................................... 33
F IGURE 3.9( A) VOLTAGE SIGNALS FOR THE CASE WITH GENERATOR 2 ANGLES SET TO 0DEGREES .................................................................................................................... 34
F IGURE 3.9( B ) C URRENT SIGNALS FOR THE CASE WITH GENERATOR 2 ANGLES SET TO 0DEGREES .................................................................................................................... 34
F IGURE 3.10( A) VOLTAGE SIGNALS FOR THE CASE WITH GENERATOR 3 ANGLES SET TO 0DEGREES .................................................................................................................... 35
F IGURE 3.10( B ) C URRENT SIGNALS FOR THE CASE WITH GENERATOR 3 ANGLES SET TO 0DEGREES .................................................................................................................... 35 viii
F IGURE 3.11( A) VOLTAGE SIGNALS FOR THE CASE WITH GENERATOR 3 ANGLES SET TO 10 DEGREES .................................................................................................................... 36
F IGURE 3.11( B ) C URRENT SIGNALS FOR THE CASE WITH GENERATOR 3 ANGLES SET TO 10 DEGREES .................................................................................................................... 36
F IGURE 3.12( A) VOLTAGE SIGNALS FOR THE CASE WITH GENERATOR 3 ANGLES SET TO 20 DEGREES .................................................................................................................... 37
F IGURE 3.12( B ) C URRENT SIGNALS FOR THE CASE WITH GENERATOR 3 ANGLES SET TO 20 DEGREES .................................................................................................................... 37
F IGURE 3.13( A) VOLTAGE SIGNALS FOR THE CASE WITH GENERATOR 3 ANGLES SET TO 30 DEGREES .................................................................................................................... 38
F IGURE 3.13( B ) C URRENT SIGNALS FOR THE CASE WITH GENERATOR 3 ANGLES SET TO 30 DEGREES .................................................................................................................... 38
F IGURE 3.14( A) VOLTAGE SIGNALS FOR THE CASE WITH GENERATOR 3 ANGLES SET TO 40 DEGREES .................................................................................................................... 39
F IGURE 3.14( B ) C URRENT SIGNALS FOR THE CASE WITH GENERATOR 3 ANGLES SET TO 40 DEGREES .................................................................................................................... 39
F IGURE 3.15( A) VOLTAGE SIGNALS FOR THE CASE WITH GENERATOR 3 ANGLES SET TO 60 DEGREES .................................................................................................................... 40
F IGURE 3.15( B ) C URRENT SIGNALS FOR THE CASE WITH GENERATOR 3 ANGLES SET TO 60 DEGREES .................................................................................................................... 40
F IGURE 4.1 A GENERAL P OWER NETWORK ....................................................................... 42
F IGURE 4.2 (A) P OSITIVE SEQUENCE NETWORK AS SEEN FROM THE FAULT POINT .............. 42
F IGURE 4.2 (B ) NEGATIVE S EQUENCE NETWORK AS SEEN FROM THE FAULT POINT ........... 43
F IGURE 4.2( C ) ZERO SEQUENCE NETWORK AS SEEN FROM THE FAULT POINT ..................... 43
F IGURE 4.2 (D) THEVENIN EQUIVALENT OF P OSITIVE SEQUENCE NETWORK AS SEEN FROM F
................................................................................................................................... 43
F IGURE 4.2 (E) THEVENIN EQUIVALENT OF N EGATIVE SEQUENCE NETWORK AS SEEN FROM F ................................................................................................................................ 44
F IGURE 4.2 ( F) THEVENIN EQUIVALENT OF ZERO SEQUENCE NETWORK AS SEEN FROM F . 44
F IGURE 4.3( A) S INGLE LINE TO GROUND FAULT AT F ....................................................... 45
F IGURE 4.3( B ) C ONNECTION OF SEQUENCE NETWORKS FOR SINGLE LINE TO GROUND FAULT
................................................................................................................................... 47
F IGURE 4.4( A) LINE TO LINE FAULT THROUGH IMPEDANCE fZ ......................................... 48
F IGURE 4.4( B ) P OSITIVE AND NEGATIVE S EQUENCE C ONNECTIONS FOR A LINE TO LINE FAULT ........................................................................................................................ 49
F IGURE 4.4( C ) THEVENIN EQUIVALENT FOR CONNECTION OF S EQUENCE NETWORKS FOR L-L FAULT ..................................................................................................................... 50
F IGURE 4.5( A) DOUBLE LINE TO GROUND FAULT THROUGH IMPEDANCE fZ .................... 51
F IGURE 4.5( B ) CONNECTION OF SEQUENCE NETWORKS FOR A DOUBLE LINE TO GROUND FAULT ........................................................................................................................ 52
F IGURE 4.5( C ) THEVENIN EQUIVALENT FOR THE SEQUENCE NETWORK CONNECTIONS FOR A LLG FAULT ................................................................................................................ 52
F IGURE 4.6 TRANSMISSION LINE C ONSIDERED FOR THE A LGORITHM .......................... 53
F IGURE 4.7 NEGATIVE S EQUENCE NETWORK DURING THE FAULT NEGLECTING S HUNT CAPACITANCE ....................................................................................................... 53 ix
F IGURE 4.8( A) VOLTAGE WAVEFORMS FOR A PHASE A TO GROUND FAULT WITH A FAULT LOCATION OF 0.2 P .U .................................................................................................. 55
F IGURE 4.8( B ) C URRENT WAVEFORMS FOR A PHASE A TO GROUND FAULT WITH A FAULT LOCATION OF 0.2 P .U .................................................................................................. 55
F IGURE 4.9( A) VOLTAGE WAVEFORMS FOR A PHASE A TO GROUND FAULT WITH A FAULT LOCATION OF 0.4 P .U .................................................................................................. 56
F IGURE 4.9( B ) C URRENT WAVEFORMS FOR A PHASE A TO GROUND FAULT WITH A FAULT LOCATION OF 0.4 P .U .................................................................................................. 56
F IGURE 4.10( A) VOLTAGE WAVEFORMS FOR A PHASE A TO GROUND FAULT WITH A FAULT LOCATION OF 0.6 P .U .................................................................................................. 57
F IGURE 4.10( B ) C URRENT WAVEFORMS FOR A PHASE A TO GROUND FAULT WITH A FAULT LOCATION OF 0.6 P .U .................................................................................................. 57
F IGURE 4.11( A) VOLTAGE WAVEFORMS FOR A PHASE A TO GROUND FAULT WITH A FAULT LOCATION OF 0.7 P .U .................................................................................................. 58
F IGURE 4.11( B ) C URRENT WAVEFORMS FOR A PHASE A TO GROUND FAULT WITH A FAULT LOCATION OF 0.7 P .U .................................................................................................. 58 1
Introduction
1.1 Background
The pressure on the power transmission network has been increasing in recent times. Some of the reasons for this which have been mentioned in are
• A deregulated energy market.
• Environmental constraints.
• Limited investment in transmission system reinforcement.
• An increased competition in order to yield greater outputs.
Hence, the Power System is forced to operate closer to the stability limit. A Major problem arising out of this is voltage instability or collapse, which causes a steady state security problem. When the loading of a Power system approaches the maximum permissible loading, at some local bus in the power transmission network, the magnitude of the voltage tends to decrease. But only by knowing the voltage magnitude of local buses, we cannot exactly assess the impending voltage collapse. The voltage magnitude decreases because of inadequate local reactive power support to meet local demand and losses. Large amounts of reactive power from other buses in the network will deteriorate the voltage profile which may lead to voltage collapse. In recent years, voltage instability has been responsible for major blackouts. The following are some examples :
• North East blackout, August 14, 2003.
• Texas blackout, September 13, 2008.
• New York Power Pool Disturbances of September 22, 1970
• Florida System Disturbance of December 28, 1982
• French System Disturbances of December 19, 1978 and January 12, 1987
• Northern Belgium System Disturbance of August 4, 1982
• Swedish System Disturbance of December 27, 1983
• Japanese System Disturbance of July 23, 1987.
Thus voltage stability studies have become of more importance than ever. 2
Another important thing to consider in this thesis is fault location. Fault location studies are very important for the transient stability limit of the system. The increased complexities of modern power systems have raised the importance of fault location research studies . Accurate and fast fault location helps in reducing the maintenance and restoration times, reduce the outage times and thus improve the power system
reliability .
1.2 Purpose of the Thesis
To operate the power system with an adequate security margin, it is essential to estimate the maximum permissible loading .The maximum power that can be transferred to the load bus in a power system can be effectively studied by estimating the Thevenin equivalent circuit of the power system Model. Thus, in one part of my Thesis, I will be calculating the Thevenin parameters of a six bus power system model. This would provide me with considerable results to calculate the maximum power that can be delivered. The Thevenin equivalent circuit parameters are useful in the applications for power system monitoring and protection. The Thevenin parameters that I obtain in the first part are used for fault location based on voltage measurements. The fault location algorithm is taken from , which are described in detail in Chapter 4.
1.3 Overview of System Stability
The stability of a system of interconnected dynamic components is its ability to return to normal or stable operation after having been subjected to some form of disturbance . In a power system, we typically deal with two forms of instability: The loss of synchronism between synchronous machines and voltage instability. Synchronous stability can be classified as steady state and transient stability and are studied in this chapter. The voltage stability is studied in the next chapter. The equations and figures in the subsequent sections have mainly been obtained from and .
As defined in , steady state stability is the ability of the power system, when operating under given load conditions, to retain synchronism when subjected to small disturbances 3
such as the continual changes in load or generation and the switching out of lines. This is also known as dynamic stability. Transient stability deals with sudden and large changes in the system. One example is faults in a Power system. During fault conditions, the stability limit is less than the steady state condition. Before we make a detailed study of steady state and transient stability, it is important to study the equation of motion of a rotating machine.
1.4 Equation of Motion of a Rotating Machine
In this section, we will be studying the Equation of Motion of a Rotating Machine and deriving the swing equation. The equations in this section have all been obtained from . Let the moment of inertia of the rotor be I and the angular acceleration is
α . T∆ is the net torque applied on the rotor.
ω is the synchronous speed of the rotor (radians/second). The kinetic energy absorbed by the rotor is given by 221
ωI Joules. The angular momentum is
ωIM = Joules-Seconds per radian. An inertia constant, H can be defined as the stored energy at synchronous speed per volt-ampere of the rating of the machine . As we know that the unit of energy used in power systems analysis is Kilojoules or Mega joules and if we consider the rating of the machine to be G Mega-Volt-Amperes, then by multiplying G with the inertia constant we get the kinetic energy of the machine.
ωω MIGH 2121 2 == is the Kinetic Energy or the stored Energy. (1.1)
f360 =ω Electrical Degrees per second where f is the system frequency in Hz. (1.2) Substituting (1.2) in (1.1)
fMGH )360 (21
= (1.3)
fGH M 180 /=⇒ Mega joule-seconds/electrical degree (1.4)
=∆T Mechanical Torque Input- Electrical Torque Output
22
dt dI
δ= (1.5) 4
( )
2/2 222
ωωωδ
xI TITdt d ∆=∆=∴ (1.6)
..2.
ExK P
ω∆= (1.7) Here electrical mech PPP −=∆ (1.8) By using Equation (1.1) in (1.7), we can write
MPdt d ∆=22
δ (1.9) There is an increase in the value of
δ when there is a negative change in the Power output in Equation (1.9). electrical mech PPP −=∆ is sometimes considered as the change in Electrical Power output. An increase in electrical P∆ will increase the value of
δ . The Power input is assumed to be constant.
M
Pdt d ∆−=⇒ 22
δ or 022
=∆+ Pdt dM
δ (1.10) Equation (1.10) is known as Swing Equation. Now that we have studied about the Equation of Motion of Rotating Machine, we can analyze Steady State and Transient Stability of a System based on this.
1.5 Steady State Stability
In this section we will be studying the steady state analysis for a power system. The equations in this section have been obtained from . The steady state stability limit of a particular circuit of a power system is defined as the maximum power that can be transmitted to the receiving end without loss of synchronism .Figure 1.1 represents a simple system for the purpose of analysis. The dynamics of this system are described by the equations (1.11) to (1.13)
em
PPdt dM −=22
δ (1.11)
fHM
π= in the Per Unit System (1.12) 5
δδ sin sin max PXVEPde == (1.13)
d
X is the direct axis reactance . The plot for equation (1.13) also known as the power angle curve is represented in Figure 1.2
+
δ∠'
E
dX ' eX
eP
00∠V
Infinite Bus
Figure 1.1 Machine Connected to Infinite Bus
The system has a steady power transfer meo PP = and the torque angle is o
δ as shown in Figure 1.2. For a small increment P∆ in the electric power with the input mP being constant, the torque angle changes to ( )δδ ∆+o . Linearizing about the operating point
),( oeo o PQ
δ , we get
δδ ∆∂∂=∆
0
ee
PP (1.14) Rewriting Equation (1.11) in the current analysis,
( ) eeeo m PPPPdt dM ∆−=∆+−=∆
22
δ (1.15) Using Equation (1.14) in (1.15)
0022
=∆∂∂+∆δδδ ePdt dM (1.16)
002 =∆ ∂∂+⇒δδ
e
PMk (1.17) 6
Where dt dk =
The stability of the system for small changes is determined by the characteristic equation 002 = ∂∂+δ
e
PMk (1.18) The roots of Equation (1.18) are given by
( ) 210/
∂∂−±= MPk e
δ (1.19)
e
P
eeo
PP ∆+ max P
eo
P
o
Q
o
δ
δδ ∆+o
δ
0
90 0180 0180 − 090 −
Generator Motor
Figure 1.2 Power Angle Curve
Now the system behavior depends on the value of 0)/(
δ∂∂ eP .
If 0)/(
δ∂∂ eP is positive, the roots are imaginary and conjugate. The system behavior is oscillatory about o
δ . But in our analysis the machine damper windings line resistance had 7
been neglected. These cause the system oscillations to decay and hence the system is stable for a small increment in power. If 0)/(
δ∂∂ eP is negative, the roots are real. One is positive and the other is negative. Though, they are equal in magnitude. For a small increment in power, the system is unstable as the synchronism is lost due to increase in torque angle with increase in power. From Equation (1.13) assuming VE , to remain constant, the system is unstable if 090 >o
δ . (1.20) The maximum power transfer without loss of stability occurs for 090 =o
δ (1.21) The maximum power transferred is therefore given by
XVEP =max (1.22) But in the analysis we had assumed that the internal machine voltage remains constant. In such a case as the loading is increased, the terminal voltage dips heavily which is practically not acceptable. In practice, we must consider the steady state stability limit by assuming that the excitation is adjusted for every load increase to keep the terminal voltage constant. The effects of governor and excitation control were not considered in the analysis.
Steady state stability limit is very important as it should be taken care that a system can operate above transient stability but not above steady state stability limit. The transient stability limit can be made to closely approach the steady state limit currently with increased speeds in fault clearing.
1.6 Methods of Improving Steady State stability Limit
The following methods can be used depending on the conditions in order to improve the steady state stability limit
• From Equation (1.22), we can say that the steady state stability limit can be improved by reducing X or by increasing either E or V or both.
• For transmission lines of high reactance, the stability limit can be increased by using two parallel lines. 8
• Use of series capacitors in the lines to get better voltage regulation raises the stability limits by decreasing the line reactance.
• Employing quick excitation systems and higher excitation voltages.
1.7 Transient Stability Limit
Transient stability limit is the maximum possible power that can be transmitted through a point in the system when the system is operating with stability during transient disturbances . The type of disturbance and the duration of disturbance affect the transient stability limit. The duration of a fault determines the amount of power that can be transmitted from one machine to another machine in a two machine system without loss of synchronism. The power limit is determined using the Equal Area Criterion. This is studied in section 1.8.
1.8 Equal Area Criterion
As we had considered one finite machine system for analysis for steady state stability, we will study the Equal Area Criterion for one finite machine swinging with an infinite bus in this section. The equations in this section have been mainly obtained from . The detailed study of Equal Area Criterion for a system with two finite machines swinging with respect to each other is discussed in and . The Swing Equation of a finite machine swinging with respect to an infinite machine is given by aem PPPdt dM =−=22
δ (1.23) Multiplying both sides of Equation (1.23) by dt d /2
δ and rearranging, we get
dt
dMPdt ddt d a
δδδ 22 22
= (1.24)
dt dMPdt ddt d a
δδ 22
= ⇒ (1.25) Upon integrating Equation (1.25) with respect to time, we get 9
∫= δδδδ
o
dPMdt d a
22
(1.26)
∫==⇒δδδωδ
o
dPMdt d a
2 (1.27)
0=ω when the machine comes to rest with respect to the infinite machine. The condition required for the stability of a single machine system connected to infinite bus is
0=∫δδδ
o
dPa (1.28) The integral in Equation (1.28) can be represented as the area between the curve mP
versus
δ and the curve eP versus
δ . This is shown in Figure 1.3. For the area to be Zero, )()( 21 emem PPAPPA <=> . Hence this method is called Equal Area criterion.
δo
δ δ
m
P
e
P
P
1
A
2A
Figure 1.3 Curve Showing the Equal Area criterion
1.9 Factors Affecting Transient Stability
The factors affecting transient stability limit mentioned in are as follows:
• Inertia Constant
• Type of Disturbance
• Fault clearing time 10
• Location of the fault
• Initial operating Condition of the system
• The way in which the fault is cleared.
Thus, in this chapter we have presented the purpose of the thesis research that has been carried out and an overview of the basic concepts related to the area of my research. An advanced study of these concepts can be made through the references that have been mentioned. 11
Power System Voltage Stability Analysis
In chapter 1, I gave an overview about the importance of voltage stability studies as voltage instability or voltage collapse may lead to a blackout. In this chapter we will be making a detailed study of the relevant concepts that could help us make the understanding of voltage stability better. Before we study voltage stability in particular, we define and classify power system stability in general. This is studied in the initial sections of this chapter. In the later sections we discuss voltage stability. These include the definitions, concepts of mathematical formulation of the voltage stability problems and some significant criteria of voltage stability studies. The equations and figures have been mainly obtained from and .
2.1
Definition and Classification of Power System stability
In a broad terminology, power system stability may be defined as that property of the power system that enables it to remain in a state of operating equilibrium under normal operating conditions and to regain an acceptable state of equilibrium after being subjected to a disturbance . The definitions of power system stability though have not been precise and do not include all practical instability scenarios . A proposal is presented in which attempts to define power system stability more precisely which includes all forms of system instability. “Power system stability is the ability of an electric power system, for a given initial operating condition, to regain a state of operating equilibrium after being subjected to a physical disturbance with most system variables bounded so that practically the entire system remains intact” .
2.2
Classification of Power System Stability
Power system stability is classified based on the following considerations
• The physical nature of the instability
• The size of the disturbance
• The time span 12
Based on the physical nature of the instability, it can be classified as rotor angle stability and voltage stability. Based on the size of the disturbance, it is classified as large disturbance and small disturbance stability. Based on the time span, it can be classified as long term and short term stability.
2.3
Voltage Stability
Voltage stability is the ability of a power system to maintain steady acceptable voltages at all buses in the system under normal operating conditions and after being subjected to a disturbance . There is voltage instability when there is voltage drop in the system or at a bus due to several reasons which include a general disturbance, a change in system condition or due to fluctuating loads. But the main reason for a voltage instability is the inadequacy of reactive power demands to be met by the system. Reactive power is injected into the system to meet the increasing demands. For a voltage stable system, as the reactive power is injected into the buses in the system, the voltage magnitude should increase. But, if the voltage magnitude decreases even at one bus in the system for increase in reactive power, the System is said to be voltage unstable. Voltage instability is a local phenomenon but its consequences may have a widespread impact . Voltage instability leads to low voltage profile in the system and it can have a cumulative effect ultimately leading to voltage collapse.
2.4
Voltage Stability Analysis
In this section we study a voltage study analysis for a simple two terminal network. Figure 2.1 represents a simple radial system. This figure and the equations in this section are taken from . 13
θ∠LN Z
~
I
~
s
E
~
R
V
φ∠LD Z
RR jQ P +
Figure 2.1 Simple radial System for Voltage Stability Analysis
s
E is the Voltage Source
LD
Z is the load impedance
LN
Z is the series impedance of the system . The Current ~
I can be expressed as
~~~~
LD LN s
ZZEI
+= (2.1)
~
I and ~
s
E are the phasors of current and source voltage. The series impedance and the load impedance phasors can be expressed as
θ∠= LN LN ZZ~
and
φ∠= LD LD ZZ~
respectively. (2.2) The magnitude of the current can then be expressed as
( ) ( )22 sin sin cos cos
φθφθ LD LN LD LN sZZZZEI
+++= (2.3) Simplifying Equation (2.3) 14
LN s
ZEZI '1
= (2.4) Where
)cos( 212'
φθ − ++=
LN LD LN LD
ZZZZZ (2.5) The Magnitude of Voltage at the receiving end can be expressed as
IZV LD R = (2.6) Substituting the value of current from Eq. (2.4) into Eq. (2.6)
sLN LD
EZZZ '1
= (2.7) Now, calculating the power delivered It is given by
φcos IVP RR = (2.8)
φcos 2' =
LN sLD
ZEZZ (2.9) Figure 2.2 shows the plots of I , RV and RP as a function of LD LN ZZ / . RP increases rapidly as the load demand i.e. LD LN ZZ / is increased. This is done by decreasing LD Z . RP
reaches a maximum value and then begins to decrease. The maximum value of RP
indicates the maximum value of active power that can be transmitted through an impedance from a constant voltage source. This power transmitted is maximum when the voltage drop in the line is equal in magnitude to RV that is when 1/ =LD LN ZZ . With a gradual decrease in LD Z there is an increase in I and decrease in RV . The reason for a rapid increase in RP initially is due to the dominant increase of I in comparison to the decrease in RV at high values of LD Z . As LD Z approaches LN Z this effect is not so dominant and hence there is a gradual change rather than a sharp increase and decrease in the values of I and RV respectively. As LD Z goes below LN Z , the decrease in RV
dominates the increase in I and hence there is a decrease in RP .15
LD LN
ZZ /
Max R
PP /
sc
II /
SR
EV /
Normal Operation Critical Value Abnormal operation
Figure 2.2 Reactive End Voltages, Power and Current as a Function of Load Demand
The normal operation takes place till the critical value is reached. This corresponds to the maximum power in the Figure 2.2. As the load demand increases i.e. as LD LN ZZ / is increasing the control of power would be unstable. This means that as the load impedance
LD
Z is decreased the power is also reduced. The load characteristics determine if the voltage collapse takes place or not. For a constant impedance static load characteristics, the system stabilizes at power and voltage levels lower than the desired values where as for a constant power load characteristic the system becomes unstable through collapse of load bus voltage . Thus it is important to analyze the relationship between RP and RV
for the purpose of voltage stability studies. 16
2.5 P-V Curves
The relationship between RP and RV is shown in Figure2.3 for a particular power factor value. But the voltage drop in the transmission lines is a function of both the active and the reactive power transfer as seen in Equations (2.7) and (2.9). Thus the load power factor affects the power voltage characteristics of the system Figure 2.4 represents the curves for RP and RV for different load power factor values.
SR
EV /
RM AX R
PP /
Critical Voltage
Figure 2.3 Power Voltage Characteristics for the System of Figure2.1
The dotted lines represent the locus of critical operating points. This means that operating points above the critical values represent satisfactory operation. A sudden reduction in power factor, which causes an increase in the reactive power delivered, can cause the system to change from a stable operating condition to an unstable condition as shown in the lower part of the curves in Figure 2.3 and Figure 2.4. 17
RMax R
PP /
SR
EV /
0.9 Lag 0.9 Lead 0.95 lead 0.95 Lag 1.0 Locus of Critical Points
Figure 2.4 Power Voltage characteristics for Different Load Power Factors
2.6 V-Q Characteristics
For purpose of analysis let us consider a simple radial system as shown in Figure 2.5. The system load end voltage can be expressed in terms of QP, as given in
( ) ( ) 2/1222222422122 +−−±+−= QPXEQX EQX V (2.10) For Reactive Power Flow RX >>
i.e. 090 ≈φ
XVXEV Q
2
cos −=⇒δ (2.11)
0cos 2 =+−⇒ QX EV V
δ (2.12)
XVEdV dQ 2cos −=δ (2.13) 18
AC
LOAD
jQ P +
jX R +E V
Figure 2.5 Simple Radial Two Bus System
1.0 0.9 0.75 0.6 0.5
R M A X R
PQ /
SR EV /
Locus of critical Points
Figure 2.6 V-Q Characteristics of the system in Figure 2.1
Figure 2.6 represents the RR QV − characteristics with different RMax R PP / ratios of a simple two-terminal system. The system is voltage stable in the region where dV dQ / is positive . The locus of critical operating points is shown in the Figure 2.6 with dotted 19
lines. The critical operating point is where the voltage stability limit is reached i.e.
0/ =dV dQ . At voltage stability limit, the limiting reactive power is given by
δ2cos 2lim X
VQ = (2.14) The parts of the curve to the right of the minima represent stable operation. There is unstable operation when 0/ <dV dQ . But stable operation in the region when
0/ <dV dQ is can also be done. This is by using regulated reactive power compensation having sufficient control range and high VQ / gain with an opposite polarity . The analysis we have presented in this section is limited to radial systems in order to make the understanding of different power system stability concepts much easy. But in complex power systems there are many factors that contribute to the system instability and can be studied at a higher level.
2.7 Some Significant Results and Criteria in Voltage Stability
In this section we present a summary of certain other significant criteria and results in voltage stability studies which have not been mentioned in the previous sections. These concepts have been mainly obtained from .
• Voltage stability limit is reached when 12 =
VYSLL
. (2.14)
S is the complex power at load bus, LL Y is the load bus admittance and V is the voltage at the load bus.
• The limit of maximum loading of a transmission line can be given by
cri cri
XVS /2
= (2.15) Where cri X is the critical reactance of the system after which voltage instability occurs. It is expressed as )sec tan (22
φφ + −= PEX cri (2.16)
• dV dE Criterion . The voltage stability limit is reached when
0sin 2cos =−+ + XEdV dP XVdV dQ
δδ (2.17) 20
• dV dZ Criterion . The value of critical impedance beyond which voltage instability occurs can found from this criteria. Voltage instability occurs when
0=
dV dZ
In this chapter we have tried to present a simple and clear understanding of the voltage stability concepts. An advanced level of analytical methods for voltage stability and rotor angle stability, which often go hand in hand, has been described in . 21
Power System Modeling and Thevenin Equivalent Circuit Parameters Estimation
In this Chapter, we will be estimating the Thevenin parameters for the power system model that has been built in MATLAB and SimPower system. The six bus power system model is represented in Figure 3.1. The various SIMULINK blocks that have been used in the model have been studied in the initial sections of this Chapter. The block parameters have been represented in tables for each block. The power system model is simulated and the results are tabulated which are used for calculating the Thevenin parameters for the power system model built for this thesis research. The maximum power transferred is also calculated. The simulation results are also represented in terms of voltage and current signal waveforms at the end of this chapter.
G2 G1 G3 L4 L5 L3 L2 L6 L1 453216
Figure 3.1 Six Bus Power System Model
3.1 Transmission Line Data
The transmission lines in the power system model have been represented by the three phase mutual inductance block in the SimPower system. It is to be noted that the block parameters in this SimPower system block are resistance and inductance whereas the data used has resistance and reactance values. So, by using, fL X
π2= (3.1) 22
For a system frequency of 60 Hz and individual reactance values as mentioned in the table, the corresponding inductance values have been calculated.
3.2 Generator Data
The generators have been represented by a three phase source block of SimPower system which is a three phase voltage source in series with RL branch. The block parameters of this block are:
(a) Phase-to-Phase rms voltage (V)
(b) Phase Angle of Phase A ( degrees)
(c) Frequency ( Hz)
(d) Source Resistance (Ohms ) and Source Inductance ( Henry)
The Values for the above parameters have been set as show in Table 3.1.
Table 3.1 Generators Block Parameter Values
Parameter Value
Phase-to-Phase rms Voltage (v) 3
Phase Angle of Phase A ( degrees) Varied from 0 to 60 for G2 ; 10 for G1 and 30 for G3
Frequency(Hz) 60
Source Resistance (Ohms ) and Source Inductance ( Henry)
0.001 (Ohms) and 0.001/377(Henry) Respectively.
The phase angle at Generator 2 is varied from 0 to 60 degrees and the three phase current and voltage waveforms have been plotted, which is shown later in this chapter. The three voltage sources are connected in Wye with a neutral connection that has been internally grounded as obtained from .
The source impedance values of the generators are represented in the table 3.2. Equation 3.1 is used to convert the reactance values into the inductance and subsequently entered in the parameter block of the SimPower system model. 23
3.3 Load Data
A three-phase series RLC load block from the SimPower system has been used. RLC elements are combined in series to implement the three phase balanced load . The parameters in this block have been set as shown in Table 3.2
Table 3.2 Load Block Parameter Values
Parameter Value
Nominal Phase to Phase Voltage Vrms (V) 1
Nominal Frequency (Hz) 60
Active Power P (W) 0.25
Inductive Reactive Power(Var) 0.1
Capacitive Reactive Power(Var) 0
The configuration of the block is set to Wye (floating). Thus, the connection of the three phases is in Wye with neutral inaccessible.
3.4 Algorithm for Thevenin Equivalent Circuit Estimation
The six bus power system model in Figure 3.1 can be represented by its Thevenin equivalent circuit as referred from load bus L3 or L5 as shown in Figure 3.2. The Thevenin equivalent parameters can be obtained by writing the network equations as seen from load bus L3 or L5.
kKSS
VIZE =− (3.2) Equation (3.2) represents the Network Equation for Figure 3.2.
S
E is the Source Voltage, SZ is the Source Impedance. 24
SE SZ kV
kI
LZ
-+
Figure 3.2 Thevenin Equivalent Circuit
At bus k , the Voltage and Current is represented by kV and kI respectively. Equation (3.2) can be written for various values of currents and voltages at the bus k .This is done by varying the phase angle at one of the generators. In the power system model shown in Figure 3.1, the generator phase angles at G2 are varied first in order to get different values of currents and voltages at the bus k . So, Equation (3.2) can be further written as
11kKSS
VIZE =− (3.3)
22kKSS
VIZE =− (3.4) For n values of generator angles, the Equation (3.2) can give n values of currents and voltages at the bus k .Expressing it in the form BAX = ,
=−−−
kn ksskn kk
VVZEIII
...1....11 121
(3.5) 25
A is an )2(nx Matrix.
X is a )12( x Matrix and B is an 1nx Matrix. The solution for Equation (3.5) is obtained using the Least Mean Squares technique.
)()( 1 BAAAX TT −=
Thus the Thevenin equivalent parameters are calculated. In the power system model represented in Figure 3.1, the phase angles at G2 are varied from 0 to 60 degrees to get the Thevenin equivalent voltage and current at the load, L3. The results are tabulated as shown in Table 3.3
Table 3.3 Voltage and Currents at Bus L3
Angle at Generator 2 ( Degrees)
Voltage at bus 3 (p.u.) Magnitude(angle in degree)
Current at bus 3 (p.u.)
Active Power P ( p.u.)
Reactive Power Q ( p.u. )
0 1.252 (12.62) 0.97(12) 1.821 0.01968
10 1.257(13.11) 0.8362(13.5) 1.577 -0.00953
20 1.26(13.63) 0.7006(13.5) 1.324 0.001912
30 1.262(14.17) 0.567 (11.3) 1.072 0.05371
40 1.261 (14.72) 0.444 (4.8) 0.8271 0.1442
60 1.254 (15.75) 0.3082 (-32) 0.3894 0.4294
At load bus L5, the currents and voltages are tabulated by varying the phase angle of the generator G1. This is represented in Table 3.4 26
Table 3.4Voltage and Current at Load Bus L5
Angle at Generator 1 ( Degrees)
Voltage at bus 5 (p.u.) Magnitude(angle in degree)
Current at bus 5 (p.u.)
Active Power P ( p.u.)
Reactive Power Q ( p.u. )
0 1.223(-5.411) 0.9163(2.452) 1.665 -0.2299
10 1.229(-1.684) 0.5881(3.134) 1.08 -0.09098
20 1.226(2.047) 0.267(-6.839) 0.4851 0.07585
30 1.1214(5.761) 0.1562(-105.2)
-0.1016 0.2655
40 1.194(9.441) 0.4544(-135.1)
-0.6624 0.4723
60 1.128(16.61) 1.109(-134.3) -1.638 0.9116
Using these values of voltage and current at the buses, the Thevenin parameters have been calculated. The Thevenin parameters for different sets of measurements are tabulated in Tables 3.5 and 3.6
Table 3.5Thevenin Parameters for 4, 5 and 6 sets of measurements at Load bus L3
Number of Sets of Measurements
S
E SZ
SR
E SM E SR SX
4 1.2273 0.3586 0.0240 0.0846
5 1.227 0.3588 0.0237 0.0849
6 1.2270 0.3587 0.0236 0.0848 27
Table 3.6Thevenin Parameters for 4, 5 and 6 sets of measurements at Load bus L5
Number of Sets of Measurements
S
E SZ
SR
E SM E SR SX
4 1.1827 0.1078 -0.0483 0.2437
5 1.2170 0.109 0.0094 0.2483
6 1.2283 0.1059 0.0351 0.2463
S
E is the Source Voltage. SZ is the Source Impedance.
SR
E and SM E represent the Thevenin Voltage real and imaginary parts respectively.
S
R and SX represent the Thevenin Resistance and Reactance respectively.
3.5 Equation for Maximum Power Delivered
From circuit theory, maximum power is delivered to a load when the source impedance value is equal to the load impedance .
SL
ZZ = , where LZ is the impedance at the load and SZ is the source impedance. Suppose load power factor angle is
α . Then,
α∠= LL ZZ .From the Network in Figure 3.3, we can write
LSs
ZZEI +=
The Power delivered is given by ). .( IIZjQ PS L=+=
2
IZ L=
2
LssL
ZZEZ +=
The power delivered at the load buses L3 and L5 for the system which we modeled is shown in Tables 3.7 and 3.8 28
SE SZ
kV
kI
SL ZZ =
-+
P Q
Figure 3.3 Equivalent Power System Model for Calculating Maximum Power Delivered
Table 3.7 Power Delivered at the Bus L3 for Different Power Factor angles
Angle P (p.u) Q (p.u)
0 7.3202 -
10 6.3869 1.1262
20 5.5158 2.0076
30 4.6906 2.7081
40 3.8972 3.2702
Table 3.8 Power Delivered at the Bus L5 for Different Power Factor angles
Angle P (p.u) Q (p.u)
0 2.6770
10 2.2949 0.4047
20 1.9511 0.7102
30 1.6358 0.9444
40 1.3414 1.1256 29
3.6 Voltage and Current Waveforms at Load Bus L3
The voltage and current waveforms are plotted by varying the phase angles at G2 and G3. This is represented in the figures in this section
020 40 60 80 100 120 140
-1.5
-1
-0.5
0
0.5
1
1.5
samples voltage
A
B
c
Figure 3.4(a) Voltage signals for the case with generator 2 angles set to 10 degrees.
020 40 60 80 100 120 140
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
samples current
A
B
C
Figure 3.4(b) Current signals for the case with generator 2 angles set to 10 degrees. 30
.
020 40 60 80 100 120 140
-1.5
-1
-0.5
0
0.5
1
1.5
samples voltage
A
B
c
Figure 3.5(a) Voltage signals for the case with generator 2 angles set to 20 degrees
020 40 60 80 100 120 140
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
samples current
A
B
C
Figure 3.5(b) Current signals for the case with generator 2 angles set to 20 degrees 31
020 40 60 80 100 120 140
-1.5
-1
-0.5
0
0.5
1
1.5
samples voltage
A
B
c
Figure 3.6(a) Voltage signals for the case with generator 2 angles set to 30 degrees
020 40 60 80 100 120 140
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
samples current
A
B
C
Figure 3.6(b) Current signals for the case with generator 2 angles set to 30 degrees 32
020 40 60 80 100 120 140
-1.5
-1
-0.5
0
0.5
1
1.5
samples voltage
A
B
c
Figure 3.7(a) Voltage signals for the case with generator 2 angles set to 40 degrees
020 40 60 80 100 120 140
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
samples current
A
B
C
Figure 3.7(b) Current signals for the case with generator 2 angles set to 40 degrees 33
020 40 60 80 100 120 140
-1.5
-1
-0.5
0
0.5
1
1.5
samples voltage
A
B
c
Figure 3.8(a) Voltage signals for the case with generator 2 angles set to 60 degrees
020 40 60 80 100 120 140
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
samples current
A
B
C
Figure 3.8(b) Current signals for the case with generator 2 angles set to 60 degrees 34
020 40 60 80 100 120 140
-1.5
-1
-0.5
0
0.5
1
1.5
samples voltage
A
B
c
Figure 3.9(a) Voltage signals for the case with generator 2 angles set to 0 degrees
020 40 60 80 100 120 140
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
samples current
A
B
C
Figure 3.9(b) Current signals for the case with generator 2 angles set to 0 degrees 35
3.7 Waveforms for Voltage and Current at the Load bus L5
050 100 150 200 250
-1.5
-1
-0.5
0
0.5
1
1.5
samples voltage
A
B
c
Figure 3.10(a) Voltage signals for the case with generator 3 angles set to 0 degrees
0 50 100 150 200 250
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
samples current
A
B
C
Figure 3.10(b) Current signals for the case with generator 3 angles set to 0 degrees 36
050 100 150 200 250
-1.5
-1
-0.5
0
0.5
1
1.5
samples voltage
A
B
c
Figure 3.11(a) Voltage signals for the case with generator 3 angles set to 10 degrees
0 50 100 150 200 250
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
samples current
A
B
C
Figure 3.11(b) Current signals for the case with generator 3 angles set to 10 degrees 37
050 100 150 200 250
-1.5
-1
-0.5
0
0.5
1
1.5
samples voltage
A
B
c
Figure 3.12(a) Voltage signals for the case with generator 3 angles set to 20 degrees
050 100 150 200 250
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
samples current
A
B
C
Figure 3.12(b) Current signals for the case with generator 3 angles set to 20 degrees 38
050 100 150 200 250
-1.5
-1
-0.5
0
0.5
1
1.5
samples voltage
A
B
c
Figure 3.13(a) Voltage signals for the case with generator 3 angles set to 30 degrees
050 100 150 200 250
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
samples current
A
B
C
Figure 3.13(b) Current signals for the case with generator 3 angles set to 30 degrees 39
050 100 150 200 250
-1.5
-1
-0.5
0
0.5
1
1.5
samples voltage
A
B
c
Figure 3.14(a) Voltage signals for the case with generator 3 angles set to 40 degrees
050 100 150 200 250
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
samples current
A
B
C
Figure 3.14(b) Current signals for the case with generator 3 angles set to 40 degrees 40
050 100 150 200 250
-1.5
-1
-0.5
0
0.5
1
1.5
samples voltage
A
B
c
Figure 3.15(a) Voltage signals for the case with generator 3 angles set to 60 degrees
050 100 150 200 250
-1.5
-1
-0.5
0
0.5
1
1.5
samples current
A
B
C
Figure 3.15(b) Current signals for the case with generator 3 angles set to 60 degrees 41
Fault Analysis and Estimation of Fault Location
In this Chapter, we will be studying the concepts of unsymmetrical fault analysis and also the algorithms that have been proposed in Dr. Liao’s work, “Fault location utilizing unsynchronized voltage measurements during fault”, Electric Power Components & Systems , vol. 34, no. 12, December 2006, pp. 1283 – 1293. One of the algorithms has been used to estimate the fault location for my power system model. The shunt capacitances have been neglected in the algorithm used for estimation in order to have simplicity and computational efficiency. The Thevenin parameter values obtained for the power system model of Chapter 3 have been used in the fault location algorithm in to calculate the fault location for the model used in this thesis. The current and voltage signal waveforms for the different values of fault location have been presented at the end of this chapter. In the initial sections of this chapter, the basic concepts of unsymmetrical fault analysis have been presented. The equations and figures of unsymmetrical fault analysis in sections (4.1-4.5) have all been obtained from and we explain it as much as possible.
4.1 Unsymmetrical Faults
The unsymmetrical faults can be classified as shunt type faults and series type faults. Shunt type faults can again be classified as :
(a) Single Line to Ground fault
(b) Line to Line Fault
(c) Double Line to Ground fault
Before we study in detail about the shunt type faults, it is important to study the symmetrical component analysis of unsymmetrical faults.
4.2 Symmetrical Component Analysis of Unsymmetrical Faults
In this section, we will analyze how a power network which is under a fault condition can be represented in terms of positive, negative and zero sequence networks as seen from the point where the fault is occurring. 42
Figure 4.1 represents a general power network . F is the point of fault occurrence. When the fault occurs in the system, the currents are represented by aI , bI , cI as shown, which flow out of the system. The voltages of lines ,a ,b c with respect to the ground are
,aV ,bV cV respectively.
aV bV cV
aI bI cI
a
F
b
c
Figure 4.1 A General Power Network
Before the fault occurs, the positive sequence voltages of all synchronous machines in the network are given by aE . This is the pre fault voltage at F. The positive, negative and zero sequence networks after the occurrence of the fault as seen from F are represented in Figures 4.2 (a), (b) and (c) .
1a
V
1aI
F
1aI
Figure 4.2 (a) Positive sequence network as seen from the fault point
43
2aV
2aI
2aI
F
Figure 4.2 (b) Negative Sequence Network as seen from the fault point
0aV
F
0a
I
0a
I
Figure 4.2(c) Zero sequence Network as seen from the fault point
The Thevenin equivalents of the sequence networks are represented in Figure 4.2 (d), (e), (f)
1aI
1aV1Z
F
+
aE
Figure 4.2 (d) Thevenin Equivalent of Positive sequence Network as seen from F 44
2aV
2a
I
2Z
F
Figure 4.2 (e) Thevenin Equivalent of Negative sequence Network as seen from F
0a
I
0aV
0
Z
F
Figure 4.2 (f) Thevenin Equivalent of Zero sequence Network as seen from F
We have mentioned, aE is present only in the positive sequence network. We can express the sequence voltages at F as shown in Equation (4.1) .
−=
021021021
00000000
aaaaaaa
IIIZZZEVVV
(4.1)
1a
V , 2aV , 0aV are the Positive, negative and Zero Sequence Voltages respectively.
1a
I , 2aI , 0aI are the Positive, negative and Zero sequence Currents respectively.
1
Z , 2Z , 0Z are the Positive, negative and Zero sequence Impedances respectively. Now, we can analyze the different types of shunt faults based on the concepts presented in section 4.2. The expressions for fault currents and voltages in the lines are derived subsequently. 45
4.3 Analysis of Single Line to Ground Fault
Figure 4.3(a) shows the power network when a single line to ground fault occurs at point F . The fault occurs on Phase a.
F
aI
0=bI
0=cIfZ
a
b
c
Figure 4.3(a) Single Line to Ground Fault at F
The equations for currents and the line to ground voltages are represented as follows :
0=bI (4.2)
0=cI (4.3)
afa
IZV = (4.4) Using the symmetrical components method, we get the fault currents expressed in terms of positive, negative and zero sequence components as shown in Equation (4.5).
=
001111131 22021 aaaa IIII
αααα
(4.5) Therefore,
aaaa
IIII 31021 === (4.6) From Eq. (4.4) and (4.6), we get
0211
3 aaaafaf VVVIZIZ ++== (4.7) 46
From the Equations (4.6) and (4.7), it can be analyzed that there is a series connection of sequence networks represented by Figure 4.3(b) . The sequence components of fault current aI and the voltages bV and cV can be found by using the Thevenin equivalents of the sequence networks which is shown in Figure 4.3(b). These are represented in Equation (4.8)-Equation (4.15).
faa
ZZZZEI 3)( 0211 +++= (4.8) The fault current aI can thus be given by
faaa
ZZZZEII 3)(330211 +++== (4.9) Also by Using Eq. (4.1), the fault current aI can be obtained.
1002211
3)()()( afaaaa IZZIZIZIE =−+−+− (4.10)
aaf
EIZZZZ =+++⇒ 1021 ]3)[( (4.11)
faa
ZZZZEI 3)( 0211 +++=⇒ (4.12) Now, making use of the method of symmetrical components and finding the voltage of line b to ground under fault conditions.
0212
aaab
VVVV ++=⇒αα (4.13) Using the values of 1aV , 2aV , 0aV from Eq. (4.1) in (4.13), we get
−+ −+ −= 333 0212 aaaabIZIZIZEV
αα (4.14) Using Equation (4.9) in Equation (4.14)
ffab
ZZZZZZZEV 3)()1()(302120222
+++−+−+=αααα (4.15) 47
1a
V
1a
I
2a
V
0a
V
F
F
F
12aa
II =
10aa
II =
aaaa
IIII 31021 ===
fa
ZI fZ30aV
2a
V
0
Z
2
Z
F
F
1a
I
1a
V1Z
F
+
a
E
12aa
II =
10aa
II =
f
Z3
fa
ZI
aaaa
IIII 31021 ===
Figure 4.3(b) Connection of sequence networks for single Line to Ground Fault
4.4 Analysis of Line to Line Fault
In the Figure 4.4 a line to line fault through fault impedance fZ is indicated as shown. The figures and equations in this section have all been obtained from . The currents are expressed as
0=aI (4.16)
0=+ cb II (4.17)
bc
II −=⇒ (4.18) 48
F
0=aI
cIfZ
a
b
c bI
Figure 4.4(a) Line to Line Fault through Impedance fZ
The voltage relationship between bV and cV is expressed as
fbcb
ZIVV =− (4.19) The positive, negative and zero sequence components of the fault currents are expressed as
−=
bbaaa
IIIII 01111131 22021
αααα
(4.20) On solving Equation (4.20), we get
12aa
II −= (4.21)
00 =aI (4.22) Now, the symmetrical components of voltages under fault at F are expressed as
−=
bfbbaaaa
IZVVVVVV
1111131 22021
αααα
(4.23) From the Equation (4.23) expressing 1aV and 2aV
bfbaa
IZVVV 221 )(3
ααα −++= (4.24)
bfbaa
IZVVV
ααα −++= )(3 22 (4.25) 49
Solving Equation (4.24) and (4.25)
bfbfaa
IZjIZVV 3)()(3 221 =−=−αα (4.26) Using Equations (4.21) and (4.22) in (4.20)
112
3)( aab IjII −=−=αα (4.27) Substituting Equation (4.27) in Equation (4.26)
121afaa
IZVV =− (4.28) From Equations (4.21) and (4.28), we can draw a parallel connection of positive and negative sequence networks through fZ as series impedance as shown in Figure 4.4(b). The zero sequence network is not connected as 00 =aI . Its Thevenin equivalent is also represented in 4.4 (c).
1aV 2aV
F
1aI 2aI
F
fZ
Figure 4.4(b) Positive and Negative Sequence Connections for a Line to Line Fault
50
2aV
2a
I
2Z
F
1aI
1Z
F
+
aE
1aV
fZ
Figure 4.4(c) Thevenin Equivalent for connection of Sequence Networks for L-L Fault
By using the Thevenin equivalent, we can write the expressions for
faa
ZZZEI ++=
211
(4.29)
facb
ZZZEjII ++−=−=
21
3 (4.30) The voltages at fault can be found out by knowing 1aV and 2aV . This can be calculated from 1aI .
4.5 Double Line to Ground Fault Analysis
In this section, we will make the analysis of a double line to ground fault. The figures and equations have all been obtained from .Figure 4.5(a) shows the double line to ground fault in a power system at a point F . The fault current for a double line to ground fault is expressed as
00 021 =++⇒= aaaa IIII (4.31) The voltage to ground at fault conditions are expressed as
0
3)( afcbfcb IZIIZVV =+== (4.32) 51
F
0=aI
cI fZ
a
b
c bI
03 aI
Figure 4.5(a) Double Line to Ground Fault through Impedance fZ
The symmetrical components of voltages under fault at F are expressed as
=
bbaaaa
VVVVVV
1111131 22021
αααα
(4.33) Thus,
[ ]baaa VVVV )(31 221
αα + +== (4.34)
)2(310 baa VVV += (4.35) From Equations (4.34) and (4.35)
0210
3)2(31 afbbaa IZVVVV ==−−=−αα (4.36)
010
3 afaa IZVV +=⇒ (4.37) Thus, the connection for the positive, negative and zero sequence networks can be drawn based on the equations obtained in this section. Figure 4.5(b) represents the connection of sequence networks for a double line to ground fault . 52
1a
V 2aV
F
1a
I 2aI
F
f
Z30aV
F
0a
I
Figure 4.5(b) connection of sequence networks for a double line to ground fault
2a
V
2a
I
2
Z
F
1a
I
1
Z
F
+
a
E
1a
V 0aV
0
Z
F
f
Z3 0aI
Figure 4.5(c) Thevenin Equivalent for the sequence network connections for a LLG fault
From Figure 4.5 (c) the following expressions can be written
)3/( )3( 020211 ffaa ZZZZZZZEI ++++=⇒ (4.38)
4.6 Fault Location Algorithm
In this section we study a fault location algorithm that has been presented in . The impedance based algorithm is studied and made use in order to calculate the fault location. This algorithm is applicable for all kind of unsymmetrical faults . For studying the algorithm we consider the transmission line represented in Figure 4.6. 53
GE HE
P Q
G a b c Z abc Z Habc Z
Figure 4.6 Transmission Line Considered for the Algorithm
4.7 Impedance Based Algorithm
G
E , Gabc Z are the Thevenin equivalent voltage source and impedance respectively at terminal P. HE and Habc Z are the Thevenin equivalent voltage source and impedance respectively at terminal Q. abc Z represents the line impedance. Assuming that an unsymmetrical fault occurs, we can make use of the symmetrical components theory and a negative sequence network is represented as shown in Figure 4.7
G
Z2 HZ22)1( Zm−
2mZ
P Q
2pI 2qI2fI
2fV
Figure 4.7 Negative Sequence Network during the fault neglecting Shunt capacitance
We have
Gpp
ZVI 222 /−= (4.39)
Hqq
ZVI 222 /−= (4.40) 54
δjqqpp
eIZmVImZ V ])1([ 222222 −−=− (4.41) Where,
2p
V , 2pI Negative sequence voltage and current during the fault at terminal P;
2q
V , 2qI Negative sequence voltage and current during the fault at terminal Q;
G
Z2 , HZ2 Negative sequence source impedance at terminal P and Q;
δ Synchronization angle between measurements at terminal P and Q;
m Per unit fault distance from terminal P;
2
Z Total negative sequence impedance of the line. In the above equations, 2pV and 2qV are known based on the measurements. Then 2pI
and 2qI can be obtained using Equations (4.39) and (4.40). Solving Equation (4.41) will result in the fault location. The Detailed method is referred to the original work presented in . The algorithm neglecting shunt capacitances, as presented above, has advantages of simplicity and computational efficiency. However, neglecting shunt capacitances may lead to certain errors for long lines. Consideration of shunt capacitances is discussed in . The following table lists the estimated fault location for phase A to ground faults with fault resistance 5 ohms. The fault resistance value does not affect the fault location .
Table 4.1Fault location estimation
Actual fault location (p.u.) Estimated fault location (p.u.)
0.2 0.2105
0.4 0.4124
0.6 0.6130
0.8 0.8120
It is shown from the table that quite accurate estimates have been obtained. The errors may be due to inaccuracy of Thevenin parameter estimates since the voltage and current phasors utilized are not precise. 55
4.8 Voltage and Current Waveforms for Different Fault Locations
The voltage and current waveforms for different fault locations are plotted which is represented in the figures in this section.
020 40 60 80 100 120 140
-1.5
-1
-0.5
0
0.5
1
1.5
samples voltage
A
B
c
Figure 4.8(a) Voltage waveforms for a phase A to ground fault with a fault location of 0.2 p.u
020 40 60 80 100 120 140
-4
-3
-2
-1
0
1
2
3
samples current
A
B
C
Figure 4.8(b) Current waveforms for a phase A to ground fault with a fault location of 0.2 p.u 56
020 40 60 80 100 120 140
-1.5
-1
-0.5
0
0.5
1
1.5
samples voltage
A
B
c
Figure 4.9(a) Voltage waveforms for a phase A to ground fault with a fault location of 0.4 p.u
0 20 40 60 80 100 120 140
-5
-4
-3
-2
-1
0
1
2
3
4
samples current
A
B
C
Figure 4.9(b) Current waveforms for a phase A to ground fault with a fault location of 0.4 p.u 57
0 10 20 30 40 50 60 70 80 90 100
-1.5
-1
-0.5
0
0.5
1
1.5
samples voltage
A
B
c
Figure 4.10(a) Voltage waveforms for a phase A to ground fault with a fault location of 0.6 p.u
020 40 60 80 100 120 140
-8
-6
-4
-2
0
2
4
6
samples current
A
B
C
Figure 4.10(b) Current waveforms for a phase A to ground fault with a fault location of 0.6 p.u 58
010 20 30 40 50 60 70 80 90 100
-1.5
-1
-0.5
0
0.5
1
1.5
samples voltage
A
B
c
Figure 4.11(a) Voltage waveforms for a phase A to ground fault with a fault location of 0.7 p.u
010 20 30 40 50 60 70 80 90 100
-8
-6
-4
-2
0
2
4
6
samples current
A
B
C
Figure 4.11(b) Current waveforms for a phase A to ground fault with a fault location of 0.7 p.u 59
Conclusion
The main objectives of this thesis were Thevenin parameters estimation, determining the maximum power that can be transferred and the estimation of fault location. In Chapter 3, we made use of MATLAB SIMULINK ® to simulate the power system model that has been used in this Thesis. The results yielded the values of Thevenin equivalent parameters. These in turn were used to calculate the maximum power that can be transferred to the load bus under different conditions. In Chapter 4 we used the values of the Thevenin parameters for fault location. The results obtained confirm the available results of the algorithms that have been used. 60
Bibliography
M H Haque, On-line monitoring of maximum permissible loading of a power system within voltage stability limits IEE Proceedings on Generation, Transmission and Distribution, Vol. 150, No. 1, January 2003 Prabha Kundur, Power System Stability and Control , McGraw-Hill, 1994 Prabha Kundur, John Paserba, Venkat Ajjarapu, Göran Andersson, Anjan Bose, Claudio Canizares, Nikos Hatziargyriou, David Hill, Alex Stankovic, Carson Taylor, Thierry Van Cutsem, and Vijay Vittal, Definition and Classification of Power System Stability , IEEE Transactions on Power Systems, Vol. 19, No. 2, May 2004 D P Kothari, I J Nagrath, Modern Power System Analysis , McGraw-Hill, 2003 Yuan Liao, Fault Location Utilizing Unsynchronized Voltage Measurements during Fault , Electric Power Components and Systems, Vol.34, pp: 1283–1293, 2006 B Milosevic and Miroslav Begovic, Voltage-Stability Protection and Control Using a Wide-Area Network of Phasor Measurements , IEEE Transactions on Power Systems, Vol. 18, No.1, February 2003 IEEE Special Publication 907H0358-2-PWR, Voltage stability of Power Systems: Concepts, Analytical Tools and Industry Experience , 1990 B. M. Weedy, B. J. Cory, Electric Power Systems , John Wiley & Sons, 1998
Khoi Vu, Miroslav M. Begovic, Damir Novosel, Murari Mohan Saha, Use of Local Measurements to Estimate Voltage-Stability Margin, IEEE Transactions on Power Systems, Vol. 14, No. 3 , August 1999 Matlab Student Version 7.0.0 19920 (R14) SimPower Systems May 2004 61
Tamer Kawady and Jürgen Stenzel , A Practical Fault Location Approach for Double Circuit Transmission Lines Using Single End Data, IEEE Transactions on Power Delivery, Vol. 18, No. 4, October 2003 Yuan Liao, Fault Location for Single-Circuit Line Based on Bus-Impedance Matrix Utilizing Voltage Measurements , IEEE Transactions on Power Delivery, Vol. 23, No. 2, April 2008 Yuan Liao, Transmission Line Fault Location Algorithms without Requiring Line Parameters, Electric Power Components and Systems, Vol. 33, No. 10, October 2005 Y Liao and S Elangovan, Improved symmetrical component-based fault Distance estimation for digital distance protection, IEE Proceedings on Generation, Transmission and Distribution, Vol. 145, No. 6, November 1998 J J Grainger and W D Stevenson Jr., Power System Analysis, McGraw Hill, 1994 62
Vita
Mohammad Museb Iftakhar was born in Karimnagar, Andhra Pradesh, India on December 14 th , 1985. He received his Bachelor of Engineering degree in Electrical and Electronics Engineering in 2006 from Osmania University, Hyderabad, India. He worked as a Research Assistant in spring 2007 in the Power Systems Research Group at the University of Kentucky, Lexington, Kentucky, U.S.A.
Mohammad M Iftakhar |
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10928 | https://www.youtube.com/watch?v=33UpHFDxF5k&pp=0gcJCfwAo7VqN5tD | Determine the Number of Subsets and Proper Subsets in a Given Set
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determine the number of subsets and proper subsets in a given set okay so first we are going to tackle subsets and proper subsets so a few notes the empty set right so the empty set is all sometimes shown like this basically meaning there's nothing in it right it's empty the empty set is a subset and a proper subset of any finite set let's call it a so no matter what set we have the empty set is considered a subset and a proper subset and a set is always a subset of itself but it is not a proper subset so if we have set a right finite set a then a is also a subset but it is not a proper subset okay so while an empty set is both a subset and a proper subset they the full set is only a subset okay so now we're gonna look at two scenarios of counting subsets so the first one here we'll call this one if we have finite set a where a is only the set of one then we have two subsets the empty set and the set that contains one so there are two total subsets if in option two set a is both one and two then we have of course still the empty set and the full set right because those are both subsets and then we also have the subset one and the subset of just two so there are four you might notice if we have one element in the set that means we have two subsets if we have two elements in this set then we have four subsets are we seeing a pattern yes we are so for the number of subsets if a has n elements so in this first one we had one element in the second one we had two elements so that's how we can figure out n is the number of elements in the set then there are two to the end n subsets right and that includes the empty set and a itself so think about it in our first one when we had one element two to the first is two that's the empty set in a itself like we showed in the second one we had two elements two squared is four we had the empty set we had a itself and then we had one and we had two right remember from the previous page now in proper subsets only the set itself is not a proper subset so if we go back to our last examples if we would take out the set itself in each one so the first one was just one we'd be down to one proper subset and we'd be down to three meaning we would just take two to the n and subtract one so these are the formulas to find the number of subsets and the number of proper subsets so let's look at that how many subsets and proper subsets does the following set have so we have the set n 0 3 6 9 12 so 1 2 3 4 5. set n has five elements so to find the number of subsets we are going to take remember two to the n power where n is the number of elements so we're going to take two to the fifth power well that's 2 times 2 times 2 times 2 times 2 which in this case is 32 so there are 32 subsets to find the proper subsets we just subtract 1 right because we are taking away the set itself so in this case there are 31 proper subsets because that is 2 to the n minus 1. okay how many subsets and proper subsets does the following set have set t consists of the colors red blue green and brown one two three four so there are four elements so to find the subsets i take 2 to the fourth power which is 16 so there are 16 subsets and to find the proper subsets i take 2 to the 4th minus 1 and 16 minus 1 is 15 proper subsets and that is how we determine the number of subsets and proper subsets in a given set |
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## Cantor’s Paradise
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Medium’s #1 Math Publication
Exploring the Monty Hall and Birthday Problem with Simulations
Understand the results of two classic probability puzzles through interactive R simulations
Robby Sneiderman
6 min readJul 28, 2024
Introduction:
In this article, we’ll explore two classical problems in probability — the Monty Hall Problem and the Birthday Problem— and we will use R simulations to visualize and verify the surprising results.
Simulations are an extremely powerful tool in Statistics and Data Science, as they can be used to verify results practically.
We’ll also provide interactive Shiny apps and links to the GitHub with the full code so you can run it yourself.
Section 1: The Monty Hall Problem
The Monty Hall problem is a probability puzzle based on the American television game show “Let’s Make a Deal.” Here’s the setup:
You are a contestant on a game show.
There are three identical looking doors: behind one is a car, and behind the other two are goats.
You are asked to pick one of three doors.
Once you have made your initial pick , the host, who knows what’s behind each door, will always open another door to reveal a goat.
The host then asks if you want to switch to the remaining door.
Try it yourself using my interactive Shiny Application
Should you switch doors?
A common answer would be “No” as it seems there should be no difference between staying or switching. But the actual answer is, yes, you should switch, in fact this will increase your odds of winning the car to 2/3.
Here is some code that generates and simulates thousands of random instances of the Monty Hall problem.
```
Monty Hall Simulation Functionmonty_hall <- function(iterations, switch) { results <- replicate(iterations, { # Step 1: Generate doors with one car and two goats doors <- c(rep('goat',2), "car") doors <- sample(doors) # Randomly place the car behind one of the doors # Step 2: Player makes a choice choice <- sample(1:3, 1) # Player's initial choice # Step 3: Host reveals a goat car_position <- which(doors == "car") available_doors <- setdiff(1:3, c(choice, car_position)) if (length(available_doors) > 1) { reveal <- sample(available_doors, 1) } else { reveal <- available_doors } # Step 4: Player decides whether to switch if (switch) { choice <- setdiff(1:3, c(choice, reveal)) } # Step 5: Check if the final choice has the car result <- doors[choice] == "car" return(as.numeric(result)) }) win_rate <- mean(results) loss_rate <- 1 - win_rate return(list(win_rate = win_rate, loss_rate = loss_rate)) # Return the win and loss rates}# Function to compare both strategies and plot densitiescompare_strategies <- function(max_iterations) { iteration_points <- c(10, 100, 500, 1000, 2000, 5000, 10000, 20000, 50000, 100000) win_rates_switch <- numeric(length(iteration_points)) loss_rates_switch <- numeric(length(iteration_points)) win_rates_stay <- numeric(length(iteration_points)) loss_rates_stay <- numeric(length(iteration_points)) for (i in seq_along(iteration_points)) { iter <- iteration_points[i] result_switch <- monty_hall(iter, TRUE) result_stay <- monty_hall(iter, FALSE) win_rates_switch[i] <- result_switch$win_rate loss_rates_switch[i] <- result_switch$loss_rate win_rates_stay[i] <- result_stay$win_rate loss_rates_stay[i] <- result_stay$loss_rate } df <- data.frame( Iterations = rep(iteration_points, 4), Rate = c(win_rates_switch, loss_rates_switch, win_rates_stay, loss_rates_stay), Strategy = rep(c("Switch (Win)", "Switch (Loss)", "Stay (Win)", "Stay (Loss)"), each = length(iteration_points)) ) # Example Usagecompare_strategies(100000)
Monty Hall Simulation Function<- function(, switch){<-(,{
Step 1: Generate doors with one car and two goats<- c(rep('goat', 2), "car")<-()
Randomly place the car behind one of the doors
Step 2: Player makes a choice<-(1: 3, 1)
Player's initial choice
Step 3: Host reveals a goat<-(== "car")<-(1: 3, c(,)) if(length()> 1){<-(, 1)} else{<-}
Step 4: Player decides whether to switch if(switch){<-(1: 3, c(,))}
Step 5: Check if the final choice has the car<-[] == "car" return(as.numeric())})<-()<- - return(list(=, =))
Return the win and loss rates}
Function to compare both strategies and plot densities<- function(){<- c(10, 100, 500, 1000, 2000, 5000, 10000, 20000, 50000, 100000)<-(length())<-(length())<-(length())<-(length()) for(in seq_along()){<-[]<-(, TRUE)<-(, FALSE)[]<-$[]<-$[]<-$[]<-$}<-(= rep(, 4), = c(,,,), = rep(c("Switch (Win)","Switch (Loss)","Stay (Win)","Stay (Loss)"), = length()))
Example Usage(100000)
```
Here is the result showing the simulation over a number of iterations (on the x axis). As we try enough games, we see that switching most definitely improves our odds of winning.
Probability Analysis
We won’t provide detail or proof here instead we showcase how using simulation can prove to us a result without needing to understand the details of the proof.
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The standard proof of this result is often given via Bayesian statistics. However it can arguably better be understood via the minimax theorem. it goes as follows:
The player’s goal is to maximize their chances of winning the car.
When the player initially picks a door, they have a 1/3 chance of picking the car and a 2/3 chance of picking a goat. The host then opens a door with a goat, introducing new information.
If you initially chose the correct door (1/3 probability) that had the car behind it, if you switch you will lose the game and switch to a door with a goat.
If you initially chose a door with a goat behind it (occurs with a 2/3 probability), the host will have no choice but to reveal the other door with the goat, and hence if you switch you will win the car.
By switching doors, the player effectively maximizes their chances of winning the car to 2/3. This is because the strategy leverages the host’s reveal to update the probabilities and minimize the worst-case scenario (ending up with a goat). Hence, in the more common case, (2/3 case) switching leads to a car every single time. I find this to be the best way of understanding the Monty Hall Problem.
Section 2: Birthday Problem
Ask yourself the following: how many people would you expect to need in a room for there to be at least a 50% chance at least two share a birthday?
The answer is the seemingly unintuitive 23 people.
Try it yourself using my interactive Shiny Application
Let us also use R to simulate thousands of selections of dates and flag those which have a match. Here is some code that helped generate the simulation and creates samples of random birthdays you can try running.
```
Load necessary librarieslibrary(ggplot2)library(ggthemes)# Function to simulate the Birthday Paradoxsimulate_birthday_paradox <- function(num_people, num_simulations) { has_shared_birthday <- function(num_people) { birthdays <- sample(1:365, num_people, replace = TRUE) return(any(duplicated(birthdays))) } results <- replicate(num_simulations, has_shared_birthday(num_people)) probability <- mean(results) return(probability)}# Function to generate random birthdates and highlight matchesgenerate_birthdates <- function(num_people) { months <- c("January", "February", "March", "April", "May", "June", "July", "August", "September", "October", "November", "December") days_in_month <- c(31, 28, 31, 30, 31, 30, 31, 31, 30, 31, 30, 31) cumulative_days <- rep(NA, length(days_in_month)) cumulative_days <- days_in_month for (i in 2:length(days_in_month)) { cumulative_days[i] <- cumulative_days[i - 1] + days_in_month[i] } birthdates <- sample(1:365, num_people, replace = TRUE) date_names <- sapply(birthdates, function(x) { month <- months[ceiling(x / 31)] day <- days[(x %% 31) + 1] return(paste(month, day)) }) unique_dates <- unique(date_names) duplicated_dates <- unique(date_names[duplicated(date_names)]) cat("Randomly generated birthdates:\n") for (i in 1:num_people) { if (date_names[i] %in% duplicated_dates) { cat("", date_names[i], "(duplicate)\n") } else { cat(date_names[i], "\n") } }}# Example usagenum_people <- 23num_simulations <- 10000# Print random birthdates with duplicates highlightedgenerate_birthdates(num_people)
Load necessary libraries()()
Function to simulate the Birthday Paradox<- function(,){<- function(){<-(1: 365,, = TRUE) return(any(()))}<-(,())<-() return()}
Function to generate random birthdates and highlight matches<- function(){<- c("January", "February", "March", "April", "May", "June", "July", "August", "September", "October", "November", "December")<- c(31, 28, 31, 30, 31, 30, 31, 31, 30, 31, 30, 31)<- rep(NA, length())<- for(in 2: length()){[]<-[- 1] +[]}<-(1: 365,, = TRUE)<-(, function(){<-[ceiling(/ 31)]<-[(%% 31) + 1] return((,))})<-()<-([()])("Randomly generated birthdates:\n") for(in 1:){if([]%in%){("",[],"(duplicate)\n")} else{([], "\n")}}}
Example usage<-<-
Print random birthdates with duplicates highlighted()
```
Here is the plot of our code ran on 10,000 simulations each on a variety of a number of people. We can also see that once the number of people in the room is at or above 23, we have a greater than 50% chance that at least two will share a birthday.
The key for understanding this unintuitive result is to realize is that each additional person introduces another set of possible combinations to compare.
Simulations are a powerful and often under utilized tool in statistics and data science. They allow researchers or those working with data to quickly test or verify results, and can be used to model and test results especially in the realm of discrete math and statistics.
Do you know any other counterintuitive or interesting results in statistics or probability that can be better understood with simulations?
Thanks for reading!
GitHub for the full code used in this project.
References:
Gardner, M. (1959a), “Mathematical Games” column, Scientific American, October 1959, 180–182. Reprinted in The Second Scientific American Book of Mathematical Puzzles and Diversions.
Gill, R.D. (2011), The Monty Hall Problem is not a Probability Puzzle — it’s a challenge in mathematical modelling, Statistica Neerlandica 65 58–71.
Data Science
Statistics
Simulation
Probability
## Published in Cantor’s Paradise
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Medium’s #1 Math Publication
## Written by Robby Sneiderman
123 followers
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-BSc Mathematics. -MSc Statistics. my apps SmartBreeds.io, Smart-Trends.io, SmartBookshelf.io and SmartBirds.io. contact via Robbysneiderman.com
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10930 | https://www.chegg.com/homework-help/questions-and-answers/given-function-2-2-sin-t-find-t-0-q84760852 | Your solution’s ready to go!
Our expert help has broken down your problem into an easy-to-learn solution you can count on.
Question: Given the function 𝑢(𝑥, 𝑦) = 𝑥2𝑦 − 𝑦2, where 𝑥 = sin 𝑡 and 𝑦 = 𝑒t, find 𝑑𝑢/𝑑𝑡 when t= 0.
Given the function 𝑢(𝑥, 𝑦) = 𝑥2𝑦 − 𝑦2,
where 𝑥 = sin 𝑡 and 𝑦 = 𝑒t, find 𝑑𝑢/𝑑𝑡 when t= 0.
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10931 | https://www.quora.com/How-do-I-compare-irrational-numbers-For-example-which-one-is-bigger-between-%C2%B3%E2%88%9A5-and-%E2%81%B4%E2%88%9A2 | How to compare irrational numbers? For example, which one is bigger between ³√5 and ⁴√2 - Quora
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How do I compare irrational numbers? For example, which one is bigger between ³√5 and ⁴√2?
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1y
To compare the two irrational numbers 3√5 5 3 (the cube root of 5) and 4√2 2 4 (the fourth root of 2), we can evaluate their approximate decimal values or compare their powers.
Step 1: Calculate approximate values
Calculate 3√5 5 3:
3√5≈1.710 5 3≈1.710
2. Calculate 4√2 2 4:
4√2≈1.189 2 4≈1.189
Step 2: Compare the values
From the calculations:
3√5≈1.710 5 3≈1.710
4√2≈1.189 2 4≈1.189
Conclusion
Since 1.710>1.189 1.710>1.189, we conclude that:
3√5>4√2 5 3>2 4
Thus, 3√5 5 3 is bigger than 4√2 2 4.
Upvote ·
Related questions
More answers below
How do I compare between irrational numbers?
How do you construct an irrational number equivalent to 6√5?
Show that 3+4√2 3+4 2 is an irrational number?
What are 5 examples of irrational numbers between 3/4 and 1/3?
How can I find 2 irrational numbers between 1.42 and 1.73?
Dave Williamson
Works at Old-style Webmaster and Programmer (94-present) · Author has 1.7K answers and 12.3M answer views
·8y
With your example, some people can quickly estimate that the fourth root of two must be the smaller number. Why is this?
When you have two radicals such as 3√5 5 3 and 4√5 5 4
where the bases are the same, the higher root is always smaller than the smaller root.
3√5 5 3>4√5 5 4
When you have two radicals such as 3√5 5 3 and 3√4 4 3 where the roots are the same, but the bases are different, the smaller base will cause a smaller result than the larger base.
3√5 5 3>3√4 4 3
Since both rules imply that 3√5 5 3>4√2 2 4, some people will immediately answer th
Continue Reading
With your example, some people can quickly estimate that the fourth root of two must be the smaller number. Why is this?
When you have two radicals such as 3√5 5 3 and 4√5 5 4
where the bases are the same, the higher root is always smaller than the smaller root.
3√5 5 3>4√5 5 4
When you have two radicals such as 3√5 5 3 and 3√4 4 3 where the roots are the same, but the bases are different, the smaller base will cause a smaller result than the larger base.
3√5 5 3>3√4 4 3
Since both rules imply that 3√5 5 3>4√2 2 4, some people will immediately answer this question based on these rules.
However, this method is not as obvious.
What if we needed to compare 5√5 5 5 and 3√3 3 3
In this case, we need to simplify both numbers by getting rid of the radicals
Let’s look at the numbers that indicate the root.
What is the least common multiple of 3 and 5? (yes, 15.)
Raise each value to this power, the 15th power:
(5√5)15(5 5)15 and (3√3)15(3 3)15
and simplify the exponents:
5 15/5[/m a t h a n d[m a t h]3 15/3 5 15/5[/m a t h a n d[m a t h]3 15/3
Compare: 5³ and 3 5 3 5
Which one is larger?
Finally, we can also fall back on our good old calcuators, if they have the exponent key (^) and other functions. We can even use variables for temporary storage for later comparison:
What I like to do, when I have time on tests, is use two different methods. I use the method that I know the teacher prefers that I use the first time I answer the question, then after I have answered all the questions on the test, I go back and use a different method to double-check my answer if there is a possibility that I got the wrong answer.
Upvote ·
9 9
9 1
Alex Heitzman
Studied at University of Nebraska-Lincoln · Author has 931 answers and 857.2K answer views
·8y
Note that
5^(1/3) > 5^(1/4)
and
5^(1/4) > 2^(1/4)
Therefore,
5^(1/3) > 2^(1/4)
Another way to think of it:
the cube root of 5 is x. the fourth root of 2 is y. If multiplying x by itself three times give you five but multiplying y by itself four times only gets you to 2 then clearly x > y.
Sometimes you can use tricks with the above method. For example,
which is bigger,
17^(1/8) or 4^(1/5) ?
Well,
17^(1/8) > 16^(1/8) = (4^2)^(1/8) = 4^(1/4) > 4^1/5
But other times you can’t do this no matter how clever you are and you’ll have to raise both sides to a common denominator power as other answers gave.
Upvote ·
Sheldon Coopa
8y
In this case you raise both numbers to the power of 12. Then we we will have two numbers, 625 and 8. 5^(1/3) is greater. If number a is greater than number b, a will still be the greatest number after this step.
Upvote ·
9 2
Related questions
More answers below
Which number is greater, -2 or -3?
Is 2√2 an irrational number?
How is the interpretation of the name number 51 in numerology?
What are the two irrational numbers between 2√2 and 3?
Is this number irrational 4√4−2√3+√97−56√3 4 4−2 3+97−56 3?
Nilesh Darsan
Sep 10
A small trick, if your radical is n√x x n you can write that as a fraction of x n x n and compare those fractions individually
Be careful, the fraction is in reverse order, x n x n not n x n x
Example:
³√5 and ⁴√2 if written as (³√5 and ⁴√2 are not literally equal to 5 3 5 3 and 2 4 2 4!) 5 3 5 3 and 2 4 2 4 and made the same denominator
= 20 12 20 12, 6 12.6 12.
and it becomes obvious that ³√5 is larger
Upvote ·
Anthony Clohesy
Maths Graduate, High School Maths teacher · Author has 165 answers and 1.3M answer views
·8y
Use the same trick as comparing fractions: convert the numbers to a more comparable form. If I want to compare 2/5 and 1/4, multiplying both by 20 gives 8 and 5 respectively.
Similarly, raising both your numbers to the same power (one which is a multiple of 3 and 4) will help: raised to the power 12 gives 5 to the 4 and 2 to the 3. 625 or 8. When you think about it, with those numbers it's a bit more obvious, but the method will work for any roots like that.
Upvote ·
9 1
Manjunath Subramanya Iyer
I am a retired bank officer teaching maths · Author has 7.2K answers and 10.4M answer views
·6y
Related
How do I compare between irrational numbers?
An irrational number is a decimal which is neither terminating nor recurring.
Some examples of irrational numbers are
2.34875183…………
0.482958351………..
Obviously the first irrational number is greater than the second, because, the first starts with 2 while the second starts with 0.
If two irrational numbers start with the same number as in 3.4295104287….. and 3.53810397… observe the second digit. The second digit is 4 in the first irrational number while it is 5 in the second irrational number. Therefore the second irrational number is greater than the first. And so on we compare the next digit if th
Continue Reading
An irrational number is a decimal which is neither terminating nor recurring.
Some examples of irrational numbers are
2.34875183…………
0.482958351………..
Obviously the first irrational number is greater than the second, because, the first starts with 2 while the second starts with 0.
If two irrational numbers start with the same number as in 3.4295104287….. and 3.53810397… observe the second digit. The second digit is 4 in the first irrational number while it is 5 in the second irrational number. Therefore the second irrational number is greater than the first. And so on we compare the next digit if the earlier digits are same in both the irrational numbers.
Some irrational numbers will be in the form of irrational roots of rational numbers. They are called surds or radicals.
For example, √5 and √12. Obviously √12 is greater than √5. For such surds (i.e., √), the order is 2. For cube root the order is 3, for fourth root it is 4, and so on.
Similarly cube root of 17 is greater than say cube root of 11.
The comparison will be a little difficult if the surds are of different order. For instance which is greater? Cube root of 7 or fourth root of 19? First, we reduce them to same order by taking LCM of the two orders. In the given example the LCM of the orders 3 and 4 is 12. Now cube root of 7 = 7^1/3 = 7 ^(4/12) = (7^4)1/12 = (2401)^1/12, and (19)1/4 = (19)^(3/12) = (19^3)^1/12 = (6859)^1/12. Now it is easy to compare. We can say fourth root of 19 is greater than cube root of 7.
Thus we can compare any two irrational numbers.
Upvote ·
9 2
9 2
Bob Gustafson
Former Mathematics Teacher at Mount Vernon High School (1988–2006) · Author has 341 answers and 335.7K answer views
·7y
Raise each expression to the twelfth power. (Twelve is the least common multiple of three and four.) You get 5^4 and 2^3. The former is larger. therefore the twelfth root of the first is greater than the twelfth root of the second.
Upvote ·
Wan Kang
b.a. in Mathematics, Nanjing University (南京大学) (Graduated 1994) · Author has 631 answers and 268.8K answer views
·8y
do you forget logarithm?
1 3 l o g 10 5=1 3(1−l o g 10 2)=(1−0.3010)/3 1 3 l o g 10 5=1 3(1−l o g 10 2)=(1−0.3010)/3
1 4 l o g 10 2=0.3010/4 1 4 l o g 10 2=0.3010/4
Upvote ·
G. E. Wolf
Former Remedial Math Instructor · Author has 268 answers and 197.6K answer views
·8y
Compare by raising both expressions to the 12th power.
(5^(1/3))^12=625
(2^(1/4))^12=8
proving 5(1/3) is larger
Upvote ·
Narayan Dalai
Studied Mathematics&Science &Electrical Engg at Diploma in Electrical Engineering (IAF) · Author has 2.5K answers and 2.6M answer views
·6y
Related
How do I compare between irrational numbers?
There is,practically, nothing to compare between two irrational numbers (Example:✔2 and ✔3):each irrational number acquires all properties attributable to irrationality; such as both are non-repeating non-terminating fractions; in other words, they are not perfect fractions.
Only you can compare which is larger or smaller. Simply you multiply each irrational by its respective multiplicative inverses(Reciprocals ) thereby converting to integers for easy comparison.
Example :- ✔21/✔2=2
✔31/✔3=3 ; 3>2; So ✔3>✔2 .
Upvote ·
Philip Lloyd
Specialist Calculus Teacher, Motivator and Baroque Trumpet Soloist. · Author has 6.8K answers and 52.8M answer views
·7y
Related
How does one show the number of irrationals (uncountable infinite) is larger than the number of rationals (countable infinite)?
Firstly, “countable” just means you can “pair off” the things being counted with our Natural Counting Numbers 1, 2, 3, 4… which, of course, will carry on for ever.
So “countable” does not imply you actually finish the counting!
FRACTIONS (or Rational Numbers)
Consider a list of all possible fractions as shown below.
Each row goes on for ever so each separate row could be “counted” by showing a one to
Continue Reading
Firstly, “countable” just means you can “pair off” the things being counted with our Natural Counting Numbers 1, 2, 3, 4… which, of course, will carry on for ever.
So “countable” does not imply you actually finish the counting!
FRACTIONS (or Rational Numbers)
Consider a list of all possible fractions as shown below.
Each row goes on for ever so each separate row could be “counted” by showing a one to one correspondence with our set of Natural numbers.
So how could we devise a method of “counting” all these rows?
Clearly, the set of all such fractions has the same infinite size as our Natural counting numbers = Aleph Naught.
The obvious question now is “How can we possibly count the negative fractions too?”
Well the following pattern shows that even including the negative fractions we still have the same sized infinity as our Natural numbers Aleph Naught!
At this point, it seems as though every set of numbers is “countable” meaning it can be put in a one to one correspondence with the Natural numbers but now let’s consider the Real Numbers, R.
In simple terms, the Real Numbers has all the Integers and Fractions (Rationals) but it includes all the numbers which cannot be expressed as fractions.
These are called the Irrationals.
These only exist in decimal form as never-ending decimals.
I will now show that the numbers 0 < x < 1 cannot be put into one to one correspondence with the natural numbers N = 1, 2, 3, 4 ……..
I will start by pretending that it is actually possible to make a list the real numbers.
Let’s imagine that we’ve done this, and we’ve written them all out in some sort of order, using their decimal expansions.
I am going to make another real number between 0 and 1, in such a way that it cannot already be on our list!
Since the list was supposed to contain all such real numbers, that will be a contradiction so this will be a proo...
Upvote ·
9 7
9 1
Lance Berg
Author has 28K answers and 54.7M answer views
·5y
Related
How can I take the square root of an irrational number like √5 without a calculator?
5 is not an irrational number. It is the ratio of two integers, 5 and 1, as you can see here, 5/1=5 Irrational means “no ratio of two integers represents this”
1/3 is rational, for example, even though there is no decimal expression for it, you just have 0.33333… with threes as far as you care to go, and more behind that.
To find the square root of a number like 5 which is not a perfect square, you have to guess, and refine your guess, until you get bored doing so… because the square root of five IS irrational.
Let’s see how this goes.
The square root of four is two. So the square root of five is
Continue Reading
5 is not an irrational number. It is the ratio of two integers, 5 and 1, as you can see here, 5/1=5 Irrational means “no ratio of two integers represents this”
1/3 is rational, for example, even though there is no decimal expression for it, you just have 0.33333… with threes as far as you care to go, and more behind that.
To find the square root of a number like 5 which is not a perfect square, you have to guess, and refine your guess, until you get bored doing so… because the square root of five IS irrational.
Let’s see how this goes.
The square root of four is two. So the square root of five is more than two.
The square of three is nine. So the square root of five is less than three.
How about the square of 2.1? That’s (2121)/100=(420+21)/100=441/100=4.41 not quite
And the square of 2.2 is (440+44)/100=480
The square of 2.3 is 5.29, oops, too high.
The square of 2.25 is 5.0625, really close. Probably, frankly, close enough for anything you’re likely to be doing. Six and a quarter hundredths off?
2.24 squared is 5.0176, even closer
2.23 squared is 4.9729, there, got it bracketed, it’s between those two. But this just keeps going until you have more digits than you require.
Checking our work, using a calculator we get 2.2360679775 which is STILL not correct, it’s just accurate down to 9 digits… the billionths, instead of the hundredths that I got to above.
Upvote ·
9 4
9 3
Related questions
How do I compare between irrational numbers?
How do you construct an irrational number equivalent to 6√5?
Show that 3+4√2 3+4 2 is an irrational number?
What are 5 examples of irrational numbers between 3/4 and 1/3?
How can I find 2 irrational numbers between 1.42 and 1.73?
Which number is greater, -2 or -3?
Is 2√2 an irrational number?
How is the interpretation of the name number 51 in numerology?
What are the two irrational numbers between 2√2 and 3?
Is this number irrational 4√4−2√3+√97−56√3 4 4−2 3+97−56 3?
How would one find the irrational numbers between 3÷5 and 4÷7?
How come in the range [0,1] we can say that there is no first irrational number, but we can say there is a last irrational number (.9999…)?
What are irrational numbers and their example?
How can you show that 5√2-2√30 is an irrational number?
Which one is greater, 1/3 or 1/4?
Related questions
How do I compare between irrational numbers?
How do you construct an irrational number equivalent to 6√5?
Show that 3+4√2 3+4 2 is an irrational number?
What are 5 examples of irrational numbers between 3/4 and 1/3?
How can I find 2 irrational numbers between 1.42 and 1.73?
Which number is greater, -2 or -3?
Is 2√2 an irrational number?
How is the interpretation of the name number 51 in numerology?
What are the two irrational numbers between 2√2 and 3?
Is this number irrational 4√4−2√3+√97−56√3 4 4−2 3+97−56 3?
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10932 | https://fiveable.me/key-terms/ap-physics-1/net-external-torque | Net external torque - (AP Physics 1) - Vocab, Definition, Explanations | Fiveable | Fiveable
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All Key Terms
AP Physics 1
Net external torque
🎡ap physics 1 review
key term - Net external torque
Citation:
MLA
Definition
Net external torque represents the sum total effect produced by all external torques acting on an object or system. It determines the rate at which an object's angular momentum changes.
Related terms
Torque arm:Torque arm refers to the perpendicular distance between the axis of rotation and where a force is applied, affecting the magnitude of torque.
Rotational equilibrium:Rotational equilibrium occurs when an object's net torque is zero, resulting in no rotational acceleration.
Lever arm:Lever arm represents the perpendicular distance between an axis of rotation and where a force is applied, influencing both torque and mechanical advantage.
"Net external torque" also found in:
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10933 | https://curriculum.illustrativemathematics.org/MS/teachers/2/4/11/index.html | Lesson 11
Percentage Contexts
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11.1: Leaving a Tip (5 minutes)
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Building On
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Building Towards
7.RP.A.3
Warm-up
The purpose of this warm-up is to help students connect their current work with percentage contexts to their prior work on percent increase and efficient ways of finding percent increase.
Launch
Consider telling students that these questions may have more than one correct answer. Students in groups of 2. 2 minutes of quiet think time followed by partner and then whole-class discussion.
Student Facing
Which of these expressions represent a 15% tip on a $20 meal? Which represent the total bill?
(15 \boldcdot 20)
(20 + 0.15 \boldcdot 20)
(1.15 \boldcdot 20)
(\frac{15}{100} \boldcdot 20)
Student Response
For access, consult one of our IM Certified Partners.
Activity Synthesis
For each expression, ask a few students to explain whether they think it represents: the total bill, the tip, or neither. For each expression, select a student to explain their reasoning.
11.2: A Car Dealership (10 minutes)
CCSS Standards
Addressing
7.RP.A.3
Routines and Materials
Instructional Routines
MLR6: Three Reads
Think Pair Share
Required Materials
Four-function calculators
Activity
The purpose of this activity is to introduce students to a context involving markups and markdowns or discounts, and to connect this to the work on percent increase and percent decrease they did earlier. The first question helps set the stage for students to see the connection to markups and percent increase. Look for students who solve the second question by finding 90% of the retail price, and highlight this approach in the discussion.
Launch
Tell students that a mark-up is a percentage that businesses often add to the price of an item they sell, and a mark-down is a percentage they take off of a given price. If helpful, review the meaning of wholesale (the price the dealership pays for the car) and retail price (the price the dealership charges to sell the car). Sometimes people call mark-downs discounts.
Provide access to calculators. Students in groups of 2. Give students 5 minutes of quiet work time, followed by partner then whole-class discussion.
Reading, Writing: MLR6 Three Reads. Use this routine to support reading comprehension of this word problem, without solving it for students. In the first read, students read the problem with the goal of comprehending the situation (e.g., A car dealership bought a car. The dealership wants to make a profit. They need to decide what price the car should be.). If needed, discuss the meaning of unfamiliar terms at this time (e.g., profit, wholesale, retail price, commission, etc.). Use the second read to identify the important quantities by asking students what can be counted or measured (e.g., wholesale price, profit or mark-up, and retail price). In the third read, ask students to brainstorm possible mathematical solution strategies to complete the task. This will help students connect the language in the word problem and the reasoning needed to solve the problem while keeping the intended level of cognitive demand in the task. Design Principle(s): Support sense-making
Student Facing
A car dealership pays a wholesale price of $12,000 to purchase a vehicle.
The car dealership wants to make a 32% profit.
By how much will they mark up the price of the vehicle?
After the markup, what is the retail price of the vehicle?
During a special sales event, the dealership offers a 10% discount off of the retail price. After the discount, how much will a customer pay for this vehicle?
Student Response
For access, consult one of our IM Certified Partners.
Student Facing
Are you ready for more?
This car dealership pays the salesperson a bonus for selling the car equal to 6.5% of the sale price. How much commission did the salesperson lose when they decided to offer a 10% discount on the price of the car?
Student Response
For access, consult one of our IM Certified Partners.
Anticipated Misconceptions
It is important throughout that students attend to the meanings of particular words and remain clear on the meaning of the different values they find. For example, "wholesale price," "retail price," and "sale price" all refer to specific dollar amounts. Help students organize their work by labeling the different quantities they find or creating a graphic organizer.
Activity Synthesis
For the first question, help students connect markups to percent increase.
Select students to share solutions to the second question. Highlight finding 90% of the retail price, and reinforce that a 10% discount is a 10% decrease.
Ask them to describe how they would find (but not actually find) . . .
"The retail price after a 12% markup?" (Multiply the retail price by 0.12, then add that answer to the retail price. Alternatively, multiply the retail price by 1.12.)
"The price after a 24% discount?" (Multiply the retail price by 0.24, then subtract that answer from the retail price. Alternatively, multiply the retail price by 0.76.)
11.3: Commission at a Gym (10 minutes)
CCSS Standards
Addressing
7.RP.A.3
Routines and Materials
Instructional Routines
MLR3: Clarify, Critique, Correct
Think Pair Share
Required Materials
Four-function calculators
Activity
The purpose of this activity is to introduce students to the concept of a commission and to solve percentage problems in that context. Students continue to practice finding percentages of total prices in a new context of commission.
Monitor for students who use equations like (c = r \boldcdot p) where (c) is the commission, (r) represents the percentage of the total that goes to the employee, and (p) is the total price of the membership.
Launch
Tell students that a commission is the money a salesperson gets when they sell an item. It is usually used as an incentive for employees to try to sell more or higher priced items than they usually would. The commission is usually a percentage of the price of the item they sell.
Provide access to calculators. Students in groups of 2. Give students 2 minutes of quiet work time. Partner then whole-class discussion.
Student Facing
For each gym membership sold, the gym keeps $42 and the employee who sold it gets $8. What is the commission the employee earned as a percentage of the total cost of the gym membership?
If an employee sells a family pass for $135, what is the amount of the commission they get to keep?
Student Response
For access, consult one of our IM Certified Partners.
Anticipated Misconceptions
Students may find the percentage of an incorrect quantity. Ask them to state, in words, what they are finding a percentage of.
Students may not understand the first question. Tell them that a membership is sold for a certain price and the money is split with \$42 going to the gym and \$8 going to the employee.
Activity Synthesis
Select students to share how they answered the questions.
During the discussion, draw attention to strategies for figuring out which operations to do with which numbers. In particular, strategies involving equations like (c = r \boldcdot p) where (c) is the commission, (r) represents the percentage of the total that goes to the employee, and (p) is the total price of the membership.
Reading, Writing: MLR3 Clarify, Critique, Correct. Present an incorrect response to the second question that reflects a possible misunderstanding from the class. For example, “For the family membership of \$135, the employee would keep \$8.” Prompt students to identify the error (e.g., ask, “Do you agree with the author’s reasoning? Why or why not?”), and then write a correct version. This will help students to understand that the employee's commission is always a rate of 16% and not a flat amount of $8. Design Principle(s): Maximize meta-awareness
11.4: Card Sort: Percentage Situations (10 minutes)
CCSS Standards
Addressing
7.RP.A.3
Routines and Materials
Instructional Routines
MLR8: Discussion Supports
Take Turns
Required Materials
Pre-printed slips, cut from copies of the blackline master
Optional activity
This activity gives students an opportunity to practice various vocabulary terms that come along with percentages. Students are asked to sort scenarios to different descriptors using the images, sentences or questions found on the scenario cards. The questions found on the scenario cards are intended to help students figure out which descriptor the scenario card belongs under.
As students work on the task, identify students that are using the vocabulary: tip, tax, gratuity, commission, markup/down, and discount. These students should be asked to share during the discussion.
Launch
Arrange students in groups of 2. Distribute the sorting cards, and explain that students will sort 8 scenarios into one of 6 categories. Demonstrate how students can take turns placing a scenario under a category and productive ways to disagree. Here are some questions they might find useful:
Which category would you sort this under?
What do you think this word means?
What words can we use as clues about where to sort this card?
Representation: Internalize Comprehension. Chunk this task into more manageable parts to differentiate the degree of difficulty or complexity by beginning with fewer cards. For example, give students a subset of the cards to start with and introduce the remaining cards once students have completed their initial set of matches. Supports accessibility for: Conceptual processing; Organization
Speaking: MLR8 Discussion Supports. Show central concepts multi-modally by using different types of sensory inputs: acting out scenarios or inviting students to do so, showing videos or images, using gestures, and talking about the context of what is happening. This will help students to produce and make sense of the language needed to communicate their own ideas. Design Principle(s): Optimize output (for explanation)
Student Facing
Your teacher will give you a set of cards. Take turns with your partner matching a situation with a descriptor. For each match, explain your reasoning to your partner. If you disagree, work to reach an agreement.
Student Response
For access, consult one of our IM Certified Partners.
Anticipated Misconceptions
Students should use the question at the bottom of the card to help them if they get stuck sorting the scenarios.
Activity Synthesis
Ask identified students to share which situations they sorted under each word. Ask them:
"What made you decide to put these situations under this descriptor?"
"Were there any situations that you were really unsure of? What made you decide on where to sort them?"
Consider asking some groups to order the situations from least to greatest in terms of the dollar amount of the increase of decrease and asking other groups to order them in terms of the percentage. Then, have them compare their results with a group that did the other ordering.
Answer students’ remaining questions about any of these contexts. Tell students there is a copy of this chart at the end of the lesson that they can use as a reference tool during future lessons. Allow them a space to take notes on their own to remember it or details from one of the activity examples.
| paid to: | how it works: |
--- |
| sales tax | the government | added to the price of the item |
| gratuity (tip) | the server | added to the cost of the meal |
| interest | the lender (or account holder) | added to the balance of the loan, credit card, or bank account |
| markup | the seller | added to the price of an item so the seller can make a profit |
| markdown (discount) | the customer | subtracted from the price of an item to encourage the customer to buy it |
| commission | the salesperson | subtracted from the payment the store collects |
Lesson Synthesis
Lesson Synthesis
In this lesson, we studied lots of different situations where people use percentages.
“What are some situations in life in which people encounter percentages?”
“Give examples of situations where you would encounter tax, tip, markup, markdown, commission.” (Lots of possible answers.)
“When an item is marked down 10%, why does it make sense to multiply the price by 0.9?” (Since there is 10% off of the price, the new cost is 90% of the original.)
“When an item is marked up 25%, why does it make sense to multiply the price by (1.25)?” (Since the item now costs 100% plus an extra 25%, the new item costs 1.25 times the original.)
11.5: Cool-down - The Cost of a Bike (5 minutes)
CCSS Standards
Addressing
7.RP.A.3
For access, consult one of our IM Certified Partners.
Student Lesson Summary
Student Facing
There are many everyday situations where a percentage of an amount of money is added to or subtracted from that amount, in order to be paid to some other person or organization:
| goes to | how it works |
--- |
| sales tax | the government | added to the price of the item |
| gratuity (tip) | the server | added to the cost of the meal |
| interest | the lender (or account holder) | added to the balance of the loan, credit card, or bank account |
| markup | the seller | added to the price of an item so the seller can make a profit |
| markdown (discount) | the customer | subtracted from the price of an item to encourage the customer to buy it |
| commission | the salesperson | subtracted from the payment that is collected |
For example,
If a restaurant bill is \$34 and the customer pays \$40, they left \$6 dollars as a tip for the server. That is 18% of $34, so they left an 18% tip. From the customer's perspective, we can think of this as an 18% increase of the restaurant bill.
If a realtor helps a family sell their home for \$200,000 and earns a 3% commission, then the realtor makes \$6,000, because ((0.03) \boldcdot 200,!000 = 6,!000), and the family gets \$194,000, because (200,!000 - 6,!000 = 194,!000). From the family's perspective, we can think of this as a 3% decrease on the sale price of the home. |
10934 | https://www.teacherspayteachers.com/Browse/Search:algebra+factoring+gcf+activity/Page:3?aref=etlgyj04 | Algebra Factoring Gcf Activity - Page 3 | TPT
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Algebra Factoring Gcf Activity - Page 3
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Factoring out the GCF Digital Pixel Art | 6th Grade GCF Factoring Activity
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Stop Hayden on Math
Use this engaging digital activity to practice factoring the greatest common factor out of expressions! This is a digital pixel art activity for use with Google Sheets. There are 14 questions for students to answer, where they have to factor the positive GCF out of an expression. When students type in the correct answer, pixels appear and eventually form a cohesive picture. Click on the product preview to see the problems larger. Students love digital pixel arts, as they get the instant gratific
5 th - 7 th
Algebra, Other (Math)
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6.EE.A.3
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Factoring Activity BUNDLE - GCF, DOTS, Trinomials, Grouping, Completely
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Tater Taught by Jen Tate
Do you want 56 no-prep, print and digital, easy to check, differentiated math activities where students practice factoring polynomials? This will include factoring out a Greatest Common Factor (GCF), factoring the Difference of Two Squares (Difference of Perfect Squares, Difference of Squares, DOTS, or DOPS), factoring Trinomials, factoring by Grouping, and factoring Completely (everything together). What activities are included?20 Riddle Activities (4 GCF, 4 DOTS, 4 Trinomials, 4 Grouping, 4 C
8 th - 11 th
Algebra, Algebra 2, Math
CCSS
HSA-SSE.A.2
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Factoring Polynomials (a=1, a>1, GCF, DOTS) Activity Bundle
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Kate Dean - Secondary Math
With this engaging and self-checking mini color by number bundle, students will get practice factoring trinomials when A equal to 1, A greater than 1, and a mix. Students will also factor out a GCF, and factor using DOTS. These no prep, ready to print activities make the perfect warm up or done early activity for your classroom. Included are 6 mini color by number activities, where students answer 6 problems and use the color code at the bottom of the page to color a fun mandala image. To
8 th - 11 th
Algebra, Algebra 2, Math
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Algebra State Test Prep Stations Activity - Spring-Themed Review Centers
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Surviving in Secondary
Algebra Keystone Test-Prep Stations/Centers ActivityLooking for a way to review for the Keystone State Test in an engaging and motivating way for your students? Searching for an activity that supports various review Algebra topics such as Greatest Common Factor, System of Equations, Linear Equations and Terminology? Engage your high school Algebra students with this gamified and inclusive testing preparation stations activity! This resource is designed to help students review basic skills and re
8 th - 11 th
Algebra, Math Test Prep
Also included in:Keystone Exams Test Prep Bundle - Algebra, Biology, and Literature Review
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Factoring by GCF Activity Domino - Binomials, Trinomials, & Polynomials w 4 term
Created by
Math And Science Sarah
Engage your students with these 3 different Factoring by GCF Activity Domino Puzzles. This resource will have your students practicing 15 questions each to find the gcf of binomials, trinomials, and polynomials with 4 terms. Use these Factoring by GCF Activity Domino Puzzles as a way to assess your student's understanding as a bell ringer/exit ticket, independent or group practice, early finishers box, stations/centers, and so much more. Allow them to do each puzzle to go deeper into their know
8 th - 11 th
Algebra, Algebra 2
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Factoring Task Cards | Algebra Practice | GCF, Trinomials, Grouping, Perfect Sq.
Created by
Bright Path Teaching
Help your students master factoring with this set of 20 algebra-level factoring task cards, perfect for independent practice, math centers, early finishers, review activities, or partner work! This engaging resource covers a wide variety of factoring strategies and is ideal for reinforcing key concepts in Grade 9 & 10 algebra, including: ✅ Common factoring (GCF) ✅ Factoring trinomials (a = 1 and a ≠ 1) ✅ Factoring by grouping ✅ Special products (difference of squares, perfect square trinomials
9 th - 11 th
Algebra, Algebra 2, Arithmetic
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HSA-SSE.A.1
, HSA-SSE.A.1a
, HSA-SSE.A.1b
+6
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Create a Store Activity, Combine Like Terms Distributive Factoring GCF, 7th
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Differentiated with Delight
Practice Combining Like Terms, Distributive Property, Factoring with the Greatest Common Factor, and Line Graphs in this Evaluating Expressions with Algebraic Terms PBL Performance Task Activity for Grade 7 4 Days of Activities with a Real-World Connection!Students create their own competing storesCreate their own prices, shipping fees, and ordersCreate and solve equations to find costs of various ordersExpressions & Equations Galore!Creating variables to represent valuesCombining Like Terms to
7 th - 8 th
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7.EE.A.1
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, 7.EE.B.4
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Factoring Polynomials Jeopardy Algebra (Review Game)
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Math Made Easy with Ms D
Factoring Polynomials Jeopardy Game – Algebra 1 Review ActivityMake factoring fun with this interactive Jeopardy-style game! Perfect for whole-class review, test prep, or a fun end-of-unit activity. ✅ Covers:Greatest Common Factor (GCF)Difference of SquaresTrinomialsGrouping Polynomials How to Play:Split students into teams and have them compete for points. One player from each team comes up to answer a question—whoever gets it right wins the points! Great for encouraging participation and r
7 th - 12 th
Algebra
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Factoring Polynomials Greatest Common Factor Differentiated Worksheets Algebra 1
Created by
mrscasiasmath
These no-prep factoring polynomials by GCF worksheets include two differentiated and leveled versions. Students factor polynomials by finding and factoring out the greatest common factor. These worksheets are perfect for your students no matter their level as the differentiation is already done for you! This product (PDF) includes:● Worksheet Version A - 12 easier problems ● Worksheet Version B - 12 harder problems ● Answer Keys for both versions There a total of 24 different
8 th - 9 th
Algebra, Math, Other (Math)
CCSS
HSA-SSE.A.2
Also included in:Factoring Polynomials Algebra 1 Differentiated Worksheets BUNDLE
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Greatest Common Factor Self Checking Millions Activity
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Math and Motivation
Engage your upper elementary or middle school math students by using this self-checking activity to practice finding the greatest common factor of two numbers. The self-checking feature in this resource helps students identify questions they may be struggling with, as well as advocate for help when they are unable to correct their errors in solving. How does this work: Students are able to see all 12 questions upon opening the assignment and will solve each question in order to see if they fin
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6.NS.B.4
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10935 | https://www.geeksforgeeks.org/aptitude/puzzle-arranging-balls-in-a-row/ | Puzzle - Arranging Balls in a row - GeeksforGeeks
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Puzzle - Arranging Balls in a row
Last Updated : 22 Feb, 2023
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We have a bag currently with 4 Red, 4 Blue,and4 Green Balls. Now a game wants us to arrange 12 balls in a row, such that it follows the below two conditions:
No two balls of the same color can be adjacent to each other.
The first and last balls in the row must be of different colors.
Can you find a solution to this puzzle?
4 Red, 4 Blue, and 4 Green Balls we have.
Solution:
First, we must determine the possible combinations of the first and last balls in the row that meet the condition that they must be of different colors. There are three possible combinations: Red-Green, Green-Blue, and Blue-Red.
Now, we can approach the problem by first arranging the 4 red balls, 4 blue balls, and 4 green balls in separate groups. Then, we can arrange the groups of balls such that no two balls of the same color are adjacent to each other.
Here's a possible arrangement of the balls:
Red - Green - Blue - Green - Red - Blue - Red - Green - Blue - Green - Red - Blue
One of the possible solutions - "RGBGRBRGBGRB"
In this arrangement, the first and last balls are Red and Blue, respectively, which is one of the possible combinations that meet the condition.
We can verify that this arrangement meets the condition that no two balls of the same color are adjacent to each other:
There are no two adjacent Red balls, Green balls, or Blue balls.
The first ball is Red and the second ball is Green, so the first and second balls are of different colors.
The eleventh ball is Red and the twelfth ball is Blue, so the eleventh and twelfth balls are of different colors.
Therefore, the arrangement meets both conditions.
There may be other possible arrangements that meet the conditions of the puzzle, but this is one example of a solution.
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10936 | https://www.youtube.com/watch?v=ZkGMe-Y-V0U | Irodov 1.249 Rotational Motion- problem-Irodov 1.249
Prakash Academy
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Irodov 1.249 Rotational Motion- problem-Irodov 1.249
5 comments
Transcript:
okay let us discuss this problem so this ISS a uniform disc of radius R is a spin to angular velocity Omega so I have a dis and this dis is rotating with angular velocity Omega and then carefully placed on a horizontal surface so what we have so this is my horizontal surface and on this horizontal surface there is a dis that is initially rotating with angular velocity Omega so let us say this is the axis by which initially it is rotated to by angular velocity Omega and now this is this dis is kept over a horizontal surface now question says how long the disk will be rotating if the friction coefficient between dis and the surface is Mu so friction for coefficient is given to you you have to find what is the time after which this will stop so this means final Omega becomes zero so what is that an initial Omega let us call this as Omega KN initial Omega is given to you Omega so this starts the motion so this dis starts the mo motion with Omega not and final angular velocity becomes zero what should be the time so you see if somehow if I apply this equation Omega isal to Omega + Alpha T and in this expression I know the Omega I know Omega if somehow I can calculate Alpha I can get the time so now Alpha is basically deceleration in this case why there will be deceleration deceleration because between this and this surface because there are some friction forces are acting so this dis has a tendency to go this direction so if I take if I consider any point so let me make another diagram I think all of you are able to see or if you see here itself so you see if you consider any a small portion of dis let us consider a ring of the disk so inside the dis if I consider ring all Point has a tendency to move this side but the horiz the horizontal table that is a static so the points on this these are the points that try to slip this side so there will be friction force in this direction so you see at all point of disk so let us say this is my Elemental ring so at all point of ring there will be some friction force that will be basically directed tangentially and this Omega is in this direction so this will act in the opposite direction are you getting and this friction force will be mu let us say the mass of this ring is DM so friction force will be mu n and n will be DM into Z so this is the friction Force are you getting now what is the torque of this friction Force about Center so I can write D to let us say this ring as distance of X so from origin this is X distance so I will have torque = to Mu mg D mg basically into X are you getting or not I can make another diagram if you have some problem so this is my Elemental dis this is my X and at X I have considered thickness of DX so this is my DX and this is my Elemental ring are you getting or not so what will be the let us say mass is DM I can find what is the DM DM will be let us a mass per so total mass of dis is M so it is given to you let us assume total mass is M so M by < r² that will give you mass per n area that is Sigma so now in this case I can find DM is Sigma into 2 pix into DX so this is my DM are you getting now D to will be mu DM DM is 2 pi Sigma X DX so this is Mu DM into Z intox now see D is basically variable so let us write 2 pi mu Sigma z x into X x² DX now you see this torque D to is not constant if you have uh increased the X torque will increase so this is a function of x if I integrate from 0 to R I will get the total torque so what you will have 2 pi mu and sigma if I want I can write Sigma is nothing but M by pi r² are you getting so this is a small M let us say a small M Pi r² are you getting or not so this is 2 pi mu Sigma is M by P R square I have already written into Z and x² will be xq by 3 and this will be r q by3 so what you will have Pi Pi goes out so you will have 2x3 R square R square goes here so we will have mu mzr so this is the torque that is acting now I can write equation that is torque is equals to I Alpha so if I use that equation so this torque now this torque is basically negative so this try to dis accelerate so I can write torque is equal to I Alpha so 2x3 mu mzr this is the torque is equal to I I is m² by 2 because this is a dis so I can assume this is Mr Square by 2 and this is Alpha but because Alpha is decelerating so let us plug the value of minus here because this torque will decrease the alpha so this torque is basically negative so I can either put the negative sign this side or this side so from here I can find Alpha so what will be the value of alpha so r r goes out so we will have 2 into 4 by 3 and then mu will have G also you will have and one R this side so this divided by R minus this is equal to Alpha now question said what is the time after which the system will come into the rest so now I will apply y Omega is equal to Omega kn+ Alpha T final Omega system comes into rest so Omega finally is zero is equal to Omega Alpha is - 4x3 mu z ided r and this into T are you getting or not so what is the time that is coming from here so if I solve for this I will have time is equal to 3 omega R by 4 muz I think all of you are getting this result so you see in this problem one important point we have is to find out the torque of friction you see I cannot assume the t is because friction force is a fun function of X so this is my distance X and now the friction Force here at Elemental ring and it will have a different torque if the same friction Force acts here and friction Force also depends upon the area because Normal reaction so if you increase the X the friction force will increase so if you increase X friction force will increase and torque due to friction force will increase and that's why I need integration are you getting getting not so that's why I need integration for D to to find Total torque I cannot find Total torque without integration because torque is a variable it depends upon distance X from the center so this will give you total number of TS you can one can also find total number of TS so this is time if somebody ask what is the number of TS in that case you have to apply Theta is equals to Theta kn s equal to U that is okay so we will have U that is Omega into T minus half a t² in this let us put plus half Alpha t² so from here you will have Theta equal to Omega is given to you and time is given to you 3 omega R by 4 mu Z so 3 omega R by 4 muz this is the time after which the system comes to the rest plus half Alpha we have already calculated this is - 4 M by 3 R so - 4 mu Z by 3 R so this is my Alpha so half a t² and t is this so 3 omega R by 4 mu Z and if you square this term are you getting or not 4 mu Z and you can solve for this are you getting you can solve for this so this will be let us try to solve 3 omega s r by 4 mu Z and he will have minus so this is 3x 4 so 9 by 16 and I will have 9 by 16 4 into 2 so this is 2 so 3 y so we will have on the top 3 16 so you will have 8 so - 3 by 8 Omega not Square R square so 1 R goes out so we will have omega Square R Square 1 R goes out now mu sare G Square One music go so you will have at M so from here if you take at mic common or LCM you will have 6 - 3 that is 3 omega s r so this is the Theta number of Revolution if somebody asked then you will have Theta by 2 pi in that case you will have 3 omega s r by 2 pi that is 16 Pi mu Z so this is the number of Ts that we will make before coming to the rest are you getting so this is a quite good problem we'll discuss the next problem sh |
10937 | https://www.quora.com/What-is-the-highest-power-of-a-prime-number-in-a-factorial-value | Something went wrong. Wait a moment and try again.
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What is the highest power of a prime number in a factorial value?
Julien Côté
Bachelors in Mathematics, University of Waterloo (Expected 2028)
·
Apr 28
There is no highest power. Consider the prime 2. Then, given an even number k=2n, k! will have at least n powers of two in it, that is 2^n divides k!. This is because every second natural number is divisible by two, so if you are multiplying 2n of them, obviously you will have n factors of 2. So, for any number n, (2n)! will have 2 to some power of at least n.
Related questions
What is the highest power of 7 in 150 factorial?
What is the value of a negative factorial number?
What happens when a negative number is used in a factorial?
Do the prime factors and/or factorials of large numbers have any interesting properties (for example, numbers over 1000)?
What is the prime factorization of 15! (15 factorial)?
Alon Amit
PhD in Mathematics; Mathcircler.
·
Author has 8.7K answers and 172.8M answer views
·
Updated 5y
Related
About how many primes are in between 1 and 2014 factorial?
Instead of talking about that number, it’s better to talk about the ratio between this number and 2014! itself.
Why? Because both 2014! and the number of primes less than 2014! are huge numbers which are pretty incomprehensible to our feeble minds. On the other hand, we can easily grasp that ratio, and we can fairly easily estimate it, even in our heads. It’s about one in 13,000.
You see, when you’re talking about numbers that large, you can’t really visualize the individual magnitudes of 105000 and 104996. Yes, those are big numbers. Really big. How big? Er, bigger than the number of elem
Instead of talking about that number, it’s better to talk about the ratio between this number and 2014! itself.
Why? Because both 2014! and the number of primes less than 2014! are huge numbers which are pretty incomprehensible to our feeble minds. On the other hand, we can easily grasp that ratio, and we can fairly easily estimate it, even in our heads. It’s about one in 13,000.
You see, when you’re talking about numbers that large, you can’t really visualize the individual magnitudes of 105000 and 104996. Yes, those are big numbers. Really big. How big? Er, bigger than the number of elementary particles in the visible universe. How bigger? Er, bigger than the number of particles in the visible universe if every particle is replaced with a whole other visible universe. How bigger? Er, do that again…? And again and again and…?
On the other hand, it’s a useful observation that 105000is exactly 10,000 times larger than 104996. That is something we can process.
So this here question is similar. I could tell you that the number of primes less than 2014 factorial is around 105777. Does this mean anything to you? As it stands, you can’t even tell if this number is less or more than 2014 factorial! Of course, thanks to the context you know that it has to be less, but you wouldn’t have known that without this context, right? And now that you know that it’s less, can you tell how much smaller it is? Is it a tenth? Half? A billionth? A quadrillionth? No idea.
So the question really should be, what is the frequency of primes up to N? Is it every tenth number? Every one hundredth? We know the answer to that, and it’s really nice: about one in ln(N) numbers between 1 and N are prime. So it’s not every tenth number, or every one hundredth: it’s gradually thinning out. The larger N is, the lower the average frequency.
So now we just need to estimate ln(2014!). It’s useful to know that n! is around (n/e)n, so ln(n!) is around n(ln(n)−1). We now just need to estimate ln(2014), which I would do by being aware of ln(2)≈0.7, and multiplying that by 11 because 211=2048. This gives 7.7, when in fact it’s more like 7.6. No big deal. I’ll stick to my rough estimate.
So ln(2014!)≈2014×6.7 which is about 13,500. This lets me assess that the number of primes less than 2014! is about 2014! divided by 13,500, which is pretty darn close.
Alon Amit
PhD in Mathematics; Mathcircler.
·
Upvoted by
Michael L. Hall
, PhD Nuclear Engineering & Mathematics, North Carolina State University at Raleigh (1988) and
Paulina Jonušaitė
, is an MSc in Applied Mathematics. · Author has 8.7K answers and 172.8M answer views
·
9y
Related
What prime number factorial have exactly one billion trailing zeros?
I'm not sure why anyone would need a prime number whose factorial has exactly one billion trailing zeros in its decimal notation, but if you need one, you really have just one option which is the prime "four billion and nineteen".
p=4×109+19
Here's why. The number of trailing zeros of n! is determined by the number of times n! is divisible by 5, since there's way more factors of 2 than factors of 5 in a factorial.
How many 5's are there? Up to and including n there are ⌊n/5⌋ multiples of 5, and then ⌊n/25⌋ multiples of 25, and then ⌊n/125⌋ multiple
I'm not sure why anyone would need a prime number whose factorial has exactly one billion trailing zeros in its decimal notation, but if you need one, you really have just one option which is the prime "four billion and nineteen".
p=4×109+19
Here's why. The number of trailing zeros of n! is determined by the number of times n! is divisible by 5, since there's way more factors of 2 than factors of 5 in a factorial.
How many 5's are there? Up to and including n there are ⌊n/5⌋ multiples of 5, and then ⌊n/25⌋ multiples of 25, and then ⌊n/125⌋ multiples of 125, and so on. Each of these adds another factor of 5 to n!, so the grand total is
val5(n!)=⌊n/5⌋+⌊n/25⌋+⌊n/125⌋+….
For example, how many 0's are there at the end of the decimal expansion of 400!? We first divide 400 by 5 to get 80, then again to get 16, then again to get 3 (remember we are rounding down), and here it ends. So 400! ends in exactly 99 zeros.
You can see that numbers like 4000! and 40000! and so on have a number of trailing zeros that's just shy of a thousand or ten thousand etc. Indeed a quick calculation shows that four billion factorial is divisible by 5 exactly 999,999,997 times.
So to get a billion zeros we need three more 5′s, which we can get by moving up to four billion and something between and . Adding will already get you past a billion zeros.
The only two numbers in this list which could possibly be prime are and . But the first of these is actually divisible by , so it's out.
There's no quick way I can see to confirm the primality of four billion and 19, but a quick computer program verifies that it is. So, there's your answer.
CMA K S Narayanan
Cost Accountant having 24 years Telecom Experience
·
Author has 7K answers and 13.4M answer views
·
7y
Related
What is the highest power of 10 that divides 1000 factorial?
Factors of 10 are 2 & 5
Find the number of factors of 2 & 5 in 1000!
We see there are 249 powers of 5 and 994 powers of 2 in 1000!.
These 249 powers of 5 will produce 249 zeroes when multiplied with powers of 2.
So, highest power of 10 that divides 1000! is 10^249
:-)
Factors of 10 are 2 & 5
Find the number of factors of 2 & 5 in 1000!
We see there are 249 powers of 5 and 994 powers of 2 in 1000!.
These 249 powers of 5 will produce 249 zeroes when multiplied with powers of 2.
So, highest power of 10 that divides 1000! is 10^249
:-)
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Eashan
I can manipulate numbers for mathematical purposes.
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What is the highest power of 56 in 1000 factorial?
56 power 164
This is the highest power of 56 that you can get in 1000!.
56 is 7×(2^3).
1000!=1000×999×998×…4×3×2×1.
When you multiply these numbers, you have 142 multiples of 7, 20 multiples of 49 (7^2) and 2 multiples of 343 (7^3).
(Multiples of 7 are 7,14,21,…994,which makes a total of 142)
(Multiples of 49 are 49, 98….980,which makes a total of 20)
(Multiples of 343 are 343 and 686).
When they get multiplied, we get 7^142 ×7^20 ×7^2 .
Note that we take multiples of 49 as 77, once as a multiple of 7 and once as a multiple of 49 so we dont have to calculate 20×2 as the total 7s, and multiples of 343
56 power 164
This is the highest power of 56 that you can get in 1000!.
56 is 7×(2^3).
1000!=1000×999×998×…4×3×2×1.
When you multiply these numbers, you have 142 multiples of 7, 20 multiples of 49 (7^2) and 2 multiples of 343 (7^3).
(Multiples of 7 are 7,14,21,…994,which makes a total of 142)
(Multiples of 49 are 49, 98….980,which makes a total of 20)
(Multiples of 343 are 343 and 686).
When they get multiplied, we get 7^142 ×7^20 ×7^2 .
Note that we take multiples of 49 as 77, once as a multiple of 7 and once as a multiple of 49 so we dont have to calculate 20×2 as the total 7s, and multiples of 343 are taken 3 times similarly.
So a total of 164 7s will require 164 8s.
2 taken thrice in a product makes 8, so 3×164 =492 is the number of 2s required
Total numbers divisble by 2 between [1,1000] are 1000/2=500.
As we have 500 2s, which is greater than 492, each 7 will get 3 2s.
This would make 56 power 164 the largest power.
John K WilliamsSon
BASIC programming expert teaching myself PYTHON !! at Home Office, Retired (2018–present)
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3y
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What is the highest number of primes a number can be divisible by?
The composite number: [math]2×3×5×7×11×13×17×19×23×29=6469693230[/math] has ten prime factors, so it can be divided by ten different prime numbers.
But you can double the number of primes that my number can be divided by, by multiplying my number times the next ten prime numbers.
557940830126698960967415390 = 2·3·5·7·11·13·17·19·23·29·31·37·41·43·47·53·59·61·67·71
Then your neighbor across the street can find a better answer by multiplying your number times ten more prime numbers.
31610054640417607788145206291543662493274686990 = 2·3·5·7·11·13·17·19·23·29·31·37·41·43·47·53·59·61·67·71·73·79·83·89·97·101·103·10
The composite number: [math]2×3×5×7×11×13×17×19×23×29=6469693230[/math] has ten prime factors, so it can be divided by ten different prime numbers.
But you can double the number of primes that my number can be divided by, by multiplying my number times the next ten prime numbers.
557940830126698960967415390 = 2·3·5·7·11·13·17·19·23·29·31·37·41·43·47·53·59·61·67·71
Then your neighbor across the street can find a better answer by multiplying your number times ten more prime numbers.
31610054640417607788145206291543662493274686990 = 2·3·5·7·11·13·17·19·23·29·31·37·41·43·47·53·59·61·67·71·73·79·83·89·97·101·103·107·109·113
But whatever you do, do not ask your grandfather who writes computer programs, because he might tell you that this number:
678629608419755514953266004896957820972161078160377361970324401521111792080121479864721936071815069425907219215791646774510151130705671056416094404541167439287735488353736963531288441938981088407654256240451529081607242659988552012480001287133802278572298314458227654950008738955663072953766341488209509227159933381319371567666804963833249523370831655778314080604712246344649628072459805028063160913071005795183295590443375991860551286230065601580359306757988823124262933259305966372664091680948986620887898883461227980556352852601733860114246410887151983493540958775872577571329277597701163671587052591794386970584444752423596023268793021595936555282977008138833858707329536639661377014042325817639809356799596347944462538427778375525904007169834445567450156949173690701738594584875536885957881452438269676946038980597530032671949818526703398270502591574889228837327819994695664173214894557366363343168494592437205324652573516528943874382178600874878024643322031797588414862315122048846223291257900756812820806739795819803783834366449110996030165071920678407750230118672657378102915524688059208755108467225277065866103666795739208709483959119145497860116133180335757702319385020561042517429031288526721801002679092058170909635701703382390753126302005323612316630558515594616479515096004453718500060291836932140612551722161051067379805065002788004096547708243964735215852734827632098700684466036892770059458754742495711074949314613079781545359495019757827538184361308856825999513366660884541936335491466045305322353749545362962683762333460252556042583248154845846566948014188971651057314058851019340282646752239847232045463969939303431371658607220786663205842510175297602195433569758123945251755043878718459161595137019904240640962465899496512410906852088532419874383895656303779315512987369934711061777117329635461569528504994783413643047392160871963795694958724055597996525917454740621526108635321204763824742430011606570436994644169759611263012712375861911682673548369764923418748711813157811279361700331599397588282864147719911156923709896847720603482450047076226728760035577410722701184878333100234780537897462936378382079055966277885316116887834607362114802378706815302650083359076798475953780285866955566883261644281750278358349579977889429105626865087038835977930842352223971442123281019745568694318200865586150762549114357677130353514342849892002965601064686292493671204318349298134598116662388818407027989992498970986262856712232401426575229549744739851333516937170071337085705197690437625282926914858257689908846227286051735284322402597283976180484905838486513162987381659809287870592690902387482033879184700359561190209417618607868793293476867624464497838299321267571049753373623085351455438610076341961842557148160442782839736179329056237366708383637405663196770746783100179128651460773512143616414356080816160456447832856222804164147618891013658880373227849181446498052320436905124576367614898030410445386643656246089772967461562154147355201124738052009172637452710027640262529821855681129322547617443299372089380860873141895162966481252930360380537684913059090577224188204179681342669502124011214018434733385892140553307905100266308832521127607403573729242486985024795253305646999864066282626291530104297235324933472771821035277094700384260778312268190937365143307612108901729316774669077441981239149913617114331308200242717771235228048768133852203532299832810943137983635951570
is divisible by one thousand different prime numbers. Here are all of its multiplicands:
2·3·5·7·11·13·17·19·23·29·31·37·41·43·47·53·59·61·67·71·73·79·83·89·97·101·103·107·109·113·127·131·137·139·149·151·157·163·167·173·179·181·191·193·197·199·211·223·227·229·233·239·241·251·257·263·269·271·277·281·283·293·307·311·313·317·331·337·347·349·353·359·367·373·379·383·389·397·401·409·419·421·431·433·439·443·449·457·461·463·467·479·487·491·499·503·509·521·523·541·547·557·563·569·571·577·587·593·599·601·607·613·617·619·631·641·643·647·653·659·661·673·677·683·691·701·709·719·727·733·739·743·751·757·761·769·773·787·797·809·811·821·823·827·829·839·853·857·859·863·877·881·883·887·907·911·919·929·937·941·947·953·967·971·977·983·991·997·1009·1013·1019·1021·1031·1033·1039·1049·1051·1061·1063·1069·1087·1091·1093·1097·1103·1109·1117·1123·1129·1151·1153·1163·1171·1181·1187·1193·1201·1213·1217·1223·1229·1231·1237·1249·1259·1277·1279·1283·1289·1291·1297·1301·1303·1307·1319·1321·1327·1361·1367·1373·1381·1399·1409·1423·1427·1429·1433·1439·1447·1451·1453·1459·1471·1481·1483·1487·1489·1493·1499·1511·1523·1531·1543·1549·1553·1559·1567·1571·1579·1583·1597·1601·1607·1609·1613·1619·1621·1627·1637·1657·1663·1667·1669·1693·1697·1699·1709·1721·1723·1733·1741·1747·1753·1759·1777·1783·1787·1789·1801·1811·1823·1831·1847·1861·1867·1871·1873·1877·1879·1889·1901·1907·1913·1931·1933·1949·1951·1973·1979·1987·1993·1997·1999·2003·2011·2017·2027·2029·2039·2053·2063·2069·2081·2083·2087·2089·2099·2111·2113·2129·2131·2137·2141·2143·2153·2161·2179·2203·2207·2213·2221·2237·2239·2243·2251·2267·2269·2273·2281·2287·2293·2297·2309·2311·2333·2339·2341·2347·2351·2357·2371·2377·2381·2383·2389·2393·2399·2411·2417·2423·2437·2441·2447·2459·2467·2473·2477·2503·2521·2531·2539·2543·2549·2551·2557·2579·2591·2593·2609·2617·2621·2633·2647·2657·2659·2663·2671·2677·2683·2687·2689·2693·2699·2707·2711·2713·2719·2729·2731·2741·2749·2753·2767·2777·2789·2791·2797·2801·2803·2819·2833·2837·2843·2851·2857·2861·2879·2887·2897·2903·2909·2917·2927·2939·2953·2957·2963·2969·2971·2999·3001·3011·3019·3023·3037·3041·3049·3061·3067·3079·3083·3089·3109·3119·3121·3137·3163·3167·3169·3181·3187·3191·3203·3209·3217·3221·3229·3251·3253·3257·3259·3271·3299·3301·3307·3313·3319·3323·3329·3331·3343·3347·3359·3361·3371·3373·3389·3391·3407·3413·3433·3449·3457·3461·3463·3467·3469·3491·3499·3511·3517·3527·3529·3533·3539·3541·3547·3557·3559·3571·3581·3583·3593·3607·3613·3617·3623·3631·3637·3643·3659·3671·3673·3677·3691·3697·3701·3709·3719·3727·3733·3739·3761·3767·3769·3779·3793·3797·3803·3821·3823·3833·3847·3851·3853·3863·3877·3881·3889·3907·3911·3917·3919·3923·3929·3931·3943·3947·3967·3989·4001·4003·4007·4013·4019·4021·4027·4049·4051·4057·4073·4079·4091·4093·4099·4111·4127·4129·4133·4139·4153·4157·4159·4177·4201·4211·4217·4219·4229·4231·4241·4243·4253·4259·4261·4271·4273·4283·4289·4297·4327·4337·4339·4349·4357·4363·4373·4391·4397·4409·4421·4423·4441·4447·4451·4457·4463·4481·4483·4493·4507·4513·4517·4519·4523·4547·4549·4561·4567·4583·4591·4597·4603·4621·4637·4639·4643·4649·4651·4657·4663·4673·4679·4691·4703·4721·4723·4729·4733·4751·4759·4783·4787·4789·4793·4799·4801·4813·4817·4831·4861·4871·4877·4889·4903·4909·4919·4931·4933·4937·4943·4951·4957·4967·4969·4973·4987·4993·4999·5003·5009·5011·5021·5023·5039·5051·5059·5077·5081·5087·5099·5101·5107·5113·5119·5147·5153·5167·5171·5179·5189·5197·5209·5227·5231·5233·5237·5261·5273·5279·5281·5297·5303·5309·5323·5333·5347·5351·5381·5387·5393·5399·5407·5413·5417·5419·5431·5437·5441·5443·5449·5471·5477·5479·5483·5501·5503·5507·5519·5521·5527·5531·5557·5563·5569·5573·5581·5591·5623·5639·5641·5647·5651·5653·5657·5659·5669·5683·5689·5693·5701·5711·5717·5737·5741·5743·5749·5779·5783·5791·5801·5807·5813·5821·5827·5839·5843·5849·5851·5857·5861·5867·5869·5879·5881·5897·5903·5923·5927·5939·5953·5981·5987·6007·6011·6029·6037·6043·6047·6053·6067·6073·6079·6089·6091·6101·6113·6121·6131·6133·6143·6151·6163·6173·6197·6199·6203·6211·6217·6221·6229·6247·6257·6263·6269·6271·6277·6287·6299·6301·6311·6317·6323·6329·6337·6343·6353·6359·6361·6367·6373·6379·6389·6397·6421·6427·6449·6451·6469·6473·6481·6491·6521·6529·6547·6551·6553·6563·6569·6571·6577·6581·6599·6607·6619·6637·6653·6659·6661·6673·6679·6689·6691·6701·6703·6709·6719·6733·6737·6761·6763·6779·6781·6791·6793·6803·6823·6827·6829·6833·6841·6857·6863·6869·6871·6883·6899·6907·6911·6917·6947·6949·6959·6961·6967·6971·6977·6983·6991·6997·7001·7013·7019·7027·7039·7043·7057·7069·7079·7103·7109·7121·7127·7129·7151·7159·7177·7187·7193·7207·7211·7213·7219·7229·7237·7243·7247·7253·7283·7297·7307·7309·7321·7331·7333·7349·7351·7369·7393·7411·7417·7433·7451·7457·7459·7477·7481·7487·7489·7499·7507·7517·7523·7529·7537·7541·7547·7549·7559·7561·7573·7577·7583·7589·7591·7603·7607·7621·7639·7643·7649·7669·7673·7681·7687·7691·7699·7703·7717·7723·7727·7741·7753·7757·7759·7789·7793·7817·7823·7829·7841·7853·7867·7873·7877·7879·7883·7901·7907·7919
But I’m better than your grandfather. I found a composite integer that is divisible by fifty thousand different prime numbers. You can see that number by running the Python program you’ll find by clicking this link:
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Brian Overland
Tutor, author, computer programmer and tech writer
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7mo
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Can a number have only prime factors greater than six (or five)?
Of course. In fact, a cornerstone of modern cryptography to use two large primes to produce a “key” that way. Two individuals each have one of the two prime numbers… but only by multiplying BOTH together does the system produce the composite number needed to decrypt a particular code.
And here is the trick — because two large PRIME numbers are involved, there is no other way to produce this composite number.
Beyond that, there are infinite examples of composite numbers that exclusively have prime factors greater than six, and none smaller. A simple example is 77, produced by multiplying 7 and 11
Of course. In fact, a cornerstone of modern cryptography to use two large primes to produce a “key” that way. Two individuals each have one of the two prime numbers… but only by multiplying BOTH together does the system produce the composite number needed to decrypt a particular code.
And here is the trick — because two large PRIME numbers are involved, there is no other way to produce this composite number.
Beyond that, there are infinite examples of composite numbers that exclusively have prime factors greater than six, and none smaller. A simple example is 77, produced by multiplying 7 and 11 together.
Let us Find X
An online community for Mathematics lovers
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2y
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What is the highest power of 7 in a 2000 factorial?
The highest power of any prime number [math]p[/math] in [math]n![/math] can be found by repeated division of [math]n[/math] by [math]p[/math].
Let us now apply the same process to solve this question.
First, divide [math]2000[/math] by [math]7[/math]. Keep on dividing the quotients by [math]7[/math] till the result becomes less than [math]7[/math]. Ignore the decimal part. We just need tho quotient for subsequent divisions.
[math]\frac{2000}{7} = 285[/math]
[math]\frac{285}{7} = 40[/math]
[math]\frac{40}{7} = 5[/math]
Since [math]5[/math] is less than [math]7[/math], we stop here and add all those results.
[math]285 + 40 + 5 = 330[/math]
[math]\therefore[/math] The highest power of [math]7[/math] in [math]2000![/math] is [math]330[/math].
The answer is [math]330[/math].
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Brian Powell
B.S. in Mathematics & Education, East Carolina University (Graduated 1981)
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David Joyce
, Ph.D. Mathematics, University of Pennsylvania (1979) · Author has 1.6K answers and 2.1M answer views
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5y
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About how many primes are in between 1 and 2014 factorial?
If a number, n, is very large, then the number of primes, P, less than n is estimated by the formula: P = n/ln(n)
Therefore, the number of primes between 1 and 2014! can be closely estimated by the expression,
2014!/(ln(2014!)
The number, 2014!, is unfathomably large, and is 5.7255 x 10^5781. Then the number of primes less than5.7255 x 10^5781 is 5.7255 x 10^5781/ln(5.7255 x 10^5781) The Prime Number Theorem also indicates that the average (mean) distance between consecutive primes, d, in a number having 5781 digits is is d = ln(10^5781). A little algebraic manipulation:
d = ln(10^5781)
= 5781 ln(
If a number, n, is very large, then the number of primes, P, less than n is estimated by the formula: P = n/ln(n)
Therefore, the number of primes between 1 and 2014! can be closely estimated by the expression,
2014!/(ln(2014!)
The number, 2014!, is unfathomably large, and is 5.7255 x 10^5781. Then the number of primes less than5.7255 x 10^5781 is 5.7255 x 10^5781/ln(5.7255 x 10^5781) The Prime Number Theorem also indicates that the average (mean) distance between consecutive primes, d, in a number having 5781 digits is is d = ln(10^5781). A little algebraic manipulation:
d = ln(10^5781)
= 5781 ln(10)
= 5781 2.302585093
= 13,311
So, the average (mean) distance between all of the primes less than 2014! is estimated at 13,311.
The number of primes less than 2014! is 5.7255 x 10^5781/13,311 or 4.301 x 10^5777
Amitabha Tripathi
four decades of Number Theory and counting
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Author has 4.7K answers and 13.8M answer views
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5y
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How do I find the exponent of a prime number in an n factorial?
Legendre's formula - Wikipedia
If we denote by [math]\nu_p\big(n!\big)[/math] the highest power of a prime number [math]p[/math] that divides [math]n![/math], then
[math]\nu_p\big(n!\big) = \displaystyle \sum_{k=1}^{\infty} \left\lfloor \dfrac{n}{p^k} \right\rfloor[/math].
Example. The highest power [math]\nu_2\big(100!\big) [/math]of [math]2[/math] dividing [math]100![/math] is
[math]\begin{align} \left\lfloor \dfrac{100}{2} \right\rfloor + \left\lfloor \dfrac{100}{4} \right\rfloor + \left\lfloor \dfrac{100}{8} \right\rfloor + \left\lfloor \dfrac{100}{16} \right\rfloor + \left\lfloor \dfrac{100}{32} \right\rfloor + \left\lfloor \dfrac{100}{64} \right\rfloor \ = 50 + 25 + 12 + 6 + 3 + 1 \ =[/math]
Legendre's formula - Wikipedia
If we denote by [math]\nu_p\big(n!\big)[/math] the highest power of a prime number [math]p[/math] that divides [math]n![/math], then
[math]\nu_p\big(n!\big) = \displaystyle \sum_{k=1}^{\infty} \left\lfloor \dfrac{n}{p^k} \right\rfloor[/math].
Example. The highest power [math]\nu_2\big(100!\big) [/math]of [math]2[/math] dividing [math]100![/math] is
[math]\begin{align} \left\lfloor \dfrac{100}{2} \right\rfloor + \left\lfloor \dfrac{100}{4} \right\rfloor + \left\lfloor \dfrac{100}{8} \right\rfloor + \left\lfloor \dfrac{100}{16} \right\rfloor + \left\lfloor \dfrac{100}{32} \right\rfloor + \left\lfloor \dfrac{100}{64} \right\rfloor \ = 50 + 25 + 12 + 6 + 3 + 1 \ = 97, \end{align}[/math]
and the highest power [math]\nu_5\big(100!\big)[/math] of [math]5[/math] dividing [math]100![/math] is
[math]\left\lfloor \dfrac{100}{5} \right\rfloor + \left\lfloor \dfrac{100}{25} \right\rfloor = 20+4 = 24[/math].
Therefore the highest power of [math]10[/math] dividing [math]100![/math] is [math]\min{24,97}=24[/math]. This is the same as saying that [math]100![/math] has [math]24[/math] trailing zeros. More generally, [math]n![/math] has [math]\nu_5\big(n!\big)[/math] trailing zeros.
WILLIAM SHAW
Honours degree 2nd class from St. Mary Magdalene Catholic School (Graduated 1963)
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Author has 453 answers and 91K answer views
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Updated 1y
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Is it possible for a prime number to have more factors than two?
By the definition of a prime number, a prime number is an integer, only divisible by unity and by itself, leaving in each case a remainder equal to zero. Dividing a prime number by some other prime number leaves an integer remainder which is non-zero. An example is 7/5 giving a quotient of one, unity and a remainder of two, since only integers are being considered, remembering that a division operation is defined as:
Dividend = Divisor x Quotient + Remainder or Remainder = Dividend - Divisor x Quotient.
The remainder is put over the divisor as a fraction. The remainder is always less then the di
By the definition of a prime number, a prime number is an integer, only divisible by unity and by itself, leaving in each case a remainder equal to zero. Dividing a prime number by some other prime number leaves an integer remainder which is non-zero. An example is 7/5 giving a quotient of one, unity and a remainder of two, since only integers are being considered, remembering that a division operation is defined as:
Dividend = Divisor x Quotient + Remainder or Remainder = Dividend - Divisor x Quotient.
The remainder is put over the divisor as a fraction. The remainder is always less then the divisor.
Example, 23/7 = 3 2/7. There is always a remainder which can be any of: 0, 1, 2, … [Divisor - unity]. An example is 391/23 = 17 and the remainder = 0.
Prime numbers cannot have prime factors which of their nature are whole numbers, integers, apart unity and the prime number itself.
The question is not precise and presumably only whole numbers are meant.
For example: 7 = Z^{2} + Z{1} + 1. That is Z^{2} + Z^{1]} - 6 = 0 Equation [ which is divisible by [Z - 2] giving the algebraic quotient: [Z + 3].
To factorize seven we need the zeros of the polynomial: Z^{2} + Z^{1} + 1 = 0 The substitution Z = 2 in this polynomial gives the value 7. Without knowing the zeros, but calling the zeros Z and Z, we have: Z^{2} + Z^{1} + 1 = [Z - Z] [Z - Z] Equation :. Then multiplying out the right hand side, we have:
Z^{2} + Z^{1} + 1 = Z^{2} - [Z[1} + Z] Z + Z [Z. From the theory of equations, the sum of the zeros, [Z + Z] is minus the coefficient of Z^{1} and product of the zeros, Z Z is the last term Z^{0} or unity. It is seen that Equation is an identity, that is Equation is valid for every value of Z.
The solutions of Equation substituted serially, one at a time, into Equation enable the prime seven to be factorized. In every case of factorizing a prime number, there are complex factors and their complex conjugates and in some cases a real irrational factor which when all multiplied together result in a r
Solving [a] Z^{2} + Z^{1} - 6 = 0 and solving [b] Z^{2} + Z^{1} + 1 = 0 enables prime seven to be factored. Equation [a] is divisible by two giving the algebraic quotient [Z + 3]
Equation [b] has the zeros - 1/2 + [i SQRT 3]/2 and - 1/2 - [i SQRT 3]/2
Factors of seven are: [2 - [ - 1/2 + [i SQRT 3]/2]] and [2 - [ - 1/2 - [i sqrt 3]/2]]
Multiplying the factors of seven gives: The other factors of seven:
[2 - [ -1/2 + [i SQRT 3]/2]]] x [2 - [ - 1/2 - [i SQRT 3]/2]]] which is
The other factors is [ - 3 - [ - 1/2 - [i SQRT 3]/2]] and [- 3 - [ -1/2 + [i SQRT 3]/2]]]. Multiplying the factors of seven gives: [ - 5/2 - i SQRT etc] x [ - 5/2 + i SQRT etc ] = 25/4 + 3/4 as with the other root, two.
[2 + 1/2 - [i SQRT 3]/2]] x [2 + 1/2 + [i SQRT 3]/2] becoming:
[5/2 - x [5/2 + which is [5/2 - [1/2] [i SQRT 3]] x [5/2 + [1/2] [i SQRT 3]]. This is [25/4 + 3/4] which is 28/4 which is 7.
Eleven can be written as the polynomial: 11 = Z^{3} + Z^{2} - Z^{1} + 1 This becomes
Z^{3} + Z^{2} - Z^{1} - 10 = 0 which is divisible by [Z - 2] giving the algebraic quotient:
Z^{2} + 3Z + 5 = 0 having the roots Z = - 3/2 + [i SQRT 11]/2 and Z = - 3/2 - [i SQRT 11]/2]
For example, the equation: [Z + 3] [Z - 2] = 0 which is Z^{2} + Z^{1} +1 = 0, does not give the value zero if Z is different from the roots - 3 and + 2. Thus this quadratic can be equated to seven for the correct value of Z.
The zeros of the polynomial: Z{3} + Z^{2} - Z^{1} + 1 = 0 are needed to complete the factorisation of eleven. The factors of 7 and eleven are complex. Primes between 16 and 32 which are 17, 19, 23, 29 and 31 might have real irrational and complex factors where N is even. See below.
Any uneven integer which includes the primes, 3, 5, 7, 11, … can be written as a polynomial similar to the above polynomials, as
Q[N] = Z^{N} + Z^{N - 1} + r Z^{N -2} + … + r[N-1] = 0 The uneven integer Q[N] lies between 2^{N} and 2^{N + 1} with N taking values: 1, 2, 3, … The r, … r[N - 1] exclusively take the values one or minus one, no other values. The above equation giving Q[N] is valid for Z = 2.
The substitution Z = - 2 is valid, but gives a different value for Q[N].
Subtracting Q[N] from the polynomial that represents the polynomial gives another polynomial having the divisor [Z - 2].
There is a lemma; Z^{K} - Z^{K - 1} - … - Z^{N} = Z^{N} which allows the two equations giving the factors as above to be augmented to a larger, but the same degree. The largest term Z^{2} in a quadratic can be augmented to Z^{3} - Z^{2} etc.
Does this answer the question as to whether a prime number has or has not more than two factors ?
There are factorisations of a prime number and the lemma gives an indefinite number of factorisations.
A simple final example - Using the lemma, prime three can be factorized. We have 3 = Z + 1, a polynomial. But 3 = Z^{2} - Z^{1} + 1. This polynomial is solved having the zeros Z and Z and left in the form of the identity:
Z^{2} - Z^[1} + 1 = [Z - Z] [Z - Z] Equation .
Then the equation Z^{2} - Z{1} + 1 = 3 Equation is divided by [Z - 2] and the solutions of Equation are substituted, one at a time, into right hand side of Equation giving the factors of prime three. The solutions of Equation are found to be complex.
In point of fact any prime number can be factorized using any polynomial.
An example of this is the polynomial: X^{2} + 3 X^{1} + 2 = 29. The polynomial set to value zero instead of 29 has the roots: X, X = [ - 3 +/ - SQRT[9 - 4x1x2]/2
X = -3/2 + 1/2 = -1 and X = - 3/2 - 1/2 = -2.
In the identity: X^{2} + 3X^{1} + 2 = [X + 1] [X + 2], any value will do for X. The values of X that make the right hand side of the identity equal to 29 can be put into the left hand side of the identity
Setting the polynomial in X to 29 gives: X^[2} + 3 X^{1} + 2 = 29 ,
That is X^{2} + 3X^{1} - 27 = 0 One of the solutions giving 29 are: - 3/2 + [SQRT[9 + 4 x 27]/2
= - 1.5 + 5.4083265 … = 3.9083265 Putting this value of X into the identity
gives: [3.9083265 + 1] [3.9083265 + 2] = 4.9083265 x 5.9083265 = 28.999995 to the accuracy of the square root.
The factors of 29 are: 4.9083265 and 5.9083265. The factors of a prime number can be any factors, real or complex or one real together with complex and their complex conjugate factors.
However, the above analysis presented with indefinite Z systematises the factorization of prime numbers providing at least one factorization and with the lemma, more but a finite number of factorizations.
Balázs Iván József
Master in Mathematics, Eötvös Loránd University (Graduated 1983)
·
Upvoted by
David Joyce
, Ph.D. Mathematics, University of Pennsylvania (1979) and
Klaus Ole Kristiansen
, M.Sc. Mathematics, University of Copenhagen (1992) · Author has 5.2K answers and 1.8M answer views
·
4y
Related
What is the largest prime factor of the factorial 49?
49! = 1 . … . 49
Now 49 = 7 7
48 = 3 16 = 2 2^4
47 is a prime, and hence the largest prime factor of 49!
Ege Erdil
Very interested in mathematics
·
Author has 273 answers and 537.9K answer views
·
9y
Related
What is the unique prime number which has the number of digits of its factorial equal to the number itself?
The number of digits in a number base 10 is given by [math]\lfloor \log_{10}(n) \rfloor + 1[/math]. Now, we have the following estimate:
[math]\displaystyle \log(n!) = \sum_{k=1}^{n} \log(k) > \int_{1}^{n} \log(x)\, dx = n\log(n) - n + 1[/math]
which entails
[math]\displaystyle \log_{10} (n!) = \frac{\log(n!)}{\log(10)} > \frac{n\log(n) - n + 1}{\log(10)}[/math]
For [math]n > 10e \approx 27.1[/math], we have [math]\log(n) > \log(10) + 1[/math] and
[math]\displaystyle \lfloor \log_{10}(n!) \rfloor + 1 > \log_{10}(n!) > n + \frac{1}{\log(10)} > n[/math]
Therefore, such a number [math]n[/math] must satisfy [math]n \leq 27[/math]. A simple brute force search then yields that [math]n=23[/math] is the only prime soluti
The number of digits in a number base 10 is given by [math]\lfloor \log_{10}(n) \rfloor + 1[/math]. Now, we have the following estimate:
[math]\displaystyle \log(n!) = \sum_{k=1}^{n} \log(k) > \int_{1}^{n} \log(x)\, dx = n\log(n) - n + 1[/math]
which entails
[math]\displaystyle \log_{10} (n!) = \frac{\log(n!)}{\log(10)} > \frac{n\log(n) - n + 1}{\log(10)}[/math]
For [math]n > 10e \approx 27.1[/math], we have [math]\log(n) > \log(10) + 1[/math] and
[math]\displaystyle \lfloor \log_{10}(n!) \rfloor + 1 > \log_{10}(n!) > n + \frac{1}{\log(10)} > n[/math]
Therefore, such a number [math]n[/math] must satisfy [math]n \leq 27[/math]. A simple brute force search then yields that [math]n=23[/math] is the only prime solution. (Not very daunting, as there are 9 primes to check.)
Related questions
What is the highest power of 7 in 150 factorial?
What is the value of a negative factorial number?
What happens when a negative number is used in a factorial?
Do the prime factors and/or factorials of large numbers have any interesting properties (for example, numbers over 1000)?
What is the prime factorization of 15! (15 factorial)?
Can power values be in factorial form?
How do you calculate the highest power of a number in a factorial?
How do you find the number of factors in a factorial?
What is the highest value of consecutive primes that can be found in a number's factorial?
Can all factorial numbers be represented and the product of power sums of 2 and 3?
What can we call if the factorial of any number is prime as a prime number?
Can there be two different numbers whose prime factorization has the same number of factors as their factorial will have terms?
What is the factorial of a number until 100?
What is the prime number n (500 > n > 10) whose factorial minus 1 is also a prime number? Hint: it has 93 nines at the end.
Can a subfactorial be larger than a factorial if the same number is used?
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Circular Permutation
The document outlines a mathematics lesson plan on circular permutations for grade 10 students. The lesson plan aims to teach students to define circular permutation, solve problems involvin…
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LESSON PLAN IN MATHEMATICS GRADE 10 (
March 02, 2023
)
I.O b j e c t i v e s A.Co nt en t St an da rd:
The learners will demonstrates understanding of key concepts of combinatorics and probability.
B.Per for man ce Sta nd ard:
The learners will able to use precise counting tec hni que an d pro bab ilit y in for mul ati ng con clu sio ns and mak ing decisions.
C.Le ar ni ng Co mp et en ci es: (M 10 SP-I Il b-1)
solve prob lems invo lving permutations
D.Lea rni ng O bje cti ves:
At the end of the lesson, the s tudents should be a ble to:a.def ine cir cul ar p erm uta tio n
b.
solve problems involving circular permutation
c.
use the derived formula in solving problems involving clockwise and counter clockwise orders
d.
relate circular permutations in real-life situations.
I I.S u b j e c t M a t t e r A.T o p i c:
Circular Permutations
B.Ma te ri al s:
Laptop, PowerPoint, Chalk, Manila Paper, Marker
C.Re fe re nc es:
Mathematics Quarter 3 – Module 1
D.Val ues Int egr ate d:
Self-reliance and attentiveness
I I I.P r o c e d u r e A.Pre pa rat or y Activ itie s
1.P r a y e r 2.G re e t i n g s 3.Ch ec ki ng of A tt en da nc e
B.Dev elo pme nta l Activi tie s Motivation
The teacher will show a picture and let the students analyse what the picture is all about. Ask them what they have observed about the picture
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1 3 2
(b)
(a)
2 1 3
C.Ac ti vi ty
Group Activity:
LET’S THINK!Direc tion:
The teacher will divide the students into 4 groups and let them answer the question.
D.An al ys is
Answer the following ques tions:1.What have you obse rved in th e ac tivity? 2.What s trate gy did you us e to ge t the p ossib le way s?
E.Ab st ra ct io n CIRCULAR PERMUTATION
Circular permutation is the arrangement of objects in a circular manner. It is the total number of ways in which n distinct object can be arranged around a fixed circle defined as,
P
=
(
n
−
1
)
!
Analyze this:
Suppose it happens that (1) Jose, (2) Wally and (3) Paolo will visit you in your house, how can you arrange them in a round table if you will prepare them a snack?Th es e ar e 3 po ss ib le ar ra ng em en ts if we wi ll ar ra ng e th em in
clockwise
position:(1)Jose, (2) Wall y, (3) Pa olo(2)Wally, (3) Pao lo, (1) Jo se(3)Paol o, (1) Jos e, (2) Wal ly These are the 3 possible arrangements if we will arrange them in
counter clockwise
position:(1)Jose, (3) Pao lo, (2) W ally
In how many ways can 3 students seated in a circular/roun d table?
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(3) Paolo, (2) Wally, (1) Jose(2)Wally, (1) Jos e, (2) Pa olo Thus, the permutation of n objects arranged in a circle is
P
=
(
n
−
1
)
!
Wherein:
n
is the no. of objects To solve the given problem lets apply the formula
Given:
n= 3 (Jose, Paolo, Wally)
Solution:
P
=
(
n
−
1
)
!P
=
(
3
−
1
)
!P
=
2
!
P
=
2
×
1
P
=
2
Therefore, there are 2 possible ways we can arrange Jose, Wally and Paolo if they will seat in a round table.
There are two types of circular permutation:
a. When clockwise and counter-clockwise orders are different b. When clockwise and counter-clockwise orders are the same
a. When clockwise and counter-clockwise orders are different/ If t h e c l o c k w i s e a n d c o u n t e r-c l o c k w i s e o r d e r s C A N b e dis tin gui she d,
the n the total num ber of cir cu lar per mut ati ons of n elements taken all together is
P
n
=
n
−
1
!
Example 1:
Suppose 7 students are sitting around a circle. Calculate the number of permutations if clockwise and anticlockwise arrangements are different.
P
n
=
n
−
1
!
Su bs tit ut e th e va lu es in th e ab ov e fo rm ul a to ge t th e nu mb er of combinations:
P
n
=
7
−
1
!
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P
n
=
6
!
P
n
=
6
⋅
5
⋅
4
⋅
3
⋅
2
⋅
1
¿
720
Hence, there are 720 possible arrangements of 7 students around a circle, given the fact that clockwise and anticlockwise arrangements are different.
Example 2:
In how many ways can 8 people be seated at a round table?
Solution:
P
n
=
n
−
1
!P
n
=
8
−
1
!P
n
=
7
!P
n
=
7
⋅
6
⋅
5
⋅
4
⋅
3
⋅
2
⋅
1
P
n
=
5040
Hence, 5040 different combinations are possible of 8 balls in a circle,given the fact that the clockwise and anticlockwise arrangements are different.
b.
Observe the arrangement of different beads in a bracelet, keys on the key rings, and the like. The clockwise and the counter-clockwise ord ers are not disti ngu ish abl e. So,
when cloc kwise and coun ter-clo ckw ise ord ers are the sam e/ If the clo ckw ise an d cou nte r-clockwise orders CANNOT be distinguished,
then the total number of circular permutations of n elements taken all together is
P
=
n
−
1
!
2
(without lock) and Bu t if br ac el et s, ke y ri ng s, an d th e li ke ha ve a lo ck, th en th e permutation becomes linear and can be denoted as,
P
=
n!
2
(with lock)
where:n
represents the number of objects in a set
Example 1:
In how many ways can 5 different beads be arranged if:a. a bracelet has no lock?b. a bracelet has a lock?
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10939 | https://sites.unimi.it/molteni/research/papers-pdf/26-molteni-Cancellation_in_a_short_exponential_sum.pdf | CANCELLATION IN A SHORT EXPONENTIAL SUM G. MOLTENI Abstract. Let q be an odd integer, let τ be the order of 2 modulo q and let ξ be a primitive qth root of unity. Upper bounds for Pτ k=1 ξ2k are proved in terms of the parameters µ and ν when q diverges along sequences Sµ,ν for which the quotient τ/ log2 q belongs to the interval [µ, ν], with 1 ≤µ and ν close enough to 1.
Journal N. Theory 130(9), 2011–2027 (2010).
1. Introduction and results Notation. We denote by ⌊x⌋the integer part of x, by ♯A the cardinality of a set A and by ζ(x) the value of the Riemann zeta function at x. Moreover, several constants appear in this paper: in those inequalities where their numerical value appears explicitly, it is always rounded up or down in such a way to produce a correct statement.
Let q be an odd integer, let τ be the order of 2 modulo q and let ξ be a primitive qth root of unity. In this paper we deal with bounds for the sum (1) s(ξ) := τ X r=1 ξ2r.
This problem and its generalizations appear in many different contexts and are the subject of an intense research: for example see [2, 3, 4, 5, 8, 11, 12, 18] and the bibliography cited therein. Roughly speaking, upper bounds of type s(ξ) ≪ τ 1−δ for some positive and explicit constant δ have been proved whenever log τ ≫log q, i.e. when the sum contains sufficiently many terms with respect to the order of ξ. A considerably smaller cancellation is expected when the condition log τ ≫log q is violated. In fact, it can be proved that max ξ: ξq=1 ξ primitive {|s(ξ)|} ≥0.3 τ when q diverges along suitable sequences. In his study of the Linnik’s problem about the representability of even integers as a sum of two primes and N 2010 Mathematics Subject Classification. Primary 11 L 07.
1 2 G. MOLTENI powers of 2, Gallagher (see Lemma 3 of ) proved that there exists a positive constant c such that (2) |s(ξ)| ≤τ −c (see also Thm. 1 in and Eq. (4.2) in ). In the values of c and N are not explicitly given. More recently, H. Li, J. Liu., M. Liu and T. Wang [13, 14, 15, 16] have proved that N ≤1906 (N ≤200 under GRH), and an essential ingredient of [13, 14, 15] is an explicit version of the argument of Gallagher, see Lemma 4 in .
This lemma implies (2) with any c < sin2(π/8) = 0.146 . . . . In this paper we are concerned with the behavior of s(ξ) when q diverges along a sequence for which the quotient τ/L belongs to the range [µ, ν] with 1 ≤µ < ν and ν −1 small enough, where L denotes the integer part of log2 q. The interest for such a type of results comes from the fact that, according to the previous discussion, along these sequences we should get the smallest cancellation for s(ξ). A simple and typical example is the sequence q = 2n −1 for which τ = n = L + 1. The following examples are less trivial.
Example 1. Let m be an odd integer not of the form 2n −1. Denote by τm the order of 2 modulo m. Let q := (2τm −1)/m and finally let τq be the order of 2 modulo q. The number τq divides τm and is equal to τm when m2 < 2τm.
Indeed, let s := (2τq −1)/q. The equality 1 + mq = 2τm = (2τq)τm/τq = (1 + sq)τm/τq shows that if τq ̸= τm we must have mq > q2 implying that m2 ≥2τm.
Since m2 < 2ϕ(m) for every m > 5, the previous criterion shows that τq = τm = ϕ(m) whenever 2 is a primitive root modulo m and m > 5.
These facts suggest the following construction: 2 is a primitive root modulo 3k for every k, therefore we take m = 3k (for k > 1) and q := (2ϕ(m) −1)/m. Then τq = τm = ϕ(m) = 2m/3 and q = (2τq −1)/(3τq/2) implying that for these numbers we have τq = log2 q + log2 log2 q + O(1) = L + log2 L + O(1) as k diverges.
Example 2. For every couple of positive integers m, n with m > 1, let q := 2mn−1 2n−1 .
The order τq of 2 modulo q is mn (since the congruence 2mn = 1 (mod q) implies that τq divides mn and the congruence 2τq = 1 (mod q) implies that (2τq −1)(2n −1) ≥2mn −1, so that τq must be greater than (m −1)n).
Moreover, the inequalities 2(m−1)n < q < 2(m−1)n+1 prove that L = (m −1)n, hence for such numbers we have τq = mn = m m−1 L.
CANCELLATION IN A SHORT EXPONENTIAL SUM 3 Thus, when n →∞and m is fixed these numbers define a sequence for which τq ∼νL holds with ν = m/(m −1).
Example 3. Let q, m, n and τ be as in the previous example, but this time let n be fixed while m diverges. For such numbers we have τq = mn = L + n, so that along this sequence τq ∼L again, but this time the difference τq −L is constant.
A possible attack to our problem is via the Vinogradov’s method (see ).
Very roughly speaking, this method provides a set of technics allowing one to obtain upper bounds for exponential sums via the study of the cardinality of sets of representations of integers as sum of numbers taken in a suitably chosen and fixed set. In this sense it is not surprising that one can deduce bounds for (1) from bounds for the number of representations of an integer as sum of powers of two. The following theorem represents an explicit and simple realization of this idea; here U(ℓ, k) denotes the number of representations of ℓas sum of k powers of two (see Section 2): Theorem 1. Let ϱ be a constant such that maxℓ{U(ℓ, k)} ≤(ϱk)k · kO(1) holds for k large enough. Let µ, ν be positive constants with 1 ≤µ < ν. Let hµ,ν(ϱ, x) := −ν −1/2 µ x2 log 2 + x log(x/ϱ) and let cµ,ν(ϱ) := maxx≥1{hµ,ν(ϱ, x)}, which exists and is positive when ν−1/2 µ < λ := (eϱ log 2)−1. Suppose that q diverges along a sequence S for which τ/L ∈ [µ, ν] with ν−1/2 µ < λ and where L = ⌊log2 q⌋. Then (3) max ξ: ξq=1 ξprimitive {|s(ξ)|} ≤τ −cµ,ν(ϱ) + oϱ,µ,ν(1).
It is evident that this proposition is useful only if we have an explicit value for ϱ and Section 2 is devoted to the proof of the following fact.
Theorem 2. For every k, maxℓ{U(ℓ, k)} ≤2.62 · (0.646661 k)k · k1/2.
In order to appreciate this result, we mention here also a second result that we will prove in the same section.
Theorem 3. For every k, maxℓ{U(ℓ, k)} ≫(0.644591 k)k.
Theorem 2 shows that 0.646661 is an admissible value for ϱ in Theorem 1.
This value gives λ = (eϱ log 2)−1 > 0.8 so that the bound in (3) applies to all sequences in Examples 1–3. For the sequence with m = 2 in Example 2 our theorem predicts the cancellation c2,2(ϱ) ≥0.1809 which is already better than that one previously known; the cancellation predicted by Theorem 1 for 4 G. MOLTENI every sequence in Example 2 becomes as better as greater is m and reaches its best (largest) result c1,1(ρ) in the limit m →∞. Besides, c1,1(ρ) is also the cancellation which is predicted for each sequence with τ/L →1 (for example the sequences in Examples 1 and 3); since c1,1(ρ) ≥1.7465, we deduce that for these sequences (4) max ξ: ξq=1 ξ primitive {|s(ξ)|} ≤τ −1.7465 + o(1), which is a sharp improvement on all previously known bounds. This cancella-tion is the strongest we can recover from Theorem 1.
The argument proving Theorem 1 produces an explicit bound for the little-o term in (4) and in the more general (3) whenever the behavior of τ with respect to L is explicitly known. For example, using the full strength of Theorem 2 this approach shows that if q = 2n −1 then max ξ: ξq=1 ξ primitive {|s(ξ)|} ≤n −1.7465 + 33.5 log n n .
This bound is non-trivial for every n > 87. We do not give here the details of its proof, the interested reader will be able to produce himself the necessary computations by following the argument in Section 3.
A final remark about the Linnik’s problem. It is possible that our result and the arguments in [14, 15] produce a bound for N lower than 1906, but we believe that such improvement will not overcome the best results (N ≤8 unconditionally and N ≤7 under GRH) that Heath-Brown and Puchta and Pintz and Ruzsa have obtained with different approaches that do not involve bounds for (1).
The paper is organized as follows: in Section 2 we first prove some facts mainly of combinatorial flavor about the representations of integers as sum of 2-powers, then we prove Theorems 2 and 3. The proof of Theorem 1 is given in Section 3.
2. Combinatorial tools Given two positive integers k and ℓ, we call k-representation of ℓa string (n1, . . . , nk) of non-negative integers such that Pk j=1 2nj = ℓ, where strings differing by the order are considered as distinct.
Moreover, we denote by U(ℓ, k) the number of k-representations of ℓ: U(ℓ, k) := ♯{(n1, . . . , nk) ∈Nk : k X j=1 2nj = ℓ}.
Let σ(ℓ) be the Hamming weight of ℓ, i.e. the number of 1’s appearing in the binary representation of ℓ, so that ℓ= Pσ(ℓ) j=1 2mj with m1 < m2 < · · · < CANCELLATION IN A SHORT EXPONENTIAL SUM 5 mσ(ℓ). For every fixed k, the behavior of U(ℓ, k) in dependence on ℓreveals a very chaotic pattern but it appears more regular when is considered along sequences of integers having the same Hamming weight. This fact suggests the introduction of the quantity W(σ, k) := max ℓ: σ(ℓ)=σ{U(ℓ, k)}.
A manifestation of the greater regularity of W(σ, k) is the following circum-stance: the Hamming weight is sub-additive, meaning that σ(ℓ1 + ℓ2) ≤ σ(ℓ1) + σ(ℓ2) for every couple of integers ℓ1 and ℓ2, so that W(σ, k) = 0 when σ is greater than k. Moreover, nothing is lost by studying W(σ, k) in place of U(ℓ, k) because it is evident that maxσ{W(σ, k)} = maxℓ{U(ℓ, k)}.
Let us consider the simpler case where also ℓis a power of two, ℓ= 2w say, so that a k-representation of ℓis actually a solution of 2w = 2n1 + · · · + 2nk. The following proposition shows an important relation between the parameters w, k and the set {nj}k j=1.
Lemma 1. Let (n1, . . . , nk) be a k-representation of 2w. Then min{nj} ≥ w −k + 1.
Note that the inequality is sharp, since (0, 0, 1, . . . , k −2) is a k-representation of 2k−1.
Proof. The string (n1 −min{nj}, . . . , nk −min{nj}) is a k-representation of 2w−min{nj}, therefore, without loss of generality, we can assume that 0 = n1 ≤ n2 ≤· · · ≤nk: under these assumptions we have to prove that w ≤k −1.
The claim is evident for k = 1 and 2, thus we suppose k ≥3. The existence of an upper bound for w becomes clear if we consider Pk j=1 2nj as an addition of binary digits, thus let ¯ w be this maximal value and let 0 = n1 ≤n2 ≤· · · ≤nk be a sequence producing it. The special sequence (0, 0, 1, . . . , k−2) shows that ¯ w ≥k −1. The congruence 0 = 2 ¯ w = Pk j=1 2nj = ♯{j : nj = 0} (mod 2) shows that the number of indexes j with nj = 0 must be even, so that certainly n2 = n1 = 0. Let r be such that n2r = 0 and n2r+1 > 0. If r > 1 the sum of the powers associated with the new sequence (0, 0, 1, . . . , 1 | {z } r −1 times , n2r+1, n2r+2, . . . , nk, ¯ w, ¯ w + 1, ¯ w + 2, . . . , ¯ w + r −2) is 2 ¯ w+r−1, contradicting the maximality of the original sequence n1, . . . , nk, so that r = 1 implying that n3 ≥1. The case n3 > 1 is impossible, since otherwise we would have both 2 ¯ w = 0 (mod 22) (because we know that ¯ w ≥k −1 and we are assuming k ≥3) and 2 ¯ w = 2n1 + 2n2 = 2 (mod 22). Hence n3 = 1, thus proving the claim if k = 3. Suppose k ≥4, then the congruence 0 = 2 ¯ w = Pk j=1 2nj (mod 22) shows that ♯{j : j ≥4, nj = 1} is even. In particular, if 6 G. MOLTENI n4 = 1 then also n5 = 1. Let r be such that n2r+1 = 1 and n2r+2 > 1. If r > 1 the sum of powers associated with the new sequence (0, 0, 1, 2, . . . , 2 | {z } r −1 times , n2r+2, n2r+3, . . . , nk, ¯ w, ¯ w + 1, ¯ w + 2, . . . , ¯ w + r −2) is 2 ¯ w+r−1, contradicting the maximality of the original sequence n1, . . . , nk, so that r = 1 implying that n4 ≥2. If n4 > 2 we have both 2 ¯ w = 0 (mod 23) and 2 ¯ w = 2n1 + 2n2 + 2n3 = 4 (mod 23): the contradiction proves that n4 = 2.
Iterating the argument we prove that nj = j −2 for every j ≥2, so that ¯ w is exactly k −1.
□ Adding 1 to each element of a k-representation of 2w we get a k-repres-entation of 2w+1, thus proving that U(2w, k) ≤U(2w+1, k). Vice versa, the lower-bound for min{nj} in Lemma 1 implies that we can subtract 1 from each element of every k-representation of 2w+1 whenever w ≥k −1, obtaining in this way a k-representation of 2w. In other words, we have obtained that (5) U(2, k) ≤U(22, k) ≤· · · ≤U(2k−1, k) = U(2k, k) = U(2k+1, k) = . . .
proving that the quantity W(k) := W(1, k) = U(2k−1, k) represents the maxi-mum number of k-representations that a power of 2 can have.
It is evident that a relation among the general function W(σ, k) and the special function W(k) must exist, because it is intuitively clear that every k-representation of an integer ℓis made of representations of its σ(ℓ) nonzero binary digits. This idea is clarified by the next formula (7), that we now prove.
We need a second lemma.
Lemma 2. Let {mj}, {nj} be finite sets of integers not necessarily distinct, with P j 2mj = P j 2nj and m1 < mj for every j ̸= 1. Then there is a set S ⊆{nj} such that P j∈S 2nj = 2m1.
Proof. Without loss of generality we can suppose that n1 ≤n2 ≤· · · . The number n1 cannot be strictly greater than m1, otherwise the congruence P j 2mj = P j 2nj (mod 2m1+1) is false (here we use the hypothesis m1 < mj for every j ̸= 1). If n1 = m1 the claim holds with S = {n1}, therefore suppose n1 < m1.
Suppose now that 2n1 + 2n2 > 2m1, then 2n1 + 2n2 > 2m1 ≥2n1+1 so that n2 is strictly larger than n1, but this is impossible because it contradicts the con-gruence P j 2mj = P j 2nj (mod 2n1+1). Hence 2n1 +2n2 ≤2m1. If the equality holds we have the claim with S = {n1, n2}. Suppose 2n1 + 2n2 < 2m1. Then n1 = n2 (otherwise the congruence P j 2mj = P j 2nj (mod 2n1+1) is false) and m1 ≥n1 + 2. Consider the sum 2n1 + 2n2 + 2n3. If this sum is greater than 2m1 we have 2n1 + 2n2 + 2n3 > 2m1 ≥2n1+2 giving n3 > n1 + 1 which is impossible because the congruence P j 2mj = P j 2nj (mod 2n1+2) would be false, hence 2n1 + 2n2 + 2n3 ≤2m1. If the equality holds here we take S = {n1, n2, n3} and CANCELLATION IN A SHORT EXPONENTIAL SUM 7 the proof terminates, otherwise we repeat the previous steps. The argument terminates after a finite number of steps, because P j 2nj = P j 2mj ≥2m1.
□ Let ℓbe an arbitrary positive integer. Iterating Lemma 2, we see that every k-representation of ℓcan be decomposed as union of representations of its σ(ℓ) nonzero digits appearing in its binary representation. Note that the orders k1, . . . , kσ(ℓ) of these representations satisfy the restriction k1 + · · · + kσ(ℓ) = k, that by definition there are W(k1) representations of order k1 for the first digit, W(k2) representations for the second, and so on for every nonzero digit, and that these representations can be permutated in k!/k1! · · · kσ(ℓ)! ways, at most; it follows that (6) U(ℓ, k) ≤ X k1,...,kσ(ℓ)≥1 k1+···+kσ(ℓ)=k W(k1) · · · W(kσ(ℓ)) · k!
k1! · · · kσ(ℓ)!.
The strict inequality can hold in (6), because different permutations of the representations of the nonzero digits can give the same k-representation of ℓ: this happens iffthere are two nonzero digits in ℓadmitting some representation with common integers. By Lemma 1 the representations of the binary digits in ℓdo not have common integers whenever the nonzero digits are separated by gaps of length k −1, at least. In other words, if Pσ(ℓ) j=1 2mj is the binary representation of ℓand mj −mj−1 ≥k for every j (with m0 := 0), then the representations of the nonzero digits do not overlap and (6) holds as equality.
Since for every integer σ there exist (infinitely many) integers ℓwith σ(ℓ) = σ and whose binary nonzero digits have gaps of length k−1 at least, we conclude that the quantities W(σ, k) and W(kj) are related by the formula (7) W(σ, k) = X k1,...,kσ≥1 k1+···+kσ=k W(k1) · · · W(kσ) · k!
k1! · · · kσ!.
Denoting by Lσ(x) the formal series P+∞ k=1 W(σ,k) k!
xk, the previous identity can be stated simply by saying that Lσ(x) = (L1(x))σ.
The identity Lσ(x) = L1(x)Lσ−1(x) immediately gives the formula W(σ, k) = k−1 X n=1 W(n) · W(σ −1, k −n) · k!
n!(k −n)!
which is particularly useful in order to compute W(σ, k) iteratively from a given set of values for W(k). For example, we have the following table: 8 G. MOLTENI σ\k 1 2 3 4 5 6 7 8 9 10 1 1 1 3 13 75 525 4347 41245 441675 5259885 2 2 6 30 190 1470 13230 135982 1565694 19959390 3 6 36 270 2340 23310 260820 3242862 44292420 4 24 240 2520 28560 355320 4823280 71147160 5 120 1800 25200 361200 5481000 88565400 6 720 15120 272160 4808160 87318000 7 5040 141120 3175200 67737600 8 40320 1451520 39916800 9 362880 16329600 10 3628800 Table 1: Values of W(σ, k) for k ≤10.
These values suggest that both W(k) and maxσ{W(σ, k)} grow as ckk! for suitable constants c; we have not been able to prove an asymptotic result, nevertheless the next section provides tight upper and lower bounds of that form. A final remark: the value of W(k) can be computed by hand only for the smallest k, but also a computer can be of little help if the computation is done in the naive way, i.e. by searching all k-representations of 2k−1 . In a recursive formula allowing one to compute W(k) very efficiently is discussed.
Remark. All the numbers W(k) appearing in Table 1 are odd, a fact which is quite surprising because we do not see any simple or combinatorial explanation for it. Actually, more is true and the congruence W(k) = 4 + (−1)k (mod 8) for k ≥3 is proved in .
2.1. The lower bound: proof of Theorem 3. For every k ≥2, the k!/2 permutations of the string (0, 0, 1, . . . , k −2) are k-representations of 2k−1, so that W(k) ≥k!/2 for every k. This simple lower bound immediately produces a lower bound for W(σ, k) of the type considered in Theorem 3.
In fact, introducing it in (7) we have W(σ, k) k!
≥ X k1,...,kσ≥1 k1+···+kσ=k 1 2σ = 1 2σ k −1 σ −1 , where we have used the combinatorial identity P k′ 1,··· ,k′ b≥0 k′ 1+···+k′ b=c 1 = b+c−1 b−1 . This result and the simple inequality maxσ{W(σ, k)} ≥1 k Pk σ=1 W(σ, k) gives the bound maxσ{W(σ, k)} ≥(3/2)k−1(k −1)! that by Stirling becomes (8) max σ {W(σ, k)} ≫(0.5518 k)k.
CANCELLATION IN A SHORT EXPONENTIAL SUM 9 We consider this lower bound as the trivial one: aim of the present section is to improve it up to the result given in Theorem 3. The corollary following the next lemma improves the lower bound for W(k).
Lemma 3. for every k we have W(k) ≥P∗ j k 2j−1 W(k −2j + 1), where the sum P∗ j is restricted to the positive integers j with 2j < k.
Proof. We fix a positive integer j with 2j < k. Let (n1, . . . , nk−2j+1) be a (k − 2j +1)-representation of 2k−j. We notice that the number of these (k −2j +1)-representations is W(k −2j +1) (by (5)) and that each ni is strictly lower than k−j. The k 2j−1 strings that we obtain by joining 2j−1 times the number k−j in all possible positions to the string (n1, . . . , nk−2j+1) are k-representations of 2k. Since every ni is strictly lower than k −j, each representation that we generate in this way is completely characterized by the position where the numbers k −j appear. In particular, they are distinct. Let Kj denote the set of representations of 2k that we obtain using the previous construction: we have just proved that ♯Kj = k 2j−1 W(k −2j + 1). Every representation in Kj contains the exponent k −j and no exponent of greater value, therefore the representations in Kj and Kj′ are distinct when j ̸= j′, and the claim is proved.
□ Corollary. For every ε > 0 we have W(k) ≫ε (η −ε)k k!, where η is the solution of P+∞ j=1 η1−2j (2j−1)! = 1. In particular, W(k) ≥0.3316 · (1.1305)k k!
∀k.
Proof. Let F∞(x) := P+∞ j=1 x1−2j (2j−1)! and Fn(x) := Pn j=1 x1−2j (2j−1)! for every n > 1.
Functions F∞, Fn decrease in R+ with F∞(x) > Fn(x), Fn(1) > 1 and F∞(2) < √e −1 < 1. Hence there exist a unique solution η of F∞(x) = 1 and a unique solution ηn of Fn(x) = 1 for every n, with η, ηn ∈(1, 2) and η > ηn. Moreover, |F ′ ∞(x)| = P+∞ j=1 x−2j/(2j −2)! > 1/4 for x ∈(1, 2). This lower-bound and the equality |F∞(ηn)−1| = |F ′ ∞(γ)||ηn −η| for a suitable γ ∈(ηn, η) ⊂(1, 2) imply that |ηn −η| ≤4|F∞(ηn) −1| = 4|F∞(ηn) −Fn(ηn)| = 4 +∞ X j=n+1 η1−2j n (2j −1)!
≤4 +∞ X m=2n η−m n m! ≤4η−2n n e1/ηn (2n)!
≤ 4e (2n)!
thus proving that ηn tends to η. Let ε > 0 be arbitrarily fixed, let n = n(ε) be an integer such that ηn ≥η −ε and let α > 0 be chosen in such a way that 10 G. MOLTENI W(k) ≥αηk n · k! holds for k ≤2n. By Lemma 3 we know that W(k) ≥ n X j=1 k 2j −1 W(k −2j + 1) whenever k > 2n. By induction on k, in order to prove that W(k) ≥αηk n · k!
for every value of k it is sufficient that n X j=1 k 2j −1 αηk−2j+1 n · (k −2j + 1)! ≥αηk n · k!.
This inequality can be written as Fn(ηn) ≥1 and is evidently satisfied by the definition of ηn, thus the first claim is proved. The second claim follows using this argument with n = 3 and the known values of W(k) for k ≤8.
□ Remark. Using the bound |ηn −η| ≤4e/(2n)! it is possible to compute η with arbitrarily large precision: η = 1.1305033 . . ..
For its frequent use in the following part of this section it is convenient to introduce the symbols α and β to denote the constants 0.3316 and 1.1305, re-spectively; with this notation, the previous corollary says that W(k) ≥αβk k!.
This bound improves considerably the bound W(k) ≥k!/2 for large values of k, nevertheless it badly underestimates W(k) for small values of k. Since also these terms affect the final result, in order to recover a lower bound for W(σ, k) from (7) we split the range for k in two sets, k ≤¯ k and k > ¯ k, where ¯ k is a parameter ≥3 that we will choose later, and we use the true value of W(k) in the first set and the bound W(k) ≥αβk k! in the second one. Decomposing (7) according to the number of variables whose index is ≤¯ k, we obtain W(σ, k) k!
= σ X h=0 X k1,...,kσ≥1 k1+···+kσ=k ♯{j: kj≤¯ k}=h σ Y j=1 W(kj) kj!
= σ X h=0 σ h X 1≤k1,...,kh≤¯ k kh+1,...,kσ>¯ k k1+···+kσ=k σ Y j=1 W(kj) kj!
.
We set wj := W(j)/(j!αβj) for j ≤¯ k so that we can bound W(kj)/kj! by wkjαβkj when kj ≤¯ k and by αβkj for kj > ¯ k, obtaining W(σ, k) k!
≥ασβk σ X h=0 σ h X k1,...,kσ≥1 k1,...,kh≤¯ k kh+1,...,kσ>¯ k k1+···+kσ=k wk1 · · · wkh = ασβk σ X h=0 σ h ¯ kh X w=h h X kh+1,...,kσ>¯ k kh+1+···+kσ=k−w 1 ih X 1≤k1,...,kh≤¯ k k1+···+kh=w wk1 · · · wkh i .
CANCELLATION IN A SHORT EXPONENTIAL SUM 11 The third sum is evaluated by using the identity P k′ 1,...,k′ b≥0 k′ 1+···+k′ b=c 1 = b+c−1 b−1 , while the last sum admits an alternative representation: for every i ∈1, . . . , ¯ k let ai := ♯{j : kj = i}, then X 1≤k1,...,kh≤¯ k k1+···+kh=w wk1 · · · wkh = X a1+···+a¯ k=h a1+2a2+3a3+···+¯ ka¯ k=w h!wa1 1 · · · w a¯ k ¯ k a1!a2! · · · a¯ k!
= X∗ a1,...,a¯ k−2≥0 h!wa1 1 · · · w a¯ k−2 ¯ k−2 wA ¯ k−1wB ¯ k a1!a2! · · · a¯ k−2!A!B!
where A := ¯ kh −w −P¯ k−2 i=1 (¯ k −i)ai, B := w −(¯ k −1)h + P¯ k−2 i=1 (¯ k −i −1)ai and the symbol P∗means that the sum is restricted to those a1, . . . , a¯ k−2 such that A, B ≥0. In this way we get the following lower bound for W(σ, k): W(σ, k) k!
≥ασβk σ X h=0 ¯ kh X w=h X∗ a1,...,a¯ k−2≥0 σ h k −w −¯ k(σ −h) −1 σ −h −1 h!wa1 1 · · · w a¯ k−2 ¯ k−2 wA ¯ k−1wB ¯ k a1!a2! · · · a¯ k−2!A!B!
.
The previous multiple sum is quite intricate; we bound it from below simply with one of its terms, i.e.
(9) W(σ, k) k!
≥ασβk σ h k −w −¯ k(σ −h) −1 σ −h −1 h!wa1 1 · · · w a¯ k−2 ¯ k−2 wA ¯ k−1wB ¯ k a1!a2! · · · a¯ k−2!A!B!
where σ, h, w and ai for every i can be arbitrary chosen but must be taken in such a way that the constraints h ∈(0, σ), w ∈(h, ¯ kh), ai ≥0 and A, B ≥0 be satisfied. Our aim is now to determine a convenient set of values for these parameters in such a way to pick up a value as large as possible for the R.H.S.
of (9). We simplify a little bit the discussion by setting σ = ⌊uk⌋, h = ⌊vk⌋, w = ⌊zk⌋and ai = ⌊sik⌋for every i, with u, v, z and si as new parameters, independent of k: in this way the dependence on k appears only to the exponent and by Stirling we deduce that (10) W(σ, k) k!kO(1) ≥ h αuβ uu(1 −z −¯ k(u −v))1−z−¯ k(u−v) (u −v)2u−2v(1 −z −(¯ k + 1)(u −v))1−z−(¯ k+1)(u−v) · ws1 1 · · · w a¯ k−2 ¯ k−2 w ¯ A ¯ k−1w ¯ B ¯ k ss1 1 · · · s s¯ k−2 ¯ k−2 ¯ A ¯ A ¯ B ¯ B ik 12 G. MOLTENI where ¯ A := ¯ kv −z − ¯ k−2 X i=1 (¯ k −i)si , ¯ B := z −(¯ k −1)v + ¯ k−2 X i=1 (¯ k −i −1)si.
The stationary points of the function of u, v, z, {si} ¯ k−2 i=1 to the R.H.S. of (10) are solutions of the system (1−z−¯ k(u −v)) ¯ k(u−v)2 = αu(1 −z −(¯ k + 1)(u−v)) ¯ k+1 (1−z−(¯ k + 1)(u−v)) ¯ k+1( ¯ A/w¯ k−1) ¯ k = (1−z−¯ k(u−v)) ¯ k(u−v)2( ¯ B/w¯ k) ¯ k−1 (1−z−(¯ k + 1)(u−v))( ¯ A/w¯ k−1) = (1 −z −¯ k(u−v))( ¯ B/w¯ k) wi( ¯ A/w¯ k−1) ¯ k−i = si( ¯ B/w¯ k) ¯ k−1−i ∀i = 1, . . . , ¯ k −2 and can be explicitly found. Let x, y, u′ and ri for every i ≤¯ k −2 be a new set of variables related to the previous ones by: x = u−v, u = u′x, z = 1−¯ kx−yx and si = xri, and let A := ¯ A/x = −1/x + ¯ ku′ + y −P¯ k−2 i=1 (¯ k −i)ri, B := ¯ B/x = 1/x −(¯ k −1)u′ −y −1 + P¯ k−2 i=1 (¯ k −i −1)ri.
After simple computations the system yields (11) u′ = y¯ k α(y−1)¯ k+1 ri = wi y¯ k−i (y−1)¯ k+1−i ∀i = 1, . . . , ¯ k −2 , ( A = w¯ k−1 y (y−1)2 B = w¯ k (y−1).
These relations and the equality A+B = u′−1−P¯ k−2 i=1 ri give a closed equation for y: (y −1) ¯ k+1 −1 α y ¯ k + ¯ k X i=1 wi y ¯ k−i(y −1)i = 0 that admits a unique solution y > 1. With this solution we can compute u′, A, B and each ri by (11) and x by the identity 1/x = −A+¯ ku′+y−P¯ k−2 i=1 (¯ k−i)ri, hence also u, v, z and each si are determined. At last, we obtain from (10) a bound of the type W(σ, k) ≥(c(¯ k) k)k · kO(1). We have computed c(¯ k) with ¯ k = 3, . . . , 1500; apparently c(¯ k) steady grows with ¯ k, with c(3) = 0.641134 and c(1500) = 0.644591. The constant c(1500) yields the claim in Theorem 3.
Remark. The use of the exact value of W(k) for small k is fundamental: if we use the inequality W(k)/k! ≥αβk for every k, our argument becomes much simpler but produces only the lower bound maxσ{W(σ, k)} ≫(0.5537 k)k which is a very modest improvement on the trivial bound (8).
CANCELLATION IN A SHORT EXPONENTIAL SUM 13 2.2. The upper bound: proof of Theorem 2. Table 1 suggests the valid-ity of some inequalities among the values of W(σ, k); one of these says that 2W(k) ≤W(2, k), another one that 24(W(k)−kW(k−1)) ≤W(4, k) for every k ≥4. Both inequalities are true and can be proved with similar arguments.
Moreover, both can be used to prove upper-bounds for W(k), but the result we obtain from the second inequality is stronger, so we prove here only the second one.
Lemma 4. The inequality 24(W(k) −kW(k −1)) ≤W(4, k) holds for every k ≥4.
Proof. We need the following general fact which is a variation of Lemma 2: let S be a finite set of nonnegative integers, suppose that P n∈S 2n is a 2-power, 2q say, and that S contains two integers at least, then there exists S′ ⊂S such that P n∈S′ 2n = 2q−1. In fact, let S0 be any proper and non-empty subset of S.
Exchanging S0 with Sc 0 if necessary, we can assume that P n∈S0 2n ≥2q−1. If the equality holds here we have done, thus we assume that P n∈S0 2n > 2q−1. Let n′ be the smallest integer in S0. The set S0 does not coincide with {n′}, otherwise n′ is equal to q (because 2n′ ≤P n∈S 2n = 2q and 2q−1 < P n∈S0 2n = 2n′) implying that Sc 0 = ∅(because P n∈S0 2n = 2n′ = 2q = P n∈S 2n), against our assumption. Let S1 := S0{n′}; we have just proved that S1 is not empty. If P n∈S1 2n < 2q−1 we have 2q−1 < X n∈S0 2n = 2n′ + X n∈S1 2n < 2n′ + 2q−1 implying that 2q−n′−1 < P n∈S0 2n−n′ < 1 + 2q−n′−1 which is evidently impos-sible, hence P n∈S1 2n ≥2q−1. If the equality holds here the claim is proved, otherwise we repeat the argument with S1 in place of S0. The argument ter-minates after a finite number of steps because the definition of S1 implies that P n∈S1 2n < P n∈S0 2n.
Let (n1, . . . , nk) be a k-representation of 2k−1. The previous remark shows that there exists a subset R ⊂{1, . . . , k} such that P j∈R 2nj = 2k−2 = P j∈Rc 2nj.
Let us assume that both R and Rc contain two integers, at least. Then we can iterate the decomposition of R as union of R0, R1, and of Rc as union of R2, R3, say, such that P j∈Ri 2nj = 2k−3 for i = 0, . . . , 3. Let π be a permutation of {0, 1, 2, 3} and consider the new string (nπ 1, . . . , nπ k), with nπ j := nj + π(i)k if j ∈Ri.
The sum sk := Pk j=1 2nπ j = 24k−3 +23k−3 +22k−3 +2k−3 is independent of π and each string (nπ 1, . . . , nπ k) is a k-representation of sk. These k-representations are distinct. In fact, let (n1, . . . , nk) and (m1, . . . , mk) be two k-representations of 2k−1 and let π, π′ be two permutations. For i = 0, . . . , 3 let Ai := {j : nπ j ∈ 14 G. MOLTENI [ik, (i + 1)k)} and A′ i := {j : mπ′ j ∈[ik, (i + 1)k)}. Suppose (nπ 1, . . . , nπ k) = (mπ′ 1 , . . . , mπ′ k ), then Ai = A′ i for every i. It follows that for every j we have nj = nπ j −ik = mπ′ j −ik = mj (where i = i(j) denotes the index i such that j ∈Ai ≡A′ i), i.e. the original k-representations (n1, . . . , nk) and (m1, . . . , mk) are equal. Under this hypothesis, the proof of the equality π = π′ is immedi-ate.
Let B(k) be the number of k-representations of 2k−1 we have considered here, i.e.
for which both R and Rc contain two numbers at least.
The argu-ment we have just discussed proves that sk admits at least 24B(k) differ-ent k-representations.
The Hamming weight of sk is four, hence we have proved that 24B(k) ≤W(4, k).
In order to terminate the proof we verify now that B(k) = W(k) −kW(k −1). The number W(k) −B(k) counts the k-representations of 2k−1 containing k −2. When k ≥3 the number k −2 appears in these representations only once, therefore these representations are exactly those ones we obtain adding k −2 to any (k −1)-representation of 2k−2. The claim follows because there are k possible places for k −2 in any (k −1)-representation and W(k −1) representations of 2k−2 as sum of (k −1) powers of 2 (by (5)).
□ For the proof of Theorem 2 we need to study the sequence χσ(b, k) := X k1...,kσ≥1 k1+···+kσ=k k k1 · · · kσ b and the constant χσ(b) := supk{χσ(b, k)}. When b > 1 the sequence converges to σζ(b)σ−1 as k diverges, therefore χσ(b) is certainly finite. For our application we need an accurate determination of the value of χσ(b), thus the next lemma not only proves the convergence but also provides an explicit inequality.
Lemma 5. Let b > 1 and let c = c(b) be the positive constant such that (1 −(σ −1)c)−b = 1 + 2(σ −1)bc. Then, for σ > 1 we have χσ(b, k) σ −ζ(b)σ−1 ≤(σ −1)ζ(b)σ−2 (b −1)(ck)b−1 + 2bχσ−1(b) k hZ (σ−1)ck σ−1 w1−bdw+ 1 (σ −1)b−1 i + c−bζ(b)σ−1 σ σ X m=2 σ m h ζ(b)−1 (b −1)(ck)(b−1) im−1 .
Proof. We decompose the sum defining χσ(b, k) according to the number m of variables which are “large”, where “large” here means greater than ck. The CANCELLATION IN A SHORT EXPONENTIAL SUM 15 symmetry of the sum allows us to write this decomposition as (12) χσ(b, k) = σ X m=0 σ m Sm with Sm := X k1,...,kσ k1,...,km>ck km+1,...,kσ≤ck k1+···+kσ=k k k1 · · · kσ b .
The term S0 is empty because a simple argument proves that the constant c is lower than 1/σ; the following argument will show that the main term comes from S1 and that the other terms contribute only at lower orders. The term S1 is S1 = X 1≤k2,...,kσ≤ck k1>ck k1+···+kσ=k k k1 · · · kσ b .
The parameters k2, . . . , kσ appearing in this sum are small with respect to k, while k1 = k −(k2 + · · · + kσ) is large, so we write S1 as: (13) S1 = ck X k2,...,kσ=1 1 kb 2 · · · kb σ + ck X k2,...,kσ=1 1 kb 2 · · · kb σ h 1 −k2 + · · · + kσ k −b −1 i .
The first sum is the σ −1 power of the sum Pck w=1 w−b converging to ζ(b).
Using the upper bound P w>ck w−b ≤ R +∞ ck w−b dw we get (14) ck X k2,...,kσ=1 1 kb 2 · · · kb σ −ζ(b)σ−1 = h ck X w=1 1 wb iσ−1 −ζ(b)σ−1 ≤(σ −1)ζ(b)σ−2 (b −1)(ck)b−1 .
In (13) every kj with j ≥2 is lower than ck, hence k2+···+kσ k ≤(σ −1)c so that by convexity we have 1 −k2+···+kσ k −b −1 ≤2b k2+···+kσ k . We deduce that (15) ck X k2,...,kσ=1 1 kb 2 · · · kb σ h 1 −k2 + · · · + kσ k −b −1 i ≤2b k ck X k2,...,kσ=1 k2 + · · · + kσ kb 2 · · · kb σ = 2b k (σ−1)ck X w=σ−1 1 wb−1 X k2,...,kσ≤ck k2+···+kσ=w wb kb 2 · · · kb σ ≤2bχσ−1(b) k (σ−1)ck X w=σ−1 1 wb−1.
The R.H.S. here tends to 0 for every b > 1, but in different ways for b ∈(1, 2), b = 2 and b > 2. We bound it simply via the integral test P(σ−1)ck w=σ−1 1 wb−1 ≤ 1 (σ−1)b−1 + R (σ−1)ck σ−1 w1−bdw. The terms Sm with m ≥2 can be bounded by 16 G. MOLTENI using k1 to control the growth of the numerator and by splitting the sum over the large variables (k2, . . . , km) and that one over the small variables (km+1, . . . , kσ): Sm = X k1,k2,...,km>ck km+1,...,kσ≤ck k1+···+kσ=k h k k1 · · · kσ ib ≤1 cb X k2,...,km>ck km+1,...,kσ≤ck h 1 k2 · · · kσ ib = 1 cb h X w>ck 1 wb im−1 · h X w≤ck 1 wb iσ−m ≤ c−bζ(b)σ−m (b −1)m−1(ck)(m−1)(b−1).
(16) The lemma follows by collecting the results in (12)–(16).
□ Remark. Lemma 5 defines c as the number such that (1 −(σ −1)c)−b = 1 + 2b(σ −1)c; this choice is evidently arbitrary and c could be defined by (1 −(σ −1)c)−b = 1 + rb(σ −1)c for any r ≥2. Actually, the new parameter r can be used to improve the final bound; for sake of simplicity we have not mentioned this fact in Lemma 5.
The importance of the sequence χσ(b, k) for our problem comes from the following result.
Lemma 6. Let b > 1, k0 ≥4 and γ(b) := max{y0(b), y(b)}, where y0(b) is the greatest value of (W(k)kb/k!)1/(k−1) for k < k0, and y(b) is the positive solution of y3 − kb 0 (k0 −1)b y2 −χ4(b) 24 = 0.
Then W(k) ≤γ(b)k−1k! · k−b for every k.
Proof. The definition of γ(b) immediately implies the claim for k < k0. Sup-pose k ≥k0. By (7) and Lemma 4 we have W(k) ≤kW(k −1) + 1 24W(4, k) = kW(k −1) + k!
24 X k1,k2,k3,k4≥1 k1+k2+k3+k4=k 4 Y j=1 W(kj) kj!
that by induction gives W(k) k!
≤γ(b)k−2 (k −1)b + γ(b)k−4 24 X k1,k2,k3,k4≥1 k1+k2+k3+k4=k 1 (k1k2k3k4)b.
The claim is proved if γ(b) satisfies γ(b)k−2 (k −1)b + γ(b)k−4 24 X k1,k2,k3,k4≥1 k1+k2+k3+k4=k 1 (k1k2k3k4)b ≤γ(b)k−1k−b.
CANCELLATION IN A SHORT EXPONENTIAL SUM 17 Since we are assuming k ≥k0, simple computations prove that this inequality holds whenever γ(b) ≥y(b).
□ We show now how Lemmas 5-6 and some numerical computations allow us to find a convenient upper bound for the growth of W(k). We have at our disposal the values of W(k) for k ≤1500; the value of y0(k) is only marginally influenced by the choice of k0 while y(b) decreases with k0, hence in all our computations we set k0 = 1501. For a given b > 1 we compute y0(b) and χ4(b, k) for small values of k (for k ≤1000, say). In this range we identify the element χ4(b, ˜ k) having the greatest value. In our numerical computations χ4(b, k) decreases for k > ˜ k and χ4(b, ˜ k) is greater than 4ζ(b)3; with these evidences it is natural to guess the equality χ4(b) = χ4(b, ˜ k). We can prove the correctness of this guess in two steps: first by using Lemma 5 to compute an index ˇ k such that |χ4(b, k) −4ζ(b)3| ≤|χ4(b, ˜ k) −4ζ(b)3| when k > ˇ k, then by verifying that χ4(b, k) ≤χ4(b, ˜ k) for every k < ˇ k with a direct numerical computation. We notice that Lemma 5 produces ˇ k only if we already know χ3(b) hence a descent process is triggered here: to compute χ3(b) we use Lemma 5 needing χ2(b), and to compute χ2(b) we employ again Lemma 5. At this point the process terminates because χ1(b) is equal to 1 for every b. Having determined χ4(b), we can compute y(b). The parameter y0(b) grows with b while the numerical computations show that y(b) decreases with b, therefore we can repeat this process several times adjusting the value for b until the difference between y0(b) and y(b) is small enough. With k0 = 1501 and b = 1.6056 this algorithm produces the following values: χ4(b) = χ4(b, 71) ≤49.95 (and χ2(b) = χ2(b, 25) ≤4.72, χ3(b) = χ3(b, 46) ≤16.33), y0(b) ≤1.71186, y(b) ≤1.71154 proving that γ(b) ≤1.71186, i.e., that W(k) ≪(0.62976 k)k.
It is interesting to remark that the upper bound for the growth of W(k) we found here is strictly lower than the lower bound we found in Theorem 2 for maxσ{W(σ, k)}: 0.62976 instead of 0.64459. This is not surprising because maxσ{W(σ, k)} is evidently larger than W(1, k) = W(k); nevertheless, being able to prove it here means that our argument producing upper and lower bounds for W(k) and maxσ{W(σ, k)} is sufficiently precise. Moreover it has an interest in itself: it proves that not only maxσ{W(σ, k)} is greater than W(k) but also that it grows exponentially faster than W(k).
Now we describe the strategy for the proof of Theorem 2. For every b > 1, from (7) and Lemma 6 we get W(σ, k) k!
≤γ(b)k−σ X k1,...,kσ≥1 k1+···+kσ=k h 1 k1 · · · kσ ib ≤γ(b)k−σζ(b)σ 18 G. MOLTENI proving that maxσ{W(σ,k)} k!
≤max{γ(b), ζ(b)}k.
A slightly more complicated argument produces a better bound. As we done in Section 2.1, we decom-pose (7) according to the number or variables whose index is ≤¯ k, where ¯ k is a parameter that we will set later: W(σ, k) k!
= σ X h=0 σ h X 1≤k1,...,kh≤¯ k kh+1,...,kσ>¯ k k1+···+kσ=k σ Y j=1 W(kj) kj!
.
We bound this sum from above by eliminating the constraint k1 +· · ·+kσ = k; moreover, we introduce the quantities wj := W(j)/(j!γ(b)j−1) for j ≤¯ k, so that each W(kj)/kj! can be estimated by wkjγ(b)kj−1 when kj ≤¯ k and by γ(b)kj−1k−b j for kj > ¯ k, obtaining W(σ, k) k!
≤γ(b)k−σ σ X h=0 σ h X k1,...,kσ≥1 k1,...,kh≤¯ k kh+1,...,kσ>¯ k wk1 · · · wkh (kh+1 . . . kσ)b = γ(b)k−σ σ X h=0 σ h h ¯ k X j=1 wj ihh ζ(b) − ¯ k X j=1 j−biσ−h = γ(b)k−σh ζ(b) − ¯ k X j=1 (j−b −wj) iσ , proving that (17) maxσ{W(σ, k)} k!
≤max{γ(b), ψ(b)}k where for convenience we have set ψ(b) := ζ(b) −P¯ k j=1(j−b −wj). The new upper bound improves the previous one because ψ(b) < ζ(b), since each wj is lower than j−b (by Lemma 6); for the same reason in these computations we choose ¯ k as large as possible: ¯ k = 1500.
In order to bound maxσ{W(σ, k)} from above we must find the smallest value for max{γ(b), ψ(b)}. We know that γ(1.6056) ≤1.71185, but ψ(1.6056) > 1.8 so a different (larger) b must be chosen. Proceeding as we have shown before we arrive to the (almost) optimal choice b = 1.6578, giving: y0(b) ≤1.75781, χ4(b) ≤44.32, y(b) ≤1.66746, γ(b) ≤1.75761 and ψ(b) ≤1.75772 so that (17) yields the bound maxσ{W(σ, k)} ≤ψ(b)kk!. We obtain the claim in Theorem 2 by using the explicit inequality k! ≤(k/e)k√ 2πke1/12k.
CANCELLATION IN A SHORT EXPONENTIAL SUM 19 3. Proof of Theorem 1 Now we can prove Theorem 1. Let ϱ, hµ,ν(ϱ, x) and cµ,ν(ϱ) be defined as in that theorem and suppose that q diverges along a sequence S for which τ/L ∈[µ, ν]. Let T (q; k) be the number of solutions of the congruence k X i=1 2ni = k X j=1 2mj (mod q), with 0 ≤ni, mj < τ for every i and j. The congruence means that (18) k X i=1 2ni = k X j=1 2mj + qw for some w ∈Z: suppose w ≥0, then w < k2τ/2q, so that there are k2τ/2q·τ k possible choices for the values of the set of parameters w and mjs; for every such choice there are ≤(ϱk)k · kO(1) solutions for n1, . . . , nk, hence we have (ϱk)k2ττ k · kO(1)/2q solutions, at most. If w < 0 we obtain the same bound by moving w to the L.H.S. in (18), hence we have proved that (19) T (q; k) ≤2τ q (ϱkτ)k · kO(1).
The second inequality of Lemma 3.1 in says that max ξ: ξq=1 ξ primitive {|s(ξ)|} ≤q1/2k2T (q; k)1/k2τ 1−2/k for every integer k ≥1, so that (19) gives max ξ: ξq=1 ξ primitive {|s(ξ)|} ≤q1/2k22τ q · (ϱkτ)k · kO(1)1/k2 τ 1−2/k.
By hypothesis, for q ∈S we have τ/L ≤ν so that 2τ/q ≤qν−1, hence max ξ: ξq=1 ξ primitive {|s(ξ)|} ≤τ exp ν −1 2 log q k2 + log(ϱk/τ) k + O(log k k2 ) .
In this inequality we set k = ⌊τ/x⌋for a constant x ≥1 that we will choose later, obtaining that max ξ: ξq=1 ξ primitive {|s(ξ)|} ≤τ exp ν −1 2 x2 L τ 2 log 2 −xlog(x/ϱ) τ + Ox,ν log(Lτ) τ 2 .
20 G. MOLTENI By hypothesis, for q ∈S we have also L/τ ≤µ−1 so that we deduce from the previous inequality that max ξ: ξq=1 ξ primitive {|s(ξ)|} ≤τ −hµ,ν(ϱ, x) + ox,µ,ν(1).
The proof concludes by choosing for x the value xµ,ν for which hµ,ν(ϱ, xµ,ν) = cµ,ν(ϱ) in the previous inequality.
Remark. At the web page www.mat.unimi.it/users/molteni/research/cancellation/ paper.html we have collected both the data files and the macros written with the PARIgp programming language which are necessary for the computa-tions contained in this paper.
Acknowledgements. The author thanks A. Languasco, A. Giorgilli and A. Perel-li for many interesting remarks and suggestions about the subject of this paper.
References G. I. Arkhipov, V. N. Chubarikov, and A. A. Karatsuba, Trigonometric sums in number theory and analysis, de Gruyter Expositions in Mathematics, vol. 39, Walter de Gruyter GmbH & Co. KG, Berlin, 2004.
J. Bourgain, New bounds on exponential sums related to the Diffie-Hellman distribu-tions, C. R. Math. Acad. Sci. Paris 338 (2004), no. 11, 825–830.
, Estimates on exponential sums related to the Diffie-Hellman distributions, Geom. Funct. Anal. 15 (2005), no. 1, 1–34.
, On an exponential sum related to the Diffie-Hellman cryptosystem, Int. Math.
Res. Not. (2006), Art. ID 61271, 15.
J. B. Friedlander, J. Hansen, and I. E. Shparlinski, Character sums with exponential functions, Mathematika 47 (2000), no. 1-2, 75–85 (2002).
P. X. Gallagher, Primes and powers of 2, Invent. Math. 29 (1975), no. 2, 125–142.
A. Giorgilli and G. Molteni, Representation of a 2-power as sum of k 2-powers: a recursive formula, (2009), preprint.
D. R. Heath-Brown and S. Konyagin, New bounds for Gauss sums derived from kth powers, and for Heilbronn’s exponential sum, Q. J. Math. 51 (2000), no. 2, 221–235.
D. R. Heath-Brown and J.-C. Puchta, Integers represented as a sum of primes and powers of two, Asian J. Math. 6 (2002), no. 3, 535–565.
S. V. Konyagin, Estimates for Gaussian sums and Waring’s problem modulo a prime, Trudy Mat. Inst. Steklov. 198 (1992), 111–124.
S. V. Konyagin and I. E. Shparlinski, Character sums with exponential functions and their applications, Cambridge Tracts in Mathematics, vol. 136, Cambridge University Press, Cambridge, 1999.
N. M. Korobov, Exponential sums and their applications, Mathematics and its Appli-cations (Soviet Series), vol. 80, Kluwer Academic Publishers Group, Dordrecht, 1992.
H. Li, The number of powers of 2 in a representation of large even integers by sums of such powers and of two primes. II, Acta Arith. 96 (2001), no. 4, 369–379.
J. Liu, M. Liu, and T. Wang, The number of powers of 2 in a representation of large even integers. I, Sci. China Ser. A 41 (1998), no. 4, 386–398.
CANCELLATION IN A SHORT EXPONENTIAL SUM 21 , The number of powers of 2 in a representation of large even integers. II, Sci.
China Ser. A 41 (1998), no. 12, 1255–1271.
, On the almost Goldbach problem of Linnik, J. Th´ eor. Nombres Bordeaux 11 (1999), no. 1, 133–147, Les XXemes Journ´ ees Arithm´ etiques (Limoges, 1997).
J. Pintz and I. Z. Ruzsa, On Linnik’s approximation to Goldbach’s problem. I, Acta Arith. 109 (2003), no. 2, 169–194.
I. E. Shparlinski, On the distribution of the Diffie-Hellman pairs, Finite Fields Appl. 8 (2002), no. 2, 131–141.
The PARI Group, Bordeaux, Pari/gp, version 2.2.10, 2004, available from http: //pari.math.u-bordeaux.fr/.
Dipartimento di Matematica, Universit a di Milano, via Saldini 50, I-20133 Milano, Italy E-mail address: giuseppe.molteni1@unimi.it |
10940 | https://www.geogebra.org/m/FKDa43gU | Exterior Angles of Polygons--Proofs without Words – GeoGebra
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Exterior Angles of Polygons--Proofs without Words
Author:Tim Brzezinski
Topic:Angles, Polygons
Choose any polygon of your choice on the right side of the screen. Polygons range from n = 3 (a triangle) to n = 8 (octagon). You can only select one polygon at a time. This applet displays the EXTERIOR ANGLES of a polygon you select (on the right side of the screen.) These exterior angles are shown in different colors.
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10941 | https://steminthemiddle.net/top-5-unplugged-coding-activities-to-teach-computer-science/ | Published Time: 2024-12-27T00:07:14+00:00
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Top 5 Unplugged Coding Activities to Teach Computer Science
December 26, 2024
Computer Science, STEM
Computer Science Lesson, Engaging Lessons, STEM Lesson Plans
By Trilby Hillenbrand
In today’s digital age, teaching computer science is more important than ever. But coding doesn’t always have to involve screens and devices. Unplugged coding activities offer a hands-on, screen-free approach to teaching fundamental computer science concepts.
Short on time? Grabready-to-go computer science lessons and activities.
By stepping away from the computer, students can focus on building essential skills like problem-solving, logic, and computational thinking through engaging and interactive experiences. These activities are especially great for introducing coding concepts to beginners, reinforcing new ideas, and creating engaging substitute lesson plans.
Let’s explore five amazing unplugged coding activities that are fun, easy to implement, and designed to deepen your students’ understanding of core computer science concepts.
1. Code the Teacher
Concepts: Algorithms, Sequencing, Debugging
Activity:Have students write a detailed algorithm (step-by-step instructions) for a simple task, such as tying a shoe, brushing their teeth, or making a sandwich. Then, you act as the “computer,” meticulously following the algorithm exactly as the students have written it.
Learning: Students will quickly realize that computers strictly adhere to the provided instructions. This activity helps them understand the importance of clear, precise instructions and the need to break down complex tasks into simple, specific steps. They’ll also experience the frustration of encountering errors in the algorithm, teaching them valuable debugging skills.
2. Directed Drawings
Concepts:Algorithms, Sequencing, Communication
Activity: Students create an algorithm for drawing a simple picture, such as a smiley face, snowman, or house. Then they exchange instructions with a partner and attempt to recreate the drawing based solely on their partner’s written directions.
Learning: This activity reinforces the importance of clear, concise instructions and the need to consider the perspective of the “computer” (in this case, their partner) when writing the algorithm.
3. Flowchart Recipes
Concepts: Flowcharts, Decision Making
Activity: Introduce students to the standard symbols used in flowcharts. Then, have them create a flowchart representing the steps involved in a simple recipe, a science experiment (like making ooblek or slime), or a classroom procedure.
Learning: This activity helps students visualize the flow of instructions and understand how decisions (e.g., “if the oven is preheated, proceed to step 3”) are incorporated into computer programs.
4. Dance Sequences
Concepts: Loops, Repetition
Activity: Challenge students to create dance choreography using loops. For example, they might design an algorithm that includes clapping twice, spinning, and repeating the sequence five times. After recording their algorithms, play some music and have the class perform the dances.
Learning: This activity introduces the concept of loops (repetition) in a fun and engaging way. You can increase the complexity by incorporating nested loops (loops within loops) for more advanced learning.
5. Conditional Card Games
Concept: Conditional statements, Decision making
Activity: Divide the class into teams. Have each team write a conditional statement related to a deck of cards, such as “If the card is red, award your team 1 point; else award the other teams 1 point.” After each team creates a rule, play a game where teams draw cards and apply the conditional statements to determine points.
Learning: This activity introduces the concept of “if-then-else” statements, a fundamental building block of programming. You can increase the complexity by having students create more intricate nested conditional statements.
By incorporating these unplugged coding activities into your classroom, you can make computer science education more engaging, accessible, and enjoyable for all students, regardless of their prior experience.
Looking for more resources to teach coding?
Grab this complete introduction to computer science unit. The bundle includes over 100 pages of resources that can be used individually where they best fit within your STEM course or taught in sequence as a 3-week middle school computer science unit. Your students will learn about computer parts, algorithms, flowcharts, code, and the role of computer science in our daily lives.
Grab these lessons today to get back your nights and weekends while knowing your students will be engaged in learning the basics of computer coding!
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10942 | https://www.osti.gov/servlets/purl/1810019 | 1
For submission to Applied Radiation and Isotopes, Version 5-11-2020
Nuclear Data for Reactor Production of 131 Ba and 133 Ba
Zakary Kulage, 1,2a Tyler Cantrell, 1,2b , Justin Griswold 1 , David Denton, 1 Marc Garland, 1c Roy Copping 1, and Saed Mirzadeh. 1
1
Isotopes and Fuel Cycle Technology Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831-6226
2
Department of Nuclear Engineering, University of Texas A&M University, College Station TX Corresponding Author: griswoldjr@ornl.gov
Current addresses:
a
Zak Kulage
b
1681 W Old Henderson Road, Columbus OH 43220
c
Department of Energy, Office of Nuclear Physics, Isotope Program, SC-26/Germantown Building, 1000 Independence Ave., SW, Washington, DC 20585, USA.
Abstract
In the fight against prostate cancer 131 Cs (t 1/2 = 9.69 d, 100% EC) is the newest tool. Generated via electron capture decay of 131 Ba (t 1/2 = 11.6 d, 100% EC), it has been used in brachytherapy for prostate cancer since 2004. The 131 Ba parent is produced through neutron capture of enriched 130 Ba in a nuclear reactor. For large-scale production of 131 Ba, an accurate knowledge of production and burn up cross-sections of 131 Ba are essential. In this paper, we report two group cross-sections (thermal and resonance integrals) for 130 Ba and
131
Ba and a new measure of the half-life of 131 Ba and 131 Cs. Targets consisting of milligram quantities of enriched 130 Ba (~35% ) were irradiated at the ORNL High Flux Isotope Reactor at thermal and resonance neutron fluxes of (1.8 - 2.0) x 10 15 and (5.8 - 7.0) x 10 13 neutrons·cm -2 ·s -1 , respectively, for durations ranging from 3 to 26 days. In addition, cadmium covered samples of 130 Ba were irradiated for one hour at 12.6% reactor power (10.7 MW). The yield of 131 Ba approaches a saturation value of ~60 GBq (~1.6 Ci) per mg of
130
Ba for 20 days irradiation at φth = 1.8x10 15 n·s -1 ·cm -2 , with thermal/epithermal ratio of ~30. Under the above experimental conditions, the two group cross-sections of 130 Ba are 6.9 ± 0.5 b (thermal, σ0) and 173 ± 7 b (resonance, I 0). These values represent the sum of cross-sections to metastable and ground state of 131 Ba. For 131 Ba, the empirically measured thermal cross-section is 200 ± 50 b assuming an I 0/ σ0 of 10. This cross section is reported for the first time. Further, the half-life of 131 Ba was re-measured to be 11.657 ± 0.008 d. Lastly, this study also resulted in the co-production of 133 Ba (t 1/2 = 10.52 y, 100% EC ). The experimental yield of 133 Ba is ~370 MBq (~10 mCi) per mg of 132 Ba (thin target) for one cycle irradiation at HFIR, and measured two-group 132 Ba cross-sections are 7.2 ± 0.2 b and 39.9 ± 1.3 b. These values also represent the sum of cross-sections to metastable and ground state of 133 Ba.
Keywords: Ba-131, Cs-131, Ba-133, nuclear reaction, neutron capture cross-section, HFIR, prostate cancer
Notice: This manuscript has been authored by UT-Battelle, LLC, under contract DE-AC05-00OR22725 with the US Department of Energy (DOE). The US government retains and the publisher, by accepting the article for publication, acknowledges that the US government retains a nonexclusive, paid-up, irrevocable, worldwide license to publish or reproduce the published form of this manuscript, or allow others to do so, for US government purposes. DOE will provide public access to these results of federally sponsored research in accordance with the DOE Public Access Plan ( 2
Introduction
Seed implantation of radionuclides for brachytherapy has a long history, starting with the therapeutic use of
226
Ra to treat prostate cancer. Other isotopes soon followed with the artificially created 198 Au and then 125 I,
103
Pd and many others [Skowronek, J., 2017]. Prostate brachytherapy employs radioactive seeds via interstitial implantation as an approach towards treating prostate cancer. Permanent prostate seed implantation is an ideal method of treatment due to its minimally invasive nature. The treatment process utilizes low energy photons (X-ray or γ-ray) from decay of radionuclides with minimal particulate emissions, and currently 125 I and 103 Pd are two the most extensively used radioisotopes for this application [Kehwar, T., 2009].
The newest radioisotope for use in prostate brachytherapy treatment is 131 Cs, introduced by Iso Ray, Richmond, WA in 2004. As the decay product of 131 Ba through electron capture decay, 131 Cs is a short-lived radioisotope with a half-life of 9.689 ± 0.016 d while the parent has a half-life of 11.50 ± 0.06 d [Firestone and Shirley, 1996]. Cs-131 can be produced in a nuclear reactor or in an accelerator directly or indirectly via the decay of 131 Ba. The accelerator approach can utilize charged particle-induced reactions such as
133
Cs[p,4n] 131 Ba [Tilbury and Kolar, 1986], 131 Xe[p,n] 131 Cs [Tárkányi, 2009] and nat Ba[p,pxn] 131 Ba [Tárkányi, 2010]. The reactor approach, utilized in our study, is limited to neutron capture of 130 Ba,
130
Ba[n,γ] 131 Ba reaction, Figure 1 [Zlokazov, 2012, Kuznetsov, and Toporov, 2009]. Once produced, the
131
Cs is chemically separated from the barium target and processed to manufacture the brachytherapy seeds [Meikrantz, 2012, Zlokazov, 2012, Tilbury and Kolar, 1986].
The natural abundance of 130 Ba is only 0.106 %. In order to make 131 Ba production feasible, 130 Ba with enrichment of up to ~37% is currently available (see for example For large-scale production of 131 Ba, using a lower enrichment target requires a longer irradiation time. Consequently, a knowledge of the 131 Ba burn-up cross-section, resulting in a loss of 131 Ba via the 131 Ba[n,γ] 132 Ba reaction, is essential in order to optimize the production parameters and costs. A systematic study of the neutron capture cross sections of Ba isotopes was reported by Dauenhauer et al., [Dauenhauer and Krane, 2012]. While determination of production cross-sections of radionuclides is rather straight forward, measurement of the burn-up cross-sections of radionuclides, especially short-lived radionuclides, is rather challenging. Burn-up cross-section of short-lived radionuclides can be measured empirically by superimposing the theoretical calculations on experimental measurements. This technique requires measurements of the yields of the radionuclide of interest for various irradiation periods at ideally constant neutron flux. The calculated yields, based on known neutron capture cross-sections, as a function of 3
irradiation time is then fitted to the experimental points by varying the burn-up cross-section while keeping all other variables constant [Ersoz et al., 2018, Broderick et al., 2018]. In this study, we report two group neutron capture cross sections for 130 Ba[n,γ] 131 Ba and 131 Ba[n,γ] 132 Ba reactions via irradiations of several highly enriched 130 Ba targets in the ORNL HFIR. The thermal and resonance cross sections of the 130 Ba[n,γ] 131 Ba reaction were measured through short irradiations of unfiltered and Cd-filtered targets. Then, longer irradiations were performed for assessing the burn-up cross-section of 131 Ba, i.e. cross sections for the 131 Ba[n,γ] 132 Ba reaction. Due to the rather short half-life of 131 Ba, the cross section for the 131 Ba[n,γ] 132 Ba reaction was evaluated empirically by superimposing the experimental 131 Ba yields on the theoretical yield curves while keeping all variables other than the 131 Ba cross section constant. We also report a new measurement for the half-life of 131 Ba. Since the enriched 131 Ba target used in these studies contained a small amounts of 132 Ba, our data also provided additional information on two group cross sections for 132 Ba[n,γ] 133 Ba reaction. Ba-133, with a half-life of 10.51 ± 0.05 years and intense γ-rays ranging from 81-384 keV, is a common standard for calibration of γ-ray spectrometers.
Material and Methods
2.1. Irradiation facility. The High Flux Isotope Reactor (HFIR), located at Oak Ridge National Laboratory (Oak Ridge, TN, USA), is a nuclear reactor with the capability and facilities for a wide range of irradiations in near steady-state thermal neutron fluxes up to 2.1 x 10 15 n·cm -2 -s -1 (at 85 MW power level). The reactor has been in use since 1965 with the original purpose of producing transuranic isotopes and performing material irradiations, and the current objective focusing on neutron scattering. HFIR currently operates at 85 MW power level with a refueling cycle of ~24 days. A hydraulic tube facility (HT) consisting of a singular bundle of nine tubes (“rabbits”) that passes through and exits the reactor, provides an on-line access to the core while reactor is on-line for allowing for partial or full cycle irradiations. Targets in the HT were cooled to ~60 ºC by water with a flow rate of ~ 24 L·m -1 . Position 5 of the HT, centered at the reactor mid-plane, provides the highest possible neutron flux of 2.1 x 10 15 n·cm -2 -s -1 [Mirzadeh et al.1992]. Detailed information with regard to neutron flux measurements and neutron spectra unfolding can be found elsewhere [Mahmood et al., 1995, and Garland et al. 2003]. With regard to neutron energy, thermal is defined as E n
=2200 m/s, epithermal is defined as E n ≥0.5 eV, and r = φth /φ epi is defined as the ratio of the thermal flux to epithermal flux in the energy range 0.5 eV and 0.1 MeV divided by the logarithm of the energy di ff erence of that energy range. This term is known known as the flux per unit lethargy (see for example Hogle et al., 2016, Mirzadeh et al., 2011, or Stoughton and Halperin, 1959). 4
2.2 Target Preparation and Irradiation. Barium-130 with an enrichment of 35.839 ± 0.007 atom%, in the form of barium carbonate, was obtained from the US Department of Energy National Isotope Development Center (www.isotopes.gov/catalog). This batch of 130 Ba contained 0.414 ± 0.001% 132 Ba. The complete isotopic distributions of the enriched 130 Ba, in atom and mass percent, used in these studies are given in Table 1. An atomic mass of 134.76 g·mol -1 was calculated for the enriched 130 Ba lot, based on the mass distribution given in Table 1, where Ba represents 68.28% of the total mass of the carbonate compound. The targets consisted of milligram quantities of enriched 130 BaCO 3, which were encapsulated in high-purity synthetic quartz ampoules (25 mm long with 3 mm OD, and 0.3 mm wall thickness Suprasil, Heraeus Amersil, Duluth, GA). The quartz ampules were cleaned in concentrated nitric acid, rinsed thoroughly with D.I. water, and dried at 120 °C. Prior to use, the ampoules were allowed to cool to room temperature in a desiccator. Then the tare weight of each ampoule was recorded. Barium carbonate was added to the ampoules and initial gross weights were measured. The loaded ampoules were placed back in the oven at 120 °C for at least an hour, and after allowing the loaded ampoule to cool to room temperature in a desiccator, the final gross weights were used to calculate the net weights of 130 BaCO 3 in each ampoule (Table 2). The quartz ampoules containing the target material were flame sealed under a slight negative pressure of He, then placed in Al irradiation capsules (6.7 x 51 mm, manufactured from Al 6061 alloy) and were welded in a He atmosphere and subjected to a He leak test to ensure weld integrity. A total of 16 targets were prepared. Targets 3-9 were grouped into pairs, designated “a” and “b”, to provide two data points for each irradiation time. To assess the contribution from epithermal neutrons, samples 1 and 2 were wrapped in Cd filters (0.26 mm). To minimize the Cd depletion, the reactor power level was maintained at 10.7 MW (~12.5%) during this irradiation. The two targets for short irradiations each contained only one glass ampoule. After irradiation, the Al capsules were transported to a hot cell, cut open, glass ampoules removed, and transported to a class C chemical hood. The quartz sample tubes were soaked in conc. HNO 3 for a few minutes, then thoroughly rinsed with D.I. water in order to remove any surface contamination on the quartz tubes and mounted on a counting card for radioactivity assay. The mass of each target (presented as BaCO 3,
130
Ba, and 132 Ba), irradiation time, hydraulic tube position, and the corresponding neutron flux are summarized in Table 2.
2.3 Radioactivity Measurement. After an appropriate cooling time, the activities of targets 5-6 and 13-16 were measured at a distance of 25 cm from a calibrated solid-state 1-cm-thick HPGe-γ-ray detector. Radioactivity of targets 1-4 and 7-12 were measured at 60 cm from a 50 cm 3detector. Both detectors, (EG&G Ortec, Oak Ridge, TN) were coupled to two independent PC-based MCA’s (Canberra Industries, INC., Meriden, CT). Dead times were kept below 5%. The γ-rays measured for 131 Ba were 123.8 (2.19%), 216.1 (20.0%), 373.2 (13.3%), and 496.3 (44.0%) keV. The γ-rays for 133 Ba were 276.4 (7.09%), 302.9 5
(18.4%), 356.0 (62.2%) and 383.8 (8.92%) keV [Firestone and Shirley, 1996]. Since multiple γ-rays were utilized for assay, the inverse square of error (σ i) associated with count rates (y i) were used as weighting factors in calculating the weighted average values (ȳ) of each sample; ȳ = ∑y i.(1/σ i2)/ ∑(1/σ i2) and σi =
[∑(1/σ i2)] -1/2 .Activities were decay-corrected to obtain the activities at the end of bombardment (EOB), and appropriate corrections were applied to convert the count rates to disintegration rate and to make small correction due to decay during counting period; , where A is activity (disintegration per sec, dps), C 𝐴 = (𝜆𝐶) [(1 ― 𝑒 ―𝜆𝑡 )𝜀 𝐼 𝛾 ]
is the uncorrected count rate (counts per sec, cps), t is count time (counting period, sec), λ is the decay constant (1/sec), I γ is γ-ray intensity, and ε is detector efficiency for a specific γ-ray. The activity at EOB per unit mass of target, representing the production yield of the radionuclide, reported in terms of MBq/mg (activity of 131 Ba/mass of 130 Ba and activity of 133 Ba/mass of 132 Ba). For half-life measurement, activities of
131
Ba in several samples were followed for several half-lives and the CLSQ code (Cummings, 1962) was used to obtain the best fit. Systematic error was calculated based on the errors associated with target mass, activity rates, detector efficiencies (3%) and neutron flux (3%). The error propagation throughout the cross-section calculations was based on the sum of squares of the systematic relative errors or the deviation from mean of the data points, whichever was greater. With only a few exceptions, the errors based on the deviations from mean were greater.
2.4 Theoretical calculations. Theoretical two-group calculations were conducted using the IsoChain code (Schmittroth, 2006; Almanza et al., 2006); the JAVA version of its FORTRAN-based predecessor LAURA. This code is used to calculate the transmutation and burnup of different nuclei during irradiation (Mirzadeh and Walsh, 1998). IsoChain uses analytical methods to solve the Bateman equations with multiple radionuclides over a user specified time interval and input flux. Further, IsoChain calculates the production yield of an isotope based on the two-group theory using the thermal and epithermal cross sections and the thermal-to-epithermal neutron flux ratio of the irradiation facility. In this notation, σeff = σ0 + (1/r) I 0, where
σ0 ≡ thermal cross section, I 0 ≡ resonance integrals, and r = φth / φepi (Mirzadeh et al. 2011). The IsoChain input data were determined according to each designed experiment. Input fluxes and flux ratios for each target are shown in Table 2. When the burn up cross-section is insignificant, the production term reduces to the first term of the Bateman equation , allowing for the direct calculation of the two-𝐴 = 𝑅(1 ― 𝑒 ―𝜆 𝑡 𝑖𝑟𝑟 )
group cross-section. In Eq (1), (denoted as saturation yield) and N 0 is number of target atoms, 𝑅 = 𝑁 0𝜑 𝑛 𝜎 𝑒𝑓𝑓
φn is the thermal neutron flux, and σeff is the effective cross section of the target atom. 6
Results and Discussion.
In this study, for an accurate measurement of production and burn up cross-sections of 131 Ba, 16 targets containing mg quantities of enriched 130 Ba were irradiated in the hydraulic tube facilities of ORNL HFIR. The target material was ~34.6% enriched in 130 Ba and contained 0.406% 132 Ba by mass. The 132 Ba content of the target material, provided additional information about production cross-sections of 133 Ba. The chemical form of the target was carbonate and each ampoule contained ~1 mg of BaCO 3 (Table 2).
The production yields of 131 Ba at EOB as a function of irradiation time at the HFIR HT are summarized in Table 3. Targets 1-3 were used to assess the contribution of epi-thermal and thermal neutrons to the total reaction rates by measuring resonance integrals and thermal cross-sections. This set of targets, consisting of two Cd-filtered targets (target 1 and 2) and one target consisting of two unfiltered samples (bare, targets 3a and 3b) were irradiated for 1 hour at 10.7 MW th power with a thermal neutron flux of 2.58 x 10 14 n·cm -2 ·s -1 .The cadmium filtered targets allowed for the direct measurement of the cross sections (resonance integrals) occurring at the epithermal neutron energies. Targets 4-9, each consisting of two unfiltered samples, were used to assess the burn-up cross section of the 131 Ba. These targets were irradiated for durations ranging from 3 to 26 days at full reactor power (85 MW th ), and were subjected to slightly different neutron fluxes ranging from 1.8-2.0 x10 15 n·cm -2 ·s -1 with a thermal to epithermal ratio of ~30, dependent upon their position within the HT array (Table 2). For visual presentation, the yields given in Column 4 of Table 3, were normalized to a thermal flux of 1.8x10 15 n·s -1 ·cm -2 (φth /φ epi ≈ 30) and plotted as a function of irradiation time in Figure 2. In this figure, the experimental data with their associated errors are shown as points, and the solid curves are the theoretical yields of 131 Ba for burn-up cross-sections ranging from 100 b to 300 b ( 130 Ba cross-section was held constant at σ0 = 6.9 b, I 0 = 173 b) As shown, under the above experimental conditions, the yield of
131
Ba approaches a saturation value of ~60 GBq (1.62 Ci) per mg of 130 Ba for 20 days irradiation at φth =1.8x10 15 n·s -1· cm -2 , with φth /φ epi ≈ 30, with an effective production cross-section of ~10 b.
The average values of the resonance integrals for 130 Ba[n,γ] 131 Ba reaction from the two cadmium filtered targets 1 and 2 was 173 ± 7 b. This value was then used to calculate a thermal neutron cross section of 6.9 ± 0.5 b for the unfiltered targets (3a and 3b). These cross-section values represent the sum of cross-sections to metastable and ground state of 131 Ba, as the 14.3-m 131m Ba decays via IT (100%) to 131g Ba (see Figure 1). The resonance cross-section of 130 Ba measured in our studies is in excellent agreement with the evaluated value of 176 ± 7 b [Mughabghab, 2006], but 12% lower than 197 ± 10 reported by Dauenhauer et. al., [Dauenhauer and Krane, 2012]. For the thermal neutron cross section of 130 Ba, our measurement is 10% lower than reported value of 7.65 ± 0.36 b [Dauenhauer and Krane], but lower than the adopted value of 8.68 ± 0.90 by 20% [Mughabghab, 2006]. 7
The effective production cross-sections (σ eff ) of 131 Ba for irradiation time of 3-26 days are given in column 5 of Table 3. The gradual decrease of the σeff as a function of irradiation time, from 12 b for 3 d irradiation to 9 b for 25 d irradiation, is an indication of significant burn up cross-section of 131 Ba, i.e. cross section of
131
Ba[n,γ] 132 Ba reaction. Depletion of the 130 Ba target nuclei only accounts for a ~6% reduction in a 26-day irradiation under our experimental conditions. As noted, we used an empirical methodology, i.e. superimposing the theoretical calculations on experimental measurements, to obtain the burn up cross-section of 131 Ba. The methodology was used earlier for estimating the burn up cross-sections of 187 W [Ersoz 2018] and 147 Pm [Broderick et al. 2018]. For the theoretical calculation, we used our measured values of 6.9 b and 173 b for the thermal and resonance cross-sections of 130 Ba to 131g+m Ba, respectively (see Figure 1) and assumed a factor of 10 for the resonance-to-thermal ratio (I o/σ 0) for 131 Ba to 132 Ba. We further assumed that the cross-section of 131m Ba to 132 Ba was 10 times smaller than cross-section of 131g Ba to 132 Ba. The magnitude of 131m Ba to 132 Ba cross-section, however, has an insignificant effect on the calculated 131 Ba yields because of the short half-life of 131m Ba. Then we varied the thermal cross-section, until the calculated activity of 131 Ba was equal to the experimental values given in Table 3, column 4, and the associated errors were at a minimum. The calculated yields of 131 Ba (in MBq/mg of 130 Ba) for thermal cross-sections ranging from 100 b to 300 b are shown in Figure 2. Using the least squares technique, (Bevington and Robinson, 2003), the best fit between experimental and theoretical yields was obtained with 200 ± 15 b cross section (I 0 / σ0 = 10) (Fig. 2). The quoted error is derived from the average of relative errors associated with each yield measurement given in Table 3 ( σrel = (1/N) SQRT [Σ (σ i/V i)2]). Figure 3 shows the goodness-of-fit, ,χ2
between experimental and theoretical data, which is calculated by following relationship: χ2 = Σ {σ ―2
𝑖
[𝑦 𝑖
}, where and are the experimentally measured yields and the calculated yields for different ― 𝑦( 𝑥 𝑖 )] 2 𝑦 𝑖 𝑦( 𝑥 𝑖 )
thermal cross-sections, , respectively, and σi in the error associated with . The approximate location of 𝑥 𝑖 𝑦 𝑖
the minimum was calculated by fitting a parabolic function through the and solving for the value of at χ2 𝑥 𝑖
the minimum. The least squares treatment of the data gave a value of 200 ± 10 b. Considering that we used an arbitrary value of 10 for I 0 to σ0 ratio, we believe an error of 50 or 20% is a more realistic value. Direct experimental measurements of resonance cross sections of 131 Ba utilizing longer irradiation times under cadmium filter, however, is not possible due to rapid depletion of 113 Cd (σ 0 = 2.8x10 5 b) at very high thermal neutron flux. Increasing the mass of Cd to compensate for depletion of 113 Cd is not an option because of the heat generation within the target, the low melting point of Cd metal, and the limited internal volume of the irradiation canisters (~1.5 cm 3). Possible use of other neutron filters with higher melting points such as Gd metal or boron nitride is beyond the scope of present work. 8
Following the decay of 131 Ba in the 16 samples for several half-lives provided a new value 11.657 ± 0.008 d for the half-life of 131 Ba. The new value, based on the weighted average of 16 independent measurements, is slightly larger than the published half-life of 11.52 ± 0.01 d [Dauenhauer and Krane, 2012]. The weighted average was calculated using the equation given in Section 2.3. The experimental yields of 133 Ba are summarized in Table 4. As shown, the yield of 133 Ba is ~300 MBq (~8 mCi) per mg of 132 Ba for one cycle irradiation at φth = 1.8x10 15 n·s -1 ·cm -2 , with φth /φ epi ≈ 30. The effective production cross-section of 133 Ba was 8.7 ± 0.1 b. The average values of the resonance integrals for the
132
Ba[n,γ] 133 Ba reaction from the two cadmium-filtered targets 1 and 2 was 39.9 ± 1.3 b. This value was then used to calculate a thermal neutron cross section of 7.2 ± 0.2 for the unfiltered targets (3-9). These values represent the sum of cross-sections to both 133 Ba metastable and ground states as the 38.9-h 133m Ba decays via 100% IT to 133 Ba ground state (see Figure 1). The cumulative thermal and resonance cross sections of
132
Ba to both states of 133 Ba measured in this study are in good agreement with the evaluated values of 7.0 ± 0.8 b and 37.8 b, respectively [Mughabghab, 2006]. Our measured values for both terms, however, are about 10% lower than that reported by Dauenhauer et. al., [Dauenhauer and Krane, 2012]. The discrepancy most likely is due to the use of neutron flux monitoring reactions. Dauenhauer et. al. exclusively used
187
Au[n,γ] 188 Au reaction, whereas our neutron flux measurement is based on the averaged values from
187
Au[n,γ] 188 Au, 109 Ag[n,γ] 110 Ag, 59 Co[n,γ] 60 Co reactions flux monitoring reactions; each exhibiting very different sensitivity to the thermal and epithermal neutrons [Mahmood et al., 1995]. The effective production cross-sections (σ eff ) of 133 Ba given in column 5 of Table 4, shows no discernable variation as a function of irradiation time which ranged from 3-26 d, and the weighted average of σeff of
133
Ba for targets 4-9 was 8.7 ± 0.1 b. The rather constant value of σeff for 133 Ba is a possible indication that the burn up cross-section of 133 Ba (i.e. cross section of the 133 Ba[n,γ] 134 Ba reaction) is not significant. However, our experiment was not sufficiently sensitive to yield a value for the 133 Ba[n,γ] 134 Ba cross-section. The highly enriched 130 Ba used in our experiment contained only 0.41% 132 Ba (Table 1), and because of the high burn-up cross-section of 131 Ba, some 132 Ba is formed from transmutation of 130 Ba through double neutron capture. Calculations indicated that the number of 132 Ba atoms in the target actually increases from 0.41% to ~1% during a 26-d irradiation under our experimental conditions. The increase in number of 132 Ba atoms in the target somewhat compensate for the 133 Ba burn-up. The assessment of the burn-up cross-section of 133 Ba, requires repeating similar study using targets highly enriched in 132 Ba with low of 130 Ba content. Alternatively, the burn-up cross-section of 133 Ba could be measured directly by irradiating a 133 Ba target and quantifying the level of 134 Ba in the target before and after irradiation by mass spectrometry. A summary of the cross-sections measured in our study together with the corresponding evaluated cross-sections 9
(Mugabghab, 2006), and those reported by Dauenhauer et. al., [Dauenhauer and Krane, 2012] are given in Table 5.
3 Conclusion
The short half-life of 131 Ba, combined with relatively large cross-sections and very high thermal neutron flux available from HFIR resulted in a saturation yield of ~60 GBq (1.62 Ci) per mg of 130 Ba for a 20-day irradiation at φth = 1.8x10 15 n·s -1 ·cm -2 , with a thermal-to-epithermal ratio of ~30. The effective production cross-section of 131 Ba was ~10 b. The two group cross-sections of 130 Ba measured in our study are 6.9 ± 0.5 b (thermal) and 173 ± 7 b (resonance). These values represent the sum of cross-sections to the metastable and ground state of 131 Ba. For the first time, we report a burn-up cross-section of 200 ± 50 b for 131 Ba. This cross-section was measured empirically assuming a ratio of 10 for the resonance to thermal cross-sections. Further, the half-life of 131 Ba was re-measured to be 11.657 ± 0.008 d. Lastly, this study also resulted in the co-production of 133 Ba. The experimental yield of 133 Ba is ~370 MBq (~10 mCi) per mg of 132 Ba (thin target) for one cycle irradiation at HFIR (~25 d), with an effective production cross-section of 8.7 ± 0.1 b. The measured two-group 132 Ba cross-sections are 7.2 ± 0.2 b and 39.9 ± 1.3 b. These values also represent the sum of cross-sections to the metastable and ground state of 133 Ba.
4 Acknowledgment
Research supported in part by the Isotope Production/Distribution Program, Office of Nuclear Physics of the US Department of Energy and under a DOE Nuclear Energy University Program Graduate Fellowship. A portion of this research was conducted at the ORNL High Flux Isotope Reactor which is sponsored at ORNL by the Office of Basic Energy Sciences, US DOE. Three students Patricia Glenn (Georgia Institute of Technology), Elizabeth Bressler (Centre College), and Thomas Khinda (University of Chicago) contributed to this study. The authors extend their gratitude to the Drs. Xxx and xxx for critical review of the manuscript, and Mr. G. Hirtz (ORNL Research Reactors Division) for coordinating reactor irradiations. 10
5 References
Almanza, C. L. J., Schmittroth, F., Lovett, H. A., Garland, M.A., Mirzadeh, S., 2006. IsoChain: A user-friendly, two-group nuclear transmutation and decay code, Transactions of the American Nuclear Society 95, 441–442.
Bevington, P. R., Robinson, D. K., 2003. Data Reduction and Error Analysis, 3rd ed., McGraw Hill, NY.
Broderick, K., Lusk, R., Hinderer, J., Griswold, J. R., Boll, R. A., Garland, M., Heilbronn, L., Mirzadeh, S. (2018). Reactor Production of Promethium-147. Applied Radiation and Isotopes, 144, 54-63.
Cumming, J. B., 1962. CLSQ, the Brookhaven decay curve analysis program. National Research Council, Nuclear Science Series, Report No. NAS-NS-3107.
Dauenhauer, A. Y., Kran, K. S., 2012. Neutron capture cross sections of 130,132,134,136,138 Ba. Physical Review C 85, 064301.
Firestone, R.B., Shirley, V.S. (Eds.), 1996. Table of Isotopes, 8 th ed., John Wiley and Sons Inc., New York.
Ersöz, O. A., Spink, R., Griswold, J. R., Yurt, F., Mirzadeh, S., 2018. Measurement of Neutron capture Cross section of 187 W for Production of 188 W. Applied Radiation and Isotopes 148, 191-196.
Kehwar, T., 2009. Use of Cesium-131 radioactive seeds in prostate permanent implants. Journal of medical physics/Association of Medical Physicists of India 34, 191.
Kuznetsov, R. A., Toporov, Yu. G., 2009. Development of radionuclide production using research reactors of RIAR, In: Proceedings of the 6th Russian Conference on Radiochemistry. October 12–16, Moscow. Abstracts, p. 352
Garland, M.A., Mirzadeh, S., Alexander, C.W., Hirtz, G.J., Hobbs, R.W., Pertmer, G.A., & Knapp, F.F., 2003. Neutron flux characterization of a peripheral target position in the High Flux Isotope Reactor. Applied Radiation and Isotopes 59, 63-72.
Hogle S., Boll R.A., Murphy K., Denton, D., Owens, A., Haverlock, T.J., Garland, M., Mirzadeh, S., 2016. Reactor Production of Thorium-229. Applied Radiation and Isotopes, 114, 19-27.
Meikrantz, D. H., Snyder, J. R., 2012. Method of separation of Cesium-131 from Barium. US Patent 8,270,554 B2, Sep. 18, 2012.
Mahmood, S.T., Mirzadeh, S., Farrell, K., Pace, J.V., Oliver, B.M., 1995. Neutron Dosimetry of the HFIR Hydraulic Facility. ORNL/TM-12831, Lockheed Martin Energy Systems, Inc., Oak Ridge National Laboratory.
Mirzadeh, S., Mausner, L. F., Garland, M. A., 2011. Reactor-Produced Medical Radionuclides. In Handbook of Nuclear Chemistry, Second Edition (Volume 4, Radiochemistry and Radiopharmaceutical Chemistry in Life Sciences, Chapter 38, PP 1857 – 1902), Vertes, A., Nagy, S., Klencsar, Z., Lovas, R. G., Roesch, F., editors. New York, NY: Springer.
Mirzadeh, S., Schenter, R.E., Callahan, A.P., Knapp, F.F., 1992. Production Capabilities in U.S. Nuclear Reactors for Medical Radioisotopes, Report no. ORNL/TM-12010. Oak Ridge National Laboratory, Oak Ridge, TN. 11
Mirzadeh, S., Walsh, P., 1998. Numerical evaluation of the production of radionuclides in a nuclear reactor (Part I & II). Applied Radiation and Isotopes 49, 379-382 and 383-395. Mughabghab, S.F., 2006. Atlas of Neutron Resonances: Resonance Parameters and Thermal Cross Sections. Z= 1-100. Elsevier. Schmittroth, F., 2006. IsoChain: A JAVA-based user-friendly nuclear transmutation and decay code based on FORTRAN code Laura. Unpublished. Skowronek, J., 2017. Current status of brachytherapy in cancer treatment – short overview. J. Contemp. Brachytherapy. 9, 581–589. doi: 10.5114/jcb.2017.72607. Stoughton, R.W., Halperin, J., 1959. Heavy nuclide cross sections of particular interest to thermal reactor operation: conventions, measurements and preferred values. Nucl. Sci. Eng. 6, 100. Tárkányi, F., et al., 2009. Cross section measurements of the 131 Xe(p,n) reaction for production of the therapeutic radionuclide 131 Cs. Applied Radiation and Isotopes, 67, 1751-1757. Tárkányi, F., et al., 2010. Study of activation cross sections of proton induced reactions on barium: Production of 131 Ba→ 131 Cs. Applied Radiation and Isotopes, 68, 1869-1877. Tilbury, R. S., Kolar, A. J., 1986. Production of Ba-131/Cs-131 for use in radiotherapy implants. System Cancer Center, University of Texas at Houston (USA). Report DOE/ER/60176-2 (1986). Zlokazov, S., Swanberg, D. J., Egorov, O., Brown, G. N., Boyce, D. E., 2012. Method for large scale production of cesium-131 with low cesium-132 content. United States Patent 2012O142993A1, Jun. 7, 2012 2. Nuclear data Sheet, NDS 72,487 (1994) 12
Captions to the figures Figure 1. Scheme for reactor production of 131 Ba and 133 Ba. Values on the top of horizontal arrows are the neutron capture cross-sections; thermal (σ 0) and resonance integrals (I 0). The bolded values are those measured in this study, representing the cumulative cross sections to the metastable and ground states of
131
Ba and 133 Ba. Other nuclear data are from [Mughabghab, 2006] and [Firestone and Shirley, 1996].
Figure 2. 131 Ba yield at EOB as a function of irradiation time at HFIR HT at nominal thermal flux of 1.8x10 15 n·s -1 ·cm -2 , and φth /φ epi ≈ 30. The points are experimental measurements, and solid curves are calculated yields for 131 Ba burn up cross-sections (thermal) of 100 - 300 b, assuming I 0/σ 0 =10.
Figure 3. Plot of goodness-of-fit, Chi-squared, as a function of IsoChain-generated thermal neutron cross-sections. Location of the minimum is calculated by fitting a parabola through the Chi-squared value of the calculated data points (100 to 300 b) closer to the minimum (200 b). 13
Table 1. Isotopic Compositions of Enriched Ba Target
Composition (%)
Isotope
Atom Mass
130
Ba 35.839 34.600
132
Ba 0.414 0.406
134
Ba 2.581 2.568
135
Ba 5.547 5.561
136
Ba 5.556 5.612
137
Ba 7.187 7.312
138
Ba 42.875 43.940 14
Table 2. Targets, irradiation times, positions in the Hydraulic tube and corresponding thermal neutron fluxes a
Target Mass (mg) Target No. BaCO 3 130 Ba 132 Ba
Irradiation Time (d)
HT Position φth /φ epi b Thermal Neutron Flux (n·s -1 ·cm -2 )
1 (F) 0.93 0.223 2.61 x 10 -3 0.04 5 31.0 2.58 x 10 14
2 (F) 0.81 0.194 2.27x 10 -3 0.04 5 31.0 2.58 x 10 14
3a 1.08 0.268 2.97 x 10 -3 0.04 5 31.0 2.58 x 10 14
3b 0.98 0.243 2.94 x 10 -3 0.04 5 31.0 2.58 x 10 14
4a 1.25 0.299 3.51 x 10 -3 2.92 6 28 1.95 x10 15
4b 1.71 0.409 4.80 x 10 -3 2.92 6 28 1.95 x10 15
5a 1.01 0.242 2.83 x 10 -3 3.00 4 26 2.00 x10 15
5b 0.70 0.168 1.97 x 10 -3 3.00 4 26 2.00 x10 15
6a 1.00 0.239 2.81 x 10 -3 6.89 3 26 1.80 x10 15
6b 0.80 0.191 2.25 x 10 -3 6.89 3 26 1.80 x10 15
7a 1.02 0.244 2.86 x 10 -3 9.89 3 26 1.80 x10 15
7b 0.71 0.170 1.96 x 10 -3 9.89 3 26 1.80 x10 15
8a 1.22 0.292 3.42 x 10 -3 19.6 4 28 2.00 x10 15
8b 1.19 0.285 3.34 x 10 -3 19.6 4 28 2.00 x10 15
9a 0.89 0.213 2.49 x 10 -3 25.9 4 28 2.00 x10 15
9b 0.79 0.177 2.07 x 10 -3 25.9 4 28 2.00 x10 15
a) All targets unfiltered expect for targets 1 and 2.
b) φth /φ epi : Thermal to Epithermal neutron flux ratio. 15
Table 3. Yields and production cross-sections of 131 Ba at EOB as a function of irradiation time at the HFIR HT irradiation facility
a
Targets 1 and 2 consisted of one sample per target and were Cd filtered. All targets were un-filtered and consisted of two samples per target. In all targets, 131 Ba enrichment was 35.8%.
b
σeff = σ0 + (1/r)I 0 , where r = φth /φ epi
Neutron Capture Cross-sections of 130 Ba, (b)
Target No. a Irradiation Time (d)
Thermal Neutron Flux (n·cm -2 ·s -1 )
131
Ba Yield at EOB (MBq/mg) Effective, σeff b Thermal, σ0 Resonance, I0
1 (F) 0.04 2.58 x 10 14 (1.53± 0.02) x 10 1 n/a - 166 ± 2
2 (F) 0.04 2.58 x 10 14 (1.65 ± 0.02) x 10 1 n/a - 179 ± 2
Average 173 ± 7
3a,b 0.04 2.58 x 10 14 (3.57 ± 0.14) x 10 2 12.5 ± 0.5 6.9 ± 0.5 173
4a,b 2.92 1.95 x 10 15 (1.66 ± 0.07) x 10 4 12.0 ± 2.2
5a,b 3.00 1.80 x 10 15 (1.76 ± 0.08) x 10 4 12.1 ± 0.5
6a,b 6.89 1.80 x 10 15 (2.89 ± 0.18) x 10 4 10.7 ± 0.7
7a,b 9.89 1.80 x 10 15 (3.61 ± 0.01) x 10 4 10.1 ± 0.1
8a,b 19.6 2.00 x 10 15 (6.02 ± 0.31) x 10 4 9.8 ± 0.5
9a,b 25.9 2.00 x 10 15 (6.13 ± 0.31) x 10 4 8.7 ± 0.4 16
Table 4. Production yield of 133 Ba at HFIR HT and Neutron Capture Cross-sections of 132 Ba a
Neutron Capture Cross-sections of 132 Ba, (b) Target No. Irradiation Time (d)
133
Ba Activity at EOB (MBq/mg) Effective b Thermal Resonance
1(F) 4.17x10 -2 (1.08 ± 0.04) x 10 -2 n/a - 38.7 ± 1.5
2(F) 4.17x10 -2 (1.15 ± 0.06) x 10 -2 n/a - 41.2 ± 2.1
Average (1.11 ± 0.04) x 10 -2 n/a - 39.9 ± 1.3
3(a, b) 4.17x10 -2 (8.25 ± 0.18) x10 -2 9.5 ± 0.6 8.2 ± 0.6 39.9
4(a, b) 2.92 (4.09 ± 0.29) x 10 1 8.9 ± 0.6 7.5 ± 0.7 39.9
5(a, b) 3.00 (3.75 ± 0.16) x 10 1 8.6 ± 0.4 7.1 ± 0.5 39.9
6(a, b) 6.89 (8.83 ± 0.41) x 10 1 8.9 ± 0.4 7.3 ± 0.5 39.9
7(a, b) 9.89 (1.22 ± 0.05) x 10 2 8.6 ± 0.1 7.0 ± 0.4 39.9
8(a, b) 19.6 (2.72 ± 0.03) x 10 2 8.7 ± 0.2 7.2 ± 0.3 39.9
9(a, b) 25.9 (3.72 ± 0.06) x 10 2 9.0 ± 0.6 7.5 ± 0.7 39.9
Weighted Average (Targets 4-9) 8.7 ± 0.1 7.2 ± 0.2
a
Targets 1 and 2 consisted of one sample per target and were Cd filtered. All other targets were un-filtered and consisted of two samples per target. In all targets, 132 Ba content was 0.414 atom %.
b
σeff = σ0 + (1/r)I 0 , where r = Ø th /Ø Epi 17
Table 5. Neutron capture cross-sections of barium-130, 131 and 132 isotopes
a
This cross-section was measured empirically assuming a I 0/ σ0 = 10; based on this assumption a value of 2000 b for I 0 given in parenthesis in column 6.
Thermal cross-section , σ0 (b) Resonance integral , I 0 (b)
Reported work Reported work Target nuclei
Product nuclei This work Mugha bghab, 2006
Dauenhauer, 2012
This work Mugha bghab, 2006
Dauenhauer, 2012
131 g
Ba 7.7 ± 0.9 7.15 ± 0.34 153 ± 7 178 ± 10
131 m
Ba 0.98 ± 0.05 0. 596 ± 0. 037 23 ± 1 19.3 ± 0.9 130 Ba
sum 6.94 ± 0.47 8.7 ± 0.9 7.75 ± 0. 34 173 ± 7 176 ± 7 197 ± 10
131
Ba 132 Ba 200 ± 50 a (2000)
133 g
Ba 6.5 ± 0.8 7.51 ± 0.32 35.0 42.5 ± 2.2
133 m
Ba 0.5 0.682 ± 0.029 2.80 4.32 ± 0.22 132 Ba
sum 7.23 ± 0.18 7.0 ± 0.8 8.19 ± 0.32 39.9 ± 1.3 37.8 46.8 ± 2.2 18
Figure 1. Scheme for reactor production of 131 Ba and 133 Ba. Values on the top of the horizontal arrows are the neutron capture cross-sections: thermal (σ 0) and resonance integrals (I 0). The bolded values are those measured in this study, representing the cumulative cross sections to the metastable and ground states of
131
Ba and 133 Ba. Other nuclear data are from [Mughabghab, 2006] and [Firestone and Shirley, 1996].
NR: Not reported
54
74
56
116
132
Ba 133Ba 134Ba
131
Cs 132Cs 133Cs
11.65d14.6m
stable
(100%)
6.475d9.69d
IT
(100%)
131
Ba
stable
(0.106%)
131
Xe
stable
(21.2%)
130
Ba
1.6210.51y
stable
(0.101%)
IT
(100%)
stable
(2.42%)
132
Xe
11.9d
IT
(100%)
stable
(26.9%)
6.94, 173 200, 2000 7.23, 39.9
87
[n,]
EC (100%) EC (100%)
EC (100%) EC (98%)
78
76
[n,]
(2%)
NR19
Irradiation time (days)
0 5 10 15 20 25
Adjusted 131Ba activity (MBq/mg of 130Ba)
104
100
125
150
175
200
220
250
275
300
2
3
4
5
6
7
8
0(b)
(toptobottom)
Figure 2. 131 Ba yield at EOB as a function of irradiation time at HFIR HT at nominal thermal flux of 1.8x10 15 n·s -1 ·cm -2 , and φth /φ epi ≈ 30. The points are experimental measurements, and solid curves are calculated yields for 131 Ba burn up cross-sections (thermal) of 100 - 300 b, assuming I 0/σ 0 =10. 20
Calculated thermal cross-section of 131Ba (b)
100 150 200 250 300
Goodnees-of-fit, 2, for 131Ba activity
2
4
6
8
10
12
14
16
Figure 3. Plot of goodness-of-fit, Chi-squared, as a function of IsoChain-generated thermal neutron cross-sections. Location of the minimum is calculated by fitting a parabola through the Chi-squared value of the calculated data points (100 to 300 b) closer to the minimum (200 b). |
10943 | https://byjus.com/magnetic-flux-formula/ | Magnetic Flux
Magnetic flux is defined as the total number of magnetic field lines through a given coil or area. It is the common component of the magnetic field which passes through the coil. Magnetic flux is denoted by ΦB where B is a magnetic field and its unit is Weber (Wb). The magnetic flux value depends on the magnetic field direction and it is a vector quantity.
The magnetic flux formula is given by,
(\begin{array}{l}\Phi_B=B\cdot A\end{array} )
(\begin{array}{l}\Phi_B = BA\ cos\ \theta\end{array} )
B = Magnetic field,
A = Surface area and
θ = Angle between the magnetic field and normal to the surface.
Solved Example
Example 1
The Dimension of a rectangular loop is 0.50m and 0.60m. B and θ are 0.02T and 45° respectively. Determine the magnetic flux through the surface.
Solution:
Given
Dimensions of rectangular loop = 0.50m and 0.60m,
B = 0.02T
θ = 45°
Magnetic flux formula is given by
ΦB = B A Cos θ
Area, A = 0.50 × 0.60
= 0.3 m2
ΦB = 0.02 × 0.3 × Cos 45
ΦB = 0.00312 Wb
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10944 | https://www.khanacademy.org/math/grade-6-math-snc-aligned/xa86a6c4721d5a691:geometry/xa86a6c4721d5a691:mathematical-reflection/v/determining-reflections | Determining reflections (video) | Geometry | Khan Academy
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Grade 6 Math -Pakistan National Curriculum
Course: Grade 6 Math -Pakistan National Curriculum>Unit 7
Lesson 6: Mathematical reflection
Reflecting points
Reflect points
Reflecting shapes
Reflect shapes
Determining reflections
Determine reflections
Reflections review
Math>
Grade 6 Math -Pakistan National Curriculum>
Geometry>
Mathematical reflection
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Determining reflections
CCSS.Math: HSG.CO.A.5
Google Classroom
About About this video Transcript
A line of reflection is an imaginary line that flips one shape onto another. We find this line by finding the halfway points between matching points on the source and image triangles. All of the halfway points are on the line. Once we find that line, it shows how one triangle reflects onto the other.
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Mohammad Zayd 6 years ago Posted 6 years ago. Direct link to Mohammad Zayd's post “I have a question. To fin...” more I have a question. To find the line of reflection for a triangle, could someone count all the spaces between the two same vertices and then divide them by two. Then add that quotient to a vertice. One example could be in the video. The distance between Triangle ABC's vertice of C and Triangle A'B'C''s vertice of C is six. So then divide six by two to get 3. Then add that 3 to Triangle A'B'C' vertice c's Y-coordinate to get 1. The line of reflection is on the Y-coordinate of 1. Sorry if this was a little confusing. It is difficult to type about Triangle A'B'C' and the different vertices. Sorry.
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Ellie 2 years ago Posted 2 years ago. Direct link to Ellie's post “Yes, you can do it that w...” more Yes, you can do it that way, although you probably figured that out by now because it's been 4 years.
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Aryanna Cortez 3 years ago Posted 3 years ago. Direct link to Aryanna Cortez's post “Do you know any tricks or...” more Do you know any tricks or like an easier way to find reflections?
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The Telepath 3 years ago Posted 3 years ago. Direct link to The Telepath's post “I use a memorization tric...” more I use a memorization trick. Let's say you are given the point (2, -7).
To reflect across the x-axis, use the rule (x, -y). This will give you (2, 7).
To reflect across the y-axis, use the rule (-x, y). This gives you (-2, -7).
To reflect across the line y=x, use the rule (y, x). This gives you (-7, 2).
To reflect across the line y=-x, use the rule (-y, -x). This gives you (7, -2).
Just memorize these formulas and you'll be good. You don't have to graph a point to find its reflection point.
Hope this helps :D
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zaksab1 2 years ago Posted 2 years ago. Direct link to zaksab1's post “i didn't understand” more i didn't understand
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joshua 2 years ago Posted 2 years ago. Direct link to joshua's post “Please specify what you d...” more Please specify what you didn't understand. To do reflection for a shape, simply reflect each point respectively, last connect it, forming the reflected shape.
To know where do you place the reflected point, simply count how many unit(s) is there from that initial point to the line of reflection. Then place the point on the other side of the line of reflection with the same number of unit(s).
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Anderson Adoral 2 years ago Posted 2 years ago. Direct link to Anderson Adoral's post “what if the line of refle...” more what if the line of reflection os oblique? is there a general rule for the points?
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Venkata 2 years ago Posted 2 years ago. Direct link to Venkata's post “One thing you could do is...” more One thing you could do is this: Consider the point given and the line of reflection (which is oblique). Now, draw a line from the point till you intersect the line of reflection. After you intersect it, draw a line perpendicular to the line you just drew, but make sure that this line is equal in length to the first line. Where your second line stops is the reflection of the point.
Observe that the idea here is to make a square with the point as one corner and the line of reflection as the diagonal.
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mohidafzal31 6 years ago Posted 6 years ago. Direct link to mohidafzal31's post “I can't seem to find it a...” more I can't seem to find it anywhere, but one of the questions in a worksheet given by my teacher, we are asked to:
Reflect at "y = -x"
Is there a video or exercise on this that I missed? if not then pls guide me
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mohidafzal31 6 years ago Posted 6 years ago. Direct link to mohidafzal31's post “Nevermind, punching y = ...” more Nevermind, punching y = -x into desmos gave me the line of reflection!
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parisah.jiwani a year ago Posted a year ago. Direct link to parisah.jiwani's post “He said that A was 5 unit...” more He said that A was 5 units from the line but in the video, it looks like 6 and A' looks like only 4 units away from the line even though he also said it was 5. Did I miss something?
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joshua a year ago Posted a year ago. Direct link to joshua's post “We are talking about the ...” more We are talking about the green line, not the x-axis.
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MartiW 2 years ago Posted 2 years ago. Direct link to MartiW's post “the dang volume isn't wor...” more the dang volume isn't working, so, I am confused on what's happening. Please help me with this!
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Storm_0891 2 years ago Posted 2 years ago. Direct link to Storm_0891's post “Sal was just attempting t...” more Sal was just attempting to find the where the line of reflection is at. He found it, then checked his work with the points by counting how many units away they were from the line.
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CamarionS 2 years ago Posted 2 years ago. Direct link to CamarionS's post “can u explain better next...” more can u explain better next time
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#1 a year ago Posted a year ago. Direct link to #1's post “All you have to do is fin...” more All you have to do is find the distance between two points and divide that number by 2. Then, you just put a line there.
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<3 2 years ago Posted 2 years ago. Direct link to <3's post “does the video cover if t...” more does the video cover if there is an uneven number of units between ABC and A’B’C’ etc ?
if not, how would that be solved?
please help!
kumi
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Mango 2 years ago Posted 2 years ago. Direct link to Mango's post “I mean, I don't think you...” more I mean, I don't think you'd usually see that in a question. And if you did, the reflection line must thus be a decimal.
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Ryan Wilson 6 years ago Posted 6 years ago. Direct link to Ryan Wilson's post “How do you explain if the...” more How do you explain if there is or is not a line of refection
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Video transcript
[Instructor] We're asked to draw the line of reflection that reflects triangle ABC, so that's this blue triangle, onto triangle A prime B prime C prime, which is this red triangle right over here. And they give us a little line drawing tool in order to draw the line of reflection. So the way I'm gonna think about it is well, when I just eyeball it, it looks like I'm just flipped over some type of a horizontal line here. But let's see if we can actually construct a horizontal line where it does actually look like the line of reflection. So let's see, C and C prime, how far apart are they from each other? So if we go one, two, three, four, five, six down. So they are six apart. So let's see if we just put this three above C prime and three below C, let's see if this horizontal line works as a line of reflection. So C, or C prime is definitely the reflection of C across this line. C is exactly three units above it, and C prime is exactly three units below it. Let's see if it works for A and A prime. A is one, two, three, four, five units above it. A prime is one, two, three, four, five units below it. So that's looking good. Now let's just check out B. So B, we can see it's at the y-coordinate here is seven. This line right over here is y is equal to one. And so what we would have here is, let's see, this looks like it's six units above this line, and B prime is six units below the line. So this indeed works. We've just constructed the line of reflection that reflects the blue triangle, triangle ABC, onto triangle A prime B prime C prime.
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10945 | https://www.youtube.com/watch?v=Rii__S8J68s | Exponents With Integer Bases - How To Evaluate an Exponent With a Negative Base
Miacademy & MiaPrep Learning Channel
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Ready to expand your knowledge about exponents? In this math lesson for 7th graders, students will review exponent basics and learn how to evaluate exponential expressions with negative bases. This lesson is from Miacademy and MiaPrep's Math: Level H course. Check out our playlists for more 7th grade math lessons!
We hope you are enjoying our large selection of engaging core & elective K-12 learning videos. New videos are added all the time - make sure you come back often to learn more! If you'd like us to cover any additional topics, please let us know. For practice, assessment, and many interactive activities that go along with each video, as well as a teacher/parent dashboard, go to Miacademy.co for Grades K-8 or Miaprep.com for Grades 9-12!
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Transcript:
Oh, hey everyone. Justin here. Guess
who just joined a science club. It's been awesome so far. Today I'm in the
lab measuring temperature changes for reactions we're studying. Come on in
and check out my lab station. [Music] I just cooled one of the solutions and took
measurements to track the temperature change. Here are my results. At 1 minute, the temperature
dropped 2 degrees Celsius. After 2 minutes, it had dropped 4 degrees, and after 3 minutes, it
had dropped a total of 8 degrees. Do you notice a pattern in how the temperature is changing
over time? Look closely at how the temperature changes from one minute to the next. The total
change is doubling each minute. Since the change doubles each time, we can describe the change
using repeated multiplication. At one minute, the change was 2 degrees. At 2 minutes, it was
2 times 2. At 3 minutes, 2 times 2 times 2. But the temperature is dropping. So each
value should be negative. To show that, we can multiply each of these by negative 1.
These expressions are getting kind of long and if the temperature keeps dropping they'll get even
longer. Is there a more concise way to represent repeated multiplication? We can use exponents. So
to rewrite our temperature change, we'll focus on the twos first since those are the repeated
values. Then we'll come back to the negative 1s later. In an exponential expression, there
are two parts. The base and the exponent. The base is the number being repeatedly multiplied.
What's the base in our temperature pattern? It's 2. The exponent represents how many times the base
gets multiplied. In the first row, after 1 minute, what's the exponent? It's one because we just have
one two. How about after 2 minutes and 3 minutes? What are the exponents? After 2 minutes, 2 was
multiplied by itself 2 times. So the exponent is 2. And after 3 minutes, 2 was multiplied three
times. So the exponent is three. We still need to show that these temperature changes are negative
since the temperature is dropping. So we'll multiply each exponential expression by negative
1 to make the value negative. We can simplify that by just putting a negative sign in front. The base
of each of these exponents is still positive 2, but multiplying by negative 1 makes each value
negative. But what if we used a negative base instead? Would we get the same values? Let me turn
off my lab equipment so we can take a closer look. If we want to write an exponential expression with
a negative base, we'll need to use parentheses around the base including the negative sign, like
this. This shows that negative 2 is the base and it's raised to the power or exponent of 1. We
could represent negative 2 as negative 1 times 2, but negative 1 is inside the parenthesis, so
it's still just part of the base. We'll just leave the base as negative 2 and work with it as
a negative number. So, negative 2 to the power of 1 means negative 2 is multiplied by itself
one time. That's just negative 2. How could we write negative 2 to the power of two as repeated
multiplication? Well, the base is negative 2. So that's the number being multiplied. The
exponent is 2. So we're multiplying the base by itself 2 times negative 2 times negative
2. What's negative 2 times negative 2? To multiply negative numbers, we ignore the signs.
multiply as if they're positive and then apply the sign at the end. So 2 times 2 is 4 and a
negative times a negative is a positive. So this equals positive 4. Whoa, that means this
is different from our temperature values since those were all negative. Let's keep going
and see what happens with larger exponents. How can we write negative 2 to the third power as
repeated multiplication? We'll multiply negative 2 by itself three times. Negative 2 times negative 2
times negative 2. To evaluate it, we just multiply from left to right. Negative 2 times negative 2 is
positive 4. Then we multiply 4 times negative 2. 4 times 2 is 8, and a positive times a negative
is a negative. So, we get negative 8. We're back to a negative number. Try the next three
exponents on your own. Write each one as repeated multiplication, then evaluate it. Pause the video
here to finish the chart in your guided notes. Here's what you should have. Feel free to pause the video here if you need to double
check or fix anything in your notes. Do you see any patterns between the
exponential expressions and their resulting values? Pause the video
to write down anything you notice. As the exponent increases by one, the value
flips between positive and negative. And did you notice how those signs relate to
the exponent? When the exponent was even, the value was positive. But when the exponent
was odd, the value was negative. Why did that happen? With even exponents, the negative base is
multiplied in pairs with none left over. Each pair makes a positive product, so the whole expression
ends up positive. With the odd exponents though, after pairing up the numbers, there's
always one negative left over. Since a positive times a negative is always a
negative, the final value is negative. So for an exponent with a negative base, if the
exponent is even, the resulting value is positive. But if the exponent is odd, the resulting
value is negative. But remember, this is different from what happened with our temperature
measurements. There the base was positive 2 and we multiplied the result by negative 1 to get a
negative value every time. Here the base itself is negative 2 and that causes the values
to alternate. Both of those are different from a base of positive 2 with no negative
1 multiplied to it, which just has positive values. The negative sign and the parentheses
really matter. Let's practice what we just learned with an example problem. Pause the video
here to evaluate these expressions. Remember, start by identifying the base and exponent and
then writing out the repeated multiplication. In part A, 6 to the power of 3 means 6 times
6 times 6. When we multiply, we get 216. In part B, what's the base? It's negative
3 because the parentheses group the negative sign with the three. That means we're
multiplying negative 3 by itself four times. We can multiply two pairs of negative
3 which each gives us positive 9. Then 9 times 9 equals 81. How about part
C? What's the base? It's positive 9. The negative is not in parenthesis. So it's
like a negative 1 is multiplied separately. This is equivalent to negative 1 times
9 times 9. That gives us negative 81. And finally in part D, the base is
negative 5 and we'll multiply it 3 times. Negative 5 times negative 5 is positive
25, and then 25 times negative 5 is negative 125. Each of these expressions showed a
different situation we've seen today. Asking yourself what the base and the exponent
are can help you keep these straight. Thanks for joining me in the lab today. Together, we
found an exponential expression to model our temperature change. And I'm so glad we explored
the difference between positive and negative bases. I'm going to head back to my station,
finish studying those reactions. While I do that, finish up your guided notes and complete levels 1
through 4 of the online practice. Until next time, keep questioning, keep calculating,
and remember, parentheses may be small, but in exponential expressions, they've
got power. See you next time. [Music] |
10946 | https://www.youtube.com/watch?v=ZoO7CRb2sxM | Nut Sort Level 1001 Walkthrough | Nuts — Color Sort 1001 solution
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Nut Sort Level 1001 Walkthrough
Nuts — Color Sort Puzzle Games
Play Brainteaser Nuts Color Sort Puzzle – a sorting game designed to engage and relax you with different levels.
In this addictive sort colors puzzle game, where every level offers an exciting challenge.
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10947 | https://sovathrothsama.wordpress.com/wp-content/uploads/2016/03/ashrae-hvac-2001-fundamentals-handbook.pdf | CONTRIBUTORS In addition to the Technical Committees, the following individuals contributed significantly to this volume. The appropriate chapter numbers follow each contributor’s name.
Charles H. Bemisderfer (1) York International Donald C. Erickson (1) Energy Concepts Co.
Hans-Martin Hellmann (1) Zent-Frenger Thomas H. Kuehn (1, 6) University of Minnesota Christopher P. Serpente (1) Carrier Corp.
Robert M. Tozer (1) Waterman-Gore M&E Consulting Engineers Anthony M. Jacobi (2) University of Illinois at Urbana-Champaign Dr. Arthur E. Bergles (3) Rensselaer Polytechnic Institute Michael M. Ohadi (3, 5) University of Maryland Steven J. Eckels (4) Kansas State University Rick J. Couvillion (5) University of Arkansas Albert C. Kent (5) Southern Illinois University Ray Rite (5) Trane Company Jason T. LeRoy (6) Trane Company Warren E. Blazier, Jr. (7) Warren Blazier Associates, Inc.
Alfred C. C. Warnock (7) National Research Council Canada Larry G. Berglund (8) U.S. Army Research Institute for Environmental Medicine Gemma Kerr (9, 12) InAir Environmental Ltd.
D.J. Marsick (9) U.S. Department of Energy D.J. Moschandreas (9) The Institute for Science, Law, and Technology Kenneth M. Wallingford (9) NIOSH Richard S. Gates (10) University of Kentucky Albert J. Heber (10) Purdue University Farhad Memarzadeh (10) National Institutes of Health Gerald L. Riskowski (10, 11) University of Illinois Yuanhui Zhang (10) University of Illinois Roger C. Brook (11) Michigan State University Joe F. Pedelty (12, 13) Holcomb Environmental Services Pamela Dalton (13) Monell Chemical Senses Center Martin Kendal-Reed (13) Florida State University Sensory Research Institute James C. Walker (13) Florida State University Research Institute Rick Stonier (14) Gray Wolf Sensing Solutions Monica Y. Amalfitano (15) P2S Engineering James J. Coogan (15) Siemens Building Technologies David M. Underwood (15) USA-CERL John J. Carter (16) CPP Inc.
Dr. David J. Wilson (16) University of Alberta William J. Coad (17) McClure Engineering Assoc.
David Grumman (17) Grumman/Butkus Associates Douglas W. DeWerth (18) Robert G. Doerr (19) The Trane Company Eric W. Lemmon (20) National Institute of Standards and Technology Mark O. McLinden (20) National Institute of Standards and Technology Steven G. Penoncello (20) University of Idaho Sherry K. Emmrich (21) Dow Chemical Lewis G. Harriman III (22) Mason-Grant Company Hugo L.S.C. Hens (23) Katholieke Universiteit Achilles N. Karagiozis (23) Oak Ridge National Laboratory Hartwig M. Kuenzel (23) Fraunhofer Institut Bauphysik Anton TenWolde (23) Forest Products Laboratory William Brown (24) Morrison Herschfield Andre Desjarlais (24) Oak Ridge National Laboratory David Roodvoets (24) DLR Consultants William B. Rose (24, 25) University of Illinois William P. Goss (25) University of Massachusetts Malcolm S. Orme (26) Air Infiltration and Ventilation Centre Andrew K. Persily (26) National Institute of Standards and Technology Brian A. Rock (26) University of Kansas Armin F. Rudd (26) Building Science Corporation Max H. Sherman (26) Lawrence Berkeley National Laboratory Iain S. Walker (26) Lawrence Berkeley National Laboratory Craig P. Wray (26) Lawrence Berkeley National Laboratory University of Nebraska-Omaha Grenville K. Yuill (26) University of Nebraska-Omaha Robert Morris (27) Environment Canada Climate and Water Products Division Raymond G. Alvine (28) Jim Norman (28) AAA Enterprises Charles F. Turdik (28) Lynn Bellenger (29) Pathfinder Engineers LLP Steven Bruning (29) Newcomb & Boyd Curt Petersen (29) Thomas Romine (29) Romine, Romine & Burgess Christopher Wilkins (29) Hallam Associates D. Charlie Curcija (30) University of Massachusetts William C. duPont (30) Lawrence Berkeley National Laboratory John F. Hogan (30) Joseph H. Klems (30) Lawrence Berkeley National Laboratory W. Ross McCluney (30) Florida Solar Energy Center M. Susan Reilly (30) Enermodal Engineering Eleanor S. Lee (30) Lawrence Berkeley National Laboratory David Tait (30) Tait Solar Leslie Norford (31) Massachusetts Institute of Technology Fred S. Bauman (32) University of California Mohammad H. Hosni (32) Kansas State University Leon Kloostra (32) TITUS, Division of Tomkins Raymond H. Horstman (33) Boeing Commercial Airplanes Herman F. Behls (34) Behls & Associates Albert W. Black (35) McClure Engineering Associates ASHRAE HANDBOOK COMMITTEE Dennis J. Wessel, Chair 2001 Fundamentals Volume Subcommittee: George Reeves, Chair David E. Claridge Frederick H. Kohloss Brian A. Rock T. David Underwood Michael W. Woodford ASHRAE HANDBOOK STAFF Jeanne Baird, Associate Editor Scott A. Zeh, Nancy F. Thysell, and Jayne E. Jackson, Publishing Services W. Stephen Comstock, Director, Communications and Publications Publisher ASHRAE Research: Improving the Quality of Life The American Society of Heating, Refrigerating and Air-Condi-tioning Engineers is the world’s foremost technical society in the fields of heating, ventilation, air conditioning, and refrigeration. Its members worldwide ideas, identify needs, support research, and write the industry’s standards for testing and practice. The result is that engineers are better able to keep indoor environments safe and productive while protecting and preserving the outdoors for gener-ations to come.
One of the ways that ASHRAE supports its members’ and indus-try’s need for information is through ASHRAE Research. Thou-sands of individuals and companies support ASHRAE Research annually, enabling ASHRAE to report new data about material properties and building physics and to promote the application of innovative technologies.
The chapters in ASHRAE Handbooks are updated through the experience of members of ASHRAE technical committees and through results of ASHRAE Research reported at ASHRAE meet-ings and published in ASHRAE special publications and in ASHRAE Transactions. For information about ASHRAE Research or to become a mem-ber contact, ASHRAE, 1791 Tullie Circle, Atlanta, GA 30329; tele-phone: 404-636-8400; www.ashrae.org. The 2001 ASHRAE Handbook The Fundamentals volume covers basic principles and provides data for the practice of HVAC&R technology. Although design data change little over time, research sponsored by ASHRAE and others continues to generate new information that meets the evolving needs of the people and industries that rely on HVAC&R technol-ogy to improve the quality of life. The ASHRAE technical commit-tees that prepare chapters strive to provide new information, clarify existing information, delete obsolete materials and reorganize chap-ters to make the Handbook more understandable and easier to use.
In this volume, some of the changes and additions are as follows: • Chapter 1, Thermodynamics and Refrigeration Cycles, includes new sections on ideal thermal and absorption cycles, multiple stage cycles, and thermodynamic representation of absorption cycles. The section on ammonia water cycles has been expanded.
• Chapter 12, Air Contaminants, has undergone major revisions.
Material has been added from the 1999 ASHRAE Handbook, Chapter 44, Control of Gaseous Indoor Air Contaminants.
Health-related material with standards and guidelines for expo-sure has been moved to Chapter 9, Indoor Environmental Health.
• Chapter 15, Fundamentals of Control now includes new or revised figures on discharge air temperature control, step input process, and pilot positioners. New are sections on networking and fuzzy logic, revised descriptions on dampers and modulating control, and text on chilled mirror humidity sensors and disper-sive infrared technology.
• Chapter 17, Energy Resources, contains new sections on sustain-ability and designing for effective energy resource use.
• Chapter 19, Refrigerants, provides information on phaseout of CFC and HCFC refrigerants and includes new data on R-143a and R-404A, R-407C, R-410A, R-507, R-508A, and R-508B blends.
• Chapter 20, Thermophysical Properties of Refrigerants, has new data on R-143a and R-245fa. Though most CFC Refrigerants have been removed from the chapter, R-12 has been retained to assist in making comparisons. Revised formulations have been used for many of the HFC refrigerants, conforming to interna-tional standards where applicable.
• Chapter 23, Thermal and Moisture Control in Insulated Assem-blies—Fundamentals, now has a reorganized section on eco-nomic insulation thickness, a revised surface condensation section, and a new section on moisture analysis models.
• Chapter 26, Ventilation and Infiltration, includes rewritten stack pressure and wind pressure sections. New residential sections dis-cuss averaging time variant ventilation, superposition methods, the enhanced (AIM-2) model, air leakage through automatic doors, and central air handler blowers in ventilation systems. The nonresidential ventilation section has also been rewritten, and now includes a commercial building envelope leakage measure-ments summary.
• Chapter 27, Climatic Design Information, now contains new monthly, warm-season design values for some United States loca-tions. These values aid in consideration of seasonal variations in solar geometry and intensity, building occupancy, and use patterns.
• Chapter 29, Nonresidential Cooling and Heating Load Calcula-tions, now contains enhanced data on internal loads, an expanded description of the heat balance method, and the new, simplified radiant time series (RTS) method.
• Chapter 30, Fenestration, now has revised solar heat gain and vis-ible transmittance sections, including information on the solar heat gain coefficients (SHGC) method. The chapter now also has a rewritten section on solar-optical properties of glazings, an expanded daylighting section, and a new section on occupant comfort and acceptance.
• Chapter 31, Energy Estimating and Modeling Methods, now con-tains improved model forms for both design and existing building performance analysis. A new section describes a simplified method for calculating heat flow through building foundations and basements. Sections on secondary equipment and bin-energy method calculations have added information, while the section on data-driven models has been rewritten and now illustrates the variable-base degree-day method.
• Chapter 32, Space Air Diffusion, has been reorganized to be more user-friendly. The section on principles of jet behavior now includes simpler equations with clearer tables and figures. Tem-perature profiles now accompany characteristics of different out-lets, with stagnant regions identified. The section on underfloor air distribution and task/ambient conditioning includes updates from recent ASHRAE-sponsored research projects.
• Chapter 33, HVAC Computational Fluid Dynamics, is a new chapter that provides an introduction to computational methods in flow modeling, including a description of computational fluid dynamics (CFD) with discussion of theory and capabilities.
• Chapter 34, Duct Design, includes revisions to duct sealing requirements from ASHRAE Standard 90.1, and has been expanded to include additional common fittings, previously included in electronic form in ASHRAE’s Duct Fitting Database.
This Handbook is published both as a bound print volume and in electronic format on a CD-ROM. It is available in two editions— one contains inch-pound (I-P) units of measurement, and the other contains the International System of Units (SI). Look for corrections to the 1998, 1999, and 2000 Handbooks on the Internet at Any changes in this volume will be reported in the 2002 ASHRAE Handbook and on the ASHRAE web site.
If you have suggestions for improving a chapter or you would like more information on how you can help revise a chapter, e-mail ashrae@ashrae.org; write to Handbook Editor, ASHRAE, 1791 Tullie Circle, Atlanta, GA 30329; or fax 404-321-5478.
ASHRAE TECHNICAL COMMITTEES AND TASK GROUPS SECTION 1.0—FUNDAMENTALS AND GENERAL 1.1 Thermodynamics and Psychrometrics 1.2 Instruments and Measurements 1.3 Heat Transfer and Fluid Flow 1.4 Control Theory and Application 1.5 Computer Applications 1.6 Terminology 1.7 Operation and Maintenance Management 1.8 Owning and Operating Costs 1.9 Electrical Systems 1.10 Energy Resources SECTION 2.0—ENVIRONMENTAL QUALITY 2.1 Physiology and Human Environment 2.2 Plant and Animal Environment 2.3 Gaseous Air Contaminants and Gas Contaminant Removal Equipment 2.4 Particulate Air Contaminants and Particulate Contaminant Removal Equipment 2.6 Sound and Vibration Control 2.7 Seismic and Wind Restraint Design TG Buildings’ Impacts on the Environment TG Global Climate Change SECTION 3.0—MATERIALS AND PROCESSES 3.1 Refrigerants and Secondary Coolants 3.2 Refrigerant System Chemistry 3.3 Refrigerant Contaminant Control 3.4 Lubrication 3.5 Desiccant and Sorption Technology 3.6 Water Treatment 3.8 Refrigerant Containment SECTION 4.0—LOAD CALCULATIONS AND ENERGY REQUIREMENTS 4.1 Load Calculation Data and Procedures 4.2 Weather Information 4.3 Ventilation Requirements and Infiltration 4.4 Building Materials and Building Envelope Performance 4.5 Fenestration 4.6 Building Operation Dynamics 4.7 Energy Calculations 4.10 Indoor Environmental Modeling 4.11 Smart Building Systems 4.12 Integrated Building Design TG Mechanical Systems Insulation SECTION 5.0—VENTILATION AND AIR DISTRIBUTION 5.1 Fans 5.2 Duct Design 5.3 Room Air Distribution 5.4 Industrial Process Air Cleaning (Air Pollution Control) 5.5 Air-to-Air Energy Recovery 5.6 Control of Fire and Smoke 5.7 Evaporative Cooling 5.8 Industrial Ventilation 5.9 Enclosed Vehicular Facilities 5.10 Kitchen Ventilation SECTION 6.0—HEATING EQUIPMENT, HEATING AND COOLING SYSTEMS AND APPLICATIONS 6.1 Hydronic and Steam Equipment and Systems 6.2 District Energy 6.3 Central Forced Air Heating and Cooling Systems 6.4 In Space Convection Heating 6.5 Radiant Space Heating and Cooling 6.6 Service Water Heating 6.7 Solar Energy Utilization 6.8 Geothermal Energy Utilization 6.9 Thermal Storage 6.10 Fuels and Combustion SECTION 7.0—PACKAGED AIR-CONDITIONING AND REFRIGERATION EQUIPMENT 7.1 Residential Refrigerators and Food Freezers 7.4 Combustion Engine Driven Heating and Cooling Equipment 7.5 Mechanical Dehumidification Equipment and Heat Pipes 7.6 Unitary and Room Air Conditioners and Heat Pumps SECTION 8.0—AIR-CONDITIONING AND REFRIGERATION SYSTEM COMPONENTS 8.1 Positive Displacement Compressors 8.2 Centrifugal Machines 8.3 Absorption and Heat Operated Machines 8.4 Air-to-Refrigerant Heat Transfer Equipment 8.5 Liquid-to-Refrigerant Heat Exchangers 8.6 Cooling Towers and Evaporative Condensers 8.7 Humidifying Equipment 8.8 Refrigerant System Controls and Accessories 8.10 Pumps and Hydronic Piping 8.11 Electric Motors and Motor Control SECTION 9.0—AIR-CONDITIONING SYSTEMS AND APPLICATIONS 9.1 Large Building Air-Conditioning Systems 9.2 Industrial Air Conditioning 9.3 Transportation Air Conditioning 9.4 Applied Heat Pump/Heat Recovery Systems 9.5 Cogeneration Systems 9.6 Systems Energy Utilization 9.7 Testing and Balancing 9.8 Large Building Air-Conditioning Applications 9.9 Building Commissioning 9.10 Laboratory Systems 9.11 Clean Spaces 9.12 Tall Buildings TG Combustion Gas Turbine Inlet Air Cooling Systems SECTION 10.0—REFRIGERATION SYSTEMS 10.1 Custom Engineered Refrigeration Systems 10.2 Automatic Icemaking Plants and Skating Rinks 10.3 Refrigerant Piping, Controls, and Accessories 10.4 Ultra-Low Temperature Systems and Cryogenics 10.5 Refrigerated Distribution and Storage Facilities 10.6 Transport Refrigeration 10.7 Commercial Food and Beverage Cooling Display and Storage 10.8 Refrigeration Load Calculations 10.9 Refrigeration Application for Foods and Beverages TG Mineral Oil Circulation 1.1 CHAPTER 1 THERMODYNAMICS AND REFRIGERATION CYCLES THERMODYNAMICS ............................................................... 1.1 First Law of Thermodynamics .................................................. 1.2 Second Law of Thermodynamics .............................................. 1.2 Thermodynamic Analysis of Refrigeration Cycles .................... 1.3 Equations of State ..................................................................... 1.3 Calculating Thermodynamic Properties ................................... 1.4 COMPRESSION REFRIGERATION CYCLES ......................... 1.6 Carnot Cycle ............................................................................. 1.6 Theoretical Single-Stage Cycle Using a Pure Refrigerant or Azeotropic Mixture ........................................................... 1.8 Lorenz Refrigeration Cycle ....................................................... 1.9 Theoretical Single-Stage Cycle Using Zeotropic Refrigerant Mixture ............................................................. 1.10 Multistage Vapor Compression Refrigeration Cycles .................................................................................. 1.10 Actual Refrigeration Systems .................................................. 1.12 ABSORPTION REFRIGERATION CYCLES .......................... 1.14 Ideal Thermal Cycle ................................................................ 1.14 Working Fluid Phase Change Constraints .......................................................................... 1.14 Working Fluids ........................................................................ 1.15 Absorption Cycle Representations .......................................... 1.16 Conceptualizing the Cycle ...................................................... 1.16 Absorption Cycle Modeling .................................................... 1.17 Ammonia-Water Absorption Cycles ........................................ 1.19 Nomenclature for Examples .................................................... 1.20 HERMODYNAMICS is the study of energy, its transforma-Ttions, and its relation to states of matter. This chapter covers the application of thermodynamics to refrigeration cycles. The first part reviews the first and second laws of thermodynamics and presents methods for calculating thermodynamic properties. The second and third parts address compression and absorption refrigeration cycles, the two most common methods of thermal energy transfer.
THERMODYNAMICS A thermodynamic system is a region in space or a quantity of matter bounded by a closed surface. The surroundings include everything external to the system, and the system is separated from the surroundings by the system boundaries. These boundaries can be movable or fixed, real or imaginary.
The concepts that operate in any thermodynamic system are entropy and energy. Entropy measures the molecular disorder of a system. The more mixed a system, the greater its entropy; con-versely, an orderly or unmixed configuration is one of low entropy.
Energy has the capacity for producing an effect and can be catego-rized into either stored or transient forms as described in the follow-ing sections. Stored Energy Thermal (internal) energy is the energy possessed by a system caused by the motion of the molecules and/or intermolecular forces.
Potential energy is the energy possessed by a system caused by the attractive forces existing between molecules, or the elevation of the system.
(1) where m = mass g = local acceleration of gravity z = elevation above horizontal reference plane Kinetic energy is the energy possessed by a system caused by the velocity of the molecules and is expressed as (2) where V is the velocity of a fluid stream crossing the system boundary.
Chemical energy is energy possessed by the system caused by the arrangement of atoms composing the molecules.
Nuclear (atomic) energy is energy possessed by the system from the cohesive forces holding protons and neutrons together as the atom’s nucleus.
Energy in Transition Heat (Q) is the mechanism that transfers energy across the boundary of systems with differing temperatures, always toward the lower temperature. Heat is positive when energy is added to the sys-tem (see Figure 1).
Work is the mechanism that transfers energy across the bound-ary of systems with differing pressures (or force of any kind), always toward the lower pressure. If the total effect produced in the system can be reduced to the raising of a weight, then nothing but work has crossed the boundary. Work is positive when energy is removed from the system (see Figure 1).
Mechanical or shaft work (W ) is the energy delivered or absorbed by a mechanism, such as a turbine, air compressor, or internal combustion engine.
Flow work is energy carried into or transmitted across the sys-tem boundary because a pumping process occurs somewhere out-side the system, causing fluid to enter the system. It can be more easily understood as the work done by the fluid just outside the The preparation of the first and second parts of this chapter is assigned to TC 1.1, Thermodynamics and Psychrometrics. The third part is assigned to TC 8.3, Absorption and Heat-Operated Machines.
PE mgz = KE mV 2 2 ⁄ = Fig. 1 Energy Flows in General Thermodynamic System 1.2 2001 ASHRAE Fundamentals Handbook (SI) system on the adjacent fluid entering the system to force or push it into the system. Flow work also occurs as fluid leaves the system.
(3) where p is the pressure and v is the specific volume, or the volume displaced per unit mass.
A property of a system is any observable characteristic of the system. The state of a system is defined by listing its properties. The most common thermodynamic properties are temperature T, pres-sure p, and specific volume v or density ρ. Additional thermody-namic properties include entropy, stored forms of energy, and enthalpy.
Frequently, thermodynamic properties combine to form other properties. Enthalpy (h), a result of combining properties, is defined as (4) where u is internal energy per unit mass.
Each property in a given state has only one definite value, and any property always has the same value for a given state, regardless of how the substance arrived at that state.
A process is a change in state that can be defined as any change in the properties of a system. A process is described by specifying the initial and final equilibrium states, the path (if identifiable), and the interactions that take place across system boundaries during the process.
A cycle is a process or a series of processes wherein the initial and final states of the system are identical. Therefore, at the conclu-sion of a cycle, all the properties have the same value they had at the beginning.
A pure substance has a homogeneous and invariable chemical composition. It can exist in more than one phase, but the chemical composition is the same in all phases.
If a substance exists as liquid at the saturation temperature and pressure, it is called saturated liquid. If the temperature of the liq-uid is lower than the saturation temperature for the existing pres-sure, it is called either a subcooled liquid (the temperature is lower than the saturation temperature for the given pressure) or a com-pressed liquid (the pressure is greater than the saturation pressure for the given temperature).
When a substance exists as part liquid and part vapor at the sat-uration temperature, its quality is defined as the ratio of the mass of vapor to the total mass. Quality has meaning only when the substance is in a saturated state; i.e., at saturation pressure and temperature.
If a substance exists as vapor at the saturation temperature, it is called saturated vapor. (Sometimes the term dry saturated vapor is used to emphasize that the quality is 100%.) When the vapor is at a temperature greater than the saturation temperature, it is super-heated vapor. The pressure and temperature of superheated vapor are independent properties, since the temperature can increase while the pressure remains constant. Gases are highly superheated vapors. FIRST LAW OF THERMODYNAMICS The first law of thermodynamics is often called the law of the conservation of energy. The following form of the first law equa-tion is valid only in the absence of a nuclear or chemical reaction.
Based on the first law or the law of conservation of energy for any system, open or closed, there is an energy balance as or Energy In – Energy Out = Increase in Energy in System Figure 1 illustrates energy flows into and out of a thermody-namic system. For the general case of multiple mass flows in and out of the system, the energy balance can be written (5) where subscripts i and f refer to the initial and final states, respectively.
The steady-flow process is important in engineering applica-tions. Steady flow signifies that all quantities associated with the system do not vary with time. Consequently, (6) where h = u + pv as described in Equation (4).
A second common application is the closed stationary system for which the first law equation reduces to (7) SECOND LAW OF THERMODYNAMICS The second law of thermodynamics differentiates and quantifies processes that only proceed in a certain direction (irreversible) from those that are reversible. The second law may be described in sev-eral ways. One method uses the concept of entropy flow in an open system and the irreversibility associated with the process. The con-cept of irreversibility provides added insight into the operation of cycles. For example, the larger the irreversibility in a refrigeration cycle operating with a given refrigeration load between two fixed temperature levels, the larger the amount of work required to oper-ate the cycle. Irreversibilities include pressure drops in lines and heat exchangers, heat transfer between fluids of different tempera-ture, and mechanical friction. Reducing total irreversibility in a cycle improves the cycle performance. In the limit of no irrevers-ibilities, a cycle will attain its maximum ideal efficiency.
In an open system, the second law of thermodynamics can be described in terms of entropy as (8) where dSsystem = total change within system in time dt during process δmisi = entropy increase caused by mass entering (incoming) δmese = entropy decrease caused by mass leaving (exiting) δQ/T = entropy change caused by reversible heat transfer between system and surroundings dI = entropy caused by irreversibilities (always positive) Equation (8) accounts for all entropy changes in the system. Re-arranged, this equation becomes (9) Flow Work (per unit mass) pv = h u pv + ≡ Net Amount of Energy Added to System Net Increase in Stored Energy of System = min u pv V2 2 ------gz + + + in ∑ mout u pv V2 2 ------gz + + + out ∑ Q W – + – mf u V2 2 ------gz + + f mi u V2 2 ------gz + + i – system = m · h V2 2 ------gz + + all streams leaving ∑ m · h V2 2 ------gz + + all streams entering ∑ – Q · W · – + 0 = Q W – m uf ui – ( ) [ ]system = dSsystem δQ T -------δmisi δmese – dI + + = δQ T δmese δmisi – ( ) dSsys dI – + [ ] = Thermodynamics and Refrigeration Cycles 1.3 In integrated form, if inlet and outlet properties, mass flow, and interactions with the surroundings do not vary with time, the general equation for the second law is (10) In many applications the process can be considered to be operat-ing steadily with no change in time. The change in entropy of the system is therefore zero. The irreversibility rate, which is the rate of entropy production caused by irreversibilities in the process, can be determined by rearranging Equation (10) (11) Equation (6) can be used to replace the heat transfer quantity.
Note that the absolute temperature of the surroundings with which the system is exchanging heat is used in the last term. If the temper-ature of the surroundings is equal to the temperature of the system, the heat is transferred reversibly and Equation (11) becomes equal to zero. Equation (11) is commonly applied to a system with one mass flow in, the same mass flow out, no work, and negligible kinetic or potential energy flows. Combining Equations (6) and (11) yields (12) In a cycle, the reduction of work produced by a power cycle or the increase in work required by a refrigeration cycle is equal to the absolute ambient temperature multiplied by the sum of the irrevers-ibilities in all the processes in the cycle. Thus the difference in the reversible work and the actual work for any refrigeration cycle, the-oretical or real, operating under the same conditions becomes (13) THERMODYNAMIC ANALYSIS OF REFRIGERATION CYCLES Refrigeration cycles transfer thermal energy from a region of low temperature TR to one of higher temperature. Usually the higher temperature heat sink is the ambient air or cooling water. This tem-perature is designated as T0, the temperature of the surroundings.
The first and second laws of thermodynamics can be applied to individual components to determine mass and energy balances and the irreversibility of the components. This procedure is illustrated in later sections in this chapter.
Performance of a refrigeration cycle is usually described by a coefficient of performance. COP is defined as the benefit of the cycle (amount of heat removed) divided by the required energy input to operate the cycle, or (14) For a mechanical vapor compression system, the net energy sup-plied is usually in the form of work, mechanical or electrical, and may include work to the compressor and fans or pumps. Thus (15) In an absorption refrigeration cycle, the net energy supplied is usually in the form of heat into the generator and work into the pumps and fans, or (16) In many cases the work supplied to an absorption system is very small compared to the amount of heat supplied to the generator, so the work term is often neglected.
Application of the second law to an entire refrigeration cycle shows that a completely reversible cycle operating under the same conditions has the maximum possible Coefficient of Performance.
A measure of the departure of the actual cycle from an ideal revers-ible cycle is given by the refrigerating efficiency: (17) The Carnot cycle usually serves as the ideal reversible refriger-ation cycle. For multistage cycles, each stage is described by a reversible cycle.
EQUATIONS OF STATE The equation of state of a pure substance is a mathematical rela-tion between pressure, specific volume, and temperature. When the system is in thermodynamic equilibrium, (18) The principles of statistical mechanics are used to (1) explore the fundamental properties of matter, (2) predict an equation of state based on the statistical nature of a particular system, or (3) propose a functional form for an equation of state with unknown parameters that are determined by measuring thermodynamic properties of a substance. A fundamental equation with this basis is the virial equation. The virial equation is expressed as an expansion in pres-sure p or in reciprocal values of volume per unit mass v as (19) (20) where coefficients B’, C’, D’, etc., and B, C, D, etc., are the virial coef-ficients. B’ and B are second virial coefficients; C’ and C are third virial coefficients, etc. The virial coefficients are functions of temperature only, and values of the respective coefficients in Equations (19) and (20) are related. For example, B’ = B/RT and C’ = (C – B2)/(RT)2.
The ideal gas constant R is defined as (21) where (pv)T is the product of the pressure and the volume along an isotherm, and Ttp is the defined temperature of the triple point of water, which is 273.16 K. The current best value of R is 8314.41 J/(kg mole·K).
The quantity pv/RT is also called the compressibility factor; i.e., Z = pv/RT or (22) An advantage of the virial form is that statistical mechanics can be used to predict the lower order coefficients and provide physical significance to the virial coefficients. For example, in Equation Sf Si – ( )system δQ T -------ms ( )in ms ( )out ∑ – I + ∑ + rev ∫ = I· m · s ( )out ∑ m · s ( )in ∑ – Q · Tsurr ------------∫ – = I· m · sout sin – ( ) hout hin – Tsurr -----------------------– = W · actual W · reversible T0 I· ∑ + = COP Useful refrigerating effect Net energy supplied from external sources -----------------------------------------------------------------------------------------------------≡ COP Qevap Wnet --------------= COP Qevap Qgen Wnet + ------------------------------= ηR COP COP ( )rev -----------------------= f p v T , ( , ) 0 = pv RT -------1 B′p C′p2 D′p3 … + + + + = pv RT -------1 B v ⁄ ( ) C v2 ⁄ ( ) D v3 ⁄ ( ) … + + + + = R pv ( )T Ttp -------------p 0 → lim = Z 1 B v ⁄ ( ) C v2 ⁄ ( ) D v3 ⁄ ( ) … + + + + = 1.4 2001 ASHRAE Fundamentals Handbook (SI) (22), the term B/v is a function of interactions between two mole-cules, C/v2 between three molecules, etc. Since the lower order interactions are common, the contributions of the higher order terms are successively less. Thermodynamicists use the partition or distri-bution function to determine virial coefficients; however, experi-mental values of the second and third coefficients are preferred. For dense fluids, many higher order terms are necessary that can neither be satisfactorily predicted from theory nor determined from exper-imental measurements. In general, a truncated virial expansion of four terms is valid for densities of less than one-half the value at the critical point. For higher densities, additional terms can be used and determined empirically.
Digital computers allow the use of very complex equations of state in calculating p-v-T values, even to high densities. The Bene-dict-Webb-Rubin (B-W-R) equation of state (Benedict et al. 1940) and the Martin-Hou equation (1955) have had considerable use, but should generally be limited to densities less than the critical value.
Strobridge (1962) suggested a modified Benedict-Webb-Rubin relation that gives excellent results at higher densities and can be used for a p-v-T surface that extends into the liquid phase.
The B-W-R equation has been used extensively for hydrocar-bons (Cooper and Goldfrank 1967): (23) where the constant coefficients are Ao, Bo, Co, a, b, c, α, γ.
The Martin-Hou equation, developed for fluorinated hydro-carbon properties, has been used to calculate the thermodynamic property tables in Chapter 20 and in ASHRAE Thermodynamic Properties of Refrigerants (Stewart et al. 1986). The Martin-Hou equation is as follows: (24) where the constant coefficients are Ai, Bi, Ci, k, b, and α.
Strobridge (1962) suggested an equation of state that was devel-oped for nitrogen properties and used for most cryogenic fluids.
This equation combines the B-W-R equation of state with an equa-tion for high density nitrogen suggested by Benedict (1937). These equations have been used successfully for liquid and vapor phases, extending in the liquid phase to the triple-point temperature and the freezing line, and in the vapor phase from 10 to 1000 K, with pres-sures to 1 GPa. The equation suggested by Strobridge is accurate within the uncertainty of the measured p-v-T data. This equation, as originally reported by Strobridge, is (25) The 15 coefficients of this equation’s linear terms are determined by a least-square fit to experimental data. Hust and Stewart (1966) and Hust and McCarty (1967) give further information on methods and techniques for determining equations of state.
In the absence of experimental data, Van der Waals’ principle of corresponding states can predict fluid properties. This principle relates properties of similar substances by suitable reducing factors; i.e., the p-v-T surfaces of similar fluids in a given region are assumed to be of similar shape. The critical point can be used to define reducing parameters to scale the surface of one fluid to the dimensions of another. Modifications of this principle, as suggested by Kamerlingh Onnes, a Dutch cryogenic researcher, have been used to improve correspondence at low pressures. The principle of corresponding states provides useful approximations, and numer-ous modifications have been reported. More complex treatments for predicting property values, which recognize similarity of fluid prop-erties, are by generalized equations of state. These equations ordi-narily allow for adjustment of the p-v-T surface by introduction of parameters. One example (Hirschfelder et al. 1958) allows for departures from the principle of corresponding states by adding two correlating parameters.
CALCULATING THERMODYNAMIC PROPERTIES While equations of state provide p-v-T relations, a thermody-namic analysis usually requires values for internal energy, enthalpy, and entropy. These properties have been tabulated for many sub-stances, including refrigerants (See Chapters 6, 20, and 38) and can be extracted from such tables by interpolating manually or with a suitable computer program. This approach is appropriate for hand calculations and for relatively simple computer models; however, for many computer simulations, the overhead in memory or input and output required to use tabulated data can make this approach unacceptable. For large thermal system simulations or complex analyses, it may be more efficient to determine internal energy, enthalpy, and entropy using fundamental thermodynamic relations or curves fit to experimental data. Some of these relations are dis-cussed in the following sections. Also, the thermodynamic relations discussed in those sections are the basis for constructing tables of thermodynamic property data. Further information on the topic may be found in references covering system modeling and thermo-dynamics (Stoecker 1989, Howell and Buckius 1992).
At least two intensive properties must be known to determine the remaining properties. If two known properties are either p, v, or T (these are relatively easy to measure and are commonly used in sim-ulations), the third can be determined throughout the range of inter-est using an equation of state. Furthermore, if the specific heats at zero pressure are known, specific heat can be accurately determined from spectroscopic measurements using statistical mechanics (NASA 1971). Entropy may be considered a function of T and p, and from calculus an infinitesimal change in entropy can be written as follows: (26) Likewise, a change in enthalpy can be written as (27) Using the relation Tds = dh − vdp and the definition of specific heat at constant pressure, cp ≡ (∂h/∂T)p, Equation (27) can be rear-ranged to yield P RT v ⁄ ( ) BoRT Ao – Co – T2 ⁄ ( ) v2 ⁄ bRT a – ( ) v3 ⁄ + + = aα ( ) v6 ⁄ c 1 γ v2 ⁄ + ( )e γ – v2 ⁄ ( ) [ ] v3T2 ⁄ + + p RT v b – -----------A2 B2T C2e kT Tc ⁄ – ( ) + + v b – ( )2 ---------------------------------------------------------A3 B3T C3e kT Tc ⁄ – ( ) + + v b – ( )3 ---------------------------------------------------------+ + = A4 B4T + v b – ( )4 ----------------------A5 B5T C5e kT Tc ⁄ – ( ) + + v b – ( )5 ---------------------------------------------------------A6 B6T + ( )eav + + + p RTρ Rn1T n2 n3 T -----n4 T 2 -----n5 T 4 -----+ + + + ρ2 + = Rn6T n7 + ( )ρ3 n8Tρ4 + + ρ3 n9 T 2 -----n10 T3 -------n11 T4 -------+ + n16 – ρ2 ( ) exp + ρ5 n12 T 2 -------n13 T3 -------n14 T4 -------+ + n16 – ρ2 ( ) exp n15ρ6 + + ds ∂s ∂T ------ p dT ∂s ∂p ------ T dp + = dh ∂h ∂T ------ p dT ∂h ∂p ------ T dp + = Thermodynamics and Refrigeration Cycles 1.5 (28) Equations (26) and (28) combine to yield (∂s/∂T)p = cp/T. Then, using the Maxwell relation (∂s/∂p)T = −(∂v/∂T)p, Equation (26) may be rewritten as (29) This is an expression for an exact derivative, so it follows that (30) Integrating this expression at a fixed temperature yields (31) where cp0 is the known zero pressure specific heat, and dpT is used to indicate that the integration is performed at a fixed temperature.
The second partial derivative of specific volume with respect to temperature can be determined from the equation of state. Thus, Equation (31) can be used to determine the specific heat at any pres-sure.
Using Tds = dh − vdp, Equation (29) can be written as (32) Equations (28) and (32) may be integrated at constant pressure to obtain (33) and (34) Integrating the Maxwell relation (∂s/∂p)T = −(∂v/∂T)p gives an equation for entropy changes at a constant temperature as (35) Likewise, integrating Equation (32) along an isotherm yields the following equation for enthalpy changes at a constant temperature (36) Internal energy can be calculated from u = h − pv.
Combinations (or variations) of Equations (33) through (36) can be incorporated directly into computer subroutines to calculate properties with improved accuracy and efficiency. However, these equations are restricted to situations where the equation of state is valid and the properties vary continuously. These restrictions are violated by a change of phase such as evaporation and condensa-tion, which are essential processes in air-conditioning and refriger-ating devices. Therefore, the Clapeyron equation is of particular value; for evaporation or condensation it gives (37) where sfg = entropy of vaporization hfg = enthalpy of vaporization vfg = specific volume difference between vapor and liquid phases If vapor pressure and liquid and vapor density data are known at saturation, and these are relatively easy measurements to obtain, then changes in enthalpy and entropy can be calculated using Equa-tion (37).
Phase Equilibria for Multicomponent Systems To understand phase equilibria, consider a container full of a liq-uid made of two components; the more volatile component is des-ignated i and the less volatile component j (Figure 2A). This mixture is all liquid because the temperature is low—but not so low that a solid appears. Heat added at a constant pressure raises the tempera-ture of the mixture, and a sufficient increase causes vapor to form, as shown in Figure 2B. If heat at constant pressure continues to be added, eventually the temperature will become so high that only vapor remains in the container (Figure 2C). A temperature-concen-tration (T-x) diagram is useful for exploring details of this situation.
Figure 3 is a typical T-x diagram valid at a fixed pressure. The case shown in Figure 2A, a container full of liquid mixture with mole fraction xi,0 at temperature T0, is point 0 on the T-x diagram.
When heat is added, the temperature of the mixture increases. The point at which vapor begins to form is the bubble point. Starting at point 0, the first bubble will form at temperature T1, designated by point 1 on the diagram. The locus of bubble points is the bubble point curve, which provides bubble points for various liquid mole fractions xi.
When the first bubble begins to form, the vapor in the bubble may not have the i mole fraction found in the liquid mixture.
Rather, the mole fraction of the more volatile species is higher in the vapor than in the liquid. Boiling prefers the more volatile spe-cies, and the T-x diagram shows this behavior. At Tl, the vapor-forming bubbles have an i mole fraction of yi,l. If heat continues to be added, this preferential boiling will deplete the liquid of species ds cp T -----dT p ∂ ∂h T v – dp T ------+ = ds cp T -----dT T ∂ ∂v pdp – = p ∂ ∂cp T T T2 2 ∂ ∂v p – = cp cpo T T2 2 ∂ ∂v pT d 0 p ∫ – = dh cpdT v T T ∂ ∂v p – dp + = s T1 p0 , ( ) s T0 p0 , ( ) cp T ----- Tp d T0 T1 ∫ + = h T1 p0 , ( ) h T0 p0 , ( ) cp T d T0 T1 ∫ + = s T0 p1 , ( ) s T0 p0 , ( ) T ∂ ∂v p pT d p0 p1 ∫ – = h T0 p1 , ( ) h T0 p0 , ( ) v T T ∂ ∂v p – p d p0 p1 ∫ + = T d dp sat sfg vfg ------hfg Tvfg ----------= = Fig. 2 Mixture of i and j Components in Constant Pressure Container 1.6 2001 ASHRAE Fundamentals Handbook (SI) i and the temperature required to continue the process will increase.
Again, the T-x diagram reflects this fact; at point 2 the i mole frac-tion in the liquid is reduced to xi,2 and the vapor has a mole fraction of yi,2. The temperature required to boil the mixture is increased to T2. Position 2 on the T-x diagram could correspond to the physical situation shown in Figure 2B.
If the constant-pressure heating continues, all the liquid eventu-ally becomes vapor at temperature T3. At this point the i mole frac-tion in the vapor yi,3 equals the starting mole fraction in the all-liquid mixture xi,1. This equality is required for mass and species conser-vation. Further addition of heat simply raises the vapor temperature.
The final position 4 corresponds to the physical situation shown in Figure 2C.
Starting at position 4 in Figure 3, the removal of heat leads to 3, and further heat removal would cause droplets rich in the less vola-tile species to form. This point is called the dew point, and the locus of dew points is called the dew-point curve. The removal of heat will cause the mixture to reverse through points 3, 2, 1, and to start-ing point 0. Because the composition shifts, the temperature required to boil (or condense) this mixture changes as the process proceeds. This mixture is therefore called zeotropic.
Most mixtures have T-x diagrams that behave as previously described, but some have a markedly different feature. If the dew point and bubble point curves intersect at any point other than at their ends, the mixture exhibits what is called azeotropic behavior at that composition. This case is shown as position a in the T-x diagram of Figure 4. If a container of liquid with a mole fraction xa were boiled, vapor would be formed with an identical mole fraction ya. The addition of heat at constant pressure would continue with no shift in composition and no temperature glide.
Perfect azeotropic behavior is uncommon, while near azeotropic behavior is fairly common. The azeotropic composition is pressure dependent, so operating pressures should be considered for their impact on mixture behavior. Azeotropic and near-azeotropic refrig-erant mixtures find wide application. The properties of an azeo-tropic mixture are such that they may be conveniently treated as pure substance properties. Zeotropic mixtures, however, require special treatment, using an equation-of-state approach with appro-priate mixing rules or using the fugacities with the standard state method (Tassios 1993). Refrigerant and lubricant blends are a zeo-tropic mixture and can be treated by these methods (see Thome 1995 and Martz et al. 1996a, b).
COMPRESSION REFRIGERATION CYCLES CARNOT CYCLE The Carnot cycle, which is completely reversible, is a perfect model for a refrigeration cycle operating between two fixed temper-atures, or between two fluids at different temperatures and each with infinite heat capacity. Reversible cycles have two important proper-ties: (1) no refrigerating cycle may have a coefficient of perfor-mance higher than that for a reversible cycle operated between the same temperature limits, and (2) all reversible cycles, when oper-ated between the same temperature limits, have the same coefficient of performance. Proof of both statements may be found in almost any textbook on elementary engineering thermodynamics.
Figure 5 shows the Carnot cycle on temperature-entropy coordi-nates. Heat is withdrawn at the constant temperature TR from the region to be refrigerated. Heat is rejected at the constant ambient temperature T0. The cycle is completed by an isentropic expansion and an isentropic compression. The energy transfers are given by Fig. 3 Temperature-Concentration (T-x) Diagram for Zeotropic Mixture Fig. 4 Azeotropic Behavior Shown on T-x Diagram Fig. 5 Carnot Refrigeration Cycle Thermodynamics and Refrigeration Cycles 1.7 Thus, by Equation (15), (38) Example 1. Determine entropy change, work, and coefficient of perfor-mance for the cycle shown in Figure 6. Temperature of the refrigerated space TR is 250 K and that of the atmosphere T0 is 300 K. Refrigeration load is 125 kJ.
Solution: Flow of energy and its area representation in Figure 6 is: The net change of entropy of any refrigerant in any cycle is always zero. In Example 1 the change in entropy of the refrigerated space is ∆SR = −125/250 = −0.5 kJ/K and that of the atmosphere is ∆So = 125/250 = 0.5 kJ/K. The net change in entropy of the isolated system is ∆Stotal = ∆SR + ∆So = 0.
The Carnot cycle in Figure 7 shows a process in which heat is added and rejected at constant pressure in a two-phase region of a refrigerant. Saturated liquid at state 3 expands isentropically to the low temperature and pressure of the cycle at state d. Heat is added isothermally and isobarically by evaporating the liquid phase refrig-erant from state d to state 1. The cold saturated vapor at state 1 is compressed isentropically to the high temperature in the cycle at state b. However the pressure at state b is below the saturation pressure corresponding to the high temperature in the cycle. The compression process is completed by an isothermal compression process from state b to state c. The cycle is completed by an isother-mal and isobaric heat rejection or condensing process from state c to state 3.
Applying the energy equation for a mass of refrigerant m yields (all work and heat transfer are positive) The net work for the cycle is and Energy kJ Area Qi 125 b Qo 150 a + b W 25 a Fig. 6 Temperature-Entropy Diagram for Carnot Refrigeration Cycle of Example 1 Q0 T0 S2 S3 – ( ) = Qi TR S1 S4 – ( ) TR S2 S3 – ( ) = = Wnet Qo Qi – = COP TR T0 TR – ------------------= ∆S S1 S4 – Qi TR ⁄ 125 250 ⁄ 0.5 kJ K ⁄ = = = = W ∆S T0 TR – ( ) 0.5 300 250 – ( ) 25 kJ = = = COP Qi Qo Qi – ( ) ⁄ Qi W ⁄ 125 25 ⁄ 5 = = = = Fig. 7 Carnot Vapor Compression Cycle W 3 d m h3 hd – ( ) = W 1 b m hb h1 – ( ) = W b c T0 Sb Sc – ( ) m hb hc – ( ) – = Q d 1 m h1 hd – ( ) Area def1d = = Wnet W 1 b W b c W 3 d – + Area d1bc3d = = COP Q d 1 Wnet -----------TR T0 TR – ------------------= = 1.8 2001 ASHRAE Fundamentals Handbook (SI) THEORETICAL SINGLE-STAGE CYCLE USING A PURE REFRIGERANT OR AZEOTROPIC MIXTURE A system designed to approach the ideal model shown in Figure 7 is desirable. A pure refrigerant or an azeotropic mixture can be used to maintain constant temperature during the phase changes by maintaining a constant pressure. Because of such concerns as high initial cost and increased maintenance requirements, a practical machine has one compressor instead of two and the expander (engine or turbine) is replaced by a simple expansion valve. The valve throttles the refrigerant from high pressure to low pressure.
Figure 8 shows the theoretical single-stage cycle used as a model for actual systems.
Applying the energy equation for a mass of refrigerant m yields (39) The constant enthalpy throttling process assumes no heat transfer or change in potential or kinetic energy through the expansion valve.
The coefficient of performance is (40) The theoretical compressor displacement CD (at 100% volumet-ric efficiency), is (41) which is a measure of the physical size or speed of the compressor required to handle the prescribed refrigeration load.
Example 2. A theoretical single-stage cycle using R-134a as the refrigerant operates with a condensing temperature of 30°C and an evaporating temperature of −20°C. The system produces 50 kW of refrigeration.
Determine (a) the thermodynamic property values at the four main state points of the cycle, (b) the coefficient of performance of the cycle, (c) the cycle refrigerating efficiency, and (d) rate of refrigerant flow.
Solution: (a) Figure 9 shows a schematic p-h diagram for the problem with numerical property data. Saturated vapor and saturated liquid proper-ties for states 1 and 3 are obtained from the saturation table for R-134a in Chapter 20. Properties for superheated vapor at state 2 are obtained by linear interpolation of the superheat tables for R-134a in Chapter 20.
Specific volume and specific entropy values for state 4 are obtained by determining the quality of the liquid-vapor mixture from the enthalpy. Fig. 8 Theoretical Single-Stage Vapor Compression Refrigeration Cycle Q 4 1 m h1 h4 – ( ) = W 1 2 m h2 h1 – ( ) = Q 2 3 m h2 h3 – ( ) = h3 h4 = COP Q 4 1 W 1 2 ---------h1 h4 – h2 h1 – -----------------= = CD m · v3 = Fig. 9 Schematic p-h Diagram for Example 2 x4 h4 hf – hg hf – ---------------241.65 173.82 – 386.66 173.82 – ---------------------------------------0.3187 = = = v4 vf x4 vg vf – ( ) + 0.0007374 0.3187 0.14744 0.0007374 – ( ) + = = 0.04749 m3/kg = Thermodynamics and Refrigeration Cycles 1.9 The property data are tabulated in Table 1. (b) By Equation (40) (c) By Equation (17) (d) The mass flow of refrigerant is obtained from an energy balance on the evaporator. Thus The saturation temperatures of the single-stage cycle have a strong influence on the magnitude of the coefficient of performance.
This influence may be readily appreciated by an area analysis on a temperature-entropy (T-s) diagram. The area under a reversible pro-cess line on a T-s diagram is directly proportional to the thermal energy added or removed from the working fluid. This observation follows directly from the definition of entropy [see Equation (8)].
In Figure 10 the area representing Qo is the total area under the constant pressure curve between states 2 and 3. The area represent-ing the refrigerating capacity Qi is the area under the constant pres-sure line connecting states 4 and 1. The net work required Wnet equals the difference (Qo − Qi), which is represented by the shaded area shown on Figure 10.
Because COP = Qi/Wnet, the effect on the COP of changes in evaporating temperature and condensing temperature may be observed. For example, a decrease in evaporating temperature TE significantly increases Wnet and slightly decreases Qi. An increase in condensing temperature TC produces the same results but with less effect on Wnet. Therefore, for maximum coefficient of perform-ance, the cycle should operate at the lowest possible condensing temperature and at the maximum possible evaporating temperature.
LORENZ REFRIGERATION CYCLE The Carnot refrigeration cycle includes two assumptions which make it impractical. The heat transfer capacity of the two external fluids are assumed to be infinitely large so the external fluid tem-peratures remain fixed at T0 and TR (they become infinitely large thermal reservoirs). The Carnot cycle also has no thermal resistance between the working refrigerant and the external fluids in the two heat exchange processes. As a result, the refrigerant must remain fixed at T0 in the condenser and at TR in the evaporator. The Lorenz cycle eliminates the first restriction in the Carnot cycle and allows the temperature of the two external fluids to vary during the heat exchange. The second assumption of negligible thermal resistance between the working refrigerant and the two external fluids remains. Therefore the refrigerant temperature must change during the two heat exchange processes to equal the chang-ing temperature of the external fluids. This cycle is completely reversible when operating between two fluids, each of which has a finite but constant heat capacity.
Figure 11 is a schematic of a Lorenz cycle. Note that this cycle does not operate between two fixed temperature limits. Heat is added to the refrigerant from state 4 to state 1. This process is assumed to be linear on T-s coordinates, which represents a fluid with constant heat capacity. The temperature of the refrigerant is increased in an isentropic compression process from state 1 to state 2. Process 2-3 is a heat rejection process in which the refrigerant temperature decreases linearly with heat transfer. The cycle is con-cluded with an isentropic expansion process between states 3 and 4.
The heat addition and heat rejection processes are parallel so the entire cycle is drawn as a parallelogram on T-s coordinates. A Car-not refrigeration cycle operating between T0 and TR would lie between states 1, a, 3, and b. The Lorenz cycle has a smaller refrig-erating effect than the Carnot cycle and more work is required.
However this cycle is a more practical reference to use than the Car-not cycle when a refrigeration system operates between two single-phase fluids such as air or water.
Table 1 Thermodynamic Property Data for Example 2 State t, °C p, kPa v, m3/kg h, kJ/kg s, kJ/(kg·K) 1 −20.0 132.68 0.14744 386.66 1.7417 2 37.8 770.08 0.02798 423.07 1.7417 3 30.0 770.08 0.00084 241.65 1.1432 4 −20.0 132.68 0.04749 241.65 1.1689 s4 sf x4 sg sf – ( ) + 0.9009 0.3187 1.7417 0.9009 – ( ) + = = 1.16886 kJ/(kg·K) = COP 386.66 241.65 – 423.07 386.66 – ---------------------------------------3.98 = = ηR COP T3 T1 – ( ) T1 ----------------------------------3.98 ( ) 50 ( ) 253.15 --------------------------0.79 or 79% = = = m · h1 h4 – ( ) qi 50 kW = = and m · Q · i h1 h4 – ( ) ---------------------50 386.66 241.65 – ( ) -------------------------------------------0.345 kg/s = = = Fig. 10 Areas on T-s Diagram Representing Refrigerating Effect and Work Supplied for Theoretical Single-Stage Cycle Fig. 11 Processes of Lorenz Refrigeration Cycle 1.10 2001 ASHRAE Fundamentals Handbook (SI) The energy transfers in a Lorenz refrigeration cycle are as fol-lows, where ∆T is the temperature change of the refrigerant during each of the two heat exchange processes.
Thus by Equation (15), (42) Example 3. Determine the entropy change, the work required, and the coefficient of performance for the Lorenz cycle shown in Figure 11 when the temperature of the refrigerated space is TR = 250 K, the ambi-ent temperature is T0 = 300 K, the ∆T of the refrigerant is 5 K and the refrigeration load is 125 kJ.
Solution: Note that the entropy change for the Lorenz cycle is larger than for the Carnot cycle at the same temperature levels and the same capacity (see Example 1). That is, the heat rejection is larger and the work requirement is also larger for the Lorenz cycle. This difference is caused by the finite temperature difference between the working fluid in the cycle compared to the bounding temperature reservoirs.
However, as discussed previously, the assumption of constant tem-perature heat reservoirs is not necessarily a good representation of an actual refrigeration system because of the temperature changes that occur in the heat exchangers.
THEORETICAL SINGLE-STAGE CYCLE USING ZEOTROPIC REFRIGERANT MIXTURE A practical method to approximate the Lorenz refrigeration cycle is to use a fluid mixture as the refrigerant and the four system components shown in Figure 8. When the mixture is not azeotropic and the phase change processes occur at constant pressure, the tem-peratures change during the evaporation and condensation pro-cesses and the theoretical single-stage cycle can be shown on T-s coordinates as in Figure 12. This can be compared with Figure 10 in which the system is shown operating with a pure simple substance or an azeotropic mixture as the refrigerant. Equations (14), (15), (39), (40), and (41) apply to this cycle and to conventional cycles with constant phase change temperatures. Equation (42) should be used as the reversible cycle COP in Equation (17).
For zeotropic mixtures, the concept of constant saturation tem-peratures does not exist. For example, in the evaporator, the refrigerant enters at T4 and exits at a higher temperature T1. The temperature of saturated liquid at a given pressure is the bubble point and the temperature of saturated vapor at a given pressure is called the dew point. The temperature T3 on Figure 12 is at the bubble point at the condensing pressure and T1 is at the dew point at the evaporating pressure.
An analysis of areas on a T-s diagram representing additional work and reduced refrigerating effect from a Lorenz cycle operating between the same two temperatures T1 and T3 with the same value for ∆T can be performed. The cycle matches the Lorenz cycle most closely when counterflow heat exchangers are used for both the condenser and the evaporator.
In a cycle that has heat exchangers with finite thermal resistances and finite external fluid capacity rates, Kuehn and Gronseth (1986) showed that a cycle which uses a refrigerant mixture has a higher coefficient of performance than a cycle that uses a simple pure sub-stance as a refrigerant. However, the improvement in COP is usu-ally small. The performance of the cycle that uses a mixture can be improved further by reducing the thermal resistance of the heat exchangers and passing the fluids through them in a counterflow arrangement.
MULTISTAGE VAPOR COMPRESSION REFRIGERATION CYCLES Multistage vapor compression refrigeration is used when several evaporators are needed at various temperatures such as in a supermar-ket or when the temperature of the evaporator becomes very low. Low evaporator temperature indicates low evaporator pressure and low refrigerant density into the compressor. Two small compressors in series have a smaller displacement and usually operate more effi-ciently than one large compressor that covers the entire pressure range from the evaporator to the condenser. This is especially true in refrigeration systems that use ammonia because of the large amount of superheating that occurs during the compression process.
The thermodynamic analysis of multistage cycles is similar to the analysis of single-stage cycles. The main difference is that the mass flow differs through various components of the system. A careful mass balance and energy balance performed on individual components or groups of components ensures the correct applica-tion of the first law of thermodynamics. Care must also be exercised when performing second law calculations. Often the refrigerating load is comprised of more than one evaporator, so the total system capacity is the sum of the loads from all evaporators. Likewise the total energy input is the sum of the work into all compressors. For multistage cycles, the expression for the coefficient of performance given in Equation 15 should be written as (43) Q0 T0 T ∆ + 2 ⁄ ( ) S2 S3 – ( ) = Qi TR T ∆ – 2 ⁄ ( ) S1 S4 – ( ) TR T ∆ – 2 ⁄ ( ) S2 S3 – ( ) = = Wnet Q0 QR – = COP TR ∆T 2 ⁄ ( ) – TO TR ∆T + – ---------------------------------= S ∆ Qi T -----4 1 ∫ Qi TR T ∆ 2 ⁄ ( ) – -------------------------------125 247.5 -------------0.5051 kJ K ⁄ = = = = QO TO T ∆ 2 ⁄ ( ) + [ ] S ∆ 300 2.5 + ( )0.5051 152.78 kJ = = = Wnet QO QR – 152.78 125 – 27.78 kJ = = = COP TR T ∆ 2 ⁄ ( ) – TO TR – T ∆ + --------------------------------250 5 2 ⁄ ( ) – 300 250 – 5 + ---------------------------------247.5 55 -------------4.50 = = = = Fig. 12 Areas on T-s Diagram Representing Refrigerating Effect and Work Supplied for Theoretical Single-Stage Cycle Using Zeotropic Mixture as Refrigerant COP Qi ∑ Wnet ⁄ = Thermodynamics and Refrigeration Cycles 1.11 When compressors are connected in series, the vapor between stages should be cooled to bring the vapor to saturated conditions before proceeding to the next stage of compression. Intercooling usu-ally minimizes the displacement of the compressors, reduces the work requirement, and increases the COP of the cycle. If the refrig-erant temperature between stages is above ambient, a simple inter-cooler that removes heat from the refrigerant can be used. If the temperature is below ambient, which is the usual case, the refrigerant itself must be used to cool the vapor. This is accomplished with a flash intercooler. Figure 13 shows a cycle with a flash intercooler installed.
The superheated vapor from compressor I is bubbled through saturated liquid refrigerant at the intermediate pressure of the cycle.
Some of this liquid is evaporated when heat is added from the superheated refrigerant. The result is that only saturated vapor at the intermediate pressure is fed to compressor II. A common assumption is to operate the intercooler at about the geometric mean of the evaporating and condensing pressures. This operating point provides the same pressure ratio and nearly equal volumetric efficiencies for the two compressors. Example 4 illustrates the ther-modynamic analysis of this cycle Example 4. Determine the thermodynamic properties of the eight state points shown in Figure 13, the mass flows, and the COP of this theoret-ical multistage refrigeration cycle when R-134a is the refrigerant. The saturated evaporator temperature is −20°C, the saturated condensing temperature is 30°C, and the refrigeration load is 50 kW. The saturation temperature of the refrigerant in the intercooler is 0°C, which is nearly at the geometric mean pressure of the cycle.
Solution: Thermodynamic property data are obtained from the saturation and superheat tables for R-134a in Chapter 20. States 1, 3, 5, and 7 are obtained directly from the saturation table. State 6 is a mixture of liquid and vapor. The quality is calculated by Then, Similarly for state 8, States 2 and 4 are obtained from the superheat tables by linear inter-polation. The thermodynamic property data are summarized in Table 2.
The mass flow through the lower circuit of the cycle is determined from an energy balance on the evaporator.
For the upper circuit of the cycle, Assuming the intercooler has perfect external insulation, an energy bal-ance on it is used to compute .
Rearranging and solving for , Examples 2 and 4 have the same refrigeration load and operate with the same evaporating and condensing temperatures. The two-stage cycle in Example 4 has a higher COP and less work input than the single-stage cycle. Also the highest refrigerant temperature leaving the compressor is about 34°C for the two-stage cycle versus about 38°C for the single-stage cycle. These differences are more pronounced for cycles operating at larger pressure ratios.
Fig. 13 Schematic and Pressure-Enthalpy Diagram for Dual-Compression, Dual-Expansion Cycle of Example 4 Table 2 Thermodynamic Property Values for Example 4 State Temperature, °C Pressure, kPa Specific Volume, m3/kg Specific Enthapy, kJ/kg Specific Entropy, kJ/(kg ·K) 1 −20.0 132.68 0.14744 386.66 1.7417 2 2.8 292.69 0.07097 401.51 1.7417 3 0.0 292.69 0.06935 398.68 1.7274 4 33.6 770.08 0.02726 418.68 1.7274 5 30.0 770.08 0.00084 241.65 1.1432 6 0.0 292.69 0.01515 241.65 1.1525 7 0.0 292.69 0.00077 200.00 1.0000 8 −20.0 132.68 0.01878 200.00 1.0043 x6 h6 h7 – h3 h7 – ----------------241.65 200 – 398.68 200 – -------------------------------0.20963 = = = v6 v7 x6 v3 v7 – ( ) + 0.000773 0.20963 0.06935 0.000773 – ( ) + = = 0.01515 m3 kg ⁄ = s6 s7 x6 s3 s7 – ( ) + 1.0 0.20963 0.7274 1.0 – ( ) + = = 1.15248 kJ kg·K ( ) ⁄ = x8 0.12300, v8 0.01878 m3/kg, s8 1.0043 kJ kg·K ( ) ⁄ = = = m · 1 Q · i h1 h8 – ----------------50 386.66 200 – -------------------------------0.2679 kg/s = = = m · 1 m · 2 m · 7 m · 8 = = = m · 3 m · 4 m · 5 m · 6 = = = m · 3 m · 6h6 m · 2h2 + m · 7h7 m · 3h3 + = m · 3 m · 3 m · 2 h7 h2 – h6 h3 – ----------------0.2679 200 401.51 – 241.65 398.68 – ---------------------------------------0.3438 kg/s = = = W · I m · 1 h2 h1 – ( ) 0.2679 401.51 386.66 – ( ) = = 3.978 kW = W · II m · 3 h4 h3 – ( ) 0.3438 418.68 398.68 – ( ) = = 6.876 kW = COP Q · i W · I W · II + ---------------------50 3.978 6.876 + ---------------------------------4.61 = = = 1.12 2001 ASHRAE Fundamentals Handbook (SI) ACTUAL REFRIGERATION SYSTEMS Actual systems operating steadily differ from the ideal cycles considered in the previous sections in many respects. Pressure drops occur everywhere in the system except in the compression process.
Heat transfers occur between the refrigerant and its environment in all components. The actual compression process differs substan-tially from the isentropic compression assumed above. The working fluid is not a pure substance but a mixture of refrigerant and oil. All of these deviations from a theoretical cycle cause irreversibilities within the system. Each irreversibility requires additional power into the compressor. It is useful to understand how these irrevers-ibilities are distributed throughout a real system. Insight is gained that can be useful when design changes are contemplated or operat-ing conditions are modified. Example 5 illustrates how the irrevers-ibilities can be computed in a real system and how they require additional compressor power to overcome. The input data have been rounded off for ease of computation.
Example 5. An air-cooled, direct-expansion, single-stage mechanical vapor-compression refrigerator uses R-22 and operates under steady conditions. A schematic drawing of this system is shown in Figure 14.
Pressure drops occur in all piping and heat gains or losses occur as indi-cated. Power input includes compressor power and the power required to operate both fans. The following performance data are obtained: Ambient air temperature, tO = 30°C Refrigerated space temperature, tR = −10°C Refrigeration load, = 7.0 kW Compressor power input, = 2.5 kW Condenser fan input, = 0.15 kW Evaporator fan input, = 0.11 kW Refrigerant pressures and temperatures are measured at the seven locations shown on Figure 14. Table 3 lists the measured and computed thermodynamic properties of the refrigerant neglecting the dissolved oil.
A pressure-enthalpy diagram of this cycle is shown in Figure 15 and is compared with a theoretical single-stage cycle operating between the air temperatures tR and tO. Compute the energy transfers to the refrigerant in each component of the system and determine the second law irreversibility rate in each component. Show that the total irreversibility rate multiplied by the absolute ambient temperature is equal to the difference between the actual power input and the power required by a Carnot cycle operating between tR and tO with the same refrigerating load.
Solution: The mass flow of refrigerant is the same through all compo-nents, so it is only computed once through the evaporator. Each compo-nent in the system is analyzed sequentially beginning with the evaporator. Equation (6) is used to perform a first law energy balance on each component and Equation (13) is used for the second law analy-sis. Note that the temperature used in the second law analysis is the absolute temperature.
Evaporator: Energy balance Second law Suction Line: Energy balance Q · evap W · comp W · CF W · EF Fig. 14 Schematic of Real, Direct-Expansion, Single-Stage Mechanical Vapor-Compression Refrigeration System Table 3 Measured and Computed Thermodynamic Properties of Refrigerant 22 for Example 5 State Measured Computed Pressure, kPa Temperature, °C Specific Enthalpy, kJ/kg Specific Entropy, kJ/(kg·K) Specific Volume, m3/kg 1 310.0 −10.0 402.08 1.7810 0.07558 2 304.0 −4.0 406.25 1.7984 0.07946 3 1450.0 82.0 454.20 1.8165 0.02057 4 1435.0 70.0 444.31 1.7891 0.01970 5 1410.0 34.0 241.40 1.1400 0.00086 6 1405.0 33.0 240.13 1.1359 0.00086 7 320.0 −12.8 240.13 1.1561 0.01910 Fig. 15 Pressure-Enthalpy Diagram of Actual System and Theoretical Single-Stage System Operating Between Same Inlet Air Temperatures TR and TO Q · 7 1 m · h1 h7 – ( ) 7.0 kW = = m · 7.0 402.08 240.13 – ( ) -------------------------------------------0.04322 kg/s = = I1 · 7 m · s1 s7 – ( ) Q · 7 1 TR --------– = 0.04322 1.7810 1.1561 – ( ) 7.0 263.15 ----------------– = 0.4074 W/K = Q · 1 2 m · h2 h1 – ( ) = 0.04322 406.25 402.08 – ( ) 0.1802 kW = = Thermodynamics and Refrigeration Cycles 1.13 Second law Compressor: Energy balance Second law Discharge Line: Energy balance Second law Condenser: Energy balance Second law Liquid Line: Energy balance Second law Expansion Device: Energy balance Second law These results are summarized in Table 4. For the Carnot cycle, The Carnot power requirement for the 7 kW load is The actual power requirement for the compressor is This result is within computational error of the measured power input to the compressor of 2.5 kW.
The analysis demonstrated in Example 5 can be applied to any actual vapor compression refrigeration system. The only required information for the second law analysis is the refrigerant thermody-namic state points and mass flow rates and the temperatures in which the system is exchanging heat. In this example, the extra compressor power required to overcome the irreversibility in each component is determined. The component with the largest loss is the compressor. This loss is due to motor inefficiency, friction losses, and irreversibilities due to pressure drops, mixing, and heat transfer between the compressor and the surroundings. The unrestrained expansion in the expansion device is also a large loss. This loss could be reduced by using an expander rather than a throttling pro-cess. An expander may be economical on large machines. All heat transfer irreversibilities on both the refrigerant side and the air side of the condenser and evaporator are included in the anal-ysis. The refrigerant pressure drop is also included. The only items not included are the air-side pressure drop irreversibilities of the two heat exchangers. However these are equal to the fan power require-ments as all the fan power is dissipated as heat.
An overall second law analysis, such as in Example 5, shows the designer those components with the most losses, and it helps deter-mine which components should be replaced or redesigned to improve performance. However, this type of analysis does not iden-tify the nature of the losses. A more detailed second law analysis in which the actual processes are analyzed in terms of fluid flow and heat transfer is required to identify the nature of the losses (Liang and Kuehn 1991). A detailed analysis will show that most irrevers-ibilities associated with heat exchangers are due to heat transfer, while pressure drop on the air side causes a very small loss and the refrigerant pressure drop causes a negligible loss. This finding indi-cates that promoting refrigerant heat transfer at the expense of increasing the pressure drop usually improves performance. This analysis does not provide the cost/benefits associated with reducing component irreversibilities. The use of a thermo-economic tech-nique is required.
I2 · 1 m · s2 s1 – ( ) Q · 1 2 TO --------– = 0.04322 1.7984 1.7810 – ( ) 0.1802 303.15 ⁄ – = 0.1575 W/K = Q · 2 3 m · h3 h2 – ( ) W · 2 3 + = 0.04322 454.20 406.25 – ( ) 2.5 – = 0.4276 kW – = I3 · 2 m · s3 s2 – ( ) Q · 2 3 TO --------– = 0.04322 1.8165 1.7984 – ( ) 0.4276 – 303.15 ⁄ ( ) – = 2.1928 kW = Q · 3 4 m · h4 h3 – ( ) = 0.04322 444.31 454.20 – ( ) 0.4274 kW – = = I4 · 3 m · s4 s3 – ( ) Q · 3 4 TO --------– = 0.04322 1.7891 1.8165 – ( ) 0.4274 – 303.15 ⁄ ( ) – = 0.2258W/K = Q · 4 5 m · h5 h4 – ( ) = 0.04322 241.4 444.31 – ( ) 8.7698 kW – = = I5 · 4 m · s5 s4 – ( ) Q · 4 5 TO --------– = 0.04322 1.1400 1.7891 – ( ) 8.7698 – 303.15 ⁄ ( ) – = 0.8747 W/K = Q · 5 6 m · h6 h5 – ( ) = 0.04322 240.13 241.40 – ( ) 0.0549 kW – = = I6 · 5 m · s6 s5 – ( ) Q · 5 6 TO --------– = 0.04322 1.1359 1.1400 – ( ) 0.0549 – 303.15 ⁄ ( ) – = 0.0039 W/K = Q · 6 7 m · h7 h6 – ( ) 0 = = I7 · 6 m · s7 s6 – ( ) = 0.04322 1.1561 1.1359 – ( ) 0.8730 W/K = = Table 4 Energy Transfers and Irreversibility Rates for Refrigeration System in Example 5 Component , kW , kW , W/K , % Evaporator 7.0000 0 0.4074 9 Suction line 0.1802 0 0.1575 3 Compressor −0.4276 2.5 2.1928 46 Discharge line −0.4274 0 0.2258 5 Condenser −8.7698 0 0.8747 18 Liquid line −0.0549 0 0.0039 ≈0 Expansion device 0 0 0.8730 18 Totals −2.4995 2.5 4.7351 Q · W · I · I · I · total ⁄ COPCarnot TR To TR – ------------------263.15 40 ----------------6.579 = = = W · Carnot Q · e COPCarnot --------------------------7.0 6.579 -------------1.064 kW = = = W · comp W · Carnot I · totalTo + = 1.064 4.7351 303.15 ( ) + 2.4994 kW = = 1.14 2001 ASHRAE Fundamentals Handbook (SI) ABSORPTION REFRIGERATION CYCLES An absorption cycle is a heat-activated thermal cycle. It ex-changes only thermal energy with its surroundings—no appreciable mechanical energy is exchanged. Furthermore, no appreciable con-version of heat to work or work to heat occurs in the cycle.
Absorption cycles find use in applications where one or more of the heat exchanges with the surroundings is the useful product. This includes refrigeration, air conditioning, and heat pumping. The two great advantages of this type of cycle in comparison to other cycles with similar product are • No large rotating mechanical equipment is required • Any source of heat can be used, including low-temperature sources (e.g., waste heat) IDEAL THERMAL CYCLE All absorption cycles include at least three thermal energy exchanges with their surroundings; that is, energy exchange at three different temperatures. The highest temperature and lowest temper-ature heat flows are in one direction, and the mid-temperature one (or two) is in the opposite direction. In the forward cycle, the extreme temperature (hottest and coldest) heat flows are into the cycle. This cycle is also called the heat amplifier, heat pump, con-ventional cycle, or Type I cycle. When the extreme temperature heat flows are out of the cycle, it is called a reverse cycle, heat trans-former, temperature amplifier, temperature booster, or Type II cycle. Figure 16 illustrates both types of thermal cycles.
This fundamental constraint of heat flow into or out of the cycle at three or more different temperatures establishes the first limita-tion on cycle performance. By the first law of thermodynamics (at steady state), (44) The second law requires that (45) with equality holding in the ideal case.
From these two laws alone (i.e., without invoking any further assumptions) it follows that, for the ideal forward cycle, (46) The heat ratio Qcold/Qhot is commonly called the coefficient of performance (COP), which is the cooling realized divided by the driving heat supplied.
Heat that is rejected to ambient may be at two different temper-atures, creating a four-temperature cycle. The ideal COP of the four-temperature cycle is also expressed by Equation (46), with Tmid signifying the entropic mean heat rejection temperature. In that case, Tmid is calculated as follows: (47) This expression results from assigning all the entropy flow to the single temperature Tmid.
The ideal COP for the four-temperature cycle requires additional assumptions, such as the relationship between the various heat quantities. Under the assumptions that Qcold = Qmid cold and Qhot = Qmid hot, the following expression results: (48) WORKING FLUID PHASE CHANGE CONSTRAINTS Absorption cycles require at least two working substances—a sorbent and a fluid refrigerant; and each substance achieves its cycle function with a phase change. Given this constraint, many combinations are not achievable. The first result of invoking the phase change constraints is that the various heat flows assume known identities. As illustrated in Figure 17, the refrigerant phase changes occur in an evaporator and a condenser, and those of the sorbent in an absorber and a desorber (generator). For the forward absorption cycle, the highest temperature heat is always supplied to the generator, (49) and the coldest heat is supplied to the evaporator: (50) Fig. 16 Thermal Cycles Qhot Qcold + Qmid – = (positive heat quantities are into the cycle) Qhot Thot -----------Qcold Tcold -------------Qmid Tmid ------------+ + 0 ≥ COPideal Qcold Qhot -------------Thot Tmid – Thot --------------------------- Tcold Tmid Tcold – -----------------------------= = Tmid Qmid hot Qmid cold + Qmid hot Tmid hot --------------------Qmid cold Tmid cold -----------------------+ ---------------------------------------------------= COPideal Thot Tmid hot – Thot ------------------------------------ Tcold Tmid cold ---------------------- Tcold Tmid hot -------------------= Qhot Qgen ≡ Fig. 17 Single-Effect Absorption Cycle Qcold Qevap ≡ Thermodynamics and Refrigeration Cycles 1.15 For the reverse absorption cycle, the highest temperature heat is rejected from the absorber, and the lowest temperature heat is rejected from the condenser.
The second result of the phase change constraint is that for all known refrigerants and sorbents over pressure ranges of interest, (51) and (52) These two relations are true because the latent heat of phase change (vapor ↔ condensed phase) is relatively constant when far removed from the critical point. Thus, each heat input can not be indepen-dently adjusted.
The ideal single-effect forward cycle COP expression is (53) Equality holds only if the heat quantities at each temperature may be adjusted to specific values, which as shown below is not possible.
The third result of invoking the phase change constraint is that only three of the four temperatures Tevap, Tcond, Tgen, and Tabs may be independently selected.
Practical liquid absorbents for absorption cycles have a signifi-cant negative deviation from behavior predicted by Raoult’s law.
This has the beneficial effect of reducing the required amount of absorbent recirculation, at the expense of reduced lift and increased sorption duty. The practical effect of the negative deviation is that for most absorbents, (54) and (55) The net result of applying the above approximations and con-straints to the ideal cycle COP for the single-effect forward cycle is (56) In practical terms, the temperature constraint reduces the ideal COP to about 0.9, and the heat quantity constraint further reduces it to about 0.8.
Another useful result is (57) where Tgen min is the minimum generator temperature necessary to achieve a given evaporator temperature.
Alternative approaches are available that lead to nearly the same upper limit on ideal cycle COP. For example, one approach equates the exergy production from a “driving” portion of the cycle to the exergy consumption in a “cooling” portion of the cycle (Tozer et al.
1997). This leads to the expression (58) Another approach derives the idealized relationship between the cycle lift (Tcond – Tevap) and drop (Tgen – Tabs), i.e., between the two temperature differences that define the cycle.
WORKING FLUIDS The working fluids for absorption cycles naturally fall into four categories, each requiring a different approach to cycle modeling and thermodynamic analysis. For the liquid absorbents, the impor-tant distinction is whether the absorbent is volatile or nonvolatile. In the latter case, the vapor phase is always pure refrigerant (neglect-ing noncondensables), and analysis is relatively straightforward.
For volatile absorbents, wherein vapor concentration is variable, the cycle and component modeling techniques must keep track of vapor concentration as well as liquid concentration.
The sorbent may be either liquid phase or solid phase. For the solid sorbents, the important distinction is whether the solid is a physisorbent (also known as adsorbent) or a chemisorbent. With the physisorbent, the sorbent temperature depends on both pressure and refrigerant loading (bivariance), the same as for the liquid absor-bents. In contrast, the chemisorbent temperature does not vary with loading, at least over small ranges, and hence a different modeling approach is required. Beyond these distinctions, various other characteristics are either necessary or desirable for suitable liquid absorbent-refrigerant pairs, as follows: Absence of Solid Phase (Solubility Field). The refrigerant-absorbent pair should not form a solid over the expected range of composition and temperature. If a solid forms, it will stop flow and cause equipment to shut down. Controls must prevent operation beyond the acceptable solubility range for the pair.
Relative Volatility. The refrigerant should be much more vola-tile than the absorbent so the two can be separated easily. Otherwise, cost and heat requirements may be excessive. Many of the absor-bents are effectively nonvolatile.
Affinity. The absorbent should have a strong affinity for the refrigerant under conditions in which absorption takes place. Affin-ity means a negative deviation from Raoult’s law and results in an activity coefficient of less than unity for the refrigerant. Strong affinity allows less absorbent to be circulated for the same refriger-ation effect, reducing sensible heat losses. A smaller liquid heat exchanger to transfer heat from the absorbent to the pressurized refrigerant-absorption solution is also a benefit of affinity. On the other hand, as affinity increases, extra heat is required in the gener-ators to separate refrigerant from the absorbent, and the COP suf-fers.
Pressure. Operating pressures, established by the thermody-namic properties of the refrigerant, should be moderate. High pres-sure requires the use of heavy-walled equipment, and significant electrical power may be required to pump the fluids from the low-pressure side to the high-pressure side. Vacuum requires the use of large-volume equipment and special means of reducing pressure drop in the refrigerant vapor paths.
Stability. High chemical stability is required because fluids are subjected to severe conditions over many years of service. Instabil-ity can cause undesirable formation of gases, solids, or corrosive substances. The purity of all components charged into the system is critical for high performance and corrosion prevention.
Corrosion. Most absorption fluids corrode materials used in construction. Therefore, corrosion inhibitors are used.
Safety. Precautions as dictated by code are followed in the cases where fluids are toxic, inflammable, or at high pressure. Codes vary according to country and region.
Transport Properties. Viscosity, surface tension, thermal diffu-sivity, and mass diffusivity are important characteristics of the refrigerant-absorbent pair. For example, low viscosity promotes heat and mass transfer and reduces pumping power.
Qevap Qcond ≈ Qgen Qabs ≈ COPideal Tgen Tabs – Tgen --------------------------- Tevap Tcond Tevap – --------------------------------- Tcond Tabs -------------≤ Qabs Qcond --------------1.2 to 1.3 ≈ Tgen Tabs – 1.2 Tcond Tevap – ( ) ≈ COPideal 1.2 TevapTcond TgenTabs ---------------------------Qcond Qabs --------------0.8 ≈ ≈ ≈ Tgen min Tcond Tabs Tevap – + = COPideal Tevap Tabs -------------≤ Tcond Tgen -------------= 1.16 2001 ASHRAE Fundamentals Handbook (SI) Latent Heat. The refrigerant latent heat should be high, so the circulation rate of the refrigerant and absorbent can be minimized.
Environmental Soundness. The two parameters of greatest concern are the global warming potential and the ozone depletion potential. No refrigerant-absorbent pair meets all requirements. Unfortu-nately, many requirements work at cross-purposes. For example, a greater solubility field goes hand-in-hand with reduced relative vol-atility. Thus, selection of a working pair is inherently a compromise. Water-lithium bromide and ammonia-water offer the best com-promises of thermodynamic performance and have no known detri-mental environmental effect (zero ozone depletion potential and zero global warming potential).
The ammonia-water pair meets most requirements, but its vola-tility ratio is low and it requires high operating pressures. Ammonia is also a Safety Code Group 2 fluid (ASHRAE Standard 15), which restricts its use indoors.
Advantages of the water-lithium bromide pair include high safety, high volatility ratio, high affinity, high stability, and high latent heat. However, this pair tends to form solids and operates at deep vacuum. Because the refrigerant turns to ice at 0°C, the pair cannot be used for low-temperature refrigeration. Lithium bromide (LiBr) crystallizes at moderate concentrations, as would be encoun-tered in air-cooled chillers, which ordinarily limits the pair to appli-cations where the absorber is water-cooled and the concentrations are lower. However, using a combination of salts as the absorbent can reduce this crystallization tendency enough to permit air cool-ing (Macriss 1968). Other disadvantages of the water-lithium bro-mide pair include the low operating pressures and high viscosity.
This is particularly detrimental to the absorption step; however, alcohols with a high relative molecular mass enhance LiBr absorp-tion. Proper equipment design and additives can overcome these disadvantages.
Other refrigerant-absorbent pairs are listed in Table 5 (Macriss and Zawacki 1989, ISHPC 1999). Several refrigerant-absorbent pairs appear suitable for certain cycles and may solve some prob-lems associated with traditional pairs. However, stability, corrosion, and property information on several is limited. Also, some of the fluids are somewhat hazardous. ABSORPTION CYCLE REPRESENTATIONS The quantities of interest to absorption cycle designers are tem-perature, concentration, pressure, and enthalpy. The most useful plots are those with linear scales and in which the key properties plot as straight lines. Some of the following plots are used: • Absorption plots embody the vapor-liquid equilibrium of both the refrigerant and the sorbent. Plots of vapor-liquid equilibrium on linear pressure-temperature coordinates have a logarithmic shape and hence are little used.
• In the van’t Hoff plot (ln P versus –1/T ), the constant concentration contours plot as nearly straight lines. Thus, it is more readily constructed (e.g., from sparse data) in spite of the awkward coordinates.
• The Dühring diagram (solution temperature versus reference temperature) retains the linearity of the van’t Hoff plot, while eliminating the complexity of nonlinear coordinates. Thus, it has found extensive use (see Figure 20). The primary drawback is the need for a reference substance.
• The Gibbs plot (solution temperature versus T ln P) retains most of the advantages of the Dühring plot (linear temperature coordinates, concentration contours are straight lines) while eliminating the recourse to a reference substance.
• The Merkel plot (enthalpy versus concentration) is used to assist thermodynamic calculations and to solve the distillation problems that arise with volatile absorbents. It has also been used for basic cycle analysis.
• Temperature-entropy coordinates are occasionally used to relate absorption cycles to their mechanical vapor compression counterparts.
CONCEPTUALIZING THE CYCLE The basic absorption cycle shown in Figure 17 must be altered in many cases to take advantage of the available energy. Examples include the following: (1) the driving heat is much hotter than the minimum required Tgen min: a multistage cycle boosts the COP; and (2) the driving heat temperature is below Tgen min: a different multi-stage cycle (half-effect cycle) can reduce the Tgen min.
Multistage means that one or more of the four basic exchangers (generator, absorber, condenser, evaporator) are present at two or more places in the cycle at different pressures or concentrations.
Multieffect is a special case of multistaging, signifying the number of times the driving heat is used as it transits the cycle. Thus there are several types of two-stage cycles: the double-effect cycle, the half-effect cycle, and the two-stage, triple-effect cycle.
Two or more single-effect absorption cycles, such as shown in Figure 17, can be combined to form a multistage cycle by coupling any of the components. Coupling implies either (1) sharing of com-ponent(s) between the cycles to form an integrated single hermetic cycle or (2) alternatively exchanging heat between components belonging to two hermetically separate cycles that operate at (nearly) the same temperature level.
Figure 18 shows a double-effect absorption cycle formed by coupling the absorbers and evaporators of two single-effect cycles into an integrated, single hermetic cycle. Heat is transferred between the high-pressure condenser and intermediate-pressure generator. The heat of condensation of the refrigerant (generated in the high-temperature generator) generates additional refrigerant in the lower temperature generator. Thus, the prime energy provided to the high-temperature generator is cascaded (used) twice in the cycle, making it a double-effect cycle. With the generation of addi-tional refrigerant from a given heat input, the cycle COP increases.
Commercial water-lithium bromide chillers normally use this cycle. The cycle COP can be further increased by coupling additional components and by increasing the number of cycles that are com-bined. This way, several different multiple-effect cycles can be Table 5 Refrigerant-Absorbent Pairs Refrigerant Absorbents H2O Salts Alkali halides LiBr LiClO3 CaCl2 ZnCl2 ZnBr Alkali nitrates Alkali thiocyanates Bases Alkali hydroxides Acids H2SO4 H3PO4 NH3 H2O Alkali thiocyanates TFE (Organic) NMP E181 DMF Pyrrolidone SO2 Organic solvents Thermodynamics and Refrigeration Cycles 1.17 combined by pressure-staging and/or concentration-staging. The double-effect cycle, for example, is formed by pressure-staging two single-effect cycles.
Figure 19 shows twelve generic triple-effect cycles identified by Alefeld and Radermacher (1994). Cycle 5 is a pressure-staged cycle, and Cycle 10 is a concentration-staged cycle. All other cycles are pressure- and concentration-staged. Cycle 1, which is called a dual loop cycle, is the only cycle consisting of two loops that doesn’t cir-culate absorbent in the low-temperature portion of the cycle.
Each of the cycles shown in Figure 19 can be made with one, two, or sometimes three separate hermetic loops. Dividing a cycle into separate hermetic loops allows the use of a different working fluid in each loop. Thus, a corrosive and/or high-lift absorbent can be restricted to the loop where it is required, and a conventional additive-enhanced absorbent can be used in other loops to reduce the system cost significantly. As many as 78 hermetic loop config-urations can be synthesized from the twelve triple-effect cycles shown in Figure 19. For each hermetic loop configuration, further variations are possible according to the absorbent flow pattern (e.g., series or parallel), the absorption working pairs selected, and vari-ous other hardware details. Thus, literally thousands of distinct vari-ations of the triple-effect cycle are possible.
The ideal analysis can be extended to these multistage cycles (Alefeld and Radermacher 1994). A similar range of cycle variants is possible for situations calling for the half-effect cycle, in which the available heat source temperature is below tgen min.
ABSORPTION CYCLE MODELING Analysis and Performance Simulation A physical-mathematical model of an absorption cycle consists of four types of thermodynamic equations: mass balances, energy balances, relations describing the heat and mass transfer, and equa-tions for the thermophysical properties of the working fluids.
As an example of simulation, Figure 20 shows a Dühring plot of a single-effect water-lithium bromide absorption chiller. The chiller is hot water driven, rejects waste heat from the absorber and the con-denser to a stream of cooling water, and produces chilled water. A simulation of this chiller starts by specifying the assumptions (Table 6) and the design parameters and the operating conditions at the design point (Table 7). Design parameters are the specified UA val-ues and the flow regime (co/counter/crosscurrent, pool, or film) of all heat exchangers (evaporator, condenser, generator, absorber, solution heat exchanger) and the flow rate of weak solution through the solution pump. One complete set of input operating parameters could be the design point values of the chilled water and cooling water temper-atures tchill in, tchill out, tcool in, tcool out, the hot water flow rate , and the total cooling capacity Qe. With this information, a cycle simulation calculates the required hot water temperatures; the cool-ing water flow rate; and the temperatures, pressures, and concentra-tions at all internal state points. Some additional assumptions are made that reduce the number of unknown parameters.
Fig. 18 Double-Effect Absorption Cycle Fig. 19 Generic Triple-Effect Cycles Table 6 Assumptions for Single-Effect Water-Lithium Bromide Model (Figure 17) Assumptions • Generator and condenser as well as evaporator and absorber are under same pressure • Refrigerant vapor leaving the evaporator is saturated pure water • Liquid refrigerant leaving the condenser is saturated • Strong solution leaving the generator is boiling • Refrigerant vapor leaving the generator has the equilibrium temperature of the weak solution at generator pressure • Weak solution leaving the absorber is saturated • No liquid carryover from evaporator • Flow restrictors are adiabatic • Pump is isentropic • No jacket heat losses • The lmtd (log mean temperature difference) expression adequately estimates the latent changes Fig. 20 Single-Effect Water-Lithium Bromide Absorption Cycle Dühring Plot m · hot 1.18 2001 ASHRAE Fundamentals Handbook (SI) With these assumptions and the design parameters and operating conditions as specified in Table 7, the cycle simulation can be con-ducted by solving the following set of equations: Mass Balances (59) (60) Energy Balances (61) (62) (63) (64) (65) Heat Transfer Equations (66) (67) (68) (69) (70) Fluid Property Equations at each state point Thermal Equations of State: hwater(t,p), hsol(t,p,ξ) Two-Phase Equilibrium: twater,sat(p), tsol,sat(p,ξ) The results are listed in Table 8.
Double-Effect Cycle Double-effect cycle calculations can be performed in a manner similar to that illustrated for the single-effect cycle. Mass and energy balances of the model shown in Figure 21 were calculated using the inputs and assumptions listed in Table 9. The results are shown in Table 10. The COP is quite sensitive to several inputs and Table 7 Design Parameters and Operating Conditions for Single-Effect Water-Lithium Bromide Absorption Chiller Design Parameters Operating Conditions Evaporator UAevap = 319.2 kW/K, countercurrent film tchill in = 12°C tchill out = 6°C Condenser UAcond = 180.6 kW/K, countercurrent film tcool out = 35°C Absorber UAabs = 186.9 kW/K, countercurrent film-absorber tcool in = 27°C Generator UAgen = 143.4 kW/K, pool-generator = 74.4 kg/s Solution UAsol = 33.8 kW/K, countercurrent General = 12 kg/s = 2148 kW Table 8 Simulation Results for Single-Effect Water-Lithium Bromide Absorption Chiller Internal Parameters Performance Parameters Evaporator tvapor,evap = 1.8°C psat,evap = 0.697 kPa = 2148 kW = 85.3 kg/s Condenser Tliq,cond = 46.2°C psat,cond = 10.2 kPa = 2322 kW = 158.7 kg/s Absorber ξweak = 59.6% tweak = 40.7°C tstrong,abs = 49.9°C = 2984 kW tcool,mean = 31.5°C Generator ξstrong = 64.6% tstrong,gen = 103.5°C tweak,gen = 92.4°C tweak,sol = 76.1°C = 3158 kW thot in = 125°C thot out = 115°C Solution tstrong,sol = 62.4°C tweak,sol = 76.1°C = 825 kW ε = 65.4% General = 0.93 kg/s = 11.06 kg/s COP = 0.68 m · refr m · strong + m · weak = m · strongξstrong m · weakξweak = Q · evap m · refr hvapor evap , hliq cond , – ( ) = m · chill hchill in hchill out – ( ) = Q · evap m · refr hvapor gen , hliq cond , – ( ) = m · cool hcool out hcool mean – ( ) = Q · abs m · refrhvapor evap , m · strong + hstrong gen , = m · weakhweak abs , Q · sol – m · cool hcool mean hcool in – ( ) = – Q · gen m · refrhvapor gen , m · strong + hstrong gen , = m · weakh – weak abs , Q · sol – m · hot hhot in hhot out – ( ) = m · hot m · weak Q · evap Q · evap m · chill Q · cond m · cool Q · abs Q · gen Q · sol m · vapor m · strong Q · sol m · strong hstrong gen , hstrong sol , – ( ) = m · weak hweak sol , hweak abs , – ( ) = Q · evap UAevap tchill in tchill out – ln tchill in tvapor evap , – tchill out tvapor evap , – ---------------------------------------------------- ---------------------------------------------------------------= Q · cond UAcond tcool out tcool mean – ln tliq cond , tcool mean – tliq cond , tcool out – -------------------------------------------------- ------------------------------------------------------------= Q · abs UAabs tstrong abs , tcool mean – ( ) tweak abs , tcool in – ( ) – ln tstrong abs , tcool mean – tweak abs , tcool in – ------------------------------------------------------- --------------------------------------------------------------------------------------------------------------------= Q · gen UAgen thot in tstrong gen , – ( ) thot out tweak gen , – ( ) – ln thot in tstrong gen , – thot out tweak gen , – --------------------------------------------- -----------------------------------------------------------------------------------------------------------= Q · sol UAsol tstrong gen , tweak sol , – ( ) tstrong sol , tweak abs , – ( ) – ln tstrong gen , tweak sol , – tstrong sol , tweak abs , – ---------------------------------------------------- ------------------------------------------------------------------------------------------------------------------------= Fig. 21 Double-Effect Water-Lithium Bromide Absorption Cycle with State Points Thermodynamics and Refrigeration Cycles 1.19 assumptions. In particular, the effectiveness of the solution heat exchangers and the driving temperature difference between the high-temperature condenser and the low-temperature generator influence the COP strongly. AMMONIA-WATER ABSORPTION CYCLES Ammonia-water absorption cycles are similar to the water-lith-ium bromide cycles, but with some important differences. The dif-ferences arise due to the lower latent heat of ammonia compared to water, the volatility of the absorbent, and the different pressure and solubility ranges. The latent heat of ammonia is only about half that of water, so, for the same duty, the refrigerant and absorbent mass circulation rates are roughly double that of water-lithium bromide.
As a result, the sensible heat loss associated with heat exchanger approaches is greater. Accordingly, ammonia-water cycles incorpo-rate more techniques to reclaim sensible heat described in Hanna et al. (1995). The refrigerant heat exchanger (RHX), also known as refrigerant subcooler, which improves COP by about 8%, is the most important (Holldorff 1979). Next is the absorber heat exchanger (AHX), accompanied by a generator heat exchanger (GHX) (Phillips 1976). These either replace or supplement the tra-ditional solution heat exchanger (SHX). These components would also benefit the water-lithium bromide cycle, except that the deep vacuum in that cycle makes them impractical there.
The volatility of the water absorbent is also key. It makes the dis-tinction between crosscurrent, cocurrent, and countercurrent mass exchange more important in all of the latent heat exchangers (Briggs 1971). It also requires a distillation column on the high-pressure side. When improperly implemented, this column can impose both cost and COP penalties. Those penalties are avoided by refluxing the column from an internal diabatic section (e.g., solution cooled rectifier [SCR]) rather than with an external reflux pump.
The high-pressure operating regime makes it impractical to achieve multieffect performance via pressure-staging. On the other hand, the exceptionally wide solubility field facilitates concentra-tion-staging. The generator-absorber heat exchange (GAX) cycle is an especially advantageous embodiment of concentration-staging (Modahl and Hayes 1988). Ammonia-water cycles can equal the performance of water-lith-ium bromide cycles. The single-effect or basic GAX cycle yields the same performance as a single-effect water-lithium bromide cycle; the branched GAX cycle (Herold et al. 1991) yields the same per-formance as a water-lithium bromide double-effect cycle; and the VX GAX cycle (Erickson and Rane 1994) yields the same perfor-mance as a water-lithium bromide triple-effect cycle. Additional advantages of the ammonia-water cycle include refrigeration capa-bility, air-cooling capability, all mild steel construction, extreme compactness, and capability of direct integration into industrial pro-cesses. Between heat-activated refrigerators, gas-fired residential air conditioners, and large industrial refrigeration plants, this tech-nology has accounted for the vast majority of absorption activity over the past century.
Figure 22 shows the diagram of a typical single-effect ammonia-water absorption cycle. The inputs and assumptions in Table 11 are used to calculate a single-cycle solution, which is summarized in Table 12. Table 9 Inputs and Assumptions for Double-Effect Water-Lithium Bromide Model Inputs Capacity 1760 kW Evaporator temperature t10 5.1°C Desorber solution exit temperature t14 170.7°C Condenser/absorber low temperature t1, t8 42.4°C Solution heat exchanger effectiveness ε 0.6 Assumptions • Steady state • Refrigerant is pure water • No pressure changes except through flow restrictors and pump • State points at 1, 4, 8, 11, 14, and 18 are saturated liquid • State point 10 is saturated vapor • Temperature difference between high-temperature condenser and low-temperature generator is 5 K • Parallel flow • Both solution heat exchangers have same effectiveness • Upper loop solution flow rate is selected such that upper condenser heat exactly matches lower generator heat requirement • Flow restrictors are adiabatic • Pumps are isentropic • No jacket heat losses • No liquid carryover from evaporator to absorber • Vapor leaving both generators is at equilibrium temperature of entering solution stream Table 10 State Point Data for Double-Effect Water-Lithium Bromide Cycle of Figure 21 Point h, kJ/kg , kg/s p, kPa Q, Fraction t, °C x, % LiBr 1 117.7 9.551 0.88 0.0 42.4 59.5 2 117.7 9.551 8.36 42.4 59.5 3 182.3 9.551 8.36 75.6 59.5 4 247.3 8.797 8.36 0.0 97.8 64.6 5 177.2 8.797 8.36 58.8 64.6 6 177.2 8.797 0.88 0.004 53.2 64.6 7 2661.1 0.320 8.36 85.6 0.0 8 177.4 0.754 8.36 0.0 42.4 0.0 9 177.4 0.754 0.88 0.063 5.0 0.0 10 2510.8 0.754 0.88 1.0 5.0 0.0 11 201.8 5.498 8.36 0.0 85.6 59.5 12 201.8 5.498 111.8 85.6 59.5 13 301.2 5.498 111.8 136.7 59.5 14 378.8 5.064 111.8 0.00 170.7 64.6 15 270.9 5.064 111.8 110.9 64.6 16 270.9 5.064 8.36 0.008 99.1 64.6 17 2787.3 0.434 111.8 155.7 0.0 18 430.6 0.434 111.8 0.0 102.8 0.0 19 430.6 0.434 8.36 0.105 42.4 0.0 COP = 1.195 qe = 1760 kW ∆t = 5 K qg = 1472 kW ε = 0.600 qshx1 = 617 kW qa = 2328 kW qshx2 = 546 kW qc = 1023 kW = 0.043 kW qc = 905 kW = 0.346 kW Q · evap m · W · p1 W · p2 Fig. 22 Single-Effect Ammonia-Water Absorption Cycle 1.20 2001 ASHRAE Fundamentals Handbook (SI) NOMENCLATURE FOR EXAMPLES cp = specific heat at constant pressure COP = coefficient of performance g = local acceleration of gravity h = enthalpy, kJ/kg I = irreversibility = irreversibility rate m = mass = mass flow, kg/s p = pressure Q = heat energy, kJ = rate of heat flow, kJ/s R = ideal gas constant s = entropy, kJ/(kg·K) S = total entropy t = temperature, °C T = absolute temperature, K u = internal energy W = mechanical or shaft work = rate of work, power v = specific volume, m3/kg V = velocity of fluid x = mass fraction (of either lithium bromide or ammonia) x = vapor quality (fraction) z = elevation above horizontal reference plane Z = compressibility factor ε = heat exchanger effectiveness η = efficiency ρ = density, kg/m3 Subscripts abs = absorber cond = condenser or cooling mode cg = condenser to generator evap = evaporator fg = fluid to vapor gen = generator gh = high-temperature generator o, 0 = reference conditions, usually ambient p = pump R = refrigerating or evaporator conditions sol = solution rhx = refrigerant heat exchanger shx = solution heat exchanger REFERENCES Alefeld, G. and R. Radermacher. 1994. Heat conversion systems. CRC Press, Boca Raton.
Benedict, M., G.B. Webb, and L.C. Rubin. 1940. An empirical equation for thermodynamic properties of light hydrocarbons and their mixtures.
Journal of Chemistry and Physics 4:334.
Benedict, M. 1937. Pressure, volume, temperature properties of nitrogen at high density, I and II. Journal of American Chemists Society 59(11): 2224.
Briggs, S.W. 1971. Concurrent, crosscurrent, and countercurrent absorption in ammonia-water absorption refrigeration. ASHRAE Transactions 77(1):171.
Cooper, H.W. and J.C. Goldfrank. 1967. B-W-R Constants and new corre-lations. Hydrocarbon Processing 46(12):141.
Erickson, D.C. and M. Rane. 1994. Advanced absorption cycle: Vapor exchange GAX. Proceedings of the International Absorption Heat Pump Conference. Chicago.
Hanna, W.T., et al. 1995. Pinch-point analysis: An aid to understanding the GAX absorption cycle. ASHRAE Technical Data Bulletin 11(2). Herold, K.E., et al. 1991. The branched GAX absorption heat pump cycle, Proceedings of Absorption Heat Pump Conference. Tokyo.
Hirschfelder, J.O. et al. 1958. Generalized equation of state for gases and liq-uids. Industrial and Engineering Chemistry 50:375.
Holldorff, G. 1979. Revisions up absorption refrigeration efficiency. Hydro-carbon Processing 58(7):149.
Howell, J.R. and R.O. Buckius. 1992. Fundamentals of engineering thermo-dynamics, 2nd ed. McGraw-Hill, New York.
Hust, J.G. and R.D. McCarty. 1967. Curve-fitting techniques and applica-tions to thermodynamics. Cryogenics 8:200.
Hust, J.G. and R.B. Stewart. 1966. Thermodynamic property computations for system analysis. ASHRAE Journal 2:64.
Kuehn, T.H. and R.E. Gronseth. 1986. The effect of a nonazeotropic binary refrigerant mixture on the performance of a single stage refrigeration cycle. Proceedings International Institute of Refrigeration Conference, Purdue University, p. 119.
Liang, H. and T.H. Kuehn. 1991. Irreversibility analysis of a water to water mechanical compression heat pump. Energy 16(6):883.
Macriss, R.A. 1968. Physical properties of modified LiBr solutions. AGA Symposium on Absorption Air-Conditioning Systems, February.
Macriss, R.A. and T.S. Zawacki. 1989. Absorption fluid data survey: 1989 update. Oak Ridge National Laboratories Report ORNL/Sub84-47989/4.
Table 11 Inputs and Assumptions for Single-Effect Ammonia/Water Cycle of Figure 22 Inputs Capacity 1760 kW High-side pressure phigh 1461 kPa Low-side pressure plow 515 kPa Absorber exit temperature t1 40.6°C Generator exit temperature t4 95°C Rectifier vapor exit temperature t7 55°C Solution heat exchanger eff. εshx 0.692 Refrigerant heat exchanger eff. εrhx 0.629 Assumptions • Steady state • No pressure changes except through flow restrictors and pump • States at points 1, 4, 8, 11, and 14 are saturated liquid • States at point 12 and 13 are saturated vapor • Flow restrictors are adiabatic • Pump is isentropic • No jacket heat losses • No liquid carryover from evaporator to absorber • Vapor leaving generator is at equilibrium temperature of entering solution stream Table 12 State Point Data for Single-Effect Ammonia/Water Cycle of Figure 22 Point h, kJ/kg , kg/s p, kPa Q, Fraction t, °C x, Fraction NH3 1 –57.2 10.65 515.0 0.0 40.56 0.50094 2 –56.0 10.65 1461 40.84 0.50094 3 89.6 10.65 1461 78.21 0.50094 4 195.1 9.09 1461 0.0 95.00 0.41612 5 24.6 9.09 1461 57.52 0.41612 6 24.6 9.09 515.0 0.006 55.55 0.41612 7 1349 1.55 1461 1.000 55.00 0.99809 8 178.3 1.55 1461 0.0 37.82 0.99809 9 82.1 1.55 1461 17.80 0.99809 10 82.1 1.55 515.0 0.049 5.06 0.99809 11 1216 1.55 515.0 0.953 6.00 0.99809 12 1313 1.55 515.0 1.000 30.57 0.99809 13 1429 1.59 1461 1.000 79.15 0.99809 14 120.4 0.04 1461 0.0 79.15 0.50094 COPc = 0.571 = 1760 kW ∆trhx = 7.24 K = 3083 kW ∆tshx = 16.68 K = 149 kW εrhx = 0.629 = 170 kW εrhx = 0.692 = 1550 kW = 2869 kW = 12.4 kW = 1862.2 kW Q · evap m · Q · evap Q · gen Q · rhx Q · r Q · shw Q · abs W · Q · cond I · m · Q · · Thermodynamics and Refrigeration Cycles 1.21 Martin, J.J. and Y. Hou. 1955. Development of an equation of state for gases.
AIChE Journal 1:142.
Martz, W.L., C.M. Burton, and A.M. Jacobi. 1996a. Liquid-vapor equilibria for R-22, R-134a, R-125, and R-32/125 with a polyol ester lubricant: Measurements and departure from ideality. ASHRAE Transactions 102(1):367-74. Martz, W.L., C.M. Burton and A.M. Jacobi. 1996b. Local composition mod-eling of the thermodynamic properties of refrigerant and oil mixtures.
International Journal of Refrigeration 19(1):25-33.
Modahl, R.J. and F.C. Hayes. 1988. Evaluation of commercial advanced absorption heat pump. Proceedings of the 2nd DOE/ORNL Heat Pump Conference. Washington.
NASA. 1971. SP-273. US Government Printing Office, Washington, D.C.
Phillips, B. 1976. Absorption cycles for air-cooled solar air conditioning.
ASHRAE Transactions 82(1):966. Dallas.
Stewart, R.B., R.T. Jacobsen, and S.G. Penoncello. 1986. ASHRAE Thermo-dynamic properties of refrigerants. ASHRAE, Atlanta, GA.
Strobridge, T.R. 1962. The thermodynamic properties of nitrogen from 64 to 300 K, between 0.1 and 200 atmospheres. National Bureau of Standards Technical Note 129.
Stoecker, W.F. 1989. Design of thermal systems, 3rd ed. McGraw-Hill, New York.
Tassios, D.P. 1993. Applied chemical engineering thermodynamics.
Springer-Verlag, New York.
Thome, J.R. 1995. Comprehensive thermodynamic approach to modeling refrigerant-lubricant oil mixtures. International Journal of Heating, Ven-tilating, Air Conditioning and Refrigeration Research 1(2):110.
Tozer, R.M. and R.W. James. 1997. Fundamental thermodynamics of ideal absorption cycles. International Journal of Refrigeration 20 (2):123-135.
BIBLIOGRAPHY Bogart, M. 1981. Ammonia absorption refrigeration in industrial processes.
Gulf Publishing Co., Houston, TX.
Herold, K.E., R. Radermacher, and S.A. Klein. 1996. Absorption chillers and heat pumps. CRC Press, Boca Raton.
Jain, P.C. and G.K. Gable. 1971. Equilibrium property data for aqua-ammo-nia mixture. ASHRAE Transactions 77(1):149.
Moran, M.J. and Shapiro, H. 1995. Fundamentals of engineering thermody-manics, 3rd Ed. John Wiley and Sons, Inc. New York.
Van Wylen, C.J. and R.E. Sonntag. 1985. Fundamentals of classical ther-modynamics, 3rd ed. John Wiley and Sons, New York.
Zawacki, T.S. 1999. Effect of ammonia-water mixture database on cycle cal-culations. Proceedings of the International Sorption Heat Pump Confer-ence. Munich.
2.1 CHAPTER 2 FLUID FLOW Fluid Properties ............................................................................................................................. 2.1 Basic Relations of Fluid Dynamics ................................................................................................ 2.1 Basic Flow Processes ..................................................................................................................... 2.3 Flow Analysis ................................................................................................................................. 2.7 Noise from Fluid Flow ................................................................................................................. 2.13 LOWING fluids in heating, ventilating, air-conditioning, and Frefrigeration systems transfer heat and mass. This chapter intro-duces the basics of fluid mechanics that are related to HVAC pro-cesses, reviews pertinent flow processes, and presents a general discussion of single-phase fluid flow analysis.
FLUID PROPERTIES Fluids differ from solids in their reaction to shearing. When placed under shear stress, a solid deforms only a finite amount, whereas a fluid deforms continuously for as long as the shear is applied. Both liquids and gases are fluids. Although liquids and gases differ strongly in the nature of molecular actions, their pri-mary mechanical differences are in the degree of compressibility and liquid formation of a free surface (interface).
Fluid motion can usually be described by one of several simpli-fied modes of action or models. The simplest is the ideal-fluid model, which assumes no resistance to shearing. Ideal flow analysis is well developed (Baker 1983, Schlichting 1979, Streeter and Wylie 1979), and when properly interpreted is valid for a wide range of applications. Nevertheless, the effects of viscous action may need to be considered. Most fluids in HVAC applications can be treated as Newtonian, where the rate of deformation is directly proportional to the shearing stress. Turbulence complicates fluid behavior, and viscosity influences the nature of the turbulent flow.
Density The density ρ of a fluid is its mass per unit volume. The densities of air and water at standard indoor conditions of 20°C and 101.325 kPa (sea level atmospheric pressure) are Viscosity Viscosity is the resistance of adjacent fluid layers to shear. For shearing between two parallel plates, each of area A and separated by distance Y, the tangential force F per unit area required to slide one plate with velocity V parallel to the other is proportional to V/Y: where the proportionality factor µ is the absolute viscosity or dynamic viscosity of the fluid. The ratio of the tangential force F to area A is the shearing stress τ, and V/Y is the lateral velocity gra-dient (Figure 1A). In complex flows, velocity and shear stress may vary across the flow field; this is expressed by the following differ-ential equation: (1) The velocity gradient associated with viscous shear for a simple case involving flow velocity in the x direction but of varying mag-nitude in the y direction is illustrated in Figure 1B.
Absolute viscosity µ depends primarily on temperature. For gases (except near the critical point), viscosity increases with the square root of the absolute temperature, as predicted by the kinetic theory.
Liquid viscosity decreases with increasing temperature. Viscosities of various fluids are given in Chapter 38.
Absolute viscosity has dimensions of force · time/length2. At standard indoor conditions, the absolute viscosities of water and dry air are In fluid dynamics, kinematic viscosity ν is the ratio of absolute viscosity to density: At standard indoor conditions, the kinematic viscosities of water and dry air are BASIC RELATIONS OF FLUID DYNAMICS This section considers homogeneous, constant-property, incom-pressible fluids and introduces fluid dynamic considerations used in most analyses.
The preparation of this chapter is assigned to TC 1.3, Heat Transfer and Fluid Flow.
ρwater 998 kg m3 ⁄ = ρair 1.20 kg m3 ⁄ = F A ⁄ µ V Y ⁄ ( ) = Fig. 1 Velocity Profiles and Gradients in Shear Flows τ µ dv dy ------= µwater 1.0 mN·s/m2 = µair 18 µN·s/m2 = ν µ ρ ⁄ = νwater 1.00 10 6 – × m2/s = νair 16 10 4 – × m2/s = 2.2 2001 ASHRAE Fundamentals Handbook (SI) Continuity Conservation of matter applied to fluid flow in a conduit requires that where v = velocity normal to the differential area dA ρ = fluid density Both ρ and v may vary over the cross section A of the conduit. If both ρ and v are constant over the cross-sectional area normal to the flow, then (2a) where is the mass flow rate across the area normal to the flow.
When flow is effectively incompressible, ρ = constant; in pipeline and duct flow analyses, the average velocity is then V = (1/A)∫vdA.
The continuity relation is (2b) where Q is the volumetric flow rate. Except when branches occur, Q is the same at all sections along the conduit.
For the ideal-fluid model, flow patterns around bodies (or in con-duit section changes) result from displacement effects. An obstruc-tion in a fluid stream, such as a strut in a flow or a bump on the conduit wall, pushes the flow smoothly out of the way, so that behind the obstruction, the flow becomes uniform again. The effect of fluid inertia (density) appears only in pressure changes.
Pressure Variation Across Flow Pressure variation in fluid flow is important and can be easily measured. Variation across streamlines involves fluid rotation (vor-ticity). Lateral pressure variation across streamlines is given by the following relation (Bober and Kenyon 1980, Olson 1980, Robert-son 1965): (3) where r = radius of curvature of the streamline z = elevation This relation explains the pressure difference found between the inside and outside walls of a bend and near other regions of conduit section change. It also states that pressure variation is hydrostatic (p + ρgz = constant) across any conduit where stream-lines are parallel.
Bernoulli Equation and Pressure Variation along Flow A basic tool of fluid flow analysis is the Bernoulli relation, which involves the principle of energy conservation along a streamline.
Generally, the Bernoulli equation is not applicable across stream-lines. The first law of thermodynamics can be applied to mechanical flow energies (kinetic and potential) and thermal energies: heat is a form of energy and energy is conserved.
The change in energy content ∆E per unit mass of flowing mate-rial is a result from the work W done on the system plus the heat Q absorbed: Fluid energy is composed of kinetic, potential (due to elevation z), and internal (u) energies. Per unit mass of fluid, the above energy change relation between two sections of the system is where the work terms are (1) the external work EM from a fluid machine (EM is positive for a pump or blower) and (2) the pressure or flow work p/ρ. Rearranging, the energy equation can be written as the generalized Bernoulli equation: (4) The term in parentheses in Equation (4) is the Bernoulli constant: (5a) In cases with no viscous action and no work interaction, B is con-stant; more generally its change (or lack thereof) is considered in applying the Bernoulli equation. The terms making up B are fluid energies (pressure, kinetic, and potential) per mass rate of fluid flow. Alternative forms of this relation are obtained through multi-plication by ρ or division by g: (5b) (5c) The first form involves energies per volume flow rate, or pres-sures; the second involves energies per mass flow rate, or heads. In gas flow analysis, Equation (5b) is often used with the ρgz term dropped as negligible. Equation (5a) should be used when density variations occur. For liquid flows, Equation (5c) is commonly used.
Identical results are obtained with the three forms if the units are consistent and the fluids are homogeneous.
Many systems of pipes or ducts and pumps or blowers can be considered as one-dimensional flow. The Bernoulli equation is then considered as velocity and pressure vary along the conduit. Analy-sis is adequate in terms of the section-average velocity V of Equa-tion (2a) or (2b). In the Bernoulli relation [Equations (4) and (5)], v is replaced by V, and variation across streamlines can be ignored; the whole conduit is now taken as one streamline. Two- and three-dimensional details of local flow occurrences are still significant, but their effect is combined and accounted for in factors.
The kinetic energy term of the Bernoulli constant B is expressed as αV2/2, where the kinetic energy factor (α > 1) expresses the ratio of the true kinetic energy of the velocity profile to that of the mean flow velocity.
For laminar flow in a wide rectangular channel, α = 1.54, and for laminar flow in a pipe, α = 2.0. For turbulent flow in a duct α ≈ 1.
Heat transfer Q may often be ignored. The change of mechanical energy into internal energy ∆u may be expressed as EL. Flow anal-ysis involves the change in the Bernoulli constant (∆B = B2 − B1) between stations 1 and 2 along the conduit, and the Bernoulli equa-tion can be expressed as ρv A d ∫ constant = m · ρVA constant = = m · Q AV constant = = ∂ ∂r ----- p ρ ---gz + v2 r -----= E ∆ W Q + = v2 2 -----gz u + + ∆ EM p ρ --- ∆ – Q + = v2 2 -----gz p ρ ---+ + ∆ u ∆ + EM Q + = p ρ ---ν2 2 -----gz + + B = p ρv2 2 --------ρgz + + ρB = p ρg ------v2 2g ------z + + B g ---= Fluid Flow 2.3 (6a) or, dividing by g, in the form as (6b) The factors EM and EL are defined as positive, where gHM = EM represents energy added to the conduit flow by pumps or blowers, and gHL = EL represents energy dissipated, that is, converted into heat as mechanically nonrecoverable energy. A turbine or fluid motor thus has a negative HM or EM. For conduit systems with branches involving inflow or outflow, the total energies must be treated, and analysis is in terms of and not B.
When real-fluid effects of viscosity or turbulence are included, the continuity relation in Equation (2b) is not changed, but V must be evaluated from the integral of the velocity profile, using time-averaged local velocities.
In fluid flow past fixed boundaries, the velocity at the boundary is zero and shear stresses are produced. The equations of motion then become complex and exact solutions are difficult to find, except in simple cases.
Laminar Flow For steady, fully developed laminar flow in a parallel-walled conduit, the shear stress τ varies linearly with distance y from the centerline. For a wide rectangular channel, where τw = wall shear stress = b (dp/ds) 2b = wall spacing s = flow direction Because the velocity is zero at the wall (y = b), the integrated result is This is the Poiseuille-flow parabolic velocity profile for a wide rectangular channel. The average velocity V is two-thirds the max-imum velocity (at y = 0), and the longitudinal pressure drop in terms of conduit flow velocity is (7) The parabolic velocity profile can also be derived for the axisym-metric conduit (pipe) of radius R but with a different constant. The average velocity is then half the maximum, and the pressure drop relation is (8) Turbulence Fluid flows are generally turbulent, involving random perturba-tions or fluctuations of the flow (velocity and pressure), character-ized by an extensive hierarchy of scales or frequencies (Robertson 1963). Flow disturbances that are not random, but have some degree of periodicity, such as the oscillating vortex trail behind bodies, have been erroneously identified as turbulence. Only flows involv-ing random perturbations without any order or periodicity are tur-bulent; the velocity in such a flow varies with time or locale of measurement (Figure 2).
Turbulence can be quantified by statistical factors. Thus, the velocity most often used in velocity profiles is the temporal average velocity , and the strength of the turbulence is characterized by the root-mean-square of the instantaneous variation in velocity about this mean. The effects of turbulence cause the fluid to diffuse momentum, heat, and mass very rapidly across the flow. The Reynolds number Re, a dimensionless quantity, gives the relative ratio of inertial to viscous forces: where L = characteristic length ν = kinematic viscosity In flow through round pipes and tubes, the characteristic length is the diameter D. Generally, laminar flow in pipes can be expected if the Reynolds number, which is based on the pipe diameter, is less than about 2300. Fully turbulent flow exists when ReD > 10 000.
Between 2300 and 10 000, the flow is in a transition state and pre-dictions are unreliable. In other geometries, different criteria for the Reynolds number exist.
BASIC FLOW PROCESSES Wall Friction At the boundary of real-fluid flow, the relative tangential veloc-ity at the fluid surface is zero. Sometimes in turbulent flow studies, velocity at the wall may appear finite, implying a fluid slip at the wall. However, this is not the case; the difficulty is in velocity mea-surement (Goldstein 1938). Zero wall velocity leads to a high shear stress near the wall boundary and a slowing down of adjacent fluid layers. A velocity profile develops near a wall, with the velocity increasing from zero at the wall to an exterior value within a finite lateral distance.
Laminar and turbulent flow differ significantly in their velocity profiles. Turbulent flow profiles are flat compared to the more pointed profiles of laminar flow (Figure 3). Near the wall, velocities of the turbulent profile must drop to zero more rapidly than those of the laminar profile, so the shear stress and friction are much greater in the turbulent flow case. Fully developed conduit flow may be characterized by the pipe factor, which is the ratio of average to maximum (centerline) velocity. Viscous velocity profiles result in pipe factors of 0.667 and 0.50 for wide rectangular and axisymmet-ric conduits. Figure 4 indicates much higher values for rectangular and circular conduits for turbulent flow. Due to the flat velocity pro-files, the kinetic energy factor α in Equation (6) ranges from 1.01 to 1.10 for fully developed turbulent pipe flow.
p ρ ---αV2 2 ------gz + + 1 EM + p ρ ---αV2 2 ------gz + + 2 EL + = p ρg ------αV2 2g ------z + + 1 HM + p ρg ------αV2 2g ------z + + 2 HL + = m · B τ y b -- τw µ dv dy ------= = v b2 y2 – 2µ ---------------- dp ds ------= dp ds ------3µV b2 ---------- – = dp ds ------8µV R2 ---------- – = Fig. 2 Velocity Fluctuation at Point in Turbulent Flow v Re VL ν ⁄ = 2.4 2001 ASHRAE Fundamentals Handbook (SI) Boundary Layer In most flows, the friction of a bounding wall on the fluid flow is evidenced by a boundary layer. For flow around bodies, this layer (which is quite thin relative to distances in the flow direction) encom-passes all viscous or turbulent actions, causing the velocity in it to vary rapidly from zero at the wall to that of the outer flow at its edge.
Boundary layers are generally laminar near the start of their forma-tion but may become turbulent downstream of the transition point (Figure 5). For conduit flows, spacing between adjacent walls is gen-erally small compared with distances in the flow direction. As a result, layers from the walls meet at the centerline to fill the conduit.
A significant boundary-layer occurrence exists in a pipeline or conduit following a well-rounded entrance (Figure 5). Layers grow from the walls until they meet at the center of the pipe. Near the start of the straight conduit, the layer is very thin (and laminar in all prob-ability), so the uniform velocity core outside has a velocity only slightly greater than the average velocity. As the layer grows in thickness, the slower velocity near the wall requires a velocity increase in the uniform core to satisfy continuity. As the flow pro-ceeds, the wall layers grow (and the centerline velocity increases) until they join, after an entrance length Le. Application of the Ber-noulli relation of Equation (5) to the core flow indicates a decrease in pressure along the layer. Ross (1956) shows that although the entrance length Le is many diameters, the length in which the pres-sure drop significantly exceeds those for fully developed flow is on the order of 10 diameters for turbulent flow in smooth pipes.
In more general boundary-layer flows, as with wall layer devel-opment in a diffuser or for the layer developing along the surface of a strut or turning vane, pressure gradient effects can be severe and may even lead to separation. The development of a layer in an adverse-pressure gradient situation (velocity v1 at edge y = δ of layer decreasing in flow direction) with separation is shown in Figure 6.
Downstream from the separation point, fluid backflows near the wall. Separation is due to frictional velocity (thus local kinetic energy) reduction near the wall. Flow near the wall no longer has energy to move into the higher pressure imposed by the decrease in v1 at the edge of the layer. The locale of this separation is difficult to predict, especially for the turbulent boundary layer. Analyses verify the experimental observation that a turbulent boundary layer is less subject to separation than a laminar one because of its greater kinetic energy.
Flow Patterns with Separation In technical applications, flow with separation is common and often accepted if it is too expensive to avoid. Flow separation may be geometric or dynamic. Dynamic separation is shown in Figure 6.
Geometric separation (Figures 7 and 8) results when a fluid stream passes over a very sharp corner, as with an orifice; the fluid gener-ally leaves the corner irrespective of how much its velocity has been reduced by friction.
For geometric separation in orifice flow (Figure 7), the outer streamlines separate from the sharp corners and, because of fluid inertia, contract to a section smaller than the orifice opening, the vena contracta, with a limiting area of about six-tenths of the ori-fice opening. After the vena contracta, the fluid stream expands rather slowly through turbulent or laminar interaction with the fluid along its sides. Outside the jet, fluid velocity is small compared to that in the jet. Turbulence helps spread out the jet, increases the losses, and brings the velocity distribution back to a more uniform profile. Finally, at a considerable distance downstream, the velocity profile returns to the fully developed flow of Figure 3.
Other geometric separations (Figure 8) occur at a sharp entrance to a conduit, at an inclined plate or damper in a conduit, and at a sudden expansion. For these, a vena contracta can be identified; for sudden expansion, its area is that of the upstream contraction. Ideal-fluid theory, using free streamlines, provides insight and predicts contraction coefficients for valves, orifices, and vanes (Robertson 1965). These geometric flow separations are large loss-producing devices. To expand a flow efficiently or to have an entrance with Fig. 3 Velocity Profiles of Flow in Pipes Fig. 4 Pipe Factor for Flow in Conduits Fig. 5 Flow in Conduit Entrance Region Fig. 6 Boundary Layer Flow to Separation Fluid Flow 2.5 minimum losses, the device should be designed with gradual con-tours, a diffuser, or a rounded entrance.
Flow devices with gradual contours are subject to separation that is more difficult to predict, because it involves the dynamics of boundary layer growth under an adverse pressure gradient rather than flow over a sharp corner. In a diffuser, which is used to reduce the loss in expansion, it is possible to expand the fluid some distance at a gentle angle without difficulty (particularly if the boundary layer is turbulent). Eventually, separation may occur (Figure 9), which is frequently asymmetrical because of irregularities. Down-stream flow involves flow reversal (backflow) and excess losses exist. Such separation is termed stall (Kline 1959). Larger area expansions may use splitters that divide the diffuser into smaller divisions less likely to have separations (Moore and Kline 1958).
Another technique for controlling separation is to bleed some low-velocity fluid near the wall (Furuya et al. 1976). Alternatively, Heskested (1965, 1970) shows that suction at the corner of a sudden expansion has a strong positive effect on geometric separation.
Drag Forces on Bodies or Struts Bodies in moving fluid streams are subjected to appreciable fluid forces or drag. Conventionally expressed in coefficient form, drag forces on bodies can be expressed as (9) where A is the projected (normal to flow) area of the body. The drag coefficient CD depends on the body’s shape and angularity and on the Reynolds number of the relative flow in terms of the body’s characteristic dimension.
For Reynolds numbers of 103 to above 105, the CD of most bodies is constant due to flow separation, but above 105, the CD of rounded bodies drops suddenly as the surface boundary layer undergoes transition to turbulence. Typical CD values are given in Table 1; Hoerner (1965) gives expanded values.
For a strut crossing a conduit, the contribution to the loss of Equation (6b) is (10) where Ac = conduit cross-sectional area A = area of the strut facing the flow Cavitation Liquid flow with gas- or vapor-filled pockets can occur if the absolute pressure is reduced to vapor pressure or less. In this case, a cavity or series of cavities forms, because liquids are rarely pure enough to withstand any tensile stressing or pressures less than vapor pressure for any length of time (John and Haberman 1980, Knapp et al. 1970, Robertson and Wislicenus 1969). Robertson and Wislicenus (1969) indicate significant occurrences in various tech-nical fields, chiefly in hydraulic equipment and turbomachines.
Initial evidence of cavitation is the collapse noise of many small bubbles that appear initially as they are carried by the flow into regions of higher pressure. The noise is not deleterious and serves as a warning of the occurrence. As flow velocity further increases or pressure decreases, the severity of cavitation increases. More bub-bles appear and may join to form large fixed cavities. The space they occupy becomes large enough to modify the flow pattern and alter performance of the flow device. Collapse of the cavities on or near solid boundaries becomes so frequent that the cumulative impact in time results in damage in the form of cavitational erosion of the sur-face or excessive vibration. As a result, pumps can lose efficiency or their parts may erode locally. Control valves may be noisy or seri-ously damaged by cavitation.
Cavitation in orifice and valve flow is indicated in Figure 10.
With high upstream pressure and a low flow rate, no cavitation occurs. As pressure is reduced or flow rate increased, the minimum pressure in the flow (in the shear layer leaving the edge of the ori-fice) eventually approaches vapor pressure. Turbulence in this layer causes fluctuating pressures below the mean (as in vortex cores) and small bubble-like cavities. These are carried downstream into the Fig. 7 Geometric Separation, Flow Development, and Loss in Flow Through Orifice Fig. 8 Examples of Geometric Separation Encountered in Flows in Conduits D CDρAV 2 2 ⁄ = Table 1 Drag Coefficients Body Shape 103 < Re < 2 × 105 Re > 3 × 105 Sphere 0.36 to 0.47 ~0.1 Disk 1.12 1.12 Streamlined strut 0.1 to 0.3 < 0.1 Circular cylinder 1.0 to 1.1 0.35 Elongated rectangular strut 1.0 to 1.2 1.0 to 1.2 Square strut ~2.0 ~2.0 Fig. 9 Separation in Flow in Diffuser HL CD A Ac ----- V 2 2g ------ = 2.6 2001 ASHRAE Fundamentals Handbook (SI) region of pressure regain where they collapse, either in the fluid or on the wall (Figure 10A). As the pressure is reduced, more vapor- or gas-filled bubbles result and coalesce into larger ones. Eventually, a single large cavity results that collapses further downstream (Figure 10B). The region of wall damage is then as many as 20 diameters downstream from the valve or orifice plate.
Sensitivity of a device to cavitation occurrence is measured by the cavitation index or cavitation number, which is the ratio of the available pressure above vapor pressure to the dynamic pressure of the reference flow: (11) where pv is the vapor pressure, and the subscript o refers to appro-priate reference conditions. Valve analyses use such an index in order to determine when cavitation will affect the discharge coeffi-cient (Ball 1957).With flow-metering devices such as orifices, ven-turis, and flow nozzles, there is little cavitation, because it occurs mostly downstream of the flow regions involved in establishing the metering action.
The detrimental effects of cavitation can be avoided by operating the liquid-flow device at high enough pressures. When this is not possible, the flow must be changed or the device must be built to withstand cavitation effects. Some materials or surface coatings are more resistant to cavitation erosion than others, but none is immune.
Surface contours can be designed to delay the onset of cavitation.
Nonisothermal Effects When appreciable temperature variations exist, the primary fluid properties (density and viscosity) are no longer constant, as usually assumed, but vary across or along the flow. The Bernoulli equation in the form of Equations (5a) through (5c) must be used, because volumetric flow is not constant. With gas flows, the thermodynamic process involved must be considered. In general, this is assessed in applying Equation (5a), written in the following form: (12) Effects of viscosity variations also appear. With nonisothermal laminar flow, the parabolic velocity profile (Figure 3) is no longer valid. For gases, viscosity increases as the square root of absolute temperature, and for liquids, it decreases with increasing tempera-ture. This results in opposite effects.
For fully developed pipe flow, the linear variation in shear stress from the wall value τw to zero at the centerline is independent of the temperature gradient. In the section on Laminar Flow, τ is defined as τ = (y/b)τw, where y is the distance from the centerline and 2b is the wall spacing. For pipe radius R = D/2 and distance from the wall y = R − r (see Figure 11), then τ = τw (R − y)/R. Then, solving Equa-tion (1) for the change in velocity gives (13) When the fluid has a lower viscosity near the wall than at the center (due to external heating of liquid or cooling of gas via heat transfer through the pipe wall), the velocity gradient is steeper near the wall and flatter near the center, so the profile is generally flat-tened. When liquid is cooled or gas is heated, the velocity profile becomes more pointed for laminar flow (Figure 11). Calculations were made for such flows of gases and liquid metals in pipes (Deissler 1951). Occurrences in turbulent flow are less apparent. If enough heating is applied to gaseous flows, the viscosity increase can cause reversion to laminar flow.
Fig. 10 Cavitation in Flows in Orifice or Valve σ 2 po pv – ( ) ρVo 2 -------------------------= Fig. 11 Effect of Viscosity Variation on Velocity Profile of Laminar Flow in Pipe p d ρ ------V2 2 ------gz + + ∫ B = dv τw R y – ( ) Rµ ----------------------- dy τw Rµ ------- – r dr = = Fluid Flow 2.7 Buoyancy effects and gradual approach of the fluid temperature to equilibrium with that outside the pipe can cause considerable variation in the velocity profile along the conduit. Thus, Colborne and Drobitch (1966) found the pipe factor for upward vertical flow of hot air at a Reynolds number less than 2000 reduced to about 0.6 at 40 diameters from the entrance, then increased to about 0.8 at 210 diameters, and finally decreased to the isothermal value of 0.5 at the end of 320 diameters.
Compressibility All fluids are compressible to some degree; their density depends on the pressure. Steady liquid flow may ordinarily be treated as incompressible, and incompressible flow analysis is satisfactory for gases and vapors at velocities below about 20 to 40 m/s, except in long conduits.
For liquids in pipelines, if flow is suddenly stopped, a severe pressure surge or water hammer is produced that travels along the pipe at the speed of sound in the liquid. This pressure surge alter-nately compresses and decompresses the liquid. For steady gas flows in long conduits, a decrease in pressure along the conduit can reduce the density of the gas significantly enough to cause the velocity to increase. If the conduit is long enough, velocities approaching the speed of sound are possible at the discharge end, and the Mach number (the ratio of the flow velocity to the speed of sound) must be considered.
Some compressible flows occur without heat gain or loss (adia-batically). If there is no friction (conversion of flow mechanical energy into internal energy), the process is reversible as well. Such a reversible adiabatic process is called isentropic, and follows the relationship where k, the ratio of specific heats at constant pressure and volume, has a value of 1.4 for air and diatomic gases.
The Bernoulli equation of steady flow, Equation (12), as an inte-gral of the ideal-fluid equation of motion along a streamline, then becomes (14) where, as in most compressible flow analyses, the elevation terms involving z are insignificant and are dropped. For a frictionless adiabatic process, the pressure term has the form (15) Then, between stations 1 and 2 for the isentropic process, (16) Equation (16) replaces the Bernoulli equation for compressible flows and may be applied to the stagnation point at the front of a body. With this point as station 2 and the upstream reference flow ahead of the influence of the body as station 1, V2 = 0. Solving Equa-tion (16) for p2 gives (17) where ps is the stagnation pressure.
Because kp/ρ is the square of the acoustic velocity a and the Mach number M = V/a, the stagnation pressure relation becomes (18) For Mach numbers less than one, (19) When M = 0, Equation (19) reduces to the incompressible flow result obtained from Equation (5a). Appreciable differences appear when the Mach number of the approaching flow exceeds 0.2. Thus a pitot tube in air is influenced by compressibility at velocities over about 66 m/s.
Flows through a converging conduit, as in a flow nozzle, venturi, or orifice meter, also may be considered isentropic. Velocity at the upstream station 1 is negligible. From Equation (16), velocity at the downstream station is (20) The mass flow rate is (21) The corresponding incompressible flow relation is (22) The compressibility effect is often accounted for in the expan-sion factor Y: (23) Y is 1.00 for the incompressible case. For air (k = 1.4), a Y value of 0.95 is reached with orifices at p2/p1 = 0.83 and with venturis at about 0.90, when these devices are of relatively small diameter (D2/D1 less than 0.5).
As p2/p1 decreases, the flow rate increases, but more slowly than for the incompressible case because of the nearly linear decrease in Y. However, the downstream velocity reaches the local acoustic value and the discharge levels off at a value fixed by upstream pres-sure and density at the critical ratio: (24) At higher pressure ratios than critical, choking (no increase in flow with decrease in downstream pressure) occurs and is used in some p ρk ⁄ constant = k cp cv ⁄ = p d ρ ------V2 2 ------+ ∫ constant = p d ρ ------1 2 ∫ k k 1 – ----------- p2 ρ2 -----p1 ρ1 -----– = p1 ρ1 -----k k 1 – ----------- p2 p1 ----- k 1 – ( ) k ⁄ 1 – V2 2 V1 2 – 2 ------------------+ 0 = ps p2 p1 1 k 1 – 2 ----------- ρ1V1 2 kp1 ------------+ k k 1 – ( ) ⁄ = = ps p1 1 k 1 – 2 ----------- M1 2 + k k 1 – ( ) ⁄ = ps p1 ρ1V1 2 2 ------------ 1 M1 4 -------2 k – 24 ----------- M1 4 … + + + + = V2 2k k 1 – ----------- p1 ρ1 ----- 1 p2 p1 ----- k 1 – ( ) k ⁄ – = m · V2A2ρ2 = = A2 2k k 1 – ----------- p1ρ1 ( ) p2 p1 ----- 2 k ⁄ p2 p1 ----- k 1 + ( ) k ⁄ – m · in A2ρ 2 p ρ ⁄ ∆ A2 2ρ p1 p2 – ( ) = = m · Ym · in A2Y 2ρ p1 p2 – ( ) = = p2 p1 -----c 2 k 1 + ------------ k k 1 – ( ) ⁄ 0.53 for air = = 2.8 2001 ASHRAE Fundamentals Handbook (SI) flow control devices to avoid flow dependence on downstream conditions.
FLOW ANALYSIS Fluid flow analysis is used to correlate pressure changes with flow rates and the nature of the conduit. For a given pipeline, either the pressure drop for a certain flow rate, or the flow rate for a certain pressure difference between the ends of the conduit, is needed. Flow analysis ultimately involves comparing a pump or blower to a con-duit piping system for evaluating the expected flow rate.
Generalized Bernoulli Equation Internal energy differences are generally small and usually the only significant effect of heat transfer is to change the density ρ. For gas or vapor flows, use the generalized Bernoulli equation in the pressure-over-density form of Equation (6a), allowing for the ther-modynamic process in the pressure-density relation: (25a) The elevation changes involving z are often negligible and are dropped. The pressure form of Equation (5b) is generally unaccept-able when appreciable density variations occur, because the volu-metric flow rate differs at the two stations. This is particularly serious in friction-loss evaluations where the density usually varies over considerable lengths of conduit (Benedict and Carlucci 1966).
When the flow is essentially incompressible, Equation (25a) is sat-isfactory.
Example 1. Specify the blower to produce an isothermal airflow of 200 L/s through a ducting system (Figure 12). Accounting for intake and fitting losses, the equivalent conduit lengths are 18 and 50 m and the flow is isothermal. The pressure at the inlet (station 1) and following the dis-charge (station 4), where the velocity is zero, is the same. The frictional losses HL are evaluated as 7.5 m of air between stations 1 and 2, and 72.3 m between stations 3 and 4.
Solution: The following form of the generalized Bernoulli relation is used in place of Equation (25a), which also could be used: (25b) The term can be calculated as follows: The term can be calculated in a similar manner.
In Equation (25b), HM is evaluated by applying the relation between any two points on opposite sides of the blower. Because condi-tions at stations 1 and 4 are known, they are used, and the location-specifying subscripts on the right side of Equation (25b) are changed to 4. Note that p1 = p4 = p, ρ1 = ρ4 = ρ, and V1 = V4 = 0. Thus, so HM = 82.2 m of air. For standard air (ρ = 1.20 kg/m3), this corre-sponds to 970 Pa.
The pressure difference measured across the blower (between sta-tions 2 and 3), is often taken as the HM. It can be obtained by calculat-ing the static pressure at stations 2 and 3. Applying Equation (25b) successively between stations 1 and 2 and between 3 and 4 gives where α just ahead of the blower is taken as 1.06, and just after the blower as 1.03; the latter value is uncertain because of possible uneven discharge from the blower. Static pressures p1 and p4 may be taken as zero gage. Thus, The difference between these two numbers is 81 m, which is not the HM calculated after Equation (25b) as 82.2 m. The apparent discrep-ancy results from ignoring the velocity at stations 2 and 3. Actually, HM is the following: The required blower energy is the same, no matter how it is evalu-ated. It is the specific energy added to the system by the machine. Only when the conduit size and velocity profiles on both sides of the machine are the same is EM or HM simply found from ∆p = p3 − p2.
Conduit Friction The loss term EL or HL of Equation (6a) or (6b) accounts for fric-tion caused by conduit-wall shearing stresses and losses from con-duit-section changes. HL is the loss of energy per unit weight (J/N) of flowing fluid.
In real-fluid flow, a frictional shear occurs at bounding walls, gradually influencing the flow further away from the boundary. A lateral velocity profile is produced and flow energy is converted into heat (fluid internal energy), which is generally unrecoverable (a loss). This loss in fully developed conduit flow is evaluated through the Darcy-Weisbach equation: (26) where L is the length of conduit of diameter D and f is the friction factor. Sometimes a numerically different relation is used with the Fanning friction factor (one-quarter of f ). The value of f is nearly constant for turbulent flow, varying only from about 0.01 to 0.05.
dp ρ ------1 2 ∫ α1 V1 2 2 ------EM + + α2 V2 2 2 ------EL + = p1 ρ1g ⁄ ( ) α1 V1 2 2g ⁄ ( ) z1 HM + + + p2 ρ2g ⁄ ( ) α2 V2 2 2g ⁄ ( ) z2 HL + + + = V1 2 2g ⁄ Fig. 12 Blower and Duct System for Example 1 A1 π D 2 ---- 2 π 0.250 2 ------------- 2 0.0491 m2 = = = V1 Q A1 ⁄ 0.200 0.0491 ----------------4.07 m s ⁄ = = = V1 2 2g ⁄ 4.07 ( )2 2 9.8 ( ) ⁄ 0.846 m = = V2 2 2g ⁄ p ρg ⁄ ( ) 0 0.61 HM + + + p ρg ⁄ ( ) 0 3 7.5 72.3 + ( ) + + + = p1 ρg ⁄ ( ) 0 0.61 0 + + + p2 ρg ⁄ ( ) 1.06 0.846 × ( ) 0 7.5 + + + = p3 ρg ⁄ ( ) 1.03 2.07 × ( ) + 0 0 + + p4 ρg ⁄ ( ) 0 3 72.3 + + + = p2 ρg ⁄ 7.8 m of air – = p3 ρg ⁄ 73.2 m of air = HM p3 ρg ⁄ ( ) α3 V3 2 2g ⁄ ( ) p2 ρg ⁄ ( ) α2 V2 2 2g ⁄ ( ) + [ ] – + = 73.2 1.03 2.07 × ( ) + 7.8 1.06 0.846 × ( ) + – [ ] – = 75.3 6.9 – ( ) – 82.2 m = = HL ( )f f L D ---- V2 2g ------ = Fluid Flow 2.9 For fully developed laminar-viscous flow in a pipe, the loss is evaluated from Equation (8) as follows: (27) where Thus, for laminar flow, the friction factor varies inversely with the Reynolds number.
With turbulent flow, friction loss depends not only on flow con-ditions, as characterized by the Reynolds number, but also on the nature of the conduit wall surface. For smooth conduit walls, empir-ical correlations give (28a) (28b) Generally, f also depends on the wall roughness ε. The variation is complex and best expressed in chart form (Moody 1944) as shown in Figure 13. Inspection indicates that, for high Reynolds numbers and relative roughness, the friction factor becomes inde-pendent of the Reynolds number in a fully-rough flow regime. Then (29a) Values of f between the values for smooth tubes and those for the fully-rough regime are represented by Colebrook’s natural rough-ness function: (29b) A transition region appears in Figure 13 for Reynolds numbers between 2000 and 10 000. Below this critical condition, for smooth walls, Equation (27) is used to determine f ; above the critical con-dition, Equation (28b) is used. For rough walls, Figure 13 or Equa-tion (29b) must be used to assess the friction factor in turbulent flow.
To do this, the roughness height ε, which may increase with conduit use or aging, must be evaluated from the conduit surface (Table 2).
Fig. 13 Relation Between Friction Factor and Reynolds Number (Moody 1944) HL ( )f L ρg ------ 8µV R2 ---------- 32LνV D2g -----------------64 VD ν ⁄ --------------- L D ---- V2 2g ------ = = = Re VD ν and f 64 Re.
⁄ = ⁄ = f 0.3164 Re0.25 ----------------= for Re 105 < f 0.0032 0.221 Re0.237 -----------------+ = for 105 Re 3 < < 106 × 1 f ---------1.14 2 log D ε ⁄ ( ) + = 1 f ---------1.14 2 log D ε ⁄ ( ) + = 2 log 1 9.3 Re ε D ⁄ ( ) f --------------------------------+ – 2.10 2001 ASHRAE Fundamentals Handbook (SI) Although the preceding discussion has focused on circular pipes and ducts, air ducts are often rectangular in cross section. The equiv-alent circular conduit corresponding to the noncircular conduit must be found before Figure 13 or Equations (28) or (29) can be used.
Based on turbulent flow concepts, the equivalent diameter is determined by (30) where A = flow area Pw = wetted perimeter of the cross section For turbulent flow, Deq is substituted for D in Equation (26) and the Reynolds number definition in Equation (27). Noncircular duct friction can be evaluated to within 5% for all except very extreme cross sections. A more refined method for finding the equivalent circular duct diameter is given in Chapter 34. With laminar flow, the loss predictions may be off by a factor as large as two.
Section Change Effects and Losses Valve and section changes (contractions, expansions and diffus-ers, elbows or bends, tees), as well as entrances, distort the fully developed velocity profiles (Figure 3) and introduce extra flow losses (dissipated as heat) into pipelines or duct systems. Valves produce such extra losses to control flow rate. In contractions and expansions, flow separation as shown in Figures 8 and 9 causes the extra loss. The loss at rounded entrances develops as the flow accel-erates to higher velocities. The resulting higher velocity near the wall leads to wall shear stresses greater than those of fully devel-oped flow (Figure 5). In flow around bends, the velocity increases along the inner wall near the start of the bend. This increased veloc-ity creates a secondary motion, which is a double helical vortex pat-tern of flow downstream from the bend. In all these devices, the disturbance produced locally is converted into turbulence and appears as a loss in the downstream region.
The return of disturbed flow to a fully developed velocity profile is quite slow. Ito (1962) showed that the secondary motion follow-ing a bend takes up to 100 diameters of conduit to die out but the pressure gradient settles out after 50 diameters.
With laminar flow following a rounded entrance, the entrance length depends on the Reynolds number: (31) At Re = 2000, a length of 120 diameters is needed to establish the parabolic profile. The pressure gradient reaches the devel-oped value of Equation (26) much sooner. The extra drop is 1.2V2/2g; the change in profile from uniform to parabolic results in a drop of 1.0V2/2g (since α = 2.0), and the rest is due to excess friction. With turbulent flow, 80 to 100 diameters following the rounded entrance are needed for the velocity profile to become fully developed, but the friction loss per unit length reaches a value close to that of the fully developed flow value more quickly. After six diameters, the loss rate at a Reynolds number of 105 is only 14% above that of fully developed flow in the same length, while at 107, it is only 10% higher (Robertson 1963). For a sharp entrance, the flow separation (Figure 8) causes a greater disturbance, but fully developed flow is achieved in about half the length required for a rounded entrance. With sudden expansion, the pressure change settles out in about eight times the diameter change (D2 = D1), while the velocity profile takes at least a 50% greater distance to return to fully developed pipe flow (Lipstein 1962).
These disturbance effects are assumed compressed (in the flow direction) into a point, and the losses are treated as locally occur-ring. Such a loss is related to the velocity by the fitting loss coeffi-cient K: (32) Chapter 35 and the Pipe Friction Manual (Hydraulic Institute 1961) have information for pipe applications. Chapter 34 gives infor-mation for airflow. The same type of fitting in pipes and ducts may give a different loss, because flow disturbances are controlled by the detailed geometry of the fitting. The elbow of a small pipe may be a threaded fitting that differs from a bend in a circular duct. For 90 screw-fitting elbows, K is about 0.8 (Ito 1962), whereas smooth flanged elbows have a K as low as 0.2 at the optimum curvature.
Table 3 gives a list of fitting loss coefficients. These values indi-cate the losses, but there is considerable variance. Expansion flows, such as from one conduit size to another or at the exit into a room or reservoir, are not included. For such occurrences, the Borda loss pre-diction (from impulse-momentum considerations) is appropriate: (33) Such expansion loss is reduced by avoiding or delaying separa-tion using a gradual diffuser (Figure 9). For a diffuser of about 7° total angle, the loss is minimal, about one-sixth that given by Equa-tion (33). The diffuser loss for total angles above 45 to 60° exceeds that of the sudden expansion, depending somewhat on the diameter ratio of the expansion. Optimum design of diffusers involves many factors; excellent performance can be achieved in short diffusers with splitter vanes or suction. Turning vanes in miter bends produce the least disturbance and loss for elbows; with careful design, the loss coefficient can be reduced to as low as 0.1.
For losses in smooth elbows, Ito (1962) found a Reynolds num-ber effect (K slowly decreasing with increasing Re) and a minimum loss at a bend curvature (bend radius to diameter ratio) of 2.5. At this optimum curvature, a 45° turn had 63%, and a 180° turn approxi-mately 120%, of the loss of a 90° bend. The loss does not vary lin-early with the turning angle because secondary motion occurs.
Use of coefficient K presumes its independence of the Reynolds number. Crane Co. (1976) found a variation with the Reynolds num-ber similar to that of the friction factor; Kittridge and Rowley (1957) observed it only with laminar flow. Assuming that K varies with Re similarly to f, it is convenient to represent fitting losses as adding to the effective length of uniform conduit. The effective length of a fitting is then (34) where fref is an appropriate reference value of the friction factor.
Deissler (1951) uses 0.028, and the air duct values in Chapter 34 are based on an fref of about 0.02. For rough conduits, appreciable errors can occur if the relative roughness does not correspond to that used when fref was fixed. It is unlikely that the fitting losses involving separation are affected by pipe roughness. The effective length method for fitting loss evaluation is still useful.
When a conduit contains a number of section changes or fittings, the values of K are added to the fL/D friction loss, or the Leff /D of the fittings are added to the conduit length L/D for evaluating the total loss HL. This assumes that each fitting loss is fully developed and its disturbance fully smoothed out before the next section Table 2 Effective Roughness of Conduit Surfaces Material ε, ft Commercially smooth brass, lead, copper, or plastic pipe 0.000005 Steel and wrought iron 0.00015 Galvanized iron or steel 0.0005 Cast iron 0.00085 Deq 4A Pw ⁄ = Le D ⁄ 0.06 Re ≈ Loss of section K V2 2g ------ = Loss at expansion V1 V2 – ( )2 2g -------------------------V1 2 2g ------ 1 A1 A2 ------– 2 = = Leff D ⁄ K fref ⁄ = Fluid Flow 2.11 change. Such an assumption is frequently wrong, and the total loss can be overestimated. For elbow flows, the total loss of adjacent bends may be over or underestimated. The secondary flow pattern following an elbow is such that when one follows another, perhaps in a different plane, the secondary flow of the second elbow may reinforce or partially cancel that of the first. Moving the second elbow a few diameters can reduce the total loss (from more than twice the amount) to less than the loss from one elbow. Screens or perforated plates can be used for smoothing velocity profiles (Wile 1947) and flow spreading. Their effectiveness and loss coefficients depend on their amount of open area (Baines and Peterson 1951).
Compressible Conduit Flow When friction loss is included, as it must be except for a very short conduit, the incompressible flow analysis previously consid-ered applies until the pressure drop exceeds about 10% of the initial pressure. The possibility of sonic velocities at the end of relatively long conduits limits the amount of pressure reduction achieved. For an inlet Mach number of 0.2, the discharge pressure can be reduced to about 0.2 of the initial pressure; for an inflow at M = 0.5, the dis-charge pressure cannot be less than about 0.45p1 in the adiabatic case and about 0.6p1 in isothermal flow.
Analysis of such conduit flow must treat density change, as eval-uated from the continuity relation in Equation (2), with the frictional occurrences evaluated from wall roughness and Reynolds number correlations of incompressible flow (Binder 1944). In evaluating valve and fitting losses, consider the reduction in K caused by com-pressibility (Benedict and Carlucci 1966). Although the analysis differs significantly, isothermal and adiabatic flows involve essen-tially the same pressure variation along the conduit, up to the limit-ing conditions.
Control Valve Characterization Control valves are characterized by a discharge coefficient Cd.
As long as the Reynolds number is greater than 250, the orifice equation holds for liquids: (35) where Ao = area of orifice opening P = absolute pressure The discharge coefficient is about 0.63 for sharp-edged configu-rations and 0.8 to 0.9 for chamfered or rounded configurations. For gas flows at pressure ratios below the choking critical [Equation (24)], the mass rate of flow is (36) where C1 = k = ratio of specific heats at constant pressure and volume R = gas constant T = absolute temperature u, d = subscripts referring to upstream and downstream positions Incompressible Flow in Systems Flow devices must be evaluated in terms of their interaction with other elements of the system, for example, the action of valves in modifying flow rate and in matching the flow-producing device (pump or blower) with the system loss. Analysis is via the general Bernoulli equation and the loss evaluations noted previously.
A valve regulates or stops the flow of fluid by throttling. The change in flow is not proportional to the change in area of the valve opening. Figures 14 and 15 indicate the nonlinear action of valves in controlling flow. Figure 14 shows a flow in a pipe discharging water from a tank that is controlled by a gate valve. The fitting loss coeffi-cient K values are those of Table 3; the friction factor f is 0.027. The degree of control also depends on the conduit L/D ratio. For a rela-tively long conduit, the valve must be nearly closed before its high K value becomes a significant portion of the loss. Figure 15 shows a con-trol damper (essentially a butterfly valve) in a duct discharging air from a plenum held at constant pressure. With a long duct, the damper does not affect the flow rate until it is about one-quarter closed. Duct length has little effect when the damper is more than half closed. The damper closes the duct totally at the 90° position (K = ∞).
Flow in a system (pump or blower and conduit with fittings) involves interaction between the characteristics of the flow-produc-ing device (pump or blower) and the loss characteristics of the pipeline or duct system. Often the devices are centrifugal, in which case the pressure produced decreases as the flow increases, except for the lowest flow rates. System pressure required to overcome losses increases roughly as the square of the flow rate. The flow rate of a given system is that where the two curves of pressure versus flow rate intersect (point 1 in Figure 16). When a control valve (or Q CdAo 2 P ρ ⁄ ∆ = m · CdAoC1 Pu Tu ----------- Pd Pu -------- 1 Pd Pu ------ k 1 – ( ) k ⁄ – = 2k R ⁄ k 1 – ( ) Fig. 14 Valve Action in Pipeline Fig. 15 Effect of Duct Length on Damper Action 2.12 2001 ASHRAE Fundamentals Handbook (SI) damper) is partially closed, it increases the losses and reduces the flow (point 2 in Figure 16). For cases of constant pressure, the flow decrease due to valving is not as great as that indicated in Figures 14 and 15.
Flow Measurement The general principles noted (the continuity and Bernoulli equa-tions) are basic to most fluid-metering devices. Chapter 14 has fur-ther details.
The pressure difference between the stagnation point (total pres-sure) and that in the ambient fluid stream (static pressure) is used to give a point velocity measurement. The flow rate in a conduit is measured by placing a pitot device at various locations in the cross section and spatially integrating the velocity profile found. A single point measurement may be used for approximate flow rate evalua-tion. When the flow is fully developed, the pipe-factor information of Figure 4 can be used to estimate the flow rate from a centerline measurement. Measurements can be made in one of two modes.
With the pitot-static tube, the ambient (static) pressure is found from pressure taps along the side of the forward-facing portion of the tube. When this portion is not long and slender, static pressure indication will be low and velocity indication high; as a result, a tube coefficient less than unity must be used. For parallel conduit flow, wall piezometers (taps) may take the ambient pressure, and the pitot tube indicates the impact (total pressure).
The venturi meter, flow nozzle, and orifice meter are flow rate metering devices based on the pressure change associated with rel-atively sudden changes in conduit section area (Figure 17). The elbow meter (also shown in Figure 17) is another differential pres-sure flowmeter. The flow nozzle is similar to the venturi in action, but does not have the downstream diffuser. For all these, the flow rate is proportional to the square root of the pressure difference resulting from fluid flow. With the area change devices (venturi, flow nozzle, and orifice meter), a theoretical flow rate relation is found by applying the Bernoulli and continuity equations in Equa-tions (6) and (2) between stations 1 and 2: (37) where β = d/D = ratio of throat (or orifice) diameter to conduit diameter.
The actual flow rate through the device can differ because the approach flow kinetic energy factor α deviates from unity and because of small losses. More significantly, the jet contraction of orifice flow is neglected in deriving Equation (37), to the extent that it can reduce the effective flow area by a factor of 0.6. The effect of all these factors can be combined into the discharge coefficient Cd: (38) Sometimes an alternate coefficient is used of the form For compressible fluid metering, the expansion factor Y as described by Equation (23) must be included, and the mass flow rate is (39) Values of Y depend primarily on the pressure ratio p2/p1, and also on the metering device and k value of the particular gas.
The general mode of variation in Cd for orifices and venturis is indicated in Figure 18 as a function of Reynolds number and, to a lesser extent, diameter ratio β. For Reynolds numbers less than 10, the coefficient varies as .
The elbow meter employs the pressure difference between inside and outside the bend as the metering signal (Murdock et al. 1964).
A momentum analysis gives the flow rate as (40) where R is the radius of curvature of the bend. Again, a discharge coefficient Cd is needed; as in Figure 18, this drops off for the lower Reynolds numbers (below 105). These devices are calibrated in pipes Table 3 Fitting Loss Coefficients of Turbulent Flow Fitting Geometry Entrance Sharp 0.50 Well-rounded 0.05 Contraction Sharp (D2/D1 = 0.5) 0.38 90° Elbow Miter 1.3 Short radius 0.90 Long radius 0.60 Miter with turning vanes 0.2 Globe valve Open 10 Angle valve Open 5 Gate valve Open 0.19 to 0.22 75% open 1.10 50% open 3.6 25% open 28.8 Any valve Closed ∞ Tee Straight through flow 0.5 Flow through branch 1.8 K p ∆ ρg ⁄ V 2 2g ⁄ -----------------= Fig. 16 Matching of Pump or Blower to System Characteristics Fig. 17 Differential Pressure Flowmeters Qtheor πd2 4 --------- 2 p ∆ ρ 1 β4 – ( ) -----------------------= Q CdQtheor Cd πd2 4 --------- 2 p ∆ ρ 1 β4 – ( ) -----------------------= = Cd 1 β4 – --------------------m · CdYρQtheor CdY πd2 4 --------- 2ρ p ∆ 1 β4 – --------------= = Re Qtheor πD2 4 ---------- R 2D ------- 2 p ∆ ρ ---------- = Fluid Flow 2.13 with fully developed velocity profiles, so they must be located far enough downstream of sections that modify the approach velocity.
Unsteady Flow Conduit flows are not always steady. In a compressible fluid, the acoustic velocity is usually high and the conduit length is rather short, so the time of signal travel is negligibly small. Even in the incompressible approximation, system response is not instanta-neous. If a pressure difference ∆p is applied between the conduit ends, the fluid mass must be accelerated and wall friction overcome, so a finite time passes before the steady flow rate corresponding to the pressure drop is achieved.
The time it takes for an incompressible fluid in a horizontal con-stant-area conduit of length L to achieve steady flow may be esti-mated by using the unsteady flow equation of motion with wall friction effects included. On the quasi-steady assumption, friction is given by Equation (26); also by continuity, V is constant along the conduit. The occurrences are characterized by the relation (41) where θ = time s = distance in the flow direction Since a certain ∆p is applied over the conduit length L, (42) For laminar flow, f is given by Equation (27), and (43) Equation (43) can be rearranged and integrated to yield the time to reach a certain velocity: (44) and (45a) For long times (θ → ∞), this indicates steady velocity as (45b) as by Equation (8). Then, Equation (45a) becomes (46) where The general nature of velocity development for starting-up flow is derived by more complex techniques; however, the temporal variation is as given above. For shutdown flow (steady flow with ∆p = 0 at θ > 0), the flow decays exponentially as e−θ.
Turbulent flow analysis of Equation (41) also must be based on the quasi-steady approximation, with less justification. Daily et al.
(1956) indicate that the frictional resistance is slightly greater than the steady-state result for accelerating flows, but appreciably less for decelerating flows. If the friction factor is approximated as constant, and, for the accelerating flow, or Because the hyperbolic tangent is zero when the independent variable is zero and unity when the variable is infinity, the initial (V = 0 at θ = 0) and final conditions are verified. Thus, for long times (θ → ∞), which is in accord with Equation (26) when f is constant (the flow regime is the fully rough one of Figure 13). The temporal velocity variation is then (47) In Figure 19, the turbulent velocity start-up result is compared with the laminar one in Figure 19, where initially the turbulent is steeper but of the same general form, increasing rapidly at the start but reaching V∞ asymptotically.
Fig. 18 Flowmeter Coefficients dV dθ -------1 ρ --- dp ds -------fV 2 2D --------+ + 0 = dV dθ -------p ∆ ρL ------fV 2 2D --------– = dV dθ -------p ∆ ρL ------32µV ρD2 -------------– A BV – = = θ θ d ∫ V d A BV – -----------------∫ 1 B ---A BV – ( ) ln – = = = V p ∆ L ------ D2 32µ --------- 1 ρL p ∆ ------32νθ – D2 ---------------- exp – = V∞ p ∆ L ------D2 32µ --------- p ∆ L ------ R2 8µ ------ = = V V∞1 ρL p ∆ ------f∞V∞θ – 2D -------------------- exp – = f∞ 64ν V∞D -----------= dV dθ -------p ∆ ρL ------fV 2 2D --------– A BV 2 – = = θ 1 AB ------------- tanh 1 – V B A ---- = V A B ---- tanh θ AB ( ) = V∞ A B ----- p ∆ ρL ⁄ f∞2D ⁄ ----------------- p ∆ ρL ------ 2D f∞ ------- = = = V V∞ f∞V∞θ 2D ⁄ ( ) tanh = 2.14 2001 ASHRAE Fundamentals Handbook (SI) NOISE FROM FLUID FLOW Noise in flowing fluids results from unsteady flow fields and can be at discrete frequencies or broadly distributed over the audible range. With liquid flow, cavitation results in noise through the col-lapse of vapor bubbles. The noise in pumps or fittings (such as valves) can be a rattling or sharp hissing sound. It is easily elimi-nated by raising the system pressure. With severe cavitation, the resulting unsteady flow can produce indirect noise from induced vibration of adjacent parts. See Chapter 46 of the 1999 ASHRAE Handbook—Applications for more information on sound control. The disturbed laminar flow behind cylinders can be an oscillat-ing motion. The shedding frequency f of these vortexes is charac-terized by a Strouhal number St = fd/V of about 0.21 for a circular cylinder of diameter d, over a considerable range of Reynolds num-bers. This oscillating flow can be a powerful noise source, particu-larly when f is close to the natural frequency of the cylinder or some nearby structural member so that resonance occurs. With cylinders of another shape, such as impeller blades of a pump or blower, the characterizing Strouhal number involves the trailing edge thickness of the member. The strength of the vortex wake, with its resulting vibrations and noise potential, can be reduced by breaking up the flow with downstream splitter plates or boundary-layer trip devices (wires) on the cylinder surface.
Noise produced in pipes and ducts, especially from valves and fittings, is associated with the loss through such elements. The sound pressure of noise in water pipe flow increases linearly with the pressure loss; the broad-band noise increases, but only in the lower frequency range. Fitting-produced noise levels also increase with fitting loss (even without cavitation) and significantly exceed noise levels of the pipe flow. The relation between noise and loss is not surprising because both involve excessive flow perturbations. A valve’s pressure-flow characteristics and structural elasticity may be such that for some operating point it oscillates, perhaps in reso-nance with part of the piping system, to produce excessive noise. A change in the operating point conditions or details of the valve geometry can result in significant noise reduction.
Pumps and blowers are strong potential noise sources. Turbo-machinery noise is associated with blade-flow occurrences. Broad-band noise appears from vortex and turbulence interaction with walls and is primarily a function of the operating point of the machine. For blowers, it has a minimum at the peak efficiency point (Groff et al. 1967). Narrow-band noise also appears at the blade-crossing frequency and its harmonics. Such noise can be very annoying because it stands out from the background. To reduce this noise, increase clearances between impeller and housing, and space impeller blades unevenly around the circumference.
REFERENCES Baines, W.D. and E.G. Peterson. 1951. An investigation of flow through screens. ASME Transactions 73:467.
Baker, A.J. 1983. Finite element computational fluid mechanics. McGraw-Hill, New York.
Ball, J.W. 1957. Cavitation characteristics of gate valves and globe values used as flow regulators under heads up to about 125 ft. ASME Trans-actions 79:1275.
Benedict, R.P. and N.A. Carlucci. 1966. Handbook of specific losses in flow systems. Plenum Press Data Division, New York.
Binder, R.C. 1944. Limiting isothermal flow in pipes. ASME Transactions 66:221.
Bober, W. and R.A. Kenyon. 1980. Fluid mechanics. John Wiley and Sons, New York.
Colborne, W.G. and A.J. Drobitch. 1966. An experimental study of non-iso-thermal flow in a vertical circular tube. ASHRAE Transactions 72(4):5.
Crane Co. 1976. Flow of fluids. Technical Paper No. 410. New York.
Daily, J.W., et al. 1956. Resistance coefficients for accelerated and deceler-ated flows through smooth tubes and orifices. ASME Transactions 78:1071.
Deissler, R.G. 1951. Laminar flow in tubes with heat transfer. National Advi-sory Technical Note 2410, Committee for Aeronautics.
Furuya, Y., T. Sate, and T. Kushida. 1976. The loss of flow in the conical with suction at the entrance. Bulletin of the Japan Society of Mechanical Engineers 19:131.
Goldstein, S., ed. 1938. Modern developments in fluid mechanics. Oxford University Press, London. Reprinted by Dover Publications, New York.
Groff, G.C., J.R. Schreiner, and C.E. Bullock. 1967. Centrifugal fan sound power level prediction. ASHRAE Transactions 73(II): V.4.1.
Heskested, G. 1965. An edge suction effect. AIAA Journal 3:1958.
Heskested, G. 1970. Further experiments with suction at a sudden enlarge-ment. Journal of Basic Engineering, ASME Transactions 92D:437.
Hoerner, S.F. 1965. Fluid dynamic drag, 3rd ed. Published by author, Mid-land Park, NJ.
Hydraulic Institute. 1961. Pipe friction manual. New York.
Ito, H. 1962. Pressure losses in smooth pipe bends. Journal of Basic Engi-neering, ASME Transactions 4(7):43.
John, J.E.A. and W.L. Haberman. 1980. Introduction to fluid mechanics, 2nd ed. Prentice Hall, Englewood Cliffs, NJ.
Kittridge, C.P. and D.S. Rowley. 1957. Resistance coefficients for laminar and turbulent flow through one-half inch valves and fittings. ASME Transactions 79:759.
Kline, S.J. 1959. On the nature of stall. Journal of Basic Engineering, ASME Transactions 81D:305.
Knapp, R.T., J.W. Daily, and F.G. Hammitt. 1970. Cavitation. McGraw-Hill, New York.
Lipstein, N.J. 1962. Low velocity sudden expansion pipe flow. ASHRAE Journal 4(7):43.
Moody, L.F. 1944. Friction factors for pipe flow. ASME Transactions 66:672.
Moore, C.A. and S.J. Kline. 1958. Some effects of vanes and turbulence in two-dimensional wide-angle subsonic diffusers. National Advisory Committee for Aeronautics, Technical Memo 4080.
Murdock, J.W., C.J. Foltz, and C. Gregory. 1964. Performance characteris-tics of elbow flow meters. Journal of Basic Engineering, ASME Trans-actions 86D:498.
Olson, R.M. 1980. Essentials of engineering fluid mechanics, 4th ed. Harper and Row, New York.
Robertson, J.M. 1963. A turbulence primer. University of Illinois (Urbana, IL), Engineering Experiment Station Circular 79.
Robertson, J.M. 1965. Hydrodynamics in theory and application. Prentice-Hall, Englewood Cliffs, NJ.
Robertson, J.M. and G.F. Wislicenus, ed. 1969 (discussion 1970). Cavitation state of knowledge. American Society of Mechanical Engineers, New York.
Ross, D. 1956. Turbulent flow in the entrance region of a pipe. ASME Trans-actions 78:915.
Schlichting, H. 1979. Boundary layer theory, 7th ed. McGraw-Hill, New York.
Streeter, V.L. and E.B. Wylie. 1979. Fluid mechanics, 7th ed. McGraw-Hill, New York.
Wile, D.D. 1947. Air flow measurement in the laboratory. Refrigerating Engineering: 515.
Fig. 19 Temporal Increase in Velocity Following Sudden Application of Pressure 3.1 CHAPTER 3 HEAT TRANSFER Heat Transfer Processes ........................................................... 3.1 Steady-State Conduction ........................................................... 3.1 Overall Heat Transfer ............................................................... 3.2 Transient Heat Flow ................................................................. 3.4 Thermal Radiation .................................................................... 3.6 Natural Convection ................................................................. 3.11 Forced Convection .................................................................. 3.13 Heat Transfer Augmentation Techniques ............................... 3.15 Extended Surface ..................................................................... 3.20 Symbols ................................................................................... 3.24 EAT is energy in transit due to a temperature difference. The Hthermal energy is transferred from one region to another by three modes of heat transfer: conduction, convection, and radia-tion. Heat transfer is among a group of energy transport phenomena that includes mass transfer (see Chapter 5), momentum transfer or fluid friction (see Chapter 2), and electrical conduction. Transport phenomena have similar rate equations, in which flux is propor-tional to a potential difference. In heat transfer by conduction and convection, the potential difference is the temperature difference.
Heat, mass, and momentum transfer are often considered together because of their similarities and interrelationship in many common physical processes.
This chapter presents the elementary principles of single-phase heat transfer with emphasis on heating, refrigerating, and air condi-tioning. Boiling and condensation are discussed in Chapter 4. More specific information on heat transfer to or from buildings or refrig-erated spaces can be found in Chapters 25 through 31 of this volume and in Chapter 12 of the 1998 ASHRAE Handbook—Refrigeration.
Physical properties of substances can be found in Chapters 18, 22, 24, and 36 of this volume and in Chapter 8 of the 1998 ASHRAE Handbook—Refrigeration. Heat transfer equipment, including evap-orators, condensers, heating and cooling coils, furnaces, and radia-tors, is covered in the 2000 ASHRAE Handbook—Systems and Equipment. For further information on heat transfer, see the section on Bibliography.
HEAT TRANSFER PROCESSES Thermal Conduction. This is the mechanism of heat transfer whereby energy is transported between parts of a continuum by the transfer of kinetic energy between particles or groups of particles at the atomic level. In gases, conduction is caused by elastic collision of molecules; in liquids and electrically nonconducting solids, it is believed to be caused by longitudinal oscillations of the lattice structure. Thermal conduction in metals occurs, like electrical con-duction, through the motion of free electrons. Thermal energy trans-fer occurs in the direction of decreasing temperature, a consequence of the second law of thermodynamics. In solid opaque bodies, ther-mal conduction is the significant heat transfer mechanism because no net material flows in the process and radiation is not a factor.
With flowing fluids, thermal conduction dominates in the region very close to a solid boundary, where the flow is laminar and par-allel to the surface and where there is no eddy motion.
Thermal Convection. This form of heat transfer involves energy transfer by fluid movement and molecular conduction (Burmeister 1983, Kays and Crawford 1980). Consider heat transfer to a fluid flowing inside a pipe. If the Reynolds number is large enough, three different flow regions exist. Immediately adjacent to the wall is a laminar sublayer where heat transfer occurs by thermal conduction; outside the laminar sublayer is a transition region called the buffer layer, where both eddy mixing and conduction effects are significant; beyond the buffer layer and extending to the center of the pipe is the turbulent region, where the dominant mechanism of transfer is eddy mixing.
In most equipment, the main body of fluid is in turbulent flow, and the laminar layer exists at the solid walls only. In cases of low-velocity flow in small tubes, or with viscous liquids such as glycol (i.e., at low Reynolds numbers), the entire flow may be laminar with no transition or turbulent region.
When fluid currents are produced by external sources (for exam-ple, a blower or pump), the solid-to-fluid heat transfer is termed forced convection. If the fluid flow is generated internally by non-homogeneous densities caused by temperature variation, the heat transfer is termed free convection or natural convection.
Thermal Radiation. In conduction and convection, heat trans-fer takes place through matter. In thermal radiation, there is a change in energy form from internal energy at the source to elec-tromagnetic energy for transmission, then back to internal energy at the receiver. Whereas conduction and convection heat transfer rates are driven primarily by temperature difference and somewhat by temperature level, radiative heat transfer rates increase rapidly with temperature levels (for the same temperature difference).
Although some generalized heat transfer equations have been mathematically derived from fundamentals, they are usually ob-tained from correlations of experimental data. Normally, the corre-lations employ certain dimensionless numbers, shown in Table 1, that are derived from dimensional analysis or analogy.
STEADY-STATE CONDUCTION For steady-state heat conduction in one dimension, the Fourier law is (1) where q = heat flow rate, W k = thermal conductivity, W/(m·K) A = cross-sectional area normal to flow, m2 dt/dx = temperature gradient, K/m Equation (1) states that the heat flow rate q in the x direction is directly proportional to the temperature gradient dt/dx and the cross-sectional area A normal to the heat flow. The proportionality factor is the thermal conductivity k. The minus sign indicates that the heat flow is positive in the direction of decreasing temperature. Conduc-tivity values are sometimes given in other units, but consistent units must be used in Equation (1).
The preparation of this chapter is assigned to TC 1.3, Heat Transfer and Fluid Flow.
q kA ( ) dt dx ------– = 3.2 2001 ASHRAE Fundamentals Handbook (SI) Equation (1) may be integrated along a path of constant heat flow rate to obtain (2) where Am = mean cross-sectional area normal to flow, m2 Lm = mean length of heat flow path, m ∆t = overall temperature difference, K R = thermal resistance, K/W Thermal resistance R is directly proportional to the mean length Lm of the heat flow path and inversely proportional to the conduc-tivity k and the mean cross-sectional area Am normal to the flow.
Equations for thermal resistances of a few common shapes are given in Table 2. Mathematical solutions to many heat conduction prob-lems are addressed by Carslaw and Jaeger (1959). Complicated problems can be solved by graphical or numerical methods such as described by Croft and Lilley (1977), Adams and Rogers (1973), and Patankar (1980).
Analogy to Electrical Conduction. Equation (2) is analogous to Ohm’s law for electrical circuits: thermal current (heat flow) in a thermal circuit is directly proportional to the thermal potential (temperature difference) and inversely proportional to the thermal resistance. This electrical-thermal analogy can be used for heat con-duction in complex shapes that resist solution by exact analytical means. The thermal circuit concept is also useful for problems involving combined conduction, convection, and radiation.
OVERALL HEAT TRANSFER In most steady-state heat transfer problems, more than one heat transfer mode is involved. The various heat transfer coefficients may be combined into an overall coefficient so that the total heat transfer can be calculated from the terminal temperatures. The solu-tion to this problem is much simpler if the concept of a thermal cir-cuit is employed.
Local Overall Heat Transfer Coefficient— Resistance Method Consider heat transfer from one fluid to another by a three-step steady-state process: from a warmer fluid to a solid wall, through the wall, then to a colder fluid. An overall heat transfer coefficient U based on the difference between the bulk temperatures t1 − t2 of the two fluids is defined as follows: (3) where A is the surface area. Because Equation (3) is a definition of U, the surface area A on which U is based is arbitrary; it should always be specified in referring to U.
The temperature drops across each part of the heat flow path are where ts1, and ts2 are the warm and cold surface temperatures of the wall, respectively, and R1, R2, and R3 are the thermal resistances.
Because the same quantity of heat flows through each thermal resis-tance, these equations combined yield the following: Table 1 Dimensionless Numbers Commonly Used in Heat Transfer Name Symbol Valuea Application Nusselt number Nu hD/k, hL/k, q″ D/∆tk, or q″ L/∆tk Natural or forced convection, boiling or condensing Reynolds number Re GD/µ or ρVL/µ Forced convection, boiling or condensing Prandtl number Pr µcp/k Natural or forced convection, boiling or condensing Stanton number St h/Gcp Forced convection Grashof number Gr L3ρ2βg∆t/µ2 or L3ρ2g∆t/Tµ2 Natural convection (for ideal gases) Fourier number Fo ατ / L2 Unsteady-state conduction Peclet number Pe GDcp/k or Re Pr Forced convection (small Pr) Graetz number Gz GD2cp/kL or Re Pr D/L Laminar forced convection aA list of the other symbols used in this chapter appears in the section on Symbols.
q k Am Lm ------- ∆t ∆t R -----= = q UA t1 t2 – ( ) = Table 2 Solutions for Some Steady-State Thermal Conduction Problems System R in Equation q = ∆t/R Flat wall or curved wall if curvature is small (wall thickness less than 0.1 of inside diameter) Radial flow through a right circular cylinder Buried cylinder Radial flow in a hollow sphere L, r, a = dimensions, m k = thermal conductivity at average material temperature, W/(m·K) A = surface area, m2 R L kA ------= R ro ri ⁄ ( ) ln 2πkL --------------------------= R (a a2 r2 – ) r ⁄ + ln 2πkL ---------------------------------------------------------= cosh 1 – a r ⁄ ( ) 2πkL ---------------------------------L 2r » ( ) = R 1 ri ⁄ 1 ro ⁄ – ( ) 4πk ---------------------------------------= t1 ts1 – qR1 = ts1 ts2 – qR2 = ts2 t2 – qR3 = Heat Transfer 3.3 (4) As shown above, the equations are analogous to those for elec-trical circuits; for thermal current flowing through n resistances in series, the resistances are additive.
(5) Similarly, conductance is the reciprocal of resistance, and for heat flow through resistances in parallel, the conductances are additive: (6) For convection, the thermal resistance is inversely proportional to the convection coefficient hc and the applicable surface area: (7) The thermal resistance for radiation is written similarly to that for convection: (8) The radiation coefficient hr is a function of the temperatures, radiation properties, and geometrical arrangement of the enclosure and the body in question.
Resistance Method Analysis. Analysis by the resistance method can be illustrated by considering heat transfer from air out-side to cold water inside an insulated pipe. The temperature gradi-ents and the nature of the resistance analysis are shown in Figure 1.
Because air is sensibly transparent to radiation, some heat trans-fer occurs by both radiation and convection to the outer insulation surface. The mechanisms act in parallel on the air side. The total transfer then passes through the insulating layer and the pipe wall by thermal conduction, and then by convection and radiation into the cold water stream. (Radiation is not significant on the water side because liquids are sensibly opaque to radiation, although water transmits energy in the visible region.) The contact resistance between the insulation and the pipe wall is assumed negligible.
The heat transfer rate qrc for a given length L of pipe may be thought of as the sum of the rates qr and qc flowing through the par-allel resistances Rr and Rc associated with the surface radiation and convection coefficients. The total flow then proceeds through the resistance R3 offered to thermal conduction by the insulation, through the pipe wall resistance R2, and into the water stream through the convection resistance R1. Note the analogy to direct-current electricity. A temperature (potential) drop is required to overcome resistances to the flow of thermal current. The total resis-tance to heat transfer Ro is the sum of the individual resistances: (9) where the resultant parallel resistance R4 is obtained from (10) If the individual resistances can be evaluated, the total resistance can be obtained from this relation. The heat transfer rate for the length of pipe L can be established by (11) For a unit length of the pipe, the heat transfer rate is (12) The temperature drop ∆t through each individual resistance may then be calculated from the relation: (13) where n = 1, 2, and 3.
Mean Temperature Difference When heat is exchanged between two fluids flowing through a heat exchanger, the local temperature difference ∆t varies along the flow path. Heat transfer may be calculated using (14) where U is the overall uniform coefficient of heat transfer from fluid to fluid, A is the area associated with the coefficient U, and ∆tm is the appropriate mean temperature difference.
For parallel flow or counterflow exchangers and for any ex-changer in which one fluid temperature is substantially constant, the mean temperature difference is (15) where ∆t1, and ∆t2 are the temperature differences between the fluids at each end of the heatexchanger. ∆tm is called the logarithmic mean temperature difference. For the special case of ∆t1 = ∆t2 (possible only with a counterflow heat exchanger with equal capacities), which leads to an indeterminate form of Equation (15), ∆tm = ∆t1 = ∆t2.
Equation (15) for ∆tm is true only if the overall coefficient and the specific heat of the fluids are constant through the heat exchanger, and no heat losses occur (often well-approximated in practice). Parker et al. (1969) give a procedure for cases with vari-able overall coefficient U.
t1 t2 – q --------------1 UA --------R1 R2 R3 + + = = Ro R1 R2 R3 … Rn + + + + = C 1 Ro ------1 R1 ------1 R2 ------1 R3 ------… 1 Rn ------+ + + + = = Rc 1 hc A ----------= Rr 1 hr A ---------= Fig. 1 Thermal Circuit Diagram for Insulated Cold Water Line Ro R1 R2 R3 R4 + + + = 1 R4 ------1 Rr -----1 Rc -----+ = qrc te t – Ro -----------= qrc L -------te t – RoL -----------= tn ∆ Rnqrc = q UA tm ∆ = tm ∆ t1 ∆ t2 ∆ – t1 ∆ t2 ∆ ⁄ ( ) ln ---------------------------------t1 ∆ t2 ∆ – 2.3 log t1 ∆ t2 ∆ ⁄ ( ) -------------------------------------------= = 3.4 2001 ASHRAE Fundamentals Handbook (SI) Calculations using Equation (14) and ∆tm are convenient when terminal temperatures are known. In many cases, however, the tem-peratures of the fluids leaving the exchanger are not known. To avoid trial-and-error calculations, an alternate method involves the use of three nondimensional parameters, defined as follows: 1. Exchanger Heat Transfer Effectiveness ε (16) where Ch = ( cp)h = hot fluid capacity rate, W/K Cc = ( cp)c = cold fluid capacity rate, W/K Cmin = smaller of capacity rates Ch and Cc th = terminal temperature of hot fluid, °C. Subscript i indicates entering condition; subscript o indicates leaving condition.
tc = terminal temperature of cold fluid, °C. Subscripts i and o are the same as for th.
2. Number of Exchanger Heat Transfer Units (NTU) (17) where A is the area used to define overall coefficient U.
3. Capacity Rate Ratio Z (18) For a given exchanger, the heat transfer effectiveness can gener-ally be expressed for a given exchanger as a function of the number of transfer units and the capacity rate ratio: (19) The effectiveness is independent of the temperatures in the exchanger. For any exchanger in which the capacity rate ratio Z is zero (where one fluid undergoes a phase change; e.g., in a condenser or evaporator), the effectiveness is (20) Heat transferred can be determined from (21) Combining Equations (16) and (21) produces an expression for heat transfer rate in terms of entering fluid temperatures: (22) The proper mean temperature difference for Equation (14) is then given by (23) The effectiveness for parallel flow exchangers is (24) For Z = 1, (25) The effectiveness for counterflow exchangers is (26) (27) Incropera and DeWitt (1996) and Kays and London (1984) show the relations of ε, NTU, and Z for other flow arrangements. These authors and Afgan and Schlunder (1974) present graphical repre-sentations for convenience.
TRANSIENT HEAT FLOW Often, the heat transfer and temperature distribution under unsteady-state (varying with time) conditions must be known.
Examples are (1) cold storage temperature variations on starting or stopping a refrigeration unit; (2) variation of external air temperature and solar irradiation affecting the heat load of a cold storage room or wall temperatures; (3) the time required to freeze a given material under certain conditions in a storage room; (4) quick freezing of objects by direct immersion in brines; and (5) sudden heating or cooling of fluids and solids from one temperature to a different temperature.
The equations describing transient temperature distribution and heat transfer are presented in this section. Numerical methods are the simplest means of solving these equations because numerical data are easy to obtain. However, with some numerical solutions and off-the-shelf software, the physics that drives the energy trans-port can be lost. Thus, analytical solution techniques are also included in this section.
The fundamental equation for unsteady-state conduction in sol-ids or fluids in which there is no substantial motion is (28) where thermal diffusivity αis the ratio k/ρcp; k is thermal conductiv-ity; ρ, density; and cp, specific heat. If α is large (high conductivity, low density and specific heat, or both), heat will diffuse faster.
One of the most elementary transient heat transfer models pre-dicts the rate of temperature change of a body or material being held at constant volume with uniform temperature, such as a well-stirred reservoir of fluid whose temperature is changing because of a net rate of heat gain or loss: (29) where M is the mass of the body, cp is the specific heat at constant pressure, and qnet is the net heat transfer rate to the substance (heat transfer into the substance is positive, and heat transfer out of the substance is negative). Equation (29) is applicable when the pres-sure around the substance is constant; if the volume of the substance is constant, cp should be replaced by the constant volume specific heat cv. It should be noted that with the density of solids and liquids being almost constant, the two specific heats are almost equal. The term qnet may include heat transfer by conduction, convection, or radiation and is the difference between the heat transfer rates into and out of the body.
ε thi tho – ( ) thi tci – ( ) -----------------------= when Ch Cmin = ε tco tci – ( ) thi tci – ( ) -----------------------= when Cc Cmin = m · m · NTU AUavg Cmin ----------------1 Cmin -----------U A ∫ dA = = Z Cmin Cmax ------------= ε f NTU, Z, flow arrangement ( ) = ε 1 exp NTU – ( ) – = q Ch thi tho – ( ) Cc tco tci – ( ) = = q εCmin thi tci – ( ) = tm ∆ thi tci – ( )ε NTU -------------------------= ε 1 exp NTU – [ – 1 Z + ( )] 1 Z + ---------------------------------------------------------= ε 1 exp 2 NTU – ( ) – 2 -------------------------------------------= ε 1 exp NTU – 1 Z – ( ) [ ] – 1 Z exp NTU – 1 Z – ( ) [ ] – -------------------------------------------------------------= ε NTU 1 NTU + --------------------- for Z 1 = = ∂t ∂τ ------α ∂2t ∂x2 --------∂2t ∂y2 --------∂2t ∂z2 --------+ + = qnet Mcp ( ) dt dτ ------= Heat Transfer 3.5 From Equations (28) and (29), it is possible to derive expressions for temperature and heat flow variations at different instants and different locations. Most common cases have been solved and presented in graphical forms (Jakob 1957, Schneider 1964, Myers 1971). In other cases, it is simpler to use numerical methods (Croft and Lilley 1977, Patankar 1980). When convective boundary con-ditions are required in the solution of Equations (28) and (29), h val-ues based on steady-state correlations are often used. However, this approach may not be valid when rapid transients are involved.
Estimating Cooling Times Cooling times for materials can be estimated (McAdams 1954) by Gurnie-Lurie charts (Figures 2, 3, and 4), which are graphical solutions for the heating or cooling of infinite slabs, infinite cylin-ders, and spheres. These charts assume an initial uniform tempera-ture distribution and no change of phase. They apply to a body exposed to a constant temperature fluid with a constant surface con-vection coefficient of h.
Using Figures 2, 3, and 4, it is possible to estimate both the tem-perature at any point and the average temperature in a homogeneous mass of material as a function of time in a cooling process. It is pos-sible to estimate cooling times for rectangular-shaped solids, cubes, cylinders, and spheres.
From the point of view of heat transfer, a cylinder insulated on its ends behaves like a cylinder of infinite length, and a rectangular solid insulated so that only two parallel faces allow heat transfer behaves like an infinite slab. A thin slab or a long, thin cylinder may be also considered infinite objects.
Consider a slab having insulated edges being cooled. If the cool-ing time is the time required for the center of the slab to reach a tem-perature of t2, the cooling time can be calculated as follows: 1. Evaluate the temperature ratio (tc − t2)/(tc − t1).
where tc = temperature of cooling medium t1 = initial temperature of product t2 = final temperature of product at center Note that in Figures 2, 3, and 4, the temperature ratio (tc −t2)/ (tc −t1) is designated as Y to simplify the equations.
2. Determine the radius ratio r/rm designated as n in Figures 2, 3, and 4.
where r = distance from centerline rm = half thickness of slab 3. Evaluate the resistance ratio k/hrm designated as m in Figures 2, 3, and 4.
where k = thermal conductivity of material h = heat transfer coefficient 4. From Figure 2 for infinite slabs, select the appropriate value of kτ/ρcprm 2 designated as Fo in Figures 2, 3, and 4.
where τ = time elapsed cp = specific heat ρ = density 5. Determine τ from the value of kτ/ρcprm 2.
Multidimensional Temperature Distribution The solution for semi-infinite slabs and cylinders (shown in Fig-ures 2, 3, and 4) can be used to find the temperatures in finite rect-angular solids or cylinders.
Fig. 2 Transient Temperatures for Infinite Slab 3.6 2001 ASHRAE Fundamentals Handbook (SI) The temperature in the finite object can be calculated from the temperature ratio Y of the infinite objects that intersect to form the finite object. The product of the temperature ratios of the infinite objects is the temperature ratio of the finite object; for example, for the finite cylinder of Figure 5, (30) where Yfc = temperature ratio of finite cylinder Yis = temperature ratio of infinite slab Yic = temperature ratio of infinite cylinder For a finite rectangular solid, (31) where Yfrs = temperature ratio of finite rectangular solid, and sub-scripts 1, 2, and 3 designate three infinite slabs. The convective heat transfer coefficients associated with one pair of parallel surfaces need not be equal to the coefficient associated with another pair.
However, the temperature of the fluid adjacent to every surface should be the same. In evaluating the resistance ratio and the Fourier number Fo, the appropriate values of the heat transfer coefficient and the characteristic dimension should be used.
Heat Exchanger Transients Determination of the transient behavior of heat exchangers is becoming increasingly important in evaluating the dynamic behav-ior of heating and air-conditioning systems. Many studies of the transient behavior of counterflow and parallel flow heat exchangers have been conducted; some are listed in the section on Bibliography.
THERMAL RADIATION Radiation, one of the basic mechanisms for energy transfer between different temperature regions, is distinguished from con-duction and convection in that it does not depend on an intermediate Fig. 3 Transient Temperatures for Infinite Cylinder Yfc YisYic = Yfrs Yis ( )1 Yis ( )2 Yis ( )3 = Heat Transfer 3.7 material as a carrier of energy but rather is impeded by the presence of material between the regions. The radiation energy transfer pro-cess is the consequence of energy-carrying electromagnetic waves that are emitted by atoms and molecules due to changes in their energy content. The amount and characteristics of radiant energy emitted by a quantity of material depend on the nature of the mate-rial, its microscopic arrangement, and its absolute temperature.
Although rate of energy emission is independent of the surround-ings, the net energy transfer rate depends on the temperatures and spatial relationships of the surface and its surroundings.
Blackbody Radiation The rate of thermal radiant energy emitted by a surface depends on its absolute temperature. A surface is called black if it can absorb all incident radiation. The total energy emitted per unit time per unit area of black surface Wb to the hemispherical region above it is given by the Stefan-Boltzmann law.
(32) Fig. 4 Transient Temperatures for Spheres Fig. 5 Finite Cylinder of Intersection from Intersection of Infinite Cylinder and Infinite Slab Wb σT4 = 3.8 2001 ASHRAE Fundamentals Handbook (SI) where Wb is the total rate of energy emission per unit area, and σis the Stefan-Boltzmann constant [5.670 × 10− 8 W/(m2·K4)].
The heat radiated by a body comprises electromagnetic waves of many different frequencies or wavelengths. Planck showed that the spectral distribution of the energy radiated by a blackbody is (33) where Wbλ = monochromatic emissive power of blackbody, W/m3 λ = wavelength, µm T = temperature, K C1 = first Planck’s law constant = 3.742 × 10− 16 W·m2 C2 = second Planck’s law constant = 0.014388 m·K Wbλ is the monochromatic emissive power, defined as the energy emitted per unit time per unit surface area at wavelength λ per unit wavelength interval around λ; that is, the energy emitted per unit time per unit surface area in the interval dλ is equal to Wbλdλ.
The Stefan-Boltzmann equation can be obtained by integrating Planck’s equation: (34) Wien showed that the wavelength of maximum emissive power multiplied by the absolute temperature is a constant: (35) where λmaxis the wavelength at which the monochromatic emissive power is a maximum and not the maximum wavelength. Equation (35) is known as Wien’s displacement law. According to this law, the maximum spectral emissive power is displaced to shorter wave-lengths with increasing temperature, such that significant emission eventually occurs over the entire visible spectrum as shorter wave-lengths become more prominent. For additional details, see Incrop-era and DeWitt (1996).
Actual Radiation Substances and surfaces diverge variously from the Stefan-Boltzmann and Planck laws. Wb and Wbλ are the maximum emissive powers at a surface temperature. Actual surfaces emit and absorb less than these maximums and are called nonblack. The emissive power of a nonblack surface at temperature T radiating to the hemi-spherical region above it is written as (36) where ε is known as the hemispherical emittance. The term emit-tance conforms to physical and electrical terminology; the suffix “ance” denotes a property of a piece of material as it exists. The ending “ivity” denotes a property of the bulk material independent of geometry or surface condition. Thus, emittance, reflectance, absorptance, and transmittance refer to actual pieces of material.
Emissivity, reflectivity, absorptivity, and transmissivity refer to the properties of materials that are optically smooth and thick enough to be opaque.
The emittance is a function of the material, the condition of its surface, and the temperature of the surface. Table 3 lists selected values; Siegel and Howell (1981) and Modest (1993) have more extensive lists.
The monochromatic emissive power of a nonblack surface is similarly written as (37) where ελ is the monochromatic hemispherical emittance. The rela-tionship between ε and ελ is given by or (38) If ελ does not depend on λ, then, from Equation (38), ε = ελ. Sur-faces with this characteristic are called gray. Gray surface charac-teristics are often assumed in calculations. Several classes of surfaces approximate this condition in some regionsofthe spectrum.
The simplicity is desirable, but care must be exercised, especially if Table 3 Emittances and Absorptances for Some Surfacesa Class Surfaces Total Normal Emittanceb Absorptance for Solar Radiation At 10 to 40°C At 500°C 1 A small hole in a large box, sphere, furnace, or enclosure .....................................................
0.97 to 0.99 0.97 to 0.99 0.97 to 0.99 2 Black nonmetallic surfaces such as asphalt, carbon, slate, paint, paper .................................
0.90 to 0.98 0.90 to 0.98 0.85 to 0.98 3 Red brick and tile, concrete and stone, rusty steel and iron, dark paints (red, brown, green, etc.) .......................................................................................................
0.85 to 0.95 0.75 to 0.90 0.65 to 0.80 4 Yellow and buff brick and stone, firebrick, fireclay................................................................
0.85 to 0.95 0.70 to 0.85 0.50 to 0.70 5 White or light cream brick, tile, paint or paper, plaster, whitewash........................................
0.85 to 0.95 0.60 to 0.75 0.30 to 0.50 6 Window glass ..........................................................................................................................
0.90 — c 7 Bright aluminum paint; gilt or bronze paint............................................................................
0.40 to 0.60 — 0.30 to 0.50 8 Dull brass, copper, or aluminum; galvanized steel; polished iron ..........................................
0.20 to 0.30 0.30 to 0.50 0.40 to 0.65 9 Polished brass, copper, monel metal .......................................................................................
0.02 to 0.05 0.05 to 0.15 0.30 to 0.50 10 Highly polished aluminum, tin plate, nickel, chromium.........................................................
0.02 to 0.04 0.05 to 0.10 0.10 to 0.40 11 Selective surfaces Stainless steel wire mesh...................................................................................................
0.23 to 0.28 — 0.63 to 0.86 White painted surface........................................................................................................
0.92 — 0.23 to 0.49 Copper treated with solution of NaClO2 and NaOH.........................................................
0.13 — 0.87 Copper, nickel, and aluminum plate with CuO coating ....................................................
0.09 to 0.21 — 0.08 to 0.93 aSee also Chapter 36, McAdams (1954), and Siegel and Howell (1981).
cAbsorbs 4 to 40% depending on its transmittance.
bHemispherical and normal emittance are unequal in many cases. The hemispherical emittance may vary from up to 30% greater for polished reflectors to 7% lower for nonconductors.
Wbλ C1λ 5 – e C2 λT ⁄ 1 – ----------------------------= Wb σT4 Wbλ λ d 0 ∞ ∫ = = λmaxT 2898 µm K ⋅ = W εWb εσT 4 = = Wλ ε λWbλ ε λ C1λ 5 – e C2 λT ⁄ 1 – ---------------------------- = = W εσT4 Wλ λ d 0 ∞ ∫ ε λWbλ λ d 0 ∞ ∫ = = = ε 1 σ T4 --------ε λWbλ λ d 0 ∞ ∫ = Heat Transfer 3.9 temperatures are high. Assumption of grayness is sometimes made because of the absence of information relating ελ and λ.
When radiant energy falls on a surface, it can be absorbed, reflected, or transmitted through the material. Therefore, from the first law of thermodynamics, (39) where α = fraction of incident radiation absorbed or absorptance τ = fraction of incident radiation transmitted or transmittance ρ = fraction of incident radiation reflected or reflectance If the material is opaque, as most solids are in the infrared, τ = 0 and α + ρ = 1. For a black surface, α= 1, ρ = 0, and τ = 0. Platinum black and gold black are as black as any actual surface and have absorptances of about 98% in the infrared. Any desired degree of blackness can be simulated by a small hole in a large enclosure.
Consider a ray of radiant energy entering the opening. It will undergo many internal reflections and be almost completely absorbed before it has a reasonable probability of passing back out of the opening.
Certain flat black paints also exhibit emittances of 98% over a wide range of conditions. They provide a much more durable sur-face than gold or platinum black and are frequently used on radia-tion instruments and as standard reference in emittance or reflectance measurements.
Kirchhoff’s law relates emittance and absorptance of any opaque surface from thermodynamic considerations; it states that for any surface where the incident radiation is independent of angle or where the surface is diffuse, ελ = αλ. If the surface is gray, or the incident radiation is from a black surface at the same temperature, then ε = αas well, but many surfaces are not gray. For most surfaces listed in Table 3, absorptance for solar radiation is different from emittance for low-temperature radiation. This is because the wave-length distributions are different in the two cases, and ελ varies with wavelength.
The foregoing discussion relates to total hemispherical radiation from surfaces. Energy distribution over the hemispherical region above the surface also has an important effect on the rate of heat transfer in various geometric arrangements.
Lambert’s law states that the emissive power of radiant energy over a hemispherical surface above the emitting surface varies as the cosine of the angle between the normal to the radiating surface and the line joining the radiating surface to the point of the hemi-spherical surface. This radiation is diffuse radiation. The Lambert emissive power variation is equivalent to assuming that radiation from a surface in a direction other than normal occurs as if it came from an equivalent area with the same emissive power (per unit area) as the original surface. The equivalent area is obtained by pro-jecting the original area onto a plane normal to the direction of radi-ation. Black surfaces obey the Lambert law exactly. The law is approximate for many actual radiation and reflection processes, especially those involving rough surfaces and nonmetallic materi-als. Most radiation analyses are based on the assumption of gray dif-fuse radiation and reflection.
In estimating heat transfer rates between surfaces of different geometries, radiation characteristics, and orientations, it is usually assumed that • All surfaces are gray or black • Radiation and reflection are diffuse • Properties are uniform over the surfaces • Absorptance equals emittance and is independent of the temperature of the source of incident radiation • The material in the space between the radiating surfaces neither emits nor absorbs radiation These assumptions greatly simplify problems, although results must be considered approximate.
Angle Factor The distribution of radiation from a surface among the surfaces it irradiates is indicated by a quantity variously called an intercep-tion, a view, a configuration, a shape factor, or an angle factor. In terms of two surfaces i and j, the angle factor Fij from surface i to surface j is the ratio of the radiant energy leaving surface i and directly reaching surface j to the total radiant energy leaving sur-face i. The angle factor from j to i is similarly defined, merely by interchanging the roles of i and j. This second angle factor is not, in general, numerically equal to the first. However, the reciprocity relation FijAi = FjiAj, where A is the surface area, is always valid.
Note that a concave surface may “see itself” (Fii ≠0), and that if n surfaces form an enclosure, (40) The angle factor F12 between two surfaces is (41) where dA1, and dA2 are elemental areas of the two surfaces, r is the distance between dA1 and dA2, and φ1 and φ2 are the angles between the respective normals to dA1 and dA2 and the connecting line r.
Numerical, graphical, and mechanical techniques can solve this equation (Siegel and Howell 1981, Modest 1993). Numerical values of the angle factor for common geometries are given in Figure 6.
Calculation of Radiant Exchange Between Surfaces Separated by Nonabsorbing Media Asurface radiatesenergy ata rateindependentof its surroundings and absorbs and reflects incident energy at a rate dependent on its surface condition. The net energy exchange per unit area is denoted by q or qj for unit area Aj. It is the rate of emission of the surface minus the total rate of absorption at the surface from all radiant effects in its surroundings, possibly including the return of some of its own emission by reflection off its surroundings. The rate at which energy must be supplied to the surface by other exchange processes if its temperature is to remain constant is q; therefore, to define q, the total radiant surroundings (in effect, an enclosure) must be specified.
Several methods have been developed to solve certain problems.
To calculate the radiation exchange at each surface of an enclosure of n opaque surfaces by simple, general equations convenient for machine calculation, two terms must be defined: G = irradiation; total radiation incident on surface per unit time and per unit area J = radiosity; total radiation that leaves surface per unit time and per unit area The radiosity is the sum of the energy emitted and the energy reflected: (42) Because the transmittance is zero, the reflectance is Thus, (43) The net energy lost by a surface is the difference between the radi-osity and the irradiation: α τ ρ 1 = + + Fij j 1 = n ∑ 1 = F12 1 A1 ------cos φ1cos φ2 πr2 ------------------------------A1 A d d A2 ∫ A1 ∫ = J εWb ρG + = ρ 1 α 1 ε – = – = J εWb 1 ε – ( )G + = 3.10 2001 ASHRAE Fundamentals Handbook (SI) (44) Substituting for G in terms of J from Equation (43), (45) Consider an enclosure of n isothermal surfaces with areas of A1, A2, …, An, emittances of ε1, ε2, …, εn, and reflectances of ρ1, ρ2, …, ρn, respectively.
The irradiation of surface i is the sum of the radiation incident on it from all n surfaces: or Substituting in Equation (44) yields the following simultaneous equations when each of the n surfaces is considered: (46) Equation (46) can be solved manually for the unknown Js if the number of surfaces is small. The solution for more complex enclo-sures requires a computer.
Once the radiosities (Js) are known, the net radiant energy lost by each surface is determined from Equation (45) as If the surface is black, Equation (45) becomes indeterminate, and an alternate expression must be used, such as Fig. 6 Radiation Angle Factor for Various Geometries q A ⁄ J G εWb 1 ε – ( )G G – + = – = q Wb J – 1 ε – ( ) εA ⁄ -------------------------------= GiAi FjiJjAj FijJjAi j 1 = n ∑ = j 1 = n ∑ = Gi FijJj j 1 = n ∑ = Ji ε iWbi 1 ε i – ( ) FijJj j 1 = n ∑ + = i 1 2 … n , , , = qi Wbi Ji – 1 ε i – ( ) ε iA ⁄ -----------------------------------= Heat Transfer 3.11 or (47) since All diffuse radiation processes are included in the aforemen-tioned enclosure method, and surfaces with special characteristics are assigned consistent properties. An opening is treated as an equivalent surface area Ae with a reflectance of zero. If energy enters the enclosure diffusely through the opening, Ae is assigned an equivalent temperature; otherwise, its temperature is taken as zero.
If the loss through the opening is desired, q2 is found. A window in the enclosure is assigned its actual properties.
A surface in radiant balance is one for which radiant emission is balanced by radiant absorption; heat is neither removed from nor supplied to the surface. Reradiating surfaces (insulated surfaces with qnet = 0), can be treated in Equation (46) as being perfectly reflective (i.e., ε = 0). The equilibrium temperature of such a sur-face can be found from once Equation (46) has been solved for the radiosities.
Use of angle factors and radiation properties as defined assumes that the surfaces are diffuse radiators—a good assumption for most nonmetals in the infrared region, but a poor assumption for highly polished metals. Subdividing the surfaces and considering the vari-ation of radiation properties with angle of incidence improves the approximation but increases the work required for a solution.
Radiation in Gases Elementary gases such as oxygen, nitrogen, hydrogen, and helium are essentially transparent to thermal radiation. Their absorption and emission bands are confined mainly to the ultravio-let region of the spectrum. The gaseous vapors of most compounds, however, have absorption bands in the infrared region. Carbon mon-oxide, carbon dioxide, water vapor, sulfur dioxide, ammonia, acid vapors, and organic vapors absorb and emit significant amounts of energy.
Radiation exchange by opaque solids is considered a surface phenomenon. Radiant energy does, however, penetrate the surface of all materials. The absorption coefficient gives the rate of ex-ponential attenuation of the energy. Metals have large absorption coefficients, and radiant energy penetrates less than 100 nm at most.
Absorption coefficients for nonmetals are lower. Radiation may be considered a surface phenomenon unless the material is transparent.
Gases have small absorption coefficients, so the path length of ra-diation through gas becomes very significant.
Beer’s law states that the attenuation of radiant energy in a gas is a function of the product pgL of the partial pressure of the gas and the path length. The monochromatic absorptance of a body of gas of thickness L is then given by (48) Because absorption occurs in discrete wavelengths, the absorp-tances must be summed over the spectral region corresponding to the temperature of the blackbody radiation passing through the gas.
The monochromatic absorption coefficient αλ is also a function of temperature and pressure of the gas; therefore, detailed treatment of gas radiation is quite complex.
Estimated emittance for carbon dioxide and water vapor in air at 24°C is a function of concentration and path length (Table 4). The values are for a hemispherically shaped body of gas radiating to an element of area at the center of the hemisphere. Among others, Mod-est (1993), Siegel and Howell (1981), and Hottel and Sarofim (1967) describe geometrical calculations in their texts on radiation transfer.
Generally, at low values of pgL, the mean path length L (or equivalent hemispherical radius for a gas body radiating to its surrounding sur-faces) is four times the mean hydraulic radius of the enclosure. A room with a dimensional ratio of 1:1:4 has a mean path length of 0.89 times the shortest dimension when considering radiation to all walls.
For a room with a dimensional ratio of 1:2:6, the mean path length for the gas radiating to all surfaces is 1.2 times the shortest dimen-sion. The mean path length for radiation to the 2 by 6 face is 1.18 times the shortest dimension. These values are for cases where the partial pressure of the gas times the mean path length approaches zero (pgL ≈0). The factor decreases with increasing values of pgL.
For average rooms with approximately 2.4 m ceilings and relative humidity ranging from 10 to 75% at 24°C, the effective path length for carbon dioxide radiation is about 85% of the ceiling height, or 2.0 m. The effective path length for water vapor is about 93% of the ceiling height, or 2.3 m. The effective emittance of the water vapor and carbon dioxide radiating to the walls, ceiling, and floor of a room 4.9 m by 14.6 m with 2.4 m ceilings is in the following tabulation.
The radiation heat transfer from the gas to the walls is then (49) The examples in Table 4 and the preceding text indicate the importance of gas radiation in environmental heat transfer prob-lems. Gas radiation in large furnaces is the dominant mode of heat transfer, and many additional factors must be considered. Increased pressure broadens the spectral bands, and interaction of different radiating species prohibits simple summation of the emittance fac-tors for the individual species. Departures from blackbody condi-tions necessitate separate calculations of the emittance and absorptance. McAdams (1954) and Hottel and Sarofim (1967) give more complete treatments of gas radiation.
NATURAL CONVECTION Heat transfer involving motion in a fluid due to the difference in density and the action of gravity is called natural convection or free convection. Heat transfer coefficients associated with gases for natural convection are generally much lower than those for forced convection, and it is therefore important not to ignore radia-tion in calculating the total heat loss or gain. Radiant transfer may be qi JiAiFij JjAjFji – j 1 = n ∑ = qi FijAi Ji Jj – ( ) j 1 = n ∑ = Fij Ai Fji Aj = Tk Jk σ ---- 0.25 = αλL 1 e αλL – – = Table 4 Emittance of CO2 and Water Vapor in Air at 24°C Path Length, m CO2, % by Volume Relative Humidity, % 0.1 0.3 1.0 10 50 100 0.03 0.06 0.09 0.06 0.17 0.22 30 0.09 0.12 0.16 0.22 0.39 0.47 300 0.16 0.19 0.23 0.47 0.64 0.70 Relative Humidity, % εg 10 0.10 50 0.19 75 0.22 q σAwε g Tg 4 Tw 4 – ( ) = 3.12 2001 ASHRAE Fundamentals Handbook (SI) of the same order of magnitude as natural convection, even at room temperatures, because wall temperatures in a room can affect human comfort (see Chapter 8).
Natural convection is important in a variety of heating and refrigeration equipment: (1) gravity coils used in high-humidity cold storage rooms and in roof-mounted refrigerant condensers, (2) the evaporator and condenser of household refrigerators, (3) baseboard radiators and convectors for space heating, and (4) cooling panels for air conditioning. Natural convection is also involved in heat loss or gain to equipment casings and intercon-necting ducts and pipes.
Consider heat transfer by natural convection between a cold fluid and a hot surface. The fluid in immediate contact with the surface is heated by conduction, becomes lighter, and rises because of the difference in density of the adjacent fluid. The vis-cosity of the fluid resists this motion. The heat transfer is influ-enced by (1) gravitational force due to thermal expansion, (2) viscous drag, and (3) thermal diffusion. Gravitational accelera-tion g, coefficient of thermal expansion β, kinematic viscosity v = µ/ρ, and thermal diffusivity α = k/ρcp affect natural convec-tion. These variables are included in the dimensionless numbers given in Equation (1) in Table 5. The Nusselt number Nu is a function of the product of the Prandtl number Pr and the Grashof number Gr. These numbers, when combined, depend on the fluid properties, the temperature difference ∆t between the surface and the fluid, and the characteristic length L of the surface. The con-stant c and the exponent n depend on the physical configuration and the nature of flow.
Natural convection cannot be represented by a single value of exponent n, but it can be divided into three regions: 1. Turbulent natural convection, for which n equals 0.33 2. Laminar natural convection, for which n equals 0.25 3. A region that has GrPr less than for laminar natural convection, for which the exponent n gradually diminishes from 0.25 to lower values Note that for wires, the GrPr is likely to be very small, so that the exponent n is 0.1 [Equation (5) in Table 5].
To calculate the natural-convection heat transfer coefficient, determine GrPr to find whether the boundary layer is laminar or tur-bulent; then apply the appropriate equation from Table 5. The cor-rect characteristic length indicated in the table must be used.
Because the exponent n is 0.33 for a turbulent boundary layer, the characteristic length cancels out in Equation (2) in Table 5, and the heat transfer coefficient is independent of the characteristic length, as seen in Equations (7), (9), and (11) in Table 5. Turbulence occurs when length or temperature difference is large. Because the length of a pipe is generally greater than its diameter, the heat transfer coef-ficient for vertical pipes is larger than for horizontal pipes.
Convection from horizontal plates facing downward when heated (or upward when cooled) is a special case. Because the hot air is above the colder air, theoretically no convection should occur.
Some convection is caused, however, by secondary influences such as temperature differences on the edges of the plate. As an approx-imation, a coefficient of somewhat less than half the coefficient for a heated horizontal plate facing upward can be used.
Because air is often the heat transport fluid, simplified equations for air are given in Table 5. Other information on natural convection is available in the section on Bibliography under Heat Transfer, General.
Observed differences in the comparison of recent experimental and numerical results with existing correlations for natural convec-tive heat transfer coefficients indicate that caution should be used when applying coefficients for (isolated) vertical plates to vertical surfaces in enclosed spaces (buildings). Bauman et al. (1983) and Table 5 Natural Convection Heat Transfer Coefficients I. General relationships Nu = c(Gr Pr)n (1) (2) Characteristic length L Vertical plates or pipes L = height Horizontal plates L = length Horizontal pipes L = diameter Spheres L = 0.5 × diameter Rectangular block, with horizontal length Lh and vertical length Lv 1/L = (1/Lh) + (1/Lv) II. Planes and pipes Horizontal or vertical planes, pipes, rectangular blocks, and spheres (excluding horizontal plates facing downward for heating and upward for cooling) (a) Laminar range, when Gr Pr is between 104 and 108 Nu = 0.56(Gr Pr)0.25 (3) (b) Turbulent range, when Gr Pr is between 108 and 1012 Nu = 0.13(Gr Pr)0.33 (4) III. Wires For horizontal or vertical wires, use L = diameter, for Gr Pr between 10− 7 and 1 Nu = (Gr Pr)0.1 (5) IV. With air Gr Pr = 1.6 × 106 L3∆t (at 21°C, L in m, ∆t in K) (a) Horizontal cylinders Small cylinder, laminar range h = 1.32(∆t/L)0.25 (6) Large cylinder, turbulent range h = 1.24(∆t)0.33 (7) (b) Vertical plates Small plates, laminar range h = 1.42(∆t/L)0.25 (8) Large plates, turbulent range h = 1.31(∆t)0.33 (9) (c) Horizontal plates, facing upward when heated or downward when cooled Small plates, laminar range h = 1.32(∆t/L)0.25 (10) Large plates, turbulent range h = 1.52(∆t)0.33 (11) (d) Horizontal plates, facing downward when heated or upward when cooled Small plates h = 0.59(∆t/L)0.25 (12) h c k L --- L3ρ2βg t ∆ µ2 ------------------------- f n µcp k -------- f n = Heat Transfer 3.13 Altmayer et al. (1983) developed improved correlations for calcu-lating natural convective heat transfer from vertical surfaces in rooms under certain temperature boundary conditions.
Natural convection can affect the heat transfer coefficient in the presenceof weak forced convection. As the forced-convection effect (i.e., the Reynolds number) increases, “mixed convection” (super-imposed forced-on-free convection) gives way to the pure forced-convection regime. In these cases, other sources describing com-bined free and forced convection should be consulted, since the heat transfer coefficient in the mixed-convection region is often larger than that calculated based on the natural- or forced-convection cal-culation alone. Metais and Eckert (1964) summarize natural-, mixed-, and forced-convection regimes for vertical and horizontal tubes. Figure 7 shows the approximate limits for horizontal tubes.
Other studies are described by Grigull et al. (1982).
FORCED CONVECTION Forced air coolers and heaters, forced air- or water-cooled con-densers and evaporators, and liquid suction heatexchangers are exam-ples of equipment that transfer heat primarily by forced convection.
When fluid flows over a flat plate, a boundary layer forms adja-cent to the plate. The velocity of the fluid at the plate surface is zero and increases to its maximum free stream value just past the edge of the boundary layer (Figure 8). Boundary layer formation is impor-tant because the temperature change from plate to fluid (thermal resistance) is concentrated here. Where the boundary layer is thick, thermal resistance is great and the heat transfer coefficient is small.
Flow within the boundary layer immediately downstream from the leading edge is laminar and is known as laminar forced convec-tion. As flow proceeds along the plate, the laminar boundary layer increases in thickness to a critical value. Then, turbulent eddies develop within the boundary layer, except for a thin laminar sub-layer adjacent to the plate.
The boundary layer beyond this point is a turbulent boundary layer, and the flow is turbulent forced convection. The region between the breakdown of the laminar boundary layer and the estab-lishment of the turbulent boundary layer is the transition region.
Because the turbulent eddies greatly enhance heat transport into the main stream, the heat transfer coefficient begins to increase rapidly through the transition region. For a flat plate with a smooth leading edge, the turbulent boundary layer starts at Reynolds numbers, based on distance from the leading edge, of about 300 000 to 500 000. In blunt-edged plates, it can start at much smaller Reynolds numbers.
For long tubes or channels of small hydraulic diameter, at suffi-ciently low flow velocity, the laminar boundary layers on each wall grow until they meet. Beyond this point, the velocity distribution does not change, and no transition to turbulent flow takes place.
This is called fully developed laminar flow. For tubes of large diameter or at higher velocities, transition to turbulence takes place and fully developed turbulent flow is established (Figure 9).
Therefore, the length dimension that determines the critical Rey-nolds number is the hydraulic diameter of the channel. For smooth circular tubes, flow is laminar for Reynolds numbers below 2100 and turbulent above 10 000.
Table 6 lists various forced-convection correlations. In the gen-eralized, dimensionless formula of Equation (1) in Table 6, heat transfer is determined by flow conditions and by the fluid proper-ties, as indicated by the Reynolds number and the Prandtl number.
This equation can be modified to Equation (4) in Table 6 to get the heat transfer factor j. The heat transfer factor is related to the fric-tion factor f by the interrelationship of the transport of momentum and heat; it is approximately f /2 for turbulent flow in straight ducts.
These factors are plotted in Figure 10.
Fig. 7 Regimes of Free, Forced, and Mixed Convection for Flow-Through Horizontal Tubes Fig. 8 Boundary Layer Buildup on Flat Plate (Vertical Scale Magnified) Fig. 9 Boundary Layer Buildup in Entry Length of Tube or Channel Fig. 10 Typical Dimensionless Representation of Forced-Convection Heat Transfer 3.14 2001 ASHRAE Fundamentals Handbook (SI) Table 6 Equations for Forced Convection Description Reference Equation Author Page Eq. No.
I. Generalized correlations Jakob 491 (23-36) (1) (a) Turbulent flow inside tubes (1) Using fluid properties based on bulk temperature t McAdams 219 (9-10a) (See Note a) (2) (2) Same as (1), except µ at surface temperature ts McAdams 219 (9-10c) (3) (3) Using fluid properties based on film temperature tf = 0.5(ts + t), except cp in Stanton modulus McAdams 219 (9-10b) (4) (4) For viscous fluids (viscosities higher than twice water), using viscosity µ at bulk temperature t and µs at surface temperature ts Jakob 547 (26-12) (5) (b) Laminar flow inside tubes (6) (1) For large D or high ∆t, the effect of natural convection should be included Jakob 544 (26-5) When (2) For very long tubes (c) Annular spaces, turbulent flow All fluid properties at bulk temperature except µs at surface temperature ts McAdams 242 (9-32c) (7) II. Simplified equations for gases, turbulent flow inside tubes [K in W/(m2·K), cp in kJ/(kg·K), G in kg/(m2·s), D in m] (a) Most common gases, turbulent flow (assuming µ = 18.8 µPa·s and µcp/k = 0.78) Obtained from Eq. (2) (8) (b) Air at ordinary temperatures Obtained from Eq. (2) (See Note b) (9) (c) Fluorinated hydrocarbon refrigerant gas at ordinary pressures Obtained from Eq. (2) (See Note b) (10) (d) Ammonia gas at approximately 65°C, 2 MPa Obtained from Eq. (2) (11) At − 18°C, 165 kPa (gage) Obtained from Eq. (2) (12) III. Simplified equations for liquids, turbulent flow inside tubes [h in W/(m2·K), G in kg/(m2·s), V in m/s, D in m, t in °C, µ in N·s/m2] (a) Water at ordinary temperatures, 4 to 93°C.
V is velocity in m/s, D is tube ID in metres.
McAdams 228 (9-19) (13) (b) Fluorinated hydrocarbon refrigerant liquid Obtained from Eq. (2) (See Note b) (14) (c) Ammonia liquid at approximately 38°C Obtained from Eq. (2) (15) (d) Oil heating, approximate equation Brown and Marco 146 (7-15) (16) (e) Oil cooling, approximate equation Brown and Marco 146 (7-15) (17) IV. Simplified equations for air (a) Vertical plane surfaces, V of 5 to 30 m/s (room temperature)c McAdams 249 (9-42) (18) (b) Vertical plane surfaces, V < 5 m/s (room temperature)c McAdams 249 (9-42) (19) (c) Single cylinder cross flow (film tempera-ture = 93°C) 1000 < GD/µf < 50 000 McAdams 261 (10-3c) (20) (d) Single sphere 17 < GD/µf < 70 000 McAdams 265 (10-6) (21) V. Gases flowing normal to pipes (dimensionless) (a) Single cylinder Re from 0.1 to 1000 McAdams 260 (10-3) (22) Re from 1000 to 50 000 McAdams 260 (10-3) (23) (b) Unbaffled staggered tubes, 10 rows. Approxi-mate equation for turbulent flowd McAdams 272 (10-11a) (24) (c) Unbaffled in-line tubes, 10 rows. Approximate equation for turbulent flowd (GmaxD/µf) from 2000 to 32 000 McAdams 272 (10-11a) (25) aMcAdams (1954) recommends this equation for heating and cooling. Others recom-mend exponents of 0.4 for heating and 0.3 for cooling, with a change in constant.
bTable 7 in Chapter 2 of the 1981 ASHRAE Handbook—Fundamentals lists values for c.
ch′ is expressed in W/(m2·K) based on initial temperature difference.
dGmax is based on minimum free area. Coefficients for tube banks depend greatly on geometrical details. These values approximate only.
hD k -------c GD µ --------- m µcp k -------- n = hD k -------0.023 GD µ --------- 0.8 µcp k -------- 0.4 = h cpG --------- cpµ k -------- 2 3 ⁄ µs µ ----- 0.14 0.023 GD µ ⁄ ( )0.2 ------------------------------= h cpG --------- cpµ k -------- f 2 3 ⁄ 0.023 GD µf ⁄ ( )0.2 -------------------------------j = = hD k -------0.027 GD µ --------- 0.8 µcp k -------- 1 3 ⁄ µ µs ----- 0.14 = hD k -------1.86 GD µ --------- cpµ k -------- D L ---- 1 3 ⁄ µ µs ----- 0.14 = GD µ --------- cpµ k -------- D L ---- 20 Eq. (6) should not be used , < h cpG --------- cpµ k -------- 2 3 ⁄ µs µ ----- 0.14 0.023 DeG µ ⁄ ( )0.2 ---------------------------------= h 3.031 cpG0.8 D0.2 ⁄ ( ) = h 155.2c G0.8 D0.2 ⁄ ( ) = h 155.2c G0.8 D0.2 ⁄ ( ) = h 6.663 G0.8 D0.2 ⁄ ( ) = h 5.323 G0.8 D0.2 ⁄ ( ) = h 1057 1.352 0.0198t + ( )V0.8 D0.2 -----------------------------------------------------------------= h 155.2c G0.8 D0.2 ⁄ ( ) = h 13.75 G0.8 D0.2 ⁄ ( ) = h 0.0047V µf 0.63 ⁄ = h 0.0035V µf 0.63 ⁄ = h′ 7.2V 0.78 = h′ 5.62 3.9V + = h 4.83 G0.6 D0.4 ⁄ ( ) = h 0.37 kf D ---- GD µf --------- 0.6 = hD kf -------0.32 0.43 GD µ --------- 0.52 + = hD kf -------0.24 GD µf --------- 0.6 = hD kf -------0.33 GmaxD µf ----------------- 0.6 µcp k -------- f 1 3 ⁄ = hD kf -------0.26 GmaxD µf ----------------- 0.6 µcp k -------- f 1 3 ⁄ = Heat Transfer 3.15 The characteristic length D is the diameter of the tube, outside or inside, or the length of the plane plate. For other shapes, the hydrau-lic diameter Dh is used. With a uniform surface temperature and assuming uniform heat transfer coefficient the inlet and exit temper-atures are related by: This reduces to twice the distance between surfaces for parallel plates or an annulus.
Simplified equations applicable to common fluids under normal operating conditions appear in Equations (8) through (25) of Table 6. Figure 11 gives graphical solutions for water. However, the value of the convective heat transfer coefficient with internal flows varies in the direction of the flow because of the temperature dependence of the properties of the fluids. In such a case a representative value of the heat transfer coefficient evaluated at the inlet and exit tem-peratures can be used in the above equation.
HEAT TRANSFER AUGMENTATION TECHNIQUES As discussed by Bergles (1998), techniques applied to augment (enhance) heat transfer can be classified as passive methods, which require no direct application of external power, or as active schemes, which require external power. Examples of passive tech-niques include rough surfaces, extended surfaces, displaced pro-moters, and vortex flow devices. Examples of active techniques include mechanical aids, surface vibration, fluid vibration, and elec-trostatic fields. The effectiveness of a given augmentation technique depends largely on the mode of heat transfer or the type of heat exchanger to which it is applied.
When augmentation is used, the dominant thermal resistances in Equation (9) should be considered; that is, do not invest in reducing an already low thermal resistance (increasing an already high heat transfer coefficient). Additionally, heat exchangers with a large number of heat transfer units (NTU) show relatively small gains in effectiveness with augmentation [see Equations (24) and (26)].
Finally, the increased friction factor that usually accompanies the heat transfer augmentation must be considered.
Passive Techniques Several examples of tubes with internal roughness or fins are shown in Figure 12. Rough surfaces of the spiral repeated rib variety are widely used to improve in-tube heat transfer with water, as in flooded chillers. The roughness may be produced by spirally indent-ing the outer wall, forming the inner wall, or inserting coils. Longi-tudinal or spiral internal fins in tubes can be produced by extrusion or forming and give a substantial increase in the surface area. The fin efficiency (see the section on Fin Efficiency on p. 3.20) can usu-ally be taken as unity. Twisted strips (vortex flow devices) can be inserted as original equipment or as retrofit devices. From a practi-cal point of view, the twisted tape width should be such that the tape can be easily inserted or removed. Ayub and Al-Fahed (1993) addressed the issue of the clearance between the twisted tape and tube inside dimension.
Microfin tubes (internally finned tubes with about 60 short fins around the circumference) are widely used in refrigerant evapora-tion and condensers. Since the gas entering the condenser in vapor-compression refrigeration is superheated, a considerable portion of the condenser acts to desuperheat the flow (i.e., it is single phase).
Some data on the single-phase performance of microfin tubes, showing considerably higher heat transfer coefficients than for plain tubes, are available (e.g., Khanpara et al. 1986, Al-Fahed et al.
1993), but the upper Reynolds numbers of about 10,000 are lower than those found in practice. This deficiency is being addressed in a current ASHRAE research project.
The increased friction factor may not require increased pumping power if the flow rate can be adjusted or if the length of the heat exchanger can be reduced. Nelson and Bergles (1986) discuss this issue of performance evaluation criteria, especially for HVAC applications.
Of concern in chilled water systems is the fouling that in some cases may seriously reduce the overall heat transfer coefficient U. In general, fouled enhanced tubesperformbetterthanfouled plaintubes, Dh 2rh 4 Cross-sectional area for flow Total wetted perimeter ---------------------------------------------------------------------× = = Fig. 11 Heat Transfer Coefficient for Turbulent Flow of Water Inside Tubes Fig. 12 Typical Tube-Side Enhancements 3.16 2001 ASHRAE Fundamentals Handbook (SI) Table 7 Equations for Augmented Forced Convection (Single Phase) Description Equation I. Turbulent in-tube flow of liquids (a) Spiral repeated riba where w = 0.67 −0.06(p/d) − 0.49 (α/90) x = 0.37 −0.157(p/d) y = − 1.66 × 10− 6(GD/µ) − 0.33(α/90) z = 4.59 + 4.11 × 10− 6(GD/µ) −0.15(p/d) (b) Finsb Note that in computing the Reynolds number for (b) and (c) there is allowance for the reduced cross-sectional area.
(c) Twisted-strip insertsc where II. Turbulent in-tube flow of gases (a), (b) Bent-strip insertsd (c) Twisted-strip insertsd Note that in computing the Reynolds number there is no allowance for the flow blockage of the insert.
(d) Bent-tab insertsd III. Offset strip fins for plate-fin heat exchangerse where h/cpG, fh, and GDh/µ are based on the hydraulic diameter, given by Dh = 4shl/[2(sl + hl + th) + ts] References: bCarnavos (1979) dJunkhan et al. (1985) aRavigururajan and Bergles (1985) cManglik and Bergles (1993) eManglik and Bergles (1990) ha hs -----1 2.64 GD µ --------- 0.036 e d --- 0.212 p d -- 0.21 – α 90 ------ 0.29 cpµ k -------- 0.024 – + 7 = 1 7 ⁄ fa fs ----1 29.1 GD µ --------- w e d -- x p d -- y α 90 ------ z 1 2.94 n ----------+ sin β 15 16 ⁄ + 16 15 ⁄ = hs k D ⁄ ( ) fs 2 ⁄ ( ) GD µ --------- cpµ k -------- 1 12.7 fs 2 ⁄ ( )0.5 cpµ k -------- 2 3 ⁄ 1 – + ------------------------------------------------------------------------------------= fs 1.58 GD µ --------- ln 3.28 – 2 – = hDh k ----------0.023 cpµ k -------- 0.4 GDh µ ----------- 0.8 AF AFi -------- 0.1 Ai A ----- 0.5 sec α ( )3 = fh 0.046 GDh µ ----------- 0.2 – AF AFi -------- 0.5 sec α ( )0.75 = hd k ------ hd k ------ y ∞ = ⁄ 1 0.769 y ⁄ + [ ] = hd k ------ y ∞ = 0.023 GD µ --------- 0.8 cpµ k --------- 0.4 π π 4δ d ⁄ – ------------------------- 0.8 π 2 2δ – + d ⁄ π 4δ – d ⁄ ---------------------------------- 0.2 φ = φ µb µw ⁄ ( )n = n 0.18 for liquid heating 0.30 for liquid cooling = f 0.0791 GD µ ⁄ ( )0.25 --------------------------------π π 4δ d ⁄ – ------------------------- 1.75 π 2 2 – δ + d ⁄ π 4δ – d ⁄ -------------------------------- 1.25 1 2.752 y1.29 -------------+ = hD k ------- Tw Tb ------ 0.45 0.258 GD µ --------- 0.6 = hD k ------- Tw Tb ------ 0.45 0.208 GD µ --------- 0.63 = hD k ------- Tw Tb ------ 0.45 0.122 GD µ --------- 0.65 = hD k ------- Tw Tb ------ 0.45 0.406 GD µ --------- 0.54 = h cpG ---------0.6522 GDh µ ----------- 0.5403 – α 0.1541 – δ0.1499γ 0.0678 – 1 5.269 10 5 – GDh µ ----------- 1.340 × α0.504δ0.456γ 1.055 – + 0.1 = fh 9.6243 GDh µ ----------- 0.7422 – α 0.1856 – δ 0.3053 – γ 0.2659 – 1 7.669 10 8 – GDh µ ----------- 4.429 × α0.920δ3.767γ 0.236 + 0.1 = Heat Transfer 3.17 as shown in studies of scaling due to cooling tower water (Knudsen and Roy 1983) and particulate fouling (Somerscales et al. 1991). A comprehensive review of fouling with enhanced surfaces is presented by Somerscales and Bergles (1997).
Fire-tube boilers are frequently fitted with turbulators to improve the turbulent convective heat transfer coefficient con-stituting the dominant thermal resistance. Also, due to the high gas temperatures, radiation from the convectively heated insert to the tube wall can represent as much as 50% of the total heat transfer.
(Note, however, that the magnitude of the convective contribution decreases as the radiative contribution increases because of the reduced temperature difference.) Two commercial bent-strip inserts, a twisted-strip insert, and a simple bent-tab insert are depicted in Figure 13. Design equations, for convection only, are included in Table 7. Beckermann and Goldschmidt (1986) present procedures to include radiation, and Junkhan et al. (1985, 1988) give friction factor data and performance evaluations.
Several enhanced surfaces for gases are depicted in Figure 14.
The offset strip fin is an example of an interrupted fin that is often found in compact plate fin heat exchangers used for heat recovery from exhaust air. Design equations are included in Table 7. These equations are comprehensive in that they apply to laminar and tran-sitional flow as well as to turbulent flow, which is a necessary fea-ture because the small hydraulic diameter of these surfaces drives the Reynolds number down. Data for other surfaces (wavy, spine, louvered, etc.) are given in the section on Bibliography.
Plastic heat exchangers have been suggested for HVAC appli-cations (Pescod 1980) and are being manufactured for refriger-ated sea water (RSW) applications. They could be made of materials impervious to corrosion, say from acidic condensate when cooling a gaseous stream (flue gas heat recovery), and could easily be manufactured with enhanced surfaces. Several companies now offer heat exchangers in plastic, including vari-ous enhancements.
Fig. 13 Turbulators for Fire-Tube Boilers Fig. 14 Enhanced Surfaces for Gases 3.18 2001 ASHRAE Fundamentals Handbook (SI) Active Techniques Unlike the passive techniques, active techniques require use of external power to sustain the enhancement mechanism.
Table 8 provides a list of the more commonly known active heat transfer augmentation techniques and the corresponding heat trans-fer mode believed most applicable to the particular technique. A listing of the various active techniques and their world-wide status is given in Table 9. The rankings in Tables 8 and 9 are based on a comprehensive review of the pertinent literature (Ohadi et al. 1996).
Table 9 shows that among the eight listed active techniques, the electrostatic/electrohydrodynamic and rotation appear to apply to almost all important heat transfer modes, at least from an enhance-ment applicability view point. The information in Table 9 suggests that aside from the mechanical aids technique, which is universally used for selected applications, most other active techniques have limited commercial use so far and are still in the development stage.
However, the significant research progress in recent years is now promising expedited commercialization for some of the active tech-niques, such as the electrostatic/electrohydrodynamic (EHD) tech-nique. The following is a brief summary of the enhancement techniques listed in Table 9.
Mechanical Aids. Augmentation by mechanical aids involves stirring the fluid by mechanical means, such as mixers, stirrers, or surface scrapers. Stirrers and mixers that scrape the surface are extensively used in the chemical processing of highly viscous flu-ids, such as the flow of highly viscous plastic with air. Heat exchangers that employ mechanical aids for enhancement are often called mechanically assisted heat exchangers. The surface scrap-ing method is widely used for viscous liquids in the chemical pro-cessing industry and can be applied to duct flow of gases. Hagge and Junkhan (1974) reported tenfold improvement in heat transfer coefficient for laminar flow of air over a flat plate. Table 10 pro-vides a listing of selected works on mechanical aids, suction, and injection.
Injection. This method involves supplying a gas to a flowing liquid through a porous heat transfer surface or injecting a fluid of a similar type upstream of the heat transfer test section. Injected bubbles produce an agitation similar to that of nucleate boiling.
Gose et al. (1957) bubbled gas through sintered or drilled heated surfaces and found that the heat transfer coefficient increased 500% in laminar flow and about 50% in turbulent flow. Wayner and Bankoff (1965) demonstrated that the heat transfer coefficient could be increased by 150% if a porous block was placed on the surface to stabilize the flow of liquid toward the surface. Tauscher et al. (1970) demonstrated up to a five-fold increase in local heat transfer coefficients by injecting a similar fluid into a turbulent tube flow, but the effect dies out at a length-to-diameter ratio of 10.
Table 8 Active Heat Transfer Augmentation Techniques and the Most Relevant Heat Transfer Modes Technique Heat Transfer Mode Forced Convection (Gases) Forced Convection (Liquids) Boiling Evaporation Condensation Mass Transfer Mechanical aids NA B C C NA B Surface vibration B B B B B A Fluid vibration C B B B D B Electrostatic/Electrohydrodynamic B B A A A A Suction/Injection C B NA NA B B Jet impingement B B NA B NA C Rotation C C A A A A Induced flow B B NA NA NA C A = Most significant B = Significant C = Somewhat significant D = Not significant NA = Not believed to be applicable Table 9 World-Wide Status of Active Techniques Technique Country or Countries Mechanical aids Universally used in selected applications Surface vibration USA; not significant Fluid vibration Sweden; mostly used for sonic cleaning Electrostatic/Electrohydrodynamic Japan, USA and UK; successful prototypes in operation in Japan Other electrical methods UK, France, and USA Suction/Injection No significant activity Jet impingement France and USA; high temperature units and aerospace applications Rotation US industry and R&D-based in UK Induced flow USA leader in the technology, particularly in combustion Table 10 Selected Studies on Mechanical Aids, Suction, and Injection Source Process Heat Transfer Surface Fluid Maximum α Valencia et al. (1996) Natural convection Fin type tube Air 0.5 Jeng et al. (1995) Natural convection/Suction Asymmetric isothermal wall Air 1.4 Inagaki and Komori (1993) Turbulent natural convection/Suction Vertical plate Air 1.8 Dhir et al. (1992) Forced convection/Injection Tube Air 1.45 Duignan et al. (1993) Forced convection/Film boiling Horizontal plate Air 2.0 Son and Dhir (1993) Forced convection/Injection Annuli Air 1.85 Malhotra and Mujumdar (1991) Water to bed/Stirring Granular bed Air 3.0 Aksan and Borak (1987) Pool of water/Stirring Tube coils Water 1.7 Hagge and Junkhan (1975) Forced convection/Scraping Cylindrical wall Air 11.0 Hu and Shen (1996) Turbulent natural convection Converging ribbed tube Air 1.0 Heat Transfer 3.19 The practical application of injection appears to be rather limited because of difficulty of cost-effective supplying and removing of the injection fluid.
Suction. The suction method involves fluid removal through a porous heated surface leading to reduced heat/mass transfer resis-tance at the surface. Kinney (1968) and Kinney and Sparrow (1970) reported that applying suction at the surface increased heat transfer coefficients for laminar film and turbulent flows, respectively. For laminar film condensation, Antonir and Tamir (1977) and Lienhard and Dhir (1972) indicated that heat transfer coefficient can be improved by several hundred percent when film thickening is reduced by suction. Jeng et al. (1995) conducted experiments on a vertical parallel channel with asymmetric, isothermal walls. A porous wall segment was embedded into a segment of the test sec-tion wall, and enhancement occurred as the hot air was sucked from the channel. The local heat transfer coefficient increased with increasing porosity. The maximum heat transfer enhancement obtained was 140%.
Fluid or Surface Vibration. Fluid or surface vibrations are natural processes that occur in most heat exchangers; however, naturally occurring vibration is rarely factored into thermal design.
Vibration equipment is expensive, and use of this technique for heat transfer enhancement does not have industrial applications at this stage of development. Vibrations of a wire in a forced convect-ing airflow enhanced the heat transfer to air up to 300% (Nesis et al. 1994), depending on the amplitude of vibrations and fre-quency. By using standing waves in a fluid, the input power was reduced by 75% compared with a fan that provides the same heat transfer rate (Woods 1992). Lower frequencies are preferable because they are less harmful to those who use this method of aug-menting heat transfer.
Rotation. Rotation is a type of heat transfer enhancement that occurs naturally in rotating electrical machinery, gas turbine blades, and some other equipment. The rotating evaporator, the rotating heat pipe, the Higee distillation column, and the Rotex absorption cycle heat pump are typical examples of previous work in this area.
In rotating evaporators, the rotation effectively distributes the liq-uid on the outside surface of the rotating surface. Rotation of the heat transfer surface also seems to be a promising method for effec-tively removing the condensate and decreasing liquid film thick-ness. Substantial increases in heat transfer coefficients have been demonstrated by using centrifugal force, which may be several times greater than the gravity force.
As shown in Table 11, the heat transfer enhancement obtained in the various studies varies from slight improvement up to 450%, depending on the system and rotation speed. The rotation technique is of particular interest for use in two-phase flows, particularly in boiling and condensation. It has been demonstrated that this tech-nique is not effective in the gas-to-gas heat recovery mode in lami-nar flow, but its application is more likely in turbulent flow. High power consumption, sealing and vibration problems, moving parts, and the expensive equipment required for rotation are some draw-backs of the rotation technique.
Electrohydrodynamics. The electrohydrodynamic (EHD) en-hancement of heat transfer refers to the coupling of an electric field with the fluid field in a dielectric fluid medium. The net ef-fect is the production of secondary motions that destabilize the thermal boundary layer near the heat transfer surface, leading to heat transfer coefficients that are often an order of magnitude higher than those achievable by any of the conventional enhance-ment techniques. Among the various active augmentation tech-niques, EHD has benefited the most from substantial research in the past two decades in Japan, the United States, and the United Kingdom (Ohadi 1991). Its applicability for heat transfer en-hancement in many applications has already been demonstrated, including for refrigeration/HVAC systems, process heat exchang-ers, waste heat recovery devices, cryogenics, aircraft environmen-tal control systems, avionic cooling systems, and space thermal systems.
Selected work on EHD heat transfer enhancement is listed in Table 12. The work has involved studies on phase change processes as well as the indicated single-phase process. In fact, EHD-enhanced boiling and condensation have attracted the most atten-tion from industrial and academic researchers.
The EHD effect is generally applied by placing wire or plate electrodes parallel and adjacent to the heat transfer surface. Figure 15 presents four electrode configurations for augmentation of forced-convection heat/mass transfer in-tube flows. A high-voltage, low-current electric field charges the electrode and establishes the electrical body force required to initiate and sustain augmentation.
When compared with other active techniques, a number of important advantages have contributed to the fast progress of the Table 11 Selected Studies on Rotation Source Process Heat Transfer Surface Fluids Rotation Speed, rpm Max. α Prakash and Zerle (1995) Natural convection Ribbed duct Air Given as a function 1.3 Mochizuki et al. (1994) Natural convection Serpentine duct Air Given as a function 3.0 Lan (1991) Solidification Vertical tube Water 400 NA McElhiney and Preckshot (1977) External condensation Horizontal tube Steam 40 1.7 Nichol and Gacesa (1970) External condensation Vertical cylinder Steam 2700 4.5 Astaf’ev and Baklastov (1970) External condensation Circular disc Steam 2500 3.4 Tang and McDonald (1971) Nucleate boiling Horizontal heated circular cylinder R-113 1400 <1.2 Marto and Gray (1971) In-tube boiling Vertical heated circular cylinder Water 2660 1.6 NA = Not Available Table 12 Selected Studies on EHD Technique Source Process Heat Transfer Surface/Electrode Fluid P/Q, % Max. α Poulter and Allen (1986) Internal flow Tube/Wire Aviation fuel-hexane NA 20 Fernandez and Poulter(1987) Internal flow Tube/Wire Transformer oil NA 23 Ohadi et al. (1996) Internal flow Smooth surface/Rod PAO (oil-based fluid) 1.2 3.2 Ohadi et al. (1991) Internal flow Tube/Wire Air 15 3.2 Owsenek and Seyed-Yagoobi(1995) Forced convection Horizontal flat plate Air NA 25 Ohadi et al. (1995) In-tube boiling Microfin tube/Helical R-134a 0.1 6.5 Ohadi et al. (1995) In-tube condensation Smooth tube/Rod wire R-134a 0.1 7.0 Wawzyniak and Seyed-Yagoobi (1996) External condensation Enhanced and smooth tubes/ Circular rods R-113 0.08 6.2 Seyed-Yagoobi et al. (1996) Pool boiling Horizontal tube/ Straight and circular wires R-123 0.1 12.6 NA = Not Available; P = EHD Power; Q = Heat Exchanger Capacity 3.20 2001 ASHRAE Fundamentals Handbook (SI) EHD technique in recent years. Rotation, injection, and vibration are generally mechanically complex and somewhat cumbersome to manufacture. Furthermore, the energy required to operate these sys-tems can be a significant fraction of the power employed in pump-ing the fluid. In these respects, the EHD method of enhancement is superior.
The safety aspect of the EHD technique may be misjudged if careful attention is not paid to the manner in which these systems operate. Although EHD systems work at high voltages, the fact that very small currents are employed reduces the hazard of EHD fields to well below that of conventional low-voltage household appli-ances and ensures that the electrical power consumed by the EHD process is extremely small (less than 1% in most cases). Although the EHD technique appears to be well ahead of other active tech-niques, a number of issues remain to be addressed before success-ful implementation of this technique in practical heat exchangers can be realized. The most important issues include (1) any long-term effects the electric field may have on the heat exchanger, working fluid, and components, (2) the development of low-cost, high-voltage power supplies, and (3) identifying manufacturing processes that can lead to inexpensive mass production of EHD-enhanced heat exchangers. The encouraging news is that research work addressing some of the issues has already been initiated by researchers in the academic and government sectors and in private companies and research institutions. Additional details on the prin-ciples, applicability, and limitations of the EHD technique can be found elsewhere (Ohadi et al. 1991; Ohadi et al. 1999; Yabe 1991).
EXTENDED SURFACE Heat transfer from a prime surface can be increased by attaching fins or extended surfaces to increase the area available for heat transfer. Fins provide a more compact heat exchanger with lower material costs for a given performance. To achieve optimum design, fins are generally located on the side of the heat exchanger where the heat transfer coefficients are low (such as the air side of an air-to-water coil). Equipment with an extended surface includes natural-and forced-convection coils and shell-and-tube evaporators and condensers. Fins are also used inside tubes in condensers and dry expansion evaporators.
Fin Efficiency As heat flows from the root of a fin to its tip, temperature drops because of the thermal resistance of the fin material. The tempera-ture difference between the fin and the surrounding fluid is therefore greater at the root than at the tip, causing a corresponding variation in the heat flux. Therefore, increases in fin length result in propor-tionately less additional heat transfer. To account for this effect, fin efficiency φ is defined as the ratio of the actual heat transferred from the fin to the heat that would be transferred if the entire fin were at its root or base temperature: (50) where φis the fin efficiency, te is the temperature of the surrounding environment, and tr is the temperature at the fin root. Fin efficiency is low for long fins, thin fins, or fins made of low thermal conduc-tivity material. Fin efficiency decreases as the heat transfer coeffi-cient increases because of the increased heat flow. For natural convection in air-cooled condensers and evaporators, where h for the air side is low, fins can be fairly large and fabricated from low-con-ductivity materials such as steel instead of from copper or aluminum.
For condensing and boiling, where large heat transfer coefficients are involved, fins must be very short for optimum use of material.
The heat transfer from a finned surface, such as a tube, which includes both finned or secondary area As and unfinned or prime area Ap is given by the following equation: (51) Assuming the heat transfer coefficients for the finned surface and prime surface are equal, a surface efficiency φs can be derived for use in Equation (52).
(52) (53) where A is the total surface area, equal to the sum of the finned and prime areas (A = As + Ap).
Temperature distribution and fin efficiencies for various fin shapes are derived in most heat transfer texts. Figures 16 through 19 show curves and equations for annular fins, straight fins, and spines. For constant thickness square fins, the efficiency of a constant thickness annular fin of the same area can be used. More accuracy, particularly with rectangular fins of large aspect ratio, can be obtained by dividing the fin into circular sectors (Rich 1966).
Rich (1966) presents results for a wide range of geometries in a compact form for equipment designers by defining a dimensionless thermal resistance Φ: (54) (55) where Φ = dimensionless thermal resistance φ = fin efficiency to = fin thickness at fin base l = length dimension = rt −ro for annular fins W = for rectangular fins Rich (1966) also developed expressions for Φ max, the maximum limiting value of Φ . Figure 20 gives Φ max for annular fins of con-stant and tapered cross section as a function of R = rt/ro (i.e., the Fig. 15 Electrode Configurations for Internal Forced-Convection Flow φ h t te – ( ) A d ∫ h tr te – ( ) A d ∫ ------------------------------------= q hpAp φhsAs + ( ) tr te – ( ) = φs 1 As A ----- 1 φ – ( ) – = q φshA tr te – ( ) = Φ Rf tok l 2 -------------= Rf 1 h -- 1 φ --1 – = Heat Transfer 3.21 ratio of the fin tip-to-root radii). Figure 21 gives Φ max for rectangu-lar fins of a given geometry as determined by the sector method.
Figure 22 gives correction factors (Φ/Φ max) for the determination of Φfrom Φ max for both annular and rectangular fins.
Example. This example illustrates the use of the fin resistance for a rect-angular fin typical of that for an air-conditioning coil.
Given: L = 18 mm to = 0.15 mm W = 12 mm h = 60 W/(m2·K) ro = 6 mm k = 170 W/(m·K) Solution: From Figure 18 at W/ro = 2.0 and L/W = 1.5, The correction factor Φ /Φ max, which is multiplied by Rf (max) to give Rf, is given in Figure 22 as a function of the fin efficiency. As a first approximation, the fin efficiency is calculated from Equation (54a) assuming Rf = Rf (max).
Fig. 16 Efficiency of Annular Fins of Constant Thickness Fig. 17 Efficiency of Annular Fins with Constant Metal Area for Heat Flow Fig. 18 Efficiency of Several Types of Straight Fin Fig. 19 Efficiency of Four Types of Spine Φmax Rf max ( )tok W2 ⁄ 1.12 = = Rf max ( ) 1.12 122 × 0.15 170 × 1000 × ----------------------------------------------0.00632 m2 K ⋅ W ⁄ = = φ 1 1 hRf + ( ) ⁄ 0.72 ≈ = 3.22 2001 ASHRAE Fundamentals Handbook (SI) Fig. 20 Maximum Fin Resistance of Annular Fins (Gardner 1945) Fig. 21 Maximum Fin Resistance of Rectangular Fins Determined by Sector Method Fig. 22 Variation of Fin Resistance with Efficiency for Annular and Rectangular Fins (Gardner 1945) Heat Transfer 3.23 Interpolating between L/W = 1 and L/ W = 2 at W/ro = 2 gives Therefore, The above steps may now be repeated using the corrected value of fin resistance.
φ = 0.745 Φ /Φ max = 0.9 Rf = 0.00569 m2·K/W Note that the improvement in accuracy by reevaluating Φ /Φ max is less than 1% of the overall thermal resistance (environment to fin base).
The error produced by using Rf(max) without correction is less than 3%.
For many practical cases where greater accuracy is not warranted, a single value of Rf, obtained by estimating Φ/Φ max, can be used over a range of heat transfer coefficients for a given fin. For approximate cal-culations, the fin resistance for other values of k and to can be obtained by simple proportion if the range covered is not excessive.
Schmidt (1949) presents approximate, but reasonably accurate, analytical expressions (for computer use) for circular, rectangular, and hexagonal fins. Hexagonal fins are the representative fin shape for the common staggered tube arrangement in finned-tube heat exchangers.
Schmidt’s empirical solution is given by where and Φis given by For circular fins, For rectangular fins, where M and L are defined by Figure 23 as a/2 or b/2, depending on which is greater.
For hexagonal fins, where ψand β are defined as above and M and L are defined by Figure 24 as a/2 or b (whichever is less) and respectively.
The section on Bibliography lists other sources of information on finned surfaces.
Thermal Contact Resistance Fins can be extruded from the prime surface (e.g., the short fins on the tubes in flooded evaporators or water-cooled condensers) or they can be fabricated separately, sometimes of a different material, and bonded to the prime surface. Metallurgical bonds are achieved by furnace-brazing, dip-brazing, or soldering. Nonmetallic bonding materials, such as epoxy resin, are also used. Mechanical bonds are obtained by tension-winding fins around tubes (spiral fins) or expanding the tubes into the fins (plate fins). Metallurgical bonding, properly done, leaves negligible thermal resistance at the joint but is not always economical. Thermal resistance of a mechanical bond may or may not be negligible, depending on the application, quality of manufacture, materials, and temperatures involved. Tests of plate fin coils with expanded tubes have indicated that substantial losses in performance can occur with fins that have cracked collars; but negligible thermal resistance was found in coils with continuous collars and properly expanded tubes (Dart 1959).
Thermal resistance at an interface between two solid materials is largely a function of the surface properties and characteristics of the solids, the contact pressure, and the fluid in the interface, if any.
Eckels (1977) models the influence of fin density, fin thickness, and tube diameter on contact pressure and compared it to data for wet and dry coils. Shlykov (1964) shows that the range of attain-able contact resistances is large. Sonokama (1964) presents data on the effects of contact pressure, surface roughness, hardness, void material, and the pressure of the gas in the voids. Lewis and Sauer (1965) show the resistance of adhesive bonds, and Kaspareck (1964) and Clausing (1964) give data on the contact resistance in a vacuum environment.
Finned-Tube Heat Transfer The heat transfer coefficients for finned coils follow the basic equations of convection, condensation, and evaporation. The ar-rangement of the fins affects the values of constants and the expo-nential powers in the equations. It is generally necessary to refer to test data for the exact coefficients.
Φ Φmax ⁄ 0.88 = Rf 0.88 0.00632 × 0.00556 m2 · K W ⁄ = = φ tanh mriΦ ( ) mriΦ ----------------------------= m 2h kt ⁄ = Φ re ri ⁄ ( ) 1 – [ ] 1 0.35 re ri ⁄ ( ) ln + [ ] = e ri ⁄ ro r ⁄ = re ri ⁄ 1.28ψ β 0.2 – , = ψ M ri ⁄ , = β L M 1 ≥ ⁄ = re ri ⁄ 1.27ψ β 0.3 – = 0.5 a2 2 ⁄ ( )2 b2, + Fig. 23 Rectangular Tube Array Fig. 24 Hexagonal Tube Array 3.24 2001 ASHRAE Fundamentals Handbook (SI) For natural-convection finned coils (gravity coils), approximate coefficients can be obtained by considering the coil to be made of tubular and vertical fin surfaces at different temperatures and then applying the natural-convection equations to each. This calculation is difficult because the natural-convection coefficient depends on the temperature difference, which varies at different points on the fin.
Fin efficiency should be high (80 to 90%) for optimum natural-convection heat transfer. A low fin efficiency reduces the tempera-ture near the tip. This reduces ∆t near the tip and also the coefficient h, which in natural convection depends on ∆t. The coefficient of heat transfer also decreases as the fin spacing decreases because of inter-fering convection currents from adjacent fins and reduced free-flow passage; 50 to 100 mm spacing is common. Generally, high coeffi-cients result from large temperature differences and small flow restriction.
Edwards and Chaddock (1963) give coefficients for several cir-cular fin-on-tube arrangements, using fin spacing δ as the charac-teristic length and in the form Nu = f (GrPrδ/Do), where Do is the fin diameter.
Forced-convection finned coils are used extensively in a wide variety of equipment. The fin efficiency for optimum performance is smaller than that for gravity coils because the forced-convection coefficient is almost independent of the temperature difference between the surface and the fluid. Very low fin efficiencies should be avoided because an inefficient surface gives a high (uneconom-ical) pressure drop. An efficiency of 70 to 90% is often used.
As fin spacing is decreased to obtain a large surface area for heat transfer, the coefficient generally increases because of higher air velocity between fins at the same face velocity and reduced equiv-alent diameter. The limit is reached when the boundary layer formed on one fin surface (Figure 8) begins to interfere with the boundary layer formed on the adjacent fin surface, resulting in a decrease of the heat transfer coefficient, which may offset the advantage of larger surface area.
Selection of the fin spacing for forced-convection finned coils usu-ally depends on economic and practical considerations, such as foul-ing, frost formation, condensate drainage, cost, weight, and volume.
Fins for conventional coils generally are spaced 1.8 to 4.2 mm apart, exceptwherefactorssuchasfrostformationnecessitatewiderspacing.
Several means are used to obtain higher coefficients with a given air velocity and surface, usually by creating air turbulence, generally with a higher pressure drop: (1) staggered tubes instead of in-line tubes for multiple-row coils; (2) artificial additional tubes, or collars or fingers made by suitably forming the fin materials; (3) corrugated fins instead of plane fins; and (4) louvered or interrupted fins.
Figure 25 shows data for one-row coils. The thermal resistances plotted include the temperature drop through the fins, based on one square metre of total external surface area.
The section on Bibliography lists other sources of information on fins.
SYMBOLS A = surface area for heat transfer AF = cross-sectional flow area C = conductance; or fluid capacity rate C1, C2 = Planck’s law constants [see Equation (33)] c = coefficient or constant cp = specific heat at constant pressure cv = specific heat at constant volume D = tube (inside) or rod diameter; or diameter of the vessel d = diameter; or prefix meaning differential E = electric field e = emissivity; or protuberance height F = angle factor [see Equations (40) and (41)] Fo = Fourier number (see Table 1 and Figures 2, 3, and 4) f = Fanning friction factor for single-phase flow; or electric body force G = mass velocity; or irradiation Gr = Grashof number g = gravitational acceleration h = heat transfer coefficient; or offset strip fin height I = modified Bessel function ID = inside diameter J = mechanical equivalent of heat; or radiosity j = heat transfer factor [see Equation (4), Table 6] k = thermal conductivity L = length; or height of liquid film l = length; or length of one module of offset strip fins M = mass; or molecular mass m = general exponent = mass rate of flow n = general number [see Equation (2) in Table 5 or Table 7 (c)]; or ratio r/rm (see Figures 2, 3, and 4); or number of blades NTU = number of exchanger heat transfer units [see Equation (17)] Nu = Nusselt number p = pressure; or fin pitch; or repeated rib pitch Pr = Prandtl number q = rate of heat transfer q″ = heat flux R = thermal resistance Re = pipe Reynolds number (GD/µ); or film Reynolds number (4Γ/η ) Re = rotary Reynolds number (D2Np/η ) r = radius s = lateral spacing of offset fin strips T = absolute temperature t = temperature; or fin thickness at base U = overall heat transfer coefficient V = linear velocity W = work; or total rate of energy emission; or fin dimension Fig. 25 Overall Air-Side Thermal Resistance and Pressure Drop for 1-Row Coils (Shepherd 1946) m · Heat Transfer 3.25 Wλ = monochromatic emissive power x, y, z = lengths along principal coordinate axes Y = temperature ratio (see Figures 2, 3, and 4) y = one-half diametrical pitch of a twisted tape: length of 180° revolution/tube diameter Z = ratio of fluid capacity rates [see Equation (18)] α = thermal diffusivity = k/ρcp [Equation (28)]; or absorptance; or spiral angle for helical fins; or aspect ratio of offset strip fins, s/h; or enhancement factor: ratio of enhanced to unenhanced heat transfer coefficient—conditions remaining the same.
β = coefficient of thermal expansion; or contact angle of rib profile Γ = mass flow of liquid per unit length γ = ratio, t/s ∆= difference between values δ = distance between fins; or ratio t/l; or thickness of twisted tape ε = hemispherical emittance; or exchanger heat transfer effectiveness [see Equation (16)]; or dielectric constant λ = wavelength µ = absolute viscosity v = kinematic viscosity ρ = density; or reflectance σ = Stefan-Boltzmann constant τ = time; or transmittance [see Equation (39)] Φ = fin resistance defined by Equation (54); Φ max is maximum limiting value of Φ φ = fin efficiency [see Equation (50)]; or angle [see Equation (41)]; or temperature correction factor [see Table 7(c)] Subscripts a = augmented b = blackbody; or based on bulk fluid temperature c = convection; or critical; or cold (fluid) e = equivalent; or environment f = film; or fin fc = finite cylinder frs = finite rectangular solid g = gas h = horizontal; or hot (fluid); or hydraulic i = inlet; or inside; or particular surface (radiation); or based on maximum inside (envelope) diameter ic = infinite cylinder if = interface is = infinite slab iso = isothermal conditions j = particular surface (radiation) k = particular surface (radiation) L = thickness l = liquid m = mean max = maximum min = minimum n = counter variable o = outside; or outlet; or overall; or at base of fin p = prime heat transfer surface r = radiation; or root (fin); or reduced s = surface; or secondary heat transfer surface; or straight or plain; or accounting for flow blockage of twisted tape st = static (pressure) t = temperature; or terminal temperature; or tip (fin) v = vapor; or vertical w = wall; or wafer λ = monochromatic ∞= bulk REFERENCES Adams, J.A. and D.F. Rogers. 1973. Computer aided heat transfer analysis.
McGraw-Hill, New York.
Afgan, N.H. and E.U. Schlunder. 1974. Heat exchangers: Design and theory sourcebook. McGraw-Hill, New York.
Aksan, D. and F. Borak. 1987. Heat transfer coefficients in coiled stirred tank systems. Can. J. Chem. Eng. 65:1013-14.
Al-Fahed, S.F., Z.H. Ayub, A.M. Al-Marafie, and B.M. Soliman. 1993. Heat transfer and pressure drop in a tube with internal microfins under turbu-lent water flow conditions. Exp. Thermal and Fluid Science 7:249-53.
Altmayer, E.F., A.J. Gadgil, F.S. Bauman, and R.C. Kammerud. 1983. Cor-relations for convective heat transfer from room surfaces. ASHRAE Transactions 89(2A):61-77.
Antonir, I. and A. Tamir. 1977. The effect of surface suction on condensation in the presence of a noncondensible gas. International Journal of Heat and Mass Transfer 99:496-99.
Astaf’ev, B.F. and A.M. Baklastov. 1970. Condensation of steam on a hori-zontal rotating disc. Teploenergetika 17:55-57.
Ayub, Z.H. and S.F. Al-Fahed. 1993. The effect of gap width between hori-zontal tube and twisted tape on the pressure drop in turbulent water flow, International Journal of Heat and Fluid Flow 14(1):64-67.
Bauman, F., A. Gadgil, R. Kammerud, E. Altmayer, and M. Nansteel. 1983.
Convective heat transfer in buildings. ASHRAE Transactions 89(1A): 215-33.
Beckermann, C. and V. Goldschmidt. 1986. Heat transfer augmentation in the flueway of a water heater. ASHRAE Transactions 92(2B):485-95.
Bergles, A.E. 1998. Techniques to enhance heat transfer. In Handbook of heat transfer, 3rd ed., 11.1-11.76. McGraw-Hill, New York.
Brown, A.I. and S.M. Marco. 1958. Introduction to heat transfer, 3rd ed.
McGraw-Hill, New York.
Burmeister, L.C. 1983. Convective heat transfer. John Wiley and Sons, New York.
Carnavos, T.C. 1979. Heat transfer performance of internally finned tubes in turbulent flow. In Advances in enhanced heat transfer, pp. 61-67. Amer-ican Society of Mechanical Engineers, New York.
Carslaw, H.S. and J.C. Jaeger. 1959. Conduction of heat in solids. Oxford University Press, England.
Clausing, A.M. 1964. Thermal contact resistance in a vacuum environment.
ASME Paper 64-HT-16, Seventh National Heat Transfer Conference.
Croft, D.R. and D.G. Lilley. 1977. Heat transfer calculations using finite difference equations. Applied Science Publishers, London.
Dart, D.M. 1959. Effect of fin bond on heat transfer. ASHRAE Journal 5:67.
Dhir, V.K., F. Chang, and G. Son. 1992. Enhancement of single-phase forced convection heat transfer in tubes and ducts using tangential flow injec-tion. Annual Report. Contract No. 5087-260135. Gas Research Institute.
Duignan, M., G. Greene, and T. Irvine. 1993. The effect of surface gas injec-tion on film boiling heat transfer. Journal of Heat Transfer 115:986-92.
Eckels, P.W. 1977. Contact conductance of mechanically expanded plate finned tube heat exchangers. AIChE-ASME Heat Transfer Conference, Salt Lake City, UT.
Edwards, J.A. and J.B. Chaddock. 1963. An experimental investigation of the radiation and free-convection heat transfer from a cylindrical disk extended surface. ASHRAE Transactions 69:313.
Fernandez, J. and R. Poulter. 1987. Radial mass flow in electrohydrodynam-ically-enhanced forced heat transfer in tubes. International Journal of Heat and Mass Transfer 80:2125-36.
Gardner, K.A. 1945. Efficiency of extended surface. ASME Transactions 67:621.
Gose, E.E., E.E. Peterson, and A. Acrivos. 1957. On the rate of heat transfer in liquids with gas injection through the boundary layer. Journal of Applied Physics 28:1509.
Grigull, U. et al. 1982. Heat transfer. Proceedings of the Seventh Interna-tional Heat Transfer Conference, Munich 3. Hemisphere Publishing, New York.
Hagge, J.K. and G.H. Junkhan. 1974. Experimental study of a method of mechanical augmentation of convective heat transfer in air. HTL3, ISU-ERI-Ames-74158, Nov. 1975. Iowa State University, Ames, Iowa.
Hottel, H.C. and A.F. Sarofim. 1967. Radiation transfer. McGraw-Hill, New York.
Hu, Z. and J. Shen. 1996. Heat transfer enhancement in a converging pas-sage with discrete ribs. Int. J. Heat Mass Trans. 39:1719.
Inagaki, T. and I. Komori. 1993. Experimental study of heat transfer enhancement in tubulent natural convection along a vertical flat plate, Part 1: The effect of injection and suction. Heat Transactions—Japanese Research 22:387.
Incropera, F.P. and D.P. DeWitt. 1996. Fundamentals of heat transfer. John Wiley and Sons, New York.
Jakob, M. 1949, 1957. Heat transfer, Vols. I and II. John Wiley and Sons, New York.
Jeng, Y., J. Chen, and W. Aung. 1995. Heat transfer enhancement in a ver-tical channel with asymmetric isothermal walls by local blowing or suc-tion. International Journal of Heat and Fluid Flow 16:25.
Junkhan, G.H. et al. 1985. Investigation of turbulators for fire tube boilers.
Journal of Heat Transfer 107:354-60.
3.26 2001 ASHRAE Fundamentals Handbook (SI) Junkhan, G.H., A.E. Bergles, V. Nirmalan, and W. Hanno. 1988. Perfor-mance evaluation of the effects of a group of turbulator inserts on heat transfer from gases in tubes. ASHRAE Transactions 94(2):1195-1212.
Kaspareck, W.E. 1964. Measurement of thermal contact conductance be-tween dissimilar metals in a vacuum. ASME Paper 64-HT-38, Seventh National Heat Transfer Conference.
Kays, W.M. and A.L. London. 1984. Compact heat exchangers, 3rd ed.
McGraw-Hill, New York.
Kays, W.M. and M. Crawford. 1980. Convective heat and mass transfer, 2nd ed. McGraw-Hill, New York.
Khanpara, J.C., A.E. Bergles, and M.B. Pate. 1986. Augmentation of R-113 in-tube condensation with micro-fin tubes. HTD 65:21-32.
Kinney, R.B. 1968. Fully developed frictional and heat transfer characteris-tics of laminar flow in porous tubes. International Journal of Heat and Mass Transfer 11:1393-1401.
Kinney, R.B. and E.M. Sparrow. 1970. Turbulent flow: Heat transfer and mass transfer in a tube with surface suction. Journal of Heat Transfer 92:117-25.
Knudsen, J.G. and B.V. Roy. 1983. Studies on scaling of cooling tower water.
Fouling of heat enhancement surfaces, pp. 517-30. Engineering Founda-tion, New York.
Lan, C.W. 1991. Effects of rotation on heat transfer fluid flow and interface in normal gravity floating zone crystal growth. Journal of Crystal Growth 114:517.
Lewis, D.M. and H.J. Sauer, Jr. 1965. The thermal resistance of adhesive bonds. ASME Journal of Heat Transfer 5:310.
Lienhard, J. and V.A. Dhir. 1972. Simple analysis of laminar film conden-sation with suction. Journal of Heat Transfer 94:334-36.
Malhotra, K. and A.S. Mujumdar. 1991. Wall to bed contact heat transfer rates in mechanically stirred granular beds. International Journal of Heat and Mass Transfer 34:724-35.
Manglik, R.M. and A.E. Bergles. 1990. The thermal-hydraulic design of the rectangular offset-strip-fin-compact heat exchanger. In Compact heat exchangers, pp. 123-49. Hemisphere Publishing, New York.
Manglik, R.M. and A.E. Bergles. 1993. Heat transfer and pressure drop cor-relation for twisted-tape insert in isothermal tubes: Part II —Transition and turbulent flows. Journal of Heat Transfer 115:890-96.
Marto, P.J. and V.H. Gray. 1971. Effects of high accelerations and heat fluxes on nucleate boiling of water in an axisymmetric rotating boiler. NASA Technical Note TN. D-6307. Washington, DC.
McAdams, W.H. 1954. Heat transmission, 3rd ed. McGraw-Hill, New York.
McElhiney, J.E. and G.W. Preckshot. 1977. Heat transfer in the entrance length of a horizontal rotating tube. International Journal of Heat and Mass Transfer 20:847-54.
Metais, B. and E.R.G. Eckert. 1964. Forced, mixed and free convection regimes. ASME Journal of Heat Transfer 86(C2)(5):295.
Mochizuki, S., J. Takamura, and S. Yamawaki. 1994. Heat transfer in ser-pentine flow passages with rotation. J. Turbomachinery 116:133.
Modest, M.F. 1993. Radiation heat transfer. McGraw-Hill, New York.
Myers, G.E. 1971. Analytical methods in conduction heat transfer. McGraw-Hill, New York.
Nelson, R.M. and A.E. Bergles. 1986. Performance evaluation for tubeside heat transfer enhancement of a flooded evaporative water chiller.
ASHRAE Transactions 92(1B):739-55.
Nesis, E.I., A.F. Shatalov, and N.P. Karmatskii. 1994. Dependence of the heat transfer coefficient on the vibration amplitude and frequency of a vertical thin heater. J. Eng. Physics and Thermophysics 67(No. 1-2).
Nichol, A.A. and M. Gacesa. 1970. Condensation of steam on a rotating ver-tical cyclinder. Journal of Heat Transfer 144-52.
Ohadi, M.M. 1991. Heat transfer enhancement in heat exchangers. ASHRAE Journal (December):42-50.
Ohadi, M.M., S. Dessiatoun, A. Singh, K. Cheung, and M. Salehi. 1995a.
EHD-Enhancement of boiling/condensation heat transfer of alternate refrigerants, Progress Report No. 6. Presented to the U.S. Department of Energy and the EHD Consortium Members, under DOE Grant No. DE-FG02-93CE23803.A000, Chicago, January 1995.
Ohadi, M.M., S.V. Dessiatoun, J. Darabi, and M. Salehi. 1996. Active aug-mentation of single-phase and phase-change heat transfer—An over-view. In Process enhanced and multiphase heat transfer: A festschrift for A.E. Bergles, R.M. Manglik and A.D. Kraus, eds. Begell House Publish-ers, pp. 277-86.
Ohadi, M.M., J. Darabi, and B. Roget. 2000. Electrode design, materials, and fabrication fundamentals. In Advances in Heat Transfer,C.L. Tien, ed. Begell House Publishers.
Ohadi, M.M. et al. 1991. Electrohydrodynamic enhancement of heat transfer in a shell-and-tube heat exchanger. Enhanced Heat Transfer 4(1):19-39.
Ohadi, M.M., R. Papar, T.L. Ng, M. Faani, and R. Radermacher. 1992. EHD enhancement of shell-side boiling heat transfer coefficients of R-123/oil mixture. ASHRAE Transactions 98(2):427-34.
Owsenek, B. and J. Seyed-Yagoobi. 1995. Experimental investigation of corona wind heat transfer enhancement with a heated horizontal flat plate. Journal of Heat Transfer 117:309.
Parker, J.D., J.H. Boggs, and E.F. Blick. 1969. Introduction to fluid mechan-ics and heat transfer. Addison Wesley Publishing, Reading, MA.
Patankar, S.V. 1980. Numerical heat transfer and fluid flow. McGraw-Hill, New York.
Pescod, D. 1980. An advance in plate heat exchanger geometry giving increased heat transfer. HTD 10:73-77, American Society of Mechanical Engineers, New York.
Poulter, R. and P.H.G. Allen. 1986. Electrohydrodynamincally augmented heat and mass transfer in the shell tube heat exchanger. Proceedings of the Eighth International Heat Transfer Conference 6:2963-68.
Prakash, C. and R. Zerle. 1995. Prediction of turbulent flow and heat transfer in a ribbed rectangular duct with and without rotation. J. Turbomachin-ery 117:255.
Ravigururajan, T.S. and A.E. Bergles. 1985. General correlations for pres-sure drop and heat transfer for single-phase turbulent flow in internally ribbed tubes. Augmentation of heat transfer in energy systems, 52:9-20.
American Society of Mechanical Engineers, New York.
Rich, D.G. 1966. The efficiency and thermal resistance of annular and rect-angular fins. Proceedings of the Third International Heat Transfer Con-ference, AIChE 111:281-89.
Schmidt, T.E. 1949. Heat transfer calculations for extended surfaces. Refrig-erating Engineering 4:351-57.
Schneider, P.J. 1964. Temperature response charts. John Wiley and Sons, New York.
Seyed-Yagoobi, J., C.A. Geppert, and L.M. Geppert. 1996. Electrohydrody-namically enhanced heat transfer in pool boiling. ASME Journal of Heat Transfer 118:233-37.
Shepherd, D.G. 1946. Performance of one-row tube coils with thin plate fins, low velocity forced convection. Heating, Piping, and Air Conditioning (April).
Shlykov, Y.P. 1964. Thermal resistance of metallic contacts. International Journal of Heat and Mass Transfer 7(8):921.
Siegel, R. and J.R. Howell. 1981. Thermal radiation heat transfer. McGraw-Hill, New York.
Somerscales, E.F.C. et al. 1991. Particulate fouling of heat transfer tubes enhanced on their inner surface. Fouling and enhancement interactions.
HTD 164:17-28. American Society of Mechanical Engineers, New York.
Somerscales, E.F.C. and A.E. Bergles. 1997. Enhancement of heat transfer and fouling mitigation. Advances in Heat Transfer 30:197-253. Aca-demic Press, San Diego.
Son, G. and V.K. Dhir. 1993. Enhancement of heat transfer in annulus using tangential flow injection. HTD-Vol. 246, Heat Transfer in Turbulent Flows, ASME.
Sonokama, K. 1964. Contact thermal resistance. Journal of the Japan Soci-ety of Mechanical Engineers 63(505):240. English translation in RSIC-215, AD-443429.
Sunada, K., A. Yabe, T. Taketani, and Y. Yoshizawa. 1991. Experimental study of EHD pseudo-dropwise condensation. Proceedings of the ASME-JSME Thermal Engineering Joint Conference 3:47-53.
Tang, S. and T.W. McDonald. 1971. A study of boiling heat transfer from a rotating horizontal cylinder. International Journal of Heat and Mass Transfer 14:1643-57.
Tauscher, W.A., E.M. Sparrow, and J.R. Lloyd. 1970. Amplification of heat transfer by local injection of fluid into a turbulent tube flow. Interna-tional Journal of Heat and Mass Transfer 13:681-88.
Uemura, M., S. Nishio, and I. Tanasawa. 1990. Enhancement of pool boiling heat transfer by static electric field. Ninth International Heat Transfer Conference, 75-80.
Valencia, A., M. Fiebig, and V.K. Mitra. 1996. Heat transfer enhancement by longitudinal vortices in a fin tube heat exchanger. Journal of Heat Trans-fer 118:209.
Wawzyniak, M. and J. Seyed-Yagoobi. 1996. Augmentation of condensa-tion heat transfer with electrohydrodynamics on vertical enhanced tubes.
ASME Journal of Heat Transfer 118:499-502.
Wayner, P.C., Jr. and S.G. Bankoff. 1965. Film boiling of nitrogen with suc-tion on an electrically heated porous plate. AIChE Journal 11:59-64.
Heat Transfer 3.27 Woods, B.G. 1992. Sonically enhanced heat transfer from a cylinder in cross flow and its impact on process power consumption. International Jour-nal of Heat and Mass Transfer 35:2367-76.
Yabe, A. and H. Maki. 1988. Augmentation of convective and boiling heat transfer by applying an electrohydrodynamical liquid jet. International Journal of Heat Mass Transfer 31(2):407-17.
BIBLIOGRAPHY Fins General Gunter, A.Y. and A.W. Shaw. 1945. A general correlation of friction factors for various types of surfaces in cross flow. ASME Transactions 11:643.
Shah, R.K. and R.L. Webb. 1981. Compact and enhanced heat exchangers, heat exchangers, theory and practice, pp. 425-68. J. Taborek et al., eds.
Hemisphere Publishing, New York.
Webb, R.L. 1980. Air-side heat transfer in finned tube heat exchangers. Heat Transfer Engineering 1(3):33-49.
Smooth Clarke, L. and R.E. Winston. 1955. Calculation of finside coefficients in lon-gitudinal finned heat exchangers. Chemical Engineering Progress 3:147.
Elmahdy, A.H. and R.C. Biggs. 1979. Finned tube heat exchanger: Correla-tion of dry surface heat transfer data. ASHRAE Transactions 85:2.
Ghai, M.L. 1951. Heat transfer in straight fins. General discussion on heat transfer. London Conference, September.
Gray, D.L. and R.L. Webb. 1986. Heat transfer and friction correlations for plate finned-tube heat exchangers having plain fins. Proceedings of Eighth International Heat Transfer Conference, San Francisco.
Wavy Beecher, D.T. and T.J. Fagan. 1987. Fin patternization effects in plate finned tube heat exchangers. ASHRAE Transactions 93:2.
Yashu, T. 1972. Transient testing technique for heat exchanger fin. Reito 47(531):23-29.
Spine Abbott, R.W., R.H. Norris, and W.A. Spofford. 1980. Compact heat exchangers for General Electric products—Sixty years of advances in design and manufacturing technologies. Compact heat exchangers— History, technological advancement and mechanical design problems.
R.K. Shah, C.F. McDonald, and C.P. Howard, eds. Book No. G00183, pp. 37-55. American Society of Mechanical Engineers, New York.
Moore, F.K. 1975. Analysis of large dry cooling towers with spine-fin heat exchanger elements. ASME Paper No. 75-WA/HT-46.
Rabas, T.J. and P.W. Eckels. 1975. Heat transfer and pressure drop perfor-mance of segmented surface tube bundles. ASME Paper No. 75-HT-45.
Weierman, C. 1976. Correlations ease the selection of finned tubes. Oil and Gas Journal 9:94-100.
Louvered Hosoda, T. et al. 1977. Louver fin type heat exchangers. Heat Transfer Jap-anese Research 6(2):69-77.
Mahaymam, W. and L.P. Xu. 1983. Enhanced fins for air-cooled heat ex-changers—Heat transfer and friction factor correlations. Y. Mori and W.
Yang, eds. Proceedings of the ASME-JSME Thermal Engineering Joint Conference, Hawaii.
Senshu, T. et al. 1979. Surface heat transfer coefficient of fins utilized in air-cooled heat exchangers. Reito 54(615):11-17.
Circular Jameson, S.L. 1945. Tube spacing in finned tube banks. ASME Transactions 11:633.
Katz, D.L. 1954-55. Finned tubes in heat exchangers; Cooling liquids with finned coils; Condensing vapors on finned coils; and Boiling outside finned tubes. Bulletin reprinted from Petroleum Refiner.
Heat Exchangers Gartner, J.R. and H.L. Harrison. 1963. Frequency response transfer func-tions for a tube in crossflow. ASHRAE Transactions 69:323.
Gartner, J.R. and H.L. Harrison. 1965. Dynamic characteristics of water-to-air crossflow heat exchangers. ASHRAE Transactions 71:212.
McQuiston, F.C. 1981. Finned tube heat exchangers: State of the art for the air side. ASHRAE Transactions 87:1.
Myers, G.E., J.W. Mitchell, and R. Nagaoka. 1965. A method of estimating crossflow heat exchanger transients. ASHRAE Transactions 71:225.
Stermole, F.J. and M.H. Carson. 1964. Dynamics of forced flow distributed parameter heat exchangers. AIChE Journal 10(5):9.
Thomasson, R.K. 1964. Frequency response of linear counterflow heat exchangers. Journal of Mechanical Engineering Science 6(1):3.
Wyngaard, J.C. and F.W. Schmidt. Comparison of methods for determining transient response of shell and tube heat exchangers. ASME Paper 64-WA/HT-20 Yang, W.J. Frequency response of multipass shell and tube heat exchangers to timewise variant flow perturbance. ASME Paper 64-HT-18.
Heat Transfer, General Bennet, C.O. and J.E. Myers. 1984. Momentum, heat and mass transfer, 3rd ed. McGraw-Hill, New York.
Chapman, A.J. 1981. Heat transfer, 4th ed. Macmillan, New York.
Holman, J.D. 1981. Heat transfer, 5th ed. McGraw-Hill, New York.
Kern, D.Q. and A.D. Kraus. 1972. Extended surface heat transfer. McGraw-Hill, New York.
Kreith, F. and W.Z. Black. 1980. Basic heat transfer. Harper and Row, New York.
Lienhard, J.H. 1981. A heat transfer textbook. Prentice Hall, Englewood Cliffs, NJ.
McQuiston, F.C. and J.D. Parker. 1988. Heating, ventilating and air-condi-tioning, analysis and design, 4th ed. John Wiley and Sons, New York.
Rohsenow, W.M. and J.P. Hartnett, eds. 1973. Handbook of heat transfer.
McGraw-Hill, New York.
Sissom, L.E. and D.R. Pitts. 1972. Elements of transport phenomena.
McGraw-Hill, New York.
Smith, E.M. 1997. Thermal design of heat exchangers. John Wiley and Sons, Chichester, UK.
Todd, J.P. and H.B. Ellis. 1982. Applied heat transfer. Harper and Row, New York.
Webb, R.L. and A.E. Bergles. 1983. Heat transfer enhancement, second gen-eration technology. Mechanical Engineering 6:60-67.
Welty, J.R. 1974. Engineering heat transfer. John Wiley and Sons, New York.
Welty, J.R., C.E. Wicks, and R.E. Wilson. 1972. Fundamentals of momen-tum, heat and mass transfer. John Wiley and Sons, New York.
Wolf, H. 1983. Heat transfer. Harper and Row, New York.
4.1 CHAPTER 4 TWO-PHASE FLOW Boiling ............................................................................................................................................ 4.1 Condensing ..................................................................................................................................... 4.7 Pressure Drop .............................................................................................................................. 4.11 Enhanced Surfaces ....................................................................................................................... 4.13 Symbols ........................................................................................................................................ 4.13 WO-PHASE FLOW is encountered extensively in the air-con-Tditioning, heating, and refrigeration industries. A combination of liquid and vapor refrigerant exists in flooded coolers, direct-expansion coolers, thermosiphon coolers, brazed and gasketed plate evaporators and condensers, and tube-in-tube evaporators and con-densers, as well as in air-cooled evaporators and condensers. In the pipes of heating systems, steam and liquid water may both be present. Because the hydrodynamic and heat transfer aspects of two-phase flow are not as well understood as those of single-phase flow, no single set of correlations can be used to predict pressure drops or heat transfer rates. Instead, the correlations are for specific thermal and hydrodynamic operating conditions.
This chapter presents the basic principles of two-phase flow and provides information on the vast number of correlations that have been developed to predict heat transfer coefficients and pressure drops in these systems.
BOILING Commonly used refrigeration evaporators are (1) flooded evapo-rators, where refrigerants at low fluid velocities boil outside or inside tubes; and (2) dry expansion shell-and-tube evaporators, where refrig-erants at substantial fluid velocities boil outside or inside tubes.
Two-phase heat and mass transport are characterized by various flow and thermal regimes, whether vaporization takes place under natural convection or in forced flow. Unlike single-phase flow systems, the heat transfer coefficient for a two-phase mixture depends on the flow regime, the thermodynamic and transport properties of both the vapor and the liquid, the roughness of the heating surface, the wetting characteristics of the surface-liquid pair, and other parameters. Therefore, it is necessary to consider each flow and boiling regime separately to determine the heat transfer coefficient.
Accurate data defining limits of regimes and determining the effects of various parameters are not available. The accuracy of cor-relations in predicting the heat transfer coefficient for two-phase flow is in most cases not known beyond the range of the test data.
Boiling and Pool Boiling in Natural Convection Systems Regimes of Boiling. The different regimes of pool boiling described by Farber and Scorah (1948) verified those suggested by Nukiyama (1934). The regimes are illustrated in Figure 1. When the temperature of the heating surface is near the fluid saturation tem-perature, heat is transferred by convection currents to the free sur-face where evaporation occurs (Region I). Transition to nucleate boiling occurs when the surface temperature exceeds saturation by a few degrees (Region II).
In nucleate boiling (Region III), a thin layer of superheated liq-uid is formed adjacent to the heating surface. In this layer, bubbles nucleate and grow from spots on the surface. The thermal resistance Fig. 1 Characteristic Pool Boiling Curve The preparation of this chapter is assigned to TC 1.3, Heat Transfer and Fluid Flow.
4.2 2001 ASHRAE Fundamentals Handbook (SI) of the superheated liquid film is greatly reduced by bubble-induced agitation and vaporization. Increased wall temperature increases bubble population, causing a large increase in heat flux.
As heat flux or temperature difference increases further and as more vapor forms, the flow of the liquid toward the surface is inter-rupted, and a vapor blanket forms. This gives the maximum heat flux, which is at the departure from nucleate boiling (DNB) at point a, Figure 1. This flux is often termed the burnout heat flux or boil-ing crisis because, for constant power-generating systems, an increase of heat flux beyond this point results in a jump of the heater temperature (to point c, Figure 1), often beyond the melting point of a metal heating surface.
In systems with controllable surface temperature, an increase beyond the temperature for DNB causes a decrease of heat flux den-sity. This is the transition boiling regime (Region IV); liquid alter-nately falls onto the surface and is repulsed by an explosive burst of vapor.
At sufficiently high surface temperatures, a stable vapor film forms at the heater surface; this is the film boiling regime (Regions V and VI). Because heat transfer is by conduction (and some radia-tion) across the vapor film, the heater temperature is much higher than for comparable heat flux densities in the nucleate boiling regime. The minimum film boiling (MFB) heat flux (point b) is the lower end of the film boiling curve.
Free Surface Evaporation. In Region I, where surface temper-ature exceeds liquid saturation temperature by less than a few degrees, no bubbles form. Evaporation occurs at the free surface by convection of superheated liquid from the heated surface. Correla-tions of heat transfer coefficients for this region are similar to those for fluids under ordinary natural convection [Equations (1) through (4) in Table 1].
Nucleate Boiling. Much information is available on boiling heat transfer coefficients, but no universally reliable method is available for correlating the data. In the nucleate boiling regime, heat flux density is not a single, valued function of the temperature but depends also on the nucleating characteristics of the surface, as illustrated by Figure 2 (Berenson 1962).
The equations proposed for correlating nucleate boiling data can be put in a form that relates heat transfer coefficient h to temperature difference (tw − tsat): (1) Exponent a is normally about 3 for a plain, smooth surface; its value depends on the thermodynamic and transport properties of the vapor and the liquid. Nucleating characteristics of the surface, including the size distribution of surface cavities and the wetting characteristics of the surface-liquid pair, affect the value of the mul-tiplying constant and the value of the exponent a in Equation (1).
A generalized correlation cannot be expected without consider-ation of the nucleating characteristics of the heating surface. A sta-tistical analysis of data for 25 liquids by Hughmark (1962) shows that in a correlation not considering surface condition, deviations of more than 100% are common.
In the following sections, correlations and nomographs for pre-diction of nucleate and flow boiling of various refrigerants are given. For most cases, these correlations have been tested for refrigerants, such as R-11, R-12, R-113, and R-114, that have now been identified as environmentally harmful and are no longer being used in new equipment. Extensive research on the thermal and fluid characteristics of alternative refrigerants/refrigerant mix-tures has taken place in recent years, and some correlations have been suggested.
Stephan and Abdelsalam (1980) developed a statistical ap-proach for estimating the heat transfer during nucleate boiling. The correlation [Equation (5) in Table 1] should be used with a fixed contact angle θ regardless of the fluid. Cooper (1984) proposed a dimensional correlation for nucleate boiling, shown as Equation (6) in Table 1. The dimensions required are listed in Table 1. This cor-relation is recommended for fluids with poorly defined physical properties.
Gorenflo (1993) proposed a nucleate boiling correlation based on a set of reference conditions and a base heat transfer coefficient as shown in Equation (7) in Table 1. The correlation was devel-oped for a reduced pressure pr of 0.1 and the reference conditions given in Table 1. Base heat transfer coefficients are given for three fluids in Table 1, and Gorenflo (1993) should be consulted for additional fluids.
In addition to correlations dependent on thermodynamic and transport properties of the vapor and the liquid, Borishansky et al.
(1962) and Lienhard and Schrock (1963) documented a correlating method based on the law of corresponding states. The properties can be expressed in terms of fundamental molecular parameters, leading to scaling criteria based on the reduced pressure, pr = p/pc, where pc is the critical thermodynamic pressure for the coolant. An example of this method of correlation is shown in Figure 3. The ref-erence pressure p was chosen as p = 0.029pc. This correlation provides a simple method for scaling the effect of pressure if data are available for one pressure level. It also has an advantage if the thermodynamic and particularly the transport properties used in several equations in Table 1 are not accurately known. In its present form, this correlation gives a value of a = 2.33 for the exponent in Equation (1) and consequently should apply for typical aged metal surfaces.
There are explicit heat transfer coefficient correlations based on the law of corresponding states for various substances (Borishansky and Kosyrev 1966), halogenated refrigerants (Danilova 1965), and flooded evaporators (Starczewski 1965). Other investigations examined the effects of oil on boiling heat transfer from diverse configurations, including boiling from a flat plate (Stephan 1963b); a 14.0 mm OD horizontal tube using an oil-R-12 mixture (Tscher-nobyiski and Ratiani 1955); inside horizontal tubes using an oil-R-12 mixture (Breber et al. 1980, Worsoe-Schmidt 1959, Green and Furse 1963); and commercial copper tubing using R-11 and R-113 with oil content to 10% (Dougherty and Sauer 1974). Additionally, Furse (1965) examined R-11 and R-12 boiling over a flat horizontal copper surface.
Fig. 2 Effect of Surface Roughness on Temperature in Pool Boiling of Pentane h constant tw tsat – ( )a = Two-Phase Flow 4.3 Table 1 Equations for Boiling Heat Transfer Description References Equations Free convection Jakob (1949 and 1957) Free convection boiling, or boiling without bubbles for low ∆t and GrPr < 108 (all properties based on liquid state) Nu = C(Gr)m(Pr)n (1) Vertical submerged surface Nu = 0.61(Gr)0.25(Pr)0.25 (2) Horizontal submerged surface Nu = 0.16(Gr)1/3(Pr)1/3 (3) Simplified equation for water h ∼ 80(∆t)1/3, where h is in W/(m2·K), ∆t in K (4) Nucleate boiling Stephan and Abdelsalam (1980) (5) where Cooper (1984) (6) where h is in W/(m2·K), q/A is in W/m2, and Rp is surface roughness in µm Gorenflo (1993) (7) where reference conditions are (q/A)o = 20 000 W/m2, Rpo = 0.4 µm (8) (9) For all fluids except water and helium.
Fluid ho, W/(m2·K) R-134a 4500 R-22 3900 Ammonia 7000 Critical heat flux Kutateladze (1951) Zuber et al. (1962) (10) For many liquids, KD varies from 0.12 to 0.16.
Recommended average value is 0.13.
Minimum heat flux in film boiling from horizontal plate Zuber (1959) (11) Minimum heat flux in film boiling from horizontal cylinders Lienhard and Wong (1963) (12) Minimum temperature difference for film boiling from horizontal plate Berenson (1961) (13) Film boiling from horizontal plate Berenson (1961) (14) Film boiling from horizontal cylinders Anderson et al.
(1966) (15) Effect of radiation Anderson et al.
(1966) Substitute hDd kL ----------0.0546 ρv ρl ----- 0.5 qDd AklTsat ------------------ 0.67 hfgDd 2 aL 2 -------------- 0.248 ρl ρv – ρl ---------------- 4.33 – = Dd 0.0208θ σ g ρl ρv – ( ) ------------------------1 2 ⁄ with θ 35° = = h 55Pr 0.12 0.4343 Rp ( ) ln – 0.4343 Pr ln – ( )M 0.5 – q A --- 0.67 = h hoFPF q A ⁄ q A ⁄ ( )o ------------------ nf Rp Rpo -------- 0.133 = FPF 1.2Pr 0.27 2.5Pr Pr 1 Pr – --------------+ + = nf 0.9 0.3Pr 0.3 – = q A ⁄ ρvhfg -------------ρ2v σtg ρl ρv – ( ) ------------------------------0.25 KD = q A ⁄ ρvhfg -------------ρl ρv + ( ) σtg ρl ρv – ( ) ------------------------------0.25 0.09 = q A ⁄ ρvhfg -------------ρl ρv + ( )2 σtg ρl ρv – ( ) ------------------------------0.25 0.114 2σt g ρl ρv – ( )D2 -------------------------------0.5 1 2σt g ρl ρv – ( )D2 -------------------------------+ 0.25 -------------------------------------------------------= tw tsat – ( ) 0.127 ρvhfg kv ------------- g ρl ρv – ( ) ρl ρv + ------------------------2 3 ⁄ = σt g ρl ρv – ( ) ------------------------0.5 µv ρl ρv – ----------------1 3 ⁄ × h 0.425 kv 3ρvhfgg ρl ρv – ( ) µv tw tsat – ( ) φt g ρl ρv – ( ) ⁄ ---------------------------------------------------------------------0.25 = h 0.62 kv 3ρvg ρl ρv – ( )hfg Dµv tw tsat – ( ) --------------------------------------------0.25 = hfg ′ hfg 1 0.4cp tw tb – hfg ---------------+ = 4.4 2001 ASHRAE Fundamentals Handbook (SI) Maximum Heat Flux and Film Boiling Maximum or critical heat flux and the film boiling region are not as strongly affected by conditions of the heating surface as the heat flux in the nucleate boiling region, making analysis of DNB and of film boiling more tractable.
Carey (1992) provides a review of the mechanisms that have been postulated to cause the DNB phenomenon in pool boiling.
Each model is based on the scenario that vapor blankets, which lead to an increased thermal resistance, exist on portions of the heat transfer surface. It has been proposed that these blankets may result from Helmholtz instabilities.
When DNB (point a, Figure 1) is assumed to be a hydrodynamic instability phenomenon, a simple relation, Equation (10) in Table 1, can be derived to predict this flux for pure, wetting liquids (Kutate-ladze 1951, Zuber et al. 1962). The dimensionless constant K varies from approximately 0.12 to 0.16 for a large variety of liquids. The effect of wettability is still in question. Van Stralen (1959) found that for liquid mixtures, DNB is a function of the concentration.
The minimum heat flux density (point b, Figure 1) in film boiling from a horizontal surface and a horizontal cylinder can be predicted by Equations (11) and (12) in Table 1. The numerical factors 0.09 and 0.114 were adjusted to fit experimental data; values predicted by two analyses were approximately 30% higher. Equation (13) in Table 1 predicts the temperature difference at minimum heat flux of film boiling.
The heat transfer coefficient in film boiling from a horizontal surface can be predicted by Equation (14) in Table 1; and from a horizontal cylinder by Equation (15) in Table 1 (Bromley 1950), which has been generalized to include the effect of surface tension and cylinder diameter, as shown in Equations (16), (17), and (18) in Table 1 (Breen and Westwater 1962).
Frederking and Clark (1962) found that for turbulent film boil-ing, Equation (19) in Table 1 agrees with data from experiments at reduced gravity (Rohsenow 1963, Westwater 1963, Kutateladze 1963, Jakob 1949 and 1957).
Flooded Evaporators Equations in Table 1 merely approximate heat transfer rates in flooded evaporators. One reason is that vapor entering the evaporator combined with vapor generated within the evaporator can produce significant forced convection effects superimposed on those caused by nucleation. Nonuniform distribution of the two-phase, vapor-liq-uid flow within the tube bundle of shell-and-tube evaporators or the tubes of vertical-tube flooded evaporators is also important.
Bundle data and design methods for plain, low fin, and enhanced tubes have been reviewed in Thome (1990) and Collier and Thome (1996).
Typical performance of vertical tube natural circulation evapo-rators, based on data for water, is shown in Figure 4 (Perry 1950).
Low coefficients are at low liquid levels because insufficient liquid covers the heating surface. The lower coefficient at high levels is the result of an adverse effect of hydrostatic pressure on temperature difference and circulation rate. Perry (1950) noted similar effects in horizontal shell-and-tube evaporators.
Effect of surface tension and of pipe diameter Breen and Westwater (1962) Λ/D < 0.8: h(Λ)0.25/F = 0.60 (16) 0.8 < Λ/D < 8: hD0.25/F= 0.62 (17) 8 < Λ/D : h(Λ)0.25/F = 0.016 (Λ/D)0.83 (18) where Turbulent film Frederking and Clark (1962) Nu = 0.15 (Ra)1/3 for Ra > 5 × 107 (19) a = local acceleration Table 1 Equations for Boiling Heat Transfer (Continued) Description References Equations Λ 2π σt g ρl ρv – ( ) ------------------------0.25 = F ρvhfgg ρl ρv – ( )kv 3 µv tw tsat – ( ) --------------------------------------------0.25 = Ra D3g ρl ρv – ( ) vv 2ρv ------------------------------- cpµ k --------- v hfg cp tw tsat – ( ) -----------------------------0.4 + a g --1 3 ⁄ = Fig. 3 Correlation of Pool Boiling Data in Terms of Reduced Pressure Fig. 4 Boiling Heat Transfer Coefficients for Flooded Evaporator Two-Phase Flow 4.5 Forced-Convection Evaporation in Tubes Flow Mechanics. When a mixture of liquid and vapor flows inside a tube, a number of flow patterns occur, depending on the mass fraction of liquid, the fluid properties of each phase, and the flow rate. In an evaporator tube, the mass fraction of liquid decreases along the circuit length, resulting in a series of changing vapor-liquid flow patterns. If the fluid enters as a subcooled liquid, the first indications of vapor generation are bubbles forming at the heated tube wall (nucleation). Subsequently, bubble, plug, churn (or semiannular), annular, spray annular, and mist flows can occur as the vapor content increases for two-phase flows in horizontal tubes.
Idealized flow patterns are illustrated in Figure 5A for a horizontal tube evaporator.
Because nucleation occurs at the heated surface in a thin sublayer of superheated liquid, boiling in forced convection may begin while the bulk of the liquid is subcooled. Depending on the nature of the fluid and the amount of subcooling, the bubbles formed can either collapse or continue to grow and coalesce (Figure 5A), as Gouse and Coumou (1965) observed for R-113. Bergles and Rohsenow (1964) developed a method to determine the point of incipient sur-face boiling.
After nucleation begins, bubbles quickly agglomerate to form vapor plugs at the center of a vertical tube, or, as shown in Figure 5A, vapor plugs form along the top surface of a horizontal tube. At the point where the bulk of the fluid reaches saturation temperature, which corresponds to local static pressure, there will be up to 1% vapor quality because of the preceding surface boiling (Guerrieri and Talty 1956).
Further coalescence of vapor bubbles and plugs results in churn, or semiannular flow. If the fluid velocity is high enough, a continu-ous vapor core surrounded by a liquid annulus at the tube wall soon forms. This annular flow occurs when the ratio of the tube cross sec-tion filled with vapor to the total cross section is approximately 85%.
With common refrigerants, this equals a vapor quality of about 0% to 30%. Vapor quality is the ratio of mass (or mass flow rate) of vapor to total mass (or mass flow rate) of the mixture. The usual flowing vapor quality or vapor fraction is referred to throughout this discus-sion. Static vapor quality is smaller because the vapor in the core flows at a higher average velocity than the liquid at the walls (see Chapter 2).
If two-phase mass velocity is high [greater than 200 kg/(s·m2) for a 12 mm tube], annular flow with small drops of entrained liquid in the vapor core (spray) can persist over a vapor quality range from about 10% to more than 90%. Refrigerant evaporators are fed from an expansion device at vapor qualities of approximately 20%, so that annular and spray annular flow predominate in most tube lengths. In a vertical tube, the liquid annulus is distributed uni-formly over the periphery, but it is somewhat asymmetric in a hor-izontal tube (Figure 5A). As vapor quality reaches about 80%, the surface dries out. Chaddock and Noerager (1966) found that in a horizontal tube, dryout occurs first at the top of the tube and progresses toward the bottom with increasing vapor quality (Figure 5A).
If two-phase mass velocity is low [less than 200 kg/(s·m2) for a 12 mm horizontal tube], liquid occupies only the lower cross section of the tube. This causes a wavy type of flow at vapor qualities above about 5%. As the vapor accelerates with increasing evaporation, the interface is disturbed sufficiently to develop annular flow (Figure 5B). Liquid slugging can be superimposed on the flow configura-tions illustrated; the liquid forms a continuous, or nearly continu-ous, sheet over the tube cross section. The slugs move rapidly and at irregular intervals. Kattan et al. (1998a) presented a general method for prediction of flow pattern transitions (i.e., a flow pattern map) based on observations for R-134a, R-125, R-502, R-402A, R-404A, R-407C, and ammonia.
Heat Transfer. It is difficult to develop a single relation to describe the heat transfer performance for evaporation in a tube over the full quality range. For refrigerant evaporators with several per-centage points of flash gas at entrance, it is less difficult because annular flow occurs in most of the tube length. The reported data are accurate only within geometry, flow, and refrigerant conditions tested; therefore, a large number of methods for calculating heat transfer coefficients for evaporation in tubes is presented in Table 2 (also see Figures 6 through 8).
Figure 6 gives heat transfer data for R-22 evaporating in a 19.6 mm tube (Gouse and Coumou 1965). At low mass velocities [below 200 kg/(s·m2)], the wavy flow regime shown in Figure 5B probably exists, and the heat transfer coefficient is nearly constant along the tube length, dropping at the tube exit as complete vaporization occurs. At higher mass velocities, the flow pattern is usually annu-lar, and the coefficient increases as vapor accelerates. As the sur-face dries and the flow reaches a 90% vapor quality, the coefficient drops sharply.
Equation (1) in Table 2 is used to estimate average heat transfer coefficients for refrigerants evaporating in horizontal tubes (Pierre Fig. 5 Flow Regimes in Typical Smooth Horizontal Tube Evaporator 4.6 2001 ASHRAE Fundamentals Handbook (SI) Table 2 Equations for Forced Convection Evaporation in Tubes Equations Comments and References Horizontal Tubes (1) Average coefficients for R-12 and R-22 evaporating in copper tubes of 12.0 and 16.0 mm ID, from 4.1 to 9.5 m long, and at evaporating temperatures from −20 to 0°C (Pierre 1955, 1957).
where C1 = 0.0009 and n = 0.5 for exit qualities < 90% C1 = 0.0082 and n = 0.4 for 6 K superheat at exit Equation (1) with C1 = 0.0225 and n = 0.375 Average coefficients for R-22 evaporating at temperatures from 4.4 to 26°C in an 8.7 mm ID tube, 2.4 m long (Altman et al. 1960b).
(2) Compiled from a data base of 3693 data points including data for R-11, R-12, R-22, R-113, R-114, and water. Useful for vertical flows and horizontal flows (Gungor and Winterton 1986).
where (3) (4) (5) (6) hncb is found from Equation (6) in Table 1, which has units W/m2. If tube is horizontal and Fr < 0.05, use following multipliers: (7) Compiled from a database of 5246 data points including data for R-11, R-12, R-22, R-113, R-114, R-152a, nitrogen, neon, and water. Tube sizes ranged from 5 mm to 32 mm (Kandlikar 1990).
where hf is found from Equations (5) and (6) shown above.
Constant Convective Nucleate Boiling C1 1.136 0.6683 C2 –0.9 –0.2 C3 667.2 1058 C4 0.7 0.7 C5 0.3 0.3 Use convective constants if Co < 0.65 and nucleate if Co > 0.65. C5 = 0 for vertical tubes and horizontal with Frl > 0.04.
Fluid Ffl R-22 2.2 R-12 1.5 R-152a 1.1 Vertical Tubes h = 3.4hl(1/Xtt)0.45 (8) Equations (8) and (9) were fitted to experimental data for vertical upflow in tubes. Both relate to forced-convection evaporation regions where nucleate boiling is suppressed (Guerrieri and Talty 1956, Dengler and Addoms 1956). A multiplying factor is recommended when nucleation is present.
h = 3.5hL(1/Xtt)0.5 (9) where hl is from (3), Xtt from (7), hL from (6) h = 0.74hL[Bo × 104 + (1/Xtt)0.67] (10) Local coefficients for water in vertical upflow in tubes with diameters from 3.0 to 11 mm and lengths of 380 to 1020 mm. The boiling number Bo accounts for nucleation effects, and the Martinelli parameter Xtt, for forced-convection effects (Schrock and Grossman 1962).
where Bo is from (5), hL from (6), Xtt from (7) h = hmic + hmac (11) Chen developed this correlation reasoning that the nucleation transfer mechanism (represented by hmic) and the convective transfer mechanism (represented by hmac) are additive. hmac is expressed as a function of the two-phase Reynolds number after Martinelli, and hmic is obtained from the nucleate boiling correlation of Forster and Zuber (1955). Sc is a suppression factor for nucleate boiling (Chen 1963).
where hmac = hlFc hmic = 0.00122 (Sc)(E )(∆t)0.24(∆p)0.75 (12) (13) Fc and Sc from Figures 7 and 8 Note: Except for dimensionless equations, inch-pound units (lbm, h, ft, °F, and Btu) must be used.
h C1 kl d ---- GD µl --------- 2 J xhfg ∆ L ----------------- n = h Ehf Shncb + = E 1 24 000Bo 1.16 1.37 1 Xtt ⁄ ( )0.86 + + = S 1 0.00000115E2Rel 1.17 + [ ] 1 – = hf 0.023Rel 0.8Prl 0.4 kl d ⁄ ( ) = Rel G 1 x – ( )d µl ------------------------= E2 FrL 0.1 2Frl – ( ) = S2 Frl 1 2 ⁄ = h C1Co C2 25FrL ( ) C3Bo C4Ffl + ( )hf = E kl 0.79 cp ( )l 0.45ρl 0.49gc 0.25 σt 0.50µl 0.29hfg 0.24ρv 0.24 -----------------------------------------------------= Two-Phase Flow 4.7 1955, 1957). A number of methods are also available to estimate local heat transfer coefficients during evaporation. Equations (2) through (6) in Table 2 summarize the Gungor and Winterton (1986) model, while Equation (7) gives the Kandlikar (1990) model. In addition, the Shah (1982) model is also recommended for estimat-ing local heat transfer coefficients during annular flow. These local models have been found accurate for a wide range of refrigerants but do not include mechanisms to model dryout. The flow pattern based model of Kattan et al. (1998b) includes specific models for each flow pattern type and has been tested with the newer refriger-ants such as R-134a and R-407C.
Heat transfer coefficients during flow in vertical tubes can be estimated with Equations (8), (9), and (10) in Table 2. The Chen correlation shown in Equations (11) through (13) in Table 2 includes terms for the velocity effect (convection) and heat flux (nucleation) and produces local heat transfer coefficients as a func-tion of local vapor quality. The method developed by Steiner and Taborek (1992) includes an asymptotic model for the convection and nucleation component of heat transfer and is recommended for most situations.
The effect of lubricant on the evaporation heat transfer coeffi-cients has been studied by a number of authors (Schalger et al. 1988, Eckels et al. 1993, and Zeurcher et al. 1999). Schalger et al. and Eckels et al. showed that the average heat transfer coefficients dur-ing evaporation of R-22 and R-134a in smooth and enhanced tubes are in general decreased by presence of lubricant (up to a 20% re-duction at a 5% lubricant concentration by mass). Slight enhance-ments at lubricant concentrations under 3% are observed with some refrigerant lubricant mixtures. Zeurcher et al. (1999) studied local heat transfer coefficients of refrigerant/lubricant mixtures in the dry wall region of the evaporator (see Figure 5) and proposed prediction methods. The effect of lubricant concentration on local heat transfer coefficients was shown to be dependent on the mass flux and vapor quality. At low mass fluxes, the oil sharply decreased performance, while at higher mass fluxes, enhancements at certain vapor qualities were seen.
CONDENSING In most applications that use the condensation process, conden-sation is initiated by removing heat at a solid-vapor interface, either through the walls of the vessel containing the saturated vapor or through the solid surface of a cooling mechanism placed within the saturated vapor. If a sufficient amount of energy is removed, the local temperature of the vapor near the interface will drop below its equilibrium saturation temperature. Because the heat removal pro-cess creates a temperature gradient with the lowest temperature near the interface, vapor droplets most likely form at this location. This defines one type of heterogeneous nucleation that can result in either dropwise condensation or film condensation, depending on the phys-ical characteristics of the solid surface and the working fluid.
Dropwise condensation occurs on the cooling solid surface when its surface free energy is relatively low compared to that of the liquid. Examples of this type of interface include highly polished or fatty acid-impregnated surfaces in contact with steam. Film con-densation occurs when a cooling surface having relatively high sur-face free energy contacts a fluid having lower surface free energy Fig. 6 Heat Transfer Coefficient Versus Vapor Fraction for Partial Evaporation Fig. 7 Reynolds Number Factor Fc Fig. 8 Suppression Factor Sc 4.8 2001 ASHRAE Fundamentals Handbook (SI) (see Isrealachvili 1991). This is the type of condensation that occurs in most systems.
The rate of heat transport depends on the condensate film thick-ness, which depends on the rate of vapor condensation and the rate of condensate removal. At high reduced pressures, the heat transfer coefficients for dropwise condensation are higher than those avail-able in the presence of film condensation at the same surface load-ing. At low reduced pressures, the reverse is true. For example, there is a reduction of 6 to 1 in the dropwise condensation coefficient of steam when saturation pressure is decreased from 90 to 16 kPa. One method for correlating the dropwise condensation heat transfer coefficient employs nondimensional parameters, including the effect of surface tension gradient, temperature difference, and fluid properties.
When condensation occurs on horizontal tubes and short verti-cal plates, the condensate film motion is laminar. On vertical tubes and long vertical plates, the film motion can become turbu-lent. Grober et al. (1961) suggest using a Reynolds number (Re) of 1600 as the critical point at which the flow pattern changes from laminar to turbulent. This Reynolds number is based on con-densate flow rate divided by the breadth of the condensing sur-face. For a vertical tube, the breadth is the circumference of the tube; for a horizontal tube, the breadth is twice the length of the tube. Re = 4Γ/µf, where Γ is the mass flow of condensate per unit of breadth, and µf is the absolute (dynamic) viscosity of the con-densate at the film temperature tf. In practice, condensation is usually laminar in shell-and-tube condensers with the vapor out-side horizontal tubes.
Vapor velocity also affects the condensing coefficient. When this is small, condensate flows primarily by gravity and is resisted by the viscosity of the liquid. When vapor velocity is high relative to the condensate film, there is appreciable drag at the vapor-liquid interface. The thickness of the condensate film, and hence the heat transfer coefficient, is affected. When vapor flow is upward, a retarding force is added to the viscous shear, increasing the film thickness. When vapor flow is downward, the film thickness decreases and the heat transfer coefficient increases. For condensa-tion inside horizontal tubes, the force of the vapor velocity causes Table 3 Heat Transfer Coefficients for Film-Type Condensation Description References Equations 1. Vertical surfaces, height L Laminar condensate flow, Re = 4Γ/µf < 1800 McAdams (1954) h = 1.13F1(hfg/L∆t)0.25 (1) McAdams (1954) h = 1.11F2(b/wl)1/3 (2) Grigull (1952) h = 0.003(F1)2(∆tL/µf2hfg)0.5 (3) Turbulent flow, Re = 4Γ/µf > 1800 McAdams (1954) h = 0.0077F2(Re)0.4(1/µf)1/3 (4) 2. Outside horizontal tubes, N rows in a vertical plane, length L, laminar flow McAdams (1954) h = 0.79F1(hfg/Nd∆t)0.25 (5) McAdams (1954) h = 1.05F2(L/wl)1/3 (6) Finned tubes Beatty and Katz (1948) h = 0.689F1(hfg/∆tDe)0.25 (7) where De is determined from with Aeff = Asφ + Ap and Lmf = af/Do 3. Simplified equations for steam Outside vertical tubes, Re = 4Γ/µf < 2100 McAdams (1954) h = 4000/(L)0.25(∆t)1/3 (8) Outside horizontal tubes, Re = 4Γ/µf < 1800 McAdams (1954) Single tube h = 3100/(d′)0.25(∆t)1/3 (9a) Multiple tubes h = 3100/(Nd′)0.25(∆t)1/3 (9b) 4. Inside vertical tubes Carpenter and Colburn (1949) (10) where 5. Inside horizontal tubes, Ackers and Rosson (1960) (11) Ackers and Rosson (1960) (12) For Ackers et al. (1959) (13) where Ge = Gv(ρl/ρv)0.5 + Gl Notes: 1. Equations (1) through (10) are dimensional; inch-pound units (Btu, h, ft, °F, and lbm) must be used.
2. tf = liquid film temperature = tsat − 0.75∆t 1 De ( )0.25 -------------------1.30 Asφ Aeff Lmf ( )0.25 ------------------------------Ap Aeff D ( )0.25 --------------------------+ = h 0.065 cpf kf ρf f ′ 2µf ρv ------------------------- 0.5 Gm = Gm Gi 2 GiGo Go 2 + + 3 ---------------------------------------- 0.5 = DGl µl ----------5000 < 1000 DGv µl ----------- ρl ρv ----- 0.5 20 000 < < hD kl -------13.8 cpµl kl ---------- 1 3 ⁄ hfg cp t ∆ ---------- 1 6 ⁄ DGv µl ----------- ρl ρv ----- 0.5 0.2 = 20 000 DGv µl ----------- ρl ρv ----- 0.5 100 000 < < hD kl -------0.1 cpµl kl ---------- 1 3 ⁄ hfg cp t ∆ ---------- 1 6 ⁄ DGv µl ----------- ρl ρv ----- 0.5 2 3 ⁄ = DGl µl ----------5000 DGv µl ----------- ρl ρv ----- 0.5 20 000 > > hD kl -------0.026 cpµl kl ---------- 1 3 ⁄ DGe µl ----------- 0.8 = Two-Phase Flow 4.9 the condensate to flow. When the vapor velocity is high, the transi-tion from laminar to turbulent flow occurs at Reynolds numbers lower than previously described [i.e., 1600 according to Grober et al. (1961)].
When superheated vapor is condensed, the heat transfer coeffi-cient depends on the surface temperature. When the surface temper-ature is below saturation temperature, using the value of h for condensation of saturated vapor that incorporates the difference between the saturation temperature and the surface temperature leads to insignificant error (McAdams 1954). If the surface temper-ature is above the saturation temperature, there is no condensation and the equations for gas convection apply.
Correlation equations for condensing heat transfer are given in Table 3. Factors F1 and F2, which depend only on the physical properties of the working fluid and which occur often in these equations, have been computed for some commonly used refriger-ants in Table 4. Refrigerant properties used in the calculations may be found in Chapter 19.
In some cases, the equations are given in two forms: one is conve-nient when the amount of refrigerant to be condensed or the condens-ing load is known; the second is useful when the difference between the vapor temperature and the condensing surface temperature is known.
Condensation on Outside Surface of Vertical Tubes For film-type condensation on the outside surface of vertical tubes and on vertical surfaces, Equations (1) and (2) in Table 3 are recommended when 4Γ/µf is less than 1800 (McAdams 1954). For these equations, fluid properties are evaluated at the mean film temperature. When 4Γ/µf is greater than 1800 (tall vertical plates or tubes), use Equation (3) or (4) in Table 3. Equations (2) and (4) in Table 3 are plotted in Figure 9. The theoretical curve for laminar film-type condensation is shown for comparison. A semitheoretical relationship for turbulent film-type condensation is also shown for Pr values of 1.0 and 5.0 (Colburn 1933-34).
Condensation on Outside Surface of Horizontal Tubes For a bank of N tubes, Nusselt’s equations, increased by 10% (Jakob 1949 and 1957), are given in Equations (5) and (6) in Table 3. Experiments by Short and Brown (1951) with R-11 suggest that drops of condensation falling from row to row cause local turbu-lence and increase heat transfer.
For condensation on the outside surface of horizontal finned tubes, Equation (7) in Table 3 is used for liquids that drain readily from the surface (Beatty and Katz 1948). For condensing steam out-side finned tubes, where liquid is retained in the spaces between the tubes, coefficients substantially lower than those given by Equation (7) in Table 3 were reported. For additional data on condensation outside finned tubes, see Katz et al. (1947). For more on this topic, refer to Webb (1994).
Fig. 9 Film-Type Condensation Table 4 Values of Condensing Coefficient Factors for Different Refrigerants (from Chapter 19) Refrigerant Film Temperature, °C tf = tsat − 0.75∆t F1 F2 Refrigerant 11 24 80.7 347.7 38 80.3 344.7 52 79.2 339.7 Refrigerant 12 24 69.8 284.3 38 64.0 257.2 52 58.7 227.6 Refrigerant 22 24 80.3 347.7 38 75.5 319.4 52 69.2 285.5 Sulfur Dioxide 24 152.1 812.2 38 156.8 846.0 52 166.8 917.9 Ammonia 24 214.5 1285.9 38 214.0 1283.8 52 214.0 1281.7 Propane 24 83.4 359.6 38 82.3 357.4 52 80.7 353.6 Butane 24 81.8 355.3 38 81.8 356.6 52 82.3 357.4 F1 kf 3ρf 2g µf --------------- 0.25 = Units: W3 kg ⋅ s m7 K3 ⋅ ⋅ -------------------------0.25 F2 kf 3ρf 2g µf --------------- 1 3 ⁄ = Units: W3 kg ⋅ s m7 K3 ⋅ ⋅ -------------------------1 3 ⁄ 4.10 2001 ASHRAE Fundamentals Handbook (SI) Simplified Equations for Steam For film-type steam condensation at atmospheric pressure and film temperature drops of 5 to 85 K, McAdams (1954) recommends Equations (8) and (9) in Table 3.
Condensation on Inside Surface of Vertical Tubes Condensation on the inside surface of tubes is generally affected by appreciable vapor velocity. The measured heat transfer coeffi-cients are as much as 10 times those predicted by Equation (4) in Table 3. For vertical tubes, Jakob (1949 and 1957) gives theoretical derivations for upward and downward vapor flow. For downward vapor flow, Carpenter and Colburn (1949) suggest Equation (10) in Table 3. The friction factor f ′ for vapor in a pipe containing conden-sate should be taken from Figure 10.
Condensation on Inside Surface of Horizontal Tubes For condensation on the inside surface of horizontal tubes (as in air-cooled condensers, evaporative condensers, and some shell-and-tube condensers), the vapor velocity and resulting shear at the vapor-liquid interface are major factors in analyzing heat transfer. Hoogen-doorn (1959) identified seven types of two-phase flow patterns. For semistratified and laminar annular flow, use Equations (11) and (12) in Table 3 (Ackers and Rosson 1960). Ackers et al. (1959) recom-mend Equation (13) in Table 3 for turbulent annular flow (vapor Reynolds number greater than 20 000 and liquid Reynolds number greater than 5000). For high mass flux [> 200 kg/(m2·s)], the method of Shah (1979) is recommended for predicting local heat transfer coefficients during condensation. A method for using a flow regime map to predict the heat transfer coefficient for condensation of pure components in a horizontal tube is presented in Breber et al. (1980).
More recently, the flow regime dependent method of Dobson and Chato (1998) provides a more accurate design approach.
Noncondensable Gases Condensation heat transfer rates reduce drastically if one or more noncondensable gases are present in the condensing vapor/gas mix-ture. In mixtures, the condensable component is termed vapor and the noncondensable component is called gas. As the mass fraction of gas increases, the heat transfer coefficient decreases in an approx-imately linear manner. In a steam chest with 2.89% air by volume, Othmer (1929) found that the heat transfer coefficient dropped from about 11.4 to about 3.4 kW/(m2·K). Consider a surface cooled to some temperature ts below the saturation temperature of the vapor (Figure 11). In this system, accumulated condensate falls or is driven across the condenser surface. At a finite heat transfer rate, a temperature profile develops across the condensate that can be esti-mated from Table 3; the interface of the condensate is at a temper-ature tif > ts. In the absence of gas, the interface temperature is the vapor saturation temperature at the pressure of the condenser.
Fig. 10 Friction Factors for Gas Flow Inside Pipes with Wetted Walls Curve parameter = Γ/ρs, where Γ = liquid flow rate, ρ = liquid density, and s = surface tension of liquid relative to water; values of gas velocity used in calculating f and Re are calculated as though no liquid were present.
Fig. 11 Origin of Noncondensable Resistance Two-Phase Flow 4.11 The presence of noncondensable gas lowers the vapor partial pressure and hence the saturation temperature of the vapor in equi-librium with the condensate. Further, the movement of the vapor toward the cooled surface implies similar bulk motion of the gas. At the condensing interface, the vapor is condensed at temperature tif and is then swept out of the system as a liquid. The gas concentra-tion rises to ultimately diffuse away from the cooled surface at the same rate as it is convected toward the surface (Figure 11). If gas (mole fraction) concentration is Yg and total pressure of the system is p, the partial pressure of the bulk gas is (2) The partial pressure of the bulk vapor is (3) As opposing fluxes of convection and diffusion of the gas increase, the partial pressure of gas at the condensing interface is pgif > pg∞.
By Dalton’s law, assuming isobaric condition, (4) Hence, pvif < pv∞.
Sparrow et al. (1967) noted that thermodynamic equilibrium exists at the interface, except in the case of very low pressures or liquid metal condensation, so that (5) where psat(t) is the saturation pressure of the vapor at temperature t.
The available ∆t for condensation across the condensate film is reduced from (t∞ − ts) to (tif − ts), where t∞ is the bulk temperature of the condensing vapor-gas mixture, caused by the additional non-condensable resistance.
The equations in Table 3 are still valid for the condensate resis-tance, but the interface temperature tif must be found. The noncon-densable resistance, which accounts for the temperature difference (t∞ − tif), depends on the heat flux (through the convecting flow to the interface) and the diffusion of gas away from the interface.
In simple cases, Sparrow et al. (1967), Rose (1969), and Spar-row and Lin (1964) found solutions to the combined energy, diffu-sion, and momentum problem of noncondensables, but they are cumbersome.
A general method given by Colburn and Hougen (1934) can be used over a wide range if correct expressions are provided for the rate equations—add the contributions of the sensible heat transport through the noncondensable gas film and the latent heat transport via condensation: (6) where h is from the appropriate equation in Table 3.
The value of the heat transfer coefficient for the stagnant gas depends on the geometry and flow conditions. For flow parallel to a condenser tube, for example, (7) where j is a known function of Re = GD/µgv.
The mass transfer coefficient KD is (8) The calculation method requires substitution of Equation (8) into Equation (6). For a given flow condition, G, Re, j, Mm, pg∞, hg, and h (or U ) are known. Assume values of tif ; calculate psat(tif) = pvif and hence pgif. If ts is not known, use the overall coefficient U to the coolant and tc in place of h and ts in Equation (6). For either case, at each location in the condenser, iterate Equation (6) until it balances, giving the condensing interface temperature and, hence, the thermal load to that point (Colburn and Hougen 1934, Colburn 1951). For more detail, refer to Chapter 10 in Collier and Thome (1996).
Other Impurities Vapor entering the condenser often contains a small percentage of impurities such as oil. Oil forms a film on the condensing surfaces, creating additional resistance to heat transfer. Some allowance should be made for this, especially in the absence of an oil separator or when the discharge line from the compressor to the condenser is short.
PRESSURE DROP Total pressure drop for two-phase flow in tubes consists of fric-tion, acceleration, and gravitational components. It is necessary to know the void fraction (the ratio of gas flow area to total flow area) to compute the acceleration and gravitational components. To com-pute the frictional component of pressure drop, either the two-phase friction factor or the two-phase frictional multiplier must be determined.
The homogeneous model provides a simple method for comput-ing the acceleration and gravitational components of pressure drop.
The homogeneous model assumes that the flow can be character-ized by average fluid properties and that the velocities of the liquid and vapor phases are equal (Collier and Thome 1996, Wallis 1969).
Martinelli and Nelson (1948) developed a method for predicting the void fraction and two-phase frictional multiplier to use with a separated flow model. This method predicts the pressure drops of boiling refrigerants reasonably well. Other methods of computing the void fraction and two-phase frictional multiplier used in a sep-arated flow model are given in (Collier and Thome 1996, Wallis 1969).
The general nature of annular gas-liquid flow in vertical, and to some extent horizontal, pipe is indicated in Figure 12 (Wallis 1970), which plots the effective gas friction factor versus the liquid fraction (1 − a). Here a is the void fraction, or fraction of the pipe cross section taken up by the gas or vapor.
The effective gas friction factor is defined as (9) where D is the pipe diameter, ρg the gas density, and Qg the gas vol-umetric flow rate. The friction factor of gas flowing by itself in the pipe (presumed smooth) is denoted by fg. Wallis’ analysis of the flow occurrences is based on interfacial friction between the gas and liquid. The wavy film corresponds to a conduit of relative roughness ε/D, about four times the liquid film thickness. Thus, the pressure drop relation of vertical flow is (10) This corresponds to the Martinelli-type analysis with when (11) pg∞ Yg∞p = pv∞ 1 Yg∞ – ( )p Yv∞p = = pgif pvif + p = pvif psat tif ( ) = hg t∞ tif – ( ) KDMvhlv pv∞ pvif – ( ) + h tif ts – ( ) U tif tc – ( ) = = j hg cp ( )gG ----------------- cp ( )gµgv KDg --------------------- 2 3 ⁄ = KD Mm --------pg∞ pgif – pg∞pgif ⁄ ( ) ln --------------------------------µgv ρgD ---------- 2 3 ⁄ j = feff a5 2 ⁄ D 2ρg 4Qg πD2 ⁄ ( ) 2 -----------------------------------------dp ds ------ – = dp ds ------– ρgg + 0.01 ρg D5 ------ 4Qg π ---------- 21 75 1 a – ( ) + a5 2 ⁄ --------------------------------= ftwo-phase φg 2 fg = φg 2 1 75 1 a – ( ) + a5 2 ⁄ --------------------------------= 4.12 2001 ASHRAE Fundamentals Handbook (SI) The friction factor fg (of the gas alone) is taken as 0.02, an appropri-ate turbulent flow value. This calculation can be modified for more detailed consideration of factors such as Reynolds number variation in friction, gas compressibility, and entrainment (Wallis 1970).
In two-phase flow inside horizontal tubes, the pressure gradient is written as the sum of frictional and momentum terms. Thus, (12) In adiabatic two-phase flow, the contribution of the momentum transfer to the overall pressure drop is negligibly small; theoreti-cally, it is nonexistent if the flow is fully developed. In condensation heat transfer, the momentum transfer term contributes to the overall pressure drop due to the mass transfer that occurs at the liquid-vapor interface.
Two basic models were used in developing frictional pressure drop correlations for two-phase adiabatic flow. In the first, the flow of both phases is assumed to be homogeneous; the gas and liquid velocities are assumed equal. The frictional pressure drop is com-puted as if the flow were single phase, except for introducing mod-ifiers to the single-phase friction coefficient. In the second model, the two phases are considered separate, and the velocities may dif-fer. Two correlations used to predict the frictional pressure drop are those of Lockhart and Martinelli (1949) and Dukler et al. (1964).
In the Lockhart-Martinelli correlation, a parameter X was defined as (13) where = frictional pressure gradient, assuming that liquid alone flows in pipe = frictional pressure gradient, assuming that gas (or vapor in case of condensation) alone flows in pipe The frictional pressure gradient due to the single-phase flow of the liquid or vapor depends on the type of flow of each phase, lam-inar or turbulent. For turbulent flow during condensation, replace X by Xtt. Thus, (14) Lockhart and Martinelli (1949) also defined φv as (15) For condensation, (16) where (17) Here fo is the friction factor for adiabatic two-phase flow.
By analyzing the pressure drop data of simultaneous adiabatic flow of air and various liquids, Lockhart and Martinelli (1949) cor-related the parameters φv and X and reported the results graphically.
Soliman et al. (1968) approximated the graphical results of φv ver-sus Xtt by (18) Fig. 12 Qualitative Pressure Drop Characteristics of Two-Phase Flow Regime dp dz ------dp dz ------ f dp dz ------ m + = X dp dz ------ l dp dz ------ v ÷ 0.5 = dp dz ------ l dp dz ------ v Xtt 1 x – x ----------- 0.9 µ l µv ------ 0.1 ρv ρl ----- 0.5 = φv dp dz ------ f dp dz ------ v ÷ 0.5 = dp dz ------ v 2fo xG ( )2 ρvDi ----------------------– = fo 0.045 GxDi µv ⁄ ( )0.2 ----------------------------------= φv 1 2.85Xtt 0.523 + = Two-Phase Flow 4.13 In the correlation of Dukler et al. (1964), the frictional pressure gradient is given by (19) where fo = single-phase friction coefficient evaluated at two-phase Reynolds number (20) (21) (22) (23) (24) (25) Because the correlations mentioned here were originally devel-oped for adiabatic two-phase flow, Luu and Bergles (1980) modified the friction coefficients in Equations (16) and (19), using the modi-fier suggested by Silver and Wallis (1965-66). The modification replaced the friction coefficient fo with the friction coefficient fco.
These terms are related by (26) where (27) Because the Lockhart-Martinelli and Dukler correlations for the frictional pressure gradient were based on the separated flow model, the momentum pressure gradient should be as well. Thus, (28) To determine (dp/dz)m, the void fraction ψ and the quality gradient must be known. A generalized expression for ψ was suggested by Butterworth (1975): (29) where Al, ql, rl, and Sl are constants and are listed for the various cor-relations in Table 5.
The quality gradient dx/dz in Equation (28) can be estimated by assuming a constant rate of cooling. In the case of complete con-densation, its value is −1/L, where L is the length of the condenser tube.
Evaporators and condensers often have valves, tees, bends, and other fittings that contribute to the overall pressure drop of the heat exchanger. Collier and Thome (1996) summarize methods predict-ing the two-phase pressure drop in these fittings.
ENHANCED SURFACES Enhanced heat transfer surfaces are used in heat exchangers to improve performance and decrease cost. Condensing heat transfer is often enhanced with circular fins attached to the external surfaces of tubes to increase the heat transfer area. Other enhancement meth-ods, such as porous coatings, integral fins, and reentrant cavities, are used to augment boiling heat transfer on the external surfaces of evaporator tubes. Webb (1981) surveys external boiling surfaces and compares the performances of several enhanced surfaces with the performance of smooth tubes. For heat exchangers, the heat transfer coefficient for the refrigerant side is often smaller than the coefficient for the water side. Thus, enhancing the refrigerant-side surface can reduce the size of the heat exchanger and improve its performance.
Internal fins increase the heat transfer coefficients during evap-oration or condensation in tubes. However, internal fins increase the refrigerant pressure drop and reduce the heat transfer rate by decreasing the available temperature difference between hot and cold fluids. Designers should carefully determine the number of parallel refrigerant passes that give optimum loading for best over-all heat transfer.
For additional information on enhancement methods in two-phase flow, consult Bergles (1976, 1985), Thome (1990), and Webb (1994).
SYMBOLS A = area Aeff = total effective area [Equation (7) in Table 3] a = local acceleration [Equation (19) in Table 1]; void fraction [Equations (9) and (10)] af = area of one side of one film Bo = boiling number [Equations (3) and (7) in Table 2] b = breadth of a condensing surface. For vertical tube, b = πd; for horizontal tube, b = 2L C = a coefficient or constant Co = convection number C1…C5 = special constants (see Table 2) cp = specific heat at constant pressure cv = specific heat at constant volume D = diameter Dd = bubble departure diameter [Equation (5) in Table 1] Di = inside tube diameter Do = outside tube diameter d = diameter; or prefix meaning differential (dp/dz) = pressure gradient dp dz ------ f 2G2foα λ ( )β DiρNS -------------------------------– = 0.0014 0.125 4m · tβ πDiµNS ------------------- 0.32 – + = α λ ( ) 1 λ ln ( ) [1.281 ⁄ – 0.478 λ ln 0.444 λ ln ( )2 + + = 0.094 λ ln ( )3 0.00843 λ ln ( )4] + + β ρl ρNS ---------- λ2 1 ψ – ( ) -----------------ρv ρNS ---------- 1 λ – ( )2 ψ -----------------+ = ρNS ρl λ ρv 1 λ – ( ) + = µNS µl λ µv 1 λ – ( ) + = λ 1 1 x 1 x – ( ) ---------------- ρv ρe ----- + ⁄ = fco fo ------ ε 2fo ------- ξ fo ---- – exp = ξ Diψ 2x ---------- dx dz ------= dp dz ------ m G2 dx dz ------ 2x ρvψ ----------2 1 x – ( ) ρl 1 ψ – ( ) -----------------------– – = ql ψ1 x – x1 ψ – ρl ---------------------x 1 ψ – ( ) ψ 1 x – ( )ρv --------------------------– + ψ 1 1 Al 1 x – ( ) x ⁄ [ ] ql ρv ρl ⁄ ( ) rl µl µv ⁄ ( ) Sl + ---------------------------------------------------------------------------------------------= Table 5 Constants in Equation (29) for Different Void Fraction Correlations Model Al ql rl Sl Homogeneous (Collier 1972) 1.0 1.0 1.0 0 Lockhart-Martinelli (1949) 0.28 0.64 0.36 0.07 Baroczy (1963) 1.0 0.74 0.65 0.13 Thom (1964) 1.0 1.0 0.89 0.18 Zivi (1964) 1.0 1.0 0.67 0 Turner-Wallis (1965) 1.0 0.72 0.40 0.08 4.14 2001 ASHRAE Fundamentals Handbook (SI) (dp/dz)f = frictional pressure gradient (dp/dz)l = frictional pressure gradient, assuming that liquid alone is flowing in pipe (dp/dz)m= momentum pressure gradient (dp/dz)v = frictional pressure gradient, assuming that gas (or vapor) alone is flowing in pipe E = special coefficient (Table 2) Fc = Reynolds number factor [Equation (12) in Table 2 and Figure 7] FPF = special coefficient [Equation (7) in Table 1] Fr = Froude number F1, F2 = condensing coefficient factors (Table 4) f = friction factor for single-phase flow f ′ = friction factor for gas flow inside pipes with wetted walls (Figure 10) fco = friction factor in presence of condensation [Equation (26)] fo = friction factor [Equations (17) and (19)] G = mass velocity Gr = Grashof number g = gravitational acceleration gc = gravitational constant h = heat transfer coefficient hf = special coefficient [Equation (7) in Table 1] hfg = latent heat of vaporization or of condensation j = Colburn j-factor KD = mass transfer coefficient k = thermal conductivity L = length Lmf = mean length of fin [Equation (7) in Table 3] ln = natural logarithm M = mass; or molecular mass Mm = mean relative molecular mass of vapor-gas mixture Mv = relative molecular mass of condensing vapor m = general exponent [Equations (1) and (6) in Table 1] = mass rate of flow N = number of tubes in vertical tier Nu = Nusselt number n = general exponent [Equations (1) and (6) in Table 1] Pr = Prandtl number p = pressure pc = critical thermodynamic pressure for coolant pr = reduced pressure Q = total heat transfer q = rate of heat transfer r = radius Ra = Rayleigh number Re = Reynolds number Rp = surface roughness, µm S = distance along flow direction Sc = suppression factor (Table 2 and Figure 8) t = temperature U = overall heat transfer coefficient V = linear velocity x = quality (i.e., vapor fraction = Mv/M); or distance in dt/dx Xtt = Martinelli parameter [Figure 7, Table 2, and Equation (14)] x,y,z = lengths along principal coordinate axes Yg = mole fraction of gas [Equations (2) and (3)] Yv = mole fraction of vapor [Equation (3)] α = thermal diffusivity = k/ρcp α(λ) = ratio of two-phase friction factor to single-phase friction factor at two-phase Reynolds number [Equation (21)] β = ratio of two-phase density to no-slip density [Equation (22)] Γ = mass rate of flow of condensate per unit of breadth (see section on Condensing) ∆= difference between values ε = roughness of interface Λ = special coefficient [Equations (16) through (19) in Table 1] λ = ratio of liquid volumetric flow rate to total volumetric flow rate [Equation (25)] µ = absolute (dynamic) viscosity µl = dynamic viscosity of saturated liquid µNS = dynamic viscosity of two-phase homogeneous mixture [Equation (24)] µv = dynamic viscosity of saturated vapor ν = kinematic viscosity ρ = density ρl = density of saturated liquid ρNS = density of two-phase homogeneous mixture [Equation (23)] ρv = density of saturated vapor phase σ = surface tension φg = fin efficiency, Martinelli factor [Equation (11)] φv = Lockhart-Martinelli parameter [Equation (15)] ψ = void fraction Subscripts and Superscripts a = exponent in Equation (1) b = bubble c = critical or cold (fluid) cg = condensing e = equivalent eff = effective f = film or fin g = gas h = horizontal or hot (fluid) or hydraulic i = inlet or inside if = interface L = liquid l = liquid m = mean mac = convective mechanism [Equations (11) through (13) in Table 2] max = maximum mic = nucleation mechanism [Equations (11) through (13) in Table 2] min = minimum ncb = nucleate boiling o = outside or outlet or overall r = root (fin) or reduced pressure s = surface or secondary heat transfer surface sat = saturation (pressure) t = temperature or terminal temperature of tip (fin) v = vapor or vertical w = wall ∞= bulk = reference REFERENCES Ackers, W.W., H.A. Deans, and O.K. Crosser. 1959. Condensing heat trans-fer within horizontal tubes. Chemical Engineering Progress Symposium Series 55(29):171-76.
Ackers, W.W. and H.F. Rosson. 1960. Condensation inside a horizontal tube.
Chemical Engineering Progress Symposium Series 56(30):145-50.
Altman, M., F.W. Staub, and R.H. Norris. 1960b. Local heat transfer and pressure drop for Refrigerant-22 condensing to horizontal tubes. Chem-ical Engineering Progress Symposium Series 56(30):151-60.
Anderson, W., D.G. Rich, and D.F. Geary. 1966. Evaporation of Refrigerant 22 in a horizontal 3/4-in. OD tube. ASHRAE Transactions 72(1):28.
Baroczy, C.J. 1963. Correlation of liquid fraction in two-phase flow with application to liquid metals. North American Aviation Report SR-8171, El Segundo, CA.
Beatty, K.O. and D.L. Katz. 1948. Condensation of vapors on outside of finned tubes. Chemical Engineering Progress 44(1):55.
Berenson, P.J. 1961. Film boiling heat transfer from a horizontal surface.
ASME Journal of Heat Transfer 85:351.
Berenson, P.J. 1962. Experiments on pool boiling heat transfer. Interna-tional Journal of Heat and Mass Transfer 5:985.
Bergles, A.E. 1976. Survey and augmentation of two-phase heat transfer.
ASHRAE Transactions 82(1):891-905.
Bergles, A.E. 1985. Techniques to augment heat transfer. In Handbook of heat transfer application, 2nd ed. McGraw-Hill, New York.
Bergles, A.E. and W.M. Rohsenow. 1964. The determination of forced con-vection surface-boiling heat transfer. ASME Journal of Heat Transfer, Series C, 86(August):365.
Borishansky, W. and A. Kosyrev. 1966. Generalization of experimental data for the heat transfer coefficient in nucleate boiling. ASHRAE Journal (May):74.
Borishansky, V.M., I.I. Novikov, and S.S. Kutateladze. 1962. Use of thermo-dynamic similarity in generalizing experimental data on heat transfer.
Proceedings of the International Heat Transfer Conference.
m · Two-Phase Flow 4.15 Breber, G., J.W. Palen, and J. Taborek. 1980. Prediction of the horizontal tubeside condensation of pure components using flow regime criteria.
ASME Journal of Heat Transfer 102(3):471-76.
Breen, B.P. and J.W. Westwater. 1962. Effects of diameter of horizontal tubes on film boiling heat transfer. AIChE Preprint No. 19, Fifth National Heat Transfer Conference, Houston, TX. Chemical Engineering Progress 58(7):67-72.
Bromley, L.A. 1950. Heat transfer in stable film boiling. Chemical Engi-neering Progress (46):221.
Butterworth, D. 1975. A comparison of some void-fraction relationships for co-current gas-liquid flow. International Journal of Multiphase Flow 1:845-50.
Carey, V.P. 1992. Liquid-vapor phase change phenomena: An introduction to the thermophysics of vaporization and condensation processes in heat transfer equipment. Hemisphere Publishing Corporation, Washington, D.C.
Carpenter, E.F. and A.P. Colburn. 1949. The effect of vapor velocity on condensation inside tubes. General discussion on Heat Transfer and Fluid Mechanics Institute, American Society of Mechanical Engineers, New York.
Chaddock, J.B. and J.A. Noerager. 1966. Evaporation of Refrigerant 12 in a horizontal tube with constant wall heat flux. ASHRAE Transactions 72(1):90.
Chen, J.C. 1963. A correlation for boiling heat transfer to saturated fluids on convective flow. ASME Paper 63-HT-34. American Society of Mechan-ical Engineers, New York.
Colburn, A.P. 1933-34. Note on the calculation of condensation when a por-tion of the condensate layer is in turbulent motion. AIChE Transactions No. 30.
Colburn, A.P. 1951. Problems in design and research on condensers of vapours and vapour mixtures. Proceedings of the Institute of Mechanical Engineers, 164:448, London.
Colburn, A.P. and O.A. Hougen. 1934. Design of cooler condensers for mix-tures of vapors with noncondensing gases. Industrial and Engineering Chemistry 26 (November):1178.
Collier, J.G. and J.R. Thome. 1996. Convective boiling and condensation, 3rd ed. Oxford University Press.
Cooper, M.G. 1984. Heat flow rates in saturated nucleate pool boiling-a wide-ranging examination using reduced properties. Advances in Heat Transfer. Academic Press, Orlando, 16, 157-239.
Danilova, G. 1965. Influence of pressure and temperature on heat exchange in the boiling of halogenated hydrocarbons. Kholodilnaya Teknika, No.
2. English abstract, Modern Refrigeration (December).
Dengler, C.E. and J.N. Addoms. 1956. Heat transfer mechanism for vapor-ization of water in a vertical tube. Chemical Engineering Progress Sym-posium Series 52(18):95.
Dobson, M.K., and J.C. Chato. 1998. Condensation in smooth horizontal tubes. Journal of Heat Transfer 120:193-213.
Dougherty, R.L. and H.J. Sauer, Jr. 1974. Nucleate pool boiling of refriger-ant-oil mixtures from tubes. ASHRAE Transactions 80(2):175.
Dukler, A.E., M. Wicks, III, and R.G. Cleveland. 1964. Frictional pressure drop in two-phase flow: An approach through similarity analysis. AIChE Journal 10(January):44-51.
Eckels, S.J., T.M. Doer, and M.B. Pate. 1994. In-tube heat transfer and pres-sure drop of R-134a and ester lubricant mixtures in a smooth tube and a micro-fin tube: Part 1 evaporation. ASHRAE Transactions 100(2)(1994): 265-82.
Farber, E.A. and R.L. Scorah. 1948. Heat transfer to water boiling under pressure. ASME Transactions (May):373.
Forster, H.K. and N. Zuber. 1955. Dynamics of vapor bubbles and boiling heat transfer. AIChE Journal 1(4):531-35.
Frederking, T.H.K. and J.A. Clark. 1962. Natural convection film boiling on a sphere. In Advances in cryogenic engineering, ed. K.D. Timmerhouse, Plenum Press, New York.
Furse, F.G. 1965. Heat transfer to Refrigerants 11 and 12 boiling over a hor-izontal copper surface. ASHRAE Transactions 71(1):231.
Gorenflo, D. 1993. Pool boiling. VDI-Heat Atlas. VDI-Verlag, Düsseldorf.
Gouse, S.W., Jr. and K.G. Coumou. 1965. Heat transfer and fluid flow inside a horizontal tube evaporator, Phase I. ASHRAE Transactions 71(2):152.
Green, G.H. and F.G. Furse. 1963. Effect of oil on heat transfer from a hori-zontal tube to boiling Refrigerant 12-oil mixtures. ASHRAE Journal (October):63.
Grigull, U. 1952. Wärmeübergang bei Filmkondensation. Forsch, Gebiete Ingenieurw. 18.
Grober, H., S. Erk, and U. Grigull. 1961. Fundamentals of heat transfer.
McGraw-Hill, New York.
Guerrieri, S.A. and R.D. Talty. 1956. A study of heat transfer to organic liq-uids in single tube boilers. Chemical Engineering Progress Symposium Series 52(18):69.
Gungor, K.E., and R.H.S. Winterton. 1986. A general correlation for flow boiling in tubes and annuli. International Journal Heat Mass Transfer 29:351-58.
Hoogendoorn, C.J. 1959. Gas-liquid flow in horizontal pipes. Chemical Engineering Sciences IX(1).
Hughmark, G.A. 1962. A statistical analysis of nucleate pool boiling data.
International Journal of Heat and Mass Transfer 5:667.
Isrealachvili, J.N. 1991. Intermolecular surface forces. Academic Press, New York.
Jakob, M. 1949 and 1957. Heat transfer, Vols. I and II. John Wiley and Sons, New York.
Kandlikar, S.G. 1990. A general correlation for saturated two-phase flow and boiling heat transfer inside horizontal and vertical tubes. Journal of Heat Transfer 112:219-28.
Kattan, N., J.R. Thome, and D. Favrat. 1998a. Flow boiling in horizontal tubes Part 1: Development of diabatic two-phase flow pattern map. Jour-nal of Heat Transfer 120(1):140-47.
Kattan, N., J.R. Thome, and D. Favrat. 1998b. Flow boiling in horizontal tubes Part 3: Development of new heat transfer model based on flow pat-terns. Journal of Heat Transfer 120(1):156-65.
Katz, D.L., P.E. Hope, S.C. Datsko, and D.B. Robinson. 1947. Condensation of Freon-12 with finned tubes. Part I, Single horizontal tubes; Part II, Multitube condensers. Refrigerating Engineering (March):211, (April): 315.
Kutateladze, S.S. 1951. A hydrodynamic theory of changes in the boiling process under free convection. Izvestia Akademii Nauk, USSR, Otdele-nie Tekhnicheski Nauk 4:529.
Kutateladze, S.S. 1963. Fundamentals of heat transfer. E. Arnold Press, London.
Lienhard, J.H. and V.E. Schrock. 1963. The effect of pressure, geometry and the equation of state upon peak and minimum boiling heat flux. ASME Journal of Heat Transfer 85:261.
Lienhard, J.H. and P.T.Y. Wong. 1963. The dominant unstable wave length and minimum heat flux during film boiling on a horizontal cylinder.
ASME Paper No. 63-HT-3. ASME-AIChE Heat Transfer Conference, Boston, August.
Lockhart, R.W. and R.C. Martinelli. 1949. Proposed correlation of data for isothermal two-phase, two-component flow in pipes. Chemical Engi-neering Progress 45(1):39-48.
Luu, M. and A.E. Bergles. 1980. Augmentation of in-tube condensation of R-113. ASHRAE Research Project RP-219.
Martinelli, R.C. and D.B. Nelson. 1948. Prediction of pressure drops during forced circulation boiling of water. ASME Transactions 70:695.
McAdams, W.H. 1954. Heat transmission, 3rd ed. McGraw-Hill, New York.
Nukiyama, S. 1934. The maximum and minimum values of heat transmitted from metal to boiling water under atmospheric pressure. Journal of the Japanese Society of Mechanical Engineers 37:367.
Othmer, D.F. 1929. The condensation of steam. Industrial and Engineering Chemistry 21(June):576.
Perry, J.H. 1950. Chemical engineers handbook, 3rd ed. McGraw-Hill, New York.
Pierre, B. 1955. S.F. Review. A.B. Svenska Flaktafabriken, Stockholm, Swe-den 2(1):55.
Pierre, B. 1957. Kylteknisk Tidskrift 3 (May):129.
Pierre, B. 1964. Flow resistance with boiling refrigerant. ASHRAE Journal (September through October).
Rohsenow, W.M. 1963. Boiling heat transfer. In Modern developments in heat transfer, ed. W. Ibele. Academic Press, New York.
Rose, J.W. 1969. Condensation of a vapour in the presence of a noncondens-able gas. International Journal of Heat and Mass Transfer 12:233.
Schlager, L.M., M.B. Pate, and A.E. Bergles. 1987. Evaporation and con-densation of refrigerant-oil mixtures in a smooth tube and micro-fin tube.
ASHRAE Transactions 93:293-316.
Schrock, V.E. and L.M. Grossman. 1962. Forced convection boiling in tubes. Nuclear Science and Engineering 12:474.
Shah, M.M. 1979. A general correlation for heat transfer during film con-densation inside pipes. International Journal Heat Mass Transfer 22:547-56.
4.16 2001 ASHRAE Fundamentals Handbook (SI) Shah, M.M. 1982. A new correlation for saturated boiling heat transfer: equations and further study. ASHRAE Transactions 88(1):185-96.
Short, B.E. and H.E. Brown. 1951. Condensation of vapors on vertical banks of horizontal tubes. American Society of Mechanical Engineers, New York.
Silver, R.S. and G.B. Wallis. 1965-66. A simple theory for longitudinal pres-sure drop in the presence of lateral condensation. Proceedings of Institute of Mechanical Engineering, 180 Part I(1):36-42.
Soliman, M., J.R. Schuster, and P.J. Berenson. 1968. A general heat transfer correlation for annular flow condensation. Journal of Heat Transfer 90:267-76.
Sparrow, E.M. and S.H. Lin. 1964. Condensation in the presence of a non-condensable gas. ASME Transactions, Journal of Heat Transfer 86C:430.
Sparrow, E.M., W.J. Minkowycz, and M. Saddy. 1967. Forced convection condensation in the presence of noncondensables and interfacial resis-tance. International Journal of Heat and Mass Transfer 10:1829.
Starczewski, J. 1965. Generalized design of evaporation heat transfer to nucleate boiling liquids. British Chemical Engineering (August).
Steiner, D., and J. Taborek. 1992. Flow boiling heat transfer in vertical tubes correlated by an asymptotic model. Heat Transfer Engineering 13(2), 43-69.
Stephan, K. 1963a. The computation of heat transfer to boiling refrigerants.
Kältetechnik 15:231.
Stephan, K. 1963b. Influence of oil on heat transfer of boiling Freon-12 and Freon-22. Eleventh International Congress of Refrigeration, I.I.R. Bulle-tin No. 3.
Stephan, K. 1963c. A mechanism and picture of the processes involved in heat transfer during bubble evaporation. Chemic. Ingenieur Technik 35:775.
Stephan, K. and M. Abdelsalam. 1980. Heat transfer correlations for natural convection boiling. International Journal of Heat and Mass Transfer 23:73-87.
Thom, J.R.S. 1964. Prediction of pressure drop during forced circulation boiling water. International Journal of Heat and Mass Transfer 7:709-24.
Thome, J.R. 1990. Enhanced boiling heat transfer. Hemisphere (Taylor and Francis), New York.
Tschernobyiski, I. and G. Ratiani. 1955. Kholodilnaya Teknika 32.
Turner, J.M. and G.B. Wallis. 1965. The separate-cylinders model of two-phase flow. Report No. NYO-3114-6. Thayer’s School of Engineering, Dartmouth College, Hanover, NH.
Van Stralen, S.J. 1959. Heat transfer to boiling binary liquid mixtures.
Chemical Engineering (British) 4(January):78.
Wallis, G.B. 1969. One-dimensional two-phase flow. McGraw-Hill, New York.
Wallis, G.C. 1970. Annular two-phase flow, Part I: A simple theory, Part II: Additional effect. ASME Transactions, Journal of Basic Engineering 92D:59 and 73.
Webb, R.L. 1981. The evolution of enhanced surface geometrics for nucle-ate boiling. Heat Transfer Engineering 2(3-4):46-69.
Webb, J.R. 1994. Enhanced boiling heat transfer. John Wiley and Sons, New York.
Westwater, J.W. 1963. Things we don’t know about boiling. In Research in Heat Transfer, ed. J. Clark. Pergamon Press, New York.
Worsoe-Schmidt, P. 1959. Some characteristics of flow-pattern and heat transfer of Freon-12 evaporating in horizontal tubes. Ingenieren, Inter-national edition, 3(3).
Zeurcher O., J.R. Thome, and D. Favrat. 1998. In-tube flow boiling of R-407C and R-407C/oil mixtures, part II: Plain tube results and predictions.
Int. J. HVAC&R Research 4(4):373-399.
Zivi, S.M. 1964. Estimation of steady-state steam void-fraction by means of the principle of minimum entropy production. Journal of Heat Transfer 86:247-52.
Zuber, N. 1959. Hydrodynamic aspects of boiling heat transfer. U.S. Atomic Energy Commission, Technical Information Service, Report AECU 4439. Oak Ridge, TN.
Zuber, N., M. Tribus, and J.W. Westwater. 1962. The hydrodynamic crisis in pool boiling of saturated and subcooled liquids. Proceedings of the Inter-national Heat Transfer Conference 2:230, and discussion of the papers, Vol. 6.
5.1 CHAPTER 5 MASS TRANSFER Molecular Diffusion ....................................................................................................................... 5.1 Convection of Mass ........................................................................................................................ 5.5 Simultaneous Heat and Mass Transfer Between Water-Wetted Surfaces and Air ......................... 5.9 Symbols ........................................................................................................................................ 5.13 ASS transfer by either molecular diffusion or convection M is the transport of one component of a mixture relative to the motion of the mixture and is the result of a concentration gradient.
In an air-conditioning process, water vapor is added or removed from the air by a simultaneous transfer of heat and mass (water vapor) between the airstream and a wetted surface. The wetted sur-face can be water droplets in an air washer, wetted slats of a cooling tower, condensate on the surface of a dehumidifying coil, surface presented by a spray of liquid absorbent, or wetted surfaces of an evaporative condenser. The performance of equipment with these phenomena must be calculated carefully because of the simulta-neous heat and mass transfer.
This chapter addresses the principles of mass transfer and pro-vides methods of solving a simultaneous heat and mass transfer problem involving air and water vapor. Emphasis is on air-condi-tioning processes involving mass transfer. The formulations pre-sented can help in analyzing the performance of specific equipment.
For a discussion on the performance of air washers, cooling coils, evaporative condensers, and cooling towers, see Chapters 19, 21, 35, and 36, respectively, of the 2000 ASHRAE Handbook—Systems and Equipment.
This chapter is divided into (1) the principles of molecular diffu-sion, (2) a discussion on the convection of mass, and (3) simultaneous heat and mass transfer and its application to specific equipment.
MOLECULAR DIFFUSION Most mass transfer problems can be analyzed by considering the diffusion of a gas into a second gas, a liquid, or a solid. In this chap-ter, the diffusing or dilute component is designated as component B, and the other component as component A. For example, when water vapor diffuses into air, the water vapor is component B and dry air is component A. Properties with subscripts A or B are local proper-ties of that component. Properties without subscripts are local prop-erties of the mixture.
The primary mechanism of mass diffusion at ordinary tempera-ture and pressure conditions is molecular diffusion, a result of density gradient. In a binary gas mixture, the presence of a concen-tration gradient causes transport of matter by molecular diffusion; that is, because of random molecular motion, gas B diffuses through the mixture of gases A and B in a direction that reduces the concen-tration gradient.
Fick’s Law The basic equation for molecular diffusion is Fick’s law.
Expressing the concentration of component B of a binary mixture of components A and B in terms of the mass fraction ρB/ρ or mole fraction CB/C, Fick’s law is (1a) (1b) where ρ = ρA + ρB and C = CA + CB.
The minus sign indicates that the concentration gradient is neg-ative in the direction of diffusion. The proportionality factor Dv is the mass diffusivity or the diffusion coefficient. The total mass flux and molar flux are due to the average velocity of the mixture plus the diffusive flux: (2a) (2b) where v is the mass average velocity of the mixture and v is the molar average velocity.
Bird et al. (1960) present an analysis of Equations (1a) and (1b).
Equations (1a) and (1b) are equivalent forms of Fick’s law. The equation used depends on the problem and individual preference.
This chapter emphasizes mass analysis rather than molar analysis.
However, all results can be converted to the molar form using the relation CB ≡ ρB /MB.
Fick’s Law for Dilute Mixtures In many mass diffusion problems, component B is dilute; the den-sity of component B is small compared to the density of the mixture.
In this case, Equation (1a) can be written as (3) when ρB << ρ and ρA ≈ ρ.
Equation (3) can be used without significant error for water vapor diffusing through air at atmospheric pressure and a temperature less than 27°C. In this case, ρB < 0.02ρ, where ρB is the density of water vapor and ρ is the density of moist air (air and water vapor mixture).
The error in JB caused by replacing ρ[d(ρB/ρ)/dy] with dρB/dy is less than 2%. At temperatures below 60°C where ρB < 0.10ρ, Equation (3) can still be used if errors in JB as great as 10% are tolerable.
Fick’s Law for Mass Diffusion Through Solids or Stagnant Fluids (Stationary Media) Fick’s law can be simplified for cases of dilute mass diffusion in solids, stagnant liquids, or stagnant gases. In these cases, ρB « ρ and v ≈ 0, which yields the following approximate result: (4) The preparation of this chapter is assigned to TC 1.3, Heat Transfer and Fluid Flow.
JB ρ – Dv d ρB ρ ⁄ ( ) di ---------------------JA – = = JB CDv d CB C ⁄ ( ) di -----------------------– JA – = = m · B ″ m · B ″ m · B ″ ρBv ρDv d ρB ρ ⁄ ( ) dy ---------------------– = m · B ″ CBv CDv d CB C ⁄ ( ) dy -----------------------– = JB Dv – dρB dy ---------= m · B ″ JB Dv – = dρB dy ---------= 5.2 2001 ASHRAE Fundamentals Handbook (SI) Fick’s Law for Ideal Gases with Negligible Temperature Gradient For cases of dilute mass diffusion, Fick’s law can be written in terms of partial pressure gradient instead of concentration gradient.
When gas B can be approximated as ideal, (5) and when the gradient in T is small, Equation (3) can be written as (6a) or (6b) If v ≈ 0, Equation (4) may be written as (7a) or (7b) The partial pressure gradient formulation for mass transfer anal-ysis has been used extensively; this is unfortunate because the pres-sure formulation [Equations (6) and (7)] applies only to cases where one component is dilute, the fluid closely approximates an ideal gas, and the temperature gradient has a negligible effect. The density (or concentration) gradient formulation expressed in Equations (1) through (4) is more general and can be applied to a wider range of mass transfer problems, including cases where neither component is dilute [Equation (1)]. The gases need not be ideal, nor the tempera-ture gradient negligible. Consequently, this chapter emphasizes the density formulation.
Diffusion Coefficient For a binary mixture, the diffusion coefficient Dv is a function of temperature, pressure, and composition. Experimental measure-ments of Dv for most binary mixtures are limited in range and accu-racy. Table 1 gives a few experimental values for diffusion of some gases in air. For more detailed tables, see the section on Bibliogra-phy at the end of this chapter.
In the absence of data, use equations developed from (1) theory or (2) theory with constants adjusted from limited experimental data. For binary gas mixtures at low pressure, Dv is inversely pro-portional to pressure, increases with increasing temperature, and is almost independent of composition for a given gas pair. Bird et al.
(1960) present the following equation, developed from kinetic the-ory and corresponding states arguments, for estimating Dv at pres-sures less than 0.1pc min: (8) where Dv = diffusion coefficient, mm2/s a = constant, dimensionless b = constant, dimensionless T = absolute temperature, K p = pressure, kPa M = relative molecular mass, kg/kg mol The subscripts cA and cB refer to the critical states of the two gases.
Analysis of experimental data gives the following values of the con-stants a and b: For nonpolar gas pairs, a = 0.1280 and b = 1.823 For water vapor with a nonpolar gas, a = 0.1697 and b = 2.334 A nonpolar gas is one for which the intermolecular forces are independent of the relative orientation of molecules, depending only on the separation distance from each other. Air, composed of nonpolar gases O2 and N2, is nonpolar.
Equation (8) is stated to agree with experimental data at atmo-spheric pressure to within about 8% (Bird et al. 1960).
The mass diffusivity Dv for binary mixtures at low pressure is predictable within about 10% by kinetic theory (Reid et al.
1987).
(9) where σAB = characteristic molecular diameter, nm ΩD, AB = temperature function, dimensionless Dv is in mm2/s, p in kPa, and T in kelvins. If the gas molecules of A and B are considered rigid spheres having diameters σA and σB [and σAB = (σA/2) + (σB/2)], all expressed in nanometres, the dimensionless function ΩD, AB equals unity. More realistic mod-els for the molecules having intermolecular forces of attraction and repulsion lead to values of ΩD, AB that are functions of tem-perature. Reid et al. (1987) present tabulations of this quantity.
These results show that Dv increases as the 2.0 power of T at low temperatures and as the 1.65 power of T at very high temperatures.
The diffusion coefficient of moist air has been calculated for Equation (8) using a simplified intermolecular potential field func-tion for water vapor and air (Mason and Monchick 1965).
The following is an empirical equation for mass diffusivity of water vapor in air up to 1100°C (Sherwood and Pigford 1952): (10) where Dv is in mm2/s, p in kPa, and T in kelvins.
Table 1 Mass Diffusivities for Gases in Aira Gas Dv, mm2/s Ammonia 27.9 Benzene 8.8 Carbon dioxide 16.5 Ethanol 11.9 Hydrogen 41.3 Oxygen 20.6 Water vapor 25.5 aGases at 25°C and 101.325 kPa.
pB ρBRuT MB ----------------CBRuT = = JB MBDv RuT --------------- dpB dy ---------– = JB Dv RuT --------- dpB dy ---------– = m · B ″ JB MBDv RuT --------------- dpB dy ---------– = = m · B ″ JB Dv RuT --------- dpB dy -----------– = = Dv a T TcA TcB + ------------------------------ b 1 MA --------1 MB -------- + = pcApcB ( )1 3 ⁄ TcATcB ( )5 12 ⁄ p ----------------------------------------------------------------× Dv 0.5320 T1.5 p σAB ( )2ΩD AB , --------------------------------------- 1 MA --------1 MB -------- + = Dv 0.926 p -------------T 2.5 T 245 + ------------------ = Mass Transfer 5.3 Diffusion of One Gas Through a Second Stagnant Gas Figure 1 shows diffusion of one gas through a second stagnant gas. Water vapor diffuses from the liquid surface into surrounding stationary air. It is assumed that local equilibrium exists through the gas mixture, that the gases are ideal, and that the Gibbs-Dalton law is valid, which implies that the temperature gradient has a negligible effect. Diffusion of water vapor is due to concentration gradient and is given by Equation (6a). There is a continuous gas phase, so the mixture pressure p is constant, and the Gibbs-Dalton law yields (11a) or (11b) The partial pressure gradient of the water vapor causes a partial pressure gradient of the air such that or (12) Air, then, diffuses toward the liquid water interface. Because it can-not be absorbed there, a bulk velocity v of the gas mixture is estab-lished in a direction away from the liquid surface, so that the net transport of air is zero (i.e., the air is stagnant): (13) The bulk velocity v transports not only air but also water vapor away from the interface. Therefore, the total rate of water vapor dif-fusion is (14) Substituting for the velocity v from Equation (13) and using Equations (11b) and (12) gives (15) Integration yields (16a) or (16b) where (17) PAm is the logarithmic mean density factor of the stagnant air.
The pressure distribution for this type of diffusion is illustrated in Figure 2. Stagnant refers to the net behavior of the air; it does not move because the bulk flow exactly offsets diffusion. The term PAm in Equation (16b) approximately equals unity for dilute mixtures such as water vapor in air at near atmospheric conditions. This con-dition makes it possible to simplify Equations (16) and implies that in the case of dilute mixtures, the partial pressure distribution curves in Figure 2 are straight lines.
Example 1. A vertical tube of 25 mm diameter is partially filled with water so that the distance from the water surface to the open end of the tube is 60 mm, as shown in Figure 1. Perfectly dried air is blown over the open tube end, and the complete system is at a constant temperature of 15°C.
In 200 h of steady operation, 2.15 g of water evaporates from the tube.
The total pressure of the system is 101.325 kPa. Using these data, (a) calculate the mass diffusivity of water vapor in air, and (b) compare this experimental result with that from Equation (10).
Solution: (a) The mass diffusion flux of water vapor from the water surface is The cross-sectional area of a 25 mm diameter tube is π(12.5)2 = 491 mm2. Therefore, = 0.00608 g/(m2·s). The partial densities are determined with the aid of the psychrometric tables.
Fig. 1 Diffusion of Water Vapor Through Stagnant Air pA pB + p constant = = ρA MA --------ρB MB --------+ p RuT ---------constant = = dpA dy --------- dpB dy ---------– = 1 MA -------- dρA dy --------- 1 MB -------- dρB dy ---------– = m · A ″ D – v dρA dy ---------ρAv + 0 = = m · B ″ D – v dρB dy ---------ρBv + = m · B ″ DvMBp ρARuT ------------------ dρA dy ---------= m · B ″ DvMBp RuT ------------------ρAL ρA0 ⁄ ( ) ln yL y0 – -------------------------------- = m · B ″ D – vPAm ρBL ρB0 – yL y0 – ------------------------ = PAm p pAL ---------ρAL ρAL ρA0 ⁄ ( ) ln ρAL ρA0 – --------------------------------≡ Fig. 2 Pressure Profiles for Diffusion of Water Vapor Through Stagnant Air m · B 2.15 200 ⁄ 0.01075 g/h = = m · B ″ ρBL 0; ρB0 12.8 g/m3 = = ρAL 1.225 kg/m3; ρA0 1.204 kg/m3 = = 5.4 2001 ASHRAE Fundamentals Handbook (SI) Because p = pAL = 101.325 kPa, the logarithmic mean density factor [Equation (17)] is The mass diffusivity is now computed from Equation (16b) as (b) By Equation (10), with p = 101.325 kPa and T = 15 + 273 = 288 K, Neglecting the correction factor PAm for this example gives a dif-ference of less than 1% between the calculated experimental and empirically predicted values of Dv.
Equimolar Counterdiffusion Figure 3 shows two large chambers, both containing an ideal gas mixture of two components A and B (e.g., air and water vapor) at the same total pressure p and temperature T. The two chambers are con-nected by a duct of length L and cross-sectional area Acs. Partial pressure pB is higher in the left chamber, and partial pressure pA is higher in the right chamber. The partial pressure differences cause component B to migrate to the right and component A to migrate to the left.
At steady state, the molar flows of A and B must be equal, but in the opposite direction, or (18) because the total molar concentration C must stay the same in both chambers if p and T remain constant. Since the molar fluxes are the same in both directions, the molar average velocity v = 0. Thus, Equation (7b) can be used to calculate the molar flux of B (or A): (19) or (20) or (21) Example 2. One large room is maintained at 22°C (295 K), 101.3 kPa, 80% rh. A 20 m long duct with cross-sectional area of 0.15 m2 connects the room to another large room at 22°C, 101.3 kPa, 10% rh. What is the rate of water vapor diffusion between the two rooms?
Solution: Let air be component A and water vapor be component B.
Equation (21) can be used to calculate the mass flow of water vapor B.
Equation (10) can be used to calculate the diffusivity.
From a psychrometric table (Table 3, Chapter 6), the saturated vapor pressure at 22°C is 2.645 kPa. The vapor pressure difference pB0 − pBL is Then, Equation (21) gives Molecular Diffusion in Liquids and Solids Because of the greater density, diffusion is slower in liquids than in gases. No satisfactory molecular theories have been developed for calculating diffusion coefficients. The limited measured values of Dv show that, unlike for gas mixtures at low pressures, the diffu-sion coefficient for liquids varies appreciably with concentration.
Reasoning largely from analogy to the case of one-dimensional diffusion in gases and employing Fick’s law as expressed by Equa-tion (4) gives (22) Equation (22) expresses the steady-state diffusion of the solute B through the solvent A in terms of the molal concentration difference of the solute at two locations separated by the distance ∆y = y2 − y1.
Bird et al. (1960), Hirschfelder et al. (1954), Sherwood and Pigford (1952), Reid and Sherwood (1966), Treybal (1980), and Eckert and Drake (1972) provide equations and tables for evaluating Dv. Hir-schfelder et al. (1954) provide comprehensive treatment of the molecular developments.
Diffusion through a solid when the solute is dissolved to form a homogeneous solid solution is known as structure-insensitive dif-fusion (Treybal 1980). This solid diffusion closely parallels diffu-sion through fluids, and Equation (22) can be applied to one-dimensional steady-state problems. Values of mass diffusivity are generally lower than they are for liquids and vary with temperature.
The diffusion of a gas mixture through a porous medium is com-mon (e.g., the diffusion of an air-vapor mixture through porous insulation). The vapor diffuses through the air along the tortuous narrow passages within the porous medium. The mass flux is a func-tion of the vapor pressure gradient and diffusivity as indicated in Equation (7a). It is also a function of the structure of the pathways within the porous medium and is therefore called structure-sensi-tive diffusion. All of these factors are taken into account in the fol-lowing version of Equation (7a): (23) where is called the permeability of the porous medium. Chapter 23 presents this topic in more depth.
PAm 1.225 1.225 1.204 ⁄ ( ) ln 1.225 1.204 – -----------------------------------------1.009 = = Dv m · B ″ – yL y0 – ( ) PAm ρBL ρB0 – ( ) ---------------------------------------0.00608 ( ) – 0.060 ( ) 106 ( ) 1.009 ( ) 0 12.8 – ( ) ------------------------------------------------------------= = 28.2 mm2 s ⁄ = Dv 0.926 101.325 ------------------- 2882.5 288 245 + ------------------------ 24.1 mm2 s ⁄ = = Fig. 3 Equimolar Counterdiffusion m · A ″ m · B ″ + 0 = m · B ″ Dv – RuT --------- dpB dy ---------= m · B AcsDv RuT --------------- pB0 pBL – L -----------------------= m · B MBAcsDv RuT ----------------------- pB0 pBL – L -----------------------= Dv 0.926 101.3 -------------2952.5 295 245 + ------------------------ 25.3 mm2/h = = pB0 pBL – 0.8 0.1 – ( )2.645 kPa 1.85 kPa = = m · b 18 0.15 25.3 106 ⁄ ( ) × 8.314 295 × ---------------------------------------------------1.85 20 ----------2.58 10 9 – × kg/s = = m · B ″ Dv ρB1 ρB2 – y2 y1 – ------------------------ = m · B ″ µ dpB dy ---------– = µ Mass Transfer 5.5 CONVECTION OF MASS Convection of mass involves the mass transfer mechanisms of molecular diffusion and bulk fluid motion. Fluid motion in the region adjacent to a mass transfer surface may be laminar or turbu-lent, depending on geometry and flow conditions.
Mass Transfer Coefficient Convective mass transfer is analogous to convective heat trans-fer where geometry and boundary conditions are similar. The anal-ogy holds for both laminar and turbulent flows and applies to both external and internal flow problems.
Mass Transfer Coefficients for External Flows. Most external convective mass transfer problems can be solved with an appropri-ate formulation that relates the mass transfer flux (to or from an interfacial surface) to the concentration difference across the boundary layer illustrated in Figure 4. This formulation gives rise to the convective mass transfer coefficient, defined as (24) where hM = local external mass transfer coefficient, m/s = mass flux of gas B from surface, kg/(m2·s) ρBi = density of gas B at interface (saturation density), kg/m3 ρB∞= density of component B outside boundary layer, kg/m3 If ρBi and ρB∞ are constant over the entire interfacial surface, the mass transfer rate from the surface can be expressed as (25) where is the average mass transfer coefficient, defined as (26) Mass Transfer Coefficients for Internal Flows. Most internal convective mass transfer problems, such as those that occur in chan-nels or in the cores of dehumidification coils, can be solved if an appropriate expression is available to relate the mass transfer flux (to or from the interfacial surface) to the difference between the con-centration at the surface and the bulk concentration in the channel, as illustrated in Figure 5. This formulation leads to the definition of the mass transfer coefficient for internal flows: (27) where hM = internal mass transfer coefficient, m/s = mass flux of gas B at interfacial surface, kg/(m2·s) ρBi = density of gas B at interfacial surface, kg/m3 ρBb ≡ = bulk density of gas B at location x ≡ = average velocity of gas B at location x, m/s Acs = cross-sectional area of channel at station x, m2 uB = velocity of component B in x direction, m/s ρB = density distribution of component B at station x, kg/m3 Often, it is easier to obtain the bulk density of gas B from (28) where = mass flow rate of component B at station x = 0, kg/s A = interfacial area of channel between station x = 0 and station x = x, m2 Equation (28) can be derived from the preceding definitions. The major problem is the determination of . If, however, the analysis is restricted to cases where B is dilute and concentration gradients of B in the x direction are negligibly small, . Component B is swept along in the x direction with an average velocity equal to the average velocity of the dilute mixture.
Analogy Between Convective Heat and Mass Transfer Most expressions for the convective mass transfer coefficient hM are determined from expressions for the convective heat transfer coefficient h.
For problems in internal and external flow where mass transfer occurs at the convective surface and where component B is dilute, it is shown by Bird et al. (1960) and Incropera and DeWitt (1996) that the Nusselt and Sherwood numbers are defined as follows: (29) (30) and (31) (32) Fig. 4 Nomenclature for Convective Mass Transfer from External Surface at Location x Where Surface is Impermeable to Gas A hM m · B ″ ρBi ρB∞ – ------------------------≡ m · ""B ″ m · B ″ hM ρBi ρB∞ – ( ) = hM hM 1 A --- hm A d A ∫ ≡ Fig. 5 Nomenclature for Convective Mass Transfer from Internal Surface Impermeable to Gas A hM m · B ″ ρBi ρBb – -----------------------≡ m · B ″ 1 uBAcs ⁄ ( ) Acs ∫ uBρB dAcs uB 1 Acs ⁄ ( ) A ∫uB dAcs ρBb m · Bo m · B ″ A d A ∫ + uBAcs -------------------------------------= m · Bo uB uB u ≈ Nu f X Y Z Pr Re , , , , ( ) = Sh f X Y Z Sc Re , , , , ( ) = Nu g Pr Re , ( ) = Sh g Sc Re , ( ) = 5.6 2001 ASHRAE Fundamentals Handbook (SI) where the function f is the same in Equations (29) and (30), and the function g is the same in Equations (31) and (32). The quantities Pr and Sc are dimensionless Prandtl and Schmidt numbers, respec-tively, as defined in the section on Symbols. The primary restric-tions on the analogy are that the surface shapes are the same and that the temperature boundary conditions are analogous to the density distribution boundary conditions for component B when cast in dimensionless form. Several primary factors prevent the analogy from being perfect. In some cases, the Nusselt number was derived for smooth surfaces. Many mass transfer problems involve wavy, droplet-like, or roughened surfaces. Many Nusselt number relations are obtained for constant temperature surfaces. Sometimes ρBi is not constant over the entire surface because of varying saturation con-ditions and the possibility of surface dryout.
In all mass transfer problems, there is some blowing or suction at the surface because of the condensation, evaporation, or transpira-tion of component B. In most cases, this blowing/suction phenom-enon has little effect on the Sherwood number, but the analogy should be examined closely if vi/u∞ > 0.01 or > 0.01, espe-cially if the Reynolds number is large.
Example 3. Use the analogy expressed in Equations (31) and (32) to solve the following problem. An expression for heat transfer from a constant temperature flat plate in laminar flow is (33) Sc = 0.35, Dv = 3.6 × 10−5 m2/s, and Pr = 0.708 for the given condi-tions; determine the mass transfer rate and temperature of the water-wetted flat plate in Figure 6 using the heat/mass transfer analogy.
Solution: To solve the problem, properties should be evaluated at film conditions. However, since the plate temperature and the interfacial water vapor density are not known, a first estimate will be obtained assuming the plate ti1 to be at 25°C. The plate Reynolds number is The plate is entirely in laminar flow, since the transitional Reynolds number is about 5 × 105. Using the mass transfer analogy given by Equations (31) and (32), Equation (33) yields From the definition of the Sherwood number, The psychrometric tables give a humidity ratio W of 0.0121 at 25°C and 60% rh. Therefore, From steam tables, the saturation density for water at 25°C is Therefore, the mass transfer rate from the double-sided plate is This mass rate, transformed from the liquid state to the vapor state, requires the following heat rate to the plate to maintain the evaporation: To obtain a second estimate of the wetted plate temperature in this type of problem, the following criteria are used. Calculate the ti neces-sary to provide a heat rate of qi1. If this temperature tiq1 is above the dew-point temperature tid, set the second estimate at ti2 = (tiq1 + ti1)/2.
If tiq1 is below the dew-point temperature, set ti2 = (tid + ti1)/2. For this problem, the dew point is tid = 14°C.
Obtaining the second estimate of the plate temperature requires an approximate value of the heat transfer coefficient.
From the definition of the Nusselt number, Therefore, the second estimate for the plate temperature is This temperature is below the dew-point temperature; therefore, The second estimate of the film temperature is The next iteration on the solution is as follows: The free stream density of the water vapor has been evaluated. The density of the water vapor at the plate surface is the saturation density at 19.5°C. This temperature is above the dew-point temperature; therefore, vi u ⁄ NuL 0.664Pr1 3 ⁄ ReL 1 2 ⁄ = Fig. 6 Water-Saturated Flat Plate in Flowing Airstream ReL1 ρu∞L µ --------------1.166 kg m3 ⁄ ( ) 10 m s ⁄ ( ) 0.1 m ( ) 1.965 10 5 – kg m s ⋅ ( ) ⁄ × [ ] ---------------------------------------------------------------------------------59 340 = = = ShL1 0.664 Sc1 3 ⁄ ReL 1 2 ⁄ = 0.664 0.35 ( )1 3 ⁄ 59 340 ( )1 2 ⁄ 114 = = hM1 ShL1Dv L ⁄ 114 ( ) 3.6 10 5 – m2 s ⁄ × ( ) 0.1 m ( ) ⁄ 0.04104 m/s = = = ρB∞ 0.0121ρA∞ 0.0121 ( ) 1.166 kg m3 ⁄ ( ) 0.01411 kg/m3 = = = ρBi1 0.02352 kg/m3 = m · B1 hM1A ρBi ρB∞ – ( ) = 0.04104 m/s ( ) 0.1 m 1.5 m 2 × × ( ) = 0.02352 kg m3 ⁄ 0.01411 kg m3 ⁄ – ( ) × 1.159 4 – ×10 kg s ⁄ 0.1159 = g/s = qi1 m · B1hfg 0.1159 g/s ( ) 2443 kJ/kg ( ) 283.1 W = = = NuL1 0.664Pr1 3 ⁄ ReL 1 2 ⁄ 0.664 0.708 ( )1 3 ⁄ 59 340 ( )1 2 ⁄ = = 144.2 = h1 NuL1k L ⁄ 144.2 ( ) 0.0261 W m K ⋅ ( ) ⁄ [ ] 0.1 m ⁄ = = 37.6 W m2 K ⋅ ( ) ⁄ = tiq1 t∞ qi1 h1A ( ) ⁄ – = 25°C 283.1 W 37.6 W m2 K ⋅ ( ) ⁄ 2 0.1 m × 1.5 m × ( ) × [ ] ⁄ { } – = 25°C 25°C – 0°C = = ti2 14°C 25°C + ( ) 2 ⁄ 19.5°C = = tf 2 ti2 t∞ + ( ) 2 ⁄ 19.5°C 25°C + ( ) 2 ⁄ 22.25°C = = = ReL2 61 010 = ShL2 0.664 0.393 ( )1 3 ⁄ 61 010 ( )1 2 ⁄ 120 = = hM2 120 ( ) 3.38 10 5 – × ( ) 0.1 ⁄ 0.04056 m/s = = ρBi2 0.01374 ( ) 1.183 kg m3 ⁄ ( ) 0.01625 kg m3 ⁄ = = A 2 0.1 × 1.5 0.3 m2 = × = m · B2 0.04056 m/s ( ) 0.3 m2 ( ) 0.01625 kg m3 ⁄ 0.01411 kg m3 ⁄ – ( ) = 2.604 10 5 – kg/s × 0.02604 g s ⁄ = = qi2 0.02604 g s ⁄ ( ) 2458 J kg ⁄ ( ) 64.01 W = = NuL2 0.664 0.709 ( )1 3 ⁄ 61 010 ( )1 2 ⁄ 146 = = h2 146 ( ) 0.02584 ( ) 0.1 ⁄ 37.73 W m2 K ⋅ ( ) ⁄ = = tiq2 25°C 64.01 W ( ) 37.73 0.3 × ( ) ⁄ [ ] – 19.34°C = = ti3 ti2 tiq2 + ( ) 2 ⁄ 19.5 19.34 + ( ) 2 ⁄ 19.42°C = = = Mass Transfer 5.7 This is approximately the same result as that obtained in the previ-ous iteration. Therefore, the problem solution is The kind of similarity between heat and mass transfer that results in Equation (29) through Equation (32) can also be shown to exist between heat and momentum transfer. Chilton and Colburn (1934) used this similarity to relate Nusselt number to friction factor by the analogy (34) where n = 2/3, St = Nu/(Re Pr) is the Stanton number, and jH is the Chilton-Colburn j-factor for heat transfer. Substituting Sh for Nu and Sc for Pr in Equations (31) and (32) gives the Chilton-Colburn j-factor for mass transfer, jD: (35) where Stm = ShPAM/(Re Sc) is the Stanton number for mass transfer.
Equations (34) and (35) are called the Chilton-Colburn j-factor analogy.
The power of the Chilton-Colburn j-factor analogy is repre-sented in Figures 7 through 10. Figure 7 plots various experimen-tal values of jD from a flat plate with flow parallel to the plate surface. The solid line, which represents the data to near perfec-tion, is actually f /2 from Blasius’ solution of laminar flow on a flat plate (left-hand portion of the solid line) and Goldstein’s solution for a turbulent boundary layer (right-hand portion). The right-hand portion of the solid line also represents McAdams’ (1954) correlation of turbulent flow heat transfer coefficient for a flat plate.
A wetted-wall column is a vertical tube in which a thin liquid film adheres to the tube surface and exchanges mass by evaporation or absorption with a gas flowing through the tube. Figure 8 illus-trates typical data on vaporization in wetted-wall columns, plotted as jD versus Re. The spread of the points with variation in µ/ρDv results from Gilliland’s finding of an exponent of 0.56, not 2/3, representing the effect of the Schmidt number. Gilliland’s equation can be written as follows: (36) Similarly, McAdams’ (1954) equation for heat transfer in pipes can be expressed as (37) This is represented by the dash-dot curve in Figure 8, which falls below the mass transfer data. The curve f /2 representing friction in smooth tubes is the upper, solid curve.
Data for the evaporation of liquids from single cylinders into gas streams flowing transversely to the cylinders’ axes are shown in Figure 9. Although the dash-dot line on Figure 9 represents the data, it is actually taken from McAdams (1954) as representative of a large collection of data on heat transfer to single cylinders placed transverse to airstreams. To compare these data with fric-tion, it is necessary to distinguish between total drag and skin fric-tion. Since the analogies are based on skin friction, the normal pressure drag must be subtracted from the measured total drag. At Re = 1000, the skin friction is 12.6% of the total drag; at Re = 31 600, it is only 1.9%. Consequently, the values of f /2 at a high Reynolds number, obtained by the difference, are subject to con-siderable error.
In Figure 10, data on the evaporation of water into air for single spheres are presented. The solid line, which best represents these data, agrees with the dashed line representing McAdams’ correla-tion for heat transfer to spheres. These results cannot be compared with friction or momentum transfer because total drag has not been allocated to skin friction and normal pressure drag. Application of these data to air-water contacting devices such as air washers and spray cooling towers is well substantiated.
When the temperature of the heat exchanger surface in contact with moist air is below the dew-point temperature of the air, vapor condensation occurs. Typically, the air dry-bulb temperature and ti 19.5°C = m · B 0.0260 g/s = Fig. 7 Mass Transfer from Flat Plate jH Nu Re Pr 1 n – ( ) --------------------------St Prn f 2 --= = = jD Sh Re Sc 1 n – ( ) ---------------------------StmScn f 2 --= = = jD 0.023Re 0.17 – µ ρDv ---------- 0.56 – = jH 0.023Re 0.20 – cpµ k --------- 0.7 – = Fig. 8 Vaporization and Absorption in Wetted-Wall Column 5.8 2001 ASHRAE Fundamentals Handbook (SI) humidity ratio both decrease as the air flows through the exchanger.
Therefore, sensible and latent heat transfer occur simultaneously.
This process is similar to one that occurs in a spray dehumidifier and can be analyzed using the same procedure; however, this is not gen-erally done.
Cooling coil analysis and design are complicated by the problem of determining transport coefficients h, hM, and f. It would be con-venient if heat transfer and friction data for dry heating coils could be used with the Colburn analogy to obtain the mass transfer coeffi-cients. However, this approach is not always reliable, and work by Guillory and McQuiston (1973) and Helmer (1974) shows that the analogy is not consistently true. Figure 11 shows j-factors for a sim-ple parallel plate exchanger for different surface conditions with sen-sible heat transfer. Mass transfer j-factors and the friction factors exhibit the same behavior. Dry surface j-factors fall below those obtained under dehumidifying conditions with the surface wet. At low Reynolds numbers, the boundary layer grows quickly; the drop-lets are soon covered and have little effect on the flow field. As the Reynolds number is increased, the boundary layer becomes thin and more of the total flow field is exposed to the droplets. The roughness caused by the droplets induces mixing and larger j-factors. The data in Figure 11 cannot be applied to all surfaces because the length of the flow channel is also an important variable. However, the water collecting on the surface is mainly responsible for breakdown of the j-factor analogy. The j-factor analogy is approximately true when the surface conditions are identical. Under some conditions, it is possi-ble to obtain a film of condensate on the surface instead of droplets.
Guillory and McQuiston (1973) and Helmer (1974) related dry sen-sible j- and f-factors to those for wetted dehumidifying surfaces.
The equality of jH, jD, and f/2 for certain streamline shapes at low mass transfer rates has experimental verification. For flow past bluff objects, jH and jD are much smaller than f/2, based on total pressure drag. The heat and mass transfer, however, still relate in a useful way by equating jH and jD.
Example 4. Using solid cylinders of volatile solids (e.g., naphthalene, camphor, dichlorobenzene) with airflow normal to these cylinders, Bedingfield and Drew (1950) found that the ratio between the heat and mass transfer coefficients could be closely correlated by the following relation: For completely dry air at 21°C flowing at a velocity of 9.5 m/s over a wet-bulb thermometer of diameter d = 7.5 mm, determine the heat and mass transfer coefficients from Figure 9 and compare their ratio with the Bedingfield-Drew relation.
Solution: For dry air at 21°C and standard pressure, ρ = 1.198 kg/m3, µ = 1.82 × 10−5 kg/(s· m), k = 0.02581 W/(m·K), and cp = 1.006 kJ/(kg·K). From Equation (10), Dv = 25.13 mm2/s. Therefore, From Figure 9 at Reda = 4700, read jH = 0.0089, jD = 0.010. From Equations (34) and (35), Fig. 9 Mass Transfer from Single Cylinders in Crossflow Fig. 10 Mass Transfer from Single Spheres Fig. 11 Sensible Heat Transfer j-Factors for Parallel Plate Exchanger h ρhM ----------1230 J kg K ⋅ ( ) ⁄ [ ] µ ρDv ---------- 0.56 = Reda ρu∞d µ ⁄ 1.198 9.5 7.5 × × 1000 1.82 × 10 5 – × ( ) ⁄ 4690 = = = Pr cpµ k ⁄ 1.006 1.82 × 10 5 – × 1000 × 0.02581 ( ) ⁄ 0.709 = = = Sc µ ρDv ⁄ 1.82 = 10 5 – × 106 × 1.198 25.13 × ( ) ⁄ 0.605 = = h jHρcpu∞ Pr ( )2 3 ⁄ ⁄ = 0.0089 1.198 × 1.006 × 9.5 × 1000 0.709 ( )2 3 ⁄ ⁄ × = 128 W m2 K ⋅ ( ) ⁄ = hM jDu∞ Sc ( ) ⁄ 2 3 ⁄ 0.010 9.5 × 0.605 ( )2 3 ⁄ ⁄ = = 0.133 m/s = h ρhM ⁄ 128 1.198 0.133 × ( ) ⁄ 803 J kg K ⋅ ( ) ⁄ = = Mass Transfer 5.9 From the Bedingfield-Drew relation, Equations (34) and (35) are call the Reynolds analogy when Pr = Sc = 1. This suggests that h/ρhM = cp = 1006 J/(kg·K). This close agreement is because the ratio Sc/Pr is 0.605/0.709 or 0.85, so that the exponent of these numbers has little effect on the ratio of the transfer coefficients.
The extensive developments for calculating heat transfer coeffi-cients can be applied to calculate mass transfer coefficients under similar geometrical and flow conditions using the j-factor analogy.
For example, Table 6 of Chapter 3 lists equations for calculating heat transfer coefficients for flow inside and normal to pipes. Each equation can be used for mass transfer coefficient calculations by equating jH and jD and imposing the same restriction to each stated in Table 6 of Chapter 3. Similarly, mass transfer experiments often replace corresponding heat transfer experiments with complex geometries where exact boundary conditions are difficult to model (Sparrow and Ohadi 1987a, 1987b).
The j-factor analogy is useful only at low mass transfer rates. As the rate of mass transfer increases, the movement of matter normal to the transfer surface increases the convective velocity. For exam-ple, if a gas is blown from many small holes in a flat plate placed parallel to an airstream, the boundary layer thickens, and resistance to both mass and heat transfer increases with increasing blowing rate. Heat transfer data are usually collected at zero or, at least, insignificant mass transfer rates. Therefore, if such data are to be valid for a mass transfer process, the mass transfer rate (i.e., the blowing) must be low.
The j-factor relationship jH = jD can still be valid at high mass transfer rates, but neither jH nor jD can be represented by data at zero mass transfer conditions. Eckert and Drake (1972) and Chap-ter 24 of Bird et al. (1960) have detailed information on high mass transfer rates.
Lewis Relation Heat and mass transfer coefficients are satisfactorily related at the same Reynolds number by equating the Chilton-Colburn j-fac-tors. Comparing Equations (34) and (35) gives Inserting the definitions of St, Pr, Stm, and Sc gives or (38) The quantity α/Dv is the Lewis number Le. Its magnitude expresses relative rates of propagation of energy and mass within a system. It is fairly insensitive to temperature variation. For air and water vapor mixtures, the ratio is (0.60/0.71) or 0.845, and (0.845)2/3 is 0.894. At low diffusion rates, where the heat-mass transfer analogy is valid, PAm is essentially unity. Therefore, for air and water vapor mixtures, (39) The ratio of the heat transfer coefficient to the mass transfer coef-ficient is equal to the specific heat per unit volume of the mixture at constant pressure. This relation [Equation (39)] is usually called the Lewis relation and is nearly true for air and water vapor at low mass transfer rates. It is generally not true for other gas mixtures because the ratio Le of thermal to vapor diffusivity can differ from unity. The agreement between wet-bulb temperature and adiabatic saturation temperature is a direct result of the nearness of the Lewis number to unity for air and water vapor.
The Lewis relation is valid in turbulent flow whether or not α/Dv equals 1 because eddy diffusion in turbulent flow involves the same mixing action for heat exchange as for mass exchange, and this action overwhelms any molecular diffusion. Deviations from the Lewis relation are, therefore, due to a laminar boundary layer or a laminar sublayer and buffer zone where molecular transport phe-nomena are the controlling factors.
SIMULTANEOUS HEAT AND MASS TRANSFER BETWEEN WATER-WETTED SURFACES AND AIR A simplified method used to solve simultaneous heat and mass transfer problems was developed using the Lewis relation, and it gives satisfactory results for most air-conditioning processes. Ex-trapolation to very high mass transfer rates, where the simple heat-mass transfer analogy is not valid, will lead to erroneous results.
Enthalpy Potential The water vapor concentration in the air is the humidity ratio W, defined as (40) A mass transfer coefficient is defined using W as the driving potential: (41) where the coefficient KM is in kg/(s·m2). For dilute mixtures, ρAi ≅ ρA∞; that is, the partial mass density of dry air changes by only a small percentage between interface and free stream conditions.
Therefore, (42) where ρAm = mean density of dry air, kg/m3. Comparing Equation (42) with Equation (24) shows that (43) The humid specific heat cpm of the airstream is, by definition (Mason and Monchick 1965), (44a) or (44b) where cpm is in kJ/(kgda·K).
Substituting from Equations (43) and (44b) into Equation (39) gives h ρhM ⁄ 1230 0.605 ( )0.56 928 J kg K ⋅ ( ) ⁄ = = St Prn f 2 --StmScn = = h ρcpu ------------ cpµ k --------- 2 3 ⁄ hMPAm u -----------------µ ρDv ---------- 2 3 ⁄ = h hMρcp ----------------PAm µ ρDv ⁄ ( ) cpµ k ⁄ ( ) ----------------------2 3 ⁄ = PAm α Dv ⁄ ( )2 3 ⁄ = h hMρcp ----------------1 ≈ W ρB ρA ------≡ m · B ″ KM Wi W∞ – ( ) = m · B ″ KM ρAm ---------- ρBi ρ∞ – ( ) = hM KM ρAm ----------= cpm 1 W∞ + ( )cp = cpm ρ ρA∞ ---------- cp = 5.10 2001 ASHRAE Fundamentals Handbook (SI) (45) since ρAm ≅ ρA∞ because of the small change in dry-air density.
Using a mass transfer coefficient with the humidity ratio as the driv-ing force, the Lewis relation becomes ratio of heat to mass transfer coefficient equals humid specific heat.
For the plate humidifier illustrated in Figure 6, the total heat transfer from liquid to interface is (46) Using the definitions of the heat and mass transfer coefficients gives (47) Assuming Equation (45) is valid gives (48) The enthalpy of the air is approximately (49) The enthalpy hs of the water vapor can be expressed by the ideal gas law as (50) where the base of enthalpy is taken as saturated water at temperature to. Choosing to = 0°C to correspond with the base of the dry-air enthalpy gives (51) If small changes in the latent heat of vaporization of water with tem-perature are neglected when comparing Equations (49) and (51), the total heat transfer can be written as (52) Where the driving potential for heat transfer is temperature dif-ference and the driving potential for mass transfer is mass concen-tration or partial pressure, the driving potential for simultaneous transfer of heat and mass in an air water-vapor mixture is, to a close approximation, enthalpy.
Basic Equations for Direct-Contact Equipment Air-conditioning equipment can be classified as (1) having direct contact between air and water used as a cooling or heating fluid or (2) having the heating or cooling fluid separated from the airstream by a solid wall. Examples of the former are air washers and cooling towers; an example of the latter is a direct-expansion refrigerant (or water) cooling and dehumidifying coil. In both cases, the airstream is in contact with a water surface. Direct contact implies contact directly with the cooling (or heating) fluid. In the dehumidifying coil, the contact with the condensate removed from the airstream is direct, but it is indirect with the refrigerant flowing inside the tubes of the coil. These two cases are treated separately because the sur-face areas of direct-contact equipment cannot be evaluated.
For the direct-contact spray chamber air washer of cross-sec-tional area Acs and length l (Figure 12), the steady mass flow rate of dry air per unit cross-sectional area is (53) and the corresponding mass flux of water flowing parallel with the air is (54) where = mass flow rate of air, kg/s Ga = mass flux or flow rate per unit cross-sectional area for air, kg/(s·m2) = mass flow rate of liquid, kg/s GL = mass flux or flow rate per unit cross-sectional area for liquid, kg/(s·m2) Because water is evaporating or condensing, GL changes by an amount dGL in a differential length dl of the chamber. Similar changes occur in temperature, humidity ratio, enthalpy, and other properties.
Because evaluating the true surface area in direct-contact equip-ment is difficult, it is common to work on a unit volume basis. If aH and aM are the area of heat transfer and mass transfer surface per unit of chamber volume, respectively, the total surface areas for heat and mass transfer are (55) The basic equations for the process occurring in the differential length dl can be written for 1. Mass transfer (56) That is, the water evaporated, the moisture increase of the air, and the mass transfer rate are all equal.
2. Heat transfer to air (57) 3. Total energy transfer to air (58) Assuming aH = aM and Le = 1, and neglecting small variations in hfg, Equation (58) reduces to (59) hρAm KMρA∞cpm ---------------------------1 h KMcpm -----------------≈ = q″ qA ″ m · B ″ hfg + = q″ h ti t∞ – ( ) KM Wi W∞ – ( )hfg + = q″ KM cpm ti t∞ – ( ) Wi W∞ – ( )hfg + = h cpat Whs + = hs cps t to – ( ) hfgo + = h cpa Wcps + ( )t Whfgo + cpmt Whfgo + = = q″ KM hi h∞ – ( ) = Fig. 12 Air Washer Spray Chamber m · a Acs ⁄ Ga = m · L Acs ⁄ GL = m · a m · L AH aHAcsl and AM aMAcsl = = dGL – Ga dW KMaM Wi W – ( )dl = = Gacpm dta haaH ti ta – ( )dl = Ga cpm dta hfgo dW + ( ) KMaM Wi W – ( )hfg haaH ti ta – ( ) + [ ] dl = Ga dh KMaM hi h – ( ) dl = Mass Transfer 5.11 The heat and mass transfer areas of spray chambers are assumed to be identical (aH = aM). Where packing materials, such as wood slats or Raschig rings, are used, the two areas may be considerably different because the packing may not be wet uniformly. The valid-ity of the Lewis relation was discussed previously. It is not neces-sary to account for the small changes in latent heat hfg after making the two previous assumptions.
4. Energy balance (60) A minus sign refers to parallel flow of air and water; a plus sign refers to counterflow (water flow in the opposite direction from airflow).
The water flow rate changes between inlet and outlet as a result of the mass transfer. For exact energy balance, the term (cLtLdGL) should be added to the right side of Equation (60). The percentage change in GL is quite small in usual applications of air-conditioning equipment and, therefore, can be ignored.
5. Heat transfer to water (61) Equations (56) to (61) are the basic relations for solution of simultaneous heat and mass transfer processes in direct-contact air-conditioning equipment.
To facilitate the use of these relations in equipment design or per-formance, three other equations can be extracted from the above set.
Combining Equations (59), (60), and (61) gives (62) Equation (62) relates the enthalpy potential for the total heat transfer through the gas film to the temperature potential for this same trans-fer through the liquid film. Physical reasoning leads to the conclu-sion that this ratio is proportional to the ratio of gas film resistance (1/KM) to liquid film resistance (1/hL). Combining Equations (57), (59), and (45) gives (63) Similarly, combining Equations (56), (57), and (45) gives (64) Equation (64) indicates that at any cross section in the spray cham-ber, the instantaneous slope of the air path dW/dta on a psychromet-ric chart is determined by a straight line connecting the air state with the interface saturation state at that cross section. In Figure 13, state 1 represents the state of the air entering the parallel flow air washer chamber of Figure 12. The washer is operating as a heating and humidifying apparatus so that the interface saturation state of the water at air inlet is the state designated 1i. Therefore, the initial slope of the air path is along a line directed from state 1 to state 1i.
As the air is heated, the water cools and the interface temperature drops. Corresponding air states and interface saturation states are indicated by the letters a, b, c, and d in Figure 13. In each instance, the air path is directed toward the associated interface state. The interface states are derived from Equations (60) and (62). Equation (60) describes how the air enthalpy changes with water tempera-ture; Equation (62) describes how the interface saturation state changes to accommodate this change in air and water conditions.
The solution for the interface state on the normal psychrometric chart of Figure 13 can be determined either by trial and error from Equations (60) and (62) or by a complex graphical procedure (Kusuda 1957).
Air Washers Air washers are direct-contact apparatus used to (1) simulta-neously change the temperature and humidity content of air passing through the chamber and (2) remove air contaminants such as dust and odors. Adiabatic spray washers, which have no external heating or chilling source, are used to cool and humidify air. Chilled spray air washers have an external chiller to cool and dehumidify air.
Heated spray air washers, whose external heating source provides additional energy for evaporation of water, are used to humidify and possibly heat air.
Example 5. A parallel flow air washer with the following design condi-tions is to be designed (see Figure 12).
Water temperature at inlet tL1 = 35°C Water temperature at outlet tL2 = 23.9°C Air temperature at inlet ta1 = 18.3°C Air wet-bulb at inlet t ′ a1 = 7.2°C Air mass flow rate per unit area Ga = 1.628 kg/(s·m2) Spray ratio GL/Ga = 0.70 Air heat transfer coefficient per cubic metre of chamber volume haaH = 1.34 kW/(m3·K) Liquid heat transfer coefficient per cubic metre of chamber volume hLaH = 16.77 kW/(m3·K) Air volumetric flow rate Q = 3.07 m3/s Solution: The air mass flow rate = 3.07 × 1.20 = 3.68 kg/s; the required spray chamber cross-sectional area is, then, Acs = /Ga = 3.68/1.628 = 2.26 m2. The mass transfer coefficient is given by the Lewis relation [Equation (45)] as Figure 14 shows the enthalpy-temperature psychrometric chart with the graphical solution for the interface states and the air path through the washer spray chamber. The solution proceeds as follows: Gadh GLcL dtL ± = G ± LcL dtL hLaH tL ti – ( ) dl = h hi – tL ti – ------------- hLaH KMaM ---------------– hL KM --------– = = dh dta -------h hi – ta ti – -------------= dW dta --------W Wi – ta ti – -----------------= Fig. 13 Air Washer Humidification Process on Psychrometric Chart m · a m · a KMaM haaH ( ) cpm ⁄ 1.34 1.005 ⁄ 1.33kg m3 s ⋅ ( ) ⁄ = = = 5.12 2001 ASHRAE Fundamentals Handbook (SI) 1. Enter bottom of chart with t ′ a1 of 7.2°C, and follow up to saturation curve to establish air enthalpy h1 of 41.1 kJ/kg. Extend this enthalpy line to intersect initial air temperature ta1 of 18.3°C (state 1 of air) and initial water temperature tL1 of 35°C at point A. (Note that the temperature scale is used for both air and water temperatures.) 2. Through point A, construct the energy balance line A-B with a slope of Point B is determined by intersection with the leaving water tem-perature tL2 = 23.9°C. The negative slope here is a consequence of the parallel flow, which results in the air-water mixture’s approach-ing, but not reaching, the common saturation state s. (The line A-B has no physical significance in representing any air state on the psychrometric chart. It is merely a construction line in the graphi-cal solution.) 3. Through point A, construct the tie-line A-1i having a slope of The intersection of this line with the saturation curve gives the initial interface state 1i at the chamber inlet. (Note how the energy balance line and tie-line, representing Equations (60) and (62), combine for a simple graphical solution on Figure 14 for the interface state.) 4. The initial slope of the air path can now be constructed, according to Equation (63), drawing line 1-a toward the initial interface state 1i.
(The length of the line 1-a will depend on the degree of accuracy required in the solution and the rate at which the slope of the air path is changing.) 5. Construct the horizontal line a-M locating the point M on the energy-balance line. Draw a new tie-line (slope of −12.6 as before) from M to ai locating interface state ai. Continue the air path from a to b by directing it toward the new interface state ai. (Note that the change in slope of the air path from 1-a to a-b is quite small, justifying the path incremental lengths used.) 6. Continue in the manner of step 5 until point 2, the final state of the air leaving the chamber, is reached. In this example, six steps are used in the graphical construction with the following results: The final state of the air leaving the washer is ta2 = 22.4°C and h2 = 73.6 kJ/kg (wet-bulb temperature ta2 ′ = 19.4°C).
7. The final step involves calculating the required length of the spray chamber. From Equation (59), The integral is evaluated graphically by plotting 1/(hi − h) versus h as shown in Figure 15. Any satisfactory graphical method can be used to evaluate the area under the curve. Simpson’s rule with four equal increments of ∆h equal to 8.2 gives The design length is, therefore, l = (1.628/1.33)(0.975) = 1.19 m.
The method used in Example 5 can also be used to predict the performance of existing direct-contact equipment and can deter-mine the transfer coefficients when performance data from test runs are available. By knowing the water and air temperatures entering and leaving the chamber and the spray ratio, it is possible, by trial and error, to determine the proper slope of the tie-line necessary to achieve the measured final air state. The tie-line slope gives the ratio Fig. 14 Graphical Solution for Air-State Path in Parallel Flow Air Washer dh dtL ------- cLGL Ga --------------– 2.95 – = = h hi – tL ti – ------------- hLaH KMaM ---------------– 16.77 1.33 -------------– 12.6 – = = = State 1 a b c d 2 tL 35 32.8 30.6 28.3 26.1 23.9 h 41.1 47.7 54.3 60.8 67.4 73.9 ti 29.2 27.9 26.7 25.4 24.2 22.9 hi 114.4 108.0 102.3 96.9 91.3 86.4 ta 18.3 19.3 20.3 21.1 21.9 22.4 Fig. 15 Graphical Solution of ∫dh/(hi − h) l Ga KMaM --------------- h d hi h – ( ) ------------------1 2 ∫ = N h d hi h – ( ) ------------------1 2 ∫ h ∆ 3 ⁄ ( ) y1 4y2 2y3 4y4 y5 + + + + ( ) ≈ = N 8.2 3 ⁄ ( )[0.0136 4 0.0167 × ( ) 2 0.0238 × ( ) + + = 4 0.0372 × ( ) 0.0800] 0.975 = + + Mass Transfer 5.13 hLaH/KMaM; KMaM is found from the integral relationship in Exam-ple 5 from the known chamber length l.
Additional descriptions of air spray washers and general perfor-mance criteria are given in Chapter 19 of the 2000 ASHRAE Hand-book—Systems and Equipment.
Cooling Towers A cooling tower is a direct-contact heat exchanger in which waste heat picked up by the cooling water from a refrigerator, air conditioner, or industrial process is transferred to atmospheric air by cooling the water. Cooling is achieved by breaking up the water flow to provide a large water surface for air, moving by natural or forced convection through the tower, to contact the water. Cooling towers may be counterflow, crossflow, or a combination of both.
The temperature of the water leaving the tower and the packing depth needed to achieve the desired leaving water temperature are of primary interest for design. Therefore, the mass and energy bal-ance equations are based on an overall coefficient K, which is based on (1) the enthalpy driving force due to h at the bulk water temper-ature and (2) neglecting the film resistance. Combining Equations (59) and (60) and using the parameters described above yields (65) or (66) Chapter 36 of the 2000 ASHRAE Handbook—Systems and Equipment covers cooling tower design in detail.
Cooling and Dehumidifying Coils When water vapor is condensed out of an airstream onto an ex-tended surface (finned) cooling coil, the simultaneous heat and mass transfer problem can be solved by the same procedure set forth for di-rect-contact equipment. The basic equations are the same, except that the true surface area of the coil A is known and the problem does not have to be solved on a unit volume basis. Therefore, if in Equations (56), (57), and (59) aM dl or aH dl is replaced by dA/Acs, these equa-tions become the basic heat, mass, and total energy transfer equations for indirect-contact equipment such as dehumidifying coils. The en-ergy balance shown by Equation (60) remains unchanged. The heat transfer from the interface to the refrigerant now encounters the com-bined resistances of the condensate film (RL = 1/hL); the metal wall and fins, if any (Rm); and the refrigerant film (Rr = A/hrAr). If this combined resistance is designated as Ri = RL + Rm + Rr = 1/Ui, Equa-tion (61) becomes, for a coil dehumidifier, (67) (plus sign for counterflow, minus sign for parallel flow).
The tie-line slope is then (68) Figure 16 illustrates the graphical solution on a psychrometric chart for the air path through a dehumidifying coil with a constant refrigerant temperature. Because the tie-line slope is infinite in this case, the energy balance line is vertical. The corresponding inter-face states and air states are denoted by the same letter symbols, and the solution follows the same procedure as in Example 5.
If the problem is to determine the required coil surface area for a given performance, the area is computed by the following relation: (69) This graphical solution on the psychrometric chart automatically determines whether any part of the coil is dry. Thus, in the example illustrated in Figure 16, the entering air at state 1 initially encounters an interface saturation state 1i, clearly below its dew-point temper-ature td1, so the coil immediately becomes wet. Had the graphical technique resulted in an initial interface state above the dew-point temperature of the entering air, the coil would be initially dry. The air would then follow a constant humidity ratio line (the sloping W = constant lines on the chart) until the interface state reached the air dew-point temperature.
Mizushina et al. (1959) developed this method not only for water vapor and air, but also for other vapor-gas mixtures. Chapter 21 of the 2000 ASHRAE Handbook—Systems and Equipment shows another related method, based on ARI Standard 410, of determining air-cooling and dehumidifying coil performance.
SYMBOLS a = constant, dimensionless; or surface area per unit volume, m2/m3 A = surface area, m2 Acs = cross-sectional area, m2 b = exponent, dimensionless cL = specific heat of liquid, kJ/(kg·K) cp = specific heat at constant pressure, kJ/(kg·K) cpm = specific heat of moist air at constant pressure, kJ/(kgda ·K) C = molal concentration of solute in solvent, mol/m3 d = diameter, m Dv = diffusion coefficient (mass diffusivity), mm2/s f = Fanning friction factor, dimensionless G = mass flux, flow rate per unit of cross-sectional area, kg/(s·m2) h = enthalpy, kJ/kg; or heat transfer coefficient, W/(m2·K) GLcL dt KMaM hi h – ( )dl Ga dh = = Ka dV h′ ha – ( ) Acs ------------------------------------= KaV m · L ----------cL t d h′ ha – ( ) ---------------------t1 t2 ∫ = m · LcL dtL ± Ui tL ti – ( )dA = h hi – tL ti – ------------- Ui KM --------+ − = Fig. 16 Graphical Solution for Air-State Path in Dehumidifying Coil with Constant Refrigerant Temperature A m · a KM --------h d hi h – ( ) ------------------1 2 ∫ = 5.14 2001 ASHRAE Fundamentals Handbook (SI) hfg = enthalpy of vaporization, kJ/kg hM = mass transfer coefficient, m/s jD = Colburn mass transfer group = Sh/(Re·Sc1/3), dimensionless jH = Colburn heat transfer group = Nu/(Re·Pr1/3), dimensionless J = diffusive mass flux, kg/(s·m2) J = diffusive molar flux, mol/(s·m2) k = thermal conductivity, W/(m·K) KM = mass transfer coefficient, kg/(s·m2) l = length, m L = characteristic length, m L/G = liquid-to-air mass flow ratio Le = Lewis number = α/Dv, dimensionless = rate of mass transfer, m/s = mass flux, kg/(s·m2) = molar flux, mol/(s·m2) M = relative molecular mass, kg/mol Nu = Nusselt number = hL/k, dimensionless p = pressure, kPa PAm = logarithmic mean density factor Pr = Prandtl number = cpµ/k, dimensionless q = rate of heat transfer, W = heat flux per unit area, W/m2 Q = volumetric flow rate, m3/s Ri = combined thermal resistance, m2·K/W RL = thermal resistance of condensate film, m2·K/W Rm = thermal resistance across metal wall and fins, m2·K/W Rr = thermal resistance of refrigerant film, m2·K/W Ru = universal gas constant = 8.314 kJ/(mol·K) Re = Reynolds number = , dimensionless Sc = Schmidt number = µ/ρDv, dimensionless Sh = Sherwood number = hML/Dv, dimensionless St = Stanton number = , dimensionless Stm = mass transfer Stanton number = , dimensionless t = temperature, °C T = absolute temperature, K u = velocity in x direction, m/s Ui = overall conductance from refrigerant to air-water interface for dehumidifying coil, W/(m2·K) v = velocity in y direction, m/s vi = velocity normal to mass transfer surface for component i, m/s V = fluid stream velocity, m/s W = humidity ratio, kg of water vapor per kg of dry air x, y, z = coordinate direction, m X,Y, Z = coordinate direction, dimensionless α = thermal diffusivity = k/ρcp, m2/s εD = eddy mass diffusivity, m2/s θ = dimensionless time parameter µ = absolute (dynamic) viscosity, kg/(m·s) = permeability, mg/(s·m·Pa) ν = kinematic viscosity, m2/s ρ = mass density or concentration, kg/m3 σ = characteristic molecular diameter, nm τ = time τi = shear stress in the x-y coordinate plane, N/m2 ω = mass fraction, kg/kg ΩD,AB = temperature function in Equation (9) Subscripts a = air property Am = logarithmic mean A = gas component of binary mixture B = the more dilute gas component of binary mixture c = critical state da = dry air property or air-side transfer quantity H = heat transfer quantity i = air-water interface value L = liquid m = mean value or metal M = mass transfer quantity o = property evaluated at 0°C s = water vapor property or transport quantity w = water vapor ∞= property of main fluid stream Superscripts = on molar basis −= average value ′ = wet bulb REFERENCES Bedingfield, G.H., Jr. and T.B. Drew. 1950. Analogy between heat transfer and mass transfer—A psychrometric study. Industrial and Engineering Chemistry 42:1164.
Bird, R.B., W.E. Stewart, and E.N. Lightfoot. 1960. Transport phenomena.
John Wiley and Sons, New York.
Chilton, T.H. and A.P. Colburn. 1934. Mass transfer (absorption) coeffi-cients—Prediction from data on heat transfer and fluid friction. Indus-trial and Engineering Chemistry 26 (November):1183.
Guillory, J.L. and F.C. McQuiston. 1973. An experimental investigation of air dehumidification in a parallel plate heat exchanger. ASHRAE Trans-actions 79(2):146.
Helmer, W.A. 1974. Condensing water vapor—Airflow in a parallel plate heat exchanger. Ph.D. thesis, Purdue University, West Lafayette, IN.
Hirschfelder, J.O., C.F. Curtiss, and R.B. Bird. 1954. Molecular theory of gases and liquids. John Wiley and Sons, New York.
Incropera, F.P. and D.P. DeWitt. 1996. Fundamentals of heat and mass transfer, 4th ed. John Wiley and Sons, New York.
Kusuda, T. 1957. Graphical method simplifies determination of aircoil, wet-heat-transfer surface temperature. Refrigerating Engineering 65:41.
Mason, E.A. and L. Monchick. 1965. Survey of the equation of state and transport properties of moist gases. Humidity and Moisture 3. Reinhold Publishing Corporation, New York.
McAdams, W.H. 1954. Heat transmission, 3rd ed. McGraw-Hill, New York.
Mizushina, T., N. Hashimoto, and M. Nakajima. 1959. Design of cooler con-densers for gas-vapour mixtures. Chemical Engineering Science 9:195.
Ohadi, M.M. and E.M. Sparrow. 1989. Heat transfer in a straight tube situ-ated downstream of a bend. International Journal of Heat and Mass Transfer 32(2):201-12.
Reid, R.C. and T.K. Sherwood. 1966. The properties of gases and liquids: Their estimation and correlation, 2nd ed. McGraw-Hill, New York, pp. 520-43.
Reid, R.C., J.M. Prausnitz, and B.E. Poling. 1987. The properties of gases and liquids, 4th ed. McGraw-Hill, New York, pp.21-78.
Sherwood, T.K. and R.L. Pigford. 1952. Absorption and extraction.
McGraw-Hill, New York, pp. 1-28.
Sparrow, E.M. and M.M. Ohadi. 1987a. Comparison of turbulent thermal entrance regions for pipe flows with developed velocity and velocity developing from a sharp-edged inlet. ASME Transactions, Journal of Heat Transfer 109:1028-30.
Sparrow, E.M. and M.M. Ohadi. 1987b. Numerical and experimental studies of turbulent flow in a tube. Numerical Heat Transfer 11:461-76.
Treybal, R.E. 1980. Mass transfer operations, 3rd ed. McGraw-Hill, New York.
BIBLIOGRAPHY Bennett, C.O. and J.E. Myers. 1982. Momentum, heat and mass transfer, 3rd ed. McGraw-Hill, New York.
DeWitt, D.P. and E.L. Cussler. 1984. Diffusion, mass transfer in fluid sys-tems. Cambridge University Press, UK.
Eckert, E.R.G. and R.M. Drake, Jr. 1972. Analysis of heat and mass transfer.
McGraw-Hill, New York.
Geankopolis, C.J. 1993. Transport processes and unit operations, 3rd ed.
Prentice Hall, Englewood Cliffs, NJ.
Kays, W.M. and M.E. Crawford. 1993. Convective heat and mass transfer.
McGraw-Hill, New York.
Mikielviez, J. and A.M.A. Rageb. 1995. Simple theroretical approach to direct-contact condensation on subcooled liquid film. International Journal of Heat and Mass Transfer 38(3):557.
m · m · ″ m · ″ q″ ρuL µ ⁄ h ρcpu ⁄ hM PAm u ⁄ µ 6.1 CHAPTER 6 PSYCHROMETRICS Composition of Dry and Moist Air ........................................... 6.1 United States Standard Atmosphere ......................................... 6.1 Thermodynamic Properties of Moist Air ................................. 6.2 Thermodynamic Properties of Water at Saturation .............................................................................. 6.2 Humidity Parameters ................................................................ 6.8 Perfect Gas Relationships for Dry and Moist Air ................................................................................ 6.8 Thermodynamic Wet-Bulb Temperature and Dew-Point Temperature ........................................................ 6.9 Numerical Calculation of Moist Air Properties ...................... 6.10 Psychrometric Charts ............................................................. 6.12 Typical Air-Conditioning Processes ....................................... 6.12 Transport Properties of Moist Air .......................................... 6.16 References for Air, Water, and Steam Properties ................... 6.16 Symbols ................................................................................... 6.17 SYCHROMETRICS deals with the thermodynamic properties Pof moist air and uses these properties to analyze conditions and processes involving moist air.
Hyland and Wexler (1983a,b) developed formulas for thermody-namic properties of moist air and water. However, perfect gas rela-tions can be used instead of these formulas in most air-conditioning problems. Kuehn et al. (1998) showed that errors are less than 0.7% in calculating humidity ratio, enthalpy, and specific volume of sat-urated air at standard atmospheric pressure for a temperature range of −50 to 50°C. Furthermore, these errors decrease with decreasing pressure.
This chapter discusses perfect gas relations and describes their use in common air-conditioning problems. The formulas developed by Hyland and Wexler (1983a) and discussed by Olivieri (1996) may be used where greater precision is required.
COMPOSITION OF DRY AND MOIST AIR Atmospheric air contains many gaseous components as well as water vapor and miscellaneous contaminants (e.g., smoke, pollen, and gaseous pollutants not normally present in free air far from pol-lution sources).
Dry air exists when all water vapor and contaminants have been removed from atmospheric air. The composition of dry air is relatively constant, but small variations in the amounts of indi-vidual components occur with time, geographic location, and altitude. Harrison (1965) lists the approximate percentage com-position of dry air by volume as: nitrogen, 78.084; oxygen, 20.9476; argon, 0.934; carbon dioxide, 0.0314; neon, 0.001818; helium, 0.000524; methane, 0.00015; sulfur dioxide, 0 to 0.0001; hydrogen, 0.00005; and minor components such as krypton, xenon, and ozone, 0.0002. The relative molecular mass of all components for dry air is 28.9645, based on the carbon-12 scale (Harrison 1965). The gas constant for dry air, based on the car-bon-12 scale, is Rda = 8314.41/28.9645 = 287.055 J/(kg·K) (1) Moist air is a binary (two-component) mixture of dry air and water vapor. The amount of water vapor in moist air varies from zero (dry air) to a maximum that depends on temperature and pres-sure. The latter condition refers to saturation, a state of neutral equilibrium between moist air and the condensed water phase (liq-uid or solid). Unless otherwise stated, saturation refers to a flat inter-face surface between the moist air and the condensed phase.
Saturation conditions will change when the interface radius is very small such as with ultrafine water droplets. The relative molecular mass of water is 18.01528 on the carbon-12 scale. The gas constant for water vapor is Rw = 8314.41/18.01528 = 461.520 J/(kg·K) (2) UNITED STATES STANDARD ATMOSPHERE The temperature and barometric pressure of atmospheric air vary considerably with altitude as well as with local geographic and weather conditions. The standard atmosphere gives a standard of reference for estimating properties at various altitudes. At sea level, standard temperature is 15°C; standard barometric pressure is 101.325 kPa. The temperature is assumed to decrease linearly with increasing altitude throughout the troposphere (lower atmosphere), and to be constant in the lower reaches of the stratosphere. The lower atmosphere is assumed to consist of dry air that behaves as a perfect gas. Gravity is also assumed constant at the standard value, 9.806 65 m/s2. Table 1 summarizes property data for altitudes to 10 000 m.
The preparation of this chapter is assigned to TC 1.1, Thermodynamics and Psychrometrics.
Table 1 Standard Atmospheric Data for Altitudes to 10 000 m Altitude, m Temperature, °C Pressure, kPa −500 18.2 107.478 0 15.0 101.325 500 11.8 95.461 1 000 8.5 89.875 1 500 5.2 84.556 2 000 2.0 79.495 2 500 −1.2 74.682 3 000 −4.5 70.108 4 000 −11.0 61.640 5 000 −17.5 54.020 6 000 −24.0 47.181 7 000 −30.5 41.061 8 000 −37.0 35.600 9 000 −43.5 30.742 10 000 −50 26.436 12 000 −63 19.284 14 000 −76 13.786 16 000 −89 9.632 18 000 −102 6.556 20 000 −115 4.328 6.2 2001 ASHRAE Fundamentals Handbook (SI) The pressure values in Table 1 may be calculated from (3) The equation for temperature as a function of altitude is given as (4) where Z = altitude, m p = barometric pressure, kPa t = temperature, °C Equations (3) and (4) are accurate from −5000 m to 11 000 m.
For higher altitudes, comprehensive tables of barometric pressure and other physical properties of the standard atmosphere can be found in NASA (1976).
THERMODYNAMIC PROPERTIES OF MOIST AIR Table 2, developed from formulas by Hyland and Wexler (1983a,b), shows values of thermodynamic properties of moist air based on the thermodynamic temperature scale. This ideal scale differs slightly from practical temperature scales used for physical measurements. For example, the standard boiling point for water (at 101.325 kPa) occurs at 99.97°C on this scale rather than at the tra-ditional value of 100°C. Most measurements are currently based on the International Temperature Scale of 1990 (ITS-90) (Preston-Thomas 1990).
The following paragraphs briefly describe each column of Table 2: t = Celsius temperature, based on thermodynamic temperature scale and expressed relative to absolute temperature T in kelvins (K) by the following relation: Ws = humidity ratio at saturation, condition at which gaseous phase (moist air) exists in equilibrium with condensed phase (liquid or solid) at given temperature and pressure (standard atmospheric pressure). At given values of temperature and pressure, humidity ratio W can have any value from zero to Ws.
vda = specific volume of dry air, m3/kg (dry air).
vas = vs − vda, difference between specific volume of moist air at satu-ration and that of dry air itself, m3/kg (dry air), at same pressure and temperature.
vs = specific volume of moist air at saturation, m3/kg (dry air).
hda = specific enthalpy of dry air, kJ/kg (dry air). In Table 2, hda has been assigned a value of 0 at 0°C and standard atmospheric pres-sure.
has = hs − hda, difference between specific enthalpy of moist air at sat-uration and that of dry air itself, kJ/kg (dry air), at same pressure and temperature.
hs = specific enthalpy of moist air at saturation, kJ/kg (dry air).
sda = specific entropy of dry air, kJ/(kg·K) (dry air). In Table 2, sda has been assigned a value of 0 at 0°C and standard atmospheric pres-sure.
sas = ss − sda, difference between specific entropy of moist air at satu-ration and that of dry air itself, kJ/(kg·K) (dry air), at same pres-sure and temperature.
ss = specific entropy of moist air at saturation kJ/(kg·K) (dry air).
hw = specific enthalpy of condensed water (liquid or solid) in equi-librium with saturated moist air at specified temperature and pressure, kJ/kg (water). In Table 2, hw is assigned a value of 0 at its triple point (0.01°C) and saturation pressure.
Note that hw is greater than the steam-table enthalpy of satu-rated pure condensed phase by the amount of enthalpy increase governed by the pressure increase from saturation pressure to 101.325 kPa, plus influences from presence of air.
sw = specific entropy of condensed water (liquid or solid) in equi-librium with saturated air, kJ/(kg·K) (water); sw differs from entropy of pure water at saturation pressure, similar to hw.
ps = vapor pressure of water in saturated moist air, kPa. Pressure ps differs negligibly from saturation vapor pressure of pure water pws for conditions shown. Consequently, values of ps can be used at same pressure and temperature in equations where pws appears. Pressure ps is defined as ps = xwsp, where xws is mole fraction of water vapor in moist air saturated with water at tem-perature t and pressure p, and where p is total barometric pres-sure of moist air.
THERMODYNAMIC PROPERTIES OF WATER AT SATURATION Table 3 shows thermodynamic properties of water at saturation for temperatures from −60 to 160°C, calculated by the formulations described by Hyland and Wexler (1983b). Symbols in the table fol-low standard steam table nomenclature. These properties are based on the thermodynamic temperature scale. The enthalpy and entropy of saturated liquid water are both assigned the value zero at the tri-ple point, 0.01°C. Between the triple-point and critical-point tem-peratures of water, two states—liquid and vapor—may coexist in equilibrium. These states are called saturated liquid and saturated vapor.
The water vapor saturation pressure is required to determine a number of moist air properties, principally the saturation humidity ratio. Values may be obtained from Table 3 or calculated from the following formulas (Hyland and Wexler 1983b).
The saturation pressure over ice for the temperature range of −100 to 0°C is given by (5) where C1 = −5.674 535 9 E+03 C2 = 6.392 524 7 E+00 C3 = −9.677 843 0 E–03 C4 = 6.221 570 1 E−07 C5 = 2.074 782 5 E−09 C6 = −9.484 024 0 E−13 C7 = 4.163 501 9 E+00 The saturation pressure over liquid water for the temperature range of 0 to 200°C is given by (6) where C8 = −5.800 220 6 E+03 C9 = 1.391 499 3 E+00 C10 = −4.864 023 9 E−02 C11 = 4.176 476 8 E−05 C12 = −1.445 209 3 E−08 C13 = 6.545 967 3 E+00 In both Equations (5) and (6), ln = natural logarithm pws = saturation pressure, Pa T = absolute temperature, K = °C + 273.15 The coefficients of Equations (5) and (6) have been derived from the Hyland-Wexler equations. Due to rounding errors in the deriva-tions and in some computers’ calculating precision, the results obtained from Equations (5) and (6) may not agree precisely with Table 3 values.
p 101.325 1 2.25577 10 5 – Z × – ( ) 5.2559 = t 15 0.0065Z – = T t 273.15 + = pws ln C1 T ⁄ C2 C3T C4T2 C5T3 + + + + = C6T4 C7 T ln + + pws ln C8 T ⁄ C9 C10T C11T2 + + + = C12T3 C13 T ln + + Psychrometrics 6.3 Table 2 Thermodynamic Properties of Moist Air at Standard Atmospheric Pressure, 101.325 kPa Temp., °C t Humidity Ratio, kg(w)/kg(da) Ws Specific Volume, m3/kg (dry air) Specific Enthalpy, kJ/kg (dry air) Specific Entropy, kJ/(kg ·K) (dry air) Condensed Water Temp., °C t Specific Enthalpy, kJ/kg hw Specific Entropy, kJ/(kg·K) sw Vapor Pressure, kPa ps vda vas vs hda has hs sda sas ss −60 0.0000067 0.6027 0.0000 0.6027 −60.351 0.017 −60.334 −0.2495 0.0001 −0.2494 −446.29 −1.6854 0.00108 −60 −59 0.0000076 0.6056 0.0000 0.6056 −59.344 0.018 −59.326 −0.2448 0.0001 −0.2447 −444.63 −1.6776 0.00124 −59 −58 0.0000087 0.6084 0.0000 0.6084 −58.338 0.021 −58.317 −0.2401 0.0001 −0.2400 −442.95 −1.6698 0.00141 −58 −57 0.0000100 0.6113 0.0000 0.6113 −57.332 0.024 −57.308 −0.2354 0.0001 −0.2353 −441.27 −1.6620 0.00161 −57 −56 0.0000114 0.6141 0.0000 0.6141 −56.326 0.028 −56.298 −0.2308 0.0001 −0.2306 −439.58 −1.6542 0.00184 −56 −55 0.0000129 0.6170 0.0000 0.6170 −55.319 0.031 −55.288 −0.2261 0.0002 −0.2260 −437.89 −1.6464 0.00209 −55 −54 0.0000147 0.6198 0.0000 0.6198 −54.313 0.036 −54.278 −0.2215 0.0002 −0.2214 −436.19 −1.6386 0.00238 −54 −53 0.0000167 0.6226 0.0000 0.6227 −53.307 0.041 −53.267 −0.2170 0.0002 −0.2168 −434.48 −1.6308 0.00271 −53 −52 0.0000190 0.6255 0.0000 0.6255 −52.301 0.046 −52.255 −0.2124 0.0002 −0.2122 −432.76 −1.6230 0.00307 −52 −51 0.0000215 0.6283 0.0000 0.6284 −51.295 0.052 −51.243 −0.2079 0.0002 −0.2076 −431.03 −1.6153 0.00348 −51 −50 0.0000243 0.6312 0.0000 0.6312 −50.289 0.059 −50.230 −0.2033 0.0003 −0.2031 −429.30 −1.6075 0.00394 −50 −49 0.0000275 0.6340 0.0000 0.6341 −49.283 0.067 −49.216 −0.1988 0.0003 −0.1985 −427.56 −1.5997 0.00445 −49 −48 0.0000311 0.6369 0.0000 0.6369 −48.277 0.075 −48.202 −0.1944 0.0004 −0.1940 −425.82 −1.5919 0.00503 −48 −47 0.0000350 0.6397 0.0000 0.6398 −47.271 0.085 −47.186 −0.1899 0.0004 −0.1895 −424.06 −1.5842 0.00568 −47 −46 0.0000395 0.6426 0.0000 0.6426 −46.265 0.095 −46.170 −0.1855 0.0004 −0.1850 −422.30 −1.5764 0.00640 −46 −45 0.0000445 0.6454 0.0000 0.6455 −45.259 0.108 −45.151 −0.1811 0.0005 −0.1805 −420.54 −1.5686 0.00721 −45 −44 0.0000500 0.6483 0.0001 0.6483 −44.253 0.121 −44.132 −0.1767 0.0006 −0.1761 −418.76 −1.5609 0.00811 −44 −43 0.0000562 0.6511 0.0001 0.6512 −43.247 0.137 −43.111 −0.1723 0.0006 −0.1716 −416.98 −1.5531 0.00911 −43 −42 0.0000631 0.6540 0.0001 0.6540 −42.241 0.153 −42.088 −0.1679 0.0007 −0.1672 −415.19 −1.5453 0.01022 −42 −41 0.0000708 0.6568 0.0001 0.6569 −41.235 0.172 −41.063 −0.1636 0.0008 −0.1628 −413.39 −1.5376 0.01147 −41 −40 0.0000793 0.6597 0.0001 0.6597 −40.229 0.192 −40.037 −0.1592 0.0009 −0.1584 −411.59 −1.5298 0.01285 −40 −39 0.0000887 0.6625 0.0001 0.6626 −39.224 0.216 −39.007 −0.1549 0.0010 −0.1540 −409.77 −1.5221 0.01438 −39 −38 0.0000992 0.6653 0.0001 0.6654 −38.218 0.241 −37.976 −0.1507 0.0011 −0.1496 −407.96 −1.5143 0.01608 −38 −37 0.0001108 0.6682 0.0001 0.6683 −37.212 0.270 −36.942 −0.1464 0.0012 −0.1452 −406.13 −1.5066 0.01796 −37 −36 0.0001237 0.6710 0.0001 0.6712 −36.206 0.302 −35.905 −0.1421 0.0014 −0.1408 −404.29 −1.4988 0.02005 −36 −35 0.0001379 0.6739 0.0001 0.6740 −35.200 0.336 −34.864 −0.1379 0.0015 −0.1364 −402.45 −1.4911 0.02235 −35 −34 0.0001536 0.6767 0.0002 0.6769 −34.195 0.375 −33.820 −0.1337 0.0017 −0.1320 −400.60 −1.4833 0.02490 −34 −33 0.0001710 0.6796 0.0002 0.6798 −33.189 0.417 −32.772 −0.1295 0.0018 −0.1276 −398.75 −1.4756 0.02772 −33 −32 0.0001902 0.6824 0.0002 0.6826 −32.183 0.464 −31.718 −0.1253 0.0020 −0.1233 −396.89 −1.4678 0.03082 −32 −31 0.0002113 0.6853 0.0002 0.6855 −31.178 0.517 −30.661 −0.1212 0.0023 −0.1189 −395.01 −1.4601 0.03425 −31 −30 0.0002346 0.6881 0.0003 0.6884 −30.171 0.574 −29.597 −0.1170 0.0025 −0.1145 −393.14 −1.4524 0.03802 −30 −29 0.0002602 0.6909 0.0003 0.6912 −29.166 0.636 −28.529 −0.1129 0.0028 −0.1101 −391.25 −1.4446 0.04217 −29 −28 0.0002883 0.6938 0.0003 0.6941 −28.160 0.707 −27.454 −0.1088 0.0031 −0.1057 −389.36 −1.4369 0.04673 −28 −27 0.0003193 0.6966 0.0004 0.6970 −27.154 0.782 −26.372 −0.1047 0.0034 −0.1013 −387.46 −1.4291 0.05175 −27 −26 0.0003533 0.6995 0.0004 0.6999 −26.149 0.867 −25.282 −0.1006 0.0037 −0.0969 −385.55 −1.4214 0.05725 −26 −25 0.0003905 0.7023 0.0004 0.7028 −25.143 0.959 −24.184 −0.0965 0.0041 −0.0924 −383.63 −1.4137 0.06329 −25 −24 0.0004314 0.7052 0.0005 0.7057 −24.137 1.059 −23.078 −0.0925 0.0045 −0.0880 −381.71 −1.4059 0.06991 −24 −23 0.0004762 0.7080 0.0005 0.7086 −23.132 1.171 −21.961 −0.0885 0.0050 −0.0835 −379.78 −1.3982 0.07716 −23 −22 0.0005251 0.7109 0.0006 0.7115 −22.126 1.292 −20.834 −0.0845 0.0054 −0.0790 −377.84 −1.3905 0.08510 −22 −21 0.0005787 0.7137 0.0007 0.7144 −21.120 1.425 −19.695 −0.0805 0.0060 −0.0745 −375.90 −1.3828 0.09378 −21 −20 0.0006373 0.7165 0.0007 0.7173 −20.115 1.570 −18.545 −0.0765 0.0066 −0.0699 −373.95 −1.3750 0.10326 −20 −19 0.0007013 0.7194 0.0008 0.7202 −19.109 1.729 −17.380 −0.0725 0.0072 −0.0653 −371.99 −1.3673 0.11362 −19 −18 0.0007711 0.7222 0.0009 0.7231 −18.103 1.902 −16.201 −0.0686 0.0079 −0.0607 −370.02 −1.3596 0.12492 −18 −17 0.0008473 0.7251 0.0010 0.7261 −17.098 2.092 −15.006 −0.0646 0.0086 −0.0560 −368.04 −1.3518 0.13725 −17 −16 0.0009303 0.7279 0.0011 0.7290 −16.092 2.299 −13.793 −0.0607 0.0094 −0.0513 −366.06 −1.3441 0.15068 −16 −15 0.0010207 0.7308 0.0012 0.7320 −15.086 2.524 −12.562 −0.0568 0.0103 −0.0465 −364.07 −1.3364 0.16530 −15 −14 0.0011191 0.7336 0.0013 0.7349 −14.080 2.769 −11.311 −0.0529 0.0113 −0.0416 −362.07 −1.3287 0.18122 −14 −13 0.0012262 0.7364 0.0014 0.7379 −13.075 3.036 −10.039 −0.0490 0.0123 −0.0367 −360.07 −1.3210 0.19852 −13 −12 0.0013425 0.7393 0.0016 0.7409 −12.069 3.327 −8.742 −0.0452 0.0134 −0.0318 −358.06 −1.3132 0.21732 −12 −11 0.0014690 0.7421 0.0017 0.7439 −11.063 3.642 −7.421 −0.0413 0.0146 −0.0267 −356.04 −1.3055 0.23775 −11 −10 0.0016062 0.7450 0.0019 0.7469 −10.057 3.986 −6.072 −0.0375 0.0160 −0.0215 −354.01 −1.2978 0.25991 −10 −9 0.0017551 0.7478 0.0021 0.7499 −9.052 4.358 −4.693 −0.0337 0.0174 −0.0163 −351.97 −1.2901 0.28395 −9 −8 0.0019166 0.7507 0.0023 0.7530 −8.046 4.764 −3.283 −0.0299 0.0189 −0.0110 −349.93 −1.2824 0.30999 −8 −7 0.0020916 0.7535 0.0025 0.7560 −7.040 5.202 −1.838 −0.0261 0.0206 −0.0055 −347.88 −1.2746 0.33821 −7 −6 0.0022811 0.7563 0.0028 0.7591 −6.035 5.677 −0.357 −0.0223 0.0224 −0.0000 −345.82 −1.2669 0.36874 −6 −5 0.0024862 0.7592 0.0030 0.7622 −5.029 6.192 1.164 −0.0186 0.0243 −0.0057 −343.76 −1.2592 0.40178 −5 −4 0.0027081 0.7620 0.0033 0.7653 −4.023 6.751 2.728 −0.0148 0.0264 −0.0115 −341.69 −1.2515 0.43748 −4 −3 0.0029480 0.7649 0.0036 0.7685 −3.017 7.353 4.336 −0.0111 0.0286 −0.0175 −339.61 −1.2438 0.47606 −3 −2 0.0032074 0.7677 0.0039 0.7717 −2.011 8.007 5.995 −0.0074 0.0310 −0.0236 −337.52 −1.2361 0.51773 −2 −1 0.0034874 0.7705 0.0043 0.7749 −1.006 8.712 7.706 −0.0037 0.0336 −0.0299 −335.42 −1.2284 0.56268 −1 0 0.0037895 0.7734 0.0047 0.7781 0.000 9.473 9.473 0.0000 0.0364 0.0364 −333.32 −1.2206 0.61117 0 0 0.003789 0.7734 0.0047 0.7781 0.000 9.473 9.473 0.0000 0.0364 0.0364 0.06 −0.0001 0.6112 0 1 0.004076 0.7762 0.0051 0.7813 1.006 10.197 11.203 0.0037 0.0391 0.0427 4.28 0.0153 0.6571 1 2 0.004381 0.7791 0.0055 0.7845 2.012 10.970 12.982 0.0073 0.0419 0.0492 8.49 0.0306 0.7060 2 3 0.004707 0.7819 0.0059 0.7878 3.018 11.793 14.811 0.0110 0.0449 0.0559 12.70 0.0459 0.7581 3 4 0.005054 0.7848 0.0064 0.7911 4.024 12.672 16.696 0.0146 0.0480 0.0627 16.91 0.0611 0.8135 4 5 0.005424 0.7876 0.0068 0.7944 5.029 13.610 18.639 0.0182 0.0514 0.0697 21.12 0.0762 0.8725 5 6 0.005818 0.7904 0.0074 0.7978 6.036 14.608 20.644 0.0219 0.0550 0.0769 25.32 0.0913 0.9353 6 7 0.006237 0.7933 0.0079 0.8012 7.041 15.671 22.713 0.0255 0.0588 0.0843 29.52 0.1064 1.0020 7 8 0.006683 0.7961 0.0085 0.8046 8.047 16.805 24.852 0.0290 0.0628 0.0919 33.72 0.1213 1.0729 8 9 0.007157 0.7990 0.0092 0.8081 9.053 18.010 27.064 0.0326 0.0671 0.0997 37.92 0.1362 1.1481 9 10 0.007661 0.8018 0.0098 0.8116 10.059 19.293 29.352 0.0362 0.0717 0.1078 42.11 0.1511 1.2280 10 11 0.008197 0.8046 0.0106 0.8152 11.065 20.658 31.724 0.0397 0.0765 0.1162 46.31 0.1659 1.3128 11 12 0.008766 0.8075 0.0113 0.8188 12.071 22.108 34.179 0.0433 0.0816 0.1248 50.50 0.1806 1.4026 12 13 0.009370 0.8103 0.0122 0.8225 13.077 23.649 36.726 0.0468 0.0870 0.1337 54.69 0.1953 1.4979 13 Extrapolated to represent metastable equilibrium with undercooled liquid.
6.4 2001 ASHRAE Fundamentals Handbook (SI) 14 0.010012 0.8132 0.0131 0.8262 14.084 25.286 39.370 0.0503 0.0927 0.1430 58.88 0.2099 1.5987 14 15 0.010692 0.8160 0.0140 0.8300 15.090 27.023 42.113 0.0538 0.0987 0.1525 63.07 0.2244 1.7055 15 16 0.011413 0.8188 0.0150 0.8338 16.096 28.867 44.963 0.0573 0.1051 0.1624 67.26 0.2389 1.8185 16 17 0.012178 0.8217 0.0160 0.8377 17.102 30.824 47.926 0.0607 0.1119 0.1726 71.44 0.2534 1.9380 17 18 0.012989 0.8245 0.0172 0.8417 18.108 32.900 51.008 0.0642 0.1190 0.1832 75.63 0.2678 2.0643 18 19 0.013848 0.8274 0.0184 0.8457 19.114 35.101 54.216 0.0677 0.1266 0.1942 79.81 0.2821 2.1979 19 20 0.014758 0.8302 0.0196 0.8498 20.121 37.434 57.555 0.0711 0.1346 0.2057 84.00 0.2965 2.3389 20 21 0.015721 0.8330 0.0210 0.8540 21.127 39.908 61.035 0.0745 0.1430 0.2175 88.18 0.3107 2.4878 21 22 0.016741 0.8359 0.0224 0.8583 22.133 42.527 64.660 0.0779 0.1519 0.2298 92.36 0.3249 2.6448 22 23 0.017821 0.8387 0.0240 0.8627 23.140 45.301 68.440 0.0813 0.1613 0.2426 96.55 0.3390 2.8105 23 24 0.018963 0.8416 0.0256 0.8671 24.146 48.239 72.385 0.0847 0.1712 0.2559 100.73 0.3531 2.9852 24 25 0.020170 0.8444 0.0273 0.8717 25.153 51.347 76.500 0.0881 0.1817 0.2698 104.91 0.3672 3.1693 25 26 0.021448 0.8472 0.0291 0.8764 26.159 54.638 80.798 0.0915 0.1927 0.2842 109.09 0.3812 3.3633 26 27 0.022798 0.8501 0.0311 0.8811 27.165 58.120 85.285 0.0948 0.2044 0.2992 113.27 0.3951 3.5674 27 28 0.024226 0.8529 0.0331 0.8860 28.172 61.804 89.976 0.0982 0.2166 0.3148 117.45 0.4090 3.7823 28 29 0.025735 0.8558 0.0353 0.8910 29.179 65.699 94.878 0.1015 0.2296 0.3311 121.63 0.4229 4.0084 29 30 0.027329 0.8586 0.0376 0.8962 30.185 69.820 100.006 0.1048 0.2432 0.3481 125.81 0.4367 4.2462 30 31 0.029014 0.8614 0.0400 0.9015 31.192 74.177 105.369 0.1082 0.2576 0.3658 129.99 0.4505 4.4961 31 32 0.030793 0.8643 0.0426 0.9069 32.198 78.780 110.979 0.1115 0.2728 0.3842 134.17 0.4642 4.7586 32 33 0.032674 0.8671 0.0454 0.9125 33.205 83.652 116.857 0.1148 0.2887 0.4035 138.35 0.4779 5.0345 33 34 0.034660 0.8700 0.0483 0.9183 34.212 88.799 123.011 0.1180 0.3056 0.4236 142.53 0.4915 5.3242 34 35 0.036756 0.8728 0.0514 0.9242 35.219 94.236 129.455 0.1213 0.3233 0.4446 146.71 0.5051 5.6280 35 36 0.038971 0.8756 0.0546 0.9303 36.226 99.983 136.209 0.1246 0.3420 0.4666 150.89 0.5186 5.9468 36 37 0.041309 0.8785 0.0581 0.9366 37.233 106.058 143.290 0.1278 0.3617 0.4895 155.07 0.5321 6.2812 37 38 0.043778 0.8813 0.0618 0.9431 38.239 112.474 150.713 0.1311 0.3824 0.5135 159.25 0.5456 6.6315 38 39 0.046386 0.8842 0.0657 0.9498 39.246 119.258 158.504 0.1343 0.4043 0.5386 163.43 0.5590 6.9988 39 40 0.049141 0.8870 0.0698 0.9568 40.253 126.430 166.683 0.1375 0.4273 0.5649 167.61 0.5724 7.3838 40 41 0.052049 0.8898 0.0741 0.9640 41.261 134.005 175.265 0.1407 0.4516 0.5923 171.79 0.5857 7.7866 41 42 0.055119 0.8927 0.0788 0.9714 42.268 142.007 184.275 0.1439 0.4771 0.6211 175.97 0.5990 8.2081 42 43 0.058365 0.8955 0.0837 0.9792 43.275 150.475 193.749 0.1471 0.5041 0.6512 180.15 0.6122 8.6495 43 44 0.061791 0.8983 0.0888 0.9872 44.282 159.417 203.699 0.1503 0.5325 0.6828 184.33 0.6254 9.1110 44 45 0.065411 0.9012 0.0943 0.9955 45.289 168.874 214.164 0.1535 0.5624 0.7159 188.51 0.6386 9.5935 45 46 0.069239 0.9040 0.1002 1.0042 46.296 178.882 225.179 0.1566 0.5940 0.7507 192.69 0.6517 10.0982 46 47 0.073282 0.9069 0.1063 1.0132 47.304 189.455 236.759 0.1598 0.6273 0.7871 196.88 0.6648 10.6250 47 48 0.077556 0.9097 0.1129 1.0226 48.311 200.644 248.955 0.1629 0.6624 0.8253 201.06 0.6778 11.1754 48 49 0.082077 0.9125 0.1198 1.0323 49.319 212.485 261.803 0.1661 0.6994 0.8655 205.24 0.6908 11.7502 49 50 0.086858 0.9154 0.1272 1.0425 50.326 225.019 275.345 0.1692 0.7385 0.9077 209.42 0.7038 12.3503 50 51 0.091918 0.9182 0.1350 1.0532 51.334 238.290 289.624 0.1723 0.7798 0.9521 213.60 0.7167 12.9764 51 52 0.097272 0.9211 0.1433 1.0643 52.341 252.340 304.682 0.1754 0.8234 0.9988 217.78 0.7296 13.6293 52 53 0.102948 0.9239 0.1521 1.0760 53.349 267.247 320.596 0.1785 0.8695 1.0480 221.97 0.7424 14.3108 53 54 0.108954 0.9267 0.1614 1.0882 54.357 283.031 337.388 0.1816 0.9182 1.0998 226.15 0.7552 15.0205 54 55 0.115321 0.9296 0.1713 1.1009 55.365 299.772 355.137 0.1847 0.9698 1.1544 230.33 0.7680 15.7601 55 56 0.122077 0.9324 0.1819 1.1143 56.373 317.549 373.922 0.1877 1.0243 1.2120 234.52 0.7807 16.5311 56 57 0.129243 0.9353 0.1932 1.1284 57.381 336.417 393.798 0.1908 1.0820 1.2728 238.70 0.7934 17.3337 57 58 0.136851 0.9381 0.2051 1.1432 58.389 356.461 414.850 0.1938 1.1432 1.3370 242.88 0.8061 18.1691 58 59 0.144942 0.9409 0.2179 1.1588 59.397 377.788 437.185 0.1969 1.2081 1.4050 247.07 0.8187 19.0393 59 60 0.15354 0.9438 0.2315 1.1752 60.405 400.458 460.863 0.1999 1.2769 1.4768 251.25 0.8313 19.9439 60 61 0.16269 0.9466 0.2460 1.1926 61.413 424.624 486.036 0.2029 1.3500 1.5530 255.44 0.8438 20.8858 61 62 0.17244 0.9494 0.2614 1.2109 62.421 450.377 512.798 0.2059 1.4278 1.6337 259.62 0.8563 21.8651 62 63 0.18284 0.9523 0.2780 1.2303 63.429 477.837 541.266 0.2089 1.5104 1.7194 263.81 0.8688 22.8826 63 64 0.19393 0.9551 0.2957 1.2508 64.438 507.177 571.615 0.2119 1.5985 1.8105 268.00 0.8812 23.9405 64 65 0.20579 0.9580 0.3147 1.2726 65.446 538.548 603.995 0.2149 1.6925 1.9074 272.18 0.8936 25.0397 65 66 0.21848 0.9608 0.3350 1.2958 66.455 572.116 638.571 0.2179 1.7927 2.0106 276.37 0.9060 26.1810 66 67 0.23207 0.9636 0.3568 1.3204 67.463 608.103 675.566 0.2209 1.8999 2.1208 280.56 0.9183 27.3664 67 68 0.24664 0.9665 0.3803 1.3467 68.472 646.724 715.196 0.2238 2.0147 2.2385 284.75 0.9306 28.5967 68 69 0.26231 0.9693 0.4055 1.3749 69.481 688.261 757.742 0.2268 2.1378 2.3646 288.94 0.9429 29.8741 69 70 0.27916 0.9721 0.4328 1.4049 70.489 732.959 803.448 0.2297 2.2699 2.4996 293.13 0.9551 31.1986 70 71 0.29734 0.9750 0.4622 1.4372 71.498 781.208 852.706 0.2327 2.4122 2.6448 297.32 0.9673 32.5734 71 72 0.31698 0.9778 0.4941 1.4719 72.507 833.335 905.842 0.2356 2.5655 2.8010 301.51 0.9794 33.9983 72 73 0.33824 0.9807 0.5287 1.5093 73.516 889.807 963.323 0.2385 2.7311 2.9696 305.70 0.9916 35.4759 73 74 0.36130 0.9835 0.5662 1.5497 74.525 951.077 1025.603 0.2414 2.9104 3.1518 309.89 1.0037 37.0063 74 75 0.38641 0.9863 0.6072 1.5935 75.535 1017.841 1093.375 0.2443 3.1052 3.3496 314.08 1.0157 38.5940 75 76 0.41377 0.9892 0.6519 1.6411 76.543 1090.628 1167.172 0.2472 3.3171 3.5644 318.28 1.0278 40.2369 76 77 0.44372 0.9920 0.7010 1.6930 77.553 1170.328 1247.881 0.2501 3.5486 3.7987 322.47 1.0398 41.9388 77 78 0.47663 0.9948 0.7550 1.7498 78.562 1257.921 1336.483 0.2530 3.8023 4.0553 326.67 1.0517 43.7020 78 79 0.51284 0.9977 0.8145 1.8121 79.572 1354.347 1433.918 0.2559 4.0810 4.3368 330.86 1.0636 45.5248 79 80 0.55295 1.0005 0.8805 1.8810 80.581 1461.200 1541.781 0.2587 4.3890 4.6477 335.06 1.0755 47.4135 80 81 0.59751 1.0034 0.9539 1.9572 81.591 1579.961 1661.552 0.2616 4.7305 4.9921 339.25 1.0874 49.3670 81 82 0.64724 1.0062 1.0360 2.0422 82.600 1712.547 1795.148 0.2644 5.1108 5.3753 343.45 1.0993 51.3860 82 83 0.70311 1.0090 1.1283 2.1373 83.610 1861.548 1945.158 0.2673 5.5372 5.8045 347.65 1.1111 53.4746 83 84 0.76624 1.0119 1.2328 2.2446 84.620 2029.983 2114.603 0.2701 6.0181 6.2882 351.85 1.1228 55.6337 84 85 0.83812 1.0147 1.3518 2.3666 85.630 2221.806 2307.436 0.2729 6.5644 6.8373 356.05 1.1346 57.8658 85 86 0.92062 1.0175 1.4887 2.5062 86.640 2442.036 2528.677 0.2757 7.1901 7.4658 360.25 1.1463 60.1727 86 87 1.01611 1.0204 1.6473 2.6676 87.650 2697.016 2784.666 0.2785 7.9128 8.1914 364.45 1.1580 62.5544 87 88 1.12800 1.0232 1.8333 2.8565 88.661 2995.890 3084.551 0.2813 8.7580 9.0393 368.65 1.1696 65.0166 88 89 1.26064 1.0261 2.0540 3.0800 89.671 3350.254 3439.925 0.2841 9.7577 10.0419 372.86 1.1812 67.5581 89 90 1.42031 1.0289 2.3199 3.3488 90.681 3776.918 3867.599 0.2869 10.9586 11.2455 377.06 1.1928 70.1817 90 Table 2 Thermodynamic Properties of Moist Air at Standard Atmospheric Pressure, 101.325 kPa (Continued) Temp., °C t Humidity Ratio, kg(w)/kg(da) Ws Specific Volume, m3/kg (dry air) Specific Enthalpy, kJ/kg (dry air) Specific Entropy, kJ/(kg ·K) (dry air) Condensed Water Temp., °C t Specific Enthalpy, kJ/kg hw Specific Entropy, kJ/(kg·K) sw Vapor Pressure, kPa ps vda vas vs hda has hs sda sas ss Psychrometrics 6.5 Table 3 Thermodynamic Properties of Water at Saturation Temp., °C t Absolute Pressure, kPa p Specific Volume, m3/kg (water) Specific Enthalpy, kJ/kg (water) Specific Entropy, kJ/(kg ·K) (water) Temp., °C t Sat. Solid vi Evap.
vig Sat. Vapor vg Sat. Solid hi Evap.
hig Sat. Vapor hg Sat. Solid si Evap.
sig Sat. Vapor sg −60 0.00108 0.001082 90942.00 90942.00 −446.40 2836.27 2389.87 −1.6854 13.3065 11.6211 −60 −59 0.00124 0.001082 79858.69 79858.69 −444.74 2836.46 2391.72 −1.7667 13.2452 11.5677 −59 −58 0.00141 0.001082 70212.37 70212.37 −443.06 2836.64 2393.57 −1.6698 13.8145 11.5147 −58 −57 0.00161 0.001082 61805.35 61805.35 −441.38 2836.81 2395.43 −1.6620 13.1243 11.4623 −57 −56 0.00184 0.001082 54469.39 54469.39 −439.69 2836.97 2397.28 −1.6542 13.0646 11.4104 −56 −55 0.00209 0.001082 48061.05 48061.05 −438.00 2837.13 2399.12 −1.6464 13.0054 11.3590 −55 −54 0.00238 0.001082 42455.57 42455.57 −436.29 2837.27 2400.98 −1.6386 12.9468 11.3082 −54 −53 0.00271 0.001083 37546.09 37546.09 −434.59 2837.42 2402.83 −1.6308 12.8886 11.2578 −53 −52 0.00307 0.001083 33242.14 33242.14 −432.87 2837.55 2404.68 −1.6230 12.8309 11.2079 −52 −51 0.00348 0.001083 29464.67 29464.67 −431.14 2837.68 2406.53 −1.6153 12.7738 11.1585 −51 −50 0.00394 0.001083 26145.01 26145.01 −429.41 2837.80 2408.39 −1.6075 12.7170 11.1096 −50 −49 0.00445 0.001083 23223.69 23223.70 −427.67 2837.91 2410.24 −1.5997 12.6608 11.0611 −49 −48 0.00503 0.001083 20651.68 20651.69 −425.93 2838.02 2412.09 −1.5919 12.6051 11.0131 −48 −47 0.00568 0.001083 18383.50 18383.51 −424.27 2838.12 2413.94 −1.5842 12.5498 10.9656 −47 −46 0.00640 0.001083 16381.35 16381.36 −422.41 2838.21 2415.79 −1.5764 12.4949 10.9185 −46 −45 0.00721 0.001984 14612.35 14512.36 −420.65 2838.29 2417.65 −1.5686 12.4405 10.8719 −45 −44 0.00811 0.001084 13047.65 13047.66 −418.87 2838.37 2419.50 −1.5609 12.3866 10.8257 −44 −43 0.00911 0.001084 11661.85 11661.85 −417.09 2838.44 2421.35 −1.5531 12.3330 10.7799 −43 −42 0.01022 0.001084 10433.85 10433.85 −415.30 2838.50 2423.20 −1.5453 12.2799 10.7346 −42 −41 0.01147 0.001084 9344.25 9344.25 −413.50 2838.55 2425.05 −1.5376 12.2273 10.6897 −41 −40 0.01285 0.001084 8376.33 8376.33 −411.70 2838.60 2426.90 −1.5298 12.1750 10.6452 −40 −39 0.01438 0.001085 7515.86 7515.87 −409.88 2838.64 2428.76 −1.5221 12.1232 10.6011 −39 −38 0.01608 0.001085 6750.36 6750.36 −508.07 2838.67 1430.61 −1.5143 12.0718 10.5575 −38 −37 0.01796 0.001085 6068.16 6068.17 −406.24 2838.70 2432.46 −1.5066 12.0208 10.5142 −37 −36 0.02004 0.001085 5459.82 5459.82 −404.40 2838.71 2434.31 −1.4988 11.9702 10.4713 −36 −35 0.02235 0.001085 4917.09 4917.10 −402.56 2838.73 2436.16 −1.4911 11.9199 10.4289 −35 −34 0.02490 0.001085 4432.36 4432.37 −400.72 2838.73 2438.01 −1.4833 11.8701 10.3868 −34 −33 0.02771 0.001085 3998.71 3998.71 −398.86 2838.72 2439.86 −1.4756 11.8207 10.3451 −33 −32 0.03082 0.001086 3610.71 3610.71 −397.00 2838.71 2441.72 −1.4678 11.7716 10.3037 −32 −31 0.03424 0.001086 3263.20 3263.20 −395.12 2838.69 2443.57 −1.4601 11.7229 10.2628 −31 −30 0.03802 0.001086 2951.64 2951.64 −393.25 2838.66 2445.42 −1.4524 11.6746 10.2222 −30 −29 0.04217 0.001086 2672.03 2672.03 −391.36 2838.63 2447.27 −1.4446 11.6266 10.1820 −29 −28 0.04673 0.001086 2420.89 2420.89 −389.47 2838.59 2449.12 −1.4369 11.4790 10.1421 −28 −27 0.05174 0.001086 2195.23 2195.23 −387.57 2838.53 2450.97 −1.4291 11.5318 10.1026 −27 −26 0.05725 0.001087 1992.15 1992.15 −385.66 2838.48 2452.82 −1.4214 11.4849 10.0634 −26 −25 0.06329 0.001087 1809.35 1809.35 −383.74 2838.41 2454.67 −1.4137 11.4383 10.0246 −25 −24 0.06991 0.001087 1644.59 1644.59 −381.34 2838.34 2456.52 −1.4059 11.3921 9.9862 −24 −23 0.07716 0.001087 1495.98 1495.98 −379.89 2838.26 2458.37 −1.3982 11.3462 9.9480 −23 −22 0.08510 0.001087 1361.94 1361.94 −377.95 2838.17 2460.22 −1.3905 11.3007 9.9102 −22 −21 0.09378 0.001087 1240.77 1240.77 −376.01 2838.07 2462.06 −1.3828 11.2555 9.8728 −21 −20 0.10326 0.001087 1131.27 1131.27 −374.06 2837.97 2463.91 −1.3750 11.2106 9.8356 −20 −19 0.11362 0.001088 1032.18 1032.18 −372.10 2837.86 2465.76 −1.3673 11.1661 9.7988 −19 −18 0.12492 0.001088 942.46 942.47 −370.13 2837.74 2467.61 −1.3596 11.1218 9.7623 −18 −17 0.13725 0.001088 861.17 861.18 −368.15 2837.61 2469.46 −1.3518 11.0779 9.7261 −17 −16 0.15068 0.001088 787.48 787.49 −366.17 2837.47 2471.30 −1.3441 11.0343 9.6902 −16 −15 0.16530 0.001088 720.59 720.59 −364.18 2837.33 2473.15 −1.3364 10.9910 9.6546 −15 −14 0.18122 0.001088 659.86 659.86 −362.18 2837.18 2474.99 −1.3287 10.9480 9.6193 −14 −13 0.19852 0.001089 604.65 604.65 −360.18 2837.02 2476.84 −1.3210 10.9053 9.5844 −13 −12 0.21732 0.001089 554.45 554.45 −358.17 2836.85 2478.68 −1.3232 10.8629 9.5497 −12 −11 0.23774 0.001089 508.75 508.75 −356.15 2836.68 2480.53 −1.3055 10.8208 9.5153 −11 −10 0.25990 0.001089 467.14 467.14 −354.12 2836.49 2482.37 −1.2978 10.7790 9.4812 −10 −9 0.28393 0.001089 429.21 429.21 −352.08 2836.30 2484.22 −1.2901 10.7375 9.4474 −9 −8 0.30998 0.001090 394.64 394.64 −350.04 2836.10 2486.06 −1.2824 10.6962 9.4139 −8 −7 0.33819 0.001090 363.07 363.07 −347.99 2835.89 2487.90 −1.2746 10.6552 9.3806 −7 −6 0.36874 0.001090 334.25 334.25 −345.93 2835.68 2489.74 −1.2669 10.6145 9.3476 −6 −5 0.40176 0.001090 307.91 307.91 −343.87 2835.45 2491.58 −2.2592 10.4741 9.3149 −5 −4 0.43747 0.001090 283.83 283.83 −341.80 2835.22 2493.42 −1.2515 10.5340 9.2825 −4 −3 0.47606 0.001090 261.79 261.79 −339.72 2834.98 2495.26 −1.2438 10.4941 9.2503 −3 −2 0.51772 0.001091 241.60 241.60 −337.63 2834.72 2497.10 −1.2361 10.4544 9.2184 −2 −1 0.56267 0.001091 223.11 223.11 −335.53 2834.47 2498.93 −1.2284 10.4151 9.1867 −1 0 0.61115 0.001091 206.16 206.16 −333.43 2834.20 2500.77 −1.2206 10.3760 9.1553 0 6.6 2001 ASHRAE Fundamentals Handbook (SI) Table 3 Thermodynamic Properties of Water at Saturation (Continued) Temp., °C t Absolute Pressure, kPa p Specific Volume, m3/kg (water) Specific Enthalpy, kJ/kg (water) Specific Entropy, kJ/(kg ·K) (water) Temp., °C t Sat. Liquid vf Evap.
vfg Sat. Vapor vg Sat. Liquid hf Evap.
hfg Sat. Vapor hg Sat. Liquid sf Evap.
sfg Sat. Vapor sg 0 0.6112 0.001000 206.141 206.143 −0.04 2500.81 2500.77 −0.0002 9.1555 9.1553 0 1 0.6571 0.001000 192.455 192.456 4.18 2498.43 2502.61 0.0153 9.1134 9.1286 1 2 0.7060 0.001000 179.769 179.770 8.39 2496.05 2504.45 0.0306 9.0716 9.1022 2 3 0.7580 0.001000 168.026 168.027 12.60 2493.68 2506.28 0.0459 9.0302 9.0761 3 4 0.8135 0.001000 157.137 157.138 16.81 2491.31 2508.12 0.0611 8.9890 9.0501 4 5 0.8725 0.001000 147.032 147.033 21.02 2488.94 2509.96 0.0763 8.9482 9.0244 5 6 0.9353 0.001000 137.653 137.654 25.22 2486.57 2511.79 0.0913 8.9077 8.9990 6 7 1.0020 0.001000 128.947 128.948 29.42 2484.20 2513.62 0.1064 8.8674 8.9738 7 8 1.0728 0.001000 120.850 120.851 33.62 2481.84 2515.46 0.1213 8.8273 8.9488 8 9 1.1481 0.001000 113.326 113.327 37.82 2479.47 2517.29 0.1362 8.7878 8.9245 9 10 1.2280 0.001000 106.328 106.329 42.01 2477.11 2519.12 0.1511 8.7484 8.8995 10 11 1.3127 0.001000 99.812 99.813 46.21 2474.74 2520.95 0.1659 8.7093 8.8752 11 12 1.4026 0.001001 93.743 93.744 50.40 2472.38 2522.78 0.1806 8.6705 8.8511 12 13 1.4978 0.001001 88.088 88.089 54.59 2470.02 2524.61 0.1953 8.6319 8.8272 13 14 1.5987 0.001001 82.815 82.816 58.78 2467.66 2526.44 0.2099 8.5936 3.8035 14 15 1.7055 0.001001 77.897 77.898 62.97 2465.30 2528.26 0.2244 8.5556 8.7801 15 16 1.8184 0.001001 73.307 73.308 67.16 2462.93 2530.09 0.2389 8.5178 8.7568 16 17 1.9380 0.001001 69.021 69.022 71.34 2460.57 2531.92 0.2534 8.4804 8.7338 17 18 2.0643 0.001002 65.017 65.018 75.53 2458.21 2533.74 0.2678 8.4431 8.7109 18 19 2.1978 0.001002 65.274 61.273 79.72 2455.85 2535.56 0.2821 8.4061 8.6883 19 20 2.3388 0.001002 57.774 57.773 83.90 2453.48 2537.38 0.2964 8.3694 8.6658 20 21 2.4877 0.001002 54.450 54.500 88.08 2451.12 2539.20 0.3107 8.3329 8.6436 21 22 2.6448 0.001002 51.433 51.434 92.27 2448.75 2541.02 0.3249 8.2967 8.6215 22 23 2.8104 0.001003 48.562 48.563 96.45 2446.39 2542.84 0.3390 8.2607 8.5996 23 24 2.9851 0.001003 45.872 45.873 100.63 2444.02 2544.65 0.3531 8.2249 8.5780 24 25 3.1692 0.001003 43.350 43.351 104.81 2441.66 2546.47 0.3672 8.1894 8.5565 25 26 3.3631 0.001003 40.985 40.986 108.99 2439.29 2548.28 0.3812 8.1541 8.5352 26 27 3.5673 0.001004 38.766 38.767 113.18 2436.92 2550.09 0.3951 8.1190 8.5141 27 28 3.7822 0.001004 36.682 36.683 117.36 2434.55 2551.90 0.4090 8.0842 8.4932 28 29 4.0083 0.001004 34.726 34.727 121.54 2432.17 2553.71 0.4229 8.0496 8.4724 29 30 4.2460 0.001004 32.889 32.889 125.72 2429.80 2555.52 0.4367 8.0152 8.4519 30 31 4.4959 0.001005 31.160 31.161 129.90 2427.43 2557.32 0.4505 7.9810 8.4315 31 32 4.7585 0.001005 29.535 29.536 134.08 2425.05 2559.13 0.4642 7.9471 8.4112 32 33 5.0343 0.001005 28.006 28.007 138.26 2422.67 2560.93 0.4779 7.9133 8.3912 33 34 5.3239 0.001006 26.567 26.568 142.44 2410.29 2562.73 0.4915 7.8790 8.3713 34 35 5.6278 0.001006 25.212 25.213 146.62 2417.91 2564.53 0.5051 7.8465 8.3516 35 36 5.9466 0.001006 23.935 23.936 150.80 2415.53 2566.33 0.5186 7.8134 8.3320 36 37 6.2810 0.001007 22.733 22.734 154.98 2413.14 2568.12 0.5321 7.7805 8.3127 37 38 6.6315 0.001007 21.599 21.600 159.16 2410.76 2569.91 0.5456 7.7479 8.2934 38 39 6.9987 0.001008 20.529 20.530 163.34 2408.37 2571.71 0.5590 7.7154 8.2744 39 40 7.3835 0.001008 19.520 19.521 167.52 2405.98 2573.50 0.5724 7.6831 8.2555 40 41 7.7863 0.001008 18.567 18.568 171.70 2403.58 2575.28 0.5857 7.6510 8.2367 41 42 8.2080 0.001009 17.667 17.668 175.88 2401.19 2577.07 0.5990 7.6191 8.2181 42 43 8.6492 0.001009 16.818 16.819 180.06 2398.79 2578.85 0.6122 7.5875 8.1997 43 44 9.1107 0.001010 16.014 16.015 184.24 2396.39 2580.63 0.6254 7.3560 8.1814 44 45 9.5932 0.001010 15.255 15.256 188.42 2393.99 2582.41 0.6386 7.5247 8.1632 45 46 10.0976 0.001010 14.537 14.538 192.60 2391.59 2584.19 0.6517 7.4936 8.1452 46 47 10.6246 0.001011 13.858 13.859 196.78 2389.18 2585.96 0.6648 7.4626 8.1274 47 48 11.1751 0.001011 13.214 13.215 200.97 2386.77 2587.74 0.6778 7.4319 8.1097 48 49 11.7500 0.001012 12.606 12.607 205.15 2384.36 2589.51 0.6908 7.4013 8.0921 49 50 12.3499 0.001012 12.029 12.029 209.33 2381.94 2591.27 0.7038 7.3709 8.0747 50 51 12.9759 0.001013 11.482 11.483 213.51 2379.53 2593.04 0.7167 7.3407 8.0574 51 52 13.6290 0.001013 10.964 10.965 217.70 2377.10 2594.80 0.7296 7.3107 8.0403 52 53 14.3100 0.001014 10.473 10.474 221.88 2374.68 2596.56 0.7424 7.2809 8.0233 53 54 15.0200 0.001014 10.001 10.008 226.06 2372.26 2598.32 0.7552 7.2512 8.0064 54 55 15.7597 0.001015 9.563 9.5663 230.25 2369.83 2600.07 0.7680 7.2217 7.9897 55 56 16.5304 0.001015 9.147 9.1468 234.43 2367.39 2601.82 0.7807 7.1924 7.9731 56 57 17.3331 0.001016 8.744 8.7489 238.61 2364.96 2603.57 0.7934 7.1632 7.9566 57 58 18.1690 0.001016 8.3690 8.3700 242.80 2362.52 2605.32 0.8061 7.1342 7.9403 58 59 19.0387 0.001017 8.0094 8.0114 246.99 2360.08 2607.06 0.8187 7.1054 7.9240 59 60 19.944 0.001017 7.6677 7.6697 251.17 2357.63 2608.80 0.8313 7.0767 7.9079 60 61 20.885 0.001018 7.3428 7.3438 255.36 2355.19 2610.54 0.8438 7.0482 7.8920 61 62 21.864 0.001018 7.0337 7.0347 259.54 2352.73 2612.28 0.8563 7.0198 7.8761 62 63 22.882 0.001019 6.7397 6.7407 263.73 2350.28 2614.01 0.8688 6.9916 7.8604 63 64 23.940 0.001019 6.4599 6.4609 267.92 2347.82 2615.74 0.8812 6.9636 7.8448 64 65 25.040 0.001020 6.1935 6.1946 272.11 2345.36 2617.46 0.8936 6.9357 7.8293 65 66 26.180 0.001020 5.9397 5.9409 276.30 2342.89 2619.19 0.9060 6.9080 7.8140 66 67 27.366 0.001021 5.6982 5.6992 280.49 2340.42 2620.90 0.9183 6.8804 7.7987 67 68 28.596 0.001022 5.4680 5.4690 284.68 2337.95 2622.62 0.9306 2.8530 7.7836 68 69 29.873 0.001022 5.2485 5.2495 288.87 2335.47 2624.33 0.9429 6.8257 7.7686 69 Psychrometrics 6.7 70 31.198 0.001023 5.0392 5.0402 293.06 2332.99 2626.04 0.9551 6.7986 7.7537 70 71 32.572 0.001023 4.8396 4.8407 297.25 2330.50 2627.75 0.9673 6.7716 7.7389 71 72 33.997 0.001024 4.6492 4.6502 301.44 2328.01 2629.45 0.9795 6.7448 7.7242 72 73 35.475 0.001025 4.4675 4.4685 305.63 2325.51 2631.15 0.9916 6.7181 7.7097 73 74 37.006 0.001025 4.2940 4.2951 309.83 2323.02 2632.84 1.0037 6.6915 7.6952 74 75 38.592 0.001026 4.1284 4.1294 314.02 2320.51 2634.53 1.0157 6.6651 7.6809 75 76 40.236 0.001026 3.9702 3.9712 318.22 2318.01 2636.22 1.0278 6.6389 7.6666 76 77 41.938 0.001027 3.8190 3.8201 322.41 2315.49 2637.90 1.0398 6.6127 7.6525 77 78 43.700 0.001028 3.6746 3.6756 326.61 2312.98 2639.58 1.0517 6.5867 7.6384 78 79 45.524 0.001028 3.5365 3.5375 330.81 2310.46 2641.26 1.0636 6.5609 7.6245 79 80 47.412 0.001029 3.4044 3.4055 335.00 2307.93 2642.93 1.0755 6.5351 7.6107 80 81 49.364 0.001030 3.2781 3.2792 339.20 2305.40 2644.60 1.0874 6.5095 7.5969 81 82 51.384 0.001030 3.1573 3.1583 343.40 2902.86 2646.26 1.0993 6.4841 7.5833 82 83 53.473 0.001031 3.0417 3.0427 347.60 2300.32 2647.92 1.1111 6.4587 7.5698 83 84 55.633 0.001032 2.9310 2.9320 351.80 2297.78 2649.58 1.1228 6.4335 7.5563 84 85 57.865 0.001032 2.8250 2.8260 356.01 2295.22 2651.23 1.1346 6.4084 7.5430 85 86 60.171 0.001033 2.7235 2.7245 350.21 2292.67 2652.88 1.1463 6.3834 7.5297 86 87 62.554 0.001034 2.6263 2.6273 364.41 2290.11 2654.52 1.1580 6.3586 7.5166 87 88 65.015 0.001035 2.5331 2.5341 368.62 2287.54 2656.16 1.1696 6.3339 7.5035 88 89 67.556 0.001035 2.4438 2.4448 372.82 2284.97 2657.79 1.1812 6.3093 7.4905 89 90 70.180 0.001036 2.3582 2.3592 377.03 2282.39 2659.42 1.1928 6.2848 7.4776 90 91 72.888 0.001037 2.2760 2.2771 381.24 2279.81 2661.04 1.2044 6.2605 7.4648 91 92 75.683 0.001037 2.1973 2.1983 385.45 2277.22 2662.66 1.2159 6.2362 7.4521 92 93 78.566 0.001038 2.1217 2.1228 389.66 2274.62 2664.28 1.2274 6.2121 7.4395 93 94 81.541 0.001039 2.0492 2.0502 393.87 2272.02 2665.89 1.2389 6.1881 7.4270 94 95 84.608 0.001040 1.9796 1.9806 398.08 2269.41 2667.49 1.2504 6.1642 7.4146 95 96 87.770 0.001040 1.9128 1.9138 402.29 2266.80 2669.09 1.2618 6.1404 7.4022 96 97 91.030 0.001041 1.8486 1.8496 406.51 2264.18 2670.69 1.2732 6.1168 7.3899 97 98 94.390 0.001042 1.7869 1.7880 410.72 2261.55 2672.28 1.2845 6.0932 7.3777 98 99 97.852 0.001044 1.7277 1.7287 414.94 2258.92 2673.86 1.2959 6.0697 7.3656 99 100 101.419 0.001044 1.6708 1.6718 419.16 2256.28 2675.44 1.3072 6.0464 7.3536 100 101 105.092 0.001044 1.6161 1.6171 423.38 2253.64 2677.02 1.3185 6.0232 7.3416 101 102 108.875 0.001045 1.5635 1.5645 427.60 2250.99 2678.58 1.3297 6.0000 7.3298 102 103 112.770 0.001046 1.5129 1.5139 431.82 2248.33 2680.15 1.3410 5.9770 7.3180 103 104 116.779 0.001047 1.4642 1.4652 436.04 2245.66 2681.71 1.3522 5.9541 7.3062 104 105 120.906 0.001047 1.4174 1.4184 440.27 2242.99 2683.26 1.3634 5.9313 7.2946 105 106 125.152 0.001048 1.3723 1.3734 444.49 2240.31 2684.80 1.3745 5.9086 7.2830 106 107 129.520 0.001049 1.3290 1.3300 448.72 2237.63 2686.35 1.3856 5.8860 7.2716 107 108 134.012 0.001050 1.2872 1.2883 452.95 2234.93 2687.88 1.3967 5.8635 7.2601 108 109 138.633 0.001051 1.2470 1.2481 457.18 2232.23 2689.41 1.4078 5.8410 7.2488 109 110 143.384 0.001052 1.2083 1.2093 461.41 2229.52 2690.93 1.4188 5.8187 7.2375 110 111 148.267 0.001052 1.1710 1.1720 465.64 2226.81 2692.45 1.4298 5.7965 7.2263 111 112 153.287 0.001053 1.1350 1.1361 469.88 2224.09 2693.96 1.4408 5.7744 7.2152 112 113 158.445 0.001054 1.1004 1.1015 474.11 2221.35 2695.47 1.4518 5.7524 7.2402 113 114 163.745 0.001055 1.0670 1.0681 478.35 2218.62 2696.97 1.4627 5.7304 7.1931 114 115 169.190 0.001056 1.0348 1.0359 482.59 2215.87 2698.46 1.4737 5.7086 7.1822 115 116 174.782 0.001057 1.0038 1.0048 486.83 2213.12 2699.95 1.4846 5.6868 7.1714 116 117 180.525 0.001058 0.9739 0.9749 491.07 2210.35 2701.43 1.4954 5.6652 7.1606 117 118 186.420 0.001059 0.9450 0.9460 495.32 2207.58 2702.90 1.5063 5.6436 7.1499 118 119 192.473 0.001059 0.9171 0.9182 499.56 2204.80 2704.37 1.5171 5.6221 7.1392 119 120 198.685 0.001060 0.8902 0.8913 503.81 2202.02 2705.83 1.5279 5.6007 7.1286 120 122 211.601 0.001062 0.8391 0.8402 512.31 2196.42 2706.73 1.5494 5.5582 7.1076 122 124 225.194 0.001064 0.7916 0.7927 520.82 2190.78 2711.60 1.5709 5.5160 7.0869 124 126 239.490 0.001066 0.7472 0.7483 529.33 2185.11 2714.44 1.5922 5.4742 7.0664 126 128 254.515 0.001068 0.7057 0.7068 537.86 2179.40 2717.26 1.6135 5.4326 7.0461 128 130 270.298 0.001070 0.6670 0.6681 546.39 2173.66 2720.05 1.6347 5.3914 7.0261 130 132 286.866 0.001072 0.6308 0.6318 554.93 2167.87 2722.80 1.6557 5.3505 7.0063 132 134 304.247 0.001074 0.5969 0.5979 563.48 2162.05 2725.53 1.6767 5.3099 6.9867 134 136 322.470 0.001076 0.5651 0.5662 572.04 2156.18 2728.22 1.6977 5.2697 6.9673 136 138 341.566 0.001078 0.5354 0.5364 580.60 2150.28 2730.88 1.7185 5.2296 6.9481 138 140 361.565 0.001080 0.5075 0.5085 589.18 2144.33 2733.51 1.7393 5.1899 6.9292 140 142 382.497 0.001082 0.4813 0.4824 597.76 2138.34 2736.11 1.7599 5.1505 6.9104 142 144 404.394 0.001084 0.4567 0.4578 606.36 2132.31 2738.67 1.7805 5.1113 6.8918 144 146 427.288 0.001086 0.4336 0.4347 614.97 2126.23 2741.19 1.8011 5.0724 6.8735 146 148 451.211 0.001088 0.4119 0.4130 623.58 2120.10 2743.68 1.8215 5.0338 6.8553 148 150 476.198 0.001091 0.3914 0.3925 632.21 2113.92 2746.13 1.8419 4.9954 6.8373 150 152 502.281 0.001093 0.3722 0.3733 640.85 2107.70 2748.55 1.8622 4.9573 6.8194 152 154 529.495 0.001095 0.3541 0.3552 649.50 2101.43 2750.93 1.8824 4.9194 6.8017 154 156 557.875 0.001097 0.3370 0.3381 658.16 2095.11 2753.27 1.9026 4.8817 6.7842 156 158 587.456 0.001100 0.3209 0.3220 666.83 2088.73 2755.57 1.9226 4.8443 6.7669 158 160 618.275 0.001102 0.3058 0.3069 675.52 2082.31 2757.82 1.9427 4.8070 6.7497 160 Table 3 Thermodynamic Properties of Water at Saturation (Continued) Temp., °C t Absolute Pressure, kPa p Specific Volume, m3/kg (water) Specific Enthalpy, kJ/kg (water) Specific Entropy, kJ/(kg ·K) (water) Temp., °C t Sat. Liquid vf Evap.
vfg Sat. Vapor vg Sat. Liquid hf Evap.
hfg Sat. Vapor hg Sat. Liquid sf Evap.
sfg Sat. Vapor sg 6.8 2001 ASHRAE Fundamentals Handbook (SI) HUMIDITY PARAMETERS Basic Parameters Humidity ratio (alternatively, the moisture content or mixing ratio) W of a given moist air sample is defined as the ratio of the mass of water vapor to the mass of dry air contained in the sample: (7) The humidity ratio W is equal to the mole fraction ratio xw/xda mul-tiplied by the ratio of molecular masses, namely, 18.01528/28.9645 = 0.62198: (8) Specific humidity γ is the ratio of the mass of water vapor to the total mass of the moist air sample: (9a) In terms of the humidity ratio, (9b) Absolute humidity (alternatively, water vapor density) dv is the ratio of the mass of water vapor to the total volume of the sample: (10) The density ρ of a moist air mixture is the ratio of the total mass to the total volume: (11) where v is the moist air specific volume, m3/kg (dry air), as defined by Equation (27).
Humidity Parameters Involving Saturation The following definitions of humidity parameters involve the concept of moist air saturation: Saturation humidity ratio Ws(t, p) is the humidity ratio of moist air saturated with respect to water (or ice) at the same temper-ature t and pressure p.
Degree of saturation µ is the ratio of the air humidity ratio W to the humidity ratio Ws of saturated moist air at the same temperature and pressure: (12) Relative humidity φ is the ratio of the mole fraction of water vapor xw in a given moist air sample to the mole fraction xws in an air sample saturated at the same temperature and pressure: (13) Combining Equations (8), (12), and (13), (14) Dew-point temperature td is the temperature of moist air satu-rated at the same pressure p, with the same humidity ratio W as that of the given sample of moist air. It is defined as the solution td(p, W) of the following equation: (15) Thermodynamic wet-bulb temperature t is the temperature at which water (liquid or solid), by evaporating into moist air at a given dry-bulb temperature t and humidity ratio W, can bring air to saturation adiabatically at the same temperature t while the total pressure p is maintained constant. This parameter is considered sep-arately in the section on Thermodynamic Wet-Bulb Temperature and Dew-Point Temperature.
PERFECT GAS RELATIONSHIPS FOR DRY AND MOIST AIR When moist air is considered a mixture of independent perfect gases (i.e., dry air and water vapor), each is assumed to obey the per-fect gas equation of state as follows: (16) (17) where pda = partial pressure of dry air pw = partial pressure of water vapor V = total mixture volume nda = number of moles of dry air nw = number of moles of water vapor R = universal gas constant, 8314.41 J/(kg mol·K) T = absolute temperature, K The mixture also obeys the perfect gas equation: (18) or (19) where p = pda + pw is the total mixture pressure and n = nda + nw is the total number of moles in the mixture. From Equations (16) through (19), the mole fractions of dry air and water vapor are, respectively, (20) and (21) From Equations (8), (20), and (21), the humidity ratio W is given by (22) The degree of saturation µ is, by definition, Equation (12): where (23) W Mw Mda ⁄ = W 0.62198xw xda ⁄ = γ Mw Mw Mda + ( ) ⁄ = γ W 1 W + ( ) ⁄ = dv Mw V ⁄ = ρ Mda Mw + ( ) V ⁄ 1 v ⁄ ( ) 1 W + ( ) = = µ W Ws ------- t p , = φ xw xws -------- t p , = µ φ 1 1 φ – ( )Ws 0.62198 ⁄ + ---------------------------------------------------------= Ws p td , ( ) W = Dry air: pdaV ndaRT = Water vapor: pwV nwRT = pV nRT = pda pw + ( )V nda nw + ( )RT = xda pda pda pw + ( ) ⁄ pda p ⁄ = = xw pw pda pw + ( ) ⁄ pw p ⁄ = = W 0.62198 pw p pw – ---------------= µ W Ws ------- t p , = Ws 0.62198 pws p pws – -----------------= Psychrometrics 6.9 The term pws represents the saturation pressure of water vapor in the absence of air at the given temperature t. This pressure pws is a function only of temperature and differs slightly from the vapor pressure of water in saturated moist air.
The relative humidity φ is, by definition, Equation (13): Substituting Equation (21) for xw and xws, (24) Substituting Equation (21) for xws into Equation (14), (25) Both φ and µ are zero for dry air and unity for saturated moist air.
At intermediate states their values differ, substantially so at higher temperatures.
The specific volume v of a moist air mixture is expressed in terms of a unit mass of dry air: (26) where V is the total volume of the mixture, Mda is the total mass of dry air, and nda is the number of moles of dry air. By Equations (16) and (26), with the relation p = pda + pw, (27) Using Equation (22), (28) In Equations (27) and (28), v is specific volume, T is absolute tem-perature, p is total pressure, pw is the partial pressure of water vapor, and W is the humidity ratio.
In specific units, Equation (28) may be expressed as where v = specific volume, m3/kg (dry air) t = dry-bulb temperature, °C W = humidity ratio, kg (water)/kg (dry air) p = total pressure, kPa The enthalpy of a mixture of perfect gases equals the sum of the individual partial enthalpies of the components. Therefore, the spe-cific enthalpy of moist air can be written as follows: (29) where hda is the specific enthalpy for dry air in kJ/kg (dry air) and hg is the specific enthalpy for saturated water vapor in kJ/kg (water) at the temperature of the mixture. As an approximation, (30) (31) where t is the dry-bulb temperature in °C. The moist air spe-cific enthalpy in kJ/kg (dry air) then becomes (32) THERMODYNAMIC WET-BULB TEMPERATURE AND DEW-POINT TEMPERATURE For any state of moist air, a temperature t exists at which liquid (or solid) water evaporates into the air to bring it to saturation at exactly this same temperature and total pressure (Harrison 1965).
During the adiabatic saturation process, the saturated air is expelled at a temperature equal to that of the injected water. In this constant pressure process, • Humidity ratio is increased from a given initial value W to the value Ws corresponding to saturation at the temperature t • Enthalpy is increased from a given initial value h to the value hs corresponding to saturation at the temperature t • Mass of water added per unit mass of dry air is (Ws −W), which adds energy to the moist air of amount (Ws −W)hw, where hw denotes the specific enthalpy in kJ/kg (water) of the water added at the temperature t Therefore, if the process is strictly adiabatic, conservation of enthalpy at constant total pressure requires that (33) The properties Ws, hw, and hs are functions only of the tem-perature t for a fixed value of pressure. The value of t, which sat-isfies Equation (33) for given values of h, W, and p, is the thermodynamic wet-bulb temperature.
The psychrometer consists of two thermometers; one ther-mometer’s bulb is covered by a wick that has been thoroughly wetted with water. When the wet bulb is placed in an airstream, water evaporates from the wick, eventually reaching an equilib-rium temperature called the wet-bulb temperature. This process is not one of adiabatic saturation, which defines the thermody-namic wet-bulb temperature, but one of simultaneous heat and mass transfer from the wet bulb. The fundamental mechanism of this process is described by the Lewis relation [Equation (39) in Chapter 5]. Fortunately, only small corrections must be applied to wet-bulb thermometer readings to obtain the thermodynamic wet-bulb temperature.
As defined, thermodynamic wet-bulb temperature is a unique property of a given moist air sample independent of measurement techniques.
Equation (33) is exact since it defines the thermodynamic wet-bulb temperature t. Substituting the approximate perfect gas rela-tion [Equation (32)] for h, the corresponding expression for hs, and the approximate relation (34) into Equation (33), and solving for the humidity ratio, (35) where t and t are in °C.
The dew-point temperature td of moist air with humidity ratio W and pressure p was defined earlier as the solution td(p, w) of Ws(p, td). For perfect gases, this reduces to (36) where pw is the water vapor partial pressure for the moist air sample and pws(td) is the saturation vapor pressure at temperature td. The saturation vapor pressure is derived from Table 3 or from Equation φ xw xws -------- t p , = φ pw pws -------- t p , = φ µ 1 1 µ – ( ) pws p ⁄ ( ) – -----------------------------------------------= v V Mda ⁄ V 28.9645nda ( ) ⁄ = = v RT 28.9645 p pw – ( ) ----------------------------------------RdaT p pw – ---------------= = v RT 1 1.6078W + ( ) 28.964p -------------------------------------------RdaT 1 1.6078W + ( ) p -------------------------------------------------= = v 0.2871 t 273.15 + ( ) 1 1.6078W + ( ) p ⁄ = h hda Whg + = hda 1.006t ≈ hg 2501 1.805t + ≈ h 1.006t W 2501 1.805t + ( ) + = h Ws W – ( ) + hw hs = hw 4.186t ≈ W 2501 2.381t – ( )Ws 1.006 t t – ( ) – 2501 1.805t 4.186t – + ---------------------------------------------------------------------------------------= pws td ( ) pw pW ( ) 0.62198 W + ( ) ⁄ = = 6.10 2001 ASHRAE Fundamentals Handbook (SI) (5) or (6). Alternatively, the dew-point temperature can be calcu-lated directly by one of the following equations (Peppers 1988): For the dew-point temperature range of 0 to 93°C, (37) For temperatures below 0°C, (38) where td = dew-point temperature, °C α = ln pw pw = water vapor partial pressure, kPa C14 = 6.54 C15 = 14.526 C16 = 0.7389 C17 = 0.09486 C18 = 0.4569 NUMERICAL CALCULATION OF MOIST AIR PROPERTIES The following are outlines, citing equations and tables already presented, for calculating moist air properties using perfect gas rela-tions. These relations are sufficiently accurate for most engineering calculations in air-conditioning practice, and are readily adapted to either hand or computer calculating methods. For more details, refer to Tables 15 through 18 in Chapter 1 of Olivieri (1996). Graphical procedures are discussed in the section on Psychrometric Charts.
SITUATION 1.
Given: Dry-bulb temperature t, Wet-bulb temperature t, Pressure p SITUATION 2.
Given: Dry-bulb temperature t, Dew-point temperature td, Pressure p SITUATION 3.
Given: Dry-bulb temperature t, Relative humidity φ, Pressure p Exact Relations for Computing Ws and φ Corrections that account for (1) the effect of dissolved gases on properties of condensed phase; (2) the effect of pressure on prop-erties of condensed phase; and (3) the effect of intermolecular force on properties of moisture itself, can be applied to Equations (23) and (25): (23a) (25a) Table 4 lists f values for a number of pressure and temperature combinations. Hyland and Wexler (1983a) give additional values.
Moist Air Property Tables for Standard Pressure Table 2 shows values of thermodynamic properties for standard atmospheric pressure at temperatures from −60 to 90°C. The prop-erties of intermediate moist air states can be calculated using the degree of saturation µ: (39) (40) (41) These equations are accurate to about 70°C. At higher temperatures, the errors can be significant. Hyland and Wexler (1983a) include charts that can be used to estimate errors for v, h, and s for standard barometric pressure.
To Obtain Use Comments pws(t) Table 3 or Equation (5) or (6) Sat. press. for temp. t Ws Equation (23) Using pws(t) W Equation (35) pws(t) Table 3 or Equation (5) or (6) Sat. press. for temp. t Ws Equation (23) Using pws(t) µ Equation (12) Using Ws φ Equation (25) Using pws(t) v Equation (28) h Equation (32) pw Equation (36) td Table 3 with Equation (36), (37), or (38) To Obtain Use Comments pw = pws(td) Table 3 or Equation (5) or (6) Sat. press. for temp. td W Equation (22) pws(t) Table 3 or Equation (5) or (6) Sat. press. for temp. td Ws Equation (23) Using pws(t) µ Equation (12) Using Ws φ Equation (25) Using pws(t) v Equation (28) h Equation (32) t Equation (23) and (35) with Table 3 or with Equation (5) or (6) Requires trial-and-error or numerical solution method td C14 C15α C16α2 C17α3 C18 pw ( )0.1984 + + + + = td 6.09 12.608α 0.4959α2 + + = To Obtain Use Comments pws(t) Table 3 or Equation (5) or (6) Sat. press. for temp. t pw Equation (24) W Equation (22) Ws Equation (23) Using pws(t) µ Equation (12) Using Ws v Equation (28) h Equation (32) td Table 3 with Equation (36), (37), or (38) t Equation (23) and (35) with Table 3 or with Equation (5) or (6) Requires trial-and-error or numerical solution method Table 4 Values of f and Estimated Maximum Uncertainties (EMUs) T, K 0.1 MPa 0.5 MPa 1 MPa f EMU E+04 f EMU E+04 f EMU E+04 173.15 1.0105 134 1.0540 66 1.1130 136 273.15 1.0039 2 1.0177 10 1.0353 19 373.15 1.0039 0.1 1.0180 4 1.0284 11 Ws 0.62198 f pws p fpws – --------------------= φ µ 1 1 µ – ( ) f pws p ⁄ ( ) – ---------------------------------------------------= Volume v vda µvas + = Enthalpy h hda µhas + = Entropy s sda µsas + = Psychrometrics 6.11 Fig. 1 ASHRAE Psychrometric Chart No. 1 6.12 2001 ASHRAE Fundamentals Handbook (SI) PSYCHROMETRIC CHARTS A psychrometric chart graphically represents the thermody-namic properties of moist air.
The choice of coordinates for a psychrometric chart is arbitrary.
A chart with coordinates of enthalpy and humidity ratio provides convenient graphical solutions of many moist air problems with a minimum of thermodynamic approximations. ASHRAE developed seven such psychrometric charts. Chart No. 1 is shown as Figure 1; the others may be obtained through ASHRAE.
Charts 1 through 4 are for sea level pressure (101.325 kPa). Chart 5 is for 750 m altitude (92.66 kPa), Chart 6 is for 1500 m altitude (84.54 kPa), and Chart 7 is for 2250 m altitude (77.04 kPa). All charts use oblique-angle coordinates of enthalpy and humidity ratio, and are consistent with the data of Table 2 and the properties com-putation methods of Goff and Gratch (1945), and Goff (1949) as well as Hyland and Wexler (1983a). Palmatier (1963) describes the geometry of chart construction applying specifically to Charts 1 and 4.
The dry-bulb temperature ranges covered by the charts are Charts 1, 5, 6, 7 Normal temperature 0 to 50°C Chart 2 Low temperature −40 to 10°C Chart 3 High temperature 10 to 120°C Chart 4 Very high temperature 100 to 200°C Psychrometric properties or charts for other barometric pressures can be derived by interpolation. Sufficiently exact values for most purposes can be derived by methods described in the section on Per-fect Gas Relationships for Dry and Moist Air. The construction of charts for altitude conditions has been treated by Haines (1961), Rohsenow (1946), and Karig (1946).
Comparison of Charts 1 and 6 by overlay reveals the following: 1. The dry-bulb lines coincide.
2. Wet-bulb lines for a given temperature originate at the intersections of the corresponding dry-bulb line and the two saturation curves, and they have the same slope.
3. Humidity ratio and enthalpy for a given dry- and wet-bulb temperature increase with altitude, but there is little change in relative humidity.
4. Volume changes rapidly; for a given dry-bulb and humidity ratio, it is practically inversely proportional to barometric pressure.
The following table compares properties at sea level (Chart 1) and 1500 m (Chart 6): Figure 1, which is ASHRAE Psychrometric Chart No. 1, shows humidity ratio lines (horizontal) for the range from 0 (dry air) to 30 g (water)/kg (dry air). Enthalpy lines are oblique lines drawn across the chart precisely parallel to each other.
Dry-bulb temperature lines are drawn straight, not precisely par-allel to each other, and inclined slightly from the vertical position.
Thermodynamic wet-bulb temperature lines are oblique lines that differ slightly in direction from that of enthalpy lines. They are straight but are not precisely parallel to each other.
Relative humidity lines are shown in intervals of 10%. The sat-uration curve is the line of 100% rh, while the horizontal line for W = 0 (dry air) is the line for 0% rh.
Specific volume lines are straight but are not precisely parallel to each other.
A narrow region above the saturation curve has been developed for fog conditions of moist air. This two-phase region represents a mechanical mixture of saturated moist air and liquid water, with the two components in thermal equilibrium. Isothermal lines in the fog region coincide with extensions of thermodynamic wet-bulb tem-perature lines. If required, the fog region can be further expanded by extension of humidity ratio, enthalpy, and thermodynamic wet-bulb temperature lines.
The protractor to the left of the chart shows two scales—one for sensible-total heat ratio, and one for the ratio of enthalpy difference to humidity ratio difference. The protractor is used to establish the direction of a condition line on the psychrometric chart.
Example 1 illustrates use of the ASHRAE Psychrometric Chart to determine moist air properties.
Example 1. Moist air exists at 40°C dry-bulb temperature, 20°C thermody-namic wet-bulb temperature, and 101.325 kPa pressure. Determine the humidity ratio, enthalpy, dew-point temperature, relative humidity, and specific volume.
Solution: Locate state point on Chart 1 (Figure 1) at the intersection of 40°C dry-bulb temperature and 20°C thermodynamic wet-bulb temper-ature lines. Read humidity ratio W = 6.5 g (water)/kg (dry air).
The enthalpy can be found by using two triangles to draw a line parallel to the nearest enthalpy line [60 kJ/kg (dry air)] through the state point to the nearest edge scale. Read h = 56.7 kJ/kg (dry air).
Dew-point temperature can be read at the intersection of W = 6.5 g (water)/kg (dry air) with the saturation curve. Thus, td = 7°C.
Relative humidity φ can be estimated directly. Thus, φ = 14%.
Specific volume can be found by linear interpolation between the volume lines for 0.88 and 0.90 m3/kg (dry air). Thus, v = 0.896 m3/kg (dry air).
TYPICAL AIR-CONDITIONING PROCESSES The ASHRAE psychrometric chart can be used to solve numer-ous process problems with moist air. Its use is best explained through illustrative examples. In each of the following examples, the process takes place at a constant total pressure of 101.325 kPa.
Moist Air Sensible Heating or Cooling The process of adding heat alone to or removing heat alone from moist air is represented by a horizontal line on the ASHRAE chart, since the humidity ratio remains unchanged.
Figure 2 shows a device that adds heat to a stream of moist air.
For steady flow conditions, the required rate of heat addition is (42) Example 2. Moist air, saturated at 2°C, enters a heating coil at a rate of 10 m3/s. Air leaves the coil at 40°C. Find the required rate of heat addition.
Chart No.
db wb h W rh v 1 40 30 99.5 23.0 49 0.920 6 40 30 114.1 28.6 50 1.111 q 1 2 m · da h2 h1 – ( ) = Fig. 2 Schematic of Device for Heating Moist Air Psychrometrics 6.13 Solution: Figure 3 schematically shows the solution. State 1 is located on the saturation curve at 2°C. Thus, h1 = 13.0 kJ/kg (dry air), W1 = 4.3 g (water)/kg (dry air), and v1 = 0.784 m3/kg (dry air). State 2 is located at the intersection of t = 40°C and W2 = W1 = 4.3 g (water)/kg (dry air). Thus, h2 = 51.6 kJ/kg (dry air). The mass flow of dry air is From Equation (42), Moist Air Cooling and Dehumidification Moisture condensation occurs when moist air is cooled to a tem-perature below its initial dew point. Figure 4 shows a schematic cooling coil where moist air is assumed to be uniformly processed.
Although water can be removed at various temperatures ranging from the initial dew point to the final saturation temperature, it is assumed that condensed water is cooled to the final air temperature t2 before it drains from the system.
For the system of Figure 4, the steady flow energy and material balance equations are Thus, (43) (44) Example 3. Moist air at 30°C dry-bulb temperature and 50% rh enters a cooling coil at 5 m3/s and is processed to a final saturation condition at 10°C. Find the kW of refrigeration required.
Solution: Figure 5 shows the schematic solution. State 1 is located at the intersection of t = 30°C and φ = 50%. Thus, h1 = 64.3 kJ/kg (dry air), W1 = 13.3 g (water)/kg (dry air), and v1 = 0.877 m3/kg (dry air).
State 2 is located on the saturation curve at 10°C. Thus, h2 = 29.5 kJ/kg (dry air) and W2 = 7.66 g (water)/kg (dry air). From Table 2, hw2 = 42.11 kJ/kg (water). The mass flow of dry air is From Equation (44), Adiabatic Mixing of Two Moist Airstreams A common process in air-conditioning systems is the adiabatic mixing of two moist airstreams. Figure 6 schematically shows the problem. Adiabatic mixing is governed by three equations: Fig. 3 Schematic Solution for Example 2 Fig. 4 Schematic of Device for Cooling Moist Air m · da 10 0.784 ⁄ 12.76 kg s (dry air) ⁄ = = q 1 2 12.76 51.6 13.0 – ( ) 492 kW = = m · dah1 m · dah2 q 1 2 m · whw2 + + = m · daW1 m · daW2 m · w + = m · w m · da W1 W2 – ( ) = Fig. 5 Schematic Solution for Example 3 q 1 2 m · da h1 h2 – ( ) W1 W2 – ( ) – hw2 [ ] = m · da 5 0.877 ⁄ 5.70 kg s (dry air) ⁄ = = q 1 2 5.70 64.3 29.5 – ( ) 0.0133 0.00766 – ( )42.11 – [ ] = 197 kW = Fig. 6 Adiabatic Mixing of Two Moist Airstreams 6.14 2001 ASHRAE Fundamentals Handbook (SI) Eliminating gives (45) according to which, on the ASHRAE chart, the state point of the resulting mixture lies on the straight line connecting the state points of the two streams being mixed, and divides the line into two seg-ments, in the same ratio as the masses of dry air in the two streams.
Example 4. A stream of 2 m3/s of outdoor air at 4°C dry-bulb temperature and 2°C thermodynamic wet-bulb temperature is adiabatically mixed with 6.25 m3/s of recirculated air at 25°C dry-bulb temperature and 50% rh. Find the dry-bulb temperature and thermodynamic wet-bulb temperature of the resulting mixture.
Solution: Figure 7 shows the schematic solution. States 1 and 2 are located on the ASHRAE chart, revealing that v1 = 0.789 m3/kg (dry air), and v2 = 0.858 m3/kg (dry air). Therefore, According to Equation (45), Consequently, the length of line segment 1–3 is 0.742 times the length of entire line 1–2. Using a ruler, State 3 is located, and the values t3 = 19.5°C and t3 = 14.6°C found.
Adiabatic Mixing of Water Injected into Moist Air Steam or liquid water can be injected into a moist airstream to raise its humidity. Figure 8 represents a diagram of this common air-conditioning process. If the mixing is adiabatic, the following equa-tions apply: Therefore, (46) according to which, on the ASHRAE chart, the final state point of the moist air lies on a straight line whose direction is fixed by the specific enthalpy of the injected water, drawn through the initial state point of the moist air.
Example 5. Moist air at 20°C dry-bulb and 8°C thermodynamic wet-bulb temperature is to be processed to a final dew-point temperature of 13°C by adiabatic injection of saturated steam at 110°C. The rate of dry air-flow is 2 kg/s (dry air). Find the final dry-bulb temperature of the moist air and the rate of steam flow.
Solution: Figure 9 shows the schematic solution. By Table 3, the enthalpy of the steam hg = 2691 kJ/kg (water). Therefore, according to Equation (46), the condition line on the ASHRAE chart connecting States 1 and 2 must have a direction: m · da1h1 m · da2h2 + m · da3h3 = m · da1 m · da2 + m · da3 = m · da1W1 m · da2W2 + m · da3W3 = m · da3 h2 h3 – h3 h1 – -----------------W2 W3 – W3 W1 – ---------------------m · da1 m · da2 ------------= = Fig. 7 Schematic Solution for Example 4 m · da1 2 0.789 ⁄ 2.535 kg s (dry air) ⁄ = = m · da2 6.25 0.858 ⁄ 7.284 kg s (dry air) ⁄ = = Line 3–2 Line 1–3 ---------------------m · da1 m · da2 ------------ or Line 1–3 Line 1–2 ---------------------m · da2 m · da3 ------------7.284 9.819 -------------0.742 = = = = Fig. 8 Schematic Showing Injection of Water into Moist Air Fig. 9 Schematic Solution for Example 5 m · dah1 m · whw + m · dah2 = m · daW1 m · w + m · daW2 = h2 h1 – W2 W1 – ---------------------h ∆ W ∆ --------hw = = h ∆ W ∆ ⁄ 2.691 kJ/g (water) = Psychrometrics 6.15 The condition line can be drawn with the ∆h/∆W protractor. First, establish the reference line on the protractor by connecting the origin with the value ∆h/∆W = 2.691 kJ/g (water). Draw a second line parallel to the reference line and through the initial state point of the moist air.
This second line is the condition line. State 2 is established at the inter-section of the condition line with the horizontal line extended from the saturation curve at 13°C (td2 = 13°C). Thus, t2 = 21°C.
Values of W2 and W1 can be read from the chart. The required steam flow is, Space Heat Absorption and Moist Air Moisture Gains Air conditioning a space is usually determined by (1) the quan-tity of moist air to be supplied, and (2) the supply air condition nec-essary to remove given amounts of energy and water from the space at the exhaust condition specified.
Figure 10 schematically shows a space with incident rates of energy and moisture gains. The quantity qs denotes the net sum of all rates of heat gain in the space, arising from transfers through boundaries and from sources within the space. This heat gain involves addition of energy alone and does not include energy con-tributions due to addition of water (or water vapor). It is usually called the sensible heat gain. The quantity Σ denotes the net sum of all rates of moisture gain on the space arising from transfers through boundaries and from sources within the space. Each kilo-gram of water vapor added to the space adds an amount of energy equal to its specific enthalpy.
Assuming steady-state conditions, governing equations are or (47) (48) The left side of Equation (47) represents the total rate of energy addition to the space from all sources. By Equations (47) and (48), (49) according to which, on the ASHRAE chart and for a given state of the withdrawn air, all possible states (conditions) for the supply air must lie on a straight line drawn through the state point of the with-drawn air, that has a direction specified by the numerical value of . This line is the condition line for the given problem.
Example 6. Moist air is withdrawn from a room at 25°C dry-bulb tempera-ture and 19°C thermodynamic wet-bulb temperature. The sensible rate of heat gain for the space is 9 kW. A rate of moisture gain of 0.0015 kg/s (water) occurs from the space occupants. This moisture is assumed as saturated water vapor at 30°C. Moist air is introduced into the room at a dry-bulb temperature of 15°C. Find the required thermo-dynamic wet-bulb temperature and volume flow rate of the supply air.
Solution: Figure 11 shows the schematic solution. State 2 is located on the ASHRAE chart. From Table 3, the specific enthalpy of the added water vapor is hg = 2555.52 kJ/kg (water). From Equation (49), With the ∆h/∆W protractor, establish a reference line of direction ∆h/∆W = 8.555 kJ/g (water). Parallel to this reference line, draw a straight line on the chart through State 2. The intersection of this line with the 15°C dry-bulb temperature line is State 1. Thus, t1 = 13.8°C.
An alternate (and approximately correct) procedure in establishing the condition line is to use the protractor’s sensible-total heat ratio scale instead of the ∆h/∆W scale. The quantity ∆Hs/∆Ht is the ratio of the rate of sensible heat gain for the space to the rate of total energy gain for the space. Therefore, Note that ∆Hs/∆Ht = 0.701 on the protractor coincides closely with ∆h/∆W = 8.555 kJ/g (water).
The flow of dry air can be calculated from either Equation (47) or (48). From Equation (47), Therefore, supply volume = = 0.856 × 0.859 = 0.735 m3/s m · w m · da W2 W1 – ( ) 2 1000 × 0.0093 0.0018 – ( ) = = 15.0 kg/s (steam) = m · w m · dah1 qs m · whw ( ) ∑ + + m · dah2 = m · daW1 m · w ∑ + m · daW2 = qs m · whw ( ) ∑ + m · da h2 h1 – ( ) = m · w ∑ m · da W2 W1 – ( ) = Fig. 10 Schematic of Air Conditioned Space h2 h1 – W2 W1 – ---------------------h ∆ W ∆ --------qs m · whw ( ) ∑ + m · w ∑ ------------------------------------= = qs Σ m · whw ( ) + [ ] Σm · w ⁄ Fig. 11 Schematic Solution for Example 6 h ∆ W ∆ --------9 0.0015 2555.52 × ( ) + 0.0015 --------------------------------------------------------8555 kJ/kg (water) = = Hs ∆ Ht ∆ ---------qs qs Σ m · whw ( ) + ----------------------------------9 9 0.0015 2555.52 × ( ) + --------------------------------------------------------0.701 = = = m · da qs Σ m · whw ( ) + h2 h1 – ----------------------------------9 0.0015 2555.52 × ( ) + 54.0 39.0 – --------------------------------------------------------= = 0.856 kg s (dry air) ⁄ = At State 1, v1 0.859 m3/kg (dry air) = m · dav1 6.16 2001 ASHRAE Fundamentals Handbook (SI) TRANSPORT PROPERTIES OF MOIST AIR For certain scientific and experimental work, particularly in the heat transfer field, many other moist air properties are important.
Generally classified as transport properties, these include diffusion coefficient, viscosity, thermal conductivity, and thermal diffusion factor. Mason and Monchick (1965) derive these properties by cal-culation. Table 5 and Figures 12 and 13 summarize the authors’ results on the first three properties listed. Note that, within the boundaries of ASHRAE Psychrometric Charts 1, 2, and 3, the vis-cosity varies little from that of dry air at normal atmospheric pres-sure, and the thermal conductivity is essentially independent of moisture content.
REFERENCES FOR AIR, WATER, AND STEAM PROPERTIES Coefficient fw (over water) at pressures from 0.5 to 110 kPa for temperatures from −50 to 60°C (Smithsonian Institution).
Coefficient fi (over ice) at pressures from 0.5 to 110 kPa for temper-atures from 0 to 100°C (Smithsonian Institution).
Compressibility factor of dry air at pressures from 1 kPa to 10 MPa and at temperatures from 50 to 3000 K (Hilsenrath et al. 1960).
Compressibility factor of moist air at pressures from 0 to 10 MPa, at values of degree of saturation from 0 to 100, and for temperatures from 0 to 60°C (Smithsonian Institution). [Note: At the time the Smithsonian Meteorological Tables were published, the value µ = W/Ws was known as relative humidity, in terms of a percent-age. Since that time, there has been general agreement to desig-nate the value µ as degree of saturation, usually expressed as a decimal and sometimes as a percentage. See Goff (1949) for more recent data and formulations.] Compressibility factor for steam at pressures from 100 kPa to 30 MPa and at temperatures from 380 to 850 K (Hilsenrath et al.
1960).
Density, enthalpy, entropy, Prandtl number, specific heat, specific heat ratio, and viscosity of dry air (Hilsenrath et al. 1960).
Density, enthalpy, entropy, specific heat, viscosity, thermal conduc-tivity, and free energy of steam (Hilsenrath et al. 1960).
Dry air. Thermodynamic properties over a wide range of tempera-ture (Keenan and Kaye 1945).
Enthalpy of saturated steam (Osborne et al. 1939).
Ideal-gas thermodynamic functions of dry air at temperatures from 10 to 3000 K (Hilsenrath et al. 1960).
Ideal-gas thermodynamic functions of steam at temperatures from 50 to 5000 K. Functions included are specific heat, enthalpy, free energy , and entropy (Hilsenrath et al. 1960).
Moist air properties from tabulated virial coefficients (Chaddock 1965).
Saturation humidity ratio over ice at pressures from 30 to 100 kPa and for temperatures from −88.8 to 0°C (Smithsonian Institu-tion).
Saturation humidity ratio over water at pressures from 6 to 105 kPa and for temperatures from −50 to 59°C (Smithsonian Institu-tion).
Saturation vapor pressure over water for temperatures from −50 to 102°C (Smithsonian Institution).
Speed of sound in dry air at pressures from 0.001 to 10 MPa for tem-peratures from 50 to 3000 K (Hilsenrath et al. 1960). At atmo-spheric pressure for temperatures from −90 to 60°C (Smithsonian Institution).
Speed of sound in moist air. Relations using the formulation of Goff and Gratch and studies by Hardy et al. (1942) give methods for calculating this speed (Smithsonian Institution).
Steam tables covering the range from –40 to 1315°C (Keenan et al.
1969).
Transport properties of moist air. Diffusion coefficient, viscosity, thermal conductivity, and thermal diffusion factor of moist air are listed (Mason and Monchick 1965). The authors’ results are summarized in Table 5 and Figures 12 and 13.
Virial coefficients and other information for use with Goff and Gratch formulation (Goff 1949).
Volume of water in cubic metres for temperatures from −10 to 250°C (Smithsonian Institution 1954).
Water properties. Includes properties of ordinary water substance for the gaseous, liquid, and solid phases (Dorsey 1940).
Table 5 Calculated Diffusion Coefficients for Water−Air at 101.325 kPa Temp., °C mm2/s Temp., °C mm2/s Temp., °C mm2/s −70 13.2 0 22.2 50 29.5 −50 15.6 5 22.9 55 30.3 −40 16.9 10 23.6 60 31.1 −35 17.5 15 24.3 70 32.7 −30 18.2 20 25.1 100 37.6 −25 18.8 25 25.8 130 42.8 −20 19.5 30 26.5 160 48.3 −15 20.2 35 27.3 190 54.0 −10 20.8 40 28.0 220 60.0 −5 21.5 45 28.8 250 66.3 Fig. 12 Viscosity of Moist Air Fig. 13 Thermal Conductivity of Moist Air Psychrometrics 6.17 SYMBOLS C1 to C18 = constants in Equations (5), (6), and (37) dv = absolute humidity of moist air, mass of water per unit volume of mixture f = enhancement factor, used in Equations (23a) and (25a) h = specific enthalpy of moist air hs = specific enthalpy of saturated moist air at thermodynamic wet-bulb temperature hw = specific enthalpy of condensed water (liquid or solid) at thermo-dynamic wet-bulb temperature and pressure of 101.325 kPa Hs = rate of sensible heat gain for space Ht = rate of total energy gain for space = mass flow of dry air, per unit time = mass flow of water (any phase), per unit time Mda = mass of dry air in moist air sample Mw = mass of water vapor in moist air sample n = nda + nw, total number of moles in moist air sample nda = moles of dry air nw = moles of water vapor p = total pressure of moist air pda = partial pressure of dry air ps = vapor pressure of water in moist air at saturation. Differs from saturation pressure of pure water because of presence of air.
pw = partial pressure of water vapor in moist air pws = pressure of saturated pure water qs = rate of addition (or withdrawal) of sensible heat R = universal gas constant, 8314.41 J/(kg mole·K) Rda = gas constant for dry air Rw = gas constant for water vapor s = specific entropy t = dry-bulb temperature of moist air td = dew-point temperature of moist air t = thermodynamic wet-bulb temperature of moist air T = absolute temperature v = specific volume vT = total gas volume V = total volume of moist air sample W = humidity ratio of moist air, mass of water per unit mass of dry air Ws = humidity ratio of moist air at saturation at thermodynamic wet-bulb temperature xda = mole-fraction of dry air, moles of dry air per mole of mixture xw = mole-fraction of water, moles of water per mole of mixture xws = mole-fraction of water vapor under saturated conditions, moles of vapor per mole of saturated mixture Z = altitude α = ln(pw), parameter used in Equations (37) and (38) γ = specific humidity of moist air, mass of water per unit mass of mixture µ = degree of saturation W/Ws ρ = moist air density φ = relative humidity, dimensionless Subscripts as = difference between saturated moist air and dry air da = dry air f = saturated liquid water fg = difference between saturated liquid water and saturated water vapor g = saturated water vapor i = saturated ice ig = difference between saturated ice and saturated water vapor s = saturated moist air t = total w = water in any phase REFERENCES Chaddock, J.B. 1965. Moist air properties from tabulated virial coefficients.
Humidity and moisture measurement and control in science and industry 3:273. A. Wexler and W.A. Wildhack, eds. Reinhold Publishing, New York.
Dorsey, N.E. 1940. Properties of ordinary water substance. Reinhold Pub-lishing, New York.
Goff, J.A. 1949. Standardization of thermodynamic properties of moist air.
Heating, Piping, and Air Conditioning 21(11):118.
Goff, J.A. and S. Gratch. 1945. Thermodynamic properties of moist air.
ASHVE Transactions 51:125.
Goff, J.A., J.R. Anderson, and S. Gratch. 1943. Final values of the interac-tion constant for moist air. ASHVE Transactions 49:269.
Haines, R.W. 1961. How to construct high altitude psychrometric charts.
Heating, Piping, and Air Conditioning 33(10):144.
Hardy, H.C., D. Telfair, and W.H. Pielemeier. 1942. The velocity of sound in air. Journal of the Acoustical Society of America 13:226.
Harrison, L.P. 1965. Fundamental concepts and definitions relating to humidity. Humidity and moisture measurement and control in science and industry 3:3. A. Wexler and W.A. Wildhack, eds. Reinhold Publish-ing, New York.
Hilsenrath, J. et al. 1960. Tables of thermodynamic and transport properties of air, argon, carbon dioxide, carbon monoxide, hydrogen, nitrogen, oxy-gen, and steam. National Bureau of Standards. Circular 564, Pergamon Press, New York.
Hyland, R.W. and A. Wexler. 1983a. Formulations for the thermodynamic properties of dry air from 173.15 K to 473.15 K, and of saturated moist air from 173.15 K to 372.15 K, at pressures to 5 MPa. ASHRAE Trans-actions 89(2A):520-35.
Hyland, R.W. and A. Wexler. 1983b. Formulations for the thermodynamic properties of the saturated phases of H2O from 173.15 K to 473.15 K.
ASHRAE Transactions 89(2A):500-519.
Karig, H.E. 1946. Psychrometric charts for high altitude calculations.
Refrigerating Engineering 52(11):433.
Keenan, J.H. and J. Kaye. 1945. Gas tables. John Wiley and Sons, New York.
Keenan, J.H., F.G. Keyes, P.G. Hill, and J.G. Moore. 1969. Steam tables. John Wiley and Sons, New York.
Kuehn, T.H., J.W. Ramsey, and J.L. Threlkeld. 1998. Thermal environ-mental engineering, 3rd ed., p. 188. Prentice-Hall, Upper Saddle River, NJ.
Kusuda, T. 1970. Algorithms for psychrometric calculations. NBS Publica-tion BSS21 (January) for sale by Superintendent of Documents, U.S.
Government Printing Office, Washington, D.C.
Mason, E.A. and L. Monchick. 1965. Survey of the equation of state and transport properties of moist gases. Humidity and moisture measurement and control in science and industry 3:257. Reinhold Publishing, New York.
NASA. 1976. U.S. Standard atmosphere, 1976. National Oceanic and Atmo-spheric Administration, National Aeronautics and Space Administration, and the United States Air Force. Available from National Geophysical Data Center, Boulder, CO.
NIST. 1990. Guidelines for realizing the international temperature scale of 1990 (ITS-90). NIST Technical Note 1265. National Institute of Tech-nology and Standards, Gaithersburg, MD.
Osborne, N.S. 1939. Stimson and Ginnings. Thermal properties of satu-rated steam. Journal of Research, National Bureau of Standards, 23(8):261.
Olivieri, J. 1996. Psychrometrics—Theory and practice. ASHRAE, Atlanta.
Palmatier, E.P. 1963. Construction of the normal temperature. ASHRAE psychrometric chart. ASHRAE Journal 5:55.
Peppers, V.W. 1988. Unpublished paper. Available from ASHRAE.
Preston-Thomas, H. 1990. The international temperature scale of 1990 (ITS-90). Metrologia 27(1):3-10.
Rohsenow, W.M. 1946. Psychrometric determination of absolute humidity at elevated pressures. Refrigerating Engineering 51(5):423.
Smithsonian Institution. 1954. Smithsonian physical tables, 9th rev. ed.
Available from the Smithsonian Institution, Washington, D.C.
Smithsonian Institution. Smithsonian meteorological tables, 6th rev. ed. Out of print, but available in many libraries. Washington, D.C.
m · da m · w 7.1 CHAPTER 7 SOUND AND VIBRATION Acoustical Design Objective ..................................................... 7.1 Characteristics of Sound ........................................................... 7.1 Measuring Sound ...................................................................... 7.3 Determining Sound Power ........................................................ 7.5 Converting from Sound Power to Sound Pressure ................... 7.6 Sound Transmission Paths ........................................................ 7.6 Typical Sources of Sound .......................................................... 7.7 Controlling Sound ..................................................................... 7.7 System Effects ............................................................................ 7.9 Human Response to Sound ........................................................ 7.9 Sound Rating Systems and Acoustical Design Goals ....................................................................... 7.11 Fundamentals of Vibration ..................................................... 7.14 Single-Degree-of-Freedom Model .......................................... 7.14 Two-Degree-of-Freedom Model ............................................. 7.16 Vibration Measurement Basics ............................................... 7.17 F FUNDAMENTAL PRINCIPLES of sound and vibration con-I trol are applied in the design, installation, and use of HVAC and refrigeration systems, unacceptably high noise and vibration levels and the consequent complaints can be avoided. This chapter intro-duces these fundamental principles, including characteristics of sound, basic definitions and terminology, human response to sound, acoustical design goals, and vibration isolation fundamen-tals. Chapter 46 of the 1999 ASHRAE Handbook—Applications and the references listed at the end of this chapter contain technical dis-cussions, tables, and design examples hellpful to HVAC designers.
ACOUSTICAL DESIGN OBJECTIVE The primary objective for the acoustical design of HVAC sys-tems and equipment is to ensure that the acoustical environment in a given space is not degraded. Sound and vibration are created by a source, are transmitted along one or more paths, and reach a receiver. Treatments and modifications can be applied to any or all of these elements to achieve an acceptable acoustical environment, although it is usually most effective and least expensive to reduce noise at the source.
CHARACTERISTICS OF SOUND Sound is a propagating disturbance in a fluid (gas or liquid) or in a solid. In fluid media, the disturbance travels as a longitudinal com-pression wave. Sound in air is called airborne sound or simply sound. It is generated by a vibrating surface or a turbulent fluid stream. In solids, sound can travel as bending waves, compressional waves, torsional waves, shear waves and others. Sound in solids is generally called structureborne sound. In HVAC system design, both airborne and structureborne sound propagation are important.
Speed The speed of a longitudinal wave in a fluid is a function of the fluid’s density and bulk modulus of elasticity. In air, at room tem-perature, the speed of sound is about 340 m/s; in water, it is about 1500 m/s. In solids, there are several different types of waves, each with a different speed and some that depend on frequency. In solids, the speed of sound is usually higher than that in air.
Sound Pressure and Sound Pressure Level Sound waves in air are variations in pressure above and below atmospheric pressure. Sound pressure is measured in pascals. The human ear responds across a broad range of sound pressures; the threshold of hearing to the threshold of pain covers a range of approximately 1014:1. Table 1 gives approximate values of sound pressure for various sources.
The range of sound pressure in Table 1 is so large that it is more convenient to use a scale that is proportional to the logarithm of this quantity. The decibel (dB) scale is such a scale and is the preferred method of presenting quantities in acoustics. The term level, when used in relation to sound power, sound intensity, or sound pressure, indicates that dB notation is being used. Numerically, the decibel is ten times the base 10 logarithm of the ratio of two like quantities proportional to acoustical power or energy. Thus, the sound pres-sure level Lp (in dB) corresponding to a sound pressure is given by (1) where p is the root mean square (rms) value of acoustic pressure in pascals. The ratio p/pref is squared to give quantities proportional to intensity or energy. The reference quantity pref is 20 µPa, which cor-responds to the approximate threshold of hearing. The decibel scale is used for many different descriptors relat-ing to sound: source strength, sound level, and sound attenuation, among others; each has a different reference quantity. For this rea-son, it is important to be aware of the context in which the term decibel or level is used. For most acoustical quantities, there is an The preparation of this chapter is assigned to TC 2.6, Sound and Vibration Control.
Table 1 Typical Sound Pressures and Sound Pressure Levels Source Sound Pressure, Pa Sound Pressure Level, dB re 20 µPa Subjective Reaction Military jet takeoff at 30 m 200 140 Extreme danger Artillery fire at 3 m 63.2 130 Passenger jet takeoff at 30 m 20 120 Threshold of pain Loud rock band 6.3 110 Threshold of discomfort Platform of subway station (steel wheels) 2 100 Unmuffled large diesel engine at 40 m 0.6 90 Very loud Computer printout room 0.2 80 Freight train at 30 m 0.06 70 Conversational speech at 1 m 0.02 60 Window air conditioner at 1 m 0.006 50 Moderate Quiet residential area 0.002 40 Whispered conversation at 2 m 0.0006 30 Buzzing insect at 1 m 0.0002 20 Perceptible Threshold of good hearing 0.00006 10 Faint Threshold of excellent youthful hearing 0.00002 0 Threshold of hearing Lp 10 p pref ⁄ ( )2 log = 7.2 2001 ASHRAE Fundamentals Handbook (SI) internationally accepted reference value. A reference quantity is always implied even if it does not appear.
Sound pressure level is relatively easy to measure and for this reason most noise codes and criteria use sound pressure level. (The human ear and microphones are pressure-sensitive devices.) Sound pressure levels for each source are also given in Table 1.
Frequency Frequency is the number of oscillations (or cycles) completed per second by a vibrating object. The international unit for fre-quency is hertz (Hz) with dimension s–1. When the motion of vibrat-ing air particles is simple harmonic, the sound is said to be a pure tone and the sound pressure p as a function of time and frequency can be described by (2) where f is frequency in hertz and θ is time in seconds. The audible frequency range for humans extends from about 20 Hz to 20 kHz. In some cases, infrasound (< 20 Hz) or ultrasound (> 20 kHz) are important, but methods and instrumentation for these frequency regions are specialized and are not considered here.
Wavelength The wavelength of sound in a medium is the distance between successive maxima or minima of a simple harmonic disturbance propagating in that medium. Wavelength, speed, and frequency are related by λ = c ⁄ f (3) where λ = wavelength, m c = speed of sound, m/s f = frequency, Hz Sound Power and Sound Power Level The sound power of a source is its rate of emission of acoustical energy and is expressed in watts. The sound power of a source depends on the operating conditions. Approximate sound power outputs for common sources are shown in Table 2 together with the corresponding sound power levels. For sound power level Lw, the power reference is 10–12 W. The definition of sound power level is therefore (4) where w is the acoustic power emitted by the source in watts. The acoustic power emitted by a source is not the same as the power consumed by the source. Only a small fraction of the consumed power is converted into sound. For example, a loudspeaker rated at 100 W may be only 1% to 5% efficient, generating only 1 to 5 W (acoustic). Most mechanical equipment is rated according to sound power so equipment can be compared according to a common reference independent of distance and acoustical conditions in the room.
AMCA Publication 303 provides guidelines for the application of sound power level ratings. In addition, AMCA Standard 301 pro-vides standard methods for developing fan sound ratings from lab-oratory test data.
Sound Intensity and Sound Intensity Level The sound intensity I at a point in a specified direction is the rate of flow of sound energy through unit area at that point. The unit area is perpendicular to the specified direction, and the units of intensity are W/m2. Sound intensity level LI is expressed in dB with a reference quantity of 10–12 W/m2, thus Combining Sound Levels Because the decibel is a logarithmic unit, two sound pressure lev-els cannot be added arithmetically. The levels must first be con-verted back to energy units, summed, and then converted to a level again. Thus, the combination of two levels, L1 and L2, produces a level Lsum given by (5) This process may be extended to as many levels as needed. A simpler and slightly less accurate method is outlined in Table 3. This method, although not exact, results in errors of 1 dB or less. The pro-cess with a series of levels may be shortened by combining the larg-est with the next largest. Then this sum is combined with the third largest, then the fourth largest, and so on until the next level added has little or no influence. The process may then be stopped.
The procedures outlined in Table 3 and in Equation (5) are valid if the individual sound levels are not highly correlated. This is true for most (but not all) sounds encountered in HVAC work. One nota-ble exception is the pure tone. If two or more noise signals contain pure tones at the same frequency, the combined sound level is a function of not only the level of each tone, but also the phase differ-ence between the tones. The combined sound levels from two tones of equal amplitude can range from zero (if the two tones are 180° out of phase) up to 6 dB greater than the level of either tone (if the two tones are exactly in phase). When two tones of similar ampli-tude are very close in frequency, but not exactly the same, the com-bined sound level oscillates as the tones move in and out of phase.
This effect creates an audible “beating” with a period equal to the inverse of the difference in frequency between the two tones. p θ f , ( ) p0 2π f θ sin = Lw 10 w 10 12 – ⁄ ( ) log = Table 2 Typical Sound Power Outputs and Sound Power Levels Source Sound Power, W Sound Power Level, dB re 10–12 W Saturn rocket 108 200 Turbojet enginea 105 170 Jet aircraft at takeoffb 104 160 Turboprop at takeoff 1000 150 Prop aircraft at takeoffc 100 140 Large pipe organ 10 130 Small aircraft engine 1 120 Noisy HVAC fan 0.1 110 Automobile at highway speed 0.01 100 Voice, shouting 0.001 90 Garbage disposal unit 10–4 80 Voice, conversation level 10–5 70 Electronic equipment ventilation fan 10–6 60 Office air diffuser 10–7 50 Small electric clock 10–8 40 Voice, soft whisper 10–9 30 Rustling leaves 10–10 20 Human breath 10–11 10 a With afterburner b Four jet engines c Four propeller engines Table 3 Combining Two Sound Levels Difference between two levels to be combined, dB 0 to 1 2 to 4 5 to 9 10 and More Number of decibels to be added to highest level to obtain combined level 3 2 1 0 LI I 10 12 – ⁄ ( ) log = Lsum 10 10 L1 10 ⁄ 10 L2 10 ⁄ + ( ) log = Sound and Vibration 7.3 MEASURING SOUND Instrumentation The basic instrument for measuring sound is a sound level meter, which includes a microphone, electronic circuitry, and a display de-vice. Sound pressure at a point is converted to sound pressure level and displayed by analog or digital meters. These devices are usually light, battery-operated, hand-held units with outputs that vary in complexity depending on cost and the level of current technology.
Time-Averaging No sounds are constant; the pressure fluctuates from moment to moment and the level can vary quickly or slowly. Sound level meters can show the time fluctuations of the sound pressure level using specified time constants (slow, fast, impulse), or can hold the maximum or minimum level recorded during some specified inter-val. All sound level meters perform some kind of time-averaging.
Some integrating sound level meters take an average of the sound pressure level over a user-definable time, then hold and display the result. As a result, an integrating meter is easier to read and more repeatable (especially if the measurement period is long). The quan-tity measured by the integrating sound level meter is the equivalent continuous sound pressure level Leq.
Spectra and Analysis Bandwidths Real sounds are much more complex than simple pure tones.
Broadband sound contains energy at many frequencies, usually covering most of the audible frequency range. All sounds, however, can be represented as a summation of pure tones with different amplitudes. This representation of a sound is called frequency or spectral analysis and is similar to spectral analysis in optics.
A constant-bandwidth analysis expresses the energy content of a sound as a spectrum where each data point represents the same spectral width, for example, 1 Hz. This kind of analysis is useful when an objectionable sound obviously contains strong tones and the frequencies need to be accurately identified before remedial action is taken. A constant-band spectrum usually contains too much information for typical noise control work.
Measurements for most HVAC noise control work are usually made with filters that extract the energy in either octave bands or one-third octave bands. An octave band is a frequency band with its upper frequency limit twice that of its lower frequency limit.
Octave and 1/3 octave bands are identified by the mid-band fre-quency, which is the geometric mean of the upper and lower band limits (ANSI Standards S1.6, S1.11). Three 1/3 octave bands make up an octave band. Table 4 lists the preferred series of octave and one-third octave bands and the upper and lower band limit frequen-cies. For HVAC sound measurements, filters for the range of 16 Hz to 8000 Hz are usually adequate.
While analysis in octave bands is usually acceptable for rating acoustical environments in rooms, 1/3 octave band analysis is often useful in product development and for remedial investigations.
Some sound level meters have octave or 1/3 octave filters for determining the frequency content of the sound. Usually, standard broadband filters that simulate the response of the average human ear to sound are provided. The most commonly used filter is the A-weighting filter (Figure 1). This filter simulates the response of the human ear. It de-emphasizes the low-frequency portions of a sound spectrum, automatically compensating for the lower sensitiv-ity of the human ear to low-frequency sound. The C-weighting filter is sometimes used to estimate whether a particular sound has excessive low-frequency energy present when a spectrum analyzer is not available. If the difference between the C- and A-weighted levels for the sound exceeds about 30 dB, then the sound is likely to be annoying because of excessive low-frequency noise. Note (in Figure 1) that the C-weighting curve attenuates significantly at low and high frequencies; the instru-ment response is not flat when this filter is used.
Sound level meters are available in several accuracy grades (ANSI Standard S1.4). A Type 1 meter has a tolerance of about ±0.7 dB. The general-purpose meter, which is less expensive, is desig-nated Type 2 with a tolerance of about ±1 dB, and is adequate for most HVAC sound measurements.
Table 4 Mid-Band and Approximate Upper and Lower Cutoff Frequencies for Octave and 1/3 Octave Band Filters Octave Bands, Hz 1/3 Octave Bands, Hz Lower Mid-Band Upper Lower Mid-Band Upper 11.2 12.5 14 11.2 16 22.4 14 16 18 18 20 22.4 22.4 25 28 22.4 31.5 45 28 31.5 35.5 35.5 40 45 45 50 56 45 63 90 56 63 71 71 80 90 90 100 112 90 125 180 112 125 140 140 160 180 180 200 224 180 250 355 224 250 280 280 315 355 355 400 450 355 500 710 450 500 560 560 630 710 710 800 900 710 1 000 1 400 900 1 000 1 120 1 120 1 250 1 400 1 400 1 600 1 800 1 400 2 000 2 800 1 800 2 000 2 240 2 240 2 500 2 800 2 800 3 150 3 550 2 800 4 000 5 600 3 550 4 000 4 500 4 500 5 000 5 600 5 600 6 300 7 100 5 600 8 000 11 200 7 100 8 000 9 000 9 000 10 000 11 200 11 200 12 500 14 000 11 200 16 000 22 400 14 000 16 000 18 000 18 000 20 000 22 400 Fig. 1 Curves Showing A- and C-Weighting Responses for Sound Level Meters 7.4 2001 ASHRAE Fundamentals Handbook (SI) Manually selecting filters sequentially to cover the frequency range from 16 Hz to 8000 Hz is time-consuming. An instrument that gives all filtered levels simultaneously is called a real-time ana-lyzer (RTA). It speeds up measurement significantly and, on most models, the information can be saved to an internal or external dig-ital storage device.
The process described in the previous section on Combining Sound Levels can be applied to a set of octave or 1/3 octave bands to calculate the overall broadband level (see Table 5 for an example).
Sound Measurement Basics The sound pressure level in an occupied space can be measured directly with a sound level meter, or estimated from published sound power data after accounting for room volume, distance from the source, and other acoustical factors. Sound level meters measure the sound pressure at the microphone location. Estimation tech-niques calculate sound pressure at a specified point in an occupied space. Measured or estimated sound pressure levels in frequency bands can then be plotted, analyzed, and compared with established criteria for acceptance.
Measurements of HVAC sound must be done carefully to ensure repeatable and accurate results. The sound levels may not be steady, particularly at low frequencies (250 Hz and lower) and can vary sig-nificantly with time. In such cases, both the peak and average levels should be recorded.
Sophisticated sound measurements and their procedures should be carried out by experienced sound professionals. At present, only a few noise standards apply to measuring the interior noise from mechanical equipment (ASTM Standard E 1574, ASTM Standard E 1573). Most manuals for sound level meters include sections on how to measure sound.
Determining the sound spectrum in a room or investigating a noise complaint usually requires measuring the sound pressure lev-els in the octave bands from 16 Hz to 8000 Hz. In cases of tonal noise or rumble, narrow bands or 1/3 octave bands should be mea-sured because their frequency resolution is higher. Whatever the measurement method, the sound pressure level is measured at a point. In a room, each measurement point often has a different sound pressure level, so the precise location of the measurement must be detailed in the report. One might survey the room and record the location and level of the loudest position. Also, one could establish a few representative locations where occupants are nor-mally situated. In general, the most appropriate height is 1.2 to 1.8 m above the floor. The exact geometric center of the room should be avoided, as should any location within 1 m of a wall.
Wherever the location, it must be defined and recorded. If the meter has an integrating-averaging function, one can use a rotating boom to sample a large area or walk slowly around the room to mea-sure the average sound pressure level over that path. However, care must be taken that no extraneous sounds are generated by micro-phone movement or by walking. Also, locations where the sound levels are notably higher than average should be recorded. The sec-tion on Measurement of Room Sound Pressure Level has more details.
The contribution of other sources (plumbing noise, business machines, nearby traffic, etc.) to the sound pressure levels must also be determined. Sound from sources other than the source to be mea-sured is designated background sound. Sometimes the sound from a particular piece of HVAC equip-ment must be measured in the presence of background sound from other sources that cannot be turned off, such as automobile traffic or certain business machines. Two sets of measurements are required to determine the sound level due to selected equipment: one set with both the HVAC equipment sound and the background sound and another set with only the background sound (the HVAC equipment is turned off). This situation might also occur, for example, when determining whether the noise exposure at the property line due to a cooling tower meets a local noise ordinance. The guidelines in Table 6 will help in determining the sound level of a particular machine in the presence of background sound.
The uncertainty associated with correcting for background sound depends on the uncertainty of the measuring instrument and the steadiness of the sounds being measured. In favorable circum-stances, it might be possible to extend Table 6. In particularly unfa-vorable circumstances, even values obtained from the table could be substantially in error.
Measuring sound emissions from a particular piece of equipment or group of equipment requires a measurement plan specific to the situation. The section on Standards lists several sound level mea-surement procedures for various laboratory and field sound mea-surement situations.
Outdoor measurements are somewhat easier to make than indoor because typically there are few or no boundary surfaces to affect sound build-up or absorption. Nevertheless, important issues such as the effect of large, nearby sound-reflecting surfaces and weather conditions such as wind, temperature, and precipitation must be considered. (In cases where measurements are made close to ex-tended surfaces, sound pressure levels can be significantly in-creased.) These effects can be estimated through guidelines in many sources such as Harris (1991).
Measurement of Room Sound Pressure Level In the commissioning of HVAC systems in buildings, it often must be demonstrated that a specified room noise criterion has been met. Measurement procedures for obtaining the data to demonstrate compliance are often not specified. This can lead to confusion when different parties make measurements using different procedures, as the results often do not agree. The problem is that most rooms exhibit significant point-to-point variation in sound pressure level.
When a noise has no audible tonal components, the differences in measured sound pressure level at several locations in a room may be as high as 3 to 5 dB. However, when audible tonal components are present, especially at low frequencies, the variations due to standing waves may exceed 10 dB. These variations are generally noticeable to the average listener when moving through the room.
Table 5 Combining Decibels to Determine Overall Sound Pressure Level Octave Band Frequency, Hz Octave Band Level Lp, dB 10 Lp /10 63 85 3.2 × 108 = 0.32 × 109 125 90 1.0 × 109 = 1.0 × 109 250 92 1.6 × 109 = 1.6 × 109 500 87 5.0 × 108 = 0.5 × 109 1000 82 1.6 × 108 = 0.16 × 109 2000 78 6.3 × 107 = 0.06 × 109 4000 65 3.2 × 106 = 0.003 × 109 8000 54 2.5 × 105 = 0.0002 × 109 3.6432 × 109 10 log (3.6 × 109) = 96 dB Table 6 Guidelines for Determining Equipment Sound Levels in the Presence of Contaminating Background Sound Measurement A minus Measurement B Correction to Measurement A to Obtain Equipment Sound Level 10 dB or more 0 dB 6 to 9 dB –1 dB 4 to 5 dB –2 dB Less than 4 dB Equipment sound level is more than 2 dB below Measurement A Measurement A = Tested equipment plus background sound Measurement B = Background sound alone Sound and Vibration 7.5 Although the commissioning procedures usually set precise lim-its for demonstrating compliance, the outcome can be controversial unless the measurement procedure has been specified in detail. At present, the industry has no general agreement regarding an acous-tical measurement procedure for commissioning HVAC systems.
However, ARI Standard 885 incorporates a “suggested procedure for field verification of NC/RC levels.” DETERMINING SOUND POWER No instrument can measure the sound power of a source directly.
Rather, sound power is calculated from several measurements of sound pressure or sound intensity created by a source in one of sev-eral test environments. The following four methods are commonly used.
1. Free-Field Method A free field is a sound field where the effects of any boundaries are negligible over the frequency range of interest. In ideal condi-tions, there are no boundaries. Free-field conditions can be approx-imated in rooms having highly sound-absorbing walls, floor, and ceiling (anechoic rooms). In a free field the sound power of a sound source can be determined from a number of measurements of sound pressure level on an imaginary spherical surface centered on and surrounding the source. This method is based on the fact that, because absorption of sound in air can be practically neglected at small distances from the sound source, the sound power generated by a source must flow through an imagined sphere with the source at its center. The intensity of the sound is determined at each of the measuring points around the source and multiplied by the area of the imagined sphere associated with the measuring points. Total sound power is the sum of these products for each point.
ANSI Standard S12.35 describes various methods used to calcu-late the sound power level under free-field conditions. Measure-ment accuracy is limited at the lower frequencies because room surface treatments do not have high sound absorption coefficients at low frequencies. For example, a glass fiber wedge structure that gives significant absorption at 70 Hz must be at least 1.2 m long.
Using values for the speed of sound for air at 20°C and 100 kPa, the relationship between sound power level and sound pressure level for a nondirectional sound source is (6) where Lw = sound power level, dB re 10–12 W Lp = sound pressure level dB re 20 µPa r = distance from nondirectional sound source, m Free-Field Over Reflecting Plane. In many cases, a completely free field is not available, and measurements can only be made in a free field over a reflecting plane. That is, the sound source is placed on a hard floor (in an otherwise sound-absorbing room) or on pave-ment outdoors. Since the sound is then radiated into a hemisphere rather than a full sphere, the relationship for Lw and Lp for a non-directional sound source becomes (7) Source Directivity. A sound source may radiate different amounts of sound power in different directions because various areas of its surface do not vibrate at the same level or in phase. A directivity pattern can be established by measuring sound pressure under free-field conditions, either in an anechoic room or over a reflecting plane in a hemi-anechoic space at several points around the source. The directivity factor Q is defined as the ratio of sound pressure at a given angle from the sound source to the sound pres-sure that would be produced by the same source radiating uniformly in all directions. Q is a function of both frequency and direction.
Chapter 46 of the 1999 ASHRAE Handbook—Applications provides more detailed information on sound source directivity.
2. Reverberation Room Method Another method to determine sound power places the sound source in a reverberation room. Standardized methods for determin-ing the sound power of HVAC equipment in reverberation rooms are given in ANSI Standard S12.31, when the sound source contains mostly broadband sound; ANSI Standard S12.32, when tonal sound is prominent; and AMCA Standard 300 for testing fans.
Sound sources that can be measured by these methods include room air conditioners, refrigeration compressors, components of central HVAC systems, and air terminal devices. AMCA Standard 300, ASHRAE Standard 130, and ARI Standard 880 establish spe-cial measuring procedures for some of these units. Large equipment that can operate on a large paved area, such as a parking lot, can also be measured under free-field conditions above a reflecting plane.
Determining the sound power of large equipment outdoors is diffi-cult; however, data may be available from some manufacturers.
A diffuse field is a sound field in which the sound intensity is the same in all directions and at every point. The reverberant field in a reverberation room, because of its sound reflecting walls, floor, and ceiling, provides an approximation to a diffuse field.
Direct Method. With the source in a reverberation room, the sound pressure level is measured at some minimum distance from the source and the surfaces of the room. The sound power level is calculated from the sound pressure level, if the rate of decay of sound (in dB/s) and the volume of the reverberation room are known.
Using the direct method, the relationship between sound power level and sound pressure level in a reverberation room is given by Lw = Lp + 10logV + 10logD – 31.8 (direct method) (8) where Lp = sound pressure level averaged over room, dB re 20 µPa V = volume of room, m3 D = decay rate, dB/s = 60/θ θ = room reverberation time (time required for a 60 dB decay), s Substitution Method. Most test standards use a calibrated ref-erence sound source (RSS) to determine the sound power level of a device under test, such as a fan. The most common reference sound source is a small, direct-drive fan impeller that has no volute housing or scroll. The impeller is a forward-curved design, and a choke plate is installed on the inlet face of the impeller. The choke plate causes the fan to operate in a rotating-stall condition that is very noisy. The reference source is designed to have a stable sound power level output from 63 Hz to 8000 Hz and a relatively uniform frequency spectrum in each octave band.
To determine the sound power level of a given source, sound pressure level measurements are first made near the location of the given source with the reference sound source operating in the test room. Then the reference source is turned off and the measure-ments are repeated with the given source in operation. The differ-ences in measured sound pressure levels between the reference source and unknown source represent differences in sound power level between the two. This procedure for sound power level deter-mination is known as the substitution method. Using this method, the relationship between sound power level and sound pressure level for the two sources is given by: Lw = Lp + (Lw – Lp)ref (substitution method) (9) where Lp = sound pressure level averaged over room, dB re 20 µPa (Lw – Lp)ref = difference between sound power level and sound pressure level of reference sound source Lw Lp 20 r log 11 + + = Lw Lp 20 r log 8 + + = 7.6 2001 ASHRAE Fundamentals Handbook (SI) 3. Progressive Wave Method By attaching a fan to one end of a duct, the sound energy is con-fined to a progressive wave field in the duct. Fan sound power can then be determined by measuring the sound pressure level inside the duct. The method is described in ASHRAE Standard 68 (AMCA Standard 330). This method is not commonly used because of dif-ficulties in constructing the required duct termination.
4. Sound Intensity Method Advances in acoustical instrumentation now permit the direct determination of sound intensity, defined as the sound power per unit area flowing through a small element of a surface surrounding a source. The average sound power radiated by the source can be determined by measuring the sound intensity over the sphere or hemisphere surrounding a sound source. One of the advantages of this method is that, with certain limitations, sound intensity (and, therefore, sound power) measurements can be made in the presence of steady background sound in ordinary rooms, thereby eliminating the need for a special testing environment. Another advantage is that by measuring sound intensity over restricted areas around a sound source, sound directivity can be determined. This procedure can be particularly useful in reducing noise of products during their development.
International and United States standards that prescribe methods for making sound power measurements with sound intensity probes have been issued (ISO Standard 9614-1, ISO Standard 9614-2, ANSI Standard S12.12). In some situations, the sound fields may be so complex that measurements become impractical. A particular concern is that small test rooms or those having somewhat flexible boundaries (sheet metal or thin drywall) can permit a reactive sound field to exist, one in which the room’s acoustical characteristics cause it to affect the sound power output of the source.
Measurement Bandwidths for Sound Power Sound power is normally determined in octave or 1/3 octave bands. Occasionally, a more detailed determination of the sound source spectrum is required. In these cases, narrow band analysis, using either constant fractional bandwidth (1/12 or 1/24 octave) or constant absolute bandwidth (e.g., 1 Hz) can be applied. The digital filter analyzer is most frequently used for constant percent band-width measurements, and the fast Fourier transform (FFT) analyzer is used for constant bandwidth measurements. Narrow band analy-sis results are used to determine the exact frequencies of pure tones and their harmonics in a sound spectrum.
CONVERTING FROM SOUND POWER TO SOUND PRESSURE The designer is often required to use the sound power level infor-mation on a source to predict the sound pressure level at a given location. The sound pressure level at a given location in a room due to a source of known sound power level depends on: (1) room vol-ume, (2) room furnishings and surface treatments, (3) magnitude of the sound source(s), and (4) distance from the sound source(s) to the point of observation.
The classic relationship between source sound power level and room sound pressure level at some frequency is (10) where Lp = sound pressure level, dB re 20µPa Lw = sound power level, dB re 10–12 W Q = directivity of the sound source (dimensionless) r = distance from the source, m R = room constant, S = sum of all surface areas, m2 = average absorption coefficient of room surfaces at given frequency If the source is outside, far from reflecting surfaces, this relation-ship simplifies to Lp = Lw + 10log (Q/4πr2) + 0.5 (11) This relationship does not account for atmospheric absorption, weather effects and barriers. Note that the r2 term is present because the sound pressure in a free field decreases as 1/r2 (the inverse-square law). Each time the distance from the source is doubled, the sound pressure level decreases by 6 dB.
For a simple source centered in a large, flat, reflecting surface, Q may be taken as 2. At the junction of two large flat surfaces, Q is 4, and in a corner, Q is 8.
In most typical rooms, the presence of acoustically absorbent surfaces and sound-scattering elements, such as furniture, creates a relationship between sound power and sound pressure level that is almost independent of the absorptive properties of the space. For example, hospital rooms, which have only a small amount of absorption, and executive offices, which have substantial absorp-tion, are similar when the comparison is based on the same room volume and distance between the source and point of observation.
Equation (12) can be used to estimate the sound pressure level at a chosen observation point in a normally furnished room. The esti-mate is accurate to +2 dB (Schultz 1985).
Lp = Lw – 5logV – 3logf – 10log r + 12 (12) where Lp = room sound pressure level at chosen reference point, dB re 20 µPa Lw = source sound power level, dB re 10−12 W V = room volume, m3 f = octave band center frequency, Hz r = distance from source to observation point, m Equation (12) applies to a single sound source in the room itself, not to sources above the ceiling. With more than one source, total sound pressure level at the observation point is obtained by adding the contribution from each source in energy or power-like units, not decibels, and then converting back to sound pressure level. War-nock (1998b), reporting on ASHRAE Research Project 755, indi-cated that sound sources above ceilings may not act as point sources, and Equation (12) may not apply. Chapter 46 of the 1999 ASHRAE Handbook—Applications provides more information on this topic.
SOUND TRANSMISSION PATHS Sound from a source is transmitted along one or more paths to a receiver. Airborne and structureborne transmission paths are of principal concern for the HVAC system designer. Sound transmis-sion between rooms occurs along both airborne and structureborne transmission paths. Chapter 46 of the 1999 ASHRAE Handbook— Applications has additional information on transmission paths.
Airborne Transmission Atmospheric transmission. Sound transmits readily through air, both indoors and outdoors. Indoor sound transmission paths include the direct, line-of-sight path between the source and the receiver, as well as reflected paths introduced by the presence of a room’s walls, floor, ceiling and furnishings, which cause multiple sound reflection paths.
Outdoors, the effects of the reflections are small, if the source is not located near large reflecting surfaces. However, sound outdoors can refract and change propagation direction, due to the presence of wind and temperature gradient effects. Sound propagation outdoors Lp Lw 10 Q 4πr2 -----------4 R ---+ log 0.5 + + = Sα 1 α – ( ) ⁄ α Sound and Vibration 7.7 follows the inverse square law. Therefore, Equations (6) and (7) can generally be used to calculate the relationship between sound power level and sound pressure level for fully free-field and hemispherical free-field conditions, respectively.
Ductborne transmission. Ductwork can provide an effective sound transmission path because the sound is primarily contained within the boundaries of the ductwork. Sound can transmit both upstream and downstream from the source. A special case of duct-borne transmission is crosstalk, where sound is transmitted from one room to another via the duct path.
Room-to-room transmission. Room-to-room sound transmis-sion generally involves both airborne and structureborne sound paths. The sound power incident on a room surface element under-goes three processes: (1) some of the sound energy is reflected from the surface element back into the room; (2) a portion of the sound energy is lost due to energy transfer into the material comprising the element, and (3) the remainder of the sound energy is transmitted through the element to the other room. Airborne sound is radiated as the element vibrates and structureborne sound can be transmitted via the studs of a partition or the floor and ceiling surfaces.
Structureborne Transmission Solid structures are efficient transmission paths for sound, which frequently originates as a vibration imposed on the transmitting structure. Typically, only a small amount of the input energy is radi-ated by the structure as airborne sound. A light structure with little inherent damping radiates more sound than a massive structure with greater damping.
Flanking Transmission Sound from the source room can bypass the primary separating element and get into the receiving room along other paths called flanking paths. Common sound flanking paths include return air plenums, doors, and windows. Less obvious paths are those along floor and adjoining wall structures. Such flanking paths can seri-ously reduce the sound isolation between rooms. Flanking can explain poor sound isolation between spaces when the partition between spaces is known to provide very good sound insulation.
Flanking can also explain sounds being heard in one room at a great distance from another room. Determining whether flanking sound transmission is important and what paths are involved can be diffi-cult. Experience with actual situations and the theoretical aspects of flanking transmission is helpful. Sound intensity methods may be useful in determining flanking paths.
TYPICAL SOURCES OF SOUND Whenever mechanical power is generated or transmitted, a frac-tion of the power is converted into sound power and is radiated into the air. Therefore, virtually any major component of an HVAC system could be considered a sound source (e.g., fans, pumps, ductwork, piping, motors, etc.). The component’s sound source characteristics depend upon its construction, its form of mechanical power and its integration with associated system components. The most important sound source characteristics include total sound power output, frequency distribution, and radiation directivity. All of these characteristics vary with frequency.
Sound sources in HVAC systems are so numerous that it is impractical to provide a complete listing here. Typical sources of sound and vibration in HVAC systems include • Rotating and reciprocating equipment such as fans, motors, pumps, and chillers.
• Air and fluid sounds, such as those associated with flow through ductwork, piping systems, grilles, diffusers, terminal boxes, man-ifolds, and pressure-reducing stations.
• Excitation of surfaces—for example, friction; movement of mechanical linkages; turbulent flow impacts on ducts, plenum panels, and pipes; and impacts within equipment, such as cams and valve slap.
• Magnetostriction (transformer hum), which becomes significant in motor laminations, transformers, switchgear, lighting ballasts, and dimmers. A characteristic of magnetostrictive oscillations is that their fundamental frequency is twice the line frequency (120 Hz in a 60 Hz system.) CONTROLLING SOUND Terminology The following noninterchangeable terms are used to describe the acoustical performance of many system components. ASTM Stan-dard C 634 defines additional terms.
Sound attenuation is a general term describing the reduction of the level of sound as it travels from a source to a receiver.
Insertion loss (IL) of a silencer or other sound-attenuating ele-ment is expressed in dB and is defined as the decrease in sound pres-sure level or sound intensity level, measured at a fixed receiver location, when the sound-attenuating element is inserted into the path between the source and the receiver. For example, if a straight, unlined piece of ductwork were replaced with a duct silencer, the sound level difference at a fixed location would be considered the silencer’s insertion loss. Measurements are typically made in either octave or 1/3 octave bands.
Sound transmission loss (TL) of a partition or other building element is equal to 10 times the logarithm (base 10) of the ratio of the airborne sound power incident on the partition to the sound power transmitted by the partition and radiated on the other side.
The quantity so obtained is expressed in decibels. Measurements are typically made in octave or 1/3 octave bands. Chapter 46 of the 1999 ASHRAE Handbook—Applications defines the special case of breakout transmission loss through duct walls.
Noise reduction (NR) is the difference between the average sound pressure levels produced in two enclosed spaces or rooms— a receiving room and a source room—by one or more sound sources in the source room. An alternate, non-ASTM definition of NR is the difference in sound pressure levels measured upstream and down-stream of a duct silencer or sound-attenuating element. Measure-ments are typically made in octave or 1/3 octave bands.
Random incidence sound absorption coefficient α is the fraction of the incident sound energy that is absorbed by a surface exposed to randomly incident sound. It is measured in a reverber-ation room using 1/3 octave bands of broadband sound (ASTM C 423). The sound absorption coefficient of a material in a spe-cific 1/3 octave band depends on the material’s thickness, airflow resistivity, stiffness, and method of attachment to the supporting structure.
Spherical spreading is the process by which sound level de-creases with distance from a point source. It occurs when the sound source is located in free space or an anechoic room. Sound propa-gation in an anechoic space follows the inverse square law—6 dB level reduction per doubling of distance from a reference position.
Scattering is the change in direction of sound propagation due to an obstacle or inhomogeneity in the transmission medium. It results in the incident sound energy being dispersed in many directions.
Enclosures and Barriers Enclosing a sound source is a common means of controlling air-borne sound transmission. This may be done using single- or dou-ble-leaf partitions.
The term single-leaf partition refers to all types of solid homo-geneous panels where both faces are rigidly connected. Examples are gypsum board, plywood, concrete block, brick, and poured 7.8 2001 ASHRAE Fundamentals Handbook (SI) concrete. The transmission loss of a single-leaf partition depends mainly on its surface mass (mass per unit area) because the heavier the partition, the less it vibrates in response to sound waves and the less sound it radiates on the side opposite the sound source.
Increased surface mass can be achieved either by an increase in the partition’s thickness or its density. The mass law is a semi-empirical expression that may be used to predict transmission loss for randomly incident sound for thin, homogeneous single-leaf panels. It is written as TL = 20log (ws f ) – 42 (13) where ws = surface mass of panel, kg/m2 f = frequency, Hz The mass law predicts that transmission loss increases by 6 dB for each doubling of surface mass or frequency. If sound is incident only perpendicularly on the panel, the TL is about 5 dB greater than that predicted by the mass law.
Transmission loss also depends on material properties, such as stiffness and internal damping. The transmission losses of three sin-gle-leaf walls are illustrated in Figure 2. For the 16 mm gypsum board, TL depends mainly on the surface mass of the wall at fre-quencies below about 1 kHz; agreement with the mass law is good.
At higher frequencies, there is a dip in the TL curve, called the coin-cidence dip because it occurs at the frequency where the wave-length of flexural vibrations in the wall coincides with the wavelength of sound in the air. The frequency where the minimum value of TL occurs in the coincidence dip is called the critical fre-quency, which depends on material stiffness and thickness. The stiffer or thicker the layer of material, the lower the critical fre-quency. For example, the 150 mm concrete slab has a surface mass of about 370 kg/m2 and a coincidence frequency at 125 Hz. Thus, over most of the frequency range shown in Figure 2, the transmis-sion loss for the 150 mm concrete slab is well below that predicted by mass law. The coincidence dip for the 25 gage (0.531 mm thick) steel sheet occurs at high frequencies not shown in the figure.
The sound transmission class (STC) rating of a partition or assembly is a single number rating often used in architecture to clas-sify sound isolation for speech. (ASTM E 90, ASTM E 413) Because the STC rating system was developed to deal with sound sources in the speech frequency range (125 to 4000 Hz), the rating should not be used as an indicator of an assembly’s ability to control sound of any source that is rich in low frequencies. Most fan sound spectra have dominant low-frequency sound; therefore, to control fan sound, walls and slabs should be selected only on the basis of 1/3 octave or octave band sound transmission loss values, particularly at low frequencies. Sound transmission loss values for ceiling tile are also inappro-priate for estimating the reduction of sound between a sound source located in a ceiling plenum and the room below. ARI Standard 885 has guidance on this topic.
Walls with identical STC ratings may not provide identical sound insulation at all frequencies. Because of the limited frequency range of most single number rating systems, designers should select par-titions and floors on the basis of their 1/3 octave or octave band sound transmission loss values rather than single number ratings, especially when frequencies below 125 Hz are important.
For a given total mass in a wall or floor, much higher values of TL can be obtained by forming a double-leaf construction where each layer is independently or resiliently supported so vibration transmission between them is minimized. As well as mass, TL for such walls depends on cavity depth. Adding sound-absorbing mate-rial in the cavity significantly increases the TL relative to the unfilled cavity case. For further information on such walls, see Chapter 46 of the 1999 ASHRAE Handbook—Applications.
If the sound fields in the rooms on each side of a panel are diffuse and the panel is the only significant path for sound between the rooms, the noise reduction NR is a function of the panel area Sp and the total sound absorption Ar in the receiving space, according to NR = TL – 10log (Sp/Ar) (14) Because the total sound absorption in a room is expressed as the equivalent area of perfect sound absorption, both Sp and Ar are expressed in consistent units, usually square metres.
The sound reduction of an enclosure may be severely compro-mised by openings or leaks in the enclosure. Ducts that lead into or through a noisy space can carry sound to many areas of a building.
Designers need to consider this factor when designing duct, piping, and electrical systems.
Attenuation of Sound in Ducts and Plenums Most ductwork, even a sheet metal duct without acoustical lining or silencers, attenuates sound to some degree. The natural attenua-tion of unlined ductwork is minimal, but can, especially for long runs of rectangular ductwork, significantly reduce ductborne sound.
Acoustic lining of ductwork can greatly attenuate the propagation of sound through ducts, particularly at mid to high frequencies. Chap-ter 46 of the 1999 ASHRAE Handbook—Applications has a de-tailed discussion of lined and unlined ductwork attenuation.
If analysis shows that lined ductwork will not reduce sound prop-agation adequately, commercially available sound attenuators (also known as sound traps or duct silencers) can be used. There are three types: dissipative, reactive, and active. The first two are commonly known as passive attenuators.
• Dissipative silencers use absorptive media such as glass or rock fiber as the principal sound-absorption mechanism. Thick, perfo-rated sheet metal baffles filled with low-density fiber insulation restrict the air passage width within the attenuator housing. The fiber is sometimes protected from the airstream by cloths or films.
This type of attenuator is most effective in reducing mid- and high-frequency sound energy.
• Reactive silencers do not use any absorptive media to dissipate sound. This attenuator is typically used in HVAC systems serving hospitals, laboratories, or other areas with strict air quality stan-dards. They are constructed only of metal, both solid and perfo-rated. Chambers of specially designed shapes and sizes behind the perforated metal are tuned as resonators or expansion cham-bers to react with and reduce the sound power at selected frequen-cies. When designed for a broad frequency range, they are usually not as effective as dissipative attenuators and so are longer and Fig. 2 Sound Transmission Loss Spectra for Single Layers of Some Common Materials Sound and Vibration 7.9 have a greater pressure drop. However, they can be highly effec-tive and compact if designed for a limited frequency range such as for a pure tone.
• Active silencer systems use microphones, loudspeakers, and appropriate electronics to reduce in-duct sound by generating inverse-phase sound waves that destructively interfere with the incident sound energy. Microphones sample the sound field in the duct and loudspeakers generate signals with the opposite phase to the noise. Controlled laboratory experiments have shown that active attenuators reduce both broadband and tonal sound, but they are typically only effective in the 31.5 Hz through 250 Hz octave bands. Insertion losses of as much as 30 dB have been achieved under controlled conditions. The microphones and loud-speakers create a negligible pressure drop because they are mounted flush with the duct wall. Because active attenuators are not effective in the presence of excessively turbulent airflow, their use is limited to relatively long, straight duct sections with an air velocity less than about 7.5 m/s.
Silencers are available for fans, cooling towers, air-cooled con-densers, compressors, gas turbines, and many other pieces of com-mercial and industrial equipment. Silencers are normally installed on the intake or the discharge side (or both) of a fan or air handling unit. Also, they may be used on the receiver side of other noise gen-erators such as terminal boxes, valves, and dampers.
Self-noise can limit an attenuator’s effective insertion loss for air velocities in excess of about 10 m/s. Use extreme caution when reviewing manufacturers’ performance data for attenuators and duct liner materials to be sure that the test conditions are comparable to the specific design conditions. Short sections (1 to 1.5 m) of insu-lated flexible duct are often very effective as attenuators. (ARI Stan-dard 885 and Chapter 46 of the 1999 ASHRAE Handbook— Applications have information on typical flexible duct attenuation factors).
End Reflections. End reflection losses due to abrupt area changes in duct cross-section are sometimes useful in controlling low frequencies. The end reflection effect can be maximized at the end of a duct run by designing the last metre or so of duct with the characteristic dimension of less than 400 mm. Low-frequency noise reduction is inversely proportional to the characteristic dimension of the duct. However, abrupt area changes can generate high fre-quency noise, especially at high flow rates.
Lined Plenums. Where space is available, a lined plenum can provide excellent attenuation across a broad frequency range. The combination of end reflection at the plenum’s entrance and exit, a large distance between the entrance and exit, and sound-absorbing lining on the plenum walls can be as effective as a sound attenuator, but with less pressure drop.
Chapter 46 of the 1999 ASHRAE Handbook—Applications has additional information on the control of sound.
Standards for Testing Duct Silencers Attenuators and duct liner materials are tested according to ASTM Standard E 477 in North America and ISO 7235 elsewhere.
These define acoustical and aerodynamic performance in terms of insertion loss, self-generated noise (or self-noise), and airflow pressure drop. While many similarities exist, the ASTM and ISO standards produce differing results because of variations in loud-speaker location, orientation, duct termination conditions, and com-putation methods. Currently, no standard test methods are available to measure the attenuation of active silencers, although it is easy to measure in the field simply by turning the system on and off.
Insertion loss is measured in the presence of both forward and reverse flows. Forward flow occurs when the air and sound move in the same direction, as in a supply air or fan discharge system; reverse flow occurs when the air and sound travel in opposite direc-tions, as in the case of a return air or fan intake system.
SYSTEM EFFECTS The way the HVAC components are assembled into a system affects the sound level generated by the system. Many engineers believe that satisfactory noise levels in occupied spaces can be achieved solely by using a manufacturer’s sound ratings as a design tool, without consideration of the system influence.
Sound data provided by most manufacturers is obtained under standard laboratory test conditions. If the equipment is installed in a manner that differs from the test configuration, different configu-rations of connected ductwork, and interactions with other compo-nents of the installation, often significantly increase the operating noise level. For example, aerodynamically clean fan inlet and outlet conditions are rarely found in typical field applications. Further-more, components such as silencers are frequently installed too close to the fan to allow a uniform velocity profile to exist at the entrance to the silencer. This results in a significantly higher than anticipated pressure drop across that component. The combination of these two effects changes the operating point on the fan curve. As a result, airflow is reduced and must be compensated for by increas-ing the fan speed, which may increase noise.
HUMAN RESPONSE TO SOUND Noise Noise may be defined as any unwanted sound. Sound becomes noise when • It is too loud—the sound is uncomfortable or makes speech diffi-cult to understand • It is unexpected (e.g., the sound of breaking glass) • It is uncontrolled (e.g., a neighbor’s lawn mower) • it happens at the wrong time (e.g., a door slamming in the middle of the night) • It contains pure tones (e.g., a whine, whistle, or hum) • It contains unwanted information or is distracting (e.g., an adja-cent telephone conversation or undesirable music) • It is unpleasant (e.g., a dripping faucet) • It connotes unpleasant experiences (e.g., a mosquito buzz or a siren wail) • It is any combination of the above examples To be noise, sound does not have to be loud, just unwanted. In addition to being annoying, loud noise can cause hearing loss, and, depending on other factors, it could affect stress level, sleep patterns and heart rate.
To increase privacy, broadband sound may be radiated into a room from a well-designed air-conditioning system to mask or hide low-level intrusive sounds from adjacent spaces. This controlled sound may be referred to as noise, but not in the context of unwanted sound; rather, it is a broadband, neutral sound that is frequently unobtrusive. Three types of broadband noise are frequently encoun-tered in acoustics: • Random noise is an oscillation, the instantaneous magnitude of which is not specified for any given instant. The instantaneous magnitudes of a random noise are specified only by probability distributions, giving the fraction of the total time that the magni-tude, or some sequence of magnitudes, lies within a specified range (ANSI Standard S1.1).
• White noise is noise with a continuous frequency spectrum with equal energy per hertz over a specified frequency range. White noise is not necessarily random. Since octave bands double in width for each successive band, for white noise the energy also doubles in each successive octave band. Thus white noise dis-played on a 1/3 octave or octave band chart increases by 3 dB per octave.
• Pink noise is noise with a continuous frequency spectrum but equal energy per constant-percentage bandwidth, such as per 7.10 2001 ASHRAE Fundamentals Handbook (SI) octave or 1/3 octave band. Thus pink noise appears on a one-third octave or octave band chart as a horizontal line.
Predicting Human Response to Sound Predicting the response of people to any given sound is, at best, only a statistical concept, and, at worst, very inaccurate. This is because response to sound is not only physiological but psycholog-ical and depends on the varying attitude of the listener. Hence, the effect of sound is often unpredictable. However, people’s response is adverse if the sound is considered too loud for the situation or if it sounds “wrong.” Therefore, most criteria are based on descriptors that account for level and spectrum shape.
Sound Quality To determine the acoustic acceptability of a space to occupants, the sound pressure levels there must be known. This, however, is often not sufficient; the sound quality is important too. Factors influencing sound quality include (1) loudness, (2) tone perception, (3) frequency spectrum, (4) harshness, (5) time and frequency fluc-tuation, and (6) vibration.
People often perceive sounds with tones (like a whine or hum) as particularly annoying. A tone can cause a relatively low level sound to be perceived as noise. Studies have been done to characterize sounds with and without pure tones.
Loudness The primary method used to determine a subjective estimate of loudness is to present sounds to a sample of human listeners under controlled conditions. To determine the loudness of a sound, listen-ers compare an unknown sound with a standard sound. (The accepted standard sound is a pure tone of 1000 Hz or a narrow band of random noise centered on 1000 Hz.) Loudness level is expressed in phons, and the loudness level of any sound in phons is equal to the sound pressure level in decibels of a standard sound deemed to be equally loud. Thus, a sound that is judged as loud as a 40 dB, 1000 Hz tone has a loudness level of 40 phons.
Average reactions of humans to tones are shown in Figure 3 (Robinson and Dadson 1956). The reaction changes when the sound is a band of random noise (Pollack 1952), rather than a pure tone (Figure 4). The figures indicate that people are most sensitive in the mid-frequency range. The contours in Figure 3 are closer together at low frequencies showing that at lower frequencies, although people are less sensitive to sound level, they are more sensitive to changes in level.
Under carefully controlled experimental conditions, humans can detect small changes in sound level. However, for humans to describe a sound as being half or twice as loud requires changes in overall sound pressure level of about 10 dB. For many people, a 3 dB change is the minimum perceptible difference. This means that halving the power output of the source causes a barely notice-able change in sound pressure level, and the power output must be reduced by a factor of 10 before humans determine that loudness has been halved. Table 7 summarizes the effect of changes in sound levels for simple sounds in the frequency range of 250 Hz and higher.
The phon scale covers the large dynamic range of the ear, but it does not fit a subjective linear loudness scale. Over most of the audible range, a doubling of loudness corresponds to a change of approximately 10 phons. To obtain a quantity proportional to the loudness sensation, a loudness scale is defined in which the unit of loudness is known as a sone. One sone equals the loudness level of 40 phons. A rating of two sones corresponds to 50 phons and so on.
The results of such work have led to the development of standard objective methods for calculating loudness. ANSI Standard S3.4 calculates loudness or loudness level by using octave-band sound pressure level data as a starting point. The loudness index for each octave band is obtained from a graph or by calculation. Total loud-ness is then calculated by combining the loudnesses for each band according to a formula given in the standard. A more complex cal-culation method using 1/3 octave band sound pressure levels by Zwicker (ISO Standard 532) or the German Standard DIN 45631 is Fig. 3 Free-Field Equal Loudness Contours for Pure Tones Table 7 Subjective Effect of Changes in Sound Pressure Level, Broadband Sounds (Frequency > 250 Hz) Subjective Change Objective Change in Sound Level (Aproximate) Much louder More than +10 dB Twice as loud +10 dB Louder +5 dB Just perceptibly louder +3 dB Just perceptibly quieter −3 dB Quieter −5 dB Half as loud −10 dB Much quieter Less than −10 dB Fig. 4 Equal Loudness Contours for Relatively Narrow Bands of Random Noise Sound and Vibration 7.11 tones. Due to its complexity, loudness has not been widely used in engineering practice in the past. However, with an increased aware-ness of sound quality and the availability of software for calculating loudness, this measure is now being used more frequently.
AMCA Publication 302 describes how the sone method is applied to rating the relative loudness of fans and ventilators. This calculation method is usually acceptable when the measured sound spectrum has no strong tonal components.
Acceptable Frequency Spectrum The most acceptable frequency spectrum for HVAC sound is a balanced or neutral spectrum. This means that it is not too “hissy” (excessive high frequency content) or too “rumbly” (excessive low-frequency content). Unfortunately, achieving a balanced sound spectrum is not always easy—there may be a multiplicity of sound sources to consider. As a guide to the designer, Figure 5 shows the more common mechanical and electrical sound sources and fre-quency regions that control the indoor sound spectrum. Chapter 46 of the 1999 ASHRAE Handbook—Applications provides more detailed information on treating some of these sound sources.
SOUND RATING SYSTEMS AND ACOUSTICAL DESIGN GOALS Several background sound rating methods are used to rate indoor sound. They include the A-weighted sound pressure level (dBA) and noise criteria (NC), the more recent room criteria (RC) and bal-anced noise criteria (NCB), and the new RC Mark II. Each sound rating method was developed from data for specific applications; not all methods are equally suitable for the rating of HVAC-related sound in the variety of applications encountered.
The degree of occupant satisfaction achieved with a given level of background sound is determined by many factors. For example, large conference rooms, auditoriums, and recording studios can tol-erate only a low level of background sound. On the other hand, higher levels of background sound are acceptable and even desir-able in certain situations, such as in open-plan offices where a cer-tain amount of speech and activity masking is essential. Therefore, the system sound control goal varies depending on the required use of the space.
To be unobtrusive, background sound should have the following properties: • A balanced distribution of sound energy over a broad frequency range • No audible tonal or other characteristics such as whine, whistle, hum, or rumble • No noticeable time-varying levels from beats or other system-induced aerodynamic instability • No fluctuations in level such as a throbbing or pulsing At present, no acceptable process easily characterizes the effects of audible tones and level fluctuations. The preferred sound rating methods generally comprise two distinct parts: a family of criteria curves (specifying sound levels by octave bands), and a compan-ion procedure for rating the calculated or measured sound data rel-ative to the criterion curves. A table of recommended design goals can be found in Chapter 46 of the 1999 ASHRAE Handbook— Applications.
Table 8 summarizes the essential differences, advantages and disadvantages of the rating methods that are used to characterize HVAC-related background sound. The text following the table gives more information on each rating. Note that all the ratings in the table consider speech interference effects and all are currently used for rating background noise.
A-Weighted Sound Level (dBA) The A-weighted sound level LA is widely used to state acoustical design goals as a single number, but its usefulness is limited because it gives no information on spectrum content. The rating is expressed as a number followed by dBA, for example 40 dBA.
A-weighted sound levels correlate well with human judgments of relative loudness, but give no information on spectral balance.
Thus, they do not necessarily correlate well with the annoyance caused by the noise. Many different-sounding spectra can have the same numeric rating, but have quite different subjective qualities.
A-weighted comparisons are best used with sounds that sound alike but differ in level. They should not be used to compare sounds with distinctly different spectral characteristics; that is, two sounds at the same sound level but with different spectral content are likely to be judged differently by the listener in terms of acceptability as a back-ground sound. One of the sounds might be completely acceptable, while the other could be objectionable because its spectrum shape was rumbly, hissy, or tonal in character.
A-weighted sound levels are used extensively in outdoor envi-ronmental noise standards.
Noise Criteria (NC) Method The NC method is a single-number rating that is somewhat sen-sitive to the relative loudness and speech interference properties of Fig. 5 Frequencies at Which Various Types of Mechanical and Electrical Equipment Generally Control Sound Spectra Table 8 Comparison of Sound Rating Methods Method Overview Evaluates Sound Quality Used For Rating of dBA • Can be determined using sound level meter • No quality assessment • Frequently used for outdoor noise ordinances No Cooling towers Water chillers Condensing units NC • Can rate components • No quality assessment • Does not evaluate low frequency rumble, frequencies <63 Hz No Air terminals Diffusers NCB • Can rate components • Some quality assessment Yes RC • Used to evaluate systems • Should not be used to evaluate components • Can be used to evaluate sound quality • Provides some diagnostic capability Yes RC Mark II • Evaluates sound quality • Provides improved diagnostics capability Yes 7.12 2001 ASHRAE Fundamentals Handbook (SI) a given sound spectrum. The method consists of a family of criteria curves extending from 63 to 8000 Hz, and a tangency rating pro-cedure (Beranek 1957). The criteria curves, shown in Figure 6, define the limits of octave band spectra that must not be exceeded to meet occupant acceptance in certain spaces. The rating is expressed as NC followed by a number. For example, the spectrum shown in Figure 6 is rated NC 45 because this is the lowest rating curve that falls entirely above the measured data. An NC 35 design goal is commonly used for private offices. The background sound level meets this goal if no portion of its spectrum lies above the desig-nated NC 35 curve.
The NC method is sensitive to level but has the disadvantage that the tangency method used to determine the rating does not require that the sound spectrum approximate the shape of the NC curves.
Thus, many different sounds can have the same numeric rating, but rank differently on the basis of subjective sound quality. In HVAC systems that do not produce excessive low frequency sound, the NC rating correlates relatively well with occupant satisfaction if sound quality is not a significant concern.
Two problems occur in using the NC procedure: (1) when the NC level is determined by a prominent peak in the spectrum, the actual level of resulting background sound may be quieter than that desired for masking unwanted speech and activity sounds, because the spectrum on either side of the tangent peak drops off too rapidly; and (2) when the measured spectrum closely matches the shape of the NC curve, the resulting sound is either rumbly or hissy or both.
The shape of the NC curve is not that of a well-balanced, neutral sound; thus, these curves should be used with caution in critical sit-uations where the background sound of an air-conditioner is required to mask speech and activity sound.
NC contours are used to calculate ratings for some HVAC com-ponents such as terminal units and diffusers. NC ratings should not be used to characterize fans and air-handling units.
Balanced Noise Criteria (NCB) Method The NCB method (Beranek 1989, ANSI Standard S12.2) is a specification or evaluation of room sound including noise due to occupant activities. The NCB criteria curves (Figure 7) are intended as replacements for the NC curves, and include both the addition of two low-frequency octave bands (16 and 31.5 Hz) and lower per-missible sound levels in the high-frequency octave bands (4000 and 8000 Hz). The NCB rating procedure is based on the speech inter-ference level (SIL), which is the arithmetic average of the sound pressure levels in the four frequency bands: 500, 1000, 2000, and 4000 Hz. Additional tests include rumble and hiss compliance. The rating is expressed as NCB followed by a number, for example, NCB 40. The NCB method is better than the NC method in determining whether a sound spectrum has a shape sufficiently unbalanced to demand corrective action. Also, it addresses the issue of low-frequency sound. The rating procedure is somewhat more compli-cated than the tangency rating procedure. Room Criteria (RC) Method For some time, the RC method (Blazier 1981a,b; ANSI Standard S12.2) was recommended as the preferred method for rating HVAC-related sound. The RC curves were intended to establish HVAC sys-tem design goals. The revised RC Mark II method (discussed below) is now preferred.
The RC method consists of a family of criteria curves and a rat-ing procedure. The shape of these curves differs from the NC curves to approximate a well-balanced neutral-sounding spectrum, and two additional octave bands (16 and 31.5 Hz) are added to deal with low-frequency sound. This rating procedure assesses background sound in spaces based on the effect of the sound on speech commu-nication, and on estimates of subjective sound quality. The rating is Fig. 6 NC (Noise Criteria) Curves and Sample Spectrum (Curve with Symbols) Fig. 7 NCB (Noise Criteria Balanced) Curves Drawn from ANSI Standard S12.2 Sound and Vibration 7.13 expressed as RC followed by a number to show the level of the sound and a letter to indicate the quality, for example RC 35(N) where N denotes neutral.
RC Mark II Room Criteria Method Based on experience and the findings from ASHRAE-sponsored research, the RC method was revised to the RC Mark II method (Blazier 1997). Like its predecessor, the RC Mark II method is intended for rating the sound performance of an HVAC system as a whole. The method can also be used as a diagnostic tool for analyz-ing sound problems in the field. The RC Mark II method is more complicated to use than the RC method, but spreadsheet macros are available to do the calculations and graphical analysis. The RC Mark II method of rating HVAC system sound com-prises three parts: • Family of criteria curves (Figure 8) • Procedure for determining the RC numerical rating and the sound spectral balance (quality) • Procedure for estimating occupant satisfaction when the spectrum does not have the shape of an RC curve (Quality Assessment Index) (Blazier 1995) The rating is expressed as RC followed by a number and a letter, for example, RC 45(N). The number is the arithmetic average rounded to the nearest integer of the sound pressure levels in the 500, 1000, and 2000 Hz octave bands (the principal speech fre-quency region). The letter is a qualitative descriptor that identifies the perceived character of the sound: (N) for neutral, (LF) for low-frequency rumble, (MF) for mid-frequency roar, and (HF) for high-frequency hiss. In addition, the low-frequency descriptor has two subcategories: (LFB), denoting a moderate but perceptible degree of sound induced ceiling/wall vibration, and (LFA), denoting a noticeable degree of sound induced vibration.
Each reference curve in Figure 8 identifies the shape of a neutral, bland-sounding spectrum, indexed to a curve number correspond-ing to the sound level in the 1000 Hz octave band. The shape of these curves is based on research by Blazier (1981a,b) and modified at 16 Hz following recommendations by Broner (1994). Regions A and B denote levels at which sound can induce vibration in light wall and ceiling constructions that can potentially cause rattles in light fixtures, furniture, etc. Curve T is the octave-band threshold of hearing as defined by ANSI Standard 12.2.
Procedure for Determining the RC Mark II Rating for a System Step 1. Determine the appropriate RC reference curve. This is done by obtaining the arithmetic average of the sound levels in the principle speech frequency range represented by the levels in the 500, 1000, and 2000 Hz octave bands. The RC reference curve is chosen as that which has the same value at 1000 Hz as the calculated average value (rounded to the nearest integer). This curve is not to be confused with the speech-interference level (SIL), which is a four-band average obtained by including the 4000 Hz octave band.
Step 2. Assign a subjective quality by calculating the Quality Assessment Index (QAI) (Blazier 1995). This index is a measure of the degree the shape of the spectrum under evaluation deviates from the shape of the RC reference curve. The procedure requires calcu-lation of the energy-average spectral deviations from the RC ref-erence curve in each of three frequency groups: low frequency, LF (16-63 Hz), medium frequency, MF (125-500 Hz), and high fre-quency, HF (1000-4000 Hz). The procedure for the LF region is given by Equation (15) and is repeated in the MF and HF regions by substituting the corresponding values at each frequency. However, when evaluating typical HVAC-related sounds, a simple arithmetic average of these deviations is often adequate if the range of values does not exceed 3 dB.
(15) where the ∆L terms are the differences between the spectrum being evaluated and the RC reference curve in each frequency band. In this way, three spectral deviation factors (∆LF, ∆MF, ∆HF), expressed in dB with either positive or negative values, are associ-ated with the spectrum being rated. QAI is the range in dB between the highest and lowest values of the spectral deviation factors. If QAI ≤ 5 dB, the spectrum is assigned a neutral (N) rating. If QAI exceeds 5 dB, the sound quality descriptor of the RC rating is the letter designation of the frequency region of the deviation factor having the highest positive value. As an example, the spectrum plot-ted in Figure 8 is processed in Table 9. The arithmetic average of the sound levels in the 500, 1000, and 2000 Hz octave bands in Figure 8 is 35 dB, so the RC 35 curve is selected as the reference for spectrum quality evaluation. The spec-tral deviation factors in the LF, MF, and HF regions are 6.6, 4.0 and –0.6, respectively, giving a QAI of 7.2. The maximum positive deviation factor occurs in the LF region, and the QAI exceeds 5, resulting in a rating of RC 35(LF). An average room occupant should perceive this spectrum as marginally rumbly in character (see Table 10). Estimating Occupant Satisfaction Using QAI The quality assessment index (QAI) is useful in estimating the probable reaction of an occupant when the system does not produce optimum sound quality. The basis for the procedure outlined here for estimating occupant satisfaction is is as follows: Sound levels in Region B may generate perceptible vibration in light wall and ceiling construction. Rattles in light fixtures, doors, windows, etc., are a slight possibility. Sound levels in Region A have a high prob-ability of generating easily perceptible sound induced vibration in light wall and ceiling construction. Audible rattling in light fixtures, doors, win-dows etc. may be anticipated. The text explains Regions LF, MF, and HF. The solid dots are octave band sound pressure levels for the exam-ple in the text. Fig. 8 Room Criteria Curves, Mark II LF ∆ 10 10 0.1 L16 ∆ 10 0.1 L31.5 ∆ 10 0.1 L63 ∆ + + 3 ⁄ [ ] log = 7.14 2001 ASHRAE Fundamentals Handbook (SI) • Changes in sound level of less than 5 dB do not cause subjects to change their ranking of sounds of similar spectral content. A QAI of 5 dB or less corresponds to a generally acceptable condition, provided that the perceived level of the sound is in a range con-sistent with the given type of space occupancy. • A QAI that exceeds 5 dB but is less than or equal to 10 dB repre-sents a marginal situation in which the acceptance by an occupant is questionable. • A QAI greater than 10 dB will likely be objectionable to the aver-age occupant. Table 10 lists sound quality descriptors and QAI values and relates them to probable occupant reaction to the sound. An exception to this rule occurs when the sound pressure levels in the 16 Hz or 31.5 Hz octave-bands exceed 65 dB. In such cases, the potential for acoustically-induced vibration in typical light office construction should be considered. If the levels in these bands exceed 75 dB, a significant problem with induced vibration is probable.
Undoubtedly situations will occur in the assessment of HVAC-related sound where the numerical part of the RC rating is less than the specified maximum for the space use, but the sound quality descriptor is other than the desirable (N). For example, a maximum of RC 40(N) is specified, but the actual sound environment turns out to be RC 35(MF). Knowledge in this area is insufficient to decide which spectrum is preferable.
Even at moderate levels, if the dominant portion of the back-ground sound occurs in the very low-frequency region, some people experience a sense of oppressiveness or depression in the environ-ment (Persson-Waye et al. 1997). In such situations, the basis for complaint may result from exposure to that environment for several hours, and thus may not be noticed during a short exposure period.
Criteria Selection Guidelines In general, these basic guidelines are important: • Sound levels below NCB 35 or RC 35 are not detrimental to good speech intelligibility; those at or above NCB 35 or RC 35 may interfere with or mask speech. • Even if the occupancy sound will be significantly higher than the anticipated background sound level generated by mechanical equipment, the sound design goal should not necessarily be raised to levels approaching the occupancy sound. This avoids occu-pants having to raise their voices uncomfortably to be heard over the noise. Table 11 gives recommended design criteria.
FUNDAMENTALS OF VIBRATION A rigidly mounted machine transmits its internal vibratory forces directly to the supporting structure. However, by inserting resilient mountings, called vibration isolators, between the machine and supporting structure, the magnitude of transmitted vibration can be reduced to only a fraction of the original. Vibration isolators can also be used to protect sensitive equipment from disturbing vibra-tions that may be present in the floor of the building structure. SINGLE-DEGREE-OF-FREEDOM MODEL The simplest example of a vibration isolation system is the sin-gle-degree-of-freedom model illustrated in Figure 9. In this instance only motion along the vertical axis is considered and damping is dis-regarded. This is the model upon which most manufacturers of vibration isolation hardware base their catalog information. The application of a single-degree-of-freedom model to the iso-lation of HVAC equipment is valid only when the stiffness of the supporting structure is large with respect to the stiffness of the vibration isolator. Under these conditions, the natural frequency fn of the system is (16) Table 9 Example Calculation of RC Mark II Rating for Sound Spectrum in Figure 8 Frequency, Hz 16 31.5 63 125 250 500 1000 2000 4000 Sound pressure 64 65 64 57 47 40 35 30 23 Average sound pressure at 500-2000 Hz 35 RC contour 60 60 55 50 45 40 35 30 25 Levels: RC contour 4 5 9 7 2 0 0 0 –2 LF MF HF Spectral deviations 6.6 4.0 –0.6 QAI 6.6 – (–0.6) = 7.2 RC Mark II rating RC 35(LF) fn 1 2π ------k M ----= Table 10 Definition of Sound Quality Descriptor and Quality Assessment Index (QAI) to Aid in Interpreting RC Mark II Ratings of HVAC-Related Sound Sound Quality Descriptor Description of Subjective Perception Magnitude of QAI Probable Occupant Evaluation, Assuming Level of Specified Criterion is Not Exceeded (N) Neutral (Bland) Balanced sound spectrum, no single frequency range dominant QAI ≤ 5 dB, L16,L31.5 ≤ 65 QAI ≤ 5 dB, L16,L31.5 > 65 Acceptable Marginal (LF) Rumble Low-frequency range dominant (16 – 63 Hz) 5 dB < QAI ≤ 10 dB QAI > 10 dB Marginal Objectionable (LFVB) Rumble, with moderately perceptible room surface vibration Low-frequency range dominant (16 – 63 Hz) QAI ≤ 5 dB, 65<L16,L31.5<75 5 dB < QAI ≤ 10 dB QAI > 10 dB Marginal Marginal Objectionable (LFVA) Rumble, with clearly perceptible room surface vibration Low-frequency range dominant (16 – 63 Hz) QAI ≤ 5 dB, L16,L31.5 > 75 5 dB < QAI ≤ 10 dB QAI > 10 dB Marginal Marginal Objectionable (MF) Roar Mid-frequency range dominant (125 – 500 Hz) 5 dB < QAI ≤ 10 dB QAI > 10 dB Marginal Objectionable (HF) Hiss High-frequency range dominant (1000 – 4000 Hz) 5 dB < QAI ≤ 10 dB QAI > 10 dB Marginal Objectionable Sound and Vibration 7.15 where k is the stiffness of the vibration isolator (force per unit deflection) and M is the mass of the equipment supported by the iso-lator. This equation simplifies to (17) where δst is the isolator static deflection in millimetres (the incre-mental distance the isolator spring compresses under the weight of the supported equipment, or k/M = g/δst). Thus, to achieve the appropriate system natural frequency for a given application, the corresponding isolator static deflection and the load to be supported at each mounting point is specified.
The transmissibility is the ratio of the amplitudes of the force transmitted to the building structure to the exciting force produced by the vibrating equipment. Transmissibility T is inversely propor-tional to the square of the ratio of the disturbing frequency fd to the system natural frequency fn, or (18) At fd = fn, resonance occurs (the denominator of Equation (18) equals zero), with theoretically infinite transmission of vibration. In practice, however, some limit on the transmission at resonance exists because inherent damping is always present to some degree.
Thus, the magnitude of vibration amplification at resonance always has a finite value. Equation (18) is plotted in Figure 10. Table 11 Design Guidelines for HVAC-Related Background Sound in Rooms Room Types RC(N); QAI ≤ 5dB Criterion a,b Residences, Apartments, Condominiums 25 – 35 Hotels/Motels Individual rooms or suites Meeting/banquet rooms Corridors, lobbies Service/support areas 25 – 35 25 – 35 35 – 45 35 – 45 Office Buildings Executive and private offices Conference rooms Teleconference rooms Open-plan offices Corridors and lobbies 25 – 35 25 – 35 25 (max) 30 – 40 40 – 45 Hospitals and Clinics Private rooms Wards Operating rooms Corridors and public areas 25 – 35 30 – 40 25 – 35 30 – 40 Performing Arts Spaces Drama theaters Concert and recital halls c Music teaching studios Music practice rooms 25 (max) 25 (max) 35 (max) Laboratories (with fume hoods) Testing/research, minimal speech communication Research, extensive telephone use, speech communication Group teaching 45 – 55 40 – 50 35 – 45 Churches, Mosques, Synagogues General assembly With critical music programsc 25 – 35 Schoolsd Classrooms up to 70 m2 Classrooms over 70 m2 Large lecture rooms, without speech amplification 40 (max) 35 (max) 35 (max) Libraries 30 – 40 Courtrooms Unamplified speech Amplified speech 25 – 35 30 – 40 Indoor Stadiums, Gymnasiums Gymnasiums, natatoriums, and large seating-capacity spaces with speech amplification e 40 – 45 aThe values and ranges are based on judgment and experience, not on quantitative evaluations of human reactions. They represent general limits of acceptability for typ-ical building occupancies. Higher or lower values may be appropriate and should be based on a careful analysis of economics, space use, and user needs. bWhen quality of sound in the space is important, specify criteria in terms of RC(N). If the quality of the sound in the space is of secondary concern, the criteria may be spec-ified in terms of NC or NCB levels of similar magnitude.
cAn experienced acoustical consultant should be retained for guidance on acoustically critical spaces (below RC 30) and for all performing arts spaces.
dHVAC-related sound criteria for schools, such as those listed in this table, may be too high and impede learning by children in primary grades whose vocabulary is limited, or whose first language is not the language of the class. Some educators and others believe that the HVAC-related background sound should not exceed RC 25(N).
eRC or NC criteria for these spaces need only be selected for the desired speech and hearing conditions.
fn 15.8 δst ------------= Fig. 9 Single-Degree-of-Freedom System T 1 1 fd fn ⁄ ( )2 – ----------------------------= Fig. 10 Vibration Transmissibility T as a Function of fd/fn 7.16 2001 ASHRAE Fundamentals Handbook (SI) Vibration isolation does not begin to occur until fd /fn > 1.4.
Above this ratio, the vibration transmissibility rapidly decreases. A frequency ratio of at least 3.5 is often specified, which corresponds to an isolation efficiency of about 90%, or 10% transmissibility.
Higher ratios may be specified, but in practice this does not gener-ally result in isolation efficiencies any greater than about 90%. The reason is that “wave-effects” and other nonlinear characteristics cause typical isolators to depart from the theoretical curve that lim-its performance.
If the mass of the equipment is increased, the resonance fre-quency decreases, thus increasing the isolation. In practice, the load-carrying capacity of isolators usually requires that their stiff-ness or their number be increased. Consequently, the static deflec-tion and the transmissibility may remain unchanged. The use of stiffer springs leads, however, to smaller vibration amplitudes—less movement of the equipment. This is one of the main reasons for placing some high-power or highly eccentric equipment on inertia pads.
For example, as shown in Figure 11, a 500 kg piece of equipment installed on isolators with stiffness k of 196 kN/m results in a 25 mm deflection and a system resonance frequency fn of 3.15 Hz. If the equipment is operated at 564 rpm (9.4 Hz) and develops a force of 5000 N, a 5000 × 0.127 = 635 N force is transmitted to the structure.
If the total mass is increased to 5000 kg by placing the equipment on a concrete inertia base and the stiffness of the springs is increased to 1960 kN/m, the deflection is still 25 mm, the resonance frequency of the system is maintained at 3.15 Hz, and the force transmitted to the structure remains at 635 N. The increased mass, however, reduces the equipment displacement.
TWO-DEGREE-OF-FREEDOM MODEL The single-degree-of-freedom model is valid only when the stiff-ness of the supporting structure is large with respect to the stiffness of the vibration isolator. This condition is usually satisfied for mechanical equipment in on-grade or basement locations. However, when heavy mechanical equipment is installed on a structural floor, and in particular on the roof of a building, the relative stiffness of the supporting system can no longer be ignored. Significantly “softer” vibration isolators are usually required than in the on-grade or base-ment case. The appropriate model for the design of vibration isola-tion in upper-floor locations is the two-degree-of-freedom model illustrated in Figure 12.
The precise behavior of this system with respect to vibration iso-lation is difficult to determine. The objective is to minimize the motion of the supporting floor Mf in response to the exciting force F. This involves evaluating the interaction between two system nat-ural frequencies and the frequency of the exciting force, which is mathematically complex. However, several engineering rules can simplify the calculations used to optimize the isolation system.
For example, the fraction of vibratory force transmitted across an isolator to the building structure (transmissibility) depends in part on the ratio between the isolator stiffness and that of the supporting floor at the point of loading. Because stiffness is inversely propor-tional to deflection under the applied load, this relationship can sometimes be expressed more conveniently as a ratio of deflections.
To optimize isolation efficiency, the static deflection of the isolator, under the applied load, must be large with respect to the incremental static deflection of the floor that occurs due to the added equipment weight. Ideally, this ratio should be on the order of 10:1 to approach an isolation efficiency of about 90% (10% transmissibility). The relationship is illustrated in Figure 13. Note that if the static deflection of the vibration isolator is similar to the incremental deflection of the supporting floor under the added weight of the equipment, 50% or more of the vibratory force will transmit directly to the building structure. This situation is a com-mon problem in the field where excessive vibration is attributable to upper floor or rooftop mechanical installations. Frequently, the floor stiffness has been neglected and the static deflection on the installed vibration isolators is inadequate because the selection was made on the basis of the single-degree-of-freedom model.
Problems of this nature can usually be avoided by asking the structural engineer to estimate the incremental static deflection of the floor due to the added weight of the equipment at the point of loading, before selecting a vibration isolator. Then, choose an iso-lator that will provide a static deflection of 8 to 10 times that of the estimated incremental floor deflection.
Fig. 11 Effect of Mass on Transmissibility Fig. 12 Two-Degree-of-Freedom System Fig. 13 Transmissibility of Two-Degree-of-Freedom System Adapted from Plunkett (1958) Sound and Vibration 7.17 VIBRATION MEASUREMENT BASICS While the control of HVAC system sound and vibration are of equal importance, the measurement of vibration is not usually nec-essary for determining the sources or transmission paths of disturb-ing sound. Because the techniques and instrumentation used for vibration measurement and analysis are specialized, designers should consult other sources (e.g., Harris 1991) for thorough descriptions of vibration measurement and analysis methods.
The typical vibrations measured are periodic motions of a sur-face. This surface displacement oscillates with one or more fre-quencies produced by mechanical means (like gears), thermal means (like combustion), or fluid-dynamic means (like airflow through a duct or fan interactions with air). The displacement is generally inversely proportional to the frequency. In other words, if the displacements are high, the frequency is low. The frequen-cies of interest for most vibration measurements are between 5 Hz and 100 Hz.
A transducer can detect displacement, velocity, or acceleration of a surface and convert the motion to electrical signals. Displace-ment is the basic measure and good for low frequencies. Velocity is good for overall measurements, but requires large transducers. For most HVAC applications, the transducer of choice is an accelerom-eter, a device that detects acceleration. Readout may be as accelera-tion level in decibels, or acceleration with modifiers of peak, peak-to-peak, or rms. The simplest measure is the overall signal as a function of time, be it acceleration, acceleration level, or another quantity. This is analogous to the unfiltered sound pressure level for sound. If a detailed frequency analysis is needed, there is a choice of filters similar to those available for sound measurements: octave band, 1/3 octave band or 1/12 octave band filters. In addition, narrow-band analyzers that use the fast Fourier transform (FFT) to analyze and filter a signal are available. While they are widely used, they should only be used by a specialist for accurate results.
The most important issues in vibration measurement include: (1) choosing a transducer with a frequency range appropriate to the measurement, (2) properly mounting the transducer to ensure that the frequency response claimed is achieved, and (3) not using hand-held probes for high frequencies where they are unreliable.
STANDARDS AMCA. 1973. Application of sone ratings for non-ducted air moving devices. Publication 302-73. Air Movement and Control Association International, Arlington Heights, IL.
AMCA. 1979. Application of sound power level ratings for fans. Publica-tion 303-79.
AMCA. 1986. Laboratory method of testing in-duct sound power measure-ment procedure for fans. Standard 330-86.
AMCA. 1990. Methods for calculating fan sound ratings from laboratory test data. Standard 301-90.
AMCA. 1996. Reverberant room method for sound testing of fans. Standard 300-96. ANSI. 1980. Procedure for computation of loudness of noise. Standard S3.4-1980 (Reaffirmed 1997). American National Standards Institute, New York.
ANSI. 1983. Specifications for sound level meters. Standard S1.4-1983 (Amendment S1.4a-85) (R-1997).
ANSI. 1984. Preferred frequencies, frequency levels, and band numbers for acoustical measurements. Standard S1.6-1984 (R 1997). ANSI. 1986. Specifications for octave-band and fractional octave-band ana-log and digital filters. Standard S1.11-1986 (R 1998).
ANSI. 1990. Precision methods for the determination of sound power levels of broadband noise sources in reverberation rooms. Standard S12.31-1990 (R 1996) (Supersedes ANSI S1.31-1980).
ANSI. 1990. Precision methods for the determination of sound power levels of discrete-frequency and narrow-band noise sources in reverberation rooms. Standard S12.32-1990 (R1996) (Supersedes ANSI S1.32-1980 and ASA 12.80).
ANSI. 1990. Precision methods for the determination of sound power levels of noise sources in anechoic and hemi-anechoic rooms. Standard S12.35-1990 (R 1996) (Supersedes ANSI S1.35-1979 and ASA 15-79).
ANSI. 1992. Engineering method for determination of sound power level of noise sources using sound intensity. Standard S12.12-1992 (R 1997). ANSI. 1995. Criteria for evaluating room noise. Standard S12.2-1995. ARI. 1998. Air terminals. Standard 880-1998. Air-Conditioning and Refrig-eration Institute, Arlington, Virginia.
ARI. 1998. Procedure for estimating occupied space sound levels in the application of air terminals and air outlets. Standard 885-1998.
ASHRAE. 1995. Methods of testing for rating ducted air terminal units.
Standard 130-1995.
ASTM. 1987. Classification for rating sound insulation. Standard E 413-87(1999). American Society for Testing and Materials, West Consho-hocken, PA.
ASTM. 1993. Test method for evaluating masking sound in open offices using a-weighted and one-third octave band sound pressure levels. Stan-dard E 1573-93(1998).
ASTM. 1998. Test method for measurement of sound in residential spaces.
Standard E 1574-98.
ASTM. 1999. Standard test method for sound absorption and sound absorp-tion coefficients by the reverberation room method. Standard C 423-99a. ASTM. 1999. Standard test method for laboratory measurement of airborne sound transmission loss of building partitions and elements. Standard E 90-99.
ASTM. 1999. Standard test method for measuring acoustical and airflow performance of duct liner materials and prefabricated silencers. Standard E 477-99.
ASTM. 2000. Standard terminology relating to environmental acoustics.
Standard C 634-00. German Standard DIN 45631. Method for predicting loudness of sound spectra with tonal qualities.
ISO. 1975. Methods for calculating loudness level. Standard 532:1975.
International Organization for Standardization, Geneva.
ISO. 1993. Determination of sound power levels of noise sources using sound intensity—Part 1: Measurements at discrete points. Standard 9614-1:1993.
ISO. 1996. Determination of sound power levels of noise sources using-sound intensity—Part 2: Measurements by scanning. Standard 9614-2:1993.
REFERENCES ASHRAE. 1997. ASHRAE RP755. Sound transmission through ceilings from air terminal devices in the plenum.
Beranek, L.L. 1957. Revised criteria for noise in buildings. Noise Control 1:19.
Beranek, L.L. 1989. Balanced noise criterion (NCB) curves. Journal of the Acoustic Society of America (86):650-54.
Blazier, W.E., Jr. 1981a. Revised noise criteria for application in the acous-tical design and rating of HVAC systems. Noise Control Eng. 16(2):64-73.
Blazier, W. E., Jr. 1981b. Revised noise criteria for design and rating of HVAC systems. ASHRAE Transactions 87(1).
Blazier, W.E., Jr. 1995. Sound quality considerations in rating noise from heating, ventilating and air-conditioning (HVAC) systems in buildings.
Noise Control Eng. J. 43(3).
Blazier, W.E., Jr. 1997. RC Mark II: A refined procedure for rating the noise of heating, ventilating and air-conditioning (HVAC) systems in build-ings. Noise Control Eng. J. 45(6), November/December.
Broner, N. 1994. Determination of the relationship between low-frequency HVAC noise and comfort in occupied spaces. ASHRAE Research Project 714 Objective.
Persson-Waye, K., et al. 1997. Effects on performance and work quality due to low-frequency ventilation noise. Journal of Sound and Vibration 205(4):467-474.
Plunkett, R. 1958. Interaction between a vibratory machine and its founda-tion. Noise Control 4(1).
Pollack, I. 1952. The loudness of bands of noise. Journal of the Acoustical Society of America 24(9):533.
Robinson, D.W. and R.S. Dadson. 1956. A redetermination of the equal loudness relations for pure tones. British Journal of Applied Physics 7(5):166. 7.18 2001 ASHRAE Fundamentals Handbook (SI) Schultz, T.J. 1985. Relationship between sound power level and sound pres-sure level in dwellings and offices. ASHRAE Transactions 91(1):124-53.
Warnock, A.C.C. 1998a. Sound pressure level vs. distance from sources in rooms. ASHRAE Transactions 104(1A):643-649.
Warnock, A.C.C. 1998b. Transmission of sound from air terminal devices through ceiling systems. ASHRAE Transactions 104(1A):650-657.
BIBLIOGRAPHY ASHRAE. 1998. Applications of manufacturers’ sound data.
Beranek, L.L. 1988. Noise and vibration control. Institute of Noise Control Engineering, Washington, D.C.
Beranek, L.L. 1986. Acoustics. American Institute of Physics, Acoustical Society of America, New York.
Harris, C.M. 1991. Handbook of acoustical measurements and noise con-trol. McGraw-Hill, New York.
Peterson, A.P.G. and E.E. Gross, Jr. 1974. Handbook of noise measurement.
GenRad, Inc., Concord, MA.
Schaffer, M.E. 1991. A practical guide to noise and vibration control for HVAC systems. ASHRAE, Atlanta.
8.1 CHAPTER 8 THERMAL COMFORT Human Thermoregulation ......................................................... 8.1 Energy Balance ......................................................................... 8.2 Thermal Exchanges with the Environment ............................... 8.3 Engineering Data and Measurements ...................................... 8.6 Conditions for Thermal Comfort ............................................ 8.12 Thermal Nonuniform Conditions and Local Discomfort ........ 8.13 Secondary Factors Affecting Comfort ..................................... 8.15 Prediction of Thermal Comfort ............................................... 8.16 Environmental Indices ............................................................ 8.19 Special Environments .............................................................. 8.22 Symbols ................................................................................... 8.26 PRINCIPAL purpose of heating, ventilating, and air-condition-Aing systems is to provide conditions for human thermal com-fort. A widely accepted definition is, “Thermal Comfort is that condition of mind that expresses satisfaction with the thermal envi-ronment” (ASHRAE Standard 55). This definition leaves open what is meant by condition of mind or satisfaction, but it correctly emphasizes that the judgment of comfort is a cognitive process involving many inputs influenced by physical, physiological, psy-chological, and other processes.
The conscious mind appears to reach conclusions about thermal comfort and discomfort from direct temperature and moisture sensa-tions from the skin, deep body temperatures, and the efforts neces-sary to regulate body temperatures (Hensel 1973, 1981; Hardy et al.
1971; Gagge 1937; Berglund 1995). In general, comfort occurs when body temperatures are held within narrow ranges, skin moisture is low, and the physiological effort of regulation is minimized.
Comfort also depends on behavioral actions that are initiated unconsciously or by the conscious mind and guided by thermal and moisture sensations to reduce discomfort. Some of the possible behavioral actions to reduce discomfort are altering clothing, alter-ing activity, changing posture or location, changing the thermostat setting, opening a window, complaining, or leaving the space.
Surprisingly, although regional climate conditions, living condi-tions, and cultures differ widely throughout the world, the temper-ature that people choose for comfort under like conditions of clothing, activity, humidity, and air movement has been found to be very similar (Fanger 1972; de Dear et al. 1991; Busch 1992).
This chapter summarizes the fundamentals of human thermoreg-ulation and comfort in terms useful to the engineer for operating sys-tems and designing for the comfort and health of building occupants.
HUMAN THERMOREGULATION The metabolic activities of the body result almost completely in heat that must be continuously dissipated and regulated to maintain normal body temperatures. Insufficient heat loss leads to overheat-ing, also called hyperthermia, and excessive heat loss results in body cooling, also called hypothermia. Skin temperature greater than 45°C or less than 18°C causes pain (Hardy et al. 1952). Skin temperatures associated with comfort at sedentary activities are 33 to 34°C and decrease with increasing activity (Fanger 1968). In con-trast, internal temperatures rise with activity. The temperature reg-ulatory center in the brain is about 36.8°C at rest in comfort and increases to about 37.4°C when walking and 37.9°C when jogging.
An internal temperature less than about 28°C can lead to serious cardiac arrhythmia and death, and a temperature greater than 46°C can cause irreversible brain damage. Therefore, the careful regula-tion of body temperature is critical to comfort and health.
The heat produced by a resting adult is about 100 W. Because most of this heat is transferred to the environment through the skin, it is often convenient to characterize metabolic activity in terms of heat production per unit area of skin. For the resting person, this is about 58 W/m2 and is called 1 met. This is based on the average male European, with a skin surface area of about 1.8 m2. For com-parison, female Europeans have an average surface area of 1.6 m2.
Systematic differences in this parameter may occur between ethnic and geographical groups. Higher metabolic rates are often described in terms of the resting rate. Thus, a person working at metabolic rate five times the resting rate would have a metabolic rate of 5 met.
The hypothalamus, located in the brain, is the central control organ for body temperature. It has hot and cold temperature sensors and is bathed by arterial blood. Since the recirculation rate of blood in the body is rapid and returning blood is mixed together in the heart before returning to the body, arterial blood is indicative of the average internal body temperature. The hypothalamus also receives thermal information from temperature sensors in the skin and per-haps other locations as well (spinal cord, gut), as summarized by Hensel (1981).
The hypothalamus controls various physiological processes of the body to regulate body temperature. Its control behavior is pri-marily proportional to deviations from set-point temperatures with some integral and derivative response aspects. The most important and often used of the physiological processes is regulating blood flow to the skin. When internal temperatures rise above a set point, an increasing proportion of the total blood is directed to the skin.
This vasodilation of skin blood vessels can increase skin blood flow by 15 times (from 1.7 mL/(s·m2) at resting comfort to 25 mL/(s·m2) in extreme heat) to carry internal heat to the skin for transfer to the environment. When body temperatures fall below the set point, skin blood flow is reduced (vasoconstricted) to conserve body heat. The effect of maximum vasoconstriction is equivalent to the insulating effect of a heavy sweater. At temperatures less than the set point, muscle tension increases to generate additional heat; where muscle groups are opposed, this may increase to visible shiv-ering. Shivering can double the resting rate of heat production.
At elevated internal temperatures, sweating occurs. This defense mechanism is a powerful way to cool the skin and increase heat loss from the core. The sweating function of the skin and its control is more advanced in humans than in other animals and is increasingly necessary for comfort at metabolic rates above resting level (Fanger 1968). Sweat glands pump perspiration onto the skin surface for evaporation. If conditions are good for evaporation, the skin can remain relatively dry even at high sweat rates with little perception of sweating. At skin conditions less favorable for evaporation, the sweat must spread out on the skin about the sweat gland until the sweat-covered area is sufficient to evaporate the sweat coming to the surface. The fraction of the skin that is covered with water to account for the observed total evaporation rate is termed skin wet-tedness (Gagge 1937).
The preparation of this chapter is assigned to TC 2.1, Physiology and Human Environment.
8.2 2001 ASHRAE Fundamentals Handbook (SI) Humans are quite good at sensing skin moisture from perspira-tion (Berglund and Cunningham 1986; Berglund 1994), and skin moisture correlates well with warm discomfort and unpleasantness (Winslow et al. 1937). It is rare for a sedentary or slightly active per-son to be comfortable with a skin wettedness greater than 25%. In addition to the perception of skin moisture, skin wettedness increases the friction between skin and fabrics, making clothing feel less pleasant and fabrics feel more coarse (Gwosdow et al. 1987).
This also occurs with architectural materials and surfaces, particu-larly smooth, nonhygroscopic surfaces.
With repeated intermittent heat exposure, the set point for the onset of sweating decreases and the proportional gain or tempera-ture sensitivity of the sweating system increases (Gonzalez et al.
1978, Hensel 1981). However, under long-term exposure to hot conditions, the set point increases, perhaps to reduce the physio-logical effort of sweating. Perspiration as secreted has a lower salt concentration than interstitial body fluid or blood plasma. After pro-longed heat exposure, sweat glands further reduce the salt concen-tration of sweat to conserve salt.
At the surface, the water in sweat evaporates while the dissolved salt and other constituents remain and accumulate. Because salt lowers the vapor pressure of water and thereby impedes its evapo-ration, the accumulating salt results in increased skin wettedness with time. Some of the relief and pleasure of washing after a warm day is related to the restoration of a hypotonic sweat film and de-creased skin wettedness. Other adaptations to heat are increased blood flow and sweating in peripheral regions where heat transfer is better. Such adaptations are examples of integral control.
The role of thermoregulatory effort in comfort is highlighted by the experiments of Chatonnet and Cabanac (1965) and observations of Kuno (1995). Chatonnet’s experiments compared the sensation of placing the subject’s hand in relatively hot or cold water (30 to 38°C) for 30 s given the subject at different thermal states. When the person was overheated or hyperthermic, the cold water was pleasant and the hot water was very unpleasant, but when the subject was in a cold or hypothermic state, the hand felt pleasant in hot water and unpleasant in cold water. Kuno (1995) describes similar observa-tions during transient whole body exposures to hot and cold envi-ronment. When a subject is in a state of thermal discomfort, any move away from the thermal stress of the uncomfortable environ-ment is perceived as pleasant during the transition.
ENERGY BALANCE Figure 1 shows the thermal interaction of the human body with its environment. The total metabolic rate of work M produced within the body is the metabolic rate required for the person’s activ-ity Mact plus the metabolic level required for shivering Mshiv (should shivering occur). A portion of the body’s energy production may be expended as external work done by the muscles W; the net heat pro-duction M −W is either stored (S), causing the body’s temperature to rise, or dissipated to the environment through the skin surface (qsk) and respiratory tract (qres).
(1) where M = rate of metabolic heat production, W/m2 W = rate of mechanical work accomplished, W/m2 qsk = total rate of heat loss from skin, W/m2 qres = total rate of heat loss through respiration, W/m2 C + R = sensible heat loss from skin, W/m2 Esk = total rate of evaporative heat loss from skin, W/m2 Cres = rate of convective heat loss from respiration, W/m2 Eres = rate of evaporative heat loss from respiration, W/m2 Ssk = rate of heat storage in skin compartment, W/m2 Scr = rate of heat storage in core compartment, W/m2 Heat dissipation from the body to the immediate surroundings occurs by several modes of heat exchange: sensible heat flow C + R from the skin; latent heat flow from the evaporation of sweat Ersw and from evaporation of moisture diffused through the skin Edif; sensible heat flow during respiration Cres; and latent heat flow due to evaporation of moisture during respiration Eres. Sensible heat flow from the skin may be a complex mixture of conduction, con-vection, and radiation for a clothed person; however, it is equal to the sum of the convection C and radiation R heat transfer at the outer clothing surface (or exposed skin).
Sensible and latent heat losses from the skin are typically ex-pressed in terms of environmental factors, skin temperature tsk, and skin wettedness w. The expressions also incorporate factors that ac-count for the thermal insulation and moisture permeability of cloth-ing. The independent environmental variables can be summarized as air temperature ta, mean radiant temperature , relative air velocity V, and ambient water vapor pressure pa. The independent personal variables that influence thermal comfort are activity and clothing.
The rate of heat storage in the body equals the rate of increase in internal energy. The body can be considered as two thermal compart-ments, the skin and the core (see the section on Two-Node Model under Prediction of Thermal Comfort). The rate of storage can be written separately for each compartment in terms of thermal capac-ity and time rate of change of temperature in each compartment: (2) (3) where αsk = fraction of body mass concentrated in skin compartment m = body mass, kg cp,b = specific heat capacity of body = 3490 J/(kg·K) AD = DuBois surface area, m2 tcr = temperature of core compartment, °C tsk = temperature of skin compartment, °C θ = time, s The fractional skin mass αsk depends on the rate of blood flow-ing to the skin surface.
M W – qsk qres S + + = C R Esk + + ( ) Cres Eres + ( ) Ssk S + cr ( ) + + = Fig. 1 Thermal Interaction of Human Body and Environment t r Scr 1 αsk – ( )mcp b , AD ----------------------------------- dtcr dθ ---------= Ssk αskmcp b , AD --------------------- dtsk dθ ---------= m · bl Thermal Comfort 8.3 THERMAL EXCHANGES WITH THE ENVIRONMENT Fanger (1967, 1970), Hardy (1949), Rapp and Gagge (1967), and Gagge and Hardy (1967) give quantitative information on calculat-ing the heat exchange between people and the environment. A sum-mary of the mathematical statements for various terms of heat exchange used in the heat balance equations (C, R, Esk, Cres, Eres) follows. Terms describing the heat exchanges associated with the thermoregulatory control mechanisms (qcr,sk, Mshiv, Ersw), values for the coefficients, and appropriate equations for Mact and AD are presented in later sections.
The mathematical description of the energy balance of the human body represents a combined rational/empirical approach to describing the thermal exchanges with the environment. Fundamen-tal heat transfer theory is used to describe the various mechanisms of sensible and latent heat exchange, while empirical expressions are used to determine the values of the coefficients describing these rates of heat exchange. Empirical equations are also used to describe the thermophysiological control mechanisms as a function of skin and core temperatures in the body.
Body Surface Area The terms in Equation (1) have units of power per unit area and refer to the surface area of the nude body. The most useful measure of nude body surface area, originally proposed by DuBois and DuBois (1916), is described by (4) where AD = DuBois surface area, m2 m = mass, kg l = height, m A correction factor fcl = Acl/AD must be applied to the heat transfer terms from the skin (C, R, and Esk) to account for the actual surface area Acl of the clothed body. This factor can be found in Table 7 for various clothing ensembles. For a 1.73 m tall, 70 kg man, AD = 1.8 m2. All terms in the basic heat balance equations are expressed per unit DuBois surface area.
Sensible Heat Loss from Skin Sensible heat exchange from the skin surface must pass through clothing to the surrounding environment. These paths are treated in series and can be described in terms of heat transfer (1) from the skin surface, through the clothing insulation, to the outer clothing surface, and (2) from the outer clothing surface to the environment.
Both convective C and radiative R heat losses from the outer sur-face of a clothed body can be expressed in terms of a heat transfer coefficient and the difference between the mean temperature tcl of the outer surface of the clothed body and the appropriate environ-mental temperature: (5) (6) where hc = convective heat transfer coefficient, W/(m2·K) hr = linear radiative heat transfer coefficient, W/(m2·K) fcl = clothing area factor Acl/AD, dimensionless The coefficients hc and hr are both evaluated at the clothing surface.
Equations (5) and (6) are commonly combined to describe the total sensible heat exchange by these two mechanisms in terms of an operative temperature to and a combined heat transfer coefficient h: (7) where (8) (9) Based on Equation (8), operative temperature to can be defined as the average of the mean radiant and ambient air temperatures, weighted by their respective heat transfer coefficients.
The actual transport of sensible heat through clothing involves conduction, convection, and radiation. It is usually most conve-nient to combine these into a single thermal resistance value Rcl, defined by (10) where Rcl is the thermal resistance of clothing in m2·K/W.
Since it is often inconvenient to include the clothing surface tem-perature in calculations, Equations (7) and (10) can be combined to eliminate tcl: (11) where to is defined in Equation (8).
Evaporative Heat Loss from Skin Evaporative heat loss Esk from skin depends on the amount of moisture on the skin and the difference between the water vapor pressure at the skin and in the ambient environment: (12) where w = skin wettedness, dimensionless psk,s = water vapor pressure at skin, normally assumed to be that of saturated water vapor at tsk, kPa pa = water vapor pressure in ambient air, kPa Re,cl = evaporative heat transfer resistance of clothing layer (analogous to Rcl), m2·kPa/W he = evaporative heat transfer coefficient (analogous to hc), W/(m2·kPa) Procedures for calculating Re,cl and he are given in the section on Engineering Data and Measurements. The skin wettedness fraction is the ratio of the actual evaporative heat loss to the maximum pos-sible evaporative heat loss Emax with the same conditions and a completely wet skin (w = 1). Skin wettedness is important in deter-mining evaporative heat loss. Maximum evaporative potential Emax occurs when w = 1.
Evaporative heat loss from the skin is a combination of the evap-oration of sweat secreted due to thermoregulatory control mecha-nisms Ersw and the natural diffusion of water through the skin Edif: (13) Evaporative heat loss by regulatory sweating is directly propor-tional to the rate of regulatory sweat generation: (14) where hfg = heat of vaporization of water = 2.43 × 106 J/kg at 30°C = rate at which regulatory sweat is generated, kg/(s·m2) AD 0.202m0.425l0.725 = C fclhc tcl ta – ( ) = R fclhr tcl t r – ( ) = C R + fclh tcl to – ( ) = to hr t r hcta + hr hc + ---------------------------= h hr hc + = C R + tsk tcl – ( ) Rcl ⁄ = C R + tsk to – Rcl 1 fclh ( ) ⁄ + ----------------------------------------= Esk w psk s , pa – ( ) Re cl , 1 fclhe ( ) ⁄ + ------------------------------------------------= Esk Ersw Edif + = Ersw m · rswhfg = m · rsw 8.4 2001 ASHRAE Fundamentals Handbook (SI) The portion wrsw of a body that must be wetted to evaporate the reg-ulatory sweat is (15) With no regulatory sweating, skin wettedness due to diffusion is approximately 0.06 for normal conditions. For large values of Emax or long exposures to low humidities, the value may drop to as low as 0.02, since dehydration of the outer skin layers alters its diffusive characteristics. With regulatory sweating, the 0.06 value applies only to the portion of skin not covered with sweat (1 −wrsw); the dif-fusion evaporative heat loss is (16) These equations can be solved for w, given the maximum evapora-tive potential Emax and the regulatory sweat generation Ersw: (17) Once skin wettedness is determined, evaporative heat loss from the skin is calculated from Equation (12), or by (18) To summarize, the following calculations determine w and Esk: Emax Equation (12), with w = 1.0 Ersw Equation (14) w Equation (17) Esk Equation (18) or (12) Although evaporation from the skin Esk as described in Equation (12) depends on w, the body does not directly regulate skin wetted-ness but, rather, regulates sweat rate [Equation (14)]. Skin wettedness is then an indirect result of the relative activity of the sweat glands and the evaporative potential of the environment. Skin wettedness of 1.0 is the upper theoretical limit. If the aforemen-tioned calculations yield a wettedness of more than 1.0, then Equa-tion (14) is no longer valid because not all the sweat is evaporated.
In this case, Esk = Emax.
Skin wettedness is strongly correlated with warm discomfort and is also a good measure of thermal stress. Theoretically, skin wetted-ness can approach 1.0 while the body still maintains thermoregula-tory control. In most situations, it is difficult to exceed 0.8 (Berglund and Gonzalez 1978). Azer (1982) recommends 0.5 as a practical upper limit for sustained activity for a healthy acclimatized person.
Respiratory Losses During respiration, the body loses both sensible and latent heat by convection and evaporation of heat and water vapor from the res-piratory tract to the inhaled air. A significant amount of heat can be associated with respiration because the air is inspired at ambient conditions and expired nearly saturated at a temperature only slightly cooler than tcr.
The total heat and moisture losses due to respiration are (19) (20) where = pulmonary ventilation rate, kg/s hex = enthalpy of exhaled air, J/kg (dry air) ha = enthalpy of inspired (ambient) air, J/kg (dry air) = pulmonary water loss rate, kg/s Wex = humidity ratio of exhaled air, kg (water vapor)/kg (dry air) Wa = humidity ratio of inspired (ambient) air, kg (water vapor)/kg (dry air) Under normal circumstances, pulmonary ventilation rate is prima-rily a function of metabolic rate (Fanger 1970): (21) where M = metabolic rate, W/m2 Kres = proportionality constant (1.43 × 10–6 kg/J) Respiratory air is nearly saturated and near the body temperature when exhaled. For typical indoor environments (McCutchan and Taylor 1951), the exhaled temperature and humidity ratio are given in terms of ambient conditions: (22) (23) where ambient ta and exhaled tex air temperatures are in °C. For extreme conditions, such as outdoor winter environments, different relationships may be required (Holmer 1984).
The humidity ratio of ambient air can be expressed in terms of total or barometric pressure pt and ambient water vapor pressure pa: (24) Respiratory heat loss is often expressed in terms of sensible Cres and latent Eres heat losses. Two approximations are commonly used to simplify Equations (22) and (23) for that purpose. First, because the dry respiratory heat loss is relatively small compared to the other terms in the heat balance, an average value for tex is determined by evaluating Equation (22) at standard conditions of 20°C, 50% rh, sea level. Second, noting in Equation (23) that there is only a weak dependence on ta, the second term in Equation (23) and the denom-inator in Equation (24) are evaluated at standard conditions. Using these approximations and substituting latent heat hfg and specific heat of air cp,a at standard conditions, Cres and Eres can be deter-mined by (25) (26) where pa is expressed in kPa and ta is in °C.
Alternative Formulations Equations (11) and (12) describe heat loss from skin for clothed people in terms of clothing parameters Rcl, Re,cl, and fcl; parameters h and he describe outer surface resistances. Other parameters and definitions are also used. Although these alternative parameters and definitions may be confusing, note that the information presented in one form can be converted to another form. Table 1 presents com-mon parameters and their qualitative descriptions. Table 2 presents equations showing their relationship to each other. Generally, parameters related to dry or evaporative heat flows are not indepen-dent because they both rely, in part, on the same physical processes.
The Lewis relation describes the relationship between convective heat transfer and mass transfer coefficients for a surface [see Equa-tion (39) in Chapter 5]. The Lewis relation can be used to relate con-vective and evaporative heat transfer coefficients defined in Equations (5) and (12) according to wrsw Ersw Emax ⁄ = Edif 1 wrsw – ( )0.06Emax = w wrsw 0.06 1 wrsw – ( ) + 0.06 0.94Ersw Emax ⁄ + = = Esk wEmax = m · rsw qres Cres Eres + = m · res hex ha – ( ) AD -----------------------------------= m · w res , m · res Wex Wa – ( ) AD ---------------------------------------= m · res m · w res , m · res KresMAD = tex 32.6 0.066ta 32Wa + + = Wex 0.0277 0.000065ta 0.2Wa + + = Wa 0.622pa pt pa – -------------------= Cres 0.0014M 34 ta – ( ) = Eres 0.0173M 5.87 pa – ( ) = Thermal Comfort 8.5 (27) where LR is the Lewis ratio and, at typical indoor conditions, equals approximately 16.5 K/kPa. The Lewis relation applies to sur-face convection coefficients. Heat transfer coefficients that include the effects of insulation layers and/or radiation are still coupled, but the relationship may deviate significantly from that for a surface.
The i terms in Tables 1 and 2 describe how the actual ratios of these parameters deviate from the ideal Lewis ratio (Woodcock 1962; Oohori et al. 1984).
Depending on the combination of parameters used, heat transfer from the skin can be calculated using several different formulations (see Tables 2 and 3). If the parameters are used correctly, the end result will be the same regardless of the formulation used.
Total Skin Heat Loss Total skin heat loss—sensible heat plus evaporative heat—can be calculated from any combination of the equations presented in Table 3. Total skin heat loss is used as a measure of the thermal envi-ronment; two combinations of parameters that yield the same total heat loss for a given set of body conditions (tsk and w) are considered to be approximately equivalent. The fully expanded skin heat loss equation, showing each parameter that must be known or specified, is as follows: (28) where to is the operative temperature and represents the temperature of a uniform environment that will transfer dry heat at the same rate as in the actual environment [to = (trhr + tahc)/(hc + hr)].
After rearranging, Equation (28) becomes (29) This equation allows the trade-off between any two or more parameters to be evaluated under given conditions. If the trade-off Table 1 Parameters Used to Describe Clothing Sensible Heat Flow Evaporative Heat Flow Rcl = intrinsic clothing insulation, the thermal resistance of a uniform layer of insulation covering the entire body that has the same effect on sensible heat flow as the actual clothing.
Re,cl = evaporative heat transfer resistance of the clothing, the impedance to transport of water vapor of a uniform layer of insulation covering the entire body that has the same effect on evaporative heat flow as the actual clothing.
Rt = total insulation, the total equivalent uniform thermal resistance between the body and the environment: clothing and boundary resistance.
Re,t = total evaporative resistance, the total equivalent uniform impedance to the transport of water vapor from the skin to the environment.
Rcle = effective clothing insulation, the increased body insulation due to clothing as compared to the nude state.
Fpcl = permeation efficiency, the ratio of the actual evaporative heat loss to that of a nude body at the same conditions, including an adjustment for the increase in surface area due to the clothing.
Ra = boundary insulation, the thermal resistance at the skin boundary for a nude body.
Ra,cl = outer boundary insulation, the thermal resistance at the outer boundary (skin or clothing).
Parameters Relating Sensible and Evaporative Heat Flows h′ = overall sensible heat transfer coefficient, overall equivalent uniform conductance between the body (including clothing) and the environment.
icl = clothing vapor permeation efficiency, the ratio of the actual evaporative heat flow capability through the clothing to the sensible heat flow capability as compared to the Lewis ratio.
h′ cl = clothing conductance, the thermal conductance of a uniform layer of insulation covering the entire body that has the same effect on sensible heat flow as the actual clothing.
im = total vapor permeation efficiency, the ratio of the actual evaporative heat flow capability between the skin and the environment to the sensible heat flow capability as compared to the Lewis ratio.
Fcle = effective clothing thermal efficiency, the ratio of the actual sensible heat loss to that of a nude body at the same conditions.
Fcl = intrinsic clothing thermal efficiency, the ratio of the actual sensible heat loss to that of a nude body at the same conditions including an adjustment for the increase in surface area due to the clothing.
ia = air layer vapor permeation efficiency, the ratio of the actual evaporative heat flow capability through the outer air layer to the sensible heat flow capability as compared to the Lewis ratio.
Table 2 Relationships Between Clothing Parameters Sensible Heat Flow Rt = Rcl + 1/(hfcl) = Rcl + Ra/fcl Rt = Rcle + 1/h = Rcle + Ra h′ cl = 1/Rcl h′ = 1/Rt h = 1/Ra Ra,cl = Ra/fcl Fcl = h′ /(hfcl) = 1/(1 + fclhRcl) Fcle = h′ /h = fcl/(1 + fclhRcl) = fclFcl Evaporative Heat Flow Re,t = Re,cl + 1/(he fcl) = Re,cl + Re,a/fcl he = 1/Re,a h′ e,cl = 1/Re,cl h′ e = 1/Re,t = fclFpclhe Fpcl = 1/(1 + fclheRe,cl) Parameters Relating Sensible and Evaporative Heat Flows iclLR = h′ e,cl/h′ cl = Rcl/Re,cl imLR = h′ e/h′ = Rt/Re,t im = (Rcl + Ra,cl)/[(Rcl/icl) + (Ra,cl/ia)] iaLR = he/h ia = hc/(hc + hr) he hc ⁄ LR = Table 3 Skin Heat Loss Equations Sensible Heat Loss C + R = (tsk −to)/[Rcl + 1/(fclh)] C + R = (tsk −to)/Rt C + R = Fcleh(tsk −to) C + R = Fcl fclh(tsk −to) C + R = h′ (tsk −to) Evaporative Heat Loss Esk = w(psk,s −pa)/[Re,cl + 1/( fclhe)] Esk = w(psk,s −pa)/Re,t Esk = wFpcl fcl he (psk,s −pa) Esk = h′ e w(psk,s −pa) Esk = h′ wimLR(psk,s −pa) qsk tsk to – Rcl Ra cl , + ---------------------------w psk s , pa – ( ) Re cl , 1 LRhcfcl ( ) ⁄ + ------------------------------------------------------+ = ta tr – ( ) qsk Fcl fclh tsk to – ( ) wLRFpcl hc psk s , pa – ( ) + = 8.6 2001 ASHRAE Fundamentals Handbook (SI) between two specific variables is to be examined, then a simplified form of the equation suffices. The trade-off between operative temperature and humidity is often of interest. Equation (28) can be written in a simpler form for this purpose (Fobelets and Gagge 1988): (30) Equation (30) can be used to define a combined temperature tcom, which reflects the combined effect of operative temperature and humidity for an actual environment: or (31) where is a vapor pressure related in some fixed way to tcom and is analogous to pwb,s for twb. The term wimLR is constant to the extent that im is constant, and any combination of to and pa that gives the same tcom will result in the same total heat loss.
Environmental indices are discussed in the section on Environ-mental Indices. Two of these, the humid operative temperature toh and the effective temperature ET, can be represented in terms of Equation (31). The humid operative temperature is that tempera-ture which at 100% rh yields the same total heat loss as for the actual environment: (32) where poh,s is saturated vapor pressure, in kPa, at toh.
The effective temperature is the temperature at 50% rh that yields the same total heat loss from the skin as for the actual environment: (33) where pET,s is saturated vapor pressure, in kPa, at ET.
The psychrometric chart in Figure 2 shows a constant total heat loss line and the relationship between these indices. This line represents only one specific skin wettedness and permeation efficiency index. The relationship between indices depends on these two parameters (see the section on Environmental Indices).
ENGINEERING DATA AND MEASUREMENTS Applying the preceding basic equations to practical problems of the thermal environment requires quantitative estimates of the body’s surface area, metabolic requirements for a given activity and the mechanical efficiency for the work accomplished, evalu-ation of the heat transfer coefficients hr and hc, and the general nature of the clothing insulation used. This section provides the necessary data and describes methods used to measure the param-eters of the heat balance equation.
Metabolic Rate and Mechanical Efficiency Maximum Capacity. In choosing optimal conditions for com-fort and health, the rate of work done during routine physical activities must be known, because metabolic power increases in proportion to exercise intensity. Metabolic rate varies over a wide range, depending on the activity, the person, and the conditions under which the activity is performed. Table 4 lists typical meta-bolic rates for an average adult (AD = 1.8 m2) for activities per-formed continuously. The highest power a person can maintain for any continuous period is approximately 50% of the maximal capacity to use oxygen (maximum energy capacity).
A unit used to express the metabolic rate per unit DuBois area is the met, defined as the metabolic rate of a sedentary person (seated, quiet): 1 met = 58.1 W/m2 = 50 kcal/(h·m2). A normal, healthy man has a maximum capacity of approximately Mact = 12 met at age 20, which drops to 7 met at age 70. Maximum rates for women are about 30% lower. Long-distance runners and trained athletes have maximum rates as high as 20 met. An aver-age 35 year old who does not exercise has a maximum rate of about 10 met, and activities with Mact > 5 met are likely to prove exhausting.
Intermittent Activity. The activity of many people consists of a mixture of activities or a combination of work-rest periods. A weighted average metabolic rate is generally satisfactory, pro-vided that activities alternate frequently (several times per hour).
For example, a person typing 50% of the time, filing while seated 25% of the time, and walking about 25% of the time would have an average metabolic rate of 0.50 × 65 + 0.25 × 70 + 0.25 × 100 = 75 W/m2 (see Table 4).
Accuracy. Estimating metabolic rates is difficult. The values given in Table 4 indicate metabolic rates only for the specific activities listed. Some entries give a range and some a single value, depending on the source of the data. The level of accuracy depends on the value of Mact and how well the activity can be defined. For well-defined activities with Mact < 1.5 met (e.g., read-ing), Table 4 is sufficiently accurate for most engineering pur-poses. For values of Mact > 3, where a task is poorly defined or where there are a variety of ways of performing a task (e.g., heavy machine work), the values may be in error by as much as ±50% for a given application. Engineering calculations should thus allow for potential variations.
Measurement. When metabolic rates must be determined more accurately than is possible with tabulated data, physiological mea-surements with human subjects may be necessary. The rate of met-abolic heat produced by the body is most accurately measured by the rate of respiratory oxygen consumption and carbon dioxide pro-duction. An empirical equation for metabolic rate is given by Nishi (1981): Fig. 2 Constant Skin Heat Loss Line and Its Relationship to toh and ET qsk h′ tsk wimLRpsk s , + ( ) to wimLRpa + ( ) – [ ] = tcom wimLRptcom + to wimLRpa + = tcom to wimLRpa wimLRptcom – + = ptcom ptcom toh to wimLR pa poh s , – ( ) + = ET to wimLR pa 0.5pET s , – ( ) + = Thermal Comfort 8.7 (34) where M = metabolic rate, W/m2 RQ = respiratory quotient; molar ratio of QCO2 exhaled to QO2 inhaled, dimensionless QO2 = volumetric rate of oxygen consumption at conditions (STPD) of 0°C, 101.325 kPa, mL/s The exact value of the respiratory quotient RQ used in Equation (34) depends on a person’s activity, diet, and physical condition. It can be determined by measuring both carbon dioxide and oxygen in the respiratory airflows, or it can be estimated with reasonable accuracy. A good estimate for the average adult is RQ = 0.83 for light or sedentary activities (M < 1.5 met), increasing proportion-ately to RQ = 1.0 for extremely heavy exertion (M = 5.0 met). Esti-mation of RQ is generally sufficient for all except precision laboratory measurements since it does not strongly affect the value of the metabolic rate. A 10% error in estimating the respiratory quotient results in an error of less than 3% in the metabolic rate.
A second, much less accurate, method of estimating metabolic rate physiologically is to measure the heart rate. Table 5 shows the relationship between heart rate and oxygen consumption at different levels of physical exertion for a typical person. Once oxygen con-sumption is estimated from heart rate information, Equation (34) can be used to estimate the metabolic rate. A number of factors other than metabolic rate affect heart rate, such as physical condition, heat, emotional factors, muscles used, etc. Astrand and Rodahl (1977) show that heart rate is only a very approximate measure of metabolic rate and should not be the only source of information where accuracy is required.
Mechanical Efficiency. In the heat balance equation, the rate W of work accomplished must be in the same units as metabolism M and expressed in terms of AD in W/m2. The mechanical work done by the muscles for a given task is often expressed in terms of the body’s mechanical efficiency µ = W/M. It is unusual for µ to be more than 0.05 to 0.10; for most activities, it is close to zero. The maximum value under optimal conditions (e.g., bicycle ergometer) is µ = 0.20 to 0.24 (Nishi 1981). It is common to assume that mechanical work is zero for several reasons: (1) the mechanical work produced is small compared to metabolic rate, especially for office activities; (2) estimates for metabolic rates can often be inaccurate; and (3) this assumption results in a more conservative estimate when designing air-conditioning equip-ment for upper comfort and health limits. More accurate calcula-tion of heat generation may require estimation of the mechanical work produced for activities where it is significant (walking on a grade, climbing a ladder, bicycling, lifting, etc.). In some cases, it is possible to either estimate or measure the mechanical work.
For example, a 90 kg person walking up a 5% grade at 1.0 m/s would be lifting an 882 N (90 kg × 9.8 N/kg) weight over a height of 0.05 m every second, for a work rate of 44 N·m/s = 44 W. This rate of mechanical work would then be subtracted from M to determine the net heat generated.
Heat Transfer Coefficients Values for the linearized radiative heat transfer coefficient, con-vective heat transfer coefficient, and evaporative heat transfer coef-ficient are required to solve the equations describing heat transfer from the body.
Radiative Heat Transfer Coefficient. The linearized radiative heat transfer coefficient can be calculated by (35) where hr = radiative heat transfer coefficient, W/(m2·K) ε = average emissivity of clothing or body surface, dimensionless σ = Stefan-Boltzmann constant, 5.67 × 10− 8 W/(m2·K4) Ar = effective radiation area of body, m2 Table 4 Typical Metabolic Heat Generation for Various Activities W/m2 meta Resting Sleeping 40 0.7 Reclining 45 0.8 Seated, quiet 60 1.0 Standing, relaxed 70 1.2 Walking (on level surface) 3.2 km/h (0.9 m/s) 115 2.0 4.3 km/h (1.2 m/s) 150 2.6 6.4 km/h (1.8 m/s) 220 3.8 Office Activities Reading, seated 55 1.0 Writing 60 1.0 Typing 65 1.1 Filing, seated 70 1.2 Filing, standing 80 1.4 Walking about 100 1.7 Lifting/packing 120 2.1 Driving/Flying Car 60 to 115 1.0 to 2.0 Aircraft, routine 70 1.2 Aircraft, instrument landing 105 1.8 Aircraft, combat 140 2.4 Heavy vehicle 185 3.2 Miscellaneous Occupational Activities Cooking 95 to 115 1.6 to 2.0 Housecleaning 115 to 200 2.0 to 3.4 Seated, heavy limb movement 130 2.2 Machine work sawing (table saw) 105 1.8 light (electrical industry) 115 to 140 2.0 to 2.4 heavy 235 4.0 Handling 50 kg bags 235 4.0 Pick and shovel work 235 to 280 4.0 to 4.8 Miscellaneous Leisure Activities Dancing, social 140 to 255 2.4 to 4.4 Calisthenics/exercise 175 to 235 3.0 to 4.0 Tennis, singles 210 to 270 3.6 to 4.0 Basketball 290 to 440 5.0 to 7.6 Wrestling, competitive 410 to 505 7.0 to 8.7 Sources: Compiled from various sources. For additional information, see Buskirk (1960), Passmore and Durnin (1967), and Webb (1964).
a1 met = 58.1 W/m2 M 21 0.23RQ 0.77 + ( )QO2 AD ---------------------------------------------------------= Table 5 Heart Rate and Oxygen Consumption at Different Activity Levels Level of Exertion Heart Rate, bpm Oxygen Consumed, mL/s Light work < 90 < 8 Moderate work 90 to 110 8 to 16 Heavy work 110 to 130 16 to 24 Very heavy work 130 to 150 24 to 32 Extremely heavy work 150 to 170 > 32 Source: Astrand and Rodahl (1977).
hr 4εσ Ar AD ------- 273.2 tcl t r + 2 -----------------+ 3 = 8.8 2001 ASHRAE Fundamentals Handbook (SI) The ratio Ar/AD is 0.70 for a sitting person and 0.73 for a standing person (Fanger 1967). Emissivity ε is close to unity (typically 0.95), unless special reflective materials are used or high-temperature sources are involved. It is not always possible to solve Equation (35) explicitly for hr, since tcl may also be an unknown. Some form of iteration may be necessary if a precise solution is required. Fortu-nately, hr is nearly constant for typical indoor temperatures, and a value of 4.7 W/(m2·K) suffices for most calculations. If the emis-sivity is significantly less than unity, the value should be adjusted by (36) where ε represents the area-weighted average emissivity for the clothing/body surface.
Convective Heat Transfer Coefficient. Heat transfer by con-vection is usually caused by air movement within the living space or by body movements. Equations for estimating hc under various conditions are presented in Table 6. Where two conditions apply (e.g., walking in moving air), a reasonable estimate can be ob-tained by taking the larger of the two values for hc. Limits have been given to all equations. If no limits were given in the source, reasonable limits have been estimated. Care should be exercised in using these values for seated and reclining persons. The heat trans-fer coefficients may be accurate, but the effective heat transfer area may be substantially reduced due to body contact with a padded chair or bed.
Quantitative values of hc are important, not only in estimating convection loss, but in evaluating (1) operative temperature to, (2) clothing parameters It and im, and (3) rational effective temperatures toh and ET. All heat transfer coefficients in Table 6 were evaluated at or near 101.33 kPa. These coefficients should be corrected as fol-lows for atmospheric pressure: (37) where hcc = corrected convective heat transfer coefficient, W/(m2·K) pt = local atmospheric pressure, kPa The combined coefficient h is the sum of hr and hc, described in Equation (35) and Table 6, respectively. The coefficient h governs exchange by radiation and convection from the exposed body sur-face to the surrounding environment.
Evaporative Heat Transfer Coefficient. The evaporative heat transfer coefficient he for the outer air layer of a nude or clothed per-son can be estimated from the convective heat transfer coefficient using the Lewis relation given in Equation (27). If the atmospheric pressure is significantly different from the reference value (101.33 kPa), the correction to the valueobtained from Equation (27) is (38) where hec is the corrected evaporative heat transfer coefficient in W/(m2·kPa).
Clothing Insulation and Permeation Efficiency Thermal Insulation. The most accurate methods for determin-ing clothing insulation are (1) measurements on heated mannequins (McCullough and Jones 1984, Olesen and Nielsen 1983) and (2) measurements on active subjects (Nishi et al. 1975). For most rou-tine engineering work, estimates based on tables and equations presented in this section are sufficient. Thermal mannequins can measure the sensible heat loss from the “skin” (C + R) in a given environment. Equation (11) can then be used to evaluate Rcl if the environmental conditions are well defined and fcl is measured.
Evaluation of clothing insulation on subjects requires measurement of tsk, tcl, and to. The clothing thermal efficiency is calculated by (39) The intrinsic clothing insulation can then be calculated from man-nequin measurements by the following relationship, provided fcl is measured and conditions are sufficiently well defined to make an accurate determination of h: (40) where q is heat loss from the mannequin in W/m2.
Clothing insulation value may be expressed in clo units. In order to avoid confusion, the symbol I is used with the clo unit instead of the symbol R. The relationship between the two is (41) or 1.0 clo is equivalent to 0.155 m2·K/W.
Because clothing insulation cannot be measured for most routine engineering applications, tablesofmeasuredvalues forvariouscloth-ing ensembles can be used to select an ensemble comparable to the one(s) in question. Table 7 gives values for typical indoor clothing ensembles. More detailed tables are presented by McCullough and Jones(1984) and Olesen and Nielsen (1983). Accuracies for Icl on the order of ±20% are typical if good matches between ensembles are found.
Often it is not possible to find an already measured clothing ensemble that matches the one in question. In this case, the ensem-ble insulation can be estimated from the insulation of individual gar-ments. Table 8 gives a list of individual garments commonly worn.
The insulation of an ensemble is estimated from the individual val-ues using a summation formula (McCullough and Jones 1984): (42) where Iclu,i is the effective insulation of garment i, and Icl, as before, is the insulation for the entire ensemble. A simpler and nearly as accurate summation formula is (Olesen 1985) Table 6 Equations for Convection Heat Transfer Coefficients Equation Limits Condition Remarks/Sources hc = 8.3V 0.6 0.2 < V < 4.0 Seated with moving air Mitchell (1974) hc = 3.1 0 < V < 0.2 hc = 2.7 + 8.7V 0.67 0.15 < V < 1.5 Reclining with moving air Colin and Houdas (1967) hc = 5.1 0 < V < 0.15 hc = 8.6V 0.53 0.5 < V < 2.0 Walking in still air V is walking speed (Nishi and Gagge 1970) hc = 5.7(M −0.8)0.39 1.1 < M < 3.0 Active in still air Gagge et al. (1976) hc = 6.5V 0.39 0.5 < V < 2.0 Walking on treadmill in still air V is treadmill speed (Nishi and Gagge 1970) hc = 14.8V 0.69 0.15 < V < 1.5 Standing person in moving air Developed from data presented by Seppanen et al.
(1972) hc = 4.0 0 < V < 0.15 Note: hc in W/(m2· K), V in m/s, and M in mets, where 1 met = 58.1 W/m2.
hr 4.7ε = hcc hc pt 101.33 ⁄ ( )0.55 = hec he 101.33 pt ⁄ ( )0.45 = Fcl tcl to – tsk to – ----------------= Rcl tsk to – q ----------------1 hfcl --------– = R 0.155I = Icl 0.835 Iclu i , i ∑ 0.161 + = Thermal Comfort 8.9 (43) Either Equation (42) or (43) gives acceptable accuracy for typical indoor clothing. The main source of inaccuracy is in determining the appropriate values for individual garments. Overall accuracies are on the order of ±25% if the tables are used carefully. If it is important to include a specific garment that is not included in Table 8, its insulation can be estimated by (McCullough and Jones 1984) (44) where xf = fabric thickness, mm AG = body surface area covered by garment, m2 Values in Table 7 may be adjusted by information in Table 8 and a summation formula. Using this method, values of Iclu,i for the selected items in Table 8 are then added to or subtracted from the ensemble value of Icl in Table 7.
When a person is sitting, the chair generally has the effect of increasing clothing insulation by up to 0.15 clo, depending on the contact area Ach between the chair and body (McCullough et al.
1994). A string webbed or beach chair has little or no contact area, and the insulation actually decreases by about 0.1 clo due likely to compression of the clothing in the contact area. In contrast, a cush-ioned executive chair has a large contact area that can increase the intrinsic clothing insulation by 0.15 clo. For other chairs, the increase in intrinsic insulation (∆Icl) can be estimated from (45) where Ach is in m2.
For example, a desk chair with a body contact area of 0.27 m2 has a ∆Icl of 0.1 clo. This amount should be added to the intrinsic insu-lation of the standing clothing ensemble to obtain the insulation of the ensemble when sitting in the desk chair.
Although sitting has the effect of increasing clothing insulation, walking decreases it (McCullough and Hong 1994). The change in Table 7 Typical Insulation and Permeation Efficiency Values for Clothing Ensembles Ensemble Descriptiona Icl (clo) It b (clo) fcl icl im b Walking shorts, short-sleeved shirt 0.36 1.02 1.10 0.34 0.42 Trousers, short-sleeved shirt 0.57 1.20 1.15 0.36 0.43 Trousers, long-sleeved shirt 0.61 1.21 1.20 0.41 0.45 Same as above, plus suit jacket 0.96 1.54 1.23 Same as above, plus vest and T-shirt 1.14 1.69 1.32 0.32 0.37 Trousers, long-sleeved shirt, long-sleeved sweater, T-shirt 1.01 1.56 1.28 Same as above, plus suit jacket and long underwear bottoms 1.30 1.83 1.33 Sweat pants, sweat shirt 0.74 1.35 1.19 0.41 0.45 Long-sleeved pajama top, long pajama trousers, short 3/4 sleeved robe, slippers (no socks) 0.96 1.50 1.32 0.37 0.41 Knee-length skirt, short-sleeved shirt, panty hose, sandals 0.54 1.10 1.26 Knee-length skirt, long-sleeved shirt, full slip, panty hose 0.67 1.22 1.29 Knee-length skirt, long-sleeved shirt, half slip, panty hose, long-sleeved sweater 1.10 1.59 1.46 Same as above, replace sweater with suit jacket 1.04 1.60 1.30 0.35 0.40 Ankle-length skirt, long-sleeved shirt, suit jacket, panty hose 1.10 1.59 1.46 Long-sleeved coveralls, T-shirt 0.72 1.30 1.23 Overalls, long-sleeved shirt, T-shirt 0.89 1.46 1.27 0.35 0.40 Insulated coveralls, long-sleeved thermal underwear, long underwear bottoms 1.37 1.94 1.26 0.35 0.39 Source: From McCullough and Jones (1984) and McCullough et al. (1989).
aAll ensembles include shoes and briefs or panties. All ensembles except those with panty hose include socks unless otherwise noted.
bFor tr = ta and air velocity less than 0.2 m/s (Ia = 0.72 clo and im = 0.48 when nude).
1 clo = 0.155 m2·K/W.
Icl Iclu i , i ∑ = Iclu i , 0.534 0.135xf + ( ) AG AD ⁄ ( ) 0.0549 – = Icl ∆ 0.748Ach 0.1 – = Table 8 Garment Insulation Values Garment Descriptiona Iclu,i, clob Garment Descriptiona Iclu,i, clob Garment Descriptiona Iclu,i, clob Underwear Long-sleeved, flannel shirt 0.34 Long-sleeved (thin) 0.25 Men’s briefs 0.04 Short-sleeved, knit sport shirt 0.17 Long-sleeved (thick) 0.36 Panties 0.03 Long-sleeved, sweat shirt 0.34 Dresses and skirtsc Bra 0.01 Trousers and Coveralls 0.06 Skirt (thin) 0.14 T-shirt 0.08 Short shorts 0.08 Skirt (thick) 0.23 Full slip 0.16 Walking shorts 0.15 Long-sleeved shirtdress (thin) 0.33 Half slip 0.14 Straight trousers (thin) 0.24 Long-sleeved shirtdress (thick) 0.47 Long underwear top 0.20 Straight trousers (thick) 0.28 Short-sleeved shirtdress (thin) 0.29 Long underwear bottoms 0.15 Sweatpants 0.30 Sleeveless, scoop neck (thin) 0.23 Footwear Overalls 0.49 Sleeveless, scoop neck (thick), i.e., jumper 0.27 Ankle-length athletic socks 0.02 Coveralls Calf-length socks 0.03 Suit jackets and vests (lined) Sleepwear and Robes Knee socks (thick) 0.06 Single-breasted (thin) 0.36 Sleeveless, short gown (thin) 0.18 Panty hose 0.02 Single-breasted (thick) 0.44 Sleeveless, long gown (thin) 0.20 Sandals/thongs 0.02 Double-breasted (thin) 0.42 Short-sleeved hospital gown 0.31 Slippers (quilted, pile-lined) 0.03 Double-breasted (thick) 0.48 Long-sleeved, long gown (thick) 0.46 Boots 0.10 Sleeveless vest (thin) 0.10 Long-sleeved pajamas (thick) 0.57 Shirts and Blouses Sleeveless vest (thick) 0.17 Short-sleeved pajamas (thin) 0.42 Sleeveless, scoop-neck blouse 0.12 Sweaters Long-sleeved, long wrap robe (thick) 0.69 Short-sleeved, dress shirt 0.19 Sleeveless vest (thin) 0.13 Long-sleeved, short wrap robe (thick) 0.48 Long-sleeved, dress shirt 0.25 Sleeveless vest (thick) 0.22 Short-sleeved, short robe (thin) 0.34 a“Thin” garments are made of light, thin fabrics worn in summer; “thick” garments are made of heavy, thick fabrics worn in winter.
b1 clo = 0.155 m2·K/W cKnee-length 8.10 2001 ASHRAE Fundamentals Handbook (SI) clothing insulation (∆Icl) can be estimated from the standing intrin-sic insulation of the ensemble (Icl) and the walking speed (Walk-speed) in steps per minute: (46) For example, the clothing insulation of a person wearing a winter business suit with a standing intrinsic insulation of 1 clo would decrease by 0.52 clo when the person walks at 90 steps per minute (about 3.7 km/h). Thus, when the person is walking, the intrinsic insulation of the ensemble would be 0.48 clo.
Permeation Efficiency. Permeation efficiency data for some clothing ensembles are presented in terms of icl and im in Table 7.
The values of im can be used to calculate Re,t using the relationships in Table 2. Ensembles worn indoors generally fall in the range 0.3 < im < 0.5, and assuming im = 0.4 is reasonably accurate (McCullough et al. 1989). This latter value may be used if a good match to ensembles in Table 7 cannot be made. The value of im or Re,t may be substituted directly into equations for body heat loss cal-culations (see Table 3). However, im for a given clothing ensemble is a function of the environment as well as the clothing properties.
Unless im is evaluated at conditions very similar to the intended application, it is more rigorous to use icl to describe the permeation efficiency of the clothing. The value of icl is not as sensitive to envi-ronmental conditions; thus, given data are more accurate over a wider range of air velocity and radiant and air temperature combi-nations for icl than for im. The relationships in Table 2 can be used to determine Re,cl from icl, and icl or Re,cl can then be used for body heat loss calculations (see Table 3). McCullough et al. (1989) found an average value of icl = 0.34 for common indoor clothing; this value can be used when other data are not available.
Measurements of im or icl may be necessary if unusual clothing (e.g., impermeable or metallized) and/or extreme environments (e.g., high radiant temperatures or high air velocities) are to be addressed. There are three different methods for measuring the per-meation efficiency of clothing: the first uses a wet mannequin to measure the effect of sweat evaporation on heat loss (McCullough 1986); the second uses permeation efficiency measurements on component fabrics as well as dry mannequin measurements (Umbach 1980); and the third uses measurements from sweating subjects (Nishi et al. 1975; Holmer 1984).
Clothing Surface Area. Many clothing heat transfer calcula-tions require that clothing area factor fcl be known. The most reli-able approach is to measure it using photographic methods (Olesen et al. 1982). Other than actual measurements, the best method is to use previously tabulated data for similar clothing ensembles. Table 7 is adequate for most indoor clothing ensembles. No good method of estimating fcl for a clothing ensemble from other information is available, although a rough estimate can be made by (McCullough and Jones 1984) (47) Total Evaporative Heat Loss The total evaporative heat loss (latent heat) from the body due to both respiratory losses and skin losses, Esk + Eres, can be measured directly from the body’s rate of mass loss as observed by a sensitive scale: (48) where hfg = latent heat of vaporization of water, J/kg m = body mass, kg θ = time, s When using Equation (48), adjustments should be made for any materials consumed (e.g., food and drink), body effluents (e.g., wastes), and metabolic mass losses. Metabolism contributes slightly to mass loss primarily because the oxygen absorbed during respiration is converted to heavier CO2 and exhaled. It can be cal-culated by (49) where dmge/dθ = rate of mass loss due to respiratory gas exchange, kg/s QO2 = oxygen uptake at STPD, mL/s RQ = respiratory quotient 1.977 = density of CO2 at STPD, kg/m3 1.429 = density of O2 at STPD, kg/m3 STPD = standard temperature and pressure of dry air at 0°C and 101.325 kPa Environmental Parameters The parameters describing the thermal environment that must be measured or otherwise quantified if accurate estimates of human thermal response are to be made are divided into two groups—those that can be measured directly and those that are calculated from other measurements.
Directly Measured. Seven of the parameters frequently used to describe the thermal environment are psychrometric and include (1) air temperature ta; (2) wet-bulb temperature twb; (3) dew-point temperature tdp; (4) water vapor pressure pa; (5) total atmospheric pressure pt; (6) relative humidity (rh); and (7) humidity ratio Wa.
These parameters are discussed in detail in Chapter 6, and methods for measuring them are discussed in Chapter 14. Two other impor-tant parameters include air velocity V and mean radiant temperature . Air velocity measurements are also discussed in Chapter 14. The radiant temperature is the temperature of an exposed surface in the environment. The temperatures of individual surfaces are usually combined into a mean radiant temperature . Finally, globe tem-perature tg, which can also be measured directly, is a good approx-imation of the operative temperature to and is also used with other measurements to calculate the mean radiant temperature.
Calculated Parameters. The mean radiant temperature is a key variable in making thermal calculations for the human body.
It is the uniform temperature of an imaginary enclosure in which radiant heat transfer from the human body equals the radiant heat transfer in the actual nonuniform enclosure. Measurements of the globe temperature, air temperature, and air velocity can be com-bined to estimate the mean radiant temperature (see Chapter 14).
The accuracy of the mean radiant temperature determined this way varies considerably depending on the type of environment and the accuracy of the individual measurements. Since the mean radiant temperature is defined with respect to the human body, the shape of the sensor is also a factor. The spherical shape of the globe ther-mometer gives a reasonable approximation of a seated person; an ellipsoid-shaped sensor gives a better approximation of the shape of a human, both upright and seated.
The mean radiant temperature can also be calculated from mea-sured values of the temperature of the surrounding walls and surfaces and their positions with respect to the person. As most building mate-rials have a high emittance ε, all the surfaces in the room can be assumed to be black. The following equation is then used: (50) where = mean radiant temperature, K TN = surface temperature of surface N, K Fp− N = angle factor between a person and surface N Icl ∆ 0.504 – Icl 0.00281 – Walkspeed ( ) 0.24 + = fcl 1.0 0.3Icl + = Esk Eres + hfg AD ------- dm dθ -------= dmge dθ ------------QO2 1.977RQ 1.429 – ( ) 106 ⁄ = tr tr tr Tr 4 T1 4Fp 1 – T2 4Fp 2 – … TN 4 Fp N – + + + = Tr Thermal Comfort 8.11 Because the sum of the angle factors is unity, the fourth power of mean radiant temperature equals the mean value of the surrounding surface temperatures to the fourth power, weighted by the respec-tive angle factors. In general, angle factors are difficult to deter-mine, although Figures 3A and 3B may be used to estimate them for rectangular surfaces. The angle factor normally depends on the position and orientation of the person (Fanger 1982).
If relatively small temperature differences exist between the sur-facesoftheenclosure,Equation(50)canbesimplifiedtoalinearform: (51) Equation (51) always gives a slightly lower mean radiant temper-ature than Equation (50), but in many cases the difference is small.
If, for example, half the surroundings (Fp− N = 0.5) has a tem-perature 5 K higher than the other half, the difference between the calculated mean radiant temperatures—according to Equations (50) and (51)—is only 0.2 K. If, however, this difference is 100 K, the mean radiant temperature calculated by Equation (51) is 10 K too low.
The mean radiant temperature may also be calculated from the plane radiant temperature tpr (defined below) in six directions (up, down, right, left, front, back) and for the projected area factors of a person in the same six directions. For a standing person, the mean radiant temperature may be estimated as (52) For aseated person, the mean radianttemperaturemay beestimated as Fig. 3 Mean Value of Angle Factor Between Seated Person and Horizontal or Vertical Rectangle when Person is Rotated Around Vertical Axis (Fanger 1982) Fig. 4 Analytical Formulas for Calculating Angle Factor for Small Plane Element tr t1Fp 1 – t2Fp 2 – … tNFp N – + + + = tr {0.08 tpr up ( ) tpr down ( ) + [ ] 0.23[tpr right ( ) + = tpr left ( )] 0.35 tpr front ( ) tpr back ( ) + [ ]} + + 2 0.08 0.23 0.35 + + ( ) [ ] ÷ 8.12 2001 ASHRAE Fundamentals Handbook (SI) (53) The plane radiant temperature tpr, first introduced by Kors-gaard (1949), is the uniform temperature of an enclosure in which the incident radiant flux on one side of a small plane element is the same as that in the actual environment. The plane radiant tempera-ture describes the thermal radiation in one direction, and its value thus depends on the direction. In comparison, the mean radiant tem-perature describes the thermal radiation for the human body from all directions. The plane radiant temperature can be calculated using Equations (50) and (51) with the same limitations. Area factors are determined from Figure 4.
The radiant temperature asymmetry ∆tpr is the difference between the plane radiant temperature of the opposite sides of a small plane element. This parameter describes the asymmetry of the radiant environment and is especially important in comfort condi-tions. Because it is defined with respect to a plane element, its value depends on the orientation of that plane. This orientation may be specified in some situations (e.g., floor to ceiling asymmetry) and not in others. If direction is not specified, the radiant asymmetry should be for the orientation that gives the maximum value.
CONDITIONS FOR THERMAL COMFORT In addition to the previously discussed independent environmen-tal and personal variables influencing thermal response and com-fort, other factors may also have some effect. These factors, such as nonuniformity of the environment, visual stimuli, age, and outdoor climate are generally considered secondary factors. Studies by Rohles and Nevins (1971) and Rohles (1973) on 1600 college-age students revealed correlations between comfort level, temperature, humidity, sex, and length of exposure. Many of these correlations are given in Table 9. The thermal sensation scale developed for these studies is called the ASHRAE thermal sensation scale: +3 hot +2 warm +1 slightly warm 0 neutral − 1 slightly cool − 2 cool − 3 cold The equations in Table 9 indicate that the women of this study were more sensitive to temperature and less sensitive to humidity than the men. But in general about a 3 K change in temperature or a 3 kPa change in water vapor pressure is necessary to change a thermal sen-sation vote by one unit or temperature category.
Current and past studies are periodically reviewed to update ASHRAE Standard 55, Thermal Environmental Conditions for Human Occupancy. This standard specifies conditions or comfort zones where 80% of sedentary or slightly active persons find the environment thermally acceptable.
Because people typically change their clothing for the seasonal weather, ASHRAE Standard 55 specifies summer and winter com-fort zones appropriate for clothing insulation levels of 0.5 and 0.9 clo (0.078 and 0.14 m2·K/W), respectively (Figure 5) (Addendum 55a to ASHRAE Standard 55). The warmer and cooler temperature borders of the comfort zones are affected by humidity and coincide with lines of constant ET. In the middle region of a zone, a typical person wearing the prescribed clothing would have a thermal sen-sation at or very near neutral. Near the boundary of the warmer zone, a person would feel about +0.5 warmer on the ASHRAE ther-mal sensation scale; near the boundary of the cooler zone, that per-son may have a thermal sensation of − 0.5.
Comfort zones for other clothing levels can be approximated by decreasing the temperature borders of the zone by 0.6 K for each 0.1 clo increase in clothing insulation and vice versa. Similarly a zone’s temperatures can be decreased by 1.4 K per met increase in activity above 1.2 met.
The upper and lower humidity levels of the comfort zones are less precise. Low humidity can lead to drying of the skin and mucous surfaces. Comfort complaints about dry nose, throat, eyes, and skin occur in low-humidity conditions, typically when the dew point is less than 0°C. Liviana et al. (1988) found eye discomfort increased with time in low-humidity environments (dew point < 2°C). Green (1982) quantified that respiratory illness and absen-teeism increase in winter with decreasing humidity and found that any increase in humidity from very low levels decreased absentee-ism in winter. In compliance with these and other discomfort obser-vations, ASHRAE Standard 55 recommends that the dew-point temperature of occupied spaces not be less than 2°C.
Table 9 Equations for Predicting Thermal Sensation (Y ) of Men, Women, and Men and Women Combined Exposure Period, h Subjects Regression Equationsa, b t = dry-bulb temperature, °C p = vapor pressure, kPa 1.0 Men Y = 0.220 t + 0.233 p −5.673 Women Y = 0.272 t + 0.248 p −7.245 Both Y = 0.245 t + 0.248 p −6.475 2.0 Men Y = 0.221 t + 0.270 p −6.024 Women Y = 0.283 t + 0.210 p −7.694 Both Y = 0.252 t + 0.240 p −6.859 3.0 Men Y = 0.212 t + 0.293 p −5.949 Women Y = 0.275 t + 0.255 p −8.622 Both Y = 0.243 t + 0.278 p −6.802 aY values refer to the ASHRAE thermal sensation scale.
bFor young adult subjects with sedentary activity and wearing clothing with a thermal resistance of approximately 0.5 clo, tr ≈ta and air velocities < 0.2 m/s.
tr {0.18 tpr up ( ) tpr down ( ) + [ ] 0.22[tpr right ( ) + = tpr left ( )] 0.30 tpr front ( ) tpr back ( ) + [ ]} + + 2 0.18 0.22 0.30 + + ( ) [ ] ÷ tr Fig. 5 ASHRAE Summer and Winter Comfort Zones (Acceptable ranges of operative temperature and humidity for people in typical summer and winter clothing during primarily sedentary activity.) Thermal Comfort 8.13 At high humidity levels, too much skin moisture tends to increase discomfort (Gagge 1937, Berglund and Cunningham 1986), partic-ularly skin moisture that is physiological in origin (water diffusion and perspiration). At high humidity levels, thermal sensation alone is not a reliable predictor of thermal comfort (Tanabe et al. 1987). The discomfort appears to be due to the feeling of the moisture itself, increased friction between skin and clothing with skin moisture (Gwosdow et al. 1986), and other factors. To prevent warm discom-fort, Nevins et al. (1975) recommended that on the warm side of the comfort zone the relative humidity not exceed 60%.
The upper humidity limits of ASHRAE Standard 55 were devel-oped theoretically from limited data. However, thermal acceptabil-ity data gathered at medium and high humidity levels at summer comfort temperatures with subjects wearing 0.55 clo corroborated the shape of the upper limit and found it corresponded to an 80% thermal acceptability level (Berglund 1995).
THERMAL NONUNIFORM CONDITIONS AND LOCAL DISCOMFORT A person may feel thermally neutral as a whole but still feel uncomfortable if one or more parts of the body are too warm or too cold. Nonuniformities may be due to a cold window, a hot surface, a draft, or a temporal variation of these. Even small variations in heat flow cause the thermal regulatory system to compensate, thus increasing the physiological effort of maintaining body tempera-tures. The boundaries of the comfort zones (Figure 5) of ASHRAE Standard 55 provide a thermal acceptability level of 90% if the environment is thermally uniform. Because the standard’s objective is to specify conditions for 80% acceptability, the standard permits nonuniformities to decrease acceptability by 10%. Fortunately for the designer and user, the effect of common thermal nonuniformi-ties on comfort is quantifiable and predictable as discussed in the following sections. Furthermore, most humans are fairly insensitive to small nonuniformities.
Asymmetric Thermal Radiation Asymmetric or nonuniform thermal radiation in a space may be caused by cold windows, uninsulated walls, cold products, cold or warm machinery, or improperly sized heating panels on the wall or ceiling. In residential buildings, offices, restaurants, etc., the most common reasons for discomfort due to asymmetric thermal radia-tion are large windows in the winter or improperly sized or installed ceiling heating panels. At industrial workplaces, the reasons include cold or warm products, cold or warm equipment, etc.
The recommendations in ISO Standard 7730 and ASHRAE Standard 55 are based primarily on studies reported by Fanger et al.
(1980). These standards include guidelines regarding the radiant temperature asymmetry from an overhead warm surface (heated ceiling) and a vertical cold surface (cold window). Among the stud-ies conducted on the influence of asymmetric thermal radiation are those by McIntyre (1974, 1976), McIntyre and Griffiths (1975), Fanger and Langkilde (1975), McNall and Biddison (1970), and Olesen et al. (1972). These studies all used seated subjects. In these studies, the subjects were always in thermal neutrality and exposed only to the discomfort resulting from excessive asymmetry.
The subjects gave their reactions on their comfort sensation, and a relationship between the radiant temperature asymmetry and the number of subjects feeling dissatisfied was established (Figure 6).
Radiant asymmetry, as defined in the section on Environmental Parameters, is the difference in radiant temperature of the environ-ment on opposite sides of the person. More precisely, radiant asym-metry is the difference in radiant temperatures seen by a small flat element looking in opposite directions.
Figure 6 shows that people are more sensitive to asymmetry caused by an overhead warm surface than by a vertical cold surface.
The influence of an overhead cold surface and a vertical warm surface is much less. These data are particularly important when applying radiant panels to provide comfort in spaces with large cold surfaces or cold windows.
Other studies of clothed persons in neutral environments found thermal acceptability unaffected by radiant temperature asymme-tries of 10 K or less (Berglund and Fobelets 1987) and comfort un-affected by radiant temperature asymmetries of 20 K or less (McIntyre 1975).
Draft Draft is an undesired local cooling of the human body caused by air movement. This is a serious problem, not only in many venti-lated buildings but also in automobiles, trains, and aircraft. Draft has been identified as one of the most annoying factors in offices.
When people sense draft, they often demand higher air temperatures in the room or that ventilation systems be stopped.
Fanger and Christensen (1986) aimed to establish the percentage of the population feeling draft when exposed to a given mean veloc-ity. Figure 7 shows the percentage of subjects who felt draft on the head region (the dissatisfied) as a function of the mean air velocity at the neck. The head region comprises head, neck, shoulders, and back. The air temperature had a significant influence on the percent-age of dissatisfied. There was no significant difference between responses of men and women to draft. The data in Figure 7 apply only to persons wearing normal indoor clothing and performing light, mainly sedentary work. Persons with higher activity levels are not as sensitive to draft (Jones et al. 1986).
A study of the effect of air velocity over the whole body found thermal acceptability unaffected in neutral environments by air speeds of 0.25 m/s or less (Berglund and Fobelets 1987). This study also found no interaction between air speed and radiant temperature asymmetry on subjective responses. This means that acceptability changes and the percent dissatisfied due to draft and radiant asym-metry are independent and additive.
Fanger et al. (1989) investigated the effect of turbulence inten-sity on sensation of draft. The turbulence intensity had a significant effect on the occurrence of draft sensation. The following model predicts the percentage of people dissatisfied because of draft inten-sity. The model can be used for quantifying draft risk in spaces and for developing air distribution systems with a low draft risk.
(54) Fig. 6 Percentage of People Expressing Discomfort due to Asymmetric Radiation PD 34 ta – ( ) V 0.05 – ( )0.62 0.37V Tu 3.14 + ( ) = 8.14 2001 ASHRAE Fundamentals Handbook (SI) where Tu is the turbulence intensity in % defined by (55) For V < 0.05 m/s, insert V = 0.05; and for PD > 100%, insert PD = 100%. Vsd is the standard deviation of the velocity measured with an omnidirectional anemometer having a 0.2 s time constant.
The model extends the Fanger and Christensen (1986) draft chart model to include turbulence intensity. In this study, Tu decreases when V increases. This means that the effect of V for the experimen-tal data to which the model is fitted are: 20 < ta < 26°C, 0.05 < V < 0.5 m/s, and 0 < Tu < 70%. Figure 8 gives more precisely the curves that result from intersections between planes of constant Tu and the surfaces of PD = 15%.
Vertical Air Temperature Difference In most spaces in buildings, the air temperature normally increases with height above the floor. If the gradient is sufficiently large, local warm discomfort can occur at the head and/or cold discomfort can occur at the feet, although the body as a whole is thermally neutral. Among the few studies of vertical air temperature differences and the influence of thermal comfort reported are Ole-sen et al. (1979), McNair (1973), McNair and Fishman (1974), and Eriksson (1975). Subjects were seated in a climatic chamber so they were individually exposed to different air temperature differences between head and ankles (Olesen et al. 1979). During the tests, the subjects were in thermal neutrality because they were allowed to change the temperature level in the test room whenever they desired; the vertical temperature difference, however, was kept unchanged. The subjects gave subjective reactions to their thermal sensation; Figure 9 shows the percentage of dissatisfied as a func-tion of the vertical air temperature difference between head (1.1 m above the floor) and ankles (0.1 m above the floor).
The case where the air temperature at head level is lower than that at ankle level will not be as critical for the occupants. Eriksson (1975) indicated that his subjects could tolerate much greater differ-ences if the head were cooler. This observation is verified in the experiments with asymmetric thermal radiation from a cooled ceil-ing (Fanger et al. 1985).
Warm or Cold Floors Due to the direct contact between the feet and the floor, local dis-comfort of the feet can often be caused by a too-high or too-low floor temperature. Also, the floor temperature has a significant influence on the mean radiant temperature in a room. The floor tem-perature is greatly influenced by the way a building is constructed (e.g., insulation of the floor, above a basement, directly on the ground, above another room, use of floor heating, floors in radiant heated areas). If a floor is too cold and the occupants feel cold dis-comfort in their feet, a common reaction is to increase the temper-ature level in the room; in the heating season, this also increases energy consumption. A radiant system, which radiates heat from the floor, can also prevent discomfort from cold floors.
The most extensive studies of the influence of floor temperature on feet comfort were performed by Olesen (1977a, 1977b), who, based on his own experiments and reanalysis of the data from Nevins and Flinner (1958), Nevins et al. (1964), and Nevins and Feyerherm (1967), recorded the following results. For floors occupied by peo-ple with bare feet (in swimming halls, gymnasiums, dressing rooms, bathrooms, and bedrooms), flooring material is important. Ranges for some typical floor materials are as follows Textiles (rugs) 21 to 28°C Pine floor 22.5 to 28°C Oak floor 24.5 to 28°C Fig. 7 Percentage of People Dissatisfied as Function of Mean Air Velocity Fig. 8 Draft Conditions Dissatisfying 15% of Population Tu 100 Vsd V --------= Fig. 9 Percentage of Seated People Dissatisfied as Function of Air Temperature Difference Between Head and Ankles Thermal Comfort 8.15 Hard linoleum 24 to 28°C Concrete 26 to 28.5°C To save energy, flooring materials with a low contact coefficient (cork, wood, carpets), radiant heated floors, or floor heating sys-tems can be used to eliminate the desire for higher ambient temper-atures caused by cold feet. These recommendations should also be followed in schools, where children often play directly on the floor.
For floors occupied by people with normal indoor footwear, flooring material is insignificant. Olesen (1977b) found an optimal temperature of 25°Cfor sedentary and 23°Cfor standing or walking persons. At the optimal temperature, 6% of the occupants felt warm or cold discomfort in the feet. Figure 10 shows the relationship between floor temperature and percentage of dissatisfied, combin-ing data from experiments with seated and standing subjects. In all experiments, the subjects were in thermal neutrality; thus, the per-centage of dissatisfied is only related to the discomfort due to cold or warm feet. No significant difference in floor temperature was preferred by females and males.
SECONDARY FACTORS AFFECTING COMFORT Temperature, air speed, humidity, their variation, and personal parameters of metabolism and clothing insulation are primary fac-tors that directly affect energy flow and thermal comfort. However, many secondary factors, some of which are discussed in this sec-tion, may more subtly influence comfort.
Day-to-Day Variations Fanger (1973) conducted an experiment with a group of subjects, where the preferred ambient temperature for each subject under identical conditions was determined on four different days. Since the standard deviation was only 0.6 K, Fanger concluded that the comfort conditions for the individual can be reproduced and will vary only slightly from day to day.
Age Because metabolism decreases slightly with age, many have stated that comfort conditions based on experiments with young and healthy subjects cannot be used for other age groups. Fanger (1982), Fanger and Langkilde (1975), Langkilde (1979), Nevins et al.
(1966), and Rohles and Johnson (1972) conducted comfort studies in Denmark and the United States on different age groups (mean age 21 to 84). The studies revealed that the thermal environments pre-ferred by older people do not differ from those preferred by younger people. The lower metabolism in older people is compensated for by a lower evaporative loss. Collins and Hoinville (1980) confirmed these results.
The fact that young and old people prefer the same thermal envi-ronment does not necessarily mean that they are equally sensitive to cold or heat. In practice, the ambient temperature level in the homes of older people is often higher than that for younger people. This may be explained by the lower activity level of elderly people, who are normally sedentary for a greater part of the day.
Adaptation Many believe that people can acclimatize themselves by expo-sure to hot or cold surroundings, so that they prefer other thermal environments. Fanger (1982) conducted experiments involving subjects from the United States, Denmark, and tropical countries.
The latter group was tested in Copenhagen immediately after their arrival by plane from the tropics where they had lived all their lives. Other experiments were conducted for two groups exposed to cold daily. One group comprised subjects who had been doing sedentary work in cold surroundings (in the meat-packing industry) for 8 h daily for at least 1 year. The other group consisted of winter swimmers who bathed in the sea daily.
Only slight differences in both the preferred ambient tempera-ture and the physiological parameters in the comfort conditions were reported for the various groups. These results indicate that people cannot adapt to preferring warmer or colder environments.
It is therefore likely that the same comfort conditions can be applied throughout the world. However, in determining the pre-ferred ambient temperature from the comfort equations, a clo-value that corresponds to the local clothing habits should be used.
A comparison of field comfort studies from different parts of the world shows significant differences in clothing habits depending on, among other things, the outdoor climate (Nicol and Hum-phreys 1972). According to these results, adaptation has little influence on the preferred ambient temperature. In uncomfortable warm or cold environments, however, adaptation will often have an influence. People used to working and living in warm climates can more easily accept and maintain a higher work performance in hot environments than people from colder climates.
Sex Previously cited experiments by Fanger (1982), Fanger and Langkilde (1975), and Nevins et al. (1966) used equal numbers of male and female subjects, so comfort conditions for the two sexes can be compared. The experiments show that men and women pre-fer almost the same thermal environments. Women’s skin tempera-ture and evaporative loss are slightly lower than those for men, and this balances the somewhat lower metabolism of women. The rea-son that women often prefer higher ambient temperatures than men may be partly explained by the lighter clothing normally worn by women.
Seasonal and Circadian Rhythms Since people cannot adapt to prefer warmer or colder environ-ments, it follows that there is no difference between comfort condi-tions in winter and in summer. McNall et al. (1968) confirmed this in an investigation where results of winter and summer experiments showed no difference. On the other hand, it is reasonable to expect the comfort conditions to alter during the day because the internal body temperature has a daily rhythm—a maximum occurring late in the afternoon, and a minimum early in the morning.
In determining the preferred ambient temperature for each of 16 subjects both in the morning and in the evening, Fanger et al. (1974) and Ostberg and McNicholl (1973) observed no difference. Further-more, Fanger et al. (1973) found only small fluctuations in the pre-ferred ambient temperature during a simulated 8 h workday (sedentary work). There is a slight tendency to prefer somewhat Fig. 10 Percentage of People Dissatisfied as Function of Floor Temperature 8.16 2001 ASHRAE Fundamentals Handbook (SI) warmer surroundings before lunch, but none of the fluctuations are significant.
PREDICTION OF THERMAL COMFORT Thermal comfort and thermal sensation can be predicted several ways. One way is to use Figure 5 and Table 9 and adjust for clothing and activity levels that differ from those of the figure. More numer-ical and rigorous predictions are possible by using the PMV-PPD and two-node models described in this section.
Steady-State Energy Balance Fanger (1982) related the comfort data to physiological vari-ables. At a given level of metabolic activity M, and when the body is not far from thermal neutrality, the mean skin temperature tsk and sweat rate Ersw are the only physiological parameters influencing the heat balance. However, heat balance alone is not sufficient to establish thermal comfort. In the wide range of environmental con-ditions where heat balance can be obtained, only a narrow range provides thermal comfort. The following linear regression equa-tions based on data from Rohles and Nevins (1971) indicate values of tsk and Ersw that provide thermal comfort.
(56) (57) At higher activity levels, sweat loss increases and the mean skin temperature decreases. Both reactions increase the heat loss from the body core to the environment. These two empirical relationships link the physiological and heat flow equations and thermal comfort perceptions. By substituting these values into Equation (11) for C + R, and into Equations (17) and (18) for Esk, Equation (1), the energy balance equation, can be used to determine combinations of the six environmental and personal parameters that optimize com-fort for steady-state conditions.
Fanger (1982) reduced these relationships to a single equation, which assumed all sweat generated is evaporated, eliminating cloth-ing permeation efficiency icl as a factor in the equation. This assumption is valid for normal indoor clothing worn in typical indoor environments with low or moderate activity levels. At higher activity levels (Mact > 3 met), where a significant amount of sweat-ing occurs even at optimum comfort conditions, this assumption may limit accuracy. The reduced equation is slightly different from the heat transfer equations developed here. The radiant heat exchange is expressed in terms of the Stefan-Boltzmann law (instead of using hr), and diffusion of water vapor through the skin is expressed as a diffusivity coefficient and a linear approximation for saturated vapor pressure evaluated at tsk. The combination of environmental and personal variables that produces a neutral sensa-tion may be expressed as follows: (58) where (59) The values of hc and fcl can be estimated from tables and equations given in the section on Engineering Data and Measurements.
Fanger used the following relationships: (60) (61) Figures 11and 12 show examples of how Equation (58) can be used.
Equation (58) is expanded to include a range of thermal sensa-tions by using a predicted mean vote (PMV) index. The PMV index predicts the mean response of a large group of people accord-ing to the ASHRAE thermal sensation scale. Fanger (1970) related PMV to the imbalance between the actual heat flow from the body in a given environment and the heat flow required for optimum comfort at the specified activity by the following equation: (62) where L is the thermal load on the body, defined as the difference between internal heat production and heat loss to the actual envi-ronment for a person hypothetically kept at comfort values of tsk and Ersw at the actual activity level. Thermal load L is then the dif-ference between the left and right sides of Equation (58) calcu-lated for the actual values of the environmental conditions. As part of this calculation, the clothing temperature tcl is found by itera-tion as (63) tsk req , 35.7 0.0275 – M W – ( ) = Ersw req , 0.42 M W – 58.15 – ( ) = M W – 3.96 10 8 – fcl tcl 273 + ( )4 t r 273 + ( ) 4 – [ ] × = fclhc tcl ta – ( ) + 3.05 5.73 0.007 M W – ( ) – pa – [ ] + 0.42 M W – ( ) 58.15 – [ ] + 0.0173M 5.87 pa – ( ) + 0.0014M 34 ta – ( ) + tcl 35.7 0.0275 M W – ( ) – = Rcl { M W – ( ) – 3.05 5.73 0.007 M W – ( ) – pa – [ ] – 0.42 M W – ( ) 58.15 – [ ] 0.0173M 5.87 pa – ( ) – – 0.0014M 34 ta – ( )} – hc 2.38 tcl ta – ( )0.25 2.38 tcl ta – ( )0.25 12.1 V > 12.1 V 2.38 tcl ta – ( )0.25 12.1 V < = fcl 1.0 0.2 Icl + Icl 0.5 clo < 1.05 0.1 Icl + Icl 0.5 clo > = Fig. 11 Air Velocities and Operative Temperatures at 50% rh Necessary for Comfort (PMV = 0) of Persons in Summer Clothing at Various Levels of Activity PMV 0.303 0.036M – ( ) exp 0.028 + [ ]L = tcl 35.7 0.028 M W – ( ) – = Rcl – {39.6 10 9 – fcl × tcl 273 + ( )4 tr 273 + ( ) 4 – [ ] + fclhc tcl ta – ( )} Thermal Comfort 8.17 After estimating the PMV with Equation (62) or another method, the predicted percent dissatisfied (PPD) with a condition can also be estimated. Fanger (1982) related the PPD to the PMV as follows: (64) where dissatisfied is defined as anybody not voting − 1, +1, or 0.
This relationship is shown in Figure 13. A PPD of 10% corresponds to the PMV range of ±0.5, and even with PMV = 0, about 5% of the people are dissatisfied.
The PMV-PPD model is widely used and accepted for design and field assessment of comfort conditions. ISO Standard 7730 includes a short computer listing that facilitates computing PMV and PPD for a wide range of parameters.
Two-Node Model The PMV model is useful only for predicting steady-state com-fort responses. The two-node model can be used to predict physio-logical responses or responses to transient situations, at least for low and moderate activity levels in cool to very hot environments (Gagge et al. 1971, 1986). The two-node model is a simplification of more complex thermoregulatory models developed by Stolwijk and Hardy (1966). The simple, lumped parameter model considers a human as two concentric thermal compartments that represent the skin and the core of the body.
The skin compartment simulates the epidermis and dermis and is about 1.6 mm thick. Its mass, which is about 10% of the total body, depends on the amount of blood flowing through it for thermoregulation. The temperature in a compartment is assumed to be uniform so that the only temperature gradients are between compartments. In a cold environment, blood flow to the extremi-ties may be reduced to conserve the heat of vital organs, resulting in axial temperature gradients in the arms, legs, hands, and feet.
Heavy exercise with certain muscle groups or asymmetric envi-ronmental conditions may also cause nonuniform compartment temperatures and limit the accuracy of the model.
All the heat is assumed to be generated in the core compart-ment. In the cold, shivering and muscle tension may generate addi-tional metabolic heat. This increase is related to skin and core temperature depressions from their set point values, or (65) where the temperature terms are set to zero if they become negative.
The core loses energy when the muscles do work on the surround-ings. Heat is also lost from the core through respiration. The rate of respiratory heat loss is due to sensible and latent changes in the respired air and the ventilation rate as in Equations (19) and (20).
In addition, heat is conducted passively from the core to the skin.
This is modeled as a massless thermal conductor [K = 5.28 W/(m2·K)]. A controllable heat loss path from the core consists of pumping variable amounts of warm blood to the skin for cooling.
This peripheral blood flow in L/h·m2 depends on skin and core temperature deviations from their respective set points: (66) The temperature terms can only be > 0. If the deviation is nega-tive, the term is set to zero. For average persons, the coefficients BFN, cdil, and Str are 6.3, 175 and 0.5. Further, skin blood flow is limited to a maximum of 90 L/(h·m2).
Dry (sensible) heat loss qdry from the skin flows through the clothing by conduction and then by parallel paths to the air and sur-rounding surfaces. Evaporative heat follows a similar path, flowing through the clothing and through the air boundary layer. Maximum evaporation Emax occurs if the skin is completely covered with sweat. The actual evaporation rate Esw depends on the size w of the sweat film: (67) where w is Ersw/Emax.
The rate of regulatory sweating Ersw (rate at which water is brought to the surface of the skin in W/m2) can be predicted by skin and core temperature deviations from their set points: (68) where tb = (1 −αsk)tcr + αsktsk and is the mean body temperature, and csw = 170 W/(m2·K). The temperature deviation terms are set to zero when negative. αsk is the fraction of the total body mass that is considered to be thermally in the skin compartment.
Fig. 12 Air Temperatures and Mean Radiant Temperatures Necessary for Comfort (PMV = 0) of Sedentary Persons in Summer Clothing at 50% rh PPD 100 95 0.03353PMV 4 0.2179PMV2 + ( ) – [ ] exp – = Fig. 13 Predicted Percentage of Dissatisfied (PPD) as Function of Predicted Mean Vote (PMV) Mshiv 19.4 34 tsk – ( ) 37 tcr – ( ) = Qbl Qbl BFN c + dil tcr 37 – ( ) 1 Str 34 tsk – ( ) + -------------------------------------------------= Qbl Esw wEmax = Ersw csw tb tbset – ( ) tsk 34 – ( ) 10.7 ⁄ – [ ] exp = 8.18 2001 ASHRAE Fundamentals Handbook (SI) (69) Regulatory sweating in the model is limited to 1 L/h·m2 or 670 W/m2. Ersw evaporates from the skin, but if Ersw is greater than Emax, the excess drips off.
An energy balance on the core yields (70) and for the skin, (71) where ccr, csk, and cp,bl are specific heats of core, skin, and blood [3500, 3500, and 4190 J/(kg· K), respectively], and SKBF is ρblQbl, where ρbl is density of blood (12.9 kg/L).
Equations (70) and (71) can be rearranged in terms of dtsk/dθ and dtcr/dθ and numerically integrated with small time steps (10 to 60 s) from initial conditions or previous values to find tcr and tsk at any time.
After calculating values of tsk, tcr, and w, the model uses empir-ical expressions to predict thermal sensation (TSENS) and thermal discomfort (DISC). These indices are based on 11-point numerical scales, where positive values represent the warm side of neutral sen-sation or comfort, and negative values represent the cool side.
TSENS is based on the same scale as PMV, but with extra terms for ±4 (very hot/cold) and ±5 (intolerably hot/cold). Recognizing the same positive/negative convention for warm/cold discomfort, DISC is defined as 5 intolerable 4 limited tolerance 3 very uncomfortable 2 uncomfortable and unpleasant 1 slightly uncomfortable but acceptable 0 comfortable TSENS is defined in terms of deviations of mean body tempera-ture tb from cold and hot set points representing the lower and upper limits for the zone of evaporative regulation: tb,c and tb,h, respec-tively. The values of these set points depend on the net rate of inter-nal heat production and are calculated by (72) (73) TSENS is then determined by (74) where η ev is the evaporative efficiency (assumed to be 0.85).
DISC is numerically equal to TSENS when tb is below its cold set point tb,c and it is related to skin wettedness when body temper-ature is regulated by sweating: (75) where Ersw,req is calculated as in Fanger’s model, using Equation (57).
Adaptive Models Adaptive models do not actually predict comfort responses but rather the almost constant conditions under which people are likely to be comfortable in buildings. In general, people naturally adapt and may also make various adjustments to themselves and their sur-roundings to reduce discomfort and physiological strain. It has been observed that, through adaptive actions, an acceptable degree of comfort in residences and offices is possible over a range of air tem-peratures from about 17 to 31°C (Humphreys and Nicol 1998).
The adaptive adjustments are typically conscious behavioral actions such as altering clothing, posture, activity schedules, activ-ity levels, rate of working, diet, ventilation, air movement, and local temperature. The adaptations may also include unconscious longer term changes to physiological set points and gains for the control of shivering, skin blood flow, and sweating, as well as adjustments to body fluid levels and salt loss. However, only limited documenta-tion and information on such changes is available.
An important driving force behind the adaptive process is the pattern of outside weather conditions and the exposure to them. This is the principal input to the adaptive models that have evolved to date, and these models predict likely comfort temperatures tc or ranges of tc from monthly mean outdoor temperatures tout. Such a model (Humphreys and Nicol 1998), based on data from a wide range of buildings, climates, and cultures is (76) The adaptive models are useful to guide design and energy deci-sions. They may also be useful to specify building temperatures set points throughout the year. A recent ASHRAE-sponsored study on adaptive models compiled an extensive database from past field studies to study, develop, and test adaptive models. For climates and buildings where cooling and central heating are not required, the study suggests the following model (de Dear and Brager 1998): (77) where toc is the operative comfort temperature.
In general, the value of using an adaptive model to specify set points or guide temperature control strategies is likely to increase with the freedom that occupants are given to adapt (e.g., by having flexible working hours, locations, or dress codes).
Zones of Comfort and Discomfort The section on Two-Node Model shows that comfort and ther-mal sensation are not necessarily the same variable, especially for a person in the zone of evaporative thermal regulation. Figures 14 and 15 show this difference for the standard combination of met-clo-air movement used in the standard effective temperature. Fig-ure 14 demonstrates that practically all basic physiological vari-ables predicted by the two-node model are functions of ambient temperature and are relatively independent of vapor pressure. All exceptions occur at relative humidities above 80% and as the iso-therms reach the ET = 41.5°C line, where regulation by evapora-tion fails. Figure 15 shows that lines of constant ET and wettedness are functions of both ambient temperature and vapor αsk 0.0418 0.745 10.8Qbl 0.585 – --------------------------------------+ = Qrsw M Mshiv + W qres K SKBFcp bl , + ( ) tcr tsk – ( ) + + = + mcrccr θ d dtcr K SKBFcp bl , + ( ) tcr tsk – ( ) qdry qevap msk + + = csk θ d dtsk tb c , 0.194 58.15 ------------- M W – ( ) 36.301 + = tb h , 0.347 58.15 ------------- M W – ( ) 36.669 + = TSENS 0.4685 tb tb c , – ( ) tb tb c , < 4.7η ev tb tb c , – ( ) tb h , tb c , – ( ) ⁄ tb c , tb tb h , ≤ ≤ 4.7η ev 0.4685 tb tb h , – ( ) + tb h , tb < = DISC 0.4685 tb tb c , – ( ) tb tb c , < 4.7 Ersw Ersw req , – ( ) Emax Ersw req , – Edif – -----------------------------------------------------tb c , tb ≤ = tc 24.2 0.43 tout 22 – ( )exp tout 22 – 24 2 -------------------- 2 – + = toc 18.9 0.255tout + = Thermal Comfort 8.19 pressure. Thus, human thermal responses are divided into two classes—those in Figure 14, which respond only to heat stress from the environment, and those in Figure 15, which respond to both the heat stress from the environment and the resultant heat strain (Stol-wijk et al. 1968).
For warm environments, any index with isotherms parallel to skin temperature is a reliable index of thermal sensation alone, and not of discomfort caused by increased humidity. Indices with iso-therms parallel to ET are reliable indicators of discomfort or dis-satisfaction with thermal environments. For a fixed exposure time to cold, lines of constant tsk, ET, and to are essentially identical, and cold sensation is no different from cold discomfort. For a state of comfort with sedentary or light activity, lines of constant tsk and ET coincide. Thus comfort and thermal sensations coincide in this region as well. The upper and lower temperature limits for comfort at these levels can be specified either by thermal sensation (Fanger 1982) or by ET, as is done in ASHRAE Standard 55, since lines of constant comfort and lines of constant thermal sensation should be identical.
ENVIRONMENTAL INDICES An environmental index combines two or more parameters (e.g., air temperature, mean radiant temperature, humidity, or air velocity) into a single variable. Indices simplify the description of the thermal environment and the stress imposed by an environment. Environ-mental indices may be classified according to how they are devel-oped. Rational indices are based on the theoretical concepts presented earlier. Empirical indices are based on measurements with subjects or on simplified relationships that do not necessarily follow theory. Indices may also be classified according to their application, generally either heat stress or cold stress.
Effective Temperature The effective temperature ET is probably the most common environmental index, and it has the widest range of application. It combines temperature and humidity into a single index, so two environments with the same ET should evoke the same thermal response even though they have different temperatures and humid-ities; but they must have the same air velocities.
The original empirical effective temperature was developed by Houghten and Yaglou (1923). Gagge et al. (1971) defined a new effective temperature using a rational approach. Defined mathemat-ically in Equation (33), this is the temperature of an environment at 50% rh that results in the same total heat loss Esk from the skin as in the actual environment.
Because the index is defined in terms of operative temperature to, it combines the effects of three parameters ( , ta, and pa) into a sin-gle index. Skin wettedness w and the permeability index im must be specified and are constant for a given ET line for a particular situ-ation. The two-node model is used to determine skin wettedness in the zone of evaporative regulation. At the upper limit of regulation, w approaches 1.0; at the lower limit, w approaches 0.06. Skin wet-tedness equals one of these values when the body is outside the zone of evaporative regulation. Since the slope of a constant ET line depends on skin wettedness and clothing moisture permeability, effective temperature for a given temperature and humidity may depend on the clothing and activity of the person. This difference is shown in Figure 16. At low skin wettedness, the air humidity has lit-tle influence, and lines of constant ET are nearly vertical. As skin wettedness increases due to activity and/or heat stress, the lines become more horizontal and the influence of humidity is much more pronounced. The ASHRAE comfort envelope shown in Fig-ure 5 is described in terms of ET.
Since ET depends on clothing and activity, it is not possible to generate a universal ET chart. Calculation of ET can also be tedious, requiring the solution of multiple coupled equations to determine skin wettedness. A standard set of conditions represen-tative of typical indoor applications is used to define a standard effective temperature SET. The standard effective temperature is then defined as the equivalent air temperature of an isothermal environment at 50% rh in which a subject, while wearing cloth-ing standardized for the activity concerned, has the same heat stress (skin temperature tsk) and thermoregulatory strain (skin wettedness w) as in the actual environment.
Humid Operative Temperature The humid operative temperature toh is the temperature of a uniform environment at 100% rh in which a person loses the same total amount of heat from the skin as in the actual envi-ronment. This index is defined mathematically in Equation (32). It is analogous to ET, the only difference being that it is defined at 100% rh and 0% rh rather than at 50% rh. Figures 2 and 16 indicate that lines of constant ET are also lines of con-stant toh. However, the values of these two indices differ for a given environment.
Fig. 14 Effect of Environmental Conditions on Physiological Variables Fig. 15 Effect of Thermal Environment on Discomfort tr 8.20 2001 ASHRAE Fundamentals Handbook (SI) Heat Stress Index Originally proposed by Belding and Hatch (1955), this rational index is the ratio of the total evaporative heat loss Esk required for thermal equilibrium (the sum of metabolism plus dry heat load) to the maximum evaporative heat loss Emax possible for the environ-ment, multiplied by 100, for steady-state conditions (Ssk and Scr are zero) and with tsk held constant at 35°C. The ratio Esk/Emax equals skin wettedness w [Equation (18)]. When heat stress index (HSI) > 100, body heating occurs; when HSI < 0, body cooling occurs. Belding and Hatch (1955) limited Emax to 700 W/m2, which corresponds to a sweat rate of approximately 280 mg/ (s·m2). When tsk is constant, loci of constant HSI coincide with lines of constant ET on a psychrometric chart. Other indices based on wettedness have the same applications (Gonzalez et al.
1978, Belding 1970, ISO Standard 7933) but differ in their treat-ment of Emax and the effect of clothing. Table 10 describes physi-ological factors associated with HSI values.
Index of Skin Wettedness Skin wettedness w is the ratio of observed skin sweating Esk to the Emax of the environment as defined by tsk, ta, humidity, air movement, and clothing in Equation (12). Except for the factor of 100, it is essentially the same as HSI. Skin wettedness is more closely related to the sense of discomfort or unpleasantness than to temperature sensation (Gagge et al. 1969a,b; Gonzalez et al. 1978).
Wet-Bulb Globe Temperature The WBGT is an environmental heat stress index that combines dry-bulb temperature tdb, a naturally ventilated (not aspirated) wet-bulb temperature tnwb, and black globe temperature tg, accord-ing to the relation (Dukes-Dobos and Henschel 1971, 1973) (78) This form of the equation is usually used where solar radiation is present. The naturally ventilated wet-bulb thermometer is left exposed to the sunlight, but the air temperature ta sensor is shaded.
In enclosed environments, Equation (78) is simplified by dropping the ta term and using a 0.3 weighting factor for tg.
The black globe thermometer responds to air temperature, mean radiant temperature, and air movement, while the naturally venti-lated wet-bulb thermometer responds to air humidity, air move-ment, radiant temperature, and air temperature. Thus, WBGT is a function of all four environmental factors affecting human environ-mental heat stress.
The WBGT is a better index of heat stress than the old ET; it shows almost as good a correlation with sweat rate as do the later corrected effective temperature (CET) and effective temperature with radiation (ETR) indices (Minard 1961); the CET and ETR both require direct measurement of wind velocity which, for accuracy, requires special instruments and trained technicians.
The WBGT index is widely used for estimating the heat stress potential of industrial environments (Davis 1976). In the United States, the National Institute of Occupational Safety and Health (NIOSH) developed criteria for a heat-stress-limiting standard (NIOSH 1986). ISO Standard 7243 also uses the WBGT. Figure 17 graphically summarizes the permissible heat exposure limits, Fig. 16 Effective Temperature ET and Skin Wettedness w [Adapted from Nishi et al. (1975) and Gonzalez et al. (1978)] Table 10 Evaluation of Heat Stress Index Heat Stress Index Physiological and Hygienic Implications of 8 h Exposures to Various Heat Stresses 0 No thermal strain.
10 Mild to moderate heat strain. If job involves higher intellectual functions, dexterity, or alertness, subtle to substantial decrements in performance may be expected. In performing heavy physical work, little decrement is expected, unless ability of individuals to perform such work under no thermal stress is marginal.
20 30 40 Severe heat strain involving a threat to health unless men are physically fit. Break-in period required for men not previously acclimatized. Some decrement in performance of physical work is to be expected. Medical selection of personnel desirable, because these conditions are unsuitable for those with cardiovascular or respiratory impairment or with chronic dermatitis. These working conditions are also unsuitable for activities requiring sustained mental effort.
50 60 70 Very severe heat strain. Only a small percentage of the population may be expected to qualify for this work. Personnel should be selected: (a) by medical examination, and (b) by trial on the job (after acclimatization). Special measures are needed to assure adequate water and salt intake. Amelioration of working conditions by any feasible means is highly desirable, and may be expected to decrease the health hazard while increasing job efficiency. Slight “indisposition,” which in most jobs would be insufficient to affect performance, may render workers unfit for this exposure.
80 90 100 The maximum strain tolerated daily by fit, acclimatized young men.
WBGT 0.7tnwb 0.2tg 0.1ta + + = Thermal Comfort 8.21 expressed as working time per hour, for a fit individual, as speci-fied for various WBGT levels. Values apply for normal perme-able clothing (0.6 clo) and must be adjusted for heavy or partly vapor-permeable clothing. The USAF has recommended adjust-ing the measured WBGT upwards by 6 K for personnel wearing chemical protective clothing or body armor. This type of clothing increases the resistance to sweat evaporation about threefold (higher if it is totally impermeable), requiring an adjustment in WBGT level to compensate for reduced evaporative cooling at the skin.
Several mathematical models are available for predicting WBGT from the environmental factors: air temperature, psychrometric wet-bulb temperature, mean radiant temperature, and air motion (Azer and Hsu 1977; Sullivan and Gorton 1976). A simpler approach, involving plotting WBGT lines on a psychrometric chart, is recommended. Isotherms of WBGT are parallel and have nega-tive slopes varying from 0.17 kPa/Kfor still air to 0.20 kPa/Kfor air motion greater than 1 m/s. By comparison, psychrometric wet-bulb lines have negative slopes of about 0.07 kPa/K, or about 35% as steep.
Wet-Globe Temperature The WGT, introduced by Botsford (1971), is a simpler approach to measuring environmental heat stress than the WBGT. The mea-surement is made with a wetted globe thermometer called a Bots-ball, which consists of a 65 mm black copper sphere covered with a fitted wet black mesh fabric, into which the sensor of a dial ther-mometer is inserted. A polished stem attached to the sphere sup-ports the thermometer and contains a water reservoir for keeping the sphere covering wet. This instrument is suspended by the stem at the indoor (or outdoor) site to be measured.
Onkaram et al. (1980) have shown that WBGT can be predicted with reasonable accuracy from WGT for temperate to warm envi-ronments with medium to high humidities. With air temperatures between 20 and 35°C, dew points ranging from 7 to 25°C(relative humidities above 30%), and wind speeds of 7 m/s or less, the experimental regression equation (r = 0.98) in °C for an outdoor environment is (79) This equation should not be used outside the experimental range given because data from hot-dry desert environments show differ-ences between WBGT and WGT that are too large (6 K and above) to be adjusted by Equation (79) (Matthew et al. 1986). At very low humidity and high wind, WGT approaches the psychrometric wet-bulb temperature, which is greatly depressed below ta. However, in the WBGT, tnwb accounts for only 70% of the index value, with the remaining 30% at or above ta.
Ciriello and Snook (1977) handle the problem by providing a series of regression equations, the choice depending on the levels of wind speed, humidity, and radiant heat. They report an accuracy of conversion from WGT to WBGT within 0.4 K (90% confidence level), if good estimates of wind speed, humidity, and radiation level are available.
Wind Chill Index The wind chill index (WCI) is an empirical index developed from cooling measurements obtained in Antarctica on a cylindri-cal flask partly filled with water (Siple and Passel 1945). The index describes the rate of heat loss from the cylinder by radia-tion and convection for a surface temperature of 33°C, as a func-tion of ambient temperature and wind velocity. As originally proposed, (80) where V and ta are in m/s and °C, respectively, and WCI units are kcal/(h·m2). Multiply WCI by 1.162 to convert to SI units of W/m2. The 33°C surface temperature was chosen to be represen-tative of the mean skin temperature of a resting human in com-fortable surroundings.
A number of valid objections have been raised about this for-mulation. Cooling rate data from which it was derived were mea-sured on a 57 mm diameter plastic cylinder, making it unlikely that WCI would be an accurate measure of heat loss from exposed flesh, which has different characteristics from the plastic (curvature, roughness, and radiation exchange properties) and is invariably below 33°C in a cold environment. Moreover, values given by the equation peak at 90 km/h, then decrease with increasing velocity.
Nevertheless, for velocities below 80 km/h, this index reliably expresses combined effects of temperature and wind on subjective discomfort. For example, if the calculated WCI is less than 1400 and actual air temperature is above − 10°C, there is little risk of frostbite during brief exposures (1 h or less), even for bare skin. However, at a WCI of 2000 or more, the probability is high that exposed flesh will begin to freeze in 1 min or less unless measures are taken to shield the exposed skin (such as a fur ruff to break up the wind around the face).
Rather than using the WCI to express the severity of a cold envi-ronment, meteorologists use an index derived from the WCI called the equivalent wind chill temperature. This is the ambient tem-perature that would produce, in a calm wind (defined for this appli-cation as 6.4 km/h), the same WCI as the actual combination of air temperature and wind velocity. Equivalent wind chill temperature teq,wc can be calculated by (81) 24 26 28 30 32 34 100 200 300 400 500 600 WET-BULB GLOBE TEMPERATURE (WBGT), °C METABOLIC HEAT, W 25% WORK, 75% REST EACH h 50% WORK, 50% REST EACH h 75% WORK, 25% REST EACH h 8 h CONTINUOUS WORK BODY MASS AND 1.8 m2 BODY SURFACE RECOMMENDED EXPOSURE LIMIT FOR STANDARD WORKER OF 70 kg Fig. 17 Recommended Heat Stress Exposure Limits for Heat Acclimatized Workers [Adapted from NIOSH (1986)] WBGT 1.044 WGT ( ) 0.187 – = WCI 10.45 10 V V – + ( ) 33 ta – ( ) in kcal m2 h ⋅ ( ) ⁄ = teq wc , 0.04544 – WCI ( ) 33 + = 8.22 2001 ASHRAE Fundamentals Handbook (SI) where teq,wc is in °C (and frequently referred to as a wind chill fac-tor), thus distinguishing it from WCI, which is given either as a cooling rate or as a plain number with no units. For velocities less than 6.4 km/h (1.8 m/s), Equation (81) does not apply, and the wind chill temperature is equal to the air temperature.
Equation (81) does not imply cooling to below ambient temper-ature, but recognizes that, because of wind, the cooling rate is increased as though it were occurring at the lower equivalent wind chill temperature. Wind accelerates the rate of heat loss, so that the skin surface is cooling faster toward the ambient temperature. Table 11 shows a typical wind chill chart, expressed in equivalent wind chill temperature.
SPECIAL ENVIRONMENTS Infrared Heating Optical and thermal properties of skin must be considered in studies concerning the effects of infrared radiation in (1) producing changes in skin temperature and skin blood flow, and (2) evoking sensations of temperature and comfort (Hardy 1961). Although the body can be considered to have the properties of water, thermal sensation and heat transfer with the environment require a study of the skin and its interaction with visible and infrared radiation.
Figure 18 shows how skin reflectance and absorptance vary for a blackbody heat source at the temperature (in K) indicated. These curves show that darkly pigmented skin is heated more by direct radiation from a high-intensity heater at 2500 K than is lightly pigmented skin. With low-temperature and low-intensity heating equipment used for total area heating, there is minimal, if any, dif-ference. Also, in practice, clothing minimizes differences.
Changes in skin temperature caused by high-intensity infrared radiation depend on the thermal conductivity, density, and specific heat of the living skin (Lipkin and Hardy 1954). Modeling of skin heating with the heat transfer theory yields a parabolic relation between exposure time and skin temperature rise for nonpenetrating radiation: (82) where tsf = final skin temperature, °C tsi = initial skin temperature, °C J = irradiance from source radiation temperatures, W/m2 θ = time, h k = specific thermal conductivity of tissue, W/(m·K) ρ = density, kg/m3 cp = specific heat, J/kg·K α = skin absorptance at radiation temperatures, dimensionless Product kρcp is the physiologically important quantity that deter-mines temperature elevation of skin or other tissue on exposure to nonpenetrating radiation. Fatty tissue, because of its relatively low specific heat, is heated more rapidly than moist skin or bone. Exper-imentally, kρcp values can be determined by plotting ∆t2 against 1.13J2θ (Figure 19). The relationship is linear, and the slopes are inversely proportional to the kρcp of the specimen. Comparing leather and water with body tissues suggests that thermal inertia val-ues depend largely on tissue water content.
Living tissues do not conform strictly to this simple mathemati-cal formula. Figure 20 compares excised skin with living skin with normal blood flow, and skin with blood flow occluded. For short exposure times, the kρcp of normal skin is the same as that in which blood flow has been stopped; excised skin heats more rapidly due to unavoidable dehydration that occurs postmortem. However, with longer exposure to thermal radiation, vasodilation increases blood flow, cooling the skin. For the first 20 s of irradiation, skin with nor-mally constricted blood vessels has a kρcp value one-fourth that for skin with fully dilated vessels.
Skin temperature is the best single index of thermal comfort. The most rapid changes in skin temperature occur during the first 60 s of Table 11 Equivalent Wind Chill Temperatures of Cold Environments Wind Speed, km/h Actual Thermometer Reading, °C 10 5 0 − 5 − 10 − 15 − 20 − 25 − 30 − 35 − 40 − 45 − 50 Equivalent Wind Chill Temperature, °C Calm 10 5 0 − 5 − 10 − 15 − 20 − 25 − 30 − 35 − 40 − 45 − 50 10 8 2 − 3 − 9 − 14 − 20 − 25 − 31 − 37 − 42 − 48 − 53 − 59 20 3 − 3 − 10 − 16 − 23 − 29 − 35 − 42 − 48 − 55 − 61 − 68 − 74 30 1 − 6 − 13 − 20 − 27 − 34 − 42 − 49 − 56 − 63 − 70 − 77 − 84 40 − 1 − 8 − 16 − 23 − 31 − 38 − 46 − 53 − 60 − 68 − 75 − 83 − 90 50 − 2 − 10 − 18 − 25 − 33 − 41 − 48 − 56 − 64 − 71 − 79 − 87 − 94 60 − 3 − 11 − 19 − 27 − 35 − 42 − 50 − 58 − 66 − 74 − 82 − 90 − 97 70 − 4 − 12 − 20 − 28 − 35 − 43 − 51 − 59 − 67 − 75 − 83 − 91 − 99 Little danger: In less than 5 h, with dry skin.
Maximum danger from false sense of security.
(WCI < 1400) Increasing danger: Danger of freezing exposed flesh within 1 min.
(1400 ≤WCI ≤2000) Great danger: Flesh may freeze within 30 s.
(WCI > 2000) Source: U.S. Army Research Institute of Environmental Medicine.
Notes: Cooling power of environment expressed as an equivalent temperature under calm conditions [Equation (81)].
Winds greater than 70 km/h have little added chill-ing effect.
tsf tsi – t ∆ 2Jα θ πkρcp ( ) ⁄ = = Fig. 18 Variation in Skin Reflection and Absorptivity for Blackbody Heat Sources Thermal Comfort 8.23 exposure to infrared radiation. During this initial period, thermal sensation and the heating rate of the skin vary with the quality of infrared radiation (color temperature in K). Because radiant heat from a gas-fired heater is absorbed at the skin surface, the same unit level of absorbed radiation during the first 60 s of exposure can cause an even warmer initial sensation than penetrating solar radia-tion. Skin heating curves tend to level off after a 60 s exposure (Fig-ure 20); this means that a relative balance is quickly created between heat absorbed, heat flow to the skin surface, and heat loss to the ambient environment. Therefore, the effects of radiant heating on thermal comfort should be examined for conditions approaching thermal equilibrium.
Stolwijk and Hardy (1966) described an unclothed subject’s response for a 2 h exposure to temperatures of 5 to 35°C. Nevins et al. (1966) showed a relation between ambient temperatures and thermal comfort of clothed, resting subjects. For any given uniform environmental temperature, both initial physiological response and degree of comfort can be determined for a subject at rest.
Physiological implications for radiant heating can be defined by two environmental temperatures: (1) mean radiant temperature and (2) ambient air temperature ta. For this discussion on radiant heat, assume that (1) relative humidity is less than 50%, and (2) air movement is low and constant, with an equivalent convection coef-ficient of 2.9 W/(m2·K).
The equilibrium equation, describing heat exchange between skin surface at mean temperature tsk and the radiant environment, is given in Equation (28), and can be transformed to give (see Table 2) (83) where M′ is the net heat production (M −W) less respiratory losses.
By algebraic transformation, Equation (83) can be rewritten as (84) where ERF = is the effective radiant field and represents the additional radiant exchange with the body when .
The last term in Equation (84) describes heat exchange with an environment uniformly heated to temperature ta. The term hr, eval-uated in Equation (35), is also a function of posture, for which factor Ar /AD can vary from 0.67 for crouching to 0.73 for standing. For preliminary analysis, a useful value for hr is 4.7 W/(m2·K), which corresponds to a normally clothed (at 24°C) sedentary subject.
Ambient air movement affects hc, which appears only in the right-hand term of Equation (84).
Although the linear radiation coefficient hr is used in Equations (83) and (84), the same definition of ERF follows if the fourth power radiation law is used. By this law, assuming emissivity of the body surface is unity, the ERF term in Equation (84) is (85) where σis the Stefan-Boltzmann constant, 5.67 × 10− 8 W/(m2·K4).
Because tr equals the radiation of several surfaces at different temperatures (T1, T2, …, Tj), (86) where ERFj = αj = absorptance of skin or clothing surface for source radiating at temperature Tj Fm −j = angle factor to subject m from source j Ta = ambient air temperature, K ERF is the sum of the fields caused by each surface Tj [e.g., T1 may be an infrared beam heater; T2, a heated floor; T3, a warm ceil-ing; T4, a cold plate glass window (T4 < Ta); etc.]. Only surfaces with Tj differing from Ta contribute to the ERF.
Comfort Equations for Radiant Heating The comfort equation for radiant heat (Gagge et al. 1967a,b) follows from definition of ERF and Equation (8): to (for comfort) = ta + ERF (for comfort)/h (87) Thus, operative temperature for comfort is the temperature of the ambient air plus a temperature increment ERF/h, a ratio that mea-sures the effectiveness of the incident radiant heating on occupants.
Higher air movement (which increases the value of h or hc) reduces the effectiveness of radiant heating systems. Clothing lowers to for comfort and for thermal neutrality.
Values for ERF and h must be determined to apply the comfort equation for radiant heating. Table 3 may be used to estimate h. One method of determining ERF is to calculate it directly from radiomet-ric data that give (1) radiation emission spectrum of the source, (2) concentration of the beam, (3) radiation from the floor, ceiling, and windows, and (4) corresponding angle factors involved. This ana-lytical approach is described in Chapter 52 of the 1999 ASHRAE Handbook—Applications.
For direct measurement, a skin-colored or black globe, 150 mm in diameter, can measure the radiant field ERF for comfort, by the following relation: (88) Fig. 19 Comparing Thermal Inertia of Fat, Bone, Moist Muscle, and Excised Skin to That of Leather and Water Fig. 20 Thermal Inertias of Excised, Bloodless, and Normal Living Skin tr M′ Esk – Fcle hr tsk t r – ( ) hc tsk to – ( ) + [ ] – 0 = M′ ERF Fcle ⋅ + Esk hr hc + ( ) tsk ta – ( ) + Fcle = hr tr ta – ( ) tr ta ≠ ERF σ Ar AD ⁄ ( ) t r 273 + ( )4 ta 273 + ( )4 – [ ]Fcle = ERF ERF ( )1 ERF ( )2 … ERF ( )j + + + = σ Ar AD ⁄ ( )αjFm j – Tj4 Ta4 – ( )Fcle ERF Ar AD ⁄ ( ) 6.1 13.6 V + [ ] tg ta – ( ) = 8.24 2001 ASHRAE Fundamentals Handbook (SI) where tg is uncorrected globe temperature in °C and V is air move-ment in m/s. The average value of Ar/AD is 0.7. For a skin-colored globe, no correction is needed for the quality of radiation. For a black globe, ERF must be multiplied by α for the exposed cloth-ing/skin surface. For a subject with 0.6 to 1.0 clo, to for comfort should agree numerically with ta for comfort in Figure 5. When to replaces ta in Figure 5, humidity is measured in vapor pressure rather than relative humidity, which refers only to air temperature.
Other methods may be used to measure ERF. The most accurate is by physiological means. In Equation (84), when M, tsk −ta, and the associated transfer coefficients are experimentally held constant, ∆E = ∆ERF (89) The variation in evaporative heat loss E (rate of mass loss) caused by changing the wattage of two T-3 infrared lamps is a mea-sure in absolute terms of the radiant heat received by the body.
A third method uses a directional radiometer to measure ERF directly. For example, radiation absorbed at the body surface [in W/m2] is ERF = α(Ai/AD)J (90) where irradiance J can be measured by a directional (Hardy-type) radiometer; α is the surface absorptance effective for the source used; and Ai is the projection area of the body normal to the direc-tional irradiance. Equation (90) can be used to calculate ERF only for the simplest geometrical arrangements. For a human subject lying supine and irradiated uniformly from above, Ai/AD is 0.3. Fig-ure 18 shows variance of αfor human skin with blackbody temper-ature (in K) of the radiating source. When irradiance J is uneven and coming from many directions, as is usually the case, the previous physiological method can be used to obtain an effective Ai/AD from the observed ∆E and ∆(αJ).
Hot and Humid Environments Tolerance limits to high temperature vary with the ability to (1) sense temperature, (2) lose heat by regulatory sweating, and (3) move heat from the body core by blood flow to the skin surface, where cooling is the most effective. Many interrelating processes are involved in heat stress (Figure 21).
Skin surface temperatures of 46°C trigger pain receptors in the skin; direct contact with metal at this temperature is painful. How-ever, since thermal insulation of the air layer around the skin is high, much higher dry air temperatures can be tolerated. For lightly clothed subjects at rest, tolerance times of nearly 50 min have been reported at 82°C dry-bulb temperature; 33 min at 93°C; 26 min at 104°C; and 24 min at 115°C. In each case, dew points were lower than 30°C. Many individuals are stimulated by brief periods of expo-sure to 85°C dry air in a sauna. Short exposures to these extremely hot environments are tolerable because of cooling by sweat evapo-ration. However, when ambient vapor pressure approaches 6.0 kPa (36°C dew point, typically found on sweating skin), tolerance is drastically reduced. Temperatures of 50°C can be intolerable if the dew-point temperature is greater than 25°C, and both deep body tem-perature and heart rate rise within minutes (Gonzalez et al. 1978).
The rate at which and length of time a body can sweat are lim-ited. The maximum rate of sweating for an average man is about 0.5 g/s. If all this sweat evaporates from the skin surface under conditions of low humidity and air movement, maximum cooling is about 675 W/m2. However, this value does not normally occur because sweat rolls off the skin surface without evaporative cool-ing or is absorbed by or evaporated within clothing. A more typi-cal cooling limit is 6 mets, 350 W/m2, representing approximately 0.3 g/s (1 L/h) of sweating for the average man.
Thermal equilibrium is maintained by dissipation of resting heat production (1 met) plus any radiant and convective load. If the environment does not limit heat loss from the body during heavy activity, decreasing skin temperature compensates for the core tem-perature rise. Therefore, mean body temperature is maintained, although the gradient from core to skin is increased. Blood flow through the skin is reduced, but muscle blood flow necessary for exercise is preserved. The upper limit of skin blood flow is about 25 g/s (Burton and Bazett 1936).
Body heat storage of 335 kJ (or a rise in tb of 1.4 K) for an aver-age-sized man represents an average voluntary tolerance limit. Con-tinuing work beyond this limit increases the risk of heat exhaustion.
Collapse can occur at about 670 kJ of storage (2.8 Krise); few indi-viduals can tolerate heat storage of 920 kJ (3.8 K above normal).
The cardiovascular system affects tolerance limits. In normal, healthy subjects exposed to extreme heat, heart rate and cardiac out-put increase in an attempt to maintain blood pressure and supply of Fig. 21 Schematic Design of Heat Stress and Heat Disorders [Modified by Buskirk from scale diagram by Belding (1967) and Leithead and Lind (1964)] Thermal Comfort 8.25 blood to the brain. At a heart rate of about 180 bpm, the short time between contractions prevents adequate blood supply to the heart chambers. As heart rate continues to increase, cardiac output drops, causing inadequate convective blood exchange with the skin and, perhaps more important, inadequate blood supply to the brain. Vic-tims of this heat exhaustion faint or black out. Accelerated heart rate can also result from inadequate venous return to the heart caused by pooling of blood in the skin and lower extremities. In this case, car-diac output is limited because not enough blood is available to refill the heart between beats. This occurs most frequently when an over-heated individual, having worked hard in the heat, suddenly stops working. The muscles no longer massage the blood back past the valves in the veins toward the heart. Dehydration compounds the problem, since fluid volume in the vascular system is reduced.
If core temperature tcr increases above 41°C, critical hypotha-lamic proteins can be damaged, resulting in inappropriate vasocon-striction, cessation of sweating, increased heat production by shivering, or some combination of these. Heat stroke damage is fre-quently irreversible and carries a high risk of fatality.
A final problem, hyperventilation, occurs predominantly in hot-wet conditions, when too much CO2 is washed from the blood. This can lead to tingling sensations, skin numbness, and vasoconstriction in the brain with occasional loss of consciousness.
Since a rise in heart rate or rectal temperature is essentially linear with ambient vapor pressure above a dew point of 25°C, these two changes can measure severe heat stress. Although individual heart rate and rectal temperature responses to mild heat stress vary, severe heatstress saturatesphysiologicalregulating systems, producing uni-form increases in heart rate and rectal temperature. In contrast, sweat production measures stress under milder conditions but becomes less useful under more severe stress. The maximal sweat rate compatible with body cooling varies with (1) degree of heat acclimatization, (2) duration of sweating, and (3) whether the sweat evaporates or merely saturates the skin and drips off. Total sweat rates in excess of 2 L/h can occur in short exposures, but about 1 L/h is an average maximum sustainable level for an acclimatized man.
Figure 22 illustrates the decline in heart rate, rectal temperature, and skin temperature when exercising subjects are exposed to 40°C over a period of days. Acclimatization can be achieved by working in the heat for 100 min each day—30% improvement occurs after the first day, 50% after 3 days, and 95% after 6 or 7 days. Increased sweat secretion while working in the heat can be induced by rest.
Although reducing salt intake during the first few days in the heat can conserve sodium, heat cramps may result. Working regularly in the heat improves cardiovascular efficiency, sweat secretion, and sodium conservation. Once induced, heat acclimatization can be maintained by as few as once-a-week workouts in the heat; other-wise, it diminishes slowly over a 2- to 3-week period and disappears.
Extreme Cold Environments Human performance in extreme cold ultimately depends on maintaining thermal balance. Subjective discomfort is reported by a 70 kg man with 1.8 m2 of body surface area when a heat debt of about 104 kJ is incurred. A heat debt of about 630 kJ is acutely uncomfortable; this represents a drop of approximately 2.6 K (or about 7% of total heat content) in mean body temperature.
This loss can occur during 1 to 2 h of sedentary activity outdoors.
A sleeping individual will awake after losing about 314 kJ, decreas-ing mean skin temperature by about 3 Kand deep body temperature by about 0.5 K. A drop in deep body temperature (e.g., rectal tem-perature) below 35°C threatens a loss of body temperature regula-tion, while 28°Cis considered critical for survival, despite recorded survivalfroma deep body temperatureof 18°C. Temperature is more crucial than rate of temperature change; Witherspoon et al. (1971) observed a rate of fall in the core temperature of 3 Kper hour in sub-jects immersed in 10°C water, without residual effect.
Activity level also affects human performance. Subjective sen-sations reported by sedentary subjects at a mean skin temperature of 33.3°C, are comfortable; at 31°C, uncomfortably cold; at 30°C, shivering cold; and at 29°C, extremely cold. The critical subjective tolerance limit (without numbing) for mean skin temperature appears to be about 25°C. However, during moderate to heavy activ-ity, subjects reported the same skin temperatures as comfortable.
Although mean skin temperature is significant, the temperature of the extremities is more frequently the critical factor for comfort in the cold. Consistent with this, one of the first responses to cold expo-sure is vasoconstriction, which reduces circulatory heat input to the hands and feet. A hand-skin temperature of 20°C causes a report of uncomfortably cold; 15°C, extremely cold; and 5°C, painful. Iden-tical verbal responses for the foot surface occur at approximately 1.5 to 2 K warmer temperatures.
An ambient temperature of − 35°C is the lower limit for useful outdoor activity, even with adequate insulative clothing. At − 50°C, almost all outdoor effort becomes exceedingly difficult; even with appropriate protective equipment, only limited exposure is possible.
Reported exposures of 30 min at − 75°C have occurred in the Ant-arctic without injury.
In response to extreme heat loss, maximal heat production becomes very important. When the less efficient vasoconstriction cannot prevent body heat loss, shivering is an automatic, more effi-cient defense against cold. This can be triggered by low deep body temperature, low skin temperature, rapid change of skin tempera-ture, or some combination of all three. Shivering is usually preceded Fig. 22 Acclimatization to Heat Resulting from Daily Exposure of Five Subjects to Extremely Hot Room (Robinson et al. 1943) 8.26 2001 ASHRAE Fundamentals Handbook (SI) by an imperceptible increase in muscle tension and by noticeable gooseflesh produced by muscle contraction in the skin. It begins slowly in small muscle groups, initially increasing total heat production by 1.5 to 2 times resting levels. As body cooling increases, the reaction spreads to additional body segments. Ulti-mately violent, whole body shivering causes maximum heat pro-duction of about 6 times resting levels, rendering the individual totally ineffective.
Given sufficient cold exposure, the body undergoes changes that indicate cold acclimatization. These physiological changes include, (1) endocrine changes (e.g., sensitivity to norepinephrine), causing nonshivering heat production by metabolism of free fatty acids released from adipose tissue; (2) improved circulatory heat flow to skin, causing an overall sensation of greater comfort; and (3) improved circulatory heat flow to the extremities, reducing the risk of injury and permitting activities at what ordinarily would be severely uncomfortable temperatures in the extremities. Generally, these physiological changes are minor and are induced only by repeated extreme exposures. Nonphysiological factors, including training, experience, and selection of adequate protective clothing, are more useful and may be safer than dependence on physiological changes.
The energy requirement for adequately clothed subjects in extreme cold is only slightly greater than that for subjects living and working in temperate climates. This greater requirement results from added work caused by (1) carrying the weight of heavy cloth-ing (energy cost for heavy protective footwear may be six times that of an equivalent weight on the torso); and (2) the inefficiency of walking in snow, snowshoeing, or skiing, which can increase energy cost up to 300%.
To achieve proper protection in low temperatures, a person must either maintain high metabolic heat production by activity or reduce heat loss by controlling the body’s microclimate with clothing.
Other protective measures include spot radiant heating, showers of hot air for work at a fixed site, and warm-air-ventilated or electri-cally heated clothing. The extremities, such as fingers and toes, pose more of a problem than the torso because, as thin cylinders, they are particularly susceptible to heat loss and difficult to insulate without increasing the surface for heat loss. Vasoconstriction can reduce cir-culatory heat input to extremities by over 90%.
Although there is no ideal insulating material for protective clothing, radiation-reflective materials are promising. Insulation is primarily a function of clothing thickness; the thickness of trapped air, rather than fibers used, determines insulation effectiveness.
Protection for the respiratory tract seems unnecessary in healthy individuals, even at − 45°C. However, asthmatics or individuals with mild cardiovascular problems may benefit from a face mask that warms inspired air. Masks are unnecessary for protecting the face because heat to facial skin is not reduced by local vasoconstric-tion, as it is for hands. If wind chill is great, there is always a risk of cold injury caused by freezing of exposed skin. Using properly designed torso clothing, such as a parka with a fur-lined hood to minimize wind penetration to the face, and 10 Wof auxiliary heat to each hand and foot, inactive people can tolerate − 55°C with a 16 km/h wind for more than 6 h. As long as the skin temperature of fingers remains above 15°C, manual dexterity can be maintained and useful work performed without difficulty.
SYMBOLS A = area, m2 Ach = contact area between chair and body Acl = surface area of clothed body AD = DuBois surface area of nude body AG = body surface area covered by garment Ar = effective radiation area of body BFN = neutral skin blood flow, L/(h·m2) c = specific heat, J/(kg· K) ccr = body core csk = skin cp = at constant pressure cp,a = of air cp,b = of body tissue cp,bl = of blood cdil = constant for skin blood flow csw = proportionality constant for sweat control C = convective heat loss, W/m2 Cres = sensible heat loss due to respiration, W/m2 C + R = total sensible heat loss from skin, W/m2 E = evaporative heat loss, W/m2 Edif = due to moisture diffusion through skin Emax = maximum possible Ersw = due to regulatory sweating Eres = due to respiration Ersw,req = required for comfort Esw = actual evaporation rate Esk = total from skin ERF = effective radiant field, W/m2 ET = original effective temperature based on 100% rh, °C ET = effective temperature, based on 50% rh, °C fcl = clothing area factor, Acl/AD, dimensionless Fcl = intrinsic clothing thermal efficiency, dimensionless Fcle = effective clothing thermal efficiency, dimensionless Fpcl = permeation efficiency, dimensionless Fm −j = angle factor to person from source j, dimensionless Fp −N = angle factor between person and source N, dimensionless h = sensible heat transfer coefficient, W/(m2·K) h = total at surface h′ = overall including clothing hc = convection at surface hcc = corrected convection at surface hr = radiation he = evaporative heat transfer coefficient, W/(m2·kPa) he = at surface hec = at surface, corrected for atmospheric pressure h′ e,cl = clothing evaporative conductance, the evaporative con-ductance of a uniform layer of insulation covering the entire body that has same effect on evaporative heat flow as the actual clothing.
= overall including clothing ha = enthalpy of ambient air, J/kg (dry air) hex = enthalpy of exhaled air, J/kg (dry air) hfg = heat of vaporization of water, J/kg i = vapor permeation efficiency, dimensionless ia = air layer icl = clothing im = total I = thermal resistance in clo units, clo All subscripts given for symbol R apply to symbol I I clu,i = effective insulation of garment i J = irradiance, W/m2 k = thermal conductivity of body tissue, W/(m·K) K = effective conductance between core and skin, W/(m2·K) l = height, m L = thermal load on body, W/m2 LR = Lewis ratio, K/kPa m = body mass, kg mcr = mass of body core msk = mass of skin mge = mass to gas exchange, kg = mass flow, kg/(s·m2) = pulmonary ventilation flow rate, kg/s = pulmonary water loss rate, kg/s M = metabolic heat production, W/m2 M = total M′ = net (M −W) Mact = due to activity Mshiv = due to shivering p = water vapor pressure, kPa pa = in ambient air pET , s = saturated at ET poh,s = saturated at toh he ′ m · m · res m · w,res Thermal Comfort 8.27 psk,s = saturated at tsk pt = atmospheric pressure, kPa ptcom = related to tcom q = heat flow, W/m2 qdry = sensible from skin qevap = latent from skin qcr,sk = from core to skin qres = total due to respiration qsk = total from the skin Qbl = blood flow between core and skin, [L/(h·m2)] QCO2 = volume rate of CO2 produced, mL/s QO2 = volume rate of O2 consumed, mL/s = rate of regulatory sweat generation, [L/(h·m2)] R = radiative heat loss from skin, W/m2 R = thermal resistance, m2·K/W Ra = air layer on nude skin Ra,cl = air layer at outer surface Rcl = clothing Rcle = change due to clothing Re = evaporative resistance, m2·kPa/W Re,a = outer boundary layer air Re,cl = clothing Re,t = total Rt = total RQ = respiratory quotient, dimensionless S = heat storage, W/m2 Scr = in core compartment Ssk = in skin compartment Str = constriction constant for skin blood flow SET = standard effective temperature, °C SKBF = skin blood flow lb/h·ft2 t = temperature, °C ta = ambient air tb = average of body tb,c = lower limit for evaporative regulation zone tb,h = upper limit for evaporative regulation zone tc = comfort tcl = clothing surface tcom = combined temperature tcr = core compartment tdb = dry bulb tdp = dew point teq,wc = equivalent wind chill temperature tex = of exhaled air tg = globe tN = of surface N tnwb = naturally ventilated wet bulb to = operative toc = operative comfort toh = humid operation tout = monthly mean outside tpr = plane radiant = mean radiant tsf = final skin tsi = initial skin tsk = skin compartment tsk,n = at neutrality tsk,req = required for comfort twb = wet bulb T = absolute temperature, K All subscripts that apply to symbol t may apply to symbol T.
Tu = turbulence intensity, % V = air velocity, m/s w = skin wettedness, dimensionless wrsw = required to evaporate regulatory sweat W = external work accomplished, W/m2 Wa = humidity ratio of ambient air, kg (water vapor)/kg (dry air) Wex = humidity ratio of exhaled air, kg (water vapor)/kg (dry air) WBGT = wet-bulb globe temperature, °C WCI = wind chill index, kcal/(h·m2) WGT = wet-globe temperature, °C α = skin absorptance, dimensionless αsk = fraction of total body mass concentrated in skin compart-ment, dimensionless ε = emissivity, dimensionless η ev = evaporative efficiency, dimensionless µ = body’s mechanical efficiency = W/M, dimensionless ρ = density, kg/m3 σ = Stefan-Boltzmann constant = 5.67 × 10− 8 W/(m2·K4) θ = time, s CODES AND STANDARDS ASHRAE. 1992. Thermal environmental conditions for human occupancy.
ANSI/ASHRAE Standard 55-1992.
ASHRAE. 1994. Addendum 55a. ANSI/ASHRAE Standard 55-1992.
ISO. 1989. Hot environments—Analytical determination and interpretation of thermal stress using calculation of required sweat rates. Standard 7933. International Organization for Standardization, Geneva.
ISO. 1989. Hot environments—Estimation of the heat stress on working man, based on the WBGT-index (wet bulb globe temperature). Standard 7243.
ISO. 1994. Moderate thermal environments—Determination of the PMV and PPD indices and specification of the conditions for thermal comfort.
Standard 7730.
REFERENCES Astrand, P. and K. Rodahl. 1977. Textbook of work physiology: Physiologi-cal bases of exercise. McGraw-Hill, New York.
Azer, N.Z. 1982. Design guidelines for spot cooling systems: Part I— Assessing the acceptability of the environment. ASHRAE Transactions 88:1.
Azer, N.Z. and S. Hsu. 1977. OSHA heat stress standards and the WBGT index. ASHRAE Transactions 83(2):30.
Belding, H.D. 1967. Heat stress. In Thermobiology, ed. A.H. Rose. Aca-demic Press, New York.
Belding, H.S. 1970. The search for a universal heat stress index. Physiolog-ical and Behavioral Temperature Regulation, eds. J.D. Hardy et al.
Springfield, IL.
Belding, H.S. and T.F. Hatch. 1955. Index for evaluating heat stress in terms of resulting physiological strains. Heating, Piping and Air Conditioning 207:239.
Berglund, L.G. 1994. Common elements in the design and operation of ther-mal comfort and ventilation systems. ASHRAE Transactions 100(1).
Berglund, L.G. 1995. Comfort criteria: Humidity and standards. Proceed-ings of Pan Pacific Symposium on Building and Urban Environmental Conditioning in Asia 2:369-82. Architecture Department, University of Nagoya, Japan.
Berglund, L.G and D.J. Cunningham. 1986. Parameters of human discom-fort in warm environments. ASHRAE Transactions 92(2):732-46.
Berglund, L.G. and A. Fobelets. 1987. A subjective human response to low level air currents and asymmetric radiation. ASHRAE Transactions 93(1):497-523.
Berglund, L.G. and R.R. Gonzalez. 1978. Evaporation of sweat from seden-tary man in humid environments. J. Appl. Physiol: Respirat, Environ, Exercise Physiol. 42(5): 767-72.
Botsford, J.H. 1971. A wet globe thermometer for environmental heat mea-surement. American Industrial Hygiene Association Journal 32:1-10.
Burton, A.C. and H.C. Bazett. 1936. A study of the average temperature of the tissues, the exchange of heat and vasomotor responses in man, by a bath calorimeter. American Journal of Physiology 117:36.
Busch, J.F. 1992. A tale of two populations: Thermal comfort in air-condi-tioned and naturally ventilated offices in Thailand. Energy and Buildings 18:235-49.
Buskirk, E.R. 1960. Problems related to the caloric cost of living. Bulletin of the New York Academy of Medicine 26:365.
Chatonnet, J. and M. Cabanac. 1965. The perception of thermal comfort.
International Journal of Biometeorology 9:183-93.
Ciriello, V.M. and S.H. Snook. 1977. The prediction of WBGT from the Botsball. American Industrial Hygiene Association Journal 38:264.
Colin, J. and Y. Houdas. 1967. Experimental determination of coefficient of heat exchange by convection of the human body. Journal of Applied Physiology 22:31.
Collins, K.J. and E. Hoinville. 1980. Temperature requirements in old age.
Building Services Engineering Research and Technology 1(4):165-72.
Davis, W.J. 1976. Typical WBGT indexes in various industrial environ-ments. ASHRAE Transactions 82(2):303.
Qrsw tr 8.28 2001 ASHRAE Fundamentals Handbook (SI) de Dear, R.J. and G.S. Brager. 1998. Developing an adaptive model of ther-mal comfort and preference. ASHRAE Technical Data Bulletin 14(1):27-49.
de Dear, R., K. Leow, and A. Ameen. 1991. Thermal comfort in the humid tropics— Part I. ASHRAE Transactions 97(1):874-79.
DuBois, D. and E.F. DuBois. 1916. A formula to estimate approximate sur-face area, if height and weight are known. Archives of Internal Medicine 17:863-71.
Dukes-Dobos, F. and A. Henschel. 1971. The modification of the WBGT index for establishing permissible heat exposure limits in occupational work. HEW, USPHE, NIOSH, TR-69.
Dukes-Dobos, F. and A. Henschel. 1973. Development of permissible heat exposure limits for occupational work. ASHRAE Journal 9:57.
Eriksson, H.A. 1975. Heating and ventilating of tractor cabs. Presented at the 1975 Winter Meeting, American Society of Agricultural Engineers, Chicago.
Fanger, P.O. 1967. Calculation of thermal comfort: Introduction of a basic comfort equation. ASHRAE Transactions 73(2):III.4.1.
Fanger, P.O. 1970. Thermal comfort analysis and applications in environ-mental engineering. McGraw-Hill, New York.
Fanger, P.O. 1972. Thermal comfort. McGraw-Hill, New York.
Fanger, P.O. 1973. The variability of man’s preferred ambient temperature from day to day. Archives des Sciences Physiologiques 27(4):A403.
Fanger, P.O. 1982. Thermal comfort. Robert E. Krieger Publishing Com-pany, Malabar, FL.
Fanger, P.O., L. Banhidi, B.W. Olesen, and G. Langkilde. 1980. Comfort limits for heated ceilings. ASHRAE Transactions 86.
Fanger, P.O. and N.K. Christensen. 1986. Perception of draught in ventilated spaces. Ergonomics 29(2):215-35.
Fanger, P.O., J. Hojbjerre, and J.O.B. Thomsen. 1974. Thermal comfort conditions in the morning and the evening. International Journal of Biometeorology 18(1):16.
Fanger, P.O., J. Hojbjerre, and J.O.B. Thomsen. 1973. Man’s preferred ambient temperature during the day. Archives des Sciences Physiologi-ques 27(4): A395-A402.
Fanger, P.O., B.M. Ipsen, G. Langkilde, B.W. Olesen, N.K. Christensen, and S. Tanabe. 1985. Comfort limits for asymmetric thermal radiation.
Energy and Buildings.
Fanger, P.O. and G. Langkilde. 1975. Interindividual differences in ambient temperature preferred by seated persons. ASHRAE Transactions 81(2): 140-47.
Fanger, P.O., A.K. Melikov, H. Hanzawa, and J. Ring, J. 1989. Turbulence and draft. ASHRAE Journal 31(4):18-25.
Fobelets, A.P.R. and A.P. Gagge. 1988. Rationalization of the ET as a mea-sure of the enthalpy of the human environment. ASHRAE Transactions 94:1.
Gagge, A.P. 1937. A new physiological variable associated with sensible and insensible perspiration. American Journal of Physiology 20(2):277-87.
Gagge, A.P., A.C. Burton, and H.D. Bazett. 1971. A practical system of units for the description of heat exchange of man with his environment. Sci-ence 94:428-30.
Gagge, A.P. and J.D. Hardy. 1967. Thermal radiation exchange of the human by partitional calorimetry. Journal of Applied Physiology 23:248.
Gagge, A.P., A.P. Fobelets, and L.G. Berglund. 1986. A standard predictive index of human response to the thermal environment. ASHRAE Trans-actions 92(2B).
Gagge, A.P., Y. Nishi, and R.G. Nevins. 1976. The role of clothing in meet-ing FEA energy conservation guidelines. ASHRAE Transactions 82(2): 234.
Gagge, A.P., G.M. Rapp, and J.D. Hardy. 1967a. The effective radiant field and operative temperature necessary for comfort with radiant heating.
ASHRAE Transactions 73(1):I.2.1.
Gagge, A.P., G.M. Rapp, and J.D. Hardy. 1967b. The effective radiant field and operative temperature necessary for comfort with radiant heating.
ASHRAE Journal 9(5):63.
Gagge, A.P., J. Stolwijk, and Y. Nishi. 1971. An effective temperature scale based on a simple model of human physiological regulatory response.
ASHRAE Transactions 77(1):247-62.
Gagge, A.P., J.A.J. Stolwijk, and B. Saltin. 1969a. Comfort and thermal sen-sation and associated physiological responses during exercise at various ambient temperatures. Environmental Research 2:209.
Gagge, A.P., J.A.J. Stolwijk, and Y. Nishi. 1969b. The prediction of thermal comfort when thermal equilibrium is maintained by sweating. ASHRAE Transactions 75(2):108.
Gonzalez, R.R., L.G. Berglund, and A.P. Gagge. 1978. Indices of thermoreg-ulatory strain for moderate exercise in the heat. Journal of Applied Phys-iology 44:889.
Green, G.H. 1982. Positive and negative effects of building humidification.
ASHRAE Transactions 88(1):1049-61.
Gwosdow, A.R., J.C. Stevens, L. Berglund, and J.A.J. Stolwijk. 1986. Skin friction and fabric sensations in neutral and warm environments. Textile Research Journal 56:574-80.
Hardy, J.D. 1949. Heat transfer. In Physiology of heat regulation and science of clothing. eds. L.H. Newburgh and W.B. Saunders Ltd., London, 78.
Hardy, J.D. 1961. Physiological effects of high intensity infrared heating.
ASHRAE Journal 4:11.
Hardy, J.D., J.A.J. Stolwijk, and A.P. Gagge. 1971. Man. In Comparative physiology of thermoregulation, Chapter 5. Charles C. Thomas, Spring-field, IL.
Hardy, J.D., H.G. Wolf, and H. Goodell. 1952. Pain sensations and reac-tions. Williams and Wilkins, Baltimore.
Hensel, H. 1973. Temperature reception and thermal comfort. Archives des Sciences Physiologiques 27:A359-A370.
Hensel, H. 1981. Thermoreception and temperature regulation. Academic Press, London.
Holmer, I. 1984. Required clothing insulation (IREQ) as an analytical index of cold stress. ASHRAE Transactions 90(1B):1116-28.
Houghten, F.C. and C.P. Yaglou. 1923. ASHVE Research Report No. 673.
ASHVE Transactions 29:361.
Humphreys, M. and J.F. Nicol. 1998. Understanding the adaptive approach to thermal comfort. ASHRAE Technical Data Bulletin 14(1):1-14.
Jones, B.W., K. Hsieh, and M. Hashinaga. 1986. The effect of air velocity on thermal comfort at moderate activity levels. ASHRAE Transactions 92:2.
Korsgaard, V. 1949. Necessity of using a directional mean radiant tempera-ture to describe thermal conditions in rooms. Heating, Piping and Air Conditioning 21(6):117-20.
Kuno, S. 1995. Comfort and pleasantness. In Proceedings of Pan Pacific Symposium on Building and Urban Environmental Conditioning in Asia, 2:383-92. Architecture Department, University of Nagoya, Japan.
Langkilde, G. 1979. Thermal comfort for people of high age. In Confort thermique: Aspects physiologiques et psychologiques, INSERM, Paris 75:187-93.
Leithead, C.S. and A.R. Lind. 1964. Heat stress and heat disorders. Cassell & Co., London, 108. (F.A. Davis, Philadelphia, 1964).
Lipkin, M. and J.D. Hardy 1954. Measurement of some thermal properties of human tissues. Journal of Applied Physiology 7:212.
Liviana, J.E., F.H. Rohles, and O.D. Bullock. 1988. Humidity, comfort and contact lenses. ASHRAE Transactions 94(1):3-11.
Matthew, W.H. et al. 1986. Botsball (WGT) performance characteristics and their impact on the implementation of existing military hot weather doc-trine. U.S. Army Reserves Institute of Environmental Medicine Techni-cal Report T 9/86, April.
McCullough, E.A. 1986. An insulation data base for military clothing. Insti-tute for Environmental Research Report 86-01, Kansas State University, Manhattan.
McCullough, E.A. and S. Hong. 1994. A data base for determining the decrease in clothing insulation due to body motion. ASHRAE Trans-actions 100(1):765.
McCullough, E.A. and B.W. Jones. 1984. A comprehensive data base for estimating clothing insulation. IER Technical Report 84-01, Institute for Environmental Research, Kansas State University, Manhattan. (Final report to ASHRAE Research Project 411-RP).
McCullough, E.A., B.W. Jones, and T. Tamura. 1989. A data base for deter-mining the evaporative resistance of clothing. ASHRAE Transactions 95(2).
McCullough, E.A., B.W. Olesen, and S.W. Hong. 1994. Thermal insulation provided by chairs. ASHRAE Transactions 100(1):795-802.
McCutchan, J.W. and C.L. Taylor. 1951. Respiratory heat exchange with varying temperature and humidity of inspired air. Journal of Applied Physiology 4:121-35.
McIntyre, D.A. 1974. The thermal radiation field. Building Science 9:247-62.
McIntyre, D.A. 1976. Overhead radiation and comfort. The Building Ser-vices Engineer 44:226-32.
McIntyre, D.A. and I.D. Griffiths. 1975. The effects of uniform and asym-metric thermal radiation on comfort. CLIMA 2000, 6th International Congress of Climatritics, Milan.
Thermal Comfort 8.29 McNall, P.E., Jr. and R.E. Biddison. 1970. Thermal and comfort sensations of sedentary persons exposed to asymmetric radiant fields. ASHRAE Transactions 76(1):123.
McNall, P.E., P.W. Ryan, and J. Jaax. 1968. Seasonal variation in comfort conditions for college-age persons in the Middle West. ASHRAE Trans-actions 74(1):IV.2.1-9.
McNair, H.P. 1973. A preliminary study of the subjective effects of vertical air temperature gradients. British Gas Corporation Report No. WH/T/ R&D/73/94, London.
McNair, H.P. and D.S. Fishman. 1974. A further study of the subjective effects of vertical air temperature gradients. British Gas Corporation Report No. WH/T/R&D/73/94, London.
Minard, D. 1961. Prevention of heat casualties in Marine Corps recruits.
Military Medicine 126:261.
Mitchell, D. 1974. Convective heat transfer in man and other animals. Heat loss from animals and man. Butterworth Publishing, London, 59.
Nevins, R., R.R. Gonzalez, Y. Nishi, and A.P. Gagge. 1975. Effect of changes in ambient temperature and level of humidity on comfort and thermal sensations. ASHRAE Transactions 81(2).
Nevins, R.G. and A.M. Feyerherm. 1967. Effect of floor surface temperature on comfort: Part IV, Cold floors. ASHRAE Transactions 73(2):III.2.1.
Nevins, R.G. and A.O. Flinner. 1958. Effect of heated-floor temperatures on comfort. ASHRAE Transactions 64:175.
Nevins, R.G., K.B. Michaels, and A.M. Feyerherm. 1964. The effect of floor surface temperature on comfort: Part 1, College age males; Part II, Col-lege age females. ASHRAE Transactions 70:29.
Nevins, R.G., F.H. Rohles, Jr., W.E. Springer, and A.M. Feyerherm. 1966.
Temperature-humidity chart for thermal comfort of seated persons.
ASHRAE Transactions 72(1):283.
Nicol, J.F. and M.A. Humphreys. 1972. Thermal comfort as part of a self-regulating system. Proceedings of CIB symposium on thermal comfort, Building Research Station, London.
NIOSH. 1986. Criteria for a recommended standard—Occupational expo-sure to hot environments, revised criteria. U.S. Dept. of Health and Human Services, USDHHS (NIOSH) Publication 86-113. Available from www.cdc.gov/NIOSH/86-113.html.
Nishi, Y. 1981. Measurement of thermal balance of man. In Bioengineering thermal physiology and comfort, K. Cena and J.A. Clark, Elsevier, eds.
New York.
Nishi, Y. and A.P. Gagge. 1970. Direct evaluation of convective heat transfer coefficient by naphthalene sublimation. Journal of Applied Physiology 29:830.
Nishi, Y., R.R. Gonzalez, and A.P. Gagge. 1975. Direct measurement of clothing heat transfer properties during sensible and insensible heat exchange with thermal environment. ASHRAE Transactions 81(2):183.
Olesen, B.W. 1977a. Thermal comfort requirements for floors. In Proceed-ings of Commissions B1, B2, E1 of the IIR, Belgrade, 337-43.
Olesen, B.W. 1977b. Thermal comfort requirements for floors occupied by people with bare feet. ASHRAE Transactions 83(2).
Olesen, B.W. and R. Nielsen. 1983. Thermal insulation of clothing measured on a moveable manikin and on human subjects. Technical University of Denmark, Lyngby, Denmark.
Olesen, B.W., M. Scholer, and P.O. Fanger. 1979. Vertical air temperature differences and comfort. In Indoor climate, eds. P.O. Fanger and O. Val-bjorn, 561-79. Danish Building Research Institute, Copenhagen.
Olesen, S., J.J. Bassing, and P.O. Fanger. 1972. Physiological comfort con-ditions at sixteen combinations of activity, clothing, air velocity and ambient temperature. ASHRAE Transactions 78(2):199.
Olesen, B.W., E. Sliwinska, T.L. Madsen, and P.O. Fanger. 1982. Effect of posture and activity on the thermal insulation of clothing. Measurement by a movable thermal manikin. ASHRAE Transactions 82(2):791-805.
Onkaram, B., L. Stroschein, and R.F. Goldman. 1980. Three instruments for assessment of WBGT and a comparison with WGT (Botsball). American Industrial Hygiene Association 41:634-41.
Oohori, T., L.G. Berglund, and A.P. Gagge. 1984. Comparison of current two-parameter indices of vapor permeation of clothing—As factors gov-erning thermal equilibrium and human comfort. ASHRAE Transactions 90(2).
Ostberg, O. and A.G. McNicholl. 1973. The preferred thermal conditions for “morning” and “evening” types of subjects during day and night—Pre-liminary results. Build International 6(1):147-57.
Passmore, R. and J.V.G. Durnin. 1967. Energy, work and leisure. Heinemann Educational Books, London.
Rapp, G. and A.P. Gagge. 1967. Configuration factors and comfort design in radiant beam heating of man by high temperature infrared sources.
ASHRAE Transactions 73(2):III.1.1.
Robinson, et al. 1943. American Journal of Physiology 140:168.
Rohles, F.H., Jr. 1973. The revised modal comfort envelope. ASHRAE Transactions 79(2):52.
Rohles, F.H., Jr. and M.A. Johnson. 1972. Thermal comfort in the elderly.
ASHRAE Transactions 78(1):131.
Rohles, F.H., Jr. and R.G. Nevins. 1971. The nature of thermal comfort for sedentary man. ASHRAE Transactions 77(1):239.
Seppanen, O., P.E. McNall, D.M. Munson, and C.H. Sprague. 1972. Ther-mal insulating values for typical indoor clothing ensembles. ASHRAE Transactions 78(1):120-30.
Siple, P.A. and C.F. Passel. 1945. Measurements of dry atmospheric cooling in subfreezing temperatures. In Proceedings of the American Philosoph-ical Society 89:177.
Stolwijk, J.A.J., A.P. Gagge, and B. Saltin. 1968. Physiological factors asso-ciated with sweating during exercise. Journal of Aerospace Medicine 39: 1101.
Stolwijk, J.A.J. and J.D. Hardy. 1966. Partitional calorimetric studies of response of man to thermal transients. Journal of Applied Physiology 21:967.
Sullivan, C.D. and R.L. Gorton. 1976. A method of calculating WBGT from environmental factors. ASHRAE Transactions 82(2):279.
Tanabe, S., K. Kimura, and T. Hara. 1987. Thermal comfort requirements during the summer season in Japan. ASHRAE Transactions 93(1):564-77.
Umbach, K.H. 1980. Measuring the physiological properties of textiles for clothing. Melliand Textilberichte (English edition) G1:543-48.
Webb, P. 1964. Bioastronautics data base, NASA.
Winslow, C.-E.A., L.P. Herrington, and A.P. Gagge. 1937. Relations between atmospheric conditions, physiological reactions and sensations of pleasantness. American Journal of Hygiene 26(1):103-15.
Witherspoon, J.M., R.F. Goldman, and J.R. Breckenridge. 1971. Heat trans-fer coefficients of humans in cold water. Journal de Physiologie, Paris, 63:459.
Woodcock, A.H. 1962. Moisture transfer in textile systems. Textile Research Journal 8:628-33.
BIBLIOGRAPHY Fanger, P.O., A. Melikov, H. Hanzawa, and J. Ring. 1988. Air turbulence and sensation of draught. Energy and Buildings 12:21-39.
9.1 CHAPTER 9 INDOOR ENVIRONMENTAL HEALTH Terminology .............................................................................. 9.1 Scope ......................................................................................... 9.1 DESCRIPTIONS OF SELECTED HEALTH SCIENCES ......... 9.1 Epidemiology and Biostatistics ................................................. 9.1 Toxicology ................................................................................. 9.2 Molecular Biology .................................................................... 9.2 Cellular Biology ........................................................................ 9.2 Genetics .................................................................................... 9.2 Industrial Hygiene .................................................................... 9.2 AIR CONTAMINANTS .............................................................. 9.4 Particulate Matter ..................................................................... 9.4 Bioaerosols ................................................................................ 9.6 Gaseous Contaminants ............................................................. 9.7 Gaseous Contaminants in Industrial Environments ................. 9.8 Gaseous Contaminants in Nonindustrial Environments ......................................................................... 9.8 PHYSICAL AGENTS ............................................................... 9.11 Thermal Environment ............................................................. 9.11 Electrical Hazards .................................................................. 9.13 Mechanical Energies ............................................................... 9.13 Electromagnetic Radiation ..................................................... 9.15 Ergonomics ............................................................................. 9.17 INTRODUCTION HIS CHAPTER INTRODUCES the field of environmental Thealth as it pertains to the indoor environment of buildings, briefly covering problem identification and control and regulations.
The role of HVAC designers in delivering clean, appropriately con-ditioned air and removing contaminants is vital, both in industrial and nonindustrial environments. Some knowledge of indoor envi-ronmental health will assist in that delivery. In many cases, archi-tectural, structural, cleaning, maintenance, materials use, and other activities that affect the environment are outside the control of that individual. Nevertheless, whenever possible, the HVAC designer should encourage features and decisions that create a healthy build-ing environment.
TERMINOLOGY The most clearly defined area of indoor environmental health is occupational health, particularly as it pertains to workplace air contaminants. Poisoning incidents as well as human and animal lab-oratory studies have generated reasonable consensus on safe and unsafe workplace exposures to about one thousand chemicals and dusts. Consequently, many countries regulate exposures of workers.
However, contaminant concentrations meeting occupational health criteria usually exceed levels found acceptable to occupants in non-industrial spaces such as offices, schools, and residences, where exposures are often longer in duration and may involve mixtures of many contaminants and a less robust population (NAS 1981).
Operational definitions of health, disease, and discomfort are controversial (Cain et al. 1995). The World Health Organization (WHO) defines health as “a state of complete physical, mental, and social well-being and not merely the absence of disease or disabil-ity.” Last (1983) defines health as “a state characterized by anatomic integrity, ability to perform personally valued family, work, and social roles; ability to deal with physical, biologic, and social stress; a feeling of well-being; and freedom from the risk of disease and untimely death.” Higgins (1983) defines an adverse health effect as a biological change that reduces the level of well-being or functional capacity. These definitions indicate that good health is a function of freedom from active ill health or disease (i.e., short- and long-term disability or impairment, freedom from risk of disease in the future resulting from current exposures, and current subjective well-being).
Definitions of comfort are also varying. Traditionally comfort refers to immediate satisfaction. It encompasses perception of the environment (hot, cold, noisy, etc.) and a value rating of affective implications (too hot, too cold, etc.). Rohles et al. (1989) noted that acceptability may represent a more useful concept as it allows pro-gression toward a concrete goal. This serves as the foundation of a number of standards including thermal comfort, acoustics, etc. Nev-ertheless, acceptability may change over time (secular drift) as expectations change.
SCOPE This chapter covers indoor environmental health issues such as effects of air contaminants, thermal comfort, acoustics, exposure to radiation, ergonomics, and electrical safety. It presents information on the mechanisms by which the environment affects the human body, gives guidance on recognizing problems, and discusses stan-dards and guidelines that provide protection. Since the number of air contaminants is very large, the introductory material on types, their characteristics, typical levels, and measurement methods is given in Chapter 12. Chapter 13 covers odors.
This chapter also covers indoor air quality (IAQ). The field of indoor air quality is distinct from that of occupational health in sev-eral ways, including locations of concern, which are mainly non-industrial and include offices, schools, public and commercial buildings, and residences; existence of both comfort and health issues; presence of multiple contaminants at low concentrations rather than single ones at high concentration; and impacts of non-contaminant-related issues such as thermal comfort.
More is known about industrial exposures to air contaminants than is known about residential and light commercial exposures.
Therefore, most of this chapter focusses on industrial environments.
DESCRIPTIONS OF SELECTED HEALTH SCIENCES Normal interactions of the human body with its surrounding environment occur in predictable fashions. Light, heat, cold, and sound at extremes of the exposure range can result in such diseases as frostbite, burns, and noise-induced hearing loss. Some transitions between normal and disease states are more difficult to delineate.
Pain from bright light, erythema from heat, and nausea from vibra-tion represent reversible effects but are interpreted by health profes-sionals as abnormal. Study of these effects includes a number of scientific disciplines.
The preparation of this chapter is assigned to the Environmental Health Committee.
9.2 2001 ASHRAE Fundamentals Handbook (SI) EPIDEMIOLOGY AND BIOSTATISTICS Epidemiology is the study of distributions and determinants of disease. It represents the application of quantitative methods to evaluate diseases or conditions of interest. The subjects may be humans, animals, or even buildings. Epidemiology is traditionally subdivided into observational and analytical components. It may be primarily descriptive, or it may attempt to identify causal associa-tions. Some classical criteria for determining causal relationships in epidemiology are consistency, temporality, plausibility, specificity, strength of the association, and dose-response relationships.
Observational studies are generally performed by defining some group of interest because of a specific exposure or risk factor. A control group is selected on the basis of similar criteria, but without the factor of interest. Observations conducted at one point in time are considered cross-sectional studies. On the other hand, a group may be defined by some criteria at a specific time, for example, all employees who worked in a certain building for at least one month in 1982. They may then be followed over time, leading to an obser-vational cohort study.
Analytical studies may be either experimental or case-control studies. In experimental studies, individuals are selectively exposed to specific agents or conditions. Such studies are generally per-formed with the consent of the participants, unless the conditions are part of their usual working conditions and known to be harmless.
Sometimes exposures cannot be controlled on an individual basis, and the intervention must be applied to entire groups. Control groups must be observed in parallel. Case-control studies are con-ducted by identifying individuals with the condition of interest and comparing risk factors between these people and individuals with-out that condition.
All factors of interest must be measured in an unbiased fashion to avoid a subconscious influence of the investigator. The method of measurement should be repeatable and the technique meaningful.
Statistical methods for data analysis follow standard procedures.
Tests of hypotheses are performed ata specific probability level. They must have adequate power; that is, if a sample size or a measurement difference between the factors or groups is too small, a statistically significant association may not be found even if one is present.
Results obtained in a specific situation (i.e., in a sample of exposed individuals) may be generalized to others only if they share the same characteristics. For example, it may not be legitimate to assume that all individuals have the same tolerance of thermal con-ditions irrespective of their heritage. Therefore, the results of studies and groups must be evaluated as they apply to a specific situation.
TOXICOLOGY Toxicology is the study of the influence of poisons on health.
Most researchers hold that essentially all substances may function as poisons and that only smaller doses prevent them from becoming harmful. Of fundamental importance is defining which component of the chemical structure predicts the harmful effect. The second issue is defining the dose-response relationships. The definition of dose may refer to delivered dose (exposure that is presented to the lungs) or absorbed dose (the dose that is actually absorbed through the lungs into the body and available for metabolism). Measures of exposure may be quite distinct from measures of effect because of internal dose modifiers (e.g., the delayed metabolism of some poi-sons because of lack of enzymes to degrade them). In addition, the mathematical characteristics of a dose may vary, depending on whether a peak dose, a geometric or arithmetic mean dose, or an integral under the dose curve is used.
Because humans often cannot be exposed in experimental con-ditions, most toxicological literature is based on animal studies.
Recent studies suggest that it is not easy to extrapolate between dose level effects from animals to man. Isolated animal systems, such as homogenized rat livers, purified enzyme systems, or other isolated living tissues, may be used to study the impact of chemicals.
MOLECULAR BIOLOGY Molecular biology is the branch of science that studies the chem-ical and physical structure of biological macromolecules. (DNA, proteins, carbohydrates, etc.). It is interested in processes on a molecular level, identifying actual mechanisms of effect or toxicity, rather on the level of cells.
CELLULAR BIOLOGY Cellular biology is the branch of science that studies cellular organelles, activities, and processes. Little indoor air quality related research has been done at this mechanistic level, but it offers the final evidence in postulated cause-and-effect relationships.
GENETICS Genetics is a branch of science that examines heredity and vari-ation among organisms at population, individual, and chromosomal levels. Newer studies in genetics appear to indicate that some indi-viduals are more susceptible to or are at greater risk from certain exposures than the rest of the population. This susceptibility would explain why not all individuals react the same way to the same exposure or lifestyle.
INDUSTRIAL HYGIENE Industrial hygiene is the science of anticipating, recognizing, evaluating, and controlling workplace conditions that may cause worker illness or injury. Important aspects of industrial hygiene included under these principles are (1) identification of toxic chemicals; (2) evaluation of the importance of the physical state and size of airborne particle to absorption by the lungs; (3) evalu-ation of the importance of airborne particle size to absorption by the lungs and the physical state of individuals; (4) evaluation of the importance of skin absorption and ingestion to exposure and absorption; (5) identification of chemicals to be collected and ana-lyzed; (6) determination of methods for collection of air samples; (7) identification of analytic methods to be used or collaboration with a chemist to develop methods to be used; (8) evaluation of results of measurements; (9) identification of physical stressors, including noise, heat stress, ionizing radiation, nonionizing radia-tion, ergonomics, and illumination; and (10) development of control measures. In addition to examining the environment, interpreting collected data, and implementing control measures, the industrial hygienist has responsibilities related to creating regulatory stan-dards for the work environment, preparing programs to comply with regulations, and collaborating in epidemiologic studies to document exposures and potential exposures to help determine occupation-related illness.
Hazard Recognition Occupational hazards generally fall into one of four classes of environmental stressors: chemical, biological, physical, and ergonomic.
Chemical Hazards. Airborne chemical hazards exist as con-centrations of mists, vapors, gases, fumes, or solids. Some are toxic through inhalation, some can irritate the skin on contact, some can be toxic by absorption through the skin or through ingestion, and some are corrosive to living tissue. The degree of risk from expo-sure to any given substance depends on the nature and potency of the toxic effects and the magnitude and duration of exposure. Air contaminants represent a very important subclass of chemical haz-ards due to their mobility and the ease of exposure through the lungs when they are inhaled. Air contaminants are commonly clas-sified as either particulate or gaseous. Common particulate contam-Indoor Environmental Health 9.3 inants include dusts, fumes, mists, aerosols, and fibers. Gaseous contaminants exist as vapors or gases. More background informa-tion on these is given in Chapter 12. Air contaminants are of con-cern in both industrial and nonindustrial environments. Table 1 summarizes diseases that have been associated with specific aspects of indoor environments, mostly in nonindustrial environ-ments. For these diseases, diagnostic criteria may be used to distin-guish between presence or absence of disease. These diseases come about because of the presence of an exposure, a susceptible host, and a vector of transmission.
Biological Hazards. These include bacteria, viruses, fungi, and other living or nonliving organisms that can cause acute and chronic infections by entering the body either directly or through breaks in the skin. In addition, some biological agents can cause allergic reac-tions or be toxic. Wastes, body parts, etc., of these organisms can also cause illness and allergic reactions.
Physical Hazards. These include excessive levels of ionizing and nonionizing electromagnetic radiation, noise, vibration, illumi-nation, temperature, and force.
Ergonomic Hazards. These include tasks that involve repetitive motions, require excessive force, or must be carried out in awkward postures, all of which can result in damage to muscles, nerves, and joints.
Hazard Evaluation Hazard evaluation determines the sources of potential problems.
Safety and health professionals research, inspect, and analyze how the particular hazard affects worker health. Assessment of such exposures relies on qualitative, semiquantitative, or quantitative ap-proaches. In many situations, air sampling will determine whether a hazardous condition exists. An appropriate sampling strategy must be used to ensure the validity of collected samples, determining worst-case (for compliance) or usual (average) exposures. Air sam-pling can be conducted to determine time-weighted average (TWA) exposures, which would cover an entire work shift; or short-term exposures, which would determine the magnitude of exposures to materials that are acutely hazardous. Samples may be collected for a single substance or a multicomponent mixture. Haz-ard evaluation also characterizes the workplace with respect to potential skin absorption or ingestion hazards. Analysis of bulk material samples and surface wipe samples could determine whether hazardous conditions exist. Physical agent characterization Table 1 Diseases Related to Buildings Disease History and Physical Examination Laboratory Testing Linkage Causes Rhinitis Sinusitis Stuffy/ runny nose, post-nasal drip, pale or erythematous mucosa Anterior and posterior rhino-manometry, acoustic rhino-metry, nasal lavage, biopsy, rhinoscopy, RAST or skin prick testing Immunologic skin prick or RAST testing, bracketed physiology Direct occupational exposures; molds in the workplace; specific occupational factors (laser toners, carbonless copy paper, cleaning agents), secondary occupational exposures; pet (e.g., cat) danders brought from home Asthma Coughing, wheezing, episodic dyspnea, wheezing on examination, chest tightness, temporal pattern at work Spirometry before and after work on Monday, peak expiratory flow diary, methacholine challenge Immunologic: skin prick or RAST testing Physiologic: related to work See rhinitis/sinusitis Hypersensitivity pneumonitis Cough, dyspnea, myalgia, weakness, rales, clubbing, feverishness DLCO, FVC, TLC, CXR, lung biopsy Immunologic: IgG ab to agents present, challenge testing Physiologic (in acute forms): spirometry, DLCO Molds, moisture Organic dust toxic syndrome Cough, dyspnea, chest tightness, feverishness DLCO, TLC Temporal pattern related to work Gram-negative bacteria Contact dermatitis (allergic) Dry skin, itching, scaling skin Scaling rash, eczema biopsy Patch testing Molds, carbonless copy paper, laser toners Contact urticaria Hives Inspection biopsy Provocative testing Office products (carbonless copy paper) Eye irritation Eye itching, irritation, dryness Tear-film break-up time, conjunctival staining (fluorescein) Temporal pattern Low relative humidity, volatile organic compounds (allergic conjunctivitis), particulates Nasal irritation Stuffy, congested nose, rhinitis Acoustic rhinometry, posterior and anterior rhinomanometry, nasal lavage, nasal biopsy Temporal pattern Low relative humidity, volatile organic compounds (allergic conjunctivitis), particulates Central nervous system symptoms Headache, fatigue, irritability, difficulty concentrating Neuropsychological testing Temporal pattern (epidemiology) Volatile organic compounds, noise, lighting, work stress, carbon monoxide, cytokines from bioaerosol exposure Legionnaires’ disease Pneumonic illness History, Legionella culture from biopsy fluids 1) Organism isolated from patient and source, 2) immunologic watch Aerosols from contaminated water sources, shower heads, water faucet aerators, humidifiers at home and at work, potable water sources (hot water heaters, etc.) (1) 10% decrement in FEV1 across workday, (2) peak flow changes suggestive of work relatedness, and (3) methacholine reactivity resolving after six weeks away from exposure RAST = radio allergen sorbent test DLCO = single breath carbon monoxide diffusing capacity FVC = forced vital capacity TLC = total lung capacity CXR = chest X-ray IgG = class G immune globulins FEV1 = forced expiratory volume in the first second 9.4 2001 ASHRAE Fundamentals Handbook (SI) may require direct-reading sampling methods. After collection and analysis, the industrial hygienist must interpret the results and determine appropriate control strategies.
Hazard Control The principles for controlling the occupational environment are substitution, isolation, ventilation, and air cleaning. Not all of these principles may be applied to all types of hazards, but all hazards can be controlled by using one of these principles. Engineering controls, work practices controls, administrative controls, and per-sonal protective equipment are used to apply these principles.
Source removal or substitution are customarily the most effective measure, but are not always feasible. Engineering controls such as ventilation and air cleaning may be effective for a range of haz-ards, but usually consume energy. Local exhaust ventilation is more effective for controlling point-source contaminants than is general ventilation. A building HVAC system is an example of a general ventilation system.
AIR CONTAMINANTS Many of the same air contaminants cause problems in industrial and nonindustrial indoor environments. These include nonbiologi-cal particles (synthetic vitreous fibers, combustion nuclei, nuisance dust, and others); bioaerosols; and gases and vapors that may be generated due to industrial processes, by building materials, fur-nishings, and equipment, by occupants and their activities in a space, or brought in from the outdoors. In industrial environments, contaminants are usually known from the type of process, and expo-sures may be determined relatively easily by air sampling. Airborne contaminants in nonindustrial environments may (1) occur from emissions and/or shedding of building materials and systems, (2) originate in outside air, and/or (3) be from building operating and maintenance programs and procedures and conditions that may fos-ter growth of biological organisms. In general, in nonindustrial environments, there are many more contaminants that may contrib-ute to problems, they are more difficult to identify, and they are usu-ally present in much smaller concentrations. More information on contaminant types, characteristics, typical levels, and measurement methods can be found in Chapter 12.
PARTICULATE MATTER Particulate matter includes airborne solid or liquid particles. Typ-ical examples of particulates include dust, smoke, fumes, and mists.
Dusts range in size from 0.1 to 25 µm, smoke particulate is typically around 0.25 µm, and fumes are usually less than 0.1 µm in diameter (Zenz 1988). Bioaerosols and particles produced from abrasions are usually smaller than 1.0 µm.
Units of Measurement The quantity of particulate matter in the air is frequently reported as the mass or the particle count in a given volume of air. Mass units are milligrams per cubic metre of air sampled (mg/m3) or micro-grams per cubic metre of air sampled (µg/ m3) . 1 mg /m3 = 1000 µg/m3. Mass units are widely used in industrial environments as these units are used to express occupational exposure limits.
Particle counts are usually quoted for volumes of 1 cubic foot, 1 litre, or 1 cubic metre and are specified for a given range of particle diameter. Particle count measurements are useful in less contami-nated environments such as office buildings and industrial clean rooms.
Particles in Industrial Environments Particulates found in the work environment are generated as a result of work-related activities (i.e., adding batch ingredients for a manufacturing process, applying asphalt in a roofing operation, or drilling an ore deposit in preparation for blasting). The engineer must recognize sources of particulate generation in order to appro-priately address exposure concerns.
The engineer or industrial hygienist determining worker expo-sure should assess particulate release by the activity, local air move-ment caused by makeup air and exhaust, and worker procedures for a complete evaluation (Burton 2000).
Health Effects of Exposure. The health effects of airborne particulates depend on several factors that include particle dimen-sion, durability, dose, and toxicity of the materials in the particle.
A particulate must first be inhalable to be potentially hazardous.
Respirable particulates vary in size from < 1 to 10 µm (Alpaugh and Hogan 1988) depending on the source of the particulate.
Methods for measuring particulates are discussed in Chapter 12.
Durability, or how long the particle can live in the biological sys-tem before it dissolves, can determine relative toxicity. Lastly, the dose or the amount of exposure encountered by the worker must be considered. In some instances, very small exposures can cause adverse health effects (hazardous exposures) and in others, seem-ingly large exposures may not cause any adverse health effects (nuisance exposures).
Safety and health professionals are primarily concerned with particles smaller than 2 µm, since this is the range of sizes most likely to be retained in the lungs (Morrow 1964). Particles larger than 8 to 10 µm in aerodynamic diameter are primarily separated and retained by the upper respiratory tract. Intermediate sizes are deposited mainly in the conducting airways of the lungs, from which they are rapidly cleared and swallowed or coughed out.
About 50% or less of the particles in inhaled air settle in the respi-ratory tract. Submicrometre particles penetrate deeper into the lungs, but many do not deposit and are exhaled.
Dusts Dusts are coarse solid particles generated by handling, crushing, or grinding. Their size range is typically 1 to 100 µm. They may become airborne during the generation process or through handling.
Any industrial process that produces dust fine enough to remain in the air long enough to be inhaled or ingested (size about 10 µm) should be regarded as hazardous until proven otherwise.
Fibers are solid particles whose length is several times greater than their diameter, such as asbestos, man-made mineral fibers, syn-thetic vitreous fibers, and refractory ceramic fibers.
Mechanisms of Health Effects. A disease associated with the inhalation of particulates in industrial settings is pneumoconiosis, a fibrous hardening of the lungs caused by the irritation created from the inhalation of dust. The most commonly known pneumoconioses are asbestosis, silicosis, and coal worker’s pneumoconiosis.
Asbestosis results from the inhalation of asbestos fibers (chryso-tile, crocidolite, amosite, actinolite, anthophyllite, and tremolite) found in the work environment. The onset of symptomatic asbestosis is uncommon under exposures encountered in the last 45 years before at least 20 to 30 years of exposure (Selikoff et al. 1965, Smith 1955). The asbestos fibers cause fibrosis (scarring) of the lung tissue, which clinically manifests itself as dyspnea (shortness of breath) and a nonproductive, irritating cough. Asbestos fiber is both dimension-ally respirable as well as durable in the respiratory system.
Silicosis is probably the most common of all industrial occupa-tional diseases of the lung. The hazard is created by inhalation of silica dust. The worker with silicosis usually is asymptomatic, and even the early stages of massive fibrosis are not associated with signs and symptoms (Leathart 1972). It is not considered a problem in nonindustrial indoor environments.
Coal worker’s pneumoconiosis (CWP) results from the inhala-tion of dust generated in coal mining operations. The dust is com-posed of a combination of carbon and varying percentages of silica (usually <10%) (Alpaugh and Hogan 1988). Due to the confined underground work environment, exposures have the potential to be Indoor Environmental Health 9.5 very high at times, thus creating very high doses. Data meanwhile show that workers may develop CWP at exposure below the current dust standard of 1 mg/m3.
Exposure Control Strategies. Particulate or dust control strate-gies include source elimination or enclosure, local exhaust, general ventilation, wetting, filtration, and the use of personal protective devices such as respirators.
The most effective means of controlling exposures to a particu-late is to totally eliminate it from the work environment. The best dust-control method is a total enclosure of the dust-producing pro-cess. A negative pressure is maintained inside the entire enclosure by exhaust ventilation (Alpaugh and Hogan 1988). This control strategy is typically found in manufacturing operations.
Local exhaust ventilation as an exposure control strategy is most frequently used where particulate is generated either at large vol-umes or with high velocities (i.e., lathe and grinding operations). In this situation, high-velocity air movement captures the particulate and removes it from the work environment. A number of recent studies show that push/pull methodology enhances the capture efficiency, but requires care in not “pushing” contaminants into the work environment. Counter airflow situations in source capture applications should be avoided.
General ventilation control of the work environment is defined as a dilution approach to reducing exposures. This type of ventilation is used when particulate sources are numerous and widely distributed over a large area. In this control strategy, the work environment is exhausted outside and resupplied with fresh air, thus diluting the work environment. Unfortunately this strategy is the least effective means of control and very costly because conditioned (warm or cold) air is exhausted and nonconditioned air is introduced.
Recirculation of indoor air through filters can be an effective method of reducing indoor particle concentrations. Filtration costs are often lower then costs of general ventilation.
The least desirable strategy used to control exposures is the use of personal protective equipment—a respirator. Respirators are appropriate as a primary control during intermittent maintenance or cleaning activities when fixed engineering (local or general ventila-tion) controls may not be feasible. Respirators can also be used as a supplement to good engineering and work practice controls to increase employee protection and comfort (Alpaugh and Hogan 1988).
Exposure Standards and Criteria. In the United States, the Occupational Safety and Health Administration (OSHA) has established Permissible Exposure Limits (PELs), which are pub-lished in the Code of Federal Regulations (CFR 1989a,b) under the authority of the Department of Labor. Table 2 lists PELs for several particulates commonly encountered in the workplace.
Synthetic Vitreous Fibers Exposures and Exposure Sources. A fiber can be defined as a slender, elongated structure with substantially parallel sides. These parameters distinguish this form of particulate from a dust, which is more spherical. Synthetic vitreous fibers (SVFs) comprise a large number of important manufactured products, such as textile fibers; insulation and ceiling tile wool, including glass fibers, slag, and rock wool fibers; refractory ceramic fibers; and certain specialty glass fibers.
Exposures to SVFs primarily occur during manufacture, fabrica-tion and installation, and demolition. Simultaneous exposures to other dusts (asbestos during manufacture, demolition products and bioaerosols during demolition) may be important as well. Facilities generally manufacture only one form. Generally, only spun glass and refractory ceramic fibers are in the respirable range. Manufac-turing operations are most easily designed to assure a clean work environment, while product application operations are more diffi-cult to control. Data on exposure likely to occur in buildings show that background levels are almost uniformly below 0.0001 fibers/cm3.
Health Effects of Exposure. The possible effects of SVFs on health include the following: Cancer. Respirable SVFs are considered to have the potential to cause carcinogenic and noncarcinogenic health effects. Although implantation studies have suggested the potential for carcinogene-sis, this route of exposure is generally not pertinent for humans.
Therefore, although SVFs are often classified as potential human carcinogens by regulatory and professional agencies and organiza-tions, reviews of epidemiology studies generally fail to find con-vincing evidence that they are associated with excess rates of human cancer. Some mortality studies have identified mild excesses of res-piratory cancer. These have been attributed to concurrent asbestos exposure and to smoking. Only refractory ceramic fibers are cur-rently considered likely to represent true human carcinogens, although other very hard fibers are likely to have similar effects.
Nonmalignant respiratory disease. Cross-sectional surveys have suggested that few measurable adverse health effects are attributable to SVFs alone. The strongest evidence suggests that SVFs may exacerbate smoking-induced obstructive lung disease; some authors consider fiberglass, no different than any other dust, to cause excess rates of chest symptoms.
Dermatitis. SVFs may cause an irritant contact dermatitis through embedding in the skin or conjunctivae with local inflamma-tion. Resin binders sometimes used to tie fibers together have, on rare occasions, been associated with allergic contact dermatitis.
Exposure Control Strategies. As with other particulates, SVF control strategies include source exclusion or enclosure, local exhaust, and the use of personal protective devices such as respira-tors. In indoor environments, SVFs may be identified in surface wipe samples. Appropriate intervention strategies focus on source control.
Exposure Standards and Criteria. At present, SVFs are regulated by OSHA as a “nuisance dust” with an 8-hour time-weighted average of 15 mg/m3 for total dust and 5 mg/m3 for respirable dust.
Combustion Nuclei Exposures and Exposure Sources. Combustion nuclei can be defined as the particulate products of the combustion process. Com-bustionproductsfromamaterialincludewatervapor,carbon dioxide, heat, oxides of carbon and nitrogen, and particulates known as com-bustion nuclei. In many situations combustion nuclei can be hazard-ous. They may contain potential carcinogens such as polycyclic aromatic hydrocarbons (PAHs).
Polycyclic aromatic compounds (PACs) are the nitrogen-, sulfur-, and oxygen-heterocyclic analogs of PAH and other related PAH deriv-atives. Depending on their relative molecular mass and vapor pres-sure, PACs are distributed between vapor and particulate phases. In Table 2 OSHA Permissible Exposure Limits (PELs) for Particulates (29 CFR 1910.1000, 29 CFR 1926.1101) Substance CAS # PEL Cadmium 7440-43-9 0.05 mg/m3 Manganese fume 7439-96-5 1.0 mg/m3 Plaster of Paris Nuisance 10.0 mg/m3 Emery Nuisance 10.0 mg/m3 Grain dust Nuisance 10.0 mg/m3 Crystalline silica (as quartz) 14808-60-7 0.1 mg/m3 Asbestos 1332-21-4 0.1 fibers/cm3 Total dust Nuisance 15.0 mg/m3 Respirable dust Nuisance 5.0 mg/m3 9.6 2001 ASHRAE Fundamentals Handbook (SI) general, combustion particulates are smaller than dusts generated by mechanical means.
Typical sources of combustion nuclei are tobacco smoke (ciga-rettes, pipes, and cigars), fossil-fuel-based heating devices such as unvented space heaters and gas ranges, and flue gas from improp-erly vented gas- or oil-fired furnaces and wood-burning fireplaces or stoves. Infiltration of outdoor combustion contaminants can also be a significant source of such contaminants in indoor air.
Combustion nuclei are thus important in both industrial and non-industrial settings.
Exposure Standards and Criteria. OSHA has established expo-sure limits for several of the carcinogens categorized as combustion nuclei (benzo(a)pyrene, cadmium, nickel, benzene, n-nitrosodimeth-ylamine). These limits are established for industrial work environ-ments and are not directly applicable to indoor air situations.
Underlying atherosclerotic heart disease in individuals may be exac-erbated by carbon monoxide (CO) exposures.
Exposure Control Strategies. Exposure control strategies for combustion nuclei are in many ways similar to those applied for other particles. For combustion nuclei derived from heating spaces, air contamination can be avoided by proper installation and venti-lation of equipment to ensure that these contaminants cannot enter the work or personal environment. Proper equipment maintenance is also essential to minimize exposures to combustion nuclei.
Changing makeup air availability, through the addition of enclo-sures, may be equally important.
Particles in Nonindustrial Environments In the nonindustrial indoor environment, the indoor aerosol will be affected greatly by the outdoor particle environment. Indoor par-ticulate sources may include cleaning, resuspension of particles from carpets and other surfaces, construction and renovation debris, paper dust, deteriorated insulation, office equipment, and combus-tion processes including cooking stoves and fires and environmen-tal tobacco smoke. In general, source control is the preferred method. If a dust problem is identified, characterization of the nature of the dust will allow the development of an appropriate intervention strategy.
Although asbestos is encountered in insulation in many build-ings, it generally does not represent a respiratory hazard except to individuals who actively disturb it in the course of maintenance and construction. School custodians, therefore, are recognized to be at risk for asbestos-related changes. Anderson et al. (1991) and Lilienfeld (1991) raise questions about risk to teachers.
The combustion nuclei of environmental tobacco smoke (ETS) consists of exhaled mainstream smoke from the smoker and side-stream smoke that is emitted from the smoldering tobacco. ETS consists of between 70 and 90% sidestream smoke and has a some-what different chemical composition from mainstream smoke.
More than 4700 compounds have been identified in laboratory-based studies, including known human toxic and carcinogenic compounds such as carbon monoxide, ammonia, formaldehyde, nicotine, tobacco-specific nitrosamines, benzo(a)pyrene, ben-zene, cadmium, nickel, and aromatic amines. Many of these toxic constituents are more concentrated in sidestream than in main-stream smoke (Glantz and Parmley 1991). In studies conducted in residences and office buildings with tobacco smoking, ETS was a substantial source of many gas and particulate PACs (Offermann et al. 1991).
Health Effects of Exposure. The health effects of exposure to combustion nuclei depend on many factors, including concentra-tion, toxicity, and individual susceptibility or sensitivity to the particular substance. Polycyclic aromatic compounds generated by combustion processes include many PAHs and nitro-PAHs that have been shown to be carcinogenic in animals (NAS 1983). Other PACs are biologically active as tumor promoters and/or cocar-cinogens. Mumford et al. (l987) reported high exposures to PAH and aza-arenes for a population in China with very high lung can-cer rates.
ETS has been shown to be causally associated with lung cancer in adults (NRC 1986, DHHS 1986), and respiratory infections, asthma exacerbations, middle ear effusion (NRC 1986, DHHS 1986), and low birth mass in children (Martin and Bracken 1986).
The U.S. Environmental Protection Agency classifies ETS as a known human carcinogen (EPA 1992). Health effects can also include headache and irritation. ETS is also a cause of sensory irri-tation and annoyance (odors and eye irritation).
Control of ETS is somewhat different in that it has been done primarily through regulatory mandates controlling the practice of tobacco smoking. Most states in the United States have passed laws to control tobacco smoking in public places such as restaurants and workplaces, and airlines have prohibited tobacco smoking on flights lasting 6 h or less. Where tobacco smoking is allowed, appropriate local and general ventilation can be used for control. OSHA has proposed that tobacco smoke in indoor environments be controlled through the use of separately venti-lated and exhausted smoking lounges. These lounges are kept under negative pressure relative to all adjacent and communicat-ing indoor spaces.
BIOAEROSOLS Aerobiology is the study of airborne microorganisms or other biologically produced particles and the effects of these aerosols on other living organisms (people, animals, vegetation, etc.). Bioaero-sols are airborne microbiological particulate matter derived from viruses, bacteria, fungi, protozoa, algae, mites, plants, insects, and their cellular or cell mass components. Bioaerosols are present in both indoor and outdoor environments. For the indoor environment, locations that provide appropriate temperature and humidity condi-tions for reproduction and a food source to support growth may become problematic.
Sources Floors and floor coverings in hospitals can be reservoirs for organ-isms that are subsequently resuspended into the air. Carpet cleaning may even promote resuspension (Cox 1987). Some viruses may per-sist up to 8 weeks on nonporous surfaces (Mbithi et al. 1991).
Nonpotable water is a well-known source of infective agents, even by aerosolization. Baylor et al. (1977) demonstrated the sequestering of small particles by foam and their subsequent dis-persal through a bubble burst phenomenon. Such dispersal may take place in surf, river sprays, or man-made sources such as whirlpools.
People are an important source of bacteria and viruses in indoor air. Contagious diseases occur when living organisms overcome the defense of the host and establish an infection in the host that may in turn infect another human. Infected humans are the pri-mary sources for contagious disease and primary disseminators as well. Virulent agents can also be released from human skin when disease produces skin lesions, or dispersed from respiratory tract infection during coughing, sneezing, or talking. Other means for release directly from infected humans include sprays of saliva and other respiratory secretions during dental and respiratory therapy procedures. Blood sprays that occur during dental and surgical procedures are of potential concern for aerosol transmission of bloodborne diseases, including HIV and hepatitis viruses. Large droplets can transmit infectious particles to those close to the dis-seminator, while smaller particles can remain airborne for short or very long distances (Moser et al. 1979).
Both the physical and biological properties of the bioaerosols need to be understood. For a microorganism to cause a building-related illness, it must be transported in sufficient dose to the breathing zone of a susceptible occupant. Airborne infectious par-Indoor Environmental Health 9.7 ticles behave physically in the same way as any other aerosol-con-taining particles with similar size, density, electrostatic charge, etc. The major difference with bioaerosols is that they must remain viable to cause infection, although nonviable particles may pro-mote an immunological response. An organism that does not remain virulent in the airborne state cannot cause infection, regardless of how many units of organisms are deposited in the human respiratory tract. Virulence depends on such factors as rel-ative humidity, temperature, oxygen, pollutants, ozone, and ultra-violet light (Burge 1995). The effect of any one factor on survival and virulence can be different for different organisms.
Although microorganisms are normally present in indoor envi-ronments, the presence of abundant moisture and nutrients in inte-rior niches amplifies the growth of some microbial agents. Thus certain types of humidifiers, water spray systems, and wet porous surfaces can be reservoirs and sites for growth of fungi, bacteria, protozoa, algae, or even nematodes (Strindehag et al. 1988, Arnow et al. 1978, Morey et al. 1986, Morey and Jenkins 1989). Excessive air moisture (Burge 1995) and floods (Hodgson et al. 1985) may cause the proliferation of microorganisms indoors. The turbulence associated with the start-up of air-handling unit plenums may also elevate concentrations of bacteria and fungi in occupied spaces (Yoshizawa et al. 1987).
Health Effects Exposure to airborne fungal spores, hyphal fragments, or metabolites can cause a variety of respiratory diseases. These range from allergic diseases including allergic rhinitis, asthma, and hypersensitivity pneumonitis to infectious diseases such as histoplasmosis, blastomycosis, and aspergillosis. In addition, acute toxicosis and cancer have been ascribed to respiratory expo-sure to mycotoxins (Levetin 1995). A large body of literature sup-ports an association between moisture indicators in the home and symptoms of coughing and wheezing (Spengler et al. 1992, Miller and Day 1997).
The presence of microorganisms in indoor environments may cause infective and/or allergic building-related illnesses (Morey and Feeley 1988, Burge 1989). Some microorganisms under certain conditions may produce volatile chemicals (Hyppel 1984) that are malodorous.
The diseases produced by the Legionella genus of bacteria are collectively called legionelloses. Presently more than 34 species of the Legionella family have been identified, of which over 20 have been isolated from both environmental and clinical sources. The diseases produced by Legionella pneumophila include the pneumo-nia form, Legionnaires’ disease, and the flu-like form, Pontiac fever. L. pneumophila serogroup 1 is the most frequently isolated from nature and most frequently associated with disease. It has also been suggested that the host relationship affects the virulence of Legionella spp.
The fungal genus Aspergillus is widely distributed and is com-mon in the soil and on decaying vegetation, dust, and other organic debris (Levitin 1995). The small spores are buoyant and remain airborne for long periods (Streifel et al. 1989). Most opportunistic fungal infections are caused by Aspergillus fumigatus. The litera-ture on aspergillosis is extensive, particularly for hospitals, and in many cases the environmental source of the infection has been identified.
Histoplasmosis, an infective illness caused by the fungus Histo-plasma capsulatum, has occurred (rarely) as a building-related ill-ness among individuals involved in the removal of bat or bird droppings in abandoned buildings (Bartlett et al. 1982) and among chicken coop cleaners. Presumably asexual spores (conidia) from this fungus were inhaled by workers who removed the droppings without adequate respiratory protection.
Outbreaks of infective illness in the indoor air may be caused by other types of microorganisms, such as viruses. For example, most of the passengers in an airline cabin developed influenza following exposure to one acutely ill person (Moser et al. 1979). In this case, the plane had been parked on a runway for several hours with the ventilation system turned off.
Allergic respiratory illness may develop due to inhalation of particulates containing microorganisms or their components, such as spores, enzymes, mite excreta, and cell wall fragments. Numer-ous cases of allergic respiratory illness (humidifier fever, hyper-sensitivity pneumonitis) report affected people manifesting acute symptoms such as malaise, fever, chills, shortness of breath, and coughing (Edwards 1980, Morey 1988). In buildings, these ill-nesses may occur as a response to microbiological contaminants originating from HVAC system components, such as humidifiers and water spray systems, or other mechanical components that have been damaged by chronic water exposure (Hodgson et al.
1985, 1987). Affected individuals usually experience relief only after having left the building for an extended period in contrast to occupants with sick building syndrome, where relief is relatively rapid.
Crandall and Sieber (1996) showed that 47 of 104 problem build-ings evaluated had water damage in occupied building areas. Other studies suggest that microorganisms in indoor air are important (Burge et al. 1987, Brundage et al. 1988, Burge 1995).
Guidelines At present, numerical guidelines for bioaerosol exposure in indoor environments are not available for the following reasons (Morey 1990): • Incomplete data on concentrations and types of microbial particulates indoors, especially as affected by geographical, seasonal, and type-of-building parameters • Absence of data relating bioaerosol exposure to building-related illness • Enormous variability in kinds of microbial particulates including viable cells, dead spores, toxins, antigens, and viruses • Large variation in human susceptibility to microbial particulates, making estimates of health risk difficult However, even in the absence of numerical guidelines, bioaero-sol sampling data can be interpreted based on such factors as • Rank order assessment of the kinds (genera/species) of microbial agents present in complainant and control locations (ACGIH 1989) • Medical or laboratory evidence that a building-related illness is caused by a microorganism (ACGIH 1989) • Indoor/outdoor concentration ratios for various microbial agents (Morey and Jenkins 1989, ACGIH 1989) For a microorganism to cause a building-related illness, it must be transported in sufficient dose to the breathing zone of a suscep-tible occupant. Thus, the concepts of reservoir, amplifier, and dis-seminator need to be considered in interpreting data. Reservoirs allow microorganisms to survive, amplifiers allow microorgan-isms to proliferate, and disseminators effectively distribute bio-aerosols. Some factors and systems may be all or only one of these. A cooling tower is all three for Legionella; that is, a cooling tower can harbor microorganisms in scale, allow them to prolifer-ate, and generate an aerosol.
GASEOUS CONTAMINANTS This category of indoor contaminants includes both true gases (which have boiling points less than room temperature) and vapors of liquids with boiling points above normal indoor temperatures. It also includes both volatile organic compounds (VOCs) and inor-ganic air contaminants.
9.8 2001 ASHRAE Fundamentals Handbook (SI) Volatile organic compounds include 4- to 16-carbon alkanes, chlorinated hydrocarbons, alcohols, aldehydes, ketones, esters, terpenes, ethers, aromatic hydrocarbons (such as benzene and tolu-ene), and heterocyclic hydrocarbons. Also included are chlorofluo-rocarbons (CFCs) and hydrochloroflurocarbons (HCFCs) used as refrigerants. More information on classifications, characteristics and measurement methods can be found in Chapter 12.
Inorganic gaseous air contaminants include ammonia, nitrogen oxides, ozone, sulfur dioxide, carbon monoxide, and carbon diox-ide. Although the last two contain carbon, they are by tradition regarded as inorganic chemicals.
The most common units of measurement for gaseous contami-nants are parts per million by volume (ppm) and milligrams per cubic metre (mg/m3). For smaller quantities, parts per billion (ppb) and micrograms per cubic metre (µg/m3) are used. The relationship between all of these is explained in Chapter 12.
GASEOUS CONTAMINANTS IN INDUSTRIAL ENVIRONMENTS The Occupational Safety and Health Administration (OSHA) sets Permissible Exposure Limits (PELs), which are the only workplace regulatory standards in the United States. These are published yearly in the Code of Federal Regulations (29 CFR 1900, Part 1900.1000 ff) and intermittently in the Federal Regis-ter. Most of the levels were recommended by the American Con-ference of Governmental Industrial Hygienists (ACGIH) and the American National Standards Institute (ANSI). The health effects on which these standards were based can be found in their publi-cations (ACGIH 1989). ACGIH reviews data on a regular basis and publishes Threshold Limit Values (TLVs) yearly.
The National Institute for Occupational Safety and Health (NIOSH) is charged with researching toxicity problems, and it influences the legally required levels. NIOSH publishes the Reg-istry of Toxic Effects and Chemical Substances (RTECS) as well as numerous Criteria for Recommended Standard for Occupa-tional Exposure to (compound). Some compounds not in the OSHA list are covered by NIOSH literature, and their Recom-mended Exposure Limits (RELs) are sometimes lower than the legal requirements set by OSHA. The NIOSH Pocket Guide to Chemical Hazards (NIOSH 1990) is a condensation of these ref-erences and is convenient for engineering purposes.
The harmful effects of gaseous pollutants on a person depend on both short-term peak concentrations and the time-integrated exposures received by the person. OSHA has defined three peri-ods for concentration averaging and has assigned allowable lev-els that may exist in these categories in workplaces for over 490 compounds, mostly gaseous contaminants. The abbreviations for concentrations for the three averaging periods are AMP = acceptable maximum peak for a short exposure ACC = acceptable ceiling concentration, not to be exceeded dur-ing an 8-h shift, except for periods where AMP applies TWA8 = time-weighted average, not to be exceeded in any 8-h shift of a 40-h week.
In non-OSHA literature, AMP is sometimes called STEL (short-term exposure limit), and TWA8 is sometimes called TLV (threshold limit value.) NIOSH (1990) also lists values for the toxic limit IDLH—immediately dangerous to life and health.
Table 3 lists values of the various exposure limits defined above for a selection of common gaseous industrial contami-nants. Other countries have also issued standards for industrial exposure.
It is of interest to compare standards for industrial and nonin-dustrial environments. A Canadian National Task Force devel-oped guideline criteria for residential indoor environments.
Similarly, the World Health Organization has published indoor air quality guidelines for Europe. Table 4 compares these guide-lines with occupational criteria for selected contaminants.
GASEOUS CONTAMINANTS IN NONINDUSTRIAL ENVIRONMENTS The gaseous contaminants that are of concern in nonindustrial environments are volatile organic compounds and inorganic gases.
Table 3 Characteristics of Selected Gaseous Air Pollutants Pollutant Allowable Concentration, mg/m3 IDLHa AMPa ACCa TWA8a Acetaldehyde 18 000 360 Acetone 4 800 3 200 2 400 Acetonitrile 7 000 105 70 Acrolein 13 0.75 0.25 Acrylonitrile 10 45 Allyl chloride 810 9 3 Ammonia 350 35 38 Benzene 10 000 25 5 Benzyl chloride 50 5 2-Butanone (MEK) 8 850 590 Carbon dioxide 90 000 54 000 9 000 Carbon monoxide 1 650 220 55 Carbon disulfide 1 500 300 90 60 Carbon tetrachloride 1 800 1 200 150 60 Chlorine 75 1.5 3 Chloroform 4 800 9.6 240 Chloroprene 1 440 3.6 90 p-Cresol 1 100 22 Dichlorodifluoromethane 250 000 4 950 Dioxane 720 360 Ethylene dibromide 3 110 271 233 155 Ethylene dichloride 4 100 818 410 205 Ethylene oxide 1 400 135 90 Formaldehyde 124 12 6 4 n-Heptane 17 000 2 000 Hydrogen chloride 140 7 7 Hydrogen cyanide 55 11 Hydrogen fluoride 13 5 2 Hydrogen sulfide 420 70 28 30 Mercury 28 0.1 Methane ASPHYb Methanol 32 500 260 Methyl chloride 59 500 1 783 1 189 Methylene chloride 7 500 3 480 1 740 Nitric acid 250 5 Nitric oxide 120 45 30 Nitrogen dioxide 90 1.8 9 Ozone 20 2 Phenol 380 60 19 Phosgene 8 0.8 0.4 Propane 36 000 Sulfur dioxide 260 13 Sulfuric acid 80 1 Tetrachloroethane 1 050 35 Tetrachloroethylene 3 430 2 060 1 372 686 o-Toluidene 440 22 Toluene 7 600 1 900 1 140 760 Toluene diisocyanate 70 0.14 0.14 1,1,1-Trichloroethane 2 250 45 Trichloroethylene 5 410 1 620 1 080 541 Vinyl chloride monomer 0.014 0.003 Xylene 43 500 870 435 aIDLH, AMP, ACC, and TWA8 are defined in the section on Gaseous Contaminants in Industrial Environments.
bASPHY = Simple asphyxiant; causes breathing problems when concentration reaches about 1/3 atmospheric pressure.
Indoor Environmental Health 9.9 The alternative Indoor Air Quality Procedure specified in ASHRAE Standard 62 sets limits for concentrations of several contaminants in nonindustrial environments and cautions that contaminants whose toxicity is well-known should be kept at or below one-tenth of the threshold limit value (TLV) specified by ACGIH (annual). When outdoor air quantities are reduced, the actual gaseous contaminant concentrations must be measured to ensure that Standard 62 is met.
Health Effects of Volatile Organic Compounds Adverse health effects potentially caused by VOCs in nonindus-trial indoor environments are not well understood, but may include (1) irritant effects, including perception of unpleasant odors, mucous membrane irritation, and exacerbation of asthma; (2) sys-temic effects, such as fatigue and difficulty concentrating; and (3) toxic, chronic effects, such as carcinogenicity (Girman et al. 1989).
The chronic adverse health effects due to VOC exposure are of concern because some VOCs commonly found in indoor air are human (benzene) or animal (chloroform, trichloroethylene, carbon tetrachloride, p-dichlorobenzene) carcinogens. Some other VOCs are also genotoxic. Theoretical risk assessment studies suggest that chronic exposure risk due to VOCs in residential indoor air is greater than that associated with exposure to VOCs in the outdoor air or in drinking water (McCann et al. 1987, Tancrede et al. 1987).
Carbon tetrachloride (CCl4) causes central nervous system (CNS) depression and significant liver and kidney damage. CCl4 has also been shown to be an animal carcinogen and is classified as a potential human carcinogen.
A biological model for acute human response to low levels of VOCs indoors is based on three mechanisms: sensory perception of the environment, weak inflammatory reactions, and environmental stress reaction (Molhave 1991). A growing body of literature sum-marizes measurement techniques for the effects of VOCs on nasal (Koren 1990, Koren et al. 1992, Meggs 1994, Ohm et al. 1992, Mol-have et al. 1993) and ocular (Kjaergard et al. 1991, Kjaergard 1992, Franck et al. 1993) mucosa. It is not well known how different sen-sory receptions to VOCs are combined into perceived comfort and the sensation of air quality. This perception is apparently inter-related to stimulation of the olfactory sense in the nasal cavity, to the gustatory sense on the tongue, and the common chemical sense (Molhave 1991, Cain 1989).
Cometto-Muñiz and Cain (1994a,b) addressed the independent contribution of the trigeminal and olfactory nerves to the detection of airborne chemicals. The sense of smell is experienced through receptors in the nose of the olfactory nerve. Nasal pungency, described as common chemical sensations including prickling, irritation, tingling, freshness, stinging, and burning among others, is experienced through the nonspecialized receptors of the trigem-inal nerve in the face. Odor and pungency thresholds follow differ-ent patterns related to chemical concentration. Odor is often detected at much lower levels. A linear correlation between pun-gency thresholds of homologous series—of alcohols, acetates, ketones, and alkylbenzenes, relatively nonreactive agents—sug-gests that nasal pungency relies on a physicochemical interaction with a susceptible biophase within the cell membrane. It is postu-lated that through this nonspecific mechanism, low, subthreshold levels of a wide variety of VOCs—as found in many polluted indoor environments—can be additive in their sensory impact to produce noticeable sensory irritation.
Table 4 Comparison of Standards Pertinent to Indoor Environments Canadian WHO/Europe NAAQS/EPAf NIOSH REL OSHA ACGIH MAKg Aldehydes Acrolein 0.02 ppma 0.1 ppm 0.25 ppm (15 min) 0.1 ppm 0.3 ppm (15 min) 0.1 ppm 0.3 ppm (15 min) 0.1 ppm 0.2 ppm (15 min) Acetaldehyde 5.0 ppm ALARAb 100 ppm 150 ppm (15 min) 100 ppm 150 ppm (15 min) 50 ppm Formaldehyde 0.1 ppmc 0.081 ppm 0.016 ppm 0.1 ppm (15 min) 0.75 ppm 2 ppm (15 min) 0.3 ppm 0.3 ppm Carbon dioxide 3500 ppm 5000 ppm 30,000 ppm (15 min) 10,000 ppm 30,000 ppm (15 min) 5000 ppm 9000 ppm (15 min) 5000 ppm 9000 ppm (15 min) Carbon monoxide 11 ppm (8 h) 25 ppm (1 h) 8.6 ppm (8 h) 25 ppm (1 h) 51 ppm (30 min) 86 ppm (15 min) 9 ppm (8 h) 35 ppm (1 h) 35 ppm (8 h) 200 ppm (15 min) 35 ppm (8 h) 200 ppm (15 min) 25 ppm (8 h) 30 ppm Nitrogen dioxide 0.05 ppm 0.25 ppm (1 h) 0.08 ppm (24 h) 0.2 ppm (1 h) 0.053 ppm (1yr) 1 ppm (15 min) 3 ppm 5 ppm (15 min) 5 ppm Ozone 0.12 ppm (1 h) no long-term level 0.08 ppm (8 h) 0.1 ppm (1 h) 0.12 ppm (1 h) 0.085 ppm (8 h) 0.1 ppm (15 min) 0.1 ppm (8 h) 0.3 ppm (15 min) 0.05 ppm (8 h) 0.2 ppm (15 min) 0.1 ppm Particulate < 2.5 MMADd 40 µg/m3 (8 h) 100 µg/m3 (1 h) 50 g/m3 (1 yr) 5 mg/m3 (8 h) (respirable dust) 3 mg/m3 (8 h) (no asbestos, <1% crystalline silica) Sulfur dioxide 0.019 ppm 0.38 ppm (5 min) 2 ppm (8 h) 5 ppm (15 min) 2 ppm (8 h) 5 ppm (15 min) 2 ppm (8 h) 5 ppm (15 min) 2 ppm Radon 800 Bq/m3e Relative humidity 30-80% (summer) 30-55% (winter) ( ) Numbers in parentheses represent averaging periods aParts per million (106) bAs low as reasonably achievable cTarget level of 0.05 ppm because of its carcinogenic effects dMass median aerodynamic diameter eMean in normal living areas fU.S. EPA National Ambient Air Quality Standards gGerman Maximale Arbeitsplatz Konzentrationen 9.10 2001 ASHRAE Fundamentals Handbook (SI) Formaldehyde is a very reactive small molecule that requires dif-ferent analytical techniques than those usually employed in VOC assessment. Primary sources include urea-formaldehyde resin-based particle and chipboard products used in indoor spaces. It is frequently encountered in indoor spaces in concentrations between 0.04 ppm, a frequently encountered lower limit of detection, and 0.1 ppm (Liu et al. 1991, Ritchie and Lehnen 1987). Many studies have demonstrated its ability to trigger mucous membrane irritation at levels below the ACGIH TLV, and even at levels below 0.1 ppm.
Standards for Volatile Organic Compounds No standards for exposure to VOCs relevant to nonindustrial indoor environments are in place. NIOSH, OSHA, and the ACGIH have published regulatory standards or recommended limits for industrial occupational exposures (NIOSH 1992, ACGIH annual).
With few exceptions, concentrations observed in nonindustrial indoor environments fall well below (100 to 1000 times lower) these published pollutant-specific occupational standards or recom-mended exposure limits. However, standards for the industrial workplace are higher than would be appropriate for the general pop-ulation, which includes the elderly, children, and people who are more sensitive to VOCs than the average industrial worker.
Total VOC (TVOC) concentration has previously been used as an indicator of the potency of VOCs to cause health effects. This approach is no longer recommended since the toxicities of individ-ual VOCs vary very widely, and concentrations differ depending on the measurement method used (Hodgson 1995). In controlled expo-sure experiments, odors are significant at 3 mg/m3. At 5 mg/m3, objective effects were seen in addition to the subjective irritation.
Exposures for 50 min to 8 mg/m3 of synthetic mixtures of 20 VOCs lead to significant irritation of mucous membranes in the eyes, nose, and throat.
Both OSHA and the ACGIH have set 8 h standards for formal-dehyde as a ceiling level. California issued a residential air quality guideline of 0.1 ppm. In the setting of occupant complaints, the tar-get guideline is 0.05 ppm.
Health Effects of Refrigerants ASHRAE Standard 34 assigns refrigerants to one of two toxic-ity classes (A or B) based on allowable exposure. Fatalities have been reported following acute exposure to fluorocarbon refriger-ants. Inhalation exposures to CFCs can cause cardiotoxicity at chronic, low-level exposures. Some are thought to be cardiac sensi-tizers to epinephrine and put occupants at risk for arrhythmias. Cen-tral nervous system (CNS) depression has been found at very high concentrations along with asphyxia. Proctor and Hughes (1991) found that volunteers exposed to 200 000 ppm of R-12 experienced significant eye irritation and CNS effects. Chronic exposure to 1000 ppm for 8 h per day for up to 17 days caused no subjective symp-toms or changes in pulmonary function.
A significant hazard exists when chlorinated hydrocarbons (R-11, for example) are used in the vicinity of open flame or heated surfaces. Phosgene gas (carbonyl chloride), an extreme irritant to the lungs, and halogen acids may be generated when chlorinated or fluorinated solvents or gases decompose in the presence of heat.
CFC-containing systems may only be serviced by certified tech-nicians. Controls for preventing exposures include selection and use of appropriate fittings and valves and insuring that compressed gas cylinders are secured when in use, in transport, and in storage.
When repairs are made to leaking or defective components in HVAC equipment, adequate dilution ventilation should be provided to the work area. CFCs should never be used in the vicinity of open flame or heated materials due to the potential for the formation of phosgene gas. ASHRAE Standard 15 establishes specific require-ments for designing, installing, operating, and servicing mechanical refrigeration equipment.
Health Effects of Inorganic Gases Carbon monoxide is a chemical asphyxiant. Inhalation of CO causes a throbbing headache brought about because CO has a competitive preference for hemoglobin (about 240 times that of oxygen) and also a shift in the oxygen dissociation curve. Carbon monoxide inhibits oxygen transport in the blood through the for-mation of carboxyhemoglobin and inhibition of cytochrome oxi-dase at the cellular level. Deaths and adverse health effects from overexposures are attributed primarily to motor vehicles. Cobb and Etzel (1991) suggested that CO poisoning at home repre-sented a major preventable disease. Moolenaar et al. (1995) sub-sequently identified similar data and suggested that motor vehicles and home furnaces were primary causes of mortality.
Girman et al. (1996) identified both fatal outcomes and episodes.
Respectively, 35.9% and 30.6% resulted from motor vehicles, 34.8% and 39.9% from appliance combustion, 4.5% and 5.2% from small appliances, 2.2% and 2.3% from camping equipment, 5.6% and 5.0% from fires, 13.4% and 13.3% from grills and hibachis, and the remainder were unknown.
Carbon dioxide can become dangerous not as a toxic agent but as a secondary asphyxiant. When concentrations exceed 35 000 ppm, central breathing receptors are triggered and cause the sensa-tion of shortness of breath. At progressively higher concentrations, central nervous system dysfunction begins due to simple displace-ment of oxygen. Concentrations of CO2 in the nonindustrial envi-ronment (office buildings and schools) are often measured in the range of 400 to 1500 depending on occupant density, ventilation distribution, and amount of outside air supplied to the occupied spaces.
Inhalation of nitric oxide (NO) causes the formation of methe-moglobin, which adversely affects the body by interfering with oxy-gen transport at the cellular level. NO exposures of 3 ppm have been compared to carbon monoxide exposures of 10 to 15 ppm (Case et al. 1979 in EPA 1991).
Nitrogen dioxide (NO2) is a corrosive gas with a pungent odor, the odor threshold of which is reported to be between 0.11 and 0.22 ppm (WHO 1987). NO2 has a low water solubility and there-fore can be inhaled into the deep lung where it causes a delayed inflammatory response. Increased airway resistance has been reported at 1.5 to 2 ppm (Bascom 1996). NO2 is reported to be a potential carcinogen by way of free radical production (Burgess and Crutchfield 1995). At high concentrations, NO2 causes lung damage directly by its oxidant properties and may cause health effects indirectly by increasing host susceptibility to respiratory infections. Health effects from exposures to ambient outdoor con-centrations or in residential situations show inconsistency, espe-cially studies relating to exposures from gas cooking stoves (Samet et al. 1987). Indoor concentrations of NO2 often exceed ambient concentrations due to the presence of strong indoor sources and a trend toward more energy efficient (tighter) homes. Acute toxicity is seldom seen from NO2 produced by unvented indoor combus-tion because of the insufficient quantities of NO2 produced.
Chronic pulmonary effects from exposure to combinations of low-level combustion pollutants are possible, however (Bascom et al.
1996).
Sulfur dioxide (SO2) is a colorless gas with a pungent odor detected at about 0.5 ppm (EPA 1991). Because SO2 is quite solu-ble in water, it can react with moisture in the upper respiratory tract to produce irritant effects on the upper respiratory mucosa. Con-comitant exposure to fine particulates, an individual's depth and rate of breathing, and the presence of preexisting disease can influ-ence the degree of SO2 toxicity.
Ozone is a pulmonary irritant and causes changes in human pulmonary function at concentrations of approximately 0.12 ppm (Bates 1989). Exposure to ozone at 60 to 80 ppb causes inflamma-tion, bronchoconstriction, and increased airway responsiveness.
Indoor Environmental Health 9.11 Inhalation exposures to the gaseous oxides of nitrogen, sulfur (NOx and SO2) and ozone (O3) can and do occur in residential and commercial buildings. These air pollutants are of considerable con-cern due to the potential for acute and chronic respiratory tract health effects in exposed individuals, particularly individuals with preexisting pulmonary disease.
Standards for Inorganic Gases Currently, there are no specific United States government stan-dards relative to nonindustrial occupational exposures to air con-taminants. Occupational exposure criteria are health-based; that is, they consider healthy workers in an environment, and not necessar-ily individuals who may be unusually responsive to the effects of chemical exposures. The EPA National Ambient Air Quality Stan-dards (NAAQS) (see Chapter 12) are also health-based standards designed to protect the general public health from the effects of haz-ardous airborne pollutants. However, there is a debate as to whether these standards truly represent health-based thresholds. Two of the six criteria, ozone and carbon monoxide, involve toxicologically based research for the development of the standards. The criteria (Table 5) are not meant to be health-based guidelines for the evalu-ation of exposures to inorganic gasses in the indoor environment.
Table 5 is included for comparative use and consideration by inves-tigators of the indoor environment with the understanding that these criteria may not be completely protective to all industrial workers.
PHYSICAL AGENTS Physical factors in the indoor environment include thermal con-ditions (temperature, moisture, air velocity, and radiant energy), mechanical energy (noise and vibration), and electromagnetic radi-ation [ionizing (radon) and nonionizing (light, radio-frequency, and extremely low frequency magnetic and electric fields)]. Physical agents can act directly upon building occupants, interact with indoor air quality factors, or affect the way humans respond to the indoor environment. Physical agents, while not categorized as indoor air quality factors, often affect human perception of the qual-ity of indoor air.
THERMAL ENVIRONMENT The thermal environment affects human health in that it affects body temperatures, both internally and externally (of the skin). In the normal, healthy, resting adult, internal or core body temperatures are very stable, with variations seldom exceeding 0.5 K. The internal temperature of a resting adult, measured orally, averages about 37°C; measured rectally, it is about 0.5 K higher (Guyton 1972). The tem-perature of the core is carefully modulated by an elaborate physio-logical control system. In contrast, the temperature of the skin is basically unregulated and can vary from about 31 to 36°C in normal environments and activities and also varies over different parts of the skin, with the greatest variation in the hands and feet.
Range of Healthy Living Conditions The environmental conditions for thermal comfort are those that minimize the effort of the physiological control systems to maintain the internal temperature. The control system regulates the internal body temperature by varying the amount of blood flowing to different skin areas, thus increasing or decreasing heat lost to the environment, by secreting and evaporating sweat from the skin in warm or hot environments, and by increasing the met-abolic heat production by shivering in the cold. For a resting per-son wearing trousers and a long-sleeved shirt, thermal comfort is experienced in a still air environment at 24°C. A zone of comfort extends about 1.5 K above and below this optimum level. An individual can minimize the need for physiological (involuntary) responses to the thermal environment that are generally per-ceived as uncomfortable, by a variety of behavioral responses. In a cool or cold environment, such responses include increased clothing, increased activity, or seeking or creating an environ-ment that is warmer. In a warm or hot environment, the amount of clothing can be reduced, the level of physical activity can be reduced, or an environment that is more conducive to increased heat loss can be created. Some of the human responses to the thermal environment are shown in Figure 1.
Cardiovascular and other diseases and the inevitable processes of aging can reduce the capacity or ability of physiological pro-cesses to maintain internal body temperature through the balancing of heat gains and heat losses. Thus, some persons are less able to deal with thermal challenges and deviations from comfortable thermally neutral conditions. Metabolic heat production tends to decrease with age, as a result of decreasing basal metabolism together with decreased physical activity. Metabolic heat produc-tion at age 80 is about 20% less than that at 20 years old, for com-parable size and mass. Persons in their eighties, therefore, prefer an environmental temperature about 1.5 K warmer than persons in their twenties. In any given environment near thermally neutral temperature, an older person is likely to have a lower core and skin temperature. Older people may have reduced capacity to secrete and evaporate sweat and to increase their skin blood flow and are therefore more likely to experience greater strain in warm and hot conditions as well as in cool and cold conditions.
Table 5 Inorganic Gas Comparative Criteria Contaminant OSHA/NIOSH TWAa EPA NAAQS 1 Std.
Nitric oxide 1 h 2 ppm (5 mg/m3) 24 h 25 ppm (30 mg/m3) None Nitrogen dioxide 1 h 5 ppm (9 mg/m3) 24 h 1 ppm (1.8 mg/m3) 0.053 ppm (100 µg/m3) Sulfur dioxide 1 h 5 ppm (13 mg/m3) 24 h 2 ppm (5 mg/m3) 0.014 ppm (365 µg/m3) Ozone 1 h 0.1 ppm (0.2 mg/m3) 24 h 0.1 ppm (0.2 mg/m3) 0.12 ppm (235 µg/m3) aThe values listed are the annual arithmetic mean unless otherwise listed. The first value listed is the 24 h average and the second value is the maximum 1 h average.
(TWA = time weighted average) Fig. 1 Related Human Sensory, Physiological, and Health Responses for Prolonged Exposure 9.12 2001 ASHRAE Fundamentals Handbook (SI) Hyperthermia Hyperthermia refers to the condition where body temperatures are above normal. A deep body temperature increase of 2 K above the normal range does not generally impair body function. For example, it is not unusual for runners to have rectal temperatures of 40°C after a long race. An elevated body temperature increases metabolism. Central nervous system function deteriorates at deep body temperatures above 41 to 42°C. Convulsions may occur above such temperatures and cells may be damaged. This condition is par-ticularly dangerous for the brain, because lost neurons are not replaced. Thermoregulatory functions of sweating and peripheral vasodilation cease at about 43°C, after which body temperatures tend to rise rapidly if external cooling is not imposed.
Seasonal Patterns Ordinary season changes in temperate climates are temporally associated with the prevalence of illness. Many acute and several chronic diseases vary in frequency or severity with time of year, and some are present only in certain seasons. Minor respiratory infec-tions, such as colds and sore throats, occur mainly in fall and winter.
More serious infections, such as pneumonia, have a somewhat shorter season in winter. Intestinal infections, such as dysentery and typhoid fever, are more prevalent in summer. Diseases transmitted by insects such as encephalitis and endemic typhus are limited to summer since insects are active in warm temperatures only.
Martinez et al. (1989), Hryhorczuk et al. (1992), and others describe a correlation between weather and seasonal illnesses; but such correlations do not necessarily establish a causal relationship.
Daily or weekly mortality and heat stress in heat waves have a strong physiological basis directly linked to outdoor temperature. In indoor environments which are well controlled with respect to tem-perature and humidity, such temperature extremes and the possible adverse effects on health are strongly attenuated.
Increased Deaths in Heat Waves The role of ambient temperature extremes produced by weather conditions in producing discomfort, incapacity, and death has been studied extensively (Katayama 1970). Military personnel, deep mine workers, and other workers occupationally exposed to extremes of high and low temperature have been studied, but the importance of thermal stress affecting both the sick and healthy is not sufficiently appreciated. Collins and Lehmann (1953) studied weekly deaths over many years in large cities in the United States and demonstrated the impact of heat waves in producing conspicu-ous periods of excess mortality. Excess mortality due to heat waves was of the same amplitude as that due to influenza epidemics, but tended to last one week instead of the 4 to 6 weeks duration of influ-enza epidemics.
Ellis (1972) reviewed heat wave related excess mortality in the United States. Mortality increases of 30% over background are commonly seen, especially in heat waves that occur early in the summer. Much of the increase occurs in the population over age 65, more of it in women than in men, and many deaths are due to car-diovascular and cerebrovascular causes. Oeschli and Buechley (1970) studied heat-related deaths in Los Angeles heat waves of 1939, 1955, and 1963. Kilbourne et al. (1982) suggested that the same risk factors (i.e., age, low income, and African-American der-ivation) persist in the heat death epidemics that continue to occur.
Hardy (1971) showed the relationship of health data to comfort on a psychrometric diagram (Figure 2). The diagram contains ASHRAE effective temperature ET lines and lines of constant skin moisture level or skin wettedness w. Skin wettedness is defined as that fraction of the skin covered with water to account for the observed evaporation rate. The ET lines are loci of constant phys-iological strain, and also correspond to constant levels of physiolog-ical discomfort—slightly uncomfortable, comfortable, and very comfortable (Gonzalez et al. 1978). Skin wettedness, as an indicator of strain (Berglund and Gonzalez 1977, Berglund and Cunningham 1986) and the fraction of the skin wet with perspiration, is fairly constant along an ET line. Numerically ET is the equivalent tem-perature at 50% rh that produces the strain and discomfort of the actual condition. The summer comfort range is between an ET of 23 and 26°C. In this region, skin wettedness is less than 0.2. Heat strokes occur generally when ET exceeds 34°C (Bridger and Helfand 1968). Thus, the ET line of 35°C is generally considered dangerous. At this point, skin wettedness will be 0.4 or higher.
The black dots in Figure 2 correspond to heat stroke deaths of healthy male U.S. soldiers assigned to sedentary duties in midwest-ern army camp offices (Shickele 1947). It is to be expected that older persons respond less well to thermal challenges than do healthy soldiers. This was apparently the case in the Illinois heat wave study mentioned earlier, where the first wave with a 33% increase in death rate and an ET of 29.5°C affected mainly the over 65-year-old group. The studies suggest that the “danger line” repre-sents a threshold of significant risk for young healthy people, and that the danger tends to move to lower values of ET with increas-ing age.
Effects of Thermal Environment on Specific Diseases Cardiovascular diseases are largely responsible for excess mor-tality during heat waves. For example, Burch and DePasquale (1962) found that the heart disease cases in whom decompensation is present are extremely sensitive to high temperatures and particu-larly to moist heat. However, both cold and hot temperature extremes are associated with increased coronary heart disease deaths and anginal symptoms (Teng and Heyer 1955).
Both acute and chronic respiratory diseases often increase in fre-quency and severity during extreme cold weather. No increase in these diseases has been noted in extreme heat. Additional studies of hospital admissions for acute respiratory illness show a negative cor-relation with temperature after removal of seasonal trends (Holland 1961). The symptoms of patients with chronic respiratory disease (bronchitis, emphysema) increase in cold weather. This is thought to be due to reflex constriction of the bronchi, adding to the obstruction already present. Greenberg (1964) revealed evidence of cold sensitivity in asthmatics; emergency room treatments for asthma increased abruptly in local hospitals with early and severe autumn cold spells. Later cold waves with even lower temperatures produced no such effects, and years without early extreme cold had no asthma Fig. 2 Isotherms for Comfort, Discomfort, Physiological Strain, Effective Temperature (ET), and Heat Stroke Danger Threshold Indoor Environmental Health 9.13 epidemics of this type. Patients with cystic fibrosis are extremely sensitive to heat because their diminished sweat gland function greatly diminishes their ability to cope with increased temperature (Kessler and Anderson 1951).
Itching and chapping of the skin is influenced by (1) atmospheric factors, particularly cold and dry air, (2) frequent washing or wet-ting of skin, and (3) low indoor humidities. Although itching of the skin is usually a winter cold climate illness in the general popula-tion, it can be caused by excessive summer air conditioning (Suss-kind and Ishihara 1965, Gaul and Underwood 1952).
People suffering from chronic illness (heart disease) or serious acute illnesses that require hospitalization often manage to avoid serious thermal stress. Katayama et al. (1970) found that countries with the most carefully regulated indoor climates (such as the Scandinavian countries and the United States) have had only small seasonal fluctuations in mortality in recent decades, while coun-tries with less space heating and cooling exhibit greater seasonal swings in seasonal mortality. For example, mandatory air condi-tioning in homes for the aged in the southwest United States has virtually eliminated previously observed mortality increases dur-ing heat waves.
Summer cooling reduces heat stress by removing both sensible and latent heat from the occupied space, but winter heating has a mixed effect. It reduces cold stress, but it usually does not increase the low water vapor pressure that occurs outdoors during the winter.
This results in very low relative humidity in the heated space, which can contribute to dehydration and discomfort and cause injury to skin, eyes, nose, throat, and mucous membranes. These dry tissues may be less resistant to infection. Animal experiments also show that infection rates increase with low levels of either ventilation or relative humidity (Schulman and Kilbourne 1962).
In various tests conducted under identical conditions except humidity level, mechanical humidification raised the relative humid-ity in one space above that in the matched space; no humidified room was higher than 50% rh (Green 1979 and 1982, Gelperin 1973, Serati and Wuthrich 1969). In each investigation, the humidified rooms showed a reduction in absenteeism and upper respiratory infection— 49% reduction in kindergarten children, 6% and 18% in office work-ers, and 8% and 18% in army recruits. Since occupants in each pair of spaces were subject to the same outdoor conditions and the same indoor air temperature, reductions were attributed to differences in humidity or a related factor (e.g., reduced dust levels and coughing).
Therefore, while low humidity does not have a direct pathological effect, it is a factor contributing to disease. A more direct effect has been indicated among users of contact lenses on long airline flights in cabins at low humidity. Here, dehydration of the eyes has been blamed for causing irritation and corneal edema or even ulceration of the corneal epithelium (Laviana et al. 1988).
Injury from Hot and Cold Surfaces The skin has cold, warm, and pain sensors to feed back thermal information about surface contacts. When the skin temperature rises above 45°C or falls below about 15°C, sensations from the skin's warm and cold receptors are replaced by those from the pain receptors to warn of thermal injury to the tissue (Guyton 1968).
The temperature of the skin depends on the temperature of the contact surface, its conductivity, and the contact time. Table 6 gives approximate temperature limits to avoid pain and injury when contacting three classes of conductors for various contact times (CEN).
ELECTRICAL HAZARDS Electrical current can cause burns, neural disturbances, and car-diac fibrillation (Billings 1975). The threshold of perception is about 5 mA for direct current, with a feeling of warmth at the con-tact site. The threshold is 1 mA for alternating current, which causes a tingling sensation.
The resistance of the current pathway through the body is a com-bination of core and skin resistance. The core is basically a saline volume conductor with very little resistance; therefore, the skin resistance provides the largest component of the resistance. The skin resistance decreases with moisture. If the skin is moist, volt-ages as low as 2 V (ac) or 5 V (dc) are sufficient to be detected, and voltages as low as 20 V (ac) or 100 V (dc) can cause a 50% loss in muscular control.
The dangerous aspect of alternating electrical current is its abil-ity to cause cardiac arrest by ventricular fibrillation. If a weak alter-nating current (100 mA for 2 s) passes through the heart (as it would in going from hand to foot), the current can force the heart muscle to fibrillate and lose the rhythmic contractions of the ventricles nec-essary to pump blood. Unconsciousness and death will soon follow if medical aid cannot rapidly restore normal rhythm.
MECHANICAL ENERGIES Vibration Vibration in a building originates from both outside and inside the building. Sources outside a building include blasting opera-tions, road traffic, overhead aircraft, underground railways, earth movements, and weather conditions. Sources inside a building include doors closing, foot traffic, moving machinery, elevators, HVAC systems, and other building services. Vibration is an omnipresent, integral part of the built environment. The effects of the vibration on building occupants depend on whether it is perceived by those persons and on factors related to the building, the location of the building, the activities of the occupants in the building, and the perceived source and magnitude of the vibra-tion. Factors influencing the acceptability of building vibration are presented in Figure 3.
The combination of hearing, seeing, or feeling vibration deter-mines human response. Components concerned with hearing and seeing are part of the visual environment of a room and can be assessed as such. The perception of mechanical vibration by feeling is generally through the cutaneous and kinesthetic senses at high frequencies, and through the vestibular and visceral senses at low frequencies. Because of this and the nature of vibration sources and building responses, building vibration may be conveniently consid-ered in two categories—low-frequency vibrations less than 1 Hz and high-frequency vibrations of 1 to 80 Hz.
Measurement and Assessment. Human response to vibration depends on the vibration of the body. The main vibrational charac-teristics are vibration level, frequency, axis (and area of the body), and exposure time. A root-mean-square (rms) averaging procedure (over the time of interest) is often used to represent vibration accel-eration (m/s2 rms). Vibration frequency is measured in cycles per second (Hz), and the vibration axis is usually considered in three orthogonal, human-centered translational directions (up-and-down, side-to-side, and fore-and-aft). Although the coordinate system is centered inside the body, in practice, vibration is measured at the human surface and measurements are directly compared with rele-vant limit values or other data concerning human response.
Rotational motions of a building in roll, pitch, and yaw are usu-ally about an axis of rotation some distance from the building Table 6 Approximate Surface Temperature Limits to Avoid Pain and Injury Material Contact Time 1 s 10 s 1 min 10 min 8 h Metal, water 149°F 133°F 124°F 118°F 109°F Glass, concrete 176°F 151°F 129°F 118°F 109°F Wood 248°F 190°F 140°F 118°F 109°F 9.14 2001 ASHRAE Fundamentals Handbook (SI) occupants. For most purposes, these motions can be considered as the translational motions of the person. For example, a roll motion in a building about an axis of rotation some distance from a seated person will have a similar effect as side-to-side translational motions of that person, etc.
Most methods assess building vibrations with rms averaging and frequency analysis. However, human response is related to the time-varying characteristics of vibration as well. For example, many stimuli are transient, such as those caused by a train passing a build-ing. The vibration event builds to a peak, followed by a decay in level over a total period of about 10 s. The nature of the time-vary-ing event and the number of occasions it occurs during a day are important factors that might be overlooked if data are treated as steady-state and continuous.
Standard Limits Low-Frequency Motion (1 Hz). The most commonly experi-enced form of slow vibration in buildings is building sway. This motion can be alarming to occupants if there is fear of building dam-age or injury. While occupants of two-story wood frame houses accept occasional creaks and motion from wind storms or a passing heavy vehicle, such events are not as accepted by occupants of high-rise buildings. Detected motion in tall buildings can cause discom-fort and alarm. The perception thresholds of normal sensitive humans to low-frequency horizontal motion are given in Figure 4 (ISO 1984, Chen and Robertson 1972). The frequency range is from 0.06 to 1 Hz or, conversely, for oscillations with periods of 1 to 17 s.
The natural frequency of sway of the Empire State Building in New York City, for example, has a period of 8.3 s (Davenport 1988). The thresholds are expressed in terms of relative acceleration, which is the actual acceleration divided by the standard acceleration of grav-ity (g = 9.8 m/s2). The perception threshold to sway in terms of building accelerations decreases with increasing frequency and ranges from 50 to 20 mm2/s.
For tall buildings, the highest horizontal accelerations generally occur near the top at the building’s natural frequency of oscillation.
Other parts of the building may have high accelerations at multi-ples of the natural frequency. Tall buildings always oscillate at their natural frequency, but the deflection is small and the motion undetectable. In general, short buildings have a higher natural fre-quency of vibration than taller ones. However, strong wind forces energize the oscillation and increase the horizontal deflection, speed, and accelerations of the structure.
ISO (1984) states that building motions are not to produce alarm and adverse comment from more than 2% of the building’s occu-pants. The level of alarm depends on the interval between events. If noticeable building sway occurs for at least 10 min at intervals of 5 years or more, the accept able acceleration limit is higher than if this sway occurs annually (Figure 4). For annual intervals, the acceptable limit is only slightly above the normal person’s threshold of percep-tion. Motion at the 5 year limit level is estimated to cause 12% to complain if it occurred annually. The recommended limits are for purely horizontal motion; rotational oscillations, wind noise, and/or visual cues of the building’s motion exaggerate the sensation of motion, and, for such factors, the acceleration limit would be lower.
The upper line in Figure 4 is intended for offshore fixed struc-tures such as oil drilling platforms. The line indicates the level of horizontal acceleration above which routine tasks by experienced personnel would be difficult to accomplish on the structure.
Fig. 3 Factors Affecting Acceptability of Building Vibration Fig. 4 Acceleration Perception Thresholds and Acceptability Limits for Horizontal Oscillations Indoor Environmental Health 9.15 High-Frequency Motion (1 to 80 Hz). Higher frequency vibra-tions in buildings are caused by machinery, elevators, foot traffic, fans, pumps, and HVAC equipment. Further, the steel structures of modern buildings are good transmitters of high-frequency vibrations.
The sensitivity to these higher frequency vibrations is indicated in Figure 5 (Parsons and Griffin 1988). Displayed are the median per-ception thresholds to vertical and horizontal vibrations in the 2 to 100 Hz frequency range. The average perception threshold for vibrations of this type is from 10 to 90 mm/s2, depending on the frequency and on whether the person is standing, sitting, or lying down.
People detect horizontal vibrations at lower acceleration levels when lying down than when standing. However, a soft bed decou-ples and isolates a person fairly well from the vibrations of the structure. The threshold to vertical vibrations is nearly constant at approximately 12 mm/s2 for both sitting and standing positions from 2 to 100 Hz. This agrees with earlier observations by Reiher and Meister (1931).
Many building spaces with critical work areas (surgery, preci-sion laboratory work) are considered unacceptable if vibration is perceived by the occupants. In other situations and activities, per-ceived vibration may be acceptable. Parsons and Griffin (1988) found that accelerations twice the threshold level would be unac-ceptable to occupants in their homes. A method of assessing vibra-tional acceptability in buildings is to compare the vibration with perception threshold values (Table 7).
Sound and Noise When the vibration of an object is transmitted to air particles, making them vibrate, a variation in normal atmospheric pressure is created. When this disturbance spreads to the eardrum, it is vibrated, and this vibration is translated into the sensation called “sound.” In general terms, sound in the physical sense is the vibration of parti-cles in a gas, a liquid, or a solid. The entire mechanical energy spec-trum includes include infrasound and ultrasound as well as audible sound (Figure 6).
Health Effects. Hearing loss is generally considered the most undesirable effect of noise exposure, although there are other effects. Tinnitus, a ringing in the ears, is really the hearing of sounds that do not exist. It often accompanies hearing loss. Paracusis is a disorder where a sound is heard incorrectly; that is, a tone is heard, but has an inappropriate pitch. Speech misperception occurs when an individual mistakenly hears one sound for another; for example, when the sound for “t” is heard as a “p.” Hearing loss can be categorized as conductive, sensory, or neu-ral. Conductive hearing loss results from a general decrease in the amount of sound transmitted to the inner ear. Excessive ear wax, a ruptured ear drum, fluid in the middle ear, or missing elements of the bone structures in the middle ear are all associated with conduc-tive hearing loss. These are generally not occupationally related and are generally reversible by medical or surgical means. Sensory hear-ing losses are associated with irreversible damage to the inner ear.
Sensory hearing loss is further classified as (1) presbycusis, loss caused as the result of aging; (2) noise-induced hearing loss (indus-trial hearing loss and sociacusis, which is caused by noise in every-day life); and (3) nosoacusis, losses attributed to all other causes.
Neural deficits are related to damage to higher centers of the audi-tory system.
Noise-induced hearing loss is believed to occur, in the most sen-sitive individuals, in those exposed for 8 h per day over a working lifetime at levels of 75 dBA and for most people similarly exposed to 85 dBA.
ELECTROMAGNETIC RADIATION Radiation energy is emitted, transmitted, or absorbed in wave or particulate form. This energy consists of electric and magnetic forces which, when disturbed in some manner, produce electromag-netic radiation. Electromagnetic radiation is grouped into a spec-trum arranged by frequency and/or wavelength. The product of frequency and wavelength is the speed of light (3 × 108 m/s). The Table 7 Acceptable to Threshold Vibration Level Ratios Place Time Continuous or Intermittent Vibration Impulse or Transient Vibration Several Times per Day Critical work areas Day or night 1 1 Residential Day/Night 2 to 4 / 1.4 30 to 90 / 1.4 to 20 Office Day or night 4 60 to 128 Workshop Day or night 8 90 to 128 Note: The ratios for continuous or intermittent vibration and repeated impulse shock are in the range of 0.7 to 1.0 for hospital operating theaters (room) and critical work-ing areas. In other situations, impulse shock can generally be much higher than when the vibration is more continuous.
Fig. 5 Median Perception Thresholds to Horizontal (Solid Lines) and Vertical (Dashed Line) Vibrations Fig. 6 Mechanical Energy Spectrum 9.16 2001 ASHRAE Fundamentals Handbook (SI) spectrum includes ionizing, ultraviolet, visible, infrared, micro-wave, radio-frequency, and extremely low frequency (Figure 7).
Table 8 presents these electromagnetic radiations by their range of energies, frequencies, and wavelengths. The regions are not sharply delineated from each other and, in fact, often overlap. It is conve-nient to divide these regions as listed in Table 8, due to the nature of the physical and biological effects.
Ionizing Radiation Ionizing radiation is that part of the electromagnetic spectrum that has very short wavelengths and high frequencies, and it has the ability to ionize matter. Such ionizations tend to be very damaging to living matter. Background radiation that occurs naturally in the environment is from cosmic rays and naturally occurring radionu-clides. It has not been established whether exposure at the low dose rate of average background levels is harmful to humans.
The basic standards for permissible air concentrations of radio-active materials are those of the National Committee on Radiation Protection, published by the National Bureau of Standards as Handbook No. 69. Industries operating under licenses from the U.S. Nuclear Regulatory Commission or state licensing agencies must meet requirements of the Code of Federal Regulations, Title 10, Part 20. Some states have additional requirements.
An important naturally occurring radionuclide is radon (222Rn), a decay product of uranium in the soil (238U). Radon, denoted by the symbol Rn, is chemically inert. Details of units of measure-ment, typical radon levels, measurement methods and control strategies can be found in Chapter 12.
Health Effects of Radon. Studies of workers in uranium and other underground mines form the principal basis for knowledge about health risks due to radon. The radioactive decay of radon produces a series of radioactive isotopes of polonium, bismuth, and lead. Unlike their chemically inert radon parent, these progeny are chemically active. They can attach to airborne particles that subsequently deposit in the lung or deposit directly in the lung without prior attachment to particles. Some of these progeny, like radon, are alpha-particle emitters, and the passage of these alpha particles through lung cells can lead to cellular changes that may initiate lung cancer (Samet 1989). Thus, adverse health effects associated with radon are due to exposures to radon decay prod-ucts, and the amount of risk is assumed to be directly related to the total exposure. Even though it is the radon progeny that present the possibility of adverse health risks, radon itself is usually measured and used as a surrogate for progeny measurements because of the expense involved in accurate measurements of radon progeny.
Standards. Many countries besides the United States have established standards for exposure to radon. International action levels are listed in Table 9.
About 6% of homes in the United States (5.8 million homes) have annual average radon concentrations exceeding the action level of 148 Bq/m3 (4 pCi/L) set by the U.S. Environmental Pro-tection Agency (Marcinowski et al. 1994).
Nonionizing Radiation Ultraviolet radiation, visible light, and infrared radiation are components of sunlight and of all artificial light sources. Micro-wave radiation and radio-frequency radiation are essential in a wide range of communication technologies and are also in widespread Fig. 7 Electromagnetic Spectrum Table 8 Energy, Wavelength, and Frequency Ranges for Electromagnetic Radiation Radiation Type Energy Range Wavelength Range Frequency Range Ionizing > 12.4 eV < 100 nm > 3.00 PHz Ultraviolet (UV) 12.40 −3.10 eV 100 −400 nm 3.00 PHz − 0.75 PHz Visible 3.10 −1.63 eV 400 −760 nm 750 THz − 395 THz Infrared (IR) 1.63 −1.24 meV 760 nm −1 mm 395 THz − 0.30 THz Microwave (MW) 1.24 meV − 1.24 eV 1 mm −1 m 300 GHz − 300 MHz Radio-frequency (RF) 1.24 eV − 1.24 peV 1 m −1 Mm 300 MHz − 300 Hz Extremely low frequency (ELF) < 1.24 peV > 1 Mm < 300 Hz Table 9 Action Levels for Radon Concentration Indoors Action Level Country/Agency Bq/m3 pCi/L Australia 200 5.4 Austria 400 10.8 Belgium 400 10.8 CEC 400 10.8 Canada 800 21.6 Czech Republic 400 10.8 P.R. China 200 5.4 Finland 400 10.8 Germany 250 6.7 ICRP 200 5.4 Ireland 200 5.4 Italy 400 10.8 Norway 400 10.8 Sweden 400 10.8 United Kingdom 200 5.4 United States 148 4.0 World Health Organization 200 5.4 Source: DOE (1995).
Indoor Environmental Health 9.17 use for heating as in microwave ovens and heat sealers, and for heat treatments of a variety of products. Power frequency fields are an essential and unavoidable consequence of the generation, transmis-sion, distribution, and use of electrical power.
Optical Radiation. Ultraviolet (UV), visible, and infrared (IR) radiation compose the optical radiation region of the electromag-netic spectrum. The wavelengths range from 100 nm in the UV to 1 mm in the IR, with 100 nm generally considered to be the bound-ary between ionizing and nonionizing. The UV region wavelengths range from 100 to 400 nm, the visible region from 400 to 760 nm, and the IR from 760 nm to 1 mm.
Optical radiation can interact with a medium by reflection, absorption, or transmission. The skin and eyes are the organs at risk in humans. Optical radiation from any of the spectral regions can cause acute and/or chronic biologic effects given appropriate energy characteristics and exposure. These effects include tanning, burning (erythema), premature “aging,” and cancer of the skin; and dryness, irritation, cataracts, and blindness in the eyes.
The region of the electromagnetic spectrum visible to humans is known as light. There can be biological, behavioral, psychological, and health effects from exposure to light. Assessment of these effects depends on the purpose and application of the illumination.
Individual susceptibility varies, with other environmental factors (air quality, noise, chemical exposures, and diet) acting as modifi-ers. It is difficult, therefore, to generalize potential hazards. Light pollution results from the presence of unwanted light.
Light penetrating the retina not only allows the exterior world to be seen, but, like food and water, it is used in a variety of metabolic processes. Light stimulates the pineal gland to secrete melatonin, which regulates the human biological clock. This, in turn, influ-ences reproductive cycles, sleeping, eating patterns, activity levels, and moods. The color of light affects the way the objects appear.
The distortion of color rendition may result in disorientation, head-ache, dizziness, nausea, and fatigue.
As the daylight shortens, the human body may experience a grad-ual slowing down, loss of energy, and a need for more sleep. It becomes harder to get to work, and depression or even withdrawal may take place. This type of seasonal depression, brought on by changes in light duration and intensity, is called seasonal affective disorder (SAD). Sufferers of this syndrome also complain of anxi-ety, irritability, headache, weight gain, and lack of concentration and motivation. Treatment of this problem is through the manipula-tion of environmental lighting (exposure to full-spectrum lighting for extended periods, 12 h/day).
Radio-Frequency Radiation. Just as the body absorbs infrared and light energy, which can affect thermal balance, it can also absorb other longer wavelength electromagnetic radiation. For comparison, visible light has wavelengths in the range 0.4 to 0.7 µm and infrared from 0.7 to 10 µm, while the wavelength of K and X band radar is 12 and 28.6 mm. The wavelength of radiation in a typ-ical microwave oven is 120 mm. Infrared is absorbed within 1 mm of the surface (Murray 1995).
The heat of the absorbed radiation raises the skin temperature and, if sufficient, is detected by the skin’s thermoreceptors, warning the person of the possible thermal danger. With increasing wave-length, the radiation penetrates deeper into the body. The energy can thus be deposited well beneath the skin’s thermoreceptors, making the person less able or slower to detect and be warned of the radia-tion (Justesen et al. 1982). Physiologically, these longer waves only heat the tissue and, because the heat may be deeper and less detect-able, the maximum power density of such waves in occupied areas is regulated (ANSI 1991) (Figure 8). The maximum permitted power densities are less than half of sensory threshold values.
ERGONOMICS Ergonomics may be defined as the scientific study of the rela-tionship between man and his work environment to achieve opti-mum adjustment in terms of efficiency, health, and well-being.
Ergonomic designs of tools, chairs, etc., help workers interact more comfortably and efficiently with their environment. In jobs that were ergonomically designed, productivity typically increased and the worker enjoyed a healthier working experience. More recently, researchers have distinguished intrinsic ergonomics from extrinsic, or traditional, ergonomics. Intrinsic ergonomics considers how the interface between an individual and the environment affects and relies on specific body parts (i.e., muscles, tendons, and bones) and work practices such as force of application, relaxation intervals, styles, and strength reserves that are not adequately considered in simple analyses of the physical environment.
The goal of ergonomic programs ranges from making work safe and humane, to increasing human efficiency, to creating human Fig. 8 Maximum Permissible Levels of Radio Frequency Radiation for Human Exposure 9.18 2001 ASHRAE Fundamentals Handbook (SI) well-being. The successful application of ergonomic factors is mea-sured by improved productivity, efficiency, safety, and acceptance of the resultant system design. The design engineer uses not only engineering skills, but also the sciences and principles of anatomy, orthopedics, physiology, medicine, psychology, and sociology to apply ergonomics to a design.
Implementing ergonomic principles in the workplace helps min-imize on-the-job stress and strain, and prevents cumulative trauma disorders or CTDs. These disorders are subtle injuries that can affect the muscles, tendons, and nerves at body joints, especially the hands, wrists, elbows, shoulders, neck, back, and knees. Carpal tunnel syndrome is an example of a CTD. CTDs most frequently occur as a result of strain from performing the same task on a con-tinuous or repetitive basis. This strain can slowly build over time, until the worker experiences pain and difficulty using the injured part of the body. Higher risks of developing CTDs are encountered when the work task requires repetitive motions, excessive force, or awkward postures. The ergonomics engineer addresses these risk factors by analyzing the task thoroughly and minimizing the repet-itive motion, excessive force, and awkward posture.
REFERENCES ACGIH. 1989. Guidelines for the assessment of bioaerosols in the indoor environment. American Conference of Government Industrial Hygien-ists, Cincinnati, OH.
ACGIH. Annual. Threshold Limit Values and Biological Exposure Indices for 1995-1996.
Alpaugh, E.L. and T.J. Hogan. 1988. Fundamentals of industrial hygiene: Particulates, 3rd Edition. National Safety Council.
Anderson, H.A., D. Higgins, L.P. Hanrahan, P. Sarow, and J. Schirmer. 1991.
Mesothelioma among employees with likely contact with in-place asbes-tos-containing building materials. Ann. N.Y. Acad. Sci. 643:550-72.
ANSI. 1991. Safety level with respect to human exposure to radio frequency electromagnetic radiation of 3KH-300GH Standard C95.1-1991. Amer-ican National Standards Institute, New York.
Arnow, P.M., J.N. Fink, D.P. Schlueter, J.J. Barboriak, G. Mallison, S.I. Said, S. Martin, G.F. Unger, G.T. Scanlon, and V.P. Kurup. 1978. American Journal of Medicine 64:237.
Bartlett, P.C., L.A. Vonbehren, R.P. Tewari, R.J. Martin, L. Eagleton, M.J.
Isaac, and P.S. Kulkarni. 1982. American Journal of Public Health 72: 1369.
Bascom, R.A. 1996. Environmental factors and respiratory hypersensitivity: The Americas. Toxicol. Lett. 86:115-30.
Bascom, R., P. Bromberg, D.L. Costa, et al. 1996. Health effects of outdoor pollution, Parts I and II. American Journal Respir. Crit. Care Med.
153:3-50, 477-89.
Bates, D.V. 1989. Ozone—Myth and reality. Environmental Research 50: 230-37.
Baylor, E.R., V. Peters, and M.B. Baylor. 1977. Water-to-air transfer of virus. Science. 1977:763.
Berglund, L.G. and D. Cunningham. 1986. Parameters of human discomfort in warm environments. ASHRAE Transactions 92(2).
Berglund, L.G. and R.R. Gonzalez. 1977. Evaporation of sweat from seden-tary man in humid environments. Journal of Applied Physiology, Respi-ratory, Environmental and Exercise Physiology 42(5):767-72.
Billings, C.E. 1975. Electrical shock. In Textbook of medicine. Saunders, Philadelphia, 72-73.
Bridger, C.A. and L.A. Helfand. 1968. Mortality from heat during July 1966 in Illinois. International Journal of Biometeorology 12:51.
Brundage, J.F., R.M. Scott, W.M. Lodnar, D.W. Smith, and R.N. Miller.
1988. JAMA 259:2108.
Burch, G.E. and N.P. DePasquale. 1962. Hot climates, man and his heart.
Charles C. Thomas, Springfield, IL.
Burge, H.A. 1995. Bioaerosols. Lewis Publishers, Chelsea, MI.
Burge, H.A. 1989. Occupational medicine: State of the art reviews. J.E.
Cone and M.J. Hodgson, eds. 4:713-21. Hanley and Belfus, Philadelphia.
Burge, S., A. Hedge, S. Wilson, J.H. Bass, and A. Robertson. 1987. Annals of Occupational Hygiene 31:493.
Burgess, J.L. and C.D. Crutchfield. 1995. Quantitative respirator fit tests of Tuscon fire fighters and measures of negative pressure excursions during exertion. Appl. Occup. Env. Hyg. 10(1):29-36.
Burton, D.J. 2000. Industrial ventilation, a self-directed learning workbook.
Carr Printing.
Cain, W.S. 1989. Perceptual characteristics of nasal irritation. National Dan-ish Institute on Occupational Health, Copenhagen.
Cain, W.S., J.M. Samet, and M.J. Hodgson. 1995. The quest for negligible health risk from indoor air. ASHRAE Journal 37(7):38.
CEN. Surface temperatures of touchable parts, a draft proposal. TC 114 N 122 D/E. European Standards Group.
CFR. 1989a. 29 CFR 1910.1000. U.S. Government Printing Office, Wash-ington, DC.
CFR. 1989b. 29 CFR 1926.1101.
Chen, P.W. and L.E. Robertson. 1972. Human perception thresholds of hor-izontal motion. ASCE Journal, Structure Division, August.
Cobb, N. and R.A. Etzel. 1991. Unintentional carbon monoxide-related deaths in the United States, 1979 through 1988. JAMA 266:659-63.
Collins, S.D. and J. Lehmann. 1953. Excess deaths from influenza and pneu-monia and from important chronic diseases during epidemic periods, 1918-51. Public Health Monograph 20.10, U.S. Public Health Service Publication No. 213.
Cometto-Muñiz, J.E. and W.S. Cain. 1994a. Sensory reactions of nasal pun-gency and odor to volatile organic compounds: The alkylbenzenes. Am.
Ind. Hyg. Assoc. J. 55(9):811-17.
Cometto-Muñiz, J.E. and W.S. Cain. 1994b. Perception of odor and nasal pungency from homologous series of volatile organic compounds.
Indoor Air 4:140-45.
Cox, C.S. 1987. The aerobiological pathway of microorganisms. John Wiley and Sons, New York.
Crandall, M.S. and W.K. Sieber. 1996. The NIOSH indoor environmental evaluation experience: Part one, building evaluations. Applied Occupa-tional and Environmental Hygiene.
Davenport, A.G. 1988. The response of supertall buildings to wind. In Sec-ond century of the skyscraper. Van Nostrand Reinhold, New York, 705-26.
DHHS. 1986. The health consequences of involuntary smoking. A report of the surgeon general. DHHS Publication No. (PHS) 87-8398. U.S.
Department of Health and Human Services, Public Health Services, Office of the Asst. Secretary for Health, Office of Smoking and Health.
DOE. 1995. Symposium Series CONF-77114, pp. 663-90. U.S. Dept. of Energy.
Edwards, J.H. 1980. Microbial and immunological investigations and reme-dial action after an outbreak of humidifier fever. British Journal of Indus-trial Medicine 37:55-62.
Ellis, F.P. 1972. Mortality from heat illness and heat aggravated illness in the United States. Environmental Research 5.
EPA. 1991. Introduction to indoor air quality, a reference manual. Report EPA 400/3-91/003 EPA. 1992. Respiratory health effects of passive smoking: Lung cancer and other disorders, review draft. EPA/6O0-6-90/006B. Office of Research and Development, Washington, DC.
Franck, C., P. Skov, and O. Bach. 1993. Prevalence of objective eye mani-festations in people working in office buildings with different preva-lences of the sick building syndrome compared with the general population. Int. Arch. Occup. Environ. Health 65:65-69.
Gaul, L.E. and G.B. Underwood. 1952. Relation of dew point and barometric pressure to chapping of normal skin. Journal of Investigative Dermatol-ogy 19:9.
Gelperin, A. 1973. Humidification and upper respiratory infection inci-dence. Heating, Piping and Air Conditioning 45:77.
Girman, J.R. 1989. Volatile organic compounds and building bake-out. In Occupational medicine: State of the art reviews—Problem buildings.
J.E. Cone and M.J. Hodgson, eds. Hanley and Belfus, Philadelphia.
4(4):695-712.
Girman, J., Y.-L. Chang, S.B. Hayward, and K.S. Liu. 1996. Causes of unin-tentional deaths from carbon monoxide poisonings in California. Unpub-lished.
Glantz, S.A. and W.W. Parmley. 1991. Passive smoking and heart disease epidemiology, physiology, and biochemistry. Circulation 83:633-42.
Gonzalez, R.R., L.G. Berglund, and A.P. Gagge. 1978. Indices of thermoreg-ulatory strain for moderate exercise. Journal of Applied Physiology: Res-piratory Environmental and Exercise Physiology 44(6):889-99.
Green, G.H. 1979. The effect of indoor relative humidity on colds. ASHRAE Transactions 85(1).
Green, G.H. 1982. The positive and negative effects of building humidifica-tion. ASHRAE Transactions 88(1):1049.
Indoor Environmental Health 9.19 Greenberg, L. 1964. Asthma and temperature change. Archives of Environ-mental Health 8:642.
Guyton, A.C. 1968. Textbook of medical physiology. Saunders, Philadelphia.
Hardy, J.D. 1971. Thermal comfort and health. ASHRAE Journal 13:43.
Higgins, I.T.T. 1983. What is an adverse health effect? APCA Journal 33:661-63.
Hodgson, A.T. 1995. A review and a limited comparison of methods for measuring total volatile organic compounds in indoor air. Indoor Air 5(4):247.
Hodgson, M.J., P.R. Morey, J.S. Simon, T.D. Waters, and J.N. Fink. 1987.
American Journal of Epidemiology 125:631.
Hodgson, M.J., P.R. Morey, M. Attfield, W. Sorenson, J.N. Fink, W.W.
Rhodes, and G.S. Visvesvara. 1985. Archives of Environmental Health 40:96.
Holland, W.W. 1961. Influence of the weather on respiratory and heart dis-ease. Lancet 2:338.
Hryhorczuk, D.O., L.J. Frateschi, J.W. Lipscomb, and R. Zhang. 1992. Use of the scan statistic to detect temporal clustering of poisonings. J. Tox-Clinical Toxicology 30:459-65.
Hyppel, A. 1984. Proceedings of the 3rd International Conference on Indoor Air Quality and Climate, Stockholm, Sweden, 3:443-47.
ISO. 1984. Guideline to the evaluation of responses of occupants of fixed structures, especially buildings and off-shore structures, to low fre-quency horizontal motion, 0.063 to 1 Hz. ISO 6897. International Orga-nization for Standardization, Geneva.
Justesen, D.R., E.R. Adair, J.C. Stevens and V. Bruce-Wolfe. 1982. A com-parative study of human sensory thresholds: 2450 MHz microwaves vs.
far-infrared radiation. Bioelectromagnetics 3:117-25.
Katayama, K. and M. Momiyana-Sakamoto. 1970. A biometeorological study of mortality from stroke and heart diseases. Meteorological Geo-physics 21:127.
Kessler, W.R. and W.R. Anderson. 1951. Heat prostration in fibrocystic dis-ease of pancreas and other conditions. Pediatrics 8:648.
Kilbourne, E.M., T.S. Jones, K. Choi, and S.B. Thacker. 1982. Risk factors for heatstroke: A case-control study. JAMA 247(24):3332-36.
Kjaergard, S., L. Molhave, and O.F. Pedersen. 1991. Human reactions to a mixture of indoor pollutants. Atmos. Environ. 25:1417-26.
Kjaergard, S. 1992. Assessment methods and causes of eye irritation in humans in indoor environments. H. Knoeppel and P. Wolkoff, eds.
Chemical, Microbiological, Health, and Comfort Aspects of Indoor Air Quality. ECSC, EEC, EAEC, Brussels 115-27.
Koren, H. 1990. The inflammatory response of the human upper airways to volatile organic compounds. In: Proceedings of Indoor Air 90 1:325-330.
Koren, H., D.E. Graham, and R.B. Devlin. 1992. Exposure of humans to a volatile organic mixture III: Inflammatory response. Arch. Environ.
Health 47:39-44.
Last, J.M. 1983. Public health and human ecology. Appleton & Lange, New York.
Laviana, J.E., F.H. Rohles, and P.E. Bullock. 1988. Humidity, comfort and contact lenses. ASHRAE Transactions 94(1).
Leathart, G.L. 1972. Clinical Aspects of Respiratory Disease Due to Mining.
Medicine in the Mining Industry.
Levetin, E. 1995. Fungi. In: Burge, H.A. 1995. Bioaerosols. CRC Press.
Lewis Publishers. Boca Raton, FL.
Lilienfeld, D.E. 1991. Asbestos-associated pleural mesothelioma in school teachers: A discussion of four cases. Ann. N.Y. Acad. Sci. 643:454-86.
Liu, K.S., J. Wesolowski, F.Y. Huang, K. Sexton, and S.B. Hayward. 1991.
Irritant effects of formaldehyde exposure in mobile homes. Environ.
Health Perspect. 94:91-94.
Marcinowski, F., R.M. Lucas, and W.M. Yeager 1994. National and regional distributions of airborne radon concentrations in U.S. homes. Health Physics 66(6):699-706.
Martin, T.R. and M.B. Braken. 1986. Association of low birth weight pas-sive smoke and exposure in pregnancy. American Journal of Epidemiol-ogy 124:633-42.
Martinez, B.F., M.L. Kirk, J.L. Annest, K.J. Lui, E.M. Kilbourne, and S.M.
Smith. 1989. Geographic distribution of heat-related deaths among eld-erly persons. Use of county-level dot maps for injury surveillance and epidemiologic research. JAMA 262:2246-50.
Mbithi, J.N., V.S. Springthorpe, and S.A. Sattar. 1991. Effect of relative humidity and air temperature on survival of hepatitis A virus on environ-mental surfaces. Appl. Environ. Microbiol. 57:1394.
McCann, J., L. Horn, J. Girman, and A.V. Nero. 1987. Short-term bioassays in the analysis of complex mixtures. Plenum Press, New York 5:325-54.
Meggs, W.J. 1994. RADS and RUDS—The toxic induction of asthma and rhinitis. J. Toxicol. Clin. Toxicol. 32:487-501.
Miller, J.D. and J. Day. 1997. Indoor mold exposure: epidemiology, conse-quences and immunotherapy. Can. J. Allergy and Clinical Immunology 2(1):25-32.
Molhave, L. 1991. Volatile organic compounds, indoor air quality and health. Indoor Air 1(4):357-76.
Molhave, L., Z. Liu, A.H. Jorgensen, O.F. Pederson, and S. Kjaergard. 1993.
Sensory and physiologic effects on humans of combined exposures to air temperatures and volatile organic compounds. Indoor Air 3:155-69.
Moolenaar, R.L., R.A. Etzel, and R.G. Parrish. 1995. Unintentional deaths from carbon monoxide poisoning in New Mexico, 1980 to 1988. A com-parison of medical examiner and national mortality data. Western Jour-nal of Medicine 163(5):431-4.
Morey, P.R. 1988. Architectural design and indoor microbial pollution.
Oxford University Press, New York, 40-80.
Morey, P.R. 1990. The practitioner’s approach to indoor air quality investi-gations. Proceedings of the Indoor Air Quality International Sympo-sium. American Industrial Hygiene Association, Akron, OH.
Morey, P.R. and J.C. Feeley. 1988. ASTM Standardization News 16:54.
Morey, P.R., M.J. Hodgson, W.G. Sorenson, G.J. Kullman, W.W. Rhodes, and G.S. Visvesvara. 1986 Environmental studies in moldy office build-ings. ASHRAE Transactions 93(1B):399-419.
Morey, P.R. and B.A. Jenkins. 1989. What are typical concentrations of fungi, total volatile organic compounds, and nitrogen dioxide in an office environment. Proceedings of IAQ ‘89, The Human Equation: Health and Comfort. ASHRAE, Atlanta, 67-71.
Morrow, P.E. 1964. Evaluation of inhalation hazards based upon the respi-rable dust concept and the philosophy and application of selective sam-pling. AIHA Journal 25:213.
Moser, M.R., T. R. Bender, H.S. Margolis, et al. 1979. An outbreak of influ-enza aboard a commercial airliner. Am. J. Epidemiol. 110:1.
Mumford, J.L., X.Z. He, R.S. Chapman, S.R. Cao, D.B. Harris, K.M. Li, Y.L. Xian, W.Z. Jiang, C.W. Xu, J.C. Chang, W.E. Wilson, and M.
Cooke. 1987. Lung cancer and indoor air pollution in Xuan Wei, China.
Science 235:217-20.
Murray, W. 1995. Non-ionizing electromagnetic energies (Chapter 14).
Patty’s Industrial Hygiene and Toxicology 3B:623-727 (644).
NAS. 1981. Indoor Pollutants. National Research Council/National Acad-emy of Sciences, Committee on Indoor Pollutants. National Academy of Sciences Press, Washington, D.C., Oct.
NAS. 1983. Polycyclic aromatic hydrocarbons: Evaluation of sources and effects. National Academy Press, Washington, DC.
NIOSH. 1990. NIOSH/OSHA pocket guide to chemical hazards. DHHS (NIOSH) Publication 90-117. U.S. Department of Labor, OSHA, Wash-ington, DC. Users should read Cohen (1993) before using this reference.
NIOSH. Annual registry of toxic effects of chemical substances. U.S.
Department of Health and Human Services, National Institute for Occu-pational Safety and Health, Washington DC.
NIOSH. 1992. NIOSH recommendations for occupational safety and health compendium of policy documents and statements. DHHS (NIOSH) Pub-lication 92-100. U.S. Dept. of Health and Human Services, Public Health Service, Centers for Disease Control, National Institute for Occupational Safety and Health, Atlanta.
NRC. 1986. Environmental tobacco smoke: Measuring exposures and assessing health effects. National Research Council. National Academy Press, Washington, DC.
Oeschli, F.W. and R.W. Beuchley. 1970. Excess mortality associated with three Los Angeles September hot spells. Environmental Research 3:277.
Ohm, M., J.E. Juto, and K. Andersson. 1992. Nasal hyperreactivity and sick building syndrome. IAQ 92: Environments for People. ASHRAE, Atlanta.
Offermann, F.J., S.A. Loiselle, A.T. Hodgson, L.A. Gundel, and J.M. Dai-sey. 1991. A pilot study to measure indoor concentrations and emission rates of polycyclic aromatic hydrocarbons. Indoor Air 4:497-512.
Parsons, K.C. and M.J. Griffin. 1988. Whole-body vibration perception thresholds. Journal of Sound and Vibration 121(2):237-58.
Proctor and Hughes’ chemical hazards in the workplace, 3rd ed. 1991. G.J.
Hathaway et al. eds. Van Nostrand Reinhold, New York.
Reiher, H. and F.J. Meister. 1931. Translation of Report Fts616RE 1946.
Forschung VDI 2, 381-86, “The sensitivities of the human body to vibra-tions.” Headquarters Air Material Command, Wright Field, Dayton, OH.
Ritchie, I.M. and R.E. Lehnen. 1987. Irritation from formaldehyde in mobile and regular homes. Am J. Public Health 77:323-28.
9.20 2001 ASHRAE Fundamentals Handbook (SI) Rohles, F.H., J.A. Woods, and P.R. Morey. 1989. Indoor environmental acceptability: Development of a rating scale. ASHRAE Transactions 95(1):23-27.
Samet, J.M., M.C. Marbury, and J.D. Spengler. 1987. Health effects and sources of indoor air pollution. Am. Rev. Respir. Disease 136:1486-1508.
Samet, J.M. 1989. Radon and lung cancer. J. Natl. Cancer Institute 81:145.
Schulman, J.H. and E.M. Kilbourne. 1962. Airborne transmission of influ-enza virus infection in mice. Nature 195:1129.
Selikoff, I.J., J. Churg, and E.C. Hammond. 1965. The occurrence of asbes-tosis among insulation workers in the United States. Ann. Rpt. NY Acad-emy of Science.
Serati, A. and M. Wuthrich. 1969. Luftfreughtigkeit und saison Kranken-heit. Schweizerische Medizinische Wochenscrift 99:46.
Shickele, E. 1947. Environment and fatal heat stroke. Military Surgeon 100:235.
Smith, K.W. 1955. Pulmonary disability in asbestos workers. Archives of Industrial Health.
Spengler, J.D., H.A. Burge, and H.J. Su. 1992. Biological agents and the home environment. In: Bugs, Mold and Rot (I): Proceedings of the Mois-ture Control Workshop, E. Bales and W.B. Rose (eds). 11-18. Building Thermal Envelope Council, National Institute of Building Sciences, Washington, DC.
Strindehag, O., I. Josefsson, and E. Hennington. 1988. Healthy Buildings ’88, 3:611-20. Stockholm, Sweden.
Streifel, A.J., D. Vesley, F.S. Rhame, and B. Murray. 1989. Control of air-borne fungal spores in a university hospital. Environ. Int. 15:221.
Susskind, R.R. and M. Ishihara. 1965. The effects of wetting on cutaneous vulnerability. Archives of Environmental Health 11:529.
Tancrede, M., R. Wilson, L. Ziese, and E.A.C. Crouch. 1987. Atmospheric Environment 21:2187.
Teng, H.C. and H.E. Heyer, eds. 1955. The relationship between sudden changes in the weather and acute myocardial infarction. American Heart Journal 49:9.
WHO. 1987. Air Quality Guidelines for Europe. European Series No. 23.
WHO, Copenhagen.
Yoshizawa, S., F. Surgawa, S. Ozawo, Y. Kohsaka, and A. Matsumae. 1987.
Proceedings 4th International Conference on Indoor Air Quality and Cli-mate, Berlin 1:627-31.
Zenz, C. 1988. Occupational safety in industry, occupational medicine, prin-ciples and practical applications.
10.1 CHAPTER 10 ENVIRONMENTAL CONTROL FOR ANIMALS AND PLANTS ANIMALS ................................................................................ 10.1 Animal Care/Welfare .............................................................. 10.2 Physiological Control Systems ............................................... 10.2 Principles for Air Contaminant Control ................................................................................ 10.5 Cattle ....................................................................................... 10.7 Sheep ....................................................................................... 10.9 Swine ..................................................................................... 10.10 Chickens ................................................................................ 10.12 Turkeys .................................................................................. 10.13 Laboratory Animals .............................................................. 10.14 PLANTS: GREENHOUSES, GROWTH CHAMBERS, AND OTHER FACILITIES ................................................ 10.15 Temperature .......................................................................... 10.15 Light and Radiation .............................................................. 10.17 Relative Humidity .................................................................. 10.19 Air Composition .................................................................... 10.20 Air Movement ........................................................................ 10.20 HERMAL conditions, air quality, lighting, noise, ion concen-Ttration, and crowding are important in designing structures for animals and plants. Thermal environment influences heat dissipa-tion by animals and chemical process rates in plants. Lighting influ-ences photoperiodism in animals and plants, and photosynthesis and regulation in plants. Air quality, noise, ion concentrations, and crowding can affect the health and/or productivity of animals or plants. This chapter summarizes the published results from various research projects and provides a concept of the physiological factors involved in controlling the environment.
ANIMALS Animal performance (growth, egg or milk production, wool growth, and reproduction) and their conversion of feed to useful products are closely tied to the thermal environment. For each homeothermic species, an optimum thermal environment permits necessary and desirable body functions with minimum energetic input (Figure 1A). The optimal thermal environment—in terms of an effective temperature that integrates the effects of dry-bulb tem-perature, humidity, air movement, and radiation—is less important to the designer than the range of conditions that provides acceptable animal performance, efficiency, well-being, and economic return for a given species. Figure 1A depicts this range as the zone of nom-inal losses, selected to limit losses in performance to a level accept-able to the livestock manager. Researchers have found that the zone of nominal losses corresponds to the welfare plateau (i.e., welfare is enhanced by maintaining environmental conditions within the zone of nominal losses). Milk and egg production by mature animals also shows an optional thermal environment zone, or zone of nominal losses (Figure 2). Developed from actual measurements of swine growth, Fig-ure 1B shows the relationships of energy, growth, and efficiency with air temperature. In the case of growing pigs in Figure 1B, the range of temperatures from 15 to 22°C, which includes both optimal productivity and efficiency levels, represents acceptable design conditions to achieve maximum performance and effi-ciency. Even beyond that temperature range, performance and efficiency do not markedly decline in the growing pig until near the lower critical temperature (LCT) or upper critical tempera-ture (UCT), and potential performance losses within the tempera-ture range from 10 to 25°C may be acceptable. Response relationships, as shown in Figure 1B, allow environmental selec-tion and design criteria to be based on penalties to performance (i.e, economic costs) and animal well-being—particularly when used with climatological information to evaluate risks for a par-ticular situation (Hahn et al. 1983). Choosing housing requires caution, because research indicates that factors such as group versus individual penning, feed intake, and floor type can affect the LCT by 5 K.
The limits of acceptable values of the LCT and UCT depend on such effects as the species, breed, genetic characteristics of an individual animal’s age, mass, sex, level of feeding and type of feed, prior conditioning, parasites, disease, social factors such as space allocation, lactation or gestation, and physical features of the environment. The LCT and UCT vary among individuals; data reported are for group means. As a result, the limits become statistical values based on animal population and altered by time-dependent factors.
Acceptable conditions are most commonly established based on temperature because an animal’s sensible heat dissipation is largely influenced by the temperature difference between the animal’s sur-face and ambient air. Humidity and air movement are sometimes included as modifiers for an effective temperature. This has been a logical development. Air movement is a secondary but influential factor in sensible heat dissipation. The importance of air velocities is species and age dependent (e.g., swine under 8 weeks of age expe-rience slower gains and increased disease susceptibility when air velocity is increased from 125 to 250 mm/s).
With warm or hot ambient temperature, elevated humidity can restrict performance severely. Relative humidity has little effect on the animal’s heat dissipation during cold temperatures, and it is usu-ally only moderately important to thermal comfort during moderate temperatures. (Information is limited on such interactions, as well as on the effects of barometric pressure, air composition, and ther-mal radiation.) Animals housed in a closed environment alter air composition by reducing oxygen content and increasing carbon dioxide and vapor content. Decomposing waste products add methane, hydro-gen sulfide, and ammonia. Animal activities and air movement add microscopic particles of dust from feed, bedding, and fecal mate-rial. Generally, a ventilation rate sufficient to remove water vapor adequately controls gases. However, improper air movement pat-terns, certain waste-handling methods, and special circumstances (e.g., disease outbreak) may indicate that more ventilation is nec-essary. In many cases, ventilation is not as effective for dust con-trol as for gas control. Alternative dust control strategies may be needed.
The preparation of this chapter is assigned to TC 2.2, Plant and Animal Environment, with cooperation of Committees SE-302 and SE-303 of the American Society of Agricultural Engineers.
10.2 2001 ASHRAE Fundamentals Handbook (SI) ANIMAL CARE/WELFARE Animal facilities that facilitate good animal care and welfare must be designed considering a wide range of environmental factors beyond thermal conditions. These include space requirements, floor-ing type, lighting, feed and water requirements, animal handling, and waste management. Facilities that meet animal care and welfare needs vary considerably by purpose, animal species, and geographic location. For additional information and guidelines for designing animal facilities to meet animal care and welfare needs, see the sec-tion on Bibliography.
PHYSIOLOGICAL CONTROL SYSTEMS An animal has a phenomenally stable control system. Despite wide variations in environmental, nutritional, or pathological con-ditions, animals can control blood pressure and composition, body temperature, respiration, and cardiac output without conscious effort. Physiological control systems react to unfavorable condi-tions to ensure survival of the animal at the expense of production or reproduction.
One physiological control system important for air-conditioning design is the homeothermic system—the means by which an animal adapts to its thermal environment. The animal strives to control body temperature by adjusting both heat production in the tissues and transfer of heat to the environment from exposed surfaces. The homeothermic system in domestic animals is a closed-loop system and can be analyzed like any closed-loop control system.
Heat Production Heat production data have been measured for many farm animals (Yeck and Stewart 1959, Longhouse et al. 1960, 1968, Bond et al.
1959). Much of the early data were obtained for a basal condition (i.e., all life processes at a minimum level). Heat production under conditions of normal metabolic activity is more useful to the engi-neer. Such data are now available for most farm animals (Figures 3 and 4). Where possible, the data show total or sensible and latent heat for animals fed in a typical housing situation. Thus, the sensible heat used to evaporate urine moisture (assuming no other moisture) appears as latent heat. For design purposes, such data are better reflections of total heat partitioning than metabolic heat production obtained by calorimetry. Scott et al. (1983) provide additional infor-mation on specific situations.
The rate of heat production is primarily a function of tempera-ture, animal species, and animal size. Heat production varies diur-nally, depending on animal activity and eating times, and may change dramatically under special circumstances (e.g., when ani-mals are disease-stressed). Building type and waste-handling sys-tem can affect the conversion of sensible to latent heat by more than 50%. Heat and moisture production of animals changes with other Fig. 1 Energetic and Performance Relationships Typical for Animals as Affected by Effective Environmental Temperatures (Hahn et al. 1983) Fig. 2 Comparative Effect of Air Temperature in Mature Animal Production (Hahn et al. 1983, Tienhoven et al. 1979) Environmental Control for Animals and Plants 10.3 variables such as their genotype and growth rate. For example, a broiler chicken grows to 4 kg in 7 weeks compared to 12 weeks 30 years ago. Faster growth rate is usually associated with higher heat production, so updated data is needed. Therefore, the ranges of ven-tilation require careful analysis.
Heat Transfer to Environment An animal can control, to some extent, the amount of heat trans-ferred from its body by chemical and physical regulations. In a cold environment, the animal’s metabolism increases, which increases the amount of heat production, offsetting heat transfer to the environ-ment. (The principal result of such chemical regulation is inefficient feed utilization.) Physical regulation is also used in a cold environ-ment; blood circulation to subcutaneous capillaries decreases, hair or feathers are erected, and animals huddle together in an attempt to reduce sensible heat loss. In a hot environment, opposite physical and physiological responses generally occur to enable transfer to the environment of heat associated with necessary and productive life processes. In addition, as sensible heat dissipation becomes more difficult, evaporative heat loss increases and air moisture content becomes a factor in heat loss. Since the production of heat is a necessary by-product of growth and useful production, environmen-tal limits to dissipating this heat cause a decrease in feed consumption and subsequent decrease in growth and production.
Most sensible heat of domestic animals is dissipated through the skin. Birds, sheep, and swine transfer most of their latent heat through the respiratory tract; cattle and horses transfer most of their latent heat through the skin. Blood transfers the heat produced by metabolism to the skin or to the lung surfaces, where it is dissipated by evaporation from the mucous layer coating the inside of the alve-oli. Inspired air reaching the alveoli is heated almost to body tem-perature and may become nearly saturated with moisture. Expired air may not be saturated at body temperature—especially during periods of heat stress—because it is a combination of air, some of which has not reached deeply into the lungs. Minor amounts of heat are also transferred by ingestion of feed and water and through excretion.
Design factors affecting animal heat loss are (1) air temperature; (2) air vapor pressure; (3) air movement; (4) configuration, emissiv-ity, absorptivity, and surface temperature of the surrounding shelter; and (5) temperature and conductivity of surfaces (e.g., floors with which the animal may be in contact).
Cyclic Conditions The physiologic sensing elements respond both to environmental conditions and to changes in those conditions. Cycles of tempera-ture, pressure, light, nutrients, parasites, magnetic fields, ionization, and other factors frequently occur with little engineering control.
Light is perhaps the earliest discovered and most important con-trolled environmental variable affecting reproductive processes (Farner 1961). However, the effects vary widely among animal species and age. Some animals grow and remain healthy with or without light (e.g., growing swine), while for others, lighting man-agement is important. Studies have shown that (1) short photoperi-ods induce or accelerate estrus development in sheep; (2) day length affects semen production in sheep and horses (Farner 1961); (3) continuous white incandescent light during incubation caused White Leghorn eggs to hatch from 16 to 24 h earlier than eggs incu-bated in darkness (Shutze et al. 1962); and (4) red, yellow, or blue lights gave comparable results. Use of lights for only one of the three weeks also reduced the time required for hatching, but differ-ences were not as marked. Percent hatchability was not affected by lighting treatments.
Light is used to delay sexual maturity in hens, which enhances subsequent production. This is done by gradually decreasing day length from hatching to 22 weeks of age or by abruptly decreasing day length to 9 h at 14 to 16 weeks of age. If pullets have previ-ously been exposed to an increasing day length, the change should take place at 14 weeks of age; if exposed to a constant or decreas-ing schedule, a change at 16 weeks is adequate. Light intensities of 10 to 20 lx measured at bird height were adequate in all cases.
Light can then be abruptly increased at 21 weeks of age to 14 or 16 h of light. The economic value of increasing day length beyond 14 h in a windowless poultry house, and 16 h in an open poultry house, has not been proved. Photoperiods such as 8 h of light (L), 10 h of darkness (D), 2 L and 4 D, and other cycles have improved feed utilization, egg production, and poultry growth (Buckland 1975, Riskowski et al. 1977).
Continuous light from hatching through 20 to 21 weeks of age markedly depressed subsequent egg production and caused a severe eye abnormality, but did not depress egg mass. (Light intensity was 10 to 30 lx, measured at bird height.) Temperature cycles have been studied in cattle (Brody et al.
1955, Kibler and Brody 1956), swine (Bond et al. 1963, Nienaber et al. 1987), and poultry (Squibb 1959). The results differ some-what among animals. Heat loss from cattle and swine can usually Fig. 3 Comparative Heat Loss of Mature, Producing Animals (Tienhoven et al. 1979) Fig. 4 Comparative Heat Loss of Growing Animals (Bond et al. 1959) 10.4 2001 ASHRAE Fundamentals Handbook (SI) be calculated from average daily environmental temperatures with sufficient accuracy for design load calculation.
Productivity is only slightly different for averaged temperatures when cyclic conditions with a range less than 10K are experienced (Squibb 1959). Above that range, productivity is depressed below that expected from the averaged temperature as determined in con-stant temperature tests, but under diurnally varying air conditions ( 15 to 25°C temperature and widely varying humidities), near-nor-mal egg production is maintained.
Air Composition and Contaminants The major contaminants in livestock housing are (1) respirable dusts from feed; manure; and animal skin, hair, and feathers; (2) microbes, both pathogenic and nonpathogenic, hosted in the respi-ratory tracts, animal wastes, or feed; and (3) several noxious gases of various concentrations. Respirable dust particles have diameters between approximately 0.5 and 5 µm. Gases are produced from the metabolic processes of animals, from the anaerobic microbial deg-radation of wastes, and from combustion processes. The gases of most concern are ammonia, hydrogen sulfide, carbon monoxide, and methane.
Dust levels in animal housing are high enough to create a nui-sance in and near animal buildings, increase labor requirements for building and equipment maintenance, and interfere with the perfor-mance of heating and ventilating equipment. Dust has been impli-cated in poultry building fires. Dust is generated primarily by feed handling and increases with animal activity and air movement caus-ing reentrainment of settled dust.
Contaminants are a concern because they predispose animals to disease and poorer performance and affect operator health. Animals experiencing stress (e.g., newborns, hot or cold, nutritionally lim-ited, and sick animals) are more sensitive, and the presence of low levels of contaminants can have adverse effects on them.
Particulates (dust, endotoxins, live and dead microorganisms) and gases (ammonia, carbon dioxide, and hydrogen sulfide) in swine buildings (Zejda et al. 1993, Senthilselvan et al. 1997) have been implicated as contributors to the increased incidence of respiratory disorders among livestock producers compared to grain farmers and non-farm workers (Donham et al. 1989, Dos-man et al. 1988). Young farmers may be at particular risk of de-veloping chronic bronchitis, coughing, wheezing, toxic organic dust syndrome, and/or occupational chronic pulmonary disease (Zejda et al. 1994). For example, 33% of pork producers in Europe suffer chronic respiratory symptoms related to poor in-door air quality (Van’t Klooster 1993). In the Netherlands, 10% of swine producers had to change jobs due to severe respiratory problems caused by poor air quality (Preller et al. 1992). In cold climates, the problems appear to be more serious because of (1) larger building size and longer working hours and (2) lower air exchange rates due to cold climates and energy conservation concerns.
Particle size is most commonly described using aerodynamic diameter, which is defined in terms of its aerodynamic behavior rather than its geometric properties. Aerodynamic diameter is a function of its equivalent geometric diameter, density, and a shape factor. The aerodynamic diameter of a particle is the diameter of a spherical water droplet having the same settling velocity as the par-ticle of concern. Size distribution of dust particles influences the dust transportation and its effect on a human respiratory system. The human eye can only see a particle larger than 50 µm under normal light. For evaluating animal buildings, particle size is often divided into the following categories: • Total dust = all sizes of particles suspended in the air • Inhalable dust = all sizes of particles that are inhaled by a sampling instrument • Respirable dust = particles smaller than 10 µm In animal facilities, ultrafine particles (< 0.5 µm) are usually excluded from the evaluation because they are primarily back-ground dust and not the concern of the indoor environment. If the dust concentration is measured in counts rather than mass, the ultra-fine particles can dominate over the counts of the other particle sizes and cause large errors (Zhang et al. 1994).
Dust in animal buildings is different from other types of build-ings in at least three aspects. First, animal building dust is biologi-cally active in that it contains a variety of bacteria, microorganisms, and fungi (Martin et al. 1996, Wiegard and Hartung 1993). Second, its concentration is high, typically more than ten times higher than office buildings and residential buildings. Third, it is an odor carrier.
Dust sources and flora are summarized in Table 1. Of those microbes, some are transmittable to people. For example, Listeria monocytogenes and Streptococcus suis are zoonotic agents that have caused fatal diseases in people (Fachlam and Carey 1985, Bor-tolussi et al. 1985).
Table 1 Dust Sources and Flora in Swine Buildings Dust Sources Feed particles: Grain dust Antibiotics Growth promoters Swine protein: Urine Dander Serum Other agents: Swine feces Mold Pollen Grain mites, insect parts Mineral ash Gram-negative bacteria Endotoxin Microbial proteases Ammonia adsorbed to particles Infectious agents Microbial flora Gram-positive cocci: Staphylococcus species (coagulase-negative) Staphylococcus haemolyticus Staphylococcus hominis Staphylococcus simulans Staphylococcus sciuri Staphylococcus warneri Micrococcus species Aerococcus species Streptococcus equines Streptococcus suis (presumptive) Enterococcus durans Gram-positive bacilli: Corynebacterium species Corynebacterium xerosis Bacillus species Gram-negative bacilli: Acinetobacter calcoaceticus Nonfermentative gram-negative bacillus: Enterobacter calcoaceticus Pasteurella species Vibrio species Fungi: Alternaria species Cladosporium species Penicillium species Source: Zhang (1995, 1996).
Environmental Control for Animals and Plants 10.5 Threshold limit values for different types of exposure are avail-able (Table 2). These limit values are likely additive. For example, someone exposed to 75% of maximum ammonia level may be exposed to no more than 25% of the maximum for any other contam-inant or combination of contaminants. Due to its biological nature, a lower limit for animal building dust may be warranted. For example, Donham et al. (1989) proposed a more stringent threshold TWA value of 0.23 mg/m3 respirable dust for farm animal buildings.
Airborne dust and microbes have been linked to respiratory disease in cattle, poultry, swine, horses, and laboratory animals.
Infectious diseases, such as viral pneumonia and diarrhea, in calf barns have caused mortality rates between 20 and 80%, with the highest mortality occurring in enclosed barns that were underven-tilated during cold weather. Many respiratory diseases of poultry have been found to be transmitted via pathogenic microbes that can travel over 50 km and remain infectious for months (Sieg-mund 1979). Respiratory diseases are also costly to the swine industry—the incidence of enzootic pneumonia in pigs has ranged from 30 to 75%.
Gaseous ammonia is frequently a contaminant causing serious problems in swine and poultry housing. Ammonia levels of 50 ppm have been shown to reduce body mass gain rates of swine (Curtis 1983), and many researchers suspect that adverse health effects begin at levels below 25 ppm (18 mg/kg). Some physiolo-gists and veterinary science researchers suggest that the level is between 5 and 10 ppm (3.5 and 7 mg/kg) (Donham 1987). Ammo-nia at concentrations greater than 60 ppm (45 mg/kg) has been implicated in reduced body mass and feed consumption, and in increased ocular and respiratory problems with broiler chickens (Carr and Nicholson 1980). Ammonia levels are influenced by animal diet, the ventilation rate, sanitation practices, and the waste-handling system. Ventilating for moisture control is usually adequate to control ammonia below 10 ppm (7 mg/kg) in build-ings sanitized monthly, where wastes are not allowed to accumu-late for more than two weeks and where manure solids are covered by water.
Hydrogen sulfide is produced mainly from the wastes and is present at toxic levels that usually cause problems only when liquid wastes are agitated or pumped and emitted gases can filter into the animal occupied area. Hydrogen sulfide production can rise to lethal levels when manure in a pit below the animals is agitated, so extra precautions are necessary to ensure that ventilation is suffi-cient to remove the contamination.
Carbon monoxide is usually produced by malfunctioning fuel-burning heaters and internal combustion engines used for spray washing equipment. Many heaters are unventilated, which, when combined with the corrosive environment in livestock housing, makes this equipment susceptible to failure. Heaters should be vented to the outdoors or sensors should be installed to ensure com-plete combustion.
PRINCIPLES FOR AIR CONTAMINANT CONTROL Reducing dust concentration by 85% or reducing dust inhalation by wearing a respirator improves human respiratory responses sig-nificantly (Senthilselvan et al. 1997, Zhang et al. 1998, Barber et al.
1999). Air contaminant control strategies include source suppres-sion, air cleaning, and effective ventilation.
Source Suppression Spreading (e.g., spraying, sprinkling, or fogging) a small quan-tity of oil, water, emulsifier or mixture of these can reduce dust concentration in the air (Takai et al. 1993, Zhang et al. 1995, 1996). Adding oil or fat to feed (Heber and Martin 1988) reduces emission of large feed dust particles. Other management prac-tices, such as wet feeding, power washing, and maintaining sani-tation in the facility also reduces the production and emission of air contaminants.
Air Cleaning Generally, air contaminants can be removed mechanically, elec-trically, chemically, or biologically. Mechanical air cleaning in-cludes filtration, scrubbing, and aerodynamic separation (cyclones or dust separators). Existing mechanical air cleaning methods, in-cluding fiber filters, require large quantities of airflow through the filtration media. For many dusty environments such as found in an-imal buildings, this type of cleaner requires frequent maintenance.
Cyclones use centrifugal force to separate particles from the main air stream. But a high pressure drop (typically higher than 500 Pa) across the cyclone is required to create the cyclone effect.
This high pressure requires a large amount of power and creates high turbulence, which reduces the particle separation efficiency.
Carpenter (1986) concluded that it is impractical to separate parti-cles smaller than those that cause health concerns in animal build-ings.
Electrical air cleaning includes ionization, electrostatic precip-itation, and ozonation. The principle of the three technologies is to charge a dust particle with polarized electrical particles (ions or electrons) so that the charged dust particles can be attracted to a neu-trally or opposite charged surface. Challenges of the ionization tech-nology for livestock buildings include dust removal efficiency, initial and maintenance costs, and a potential problem of static elec-tricity when used under cold and dry weather conditions.
The air cleaning efficiency of an electrostatic precipitator is pro-portional to the airflow rate through the precipitator. A large airflow rate in an airspace can reentrain a large amount of settled dust and thus create a high dust concentration in the air. Dust reduction effi-ciency, airflow rate, and power consumption of the air delivery through the precipitator must be balanced for optimum perfor-mance. Similar to air filters, an electrostatic precipitator requires frequent cleaning and maintenance. When dust particles accumulate on the precipitation surfaces of the device, the air cleaning effi-ciency rapidly decreases.
Ozone can effectively reduce airborne microorganisms in an ani-mal building. Care must be taken to maintain an appropriate con-centration because ozone is a potential irritant to humans and animals.
Ventilation Requirement Ventilation is the primary method for controlling air quality in confinement buildings as it brings in fresh air to dilute the contam-inants. The logic of ventilation is illustrated in Figure 5. Unlike res-Table 2 Gas and Dust Threshold Values for People Exposure Contaminant Specific Densitya Odor TWAb STEVb FEVc Toxic gases (ppm in volume) Ammonia 0.6 sharp, pungent 25 35 300 Hydrogen sulfide 1.19 rotten eggs, nauseating 10 15 150 Methane 0.5 none n/a n/a 500 000 Carbon dioxide 1.53 none 5 000 30 000 50 000 Dust (mg/m3) Respirable 175 n/a n/a aRelative to the density of air.
bFrom American Conference of Governmental Industrial Hygienists (ACGIH 1998).
cFrom Canada Plan Service, M-8710 (1985), M-9707 (1992).
n/a = currently not available TWA = time-weighted average, for 40 hours per week STEV = short-term exposure value for not exceeding 30 minutes FEV = fatal exposure values 10.6 2001 ASHRAE Fundamentals Handbook (SI) idential or office buildings, the minimum ventilation for animal buildings is often determined by humidity or contaminant (such as ammonia) balances. Therefore, ventilation requirements must be calculated for sensible heat, humidity, and a given contaminant to determine the minimum ventilation for an animal building.
Volumetric ventilation requirement for sensible heat balance Vs, moisture balance Vw, and contaminant balance Vc (in m3/s) can be calculated as follows: (1) (2) (3) where ν = specific volume of air, m3/kg qn = net heat transfer rate through the building shelter, kW hs = sensible heat content of room air, kJ/kg hso = sensible heat content of outside air or supply air, kJ/kg w = moisture content of room air, kg/kg of air wo = moisture content of outside air, kg/kg of air Wn = production rate of moisture within the room, kg/s c = contaminant concentration of room air, kg/kg of air co = contaminant concentration of outside or supply air, kg/kg of air Mc = production rate of the contaminant in the room, kg/s Ventilation Effectiveness Two terms are often used to evaluate a ventilation system: ventilation efficiency and ventilation effectiveness. Ventilation efficiency is a criterion for energy and fan performance; it is not directly related to ventilation effectiveness and air quality control.
Ventilation efficiency refers to the mass of air delivered per unit of power consumed by the ventilation system at a given pressure.
The more air delivered per unit of power, the more efficient the ventilation system. It is also called the ventilation efficiency ra-tio (VER) or energy efficiency ratio (EER) (Pratt 1983).
Zhang et al. (2000) proposed a ventilation effectiveness factor (VEF) to quantitatively describe the ventilation effectiveness for controlling indoor air quality. The VEF can be defined as the ratio of two contaminant concentration differentials: (4) where Cx is the contaminant concentration under complete mixing conditions, which can be determined based on the mass balance of the contaminant of concern. For a ventilated airspace at steady state, the amount of contaminant removed by the exhaust air must equal the sum of the amount of contaminant brought in by the supply air plus the contaminant produced in the airspace, regardless of com-plete or incomplete mixing conditions. The contaminant concentra-tion of the exhaust air Ce can be measured. Under complete mixing conditions, Ce is the same as the Cx; thus, (5) Cs is the contaminant concentration of the supply air, Cm is the mean contaminant concentration of the airspace at n measured locations.
(6) Substituting Equations (5) and (6) into Equation (4), ventilation effectiveness factor becomes (7) The ventilation effectiveness factor is nondimensional and inde-pendent of the ventilation rate. The higher the value of the factor, the more effective the ventilation system for contaminant removal.
When Ce < Cs, the airspace becomes a settling chamber or an air cleaner and the air cleaning efficiency ξ = 1 − Ce/Cs should be used to evaluate the system. When the contaminant concentration of sup-ply air is negligible, i.e., Cs ≈ 0, the ventilation effectiveness factor becomes (8) One useful feature of VEF is that the system being evaluated can be its own control. This feature is particularly valuable for evaluat-ing or troubleshooting in the field. From Equation (4) and Equation (8), the VEF compares the actual contaminant concentration with its own concentration under complete mixing conditions in the same airspace. The existing methods of evaluating ventilation effective-ness often require comparing two systems. In practice, it is usually difficult to find an identical system for comparison.
A ventilation effectiveness map (VEM) such as shown in Fig-ure 6 is a useful tool for design, analysis, and modification of a ven-tilation system. It is a contour plot of the ventilation effectiveness factors calculated using Equation (8) for the entire airspace. Dust mass was measured concurrently at several locations throughout the building cross section and the VEF values were calculated from that Fig. 5 Logic for Selecting the Appropriate Ventilation Rate in Livestock Buildings (Christianson and Fehr 1983) Vs qnν hs hso – ( ) -----------------------= Vw Wnν w wo – ( ) ---------------------= Vc Mcν c co – ( ) ------------------= VEF Cx Cs – Cm Cs – -------------------= Cm Cs and Cx Cs > > ( ) Cx Ce = Cm 1 N ---Ci i=1 N ∑ = VEF N Ce Cs – ( ) Ci NCs – i=1 N ∑ -----------------------------= Ce Cs > ( ) VEF Ce Cm -------= Environmental Control for Animals and Plants 10.7 data. The higher the VEF values in the map, the more effective the ventilation system is in that area.
Improved ventilation effectiveness is an important strategy to reduce contaminant concentration in animal buildings. It is very expensive to clean large quantities of air and it is difficult to move the air through central cleaning equipment without creating drafts.
Incomplete mixing ventilation such as displacement and zonal ventilation has great potential in creating a cleaner zone in a dusty airspace without changing the overall ventilation rates.
Interactive Stressors Thermal (air temperature, air humidity, air velocity, and sur-rounding surface temperatures) and air quality (relative absence of dust, microbes, and contaminant gases) are the two stressors of most concern to ventilating a livestock structure. However, health status, animal age, stage of production, nutrition, and social conditions interact with the thermal and air quality conditions to determine ani-mal health and performance.
Whether stressors are linearly additive or otherwise is an impor-tant research area. McFarlane (1987) found that ammonia, heat, acoustic noise, disease, and beak-trimming stressor effects on chickens were linearly additive in effect on the feed efficiencies and daily gain rates.
Comparing Individual Animal and Room Heat Production Data Liquid wastes from urine, manure, waterer spillage, and clean-ing affect the sensible and latent heat fraction in a room (ASAE 1991). More water on the floors requires more thermal energy from the animals or the building heating system to evaporate the mois-ture. This evaporated moisture must then be removed by ventila-tion to prevent condensation on building surfaces and equipment and minimize adverse health effects, which can occur when humid-ities exceed 80%.
Floor flushing or excessive water spillage by animals can increase latent heat production by one-third from animal moisture production data, with a corresponding reduction in sensible heat production on a per-animal basis (ASAE 1991). Partially slatted floors with underfloor storage of wastes reduce the room moisture production by approximately 35%, compared to solid floor systems.
Rooms with slatted floors throughout the building may have mois-ture production rates as much as 50% lower than solid floors. There-fore, slatted floors reduce the ventilation rate required for moisture removal, while excessive water spillage increases the moisture removal ventilation rate.
Influence of Genetic Change and Breed on Heat Production Researchers have noted significant genetic correlations with per-formance. For example, swine breeds with high fat content charac-teristics outgained low fat content breeds, but the low fat content lines used feed more efficiently (Bereskin et al. 1975). Barrows (male hogs castrated before sexual maturity) ate 6% more than gilts (young sows that have not farrowed) to sustain a 7% faster growth rate.
Genetic improvements within breeds suggest that performance data should be updated periodically. Between 1962 and 1977, for example, the average gain rate for swine increased 70 g per day, and feed efficiency improved 15% for Missouri hogs (Thomeczek et al.
1977).
CATTLE Growth Figure 7 shows the general growth rate for both beef and dairy breeds. Efficiency of beef calf growth is of economic importance; the dairy calf is developed for adult productive and reproductive capacity. Fattening calves and yearlings could be fed increasing amounts of grain and hay, with an expected gain of about 0.9 kg per day. Figure 8 shows the effect of temperature on the growth rate of several breeds of beef calves.
Food and energy requirements for growth can be computed by taking the difference between the energy in the available ration con-sumed and the energy required for maintenance and growth. Energy requirements for maintenance at an air temperature of 20°C has been reported to be 9.13 kJ/h per kilogram of body mass (Blaxter and Wood 1951), 8.42 kJ/h (Bryant et al. 1967), and 8.80 kJ/h (Gebremedhin et al. 1983). Using 3413 kJ/kg body tissue for con-version of feed energy to body tissue (Brody 1945), the growth rate of a calf at 21°C (0.54 g/h gain per kilogram of bodymass) can be nearly 30 times that of a calf being kept at 3°C [18 mg/(h·kg)].
Alternatively, to get the same growth rate at 3°C as at 21°C, the same ration must be fed at the rate of 11.6% of body mass each day, a 16% increase (Gebremedhin et al. 1981).
Lactation Figure 9 illustrates the effect of high temperature on the milk production of one Holstein in a test designed to study the combined 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 X, m 0.0 0.5 1.0 1.5 2.0 INLET OUTLET Y, m Fig. 6 Typical Ventilation Effectiveness Map for a Room (Room is 5.5 m long by 3.7 m wide by 2.4 m high with slot air inlet at top of one wall and air outlet near bottom of same wall.) Fig. 7 General Growth Curves for Calves Fed Grain (Johnson et al. 1958, Ragsdale 1960) 10.8 2001 ASHRAE Fundamentals Handbook (SI) effects of temperature and humidity. The dotted line represents the normal decline in milk production (persistency) from an advancing stage of lactation.
The ideal environment for Holstein cattle should not exceed 24°C. Jerseys are somewhat more heat tolerant, and their limit can be 27°C (Yeck and Stewart 1959). At the lower end of the temperature scale, production decreases may be expected at −1°C for Jerseys and below −12°C for Holsteins and Brown Swiss.
A temperature-humidity index (THI) expresses the relationship of temperature and humidity to milk production. Temperature-humidity indexes have a greater effect on cows with a genetic potential for high milk production than for those with a lower potential (Figure 10).
Under high temperatures, a cooling system can help maintain milk production and reproductive function. In field studies con-ducted by Bucklin and Turner (1991) in both warm and hot, humid climates, a system providing evaporative cooling by forced air movement and direct sprinkling of lactating cows increased milk production by a range of 7 to 15% over cows not exposed to cooling.
Reproduction Prolonged low temperatures, even those well below freezing, do not affect the reproductive performance of farm livestock, but breeding efficiency of both sexes decreases under summer condi-tions. Sustained temperatures above 30°C may decrease fertility, sperm production, and semen quality of males, and increase anestrus and embryonic death in females. In bulls, temperatures above 24°C decrease spermatogenesis, and long exposures at 30°C or above cause temporary sterility. The extent of the reaction depends on temperature rise and exposure duration.
In females, Guazdauskas (1985) observed that conception rates decreased from a range of 40 to 80% conception in thermoneutral environments (10 to 22°C) to a range of 10 to 51% conception in hot environments (>27.5°C). A cooling system involving a cooled shade and artificially induced air movement increased breeding efficiency by approximately 100% in a hot climate (Wiersma and Stott 1969).
Heat and Moisture Production Sensible and latent heat production from individual animals (Figure 11) differ from the stable heat and moisture production of animals (Figures 12 and 13). The data in Figures 12 and 13 were obtained while ambient conditions were at constant temperatures and relative humidities were between 55 and 70%. The effects of evaporation from feces and urine are included in these data.
The rate of cutaneous water loss is very small at colder temper-atures but rises sharply above 18°C. Cutaneous evaporation as a means of heat loss in calves becomes increasingly important as the air temperature rises above 24°C. As the air temperature rises, the Fig. 8 Daily Mass Gain of Beef Calves (Longhouse et al. 1960) Fig. 9 Milk Production Decline of One Holstein During Temperature-Humidity Test (Yeck and Stewart 1959) Fig. 10 Effect of Milk Production Level on Average Decline in Milk Production, Versus Temperature-Humidity Index (Johnson et al. 1962) Environmental Control for Animals and Plants 10.9 proportion of nonevaporative cooling decreases. Above 30°C, about 80% of the heat transferred is by evaporative cooling.
Gebremedhin et al. (1981) observed that cutaneous water loss from calves varied between 0.06 and 0.4 g/h per kilogram of body mass for air temperatures between 0 and 18°C and increases steadily beyond 18°C (Figure 14). Water loss by respiration, the other ave-nue of loss, is shown in Figure 15.
SHEEP Growth In normal environments, an average daily gain of 0.25 kg per day can be expected of lambs marketed at 40 to 60 kg, depending on breed. Hampshires often gain more than 0.5 kg per day, and lambs can be fed to have a mass over 90 kg while still under a year old. Variations exist among breeds and strains of sheep in their ability to adjust to environmental changes. Temperature effects on sheep growth suggest a lower rate at elevated temperatures. A South Afri-can study shows that lack of shade in warm climates reduces growth rate. Australian studies indicate that Merino lambs survive for only about 2 h in an air temperature of 38°C.
Wool Production The amount and quality of wool produced varies considerably among breeds, ranging from about 1.4 kg of poor quality wool from Fig. 11 Sensible and Latent Heat Production per Unit Livestock Mass (Hellickson et al. 1974) Fig. 12 Stable Heat and Moisture Dissipation Rates Dairy Cattle Stanchioned in Enclosed Stables (Yeck and Stewart 1959) Fig. 13 Stable Heat and Moisture Loads for Beef and Dairy Calves in Pens (Yeck 1957, Yeck and Stewart 1960) 10.10 2001 ASHRAE Fundamentals Handbook (SI) Dorset ewes (a breed developed primarily for mutton) to well over 4.5 kg of high quality fleece from dual-purpose breeds. Rambouil-let, Merino, and Columbia-Southdale breeds supplied 3.2 kg per year, and about 2.0 kg was grown by a Hampshire, where monthly shearing increased wool production by about 0.5 kg per year.
Environmental factors such as photoperiod, nutritional level, and temperature also affect wool growth. Skin temperature is consid-ered a dominant factor; high wool growth is associated with high skin temperatures. In a related study, low subcutaneous blood cir-culation limited wool growth. A thick fleece appears to limit radiant heat loads.
Reproduction Sheep mate only during certain periods of the year, but they can sometimes be induced to mate outside the normal season if the natural environment is modified. Ewes may breed if exposed to an air temperature of 7°C before mating is attempted.
In studies on the effects of high temperatures on sheep reproduc-tion, air temperature was reported to affect the spermatogenesis rate. Rams kept at 32°C environmental temperature showed a reduced rate of spermatogenesis, although a 3-week period at 21°C effected recovery to the normal rate. Later reports suggest that high air temperatures may cause lowered fertility for other reasons. Sub-jecting ewes to 38°C just before mating causes fertilization failure because of some form of ovum structure degeneration.
Under practical conditions, early embryonic death appears to be the greatest loss in potential offspring conceived in a high-temper-ature environment. Degree of susceptibility to high temperature is greatest near mating time but generally decreases as the length of time after mating increases. A temperature of 32°C and 65% rh at mating kills most embryos at an early age. The same conditions applied later in gestation do not cause death, but do cause the birth of small, weak lambs. The longer the high-temperature period, especially during the last third of gestation, the greater the number of weak lambs born. All reproductive processes appear to be more adversely affected by high air temperature when accompanied by high relative humidity.
Heat Production Only a few heat-production tests have been conducted, mostly for correlation with other physiological data. A lamb’s ability to generate heat is important because it is normally born during the most severe climatic season. One study indicated newborn lambs were limited in heat production to 19.4 W/kg—five times the basal level. Test conditions were from −10 to 0°C with a 19 km/h wind, but lambs are capable of withstanding temperatures as low as − 40°C. Table 3 provides heat production data for sheep; while the total heat production data are reliable, the latent heat proportions do not reflect the portions of sensible heat used for water and urine evaporation from flooring and bedding.
SWINE Growth Ambient air temperature affects the feed conversion and daily mass of growing swine. As shown in Figure 1B, a temperature range of 16 to 21°C produces maximum rates of gain and feed use for 70 to 100 kg hogs, while a broader range of 10 to 24°C reduces performance only slightly. For 20 to 59 kg animals, the optimal and nominal loss ranges are about 17 to 23°C and 13 to 24°C, respectively (Kibler and Brody 1956). Younger animals require temperatures of 23 to 28°C for best performance, and piglets from 3 days to 2 weeks of age should have 30 to 32°C conditions (Hahn 1983).
Daily air temperature cycles of more than 5 K on either side of 21°C result in reduced daily gain by pigs and an increased feed requirement per unit of gain (Nienaber et al. 1987). Reasonably constant air temperatures are desirable.
The level of air temperature in which swine grow affects deposi-tion and retention of protein (carcass quality). Lean meat formation is reportedly highest in pigs raised between 16 and 21°C (Mount 1963). However, the ratio of protein to fat decreases at air tempera-tures above 16°C.
Swine gains and feed conversion rates are highly sensitive to air velocities and, in many cases, are affected adversely at velocities as low as 0.25 m/s. Swine less than 8 weeks of age should not be Fig. 14 Cutaneous Water Loss per Unit Body Mass of Holstein Calves (Gebremedhin et al. 1981) Fig. 15 Respiratory Water Loss per Unit Body Mass of Holstein Calves (Gebremedhin et al. 1981) Table 3 Heat Production of Sheep Fleece At 8°C At 20°C At 32°C Length, mm Totala Latentb Totala Latentb Totala Latentb Mature, maintenance-fed Shorn 2.6 8% 1.7 12% 1.3 38% 30 1.4 29% 1.3 28% 1.3 65% 60 1.3 23% 1.2 43% 1.2 76% Lambs, 1 to 14 days Normal 6.7 — 5.0 — — — Source: Scott et al. (1983).
aW/kg of body mass bPercent of total heat Environmental Control for Animals and Plants 10.11 exposed to velocities greater than 0.25 m/s; a velocity lower than 0.13 m/s is preferred when temperatures are in the recommended range. Even in hot conditions, pigs are affected adversely by air velocities greater than 1 m/s (Bond et al. 1965, Gunnarson et al.
1967, Riskowski and Bundy 1991). Feed utilization and gain are much better at low air velocities (0.18 m/s) than at high (1.5 m/s) air velocities when temperatures are optimal (Figure 16).
Reproduction Merkle and Hazen (1967) and Heard et al. (1986) have shown that sows benefit from some type of cooling in hot weather. In these studies, cool, dry air was directed at the sow to relieve heat stress by increasing evaporative and convective heat dissipation. Field studies of breeding problems and resultant small litters have established that both sexes suffer losses from high temperature. Sprinkling the sow during hot weather at breeding time and shortly after resulted in more live births than with unsprinkled sows. Sprinkling boars before mat-ing also increased the number of live births per litter. Conception rate varied from about 100% of normal at 21°C to only 70% at 32°C.
Breeding difficulties and a decrease in live embryos were observed in tests with controlled temperatures and relative humidities (Roller and Teague 1966). In some species, spontaneous abortion under severe heat stress may save the mother’s life; however, sows appear to die of heat prostration, due to extra metabolic heat generation in late pregnancy, before spontaneous abortion occurs.
Little information is available on the value of temperature control for sows during cold weather. Forcing sows to produce the necessary heat to maintain body temperature may require rigid nutritional management to avoid the animals from becoming overweight—a condition resulting in poorer conception and smaller litters. Heat and Moisture Production Table 4 (Butchbaker and Shanklin 1964, Ota et al. 1982) shows heat production for piglets. Figure 17 and Figure 18 (Bond et al.
1959) show heat and moisture that must be accounted for in venti-lating or air conditioning older swine housing for swine between 20 and 180 kg, housed at temperatures from 10 to 32°C. These data are the sensible and latent heat levels measured in a room containing hogs. For design purposes, theroom heat data reflect swine housing conditions rather than metabolic heat production from an individual animal.
The same data can be applied even where building temperatures are cycling, if the average air temperature during the cycle is used for design. The total animal heat load and the latent load that the ventilation system must remove are greater than the values in Figure 17 and Figure 18, if the air velocity around the animals is more than about 0.25 m/s.
The lower critical temperature (the ambient temperature below which heat production increases) is about 25 to 30°C for a group of newborn pigs; for a single piglet, the critical temperature is about 34 to 35°C. As the pigs grow, the critical temperature falls—for 2 to 4 kg pigs, it is between 30 and 35°C; for 4 to 8 kg pigs, between 25 and 30°C. The floor bedding materials for growing piglets influence these critical temperatures. For a group of nine pigs averaging 40 kg, the critical temperature is 12 to 13°C on straw, 14 to 15°C on asphalt, and 19 to 20°C on concrete slabs (Butchbaker and Shanklin 1964).
Table 4 Heat Production of Grouped Nursery Pigs Weight Range, kg Temperature, °C Total Heat Production, W/kg Latent Heat, % of Total 4 to 6 29 5.1 33 6 to 11 24 7.0 31 11 to 17 18 7.8 30 Source: Ota et al. (1982).
Fig. 16 Swine Response to Air Velocity (Riskowski and Bundy 1986, Bond et al. 1965) Fig. 17 Room Sensible Heat in Hog House (Bond et al. 1959) Fig. 18 Room Latent Heat in Hog House (Bond et al. 1959) 10.12 2001 ASHRAE Fundamentals Handbook (SI) CHICKENS Growth The tremendous worldwide production of broiler chickens is a result of improvements in genetics, nutrition, and housing. Two-kilogram broilers are produced in less than 6 weeks. Intermittent light and darkness (photoperiod) improve growth and conserve feed and energy (McDaniel and Brewer 1975).
Supplemental heat is generally needed for newly hatched chicks.
Broiler strains respond well to brooding temperatures of 33°C for the first few days. Thereafter, air temperatures may be decreased at a rate of 4 K per week until 21°C is reached. Relative humidities of 65 to 70% promote good feathering and market-quality broilers.
Effects of air velocity on heat dissipation of broilers is widely acknowledged, although not completely quantified, and has led to the widespread adoption of tunnel ventilation in new housing. Some benefit from increased velocity (up to 2.5 m/s) is obtained at air tem-peratures from 24 to 35°C (Drury 1966). High-velocity air, at a tem-perature above the feather temperatures, causes more, not less, heat stress. Observations showed that 28-day-old broilers rested in areas with airflow temperature combinations of 0.28 m/s at 15°C, 0.5 m/s at 21°C, and 0.75 m/s at 24°C.
Reproduction Adverse effects of high thermal environments (above 29°C) on egg production include fewer eggs, reduced egg mass, and thinner shells. Hens over 1 year of age are more adversely affected by high thermal environments than younger hens. Larger hens are more adversely affected than smaller hens. Hatchability also declines as temperatures increase. Although the fertility rate is good at 21°C, it declines at 30°C. Feed requirements increase markedly below 7°C; activity and productivity decline below 0°C. The suggested ideal environment is between 13 and 24°C. Increasing relative humidity above 80%, at 29°C dry-bulb temperature, produces an increased respiration rate and drooping wings. After 7 to 10 days, acclimatization may reduce the respi-ration rate, but production will still be less than optimal. Gener-ally, relative humidity can vary from around 40% to 75% with little impact on bird performance. Lower relative humidities may be associated with dusty conditions and signal excessive mini-mum ventilation; higher relative humidities are conducive to mold growth and building deterioration from excess moisture.
Heat and Moisture Production Figure 19 shows the total sensible and latent heat produced by laying hens during day and night at various temperatures. Figure 20 Fig. 19 Heat and Moisture Loads for Caged Laying Hens at Various Air Temperatures (Ota and McNally 1961) Fig. 20 Sensible, Latent, and Total Heat for Chicks Brooded on Litter (Reece and Lott 1982) Fig. 21 Sensible and Latent Heat for Broilers Raised at 27°C on Litter (Reece and Lott 1982) Environmental Control for Animals and Plants 10.13 shows the same data for chicks. Figure 21 and Figure 22 show sen-sible and latent heat production for broilers grown on litter at typical stocking rates at two temperatures. The effects of heat and moisture absorption or release by the litter are included in these figures.
(Note: modern bird growth rates have increased approximately threefold since the work reported in Figures 20, 21, and 22 was per-formed.) The continued improvement in nutrition, genetics, and housing for poultry means that currently available data for heat and moisture production is outdated. Research by Gates et al. (1996) suggests that total specific heat production (i.e., expressed per kilogram of body mass) may be reasonably similar, but that partitioning between sen-sible and latent components has changed. In general, latent loads have increased during brooding and decreased during growout, pre-sumably because of new water delivery systems with fewer leaks.
TURKEYS Growth The turkey poult has the best rate and efficiency of gain at a brooder room temperature of 21 to 24°C for the first 2 weeks, and 18°C thereafter, with about 70% rh. Initial brooder temperature should be 38°C, reduced 3 K per week until room temperature is reached. Proper control of photoperiod stimulates the growth rate in turkeys. Table 5 shows weekly cumulative average live mass for both sexes (Sell 1990). Figure 23 provides heat production data for medium breed turkeys. Reproduction Mature turkeys tolerate temperature and humidity over a range of at least −7 to 32°C and 35 to 85% rh. Since mature birds are kept only for fertile egg production, lighting for off-season egg production becomes very important. The young stock are raised to 20 weeks on natural light. At 20 weeks, females are placed in a Fig. 22 Sensible and Latent Heat for Broilers Raised at 27°C on Litter (Reece and Lott 1982) Fig. 23 Total Heat Production of Medium Breed Turkeys Versus Age During Daytime and Nighttime (Buffington et al. 1974) Table 5 Weekly Average Live Mass of Male and Female Turkeys Age, Weeks Male, kg Female, kg 0 0.03 0.03 1 0.15 0.14 2 0.30 0.29 3 0.52 0.50 4 0.84 0.80 5 1.27 1.17 6 1.78 1.61 7 2.39 2.11 8 3.06 2.67 9 3.78 3.26 10 4.58 3.88 11 5.42 4.52 12 6.35 5.17 13 7.29 5.81 14 8.28 6.43 15 9.28 7.04 16 10.28 7.62 17 11.28 8.16 18 12.27 8.68 19 13.26 9.16 20 14.24 9.60 21 15.20 22 16.12 23 17.01 24 17.88 Source: Sell (1990).
Table 6 Heat Loss of Growing Turkeys at Various Air Temperatures and Relative Humidities Age, Days Mass of Poultry, kg Dry Bulb, °C Relative Humidity, % Heat Loss, W/kg Total Live Mass, kg Sensible Latent 15 0.221 38 23 1.48 10.64 12.12 6 0.106 35 26 4.97 11.74 16.71 19 0.364 35 26 2.13 6.51 8.64 14 0.235 32 31 5.93 7.03 12.96 29 0.740 32 31 3.61 3.93 7.54 7 0.111 29 36 7.87 9.09 16.96 21 0.419 29 36 4.97 5.16 10.13 36 0.908 29 36 4.39 2.77 7.16 28 0.629 27 42 5.81 2.64 8.45 23 0.437 24 50 6.97 4.00 10.97 27 0.568 24 50 7.55 2.00 9.55 35 0.962 24 50 6.13 1.68 7.81 Source: DeShazer et al. (1974).
10.14 2001 ASHRAE Fundamentals Handbook (SI) totally darkened pen and given 8 h per day illumination at an intensity of 20 to 50 lux at bird’s-eye level. At 30 weeks, the sexes are mixed, and the lighting period is increased to 13 to 15 h. The breeding season then continues for 12 to 26 weeks.
Heat and Moisture Production Table 6 gives limited heat loss data on large breed growing tur-keys (DeShazer et al. 1974); however, an estimate from data on heavy chickens, applied to turkeys on a live-mass basis, should be satisfactory for design purposes. The reduction in heat loss during the dark period of the day is 25% for large breeds (Buffington et al.
1974) and between 5 and 40% for small breeds.
The latent heat production of turkeys may be significantly higher than the 1970s data show if the growth rate or feed consumption is significantly higher in the 1990s than it was in the 1970s (see the section on Chickens, Heat and Moisture Production).
LABORATORY ANIMALS Significant environmental conditions for facilities that house laboratory animals include temperature, humidity, air motion, illu-mination, noise, and gaseous and viable particulate contaminants (Moreland 1975). Design conditions vary widely, depending on whether the animals are experiencing disease-induced stress (which alters environmental needs), subjected to test environments, or sim-ply housed (Besch 1975, Murakami 1971, Nienaber and Hahn 1983). This fact reflects differing housing and ventilation guidelines for animals used in research (NIH 1985), compared to recommen-dations by the ASAE (1991) and Midwest Plan Service (MWPS 1983) for production agriculture. Little is known about the influ-ence of disease on environmental requirements, animal perfor-mance, and well-being. Because significantly different conditions may exist between animal cages and the animal room (macroenvi-ronment), control of cage environments is essential to ensure the animal’s physiological well-being. Temperature and velocity gradi-ent controls require low supply air-to-room air temperature differ-ential, overhead high induction diffusion, uniform horizontal and vertical air distribution, and low return outlets. Heat and Moisture Production Memarazadeh (1998) reported CO2, H2O and NH3 generation rates, O2 consumption rates, and heat generation rates of labora-tory mice. The rates were measured at two room relative humidi-ties (35% and 75%), and during light and dark periods. The production rate of CO2 and H2O and the consumption of O2 were constant over the 10 day period once the mice became acclimatized to cage conditions (Table 7). However, the generation rate of ammonia depended on the cage relative humidity, light phase, and day in the experiment (Figure 24). Table 8 lists the approximate amount of heat released by labo-ratory animals at rest and during normal activity. For load calcula-tion purposes, heat gain from all laboratory animal species can be estimated (Wood et al. 1972, Gordon et al. 1976) with an accept-able level of error from ATHG = 2.5M where M = 3.5W0.75 (9) and where ATHG = average total heat gain, W per animal M = basal metabolic rate of animal, W per animal W = mass of animal, kg Conditions in animal rooms must be maintained continuously.
This requires year-round availability of refrigeration and, in some cases, dual air-conditioning facilities and emergency power for motor drives and control instruments. Chapter 21 of the 1999 ASHRAE Handbook—Applications has additional information on laboratory animal facilities.
0.00 0.02 0.04 0.06 0.08 0.10 0 2 4 6 8 10 12 75% RH, LIGHTS OFF 75% RH, LIGHTS ON 35% RH, LIGHTS OFF 35% RH, LIGHTS ON AMMONIA PRODUCTION, g/(h·kg) BODY MASS BEDDING AGE, DAYS Fig. 24 Ammonia Generated by Laborator Mice Table 7 Average Production/Consumption Rates for Laboratory Mice Production/Consumption Rates per Kilogram of Body Mass Light Period Dark Period Light/Dark Average Total heat, W 23 27 — Water generated, kg/h — — 0.08 CO2 generated, kg/h 0.0068 0.0090 — O2 consumed, kg/h 0.0059 0.0066 — Notes: 1. Mice were white laboratory females in shoebox cages with chipped hardwood bed-ding.
2. Heat production rates were calculated from O2 consumption data. A mixed respira-tory quotient (RQ) was assumed (RQ = 0.82), and heat production was based on a rate of 0.020 kJ/m3 of O2 consumed.
Table 8 Heat Generated by Laboratory Animals Animal Mass, kg Heat Generation, Watts per Animal Basala Normally Activeb,c Sensible Latent Total Mouse 0.021 0.19 0.33 0.16 0.49 Hamster 0.118 0.70 1.18 0.58 1.76 Rat 0.281 1.36 2.28 1.12 3.40 Guinea pig 0.41 1.79 2.99 1.47 4.46 Rabbit 2.46 6.86 11.49 5.66 17.15 Cat 3.00 7.97 13.35 6.58 19.93 Primate 5.45 12.47 20.88 10.28 31.16 Dog 10.3 20.11 30.71 16.53 47.24 Dog 22.7 36.36 67.60 36.39 103.99 Goat 36 51.39 86.08 42.40 128.48 Sheep 45 60.69 101.66 50.07 151.73 Pig 68 82.65 108.70 85.56 194.26 Chicken 1.82 5.47 3.78 6.42 10.20 aBased on standard metabolic rate M = 3.5W 0.75 watt per animal (Kleiber 1961) or appropriate reference (W = animal mass, kg).
bReferenced according to availability of heat generation data. Otherwise, heat genera-tions is calculated on basis of ATHG = 2.5M (Gordon et al. 1976). Latent heat is assumed to be 33% of total heat and sensible heat is 67% of total heat (Besch 1973, Woods et al. 1972).
cData taken from Runkle (1964), Kleiber (1961), Besch (1973), Woods and Besch (1974), Woods et al. (1972), Bond et al. (1959), and Ota and McNally (1961).
Environmental Control for Animals and Plants 10.15 PLANTS: GREENHOUSES, GROWTH CHAMBERS, AND OTHER FACILITIES Most agronomically important plant crops are produced out-doors in favorable climates and seasons. Greenhouses and other indoor facilities are used for the out-of-season production of horti-cultural crops for both commercial sales and research purposes, and for producing food, floricultural, and other crops in conditions that permit the highest quality by buffering the crops from the vagaries of weather. The industry that produces crops in greenhouses may be termed controlled environment agriculture (CEA).
Historically, many cold-climate commercial greenhouses were operated only from late winter into early summer, and during autumn. Greenhouses were too warm during midsummer; during winter in some cold-climate locations, light levels were too low and the day length inadequate for many crops. Mechanical ventilation, evaporative cooling, centralized heating systems, movable insula-tions, carbon dioxide enrichment, and supplemental lighting have extended the use of greenhouses to year-round cropping on a rela-tively large scale.
Growth chambers, growth rooms, and propagation units are environmentally controlled spaces used for either research or com-mercial crop production. Environmentally controlled chambers may include highly sophisticated facilities used for micropropaga-tion (e.g., tissue culture), or may be simple boxes in which air temperature and lights are controlled. Indoor facilities having con-trolled temperature and humidity environments may be used as warehouses to hold plants and plant products prior to commercial sale. Often these are simple refrigerated storage rooms or chambers.
Primary atmospheric requirements for plant production include: (1) favorable temperatures, (2) adequate light intensity and suitable radiation spectrum, and (3) favorable air composition and circula-tion. Engineering design to meet these requirements typically is based on steady-state assumptions. The thermal and ventilation time constants of most greenhouses are sufficiently short that transient conditions are seldom considered.
TEMPERATURE Plant Requirements Leaf and root temperatures are dominant environmental factors for plant growth and flowering. Factors in the energy balance of a plant canopy include air temperature, relative humidity, air move-ment, thermal radiation exchange, and convective exchange coeffi-cients for sensible and latent heat. Therefore, leaf temperature is affected by such environmental factors as the type of heating and ventilating systems, supplemental lighting, light transmittance char-acteristics of the greenhouse cover, misting or evaporative cooling, location of the leaf on the plant, and the geometry of the surrounding leaf canopy.
Most information on plant responses to temperature is based on air temperature rather than plant temperature. Leaf temperature is difficult to measure, and one or several leaves represent neither the average nor the extreme temperatures of the plant. Since plants can-not actively regulate their cell and tissue temperatures in response to changing ambient conditions (passive regulation by opening and closing leaf stomata, which controls evapotranspiration, provides a small degree of control), their leaves and stems are usually within a few degrees of the surrounding air temperature (above during times of solar insolation, below at other times due to thermal reradiation and evapotranspiration).
All plants have minimum, optimum, and maximum temperatures for growth. Optimum temperature depends on the physiological process desired. Thermoperiodic species have different optimum day and night temperatures for each stage of growth, and each stage of plant growth may have its own unique optimum temperature influenced by radiant flux density, the ambient carbon dioxide level, and water and nutrient availability.
Historically, plants have been grown with night temperatures lower than day temperatures. In practice, many greenhouse crops are grown at standard (blueprint) night temperatures. Day tempera-tures are increased from 5 to 10 K (depending on solar intensity) above night temperatures. Table 9 presents recommended ranges of night temperatures for a selection of greenhouse crops.
Night temperature (NT) and day temperature (DT) are manipu-lated to provide nonchemical means to control plant height and development in some greenhouse crops. The effects of both DT and NT and their difference (DIF = DT – NT) are being investigated.
Stem elongation, internode spacing and elongation, rate of elonga-tion, photosynthesis rate, mature stem length, and fruit yield can be affected with DIF control (Jacobson et al. 1998, Neily et al. 1997, Erwin et al. 1991, Moe and Heins 1990). Typically, a negative DIF results in shorter crops. For example, Gent and Ma (1998) increased the yield of greenhouse tomatoes with a larger positive DIF (14 K) as compared to a lower positive DIF (5 K).
Heating Greenhouses Heat loss from greenhouses is caused primarily by conduction through the structural cover and infiltration of outdoor air. The heat-ing system is designed to meet the sum of the two. Perimeter heat loss is generally only a few percent of the total and is often neglected in design. When movable night insulation is used to con-serve energy, heating systems are still designed to match conduction and infiltration losses without insulation because movable insula-tion may be opened early in the morning when outdoor air temper-ature is near its minimum, and excess heating capacity may be useful to melt snow in climatic regions where this occurs. Green-houses are not designed to carry heavy snow loads.
Energy Balance Radiation energy exchange. Solar gain can be estimated using procedures presented in Chapter 30. Not all insolation appears as sensible heat, however. As a general rule, from two-thirds to three-quarters of ambient insolation is available inside a typical commer-cial greenhouse. (Highly detailed models for calculating solar trans-mittance may be found in the literature.) If a greenhouse is filled with mature plants, approximately one-half of the available insola-tion (transmitted) may be converted to latent heat, one-quarter to one-third released as sensible heat, and the rest either reflected back outdoors or converted through photosynthesis (perhaps 3%).
Supplemental lighting can add significantly to the thermal load in a greenhouse. If movable night insulation is used, venting may be required during times of lighting, even during cold weather. The components of heat addition from supplemental lighting are divided between sensible and latent loads, with approximately one-quarter to one-third of the total heat load in latent form.
Reradiation heat loss from greenhouses comprises complex pro-cesses and may involve both reradiation from the structural cover and reradiation from inside the greenhouse if the cover is not ther-mally opaque. Glass is nearly thermally opaque, but many plastics are not. Newer plastic films may contain IR-inhibiting substances and can save a significant amount of heating energy, while adding only slightly to summer ventilation needs. Condensation on plastic films also reduces transmittance of thermal radiation, while diffus-ing but not seriously impairing transmittance of solar radiation.
Generally, heat loss coefficients used in greenhouse design include the effects of thermal radiation exchange by the structural cover.
Structural heat loss. Conduction qc plus infiltration qi deter-mine total heating requirements qt. While infiltration heat loss is most accurately calculated using enthalpy differences, in practice only air temperature changes are considered, but with the apparent specific heat of air adjusted upward to account for the latent heat component. 10.16 2001 ASHRAE Fundamentals Handbook (SI) Table 9 Recommended Night Temperatures for Greenhouse Crops Crop Species Night Temperatures, °C Remarks Crop Species Night Temperatures, °C Remarks Aster Callistephus chinensis 10-13 Long days during early stages of growth Gloxinia Sinningia speciosa 18-21 Lower temperatures increase bud brittleness Azalea Hydrangea Rhododendron spp.
16-18 Vegetative growth and forcing specific tempera-tures required for flower initiation and development H. macrophylla 13-16 16-17 (forcing) Specific temperature for flower initiation and development Iris Calceolaria I. tingitana 7-16 (forcing) Forcing temperature 13 to 14°C for 10/11 bulbs; 10 to 12°C for 9/10 bulbs C. herbeohydrida 16 10 Vegetative growth Flower initiation and devel-opment; initiation also occurs with long days and high temperatures if photon flux density is high (Wedgewood) Kalanchoe (K. Blossfeldiana) 16 Temperatures influence rate of flower development and incidence of powdery mildew Calendula C. officinallis 4-7 Lily Calla Lilium longiflorum 16 Temperatures manipulated to alter rate of flower development; specific temperatures for flower initiation Zantedeschia spp.
13-16 Decrease to 13K as plants bloom Carnation Dianthus caryophyllus 10-11 winter 13 spring 13-16 summer Night temperatures adjusted seasonally in relation to photon energy flux density Orchida Cattleya spp.
16 Temperature requirement of hybrids related in parental species Chrysanthemum C. morifolium 16 cut flowers 17-18 pot plants Temperatures during flower initiation especially critical; uniform initiation very important for pot mums; cultivars classified on basis of temperature for development Orchids Phalaenopsis spp.
Cymbidium spp.
Cypripedium 18 10 10-13 Poinsettia Euphorbia pulcherrima 18 16-17 Vegetative growth Photoperiod requirement changes with temperature; bract development influenced by temperature Cineraria Senecio cruentus 16 9-10 Vegetative growth Flower initiation and devel-opment; plant quality best at low temperatures Roses Rosa spp.
16-17 Crossandra C. infundibuliforms 24-27 18 Germination Growth and flowering Saintpaulia S. ionantha 18-21 Below 16°C, growth is slow, hard and brittle Cyclamen C. indicum 16-18 13 10-11 Germination Seedlings Growth and flowering Snapdragon Antirrhinum maias 9-10 13-16 Winter Spring and Fall seedlings benefit from 16 to 18°C temperatures Foliage plants 18-21 Species differ in their temperature and radiant energy requirements Stock Matthiola incana 7-10 Buds fail to set if temperatures are above 18°C for 6 h or more per day. Grown mainly as field crop Fuchsia F. hydrida 11-16 Long days for flower initiation Geranium Pelargonium hortorum 11-16 16 to 18°C for fast crops at high radiant energy flux Tomato 16-19 Dry temperatures from 21 to 27°C on sunny days Gardenia Lettuce 13 17 to 18°C on cloudy days 21 to 26°C on sunny days G. grandiflora G. jasminoides 16-17 16-17 Lower temperatures result in iron chlorosis; higher temperatures increase bud abscission Cucumber 18 24°C on cloudy days 27°C on sunny days Environmental Control for Animals and Plants 10.17 (10) (11) (12) where U = heat loss coefficient, W/(m2 ·K) (Table 10) A = exposed surface area, m2 ∆t = inside minus outside air temperature, °C cp = volumetric specific heat of air (adjusted upward to account for latent heat component), 0.5 kJ/(m3·K) V = greenhouse internal volume, m3 N = number of air exchanges per hour (Table 11) Σ = summation of all exposed surfaces of the greenhouse and perimeter heat losses When design conditions are assumed for indoor and outdoor air temperatures and air exchange rate, the resulting heat loss may be assumed equal to the peak heating requirement for the greenhouse.
No universally accepted method exists to determine season-long heating needs for greenhouses. The heating degree-day method may be applied, but heating degree-day data must be adjusted to a base lower than 18.3°C because of the significant passive solar heating effect in greenhouses. The proper base must be determined locally to reflect the expected solar climate of the region and the expected greenhouse operating temperature. These difficulties often lead designers to obtain season-long heating data from comparable, existing greenhouses in the region, and apply them to new designs.
LIGHT AND RADIATION Plant Requirements Light (400 to 700 nm) is essential for plant vegetative growth and reproduction (Figure 25 and Figure 26). Intensity integrated over time provides the energy for growth and development, while duration (either long or short, depending on species) may be essen-tial for certain physiological processes such as flowering. High light intensity may exceed the ability of individual leaves to photo-synthesize. However, if there is a dense canopy, excess light may be beneficial to lower leaves even when upper leaves are light saturated. The intensity at which light saturates a leaf depends on various environmental factors, such as the concentration of carbon dioxide in the ambient air, as well as biological factors (Figure 27).
Spectral distribution of light can affect plant development, but sunlight’s spectral distribution need not be duplicated by artificial lighting to have suitable growth and development. Certain repro-Table 10 Suggested Heat Transmission Coefficients U, W/(m2 · K) Glass Single-glazing 6.4 Double-glazing 4.0 Plastic film Manufacturer’s Data Single filma 6.8 Double film, inflated 4.0 Single film over glass 4.8 Double film over glass 3.4 Corrugated glass fiber Reinforced panels 6.8 Plastic structured sheetb 16 mm thick 3.3 8 mm thick 3.7 6 mm thick 4.1 aInfrared barrier polyethylene films reduce heat loss; however, use this coefficient when designing heating systems because the structure could occasionally be covered with non-IR materials.
bPlastic structured sheets are double-walled, rigid plastic panels.
Table 11 Typical Infiltration Rates for Various Construction Types (ach) Metal frame and glazing system, 400 to 600 mm spacing 1.08 Metal frame and glazing system, 1200 mm spacing 1.05 Fiberglass on metal frame 1.03 Film plastic on metal frame 1.02 Film or fiberglass on wood 1.00 qt qc qi + = qc U ∑ A t ∆ = qi cpV N 3600 ------------ t ∆ = Fig. 25 Traditional Photosynthesis Action Spectra Based on Chlorophyll Absorption Fig. 26 Relative Photosynthetic Response Fig. 28 Phytochrome Action Spectra 10.18 2001 ASHRAE Fundamentals Handbook (SI) ductive changes are initiated by red (660 nm) and far red (730 nm) light (Figure 28), and excessive ultraviolet light (290 to 390 nm) may be detrimental to growth.
Plants that respond to the durations of light and dark periods are termed photoperiodic (photoperiodic effects generally relate to flowering). Some plant species are long-day obligates, some are short-day obligates, some are day length-intermediate, and others are day-neutral. Such responses are usually (relatively) independent of light intensity. Photoperiodic effects can be initiated by very low light levels (less than 1 W/m2), such as that provided to chrysanthe-mums by incandescent lights for a short period during the middle of the night to promote vegetative growth and inhibit flowering (until a suitable size has been attained) during the winter. Some plant spe-cies can tolerate continuous light, but others require some period of darkness for proper growth and development.
Sunlight is the most common source of photosynthetically active radiation (PAR, 400 to 700 nm). Although specially designed lamp sources may provide light similar to sunlight, no single source or combination of sources has spectral radiation exactly like the emis-sion of the sun from 300 to 2700 nm. Table 12 summarizes the spec-tral distribution of various light sources. Three systems of measuring radiation are as follows: 1. Radiometric units (irradiance) in watts per square metre (W/m2), with specified wavelength intervals.
2. Quantum units as photon flux density in µmol/(s·m2) (at 400 to 700 nm unless otherwise specified). A mole of photons delivered in one second over a square meter may be called an einstein.
3. Photometric units (illuminance) as one lumen per square metre, or lux (lx). Plant scientists use photosynthetic photon flux density (PPFD) in µmol/(s·m2) (400 to 700 nm). Engineering organizations and manufacturers of light sources use photometric and radiometric units. Because of the variation in spectral power distribution, con-version from one system of units to another must be made indi-vidually for each light source for the wavelength interval included (Table 13). To obtain comparable plant growth from different light sources, the same radiation levels (PAR) and red/far-red ratios must be maintained. Radiation Levels for Plant Growth Display 0.3 W/m2. For display purposes, plants can exist at an irradiance of 0.3 W/m2. The preferred lamp has changed with technological advances in efficiency and distribution. The empha-sis, however, has always been on color rendering and the type of atmosphere created in the display space. Low-wattage incandescent and fluorescent lamps are preferred. At this irradiance, plants can be displayed (seen), but little or no significant positive effect on plants can be expected. Extended holding in such low light conditions will have a negative effect on many plant species. Timing (light-dark durations) and temperature interaction are not a concern.
Photoperiod Response 0.9 W/m2 (4 to 6 h). For a photoperiod response, plant growth can be regulated at an irradiance of 0.9 W/m2 for as little time as 1 h. This irradiance is called a low light intensity system. The range of plant responses (promote or delay flowering, promote growth) that can be regulated is extensive, and this lighting is widely used by commercial growers.
Survival 3 W/m2 (8 h). Plants can survive at an irradiance of 3 W/m2 for 8 or more hours daily. This level enables many green plants to maintain their color. However, stem lengthening (eteliola-tion) and reduction in new leaf size and thickness occur under this irradiance level. In time, the overall development of the plants falls behind that of other plants grown under higher radiation levels. Pho-toperiod responses do not function well at this irradiance. However, strong interactions occur between this irradiance and temperature, watering frequency, and nutrition. Cooler temperatures (less than 17°C) help conserve previously stored material, while frequent watering and fertilization aggravate stem lengthening and senes-cence of older foliage.
Growth Maintenance 9 W/m2 (12 h). Plants maintain growth over many months when exposed to an irradiance of 9 W/m2 of 12-h duration daily. This is the intensity at which many indoor gardeners (professional or hobbyists) grow their plants when Fig. 27 Photosynthesis of Cucumber Leaf at Limiting and Saturating Carbon Dioxide Concentrations under Incandescent Light Table 12 Radiation Power Distribution of Light Sources Light Sources UV 300-400 nm PAR + FAR 400-850 nm IR 850-2700 nm Thermal 2700 + nm Total Radiation FCW 2 36 1 61 100 HG/DX 3 19 18 60 100 MH 4 41 8 47 100 HPS 0.4 50 12 38 100 LPS 0.1 56 3 41 100 INC 0.2 17 74 9 100 SUN 6 59 33 2 100 Values are in watts per 100 W of radiation.
Table 13 Light Conversion Factors Multiply W/m2 (400-850 nm) to Obtain: Divide lux by Constant to Obtain: Light Source µmol/(s·m2) (400-700 nm) µmol/(s·m2) (400-700 nm) µmol/(s·m2) (400-850 nm) Sun and sky, daylight 4.57 54 36 Blue sky only 4.24 52 41 High-pressure sodium 4.98 82 54 Metal halide 4.59 71 61 Mercury deluxe 4.52 84 77 Warm white fluorescent 4.67 76 74 Cool white fluorescent 4.59 74 72 Plant growth fluorescent A 4.80 33 31 Plant growth fluorescent B 4.69 54 47 Incandescent 5.00 50 20 Low-pressure sodium 4.92 106 89 Adapted from Thimijan and Heins (1983).
Environmental Control for Animals and Plants 10.19 starting them from seeds, cuttings, or meristems. Interactions with the environment (temperature, airflow, relative humidity, and pollutants) can vary among installations. Simple facilities with good air exchange and limited lamp concentration can grow a wide range of plant species. The rate of development, particu-larly as the plants grow in size, can be slow compared to plants grown at higher irradiances.
Propagation 18 W/m2 (6 to 8 h). Plants propagate rapidly when exposed to an irradiance of 18 W/m2 for a minimum of 6 to 8 h daily, but they prefer 12 h. Above this level, many propagators attempt to shade their greenhouses with one or several layers of neutral filters (painted films on glazing, or movable or semipermanent plastic or other fabric shade cloth materials) to restrict light (and heat) in the propagation area.
Cuttings rooted at this intensity maintain a growth rate much like that of similar tissue on a stock plant. Stem length, branching, and leaf color, however, can be regulated by manipulating temperature, moisture, stress, and nutrients. Most plants grown for their flowers and fruits can be brought to maturity by increasing the day length to 16 to 18 h for flower initiation (or rapid growth) and then reducing the day length to 8 to 12 h for development. The growth rate, how-ever, is relatively slow. For quickest development (leaf number, number of branches, and early flower initiation), the plants must be transferred to a higher lighting regime—24 to 50 W/m2.
Greenhouse Supplemental Light 24 W/m2 (8 to 16 h). When natural light is inadequate, it may be supplemented up to approxi-mately 24 W/m2 for 8 to 16 h daily. When coupled with the ambient sunlight (shaded by clouds, greenhouse structures, and lamp fix-tures), this irradiance simulates many of the growth responses and rates associated with growth chamber studies. Plants grown in green-houses without supplemental lighting grow slower and flower later than lighted ones in cloudy regions or in northern areas during win-ter. Duration (in hours) and timing (day-night) of lighting is critical.
Supplemental lighting for 8 h, particularly during the day (0800 to 1600) may not be as cost-effective as lighting at night (2000 to 0400) if off-peak electric rates are available. Neither of these light-ing regimes, however, is as effective as lighting for 16 h from morn-ing to midnight.
Lighting short-day plants, such as chrysanthemums and poinset-tias, is relatively inefficient because they can be lighted only during the 8- to 12-h day, followed by an obligatory 12- to 16-h daily dark period.
Growth Chambers 50 W/m2 (8 to 24 h). Plants grow in growth chambers or growth rooms if the light irradiance is a minimum of 50 W/m2 for 8 to 24 h daily. This irradiance is approximately one-fourth that of outdoor sunlight. Cool, white fluorescent lamps, combined with incandescent lamps, are widely used. More recently, HID lamps have been substituted for fluorescent lamps. For consis-tent results, all require a barrier of glass or other material between the lamp and the plants, and a separate ventilating system to remove the heat from such enclosed spaces.
Since filters cannot remove infrared completely, chambers are difficult to standardize. This often leads to confusing information on plant growth and flowering of plants grown in greenhouses and out-doors. When the total irradiance is 50 W/m2 and 10 to 20% of the total radiation is provided by incandescent lamps, most kinds of plants can be grown. In typical plant forms, flowering and fruiting responses occur when the plants are subjected to the following parameters: • Day length, 8 to 24 h • Temperature, 10 to 35°C • Carbon dioxide, 300 to 2000 ppm (540 to 3600 mg/m3) • Relative humidity, 20 to 80% Photoperiod Day length affects the performance of some plants. There are four basic day length plant groups: 1. Short-day plants flower only when the length of the daily light period is less than the critical number of hours. Daily light periods longer than the critical length inhibit flowering.
2. Long-day plants flower only when the daily light period is longer than the critical number of hours. They become dormant or remain vegetative when the daily light period is shorter than the critical length.
3. Day length-intermediate plants flower only within a narrow range of day length, usually between about 10 and 14 h. If the day length is shorter than the optimum day length for flowering, the plants stop growing.
4. Day-neutral plants continue in vegetative growth or flower regardless of the day length.
Continuous light inhibits flowering and promotes vegetative growth of short-day plants, but encourages continued vegetative growth and early flowering of long-day plants, blocks the flowering of day length-intermediate plants, and in many instances, increases the stem length of day-neutral plants. Plants vary in their respon-siveness to light source, duration, and intensity. The technology that has evolved to control the photoperiod of plants is based primarily on the incandescent-filament lamp. Of all the light sources avail-able, this lamp creates the regulating mechanism most similar to that of sunlight. This is because the red/far-red wavelength ratio of light from an incandescent lamp is similar to the ratio of sunlight.
The effectiveness for photoperiod response in plants peaks at wavelengths of 660 nm (red) and 730 nm (far-red). The relative order of activity in regulating photoperiod responses by lamp type is as follows: Incandescent (INC) > High-pressure sodium (HPS) >> Metal halide (MN) = Cool white fluorescent (F) = Low-pressure sodium (LPS) >> Clear mercury (Hg). Photoperiod lighting is always used in combination with daylight or another main light source. Short days (less than normal day length) are created in the greenhouse with opaque materials that surround the plants.
RELATIVE HUMIDITY Relative humidity affects the rate at which plants take water up, the rate of latent heat transfer, and certain diseases. Normal plant growth generally occurs at relative humidities between 20 and 80% if the plants have a well-developed root system, although relative humidities above 40% are preferred to avoid water stress conditions.
Transpiration, the movement of water vapor and gases from the plant to its surroundings, is controlled by the plant’s stomatal open-ings. It is a function of air velocity and the vapor pressure difference between water at saturation at the leaf temperature and the actual water vapor partial pressure in the air. Generally, as relative humidity decreases, transpiration increases. Very low relative humidities (less than 20%) can cause wilting, since evaporation losses may be higher than the plant can replace, especially when light intensity is high.
High humidity provides a good environment for pathogenic organisms. Many pathogenic spores do not germinate unless rela-tive humidity is 96% or more and many require a film of water on the leaves. Somewhat lower relative humidities may support other pathogen growth stages.
Still air surrounding a plant may be much wetter than the general atmosphere because evapotranspiration from the leaves raises the relative humidity in interfoliage air. The lower leaves, which stay moist longer, are more susceptible to disease. The upper leaves are dried by radiation and air currents.
10.20 2001 ASHRAE Fundamentals Handbook (SI) AIR COMPOSITION Carbon dioxide, which comprises about 0.035% of ambient air, is essential for plant growth. There are basically three ways to obtain carbon dioxide for greenhouse enrichment: pure in solid, liq-uid, or gaseous form; from burning fuels such as propane, natural gas, or kerosene; and by the aerobic breakdown of organic matter.
The three ways are listed in order of purity and reliability. Carbon dioxide enters plants through stomata and is converted to carbohy-drates through photosynthesis. The carbon dioxide concentration in air surrounding a plant, as well as light level, affects the rate of pho-tosynthesis. The concentration for maximum growth depends on many factors, including the stage of growth, leaf area, light inten-sity, temperature, and air velocity past the stomatal openings.
An important relationship exists between light level and carbon dioxide uptake (Figure 27). As light level increases, carbon dioxide concentration must increase concurrently to take maximum advan-tage of the greater photosynthetic potential. In plastic greenhouses, and in glass greenhouses sealed against infiltration, the carbon diox-ide level can drop below 360 mg/m3 when the weather is cold, light levels are moderate, and the greenhouse is not ventilated. Carbon dioxide enrichment just to maintain normal levels can then be bene-ficial. During times of high light levels, carbon dioxide enrichment gives maximum benefit from the available light and may even be economically desirable when greenhouse ventilation is modest.
However, carbon dioxide concentrations above 2700 mg/m3 are sel-dom recommended; levels between 1400 and 2200 mg/m3 are typi-cally used.
The effects of enrichment are not always positive. Without proper crop management, the yield, quality, or both may decrease, and timing of crop maturity may change.
Pollutants Plants are sensitive to atmospheric pollutants such as ethylene, ammonia, gaseous fuels, ozone, fluorides, photochemical smog, and oxidants (from nitrogen and sulfur). Pollution damage can range from small spots on leaves, to yellowing of leaves, to severe foliage burn, and, ultimately, to plant death in extreme but not rare situations. The effect occurs both outdoors and in greenhouses; however, this is more common in greenhouses, because of their closed nature. Pollutants indoors can be removed by activated char-coal filters in the ventilation system; however, these are seldom used in commercial greenhouses. Economically, the more feasible approach is to limit pollutant production within, or introduction into, the greenhouse air space.
Ethylene is produced naturally by plants and leads to flower and whole plant senescence. It is also produced by combustion of gas-eous and liquid fuels and can rapidly cause plant damage. Concen-trations above 230 µg/m3 can have a detrimental effect on plant growth. Unvented heaters, air currents that bring vented combustion products back into the greenhouse, and burners for carbon dioxide production are common sources of ethylene injury. Liquefied car-bon dioxide may be used to supplement natural levels rather than combustion, specifically to avoid introducing ethylene into the greenhouse air, but even liquefied carbon dioxide should be care-fully selected to avoid residual amounts of ethylene that may be contained within it.
Nitrogen oxides, common components of air pollution, can cause serious plant damage. Greenhouse locations near highways, nearby industrial complexes, and even a truck left running for an extended time near a greenhouse air intake vent may lead to leaf damage from NO and NO2.
Sulfur dioxide, produced by the burning of sulfur containing fuels, causes injury to many plants within a short time. Sources of sulfur dioxide may be nearby, such as an industrial area, or may be within the greenhouse complex, such as the vented combustion products from a central heating facility, combustion products from carbon dioxide burners (using kerosene as a fuel, for example), and sulfur burned for mildew control.
Ozone is widely recognized as a serious pollutant affecting the production of many agronomic crops. Although few research results exist to quantify the effect of ozone on greenhouse crops, damage is likely to occur when greenhouses are located near ozone sources.
Phenolics and certain other organic vapors are phytotoxic. Pheno-lics, as volatiles from certain wood preservatives (creosote and pen-tachlorophenol), can cause leaf and petal damage. Vapors from some paints can also be damaging. Misuse of herbicides and pesticides can lead to plant injury, either through spray drift or volatilization.
Covering and sealing greenhouses for energy conservation can increase concentrations of ethylene and other air pollutants if their sources are within the air space, since infiltration and ventilation are decreased. Sealing to reduce infiltration can also lead to rapid car-bon dioxide depletion and inhibited plant growth during cold tem-peratures when, even with bright light, ventilation is not required.
AIR MOVEMENT Air movement influences transpiration, evaporation, and the availability of carbon dioxide. Air speed affects the thickness of the boundary layer at the leaf surface, which in turn influences the trans-port resistance between the ambient air and the leaf stomatal cavi-ties. Air speed of 0.5 to 0.7m/s is commonly accepted as suitable for plant growth under CEA conditions. Air speeds across the leaf of 0.03 to 0.1 m/s are needed to facilitate carbon dioxide uptake. Air speeds above 1 m/s can induce excessive transpiration, cause the sto-matal guard cells to close, reduce carbon dioxide uptake, and inhibit plant growth. Air speeds above 5 m/s may cause physical damage to plants. Generally, if plants within a greenhouse move noticeably due to ventilation, air speed is excessive.
Air circulation within greenhouses may be created to reduce thermal stratification and maintain suitable levels of carbon dioxide within the leaf canopy. Horizontal air flow, produced by small pro-peller fans that move air around the greenhouse in a racetrack pattern, has been found to be effective. Such fans are approximately 350 mm in diameter, with approximately 0.2 kW motors, spaced at approximately 15 m intervals. Total fan capacity in metres per sec-ond (m3/s) should equal about 25% of the greenhouse volume in cubic metres.
REFERENCES ASAE. 1991. Design of ventilation systems for poultry and livestock shel-ters. ASAE Standard D270.5. American Society of Agricultural Engi-neers, St. Joseph, MI.
Barber, E.M., A. Senthilselvan, S. Kirychuk, S. Lemay, J.A. Dosman, Y.
Zhang, L. Holfeld, D. Bono, and Y. Comier. 1999. Evaluation of two strategies for reducing exposure of swine barn workers to dust. Proceed-ings International Symposium on Dust Control for Animal Production Facilities. Aarhus, Denmark.
Bereskin, B., R.J. Davey, W.H. Peters, and H.O. Hetzer. 1975. Genetic and environmental effects and interactions in swine growth and feed utiliza-tion. Journal of Animal Science 40(1):53.
Besch, E.L. 1973. Final report to Animal Resources Branch, Division of Research Resources, National Institutes of Health. NIH Contract 71-2511.
Besch, E.L. 1975. Animal cage room dry-bulb and dew point temperature differentials. ASHRAE Transactions 81(2):549.
Blaxter, K.L.and W.A. Wood. 1951. The nutrition of the young Ayrshire calf, IV some factors affecting the biological value of protein determined by nitrogen-balance methods. Brit. J. Nutrition 5:55.
Bond, T.E., H. Heitman, Jr., and C.F. Kelly. 1965. Effects of increased air velocities on heat and moisture loss and growth of swine. Transactions of ASAE 8:167.
Bond, T.E., C.F. Kelly, and H. Heitman, Jr. 1959. Hog house air conditioning and ventilation data. Transactions of ASAE 2:1.
Bond, T.E., C.F. Kelly, and H. Heitman, Jr. 1963. Effect of diurnal temper-ature upon swine heat loss and well-being. Transactions of ASAE 6:132.
Environmental Control for Animals and Plants 10.21 Bortolussi, R., W.F. Schlech, and W.L Albritton, eds. 1985. Listeria. In: Manual of clinical microbiology. Amer. Soc. Microbiology, Washington D.C.
Brody, S. 1945. Bioenergetics and growth. Reinhold Publishing Co., New York.
Brody, S., A.C. Ragsdale, R.G. Yeck, and D. Worstell. 1955. Milk produc-tion, feed and water consumption, and body weight of Jersey and Hol-stein cows in relation to several diurnal temperature rhythms. University of Missouri Agricultural Experiment Station Research Bulletin No. 578.
Bryant, J.M., C.F. Foreman, N.L. Jacobson, and A.D. McGilliard. 1967. Pro-tein and energy requirementsof the young calf. Journal of Dairy Science 50(10):1645-1653.
Buckland, R.B. 1975. The effect of intermittent lighting programmes on the production of market chickens and turkeys. World Poultry Science Jour-nal 31(4):262.
Bucklin, R.A. and L.W. Turner. 1991. Methods to relieve heat stress in hot, humid climates. Applied Engineering in Agriculture 7. ASAE.
Buffington, D.E., K.A. Jordan, W.A. Junnila, and L.L. Boyd. 1974. Heat production of active, growing turkeys. Transactions of ASAE 17:542.
Bundy, D.S. 1984. Rate of dust decay as affected by relative humidity, ion-ization and air movement. Transactions of ASAE 27(3):865-70.
Bundy, D.S. 1986. Sound preventive measures to follow when working in confinement buildings. Presented at the American Pork Congress, St.
Louis, MO.
Butchbaker, A.F. and M.D. Shanklin. 1964. Partitional heat losses of new-born pigs as affected by air temperature, absolute humidity, age and body weight. Transactions of ASAE 7(4):380.
Carpenter, G.A. 1986. Dust in livestock buildings–Review of some aspects.
J. Agr. Engng. Res. 33:227-41.
Carr, L.E. and J.L. Nicholson. 1980. Broiler response to three ventilation ranges. Transactions of ASAE 22(2):414-18.
Christianson, L.L. and R.L. Fehr. 1983. Ventilation energy and economics.
In Ventilation of agricultural structures, M.A. Hellickson and J.N.
Walker, eds. ASAE Nomograph No. 6, 336.
Curtis, S.E. 1983. Environmental management in animal agriculture. Iowa State University Press, Ames, IA.
DeShazer, J.A., L.L. Olson, and F.B. Mather. 1974. Heat losses of large white turkeys—6 to 36 days of age. Poultry Science 53(6):2047.
Donham, K.J. 1987. Human health and safety for workers in livestock hous-ing. CIGR Proceedings. ASAE Special Public. 6-87.
Donham, K.J., P. Haglind, Y. Peterson, R. Rylander, and L. Belin. 1989.
Environmental and health studies of workers in Swedish swine confine-ment buildings. British J. Ind. Med. 40:31-37.
Donham, J.K., P. Haglind, Y. Peterson, R. Rylander, and L. Belin. 1989.
Environmental and health studies of workers in Swedish swine confine-ment buildings. British Journal of Industrial Medicine 40:31-37.
Dosman, J.A., B.L. Graham, D. Hall, P. Pahwa, H.H. Mcduffie, and M.
Lucewicz. 1988. Respiratory symptoms and alterations in pulmonary function test in swine producers in Sakstchewan: Results of a survey of farmers. J. Occup. Med. 30:715-20.
Drury, L.N. 1966. The effect of air velocity of broiler growth in a diurnally cycling hot humid environment. Transactions of ASAE 9:329.
Erwin, J.E., R.D. Heins, and R. Moe. 1991. Temperature and photoperiod effects of Fuschia X hybrida morphology. J. Am. Soc. Hortic. Sci 116(6): 955-60.
Fachlam, R.R. and R.B. Carey. 1985. Streptococci and aerococci. In: Man-ual of clinical microbiology. American Society of Microbiology. Farner, D.S. 1961. Comparative physiology: Photoperiodicity. Annual Review of Physiology 23:71.
Gates, R.S., D.G. Overhults, and S.H. Zhang. 1996 Minimum ventilation for modern broiler facilities. Transactions of ASAE 38(5):1479-86.
Gebremedhin, K.G., C.O. Cramer, and W.P. Porter. 1981. Predictions and measurements of heat production and food and water requirements of Holstein calves in different environments. Transactions of ASAE 24(3): 715-20.
Gebremedhin, K G., W.P. Porter, and C.O. Cramer. 1983. Quantitative anal-ysis of the heat exchange through the fur layer of Holstein calves. Trans-actions of ASAE 26(1):188-93.
Gent, M.P.N. and Y.Z. Ma. 1998. Diurnal temperature variation of the root and shoot affects yield of greenhouse tomato. HortScience 33(1):47-51.
Gordon, R.L., J.E. Woods, and E.L. Besch. 1976. System load characteris-tics and estimation of animal heat loads for laboratory animal facilities.
ASHRAE Transactions 82(1):107.
Guazdauskas, F.C. 1985. Effects of climate on reproduction in cattle. Jour-nal of Dairy Science 68:1568-78.
Gunnarson, H.J. et al. 1967. Effect of air velocity, air temperatures and mean radiant temperature on performance of growing-finishing swine. Trans-actions of ASAE 10:715.
Hahn, G.L. 1983. Management and housing of farm animals environments.
Stress physiology in livestock. CRC Press, Boca Raton, FL.
Hahn, G.L., A. Nygaard, and E. Simensen. 1983. Toward establishing ratio-nal criteria for selection and design of livestock environments. ASAE Paper 83-4517.
Heard, L., D. Froehlich, L. Christianson, R. Woerman, and W. Witmer. 1986.
Snout cooling effects on sows and litters. Transactions of ASAE 29(4):1097.
Heber, A.J., M. Stroik, J.L. Nelssen, and D.A. Nichols. 1988. Influence of environmental factors on concentrations and inorganic content of aerial dust in swine finishing buildings. Transactions of ASAE 31(3):875-881.
Heber, A.J. and C.R. Martin. 1988. Effect of additives on aerodynamic seg-regation of dust from swine feed. Transactions of ASAE 31(2):558-563.
Hellickson, M.A., H.G. Young, and W.B. Witmer. 1974. Ventilation design for closed beef buildings. Proceedings of the International Livestock Environment Symposium, SP-0174, 123.
Hillman, P.E., K.G. Gebremedhin, and R.G. Warner. 1992. Ventilation sys-tem to minimize airborne bacteria, dust, humidity, and ammonia in calf nurseries. Journal of Dairy Science 75:1305-12.
Jacobson, B.M., D.H. Willits, and P.V. Nelson. 1998. Dynamic internode elongation in chrysanthemum. Appl. Engr. Agric. 14(3):311-16.
Janni, K.A., P.T. Redig, J. Newmen, and J. Mulhausen. 1984. Respirable aerosol concentration in turkey grower building. ASAE Paper No. 84-4522.
Johnson, H.D., A.C. Ragsdale, and R.G. Yeck. 1958. Effects of constant environmental temperature of 50°F and 80°F on the feed and water con-sumption of Brahman, Santa Gertrudis and Shorthorn calves during growth. University of Missouri Agricultural Experiment Station Research Bulletin 683.
Johnson, H.D., A.C. Ragsdale, I.L. Berry, and M.D. Shanklin. 1962. Effect of various temperature-humidity combinations on milk production of Holstein cattle. University of Missouri Agricultural Experiment Station Research Bulletin 791.
Kibler, H.H. and S. Brody. 1956. Influence of diurnal temperature cycles on heat production and cardiorespirativities in Holstein and Jersey cows.
University of Missouri Agricultural Experiment Station Research Bulle-tin 601.
Kleiber, M. 1961. The fire of life: An introduction to animal energetics. John Wiley and Sons, New York.
Longhouse, A.D. et al. 1968. Heat and moisture design data for broiler houses. Transactions of ASAE 41(5):694.
Longhouse, A.D., H. Ota, and W. Ashby. 1960. Heat and moisture design data for poultry housing. Agricultural Engineering 41(9):567.
Martin, W.T., Y. Zhang, P.J. Willson, T.P. Archer, C. Kinahan, and E.M. Bar-ber. 1996. Bacterial and Fungal flora of dust deposits in a swine building.
British J. Occupational and Environmental Medicine 53:484-487.
McDaniel, G.R. and R.N. Brewer. 1975. Intermittent light speeds broiler growth and improves efficiency. Highlights of Agricultural Research 4:9. Auburn University, Auburn, AL.
McFarlane, J. 1987. Linear additivity of multiple concurrent environmental stressors effects on chick performance, physiology, histopathology and behavior. Ph.D. thesis, Animal Sciences Dept., Univ. of Illinois, Urbana.
Memarzadeh, F. 1998. Design handbook on animal research facilities using static microisolators, Volumes I and II. National Institute of Health.
Bethesda, MD.
Merkle, J.A. and T.E. Hazen. 1967. Zone cooling for lactating sows.Trans-actions of ASAE 10:444.
Moe, R. and R. Heins. 1990. Control of plant morphogenesis and flowering by light quality and temperature. Acta Hortic. (July): 81-89.
Moreland, A.F. 1975. Characteristics of the research animal bioenviron-ment. ASHRAE Transactions 81(2):542.
Mount, L.E. 1963. Food, meat, and heat conservation. Pig Industry Devel-opment Authority Conference Circulat. Buxton, Derbyshire, England.
Murakami, H. 1971. Differences between internal and external environ-ments of the mouse cage. Laboratory Animal Science 21:680.
MWPS. 1983. Structures and environment handbook. Midwest Plan Ser-vice, Ames, IA.
Neily, W.G., P.R. Hickleton, and D.N. Kristie. 1997. Temperature and devel-opmental state influence diurnal rhythms of stem elongation in snap-dragon and zinnia. J. Am. Soc. Hortic. Sci. 122(6):778-83.
10.22 2001 ASHRAE Fundamentals Handbook (SI) Nienaber, J.A. and G.L. Hahn. 1983. Temperature distribution within con-trolled-environment animal rooms. Transactions of ASAE 26:895.
Nienaber, J.A., G.L. Hahn, H.G. Klencke, B.A. Becker, and F. Blecha. 1987.
Cyclic temperature effects on growing-finishing swine. CIGR Proceed-ings. ASAE Special Public. 687:312.
NIH. 1985. Guide for the care and use of laboratory animals. National Insti-tutes of Health Publication 85-23. Bethesda, MD.
OSHA. 1985. OSHA Safety and Health Standard. U.S. Department of Labor Code 1910.1000, 653-59.
Ota, H.J. and E.H. McNally. 1961. Poultry respiration calorimetric studies of laying hens. ARS-USDA 42-13, June.
Ota, H., J.A. Whitehead, and R.J. Davey. 1982. Heat production of male and female piglets. Proceedings of Second International Livestock Environ-ment Symposium, SP-03-82. ASAE.
Pratt, G.L. 1983. Ventilation equipment and controls. In: Ventilation of agri-cultural structures, eds. M.A. Hellickson and J.N. Walker. ASAE, pp. 54.
Preller, L., J. Boley, D. Heederik, S. Gielen, and M. Tielen. 1992. Proceed-ings 3rd Inter. Symp: Issues in Health, Safety and Agriculture. Center for Agr. Medicine, University of Saskatchewan, Saskatoon, pp. 23-24. Reece, F.N. and B.D. Lott. 1982. Heat and moisture production of broiler chickens. Proceedings of the Second International Livestock Environ-ment Symposium, SP-03-82. ASAE.
Riskowski, G.L., J.A. DeShazer, and F.B. Mather. 1977. Heat losses of White Leghorn hens as affected by intermittent lighting schedules.
Transactions of ASAE 20(4):727-731.
Riskowski, G.L. and D.S. Bundy. 1991. Response of young pigs to various air temperatures and velocities. ASHRAE Transactions 97(2):543-549.
Roller, W.L. and H.S. Teague. 1966. Effect of controlled thermal environ-ment on reproductive performance of swine. Proceedings of the Fourth International Biometeorological Congress. Rutgers University, New Brunswick, NJ.
Runkle, R.S. 1964. Laboratory animal housing, Part II. AIA Journal 4:73.
Scott, N.R., J.A. DeShazer, and W.L. Roller. 1983. Effects of the thermal and gaseous environment on livestock. ASAE Monograph No. 6. ASAE.
Sell, J.L. 1990. Faster growing, more efficient turkeys in 1989. 66(1):12.
Senthilselvan, A., Y. Zhang, J.A. Dosman, E.M. Barber, L.E. Holfeld, S.P.
Kirychuk, Y. Comier, T.S. Hurst, and C.S. Rhodes. 1997. Positive human health effects of dust suppression with canola oil in swine barns. Amer-ican Journal for Respiratory and Critical Care Medicine 156:410-17.
Shutze, J.V., J.K. Lauber, M. Kato, and W.O. Wilson. 1962. Influence of incandescent and colored light on chicken embryos during incubation.
Nature 196(4854):594.
Siegmund, H., ed. 1979. The Merck veterinary manual: A handbook of diag-nosis and therapy for the veterinarian. Merck & Co., Inc., Rahway, NJ.
Squibb, R.L. 1959. Relation of diurnal temperature and humidity ranges to egg production and feed efficiency of New Hampshire hens. Journal of Agricultural Science 52(2):217.
Sutton, A.L., S.R. Nichols, D.D. Jones, D.T. Kelley, and A.B. Scheidt. 1987.
Survey of seasonal atmospheric changes in confinement farrowing houses. In Latest developments in livestock housing. International Com-mission of Agricultural Engineering (CIGR).
Takai, H., F. Moller, M. Iversen, S.E. Jorsal, and V. Bille-Hansen. 1993. Dust control in swine buildings by spraying of rapeseed oil. Proceedings of the Fourth Livestock Environment Symposium. ASAE, pp. 726-33.
Thomeczek, F.J., M.R. Ellersieck, R.K. Leavitt, and J.F. Lasley. 1977.
Trends in economic traits of production tested boars in the Missouri Evaluation Station. University of Missouri Research Bulletin No. 1021.
Tienhoven, A.W., N.R. Scott, and P.E. Hillman. 1979. The hypothalamus and thermoregulation: A review. Poultry Science 52(6):1633.
Van’t Klooster, C.E. and J.A.M. Voermans. 1993. European perspectives— How are they solving their problems? Symposium “Meeting the Environ-mental Challenge.” National Pork Producers Council. Wiegard, B. and J. Hartung. 1993. Bacterial contaminant of air and floor sur-faces in an animal house of a cattle clinic. Proceedings of 4th Interna-tional Livestock Environment Symposium ASAE, St. Joseph, MI, pp.
643-49.
Wiersma, F. and G.H. Stott. 1969. New concepts in the physiology of heat stress in dairy cattle of interest of engineers. Transactions of ASAE 12(1): 130-32.
Woods, J.E. and E.L. Besch. 1974. Influence of group size on heat dissipa-tion from dogs in a controlled environment. Laboratory Animal Science 24:72.
Woods, J.E., E.L. Besch, and R.G. Nevins. 1972. A direct calorimetric anal-ysis of heat and moisture dissipated from dogs. ASHRAE Transactions 78(2):170-83.
Yeck, R.G. 1957. Stable heat and moisture dissipation with beef calves at temperatures of 50° and 80°F. University of Missouri Agricultural Experiment Station Research Bulletin 645.
Yeck, R.G. and R.E. Stewart. 1959. A ten-year summary of the psychroen-ergetic laboratory dairy cattle research at the University of Missouri.
Transactions of ASAE 2(1):71.
Yeck, R.G. and R.E. Stewart. 1960. Stable heat and moisture dissipation with dairy calves at temperatures of 50° and 80°F. University of Missouri Research Bulletin No. 759.
Zejda, J.E, T.S. Hurst, C.S. Rhodes, E.M. Barber, H.H. Mcduffie, and J.A.
Dosman. 1993. Respiratory health of swine producers, focus on young workers. CHEST 103:702-09.
Zejda, J.E., E.M. Barber, J.A. Dosman, S.A. Olenchock, H.H. McDuffie, C.S. Rhodes, and T.S. Hurst. 1994. Respiratory health status in swine producers relates to endotoxin exposure in the presence of low dust lev-els. J. Occup. Med. 36(1):49-56.
Zhang, Y. 1994. Swine building ventilation, pp.14-15, Prairie Swine Centre, Saskatoon, SK, Canada.
Zhang, Y., E.M. Barber, J.J.R. Feddes, and J.F. Patience. 1995. Identification of oils to be sprinkled in swine barns to reduce dust. ASHRAE Trans-actions 101(2):1179-91. Zhang, Y., A. Tanaka, E.M. Barber, and J.J.R. Feddes. 1996. Effect of fre-quency and quantity of sprinkling canola oil on dust reduction in swine buildings. Transactions of ASAE 39(3):1077-81.
Zhang, Y., A. Tanaka, J.A. Dosman, A. Senthilselvan, E.M. Barber, S. Kiry-chuk, and T. Hurst. 1998. Acute respiratory responses of human subjects to air quality in a swine building. J. Agr. Engng. Res. 70:367-73.
Zhang, Y., X. Wang, G.L. Riskowski, and L.L. Christianson. 2000. A method to quantify ventilation effectiveness for air quality control. Proceedings 2nd International Conference on Air Pollution in Agricultural Operations, Des Moines, IA.
Zulovich, J.M., M.B. Manbeck, and W.B. Roush. 1987. Whole-house heat and moisture production of young floor brood layer pullets. Transactions of ASAE 30(2)455-58.
BIBLIOGRAPHY ASAE. 1991. Design of ventilation systems for poultry and livestock shel-ters. ASAE Standard D270.5. American Society of Agricultural Engi-neers, St. Joseph, MI.
Consortium for Developing a Guide for the Care and Use of Agricultural Animals in Agricultural Research and Teaching. 1988. Guide for the care and use of agricultural animals in agricultural research and teaching.
Agricultural Animal Care Guide Division of Agriculture, NASULGC, Washington, D.C. 20036-1191.
Esmay, M.E. and J.E. Dixon. 1986. Environmental control for agricultural buildings. AVI Publications, Westport, CT.
Hellickson, M.A. and J.N. Walker, eds. 1983. Ventilation of agricultural structures. ASAE Monograph No. 6. ASAE.
MWPS. 2000. The Midwest Plan Service handbooks and plans series. Mid-west Plan Service, Ames, IA.
11.1 CHAPTER 11 PHYSIOLOGICAL FACTORS IN DRYING AND STORING FARM CROPS Factors Determining Safe Storage ............................................................................................... 11.1 Moisture Measurement ................................................................................................................. 11.5 Prevention of Deterioration ......................................................................................................... 11.6 Drying Theory .............................................................................................................................. 11.8 Drying Specific Crops ................................................................................................................ 11.11 HIS CHAPTER focuses on the drying and storage of grains, Toil-seeds, hay, cotton, and tobacco. However, the primary focus is on grains and oilseeds, collectively referred to as grain. Major causes of postharvest losses in these products are fungi, insects, and rodents. Substantial deterioration of grain can occur in storage.
However, where the principles of good grain storage are applied, losses are minimal.
Preharvest invasion of grains by storage insects is usually not a problem in the midwestern United States. Field infestations can occur in grains when they are dried in the field at warm tempera-tures during harvest. Preharvest invasion by storage fungi is possi-ble and does occur if appropriate weather conditions prevail when the grain is ripening. For example, preharvest invasion of corn by Aspergillus flavus occurs when hot weather is prevalent during grain ripening; it is, therefore, more common in the southeastern United States (McMillan et al. 1985). Invasion of wheat, soybeans, and corn by other fungi can occur when high ambient relative humidities prevail during grain ripening (Christensen and Mero-nuck 1986). However, the great majority of damage occurs during storage, due to improper conditions that permit storage fungi or insects to develop.
Deterioration from fungi during storage is prevented or mini-mized by (1) reduction of grain moisture content to below limits for growth of fungi, (2) maintenance of low grain temperatures throughout the storage period, (3) chemical treatment to prevent the development of fungi or to reduce the rate of fungal growth while the grain moisture content is being lowered to a safe level, and (4) airtight storage in which initial microbial and seed respiration reduces the oxygen level so that further activity by potentially harm-ful aerobic fungi is reduced.
Reduction of moisture by artificial drying is the most commonly used technique. The longer grain is stored, the lower its storage moisture should be. Some of the basic principles of grain drying and a summary of methods for predicting grain drying rate are included in the section on Drying Theory.
Reduction of grain temperature by aeration is practical in tem-perate climates and for grains that are harvested during cooler sea-sons. Fans are operated when ambient temperature is lower than grain temperature. Basic information on aeration is summarized in the section on Drying Theory. Use of refrigeration systems to reduce temperature is not generally cost-effective for feed grains but may have application for higher value food grains.
Chemical treatment of grain is becoming more common and is briefly described in the section on Prevention of Deterioration.
When grain is placed in airtight silos, the oxygen level is rapidly reduced, and carbon dioxide increases. Although many fungi will not grow under ideal hermetic conditions, some will grow initially in imperfectly sealed bins, and this growth can reduce the feeding value of the grain for some animals. Partially emptied bins may sup-port harmful mold, yeast, and bacterial growth, which makes the grain unsuitable for human consumption. Airtight storage is briefly addressed in the section on Oxygen and Carbon Dioxide under Fac-tors Determining Safe Storage.
Deterioration from insects can also be prevented by a combina-tion of reducing moisture and lowering temperature. Lowering of temperatures is best achieved by aeration with cool ambient air dur-ing cool nights and periods of cool weather. Both the use of clean storage structures and the segregation of new crop grain from car-ryover grain or grain contaminated with insects are important. If insect infestation has already occurred, fumigation is often used to kill the insects. Aeration with cold air may retard the development of the insect population. Prevention and control of insect infesta-tions are addressed in the section on Prevention of Deterioration.
For information on rodent problems, see the section on Preven-tion of Deterioration.
Moisture content is the most important factor determining suc-cessful storage. Although some grains are harvested at safe storage moistures, other grains (notably corn, rice, and most oilseeds) must usually be artificially dried prior to storage. During some harvest seasons, wheat and soybeans are harvested at moistures above those safe for storage and, therefore, also require drying.
Sauer (1992), Brooker et al. (1992), Hall (1980), Christensen and Meronuck (1986), and Gunasekaran (1986) summarize the basic aspects of grain storage and grain drying. Chapter 22 of the 1999 ASHRAE Handbook—Applications covers crop-drying equipment and aeration systems.
FACTORS DETERMINING SAFE STORAGE Moisture Content Grain is bought and sold on the basis of characteristics of repre-sentative samples. Probes or samplers, such as diverters, are used to obtain representative subsamples. Often representative subsamples must be taken from a large quantity (several tonnes) of grain. Manis (1992) summarizes sampling procedures and equipment. For safe storage, it is necessary to know the range in moisture content within a given bulk and whether any of the grain in the bulk has a moisture content high enough to permit damaging fungal growth. This range can be determined by taking probe samples from different portions of the bulk. Commonly, in large quantities of bulk-stored grain, some portions have moisture contents 2 to 3% higher than the aver-age (Brusewitz 1987). If the moisture content anywhere in the bulk is too high, fungi will grow, regardless of the average. Therefore, the moisture content of a single representative sample is not a reliable measure of storage risk or spoilage hazard. Measurement of mois-ture content and the precision of various moisture-measuring meth-ods are covered in the section on Moisture Measurement.
Table 1 summarizes recommended safe storage moistures for several common grains. Note that for long-term storage, lower The preparation of this chapter is assigned to TC 2.2, Plant and Animal Environment.
11.2 2001 ASHRAE Fundamentals Handbook (SI) moistures are recommended. Most storage fungi will not grow in environments where the relative humidity of the air between kernels is lower than 60%. The relationship between grain moisture and the relative humidity of air between kernels is addressed in the section on Equilibrium Moisture. Table 2 summarizes the relative humidi-ties and temperatures that permit the growth of common storage fungi. Table 3 summarizes the relative humidities and temperatures that permit growth of common storage insects.
Moisture Transfer If temperatures vary within bulk-stored grain, moisture migrates from warmer to cooler portions. The rate of movement depends on the gradients in moisture content and temperature. Sellam and Christensen (1976) studied moisture transfer in a sample of 28.3 L of shelled corn initially at 15.5% moisture. They used heat lamps to produce a temperature differential of 10 K along the length of a sealed plastic container 370 mm long. After 2 days, this gradient (approximately 27 K/m) caused the moisture content at the cool end to increase by 1.2% and the moisture content at the warm end to decrease by 1.1%.
Thorpe (1982) developed an equation to describe moisture trans-fer caused by a temperature gradient. The equation was solved numerically, and laboratory experiments of moisture transfer in wheat were successfully modeled initially at 12% moisture content.
In the experiments, a 10 K temperature gradient was developed across a column of wheat 0.2 mm thick (equivalent to a gradient of 50 K/m). After one month, the moisture content of the warmest grain dropped to 10.6%, while the moisture content of the coolest grain increased to 14%.
Smith and Sokhansanj (1990a) provided a method of approxi-mate analysis of the energy and velocity equations of the natural convection in grain bins. They showed that for small cereals such as wheat, the influence of convection on temperature gradients may not be significant, whereas for larger cereals such as corn, the effect of convection is more noticeable. Smith and Sokhansanj (1990b) also showed that convection flows in a grain bin are significant if the radius of the storage bin is approximately equal to the height of the bin.
Christensen and Meronuck (1986) cite an example of heating that developed in wheat initially at 13.2% in a nonaerated bin. Spe-cially prepared samples were placed at various positions in the bin at the time the bin was filled. After 3 months, the grain began to heat from fungal activity. Moisture content in some of the samples had increased to 18%, while in others it had decreased to 10%.
These examples illustrate the importance of aeration in long-term storage. Aeration is generally required for storage structures with capacities exceeding 70 m3 or 45 t. Moisture migration can ini-tiate fungal and insect growth, and the heat of respiration generated by these organisms accelerates their growth and leads to spoilage.
Studies suggest that temperature gradients could promote spoilage of grain loaded into a ship or barge—even if the grain is initially at a uniform moisture. Most shipments do not spoil because they remain in the ship or barge for a short time and because large tem-perature gradients do not develop. Christensen and Meronuck (1986) report studies of grain quality in barges and ships.
Table 1 Safe Storage Moisture for Aerated Good-Quality Grain Grain Maximum Safe Moisture Content, % wet basis Shelled corn and sorghum To be sold as #2 grain or equivalent by spring 15 To be stored up to 1 year 14 To be stored more than 1 year 13 Soybeans To be sold by spring 14 To be stored up to 1 year 12 Wheat 13 Small grain (oats, barley, etc.) 13 Sunflower To be stored up to 6 months 10 To be stored up to 1 year 8 Source: McKenzie (1980).
Table 2 Approximate Temperature and Relative Humidity Requirements for Spore Germination and Growth of Fungi Common on Corn Kernels Fungus Minimum Relative Humidity for Spore Germination,b % Grain Moisture,a % w.b.
Growth Temperature, °C Lower Limit Optimum Upper Limit Alternaria 91 19 − 4 20 36 to 40 Aspergillus glaucus 70 to 72 13.5 to 14 8 24 38 Aspergillus flavus 82 16 to 17 6 to 8 36 to 38 44 to 46 Aspergillus fumigatus 82 16 to 17 12 40 to 42 55 Cephalosporium acremoniumc 97 22 8 24 40 Cladosporium 88 18 − 5 24 to 25 30 to 32 Epicoccum 91 19 − 4 24 28 Fusarium moniliforme 91 19 4 28 36 Fusarium graminearum, 94 20 to 21 4 24 32 Fusarium roseum (Gibberella zeae) Mucor 91 19 − 4 28 36 Nigrospora oryzaec 91 19 4 28 32 Penicillium funiculosumc (field) 91 19 8 30 36 Penicillium oxalicumc (field) 86 17 to 18 8 30 36 Penicillium brevicompactum (storage) 81b 16 − 2 23 30 Penicillium cyclopium (storage) 81b 16 − 2 23 30 Penicillium viridicatum (storage) 81b 16 − 2 23 36 Source: Stroshine et al. (1984).
a Approximate corn moisture content at 25°C, which gives an interseed relative humid-ity equal to the minimum at which fungus can germinate. It is probably below the moisture content at which the fungus would be able to compete with other fungi on grain, except for Aspergillus glaucus. The latter has no real competitor at 72% rh, except occasionally Aspergillus restrictus.
b Approximately 5% or more of the population can germinate at this relative humidity.
c Rarely found growing in stored grain, regardless of moisture and temperature.
Physiological Factors in Drying and Storing Farm Crops 11.3 Temperature Most processes that cause spoilage in stored grains are accompa-nied by a temperature rise. Therefore, temperatures should be mon-itored throughout the bulk. Temperature monitoring is commonly done by attaching thermocouples to cables that extend through the bulk from the top to the bottom, with thermocouples about 0.9 to 1.8 m apart on each cable. Single cables are used in the center of cir-cular bins up to 7.6 m in diameter. In large bins or flat storage struc-tures, cables are spaced 6 to 7.6 m apart. Relatively dry grain is a good insulator, so a hot spot can develop without being detected immediately (Foster and Mayes 1962). However, when these ther-mocouple spacings are used, extensive spoilage can usually be detected by a temperature rise at a nearby thermocouple. A temper-ature rise of even a few degrees is evidence that grain has spoiled or is spoiling. Forced aeration maintains a uniform and preferably low temperature throughout the bulk.
Table 2 summarizes minimum, optimum, and maximum temper-atures for the growth of some common storage fungi. Storage molds grow slowly at 0 to 4.5°C. However, at higher moisture contents, some species of Penicillium will grow when the temperature is slightly below freezing. Grains with a moisture content high enough for invasion by Aspergillus glaucus will deteriorate rapidly at tem-peratures of 24 to 29°C but can be kept for months without damage at 4.5 to 10°C. Most grain-infesting insects become inactive below about 10°C. Mites remain active but cannot develop rapidly below about 4.5°C. Control of fungi and insects is described further in the section on Prevention of Deterioration.
Oxygen and Carbon Dioxide Only a few fungi that cause stored grain deterioration can grow in an atmosphere containing only 0.1 to 0.2% oxygen or more than 60% carbon dioxide. Some yeasts can grow in grain stored in air-tight storage at moisture contents above 18 to 19% and temperatures above 4.5°C, producing flavors that make the grain unsuitable as food. However, the grain remains suitable feed for cattle and swine (Bell and Armitage 1992), and its nutritional value may be enhanced (Beeson and Perry 1958).
Dry grain is stored in airtight underground or earth-sheltered structures in many parts of the world (Dunkel 1985, Bell and Armit-age 1992). Janardhana et al. (1998) reported that while storing shelled corn at 15 to 20% moisture in a warm, high-humidity envi-ronment, visible molding and the loss of food reserves can be post-poned up to 45 days by using high carbon dioxide in the storage.
Alagusundarum et al. (1996) studied the diffusion of carbon dioxide introduced into bulk stored grain. Bell and Armitage (1992) and Shejbal (1980) cover controlled atmosphere storage in more detail.
Insects present when dry grain is put into storage usually die when oxygen has been depleted, and do not usually reproduce if grain is sufficiently dry and in good condition. Insects can also be controlled in conventional storage structures by forcing carbon dioxide or other gases such as nitrogen through the grain (Jay 1980, Ripp 1984, White and Jayas 1991). Generally, carbon dioxide envi-ronments are effective for insect control only when concentrations are greater than 40% for long periods. Paster et al. (1990, 1991) investigated biogenerated modified atmospheres for insect control.
Athie et al. (1998) reported that using the toxic chemical phosphine with a high carbon dioxide environment increased the effectiveness of the phosphine, particularly in resistant populations. However, the costs of controlled-atmosphere storage may be uneconomic unless the structure can be inexpensively sealed or the gases can be inex-pensively generated or purchased.
Grain Condition Grain that has been stored for several months may already be invaded by storage fungi and partly deteriorated, whether or not this is evident to the naked eye. Molding occurs more rapidly in partially deteriorated grain than in sound grain when the grain is exposed to conditions favorable to mold growth. Microscopic examination and plating techniques can often reveal the fungal infection of grain in its early stages (Sauer et al. 1992, Christensen and Meronuck 1986, Stroshine et al. 1984). Accelerated storage tests, in which samples of grain are stored at a moisture content in equilibrium with air at 80% rh and 29°C and examined periodically, are useful in evaluat-ing storability. These tests enable a manager to estimate the risk of spoilage during storage and to take appropriate action.
Equilibrium Moisture If air remains in contact with a product for sufficient time, the partial pressure of the water vapor in the air reaches equilibrium Table 3 Estimates of Optimum and Minimum Temperatures and Relative Humidity Conditions for Population Increase of Grain-Infesting Insects Insect Type Species Temperature, °C Minimum Relative Humidity, % In Regard to Temperature In Regard to Relative Humidity Minimum Optimum Species Needing High Temperatures Cold hardy Tolerant of low Trogoderma granarium 24 33 to 37 1 Cryptolestes ferrugineus 23 32 to 35 10 Oryzaephilus surinamensis 21 31 to 34 10 Need moderate Plodia interpunctella 18 28 to 32 40 Need high Cryptolestes turcicus 21 30 to 33 50 Moderately cold hardy Tolerant of low Tribolium confusum 21 30 to 33 1 Need moderate Rhyzopertha dominica 23 32 to 35 30 Lasioderma serricorne 22 32 to 35 30 Cold susceptible Tolerant of low Tribolium castaneum 22 32 to 35 1 Oryzaephilus mercator 20 31 to 34 10 Need high Cryptolestes pusillus 22 28 to 33 60 Species Thriving at Moderate Temperatures Cold hardy Need moderate Sitotroga cerealella 16 26 to 30 30 Need high Sitophilus granarius 15 26 to 30 50 Stegobium paniceum 17 25 to 28 60 Acarus siro 7 21 to 27 65 Moderately cold hardy Need high Sitophilus oryzae 17 27 to 31 60 Source: Pederson (1992). Reprinted with permission.
11.4 2001 ASHRAE Fundamentals Handbook (SI) with the partial pressure of the water vapor in the material. The rel-ative humidity of the air at equilibrium with a material of a given moisture is the equilibrium relative humidity. The moisture con-tent of a hygroscopic material in equilibrium with air of a given rel-ative humidity is the equilibrium moisture content Me.
Several theoretical, semitheoretical, and empirical models have been proposed for calculating the Me of grains. Morey et al. (1978) report that the modified Henderson equation is among the best equa-tions available: (1) where Me = equilibrium moisture content, decimal, dry basis t = temperature, °C φ = relative humidity, decimal equivalent K, N, C = empirical constants Table 4 lists values for K, N, and C for various crops. ASAE Standard D245.5 also gives the Chung-Pfost equation, another equation used to predict Me. Figure 1, based on the Chung-Pfost equation, shows equilibrium moisture content curves for shelled corn, wheat, soybeans, and rice. Note that equilibrium moisture depends strongly on temperature. ASAE Standard D245.5 gives additional curves drawn from the Chung-Pfost equation and tabu-lated experimental data. Pfost et al. (1976) summarize variations in reported values of Me for several grains. Locklair et al. (1957) give data for tobacco.
The modified Henderson and the Chung-Pfost equations give only approximate values of Me and are for desorption. When grain is rewetted after it has been dried to a low moisture, the value of Me is generally lower for a given relative humidity. Sun and Woods (1993) reviewed the equilibrium relationship between wheat mois-ture content and air relative humidity. They reported that the hyster-esis effect was greatest at 20 to 40% rh. They also observed that the modified Henderson equation was least effective for wheat.
Me 1 100 ---------1.0 φ – ( ) ln K – t C + ( ) --------------------------1 N ⁄ = Fig. 1 Equilibrium Moisture Relationships for Certain Crops (ASAE Standard D245.5) Physiological Factors in Drying and Storing Farm Crops 11.5 Variations of as much as 0.5 to 1.0% can result from differences in variety; maturity; and relative starch, protein, and oil content.
Shivhare et al. (1992) reported that, under a microwave drying regime, Me increased with drying air velocity and decreased with microwave power.
High-temperature drying can decrease the Me of shelled corn by 0.5 to 1.0% for a given relative humidity (Tuite and Foster 1963).
Sun and Woods (1994) reported that the equilibrium moisture con-tent for wheat dried at low temperatures was generally higher than the data reported in the literature. Chung and Verma (1991) studied the Me of rice during drying and storage.
MOISTURE MEASUREMENT Rapid and accurate measurement of moisture of grains, seeds, and other farm crops determines whether they can be safely stored.
Allowable upper limits for moisture are set by the market, and dis-counts and/or drying charges are usually imposed for higher mois-tures. Drying the grain to moistures below the accepted market limit or the limit for safe storage moisture results in additional drying expense and may actually decrease the value of the grain.
If shelled corn is dried to 12% moisture, it becomes brittle and breaks more easily during handling. The moisture removal also reduces the total mass of grain. After drying 1 kg of shelled corn at 18% moisture to 15%, only 0.965 kg remains. However, at high moistures, seed respiration and fungal growth can cause greater loss in value.
Moisture content can be expressed on a wet or dry basis. The wet basis is used by farmers and the grain trade, while dry-basis mois-tures are often used by engineers and scientists to describe drying rates. Unless otherwise noted, moisture contents in this chapter are on a wet basis and are calculated by dividing the mass of water in the material by the total mass. The dry basis is calculated by divid-ing the mass of water by the mass of dry matter.
(2) (3) where Ww = mass of water Wd = mass of dry matter Percent moisture on a wet basis Mw can be converted to percent moisture on a dry basis Md and vice versa by the following formulas: (4) (5) The mass change resulting from a change in moisture can be determined by assuming that the mass of dry matter is constant. The dry matter is calculated by multiplying the mass of grain by the quantity (1 −Mw /100). For example, 1000 kg of wheat at 15% mois-ture has 1000 (1 −15/100) = 850 kg of dry matter and 150 kg water.
As grain dries, the dry matter remains constant and the mass of water is reduced. If the dry matter is constant, the mass or moisture content of the dried grain is (6) The mass Wd of the dried grain after it reaches a moisture content of Md is (7) For example, if 1000 kg of grain is dried from 15% to 13% moisture, the dried mass is Similarly, to find the moisture content Md of grain after it reaches a dried mass of Wd (8) For example, 100 kg of grain at 15% is dried to a mass of 950 kg.
The dried moisture content is If two quantities of grain at differing moistures are mixed, the final moisture of the mixture can be determined by calculating the mass of water in each, adding these together, and dividing by the total mass. The mass of water is the product of the decimal equiva-lent of Mw and the total mass. For example, if 500 kg of grain at 16% moisture is mixed with 1000 kg of grain at 14% moisture, the mass of water in each sample is The moisture content after mixing will be Either a direct method or an indirect method is used to deter-mine moisture content. Direct methods involving the use of an oven determine moisture content based on the loss in product mass caused by evaporation of the water. The Karl Fischer method, a Table 4 Desorption Equilibrium Moisture Constants for Modified Henderson Equation [Equation (1)] for Various Crops Product K N C Barley 0.000022919 2.0123 195.267 Beans, edible 0.000020899 1.8812 254.230 Canola (rapeseed) 0.000505600 1.5702 40.1205 Corn, yellow dent 0.000086541 1.8634 49.810 Peanut, kernel 0.000650413 1.4984 50.561 Peanut, pod 0.000066587 2.5362 23.318 Rice, rough 0.000019187 2.4451 51.161 Sorghum 0.000085320 2.4757 113.725 Soybean 0.000305327 1.2164 134.136 Wheat, durum 0.000025738 2.2110 70.318 Wheat, hard 0.000023007 2.2857 55.815 Wheat, soft 0.000012299 2.5558 64.346 Source: ASAE Standard D245.5.
% Moisture (wet basis): Mw 100Ww Ww Wd + ----------------------= % Moisture (dry basis): Md 100Ww Wd -----------------= Md 100Mw 100 Mw – -----------------------= Mw 100Md 100 Md + ----------------------= Ww 1 Mw 100 ---------– Wd 1 Md 100 ---------– = Wd Ww 100 Mw – 100 Md – ----------------------- = Wd 1000 100 15 – 100 13 – --------------------- 977 kg = = Md 100 Ww Wd --------– 100 Mw – ( ) = Md 100 1000 950 ------------ 100 15 – ( ) – 10.5% = = 500 kg ( ) 0.16 ( ) 80 kg = 1000 kg ( ) 0.14 ( ) 140 kg = 80 140 + 500 1000 + ---------------------------100 × 14.7% = 11.6 2001 ASHRAE Fundamentals Handbook (SI) basic reference method involving a chemical reaction of water and a reagent, is classified as a direct method.
Indirect methods such as moisture meters measure the properties of the material that are functions of moisture content. Indirect mois-ture measurement methods are used in commercial practice. Direct methods are used in research and to calibrate indirect methods.
Christensen et al. (1992) summarize approved methods used in Europe and the United States.
Direct Methods Christensen et al. (1992) describe the fundamental or basic meth-ods of moisture determination as (1) drying in a vacuum with a des-iccant and (2) titration with a Karl Fischer reagent. It is assumed that these methods measure the true water content and can be used to verify measurements obtained with routine reference methods, including oven drying and the Brown-Duval distillation method.
The Brown-Duval method, not commonly used, involves heating the grain in a special apparatus and condensing and collecting the vaporized water.
Oven techniques use either forced-convection air ovens or vac-uum ovens and either ground or whole kernels. Drying times and temperatures vary considerably, and the different techniques can give significantly different results. Oven techniques are used to cal-ibrate moisture meters (see the section on Indirect Methods). As a result, during the export of grain, the meter moisture measurements can vary between arrival and destination if the importing country uses a different standard oven technique than the exporting country.
ASAE Standard S352.2 is a widely used standard that recom-mends heating temperatures and times for various grains. The tem-peratures may be either 103°C (shelled corn, soybeans, sunflower) or 130°C (wheat, barley, onion). Heating times vary between 50 min (onion seeds) and 72 h (soybeans, shelled corn).
Grinding samples and using a vacuum oven reduce heating time.
When initial moistures are high, a two-stage method may be used (USDA 1971). A weighed sample of whole grain is partially dried to a moisture content of 13% or below, weighed, and then ground and completely dried as in the one-stage method. The mass lost in both stages is used to calculate moisture content.
Indirect Methods Electronic moisture meters are simple to operate and give read-ings within minutes. Direct methods of moisture measurement are used to calibrate the meters for each type of grain. Meters are sen-sitive to grain temperature, and calibration must include a tempera-ture correction factor. The newer automatic meters or moisture computers sense and correct for sample temperature and print or display the corrected moisture.
Near infrared reflectance (NIR) instruments have been devel-oped that measure moisture, protein, starch, and oil content of ground samples (Butler 1983, Cooper 1983, Watson 1977). Near infrared transmittance (NIT) instruments measure the properties of whole seeds.
Conductance meters measure resistance, which varies with grain moisture. The practical range of moisture content measurable by conductance meters is approximately 7 to 23%. For up to 72 h after moisture addition or removal, the moisture at the surface of the kernels differs from the moisture in the interior. Therefore, recently dried grain reads low and recently wetted grain reads high. Mixing wet and dry grain and mixing good grain with partially deteriorated grain also result in erroneous readings. Martin et al. (1986) mea-sured the signal from the conductance Tag-Heppenstall meter and related this to individual kernel moisture variations in mixtures of wet and dry corn.
The dielectric properties of products depend largely on moisture content. The capacitance meter uses this relationship by using grain as the dielectric in a capacitor in a high-frequency electrical circuit. Although the capacitive reactance is the primary portion of the overall impedance measured, the resistive component is also significant in many capacitance meters. At higher frequencies and in instruments with insulated electrodes, the relative effect of the resistance is reduced, which is important in reducing errors intro-duced by unusual product surface conditions. Capacitance meters are affected less than conductance meters by uneven moisture dis-tribution within kernels. Sokhansanj and Nelson (1988a) showed that the capacitance meters give low and high readings, respec-tively, on recently dried or rewetted grain. The range of measurable moisture content is slightly wider than that for conductance meters.
Moisture measurement by capacitance meters is sensitive to tem-perature, product mass, and product density (Sokhansanj and Nel-son 1988b). To reduce these sources of error, a weighed sample is introduced into the measuring cell by reproducible mechanical means. Calibration, including temperature correction, is required.
At least one commercially available unit measures bulk density and corrects for this factor as well as temperature. Tests of moisture meter accuracy have been reported by Hurburgh et al. (1985, 1986).
Accuracy of moisture readings can be improved by taking multiple samples from a grain lot and averaging the meter measurements.
Equipment for continuous measurement of moisture in flowing grain is available commercially but is not widely used in the grain trade.
Equilibrium relative humidity (described in the section on Equilibrium Moisture) can be used to indicate moisture content. It also indicates storability independent of the actual moisture content because the equilibrium relative humidity of the air surrounding the grain, to a large extent, determines whether mold growth can occur (see Table 2). Measurement of equilibrium relative humidity at spe-cific points within a grain mass requires specialized sampling equipment and has been used primarily for research.
Hay moisture content does not receive the consideration devoted to grains. Oven methods (ASAE Standard S358.2) are used exten-sively, as well as some commercial hay and forage moisture meters.
Several conductance moisture meters are available for both hay and forages, but the extreme variability of the moisture and density of the material tested lead to great variability in the readings obtained.
A reasonable indication of the average moisture content of a mass of hay can be obtained if many (25 or more) measurements are taken and averaged.
Microwave radiation can be used to sense properties of grains.
Kraszewski and Nelson (1992) reviewed the use of resonant micro-wave cavities to determine simultaneously both mass and moisture content in grain kernels.
PREVENTION OF DETERIORATION Fungal Growth and Mycotoxins Fungal growth is the most important limitation to successful dry-ing and storage of grain. Sauer et al. (1992) provide a good review of microflora in grains. Early detection of mold growth would pro-vide the storage manager with a management tool. Magan (1993) reviews early detection methods, including enzyme and biochemi-cal tests, fungal volatiles, and respiration activity.
Mycotoxins (fungal metabolites) may affect the marketability and use of moldy grains. Wicklow (1988) reported that climate and other natural processes influenced the distribution of aflatoxigenic strains of microflora. Choudhary and Sinha (1993) reported that aflatoxigenic Aspergillus species were positively correlated with grain moisture content in the field and afternoon relative humidity.
In the United States, corn is one of the major crops that must be harvested above safe storage moistures. Shelled corn can be held at these higher moistures for a limited time before it must be dried.
Mold growth produces carbon dioxide (CO2). Allowable storage time at moistures above those for safe storage can be estimated by measuring CO2 production of samples. By assuming that a simple Physiological Factors in Drying and Storing Farm Crops 11.7 sugar is being oxidized by microbial respiration, CO2 production can be expressed in terms of dry matter loss in percent by mass.
Saul and Steele (1966) and Steele et al. (1969) studied the pro-duction of CO2 in shelled corn, mostly on samples above 18%.
Based on changes in the official grade of shelled corn, Saul and Steele (1966) established a criterion for acceptable deterioration of quality as 0.5% dry matter loss. This is equivalent to the production of 0.00735 kg of CO2 per kilogram of dry matter. Thompson (1972) expressed Saul’s data on dry matter loss per kilogram of dry matter as a function of moisture, time, and temperature using the following mathematical expression: (9) with (10) (11) where DML = dry matter loss per kilogram of dry matter, kg/kg θ = time in storage, h t = grain temperature, °C Md = moisture content, % dry basis Table 5 lists values for A, B, and C. According to Steele (1967), the damage level effect can be determined for dry matter losses of 0.1, 0.5, and 1.0% by multiplying θ from Equation (9) by Kd, where Kd is calculated as follows: (12a) (12b) (12c) where d = mechanical damage, % by mass.
Seitz et al. (1982a, 1982b) found unacceptable levels of aflatoxin production prior to the time when 0.5% dry matter loss occurred.
Nevertheless, Equations (9) through (12) give approximate predic-tions of mold activity, and they have been used in several computer simulation studies (Thompson 1972, Pierce and Thompson 1979, Brooker and Duggal 1982). Based on a simulation, Thompson (1972) concluded that for airflow rates between 7 and 27 L/(s·m3), grain deterioration in the top layer during low-temperature drying is doubled when the airflow rate is halved. Thompson also concluded that weather variations during harvest and storage seasons can cause up to a twofold difference in deterioration. Pierce and Thompson (1979) recommend airflow rates for several common low-tempera-ture drying systems and for various locations in the midwestern United States.
Acceptable dry matter losses for wheat and barley are much lower than those for shelled corn—0.085% and 0.10%, respectively (Brook 1987). Hamer et al. (1992) reported visible mold growth at 0.15% dry matter loss. Brook reported reasonable agreement with published experimental data for the following equation (Frazer and Muir 1981) for allowable storage time as a function of percent wet basis moisture and temperature based on the development of visible mold: (13) where θD = allowable storage time, days t = temperature, °C A,B,C = empirical constants, defined as follows: Brook (1987) also reported that an adaptation of Equation (9) by Morey et al. (1981) gave reasonable results for storage time of wheat. Morey’s method predicts dry matter loss by adjusting Md for differences between corn and wheat equilibrium relative humidities.
Table 2 can be used to gain insight into the deterioration of stored grain. Aspergillus and Penicillium sp. are primarily respon-sible for deterioration because some of their species can grow at storage moistures and temperatures frequently encountered in commercial storage. In temperate climates, shelled corn is often harvested at relatively high moistures; during the harvest and stor-age season, ambient temperatures can be relatively low. Aeration of the grain during cold weather and cool nights can reduce the temperature of the grain to 4 to 16°C. This is below the optimum temperature for growth of Aspergillus sp. (Table 2). However, Pen-icillium sp. can still grow if grain moisture is above 16 to 17%; therefore, its growth is a persistent problem in temperate climates.
If hot weather prevails prior to harvest, Aspergillus flavus, which competes effectively at warmer temperatures and higher moistures, can begin to grow in the field and continue to grow in stored shelled corn. In growing seasons when shelled corn must be har-vested at moistures above 22%, Fusarium, Alternaria, Epicoccum and Mucor can compete with Penicillium sp.
Chemical Treatment. Application of chemicals slows deterio-ration until grain can be either dried or fed to animals. Preservatives include propionic acid, acetic acid, isobutyric acid, butyric acid (Sauer and Burroughs 1974), a combination of sorbic acid and car-bon dioxide (Danziger et al. 1973), ammonia (Peplinski et al. 1978), and sulfur dioxide (Eckhoff et al. 1984). Propionic acid (Hall et al.
1974) or propionic-acetic acid mixtures, although not extensively used, are perhaps the most popular in the United States with high-moisture corn. Acetic acid and formic acid are popular in Europe.
Grain treated with propionic acid can be used only as animal feed.
Hertung and Drury (1974) summarize fungicidal levels needed to preserve grain at various moistures. Both ammonia (Nofsinger et al. 1979, Nofsinger 1982) and sulfur dioxide (Eckhoff et al. 1984, Tuite et al. 1986) treatments require considerable management.
Attention must be given to uniform application of the chemicals to the entire quantity of stored grain.
Antimicrobial properties occurring naturally in plants have been studied (Beuchar and Golden 1989, Shelef 1984). Some of these also inhibit mycotoxin formation (Bullerman et al. 1984, Rusal and Marth 1988). The essential oils of oregano and thyme were tested as fumigants against Aspergillus species and natural microflora of wheat (Paster et al. 1995). Oregano oil provided complete control at 2.0 mL/m3; thyme oil was not completely effective at 4.0 mL/m3 and affected seed germination at 5.0 mL/m3.
Table 5 Constants for Dry Matter Loss of Shelled Corn [Equation (11)] Temperature Range, °C Moisture Range, % Wet Bulb.
A B C t < 15 All moistures 128.76 − 4.68 0 t ≥15 Mw ≤19 32.3 − 3.48 0 t ≥15 19 < Mw ≤28 32.3 − 3.48 (Mw −19)/100 t ≥15 Mw >28 32.3 − 3.48 0.09 DML 1.3 0.006θ KmKt ----------------- 1.0 – exp 0.015θ KmKt -----------------+ = Km 0.103 455 Md 1.53 --------------- 0.00845Md 1.558 + – exp = Kt A B 0.03t + 0.533 ( ) [ ] C + 0.0183t 0.285 – ( ) exp exp = 0.1% DML: Kd 1.82 0.0143d – ( ) exp = 0.5% DML: Kd 2.08 0.0239d – ( ) exp = 1.0% DML: Kd 2.17 0.0254d – ( ) exp = Moisture Range, % w.b.
A B C 12.0 < Mw ≤19.0 6.234 − 0.2118 − 0.0527 19.0 < Mw < 24.0 4.129 − 0.0997 − 0.0567 θD log A BMw Ct + + = 11.8 2001 ASHRAE Fundamentals Handbook (SI) Insect Infestation Insects cause major losses of stored grain. Grain containing live insects or insect fragments in sufficient numbers is unsuitable for human food. When grain is stored for long periods (a year or more), insects can infest the grain and cause significant amounts of deteri-oration. Traps and chemical attractants have been developed that monitor insects in storage facilities (Barak and Harein 1982, Barak and Burkholder 1985, Burkholder and Ma 1985). Detection in sam-ples of grain taken for grading and inspection is often difficult.
Many of the insects are relatively small and can be seen easily only with a magnifying lens. Many of the insect larvae develop within the kernels and cannot be detected without staining techniques or grinding of the grain sample. Infested grain mixed with good grain in marketing channels compounds the infestation problem.
Sanitation is one of the most effective methods of insect control.
Cleaning of bins after removal of old-crop grain and prior to filling with new-crop grain is essential. In bins containing perforated floors, fine material that collects beneath the floors can harbor insects, which infest new-crop grain when it is added. Control by aeration is feasible in temperate climates because insect activity is reduced greatly at temperatures below 10°C. The effectiveness of temperature control has been documented by Bloome and Cuperus (1984) and Epperly et al. (1987). Chemicals have frequently been used to control live insects in grain, and methods are described by Harein and Davis (1982). Thermal treatments have also been inves-tigated (Lapp et al. 1986). Pederson (1992) summarizes the types of grain insects, the ecology of insect growth, and the methods of detecting insects in samples of grain. Athie et al. (1998) reviewed the status and future of chemical grain protectants. Armitage et al.
(1994) proposed an integrated pest management strategy combining surface insecticide treatment and aeration. Control of insects in farm-stored grain is detailed by Storey et al. (1979), Quinlan (1982), and Harein and Davis (1992).
Rodents The shift from ear corn harvesting and storage to field shelling and the introduction of metal bins have helped to reduce rodent problems. However, significant problems can arise when rodents consume grain and contaminate it with their hair and droppings.
Storage structures should be made rodent-proof whenever possible.
Rats can reach 330 mm up a wall, so storage structures should have concrete foundations and metal sides that resist gnawing.
In some countries, smaller on-farm storage structures are often elevated 460 mm to give protection from rodents. Double-wall con-struction and false ceilings should be avoided, and vents and holes should be covered with wire grates. Proper sanitation can help pre-vent rodent problems by eliminating areas where rodents can nest and hide. Rodents need water to survive, so elimination of available water is also effective. Techniques for killing rodents include trap-ping, poisoning with bait, and fumigation. Harris and Bauer (1992) address rodent problems and control in more detail.
DRYING THEORY In ordinary applications, drying is a heat and mass transfer pro-cess that vaporizes liquid water, mixes the vapor with the drying air, and removes the vapor by carrying away the mixture mechanically.
In forced-convection drying, sufficient heat for vaporization of product moisture (about 2560 kJ/kg of water) comes from the sen-sible heat in the drying air.
The most common mode of drying uses the sensible heat content of the air. The method can be diagrammed on the psychrometric chart by locating the state points for the air as it is heated from ambi-ent temperature to plenum temperature and then exhausted from the grain. The process is assumed to be adiabatic (i.e., all the sensible heat lost by the air is used for moisture vaporization and converted to latent heat of the water vapor in the drying air). Therefore, the state point of the air can be considered to move along adiabatic sat-uration lines on the psychrometric chart. In the simplified psychro-metric chart in Figure 2, the ambient air at dry-bulb temperature ta and dew-point temperature tdp is heated to drying air temperature td, where it has a relative humidity φ1. As the air passes through the grain, its sensible heat provides the latent heat of vaporization of the water. When the air exits from the grain, its temperature has dropped to te, and its relative humidity has increased to φ2. The moisture gained by each kilogram of drying air is the difference W2 −W1 in humidity ratio. If the air has sufficient contact time with the grain, the value for φ2 will be the equilibrium relative humidity of the grain at that moisture and temperature te.
Example 1. Shelled corn at 20% moisture content is dried with air heated to 70°C. The air has an ambient temperature of 20°C with a dew point of 10°C. The air is observed to exhaust from the shelled corn at 30°C.
Find the amount of energy needed to heat the air and the amount of water removed per kilogram of dry air.
Solution: Estimate the psychrometric air conditions, using information contained in Chapter 6 and assuming a standard atmospheric pressure of 101.325 kPa.
At 20°C, the enthalpy of the dry air is ha = 20.121 kJ/kg (Table 2 in Chapter 6) and the saturation vapor pressure pws = 2.3388 kPa (Table 3 in Chapter 6). At 10°C dew point, the vapor pressure pw = 1.2280 kPa, and the enthalpy of the water vapor hg = 2519.12 kJ/kg (Table 3 in Chapter 6). The relative humidity is then φ = 53% [Equation (24) in Chapter 6]; humidity ratio W = 0.0076 kg/kg [Equation (22) in Chapter 6]; and enthalpy h = 39.3 kJ/kg [Equation (29) in Chapter 6]. As the air is heated, the humidity ratio is assumed to remain constant. At 70°C, the enthalpy of the dry air is ha = 70.489 kJ/kg (Table 2 in Chapter 6) and the saturation vapor pressure pws = 31.198 kPa (Table 3 in Chapter 6). The relative humidity has been reduced to φ = 4% [Equation (24) in Chapter 6]; enthalpy increased to h = 89.6 kJ/kg [Equation (29) in Chapter 6]; and the wet-bulb temperature of the drying air is t = 29°C [iterative solution to Equation (35) in Chapter 6]. The amount of energy needed to heat each kilogram of dry air is then 89.6 −39.3 = 50.3 kJ.
As the heated air passes through the grain, it increases in moisture and decreases in temperature until it comes into equilibrium with the corn at the point of air exhaust (initially 20%). The exhaust air relative humidity can be estimated by reading the equilibrium relative humidity from the curve for shelled corn shown in Figure 1. Enter the curve for shelled corn at 20% equilibrium moisture content and a temperature of 30°C. The equilibrium relative humidity is approximately 92%. At 30°C, the saturation vapor pressure of the air pws = 4.2460 kPa (Table 3 in Chapter 6); the vapor pressure pw = 3.9063 kPa [Equation (24) in Chapter 6]; and the humidity ratio W = 0.0249 kg/kg [Equation (22) in Fig. 2 Drying Process Diagrammed on Psychrometric Chart Showing Adiabatic Evaporation of Moisture from Grain Physiological Factors in Drying and Storing Farm Crops 11.9 Chapter 6]. Each kilogram of dry air carries with it 0.0249 −0.0076 = 0.0173 kg of water from the grain.
After the grain at the air exhaust has dried to 15%, the equilibrium moisture content curve from Figure 1 can be used to estimate the exhaust air relative humidity. If the temperature of the air were 30°C, then the equilibrium relative humidity would be approximately 76%; if the temperature of the air were 40°C, then the equilibrium relative humidity would be approximately 81%. From Equation (35) in Chapter 6, the wet-bulb temperatures associated with these two points are 27°C and 46°C, respectively. A linear interpolation between these two points results in an air temperature of 32°C and an equilibrium relative humid-ity of 77%. At 32°C, the saturation vapor pressure of the air pws = 4.7585 kPa (Table 3 in Chapter 6); the vapor pressure pw = 3.664 kPa [Equation (24) in Chapter 6]; and the humidity ratio W = 0.0236 kg/kg [Equation (22) in Chapter 6]. Each kilogram of dry air carries with it 0.0236 −0.0076 = 0.016 kg of water from the grain.
Thin Layer Drying A thin layer of grain is a layer of grain no more than several ker-nels deep. The ratio of grain to air is such that there is only a small change in temperature and relative humidity of the drying air when it exits the grain. The maximum rate (dM/dθ) at which a thin layer of a granular hygroscopic material (such as grain) transfers moisture to or from air can be approximated by the following equation (Hukill 1947): (14) where C = constant representing vapor conductivity of kernel and surrounding air film pg = partial pressure of water vapor in grain pa = partial pressure of water vapor in drying air If pg > pa, drying takes place. If pg = pa, moisture equilibrium exists and no moisture transfer occurs. If pg < pa, wetting occurs.
The assumption of a linear relationship between (1) water vapor pressure and equilibrium relative humidity and (2) equilibrium rel-ative humidity and moisture content over the range in which drying occurs lead to the following equation: (15) where M = moisture content (dry basis) of material at time θ Me = equilibrium moisture content (dry basis) of material in reference to drying air k = constant dependent on material The solution to this differential equation is (16) where Mo = moisture content, dry basis, when θ = 0.
In later work (Hukill and Schmidt 1960, Troeger and Hukill 1971), Hukill recognized that Equation (16) did not describe the drying rate of grain adequately. Misra and Brooker (1980) identified the following model as more promising for shelled corn: (17) They give an equation for K, which is a function of drying air tem-perature and velocity, and another equation for N as a function of drying air relative humidity and initial grain moisture. Their equa-tions are valid for drying air temperatures of 2.2 to 71°C, drying air relative humidities of 3 to 83%, drying air velocities of 0.025 to 2.33 m/s, and initial moistures of 18 to 60% (dry basis).
Li and Morey (1984) also fit their data to Equation (17) and found that within the limits of drying airflow rates and air relative humidities used, K and N can be expressed as functions of air tem-perature and initial grain moisture only. Their equations for K and N apply to air temperatures ranging from 27 to 115°C, initial grain moistures of 23 to 36% dry basis, airflows of 0.27 to 1.34 m3/(s·m3), and air relative humidities of 5 to 40%.
Other forms of the thin layer drying equation have also been pro-posed. Thompson et al. (1968) fitted data for shelled corn to the fol-lowing equation, which is applicable in the range of 60 to 150°C: (18) where A = − 1.70562 + 0.00878t B = 427.3740 exp (− 1.0563 −0.05942t) MR = (M −Me)/(Mo −Me) θ = time, h t = temperature, °C Martins and Stroshine (1987) describe the effects of hybrid and damage on the thin layer drying rate and give values for constants A and B in Equation (18) for several hybrids and damage levels.
Results of thin layer drying tests for other grains have also been reported. Data are available for the following grains: • Wheat (Watson and Bhargava 1974, Sokhansanj et al. 1984, Bruce and Sykes 1983) • Soybeans (Hukill and Schmidt 1960, Overhults et al. 1973, Sabbah et al. 1976) • Barley (O’Callaghan et al. 1971, Sokhansanj et al. 1984, Bruce 1985) • Sorghum (Hukill and Schmidt 1960, Paulsen and Thompson 1973) • Rice (Agrawal and Singh 1977, Noomhorm and Verma 1986, Banaszek and Siebenmorgen 1990) • Sunflower (Syarief et al. 1984, Li et al. 1987) • Canola (Sokhansanj et al. 1984, Pathak et al. 1991) • Oats (Hukill and Schmidt 1960) • Lentil seeds (Tang et al. 1989) Sokhansanj and Bruce (1987) developed more rigorous thin layer drying equations based on simultaneous heat and mass trans-fer through a single kernel and demonstrated that such a model accurately predicts the temperature and moisture content of the grain throughout the drying process. Jayas et al. (1991) reviewed thin-layer drying models, and Parti (1993) presented comparisons of models under different conditions.
Equations (16) through (18) do not describe the usual drying pro-cess, where grain is in a deep bed and where drying air changes con-dition but does not necessarily reach moisture equilibrium with the grain. Those models, which are formulated using thin layer drying equations such as these, are summarized in the section on Deep Bed Drying.
Airflow Resistance Data on resistance of grain to airflow are used for a variety of design calculations such as selecting fans, determining optimum depths for drying bins, predicting airflow paths in bins with aeration ducts, and determining the practical limitations on airflow caused by fan power requirements. For a given fan and dryer or bin, airflow resistance can change with the type of grain being dried, the depth of grain, and the amount of fine material in the grain. In many grain-drying applications, such as when air is forced through a grain bin that has a uniform grain depth and a full perforated floor, airflow is one-dimensional and the pressure drop per unit depth of grain can be assumed to be constant. Shedd (1953) determined pressure drop per dM dθ --------C – pg pa – ( ) = dM dθ --------k – M Me – ( ) = M Me – Mo Me – --------------------kθ – ( ) exp = M Me – Mo Me – --------------------KθN – ( ) exp = θ A MR ln B MR ln ( )2 + = 11.10 2001 ASHRAE Fundamentals Handbook (SI) unit depth versus airflow for a number of grains and seeds and sum-marized by plotting them on logarithmic axes. These curves (com-monly called Shedd’s curves) are included in ASAE Standard D272.3. They can also be calculated from the following equation (ASAE 1996, Sokhansanj and Yand 1996): (19) where ∆p = pressure, Pa L = bed depth, m Q = airflow rate, m3/(s·m2) a, b = empirical constants Table 6 summarizes the constants for Equation (19) for some of the more common grains. Constants for grass seeds and some veg-etables are included in ASAE Standard D272.3. Jayas and Mann (1994) reviewed the presentation of airflow resistance data for 22 different seeds, including grains. They reported that the mean rela-tive percent error for each grain could be significantly reduced if the airflow range were divided into two subranges: 0.004 to 0.05 m3/(s·m2) and 0.05 to 0.35 m3/(s·m2).
Equation (19) gives the airflow resistance for clean, dry grain when the bin is loaded by allowing the grain to flow into the bin through a chute from a relatively low height. Kumar and Muir (1986) reported on the effect of filling method on the airflow resis-tance of wheat and barley. Jayas et al. (1987) showed that the resis-tance of canola to airflow in a horizontal direction was 0.5 to 0.7 times the resistance to airflow for the vertical direction.
The presence of fine material in grain can significantly alter the airflow resistance. Fine material is generally defined as broken ker-nels and other matter that can pass through a round hole sieve with a hole size slightly less than the kernel size. For shelled corn, a 4.76 mm round hole sieve is used to measure fines.
The pressure drop per meter is routinely increased by multiply-ing the value from Equation (19) by a packing factor. A factor of 1.5 is used for shelled corn; 1.2 for other grains. Haque et al. (1978) developed a correction factor for airflow resistance in shelled corn with fine material fractions from 0 to 20% Grama et al. (1984) reported the effect of fine material particle size distribution on resistance in shelled corn. They also reported the effect of the increased resistance from fines on fan power requirements. Kumar and Muir (1986) reported the effects of fines in wheat. Bern and Hurburgh (1992) reviewed the characteristics of fines in shelled corn, including their composition, size distribution, density, airflow resistance, and nutritive and economic value. They concluded that fines can increase airflow resistance up to 200%.
Bulk density can have a significant effect on airflow resistance.
For moderate heights of 4 to 7.5 m, drop height does not affect bulk density in bins filled with a spout (Chang et al. 1986). Bern et al.
(1982) reported that auger stirring can decrease the bulk density of bins filled with a grain spreader but has no effect on or increases bulk density in bins filled by gravity. Magnitudes of the increase in bulk density caused by grain spreaders have been reported by Stephens and Foster (1976b, 1978) and Chang et al. (1983).
Moisture content also affects airflow resistance. Its effect may, in part, be caused by its influence on bulk density. Shedd’s curves include a footnote recommending that for loose fill of clean grain, airflow resistance should be multiplied by 0.80 if the grain is in equilibrium with air at relative humidities greater than 85% (ASAE Standard D272.2). At 21°C, this corresponds to a moisture of 18% or more for shelled corn (Figure 1). Haque et al. (1982) give equa-tions that correct for the effects of moisture content of shelled corn, sorghum, and wheat.
Li and Sokhansanj (1994) argued that a generalized equation for airflow resistance, a modification of Leva’s equation (Leva 1959), could account for airflow resistance differences due to variations in grain density, moisture content, and fines. Supporting constants for nine seeds are presented. Giner and Denisienia (1996) proposed a modified Ergun equation for the quadratic moisture effects and lin-ear fines effects in wheat.
When the flow lines are parallel and airflow is linear (as is the case in a drying bin with a full perforated floor), calculation of the airflow is a straightforward application of Equation (16). For a given fan attached to a particular bin filled to a uniform depth with grain, the operating point of the fan can be determined as follows. A curve is plotted showing the total static pressure in the bin plenum versus airflow to the bin. Airflow rate is calculated by dividing the total air volume supplied to the plenum by the cross-sectional area of the bin. Using Equation (16), the pressure drop per unit depth can be calculated and multiplied by the total depth of grain in the bin to give total static pressure in the plenum. The fan curve showing air delivery volume versus static pressure can be plotted on the same axes. The intersection of the curves is the operating point for the fan.
These calculations can also be done on a computer, and the point of intersection of the curves can be determined using appropriate numerical methods. McKenzie et al. (1980) and Hellevang (1983) summarize airflow resistances for various bin and fan combinations in tabular and graphical form. Sokhansanj and Woodward (1991) developed a design procedure for use on personal computers to select fans for near-ambient drying of grain.
In cases where airflow is nonlinear, as in conical piles or systems with air ducts, computation is complex (Miketinac and Sokhansanj 1985). Numerical methods for predicting airflow patterns have been developed and applied to bins aerated with ducts (Brooker 1969, Segerlind 1982, Khompos et al. 1984), conical-shaped piles (Jindal and Thompson 1972), and bins in which porosity varies within the bed (Lai 1980). Lai’s study applies to bins in which filling methods have created differences in bulk density within the bin or where fine material is unevenly distributed. Alagusundaram et al. (1994) stud-ied airflow patterns through wheat, barley, and canola in bins with different patterns of partially perforated floors.
Analysis of Deep Bed Drying The ability to predict the rate at which grain dries in a given type of dryer operating in specific weather conditions with a specified Table 6 Constants for Airflow Resistance [Equation (19)] Material Value of a, Pa·s2/m3 Value of b, m2·s/m3 Range of Q, m3/(s·m2) Barley 2.14 × 104 13.2 0.0056 to 0.203 Canola (rapeseed) 5.22 × 104 7.27 0.0243 to 0.2633 Ear corn 1.04 × 104 325 0.051 to 0.353 Lentils 5.43 × 104 36.79 0.0028 to 0.5926 Oats 2.41 × 104 13.9 0.0056 to 0.203 Peanuts 3.80 × 103 111 0.030 to 0.304 Popcorn, white 2.19 × 104 11.8 0.0056 to 0.203 Popcorn, yellow 1.78 × 104 17.6 0.0056 to 0.203 Rice, rough 2.57 × 104 13.2 0.0056 to 0.152 Rice, long brown 2.05 × 104 7.74 0.0055 to 0.164 Rice, long milled 2.18 × 104 8.34 0.0055 to 0.164 Rice, medium brown 3.49 × 104 10.9 0.0055 to 0.164 Rice, medium milled 2.90 × 104 10.6 0.0055 to 0.164 Shelled corn 2.07 × 104 30.4 0.0056 to 0.304 Shelled corn, low airflow 9.77 × 103 8.55 0.00025 to 0.0203 Sorghum 2.12 × 104 8.06 0.0056 to 0.203 Soybeans 1.02 × 104 16.0 0.0056 to 0.304 Sunflower, confectionery 1.10 × 104 18.1 0.055 to 0.178 Sunflower, oil 2.49 × 104 23.7 0.025 to 0.570 Wheat 2.70 × 104 8.77 0.0056 to 0.203 Wheat, low airflow 8.41 × 103 2.72 0.00025 to 0.0203 Source: ASAE Standard D272.3.
p ∆ L ------aQ2 1 bQ + ( ) ln ---------------------------= Physiological Factors in Drying and Storing Farm Crops 11.11 airflow and air temperature can assist designers in developing dry-ers for maximum efficiency. It can also guide operators in finding the optimum way to operate their particular dryers for given weather conditions. Computer simulations have helped researchers under-stand the mechanisms and processes involved in drying.
Two relatively simple prediction equations can be solved on a hand calculator. Hukill (1947) developed a widely known and used method that predicts the moisture distribution in a bed of grain dur-ing drying. A graphical presentation of one of the equations, which further simplifies calculations, is available. Hukill’s method is sum-marized by Brooker et al. (1992), who give an example calculation for shelled corn drying. Barre et al. (1971) made further adaptations of Hukill’s method, and Foster (1986) gives a historical perspective on the development and utility of the method. Brooker et al. (1992) also present a technique called the heat balance equation, which equates the heat available in the air for drying with the amount of heat needed to evaporate the desired amount of water from the grain. Both of the above methods take into account airflow, drying air temperature and relative humidity, exit air conditions, grain moisture, and the amount of grain to be dried.
Thompson et al. (1968) considered a deep bed of grain as a series of thin layers of grain stacked one on top of another. Algebraic heat and mass balances were applied to each layer, with the exit air con-ditions of one layer becoming the input conditions of the next layer.
Thompson et al. (1969) used the model to predict concurrent-flow, crossflow, and counterflow drying of shelled corn. Paulsen and Thompson (1973) used it to evaluate crossflow drying of sorghum.
Stephens and Thompson (1976) and Pierce and Thompson (1981) used the model to make recommendations about optimum design of high-temperature grain dryers.
Bakker-Arkema et al. (1978) used simultaneous heat and mass transfer equations in a series of coupled partial differential equations to describe deep bed drying. The equations, solved using a finite dif-ference technique, predict grain temperature, grain moisture con-tent, and air temperature and humidity ratio. Bakker-Arkema et al.
(1979, 1984) give solutions for in-bin, batch, continuous crossflow, and continuous concurrent-flow dryers. Morey et al. (1976) used the model to evaluate energy requirements for drying. Morey and Li (1984) and Bakker-Arkema et al. (1983) demonstrated the effect of thin layer drying rate on the model predictions. Bridges et al. (1980) used the Thompson model for simulation of batch-in-bin drying.
Morey et al. (1978) and Parry (1985) review many of the mathemat-ical models used for high-temperature grain drying.
Computer simulations have also been developed for low-temper-ature and solar drying. Some of these models have been referenced in the section on Fungal Growth and Mycotoxins under Prevention of Deterioration. Thompson (1972) developed a model that was later used by Pierce and Thompson (1979) to make recommenda-tions on airflow in solar grain drying and by Pierce (1986) to eval-uate natural air drying. Sabbah et al. (1979) used the logarithmic model of Barre et al. (1971) for simulation of solar grain drying.
Bridges et al. (1984) used a model to evaluate the economics of stir-ring devices in in-bin drying systems. Morey et al. (1979), Frazer and Muir (1981), Bowden et al. (1983), and Smith and Bailey (1983) have also modeled low-temperature drying. Sharp (1982) reviewed low-temperature drying simulation models.
Aeration of Grain Aeration involves forcing small amounts of air through the stored grain to maintain a uniform temperature. Prior to the devel-opment of this concept, grain was turned by moving it from one storage bin to another. Foster (1986) credits Hukill (1953) with developing the concept of aeration. As mentioned in the sections on Fungal Growth and Mycotoxins and Insect Infestation, lowering of the grain temperature during winter in temperate climates can reduce the rate of deterioration from molds and insects. Aeration can also prevent temperature gradients from developing within the grain mass. Such gradients can cause moisture migration, which results in unacceptably high moistures in certain portions of the bin.
Aeration is used to cool stored grain in the fall. A typical practice is to aerate the grain when the difference between grain temperature and the average daily outside temperature exceeds 5.5 K. In the United States, grain is usually not warmed in the spring unless it is to be stored past early June. Foster and McKenzie (1979) and McK-enzie (1980) give practical recommendations for aeration of grain.
Airflow rates of 0.3 to 6.7 L/(s·m3) are normally used. Air is usually distributed through the bottom of the bin using ducts. Duct spacing and fan selection are related to bin size and shape and to the airflow rate. Foster and Tuite (1992) give an overview of the topic and include information and charts used for design of such systems.
Peterson (1982) gives recommendations for duct spacing in flat storages.
Several computer simulations have been developed to study the effects of heat buildup from microbial activity with and without aer-ation (Thompson 1972, Brooker and Duggal 1982, Metzger and Muir 1983, Lissik 1986). Aldis and Foster (1977) and Schultz et al.
(1984) studied the effect of aeration on grain moisture changes.
DRYING SPECIFIC CROPS Hay Forage crops can be either harvested, dried, and stored as hay or harvested and stored under anaerobic conditions as silage. Hay quality can be judged by its color, leafiness, and appearance. Labo-ratory tests and feeding trials give a more detailed picture of hay quality. The traditional method of making hay is to mow the forage and allow it to field cure or dry in the swath and windrow. Harvest-ing at higher moistures with subsequent artificial drying may be economically feasible, depending on the local weather conditions.
Basic principles of hay drying and storage are covered by Hall (1980), FEC (1985), and Schuler et al. (1986). Forage must be har-vested in the proper stage of maturity to attain maximum feeding value. Leaf loss from alfalfa is high when it is handled at moistures below 39%. Therefore, if it is baled at 40% moisture and dried arti-ficially to the recommended storage moisture of 20% (Schuler et al.
(1986), a significantly higher feeding value can be achieved. Both Schuler et al. (1986) and Hall (1980) give sketches for batch and in-storage hay dryers. They recommend airflows of 75 to 100 L/s per square metre of mow floor area.
Dehydrated alfalfa meal supplies provitamin A (carotene), vita-min E, xanthophylls (poultry pigmenting factors), vitamin K, vita-min C, and B vitamins. Figure 3 shows losses from field drying of hay found in tests conducted by Shepherd (1954). The rapid loss of carotene immediately after the forage is cut indicates the need for Fig. 3 Time in Swath and Windrow Versus Field Losses of Leaves, Dry Matter, Protein, and Carotene for Hay Drying 11.12 2001 ASHRAE Fundamentals Handbook (SI) rapid transport to the dehydrator when alfalfa meal with high vita-min content is desired.
Several factors influence retention of vitamins during storage, including the starting plant material, dehydration conditions, addi-tion of stabilizers, and storage conditions. Lowering the tempera-ture reduces the loss rate. Inert gas atmosphere in storage also reduces losses (Hoffman et al. 1945). According to Shepherd (1954), blanching of fresh alfalfa before drying does not alter the storage stability of carotene. Table 7 shows Shepherd’s results on the effect of prolonged heating at 100°C. The alfalfa was dried after 45 min; heating beyond this time represented excessive exposure to this temperature. Carotene retention in the intact meal at 65°C storage temperature was considered a measure of storage stability.
Normal storage moisture is 8 to 9%. Thompson et al. (1960) sum-marize the effects of over- and underdrying on carotene stability.
Drying and handling of large round bales has been researched.
These bales may have a mass of 385 to 680 kg and are handled indi-vidually with forklifts. Verma and Nelson (1983) studied storage of large round bales and found that dry matter loss was the primary component of the total storage losses. Bales stored so that they were protected from the weather had lower losses of dry matter than bales exposed to the weather. They were also higher in total protein. Jones et al. (1985) found significant dry matter loss in large round bales of mature fescue hay. Harrison (1985) found that addition of sulfur dioxide at the rate of 1% of dry matter had little effect on dry matter loss and nutrient contents for a mixture of alfalfa and bromegrass.
However, bales protected with plastic bags did have significantly lower dry matter loss. Jones et al. (1985) found that bales of mature fescue hay stored inside and bales treated with ammonia had less dry matter loss and higher in vitro dry matter digestibility. Henry et al. (1977) and Frisby et al. (1985) developed and tested solar dryers for large round bales.
Grain The physiological factors involved in drying and storing grain are different from those of forages. Grain is the end product of plant growth, and most physiological activity within the grain or seed is approaching a low level when harvested. With forage, the biological activity within the plant is at or near its peak at the time of harvest.
Both the deterioration of grain harvested at moistures above those safe for storage and the chemical preservation of grain are addressed in the section on Fungal Growth and Mycotoxins. Pres-ervation by ensiling or airtight storage is addressed in the section on Oxygen and Carbon Dioxide.
For more information on grain drying, see Chapter 22 of the 1999 ASHRAE Handbook—Applications and Brook (1992).
Corn Shelled field corn is used primarily as livestock feed, but some is used by milling or processing industries for manufacturing starch, corn oil, and other products. Little information is available on the relationship between the drying method and the feed value of corn.
Market grade, as established by the Agricultural Marketing Service of the United States Department of Agriculture (USDA), is the pri-mary criterion for determining corn value. Tests by Cabell et al.
(1958) indicated that shelled corn with a moisture content of 29 to 32% can be dried without loss of protein nutritive value by air with temperatures as high as 115°C, provided the airflow rate is approx-imately 1.5 m3/(s·m3).
Breakage Susceptibility. The market grade of dry corn is affected more by the amount of fine material than by other grading factors. Fine material is defined as the broken grain and other mate-rial that passes through a 4.76 mm round-hole sieve. The physical damage done to wet corn or the brittleness imparted to the corn during drying causes it to break each time it is handled. The propen-sity of corn to break during subsequent handling, called breakage susceptibility, can be measured with a multiple-impact device called the Stein breakage tester. Stephens and Foster (1976a) dem-onstrated that corn breakage in the tester was correlated with damage during handling. Watson et al. (1986) give a standardized procedure for using the Stein breakage tester, and Watson and Herum (1986) describe and compare other devices developed for measurement of breakage susceptibility. They concluded that a device developed by Singh and Finner (1983) offers great potential for testing of grain for breakage susceptibility in commercial situations.
Paulsen et al. (1983) found significant variations in breakage susceptibility among hybrids. Corn dried with air at high tempera-tures (60°C) was two to six times more susceptible to breakage than corn dried at near-ambient temperatures. Gustafson and Morey (1979) found that delayed cooling (maintaining the corn at or near its temperature at the end of drying for 6 to 12 h) reduced breakage susceptibility and improved the test mass.
In a study of combination drying, Gustafson et al. (1978) found that combination drying (high-temperature drying to 18% followed by low-temperature drying to 16.6% moisture or below) signifi-cantly reduced the increase in breakage susceptibility normally caused by high-temperature drying.
Quality. Both drying temperature and corn hybrid can affect the quality of shelled corn for specific end uses. Brekke et al. (1973) found that drying at temperatures above 60°C reduced the quality of the corn for dry milling. Peplinski et al. (1982) found that optimum dry milling quality could be achieved by harvesting corn at mois-tures below 25%, minimizing machinery-induced damage to the kernels, and drying at air temperatures below 82°C. Paulsen and Hill (1985) found that the yield of flaking grits from dry milling of corn was significantly greater for corn that had a high test mass and relatively low breakage susceptibility. Weller et al. (1987) found that corn variety affected wet milling quality. At drying tempera-tures between 49 and 71°C, protein conformational changes occurred and decreased the ethanol soluble protein. Hybrids differ in resistance to storage mold (Tuite and Foster 1979), thin layer dry-ing rate, and dry milling quality (Stroshine et al. 1986). Watson (1987) gives an extensive summary of measurement and mainte-nance of quality of corn, and Foster (1975) summarizes approaches to reducing damage during harvesting, handling, and drying.
Cotton The lint moisture content for best results in ginning cotton appears to be 5 to 7%, with an optimum moisture content of 6% (Franks and Shaw 1962). Cotton, like grain, is hygroscopic and should be dried just prior to ginning. The wide variation in incoming moisture content usually requires different amounts of drying for each load. Rapid changes in the amount of drying required can best be handled by using a multipath drying tower in which the cotton is exposed for various lengths of time (2 to 10 s) at temperatures not exceeding 175°C. The air-to-cotton ratio can range from 40 to 100 m3/(s·t) of cotton (Franks and Shaw 1962). Laird and Baker (1983) found that substantial amounts of heat could be reclaimed and used for drying in commercial cotton gin plants. Equilibrium moisture Table 7 Effect of Heating Chopped Alfalfa on Carotene Loss During Subsequent Storage of Meal Hours in Oven at 100°C Initial Carotene, ppm Carotene Retained 7 Days at 65°C, % 0.75 229 37 1 228 37 2 197 37 3 176 28 4 149 21 5.5 112 18 7.5 86 15 Source: Thompson et al. (1960).
Note: Alfalfa is fresh frozen from Ryer Island, CA.
Physiological Factors in Drying and Storing Farm Crops 11.13 content data for newly harvested cotton fibers are given by Griffin (1974). Anthony (1982) studied moisture gain of cotton bales dur-ing storage.
Cottonseed removed from the fibers is also dried. The germina-tion of cottonseed is unimpaired by drying if the internal cottonseed temperatures do not exceed 60°C (Shaw and Franks 1962). This temperature is not exceeded in the tower dryer described previously.
However, the moisture content of the seed can be above the recom-mended level of 12% following the multipath tower drying. Drying seed in a triple-pass drum at 120 to 150°C with an exposure time of 4 min, followed by cooling, reduces moisture content, inhibits the formation of free fatty acids, and improves germination compared to undried seed. Anthony (1983) dried cottonseed in a vacuum microwave dryer. The cottonseed would not germinate, but its oil properties were not harmed as long as lower temperatures were used. The drying rate was increased by reducing pressure below atmospheric. Rayburn et al. (1978) studied preservation of high-moisture cottonseed with propionic acid.
Peanuts Peanuts in the shell normally have a moisture content of about 50% at the time of digging. Allowing peanuts to dry on the vines in the windrows for a few days removes much of this water. However, peanuts normally contain 20 to 30% moisture when removed from the vines, and some artificial drying in the shell is necessary. Drying should begin within 6 h after harvesting in order to prevent peanuts from self-heating. The maximum temperature and rate of drying must be controlled to maintain quality. High temperatures result in off-flavor or bitterness. Overly rapid drying without high tempera-tures results in blandness or inability to develop flavor when roasted (Bailey et al. 1954). High temperatures and rapid or excessive dry-ing also cause the skin to slip easily and the kernels to become brit-tle. These conditions result in damage in the shelling operation and can be avoided if the moisture removal rate does not exceed 0.5% per hour. Because of these limitations, continuous-flow drying is not usually recommended.
Young (1984) found energy savings up to 26% when comparing recirculating dryers with conventional peanut dryers. Smith and Davidson (1982) and Smith et al. (1985) address the aeration of pea-nuts during warehouse storage.
Rice Of all grains, rice is possibly the most difficult to process without quality loss. Rice containing more than 12.5% moisture cannot be stored safely for long periods, yet the recommended harvest mois-ture content for best milling and germination ranges from 20 to 26% (Kramer 1951). If the rice is harvested at this moisture content, dry-ing must begin promptly to prevent heat-related damage, which can result in “stack-burn,” a yellowing of the kernel. To prevent exces-sive internal fissuring, which results in broken kernels during mill-ing, multiple-pass drying is usually necessary (Calderwood and Webb 1971). Kunze and Calderwood (1980) summarize rice-drying techniques.
Because the market demands polished whole kernels of rice, it is necessary to prevent damage in the form of fissures. Rapid moisture removal or addition can create moisture gradients within kernels.
According to Kunze (1984), gradients can develop in the field on a humid night before harvest, in a hopper containing a mixture of rice kernels at varying moistures, and in certain types of dryers. Ban-aszek and Siebenmorgen (1990) quantified the rate at which mois-ture absorption reduces head rice yields. Velupillai and Verma (1986) report that drying at 93°C followed by tempering in a sealed container for 24 h gave good kernel strength and head rice yields.
They also found that storing the rice after drying for 3 weeks gave optimum grain quality. Bakker-Arkema et al. (1984) achieved good rice quality with concurrent-flow drying of rice.
Soybean, Sunflower, and Edible Beans Prolonged periods of extremely wet weather during the harvest season can make artificial drying of soybeans necessary. Like pea-nuts and other oilseeds, soybeans cannot be dried satisfactorily with the high-temperature, high-speed methods used for cereal grains. Because of the different seed structure, rapid drying splits the seed coat and reduces quality and storage life. Overhults et al.
(1975) reported a significant decrease in the quality of oil extracted from soybeans dried at temperatures above 71°C. Soybeans have one of the slowest thin layer drying rates of commonly grown cere-als and oilseeds (Bakker-Arkema et al. 1983). Therefore, they dry more slowly and require more energy when dried in continuous-flow dryers.
Sunflower is a major crop in some areas of the United States.
Hellevang (1987) recommends maximum drying temperatures of 93°C for continuous-flow drying of oil sunflower and 82°C for non-oil sunflower to prevent scorching of the seed meat. Schuler (1974) gives data on equilibrium moisture, airflow resistance, and specific heat of sunflower seeds. Because sunflower is about half the density of shelled corn, moisture can be removed more rapidly, and there is a tendency to overdry. This factor, along with accumulation of for-eign material when drying, causes an increased fire hazard (Helle-vang 1982). Schmidt and Backer (1980) attribute most of the problems encountered with storage of sunflower seed to improper drying and/or aeration.
Edible beans, a major crop in several states, should be dried with air at relative humidities above 40% to prevent stress cracking. Nat-ural air or low-temperature drying is best (Hellevang 1987). If dried at high rates, seed coats may crack, and beans may split during sub-sequent handling (Otten et al. 1984, Radajewski et al. 1992). Broken beans can develop a bitter or undesirable flavor and spoil more eas-ily during storage (Uebersax and Bedford 1980).
Wheat and Barley In northern regions of the United States, wheat and barley may be harvested above safe storage moistures to prevent excessive field losses. Bruce (1992) modeled the effect of heated-air drying on the bread baking quality of wheat. The quality indicator was loaf vol-ume. Moilanen et al. (1973) recommended that hard red spring wheat be dried at temperatures below 70, 60, and 50°C, respec-tively, for harvest moistures of 16, 20, and 24% wet basis. These data assumed airflow of 0.5 to 0.75 m3/(s·m2). For airflow of 0.25 m3/(s·m2), the authors recommended that the drying air temperature be reduced by 5 to 8 K. In the case of barley used for malting, the seed must be able to germinate. Therefore, the maximum recom-mended drying air temperature is 43 to 50°C (Hellevang 1987, Jilak 1993). Watson et al. (1962) studied the effects of harvest moisture and drying temperature on barley malting quality and recommended harvesting below 20% moisture. If wheat or barley is used for seed, the maximum recommended drying air temperature is 43°C.
In regions where soft wheat is grown, it may be economical to harvest at 20 to 24% moisture to allow double cropping with soy-beans; this allows wheat harvest to begin 5 to 7 days earlier than normal and increases the yield of the soybeans (Swearingen 1979).
In areas where double cropping is feasible, soft wheat can be dried using low-temperature solar drying or ambient drying with intermit-tent fan operation (Barrett et al. 1981). High-speed and continuous-flow systems with reduced drying air temperatures can also be used (Parsons et al. 1979). Kirleis et al. (1982) harvested soft red winter wheat at moistures of 25% or below and dried with air temperatures of 65°C or below without adverse effects on milling or cookie bak-ing quality.
In high-temperature continuous-flow dryers, wheat and barley reduce airflow because they have a high airflow resistance. Bakker-Arkema et al. (1983) report that thin layer drying rates for barley and wheat are much faster than for corn. Barley dries more slowly 11.14 2001 ASHRAE Fundamentals Handbook (SI) than wheat, presumably because the kernels are larger. In their com-puter simulations of a concurrent-flow dryer, wet bushel capacity for wheat was about 80% of the capacity for shelled corn when moisture content was reduced by 4.7%. The drying capacity differ-ence was probably caused by a decrease in airflow.
Tobacco (Curing) Tobacco leaves normally have a moisture content of about 85% at harvest. The major methods of tobacco drying are air curing and flue curing (Johnson et al. 1960).
For air curing, whole plants are cut and allowed to wilt in the field until the leaves reach about 70% moisture (Walton et al. 1994).
The plants are then hung in open barns, where temperatures range from 15 to 32°C and humidities from 65 to 70%. The curing period is 28 to 56 days (Jefries 1940). The desired end product for air cur-ing is a tan leaf. Overdrying at low temperatures results in green color and low sugar content; overdrying at high temperatures results in yellow color (Walton and Henson 1971). Both conditions are undesirable because the normal chemical changes are arrested pre-maturely. Subsequent drying at optimum rates can reverse some damage. Underdrying at all temperatures results in undesirable dark color and damage from mold and bacterial growth (Walton et al.
1973).
Flue curing uses artificial heat. The leaves are harvested and hung in closed barns where temperatures are increased gradually during the curing period. Normally, 3 days of drying at temperatures of 32 to 50°C brings about yellowing. For the next 2 days, temper-atures of 50 to 60°C are used for leaf drying; then, stems are dried at 77°C for 1 to 2 days. A bright yellow to orange color is desirable in flue-cured or bright-leaf tobacco.
REFERENCES Agrawal, Y.C. and R.P. Singh. 1977. Thin-layer drying studies on short grain rice. Paper 77-3531. American Society of Agricultural Engineers, St.
Joseph, MI.
Alagusundaram, K., D.S. Jayas, O.H. Friesen, and N.D.G. White. 1994. Air-flow patterns through wheat, barley and canola in bins with partially per-forated floors: an experimental investigation. Applied Engineering in Agriculture 10(6):791-96.
Alagusundaram, K., D.S. Jayas, W.E. Muir, and N.D.G. White. 1996. Con-vective-diffusive transport of carbon dioxide through stored-grain bulks.
Transactions of ASAE 39(4):1505-10.
Aldis, D.F. and G.H. Foster. 1977. Moisture changes in grain from exposure to ambient air. Paper 77-3524. ASAE.
Anthony, W.S. 1982. Moisture gain and resilient forces of cotton bales dur-ing equilibration. Transactions of ASAE 25(4):1066-70.
Anthony, W.S. 1983. Vacuum microwave drying of cotton: Effect on cotton-seed. Transactions of ASAE 26(1):275-78.
Armitage, D.M., P.M. Cogan, and D.R. Wilkin. 1994. Integrated pest man-agement in stored grain: Combining surface insecticide treatments with aeration. Journal of Stored Products Research 30(4):303-19.
Arthur, F.H. 1996. Grain protectants: Current status and prospects for the future. Journal of Stored Products Research 32(4):293-302.
ASAE. 1995. Moisture relationships of grains. Standard D245.5. American Society of Agricultural Engineers, St. Joseph, MI.
ASAE. 1996. Resistance to airflow of grains, seeds, other agricultural prod-ucts, and perforated metal sheets. Standard D272.3.
ASAE. 1997. Moisture measurement—Unground grain and seeds. Standard S352.2.
ASAE. 1998. Moisture measurement—Forages. Standard S358.2.
Athie, I., R.A.R. Gomes, S. Bolonhezi, S.R.T. Valentini, and M.F.P.M. de Castro. 1998. Effects of carbon dioxide and phosphine mixtures on resis-tant populations of stored-grain insects. Journal of Stored Products Research 34(1):27-32.
Bailey, W.K., T.A. Pickett, and J.G. Futral. 1954. Rapid curing adversely affects quality of peanuts. Peanut Journal and Nut World 33(8):37-39.
Bakker-Arkema, F.W., R.C. Brook, and L.E. Lerew. 1978. Cereal grain dry-ing. In Advances in cereal science and technology, ed. Y. Pomeranz, pp. 1-90. American Association of Cereal Chemists, St. Paul, MN.
Bakker-Arkema, F.W., S. Fosdick, and J. Naylor. 1979. Testing of commer-cial crossflow dryers. Paper 79-3521. ASAE.
Bakker-Arkema, F.W., C. Fontana, R.C. Brook, and C.W. Westlake. 1984.
Concurrent flow rice drying. Drying Technology 1(2):171-91.
Bakker-Arkema, F.W., C. Fontana, G.L. Fedewa, and I.P. Schisler. 1983. A comparison of drying rates of different grains. Paper 83-3009. ASAE.
Banaszek, M.M. and T.J. Siebenmorgen. 1990. Head rice yield reduction rate caused by moisture absorption. Transactions of ASAE 33(4):1263-69.
Barak, A.V. and W.E. Burkholder. 1985. A versatile and effective trap for detecting and monitoring stored-product coleoptera. Agricultural Eco-systems and Environment 12:207-18.
Barak, A.V. and P.K. Harein. 1982. Trap detection of stored-grain insects in farm-stored shelled corn. Journal of Economic Entomology 75(1):108-11.
Barre, H.J., G.R. Baughman, and M.Y. Hamdy. 1971. Application of the log-arithmic model to cross-flow deep-bed grain drying. Transactions of ASAE 14(6):1061-64.
Barrett, Jr., J.R., M.R. Okos, and J.B. Stevens. 1981. Simulation of low tem-perature wheat drying. Transactions of ASAE 24(4):1042-46.
Beeson, W.M. and T.W. Perry. 1958. The comparative feeding value of high moisture corn and low moisture corn with different feed additives for fat-tening beef cattle. Journal of Animal Science 17(2):368-73.
Bell, C.H. and D.M. Armitage. 1992. Alternative storage practices. In Stor-age of cereal grains and their products, ed. D.B. Sauer, pp. 249-312.
Bern, C.J. and L.F. Charity. 1975. Airflow resistance characteristics of corn as influenced by bulk density. Paper 75-3510. ASAE.
Bern, C.J. and C.R. Hurburgh, Jr. 1992. Characteristics of fines in corn: Review and analysis. Transactions of ASAE 35(6):1859-67.
Bern, C.J., M.E. Anderson, W.F. Wilcke, and C.R. Hurburgh. 1982. Auger-stirring wet and dry corn—Airflow resistance and bulk density effects.
Transactions of ASAE 25(1):217-20.
Beuchar, L.R. and D.A. Golden. 1989. Antimicrobials occurring naturally in foods. Food Technology 43:135-42.
Bloome, P.D. and G.W. Cuperus. 1984. Aeration for management of stored grain insects in wheat. Paper 84-3517. ASAE.
Bowden, P.J., W.J. Lamond, and E.A. Smith. 1983. Simulation of near-ambi-ent grain drying: I, Comparison of simulations with experimental results.
Journal of Agricultural Engineering Research 28:279-300.
Brekke, O.L., E.L. Griffin, Jr., and G.C. Shove. 1973. Dry milling of corn artificially dried at various temperatures. Transactions of ASAE 16(4):761-65.
Bridges, T.C., D.G. Colliver, G.M. White, and O.J. Loewer. 1984. A com-puter aid for evaluation of on-farm stir drying systems. Transactions of ASAE 27(5):1549-55.
Bridges, T.C., I.J. Ross, G.M. White, and O.J. Loewer. 1980. Determination of optimum drying depth for batch-in bin corn drying systems. Trans-actions of ASAE 23(1):228-33.
Brook, R.C. 1987. Modelling grain spoilage during near-ambient grain dry-ing. Divisional Note DN 1388, AFRC Institute of Engineering Research, Wrest Park, Silsoe, Bedford, MK45 4HS, England, 20 p.
Brook, R.C. 1992. Drying cereal grains. In Storage of cereal grains and their products, ed. D.B. Sauer, pp. 183-218.
Brooker, D.B. 1969. Computing air pressure and velocity distribution when air flows through a porous medium and nonlinear velocity-pressure rela-tionships exist. Transactions of ASAE 12(1):118-20.
Brooker, D.B. and A.K. Duggal. 1982. Allowable storage time of corn as affected by heat buildup, natural convection and aeration. Transactions of ASAE 25(3):806-10.
Brooker, D.B., F.W. Bakker-Arkema, and C.W. Hall. 1992. Drying and stor-age of grains and oilseeds. Van Nostrand Reinhold, New York.
Bruce, D.M. 1985. Exposed-layer barley drying: Three models fitted to new data up to 150°C. Journal of Agricultural Engineering Research 32:337-47.
Bruce, D.M. 1992. A model of the effect of heated-air drying on the bread baking quality of wheat. Journal of Agricultural Engineering Research 52(1):53-76.
Bruce, D.M. and R.A. Sykes. 1983. Apparatus for determining mass transfer coefficients at high temperatures for exposed particulate crops, with ini-tial results for wheat and hops. Journal of Agricultural Engineering Research 28:385-400.
Brusewitz, G.H. 1987. Corn moisture variability during drying, mixing and storage. Journal of Agricultural Engineering Research 38:281-88.
Physiological Factors in Drying and Storing Farm Crops 11.15 Bullerman, L.B., L.L. Schroeder, and K. Park. 1984. Formation and control of mycotoxins in food. Journal of Food Protection 47:637-46.
Burkholder, W.E. and M. Ma. 1985. Pheromones for monitoring and control of stored-product insects. Annual Review of Entomology 30:257-72.
Butler, L.A. 1983. The history and background of NIR. Cereal Foods World 28(4):238-40.
Cabell, C.A., R.E. Davis, and R.A. Saul. 1958. Relation of drying air tem-perature, time and air flow rate to the nutritive value of field-shelled corn.
Technical Progress Report 1957-58 ARS 44-41. USDA, Washington, D.C.
Calderwood, D.L. and B.D. Webb. 1971. Effect of the method of dryer oper-ation on performance and on the milling and cooking characteristics of rice. Transactions of ASAE 14(1):142-46.
Chang, C.S., H.H. Converse, and F.S. Lai. 1986. Technical Notes: Distribu-tion of fines and bulk density of corn as affected by choke-flow, spout-flow, and drop-height. Transactions of ASAE 29(2):618-20.
Chang, C.S., H.H. Converse, and C.R. Martin. 1983. Bulk properties of grain as affected by self-propelled rotational type grain spreaders. Trans-actions of ASAE 26(5):1543-50.
Choudhary, A.K. and K.K. Sinha. 1993. Competition between a toxigenic Aspergillus flavus strain and other fungi on stored maize kernels. Journal of Stored Products Research 29(1):75-80.
Christensen, C.M. and R.A. Meronuck. 1986. Quality maintenance in stored grains and seeds. University of Minnesota Press, Minneapolis.
Christensen, C.M., B.S. Miller, and J.A. Johnston. 1992. Moisture and its measurement. In Storage of cereal grains and their products, ed. D.B.
Sauer, pp. 39-54.
Chung, J.H. and L.R. Verma. 1991. Dynamic and quasi-static rice moisture models using humidity sensors. Transactions of ASAE 34(6):2477-83.
Colliver, D.G., R.M. Peart, R.C. Brook, and J.R. Barrett, Jr. 1983. Energy usage for low temperature grain drying with optimized management.
Transactions of ASAE 26(2):594-600.
Cooper, P.J. 1983. NIR analysis for process control. Cereal Foods World 28(4):241-45.
Danziger, M.T., M.P. Steinberg, and A.I. Nelson. 1973. Effect of CO2, mois-ture content, and sorbate on safe storage of wet corn. Transactions of ASAE 16(4):679-82.
Dunkel, F.V. 1985. Underground and earth sheltered food storage: Histori-cal, geographic, and economic considerations. Underground Space 9:310-15.
Eckhoff, S.R., J. Tuite, G.H. Foster, R.A. Anderson, and M.R. Okos. 1984.
Inhibition of microbial growth during ambient air corn drying using sul-fur dioxide. Transactions of ASAE 27(3):907-14.
Epperly, D.R., R.T. Noyes, G.W. Cuperus, and B.L. Clary. 1987. Control stored grain insects by grain temperature management. Paper 87-6035.
ASAE.
FEC. 1985. Hay drying: A guide to the practical design of installations. Farm Electric Center, Kenilworth, Warwickshire, England.
Foster, G.H. 1975. Causes and cures of physical damage to corn. In Corn quality in world markets, L.D. Hill, ed. Interstate Printers and Publishers, Danville, IL.
Foster, G.H. 1986. William V. Hukill, a pioneer in crop drying and storage.
Drying Technology 4(3):461-71.
Foster, G.H. and H.F. Mayes. 1962. Temperature effects of an artificial hotspot embedded in stored grain. AMS-479. U.S. Department of Agri-culture, Washington, D.C.
Foster, G.H. and B.A. McKenzie. 1979. Managing grain for year-round stor-age. AE-90. Cooperative Extension Service, Purdue University, West Lafayette, IN.
Foster, G.H. and J. Tuite. 1992. Aeration and stored grain management. In Storage of cereal grains and their products, ed. D.B. Sauer, pp. 219-48.
Franks, G.N. and C.S. Shaw. 1962. Multipath drying for controlling moisture in cotton. ARS 42-69. USDA, Washington, D.C.
Frazer, B.M. and W.E. Muir. 1981. Airflow requirements for drying grain with ambient and solar-heated air in Canada. Transactions of ASAE 24(1):208-10.
Frisby, J.C., J.T. Everett, and R.M. George. 1985. A solar dryer for large, round alfalfa bales. Applied Engineering in Agriculture 1(2):50-52.
Giner, S.A. and E. Denisienia. 1996. Pressure drop through wheat as affected by air velocity, moisture content and fines. Journal of Agricul-tural Engineering Research 63(1):73-85.
Grama, S.N., C.J. Bern, and C.R. Hurburgh, Jr. 1984. Airflow resistance of moistures of shelled corn and fines. Transactions of ASAE 27(1):268-72.
Griffin, A.C., Jr. 1974. The equilibrium moisture content of newly harvested cotton fibers. Transactions of ASAE 17(2):327-28.
Gunasekaran, S. 1986. Optimal energy management in grain drying. Critical Reviews in Food Science and Nutrition 25(1):1-48.
Gustafson, R.J. and R.V. Morey. 1979. Study of factors affecting quality changes during high-temperature drying. Transactions of ASAE 22(4): 926-32.
Gustafson, R.J., R.V. Morey, C.M. Christensen, and R.A. Meronuck. 1978.
Quality changes during high-low temperature drying. Transactions of ASAE 21(1):162-69.
Hall, C.W. 1980. Drying and storage of agricultural crops. AVI Publishing Company, Westport, CT.
Hall, G.E., L.D. Hill, E.E. Hatfield, and A.H. Jenson. 1974. Propionic-acetic acid for high-moisture preservation. Transactions of ASAE 17(2):379-82, 387.
Hamer, A., J. Lacey, and N. Magan. 1992. Respiration and fungal deteriora-tion of stored grains. eds. D.S. Jayas et al., pp. 65-66.
Haque, E., Y.N. Ahmed, and C.W. Deyoe. 1982. Static pressure drop in a fixed bed of grain as affected by grain moisture content. Transactions of ASAE 25(4):1095-98.
Haque, E., G.H. Foster, D.S. Chung, and F.S. Lai. 1978. Static pressure drop across a bed of corn mixed with fines. Transactions of ASAE 21(5):997-1000.
Harein, P.K. and R. Davis. 1992. Control of stored grain insects. In Storage of cereal grains and their products, ed. D.B. Sauer, pp. 491-534.
Harris, K.L. and F.J. Bauer. 1992. Rodents. In Storage of cereal grains and their products, ed. D.B. Sauer, pp. 393-434.
Harrison, H.P. 1985. Preservation of large round bales at high moisture.
Transactions of ASAE 28(3):675-79, 686.
Hellevang, K.J. 1982. Crop dryer fires while drying sunflower. Paper 82-3563. ASAE.
Hellevang, K.J. 1983. Natural air/low temperature crop drying. Bulletin 35.
Cooperative Extension Service, North Dakota State University, Fargo, ND.
Hellevang, K.J. 1987. Grain drying. Publication AE-701. Cooperative Extension Service, North Dakota State University, Fargo, ND.
Henry, Z.A., B.L. Bledsoe, and D.D. Eller. 1977. Drying of large hay pack-ages with solar heated air. Paper 77-3001. ASAE.
Hertung, D.C. and E.E. Drury. 1974. Antifungal activity of volatile fatty acids on grains. Cereal Chemistry 51(1):74-83.
Hoffman, E.J., G.F. Lum, and A.L. Pitman. 1945. Retention of carotene in alfalfa stored in atmospheres of low oxygen content. Journal of Agricul-tural Research 71:361-73.
Hukill, W.V. 1947. Basic principles in drying corn and grain sorghum. Agri-cultural Engineering 28(8):335-38, 340.
Hukill, W.V. 1953. Grain cooling by air. Agricultural Engineering 34(7): 456-58.
Hukill, W.V. and J.L. Schmidt. 1960. Drying rate of fully exposed grain ker-nels. Transactions of ASAE 3(2): 71-77, 80.
Hurburgh, C.R., T.E. Hazen, and C.J. Bern. 1985. Corn moisture measure-ment accuracy. Transactions of ASAE 28(2):634-40.
Hurburgh, C.R., L.N. Paynter, S.G. Schmitt, and C.J. Bern. 1986. Perfor-mance of farm-type moisture meters. Transactions of ASAE 29(4): 1118-23.
Janardhana, G.R., K.A. Raveesha, and H.S. Shetty. 1998. Modified atmo-sphere storage to prevent mould-induced nutritional moss in maize. J.
Sci. Food Agric. 76(4):573-78.
Jay, E. 1980. Methods of applying carbon dioxide for insect control in stored grain. Science and Education Administration, Advances in Agricultural Technology, Southern Series, AAT-S-13, Agricultural Research (South-ern Region), SEA, USDA, P.O. Box 53326, New Orleans, LA 70153.
Jayas, D.S. and D.D. Mann. 1994. Presentation of airflow resistance data of seed bulks. Appl. Eng. Agric. 10(1):79-83.
Jayas, D.S. and S. Sokhansanj. 1989. Design data on the airflow resistance to canola (rapeseed). Transactions of ASAE 32(1):295-96.
Jayas, D.S., S. Sokhansanj, E.B. Moysey, and E.M. Barber. 1987. The effect of airflow direction on the resistance of canola (rapeseed) to airflow.
Canadian Agricultural Engineering 29(2):189-92.
Jayas, D.S., S. Cenkowsi, S. Pabis, and W.E. Muir. 1991. Review of thin-layer drying and wetting equations. Drying Tech. 9(3):551-88.
Jayas, D.S., N.D.G. White, W.E. Muir, and R.N. Sinha, eds. 1992. Interna-tional Symposium on Stored Grain Ecosystems. University of Manitoba, Winnipeg.
11.16 2001 ASHRAE Fundamentals Handbook (SI) Jefries, R.N. 1940. The effect of temperature and relative humidity during curing upon the quality of white burley tobacco. Bulletin No. 407. Ken-tucky Agricultural Experiment Station, Lexington, KY.
Jilek, J. 1993. Critical temperature for barley drying. Drying Tech.
11(1):183-193.
Jindal, V.K. and T.L. Thompson. 1972. Air pressure patterns and flow paths in two-dimensional triangular-shaped piles of sorghum using forced con-vection. Transactions of ASAE 15(4):737-44.
Johnson, W.H., W.H. Henson, Jr., F.J. Hassler, and R.W. Watkins. 1960.
Bulk curing of bright-leaf tobacco. Agricultural Engineering 41(8):511-15, 517.
Jones, A.L., R.E. Morrow, W.G. Hires, G.B. Garner, and J.E. Williams. 1985.
Quality evaluation of large round bales treated with sodium diacetate or anhydrous ammonia. Transactions of ASAE 28(4):1043-45.
Khompos, V., L.J. Segerlind, and R.C. Brook. 1984. Pressure patterns in cylindrical grain storages. Paper 84-3011. ASAE.
Kirleis, A.W., T.L. Housley, A.M. Emam, F.L. Patterson, and M.R. Okos.
1982. Effect of preripe harvest and artificial drying on the milling and baking quality of soft red winter wheat. Crop Science 22:871-76.
Kramer, H.A. 1951. Engineering aspects of rice drying. Agricultural Engi-neering 32(1):44-45, 50.
Kraszewski, A.W. and S.O. Nelson. 1992. Resonant microwave cavities for sensing properties of agricultural products. Transactions of ASAE 35(4):1315-1321.
Kumar, A. and W.E. Muir. 1986. Airflow resistance of wheat and barley affected by airflow direction, filling method and dockage. Transactions of ASAE 29(5):1423-26.
Kunze, O.R. 1984. Physical properties of rice related to drying the grain.
Drying Technology 2(3):369-87.
Kunze, O.R. and D.L. Calderwood. 1980. Systems for drying of rice. In Dry-ing and storage of agriculture crops, C.W. Hall. AVI Publishing Com-pany, Westport, CT.
Lai, F.S. 1980. Three dimensional flow of air through nonuniform grain beds. Transactions of ASAE 23(3):729-34.
Laird, W. and R.V. Baker. 1983. Heat recapture for cotton gin drying sys-tems. Transactions of ASAE 26(3):912-17.
Lapp, H.M., F.J. Madrid, and L.B. Smith. 1986. A continuous thermal treat-ment to eradicate insects from stored wheat. Paper 86-3008. ASAE.
Leva, M. 1959. Fluidization. McGraw-Hill, Inc. New York.
Li, H. and R.V. Morey. 1984. Thin-layer drying of yellow dent corn. Trans-actions of ASAE 27(2):581-85.
Li, W. and S. Sokhansanj. 1994. Generalized equation for airflow resistance of bulk grains with variable density, moisture content and fines. Drying Tech. 12(3):649-667.
Li, Y., R.V. Morey, and M. Afinrud. 1987. Thin-layer drying rates of oilseed sunflower. Transactions of ASAE 30(4):1172-75, 1180.
Lissik, E.A. 1986. A model for the removal of heat in respiring grains. Paper 86-6509. ASAE.
Locklair, E.E., L.G. Veasey, and M. Samfield. 1957. Equilibrium desorption of water vapor on tobacco. Journal of Agricultural and Food Chemistry 5:294-98.
Magan, N. 1993. Early detection of fungi in stored grain. Int. J. Biodeterio-ration and Biodegredation 32 (1/3):145-160.
Manis, J.M. 1992. Sampling, inspection and grading. In Storage of cereal grains and their products, ed. D.B. Sauer, pp. 563-88 Martin, C.R., Z. Czuchajowska, and Y. Pomeranz. 1986. Aquagram standard deviations of moisture in mixtures of wet and dry corn. Cereal Chemistry 63(5):442-45.
Martins, J. and R.L. Stroshine. 1987. Difference in drying efficiencies among corn hybrids dried in a high-temperature column-batch dryer.
Paper 87-6559. ASAE.
McKenzie, B.A. 1980. Managing dry grain in storage. AED-20. Midwest Plan Service, Iowa State University, Ames, IA.
McKenzie, B.A., G.H. Foster, and S.S. DeForest. 1980. Fan sizing and appli-cation for bin drying/cooling of grain. AE-106. Cooperative Extension Service, Purdue University, West Lafayette, IN.
McMillan, W.W., D.M. Wilson, and N.W. Widstrom. 1985. Aflatoxin con-tamination of preharvest corn in Georgia—A six-year study of insect damage and visible Aspergillus flavus. Journal of Environmental Quality 14:200-02.
Metzger, J.F. and W.E. Muir. 1983. Computer model of two-dimensional conduction and forced convection in stored grain. Canadian Agricultural Engineering 25:119-25.
Miketinac, M.J. and S. Sokhansanj. 1985. Velocity-pressure distribution in grain bins—Brooker model. International Journal of Applied Numerical Analysis in Engineering 21:1067-75.
Misra, M.K. and D.B. Brooker. 1980. Thin-layer drying and rewetting equa-tions for shelled yellow corn. Transactions of ASAE 23(5):1254-60.
Moilanen, C.W., R.T. Schuler, and E.R. Miller. 1973. Effect on wheat quality of air flow and temperatures in mechanical dryers. North Dakota Farm Research 30(6):15-19.
Morey, R.V. and H.A. Cloud. 1980. Combination high-speed, natural-air corn drying. M-163. Agricultural Extension Service, University of Min-nesota, St. Paul.
Morey, R.V. and H. Li. 1984. Thin-layer equation effects on deep-bed drying prediction. Transactions of ASAE 27(6):1924-28.
Morey, R.V., H.A. Cloud, and D.J. Hansen. 1981. Ambient air wheat drying.
Transactions of ASAE 24(5):1312-16.
Morey, R.V., H.A. Cloud, and W.E. Lueschen. 1976. Practices for the effi-cient utilization of energy from drying corn. Transactions of ASAE 19(1): 151-55.
Morey, R.V., H.A. Cloud, R.J. Gustafson, and D.W. Peterson. 1979. Man-agement of ambient air drying systems. Transactions of ASAE 22(6): 1418-25.
Morey, R.V., H.M. Keener, T.L. Thompson, G.M. White, and F.W. Bakker-Arkema. 1978. The present status of grain drying simulation. Paper 78-3009. ASAE.
Nofsinger, G.W. 1982. The trickle ammonia process—An update. Grain Conditioning Conference Proceedings, Agricultural Engineering De-partment, University of Illinois, Champaign-Urbana.
Nofsinger, G.W., R.J. Bothast, and R.A. Anderson. 1979. Field trials using extenders for ambient-conditioning high-moisture corn. Transactions of ASAE 22(5):1208-13.
Noomhorm, A. and L.R. Verma. 1986. Generalized single-layer rice drying models. Transactions of ASAE 29(2):587-91.
O’Callaghan, J.R., D.J. Menzies, and P.H. Bailey. 1971. Digital simulation of agricultural dryer performance. Journal of Agricultural Engineering Research 16:223-44.
Otten, L., R. Brown, and W.S. Reid. 1984. Drying of white beans—Effects of temperature and relative humidity on seed coat damage. Canadian Agricultural Engineering 26(2):101-04.
Overhults, D.G., G.M. White, M.E. Hamilton, and I.J. Ross. 1973. Drying soybeans with heated air. Transactions of ASAE 16(1):112-13.
Overhults, D.G., G.M. White, M.E. Hamilton, I.J. Ross, and J.D. Fox. 1975.
Effect of heated air drying on soybean oil quality. Transactions of ASAE 16(1):112-13.
Parry, J.L. 1985. Mathematical modelling and computer simulation of heat and mass transfer in agricultural grain drying: A review. Journal of Agri-cultural Engineering Research 32:1-29.
Parsons, S.D., B.A. McKenzie, and J.R. Barrett, Jr. 1979. Harvesting and drying high-moisture wheat. In Double cropping winter wheat and soy-beans in Indiana. ID 96, Cooperative Extension Service, Purdue Univer-sity, West Lafayette, IN.
Parti, M. 1993. Selection of mathematical models for drying grain in thin-layers. Journal of Agricultural Engineering Research 54(4):339-52.
Paster, N., M. Calderon, M. Menasherov, and M. Mora. 1990. Biogeneration of modified atmospheres in small storage containers using plant wastes.
Crop Protection 9:235-238.
Paster, N., M. Calderon, M. Menasherov, V. Barak, and M. Mora. 1991.
Application of biogenerated modified atmospheres for insect control in small grain bins. Tropical Sci. 31(4):355-358.
Paster, N., M. Menasherov, U. Ravid, and B. Juven. 1995. Antifungal activ-ity of oregano and thyme essential oils applied as fumigants against fungi attaching stored grain. J. Food Protect. 58(1):81-85.
Pathak, P.K., Y.C. Agrawal, and B.P.N. Singh. 1991. Thin-layer drying model for rapeseed. Trans ASAE. 34(6):2505-2508.
Paulsen, M.R. and L.D. Hill. 1985. Corn quality factors affecting dry milling performance. Journal of Agricultural Engineering Research 31:255-63.
Paulsen, M.R. and T.L. Thompson. 1973. Drying analysis of grain sorghum.
Transactions of ASAE 16(3):537-40.
Paulsen, M.R., L.D. Hill, D.G. White, and G.F. Sprague. 1983. Breakage sus-ceptibility of corn-belt genotypes. Transactions of ASAE 26(6):1830-36.
Pederson, J.R. 1992. Insects: Identification, damage, and detection. In Stor-age of cereal grains and their products, ed. D.B. Sauer, pp. 435-90.
Peplinski, A.J., R.A. Anderson, and O.L. Brekke. 1982. Corn dry milling as influenced by harvest and drying conditions. Transactions of ASAE 25(4):1114-17.
Physiological Factors in Drying and Storing Farm Crops 11.17 Peplinski, A.J., O.L. Brekke, R.J. Bothast, and L.T. Black. 1978. High mois-ture corn—An extended preservation trial with ammonia. Transactions of ASAE 21(4): 773-76, 781.
Peterson, W.H. 1982. Design principles for grain aeration in flat storages.
Illinois Farm Electrification Council Fact Sheet No. 9. University of Illi-nois, Agricultural Engineering Department, Urbana.
Pfost, H.B., S.G. Mauer, D.S. Chung, and G.A. Milliken. 1976. Summarizing and reporting equilibrium moisture data for grains. Paper 76-3520.
ASAE.
Pierce, R.O. 1986. Economic consideration for natural air corn drying in Nebraska. Transactions of ASAE 29(4):1131-35.
Pierce, R.O. and T.L. Thompson. 1979. Solar grain drying in the North Cen-tral Region—Simulation results. Transactions of ASAE 15(1):178-87.
Pierce, R.O. and T.L. Thompson. 1981. Energy use and performance related to crossflow dryer design. Transactions of ASAE 24 (1):216-20.
Quinlan, J.K. 1982. Grain protectants for insect control. Marketing Bulletin 72. Agricultural Research Service, U.S. Department of Agriculture, Washington, D.C.
Radajewski, W., T. Jensen, G.Y. Abawi, and E.J. McGahan. 1992. Drying rate and damage to navy beans. Transactions of ASAE 35(2):583-90.
Rayburn, S.T., A.C. Griffin, Jr., and M.E. Whitten. 1978. Storing cottonseed with propionic acid. Transactions of ASAE 21(5):990-92.
Ripp, B.E., ed. 1984. Controlled atmosphere and fumigation in grain stor-ages. Proceedings of an International Symposium, Practical Aspects of Controlled Atmosphere and Fumigation in Grain Storages, in Perth, Western Australia. Elsevier Science Publishing Company, New York.
Rusal, G. and E.H. Marth. 1988. Food additives and plant components con-trol growth and aflatoxin production in toxigenic Aspergilli: A review.
Mycopathologia 101:13-23.
Sabbah, M.A., H.M. Keener, and G.E. Meyer. 1979. Simulation of solar dry-ing of shelled corn using the logarithmic model. Transactions of ASAE 22(3):637-43.
Sabbah, M.A., G.E. Meyer, H.M. Keener, and W.L. Roller. 1976. Reversed-air drying for fixed bed of soybean seed. Paper 76-3023. ASAE.
Sauer, D.B., ed. 1992. Storage of cereal grains and their products. American Association of Cereal Chemists, St. Paul, MN.
Sauer, D.B. and R. Burroughs. 1974. Efficacy of various chemicals as grain mold inhibitors. Transactions of ASAE 17(3):557-59.
Sauer, D.B., R.A. Meronuck, and C.M. Christensen. 1992. Microflora. In Storage of cereal grains and their products, ed. D.B. Sauer, pp. 313-40.
Saul, R.A. and J.L. Steele. 1966. Why damaged shelled corn costs more to dry. Agricultural Engineering 47:326-29, 337.
Schmidt, B.J. and L.F. Backer. 1980. Results of a sunflower storage moni-toring program in North Dakota. Paper 80-6033, ASAE.
Schuler, R.T. 1974. Drying related properties of sunflower seeds. Paper 74-3534. ASAE.
Schuler, R.T., B.J. Holmes, R.J. Straub, and D.A. Rohweder. 1986. Hay dry-ing. Publication A3380. Cooperative Extension Service, University of Wisconsin, Madison.
Schultz, L.J., M.L. Stone, and P.D. Bloome. 1984. A comparison of simula-tion techniques for wheat aeration. Paper 84-3012. ASAE.
Segerlind, L.J. 1982. Solving the nonlinear airflow equation. Paper 82-3017. ASAE.
Seitz, L.M., D.B. Sauer, and H.E. Mohr. 1982a. Storage of high-moisture corn: Fungal growth and dry matter loss. Cereal Chemistry 59(2):100-105.
Seitz, L.M., D.B. Sauer, H.E. Mohr, and D.F. Aldis. 1982b. Fungal growth and dry matter loss during bin storage of high-moisture corn. Cereal Chemistry 59(1):9-14.
Sellam, M.A. and C.M. Christensen. 1976. Temperature differences, mois-ture transfer and spoilage in stored corn. Feedstuffs 48(36):28, 33.
Sharp, J.R. 1982. A review of low-temperature drying simulation models.
Journal of Agricultural Engineering Research 27(3):169-90.
Shaw, C.S. and G.N. Franks. 1962. Cottonseed drying and storage at cotton gins. Technical Bulletin 1262. USDA, ARS, Washington, D.C.
Shedd, C.K. 1953. Resistance of grains and seeds to air flow. Agricultural Engineering 34(9):616-18.
Shejbal, J., ed. 1980. Controlled atmosphere storage of grains. Elsevier Sci-ence Publishing Company, New York.
Shelef, L.A. 1984. Antimicrobial affects of spices. Journal of Food Safety 6:29-44.
Shepherd, J.B. 1954. Experiments in harvesting and preserving alfalfa for dairy cattle feed. Technical Bulletin 1079. USDA, Washington, D.C.
Shivhare, U.S., G.S.V. Raghaven, and R.G. Bosisio. 1992. Microwave dry-ing of corn. I. Equilibrium moisture content. Transactions of ASAE 35(3):947-50.
Singh, S.S. and M.F. Finner. 1983. A centrifugal impacter for damage sus-ceptibility evaluation of shelled corn. Transactions of ASAE 26(6):1858-63.
Smith, E.A. and P.H. Bailey. 1983. Simulation of near-ambient grain drying.
II, Control strategies for drying barley in Northern Britain. Journal of Agricultural Engineering Research 28:301-17.
Smith, E.A. and S. Sokhansanj. 1990a. Moisture transport due to natural convection in grain stores. Journal of Agricultural Engineering Research 47:23-34.
Smith, E.A. and S. Sokhansanj. 1990b. Natural convection and temperature of stored products—A theoretical analysis. Canadian Agricultural Engi-neering 32(1):91-97.
Smith, J.S., Jr. and J.I. Davidson, Jr. 1982. Psychrometrics and kernel mois-ture content as related to peanut storage. Transactions of ASAE 25(1): 231-36.
Smith, J.S., Jr., J.I. Davidson, Jr., T.H. Sanders, and R.J. Cole. 1985. Storage environment in a mechanically ventilated peanut warehouse. Transac-tions of ASAE 28(4):1248-52.
Sokhansanj, S. and D.M. Bruce. 1987. A conduction model to predict grain drying simulation. Transactions of ASAE 30(4):1181-84.
Sokhansanj, S. and S.O. Nelson. 1988a. Dependence of dielectric properties of whole-grain wheat on bulk density. Journal of Agricultural Engineer-ing Research 39:173-79.
Sokhansanj, S. and S.O. Nelson. 1988b. Transient dielectric properties of wheat associated with non-equilibrium kernel moisture conditions.
Transactions of ASAE 31(4):1251-54.
Sokhansanj, S. and G.E. Woodward. 1991. Computer assisted fan selection for natural grain drying—A teaching and extension tool. Applied Engi-neering in Agriculture 6(6):782-84.
Sokhansanj, S., D. Singh, and J.D. Wasserman. 1984. Drying characteristics of wheat, barley and canola subjected to repetitive wetting and drying cycles. Transactions of ASAE 27(3):903-906, 914.
Sokhansanj, S., W. Zhijie, D.S. Jayas, and T. Kameoka. 1986. Equilibrium relative humidity moisture content of rapeseed from 5 to 25°C. Trans-actions of ASAE 29(3):837-39.
Sokhansanj, S., A.A. Falacinski, F.W. Sosulski, D.S. Jayas, and J. Tang.
1990. Resistance of bulk lentils to airflow. Transactions of ASAE 33(4):1281-85.
Steele, J.L. 1967. Deterioration of damaged shelled corn as measured by car-bon dioxide production. Unpublished Ph.D. diss., Department of Agri-cultural Engineering, Iowa State University, Ames, IA.
Steele, J.L., R.A. Saul, and W.V. Hukill. 1969. Deterioration of shelled corn as measured by carbon dioxide production. Transactions of ASAE 12(5):685-89.
Stephens, G.R. and T.L. Thompson. 1976. Improving crossflow grain dryer design using simulation. Transactions of ASAE 19(4):778-81.
Stephens, L.E. and G.H. Foster. 1976a. Breakage tester predicts handling damage in corn. ARS-NC-49. ARS, USDA, Washington, D.C.
Stephens, L.E. and G.H. Foster. 1976b. Grain bulk properties as affected by mechanical grain spreaders. Transactions of ASAE 19(2):354-58.
Stephens, L.E. and G.H. Foster. 1978. Bulk properties of wheat and grain sorghum as affected by a mechanical grain spreader. Transactions of ASAE 21(6):1217-18.
Storey, C.L., R.D. Speirs, and L.S. Henderson. 1979. Insect control in farm-stored grain. Farmers Bulletin 2269. USDA-SEA (Available for sale from Superintendent of Documents, U.S. Government Printing Office, Washington, D.C. 20402).
Stroshine, R.L., A.W. Kirleis, J.F. Tuite, L.F. Bauman, and A. Emam. 1986.
Differences in grain quality among selected corn hybrids. Cereal Foods World 31(4):311-16.
Stroshine, R.L., J. Tuite, G.H. Foster, and K. Baker. 1984. Self-study guide for grain drying and storage. Department of Agricultural Engineering, Purdue University, West Lafayette, IN.
Sun, D.W. and J.L. Woods. 1993. The moisture content/relative humidity equilibrium relationship of wheat: A review. Drying Tech. 11(7):1523-51.
Sun, D.W. and J.L. Woods. 1994. Low temperature moisture transfer char-acteristics of wheat in thin layers. Transactions of ASAE 37(6):1919-26.
Swearingen, M.L. 1979. A practical guide to no-till double cropping. In Double cropping winter wheat and soybeans in Indiana. ID 96. Cooper-ative Extension Service, Purdue University, West Lafayette, IN.
11.18 2001 ASHRAE Fundamentals Handbook (SI) Syarief, A.M., R.V. Morey, and R.J. Gustafson. 1984. Thin-layer drying rates of sunflower seed. Transactions of ASAE 27(1):195-200.
Tang, J., S. Sokhansanj, and F.W. Sosulski. 1989. Thin-layer drying of lentil.
Paper 89-6607. ASAE.
Thompson, C.R., E.M. Bickoff, G.R. VanAtta, G.O. Kohler, J. Guggolz, and A.L. Livingston. 1960. Carotene stability in alfalfa as affected by labo-ratory- and industrial-scale processing. Technical Bulletin 1232. ARS, USDA, Washington, D.C.
Thompson, T.L. 1972. Temporary storage of high-moisture shelled corn using continuous aeration. Transactions of ASAE 15(2):333-37.
Thompson, T.L., G.H. Foster, and R.M. Peart. 1969. Comparison of concur-rent-flow, crossflow and counterflow grain drying methods. Marketing Research Report 841. USDA-ARS, Washington, D.C.
Thompson, T.L., R.M. Peart, and G.H. Foster. 1968. Mathematical simula-tion of corn drying—A new model. Transactions of ASAE 11(4):582-86.
Thorpe, G.R. 1982. Moisture diffusion through bulk grain subjected to a temperature gradient. Journal of Stored Products Research 18:9-12.
Troeger, J.M. and W.V. Hukill. 1971. Mathematical description of the dry-ing rate of fully exposed corn. Transactions of ASAE 14(6):1153-56, 1162.
Tuite, J. and G.H. Foster. 1963. Effect of artificial drying on the hygroscopic properties of corn. Cereal Chemistry 40:630-37.
Tuite, J. and G.H. Foster. 1979. Control of storage diseases of grain. Annual Review of Phytopathology 17:343.
Tuite, J., G.H. Foster, S.R. Eckhoff, and O.L. Shotwell. 1986. Sulfur dioxide treatment to extend corn drying time (note). Cereal Chemistry 63(5): 462-64.
Uebersax, M.A. and C.L. Bedford. 1980. Navy bean processing: Effect of storage and soaking methods on quality of canned beans. Research Report 410. Agricultural Experiment Station, Michigan State University, East Lansing.
USDA. 1971. Oven methods for determining moisture content of grain and related agricultural commodities, Chapter 12. Equipment manual, GR Instruction 916-6. U.S. Department of Agriculture, Consumer and Mar-keting Service, Grain Division, Hyattsville, MD.
Velupillai, L. and L.R. Verma. 1986. Drying and tempering effects on par-boiled rice quality. Transactions of ASAE 29(1):312-19.
Verma, L.R. and B.D. Nelson. 1983. Changes in round bales during storage.
Transactions of ASAE 26(2):328-32.
Walton, L.R. and W.H. Henson, Jr. 1971. Effect of environment during cur-ing on the quality of burley tobacco: Effect of low humidity curing on support price. Tobacco Science 15:54-57.
Walton, L.R., W.H. Henson, Jr., and J.M. Bunn. 1973. Effect of environment during curing on the quality of burley tobacco: Effect of high humidity curing on support price. Tobacco Science 17:25-27.
Walton, L.R., L.D. Swetnam, and J.H. Casada. 1994. Curing burley tobacco in a field curing structure. Applied Engineering in Agric. 10(3):385-89.
Watson, C.A. 1977. Near infrared reflectance spectro-photometric analysis of agricultural products. Analytical Chemistry 49(9):835A-40A.
Watson, C.A., O.J. Banasick, and G.L. Pratt. 1962. Effect of drying temper-ature on barley malting quality. Brewers Digest 37(7):44-48.
Watson, E.L. and V.K. Bhargava. 1974. Thin-layer drying studies on wheat.
Canadian Agricultural Engineering 16(1):18-22.
Watson, S.A. 1987. Measurement and maintenance of quality. In Corn: Chemistry and technology, eds. S.A. Watson and P.E. Ramstad, pp. 125-83. American Association of Cereal Chemists, St. Paul, MN.
Watson, S.A. and F.L. Herum. 1986. Comparison of eight devices for mea-suring breakage susceptibility of shelled corn. Cereal Chemistry 63(2):139-42.
Watson, S.A. and P.E. Ramstad. 1987. Corn: Chemistry and technology.
American Association of Cereal Chemists, St. Paul, MN.
Watson, S.A., L.L. Darrah, and F.L. Herum. 1986. Measurement of corn breakage susceptibility with the Stein breakage tester: A collaborative study. Cereal Foods World 31(5):366-72.
Weller, C.L., M.R. Paulsen, and M.P. Steinberg. 1987. Varietal, harvest moisture and drying air temperature effects on quality factors affecting corn wet milling. Paper 87-6046. ASAE.
White, N.D.G. and D.S. Jayas. 1991. Control of insects and mites with car-bon dioxide in wheat stored at cool temperatures in nonairtight bins. J.
Econ. Entomology 84(6):1933-42.
Wicklow, D.T. 1988. Patterns of fungal association within maize kernels harvested in North Carolina. Plant Disease 72:113-15.
Young, J.H. 1984. Energy conservation by partial recirculation of peanut drying air. Transactions of ASAE 27(3):928-34.
12.1 CHAPTER 12 AIR CONTAMINANTS Classes of Air Contaminants ................................................... 12.1 PARTICULATE CONTAMINANTS ........................................ 12.2 Particulate Matter ................................................................... 12.2 Bioaerosols ............................................................................. 12.5 GASEOUS CONTAMINANTS ................................................ 12.7 Volatile Organic Compounds ................................................. 12.9 Inorganic Gases .................................................................... 12.11 SPECIAL TYPES OF AIR CONTAMINANTS ...................... 12.12 Outdoor Air Contaminants .................................................... 12.12 Industrial Air Contaminants ................................................. 12.14 Nonindustrial Indoor Air Contaminants ............................... 12.14 Flammable Gases and Vapors .............................................. 12.15 Combustible Dusts ................................................................ 12.16 Radioactive Air Contaminants .............................................. 12.16 Soil Gases .............................................................................. 12.17 IR IS COMPOSED mainly of gases. The major gaseous com-A ponents of clean, dry air near sea level are approximately 21% oxygen, 78% nitrogen, 1% argon, and 0.04% carbon dioxide.
Normal outdoor air contains varying amounts of foreign mate-rials (permanent atmospheric impurities). These materials can arise from natural processes such as wind erosion, sea spray evap-oration, volcanic eruption, and metabolism or decay of organic matter. The natural contaminant concentrations in the air that we breathe vary but are usually lower than those caused by human activity.
Man-made outdoor contaminants are many and varied, origi-nating from numerous types of human activity. Electric power-generating plants, various modes of transportation, industrial pro-cesses, mining and smelting, construction, and agriculture gener-ate large amounts of contaminants.
Contaminants that present particular problems in the indoor environment include, among others, tobacco smoke, radon, and formaldehyde.
Air composition may be changed accidentally or deliberately.
In sewers, sewage treatment plants, tunnels, and mines, the oxy-gen content of air can become so low that people cannot remain conscious or survive. Concentrations of people in confined spaces (theaters, survival shelters, submarines) require that carbon diox-ide given off by normal respiratory functions be removed and replaced with oxygen. Pilots of high-altitude aircraft, breathing at greatly reduced pressure, require systems that increase oxygen concentration. Conversely, for divers working at extreme depths, it is common to increase the percentage of helium in the atmo-sphere and reduce nitrogen and sometimes oxygen concentrations.
At atmospheric pressure, oxygen concentrations less than 12% or carbon dioxide concentrations greater than 5% are dangerous, even for short periods. Lesser deviations from normal composi-tion can be hazardous under prolonged exposures. Chapter 9 fur-ther details environmental health issues.
CLASSES OF AIR CONTAMINANTS The major classes of air contaminants are particulate and gas-eous. The particulate class covers a vast range of particle sizes from dust large enough to be visible to the eye to submicroscopic particles that elude most filters. Particulates may be solid or liquid.
The following traditional contaminant classifications are subclasses of particulates: • Dusts, fumes, and smokes, which are mostly solid particulate matter, although smoke often contains liquid particles • Mists, fogs, and smogs, which are mostly suspended liquid particles smaller than those in dusts, fumes, and smokes • Bioaerosols, including viruses, bacteria, fungal spores, and pollen, whose primary impact is related to their biological origin • Particle size definitions such as coarse or fine, visible or invisible, and macroscopic, microscopic, or submicroscopic • Definitions that relate to particle interaction with the human respiratory system, such as inhalable and respirable These classes, their characteristics, units of measurement, and mea-surement methods are discussed in more detail in this chapter.
The gaseous class covers chemical contaminants that can exist as free molecules or atoms in air. Molecules and atoms are smaller than particles and may behave differently as a result. This class cov-ers two important subclasses: • Gases, which are naturally gaseous under ambient indoor or outdoor conditions • Vapors, which are normally solid or liquid under ambient indoor or outdoor conditions, but which evaporate readily Through evaporation, liquids change into vapors and mix with the surrounding atmosphere. Like gases, they are formless fluids that expand to occupy the space or enclosure in which they are con-fined. Typical gaseous contaminants, their characteristics, units of measurement, and measurement methods are discussed in detail later in this chapter.
Air contaminants can also be classified according to their sources; their properties; or the health, safety, and engineering issues faced by people exposed to them. Any of these can form a convenient classification system because they allow grouping of applicable standards, guidelines, and control strategies. Most of the following classes include both particulate and gaseous contami-nants. This chapter covers the background information for these classes, while Chapter 9 deals with the applicable indoor health and comfort regulations. The classes are • Industrial air contaminants • Nonindustrial indoor air contaminants (including indoor air quality) • Flammable gases and vapors • Combustible dusts • Radioactive contaminants • Soil gases The preparation of this chapter is assigned to TC 2.3, Gaseous Air Contam-inants and Gas Contaminant Removal Equipment, in conjunction with TC 2.4, Particulate Air Contaminants and Particulate Contaminant Removal Equipment.
12.2 2001 ASHRAE Fundamentals Handbook (SI) PARTICULATE CONTAMINANTS PARTICULATE MATTER Airborne particulate matter is not a single substance but a com-plex mixture of many different components, usually from several different sources. Particles can be anthropogenic or natural in ori-gin. Anthropogenic particles are those produced by human activi-ties, including fossil fuel combustion, industrial processes, and road dust. Particles from natural sources do not involve human activity.
They include wind-blown dust and smoke from forest fires.
Particles can be generated by primary or secondary processes.
Particles from primary processes are those emitted directly into the air. Particles from secondary processes are those formed from con-densable vapors and chemical reactions.
Details on the different particle types are given below.
Types of Solid Particles Dusts are solid particles projected into the air by natural forces such as wind, volcanic eruption, or earthquakes, or by mechanical processes including crushing, grinding, demolition, blasting, drill-ing, shoveling, screening, and sweeping. Some of these forces pro-duce dusts by reducing larger masses, while others disperse materials that have already been reduced. Particles are not consid-ered to be dust unless they are smaller than about 100 µm. Dusts can be mineral, such as rock, metal, or clay; vegetable, such as grain, flour, wood, cotton, or pollen; or animal, including wool, hair, silk, feathers, and leather.
Fumes are solid particles formed by condensation of vapors of solid materials. Metallic fumes are generated from molten metals and usually occur as oxides because of the highly reactive nature of finely divided matter. Fumes can also be formed by sublimation, distillation, or chemical reaction. Such processes create airborne particles smaller than 1 µm. Fumes permitted to age may agglom-erate into larger clusters.
Bioaerosols include airborne viruses, bacteria, pollen, and fun-gus spores. Viruses range in size from 0.003 to 0.06 µm, although they usually occur in colonies or attached to other particles. Most bacteria range between 0.4 and 5 µm and are usually associated with large particles. Fungus spores are usually 2 to 10 µm, while pollen grains are 10 to 100 µm, with many common varieties in the 20 to 40 µm range.
Types of Liquid Particles Mists are small airborne droplets of materials that are ordinarily liquid at normal temperatures and pressure. They can be formed by atomizing, spraying, mixing, violent chemical reactions, evolution of gas from liquid, or escape as a dissolved gas when pressure is released. Small droplets expelled or atomized by sneezing consti-tute mists.
Fogs are fine airborne droplets, usually formed by condensation of vapor, which remain airborne longer than mists. Fog nozzles are named for their ability to produce extra fine droplets, as compared with mists from ordinary spray devices. Many droplets in fogs or clouds are microscopic and submicroscopic and serve as a transition stage between larger mists and vapors.
The volatile nature of most liquids reduces the size of their air-borne droplets from the mist to the fog range and eventually to the vapor phase, until the air becomes saturated with that liquid. If solid material is suspended or dissolved in the liquid droplet, it remains in the air as particulate contamination. For example, sea spray evapo-rates fairly rapidly, generating a large number of fine salt particles that remain suspended in the atmosphere.
Complex Particles Smokes are small solid and/or liquid particles produced by incomplete combustion of organic substances such as tobacco, wood, coal, oil, and other carbonaceous materials. The term smoke is applied to a mixture of solid, liquid, and gaseous products, although technical literature distinguishes between such compo-nents as soot or carbon particles, fly ash, cinders, tarry matter, unburned gases, and gaseous combustion products. Smoke particles vary in size, the smallest being much less than 1 µm in diameter. The average is often in the range of 0.1 to 0.3 µm.
Environmental tobacco smoke consists of a suspension of 0.01 to 1.0 µm (mass median diameter of 0.3 µm) liquid particles that form as the superheated vapors leaving the burning tobacco con-dense. Also produced are numerous gaseous contaminants includ-ing carbon monoxide.
Smog commonly refers to air pollution; it implies an air mixture of smoke particles, mists, and fog droplets of such concentration and composition as to impair visibility, in addition to being irritating or harmful. The composition varies among different locations and at different times. The term is often applied to haze caused by a sun-light-induced photochemical reaction involving the materials in automobile exhausts. Smog is often associated with temperature inversions in the atmosphere that prevent normal dispersion of con-taminants.
Sizes of Airborne Particles Particle size can be defined in several different ways. These depend, for example, on the source or method of generation, the vis-ibility, or the effects.
Particles can be classified as coarse or fine. Coarse particles are larger, and are generally formed by mechanical breaking up of sol-ids. They generally have a minimum size of 1 to 3 µm (EPA 1996).
Fine particles are generally formed from chemical reactions or con-densing gases. These particles have a maximum size of about 1 to 3 µm. Fine particles are usually more chemically complex, anthropo-genic, and secondary in origin, while the coarse particles are pre-dominantly primary, natural, and chemically inert. Coarse particles also include bioaerosols such as mold spores, pollen, animal dander, and dust mite particles that can affect the immune system.
The differences between fine and coarse particles lead to a bimo-dal distribution of particles in most environments with concentra-tion peaks at about 0.25 µm and 5 µm. Figure 1 shows a typical urban distribution including the chemical species present in each mode.
The size of a particle determines where in the human respiratory system the particle is deposited. The inhalable mass is made up of Fig. 1 Typical Urban Aerosol Composition by Particle Size Fraction (EPA 1982, Willeke and Baron 1993) Air Contaminants 12.3 particles that may deposit anywhere in the respiratory system and is represented by a sample with a median cut point of 100 µm. The thoracic particle mass is that fraction which can penetrate to the lung airways and is represented by a sample with a median cut point of 10 µm. The respirable particulate mass is that fraction that can penetrate to the gas-exchange region of the lungs and is represented by a sample with a median cut point of 4 µm (ACGIH 1998). Figure 2 illustrates the relative deposition efficiencies of various sizes of particles. The characteristics described above were used by the EPA in development of their outdoor PM10 and PM2.5 standards.
Most particles are irregular in shape, and it is useful to charac-terize their size in terms of some standard particle. For example, the aerodynamic (equivalent) diameter of a particle is defined as the diameter of a unit-density sphere having the same gravitational settling velocity as the particle in question (Willeke and Baron 1993).
The tendency of particles to settle on surfaces is a property of interest. Figure 3 shows the sizes of typical indoor airborne solid and liquid particles. Particles smaller than 0.1 µm behave like gas molecules exhibiting Brownian motion due to collisions with air Fig. 3 Sizes of Indoor Particles (Owen et al. 1992) Fig. 2 Relative Deposition Efficiencies of Different Sized Particles in the Three Main Regions of the Human Respiratory System, Calculated for Moderate Activity Level (Task Group on Lung Dynamics 1966) 12.4 2001 ASHRAE Fundamentals Handbook (SI) molecules and having no measurable settling velocity. Particles in the range from 0.1 to 1 µm have settling velocities that can be cal-culated but are so low that settling is usually negligible, since nor-mal air currents counteract any settling. By number, over 99.9% of the particles in a typical atmosphere are below 1 µm (this means fewer than 1 particle in every 1000 is larger than 1 µm.).
Particles in the 1 to 10 µm range settle in still air at constant and appreciable velocity. However, normal air currents keep them in suspension for appreciable periods.
Particles larger than 10 µm settle fairly rapidly and can be found suspended in air only near their source or under strong wind condi-tions. Exceptions are lint and other light fibrous materials, such as portions of certain weed seeds, which remain suspended longer.
Table 1 shows settling times for various types of particles.
Most individual particles 10 µm or larger are visible to the naked eye under favorable conditions of lighting and contrast. Smaller particles are visible only in high concentrations. Cigarette smoke (with an average particle size less than 0.5 µm) and clouds are com-mon examples.
Direct fallout in the vicinity of the dispersing stack or flue and other nuisance problems of air pollution involve larger particles.
Smaller particles, as well as mists, fogs, and fumes, remain in sus-pension longer. In this size range, meteorology and topography are more important than physical characteristics of the particles. Since settling velocities are small, the ability of the atmosphere to disperse these small particles depends largely on local weather conditions.
Comparison is often made to screen sizes used for grading useful industrial dusts and granular materials. Table 2 illustrates the rela-tionship of U.S. standard sieve mesh to particle size in micrometres.
Particles above 40 µm are the screen sizes, and those below are the subscreen or microscopic sizes.
Particle Size Distribution The particle size distribution in any sample can be expressed as the percentage of the number of particles smaller than a specified size, area, or mass. The upper curve of Figure 4 shows these data plotted for typical atmospheric contamination. The middle curve shows the percentage of the total projected area of the particles con-tributed by particles less than a specified size. The lower curve shows the percentage of the total particle mass contributed by those particles less than a given size. Thus, for particles 10 µm and larger, 4% by mass, a 90% arrestance filter essentially removes 90% of 4% for an efficiency by mass of 3.6%.
The differences among values presented by the three curves should be noted. For example, particles 0.1 µm or less in diameter (but still above electron microscope minimum detection size of about 0.005 µm) make up 80% of the number of particles in the atmosphere but contribute only 1% of the mass. Also, the 0.1% of particles by number larger than 1 µm carry 70% of the total mass, which is the direct result of the mass of a spherical particle increas-ing as the cube of its diameter. Although most of the mass is con-tributed by intermediate and larger particles, over 80% of the area (staining) contamination is supplied by particles less than 1 µm in diameter, which is in the center of the respirable particle size range and is the size most likely to remain in the lungs. (See Chapter 9.) Of possible concern to the HVAC industry is the fact that most of the staining effect on ceilings, walls, windows, and light fixtures, as well as fouling of heat transfer devices and rotating equipment, results from particles less than 1 µm in diameter. Suspended parti-cles in urban air are predominantly smaller than 1 µm (aerodynamic diameter) and have a distribution that is approximately log-normal.
Units of Measurement The quantity of particulate matter in the air can be determined as mass or particle count in a given volume of air. Mass units are mil-ligrams per cubic metre of air sampled (mg/m3) or micrograms per cubic metre of air sampled (µg/m3). 1 mg/m3 = 1000 µg/m3. Particle counts are usually quoted for volumes of 0.1 ft3, 1 ft3, or 1 litre and are specified for a given range of particle diameter.
Measurement of Airborne Particles Suitable methods for determining the quantity of particulate mat-ter in the air vary depending on the amount present and on the size of particles involved.
The particle mass (total or respirable) is easily determined by measuring the mass of a filter before and after drawing a known vol-ume of dusty air through it. This method is widely used in industrial workplaces, where there may be significant numbers of large parti-cles, but is not sensitive enough for evaluating office environments.
Optical particle counters are widely used and likely to become more so since ASHRAE adopted Standard 52.2. This standard defines a laboratory method for assessing the performance of media Table 1 Approximate Particle Sizes and Their Times to Settle One Metre Type of Particle Diameter, µm Settling Time Human hair 100 to 150 5 s Skin flakes 20 to 40 Observable dust in air >10 Common pollens 15 to 25 Mite allergens 10 to 20 5 min Common spores 2 to 10 Bacteria 1 to 5 Cat dander 1+0.5 10 h Tobacco smoke 0.1 to 1 Metal and organic fumes <0.1 to 1 Cell debris 0.01 to 1 Viruses <0.1 10 days Source: J.D. Spengler, Harvard School of Public Health.
Table 2 Relation of Screen Mesh to Particle Size U.S. Standard sieve mesh 400 325 200 140 100 60 35 18 Nominal sieve opening, µm 37 44 74 105 149 250 500 1000 Fig. 4 Particle Size Distribution of Atmospheric Dust (Whitby et al. 1955, 1957) Air Contaminants 12.5 filters using an optical particle counter to measure particle counts upstream and downstream of the filter in 12 size ranges between 0.3 and 10 µm. Filters are then given a minimum efficiency reporting value (MERV) rating based on the count data.
Counters are also used to test cleanrooms for compliance with Federal Standard 209E and ISO Standard 14644-1. Cleanrooms are defined in terms of the number of particles in certain size ranges that they contain. Chapter 15 of the 1999 ASHRAE Handbook—Appli-cations has further information on cleanrooms.
Optical particle counters use laser light-scattering to continu-ously count and size airborne particles and can detect particles down to 0.1 µm (ASTM Standard F 50). A condensation nucleus counter can count particles to below 0.01 µm. These particles, present in great numbers in the atmosphere, serve as nuclei for con-densation of water vapor (Scala 1963).
Another indirect method measures the optical density of the collected dust based on the projected area of the particles. Dust par-ticles can be sized with graduated scales or optical comparisons using a standard microscope. The lower limit for sizing with the light field microscope is approximately 0.9 µm, depending on the vision of the observer, the dust color, and the contrast available.
This size can be reduced to about 0.4 µm by using oil immersion objective techniques. Dark field microscopic techniques reveal par-ticles smaller than these, to a limit of approximately 0.1 µm.
Smaller submicroscopic dusts can be sized and compared with the aid of an electron microscope. Other sizing techniques may take into account velocity of samplings in calibrated devices and actual settlement measurements in laboratory equipment. The electron microscope and various sampling instruments such as the cascade impactor have been successful in sizing particulates, including fogs and mists.
Each of the various methods of measuring particle size distribu-tion gives a different value for the same size particle, since different properties are actually measured. For example, a microscopic tech-nique may measure longest dimension, while impactor results are based on aerodynamic behavior (ACGIH 2001).
Typical Particle Levels Particle counters, which detect particles larger than about 0.1 µm, have indicated that the number of suspended particles is enor-mous. A room with heavy cigarette smoke has a particle concen-tration of 109 particles per cubic metre. Even clean air typically contains over 35 × 106 particles/m3. If smaller particles detectable by other means, such as an electron microscope or condensation nucleus counter, are also included, the total particle concentration would be greater than the above concentrations by a factor of 10 to 100.
Extensive measurements have been made of outdoor pollution, but limited data have been gathered on indoor pollution not associ-ated with specific industrial processes. Indoor levels are influenced by the number of people and their activities, building materials and construction, outside conditions, ventilation rate, and the air-condi-tioning and filtration system. For further information, see the sec-tion on Nonindustrial Indoor Air Contaminants, Spengler et al.
(1982), and NRC (1981).
BIOAEROSOLS Bioaerosols are airborne microbiological particulate matter derived from fungi, bacteria, viruses, protozoa, algae, pollen, mites, and their cellular or cell mass components. Bioaerosols are univer-sally present in both indoor and outdoor environments. Problems of concern to engineers occur when microorganisms grow and repro-duce indoors.
Microorganisms break down complex molecules found in dead organic materials to simple substances such as carbon dioxide, water, and nitrates. These components are then used by photosyn-thetic organisms such as plants and algae. Thus, the presence of bac-teria and fungi in soil, water, and atmospheric habitats is normal.
Spores of Cladosporium, a fungus commonly found on leaves and dead vegetation, are almost always found in outdoor air samples.
They are found in variable numbers in indoor air, depending on the amount of outdoor air that infiltrates into interior spaces or is brought in by the HVAC system. Outdoor microorganisms can also enter on shoes and clothing and be transferred to other surfaces in buildings.
Public interest has focused on airborne microorganisms respon-sible for diseases and infections, primarily bacteria and viruses.
This is discussed in more detail in Chapter 9. A variety of airborne microorganisms are of economic significance and can cause prod-uct contamination or loss. In the food processing industry, yeast and mold can reduce the shelf life of some products. Refined syrups can be damaged by mold scums. Wild yeast can destroy a batch of beer.
Antibiotic yields can be reduced by foreign organisms in the culture mix.
Fungi Much attention has been given to fungi, which include yeasts, molds, and mildews, as well as large mushrooms, puffballs, and bracket fungi. All fungi depend on external sources of organic mate-rial for both energy requirements and carbon skeletons. Thus, they cannot increase in number unless supplied with a suitable food source such as small quantities of dust, paper, or wood. Reproduc-tion also requires appropriate temperatures (20 to 40°C) and the presence of water, high air humidity (typically greater than 60%), and/or high moisture content in the food source.
Bacteria Cooling towers, evaporative condensers, and domestic water ser-vice systems all provide water and nutrients for amplification of microorganisms such as Legionella pneumophila. Growth of micro-bial populations to excessive concentrations is generally associated with inadequate preventive maintenance of these systems, at least in cooling towers. A body of literature has identified characteristics of indoor plumbing and heating systems associated with frequent iso-lation of Legionella species, including blind ends, scale, upright electric water heaters, and lower water temperatures. The survival of Legionella is enhanced by a variety of parameters including but not limited to warm temperatures, particular algal and protozoan associations, and symbiotic relationships with certain aquatic plants (Fliermans 1985). Evidence has indicated that amoebae and other protozoa act as natural hosts and amplifiers for Legionella in the environment (Barbaree et al. 1986). ASHRAE (1989) has further information on the topic.
Bacteria from the soil are likely to be spore-formers and are capable of surviving in hostile environments. Other airborne bacte-ria, especially within closed occupied spaces, may originate from droplet nuclei caused by actions such as sneezing or be carried on human or animal skin scales.
Pollen Pollen grains discharged by weeds, grasses, and trees (Hewson et al. 1967, Jacobson and Morris 1977, Solomon and Mathews 1978) and capable of causing hay fever have properties of special interest to air-cleaning equipment designers (see Chapter 24 of the 2000 ASHRAE Handbook—Systems and Equipment). Whole grains and fragments transported by air range between 10 and 50 µm; how-ever, some measure as small as 5 µm, and others measure over 100 µm in diameter. Ragweed pollen grains are fairly uniform in size, ranging from 15 to 25 µm.
Most pollen grains are hygroscopic and, therefore, vary in mass with the humidity. Illustrations and data on pollen grains are avail-12.6 2001 ASHRAE Fundamentals Handbook (SI) able in botanical literature. Geographical distribution of plants that produce hay fever is also recorded.
The quantity of pollen in the air is generally estimated by expos-ing an adhesive-coated glass plate outdoors for 24 h, then counting calibrated areas under the microscope. Methods are available for determining the number of pollen grains in a measured volume of air. However, despite their greater accuracy, these methods have not replaced the simpler gravity slide method used for most pollen counts. Counting techniques vary, but daily pollen counts reported in local newspapers during hay fever season usually represent the number of grains found on 180 mm2 of a 24 h gravity slide.
Hay fever sufferers may experience the first symptoms when the pollen count is 10 to 25; in some localities, maximum figures for the seasonal peak may approach 1000 or more for a 24 h period, depending on the sampling and reporting methods used by the laboratory. Translation of gravity counts by special formulas to a volumetric basis (i.e., number of grains per unit volume of air) is unreliable, due to the complexity of the modifying factors.
When such information is important, it should be obtained directly by a volumetric instrument. The number of pollen grains per cubic yard of air varies from 2 to 20 times the number found on 1 cm2 of a 24 h gravity slide, depending on grain diameter, shape, specific gravity, wind velocity, humidity, and physical placement of the collecting plate.
Pollen grains can be removed from the air more readily than the dust particles prevalent in outdoor air or those produced by dusty processes, since a larger fraction of them are in the size range easily removed by building HVAC filters.
Whole-grain pollens are easily removed from the outside air entering a ventilation system with medium-efficiency [35 to 45% (MERV 9 to 11)] filters selected to remove 99% of particles 10 µm and greater. Once they have entered a building, whole-grain pollens settle rapidly, reducing concentrations without the need for air cleaning. On resuspension from occupant activities, the whole-grain pollens may disintegrate into fragments, which may possibly be controlled effectively with a high-efficiency [70 to 95% (MERV 12 to 14)] filter capable of removing a high percentage of particles in the 0.3 to 3.0 µm range.
Sampling for Bioaerosols Sampling for biological agents such as fungi and bacteria can include visual observation of colonies, collection of bulk or surface samples, or air sampling. The principles of sampling and analysis for microorganisms are reviewed by Chatigny (1983). ACGIH (1989) and AIHA (1996) have developed assessment guidelines for the collection of microbiological particulates. An introduction is given below.
Preassessment. Sampling for microorganisms should be under-taken when medical evidence indicates the occurrence of diseases such as humidifier fever, hypersensitivity pneumonitis, allergic asthma, and allergic rhinitis. A walk-through examination of the indoor environment for visual detection of possible microbial res-ervoirs and amplification sites should be performed before sam-pling. Note that a visual examination will miss reservoirs that are behind walls. If a reservoir or amplifier is visually identified, it is useful to obtain bulk or source samples from it. Also, removal of clearly identified reservoirs and amplifiers is preferable to compli-cated and costly air-sampling procedures.
Air Sampling. The same principles that affect the collection of an inert particulate aerosol also govern air sampling for microor-ganisms. Air sampling is not likely to yield useful data and infor-mation unless the sample collected is representative of exposure.
The most representative samples are those that are collected in breathing zones over the range of aerosol concentrations. Pres-ently, no personal sampling method has been proposed that is sen-sitive enough for any bioaerosol (Burge 1995). Thus, ambient sampling designs that obtain reasonable estimates of exposures of given populations over representative periods are necessary.
Because Legionella requires special nutrients for growth and does not produce resistant spores, this bacterium is difficult to recover from air.
Concentrations of microorganisms in the atmosphere vary from a few to several hundred per cubic metre, depending on many fac-tors. The sampling method for microorganisms has an effect on the measured count. Collection on dry filter paper can cause count degradation because of the dehydration loss of some organisms.
Glass impingers may give high counts because agitation can cause clusters to break up into smaller individual organisms. Slit samples may give a more accurate colony count.
The viability and/or antigenicity of the microbial particulates must be protected during sampling. In general, culture plate impactors, including multiple- and single-stage devices as well as slit-to-agar samplers, are most useful in office environments where low concentrations of bacteria and fungi are expected.
Because not all microorganisms will grow on the same media, liq-uid impingement subculturing may be more suitable. Filter cas-sette samplers are useful for microorganisms or components of microorganisms (i.e., endotoxins), although binding to glass and plastic has been reported (Milton et al. 1990). Filter cassettes can also be used for spore counts. Area sampling is often used. Some investigators attempt to replicate exposure conditions through dis-turbance of the environment (semiaggressive sampling) such as occurs through walking on carpets, slamming doors, and opening books or file cabinets.
Viruses, many bacteria, algae, and protozoa are more difficult to culture than fungi, and air-sampling methodology for these agents is less well known and defined (ASTM 1990).
Data Interpretation. Rank order assessment is a method used to interpret air-sampling data for microorganisms (ACGIH 1989).
Individual organisms are listed in descending order of abundance for a complainant indoor site and for one or more control locations.
The predominance of one or more microbes in the complainant site, but not in the control sites or outdoors, suggests the presence of an amplifier for that organism. In the example in Table 3, Tritirachium and Aspergillus were the predominant fungi represented in com-plainant locations in an office building, where Cladosporium and Fusarium dominated outdoor collections. In this case, Tritirachium and Aspergillus were being amplified in the building. In addition to comparing individual organisms, indoor-outdoor ratios of overall quantities of culturable microorganisms are useful.
Control of Bioaerosols When maximum removal of airborne microorganisms is either necessary or desirable, high-efficiency particulate air (HEPA) or ultralow penetration air (ULPA) filters are used. These filters create essentially sterile atmospheres and are more frequently employed than chemical scrubbers and ultraviolet radiation for control of air-borne microorganisms. They have been used to prevent cross-infec-tion in hospitals, to protect clean rooms from contamination, and to assemble and launch space probes under sterile conditions.
Table 3 Example Case of Airborne Fungi in Building and in Outdoor Air Location cfu/m3 Rank Order Taxa Outdoors 210 Cladosporium > Fusarium > Epicoccum > Aspergillus Complainant Office #1 2500 Tritirachium > Aspergillus > Cladosporium Complainant Office #2 3000 Tritirachium > Aspergillus > Cladosporium Notes: cfu/m3 = Colony-forming units per cubic metre of air.
Culture media was malt extract agar (ACGIH 1989).
Air Contaminants 12.7 In many situations, total control of airborne microorganisms is not required. For these applications, there are various other types of high-efficiency dry media extended-surface filters that will provide the necessary efficiency. These filters have lower pressure differen-tials than HEPA filters operating at the same face velocity and, when properly selected, will remove the contaminants of concern.
GASEOUS CONTAMINANTS The terms gas and vapor are both used to describe the gaseous state of a substance. Gas is the correct term for describing any pure substance or mixture that naturally exists in the gaseous state at nor-mal atmospheric conditions. Examples are oxygen, helium, ammo-nia, and nitrogen. Vapor is used to describe a substance in the gaseous state whose natural state is a liquid or solid at normal atmo-spheric conditions. Examples include benzene, carbon tetrachlo-ride, and water. There are differences between the two classes that reflect their preferred states: • The concentration of gases in air is limited only by the strength of the source, so that gases can completely fill a space, driving out the oxygen necessary for survival.
• Vapors can never exceed their saturated vapor pressure in air. The most familiar example of a vapor is water, with relative humidity expressing the air concentration as a percentage of the saturated vapor pressure.
• Vapors, because their natural state is liquid or solid, have a tendency to condense on surfaces and be adsorbed.
Gaseous contaminants can also be divided into organic and inor-ganic types. Organic compounds include all chemicals based on a skeleton of carbon atoms. Since carbon atoms easily combine to form chain, branched, and ring structures, there is a bewildering variety of organic compounds. They include gases such as methane, but the majority are vapors. All other gaseous contaminants are classified as inorganic. Most of the inorganic air contaminants of interest to engineers are gases.
Major chemical families of gaseous pollutants and examples of specific compounds are shown in Table 4. The Merck Index (Budavi 1996), the Toxic Substances Control Act Chemical Substance Inven-tory (EPA 1979), and Dangerous Properties of Industrial Materials (Sax and Lewis 1988) are all useful in identifying contaminants, Table 4 Major Chemical Families of Gaseous Air Pollutants (with Examples) Inorganic Pollutants 11. Chlorinated hydrocarbons 19. Aromatic hydrocarbons carbon tetrachloride benzene 1. Single-element atoms and molecules chloroform toluene chlorine 1,1,1-trichloroethane p-xylene radon tetrachloroethylene naphthalene mercury dichlorobenzene benz-α-pyrene 2. Oxidants 12. Halide compounds 20. Terpenes ozone methyl bromide 2-pinene nitrogen dioxide methyl iodide limonene 3. Reducing agents 13. Alcohols 21. Heterocylics carbon monoxide methanol ethylene oxide 4. Acid gases ethanol tetrahydrofuran carbon dioxide 2-propanol (isopropanol) 1, 4-dioxane sulfur dioxide phenol pyrrole sulfuric acid ethylene glycol pyridine hydrochloric acid 14. Ethers nicotine hydrogen sulfide ethyl ether caffeine nitric acid methoxyvinyl ether 22. Organophosphates 5. Nitrogen compounds n-butoxyethanol malathion ammonia 15. Aldehydes tabun hydrazine formaldehyde sarin nitrous oxide acetaldehyde soman 6. Miscellaneous acrolein 23. Amines arsine benzaldehyde trimethylamine 16. Ketones ethanolamine Organic Pollutants 2-propanone (acetone) cyclohexylamine 2-butanone (MEK) morpholine 7. n-Alkanes methyl isobutyl ketone (MIBK) 24. Monomers methane chloroacetophenone vinyl chloride n-butane 17. Esters methyl formate n-hexane ethyl acetate ethylene n-octane vinyl acetate methyl methacrylate n-hexadecane methyl formate 25. Mercaptans and other sulfur compounds 8. Branched alkanes dioctyl phthalate (DOP) bis-2-chloroethyl sulfide (mustard gas) 2-methyl pentane 18. Nitrogen compounds other than amines ethyl mercaptan 2-methyl hexane carbon disulfide 9. Alkenes and cyclic hydrocarbons nitromethane carbonyl sulfide butadiene acetonitrile 26. Organic acids 1-octene acrylonitrile formic acid cyclohexane urea acetic acid 4-phenyl cyclohexene (4-PC) uric acid butyric acid 10. Chlorofluorocarbons skatole 27. Miscellaneous R-11 (trichlorofluoromethane) putrescine phosgene R-114 (dichlorotetrafluoroethane) hydrogen cyanide peroxyacetal nitrite (PAN) 12.8 2001 ASHRAE Fundamentals Handbook (SI) including some known by trade names only. Chemical and physical properties can be found in reference books such as the Handbook of Chemistry and Physics (Lide 1996). Note that a single chemical compound, especially if it is an organic compound, may have sev-eral scientific names. To reduce confusion, the Chemical Abstracts Service (CAS) has assigned each chemical a unique identifier num-ber containing five to nine digits.
Harmful Effects of Gaseous Contaminants Harmful effects may be divided into four categories: toxicity, odor, irritation, and material damage.
Toxicity. The harmful effects of gaseous pollutants on a person depend on both short-term peak concentrations and the time-inte-grated exposure received by the person. Toxic effects are generally considered to be proportional to the exposure dose, although indi-vidual response variation can obscure the relationship. The allow-able concentration for short exposures is higher than that for long exposures. Safe exposure limits have been set for a number of com-mon gaseous contaminants in industrial settings. This topic is cov-ered in more detail in the section on Industrial Air Contaminants and in Chapter 9.
Irritation. Although gaseous pollutants may have no discernible continuing health effects, exposure may cause physical irritation to building occupants. This phenomenon has been studied principally in laboratories and nonindustrial work environments and is dis-cussed in more detail in the section on Nonindustrial Indoor Air Contaminants and in Chapter 9.
Odors. Gaseous contaminant problems often appear as com-plaints about odors, and these usually are the result of concentra-tions considerably below industrial exposure limits. Odors are discussed in more detail in Chapter 13.
Damage to Materials. Material damage from gaseous pollutants may take such forms as corrosion, embrittlement, or discoloration.
Because such effects usually involve chemical reactions that need water, material damage from air pollutants is less severe in the rel-atively dry indoor environment than outdoors, even at similar gas-eous contaminant concentrations. To maintain this advantage, indoor condensation should be avoided. However, some dry mate-rials can be significantly damaged. These effects are most serious in museums, as any loss of color or texture changes the essence of the object. Libraries and archives are also vulnerable, as are pipe organs and textiles. Ventilation is often a poor method of protecting collec-tions of rare objects because these facilities are usually located in the centers of cities that have relatively polluted ambient outdoor air. Gaseous compounds known to be harmful include ozone, nitrous oxides, sulfur dioxide, hydrochloric acid, many volatile organic compounds, and hydrogen sulfide. Various concerns about material damage by indoor air pollutants are discussed by Lull (1995), Walsh et al. (1977), Chiarenzelli and Joba (1966), NTIS (1982, 1984), Braun and Wilson (1970), Jaffe (1967), Grosjean et al. (1987), American Guild of Organists (1966), Mathey et al.
(1983), Haynie (1978), Graminski et al. (1978), and Thomson (1986).
Unlike the human body, conservation items have no repair mech-anism, yet they are expected to have a longer life than people. This is one reason the acceptable levels of contaminants in museums, archives, and other conservation applications are very low—1 to 2 orders of magnitude below U.S. EPA standards (Mathey et al. 1983, Thomson 1986). Achieving these levels and measuring perfor-mance is difficult. Lull (1995) discusses ventilation and air cleaning specifically for conservation environments.
Units of Measurement Concentrations of gaseous contaminants are usually expressed in the following units: ppm = parts of contaminant by volume per million parts of air by volume ppb = parts of contaminant by volume per billion parts of air by volume 1000 ppb = 1 ppm mg/m3 = milligrams of contaminant per cubic metre of air µg/m3 = micrograms of contaminant per cubic metre of air The conversions between ppm and mg/m3 are ppm = [8.309(273.15 + t)/Mp] mg/m3 (1) mg/m3 = [0.1204Mp/(273.15 + t)] (ppm) (2) where M = relative molecular mass of contaminant p = mixture pressure, kPa t = mixture temperature, °C Concentration data is often reduced to standard temperature and pressure (i.e., 25°C and 101.325 kPa), in which case ppm = (24.45/M)(mg/m3) (3) Measurement of Gaseous Contaminants The concentration of contaminants in the air needs to be mea-sured to determine whether the indoor air quality conforms to occu-pational health standards (in industrial environments) and is acceptable (in nonindustrial environments).
Measurement methods for airborne chemicals that are important industrially have been published by several organizations including OSHA (1995) and NIOSH (1994). Methods typically involve sam-pling air with pumps for several hours to capture the contaminant on a filter or in an adsorbent tube, followed by laboratory analysis for detection and determination of contaminant concentration. Concen-trations measured in this way can usefully be compared to 8 h indus-trial exposure limits.
The measurement of gaseous contaminants at the lower levels acceptable for indoor air is not always as straightforward. Rela-tively costly analytical equipment may be needed, and it must be calibrated and operated by experienced personnel.
Currently available sampling techniques are listed in Table 5 with details of their advantages and disadvantages. Analytical tech-niques are shown in Table 6 with information on the types of con-taminants to which they apply.
The first two techniques in Table 5 combine sampling and anal-ysis in one piece of equipment and give immediate results on-site.
The other sampling methods require laboratory analysis following the field work. Equipment employing the first technique in Table 5 can be coupled with a data logger to perform continuous monitoring and to obtain average concentrations over a time period. Most of the sampling techniques can capture several contaminants. Several allow pollutants to be accumulated or concentrated over time so that very low concentrations can be measured.
Some analytical techniques are specific for a single pollutant, while others are capable of providing concentrations for many con-taminants simultaneously. Note that formaldehyde requires differ-ent measurement methods from other volatile organic compounds.
Measurement instruments used in industrial situations should be able to detect contaminants of interest at about one-tenth of thresh-old limit value (TLV) levels. If odors are of concern, detection sen-sitivity must be at odor threshold levels. Procedures for evaluating odor levels are given in Chapter 13.
When sampling and analytical procedures appropriate to the application have been selected, a pattern of sampling locations and times must be carefully planned. The building and air-handling sys-tem layout and the space occupancy and use patterns must be con-Air Contaminants 12.9 sidered so that representative concentrations will be measured.
Traynor (1987) and Nagda and Rector (1983) offer guidance in planning such surveys.
VOLATILE ORGANIC COMPOUNDS The entire range of organic indoor pollutants has been catego-rized as indicated in Table 7 (WHO 1989). No sharp limits exist between the categories, which are defined by boiling-point ranges.
Volatile organic compounds (VOCs) have attracted considerable attention in nonindustrial environments. They have boiling points in the range approximately 50 to 250°C, and vapor pressures greater than about 10− 3 to 10− 4 mm Hg.
Sources of VOCs include solvents, reagents, and degreasers in industrial environments; and furniture, furnishings, wall and floor finishes, cleaning and maintenance products, and office and hobby activities in nonindustrial environments.
Berglund et al. (1988) found that the sources of VOCs in nonin-dustrial indoor environments are confounded by the variable nature of emissions from potential sources. Emissions of VOCs from indoor sources can be classified by their presence and rate patterns.
For example, emissions are continuous and regular from building materials and furnishings (e.g., carpet and composite-wood furni-ture); whereas emissions from other sources can be continuous but irregular (e.g., paints used in renovation work), intermittent and regular (e.g., VOCs in combustion products from gas stoves or cleaning products), or intermittent and irregular (e.g., VOCs from carpet shampoos) (Morey and Singh 1991).
Many “wet” emission sources (paints and adhesives) have very high emission rates immediately after application, and rates drop quite steeply with time until the product has cured or dried. New “dry” materials (carpets, wall coverings, and furnishings) also emit chemicals at higher rates until aged. Decay of these elevated VOC concentrations to what can be considered normal constant-source levels can take weeks to months, depending on emission rates, sur-face areas of materials, and ventilation protocols. Renovation activ-ities can cause similar increases of somewhat lower magnitude. The total VOC concentration in new office buildings at the time of initial occupancy can be 50 to 100 times that present in outdoor air (Shel-don et al. 1988a, 1988b). In new office buildings with adequate out-door air ventilation, these ratios often fall to less than 5:1 after 4 or 5 months of aging. In older buildings with continuous, regular, and irregular emission sources, indoor/outdoor ratios of total VOCs may vary from nearly 1:1 when maximum amounts of outdoor air are being used in HVAC systems to greater than 10:1 during winter and summer months when minimum amounts of outdoor air are being used (Morey and Jenkins 1989, Morey and Singh 1991).
While direct VOC emissions are usually the primary source of VOCs in a space, some materials act as sinks for emissions and then become secondary sources as they reemit adsorbed chemicals (Ber-glund et al. 1988). Adsorption may lower the peak concentrations achieved, but the subsequent desorption prolongs the presence of indoor air pollutants. Sink materials include carpet, fabric partitions, and other fleecy materials, as well as ceiling tiles and wallboard. The type of material and compound affects the rate of adsorption and des-orption (Colombo et al. 1991). Indoor air quality models using empirically derived adsorption and desorption rates have been devel-oped to predict the behavior of sinks. Experiments conducted in an IAQ test house confirmed the importance of sinks when trying to control the level of indoor VOCs (Tichenor et al. 1991). Longer peri-ods of increased ventilation lessen sink and reemission effects.
Because of the large number of VOCs usually found indoors and the impossibility of identifying all of them in samples, the concept of total VOC or TVOC was developed. Some researchers have used it to represent the sum of all detected VOCs. TVOC concentrations Table 5 Gaseous Contaminant Sampling Techniques Techniquea Advantages Disadvantages 1. Direct flow to detectors Real-time readout, continuous monitoring possible Several pollutants possible with one sample (when coupled with chromatograph, spectroscope, or multiple detectors) Average concentration must be determined by integration No preconcentration possible before detector; sensitivity may be inadequate On-site equipment complicated, expensive, intrusive 2. Colorimetric detector tubes Very simple, relatively inexpensive equipment and materials Immediate readout Integration over time Rather long sampling period normally required One pollutant per sample Relatively high detection limit Poor precision 3. Capture by pumped flow through solid absorbent; subsequent desorption for concentration measurement On-site sampling equipment relatively simple and inexpensive Preconcentration and integration over time inherent in method Several pollutants possible with one sample Sampling media and desorption techniques are compound-specific Interaction between captured compounds and between compounds and sampling media; bias may result Gives only average over sampling period, no peaks 4. Collection in evacuated containers Very simple on-site equipment No pump (silent) Several pollutants possible with one sample Gives only average over sampling period, no peaks 5. Collection in nonrigid containers (plastic bags) held in an evacuated box Simple, inexpensive on-site equipment (pumps required) Several pollutants possible with one sample Cannot hold same pollutants 6. Cryogenic condensation Wide variety of organic pollutants can be captured Minimal problems with interferences and media interaction Several pollutants possible with one sample Water vapor interference 7. Passive diffusional samplers Simple, unobtrusive, inexpensive No pumps; mobile; may be worn by occupants to determine average exposure Gives only average over sampling period, no peaks 8. Liquid impingers (bubblers) Integration over time Several pollutants possible with one sample May be noisy Sources: NIOSH (1977, 1994), Lodge (1988), Taylor et al. (1977), and ATC (1990).
aAll techniques except 1 and 2 require laboratory work after completion of field sampling. Only technique 1 is adaptable to continuous monitoring and able to detect short-term excursions.
12.10 2001 ASHRAE Fundamentals Handbook (SI) can also be determined by using methods such as photoionization and flame ionization that do not separate chemicals. All methods for TVOC determination are intrinsically of low to moderate accuracy due to variations in detector response to different classes of VOCs.
Both theoretical and practical limitations of the TVOC approach have been discussed (Hodgson 1995, Otson and Fellin 1993).
Wallace et al. (1991) showed that individual VOC concentra-tions in homes and buildings are 2 to 5 times those of outdoors, and personal TVOC exposures were estimated to be 2 to 3 times greater than general indoor air concentrations. Normal daily activities car-ried out by individuals are the cause of these higher personal expo-sures. Personal activities frequently bring individuals close to air contaminant sources. The degree of exposure also depends on how air flows around the body due to convective forces, air turbulence, and obstructions nearby (Rodes et al. 1991).
Individual organic compounds seldom exceed 0.05 mg/m3 (50 µg/m3) in air. An upper extreme average concentration of TVOCs in normally occupied houses is approximately 20 mg/m3.
The Large Buildings Study by the U.S. EPA (Brightman et al. 1996) developed the VOC sample target list shown in Table 8 to identify common VOCs that should be measured. Lists of common indoor VOCs prepared by other organizations are similar.
A VOC enrichment factor (VEF) has been described to quantify VOC concentrations due to building sources beyond those expected as bioeffluents (Batterman and Peng 1995). A VEF > 1 describes the overabundance, while a VEF < 1 denotes the depletion of VOCs compared to VOCs expected as bioeffluents. Bioeffluents alone should produce a VEF of 1. The VEF ranged from 0.6 to 17.1 for the 20 office building studies reviewed.
Because chlorofluorocarbons (CFCs) are hydrocarbons with some hydrogen atoms replaced by chlorine, bromine, and fluorine atoms, they are classed as organic chemicals. They have been widely used as heat transfer gases in refrigeration applications, blowing agents, and propellants in aerosol products (including med-ications and consumer products) and as expanders in plastic foams.
CFCs are discussed in Chapter 19 and in ASHRAE Standards 15 and 34. During the 1970s and 1980s, strong evidence was found linking use of CFCs to depletion of the earth’s stratospheric ozone layer. CFCs add chlorine and bromine atoms to the atmosphere, which accelerate the natural ozone destruction rate. Ozone deple-Table 6 Gaseous Contaminant Concentration Measurement Methods Method Description Typical Application 1. Gas chromatography (Using the following detectors) Flame ionization Flame photometry Photoionization Electronic capture Mass spectroscopy Separation of gas mixtures by time of passage down an absorption column Change in flame electrical resistance due to ions of pollutant Measures light produced when pollutant is ionized by a flame Measures ion current for ions created by ultraviolet light Radioactively generated electrons attach to pollutant atoms; current measured Pollutant molecules are charged, passed through electrostatic magnetic fields in a vacuum; path curvature depends on mass of molecule, allowing separation and counting of each type Volatile, nonpolar organics Sulfur, phosphorous compounds Most organics (except methane) Halogenated organics Nitrogenated organics Volatile organics 2. Infrared spectroscopy and Fourier transform infrared (FTIR) spectroscopy Absorption of infrared light by pollutant gas in a transmission cell; a range of wavelengths is used, allowing identification and measurement of individual pollutants Acid gases Many organics 3. High-performance liquid chromatography (HPLC) Pollutant is captured in a liquid, which is then passed through a liquid chromatograph (analogous to a gas chromatograph) Aldehydes, ketones Phosgene Nitrosamines Cresol, phenol 4. Colorimetry Chemical reaction with pollutant in solution yields a colored product whose light absorption is measured Ozone Oxides of nitrogen Formaldehyde 5. Fluorescence and pulsed fluorescence Pollutant atoms are stimulated by a monochromatic light beam, often ultraviolet; they emit light at characteristic fluorescent wavelengths, whose intensity is measured Sulfur dioxide Carbon monoxide 6. Chemiluminescence Reaction (usually with a specific injected gas) results in photon emission proportional to concentration Ozone Nitrogen compounds Several organics 7. Electrochemical Pollutant is bubbled through reagent/water solution, changing its conductivity or generating a voltage Ozone Hydrogen sulfide Acid gases 8. Titration Pollutant is absorbed into water Acid gases 9. Ultraviolet absorption Absorption of UV light by a cell through which the polluted air passes Ozone Aromatics Sulfur dioxide Oxides of nitrogen Carbon monoxide 10. Atomic absorption Contaminant is burned in a hydrogen flame; a light beam with a spectral line specific to the pollutant is passed through the flame; optical absorption of the beam is measured Mercury vapor Sources: NIOSH (1977,1994), Lodge (1988), Taylor et al (1977), and ATC (1990).
Air Contaminants 12.11 tion is linked to increases in the amount of ultraviolet radiation (UV-B) reaching the earth’s surface. Increasing UV-B can adversely affect human health, most notably by causing malignant melanoma and cataracts, harm the environment, and decrease crop production.
As a result, CFCs are now being phased out. New refrigerants with lower ozone reduction potential are being used, including hydro-chlorofluorocarbons (HCFCs). In addition, there has been a return to refrigerants such as ammonia and propellants such as butane and hexane.
Exposure to CFCs and HCFCs occurs mainly through inhalation and can occur from leaks in refrigeration equipment or during ser-vicing of HVAC systems.
Controlling Exposures to Volatile Organic Compounds Much can be done to reduce building occupants’ exposures to emissions of VOCs from building materials and products and to prevent outdoor VOCs from being brought into buildings. The control principles of substitution, isolation, and ventilation apply.
Control measures include careful planning; specifications; and selection, modification, and treatment of products, as well as spe-cial installation procedures and proper ventilation system opera-tion. Chapter 44 of the 1999 ASHRAE Handbook —Applications provides full details.
Levin (1989, 1991) has written extensively about designing new buildings for good indoor air quality. Reducing VOC emissions by careful selection and installation of building materials and furnish-ings is a most effective strategy for controlling IAQ. Advances in product formulation and emission testing are leading to products claimed to be low-polluting, nontoxic, and environmentally safe.
Requiring the submission of emission testing data by manufacturers for building products, whether for a new building, for a building renovation or remodeling, or for substitution of a consumable prod-uct (housekeeping supplies), is becoming accepted practice. Elimi-nating the sources of VOCs prevents them from becoming a problem in the first place.
Gas-phase air filtration has been applied to control industrial gaseous contaminants for many years. The application of this tech-nology to nonindustrial building HVAC is of interest for improving IAQ, whether it is to provide ventilation without the need to use more outdoor air or to help clean poor-quality outdoor air. Activated carbon and potassium permanganate-impregnated alumina are effective adsorbents that can be used, based on the contaminant mixture present (Liu and Huza 1995, Muller and England 1995, VanOsdell and Sparks 1995). Portable air cleaners with sorbent sec-tions are only marginally effective (Shaughnessy et al. 1994). Pho-tocatalytic reactors capable of destroying VOCs are being studied (Peral et al. 1997). These reactors use ultraviolet light and a catalytic surface, such as titanium dioxide, to convert organic pollutants to CO2 and water.
Ventilation has traditionally been considered the primary means for controlling indoor VOC contaminants. Dilution ventilation is an effective way to control normal constant-emission sources present in buildings, assuming no unusually strong sources. Compliance with ASHRAE Standard 62 should satisfy indoor dilution ventila-tion requirements. Local exhaust ventilation is effective for control-ling known, unavoidable point emissions sources. It is prudent to isolate office machines, such as photocopiers and laser printers, food service equipment, such as microwave ovens and coffee mak-ers, and work areas, such as graphics and photographic labs, using dedicated local exhaust systems that vent to the outside and away from outdoor air intakes.
A good ventilation protocol during renovation or remodeling includes using a single-pass (100% outdoor air) system during and at the finish of these activities, continuing until enough time has passed to lower emitted concentrations to near background. This practice minimizes sink effects and secondary emissions.
Prudent practice and administrative control should be used to minimize the generation of VOCs in indoor air during occupied hours whenever possible. Scheduling the use of volatile organic products, housekeeping activities, and pesticide application when occupant density is lowest should be considered. VOC-containing supplies should be stored in well-ventilated areas other than HVAC mechanical rooms or plenums.
INORGANIC GASES Several inorganic gases are of concern because of their effects on human health and comfort and on materials. These include carbon dioxide, carbon monoxide, oxides of nitrogen, sulfur dioxide, ozone, and ammonia. Most have both outdoor and indoor sources.
Carbon dioxide (CO2) or carbonic acid gas is produced by human respiration. It is not normally considered to be a toxic air contaminant, but it can be a simple asphyxiant (by oxygen displace-ment) in confined spaces such as submarines. CO2 is found in the ambient environment at 330 to 370 ppm. Levels in the urban envi-ronment may be higher due to emissions from gasoline and, more often, diesel engines. Measurement of CO2 in occupied spaces has been widely used to evaluate the amount of outdoor air supplied to indoor spaces. In ASHRAE Standard 62, a level of 1000 ppm (or 650 ppm above outdoor air) has been suggested as being represen-tative of delivery rates of 7 L/s per person of outside air when CO2 is measured at equilibrium concentrations and at occupant densities of 10 people per 100 m2 of floor space. Measuring CO2 level before it has reached steady-state conditions can lead to inaccurate conclu-sions regarding the amount of outside air used in the building.
Table 7 Classification of Indoor Organic Pollutants Description Abbreviation Boiling Point Range, °C Very volatile (gaseous) organic compounds VVOC <0 to 50–100 Volatile organic compounds VOC 50–100 to 240–260 Semivolatile organics (pesticides, polynuclear aromatic compounds, plasticizers) SVOC 240–260 to 380–400 Source: WHO (1989).
Note: Polar compounds and VOCs with higher relative molecular masses appear at the higher end of each boiling-point range.
Table 8 Example Sample Contaminant Target List benzene styrene m-,p-xylene p-dichlorobenzene 1,2,4-trimethylbenzene n-undecane n-octane n-nonane n-decane ethylacetate n-dodecane dichloromethane butylacetate 1,1,1-trichloroethane chloroform tetrachloroethylene trichloroethylene carbon disulfide trichlorofluoromethane acetone dimethyl disulfide 2-butanone 4-methyl-2-pentanone methyl tertiary butyl ether limonene naphthalene α-,β-pinene 4-phenylcyclohexene propane butane butyl cellosolve ethanol isopropanol phenol formaldehyde siloxanes toluene 12.12 2001 ASHRAE Fundamentals Handbook (SI) Carbon monoxide (CO) is an odorless, colorless, and taste-less gas produced by the incomplete combustion of hydrocar-bons. It is a common ambient air pollutant and is very toxic.
Common indoor sources of CO include gas stoves, kerosene lan-terns and heaters, mainstream and sidestream tobacco smoke, woodstoves, and unvented or improperly vented combustion sources. Building makeup air intakes located at street level or near parking garages can entrain CO from automobiles and carry it to the indoor environment.
The major predictors of indoor CO concentrations are indoor fossil fuel sources, such as gas furnaces, hot water heaters, and other combustion appliances; attached garages; and weather inversions.
Levels in homes only rarely exceed 5 ppm. In one sample of ran-domly selected homes, 10% failed a backdrafting test (Conibear et al. 1996). Under backdrafting conditions, indoor CO sources may contribute to much higher, dangerous levels of CO.
Oxides of nitrogen (NOx) indoors result mainly from cooking appliances, pilot lights, and unvented heaters. Sources generating carbon monoxide (CO) often produce nitric oxide (NO) as well.
Underground or attached parking garages can also contribute to indoor concentrations of NOx. An unvented gas cookstove contrib-utes approximately 0.025 ppm of nitrogen dioxide (NO2) to a home.
During cooking, 0.2 ppm to 0.4 ppm peak levels may be reached (Samet et al. 1987). Ambient air pollution from vehicle exhausts in urban locations can contribute NOx to the indoor environment in makeup air. Oxides of nitrogen also are present in mainstream and sidestream tobacco smoke. Nitric oxide and nitrogen dioxide are of most concern.
Sulfur dioxide (SO2) can result from the emissions of kerosene space heaters; the combustion of fossil fuels such as coal, heating oil, and gasoline; or burning any material containing sulfur. As a result, sulfur dioxide is a common ambient air pollutant in many urban areas.
Ozone (O3) is a photochemical oxidant that forms at ground level when hydrocarbons and oxides of nitrogen react with ultravi-olet radiation in sunlight to produce photochemical smog. Ozone can be emitted by the electrical or coronal discharges from office equipment including laser printers and photocopiers. It can also be generated when ozone-generating devices—often marketed as por-table air cleaners and ionizers—are used in the indoor environment (Esswein and Boeniger 1994).
Controlling Exposures to Inorganic Gases Three methods of control for inorganic gaseous contaminants should be considered: source control, ventilation control, and removal by filters. Source control involves limiting (or removing) the source of the problem; for example, gas cookstoves should not be used for space heating (often a problem in low-income urban res-idences). Another example is limiting automobile parking around building makeup air intakes. Source control should always be the primary consideration. But source control is not always feasible when there are many diverse contaminant sources as in new build-ings where the building itself or building furnishings may be the prime contributors to the problem.
Ventilation control involves bringing clean dilution air into the occupied space or directly exhausting air contaminants at the point of generation. ASHRAE Standard 62 provides guidance in applying the Ventilation Rate Procedure and the Indoor Air Quality Proce-dure for ventilation control.
Where neither source control nor ventilation control appear likely to control gaseous air contaminants, air filtration should be investigated. Gas-phase air filtration involves dry scrubbing to remove contaminants by adsorption onto several sorbents, includ-ing granular activated carbon (GAC), potassium permanganate-impregnated alumina (PIA), and alkaline-impregnated carbon fil-ters. Muller and England (1995), Liu and Huza (1995), VanOsdell (1995), and Coutant et al. (1994) review various filtration proce-dures.
No one media is effective for the broad range of gaseous contam-inants found indoors. Granular activated charcoal is generally an agent of choice for nonpolar compounds and is a suitable choice for O3 and NO2 but not for SOx and NO. Permanganate-impregnated alumina is more appropriate for SOx and NO.
Ozone can be best controlled by local exhaust ventilation for demonstrated sources of ozone, such as photocopiers and equip-ment creating coronal discharges. Routine cleaning of attractor plates in an electrostatic precipitator and ensuring adequate prefil-ters can reduce ozone generation and limit arcing in this type of par-ticle removal equipment. The use of ozone-generating devices as a means of air cleaning or purification has not been documented as a prudent means of air contaminant control considering the potential health effects of the use of ozone indoors (Esswein and Boeniger 1994). The Food and Drug Administration (FDA 1990) specifically limits the use of ozone in concentrations greater than 50 ppb in areas intended for continuous occupancy, such as residences, offices, schools, and hospitals.
Carbon monoxide exposure control strategies primarily involve identification and control of CO emissions directly at their source. Local exhaust ventilation is an appropriate and effective control in most cases. For example, automobile repair garages commonly use a tailpipe exhaust extension to control the CO exposure of the mechanics working in the repair bays. Relocating building makeup air intakes or limiting vehicle access are reason-able means to prevent entrainment of automobile exhausts into building HVAC systems.
Carbon monoxide, however, is a common pollutant of ambient air. As a result, direct control by dilution may not be feasible if ambient air is heavily contaminated with CO. Diesel or natural gas may be substituted for gasoline engines to reduce CO where spe-cific sources from engine exhaust are identified or are a concern.
Adequate venting of any combustion sources is critical to prevent the buildup of CO indoors. CO may be monitored by a properly calibrated, direct-reading CO monitor, colorimetric indicator tubes, or passive diffusion sampling badges.
Exposure controls for carbon dioxide are generally limited to situations where exposure concentrations are expected to exceed 3 to 5%. CO2 is not encountered at levels harmful to humans in the ambient environment. It is normally present at 350 to 375 ppm, and slightly higher in congested cities. With the exception of an intentional or accidental CO2 “dump” from a fire suppression sys-tem or in a dry ice manufacturing facility, CO2 is not encountered in significant concentrations that require specific engineering con-trols. However, CO2 is denser than air and can persist for some time in low areas such as trenches, depressions, and pits. This characteristic creates a simple asphyxiation hazard because CO2 displaces oxygen.
SPECIAL TYPES OF AIR CONTAMINANTS OUTDOOR AIR CONTAMINANTS The total amount of suspended particulate matter in the atmo-sphere can influence the loading rate of air filters and their selection.
The amount of soot that falls in U.S. cities ranges from 7 to 70 Mg/km2 per month. Soot fall data indicate effectiveness of smoke abatement and proper combustion methods and serve as comparative indices of such control programs. However, the data are of limited value to the ventilating and air-conditioning engineer, since they do not accurately represent airborne soot concentrations.
Concentrations of outdoor pollutants are important, as they may determine indoor concentrations in the absence of indoor sources.
Table 9 presents typical urban outdoor concentrations of some com-Air Contaminants 12.13 mon gaseous pollutants. Higher levels might be found if the build-ing under consideration were located near a major source of contamination such as a power plant, a refinery, or a sewage treat-ment plant. Note that levels of sulfur dioxide and nitrogen dioxide, which are often attached to particles, may be reduced by about half by building filtration systems. Also, ozone is a reactive gas that can be significantly reduced by contact with ventilation system compo-nents (Weschler et al. 1989).
The U.S. Environmental Protection Agency has identified sev-eral important outdoor contaminants as criteria pollutants. The list includes suspended particulate matter, lead particulate matter, ozone, nitrogen dioxide, sulfur dioxide, carbon monoxide, and total hydrocarbons. Standards have been set for these contaminants, as shown in Table 10, and levels measured at a large number of loca-tions in the United States are published by the EPA each year (40 CFR 50).
Daily concentrations of VOCs in outdoor air can vary drastically (Ekberg 1994). These variations are due to vehicle traffic density, wind direction, industrial emissions, and photochemical reactions.
Table 9 Typical Outdoor Concentrations of Selected Gaseous Air Pollutants Pollutant Typical Concentration, µg/m3 Pollutant Typical Concentration, µg/m3 Acetaldehyde 20 Methylene chloride 2.4 Acetone 3 Nitric acid 6 Ammonia 1.2 Nitric oxide 10 Benzene 8 Nitrogen dioxide 51 2-Butanone (MEK) 0.3 Ozone 40 Carbon dioxide 612 000 a Phenol 20 Carbon monoxide 3 000 Propane 18 Carbon disulfide 310 Sulfur dioxide 240 Carbon tetrachloride 2 Sulfuric acid 6 Chloroform 1 Tetrachloroethylene 2.5 Ethylene dichloride 10 Toluene 20 Formaldehyde 20 1,1,1-Trichloroethane 4 n-Heptane 29 Trichloroethylene 15 Mercury (vapor) 0.005 Vinyl chloride monomer 0.8 Methane 1 100 Methyl chloride 9 Xylene 10 aNormal concentration of carbon dioxide in air. The concentration in occupied spaces should be maintained at no greater than three times this level (1000 ppm).
Sources: Braman and Shelley (1980), Casserly and O’Hara (1990), Chan et al. (1990), Cohen et al. (1989), Coy (1987), Fung and Wright (1990), Hakov et al. (1987), Hartwell et al. (1985), Hollowell et al. (1982), Lonnemann et al. (1974), McGrath and Stele (1987), Nelson et al. (1987), Sandalls and Penkett (1977), Shah and Singh (1988), Singh et al. (1981), Wallace et al. (1983), and Weschler and Shields (1989).
Table 10 Primary Ambient Air Quality Standards for the United States Contaminant Long Term Short Term Concentration, µg/m3 Averaging Period Concentration, µg/m3 Averaging Period, h Sulfur dioxide 80 1 year 365 24 Carbon monoxide 10 000 40 000 1 8 Nitrogen dioxide 100 1 year Ozonea 235 1 Hydrocarbons Total particulate (PM10 )b 75 1 year 260 24 Lead particulate 1.5 3 months aStandard is met when the number of days per year with maximum hour-period concen-tration above 235 µg/m3 is less than one.
bPM10 = particulates below 10 µm diameter Table 11 Characteristics of Selected Gaseous Air Pollutants Pollutant Chemical and Physical Properties Familya BPb,°C Mc Acetaldehyde 15 21 44 Acetone 16 56 58 Acetonitrile 18 82 41 Acrolein 15 52 56 Acrylonitrile 18 77 53 Allyl chloride 12 44 77 Ammonia 5 –33 17 Benzene 19 80 78 Benzyl chloride 12 179 127 2-Butanone (MEK) 16 79 72 Carbon dioxide 4 –78 44 Carbon monoxide 3 –192 28 Carbon disulfide 25 46 76 Carbon tetrachloride 11 77 154 Chlorine 1 –34 71 Chloroform 11 124 119 Chloroprene 12 120 89 p-Cresol 13 305 108 Dichlorodifluoromethane 10 –30 121 Dioxane 21 100 68 Ethylene dibromide 12 131 188 Ethylene dichloride 12 84 99 Ethylene oxide 21 10 44 Formaldehyde 15 97 30 n-Heptane 7 98 100 Hydrogen chloride 4 –121 37 Hydrogen cyanide 18 26 27 Hydrogen fluoride 4 19 20 Hydrogen sulfide 4 –60 34 Mercury 1 357 201 Methane 7 –164 16 Methanol 13 64 32 Methyl chloride 12 74 133 Methylene chloride 12 40 85 Nitric acid 4 84 63 Nitric oxide 2 –152 30 Nitrogen dioxide 2 21 46 Ozone 2 –112 48 Phenol 13 182 94 Phosgene 27 8 90 Propane 7 –42 44 Sulfur dioxide 4 –10 64 Sulfuric acid 4 270 98 Tetrachloroethane 11 146 108 Tetrachloroethylene 11 121 166 o-Toluidene 23 199 107 Toluene 19 111 92 Toluene diisocyanate 18 251 174 1,1,1-Trichloroethane 11 113 133 Trichloroethylene 11 87 131 Vinyl chloride monomer 24 –14 63 Xylene 19 137 106 Source: See Table 2, Chapter 44, 1999 ASHRAE Handbook.
aChemical family numbers are as given in Table 4.
bBP = boiling point at 101.325 kPa (1 atm) pressure cM = molecular mass 12.14 2001 ASHRAE Fundamentals Handbook (SI) INDUSTRIAL AIR CONTAMINANTS Many industrial processes produce significant quantities of air contaminants in the form of dusts, fumes, smokes, mists, vapors, and gases. Table 11 lists chemical and physical properties of some common gaseous industrial contaminants.
Particulate and gaseous contaminants are best controlled at the source so that they are neither dispersed through the factory nor allowed to increase to toxic concentration levels. Dilution ventila-tion is much less effective than local exhaust for reducing contam-ination from point source emissions and is used for control only when sources are distributed and not amenable to capture by an exhaust hood. For sources generating high levels of contaminants, it may also be necessary to provide equipment that reduces the amount of material discharged to the atmosphere (e.g., a dust col-lector for particulate contaminants and/or a high dwell time gas-phase media bed for solvent vapors). Control methods are covered in Chapters 24 and 25 of the 2000 ASHRAE Handbook—Systems and Equipment and Chapters 29 and 44 of the 1999 ASHRAE Hand-book—Applications.
Zero concentration of all contaminants is not economically feasible. Absolute control of all contaminants cannot be main-tained, and workers can assimilate small quantities of various toxic materials without injury. Industrial hygiene science is based on the fact that most air contaminants become toxic only if their concentration exceeds a maximum allowable limit for a specified period. Allowable limits in industrial environments are covered in Chapter 9.
Although the immediately dangerous to life and health (IDLH) toxicity limit is rarely a factor in HVAC design, HVAC engineers should consider it when deciding how much recirculation is safe in a given system. Ventilation airflow must never be so low that the concentration of any gaseous contaminant could rise to the IDLH level. Another toxic effect that may influence design is the loss of sensory acuity due to gaseous contaminant exposure. For example, high concentrations of hydrogen sulfide, which has a very unpleas-ant odor, effectively eliminate a person’s ability to smell the gas.
Carbon monoxide, which has no odor to alert people to its presence, affects psychomotor responses and could be a problem in areas such as air traffic control towers. Clearly, waste anesthetic gases should not be allowed to reach levels in operating suites such that the alert-ness of any of the personnel is affected. NIOSH recommendations are frequently based on such subtle effects.
NONINDUSTRIAL INDOOR AIR CONTAMINANTS Indoor air quality in residences, offices, and other indoor, non-industrial environments has become a widespread concern (Spen-gler et al. 1982, NRC 1981). Exposure to indoor pollutants can be as important as exposure to outdoor pollutants because a large portion of the population spends up to 90% of their time indoors and because indoor pollutant concentrations are frequently higher than corresponding outdoor contaminant levels.
Symptoms of exposure include coughing; sneezing; eye, throat, and skin irritation; nausea; breathlessness; drowsiness; headaches; and depression. Rask (1988) suggests that when 20% of a single building’s occupants suffer such irritations, the structure is suffering from sick building syndrome (SBS). Case studies of such occur-rences have consisted of analyses of questionnaires submitted to building occupants, measurements of contaminant levels, or both of these. Some attempts to relate irritations to gaseous contaminant concentrations are reported (Lamm 1986, Cain et al. 1986, Berglund et al. 1986, Molhave et al. 1982). The correlation of reported com-plaints with gaseous pollutant concentrations is not strong; many factors affect these less serious responses to pollution. In general, physical irritation does not occur at odor threshold concentrations.
Characterization of indoor air quality has been the subject of numerous recent studies. ASHRAE Indoor Air Quality (IAQ) Con-ference Proceedings discuss indoor air quality problems and some practical controls. ASHRAE Standard 62 addresses many indoor air quality concerns. Table 12 illustrates the sources, levels, and indoor-to-outdoor concentration ratios of several contaminants found in indoor environments. Chapter 9 has further information on indoor health issues.
Table 12 Sources, Possible Concentrations, and Indoor-to-Outdoor Concentration Ratios of Some Indoor Pollutants Pollutant Sources of Indoor Pollution Possible Indoor Concentration I/O Concentration Ratio Location Carbon monoxide Combustion equipment, engines, faulty heating systems 100 mg/kg >>1 Skating rinks, offices, homes, cars, shops Respirable particles Stoves, fireplaces, cigarettes, condensation of volatiles, aerosol sprays, resuspension, cooking 100 to 500 µg/m3 >>1 Homes, offices, cars, public facilities, bars, restaurants Organic vapors Combustion, solvents, resin products, pesticides, aerosol sprays NA >1 Homes, restaurants, public facilities, offices, hospitals Nitrogen dioxide Combustion, gas stoves, water heaters, driers, cigarettes, engines 200 to 1000 µg/m3 >>1 Homes, skating rinks Sulfur dioxide Heating system 20 µg/m3 <1 Removal inside Total suspended particles without smoking Combustion, resuspension, heating system 100 µg/m3 1 Homes, offices, transportation, restaurants Sulfate Matches, gas stoves 5 µg/m3 <1 Removal inside Formaldehyde Insulation, product binders, particleboard 0.05 to 1.0 mg/kg >>1 Homes, offices Radon and progeny Building materials, groundwater, soil 0.1 to 100 nCi/m3 >>1 Homes, buildings Asbestos Fireproofing <106 fiber/m3 1 Homes, schools, offices Mineral and synthetic fibers Products, cloth, rugs, wallboard NA — Homes, schools, offices Carbon dioxide Combustion, humans, pets 3000 mg/kg >>1 Homes, schools, offices Viable organisms Humans, pets, rodents, insects, plants, fungi, humidifiers, air conditioners NA >1 Homes, hospitals, schools, offices, public facilities Ozone Electric arcing Ultraviolet light sources 20 µg/kg 200 µg/kg <1 >1 Airplanes Offices Source: NRC (1981).
aConcentrations listed are only those reported indoors. Both higher and lower concentrations have been measured. No averaging times are given. NA indicates that it is not appro-priate to list a concentration.
Air Contaminants 12.15 A knowledge of sources frequently present in different types of buildings can be useful during investigations of the causes of SBS.
Common nonindustrial indoor sources are discussed in some detail below. Technical advances have allowed generation rates to be mea-sured for a number of these sources. These rates are necessary inputs for design of control equipment, and full details are given in Chapter 44 of the 1999 ASHRAE Handbook—Applications.
Building materials and furnishing sources have been well stud-ied. Particleboard, which is usually made from wood chips bonded with a phenol-formaldehyde or other resin, is widely used in current construction, especially for mobile homes, carpet underlay, and case goods. These materials, along with ceiling tiles, carpeting, wall cov-erings, office partitions, adhesives, and paint finishes emit formal-dehyde and other VOCs. Latex paints containing mercury have been shown to emit mercury vapor. While the emission rates for these materials decline steadily with age, the half-life of emissions is surprisingly long. Black and Bayer (1986), Nelms et al. (1986), and Molhave et al. (1982) report on these sources.
Ventilation systems may be a source of VOCs (Molhave and Thorsen 1990). The interior of the HVAC system can have large areas of porous material used as acoustical liner that can adsorb odorous compounds. This material can also hold nutrients and, with moisture, can become a reservoir for microorganisms. Microbial contaminants produce characteristic VOCs, called microbial VOCs (MVOCs), associated with their metabolism. Other HVAC compo-nents, such as condensate drain pans, fouled cooling coils, and some filter media, may support microbiological life. Deodorants, seal-ants, and encapsulants are also sources of VOCs in HVAC systems.
Equipment sources in commercial and residential spaces have generation rates that are usually substantially lower than in the industrial environment. As these sources are rarely hooded, emis-sions go directly to the occupants. In commercial spaces, the chief sources of gaseous contaminants are office equipment, including dry-process copiers (ozone), liquid-process copiers (VOCs), diazo printers (ammonia and related compounds), carbonless copy paper (formaldehyde), correction fluids, inks, and adhesives (various VOCs), spray cans, cosmetics, and so forth (Miksch et al. 1982).
Medical and dental activities generate pollutants from the escape of anesthetic gases (nitrous oxide and isoflurene) and from sterilizers (ethylene oxide). The potential for asphyxiation is always a concern when compressed gases are present, even if that gas is nitrogen.
In residences, the main sources of equipment-derived pollutants are gas ranges, wood stoves, and kerosene heaters. Venting is help-ful, but some pollutants escape into the occupied area. The pollutant contribution by gas ranges is somewhat mitigated by the fact that they operate for shorter periods than heaters. The same is true of showers, which can contribute to radon and halocarbon concentra-tions indoors.
Cleaning agents and other consumer products can act as con-taminant sources. Commonly used liquid detergents, waxes, pol-ishes, spot removers, and cosmetics contain organic solvents that volatilize slowly or quickly. Mothballs and other pest control agents emit organic vapors. Knoeppel and Schauenburg (1989), Black and Bayer (1986), and Tichenor (1989) report data on the release of these volatile organic compounds (VOCs). Field studies have shown that such products contribute significantly to indoor pollu-tion; however, a large variety of compounds is in use, and few stud-ies have been made that allow calculation of typical emission rates.
Pesticides, both those applied indoors and those applied outdoors to control termites, also pollute building interiors.
Tobacco smoke is a prevalent and potent source of indoor air pollutants. Almost all tobacco smoke arises from cigarette smoking.
Environmental tobacco smoke (ETS), sometimes called second-hand smoke, is the aged and diluted combination of sidestream smoke (the smoke from the lit end of a cigarette and the smoke that escapes from the filter between puffs) and mainstream smoke (the smoke exhaled by a smoker). Emission factors for ETS compo-nents, the ratio of ETS components to marker compounds, and apportionment of ETS components in indoor air are reported in the literature by Heavner et al. (1996), Hodgson et al. (1996), Martin et al. (1996), and Nelson et al. (1994).
Occupants, both humans and animals, emit a wide array of pol-lutants by breath, sweat, and flatus. Some of these emissions are conversions from solids or liquids within the body. Many volatile organics emitted are, however, reemissions of pollutants inhaled earlier, with the tracheobronchial system acting like a physical adsorber.
Floor dust, which is different from the dust in the air, has been found to be a sink (adsorption medium) and secondary emission source for VOCs. Floor dust is a mixture of organic and inorganic particles, hair and skin scales, and textile fibers. The fiber portion of floor dust has been shown to contain 169 mg/kg TVOC, and the par-ticle portion 148 mg/kg (Gyntelberg et al. 1994). These VOCs were correlated to the prevalence of irritative (sore throat) and cognitive (concentration problems) symptoms among building occupants.
One hundred eighty-eight compounds were identified from thermal desorption of office dust at 121°C (Wilkins et al. 1993). Household dust was found to be similar in composition (Wolkoff and Wilkins 1994).
Contaminants from other sources include chloroform from water; tetrachloroethylene and 1,1,1-trichloroethane from cleaning solvents; methylene chloride from paint strippers, fresheners, clean-ers, and polishers; a-pinene and limonene from floor waxes; and 1-methoxy-2-propanol from spray carpet cleaners. Formaldehyde, a major VOC, has many sources, but pressed wood products appear to be the most significant.
FLAMMABLE GASES AND VAPORS The use of flammable materials is widespread. Flammable gases and vapors (NFPA Standard 325M) can be found in sewage treat-ment plants, sewage and utility tunnels, dry-cleaning plants, auto-mobile garages, and industrial finishing process plants.
A flammable liquid’s vapor pressure and volatility or rate of evaporation determine its ability to form an explosive mixture.
These properties can be expressed by the flash point, which is the temperature to which a combustible liquid must be heated to pro-duce a flash when a small flame is passed across the surface of the liquid. Depending on the test methods, either the open cup or closed cup flash point may be listed. The higher the flash point, the more safely the liquid can be handled. Liquids with flash points under 21°C should be regarded as highly flammable.
In addition to having a low flash point, the air-vapor or air-gas mixture must have a concentration in the explosive range before it can be ignited. The explosive range is the range between the upper and lower explosive limits, expressed as percent by volume in air.
Concentrations of material above the higher range or below the lower range will not explode. The range for many chemicals is found in National Fire Code bulletins published by the National Fire Protection Association (NFPA). Some representative limits of flammability are listed in Table 13.
In designing ventilation systems to control flammable gases and vapors, the engineer must consider the following: Most safety authorities and fire underwriters prefer to limit con-centrations to 20 to 25% of the lower explosive limit of a material.
The resulting safety factor of 4 or 5 allows latitude for imperfec-tions in air distribution and variations of temperature or mixture and guards against unpredictable or unrecognized sources of igni-tion. Operation at concentrations above the upper explosive limit should be resorted to only in rare instances. To reach the upper explosive limit, the flammable gas or vapor must pass through the active explosive range, in which any source of ignition can cause an explosion. In addition, a drop in gas concentration due to unfore-12.16 2001 ASHRAE Fundamentals Handbook (SI) seen dilution or reduced evaporation rate may place a system in the dangerous explosive range.
In occupied places where ventilation is applied for proper health control, the danger of an explosion is minimized. In most instances, flammable gases and vapors are also toxic, and the maximum allow-able concentrations are far below the lower explosive limit of the material. For example, proper ventilation for acetone vapors keeps the concentration below 1000 mg/m3. This is equivalent to 0.1% by mass. The lower explosive limit for acetone is 2.5% by volume.
Proper location of exhaust and supply ventilation equipment depends primarily on how a contaminant is given off and on other problems of the process, and secondarily on the relative density of flammable vapor.
If the specific density of the explosive mixture is the same as that of air, cross drafts, equipment movement, and temperature differen-tials may cause sufficient mixing to produce explosive concentra-tions and disperse these throughout the atmosphere. In reasonably still air, heavier-than-air vapors may pool at floor level. Therefore, the engineer must either provide proper exhaust and supply air pat-terns to control the hazardous material, preferably at its source, or offset the effects of drafts, equipment movement, and convective forces by providing good distribution of exhaust and supply air for general dilution and exhaust. The intake duct should be positioned so that it does not bring in exhaust gases or emissions from ambient sources.
Adequate ventilation minimizes the risk of or prevents fires and explosions and is necessary, regardless of other precautions, such as elimination of the ignition sources, safe building construction, and the use of automatic alarm and extinguisher systems.
Chapter 29 of the 1999 ASHRAE Handbook—Applications gives more details about equipment for control of combustible materials.
COMBUSTIBLE DUSTS Many organic and some mineral dusts can produce dust explo-sions (Hartmann 1958). Often, a primary explosion results from a small amount of dust in suspension that has been exposed to a source of ignition; the pressure and vibration created can dislodge large accumulations of dust on horizontal surfaces, creating a larger secondary explosion.
For ignition, dust clouds require high temperatures and sufficient dust concentration. These temperatures and concentrations and the minimum spark energy can be found in Avallone and Baumeister (1987).
Explosive dusts are potential hazards whenever uncontrolled dust escapes, disperses in the atmosphere or settles on horizontal surfaces such as beams and ledges. Proper exhaust ventilation design involves the principles covered in Chapter 29 of the 1999 ASHRAE Handbook—Applications. The ventilation systems and equipment chosen must prevent the pocketing of dust inside the equipment. When local exhaust ventilation is used, separation equipment should be installed as close to the dust source as possible to prevent transport of dust in the exhaust system.
RADIOACTIVE AIR CONTAMINANTS Radioactive contaminants (Jacobson and Morris 1977) can be particulate or gaseous and are similar to ordinary industrial contam-inants. Many radioactive materials would be chemically toxic if present in high concentrations; however, in most cases, the radioac-tivity necessitates limiting their concentration in air.
Most radioactive air contaminants affect the body when they are absorbed and retained. This is known as the internal radiation haz-ard. Radioactive particulates may settle to the ground, where they contaminate plants and eventually enter the food chain and the human body. Deposited material on the ground increases external radiation exposure. However, except for fallout from nuclear weaponsora serious reactoraccident, suchexposureis insignificant.
Radioactive air contaminants can emit alpha, beta, or gamma rays. The alpha rays penetrate poorly and present no hazard, except when the material is deposited inside or on the body. Beta rays are somewhat more penetrating and can be both an internal and an external hazard. The penetration of gamma rays depends on their energy, which varies from one type of radioactive element or iso-tope to another. A distinction should be made between the radioac-tive material itself and the radiation it gives off. Radioactive particles can be removed from air by devices such as HEPA and ULPA filters, and radioactive gases by impregnated carbon or alu-mina (radioactive iodine) and absorption traps, but the gamma radi-ation from such material is capable of penetrating solid materials.
This distinction is frequently overlooked. The amount of radioac-tive material in air is measured in becquerels per cubic metre (1 bec-querel equals 2.702702 × 10− 11 curies), while the dose of radiation from deposited material is measured in rads.
Radioactive materials present problems that make them distinc-tive. High concentrations of radioactivity can generate enough heat to damage filtration equipment or ignite the material spontaneously.
The concentrations at which most radioactive materials are hazard-ous are much lower than those of ordinary materials; as a result, spe-cial electronic instruments that respond to radioactivity must be used to detect these hazardous levels.
The ventilation engineer faces difficulty in dealing with radioac-tive air contamination because of the extremely low permissible concentrations for radioactive materials. For certain sensitive indus-trial plants, such as those in the photographic industry, contaminants must be kept from entering the plant. If radioactive materials are handled inside the plant, the problem is to collect the contaminated air as close to the source as possible, and then remove the contam-inant from the air with a high degree of efficiency, before releasing it to the outdoors. Filters are generally used for particulate materials, but venturi scrubbers, wet washers, and other devices can be used as prefilters to meet special needs.
The design of equipment and systems for control of radioactive particulates and gases in nuclear laboratories, power plants, and fuel-processing facilities is a highly specialized technology. Careful attention must be given to the reliability, as well as the contaminant-removal ability, of the equipment under the special environmental Table 13 Flammable Limits of Some Gases and Vapors Gas or Vapor Flash Point, °C Flammable Limits, % by Volume Lower Upper Acetone –18 2.6 12.8 Ammonia Gas 16 25 Benzene (benzol) –11 1.3 7.5 n-Butane Gas 1.9 8.5 Carbon disulfide –30 1.3 44 Carbon monoxide Gas 12.5 74 1,2-Dichloroethylene 6 9.7 12 Diethylether –45 1.9 48 Ethyl alcohol 13 4.3 19 Ethylene Gas 3.1 32 Gasoline –43 1.4 7.6 Hydrogen Gas 4.0 75 Hydrogen sulfide Gas 4.3 45 Isopropyl alcohol 11.7 2.0 12 Methyl alcohol 11 7.3 36 Methyl ethyl ketone –6 1.8 10 Natural gas (variable) Gas 3.8 17 Naphtha (benzine) 10 0.9 6.7 Propane Gas 2.2 9.5 Toluene (toluol) 4 1.2 7.1 o-Xylene 32 1.0 6.0 Air Contaminants 12.17 stresses involved. Various publications of the U.S. Department of Energy can provide guidance in this field.
Radon A major source of airborne radioactive exposure to the popula-tion comes from radon. Radon (Rn) is a naturally occurring, chem-ically inert, colorless, odorless, tasteless radioactive gas. It is produced from the radioactive decay of radium, which is formed through several intermediate steps of the decay of uranium and tho-rium. Radon is widely found in the natural environment, as uranium salt precursors are widespread. Radon-222 is the most common iso-tope of radon. Before it decays, radon can move limited distances through very small spaces, such as those between particles of soil and rock, and enter indoor environments (Nazaroff et al. 1988, Tan-ner 1980). Additional but secondary sources of indoor radon include groundwater (radon is quite soluble in water) and radium-containing building materials.
Radon gas enters a house or building primarily through leakage paths in the foundation and is transported by pressure-driven flow.
Entry occurs through cracks, joints, and other holes in concrete foundations; directly through porous concrete blocks; through the joints and openings in crawl space ceilings; and through leakage points in HVAC ductwork that is embedded in slab floors or located in crawl spaces. Pressure-driven flow is the dominant radon entry mechanism in houses with elevated radon concentrations (Nazaroff et al. 1987). Pressure differences are caused by several factors, including the thermal stack effect, wind, and operation of HVAC equipment. In addition to pressure-driven radon entry, Rn can also diffuse directly through substructural materials (e.g., concrete). The diffusive Rn entry rate is often a significant portion of the total entry rate in houses with low Rn concentrations.
Measurement. Indoor concentrations of radon can vary hourly, daily, and seasonally, in some cases by as much as a factor of 10 to 20 on a daily basis (Turk et al. 1990). Thus, long-term measure-ments (3 months to 1 year) made during normal home activities gen-erally provide more reliable estimates of the average indoor concentration than do short-term measurements. Two techniques widely used for homeowner measurements are the short-term char-coal canister (up to 7 days), and the long-term alpha-track methods (90 days to 1 year). Generally, short-term measurements should only be used as a screening technique to determine whether a long-term measurement is necessary. The great uncertainties in measure-ment accuracy with these devices, up to 50% at the radon levels typ-ically found in homes, as well as the natural variability of radon concentrations should be considered in interpreting the results.
Ideally, long-term measurements should be the basis for deci-sions on installation of radon mitigation systems, and short-term measurements should only be used as a screening method to identify buildings with Rn concentrations that are very high, justifying immediate remedial action. In practice, short-term measurements at the time a building is sold are the basis for most decisions about remedial action.
Typical Levels. The outdoor radon concentration is about 15 Bq/m3 (0.4 pCi/L). The annual average concentration of radon in homes in the United States is about 46 Bq/m3 (1.25 pCi/L) (EPA 1989). While several sources of radon may contribute to the annual indoor average, pressure-driven flow of soil gas constitutes the prin-cipal source for elevated concentrations; nonmunicipal water sup-plies can be a source of elevated indoor radon, but only in isolated instances.
Control. Exposure to indoor Rn may be reduced by (1) inhibit-ing Rn entry into the building or (2) removing or diluting Rn decay products in indoor air. The most effective and energy-efficient con-trol measures are generally those that reduce Rn entry rates (Hen-schel 1993). One of the most common effective techniques is active subslab depressurization, in which a fan and piping system draw soil gas from beneath the slab and exhaust the gas outside. This technique reduces or reverses the pressure gradient that normally draws soil gas and Rn into the building and often reduces indoor Rn concentrations by a large factor (e.g., 5 to 10). Passive control meth-ods such as Rn-resistant construction techniques and/or passive stack subslab depressurization systems are also used; however, the performance of these control methods is not well characterized, and average reductions in Rn concentrations may be 50% or less. Seal-ing cracks and joints in slab floors improve the performance of sub-slab depressurization systems. Sealing by itself is often not very effective in reducing indoor Rn.
In houses with crawl spaces, active (fan-forced) or passive crawl space ventilation is often effective in maintaining low indoor Rn concentrations, although other techniques are also used (Henschel 1988, 1993).
SOIL GASES The radioactive gas radon (Rn) is the best-known soil gas. How-ever, gaseous contaminants other than radon may enter buildings from surrounding soil. Methane from landfills has reached explo-sive levels in some buildings. Potentially toxic or carcinogenic VOCs in the soil as a consequence of spills, improper disposal, leaks from storage tanks, and disposal in landfills can also be transported into buildings (Wood and Porter 1987, Hodgson et al. 1992, Garbesi and Sextro 1989, Kullman and Hill 1990). Pesticides applied to the soil beneath or adjacent to houses have also been detected in indoor air (Livingston and Jones 1981, Wright and Leidy 1982), and pres-sure-driven flow is a suspected entry mechanism. The broad signif-icance of the health effects due to exposure to these soil contaminants is not well understood. Techniques that reduce Rn entry from the soil should also be effective in reducing the entry of these contaminants into buildings. Another approach may be to increase ventilation in the building, such as by slightly opening a window. That change in home-use behavior can also help reduce the negative pressure in the house, with respect to the soil gas pressure, created by the stack effect.
REFERENCES ACGIH. 1985. Particle size-selective sampling in the workplace. American Conference of Governmental Industrial Hygienists, Cincinnati, OH.
ACGIH. 1989. Guidelines for the assessment of bioaerosols in the indoor environment.
ACGIH. 1999. TLVs and BEIs: Threshold limit values for chemical sub-stances and physical agents.
ACGIH. 2001. Air sampling instruments, 9th ed.
AIHA. 1996. Field guide for the determination of biological contaminants in environmental samples. American Industrial Hygiene Association, Fairfax, VA.
American Guild of Organists. 1966. Air pollution and organ leathers: Panel discussion. AGO Quarterly II(2):62-73.
ASHRAE. 1989. Legionellosis position paper.
ASHRAE. 1994. Safety code for mechanical refrigeration. ANSI/ASHRAE Standard 15-1994.
ASHRAE. 1997. Designation and safety classification of refrigerants.
ANSI/ASHRAE Standard 34-1997.
ASHRAE. 1999. Method of testing general ventilation air-cleaning devices for removal efficiency by particle size. ANSI/ASHRAE Standard 52.2-1999.
ASHRAE.
1999.
Ventilation for acceptable indoor air quality.
ANSI/ASHRAE Standard 62-1999.
ASTM. 1983. Practice for continuous sizing and counting of airborne parti-cles in dust-controlled areas using instruments based upon light-scatter-ing principles. ASTM Standard F 50-83. American Society for Testing and Materials, West Conshohocken, PA.
ASTM. 1990. Biological contaminants in indoor environments. STP 1071.
ATC. 1990. Technical assistance document for sampling and analysis of toxic organic compounds in ambient air. Environmental Protection Agency, Research Triangle Park, NC.
Avallone, E.A. and T. Baumeister. 1987. Marks’ standard handbook for mechanical engineers. McGraw-Hill Book Company, New York.
12.18 2001 ASHRAE Fundamentals Handbook (SI) Barbaree, J.M., B.S. Fields, J.C. Feeley, G.W. Gorman, and W.T. Martin.
1986. Isolation of protozoa from water associated with a Legionellosis outbreak and demonstration of intracellular multiplication of Legionella pneumophila. Applied Environmental Microbiology 51:422-24.
Batterman, S. and C. Peng. 1995. TVOC and CO2 as indicators in indoor air quality studies. American Industrial Hygiene Association Journal 56(1):55-65.
Berglund, B., U. Berglund, and T. Lindvall. 1986. Assessment of discomfort and irritation from the indoor air. In IAQ ’86: Managing Indoor Air for Health and Energy Conservation, 138-49. ASHRAE, Atlanta.
Berglund, B., I. Johansson, and T. Lindvall. 1988. Healthy Buildings 88, 3:299-309. Stockholm, Sweden.
Black, M.S. and C.W. Bayer. 1986. Formaldehyde and other VOC exposures from consumer products. In IAQ ’86: Managing Indoor Air for Health and Energy Conservation. ASHRAE, Atlanta.
Braman, R.S. and T.J. Shelley. 1980. Gaseous and particulate and ammonia and nitric acid concentrations. Columbus, Ohio area—Summer 1980. PB 81-125007. National Technical Information Service, Springfield, VA.
Braun, R.C. and M.J.G. Wilson. 1970. The removal of atmospheric sulphur by building stones. Atmospheric Environment 4:371-78.
Brightman, H.S., S.E. Womble, E.L. Ronca, and J.R. Girman. 1996. Base-line information on indoor air quality in large buildings (BASE ‘95). In Proceedings of Indoor Air ‘96, Vol. 3:1033-38.
Budavi, S., ed. 1996. The Merck index, 12th ed. Merck and Company, White Station, NJ.
Burge, H.A. 1995. Bioaerosols. Lewis Publishers, Chelsea, MA.
Cain, W.S., L.C. See, and T. Tosun. 1986. Irritation and odor from formal-dehyde chamber studies, 1986. In IAQ ’86: Managing Indoor Air for Health and Energy Conservation. ASHRAE, Atlanta.
Casserly, D.M. and K.K. O’Hara. 1987. Ambient exposures to benzene and toluene in southwest Louisiana. Paper 87-98.1. Air and Waste Manage-ment Association, Pittsburgh, PA.
Chan, C.C., L. Vanier, J.W. Martin, and D.T. William. 1990. Determination of organic contaminants in residential indoor air using an adsorption-thermal desorption technique. Journal of the Air and Waste Management Association 40(1):62-67.
Chatigny, M. 1983. Air sampling instruments, 6th ed. American Conference of Governmental Industrial Hygienists. Cincinnati, OH.
Chiarenzelli, R.V. and E.L. Joba. 1966. The effects of air pollution on elec-trical contact materials: A field study. Journal of the Air Pollution Con-trol Association 16(3):123-27.
Code of Federal Regulations. National primary and secondary ambient air quality standards. 40 CFR 50. U.S. Government Printing Office, Wash-ington, D.C. Revised annually.
Code of Federal Regulations. 29 CFR 1900. U.S. Government Printing Office, Washington, D.C. Revised annually.
Cohen, M.A., P.B. Ryan, Y. Yanigisawa, J.D. Spengler, H. Ozkaynak, and P.S. Epstein. 1989. Indoor/outdoor measurements of volatile organic compounds in the Kanawha Valley of West Virginia. Journal of the Air Pollution Control Association 39(8):1986-93.
Colombo, A., M. DeBortoli, H. Knöppel, H. Schauenburg, and H. Vissers.
1991. Small chamber tests and headspace analysis of volatile organic compounds emitted from household products. Indoor Air 1:13-21.
Conibear, S., S. Geneser, and B.W. Carnow. 1996. Carbon monoxide levels and sources found in a random sample of households in Chicago during the 1994-1995 heating season. Proceedings of IAQ 95. ASHRAE, 111-18.
Coutant, R.W., G.F. Ward, C.W. Spicer, et al. 1994. Control of NO2 and nitrogen acids using alkaline-impregnated carbon filters. Proceedings of IAQ 94. ASHRAE, 139-45.
Coy, C.A. 1987. Regulation and control of air contaminants during hazard-ous waste site redemption. Paper 87-18.1. Air and Waste Management Association, Pittsburgh, PA.
Ekberg, L.E. 1994. Outdoor air contaminants and indoor air quality under transient conditions. Indoor Air 4:189-196.
EPA. 1979. Toxic substances control act chemical substance inventory, Vol-umes I-IV. Environmental Protection Agency, Office of Toxic Sub-stances, Washington, D.C.
EPA. 1982. Air quality criteria for particulate matter and sulfur. EPA-600/8-82-029b.
EPA. 1989. Radon and radon reduction technology. EPA-600/9-89/006a, 1:4-15.
EPA. 1996. Air quality criteria for particulate matter. Office of Research and Development.
Esswein, E.J. and M.F. Boeniger. 1994. Effect of an ozone generating air purifying device on reducing concentrations of formaldehyde in air.
Applied Occupational Environmental Hygiene 9(2).
FDA. 1990. Maximum acceptable level of ozone. 21 CFR Ch. 1. Sec.
801.415: 24-25. U.S. Food and Drug Administration.
Federal Standard 209E. 1992. Airborne particulate cleanliness classes in cleanrooms and clean zones.
Fliermans, C.B. 1985. Ecological niche of Legionella pneumophila. In: Crit-ical Reviews of Microbiology, 75-116.
Fung, K. and B. Wright. 1990. Measurement of formaldehyde and acetalde-hyde using 2-4 dinitrophenylhydrazine-impregnated cartridges. Aerosol Science and Technology 12(1):44-48.
Garbesi, K. and R.G. Sextro. 1989. Modeling and field evidence of pressure-driven entry of soil gas into a home through permeable below-grade walls. Environment Science and Technology 23:1481-87.
Graminski, E.L., E.J. Parks, and E.E. Toth. 1978. The effects of temperature and moisture on the accelerated aging of paper. NBSIR 78-1443.
National Institute of Standards and Technology, Gaithersburg, MD.
Grosjean, D., P.M. Whitmore, C.P. de Moor, G.R. Cass, and J.R. Druzik.
1987. Fading of alizarinad-related artist’s pigments by atmospheric ozone. Environmental Science and Technology 21:635-43.
Gyntelberg, F., P. Suadicami, J.W. Nielsen, P. Skov, O. Valbjorn, T. Nielsen, T.O. Schneider, O. Jorgenson, P. Wolkoff, C. Wilkins, S. Gravesen, and S. Nom. 1994. Dust and the sick-building syndrome. Indoor Air 4: 223-38.
Hakov, R.J., J. Kemlis, and C. Ruggeri. 1987. Volatile organic compounds in the air near a regional sewage treatment plant in New Jersey. Paper 87-95.1. Air and Waste Management Association, Pittsburgh, PA.
Hartmann, I. 1958. Explosion and fire hazards of combustible dusts. Indus-trial Hygiene and Toxicology 1:2 Interscience Publishers, New York.
Hartwell, TD., J.H. Crowder, L.S. Sheldon, and E.D. Pellizzari. 1985. Lev-els of volatile organics in indoor air. Paper 85-30B.3. Air and Waste Management Association, Pittsburgh, PA.
Haynie, F.H. 1978. Theoretical air pollution and climate effects on materials confirmed by zinc corrosion data. In Durability of building materials and components (ASTM STP 691). American Society for Testing and Mate-rials, West Conshohocken, PA.
Heavner, D.L., W.T. Morgan, and M.W. Ogden. 1996. Determination of vol-atile organic compounds and respirable particulate matter in New Jersey and Pennsylvania homes and workplaces. Environment International 22:159-183.
Henschel, D.B. 1988. Radon reduction techniques for detached houses— Technical guidance, 2nd ed. EPA/625/5-87/019.
Henschel, D.B. 1993. Radon reduction techniques for existing detached houses—Technical guidance, 3rd ed. EPA/625/R-93/011.
Hewson, E.W. et al. 1967. Air pollution by ragweed pollen. Journal of the Air Pollution Control Association 17(10):651.
Hodgson, A.T., K. Garbesi, R.G. Sextro, and J.M. Daisey. 1988. Transport of volatile organic compounds from soil into a residential basement. Paper 88-95B.1, Proceedings of the 81st Annual Meeting of the Air Pollution Control Association. Also LBL Report No. 25267.
Hodgson, A.T., K. Garbesi, R.G. Sextro, and J.M. Daisey 1992. Soil gas con-tamination and entry of volatile organic compounds into a house near a landfill. Journal of the Air and Waste Management Association 42: 277-83.
Hodgson, A.T. 1995. A review and a limited comparison of methods for measuring total volatile organic compounds in indoor air. Indoor Air 5(4):247.
Hodgson, A.T., J.M. Daisey, K.R.R. Mahanama, J.T. Brinke, and L.E.
Alevantis. 1996. Use of volatile tracers to determine the contribution of environmental tobacco smoke to concentrations of volatile organic com-pounds in smoking environments. Environment International 22:295-307.
Hollowell, C.D., R.A. Young, J.V. Berk, and S.R. Brown. 1982. Energy-con-serving retrofits and indoor air quality in residential housing. ASHRAE Transactions 88(l):875-93.
ISO. 1999. Cleanrooms and associated controlled environments—Part 1: Classification of air cleanliness. Standard 14644-1. International Orga-nization for Standardization, Geneva.
Jacobson, A.R. and S.C. Morris. 1977. The primary pollutants, viable par-ticulates, their occurrence, sources and effects. In Air pollution, 3rd ed., Academic Press, New York, 169.
Jaffe, L.S. 1967. The effects of photochemical oxidants on materials. Jour-nal of the Air Pollution Control Association 17(6):375-78.
Air Contaminants 12.19 Knoeppel, H. and H. Schauenburg. 1989. Screening of household products for the emission of volatile organic compounds. Environment Interna-tional 15:413-18.
Kullman, G.J. and R.A. Hill 1990. Indoor air quality affected by abandoned gasoline tanks. Applied Occupational Environmental Hygiene 5:36-37.
Lamm, S.H. 1986. Irritancy levels and formaldehyde exposures in U.S.
mobile homes. In Indoor Air Quality in Cold Climates, 137-47. Air and Waste Management Association, Pittsburgh, PA.
Levin, H. 1989. Building materials and indoor air quality. In: Occupational Medicine: State of the Art Reviews—Problem Buildings, J.E. Cone and M.J. Hodgson, eds. Hanley & Belfus, Philadelphia. 4(4):667-94.
Levin, H. 1991. Critical building design factors for indoor air quality and cli-mate: current status and predicted trends. Indoor Air 1(1):79-92.
Lide, D.R., ed. 1996. Handbook of chemistry and physics, 77th ed. Chemical Rubber Company Press, Boca Raton, FL.
Liu, R. and M.A. Huza. 1995. Filtration and indoor air quality: A practical approach. ASHRAE Journal 37(2):18-23.
Livingston, J.M. and C.R. Jones. 1981. Living area contamination by chlo-rdane used for termite treatment. Bulletin of Environmental Contaminant Toxicology 27:406-11.
Lodge, J.E., ed. 1988. Methods of air sampling and analysis, 3rd ed. Lewis Publishers, Chelsea, MD.
Lonnemann, W.A., S.L. Kopczynski, P.E. Darley, and P.D. Sutterfield. 1974.
Hydrocarbon composition of urban air pollution. Environmental Science and Technology 8(3):229-35.
Lull, W.P. 1995. Conservation environment guidelines for libraries and archives. Canadian Council of Archives.
Martin, P., D.L. Heavner, P.R. Nelson, K.C. Maiolo, C.H. Risner, P.S. Sim-mons, W.T. Morgan, and M.W. Ogden. 1996. Environment International 22.
Mathey, R.G., T.K. Faison, and S. Silberstein. 1983. Air quality criteria for storage of paper-based archival records. NBSIR 83-2795. National Insti-tute of Standards and Technology, Gaithersburg, MD.
McGrath, T.R. and D.B. Stele. 1987. Characterization of phonemic odors in a residential neighborhood. Paper 87-75A.5. Air and Waste Manage-ment Association, Pittsburgh, PA.
Miksch, R.R., C.D. Hollowell, and H.E. Schmidt. 1982. Trace organic chemical contaminants in office spaces. Atmospheric Environment 8:129-37.
Milton, D.K., R.J. Gere, H.A. Feldman, and I.A. Greaves. 1990. Endotoxin measurment: Aerosol sampling and application of a new limulus method.
American Industrial Hygiene Association Journal 51:331.
Molhave, L. and M. Thorsen. 1990. A model for investigations of ventilation systems as sources for volatile organic compounds in indoor climate.
Atmospheric Environment 25A:241-49.
Molhave, L., L. Anderson, G.R. Lundquist, and O. Nielson. 1982. Gas emis-sion from building materials. Report 137. Danish Building Research Institute, Copenhagen.
Morey, P.R. and B.A. Jenkins. 1989. What are typical concentrations of fungi, total volatile organic compounds, and nitrogen dioxide in an office environment. Proceedings of IAQ ’89, The Human Equation: Health and Comfort. ASHRAE, Atlanta, 67-71.
Morey, P.R. and J. Singh. 1991. Indoor air quality in non-industrial occupa-tional environments. In: Patty’s industrial hygiene and toxicology, 4th ed. 1(1).
Muller, C.O. and W.G. England. 1995. Achieving your indoor air quality goals: Which filtration system works best? ASHRAE Journal 27(2):24-32.
Nagda, N.L. and H.E. Rector. 1983. Guidelines for monitoring indoor-air quality. EPA 600/4-83-046. Environmental Protection Agency, Research Triangle Park, NC.
Nazaroff, W.W., S.R. Lewis, S.M. Doyle, B.A. Moed, and A.V. Nero. 1987.
Experiments on pollutant transport from soil into residential basements by pressure-driven air flow. Environment Science and Technology 21:459.
Nazaroff, W.W., B.A. Moed, and R.G. Sextro. 1988. Soil as a source of indoor radon: Generation, migration and entry. In: Radon and its decay products in indoor air. Wiley, New York, 57-112.
Nelms, L.H., M.A. Mason, and B.A. Tichenor. 1986. The effects of ventila-tion rates and product loading on organic emission rates from particle-board. In IAQ ’86: Managing Indoor Air for Health and Energy Conservation, 469-85. ASHRAE.
Nelson, E.D.P., D. Shikiya and C.S. Liu. 1987. Multiple air toxins exposure and risk assessment in the south coast air basin. Paper 87-97.4. Air and Waste Management Association, Pittsburgh, PA.
Nelson, P.R., P. Martin, M.W. Ogden, D.L. Heavner, C.H. Risner, K.C.
Maiolo, P.S. Simmons, and W.T. Morgan. 1994. Environmental tobacco smoke characteristics of different commercially available cigarettes.
Proceedings of the Fourth International Aerosol Conference, Vol. 1: 454-55.
NFPA. 1984. Fire hazard properties of flammable liquids, gases and volatile solids. NFPA Standard 325M. National Fire Protection Association, Quincy, MA.
NIOSH. 1977. NIOSH manual of sampling data sheets. U.S. Department of Health and Human Services, National Institute for Occupational Safety and Health, Washington, D.C.
NIOSH. 1994. NIOSH manual of analytical methods, 4th ed. Cassellini, M.E. and P.F. O’Connor, eds. DHHS (NIOSH) Publication 94-113.
NRC. 1981. Indoor pollutants. National Research Council, National Acad-emy Press, Washington, D.C.
NTIS. 1982. Air pollution effects on materials. Published Search PB 82-809427. National Technical Information Service, Springfield, VA.
NTIS. 1984. Air pollution effects on materials. Published Search PB 84-800267.
OSHA. 1995. OSHA computerized information system chemical sampling information. U.S. Government Printing Office, Washington, D.C.
Otson, R. and P. Fellin. 1993. TVOC measurements: Relevance and limita-tions. Proceedings of Indoor Air ’93, Vol. 2:281-85.
Owen et al. 1992. Airborne particle sizes and sources found in indoor air.
Atmospheric Environment 26A(12):2149-62.
Peral, J., X. Domenech, and D.F. Ollis. 1997. Heterogeneous photocatalysis for purification, decontamination and deodorization of air. J. Chem.
Technol. Biotechnol. 70:117-40.
Rask, D. 1988. Indoor air quality and the bottom line. Heating, Piping and Air Conditioning 60(10).
Rodes, C.E., R.M. Kamens, and R.W. Wiener. 1991. The significance and characteristics of the personal activity cloud on exposure assessment methods for indoor contaminants. Indoor Air 2:123-145.
Samet, J.M., M.C. Marbury, and J.D. Spengler. 1987. Health effects and sources of indoor air pollution. Am. Rev. Respir. Disease 136:1486-1508.
Sandalls, E.J. and S.A. Penkett. 1977. Measurements of carbonyl sulfide and carbon disulphide in the atmosphere. Atmospheric Environment 11:197-99.
Sax, N.I. and R.J. Lewis, Sr. 1988. Dangerous properties of industrial mate-rials, 6th ed, 3 vols. Van Nostrand Reinhold, New York.
Scala, G.F. 1963. A new instrument for the continuous measurement of con-densation nuclei. Analytical Chemistry 35(5):702.
Shah, J.J. and H.B. Singh. 1988. Distribution of volatile organic chemicals in outdoor and indoor air. Environmental Science and Technology 22(12):1391-88.
Shaughnessy, R.J., E. Levetin, J. Glocker, and K.L. Sublette. 1994. Effec-tiveness of portable indoor air cleaners: Sensory testing results. Indoor Air 4:179-88.
Sheldon, L., R.W. Handy, T. Hartwell, R.W. Whitmore, H. Zelon, and E.D.
Pellizzari. 1988a. Indoor air quality in public buildings, Vol. I. EPA/600/ S6-88/009a. Environmental Protection Agency, Washington, D.C.
Sheldon, L., H. Zelon, J. Sickles, C. Easton, T. Hartwell, and L. Wallace.
1988b. Indoor air quality in public buildings, Vol. II. EPA/600/S6-88/ 009b. Environmental Protection Agency, Research Triangle Park, NC.
Singh. H.B., L.J. Salas, A. Smith, R. Stiles, and H. Shigeishi. 1981. Atmo-spheric measurements of selected hazardous organic chemicals. PB 81200628. National Technical Information Service, Springfield, VA.
Solomon, W.R. and K.P. Mathews. 1978. Aerobiology and inhalant aller-gens. Allergy, principles and practices, St. Louis, MO.
Spengler, J., C. Hallowell, D. Moschandreas, and O. Fanger. 1982. Environ-ment international. Indoor air pollution. Pergamon Press, Oxford, England.
Tanner, A.B. 1980. Radon migration in the ground: A supplementary review.
In: Natural radiation environment III. U.S. Department of Commerce, NTIS, Springfield, VA.
Task Group on Lung Dyamics. 1966. Deposition and retention models for internal dosimetry of the human respiratory tract. Health Physics 12:173-207.
Taylor, D.G., R.E. Kupel, and J.M. Bryant. 1977. Documentation of the NIOSH validation tests.
12.20 2001 ASHRAE Fundamentals Handbook (SI) Thomson, G. 1986. The museum environment, 2nd ed. Butterworths Publish-ers, Boston.
Tichenor, B.A. 1989. Measurement of organic compound emissions using small test chambers. Environment International 15:389-96.
Tichenor, B.A., G. Guo, J.E. Dunn, L.E. Sparks, and M.A. Mason. 1991. The interaction of vapour phase organic compounds with indoor sinks.
Indoor Air 1:23-35.
Traynor, G.W. 1987. Field monitoring design considerations for assessing indoor exposures to combustion pollutants. Atmospheric Environment 21(2):377-83.
Turk, B.H., R.J. Prill, D.T. Grimsrud, B.A. Moed, and R.G. Sextro. 1990.
Characterizing the occurrence, sources and variability of radon in Pacific Northwest homes. Journal of the Air and Waste Management Associa-tion 40:498-506.
VanOsdell, D.W. and L.E. Sparks. 1995. Carbon absorption for indoor air cleaning. ASHRAE Journal 27(2):34-40.
Wallace, L.A., E.D. Pellizari, T.D. Hartwell, C. Sparacino, and H. Zelon.
1983. Personal exposure to volatile organic and other compounds indoors and outdoors—the team study. APCA Meeting Paper, and later Report.
Wallace, L.A., E. Pellizzari, and C. Wendel. 1991. Total volatile organic concentrations in 2700 personal, indoor, and outdoor air samples col-lected in the US EPA Team Studies. Indoor Air 4:465-77.
Walsh, M., A. Black, A. Morgan, and G.H. Crawshaw. 1977. Sorption of SO2 by typical indoor surfaces including wool carpets, wallpaper and paint.
Atmospheric Environment 11:1107-11.
Weschler, C.J. and H.C. Shields. 1989. The effects of ventilation, filtration, and outdoor air on the composition of indoor air at a telephone office building. Environment International 15:593-604.
Weschler, C.J., H.C. Shields, and D.V. Naik. 1989. Indoor ozone exposures.
Journal of the Air Pollution Control Association 39:1562-68.
Whitby, K.T., A.B. Algren, and R.C. Jordan. 1955. Size distribution and concentration of airborne dust. Heating, Piping and Air Conditioning 27:121.
Whitby, K.T., A.B. Algren, and R.C. Jordan. 1957. The ASHRAE airborne dust survey. Heating, Piping and Air Conditioning 29(11):185.
WHO. 1989. Indoor air quality: Organic pollutants. Report on a WHO-meet-ing, Euro Report and Studies III. WHO Regional Office for Europe, Copenhagen.
Wilkins, C.K., P. Wolkoff, F. Gyntelberg, P. Skov, and O. Valbjørn. 1993.
Characterization of office dust by VOCs and TVOC release—Identifi-cation of potential irritant VOCs by partial least squares analysis.
Indoor Air 3:283-90.
Willeke, K. and P.A. Baron, eds. 1993. Aerosol measurement—Principles, techniques and applications. Van Nostrand Reinhold, New York.
Wolkoff, P. and C.K. Wilkins. 1994. Indoor VOCs from household floor dust: Comparison of headspace with desorbed VOCs; method for VOC release determination. Indoor Air 4:248-54.
Wood, J.A. and M.L. Porter. 1987. Hazardous pollutants in Class II landfills.
Journal of the Air Pollution Control Association 37:609-15.
Wright, C.G. and R.B. Leidy. 1982. Chlordane and heptachlor in the ambi-ent air of houses treated for termites. Bulletin of Environmental Con-taminant Toxicology 28:617-23.
13.1 CHAPTER 13 ODORS Odor Sources ................................................................................................................................ 13.1 Sense of Smell ............................................................................................................................... 13.1 Factors Affecting Odor Perception .............................................................................................. 13.2 Odor Sensation Attributes ............................................................................................................ 13.3 Dilution of Odors by Ventilation .................................................................................................. 13.5 Odor Concentration ..................................................................................................................... 13.5 Olf Units ....................................................................................................................................... 13.6 ARIOUS factors make odor control a primary consideration Vin ventilation engineering: (1) contemporary construction methods result in buildings that permit less air infiltration through the building envelope; (2) indoor sources of odors associated with modern building materials, furnishings, and office equipment have increased; (3) outdoor air is often polluted; and (4) energy costs have encouraged ventilation rate reductions at a time when requirements for a relatively odor-free environment are greater than ever.
Since Yaglou’s classic studies (1936), the philosophy behind the ventilation of nonindustrial buildings has mainly been to pro-vide indoor air that is acceptable to occupants. Air is evaluated by the olfactory sense, although the general chemical sense, which is sensitive to irritants in the air, also plays a role.
This chapter reviews how odoriferous substances are per-ceived. Chapter 44 of the 1999 ASHRAE Handbook—Applica-tions covers control methods. Chapter 9 of this volume has more information on indoor environmental health.
ODOR SOURCES Outdoor sources of odors include automotive and diesel exhausts, hazardous waste sites, sewage treatment plants, com-post piles, refuse facilities, printing plants, refineries, chemical plants, and many other stationary and mobile sources. These sources produce both inorganic compounds (e.g., ammonia and hydrogen sulfide) and volatile organic compounds (VOCs), including some which evaporate from solid or liquid particulate matter. Odors emitted by outdoor sources eventually enter the indoor environment.
Indoor sources also emit odors. Sources include tobacco prod-ucts; bathrooms and toilets; building materials (adhesives, paints, caulks, processed wood, carpets, plastic sheeting, and insulation board); consumer products (e.g., food, toiletries, cleaning materi-als, and polishes); hobby materials; fabrics; and foam cushions.
In offices, offset printing processes, copiers, and computer print-ers may produce odors. If electrostatic processes are involved, ozone may be emitted. Humans emit a wide range of odorants, including acetaldehyde, ammonia, ethanol, hydrogen sulfide, and mercaptans.
Mildew and other processes of decay often produce odors in occupied spaces (home and office), damp basements, and ventila-tion systems (e.g., from wetted air-conditioning coils and spray dehumidifiers).
Chapter 44 of the 1999 ASHRAE Handbook—Applications gives further information on contaminant sources and generation rates.
SENSE OF SMELL Olfactory Stimuli Among organic substances, those with relative molecular masses greater than 300 are generally odorless. Some substances with rel-ative molecular masses less than 300 are such potent olfactory stim-uli that they can be perceived at concentrations too low to be detected with direct-reading instruments. Trimethylamine, for example, can be recognized as a fishy odor by a human observer at a concentration of about 10− 4 ppm.
Table 1 shows threshold concentrations for selected com-pounds. These threshold values are not precise numbers and may vary by several orders of magnitude. The threshold limit value (TLV) is the concentration of a compound that should have no adverse health consequences if a worker is regularly exposed for 8 h periods (ACGIH, revised annually). Table 1 also includes the ratio The preparation of this chapter is assigned to TC 2.3, Gaseous Air Contam-inants and Gas Contaminant Removal Equipment.
Table 1 Odor Thresholds, ACGIH TLVs, and TLV:Threshold Ratios of Selected Gaseous Air Pollutants Compound Odor Threshold,a ppmv TLV, ppmv Ratio Acetaldehyde 0.067 40 597 Acetone 62 500 8.1 Acetonitrile 1600 40 0.025 Acrolein 1.8 0.1 0.06 Ammonia 17 25 1.5 Benzene 61 0.5 0.01 Benzyl chloride 0.041 1 24 Carbon tetrachloride 250 5 0.02 Chlorine 0.08 0.5 6 Chloroform 192 10 0.05 Dioxane 12 25 2 Ethylene dichloride 26 10 0.4 Hydrogen sulfide 0.0094 10 1064 Methanol 160 200 1.25 Methylene chloride 160 50 0.3 Methyl ethyl ketone 16 200 12.5 Phenol 0.06 5 83 Sulfur dioxide 2.7 2 0.74 Tetrachloroethane 7.3 1 0.14 Tetrachloroethylene 47 25 0.5 Toluene 1.6 50 31 1,1,1-Trichloroethane 390 350 0.9 Trichloroethylene 82 50 0.6 Xylene (isomers) 20 100 5 Sources: AIHA (1989), ACGIH (1998).
aAll thresholds are detection thresholds (ED50).
13.2 2001 ASHRAE Fundamentals Handbook (SI) of the TLV to the odor threshold for each compound. For ratios greater than 1, most occupants can detect the odor and leave the area long before the compound becomes a health risk. As the ratio increases, the safety factor provided by the odor also increases.
Table 1 is not a comprehensive list of the chemicals found in indoor air. AIHA (1989) and EPA (1992) have published lists of odor thresholds for selected chemicals.
Olfactory sensitivity often makes it possible to detect potentially harmful substances at concentrations below dangerous levels so that they can be eliminated. Foul-smelling air is often assumed to be unhealthy.In reality,however, there is little correlation between odor perception and toxicity, and there is considerable individual varia-tion in the perception of pleasantness/unpleasantness of odors.
When symptoms such as nausea, headache, and loss of appetite are caused by an unpleasant odor, it may not matter whether the air is toxic but whether the odor is perceived to be unpleasant, associated with an unpleasant experience, or simply felt to be out of appropriate context. The magnitude of the symptoms is related to the magnitude of the odor. Even a room with a low but recognizable odor can arouse uneasiness among occupants. Several papers review sensory irrita-tion and its relation to indoorairpollution (Cain and Cometto-Muñiz 1995; Cometto-Muñiz and Cain 1992; Cometto-Muñiz et al. 1997; Shams Esfandabad 1993).
Anatomy and Physiology The olfactory receptors lie in the olfactory cleft, which is high in the nasal cavity. About five million olfactory neurons (a small cluster of nerve cells inside the nasal cavity above the bridge of the nose) each send an axon (an extension of the neuron) into the olfac-tory bulb of the brain. Information received from the receptors is passed to various central structures of the brain (e.g., the olfactory cortex, the hippocampus, and the amygdala). One sniff of an odor-ant can often evoke a complex, emotion-laden memory, such as a scene from childhood.
The surrounding nasal tissue contains other diffusely distrib-uted nerve endings of the trigeminal nerve that also respond to airborne vapors. These receptors mediate the chemosensory responses such as tickling, burning, cooling, and, occasionally, painful sensations that accompany olfactory sensations. Most odorous substances at sufficient concentration will also stimulate these nerve endings.
Olfactory Acuity The olfactory acuity of the population is normally distributed.
Most people have an average ability to smell substances or to respond to odoriferous stimuli, a few people are very sensitive or hypersensitive, and a few others are insensitive, including some who are totally unable to smell (anosmic). The olfactory acuity of an individual varies with the odorant.
Hormonal factors, which often influence emotional states, can modulate olfactory sensitivity. Although the evidence is not uni-formly compelling, research has found that (1) the sensitivity of females varies during the menstrual cycle, reaching a peak just before and during ovulation (Schneider 1974); (2) females are gen-erally more sensitive than males, but this difference only emerges around the time of sexual maturity (Koelega and Koster 1974); (3) sensitivity is altered by certain diseases (Schneider 1974); and (4) various hormones and drugs (e.g., estrogen and alcohol) alter sensitivity (Schneider 1974; Engen et al. 1975).
Other factors that may affect olfactory perception include the olfactory acuity of an individual, the magnitude of the flow rate toward the olfactory receptors, the temperature, and the relative humidity. Olfactory acuity can also vary with age (Stevens et al.
1989; Wysocki and Gilbert 1989), genetics (Wysocki and Beau-champ 1984), exposure history (Dalton and Wysocki 1996; Wysocki et al. 1997), and disease or injury (Cowart et al. 1993, 1997). Humans are able to perceive a large number of odors, yet untrained individuals are able to name only a few (Ruth 1986).
Individuals exhibiting a total inability to detect odors are rela-tively rare (Cowart et al. 1997). A more common occurrence is an inability to detect one or a very limited number of odors, a condition known as specific anosmia. Although the huge number of possible chemicals makes for an untestable hypothesis, it has been posited that most, if not all, individuals have a specific anosmia to one or more compounds (Wysocki and Beauchamp 1984). The fact that individuals with specific anosmias have normal olfactory acuity for all other odors suggests that such anosmias may be due to genetic differences.
Olfactory science uses the term adaptation to refer to decreased sensitivity or responsiveness to an odor as a function of prolonged exposures. Such exposure can selectively impair the perception of the exposure odorant, but there are also examples of cross-adaptation, where exposure to one odorant can result in adaptation to other odors as well. Adaptation can occur in the short term, where the perception of a room’s odor begins to fade within seconds of entering the room (Cometto-Muñiz and Cain 1995; Pierce et al. 1996). With long-term adaptation, an individual who habitually returns to the same environment does not smell odors that are quite discernible to a naive observer. This effect appears to shift both the threshold and the suprathreshold (stimuli above the threshold level) response to the odor (Dalton and Wysocki 1996).
This is an important phenomenon for indoor air quality (IAQ) per-sonnel to be aware of because it is often one of the biggest reasons for the variation in detectability or response in real-world environ-ments and makes the choice of test population or panelists for air quality evaluations a critical one.
FACTORS AFFECTING ODOR PERCEPTION Humidity and Temperature Temperature and humidity can both affect the perception of odors. Cain et al. (1983) reported that a combination of high tem-perature (25.5°C) and high humidity exacerbates odor problems.
Berglund and Cain (1989) found that air was generally perceived to be fresher and less stuffy with decreasing temperature and humidity.
Fang et al. (1998a,b) and Toftum (1998) found little or no increase in odor intensity with increasing enthalpy (temperature and humid-ity) but reported a very significant decrease in odor acceptability with increasing enthalpy.
Not all researchers have supported these findings. Kerka and Humphreys (1956) reported a decrease in odor intensity with increasing humidity. Berg-Munch and Fanger (1982) found no increase in odor intensity with increasing temperature (23 to 32°C).
Clausen et al. (1985) found no significant change in odor intensity with increasing relative humidity (30% to 80%).
While the findings are not homogeneous, they do show that tem-perature and humidity can act together to affect one’s perception of odors. Air that is cooler and drier is generally perceived to be fresher and more acceptable even if the odor intensity is not affected.
Sorption and Release of Odors Due to sorption of odors by furnishings and interior surfaces dur-ing occupancy, with later desorption, spaces frequently retain nor-mal occupancy odor levels long after occupancy has ceased. This phenomenon is observed when furnaces or radiators, after a long shutdown, are heated at winter start-up and when evaporator coils warm up. The rate of desorption can be decreased by decreasing temperature and relative humidity, and increased (as for cleaning) by the reverse.
Environmental tobacco smoke may desorb from surfaces long after smoking has taken place. This phenomenon has caused many hotels to establish nonsmoking rooms.
Odors 13.3 Where the odor source is intrinsic to the materials (as in lino-leum, paint, rubber, and upholstery), reducing the relative humidity decreases the rate of odor release. Quantitative values should not be used without considering the sorption-desorption phenomenon.
Emotional Responses to Odors There can be considerable variation between individuals regard-ing the perceived pleasantness or unpleasantness of a given odor.
Responses to odors may be determined by prior experiences and can include strong emotional reactions. This is because one of the brain structures involved in the sense of smell is the amygdala, a regula-tor of emotional behaviors (Frey 1995). Some IAQ complaints can involve emotional responses completely out of proportion to the concentration of the odorant or the intensity of the odor it produces.
Two theories describe physiological reasons for these strong responses. One of these is kindling, in which repeated, intermittent stimuli produce an amplification of nerve responses. The other is response facilitation, in which an initial stimulus perceived as strong is facilitated (becomes greater) rather than adapted to (Frey 1995).
Because of this emotional aspect, IAQ complaints involving odors can be very difficult to solve, especially if they are coming from a few sensitized individuals. It is important to respond quickly to complaints in order to minimize the risk of kindling or response facilitation.
ODOR SENSATION ATTRIBUTES Odor sensation has four components or attributes: detectability, intensity, character, and hedonic tone.
Detectability (or threshold) is the minimum concentration of an odorant that provokes detection by some predetermined segment of the population. Two types of thresholds exist: the detection thresh-old and the recognition threshold.
The detection threshold is the lowest level that elicits response by a segment of the population. If that segment is 50%, the detection threshold is denoted by ED50. Recognition threshold is the lowest level at which a segment of the population can recognize a given odor. Thresholds can be attributed to 100%, which includes all olfactory sensitivities, or to 10%, which includes only the most sen-sitive segment of the population. Threshold values are not physical constants; rather they are statistical measurements of best estimates.
Intensity is a quantitative aspect of a descriptive analysis, stating the degree or magnitude of the perception elicited. Intensity of the perceived odor is, therefore, the strength of the odoriferous sensa-tion. Detection threshold values and, most often, odor intensity determine the need for indoor odor controls.
Character defines the odor as similar to some familiar smell (e.g., fishy, sour, flowery, and the like). Hedonics, or the hedonic tone of an odor, is the degree to which an odor is perceived as pleas-ant or unpleasant. Hedonic judgments include both a category judg-ment (pleasant, neutral, unpleasant) and a magnitude judgment (very unpleasant, slightly pleasant).
Important questions are 1. What is the minimum concentration of odorant that can be detected?
2. How does perceived odor magnitude grow with concentration above the threshold?
No universal method has been accepted to measure either the threshold or the perceived magnitude of the odor above threshold.
However, certain guidelines and conventions simplify the choice of methods.
Detectability The perception of weak odoriferous signals is probabilistic: at one moment odor may be perceptible, and at the next moment it may not.
Factors affecting this phenomenon include the moment-to-moment variability in the number of molecules striking the olfactory recep-tors, the variability in which of the receptors are stimulated, the con-centration of the odor, the individual’s style of breathing, and the individual’s previous experience with the odor. The combined effect of these factors may prevent an individual from perceiving an odor during the entire time of the stimulus. During odor evaluation, dilu-tionto detectionorrecognitionthresholdvaluesallowsdetermination ofthe largestnumberofdilutionsthatstillpermitshalfofthepanelists to detect or recognize the odor.
Determination of Odor Thresholds. Odor threshold testing is over a century old. The process is complex, and several different methods are used. Partly due to variations in measurement tech-niques, reported threshold values can vary by several orders of mag-nitude for a given substance. In order to minimize variation due to experimental techniques, a standard set of criteria has been devel-oped. These criteria include the panel, the presentation apparatus, and the presentation method (AIHA 1989; EPA 1992).
The panel should • Include at least six members per group.
• Be selected based on odor sensitivity. Factors to be considered include anosmia, pregnancy, drug use, and smoking.
• The panel should be calibrated to document individual and group variability.
Considerations concerning the presentation apparatus include • Vapor modality—choice of a gas-air mixture, water vapor, or other substance.
• Diluent—choice of diluent (e.g., air, nitrogen), how it is treated, and what its source is.
• Presentation mode—delivery systems can be nose ports, vents into which the head is inserted, flasks, whole rooms.
• Analytic measurement of odorant concentration.
• System calibration—flow rate and face velocity. Flow rate should be approximately 0.05 L/s. Face velocity should be low enough to be barely perceptible to the panelists.
Criteria for the presentation method include • Threshold type—detection or recognition.
• Concentration presentation—this must take adaptation into account. Presenting ascending concentrations or allowing longer periods between concentrations helps avoid adaptation.
• Number of trials—test-retest reliability for thresholds is low.
Increasing the number of trials helps correct for this.
• Forced-choice procedure—panelists must choose between the stimuli and one or two blanks. This helps eliminate false positive responses.
• Concentration steps—odorant should be presented successively at concentrations no more than three times the preceding one.
For more details regarding psychophysical procedures, ways to sample odoriferous air, handling samples, means of stimulus pre-sentation, and statistical procedures, consult ASTM STP 434.
Intensity Psychophysical Power Law. The relation between perceived odor magnitude S and concentration C conforms to a power function: (1) where S = perceived intensity (magnitude) of sensation k = characteristic constant C = odorant concentration n = exponent of psychophysical function (slope on a log-log scale) S kC n = 13.4 2001 ASHRAE Fundamentals Handbook (SI) This exemplifies the psychophysical power law, also called Stevens’ law (Stevens 1957). In the olfactory realm, n < 1.0.
Accordingly, a given percentage change in odorant concentration causes a smaller percentage change in perceived odor magnitude.
Scaling Methods. Of the various ways to scale perceived mag-nitude, a category scale, which can be either number- or word-cat-egorized, is commonly used. Numerical values on this scale do not reflect ratio relations among odor magnitudes (e.g., a value of 2 does not represent a perceived magnitude twice as great as a value of 1). Table 2 gives four examples of category scales.
Although category scaling procedures can be advantageous in field situations, ratio scaling techniques are used frequently in the laboratory (Cain and Moskowitz 1974). Ratio scaling procedures require observers to assign numbers proportional to perceived magnitude. For example, if the observer is instructed to assign the number 10 to one concentration and a subsequently presented con-centration seems three times as strong, the observer calls it 30; if another seems one-half as strong, the observer assigns it 5. This ratio scaling procedure, called magnitude estimation, was used to derive the power function for butanol (Figure 1). Ratio scaling tech-niques allow for such relationships because they require subjects to produce numbers to match perceived sensations in which the num-bers emitted reflect the ratio relations among the sensations.
More recently, a hybrid of category and ratio scales known as the labeled magnitude scale has been developed (Green et al. 1996).
This scale is intended to yield ratio-level data with a true zero and an orderly relationship among the scale values, such that any stimulus can be expressed as being proportionately more or less intense than another. Because it allows subjects to use natural language descrip-tors to scale perceived experience, it often requires less training than ratio scales and produces absolute intensity estimates of perceived sensation (Figure 2).
A fourth way to measure suprathreshold odor intensity is to match the intensity of odorants. An observer can be given a con-centration series of a matching odorant (e.g., 1-butanol) to choose the member that matches most closely the intensity of an unknown odorant. The matching odorant can be generated by a relatively inexpensive olfactometer such as that shown in Figure 3. Figure 4 shows, in logarithmic coordinates, functions for various odorants obtained by the matching method (Dravnieks and Laffort 1972).
The left-hand ordinate expresses intensity in terms of concentration of butanol, and the right-hand ordinate expresses intensity in terms of perceived magnitude. The two ordinates are related by the func-tion in Figure 1, the standardized function for butanol. The match-ing method illustrated here has been incorporated into ASTM Table 2 Examples of Category Scales Number Category Word Category Scale I Scale II Scale I Scale II 0 0 None None at all 1 1 Threshold Just detectable 2 2.5 Very slight Very mild 3 5 Slight Mild 4 7.5 Slight-moderate Mild-distinct 5 10 Moderate Distinct 6 12.5 Moderate-strong Distinct-strong 7 15 Strong Strong Source: Meilgaard et al. (1987).
Fig. 1 Standardized Function Relating Perceived Magnitude to Concentration of 1-Butanol (Moskowitz et al. 1974) Fig. 2 Labeled Magnitude Scale Fig. 3 Panelist Using Dravnieks Binary Dilution Olfactometer (Dravnieks 1975) Odors 13.5 Standard E 544, Standard Practices for Referencing Suprathreshold Odor Intensity.
Character The quality or character of an odor is difficult to assess quantita-tively. A primary difficulty is that odors can vary along many dimensions. One way to assess quality is to ask panelists to judge the similarity between a test sample and various reference samples, using a five-point category scale. For certain applications, reference odorants can be chosen to represent only that portion of the qualita-tive range relevant to the odor problem under investigation (e.g., animal odors). Another procedure is to ask panelists to assess the degree of association between a test sample’s quality and certain verbal descriptors (e.g., sweaty, woody, chalky, sour).
The number of odorant descriptors and the descriptors to be used have been subjects of disagreement (Harper et al. 1968). The num-ber of descriptors varies from a minimum of seven (Amoore 1962) to as many as 830 used by an ASTM subcommittee. An atlas of odor characters, containing 146 descriptors, was compiled for 180 chem-icals by ASTM (1985).
An odor can be characterized either by an open-ended word description or by multidimensional scaling. Multidimensional scaling is based on similarity and dissimilarity judgments in com-parison to a set of standard odors or to various descriptors.
In some instances, the interest may be merely whether an odor’s quality has changed as a result of some treatment (e.g., use of a bac-teriostat). Under these circumstances, samples of air taken before and after treatment can be compared directly (using a simple scale of similarity) or indirectly (with appropriate verbal descriptors).
Hedonics The acceptability or pleasantness of an odor can be measured psychophysically in the same way as odor intensity. Both ratio and category scaling procedures can be adapted to odor acceptability.
Odors do not always cause adverse reactions. Products are man-ufactured to elicit favorable responses. Acceptance tests may in-volve product comparison (frequently used in the perfume industry) or a hedonic scale. The premise of acceptance tests is that the larger the segment of subjects accepting the odor, the better the odor. A he-donic scale that allows for negative as well as positive responses is likely to better answer the question of how acceptable the odor is.
All persons exposed to a given odor are not likely to agree on its acceptability. Acceptability of a given odor to a person is based on a complex combination of associations and is not simply a charac-teristic of the odor itself (Beck and Day 1991). Responses to odors are determined by both bottom-up factors (attributes or properties of the odorant) and top-down factors (expectations, attitudes, and associations from prior experience stored in memory, and appropri-ateness of the odor in its present context). Both of these factors are activated when an individual detects an odor, and the individual’s ultimate response (e.g., perception of intensity, hedonics, irritation, or symptoms) is a joint function of both (Dalton 1996; Dalton et al.
1997). In some cases, the interpretation provided by the top-down process appears to override the outcome from the bottom-up pro-cess, resulting in complaints, symptoms, and reports of illness.
DILUTION OF ODORS BY VENTILATION The size of the exponent n in Stevens’ law [Equation (1)] varies from one odorant to another, ranging from less than 0.2 to about 0.7 (Cain and Moskowitz 1974). This determines the slope or dose response of the odor intensity/odorant concentration function and has important consequences for malodor control. A low slope value indicates an odor that requires greater relative dilution for the odor to dissipate; a high slope value indicates an odor that can be more quickly reduced by ventilation. For example, an exponent of 0.7 implies that in order to reduce perceived intensity by a factor of 5, the concentration must be reduced by a factor of 10; an exponent of 0.2 implies that a reduction in perceived magnitude by a factor of 5 would require a reduction in concentration by a factor of more than 3000. Examples of compounds with low slope values include hydrogen sulfide, butyl acetate and amines. Compounds with high slope values include ammonia and aldehydes.
The ability of ventilation to control odors also depends on the strength of the source generating the odorant(s) and the nature of the odor. An odorant with a stronger source requires proportionately more ventilation to achieve the same reduction in concentration.
Odors that are perceived as unpleasant may require substantially greater reduction before being perceived as acceptable. In addition, some sources, such as painted walls and flooring materials, may show increased emission rates in response to increased ventilation rates, which further complicates this picture (Gunnarsen 1997).
ODOR CONCENTRATION Analytical Measurement Performance data on the control of specific odorants can be obtained using suitable analytical methods. Detectors permit detec-tion of substances in amounts as little as 1 ng. Air contains many minor components, so gas chromatographic separation of the com-ponents must precede detection. Because odor thresholds for some compounds are low, preconcentration of the minor components is necessary. Preconcentration consists of adsorption or absorption by a stable, sufficiently nonvolatile material, followed by thermal desorption or extraction. The state of the art for sampling and anal-ysis of VOCs in indoor air is reviewed in NIOSH (1993).
Mass spectrometry can be used with gas chromatography to identify constituents of complex mixtures. The chromatograph resolves a mixture into its constituents, and the spectrometer pro-vides identification and concentration of selected constituents.
Several other detectors are sufficiently sensitive and specific to detect resolved components. Hydrogen flame ionization detectors respond adequately and nearly mass-proportionally to almost all organic compounds. Flame photometric detectors can pinpoint with equal sensitivity compounds that contain sulfur; many sulfur compounds are strongly odorous and are of interest in odor work. A Coulson conductometric detector is specifically and adequately sensitive to ammonia and organic nitrogen compounds. Thermal conductivity detectors are generally not sensitive enough for ana-lytical work on odors.
Frequently, a sniffing port (Dravnieks and O’Donnell 1971; Dravnieks and Krotoszynski 1969) is installed in parallel with the detector(s). Part of the resolved effluent exhausts through the port and allows the components that are particularly odorous or carry Fig. 4 Matching Functions Obtained with Dravnieks Olfactometer (Dravnieks and Laffort 1972; Cain 1978) 13.6 2001 ASHRAE Fundamentals Handbook (SI) some relevant odor quality to be annotated. Usually, only a fraction of all components studied exhibits odors.
Airborne VOCs cause odors, but the correlation between indoor VOC concentrations and odor complaints in indoor envi-ronments is poor. Considerable work is being done on artificial noses, which may be the future of objective determination of odor-ants (Bartlett et al. 1997; Freund and Lewis 1995; Moy et al. 1994; Taubes 1996). However, because the physicochemical correlates of olfaction are poorly understood, no simple analytical means to predict the perceived quality and intensity of an odorant exists.
Moreover, since acceptability of an odorant depends strongly on context, it is unlikely that analytical instruments will supplant human evaluation.
Odor Units Odor concentration can be expressed as the number of unit vol-umes that a unit volume of odorous sample occupies when diluted to the odor threshold with nonodorous air. If a sample of odorous air can be reduced to threshold by a tenfold dilution with pure air, the concentration of the original sample is said to be 10 odor units.
Hence, odor units are equivalent to multiples of threshold concen-trations. Odor units are not units of perceived magnitude.
Odor units are widely used to express legal limits for emission of odoriferous materials. For example, the law may state that a fac-tory operation may not cause the ambient odor level to exceed 15 odor units. For every odorant (chemical), odor units and parts per million (ppm) are proportional. The proportionality constant varies from one odorant to another, depending on the number of ppm needed to evoke a threshold response. Perceived odor magnitude (intensity), however, does not grow proportionally with concentra-tion expressed in ppm. Therefore, it cannot grow proportionally with concentration expressed in odor units. For example, a sample of 20 odor units is always perceived as less than twice as strong as a sample of 10 odor units. Moreover, because the psychophysical function (slope) varies from one odorant to another, samples of two odorants, each at 20 odor units, may have unequal perceived intensities.
Although odor units are not equivalent to units of perceived mag-nitude, they can be useful. Most indoor and outdoor contaminants are complex mixtures, so that the actual concentration of the odor-iferous portion of a sample cannot be expressed with certainty.
Thus, the odor unit is a useful measure of concentration of the mix-ture when evaluating, for example, the efficiency of a filter or ven-tilation system to remove or dilute the odor.
OLF UNITS Sometimes IAQ scientists cannot successfully resolve com-plaints about the air in offices, schools, and other nonindustrial environments. Customarily, complaints are attributed to elevated pollutant concentrations; frequently, however, such high concentra-tions are not found, yet complaints persist.
On the assumption that the inability to find a difference between air pollutant levels in buildings with registered com-plaints and those without complaints is due to inadequacies of pre-vailing measurement techniques, Fanger and others changed the focus from chemical analysis to sensory analysis (Fanger 1987, 1988; Fanger et al. 1988). Fanger quantified air pollution sources by comparing them with a well-known source: a sedentary person in thermal comfort. A new unit, the olf, was introduced. An olf is defined as the emission rate of air pollutants (bioeffluents) from a standard person. A decipol is one olf ventilated at a rate of 10 L/s of unpolluted air.
To use these units, Fanger generated a curve that relates the per-centage of persons dissatisfied with air polluted by human bioefflu-ents as a function of the outdoor air ventilation rate and obtained the following expression: (2) where D = percentage of persons dissatisfied q = ventilation-emission ratio, L/s per olf This curve, which is shown in Figure 5, is based on experiments involving more than 1000 European subjects (Fanger and Berg-Munch 1983). Experiments with American (Cain et al. 1983) and Japanese (Iwashita et al. 1990) subjects show very similar results.
The idea behind the olf concept is that sensory sources other than humans also be expressed in equivalent standard persons (i.e., in olfs). A room should therefore be ventilated to handle the total sensory load from persons and building. The olf concept is used in European publications for ventilation (CEN 1998; ECA 1992) to determine the required ventilation and in several national standards (DIN 1994; Norwegian Building Code 1996). Table 3 shows the sensory loads from different pollution sources used in CEN (1998).
Example. Office, low-polluting building, occupancy 0.07 persons/m2 30% dissatisfied requires 4 L/s per olf ventilation rate (Figure 5) Required ventilation: 4 × 0.17 = 0.7 L/(s·m2) Table 3 Sensory Pollution Load from Different Pollution Sources Source Sensory Load Sedentary person (1 to 1.5 met) 1 olf Person exercising Low level (3 met) 4 olf Medium level (6 met) 10 olf Children, kindergarten (3 to 6 yrs) 1.2 olf Children, school (4 to 16 yrs) 1.3 olf Low-polluting building 0.1 olf/m2 Non-low-polluting building 0.2 olf/m2 Source: CEN (1998).
Occupants 0.07 olf/m2 Building 0.1 olf/m2 Total sensory load 0.17 olf/m2 Fig. 5 Percentage of Dissatisfied Persons as a Function of Ventilation Rate per Standard Person (i.e., per Olf) (CEN 1998) D 395 1.83 – q0.26 ( ) for q 0.332 ≥ exp = D 100 for q 0.332 < = Odors 13.7 The sensory load on the air in a space can be determined from Figure 5 by measuring the outdoor ventilation rate and determining the percent dissatisfied, using an untrained panel with a minimum of 20 impartial persons (ASHRAE Standard 62; Gunnarsen and Fanger 1992). The panel judges the acceptability of the air just after entering the space. The required ventilation rate depends on the desired per-centage of occupant satisfaction. In ASHRAE Standard 62, 80% acceptability (20% dissatisfied) is the goal; European guidelines offer three quality levels: 15%, 20%, and 30% dissatisfied.
Although this system has much to offer from a theoretical stand-point, its use is controversial in some areas. Problems have been found in cultural differences among panel members and access to outdoor air for dilution (Aizlewood et al. 1996). The trend is now to use untrained panels as described in the previous paragraph. Knud-sen et al. (1998) have shown that the curve giving the relation between percent dissatisfied and ventilation rate for some building materials is less steep than that in Figure 5, while other materials show a steeper curve. The sensory load in this case depends on the ventilation rate. The constant sensory loads given in Table 3 should therefore be seen as a first approximation.
CODES AND STANDARDS ASHRAE.
1999.
Ventilation for acceptable indoor air quality.
ANSI/ASHRAE Standard 62-1989.
ASTM. 1997. Standard practices for referencing suprathreshold odor inten-sity. ASTM Standard E 544-75 (R 1993). American Society for Testing and Materials, West Conshohocken, PA.
DIN. 1994. Raumlufttechnik; Gesundheitstechnische Anforderungen, VDI-Lüftungsregeln. DIN Standard 1946-2. Deutsches Institut für Normung, Berlin.
Norwegian Building Code. 1996. Oslo, Statens Hygningstekniske Eur.
REFERENCES ACGIH. 1998 (revised annually). Threshold limit values for chemical sub-stances and physical agents and biological exposure indices. American Conference of Governmental Industrial Hygienists, Cincinnati, OH.
AIHA. 1989. Odor thresholds for chemicals with established occupational health standards. American Industrial Hygiene Association, Akron, OH.
Aizlewood, C.E., G.J Raw, and N.A. Oseland. 1996. Decipols: Should we use them? Indoor Built Environment 5:263-69.
Amoore, J.E. 1962. The stereochemical theory of olfaction 1, Identification of seven primary odors. Proc. Sci. Sect. Toilet Goods Assoc. 37:1-12.
ASTM. 1985. Atlas of odor character profiles. Data Series 61. American Society for Testing and Materials, West Conshohocken, PA.
ASTM. 1996. Manual on sensory testing methods. Special Technical Publi-cation STP 434-96.
Bartlett, P., J. Elliott, and J. Gardner. 1997. Electronic noses and their appli-cations in the food industry. Food Technology 51(12):44-48.
Beck, L. and V. Day. 1991. New Jersey’s approach to odor problems. From Transactions: Recent developments and current practices and odor reg-ulations control and technology, D.R. Derenzo and A. Gnyp, eds. Air and Waste Management Association, Pittsburgh, PA.
Berglund, L. and W.S. Cain. 1989. Perceived air quality and the thermal environment. In Proceedings of IAQ ’89: The Human Equation: Health and Comfort, San Diego, pp. 93-99.
Berg-Munch, B. and P.O. Fanger. 1982. The influence of air temperature on the perception of body odour. Environment International 8:333-35.
Cain, W.S. 1978. The odoriferous environment and the application of olfac-tory research. In Handbook of perception, Vol. 6, Tasting, smelling, feel-ing, and hurting, eds. Carterette and Friedman. Academic Press, New York.
Cain, W.S. and J.E. Cometto-Muñiz. 1995. Irritation and odors as indicators of indoor pollution. Occupational Medicine 10(1):133-35.
Cain, W.S. and H.R. Moskowitz. 1974. Psychophysical scaling of odor. In Human responses to environmental odors, eds. Turk, Johnson, and Moulton. Academic Press, New York.
Cain, W.S., B.P. Leaderer, R. Isseroff, L.G. Berglund, R.I. Huey, E.D. Lipsitt, and D. Perlman. 1983. Ventilation requirements in buildings—I: Control of occupancy odor and tobacco smoke odor. Atmospheric Environment 17:1183-97.
CEN. 1998. Ventilation for buildings: Design criteria for the indoor environ-ment. Technical Report CR1752. European Committee for Standardiza-tion, Brussels.
Clausen, G., P.O Fanger, W.S. Cain, and B.P. Leaderer. 1985. The influence of aging particle filtration and humidity on tobacco smoke odour. In Pro-ceedings of CLIMA 2000 4:345-49. Copenhagen.
Cometto-Muñiz, J.E. and W.S. Cain. 1992. Sensory irritation, relation to indoor air pollution in sources of indoor air contaminants—characteriz-ing emissions and health effects. Annals of the New York Academy of Sci-ences, Vol. 641.
Cometto-Muñiz, J.E. and W.S. Cain. 1995. Olfactory adaptation. In Hand-book of Olfaction and Gustation, ed. R.L. Doty. Marcel Dekker, New York.
Cometto-Muñiz, J.E., W.S. Cain, and H.K. Hudnell. 1997. Agonistic sen-sory effects of airborne chemicals in mixtures: Odor, nasal pungency and eye irritation. Perception and Psychophysics 59(5):665-74.
Cowart, B.J., K. Flynn-Rodden, S.J. McGeady, and L.D. Lowry. 1993.
Hyposmia in allergic rhinitis. Journal of Allergy and Clinical Immunol-ogy 91:747-51.
Cowart, B.J., I.M. Young, R.S. Feldman, and L.D. Lowry. 1997. Clinical disorders of smell and taste. Occupational Medicine 12:465-83.
Dalton, P. 1996. Odor perception and beliefs about risk. Chemical Senses 21:447-58.
Dalton, P. and C.J. Wysocki. 1996. The nature and duration of adaptation following long-term exposure to odors. Perception & Psychophysics 58(5):781-92.
Dalton, P., C.J. Wysocki, M.J. Brody, and H.J. Lawley. 1997. The influence of cognitive bias on the perceived odor, irritation and health symptoms from chemical exposure. International Archives of Occupational and Environmental Health 69:407-17.
Dravnieks, A. 1975. Evaluation of human body odors, methods and inter-pretations. Journal of the Society of Cosmetic Chemists 26:551.
Dravnieks, A. and B. Krotoszynski. 1969. Analysis and systematization of data for odorous compounds in air. ASHRAE Symposium Bulletin, Odor and odorants: The engineering view.
Dravnieks, A. and P. Laffort. 1972. Physicochemical basis of quantitative and qualitative odor discrimination in humans. Olfaction and Taste IV:142, ed. Schneider. Wissenschaftliche Verlagsgesellschaft mbH, Stuttgart.
Dravnieks, A. and A. O’Donnell. 1971. Principles and some techniques of high resolution headspace analysis. Journal of Agricultural and Food Chemistry 19:1049.
ECA. 1992. Guidelines for ventilation requirements in buildings. European Collaborative Action Indoor Air Quality and its Impact on Man. Report No. 11. EUR 14449 EN. Office for Official Publications of the European Committees, Luxembourg.
Engen, T., R.A. Kilduff, and N.J. Rummo. 1975. The influence of alcohol on odor detection. Chemical Senses and Flavor 1:323.
EPA. 1992. Reference guide to odor thresholds for hazardous air pollutants listed in the Clean Air Act Amendments of 1990. EPA/600/R-92/047.
Office of Research and Development, U.S. Environmental Protection Agency, Washington, D.C.
Fang, L., G. Clausen, and P.O. Fanger. 1998a. Impact of temperature and humidity on the perception of indoor air quality. Indoor Air 8(2):80-90.
Fang, L., G. Clausen, and P.O. Fanger. 1998b. Impact of temperature and humidity on perception of indoor air quality during immediate and longer whole-body exposures. Indoor Air 8(4):276-84.
Fanger, P.O. 1987. A solution to the sick building mystery. In Indoor Air ’87, Proceedings of the International Conference on Indoor Air and Climate.
Institute of Water, Soil and Air Hygiene, Berlin.
Fanger, P.O. 1988. Introduction of the olf and decipol units to quantify air pollution perceived by humans indoors and outdoors. Energy and Build-ings 12:1-6.
Fanger, P.O. and B. Berg-Munch. 1983. Ventilation and body odor. In Pro-ceedings of Engineering Foundation Conference on Management of Atmospheres in Tightly Enclosed Spaces, pp. 45-50. ASHRAE, Atlanta.
Fanger, P.O., J. Lauridsen, P. Bluyssen, and G. Clausen. 1988. Air pollution sources in offices and assembly halls quantified by the olf unit. Energy and Buildings 12:7-19.
Freund, M.S. and N.S. Lewis. 1995. A chemically diverse conduction poly-mer-based “electronic nose”. Proceedings of the National Academy of Sciences 92:2652-56.
Frey, A.F. 1995. A review of the nature of odour perception and human response. Indoor Environment 4:302-05.
13.8 2001 ASHRAE Fundamentals Handbook (SI) Green, B.G., P. Dalton, B. Cowart, G. Shaffer, K.R. Rankin, and J. Higgins.
1996. Evaluating the “labeled magnitude scale” for measuring sensations of taste and smell. Chemical Senses 21(3):323-34.
Gunnarsen, L. 1997. The influence of area-specific rate on the emissions from construction products. Indoor Air 7:116-20.
Gunnarsen, L. and P.O. Fanger. 1992. Adaptation to indoor air pollution.
Energy and Buildings 18:43-54.
Harper, R., E.C. Bate Smith, and D.G. Land. 1968. Odour description and odour classification. American Elsevier Publishing, New York. Distrib-utors for Churchill Livingston Publishing, Edinburgh, Scotland.
Iwashita, G., K. Kimura, S. Tanabe, S. Yoshizawa, and K. Ikeda. 1990.
Indoor air quality assessment based on human olfactory sensation. Jour-nal of Architecture, Planning and Environmental Engineering 410:9-19.
Kerka, W.F. and C.M. Humphreys. 1956. Temperature and humidity effect on odour perception. ASHRAE Transactions 62:531-52.
Knudsen, H.N. et al. 1998. Determination of exposure-response relation-ships for emissions from building products. Indoor Air 8(4):264-75.
Koelega, H.S. and E.P. Koster. 1974. Some experiments on sex differences in odor perception. Annals of the New York Academy of Sciences 237:234.
Meilgaard, M., G.V. Civille, and B.T. Carr. 1987. Sensory evaluation tech-niques. CRC Press, Boca Raton, FL.
Moskowitz, H.R., A. Dravnieks, W.S. Cain, and A. Turk. 1974. Standard-ized procedure for expressing odor intensity. Chemical Senses and Fla-vor 1:235.
Moy, L., T. Tan, and J.W. Gardner. 1994. Monitoring the stability of perfume and body odors with an “electronic nose”. Perfumer and Flavorist 19:11-16.
NIOSH. 1993. Case studies—Indoor environmental quality “from the ground up.” Applied Occupational and Environmental Hygiene 8:677-80.
Pierce, J.J.D., C.J. Wysocki, E.V. Aronov, J.B. Webb, and R.M. Boden. 1996.
The role of perceptual and structural similarity in cross adaptation. Chem-ical Senses 21:223-27.
Ruth, J.H. 1986. Odor thresholds and irritation levels of several chemical substances: A review. American Industrial Hygiene Association Journal 47:142-51.
Schneider, R.A. 1974. Newer insights into the role and modifications of olfaction in man through clinical studies. Annals of the New York Acad-emy of Sciences 237:217.
Shams Esfandabad, H. 1993. Perceptual analysis of odorous irritants in indoor air. Ph.D. diss., Department of Psychology, Stockholm Univer-sity.
Stevens, J.C., W.S. Cain, F.T. Schiet, and M.W. Oatley. 1989. Olfactory adaptation and recovery in old age. Perception 18(22):265-76.
Stevens, S.S. 1957. On the psychophysical law. Psychological Review 64:153.
Taubes, G. 1996. The electronic nose. Discover (September):40-50.
Toftum, J. et al. 1998. Upper limits for air humidity for preventing warm res-piratory discomfort. Energy and Buildings 28(3):15-23.
Wysocki, C.J. and G.K. Beauchamp. 1984. Ability to smell androstenone is genetically determined. In Proceeding of the National Academy of Sci-ences of the United States of America 81:4899-4902.
Wysocki, C.J. and A.N. Gilbert. 1989. National geographic smell survey: Effects of age are heterogeneous. In Nutrition and the Chemical Senses in Aging: Recent Advances and Current Research, eds. C.L. Murphy, W.S. Cain, and D.M. Hegsted, pp. 12-28. New York Academy of Sci-ences, New York.
Wysocki, C.J., P. Dalton, P., M.J. Brody, and H.J. Lawley. 1997. Acetone odor and irritation thresholds obtained from acetone-exposed factory workers and from control (occupationally non-exposed) subjects. Amer-ican Industrial Hygiene Association Journal 58:704-12.
Yaglou, C.P., E.C. Riley, and D.I. Coggins. 1936. Ventilation requirements.
ASHRAE Transactions 42:133-62.
BIBLIOGRAPHY ACGIH. 1988. Advances in air sampling. American Conference of Govern-ment Industrial Hygienists, Cincinnati, OH.
Clemens, J.B. and R.G. Lewis. 1988. Sampling for organic compounds. In Principles of environmental sampling 20:147-57. American Chemical Society, Washington, D.C.
Moschandreas, D.J. and S.M. Gordon. 1991. Volatile organic compounds in the indoor environment: Review of characterization methods and indoor air quality studies. In Organic chemistry of the atmosphere. CRC Press, Boca Raton, FL.
14.1 CHAPTER 14 MEASUREMENT AND INSTRUMENTS Terminology ............................................................................ 14.1 Uncertainty Analysis ............................................................... 14.2 Temperature Measurement ..................................................... 14.3 Humidity Measurement ........................................................... 14.9 Pressure Measurement .......................................................... 14.12 Velocity Measurement ........................................................... 14.14 Flow Rate Measurement ....................................................... 14.17 Air Infiltration, Airtightness, and Outdoor Air Ventilation Rate Measurement .......................................... 14.21 Carbon Dioxide Measurement .............................................. 14.22 Electric Measurement ........................................................... 14.23 Rotative Speed Measurement ................................................ 14.24 Sound and Vibration Measurement ....................................... 14.24 Lighting Measurement .......................................................... 14.27 Thermal Comfort Measurement ............................................ 14.28 Moisture Content and Transfer Measurement ...................... 14.29 Heat Transfer Through Building Materials .......................... 14.30 Air Contaminant Measurement ............................................. 14.30 Combustion Analysis ............................................................. 14.31 Data Acquisition and Recording ........................................... 14.31 EATING, refrigerating, and air-conditioning engineers and H technicians require instruments for both laboratory work and fieldwork. Precision is more essential in the laboratory, where research and development are undertaken, than in the field, where acceptance and adjustment tests are conducted. This chapter describes the characteristics and uses of some of these instruments.
TERMINOLOGY The following definitions are generally accepted.
Accuracy. The capability of an instrument to indicate the true value of measured quantity. This is often confused with inaccuracy, which is the departure from the true value to which all causes of error (e.g., hysteresis, nonlinearity, drift, temperature effect, and other sources) contribute.
Amplitude. The magnitude of variation from its zero value in an alternating quantity.
Average. The sum of a number of values divided by the number of values.
Bandwidth. The range of frequencies over which a given device is designed to operate within specified limits.
Bias. The tendency of an estimate to deviate in one direction from a true value (a systematic error).
Calibration. (1) The process of comparing a set of discrete mag-nitudes or the characteristic curve of a continuously varying magni-tude with another set or curve previously established as a standard.
Deviation between indicated values and their corresponding stan-dard values constitutes the correction (or calibration curve) for inferring true magnitude from indicated magnitude thereafter; (2) the process of adjusting an instrument to fix, reduce, or eliminate the deviation defined in (1).
Calibration curve. (1) The path or locus of a point that moves so that its coordinates on a graph correspond to values of input signals and output deflections; (2) the plot of error versus input (or output).
Confidence. The degree to which a statement (measurement) is believed to be true.
Dead band. The range of values of the measured variable to which an instrument will not effectively respond.
Deviate. Any item of a statistical distribution that differs from the selected measure of control tendency (average, median, mode).
Deviation, standard. The square root of the average of the squares of the deviations from the mean (root mean square devia-tion). A measure of dispersion of a population.
Distortion. An unwanted change in wave form. Principal forms of distortion are inherent nonlinearity of the device, nonuniform response at different frequencies, and lack of constant proportional-ity between phase-shift and frequency. (A wanted or intentional change might be identical, but it is called modulation.) Drift. A gradual, undesired change in output over a period of time that is unrelated to input, environment, or load. Drift is gradual; if variation is rapid, the fluctuation is referred to as cycling.
Dynamic error band. The spread or band of output-amplitude deviation incurred by a constant amplitude sine wave as its fre-quency is varied over a specified portion of the frequency spectrum (see Static error band).
Error. The difference between the true or actual value to be mea-sured (input signal) and the indicated value (output) from the mea-suring system. Errors can be systematic or random.
Error, accuracy. See Error, systematic.
Error, fixed. See Error, systematic.
Error, instrument. The error of an instrument’s measured value that includes random or systematic errors.
Error, precision. See Error, random.
Error, probable. An error with a 50% or higher chance of occur-rence. A statement of probable error is of little value.
Error, random. A statistical error caused by chance and not recurring. This term is a general category for errors that can take values on either side of an average value. To describe a random error, its distribution must be known.
Error, root mean square, or RMS. An accuracy statement of a system comprising several items. For example, a laboratory poten-tiometer, volt box, null detector, and reference voltage source have individual accuracy statements assigned to them. These errors are generally independent of one another, so a system of these units dis-plays an accuracy given by the square root of the sum of the squares of the individual limits of error. For example, four individual errors of 0.1% yield a calibrated accuracy of 0.4% but an RMS error of only 0.2%.
Error, systematic. A persistent error not due to chance; system-atic errors are causal. A systematic error is likely to have the same magnitude and sign for every instrument constructed with the same components and procedures. Errors in calibrating equipment cause systematic errors because all instruments calibrated are biased in the direction of the calibrating equipment error. Voltage and resistance drifts over time are generally in one direction and are classed as sys-tematic errors.
Frequency response (flat). The portion of the frequency spec-trum over which the measuring system has a constant value of amplitude response and a constant value of time lag. Input signals that have frequency components within this range are indicated by the measuring system (without distortion).
The preparation of this chapter is assigned to TC 1.2, Instruments and Measurements.
14.2 2001 ASHRAE Fundamentals Handbook (SI) Hysteresis. The summation of all effects, under constant envi-ronmental conditions, that cause the output of an instrument to assume different values at a given stimulus point when that point is approached with increasing stimulus and with decreasing stim-ulus. Hysteresis includes backlash. It is usually measured as a per-cent of full scale when input varies over the full increasing and decreasing range. In instrumentation, hysteresis and dead band are similar quantities.
Linearity. The straight-lineness of the transfer curve between an input and an output; that condition prevailing when output is directly proportional to input (see Nonlinearity).
Loading error. A loss of output signal from a device caused by a current drawn from its output. It increases the voltage drop across the internal impedance, where no voltage drop is desired.
Mean. See Average.
Median. The middle value in a distribution, above and below which lie an equal number of values.
Noise. Any unwanted disturbance or spurious signal that modi-fies the transmission, measurement, or recording of desired data.
Nonlinearity. The prevailing condition (and the extent of its measurement) under which the input-output relationship (known as the input-output curve, transfer characteristic, calibration curve, or response curve) fails to be a straight line. Nonlinearity is measured and reported in several ways, and the way, along with the magni-tude, must be stated in any specification.
Minimum-deviation-based nonlinearity. The maximum depar-ture between the calibration curve and a straight line drawn to give the greatest accuracy; expressed as a percent of full-scale deflection.
Slope-based nonlinearity. The ratio of maximum slope error any-where on the calibration curve to the slope of the nominal sensitivity line; usually expressed as a percent of nominal slope.
Most variations beyond these two definitions result from the many ways in which the straight line can be arbitrarily drawn. All are valid as long as construction of the straight line is explicit.
Population. A group of individual persons, objects, or items from which samples may be taken for statistical measurement.
Precision. The repeatability of measurements of the same quan-tity under the same conditions; not a measure of absolute accuracy.
The precision of a measurement is used here to describe the relative tightness of the distribution of measurements of a quantity about their mean value. Therefore, precision of a measurement is associ-ated more with its repeatability than its accuracy. It combines uncer-tainty caused by random differences in a number of identical measurements and the smallest readable increment of the scale or chart. Precision is given in terms of deviation from a mean value.
Primary calibration. A calibration procedure in which the instrument output is observed and recorded while the input stimulus is applied under precise conditions—usually from a primary exter-nal standard traceable directly to the National Institute of Standards and Technology (NIST).
Range. A statement of the upper and lower limits between which an instrument’s input can be received and for which the instrument is calibrated.
Reliability. The probability that an instrument’s precision and accuracy will continue to fall within specified limits.
Repeatability. See Precision.
Reproducibility. In instrumentation, the closeness of agreement among repeated measurements of the output for the same value of input made under the same operating conditions over a period of time, approaching from both directions; it is usually measured as a nonreproducibility and expressed as reproducibility in percent of span for a specified time period. Normally, this implies a long period of time, but under certain conditions, the period may be a short time so that drift is not included. Reproducibility includes hys-teresis, dead band, drift, and repeatability. Between repeated mea-surements, the input may vary over the range, and operating conditions may vary within normal limits.
Resolution. The smallest change in input that produces a detect-able change in instrument output. Resolution differs from precision in that it is a psychophysical term referring to the smallest increment of humanly perceptible output (rated in terms of the corresponding increment of input). The precision, the resolution, or both may be bet-ter than the accuracy. An ordinary six-digit (or dial) instrument has a resolution of one part per million (ppm) of full scale; however, it is possible that the accuracy is no better than 25 ppm (0.0025%). Note that the practical resolution of an instrument cannot be any better than the resolution of the indicator or detector, whether internal or external.
Scale factor. (1) The amount by which a measured quantity must change to produce unity output; (2) the ratio of real to analog values.
Sensitivity. The property of an instrument that determines scale fac-tor. The word is often short for maximum sensitivity or the minimum scale factor with which an instrument can respond. The minimum input signal strength required to produce a desired value of output signal (e.g., full scale or unit output or the ratio of output to input values).
Sensitivity inaccuracy. The maximum error in sensitivity dis-played as a result of the summation of the following: frequency response; attenuator inaccuracy; hysteresis or dead band; amplitude distortion (sensitivity nonlinearity); phase distortion (change in phase relationship between input signal and output deflection); and gain instability. Only by taking into account all these factors can nominal sensitivity, as indicated by the numeral on the attenuator readout, be discounted for accurate interpretation.
Stability. (1) Independence or freedom from changes in one quantity as the result of a change in another; (2) the absence of drift.
Static error band. (1) The spread of error present if the indicator (pen, needle) stopped at some value (e.g., at one-half of full scale).
It is normally reported as a percent of full scale; (2) a specification or rating of maximum departure from the point where the indicator must be when an on-scale signal is stopped and held at a given sig-nal level. This definition stipulates that the stopped position can be approached from either direction in following any random wave-form. Therefore, it is a quantity that includes hysteresis and nonlin-earity but excludes items such as chart paper accuracy or electrical drift (see Dynamic error band).
Step-function response. The characteristic curve or output plot-ted against time resulting from the input application of a step func-tion (a function that is zero for all values of time before a certain instant, and a constant for all values of time thereafter).
Threshold. The smallest stimulus or signal that results in a detectable output.
Time constant. The time required for an exponential quantity to change by an amount equal to 0.632 times the total change required to reach steady state for first-order systems.
Transducer. A device for translating the changing magnitude of one kind of quantity into corresponding changes of another kind of quantity. The second quantity often has dimensions different from the first and serves as the source of a useful signal. The first quantity may be considered an input and the second an output. Significant energy may or may not transfer from the transducer’s input to output.
Uncertainty. An estimated value for the error (i.e., what an error might be if it were measured by calibration). Although uncertainty may be the result of both systematic and precision errors, only pre-cision error can be treated by statistical methods.
Zero shift. Drift in the zero indication of an instrument without any change in the measured variable.
UNCERTAINTY ANALYSIS Uncertainty Sources Measurement generally consists of a sequence of operations or steps. Virtually every step introduces a conceivable source of uncer-tainty, the effect of which must be assessed. The following list is rep-resentative of the most common, but not all, sources of uncertainty.
Measurement and Instruments 14.3 • Inaccuracy in the mathematical model that describes the physical quantity • Inherent stochastic variability of the measurement process • Uncertainties in measurement standards and calibrated instru-mentation • Time-dependent instabilities due to gradual changes in standards and instrumentation • Effects of environmental factors such as temperature, humidity, and pressure • Values of constants and other parameters obtained from outside sources • Uncertainties arising from interferences, impurities, inhomoge-neity, inadequate resolution, and incomplete discrimination • Computational uncertainties and data analysis • Incorrect specifications and procedural errors • Laboratory practice, including handling techniques, cleanliness, and operator techniques, etc.
• Uncertainty in corrections made for known effects, such as installation effect corrections Uncertainty of a Measured Variable For a measured variable Xi, the total error is caused by both pre-cision (random) and systematic (bias) errors. This relationship is shown in Figure 1. The possible measurement values of the vari-able are scattered in a distribution around the parent population mean µi (Figure 1A). The curve is the normal or Gaussian distri-bution and is the theoretical distribution function for the infinite population of measurements that generated Xi. The parent popula-tion mean differs from (Xi)true by an amount called the systematic (or bias) error βi (Figure 1B). The quantity βi is the total fixed error that remains after all calibration corrections have been made. In general, there are several sources of bias error, such as calibration standard errors, data acquisition errors, data reduction errors, and test technique errors. There is usually no direct way to measure these errors. These errors are unknown and are assumed to be zero; otherwise, an additional correction would be applied to reduce them to as close to zero as possible.
The precision uncertainty for a variable, which is an estimate of the possible error associated with the repeatability of a particular measurement, is determined from the sample standard deviation, or the estimate of the error associated with the repeatability of a par-ticular measurement. Unlike the systematic error, the precision error varies from reading to reading. As the number of readings of a par-ticular variable tends to infinity, the distribution of these possible errors becomes Gaussian.
For each bias error source, the experimenter must estimate a sys-tematic uncertainty. Systematic uncertainties are usually estimated from previous experience, calibration data, analytical models, and engineering judgment. For a discussion on estimating systematic uncertainties (bias limits), see Coleman and Steele (1989).
For further information on measurement uncertainty, see ASME Standards MFC-2M and PTC 19.1 and Coleman and Steele (1995).
TEMPERATURE MEASUREMENT Instruments for measuring temperature are listed in Table 1.
Temperature sensor output must be related to an accepted tempera-ture scale. This is achieved by manufacturing the instrument according to certain specifications or by calibrating it against a tem-perature standard. To help users conform to standard temperatures and temperature measurements, the International Committee of Weights and Measures (CIPM) has adopted the International Tem-perature Scale of 1990 (ITS-90).
The unit of temperature of the ITS-90 is the kelvin (K) and has a size equal to the fraction 1/273.16 of the thermodynamic tempera-ture of the triple point of water.
The ITS-90 is maintained in the United States by the National Institute of Standards and Technology (NIST), and any laboratory may obtain calibrations from NIST based on this scale.
Benedict (1984), Considine (1985), Quinn (1990), Schooley (1986, 1992), and DeWitt and Nutter (1988) cover temperature measurement in more detail.
Sampling and Averaging Although temperature is usually measured within, and is associ-ated with, a relatively small volume (depending on the size of the thermometer), it can also be associated with an area (e.g., on a sur-face or in a flowing stream). To determine average stream temper-ature, the cross section must be divided into smaller areas and the temperature of each area measured. The temperatures measured are then combined into a weighted mass flow average by either (1) using equal areas and multiplying each temperature by the fraction of total mass flow in its area or (2) using areas of size inversely pro-portional to mass flow and taking a simple arithmetic average of the temperatures in each. A means of mixing or selective sampling may be preferable to these cumbersome procedures. While mixing can occur from turbulence alone, transposition is much more effective.
In transposition, the stream is divided into parts determined by the type of stratification, and alternate parts pass through one another.
Static Temperature Versus Total Temperature When a fluid stream impinges on a temperature-sensing element such as a thermometer or thermocouple, the element is at a temper-ature greater than the true stream temperature. The difference is a fraction of the temperature equivalent of the stream velocity te.
(1) where te = temperature equivalent of stream velocity, °C V = velocity of stream, m/s J = mechanical equivalent of heat, 1000 N·m/kJ cp = specific heat of stream at constant pressure, kJ/(kg·K) Fig. 1 Errors in the Measurement of a Variable X te V 2 2Jcp -----------= 14.4 2001 ASHRAE Fundamentals Handbook (SI) Table 1 Temperature Measurement Measurement Means Application Approximate Range, °C Uncertainty, K Limitations Liquid-in-glass thermometers Mercury-in-glass Temperature of gases and liquids by contact −38/550 0.03 to 2 In gases, accuracy affected by radiation Organic Temperature of gases and liquids by contact −200/200 0.03 to 2 In gases, accuracy affected by radiation Gas thermometer Primary standard −271/665 Less than 0.01 Requires considerable skill to use Resistance thermometers Platinum Precision; remote readings; temper-ature of fluids or solids by contact −259/1000 Less than 0.0001 to 0.1 High cost; accuracy affected by radiation in gases Rhodium-iron Transfer standard for cryogenic applications −273/−243 0.0001 to 0.1 High cost Nickel Remote readings; temperature by contact −250/200 0.01 to 1 Accuracy affected by radiation in gases Germanium Remote readings; temperature by contact −273/−243 0.0001 to 0.1 Thermistors Remote readings; temperature by contact Up to 200 0.0001 to 0.1 Thermocouples Pt-Rh/Pt (type S) Standard for thermocouples on IPTS-68, not on ITS-90 0/1450 0.1 to 3 High cost Au/Pt Highly accurate reference ther-mometer for laboratory applications −50/1000 0.05 to 1 High cost Types K and N General testing of high temperature; remote rapid readings by direct contact Up to 1250 0.1 to 10 Less accurate than listed above thermocouples Iron/Constantan (type J) Same as above Up to 750 0.1 to 6 Subject to oxidation Copper/Constantan (type T) Same as above, especially suited for low temperature Up to 350 0.1 to 3 Ni-Cr/Constantan (type E) Same as above, especially suited for low temperature Up to 900 0.1 to 7 Beckman thermometers (metastatic) For differential temperature in same applications as in glass-stem thermometer 0 to 100 0.005 Must be set for temperature to be measured Bimetallic thermometers For approximate temperature −20/660 1, usually much more Time lag; unsuitable for remote use Pressure-bulb thermometers Gas-filled bulb Remote testing −75/660 2 Caution must be exercised so that installation is correct Vapor-filled bulb Remote testing −5/250 2 Caution must be exercised so that installation is correct Liquid-filled bulb Remote testing −50/1150 2 Caution must be exercised so that installation is correct Optical pyrometers For intensity of narrow spectral band of high-temperature radiation (remote) 800 and up 15 Radiation pyrometers For intensity of total high-tempera-ture radiation (remote) Any range Seger cones (fusion pyrometers) Approximate temperature (within temperature source) 660/2000 50 Triple points, freezing/melting points, and boiling points of materials Standards All except ex-tremely high temperatures Extremely precise For laboratory use only Measurement and Instruments 14.5 This fraction of the temperature equivalent of the velocity is the recovery factor, and it varies from 0.3 to 0.4 K for bare thermom-eters to 0.5 K for aerodynamically shielded thermocouples. For pre-cise temperature measurement, each temperature sensor must be calibrated to determine its recovery factor. However, for most appli-cations where air velocities are below 10 m/s, the recovery factor can be omitted.
Various temperature sensors are available for temperature mea-surement in fluid streams. The principal sensors are the static tem-perature thermometer, which indicates true stream temperature but is cumbersome, and the thermistor, used for accurate tempera-ture measurement within a limited range.
LIQUID-IN-GLASS THERMOMETERS Any device that changes monotonically with temperature is a thermometer; however, the term usually signifies an ordinary liquid-in-glass temperature-indicating device. Mercury-filled ther-mometers have a useful range from −38.8°C, the freezing point of mercury, to about 550°C, near which the glass usually softens.
Lower temperatures can be measured with organic-liquid-filled thermometers (e.g., alcohol-filled), with ranges of −200 to 200°C.
During manufacture, thermometers are roughly calibrated for at least two temperatures, often the freezing and boiling points of water; space between the calibration points is divided into desired scale divisions. Thermometers that are intended for precise mea-surement applications have scales etched into the glass that forms their stems. The probable error for as-manufactured, etched-stem thermometers is ±1 scale division. The highest quality mercury thermometers may have uncertainties of ±0.03 to ±2 K if they have been calibrated by comparison against primary reference stan-dards.
Liquid-in-glass thermometers are used for many applications within the HVAC industry. Some of these uses include local temper-ature indication of process fluids related to HVAC systems, such as cooling and heating fluids and air. The use of mercury-in-glass thermometers as temperature measurement standards is fairly common because of their rela-tively high accuracy and low cost. Such thermometers used as ref-erences must be calibrated on the ITS-90 by comparison in a uniform bath with a standard platinum resistance thermometer that has been calibrated either by the appropriate standards agency or by a laboratory that has direct traceability to the standards agency and the ITS-90. Such a calibration is necessary in order to deter-mine the proper corrections to be applied to the scale readings. For application and calibration of liquid-in-glass thermometers, refer to NIST (1976, 1986).
Liquid-in-glass thermometers are calibrated by the manufacturer for total or partial stem immersion. If a thermometer calibrated for total immersion is used at partial immersion (i.e., with a portion of the liquid column at a temperature different from that of the bath), an emergent stem correction must be made. This correction can be calculated as follows: (2) where K = differential expansion coefficient of mercury or other liquid in glass. K is 0.00016 for Celsius mercurial thermometers. For K values for other liquids and specific glasses, refer to Schooley (1992).
n = number of degrees that liquid column emerges from bath tb = temperature of bath, °C ts = average temperature of emergent liquid column of n degrees, °C Sources of Thermometer Errors A thermometer measuring gas temperatures can be affected by radiation from surrounding surfaces. If the gas temperature is approximately the same as that of the surrounding surfaces, radia-tion effects can be ignored. If the temperature differs considerably from that of the surroundings, radiation effects should be minimized by shielding or aspiration (ASME Standard PTC 193). Shielding may be provided by highly reflective surfaces placed between the thermometer bulb and the surrounding surfaces such that air move-ment around the bulb is not appreciably restricted (Parmelee and Huebscher 1946). Improper shielding can increase errors. Aspira-tion results from passing a high-velocity stream of air or gas over the thermometer bulb.
When a thermometer well within a container or pipe under pres-sure is required, the thermometer should fit snugly and be sur-rounded with a high thermal conductivity material (oil, water, or mercury, if suitable). Liquid in a long, thin-walled well is advanta-geous for rapid response to temperature changes. The surface of the pipe or container around the well should be insulated to eliminate heat transfer to or from the well.
Industrial thermometers are available for permanent installation in pipes or ducts. These instruments are fitted with metal guards to prevent breakage and are useful for many other purposes. The con-siderable heat capacity and conductance of the guards or shields can cause errors, however.
Allowing ample time for the thermometer to attain temperature equilibrium with the surrounding fluid prevents excessive errors in temperature measurements. When reading a liquid-in-glass ther-mometer, the eye should be kept at the same level as the top of the liquid column to avoid parallax.
RESISTANCE THERMOMETERS Resistance thermometers depend on a change of the electrical resistance of a sensing element (usually metal) with a change in temperature; resistance increases with increasing temperature. The use of resistance thermometers largely parallels that of thermocou-ples, although readings are usually unstable above about 550°C.
Two-lead temperature elements are not recommended because they do not permit correction for lead resistance. Three leads to each resistor are necessary to obtain consistent readings, and four leads are preferred. Wheatstone bridge circuits or 6-1/2-digit multimeters can be used for measurements.
A typical circuit used by several manufacturers is shown in Fig-ure 2. In this design, a differential galvanometer is used in which coils L and H exert opposing forces on the indicating needle. Coil L is in series with the thermometer resistance AB, and coil H is in series with the constant resistance R. As the temperature falls, the resistance of AB decreases, allowing more current to flow through coil L than through coil H. This causes an increase in the force exerted by coil L, pulling the needle down to a lower reading. Like-wise, as the temperature rises, the resistance of AB increases, caus-ing less current to flow through coil L than through coil H. This forces the indicating needle to a higher reading. Rheostat S must be adjusted occasionally to maintain a constant current.
The resistance thermometer is more costly to make and likely to have considerably longer response times than thermocouples. A resistance thermometer gives best results when used to measure steady or slowly changing temperature.
Resistance Temperature Devices Resistance temperature devices (RTDs) are typically constructed from platinum, rhodium-iron, nickel, nickel-iron, tungsten, or cop-per. These devices are further characterized by their simple circuit designs, high degree of linearity, good sensitivity, and excellent sta-bility. The choice of materials for an RTD usually depends on the intended application; temperature range, corrosion protection, mechanical stability, and cost are some of the selection criteria.
Platinum RTDs. Presently, for HVAC applications, RTDs con-structed of platinum are the most widely used. Platinum is extremely Stem correction Kn tb ts – ( ) = 14.6 2001 ASHRAE Fundamentals Handbook (SI) stable and corrosion-resistant. Platinum RTDs are highly malleable and can thus be drawn into fine wires; they can also be manufactured at low cost as thin films. They have a high melting point and can be refined to a high degree of purity, thus attaining highly reproducible results. Due to these properties, platinum RTDs are used to define the ITS-90 for the range of 13.8033 K (triple point of equilibrium hydrogen) to 1234.93 K (freezing point of silver).
Platinum resistance temperature devices can measure the widest range of temperatures and are the most accurate and stable temper-ature sensors. Their resistance-temperature relationship is one of the most linear. The higher the purity of the platinum, the more sta-ble and accurate the sensor. With high-purity platinum, primary grade platinum RTDs are capable of achieving reproducibility of ±0.00001 K, whereas the minimum uncertainty of a recently cali-brated thermocouple is ±0.2 K.
Platinum RTD Design. The most widely used RTD is designed with a resistance of 100 Ω at 0°C (R0 = 100 Ω). Other RTDs are available that use lower resistances at temperatures above 600°C.
The lower the resistance value, the faster the response time for sen-sors of the same size.
Thin-Film Platinum RTDs. Thin-film 1000 Ω platinum RTDs are readily available. They have the excellent linear properties of lower resistance platinum RTDs and are more cost-effective because they are mass produced and have lower platinum purity.
However, the problem with many platinum RTDs with R0 values of greater than 100 Ω is the difficulty in obtaining transmitters or elec-tronic interface boards from sources other than the RTD manufac-turer. In addition to a nonstandard interface, higher R0 value platinum RTDs may have higher self-heating losses if the excitation current is not controlled properly.
Thin-film RTDs have the advantages of lower cost and smaller sensor size. They are specifically adapted to surface mounting.
Thin-film sensors tend to have an accuracy limitation of ±0.1% or ±0.1 K. This may prove to be adequate for most HVAC applications; only in tightly controlled facilities may users wish to install the standard wire-wound platinum RTDs with accuracies of 0.01% or ±0.01 K (these are available upon special request for certain tem-perature ranges).
Assembly and Construction. Regardless of the R0 resistance value of RTDs, their assembly and construction are relatively sim-ple. The electrical connections come in three basic types, depending on the number of wires to be connected to the resistance measure-ment circuitry. Two, three, or four wires are used for electrical con-nection using a Wheatstone bridge or a variation of it (Figure 3).
In the basic two-wire configuration, the resistance of the RTD is measured through the two connecting wires. Because the connect-ing wires extend from the site of the temperature measurement, any additional changes in resistivity due to a change in temperature may affect the measured resistance. Three- and four-wire assemblies are built to compensate for the connecting lead resistance values. The original three-wire circuit improved the resistance measurement by adding a compensating wire to the voltage side of the circuit. This helps reduce part of the connecting wire resistance. When more accurate measurements (better than ±0.1 K) are required, the four-wire bridge is recommended. The four-wire bridge eliminates all connecting wire resistance errors.
All the bridges discussed here are direct current (dc) circuits and were used extensively until the advent of precision alternating cur-rent (ac) circuits using microprocessor-controlled ratio transform-ers, dedicated analog-to-digital converters, and other solid-state Fig. 2 Typical Resistance Thermometer Circuit Fig. 3 Typical Resistance Temperature Device Bridge Circuits Measurement and Instruments 14.7 devices that measure resistance with uncertainties of less than 1 ppm. Resistance measurement technology now allows more por-table thermometers, lower cost, ease of use, and high-precision tem-perature measurement in industrial uses.
Thermistors Certain semiconductor compounds (usually sintered metallic oxides) exhibit large changes in resistance with temperature, usu-ally decreasing as the temperature increases. For use, the thermistor element may be connected by lead wires into a galvanometer bridge circuit and calibrated. Alternatively, a 6-1/2-digit multimeter and a constant-current source with a means for reversing the current to eliminate thermal electromotive force (emf) effects may also be used. This method of measurement is easier and faster, and it may be more precise and accurate. Thermistors are usually applied to elec-tronic temperature compensation circuits, such as thermocouple ref-erence junction compensation, or to other applications where high resolution and limited operating temperature ranges exist. Figure 4 illustrates a typical thermistor circuit.
Semiconductor Devices In addition to the positive resistance coefficient RTDs and the negative resistance coefficient thermistor, there are two other types of devices that vary resistance or impedance with temperature.
Although the principle of their operation has long been known, their reliability was questioned due to imprecise manufacturing tech-niques. Improvements in silicon microelectronics manufacturing techniques have brought semiconductors to the point where low-cost, precise temperature sensors are commercially available.
Elemental Semiconductors. Due to controlled doping of impu-rities into elemental germanium, a germanium semiconductor is a reliable temperature sensor for cryogenic temperature measurement in the range of 1 to 84 K.
Junction Semiconductors. The first simple junction semicon-ductor device consisted of a single diode or transistor, in which the forward-connected base emitter voltage is very sensitive to temper-ature. Today the more common form is a pair of diode-connected transistors, which make the device suitable for ambient temperature measurement. Applications include thermocouple reference junc-tion compensation.
The primary advantages of silicon transistor temperature sen-sors are their extreme linearity and exact R0 value. Another advan-tage is the incorporation of signal conditioning circuitry into the same device as the sensor element. As with thermocouples, these semiconductors require highly precise manufacturing techniques, extremely precise voltage measurements, multiple-point calibra-tion, and temperature compensation to achieve an accuracy as high as ±0.01 K, but with a much higher cost. Lower cost devices achieve accuracies of ±0.1 K using mass manufacturing techniques and single-point calibration. A mass-produced silicon temperature sensor can be interchanged easily. If one device fails, only the sen-sor element need be changed. Electronic circuitry can be used to recalibrate the new device.
Winding Temperature. The winding temperature of electrical operating equipment is usually determined from the resistance change of these windings in operation. With copper windings, the relation between these parameters is (3) where R1 = winding resistance at temperature t1, Ω R2 = winding resistance at temperature t2, Ω t1, t2 = winding temperatures, °C The classical method of determining winding temperature is to measure the equipment when it is inoperative and temperature-sta-bilized at room temperature. After the equipment has operated suf-ficiently to cause temperature stabilization under load conditions, the winding resistance should be measured again. The latter value is obtained by taking resistance measurements at known short time intervals after shutdown. These values may be extrapolated to zero time to indicate the winding resistance at the time of shutdown. The obvious disadvantage of this method is that the device must be shut down to determine winding temperature. A circuit described by Seely (1955), however, makes it possible to measure resistances while the device is operating.
THERMOCOUPLES When two wires of dissimilar metals are joined by soldering, welding, or twisting, they form a thermocouple junction or thermo-junction. An emf that depends on the wire materials and the junc-tion temperature exists between the wires. This is known as the Seebeck voltage.
Thermocouples for temperature measurement yield less precise results than platinum resistance thermometers, but except for glass thermometers, thermocouples are the most common instruments of temperature measurement for the range of 0 to 1000°C. Due to their low cost, moderate reliability, and ease of use, thermocouples con-tinue to maintain widespread acceptance.
The most commonly used thermocouples in industrial applica-tions are assigned letter designations. The tolerances of such com-mercially available thermocouples are given in Table 2.
Because the measured emf is a function of the difference in tem-perature and the type of dissimilar metals used, a known tempera-ture at one junction is required, whereas the remaining junction temperature may be calculated. It is common practice to call the one with a known temperature the (cold) reference junction and the one with the unknown temperature the (hot) measured junction. The reference junction is typically kept at a reproducible temperature, such as the ice point of water.
Various systems are used to maintain the reference junction tem-perature—a mixture of ice and water contained in an insulated flask or commercially available thermoelectric coolers to maintain the ice point temperature automatically within a reference chamber. When Fig. 4 Basic Thermistor Circuit R1 R2 ------100 t1 + 100 t2 + -------------------= 14.8 2001 ASHRAE Fundamentals Handbook (SI) these systems cannot be used in an application, measuring instru-ments with automatic reference junction temperature compensation may be used.
As previously described, the principle for measuring tempera-ture with a thermocouple is based on the accurate measurement of the Seebeck voltage. The acceptable dc voltage measurement meth-ods are (1) millivoltmeter, (2) millivolt potentiometer, and (3) a high-input impedance digital voltmeter. Many digital voltmeters include built-in software routines for the direct calculation and dis-play of temperature. Regardless of the method selected, many options to simplify the measurement process are available.
Solid-state digital readout devices in combination with a milli-volt- or microvoltmeter, as well as packaged thermocouple readouts with built-in cold junction and linearization circuits, are available.
The latter requires a proper thermocouple to provide direct meter reading of temperature. Accuracy approaching or surpassing that of potentiometers can be attained, depending on the instrument qual-ity. This method is popular because it eliminates the null balancing requirement and reads temperature directly in a digital readout.
Wire Diameter and Composition Thermocouple wire is selected by considering the temperature to be measured, the corrosion protection afforded to the thermocouple, and the precision and service life required. Type T thermocouples are suitable for temperatures up to 350°C; type J, up to 750°C; and types K and N, up to 1250°C. Higher temperatures require noble metal thermocouples (type S, R, or B), which have a higher initial cost and do not develop as high an emf as the base metal thermo-couples. Thermocouple wires of the same type have small compo-sitional variation from lot to lot from the same manufacturer and especially among different manufacturers. Consequently, calibrat-ing samples from each wire spool is essential for precision. Calibra-tion data on wire may be obtained from the manufacturer.
Reference functions are available for relating temperature and emf of letter-designated thermocouple types. Such functions are easy to use with computers. The functions depend on thermocou-ple type and on the range of temperature; they are used to generate reference tables of emf as a function of temperature but are not well suited for calculating temperatures directly from values of emf. Approximate inverse functions are available, however, for calculating temperature and are of the form (4) where t = temperature, a = thermocouple constant, and E = voltage.
Burns et al. (1992) give the reference functions and approximate inverses for all letter-designated thermocouples.
The emf of a thermocouple, as measured with a high-input impedance device, is independent of the diameters of its constituent wires. Thermocouples with small-diameter wires respond faster to temperature changes and are less affected by radiation than larger ones. Large-diameter wire thermocouples, however, are necessary for high-temperature work when wire corrosion is a problem. For use in heated air or gases, thermocouples are often shielded and sometimes aspirated. An arrangement for avoiding error due to radi-ation involves using several thermocouples of different wire sizes and estimating the true temperature by extrapolating readings to zero diameter.
With thermocouples, temperatures can be indicated or recorded remotely on conveniently located instruments. Because thermocou-ples can be made of small-diameter wire, they can be used to mea-sure temperatures within thin materials, within narrow spaces, or in otherwise inaccessible locations.
Multiple Thermocouples Thermocouples in series, with alternate junctions maintained at a common temperature, produce an emf that, when divided by the number of thermocouples, gives the average emf corresponding to the temperature difference between two sets of junctions. This series arrangement of thermocouples, often called a thermopile, is used to increase sensitivity and is often used for measuring small temperature changes and differences.
Table 2 Thermocouple Tolerances on Initial Values of Electromotive Force Versus Temperature Thermocouple Type Material Identification Temperature Range, °C Reference Junction Tolerance at 0°C a Standard Tolerance (whichever is greater) Special Tolerance (whichever is greater) T Copper versus Constantan 0 to 350 ±1 K or ±0.75% ±0.5 K or ±0.4% J Iron versus Constantan 0 to 750 ±2.2 K or ±0.75% ±1.1 K or ±0.4% E Nickel-10% Chromium versus Constantan 0 to 900 ±1.7 K or ±0.5% ±1 K or ±0.4% K Nickel-10% Chromium versus 5% Aluminum, Silicon 0 to 1250 ±2.2 K or ±0.75% ±1.1 K or ±0.4% N Nickel-14% Chromium, 1.5% Silicon versus Nickel-4.5% Silicon, 0.1% Magnesium 0 to 1250 ±2.2 K or ±0.75% ±1.1 K or ±0.4% R Platinum-13% Rhodium versus Platinum 0 to 1450 ±1.5 K or ±0.25% ±0.6 K or ±0.1% S Platinum-10% Rhodium versus Platinum 0 to 1450 ±1.5 K or ±0.25% ±0.6 K or ±0.1% B Platinum-30% Rhodium versus Platinum-6% Rhodium 870 to 1700 ±0.5% ±0.25% Tb Copper versus Constantan −200 to 0 ±1 K or ±1.5% c Eb Nickel-10% Chromium versus Constantan −200 to 0 ±1.7 K or ±1% c Kb Nickel-10% Chromium versus 5% Aluminum, Silicon −200 to 0 ±2.2 K or ±2% c Source: ASTM Standard E 230, Temperature-Electromotive Force (EMF) Tables for Standardized Thermocouples.
aTolerances in this table apply to new thermocouple wire, normally in the size range of 0.25 to 3 mm diameter and used at temperatures not exceeding the recommended limits. Thermocouple wire is available in two grades: standard and special.
bThermocouples and thermocouple materials are normally supplied to meet the tolerance specified in the table for temperatures above 0°C. The same materials, however, may not fall within the tolerances given in the second section of the table when operated below freezing (0°C). If materials are required to meet tolerances at subfreezing temperatures, the purchase order must state so.
cLittle information is available to justify establishing special tolerances for below-freezing temperatures. Limited experience suggests the following special tolerances for types E and T thermocouples: Type E −200 to 0°C; ±1 K or ±0.5% (whichever is greater) Type T −200 to 0°C; ±0.5 K or ±0.8% (whichever is greater) These tolerances are given only as a guide for discussion between purchaser and supplier.
t aiEi i 0 = n ∑ = Measurement and Instruments 14.9 Connecting a number of thermocouples of the same type in par-allel with a common reference junction is useful for obtaining an average temperature of an object or volume. In such measurements, however, it is important that the electrical resistances of the individ-ual thermocouples be the same. The use of thermocouples in series and parallel arrangements is discussed in ASTM Manual 12.
Surface Temperature Measurement The thermocouple is useful in determining surface temperature. It can be attached to a metal surface in several ways. For permanent installations, soldering, brazing, or peening is suggested. For peening, a small hole is drilled and the thermocouple measuring junction is driven into it. For temporary arrangements, thermocouples can be attached by tape, adhesive, or putty-like material. For boiler or furnace surfaces, furnace cement should be used. To minimize the possibility of error caused by heat conduction along wires, a surface thermocou-ple should be made of fine wires placed in close contact with the sur-face being measured for about 25 mm from the junction to ensure good thermal contact. The wires must be insulated electrically from each other and from the metal surface (except at the junction).
Thermocouple Construction The thermocouple wires are typically insulated with fibrous glass, fluorocarbon resin, or ceramic insulators. In another form of thermocouple, the thermocouple wires are insulated with com-pacted ceramic insulation inside a metal sheath. This form of ther-mocouple provides both mechanical protection and protection from stray electromagnetic fields. The measuring junction may be ex-posed or enclosed within the metal sheath. An enclosed junction may be either grounded or ungrounded to the metal sheath.
For the exposed junction type, the measuring junction is in direct contact with the process stream; it is therefore subject to corrosion or contamination but provides a fast temperature response. The grounded enclosed junction type, in which the thermocouple wires are welded to the metal sheath, provides electrical grounding, as well as mechanical and corrosion protection. This type, however, has a slower response time than the exposed junction type. With the ungrounded enclosed junction construction, the response time is even slower, but the thermocouple wires are isolated electrically and are less susceptible to some forms of mechanical strain than those with grounded construction.
INFRARED RADIOMETERS Infrared radiation thermometers, also known as remote temper-ature sensors (Hudson 1969), permit noncontact measurement of surface temperature over a wide range. In these instruments, radiant flux from the observed object is focused by an optical system onto an infrared detector that generates an output signal proportional to the incident radiation that can be read from a meter or display unit.
Point and scanning radiometers are available; the latter are able to display the temperature variation existing within the field of view.
Radiometers are usually classified according to the detector used—either thermal or photon. In thermal detectors, a change in electrical property is caused by the heating effect of the incident radiation. Examples of thermal detectors are the thermocouple, the thermopile, and metallic and semiconductor bolometers. In photon detectors, a change in electrical property is caused by the surface absorption of incident photons. Because these detectors do not require an increase in temperature for activation, their response time is much shorter than that of thermal detectors. Scanning radiometers usually use photon detectors.
A radiometer only measures the power level of the radiation inci-dent on the detector; this incident radiation is a combination of the thermal radiation emitted by the object and the surrounding back-ground radiation reflected from the surface of the object. An accu-rate measurement of the temperature, therefore, requires knowledge of the long-wavelength emissivity of the object as well as the effec-tive temperature of the thermal radiation field surrounding the object. Calibration against an internal or external source of known temperature and emissivity is required in order to obtain true surface temperature from the radiation measurements.
The temperature resolution of a radiometer decreases as the object temperature decreases. For example, a radiometer that can resolve a temperature difference of 0.25 K on an object near 20°C may only resolve a difference of 1 K on an object at 0°C.
INFRARED THERMOGRAPHY Infrared thermography is the discipline concerned with the acquisition and analysis of thermal information in the form of images from an infrared imaging system. An infrared imaging sys-tem consists of (1) an infrared television camera and (2) a display unit. The infrared television camera scans a surface and senses the self-emitted and reflected radiation viewed from the surface. The display unit contains either a cathode-ray tube (CRT) that displays a gray-tone or color-coded thermal image of the surface or a color liquid crystal display (LCD) screen. A photograph of the image on the CRT is called a thermogram. An introductory treatise on infra-red thermography is given by Paljak and Pettersson (1972).
Thermography has been used successfully to detect missing insulation and air infiltration paths in building envelopes (Burch and Hunt 1978). Standard practices for conducting thermographic inspections of buildings are given in ASTM Standard C 1060. A technique for quantitatively mapping the heat loss in building enve-lopes is given by Mack (1986).
Aerial infrared thermography of buildings is effective in identi-fying regions of an individual built-up roof that have wet insulation (Tobiasson and Korhonen 1985), but it is ineffective in ranking a group of roofs according to their thermal resistance (Goldstein 1978, Burch 1980). In this latter application, the emittances of the separate roofs and outdoor climate (i.e., temperature and wind speed) throughout the microclimate often produce changes in the thermal image that may be incorrectly attributed to differences in thermal resistance.
Industrial applications include locating defective or missing pipe insulation in buried heat distribution systems, surveys of manufac-turing plants to quantify energy loss from equipment, and locating defects in coatings (Bentz and Martin 1987).
HUMIDITY MEASUREMENT Any instrument capable of measuring the humidity or psychro-metric state of air is a hygrometer, and many are available. The indi-cation sensors used on the instruments respond to different moisture property contents. These responses are related to factors such as wet-bulb temperature, relative humidity, humidity (mixing) ratio, dew point, and frost point.
Table 3 lists instruments for measuring humidity. Each is capable of accurate measurement under certain conditions and within spe-cific limitations. The following sections describe various instru-ments used to measure humidity.
PSYCHROMETERS A typical industrial psychrometer consists of a pair of matched electrical or mechanical temperature sensors, one of which is kept wet with a moistened wick. A blower aspirates the sensor, which lowers the temperature at the moistened temperature sensor. The lowest temperature depression occurs when the evaporation rate required to saturate the moist air adjacent to the wick is constant.
This is a steady-state, open-loop, nonequilibrium process, which depends on the purity of the water, the cleanliness of the wick, the ventilation rate, radiation effects, the size and accuracy of the tem-perature sensors, and the transport properties of the gas.
14.10 2001 ASHRAE Fundamentals Handbook (SI) ASHRAE Standard 41.6 recommends an airflow over both the wet and dry bulbs of 3 to 5 m/s for transverse ventilation and 1.5 to 2.5 m/s for axial ventilation.
The sling psychrometer consists of two thermometers mounted side by side in a frame fitted with a handle for whirling the device through the air. The thermometers are spun until their readings become steady. In the ventilated or aspirated psychrometer, the thermometers remain stationary, and a small fan, blower, or syringe moves the air across the thermometer bulbs. Various designs are used in the laboratory, and commercial models are available.
Other temperature sensors, such as thermocouples and ther-mistors, are also used and can be adapted for recording temperatures or for use where a small instrument is required. Small-diameter wet-bulb sensors operate with low ventilation rates.
Charts and tables showing the relationship between the temper-atures and humidity are available. Data are usually based on a barometric pressure equal to one standard atmosphere. To meet special needs, charts can be produced that apply to nonstandard pressure (e.g., the ASHRAE 2250 m psychrometric chart). Alter-natively, mathematical calculations can be made (Kusuda 1965).
Uncertainties of 3 to 7% rh are typical for psychrometer-based derivation. The degree of uncertainty is a function of the accuracy of the temperature measurements, wet and dry bulb, knowledge of the barometric pressure, and conformance to accepted operational procedures such as those outlined in ASHRAE Standard 41.6.
In air temperatures below 0°C, the water on the wick may either freeze or supercool. Because the wet-bulb temperature is different for ice and water, the state must be known and the proper chart or table used. Some operators remove the wick from the wet-bulb for freezing conditions and dip the bulb in water a few times; this allows water to freeze on the bulb between dips, forming a film of ice. Because the wet-bulb depression is slight at low temperatures, Table 3 Humidity Sensor Properties Type of Sensor Sensor Category Method of Operation Approximate Range Some Uses Approximate Accuracy Psychrometer Evaporative cooling Temperature measurement of wet bulb 0 to 80°C Measurement, standard ±3 to ±7% rh Adiabatic saturation psychrometer Evaporative cooling Temperature measurement of thermodynamic wet bulb 5 to 30°C Measurement, standard ±0.2 to ±2% rh Chilled mirror Dew point Optical determination of moisture formation −75 to 95°C dp Measurement, control, meteorology ±0.2 to ±2 K Heated saturated salt solution Water vapor pressure Vapor pressure depression in salt solution −30 to 70°C dp Measurement, control, meteorology ±1.5 K Hair Mechanical Dimensional change 5 to 100% rh Measurement, control ±5% rh Nylon Mechanical Dimensional change 5 to 100% rh Measurement, control ±5% rh Dacron thread Mechanical Dimensional change 5 to 100% rh Measurement ±7% rh Goldbeater’s skin Mechanical Dimensional change 5 to 100% rh Measurement ±7% rh Cellulosic materials Mechanical Dimensional change 5 to 100% rh Measurement, control ±5% rh Carbon Mechanical Dimensional change 5 to 100% rh Measurement ±5% rh Dunmore type Electrical Impedance 7 to 98% rh at 5 to 60°C Measurement, control ±1.5% rh Ion exchange resin Electrical Impedance or capacitance 10 to 100% rh at −40 to 90°C Measurement, control ±5% rh Porous ceramic Electrical Impedance or capacitance Up to 200°C Measurement, control ±1 to ±1.5% rh Aluminum oxide Electrical Capacitance 5 to 100% rh Measurement, control ±3% rh Aluminum oxide Electrical Capacitance −80 to 60°C dp Trace moisture measurement, control ±1 K dp Electrolytic hygrometer Electrical Capacitance Coulometric Electrolytic cell Electrolyzes due to adsorbed moisture 1 to 1000 ppm Measurement Infrared laser diode Electrical Optical diodes 0.1 to 100 ppm Trace moisture measurement ±0.1 ppm Surface acoustic wave Electrical SAW attenuation 85 to 98% rh Measurement, control ±1% rh Piezoelectric Mass sensitive Mass changes due to adsorbed moisture −75 to −20°C Trace moisture measurement, control ±1 to ±5 K dp Radiation absorption Moisture absorption Moisture absorption of UV or IR radiation −20 to 80°C dp Measurement, control, meteorology ±2 K dp, ±5% rh Gravimetric Direct mea-surement of mixing ratio Comparison of sample gas with dry airstream 120 to 20 000 ppm mixing ratio Primary standard, research and laboratory ±0.13% of reading Color change Physical Color changes 10 to 80% rh Warning device ±10% rh Notes: 1. This table does not encompass all of the available technology for the measurement of humidity.
2. The approximate range for the device types listed is based on surveys of device manufacturers.
3. The approximate accuracy is based on manufacturers’ data.
4. Presently, the National Institute of Standards and Technology (NIST) will only certify instruments whose operating range is within −75 to 100°C dew point.
Measurement and Instruments 14.11 precise temperature readings are essential. A psychrometer can be used at high temperatures, but if the wet-bulb depression is large, the wick must remain wet and water supplied to the wick must be cooled so as not to influence the wet-bulb temperature by carrying sensible heat to it (Richardson 1965, Worrall 1965).
Greenspan and Wexler (1968) and Wentzel (1961) developed devices to measure adiabatic saturation temperature.
DEW-POINT HYGROMETERS Condensation Dew-Point Hygrometers The condensation (chilled mirror) dew-point hygrometer is an accurate and reliable instrument with a wide humidity range. How-ever, these features are obtained through an increase in complexity and cost compared to the psychrometer. In the condensation hygrometer, a surface is cooled (thermoelectrically, mechanically, or chemically) until dew or frost begins to condense out. The con-densate surface is maintained electronically in vapor pressure equi-librium with the surrounding gas, while surface condensation is detected by optical, electrical, or nuclear techniques. The measured surface temperature is then the dew-point temperature.
The largest source of error in a condensation hygrometer stems from the difficulty in measuring condensate surface temperature accurately. Typical industrial versions of the instrument are accurate to ±0.5 K over wide temperature spans. With proper attention to the condensate surface temperature measuring system, errors can be reduced to about ±0.2 K. Condensation hygrometers can be made surprisingly compact using solid-state optics and thermoelectric cooling.
Wide span and minimal errors are two of the main features of this instrument. A properly designed condensation hygrometer can mea-sure dew points from 95°C down to frost points of −75°C. Typical condensation hygrometers can cool to 80 K below the ambient tem-perature, establishing lower limits of the instrument to dew points corresponding to approximately 0.5% rh. Accuracies for measure-ments above −40°C can be ±1 K or better, deteriorating to ±2 K at lower temperatures.
The response time of a condensation dew-point hygrometer is usually specified in terms of its cooling/heating rate, typically 2 K/s for thermoelectric cooled mirrors. This makes it somewhat faster than a heated salt hygrometer. Perhaps the most significant feature of the condensation hygrometer is its fundamental measuring tech-nique, which essentially renders the instrument self-calibrating. For calibration, it is necessary only to manually override the surface cooling control loop, causing the surface to heat, and witness that the instrument recools to the same dew point when the loop is closed. Assuming that the surface temperature measuring system is correct, this is a reasonable check on the instrument’s performance.
Although condensation hygrometers can become contaminated, they can easily be cleaned and returned to service with no impair-ment to performance.
Salt-Phase Heated Hygrometers Another instrument in which the temperature varies with ambi-ent dew-point temperature is variously designated as a self-heating salt-phase transition hygrometer or a heated electrical hygrometer.
This device usually consists of a tubular substrate covered by glass fiber fabric, with a spiral bifilar winding for electrodes. The surface is covered with a salt solution, usually lithium chloride. The sensor is connected in series with a ballast and a 24 V (ac) supply. When the instrument is in operation, electrical current flowing through the salt film heats the sensor. The electrical resistance characteristics of the salt are such that a balance is reached with the salt at a critical moisture content corresponding to a saturated solution. The sensor temperature adjusts automatically so that the water vapor pressures of the salt film and ambient atmosphere are equal.
With lithium chloride, this sensor cannot be used to measure rel-ative humidity below approximately 12% (the equilibrium relative humidity of this salt), and it has an upper dew-point limit of about 70°C. The regions of highest precision are between −23 and 34°C, and above 40°C dew point. Another problem is that the lithium chloride solution can be washed off when exposed to water. In addi-tion, this type of sensor is subject to contamination problems, which limits its accuracy. Its response time is also very slow; it takes approximately 2 min for a 67% step change.
MECHANICAL HYGROMETERS Many organic materials change in dimension with changes in humidity; this action is used in a number of simple and effective humidity indicators, recorders, and controllers (see Chapter 15).
They are coupled to pneumatic leak ports, mechanical linkages, or electrical transduction elements to form hygrometers.
Commonly used organic materials are human hair, nylon, Dacron, animal membrane, animal horn, wood, and paper. Their inherent nonlinearity and hysteresis must be compensated for within the hygrometer. These devices are generally unreliable below 0°C. The response is generally inadequate for monitoring a changing process. Responses can be affected significantly by expo-sure to extremes of humidity. Mechanical hygrometers require ini-tial calibration and frequent recalibration; however, they are useful because they can be arranged to read relative humidity directly, and they are simpler and less expensive than most other types.
ELECTRICAL IMPEDANCE AND CAPACITANCE HYGROMETERS Many substances adsorb or lose moisture with changing relative humidity and exhibit corresponding changes in electrical imped-ance or capacitance.
Dunmore Hygrometers This sensor consists of dual electrodes on a tubular or flat sub-strate; it is coated with a film containing salt, such as lithium chlo-ride, in a binder to form an electrical connection between windings.
The relation of sensor resistance to humidity is usually represented by graphs. Because the sensor is highly sensitive, the graphs are a series of curves, each for a given temperature, with intermediate values found by interpolation. Several resistance elements, called Dunmore elements, cover a standard range. Systematic calibration is essential because the resistance grid varies with time and contam-ination as well as with exposure to temperature and humidity extremes.
Polymer Film Electronic Hygrometers These devices consist of a hygroscopic organic polymer depos-ited by means of thin or thick film processing technology on a water-permeable substrate. Both capacitance and impedance sen-sors are available. The impedance devices may be either ionic or electronic conduction types. These hygrometers typically have inte-grated circuits that provide temperature correction and signal con-ditioning. The primary advantages of this sensor technology are small size; low cost; fast response times (on the order of 1 to 120 s for 64% change in relative humidity); and good accuracy over the full range, including the low end (1 to 15% h), where most other devices are less accurate.
Ion Exchange Resin Electric Hygrometers A conventional ion exchange resin consists of a polymer having a high relative molecular mass and polar groups of positive or neg-ative charge in cross-link structure. Associated with these polar groups are ions of opposite charge that are held by electrostatic forces to the fixed polar groups. In the presence of water or water 14.12 2001 ASHRAE Fundamentals Handbook (SI) vapor, the electrostatically held ions become mobile; thus, when a voltage is impressed across the resin, the ions are capable of elec-trolytic conduction. The Pope cell is one example of an ion exchange element. It is a wide-range sensor, typically covering 15 to 95% rh; therefore, one sensor can be used where several Dun-more elements would be required. The Pope cell, however, has a nonlinear characteristic from approximately 1000 Ω at 100% rh to several megohms at 10% rh.
Impedance-Based Porous Ceramic Electronic Hygrometers Using the adsorption characteristics of oxides, humidity-sensi-tive ceramic oxide devices employ either ionic or electronic mea-surement techniques to relate adsorbed water to relative humidity.
Ionic conduction is produced by dissociation of water molecules forming surface hydroxyls. The dissociation causes migration of protons such that the impedance of the device decreases with increasing water content. The ceramic oxide is sandwiched between porous metal electrodes that connect the device to an impedance-measuring circuit for linearizing and signal conditioning. These sensors have excellent sensitivity, are resistant to contamination and high temperature (up to 200°C), and may get fully wet without sen-sor degradation. These sensors are accurate to about ±1.5% rh, and ±1% rh when temperature compensated. These sensors have a mod-erate cost.
Aluminum Oxide Capacitive Sensor This sensor consists of an aluminum strip that is anodized by a process that forms a porous oxide layer. A very thin coating of cracked chromium or gold is then evaporated over this structure.
The aluminum base and the cracked chromium or gold layer form the two electrodes of what is essentially an aluminum oxide capacitor.
Water vapor is rapidly transported through the cracked chro-mium or gold layer and equilibrates on the walls of the oxide pores in a manner functionally related to the vapor pressure of water in the atmosphere surrounding the sensor. The number of water molecules adsorbed on the oxide structure determines the capacitance between the two electrodes.
ELECTROLYTIC HYGROMETERS In electrolytic hygrometers, air is passed through a tube, where moisture is adsorbed by a highly effective desiccant (usually phos-phorous pentoxide) and electrolyzed. The airflow is regulated to 1.65 mL/s at a standard temperature and pressure. As the incoming water vapor is absorbed by the desiccant and electrolyzed into hydrogen and oxygen, the current of electrolysis determines the mass of water vapor entering the sensor. The flow rate of the enter-ing gas is controlled precisely to maintain a standard sample mass flow rate into the sensor. The instrument is usually designed for use with moisture-air ratios in the range of less than 1 ppm to 1000 ppm but can be used with higher humidities.
PIEZOELECTRIC SORPTION This hygrometer compares the changes in frequency of two hygroscopically coated quartz crystal oscillators. As the mass of the crystal changes due to the absorption of water vapor, the frequency changes. The amount of water sorbed on the sensor is a function of relative humidity (i.e., partial pressure of water as well as ambient temperature).
A commercial version uses a hygroscopic polymer coating on the crystal. The humidity is measured by monitoring the change in the vibration frequency of the quartz crystal when the crystal is alternately exposed to wet and dry gas.
SPECTROSCOPIC (RADIATION ABSORPTION) HYGROMETERS Radiation absorption devices operate on the principle that selec-tive absorption of radiation is a function of frequency for different media. Water vapor absorbs infrared radiation at 2 to 3 µm wave-lengths and ultraviolet radiation centered about the Lyman-alpha line at 0.122 µm. The amount of absorbed radiation is directly related to the absolute humidity or water vapor content in the gas mixture according to Beer’s law. The basic unit consists of an energy source and optical system for isolating wavelengths in the spectral region of interest and a measurement system for determin-ing the attenuation of radiant energy caused by the water vapor in the optical path. The absorbed radiation is measured extremely quickly and independent of the degree of saturation of the gas mix-ture. Response times of 0.1 to 1 s for 90% change in moisture con-tent are common. Spectroscopic hygrometers are primarily used where a noncontact application is required; this may include atmo-spheric studies, industrial drying ovens, and harsh environments.
The primary disadvantages of this device are its high cost and rela-tively large size.
GRAVIMETRIC HYGROMETERS Humidity levels can be measured by extracting and finding the mass of water vapor in a known quantity or atmosphere. For precise laboratory work, powerful desiccants, such as phosphorous pentox-ide and magnesium perchlorate, are used for the extraction process; for other purposes, calcium chloride or silica gel is satisfactory.
When the highest level of accuracy is required, the gravimetric hygrometer, developed and maintained by NIST, is the ultimate in the measurement hierarchy. The gravimetric hygrometer gives the absolute water vapor content, where the mass of the absorbed water and the precise measurement of the gas volume associated with the water vapor determine the mixing ratio or absolute humidity of the sample. This system has been chosen as the primary standard because the required measurements of mass, temperature, pressure, and volume can be made with extreme precision. However, its com-plexity and required attention to detail limit the usefulness of the gravimetric hygrometer.
CALIBRATION For many hygrometers, the need for recalibration depends on the accuracy required, the stability of the sensor, and the condi-tions to which the sensor is being subjected. Many hygrometers should be calibrated regularly by exposure to an atmosphere main-tained at a known humidity and temperature, or by comparison with a transfer standard hygrometer. Complete calibration usually requires observation of a series of temperatures and humidities.
Methods for producing known humidities include saturated salt solutions (Greenspan 1977, Huang and Whetstone 1985); sulfuric acid solutions, and mechanical systems, such as the divided flow, two-pressure (Amdur 1965); two-temperature (Till and Handegord 1960); and NIST two-pressure humidity generator (Hasegawa 1976). All these systems rely on precise methods of temperature and pressure control within a controlled environment to produce a known humidity, usually with accuracies of 0.5 to 1.0%. The oper-ating range for the precision generator is typically 5 to 95% rh.
PRESSURE MEASUREMENT Pressure is the force exerted per unit area by a medium, generally a liquid or gas. Pressure so defined is sometimes called absolute pressure. Thermodynamic and material properties are expressed in terms of absolute pressures; thus, the properties of a refrigerant will be given in terms of absolute pressures. Vacuum refers to pressures below atmospheric.
Measurement and Instruments 14.13 Differential pressure is the difference between two absolute pressures. In many cases, the differential pressure can be very small compared to either of the absolute pressures (these are often referred to as low-range, high-line differential pressures). A com-mon example of differential pressure is the pressure drop, or differ-ence between inlet and outlet pressures, across a filter or flow element.
Gage pressure is a special case of differential pressure where one of the pressures (the reference pressure) is atmospheric pres-sure. Many pressure gages, including most refrigeration test sets, are designed to make gage pressure measurements, and there are probably more gage pressure measurements made than any other.
Gage pressure measurements are often used as surrogates for abso-lute pressures. However, because of variations in atmospheric pres-sure due to elevation (atmospheric pressure in Denver, Colorado, is about 81% of sea-level pressure) and weather changes, the measure-ment of gage pressures to determine absolute pressures can signifi-cantly restrict the accuracy of the measured pressure, unless corrections are made for the local atmospheric pressure at the time of the measurement.
Pressures can be further classified as static or dynamic. Static pressures have a small or undetectable change with time; dynamic pressures include a significant pulsed, oscillatory, or other time-dependent component. Static pressure measurements are the most common, but equipment such as blowers and com-pressors can generate significant oscillatory pressures at discrete frequencies. Flow in pipes and ducts can generate resonant pres-sure changes, as well as turbulent “noise” that can span a wide range of frequencies.
Units A plethora of pressure units, many of them poorly defined, are in common use. The international (SI) unit is the newton per square metre, called the pascal (Pa). While the bar and the standard atmo-sphere are used, they should not be introduced where they are not used at present.
Types of Pressure-Measuring Instruments Broadly speaking, pressure instruments can be divided into three different categories—standards, mechanical gages, and electrome-chanical transducers. Standards instruments are used for the most accurate calibrations. The liquid-column manometer, which is the most common and potentially the most accurate standard, is used for a variety of applications, including field applications. Mechani-cal pressure gages are generally the least expensive and the most common pressure instruments. However, electromechanical trans-ducers have become much less expensive and are easier to use, so they are being used more often.
PRESSURE STANDARDS Liquid-column manometers measure pressure by determining the vertical displacement of a liquid of known density in a known gravitational field. Typically they are constructed as a U-tube of transparent material (glass or plastic). The pressure to be measured is applied to one side of the U-tube. If the other (reference) side is evacuated (zero pressure), the manometer measures absolute pres-sure; if the reference side is open to the atmosphere, it measures gage pressure; if the reference side is connected to some other pres-sure, the manometer measures the differential between the two pres-sures. Manometers filled with water and different oils are often used to measure low-range differential pressures. In some low-range instruments, one tube of the manometer is inclined in order to enhance the readability. Mercury-filled manometers are used for higher range differential and absolute pressure measurements. In the latter case, the reference side is evacuated, generally with a mechanical vacuum pump. Typical full-scale ranges for manome-ters vary from 2.5 kPa (250 mm of water) to 300 kPa.
For pressures above the range of manometers, standards are generally of the piston-gage, pressure-balance, or deadweight-tester type. These instruments apply pressure to the bottom of a vertical piston, which is surrounded by a close-fitting cylinder (typical clearances are micrometres). The pressure generates a force approximately equal to the pressure times the area of the pis-ton. This force is balanced by weights stacked on the top of the pis-ton. If the mass of the weights, the local acceleration of gravity, and the area of the piston (or more properly, the “effective area” of the piston and cylinder assembly) are known, the applied pressure can be calculated. Piston gages generally generate gage pressures with respect to the atmospheric pressure above the piston. They can be used to measure absolute pressures either indirectly by sep-arately measuring the atmospheric pressure and adding it to the gage pressure determined by the piston gage, or directly by sur-rounding the top of the piston and weights with an evacuated bell jar. Piston gage full-scale ranges vary from 35 kPa to 1.4 GPa.
At the other extreme, very low absolute pressures (below about 100 Pa), a number of different types of standards are used. These tend to be specialized and expensive instruments found only in major standards laboratories. However, one low-pressure standard, the McLeod gage, has been used for field applications. Unfortu-nately, although the theory of the McLeod gage is simple and straightforward, it is difficult to make accurate measurements with this instrument, and major errors can occur when it is used to mea-sure gases that condense or are adsorbed (e.g., water). In general, gages other than the McLeod gage should be used for most low-pressure or vacuum applications.
MECHANICAL PRESSURE GAGES Mechanical pressure gages couple a pressure sensor to a mechanical readout, typically a pointer and dial. The most com-mon type employs a Bourdon tube sensor, which is essentially a coiled metal tube of circular or elliptical cross section. Increasing pressure applied to the inside of the tube causes it to uncoil. A mechanical linkage translates the motion of the end of the tube to the rotation of a pointer. In most cases, the Bourdon tube is sur-rounded by atmospheric pressure, so that the gages measure gage pressure. A few instruments surround the Bourdon tube with a sealed enclosure that can be evacuated for absolute measurements or connected to another pressure for differential measurements.
Available instruments vary widely in cost, size, pressure range, and accuracy. Full-scale ranges can vary from 35 kPa to 700 MPa.
Accuracy of properly calibrated and used instruments can vary from 0.1 to 10% of full scale. Generally there is a strong correla-tion between size, accuracy, and price; larger instruments are more accurate and expensive.
To achieve better sensitivity, some low-range mechanical gages, sometimes called aneroid gages, employ corrugated diaphragms or capsules as sensors. The capsule is basically a short bellows sealed with end caps. These sensors are more compliant than a Bourdon tube, and a given applied pressure will cause a larger deflection of the sensor. The inside of a capsule can be evacuated and sealed in order to measure absolute pressures or connected to an external fit-ting to allow differential pressures to be measured. Typically, these gages are used for low-range measurements of 100 kPa or less. In instruments of better quality, accuracies of 0.1% of reading or better can be achieved.
ELECTROMECHANICAL TRANSDUCERS Mechanical pressure gages are generally limited by inelastic behavior of the sensing element, friction in the readout mechanism, and limited resolution of the pointer and dial. These effects can be eliminated or reduced by using electronic techniques to sense the 14.14 2001 ASHRAE Fundamentals Handbook (SI) distortion or stress of a mechanical sensing element and electroni-cally convert that stress or distortion to a pressure reading. A wide variety of sensors is used, including Bourdon tubes, capsules, dia-phragms, and different resonant structures whose vibration fre-quency varies with the applied pressure. Capacitive, inductive, and optical lever sensors are used to measure the displacement of the sen-sor element. In some cases, feedback techniques may be used to con-strain the sensor in a null position, minimizing distortion and hysteresis of the sensing element. Temperature control or compen-sation is often included. Readout may be in the form of a digital dis-play, analog voltage or current, or a digital code. Size varies, but in the case of transducers employing a diaphragm fabricated as part of a silicon chip, the sensor and signal-conditioning electronics can be contained in a small transistor package, and the largest part of the device is the pressure fitting. The best of these instruments achieve long-term instabilities of 0.01% or less of full scale, and correspond-ing accuracies when properly calibrated. Performance of the less expensive instruments can be more on the order of several percent.
While the dynamic response of most mechanical gages is limited by the sensor and readout, the response of some electromechanical transducers can be much faster, allowing measurements of dynamic pressures at frequencies up to 1 kHz and beyond in the case of trans-ducers specifically designed for dynamic measurements. Manufac-turers’ literature should be consulted as a guide to the dynamic response of specific instruments.
As the measured pressure is reduced below about 10 kPa, it becomes increasingly difficult to sense mechanically. A variety of gages have been developed that measure some other property of the gas that is related to the pressure. In particular, thermal conductivity gages, known as thermocouple, thermistor, Pirani, and convection gages, are used for pressures down to about 0.1 Pa. These gages have a sensor tube with a small heated element and a temperature sensor; the temperature of the heated element is determined by the thermal conductivity of the gas, and the output of the temperature sensor is displayed on an analog or digital electrical meter contained in an attached electronics unit. The accuracy of thermal conductiv-ity gages is limited by their nonlinearity, dependence on gas species, and tendency to read high when contaminated. Oil contamination is a particular problem. However, these gages are small, reasonably rugged, and relatively inexpensive; in the hands of a typical user, they will give far more reliable results than a McLeod gage. They can be used to check the base pressure in a system that is being evac-uated prior to being filled with refrigerant. They should be checked periodically for contamination by comparing the reading with that from a new, clean sensor tube.
GENERAL CONSIDERATIONS Accurate values of atmospheric or barometric pressure are required for weather prediction and aircraft altimetry. In the United States, a network of calibrated instruments, generally accurate to within 0.1% of reading and located at airports, is maintained by the National Weather Service, the Federal Aviation Administration, and local airport operating authorities. These agencies are gener-ally cooperative in providing current values of atmospheric pres-sure that can be used to check the calibration of absolute pressure gages or to correct gage pressure readings to absolute pressures.
However, the pressure readings generally reported for weather and altimetry purposes are not the true atmospheric pressure, but rather a value adjusted to an equivalent sea level pressure. Therefore, unless the location is near sea level, it is important to ask for the sta-tion or true atmospheric pressure rather than using the adjusted val-ues broadcast by radio stations. Further, the atmospheric pressure decreases with increasing elevation at a rate (near sea level) of about 10 Pa/m, and corresponding corrections should be made to account for the difference in elevation between the instruments being compared.
As noted before, gage-pressure instruments are sometimes used to measure absolute pressures, and the accuracy of these measurements can be compromised by uncertainties in the atmospheric pressure.
This error can be particularly serious when gage-pressure instruments are used to measure a vacuum (negative gage pressures). For all but the most crude measurements, absolute-pressure gages should be used for vacuum measurements; for pressures below about 100 Pa, a thermal conductivity gage should be used.
All pressure gages are susceptible to temperature errors. Sev-eral techniques are used to minimize these errors—sensor materi-als are generally chosen to minimize temperature effects, mechanical readouts can include temperature compensation ele-ments, electromechanical transducers may include a temperature sensor and compensation circuit, and some transducers are oper-ated at a controlled temperature. Clearly, temperature effects are of greater concern for field applications, and it is prudent to check the manufacturers’ literature for the temperature range over which the specified accuracy can be maintained. Abrupt temperature changes can also cause large transient errors that may take some time to decay.
The readings of some electromechanical transducers with a res-onant or vibrating sensor can depend on the gas species. Although some of these units can achieve calibrated accuracies of the order of 0.01% of reading, they are typically calibrated with dry air or nitro-gen, and the readings for other gases can be in error by several per-cent, quite possibly much more for refrigerants and other high-density gases. High-accuracy readings can be maintained by cali-brating these devices with the gas to be measured. Manufacturer’s literature should be consulted. The measurement of dynamic pressures is limited not just by the frequency response of the pressure gage, but also by the hydraulic or pneumatic time constant of the connection between the gage and the system to be monitored. As a general rule, the longer the connecting lines and the smaller their diameter, the lower the frequency response of the system. Further, even if only the static component of the pressure is of interest, and a gage with a low-frequency response is used, a significant pulsating or oscil-lating pressure component can cause significant errors in pressure gage readings and, in some cases, can damage the gage, particu-larly gages with a mechanical readout mechanism. In these cases, a filter or snubber should be used to reduce the higher frequency components.
VELOCITY MEASUREMENT HVAC engineers measure the flow of air more often than any other gas, and the air is usually measured at or near atmospheric pressure. Under this condition, the air can be treated as an incom-pressible fluid, and simple formulas give sufficient precision to solve many problems. Instruments that measure fluid velocity and their application range and precision are listed in Table 4.
AIRBORNE TRACER TECHNIQUES Tracer techniques are suitable for measuring velocity in an open space. Typical tracers include smoke, feathers, pieces of lint, and radioactive or nonradioactive gases. Measurements are made by timing the rate of movement of solid tracers or by monitoring the change in concentration level of gas tracers.
Smoke is a useful qualitative tool in studying air movements.
Smoke can be obtained from titanium tetrachloride (irritating to nasal membranes) or by mixing potassium chlorate and powdered sugar (a nonirritating smoke) and firing the mixture with a match.
The latter process produces considerable heat and should be con-fined to a pan away from flammable materials. Titanium tetrachlo-ride smoke works well for spot tests, particularly for leakage Measurement and Instruments 14.15 through casings and ducts, because it can be handled easily in a small, pistol-like ejector.
The fumes of ammonia water and sulfuric acid, if permitted to mix, form a white precipitate. Two bottles, one containing ammonia water and the other containing acid, are connected to a common nozzle by rubber tubing. A syringe forces air over the liquid sur-faces in the bottles; the two streams mix at the nozzle and form a white cloud.
A satisfactory test smoke also can be made by bubbling an air-stream through ammonium hydroxide and then hydrochloric acid (Nottage et al. 1952). Smoke tubes, smoke candles, and smoke bombs are available for studying airflow patterns.
ANEMOMETERS Deflecting Vane Anemometers The deflecting vane anemometer consists of a pivoted vane enclosed in a case. Air exerts pressure on the vane as it passes through the instrument from an upstream to a downstream opening.
A hair spring and a damping magnet resist vane movement. The instrument gives instantaneous readings of directional velocities on an indicating scale. With fluctuating velocities, it is necessary to average the needle swings visually to obtain average velocities.
This instrument is useful for studying air motion in a room; locating objectionable drafts; measuring air velocities at supply and return diffusers and grilles; and measuring laboratory hood face velocities.
Propeller or Revolving Vane Anemometers The propeller anemometer consists of a light, revolving wind-driven wheel connected through a gear train to a set of recording dials that read linear metres of air passing in a measured length of time. It is made in various sizes—75, 100, and 150 mm are the most common. Each instrument requires individual calibration. At low velocities, the friction drag of the mechanism is considerable. To compensate for this, a gear train that overspeeds is commonly used.
For this reason, the correction is often additive at the lower range and subtractive at the upper range, with the least correction in the middle range of velocities. The best of these instruments have start-ing speeds of 0.25 m/s or higher; therefore, they cannot be used below that air speed. Electronic revolving vane anemometers, with optical or magnetic pickups to sense the rotation of the vane, are available. Sizes for the vanes range as small as 13 mm in diameter for the electronic versions.
Cup Anemometers The cup anemometer is primarily used to measure outdoor, meteorological wind speeds. It consists of three or four hemispher-ical cups mounted radially from a vertical shaft. Wind from any direction with a vector component in the plane of cup rotation causes the cups and shaft to rotate. Because the primary use of this anemometer is to make meteorological wind speed measurements, the instrument is usually constructed so that wind speeds can be recorded or indicated electrically at a remote point.
Thermal Anemometers The thermal or hot-wire anemometer consists of a heated RTD, thermocouple junction, or thermistor sensor constructed at the end of a probe; it is designed to provide a direct, simple method of deter-mining air velocity at a point in the flow field. The probe is placed into an airstream, and the movement of air past the electrically heated velocity sensor tends to cool the sensor in proportion to the speed of the airflow. The electronics and sensor are commonly com-bined into a portable, hand-held device that interprets the sensor sig-nal and provides a direct reading of air velocity in either analog or digital display format. Often the sensor probe also incorporates an ambient temperature-sensing RTD or thermistor, in which case the indicated air velocity is “temperature compensated” to “standard” air density conditions (typically 1.20 kg/m3).
Hot-wire anemometers have long been used in the fluid flow re-search field. Research anemometer sensors have been constructed using very fine wires in configurations that allow the researcher to characterize fluid flows in one, two, and three dimensions with Table 4 Velocity Measurement Measurement Means Application Range, m/s Precision Limitations Smoke puff or airborne solid tracer Low air velocities in rooms; highly directional 0.025 to 0.25 10 to 20% Awkward to use but valuable in tracing air movement Deflecting vane anemometer Air velocities in rooms, at outlets, etc.; directional 0.15 to 120 5% Needs periodic check calibration Revolving vane anemometer Moderate air velocities in ducts and rooms; somewhat directional 0.5 to 15 2 to 5% Extremely subject to error with variations in velocities with space or time; easily damaged; needs periodic calibration Hot-wire anemometer a. Low air velocities; directional and nondirectional available 0.005 to 5 2 to 5% Requires accurate calibration at frequent intervals. Some are relatively costly.
b. High air velocities Up to 300 0.2 to 5% c. Transient velocity and turbulence Pitot tube Standard instrument for measuring duct velocities 0.9 to 50 with micromanometer; 3 to 50 with draft gages; 50 up with manometer 1 to 5% Accuracy falls off at low end of range Impact tube and sidewall or other static tap High velocities, small tubes and where air direction may be variable 0.6 to 50 with micromanometer; 3 to 50 with draft gages; 50 up with manometer 1 to 5% Accuracy depends on constancy of static pressure across stream section Cup anemometer Meteorological Up to 60 2 to 5% Poor accuracy at low air velocity (<2.5 m/s) Laser Doppler velocimeter Calibration of air velocity instruments 0.005 to 30 1 to 3% High cost and complexity limit LDVs to laboratory applications 14.16 2001 ASHRAE Fundamentals Handbook (SI) sensor/electronics response rates up to several hundred kilohertz.
This technology has been incorporated into more ruggedized sensors suitable for measurements in the HVAC field, primarily for unidirectional airflow measurement. Omnidirectional sensing in-struments suitable for thermal comfort studies are also available.
The principal advantages of thermal anemometers are their wide dynamic range and their ability to sense extremely low velocities.
Typical accuracy (including repeatability) of 2 to 5% of reading over the entire velocity range is often achieved in commercially available portable instruments.
Among the limitations of thermal anemometers are the follow-ing: (1) the unidirectional sensor must be carefully aligned in the airstream (typically to within ±20° rotation) to achieve accurate results; (2) the velocity sensor must be kept clean because contam-inant buildup will cause the calibration to change; and (3) due to the inherent high speed of response of thermal anemometers, measure-ments in turbulent flows can yield fluctuating velocity measure-ments. Electronically controlled time-integrated functions are now available in many digital air velocity meters to help smooth these turbulent flow measurements.
In the HVAC field, thermal anemometers are suitable for use in a variety of applications. They are particularly well-suited to the low velocities associated with laboratory fume hood face velocity measurements (typically in the 0.3 to 1 m/s range). Thermal ane-mometers can also be used for taking multipoint traverse measure-ments in ventilation ductwork.
Laser Doppler Velocimeters (or Anemometers) The laser Doppler velocimeter (LDV) or laser Doppler anemom-eter (LDA) is an extremely complex system that collects scattered light produced by a particle passing through the intersection volume of two intersecting laser beams of the same light frequency (Mease et al. 1992). The scattered light consists of bursts containing a reg-ularly spaced fringe pattern whose frequency is linearly propor-tional to the speed of the particle. Due to the cost and complexity of these systems, they are usually not suitable for in situ field measure-ments. Rather, the primary application of LDV systems in the HVAC industry is the calibration of systems used to calibrate other air velocity instruments.
The greatest advantage of an LDV is its performance at low air speeds. It is capable of reading air speeds as low as 0.075 m/s with uncertainty levels of 1% or less (Mease et al. 1992). In addition, it is nonintrusive in the flow—only optical access is required. It can be used to measure fluctuating components as well as mean speeds and is available in one-, two-, and even three-dimensional configura-tions. Its biggest disadvantages are its high cost and extreme tech-nological complexity, which requires highly skilled operators.
Modern fiber optic systems require less-skilled operators but at a considerable increase in cost.
PITOT-STATIC TUBES The pitot-static tube, in conjunction with a suitable manometer or differential pressure transducer, provides a simple method of determining air velocity at a point in a flow field. Figure 5 shows the construction of a standard pitot tube (ASHRAE Standard 51) and the method of connecting it with inclined manometers to display both static pressure and velocity pressure. The equation for deter-mining air velocity from measured velocity pressure is (5) where V = velocity, m/s pw = velocity pressure (pitot-tube manometer reading), Pa ρ = density of air, kg/m3 The type of manometer or differential pressure transducer used with a pitot-static tube depends on the magnitude of velocity pres-sure being measured and on the desired accuracy. At velocities greater than 7.5 m/s, a draft gage of appropriate range is usually sat-isfactory. If the pitot-static tube is used to measure air velocities lower than 7.5 m/s, a precision manometer or comparable pressure differential transducer is essential.
Other pitot-static tubes have been used and calibrated. To meet special conditions, various sizes of pitot-static tubes geo-metrically similar to the standard tube can be used. For relatively high velocities in ducts of small cross-sectional area, total pres-sure readings can be obtained with an impact (pitot) tube. Where static pressure across the stream is relatively constant, as in tur-bulent flow in a straight duct, a sidewall tap to obtain static pres-sure can be used with the impact tube to obtain the velocity pressure. One form of impact tube is a small streamlined tube with a fine hole in its upstream end and with its axis placed paral-lel to the stream.
If the Mach number of the flow is greater than about 0.3, the effects of compressibility should be included in the computation of the air speed from pitot-static and impact (stagnation or pitot) tube measurements (Mease et al. 1992).
MEASURING FLOW IN DUCTS Because velocity in a duct is seldom uniform across any section, and a pitot tube reading or thermal anemometer indicates velocity at only one location, a traverse is usually made to determine average velocity. Generally, velocity is lowest near the edges or corners and greatest at or near the center.
To determine the velocity in a traverse plane, a straight average of individual point velocities will give satisfactory results when point velocities are determined by the log-Tchebycheff rule (ISO Standard 3966) or, if care is taken, by the equal area method. Fig-ure 6 shows suggested sensor locations for traversing round and rectangular ducts. The log-Tchebycheff rule provides the greatest accuracy because its location of traverse points accounts for the effect of wall friction and the fall-off of velocity near wall ducts.
This method is now recommended for rectangular ducts, although for circular ducts the log-Tchebycheff and log-linear traverse meth-ods are similar. Log-Tchebycheff minimizes the positive error (measured greater than actual) caused by the failure to account for V 2pw ρ ---------= Fig. 5 Standard Pitot Tube Measurement and Instruments 14.17 losses at the duct wall. This error can occur when using the older method of equal subareas to traverse rectangular ducts.
For a rectangular duct traverse, a minimum of 25 points should be measured. For a duct size less than 450 mm, the points should be located at the center of equal areas not more than 150 mm apart, and a minimum of 2 points per side should be used. For a duct side greater than 1400 mm, the maximum distance between points is 200 mm. For a circular duct traverse, the log-linear rule and three symmetrically disposed diameters may be used (Figure 6). Points on two perpendicular diameters may be used where access is limited.
If possible, measuring points should be located at least 7.5 diam-eters downstream and 3 diameters upstream from a disturbance (e.g., caused by a turn). Compromised traverses as close as 2 diameters downstream and 1 diameter upstream can be performed with an increase in measurement error. Because field-measured airflows are rarely steady and uniform, particularly near disturbances, accuracy can be improved by increasing the number of measuring points.
Straightening vanes (ASHRAE Standard 51) located 1.5 duct diam-eters ahead of the traverse plane improve measurement precision.
When velocities at a traverse plane fluctuate, the readings should be averaged on a time-weighted basis. Two traverse readings in short succession also help to average out velocity variations that occur with time. If negative velocity pressure readings are encoun-tered, they are considered a measurement value of zero and calcu-lated in the average velocity pressure. ASHRAE Standard 111 has further information on measuring flow in ducts.
FLOW RATE MEASUREMENT Various means of measuring fluid flow rate are listed in Table 5. The values for volumetric or mass flow rate measure-ment (ASME Standard PTC 19.5, Benedict 1984) are often determined by measuring pressure difference across an orifice, nozzle, or venturi tube. The various meters have different advan-tages and disadvantages. For example, the orifice plate is more easily changed than the complete nozzle or venturi tube assem-bly. However, the nozzle is often preferred to the orifice because its discharge coefficient is more precise. The venturi tube is a nozzle followed by an expanding recovery section to reduce net pressure loss. Differential pressure-type flow measurement has benefited through workshops addressing fundamental issues, textbooks, research, and improved standards (Miller 1983, DeCarlo 1984, Mattingly 1984, ASME Standard B40.1, ASME Standard MFC-9M, ASME Standard MFC-10M, ASME Stan-dard MFC-1M).
Fluid meters use a wide variety of physical techniques to make flow measurements (ASME Standard PTC 19.5, Miller 1983, DeCarlo 1984); those more prevalently used are described in this section. To assure and validate the accuracy of flow rate measure-ment instruments, appropriate calibration procedures should in-clude documentation of traceability to the calibration facility. The calibration facility should, in turn, provide documentation of trace-ability to national standards.
No. of Points for Traverse Lines Position Relative to Inner Wall No. of Measuring Points per Diameter Position Relative to Inner Wall 5 0.074, 0.288, 0.500, 0.712, 0.926 6 0.032, 0.135, 0.321, 0.679, 0.865, 0.968 6 0.061, 0.235, 0.437, 0.563, 0.765, 0.939 8 0.021, 0.117, 0.184, 0.345, 0.655, 0.816, 0.883, 0.981 7 0.053, 0.203, 0.366, 0.500, 0.634, 0.797, 0.947 10 0.019, 0.077, 0.153, 0.217, 0.361, 0.639, 0.783, 0.847, 0.923, 0.981 Log-Tchebycheff rule for rectangular ducts Log-linear rule for circular ducts Fig. 6 Measuring Points for Rectangular and Round Duct Traverse 14.18 2001 ASHRAE Fundamentals Handbook (SI) Flow Measurement Methods Direct. Both gas and liquid flow can be measured quite accurately by timing a collected amount of fluid that is measured gravimetrically or volumetrically. While this method is commonly used for calibrat-ing other metering devices, it is particularly useful where the flow rate is low or intermittent and where a high degree of accuracy is required.
These systems are generally large and slow, but in their simplicity, they can be considered primary devices.
The variable area meter or rotameter is a convenient direct-read-ing flowmeter for liquids and gases. This is a vertical, tapered tube in which the flow rate is indicated by the position of a float sus-pended in the upward flow. The position of the float is determined by its buoyancy and the upwardly directed fluid drag.
Displacement meters measure total liquid or gas flow over time.
The two major types of displacement meters used for gases are the conventional gas meter, which uses a set of bellows, and the wet test meter, which uses a water displacement principle.
Indirect. The Thomas meter is used in laboratories to mea-sure high gas flow rates with low pressure losses. The gas is heated by electric heaters, and the temperature rise is measured by two resistance thermometer grids. When the heat input and tem-perature rise are known, the mass flow of gas is calculated as the quantity of gas that will remove the equivalent heat at the same temperature rise.
A velocity traverse (made using a pitot tube or other velocity-measuring instrument) measures airflow rates in the field or cali-brates large nozzles. This method can be imprecise at low velocities and impracticable where many test runs are in progress.
Another field-estimating method measures the pressure drop across elements with known pressure drop characteristics, such as heating and cooling coils or fans. If the pressure drop/flow rate relationship has been calibrated, the results can be precise. If the method depends on rating data, it should be used for check pur-poses only.
VENTURI, NOZZLE, AND ORIFICE FLOWMETERS Flow in a pipeline can be measured by a venturi meter (Figure 7), flow nozzle (Figure 8), or orifice plate (Figure 9). American Society of Mechanical Engineers (ASME) Standard MFC-3M describes measurement of fluid flow in pipes using the orifice, nozzle, and venturi; ASME Standard PTC 19.5 specifies their construction.
Assuming an incompressible fluid (liquid or slow-moving gas), uniform velocity profile, frictionless flow, and no gravitational Table 5 Volumetric or Mass Flow Rate Measurement Measurement Means Application Range Precision Limitations Orifice and differential pressure measurement system Flow through pipes, ducts, and plenums for all fluids Above Reynolds number of 5000 1 to 5% Discharge coefficient and accuracy influenced by installation conditions Nozzle and differential pressure measurement system Flow through pipes, ducts, and plenums for all fluids Above Reynolds number of 5000 0.5 to 2.0% Discharge coefficient and accuracy influenced by installation conditions Venturi tube and differential pressure measurement system Flow through pipes, ducts, and plenums for all fluids Above Reynolds number of 5000 0.5 to 2.0% Discharge coefficient and accuracy influenced by installation conditions Timing given mass or volumetric flow Liquids or gases; used to calibrate other flowmeters Any 0.1 to 0.5% System is bulky and slow Rotameters Liquids or gases Any 0.5 to 5.0% Should be calibrated for fluid being metered Displacement meter Relatively small volumetric flow with high pressure loss As high as 500 L/s depending on type 0.1 to 2.0% depending on type Most types require calibration with fluid being metered Gasometer or volume displacement Short-duration tests; used to calibrate other flowmeters Total flow limited by avail-able volume of containers 0.5 to 1.0% — Thomas meter (temperature rise of stream due to electrical heating) Elaborate setup justified by need for good accuracy Any 1% Uniform velocity; usually used with gases Element of resistance to flow and differential pressure measurement system Used for check where system has calibrated resistance element Lower limit set by readable pressure drop 1 to 5% Secondary reading depends on accuracy of calibration Turbine flowmeters Liquids or gases Any 0.25 to 2.0% Uses electronic readout Instrument for measuring velocity at point in flow Primarily for installed systems with no special provision for flow measurement Lower limit set by accuracy of velocity measurement 2 to 4% Accuracy depends on uniformity of flow and completeness of traverse Heat input and temperature changes with steam and water coil Check value in heater or cooler tests Any 1 to 3% — Laminar flow element and differential pressure measurement system Measure liquid or gas volumetric flow rate; nearly linear relationship with pressure drop; simple and easy to use 50 mm3/s to 1 m3/s 1% Fluid must be free of dirt, oil, and other impurities that could plug meter or affect its calibration Magnetohydrodynamic flowmeter (electromagnetic) Measures electrically conductive fluids, slurries; meter does not obstruct flow; no moving parts 0.006 to 600 L/s 1% At present state of the art, conductivity of fluid must be greater than 5 µmho/cm Swirl flowmeter and vortex shedding meter Measure liquid or gas flow in pipe; no moving parts Above Reynolds number of 104 1% — Measurement and Instruments 14.19 effects, the principle of conservation of mass and energy can be applied to the venturi and nozzle geometries to give (6) where w = mass flow rate, kg/s V = velocity of stream, m/s A = flow area, m2 ρ = density of fluid, kg/m3 p = absolute pressure, Pa β = (D2/D1) for venturi and sharp edge orifice and d/D for flow nozzle Note: Subscript 1 refers to the entering conditions; subscript 2 refers to the throat conditions.
Because the flow through the meter is not frictionless, a correc-tion factor C is defined to account for friction losses. If the fluid is at a high temperature, an additional correction factor Fa should be included to account for thermal expansion of the primary element.
Because this amounts to less than 1% at 260°C, it can usually be omitted. Equation (6) then becomes (7) where C is the friction loss correction factor.
The factor C is a function of geometry and Reynolds number.
Values of C are given in ASME Standard PTC 19.5. The jet passing through an orifice plate contracts to a minimum area at the vena contracta located a short distance downstream from the orifice plate. The contraction coefficient, friction loss coefficient C, and approach factor 1/(1 − β4)0.5 can be combined into a single constant K, which is a function of geometry and Reynolds number. The ori-fice flow rate equations then become (8) where Q = discharge flow rate, m3/s A2 = orifice area, m2 p1 − p2 = pressure drop as obtained by pressure taps, Pa Values of K are shown in ASME Standard PTC 19.5.
Valves, bends, and fittings upstream from the flowmeter can cause errors. Long, straight pipes should be installed upstream and downstream from the flow devices to assure fully developed flow for proper measurement. ASHRAE Standard 41.8 specifies upstream Fig. 7 Typical Herschel Type Venturi Meter w ρV1A1 ρV2A2 A2 2ρ p1 p2 – ( ) 1 β4 – -----------------------------= = = w CA2 2ρ p1 p2 – ( ) 1 β4 – -----------------------------= Fig. 8 Dimensions of ASME Long-Radius Flow Nozzles From ASME PTC 19.5. Reprinted with permission of ASME.
Q KA2 2 p1 p2 – ( ) ρ -------------------------= 14.20 2001 ASHRAE Fundamentals Handbook (SI) and downstream pipe lengths for measuring flow of liquids with an orifice plate. ASME Standard PTC 19.5 gives the piping require-ments between various fittings and valves and the venturi, nozzle, and orifice. If these conditions cannot be met, flow conditioners or straightening vanes can be used (ASME Standard PTC 19.5, ASME Standard MFC-10M, Mattingly 1984, Miller 1983).
Compressibility effects must be considered for gas flow if the pressure drop across the measuring device is more than a few per-cent of the initial pressure.
Nozzles are sometimes arranged in parallel pipes from a com-mon manifold; thus, the capacity of the testing equipment can be changed by shutting off the flow through one or more nozzles. An apparatus designed for testing airflow and capacity of air-condition-ing equipment is described by Wile (1947), who also presents per-tinent information on nozzle discharge coefficients, Reynolds numbers, and resistance of perforated plates. Some laboratories refer to this apparatus as a code tester.
VARIABLE AREA FLOWMETERS (ROTAMETERS) In permanent installations where high precision, ruggedness, and operational ease are important, the variable area flowmeter is satis-factory. It is frequently used to measure liquids or gases in small-diameter pipes. For ducts or pipes over 150 mm in diameter, the expense of this meter may not be warranted. In larger systems, how-ever, the meter can be placed in a bypass line and used with an ori-fice.
The variable area meter (Figure 10) commonly consists of a float that is free to move vertically in a transparent tapered tube. The fluid to be metered enters at the narrow bottom end of the tube and moves upward, passing at some point through the annulus formed between the float and the inside wall of the tube. At any particular flow rate, the float assumes a definite position in the tube; a calibrated scale on the tube shows the float’s location and the fluid flow rate.
The position of the float is established by a balance between the fluid pressure forces across the annulus and gravity on the float. The buoyant force supporting the float, νf (ρf − ρ)g, is balanced by the pressure difference acting on the cross-sectional area of the float Af∆p, where ρf, Af, and νf are, respectively, the float density, float cross-sectional area, and float volume. The pressure difference across the annulus is (9) The mass flow follows from Equation (8) as (10) The flow for any selected fluid is nearly proportional to the area, so that calibration of the tube is convenient. To use the meter for dif-ferent fluids, the flow coefficient variation for any float must be known. Float design can reduce variation of the flow coefficient with Reynolds number; float materials can reduce the dependence of mass flow calibration on fluid density.
POSITIVE DISPLACEMENT METERS Many positive displacement meters are available for measuring total liquid or gas volumetric flow rates. The fluid measured in these meters flows progressively into compartments of definite size. As the compartments are filled, they are rotated so that the fluid dis-charges from the meter. The flow rate through the meter is equal to the product of the compartment volume, the number of compart-ments, and the rotation rate of the rotor. Most of these meters have a mechanical register calibrated to show total flow.
TURBINE FLOWMETERS Turbine flowmeters are volumetric flow rate sensing meters with a magnetic stainless steel turbine rotor suspended in the flow stream of a nonmagnetic meter body. The fluid stream exerts a force on the blades of the turbine rotor, setting it in motion and converting the fluid’s linear velocity to an angular velocity.
Design motivation for turbine meters is to have the rotational speed of the turbine proportional to the average fluid velocity and Fig. 9 Sharp Edge Orifice with Pressure Tap Locations From ASME PTC 19.5. Reprinted with permission of ASME.
Fig. 10 Variable Area Flowmeter p ∆ νf ρf ρ – ( )g Af ----------------------------= w KA2 2νf ρf ρ – ( )gρ Af -----------------------------------= Measurement and Instruments 14.21 thus to the volume rate of fluid flow (Miller 1983, DeCarlo 1984, Mattingly 1992).
The rotational speed of the rotor is monitored by an externally mounted pickoff assembly. Magnetic and radio frequency are the most commonly used pickoffs. The magnetic pickoff contains a per-manent magnet and coil. As the turbine rotor blades pass through the field produced by the permanent magnet, a shunting action induces ac voltage in the winding of the coil wrapped around the magnet. A sine wave with a frequency proportional to the flow rate develops. With the radio frequency pickoff, an oscillator applies a high-frequency carrier signal to a coil in the pickoff assembly. The rotor blades pass through the field generated by the coil and modu-late the carrier signal by shunting action on the field shape. The car-rier signal is modulated at a rate corresponding to the rotor speed, which is proportional to the flow rate. With both pickoffs, frequency of the pulses generated becomes a measure of flow rate, and the total number of pulses measures total volume (Woodring 1969, Shafer 1961, Mattingly 1992).
Because output frequency of the turbine flowmeter is propor-tional to flow rate, every pulse from the turbine meter is equivalent to a known volume of fluid that has passed through the meter; the sum of these pulses yields total volumetric flow. Summation is accomplished by electronic counters designed for use with turbine flowmeters; they combine a mechanical or electronic register with the basic electronic counter.
Turbine flowmeters should be installed with straight lengths of pipe upstream and downstream from the meter. The length of the inlet and outlet pipes should be according to manufacturers’ recom-mendations or pertinent standards. Where recommendations of standards cannot be accommodated, the meter installation should be calibrated. Some turbine flowmeters can be used in bidirectional flow applications. A fluid strainer, used with liquids of poor or mar-ginal lubricity, minimizes bearing wear.
The lubricity of the process fluid and the type and quality of rotor bearings determine whether the meter is satisfactory for the partic-ular application. When choosing turbine flowmeters for use with fluorocarbon refrigerants, attention must be paid to the type of bear-ings used in the meter and to the oil content of the refrigerant. For these applications, sleeve-type rather than standard ball bearings are recommended. The amount of oil in the refrigerant can severely affect calibration and bearing life.
In metering liquid fluorocarbon refrigerants, the liquid must not flash to a vapor (cavitate). This would cause a tremendous increase in flow volume. Flashing results in erroneous measurements and rotor speeds that can damage the bearings or cause a failure. Flash-ing can be avoided by maintaining an adequate back pressure on the downstream side of the meter (Liptak 1972).
AIRFLOW-MEASURING HOODS Flow-measuring hoods are portable instruments designed to measure supply or exhaust airflow through diffusers and grilles in HVAC systems. A flow-measuring hood assembly typically con-sists of a fabric hood section, a plastic or metal base, an airflow-measuring manifold, a meter, and handles for carrying and holding the hood in place.
For volumetric airflow measurements, the flow-measuring hood is placed over a diffuser or grille. The fabric hood captures and directs airflow from the outlet or inlet across the flow-sensing man-ifold in the base of the instrument. The manifold consists of a num-ber of tubes containing upstream and downstream holes in a grid pattern designed to simultaneously sense and average multiple velocity points across the base of the flow-measuring hood. Air from the upstream holes flows through the tubes past a sensor and then exits through the downstream holes. Sensors employed by dif-ferent manufacturers include swinging vane anemometers, elec-tronic micromanometers, and thermal anemometers. In the case of the electronic micromanometer sensor, air does not actually flow through the manifold, but the airtight sensor senses the pressure dif-ferential from the upstream to downstream series of holes. The meter on the base of the flow-measuring hood interprets the signal from the sensor and provides a direct reading of volumetric flow in either an analog or digital display format.
As a performance check in the field, the indicated flow of a mea-suring hood can be compared to a duct traverse flow measurement (using a pitot-tube or a thermal anemometer). All flow-measuring hoods induce some back pressure on the air-handling system because the hood restricts the flow coming out of the diffuser. This added resistance alters the true amount of air coming out of the dif-fuser. In most cases, this error is negligible and is less than the accu-racy of the instrument. For proportional balancing, this error need not be taken into account because all similar diffusers will have about the same amount of back pressure. To determine whether back pressure is significant, a velocity traverse can be made in the duct ahead of the diffuser with and without the flow-measuring hood in place. The difference in the average velocity of the traverse indi-cates the degree of back-pressure compensation required on similar diffusers in the system. For example, if the average velocity is 4.0 m/s with the hood in place and 4.1 m/s without the hood, the indicated flow reading can be multiplied by 1.025 on similar diffus-ers in the system (4.1/4.0 = 1.025). As an alternative, the designer of the air-handling system can predict the reduction in airflow due to the additional pressure of the hood by using a curve supplied by the flow-measuring hood manufacturer. This curve indicates the pres-sure drop through the hood for different flow rates.
AIR INFILTRATION, AIRTIGHTNESS, AND OUTDOOR AIR VENTILATION RATE MEASUREMENT Two major characteristics describe air infiltration in buildings— air exchange rate and envelope air leakage. The measurement approaches used to determine these factors are described in Chapter 26. The air exchange rate of a building refers to the rate at which out-door air enters the building under normal weather and ventilation system operation. In general, the air change rate includes both out-door air taken in through the air handlers and air leakage through the building envelope (infiltration). The outdoor air intake rate is deter-mined by the design, installation, and operation of the mechanical ventilation system. Infiltration is determined by the extent and dis-tribution of leaks over the building envelope and the pressure differ-ences across these leaks. These pressure differences are induced by wind, inside-outside temperature differences, and the operation of building mechanical equipment. To fully characterize the air exchange performance of a building, the air exchange rate must be measured over a range of weather and equipment operation.
The outdoor air ventilation rate is an indicator of the rate of dilu-tion of occupant- and building-generated contaminants. Building air exchange rates can be measured by injecting a tracer gas into a building and monitoring and analyzing the tracer gas concentration response. The equipment required for tracer testing includes (1) a means of injecting the tracer gas and (2) a tracer gas monitor. A vari-ety of tracer gas techniques are used. They are distinguished by their injection strategy and analysis approach. These techniques include the constant concentration (equilibrium tracer), tracer decay (ASTM Standard E 741), and outside air fraction (air ratio) meth-ods. Tracer decay is the simplest and most accurate of these tech-niques (as per Chapter 26), but the other methods may be satisfactory if care is taken. Carbon dioxide is often used as a tracer gas because CO2 gas monitors are relatively inexpensive and easy to use, and occupant-generated CO2 can be used for most tracer gas techniques. Bottled CO2 or CO2 fire extinguishers are also readily available for tracer gas injection.
14.22 2001 ASHRAE Fundamentals Handbook (SI) The airtightness of a building envelope can be measured rela-tively quickly using building pressurization techniques, which are described in Chapter 26. In the pressurization technique, a large fan or blower mounted in a door or window induces a large and roughly uniform pressure difference across the building shell. The airflow required to maintain this pressure difference is then measured. The more leakage in the building, the more airflow is required to induce a specific indoor-outdoor pressure difference. The building airtight-ness is characterized by the airflow rate at a reference pressure, nor-malized by the building volume or surface area. Under proper test conditions, the results of a pressurization test are independent of weather conditions. The instrumentation requirements for pressur-ization testing include air-moving equipment, a device to measure airflow, and a differential pressure gage.
CARBON DIOXIDE MEASUREMENT Carbon dioxide has become an important measurement parame-ter for air-conditioning, heating, and refrigerating engineers, partic-ularly for use in indoor air quality (IAQ) applications. Although CO2 is generally not of concern as a specific toxin in indoor air, it is used as a surrogate indicator of odor related to human occupancy.
ASHRAE Standard 62 states that maintaining CO2 concentrations below 1000 ppm (based on a differential of 700 ppm between indoor and outdoor CO2 concentrations) usually results in conditions con-ducive to comfort and reduced odor from human-generated pollut-ants. Standard 62 also recommends specific minimum outdoor air ventilation rates to ensure adequate indoor air quality. Carbon diox-ide is often used as a tracer gas when quantifying outdoor air venti-lation rates. Carbon dioxide sensors are also used in building control strategies to optimize ventilation as a function of occupancy. NONDISPERSIVE INFRARED CO2 DETECTORS The technology in most widespread use for IAQ applications is the nondispersive infrared (NDIR) sensor (Figure 11). This device makes use of the strong absorption band that CO2 produces at 4.2 µm when excited by an infrared light source. Indoor air quality-specific NDIR instruments, when calibrated between 0 and 5000 ppm, are typically accurate within 150 ppm, but the accuracy of some sensors can be improved to within 50 ppm if the instrument is calibrated for a nar-rower range. Portable NDIR meters are available with direct-reading digital displays; however, response time varies significantly among different instruments. While most NDIR cell designs facilitate very rapid CO2 sample diffusion, some of the instruments now in wide-spread use for indoor air quality measurement exhibit slower sensor response, resulting in stabilization times greater than 5 min (up to 15 min), which may complicate walk-through inspections.
Calibration In a clean, stable environment, NDIR sensors can hold calibra-tion for months, but condensation, dust, dirt, and mechanical shock may offset calibration. As with all other CO2 sensor technologies, NDIR sensor readings are proportional to pressure due to the change in density of the gas molecules that results from a change in the sam-ple pressure. This leads to errors in CO2 readings when the baromet-ric pressure changes from the calibration pressure. Weather-induced errors will be small, but all CO2 instruments should be recalibrated if used at an altitude that is significantly different from the calibra-tion altitude. Some NDIR sensors are sensitive to cooling effects when placed in an airstream. This is an important consideration when locating a fixed sensor or when using a portable system to evaluate air-handling system performance because airflow in sup-ply and return ducts may significantly shift readings.
Applications Nondispersive infrared sensors are well suited for equilibrium tracer and tracer decay ventilation studies, and faster response mod-els are ideal for a quick, basic evaluation of human-generated pollu-tion and ventilation adequacy. When properly located, these sensors are also appropriate for continuous monitoring and for control strat-egies using equilibrium tracer and air fraction tracer calculations.
AMPEROMETRIC ELECTROCHEMICAL CO2 DETECTORS Amperometric electrochemical CO2 sensors (Figure 12) use a measured current driven between two electrodes by the reduction of CO2 that diffuses across a porous membrane. Unlike NDIR sensors, which normally last the lifetime of the instrument, electrochemical CO2 sensors may change in electrolyte chemistry over time (typi-cally 12 to 18 months); so sensors should be replaced periodically.
These sensors typically hold their calibration for several weeks, but they may drift more if exposed to low humidity; this drift makes them less suitable for continuous monitoring applications. At low humidity (below 30% rh), the sensors must be kept moist to main-tain specified accuracy.
Amperometric electrochemical sensors have a lower power requirement than NDIR sensors, usually operating continuously for weeks where NDIR instruments typically operate for 6 hours (older models) to 150 hours (newer models). The longer battery life can be advantageous for spot checks and walk-throughs and for measuring CO2 distribution throughout a building and within a zone. Unlike most NDIR sensors, amperometric electrochemical sensors are not affected by high humidity, although readings may be affected if con-densate is allowed to form on the sensor.
Fig. 11 Nondispersive Infrared Carbon Dioxide Sensor Fig. 12 Amperometric Carbon Dioxide Sensor Measurement and Instruments 14.23 PHOTOACOUSTIC CO2 DETECTORS Both open- and closed-cell photoacoustic sensors are available.
Open-Cell Sensors Open-cell photoacoustic CO2 sensors (Figure 13) operate as air diffuses through a permeable membrane into a chamber that is pulsed with filtered light at the characteristic CO2 absorption frequency of 4.2 µm. The light energy absorbed by the CO2 heats the sample chamber, causing a pressure pulse, which is sensed by a piezoresis-tor. Open-cell photoacoustic CO2 sensors are presently unavailable in portable instruments, in part because any vibration that might occur during transportation would affect calibration and might affect the signal obtained for a given concentration of CO2. Ambient acous-tical noise may also influence readings. For continuous monitoring, vibration is a concern, as are temperature and airflow cooling effects.
However, if a sensor is located properly and if the optical filter is kept relatively clean, photoacoustic CO2 sensors may be very stable.
Commercially available open-cell photoacoustic transmitters do not allow recalibration to adjust for pressure differences, so an offset should be incorporated in any control system using these sensors at an altitude or duct pressure other than calibration conditions.
Closed-Cell Sensors Closed-cell photoacoustic sensors (Figure 14) operate under the same principle as the open-cell version, except that samples are pumped into a sample chamber that is sealed and environmentally stabilized. Two acoustic sensors are sometimes used in the chamber to minimize vibration effects. Closed-cell units, available as porta-ble or fixed monitors, come with particle filters that are easily replaced (typically at 3- to 6-month intervals) if dirt or dust accu-mulates on them. Closed-cell photoacoustic monitors permit recal-ibration to correct for drift, pressure effects, or other environmental factors that might influence accuracy.
POTENTIOMETRIC ELECTROCHEMICAL CO2 DETECTORS Potentiometric electrochemical CO2 sensors use a porous fluoro-carbon membrane that is permeable to CO2, which diffuses into a carbonic acid electrolyte, changing the electrolyte pH. This pH change is monitored by a pH electrode inside the cell. The pH elec-trode isopotential drift prohibits long-term monitoring to the accu-racy and resolution required for continuous measurement or control or for detailed IAQ evaluations, although accuracy within 100 ppm, achievable short-term over the 2000 ppm range, may be adequate for basic ventilation and odor evaluations. In addition, this type of sensor exhibits slow response, which increases the operator time necessary for field applications or for performing a walk-through of a building.
COLORIMETRIC DETECTOR TUBES Colorimetric detector tubes contain a chemical compound that discolors in the presence of CO2 gas, with the amount of discolora-tion related to the CO2 concentration. These detector tubes are often used to spot check CO2 levels; when used properly, they are accu-rate to within 25%. If numerous samples are taken (i.e., six or more), uncertainty may be reduced; this may render the tubes, if used in the late afternoon, adequate as a very basic determination of odor and discomfort related to human occupation. However, CO2 detector tubes are generally not appropriate for specific ventilation assess-ment because of their inaccuracy and inability to record concentra-tion changes over time.
LABORATORY MEASUREMENTS Laboratory techniques for measuring CO2 concentration include mass spectroscopy, thermal conductivity, infrared spectroscopy, and gas chromatography. These techniques typically require taking on-site grab samples for laboratory analysis. Capital costs for each piece of equipment are high, and significant training is required. A considerable drawback to grab sampling is that CO2 levels change significantly during the day and over the course of a week, making it sensible to place sensors on site with an instrument capable of recording or data logging measurements continuously over the course of a workweek. An automated grab sampling system captur-ing many samples of data would be quite cumbersome and expen-sive if designed to provide CO2 trend information over time.
However, an advantage to laboratory techniques is that they can be highly accurate. A mass spectrometer, for example, can measure CO2 concentration to within 5 ppm from 0 to 2000 ppm. All labo-ratory measurement techniques are subject to errors resulting from interfering agents. A gas chromatograph is typically used in con-junction with the mass spectrometer to eliminate interference from nitrous oxide (N2O), which has an equivalent mass, if samples are collected in a hospital or in another location where N2O might be present.
ELECTRIC MEASUREMENT Ammeters Ammeters are low-resistance instruments for measuring current.
They should be connected in series with the circuit being measured (Figure 15). Ideally, they have the appearance of a short circuit, but in practice, all ammeters have a nonzero input impedance that influ-ences the measurement to some extent.
Ammeters often have several ranges, and it is good practice when measuring unknown currents to start with the highest range and then reduce the range to the appropriate value to obtain the most sensitive reading. Ammeters with range switches maintain circuit continuity during switching. On some older instruments, it Fig. 13 Open-Cell Photoacoustic Carbon Dioxide Sensor Fig. 14 Closed-Cell Photoacoustic Carbon Dioxide Sensor 14.24 2001 ASHRAE Fundamentals Handbook (SI) may be necessary to short-circuit the ammeter terminals when changing the range.
Current transformers are often used to increase the operating range of ammeters. They may also provide isolation and thus protection from a high-voltage line. Current transformers have at least two separate windings on a magnetic core (Figure 16).
The primary winding is connected in series with the circuit in which the current is measured. In the case of a clamp-on probe, the transformer core is actually opened and then connected around a single conductor carrying the current to be measured.
That conductor serves as the primary winding. The secondary winding carries a scaled-down version of the primary current, which is connected to an ammeter. Depending on the type of instrument, the ammeter reading may have to be multiplied by the ratio of the transformer.
When using an auxiliary current transformer, the secondary circuit must not be open when current is flowing in the primary winding; dangerous high voltage may exist across the secondary terminals. A short-circuiting blade between the secondary termi-nals should be closed before the secondary circuit is opened at any point.
Transformer accuracy can be impaired by the residual magne-tism in the core when the primary circuit is opened at an instant when the flux is large. The transformer core may be left magne-tized, resulting in ratio and phase angle errors. The primary and secondary windings should be short-circuited before making changes.
Voltmeters Voltmeters are high-resistance instruments that should be con-nected across the load (in parallel), as shown in Figure 17. Ideally they have the appearance of an open circuit, but in practice all volt-meters have some finite impedance that influences the measurement to some extent.
Voltage transformers are often used to increase the operating range of a voltmeter (Figure 18). They also provide isolation from high voltages and prevent injury to the operator. Like current trans-formers, voltage transformers consist of two or more windings on a magnetic core. The primary winding is generally connected across the high voltage to be measured, and the secondary winding is con-nected to the voltmeter. It is important not to short-circuit the sec-ondary winding of a voltage transformer.
Wattmeters Wattmeters are instruments that measure the active power of an ac circuit, which equals the voltage multiplied by that part of the current in phase with the voltage. There are generally two sets of terminals—one to connect the load voltage and the other to con-nect in series with the load current. Current and voltage transform-ers can be used to extend the range of a wattmeter or to isolate it from high voltage. Figure 19 and Figure 20 show connections for single-phase wattmeters, and Figure 21 shows use of current and voltage transformers with a single-phase wattmeter.
Wattmeters with multiple current and voltage elements are available to measure polyphase power. Polyphase wattmeter con-nections are shown in Figure 22 and Figure 23.
Power-Factor Meters Power-factor meters measure the ratio of the active power to the apparent power (product of the voltage and current). The con-nections for power-factor meters and wattmeters are similar, and current and voltage transformers can be used to extend their range.
Connections for single-phase and polyphase power-factor meters are shown in Figure 24 and Figure 25, respectively.
ROTATIVE SPEED MEASUREMENT Tachometers Tachometers, or direct-measuring rpm counters, vary from hand-held mechanical or electric meters to shaft-driven and electronic pulse counters. They are used in general laboratory and shop work to check the rotative speeds of motors, engines, and turbines.
Stroboscopes Optical rpm counters work by producing a controlled high-speed electronic flashing light. The operator directs the light on a rotating member and increases the rate of flashes until the optical effect of stopping rotation of the member is achieved. At this point, the rpm measured is equal to the flashes per minute emitted by the strobe unit. Care must be taken to start at the bottom of the instrument scale and work up because multiples of the rpm pro-duce almost the same optical effect as true synchronism. Multiples can be indicated by positioning suitable marks on the shaft, such as a bar on one side and a circle on the opposite side. If, for exam-ple, the two are seen superimposed, then the strobe light is flashing at an even multiple of the true rpm.
AC Tachometer-Generators A tachometer-generator consists of a rotor and a stator. The rotor is a permanent magnet driven by the equipment. The stator is a winding with a hole through the center for the rotor. Concentricity is not critical; bearings are not required between rotor and stator.
The output can be a single-cycle-per-revolution signal whose volt-age is a linear function of rotor speed. The polypole configuration that generates 10 cycles per revolution permits measurement of speeds as low as 20 rpm without causing the indicating needle to flutter. The output of the ac tachometer-generator is rectified and connected to a dc voltmeter.
SOUND AND VIBRATION MEASUREMENT Measurement systems for determining sound pressure, sound intensity, and mechanical vibration generally involve the use of transducers to convert mechanical signals into electrical signals, which are then processed electronically in order to characterize the measured mechanical signals. These measurement systems contain one or more of the following elements, which may or may not be contained in a single instrument: 1. A transducer, or an assembly of transducers, to convert sound pressure, sound intensity, or mechanical vibration (time-varying strain, displacement, velocity, acceleration, or force) into an electrical signal that is quantitatively related to the mechanical quantity being measured.
2. Amplifiers and networks to provide such functions as electrical impedance matching, signal conditioning, integration, differen-tiation, frequency weighting, and gain.
3. Signal-processing equipment to detect and quantify those aspects of the signal that are being measured (peak value, rms value, time-weighted average level, power spectral density, or magnitude or phase of a complex linear spectrum or transfer function).
4. A device such as a meter, oscilloscope, digital display, or level recorder to display the signal or the aspects of it that are being quantified.
The relevant range of sound and vibration signals can vary over more than 12 orders of magnitude in amplitude and more than 8 orders of magnitude in frequency, depending on the application.
References on instrumentation, measurement procedures, and sig-nal analysis are given in the section on Bibliography. Product and Measurement and Instruments 14.25 Fig. 15 Ammeter Connected in Power Circuit Fig. 16 Ammeter with Current Transformer Fig. 17 Voltmeter Connected Across Load Fig. 18 Voltmeter with Potential Transformer Fig. 19 Wattmeter in Single-Phase Circuit Measuring Power Load plus Loss in Current-Coil Circuit Fig. 20 Wattmeter in Single-Phase Circuit Measuring Power Load plus Loss in Potential-Coil Circuit Fig. 21 Wattmeter with Current and Potential Transformer Fig. 22 Polyphase Wattmeter in Two-Phase, Three-Wire Circuit with Balanced or Unbalanced Voltage or Load Fig. 23 Polyphase Wattmeter in Three-Phase, Three-Wire Circuit Fig. 24 Single-Phase Power-Factor Meter Fig. 25 Three-Wire, Three-Phase Power-Factor Meter 14.26 2001 ASHRAE Fundamentals Handbook (SI) application notes, technical reviews, and books published by instrumentation manufacturers are an excellent source of addi-tional reference material. See Chapter 46 of the 1999 ASHRAE Handbook—Applications and Chapter 7 of this volume for further information on sound and vibration.
SOUND MEASUREMENT Microphones A microphone is an electroacoustical transducer that trans-forms an acoustical signal into an electrical signal. The two pre-dominant transduction principles used in the measurement of sound (as opposed to broadcasting or recording) are the electro-static and the piezoelectric. Electrostatic (capacitor) micro-phones are available either as electric microphones, which do not require an external polarizing voltage, or as condenser micro-phones, which do require an external polarizing voltage, typically in the range of 28 to 200 V (dc). Piezoelectric microphones may be manufactured using either natural piezoelectric crystals or poled ferroelectric crystals. The types of response characteristics of measuring microphones are pressure, free field, and random incidence (diffuse field).
A microphone with a uniform pressure response characteristic maintains uniform sensitivity over its operating frequency range when exposed to a sound pressure that is uniform over the surface of the sensing element. A microphone with a uniform free-field response characteristic maintains uniform sensitivity over its operating frequency range when exposed to a plane progressive sound wave at a specified angle of incidence to the surface of the sensing element. A microphone with a uniform random-incidence response characteristic maintains uniform sensitivity over its operating frequency range when exposed to a diffuse sound field.
The sensitivity and the frequency range over which the micro-phone has uniform sensitivity (flat frequency response) vary with both the diameter (surface area) of the sensing element and the microphone type. Other factors that may critically affect the per-formance or response of a measuring microphone and preampli-fier in a given measurement application are atmospheric pressure, temperature, relative humidity, external magnetic and electrostatic fields, mechanical vibration, and radiation. A microphone should be selected based on its long- and short-term stability; the match between its performance characteristics (e.g., sensitivity, fre-quency response, amplitude linearity, self-noise) and the expected amplitude of sound pressure, frequency, range of analysis, and expected environmental conditions of measurement; and any other pertinent considerations, such as size and directional char-acteristics.
Sound Measurement Systems Microphone preamplifiers, amplifiers, weighting networks (see Chapter 7), filters, and displays are available either separately or integrated into a measuring instrument such as a sound level meter, personal noise exposure meter, measuring amplifier, or real-time fractional octave or Fourier [e.g., fast Fourier transform (FFT)] sig-nal analyzer. The instrument(s) included in a sound measurement system depends on the purpose of the measurement and the fre-quency range and resolution of signal analysis. In the case of com-munity and industrial noise measurements for regulatory purposes, the instrument, signal processing, and quantity to be measured are usually dictated by the pertinent regulation. The optimal set of instruments generally varies for measurement of different charac-teristics such as sound power in HVAC ducts, sound power emitted by machinery, noise criteria (NC) numbers, sound absorption coef-ficients, sound transmission loss of building partitions, and rever-beration times (T60).
Frequency Analysis Measurement criteria often dictate the use of filters to analyze the signal in order to indicate the spectrum of the sound being mea-sured. Filters of different bandwidths for different purposes include fractional octave band (one, one-third, one-twelfth, etc.), constant percentage bandwidth, and constant (typically narrow) bandwidth.
The filters may be analog or digital and, if digital, may or may not be capable of real-time data acquisition during the measurement period, depending on the bandwidth of frequency analysis. FFT sig-nal analyzers are generally used in situations that require very nar-row band signal analysis when the amplitudes of the sound spectra vary significantly with respect to frequency. This may occur in regions of resonance or when it is necessary to identify narrow-band or discrete sine-wave signal components of a spectrum in the pres-ence of other such components or of broadband noise.
Sound Chambers Special rooms and procedures are required in order to character-ize and calibrate sound sources and receivers. The rooms are gener-ally classified into three types—anechoic, semianechoic, and reverberant. The ideal anechoic room or chamber would have boundary surfaces that completely absorb sound energy at all fre-quencies. The ideal semianechoic room or chamber would be iden-tical to the ideal anechoic room, except that one surface would totally reflect sound energy at all frequencies. The ideal reverber-ant room or chamber would have boundary surfaces that totally reflect sound energy at all frequencies.
Anechoic chambers are used to perform measurements under conditions approximating those of a free sound field. They can be used in calibrating and characterizing individual microphones, microphone arrays, acoustic intensity probes, reference sound power sources, loudspeakers, sirens, and other individual or com-plex sources of sound.
Semianechoic chambers are built with a hard reflecting floor in order to accommodate heavy machinery or to simulate large factory floor or outdoor conditions. They can be used in calibrating and char-acterizing reference sound power sources, obtaining sound power levels of noise sources, and characterizing the sound output of emer-gency vehicle sirens when mounted on an emergency motor vehicle.
Reverberation chambers are used to perform measurements under conditions approximating those of a diffuse sound field. They can be used in calibrating and characterizing random-incidence microphones and reference sound power sources, obtaining sound power ratings of equipment and sound power levels of noise sources, measuring sound absorption coefficients of building mate-rials and panels, and measuring the transmission loss through build-ing partitions and components such as doors and windows.
Calibration A measurement system should be calibrated as a system from microphone or probe to indicating device before it is used to per-form absolute measurements of sound. Acoustic calibrators and pis-tonphones of fixed or variable frequency and amplitude are available for this purpose. These calibrators should be used at a fre-quency low enough that the pressure, free-field, and random-inci-dence response characteristics of the measuring microphone(s) are, for practical purposes, equivalent, or at least related in a known quantitative manner for that specific measurement system. In gen-eral, the sound pressure produced by these calibrators may vary, depending on the microphone type, whether the microphone has a protective grid, atmospheric pressure, temperature, and relative humidity. Correction factors and coefficients are required for con-ditions of use that differ from those existing during the calibration of the acoustic calibrator or pistonphone. For demanding applications, precision sound sources and measuring microphones should period-Measurement and Instruments 14.27 ically be sent to the manufacturer, a private testing laboratory, or a national standards laboratory for calibration.
VIBRATION MEASUREMENT With the exception of seismic instruments that record or indicate vibration directly via a mechanical or optomechanical device con-nected to the test surface, vibration measurements involve the use of an electromechanical or interferometric vibration transducer. Here, the term vibration transducer refers to a generic mechanical vibra-tion transducer. Electromechanical and interferometric vibration transducers belong to a large and varied group of transducers that detect mechanical motion and furnish an electrical signal that is quantitatively related to a particular physical characteristic of the motion. Depending on the design of the transducer, the electrical signal may be related to mechanical strain, displacement, velocity, acceleration, or force. The operating principles of vibration trans-ducers may involve optical interference; electrodynamic coupling; piezoelectric (including poled ferroelectric) or piezoresistive crys-tals; or variable capacitance, inductance, reluctance, or resistance.
A considerable variety of vibration transducers with a wide range of sensitivities and bandwidths is commercially available. Vibration transducers may be contacting (e.g., seismic transducers) or non-contacting (e.g., interferometric or capacitive).
Transducers Seismic transducers use a spring mass resonator within the trans-ducer. At frequencies much greater than the fundamental natural frequency of the mechanical resonator, the relative displacement between the base and the seismic mass of the transducer is nearly proportional to the displacement of the transducer base. At frequen-cies much lower than the fundamental resonant frequency, the rela-tive displacement between the base and the seismic mass of the transducer is nearly proportional to the acceleration of the trans-ducer base. Therefore, seismic displacement transducers and seis-mic electrodynamic velocity transducers tend to have a relatively compliant suspension with a low resonant frequency; piezoelectric accelerometers and force transducers have a relatively stiff suspen-sion with a high resonant frequency.
Strain transducers include the metallic resistance gage and the piezoresistive strain gage. For dynamic strain measurements, these are usually of the bonded type, where the gages are bonded directly to the test surface. The accuracy with which a bonded strain gage replicates strain occurring in the test structure is largely a function of how well the strain gage was oriented and bonded to the test surface.
Displacement transducers include the capacitance gage, fringe-counting interferometer, seismic displacement transducer, and lin-ear variable differential transformer (LVDT). Velocity transducers include the reluctance (magnetic) gage, laser Doppler interferome-ter, and seismic electrodynamic velocity transducer. Accelerome-ters and force transducers include the piezoelectric, piezoresistive, and force-balance servo.
Vibration Measurement Systems The sensitivity, frequency limitations, bandwidth, and amplitude linearity of vibration transducers vary greatly with the transduction mechanism and the manner in which the transducer is applied in a given measurement apparatus. The performance of contacting transducers can be significantly affected by the mechanical mount-ing methods and points of attachment of the transducer and connect-ing cable and by the mechanical impedance of the structure loading the transducer. Amplitude linearity varies significantly over the operating range of the transducer, with some transducer types or configurations being inherently more linear than others. Other factors that may critically affect the performance or response of a vibration transducer in a given measurement application are temperature; relative humidity; external acoustic, magnetic, and electrostatic fields; transverse vibration; base strain; chemicals; and radiation. A vibration transducer should be selected based on its long- and short-term stability; the match between its performance characteristics (e.g., sensitivity, frequency response, amplitude lin-earity, self-noise) and the expected amplitude of vibration, fre-quency range of analysis, and expected environmental conditions of measurement; and any other pertinent considerations (e.g., size, mass, and resonant frequency).
Vibration exciters, or shakers, are used in structural analysis, vibration analysis of machinery, fatigue testing, mechanical imped-ance measurements, and vibration calibration systems. Vibration exciters have a table or moving element with a drive mechanism that may be mechanical, electrodynamic, piezoelectric, or hydraulic.
They range from relatively small, low-power units for calibrating transducers such as accelerometers to relatively large, high-power units for structural and fatigue testing.
Conditioning amplifiers, power supplies, preamplifiers, charge amplifiers, voltage amplifiers, power amplifiers, filters, controllers, and displays are available either separately or integrated into a mea-suring instrument or system, such as a structural analysis system, vibration analyzer, vibration monitoring system, vibration meter, measuring amplifier, multichannel data-acquisition and modal anal-ysis system, or real-time fractional-octave or FFT signal analyzer.
The choice of instrument(s) to include in a vibration measurement system depends on the mechanical quantity to be determined, the purpose of the measurement, and the frequency range and resolution of signal analysis. In the case of vibration measurements, the signal analysis is relatively narrow in bandwidth and may be relatively low in frequency in order to accurately characterize structural reso-nances. Accelerometers with internal integrated circuitry are avail-able to provide impedance matching or servo control for measuring very low frequency acceleration (servo accelerometers). Analog integration and differentiation of vibration signals is available through integrating and differentiating networks and amplifiers, and digital is available through FFT analyzers. Vibration measurements made for different purposes (e.g., machinery diagnostics and health monitoring, balancing rotating machinery, analysis of torsional vibration, analysis of machine-tool vibration, modal analysis, anal-ysis of vibration isolation, stress monitoring, industrial control) will generally each dictate different mechanical measurement require-ments and a different optimal set of instrumentation.
Calibration Because of their inherent long- and short-term stability, ampli-tude linearity, wide bandwidth, wide dynamic range, low noise, and wide range of sensitivities, seismic accelerometers have tra-ditionally been used as a reference standard for dynamic mechan-ical measurements. A measurement system should be calibrated as a system from transducer to indicating device before it is used to perform absolute dynamic measurements of mechanical quanti-ties. Calibrated reference vibration exciters, standard reference accelerometers, precision conditioning amplifiers, and precision calibration exciters are available for this purpose. These exciters and standard reference accelerometers can be used to transfer a calibration to another transducer. For demanding applications, either a calibrated exciter or a standard reference accelerometer with connecting cable and conditioning amplifier should periodi-cally be sent to the manufacturer, a private testing laboratory, or a national standards laboratory for calibration.
LIGHTING MEASUREMENT Light level, or illuminance, is usually measured with a photo-cell made from a semiconductor such as silicon or selenium. Such photocells produce an output current proportional to incident luminous flux; when linked with a microammeter, color- and 14.28 2001 ASHRAE Fundamentals Handbook (SI) cosine-corrected filters, and multirange switches, they are used in inexpensive hand-held light meters and more precise instruments.
Different cell heads allow multirange use in precision meters.
Cadmium sulfide photocells, in which the resistance varies with illumination, are also used in light meters. Both gas-filled and vac-uum photoelectric cells are in use.
Small survey-type meters are not as accurate as laboratory meters; their readings should be considered approximate, although consistent, for a given condition. Their range is usually from 50 to 50 000 lux. Precision low-level meters have cell heads with ranges down to 0 to 20 lux.
A photometer installed in a revolving head is called a gonio-photometer and is used to measure the distribution of light sources or luminaires. To measure total luminous flux, the luminaire is placed in the center of a sphere painted inside with a high-reflec-tance white with a near perfect diffusing matte surface. Total light output is measured through a small baffled window in the sphere wall.
To measure irradiation from germicidal lamps, a filter of fused quartz with fluorescent phosphor is placed over the light meter cell.
If meters are used to measure the number of lumens per unit area diffusely leaving a surface, luminance (cd/m2) instead of illumina-tion (lux) is read. Light meters can be used to measure luminance; or electronic lux meters containing a phototube, an amplifier, and a microammeter can read luminance directly.
THERMAL COMFORT MEASUREMENT Thermal comfort depends on the combined influence of cloth-ing, activity, air temperature, air velocity, mean radiant tempera-ture, and air humidity. Thermal comfort is influenced by heating or cooling of particular body parts. This is due to radiant temper-ature asymmetry (plane radiant temperature), draft (air tempera-ture, air velocity, turbulence), vertical air temperature differences, and floor temperature (surface temperature).
A general description of thermal comfort is given in Chapter 8, and guidelines for an acceptable thermal environment are given in ASHRAE Standard 55 and ISO Standard 7730. ASHRAE Stan-dard 55 also includes required measuring accuracy. In addition to specified accuracy, ISO Standard 7726 includes recommended measuring locations and a detailed description of instruments and methods.
Clothing and Activity Level These values are estimated from tables (Chapter 8, ISO Standard 9920, ISO Standard 8996). The thermal insulation of clothing (m2·K/W) can be measured on a thermal mannequin (McCullough et al. 1985, Olesen 1985). The activity (W/m2) can be estimated from measuring CO2 and O2 in a person’s expired air.
Air Temperature Various types of thermometers may be used to measure air tem-perature. Placed in a room, the sensor registers a temperature between air temperature and mean radiant temperature. One way of reducing the radiant error is to make the sensor as small as possible because the convective heat transfer coefficient increases as the size decreases while the radiant heat transfer coefficient is constant. A smaller sensor also provides a favorably low time constant. The radiant error can also be reduced by using a shield (an open, pol-ished aluminum cylinder) around the sensor, by using a sensor with a low-emittance surface, or by artificially increasing the air velocity around the sensor (aspirating air through a tube in which the sensor is placed).
Air Velocity In occupied zones, air velocities are usually small (0 to 0.5 m/s) but have an effect on human thermal sensation. Because the velocity fluctuates, the mean value should be measured over a suitable period, typically 3 min. Velocity fluctuations with frequencies up to 1 Hz significantly increase human discomfort due to draft, which is a function of air temperature, mean air velocity, and turbulence (see Chapter 8). The fluctuations can be given as the standard deviation of the air velocity over the measuring period (3 min) or as the tur-bulence intensity (standard deviation divided by mean air velocity).
Velocity direction may change and is difficult to identify at low air velocities. An omnidirectional sensor with a short response time should be used. A thermal anemometer is suitable. If a hot-wire ane-mometer is used, the direction of the flow being measured must be perpendicular to the hot wire. Smoke puffs can be used to identify the direction.
Plane Radiant Temperature This refers to the uniform temperature of an enclosure in which the radiant flux on one side of a small plane element is the same as in the actual nonuniform environment. It describes the radiation in one direction. The plane radiant temperature can be calculated from the surface temperatures of the environment (half-room) and the angle factors between the surfaces and a plane element (ASHRAE Standard 55). The plane radiant temperature may also be measured by a net-radiometer or a radiometer with a sensor consisting of a reflective disk (polished) and an absorbent disk (painted black) (Olesen et al. 1989).
Mean Radiant Temperature This is the uniform temperature of an imaginary black enclosure in which an occupant would exchange the same amount of radiant heat as in the actual nonuniform enclosure. The mean radiant tem-perature can be calculated from measured surface temperatures and the corresponding angle factors between the person and the surfaces (Chapter 8).
The mean radiant temperature can also be determined from the plane radiant temperature in six opposite directions weighted according to the projected area factors for a person (Chapter 8).
Because of its simplicity, the instrument most commonly used to determine the mean radiant temperature is a black globe ther-mometer (Vernon 1932, Bedford and Warmer 1935). This ther-mometer consists of a hollow sphere usually 150 mm in diameter coated in flat black paint with a thermocouple or thermometer bulb at its center. The temperature assumed by the globe at equi-librium results from a balance between heat gained and lost by radiation and convection.
The mean radiant temperatures are calculated from (11) where = mean radiant temperature, °C tg = globe temperature, °C Va = air velocity, m/s ta = air temperature, °C D = globe diameter, m ε = emissivity (0.95 for black globe) According to Equation (11), air temperature and air velocity around the globe must also be determined. The globe thermometer is spherical, while mean radiant temperature is defined in relation to the human body. For sedentary people, the globe represents a good approximation. For people who are standing, the globe, in a radiant nonuniform environment, overestimates the radiation from floor or tr tg 273 + ( )4 1.10 108Va 0.6 × εD0.4 ----------------------------------- tg ta – ( ) + 1 4 ⁄ 273 – = tr Measurement and Instruments 14.29 ceiling. An ellipsoid-shaped sensor gives a closer approximation to the human shape. A black globe will also overestimate the influence of short-wave radiation (e.g., sunshine). A flat gray color better rep-resents the radiant characteristic of normal clothing (Olesen et al.
1989). The hollow sphere is usually made of copper, which results in an undesirable high time constant. This can be overcome by using lighter materials (e.g., a thin plastic bubble).
Air Humidity The water vapor pressure (absolute humidity) is usually uniform in the occupied zone of a space; therefore, it is sufficient to measure absolute humidity at one location. Many of the instruments listed in Table 3 are applicable. At ambient temperatures that provide com-fort or slight discomfort, the thermal effect of humidity is only mod-erate, and highly accurate humidity measurements are unnecessary.
CALCULATING THERMAL COMFORT When the thermal parameters have been measured, their com-bined effect can be calculated by the thermal indices in Chapter 8.
For example, the effective temperature (Gagge et al. 1971) can be determined from air temperature and humidity. Based on the four environmental parameters and an estimation of clothing and activ-ity, the predicted mean vote (PMV) can be determined with the aid of tables (Fanger 1982, ISO Standard 7730, Chapter 8). The PMV is an index predicting the average thermal sensation that a group of occupants may experience in a given space.
For certain types of normal activity and clothing, the environ-mental parameters measured can be compared directly with those described in ASHRAE Standard 55 or ISO Standard 7730.
INTEGRATING INSTRUMENTS Several instruments have been developed to evaluate the com-bined effect of two or more thermal parameters on human comfort.
Madsen (1976) developed an instrument that gives information on the occupants’ expected thermal sensation by direct measurement of the PMV value. The comfort meter has a heated ellipsoid-shaped sensor that simulates the body (Figure 26). The estimated clothing (insulation value), activity in the actual space, and humid-ity are set on the instrument. The sensor then integrates the thermal effect of the air temperature, mean radiant temperature, and air velocity in approximately the same way the body does. The elec-tronic instrument gives the measured operative and equivalent temperature, calculated PMV, and predicted percentage of dissatis-fied (PPD). MOISTURE CONTENT AND TRANSFER MEASUREMENT Little off-the-shelf instrumentation exists to measure the mois-ture content of porous materials or the moisture transfer through those materials. However, many measurements can be set up with a small investment of time and money. Three moisture properties are most commonly sought—(1) the sorption isotherm, a measure of the amount of water vapor a hygroscopic material will adsorb from humid air; (2) vapor permeability, a measure of the rate at which water vapor will pass through a given material; and (3) liquid diffu-sivity, a measure of the rate at which liquid water will pass through a porous material.
Sorption Isotherm A sorption isotherm relates the equilibrium moisture content (EMC) of a hygroscopic material to the ambient relative humidity under conditions of constant temperature. Moisture content is the ratio of the total mass of water in a sample to the dry mass of the sample. Determining a sorption isotherm involves exposing a sam-ple of material to a known relative humidity at a known temperature and then measuring the sample’s moisture content after a sufficient period of time has elapsed for the sample to reach equilibrium with its surroundings. Hysteresis in the sorption behavior of most hygro-scopic materials requires that measurements be made both for increasing relative humidity (the adsorption isotherm) and for decreasing relative humidity (the desorption isotherm).
The ambient relative humidity can be controlled using satu-rated salt solutions or mechanical refrigeration equipment (Tveit 1966, Cunningham and Sprott 1984, Carotenuto et al. 1991). Pre-cise measurements of the relative humidity produced by various salt solutions have been reported by Greenspan (1977). ASTM Standard E 104 describes the use of saturated salt solutions. The EMC of a sample is usually determined gravimetrically using a precision balance. The sample dry mass, necessary to calculate moisture content, can be found by oven drying or desiccant dry-ing. Oven dry mass may be lower than desiccant dry mass because of the loss of volatiles other than water in the oven (Richards et al.
1992).
A major difficulty in the measurement of the sorption isotherms of engineering materials is the long time required for many materi-als to reach equilibrium—often as long as weeks or months. The rate-limiting mechanism for these measurements is usually the slow process of vapor diffusion into the pores of the material. The use of smaller samples can help reduce the diffusion time.
Vapor Permeability The diffusive transfer of water vapor through porous materials is often described by a modified form of Fick’s law: (12) where w″ v = mass of vapor diffusing through unit area in unit time, mg/(s·m2) dp/dx = vapor pressure gradient, kPa/m µ = vapor permeability, mg/(s·m·kPa) In engineering practice, permeance may be used instead of perme-ability. Permeance is simply permeability divided by the material thickness in the direction of the flow of vapor; thus, while permeabil-ity is a material property, permeance depends on thickness.
Measurement of permeability is made with wet-cup, dry-cup, or modified cup tests. Wet- and dry-cup tests are described in Chapter 23.
For many engineering materials, vapor permeability is a strong function of mean relative humidity. Wet and dry cups cannot ade-quately characterize this dependence on relative humidity. Instead, Fig. 26 Madsen’s Comfort Meter (Madsen 1976) w″ v µ – dp dx ------= 14.30 2001 ASHRAE Fundamentals Handbook (SI) a modified cup method can be used (McLean et al. 1990, Burch et al. 1992). In the modified cup method, the pure water or desiccant within a cup is replaced with a saturated salt solution. A second sat-urated salt solution is used to condition the environment external to the cup. With such an arrangement, the relative humidities on both sides of the sample material can be varied from 0 to 100%. Several cups with a range of mean relative humidities are used to map out the dependence of vapor permeability on relative humidity.
In measuring materials of high permeability, the finite rate of vapor diffusion through air in the cup may become a factor. The air-film resistance could then be a significant fraction of the resistance to vapor flow presented by the sample material. An accurate mea-surement of high-permeability materials may require an accounting of diffusive rates across all air gaps (Fanney et al. 1991).
Liquid Diffusivity The transfer of liquid water through porous materials may be characterized as a diffusion-like process: (13) where w″ l = mass of liquid transferred through unit area per unit time, kg/(s·m2) ρ = liquid density, kg/m3 Dl = liquid diffusivity, m2/s dγ/dx = moisture content gradient, m−1 Dl typically shows a strong dependence on moisture content.
Transient measurement methods deduce the functional form of Dlγ by observing the evolution of a one-dimensional moisture con-tent profile over time. An initially dry specimen is brought into con-tact with liquid water. The free water will migrate into the specimen, drawn in by surface tension. The resulting moisture content profile, which changes with time, must be differentiated to find the liquid diffusivity of the material (Bruce and Klute 1956).
Determining the transient moisture content profile typically involves the use of a noninvasive and nondestructive method of measuring local moisture content. Gamma ray absorption (Freitas et al. 1991, Kumaran and Bomberg 1985, Quenard and Sallee 1989), X-ray radiography (Ambrose et al. 1990), neutron radiogra-phy (Prazak et al. 1990), and nuclear magnetic resonance (NMR) (Gummerson et al. 1979) have all been employed.
Uncertainty in the resulting measurement of the liquid diffusivity is often large because of the necessity to differentiate noisy experi-mental data.
HEAT TRANSFER THROUGH BUILDING MATERIALS Thermal Conductivity The thermal conductivity of a heat insulator, as defined in Chap-ter 23, is a unit heat transfer factor. Two methods of determining the thermal conductivity of flat insulation are the guarded hot plate and the heat flow meter apparatus, according to ASTM Standards C 177 and C 518, respectively. Both methods use parallel isothermal plates to induce a steady temperature gradient across the thickness of the specimen(s). The guarded hot plate is considered an absolute method for determining thermal conductivity. The heat flow meter apparatus requires calibration with a specimen having a known ther-mal conductivity, usually determined in the guarded hot plate. The heat flow meter apparatus is calibrated by determining the voltage output of its heat flux transducer(s) as a function of the heat flux through the transducer(s).
The basic design of the guarded hot plate consists of an elec-trically heated plate and two liquid-cooled plates. Two similar specimens of a material are required for a test; one is mounted on each side of the hot plate. A cold plate is then pressed against the outside of each specimen by a clamp screw. The heated plate consists of two sections separated by a small gap. During tests, the central (metering) section and the outer (guard) section are maintained at the same temperature to minimize errors caused by edge effects. The electric energy required to heat the metering section is measured carefully and converted to heat flow. The thermal conductivity of the material can be calculated under steady-state conditions using this heat flow quantity, the area of the metering section, the temperature gradient, and the specimen thickness. The thermal conductivity of cylindrical or pipe insula-tion (Chapter 23) is determined in a similar manner, but an equiv-alent thickness must be calculated to account for the cylindrical shape (ASTM Standard C 335). Transient methods have been developed by D’Eustachio and Schreiner (1952), Hooper and Lepper (1950), and Hooper and Chang (1953) using a line heat source within a slender probe. These instruments are available commercially and have the advantages of rapidity and a small test specimen requirement. The probe is a useful research and development tool, but it has not been as accepted as the guarded hot plate, heat flow meter apparatus, or pipe insulation apparatus.
Thermal Conductance and Resistance Thermal conductances (C-factors) and resistances (R-values) of many building assemblies can be calculated from the conductivities and dimensions of their components, as described in Chapter 25. Test values can also be determined experimentally using large specimens representative of the building assemblies tested in the hot box appa-ratus described in ASTM Standards C 236 and C 976. This laboratory apparatus allows measurement of heat transfer through a specimen under controlled air temperature, air velocity, and radiation condi-tions. It is specially suited for large nonhomogeneous specimens.
For in situ measurements, heat flux and temperature transducers are useful in measuring the dynamic or steady-state behavior of opaque building components (ASTM Standard C 1046). A heat flux transducer is simply a differential thermopile within a core or sub-strate material. Two types of construction are used: (1) multiple thermocouple junctions wrapped around a core material, or (2) printed circuits with a uniform array of thermocouple junctions. The transducer is calibrated by determining its voltage output as a func-tion of the heat flux through the transducer. For in situ measure-ments, the transducer is installed in either the wall or roof, or mounted on an exterior surface with tape or adhesive. The data obtained can be used to compute the thermal conductance or resis-tance of the building component (ASTM Standard C 1155).
AIR CONTAMINANT MEASUREMENT Three measures of particulate air contamination include the number, projected area, and mass of particles per unit volume of air (ASTM Volume 11.03). Each requires an appropriate sampling technique.
Particles are counted by capturing them in impingers, impactors, membrane filters, or thermal or electrostatic precipitators. Counting may be done by microscope, using stage counts if the sample covers a broad range of sizes.
Electronic particle counters can give rapid data on particle size distribution and concentration. However, their accuracy depends on careful calibration, appropriate maintenance, and proper appli-cation. Particle counters have been used in indoor office environ-ments as well as in clean rooms.
wl″ ρDl dγ dx ------– = Measurement and Instruments 14.31 Projected area determinations are usually made by sampling onto a filter paper and comparing the light transmitted or scattered by this filter to a standard filter. The staining ability of dusts depends on the projected area and refractive index per unit volume. For sampling, filters must collect the minimum sized particle of interest. In this respect, membrane or glass fiber filters are recommended.
To determine particle mass, a measured quantity of air is drawn through filters, preferably of membrane or glass fiber, and the filter mass is compared to the mass before sampling. Electrostatic or ther-mal precipitators and various impactors have also been used. For further information, see ACGIH (1983), Lundgren et al. (1979), and Lodge (1989).
Chapter 44 of the 1999 ASHRAE Handbook—Applications pre-sents information on measuring and monitoring gaseous contami-nants. Relatively costly analytical equipment, which must be calibrated and operated carefully by experienced personnel, is needed. Numerous methods of sampling the contaminants, as well as the laboratory analysis techniques used after sampling, are spec-ified. Some of the analytical methods are specific to a single pollut-ant; others are capable of presenting a concentration spectrum for many compounds simultaneously.
COMBUSTION ANALYSIS Two approaches are used to measure the thermal output or capac-ity of a boiler, furnace, or other fuel-burning device. The direct or calorimetric test measures change in enthalpy or heat content of the fluid, air, or water heated by the device, and multiplies this by the flow rate to arrive at the unit’s capacity. The indirect test or flue gas analysis method determines the heat losses in the flue gases and the jacket and deducts them from the heat content (higher heating value) of the measured fuel input to the appliance. A heat balance simultaneously applies both tests to the same device. The indirect test usually indicates the greater capacity, and the difference is cred-ited to radiation from the casing or jacket and unaccounted-for losses.
With small equipment, the expense of the direct test is usually not justified, and the indirect test is used with an arbitrary radiation and unaccounted-for loss factor.
FLUE GAS ANALYSIS The flue gases from burning fossil fuels generally contain carbon dioxide (CO2), water, and hydrogen (H2) with some small amounts of carbon monoxide (CO), nitrogen oxides (NOx), sulfur oxides (SOx), and unburned hydrocarbons. However, generally only CO2 (or O2) and CO are measured to determine completeness of com-bustion and efficiency.
In the laboratory, the instruments most commonly used to mea-sure CO and CO2 are nondispersive infrared (NDIR) analyzers. The NDIR instruments have several advantages: (1) they are not very sensitive to flow rate, (2) no wet chemicals are required, (3) they have a relatively fast response, (4) measurements can be made over a wide range of concentrations, and (5) they are not sensitive to the presence of contaminants in the ambient air.
In the laboratory, oxygen is generally measured with an instru-ment that makes use of the paramagnetic properties of oxygen. The paramagnetic instruments are generally used because of their excel-lent accuracy and because they can be made specific to the measure-ment of oxygen.
For field testing and burner adjustment, portable combustion testing equipment is available. These instruments generally mea-sure O2 and CO with electrochemical cells. The CO2 is then calculated by an on-board microprocessor and, together with tem-perature, is used to calculate thermal efficiency. If a less expen-sive approach is required, a portable Orsat apparatus can be used to measure CO2, and a length-of-stain tube to measure CO.
DATA ACQUISITION AND RECORDING Almost every type of transducer and sensor is available with the necessary interface system to make it computer-compatible. The transducer itself begins to lose its identity when integrated into a system that incorporates such features as linearization, offset cor-rection, self-calibration, and so forth. This has eliminated the con-cern regarding the details of signal conditioning and amplification of basic transducer outputs. The personal computer is integrated into every aspect of data recording, including sophisticated graph-ics, acquisition and control, and analysis. Modems connected to the Internet or an internal network allow easy access to remote personal computer-based data-recording systems from virtually any locale.
Other means of recording, such as chart recorders, which can be either multipurpose or specifically designed for a given sensor, are available. Chart recorders provide a visual indication and a hard copy record of the data. Rarely is the output of a chart recorder used to process data. Simple indicators and readouts are used mostly to monitor the output of a sensor visually. In most situations, analog indicators such as d’Arsonval movement meters have been rendered obsolete by modern digital indicators. Industrial environments com-monly employ signal transmitters for control or computer data-han-dling systems to convert the signal output of the primary sensor into a compatible common signal span (e.g., the standard 4-20 mA cur-rent loop). All signal conditioning (ranging, zero suppression, refer-ence-junction compensation) is provided at the transmitter. Thus, all recorders and controllers in the system can have an identical electri-cal span, with variations only in charts and scales offering the advan-tages of interchangeability and economy in equipment cost. Long signal transmission lines can be used, and receiving devices can be added to the loop without degrading performance.
The vast selection of available hardware, an often confusing set of terminology, and the challenge of optimizing the performance/ cost ratio for a specific application make the task of configuring a data acquisition system difficult. A system specifically configured to meet a particular measurement need can quickly become obsolete if it has inadequate flexibility. Memory size, recording speed, and signal processing capability are major considerations in determin-ing the correct recording system. Thermal, mechanical, electromag-netic interference, portability, and meteorological factors also influence the selection.
Digital Recording A digital data acquisition system must contain an interface, which is a system involving one or several analog-to-digital con-verters, and in the case of multichannel inputs, circuitry for multi-plexing. The interface may also provide excitation for transducers, calibration, and conversion of units. The digital data are arranged into one or several standard digital bus formats. Many data acquisi-tion systems are designed to acquire data rapidly and store large records of data for later recording and analysis. Once the input sig-nals have been digitized, the digital data are essentially immune to noise and can be transmitted over great distances. Information is transferred to a computer/recorder from the inter-face as a pulse train, which can be transmitted as 4-, 8-, 12-, 16-, or 32-bit words. An 8-bit word is a byte; many communications meth-ods are rated according to their bytes per second transfer rate. Digital data are transferred in either serial or parallel mode. Serial transmis-sion means that the data are sent as a series of pulses, one bit at a time. Although slower than parallel systems, serial interfaces require only two wires, which lowers their cabling cost. The speed of serial transmissions is rated according to the symbols per second rate, or baud rate. In parallel transmission, the entire data word is transmitted at one time. To do this, each bit of a data word has to have its own transmission line; other lines are needed for clocking 14.32 2001 ASHRAE Fundamentals Handbook (SI) and control. Parallel mode is used for short distances or when high data transmission rates are required. Serial mode must be used for long-distance communications where wiring costs are prohibitive.
The two most popular interface bus standards currently used for data transmission are the IEEE 488, or general-purpose interface bus (GPIB), and the RS232 serial interface. The IEEE 488 bus sys-tem feeds data down eight parallel wires, one data byte at a time.
This parallel operation allows it to transfer data rapidly at up to 1 million characters per second. However, the IEEE 488 bus is limited to a cable length of 20 m and requires an interface connection on every meter for proper termination. The RS232 system feeds data serially down two wires, one bit at a time. An RS232 line may be over 300 m long. For longer distances, it may feed a modem to send data over standard telephone lines. A local area network (LAN) may be available in a facility for transmitting information. With appro-priate interfacing, transducer data are available to any computer connected to the network.
Bus measurements can greatly simplify three basic applica-tions—data gathering, automated limit testing, and computer-con-trolled processes. In data gathering applications, readings are collected over time. The most common applications include aging tests in quality control, temperature tests in quality assurance, and testing for intermittents in service. A controller can monitor any output indefinitely and then display the data directly on its screen or record it on magnetic tape or disks for future use.
In automated limit testing, the computer simply compares each measurement with programmed limits. The controller converts the readings to a good/bad readout. Automatic limit testing becomes highly cost-effective when working with large number of parame-ters of a particular unit under test.
In computer-controlled processes, the IEEE 488 bus system becomes a permanent part of a larger, completely automated sys-tem. For example, a large industrial process may require many elec-trical sensors that feed a central computer controlling many parts of the manufacturing process. An IEEE 488 bus controller collects readings from several sensors and saves the data until asked to dump an entire batch of readings to a larger central computer at one time.
Used in this manner, the IEEE 488 bus controller serves as a slave of the central computer.
Dynamic range and accuracy are two important parameters that must be considered in a digital recording system. Dynamic range refers to the ratio of the maximum input signal for which the system is useful to the noise floor of the system. The accuracy figure for a system is impacted by the signal noise level, nonlinearity, tempera-ture, time, crosstalk, and so forth. In selecting an 8-, 12-, or 16-bit analog-to-digital converter, the designer cannot assume that the sys-tem accuracy will necessarily be determined by the resolution of the encoders (i.e., 0.4%, 0.025%, and 0.0016%, respectively. If the sen-sor preceding the converter is limited to 1% full-scale accuracy, for example, no significant benefits are gained by using a 12-bit system over an 8-bit system and suppressing the least significant bit. How-ever, a greater number of bits may be required to cover a larger dynamic range.
Data Logging Devices Data loggers digitally store electrical signals (analog or digital) to an internal memory storage component. The signal from con-nected sensors is typically stored to memory at timed intervals rang-ing from MHz to hourly sampling. Some data loggers store data based on an event (e.g., button push, contact closure). Many data loggers can perform linearization, scaling, or other signal condition-ing and permit logged readings to be either instantaneous or aver-aged values. Most data loggers have built-in clocks that record the time and date together with transducer signal information. Data log-gers range from single-channel input to 256 or more channels. Some are general-purpose devices that will accept a multitude of analog and/or digital inputs, while others are more specialized to a specific measurement (e.g., a portable anemometer with built-in data log-ging capability) or for a specific application (e.g., a temperature, rel-ative humidity, CO2, and CO monitor with data logging for IAQ applications). Stored data are generally downloaded using a serial interface with a temporary direct connection to a personal computer.
Remote data loggers may also download via modem through tele-phone lines. Some data loggers are designed to allow downloading directly to a printer.
With the reduction in size of personal computers (laptops, note-books, hand-held PCs, and palmtops), the computer itself is now being used as the data logger. These “mobile” computers may be left in the field storing measurements from sensors directly interfaced into the computer.
Chart Recorders Chart recorders convert electrical signals (analog or digital) to records of the data versus time on a hard copy, usually paper.
Mechanical styluses use ink, hot wire, pressure, or electrically sen-sitive paper to provide a continuous trace. They are useful up to a few hundred hertz. Thermal and ink recorders are confined to chart speeds of several centimetres per second for recording relatively slow processes. Newer advances in portable recorders provide mul-tichannel inputs and up to 25 kHz real-time frequency response without using a pen or pen motor. Both x-y recorders and plotters allow two variables to be recorded with respect to one another. Their response times are generally limited to that of thermal and ink recorders. Oscillographic recorders have largely been made obso-lete by digital oscilloscopes.
STANDARDS ASA. 1980. Techniques of machinery vibration measurement. ANSI Stan-dard S2.17-80 (R 1986). Acoustical Society of America, New York.
ASA. 1984. Mechanical vibration of rotating and reciprocating machin-ery—Requirements for instruments for measuring vibration severity.
ANSI Standard S2.40-84 (R 1990).
ASA. 1984. Specification for acoustical calibrators. ANSI Standard S1.40-84 (R 1994).
ASA. 1985. Statistical methods for determining and verifying stated noise emission values of machinery and equipment. ANSI Standard S12.3-85 (R 1990).
ASA. 1987. Methods for determination of insertion loss of outdoor noise barriers. ANSI Standard S12.8-87.
ASA. 1989. Guide to the mechanical mounting of accelerometers. ANSI Standard S2.61-89 (R 1991).
ASA. 1989. Method for the designation of sound power emitted by machin-ery and equipment. ANSI Standard S12.23-89.
ASA. 1989. Reference quantities for acoustical levels. ANSI Standard S1.8-89.
ASA. 1990. Survey methods for the determination of sound power levels of noise sources. ANSI Standard S12.36-90.
ASA. 1990. Vibrations of buildings—Guidelines for the measurements of vibrations and evaluation of their effects on buildings. ANSI Standard S2.47-90.
ASA. 1995. Measurement of sound pressure levels in air. ANSI Standard S1.13-95.
ASHRAE. 1984. Standard method for measurement of flow of gas. ANSI/ ASHRAE Standard 41.7-1984 (RA 91).
ASHRAE. 1985. Laboratory methods of testing fans for rating.
ANSI/ASHRAE Standard 51-1985, also ANSI/AMCA Standard 210-85.
ASHRAE. 1986. Engineering analysis of experimental data. Guideline 2-1986 (RA 96).
ASHRAE. 1986. Laboratory method of testing in-duct sound power mea-surement procedure for fans. ANSI/ASHRAE Standard 68-1986, also ANSI/AMCA Standard 330-86.
ASHRAE. 1986. Standard method for temperature measurement. ANSI/ ASHRAE Standard 41.1-1986 (RA 91).
ASHRAE. 1987. Standard methods for laboratory air flow measurement.
ANSI/ASHRAE Standard 41.2-1987 (RA 92).
Measurement and Instruments 14.33 ASHRAE. 1988. Practices for measurement, testing, adjusting, and balanc-ing of building heating, ventilation, air-conditioning and refrigeration systems. ANSI/ASHRAE Standard 111-1988.
ASHRAE. 1988. A standard calorimeter test method for flow measurement of a volatile refrigerant. ANSI/ASHRAE Standard 41.9-1988.
ASHRAE. 1989. Standard method for pressure measurement.
ANSI/ASHRAE Standard 41.3-1989.
ASHRAE. 1989. Standard methods of measurement of flow of liquids in pipes using orifice flowmeters. ANSI/ASHRAE Standard 41.8-1989.
ASHRAE. 1992. Thermal environmental conditions for human occupancy.
ANSI/ASHRAE Standard 55-1992.
ASHRAE. 1994. Standard method for measurement of moist air properties.
ANSI/ASHRAE Standard 41.6-1994.
ASHRAE. 1996. Method for measurement of proportion of lubricant in liq-uid refrigerant. Standard 41.4-1996.
ASME. 1972. Application of fluid meters, Part II, 6th ed. (Interim Supple-ment 19.5 on Instruments and Apparatus) Standard PTC 19.5-72. Amer-ican Society of Mechanical Engineers, New York.
ASME. 1974. Temperature measurement instruments and apparatus. ANSI/ ASME Standard PTC 19.3-74 (R 1986).
ASME. 1983. Measurement uncertainty for fluid flow in closed conduits.
ANSI/ASME Standard MFC-2M-83.
ASME. 1985. Part I: Measurement uncertainty instruments and apparatus.
ANSI/ASME Standard PTC 19.1-85.
ASME. 1988. Measurement of liquid flow in closed conduits by weighing methods. ANSI/ASME Standard MFC-9M-88.
ASME. 1989. Measurement of fluid flow in pipes using orifice, nozzle, and venturi. Standard MFC-3M-85.
ASME. 1991. Gauges—Pressure indicating dial type—Elastic elements.
ANSI/ASME Standard B40.1-91.
ASME. 1991. Glossary of terms used in the measurement of fluid flow in pipes. ANSI/ASME Standard MFC-1M-91.
ASME. 1994. Method for establishing installation effects on flowmeters.
ANSI/ASME Standard MFC-10M-94.
ASTM. 1985. Standard practice for maintaining constant relative humidity by means of aqueous solutions. Standard E 104-85 (R 1996). American Society for Testing and Materials, West Conshohocken, PA.
ASTM. 1989. Standard test method for steady-state thermal performance of building assemblies by means of a guarded hot box. Standard C 236-89 (1993)e1.
ASTM. 1990. Standard practice for thermographic inspection of insulation installations in envelope cavities of frame buildings. Standard C 1060-90 (1997)e1.
ASTM. 1990. Standard test method for thermal performance of building assemblies by means of a calibrated hot box. Standard C 976-90 (1996)e1.
ASTM. 1993. Temperature-electromotive force (EMF) tables for standard-ized thermocouples. Standard E 230-93.
ASTM. 1995. Standard practice for in-situ measurement of heat flux and temperature on building envelope components. Standard C 1046-95.
ASTM 1995. Standard test method for determining air change in a single zone by means of gas dilution. Standard E 741-95.
ASTM. 1995. Standard test method for steady-state heat transfer properties of horizontal pipe insulation. Standard C 335-95.
ASTM. 1995. Standard practice for determining thermal resistance of build-ing envelope components from the in-situ data. Standard C 1155-95.
ASTM. 1995. Standard test methods for water vapor transmission of mate-rials. Standard E 96-95.
ASTM. 1997. Standard test method for steady-state heat flux measurements and thermal transmission properties by means of the guarded-hot-plate apparatus. Standard C 177-97.
ASTM. 1998. Standard test method for steady-state thermal transmission properties by means of the heat flow meter apparatus. Standard C 518-98.
ASTM. 2000. Atmospheric analysis; occupational health and safety; pro-tective clothing. Vol. 11.03. (182 standards).
ISO. 1977. Measurement of fluid flow in closed conduits—Velocity area method using Pitot static tubes. Standard 3966. International Organiza-tion for Standardization, Geneva.
ISO. 1985. Thermal environments—Instruments and methods for measur-ing physical quantities. Standard 7726.
ISO. 1990. Ergonomics—Determination of metabolic heat production.
Standard 8996.
ISO. 1994. Moderate thermal environments—Determination of the PMV and PPD indices and specification of the conditions for thermal comfort.
Standard 7730.
ISO. 1995. Ergonomics of the thermal environment—Estimation of the ther-mal insulation and evaporative resistance of a clothing ensemble. Stan-dard 9920.
REFERENCES ACGIH. 1983. Air sampling instruments for evaluation of atmospheric con-taminants, 6th ed. American Conference of Governmental Industrial Hygienists, Cincinnati, OH.
ASTM. 1993. Manual on the use of thermocouples in temperature measure-ment. Manual 12.
Ambrose, J.H., L.C. Chow, and J.E. Beam. 1990. Capillary flow properties of mesh wicks. AIAA Journal of Thermophysics 4:318-24.
Amdur, E.J. 1965. Two-pressure relative humidity standards. In Humidity and Moisture 3:445. Reinhold Publishing Corporation, New York.
Bedford, T. and C.G. Warmer. 1935. The globe thermometer in studies of heating and ventilating. Journal of the Institution of Heating and Venti-lating Engineers 2:544.
Benedict, R.P. 1984. Fundamentals of temperature, pressure and flow mea-surements, 3rd ed. John Wiley and Sons, New York.
Bentz, D.P. and J.W. Martin. 1987. Using the computer to analyze coating defects. Journal of Protective Coatings and Linings 4(5).
Bruce, R.R. and A. Klute. 1956. The measurement of soil moisture diffusiv-ity. Proceedings of the Soil Science Society of America 20:458-62.
Burch, D.M. 1980. Infrared audits of roof heat loss. ASHRAE Transactions 86(2).
Burch, D.M. and C.M. Hunt. 1978. Retrofitting an existing residence for energy conservation—An experimental study. Building Science Series 105. National Institute of Standards and Technology, Gaithersburg, MD.
Burch, D.M., W.C. Thomas, and A.H. Fanney. 1992. Water vapor perme-ability measurements of common building materials. ASHRAE Trans-actions 98(1).
Burns, G.W., M.G. Scroger, G.F. Strouse, M.C. Croarkin, and W.F. Guthrie.
1992. Temperature-electromotive force reference functions and tables for the letter-designated thermocouple types based on the ITS-90. NIST Monograph 175. U.S. Government Printing Office, Washington, D.C.
Carotenuto, A., F. Fucci, and G. LaFianzi. 1991. Adsorption phenomena in porous media in the presence of moist air. International Journal of Heat and Mass Transfer 18:71-81.
Cohen, E.R. 1990. The expression of uncertainty in physical measurements.
1990 Measurement Science Conference Proceedings, Anaheim, CA.
Coleman, H.W. and W.G. Steele. 1989. Experimentation and uncertainty analysis for engineers. John Wiley and Sons, New York.
Coleman, H.W. and W.G. Steele. 1995. Engineering application of experi-mental uncertainty analysis. AIAA Journal 33(10).
Considine, D.M. 1985. Process instruments and controls handbook, 3rd ed.
McGraw-Hill, New York.
Cunningham, M.J. and T.J. Sprott. 1984. Sorption properties of New Zea-land building materials. Building Research Association of New Zealand Research Report No. 45, Judgeford.
D’Eustachio, D. and R.E. Schreiner. 1952. A study of transient heat method for measuring thermal conductivity. ASHVE Transactions 58:331.
DeCarlo, J.P. 1984. Fundamentals of flow measurement. Instrumentation Society of America, Research Triangle Park, NC.
DeWitt, D.P. and G.D. Nutter. 1988. Theory and practice of radiation ther-mometry. John Wiley and Sons, New York.
Fanger, P.O. 1982. Thermal comfort. Robert E. Krieger, Malabar, FL.
Fanney, A.H., W.C. Thomas, D.M. Burch, and L.R. Mathena. 1991. Mea-surements of moisture diffusion in building materials. ASHRAE Trans-actions 97:99-113.
Freitas, V., P. Crausse, and V. Abrantes. 1991. Moisture diffusion in thermal insulating materials. In Insulation materials: Testing and applications, Vol. 2. ASTM Special Technical Publication STP 1116. American Soci-ety for Testing and Materials, West Conshohocken, PA.
Gagge, A.P., J.A.J. Stolwijk, and Y. Nishi. 1971. An effective temperature scale based on a simple model of human physiological regulatory response. ASHRAE Transactions 77(1).
Goldstein, R.J. 1978. Application of aerial infrared thermography. ASHRAE Transactions 84(1).
Greenspan, L. 1977. Humidity fixed points of binary saturated aqueous solutions. Journal of Research of the National Bureau of Standards 81A:89-95.
14.34 2001 ASHRAE Fundamentals Handbook (SI) Greenspan, L. and A. Wexler. 1968. An adiabatic saturation psychrometer.
Journal of Research of the National Bureau of Standards 72C(1):33.
Gummerson, R.J., C. Hall, W.D. Hoff, R. Hawkes, G.N. Holland, and W.S.
Moore. 1979. Unsaturated water flow within porous materials observed by NMR imaging. Nature 281:56-57.
Hasegawa, S. 1976. The NBS two-pressure humidity generator, Mark 2.
Journal of Research of the National Bureau of Standards 81A:81.
Hooper, F.C. and F.C. Lepper. 1950. Transient heat flow apparatus for the determination of thermal conductivity. ASHVE Transactions 56:309.
Hooper, F.C. and S.C. Chang. 1953. Development of thermal conductivity probe. ASHVE Transactions 59:463.
Hudson, R.D., Jr. 1969. Infrared system engineering. John Wiley & Sons, NY.
Kumaran, M.K. and M. Bomberg. 1985. A gamma-spectrometer for deter-mination of density distribution and moisture distribution in building materials. Proceedings of the International Symposium on Moisture and Humidity, Washington, D.C., pp. 485-90.
Kusuda, T. 1965. Calculation of the temperature of a flat-plate wet surface under adiabatic conditions with respect to the Lewis relation. Humidity and Moisture I:16. Reinhold Publishing Corporation, New York.
Liptak, B.G., ed. 1972. Instrument engineers handbook, Vol. 1. Chilton Book Company, Philadelphia, PA.
Lodge, J.P., ed. 1989. Methods of air sampling and analysis, 3rd ed. Lewis Publishers, MI.
Lundgren, D.A. et al., eds. 1979. Aerosol measurement. University Presses of Florida, Gainesville.
Mack, R.T. 1986. Energy loss profiles: Foundation for future profit in ther-mal imager, sales, and service. Proceedings of the 5th Infrared Informa-tion Exchange, Book 1, AGEMA Infrared Systems, Secaucus, NJ.
Madsen, T.L. 1976. Thermal comfort measurements. ASHRAE Transactions 82(1).
Mattingly, G.E. 1984. Workshop on fundamental research issues in orifice metering. GRI Report 84/0190. Gas Research Institute, Chicago, IL.
Mattingly, G.E. 1992. The characterization of a piston displacement-type flowmeter calibration facility and the calibration and use of pulsed out-put type flowmeters. Journal of Research of the National Institute of Standards and Technology 97(5):509.
McCullough, E.A., B.W. Jones, and J. Huck. 1985. A comprehensive data base for estimating clothing insulation. ASHRAE Transactions 92:29-47.
McLean, R.C., G.H. Galbraith, and C.H. Sanders. 1990. Moisture transmis-sion testing of building materials and the presentation of vapour perme-ability values. Building Research and Practice. 82-103.
Mease, N.E., W.G. Cleveland, Jr., G.E. Mattingly, J.M. Hall. 1992. Air speed calibrations at the National Institute of Standards and Technology. Pro-ceedings of the 1992 Measurement Science Conference, Anaheim, CA.
Miller, R.W. 1983. Measurement engineering handbook. McGraw-Hill, NY.
NIST. 1976. Liquid-in-glass thermometry. NIST Monograph 150. National Institute of Standards and Technology, Gaithersburg, MD.
NIST. 1986. Thermometer calibrations. NIST Monograph 174.
Nottage, H.B., J.G. Slaby, and W.P. Gojsza. 1952. A smoke-filament tech-nique for experimental research in room air distribution. ASHVE Trans-actions 58:399.
Olesen, B.W. 1985. A new and simpler method for estimating the thermal insulation of a clothing ensemble. ASHRAE Transactions 92:478-92.
Olesen, B.W., J. Rosendahl, L.N. Kalisperis, L.H. Summers, and M. Stein-man. 1989. Methods for measuring and evaluating the thermal radiation in a room. ASHRAE Transactions 95(1).
Paljak, I. and B. Pettersson. 1972. Thermography of buildings. National Swedish Institute for Materials Testing, Stockholm.
Parmelee, G.V. and R.G. Huebscher. 1946. The shielding of thermocouples from the effects of radiation. ASHVE Transactions 52:183.
Prazak, J., J. Tywoniak, F. Peterka, and T. Slonc. 1990. Description of trans-port of liquid in porous media—A study based on neutron radiography data. International Journal of Heat and Mass Transfer 33:1105-20.
Quenard, D. and H. Sallee. 1989. A gamma-ray spectrometer for measure-ment of the water diffusivity of cementitious materials. Proceedings of the Materials Research Society Symposium, Vol. 137.
Quinn, T.J. 1990. Temperature, 2nd ed. Academic Press, New York.
Richards, R.F., D.M. Burch, W.C. Thomas. 1992. Water vapor sorption mea-surements of common building materials. ASHRAE Transactions 98(1).
Richardson, L. 1965. A thermocouple recording psychrometer for measure-ment of relative humidity in hot, arid atmosphere. In Humidity and Mois-ture I:101. Reinhold Publishing Corporation, New York.
Schooley, J.F. 1986. Thermometry. CRC Press, Boca Raton, FL.
Schooley, J.F., ed. 1992. Temperature: Its measurement and control in sci-ence and in industry, Vol. 6. American Institute of Physics, New York.
Seely, R.E. 1955. A circuit for measuring the resistance of energized A-C windings. AIEE Transactions, 214.
Shafer, M.R. 1961. Performance characteristics of turbine flowmeters. Pro-ceedings of the Winter Annual Meeting, Paper No. 61-WA-25. Ameri-can Society of Mechanical Engineers, New York.
Till, C.E. and G.E. Handegord. 1960. Proposed humidity standard. ASHRAE Transactions 66:288.
Tobiasson, W. and C. Korhonen. 1985. Roofing moisture surveys: Yester-day, today, and tomorrow. Proceedings of the Second International Sym-posium on Roofing Technology, Gaithersburg, MD.
Tveit, A. 1966. Measurement of moisture sorption and moisture permeabil-ity of porous materials. Report No. 45. Norwegian Building Research Institute, Oslo.
Vernon, H.M. 1932. The globe thermometer. Proceedings of the Institution of Heating and Ventilating Engineers 39:100.
Wentzel, J.D. 1961. An instrument for measurement of the humidity of air.
ASHRAE Journal 11:67.
Wile, D.D. 1947. Air flow measurement in the laboratory. Refrigerating Engineering 6:515.
Woodring, E.D. 1969. Magnetic turbine flowmeters. Instruments and Con-trol Systems 6:133.
Worrall, R.W. 1965. Psychrometric determination of relative humidities in air with dry-bulb temperatures exceeding 212°F. In Humidity and Mois-ture I:105. Reinhold Publishing Corporation, New York.
BIBLIOGRAPHY Beranek, L.L. 1988. Acoustical measurements. Published for the Acoustical Society of America by the American Institute of Physics, New York.
Beranek, L.L. 1989. Noise and vibration control. Institute of Noise Control Engineering, Poughkeepsie, NY.
EPA. 1991. Introduction to indoor air quality: A self-paced learning module.
EPA/400/3-91/002, U.S. Environmental Protection Agency, Washington.
Harris, C.M. 1987. Shock and vibration handbook, 3rd ed. McGraw-Hill, New York.
IEEE. 1987. Standard digital interface for programmable instrumentation.
ANSI/IEEE Standard 488.1-87 (R 1994).
IESNA. 1999. Lighting handbook, 9th ed. Illuminating Engineering Society of North America, New York.
Lord, H.W. et al. 1987. Noise control for engineers. Krieger Publishing, Melbourne, FL.
Morrison, R. 1986. Grounding and shielding techniques in instrumentation, 3rd ed. John Wiley and Sons, New York.
Spitzer, D.W., ed. 1991. Flow measurement. Instrumentation Society of America, Research Triangle Park, NC.
Steele, W.G., R.A. Ferguson, R.P. Taylor, and H.W. Coleman. 1994. Com-parison of ANSI/ASME and ISO models for calculation of uncertainty.
ISA Transactions 33:339-52.
Tilford, C.R. 1992. Pressure and vacuum measurements. In Physical meth-ods of chemistry, 2nd ed., Vol. 6, pp. 106-73. John Wiley and Sons, New York.
15.1 CHAPTER 15 FUNDAMENTALS OF CONTROL Terminology ............................................................................ 15.1 Types of Control Action .......................................................... 15.2 Classification by Energy Source ............................................. 15.4 Computers for Automatic Control .......................................... 15.4 CONTROL COMPONENTS ................................................... 15.4 Controlled Devices ................................................................. 15.4 Sensors .................................................................................... 15.8 Controllers ............................................................................ 15.10 Auxiliary Control Devices .................................................... 15.11 COMMUNICATION NETWORKS FOR BUILDING AUTOMATION SYSTEMS ............................. 15.12 Communication Protocols ..................................................... 15.12 The OSI Network Model ........................................................ 15.12 Network Structure ................................................................. 15.13 Specifying BAS Networks ...................................................... 15.15 Approaches to Interoperability ............................................. 15.15 COMMISSIONING ............................................................... 15.15 Tuning ................................................................................... 15.16 UTOMATIC HVAC control systems are designed to maintain Atemperature, humidity, pressure, flow, power, lighting levels, and safe levels of indoor contaminants. Automatic control primarily modulates, stages, or sequences mechanical and electrical equip-ment to meet load requirements and provide safe operation of the equipment. It can use digital, pneumatic, mechanical, electrical, and electric control devices; human intervention is limited to starting and stopping equipment and adjusting control set points.
This chapter focuses on the fundamental concepts and devices normally used by a control system designer. It covers (1) control fundamentals, including terminology; (2) the types of control com-ponents; (3) the methods of connecting these components to form various individual control loops or subsystems; and (4) commis-sioning, operation, and maintenance. Chapter 45 of the 1999 ASHRAE Handbook—Applications discusses the design of controls for specific HVAC applications.
TERMINOLOGY A closed loop or feedback control measures actual changes in the controlled variable and actuates the controlled device to bring about a change. The corrective action continues until the variable is brought to a desired value within the design limitations of the con-troller. This arrangement of having the controller sense the value of the controlled variable is known as feedback. Figure 1 shows a feedback control.
An open loop control does not have a direct link between the value of the controlled variable and the controller. An open loop control anticipates the effect of an external variable on the system and adjusts the set point to avoid excessive offset. An example is an outdoor thermostat arranged to control heat to a building in propor-tion to the calculated load caused by changes in outdoor tempera-ture. In essence, the designer presumes a fixed relationship between outside air temperature and the heat requirement of the building and specifies control action based on the outdoor air temperature. The actual space temperature has no effect on this controller. Because there is no feedback on the controlled variable (space temperature), the control is an open loop.
Figure 1 illustrates the components of the typical control loop.
The sensor measures the controlled variable and transmits to the controller a signal (pneumatic, electric, or electronic) having a pressure, voltage, or current value proportional to the value of the variable being measured. The controller compares this value with the set point and signals to the controlled device for correc-tive action. A controller can be hardware or software. A hard-ware controller is an analog device that continuously receives and acts on data. Thermostats, humidistats, and pressure controls are examples of hardware controllers. A software controller is a digital device that receives and acts on data on a sample-rate basis. Digital algorithms are examples of software controllers.
The set point is the desired value of the controlled variable. The controller seeks to maintain this set point. The controlled device reacts to signals received from the controller to vary the control agent. The controlled device may be a valve, a damper, a heating element, or a motor driving a pump or a fan.
The control agent is the medium manipulated by the controlled device. It may be air or gas flowing through a damper; gas, steam, or water flowing through a valve; or an electric current.
The process is the air-conditioning apparatus being controlled. It reacts to the output of the control agent and effects the change in the controlled variable. It may be a coil, fan, or humidifier. The controlled variable is the temperature, humidity, pressure, etc., being controlled.
A control loop can be represented in the form of a block dia-gram, in which each component is modeled and represented in its own block. Figure 2 is a block diagram of the control loop shown in Figure 1. The flow of information from one component to the next is shown by lines between the blocks. The figure shows the set point being compared to the controlled variable. The difference is the off-set error, also known as offset drift, deviation, droop, or steady-state error. The offset error is fed into the controller, which sends a control signal to the controlled device. In this case, the controlled device is a valve that can change the amount of steam flow through the coil of Figure 1. The amount of steam flow is the input to the next block, which represents the process. From the process block comes the controlled variable, which is temperature. The controlled variable is sensed by the sensing element and fed to the controller as feedback, completing the loop.
The preparation of this chapter is assigned to TC 1.4, Control Theory and Application.
Fig. 1 Discharge Air Temperature Control (Example of Feedback Control) 15.2 2001 ASHRAE Fundamentals Handbook (SI) Each component of Figure 2 can be represented by a transfer function, which is an idealized mathematical representation of the relationship between the input and the output variables of the component. The transfer function must be sufficiently detailed to cover both the dynamic and static characteristics of the device.
The dynamics of the component are represented in the time domain by a differential equation. In environmental control, the transfer function of many of the components can be adequately described by a first-order differential equation, implying that the dynamic behavior is dominated by a single capacitance factor. For a solution, the differential equation is converted to its Laplace transform or z-transform.
The time constant is defined as the time it takes for the output to reach 63.2% of its final value when a step change in the input is effected. Components with small time constants alter their output rapidly to reflect changes in the input; conversely, components with a larger time constant are sluggish in responding to changes in the input.
Dead time is a phase shift that can cause control and modeling problems. Dead time (or time lag) is the time between a change in the process input and when the change affects the output of the pro-cess. Dead time can occur in the control loop of Figure 1 due to the transportation time of the air from the coil to the space. After a coil temperature is changed, there is dead time while the affected supply air travels the distribution system and finally reaches the sensor in the space. The mass of air within the space further delays detection of the coil temperature change. Dead time can also be caused by a slow sensor, or a time lag in the signal from the controller. If the dead time is small, it may be ignored in the model of the control; if it is significant, it must be considered.
The gain of a transfer function is the amount the output of the component changes for a given change of input under steady-state conditions. If the element is linear, its gain remains constant. How-ever, many control components are nonlinear, and have gains that depend on the operating conditions. Figure 3 shows the response of the first-order-plus-dead-time process to a step change of the input signal. Notice that the process shows no reaction during the dead time, followed by a response that resembles a first order exponential.
TYPES OF CONTROL ACTION Closed loop controls are commonly classified by the type of cor-rective action the controller is programmed to take when it senses a deviation of the controlled variable from the set point. Both hard-ware and software controllers can be classified according to the fol-lowing most common types of control action.
Two-Position Action The control device shown in Figure 4 can be positioned only to a maximum or minimum state (e.g., on or off). Because two-posi-tion control is simple and inexpensive, it is used extensively for both industrial and commercial control. A typical home thermo-stat that starts and stops a furnace is an example of two-position action.
Controller differential, as it applies to two-position control action, is the difference between a setting at which the controller operates to one position and a setting at which it operates to the other. Thermostat ratings usually refer to the differential (in degrees) that becomes apparent by raising and lowering the dial set-ting. This differential is known as the manual differential of the thermostat. When the same thermostat is applied to an operating system, the total change in temperature that occurs between a “turn-on” state and a “turn-off” state is usually different from the mechan-ical differential. The operating differential may be greater due to thermostat lag or hysteresis or to heating or cooling anticipators built into the thermostat.
Anticipation Applied to Two-Position Action. This common variation of strictly two-position action is often used on room ther-mostats to reduce the operating differential. In heating thermostats, a heater element in the thermostat is energized during “on” periods, thus shortening the on-time because the heater warms the thermo-stat. This is known as heat anticipation. The same anticipation action can be obtained in cooling thermostats by energizing a heater thermostat at “off” periods. In both cases, the percentage of on-time is varied in proportion to the load, while the total cycle time remains relatively constant.
Timed Two-Position Action. This action occurs when a heat-ing or cooling element is turned on for a time interval proportional to the deviation from set point. For example, an element may be turned on for two minutes and off for one minute when the devia-tion from set point is 1 K. This is similar to incremental action applied to floating control, except the time interval is usually shorter for incremental action.
Fig. 2 Block Diagram of Discharge Air Temperature Control Fig. 3 Process Subjected to a Step Input Fig. 4 Two-Position Control Fig. 5 Floating Control Showing Variations in Controlled Variable as Load Changes Fundamentals of Control 15.3 Floating Action In floating action, the controller can perform only two opera-tions—moving the controlled device toward either its open or closed position, usually at a constant rate (Figure 5). Generally, a neutral zone between the two positions allows the controlled device to stop at any position when the controlled variable is within the dif-ferential of the controller. When the controlled variable falls outside the differential of the controller, the controller moves the controlled device in the proper direction. In order to function properly, the sensing element must react faster than the actuator drive time. If not, the control functions the same as a two-position control.
Incremental Action. This action is a variation of floating con-trol. Incremental action varies the pulse action to open or close an actuator depending on how close the controlled variable is to the set point. As the controlled variable comes close to the set point, the pulses become shorter. This action allows closer control using float-ing motor actuators.
Modulating Control With modulating control, the output of the controller can vary infinitely over the range of the controller. The following terms are used to describe this type of control: • Throttling range is the amount of change in the controlled variable required to cause the controller to move the controlled device from one extreme to the other. It can be adjusted to meet job requirements. Throttling range is inversely proportional to proportional gain.
• Control point is the actual value of the controlled variable at which the instrument is controlling. It varies within the throttling range of the controller and changes with changing load on the system and other variables.
• Offset, or error signal, is the difference between the set point and the actual control point under stable conditions. This is sometimes called drift, deviation, droop, or steady-state error.
The following are the three typical modulating control modes: Proportional Action. In proportional action, the controlled device is positioned proportionally in response to changes in the controlled variable (Figure 6). A proportional controller can be described mathematically by (1) where Vp = output of controller Kp = proportional gain (inversely proportional to throttling range) e = error signal or offset Vo = offset adjustment parameter The output of the controller is proportional to the difference between the sensed value, the controlled variable, and its set point.
The controlled device is normally adjusted to be in the middle of its control range at set point by using an offset adjustment. This control is similar to that shown in Figure 1. Proportional plus Integral (PI) Action. This type of control improves on simple proportional control by adding another compo-nent to the control action that eliminates the offset typical of pro-portional control (Figure 7). Reset action may be described by (2) where Ki = integral gain θ = time The second term in Equation (2) implies that the longer error e exists, the more the controller output will change in attempting to eliminate the error. Proper selection of proportional and integral gain constants increases stability, and eliminates offset, giving greater control accuracy. PI control can also improve energy effi-ciency in applications such as VAV fan control, chiller control, and hot and cold deck control of the air handler.
Proportional-Integral-Derivative (PID) Action. This type of control is PI control with a derivative term added to the controller.
It varies with the value of the derivative of the error. The equation for PID control is (3) where Ka = derivative gain of controller de/dθ = time derivative of error Adding the derivative term gives some anticipatory action to the controller, which results in a faster response and greater stability.
However, the derivative term also makes the controller more sensi-tive to noisy signals and harder to tune than a PI controller. Most HVAC control loops perform satisfactorily with PI control alone.
Adaptive control, or self tuning, is a form of digital PID control, where the gain factors (Kp, Ki, and Ka) are continuously or periodi-cally modified automatically to compensate for the control loop offset.
Fuzzy Logic Fuzzy logic control offers an alternative to traditional control algorithms. A fuzzy logic controller uses a series of “if-then” rules that emulates the way a human operator might control the process.
Examples of fuzzy logic might include 1. IF the room temperature is high AND the rate of change is decreasing, THEN increase cooling a little.
2. IF the room temperature is high AND the rate of change is increasing, THEN increase cooling a lot.
Vp Kpe Vo + = Vp Kpe Ki e θ Vo + d ∫ + = Fig. 6 Proportional Control Showing Variations in Controlled Variable as Load Changes Fig. 7 Proportional plus Integral (PI) Control Vp Kpe Ki e θ Ka + d θ d de Vo + ∫ + = 15.4 2001 ASHRAE Fundamentals Handbook (SI) The designer of a fuzzy logic controller must first define the rules and then define such terms as high, increasing, decreasing, a lot, and a little. The room temperature, for instance, might be mapped into a series of functions that include very low, low, OK, high, and very high. The “fuzzy” element is introduced when the functions overlap and the room temperature is, for example, 70% high and 30% OK. In this case, multiple rules are combined to determine the appropriate control action.
CLASSIFICATION BY ENERGY SOURCE Control components may be classified according to the primary source of energy as follows: • Pneumatic components use compressed air, usually at a pressure of 100 to 140 kPa (gage), as an energy source. The air is generally supplied to the controller, which regulates the pressure supplied to the controlled device.
• Electric components use electrical energy, either low- or line-voltage, as the energy source. The controller regulates electrical energy supplied to the controlled device. Controlled devices in this category include relays; electromechanical, electromagnetic, and hydraulic actuators; and solid-state regulating devices. Components that include signal conditioning, modulation, and amplification in their operation are classified as electronic.
A digital controller receives electronic signals from the sen-sors, converts the electronic signals to numbers, and performs mathematical operations on these numbers inside a microproces-sor. The output from the digital controller takes the form of a number, which is then converted to an electronic signal to operate the actuator. The digital controller must sample its data because the microprocessor requires time for other operations than read-ing data. If the sampling interval for the digital controller is prop-erly chosen to avoid second- and third-order harmonics, there will be no significant degradation in control performance due to sam-pling.
• Self-powered components apply the power of the measured system to induce the necessary corrective action. The measuring system derives its energy from the process under control, without any auxiliary source of energy. Temperature changes at the sensor result in pressure or volume changes of the enclosed media that are transmitted directly to the operating device of the valve or damper. A component using a thermopile in a pilot flame to generate electrical energy is also self-powered.
This method of classification can be extended to individual con-trol loops and to complete control systems. For example, the room temperature control for a particular room that includes a pneumatic room thermostat and a pneumatically actuated reheat coil would be referred to as a pneumatic control loop. Many control systems use a combination of controls and are called hybrid systems.
COMPUTERS FOR AUTOMATIC CONTROL Computers can perform the control described in this chapter.
Chapter 38 of the 1999 ASHRAE Handbook—Applications covers computer components and some of the ways computers are being used in the HVAC control industry.
CONTROL COMPONENTS This section groups components by their function in a complete control system. The section on Controlled Devices considers the controlled device or final control element, examples of which are relays, valves, and dampers. Actuators, which are used to drive the valve or damper assembly, are also covered.
The section on Sensors considers the sensing element that mea-sures changes in the controlled variable. Some of the sensor types included are temperature, humidity, flow, and pressure. While many other sensors are available, these represent the majority of those found in HVAC control systems. The section on Controllers reviews various controllers. Control-lers are classified according to energy source; pneumatic, elec-tric/electronic, and digital. They are also classified according to the control action they cause to maintain the desired condition (set point)—two-position, floating, proportional, proportional plus inte-gral (PI), or proportional plus integral plus derivative (PID). Ther-mostats (devices that combine a temperature sensor and controller in a single unit) are also described as well.
Fundamental control systems can be constructed using only the components described in the first three subsections. In practice, however, a fourth group is sometimes necessary. The section on Auxiliary Control Devices covers transducers, switches, power sup-plies, and air compressors.
CONTROLLED DEVICES A controlled device regulates the flow of steam, water, electric-ity, or air in an HVAC system. Water and steam flow regulators are known as valves, and airflow control devices are called dampers; both devices perform essentially the same function and must be properly sized and selected for the particular application. The con-trol link to the valve or damper is called an actuator or operator.
This device uses electricity, compressed air, or hydraulic fluid to power the motion of the valve stem or damper linkage through its operating range.
Valves An automatic valve is designed to control the flow of steam, water, gas, or other fluids. It can be considered a variable orifice positioned by an electric or pneumatic actuator in response to impulses or signals from the controller. It may be equipped with a throttling plug or V-port specially designed to provided a desired flow characteristic.
Renewable composition disks are common. They are made of materials best suited to the media handled by the valve, the operat-ing temperature, and the pressure. For high pressure or for super-heated steam, metal disks are often used. Internal parts, such as the seat ring, throttling plug, or V-port skirt, disk holder, and stem, are sometimes made of stainless steel or other hard and corrosion-resis-tant metal for use in severe service.
Various types of automatic valves include the following: A single-seated valve (Figure 8A) is designed for tight shutoff.
Appropriate disk materials for various pressures and media are used.
A double-seated or balanced valve (Figure 8B) is designed so that the media pressure acting against the valve disk is essentially balanced, reducing the actuator force required. It is widely used where fluid pressure is too high to permit a single-seated valve to close. It cannot be used where a tight shutoff is required.
Fig. 8 Typical Single- and Double-Seated Two-Way Valves Fundamentals of Control 15.5 A three-way mixing valve (Figure 9A) has two inlet connec-tions and one outlet connection and a double-faced disk operating between two seats. It is used to mix two fluids entering through the inlet connections and leaving through the common outlet, according to the position of the valve stem and disk.
A three-way diverting valve (Figure 9B) has one inlet connec-tion and two outlet connections and two separate disks and seats. It is used to divert the flow to either of the outlets or to proportion the flow to both outlets.
A butterfly valve consists of a heavy ring enclosing a disk that rotates on an axis at or near its center and is similar to a round single-blade damper. In principle, the disk seats against a ring machined within the body or a resilient liner in the body. Two butterfly valves can be used together to act like a three-way valve for mixing or diverting.
Flow Characteristics. The performance of a valve is expressed in terms of its flow characteristics as it operates through its stroke, based on a constant pressure drop. Three common characteristics are shown in Figure 10 and are defined as follows: • Quick opening. Maximum flow is approached rapidly as the device begins to open. • Linear. Opening and flow are related in direct proportion. • Equal percentage. Each equal increment of opening increases the flow by an equal percentage over the previous value.
Because the pressure drop across a valve seldom remains con-stant as its opening changes, actual performance usually deviates from the published characteristic curve. The magnitude of the devi-ation is determined by the overall design. For example, in a system arranged so that control valves or dampers can shut off all flow, the pressure drop across a controlled device increases from a minimum at design conditions to the total pressure drop at no flow. Figure 11 shows the extent of the resulting deviations for a valve or damper designed with a linear characteristic, when selection is based on var-ious percentages of total system pressure drop. To allow for ade-quate control by valve or damper, the design pressure drop should be a reasonably large percentage of the total system pressure drop, or the system should be designed and controlled so that the pressure drop remains relatively constant.
Selection and Sizing. Higher pressure drops for controlled devices are obtained by using smaller sizes with a possible increase in size of other equipment in the system. Because sizing techniques are different for steam, water, and air, each is discussed separately.
Steam Valves. Steam-to-water and steam-to-air heat exchangers are typically controlled through regulation of steam flow using a two-way throttling valve. One-pipe steam systems require a line-size, two-position valve for proper condensate drainage and steam flow, while two-pipe steam systems can be controlled by two-posi-tion or modulating (throttling) valves.
Water Valves. Valves for water service may be two- or three-way and two-position or proportional. Proportional valves are used most often, but two-position valves are not unusual and are some-times essential. While it is possible to design a water system in which the pressure differential from supply to return is kept con-stant, it is seldom done. It is safer to assume that the pressure drop across the valve increases as it modulates from fully open to fully closed. Figure 12 shows the effect in a simple system with one pump, one two-way control valve, and a heat exchanger. The sys-tem curve represents the pressure loss in the piping and heat Fig. 9 Typical Three-Way Mixing and Diverting Valves Fig. 10 Typical Flow Characteristics of Valves Fig. 11 Typical Performance Curves for Linear Devices at Various Percentages of Total System Pressure Drop 15.6 2001 ASHRAE Fundamentals Handbook (SI) exchanger at various flow rates. The pump curve is the typical curve for a centrifugal pump. At design flow rates, the valve is selected for a specific pressure drop A – A′. At part load, the valve must partially close to provide a higher pressure drop B –B′. The ratio between the design pressure drop A – A′ and the zero Flow pressure drop C – C′ influences the control capability of the valve.
Equal percentage valves provide better control at part load, par-ticularly in hot water coils where the heat output of the coil is not linearly related to flow. As flow is reduced, a greater amount of heat is transferred from each unit of water, counteracting the reduction in flow. Equal percentage valves are used in an attempt to linearize the heat transfer from the coil with respect to the control signal.
Two-way control valves should be sized to provide from 20 to 60% of the total system pressure drop. The valve operator should be sized to close the valve against the full pump head pressure to ensure complete shut off during no-flow condition.
For additional information on control valve sizing and selection, see Chapters 12 and 42 of the 2000 ASHRAE Handbook—Systems and Equipment.
Actuators. Valve actuators include the following general types: • A pneumatic valve actuator consists of a spring-opposed, flexible diaphragm or bellows attached to the valve stem. An increase in air pressure above the minimum point of the spring range compresses the spring and simultaneously moves the valve stem. Springs of various pressure ranges can sequence the operation of two or more devices, if properly selected or adjusted.
For example, a chilled water valve actuator may modulate the valve from fully closed to fully open over a spring range of 20 to 60 kPa, while a sequenced steam valve may actuate from 60 to 90 kPa.
Two-position pneumatic control is accomplished using a two-position pneumatic relay to apply either full air pressure or no pressure to the valve actuator. Pneumatic valves and valves with spring-return electric actuators can be classified as normally open or normally closed.
A normally open valve assumes an open position, providing full flow, when all actuating force is removed.
A normally closed valve assumes a closed position, stopping flow, when all actuating force is removed.
• Springless pneumatic valve actuators, which use two opposed diaphragms or two sides of a single diaphragm, are generally limited to special applications involving large valves or high fluid pressure.
• An electric-hydraulic valve actuator is similar to a pneumatic one, except that it uses an incompressible fluid circulated by an internal electric pump.
• A solenoid consists of a magnetic coil operating a movable plunger. Most are for two-position operation, but modulating solenoid valves are available with a pressure equalization bellows or piston to achieve modulation. Solenoid valves are generally limited to relatively small sizes (up to 100 mm).
• An electric motor actuates the valve stem through a gear train and linkage. Electric motor actuators are classified in the following three types: Unidirectional—for two-position operation. The valve opens during one half-revolution of the output shaft and closes during the other half-revolution. Once started, it continues until the half-revolution is completed, regardless of subsequent action by the controller. Limit switches in the actuator stop the motor at the end of each stroke. If the controller has been satisfied during this inter-val, the actuator continues to the other position.
Spring-return—for two-position operation. Electric energy drives the valve to one position and a spring returns the valve to its normal position. Reversible—for floating and proportional operation. The motor can run in either direction and can stop in any position. It is sometimes equipped with a return spring. In proportional con-trol applications, a feedback potentiometer for rebalancing the control circuit is also driven by the motor.
Dampers Types and Characteristics. Automatic dampers are used in air-conditioning and ventilation to control airflow. They may be used (1) for modulating control to maintain a controlled variable such as mixed air temperature or supply air duct static pressure; or (2) for two-position control to initiate operation such as opening minimum outside air dampers when a fan is started.
Multiblade dampers are available in two arrangements—paral-lel blade and opposed blade (Figure 13). They are used to control flow through large openings typical of those in air handlers. Paral-lel-blade dampers are adequate for two-position control and can be used for modulating control when the pressure drop remains rela-tively constant (i.e., outdoor air and return air dampers on air-han-dling unit mixing boxes). However, opposed-blade dampers are Fig. 12 Pump and System Curves with Valve Control Fig. 13 Typical Multiblade Dampers Fundamentals of Control 15.7 preferable for throttling control because they normally provide bet-ter control when the ratio of the pressure drop between closed and fully open is large (Figures 14 and 15). In these figures, α is the ratio of the system pressure drop to the drop across the damper at maxi-mum (fully open) flow. Single-blade dampers are typically used for flow control at the zone.
Damper leakage is a concern, particularly where tight shutoff is necessary to reduce energy consumption significantly. Also, out-door air dampers in cold climates must close tightly to prevent coils and pipes from freezing. Low-leakage dampers cost more and require larger actuators because of the friction of the seals in the closed position; therefore, they should be used only as necessary.
Actuators. Either electricity or compressed air is used to actuate dampers.
• Pneumatic damper actuators are similar to pneumatic valve actuators, except that they have a longer stroke or the stroke is increased by a multiplying lever. Increasing the air pressure produces a linear motion of the shaft, which, through a linkage, moves the crank arm to open or close the dampers.
• Electric damper actuators can be unidirectional, spring-return, or reversible. A reversible actuator, which has two sets of motor windings, is frequently used for accurate control in modulating damper applications. Energizing one set of windings turns the actuator output shaft clockwise; energizing the other turns the shaft counter-clockwise.
When neither set of windings is energized, the shaft remains in its last position. The simplest form of control for this actuator is a floating point controller, which causes a contact closure to drive the motor clockwise or counter-clockwise. This type of actuator is available with a wide range of options for rotational shaft travel (expressed in degrees of rotation) and timing (expressed in the number of seconds to move through the rotational range). In addi-tion, a variety of standard electronics signals from electronic con-trollers, such as 4–20 mA (dc) or 0–10 V (dc), can be used to control this type of modulating actuator.
A two-position spring-return actuator moves in one direc-tion when power is applied to its internal windings; when no power is present, the actuator returns (via spring force) to its normal position. Depending on how the actuator is connected to the dampers, this action opens or closes the dampers. A modu-lating actuator may also have spring-return action.
Actuator Mounting. Damper actuators are mounted in different ways, depending on the size, and accessibility of the damper, and the power required to move the damper. The most common method of mounting electric actuators is directly over the damper shaft with no external linkage. Actuators can also be mounted in the airflow on the damper frame and be linked directly to a damper blade; or they can be mounted outside the duct and connected to a crank arm attached to a shaft extension of one of the blades. On large dampers, two or more actuators may be needed. In this case, they are usually mounted at separate points on the damper. An alternative is to install the damper in two or more sections, each section being con-trolled by a single damper actuator; however, proper flow control is easier with a single modulating damper. Positive positioners may be required for proper sequencing. A small damper with a two-posi-tion spring-return actuator may be used for minimum outside flow, with a large damper being independently controlled for economy cycle cooling.
Positive Positioners A pneumatic actuator may not respond quickly or accurately enough to small changes in control pressure due to friction in the actuator or load, or to changing load conditions such as wind acting on a damper blade. Where accurate positioning of a modulating damper or valve in response to load is required, positive positioners should be used. A positive positioner provides up to full supply con-trol air pressure to the actuator for any change in position required by the controller (Figure 16). An increase in branch pressure from the controller (A) moves the relay lever (B), which opens the supply valve (C). This allows supply air to flow into the relay chamber and the actuator cylinder, moving the pistons. A linkage and spring (D) transmit the piston movement to the other end of lever (B), and when the force due to movement balances the control force, the sup-ply valve closes, leaving the actuator in the new position. A decrease in control pressure allows the exhaust valve (E) to open until a new balance is obtained.
Fig. 14 Characteristic Curves of Installed Parallel-Blade Dampers Fig. 15 Characteristic Curves of Installed Opposed-Blade Dampers 15.8 2001 ASHRAE Fundamentals Handbook (SI) A positive positioner provides finite and repeatable positioning change and permits adjustment of the control range (spring range of the actuator) to provide a proper sequencing control of two or more controlled devices.
SENSORS A sensor is a device that responds to a change in the controlled variable. The response, which is a change in some physical or electrical property of the primary sensing element, is available for translation or amplification by mechanical or electrical signal. This signal is sent to the controller.
Chapter 45 of the 1999 ASHRAE Handbook—Applications and manufacturer’s catalogs and tutorials include information on spe-cific applications. In selecting a sensor for a specific application, the following should be considered: • Operating range of controlled variable. The sensor must be capable of providing an adequate change in its output signal over the expected input range.
• Compatibility of controller input. Electronic and digital controllers accept various ranges and types of electronic signals.
The specific controller to be used must be considered in the selection of an electronic sensor; if this is not known, an industry standard signal, such as 4–20 mA (dc) or 0–10 V (dc), should be used.
• Accuracy and repeatability. For some control applications, the controlled variable must be maintained within a narrow band around a desired set point. Both the accuracy and the sensitivity of the sensor selected must reflect this requirement. However, even an accurate sensor can not maintain the set point if (1) the controller is unable to resolve the input signal, (2) the controlled device can not be positioned accurately, (3) the controlled device exhibits excessive hysteresis, or (4) disturbances drive its system faster than the controls can regulate it. • System response time (or process dynamics). Associated with a sensor/transducer arrangement is a response curve, which describes the response of the sensor output to change in the controlled variable. If the time constant of the process being controlled is short, and stable accurate control is important, the sensor selected must have a fast response time.
• Control agent properties and characteristics. The control agent is the medium to which the sensor is exposed, or with which it comes in contact, for measuring a controlled variable such as temperature or pressure. If the agent corrodes the sensor or otherwise degrades its performance, a different sensor should be selected, or the sensor must be isolated or protected from direct contact with the control agent.
• Ambient environment characteristics. Even when the sensor’s components are isolated from direct contact with the control agent, the ambient environment must be considered. The temperature and humidity range of the ambient environment must not reduce the accuracy of the sensor. Likewise, the presence of certain gases, chemicals, and electromagnetic interference (EMI) can cause component degradation. In such cases, a special sensor or transducer housing can be used to protect the element, while ensuring a true indication of the controlled variable.
Temperature Sensors Temperature-sensing elements fall into three general categories: (1) those that use a change in relative dimension due to differences in thermal expansion, (2) those that use a change in state of a vapor or liquid, and (3) those that use a change in some electrical property.
Within each category, there are a variety of sensing elements to measure room, duct, water, and surface temperatures. Temperature-sensing technologies commonly used in HVAC applications are as follows: • A bimetal element is composed of two thin strips of dissimilar metals fused together. Because the two metals have different coefficients of thermal expansion, the element bends and changes position as the temperature varies. Depending on the space available and the movement required, it may be straight, U-shaped, or wound into a spiral. This element is commonly used in room, insertion, and immersion thermostats.
• A rod-and-tube element consists of a high-expansion metal tube containing a low-expansion rod. One end of the rod is attached to the rear of the tube. The tube changes length with changes in temperature, causing the free end of the rod to move. This element is commonly used in certain insertion and immersion thermostats. • A sealed bellows element is either vapor-, gas-, or liquid-filled.
Temperature changes vary the pressure and volume of the gas or liquid, resulting in a change in force or a movement. • A remote bulb element is a bulb or capsule that is connected to a sealed bellows or diaphragm by a capillary tube; the entire system is filled with vapor, gas, or liquid. Temperature changes at the bulb cause volume or pressure changes that are conveyed to the bellows or diaphragm through the capillary tube. The remote bulb element is useful where the temperature measuring point is remote from the desired thermostat location. • A thermistor is a semiconductor that changes electrical resistance with temperature. It has a negative temperature coefficient (i.e., the resistance decreases as the temperature increases). Its characteristic curve of temperature versus resistance is nonlinear over a wide range. Several techniques are used to convert its response to a linear change over a particular temperature range.
With digital control, one technique is to store a computer “look-up table” that maps the temperature corresponding to the measured resistance. The table breaks the curve into small segments, and each segment is assumed to be linear over its range. Thermistors are used because of their relatively low cost and the large charge in resistance possible for a small change in temperature.
• A resistance temperature device (RTD) is another sensor that changes resistance with temperature. Most metallic materials increase in resistance with increasing temperature. Over limited ranges, this variation is linear for certain metals such as platinum, Fig. 16 Pilot Positioner (Positive Positioner) Fundamentals of Control 15.9 copper, tungsten, and nickel/iron alloys. Platinum, for example, is linear within ±0.3% from −20 to 150°C. The RTD sensing element is available in several forms for surface or immersion mounting. Flat grid windings are used for measurements of sur-face temperatures. For direct measurement of fluid temperatures, the windings are encased in a stainless steel bulb to protect them from corrosion.
Humidity Sensors Humidity sensors, or hygrometers, are used to measure relative humidity, dew point, or absolute humidity of ambient or moving air.
Two types that detect relative humidity are mechanical hygrometers and electronic hygrometers.
A mechanical hygrometer operates on the principle that a hygroscopic material, usually a moisture-sensitive nylon or bulk polymer material, retains moisture and expands when exposed to water vapor. The change in size or form is detected by a mechanical linkage and converted to a pneumatic or electronic signal. Mechan-ical sensors using hair, wood, paper, or cotton do not match the per-formance of moisture-sensitive nylon or bulk polymer sensors and are not widely used.
Electronic hygrometers can use either resistance or capacitance sensing elements. The resistance element is a conductive grid coated with a hygroscopic (water-absorbent) substance. The conductivity of the grid varies with the water retained; thus, the resistance varies according to the relative humidity. The conductive element is arranged in an alternating current excited Wheatstone bridge and responds rapidly to humidity changes.
The capacitance element is a stretched membrane of nonconduc-tive film. It is coated on both sides with metal electrodes and mounted in a perforated plastic capsule. The response of the sensor’s capacity to rising relative humidity is nonlinear. The signal is linearized and temperature is compensated in the amplifier circuit to provide an out-put signal as the relative humidity changes from 0 to 100%.
The chilled mirror humidity sensor determines dew point rather than relative humidity. Air flows across a small mirror in the sensor. A thermoelectric cooler lowers the surface temperature of the mirror. The mirror surface condenses until it reaches the dew point of the air. The condensation from the surface reduces the amount of light reflected from the mirror compared to a reference light level.
Dispersive infrared (DIR) technology can be used to sense absolute humidity or dew point. It is similar to technology used to sense carbon dioxide or other gases. Infrared water vapor sen-sors are optical sensors that detect the amount of water vapor in air based on the infrared light absorption characteristics of water mol-ecules. Light absorption is proportional to the number of mole-cules present. The output of an infrared hygrometer is typically a value of absolute humidity or dew point. They can operate in diffu-sion or flow-through sample mode. This type of humidity sensor is unique in that the sensing element (a light detector and an infrared filter) is behind a transparent window that is never exposed directly to the sample environment. As a result, this sensor has excellent long-term stability and life and fast response time, is not subject to saturation, and operates equally well in very high or low humidity. Previously used solely for high-end applications, infra-red hygrometers are now commonly used in HVAC applications because they cost about the same as mid-range accuracy (1 to 3%) humidity sensors.
Pressure Transmitters and Transducers A pneumatic pressure transmitter converts a change in absolute, gage, or differential pressure to a mechanical motion using a bel-lows, diaphragm, or Bourdon tube mechanism. When corrected through appropriate links, this mechanical motion produces a change in the air pressure to a controller. In some instances, the sensing and control functions are combined in a single component, a pressure controller.
An electronic pressure transducer may use the mechanical actuation of a diaphragm or Bourdon tube to operate a potenti-ometer or differential transformer. Another type of transducer uses a strain gage bonded to a diaphragm. The strain gage detects the displacement resulting from the force applied to the dia-phragm. Electronic circuits provide temperature compensation and amplification to produce a standard output signal.
Flow Rate Sensors Orifice plate, pitot tube, venturi, turbine, magnetic flow, vortex shedding, and Doppler effect meters are used to sense fluid flow.
In general, the pressure differential devices (orifice plates, venturi and pitot tubes) are less expensive and simpler to use but have lim-ited range; thus, their accuracy depends on how they are applied and where in a system they are located.
More sophisticated flow devices, such as turbine, magnetic, and vortex shedding meters, usually have better range and are more accurate over a wide range. If an existing piping system is being considered for retrofit with a flow device, the expense of shutting down the system and cutting into a pipe must be considered. In this case, a noninvasive meter, such as a Doppler effect meter, can be cost-effective.
Indoor Air Quality Sensors Indoor air quality control can be divided into two categories— ventilation control and contamination protection. Ventilation control measures levels of carbon dioxide or other contaminants in a space and controls the amount of outdoor air introduced into the occupied space. Demand control of ventilation helps maintain proper ventilation rates at all levels of occupancy. Typical control set point levels for carbon dioxide are 800 to 1000 ppm (1400 to 1800 mg/m3). ASHRAE Standard 62 provides further informa-tion on ventilation for acceptable indoor air quality.
Contamination protection sensors monitor levels of hazardous or toxic substances and issue warning signals and/or initiate cor-rective actions through the building automation system (BAS).
Sensors are available for many different gases. The carbon mon-oxide (CO) sensor is one of the most common. The CO sensor is used to control and alarm CO levels in parking garages. Oxygen depletion sensors are used to measure, alarm, and initiate ventila-tion purging in enclosed spaces that house refrigeration equip-ment to prevent suffocation of occupants upon a refrigeration leak. The application of these sensors determines the type selected, the substances monitored, and the action taken.
Lighting Level Sensors Analog lighting level transmitters packaged in various config-urations allow control of ambient lighting levels using building automation strategies for energy conservation. Some examples include ceiling-mounted indoor light sensors used to measure room lighting levels; outdoor ambient lighting sensors used to control parking, general exterior, security, and sign lighting; and interior skylight sensors used to monitor and control light levels in skylight wells and other atrium spaces.
Power Sensing and Transmission Passive electronic devices that sense the magnetic field around a conductor carrying current allow low-cost instrumentation of power circuits. A wire in the sensor forms an inductive coupling that pow-ers the internal function and senses the level of the power signal.
These devices can provide an analog output signal to monitor cur-rent flow or operate a switch at a user-set level to turn on an alarm or other device.
15.10 2001 ASHRAE Fundamentals Handbook (SI) CONTROLLERS A controller compares the sensor’s signal with a desired set point and regulates an output signal to a controlled device. Digital con-trollers perform the control function using a microprocessor and control algorithm. The sensor and controller can be combined in a single instrument, such as a room thermostat, or they may be two separate devices.
Pneumatic Receiver-Controllers Pneumatic receiver-controllers are normally combined with pneu-matic elements that use a force or position reaction to the sensed vari-able to obtain a variable output air pressure. The control mode is usually proportional, but other modes such as proportional-plus-inte-gral can be used. These controllers are generally classified as nonre-lay, relay direct, or reverse-acting.
The nonrelay pneumatic controller uses low-volume output. A relay-type pneumatic controller actuates a relay device that ampli-fies the air volume available for control. The relay provides quicker response to a variable change.
Direct-acting controllers increase the output signal as the con-trolled variable increases. Reverse-acting controllers increase the output signal as the controlled variable decreases. A reverse-acting thermostat increases output pressure when the temperature drops.
Electric/Electronic Controllers For two-position control, the controller output may be a simple electrical contact that starts a burner or pump, or one that actuates a spring-return valve or damper actuator. Single-pole, double-throw (SPDT) switching circuits are used to control a three-wire unidirec-tional motor actuator. SPDT circuits are also used for heating and cooling applications. Both single-pole, single-throw (SPST) and SPDT circuits can be modified for timed two-position action.
Output for floating control is a SPDT switching circuit with a neutral zone where neither contact is made. This control is used with reversible motors; it has slow response and a wide throttling range.
Pulse modulation control is an improvement over floating con-trol. It provides closer control by varying the duration of the contact closure. As the actual condition moves closer to the set point, the pulse duration shortens for closer control. As the actual condition moves farther from the set point, the pulse duration lengthens.
Proportional control gives continuous or incremental changes in output signal to position an electrical actuator or controlled device.
Digital Controllers A microprocessor in the digital controller executes control algo-rithms on one or multiple control loops. This controller is funda-mentally different from pneumatic or electronic controllers in that the control algorithm is stored as a set of program instructions in memory (software or firmware). The controller itself calculates the proper control signals digitally rather than using an analog circuit or mechanical change.
A digital controller can be either single-loop or multiloop. Inter-face hardware allows the digital computer to process signals from various input devices such as electronic temperature, humidity, or pressure sensors described in the section on Sensors. Based on dig-itized equivalents of the input voltage or current signals, the control software calculates the required state of the output devices, such as valve and damper actuators and fan starters. The output devices are then positioned to the calculated state via interface hardware that converts the digital signal from the computer to an analog voltage or current required to position the actuator or energize a relay. It is common in both new and retrofit applications to use an additional interface device to convert the analog voltage or current into a pneu-matic signal, which operates a pneumatic actuator. Such devices are called electronic-to-pneumatic (E/P) transducers.
The operator enters parameters such as set points, proportional or integral gains, minimum on and off times, or high and low limits, but the control algorithms stored in the computer’s memory make the control decisions. The computer scans the input devices, exe-cutes the control algorithms, and then positions the output device(s), in a stepwise scheme. Digital controllers can be classified with regard to the way control algorithms are stored in memory (such as in firmware and software) and their ability to communicate to higher level devices such as terminals and computers.
Firmware and Software. Preprogrammed control routines, known as firmware, are typically stored in permanent memory such as electrically erasable programmable read-only memory (EEPROM). The operator can modify parameters such as set points, limits, and minimum off times within the control routines, but the program logic cannot be changed without replacing the memory chips.
User-programmable controllers allow the algorithms to be changed by the user. The programming language provided with the controller can vary from (1) a derivation of a standard language (such as Pascal or BASIC) to (2) a custom language developed by the controller’s manufacturer to (3) graphically based program-ming. Preprogrammed routines for proportional, proportional plus integral, Boolean logic, timers, and so forth, are typically included in the language. Standard energy management routines may also be preprogrammed and may interact with other control loops where appropriate.
Digital controllers can be furnished with both preprogrammed firmware and user-programmed routines. These routines can auto-matically modify the parameters of the firmware according to user-defined conditions to accomplish the sequence of control designed by the control engineer.
Operator Interface. Some digital controllers are designed for dedicated purposes and are adjustable only through manual switches and potentiometers mounted on the controller. This type of controller cannot be networked with other controllers. An example is the programmable room thermostat. A direct digital controller can have manual adjustable features, but it more typically is adjusted through a built-in LED or LCD display, a hand-held device, or a terminal or computer. The DDC controller can commu-nicate digitally, which allows remote connection to other controllers and to higher level computing devices and host operating stations.
A terminal allows the user to communicate with the controller and, where applicable, modify the program in the controller. Termi-nals can range from hand-held units with an LCD display and sev-eral buttons to a full-size console with a video monitor and keyboard. The terminal can be limited in function to allow only the display of sensor and parameter values or powerful enough to allow changing or reprogramming the control strategies. In some instances, a terminal can communicate remotely with one or more controllers, thus allowing central displays, alarms, and commands.
Usually, hand-held terminals are used by technicians for trouble-shooting, and full consoles at a fixed location are used to monitor the entire digital control system.
Thermostats Thermostats combine sensing and control functions in a single device. Microprocessor-based thermostats have many of the fea-tures described in the following paragraphs.
• The occupied-unoccupied or dual-temperature room thermostat controls at a reduced temperature at night. It may be indexed (changed from occupied to unoccupied) individually or in groups by a manual switch or time switch from a remote point. Some electric units have an individual clock and switch built into the thermostat.
• The pneumatic day-night thermostat uses a two-pressure air supply system—the two pressures often being 90 and 120 kPa (gage), or 100 and 140 kPa (gage). Changing the pressure at a Fundamentals of Control 15.11 central point from one value to the other actuates switching devices in the thermostat that index it from occupied to unoccupied or vice versa. • The heating-cooling or summer-winter thermostat can have its action reversed and its set point changed by indexing. It is used to actuate controlled devices, such as valves or dampers, that regulate a heating source at one time and a cooling source at another. • Multistage thermostats are arranged to operate two or more successive steps in sequence.
• A submaster thermostat has its set point raised or lowered over a predetermined range in accordance with variations in output from a master controller. The master controller can be a thermostat, manual switch, pressure controller, or similar device.
• A dead band thermostat has a wide differential over which the thermostat remains neutral, requiring neither heating nor cooling.
This differential may be adjustable up to 5 K. The thermostat then controls to maximum or minimum output over a small differential at the end of each dead band (Figure 17).
AUXILIARY CONTROL DEVICES Auxiliary control devices for electric systems include • Transformers to provide current at the required voltage.
• Occupancy sensors to automatically adjust controlled variables based on occupancy such as lighting, ventilation rate, and temperature.
• Signal transducers to change one standard signal into another.
The popularity of digital control and other electric-based control systems has generated a variety of transducers. The variables usually transformed include voltage [0–10, 0–5, 2–10 V (dc)], current [4–20 mA], resistance [0–135 Ω], pressure (20–100, 0– 140 kPa), phase cut voltage [0–20 V (dc)], pulse-width modulation, and time duration pulse. Signal transducers allow the use of an existing control device in a retrofit application.
• Electric relays to control electric heaters or to start and stop burners, compressors, fans, pumps, or other apparatus for which the electrical load is too large to be handled directly by the controller. Other uses include time-delay and circuit-interlocking safety applications.
• Potentiometers for manual positioning of proportional control devices, for remote set point adjustment of electronic controllers, and for feedback.
• Manual switches, either two-position or multiple-position with single or multiple poles.
• Auxiliary switches on valve and damper actuators for selecting a sequence of operation.
Auxiliary control devices for pneumatic systems include • Air compressors and accessories, including dryers and filters, to provide a source of clean, dry air at the required pressure.
• Electropneumatic relays, electrically actuated air valves for operating pneumatic equipment according to variations in electrical input to the relay. • Pneumatic-electric switches, which are actuated by the pressure from a controller to permit a controller actuating a proportional device to also actuate one or more two-position devices.
• Pneumatic transducers, which are used to reverse the action of a proportional controller, select the higher or lower of two or more pressures, average two or more pressures, respond to the difference between two pressures, add or subtract pressures, and amplify or retard pressure changes.
• Positive positioning relays to ensure accurate positioning of a valve or damper actuator in response to changes in pressure from a controller. • Switching relays, which are pneumatically operated air valves used to divert air from one circuit to another or to open and close air circuits.
• Pneumatic switches, which are manually operated devices used to divert air from one circuit to another or to open and close air circuits. They can be two-position or multiple- position.
• Gradual switches, which are proportional devices used to manually vary air pressure in a circuit.
Auxiliary control devices common to both electric and pneu-matic systems include the following: • Step controllers to operate several switches in sequence by means of a proportional electric or pneumatic actuator. They are commonly used to control several steps of refrigeration capacity.
They may be arranged to prevent simultaneous starting of compressors and to alternate the sequence to equalize wear. These controllers may also be used for sequenced operation of electric heating elements and other equipment.
• Power controllers to control electric power input to resistance heating elements. The final controlled device may be a variable autotransformer, a saturable-core reactor, or a solid-state power controller. They are available with various ratings for single- or three-phase heater loads and are usually arranged to regulate power input to the heater in response to the demands of the proportional electronic or pneumatic controllers. However, solid-state controllers may also be used in two-position control modes. • Clocks or timers to turn apparatus on and off at predetermined times, to switch control systems from day to night operation, and to regulate other time sequence functions.
• Transducers, which consist of combinations of electric or pneumatic control devices, may be required. For these applications, transducers are used to convert electric signals to pneumatic output or vice versa.
Transducers may convert proportional input to either proportional or two-position output.
The electronic-to-pneumatic (E/P) transducer is used in many applications. It converts a proportional electronic output signal into a proportional pneumatic signal (as illustrated in Figure 18) and can be used to combine electronic and pneumatic control components to form a control loop, as illustrated in Figure 19. Electronic compo-nents are used for sensing and signal conditioning, while pneumatic components are used for actuation. The electronic controller can be either analog or digital.
The E/P transducer presents a special option for retrofit applica-tions. An existing HVAC system with pneumatic controls can be ret-Fig. 17 Dead Band Thermostat 15.12 2001 ASHRAE Fundamentals Handbook (SI) rofitted with electronic sensors and controllers while retaining the existing pneumatic actuators (Figure 20).
COMMUNICATION NETWORKS FOR BUILDING AUTOMATION SYSTEMS A building automation system (BAS) is a centralized control and/or monitoring system for many or all building systems (e.g., HVAC, electrical, life safety, security). A BAS may link together information from control systems actuated by different technologies.
One important characteristic of direct digital control (DDC) is the ability to share information. Information is transferred (1) be-tween controllers to coordinate their action, (2) between controllers and building operator interfaces to monitor and command systems, and (3) between controllers and other computers for off-line calcu-lation. This information is typically shared over communication networks. DDC systems nearly always involve at least one network and commonly involve more than one. A network is a set of con-nections between controllers, routers, bridges, and computers that enables them to exchange digital information.
COMMUNICATION PROTOCOLS A protocol is a set of rules that define the communication behav-ior of each element in a communication network. The word may describe the communication at one layer of the network [e.g., Inter-net protocol (IP), which defines the network layer in the Internet suite of protocols], or it may refer to the entire communication pro-cess. In discussions of BASs, most communication needs are described in terms of the entire process, but it is sometimes neces-sary to discuss the protocols at a particular layer.
There is great interest in open protocols for BASs to facilitate communication among devices from different suppliers. Although there is no commonly accepted definition of “openness,” the Insti-tute of Electrical and Electronics Engineers defines three classes of protocols (IEEE Standard 802): • Standard protocol. Published and controlled by a standards body. Examples include BACnet by ASHRAE, LonTalk by Electronic Industries Alliance, and TCP/IP by Internet Engi-neering Task Force.
• Public protocol. Published but controlled by a private organization.
• Private protocol. Unpublished; use and specification controlled by a private organization. Examples include the proprietary communications used by many building automation devices.
Multivendor communication is possible with any of these three classes, but the challenges vary. Specifying a common protocol does not ensure that the end user’s requirement for interoperability is met. The engineer may select an open protocol and specify the interaction between devices. This limits the bidders, but assures the engineer of certain communication characteristics. On the other hand, the engineer can specify the required interoperation and put the burden on the suppliers to select combinations of products that meet the need. This is likely to result in a wider range of options, but they may be more difficult to compare.
THE OSI NETWORK MODEL ISO Standard 7498-1 presents a seven-layer model of informa-tion exchange called the Open Systems Interconnection (OSI) Reference Model (Figure 21). Most descriptions of computer net-works, especially open networks, are based on this reference model.
The layers can be thought of as steps in the translation of a message from something with meaning at the application layer, to something measurable at the physical layer, and back to meaningful informa-tion at the application layer.
Fig. 18 Response of Electronic-to-Pneumatic (E/P) Transducer Fig. 19 Electronic and Pneumatic Control Components Combined with Electronic-to-Pneumatic (E/P) Transducer Fig. 20 Retrofit of Existing Pneumatic Control with Electronic Sensors and Controllers Fundamentals of Control 15.13 The layered approach to network design is valuable because it allows developers to take advantage of existing standards such as IP at the network layer or Electronic Industries Association (EIA) Standard 485, a signaling standard, at the physical layer without becoming tied to one technology.
The full seven-layer model does not apply to every network, but it is still used to describe the aspects that do fit. When describing DDC networks that use the same technology throughout the system, the seven-layer model is relatively unimportant. For systems that employ various technologies at different points in the network, the model helps describe where and how the pieces are bound together.
The portion of network shown in Figure 22 illustrates how the OSI Reference Model describes communications. The system includes a number of controllers on a BACnet network and several operator workstations connected by an ISO 8802-3 backbone or local area network (LAN), in this case an Ethernet. A BACnet router links the BACnet controllers to the Ethernet.
Figure 23 shows the network layers in each device that make communication possible. Workstation 2 and the BACnet controller both show communication stacks with BACnet at the application layer; however, these devices do not communicate directly because they use different protocols at the lower layers. If the stacks are dif-ferent at any layer, a device that bridges the gap is required. The BACnet router shows two stacks. One matches the physical, data link, and network layers of the BACnet controllers; the other matches the bottom three layers of the BACnet stack on Worksta-tion 2. The BACnet network layer is the common layer that delivers BACnet messages from one end of the router to the other. When a message reaches the router, it travels up through the stack. The router reformats it and sends the same message out through the other stack to the other network. This makes it possible for application data to pass between the controller and BACnet devices on the Ethernet. Bushby (1998) described this process more fully.
Both workstations contain an OPC stack. OPC (OLE for process control) is a standard for communicating automation data between computers or processes; it is based on distributed component object model (DCOM) technology. DCOM is a software development standard that facilitates interchangeability of software components.
The workstations communicate with each other over the Ether-net with OPC messages. Workstation 2 and the BACnet router com-municate with each other over the Ethernet using BACnet messages. Although they are on the same network, Workstation 1 and the BACnet router do not communicate directly with each other because their protocol stacks do not match at the upper layers. If BACnet data is required at Workstation 1, it must pass through the OPC server on Workstation 2.
NETWORK STRUCTURE Often, a single DDC system applies several different network technologies at different points in the system. For example, a rela-tively low-speed, inexpensive network with relatively primitive functions may link a group of room controllers to a larger equip-ment controller. A faster, more sophisticated network links the large controller to its peers and to an operator’s workstation.
Several workstations communicate over a high-speed, relatively expensive, general-purpose office automation network. Figure 24 illustrates this sort of high-speed hierarchical network. Structures like it have been popular in DDC for years. Frequently, the net-work hierarchy corresponds roughly to a hierarchy related to the control function of the devices. Variations on this hierarchical structure will continue to emerge. The opposite extreme is a com-pletely flat network architecture. A flat architecture links all the devices through the same network without altering any other hier-archy that exists among the devices. A flat architecture is more Fig. 21 OSI Reference Model Fig. 22 Portion of a BAS Network Fig. 23 Network Layers in BAS Devices 15.14 2001 ASHRAE Fundamentals Handbook (SI) viable in small systems than in large ones due to the consider-ations discussed in the following paragraph.
Network structure sometimes affects the cost, operation, and opportunities for expansion of a BAS. The structure can affect the reliability and failure modes of the system. It may be appropriate to separate sections of a network in order to isolate failures. Structure can affect the way devices load the information carrying capacity of the network. It can isolate one busy branch from the rest of the sys-tem. It can isolate branches from the high-speed backbone. Struc-ture affects the cost of the system because it determines the mix of low-speed and high-speed devices. Network structure can influence system data security and access control. The relative merits of one structure versus another depend on the communication functions required, the hardware and software available for the task, and cost.
For a given job, there is probably more than one suitable structure.
Product capabilities change quickly. Engineers who choose to spec-ify network structure must be aware of new technologies to take advantage of the most cost-effective solutions.
Connections Between Networks and Network Segments Some BAS networks use other networks to connect segments of the BAS. This occurs • Within a building, using the information technology network • Between buildings, using telephone lines • Between buildings, using the Internet In each case, the link between BAS segments must be considered part of the BAS network when evaluating function, security, and performance. The link also raises new issues. The connecting seg-ment is likely to be outside the control of the owner of the BAS, which could affect availability of service. It may mean that traffic and bandwidth issues have to be addressed outside the facilities department.
The connection may be switched (dial-up) or dedicated. In the case of a switched connection, the function of the network depends on which segment may dial the other and the circumstances that trigger the call. Switched connections are most commonly used to handle remote buildings or to serve a remote operator (i.e., one who is on call over the weekend).
Transmission Media The transmission medium is the foundation of the network. It is usually, but not always, cable. In cases where physical cable connection is not possible or practical, devices may transfer information using wireless technologies, such as radio waves or infrared light. However, this section covers only physical cabling media.
Twisted-Pair Copper Cable. A twisted-pair cable consists of multiple twisted pairs (typically 24 AWG) of wire covered by an overall sheath or jacket. Varying the number of twists for each pair relative to the other pairs in the cable can greatly reduce crosstalk (interference between signals on different pairs).
In shielded twisted pair (STP) cable, each wire pair, as well as the combined grouping of all pairs, is covered with a layer of shielding to minimize interference-related problems. STP cable performs better than unshielded twisted pair (UTP) cable in environments where a high level of immunity and/or a low level of emissions is critical. It also allows less crosstalk than UTP.
However, STP requires a more labor-intensive installation, and any break or improper grounding of the shield reduces its overall effectiveness.
Category 5 (defined by TIA/EIA Standard 568-A) UTP cable is currently the most common medium. RJ-45 jacks and plugs are spec-ified and have standard pinouts. This cable is rated at up to 1 gigabit per second for 100 m and can be used over much longer distances for lower speed applications (e.g., EIA Standard 485).
Fiber Optic Cable. Fiber optic cable uses glass or plastic fibers to transfer data in the form of light pulses, which are typically gen-erated by either a laser or an LED. Fiber optic cable systems are classified as either single-mode fiber or multimode fiber systems.
Table 1 compares their characteristics.
Light in a fiber optic system experiences less energy loss than electrical signals traveling through copper and no capacitance. This translates into greater transmission distances and dramatically higher data transfer rates. With the rapid advances in this technol-ogy, the data transfer rate of a fiber cable imposes no limits on a BAS. Fiber optics also have exceptional noise immunity. However, the necessary conversions between light-based signaling and elec-tricity-based computing make fiber optics more expensive per device, which sometimes offsets the other advantages.
Structured Cabling. TIA/EIA Standard 568-A, Commercial Building Telecommunications Cabling Standard, permits cable planning and installation to begin before the network engineering is finalized. It supports both voice and data. The standard was written for the telecommunications industry, but cabling is gaining recog-nition as building infrastructure, and the standard is being applied to BAS networks as well.
TIA/EIA Standard 568-A specifies star topology (each device individually cabled to a hub) because connectivity is more robust and management is simpler than for busses and rings. If the wires in a leg are shorted, only that leg fails, making fault isolation easier; with a bus, all drops would fail.
The basic structure specified is a backbone, which typically runs from floor to floor within a building and possibly between buildings.
The horizontal cabling runs between the distribution frames on each floor and the information outlets in the work areas. The maximum length of horizontal cabling recommended is 100 m.
Fig. 24 Hierarchical Network Table 1 Comparison of Fiber Optic Technology Multimode Fiber Single-Mode Fiber Light source LED Laser Cable designation (core/cladding diameter) 62.6/125 8.3/125 Transmission distance 2000 m 3000 m Data rate >10 gigabit/s and increasing Even higher Relative cost Less per connection, more per data rate More per connection, less per data rate Fundamentals of Control 15.15 SPECIFYING BAS NETWORKS Specifying a DDC system includes specifying a network. The many network technologies available deliver many performance levels at many different prices. A rational selection requires an assessment of the requirements (i.e., what information will pass between devices and at what rates). In some cases, the new equip-ment is required to interface with existing devices, which may limit networking options.
Specification Method As with other aspects of an HVAC system, an engineer must choose a method of specification. The Construction Specification Institute lists four methods. The following list relates those methods to BAS networks.
• Descriptive. Calls out the exact properties of the products. Properties could include communication protocols and data transfer rates.
• Performance. Tells what result is required and the criteria by which performance will be verified. Allows bidders to propose products to meet the need.
• Reference standard. Requires products to conform to an established standard. Does not oblige contractor to meet end user’s needs not addressed in the standard.
• Proprietary. Calls out brand names. May be necessary in expansion of existing systems.
To write a descriptive network specification, the designer must know the details of network technology. To succeed with any spec-ification, the designer must articulate the end user’s needs. Typi-cally, a performance-based specification results in the best value for the customer (Ehrlich and Pittel 1999).
Communication Tasks Determining network performance requirements means identify-ing and quantifying the communication functions required. Ehrlich and Pittel (1999) identified five basic communication tasks. To establish network requirements, the specifier must elaborate on each basic task. They are listed here along with some of the ques-tions an engineer can use to identify the client’s needs.
Data Exchange. What data passes between which devices?
What control and optimization data passes between controllers?
What update rates are required? What data does an operator need to reach? How much delay is acceptable in retrieving values? What update rates are required on “live” data displays? Within one sys-tem, the answers may vary according to the use of the data. Which set points and control parameters do operators need to adjust over the network?
Alarms and Events. Where do alarms originate? Where are they logged and displayed? How much delay is acceptable? Where are they acknowledged? What information must be delivered along with the alarm? (Depending on the design of the system, alarm mes-sages may be passed over the network along with the alarms.) Where are alarm summary reports required? How and where do operators need to adjust alarm limits, etc?
Schedules. For the HVAC equipment that runs on schedules, where can the schedules be read? Where can they be modified?
Trends. Where does trend data originate? Where is it stored?
How much will be transmitted? Where is it displayed and pro-cessed? Which user interfaces can set and modify trend collection parameters?
Network Management. What network diagnostic and mainte-nance functions are required at which user interfaces? Data access and security functions may be handled as network management functions.
Bushby et al. (1999) refer to the same five communication tasks as interoperability areas and list many more specific consider-ations in each area.
APPROACHES TO INTEROPERABILITY In the surge toward interoperability, many approaches have been proposed and applied, each with varying degrees of success under various circumstances. The field changes quickly as product lines emerge and standards develop and gain acceptance. The building automation world continues to evaluate the options project by project.
Typically, an interoperable system uses one of two approaches: standard protocols or special-purpose gateways. With a standard, the supplier is responsible for compliance with the standard; the sys-tem specifier or integrator is responsible for interoperation. With a gateway, the supplier takes responsibility for interoperation. The majority of integrated building automation systems currently depend on gateways, especially where the job requires interopera-tion with existing equipment. Bushby (1998) addressed this issue and some of the limitations associated with gateways. To date, interoperability by any method requires solid field engineering and capable system integration; the issues extend well beyond the selec-tion of a communication protocol.
Standard Protocols Several standard protocols have been applied successfully in building automation systems. Their different characteristics make some more suited to particular tasks than others. The European Committee for Standardization (CEN) discusses characteristics of protocols appropriate for different building functions (CEN 1999).
Table 2 lists some of the applicable standard protocols.
Gateways and Interfaces Rather than conforming to a published standard, a supplier can design a specific device to exchange data with another specific device. This typically requires cooperation between two manufac-turers. It can be simpler and more cost-effective than for both man-ufacturers to conform to an agreed-upon standard. Sometimes the device is developed for one particular installation; other times it is an off-the-shelf product. In either case, the communication tasks must be carefully specified to ensure that the gateway performs as needed.
Choosing a system that supports a variety of gateways may be a way to maintain a flexible position as products and standards con-tinue to develop.
COMMISSIONING A successful control system requires a proper start-up and test-ing, not merely the adjustment of a few parameters (set points and throttling ranges) and a few quick checks. With the services of an experienced control professional, the typical DDC system can be used effectively in the commissioning process to test and document the peformance of the HVAC system. In general, the increased use of VAV systems and digital controls has increased the importance of and need for commissioning.
Table 2 Some Standard Communication Protocols Applicable to BAS Protocol Definition BACnet ASHRAE Standard 135 LonTalk EIA Standard 709.1 PROFIBUS FMS EN 50170:1996 Volume 2 EIB ENV 13154-2 Annex C EIBnet ENV 18321-2 15.16 2001 ASHRAE Fundamentals Handbook (SI) Design and construction specifications should include specific commissioning procedures. In addition, commissioning should be coordinated with testing, adjusting, and balancing (TAB) because each affects the other. The TAB procedure begins by checking each control device to ensure that it is installed and connected according to approved drawings. Each electrical and pneumatic connection is verified, and all interlocks to fan and pump motors and primary heating and cooling equipment are checked. ASHRAE Guideline 1 explains how commissioning starts with project conception and continues for the life of the building.
TUNING The systematic tuning of controllers improves the performance of all controls and is particularly important for digital control. First, the controlled process should be controlled manually between var-ious set points to evaluate the following questions: • Is the process noisy (rapid fluctuations in controlled variable)?
• Is there appreciable hysteresis (backlash) in the actuator?
• How easy (or difficult) is it to maintain and change set point?
• In which operating region is the process most sensitive (highest gain)?
If the process cannot be controlled manually, the reason should be identified and corrected before the controller is tuned.
Tuning selects control parameters that determine the steady-state and transient characteristics of the control system. HVAC pro-cesses are nonlinear, and characteristics change on a seasonal basis.
Controllers tuned under one operating condition may become unsta-ble as conditions change. A well-tuned controller (1) minimizes the steady-state error for set point, (2) responds quickly to disturbances, and (3) remains stable under all operating conditions. Tuning pro-portional controllers is a compromise between minimizing steady-state error and maintaining margins of stability. Proportional plus integral (PI) control minimizes this compromise because the inte-gral action reduces steady-state error, while the proportional term determines the controller’s response to disturbances.
Tuning Proportional, PI, and PID Controllers Popular methods of determining proportional, PI, and PID con-troller tuning parameters include closed- and open-loop process identification methods and trial-and-error methods. Two of the most widely used techniques for tuning these controllers are ultimate oscillation and first order plus dead time. There are many optimiza-tion calculations for these two techniques, but the most widely used is the Ziegler-Nichols, which is given here.
Ultimate Oscillation (Closed-Loop) Method. The closed-loop method increases the gain of the controller in proportional-only mode until the equipment continuously cycles after a set point change (Figure 25, where Kp = 40). Proportional and integral terms are then computed from the cycle’s period of oscillation and the Kp value that caused cycling. The ultimate oscillation method is as follows: 1. Adjust control parameters so that all are essentially off. This cor-responds to a proportion band (gain) at its maximum (mini-mum), the reset (repeats per minute) to maximum (minimum), and derivative to its minimum.
2. Adjust the manual output of the controller to give a measurement as close to midscale as possible.
3. Put the controller in automatic.
4. Slowly and gradually increase the proportional constant effect (this corresponds to reducing the proportional band or increasing the proportional gain) until the observed oscillations neither grow nor diminish in amplitude. If the response saturates at either extreme, start over at Step 2 to obtain a stable response. If no oscillations are observed, change the set point and try again.
5. Record the proportional band as PBu and the period of the oscillations as Tu.
6. Use the recorded proportional band and oscillation period to calculate controller settings as follows: Proportional only: PB = 1 .8(PBu) percent (4) Proportional plus integral (PI): PB = 2.22(PBu) percent (5) Ti = 0.83Tu minute per repeat (6) Proportional plus integral plus derivative (PID): PB = 1.67(PBu) percent (7) Ti = 0.50Tu minute per repeat (8) Td = 0.125Tu minute (9) First-Order-plus-Dead-Time (Open-Loop) Method. The open-loop method introduces a step change in input into the opened control loop. A graphical technique is used to estimate the process transfer function parameters. Proportional and integral terms are calculated from the estimated process parameters using a series of equations.
The value of the process variable must be recorded over time, and the dead time and time constant must be determined from it.
This can be accomplished graphically as seen in Figure 26. The first-order-plus-dead-time method is as follows: 1. Adjust the controller manual output to give a midscale measurement.
2. Arrange for the recording of the process variable over time.
3. Move the manual output of the controller by 10% as rapidly as possible to approximate a step change.
4. Record the value of the process variable over time until it reaches a new steady state value.
5. Determine the dead time and time constant.
6. Use the dead time (TD) and time constant (TC) values to calculate PID values as follows: (10) Fig. 25 Response of Discharge Air Temperature to Step Change in Set Points at Various Proportional Constants with No Integral Action ain % change in controlled variable % change in control signal ----------------------------------------------------------------------------= Fundamentals of Control 15.17 Proportional only: PB = Gain/(TC/TD) (11) Proportional plus integral (PI): PB = 0.9(Gain)/(TC/TD) (12) Ti = 3.33(TD) (13) Proportional-integral-derivative (PID): PB = 1.2(Gain)/(TC/TD) (14) Ti = 2(TD) (15) Td = 0.5(TD) (16) Trial and Error Method. This method involves adjusting the gain of the proportion-only controller until the desired response to a set point is observed. Conservative tuning dictates that this response should have a small initial overshoot and quickly damp to steady-state conditions. Set point changes should be made in the range where controller saturation, or output limit, is avoided. The integral term is then increased until changes in set point produce the same dynamic response as the controller under proportional control, but with the response now centered about the set point (Figure 27).
Tuning Digital Controllers In tuning digital controllers, additional parameters may need to be specified. The digital controller sampling interval is critical because it can introduce harmonic distortion if not selected prop-erly. This sampling interval is usually set at the factory and may not be adjustable. A controller sampling interval of about one-half the time constant of the controlled process usually provides adequate control. Many digital control algorithms include an error dead band to eliminate unnecessary control actions when the process is near set point. Hysteresis compensation is possible with digital controllers, but it must be carefully applied because overcompensation can cause continuous cycling of the control loop.
CODES AND STANDARDS ASHRAE. 1995. BACnet—A data communication protocol for building automation and control networks. ANSI/ASHRAE Standard 135-1995.
EIA. 1995. Commercial building telecommunications cabling standard. TIA/ EIA Standard 568-A-95. Electronic Industries Alliance, Arlington, VA.
EIA. 1998. Electrical characteristics of generators and receivers for use in balanced digital multipoint systems. TIA/EIA Standard 485-98.
EIA. 1999. Control network protocol specification. ANSI/EIA Standard 709.1-99 IEEE. 1990. Local and metropolitan area networks: Overview and architec-ture. IEEE Standard 802-1990. Institute of Electrical and Electronic Engineers, Piscataway, NJ.
ISO. 1994. Information processing systems—Open systems interconnec-tion—Basic reference model: The basic model. ISO/IEC 7498-1:1994.
International Organization for Standardization, Geneva, Switzerland.
REFERENCES ASHRAE. 1996. The HVAC commissioning process. Guideline 1-1996.
Avery, G. 1989. Updating the VAV outside air economizer controls.
ASHRAE Journal (April).
Avery, G. 1992. The instability of VAV systems. Heating, Piping and Air Conditioning (February).
BICSI. 1999. LAN and internetworking design manual, 3rd ed. BICSI: A Telecommunications Association, Tampa, FL.
Bushby, S.T. 1998. Friend or foe? Communication gateways. ASHRAE Journal 40(4):50-53.
Bushby, S.T., H.M. Newman, and M.A. Applebaum. 1999. GSA guide to specifying interoperable building automation and control systems using ANSI/ASHRAE Standard 135-1995, BACnet. NISTIR 6392.
National Institute of Standards and Technology, Gaithersburg, MD.
Available from National Technical Information Service, Springfield, VA.
CEN. 1999. Building control systems, Part 1: Overview and definitions.
prEN ISO16484-1.
Ehrlich, P. and O. Pittel. 1999. Specifying interoperability. ASHRAE Journal 41(4):25-29.
Haines, R.W. and D.C. Hittle. 1993. Control systems for heating, ventilating and air conditioning, 5th ed. Kluwer Academic Publishing. Hartman, T.B. 1993. Direct digital controls for HVAC systems. McGraw-Hill, New York.
Kettler, J.P. 1998. Controlling minimum ventilation volume in VAV sys-tems. ASHRAE Journal (May).
Levenhagen, J.I. and D.H. Spethmann. 1993. HVAC controls and systems.
McGraw-Hill, New York.
LonMark. 1998. LonMark application layer interoperability guidelines.
LonMark Interoperability Association, Sunnyvale, CA.
LonMark. 1999. LonMark functional profile: Space comfort controller.
8500-10.
Newman, H.M. 1994. Direct digital control for building systems, theory and Practice. John Wiley and Sons, New York.
OPC Foundation. 1998. OPC overview. OPC Foundation, Boca Raton, FL.
Rose, M.T. 1990. The open book: A practical perspective on OSI. Prentice-Hall, Englewood Cliffs, NJ.
Seem, J.E., J.M. House, and R.H. Monroe. 1999. On-line monitoring and fault detection, ASHRAE Journal (July).
Starr, R. 1999. Pneumatic controls in a digital age. Heating, Piping and Air Conditioning (November).
Tillack, L. and J.B. Rishel. 1998. Proper control of HVAC variable speed pumps. ASHRAE Journal (November).
Fig. 26 Open Loop Step Response Versus Time Fig. 27 Response of Discharge Air Temperature to Step Change in Set Points at Various Integral Constants with Fixed Proportional Constant 16.1 CHAPTER 16 AIRFLOW AROUND BUILDINGS Flow Patterns ............................................................................................................................... 16.1 Wind Pressure on Buildings ......................................................................................................... 16.3 Wind Effects on System Operation ............................................................................................... 16.7 Building Internal Pressure and Flow Control ............................................................................. 16.9 Scale Model Simulation and Testing ............................................................................................ 16.9 Symbols ...................................................................................................................................... 16.11 IRFLOW around buildings affects worker safety, process and A building equipment operation, weather and pollution protec-tion at inlets, and the ability to control environmental factors of tem-perature, humidity, air motion, and contaminants. Wind causes variable surface pressures on buildings that change intake and exhaust system flow rates, natural ventilation, infiltration and exfil-tration, and interior pressures. The mean flow patterns and turbu-lence of wind passing over a building can cause recirculation of exhaust gases to air intakes. This chapter contains information for evaluating flow patterns, estimating wind pressures, and identifying problems caused by the effects of wind on intakes, exhausts, and equipment. Related information can be found in Chapters 12, 14, 26, and 27 of this volume; in Chapters 28, 29, 43, and 51 of the 1999 ASHRAE Handbook—Applications; and in Chapters 25, 30, and 36 of the 2000 ASHRAE Handbook—Systems and Equipment.
FLOW PATTERNS Buildings having even moderately complex shapes, such as L- or U-shaped structures formed by two or three rectangular blocks, can generate flow patterns too complex to generalize for design. To determine flow conditions influenced by surrounding buildings or topography, wind tunnel or water channel tests of scale models or tests of existing buildings are required. However, if a building is ori-ented perpendicular to the wind, it can be considered as consisting of several independent rectangular blocks. Only isolated rectangu-lar block buildings are discussed here. Saunders and Melbourne (1979), Hosker (1984, 1985), Walker et al. (1996), and English and Fricke (1997) review the effects of nearby buildings.
The mean speed of wind UH approaching a building increases with height H above the ground (Figure 1). Both the upwind veloc-ity profile shape and its turbulence intensity strongly influence flow patterns and surface pressures (Melbourne 1979). A stagnation zone exists on the upwind wall. The flow separates at the sharp edges to generate recirculating flow zones that cover the down-wind surfaces of the building (roof, sides, and leeward walls) and extend for some distance into the wake. If the building has suffi-cient length L in the windward direction, the flow will reattach to the building (Figure 2) and may generate two distinct regions of sepa-rated recirculating flow—on the building and in its wake.
Surface flow patterns on the upwind wall are largely influenced by approach wind characteristics. Higher wind speed at roof level causes a larger stagnation pressure on the upper part of the wall than near the ground, which leads to downwash on the lower one-half to two-thirds of the building (Figure 1). On the upper one-quar-ter to one-third of the building, the surface flow is directed upward over the roof. For a building whose height H is three or more times the width W of the upwind face, an intermediate zone can exist between the upwash and downwash regions, where the surface streamlines pass horizontally around the building. The downwash on the lower surface of the upwind face separates from the building before it reaches ground level and moves upwind to form a vortex that can generate high velocities close to the ground. This ground level upwind vortex is carried around the sides of the building in a U shape (Figure 1) and is responsible for the suspension of dust and debris that can contaminate air intakes close to ground level.
For wind perpendicular to a building wall, the height H and width W of the upwind building face determine the flow patterns shown in Figure 2. According to Wilson (1979), the scaling length R is (1) where BS = smaller of upwind building face dimensions H and W BL = larger of upwind building face dimensions H and W When BL is larger than 8BS, use BL = 8BS in Equation (1). For build-ings with varying roof levels or with wings separated by at least a distance BS, only the height and width of the building face below the portion of the roof in question should be used to calculate R.
Streamline patterns are independent of wind speed and depend mainly on building shape and upwind conditions. Because of the three-dimensional flow around a building, the shape and size of the recirculation airflow is not constant over the surface. The airflow reattaches closer to the upwind building face along the edges of the building than it does near the middle of the roof and sidewalls (Fig-ure 2). The height Hc of the recirculation region (Figures 1 and 3) also decreases near roof edges.
The wind above the roof recirculation region is affected by the presence of the building. The flow accelerates as the streamlines curve upward over the roof and decelerates as they curve downward over the wake on the downwind side of the building. The distance above roof level where a building influences the flow is approxi-mately 1.5R (Figure 1). The roof pitch begins to affect flow when it exceeds about 15° (1:4). When roof pitch reaches 20° (1:3), the flow remains attached to the upwind pitched roof and produces a recir-culation region downwind of the roof ridge that is larger than that for a flat roof.
The downwind wall of a building exhibits a region of low aver-age velocity and high turbulence. Velocities near the downwind wall are typically one-quarter of those at the corresponding upwind wall location. Figures 1 through 3 show that an upward flow exists over most of the downwind walls. A flow recirculation region extends for an approximate distance Lr = 1.0R downwind.
If the angle of the approach wind is not perpendicular to the upwind face, complex flow patterns result. Strong vortices develop from the upwind edges of a roof, causing a strong downwash into the building wake above the roof. High speeds in these vortices cause large negative pressures near roof corners that can be a hazard to roof-mounted equipment during high winds. When the angle The preparation of this chapter is assigned to TC 2.5, Air Flow Around Buildings.
R Bs 0.67BL 0.33 = 16.2 2001 ASHRAE Fundamentals Handbook (SI) Fig. 1 Flow Patterns Around Rectangular Building Fig. 2 Surface Flow Patterns and Building Dimensions Fig. 3 Flow Recirculation Regions and Exhaust-to-Intake Stretched-String Distances (Wilson 1982) Airflow Around Buildings 16.3 between the wind direction and the upwind face of the building is less than about 70°, the downwash-upwash patterns on the upwind face of the building are less pronounced, as is the ground level vor-tex shown in Figure 1. For an approach flow angle of 45°, stream-lines remain close to the horizontal in their passage around the sides of the building (Figure 2), except near roof level where the flow is sucked upward into the roof edge vortices (Cochran 1992).
WIND PRESSURE ON BUILDINGS In addition to the flow patterns described previously, the turbu-lence or gustiness of the approaching wind and the unsteady char-acter of the separated flows cause surface pressures to fluctuate. The pressures discussed here are time-averaged values, with an aver-aging period of about 600 s. Instantaneous pressures may vary significantly above and below these averages, and peak pressures two or three times the mean values are possible. Although peak pressures are important with regard to structural loads, mean values are more appropriate for computing infiltration and ventilation rates. The time-averaged surface pressures are proportional to the wind velocity pressure pv given by Bernoulli’s equation: (2) where UH = approach wind speed at upwind wall height H ρa = ambient (outdoor) air density The difference ps between the pressure on the building surface and the local outdoor atmospheric pressure at the same level in an undisturbed wind approaching the building is (3) where Cp is the local wind pressure coefficient for the building surface.
The local wind speed UH at the top of the wall that is required for Equation (2) is estimated by applying terrain and height corrections to the hourly wind speed Umet from a nearby meteorological station.
Umet is generally measured in flat, open terrain. The anemometer that records Umet is located at a height Hmet, usually 10 m above ground level. The hourly average wind speed UH at wall height H in the undisturbed wind approaching a building in its local terrain (Figures 1 and 3) can be calculated from Umet as follows: (4) The wind boundary layer thickness δ and exponent a for the local building terrain and amet and δmet for the meteorological station are determined from Table 1. Typical values for meteoro-logical stations located in flat, open terrain (Category 3 in Table 1) are amet = 0.14 and δmet = 270 m. The values and terrain cate-gories in Table 1 are consistent with those adopted in other engi-neering applications, for example ASCE Standard 7. Equation (4) gives the wind speed at height H above the average height of local obstacles, such as buildings and vegetation, weighted by the plan-area. At heights at or below this average obstacle height (e.g., at roof height in densely built-up suburbs), the speed depends on the geometrical arrangement of the buildings, and Equation (4) is less reliable.
An alternative mathematical description of the atmospheric boundary layer, which uses a logarithmic function, is given by Deaves and Harris (1978). While this model is more complicated than the power law used in Equation (4), it more closely models the real physics of the atmosphere and has been adopted by several for-eign codes (e.g., SSA Standard AS-1170 from Australia).
Local Wind Pressure Coefficients Values of the mean local wind pressure coefficient Cp used in Equation (3) depend on building shape, wind direction, and the influence of nearby buildings, vegetation, and terrain features.
Accurate determination of Cp can be obtained only from wind tun-nel model tests of the specific site and building. Ventilation rate cal-culations for single, unshielded rectangular buildings can be reasonably estimated using existing wind tunnel data. Many wind load codes (e.g., ASCE 7-98, AS-1170) give mean pressure coeffi-cients for common building shapes.
Figure 4 shows pressure coefficients for walls of a tall rectangu-lar cross section building (high-rise) sited in urban terrain (Daven-port and Hui 1982). Figure 5 shows pressure coefficients for walls of a low-rise building (Holmes 1986). Generally, high-rise build-ings are those where the height H is more than three times the cross-wind width W. For H > 3W, use Figure 4, and for H < 3W, use Figure 5. At a wind angle θ = 0° (e.g., wind perpendicular to the face in question), the pressure coefficients are positive, and their magni-tudes decrease near the sides and the top as the flow velocities increase.
As can be seen in Figure 4, Cp generally increases with height, which reflects the increasing velocity pressure in the approach flow as wind speed increases with height. As the wind direction moves off normal (θ = 0°), the region of maximum pressure occurs closer to the upwind edge (B in Figure 4) of the building. At a wind angle of θ = 45°, the pressures become negative at the downwind edge (A in Figure 4) of the front face. At some angle θ between 60° and 75°, the pressures become negative over the whole front face. For θ = 90°, maximum suction (negative) pressure occurs near the upwind edge (B in Figure 4) of the building side and then recovers towards Cp = 0 towards the downwind edge (A in Figure 4). The degree of this recovery depends on the length of the side in relation to the width W of the structure. For wind angles larger than θ = 100°, the side is completely within the separated flow of the wake and the spatial variations in pressure over the face are not as great. The aver-age pressure on a face is positive for wind angles from θ = 0° to almost 60° and negative (suction) for θ = 60° to 180°. Table 1 Atmospheric Boundary Layer Parameters Terrain Category Description Exponent a Layer Thickness δ, m 1 Large city centers, in which at least 50% of buildings are higher than 21 m, over a distance of at least 2000 m or 10 times the height of the structure upwind, whichever is greater 0.33 460 2 Urban and suburban areas, wooded areas, or other terrain with numerous closely spaced obstructions having the size of single-family dwellings or larger, over a distance of at least 2000 m or 10 times the height of the structure upwind, whichever is greater 0.22 370 3 Open terrain with scattered obstructions having heights generally less than 10 m, including flat open country typical of meteorological station surroundings 0.14 270 4 Flat, unobstructed areas exposed to wind flowing over water for at least 1.6 km, over a distance of 500 m or 10 times the height of the structure inland, whichever is greater 0.10 210 pv ρaUH 2 2 --------------= ps Cppv = UH Umet δmet Hmet ------------ amet H δ ---- a = 16.4 2001 ASHRAE Fundamentals Handbook (SI) A similar pattern of behavior in the wall pressure coefficients for a low-rise building is shown in Figure 5. Here, the recovery from the strong suction with distance from the upwind edge is more rapid.
Surface Averaged Wall Pressures Surface averaged pressure coefficients may be used in determin-ing ventilation and/or infiltration rates, as discussed in Chapter 26.
Figure 6 shows the surface pressure coefficient Cs averaged over a complete wall of a low-rise building (Swami and Chandra 1987).
The figure also includes the values calculated from the pressure dis-tributions shown in Figure 5. Similar results for a tall building are shown in Figure 7 (Akins et al. 1979).
The wind-induced indoor-outdoor pressure difference is found using the coefficient Cp (in-out), which is defined as (5) where Cin is the internal wind-induced pressure coefficient. For uni-formly distributed air leakage sites in all the walls, Cin is about −0.2. Roof Pressures Surface pressures on the roof of a low-rise building depend strongly on roof slope. Figure 8 shows typical distributions for a wind direction normal to a side of the building. For very low slopes, the pressures are negative over the whole roof surface. The magnitude is greatest within the separated flow zone near the lead-ing edge and recovers toward the free stream pressure downwind of the edge. For steeper slopes, the pressures are weakly positive on the windward slope and negative within the separated flow over the leeward slope. With a wind angle of about 45°, the vortices originating at the leading corner of a roof with a low slope can induce very large localized negative pressures (Figure 2). A similar vortex forms on the downwind side of a leading ridge end on a steep roof. A discussion of roof corner vortices and how to disrupt their influence may be found in Cochran and Cermak (1992) and Cochran and English (1997), respectively. Figure 9 shows the average pressure coefficient over the roof of a tall building (Akins et al. 1979).
The section on Combining Driving Forces in Chapter 26 dis-cusses the effect of stack pressure and mechanical systems on infil-tration and ventilation in a building.
Interference and Shielding Effects on Pressures Nearby structures strongly influence surface pressures on both high- and low-rise buildings. These effects are very strong for spacing-to-height ratios less than five, where the distributions of pressure shown in Figures 4 through 9 do not apply. Although the effect of shielding is for low-rise buildings still significant at larger spacing, it is largely accounted for by the reduction in pv with increased terrain roughness. Saunders and Melbourne (1979), Sherman and Grimsrud (1980), Bailey and Kwok (1985), and Walker et al. (1996) discuss interference. English and Fricke (1997) discuss shielding through use of an interference index, while Walker et al. (1996) present a wind shadow model for pre-dicting shelter factors. Chapter 26 gives shielding classes for air infiltration and ventilation applications. Fig. 4 Local Pressure Coefficients (Cp × 100) for Tall Building with Varying Wind Direction (Davenport and Hui 1982) Cp in-out ( ) Cp Cin – = Airflow Around Buildings 16.5 Sources of Wind Data In order to design for the effects of airflow around buildings, wind speed and direction frequency data should be obtained. The simplest forms of wind data are tables or charts of climatic normals, which give hourly average wind speeds, prevailing wind directions, and peak gust wind speeds for each month of the year. This infor-mation can be found in sources such as The Weather Almanac (Bair 1992) and the Climatic Atlas of the United States (DOC 1968). Cli-matic design information, including wind speed at various frequen-cies of occurrence, is included in Chapter 27. A current source, which contains information on wind speed and direction frequen-cies, is the International Station Meteorological Climatic Summary available in CD-ROM format from the National Climatic Data Cen-ter (NCDC) in Asheville, North Carolina. Where more detailed information is required, digital records of hourly winds and other meteorological parameters are available (on magnetic tape or CD-ROM) from the NCDC for stations throughout the world. Most countries also have weather services that provide data. For example, in Canada, the Atmospheric Environment Service in Downsview, Ontario, provides hourly meteorological data and summaries.
When an hourly wind speed Umet at a specified probability level (e.g., the wind speed that is exceeded 1% of the time) is desired, but only the average annual wind speed Uannual is available for a given meteorological station, Umet may be estimated from Table 2. The Fig. 5 Local Pressure Coefficients for Walls of Low-Rise Building with Varying Wind Direction (Holmes 1986) Fig. 6 Variation of Surface Averaged Wall Pressure Coefficients for Low-Rise Buildings (Swami and Chandra 1987) Fig. 7 Surface Averaged Wall Pressure Coefficients for Tall Buildings (Akins et al. 1979) Fig. 8 Local Roof Pressure Coefficients for Roof of Low-Rise Buildings (Holmes 1986) 16.6 2001 ASHRAE Fundamentals Handbook (SI) ratios Umet /Uannual are based on long-term data from 24 weather sta-tions widely distributed over North America. At these stations, Umet ranges from 3.1 to 6.3 m/s. The uncertainty ranges listed in Table 2 are one standard deviation of the wind speed ratios. The following example demonstrates the use of Table 2.
Example 1. The wind speed Umet that is exceeded 1% of the time (88 hours per year) is needed for a building pressure or exhaust dilution calcula-tion. If Uannual = 4 m/s, find Umet.
Solution: From Table 2, the wind speed Umet exceeded 1% of the time is 2.5 ± 0.4 times Uannual. For Uannual = 4 m/s, Umet is 10 m/s with an uncertainty range of 8 to 12 m/s at one standard deviation.
Using a single prevailing wind direction for design can cause serious errors. For any set of wind direction frequencies, one direc-tion always has a somewhat higher frequency of occurrence. Thus, it is often called the prevailing wind, even though winds from other directions may be almost as frequent.
When using long-term meteorological records, check the ane-mometer location history as the instrument may have been relocated and its height varied. This can affect its directional exposure and the recorded wind speeds. Equation (4) can be used to correct wind data collected at different mounting heights. Poor anemometer exposure due to obstructions or mounting on top of a building cannot be eas-ily corrected, and the records for that period should be deleted.
If an estimate of the probability of an extreme wind speed outside the range of the recorded values at a site is required, the observa-tions may be fit to an appropriate probability distribution (e.g., a Weibull distribution) and the particular probabilities calculated from the resulting function (see Figure 10). This process is usually repeated for each of 16 wind directions (e.g., 22.5° intervals).
Where estimates at extremely low probability (high wind speed) are required, curve fitting at the tail of the probability distribution is very important and may require special statistical techniques appli-cable to extreme values (see Chapter 27). Building codes for wind loading on structures contain information on estimating extreme wind conditions. For ventilation applications, extreme winds are usually not required, and the 99 percentile limit can be accurately estimated from airport data averaged over less than 10 years.
Estimating Wind at Sites Remote from Recording Stations Many building sites are located far from the nearest long-term wind recording site, which is usually an airport meteorological sta-tion. To estimate wind conditions at such sites, the terrain surround-ing both the anemometer site and the building site should be checked. In the simplest case of flat or slightly undulating terrain with few obstructions extending for large distances around and between the anemometer site and building site, recorded wind data Table 2 Typical Relationship of Hourly Wind Speed Umet to Annual Average Wind Speed Uannual Percentage of Hourly Values That Exceed Umet Wind Speed Ratio Umet/Uannual 90% 0.2 ± 0.1 75% 0.5 ± 0.1 50% 0.8 ± 0.1 25% 1.2 ± 0.15 10% 1.6 ± 0.2 5% 1.9 ± 0.3 1% 2.5 ± 0.4 Fig. 9 Surface Averaged Roof Pressure Coefficients for Tall Buildings (Akins et al. 1979) Fig. 10 Frequency Distribution of Wind Speed and Direction Airflow Around Buildings 16.7 can be assumed to be representative of that at the building site. Wind direction occurrence frequency at a building site should be inferred from airport data only if the two locations are on the same terrain, with no terrain features that could alter wind direction between them.
In cases where the only significant difference between the ane-mometer site terrain and the building site terrain is surface rough-ness, the mean wind speed can be adjusted using Equation (4) and Table 1, to yield approximate wind velocities at the building site.
Wind direction frequencies at the site are assumed to be the same as at the recording station.
In using Equation (4), cases may be encountered where, for a given wind direction, the terrain upwind of either the building site or the recording site does not fall into just one of the categories in Table 1. The terrain immediately upwind of the site may fall into one cat-egory, while that somewhat further upwind falls into a different cat-egory. For example, at a downtown airport the terrain may be flat and open (Category 3) immediately around the recording instru-ment, but urban or suburban (Category 2) a relatively short distance away. This difference in terrains also occurs when a building site or recording site is in an urban area near open water or at the edge of town. In these cases, the suggested approach is to use the terrain cat-egory that is most representative of the average condition within approximately 1.6 km upwind of the site (Deaves 1981). If the aver-age condition is somewhere between two categories described in Table 1, the values of a and δ can be interpolated from those given in the table.
A rough guideline is that only wind speeds UH of 4 m/s or greater at the building site can be estimated reliably using Equation (4) and Table 2 for building and meteorological stations in different terrain categories. In addition to changes in surface roughness, several other fac-tors are important in causing the wind speed and direction at a build-ing site to differ from values recorded at a nearby meteorological station. Wind speeds for buildings on hillcrests or in valleys where the wind is accelerated or channeled can be 1.5 times higher than meteorological station data. Wind speeds for buildings sheltered in the lee of hills and escarpments can be reduced to 0.5 times the val-ues at nearby flat meteorological station terrain.
Solar heating of valley slopes can cause light winds of 1 to 4 m/s to occur as warm air flows upslope. At night, radiant cooling of the ground can produce similar speeds as cold air drains downslope. In general, rolling terrain experiences a smaller fraction of low speeds than nearly flat terrain.
When the wind is calm or light in the rural area surrounding a city, urban air tends to rise in a buoyant plume over the city center.
This rising air, heated by man-made sources and higher solar absorption in the city, is replaced by air pushed toward the city cen-ter from the edges. In this way, the urban heat island can produce light wind speeds and direction frequencies significantly different than those at a rural meteorological station.
In more complex terrain, both wind speed and direction may be significantly different from those at the distant recording site. In these cases, building site wind conditions should not be estimated from airport data. Options are either to establish an on-site wind recording station or to commission a detailed wind tunnel correla-tion study between the building site and long-term meteorological station wind observations.
WIND EFFECTS ON SYSTEM OPERATION With few exceptions, building intakes and exhausts cannot be located or oriented such that a prevailing wind ensures ventilation and air-conditioning system operation. Wind can assist or hinder inlet and exhaust fans, depending on their positions on the building, but even in locations with a predominant wind direction, the venti-lating system must perform adequately for all other directions. To avoid variability in system flow rates, use Figures 4, 5, and 8 as a guide to placing inlets and exhausts in locations where the surface pressure coefficients do not vary greatly with the wind direction.
Cooling towers and similar equipment should be oriented to take advantage of prevailing wind directions, based on careful study of the meteorological data and flow patterns on the building for the area and time of year involved.
A building with only upwind openings is under a positive pres-sure (Figure 11). Building pressures are negative when there are only downwind openings. A building with internal partitions and openings is under various pressures depending on the relative sizes of the openings and the wind direction. With larger openings on the windward face, the building interior tends to remain under positive pressure; the reverse is also true (see Figures 4 through 9, and Chap-ter 26).
Airflow through a wall opening results from differential pres-sures, which may exceed 125 Pa during high winds. Supply and exhaust systems, openings, dampers, louvers, doors, and windows make the building flow conditions too complex for direct calcula-tion. Iterative calculations are required because of the nonlinear dependence of volume flow rate on the differential pressure across an opening. Several multizone airflow models are available for these iterative calculations (Walton 1997, Feustel and Dieris 1992).
The opening and closing of doors and windows by building occu-pants add further complications. In determining Cp(in-out) from Equation (5), the wind direction is more important than the position of an opening on a wall, as shown in Figures 4 and 5. Please see Chapter 26 for more detailed information regarding wind effects on building ventilation, including natural and mechanical systems.
Natural and Mechanical Ventilation With natural ventilation, wind may augment, impede, or some-times reverse the airflow through a building. For large roof areas (Figure 2), the wind can reattach to the roof downwind of the lead-ing edge. Thus, any natural ventilation openings could see either a positive or negative pressure, dependent on wind speed and wind direction. Positive pressure existing where negative pressures were expected could lead to a reversal of expected natural ventilation.
These reversals can be avoided by using stacks, continuous roof ventilators, or other exhaust devices in which the flow is augmented by the wind.
Fig. 11 Sensitivity of System Volume to Locations of Building Openings, Intakes, and Exhausts 16.8 2001 ASHRAE Fundamentals Handbook (SI) Mechanical ventilation is also affected by wind conditions.
A low-pressure wall exhaust fan (12 to 25 Pa) can suffer drastic reduction in capacity. Flow can be reversed by wind pressures on windward walls, or its rate can be increased substantially when subjected to negative pressures on the lee and other sides. Clarke (1967), when measuring medium-pressure air-conditioning sys-tems (250 to 370 Pa), found flow rate changes of 25% for wind blowing into intakes on an L-shaped building compared to wind blowing away from intakes. Such changes in flow rate can cause noise at the supply outlets and drafts in the space served.
For mechanical systems, the wind can be thought of as an addi-tional pressure source in series with a system fan, either assisting or opposing it (Houlihan 1965). Where system stability is essential, the supply and exhaust systems must be designed for high pressures (about 750 to 1000 Pa) or must use devices to actively minimize unacceptable variations in flow rate. To conserve energy, the system pressure selected should be consistent with system needs.
Quantitative estimates of the effect of wind on a mechanical ven-tilation system can be made by using the pressure coefficients in Figures 4 through 9 to calculate the wind pressure on air intakes and exhausts. A simple worst-case estimate is to assume a system with 100% makeup air supplied by a single intake and exhausted from a single outlet. The building is treated as a single zone, with an exhaust-only fan as shown in Figure 12. This will overestimate the effect of wind on system volume flow.
Combining Equations (2) and (3), the wind pressures at the air intake and exhaust locations are (6) (7) For the single zone building shown in Figure 12, a worst-case estimate of wind effect neglects any flow resistance in the intake grill and duct, making the interior building pressure pinterior equal to the outdoor wind pressure on the intake, pinterior = ps intake. Then, with all the system flow resistance assigned to the exhaust duct in Figure 12, and a pressure rise ∆pfan across the fan, the pressure drop from outdoor intake to outdoor exhaust yields (8) where Fsys is the system flow resistance, AL is the flow leakage area, and Q is the system volume flow rate. This result shows that for the worst case estimate, the wind induced pressure difference simply adds to or subtracts from the fan pressure rise. With the inlet and exhaust pressures from Equations (6) and (7), the effective fan pres-sure rise ∆pfan eff is (9) where (10) The fan will be wind-assisted when Cp intake > Cp exhaust and wind-opposed when the wind direction changes causing Cp intake < Cp exhaust. The effect of wind-assisted and wind-opposed pressure differences is illustrated in Figure 13. Example 2. Make a worst-case estimate for the effect of wind on the sup-ply fan for a low-rise building with a height H = 15 m located in a city suburb. Use the hourly average wind speed that will be exceeded only 1% of the time and assume an annual hourly average speed of Uannual = 4 m/s measured on a meteorological tower at height Hmet = 10 m at a nearby airport. Outdoor air density is ρa = 1.2 kg/m3.
Solution: From Table 2 the wind speed that is exceeded only 1% of the hours each year will be a factor of 2.5 ± 0.4 higher than the annual average of 4 m/s, so the 1% maximum speed at the airport meteorolog-ical station is Umet = 2.5 × 4 = 10 m/s From Table 1, the airport meteorological station is in terrain cate-gory 3, with a boundary layer thickness δmet = 270 m and a velocity profile exponent a = 0.14. The suburban location of the building places it in terrain category 2 in Table 2, with δ = 370 m and a = 0.22. Using Equation (4) to determine the wind speed UH at roof level H = 15 m in the flow approaching the building, A worst-case estimate of wind effect must assume intake and exhaust locations on the building that produce the largest difference (Cp intake − Cp exhaust) in Equations (9) and (10). From Figure 5, the largest difference occurs for the intake on the upwind wall AB and the exhaust on the downwind wall CD, with a wind angle θAB = 0°. For this worst case, Cp intake = +0.8 on the upwind wall and Cp exhaust = −0.43 on the down-wind wall. Using these coefficients in Equations (9) and (10) to eval-uate the effective fan pressure ∆pfan eff , Fig. 12 Intake and Exhaust Pressures on Exhaust Fan in Single Zone Building ps intake Cp intake ρaUh 2 2 -------------= ps exhaust Cp exhaust ρaUh 2 2 -------------= ps intake ps exhaust – ( ) ∆pfan + Fsys ρQ2 AL 2 ----------= Fig. 13 Effect of Wind-Assisted and Wind-Opposed Flow ∆pfan eff ∆pfan ∆pwind + = ∆pwind Cp intake Cp exhaust – ( ) ρaUh 2 2 -------------= UH 10 270 10 --------- 0.14 15 370 --------- 0.22 7.8 m/s = = Airflow Around Buildings 16.9 This wind-assisted hourly averaged pressure is exceeded only 1% of the time (88 hours per year). When the wind direction reverses, the outlet will be on the upwind wall and the inlet on the downwind wall, producing wind-opposed flow, changing the sign from +44.9 Pa to −44.9 Pa. The importance of these pressures depends on their size relative to the fan pressure rise ∆pfan, as shown in Figure 13.
Building Pressure Balance Proper building pressure balance avoids flow conditions that make doors hard to open, cause drafts, and prevent the confinement of contaminants to specific areas. Although the supply and exhaust systems in an area may be in nominal balance, wind can upset this balance, not only because of the changes in fan capacity but also by superimposing infiltrated or exfiltrated air or both on the area.
These effects can make it impossible to control environmental con-ditions. Where building balance and minimum infiltration are important, consider the following: • Design HVAC system with pressure adequate to minimize wind effects • Include controls to regulate flow rate or pressure or both • Separate supply and exhaust systems to serve each building area requiring control or balance • Effect of doors (possibly self-closing) or double-door air locks to noncontrolled adjacent areas, particularly outside doors • Sealing of windows and other leakage sources and closing natural ventilation openings.
System volume and pressure control is described in Chapter 45 of the 1999 ASHRAE Handbook—Applications. This control is not possible without adequate system pressure for both the supply and exhaust systems to overcome wind effects. Such a control system may require fan inlet or discharge dampers, fan speed or pitch con-trol, or both.
Fume Hood Operation Wind effects can interfere with safe fume hood operation. Supply volume variations can cause both disturbances at hood faces and a lack of adequate fume hood makeup air. Volume surges, due to fluc-tuating wind pressures acting on the exhaust system, can cause momentary inadequate hood exhaust. If highly toxic contaminants are involved, surging is unacceptable. The system should be designed to eliminate this condition. On low-pressure exhaust sys-tems, it is impossible to test the hoods under wind-induced, surging conditions. These systems should be tested during calm conditions for safe flow into the hood faces; they should be rechecked by smoke tests during high wind conditions. For more information, see Chapter 13 of the 1999 ASHRAE Handbook—Applications.
Minimizing Wind Effect on System Volume Wind effect can be reduced by careful selection of inlet and exhaust locations. Because wall surfaces are subject to a wide vari-ety of positive and negative pressures, wall openings should be avoided when possible. When they are required, wall openings should be away from corners formed by building wings (Figure 11).
Mechanical ventilation systems should operate at a pressure high enough to minimize wind effect. Low-pressure systems and pro-peller exhaust fans should not be used with wall openings unless their ventilation rates are small or they are used in noncritical ser-vices such as storage areas.
Although roof air intakes in flow recirculation zones best mini-mize wind effect on system flow rates, current and future air quality in these zones must be considered. These locations should be avoided if a source of contamination exists or may be added in the future. The best area is near the middle of the roof because the neg-ative pressure there is small and least affected by changes in wind direction (Figure 8). Avoid the edges of the roof and walls, where large pressure fluctuations occur. Either vertical or horizontal (mushroom) openings can be used. On roofs having large areas, where the intake may be outside the roof recirculation zone, mush-room or 180° gooseneck designs minimize impact pressure from wind flow. The 135° gooseneck that is frequently used or vertical louvered openings are undesirable for this purpose or for rain pro-tection.
Heated air or contaminants should be exhausted vertically through stacks, above the roof recirculation zone. Horizontal, lou-vered (45° down), and 135° gooseneck discharges are undesirable, even for heat removal systems, because of their sensitivity to wind effects. A 180° gooseneck for systems handling hot air may be undesirable because of air impingement on tar and felt roofs. Verti-cally discharging stacks located in a recirculation region (except near a wall) have the advantage of being subjected only to negative pressure created by wind flow over the tip of the stack. See Chapter 43 of the 1999 ASHRAE Handbook—Applications for information regarding stack design.
BUILDING INTERNAL PRESSURE AND FLOW CONTROL In air-conditioning and ventilation systems for a building con-taining airborne contaminants, the correct internal airflow is toward the contaminated areas. Airflow direction is maintained by control-ling pressure differentials between spaces. In a laboratory building, for example, peripheral rooms such as offices and conference rooms are maintained at a positive pressure, and laboratories at a negative pressure, both with reference to corridor pressure. Pressure differ-entials between spaces are normally obtained by balancing the air-conditioning and ventilation supply system airflows in the spaces in conjunction with the exhaust systems in the laboratories, with dif-ferential pressure instrumentation to control the airflow. Chapter 45 of the 1999 ASHRAE Handbook—Applications has further informa-tion on controls.
Airflow in corridors is sometimes controlled by an outdoor ref-erence probe that senses static pressure at doorways and air intakes.
The differential pressure measured between the corridor and the outside may then signal a controller to increase or decrease airflow to the corridor. Unfortunately, it is difficult to locate an external probe where it will sense the proper external static pressure. High wind velocity and resulting pressure changes around entrances can cause great variations in pressure. Care must be taken to ensure that the probe is unaffected by wind pressure.
The pressure differential for a room adjacent to a corridor can be controlled using the corridor pressure as the reference. Outdoor pressure cannot control pressure differentials within rooms, even during periods of relatively constant wind velocity (wind-induced pressure). A single pressure sensor can measure the outside pressure at one point only and may not be representative of pressures else-where.
SCALE MODEL SIMULATION AND TESTING For many routine design applications, the flow patterns and wind pressures can be estimated using the data and equations presented in the previous sections. Exhaust dilution for simple building geome-tries located in homogeneous terrain environments (e.g., no larger buildings or terrain features nearby) can be estimated using the data and equations presented in the previous sections and in Chapter 43 of the 1999 ASHRAE Handbook—Applications. However, in criti-cal applications, such as where health and safety are of concern, physical modeling or full-scale field evaluations may be required ∆pfan eff ∆pfan 0.8 0.43 – ( ) – [ ]1.2 7.8 ( )2 2 ----------------------+ = ∆pfan 44.9 Pa + = 16.10 2001 ASHRAE Fundamentals Handbook (SI) to obtain more accurate estimates. Measurements on small-scale models in wind tunnels or water channels can provide information for design prior to construction. These measurements can also be used as an economical method of performance evaluation for exist-ing facilities. Full-scale testing is not generally useful in the initial design phase because of the time and expense required to obtain meaningful information. On the other hand, full-scale testing is use-ful for verifying data derived from physical modeling and for plan-ning remedial changes to improve existing facilities.
Detailed accounts of physical modeling, field measurements and applications, and engineering problems resulting from atmospheric flow around buildings are available in the proceedings of confer-ences on wind engineering (see the section on Bibliography).
The wind tunnel is the main tool used to assess and understand the airflow around buildings. Water channels or tanks can also be used. However, the water methods are more difficult to implement and give only qualitative results for some cases. Models of build-ings, complexes, and the local surrounding topography are con-structed and tested in a simulated turbulent atmospheric boundary layer. The airflow, wind pressures, snow loads, structural response, or pollutant concentrations can then be measured directly by prop-erly scaling the wind, building geometry, and exhaust flow charac-teristics. Weil et al. (1981), Petersen (1987a), and Dagliesh (1975) found generally good agreement between the results of wind tunnel simulations and corresponding full-scale data. Cochran (1992) and Cochran and Cermak (1992) have found good agreement between the model- and full-scale measurements of low-rise architectural aerodynamics and cladding pressures, respectively.
Similarity Requirements Physical modeling is most appropriate for applications involving small-scale atmospheric motions, such as recirculation of exhaust downwind of a laboratory, wind loads on structures, wind speeds around building clusters, snow loads on roofs, and airflow over hills or other terrain features. Winds associated with tornadoes, thunder-storms, and large-scale atmospheric motion cannot currently be simulated accurately.
Snyder (1981) gives guidelines for fluid modeling of atmo-spheric diffusion. This report contains explicit directions and should be used whenever designing wind tunnel studies to assess concen-tration levels due to air pollutants. ASCE Standard 7 and ASCE Manual of Practice 67 (ASCE 1999) also provide guidance when wind tunnels are used for evaluating wind effects on structures.
A complete and exact simulation of the airflow over buildings and the resulting concentration or pressure distributions cannot be achieved in a physical model. However, this is not a serious limita-tion. Cermak (1971, 1975, 1976a,b), Snyder (1981), and Petersen (1987a,b) found that an accurate simulation of the transport and dis-persion of laboratory exhaust can be achieved if the following cri-teria are met in the model and full scale: 1. Match exhaust velocity to wind speed ratios, Ve /UH.
2. Match exhaust to ambient air density ratios, ρe /ρa.
3. Match exhaust Froude numbers. Fr2 = ρaVe 2/[(ρe − ρa)gd], where d is the effective exhaust stack diameter.
4. Ensure fully turbulent stack gas flow by ensuring stack flow Reynolds number (Res = Ve d/ν) is greater than 2000 (where ν is the kinematic viscosity of ambient (outdoor) air), or by placing an obstruction inside the stack to enhance turbulence.
5. Ensure fully turbulent wind flow.
6. Scale all dimensions and roughness by a common factor.
7. Match atmospheric stability by the bulk Richardson number (Cermak 1975). For most applications related to airflow around buildings, neutral stratification is assumed, and no Richardson number matching is required.
8. Match mean velocity and turbulence distributions in the wind.
9. Ensure building wind Reynolds number (Reb = UHR/ν) is greater than 11 000 for sharp-edged structures, or greater than 90 000 for round-edged structures.
10. Ensure less than 5% blockage of wind tunnel cross section.
For wind speeds, flow patterns, or pressure distributions around buildings, only Conditions 5 through 10 are necessary. Usually, each wind tunnel study requires a detailed assessment to determine the appropriate parameters to match in the model and full scale.
In wind tunnel simulations of exhaust gas recirculation, the buoyancy of the exhaust gas (Condition 3) is often not modeled.
This allows using a high wind tunnel speed or a smaller model to achieve high enough Reynolds numbers (Conditions 4, 5, and 9).
Neglecting buoyancy is justified if the density of building exhaust air is within 10% of the ambient (outdoor) air. Also, critical mini-mum dilution Dcrit occurs at wind speeds high enough to produce a well-mixed, neutrally stable atmosphere, allowing stability match-ing (Condition 7) to be neglected (see Chapter 43 of the 1999 ASHRAE Handbook—Applications for discussion of Dcrit). Omis-sion of Conditions 3 and 7 simplifies the test procedure consider-ably, reducing both testing time and cost.
Buoyancy should be properly simulated for high-temperature exhausts such as boilers and diesel generators. Equality of model and prototype Froude numbers (Condition 3) requires tunnel speeds of less than 0.5 m/s for testing. However, greater tunnel speeds may be needed to meet the minimum building Reynolds number require-ment (Condition 4).
Wind Simulation Facilities Boundary layer wind tunnels are required for conducting most wind studies. The wind tunnel test section should be long enough so that a deep boundary layer that slowly changes with downwind dis-tance can be established upwind of the model building.
Other important wind tunnel characteristics include the width and height of the test section, range of wind speeds, roof adjustabil-ity, and temperature control. Larger models can be used in tunnels that are wider and taller, which, in turn, give better measurement resolution. Model blockage effects can be minimized by an adjust-able roof height. Temperature control of the tunnel surface and air-flow is required when atmospheric conditions other than neutral stability are to be simulated. Boundary layer characteristics appro-priate for the site are established by using roughness elements on the tunnel floor that produce mean velocity and turbulence intensity profiles characteristic of the full scale.
Water can also be used for the modeling fluid if an appropriate flow facility is available. Flow facilities may be in the form of a tun-nel, tank, or open channel. Water tanks with a free surface ranging in size up to that of a wind tunnel test section have been used by tow-ing a model (upside down) through the nonflowing fluid. Stable stratification can be obtained by adding a salt solution. This tech-nique (towed model in a tank) does not permit development of a boundary layer and therefore yields only approximate, qualitative information on flow around buildings. Water channels can be designed to develop thick turbulent boundary layers similar to those developed in the wind tunnel. One advantage of such a flow system is ease of flow visualization, but this is offset by a greater difficulty in developing the correct turbulence structure and the measurement of flow variables and concentrations.
Designing Model Test Programs The first step in planning a test program is selection of the model length scale. Choice of this scale depends on cross-sectional dimen-sions of the test section, dimensions of the buildings to be included in the model, and/or topographic features and thickness of the sim-ulated atmospheric boundary layer. Typical geometric scales range from about 120:1 to 1000:1.
Airflow Around Buildings 16.11 Because a large model size is desirable to meet minimum Rey-nolds number and Froude number requirements, a wide test section is advantageous. In general, the model at any section should be small compared to the test section area so that blockage is less than 5% (Melbourne 1982).
The test program must include specifications of the meteorolog-ical variables to be considered. These include wind direction, wind speed, and thermal stability. Data taken at the nearest meteorologi-cal station should be reviewed to obtain a realistic assessment of wind climate for a particular site. Ordinarily, local winds around a building, pressures, and/or concentrations are measured for 16 wind directions (e.g., 22.5° intervals). This is easily accomplished by mounting the building model and its nearby surroundings on a turn-table. More than 16 wind directions are required for highly toxic exhausts or for finding peak fluctuating pressures on a building. If only local wind information and pressures are of interest, testing at one wind speed with neutral stability is sufficient.
SYMBOLS AL = flow leakage area, Equation (8), m2 a = exponent in power law wind speed profile for local building terrain, Equation (4) and Table 1, dimensionless amet = exponent a for the meteorological station, Equation (4) and Table 1, dimensionless BL = larger of the two upwind building face dimensions H and W, Equation (1), m Bs = smaller of the two upwind building face dimensions H and W, Equation (1), m Cp in = internal wind-induced pressure coefficient, Equation (5), dimensionless Cp = local wind pressure coefficient for building surface, Equation (3), dimensionless Cp(in-out)= difference between outdoor and indoor pressure coefficients, Equation (5), dimensionless Cs = surface-averaged pressure coefficient, Figure 6, dimensionless d = effective stack diameter, m Dcrit = critical dilution factor at roof level for uncapped vertical exhaust at critical wind speed (see Chapter 43 of the 1999 ASHRAE Handbook—Applications), dimensionless Fr = Froude number, dimensionless Fsys = system flow resistance, Equation (8), dimensionless g = acceleration of gravity, 9.8 m/s2 H = wall height above ground on upwind building face, Equation (4) and Figure 1, m Hc = maximum height above roof level of upwind roof edge flow recirculation zone, Figures 1 and 3, m Hmet = height of anemometer at meteorological station, Equation (4), m L = length of building in wind direction, Figures 1 and 2, m Lc = length of upwind roof edge recirculation zone, Figure 3, m Lr = length of flow recirculation zone behind rooftop obstacle or building, Figures 1 and 3, m ps = wind pressure difference between exterior building surface and local ambient (outdoor) atmospheric pressure at the same elevation in an undisturbed approach wind, Equation (3), Pa pv = wind velocity pressure at roof level, Equation (2), Pa Q = volumetric flow rate, Equation (8), m3/s R = scaling length for roof flow patterns, Equation (1), m Reb = building Reynolds number, dimensionless Res = stack flow Reynolds number, dimensionless Uannual = annual average of hourly wind speeds Umet , Table 2, m/s UH = mean wind speed at height H of upwind wall in undisturbed flow approaching building, Equation (2) and Figures 1, 2 and 3, m/s Umet = meteorological station hourly wind speed, measured at height Hmet above ground in smooth terrain, Equation (4) and Table 2, m/s Ve = exhaust face velocity, m/s W = width of upwind building face, Figure 2, m δ = fully developed atmospheric boundary layer thickness, Equation (4) and Table 1, m δmet = atmospheric boundary layer thickness at meteorological station, Equation (4) and Table 1, m ∆pfan = pressure rise across fan, Equation (8), Pa ∆pfan eff = effective pressure rise across fan, Equation (9), Pa ∆pwind = wind-induced pressure, Equations (9) and (10), Pa ν = kinematic viscosity of ambient (outdoor) air, m2/s ρa = ambient (outdoor) air density, Equation (2), kg/m3 ρe = density of exhaust gas mixture, kg/m3 θ = angle between perpendicular line from upwind building face and wind direction, Figures 4 through 7, degrees REFERENCES AIHA. 1992. Laboratory ventilation. ANSI/AIHA Standard Z9.5-1992.
American Industrial Hygiene Association, Fairfax, VA.
Akins, R.E., J.A. Peterka, and J.E. Cermak. 1979. Averaged pressure coef-ficients for rectangular buildings. Wind Engineering. Proceedings of the Fifth International Conference 7:369-80, Fort Collins, CO. Pergamon Press, NY.
ASCE. 1998. Minimum design loads for buildings and other structures.
Standard 7-1998. American Society of Civil Engineers, New York.
ASCE. 1999. Wind tunnel model studies of buildings and structures. Manual of Practice 67. Bailey, P.A. and K.C.S. Kwok. 1985. Interference excitation of twin fall buildings. Wind Engineering and Industrial Aerodynamics 21:323-338.
Bair, F.E. 1992. The weather almanac, 6th ed. Gale Research Inc., Detroit, MI.
Cermak, J.E. 1971. Laboratory simulation of the atmospheric boundary layer. AIAA Journal 9(9):1746.
Cermak, J.E. 1975. Applications of fluid mechanics to wind engineering.
Journal of Fluid Engineering, Transactions of ASME 97:9.
Cermak, J.E. 1976a. Nature of airflow around buildings. ASHRAE Transac-tions 82(1):1044-60.
Cermak, J.E. 1976b. Aerodynamics of buildings. Annual Review of Fluid Mechanics 8:75.
Clarke, J.H. 1967. Airflow around buildings. Heating Piping and Air Con-ditioning 39(5):145.
Cochran, L.S. 1992. Low-rise architectural aerodynamics: The Texas Tech University experimental building. Architectural Science Review 35(4):131-36.
Cochran, L.S. and J.E. Cermak. 1992. Full and model scale cladding pres-sures on the Texas Tech University experimental building. Journal of Wind Engineering and Industrial Aerodynamics 41-44:1589-1600.
Cochran, L.S. and E.C. English. 1997. Reduction of wind loads by architec-tural features. Architectural Science Review 40(3):79-87.
Dagliesh, W.A. 1975. Comparison of model/full-scale wind pressures on a high-rise building. J. Industrial Aerodynamics 1:55-66.
Davenport, A.G. and H.Y.L. Hui. 1982. External and internal wind pressures on cladding of buildings. Boundary Layer Wind Tunnel Laboratory, Uni-versity of Western Ontario, London, Ontario, Canada. BLWT-820133.
Deaves, D.M. 1981. Computations of wind flow over changes in surface roughness. J. Wind Engineering and Industrial Aerodynamics 7:65-94.
Deaves, D.M. and R.I. Harris. 1978. A mathematical model of the structure of strong winds. Report 76. Construction Industry Research and Infor-mation Association (UK). DOC. 1968. Climatic atlas of the United States. U.S. Department of Com-merce, Washington, D.C.
English, E.C. and F.R. Fricke. 1997. The interference index and its predic-tion using a neural network analysis of wind tunnel data. Fourth Asia-Pacific Symposium on Wind Engineering. APSOWE IV University of Queensland 363-366.
Feustel, H.E. and J. Dieris. 1992. A survey of airflow models for multizone buildings. Energy and Buildings 18:79-100.
Holmes, J.D. 1986. Wind loads on low-rise buildings: The structural and environmental effects of wind on buildings and structures, Chapter 12.
Faculty of Engineering, Monash University, Melbourne, Australia.
Hosker, R.P. 1984. Flow and diffusion near obstacles. Atmospheric science and power production. U.S. Department of Energy DOE/TIC-27601 (DE 84005177).
Hosker, R.P. 1985. Flow around isolated structures and building clusters: A review. ASHRAE Transactions 91(2b):1671-92.
Houlihan, T.F. 1965. Effects of relative wind on supply air systems.
ASHRAE Journal 7(7):28.
Melbourne, W.H. 1979. Turbulence effects on maximum surface pressures; A mechanism and possibility of reduction. Proceedings of Fifth Interna-tional Conference on Wind Engineering. J.E. Cermak, ed. Fort Collins, Colorado. 541-551.
16.12 2001 ASHRAE Fundamentals Handbook (SI) Melbourne, W.H. 1982. Wind tunnel blockage effects and corrections. Pro-ceedings of the International Workshop on Wind Tunnel Modeling Cri-teria and Techniques in Civil Engineering Applications. T.A. Reinhold, ed. Maryland, USA. 197-216.
Meroney, R.N., and B. Bienkiewicz. 1997. Computational wind engineering 2. Elsevier, Amsterdam.
NCDC. Updated periodically. International station meteorological climatic summary (CD-ROM). National Climatic Data Center, Asheville, NC.
Published jointly with U.S. Air Force and U.S. Navy.
Petersen, R.L. 1987a. Wind tunnel investigation of the effect of platform-type structures on dispersion of effluents from short stacks. Journal of Air Pollution Control Association 36:1347-52.
Petersen, R.L. 1987b. Designing building exhausts to achieve acceptable concentrations of toxic effluent. ASHRAE Transactions 93(2):2165-85.
SAA. 1989. Minimum design loads on structures – Part 2: Wind loads. Stan-dard AS-1170 (known as the SAA Loading Code). Standards Associa-tion of Australia, North Sydney, NSW.
Saunders, J.W. and W.H. Melbourne. 1979. Buffeting effect of upwind buildings. Fifth International Conference on Wind Engineering. Pergam-mon Press.
Sherman, M.H. and D.T. Grimsrud. 1980. The measurement of infiltration using fan pressurization and weather data. Report # LBL-10852.
Lawrence Berkeley Laboratory, University of California, Berkeley.
Snyder, W.H. 1981. Guideline for fluid modeling of atmospheric diffusion.
Environmental Protection Agency Report EPA-600/881-009.
Swami, H.V. and S. Chandra. 1987. Procedures for calculating natural ven-tilation airflow rates in buildings. Final Report FSEC-CR-163-86. Flor-ida Solar Energy Center, Cape Canaveral.
Walker, I.S., D.J. Wilson, and T.W. Forest. 1996. Wind shadow model for air infiltration sheltering by upwind obstacles. Int. J. of HVAC&R Research 2(4):265-283.
Walton, G.N. 1997. CONTAM96 user manual. National Institute of Stan-dards and Technology NISTIR 6056. Gaithersburg, Maryland.
Wilson, D.J. 1979. Flow patterns over flat roofed buildings and application to exhaust stack design. ASHRAE Transactions 85:284-95.
BIBLIOGRAPHY ASCE. 1987. Wind tunnel model studies of building and structures. ASCE Manuals and Reports on Engineering Practice No. 67. American Society of Civil Engineers, New York.
Cermak, J.E. 1977. Wind-tunnel testing of structures. Journal of the Engi-neering Mechanics Division, ASCE 103, EM6:1125.
Cermak, J.E., ed. 1979. Wind engineering. Proceedings of Fifth Interna-tional Conference, Fort Collins, CO. Pergamon Press, New York.
Clarke, J.H. 1965. The design and location of building inlets and outlets to minimize wind effect and building reentry of exhaust fumes. Journal of American Industrial Hygiene Association 26:242.
Cochran, L.S. and Cermak, J.E. 1992. Full and model-scale cladding pres-sures on the Texas Tech University experimental building. Journal of Wind Engineering and Industrial Aerodynamics 41-44, 1589-1600.
Defant, F. 1951. Local winds. Compendium of Meteorology, pp. 655-72.
American Meteorology Society, Boston.
Elliot, W.P. 1958. The growth of the atmospheric internal boundary layer.
Transactions of the American Geophysical Union 39:1048-54.
ESDU. 1990. Strong winds in the atmospheric boundary layer. Part 1: Mean hourly wind speeds, pp. 15-17. Engineering Science Data Unit, Item 82-26, London, UK.
Geiger, R. 1966. The climate near the ground. Harvard University Press, Cambridge, MA.
Houghton, E.L. and N.B. Carruthers. 1976. Wind forces on buildings and structures: An introduction. Edward Arnold, London.
Landsberg, H. 1981. The urban climate. Academic Press, New York.
Panofsky, H.A. and J.A. Dutton. 1984. Atmospheric turbulence: Models and methods for engineering applications. John Wiley and Sons, New York.
Proceedings of the Fifth National Conference on Wind Engineering. 1985.
Texas Tech University, Lubbock, TX.
Simiu, V. and R. Scanlan. 1986. Wind effects on structures: An introduction to wind engineering, 2nd ed. Wiley Interscience, New York.
17.1 CHAPTER 17 ENERGY RESOURCES CHARACTERISTICS OF ENERGY AND ENERGY RESOURCE FORMS .......................................... 17.1 SUSTAINABILITY ................................................................... 17.2 DESIGNING FOR EFFECTIVE ENERGY RESOURCE UTILIZATION ................................................ 17.2 The Energy Ethic: Resource Conservation Design Principles ................................................................ 17.2 The Energy-Efficient Design Process ..................................... 17.3 Building Energy Use Elements ............................................... 17.3 On-Site Energy/Energy Resource Relationships ..................... 17.5 Summary .................................................................................. 17.6 OVERVIEW OF GLOBAL ENERGY RESOURCES ....................................................................... 17.6 World Energy Resources ......................................................... 17.6 United States Energy Use ....................................................... 17.9 ENERGY RESOURCE PLANNING ...................................... 17.11 TRADABLE EMISSION CREDITS ....................................... 17.11 Agencies and Associations in the United States .................... 17.12 UILDINGS and facilities of various types may be heated, ven-Btilated, air conditioned, and refrigerated—using systems and equipment designed for that purpose and using the site energy forms commonly available—without concern for the original energy resources from whence those energy forms came. Since the energy used in buildings and facilities comprises a significant amount of the total energy used for all purposes, and since the use of this energy has an impact on energy resources, ASHRAE recognizes the “effect of its technology on the environment and natural resources to protect the welfare of posterity” (ASHRAE 1990).
Many governmental agencies regulate energy conservation leg-islation for obtaining building permits (Conover 1984). The appli-cation of specific values to building energy use situations has a considerable effect on the selection of HVAC&R systems and equipment and how they are applied.
CHARACTERISTICS OF ENERGY AND ENERGY RESOURCE FORMS The HVAC&R industry deals with energy forms as they occur on or arrive at a building site. Generally, these forms are fossil fuels (natural gas, oil, and coal) and electricity. Solar energy and wind energy are also available at most sites, as is low-level geothermal energy (energy source for heat pumps). Direct-use (high-tempera-ture) geothermal energy is available at some. These are the prime forms of energy used to power or heat the improvements on a site.
Forms of On-Site Energy Fossil fuels and electricity are commodities that are usually metered or measured for payment by the facility owner or operator.
On the other hand, solar or wind, each of which might be considered a dispersed energy form in its natural state (i.e., requiring neither central processing nor a distribution network) costs nothing for the commodity itself, but does incur cost for the means to make use of it. High-temperature geothermal energy, which is not universally available, may or may not be a sold commodity, depending on the particular locale and local regulations. Chapter 31 of the 1999 ASHRAE Handbook—Applications has more information on geo-thermal energy.
Some prime on-site energy forms require further processing or conversion into other forms more directly suited for the particular systems and equipment needed in a building or facility. For instance, natural gas or oil is burned in a boiler to produce steam or hot water, a form of thermal energy which is then distributed to various use points (such as heating coils in air-handling systems, unit heaters, convectors, fin-tube elements, steam-powered cool-ing units, humidifiers, and kitchen equipment) throughout the building. Although electricity is not converted in form on-site, it is nevertheless used in a variety of ways, including lighting, running motors for fans and pumps, powering electronic equipment and office machinery, and space heating. While the methods and effi-ciencies with which these processes take place fall within the scope of the HVAC&R designer, the process by which a prime energy source arrives at a given facility site is not under direct control of the professional. On-site energy choices, if available, may be controlled by the designer based in part on the present and future availability of the associated resource commodities.
The basic energy source for heating may be natural gas, oil, coal, or electricity. Cooling may be produced by electricity, thermal energy, or natural gas. If electricity is generated on-site, the gener-ator may be turned by an engine using natural gas or oil, or by a tur-bine using steam or gas directly.
The term energy source refers to on-site energy in the form in which it arrives at or occurs on a site (e.g., electricity, gas, oil, or coal). Energy resource refers to the raw energy, which (1) is extracted from the earth (wellhead or mine-mouth), (2) is used in the generation of the energy source delivered to a building site (coal used to generate electricity), or (3) occurs naturally and is available at a site (solar, wind, or geothermal energy).
Nonrenewable and Renewable Energy Resources From the standpoint of energy conservation, energy resources may be classified in two broad categories: (1) nonrenewable (or discontinuous) resources, which have definite, although sometimes unknown, limitations; and (2) renewable (or continuous) resources, which can generally be freely used without depletion or have the potential to renew in a reasonable period. Resources used most in industrialized countries, both now and in the past, are nonrenewable (Gleeson 1951).
Nonrenewable resources of energy include • Coal • Crude oil • Natural gas • Uranium-235 (atomic energy) Renewable resources of energy include • Hydropower • Solar • Wind • Earth heat (geothermal) • Biomass (wood, wood wastes, and municipal solid waste) • Tidal power The preparation of this chapter is assigned to TC 1.10, Energy Resources.
17.2 2001 ASHRAE Fundamentals Handbook (SI) • Ocean thermal • Atmosphere or large body of water (as used by the heat pump) • Crops (for alcohol production) Characteristics of Fossil Fuels and Electricity Most on-site energy for buildings in developed countries involves electricity and fossil fuels as the prime on-site energy sources. Both fossil fuels and electricity can be described in terms of their energy content (joules). This implies that the two energy forms are comparable and that an equivalence can be established. In real-ity, however, fossil fuels and electricity are only comparable in energy terms when they are used to generate heat. Fossil fuels, for example, cannot directly drive motors or energize light bulbs. Con-versely, electricity gives off heat as a byproduct regardless of whether it is used for running a motor or lighting a light bulb, and regardless of whether that heat is needed. Thus, electricity and fossil fuels have different characteristics, uses, and capabilities aside from any differences relating to their derivation.
Beyond the building site, further differences between these energy forms may be observed, such as methods of extraction, transformation, transportation, and delivery, and the characteristics of the resource itself. Natural gas arrives at the site in virtually the same form in which it was extracted from the earth. Oil is processed (distilled) before arriving at the site; having been extracted as crude oil, it arrives at a given site as, for example, No. 2 oil or diesel fuel.
Electricity is created (converted) from a different energy form, often a fossil fuel, which itself may first be converted to a thermal form.
The total electricity conversion, or generation, process includes energy losses governed largely by the laws of thermodynamics.
Fuel cells, which are used only on a small scale, convert a fossil fuel to electricity by chemical means.
Fossil fuels undergo a conversion process by combustion (oxi-dation) and heat transfer to thermal energy in the form of steam or hot water. The conversion equipment used is a boiler or a furnace in lieu of a generator, and conversion usually occurs on a project site rather than off-site. (District heating is an exception.) Inefficiencies of the fossil fuel conversion occur on-site, while the inefficiencies of most electricity generation occur off-site, before the electricity arrives at the building site. (Cogeneration is an exception.) SUSTAINABILITY As the world has increased in population and developed techno-logically, the consequences of uncontrolled growth are being recog-nized: pollution, toxic waste creation, waste disposal, global climate change, ozone depletion, deforestation, and resource depletion.
Continuation of current trends without implementation of mitiga-tion strategies will adversely impact the ability of the earth’s eco-system to regenerate and remain viable for future generations.
The built environment contributes significantly to these effects, accounting for one-sixth of the world’s fresh water use, one-quarter of its wood harvest, and two-fifths of its material and energy flows.
Other impacts include air quality, transportation patterns, and watersheds. The resources required to serve this sector are consid-erable—and many of them are diminishing (Gottfried 1996).
The building industry’s recognition of the impacts of its activi-ties is changing the way it approaches the design, construction, op-eration, maintenance, reuse, and demolition of what it creates— namely, toward addressing the environmental and long-term eco-nomic consequences of its actions. While this sustainable design ethic—or sustainability—covers things beyond the purview of the heating, ventilating, air-conditioning (HVAC) industry alone, de-sign for the efficient use of energy resources would certainly be a key element of any sustainable design, and it is certainly very much under the control of the HVAC designer.
The following section of this chapter provides guidance in achieving a sustainable design.
DESIGNING FOR EFFECTIVE ENERGY RESOURCE UTILIZATION The preponderance of energy used in buildings is from nonre-newable energy resources, the cost of which historically has not included considerations of replenishment or environmental impact.
As a result, energy use resulting from many building system designs has been based primarily upon economic considerations, which unfortunately are biased to encourage more rather than less use.
Astheseresourcesbecomelessreadilyavailableandpractitioners look toward the use of more exotic and replenishable sources, the need to operate buildings effectively using less energy becomes par-amount. Extensive study since the mid-1970s has revealed that sig-nificant reductions in building energy use can be achieved by the application of some fundamental principles.
THE ENERGY ETHIC: RESOURCE CONSERVATION DESIGN PRINCIPLES The basic approach to achieving an energy-efficient design is reducing loads (power), improving transport systems, and provid-ing efficient components and “intelligent” controls. Design con-cepts applicable to successful energy-efficient design include understanding the relationship between energy and power, main-taining simplicity, using self-imposed budgets, and applying energy-smart design practices.
Energy and Power From the standpoint of economics, more energy-efficient sys-tems need not be more expensive than less efficient systems; quite the opposite is true. This observation results from the simple rela-tionship between energy and power in which power is simply the time rate of energy use (or, conversely, energy is power times time).
Power, in turn, describes the size of something. For example, power terms such as watt or kilowatt might be used in expressing the size of a motor, chiller, boiler, or transformer. Of course, the smaller something is, the less it costs; then, other things being equal, as the unit of a smaller size operates over time, it consumes less energy.
Thus, in designing for energy efficiency, the first objective is always to reduce the power required to the bare minimum necessary to pro-vide the desired performance—starting with the building’s heating and cooling loads (a power term, in kW) and working through the various systems and subsystems.
Simplicity Complex designs to save energy will seldom function in the manner intended unless the systems are continually managed and operated by technically skilled individuals. Experience in designing energy-efficient systems has shown that achieving energy-efficient performance over a long period of time with a complex system is seldom achievable; further, when these complex systems are oper-ated by minimally skilled individuals, not only are energy efficien-cies not achieved but performance suffers as well. The majority of the techniques discussed subsequently can be implemented with a high degree of simplicity.
Self-Imposed Budgets Just as an engineer must work to a cost budget with most designs, self-imposed power budgets can be similarly helpful in achieving an energy-efficient design. Examples of some budgets that designers have set for themselves for office buildings in a typical midwestern or northeastern temperate climate are Installed lighting (overall) 14 W/m2 Space sensible cooling 63 W/m2 Space heating load 47 W/m2 Energy Resources 17.3 Fan system pressure 1 kPa Air circulation rates 5 L/s·m2 Electric power (overall) 48 W/m2 Thermal power (overall) 95 W/m2 Hydronic system pressure 210 kPa Water chiller (water cooled) 0.17 kW/kW cooling Chilled water system auxiliaries 0.04 kW/kW cooling Unitary air-conditioning systems 0.28 kW/kW cooling Annual electric energy 970 MJ/m2 · yr Annual thermal energy 205 kJ/m2 · yr · K·day Then, as the building and its systems are designed, all decisions become interactive as the result of each subsystem’s power or energy peformance being continually compared to the “budget.” THE ENERGY-EFFICIENT DESIGN PROCESS Energy efficiency should be considered at the beginning of the building design process because energy-efficient features are most easily and effectively incorporated at that time. Active participation of all members of the design team—including owner, architect, engineer, and often the contractor—should be sought early in the design process. Consider building attributes such as building func-tion, form, orientation, window/wall ratio, and HVAC system types early as each has major energy implications.
Address a building’s energy requirements in the following sequence: 1. Minimize the impact of the building’s functional require-ments by analyzing how the building relates to its external envi-ronment. Advocate changes in building form, aspect ratio, and other attributes that reduce, redistribute, or delay (shift) loads.
The load calculation should be interactive so that the impact of those factors can be seen immediately.
2. Minimize loads by analyzing the external and internal loads imposed on the building energy-using subsystems, both for peak-load and part-load conditions.
3. Maximize subsystem efficiency by analyzing the diversified energy and power requirements of each energy-using subsystem serving the building’s functional requirements. Consider static and dynamic efficiencies of energy conversion and energy trans-port subsystems, and consider opportunities to reclaim, redis-tribute, and store energy for later use.
4. Study alternate ways to integrate subsystems into the building by considering both power and time components of energy use.
Identify, evaluate, and design each of these components to con-trol overall design energy consumption. The following should be considered when integrating major building subsystems: • Address more than one problem at time when developing de-sign solutions, and make maximum use of building compo-nents incorporated for nonenergy reasons (e.g., windows, structural mass).
• Examine design solutions that consider time (i.e., when the en-ergy use takes place) since sufficient energy may already be present from the environment (e.g., solar heat, night cooling) or from internal equipment (e.g., lights, computers) but avail-able at times different from when needed. Thus, active (e.g., heat pumps with water tanks) and passive (e.g., building mass) storage techniques may be considered.
• Examine design solutions that consider the anticipated use of space. For example, in large but relatively unoccupied spaces, task or zone lighting may be considered. Transporting excess energy (light and heat) from locations of production and avail-ability to locations of need may be considered instead of pur-chasing additional energy.
• Never reject waste energy at temperatures usable for space con-ditioning or other practical purposes without calculating the economic benefit of energy recovery or treatment for reuse.
• Consider or advocate design solutions that provide more com-fortable surface temperatures or increase the availability of controlled daylight in buildings where human occupancy is a primary function.
• Use easily understood design solutions as they have a greater probability of use by building operators and occupants.
• Where the functional requirements of a building are known to be likely to change over time, design the installed environmen-tal system to adapt to meet those changes that can be antici-pated as well as to provide flexibility in meeting future changes in use, occupancy, or other functions.
BUILDING ENERGY USE ELEMENTS Envelope • Control thermal conductivity through the use of insulation (including movable insulation), thermal mass, and/or phase change thermal storage at levels that minimize net heating and cooling loads on a time-integrated (annual) basis.
• Minimize unintentional or uncontrolled thermal bridges, and con-sider them in energy-related calculations because they can radi-cally alter the conductivity of a building envelope. Examples include wall studs, balconies, ledges, and extensions of building slabs.
• Minimize infiltration so that it approaches zero. (An exception is when infiltration provides the sole means of ventilation such as in small residential units.) This effort will minimize fan energy consumption in pressurized buildings during occupied periods and minimize heat loss (or unwanted heat gain in warm climates) during unoccupied periods. An additional benefit of a tight envelope in warm humid climates is that it improves indoor air quality. Infiltration should be reduced through design details that enhance the fit and integrity of building envelope joints in ways that may be readily achieved during construction. This includes infiltration control by caulking, weatherstripping, ves-tibule doors, and/or revolving doors—with construction meet-ing accepted specifications.
• Consider operable windows to allow occupant-controlled venti-lation. If this is done, the design of the building’s mechanical sys-tem must be carefully executed to minimize unnecessary HVAC energy consumption, and building operators and occupants should be cautioned about how operable windows may be improperly used.
• Strive to maintain occupant radiant comfort regardless of whether the building envelope is designed to be a static or dynamic mem-brane. Opaque surfaces should be designed so that the average inside surface temperatures remain within 3 K of room tempera-ture in the coldest anticipated weather (i.e., winter design condi-tions) and so that the coldest inside surface will remain within 14 K of room temperature (but always above the indoor dew point). In a building with time-varying internal heat generation, consider thermal mass for controlling radiant comfort. In the perimeter zone, thermal mass is more effective when it is posi-tioned inside the envelope’s insulation.
• Effective control of solar radiation is critical to the design of energy-efficient buildings due to the high level of internal heat production already present in most commercial buildings. In some climates, lighting energy consumption savings due to day-lighting techniques can be greater than the heating and cooling energy penalties that result from additional glazed surface area required, provided that the building envelope is properly designed for daylighting and that lighting controls are installed and used. (In other climates, such net savings may not be real-ized.) Daylighting designs are most effective if direct solar beam radiation is not allowed to cause glare in building spaces.
• Design the transparent portions of the building envelope to prevent solar radiant gain above that necessary for effective daylighting 17.4 2001 ASHRAE Fundamentals Handbook (SI) and solar heating. On south-facing facades, the use of low shading coefficients is generally not as effective as external physical shad-ing devices in achieving this balance. Low-emissivity, high-visi-ble-transmittance glazings may be considered for effective control of radiant heat gains and losses. For shading control, designers may consider the judicious use of vegetation to block excess gain year-round or seasonally, depending on the plant species chosen.
Lighting Lighting is both a major energy end use in commercial buildings (especially office buildings) and a major contributor to internal loads by increasing cooling loads and decreasing heating loads.
Therefore, it is important to produce (or advocate) a design that meets the lighting functional criteria of the space as well as one that minimizes energy use. The IESNA Lighting Handbook (IESNA 2000) recommends illuminance levels for visual tasks and sur-rounding lighted areas. Principles of energy-conserving design within that context are described as follows: • Energy use is determined by the lighting load (demand power) and its duration of use (time). Minimize the actual demand load rather than just the apparent connected load. Control the load rather than just area switching, if switching may adversely affect the quality of the luminous environment.
• Consider daylighting along with the proper use of controls to reduce costs of electric lighting. Design should be sensitive to window glare, sudden changes in luminances, and general user acceptance of daylighting controls. Window treatment (blinds, drapes, and shades) and glazing should be carefully selected to control direct solar penetration and luminance extremes while still maintaining the view and daylight penetration.
• Design the lighting system so that illumination required for tasks is primarily limited to the location of the task and comes from a direction that minimizes direct glare and veiling reflections on the task. When the design concept is based on nonuniform illumi-nance, walls should be a light to medium color or otherwise illu-minated to provide visual comfort. In densely occupied work spaces, uniform distribution of general lighting may be most appropriate. Where necessary, provide supplementary task illu-mination. General ambient illumination should not be lower than a third of the luminance required for the task; this will help main-tain visually comfortable luminance ratios.
• Use local task lighting to accommodate the need for higher light-ing levels due to task visual difficulty, glare, intermittently chang-ing requirements, or individual visual differences (poor or aging eyesight).
• Group similar activities so that high illuminance or special light-ing for particular tasks can be localized in certain rooms or areas, and so that less efficient fixtures required for critical glare control do not have to be installed uniformly when they are only required sparsely.
• Use lighting controls throughout so lighting is available when and where it is needed, but not wasted during those times when tasks being performed are less critical or spaces are not fully occupied.
Also consider user acceptance of control strategies to maximize energy saving.
• Limit use of lower efficiency lamps (such as incandescent) to those applications where their color, lumens, or distribution characteristics cannot be duplicated by other sources. Due to their lower efficiency, limit the use of extended service incandes-cent lamps to those applications where fixtures are difficult to reach and/or maintenance costs for replacing lamps would be excessive.
• Continue carrying through the lighting design process as the building’s interior design is occurring. Reduced light absorption may be achieved by using lighter finishes, particularly on ceil-ings, walls, and partitions.
Other Loads • Minimize the thermal impact of equipment and appliances on HVAC systems by the use of hoods, radiation shields, or other confining techniques, and by using controls to ensure that such equipment is turned off when not needed. In addition, locate, where practical, major heat-generating equipment where it can balance other heat losses. Computer centers or kitchen areas usu-ally have separate, dedicated HVAC equipment. In addition, con-sider heat recovery for this equipment.
• Use storage techniques to level or distribute loads that vary on a time or spatial basis to allow operation of a device at maximum (often full-load) efficiency.
HVAC System Design • Consider separate HVAC systems to serve areas expected to oper-ate on widely differing operating schedules or design conditions.
For instance, systems serving office areas should generally be separate from those serving retail areas.
• Arrange systems so that spaces with relatively constant and weather-independent loads are served with systems separate from those serving perimeter spaces. Areas with special temperature or humidity requirements, such as computer rooms, should be served by systems separate from those serving areas that require comfort heating and cooling only. Alternatively, these areas may be served by supplementary or auxiliary systems.
• Sequence the supply of zone cooling and heating to prevent the simultaneous operation of heating and cooling systems for the same space to the extent possible. Where this is not possible due to ventilation, humidity control, or air circulation requirements, reduce air quantities as much as possible before incorporating reheating, recooling, or mixing hot and cold airstreams. As an example, if reheat is necessary to provide dehumidification and prevent overcooling, only the ventilation air need be so treated rather than the entire recirculated air quantity. Finally, reset the supply air temperature up to the extent possible to reduce reheat-ing, recooling, or mixing losses.
• Provide controls to allow operation in an occupied mode and an unoccupied mode. In the occupied mode, controls may pro-vide for a gradually changing control point as system demands change from cooling to heating. In the unoccupied mode, ven-tilation and exhaust systems should be shut off if possible, and comfort heating and cooling systems should be shut off except to maintain space conditions ready for the next occupancy cycle.
• In geographical areas where diurnal temperature swings and humidity levels permit, consider the judicious coupling of air dis-tribution and building structural mass to allow nighttime cooling to reduce the requirement for daytime mechanical cooling.
• High ventilation rates, where required for special applications, can impose enormous heating and cooling loads on HVAC equip-ment. In these cases, consider recirculation of filtered and cleaned air to the extent possible rather than 100% outside air. Also, con-sider preheating outside air with reclaimed heat from other sources.
HVAC Equipment Selection • To allow for HVAC equipment operation at the highest efficien-cies, match conversion devices to load increments, and sequence the operation of modules. Oversized or large-scale systems should never serve small seasonal loads (e.g., a large heating boiler serving a summer service water-heated load). Include spe-cific low-load units and auxiliaries where prolonged use at mini-mal capacities is expected.
• Select the most efficient (or highest COP) equipment practical at both design and reduced capacity (part-load) operating conditions.
Energy Resources 17.5 • In the selection of large power devices such as chillers (including their auxiliary energy burdens), seriously consider life-cycle pur-chasing techniques.
• Keep fluid temperatures for heating equipment devices as low as practical and for cooling equipment as high as practical, while still meeting loads and minimizing flow quantities.
Energy Transport Systems Energy should be transported by the most energy-efficient means possible. The following options are listed in order of efficiency from the lowest energy transport burden (most efficient) to the highest (least efficient): 1. Electric wire or fuel pipe 2. Two-phase fluid pipe (steam or refrigerant) 3. Single-phase liquid/fluid pipe (water, glycol, etc.) 4. Air duct Select a distribution system that complements other parameters such as control strategies, storage capabilities, conversion effi-ciency, and utilization efficiency.
The following are specific design techniques that may be applied to thermal energy transport systems: Steam Systems • Include provisions for seasonal or non-use shutdown.
• Minimize the venting of steam and ingestion of air with the design directed toward full-vapor performance.
• Avoid subcooling, if practical.
• Return condensate to boilers or source devices at the highest pos-sible temperature.
Hydronic Systems • Minimize flow quantity by designing for the maximum practical temperature range (∆t).
• Vary flow quantity with load where possible.
• Design for the lowest practical pressure rise (or drop).
• Provide OPERATING and IDLE control modes.
• When locating equipment, identify the critical pressure path and size runs for the minimum reasonable pressure drop.
Air Systems • Minimize airflow by careful load analysis and an effective distri-bution system. If the application allows, the supply air quantity should vary with the sensible load (i.e., VAV systems). Hold the fan pressure requirement to the lowest practical value and avoid using fan pressure as a source for control power.
• Provide NORMAL and IDLE control modes for the fan systems as well as the psychrometric systems.
• Keep duct runs as short as possible, and keep runs on the critical pressure path sized for minimum practical pressure drop.
Power Distribution • Size transformers and generating units as close as possible to the actual anticipated load (i.e., avoid oversizing to minimize fixed thermal losses).
• Consider distribution of electric power at the highest practical voltage and load selection at the maximum power factor consis-tent with safety.
• Consider tenant submetering in commercial and multifamily buildings as a cost-effective energy conservation measure. (A large portion of energy use in tenant facilities occurs simply because there is no economic incentive to conserve.) Domestic Hot Water Systems • Choose shower heads that provide and maintain user comfort and energy savings. They should not have removable flow-restricting inserts to meet flow limitation requirements.
• Consider point-of-use water heaters where their use will reduce energy consumption and annual energy cost.
• Consider using storage facilitate heat recovery when the heat to be recovered is out of phase with the demand for hot water or when energy use for water heating can be shifted to take advan-tage of off-peak rates.
Controls • A well-designed digital control system provides information to managers and operators as well as to the data processor that serves as the intelligent controller. Include the energy saving concepts discussed previously throughout the operating sequences and control logic. However, energy conservation should not be sought at the expense of inadequate performance; in a well-designed sys-tem, these two parameters are compatible.
ON-SITE ENERGY/ENERGY RESOURCE RELATIONSHIPS An HVAC&R designer sooner or later must consider the use of one or more forms of prime energy. Most likely, these would be fos-sil fuels and electricity, although installations are sometimes designed using a single energy source (e.g., only a fossil fuel or only electricity).
Solar energy normally impinges on the site (and on the facilities to be put there), so it has an impact on the energy consumption of the facility. The designer must account for this impact and may have to decide whether to make active use of solar energy. Other naturally occurring and distributed renewable forms such as wind power and earth heat (if available) might also be considered.
The designer should be aware of the relationship between on-site energy sources and raw energy resources—including how these resources are used and what they are used for. The relationship between energy sources and energy resources involves two parts: (1) quantifying the energy resource units expended and (2) consid-ering the societal impact of the depletion of one energy resource (caused by on-site energy use) with respect to others. The following two sections describe those parts in more specific terms.
Quantifiable Relationships As on-site energy sources are consumed, a corresponding amount of resources are consumed to produce that on-site energy.
For instance, for every volume of No. 2 oil consumed by a boiler at a building site, some greater volume of crude oil is extracted from the earth. On leaving the well, the crude oil is transported and pro-cessed into its final form, perhaps stored, and then transported to the site where it will be used.
Even though gas requires no significant processing, it is trans-ported, often over long distances, to reach its final destination, which causes some energy loss. Electricity may have as its raw energy resource a fossil fuel, uranium, or an elevated body of water (hydroelectric generating plant).
Data to assist in determining the amount of resource use per deliv-ered on-site energy source unitare available. In the United States, data are available from entities within the U.S. Department of Energy and from the agencies and associations listed at the end of this chapter.
A resource utilization factor (RUF) is the ratio of resources consumed to energy delivered (for each form of energy) to a build-ing site. Specific RUFs may be determined for various energy sources normally consumed on-site, including nonrenewable sources such as coal, gas, oil, and electricity, and renewable sources such as solar, geothermal, waste, and wood energy. With electricity, which may derive from several resources depending on the partic-ular fuel mix of the generating stations in the region served, the overall RUF is the weighted combination of individual factors applicable to electricity and a particular energy resource. Grumman (1984) gives specific formulas for calculating RUFs.
17.6 2001 ASHRAE Fundamentals Handbook (SI) While a designer is usually not required to determine the amount of energy resources attributable to a given building or building site for its design or operation, this information may be helpful when assessing the long-range availability of energy for a building or the building’s impact on energy resources. Factors or fuel-quantity-to-energy resource ratios or factors are often used, which suggests that energy resources are of concern to the HVAC&R industry.
Intangible Relationships Energy resources should not simply be converted into common energy units [e.g., gigajoule] because the commonality gives a mis-leading picture of the equivalence of these resources. Other differ-ences and limitations of each of the resources defy easy quantification but are nonetheless real. For instance, electricity that arrives and is used on a site can be generated from coal, oil, natural gas, uranium, or hydropower. The end result is the same: electricity at x kV, y Hz. However, is a megajoule of electricity generated by hydropower equal in societal impact to that same megajoule gener-ated by coal, by uranium, by domestic oil, or by imported oil?
Intangible factors such as safety, environmental acceptability, availability, and national interest also are affected in different ways by the consumption of each resource. Heiman (1984) proposes a procedure for weighting the following intangible factors: National/Global Considerations • Balance of trade • Environmental impacts • International policy • Employment • Minority employment • Availability of supply • Alternative uses • National defense • Domestic policy • Effect on capital markets Local Considerations • Exterior environmental impact—air • Exterior environmental impact—solid waste • Exterior environmental impact—water resources • Local employment • Local balance of trade • Use of distribution infrastructure • Local energy independence • Land use • Exterior safety Site Considerations • Reliability of supply • Indoor air quality • Aesthetics • Interior safety • Anticipated changes in energy resource prices SUMMARY In designing HVAC&R systems, the need to address immediate issues such as economics, performance, and space constraints often prevents designers from fully considering the energy resources affected. Today’s energy resources are less certain because of issues such as availability, safety, national interest, environmental con-cerns, and the world political situation. As a result, the reliability, economics, and continuity of many common energy resources over the potential life of a building being designed are unclear. For this reason, the designer of building energy systems must consider the energy resources on which the long-term operation of the building will depend. If the continued viability of those resources is reason for concern, the design should provide for, account for, or address such an eventuality.
OVERVIEW OF GLOBAL ENERGY RESOURCES WORLD ENERGY RESOURCES Production Energy production trends for the world, by leading producers and world areas, from 1989 to 1998, are shown in Figure 1. World primary energy production, following essentially no increase in the Fig. 1 World Primary Energy Production (Basis: Energy Information Administration/Annual Energy Review 1999) Energy Resources 17.7 early 1990s, has risen subsequently at about 1.7% per year. The largest total energy producers in 1998 were the United States (20%), Russia (11%), and China (9%). Saudi Arabia (5%+) was the world’s fourth largest producer. Together these producers accounted for almost half of the world’s energy. Total world energy production by resource type is shown in Figure 2.
Crude Oil. World crude oil production was 10.7 × 106 m3 per day in 1999—up 18% since 1973. The biggest crude oil producers in 1999 were the eight nations comprising the Organization of Petroleum Exporting Countries (OPEC) at 42% (including Saudi Arabia at 12%+), Russia (9%), and the United States (9%). Oil pro-duction declined most noticeably since 1992 in Russia (20%) and in the United States (17%). The other primary non-OPEC producers were China, Mexico, Norway, the United Kingdom, and Canada.
Natural Gas. World production reached 2.3 × 1012 m3 in 1998— up 15% from the 1989 level. The biggest producers in 1998 were Russia (25%) and the United States (23%).
Coal. At 4.5 × 109 Mg in 1998, coal production was lower by 0.3% than in 1989 and comprised 23% of the world’s energy production. The leading producers of coal were China (27%), the United States (22%), India (7%), Russia (5%+), Germany (5%), and Poland (4%).
Reserves On January 1, 1999, the estimated world reserves of crude oil and gas were distributed by world region as shown in Figures 3 and 4. Saudi Arabia was estimated to have 39% of the Middle Eastern crude oil reserves. Iraq, the United Arab Emirates, Kuwait, and Iran were each estimated to have more crude oil reserves than any world region outside the Middle East. Outside the Middle East, the biggest reserves were estimated to be in Venezuela, Rus-sia, and Mexico. The country with the largest gas reserves by far was Russia.
World coal reserves as of January 1, 1996, are shown by world area in Figure 5 (note that the U.S. data are as of January 1, 1999).
The countries with the most plentiful reserves, as a percent of total, were Russia (16%), the United States (25%), China (12%), and Aus-tralia (9%).
Fig. 2 World Primary Energy Production by Resource: 1998 Fig. 3 World Crude Oil Reserves: January 1, 1999 (Basis: Oil and Gas Journal) Fig. 4 World Natural Gas Reserves: January 1, 1999 (Basis: Oil and Gas Journal) Fig. 5 World Recoverable Coal Reserves: December 31, 1996 and January 1, 1999 (U.S. Only) (Basis: Energy Information Administration) 17.8 2001 ASHRAE Fundamentals Handbook (SI) Consumption Data on world energy consumption are available only by type of resource rather than by total energy consumed.
Petroleum. The consumption trends of the leading consumers from 1960 to 1998 are depicted in Figure 6. In 1998, the United States consumed far more than any other country—26% of world consumption and 40% of the consumption of Organization for Economic Cooperation and Development (OECD) countries. By contrast, Japan, the second largest consumer among OECD coun-tries, consumed just 7.5% of the world total and 12% of that of the OECD countries. Of the non-OECD countries, Russia and China were the biggest consumers (3% and 6%, respectively, of world consumption).
Natural Gas. Transport pipelines notwithstanding, this energy resource tends to be consumed close to the site of production; and indeed, in 1998 the two biggest natural gas producers were also the two biggest consumers. Figure 7 depicts natural gas consumption by the leading consumer countries as a percentage of world con-sumption. Of the major consumers, the United States consumed more than it produced (114%), and the former U.S.S.R. consumed less (67%). Germany and the United Kingdom, the third and fourth largest consumers, produced very little. Canadian consumption was 49% of its production. World consumption of natural gas increased 55% between 1980 and 1998, with the three major gas-consuming states of the former U.S.S.R. (Russia, Ukraine, and Uzbekistan) up 35% and the United States up 7%. After the United States and Russia, no single country consumed more than 5% of the world total.
Coal. Here, the three largest producers were also the three larg-est consumers. Figure 8 depicts the percentage of world consump-tion by the leading consumers during 1998. Since 1980, world coal consumption has increased 21%, with most of that increase in the early 1980s. Over the same period, consumption by China in-creased 93%, the United States 48%, India 185%, and Japan 43%.
Significant drops occurred in Germany, Poland, and Russia.
Electricity. Figure 9 shows the world’s electricity generation by energy resource in 1998. Figure 10 shows installed capacity for the same resources at the beginning of 1998. Both net generation and installed capacity were dominated by the United States (26%+ and 25%, respectively). Comparable figures for the next largest are China (8% for both), Japan (7% and 7+%), and Russia (6% and 7%).
China, which has grown almost four times in net electric generating capacity since 1980, has surpassed Japan in the last few years.
Electricity generated by hydroelectric means increased in the world by 47% between 1980 and 1998, with the largest increase occurring in Central and South America. The top countries for hydroelectric generation in 1998 were the United States, Canada, and Brazil—collectively accounting for 37% of the world total quantity of electricity generated by that means.
Total world electricity generation from nuclear resources increased 239% between 1980 and 1998, with higher-than-average increases occurring in the Far East/Oceania, Western Europe, and Africa. The top-generating countries in 1998 were the United States (27% of world total), China (8%), and Japan (7%).
Fig. 6 World Petroleum Consumption: 1960-1998 (Basis: Energy Information Administration/Annual Energy Review 1999) Fig. 7 World Natural Gas Consumption: 1998 (Basis: Energy Information Administration) Fig. 8 World Coal Consumption: 1998 (Basis: Energy Information Administration) Energy Resources 17.9 Per Capita. Figure 11 compares the per capita commercial energy consumption of selected developed countries for 1998. As is apparent, the per capita energy consumption in cold-climate coun-tries tends to be highest; also, the level in more developed countries is vastly different from that in less developed countries and differs considerably even among the more developed countries.
UNITED STATES ENERGY USE Per Capita Energy Consumption Figure 12, which is based on EIA (1999), presents a capsule overview of past U.S. energy use intensity by relating the growth in per capita energy use since 1949 to population growth. The growth in per capita energy use has varied significantly from the population growth rate. The 1960s experienced a sharp increase in the per capita energy use growth rate, which leveled off during the 1970s due to higher energy prices and the emphasis on energy con-servation. [No change in the intensity of energy use would result in a flat (horizontal) energy-per-capita curve]. In the early 1980s, however, a significant drop in the per capita energy use growth rate occurred as industrial output decreased, efficiency of use improved, and global economic pressures mounted. The period since the mid-1980s shows a resumption of per capita energy use growth at a rate at first paralleling that of the population.
The Annual Energy Outlook is the basic source of data for pro-jecting the use of energy in the United States (EIA 2000). Figure 13 and Figure 14 are summaries of data from this source.
EIA (2000) presents forecasts of energy trends that are based on macroeconomic growth scenarios prepared by several outside sources. EIA has also added a low and a high world oil price sce-nario to its mid-growth economic scenario. Thus, the National Energy Modeling System (NEMS), which EIA uses for these fore-casts, has yielded three scenarios. Figure 13 and Figure 14 present the baseline or reference case. It assumes average annual growth of the real gross domestic product (GDP) at 2.2%, of the labor force at 0.9%, and of productivity at 1.3%. The forecast, in order to be Fig. 9 World Electricity Generation by Resource: 1998 Fig. 10 World Installed Electricity Generation Capacity by Resource: January 1, 1998 Fig. 11 Per Capita Energy Consumption by Selected Developed Countries in 1998 Fig. 12 Per Capita End-Use Energy Consumption Trends in the United States 17.10 2001 ASHRAE Fundamentals Handbook (SI) policy-neutral, also assumes that all federal, state, and local laws and regulations in effect as of July 1, 1999, remain unchanged through 2020.
Projected Overall Energy Consumption Figure 13 shows energy use by end-use sector, with the major end-use sectors being residential, commercial, industrial, and trans-portation. HVAC&R engineers are primarily concerned with the first three sectors. Figure 14 shows energy consumption by type of resource. Figure 13 shows less total energy consumption than Figure 14 primarily because it excludes the thermodynamic losses of elec-tricity generation and the processing and delivery burdens of vari-ous energy forms.
The following observations apply to the overall picture of pro-jected energy use in the United States over the next two decades (Figures 13 and 14): • Although a major issue in energy markets is carbon emissions, impacts of programs to reduce them are not reflected in these forecasts because no specific policies for carbon reductions have been enacted.
• Given the above premise, carbon emissions from energy use will increase by an average of 1.3% per year through 2020 due to ris-ing energy demand, declining nuclear power generation, improvements in efficiency, and slow growth in the use of renew-able sources. The 2020 level of carbon emissions will thus be almost 33% higher than the 1998 levels.
• Though average crude oil prices are expected to remain low over the next several years, unanticipated changes in supply or politi-cal events could change that scenario.
• The wellhead price of natural gas is expected to increase at an average annual rate of 1.7% up to 2020, reflecting exploration and production improvements balanced by greater demand.
• The price of coal, on the other hand, is expected to decline by 28% over the same period as a result of better productivity, more low-cost western coal production, and competitive labor pressures.
• Ongoing changes in the financial structure of the electricity indus-try and resulting cost reductions due to increased competition are reflected. Transitions to retail competitive pricing have been assumed in areas where legislation has been enacted, and stranded cost recovery will be phased out by 2008.
• Electricity prices will decline an average of 19% per megajoule between 1997 and 2020 due to lower operating, maintenance, and administrative costs resulting from industry restructuring.
• More than half of the nuclear electricity generating capacity exist-ing in 1998 will be retired by 2020 as operating licenses expire with no new plants having been built.
• Offsetting this decline, electricity generation from both coal and natural gas is projected to increase significantly, with the share of natural gas in 2020 more than doubling the 1998 level.
• Electricity generation using renewable sources (which includes cogenerators) will increase by only 0.5% per year, held back in large measure by the fact that electric industry restructuring tends to favor natural gas over renewables or coal.
• Petroleum consumption will grow by 1.3% annually, led by the transportation sector, which is where most of it (70%) is used.
• The share of petroleum consumption met by net imports will reach 64% in 2020. (This is up from 52% in 1998.) Over this period, U.S. crude oil production will decline at an annual rate of 0.8%, with declining resources overshadowing improvements in exploration and processing technologies.
• Natural gas consumption will increase by 1.8% per year, with demand increases in all sectors and led by its use for electricity generation.
• Coal consumption will increase at an average annual rate of 0.9%.
Most of it (90%) will be used for electricity generation, but its share for that purpose will decline by 2020.
• Consumption of renewable energy (including using ethanol in gasoline) will increase by only 0.8% per year, however, with about 60% going toward electricity generation.
• Electricity demand will grow by 1.4% annually, with efficiency gains offset by growth in the use of new electricity-using equipment.
• Total energy demand in the commercial and residential sectors will grow at 0.9% and 0.8% per year, respectively. This results from greater use of computers, telecommunications, and other office appliances, but it is offset by somewhat improved building and building equipment efficiencies.
• Energy use by the transportation sector will grow at an average of 1.7% per year, with variations from this average very dependent on prevailing fuel prices.
• Per capita energy use will remain stable through 2020 as overall efficiency gains offset higher per capita demand.
Fig. 13 Projected Total U.S. Energy Consumption by End-Use Sector Fig. 14 Projected Total U.S. Energy Consumption by Resource Energy Resources 17.11 • Total energy use per dollar of gross domestic product (energy intensity), however, will continue to fall at an average rate of about 1.1% per year through 2020.
Outlook Summary In general, the following key issues will dominate energy matters in the next two decades: • Rising dependency of the United States on imported oil • Expected retirement of more than half of the existing nuclear power plants • Role of technology developments, including energy conservation and energy efficiency as alternatives to energy production • Deregulation of the utility industry and the growth of independent power producers • Dealing with the increasing rate of carbon emissions, especially in light of international agreements with, and moral pressure from, others of the global community • Growth of population, coupled with the shift of “baby boomers” into retirement ENERGY RESOURCE PLANNING The energy supplier (or suppliers) in a particular jurisdiction must plan for the future energy needs of that jurisdiction. For com-petitive energy markets where these decisions do not have high societal costs, these plans are made by the energy suppliers as part of their business. These decisions are not revealed to governmental authorities or the public more than is absolutely necessary, because significant competitive advantages could be gained by knowledge of a competitor’s future plans. For electricity (and to a lesser extent natural gas) significant societal issues are involved in energy resource planning decisions, and these decisions cannot be made by energy suppliers without approval by many different groups and entities. These issues include: • Reliability, which is affected by the diversity of supply sources available. This would include the number of geographic supply sources and pipelines in the case of gas and would include the per-centage of generation from various fuel sources in the case of electricity. Due consideration should be given to the projected future supply and reliability of energy resources, including the possibility of supply disruption due to natural or political events, and to the likelihood of future supply shortages, which could reduce reliability.
• Reserve margins, or the ratio of total supply sources to expected peak supply source needs. Reserve levels that are too high result in the waste of resources, higher environmental costs, and possi-bly poor financial health of the energy suppliers. Reserves that are too low result in volatile and very high peak energy prices and reduced reliability.
• The production and transmission of energy often requires govern-mental cooperation to condemn private property for use in energy production and transmission facilities. The construction and maintenance of this property are also regulated due to wetlands, prevention of toxic waste releases, and other environmental issues.
Integrated Resource Planning Integrated resource planning (IRP) is a process commonly used for the planning of significant new energy facilities. The steps in this planning process include: (1) forecasting the amount of new resources needed and (2) determining the type of resource to be provided and who will provide this resource. Traditionally, the local utility provider forecasts the future needs of a given energy resource, then either builds the necessary facility with the approval of regulators or uses a standard offer bid to determine what non-utility provider (or the utility itself) would provide the new energy resource. However, with increasing deregulation of energy providers, a single energy provider may no longer be responsible for maintaining an adequate supply. This forecasting task may be provided by a regulatory body, or it could be by a consensus decision of the various competing energy suppliers, with oversight and approval of regulators.
Supplying new energy resources through either a standard bid process by a supplier or traditional utility regulation will usually result in the selection of the lowest cost supply option, without regard for environmental costs or other societal needs. IRP allows for the choice of a greater variety of resource options and allows environmental costs and other indirect costs to society to be given greater consideration.
IRP addresses a wider population of stakeholders then do most other planning processes. Many regulatory agencies involve the public in the formulation and review of integrated resource plans.
Customers, environmentalists, and other public interest groups are often prominent in these proceedings.
Prior to the deregulation of energy suppliers, demand-side man-agement (DSM) was a common option for providing new energy resources, especially for electricity. These are actions taken to reduce the demand for energy, rather than increase the supply of energy. DSM is desirable because the environmental costs of DSM are almost always lower than the environmental costs of building new energy facilities. However, several factors have caused a decline in the number of DSM programs: • Lower cost supply options, especially for natural gas-fired gener-ation, make it more difficult to implement DSM options that do not result in higher total costs.
• Building and equipment codes and standards are a highly efficient form of DSM, providing reduced energy use with much lower administrative costs than programs that reward the installation of more efficient equipment at a single site. However, they are more subtle than traditional DSM programs and may not always be rec-ognized as a form of DSM.
• The opening of markets to competing suppliers makes it more dif-ficult to administer and implement DSM programs. However, they are still possible if the regulators wish to continue them, and set appropriate rules and regulations for the energy market to allow the implementation of DSM programs.
Many participants in IRP processes may be interested in only one aspect of the process. For example, the energy industry’s main interest may be cost minimization, while environmentalists may want to minimize pollutant emissions and prevent environmental damage from the construction of energy facilities. Participation by all affected interest groups helps provide the best overall solution for society, including the indirect costs and benefits from these energy resource decisions.
TRADABLE EMISSION CREDITS Increasingly, quotas and limits apply to the emissions of various pollutants. These currently include sulphur dioxide (SO2) and nitrogen oxides (NOx) but may someday include carbon dioxide (CO2). Often a market-based system of tradable credits is used with these quotas. A company is given the right to produce a given level of emissions and it earns a credit, which can be sold to others, if it produces fewer emissions than that level. If one company can reduce its emissions at a lower cost than another, it can do so and sell the emissions credit to the second company and earn a profit from its pollution control efforts. To date, this type of activity has largely involved large industrial plants, but it can also involve commercial buildings with on-site emissions, such as generation equipment or gas engine-driven cooling.
17.12 2001 ASHRAE Fundamentals Handbook (SI) Designers must be aware of any regulations concerning pollutant emissions; failure to comply with these regulations may result in civil or criminal penalties for designers or their clients. However, designers should understand the options available under these reg-ulations. The purchase or sale of emissions credits may allow reduced construction or building operations costs if the equipment can “overcomply” at a lower cost than the cost of another source of emissions to comply, or vice versa. In some cases, documentation of energy savings beyond what codes and regulations require can result in receiving emissions credits that may be sold later.
AGENCIES AND ASSOCIATIONS IN THE UNITED STATES American Gas Association (AGA), Arlington, VA American Petroleum Institute (API), Washington, D.C.
Bureau of Mines, Department of Interior, Washington, D.C.
Council on Environmental Quality (CEQ), Washington, D.C.
Edison Electric Institute (EEI), Washington, D.C.
Electric Power Research Institute (EPRI), Palo Alto, CA Energy Information Administration (EIA), Washington, D.C.
Gas Research Institute (GRI), Chicago, IL National Coal Association (NCA), Washington, D.C.
North American Electric Reliability Council (NAERC), Princeton, NJ Organization of Petroleum Exporting Countries (OPEC), Vienna, Austria United States Green Building Council (USGBC), San Francisco, CA REFERENCES ASHRAE. 1990. Energy position statement.
Conover, D.R. 1984. Accounting for energy resource use in building regula-tions. ASHRAE Transactions 90(1B):547-63.
EIA. 1999. Annual energy review. DOE/EIA—0384(99). Energy Information Administration, Office of Energy Markets and End Use, U.S. Department of Energy, Washington, D.C.
EIA. 2000. Annual energy outlook 2000. DOE/EIA—0383(2000). Energy Information Administration, U.S. Department of Energy, Washington, D.C.
Gleeson, G.W. 1951. Energy—Choose it wisely today for safety tomorrow.
ASHVE Transactions 57:523-40.
Gottfried, D.A. 1996. Sustainable building technical manual. U.S. Green Building Council, San Francisco, CA.
Grumman, D.L. 1984. Energy resource accounting: ASHRAE Standard 90C-1977R. ASHRAE Transactions 90(1B):531-46.
Heiman, J.L. 1984. Proposal for a simple method for determining resource impact factors. ASHRAE Transactions 90(1B):564-70.
IESNA. 2000. The IESNA lighting handbook. IESNA, New York.
BIBLIOGRAPHY DOE. 1979. Impact assessment of a mandatory source-energy approach to energy conservation in new construction. U.S. Department of Energy, Washington, D.C.
Pacific Northwest Laboratory. 1987. Development of whole-building energy design targets for commercial buildings phase 1 planning. PNL-5854, Vol. 2. U.S. Department of Energy, Washington, D.C.
USGBC. 1999. LEED™reference guide. U.S. Green Building Council, San Francisco.
18.1 CHAPTER 18 COMBUSTION AND FUELS Principles of Combustion ........................................................ 18.1 Fuel Classification .................................................................. 18.3 Gaseous Fuels ......................................................................... 18.4 Liquid Fuels ............................................................................ 18.4 Solid Fuels ............................................................................... 18.6 Combustion Calculations ........................................................ 18.8 Efficiency Calculations ......................................................... 18.12 Combustion Considerations .................................................. 18.13 PRINCIPLES OF COMBUSTION OMBUSTION is a chemical reaction in which an oxidant Creacts rapidly with a fuel to liberate stored energy as thermal energy, generally in the form of high-temperature gases. Small amounts of electromagnetic energy (light), electric energy (free ions and electrons), and mechanical energy (noise) are also produced during combustion. Except in special applications, the oxidant for combustion is oxygen in the air.
Conventional hydrocarbon fuels contain primarily hydrogen and carbon, in elemental form or in various compounds. Their complete combustion produces mainly carbon dioxide (CO2) and water (H2O); however, small quantities of carbon monoxide (CO) and partially reacted flue gas constituents (gases and liquid or solid aerosols) may form. Most conventional fuels also contain small amounts of sulfur, which is oxidized to sulfur dioxide (SO2) or sul-fur trioxide (SO3) during combustion, and noncombustible sub-stances such as mineral matter (ash), water, and inert gases. Flue gas is the product of complete or incomplete combustion and includes excess air (if present), but not dilution air.
Fuel combustion rate depends on (1) the rate of the chemical reaction of the combustible fuel constituents with oxygen, (2) the rate at which oxygen is supplied to the fuel (the mixing of air and fuel), and (3) the temperature in the combustion region. The reac-tion rate is fixed by fuel selection. Increasing the mixing rate or tem-perature increases the combustion rate.
With complete combustion of hydrocarbon fuels, all hydrogen and carbon in the fuel are oxidized to H2O and CO2. Generally, for complete combustion, excess oxygen or excess air must be supplied beyond the amount theoretically required to oxidize the fuel. Excess air is usually expressed as a percentage of the air required to com-pletely oxidize the fuel.
In stoichiometric combustion of a hydrocarbon fuel, fuel is reacted with the exact amount of oxygen required to oxidize all car-bon, hydrogen, and sulfur in the fuel to CO2, H2O, and SO2. There-fore, exhaust gas from stoichiometric combustion theoretically contains no incompletely oxidized fuel constituents and no unre-acted oxygen (i.e., no carbon monoxide and no excess air or oxy-gen). The percentage of CO2 contained in products of stoichiometric combustion is the maximum attainable and is referred to as the stoichiometric CO2, ultimate CO2, or maximum theoretical percentage of CO2.
Stoichiometric combustion is seldom realized in practice because of imperfect mixing and finite reaction rates. For economy and safety, most combustion equipment should operate with some excess air. This ensures that fuel is not wasted and that combustion is complete despite variations in fuel properties and in the supply rates of fuel and air. The amount of excess air to be supplied to any combustion equipment depends on (1) expected variations in fuel properties and in fuel and air supply rates, (2) equipment applica-tion, (3) degree of operator supervision required or available, and (4) control requirements. For maximum efficiency, combustion at low excess air is desirable.
Incomplete combustion occurs when a fuel element is not com-pletely oxidized during combustion. For example, a hydrocarbon may not completely oxidize to carbon dioxide and water, but may form partially oxidized compounds, such as carbon monoxide, alde-hydes, and ketones. Conditions that promote incomplete combus-tion include (1) insufficient air and fuel mixing (causing local fuel-rich and fuel-lean zones), (2) insufficient air supply to the flame (providing less than the required quantity of oxygen), (3) insuffi-cient reactant residence time in the flame (preventing completion of combustion reactions), (4) flame impingement on a cold surface (quenching combustion reactions), or (5) flame temperature that is too low (slowing combustion reactions). Incomplete combustion uses fuel inefficiently, can be hazardous because of carbon monoxide production, and contributes to air pollution.
Combustion Reactions The reaction of oxygen with the combustible elements and com-pounds in fuels occurs according to fixed chemical principles, including • Chemical reaction equations • Law of matter conservation: the mass of each element in the reaction products must equal the mass of that element in the reactants • Law of combining masses: chemical compounds are formed by elements combining in fixed mass relationships • Chemical reaction rates Oxygen for combustion is normally obtained from air, which is a physical mixture of nitrogen, oxygen, small amounts of water vapor, carbon dioxide, and inert gases. For practical combustion calculations, dry air consists of 20.95% oxygen and 79.05% inert gases (nitrogen, argon, and so forth) by volume, or 23.15% oxygen and 76.85% inert gases by mass. For calculation purposes, nitrogen is assumed to pass through the combustion process unchanged (although small quantities of nitrogen oxides form). Table 1 lists oxygen and air requirements for stoichiometric combustion and the products of stoichiometric combustion of some pure combustible materials (or constituents) found in common fuels. Flammability Limits Fuel burns in a self-sustained reaction only when the volume percentages of fuel and air in a mixture at standard temperature and pressure are within the upper and lower flammability limits or explosive limits (UEL and LEL). See Table 2. Both temperature and pressure affect these limits. As the temperature of the mixture increases, the upper limit increases and the lower limit decreases.
As the pressure of the mixture decreases below atmospheric pres-sure, the upper limit decreases and the lower limit increases.
The preparation of this chapter is assigned to TC 6.10, Fuels and Combus-tion.
18.2 2001 ASHRAE Fundamentals Handbook (SI) However, as pressure increases above atmospheric pressure, the upper limit increases and the lower limit is relatively constant.
Ignition Temperature Ignition temperature is the lowest temperature at which heat is generated by combustion faster than heat is lost to the surroundings and combustion becomes self-propagating. See Table 2. The fuel-air mixture will not burn freely and continuously below the ignition temperature unless heat is supplied, but chemical reaction between the fuel and air may occur. Ignition temperature is affected by a large number of factors. The ignition temperature and flammability limits of a fuel-air mixture, together, are a measure of the potential for ignition (Gas Engineers Handbook 1965).
Combustion Modes Combustion reactions occur in either continuous or pulse flame modes. Continuous combustion burns fuel in a sustained manner as long as fuel and air are continuously fed to the combustion zone and the fuel-air mixture is within the flammability limits. Continu-ous combustion is more common than pulse combustion and is used in most fuel-burning equipment.
Pulse combustion is an acoustically resonant process that burns various fuels in small, discrete fuel-air mixture volumes in a very rapid series of combustions.
The introduction of fuel and air into the pulse combustor is con-trolled by mechanical or aerodynamic valves. Typical combustors consist of one or more valves, a combustion chamber, an exit pipe, and a control system (ignition means, fuel-metering devices, etc.).
Typically, combustors for warm air furnace, hot water boiler, and Table 1 Combustion Reactions of Common Fuel Constituents Constituent Mol-ecular Formula Combustion Reactions Stoichiometric Oxygen and Air Requirements Flue Gas from Stoichiometric Combustion with Air kg/kg Fuela m3/m3 Fuel Ulti-mate CO2, % Dew Point,c °C m3/m3 Fuel kg/kg Fuel O2 Air O2 Air CO2 H2O CO2 H2O Carbon (to CO) C C + 0.5O2 → CO 1.33 5.75 b b — — — — — — Carbon (to CO2) C C + O2 → CO2 2.66 11.51 b b 29.30 — — — 3.664 — Carbon monoxide CO CO + 0.5O2 → CO 0.57 2.47 0.50 2.39 34.70 — 1.0 — 1.571 — Hydrogen H2 H2 + 0.5O2 → H2O 7.94 34.28 0.50 2.39 — 72 — 1.0 — 8.937 Methane CH4 CH4 + 2O2 → CO2 + 2H2O 3.99 17.24 2.00 9.57 11.73 59 1.0 2.0 2.744 2.246 Ethane C2H6 C2H6 + 3.5O2 → 2CO2 + 3H2O 3.72 16.09 3.50 16.75 13.18 57 2.0 3.0 2.927 1.798 Propane C3H8 C3H8 + 5O2 → 3CO2 + 4H2O 3.63 15.68 5.00 23.95 13.75 55 3.0 4.0 2.994 1.634 Butane C4H10 C4H10 + 6.5O2 → 4CO2 + 5H2O 3.58 15.47 6.50 31.14 14.05 54 4.0 5.0 3.029 1.550 Alkanes CnH2n + 2 CnH2n + 2 + (1.5n + 0.5)O2 → nCO2 + (n + 1)H2O — — 1.5n + 0.5 7.18n + 2.39 — 53 n n + 1 Ethylene C2H4 C2H4 + 3O2 → 2CO2 + 2H2O 3.42 14.78 3.00 14.38 15.05 52 2.0 2.0 3.138 1.285 Propylene C3H6 C3H6 + 4.5O2 → 3CO2 + 3H2O 3.42 14.78 4.50 21.53 15.05 52 3.0 3.0 3.138 1.285 Alkenes CnH2n CnH2n + 1.5nO2 → nCO2 + nH2O 3.42 14.78 1.50n 7.18n 15.05 52 n n 3.138 1.285 Acetylene C2H2 C2H2 + 2.5O2 → 2CO2 + H2O 3.07 13.27 2.50 11.96 17.53 39 2.0 1.0 3.834 0.692 Alkynes CnH2m CnH2m + (n + 0.5m)O2 → nCO2 + mH2O — — n + 0.5m 4.78n +2.39m — — n m SOx H2O SOx H2O Sulfur (to SO2) S S + O2 → SO2 1.00 4.31 b b — — 1.0 SO2 — 1.998 (SO2) — Sulfur (to SO3) S S + 1.5O2 → SO3 1.50 6.47 b b — — 1.0 SO3 — 2.497 (SO3) — Hydrogen sulfide H2S H2S + 1.5O2 → SO2 + H2O 1.41 6.08 1.50 7.18 — 52 1.0 SO2 1.0 1.880 (SO2) 0.528 Adapted, in part, from Gas Engineers Handbook (1965) aAtomic masses: H = 1.008, C = 12.01, O = 16.00, S = 32.06.
bVolume ratios are not given for fuels that do not exist in vapor form at reasonable temperatures or pressure.
c Dew point is determined from Figure 2.
Table 2 Flammability Limits and Ignition Temperatures of Common Fuels in Fuel-Air Mixtures Substance Molecular Formula Lower Flammability Limit, % Upper Flammability Limit, % Ignition Temperature, °C References Carbon (activated coke) C — — 660 Hartman (1958) Carbon monoxide CO 12.5 74 609 Scott et al. (1948) Hydrogen H2 4.0 75.0 520 Zabetakis (1956) Methane CH4 5.0 15.0 705 Gas Engineers Handbook (1965) Ethane C2H6 3.0 12.5 520 to 630 Trinks (1947) Propane C3H8 2.1 10.1 466 NFPA (1962) n-Butane C4H10 1.86 8.41 405 NFPA (1962) Ethylene C2H4 2.75 28.6 490 Scott et al. (1948) Propylene C3H6 2.00 11.1 450 Scott et al. (1948) Acetylene C2H2 2.50 81 406 to 440 Trinks (1947) Sulfur S — — 190 Hartman (1958) Hydrogen sulfide H2S 4.3 45.50 292 Scott et al. (1948) Flammability limits adapted from Coward and Jones (1952). All values corrected to 16°C, 104 kPa, dry.
44.01n 14.026n 2.016 + ---------------------------------------18.01 n 1 + ( ) 14.026n 2.016 + ---------------------------------------22.005n 6.005n 1.008m + -----------------------------------------9.008m 6.005n 1.008m + -----------------------------------------Combustion and Fuels 18.3 commercial cooking equipment use mechanical valves. Aerody-namic valves are usually used in higher pressure applications, such as thrust engines. Separate valves for air and fuel, a single valve for premixed air and fuel, or multiple valves of either type can be used.
Premix valve systems may require a flame trap at the combustion chamber entrance to prevent flashback.
In a mechanically valved pulse combustor, air and fuel are forced into the combustion chamber through the valves under pressures less than 3.5 kPa. An ignition source, such as a spark, ignites the fuel-air mixture, causing a positive pressure buildup in the combus-tion chamber. The positive pressure causes the valves to close, leav-ing only the exit pipe of the combustion chamber as a pressure relief opening. The combustion chamber and exit pipe geometry deter-mine the resonant frequency of the combustor. The pressure wave from the initial combustion travels down the exit pipe at sonic velocity. As this wave exits the combustion cham-ber, most of the flue gases present in the chamber are carried with it into the exit pipe. Flue gases remaining in the combustion chamber begin to cool immediately. The contraction of the cooling gases and the momentum of gases in the exit pipe create a vacuum inside the chamber that opens the valves and allows more fuel and air into the chamber. While the fresh charge of fuel-air enters the chamber, the pressure wave reaches the end of the exit pipe and is partially reflected from the open end of the pipe. The fresh fuel-air charge is ignited by residual combustion and/or heat. The resulting combus-tion starts another cycle.
Typical pulse combustors operate at 30 to 100 cycles per second and emit resonant sound, which must be considered in their appli-cation. The pulses produce high convective heat transfer rates.
Heating Value Combustion produces thermal energy or heat. The quantity of heat generated by complete combustion of a unit of specific fuel is constant and is termed the heating value or heat of combustion of that fuel. The heating value of a fuel can be determined by measuring the heat evolved during combustion of a known quantity of the fuel in a calorimeter, or it can be estimated from chemical analysis of the fuel and the heating values of the various chemical elements in the fuel. For information on calculating heating values, see the sections on Characteristics of Fuel Oils and Characteristics of Coals. Higher heating value, gross heating value, or total heating value includes the latent heat of vaporization and is determined when water vapor in the fuel combustion products is condensed.
Conversely, lower heating value or net heating value is obtained when the latent heat of vaporization is not included. When the heat-ing value of a fuel is specified without designating higher or lower, it generally means the higher heating value in the United States.
(Lower heating value is mainly used for internal combustion engine fuels.) Heating values are usually expressed in kilojoules per litre or megajoules per cubic metre for gaseous fuels, megajoules per litre for liquid fuels, and megajoules per kilogram for solid fuels. Heat-ing values are always given in relation to a certain reference tem-perature and pressure, usually 16, 20, or 25°C and 101.325 kPa, depending on the particular industry practice. Heating values of several substances in common fuels are listed in Table 3.
With incomplete combustion, not all fuel is completely oxidized, and the heat produced is less than the heating value of the fuel.
Therefore, the quantity of heat produced per unit of fuel consumed decreases, implying lower combustion efficiency.
Not all heat produced during combustion can be used effec-tively. The greatest heat loss is the thermal energy of the increased temperature of hot exhaust gases above the temperature of incom-ing air and fuel. Other heat losses include radiation and convection heat transfer from the outer walls of combustion equipment to the environment. Altitude Compensation Air at altitudes above sea level is less dense and has less oxygen per unit volume. Therefore, combustion at altitudes above sea level has less available oxygen to burn with the fuel unless compensation is made for the altitude. Combustion occurs, but the amount of excess air is reduced. If excess air is reduced enough by an increase in altitude, combustion is incomplete or ceases.
Altitude compensation is achieved by matching the fuel and air supply rates to attain complete combustion without too much excess air or too much fuel. Fuel and air supply rates can be matched by increasing the air supply rate to the combustion zone or by decreas-ing the fuel supply rate to the combustion zone. The air supply rate can be increased with a combustion air blower, and the fuel supply rate can be reduced by decreasing the fuel input (derating).
Power burners use combustion air blowers and can increase the air supply rate to compensate for altitude. The combustion zone can be pressurized to attain the same air density in the combustion chamber as that attained at sea level.
Derating can be used as an alternative to power combustion. In the United States, the fuel gas codes generally do not require derat-ing of nonpower burners at altitudes up to 600 m. At altitudes above 600 m, burners should be derated 4% for each 300 m above sea level (NFPA/IAS National Fuel Gas Code). Chimney or vent operation also must be considered at high altitudes (see Chapter 30 of the 2000 ASHRAE Handbook—Systems and Equipment).
FUEL CLASSIFICATION Generally, hydrocarbon fuels are classified according to physical state (gas, liquid, or solid). Different types of combustion equip-ment are usually needed to burn fuels in the different physical states.
Gaseous fuels can be burned in premix or diffusion burners. Liquid fuel burners must include a means for atomizing or vaporizing fuel into small droplets or a vapor and must provide adequate mixing of fuel and air. Solid fuel combustion equipment must (1) heat fuel to vaporize sufficient volatiles to initiate and sustain combustion, (2) provide residence time to complete combustion, and (3) provide space for ash containment.
Principal fuel applications include space heating and cooling of residential, commercial, industrial, and institutional buildings; ser-vice water heating; steam generation; and refrigeration. Major fuels for these applications are natural and liquefied petroleum gases, fuel Table 3 Heating Values of Substances Occurring in Common Fuels Substance Molecular Formula Higher Heating Values,a MJ/kg Lower Heating Values,a MJ/kg Density,b kg/m3 Carbon (to CO) C 9.188 9.188 — Carbon (to CO2) C 32.780 32.780 — Carbon monoxide CO 10.111 10.111 1.187 Hydrogen H2 142.107 120.075 0.085 Methane CH4 55.533 49.997 0.679 Ethane C2H6 51.923 47.492 1.28 Propane C3H8 50.402 46.373 1.92 Butane C4H10 49.593 45.771 2.53 Ethylene C2H4 50.325 47.160 — Propylene C3H6 48.958 45.792 1.78 Acetylene C2H2 50.014 48.309 1.120 Sulfur (to SO2) S 9.257 9.257 — Sulfur (to SO3) S 13.816 13.816 — Hydrogen sulfide H2S 16.508 15.205 1.456 Adapted from Gas Engineers Handbook (1965).
aAll values corrected to 16°C, 101.4 kPa, dry. For gases saturated with water vapor at 16°C, deduct 1.74% of the value to adjust for gas volume displaced by water vapor.
bAt 0°C and 101.3 kPa.
18.4 2001 ASHRAE Fundamentals Handbook (SI) oils, diesel and gas turbine fuels (for on-site energy applications), and coal.
Fuels of limited use, such as manufactured gases, kerosene, bri-quettes, wood, and coke, are not discussed here. Fuel choice is based on one or more of the following: Fuel factors • Availability, including dependability of supply • Convenience of use and storage • Economy • Cleanliness Combustion equipment factors • Operating requirements • Cost • Service requirements • Ease of control GASEOUS FUELS Although various gaseous fuels have been used as energy sources in the past, heating and cooling applications are presently limited to natural gas and liquefied petroleum gases.
Types and Properties Natural gas is a nearly odorless and colorless gas that accumu-lates in the upper parts of oil and gas wells. Raw natural gas is a mix-ture of methane (55 to 98%), higher hydrocarbons (primarily ethane), and noncombustible gases. Some constituents, principally water vapor, hydrogen sulfide, helium, and gases for liquefied petroleum gases and gasoline are removed prior to distribution.
Natural gas used as fuel typically contains methane, CH4 (70 to 96%); ethane, C2H6 (1 to 14%); propane, C3H8 (0 to 4%); butane, C4H10 (0 to 2%); pentane, C5H12 (0 to 0.5%); hexane, C6H14 (0 to 2%); carbon dioxide, CO2 (0 to 2%); oxygen, O2 (0 to 1.2%); and nitrogen, N2 (0.4 to 17%).
The composition of natural gas depends on its geographical source. Because the gas is drawn from various sources, the compo-sition of gas distributed in a given location can vary slightly, but a fairly constant heating value is usually maintained for control and safety. Local gas utilities are the best sources of current gas compo-sition data for a particular area.
Heating values of natural gases vary from 34 to 45 MJ/m3; the usual range is 37.3 to 39.1 MJ/m3 at sea level. The heating value for a particular gas can be calculated from the composition data and val-ues in Table 3.
For safety purposes, odorants (such as mercaptans) are added to natural gas and LPG to give them noticeable odors.
Liquefied petroleum gases (LPG) consist primarily of propane and butane, and are usually obtained as a byproduct of oil refinery operations or by stripping natural gas. Propane and butane are gas-eous under usual atmospheric conditions, but can be liquefied under moderate pressures at normal temperatures.
Commercial propane consists primarily of propane but gener-ally contains about 5 to 10% propylene. It has a heating value of about 50.15 MJ/kg, about 93 MJ/m3of gas, or about 25.4 GJ/m3 of liquid propane. At atmospheric pressure, commercial propane has a boiling point of about −40°C. The low boiling point of propane allows it to be used during winter in the northern United States and in Canada. Tank heaters and vaporizers permit its use in colder cli-mates and where high fuel flow rates are required. Propane is avail-able in cylinders, bottles, tank trucks, or tank cars.
Propane-air mixtures are used in place of natural gas in small communities and by natural gas companies to supplement normal supplies at peak loads. Table 4 lists heating values and densities for various fuel-air ratios. Commercial butane consists primarily of butane but may con-tain up to 5% butylene. It has a heating value of about 49.3 MJ/kg, about 120 MJ/m3 of gas, or about 28.4 GJ/m3 of liquid butane. At atmospheric pressure, commercial butane has a relatively high boil-ing point of about 0°C. Therefore, butane cannot be used in cold weather unless the gas temperature is maintained above 0°C or the partial pressure is decreased by dilution with a gas having a lower boiling point. Butane is usually available in bottles, tank trucks, or tank cars, but not in cylinders.
Butane-air mixtures are used in place of natural gas in small communities and by natural gas companies to supplement normal supplies at peak loads. Table 4 lists heating values and densities for various fuel-air ratios.
Commercial propane-butane mixtures with various ratios of propane and butane are available. Their properties generally fall between those of the unmixed fuels.
Manufactured gases are combustible gases produced from coal, coke, oil, liquefied petroleum gases, or natural gas. For more detailed information, see Gas Engineers Handbook (1965). These fuels are used primarily for industrial in-plant operations or as spe-cialty fuels (e.g., acetylene for welding).
LIQUID FUELS Significant liquid fuels include various fuel oils for firing com-bustion equipment and engine fuels for on-site energy systems. Liq-uid fuels, with few exceptions, are mixtures of hydrocarbons derived by refining crude petroleum. In addition to hydrocarbons, crude petroleum usually contains small quantities of sulfur, oxygen, nitrogen, vanadium, other trace metals, and impurities such as water and sediment. Refining produces a variety of fuels and other prod-ucts. Nearly all lighter hydrocarbons are refined into fuels (e.g., liq-uefied petroleum gases, gasoline, kerosene, jet fuels, diesel fuels, and light heating oils). Heavy hydrocarbons are refined into residual fuel oils and other products (e.g., lubricating oils, waxes, petroleum coke, and asphalt).
Crude petroleums from different oil fields vary in hydrocarbon molecular structure. Crude is paraffin-base (principally chain-struc-tured paraffin hydrocarbons), naphthene- or asphaltic-base (con-taining relatively large quantities of saturated ring-structural naphthenes), aromatic-base (containing relatively large quantities of unsaturated, ring-structural aromatics), or mixed- or intermedi-ate-base (between paraffin- and naphthene-base crudes). Except for heavy fuel oils, the crude type has little significant effect on result-ant products and combustion applications.
Table 4 Propane-Air and Butane-Air Gas Mixtures Heating Value, MJ/m3 Propane-Aira Butane-Airb % Gas % Air Density, kg/m3 % Gas % Air Density, kg/m3 18 19.16 80.84 1.41 14.81 85.19 1.48 22 23.41 76.59 1.44 18.11 81.89 1.52 26 27.67 72.33 1.46 21.40 78.60 1.56 30 31.93 68.07 1.49 24.69 75.31 1.60 34 36.18 63.82 1.52 27.98 72.02 1.64 38 40.44 59.56 1.54 31.27 68.73 1.68 42 44.70 55.30 1.57 34.57 65.43 1.72 46 48.95 51.05 1.60 37.86 62.14 1.76 50 53.21 46.79 1.63 41.15 58.85 1.80 54 57.47 42.53 1.65 44.44 55.56 1.84 58 61.72 38.28 1.68 47.74 52.26 1.88 62 65.98 34.02 1.71 51.03 48.97 1.92 66 70.24 29.76 1.73 54.32 45.68 1.96 Adapted from Gas Engineers Handbook (1965).
Air density at 0°C and 101.325 kPa is 1.292 kg/m3.
aValues used for calculation: 93.97 MJ/m3; propane = 1.92 kg/m3.
bValues used for calculation: 121.5 MJ/m3; butane = 2.53 kg/m3.
Combustion and Fuels 18.5 Types of Fuel Oils Fuel oils for heating are broadly classified as distillate fuel oils (lighter oils) or residual fuel oils (heavier oils). ASTM Standard D 396 has specifications for fuel oil properties that subdivide the oils into various grades. Grades No. 1 and 2 are distillate fuel oils.
Grades 4, 5 (Light), 5 (Heavy), and 6 are residual fuel oils. Specifi-cations for the grades are based on required characteristics of fuel oils for use in different types of burners.
Grade No. 1 is a light distillate intended for vaporizing-type burners. High volatility is essential to continued evaporation of the fuel oil with minimum residue.
Grade No. 2 is a heavier distillate than No. 1. It is used primarily with pressure-atomizing (gun) burners that spray the oil into a com-bustion chamber. The atomized oil vapor mixes with air and burns.
This grade is used in most domestic burners and many medium-capacity commercial-industrial burners. A dewaxed No. 2 oil with a pour point of −50°C is supplied only to areas where regular No. 2 oil would jell.
Grade No. 4 is an intermediate fuel that is considered either a heavy distillate or a light residual. Intended for burners that atomize oils of higher viscosity than domestic burners can handle, its per-missible viscosity range allows it to be pumped and atomized at rel-atively low storage temperatures.
Grade No. 5 (Light) is a residual fuel of intermediate viscosity for burners that handle fuel more viscous than No. 4 without pre-heating. Preheating may be necessary in some equipment for burn-ing and, in colder climates, for handling.
Grade No. 5 (Heavy) is a residual fuel more viscous than No. 5 (Light), but intended for similar purposes. Preheating is usually nec-essary for burning and, in colder climates, for handling.
Grade No. 6, sometimes referred to as Bunker C, is a high-vis-cosity oil used mostly in commercial and industrial heating. It requires preheating in the storage tank to permit pumping, and addi-tional preheating at the burner to permit atomizing.
Low-sulfur residual oils are marketed in many areas to permit users to meet sulfur dioxide emission regulations. These fuel oils are produced (1) by refinery processes that remove sulfur from the oil (hydrodesulfurization), (2) by blending high-sulfur residual oils with low-sulfur distillate oils, or (3) by a combination of these methods. These oils have significantly different characteristics from regular residual oils. For example, the viscosity-temperature rela-tionship can be such that low-sulfur fuel oils have viscosities of No.
6 fuel oils when cold, and of No. 4 when heated. Therefore, normal guidelines for fuel handling and burning can be altered when using these fuels.
Fuel oil grade selection for a particular application is usually based on availability and economic factors, including fuel cost, clean air requirements, preheating and handling costs, and equip-ment cost. Installations with low firing rates and low annual fuel consumption cannot justify the cost of preheating and other methods that use residual fuel oils. Large installations with high annual fuel consumption cannot justify the premium cost of distillate fuel oils.
Characteristics of Fuel Oils Characteristics that determine grade classification and suit-ability for given applications are (1) viscosity, (2) flash point, (3) pour point, (4) water and sediment content, (5) carbon residue, (6) ash, (7) distillation qualities or distillation temperature ranges, (8) density, (9) sulfur content, (10) heating value, and (11) carbon-hydrogen content. Not all of these are included in ASTM Standard D 396.
Viscosity is an oil’s resistance to flow. It is significant because it indicates the ease at which oil flows or can be pumped and the ease of atomization. Differences in fuel oil viscosities are caused by vari-ations in the concentrations of fuel oil constituents and different refining methods. Approximate viscosities of fuel oils are shown in Figure 1.
Flash point is the lowest temperature to which an oil must be heated for its vapors to ignite in a flame. Minimum permissible flash point is usually prescribed by state and municipal laws.
Pour point is the lowest temperature at which a fuel can be stored and handled. Fuels with higher pour points can be used when heated storage and piping facilities are provided.
Water and sediment content should be low to prevent fouling the facilities. Sediment accumulates on filter screens and burner parts. Water in distillate fuels can cause tanks to corrode and emul-sions to form in residual oil.
Carbon residue is obtained by a test in which the oil sample is destructively distilled in the absence of air. When commercial fuels are used in proper burners, this residue has almost no relationship to soot deposits, except indirectly when deposits are formed by vapor-izing burners.
Ash is the noncombustible material in an oil. An excessive amount indicates the presence of materials that cause high wear on burner pumps.
The distillation test shows the volatility and ease of vaporization of a fuel.
Relative density is the ratio of the density of a fuel oil to the den-sity of water at a specific temperature. Relative densities cover a range in each grade, with some overlap between distillate and resid-ual grades.
Air pollution considerations are important in determining the allowable sulfur content of fuel oils. Sulfur content is frequently limited by legislation aimed at reducing sulfur oxide emissions from combustion equipment. These laws require sulfur content to be below a certain level, usually 1.0, 0.5, or 0.3%. Table 5 lists sulfur levels of some marketed fuel oils. Sulfur in fuel oils is also undesirable because of the corrosive-ness of sulfur compounds in the flue gas. Although low-temperature corrosion can be minimized by maintaining the stack at tempera-tures above the dew point of the flue gas, this limits the overall ther-mal efficiency of combustion equipment.
For certain industrial applications, the sulfur content of a fuel must be limited because of adverse effects on product quality. The Fig. 1 Approximate Viscosity of Fuel Oils 18.6 2001 ASHRAE Fundamentals Handbook (SI) applications include direct-fired metallurgy where work is per-formed in the combustion zone.
Heating value is an important property, although ASTM Stan-dard D 396 does not list it as one of the criteria for fuel oil classifi-cation. Heating value can generally be correlated with the API gravity. Table 6 shows the relationship between heating value, API gravity, and density for several oil grades. In the absence of more specific data, heating values can be calculated as derived from the North American Combustion Handbook (1978): (1) where ρ is oil density in kilograms per cubic metre.
Distillate fuel oils (Grades 1 and 2) have a carbon-hydrogen con-tent of 84 to 86% carbon, with the remainder predominantly hydro-gen. The heavier residual fuel oils (Grades 4, 5, and 6) may contain up to 88% carbon and as little as 11% hydrogen. An approximate relationship for determining the hydrogen content of fuel oils is (2) ASTM Standard D 396 is more a classification than a specifica-tion, distinguishing between six generally nonoverlapping grades, one of which characterizes any commercial fuel oil. Quality is not defined, as a refiner might control it; for example, the standard lists the distillation temperature 90% point for Grade No. 2 as having a maximum of 338°C, whereas commercial practice rarely exceeds 315°C. Types and Properties of Liquid Fuels for Engines The primary stationary engine fuels are diesel and gas turbine oils, natural gases, and liquefied petroleum gases. Other fuels include sewage gas, manufactured gas, and gas mixtures. Gasoline and the JP series of gas turbine fuels are rarely used for stationary engines.
Only properties of diesel and gas turbine fuel oils are covered here; properties of natural and liquefied petroleum gases are found in the section on Gaseous Fuels. For properties of gasolines and JP turbine fuel, consult texts on internal combustion engines and gas turbines. Properties of currently marketed gasolines can be found in the latest volumes of Mineral Industry Surveys, Motor Gaso-lines, issued semiannually by the U.S. Bureau of Mines.
Properties of the three grades of diesel fuel oils (1-D, 2-D, and 4-D) are listed in ASTM Standard D 975.
Grade No. 1-D includes the class of volatile fuel oils from kerosene to intermediate distillates. These fuels are used in high-speed engines with frequent and relatively wide variations in loads and speeds and where abnormally low fuel temperatures are encountered.
Grade No. 2-D includes the class of lower volatility distillate gas oils. These fuels are used in high-speed engines with relatively high loads and uniform speeds, or in engines not requiring fuels with the higher volatility or other properties specified for Grade No. 1-D.
Grade No. 4-D covers the class of more viscous distillates and blends of these distillates with residual fuel oils. These fuels are used in low- and medium-speed engines involving sustained loads at essentially constant speed.
Property specifications and test methods for Grade No. 1-D, 2-D, and 4-D diesel fuel oils are essentially identical to specifications of Grade No. 1, 2, and 4 fuel oils, respectively. However, diesel fuel oils have an additional specification for cetane number, which measures ignition quality and influences combustion roughness.
Cetane number requirements depend on engine design, size, speed and load variations, and starting and atmospheric conditions. An increase in cetane number over values actually required does not improve engine performance. Thus, the cetane number should be as low as possible to assure maximum fuel availability. ASTM Stan-dard D 975 provides several methods for estimating cetane number from other fuel oil properties.
ASTM Standard D 2880 for gas turbine fuel oils relates gas tur-bine fuel oil grades to fuel and diesel fuel oil grades. Test methods for determining properties of gas turbine fuel oils are essentially identical to those for fuel oils. However, gas turbine specifications contain quantity limits on some trace elements that may be present.
These limits are intended to prevent excessive corrosion in gas tur-bine engines. For a detailed discussion of fuels for gas turbines and combustion in gas turbines, see Chapters 5 and 9, respectively, in Hazard (1971).
SOLID FUELS Solid fuels include coal, coke, wood, and waste products of industrial and agricultural operations. Of these, only coal is widely used for heating and cooling applications.
The complex composition of coal makes classification difficult.
Chemically, coal consists of carbon, hydrogen, oxygen, nitrogen, sulfur, and a mineral residue, ash. Chemical analysis provides some indication of coal quality, but does not define its burning character-istics sufficiently. The coal user is principally interested in the avail-able heat per unit mass of coal and the amount of ash and dust produced, but is also interested in burning characteristics and han-dling and storing properties. A description of coal qualities and their characteristics can be obtained from the U.S. Bureau of Mines. Types of Coals Commonly accepted definitions for classifying coals are listed in Table 7. This classification is arbitrary because there are no distinct demarcation lines between coal types.
Anthracite is a clean, dense, hard coal that creates little dust in handling. It is comparatively hard to ignite, but burns freely once Table 5 Sulfur Content of Marketed Fuel Oils Grade of Oil No. 1 No. 2 No. 4 No. 5 (Light) No. 5 (Heavy) No. 6 Total fuel samples 31 61 13 15 16 96 Sulfur content, % mass minimum 0.001 0.03 0.46 0.90 0.57 0.32 maximum 0.120 0.50 1.44 3.50 2.92 4.00 average 0.023 0.20 0.83 1.46 1.46 1.41 No. samples with S over 0.3% 0 17 13 15 16 96 over 0.5% 0 2 11 15 16 93 over 1.0% 0 0 3 9 11 60 over 3.0% 0 0 0 2 0 8 Data for No. 1 and No. 2 oil derived from Dickson and Sturm (1994).
Data for No. 4, 5, and 6 oil derived from Shelton (1974).
Table 6 Typical Density and Higher Heating Value of Standard Grades of Fuel Oil Grade No.
Density, kg/m3 Higher Heating Value, GJ/m3 1 833 to 800 38.2 to 37.0 2 874 to 834 39.5 to 38.2 4 933 to 886 41.3 to 39.9 5L 951 to 921 41.8 to 40.9 5H 968 to 945 42.4 to 41.6 6 1012 to 965 43.5 to 42.2 Higher heating value, MJ/kg 51.92 8.79 10 6 – ρ2 × – = Hydrogen, % 26 15ρ 1000 ------------ – = Combustion and Fuels 18.7 started. It is noncaking and burns uniformly and smokelessly with a short flame.
Semianthracite has a higher volatile content than anthracite. It is not as hard and ignites more easily. Otherwise, its properties are similar to those of anthracite.
Bituminous coal includes many types of coal with distinctly dif-ferent compositions, properties, and burning characteristics. Coals range from high-grade bituminous, such as those found in the east-ern United States, to low-rank coals, such as those found in the west-ern United States. Caking properties range from coals that melt or become fully plastic, to those from which volatiles and tars are dis-tilled without changing form (classed as noncaking or free-burn-ing). Most bituminous coals are strong and nonfriable enough to permit screened sizes to be delivered free of fines. Generally, they ignite easily and burn freely. Flame length is long and varies with different coals. If improperly fired, much smoke and soot are possi-ble, especially at low burning rates.
Semibituminous coal is soft and friable, and handling creates fines and dust. It ignites slowly and burns with a medium-length flame. Its caking properties increase as volatile matter increases, but the coke formed is weak. With only half the volatile matter content of bituminous coals, burning produces less smoke; hence, it is sometimes called smokeless coal.
Subbituminous coal, such as that found in the western United States, is high in moisture when mined and tends to break up as it dries or is exposed to the weather; it is likely to ignite spontaneously when piled or stored. It ignites easily and quickly, has a medium-length flame, and is noncaking and free-burning. The lumps tend to break into small pieces if poked. Very little smoke and soot are formed.
Lignite is woody in structure, very high in moisture when mined, of low heating value, and clean to handle. It has a greater tendency than subbituminous coals to disintegrate as it dries and is also more likely to ignite spontaneously. Because of its high moisture, freshly mined lignite ignites slowly and is noncaking. The char left after moisture and volatile matter are driven off burns very easily, like charcoal. The lumps tend to break up in the fuel bed and pieces of char that fall into the ash pit continue to burn. Very little smoke or soot forms.
Characteristics of Coal The characteristics of coals that determine classification and suitability for given applications are the proportions of (1) volatile matter, (2) fixed carbon, (3) moisture, (4) sulfur, and (5) ash. Each of these is reported in the proximate analysis. Coal analyses can be reported on several bases: as-received, moisture-free (or dry), and mineral-matter-free (or ash-free). As-received is applicable for combustion calculations; moisture-free and mineral-matter-free, for classification purposes.
Volatile matter is driven off as gas or vapor when the coal is heated according to a standard temperature test. It consists of a variety of organic gases, generally resulting from distillation and decomposition. Volatile products given off by coals when heated differ materially in the ratios (by mass) of the gases to oils and tars.
No heavy oils or tars are given off by anthracite, and very small quantities are given off by semianthracite. As volatile matter in the coal increases to as much as 40% of the coal (dry and ash-free basis), increasing amounts of oils and tars are released. However, for coals of higher volatile content, the quantity of oils and tars decreases and is relatively low in the subbituminous coals and in lignite.
Fixed carbon is the combustible residue left after the volatile matter is driven off. It is not all carbon. Its form and hardness are an indication of fuel coking properties and, therefore, guide the choice of combustion equipment. Generally, fixed carbon represents that portion of fuel that must be burned in the solid state.
Moisture is difficult to determine accurately because a sample can lose moisture on exposure to the atmosphere, particularly when Table 7 Classification of Coals by Ranka Class Group Limits of Fixed Carbon or Energy Content, Mineral-Matter-Free Basis Requisite Physical Properties I Anthracite 1.
Metaanthracite Dry FC, 98% or more (Dry VM, 2% or less) Nonagglomerating 2.
Anthracite Dry FC, 92% or more, and less than 98% (Dry VM, 8% or less, and more than 2%) 3.
Semianthracite Dry FC, 86% or more, and less than 92% (Dry VM, 14% or less, and more than 8%) II Bituminousd 1.
Low-volatile bituminous coal Dry FC, 78% or more, and less than 86% (Dry VM, 22% or less, and more than 14%) Either agglomeratingb or nonweatheringf 2.
Medium-volatile bituminous coal Dry FC, 69% or more, and less than 78% (Dry VM, 31% or less, and more than 22%) 3.
High-volatile A bituminous coal Dry FC, less than 69% (Dry VM, more than 31%), and moistc, about 32.6 MJ/kge or more 4.
High-volatile B bituminous coal Moistc, about 30.2 MJ/kg or more, and less than 32.6 MJ/kge 5.
High-volatile C bituminous coal Moistc, about 25.6 MJ/kg or more, and less than 30.2 MJ/kge III Subbituminous 1.
Subbituminous A coal Moistc, about 25.6 MJ/kg or more, and less than 30.2 MJ/kge Both weathering and nonagglomeratingb 2.
Subbituminous B coal Moistc, about 22.1 MJ/kg or more, and less than 25.6 MJ/kge 3.
Subbituminous C coal Moistc, about 19.3 MJ/kg or more, and less than 22.1 MJ/kge IV Lignitic 1.
Lignite Moistc, less than 19.3 MJ/kg Consolidated 2.
Brown coal Moistc, less than 19.3 MJ/kg Unconsolidated Source: Adapted from ASTM Standard D 388, Standard Classification of Coals by Rank.
FC = Fixed Carbon; VM = Volatile Matter.
aThis classification does not include a few coals of unusual physical and chemical properties which come within the limits of fixed carbon or Btu of high-volatile bituminous and subbi-tuminous ranks. All these coals either contain less than 48% dry, mineral-matter-free fixed carbon, or have more than about 36.1 MJ/kg, which is moist, mineral-matter-free.
bIf agglomerating, classify in low-volatile group of the bituminous class.
cMoist refers to coal containing its natural bed moisture but not including visible water on the coal surface.
dThere may be noncaking varieties in each group of the bituminous class.
eCoals having 69% or more fixed carbon on the dry, mineral-matter-free basis shall be classified according to fixed carbon, regardless of energy content.
fThere are three varieties of coal in the high-volatile C bituminous coal group: Vari-ety 1, agglomerating and nonweathering; Variety 2, agglomerating and weather-ing; and Variety 3, nonagglomerating and nonweathering.
18.8 2001 ASHRAE Fundamentals Handbook (SI) reducing the sample size for analysis. To correct for this loss, total moisture content of a sample is customarily determined by adding the moisture loss obtained when air-drying the sample to the mea-sured moisture content of the dried sample. Moisture does not rep-resent all of the water present in coal; water of decomposition (combined water) and of hydration are not given off under standard-ized test conditions.
Ash is the noncombustible residue remaining after complete coal combustion. Generally, the mass of ash is slightly less than that of mineral matter before burning.
Sulfur is an undesirable constituent in coal, because the sulfur oxides formed when it burns contribute to air pollution and cause combustion system corrosion. Table 8 lists the sulfur content of typ-ical coals. Legislation has limited the sulfur content of coals burned in certain locations.
Heating value may be reported on an as-received, dry, dry and mineral-matter-free, or moist and mineral-matter-free basis.
Higher heating values of coals are frequently reported with their proximate analysis. When more specific data are lacking, the higher heating value of higher quality coals can be calculated by the Dulong formula: (3) where C, H, O, and S are the mass fractions of carbon, hydrogen, oxygen, and sulfur in the coal.
Other important parameters in judging coal suitability include 1. Ultimate analysis, which is another method of reporting coal composition. Percentages of C, H, O, N, S, and ash in the coal sample are reported. Ultimate analysis is used for detailed fuel studies and for computing a heat balance when required in heating device testing. Typical ultimate analyses of various coals are shown in Table 8.
2. Ash-fusion temperature, which indicates the fluidity of the ash at elevated temperatures. It is helpful in selecting coal to be burned in a particular furnace and in estimating the possibility of ash handling and slagging problems.
3. The grindability index, which indicates the ease with which a coal can be pulverized and is helpful in estimating ball mill capacity with various coals. There are two common methods for determining the index—Hardgrove and ball mill.
4. The free-swelling index, which denotes the extent of coal swelling on combustion on a fuel bed and indicates the coking characteristics of coal.
COMBUSTION CALCULATIONS Calculations of the quantity of air required for combustion and the quantity of flue gas products generated during combustion are frequently needed for sizing system components and as input to effi-ciency calculations. Other calculations, such as values for excess air and theoretical CO2, are useful in estimating combustion system performance.
Frequently, combustion calculations can be simplified by using relative molecular mass. The relative molecular mass of a com-pound equals the sum of the atomic masses of the elements in the compound. Molecular mass can be expressed in any mass units. The gram molecular mass or gram mole is the molecular mass of the compound expressed in grams. The molecular mass of any sub-stance contains the same number of molecules as the molecular mass of any other substance.
Corresponding to measurement standards common to the indus-tries, calculations involving gaseous fuels are generally based on volume, and calculations involving liquid and solid fuels are gener-ally based on mass.
Some calculations described here require data on concentrations of carbon dioxide, carbon monoxide, and oxygen in the flue gas.
Gas analyses for CO2, CO, and O2 can be obtained by volumetric chemical analysis and other analytical techniques, including elec-tromechanical cells used in portable electronic flue gas analyzers.
Air Required for Combustion Stoichiometric or theoretical air is the exact quantity of air required to provide oxygen for complete combustion.
The three most prevalent components in hydrocarbon fuels (C, H2, and S) are completely combusted as in the following reactions: In the reactions, C, H2, and S can be taken to represent 1 kg mole of carbon, hydrogen, and sulfur, respectively. Using approximate atomic masses (C = 12, H = 1, S = 32, and O = 16), 12 kg of C are oxidized by 32 kg of O2 to form 44 kg of CO2, 2 kg of H2 are oxi-dized by 16 kg of O2 to form 18 kg of H2O, and 32 kg of S are oxi-dized by 32 kg of O2 to form 64 kg of SO2. These relationships can be extended to include hydrocarbons.
The mass of dry air required to supply a given quantity of oxygen is 4.32 times the mass of the oxygen. The mass of air required to oxi-dize the fuel constituents listed in Table 1 was calculated on this basis. Oxygen contained in the fuel, except that contained in ash, should be deducted from the amount of oxygen required, because this oxygen is already combined with fuel components. In addition, when calculating the mass of air to be supplied for combustion, allowance should be made for water vapor, which is always present in atmospheric air.
As stated previously, combustion calculations for gaseous fuels are based on volume. Avogadro’s law states that, for any gas, one mole occupies the same volume at a given temperature and pres-sure. Therefore, in reactions involving gaseous compounds, the gases react in volume ratios identical to the pound mole ratios.
That is, for the oxidation of hydrogen in the above reaction, one volume (or one kg mole) of hydrogen reacts with one-half volume (or one-half kg mole) of oxygen to form one volume (or one kg mole) of water vapor.
Table 8 Typical Ultimate Analyses for Coals Rank As Received, MJ/kg Constituents, % by Mass Oxy-gen Hydro-gen Car-bon Nitro-gen Sul-fur Ash Anthracite 29.5 5.0 2.9 80.0 0.9 0.7 10.5 Semianthracite 31.6 5.0 3.9 80.4 1.1 1.1 8.5 Low-volatile bituminous 33.4 5.0 4.7 81.7 1.4 1.2 6.0 Medium-volatile bituminous 32.6 5.0 5.0 81.4 1.4 1.5 6.0 High-volatile bituminous A 32.1 9.3 5.3 75.9 1.5 1.5 6.5 High-volatile bituminous B 29.1 13.8 5.5 67.8 1.4 3.0 8.5 High-volatile bituminous C 25.6 20.6 5.8 59.6 1.1 3.5 9.4 Subbituminous B 20.9 29.5 6.2 52.5 1.0 1.0 9.8 Subbituminous C 19.8 35.7 6.5 46.4 0.8 1.0 9.6 Lignite 16.0 44.0 6.9 40.1 0.7 1.0 7.3 Higher heating value, MJ/kg 33.829C 144.28[H O 8 ⁄ ( )] – 9.42S + + = C O2 + CO2 → H2 0.5O2 + H2O → S O2 + SO2 → Combustion and Fuels 18.9 The volume of air required to supply a given volume of oxygen is 4.78 times the volume of oxygen. The volumes of dry air required to oxidize the fuel constituents listed in Table 1 were calculated on this basis. Volume ratios are not given for fuels that do not exist in vapor form at reasonable temperatures or pressures. Again, oxygen contained in the fuel should be deducted from the quantity of oxy-gen required, because this oxygen is already combined with fuel components. Allowance should be made for water vapor, which increases the volume of dry air by 1 to 3%.
From the relationships just described, the theoretical mass ma of dry air required for stoichiometric combustion of a unit mass of any hydrocarbon fuel is (4) where C, H, S, and O are the mass percentages of carbon, hydrogen, sulfur, and oxygen in the fuel.
Analyses of gaseous fuels are generally based on hydrocarbon components rather than elemental content.
If the fuel analysis is based on mass, the theoretical mass ma of dry air required for stoichiometric combustion of a unit mass of gas-eous fuel is (5) If the fuel analysis is reported on a volumetric or molecular basis, it is simplest to calculate air requirements based on volume and, if necessary, convert to mass. The theoretical volume Va of air required for stoichiometric combustion of a unit volume of gaseous fuels is (6) where CO, H2, and so forth are the volumetric fractions of each con-stituent in the fuel gas.
Illuminants include a variety of compounds not separated by usual gas analysis. In addition to ethylene (C2H4) and acetylene (C2H2), the principal illuminants included in Equation (7), and the dry air required for combustion, per unit volume of each gas, are: propylene (C3H6), 21.44; butylene (C4H8), 28.58; pentene (C5H10), 35.73; benzene (C6H6); 35.73, toluene (C7H8), 42.88; and xylene (C8H10), 50.02. Because toluene and xylene are normally scrubbed from the gas before distribution, they can be disregarded in comput-ing air required for combustion of gaseous fuels. The percentage of illuminants present in gaseous fuels is small, so the values can be lumped together, and an approximate value of 30 unit volumes of dry air per unit volume of gas can be used. If ethylene and acetylene are included as illuminants, a value of 20 unit volumes of dry air per unit volume of gaseous illuminants can be used.
For many combustion calculations, only approximate values of air requirements are necessary. If approximate values for theoretical air are sufficient, or if complete information on the fuel is not avail-able, the values in Tables 9 and 10 can be used. Another value used for estimating air requirements is 0.24 m3 of air for 1 MJ of fuel.
In addition to the amount theoretically required for combustion, excess air must be supplied to most practical combustion systems to ensure complete combustion.
(7) The excess air level at which a combustion process operates sig-nificantly affects its overall efficiency. Too much excess air dilutes flue gas excessively, lowering its heat transfer temperature and increasing sensible flue gas loss. Conversely, if the level of excess air is too low, incomplete combustion and loss of unburned combus-tible gases from the equipment can result. The highest combustion efficiency is usually obtained when just enough excess air is sup-plied and properly mixed with combustible gases to ensure com-plete combustion. The general practice is to supply from 5 to 50% excess air, the exact amount depending on the type of fuel burned, combustion equipment, and other factors.
The amount of dry air supplied per unit mass of fuel burned can be obtained from the following equation, which is reasonably pre-cise for most solid and liquid fuels.
(8) where Dry air supplied = unit mass per unit mass of fuel C = unit mass of carbon burned per unit mass of fuel, corrected for carbon in the ash CO2, CO, N2 = percentages by volume from the flue gas analysis These values of dry air supplied and theoretical air can be used in Equation (7) to determine excess air.
ma 0.0144 8C 24H 3S 3O – + + ( ) = ma 2.47CO 34.28H2 17.24CH4 16.09C2H6 + + + = 15.68C3H8 15.47C4H10 13.27C2H2 + + + 14.78C2H4 6.08H2S 4.32O2 – + + Va 2.39CO 2.39H2 9.57CH4 16.75C2H6 + + + = 23.95C3H8 31.14C4H10 11.96C2H2 + + + 14.38C2H4 7.18H2S 4.78O2 – + + 30.47 illuminants + Table 9 Approximate Air Requirements for Stoichiometric Combustion of Fuels Type of Fuel Air Required Approx. Precision, % Exceptions kg/kg Fuel m3/Unit Fuela Solid MJ/kg × 0.314 MJ/kg × 0.26 3 Fuels containing more than 30% water Liquid MJ/kg × 0.305 MJ/kg × 0.35 3 Results low for gasoline and kerosene Gas MJ/kg × 0.288 MJ/m3 × 0.24 5 11.2 MJ/m3 or less Source: Data based on Shnidman (1954).
aUnit fuel for solid and liquid fuels in kg, for gas in m3.
Table 10 Approximate Air Requirements for Stoichiometric Combustion of Various Fuels Type of Fuel Theoretical Air Required for Combustion Solid fuels kg/kg fuel Anthracite 9.6 Semibituminous 11.2 Bituminous 10.3 Lignite 6.2 Coke 11.2 Liquid fuels Mg/m3 fuel No. 1 fuel oil 12.34 No. 2 fuel oil 12.70 No. 5 fuel oil 13.42 No. 6 fuel oil 13.66 Gaseous fuels m3/m3 fuel Natural gas 9.6 Butane 31.1 Propane 24.0 Excess air, % Air supplied Theoretical air – Theoretical air ------------------------------------------------------------------------= Dry air supplied C 3.04N2 ( ) CO2 CO + ----------------------------= 18.10 2001 ASHRAE Fundamentals Handbook (SI) Excess air can also be calculated from unit volumes of stoichio-metric combustion products and air, and from volumetric analysis of the flue gas: (9) where U = ultimate carbon dioxide of flue gases resulting from stoichiometric combustion, % CO2 = carbon dioxide content of flue gases, % P = dry products from stoichiometric combustion, unit volume per unit volume of gas burned A = air required for stoichiometric combustion, unit volume per unit volume of gas burned As the ratio P/A is approximately 0.9 for most natural gases, a value of 90 can be substituted for 100 (P/A) in Equation (9) for rough calculation.
Because excess air calculations are almost invariably made from flue gas analysis results and theoretical air requirements are not always known, another convenient method of expressing the rela-tion of Equation (7) is (10) where O2, CO, and N2 are percentages by volume from the flue gas analysis.
Theoretical CO2 The theoretical CO2, ultimate CO2, stoichiometric CO2, or max-imum CO2 concentration attainable in the products from the com-bustion of a hydrocarbon fuel with air is obtained when the fuel is completely burned with the theoretical quantity of air and zero excess air. Theoretical CO2 varies with the carbon-hydrogen ratio of the fuel. For combustion with excess air present, theoretical CO2 values can be calculated from the flue gas analysis: (11) where CO2 and O2 are percentages by volume from the flue gas analysis.
Table 11 gives approximate theoretical CO2 values for stoichio-metric combustion of several common types of fuel, as well as CO2 values attained with different amounts of excess air. In practice, desirable CO2 values depend on the excess air, fuel, firing method, and other considerations.
Quantity of Flue Gas Produced The mass of dry flue gas produced per mass of fuel burned is required in heat loss and efficiency calculations. This mass is equal to the sum of the mass of (1) fuel (minus ash retained in the furnace), (2) air theoretically required for combustion, and (3) excess air. For solid fuels, this mass, determined from the flue gas analysis, is (12) where Dry flue gas = kg/kg of fuel C = kg of carbon burned per kg of fuel, corrected for carbon in the ash CO2, O2, CO, N2 = percentages by volume from flue gas analysis The total dry gas volume of flue gases from combustion of one unit volume of gaseous fuels for various percentages of CO2 is (13) where Dry flue gas = unit volume per unit volume of gaseous fuel CO2 = percentage by volume from the flue gas analysis Excess air quantity can be estimated by subtracting the quantity of dry flue gases resulting from stoichiometric combustion from the total volume of flue gas. Water Vapor and Dew Point of Flue Gas Water vapor in flue gas is the total of (1) the water contained in the fuel; (2) the water contained in the stoichiometric and excess air; and (3) the water produced from the combustion of hydrogen or hydrocarbons in the fuel. The amount of water vapor in the stoichi-Excess air, % 100 P A --- U CO2 – CO2 --------------------- = Excess air, % 100 O2 CO 2 ⁄ ( ) – [ ] 0.264N2 O2 CO 2 ⁄ ( ) – [ ] – ----------------------------------------------------------------= Theoretical CO2, % U CO2 1 O2 20.95 ⁄ ( ) – --------------------------------------= = Dry flue gas 11CO2 8O2 7 CO N2 + ( ) + + 3 CO2 CO + ( ) ---------------------------------------------------------------------- C = Table 11 Approximate Maximum Theoretical (Stoichiometric) CO2 Values, and CO2 Values of Various Fuels with Different Percentages of Excess Air Type of Fuel Theoretical or Maximum CO2, % Percent CO2 at Given Excess Air Values 20% 40% 60% Gaseous Fuels Natural gas 12.1 9.9 8.4 7.3 Propane gas (commercial) 13.9 11.4 9.6 8.4 Butane gas (commercial) 14.1 11.6 9.8 8.5 Mixed gas (natural and carbureted water gas) 11.2 12.5 10.5 9.1 Carbureted water gas 17.2 14.2 12.1 10.6 Coke oven gas 11.2 9.2 7.8 6.8 Liquid Fuels No. 1 and 2 fuel oil 15.0 12.3 10.5 9.1 No. 6 fuel oil 16.5 13.6 11.6 10.1 Solid Fuels Bituminous coal 18.2 15.1 12.9 11.3 Anthracite 20.2 16.8 14.4 12.6 Coke 21.0 17.5 15.0 13.0 Dry flue gas vol. of CO2 produced unit vol. of gas burned ------------------------------------------------------- 100 CO2 ------------ = Fig. 2 Water Vapor and Dew Point of Flue Gas Adapted from Gas Engineers Handbook (1965). Printed with permission of Industrial Press and American Gas Association.
Combustion and Fuels 18.11 ometric combustion products may be calculated from the fuel burned by using the water data in Table 1.
The dew point is the temperature at which condensation begins and can be determined using Figure 2. The volume fraction of water vapor in the flue gas can be determined as follows: (14) where Vw = total water vapor volume (from fuel; from stoichiometric, excess, and dilution air; and from combustion) Vc = unit volume of CO2 produced per unit volume of gaseous fuel Pc = percent CO2 in flue gas Using Figure 3, the dew points of solid, liquid, or gaseous fuels may be estimated. For example, to find the dew point of flue gas resulting from the combustion of a solid fuel with a mass ratio (hydrogen to carbon-plus-sulfur) of 0.088 and sufficient excess air to produce 11.4% oxygen in the flue gas, start with the mass ratio of 0.088. Proceed vertically to the intersection of the solid fuels curve and then to the theoretical dew point of 46°C on the dew-point scale (see dotted lines in Figure 3). Follow the curve fixed by this point (down and to the right) to 11.4% oxygen in the flue gas (on the abscissa). The actual dew point is 34°C and is found on the dew-point scale.
An estimation can be made of the dew point of the flue gas from natural gas having a higher heating value (HHV) of 38 MJ/m3 with 6.3% oxygen or 31.5% air. Start with 38 MJ/m and proceed verti-cally to the intersection of the gaseous fuels curve and then to the theoretical dew point of 59°C on the dew-point scale. Follow the curve fixed by this point to 6.3% oxygen or 31.5% air in the flue gas.
The actual dew point is 53°C.
The presence of sulfur dioxide, and particularly sulfur trioxide, influences the vapor pressure of condensate in flue gas, and the dew point can be raised by as much as 14 to 42 K, as shown in Figure 4.
To illustrate the use of Figure 4, for a manufactured gas with a HHV of 20.5 MJ/m3 containing 340 mg of sulfur per cubic metre being burned with 40% excess air, the proper curve in Figure 4 is deter-mined as follows: (15) This curve lies between the 0 and 20 curves and is close to the 20 curve. The dew point for any percentage of excess air from zero to 100% can be determined on this curve. For this flue gas with 40% excess air, the dew point is about 71°C, instead of 53°C for zero sul-fur at 40% excess air.
Sample Combustion Calculations Example 1. Analysis of flue gases from the burning of a natural gas shows 10.0% CO2, 3.1% O2, and 86.9% N2 by volume. Analysis of the fuel is 90% CH4, 5% N2, and 5% C2H6 by volume. Find U (maximum theo-retical percent CO2), and the percentage of excess air.
Solution: From Equation (11), From Equation (9), using 100 (P/A) = 90, Pwv Vw 100Vc Pc ⁄ ( ) Vw + -------------------------------------------= Fig. 3 Theoretical Dew Points of Combustion Products of Industrial Fuels Adapted from Gas Engineers Handbook (1965). Printed with permission of Industrial Press and American Gas Association.
Mass of sulfur in fuel, mg/m3 Fuel heating value, MJ/m3 -----------------------------------------------------------------------340 20.5 ----------16.6 = = U 10.0 1 3.1 20.95 ⁄ ( ) – --------------------------------------11.74% CO2 = = 18.12 2001 ASHRAE Fundamentals Handbook (SI) Example 2. For the same analysis as in Example 1, find, per cubic metre of fuel gas, the volume of dry air required for combustion, the volume of each constituent in the flue gases, and the total volume of dry and wet flue gases.
Solution: From Equation (6), the volume of dry air required for com-bustion is: (The volume of dry air may also be calculated using Table 10.) From Table 1, the cubic metres of flue gas constituents per cubic metre of fuel gas are as follows: EFFICIENCY CALCULATIONS In analyzing heating appliance efficiency, an energy balance is made that accounts (as much as possible) for disposition of all ther-mal energy released by combustion of the fuel quantity consumed.
The various components of this balance are generally expressed in terms of megajoules per kilogram of fuel burned or as a percentage of its higher heating value. The following are major components of an energy balance and their calculation methods: 1. Useful heat, or heat transferred to the heated medium; for convection heating equipment, this value q1 is computed as the product of the mass rate of flow and enthalpy change.
2. Heat loss as sensible heat in the dry flue gases (16) where mg is calculated as in Equation (12).
3. Heat loss in water vapor in products formed by combustion of hydrogen (17) 4. Heat loss in water vapor in the combustion air (18) where ma is calculated as in Equations (4) and (5).
5. Heat loss from incomplete combustion of carbon (19) 6. Heat loss from unburned carbon in the ash or refuse (20) 7. Unaccounted-for heat losses, q7 The following symbols are used in Equations (16) through (20): q1 = useful heat, kJ/kg of fuel q2 = heat loss in dry flue gases, kJ/kg of fuel q3 = heat loss in water vapor from combustion of hydrogen, kJ/kg of fuel q4 = heat loss in water vapor in combustion air, kJ/kg of fuel q5 = heat loss from incomplete combustion of carbon, kJ/kg of fuel q6 = heat loss from unburned carbon in ash, kJ/kg of fuel q7 = unaccounted-for heat losses, kJ/kg of fuel cpg = mean specific heat of flue gases at constant pressure (cpg ranges from 1.01 to 1.06 kJ/(kg·K) for flue gas temperatures from 150 to 540°C), kJ/(kg·K) (h)tg = enthalpy of superheated steam at flue gas temperature and 101.3 kPa, kJ/kg (hf)ta = enthalpy of saturated water vapor at air temperature, kJ/kg (hg)ta = enthalpy of saturated steam at combustion air temperature, kJ/kg ma = mass of combustion air per mass of fuel used, kg/kg of fuel mg = mass of dry flue gas per mass of fuel, kg/kg of fuel ta = temperature of combustion air, °C tg = temperature of flue gases at exit of heating device, °C H2 = hydrogen in fuel, % by mass (from ultimate analysis of fuel) M = humidity ratio of combustion air, mass of water vapor per mass of dry air CO, CO2 = carbon monoxide and carbon dioxide in flue gases, % by volume C = mass of carbon burned per unit of mass of fuel, corrected for carbon in ash, kg/kg of fuel Nitrogen, N2 From methane (0.9CH4)(9.57 − 2.0) = 6.81 From ethane (0.05C2H6)(16.75 − 3.5) = 0.66 Nitrogen in fuel = 0.05 Nitrogen in excess air 0.791 × 0.157 × 9.45 = 1.17 Total nitrogen = 8.69 m3 Oxygen, O2 In excess air 0.209 × 0.157 × 9.45 = 0.31 m3 Carbon dioxide, CO2 From methane (0.9CH4)(1.0) = 0.90 From ethane (0.05C2H6)(2.0) = 0.10 Total carbon dioxide = 1.00 m3 Water vapor, H2O (does not appear in some flue gas analyses) From methane (0.9CH4)(2.0) = 1.8 From ethane (0.05C2H6)(3.0) = 0.15 Total water vapor = 1.95 m3 Total volume of dry gas per cubic metre of fuel gas 8.69 + 0.31 + 1.00 = 10.0 m3 Total volume of wet gases per cubic metre of fuel gas (neglecting water vapor in combustion air) 10.0 + 1.95 = 11.95 m3 The cubic metres of dry flue gas per cubic metre of fuel gas can also be computed from Equation (13): (1.00)(100)/10.0 = 10.0 m3 Fig. 4 Influence of Sulfur Oxides on Flue Gas Dew Point (Stone 1969) Excess air 11.74 10.0 – ( )90 10 -----------------------------------------15.7% = = 9.57CH4 16.75C2H6 + 9.57 0.90 × ( ) 16.75 0.05 × ( ) + = 9.45 m3 per m3 of fuel gas = q2 mgcpg tg ta – ( ) = q3 9H2 100 ⁄ ( ) h ( )tg hf ( )ta – [ ] = q4 Mma h ( )tg hg ( )ta – [ ] = q5 23 591 C CO CO2 CO + ------------------------- = q6 33 957 Cu 100 ⁄ ( ) C – [ ] = Combustion and Fuels 18.13 (21) where Cu = percentage of carbon in fuel by mass from ultimate analysis Wa = mass of ash and refuse Ca = percent of combustible in ash by mass (combustible in ash is usually considered to be carbon) W = mass of fuel used Useful heat (item 1) is generally measured for a particular piece of combustion equipment.
Flue gas loss is the sum of items 2 through 6. However, for clean-burning gas- and oil-fired equipment, items 5 and 6 are usually neg-ligible and flue gas loss is the sum of items 2, 3, and 4.
Flue gas losses (the sum of items 2, 3, and 4) can be determined with sufficient precision for most purposes from the curves in Fig-ure 5, if O2 content and flue gas temperature are known. Values of the losses were computed from typical ultimate analyses, assuming 1% water vapor (by mass) in the combustion air. Curves for medium-volatile bituminous coal can be used for high-volatile bitu-minous coal with no appreciable error.
Generally, item 5 is negligible for modern combustion equip-ment in good operating condition.
Item 6 is generally negligible for gas and oil firing, but should be determined for coal-firing applications.
Item 7 consists primarily of radiation and convection losses from combustion equipment surfaces and losses caused by incomplete combustion not included in items 5 and 6. Heat loss from incom-plete combustion is determined by subtracting the sum of items 1 through 6 from the fuel heating value.
Radiation and convection losses are not usually determined by direct measurement. But if the heating appliance is located within the heated space, radiation and convection losses can be considered useful heat rather than lost heat and can be omitted from heat loss calculations or added to item 1.
If CO is present in flue gases, small amounts of unburned hydro-gen and hydrocarbons may also be present. The small losses caused by incomplete combustion of these gases would be included in item 7, if item 7 was determined by subtracting items 1 through 6 from the fuel heating value.
The overall thermal efficiency of combustion equipment is defined as (22) The following equation can be used to estimate efficiency for equipment where item 7 is small or radiation and convection are useful heat: Thermal efficiency, % = (23) Using heating values based on gas volume, the thermal effi-ciency of a gas appliance can be computed with sufficient precision by the following equation: (24) where η = thermal efficiency, % Qh = higher heating value of fuel gas per unit volume Qfl = flue gas losses per unit volume of fuel gas To produce heat efficiently by burning any common fuel, flue gas losses must be minimized by (1) providing adequate heat-absorbing surface in the appliance, (2) maintaining clean heat trans-fer surfaces on both fire and water or air sides, and (3) reducing excess air to the minimum level consistent with complete combus-tion and discharge of combustion products.
Seasonal Efficiency The method just presented is useful for calculating the steady-state efficiency of a heating system or device. Unfortunately, the seasonal efficiency of a combustion heating system can be signifi-cantly different from the steady-state efficiency. The primary factor affecting the seasonal efficiency is flue loss during the burner-off period. The warm stack that exists at the end of the firing period can cause airflow in the stack while the burner is off. This airflow can remove heat from furnace and heat exchanger components, from the structure itself, and from pilot flames. Also, if combustion air is drawn from the heated space within the structure, the heated air lost must be at least partly replaced with cold infiltrated air. For further discussion of seasonal efficiency, see Chapter 9 of the 2000 ASHRAE Handbook—Systems and Equipment.
COMBUSTION CONSIDERATIONS Air Pollution Combustion processes constitute the largest single source of air pollution. Pollutants can be grouped into four categories: 1. Products of incomplete fuel combustion • Combustible aerosols (solid and liquid), including smoke, soot, and organics, but excluding ash • Carbon monoxide, CO • Gaseous hydrocarbons 2. Oxides of nitrogen (generally grouped and referred to as NOx) • Nitric oxide, NO • Nitrogen dioxide, NO2 3. Emissions resulting from fuel contaminants • Sulfur oxides, primarily sulfur dioxide, SO2, and small quantities of sulfur trioxide, SO3 • Ash • Trace metals 4. Emissions resulting from additives • Combustion-controlling additives • Other additives Emission levels of nitrogen oxides and products of incomplete combustion are directly related to the combustion process and can be controlled, to some extent, by process modification. Emissions due to fuel contaminants are related to fuel selection and are slightly affected by the combustion process. Emissions due to additives must be considered in the overall evaluation of the merits of using additives.
Nitrogen oxides are produced during the combustion process, either (1) by thermal fixation (reaction of nitrogen and oxygen at high combustion temperatures), or (2) from fuel nitrogen (oxidation of organic nitrogen in fuel molecules). Unfortunately, high excess air and high flame temperature techniques, which ensure complete fuel combustion, tend to promote NOx formation. Table 12 lists NOx emission factors for uncontrolled fuel-burn-ing equipment (i.e., equipment that does not have exhaust gas recir-culation, low-NOx burners, or other emission controls). Differences in the NOx emissions of fuels are caused by the flame temperature and different levels of fuel nitrogen. The data in Table 12 are adapted from EPA (1993), Compilation of Air Pollutant Emission Factors, which lists emission factors of a wide variety of equip-ment, as well as emission reduction options. Carbon monoxide emissions are less dependent on fuel type and typically range from C WCu WaCa – 100W --------------------------------= Thermal efficiency, % 100 Useful heat Heating value of fuel --------------------------------------------------= 100 Heating value of fuel q2 q3 q4 q5 q6 + + + + ( ) – Heating value of fuel -------------------------------------------------------------------------------------------------------------------η 100 Qh Qfl – ( ) Qh ----------------------------------= 18.14 2001 ASHRAE Fundamentals Handbook (SI) Fig. 5 Flue Gas Losses with Various Fuels (Flue gas temperature rise shown. Loss is based on 18°C room temperature.) Combustion and Fuels 18.15 13 to 17 mg/MJ of heat input. For gas-fired commercial and indus-trial boilers, particulate emissions range from 2.2 to 2.6 mg/ MJ.
For distillate-oil-fired commercial and industrial boilers, particu-lates are typically 6.0 mg/MJ. For residential oil-fired equipment, particulate emission factors are 1.3 mg/MJ. For residual-oil-fired equipment, particulate emissions depend on the sulfur content. For a sulfur content of 1%, the particulate emission rate is typically 36 mg/MJ.
Emission levels of products of incomplete fuel combustion can be reduced by reducing burner cycling, ensuring adequate excess air, improving the mixing of air and fuel (by increasing turbulence, improving distribution, and improving liquid fuel atomization), increasing residence time in the hot combustion zone (possibly by decreasing the firing rate), increasing combustion zone tempera-tures (to speed reactions), and avoiding quenching the flame before reactions are completed.
The relative contribution of each of these mechanisms to the total NOx emissions depends on the amount of organic nitrogen in the fuel. Natural gas contains very little nitrogen. Virtually all NOx emissions with gas firing are due to the thermal mechanism. The nitrogen content of distillate oil varies, but an average of 20 ppm of fuel NOx is produced (about 20 to 30% of the total NOx). The fuel nitrogen in residual oil can be significantly higher, with fuel NOx contributing heavily to the total emissions.
Thermal fixation is strongly dependent on flame maximum tem-perature. For example, increasing the flame temperature from 1400 to 1500°C increases thermal NOx tenfold. Therefore, methods to control thermal NOx are based on methods to reduce the maximum flame temperature. Flue gas recirculation is perhaps the most effec-tive method of reducing thermal NOx in commercial and industrial boilers. In gas-fired boilers, NOx reductions of 70% can be realized with 15-20% recirculation of flue gas into the flame. The NOx reduction decreases with increasing fuel nitrogen content. With dis-tillate-oil firing, reductions of 60-70% can be achieved. In residual-oil-fired boilers, flue gas recirculation can reduce NOx emissions by 15 to 30%. The maximum rate of flue gas recirculation is limited by combustion instability and CO production.
Two-stage firing is the only technique that reduces NOx produced both by thermal fixation and fuel nitrogen in industrial and utility applications. The fuel-rich or air-deficient primary combustion zone retards NOx formation early in the combustion process (when NOx forms most readily from fuel nitrogen), and avoids peak tempera-tures, reducing thermal NOx. Retrofit low-NOx burners that control air distribution and fuel air mixing in the flame zone can be used to achieve staged combustion. With oil firing, NOx reductions of 20 to 50% can be obtained with low-NOx burners. The application of flue gas recirculation and other control methods to residential, oil-fired warm air furnaces was reviewed by Butcher et al. (1994).
The following are some methods of reducing NOx emissions from gas-fired appliances (Murphy and Putnam 1985): • Burner adjustment • Flame inserts (radiation screens or rods) • Staged combustion and delayed mixing • Secondary air baffling • Catalytic and radiant burners • Total premix • Pulse Radiation screens or rods (flame inserts) surrounding or inserted into the flame absorb radiation to reduce flame temperature and retard NOx formation. Proprietary appliance burners with no flame inserts have been developed and produced to comply with the very strict NOx emission limitations of California’s Air Quality Manage-ment Districts.
The U.S. EPA sets limits on air pollutant emissions (Source Performance Standards) from boilers larger than 3 MW of heat input. In addition, states set emission regulations that are at least as strict at the federal limits and may apply to smaller equipment.
The EPA’s automobile emission standard is 0.62 g of NO2 per kilometre, which is equivalent to 750 ng/J of NOx emission. Cali-fornia’s maximum is 0.25 g/km, equivalent to 300 ng/J. California’s Air Quality Management Districts for the South Coast (Los Ange-les) and the San Francisco Bay Area limit NOx emission to 40 ng/J of useful heat for some natural gas-fired residential heating appli-ances.
For further discussion of air pollution aspects of fuel combus-tion, see EPA (1971a and 1971b).
Condensation and Corrosion Fuel-burning systems that cycle on and off to meet demand cool down during the off-cycle. When the appliance starts again, conden-sate forms briefly on surfaces until they are heated above the dew-point temperature. Low-temperature corrosion occurs in system components (heat exchangers, flues, vents, chimneys) when their surfaces remain below the dew-point temperature of flue gas con-stituents (water vapor, sulfides, chlorides, fluorides, etc.) long enough to cause condensation. Corrosion increases as condensate dwell time increases.
Acids in the flue gas condensate are the principle substances responsible for low-temperature corrosion in fuel-fired systems.
Sulfuric, hydrochloric, and other acids are formed when acidic com-pounds in fuel and air combustion products combine with con-densed moisture in appliance heat exchangers, flues, or vents.
Corrosion can be avoided by maintaining these surfaces above the flue gas dew point.
In high-efficiency, condensing-type appliances and economiz-ers, flue gas temperatures are intentionally reduced below the flue gas dew-point temperatures to achieve efficiencies approaching 100%. In these systems, the surfaces subjected to condensate must be made of corrosion-resistant materials. The most corrosive condi-tions exist at the leading edge of the condensing region, especially those areas that experience evaporation during each cycle (Strick-land et al. 1987). Drainage of condensate retards the concentration of acids on system surfaces. Regions from which condensate par-tially or completely drains away before evaporation are less severely attacked than regions from which condensate does not drain before evaporation.
The metals most resistant to condensate corrosion are stainless-steel alloys with high chromium and molybdenum content, and nickel-chromium alloys with high molybdenum content (Stickford et al. 1988). Aluminum experiences general corrosion rather than pitting when exposed to flue gas condensate. If applied in suffi-ciently thick cross section to allow for metal loss, aluminum can be used in condensing regions. Most ceramic and high-temperature polymer materials resist the corrosive effects of flue gas conden-sate. These materials may have application in the condensing regions, if they can meet the structural and temperature require-ments of a particular application.
Table 12 NOx Emission Factors for Combustion Sources Without Emission Controls Source NOx Emission Factor, mg/MJ of Heat Input Gas-Fired Equipment Small industrial boilers 60 Commercial boilers 43 Residential furnaces 39 Distillate-oil-fired small industrial boilers, commercial boilers, and residential furnaces 60 Residual oil-fired small industrial boilers and commercial boilers 160 18.16 2001 ASHRAE Fundamentals Handbook (SI) In coal-fired power plants, the rate of corrosion for carbon steel condensing surfaces by the mixed acids (primarily sulfuric and hydrochloric) is reported to be maximum at about 50 ± 10°C (Davis 1987). Mitigation techniques include (1) acid neutralization with a base such as NH3 or Ca(OH)2; (2) use of protective linings of glass-filled polyester or coal-tar epoxy; and (3) replacement of steel with molybdenum-bearing stainless steels, nickel alloys, polymers, or other corrosion-resistant materials. Other elements in residual fuel oils and coals that contribute to high-temperature corrosion include sodium, potassium, and vanadium. Each fuel-burning system com-ponent should be evaluated during installation, or when modified, to determine the potential for corrosion and the means to retard corro-sion (Paul et al. 1988).
Soot Soot deposits on flue surfaces of a boiler or heater act as an insu-lating layer over the surface, reducing heat transfer to the water or air. Soot can also clog flues, reduce draft and available air, and pre-vent proper combustion. Proper burner adjustment can minimize soot accumulation. The use of off-specification fuel can contribute to the generation of soot.
REFERENCES ASTM. 1995. Standard specification for fuel oils. ANSI/ASTM Standard D 396-95. American Society for Testing and Materials, West Consho-hocken, PA.
ASTM. 1996. Standard specification for diesel fuel oils. ANSI/ASTM Stan-dard D 975-96.
ASTM. 1996. Standard specification for gas turbine fuel oils. ANSI/ASTM Standard D 2880-96.
ASTM. 1995. Standard classification of coals by rank. ASTM Standard D 388-95.
Butcher, T.A., L. Fisher, B. Kamath, T. Kirchstetter, and J. Batey. 1994.
Nitrogen oxides (NOx) and oil burners. Proceedings of the 1994 Oil Heat Technology Conference and Workshops. BNL Report No. 52430.
Brookhaven National Laboratory, Upton, NY.
Coward, H.F. and G.W. Jones. 1952. Limits of flammability of gases and vapors. Bulletin 503. U.S. Bureau of Mines, Washington, D.C.
Davis, J.R., ed. 1987. Metals handbook, 9th ed., Vol. 13. ASM International, Metals Park, OH.
Dickson, C.L. and G.P. Sturm, Jr. 1994. Heating oils. National Institute for Petroleum and Energy Research, Bartlesville, Oklahoma.
EPA. 1971a. Standards of performance for new stationary sources, Group I, Federal Register 36, August 17. U.S. Environmental Protection Agency, Washington, D.C.
EPA. 1971b. Standards of performance for new stationary sources, Group I, Part II, Federal Register 36, December 23. U.S. Environmental Protec-tion Agency, Washington, D.C.
EPA. 1993. Compilation of air pollutant emission factors. Report AP-42.
U.S. Environmental Protection Agency, Washington, D.C.
Gas engineers handbook. 1965. The Industrial Press, New York.
Hartman, I. 1958. “Dust explosions.” In Mechanical engineers’ handbook, 6th ed., Section 7, pp. 41-48. McGraw-Hill, New York.
Hazard, H.R. 1971. “Gas turbine fuels.” In Gas turbine handbook, Gas Tur-bine Publications, Stamford, CT.
Murphy, M.J. and A.A. Putnam. 1985. Burner technology bulletin: Control of NOx emissions from residential gas appliances. Report GRI-85/0132.
Battelle Columbus Division for Gas Research Institute.
NFPA. 1962. Fire-hazard properties of flammable liquids, gases and volatile solids, Tables 6-126, pp. 6-131 ff. In Fire protection handbook, 12th ed., National Fire Protection Association, Quincy, MA.
NFPA/IAS. 1999. ANSI/NFPA Standard 54-1992. National Fuel Gas Code, Section 8.1.2. National Fire Protection Association, Quincy, MA. ANSI/ IAS Standard Z223.1-1999. American Gas Association, Arlington, VA.
North American combustion handbook. 1978. The North American Manu-facturing Co., Cleveland, OH.
Paul, D.D., A.L. Rutz, S.G. Talbert, J.J. Crisafolli, G.R. Whitacre, and R.D.
Fischer. 1988. User’s manual for Vent-II Ver. 3.0—A dynamic micro-computer program for analyzing gas venting systems. Report GRI-88/0304. Battelle Columbus Division for Gas Research Institute.
Scott, G.S., et al. 1948. Determination of ignition temperatures of combus-tible liquids and gases. Analytical Chemistry 20:238-41.
Shelton, E.M. 1974. Burner oil fuels. Petroleum Products Survey 86. U.S.
Bureau of Mines, Washington, D.C.
Shnidman, L. 1954. Gaseous fuels. American Gas Association, Arlington, VA.
Stickford, G.H., S.G. Talbert, B. Hindin, and D.W. Locklin. 1988. Re-search on corrosion-resistant materials for condensing heat exchang-ers. Proceedings of the 39th Annual International Appliance Technical Conference.
Stone, R.L. 1969. Fireplace operation depends upon good chimney design.
ASHRAE Journal 11(February):63-69.
Trinks, W. 1947. Simplified calculation of radiation from non-luminous fur-nace gases. Industrial Heating 14:40-46.
U.S. Bureau of Mines. Semiannually. Mineral Industry Surveys, Motor Gas-olines. Washington, D.C.
Zabetakis, M.G. 1956. Research on the combustion and explosion hazards of hydrogen-water vapor-air mixtures. Division of Explosives Technology, Progress Report 1. U.S. Bureau of Mines, Washington, D.C.
BIBLIOGRAPHY Bonne, U. and A. Patani. 1982. Combustion system performance analysis and simulation study. Report GRI-81/0093 (PB 83-161 406). Honeywell SSPL, Bloomington, MN.
Gas Appliance Technology Center, Gas Research Institute, Manufacturer update on status of GATC research on heat-exchanger corrosion, May 1984. Battelle Columbus Laboratories and American Gas Association Laboratories.
Lewis, B. and G. von Elbe. 1987. Combustion, flames, and explosion of gases, 3rd ed. Academic Press, New York.
Stickford, G.H., S.G. Talbert, and D.W. Locklin. 1987. Condensate corrosiv-ity in residential condensing appliances. Proceedings of the International Symposium on Condensing Heat Exchangers, Paper 3, BNL Report No.
52068, 1 and 2. Brookhaven National Laboratory, Upton, NY.
19.1 CHAPTER 19 REFRIGERANTS Phaseout of Refrigerants .............................................................................................................. 19.1 Refrigerant Properties .................................................................................................................. 19.4 Refrigerant Performance ............................................................................................................. 19.6 Safety ............................................................................................................................................ 19.6 Leak Detection ............................................................................................................................. 19.7 Effect on Construction Materials ................................................................................................ 19.11 EFRIGERANTS are the working fluids in refrigeration, air-Rconditioning, and heat pumping systems. They absorb heat from one area, such as an air-conditioned space, and reject it into another, such as outdoors, usually through evaporation and conden-sation, respectively. These phase changes occur both in absorption and mechanical vapor compression systems, but they do not occur in systems operating on a gas cycle using a fluid such as air. (See Chapter 1 for more information on refrigeration cycles.) The design of the refrigeration equipment depends strongly on the properties of the selected refrigerant. Table 1 lists ASHRAE standard refrigerant designations from ASHRAE Standard 34.
Refrigerant selection involves compromises between conflicting desirable thermodynamic properties. A refrigerant must satisfy many requirements, some of which do not directly relate to its abil-ity to transfer heat. Chemical stability under conditions of use is the most important characteristic. Safety codes may require a nonflam-mable refrigerant of low toxicity for some applications. Cost, avail-ability, efficiency, and compatibility with compressor lubricants and materials with which the equipment is constructed are other concerns.
The environmental consequences of a refrigerant that leaks from a system must also be considered. Because of their great stability, fully halogenated compounds, such as chlorofluorocarbons (CFCs), persist in the atmosphere for many years and eventually diffuse into the stratosphere. The molecules of CFCs, such as R-11 and R-12, contain only carbon and the halogens chlorine and fluo-rine. Once in the upper atmosphere, CFC molecules break down and release chlorine, which destroys ozone (ozone depletion). In the lower atmosphere, these molecules absorb infrared radiation, which may contribute to the warming of the earth. Substitution of a hydro-gen atom for one or more of the halogens in a CFC molecule greatly reduces its atmospheric lifetime and lessens its environmental impact. These compounds are called hydrochlorofluorocarbons (HCFCs). A similar class of compounds used as fire extinguishing agents and called halons also cause ozone depletion. Halons are compounds containing bromine, fluorine, and carbon. Like CFCs, halons break down, but release bromine, which is even more destructive to stratospheric ozone than chlorine.
Latent heat of vaporization is another important property. On a molar basis, fluids with similar boiling points have almost the same latent heat. Since the compressor operates on volumes of gas, refrig-erants with similar boiling points produce similar capacities in a given compressor. On a mass basis, latent heat varies widely among fluids. The maximum efficiency of a theoretical vapor compression cycle is achieved by fluids with low vapor heat capacity. This prop-erty is associated with fluids having a simple molecular structure and low molecular weight.
Transport properties of thermal conductivity and viscosity affect the performance of heat exchangers and piping. High thermal con-ductivity and low viscosity are desirable.
No single fluid satisfies all the attributes desired of a refrigerant; as a result, a variety of refrigerants is used. This chapter describes the basic characteristics of various refrigerants, and Chapter 20 lists thermophysical properties.
PHASEOUT OF REFRIGERANTS The Montreal Protocol is an international treaty that controls the production of ozone-depleting substances, including refrigerants containing chlorine and/or bromine (U.N. 1994, 1996). The original Protocol was signed September 16, 1987, by the European Eco-nomic Community (currently the European Union) and 24 nations, including the United States. It entered into force on January 1, 1989, and limits the 1998 production of specified CFCs to 50% of their 1986 levels. Starting in 1992, the production of specified halons (including R-13B1) was frozen at 1986 levels. Developing coun-tries were granted additional time to meet these deadlines.
The original Protocol contained provisions for periodic revision.
Four such revisions, referred to as the London, Copenhagen, Mon-treal, and Beijing Amendments, were agreed to in 1990, 1992, 1997 and 1999, respectively. As of February, 2000, the Montreal Protocol had been ratified by 172 parties, the London Amendment by 138 parties, and the Copenhagen Amendment by 104 parties; the Beijing amendment has yet to be ratified.
The Copenhagen Amendment entered into force on June 14, 1994. It called for a complete cessation of the production of CFCs by January 1, 1996, and of halons by January 1, 1994. Continued use from existing (reclaimed or recycled) stock is permitted. Allow-ance is also provided for continued production for very limited essential uses. In addition, HCFCs (such as R-22 and R-123) are to be phased out relative to a 1989 reference level for developed coun-tries. Production was frozen at the reference level on January 1, 1996. Production will be limited to 65% of the reference level by January 1, 2004; to 35% by January 1, 2010; to 10% by January 1, 2015; and to 0.5% of the reference level by January 1, 2020. Com-plete cessation of the production of HCFCs is called for by January 1, 2030. In addition to the international agreement, individual coun-tries may have domestic regulations for ozone-depleting com-pounds.
The Beijing Amendment will regulate the production of HCFCs in developed countries. A production cap will begin in 2004 and will be equal to the original HCFC use cap plus an additional 15% allowance to meet developing country needs. At this time, there is no provision for reductions to this production cap. The production and use of hydrofluorocarbon (HFC) refrigerants (such as R-32, R-125, R-134a, R-143a, and their mixtures, includ-ing R-404A, R-407C, and R-410A) are not regulated by the Mont-real Protocol. The preparation of this chapter is assigned to TC 3.1, Refrigerants and Sec-ondary Coolants.
19.2 2001 ASHRAE Fundamentals Handbook Table 1 Standard Designation of Refrigerants (ASHRAE Standard 34) Refrigerant Number Chemical Name or Composition (% by mass) Chemical Formula Refrigerant Number Chemical Name or Composition (% by mass) Chemical Formula Methane Series 403A R-290/22/218 (5/75/20) 10 tetrachloromethane (carbon tetrachloride) CCl4 403B R-290/22/218 (5/56/39) 11 trichlorofluoromethane CCl3F 404A R-125/143a/134a (44/52/4) 12 dichlorodifluoromethane CCl2F2 405A R-22/152a/142b/C318 (45/7/5.5/42.5) 12B1 bromochlorodifluoromethane CBrClF2 406A R-22/600a/142b (55/4/41) 12B2 dibromodifluoromethane CBr2F2 407A R-32/125/134a (20/40/40) 13 chlorotrifluoromethane CClF3 407B R-32/125/134a (10/70/20) 13B1 bromotrifluoromethane CBrF3 407C R-32/125/134a (23/25/52) 14 tetrafluoromethane (carbon tetrafluoride) CF4 407D R-32/125/134a (15/15/70) 20 trichloromethane (chloroform) CHCl3 408A R-125/143a/22 (7/46/47) 21 dichlorofluoromethane CHCl2F 409A R-22/124/142b (60/25/15) 22 chlorodifluoromethane CHClF2 409B R-22/124/142b (65/25/10) 22B1 bromodifluoromethane CHBrF2 410A R-32/125 (50/50) 23 trifluoromethane CHF3 410B R-32/125 (45/55) 30 dichloromethane (methylene chloride) CH2Cl2 411A R-1270/22/152a (1.5/87.5/11.0) 31 chlorofluoromethane CH2ClF 411B R-1270/22/152a (3/94/3) 32 difluoromethane (methylene fluoride) CH2F2 412A R-22/218/142b (70/5/25) 40 chloromethane (methyl chloride) CH3Cl 413A R-218/134a/600a (9/88/3) 41 fluoromethane (methyl fluoride) CH3F Azeotropic Blends (% by mass) 50 methane CH4 500 R-12/152a (73.8/26.2) Ethane Series 501 R-22/12 (75.0/25.0) 110 hexachloroethane CCl3CCl3 502 R-22/115 (48.8/51.2) 111 pentachlorofluoroethane CCl3CCl2F 503 R-23/13 (40.1/59.9) 112 1,1,2,2-tetrachloro-1,2-difluoroethane CCl2FCCl2F 504 R-32/115 (48.2/51.8) 112a 1,1,1,2-tetrachloro-2,2-difluoroethane CCl3CClF2 505 R-12/31 (78.0/22.0) 113 1,1,2-trichloro-1,2,2-trifluoroethane CCl2FCClF2 506 R-31/114 (55.1/44.9) 113a 1,1,1-trichloro-2,2,2-trifluoroethane CCl3CF3 507A R-125/143a (50/50) 114 1,2-dichloro-1,1,2,2-tetrafluoroethane CClF2CClF2 508A R-23/116 (39/61) 114a 1,1-dichloro-1,2,2,2-tetrafluoroethane CCl2FCF3 508B R-23/116 (46/54) 114B2 1,2-dibromo-1,1,2,2-tetrafluoroethane CBrF2CBrF2 509A R-22/218 (44/56) 115 chloropentafluoroethane CClF2CF3 Miscellaneous Organic Compounds 116 hexafluoroethane CF3CF3 Hydrocarbons 120 pentachloroethane CHCl2CCl3 600 butane CH3CH2CH2CH3 123 2,2-dichloro-1,1,1-trifluoroethane CHCl2CF3 600a 2-methyl propane (isobutane) CH(CH3)3 123a 1,2-dichloro-1,1,2-trifluoroethane CHClFCClF2 Oxygen Compounds 124 2-chloro-1,1,1,2-tetrafluoroethane CHClFCF3 610 ethyl ether C2H5OC2H5 124a 1-chloro-1,1,2,2-tetrafluoroethane CHF2CClF2 611 methyl formate HCOOCH3 125 pentafluoroethane CHF2CF3 Sulfur Compounds 133a 2-chloro-1,1,1-trifluoroethane CH2ClCF3 620 (Reserved for future assignment) 134a 1,1,1,2-tetrafluoroethane CH2FCF3 Nitrogen Compounds 140a 1,1,1-trichloroethane (methyl chloroform) CH3CCl3 630 methyl amine CH3NH2 141b 1,1-dichloro-1-fluoroethane CCl2FCH3 631 ethyl amine C2H5NH2 142b 1-chloro-1,1-difluoroethane CClF2CH3 Inorganic Compounds 143a 1,1,1-trifluoroethane CF3CH3 702 hydrogen H2 150a 1,1-dichloroethane CHCl2CH3 704 helium He 152a 1,1-difluoroethane CHF2CH3 717 ammonia NH3 160 chloroethane (ethyl chloride) CH3CH2Cl 718 water H2O 170 ethane CH3CH3 720 neon Ne Propane Series 728 nitrogen N2 216ca 1,3-dichloro-1,1,2,2,3,3-hexafluoropropane CClF2CF2CClF2 732 oxygen O2 218 octafluoropropane CF3CF2CF3 740 argon Ar 245cb 1,1,1,2,2-pentafluoropropane CF3CF2CH3 744 carbon dioxide CO2 290 propane CH3CH2CH3 744A nitrous oxide N2O Cyclic Organic Compounds 764 sulfur dioxide SO2 C316 1,2-dichloro-1,2,3,3,4,4-hexafluorocyclobutane C4Cl2F6 Unsaturated Organic Compounds 1112a 1,1-dichloro-2,2-difluoroethene CCl2=CF2 C317 chloroheptafluorocyclobutane C4ClF7 1113 1-chloro-1,2,2-trifluoroethene CClF=CF2 C318 octafluorocyclobutane C4F8 1114 tetrafluoroethene CF2=CF2 Zeotropic Blends (% by mass) 1120 trichloroethene CHCl=CCl2 400 R-12/114 (must be specified) 1130 1,2-dichloroethene (trans) CHCl=CHCl 401A R-22/152a/124 (53/13/34) 1132a 1,1 difluoroethene (vinylidene fluoride) CF2=CH2 401B R-22/152a/124 (61/11/28) 1140 1-chloroethene (vinyl chloride) CHCl=CH2 401C R-22/152a/124 (33/15/52) 1141 1-fluoroethene (vinyl fluoride) CHF=CH2 402A R-125/290/22 (60/2/38) 1150 ethene (ethylene) CH2=CH2 402B R-125/290/22 (38/2/60) 1270 propene (propylene) CH3CH=CH2 The exact composition of this azeotrope is in question.
Refrigerants 19.3 Table 2 Physical Properties of Selected Refrigerantsa Refrigerant Chemical Formula Molecular Mass Boiling Pt. (NBP) at 14.696 psia, °F Freezing Point, °F Critical Temper-ature, °F Critical Pressure, psia Critical Volume, ft3/lb Refractive Index of Liquidb,c No.
Chemical Name or Composition (% by mass) 704 Helium He 4.0026 −452.1 None −450.3 33.21 0.2311 1.021 (NBP) 5461 Å 702p Hydrogen, para H2 2.0159 −423.2 −434.8 −400.3 187.5 0.5097 1.09 (NBP)f 702n Hydrogen, normal H2 2.0159 −423.0 −434.5 −399.9 190.8 0.5320 1.097 (NBP) 5791 Å 720 Neon Ne 20.183 −410.9 −415.5 −379.7 493.1 0.03316 — 728 Nitrogen N2 28.013 −320.4 −346.0 −232.4 492.9 0.05092 1.205 (83 K) 5893 Å 729 Air — 28.97 −317.8 — −220.95 548.9 0.0530 — −221.1 546.3 0.05007 — 740 Argon Ar 39.948 −302.55 −308.7 −188.48 704.9 0.0301 1.233 (84 K) 5893 Å 732 Oxygen O2 31.9988 −297.332 −361.8 −181.424 731.4 0.03673 1.221 (92 K) 5893 Å 50 Methane CH4 16.04 −258.7 −296 −116.5 673.1 0.099 — 14 Tetrafluoromethane CF4 88.01 −198.3 −299 −50.2 543 0.0256 — 1150 Ethylene C2H4 28.05 −154.7 −272 48.8 742.2 0.070 1.363(−148)1 744A 2 Nitrous oxide N2O 44.02 −129.1 −152 97.7 1048 0.0355 — 170 Ethane C2H6 30.07 −127.85 −297 90.0 709.8 0.0830 — 503 R-23/13 (40.1/59.9) — 87.5 −127.6 — 67.1 607 0.0326 — 508A 9 R-23/116 (39/61) — 100.1 −125.34 — 51.82 536.78 0.0279 — 508B 9 R-23/116 (46/54) — 95.39 −125.28 — 53.71 556.07 0.0280 — 23 Trifluoromethane CHF3 70.02 −115.7 −247 78.1 701.4 0.0311 — 13 Chlorotrifluoromethane CClF3 104.47 −114.6 −294 83.9 561 0.0277 1.146 (77)4 744 Carbon dioxide CO2 44.01 −109.2d −69.9e 87.9 1070.0 0.0342 1.195 (59) 13B1 Bromotrifluoromethane CBrF3 148.93 −71.95 −270 152.6 575 0.0215 1.239 (77)4 504 R-32/115 (48.2/51.8) — 79.2 −71.0 — 151.5 690.5 0.0324 — 32 Difluoromethane CH2F2 52.02 −61.1 −213 173.14 845.6 0.03726 — 410A 9 R-32/125 (50/50) — 72.6 −60.83 — 158.4 694.87 0.0293 — 125 Pentafluoroethane C2HF5 120.03 −55.43 −153.67 151.34 526.57 — — 1270 Propylene C3H6 42.09 −53.86 −301 197.2 670.3 0.0720 1.3640 (−58)1 143a 9 Trifluoroethane CH3CF3 84 −53.039 −169.26 162.87 545.49 0.0372 — 507A 9 R-125/143a (50/50) — 98.9 −52.80 — 159.34 538.97 0.0325 — 404A 9 R-125/143a/134a (44/52/4) — 97.6 −51.66 — 162.5 597.5 0.0279 — 502 5 R-22/115 (48.8/51.2) — 111.63 −49.8 — 179.9 591.0 0.0286 — 407C 9 R-32/125/134a (23/25/52) — 86.2 −46.22 — 186.9 672.2 0.0317 — 290 Propane C3H8 44.10 −43.76 −305.8 206.1 616.1 0.0726 1.3397 (−43) 22 Chlorodifluoromethane CHClF2 86.48 −41.36 −256 204.8 721.9 0.0305 1.234 (77)4 115 Chloropentafluoroethane CClF2CF3 154.48 −38.4 −159 175.9 457.6 0.0261 1.221 (77)4 500 R-12/152a (73.8/26.2) — 99.31 −28.3 −254 221.9 641.9 0.0323 — 717 Ammonia NH3 17.03 −28.0 −107.9 271.4 1657 0.068d 1.325 (61.7) 12 Dichlorodifluoromethane CCl2F2 120.93 −21.62 −252 233.6 596.9 0.0287 1.288 (77)4 134a Tetrafluoroethane CF3CH2F 102.03 −15.08 −141.9 214.0 589.8 0.029 — 152a Difluoroethane CHF2CH3 66.05 −13.0 −178.6 236.3 652 0.0439 — 40 2 Methyl chloride CH3Cl 50.49 −11.6 −144 289.6 968.7 0.0454 — 124 Chlorotetrafluoroethane CHClFCF3 136.47 8.26 −326.47 252.5 530.84 — — 600a Isobutane C4H10 58.13 10.89 −255.5 275.0 529.1 0.0725 1.3514 (−13)1 764 6 Sulfur dioxide SO2 64.07 14.0 −103.9 315.5 1143 0.0306 — 142b Chlorodifluoroethane CClF2CH3 100.5 14.4 −204 278.8 598 0.0368 — 630 6 Methyl amine CH3NH2 31.06 19.9 −134.5 314.4 1082 1.432 (63.5) C318 Octafluorocyclobutane C4F8 200.04 21.5 −42.5 239.6 403.6 0.0258 — 600 Butane C4H10 58.13 31.1 −217.3 305.6 550.7 0.0702 1.3562 (5)1 114 Dichlorotetrafluoroethane CClF2CClF2 170.94 38.8 −137 294.3 473 0.0275 1.294 (77) 21 7 Dichlorofluoromethane CHCl2F 102.92 47.8 −211 353.3 750 0.0307 1.332 (77)4 160 2 Ethyl chloride C2H5Cl 64.52 54.32 −216.9 369.0 764.4 0.0485 — 631 6 Ethyl amine C2H5NH2 45.08 61.88 −113 361.4 815.6 — — 11 Trichlorofluoromethane CCl3F 137.38 74.87 −168 388.4 639.5 0.0289 1.362 (77)4 123 Dichlorotrifluoroethane CHCl2CF3 152.93 82.17 −160.87 362.82 532.87 — — 6116 Methyl formate C2H4O2 60.05 89.2 −146 417.2 870 0.0459 — 141b Dichlorofluoroethane CCl2FCH3 116.95 89.6 — 399.6 616.4 — — 610 6 Ethyl ether C4H10O 74.12 94.3 −177.3 381.2 523 0.0607 1.3526 (68) 216ca Dichlorohexafluoropropane C3Cl2F6 220.93 96.24 −193.7 356.0 399.5 0.0279 — 30 6 Methylene chloride CH2Cl2 84.93 104.4 −142 458.6 882 — 1.4244 (68)3 113 Trichlorotrifluoroethane CCl2FCClF2 187.39 117.63 −31 417.4 498.9 0.0278 1.357 (77)4 1130 8 Dichloroethylene CHCl=CHCl 96.95 118 −58 470 795 — — 1120 6 Trichloroethylene CHCl=CCl2 131.39 189.0 −99 520 728 — 1.4782(68)3 718 6 Water H2O 18.02 212 32 705.18 3200 0.0498 — 19.4 2001 ASHRAE Fundamentals Handbook Notes for Table 2 a Data from ASHRAE Thermodynamic Properties of Refrigerants (Stewart et al. 1986) or from McLinden (1990), unless otherwise noted.
b Temperature of measurement (°F, unless kelvin is noted) shown in paren-theses. Data from CRC Handbook of Chemistry and Physics (CRC 1987), unless otherwise noted.
c For the sodium D line.
d Sublimes.
e At 76.4 psia.
f Dielectric constant data.
References 1 Kirk and Othmer (1956).
2 Matheson Gas Data Book (1966).
3 Electrochemicals Department, E.I. duPont de Nemours & Co.
4 Bulletin B-32A (duPont).
5 Bulletin T-502 (duPont 1980).
6 Handbook of Chemistry (1967).
7 Bulletin G-1 (duPont).
8 CRC Handbook of Chemistry and Physics (CRC 1987).
9 NIST Standard Reference Database 23, Version 6.01.
REFRIGERANT PROPERTIES Physical Properties Table 2 lists some physical properties of commonly used refrig-erants, a few very low-boiling cryogenic fluids, some newer refrig-erants, and some older refrigerants of historical interest. These refrigerants are arranged in increasing order of atmospheric boiling point, from helium at −452.1°F to water at 212°F. Table 2 also includes the freezing point, critical properties, and refractive index. Of these properties, the boiling point is most impor-tant because it is a direct indicator of the temperature level at which a refrigerant can be used. The freezing point must be lower than any contemplated usage. The critical properties describe a material at the point where the distinction between liquid and gas is lost. At higher temperatures, no separate liquid phase is possible. In refrigeration cycles involving condensation, a refrigerant must be chosen that allows this change of state to occur at a temperature somewhat below the critical. Cycles that reject heat at supercritical temperatures (such as cycles using carbon dioxide) are also possible. Fig. 1 Specific Gravity of Aqueous Solutions of Lithium Bromide Fig. 2 Specific Heat of Aqueous Lithium Bromide Solutions Fig. 3 Viscosity of Aqueous Solutions of Lithium Bromide Refrigerants 19.5 Lithium Bromide-Water and Ammonia-Water Solutions.
These are the most commonly used working fluids in absorption refrigeration systems. Figure 1 shows specific gravity, Figure 2 shows specific heat, and Figure 3 shows viscosity of lithium bro-mide-water solutions. Chapter 20 has an enthalpy-concentration dia-gram and a vapor pressure diagram for lithium bromide-water solutions. Chapter 20 also has equilibrium properties of water-ammonia solutions.
Electrical Properties Table 3 and Table 4 list the electrical characteristics of refriger-ants that are especially important in hermetic systems. Table 3 Electrical Properties of Liquid Refrigerants Refrigerant Temp., °F Dielectric Constant Volume Resistivity, MΩ·m Ref.
No.
Chemical Name or Composition (% by mass) 11 Trichlorofluoromethane 84 2.28 1 a 1.92 63680 2 77 2.5 90 3 12 Dichlorodifluoromethane 84 2.13 1 a 1.74 53900 2 77 2.1 > 120 3 77 2.100 4 13 Chlorotrifluoromethane −22 2.3 120 4 68 1.64 22 Chlorodifluoromethane 75 6.11 1 a 6.12 0.83 2 77 6.6 75 3 23 Trifluoromethane −22 6.3 3 68 5.51 4 32 Difluoromethane a 14.27 -6 113 Trichlorotrifluoroethane 86 2.44 1 a 1.68 45490 2 77 2.6 > 120 3 114 Dichlorotetrafluoroethane 88 2.17 1 a 1.83 66470 2 77 2.2 > 70 3 123 2,2-dichloro-1,1,1-trifluoroethane a 4.50 14700 7 124a Chlorotetrafluoroethane 77 4.0 50 3 125 Pentafluoroethane 68 4.94 -8 134a 1,1,1,2-tetrafluoroethane a 9.51 17700 7 290 Propane a 1.27 73840 2 404A R-125/143a/134a (44/52/4) a 7.58 8450 9 407C R-32/125/134a (23/25/52) a 8.74 7420 9 410A R-32/125 (50/50) a 7.78 3920 9 500 R-12/152a (73.8/26.2) a 1.80 55750 2 507A R-125/143a (50/50) a 6.97 5570 9 508A R-23/116 (39/61) −22 6.60 -1 32 5.02 1 508B R-23/116 (46/54) −22 7.24 -1 32 5.48 1 717 Ammonia 69 15.5 5 744 Carbon dioxide 32 1.59 5 a = ambient temperature References: 1 Data from E.I. duPont de Nemours & Co., Inc. Used by permission.
2 Beacham and Divers (1955) 3 Eiseman (1955) 4 Makita et al. (1976) 5 CRC Handbook of Chemistry and Physics (CRC 1987).
6 Bararo et al. (1997) 7 Fellows et al. (1991) 8 Pereira et al. (1999) 9 Meurer et al. (2000) Table 4 Electrical Properties of Refrigerant Vapors Refrigerant Pres-sure, atm.
Temp., °F Dielec-tric Con-stant Relative Dielectric Strength, Nitrogen = 1 Volume Resis-tivity, GΩ·m Ref.
No.
Chemical Name or Composition (% by mass) 11 Trichlorofluoro-methane 0.5 79 1.0019 3 a b 1.009 74.35 2 1.0 73 3.1 4 12 Dichlorodifluoro-methane 0.5 84 1.0016 3 a b 1.012 452c 72.77 2 1.0 73 2.4 4 4.9 68 1.019 5 13 Chlorotrifluoro-methane 0.5 84 1.0013 3 1.0 73 1.4 4 4.9 68 1.013 5 19.5 90 1.055 6 14 Tetrafluoro-methane 0.5 76 1.0006 3 1.0 73 1.0 4 22 Chlorodifluoro-methane 0.5 78 1.0035 3 a b 1.004 460c 2113 2 1.0 73 1.3 4 4.9 68 1.033 5 23 Trifluoromethane 4.9 68 1.042 5 113 Trichlorotri-fluoroethane a b 1.010 440c 94.18 2 0.4 73 2.6 4 114 Dichlorotetra-fluoroethane 0.5 80 1.0021 3 a b 1.002 295c 148.3 2 1.0 73 2.8 4 116 Hexafluoroethane 0.94 73 1.002 3 133a Chlorotri-fluoroethane 0.94 80 1.010 3 142b Chlorodi-fluoroethane 0.93 81 1.013 3 143a Trifluoroethane 0.85 77 1.013 3 170 Ethane 1.0 32 1.0015 1 290 Propane a b 1.009 440c 105.3 2 500 R-12/152a (73.8/26.2) a b 1.024 470c 76.45 2 508A R-23/116 (39/61) a −22 1.12 7 a 32 1.31 7 508B R-23/116 (46/54) a −22 1.13 7 a 32 1.34 7 717 Ammonia 1.0 32 1.0072 1 a 32 0.82 4 729 Air 1.0 32 1.00059 1 744 Carbon dioxide 1.0 32 1.00099 1 1.0 b 0.88 4 1150 Ethylene 1.0 32 1.00144 1 1.0 73 1.21 4 Notes: a = saturation vapor pressure b = ambient temperature c = measured breakdown voltage, volts/mil References: 1 CRC Handbook of Chemistry and Physics (CRC 1987) 2 Beacham and Divers (1955) 3 Fuoss (1938) 4 Charlton and Cooper (1937) 5 Makita et al. (1976) 6 Hess et al. (1962) 7 Data from E.I. duPont de Nemours & Co., Inc. Used by permission.
19.6 2001 ASHRAE Fundamentals Handbook Sound Velocity Table 5 gives examples of the velocity of sound in the vapor phase of various fluorinated refrigerants. Chapter 20 has sound velocity data for many refrigerants. The velocity increases when the temperature is increased and decreases when the pressure is increased. The velocity of sound can be calculated from the equation (1) where V a = sound velocity, ft/s gc = gravitational constant = 32.1740 lbm·ft/lbf·s2 p = absolute pressure, lbf/ft2 ρ = density, lbm/ft2 γ = cp/cv = ratio of specific heats S = entropy, Btu/lb·°R T = temperature, °R The sound velocity can be estimated from the tables of thermo-dynamic properties. The change in pressure with a change in density (dp/dρ) can be estimated either at constant entropy or at constant temperature. It is simpler to estimate at constant temperature but then the ratio of specific heats must also be known. The practical velocity of a gas in piping or through openings is limited by the velocity of sound in the gas.
Latent Heat of Vaporization An empirical rule of chemistry (Trouton’s rule) states that the latent heat of vaporization at the boiling point on a molar basis, divided by the temperature in absolute units, is a constant for most materials. This rule is applied to refrigerants in Table 6. It applies fairly well to these refrigerants, although the result is not entirely constant. The rule helps in comparing different refrigerants and in understanding the operation of refrigeration systems.
REFRIGERANT PERFORMANCE Chapter 1 describes several methods of calculating refrigerant performance, and Chapter 20 includes tables of thermodynamic properties of the various refrigerants.
Table 7 shows the theoretical calculated performance of a num-ber of refrigerants for the U.S. standard cycle of 5°F evaporation and 86°F condensation. Calculated data for other conditions are given in Table 8. The tables can be used to compare the properties of different refrigerants, but actual operating conditions are some-what different from the calculated data. In most cases, the suction vapor is assumed to be saturated, and the compression is assumed adiabatic or at constant entropy. For R-113 and R-114, these assumptions would cause some liquid in the discharge vapor. In these cases, it is assumed that the discharge vapor is saturated and that the suction vapor is slightly superheated. In Section F of Table 8, the temperature of the suction gas is assumed to be 65°F (−10°F saturated evaporating plus 75°F superheat). Comparison with Sec-tion E illustrates the effect of suction gas superheating on refrigerant performance.
SAFETY Table 9 summarizes the toxicity and flammability characteristics of many refrigerants. In ASHRAE Standard 34, refrigerants are classified according to the hazard involved in their use. The toxicity and flammability classifications yield six safety groups (A1, A2, A3, B1, B2, and B3) for refrigerants. Group A1 refrigerants are the least hazardous, Group B3 the most hazardous.
The safety classification in ASHRAE Standard 34 consists of a capital letter and a numeral. The capital letter designates the toxicity of the refrigerant at concentrations below 400 ppm by volume: V a gc dp dρ ------ S 0.5 γgc dp dρ ------ T 0.5 = = Table 5 Velocity of Sound in Refrigerant Vapors Refrigerant Pressure, psia Temperature, °F 50 100 150 Velocity of Sound, ft/s 11 10 b 469 490 12 10 480 503 525 100 b 457 490 200 b b 442 22 10 583 610 635 100 b 574 607 200 b 523 572 23 10 657 685 712 100 631 666 699 200 600 644 682 32 10 775 809 840 100 726 774 815 200 b 730 784 113 10 b 435 456 114 10 391 411 430 123 10 b 435 456 100 b b b 200 b b b 124 10 443 465 486 100 b b 439 200 b b b 125 10 477 500 521 100 b 466 497 200 b 420 467 134a 10 517 543 566 100 b 490 528 200 b b 476 143a 10 576 603 629 100 513 558 595 200 b 495 552 290 10 799 835 870 100 b 771 820 200 b b 754 404A 10 532 557 581 100 473 515 549 200 b 456 509 407C 10 573 600 625 100 b 558 594 200 b 500 555 410A 10 635 664 691 100 587 629 665 200 b 585 634 502 10 501 525 547 100 450 488 519 200 b 435 483 507A 10 529 554 577 100 471 512 546 200 b 454 507 508A 10 n.a.
n.a.
n.a.
100 538 538 538 200 549 549 549 508B 10 n.a.
572 595 100 n.a.
553 581 200 489 531 565 600 10 678 712 743 100 b b 652 200 b b b 600a 10 680 713 744 100 b b 666 200 b b b 717 10 1388 1453 1513 100 b 1403 1477 200 b 1336 1432 744 10 862 899 935 100 843 885 924 200 820 869 912 Source: NIST Standard Reference Database 23, Version 6.01 (NIST 1996).
b = Below saturation temperature. n.a. = Not available Refrigerants 19.7 • Class A Toxicity not identified • Class B Evidence of toxicity identified The numeral denotes the flammability of the refrigerant: • Class 1 No flame propagation in air at 65°F and 14.7 psia • Class 2 Lower flammability limit (LFL) greater than 0.00625 lb/ft3 at 70°F and 14.7 psia and heat of combustion less than 8174 Btu/lb • Class 3 Highly flammable as defined by LFL less than or equal to 0.00625 lb/ft3 at 70°F and 14.7 psia or heat of combustion greater than or equal to 8174 Btu/lb LEAK DETECTION Leak detection in refrigeration equipment is a major problem for manufacturers and service engineers. The following sections describe several leak detection methods.
Electronic Detection The electronic detector is widely used in the manufacture and assembly of refrigeration equipment. Instrument operation depends on the variation in current flow caused by ionization of decomposed refrigerant between two oppositely charged platinum electrodes.
This instrument can detect any of the halogenated refrigerants except R-14; however, it is not recommended for use in atmo-spheres that contain explosive or flammable vapors. Other vapors, such as alcohol and carbon monoxide, may interfere with the test.
The electronic detector is the most sensitive of the various leak detection methods, reportedly capable of sensing a leak of 1/100 oz of R-12 per year. A portable model is available for field testing.
Other models are available with automatic balancing systems that correct for refrigerant vapors that might be present in the atmo-sphere around the test area. Halide Torch The halide torch is a fast and reliable method of detecting leaks of chlorinated refrigerants. Air is drawn over a copper element heated by a methyl alcohol or hydrocarbon flame. If halogenated vapors are present, they decompose, and the color of the flame changes to bluish-green. Although not as sensitive as the electronic detector, this method is suitable for most purposes.
Bubble Method The object to be tested is pressurized with air or nitrogen. A pres-sure corresponding to operating conditions is generally used. The object is immersed in water, and any leaks are detected by observing bubbles in the liquid. Adding a detergent to the water decreases the surface tension, prevents escaping gas from clinging to the side of the object, and promotes the formation of a regular stream of small bubbles. Kerosene or other organic liquids are sometimes used for the same reason. A solution of soap or detergent can be brushed or poured onto joints or other spots where leakage is suspected. Leak-ing gas forms soap bubbles that can be readily detected.
Leaks can also be determined by pressurizing or evacuating and observing the change in pressure or vacuum over a period of time.
This is effective in checking the tightness of the system but does not locate the point of leakage.
Ammonia and Sulfur Dioxide Leaks Ammonia can be detected by burning a sulfur candle in the vicin-ity of the suspected leak or by bringing a solution of hydrochloric acid near the object. If ammonia vapor is present, a white cloud or smoke of ammonium sulfite or ammonium chloride forms. Ammo-nia can also be detected with indicator paper that changes color in the presence of a base.
Sulfur dioxide can be detected by the appearance of white smoke when aqueous ammonia is brought near the leak. Table 6 Latent Heat of Vaporization Versus Boiling Point Refrigerant Normal Boiling Point, °F Latent Heat λ at NBP, Btu/lb·mol Trouton Constant, λ/°R b Ref.
No.
Chemical Name or Composition (% by mass) 717 Ammonia −28.0 10,036 23.256 1 630 Methyl amine a 23.0 11,141 23.086 4 764 Sulfur dioxide 13.6 10,705 22.626 2 631 Ethyl amine 68.0 11,645 22.076 4 611 Methyl formate a 100.0 12,094 21.616 4 134a Tetrafluoroethane −15.07 9,531 21.44 5 504 R-32/115 (48.2/51.8) −71.0 8,282 21.316 1 23 Trifluoromethane −115.7 7,325 21.29 1 124 Chlorotetrafluoroethane 8.26 9,742 20.82 5 C318 Octafluorocyclobutane 21.5 10,017 20.81 1 21 Dichlorofluoromethane 47.8 10,557 20.80 3 22 Chlorodifluoromethane −41.4 8,687 20.76 1 40 Methyl chloride −10.8 9,305 20.73 3 123 Dichlorotrifluoroethane 82.17 11,215 20.70 5 506 R-31/114 (55.1/44.9) 9.9 9,644 20.54 3 125 Pentafluoroethane −55.43 8,295 20.52 5 113 Trichlorotrifluoroethane 117.6 11,828 20.49 1 152a Difluoroethane −13.0 9,045 20.25 1 502 R-22/115 (48.8/51.2) −49.9 8,280 20.21 3 114 Dichlorotetrafluoroethane 38.8 10,005 20.07 1 216ca Dichlorohexafluoropropane 96.2 11,154 20.07 1 505 R-12/31 (78.0/22.0) c −21.8 8,735 19.95 3 11 Trichlorofluoromethane 74.9 10,648 19.92 1 500 R-12/152a (73.8/26.2) −28.3 8,588 19.91 1 14 Tetrafluoromethane −198.3 5,146 19.69 1 30 Methylene chloride a 120.0 11,398 19.66 4 600 Butane 31.1 9,641 19.64 1 13B1 Bromotrifluoromethane −72.0 7,607 19.62 1 12 Dichlorodifluoromethane −21.6 8,591 19.61 1 142b Chlorodifluoroethane 14.4 9,297 19.61 1 115 Chloropentafluoroethane −38.4 8,245 19.57 1 1270 Propylene −53.9 7,931 19.55 1 503 R-23/13 (40.1/59.9) −126.1 6,483 19.43 1 600a Isobutane 10.9 9,103 19.34 1 13 Chlorotrifluoromethane −114.6 6,670 19.33 1 290 Propane −43.7 8,026 19.29 1 1150 Ethylene −154.7 5,793 19.00 1 170 Ethane −127.9 6,296 18.98 1 50 Methane −258.7 3,521 17.52 1 Notes: a Not at normal atmospheric pressure b Normal boiling temperatures c The exact composition of this azeotrope is in question.
References: 1 ASHRAE Thermodynamic Properties of Refrigerants (Stewart et al. 1986) 2 CRC Handbook of Chemistry and Physics (CRC 1987) 3 ASHRAE (1977) 4 Chemical Engineer’s Handbook (1973) 5 NIST Standard Reference Database 23 (NIST 1996) 19.8 2001 ASHRAE Fundamentals Handbook Table 7 Comparative Refrigerant Performance per Ton of Refrigeration Refrigerant Evapo-rator Pressure, psia Con-denser Pressure, psia Com-pression Ratio Net Refriger-ating Effect, Btu/lbm Refrig-erant Circu-lated, lbm/min Liquid Circu-lated, in3/min Specific Volume of Suction Gas, ft3/lbm Com-pressor Displace-ment, cfm Power Con-sump-tion, hp Coeffi-cient of Perfor-mance Comp. Dis-charge Temp., °F No.
Chemical Name or Composition (% by mass) 170 Ethane 236.41 674.71 2.85 69.27 2.887 289.13 0.534 1.543 1.73 2.72 123 744 Carbon dioxide 332.38 1045.36 3.15 57.75 3.463 158.53 0.264 0.914 1.68 2.81 156 13B1 Bromotrifluoromethane 77.82 264.13 3.39 28.45 7.029 129.78 0.380 2.669 1.13 4.16 104 1270 Propylene 52.70 189.44 3.59 123.15 1.624 90.71 2.049 3.327 1.04 4.56 108 290 Propane 42.37 156.82 3.70 120.30 1.663 95.04 2.459 4.088 1.03 4.57 98 502 R-22/115 (48.8/51.2) 50.56 191.29 3.78 44.91 4.453 103.35 0.802 3.569 1.07 4.42 98 507A R125/R-143a (50/50) 55.3 212.4 3.84 47.28 4.230 114.03 0.810 3.427 1.13 4.18 95 125 Pentafluoroethane 58.87 228.11 3.87 37.69 5.306 126.82 0.628 3.333 1.28 3.67 108 404A R125/143a/134a (44/52/4) 53.3 206.8 3.88 48.98 4.083 110.79 0.856 3.494 1.12 4.21 96 410A R-32/125 (50/50) 69.7 272.6 3.91 72.09 2.775 74.33 0.868 2.409 1.07 4.41 124 22 Chlorodifluoromethane 42.94 172.63 4.02 70.46 2.838 66.88 1.258 3.573 1.02 4.65 129 12 Dichlorodifluoromethane 26.51 107.99 4.07 50.25 3.980 85.23 1.465 5.830 0.99 4.75 100 500 R-12/152a (73.8/26.2) 31.06 127.50 4.10 60.64 3.298 80.19 1.502 4.955 1.01 4.69 105 407C R-32/125/134a (23/25/52) 42.0 183.4 4.37 69.77 2.867 70.34 1.280 3.656 1.05 4.51 117 600a Isobutane 12.92 59.29 4.59 113.00 1.770 90.01 6.419 11.361 1.07 4.41 80 134a Tetrafluoroethane 23.79 111.62 4.69 64.51 3.100 72.37 1.959 6.076 1.02 4.60 96 717 Ammonia 34.17 168.80 4.94 474.20 0.422 19.61 8.179 3.450 0.99 4.77 210 124 Chlorotetrafluoroethane 12.96 64.59 4.98 50.93 3.927 81.16 2.714 10.658 1.05 4.47 90 600 Butane 8.18 41.19 5.04 125.55 1.593 77.78 10.206 16.258 0.95 4.95 88 114 Dichlorotetrafluoroethane a 6.75 36.49 5.41 43.02 4.649 89.56 4.340 20.176 1.02 4.65 86 11 Trichlorofluoromethane 2.94 18.32 6.24 67.21 2.976 56.26 12.240 36.425 0.94 5.02 110 123 Dichlorotrifluoroethane 2.26 15.93 7.06 61.42 3.256 62.19 14.337 46.684 0.97 4.86 90 113 Trichlorotrifluoroethane a 1.01 7.88 7.83 52.08 3.840 68.60 26.285 100.945 1.11 4.27 86 Notes: Data based on 5°F evaporation, 86°F condensation, 0°F subcool, and 0°F superheat.
a Saturated suction except R-113 and R-114. Enough superheat was added to give saturated discharge.
Table 8 Comparative Refrigerant Performance per Ton at Various Evaporating and Condensing Temperatures Refrigerant Suction Temp., °F Evapo-rator Pressure, psia Con-denser Pressure, psia Com-pression Ratio Net Refrig-erating Effect, Btu/lbm Refrig-erant Circu-lated, lbm/min Specific Volume of Suction Gas, ft3/lbm Com-pressor Displace-ment, cfm Power Consump-tion, hp No.
Chemical Name or Composition (% by mass) A. −130°F Saturated Evaporating, 0°F Suction Superheat, −40°F Saturated Condensing 1150 Ethylene −130 30.89 210.67 6.82 142.01 1.408 3.853 5.43 1.756 170 Ethane −130 13.62 112.79 8.28 156.58 1.277 8.357 10.68 1.633 13 Chlorotrifluoromethane −130 9.06 88.04 9.72 45.82 4.365 3.625 15.82 1.685 23 Trifluoromethane −130 9.06 103.03 11.37 79.38 2.520 5.458 13.75 1.753 508A R-23/116 (39/61) −130 12.6 122.2 9.70 44.12 4.533 2.68 12.15 1.738 508B R-23/116 (46/54) −130 12.4 122.8 9.90 47.50 4.210 2.86 12.05 1.734 B. −100°F Saturated Evaporating, 0°F Suction Superheat, −30°F Saturated Condensing 170 Ethane −100 31.27 134.73 4.31 157.76 1.268 3.867 4.90 1.118 23 Trifluoromethane −100 23.74 125.99 5.31 79.37 2.520 2.219 5.59 1.178 13 Chlorotrifluoromethane −100 22.28 106.29 4.77 46.23 4.326 1.563 6.76 1.153 125 Pentafluoroethane −100 3.78 27.76 7.34 56.43 3.544 8.390 29.73 1.101 22 Chlorodifluoromethane −100 2.38 19.63 8.25 90.75 2.204 18.558 40.90 1.074 508A R-23/116 (39/61) −100 31.0 147.5 4.76 44.28 4.517 1.15 5.18 1.180 508B R-23/116 (46/54) −100 30.9 148.4 4.80 47.53 4.208 1.21 5.10 1.173 C. −76°F Saturated Evaporating, 0°F Suction Superheat, 5°F Saturated Condensing 1150 Ethylene −76 109.37 416.24 3.81 116.95 1.710 1.167 1.99 1.478 170 Ethane −76 54.63 235.44 4.31 322.65 0.620 2.291 1.42 0.566 23 Trifluoromethane −76 45.41 237.18 5.22 69.60 2.874 1.203 3.46 1.394 13 Chlorotrifluoromethane −76 40.87 192.14 4.70 39.42 5.074 0.880 4.47 1.382 125 Pentafluoroethane −76 8.21 58.87 7.17 50.62 3.951 4.072 16.09 1.277 290 Propane −76 6.15 42.37 6.89 147.39 1.357 14.856 20.16 1.196 22 Chlorodifluoromethane −76 5.44 42.96 7.90 84.24 2.374 8.593 20.40 1.195 717 Ammonia −76 3.18 34.26 10.79 540.63 0.37 75.784 28.04 1.247 12 Dichlorodifluoromethane −76 3.28 26.50 8.09 58.61 3.412 10.245 34.96 1.191 134a Tetrafluoroethane −76 2.3 23.77 10.32 78.1 2.561 17.304 44.32 1.182 410A R-32/125 (50/50) −76 9.4 69.7 7.41 92.86 2.153 5.84 12.57 1.204 407C R-32/125/134a (23/25/52) −76 4.9 43.5 8.88 86.88 2.302 9.74 22.43 1.200 404A R125/143a/134a (44/52/4) −76 7.2 53.4 7.42 65.25 3.065 5.73 17.55 1.219 Refrigerants 19.9 C. −76°F Saturated Evaporating, 0°F Suction Superheat, 5°F Saturated Condensing (Concluded) 507A R125/143a (50/50) −76 7.6 55.3 7.28 63.43 3.153 5.34 16.84 1.219 508A R-23/116 (39/61) −76 56.8 267.8 4.71 35.23 5.677 0.642 3.64 1.468 508B R-23/116 (46/54) −76 56.8 270.0 4.75 38.22 5.233 0.675 3.53 1.447 D. −40°F Saturated Evaporating, 0°F Suction Superheat, 68°F Saturated Condensing 744 Carbon dioxide −40 145.77 830.5 5.70 77.22 2.590 0.613 1.59 2.208 23 Trifluoromethane −40 103.03 597.9 5.80 45.67 4.379 0.545 2.39 2.442 125 Pentafluoroethane −40 21.84 175.1 8.02 37.44 5.342 1.625 8.68 1.962 290 Propane −40 16.01 121.6 7.55 119.33 1.676 6.083 10.20 1.670 22 Chlorodifluoromethane −40 15.27 132.0 8.65 70.65 2.831 3.280 9.29 1.606 717 Ammonia −40 10.4 124.3 11.95 486.55 0.411 25.144 10.33 1.576 12 Dichlorodifluoromethane −40 9.30 82.3 8.84 49.44 4.046 3.887 15.73 1.596 134a Tetrafluoroethane −40 7.42 83.0 11.19 63.17 3.166 5.790 18.33 1.597 410A R-32/125 (50/50) −40 25.5 208.9 8.19 74.42 2.69 2.28 6.12 1.643 407C R-32/125/134a (23/25/52) −40 14.1 138.7 9.84 70.34 2.84 3.60 10.23 1.624 404A R125/143a/134a (44/52/4) −40 19.5 158.9 8.15 49.36 4.05 2.24 9.07 1.735 507A R125/143a (50/50) −40 20.4 163.4 8.01 47.68 4.19 2.10 8.82 1.747 E. −10°F Saturated Evaporating, 0°F Suction Superheat, 100°F Saturated Condensing 124 Chlorotetrafluoroethane −10 8.95 80.9 9.04 44.99 4.445 3.841 17.08 1.649 134a Tetrafluoroethane −10 16.62 139.0 8.36 56.57 3.535 2.711 9.59 1.589 12 Dichlorodifluoromethane −10 19.20 131.7 6.86 44.89 4.456 1.980 8.82 1.606 717 Ammonia −10 23.73 212.0 8.93 461.25 0.434 11.677 5.07 1.494 22 Chlorodifluoromethane −10 31.23 210.7 6.75 64.07 3.122 1.676 5.23 1.602 502 R-22/115 (48.8/51.2) −10 37.26 230.9 6.20 39.05 5.122 1.073 5.49 1.904 125 Pentafluoroethane −10 43.32 277.0 6.39 31.09 6.433 0.846 5.44 2.172 410A R-32/125 (50/50) −10 51.1 331.6 6.49 64.70 3.09 1.17 3.63 1.672 407C R-32/125/134a (23/25/52) −10 29.7 225.0 7.58 62.45 3.20 1.78 5.69 1.627 404A R125/143a/134a (44/52/4) −10 39.0 251.0 6.44 41.62 4.80 1.16 5.55 1.814 507A R125/143a (50/50) −10 40.7 257.6 6.33 39.96 5.00 1.09 5.46 1.834 F. −10°F Saturated Evaporating, 75°F Suction Superheat (Included in Refrigeration Effect), 100°F Saturated Condensing 123 Dichlorotrifluoroethane 65 1.48 20.8 14.07 67.3 2.972 24.797 73.70 1.387 11 Trichlorofluoromethane 65 1.92 23.4 12.2 71.88 2.783 21.276 59.21 1.403 124 Chlorotetrafluoroethane 65 8.95 80.9 9.04 57.33 3.489 4.531 15.81 1.506 134a Tetrafluoroethane 65 16.62 139.0 8.36 71.25 2.807 3.236 9.08 1.513 12 Dichlorodifluoromethane 65 19.20 131.7 6.86 55.83 3.583 2.360 8.45 1.539 717 Ammonia 65 23.73 212.0 8.93 498.44 0.401 13.751 5.51 1.612 22 Chlorodifluoromethane 65 31.23 210.7 6.75 75.95 2.633 2.012 5.30 1.623 502 R-22/115 (48.8/51.2) 65 37.26 230.9 6.20 51.23 3.904 1.302 5.08 1.761 125 Pentafluoroethane 65 43.32 277.0 6.39 45.13 4.432 1.028 4.56 1.773 410A R-32/125 (50/50) 65 51.1 331.6 6.49 80.50 2.48 1.44 3.58 1.657 407C R-32/125/134a (23/25/52) 65 29.7 225.0 7.58 77.48 2.58 2.14 5.51 1.585 404A R125/143a/134a (44/52/4) 65 39.0 251.0 6.44 57.29 3.49 1.41 4.91 1.636 507A R125/143a (50/50) 65 40.7 257.6 6.33 55.56 3.60 1.33 4.79 1.644 G. 40°F Saturated Evaporating, 0°F Suction Superheat, 100°F Saturated Condensing 125 Pentafluoroethane 40 111.7 275.7 2.47 35.85 5.58 0.331 1.84 0.788 290 Propane 40 78.6 188.6 2.40 119.47 1.67 1.356 2.27 0.692 22 Chlorodifluoromethane 40 83.25 210.7 2.53 68.71 2.911 0.656 1.91 0.696 717 Ammonia 40 73.3 212.0 2.89 480.33 0.416 4.084 1.70 0.653 500 R-12/152a (73.8/26.2) 40 60.72 155.8 2.57 60.54 3.303 0.792 2.62 0.692 12 Dichlorodifluoromethane 40 51.71 131.7 2.55 50.50 3.960 0.778 3.08 0.689 134a Tetrafluoroethane 40 49.77 139.0 2.79 63.72 3.139 0.952 2.99 0.679 124 Chlorotetrafluoroethane 40 27.89 80.9 2.90 52.06 3.842 1.318 5.06 0.698 600a Isobutane 40 26.75 73.4 2.74 115.83 1.727 3.256 5.62 0.693 600 Butane 40 17.68 51.7 2.92 129.22 1.548 4.975 7.70 0.669 11 Trichlorofluoromethane 40 6.99 23.4 3.34 68.04 2.939 5.455 16.03 0.624 123 Dichlorotrifluoroethane 40 5.79 20.8 3.59 62.82 3.184 5.921 18.85 0.635 113 Trichlorotrifluoroethane 40 2.7 10.5 3.89 54.58 3.67 10.5 38.44 0.638 410A R-32/125 (50/50) 40 132.9 331.6 2.50 69.08 2.89 0.456 1.32 0.721 407C R-32/125/134a (23/25/52) 40 84.2 225.0 2.67 68.60 2.92 0.648 1.89 0.699 Table 8 Comparative Refrigerant Performance per Ton at Various Evaporating and Condensing Temperatures (Continued) Refrigerant Suction Temp., °F Evapo-rator Pressure, psia Con-denser Pressure, psia Com-pression Ratio Net Refrig-erating Effect, Btu/lbm Refrig-erant Circu-lated, lbm/min Specific Volume of Suction Gas, ft3/lbm Com-pressor Displace-ment, cfm Power Consump-tion, hp No.
Chemical Name or Composition (% by mass) 19.10 2001 ASHRAE Fundamentals Handbook Table 9 Comparison of Safety Group Classifications in ASHRAE Standard 34-1989 and ASHRAE Standard 34-1997 Refrigerant Number Chemical Formula Safety Group Old New 10 CCl4 2 B1 11 CCl3F 1 A1 12 CCl2F2 1 A1 13 CClF3 1 A1 13B1 CBrF3 1 A1 14 CF4 1 A1 21 CHCl2F 2 B1 22 CHClF2 1 A1 23 CHF3 A1 30 CH2Cl2 2 B2 32 CH2F2 A2 40 CH3Cl 2 B2 50 CH4 3a A3 113 CCl2FCClF2 1 A1 114 CClF2CClF2 1 A1 115 CClF2CF3 1 A1 116 CF3CF3 A1 123 CHCl2CF3 B1 124 CHClFCF3 A1 125 CHF2CF3 A1 134a CF3CH2F A1 142b CClF2CH3 3b A2 143a CF3CH3 A2 152a CHF2CH3 3b A2 170 CH3CH3 3a A3 218 CF3CF2CF3 A1 290 CH3CH2CH3 3a A3 C318 C4F8 1 A1 400 R-12/114 (must be specified) 1 A1/A1 500 R-12/152a (73.8/26.2) 1 A1 501 R-22/12 (75.0/25.0) 1 A1 502 R-22/115 (48.8/51.2) 1 A1 507A R-125/143a (50/50) A1 508A R-23/116 (39/61) A1 508B R-23/116 (46/54) A1/A1 509A R-22/218 (44/56) A1 600 CH3CH2CH2CH3 3a A3 600a CH(CH3)3 3a A3 611 HCOOCH3 2 B2 702 H2 A3 704 He A1 717 NH3 2 B2 718 H2O A1 720 Ne A1 728 N2 A1 740 Ar A1 744 CO2 1 A1 764 SO2 2 B1 1140 CHCl=CH2 B3 1150 CH2=CH2 3a A3 1270 CH3CH=CH2 3a A3 The exact composition of this azeotrope is in question.
Table 10 Swelling of Elastomers in Liquid Refrigerants at Room Temperature Linear Swell, % Refrig.
No.
Buna N Buna S (GR-S) Butyl (GR-1) Natural Rubber Neo-prene GN Thiokol FA Viton B Silicone 11 6 21 41 23 17 2 6 38 12 2 3 6 6 0 1 9 — 13 1 1 0 1 0 0 4 — 13B1 1 1 2 1 2 — 7 — 21 48 49 24 34 28 28 22 — 22 26 4 1 6 2 4 20 20 30 52 26 23 34 37 59 — — 40 35 20 16 26 22 11 — — 113 1 9 21 17 3 1 7 34 114 0 2 2 2 0 0 9 — 502 7 3 — 4 1 — — — 600 1 8 20 16 3 0 — — Adapted from Eiseman (1949).
Table 11 Diffusion of Water and R-22 Through Elastomers Elastomer Diffusion Rate Watera R-22b Neoprene 0.717 1.31 Buna N 0.109 19.7 Hypalon 40 0.457 0.52 Butyl 0.043 0.30 Viton — 3.61 Polyethylene 0.123 — Natural 1.428 — Adapted from Eiseman (1966).
a 0.003 in. film, 100% rh at 100°F. Water diffusion rate is in pounds per hour per 1000 ft2 of elastomer.
b Film thickness = 0.001 in.; temperature = 77°F. Gas at 1 atm. and 32°F. Diffusion rate per day in ft3 of gas per ft2 of elastomer.
Table 12 Swelling of Plastics in Liquid Refrigerants at Room Temperature Plastic Linear Swell, % Refrigerant 11 12 21 30 113 114a 22 Phenol formaldehyde resin 0 0 0 0 −0.2 −0.2 n.a.
Cellulose acetate 0.4 0 b b 0 −0.1 n.a.
Cellulose nitrate 0.6 0 b b 0 −0.1 n.a.
Nylon 0 0 0 0 0 −0.2 1 Methyl methacrylate resin 0 −0.1 b b −0.2 −0.2 a Polyethylene 6.7 0.4 4.5 4.6 2.3 0.6 2 Polystyrene b −0.1 b b −0.2 −0.2 n.a.
Polyvinyl alcohol 0.3 −0.7 12.9 9.1 −0.1 0.2 n.a.
Polyvinyl chloride 0 0 15.1 b 0 0.1 n.a.
Polyvinylidene chloride −0.2 0 1.0 2.4 −0.1 0 4 Polytetrafluoroethylene 0 −0.7 0.1 0 0 −0.3 1 Adapted from Brown (1960) n.a. = data not available a = sample completely disintegrated Refrigerants 19.11 EFFECT ON CONSTRUCTION MATERIALS Metals Halogenated refrigerants can be used satisfactorily under normal conditions with most common metals, such as steel, cast iron, brass, copper, tin, lead, and aluminum. Under more severe conditions, var-ious metals affect such properties as hydrolysis and thermal decom-position in varying degrees. The tendency of metals to promote thermal decomposition of halogenated compounds is in the follow-ing order: (least decomposition) Inconel < 18-8 stainless steel < nickel < copper < 1340 steel < aluminum < bronze < brass < zinc < silver (most decomposition) This order is only approximate, and exceptions may be found for individual compounds or for special use conditions. The effect of metals on hydrolysis is probably similar.
Magnesium, zinc, and aluminum alloys containing more than 2% magnesium are not recommended for use with halogenated compounds where even trace amounts of water may be present.
Warning: Never use methyl chloride with aluminum in any form. A highly flammable gas is formed, and the explosion hazard is great.
Ammonia should never be used with copper, brass, or other alloys containing copper. When water is present in sulfur dioxide systems, sulfurous acid is formed and can attack iron or steel rapidly and other metals at a slower rate.
Further discussion of the compatibility of refrigerants and lubri-cants with construction materials may be found in Chapter 5 of the 1998 ASHRAE Handbook—Refrigeration.
Elastomers The linear swelling of some elastomers in the liquid phase of var-ious refrigerants is shown in Table 10. Swelling data can be used to a limited extent in comparing the effect of refrigerants on elas-tomers. However, other factors, such as the amount of extraction, tensile strength, and degree of hardness of the exposed elastomer must be considered. When other fluids are present in addition to the refrigerant, the combined effect on elastomers should be deter-mined. In some instances, somewhat higher swelling of elastomers is found in mixtures of R-22 and lubricating oil than in either fluid alone. Table 11 shows the diffusion rate of water and R-22 through elastomers.
Plastics The effect of a refrigerant on a plastic material should be thor-oughly examined under the conditions of intended use. Plastics are often mixtures of two or more basic types, and it is difficult to pre-dict the effect of the refrigerant. The linear swelling of some plastic materials in refrigerants is shown in Table 12. Swelling data can be used as a guide but, as with elastomers, the effect on the properties of the plastic should also be examined. Comparable data for R-22 is limited, but the effect on plastics is generally more severe than that of R-12, but not as severe as that of R-21. The effect of R-114 is very similar to that of R-114a.
REFERENCES ASHRAE. 1977. ASHRAE Handbook—Fundamentals, Chapter 16.
ASHRAE. 1997. Number designation and safety classification of refriger-ants. ANSI/ASHRAE Standard 34-1997.
Bararo, M.T., U.V. Mardolcar, and C.A. Nieto de Castro. 1997. Molecular properties of alternative refrigerants derived from dielectric-constant measurements. Journal of Thermophysics 18(2):419-438.
Beacham, E.A. and R.T. Divers. 1955. Some aspects of the dielectric prop-erties of refrigerants. Refrigerating Engineering 7:33.
Brown, J.A. 1960. Effect of propellants on plastic valve components. Soap and Chemical Specialties 3:87.
Charlton, E.E. and F.S. Cooper. 1937. Dielectric strengths of insulating flu-ids. General Electric Review 865(9):438.
Chemical engineer’s handbook, 5th ed. 1973. McGraw-Hill, New York.
CRC Handbook of chemistry and physics, 68th ed. 1987. CRC Press, Boca Raton, FL.
duPont. Bulletin B-32A. Freon Products Division. E.I. duPont de Nemours & Co., Inc., Wilmington, DE.
duPont. Bulletin G-1. Freon Products Division. E.I. duPont de Nemours & Co., Inc., Wilmington, DE.
duPont. 1980. Bulletin T-502. Freon Products Division. E.I. duPont de Nem-ours & Co., Inc., Wilmington, DE.
Eiseman, B.J., Jr. 1949. Effect on elastomers of Freon compounds and other halohydrocarbons. Refrigerating Engineering 12:1171.
Eiseman, B.J., Jr. 1955. How electrical properties of Freon compounds affect hermetic system’s insulation. Refrigerating Engineering 4:61.
Fellows, B.R., R.G. Richard, and I.R. Shankland. 1991. Electrical character-ization of alternate refrigerants. Actes Congr. Int. Froid, 18th, Vol. 2.
International Institute of Refrigeration, Paris.
Fuoss, R.M. 1938. Dielectric constants of some fluorine compounds. Jour-nal of the American Chemical Society, 1633.
Handbook of chemistry, 10th ed. 1967. McGraw-Hill, New York.
Handbook of chemistry and physics, 41st ed. 1959-60. The Chemical Rub-ber Publishing Co., Cleveland, OH.
Kirk and Othmer. 1956. The encyclopedia of chemical technology. The Interscience Encyclopedia, Inc., New York.
Matheson gas data book. 1966. The Matheson Company, Inc., East Ruther-ford, NJ.
McLinden, M.O. 1990. International Journal of Refrigeration 13:149-62.
Meurer C., G. Pietsch, and M. Haacke M. 2000. Electrical properties of CFC- and HCFC-substitutes. Int. Journal of Refrigeration.
NIST. 1996. Standard Reference Database 23, Version 6.01. National Insti-tute of Standards and Technology, Gaithersburg, MD.
Pereira L.F., F.E. Brito, A.N. Gurova, U.V. Mardolcar, and C.A. Nieta de Castro. 1999. Dipole moment, expansivity and compressibility coeffi-cients of HFC 125 derived from dielectric constant measurements. 1st Int. Workshop on Thermochemical, Thermodynamic and Transport Properties of Halogenated Hydrocarbons and Mixtures, Pisa, Italy.
Stewart, R.B., R.T. Jacobsen, and S.G. Penoncello. 1986. ASHRAE Thermo-dynamic properties of refrigerants. ASHRAE.
U.N. 1994. 1994 Report of the refrigeration, air conditioning, and heat pumps technical options committee. United Nations Environment Pro-gramme, Nairobi, Kenya. ISBN 92-807-1455-4.
U.N. 1996. OzonAction (The Newsletter of the United Nations Environment Programme Industry and Environment OzonAction Programme). Octo-ber (No. 20):10.
20.1 CHAPTER 20 THERMOPHYSICAL PROPERTIES OF REFRIGERANTS HIS CHAPTER presents tabular data for the thermodynamic Tand transport properties of refrigerants arranged for the occasional user. Most of the refrigerants have a thermodynamic property chart on pressure-enthalpy coordinates with an abbrevi-ated set of tabular data for the saturated liquid and vapor on the facing page. In addition, tabular data in the superheated vapor region are given for R134a to assist students working on com-pression cycle examples.
For each of the cryogenic fluids, a second table of properties is provided for the vapor at a pressure of one standard atmo-sphere; these tables provide data needed when these gases are used in heat transfer or purge gas applications. For the zeotropic blends, including R-729 (air), tables are incremented in pressure with properties given for the liquid on the bubble line and vapor on the dew line. This arrangement is chosen because pressure is more commonly measured in the field while servicing equip-ment; it also highlights the difference between the bubble and dew point temperatures—the so-called temperature glide experi-enced with blends.
For a few fluids there are gaps in some properties at low temper-atures due to limitations in the data and/or the models used.
New for the 2001 ASHRAE Handbook are R-143a and R-245fa.
Most of the CFC refrigerants have been deleted. Tables for R-11, R-13, R-113, R-114, R-141b, R-142b, R-500, R-502, R-503, and R-720 (neon) may be found in the 1997 ASHRAE Handbook. R-12 has been retained to assist in making comparisons. Revised formu-lations have been used for many of the HFC refrigerants; these con-form to international standards, where applicable. The formulations used are detailed in the References section.
The reference states used for most of the refrigerants correspond to the international convention of 200 kJ/kg for enthalpy and 1 kJ/(kg·K) for entropy, both for the saturated liquid at 0°C. The exceptions are water and fluids that have very low critical temper-atures, such as ethylene and the cryogens.
These data are intended to help engineers make preliminary comparisons among unfamiliar fluids. For greater detail and a wider range of data, consult the sources listed in the References section.
Refrigerant Page Refrigerant Page Halocarbon Refrigerants R-718 (water/steam) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.32 Methane Series R-744 (carbon dioxide) . . . . . . . . . . . . . . . . . . . . . . . . . . 20.34 R-12 (dichlorodifluoromethane). . . . . . . . . . . . . . . . . . . 20.2 R-22 (chlorodifluoromethane) . . . . . . . . . . . . . . . . . . . . 20.4 Hydrocarbon Refrigerants R-23 (trifluoromethane) . . . . . . . . . . . . . . . . . . . . . . . . . 20.6 R-50 (methane) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.36 R-32 (difluoromethane) . . . . . . . . . . . . . . . . . . . . . . . . . 20.8 R-170 (ethane). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.38 Ethane Series R-290 (propane) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.40 R-123 (2,2-dichloro-1,1,1-trifluoroethane) . . . . . . . . . . 20.10 R-600 (n-butane). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.42 R-124 (2-chloro-1,1,1,2-tetrafluoroethane) . . . . . . . . . . 20.12 R-600a (isobutane) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.44 R-125 (pentafluoroethane) . . . . . . . . . . . . . . . . . . . . . . . 20.14 R-1150 (ethylene) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.46 R-134a (1,1,1,2-tetrafluoroethane). . . . . . . . . . . . . . . . . 20.16 R-1270 (propylene). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.48 R-152a (1,1-difluoroethane). . . . . . . . . . . . . . . . . . . . . . 20.20 R-143a (1,1,1-trifluoroethane) . . . . . . . . . . . . . . . . . . . . 20.22 Cryogenic Fluids Propane Series R-702 (normal hydrogen) . . . . . . . . . . . . . . . . . . . . . . . . 20.50 R-245fa (1,1,1,3,3-pentafluoropropane). . . . . . . . . . . . . 20.23 R-702p (parahydrogen). . . . . . . . . . . . . . . . . . . . . . . . . . 20.52 Zeotropic Blends (% by mass) R-704 (helium) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.54 R-404A [R-125/143a/134a (44/52/4)] . . . . . . . . . . . . . . 20.24 R-728 (nitrogen) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.56 R-407C [R-32/125/134a (23/25/52)] . . . . . . . . . . . . . . . 20.26 R-729 (air). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.58 R-410A [R-32/125 (50/50)] . . . . . . . . . . . . . . . . . . . . . . 20.28 R-732 (oxygen) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.60 Azeotropic Blends R-740 (argon) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.62 R-507A [R-125/143a (50/50)] . . . . . . . . . . . . . . . . . . . . 20.29 Absorption Solutions Inorganic Refrigerants Ammonia-Water . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.65 R-717 (ammonia) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.30 Water-Lithium Bromide . . . . . . . . . . . . . . . . . . . . . . . . . 20.65 The preparation of this chapter is assigned to TC 3.1, Refrigerants and Secondary Coolants.
20.2 2001 ASHRAE Fundamentals Handbook (SI) Fig. 1 Pressure-Enthalpy Diagram for Refrigerant 12 Thermophysical Properties of Refrigerants 20.3 Refrigerant 12 (Dichlorodifluoromethane) Properties of Saturated Liquid and Saturated Vapor Temp., °C Pres-sure, MPa Density, kg/m3 Liquid Volume, m3/kg Vapor Enthalpy, kJ/kg Entropy, kJ/(kg·K) Specific Heat cp, kJ/(kg·K) cp/cv Velocity of Sound, m/s Viscosity, µPa·s Thermal Cond., mW/(m·K) Surface Tension, mN/m Temp., °C Liquid Vapor Liquid Vapor Liquid Vapor Vapor Liquid Vapor Liquid Vapor Liquid Vapor –100.00 0.00119 1679.1 10.004 113.32 306.09 0.6077 1.7210 0.819 0.449 1.182 1035.
118.5 1005.
6.78 116.7 4.27 26.48 –100.00 –90.00 0.00286 1652.8 4.3948 121.53 310.59 0.6538 1.6861 0.824 0.465 1.176 990.
121.4 819.0 7.18 112.0 4.67 24.90 –90.00 –80.00 0.00619 1626.3 2.1355 129.81 315.19 0.6978 1.6576 0.831 0.481 1.172 945.
124.1 684.9 7.58 107.4 5.08 23.35 –80.00 –70.00 0.01228 1599.5 1.1286 138.17 319.87 0.7400 1.6344 0.840 0.497 1.168 902.
126.7 584.0 7.97 103.0 5.50 21.81 –70.00 –60.00 0.02261 1572.3 0.63992 146.62 324.61 0.7806 1.6156 0.850 0.513 1.166 859.
129.1 505.1 8.37 98.8 5.93 20.30 –60.00 –50.00 0.03911 1544.7 0.38494 155.18 329.39 0.8197 1.6004 0.861 0.530 1.165 816.
131.2 441.8 8.76 94.7 6.38 18.81 –50.00 –40.00 0.06409 1516.5 0.24342 163.86 334.18 0.8577 1.5882 0.873 0.548 1.166 775.
133.0 389.8 9.16 90.7 6.84 17.35 –40.00 –30.00 0.10026 1487.7 0.16057 172.67 338.94 0.8946 1.5784 0.886 0.566 1.169 733.
134.5 346.2 9.55 86.9 7.32 15.91 –30.00 –29.75b 0.10133 1487.0 0.15900 172.89 339.06 0.8955 1.5782 0.887 0.567 1.169 732.
134.5 345.2 9.56 86.8 7.33 15.88 –29.75 –28.00 0.10910 1481.9 0.14841 174.44 339.89 0.9019 1.5767 0.889 0.570 1.170 725.
134.7 338.3 9.63 86.1 7.41 15.63 –28.00 –26.00 0.11854 1476.0 0.13736 176.23 340.83 0.9091 1.5751 0.892 0.574 1.171 717.
135.0 330.6 9.71 85.3 7.51 15.35 –26.00 –24.00 0.12860 1470.1 0.12731 178.02 341.78 0.9163 1.5735 0.895 0.578 1.171 709.
135.2 323.2 9.79 84.6 7.61 15.06 –24.00 –22.00 0.13931 1464.1 0.11815 179.81 342.72 0.9234 1.5720 0.898 0.582 1.172 701.
135.4 316.0 9.87 83.8 7.71 14.78 –22.00 –20.00 0.15070 1458.1 0.10978 181.62 343.65 0.9305 1.5706 0.901 0.586 1.174 693.
135.6 309.0 9.95 83.1 7.80 14.50 –20.00 –18.00 0.16279 1452.1 0.10213 183.42 344.59 0.9376 1.5693 0.904 0.590 1.175 684.
135.8 302.2 10.03 82.4 7.90 14.23 –18.00 –16.00 0.17562 1446.1 0.09512 185.24 345.52 0.9447 1.5680 0.907 0.594 1.176 676.
136.0 295.6 10.10 81.6 8.01 13.95 –16.00 –14.00 0.18920 1440.0 0.08870 187.06 346.44 0.9517 1.5667 0.910 0.598 1.178 668.
136.1 289.2 10.18 80.9 8.11 13.67 –14.00 –12.00 0.20358 1433.8 0.08280 188.89 347.37 0.9587 1.5655 0.913 0.602 1.179 660.
136.3 283.0 10.26 80.2 8.21 13.40 –12.00 –10.00 0.21878 1427.6 0.07737 190.72 348.29 0.9656 1.5644 0.917 0.607 1.181 652.
136.4 276.9 10.34 79.4 8.31 13.12 –10.00 –8.00 0.23483 1421.4 0.07237 192.56 349.20 0.9726 1.5633 0.920 0.611 1.183 644.
136.5 271.0 10.42 78.7 8.41 12.85 –8.00 –6.00 0.25176 1415.1 0.06777 194.41 350.11 0.9795 1.5623 0.923 0.616 1.184 636.
136.6 265.2 10.50 78.0 8.52 12.58 –6.00 –4.00 0.26960 1408.8 0.06352 196.27 351.01 0.9863 1.5613 0.927 0.620 1.186 628.
136.6 259.6 10.58 77.3 8.62 12.31 –4.00 –2.00 0.28839 1402.5 0.05959 198.13 351.91 0.9932 1.5603 0.930 0.625 1.189 620.
136.7 254.1 10.66 76.6 8.73 12.04 –2.00 0.00 0.30815 1396.1 0.05595 200.00 352.81 1.0000 1.5594 0.934 0.630 1.191 612.
136.7 248.7 10.74 75.9 8.84 11.77 0.00 2.00 0.32891 1389.6 0.05258 201.88 353.69 1.0068 1.5586 0.938 0.635 1.193 604.
136.7 243.5 10.82 75.1 8.95 11.51 2.00 4.00 0.35071 1383.1 0.04946 203.76 354.57 1.0136 1.5577 0.942 0.640 1.196 596.
136.7 238.4 10.90 74.4 9.06 11.24 4.00 6.00 0.37358 1376.5 0.04656 205.65 355.45 1.0203 1.5569 0.946 0.645 1.199 588.
136.7 233.4 10.98 73.7 9.17 10.98 6.00 8.00 0.39756 1369.9 0.04386 207.56 356.32 1.0270 1.5561 0.950 0.650 1.202 580.
136.6 228.6 11.07 73.0 9.28 10.71 8.00 10.00 0.42267 1363.2 0.04135 209.46 357.18 1.0337 1.5554 0.954 0.656 1.205 572.
136.5 223.8 11.15 72.3 9.39 10.45 10.00 12.00 0.44895 1356.5 0.03901 211.38 358.03 1.0404 1.5547 0.958 0.661 1.208 564.
136.5 219.1 11.23 71.6 9.51 10.19 12.00 14.00 0.47643 1349.7 0.03683 213.31 358.88 1.0471 1.5540 0.962 0.667 1.211 556.
136.3 214.6 11.31 70.9 9.62 9.94 14.00 16.00 0.50514 1342.8 0.03480 215.24 359.71 1.0537 1.5533 0.967 0.672 1.215 548.
136.2 210.1 11.40 70.2 9.74 9.68 16.00 18.00 0.53513 1335.9 0.03290 217.18 360.54 1.0603 1.5527 0.971 0.678 1.219 540.
136.1 205.7 11.48 69.6 9.86 9.42 18.00 20.00 0.56642 1328.9 0.03112 219.14 361.36 1.0669 1.5521 0.976 0.685 1.223 532.
135.9 201.4 11.57 68.9 9.98 9.17 20.00 22.00 0.59905 1321.8 0.02946 221.10 362.17 1.0735 1.5515 0.981 0.691 1.228 524.
135.7 197.2 11.65 68.2 10.10 8.92 22.00 24.00 0.63305 1314.6 0.02790 223.07 362.97 1.0801 1.5509 0.986 0.697 1.232 516.
135.5 193.1 11.74 67.5 10.23 8.67 24.00 26.00 0.66846 1307.4 0.02643 225.05 363.76 1.0866 1.5503 0.991 0.704 1.237 508.
135.2 189.0 11.83 66.8 10.36 8.42 26.00 28.00 0.70531 1300.1 0.02506 227.04 364.54 1.0932 1.5498 0.997 0.711 1.242 499.
134.9 185.0 11.92 66.1 10.49 8.17 28.00 30.00 0.74365 1292.7 0.02377 229.04 365.31 1.0997 1.5492 1.002 0.718 1.248 491.
134.7 181.1 12.01 65.4 10.62 7.92 30.00 32.00 0.78350 1285.2 0.02256 231.06 366.07 1.1062 1.5487 1.008 0.726 1.254 483.
134.3 177.3 12.10 64.8 10.75 7.68 32.00 34.00 0.82491 1277.6 0.02142 233.08 366.81 1.1127 1.5481 1.014 0.734 1.260 475.
134.0 173.5 12.19 64.1 10.89 7.43 34.00 36.00 0.86791 1269.9 0.02034 235.12 367.54 1.1192 1.5476 1.020 0.742 1.267 467.
133.6 169.8 12.28 63.4 11.03 7.19 36.00 38.00 0.91253 1262.2 0.01933 237.16 368.26 1.1257 1.5470 1.026 0.750 1.274 459.
133.2 166.1 12.38 62.7 11.18 6.95 38.00 40.00 0.95882 1254.3 0.01838 239.22 368.96 1.1322 1.5465 1.033 0.759 1.282 450.
132.8 162.5 12.48 62.1 11.33 6.72 40.00 42.00 1.0068 1246.3 0.01748 241.29 369.65 1.1387 1.5459 1.040 0.768 1.290 442.
132.4 159.0 12.57 61.4 11.48 6.48 42.00 44.00 1.0566 1238.1 0.01662 243.38 370.33 1.1451 1.5454 1.048 0.778 1.299 434.
131.9 155.5 12.67 60.7 11.63 6.25 44.00 46.00 1.1081 1229.9 0.01582 245.47 370.98 1.1516 1.5448 1.055 0.788 1.308 426.
131.4 152.0 12.78 60.0 11.79 6.01 46.00 48.00 1.1614 1221.5 0.01505 247.59 371.62 1.1580 1.5443 1.063 0.798 1.318 417.
130.9 148.6 12.88 59.4 11.96 5.78 48.00 50.00 1.2166 1213.0 0.01433 249.71 372.24 1.1645 1.5437 1.072 0.810 1.329 409.
130.3 145.3 12.99 58.7 12.13 5.55 50.00 52.00 1.2737 1204.4 0.01365 251.85 372.85 1.1710 1.5431 1.081 0.821 1.340 400.
129.7 141.9 13.10 58.0 12.31 5.33 52.00 54.00 1.3327 1195.6 0.01300 254.01 373.43 1.1774 1.5425 1.090 0.834 1.353 392.
129.1 138.7 13.21 57.3 12.49 5.10 54.00 56.00 1.3938 1186.6 0.01238 256.18 373.99 1.1839 1.5418 1.100 0.847 1.366 383.
128.4 135.4 13.33 56.7 12.68 4.88 56.00 58.00 1.4568 1177.5 0.01180 258.38 374.53 1.1904 1.5411 1.111 0.861 1.381 375.
127.7 132.2 13.45 56.0 12.87 4.66 58.00 60.00 1.5219 1168.1 0.01124 260.58 375.05 1.1969 1.5404 1.122 0.876 1.397 366.
127.0 129.1 13.57 55.3 13.08 4.44 60.00 62.00 1.5892 1158.6 0.01071 262.81 375.54 1.2033 1.5397 1.135 0.892 1.414 357.
126.3 125.9 13.70 54.7 13.29 4.23 62.00 64.00 1.6586 1148.9 0.01021 265.06 376.00 1.2099 1.5389 1.148 0.910 1.433 348.
125.5 122.8 13.83 54.0 13.51 4.01 64.00 66.00 1.7302 1139.0 0.00973 267.33 376.44 1.2164 1.5381 1.162 0.929 1.453 339.
124.6 119.7 13.96 53.3 13.75 3.80 66.00 68.00 1.8041 1128.8 0.00927 269.62 376.84 1.2229 1.5372 1.177 0.949 1.476 330.
123.8 116.7 14.11 52.6 13.99 3.59 68.00 70.00 1.8802 1118.3 0.00883 271.94 377.22 1.2295 1.5363 1.193 0.971 1.501 321.
122.9 113.6 14.26 52.0 14.25 3.39 70.00 75.00 2.0811 1090.9 0.00782 277.84 377.99 1.2461 1.5337 1.241 1.037 1.576 298.
120.4 106.1 14.66 50.3 14.96 2.88 75.00 80.00 2.2975 1061.4 0.00691 283.94 378.48 1.2629 1.5306 1.302 1.122 1.677 274.
117.7 98.6 15.11 48.7 15.80 2.40 80.00 85.00 2.5304 1029.1 0.00608 290.27 378.64 1.2801 1.5268 1.384 1.239 1.817 249.
114.7 91.1 15.65 47.2 16.82 1.93 85.00 90.00 2.7808 993.2 0.00533 296.91 378.35 1.2978 1.5220 1.501 1.410 2.026 224.
111.4 83.4 16.29 45.9 18.11 1.49 90.00 95.00 3.0501 952.2 0.00463 303.95 377.45 1.3163 1.5159 1.679 1.683 2.362 197.
107.8 75.6 17.11 45.1 19.81 1.07 95.00 100.00 3.3399 903.8 0.00396 311.58 375.60 1.3360 1.5076 1.996 2.192 2.990 169.
103.7 67.3 18.20 45.7 22.27 0.69 100.00 105.00 3.6525 842.2 0.00330 320.24 372.08 1.3581 1.4952 2.754 3.458 4.544 139.
99.3 58.1 19.87 51.7 26.34 0.35 105.00 110.00 3.9924 742.7 0.00252 331.82 363.95 1.3874 1.4712 7.81 11.44 14.14 105.
94.0 46.3 23.46 113.7 39.46 0.07 110.00 111.97c 4.1361 565.0 0.00177 347.76 347.76 1.4283 1.4283 ∞ ∞ ∞ 0 0.0 — — ∞ ∞ 0.00 111.97 Temperatures are on the ITS-90 scale b = normal boiling point c = critical point 20.4 2001 ASHRAE Fundamentals Handbook (SI) Fig. 2 Pressure-Enthalpy Diagram for Refrigerant 22 Thermophysical Properties of Refrigerants 20.5 Refrigerant 22 (Chlorodifluoromethane) Properties of Saturated Liquid and Saturated Vapor Temp., °C Pres-sure, MPa Density, kg/m3 Liquid Volume, m3/kg Vapor Enthalpy, kJ/kg Entropy, kJ/(kg·K) Specific Heat cp, kJ/(kg·K) cp/cv Velocity of Sound, m/s Viscosity, µPa·s Thermal Cond., mW/(m·K) Surface Tension, mN/m Temp., °C Liquid Vapor Liquid Vapor Liquid Vapor Vapor Liquid Vapor Liquid Vapor Liquid Vapor –100.00 0.00201 1571.3 8.2660 90.71 358.97 0.5050 2.0543 1.061 0.497 1.243 1127.
143.6 845.8 7.25 143.1 4.46 28.12 –100.00 –90.00 0.00481 1544.9 3.6448 101.32 363.85 0.5646 1.9980 1.061 0.512 1.237 1080.
147.0 699.4 7.67 137.8 4.84 26.36 –90.00 –80.00 0.01037 1518.2 1.7782 111.94 368.77 0.6210 1.9508 1.062 0.528 1.233 1033.
150.3 591.0 8.09 132.6 5.25 24.63 –80.00 –70.00 0.02047 1491.2 0.94342 122.58 373.70 0.6747 1.9108 1.065 0.545 1.231 986.
153.3 507.6 8.52 127.6 5.68 22.92 –70.00 –60.00 0.03750 1463.7 0.53680 133.27 378.59 0.7260 1.8770 1.071 0.564 1.230 940.
156.0 441.4 8.94 122.6 6.12 21.24 –60.00 –50.00 0.06453 1435.6 0.32385 144.03 383.42 0.7752 1.8480 1.079 0.585 1.232 893.
158.3 387.5 9.36 117.8 6.59 19.58 –50.00 –48.00 0.07145 1429.9 0.29453 146.19 384.37 0.7849 1.8428 1.081 0.589 1.233 884.
158.7 377.8 9.45 116.9 6.69 19.25 –48.00 –46.00 0.07894 1424.2 0.26837 148.36 385.32 0.7944 1.8376 1.083 0.594 1.234 875.
159.1 368.6 9.53 115.9 6.79 18.92 –46.00 –44.00 0.08705 1418.4 0.24498 150.53 386.26 0.8039 1.8327 1.086 0.599 1.235 865.
159.5 359.6 9.62 115.0 6.89 18.59 –44.00 –42.00 0.09580 1412.6 0.22402 152.70 387.20 0.8134 1.8278 1.088 0.603 1.236 856.
159.9 351.0 9.70 114.0 6.99 18.27 –42.00 –40.81b 0.10132 1409.2 0.21260 154.00 387.75 0.8189 1.8250 1.090 0.606 1.236 851.
160.1 346.0 9.75 113.5 7.05 18.08 –40.81 –40.00 0.10523 1406.8 0.20521 154.89 388.13 0.8227 1.8231 1.091 0.608 1.237 847.
160.3 342.6 9.79 113.1 7.09 17.94 –40.00 –38.00 0.11538 1401.0 0.18829 157.07 389.06 0.8320 1.8186 1.093 0.613 1.238 838.
160.6 334.5 9.87 112.2 7.19 17.62 –38.00 –36.00 0.12628 1395.1 0.17304 159.27 389.97 0.8413 1.8141 1.096 0.619 1.239 828.
160.9 326.7 9.96 111.2 7.29 17.30 –36.00 –34.00 0.13797 1389.1 0.15927 161.47 390.89 0.8505 1.8098 1.099 0.624 1.241 819.
161.2 319.1 10.04 110.3 7.40 16.98 –34.00 –32.00 0.15050 1383.2 0.14682 163.67 391.79 0.8596 1.8056 1.102 0.629 1.242 810.
161.5 311.7 10.12 109.4 7.51 16.66 –32.00 –30.00 0.16389 1377.2 0.13553 165.88 392.69 0.8687 1.8015 1.105 0.635 1.244 800.
161.8 304.6 10.21 108.5 7.61 16.34 –30.00 –28.00 0.17819 1371.1 0.12528 168.10 393.58 0.8778 1.7975 1.108 0.641 1.246 791.
162.0 297.7 10.29 107.5 7.72 16.02 –28.00 –26.00 0.19344 1365.0 0.11597 170.33 394.47 0.8868 1.7937 1.112 0.646 1.248 782.
162.3 291.0 10.38 106.6 7.83 15.70 –26.00 –24.00 0.20968 1358.9 0.10749 172.56 395.34 0.8957 1.7899 1.115 0.653 1.250 772.
162.5 284.4 10.46 105.7 7.94 15.39 –24.00 –22.00 0.22696 1352.7 0.09975 174.80 396.21 0.9046 1.7862 1.119 0.659 1.253 763.
162.7 278.1 10.55 104.8 8.06 15.07 –22.00 –20.00 0.24531 1346.5 0.09268 177.04 397.06 0.9135 1.7826 1.123 0.665 1.255 754.
162.8 271.9 10.63 103.9 8.17 14.76 –20.00 –18.00 0.26479 1340.3 0.08621 179.30 397.91 0.9223 1.7791 1.127 0.672 1.258 744.
163.0 265.9 10.72 103.0 8.29 14.45 –18.00 –16.00 0.28543 1334.0 0.08029 181.56 398.75 0.9311 1.7757 1.131 0.678 1.261 735.
163.1 260.1 10.80 102.1 8.40 14.14 –16.00 –14.00 0.30728 1327.6 0.07485 183.83 399.57 0.9398 1.7723 1.135 0.685 1.264 726.
163.2 254.4 10.89 101.1 8.52 13.83 –14.00 –12.00 0.33038 1321.2 0.06986 186.11 400.39 0.9485 1.7690 1.139 0.692 1.267 716.
163.3 248.8 10.98 100.2 8.65 13.52 –12.00 –10.00 0.35479 1314.7 0.06527 188.40 401.20 0.9572 1.7658 1.144 0.699 1.270 707.
163.3 243.4 11.06 99.3 8.77 13.21 –10.00 –8.00 0.38054 1308.2 0.06103 190.70 401.99 0.9658 1.7627 1.149 0.707 1.274 697.
163.4 238.1 11.15 98.4 8.89 12.91 –8.00 –6.00 0.40769 1301.6 0.05713 193.01 402.77 0.9744 1.7596 1.154 0.715 1.278 688.
163.4 233.0 11.24 97.5 9.02 12.60 –6.00 –4.00 0.43628 1295.0 0.05352 195.33 403.55 0.9830 1.7566 1.159 0.722 1.282 679.
163.4 227.9 11.32 96.6 9.15 12.30 –4.00 –2.00 0.46636 1288.3 0.05019 197.66 404.30 0.9915 1.7536 1.164 0.731 1.287 669.
163.4 223.0 11.41 95.7 9.28 12.00 –2.00 0.00 0.49799 1281.5 0.04710 200.00 405.05 1.0000 1.7507 1.169 0.739 1.291 660.
163.3 218.2 11.50 94.8 9.42 11.70 0.00 2.00 0.53120 1274.7 0.04424 202.35 405.78 1.0085 1.7478 1.175 0.748 1.296 650.
163.2 213.5 11.59 93.9 9.56 11.40 2.00 4.00 0.56605 1267.8 0.04159 204.71 406.50 1.0169 1.7450 1.181 0.757 1.301 641.
163.1 208.9 11.68 93.1 9.70 11.10 4.00 6.00 0.60259 1260.8 0.03913 207.09 407.20 1.0254 1.7422 1.187 0.766 1.307 632.
163.0 204.4 11.77 92.2 9.84 10.81 6.00 8.00 0.64088 1253.8 0.03683 209.47 407.89 1.0338 1.7395 1.193 0.775 1.313 622.
162.8 200.0 11.86 91.3 9.99 10.51 8.00 10.00 0.68095 1246.7 0.03470 211.87 408.56 1.0422 1.7368 1.199 0.785 1.319 613.
162.6 195.7 11.96 90.4 10.14 10.22 10.00 12.00 0.72286 1239.5 0.03271 214.28 409.21 1.0505 1.7341 1.206 0.795 1.326 603.
162.4 191.5 12.05 89.5 10.29 9.93 12.00 14.00 0.76668 1232.2 0.03086 216.70 409.85 1.0589 1.7315 1.213 0.806 1.333 594.
162.2 187.3 12.14 88.6 10.45 9.64 14.00 16.00 0.81244 1224.9 0.02912 219.14 410.47 1.0672 1.7289 1.220 0.817 1.340 584.
161.9 183.2 12.24 87.7 10.61 9.35 16.00 18.00 0.86020 1217.4 0.02750 221.59 411.07 1.0755 1.7263 1.228 0.828 1.348 575.
161.6 179.2 12.33 86.8 10.77 9.06 18.00 20.00 0.91002 1209.9 0.02599 224.06 411.66 1.0838 1.7238 1.236 0.840 1.357 565.
161.3 175.3 12.43 85.9 10.95 8.78 20.00 22.00 0.96195 1202.3 0.02457 226.54 412.22 1.0921 1.7212 1.244 0.853 1.366 555.
161.0 171.5 12.53 85.0 11.12 8.50 22.00 24.00 1.0160 1194.6 0.02324 229.04 412.77 1.1004 1.7187 1.252 0.866 1.375 546.
160.6 167.7 12.63 84.1 11.30 8.22 24.00 26.00 1.0724 1186.7 0.02199 231.55 413.29 1.1086 1.7162 1.261 0.879 1.385 536.
160.2 163.9 12.74 83.2 11.49 7.94 26.00 28.00 1.1309 1178.8 0.02082 234.08 413.79 1.1169 1.7136 1.271 0.893 1.396 527.
159.7 160.3 12.84 82.3 11.69 7.66 28.00 30.00 1.1919 1170.7 0.01972 236.62 414.26 1.1252 1.7111 1.281 0.908 1.408 517.
159.2 156.7 12.95 81.4 11.89 7.38 30.00 32.00 1.2552 1162.6 0.01869 239.19 414.71 1.1334 1.7086 1.291 0.924 1.420 507.
158.7 153.1 13.06 80.5 12.10 7.11 32.00 34.00 1.3210 1154.3 0.01771 241.77 415.14 1.1417 1.7061 1.302 0.940 1.434 497.
158.2 149.6 13.17 79.6 12.31 6.84 34.00 36.00 1.3892 1145.8 0.01679 244.38 415.54 1.1499 1.7036 1.314 0.957 1.448 487.
157.6 146.1 13.28 78.7 12.54 6.57 36.00 38.00 1.4601 1137.3 0.01593 247.00 415.91 1.1582 1.7010 1.326 0.976 1.463 478.
157.0 142.7 13.40 77.8 12.77 6.30 38.00 40.00 1.5336 1128.5 0.01511 249.65 416.25 1.1665 1.6985 1.339 0.995 1.480 468.
156.4 139.4 13.52 76.9 13.02 6.04 40.00 42.00 1.6098 1119.6 0.01433 252.32 416.55 1.1747 1.6959 1.353 1.015 1.498 458.
155.7 136.1 13.64 76.0 13.28 5.77 42.00 44.00 1.6887 1110.6 0.01360 255.01 416.83 1.1830 1.6933 1.368 1.037 1.517 448.
155.0 132.8 13.77 75.1 13.55 5.51 44.00 46.00 1.7704 1101.4 0.01291 257.73 417.07 1.1913 1.6906 1.384 1.061 1.538 437.
154.2 129.5 13.90 74.1 13.83 5.25 46.00 48.00 1.8551 1091.9 0.01226 260.47 417.27 1.1997 1.6879 1.401 1.086 1.561 427.
153.4 126.3 14.04 73.2 14.13 5.00 48.00 50.00 1.9427 1082.3 0.01163 263.25 417.44 1.2080 1.6852 1.419 1.113 1.586 417.
152.6 123.1 14.18 72.3 14.45 4.74 50.00 52.00 2.0333 1072.4 0.01104 266.05 417.56 1.2164 1.6824 1.439 1.142 1.614 407.
151.7 120.0 14.32 71.4 14.78 4.49 52.00 54.00 2.1270 1062.3 0.01048 268.89 417.63 1.2248 1.6795 1.461 1.173 1.644 396.
150.8 116.9 14.47 70.4 15.14 4.24 54.00 56.00 2.2239 1052.0 0.00995 271.76 417.66 1.2333 1.6766 1.485 1.208 1.677 386.
149.8 113.8 14.63 69.5 15.52 4.00 56.00 58.00 2.3240 1041.3 0.00944 274.66 417.63 1.2418 1.6736 1.511 1.246 1.714 375.
148.8 110.7 14.80 68.6 15.92 3.75 58.00 60.00 2.4275 1030.4 0.00896 277.61 417.55 1.2504 1.6705 1.539 1.287 1.755 364.
147.7 107.6 14.98 67.6 16.36 3.51 60.00 65.00 2.7012 1001.4 0.00785 285.18 417.06 1.2722 1.6622 1.626 1.413 1.881 337.
144.9 100.0 15.46 65.3 17.61 2.92 65.00 70.00 2.9974 969.7 0.00685 293.10 416.09 1.2945 1.6529 1.743 1.584 2.056 309.
141.7 92.4 16.02 62.9 19.16 2.36 70.00 75.00 3.3177 934.4 0.00595 301.46 414.49 1.3177 1.6424 1.913 1.832 2.315 280.
138.1 84.6 16.70 60.6 21.16 1.82 75.00 80.00 3.6638 893.7 0.00512 310.44 412.01 1.3423 1.6299 2.181 2.231 2.735 249.
134.2 76.6 17.55 58.6 23.87 1.30 80.00 85.00 4.0378 844.8 0.00434 320.38 408.19 1.3690 1.6142 2.682 2.984 3.532 215.
129.7 68.1 18.71 57.4 27.82 0.83 85.00 90.00 4.4423 780.1 0.00356 332.09 401.87 1.4001 1.5922 3.981 4.975 5.626 177.
124.6 58.3 20.48 59.3 34.55 0.40 90.00 95.00 4.8824 662.9 0.00262 349.56 387.28 1.4462 1.5486 17.31 25.29 26.43 128.
118.0 44.4 24.76 83.5 59.15 0.05 95.00 96.15c 4.9900 523.8 0.00191 366.90 366.90 1.4927 1.4927 ∞ ∞ ∞ 0.
0.0 — — ∞ ∞ 0.00 96.15 Temperatures are on the ITS-90 scale b = normal boiling point c = critical point 20.6 2001 ASHRAE Fundamentals Handbook (SI) Fig. 3 Pressure-Enthalpy Diagram for Refrigerant 23 Thermophysical Properties of Refrigerants 20.7 Refrigerant 23 (Trifluoromethane) Properties of Saturated Liquid and Saturated Vapor Temp., °C Pres-sure, MPa Density, kg/m3 Liquid Volume, m3/kg Vapor Enthalpy, kJ/kg Entropy, kJ/(kg·K) Specific Heat cp, kJ/(kg·K) cp/cv Velocity of Sound, m/s Viscosity, µPa·s Thermal Cond., mW/(m·K) Surface Tension, mN/m Temp., °C Liquid Vapor Liquid Vapor Liquid Vapor Vapor Liquid Vapor Liquid Vapor Liquid Vapor –155.13a 0.00006 1682.3 238.15 –3.08 289.40 –0.0682 2.4101 1.245 0.497 1.315 1002.7 135.74 1749.9 5.35 257.6 3.78 32.34 –155.13 –150.00 0.00014 1666.7 104.65 3.24 291.93 –0.0157 2.3285 1.224 0.502 1.312 1012.7 138.45 1442.8 5.64 240.8 4.07 31.28 –150.00 –140.00 0.00061 1635.6 26.092 15.35 296.89 0.0788 2.1932 1.199 0.514 1.306 1017.8 143.48 1034.0 6.22 214.0 4.61 29.20 –140.00 –130.00 0.00207 1603.5 8.1690 27.28 301.84 0.1652 2.0832 1.190 0.530 1.300 1006.8 148.15 776.6 6.80 192.9 5.15 27.13 –130.00 –120.00 0.00591 1570.4 3.0565 39.18 306.74 0.2455 1.9926 1.190 0.553 1.296 983.3 152.42 605.4 7.37 176.0 5.70 25.07 –120.00 –110.00 0.01452 1536.4 1.3172 51.11 311.52 0.3209 1.9171 1.196 0.582 1.295 950.3 156.26 486.3 7.93 162.0 6.25 23.02 –110.00 –100.00 0.03165 1501.5 0.63560 63.13 316.12 0.3924 1.8534 1.207 0.619 1.296 910.2 159.60 399.9 8.48 150.2 6.82 20.98 –100.00 –90.00 0.06255 1465.5 0.33598 75.28 320.47 0.4604 1.7992 1.221 0.665 1.303 864.9 162.39 335.1 9.03 140.1 7.42 18.96 –90.00 –82.06b 0.10132 1436.2 0.21367 85.03 323.71 0.5124 1.7615 1.235 0.707 1.311 826.2 164.18 294.5 9.47 133.0 7.92 17.38 –82.06 –80.00 0.11402 1428.4 0.19121 87.59 324.52 0.5257 1.7524 1.238 0.718 1.314 815.8 164.58 285.1 9.58 131.3 8.05 16.97 –80.00 –70.00 0.19431 1389.9 0.11556 100.10 328.22 0.5886 1.7115 1.260 0.779 1.332 763.7 166.08 245.3 10.12 123.4 8.73 15.00 –70.00 –68.00 0.21464 1382.1 0.10513 102.63 328.91 0.6009 1.7039 1.265 0.792 1.336 753.0 166.30 238.3 10.23 121.9 8.87 14.61 –68.00 –66.00 0.23659 1374.1 0.09583 105.17 329.59 0.6131 1.6965 1.270 0.806 1.341 742.2 166.48 231.6 10.34 120.5 9.02 14.22 –66.00 –64.00 0.26024 1366.1 0.08750 107.72 330.25 0.6253 1.6893 1.275 0.820 1.346 731.3 166.63 225.1 10.45 119.0 9.17 13.87 –64.00 –62.00 0.28567 1358.0 0.08003 110.28 330.89 0.6374 1.6822 1.280 0.834 1.351 720.4 166.75 218.9 10.56 117.6 9.32 13.45 –62.00 –60.00 0.31297 1349.8 0.07333 112.86 331.51 0.6495 1.6753 1.286 0.848 1.357 709.3 166.84 212.9 10.67 116.2 9.48 13.07 –60.00 –58.00 0.34223 1341.5 0.06729 115.44 332.11 0.6614 1.6685 1.292 0.863 1.363 698.2 166.90 207.1 10.78 114.9 9.64 12.69 –58.00 –56.00 0.37354 1333.1 0.06184 118.04 332.70 0.6733 1.6619 1.299 0.878 1.370 687.1 166.92 201.5 10.89 113.5 9.80 12.31 –56.00 –54.00 0.40699 1324.7 0.05691 120.65 333.26 0.6852 1.6553 1.305 0.894 1.377 675.8 166.90 196.1 11.00 112.2 9.96 11.93 –54.00 –52.00 0.44268 1316.2 0.05245 123.28 333.81 0.6970 1.6490 1.312 0.910 1.384 664.5 166.85 190.9 11.11 110.9 10.13 11.55 –52.00 –50.00 0.48069 1307.5 0.04840 125.92 334.33 0.7087 1.6427 1.320 0.926 1.392 653.2 166.77 185.8 11.23 109.6 10.31 11.18 –50.00 –49.00 0.50060 1303.2 0.04652 127.24 334.58 0.7146 1.6396 1.324 0.934 1.396 647.5 166.71 183.4 11.28 108.9 10.40 10.99 –49.00 –48.00 0.52113 1298.8 0.04472 128.57 334.83 0.7204 1.6365 1.327 0.943 1.401 641.8 166.65 180.9 11.34 108.3 10.49 10.81 –48.00 –47.00 0.54230 1294.4 0.04301 129.90 335.07 0.7263 1.6335 1.332 0.951 1.405 636.0 166.57 178.5 11.40 107.7 10.58 10.62 –47.00 –46.00 0.56410 1289.9 0.04137 131.24 335.30 0.7321 1.6305 1.336 0.960 1.410 630.3 166.49 176.2 11.45 107.0 10.67 10.44 –46.00 –45.00 0.58656 1285.4 0.03981 132.58 335.53 0.7379 1.6275 1.340 0.969 1.414 624.5 166.39 173.9 11.51 106.4 10.77 10.25 –45.00 –44.00 0.60969 1280.9 0.03831 133.93 335.76 0.7437 1.6245 1.344 0.978 1.419 618.7 166.29 171.6 11.57 105.8 10.86 10.07 –44.00 –43.00 0.63350 1276.4 0.03688 135.28 335.97 0.7495 1.6215 1.349 0.987 1.424 612.9 166.17 169.3 11.63 105.2 10.96 9.89 –43.00 –42.00 0.65801 1271.8 0.03552 136.63 336.18 0.7553 1.6186 1.354 0.996 1.430 607.1 166.05 167.1 11.69 104.5 11.06 9.70 –42.00 –41.00 0.68322 1267.2 0.03421 137.99 336.39 0.7611 1.6157 1.358 1.006 1.435 601.3 165.91 164.9 11.74 103.9 11.16 9.52 –41.00 –40.00 0.70915 1262.6 0.03296 139.35 336.58 0.7669 1.6128 1.363 1.016 1.441 595.5 165.77 162.7 11.80 103.3 11.26 9.34 –40.00 –39.00 0.73581 1257.9 0.03177 140.72 336.77 0.7726 1.6099 1.369 1.025 1.447 589.6 165.61 160.6 11.86 102.7 11.36 9.16 –39.00 –38.00 0.76322 1253.2 0.03062 142.09 336.95 0.7784 1.6070 1.374 1.035 1.453 583.7 165.45 158.5 11.92 102.1 11.46 8.98 –38.00 –37.00 0.79138 1248.5 0.02952 143.47 337.13 0.7841 1.6042 1.379 1.046 1.459 577.9 165.27 156.4 11.98 101.5 11.57 8.80 –37.00 –36.00 0.82032 1243.7 0.02847 144.86 337.30 0.7899 1.6014 1.385 1.056 1.465 572.0 165.08 154.3 12.04 100.8 11.68 8.62 –36.00 –35.00 0.85005 1238.9 0.02746 146.24 337.46 0.7956 1.5985 1.391 1.067 1.472 566.0 164.89 152.3 12.11 100.2 11.78 8.44 –35.00 –34.00 0.88057 1234.0 0.02650 147.64 337.61 0.8014 1.5957 1.397 1.077 1.479 560.1 164.68 150.3 12.17 99.6 11.89 8.27 –34.00 –33.00 0.91191 1229.1 0.02557 149.04 337.76 0.8071 1.5929 1.403 1.088 1.486 554.2 164.46 148.3 12.23 99.0 12.01 8.09 –33.00 –32.00 0.94407 1224.2 0.02468 150.44 337.89 0.8128 1.5901 1.409 1.100 1.494 548.2 164.23 146.3 12.29 98.4 12.12 7.91 –32.00 –31.00 0.97707 1219.2 0.02382 151.86 338.02 0.8186 1.5874 1.416 1.111 1.501 542.2 163.98 144.4 12.36 97.8 12.24 7.74 –31.00 –30.00 1.0109 1214.2 0.02300 153.27 338.14 0.8243 1.5846 1.423 1.123 1.509 536.3 163.73 142.5 12.42 97.2 12.36 7.56 –30.00 –29.00 1.0456 1209.1 0.02221 154.70 338.25 0.8300 1.5818 1.430 1.135 1.518 530.3 163.46 140.6 12.49 96.6 12.48 7.39 –29.00 –28.00 1.0813 1204.0 0.02145 156.13 338.35 0.8357 1.5791 1.437 1.148 1.527 524.2 163.18 138.7 12.55 96.0 12.60 7.22 –28.00 –27.00 1.1178 1198.8 0.02072 157.57 338.45 0.8415 1.5763 1.445 1.161 1.536 518.2 162.89 136.8 12.62 95.4 12.72 7.05 –27.00 –26.00 1.1552 1193.5 0.02002 159.01 338.53 0.8472 1.5736 1.452 1.174 1.545 512.2 162.59 135.0 12.68 94.8 12.85 6.87 –26.00 –25.00 1.1935 1188.3 0.01934 160.46 338.60 0.8529 1.5708 1.461 1.187 1.555 506.1 162.28 133.2 12.75 94.2 12.98 6.70 –25.00 –24.00 1.2328 1182.9 0.01869 161.92 338.66 0.8587 1.5680 1.469 1.201 1.565 500.0 161.95 131.4 12.82 93.5 13.11 6.53 –24.00 –23.00 1.2730 1177.5 0.01807 163.39 338.72 0.8644 1.5653 1.478 1.215 1.576 493.9 161.61 129.6 12.89 92.9 13.24 6.37 –23.00 –22.00 1.3142 1172.1 0.01746 164.86 338.76 0.8701 1.5625 1.487 1.230 1.587 487.8 161.26 127.8 12.96 92.3 13.38 6.20 –22.00 –21.00 1.3564 1166.5 0.01688 166.34 338.79 0.8759 1.5598 1.497 1.245 1.598 481.7 160.89 126.1 13.03 91.7 13.52 6.03 –21.00 –20.00 1.3996 1161.0 0.01632 167.83 338.81 0.8816 1.5570 1.507 1.261 1.611 475.6 160.51 124.3 13.11 91.1 13.66 5.86 –20.00 –19.00 1.4438 1155.3 0.01578 169.33 338.82 0.8874 1.5543 1.517 1.277 1.623 469.4 160.12 122.6 13.18 90.5 13.81 5.70 –19.00 –18.00 1.4890 1149.6 0.01526 170.84 338.81 0.8932 1.5515 1.528 1.294 1.636 463.2 159.72 120.9 13.26 89.9 13.96 5.54 –18.00 –17.00 1.5353 1143.8 0.01476 172.36 338.79 0.8989 1.5487 1.539 1.312 1.650 457.0 159.30 119.2 13.33 89.3 14.11 5.37 –17.00 –16.00 1.5826 1137.9 0.01427 173.89 338.76 0.9047 1.5459 1.551 1.330 1.665 450.8 158.86 117.5 13.41 88.6 14.26 5.21 –16.00 –14.00 1.6806 1125.9 0.01335 176.97 338.66 0.9164 1.5403 1.576 1.368 1.696 438.4 157.96 114.1 13.57 87.4 14.58 4.89 –14.00 –12.00 1.7829 1113.6 0.01249 180.10 338.50 0.9280 1.5346 1.604 1.409 1.731 425.9 156.99 110.8 13.74 86.1 14.92 4.57 –12.00 –10.00 1.8899 1101.0 0.01169 183.27 338.27 0.9398 1.5288 1.634 1.455 1.770 413.3 155.96 107.6 13.92 84.8 15.27 4.26 –10.00 –8.00 2.0016 1087.9 0.01094 186.50 337.98 0.9516 1.5229 1.668 1.504 1.813 400.7 154.87 104.3 14.10 83.5 15.64 3.95 –8.00 –6.00 2.1181 1074.4 0.01023 189.78 337.61 0.9635 1.5169 1.706 1.559 1.861 388.0 153.72 101.1 14.30 82.2 16.03 3.65 –6.00 –4.00 2.2396 1060.4 0.00957 193.12 337.16 0.9755 1.5107 1.748 1.620 1.916 375.2 152.50 97.9 14.50 80.9 16.44 3.35 –4.00 –2.00 2.3663 1045.8 0.00895 196.52 336.62 0.9877 1.5044 1.795 1.689 1.978 362.3 151.22 94.7 14.72 79.5 16.87 3.06 –2.00 0.00 2.4983 1030.7 0.00836 200.00 335.99 1.0000 1.4978 1.849 1.766 2.050 349.4 149.87 91.6 14.96 78.1 17.33 2.78 0.00 2.00 2.6358 1014.9 0.00781 203.56 335.24 1.0125 1.4911 1.911 1.855 2.133 336.2 148.44 88.4 15.21 76.7 17.83 2.50 2.00 4.00 2.7790 998.40 0.00729 207.21 334.37 1.0252 1.4840 1.983 1.958 2.231 322.9 146.94 85.3 15.49 75.2 18.36 2.23 4.00 6.00 2.9279 981.01 0.00679 210.96 333.36 1.0381 1.4766 2.069 2.079 2.347 309.3 145.36 82.1 15.79 73.6 18.93 1.96 6.00 8.00 3.0829 962.61 0.00632 214.83 332.19 1.0514 1.4688 2.174 2.224 2.488 295.1 143.70 78.9 16.12 72.0 19.54 1.71 8.00 10.00 3.2442 942.97 0.00587 218.84 330.83 1.0650 1.4605 2.304 2.400 2.667 280.3 141.96 75.6 16.48 70.4 20.22 1.46 10.00 20.00 4.1519 806.58 0.00385 242.91 319.30 1.1449 1.4054 — — — — — — — — — 0.40 20.00 26.13c 4.8273 525.90 0.00190 280.72 280.72 1.2689 1.2689 ∞ ∞ ∞ 0.0 0.00 — — ∞ ∞ 0.00 26.13 Temperatures are on the ITS-90 scale a = triple point b = normal boiling point c = critical point 20.8 2001 ASHRAE Fundamentals Handbook (SI) Fig. 4 Pressure-Enthalpy Diagram for Refrigerant 32 Thermophysical Properties of Refrigerants 20.9 Refrigerant 32 (Difluoromethane) Properties of Saturated Liquid and Saturated Vapor Temp., °C Pres-sure, MPa Density, kg/m3 Liquid Volume, m3/kg Vapor Enthalpy, kJ/kg Entropy, kJ/(kg·K) Specific Heat cp, kJ/(kg·K) cp/cv Velocity of Sound, m/s Viscosity, µPa·s Thermal Cond., mW/(m·K) Surface Tension, mN/m Temp., °C Liquid Vapor Liquid Vapor Liquid Vapor Vapor Liquid Vapor Liquid Vapor Liquid Vapor –136.81a 0.00005 1429.3 453.85 –19.07 444.31 –0.1050 3.2937 1.592 0.660 1.321 1414.
169.6 — — — — 39.01 –136.81 –130.00 0.00013 1412.7 174.36 –8.26 448.77 –0.0276 3.1651 1.583 0.665 1.318 1378.
173.6 — — — — 37.47 –130.00 –120.00 0.00048 1388.4 51.184 7.52 455.33 0.0790 3.0030 1.573 0.674 1.315 1326.
179.2 — — — — 35.23 –120.00 –110.00 0.00145 1363.8 17.907 23.20 461.86 0.1782 2.8668 1.565 0.686 1.312 1273.
184.5 678.6 6.80 229.8 5.69 33.02 –110.00 –100.00 0.00381 1339.0 7.2220 38.83 468.31 0.2711 2.7515 1.560 0.703 1.310 1221.
189.5 562.0 7.22 222.9 6.09 30.83 –100.00 –90.00 0.00887 1313.9 3.2721 54.42 474.61 0.3586 2.6529 1.559 0.725 1.310 1169.
194.1 475.3 7.64 216.1 6.51 28.68 –90.00 –80.00 0.01865 1288.4 1.6316 70.02 480.72 0.4415 2.5679 1.561 0.754 1.311 1118.
198.3 408.4 8.06 209.3 6.95 26.56 –80.00 –70.00 0.03607 1262.4 0.88072 85.66 486.57 0.5204 2.4939 1.566 0.790 1.314 1066.
202.0 355.0 8.48 202.5 7.40 24.48 –70.00 –60.00 0.06496 1235.7 0.50786 101.38 492.11 0.5958 2.4289 1.576 0.833 1.320 1014.
205.1 311.5 8.90 195.8 7.88 22.42 –60.00 –51.65b 0.10133 1212.9 0.33468 114.59 496.45 0.6565 2.3805 1.587 0.875 1.328 971.
207.4 280.9 9.25 190.1 8.30 20.74 –51.65 –50.00 0.11014 1208.4 0.30944 117.22 497.27 0.6683 2.3714 1.589 0.883 1.329 962.
207.7 275.3 9.32 189.0 8.39 20.41 –50.00 –40.00 0.17741 1180.2 0.19743 133.23 502.02 0.7382 2.3200 1.608 0.940 1.343 910.
209.7 244.6 9.75 182.2 8.93 18.44 –40.00 –38.00 0.19409 1174.4 0.18134 136.45 502.91 0.7519 2.3103 1.612 0.952 1.347 900.
210.1 239.0 9.84 180.8 9.05 18.05 –38.00 –36.00 0.21197 1168.6 0.16680 139.69 503.78 0.7655 2.3008 1.616 0.965 1.350 889.
210.4 233.6 9.92 179.4 9.16 17.66 –36.00 –34.00 0.23111 1162.8 0.15365 142.93 504.63 0.7791 2.2916 1.621 0.977 1.354 879.
210.6 228.3 10.01 178.0 9.28 17.27 –34.00 –32.00 0.25159 1156.9 0.14173 146.18 505.47 0.7926 2.2824 1.626 0.990 1.358 868.
210.9 223.2 10.10 176.7 9.40 16.89 –32.00 –30.00 0.27344 1151.0 0.13091 149.45 506.27 0.8060 2.2735 1.631 1.004 1.363 858.
211.1 218.2 10.19 175.3 9.52 16.50 –30.00 –28.00 0.29675 1145.0 0.12107 152.72 507.06 0.8193 2.2647 1.637 1.017 1.367 847.
211.3 213.4 10.27 173.9 9.65 16.12 –28.00 –26.00 0.32157 1138.9 0.11211 156.01 507.83 0.8326 2.2561 1.642 1.031 1.372 837.
211.4 208.7 10.36 172.5 9.78 15.74 –26.00 –24.00 0.34796 1132.9 0.10393 159.31 508.57 0.8458 2.2476 1.648 1.045 1.377 826.
211.5 204.1 10.45 171.1 9.91 15.36 –24.00 –22.00 0.37600 1126.7 0.09646 162.62 509.28 0.8589 2.2392 1.654 1.060 1.383 816.
211.6 199.6 10.54 169.7 10.04 14.99 –22.00 –20.00 0.40575 1120.6 0.08963 165.94 509.97 0.8720 2.2310 1.661 1.075 1.389 805.
211.7 195.2 10.63 168.3 10.18 14.61 –20.00 –18.00 0.43728 1114.3 0.08337 169.28 510.64 0.8850 2.2229 1.668 1.090 1.395 794.
211.7 191.0 10.73 166.9 10.32 14.24 –18.00 –16.00 0.47067 1108.0 0.07762 172.63 511.28 0.8979 2.2149 1.675 1.106 1.401 784.
211.7 186.8 10.82 165.5 10.46 13.87 –16.00 –14.00 0.50597 1101.7 0.07234 175.99 511.89 0.9109 2.2070 1.682 1.122 1.408 773.
211.7 182.7 10.91 164.1 10.61 13.50 –14.00 –12.00 0.54327 1095.2 0.06749 179.37 512.47 0.9237 2.1992 1.690 1.139 1.416 762.
211.6 178.8 11.01 162.7 10.76 13.14 –12.00 –10.00 0.58263 1088.8 0.06301 182.76 513.02 0.9365 2.1915 1.698 1.156 1.423 751.
211.5 174.9 11.10 161.3 10.92 12.77 –10.00 –8.00 0.62414 1082.2 0.05889 186.18 513.54 0.9493 2.1839 1.706 1.174 1.432 741.
211.4 171.1 11.20 159.8 11.08 12.41 –8.00 –6.00 0.66786 1075.6 0.05508 189.60 514.03 0.9620 2.1764 1.715 1.192 1.440 730.
211.2 167.4 11.30 158.4 11.25 12.05 –6.00 –4.00 0.71388 1068.9 0.05155 193.05 514.49 0.9747 2.1690 1.725 1.211 1.450 719.
211.0 163.8 11.40 157.0 11.42 11.69 –4.00 –2.00 0.76226 1062.1 0.04829 196.52 514.91 0.9874 2.1616 1.735 1.231 1.460 708.
210.8 160.2 11.50 155.5 11.60 11.34 –2.00 0.00 0.81310 1055.3 0.04527 200.00 515.30 1.0000 2.1543 1.745 1.251 1.470 697.
210.5 156.7 11.60 154.1 11.79 10.99 0.00 2.00 0.86647 1048.3 0.04246 203.50 515.65 1.0126 2.1471 1.756 1.272 1.481 686.
210.2 153.3 11.71 152.6 11.98 10.63 2.00 4.00 0.92245 1041.3 0.03986 207.03 515.96 1.0252 2.1399 1.767 1.294 1.493 675.
209.8 149.9 11.81 151.1 12.19 10.29 4.00 6.00 0.98113 1034.2 0.03743 210.58 516.24 1.0377 2.1327 1.779 1.317 1.506 664.
209.4 146.7 11.92 149.6 12.40 9.94 6.00 8.00 1.0426 1027.0 0.03518 214.15 516.47 1.0503 2.1256 1.792 1.341 1.519 652.
209.0 143.4 12.03 148.2 12.62 9.60 8.00 10.00 1.1069 1019.7 0.03308 217.74 516.66 1.0628 2.1185 1.806 1.367 1.534 641.
208.5 140.3 12.14 146.7 12.86 9.25 10.00 12.00 1.1742 1012.2 0.03112 221.36 516.80 1.0753 2.1114 1.820 1.393 1.549 630.
208.0 137.1 12.25 145.1 13.11 8.91 12.00 14.00 1.2445 1004.7 0.02929 225.01 516.90 1.0878 2.1043 1.835 1.421 1.565 618.
207.5 134.1 12.37 143.6 13.38 8.58 14.00 16.00 1.3179 997.1 0.02758 228.68 516.95 1.1003 2.0972 1.851 1.450 1.583 607.
206.9 131.1 12.48 142.1 13.67 8.24 16.00 18.00 1.3946 989.3 0.02598 232.39 516.95 1.1128 2.0902 1.868 1.481 1.602 595.
206.3 128.1 12.60 140.5 13.97 7.91 18.00 20.00 1.4746 981.4 0.02448 236.12 516.90 1.1253 2.0831 1.886 1.514 1.622 584.
205.6 125.2 12.73 139.0 14.30 7.59 20.00 22.00 1.5579 973.3 0.02307 239.89 516.79 1.1378 2.0760 1.905 1.548 1.644 572.
204.9 122.3 12.85 137.4 14.65 7.26 22.00 24.00 1.6448 965.2 0.02175 243.69 516.62 1.1503 2.0688 1.926 1.585 1.668 560.
204.1 119.5 12.98 135.8 15.03 6.94 24.00 26.00 1.7353 956.8 0.02051 247.53 516.39 1.1629 2.0616 1.948 1.624 1.693 548.
203.3 116.7 13.11 134.2 15.44 6.62 26.00 28.00 1.8295 948.3 0.01935 251.40 516.09 1.1755 2.0544 1.972 1.667 1.721 536.
202.4 113.9 13.25 132.6 15.88 6.30 28.00 30.00 1.9275 939.6 0.01826 255.32 515.72 1.1881 2.0471 1.997 1.712 1.750 524.
201.5 111.2 13.39 131.0 16.36 5.99 30.00 32.00 2.0294 930.7 0.01722 259.28 515.29 1.2007 2.0397 2.025 1.760 1.783 512.
200.6 108.5 13.53 129.3 16.89 5.68 32.00 34.00 2.1353 921.7 0.01625 263.28 514.77 1.2134 2.0322 2.055 1.813 1.819 499.
199.6 105.8 13.68 127.6 17.47 5.37 34.00 36.00 2.2454 912.4 0.01533 267.34 514.17 1.2262 2.0246 2.088 1.870 1.858 487.
198.5 103.2 13.84 126.0 18.10 5.07 36.00 38.00 2.3597 902.8 0.01447 271.45 513.49 1.2391 2.0169 2.124 1.933 1.901 474.
197.4 100.5 14.00 124.2 18.80 4.77 38.00 40.00 2.4783 893.0 0.01365 275.61 512.71 1.2520 2.0091 2.163 2.001 1.948 461.
196.2 97.9 14.16 122.5 19.58 4.47 40.00 42.00 2.6014 883.0 0.01287 279.84 511.82 1.2650 2.0011 2.206 2.077 2.001 448.
194.9 95.4 14.34 120.7 20.44 4.18 42.00 44.00 2.7292 872.6 0.01214 284.13 510.83 1.2781 1.9929 2.255 2.160 2.060 435.
193.6 92.8 14.52 118.9 21.40 3.89 44.00 46.00 2.8616 861.9 0.01144 288.50 509.72 1.2914 1.9845 2.309 2.254 2.126 421.
192.3 90.2 14.71 117.1 22.48 3.61 46.00 48.00 2.9989 850.8 0.01078 292.95 508.48 1.3048 1.9759 2.369 2.358 2.201 408.
190.8 87.7 14.91 115.2 23.69 3.33 48.00 50.00 3.1412 839.3 0.01015 297.49 507.10 1.3183 1.9670 2.439 2.477 2.287 394.
189.3 85.1 15.13 113.3 25.06 3.06 50.00 52.00 3.2887 827.3 0.00955 302.12 505.57 1.3321 1.9578 2.518 2.613 2.385 379.
187.7 82.6 15.36 111.4 26.61 2.79 52.00 54.00 3.4415 814.8 0.00897 306.87 503.86 1.3461 1.9482 2.609 2.771 2.499 365.
186.0 80.0 15.60 109.4 28.39 2.52 54.00 56.00 3.5997 801.7 0.00843 311.74 501.95 1.3603 1.9382 2.717 2.956 2.633 350.
184.3 77.4 15.86 107.3 30.43 2.26 56.00 58.00 3.7635 787.9 0.00790 316.75 499.82 1.3749 1.9277 2.845 3.175 2.793 335.
182.4 74.8 16.15 105.2 32.82 2.01 58.00 60.00 3.9332 773.3 0.00740 321.93 497.44 1.3898 1.9166 3.001 3.441 2.987 320.
180.4 72.2 16.46 103.0 35.62 1.76 60.00 62.00 4.1089 757.8 0.00691 327.30 494.76 1.4052 1.9048 3.193 3.771 3.228 304.
178.3 69.5 16.81 100.8 38.98 1.52 62.00 64.00 4.2909 741.1 0.00644 332.90 491.73 1.4211 1.8922 3.438 4.190 3.535 288.
176.1 66.8 17.20 98.4 43.10 1.29 64.00 66.00 4.4793 723.0 0.00598 338.78 488.26 1.4377 1.8785 3.761 4.743 3.938 271.
173.7 63.9 17.64 96.0 48.32 1.06 66.00 68.00 4.6745 703.2 0.00553 345.02 484.25 1.4553 1.8634 4.207 5.508 4.495 254.
171.2 61.0 18.16 93.4 55.31 0.85 68.00 70.00 4.8768 680.9 0.00508 351.73 479.52 1.4740 1.8464 4.865 6.639 5.316 236.
168.4 57.8 18.78 90.6 65.51 0.64 70.00 75.00 5.4168 605.9 0.00391 372.39 461.72 1.5314 1.7880 10.13 15.60 11.72 186.
159.6 48.5 21.28 83.1 343.33 0.19 75.00 78.11c 5.7820 424.0 0.00236 414.15 414.15 1.6486 1.6486 ∞ ∞ ∞ 0.
0.0 — — ∞ ∞ 0.00 78.11 Temperatures are on the ITS-90 scale a = triple point b = normal boiling point c = critical point 20.10 2001 ASHRAE Fundamentals Handbook (SI) Fig. 5 Pressure-Enthalpy Diagram for Refrigerant 123 Thermophysical Properties of Refrigerants 20.11 Refrigerant 123 (2,2-Dichloro-1,1,1-trifluoroethane) Properties of Saturated Liquid and Saturated Vapor Temp., °C Pres-sure, MPa Density, kg/m3 Liquid Volume, m3/kg Vapor Enthalpy, kJ/kg Entropy, kJ/(kg·K) Specific Heat cp, kJ/(kg·K) cp/cv Velocity of Sound, m/s Viscosity, µPa·s Thermal Cond., mW/(m·K) Surface Tension, mN/m Temp., °C Liquid Vapor Liquid Vapor Liquid Vapor Vapor Liquid Vapor Liquid Vapor Liquid Vapor –80.00 0.00013 1709.6 83.667 123.92 335.98 0.6712 1.7691 0.924 0.520 1.117 1133.
108.3 2093.
6.68 107.4 3.22 28.42 –80.00 –70.00 0.00034 1687.4 32.842 133.17 341.25 0.7179 1.7422 0.927 0.537 1.113 1091.
110.8 1680.
7.09 104.8 3.79 27.09 –70.00 –60.00 0.00081 1665.1 14.333 142.46 346.66 0.7625 1.7206 0.932 0.553 1.110 1049.
113.3 1383.
7.50 102.0 4.35 25.78 –60.00 –50.00 0.00177 1642.6 6.8460 151.81 352.21 0.8054 1.7034 0.939 0.569 1.107 1006.
115.6 1160.
7.91 99.1 4.92 24.48 –50.00 –40.00 0.00358 1620.0 3.5319 161.25 357.88 0.8468 1.6901 0.948 0.585 1.105 964.
117.9 986.4 8.31 96.1 5.49 23.19 –40.00 –30.00 0.00675 1597.0 1.9470 170.78 363.65 0.8868 1.6800 0.958 0.601 1.103 923.
120.0 848.0 8.70 93.0 6.05 21.92 –30.00 –20.00 0.01200 1573.8 1.1364 180.41 369.52 0.9256 1.6726 0.968 0.617 1.102 881.
122.0 735.4 9.09 89.8 6.61 20.66 –20.00 –10.00 0.02025 1550.1 0.69690 190.15 375.45 0.9633 1.6675 0.979 0.634 1.102 841.
123.8 642.4 9.47 86.7 7.18 19.41 –10.00 0.00 0.03265 1526.1 0.44609 200.00 381.44 1.0000 1.6642 0.990 0.651 1.102 801.
125.4 564.6 9.84 83.7 7.74 18.18 0.00 2.00 0.03574 1521.3 0.40991 201.98 382.64 1.0072 1.6638 0.993 0.654 1.103 793.
125.7 550.6 9.91 83.1 7.86 17.94 2.00 4.00 0.03907 1516.4 0.37720 203.97 383.84 1.0144 1.6634 0.995 0.658 1.103 785.
126.0 537.0 9.99 82.5 7.97 17.70 4.00 6.00 0.04264 1511.5 0.34759 205.97 385.05 1.0216 1.6631 0.997 0.661 1.103 777.
126.3 523.8 10.06 81.9 8.08 17.45 6.00 8.00 0.04647 1506.6 0.32075 207.96 386.25 1.0287 1.6628 0.999 0.665 1.103 769.
126.6 511.1 10.13 81.3 8.20 17.21 8.00 10.00 0.05057 1501.6 0.29637 209.97 387.46 1.0358 1.6626 1.002 0.668 1.104 761.
126.8 498.8 10.20 80.7 8.31 16.97 10.00 12.00 0.05495 1496.7 0.27420 211.97 388.66 1.0428 1.6625 1.004 0.672 1.104 754.
127.1 486.8 10.28 80.1 8.43 16.73 12.00 14.00 0.05963 1491.7 0.25401 213.99 389.87 1.0499 1.6624 1.006 0.675 1.104 746.
127.3 475.3 10.35 79.5 8.54 16.49 14.00 16.00 0.06463 1486.7 0.23559 216.00 391.08 1.0569 1.6623 1.009 0.679 1.105 738.
127.6 464.0 10.42 79.0 8.66 16.25 16.00 18.00 0.06995 1481.7 0.21877 218.02 392.29 1.0638 1.6623 1.011 0.682 1.105 730.
127.8 453.2 10.49 78.4 8.77 16.01 18.00 20.00 0.07561 1476.6 0.20338 220.05 393.49 1.0707 1.6624 1.014 0.686 1.106 723.
128.0 442.6 10.56 77.8 8.89 15.77 20.00 22.00 0.08163 1471.5 0.18929 222.08 394.70 1.0776 1.6625 1.016 0.690 1.106 715.
128.2 432.4 10.63 77.3 9.01 15.53 22.00 24.00 0.08802 1466.4 0.17637 224.12 395.91 1.0845 1.6626 1.018 0.693 1.107 707.
128.4 422.4 10.70 76.7 9.12 15.30 24.00 26.00 0.09480 1461.3 0.16451 226.16 397.12 1.0913 1.6628 1.021 0.697 1.107 700.
128.6 412.8 10.77 76.1 9.24 15.06 26.00 27.82b 0.10133 1456.6 0.15453 228.03 398.22 1.0975 1.6630 1.023 0.701 1.108 693.
128.7 404.2 10.84 75.6 9.35 14.84 27.82 28.00 0.10198 1456.2 0.15360 228.21 398.32 1.0981 1.6630 1.023 0.701 1.108 692.
128.7 403.4 10.84 75.6 9.36 14.82 28.00 30.00 0.10958 1451.0 0.14356 230.26 399.53 1.1049 1.6633 1.026 0.705 1.109 684.
128.9 394.3 10.91 75.0 9.48 14.59 30.00 32.00 0.11762 1445.8 0.13431 232.31 400.73 1.1116 1.6635 1.028 0.709 1.109 677.
129.0 385.4 10.98 74.5 9.60 14.35 32.00 34.00 0.12611 1440.6 0.12577 234.38 401.93 1.1183 1.6639 1.031 0.712 1.110 669.
129.1 376.8 11.05 74.0 9.72 14.12 34.00 36.00 0.13507 1435.4 0.11789 236.44 403.14 1.1250 1.6642 1.033 0.716 1.111 662.
129.3 368.4 11.12 73.4 9.84 13.89 36.00 38.00 0.14452 1430.1 0.11060 238.51 404.34 1.1317 1.6646 1.036 0.720 1.112 654.
129.4 360.3 11.19 72.9 9.96 13.66 38.00 40.00 0.15447 1424.8 0.10385 240.59 405.54 1.1383 1.6651 1.038 0.724 1.113 647.
129.5 352.4 11.26 72.4 10.08 13.43 40.00 42.00 0.16495 1419.4 0.09759 242.67 406.73 1.1449 1.6655 1.041 0.728 1.114 639.
129.5 344.7 11.33 71.8 10.20 13.20 42.00 44.00 0.17597 1414.1 0.09179 244.76 407.93 1.1515 1.6660 1.044 0.732 1.115 632.
129.6 337.2 11.40 71.3 10.32 12.97 44.00 46.00 0.18755 1408.7 0.08641 246.86 409.12 1.1581 1.6665 1.046 0.736 1.116 624.
129.7 329.9 11.46 70.8 10.45 12.74 46.00 48.00 0.19971 1403.3 0.08140 248.95 410.31 1.1646 1.6670 1.049 0.741 1.117 617.
129.7 322.8 11.53 70.3 10.57 12.51 48.00 50.00 0.21246 1397.8 0.07674 251.06 411.50 1.1711 1.6676 1.052 0.745 1.119 610.
129.7 315.9 11.60 69.8 10.70 12.28 50.00 52.00 0.22584 1392.3 0.07240 253.17 412.69 1.1776 1.6682 1.055 0.749 1.120 602.
129.7 309.1 11.67 69.3 10.82 12.05 52.00 54.00 0.23985 1386.8 0.06836 255.28 413.87 1.1840 1.6688 1.058 0.753 1.121 595.
129.7 302.6 11.74 68.8 10.95 11.83 54.00 56.00 0.25451 1381.2 0.06458 257.41 415.05 1.1905 1.6694 1.060 0.758 1.123 588.
129.7 296.2 11.80 68.3 11.08 11.60 56.00 58.00 0.26985 1375.6 0.06106 259.53 416.23 1.1969 1.6701 1.063 0.762 1.124 580.
129.7 289.9 11.87 67.8 11.21 11.38 58.00 60.00 0.28589 1370.0 0.05777 261.67 417.40 1.2033 1.6707 1.066 0.767 1.126 573.
129.6 283.9 11.94 67.3 11.34 11.16 60.00 62.00 0.30264 1364.3 0.05469 263.81 418.57 1.2096 1.6714 1.069 0.771 1.127 566.
129.6 277.9 12.01 66.8 11.47 10.93 62.00 64.00 0.32013 1358.6 0.05180 265.95 419.73 1.2160 1.6721 1.072 0.776 1.129 558.
129.5 272.1 12.07 66.3 11.61 10.71 64.00 66.00 0.33838 1352.8 0.04910 268.10 420.89 1.2223 1.6728 1.076 0.781 1.131 551.
129.4 266.5 12.14 65.9 11.74 10.49 66.00 68.00 0.35740 1347.0 0.04656 270.26 422.05 1.2286 1.6735 1.079 0.785 1.133 544.
129.3 261.0 12.21 65.4 11.88 10.27 68.00 70.00 0.37722 1341.2 0.04418 272.42 423.20 1.2349 1.6743 1.082 0.790 1.135 536.
129.2 255.6 12.28 64.9 12.01 10.05 70.00 72.00 0.39787 1335.3 0.04195 274.60 424.35 1.2411 1.6750 1.085 0.795 1.137 529.
129.0 250.4 12.35 64.5 12.15 9.84 72.00 74.00 0.41936 1329.3 0.03985 276.77 425.50 1.2474 1.6758 1.089 0.800 1.139 522.
128.9 245.2 12.42 64.0 12.29 9.62 74.00 76.00 0.44171 1323.4 0.03787 278.96 426.63 1.2536 1.6766 1.092 0.806 1.142 515.
128.7 240.2 12.49 63.5 12.44 9.40 76.00 78.00 0.46494 1317.3 0.03601 281.15 427.77 1.2598 1.6774 1.096 0.811 1.144 507.
128.5 235.3 12.55 63.1 12.58 9.19 78.00 80.00 0.48909 1311.2 0.03426 283.35 428.89 1.2660 1.6781 1.100 0.816 1.147 500.
128.3 230.5 12.63 62.6 12.73 8.97 80.00 82.00 0.51416 1305.1 0.03261 285.55 430.01 1.2722 1.6789 1.103 0.822 1.150 493.
128.1 225.9 12.70 62.2 12.87 8.76 82.00 84.00 0.54019 1298.9 0.03105 287.77 431.13 1.2783 1.6797 1.107 0.827 1.152 486.
127.8 221.3 12.77 61.7 13.02 8.55 84.00 86.00 0.56720 1292.6 0.02958 289.99 432.23 1.2845 1.6806 1.111 0.833 1.156 478.
127.6 216.8 12.84 61.3 13.17 8.34 86.00 88.00 0.59520 1286.3 0.02819 292.22 433.33 1.2906 1.6814 1.115 0.839 1.159 471.
127.3 212.5 12.91 60.8 13.33 8.13 88.00 90.00 0.62423 1279.9 0.02687 294.45 434.43 1.2967 1.6822 1.120 0.845 1.162 464.
127.0 208.2 12.98 60.4 13.48 7.92 90.00 92.00 0.65430 1273.5 0.02563 296.70 435.51 1.3028 1.6830 1.124 0.851 1.166 457.
126.6 204.0 13.06 59.9 13.64 7.71 92.00 94.00 0.68544 1266.9 0.02445 298.95 436.59 1.3089 1.6838 1.129 0.858 1.169 449.
126.3 199.9 13.14 59.5 13.80 7.50 94.00 96.00 0.71768 1260.3 0.02334 301.21 437.66 1.3150 1.6846 1.133 0.864 1.173 442.
125.9 195.9 13.21 59.1 13.96 7.30 96.00 98.00 0.75103 1253.7 0.02228 303.49 438.72 1.3211 1.6854 1.138 0.871 1.177 435.
125.5 191.9 13.29 58.6 14.13 7.09 98.00 100.00 0.78553 1246.9 0.02128 305.77 439.77 1.3271 1.6862 1.143 0.878 1.182 427.
125.1 188.1 13.37 58.2 14.29 6.89 100.00 110.00 0.97603 1211.9 0.01697 317.32 444.88 1.3572 1.6902 1.172 0.917 1.208 391.
122.8 169.9 13.80 56.0 15.17 5.88 110.00 120.00 1.1990 1174.4 0.01361 329.15 449.67 1.3872 1.6938 1.207 0.964 1.243 354.
119.8 153.4 14.29 53.9 16.14 4.91 120.00 130.00 1.4578 1133.6 0.01094 341.32 454.07 1.4173 1.6969 1.254 1.026 1.294 317.
116.0 138.1 14.89 51.7 17.22 3.98 130.00 140.00 1.7563 1088.3 0.00879 353.92 457.94 1.4475 1.6992 1.318 1.111 1.369 279.
111.5 123.8 15.65 49.5 18.44 3.09 140.00 150.00 2.0987 1036.8 0.00703 367.10 461.05 1.4782 1.7003 1.415 1.240 1.493 239.
106.0 110.2 16.68 47.2 19.87 2.24 150.00 160.00 2.4901 975.7 0.00555 381.13 463.01 1.5101 1.6991 1.584 1.473 1.726 198.
99.3 96.8 18.19 44.8 21.63 1.45 160.00 170.00 2.9372 896.9 0.00425 396.61 462.89 1.5443 1.6939 1.979 2.033 2.309 154.
91.1 82.7 20.71 42.3 24.05 0.74 170.00 180.00 3.4506 765.9 0.00292 416.22 456.82 1.5867 1.6763 4.549 5.661 6.158 102.
80.6 64.3 26.59 39.7 28.82 0.15 180.00 183.68c 3.6618 550.0 0.00182 437.39 437.39 1.6325 1.6325 ∞ ∞ ∞ 0.
0.0 — — ∞ ∞ 0.00 183.68 Temperatures are on the ITS-90 scale b = normal boiling point c = critical point 20.12 2001 ASHRAE Fundamentals Handbook (SI) Fig. 6 Pressure-Enthalpy Diagram for Refrigerant 124 Thermophysical Properties of Refrigerants 20.13 Refrigerant 124 (2-Chloro-1,1,1,2-tetrafluoroethane) Properties of Saturated Liquid and Saturated Vapor Temp., °C Pres-sure, MPa Density, kg/m3 Liquid Volume, m3/kg Vapor Enthalpy, kJ/kg Entropy, kJ/(kg·K) Specific Heat cp, kJ/(kg·K) cp/cv Velocity of Sound, m/s Viscosity, µPa·s Thermal Cond., mW/(m·K) Surface Tension, mN/m Temp., °C Liquid Vapor Liquid Vapor Liquid Vapor Vapor Liquid Vapor Liquid Vapor Liquid Vapor –100.00 0.00024 1714.6 44.375 98.87 302.29 0.5417 1.7165 0.953 0.534 1.129 1052.
109.1 — — — — 26.10 –100.00 –90.00 0.00067 1688.6 16.531 108.45 307.68 0.5954 1.6832 0.962 0.551 1.125 1007.
111.9 — — — — 24.69 –90.00 –80.00 0.00169 1662.3 6.9645 118.13 313.21 0.6469 1.6569 0.972 0.569 1.122 963.
114.6 — — — — 23.30 –80.00 –70.00 0.00379 1635.9 3.2523 127.91 318.86 0.6962 1.6362 0.983 0.586 1.119 919.
117.2 972.1 7.84 101.3 6.25 21.92 –70.00 –60.00 0.00779 1609.1 1.6560 137.80 324.62 0.7437 1.6202 0.995 0.605 1.117 875.
119.6 787.9 8.23 97.4 6.75 20.56 –60.00 –50.00 0.01482 1582.0 0.90713 147.81 330.46 0.7896 1.6081 1.007 0.624 1.116 832.
121.8 655.1 8.62 93.5 7.26 19.21 –50.00 –40.00 0.02642 1554.3 0.52863 157.95 336.36 0.8340 1.5993 1.020 0.644 1.116 790.
123.8 554.9 9.01 89.8 7.78 17.88 –40.00 –30.00 0.04452 1526.1 0.32470 168.23 342.29 0.8772 1.5930 1.034 0.665 1.118 748.
125.5 476.6 9.39 86.2 8.32 16.56 –30.00 –20.00 0.07145 1497.3 0.20856 178.66 348.23 0.9191 1.5890 1.049 0.688 1.120 706.
126.9 413.7 9.77 82.7 8.88 15.26 –20.00 –18.00 0.07813 1491.4 0.19180 180.76 349.42 0.9274 1.5884 1.052 0.693 1.121 698.
127.1 402.6 9.85 82.0 9.00 15.00 –18.00 –16.00 0.08529 1485.5 0.17665 182.87 350.61 0.9356 1.5879 1.056 0.698 1.122 690.
127.3 391.9 9.93 81.3 9.11 14.74 –16.00 –14.00 0.09296 1479.6 0.16293 184.99 351.79 0.9438 1.5874 1.059 0.703 1.123 682.
127.5 381.6 10.00 80.7 9.22 14.49 –14.00 –12.00 0.10117 1473.6 0.15048 187.11 352.97 0.9519 1.5870 1.062 0.708 1.124 673.
127.7 371.7 10.08 80.0 9.34 14.23 –12.00 –11.96b 0.10133 1473.5 0.15026 187.15 352.99 0.9521 1.5870 1.062 0.708 1.124 673.
127.7 371.5 10.08 80.0 9.34 14.23 –11.96 –10.00 0.10993 1467.6 0.13917 189.24 354.15 0.9600 1.5867 1.065 0.713 1.125 665.
127.9 362.1 10.15 79.3 9.46 13.98 –10.00 –8.00 0.11928 1461.6 0.12888 191.38 355.33 0.9681 1.5864 1.069 0.718 1.126 657.
128.1 352.8 10.23 78.7 9.57 13.72 –8.00 –6.00 0.12923 1455.5 0.11950 193.52 356.51 0.9761 1.5862 1.072 0.723 1.127 649.
128.2 343.9 10.31 78.0 9.69 13.47 –6.00 –4.00 0.13983 1449.4 0.11093 195.68 357.68 0.9841 1.5860 1.076 0.729 1.128 641.
128.3 335.3 10.38 77.3 9.81 13.22 –4.00 –2.00 0.15108 1443.2 0.10310 197.83 358.86 0.9921 1.5859 1.079 0.734 1.129 633.
128.5 326.9 10.46 76.7 9.93 12.97 –2.00 0.00 0.16303 1437.0 0.09593 200.00 360.02 1.0000 1.5858 1.083 0.740 1.131 625.
128.6 318.9 10.53 76.0 10.05 12.72 0.00 2.00 0.17570 1430.8 0.08936 202.17 361.19 1.0079 1.5858 1.087 0.746 1.132 617.
128.6 311.0 10.61 75.4 10.17 12.47 2.00 4.00 0.18911 1424.5 0.08333 204.35 362.35 1.0158 1.5858 1.090 0.751 1.134 609.
128.7 303.5 10.69 74.7 10.30 12.22 4.00 6.00 0.20331 1418.1 0.07779 206.54 363.51 1.0236 1.5859 1.094 0.757 1.135 601.
128.7 296.1 10.76 74.1 10.42 11.97 6.00 8.00 0.21830 1411.8 0.07268 208.74 364.67 1.0314 1.5860 1.098 0.763 1.137 593.
128.8 289.0 10.84 73.5 10.54 11.72 8.00 10.00 0.23414 1405.3 0.06798 210.94 365.82 1.0392 1.5861 1.102 0.769 1.139 585.
128.8 282.1 10.92 72.8 10.67 11.48 10.00 12.00 0.25084 1398.9 0.06364 213.15 366.97 1.0469 1.5863 1.106 0.776 1.141 577.
128.7 275.3 10.99 72.2 10.80 11.23 12.00 14.00 0.26844 1392.3 0.05964 215.37 368.11 1.0546 1.5865 1.110 0.782 1.143 569.
128.7 268.8 11.07 71.6 10.93 10.99 14.00 16.00 0.28696 1385.7 0.05593 217.60 369.25 1.0623 1.5868 1.114 0.788 1.145 561.
128.7 262.5 11.15 70.9 11.05 10.74 16.00 18.00 0.30644 1379.1 0.05250 219.84 370.38 1.0700 1.5870 1.119 0.795 1.148 553.
128.6 256.3 11.23 70.3 11.18 10.50 18.00 20.00 0.32692 1372.4 0.04932 222.09 371.51 1.0776 1.5873 1.123 0.802 1.150 545.
128.5 250.3 11.30 69.7 11.32 10.26 20.00 22.00 0.34842 1365.6 0.04636 224.34 372.63 1.0852 1.5876 1.128 0.809 1.153 537.
128.4 244.4 11.38 69.1 11.45 10.02 22.00 24.00 0.37097 1358.8 0.04362 226.60 373.75 1.0928 1.5880 1.132 0.816 1.156 529.
128.2 238.7 11.46 68.5 11.58 9.78 24.00 26.00 0.39462 1351.9 0.04107 228.88 374.85 1.1004 1.5883 1.137 0.823 1.159 521.
128.1 233.2 11.54 67.8 11.72 9.54 26.00 28.00 0.41938 1345.0 0.03870 231.16 375.96 1.1079 1.5887 1.142 0.830 1.162 513.
127.9 227.8 11.62 67.2 11.86 9.30 28.00 30.00 0.44530 1337.9 0.03648 233.45 377.05 1.1154 1.5891 1.147 0.838 1.165 505.
127.7 222.5 11.70 66.6 12.00 9.06 30.00 32.00 0.47241 1330.8 0.03442 235.75 378.14 1.1229 1.5895 1.152 0.845 1.169 497.
127.5 217.3 11.78 66.0 12.14 8.83 32.00 34.00 0.50075 1323.7 0.03249 238.07 379.22 1.1304 1.5900 1.157 0.853 1.172 489.
127.2 212.3 11.87 65.4 12.28 8.59 34.00 36.00 0.53034 1316.4 0.03069 240.39 380.29 1.1379 1.5904 1.163 0.861 1.176 481.
126.9 207.3 11.95 64.8 12.43 8.36 36.00 38.00 0.56123 1309.1 0.02901 242.72 381.36 1.1453 1.5909 1.168 0.870 1.180 473.
126.6 202.5 12.03 64.2 12.57 8.13 38.00 40.00 0.59345 1301.6 0.02743 245.07 382.41 1.1528 1.5913 1.174 0.878 1.185 465.
126.3 197.8 12.12 63.6 12.72 7.90 40.00 42.00 0.62704 1294.1 0.02596 247.43 383.45 1.1602 1.5918 1.180 0.887 1.189 457.
125.9 193.2 12.20 63.0 12.88 7.67 42.00 44.00 0.66202 1286.5 0.02457 249.79 384.49 1.1676 1.5923 1.187 0.897 1.194 449.
125.6 188.7 12.29 62.4 13.03 7.44 44.00 46.00 0.69845 1278.8 0.02327 252.17 385.51 1.1750 1.5928 1.193 0.906 1.199 441.
125.2 184.3 12.38 61.8 13.19 7.21 46.00 48.00 0.73635 1271.0 0.02205 254.56 386.52 1.1824 1.5933 1.200 0.916 1.205 433.
124.7 179.9 12.47 61.2 13.35 6.98 48.00 50.00 0.77577 1263.1 0.02090 256.97 387.53 1.1897 1.5937 1.207 0.926 1.211 425.
124.3 175.7 12.56 60.6 13.51 6.76 50.00 52.00 0.81675 1255.1 0.01982 259.39 388.51 1.1971 1.5942 1.214 0.937 1.217 417.
123.8 171.5 12.65 60.0 13.68 6.53 52.00 54.00 0.85931 1247.0 0.01880 261.82 389.49 1.2044 1.5947 1.221 0.948 1.224 409.
123.3 167.4 12.75 59.4 13.85 6.31 54.00 56.00 0.90350 1238.7 0.01784 264.26 390.45 1.2118 1.5951 1.229 0.959 1.231 401.
122.7 163.4 12.85 58.8 14.02 6.09 56.00 58.00 0.94937 1230.3 0.01693 266.73 391.39 1.2191 1.5956 1.238 0.971 1.239 393.
122.1 159.5 12.95 58.2 14.20 5.87 58.00 60.00 0.99695 1221.8 0.01607 269.20 392.33 1.2265 1.5960 1.246 0.984 1.247 385.
121.5 155.6 13.05 57.6 14.38 5.65 60.00 62.00 1.0463 1213.2 0.01527 271.69 393.24 1.2338 1.5965 1.255 0.997 1.256 376.
120.9 151.8 13.15 57.0 14.57 5.43 62.00 64.00 1.0974 1204.3 0.01450 274.20 394.14 1.2411 1.5969 1.265 1.011 1.266 368.
120.2 148.0 13.26 56.4 14.76 5.21 64.00 66.00 1.1504 1195.4 0.01378 276.72 395.01 1.2485 1.5972 1.275 1.026 1.276 360.
119.5 144.3 13.37 55.9 14.96 5.00 66.00 68.00 1.2052 1186.2 0.01309 279.27 395.87 1.2558 1.5976 1.286 1.042 1.287 351.
118.7 140.7 13.48 55.3 15.17 4.79 68.00 70.00 1.2620 1176.9 0.01244 281.83 396.71 1.2631 1.5979 1.297 1.058 1.299 343.
117.9 137.1 13.60 54.7 15.38 4.58 70.00 72.00 1.3207 1167.4 0.01183 284.41 397.52 1.2705 1.5982 1.310 1.076 1.313 334.
117.1 133.5 13.72 54.1 15.60 4.37 72.00 74.00 1.3815 1157.6 0.01124 287.01 398.31 1.2779 1.5985 1.323 1.095 1.327 326.
116.2 130.0 13.85 53.5 15.83 4.16 74.00 76.00 1.4443 1147.7 0.01069 289.63 399.08 1.2852 1.5987 1.337 1.115 1.343 317.
115.3 126.6 13.98 52.9 16.07 3.95 76.00 78.00 1.5093 1137.5 0.01016 292.28 399.81 1.2926 1.5989 1.352 1.137 1.360 309.
114.4 123.2 14.11 52.3 16.32 3.75 78.00 80.00 1.5764 1127.0 0.00965 294.95 400.52 1.3001 1.5990 1.368 1.160 1.379 300.
113.4 119.8 14.26 51.7 16.59 3.54 80.00 85.00 1.7540 1099.6 0.00849 301.75 402.14 1.3187 1.5990 1.415 1.229 1.436 278.
110.7 111.5 14.64 50.2 17.31 3.05 85.00 90.00 1.9462 1070.1 0.00746 308.74 403.51 1.3376 1.5986 1.475 1.318 1.513 255.
107.8 103.3 15.08 48.7 18.16 2.56 90.00 95.00 2.1542 1037.8 0.00653 315.97 404.56 1.3568 1.5975 1.554 1.437 1.618 231.
104.4 95.2 15.60 47.3 19.17 2.09 95.00 100.00 2.3787 1001.8 0.00569 323.50 405.20 1.3766 1.5955 1.665 1.607 1.773 207.
100.7 87.1 16.21 45.8 20.44 1.64 100.00 105.00 2.6212 960.8 0.00492 331.44 405.26 1.3971 1.5923 1.835 1.873 2.022 182.
96.6 78.8 16.99 44.5 22.11 1.21 105.00 110.00 2.8831 912.1 0.00419 339.99 404.46 1.4188 1.5870 2.135 2.357 2.478 156.
91.9 70.1 18.02 43.7 24.46 0.80 110.00 115.00 3.1662 849.5 0.00347 349.58 402.13 1.4428 1.5782 2.837 3.513 3.576 127.
86.6 60.4 19.57 44.3 28.26 0.43 115.00 120.00 3.4739 749.1 0.00267 361.94 395.71 1.4734 1.5593 6.828 9.976 9.676 93.
80.2 48.0 22.76 52.5 38.39 0.11 120.00 122.28c 3.6243 560.0 0.00179 378.79 378.79 1.5156 1.5156 ∞ ∞ ∞ 0.
0.0 — — ∞ ∞ 0.00 122.28 Temperatures are on the ITS-90 scale b = normal boiling point c = critical point 20.14 2001 ASHRAE Fundamentals Handbook (SI) Fig. 7 Pressure-Enthalpy Diagram for Refrigerant 125 Thermophysical Properties of Refrigerants 20.15 Refrigerant 125 (Pentafluoroethane) Properties of Saturated Liquid and Saturated Vapor Temp., °C Pres-sure, MPa Density, kg/m3 Liquid Volume, m3/kg Vapor Enthalpy, kJ/kg Entropy, kJ/(kg·K) Specific Heat cp, kJ/(kg·K) cp/cv Velocity of Sound, m/s Viscosity, µPa·s Thermal Cond., mW/(m·K) Surface Tension, mN/m Temp., °C Liquid Vapor Liquid Vapor Liquid Vapor Vapor Liquid Vapor Liquid Vapor Liquid Vapor –100.63a 0.00294 1690.8 4.0484 87.33 277.25 0.4910 1.5919 1.043 0.569 1.142 955.
116.4 1218.
7.43 120.4 5.72 21.79 –100.63 –100.00 0.00312 1688.8 3.8337 87.99 277.60 0.4948 1.5899 1.043 0.570 1.141 951.
116.6 1194.
7.46 120.0 5.76 21.69 –100.00 –90.00 0.00734 1656.1 1.7168 98.44 283.22 0.5535 1.5624 1.048 0.593 1.138 892.
119.4 905.2 7.90 114.1 6.31 20.08 –90.00 –80.00 0.01556 1623.1 0.85019 108.97 288.92 0.6094 1.5411 1.057 0.618 1.137 837.
121.9 718.7 8.33 108.5 6.88 18.50 –80.00 –70.00 0.03021 1589.6 0.45734 119.61 294.68 0.6631 1.5249 1.070 0.645 1.137 785.
124.1 588.2 8.76 103.2 7.48 16.94 –70.00 –60.00 0.05450 1555.5 0.26343 130.40 300.44 0.7149 1.5126 1.087 0.675 1.140 735.
126.0 491.5 9.19 98.2 8.10 15.41 –60.00 –58.00 0.06085 1548.5 0.23760 132.57 301.58 0.7250 1.5106 1.090 0.681 1.140 725.
126.3 475.1 9.28 97.2 8.23 15.11 –58.00 –56.00 0.06777 1541.6 0.21477 134.76 302.73 0.7351 1.5086 1.094 0.688 1.141 715.
126.6 459.4 9.37 96.2 8.36 14.81 –56.00 –54.00 0.07531 1534.5 0.19454 136.95 303.88 0.7451 1.5068 1.098 0.695 1.142 706.
126.9 444.5 9.45 95.2 8.49 14.51 –54.00 –52.00 0.08350 1527.5 0.17657 139.16 305.02 0.7551 1.5051 1.102 0.701 1.143 696.
127.1 430.3 9.54 94.2 8.62 14.21 –52.00 –50.00 0.09237 1520.4 0.16058 141.37 306.16 0.7650 1.5035 1.106 0.708 1.145 686.
127.4 416.8 9.62 93.3 8.75 13.91 –50.00 –48.13b 0.10133 1513.7 0.14720 143.44 307.22 0.7743 1.5021 1.110 0.715 1.146 677.
127.6 404.6 9.70 92.4 8.87 13.63 –48.13 –48.00 0.10198 1513.2 0.14631 143.59 307.30 0.7749 1.5020 1.110 0.715 1.146 677.
127.6 403.8 9.71 92.3 8.88 13.61 –48.00 –46.00 0.11236 1506.0 0.13355 145.81 308.43 0.7847 1.5006 1.114 0.723 1.147 667.
127.8 391.4 9.80 91.4 9.02 13.32 –46.00 –44.00 0.12355 1498.7 0.12211 148.05 309.56 0.7945 1.4993 1.119 0.730 1.149 658.
128.0 379.5 9.88 90.4 9.15 13.02 –44.00 –42.00 0.13559 1491.4 0.11184 150.29 310.69 0.8042 1.4981 1.123 0.738 1.151 648.
128.1 368.1 9.97 89.5 9.29 12.73 –42.00 –40.00 0.14853 1484.0 0.10260 152.55 311.81 0.8139 1.4970 1.128 0.745 1.153 639.
128.3 357.1 10.06 88.5 9.43 12.44 –40.00 –38.00 0.16241 1476.6 0.09427 154.81 312.93 0.8235 1.4959 1.132 0.753 1.155 630.
128.4 346.6 10.14 87.6 9.57 12.15 –38.00 –36.00 0.17728 1469.1 0.08675 157.09 314.04 0.8331 1.4949 1.137 0.761 1.157 620.
128.5 336.4 10.23 86.7 9.71 11.86 –36.00 –34.00 0.19318 1461.6 0.07994 159.37 315.15 0.8426 1.4940 1.142 0.769 1.159 611.
128.6 326.7 10.32 85.7 9.85 11.57 –34.00 –32.00 0.21017 1453.9 0.07377 161.66 316.25 0.8521 1.4932 1.148 0.777 1.161 601.
128.6 317.2 10.41 84.8 10.00 11.28 –32.00 –30.00 0.22829 1446.2 0.06817 163.97 317.35 0.8616 1.4924 1.153 0.786 1.164 592.
128.6 308.2 10.50 83.9 10.14 11.00 –30.00 –28.00 0.24758 1438.5 0.06307 166.28 318.44 0.8710 1.4917 1.158 0.794 1.167 583.
128.6 299.4 10.59 83.0 10.29 10.72 –28.00 –26.00 0.26810 1430.6 0.05843 168.61 319.52 0.8804 1.4910 1.164 0.803 1.170 573.
128.6 290.9 10.68 82.1 10.44 10.43 –26.00 –24.00 0.28990 1422.7 0.05419 170.95 320.60 0.8898 1.4904 1.169 0.812 1.173 564.
128.5 282.7 10.77 81.2 10.59 10.15 –24.00 –22.00 0.31304 1414.7 0.05032 173.30 321.66 0.8991 1.4899 1.175 0.821 1.176 555.
128.5 274.8 10.86 80.3 10.75 9.87 –22.00 –20.00 0.33755 1406.6 0.04678 175.66 322.72 0.9084 1.4894 1.181 0.830 1.179 545.
128.4 267.1 10.95 79.4 10.90 9.60 –20.00 –18.00 0.36350 1398.4 0.04353 178.03 323.77 0.9177 1.4889 1.188 0.840 1.183 536.
128.2 259.6 11.05 78.5 11.06 9.32 –18.00 –16.00 0.39093 1390.1 0.04054 180.42 324.81 0.9269 1.4884 1.194 0.849 1.187 527.
128.1 252.4 11.14 77.6 11.22 9.05 –16.00 –14.00 0.41991 1381.8 0.03780 182.82 325.85 0.9361 1.4880 1.201 0.859 1.191 517.
127.9 245.4 11.24 76.7 11.38 8.77 –14.00 –12.00 0.45049 1373.3 0.03528 185.23 326.87 0.9453 1.4877 1.207 0.869 1.196 508.
127.6 238.6 11.33 75.8 11.55 8.50 –12.00 –10.00 0.48272 1364.7 0.03295 187.66 327.88 0.9545 1.4873 1.214 0.880 1.201 499.
127.4 232.0 11.43 75.0 11.72 8.23 –10.00 –8.00 0.51666 1356.0 0.03081 190.09 328.88 0.9636 1.4870 1.222 0.891 1.206 489.
127.1 225.5 11.53 74.1 11.89 7.97 –8.00 –6.00 0.55237 1347.2 0.02882 192.55 329.86 0.9727 1.4867 1.229 0.902 1.211 480.
126.8 219.2 11.63 73.2 12.07 7.70 –6.00 –4.00 0.58990 1338.3 0.02699 195.02 330.84 0.9818 1.4865 1.237 0.913 1.217 471.
126.4 213.1 11.73 72.3 12.25 7.44 –4.00 –2.00 0.62932 1329.2 0.02528 197.50 331.80 0.9909 1.4862 1.245 0.925 1.224 461.
126.0 207.2 10.61 71.4 12.41 7.17 –2.00 0.00 0.67068 1320.0 0.02371 200.00 332.74 1.0000 1.4860 1.254 0.938 1.230 452.
125.6 201.4 10.99 70.6 12.58 6.91 0.00 2.00 0.71405 1310.7 0.02224 202.52 333.67 1.0091 1.4857 1.262 0.950 1.238 442.
125.2 195.7 11.23 69.7 12.75 6.65 2.00 4.00 0.75948 1301.2 0.02088 205.05 334.59 1.0181 1.4855 1.272 0.964 1.245 433.
124.7 190.2 11.42 68.8 12.93 6.40 4.00 6.00 0.80703 1291.5 0.01961 207.60 335.49 1.0271 1.4853 1.281 0.978 1.254 423.
124.1 184.8 11.60 68.0 13.12 6.14 6.00 8.00 0.85678 1281.7 0.01843 210.17 336.36 1.0362 1.4850 1.291 0.992 1.263 413.
123.6 179.5 11.76 67.1 13.31 5.89 8.00 10.00 0.90879 1271.7 0.01733 212.75 337.22 1.0452 1.4848 1.302 1.008 1.273 404.
123.0 174.3 11.91 66.2 13.51 5.64 10.00 12.00 0.96312 1261.5 0.01630 215.36 338.06 1.0542 1.4845 1.313 1.024 1.283 394.
122.3 169.2 12.07 65.4 13.72 5.39 12.00 14.00 1.0198 1251.1 0.01533 217.99 338.87 1.0632 1.4842 1.325 1.041 1.295 384.
121.6 164.3 12.22 64.5 13.94 5.14 14.00 16.00 1.0790 1240.5 0.01443 220.63 339.66 1.0723 1.4839 1.338 1.059 1.307 375.
120.9 159.4 12.37 63.6 14.16 4.90 16.00 18.00 1.1407 1229.7 0.01359 223.31 340.43 1.0813 1.4836 1.351 1.079 1.321 365.
120.1 154.6 12.53 62.8 14.39 4.66 18.00 20.00 1.2050 1218.6 0.01279 226.00 341.17 1.0903 1.4832 1.365 1.100 1.336 355.
119.3 149.9 12.69 61.9 14.63 4.42 20.00 22.00 1.2720 1207.3 0.01205 228.72 341.87 1.0994 1.4828 1.381 1.122 1.353 345.
118.4 145.3 12.85 61.0 14.89 4.18 22.00 24.00 1.3417 1195.7 0.01135 231.46 342.55 1.1085 1.4823 1.397 1.146 1.371 335.
117.5 140.8 13.01 60.2 15.15 3.95 24.00 26.00 1.4142 1183.8 0.01069 234.24 343.19 1.1176 1.4818 1.415 1.173 1.391 324.
116.5 136.3 13.19 59.3 15.43 3.72 26.00 28.00 1.4896 1171.5 0.01007 237.04 343.79 1.1267 1.4812 1.435 1.201 1.414 314.
115.5 131.9 13.37 58.4 15.73 3.49 28.00 30.00 1.5680 1158.9 0.00949 239.88 344.35 1.1359 1.4805 1.456 1.233 1.439 304.
114.4 127.6 13.55 57.5 16.05 3.26 30.00 32.00 1.6495 1145.9 0.00893 242.75 344.87 1.1451 1.4797 1.480 1.268 1.468 293.
113.3 123.3 13.75 56.7 16.38 3.04 32.00 34.00 1.7342 1132.4 0.00841 245.65 345.33 1.1543 1.4788 1.506 1.307 1.500 283.
112.1 119.1 13.95 55.8 16.74 2.82 34.00 36.00 1.8220 1118.5 0.00791 248.60 345.75 1.1636 1.4779 1.535 1.350 1.537 272.
110.8 114.9 14.17 54.9 17.14 2.60 36.00 38.00 1.9133 1104.0 0.00744 251.59 346.10 1.1730 1.4767 1.567 1.400 1.579 261.
109.5 110.7 14.40 54.1 17.56 2.39 38.00 40.00 2.0079 1089.0 0.00700 254.63 346.39 1.1825 1.4755 1.605 1.457 1.628 250.
108.1 106.5 14.64 53.2 18.02 2.18 40.00 42.00 2.1061 1073.3 0.00657 257.73 346.60 1.1920 1.4740 1.648 1.522 1.686 239.
106.7 102.4 14.91 52.3 18.54 1.97 42.00 44.00 2.2080 1056.7 0.00617 260.88 346.72 1.2017 1.4724 1.698 1.600 1.754 228.
105.2 98.3 15.19 51.4 19.10 1.77 44.00 46.00 2.3136 1039.4 0.00578 264.10 346.75 1.2115 1.4705 1.757 1.692 1.837 216.
103.6 94.1 15.51 50.6 19.74 1.57 46.00 48.00 2.4231 1020.9 0.00541 267.41 346.66 1.2215 1.4683 1.828 1.805 1.938 204.
101.9 90.0 15.85 49.7 20.46 1.38 48.00 50.00 2.5367 1001.3 0.00505 270.80 346.44 1.2317 1.4657 1.917 1.945 2.065 192.
100.1 85.8 16.23 48.9 21.28 1.19 50.00 52.00 2.6544 980.2 0.00470 274.30 346.06 1.2421 1.4628 2.029 2.124 2.228 180.
98.2 81.5 16.66 48.2 22.24 1.01 52.00 54.00 2.7765 957.2 0.00437 277.94 345.48 1.2529 1.4593 2.177 2.363 2.447 168.
96.3 77.2 17.16 47.5 23.37 0.84 54.00 56.00 2.9032 931.9 0.00404 281.75 344.65 1.2641 1.4552 2.382 2.696 2.753 155.
94.2 72.6 17.74 47.0 24.74 0.67 56.00 58.00 3.0346 903.4 0.00371 285.79 343.48 1.2759 1.4501 2.689 3.196 3.212 141.
92.0 67.9 18.44 46.8 26.45 0.51 58.00 60.00 3.1711 870.3 0.00338 290.15 341.84 1.2886 1.4437 3.200 4.028 3.976 127.
89.6 62.8 19.32 47.3 28.70 0.35 60.00 62.00 3.3130 829.8 0.00303 295.04 339.42 1.3027 1.4351 4.238 5.696 5.502 112.
86.9 57.2 20.53 49.3 31.93 0.21 62.00 64.00 3.4609 773.5 0.00264 301.03 335.43 1.3200 1.4220 7.51 10.75 10.08 96.
83.8 50.3 22.45 54.7 38.57 0.09 64.00 66.02c 3.6290 568.0 0.00176 318.39 318.39 1.3706 1.3706 ∞ ∞ ∞ 0.
0.0 — — ∞ ∞ 0.00 66.02 Temperatures are on the ITS-90 scale a = triple point b = normal boiling point c = critical point 20.16 2001 ASHRAE Fundamentals Handbook (SI) Fig. 8 Pressure-Enthalpy Diagram for Refrigerant 134a Thermophysical Properties of Refrigerants 20.17 Refrigerant 134a (1,1,1,2-Tetrafluoroethane) Properties of Saturated Liquid and Saturated Vapor Temp., °C Pres-sure, MPa Density, kg/m3 Liquid Volume, m3/kg Vapor Enthalpy, kJ/kg Entropy, kJ/(kg·K) Specific Heat cp, kJ/(kg·K) cp/cv Velocity of Sound, m/s Viscosity, µPa·s Thermal Cond., mW/(m·K) Surface Tension, mN/m Temp., °C Liquid Vapor Liquid Vapor Liquid Vapor Vapor Liquid Vapor Liquid Vapor Liquid Vapor –103.30a 0.00039 1591.1 35.496 71.46 334.94 0.4126 1.9639 1.184 0.585 1.164 1120.
126.8 2175.
6.46 145.2 3.08 28.07 –103.30 –100.00 0.00056 1582.4 25.193 75.36 336.85 0.4354 1.9456 1.184 0.593 1.162 1103.
127.9 1893.
6.60 143.2 3.34 27.50 –100.00 –90.00 0.00152 1555.8 9.7698 87.23 342.76 0.5020 1.8972 1.189 0.617 1.156 1052.
131.0 1339.
7.03 137.3 4.15 25.79 –90.00 –80.00 0.00367 1529.0 4.2682 99.16 348.83 0.5654 1.8580 1.198 0.642 1.151 1002.
134.0 1018.
7.46 131.5 4.95 24.10 –80.00 –70.00 0.00798 1501.9 2.0590 111.20 355.02 0.6262 1.8264 1.210 0.667 1.148 952.
136.8 809.2 7.89 126.0 5.75 22.44 –70.00 –60.00 0.01591 1474.3 1.0790 123.36 361.31 0.6846 1.8010 1.223 0.692 1.146 903.
139.4 663.1 8.30 120.7 6.56 20.80 –60.00 –50.00 0.02945 1446.3 0.60620 135.67 367.65 0.7410 1.7806 1.238 0.720 1.146 855.
141.7 555.1 8.72 115.6 7.36 19.18 –50.00 –40.00 0.05121 1417.7 0.36108 148.14 374.00 0.7956 1.7643 1.255 0.749 1.148 807.
143.6 472.2 9.12 110.6 8.17 17.60 –40.00 –30.00 0.08438 1388.4 0.22594 160.79 380.32 0.8486 1.7515 1.273 0.781 1.152 760.
145.2 406.4 9.52 105.8 8.99 16.04 –30.00 –28.00 0.09270 1382.4 0.20680 163.34 381.57 0.8591 1.7492 1.277 0.788 1.153 751.
145.4 394.9 9.60 104.8 9.15 15.73 –28.00 –26.07b 0.10133 1376.7 0.19018 165.81 382.78 0.8690 1.7472 1.281 0.794 1.154 742.
145.7 384.2 9.68 103.9 9.31 15.44 –26.07 –26.00 0.10167 1376.5 0.18958 165.90 382.82 0.8694 1.7471 1.281 0.794 1.154 742.
145.7 383.8 9.68 103.9 9.32 15.43 –26.00 –24.00 0.11130 1370.4 0.17407 168.47 384.07 0.8798 1.7451 1.285 0.801 1.155 732.
145.9 373.1 9.77 102.9 9.48 15.12 –24.00 –22.00 0.12165 1364.4 0.16006 171.05 385.32 0.8900 1.7432 1.289 0.809 1.156 723.
146.1 362.9 9.85 102.0 9.65 14.82 –22.00 –20.00 0.13273 1358.3 0.14739 173.64 386.55 0.9002 1.7413 1.293 0.816 1.158 714.
146.3 353.0 9.92 101.1 9.82 14.51 –20.00 –18.00 0.14460 1352.1 0.13592 176.23 387.79 0.9104 1.7396 1.297 0.823 1.159 705.
146.4 343.5 10.01 100.1 9.98 14.21 –18.00 –16.00 0.15728 1345.9 0.12551 178.83 389.02 0.9205 1.7379 1.302 0.831 1.161 695.
146.6 334.3 10.09 99.2 10.15 13.91 –16.00 –14.00 0.17082 1339.7 0.11605 181.44 390.24 0.9306 1.7363 1.306 0.838 1.163 686.
146.7 325.4 10.17 98.3 10.32 13.61 –14.00 –12.00 0.18524 1333.4 0.10744 184.07 391.46 0.9407 1.7348 1.311 0.846 1.165 677.
146.8 316.9 10.25 97.4 10.49 13.32 –12.00 –10.00 0.20060 1327.1 0.09959 186.70 392.66 0.9506 1.7334 1.316 0.854 1.167 668.
146.9 308.6 10.33 96.5 10.66 13.02 –10.00 –8.00 0.21693 1320.8 0.09242 189.34 393.87 0.9606 1.7320 1.320 0.863 1.169 658.
146.9 300.6 10.41 95.6 10.83 12.72 –8.00 –6.00 0.23428 1314.3 0.08587 191.99 395.06 0.9705 1.7307 1.325 0.871 1.171 649.
147.0 292.9 10.49 94.7 11.00 12.43 –6.00 –4.00 0.25268 1307.9 0.07987 194.65 396.25 0.9804 1.7294 1.330 0.880 1.174 640.
147.0 285.4 10.57 93.8 11.17 12.14 –4.00 –2.00 0.27217 1301.4 0.07436 197.32 397.43 0.9902 1.7282 1.336 0.888 1.176 631.
147.0 278.1 10.65 92.9 11.34 11.85 –2.00 0.00 0.29280 1294.8 0.06931 200.00 398.60 1.0000 1.7271 1.341 0.897 1.179 622.
146.9 271.1 10.73 92.0 11.51 11.56 0.00 2.00 0.31462 1288.1 0.06466 202.69 399.77 1.0098 1.7260 1.347 0.906 1.182 612.
146.9 264.3 10.81 91.1 11.69 11.27 2.00 4.00 0.33766 1281.4 0.06039 205.40 400.92 1.0195 1.7250 1.352 0.916 1.185 603.
146.8 257.6 10.90 90.2 11.86 10.99 4.00 6.00 0.36198 1274.7 0.05644 208.11 402.06 1.0292 1.7240 1.358 0.925 1.189 594.
146.7 251.2 10.98 89.4 12.04 10.70 6.00 8.00 0.38761 1267.9 0.05280 210.84 403.20 1.0388 1.7230 1.364 0.935 1.192 585.
146.5 244.9 11.06 88.5 12.22 10.42 8.00 10.00 0.41461 1261.0 0.04944 213.58 404.32 1.0485 1.7221 1.370 0.945 1.196 576.
146.4 238.8 11.15 87.6 12.40 10.14 10.00 12.00 0.44301 1254.0 0.04633 216.33 405.43 1.0581 1.7212 1.377 0.956 1.200 566.
146.2 232.9 11.23 86.7 12.58 9.86 12.00 14.00 0.47288 1246.9 0.04345 219.09 406.53 1.0677 1.7204 1.383 0.967 1.204 557.
146.0 227.1 11.32 85.9 12.77 9.58 14.00 16.00 0.50425 1239.8 0.04078 221.87 407.61 1.0772 1.7196 1.390 0.978 1.209 548.
145.7 221.5 11.40 85.0 12.95 9.30 16.00 18.00 0.53718 1232.6 0.03830 224.66 408.69 1.0867 1.7188 1.397 0.989 1.214 539.
145.5 216.0 11.49 84.1 13.14 9.03 18.00 20.00 0.57171 1225.3 0.03600 227.47 409.75 1.0962 1.7180 1.405 1.001 1.219 530.
145.1 210.7 11.58 83.3 13.33 8.76 20.00 22.00 0.60789 1218.0 0.03385 230.29 410.79 1.1057 1.7173 1.413 1.013 1.224 520.
144.8 205.5 11.67 82.4 13.53 8.48 22.00 24.00 0.64578 1210.5 0.03186 233.12 411.82 1.1152 1.7166 1.421 1.025 1.230 511.
144.5 200.4 11.76 81.6 13.72 8.21 24.00 26.00 0.68543 1202.9 0.03000 235.97 412.84 1.1246 1.7159 1.429 1.038 1.236 502.
144.1 195.4 11.85 80.7 13.92 7.95 26.00 28.00 0.72688 1195.2 0.02826 238.84 413.84 1.1341 1.7152 1.437 1.052 1.243 493.
143.6 190.5 11.95 79.8 14.13 7.68 28.00 30.00 0.77020 1187.5 0.02664 241.72 414.82 1.1435 1.7145 1.446 1.065 1.249 483.
143.2 185.8 12.04 79.0 14.33 7.42 30.00 32.00 0.81543 1179.6 0.02513 244.62 415.78 1.1529 1.7138 1.456 1.080 1.257 474.
142.7 181.1 12.14 78.1 14.54 7.15 32.00 34.00 0.86263 1171.6 0.02371 247.54 416.72 1.1623 1.7131 1.466 1.095 1.265 465.
142.1 176.6 12.24 77.3 14.76 6.89 34.00 36.00 0.91185 1163.4 0.02238 250.48 417.65 1.1717 1.7124 1.476 1.111 1.273 455.
141.6 172.1 12.34 76.4 14.98 6.64 36.00 38.00 0.96315 1155.1 0.02113 253.43 418.55 1.1811 1.7118 1.487 1.127 1.282 446.
141.0 167.7 12.44 75.6 15.21 6.38 38.00 40.00 1.0166 1146.7 0.01997 256.41 419.43 1.1905 1.7111 1.498 1.145 1.292 436.
140.3 163.4 12.55 74.7 15.44 6.13 40.00 42.00 1.0722 1138.2 0.01887 259.41 420.28 1.1999 1.7103 1.510 1.163 1.303 427.
139.7 159.2 12.65 73.9 15.68 5.88 42.00 44.00 1.1301 1129.5 0.01784 262.43 421.11 1.2092 1.7096 1.523 1.182 1.314 418.
138.9 155.1 12.76 73.0 15.93 5.63 44.00 46.00 1.1903 1120.6 0.01687 265.47 421.92 1.2186 1.7089 1.537 1.202 1.326 408.
138.2 151.0 12.88 72.1 16.18 5.38 46.00 48.00 1.2529 1111.5 0.01595 268.53 422.69 1.2280 1.7081 1.551 1.223 1.339 399.
137.4 147.0 13.00 71.3 16.45 5.13 48.00 50.00 1.3179 1102.3 0.01509 271.62 423.44 1.2375 1.7072 1.566 1.246 1.354 389.
136.6 143.1 13.12 70.4 16.72 4.89 50.00 52.00 1.3854 1092.9 0.01428 274.74 424.15 1.2469 1.7064 1.582 1.270 1.369 379.
135.7 139.2 13.24 69.6 17.01 4.65 52.00 54.00 1.4555 1083.2 0.01351 277.89 424.83 1.2563 1.7055 1.600 1.296 1.386 370.
134.7 135.4 13.37 68.7 17.31 4.41 54.00 56.00 1.5282 1073.4 0.01278 281.06 425.47 1.2658 1.7045 1.618 1.324 1.405 360.
133.8 131.6 13.51 67.8 17.63 4.18 56.00 58.00 1.6036 1063.2 0.01209 284.27 426.07 1.2753 1.7035 1.638 1.354 1.425 350.
132.7 127.9 13.65 67.0 17.96 3.95 58.00 60.00 1.6818 1052.9 0.01144 287.50 426.63 1.2848 1.7024 1.660 1.387 1.448 340.
131.7 124.2 13.79 66.1 18.31 3.72 60.00 62.00 1.7628 1042.2 0.01083 290.78 427.14 1.2944 1.7013 1.684 1.422 1.473 331.
130.5 120.6 13.95 65.2 18.68 3.49 62.00 64.00 1.8467 1031.2 0.01024 294.09 427.61 1.3040 1.7000 1.710 1.461 1.501 321.
129.4 117.0 14.11 64.3 19.07 3.27 64.00 66.00 1.9337 1020.0 0.00969 297.44 428.02 1.3137 1.6987 1.738 1.504 1.532 311.
128.1 113.5 14.28 63.4 19.50 3.05 66.00 68.00 2.0237 1008.3 0.00916 300.84 428.36 1.3234 1.6972 1.769 1.552 1.567 301.
126.8 109.9 14.46 62.6 19.95 2.83 68.00 70.00 2.1168 996.2 0.00865 304.28 428.65 1.3332 1.6956 1.804 1.605 1.607 290.
125.5 106.4 14.65 61.7 20.45 2.61 70.00 72.00 2.2132 983.8 0.00817 307.78 428.86 1.3430 1.6939 1.843 1.665 1.653 280.
124.0 102.9 14.85 60.8 20.98 2.40 72.00 74.00 2.3130 970.8 0.00771 311.33 429.00 1.3530 1.6920 1.887 1.734 1.705 269.
122.6 99.5 15.07 59.9 21.56 2.20 74.00 76.00 2.4161 957.3 0.00727 314.94 429.04 1.3631 1.6899 1.938 1.812 1.766 259.
121.0 96.0 15.30 59.0 22.21 1.99 76.00 78.00 2.5228 943.1 0.00685 318.63 428.98 1.3733 1.6876 1.996 1.904 1.838 248.
119.4 92.5 15.56 58.1 22.92 1.80 78.00 80.00 2.6332 928.2 0.00645 322.39 428.81 1.3836 1.6850 2.065 2.012 1.924 237.
117.7 89.0 15.84 57.2 23.72 1.60 80.00 85.00 2.9258 887.2 0.00550 332.22 427.76 1.4104 1.6771 2.306 2.397 2.232 207.
113.1 80.2 16.67 54.9 26.22 1.14 85.00 90.00 3.2442 837.8 0.00461 342.93 425.42 1.4390 1.6662 2.756 3.121 2.820 176.
107.9 70.9 17.81 52.8 29.91 0.71 90.00 95.00 3.5912 772.7 0.00374 355.25 420.67 1.4715 1.6492 3.938 5.020 4.369 141.
101.9 60.4 19.61 51.7 36.40 0.33 95.00 100.00 3.9724 651.2 0.00268 373.30 407.68 1.5188 1.6109 17.59 25.35 20.81 101.
94.0 45.1 24.21 59.9 60.58 0.04 100.00 101.06c 4.0593 511.9 0.00195 389.64 389.64 1.5621 1.5621 ∞ ∞ ∞ 0.
0.0 — — ∞ ∞ 0.00 101.06 Temperatures are on the ITS-90 scale a = triple point b = normal boiling point c = critical point 20.18 2001 ASHRAE Fundamentals Handbook (SI) Refrigerant 134a Properties of Superheated Vapor Pressure = 0.101325 MPa Saturation temperature = −26.07°C Pressure = 0.200 MPa Saturation temperature = −10.07°C Pressure = 0.400 MPa Saturation temperature = 8.94°C Temp., °C Density, kg/m3 Enthalpy, kJ/kg Entropy, kJ/(kg·K) Vel. Sound, m/s Temp., °C Density, kg/m3 Enthalpy, kJ/kg Entropy, kJ/(kg·K) Vel. Sound, m/s Temp., °C Density, kg/m3 Enthalpy, kJ/kg Entropy, kJ/(kg·K) Vel. Sound, m/s Saturated Saturated Saturated Liquid 1374.34 166.07 0.8701 747.1 Liquid 1325.78 186.69 0.9506 672.8 Liquid 1263.84 212.08 1.0432 583.8 Vapor 5.26 382.90 1.7476 145.7 Vapor 10.01 392.71 1.7337 146.9 Vapor 19.52 403.80 1.7229 146.6 −20.00 5.11 387.68 1.7667 147.8 −10.00 4.89 395.65 1.7976 151.0 −10.00 10.01 392.77 1.7339 147.0 0.00 4.69 403.74 1.8278 154.2 0.00 9.54 401.21 1.7654 150.6 10.00 4.50 411.97 1.8574 157.2 10.00 9.13 409.73 1.7961 154.0 10.00 19.41 404.78 1.7263 147.0 20.00 4.34 420.34 1.8864 160.1 20.00 8.76 418.35 1.8260 157.3 20.00 18.45 414.00 1.7583 151.2 30.00 4.18 428.85 1.9150 162.9 30.00 8.42 427.07 1.8552 160.4 30.00 17.61 423.21 1.7892 155.0 40.00 4.04 437.52 1.9431 165.7 40.00 8.12 435.90 1.8839 163.4 40.00 16.87 432.46 1.8192 158.6 50.00 3.91 446.33 1.9708 168.4 50.00 7.83 444.87 1.9121 166.3 50.00 16.20 441.76 1.8485 162.0 60.00 3.78 455.30 1.9981 171.0 60.00 7.57 453.97 1.9398 169.2 60.00 15.60 451.15 1.8771 165.3 70.00 3.67 464.43 2.0251 173.6 70.00 7.33 463.20 1.9671 171.9 70.00 15.05 460.63 1.9051 168.4 80.00 3.56 473.70 2.0518 176.1 80.00 7.11 472.57 1.9940 174.6 80.00 14.54 470.21 1.9326 171.4 90.00 3.46 483.13 2.0781 178.6 90.00 6.89 482.08 2.0206 177.2 90.00 14.08 479.91 1.9597 174.3 100.00 3.36 492.71 2.1041 181.0 100.00 6.70 491.74 2.0468 179.7 100.00 13.65 489.72 1.9864 177.1 110.00 3.27 502.44 2.1298 183.4 110.00 6.51 501.53 2.0727 182.2 110.00 13.24 499.65 2.0126 179.8 120.00 3.19 512.32 2.1553 185.7 120.00 6.34 511.47 2.0983 184.7 120.00 12.87 509.71 2.0386 182.4 130.00 3.11 522.35 2.1805 188.1 130.00 6.17 521.55 2.1236 187.1 130.00 12.51 519.90 2.0641 185.0 140.00 3.03 532.52 2.2054 190.3 140.00 6.01 531.76 2.1486 189.4 140.00 12.18 530.21 2.0894 187.5 150.00 2.96 542.83 2.2301 192.6 150.00 5.87 542.12 2.1734 191.7 150.00 11.87 540.66 2.1144 190.0 Pressure = 0.600 MPa Saturation temperature = 21.58°C Pressure = 0.800 MPa Saturation temperature = 31.33°C Pressure = 1.000 MPa Saturation temperature = 39.39°C Temp., °C Density, kg/m3 Enthalpy, kJ/kg Entropy, kJ/(kg·K) Vel. Sound, m/s Temp., °C Density, kg/m3 Enthalpy, kJ/kg Entropy, kJ/(kg·K) Vel. Sound, m/s Temp., °C Density, kg/m3 Enthalpy, kJ/kg Entropy, kJ/(kg·K) Vel. Sound, m/s Saturated Saturated Saturated Liquid 1219.08 229.62 1.1035 524.0 Liquid 1181.92 243.58 1.1495 477.4 Liquid 1149.06 255.44 1.1874 438.6 Vapor 29.13 410.67 1.7178 145.0 Vapor 38.99 415.58 1.7144 142.9 Vapor 49.16 419.31 1.7117 140.6 30.00 27.79 418.97 1.7455 149.0 40.00 26.41 428.72 1.7772 153.4 40.00 36.98 424.61 1.7437 147.6 40.00 48.95 419.99 1.7139 141.0 50.00 25.21 438.44 1.8077 157.4 50.00 35.03 434.85 1.7758 152.4 50.00 45.86 430.91 1.7482 146.9 60.00 24.16 448.16 1.8374 161.2 60.00 33.36 444.98 1.8067 156.8 60.00 43.34 441.56 1.7807 152.0 70.00 23.22 457.93 1.8662 164.7 70.00 31.90 455.08 1.8366 160.8 70.00 41.21 452.05 1.8117 156.7 80.00 22.37 467.75 1.8944 168.0 80.00 30.62 465.17 1.8656 164.6 80.00 39.36 462.47 1.8416 160.9 90.00 21.59 477.65 1.9221 171.2 90.00 29.46 475.30 1.8939 168.1 90.00 37.74 472.86 1.8706 164.9 100.00 20.88 487.64 1.9492 174.3 100.00 28.41 485.49 1.9215 171.5 100.00 36.29 483.26 1.8989 168.6 110.00 20.22 497.72 1.9759 177.3 110.00 27.46 495.74 1.9486 174.7 110.00 34.99 493.69 1.9265 172.1 120.00 19.61 507.92 2.0022 180.1 120.00 26.58 506.07 1.9753 177.8 120.00 33.80 504.19 1.9535 175.4 130.00 19.04 518.22 2.0280 182.9 130.00 25.77 516.50 2.0015 180.8 130.00 32.71 514.75 1.9800 178.6 140.00 18.51 528.63 2.0536 185.6 140.00 25.01 527.03 2.0272 183.7 140.00 31.70 525.39 2.0061 181.7 150.00 18.01 539.17 2.0787 188.2 150.00 24.31 537.66 2.0527 186.4 150.00 30.76 536.12 2.0318 184.6 160.00 17.54 549.82 2.1036 190.8 160.00 23.65 548.40 2.0777 189.2 160.00 29.90 546.95 2.0571 187.5 170.00 17.10 560.59 2.1282 193.3 170.00 23.03 559.24 2.1025 191.8 170.00 29.08 557.88 2.0820 190.3 180.00 16.68 571.48 2.1525 195.8 180.00 22.45 570.20 2.1270 194.4 180.00 28.32 568.91 2.1066 193.0 190.00 16.29 582.50 2.1766 198.2 190.00 21.89 581.28 2.1511 196.9 190.00 27.60 580.05 2.1309 195.6 200.00 15.91 593.63 2.2003 200.6 200.00 21.37 592.46 2.1750 199.4 200.00 26.92 591.29 2.1550 198.2 Pressure = 1.200 MPa Saturation temperature = 46.32°C Pressure = 1.400 MPa Saturation temperature = 52.43°C Pressure = 1.600 MPa Saturation temperature = 57.91°C Temp., °C Density, kg/m3 Enthalpy, kJ/kg Entropy, kJ/(kg·K) Vel. Sound, m/s Temp., °C Density, kg/m3 Enthalpy, kJ/kg Entropy, kJ/(kg·K) Vel. Sound, m/s Temp., °C Density, kg/m3 Enthalpy, kJ/kg Entropy, kJ/(kg·K) Vel. Sound, m/s Saturated Saturated Saturated Liquid 1118.89 265.91 1.2200 405.0 Liquid 1090.50 275.38 1.2488 375.1 Liquid 1063.28 284.11 1.2748 348.1 Vapor 59.73 422.22 1.7092 138.2 Vapor 70.76 424.50 1.7068 135.6 Vapor 82.34 426.27 1.7042 132.9 50.00 58.09 426.51 1.7226 140.7 60.00 54.32 437.83 1.7571 146.9 60.00 66.61 433.69 1.7347 141.2 60.00 80.74 428.99 1.7124 134.7 70.00 51.26 448.81 1.7896 152.3 70.00 62.25 445.31 1.7691 147.5 70.00 74.43 441.47 1.7493 142.3 80.00 48.69 459.61 1.8206 157.1 80.00 58.74 456.56 1.8014 153.0 80.00 69.61 453.30 1.7833 148.7 90.00 46.49 470.30 1.8504 161.5 90.00 55.79 467.60 1.8322 158.0 90.00 65.71 464.76 1.8153 154.2 100.00 44.55 480.94 1.8794 165.6 100.00 53.24 478.53 1.8619 162.5 100.00 62.43 476.01 1.8458 159.2 110.00 42.83 491.58 1.9075 169.4 110.00 51.03 489.39 1.8906 166.6 110.00 59.62 487.13 1.8753 163.8 120.00 41.28 502.25 1.9350 173.0 120.00 49.05 500.25 1.9186 170.5 120.00 57.14 498.19 1.9038 168.0 130.00 39.87 512.95 1.9619 176.4 130.00 47.28 511.11 1.9459 174.2 130.00 54.95 509.23 1.9315 171.9 140.00 38.58 523.72 1.9882 179.7 140.00 45.67 522.02 1.9726 177.7 140.00 52.98 520.28 1.9586 175.6 150.00 37.39 534.56 2.0142 182.8 150.00 44.19 532.97 1.9988 181.0 150.00 51.18 531.36 1.9851 179.1 160.00 36.29 545.48 2.0397 185.8 160.00 42.83 544.00 2.0246 184.2 160.00 49.54 542.49 2.0111 182.5 170.00 35.26 556.50 2.0648 188.8 170.00 41.57 555.10 2.0499 187.2 170.00 48.03 553.68 2.0366 185.7 180.00 34.31 567.60 2.0896 191.6 180.00 40.41 566.28 2.0748 190.2 180.00 46.63 564.94 2.0617 188.8 190.00 33.40 578.80 2.1141 194.4 190.00 39.31 577.55 2.0994 193.1 190.00 45.32 576.29 2.0865 191.8 200.00 32.56 590.11 2.1382 197.1 200.00 38.28 588.92 2.1237 195.9 200.00 44.10 587.71 2.1109 194.7 210.00 31.76 601.51 2.1621 199.7 210.00 37.32 600.38 2.1477 198.6 210.00 42.96 599.23 2.1350 197.6 220.00 31.01 613.02 2.1856 202.3 220.00 36.41 611.94 2.1714 201.3 220.00 41.88 610.84 2.1588 200.3 230.00 30.29 624.64 2.2090 204.8 230.00 35.55 623.60 2.1948 203.9 230.00 40.87 622.55 2.1823 203.0 240.00 29.61 636.36 2.2320 207.2 240.00 34.73 635.35 2.2179 206.4 240.00 39.91 634.35 2.2055 205.6 250.00 28.96 648.18 2.2548 209.7 250.00 33.96 647.22 2.2408 208.9 250.00 39.00 646.25 2.2285 208.2 Temperatures are on the ITS-90 scale Thermophysical Properties of Refrigerants 20.19 Refrigerant 134a Properties of Superheated Vapor (Concluded) Pressure = 1.800 MPa Saturation temperature = 62.90°C Pressure = 2.000 MPa Saturation temperature = 67.49°C Pressure = 2.200 MPa Saturation temperature = 71.74°C Temp., °C Density, kg/m3 Enthalpy, kJ/kg Entropy, kJ/(kg·K) Vel. Sound, m/s Temp., °C Density, kg/m3 Enthalpy, kJ/kg Entropy, kJ/(kg·K) Vel. Sound, m/s Temp., °C Density, kg/m3 Enthalpy, kJ/kg Entropy, kJ/(kg·K) Vel. Sound, m/s Saturated Saturated Saturated Liquid 1036.81 292.26 1.2987 323.2 Liquid 1010.74 299.96 1.3209 300.1 Liquid 984.76 307.32 1.3417 278.4 Vapor 94.53 427.59 1.7014 130.1 Vapor 107.46 428.52 1.6983 127.2 Vapor 121.25 429.08 1.6948 124.3 70.00 88.23 437.17 1.7296 136.5 70.00 104.37 432.22 1.7091 129.9 80.00 81.54 449.76 1.7657 144.0 80.00 94.85 445.86 1.7483 138.9 80.00 110.03 441.49 1.7303 133.3 90.00 76.38 461.74 1.7992 150.3 90.00 87.97 458.49 1.7835 146.2 90.00 100.70 454.98 1.7680 141.8 100.00 72.17 473.36 1.8308 155.9 100.00 82.58 470.57 1.8164 152.4 100.00 93.78 467.61 1.8023 148.7 110.00 68.64 484.78 1.8610 160.8 110.00 78.17 482.32 1.8474 157.8 110.00 88.25 479.75 1.8344 154.7 120.00 65.60 496.06 1.8900 165.4 120.00 74.44 493.86 1.8772 162.7 120.00 83.70 491.59 1.8649 160.0 130.00 62.91 507.29 1.9183 169.6 130.00 71.18 505.30 1.9059 167.2 130.00 79.79 503.25 1.8942 164.9 140.00 60.53 518.50 1.9457 173.5 140.00 68.33 516.68 1.9338 171.4 140.00 76.41 514.81 1.9226 169.3 150.00 58.37 529.71 1.9725 177.3 150.00 65.78 528.03 1.9609 175.4 150.00 73.40 526.32 1.9501 173.5 160.00 56.42 540.95 1.9988 180.8 160.00 63.47 539.39 1.9875 179.1 160.00 70.71 537.81 1.9769 177.4 170.00 54.62 552.24 2.0246 184.2 170.00 61.37 550.79 2.0135 182.6 170.00 68.28 549.31 2.0032 181.1 180.00 52.97 563.59 2.0499 187.4 180.00 59.45 562.23 2.0390 186.0 180.00 66.06 560.84 2.0289 184.6 190.00 51.44 575.01 2.0748 190.6 190.00 57.67 573.72 2.0641 189.3 190.00 64.02 572.42 2.0542 188.0 200.00 50.01 586.50 2.0993 193.6 200.00 56.02 585.28 2.0888 192.4 200.00 62.13 584.06 2.0790 191.3 210.00 48.68 598.08 2.1236 196.5 210.00 54.49 596.92 2.1131 195.5 210.00 60.38 595.76 2.1035 194.4 220.00 47.43 609.74 2.1475 199.4 220.00 53.05 608.64 2.1371 198.4 220.00 58.74 607.53 2.1276 197.5 230.00 46.25 621.50 2.1710 202.1 230.00 51.70 620.44 2.1608 201.3 230.00 57.21 619.38 2.1514 200.4 240.00 45.14 633.34 2.1944 204.9 240.00 50.43 632.33 2.1842 204.1 240.00 55.77 631.31 2.1749 203.3 250.00 44.09 645.28 2.2174 207.5 250.00 49.23 644.30 2.2073 206.8 250.00 54.42 643.33 2.1981 206.1 Pressure =2.400 MPa Saturation temperature = 75.70°C Pressure = 2.600 MPa Saturation temperature = 79.41°C Pressure = 2.800 MPa Saturation temperature = 82.90°C Temp., °C Density, kg/m3 Enthalpy, kJ/kg Entropy, kJ/(kg·K) Vel. Sound, m/s Temp., °C Density, kg/m3 Enthalpy, kJ/kg Entropy, kJ/(kg·K) Vel. Sound, m/s Temp., °C Density, kg/m3 Enthalpy, kJ/kg Entropy, kJ/(kg·K) Vel. Sound, m/s Saturated Saturated Saturated Liquid 958.58 314.40 1.3616 257.9 Liquid 931.88 321.29 1.3806 238.2 Liquid 904.29 328.05 1.3990 219.1 Vapor 136.07 429.27 1.6908 121.4 Vapor 152.12 429.08 1.6863 118.3 Vapor 169.71 428.50 1.6812 115.3 80.00 127.96 436.42 1.7112 126.9 80.00 150.48 430.22 1.6895 119.3 90.00 114.90 451.12 1.7523 137.0 90.00 131.08 446.81 1.7359 131.7 90.00 150.13 441.84 1.7183 125.9 100.00 105.89 464.44 1.7885 144.8 100.00 119.15 461.03 1.7745 140.8 100.00 133.85 457.32 1.7603 136.4 110.00 99.00 477.04 1.8218 151.5 110.00 110.50 474.19 1.8093 148.1 110.00 122.89 471.16 1.7970 144.6 120.00 93.44 489.22 1.8532 157.2 120.00 103.72 486.75 1.8417 154.4 120.00 114.63 484.17 1.8305 151.5 130.00 88.79 501.14 1.8831 162.4 130.00 98.17 498.96 1.8724 160.0 130.00 108.00 496.70 1.8620 157.5 140.00 84.77 512.90 1.9119 167.2 140.00 93.46 510.94 1.9017 165.0 140.00 102.49 508.93 1.8919 162.9 150.00 81.27 524.57 1.9398 171.6 150.00 89.39 522.79 1.9301 169.7 150.00 97.78 520.97 1.9207 167.8 160.00 78.15 536.20 1.9670 175.7 160.00 85.80 534.57 1.9576 174.0 160.00 93.66 532.90 1.9486 172.3 170.00 75.35 547.82 1.9935 179.6 170.00 82.59 546.30 1.9844 178.1 170.00 90.01 544.77 1.9757 176.5 180.00 72.81 559.45 2.0195 183.3 180.00 79.70 558.04 2.0106 181.9 180.00 86.74 556.61 2.0021 180.5 190.00 70.48 571.11 2.0449 186.8 190.00 77.07 569.79 2.0362 185.5 190.00 83.78 568.45 2.0279 184.3 200.00 68.34 582.82 2.0699 190.2 200.00 74.65 581.57 2.0614 189.0 200.00 81.08 580.31 2.0533 187.9 210.00 66.36 594.58 2.0945 193.4 210.00 72.43 593.40 2.0861 192.4 210.00 78.59 592.21 2.0782 191.4 220.00 64.51 606.41 2.1188 196.6 220.00 70.36 605.29 2.1105 195.6 220.00 76.29 604.16 2.1027 194.7 230.00 62.79 618.31 2.1427 199.6 230.00 68.44 617.24 2.1345 198.8 230.00 74.15 616.17 2.1268 198.0 240.00 61.18 630.29 2.1662 202.5 240.00 66.64 629.27 2.1581 201.8 240.00 72.16 628.25 2.1505 201.1 250.00 59.66 642.35 2.1895 205.4 250.00 64.95 641.37 2.1815 204.8 250.00 70.30 640.39 2.1740 204.1 Pressure = 3.000 MPa Saturation temperatue = 86.20°C Pressure = 4.000 MPa Saturation temperature = 100.35°C Pressure = 6.00 MPa Saturation temperature = n/a (supercritical) Temp., °C Density, kg/m3 Enthalpy, kJ/kg Entropy, kJ/(kg·K) Vel. Sound, m/s Temp., °C Density, kg/m3 Enthalpy, kJ/kg Entropy, kJ/(kg·K) Vel. Sound, m/s Temp., °C Density, kg/m3 Enthalpy, kJ/kg Entropy, kJ/(kg·K) Vel. Sound, m/s Saturated Saturated Saturated Liquid 875.30 334.75 1.4171 200.4 Liquid 626.95 376.48 1.5272 101.3 Liquid Vapor 189.25 427.47 1.6752 112.2 Vapor 396.29 404.57 1.6024 93.4 Vapor 90.00 173.82 435.84 1.6983 119.1 100.00 150.47 453.20 1.7455 131.8 110.00 136.36 467.93 1.7845 141.0 110.00 233.68 446.28 1.7131 119.8 110.00 762.66 375.61 1.5174 173.6 120.00 126.23 481.47 1.8194 148.5 120.00 199.79 465.29 1.7621 132.5 120.00 591.77 405.75 1.5950 127.4 130.00 118.34 494.36 1.8518 155.0 130.00 179.83 481.11 1.8018 142.0 130.00 418.90 439.87 1.6807 120.4 140.00 111.89 506.86 1.8824 160.7 140.00 165.73 495.51 1.8371 149.7 140.00 333.91 465.19 1.7428 130.1 150.00 106.45 519.11 1.9117 165.9 150.00 154.89 509.13 1.8697 156.4 150.00 289.37 484.69 1.7894 139.9 160.00 101.75 531.21 1.9399 170.6 160.00 146.10 522.25 1.9004 162.4 160.00 260.70 501.52 1.8288 148.3 170.00 97.62 543.21 1.9673 175.0 170.00 138.74 535.07 1.9296 167.8 170.00 239.96 516.92 1.8639 155.7 180.00 93.94 555.16 1.9940 179.2 180.00 132.41 547.69 1.9578 172.7 180.00 223.87 531.45 1.8963 162.2 190.00 90.62 567.10 2.0201 183.1 190.00 126.88 560.17 1.9850 177.4 190.00 210.82 545.43 1.9269 168.1 200.00 87.61 579.05 2.0456 186.8 200.00 121.97 572.58 2.0115 181.7 200.00 199.88 559.04 1.9559 173.6 210.00 84.84 591.02 2.0706 190.4 210.00 117.55 584.95 2.0374 185.8 210.00 190.50 572.39 1.9839 178.6 220.00 82.30 603.03 2.0952 193.8 220.00 113.56 597.30 2.0627 189.7 220.00 182.31 585.57 2.0109 183.4 230.00 79.94 615.10 2.1195 197.2 230.00 109.90 609.66 2.0875 193.4 230.00 175.06 598.64 2.0371 187.8 240.00 77.74 627.22 2.1433 200.4 240.00 106.55 622.05 2.1119 197.0 240.00 168.56 611.63 2.0626 192.1 250.00 75.69 639.41 2.1668 203.4 250.00 103.44 634.47 2.1359 200.5 250.00 162.68 624.57 2.0876 196.2 260.00 73.77 651.66 2.1900 206.5 260.00 100.56 646.93 2.1595 203.8 260.00 157.33 637.50 2.1121 200.0 270.00 71.96 664.00 2.2130 209.4 270.00 97.87 659.45 2.1827 207.1 270.00 152.41 650.43 2.1361 203.8 280.00 70.25 676.41 2.2356 212.2 280.00 95.35 672.03 2.2057 210.2 280.00 147.88 663.38 2.1598 207.4 290.00 68.63 688.89 2.2580 215.0 290.00 92.98 684.67 2.2283 213.3 290.00 143.67 676.35 2.1830 210.9 300.00 67.10 701.46 2.2801 217.8 300.00 90.75 697.38 2.2507 216.2 300.00 139.75 689.36 2.2059 214.3 Temperatures are on the ITS-90 scale 20.20 2001 ASHRAE Fundamentals Handbook (SI) Fig. 9 Pressure-Enthalpy Diagram for Refrigerant 152a Thermophysical Properties of Refrigerants 20.21 Refrigerant 152a (1,1-Difluoroethane) Properties of Saturated Liquid and Saturated Vapor Temp., °C Pres-sure, MPa Density, kg/m3 Liquid Volume, m3/kg Vapor Enthalpy, kJ/kg Entropy, kJ/(kg·K) Specific Heat cp, kJ/(kg·K) cp/cv Velocity of Sound, m/s Viscosity, µPa·s Thermal Cond., mW/(m·K) Surface Tension, mN/m Temp., °C Liquid Vapor Liquid Vapor Liquid Vapor Vapor Liquid Vapor Liquid Vapor Liquid Vapor –118.59a 0.00006 1192.9 303.29 13.79 419.32 0.1119 2.7357 1.477 0.699 1.220 1401.
154.0 — — 176.3 0.10 31.65 –118.59 –110.00 0.00019 1177.1 107.74 26.62 425.38 0.1927 2.6368 1.505 0.717 1.214 1337.
157.8 — — 170.2 0.94 30.23 –110.00 –100.00 0.00058 1158.7 37.617 41.75 432.59 0.2827 2.5399 1.518 0.740 1.207 1275.
162.0 1714.
5.81 163.4 1.91 28.58 –100.00 –90.00 0.00153 1140.1 15.052 56.96 439.97 0.3681 2.4593 1.524 0.763 1.201 1219.
166.0 1222.
6.14 156.9 2.89 26.96 –90.00 –80.00 0.00359 1121.3 6.7438 72.23 447.48 0.4492 2.3920 1.530 0.789 1.196 1166.
169.8 934.8 6.47 150.7 3.86 25.34 –80.00 –70.00 0.00765 1102.4 3.3197 87.57 455.08 0.5266 2.3357 1.539 0.816 1.192 1115.
173.4 747.5 6.81 144.7 4.84 23.75 –70.00 –60.00 0.01500 1083.2 1.7682 103.02 462.74 0.6009 2.2885 1.551 0.845 1.190 1066.
176.6 615.9 7.14 139.0 5.82 22.18 –60.00 –50.00 0.02742 1063.7 1.0064 118.62 470.40 0.6723 2.2487 1.567 0.877 1.189 1016.
179.5 518.4 7.48 133.5 6.81 20.63 –50.00 –40.00 0.04721 1043.8 0.60583 134.40 478.02 0.7414 2.2152 1.587 0.913 1.190 967.
182.1 443.3 7.82 128.2 7.80 19.09 –40.00 –30.00 0.07718 1023.5 0.38242 150.39 485.55 0.8085 2.1868 1.610 0.952 1.193 919.
184.2 339.7 8.21 123.1 8.81 17.58 –30.00 –28.00 0.08469 1019.4 0.35056 153.62 487.04 0.8216 2.1817 1.615 0.960 1.194 909.
184.5 325.3 8.29 122.1 9.01 17.29 –28.00 –26.00 0.09276 1015.3 0.32186 156.86 488.52 0.8348 2.1767 1.620 0.968 1.195 899.
184.9 312.5 8.36 121.2 9.21 16.99 –26.00 –24.02b 0.10133 1011.2 0.29622 160.07 489.98 0.8477 2.1719 1.625 0.977 1.196 889.
185.2 301.2 8.44 120.2 9.41 16.69 –24.02 –24.00 0.10142 1011.1 0.29595 160.11 490.00 0.8478 2.1719 1.625 0.977 1.196 889.
185.2 301.0 8.44 120.2 9.42 16.69 –24.00 –22.00 0.11072 1006.9 0.27253 163.37 491.47 0.8608 2.1672 1.630 0.985 1.197 880.
185.5 290.7 8.52 119.2 9.62 16.39 –22.00 –20.00 0.12068 1002.7 0.25131 166.64 492.94 0.8737 2.1627 1.635 0.994 1.199 870.
185.8 281.2 8.59 118.2 9.83 16.10 –20.00 –18.00 0.13133 998.5 0.23206 169.92 494.40 0.8866 2.1583 1.641 1.003 1.200 860.
186.1 272.4 8.67 117.3 10.03 15.80 –18.00 –16.00 0.14271 994.2 0.21457 173.21 495.85 0.8994 2.1541 1.647 1.013 1.202 850.
186.3 264.4 8.74 116.4 10.24 15.51 –16.00 –14.00 0.15484 989.9 0.19865 176.52 497.29 0.9122 2.1500 1.653 1.022 1.203 841.
186.5 256.8 8.82 115.4 10.45 15.22 –14.00 –12.00 0.16777 985.6 0.18414 179.83 498.72 0.9249 2.1460 1.658 1.032 1.205 831.
186.7 249.8 8.90 114.5 10.66 14.93 –12.00 –10.00 0.18152 981.3 0.17090 183.16 500.15 0.9375 2.1421 1.665 1.041 1.207 821.
186.9 243.1 8.97 113.6 10.87 14.64 –10.00 –8.00 0.19614 976.9 0.15879 186.50 501.56 0.9501 2.1383 1.671 1.051 1.209 811.
187.0 236.9 9.05 112.6 11.08 14.35 –8.00 –6.00 0.21166 972.5 0.14770 189.86 502.96 0.9627 2.1347 1.677 1.062 1.211 801.
187.1 230.9 9.12 111.7 11.30 14.06 –6.00 –4.00 0.22812 968.1 0.13754 193.22 504.36 0.9752 2.1311 1.684 1.072 1.213 792.
187.2 225.2 9.20 110.8 11.51 13.77 –4.00 –2.00 0.24555 963.6 0.12821 196.61 505.74 0.9876 2.1277 1.690 1.083 1.215 782.
187.3 219.8 9.27 109.9 11.73 13.48 –2.00 0.00 0.26399 959.1 0.11963 200.00 507.11 1.0000 2.1243 1.697 1.094 1.218 772.
187.4 214.6 9.35 109.0 11.94 13.20 0.00 2.00 0.28349 954.6 0.11174 203.41 508.47 1.0124 2.1211 1.704 1.105 1.221 762.
187.4 209.7 9.42 108.1 12.16 12.91 2.00 4.00 0.30407 950.0 0.10447 206.83 509.82 1.0247 2.1179 1.711 1.116 1.224 752.
187.4 204.9 9.50 107.2 12.39 12.63 4.00 6.00 0.32578 945.4 0.09776 210.27 511.16 1.0370 2.1148 1.719 1.128 1.227 742.
187.4 200.3 9.57 106.3 12.61 12.35 6.00 8.00 0.34867 940.8 0.09156 213.72 512.48 1.0492 2.1118 1.726 1.139 1.230 733.
187.3 195.8 9.65 105.4 12.83 12.07 8.00 10.00 0.37277 936.1 0.08583 217.19 513.78 1.0614 2.1089 1.734 1.152 1.233 723.
187.2 191.5 9.73 104.5 13.06 11.79 10.00 12.00 0.39812 931.3 0.08052 220.67 515.08 1.0736 2.1060 1.742 1.164 1.237 713.
187.1 187.3 9.80 103.7 13.29 11.51 12.00 14.00 0.42476 926.6 0.07560 224.17 516.36 1.0857 2.1032 1.750 1.177 1.240 703.
187.0 183.3 9.88 102.8 13.52 11.23 14.00 16.00 0.45275 921.8 0.07104 227.69 517.62 1.0978 2.1005 1.759 1.190 1.244 693.
186.8 179.3 9.95 101.9 13.76 10.96 16.00 18.00 0.48211 916.9 0.06680 231.22 518.86 1.1098 2.0978 1.768 1.203 1.249 683.
186.6 175.5 10.03 101.1 13.99 10.68 18.00 20.00 0.51291 912.0 0.06286 234.77 520.09 1.1219 2.0952 1.776 1.217 1.253 673.
186.4 171.8 10.11 100.2 14.24 10.41 20.00 22.00 0.54517 907.0 0.05919 238.34 521.30 1.1339 2.0926 1.786 1.231 1.258 663.
186.1 168.1 10.18 99.4 14.48 10.14 22.00 24.00 0.57894 902.0 0.05577 241.93 522.50 1.1459 2.0901 1.795 1.246 1.263 653.
185.8 164.5 10.26 98.5 14.73 9.87 24.00 26.00 0.61428 896.9 0.05258 245.53 523.67 1.1578 2.0876 1.805 1.261 1.268 643.
185.5 161.0 10.34 97.7 14.98 9.60 26.00 28.00 0.65122 891.8 0.04960 249.16 524.83 1.1698 2.0852 1.815 1.277 1.274 633.
185.1 157.6 10.42 96.8 15.23 9.33 28.00 30.00 0.68982 886.6 0.04682 252.80 525.96 1.1817 2.0828 1.826 1.293 1.280 623.
184.7 154.3 10.50 96.0 15.49 9.06 30.00 32.00 0.73012 881.4 0.04422 256.47 527.07 1.1936 2.0804 1.837 1.309 1.286 613.
184.3 151.0 10.57 95.1 15.76 8.80 32.00 34.00 0.77216 876.0 0.04179 260.16 528.16 1.2055 2.0780 1.848 1.326 1.293 602.
183.9 147.8 10.66 94.3 16.03 8.54 34.00 36.00 0.81600 870.7 0.03951 263.86 529.23 1.2174 2.0757 1.860 1.344 1.300 592.
183.4 144.6 10.74 93.5 16.30 8.27 36.00 38.00 0.86169 865.2 0.03737 267.60 530.27 1.2292 2.0734 1.872 1.362 1.307 582.
182.9 141.5 10.82 92.6 16.58 8.01 38.00 40.00 0.90927 859.7 0.03536 271.35 531.28 1.2411 2.0711 1.885 1.381 1.315 572.
182.3 138.5 10.90 91.8 16.87 7.75 40.00 42.00 0.95879 854.1 0.03348 275.13 532.27 1.2529 2.0689 1.898 1.401 1.324 561.
181.7 135.5 10.99 91.0 17.17 7.50 42.00 44.00 1.0103 848.4 0.03170 278.93 533.23 1.2648 2.0666 1.912 1.421 1.333 551.
181.1 132.5 11.07 90.2 17.47 7.24 44.00 46.00 1.0639 842.6 0.03004 282.76 534.16 1.2766 2.0643 1.926 1.443 1.342 541.
180.4 129.6 11.16 89.3 17.78 6.99 46.00 48.00 1.1196 836.7 0.02846 286.62 535.06 1.2884 2.0620 1.941 1.465 1.353 530.
179.7 126.7 11.25 88.5 18.10 6.73 48.00 50.00 1.1774 830.8 0.02699 290.50 535.93 1.3003 2.0598 1.957 1.489 1.364 520.
178.9 123.9 11.34 87.7 18.43 6.48 50.00 52.00 1.2374 824.7 0.02559 294.41 536.77 1.3121 2.0575 1.974 1.513 1.375 509.
178.2 121.1 11.44 86.9 18.77 6.23 52.00 54.00 1.2997 818.6 0.02427 298.35 537.56 1.3240 2.0552 1.992 1.539 1.388 499.
177.3 118.4 11.53 86.1 19.12 5.98 54.00 56.00 1.3643 812.3 0.02303 302.33 538.32 1.3358 2.0528 2.010 1.566 1.402 488.
176.4 115.6 11.63 85.3 19.48 5.74 56.00 58.00 1.4313 805.9 0.02185 306.34 539.04 1.3477 2.0504 2.030 1.595 1.416 478.
175.5 113.0 11.73 84.5 19.86 5.50 58.00 60.00 1.5007 799.4 0.02074 310.38 539.72 1.3596 2.0480 2.051 1.626 1.432 467.
174.6 110.3 11.84 83.6 20.25 5.25 60.00 62.00 1.5726 792.7 0.01968 314.45 540.35 1.3716 2.0456 2.073 1.658 1.450 456.
173.6 107.7 11.95 82.8 20.66 5.01 62.00 64.00 1.6471 785.9 0.01868 318.57 540.94 1.3835 2.0431 2.097 1.693 1.468 445.
172.5 105.1 12.06 82.0 21.08 4.78 64.00 66.00 1.7242 779.0 0.01774 322.72 541.47 1.3955 2.0405 2.122 1.730 1.488 435.
171.4 102.5 12.17 81.2 21.53 4.54 66.00 68.00 1.8039 771.9 0.01684 326.92 541.95 1.4076 2.0379 2.150 1.769 1.511 424.
170.2 99.9 12.30 80.4 22.00 4.31 68.00 70.00 1.8864 764.6 0.01598 331.16 542.37 1.4196 2.0351 2.179 1.812 1.535 413.
169.0 97.4 12.42 79.6 22.49 4.08 70.00 72.00 1.9717 757.2 0.01517 335.45 542.73 1.4318 2.0323 2.211 1.859 1.562 402.
167.8 94.9 12.56 78.8 23.01 3.85 72.00 74.00 2.0599 749.5 0.01440 339.79 543.02 1.4440 2.0294 2.245 1.909 1.591 390.
166.5 92.4 12.70 78.0 23.56 3.62 74.00 76.00 2.1510 741.6 0.01366 344.18 543.24 1.4562 2.0264 2.283 1.964 1.624 379.
165.1 89.9 12.85 77.2 24.15 3.40 76.00 78.00 2.2452 733.5 0.01296 348.63 543.38 1.4686 2.0232 2.324 2.025 1.661 368.
163.7 87.5 13.01 76.4 24.77 3.18 78.00 80.00 2.3424 725.2 0.01228 353.15 543.43 1.4810 2.0198 2.370 2.092 1.702 356.
162.2 85.0 13.18 75.6 25.44 2.96 80.00 90.00 2.8780 678.5 0.00933 376.87 542.06 1.5451 2.0000 2.703 2.586 2.016 297.
153.7 72.8 14.23 71.6 29.64 1.91 90.00 100.00 3.5050 618.5 0.00686 403.59 536.28 1.6151 1.9707 3.495 3.776 2.805 233.
143.1 60.3 15.92 68.0 36.59 0.96 100.00 110.00 4.2432 517.4 0.00446 439.22 517.31 1.7058 1.9096 9.26 12.22 8.53 158.
129.1 44.7 19.94 69.5 55.48 0.17 110.00 113.26c 4.5168 368.0 0.00272 477.55 477.55 1.8037 1.8037 ∞ ∞ ∞ 0.
0.0 — — ∞ ∞ 0.00 113.26 Temperatures are on the ITS-90 scale a = triple point b = normal boiling point c = critical point 20.22 2001 ASHRAE Fundamentals Handbook (SI) Refrigerant 143a (1,1,1-Trifluoroethane) Properties of Saturated Liquid and Saturated Vapor Temp., °C Pres-sure, MPa Density, kg/m3 Liquid Volume, m3/kg Vapor Enthalpy, kJ/kg Entropy, kJ/(kg·K) Specific Heat cp, kJ/(kg·K) cp/cv Velocity of Sound, m/s Viscosity, µPa·s Thermal Cond., mW/(m·K) Surface Tension, mN/m Temp., °C Liquid Vapor Liquid Vapor Liquid Vapor Vapor Liquid Vapor Liquid Vapor Liquid Vapor –111.81a 0.00107 1330.5 14.807 52.52 319.59 0.3142 1.9695 1.211 0.630 1.192 1058.
137.6 912.1 5.91 137.0 4.90 13.72 –111.81 –110.00 0.00129 1326.2 12.430 54.71 320.68 0.3277 1.9579 1.212 0.635 1.191 1049.
138.2 867.9 5.97 135.8 5.00 13.75 –110.00 –100.00 0.00333 1301.9 5.1127 66.87 326.81 0.4000 1.9012 1.220 0.664 1.185 1002.
141.7 680.1 6.34 129.5 5.53 13.80 –100.00 –90.00 0.00761 1277.2 2.3596 79.13 333.06 0.4688 1.8553 1.233 0.694 1.181 954.
144.9 553.4 6.72 123.5 6.10 13.71 –90.00 –80.00 0.01572 1252.2 1.1971 91.55 339.40 0.5348 1.8180 1.250 0.726 1.178 907.
147.8 462.1 7.08 117.8 6.70 13.48 –80.00 –70.00 0.02991 1226.7 0.65675 104.16 345.80 0.5984 1.7879 1.270 0.759 1.177 859.
150.4 393.0 7.45 112.5 7.34 13.11 –70.00 –60.00 0.05307 1200.6 0.38446 116.99 352.21 0.6599 1.7635 1.293 0.794 1.178 811.
152.5 338.8 7.82 107.4 8.02 12.61 –60.00 –50.00 0.08874 1173.9 0.23754 130.05 358.58 0.7197 1.7438 1.318 0.833 1.182 764.
154.2 294.9 8.19 102.5 8.73 12.00 –50.00 –48.00 0.09773 1168.5 0.21695 132.69 359.85 0.7314 1.7403 1.323 0.841 1.184 754.
154.5 287.1 8.26 101.6 8.88 11.86 –48.00 –47.24b 0.10133 1166.4 0.20971 133.70 360.33 0.7359 1.7391 1.325 0.844 1.184 751.
154.6 284.2 8.29 101.2 8.94 11.81 –47.24 –46.00 0.10742 1163.0 0.19849 135.35 361.11 0.7431 1.7370 1.328 0.850 1.185 745.
154.7 279.6 8.33 100.6 9.03 11.72 –46.00 –44.00 0.11786 1157.5 0.18191 138.01 362.37 0.7548 1.7339 1.334 0.858 1.186 735.
154.9 272.3 8.41 99.7 9.18 11.57 –44.00 –42.00 0.12907 1152.0 0.16697 140.69 363.62 0.7664 1.7308 1.339 0.867 1.188 726.
155.1 265.3 8.48 98.8 9.33 11.42 –42.00 –40.00 0.14109 1146.4 0.15350 143.38 364.86 0.7779 1.7279 1.345 0.876 1.189 716.
155.3 258.6 8.56 97.8 9.49 11.27 –40.00 –38.00 0.15398 1140.8 0.14133 146.08 366.10 0.7894 1.7251 1.351 0.885 1.191 707.
155.5 252.0 8.63 96.9 9.65 11.11 –38.00 –36.00 0.16775 1135.1 0.13031 148.79 367.34 0.8008 1.7224 1.357 0.894 1.193 697.
155.6 245.7 8.70 96.0 9.81 10.95 –36.00 –34.00 0.18247 1129.4 0.12032 151.52 368.56 0.8122 1.7198 1.363 0.904 1.195 688.
155.7 239.6 8.78 95.1 9.97 10.79 –34.00 –32.00 0.19816 1123.7 0.11124 154.25 369.78 0.8236 1.7173 1.369 0.913 1.198 678.
155.8 233.6 8.85 94.2 10.13 10.62 –32.00 –30.00 0.21488 1117.9 0.10297 157.00 370.99 0.8348 1.7149 1.375 0.923 1.200 669.
155.8 227.8 8.93 93.3 10.30 10.44 –30.00 –28.00 0.23267 1112.1 0.09544 159.77 372.19 0.8461 1.7126 1.382 0.933 1.203 659.
155.9 222.3 9.18 92.4 10.49 10.27 –28.00 –26.00 0.25156 1106.2 0.08857 162.54 373.39 0.8573 1.7104 1.388 0.944 1.206 650.
155.9 216.8 9.26 91.6 10.66 10.09 –26.00 –24.00 0.27161 1100.3 0.08228 165.33 374.57 0.8685 1.7083 1.395 0.955 1.209 640.
155.8 211.6 9.33 90.7 10.83 9.90 –24.00 –22.00 0.29286 1094.3 0.07652 168.13 375.74 0.8796 1.7062 1.402 0.966 1.212 630.
155.8 206.4 9.41 89.8 11.01 9.72 –22.00 –20.00 0.31535 1088.3 0.07125 170.95 376.91 0.8907 1.7043 1.409 0.977 1.216 621.
155.7 201.4 9.48 89.0 11.19 9.53 –20.00 –18.00 0.33915 1082.2 0.06640 173.78 378.06 0.9018 1.7024 1.417 0.988 1.219 611.
155.6 196.6 9.56 88.1 11.37 9.33 –18.00 –16.00 0.36428 1076.0 0.06194 176.63 379.20 0.9128 1.7005 1.424 1.000 1.223 602.
155.4 191.8 9.64 87.2 11.55 9.14 –16.00 –14.00 0.39081 1069.8 0.05784 179.49 380.33 0.9238 1.6987 1.432 1.012 1.227 592.
155.2 187.2 9.71 86.4 11.74 8.94 –14.00 –12.00 0.41877 1063.6 0.05405 182.37 381.44 0.9347 1.6970 1.440 1.025 1.232 582.
155.0 182.7 9.79 85.5 11.94 8.74 –12.00 –10.00 0.44823 1057.2 0.05056 185.27 382.54 0.9457 1.6953 1.449 1.038 1.237 573.
154.8 178.4 9.87 84.7 12.13 8.53 –10.00 –8.00 0.47923 1050.8 0.04733 188.18 383.63 0.9566 1.6937 1.457 1.051 1.242 563.
154.5 174.1 9.95 83.9 12.33 8.32 –8.00 –6.00 0.51182 1044.3 0.04434 191.11 384.70 0.9675 1.6921 1.466 1.065 1.247 553.
154.2 169.9 10.04 83.0 12.53 8.12 –6.00 –4.00 0.54606 1037.7 0.04158 194.05 385.75 0.9783 1.6906 1.476 1.079 1.253 544.
153.9 165.8 10.12 82.2 12.74 7.90 –4.00 –2.00 0.58199 1031.0 0.03901 197.02 386.79 0.9892 1.6890 1.485 1.093 1.260 534.
153.5 161.8 10.21 81.4 12.96 7.69 –2.00 0.00 0.61967 1024.3 0.03662 200.00 387.81 1.0000 1.6876 1.495 1.109 1.266 524.
153.1 157.9 10.29 80.5 13.17 7.47 0.00 2.00 0.65916 1017.4 0.03440 203.00 388.81 1.0108 1.6861 1.505 1.124 1.273 515.
152.6 154.1 10.38 79.7 13.40 7.25 2.00 4.00 0.70051 1010.5 0.03234 206.03 389.79 1.0216 1.6846 1.516 1.141 1.281 505.
152.1 150.3 10.47 78.9 13.63 7.03 4.00 6.00 0.74378 1003.5 0.03042 209.07 390.75 1.0324 1.6832 1.528 1.158 1.289 495.
151.6 146.6 10.57 78.1 13.87 6.81 6.00 8.00 0.78901 996.3 0.02862 212.13 391.68 1.0432 1.6818 1.539 1.176 1.298 485.
151.0 143.0 10.66 77.2 14.12 6.59 8.00 10.00 0.83628 989.1 0.02695 215.22 392.60 1.0539 1.6804 1.552 1.194 1.307 475.
150.4 139.5 10.76 76.4 14.38 6.36 10.00 12.00 0.88564 981.7 0.02538 218.33 393.48 1.0647 1.6790 1.565 1.214 1.317 465.
149.8 136.0 10.86 75.6 14.64 6.14 12.00 14.00 0.93714 974.2 0.02392 221.47 394.35 1.0755 1.6775 1.578 1.234 1.328 455.
149.1 132.6 10.96 74.8 14.92 5.91 14.00 16.00 0.99085 966.5 0.02255 224.63 395.18 1.0863 1.6761 1.593 1.256 1.340 445.
148.4 129.3 11.07 74.0 15.21 5.68 16.00 18.00 1.0468 958.7 0.02126 227.81 395.98 1.0970 1.6747 1.608 1.278 1.353 435.
147.6 126.0 11.18 73.2 15.51 5.45 18.00 20.00 1.1052 950.8 0.02005 231.02 396.76 1.1078 1.6732 1.624 1.302 1.366 425.
146.8 122.7 11.29 72.4 15.82 5.22 20.00 22.00 1.1659 942.7 0.01892 234.27 397.50 1.1186 1.6717 1.641 1.328 1.381 415.
145.9 119.5 11.40 71.5 16.15 4.99 22.00 24.00 1.2290 934.4 0.01785 237.54 398.20 1.1295 1.6701 1.659 1.355 1.398 405.
145.0 116.4 11.52 70.7 16.50 4.76 24.00 26.00 1.2947 926.0 0.01685 240.84 398.87 1.1403 1.6685 1.679 1.384 1.416 394.
144.0 113.3 11.64 69.9 16.87 4.53 26.00 28.00 1.3630 917.3 0.01591 244.18 399.49 1.1512 1.6669 1.699 1.416 1.435 384.
143.0 110.2 11.77 69.1 17.26 4.30 28.00 30.00 1.4340 908.4 0.01501 247.56 400.07 1.1621 1.6652 1.722 1.449 1.457 374.
141.9 107.2 11.91 68.3 17.67 4.07 30.00 32.00 1.5077 899.3 0.01417 250.97 400.61 1.1730 1.6634 1.746 1.486 1.480 363.
140.8 104.2 12.04 67.5 18.11 3.84 32.00 34.00 1.5842 890.0 0.01338 254.42 401.09 1.1840 1.6616 1.772 1.526 1.507 352.
139.6 101.2 12.19 66.6 18.58 3.61 34.00 36.00 1.6636 880.4 0.01262 257.91 401.52 1.1951 1.6596 1.801 1.570 1.536 342.
138.4 98.3 12.34 65.8 19.09 3.38 36.00 38.00 1.7460 870.5 0.01191 261.45 401.89 1.2062 1.6575 1.832 1.618 1.569 331.
137.1 95.4 12.50 65.0 19.63 3.16 38.00 40.00 1.8314 860.3 0.01123 265.04 402.19 1.2174 1.6553 1.867 1.671 1.606 320.
135.7 92.5 12.67 64.2 20.22 2.93 40.00 42.00 1.9200 849.7 0.01059 268.68 402.42 1.2286 1.6530 1.906 1.732 1.648 309.
134.3 89.7 12.85 63.3 20.87 2.71 42.00 44.00 2.0117 838.7 0.00998 272.39 402.56 1.2400 1.6505 1.949 1.799 1.696 298.
132.8 86.8 13.03 62.5 21.57 2.49 44.00 46.00 2.1068 827.3 0.00940 276.15 402.62 1.2515 1.6478 1.998 1.877 1.752 286.
131.2 84.0 13.24 61.7 22.35 2.27 46.00 48.00 2.2053 815.4 0.00884 279.98 402.58 1.2631 1.6448 2.054 1.966 1.817 275.
129.6 81.1 13.45 60.8 23.20 2.06 48.00 50.00 2.3073 803.0 0.00831 283.90 402.43 1.2748 1.6416 2.118 2.070 1.894 263.
127.9 78.3 13.69 60.0 24.16 1.85 50.00 52.00 2.4130 789.9 0.00780 287.90 402.15 1.2868 1.6381 2.194 2.194 1.985 251.
126.1 75.4 13.94 59.1 25.23 1.64 52.00 54.00 2.5224 776.1 0.00731 292.00 401.72 1.2989 1.6343 2.285 2.343 2.097 238.
124.2 72.5 14.23 58.2 26.45 1.44 54.00 56.00 2.6357 761.5 0.00684 296.22 401.12 1.3113 1.6300 2.395 2.528 2.236 226.
122.2 69.6 14.54 57.4 27.84 1.24 56.00 58.00 2.7530 745.8 0.00639 300.57 400.31 1.3240 1.6252 2.534 2.762 2.413 213.
120.1 66.6 14.89 56.5 29.45 1.05 58.00 60.00 2.8744 728.9 0.00594 305.09 399.24 1.3371 1.6197 2.714 3.069 2.647 200.
117.9 63.6 15.29 55.6 31.37 0.87 60.00 65.00 3.1977 678.3 0.00486 317.45 394.94 1.3726 1.6018 3.564 4.532 3.763 164.
111.8 55.4 16.64 53.6 38.41 0.45 65.00 70.00 3.5527 600.8 0.00370 333.19 385.42 1.4172 1.5694 7.72 11.50 9.04 122.
104.2 45.1 19.30 53.2 55.97 0.11 70.00 72.71c 3.7610 431.0 0.00232 358.91 358.91 1.4906 1.4906 ∞ ∞ ∞ 0.
0.0 — — ∞ ∞ 0.00 72.71 Temperatures are on the ITS-90 scale a = triple point b = normal boiling point c = critical point Thermophysical Properties of Refrigerants 20.23 Refrigerant 245fa (1,1,1,3,3-Pentafluoropropane) Properties of Saturated Liquid and Saturated Vapor Temp., °C Pres-sure, MPa Density, kg/m3 Liquid Volume, m3/kg Vapor Enthalpy, kJ/kg Entropy, kJ/(kg·K) Specific Heat cp, kJ/(kg·K) cp/cv Velocity of Sound, m/s Viscosity, µPa·s Thermal Cond., mW/(m·K) Surface Tension, mN/m Temp., °C Liquid Vapor Liquid Vapor Liquid Vapor Vapor Liquid Vapor Liquid Vapor Liquid Vapor –50.00 0.00286 1523.5 4.8232 137.62 368.35 0.7482 1.7822 1.196 0.718 1.097 1054.
122.8 1749.
7.71 107.2 7.50 23.48 –50.00 –40.00 0.00582 1500.5 2.4705 149.72 375.45 0.8012 1.7694 1.221 0.740 1.095 987.
125.1 1301.
8.06 103.2 8.06 22.26 –40.00 –30.00 0.01104 1477.1 1.3545 162.03 382.69 0.8529 1.7604 1.240 0.763 1.094 929.
127.3 1020.
8.41 99.5 8.64 21.03 –30.00 –20.00 0.01967 1453.3 0.78749 174.52 390.03 0.9032 1.7545 1.257 0.787 1.094 877.
129.3 827.5 8.76 95.8 9.24 19.78 –20.00 –10.00 0.03324 1428.9 0.48154 187.18 397.45 0.9522 1.7513 1.273 0.811 1.094 829.
131.0 687.7 9.12 92.3 9.86 18.52 –10.00 0.00 0.05359 1404.0 0.30756 200.00 404.93 1.0000 1.7502 1.290 0.837 1.095 783.
132.5 581.6 9.47 88.8 10.49 17.25 0.00 2.00 0.05866 1399.0 0.28251 202.59 406.43 1.0094 1.7503 1.294 0.842 1.096 774.
132.7 563.4 9.53 88.2 10.62 17.00 2.00 4.00 0.06411 1393.9 0.25987 205.18 407.93 1.0188 1.7504 1.297 0.848 1.096 765.
133.0 546.1 9.60 87.5 10.75 16.74 4.00 6.00 0.06996 1388.8 0.23939 207.78 409.44 1.0281 1.7505 1.301 0.853 1.097 756.
133.2 529.5 9.67 86.8 10.88 16.48 6.00 8.00 0.07622 1383.7 0.22082 210.39 410.94 1.0374 1.7508 1.305 0.859 1.097 747.
133.4 513.6 9.74 86.2 11.01 16.23 8.00 10.00 0.08293 1378.5 0.20397 213.00 412.45 1.0467 1.7511 1.309 0.864 1.098 738.
133.6 498.5 9.81 85.5 11.15 15.97 10.00 12.00 0.09009 1373.3 0.18864 215.63 413.95 1.0559 1.7514 1.312 0.870 1.098 729.
133.8 483.9 9.88 84.9 11.28 15.72 12.00 14.00 0.09774 1368.1 0.17469 218.26 415.46 1.0651 1.7518 1.316 0.875 1.099 721.
134.0 470.0 9.95 84.2 11.41 15.46 14.00 14.90b 0.10133 1365.7 0.16885 219.44 416.13 1.0692 1.7520 1.318 0.878 1.099 717.
134.1 464.0 9.98 83.9 11.47 15.34 14.90 16.00 0.10589 1362.8 0.16197 220.90 416.97 1.0742 1.7523 1.320 0.881 1.100 712.
134.2 456.7 10.02 83.6 11.55 15.20 16.00 18.00 0.11457 1357.5 0.15035 223.54 418.47 1.0833 1.7528 1.324 0.887 1.100 703.
134.3 443.8 10.09 82.9 11.69 14.95 18.00 20.00 0.12380 1352.2 0.13972 226.20 419.98 1.0924 1.7534 1.328 0.893 1.101 695.
134.4 431.5 10.16 82.3 11.82 14.69 20.00 22.00 0.13360 1346.9 0.13000 228.86 421.48 1.1014 1.7540 1.332 0.899 1.102 686.
134.6 419.6 10.23 81.7 11.96 14.43 22.00 24.00 0.14400 1341.5 0.12108 231.54 422.99 1.1104 1.7547 1.337 0.905 1.103 677.
134.7 408.2 10.30 81.0 12.10 14.18 24.00 26.00 0.15503 1336.1 0.11289 234.22 424.49 1.1194 1.7554 1.341 0.911 1.104 669.
134.7 397.2 10.37 80.4 12.24 13.92 26.00 28.00 0.16670 1330.6 0.10536 236.91 425.99 1.1283 1.7562 1.345 0.917 1.105 660.
134.8 386.6 10.44 79.8 12.38 13.66 28.00 30.00 0.17904 1325.1 0.09843 239.60 427.49 1.1372 1.7570 1.350 0.923 1.106 652.
134.9 376.4 10.51 79.2 12.52 13.41 30.00 32.00 0.19209 1319.6 0.09205 242.31 428.99 1.1461 1.7578 1.354 0.929 1.107 643.
134.9 366.5 10.59 78.6 12.66 13.15 32.00 34.00 0.20586 1314.0 0.08616 245.03 430.49 1.1549 1.7587 1.359 0.936 1.108 635.
134.9 356.9 10.66 77.9 12.81 12.89 34.00 36.00 0.22038 1308.4 0.08072 247.75 431.99 1.1637 1.7597 1.364 0.942 1.110 626.
134.9 347.7 10.73 77.3 12.95 12.64 36.00 38.00 0.23568 1302.7 0.07569 250.49 433.48 1.1725 1.7606 1.368 0.949 1.111 618.
134.9 338.8 10.80 76.7 13.10 12.38 38.00 40.00 0.25179 1297.0 0.07103 253.24 434.97 1.1813 1.7616 1.373 0.956 1.112 609.
134.9 330.1 10.87 76.1 13.24 12.13 40.00 42.00 0.26873 1291.2 0.06671 255.99 436.46 1.1900 1.7626 1.378 0.962 1.114 601.
134.8 321.8 10.94 75.5 13.39 11.87 42.00 44.00 0.28654 1285.4 0.06271 258.76 437.95 1.1987 1.7637 1.383 0.969 1.115 592.
134.7 313.7 11.01 74.9 13.54 11.62 44.00 46.00 0.30523 1279.6 0.05899 261.53 439.43 1.2074 1.7648 1.388 0.976 1.117 584.
134.6 305.8 11.09 74.3 13.69 11.36 46.00 48.00 0.32485 1273.7 0.05554 264.32 440.91 1.2160 1.7659 1.394 0.984 1.119 575.
134.5 298.2 11.16 73.7 13.84 11.11 48.00 50.00 0.34542 1267.7 0.05232 267.11 442.38 1.2246 1.7670 1.399 0.991 1.121 567.
134.4 290.8 11.23 73.1 13.99 10.85 50.00 52.00 0.36696 1261.7 0.04933 269.92 443.85 1.2333 1.7682 1.405 0.998 1.123 559.
134.2 283.6 11.31 72.5 14.15 10.60 52.00 54.00 0.38951 1255.6 0.04653 272.74 445.32 1.2418 1.7694 1.410 1.006 1.125 550.
134.0 276.6 11.38 72.0 14.30 10.35 54.00 56.00 0.41311 1249.5 0.04393 275.57 446.78 1.2504 1.7706 1.416 1.013 1.127 542.
133.8 269.8 11.46 71.4 14.46 10.10 56.00 58.00 0.43777 1243.3 0.04149 278.41 448.24 1.2590 1.7718 1.422 1.021 1.129 533.
133.6 263.1 11.53 70.8 14.62 9.84 58.00 60.00 0.46353 1237.0 0.03922 281.26 449.69 1.2675 1.7730 1.428 1.029 1.131 525.
133.4 256.7 11.61 70.2 14.78 9.59 60.00 62.00 0.49043 1230.7 0.03709 284.13 451.13 1.2760 1.7743 1.434 1.038 1.134 516.
133.1 250.4 11.69 69.6 14.94 9.34 62.00 64.00 0.51849 1224.3 0.03509 287.01 452.57 1.2845 1.7756 1.441 1.046 1.137 508.
132.8 244.3 11.76 69.0 15.11 9.09 64.00 66.00 0.54774 1217.8 0.03322 289.90 454.00 1.2930 1.7768 1.447 1.055 1.140 500.
132.5 238.3 11.84 68.5 15.27 8.85 66.00 68.00 0.57823 1211.3 0.03147 292.80 455.43 1.3014 1.7781 1.454 1.063 1.143 491.
132.1 232.5 11.92 67.9 15.44 8.60 68.00 70.00 0.60998 1204.7 0.02982 295.71 456.85 1.3099 1.7794 1.461 1.072 1.146 483.
131.8 226.8 12.01 67.3 15.61 8.35 70.00 72.00 0.64302 1198.0 0.02828 298.64 458.25 1.3183 1.7807 1.468 1.082 1.149 474.
131.4 221.3 12.09 66.8 15.79 8.10 72.00 74.00 0.67739 1191.2 0.02682 301.59 459.66 1.3267 1.7820 1.476 1.091 1.153 466.
131.0 215.9 12.17 66.2 15.96 7.86 74.00 76.00 0.71313 1184.3 0.02545 304.55 461.05 1.3351 1.7834 1.483 1.101 1.157 457.
130.5 210.6 12.26 65.6 16.14 7.61 76.00 78.00 0.75026 1177.3 0.02416 307.52 462.43 1.3435 1.7847 1.491 1.111 1.161 449.
130.0 205.4 12.34 65.0 16.32 7.37 78.00 80.00 0.78882 1170.3 0.02295 310.50 463.80 1.3519 1.7860 1.499 1.122 1.165 441.
129.5 200.3 12.43 64.5 16.51 7.13 80.00 82.00 0.82886 1163.1 0.02180 313.51 465.16 1.3603 1.7873 1.508 1.133 1.169 432.
129.0 195.4 12.52 63.9 16.69 6.89 82.00 84.00 0.87039 1155.8 0.02072 316.53 466.51 1.3687 1.7886 1.517 1.144 1.174 424.
128.4 190.5 12.61 63.3 16.89 6.65 84.00 86.00 0.91347 1148.4 0.01970 319.56 467.85 1.3770 1.7899 1.526 1.156 1.180 415.
127.8 185.8 12.71 62.8 17.08 6.41 86.00 88.00 0.95813 1140.9 0.01873 322.61 469.17 1.3854 1.7912 1.535 1.168 1.185 407.
127.2 181.1 12.80 62.2 17.28 6.17 88.00 90.00 1.0044 1133.3 0.01782 325.68 470.48 1.3938 1.7925 1.545 1.180 1.191 398.
126.5 176.5 12.90 61.7 17.49 5.94 90.00 92.00 1.0523 1125.6 0.01695 328.77 471.77 1.4021 1.7938 1.556 1.194 1.198 390.
125.8 172.0 13.00 61.1 17.69 5.70 92.00 94.00 1.1020 1117.7 0.01613 331.88 473.05 1.4105 1.7950 1.567 1.208 1.205 381.
125.1 167.6 13.11 60.5 17.91 5.47 94.00 96.00 1.1533 1109.6 0.01535 335.00 474.31 1.4189 1.7962 1.578 1.222 1.212 373.
124.4 163.3 13.22 60.0 18.13 5.24 96.00 98.00 1.2064 1101.5 0.01461 338.15 475.55 1.4272 1.7974 1.590 1.237 1.220 364.
123.6 159.0 13.33 59.4 18.36 5.01 98.00 100.00 1.2614 1093.1 0.01391 341.31 476.77 1.4356 1.7986 1.603 1.254 1.229 356.
122.7 154.8 13.44 58.8 18.59 4.79 100.00 105.00 1.4070 1071.5 0.01231 349.33 479.73 1.4566 1.8014 1.638 1.299 1.255 334.
120.4 144.6 13.74 57.4 19.21 4.23 105.00 110.00 1.5648 1048.7 0.01089 357.50 482.53 1.4777 1.8040 1.679 1.352 1.286 312.
117.9 134.8 14.08 56.0 19.89 3.68 110.00 115.00 1.7358 1024.4 0.00962 365.85 485.13 1.4989 1.8062 1.729 1.416 1.327 290.
115.1 125.3 14.45 54.6 20.65 3.15 115.00 120.00 1.9205 998.4 0.00849 374.40 487.49 1.5203 1.8079 1.790 1.497 1.381 267.
112.0 116.0 14.87 53.2 21.52 2.64 120.00 125.00 2.1200 970.1 0.00748 383.19 489.54 1.5420 1.8091 1.868 1.603 1.455 244.
108.5 106.9 15.35 51.8 22.54 2.15 125.00 130.00 2.3351 939.0 0.00656 392.29 491.19 1.5642 1.8095 1.975 1.749 1.560 221.
104.6 97.8 15.92 50.4 23.77 1.68 130.00 135.00 2.5671 904.0 0.00571 401.79 492.30 1.5869 1.8087 2.131 1.967 1.723 196.
100.2 88.7 16.62 49.0 25.30 1.24 135.00 140.00 2.8173 863.4 0.00492 411.85 492.60 1.6108 1.8062 2.389 2.331 2.002 170.
95.4 79.4 17.52 47.9 27.33 0.84 140.00 145.00 3.0874 813.4 0.00416 422.82 491.58 1.6364 1.8008 2.908 3.071 2.582 143.
89.8 69.4 18.79 47.4 30.29 0.47 145.00 150.00 3.3802 743.3 0.00336 435.71 487.81 1.6661 1.7892 4.561 5.410 4.433 113.
83.5 57.8 20.97 49.3 35.69 0.17 150.00 154.05c 3.6400 517.0 0.00193 463.06 463.06 1.7294 1.7294 ∞ ∞ ∞ 0.
0.0 — — ∞ ∞ 0.00 154.05 Temperatures are on the ITS-90 scale b = normal boiling point c = critical point 20.24 2001 ASHRAE Fundamentals Handbook (SI) Fig. 10 Pressure-Enthalpy Diagram for Refrigerant 404A Reprinted with permission from E.I. duPont de Nemours Thermophysical Properties of Refrigerants 20.25 Refrigerant 404A [R-125/143a/134a (44/52/4)] Properties of Liquid on the Bubble Line and Vapor on the Dew Line Pres-sure, MPa Temperature, °C Density, kg/m3 Liquid Volume, m3/kg Vapor Enthalpy, kJ/kg Entropy, kJ/(kg·K) Specific Heat cp, kJ/(kg·K) cp/cv Velocity of Sound, m/s Viscosity, µPa·s Thermal Cond., mW/(m·K) Surface Tension, mN/m Pres-sure, MPa Bubble Dew Liquid Vapor Liquid Vapor Liquid Vapor Vapor Liquid Vapor Liquid Vapor Liquid Vapor 0.00500 –94.18 –93.00 1444.4 3.05033 83.20 310.67 0.4810 1.7496 1.147 0.637 1.163 959.
132.8 760.4 7.20 123.6 6.00 17.77 0.00500 0.00600 –91.96 –90.80 1438.3 2.57089 85.75 311.99 0.4952 1.7415 1.149 0.643 1.162 947.
133.4 723.1 7.29 122.3 6.12 17.57 0.00600 0.00700 –90.03 –88.89 1432.9 2.22501 87.98 313.15 0.5074 1.7347 1.151 0.649 1.161 937.
134.0 693.2 7.37 121.2 6.23 17.39 0.00700 0.00800 –88.31 –87.19 1428.1 1.96336 89.95 314.17 0.5181 1.7290 1.153 0.653 1.161 928.
134.5 668.3 7.44 120.2 6.33 17.23 0.00800 0.00900 –86.77 –85.67 1423.8 1.75831 91.74 315.10 0.5277 1.7240 1.155 0.657 1.160 920.
134.9 647.2 7.50 119.3 6.42 17.08 0.00900 0.01000 –85.36 –84.27 1419.9 1.59315 93.36 315.94 0.5364 1.7196 1.157 0.661 1.160 913.
135.3 628.9 7.56 118.5 6.50 16.95 0.01000 0.02000 –75.43 –74.45 1392.0 0.83310 104.92 321.94 0.5963 1.6923 1.171 0.690 1.159 863.
137.8 520.6 7.97 113.1 7.10 15.99 0.02000 0.04000 –64.18 –63.29 1359.7 0.43580 118.21 328.80 0.6617 1.6680 1.191 0.725 1.159 807.
140.3 429.8 8.43 107.2 7.77 14.84 0.04000 0.06000 –56.87 –56.03 1338.3 0.29818 126.98 333.25 0.7028 1.6553 1.206 0.749 1.161 772.
141.7 383.1 8.73 103.5 8.25 14.07 0.06000 0.08000 –51.30 –50.50 1321.7 0.22768 133.73 336.63 0.7336 1.6470 1.218 0.769 1.164 745.
142.5 352.4 8.96 100.8 8.65 13.47 0.08000 0.10000 –46.75 –45.98 1308.0 0.18460 139.30 339.37 0.7584 1.6410 1.228 0.786 1.166 723.
143.1 329.8 9.14 98.6 8.97 12.97 0.10000 0.10132b –46.48 –45.71 1307.2 0.18233 139.64 339.53 0.7599 1.6406 1.229 0.787 1.166 722.
143.2 328.5 9.15 98.4 8.99 12.94 0.10132 0.12000 –42.87 –42.12 1296.1 0.15547 144.09 341.70 0.7793 1.6364 1.238 0.801 1.169 705.
143.5 312.1 9.28 96.7 9.25 12.54 0.12000 0.14000 –39.47 –38.74 1285.5 0.13440 148.33 343.72 0.7975 1.6327 1.246 0.815 1.172 689.
143.8 297.6 9.42 95.1 9.51 12.16 0.14000 0.16000 –36.42 –35.72 1276.0 0.11844 152.14 345.52 0.8136 1.6296 1.254 0.828 1.175 674.
144.0 285.4 9.54 93.7 9.74 11.81 0.16000 0.18000 –33.66 –32.97 1267.2 0.10591 155.62 347.14 0.8282 1.6270 1.262 0.840 1.177 661.
144.2 274.9 9.65 92.4 9.95 11.50 0.18000 0.20000 –31.13 –30.45 1259.1 0.09580 158.83 348.61 0.8414 1.6248 1.269 0.851 1.180 649.
144.3 265.7 9.75 91.3 10.15 11.20 0.20000 0.22000 –28.79 –28.12 1251.5 0.08747 161.81 349.97 0.8536 1.6228 1.276 0.861 1.183 638.
144.3 257.5 9.84 90.2 10.33 10.93 0.22000 0.24000 –26.61 –25.95 1244.4 0.08049 164.61 351.22 0.8650 1.6211 1.282 0.871 1.186 628.
144.3 250.2 9.93 89.2 10.51 10.68 0.24000 0.26000 –24.56 –23.91 1237.6 0.07454 167.25 352.39 0.8756 1.6196 1.288 0.881 1.189 618.
144.3 243.5 10.02 88.3 10.66 10.44 0.26000 0.28000 –22.63 –21.99 1231.2 0.06941 169.75 353.48 0.8855 1.6182 1.294 0.891 1.192 609.
144.3 237.4 10.10 87.4 10.83 10.22 0.28000 0.30000 –20.80 –20.17 1225.1 0.06494 172.13 354.51 0.8949 1.6169 1.300 0.900 1.195 601.
144.2 231.8 10.13 86.6 11.11 10.00 0.30000 0.32000 –19.06 –18.44 1219.2 0.06101 174.40 355.48 0.9038 1.6158 1.306 0.908 1.198 593.
144.1 226.6 10.21 85.9 11.26 9.80 0.32000 0.34000 –17.40 –16.80 1213.5 0.05753 176.57 356.40 0.9123 1.6147 1.312 0.917 1.201 585.
144.0 221.8 10.29 85.1 11.40 9.60 0.34000 0.36000 –15.82 –15.22 1208.0 0.05442 178.65 357.27 0.9203 1.6138 1.317 0.925 1.204 577.
143.9 217.3 10.36 84.4 11.54 9.41 0.36000 0.38000 –14.30 –13.71 1202.8 0.05163 180.66 358.09 0.9280 1.6129 1.322 0.934 1.207 570.
143.8 213.1 10.43 83.8 11.67 9.23 0.38000 0.40000 –12.84 –12.26 1197.7 0.04910 182.60 358.88 0.9354 1.6120 1.328 0.942 1.210 563.
143.7 209.1 10.49 83.2 11.80 9.06 0.40000 0.42000 –11.44 –10.86 1192.7 0.04681 184.47 359.64 0.9425 1.6113 1.333 0.949 1.213 557.
143.5 205.4 10.56 82.5 11.93 8.89 0.42000 0.44000 –10.09 –9.51 1187.9 0.04472 186.28 360.36 0.9494 1.6105 1.338 0.957 1.216 550.
143.4 201.8 10.62 82.0 12.05 8.73 0.44000 0.46000 –8.78 –8.21 1183.2 0.04280 188.04 361.05 0.9560 1.6098 1.343 0.965 1.219 544.
143.2 198.5 10.68 81.4 12.18 8.57 0.46000 0.48000 –7.51 –6.95 1178.6 0.04104 189.75 361.71 0.9624 1.6092 1.348 0.972 1.222 538.
143.0 195.3 10.74 80.9 12.30 8.42 0.48000 0.50000 –6.28 –5.73 1174.1 0.03941 191.41 362.35 0.9685 1.6085 1.353 0.980 1.225 532.
142.9 192.2 10.79 80.3 12.41 8.28 0.50000 0.55000 –3.37 –2.83 1163.4 0.03584 195.37 363.84 0.9831 1.6071 1.365 0.998 1.234 518.
142.4 185.1 10.93 79.1 12.70 7.93 0.55000 0.60000 –0.65 –0.12 1153.1 0.03285 199.10 365.21 0.9968 1.6058 1.377 1.016 1.242 505.
141.9 178.8 11.06 77.9 12.97 7.60 0.60000 0.65000 1.90 2.42 1143.4 0.03030 202.64 366.46 1.0095 1.6046 1.389 1.033 1.250 493.
141.3 172.9 11.18 76.9 13.24 7.29 0.65000 0.70000 4.32 4.82 1134.0 0.02810 206.00 367.62 1.0215 1.6036 1.400 1.051 1.259 481.
140.8 167.6 11.30 75.8 13.50 7.00 0.70000 0.75000 6.60 7.10 1124.9 0.02619 209.21 368.69 1.0329 1.6025 1.412 1.068 1.268 470.
140.2 162.7 11.41 74.9 13.75 6.73 0.75000 0.80000 8.77 9.26 1116.1 0.02450 212.29 369.69 1.0437 1.6016 1.424 1.085 1.277 460.
139.6 158.1 11.52 74.0 14.01 6.47 0.80000 0.85000 10.85 11.33 1107.6 0.02301 215.26 370.62 1.0540 1.6007 1.435 1.102 1.287 450.
139.0 153.9 11.63 73.1 14.26 6.22 0.85000 0.90000 12.83 13.30 1099.3 0.02167 218.11 371.48 1.0639 1.5998 1.447 1.120 1.296 440.
138.4 149.9 11.74 72.3 14.51 5.98 0.90000 0.95000 14.74 15.20 1091.2 0.02047 220.87 372.29 1.0733 1.5989 1.459 1.137 1.306 431.
137.7 146.1 11.84 71.5 14.76 5.75 0.95000 1.00000 16.57 17.02 1083.3 0.01939 223.54 373.04 1.0824 1.5981 1.471 1.155 1.317 422.
137.1 142.6 11.94 70.7 15.00 5.53 1.00000 1.10000 20.03 20.47 1068.0 0.01750 228.65 374.41 1.0996 1.5965 1.495 1.190 1.339 404.
135.8 136.0 12.14 69.3 15.50 5.12 1.10000 1.20000 23.27 23.69 1053.1 0.01592 233.50 375.60 1.1158 1.5949 1.520 1.228 1.363 388.
134.4 130.0 12.34 67.9 16.00 4.74 1.20000 1.30000 26.31 26.72 1038.7 0.01457 238.12 376.65 1.1309 1.5933 1.547 1.266 1.388 373.
133.0 124.5 12.54 66.7 16.51 4.38 1.30000 1.40000 29.18 29.58 1024.5 0.01340 242.54 377.55 1.1453 1.5916 1.574 1.307 1.416 358.
131.6 119.5 12.74 65.5 17.04 4.05 1.40000 1.50000 31.90 32.29 1010.7 0.01238 246.80 378.34 1.1590 1.5900 1.603 1.350 1.446 344.
130.2 114.8 12.93 64.3 17.58 3.74 1.50000 1.60000 34.49 34.86 997.0 0.01148 250.91 379.00 1.1721 1.5882 1.634 1.397 1.480 330.
128.7 110.4 13.13 63.3 18.14 3.44 1.60000 1.70000 36.96 37.32 983.4 0.01068 254.89 379.55 1.1847 1.5864 1.667 1.447 1.516 317.
127.2 106.2 13.34 62.2 18.73 3.16 1.70000 1.80000 39.32 39.67 970.0 0.00996 258.76 380.00 1.1968 1.5846 1.703 1.501 1.556 305.
125.7 102.3 13.54 61.2 19.35 2.90 1.80000 1.90000 41.59 41.92 956.5 0.00932 262.53 380.35 1.2085 1.5826 1.741 1.560 1.601 292.
124.2 98.6 13.75 60.3 20.00 2.65 1.90000 2.00000 43.76 44.09 943.1 0.00873 266.22 380.59 1.2198 1.5805 1.784 1.625 1.651 280.
122.6 95.0 13.97 59.3 20.69 2.42 2.00000 2.10000 45.86 46.17 929.6 0.00819 269.83 380.74 1.2308 1.5783 1.830 1.697 1.708 268.
121.0 91.6 14.20 58.4 21.43 2.20 2.10000 2.20000 47.87 48.18 916.0 0.00770 273.38 380.79 1.2416 1.5760 1.882 1.778 1.772 257.
119.5 88.3 14.43 57.6 22.21 1.98 2.20000 2.30000 49.82 50.11 902.2 0.00724 276.87 380.73 1.2521 1.5735 1.940 1.869 1.845 246.
117.8 85.1 14.68 56.7 23.06 1.78 2.30000 2.40000 51.71 51.99 888.2 0.00682 280.32 380.57 1.2624 1.5709 2.006 1.974 1.929 235.
116.2 82.0 14.93 55.9 23.97 1.60 2.40000 2.50000 53.53 53.80 874.0 0.00643 283.74 380.31 1.2725 1.5680 2.081 2.095 2.027 224.
114.6 79.0 15.21 55.1 24.97 1.42 2.50000 2.60000 55.30 55.56 859.4 0.00606 287.14 379.92 1.2826 1.5649 2.169 2.238 2.143 213.
112.9 76.1 15.50 54.4 26.05 1.25 2.60000 2.70000 57.02 57.26 844.4 0.00571 290.53 379.42 1.2925 1.5616 2.272 2.407 2.282 202.
111.2 73.2 15.81 53.7 27.24 1.09 2.70000 2.80000 58.68 58.91 828.8 0.00538 293.92 378.77 1.3024 1.5580 2.397 2.613 2.451 191.
109.5 70.3 16.15 53.0 28.56 0.94 2.80000 2.90000 60.30 60.52 812.5 0.00507 297.32 377.97 1.3122 1.5540 2.552 2.868 2.662 180.
107.8 67.5 16.52 52.3 30.05 0.79 2.90000 3.00000 61.87 62.08 795.4 0.00476 300.76 376.99 1.3222 1.5496 2.748 3.195 2.931 170.
106.0 64.6 16.93 51.8 31.74 0.66 3.00000 3.20000 64.88 65.07 757.6 0.00419 307.87 374.35 1.3425 1.5391 3.366 4.224 3.781 148.
102.3 58.7 17.92 50.9 36.05 0.42 3.20000 3.40000 67.74 67.89 711.4 0.00362 315.62 370.30 1.3645 1.5249 4.792 6.551 5.695 126.
98.4 52.4 19.31 51.0 42.77 0.22 3.40000 3.775c 72.5 72.5 574.
0.00174 334.8 334.8 1.419 1.419 — — — — — — — — — 0.00 3.775 Temperatures are on the ITS-90 scale b = bubble and dew points at one standard atmosphere c = critical point 20.26 2001 ASHRAE Fundamentals Handbook (SI) Fig. 11 Pressure-Enthalpy Diagram for Refrigerant 407C Reprinted with permission from E.I. duPont de Nemours Thermophysical Properties of Refrigerants 20.27 Refrigerant 407C [R-32/125/134a (23/25/52)] Properties of Liquid on the Bubble Line and Vapor on the Dew Line Pres-sure, MPa Temperature, °C Density, kg/m3 Liquid Volume, m3/kg Vapor Enthalpy, kJ/kg Entropy, kJ/(kg·K) Specific Heat cp, kJ/(kg·K) cp/cv Velocity of Sound, m/s Viscosity, µPa·s Thermal Cond., mW/(m·K) Surface Tension, mN/m Pres-sure, MPa Bubble Dew Liquid Vapor Liquid Vapor Liquid Vapor Vapor Liquid Vapor Liquid Vapor Liquid Vapor 0.01000 –82.82 –74.96 1496.6 1.89611 91.52 365.89 0.5302 1.9437 1.246 0.667 1.181 1000.
149.0 722.4 8.22 148.0 6.43 24.80 0.01000 0.02000 –72.81 –65.15 1468.1 0.98986 104.03 371.89 0.5942 1.9071 1.255 0.692 1.180 948.
151.8 593.0 8.63 142.1 7.06 22.98 0.02000 0.04000 –61.51 –54.07 1435.2 0.51699 118.30 378.64 0.6635 1.8730 1.268 0.725 1.181 891.
154.6 486.9 9.09 135.7 7.79 20.95 0.04000 0.06000 –54.18 –46.89 1413.5 0.35346 127.63 382.97 0.7068 1.8543 1.278 0.748 1.184 854.
156.1 433.2 9.39 131.5 8.28 19.65 0.06000 0.08000 –48.61 –41.44 1396.8 0.26976 134.78 386.21 0.7389 1.8416 1.287 0.767 1.186 827.
157.1 398.2 9.62 128.5 8.65 18.68 0.08000 0.10000 –44.06 –36.98 1382.9 0.21867 140.65 388.83 0.7648 1.8321 1.295 0.783 1.189 805.
157.8 372.6 9.81 126.0 8.96 17.89 0.10000 0.10132b –43.79 –36.71 1382.1 0.21597 141.01 388.99 0.7663 1.8315 1.295 0.784 1.189 803.
157.8 371.1 9.82 125.9 8.98 17.84 0.10132 0.12000 –40.19 –33.19 1371.0 0.18413 145.69 391.04 0.7865 1.8245 1.302 0.798 1.192 786.
158.3 352.6 9.97 123.9 9.23 17.23 0.12000 0.14000 –36.80 –29.87 1360.4 0.15918 150.12 392.95 0.8053 1.8183 1.308 0.811 1.195 769.
158.7 336.3 10.11 122.1 9.47 16.65 0.14000 0.16000 –33.77 –26.90 1350.9 0.14027 154.10 394.64 0.8220 1.8130 1.314 0.823 1.198 755.
159.0 322.6 10.23 120.5 9.68 16.13 0.16000 0.18000 –31.02 –24.21 1342.2 0.12544 157.73 396.15 0.8370 1.8084 1.320 0.835 1.201 741.
159.2 310.9 10.34 119.0 9.88 15.67 0.18000 0.20000 –28.50 –21.74 1334.1 0.11348 161.07 397.52 0.8507 1.8043 1.326 0.845 1.203 729.
159.4 300.6 10.45 117.7 10.06 15.25 0.20000 0.22000 –26.17 –19.46 1326.6 0.10363 164.17 398.78 0.8632 1.8007 1.331 0.856 1.206 718.
159.6 291.5 10.55 116.5 10.23 14.86 0.22000 0.24000 –24.00 –17.34 1319.5 0.09537 167.07 399.94 0.8748 1.7974 1.336 0.865 1.209 708.
159.6 283.3 10.64 115.3 10.39 14.50 0.24000 0.26000 –21.96 –15.35 1312.8 0.08834 169.80 401.01 0.8857 1.7945 1.341 0.875 1.212 698.
159.7 275.9 10.72 114.3 10.54 14.17 0.26000 0.28000 –20.05 –13.47 1306.5 0.08228 172.38 402.01 0.8959 1.7918 1.346 0.884 1.215 689.
159.7 269.1 10.80 113.3 10.68 13.85 0.28000 0.30000 –18.23 –11.70 1300.4 0.07700 174.83 402.95 0.9055 1.7893 1.351 0.892 1.218 680.
159.8 262.9 10.88 112.3 10.82 13.56 0.30000 0.32000 –16.51 –10.01 1294.6 0.07236 177.17 403.83 0.9145 1.7869 1.355 0.901 1.221 672.
159.7 257.2 10.96 111.4 10.95 13.28 0.32000 0.34000 –14.86 –8.41 1289.0 0.06824 179.41 404.67 0.9232 1.7848 1.360 0.909 1.224 664.
159.7 251.9 11.03 110.6 11.08 13.01 0.34000 0.36000 –13.29 –6.87 1283.7 0.06457 181.55 405.45 0.9314 1.7827 1.364 0.917 1.226 656.
159.7 246.9 11.10 109.8 11.20 12.76 0.36000 0.38000 –11.79 –5.40 1278.5 0.06127 183.61 406.20 0.9392 1.7808 1.369 0.925 1.229 649.
159.6 242.3 11.16 109.0 11.32 12.51 0.38000 0.40000 –10.34 –3.99 1273.5 0.05829 185.60 406.91 0.9468 1.7790 1.373 0.932 1.232 642.
159.6 237.9 11.22 108.2 11.44 12.28 0.40000 0.42000 –8.95 –2.63 1268.7 0.05559 187.52 407.59 0.9540 1.7773 1.377 0.940 1.235 635.
159.5 233.8 11.29 107.5 11.55 12.06 0.42000 0.44000 –7.61 –1.32 1264.0 0.05312 189.37 408.24 0.9609 1.7757 1.382 0.947 1.238 629.
159.4 229.9 11.34 106.8 11.66 11.85 0.44000 0.46000 –6.31 –0.05 1259.4 0.05086 191.17 408.85 0.9676 1.7741 1.386 0.954 1.241 623.
159.3 226.2 11.40 106.2 11.76 11.64 0.46000 0.48000 –5.06 1.17 1255.0 0.04878 192.91 409.44 0.9741 1.7726 1.390 0.961 1.244 616.
159.2 222.7 11.46 105.5 11.87 11.44 0.48000 0.50000 –3.84 2.36 1250.6 0.04687 194.61 410.01 0.9803 1.7712 1.394 0.968 1.247 611.
159.1 219.3 11.51 104.9 11.97 11.25 0.50000 0.55000 –0.96 5.17 1240.2 0.04266 198.65 411.33 0.9951 1.7679 1.404 0.985 1.254 597.
158.8 211.6 11.65 103.4 12.22 10.80 0.55000 0.60000 1.73 7.79 1230.4 0.03913 202.45 412.54 1.0088 1.7649 1.414 1.002 1.262 584.
158.4 204.6 11.77 102.0 12.45 10.39 0.60000 0.65000 4.26 10.25 1221.0 0.03613 206.04 413.64 1.0217 1.7622 1.423 1.018 1.270 571.
158.1 198.3 11.89 100.7 12.68 10.00 0.65000 0.70000 6.65 12.58 1212.0 0.03355 209.45 414.64 1.0338 1.7596 1.433 1.034 1.277 560.
157.7 192.5 12.00 99.5 12.89 9.63 0.70000 0.75000 8.91 14.78 1203.3 0.03129 212.71 415.57 1.0452 1.7572 1.443 1.050 1.285 549.
157.3 187.2 12.11 98.4 13.11 9.29 0.75000 0.80000 11.06 16.87 1195.0 0.02931 215.82 416.43 1.0561 1.7549 1.452 1.066 1.293 538.
156.8 182.3 12.22 97.3 13.32 8.97 0.80000 0.85000 13.11 18.86 1186.9 0.02755 218.81 417.23 1.0664 1.7528 1.462 1.081 1.302 528.
156.4 177.6 12.33 96.2 13.52 8.66 0.85000 0.90000 15.07 20.77 1179.1 0.02598 221.69 417.97 1.0763 1.7507 1.471 1.097 1.310 519.
155.9 173.3 12.43 95.2 13.72 8.37 0.90000 0.95000 16.95 22.59 1171.5 0.02457 224.47 418.65 1.0857 1.7488 1.481 1.112 1.319 509.
155.5 169.3 12.53 94.2 13.92 8.09 0.95000 1.00000 18.76 24.35 1164.1 0.02330 227.15 419.29 1.0948 1.7469 1.490 1.127 1.327 501.
155.0 165.5 12.63 93.3 14.12 7.83 1.00000 1.10000 22.19 27.67 1149.8 0.02109 232.28 420.44 1.1120 1.7433 1.510 1.158 1.345 484.
154.0 158.4 12.82 91.5 14.51 7.33 1.10000 1.20000 25.39 30.77 1136.0 0.01923 237.13 421.44 1.1281 1.7400 1.530 1.190 1.365 468.
152.9 152.1 13.00 89.9 14.90 6.87 1.20000 1.30000 28.40 33.68 1122.8 0.01765 241.74 422.30 1.1431 1.7367 1.550 1.222 1.385 452.
151.9 146.3 13.18 88.3 15.29 6.45 1.30000 1.40000 31.24 36.42 1109.9 0.01629 246.15 423.04 1.1574 1.7337 1.571 1.255 1.406 438.
150.8 140.9 13.35 86.8 15.69 6.05 1.40000 1.50000 33.94 39.02 1097.4 0.01510 250.38 423.68 1.1709 1.7307 1.592 1.289 1.428 424.
149.7 136.0 13.53 85.4 16.09 5.68 1.50000 1.60000 36.50 41.49 1085.1 0.01405 254.44 424.21 1.1838 1.7277 1.615 1.324 1.452 411.
148.6 131.3 13.70 84.1 16.51 5.33 1.60000 1.70000 38.95 43.84 1073.1 0.01312 258.38 424.66 1.1961 1.7248 1.638 1.360 1.477 398.
147.5 127.0 13.87 82.8 16.93 5.00 1.70000 1.80000 41.29 46.09 1061.3 0.01229 262.18 425.02 1.2080 1.7220 1.662 1.398 1.504 386.
146.3 122.9 14.04 81.5 17.37 4.69 1.80000 1.90000 43.54 48.25 1049.6 0.01154 265.88 425.31 1.2194 1.7191 1.688 1.438 1.532 374.
145.2 119.1 14.21 80.3 17.83 4.40 1.90000 2.00000 45.70 50.31 1038.1 0.01087 269.48 425.51 1.2304 1.7163 1.715 1.481 1.563 363.
144.0 115.4 14.38 79.2 18.30 4.12 2.00000 2.10000 47.79 52.30 1026.7 0.01025 273.00 425.65 1.2411 1.7135 1.743 1.526 1.596 352.
142.8 111.9 14.56 78.0 18.80 3.86 2.10000 2.20000 49.80 54.22 1015.3 0.00969 276.43 425.71 1.2515 1.7106 1.774 1.573 1.632 341.
141.6 108.6 14.74 76.9 19.32 3.60 2.20000 2.30000 51.74 56.07 1004.0 0.00917 279.80 425.70 1.2616 1.7077 1.806 1.624 1.670 330.
140.4 105.4 14.92 75.9 19.87 3.36 2.30000 2.40000 53.63 57.86 992.7 0.00869 283.10 425.63 1.2714 1.7048 1.841 1.679 1.712 320.
139.2 102.4 15.10 74.8 20.45 3.13 2.40000 2.50000 55.45 59.58 981.4 0.00825 286.35 425.48 1.2810 1.7018 1.878 1.738 1.757 310.
138.0 99.4 15.29 73.8 21.06 2.91 2.50000 2.60000 57.22 61.26 970.0 0.00784 289.55 425.27 1.2904 1.6988 1.918 1.802 1.806 300.
136.7 96.6 15.48 72.8 21.71 2.70 2.60000 2.70000 58.94 62.88 958.6 0.00746 292.71 425.00 1.2996 1.6957 1.962 1.872 1.861 290.
135.5 93.8 15.68 71.8 22.39 2.50 2.70000 2.80000 60.62 64.45 947.1 0.00710 295.83 424.65 1.3087 1.6925 2.009 1.948 1.920 280.
134.2 91.1 15.89 70.9 23.13 2.31 2.80000 2.90000 62.25 65.98 935.5 0.00676 298.92 424.23 1.3176 1.6892 2.062 2.032 1.987 270.
133.0 88.5 16.11 70.0 23.91 2.13 2.90000 3.00000 63.84 67.47 923.8 0.00644 301.99 423.74 1.3264 1.6858 2.120 2.125 2.060 261.
131.7 85.9 16.33 69.1 24.75 1.95 3.00000 3.20000 66.90 70.32 899.7 0.00586 308.08 422.52 1.3438 1.6786 2.258 2.345 2.236 242.
129.1 81.0 16.81 67.3 26.62 1.62 3.20000 3.40000 69.83 73.02 874.5 0.00533 314.14 420.96 1.3609 1.6709 2.435 2.628 2.463 223.
126.5 76.1 17.35 65.6 28.81 1.31 3.40000 3.60000 72.63 75.57 847.8 0.00484 320.25 419.00 1.3779 1.6623 2.673 3.007 2.768 205.
123.8 71.4 17.96 64.1 31.44 1.04 3.60000 3.80000 75.31 78.00 819.0 0.00439 326.49 416.54 1.3952 1.6526 3.013 3.543 3.200 186.
121.1 66.7 18.67 62.7 34.68 0.78 3.80000 4.00000 77.90 80.30 787.0 0.00396 332.98 413.42 1.4130 1.6414 3.544 4.363 3.860 168.
118.2 61.8 19.54 61.6 38.88 0.55 4.00000 4.20000 80.40 82.46 749.8 0.00354 339.95 409.31 1.4321 1.6277 4.497 5.782 4.996 149.
115.2 56.7 20.67 61.2 44.76 0.35 4.20000 4.635c 86.1 86.1 506.
0.00198 375.0 375.0 1.528 1.528 — — — — — — — — — 0.00 4.63468 Temperatures are on the ITS-90 scale b = bubble and dew points at one standard atmosphere c = critical point 20.28 2001 ASHRAE Fundamentals Handbook (SI) Refrigerant 410A [R-32/125 (50/50)] Properties of Liquid on the Bubble Line and Vapor on the Dew Line Pres-sure, MPa Temperature, °C Density, kg/m3 Liquid Volume, m3/kg Vapor Enthalpy, kJ/kg Entropy, kJ/(kg·K) Specific Heat cp, kJ/(kg·K) cp/cv Velocity of Sound, m/s Viscosity, µPa·s Thermal Cond., mW/(m·K) Surface Tension, mN/m Pres-sure, MPa Bubble Dew Liquid Vapor Liquid Vapor Liquid Vapor Vapor Liquid Vapor Liquid Vapor Liquid Vapor 0.01000 –88.54 –88.50 1462.0 2.09550 78.00 377.63 0.4650 2.0879 1.313 0.666 1.227 1031.
159.6 560.3 8.20 168.6 6.60 23.95 0.01000 0.02000 –79.05 –79.01 1434.3 1.09540 90.48 383.18 0.5309 2.0388 1.317 0.695 1.227 979.
162.7 473.2 8.63 162.6 7.06 22.17 0.02000 0.04000 –68.33 –68.29 1402.4 0.57278 104.64 389.31 0.6018 1.9916 1.325 0.733 1.230 923.
165.8 398.9 9.11 155.9 7.61 20.20 0.04000 0.06000 –61.39 –61.35 1381.4 0.39184 113.86 393.17 0.6461 1.9650 1.333 0.761 1.234 888.
167.5 360.2 9.43 151.7 7.97 18.94 0.06000 0.08000 –56.13 –56.08 1365.1 0.29918 120.91 396.04 0.6789 1.9465 1.340 0.785 1.238 861.
168.6 334.4 9.66 148.5 8.26 18.00 0.08000 0.10000 –51.83 –51.78 1351.7 0.24259 126.69 398.33 0.7052 1.9324 1.347 0.805 1.242 839.
169.4 315.4 9.86 146.0 8.50 17.23 0.10000 0.10132b –51.57 –51.52 1350.9 0.23961 127.04 398.47 0.7068 1.9316 1.348 0.806 1.242 838.
169.5 314.3 9.87 145.8 8.51 17.19 0.10132 0.12000 –48.17 –48.12 1340.1 0.20433 131.64 400.24 0.7273 1.9211 1.353 0.823 1.246 821.
170.0 300.3 10.02 143.8 8.71 16.59 0.12000 0.14000 –44.96 –44.91 1329.9 0.17668 136.00 401.89 0.7464 1.9116 1.359 0.839 1.250 805.
170.5 288.0 10.17 141.9 8.89 16.03 0.14000 0.16000 –42.10 –42.05 1320.7 0.15572 139.90 403.33 0.7634 1.9034 1.365 0.854 1.254 790.
170.9 277.5 10.30 140.2 9.06 15.53 0.16000 0.18000 –39.51 –39.45 1312.2 0.13928 143.46 404.62 0.7786 1.8963 1.371 0.868 1.257 777.
171.2 268.5 10.42 138.7 9.22 15.09 0.18000 0.20000 –37.13 –37.07 1304.4 0.12602 146.73 405.78 0.7925 1.8900 1.376 0.881 1.261 766.
171.4 260.5 10.52 137.3 9.36 14.68 0.20000 0.22000 –34.93 –34.87 1297.1 0.11510 149.76 406.84 0.8052 1.8843 1.381 0.894 1.265 755.
171.6 253.4 10.62 136.0 9.50 14.30 0.22000 0.24000 –32.89 –32.83 1290.3 0.10593 152.60 407.81 0.8170 1.8791 1.386 0.906 1.268 744.
171.7 247.0 10.72 134.8 9.63 13.95 0.24000 0.26000 –30.97 –30.90 1283.9 0.09813 155.27 408.71 0.8280 1.8744 1.391 0.917 1.272 735.
171.8 241.2 10.81 133.7 9.75 13.63 0.26000 0.28000 –29.16 –29.10 1277.7 0.09141 157.79 409.54 0.8383 1.8700 1.396 0.928 1.276 726.
171.9 235.9 10.89 132.7 9.87 13.33 0.28000 0.30000 –27.45 –27.38 1271.9 0.08556 160.19 410.31 0.8481 1.8659 1.401 0.938 1.279 717.
171.9 231.0 10.97 131.7 9.98 13.04 0.30000 0.32000 –25.83 –25.76 1266.3 0.08041 162.47 411.04 0.8573 1.8622 1.405 0.948 1.283 709.
172.0 226.4 11.04 130.7 10.09 12.77 0.32000 0.34000 –24.28 –24.21 1260.9 0.07584 164.66 411.72 0.8660 1.8586 1.410 0.958 1.287 701.
172.0 222.2 11.11 129.8 10.20 12.51 0.34000 0.36000 –22.80 –22.73 1255.8 0.07177 166.75 412.36 0.8743 1.8553 1.414 0.968 1.290 694.
172.0 218.2 11.18 128.9 10.30 12.27 0.36000 0.38000 –21.39 –21.31 1250.8 0.06811 168.76 412.96 0.8823 1.8521 1.419 0.977 1.294 687.
171.9 214.5 11.25 128.1 10.40 12.03 0.38000 0.40000 –20.03 –19.95 1246.0 0.06481 170.70 413.54 0.8899 1.8491 1.423 0.986 1.298 680.
171.9 211.0 11.31 127.3 10.49 11.81 0.40000 0.42000 –18.72 –18.64 1241.3 0.06180 172.57 414.08 0.8972 1.8463 1.427 0.995 1.301 673.
171.9 207.7 11.31 126.5 10.64 11.59 0.42000 0.44000 –17.45 –17.38 1236.8 0.05907 174.38 414.60 0.9042 1.8436 1.432 1.004 1.305 667.
171.8 204.6 11.32 125.8 10.78 11.39 0.44000 0.46000 –16.24 –16.16 1232.4 0.05656 176.13 415.09 0.9110 1.8410 1.436 1.012 1.308 661.
171.7 201.6 11.38 125.1 10.87 11.19 0.46000 0.48000 –15.06 –14.98 1228.1 0.05425 177.83 415.56 0.9175 1.8385 1.440 1.021 1.312 655.
171.7 198.7 11.43 124.4 10.96 11.00 0.48000 0.50000 –13.91 –13.83 1223.9 0.05212 179.48 416.00 0.9238 1.8361 1.444 1.029 1.316 649.
171.6 196.0 11.49 123.7 11.05 10.81 0.50000 0.55000 –11.20 –11.12 1214.0 0.04746 183.41 417.04 0.9388 1.8305 1.455 1.049 1.325 635.
171.3 189.7 11.63 122.1 11.28 10.38 0.55000 0.60000 –8.68 –8.59 1204.5 0.04354 187.11 417.96 0.9527 1.8254 1.465 1.068 1.334 623.
171.0 184.0 11.76 120.6 11.49 9.98 0.60000 0.65000 –6.30 –6.22 1195.5 0.04021 190.60 418.80 0.9657 1.8207 1.475 1.088 1.344 610.
170.7 178.8 11.88 119.2 11.70 9.60 0.65000 0.70000 –4.07 –3.98 1186.9 0.03734 193.92 419.56 0.9779 1.8163 1.485 1.106 1.353 599.
170.4 174.1 12.01 117.9 11.91 9.25 0.70000 0.75000 –1.95 –1.86 1178.6 0.03484 197.08 420.25 0.9894 1.8122 1.495 1.125 1.363 588.
170.0 169.7 12.12 116.6 12.11 8.92 0.75000 0.80000 0.07 0.16 1170.6 0.03264 200.10 420.88 1.0004 1.8083 1.505 1.143 1.373 577.
169.6 165.6 12.24 115.4 12.31 8.61 0.80000 0.85000 1.99 2.08 1162.9 0.03069 203.00 421.45 1.0108 1.8046 1.515 1.161 1.383 567.
169.2 161.8 12.34 114.3 12.51 8.31 0.85000 0.90000 3.83 3.92 1155.5 0.02894 205.79 421.97 1.0207 1.8011 1.525 1.179 1.393 558.
168.8 158.2 12.45 113.2 12.71 8.03 0.90000 0.95000 5.59 5.69 1148.2 0.02738 208.49 422.45 1.0303 1.7978 1.535 1.197 1.403 549.
168.4 154.8 12.56 112.1 12.91 7.77 0.95000 1.00000 7.28 7.38 1141.2 0.02597 211.09 422.89 1.0394 1.7946 1.545 1.215 1.414 540.
168.0 151.6 12.66 111.1 13.11 7.51 1.00000 1.10000 10.48 10.59 1127.6 0.02351 216.06 423.64 1.0568 1.7885 1.565 1.251 1.435 522.
167.1 145.7 12.87 109.1 13.51 7.04 1.10000 1.20000 13.48 13.58 1114.5 0.02145 220.76 424.27 1.0729 1.7828 1.586 1.287 1.458 506.
166.1 140.3 13.06 107.3 13.92 6.60 1.20000 1.30000 16.28 16.39 1102.0 0.01970 225.22 424.78 1.0881 1.7774 1.607 1.324 1.482 491.
165.1 135.4 13.25 105.5 14.35 6.19 1.30000 1.40000 18.93 19.04 1089.8 0.01818 229.48 425.18 1.1024 1.7723 1.629 1.362 1.507 476.
164.2 130.9 13.43 103.9 14.79 5.81 1.40000 1.50000 21.44 21.55 1078.0 0.01686 233.56 425.49 1.1160 1.7674 1.651 1.402 1.533 462.
163.1 126.7 13.60 102.3 15.25 5.45 1.50000 1.60000 23.83 23.94 1066.5 0.01570 237.49 425.72 1.1290 1.7627 1.675 1.442 1.561 449.
162.1 122.8 13.78 100.8 15.73 5.12 1.60000 1.70000 26.11 26.22 1055.3 0.01467 241.29 425.86 1.1414 1.7581 1.699 1.485 1.590 436.
161.1 119.1 13.95 99.3 16.23 4.80 1.70000 1.80000 28.29 28.40 1044.2 0.01375 244.96 425.93 1.1533 1.7536 1.725 1.529 1.622 423.
160.0 115.6 14.12 97.9 16.76 4.51 1.80000 1.90000 30.37 30.49 1033.3 0.01292 248.52 425.93 1.1648 1.7492 1.751 1.576 1.655 411.
159.0 112.3 14.28 96.5 17.32 4.22 1.90000 2.00000 32.38 32.49 1022.6 0.01217 251.99 425.87 1.1759 1.7448 1.779 1.625 1.690 399.
157.9 109.2 14.45 95.2 17.91 3.96 2.00000 2.10000 34.31 34.43 1012.0 0.01149 255.37 425.74 1.1866 1.7406 1.809 1.677 1.728 387.
156.8 106.2 14.62 93.9 18.53 3.70 2.10000 2.20000 36.18 36.29 1001.4 0.01087 258.68 425.54 1.1970 1.7363 1.840 1.732 1.768 376.
155.7 103.4 14.79 92.7 19.19 3.46 2.20000 2.30000 37.98 38.09 991.0 0.01030 261.91 425.29 1.2071 1.7321 1.874 1.790 1.812 365.
154.6 100.6 14.96 91.5 19.90 3.23 2.30000 2.40000 39.72 39.83 980.5 0.00977 265.08 424.98 1.2169 1.7279 1.909 1.853 1.858 354.
153.5 98.0 15.13 90.3 20.65 3.01 2.40000 2.50000 41.40 41.51 970.1 0.00928 268.20 424.61 1.2265 1.7237 1.947 1.920 1.909 343.
152.3 95.5 15.30 89.1 21.44 2.80 2.50000 2.60000 43.04 43.15 959.7 0.00883 271.27 424.18 1.2359 1.7194 1.988 1.993 1.964 333.
151.2 93.0 15.48 88.0 22.30 2.60 2.60000 2.70000 44.62 44.73 949.3 0.00840 274.29 423.69 1.2451 1.7152 2.032 2.072 2.024 322.
150.0 90.7 15.66 86.8 23.20 2.41 2.70000 2.80000 46.17 46.27 938.8 0.00801 277.27 423.14 1.2541 1.7109 2.080 2.158 2.089 312.
148.9 88.4 15.84 85.7 24.17 2.23 2.80000 2.90000 47.67 47.77 928.3 0.00764 280.23 422.53 1.2630 1.7065 2.133 2.252 2.161 302.
147.7 86.1 16.04 84.6 25.21 2.05 2.90000 3.00000 49.13 49.23 917.7 0.00729 283.15 421.85 1.2718 1.7021 2.190 2.356 2.240 292.
146.5 83.9 16.23 83.6 26.33 1.88 3.00000 3.20000 51.94 52.04 896.0 0.00665 288.94 420.30 1.2890 1.6930 2.323 2.598 2.426 273.
144.1 79.7 16.65 81.4 28.81 1.57 3.20000 3.40000 54.61 54.71 873.7 0.00607 294.67 418.47 1.3059 1.6835 2.490 2.904 2.662 253.
141.7 75.6 17.10 79.4 31.70 1.28 3.40000 3.60000 57.17 57.26 850.4 0.00555 300.41 416.29 1.3226 1.6734 2.707 3.305 2.971 234.
139.1 71.6 17.60 77.5 35.11 1.01 3.60000 3.80000 59.61 59.69 825.8 0.00506 306.20 413.72 1.3394 1.6624 3.002 3.855 3.393 215.
136.5 67.7 18.16 76.5 39.20 0.77 3.80000 4.00000 61.94 62.02 799.1 0.00461 312.13 410.64 1.3564 1.6503 3.431 4.661 4.010 196.
133.8 63.7 18.81 76.3 44.24 0.56 4.00000 4.20000 64.18 64.25 769.5 0.00417 318.33 406.86 1.3741 1.6365 4.129 5.970 5.004 177.
130.9 59.5 19.61 78.2 50.75 0.36 4.20000 4.790c 70.2 70.2 548.
0.00183 352.5 352.5 1.472 1.472 — — — — — — — — — 0.00 4.790 Temperatures are on the ITS-90 scale b = bubble and dew points at one standard atmosphere c = critical point Thermophysical Properties of Refrigerants 20.29 Refrigerant 507A [R-125/143a (50/50)] Properties of Saturated Liquid and Saturated Vapor Temp., °C Pres-sure, MPa Density, kg/m3 Liquid Volume, m3/kg Vapor Enthalpy, kJ/kg Entropy, kJ/(kg·K) Specific Heat cp, kJ/(kg·K) cp/cv Velocity of Sound, m/s Viscosity, µPa·s Thermal Cond., mW/(m·K) Surface Tension, mN/m Temp., °C Liquid Vapor Liquid Vapor Liquid Vapor Vapor Liquid Vapor Liquid Vapor Liquid Vapor –100.00 0.00310 1471.4 4.6772 77.43 303.23 0.4473 1.7515 1.134 0.617 1.164 984.
129.6 858.0 7.03 125.6 5.62 18.29 –100.00 –95.00 0.00479 1457.5 3.1090 83.11 306.19 0.4796 1.7319 1.137 0.630 1.162 957.
131.2 759.3 7.23 122.6 5.90 17.82 –95.00 –90.00 0.00720 1443.4 2.1231 88.81 309.17 0.5112 1.7144 1.142 0.643 1.161 930.
132.6 678.9 7.43 119.7 6.18 17.35 –90.00 –85.00 0.01055 1429.3 1.4856 94.54 312.18 0.5420 1.6988 1.148 0.657 1.159 904.
134.1 612.1 7.63 116.9 6.47 16.87 –85.00 –80.00 0.01509 1415.1 1.0628 100.29 315.20 0.5722 1.6849 1.155 0.671 1.158 878.
135.4 555.7 7.83 114.1 6.77 16.37 –80.00 –75.00 0.02113 1400.8 0.77578 106.09 318.24 0.6018 1.6725 1.162 0.686 1.158 853.
136.6 507.4 8.02 111.4 7.07 15.87 –75.00 –70.00 0.02903 1386.4 0.57673 111.93 321.29 0.6309 1.6614 1.171 0.702 1.158 828.
137.8 465.5 8.22 108.8 7.39 15.35 –70.00 –65.00 0.03916 1371.8 0.43596 117.81 324.33 0.6594 1.6516 1.180 0.718 1.159 804.
138.9 428.8 8.42 106.2 7.71 14.83 –65.00 –60.00 0.05198 1357.0 0.33458 123.74 327.38 0.6875 1.6429 1.190 0.734 1.160 780.
139.8 396.3 8.62 103.7 8.03 14.29 –60.00 –55.00 0.06795 1342.1 0.26036 129.72 330.42 0.7152 1.6352 1.200 0.752 1.162 755.
140.7 367.4 8.84 101.2 8.38 13.75 –55.00 –50.00 0.08759 1326.9 0.20517 135.75 333.44 0.7425 1.6284 1.211 0.770 1.164 731.
141.4 341.5 9.04 98.8 8.73 13.20 –50.00 –48.00 0.09661 1320.8 0.18713 138.18 334.64 0.7533 1.6259 1.216 0.778 1.165 722.
141.6 331.8 9.11 97.9 8.87 12.98 –48.00 –47.01b 0.10132 1317.7 0.17894 139.39 335.24 0.7586 1.6247 1.218 0.782 1.166 717.
141.8 327.2 9.15 97.4 8.94 12.87 –47.01 –46.00 0.10634 1314.6 0.17099 140.62 335.85 0.7641 1.6235 1.221 0.786 1.167 712.
141.9 322.5 9.19 97.0 9.01 12.75 –46.00 –44.00 0.11682 1308.4 0.15650 143.07 337.04 0.7748 1.6212 1.226 0.793 1.168 703.
142.1 313.6 9.26 96.0 9.16 12.53 –44.00 –42.00 0.12809 1302.1 0.14348 145.53 338.23 0.7854 1.6191 1.231 0.801 1.170 693.
142.3 305.0 9.34 95.1 9.30 12.30 –42.00 –40.00 0.14020 1295.8 0.13175 148.00 339.42 0.7960 1.6170 1.236 0.810 1.171 684.
142.4 296.7 9.41 94.2 9.45 12.08 –40.00 –38.00 0.15318 1289.5 0.12116 150.49 340.60 0.8066 1.6151 1.241 0.818 1.173 674.
142.6 288.8 9.49 93.2 9.60 11.85 –38.00 –36.00 0.16708 1283.1 0.11159 152.98 341.78 0.8171 1.6132 1.246 0.827 1.175 665.
142.7 281.0 9.57 92.3 9.75 11.62 –36.00 –34.00 0.18194 1276.6 0.10292 155.48 342.95 0.8275 1.6115 1.252 0.835 1.177 655.
142.8 273.6 9.64 91.4 9.90 11.39 –34.00 –32.00 0.19780 1270.1 0.09505 157.99 344.11 0.8380 1.6098 1.258 0.844 1.180 646.
142.9 266.4 9.72 90.5 10.06 11.15 –32.00 –30.00 0.21471 1263.6 0.08791 160.52 345.27 0.8483 1.6082 1.263 0.853 1.182 637.
142.9 259.4 9.80 89.6 10.21 10.92 –30.00 –28.00 0.23271 1257.0 0.08140 163.06 346.42 0.8587 1.6067 1.269 0.862 1.185 627.
142.9 252.7 9.88 88.7 10.37 10.69 –28.00 –26.00 0.25185 1250.3 0.07546 165.61 347.57 0.8690 1.6052 1.275 0.872 1.188 618.
142.9 246.1 9.96 87.8 10.53 10.45 –26.00 –24.00 0.27218 1243.6 0.07004 168.17 348.70 0.8792 1.6038 1.281 0.882 1.191 608.
142.9 239.8 10.05 87.0 10.69 10.22 –24.00 –22.00 0.29374 1236.8 0.06508 170.74 349.82 0.8894 1.6025 1.288 0.891 1.194 599.
142.8 233.6 10.01 86.1 10.99 9.98 –22.00 –20.00 0.31658 1229.9 0.06054 173.33 350.94 0.8996 1.6012 1.294 0.902 1.197 589.
142.8 227.6 10.11 85.2 11.16 9.74 –20.00 –18.00 0.34076 1223.0 0.05638 175.93 352.05 0.9098 1.6001 1.301 0.912 1.201 580.
142.6 221.8 10.20 84.3 11.33 9.51 –18.00 –16.00 0.36631 1216.0 0.05255 178.55 353.14 0.9199 1.5989 1.308 0.923 1.205 570.
142.5 216.2 10.29 83.5 11.50 9.27 –16.00 –14.00 0.39329 1208.9 0.04903 181.18 354.23 0.9300 1.5978 1.315 0.933 1.209 561.
142.3 210.7 10.39 82.6 11.68 9.03 –14.00 –12.00 0.42176 1201.8 0.04579 183.82 355.30 0.9401 1.5968 1.323 0.945 1.213 552.
142.1 205.3 10.48 81.8 11.86 8.79 –12.00 –10.00 0.45176 1194.5 0.04279 186.48 356.37 0.9501 1.5957 1.330 0.956 1.218 542.
141.9 200.1 10.58 80.9 12.04 8.55 –10.00 –8.00 0.48335 1187.2 0.04003 189.15 357.41 0.9601 1.5948 1.338 0.968 1.223 533.
141.6 195.0 10.67 80.0 12.23 8.31 –8.00 –6.00 0.51658 1179.8 0.03748 191.84 358.45 0.9701 1.5938 1.346 0.980 1.228 523.
141.3 190.1 10.77 79.2 12.42 8.07 –6.00 –4.00 0.55150 1172.3 0.03511 194.54 359.47 0.9801 1.5929 1.355 0.993 1.234 514.
141.0 185.2 10.87 78.3 12.61 7.83 –4.00 –2.00 0.58817 1164.6 0.03292 197.26 360.48 0.9901 1.5920 1.363 1.006 1.240 504.
140.6 180.5 10.97 77.5 12.81 7.59 –2.00 0.00 0.62665 1156.9 0.03089 200.00 361.47 1.0000 1.5912 1.372 1.019 1.247 494.
140.2 175.9 11.07 76.7 13.02 7.35 0.00 2.00 0.66698 1149.1 0.02900 202.76 362.44 1.0099 1.5903 1.382 1.033 1.254 485.
139.8 171.4 11.17 75.8 13.23 7.11 2.00 4.00 0.70924 1141.1 0.02724 205.53 363.40 1.0198 1.5895 1.392 1.048 1.261 475.
139.3 167.0 11.28 75.0 13.45 6.87 4.00 6.00 0.75347 1133.1 0.02560 208.32 364.33 1.0297 1.5886 1.402 1.063 1.269 465.
138.8 162.6 11.38 74.1 13.67 6.63 6.00 8.00 0.79973 1124.9 0.02408 211.13 365.25 1.0396 1.5878 1.413 1.079 1.278 456.
138.2 158.4 11.49 73.3 13.90 6.39 8.00 10.00 0.84809 1116.6 0.02265 213.97 366.15 1.0495 1.5870 1.424 1.096 1.287 446.
137.6 154.3 11.60 72.5 14.14 6.15 10.00 12.00 0.89860 1108.1 0.02132 216.82 367.02 1.0594 1.5862 1.436 1.113 1.297 436.
137.0 150.2 11.71 71.6 14.39 5.91 12.00 14.00 0.95133 1099.4 0.02008 219.69 367.87 1.0693 1.5853 1.448 1.131 1.307 426.
136.3 146.2 11.83 70.8 14.65 5.67 14.00 16.00 1.0063 1090.7 0.01892 222.59 368.70 1.0792 1.5845 1.461 1.151 1.319 416.
135.6 142.3 11.94 70.0 14.91 5.43 16.00 18.00 1.0637 1081.7 0.01783 225.52 369.49 1.0890 1.5836 1.475 1.171 1.332 407.
134.9 138.5 12.06 69.1 15.19 5.19 18.00 20.00 1.1235 1072.6 0.01680 228.46 370.26 1.0989 1.5827 1.490 1.193 1.346 397.
134.1 134.7 12.19 68.3 15.49 4.95 20.00 22.00 1.1857 1063.2 0.01584 231.44 371.00 1.1089 1.5817 1.506 1.216 1.360 386.
133.2 131.0 12.32 67.5 15.80 4.72 22.00 24.00 1.2505 1053.7 0.01494 234.44 371.71 1.1188 1.5808 1.522 1.241 1.377 376.
132.3 127.3 12.45 66.7 16.12 4.48 24.00 26.00 1.3179 1043.9 0.01409 237.47 372.38 1.1287 1.5797 1.540 1.267 1.395 366.
131.4 123.7 12.59 65.8 16.46 4.25 26.00 28.00 1.3880 1033.9 0.01329 240.53 373.01 1.1387 1.5787 1.560 1.296 1.415 356.
130.4 120.1 12.73 65.0 16.83 4.02 28.00 30.00 1.4608 1023.6 0.01253 243.62 373.60 1.1487 1.5775 1.581 1.327 1.437 346.
129.3 116.6 12.88 64.2 17.22 3.79 30.00 32.00 1.5365 1013.0 0.01182 246.75 374.15 1.1587 1.5763 1.603 1.361 1.461 335.
128.2 113.2 13.03 63.3 17.63 3.56 32.00 34.00 1.6151 1002.2 0.01115 249.91 374.65 1.1688 1.5750 1.628 1.398 1.488 325.
127.1 109.7 13.19 62.5 18.07 3.33 34.00 36.00 1.6967 991.0 0.01051 253.12 375.10 1.1790 1.5735 1.655 1.439 1.518 314.
125.9 106.3 13.36 61.6 18.55 3.11 36.00 38.00 1.7814 979.4 0.00991 256.37 375.49 1.1892 1.5720 1.685 1.485 1.553 303.
124.6 103.0 13.54 60.8 19.07 2.89 38.00 40.00 1.8692 967.5 0.00933 259.66 375.81 1.1994 1.5704 1.719 1.536 1.592 292.
123.2 99.6 13.73 60.0 19.63 2.67 40.00 42.00 1.9603 955.0 0.00879 263.01 376.07 1.2098 1.5686 1.757 1.594 1.637 281.
121.8 96.3 13.93 59.1 20.24 2.45 42.00 44.00 2.0547 942.1 0.00827 266.41 376.25 1.2202 1.5666 1.799 1.660 1.689 270.
120.4 93.0 14.15 58.2 20.92 2.24 44.00 46.00 2.1526 928.7 0.00778 269.87 376.35 1.2308 1.5644 1.848 1.737 1.749 259.
118.8 89.7 14.38 57.4 21.66 2.03 46.00 48.00 2.2541 914.5 0.00731 273.40 376.34 1.2415 1.5620 1.905 1.826 1.821 247.
117.2 86.4 14.63 56.5 22.49 1.82 48.00 50.00 2.3592 899.7 0.00685 277.01 376.22 1.2523 1.5593 1.971 1.932 1.907 235.
115.5 83.1 14.90 55.7 23.41 1.62 50.00 55.00 2.6389 858.6 0.00580 286.46 375.31 1.2803 1.5511 2.206 2.313 2.219 205.
110.9 74.7 15.71 53.5 26.33 1.14 55.00 60.00 2.9446 808.9 0.00482 296.77 373.11 1.3104 1.5396 2.651 3.051 2.832 172.
105.6 65.8 16.82 51.5 30.63 0.70 60.00 65.00 3.2795 742.4 0.00386 308.71 368.41 1.3447 1.5212 3.892 5.120 4.552 136.
99.4 55.7 18.61 50.4 38.26 0.31 65.00 70.74c 3.7091 492.5 0.00203 340.07 340.07 1.4345 1.4345 ∞ ∞ ∞ 0.
0.0 — — ∞ ∞ 0.00 70.74 Temperatures are on the ITS-90 scale Small deviations from azeotropic behavior occur at some conditions; tabulated pressures are the average of the bubble and dew point pressures b = normal boiling point c = critical point 20.30 2001 ASHRAE Fundamentals Handbook (SI) Fig. 12 Pressure-Enthalpy Diagram for Refrigerant 717 (Ammonia) Note: The reference states for enthalpy and entropy differ from those in the table.
Thermophysical Properties of Refrigerants 20.31 Refrigerant 717 (Ammonia) Properties of Saturated Liquid and Saturated Vapor Temp., °C Pres-sure, MPa Density, kg/m3 Liquid Volume, m3/kg Vapor Enthalpy, kJ/kg Entropy, kJ/(kg·K) Specific Heat cp, kJ/(kg·K) cp/cv Velocity of Sound, m/s Viscosity, µPa·s Thermal Cond., mW/(m·K) Surface Tension, mN/m Temp., °C Liquid Vapor Liquid Vapor Liquid Vapor Vapor Liquid Vapor Liquid Vapor Liquid Vapor –77.65a 0.00609 732.9 15.602 –143.15 1341.23 –0.4716 7.1213 4.202 2.063 1.325 2124.
354.1 559.6 6.84 819.0 19.64 62.26 –77.65 –70.00 0.01094 724.7 9.0079 –110.81 1355.55 –0.3094 6.9088 4.245 2.086 1.327 2051.
360.5 475.0 7.03 792.1 19.73 59.10 –70.00 –60.00 0.02189 713.6 4.7057 –68.06 1373.73 –0.1040 6.6602 4.303 2.125 1.330 1967.
368.4 391.3 7.30 757.0 19.93 55.05 –60.00 –50.00 0.04084 702.1 2.6277 –24.73 1391.19 0.0945 6.4396 4.360 2.178 1.335 1890.
375.6 328.9 7.57 722.3 20.24 51.11 –50.00 –40.00 0.07169 690.2 1.5533 19.17 1407.76 0.2867 6.2425 4.414 2.244 1.342 1816.
382.2 281.2 7.86 688.1 20.64 47.26 –40.00 –38.00 0.07971 687.7 1.4068 28.01 1410.96 0.3245 6.2056 4.424 2.259 1.343 1802.
383.4 273.1 7.92 681.4 20.73 46.51 –38.00 –36.00 0.08845 685.3 1.2765 36.88 1414.11 0.3619 6.1694 4.434 2.275 1.345 1787.
384.6 265.3 7.98 674.6 20.83 45.75 –36.00 –34.00 0.09795 682.8 1.1604 45.77 1417.23 0.3992 6.1339 4.444 2.291 1.347 1773.
385.8 257.9 8.03 667.9 20.93 45.00 –34.00 –33.33b 0.10133 682.0 1.1242 48.76 1418.26 0.4117 6.1221 4.448 2.297 1.348 1768.
386.2 255.5 8.05 665.7 20.97 44.75 –33.33 –32.00 0.10826 680.3 1.0567 54.67 1420.29 0.4362 6.0992 4.455 2.308 1.349 1759.
387.0 250.8 8.09 661.3 21.04 44.26 –32.00 –30.00 0.11943 677.8 0.96396 63.60 1423.31 0.4730 6.0651 4.465 2.326 1.351 1744.
388.1 244.1 8.15 654.6 21.15 43.52 –30.00 –28.00 0.13151 675.3 0.88082 72.55 1426.28 0.5096 6.0317 4.474 2.344 1.353 1730.
389.2 237.6 8.21 648.0 21.26 42.78 –28.00 –26.00 0.14457 672.8 0.80614 81.52 1429.21 0.5460 5.9989 4.484 2.363 1.355 1716.
390.2 231.4 8.27 641.5 21.38 42.05 –26.00 –24.00 0.15864 670.3 0.73896 90.51 1432.08 0.5821 5.9667 4.494 2.383 1.358 1702.
391.2 225.5 8.33 634.9 21.51 41.32 –24.00 –22.00 0.17379 667.7 0.67840 99.52 1434.91 0.6180 5.9351 4.504 2.403 1.360 1687.
392.2 219.8 8.39 628.4 21.63 40.60 –22.00 –20.00 0.19008 665.1 0.62373 108.55 1437.68 0.6538 5.9041 4.514 2.425 1.363 1673.
393.2 214.4 8.45 622.0 21.77 39.88 –20.00 –18.00 0.20756 662.6 0.57428 117.60 1440.39 0.6893 5.8736 4.524 2.446 1.365 1659.
394.1 209.2 8.51 615.5 21.90 39.16 –18.00 –16.00 0.22630 660.0 0.52949 126.67 1443.06 0.7246 5.8437 4.534 2.469 1.368 1645.
395.0 204.2 8.57 609.1 22.05 38.45 –16.00 –14.00 0.24637 657.3 0.48885 135.76 1445.66 0.7597 5.8143 4.543 2.493 1.371 1631.
395.8 199.3 8.63 602.8 22.19 37.74 –14.00 –12.00 0.26782 654.7 0.45192 144.88 1448.21 0.7946 5.7853 4.553 2.517 1.375 1616.
396.7 194.7 8.69 596.4 22.35 37.04 –12.00 –10.00 0.29071 652.1 0.41830 154.01 1450.70 0.8293 5.7569 4.564 2.542 1.378 1602.
397.5 190.2 8.75 590.1 22.50 36.34 –10.00 –8.00 0.31513 649.4 0.38767 163.16 1453.14 0.8638 5.7289 4.574 2.568 1.382 1588.
398.2 185.9 8.81 583.9 22.67 35.65 –8.00 –6.00 0.34114 646.7 0.35970 172.34 1455.51 0.8981 5.7013 4.584 2.594 1.385 1574.
398.9 181.7 8.87 577.7 22.83 34.96 –6.00 –4.00 0.36880 644.0 0.33414 181.54 1457.81 0.9323 5.6741 4.595 2.622 1.389 1559.
399.6 177.7 8.93 571.5 23.00 34.27 –4.00 –2.00 0.39819 641.3 0.31074 190.76 1460.06 0.9662 5.6474 4.606 2.651 1.393 1545.
400.2 173.8 8.99 565.3 23.18 33.59 –2.00 0.00 0.42938 638.6 0.28930 200.00 1462.24 1.0000 5.6210 4.617 2.680 1.398 1531.
400.8 170.1 9.06 559.2 23.37 32.91 0.00 2.00 0.46246 635.8 0.26962 209.27 1464.35 1.0336 5.5951 4.628 2.710 1.402 1516.
401.4 166.5 9.12 553.1 23.55 32.24 2.00 4.00 0.49748 633.1 0.25153 218.55 1466.40 1.0670 5.5695 4.639 2.742 1.407 1502.
401.9 162.9 9.18 547.1 23.75 31.57 4.00 6.00 0.53453 630.3 0.23489 227.87 1468.37 1.1003 5.5442 4.651 2.774 1.412 1487.
402.4 159.5 9.24 541.1 23.95 30.91 6.00 8.00 0.57370 627.5 0.21956 237.20 1470.28 1.1334 5.5192 4.663 2.807 1.417 1473.
402.8 156.2 9.30 535.1 24.15 30.24 8.00 10.00 0.61505 624.6 0.20543 246.57 1472.11 1.1664 5.4946 4.676 2.841 1.422 1458.
403.2 153.0 9.36 529.1 24.37 29.59 10.00 12.00 0.65866 621.8 0.19237 255.95 1473.88 1.1992 5.4703 4.689 2.877 1.428 1443.
403.6 149.9 9.43 523.2 24.58 28.94 12.00 14.00 0.70463 618.9 0.18031 265.37 1475.56 1.2318 5.4463 4.702 2.913 1.434 1429.
403.9 146.9 9.49 517.3 24.81 28.29 14.00 16.00 0.75303 616.0 0.16914 274.81 1477.17 1.2643 5.4226 4.716 2.951 1.440 1414.
404.2 144.0 9.55 511.5 25.04 27.65 16.00 18.00 0.80395 613.1 0.15879 284.28 1478.70 1.2967 5.3991 4.730 2.990 1.446 1399.
404.4 141.1 9.61 505.6 25.27 27.01 18.00 20.00 0.85748 610.2 0.14920 293.78 1480.16 1.3289 5.3759 4.745 3.030 1.453 1384.
404.6 138.3 9.68 499.9 25.52 26.38 20.00 22.00 0.91369 607.2 0.14029 303.31 1481.53 1.3610 5.3529 4.760 3.071 1.460 1370.
404.8 135.6 9.74 494.1 25.77 25.75 22.00 24.00 0.97268 604.3 0.13201 312.87 1482.82 1.3929 5.3301 4.776 3.113 1.468 1355.
404.9 133.0 9.80 488.4 26.03 25.12 24.00 26.00 1.0345 601.3 0.12431 322.47 1484.02 1.4248 5.3076 4.793 3.158 1.475 1340.
404.9 130.4 9.87 482.7 26.29 24.50 26.00 28.00 1.0993 598.2 0.11714 332.09 1485.14 1.4565 5.2853 4.810 3.203 1.484 1324.
405.0 127.9 9.93 477.0 26.57 23.89 28.00 30.00 1.1672 595.2 0.11046 341.76 1486.17 1.4881 5.2631 4.828 3.250 1.492 1309.
404.9 125.5 10.00 471.4 26.85 23.28 30.00 32.00 1.2382 592.1 0.10422 351.45 1487.11 1.5196 5.2412 4.847 3.299 1.501 1294.
404.8 123.1 10.06 465.7 27.14 22.67 32.00 34.00 1.3124 589.0 0.09840 361.19 1487.95 1.5509 5.2194 4.867 3.349 1.510 1279.
404.7 120.7 10.13 460.1 27.43 22.07 34.00 36.00 1.3900 585.8 0.09296 370.96 1488.70 1.5822 5.1978 4.888 3.401 1.520 1263.
404.5 118.4 10.19 454.6 27.74 21.47 36.00 38.00 1.4709 582.6 0.08787 380.78 1489.36 1.6134 5.1763 4.909 3.455 1.530 1248.
404.3 116.2 10.26 449.1 28.05 20.88 38.00 40.00 1.5554 579.4 0.08310 390.64 1489.91 1.6446 5.1549 4.932 3.510 1.541 1232.
404.0 114.0 10.33 443.5 28.38 20.29 40.00 42.00 1.6435 576.2 0.07863 400.54 1490.36 1.6756 5.1337 4.956 3.568 1.553 1216.
403.7 111.9 10.39 438.0 28.71 19.71 42.00 44.00 1.7353 572.9 0.07445 410.48 1490.70 1.7065 5.1126 4.981 3.628 1.565 1201.
403.3 109.8 10.46 432.6 29.06 19.13 44.00 46.00 1.8310 569.6 0.07052 420.48 1490.94 1.7374 5.0915 5.007 3.691 1.577 1185.
402.9 107.8 10.53 427.1 29.41 18.56 46.00 48.00 1.9305 566.3 0.06682 430.52 1491.06 1.7683 5.0706 5.034 3.756 1.591 1169.
402.4 105.8 10.60 421.7 29.78 17.99 48.00 50.00 2.0340 562.9 0.06335 440.62 1491.07 1.7990 5.0497 5.064 3.823 1.605 1153.
401.9 103.8 10.67 416.3 30.16 17.43 50.00 55.00 2.3111 554.2 0.05554 466.10 1490.57 1.8758 4.9977 5.143 4.005 1.643 1112.
400.3 99.0 10.86 402.9 31.16 16.04 55.00 60.00 2.6156 545.2 0.04880 491.97 1489.27 1.9523 4.9458 5.235 4.208 1.687 1070.
398.3 94.5 11.05 389.6 32.26 14.69 60.00 65.00 2.9491 536.0 0.04296 518.26 1487.09 2.0288 4.8939 5.341 4.438 1.739 1028.
396.0 90.1 11.25 376.4 33.47 13.37 65.00 70.00 3.3135 526.3 0.03787 545.04 1483.94 2.1054 4.8415 5.465 4.699 1.799 984.
393.3 85.9 11.47 363.2 34.80 12.08 70.00 75.00 3.7105 516.2 0.03342 572.37 1479.72 2.1823 4.7885 5.610 5.001 1.870 940.
390.1 81.9 11.70 350.2 36.30 10.83 75.00 80.00 4.1420 505.7 0.02951 600.34 1474.31 2.2596 4.7344 5.784 5.355 1.955 895.
386.5 78.0 11.95 337.1 38.00 9.61 80.00 85.00 4.6100 494.5 0.02606 629.04 1467.53 2.3377 4.6789 5.993 5.777 2.058 848.
382.5 74.2 12.23 324.1 39.95 8.44 85.00 90.00 5.1167 482.8 0.02300 658.61 1459.19 2.4168 4.6213 6.250 6.291 2.187 800.
377.9 70.5 12.55 311.0 42.24 7.30 90.00 95.00 5.6643 470.2 0.02027 689.19 1449.01 2.4973 4.5612 6.573 6.933 2.349 751.
372.7 66.8 12.91 297.9 44.99 6.20 95.00 100.00 6.2553 456.6 0.01782 721.00 1436.63 2.5797 4.4975 6.991 7.762 2.562 701.
367.0 63.2 13.32 284.8 48.36 5.15 100.00 105.00 6.8923 441.9 0.01561 754.35 1421.57 2.6647 4.4291 7.555 8.877 2.851 649.
360.5 59.6 13.82 271.5 52.65 4.15 105.00 110.00 7.5783 425.6 0.01360 789.68 1403.08 2.7533 4.3542 8.36 10.46 3.26 594.
353.3 56.0 14.42 258.1 58.33 3.20 110.00 115.00 8.3170 407.2 0.01174 827.74 1379.99 2.8474 4.2702 9.63 12.91 3.91 538.
345.0 52.3 15.19 244.6 66.28 2.31 115.00 120.00 9.1125 385.5 0.00999 869.92 1350.23 2.9502 4.1719 11.94 17.21 5.04 477.
335.4 48.3 16.21 231.2 78.40 1.50 120.00 125.00 9.9702 357.8 0.00828 919.68 1309.12 3.0702 4.0483 17.66 27.00 7.62 411.
323.6 43.8 17.73 219.1 100.01 0.77 125.00 130.00 10.8977 312.3 0.00638 992.02 1239.32 3.2437 3.8571 54.21 76.49 20.66 334.
306.6 37.3 20.63 221.9 160.39 0.18 130.00 132.25c11.3330 225.0 0.00444 1119.22 1119.22 3.5542 3.5542 ∞ ∞ ∞ 0.
0.0 — — ∞ ∞ 0.00 132.25 Temperatures are on the ITS-90 scale a = triple point b = normal boiling point c = critical point 20.32 2001 ASHRAE Fundamentals Handbook (SI) Fig. 13 Pressure-Enthalpy Diagram for Refrigerant 718 (Water/Steam) Thermophysical Properties of Refrigerants 20.33 Refrigerant 718 (Water/Steam) Properties of Saturated Liquid and Saturated Vapor Temp., °C Pres-sure, MPa Density, kg/m3 Liquid Volume, m3/kg Vapor Enthalpy, kJ/kg Entropy, kJ/(kg·K) Specific Heat cp, kJ/(kg·K) cp/cv Velocity of Sound, m/s Viscosity, µPa·s Thermal Cond., mW/(m·K) Surface Tension, mN/m Temp., °C Liquid Vapor Liquid Vapor Liquid Vapor Vapor Liquid Vapor Liquid Vapor Liquid Vapor 0.01a 0.00061 999.8 205.99 0.00 2500.92 0.0000 9.1555 4.220 1.884 1.329 1402.
409.0 1791.2 9.22 561.0 17.07 75.65 0.01 5.00 0.00087 999.9 147.01 21.02 2510.06 0.0763 9.0248 4.205 1.889 1.328 1426.
412.6 1518.3 9.34 570.5 17.34 74.94 5.00 10.00 0.00123 999.7 106.30 42.02 2519.21 0.1511 8.8998 4.196 1.895 1.328 1447.
416.2 1306.0 9.46 580.0 17.62 74.22 10.00 15.00 0.00171 999.1 77.875 62.98 2528.33 0.2245 8.7803 4.189 1.900 1.328 1466.
419.7 1137.6 9.59 589.3 17.92 73.49 15.00 20.00 0.00234 998.2 57.757 83.91 2537.43 0.2965 8.6660 4.184 1.906 1.327 1482.
423.2 1001.6 9.73 598.4 18.23 72.74 20.00 25.00 0.00317 997.0 43.337 104.83 2546.51 0.3672 8.5566 4.182 1.912 1.327 1497.
426.6 890.1 9.87 607.2 18.55 71.97 25.00 30.00 0.00425 995.6 32.878 125.73 2555.55 0.4368 8.4520 4.180 1.918 1.327 1509.
430.0 797.4 10.01 615.5 18.89 71.19 30.00 35.00 0.00563 994.0 25.205 146.63 2564.55 0.5051 8.3517 4.180 1.925 1.327 1520.
433.4 719.3 10.16 623.3 19.24 70.40 35.00 40.00 0.00738 992.2 19.515 167.53 2573.51 0.5724 8.2555 4.180 1.931 1.327 1529.
436.7 653.0 10.31 630.6 19.60 69.60 40.00 45.00 0.00959 990.2 15.252 188.43 2582.43 0.6386 8.1633 4.180 1.939 1.327 1536.
440.0 596.1 10.46 637.3 19.97 68.78 45.00 50.00 0.01235 988.0 12.027 209.34 2591.29 0.7038 8.0748 4.182 1.947 1.328 1542.
443.2 546.8 10.62 643.6 20.36 67.94 50.00 55.00 0.01576 985.7 9.5643 230.26 2600.09 0.7680 7.9898 4.183 1.955 1.328 1547.
446.4 504.0 10.77 649.2 20.77 67.10 55.00 60.00 0.01995 983.2 7.6672 251.18 2608.83 0.8313 7.9081 4.185 1.965 1.328 1551.
449.5 466.4 10.93 654.3 21.19 66.24 60.00 65.00 0.02504 980.5 6.1935 272.12 2617.50 0.8937 7.8296 4.187 1.975 1.329 1553.
452.6 433.2 11.10 659.0 21.62 65.37 65.00 70.00 0.03120 977.7 5.0395 293.07 2626.10 0.9551 7.7540 4.190 1.986 1.330 1555.
455.6 403.9 11.26 663.1 22.07 64.48 70.00 75.00 0.03860 974.8 4.1289 314.03 2634.60 1.0158 7.6812 4.193 1.999 1.331 1555.
458.5 377.7 11.43 666.8 22.53 63.58 75.00 80.00 0.04741 971.8 3.4052 335.01 2643.02 1.0756 7.6111 4.197 2.012 1.332 1554.
461.4 354.3 11.59 670.0 23.01 62.67 80.00 85.00 0.05787 968.6 2.8258 356.01 2651.33 1.1346 7.5434 4.201 2.027 1.333 1553.
464.2 333.3 11.76 672.8 23.51 61.75 85.00 90.00 0.07018 965.3 2.3591 377.04 2659.53 1.1929 7.4781 4.205 2.043 1.334 1550.
466.9 314.4 11.93 675.3 24.02 60.82 90.00 95.00 0.08461 961.9 1.9806 398.09 2667.61 1.2504 7.4151 4.210 2.061 1.335 1547.
469.6 297.3 12.10 677.3 24.55 59.87 95.00 99.97b 0.10133 958.4 1.6732 419.06 2675.53 1.3069 7.3544 4.216 2.080 1.337 1543.
472.2 281.8 12.27 679.1 25.09 58.92 99.97 100.00 0.10142 958.3 1.6718 419.17 2675.57 1.3072 7.3541 4.216 2.080 1.337 1543.
472.2 281.7 12.27 679.1 25.10 58.91 100.00 105.00 0.12090 954.7 1.4184 440.27 2683.39 1.3633 7.2952 4.222 2.101 1.339 1538.
474.7 267.6 12.44 680.5 25.66 57.94 105.00 110.00 0.14338 950.9 1.2093 461.42 2691.06 1.4188 7.2381 4.228 2.124 1.341 1533.
477.1 254.7 12.61 681.7 26.24 56.96 110.00 115.00 0.16918 947.1 1.0358 482.59 2698.58 1.4737 7.1828 4.236 2.150 1.343 1527.
479.5 242.9 12.78 682.6 26.85 55.97 115.00 120.00 0.19867 943.1 0.89121 503.81 2705.93 1.5279 7.1291 4.244 2.177 1.346 1520.
481.7 232.1 12.96 683.2 27.47 54.97 120.00 125.00 0.23224 939.0 0.77003 525.07 2713.10 1.5816 7.0770 4.252 2.207 1.349 1512.
483.9 222.1 13.13 683.6 28.11 53.96 125.00 130.00 0.27028 934.8 0.66800 546.38 2720.08 1.6346 7.0264 4.261 2.239 1.352 1504.
486.0 212.9 13.30 683.7 28.76 52.93 130.00 135.00 0.31323 930.5 0.58173 567.74 2726.87 1.6872 6.9772 4.272 2.274 1.355 1496.
487.9 204.4 13.47 683.6 29.44 51.90 135.00 140.00 0.36154 926.1 0.50845 589.16 2733.44 1.7392 6.9293 4.283 2.311 1.359 1486.
489.8 196.5 13.65 683.3 30.14 50.86 140.00 145.00 0.41568 921.6 0.44596 610.64 2739.80 1.7907 6.8826 4.294 2.351 1.363 1476.
491.6 189.2 13.82 682.8 30.86 49.80 145.00 150.00 0.47616 917.0 0.39245 632.18 2745.93 1.8418 6.8371 4.307 2.394 1.368 1466.
493.3 182.5 13.99 682.0 31.60 48.74 150.00 155.00 0.54350 912.3 0.34646 653.79 2751.81 1.8924 6.7926 4.321 2.440 1.373 1455.
494.8 176.1 14.16 681.1 32.35 47.67 155.00 160.00 0.61823 907.4 0.30678 675.47 2757.44 1.9426 6.7491 4.335 2.488 1.379 1443.
496.3 170.2 14.34 680.0 33.13 46.59 160.00 165.00 0.70093 902.5 0.27243 697.24 2762.81 1.9923 6.7066 4.351 2.540 1.385 1431.
497.6 164.7 14.51 678.6 33.93 45.50 165.00 170.00 0.79219 897.5 0.24259 719.08 2767.90 2.0417 6.6650 4.368 2.594 1.392 1419.
498.9 159.6 14.68 677.0 34.75 44.41 170.00 175.00 0.89260 892.3 0.21658 741.02 2772.71 2.0906 6.6241 4.386 2.652 1.399 1405.
500.0 154.7 14.85 675.3 35.59 43.30 175.00 180.00 1.0028 887.0 0.19384 763.05 2777.21 2.1392 6.5840 4.405 2.713 1.407 1392.
501.0 150.1 15.03 673.3 36.45 42.19 180.00 185.00 1.1235 881.6 0.17390 785.19 2781.41 2.1875 6.5447 4.425 2.777 1.416 1378.
501.9 145.8 15.20 671.1 37.33 41.07 185.00 190.00 1.2552 876.1 0.15636 807.43 2785.28 2.2355 6.5059 4.447 2.844 1.425 1363.
502.7 141.8 15.37 668.8 38.24 39.95 190.00 195.00 1.3988 870.4 0.14089 829.79 2788.82 2.2832 6.4678 4.471 2.915 1.436 1348.
503.4 137.9 15.54 666.1 39.16 38.81 195.00 200.00 1.5549 864.7 0.12721 852.27 2792.01 2.3305 6.4302 4.496 2.990 1.447 1332.
503.9 134.3 15.71 663.3 40.11 37.67 200.00 205.00 1.7243 858.8 0.11508 874.88 2794.83 2.3777 6.3930 4.523 3.068 1.459 1316.
504.3 130.9 15.89 660.3 41.09 36.53 205.00 210.00 1.9077 852.7 0.10429 897.63 2797.27 2.4245 6.3563 4.551 3.150 1.472 1299.
504.6 127.6 16.06 657.0 42.09 35.38 210.00 215.00 2.1058 846.5 0.09468 920.53 2799.32 2.4712 6.3200 4.582 3.237 1.486 1282.
504.8 124.5 16.24 653.4 43.11 34.23 215.00 220.00 2.3196 840.2 0.08609 943.58 2800.95 2.5177 6.2840 4.615 3.329 1.501 1264.
504.8 121.5 16.41 649.7 44.17 33.07 220.00 225.00 2.5497 833.7 0.07840 966.80 2802.15 2.5640 6.2483 4.650 3.426 1.518 1246.
504.6 118.7 16.59 645.6 45.26 31.90 225.00 230.00 2.7971 827.1 0.07150 990.19 2802.90 2.6101 6.2128 4.688 3.528 1.536 1228.
504.4 116.0 16.76 641.3 46.38 30.74 230.00 235.00 3.0625 820.3 0.06530 1013.77 2803.17 2.6561 6.1775 4.728 3.638 1.556 1209.
503.9 113.4 16.94 636.7 47.53 29.57 235.00 240.00 3.3469 813.4 0.05970 1037.55 2802.96 2.7020 6.1423 4.772 3.754 1.578 1189.
503.3 110.9 17.12 631.8 48.73 28.39 240.00 245.00 3.6512 806.2 0.05465 1061.55 2802.22 2.7478 6.1072 4.819 3.878 1.601 1169.
502.6 108.4 17.31 626.7 49.97 27.22 245.00 250.00 3.9762 798.9 0.05008 1085.77 2800.93 2.7935 6.0721 4.870 4.011 1.627 1148.
501.6 106.1 17.49 621.2 51.26 26.04 250.00 255.00 4.3229 791.4 0.04594 1110.23 2799.07 2.8392 6.0369 4.925 4.153 1.655 1127.
500.5 103.9 17.68 615.4 52.61 24.87 255.00 260.00 4.6923 783.6 0.04217 1134.96 2796.60 2.8849 6.0016 4.986 4.308 1.686 1105.
499.2 101.7 17.88 609.2 54.03 23.69 260.00 265.00 5.0853 775.7 0.03875 1159.96 2793.49 2.9307 5.9661 5.051 4.475 1.720 1083.
497.7 99.6 18.07 602.8 55.53 22.51 265.00 270.00 5.5030 767.5 0.03562 1185.27 2789.69 2.9765 5.9304 5.123 4.656 1.757 1060.
496.0 97.5 18.28 595.9 57.11 21.34 270.00 275.00 5.9464 759.0 0.03277 1210.90 2785.17 3.0224 5.8944 5.202 4.855 1.798 1037.
494.1 95.5 18.48 588.7 58.80 20.16 275.00 280.00 6.4166 750.3 0.03015 1236.88 2779.87 3.0685 5.8579 5.289 5.073 1.845 1013.
491.9 93.5 18.70 581.1 60.61 18.99 280.00 285.00 6.9147 741.3 0.02776 1263.25 2773.73 3.1147 5.8209 5.385 5.314 1.896 988.
489.5 91.6 18.92 573.2 62.57 17.83 285.00 290.00 7.4418 731.9 0.02555 1290.03 2766.70 3.1612 5.7834 5.493 5.582 1.954 962.
486.9 89.7 19.15 565.0 64.71 16.66 290.00 295.00 7.9991 722.2 0.02353 1317.27 2758.70 3.2080 5.7451 5.614 5.882 2.019 936.
483.9 87.8 19.40 556.3 67.05 15.51 295.00 300.00 8.5879 712.1 0.02166 1345.01 2749.64 3.2552 5.7059 5.750 6.220 2.094 909.
480.7 85.9 19.65 547.4 69.65 14.36 300.00 310.00 9.8651 690.7 0.01833 1402.22 2727.95 3.3510 5.6244 6.085 7.045 2.277 853.
473.3 82.2 20.21 528.7 75.84 12.09 310.00 320.00 11.2843 667.1 0.01547 1462.22 2700.59 3.4494 5.5372 6.537 8.159 2.528 793.
464.4 78.4 20.85 509.2 83.91 9.86 320.00 330.00 12.8581 640.8 0.01298 1525.87 2666.03 3.5518 5.4422 7.186 9.753 2.889 729.
453.7 74.5 21.61 489.1 94.94 7.70 330.00 340.00 14.6007 610.7 0.01078 1594.53 2621.85 3.6601 5.3356 8.21 12.24 3.45 658.
440.7 70.4 22.55 468.5 110.91 5.63 340.00 350.00 16.5294 574.7 0.00880 1670.89 2563.64 3.7784 5.2110 10.12 16.69 4.46 578.
424.4 65.9 23.82 447.4 135.95 3.67 350.00 360.00 18.6660 527.6 0.00695 1761.66 2481.49 3.9167 5.0536 15.00 27.36 6.83 480.
402.4 60.3 25.72 425.7 181.51 1.88 360.00 370.00 21.0436 451.4 0.00495 1890.69 2334.52 4.1112 4.8012 45.16 96.60 21.15 360.
362.8 52.1 29.68 425.0 323.84 0.39 370.00 373.95c22.0640 322.0 0.00311 2084.26 2084.26 4.4070 4.4070 ∞ ∞ ∞ 0.
0.0 — — ∞ ∞ 0.00 373.95 Temperatures are on the ITS-90 scale a = triple point b = normal boiling point c = critical point 20.34 2001 ASHRAE Fundamentals Handbook (SI) Fig. 14 Pressure-Enthalpy Diagram for Refrigerant 744 (Carbon Dioxide) Note: The refe.rence states for enthalpy and entropy differ from those in the table.
Thermophysical Properties of Refrigerants 20.35 Refrigerant 744 (Carbon Dioxide) Properties of Saturated Liquid and Saturated Vapor Temp., °C Pres-sure, MPa Density, kg/m3 Liquid Volume, m3/kg Vapor Enthalpy, kJ/kg Entropy, kJ/(kg·K) Specific Heat cp, kJ/(kg·K) cp/cv Velocity of Sound, m/s Viscosity, µPa·s Thermal Cond., mW/(m·K) Surface Tension, mN/m Temp., °C Liquid Vapor Liquid Vapor Liquid Vapor Vapor Liquid Vapor Liquid Vapor Liquid Vapor –56.56a 0.51796 1178.5 0.07267 80.04 430.42 0.5213 2.1390 1.953 0.909 1.444 976.
222.8 256.7 10.95 180.6 11.01 17.16 –56.56 –50.00 0.68234 1154.6 0.05579 92.94 432.68 0.5794 2.1018 1.971 0.952 1.468 928.
223.4 229.3 11.31 172.1 11.58 15.53 –50.00 –48.00 0.73949 1147.1 0.05162 96.90 433.29 0.5968 2.0909 1.978 0.967 1.477 914.
223.5 221.6 11.42 169.5 11.76 15.04 –48.00 –46.00 0.80015 1139.6 0.04782 100.88 433.86 0.6142 2.0801 1.985 0.982 1.486 900.
223.6 214.3 11.53 166.9 11.95 14.56 –46.00 –44.00 0.86445 1132.0 0.04435 104.87 434.39 0.6314 2.0694 1.993 0.998 1.496 885.
223.6 207.2 11.64 164.4 12.14 14.07 –44.00 –42.00 0.93252 1124.2 0.04118 108.88 434.88 0.6486 2.0589 2.002 1.015 1.507 871.
223.6 200.3 11.75 161.8 12.34 13.60 –42.00 –40.00 1.0045 1116.4 0.03828 112.90 435.32 0.6656 2.0485 2.012 1.033 1.518 856.
223.5 193.8 11.87 159.3 12.54 13.12 –40.00 –38.00 1.0805 1108.5 0.03562 116.95 435.72 0.6826 2.0382 2.022 1.052 1.530 842.
223.4 187.4 11.98 156.8 12.75 12.65 –38.00 –36.00 1.1607 1100.5 0.03318 121.01 436.07 0.6995 2.0281 2.033 1.072 1.544 827.
223.2 181.3 12.10 154.3 12.97 12.18 –36.00 –34.00 1.2452 1092.4 0.03093 125.10 436.37 0.7163 2.0180 2.045 1.094 1.558 813.
223.1 175.4 12.22 151.8 13.20 11.72 –34.00 –32.00 1.3342 1084.1 0.02886 129.20 436.62 0.7331 2.0079 2.059 1.116 1.573 798.
222.8 169.7 12.34 149.3 13.43 11.26 –32.00 –30.00 1.4278 1075.7 0.02696 133.34 436.82 0.7498 1.9980 2.073 1.141 1.590 783.
222.5 164.2 12.46 146.9 13.68 10.80 –30.00 –28.00 1.5261 1067.2 0.02519 137.50 436.96 0.7665 1.9880 2.089 1.166 1.608 768.
222.2 158.9 12.59 144.4 13.94 10.35 –28.00 –26.00 1.6293 1058.6 0.02356 141.69 437.04 0.7831 1.9781 2.105 1.194 1.627 753.
221.8 153.8 12.72 141.9 14.20 9.90 –26.00 –24.00 1.7375 1049.8 0.02205 145.91 437.06 0.7997 1.9683 2.124 1.223 1.648 738.
221.4 148.8 12.85 139.5 14.49 9.46 –24.00 –22.00 1.8509 1040.8 0.02065 150.16 437.01 0.8163 1.9584 2.144 1.255 1.671 723.
220.9 144.0 12.98 137.1 14.78 9.02 –22.00 –20.00 1.9696 1031.7 0.01934 154.45 436.89 0.8328 1.9485 2.165 1.289 1.696 708.
220.4 139.3 13.12 134.6 15.09 8.59 –20.00 –19.00 2.0310 1027.0 0.01873 156.61 436.81 0.8411 1.9436 2.177 1.307 1.709 700.
220.1 137.1 13.18 133.4 15.25 8.37 –19.00 –18.00 2.0938 1022.3 0.01813 158.77 436.70 0.8494 1.9386 2.189 1.326 1.723 692.
219.8 134.8 13.26 132.2 15.42 8.16 –18.00 –17.00 2.1581 1017.6 0.01756 160.95 436.58 0.8576 1.9337 2.201 1.346 1.738 684.
219.5 132.6 13.33 131.0 15.59 7.95 –17.00 –16.00 2.2237 1012.8 0.01700 163.14 436.44 0.8659 1.9287 2.215 1.366 1.753 676.
219.2 130.4 13.40 129.8 15.77 7.74 –16.00 –15.00 2.2908 1008.0 0.01647 165.34 436.27 0.8742 1.9237 2.228 1.388 1.768 668.
218.8 128.3 13.47 128.6 15.95 7.53 –15.00 –14.00 2.3593 1003.1 0.01595 167.55 436.09 0.8825 1.9187 2.243 1.410 1.785 660.
218.5 126.2 13.55 127.4 16.14 7.32 –14.00 –13.00 2.4294 998.1 0.01545 169.78 435.89 0.8908 1.9137 2.258 1.433 1.802 651.
218.1 124.1 13.63 126.2 16.34 7.11 –13.00 –12.00 2.5010 993.1 0.01497 172.01 435.66 0.8991 1.9086 2.273 1.457 1.821 643.
217.7 122.0 13.70 125.0 16.54 6.90 –12.00 –11.00 2.5740 988.1 0.01450 174.26 435.41 0.9074 1.9036 2.290 1.483 1.840 635.
217.4 120.0 13.78 123.8 16.74 6.70 –11.00 –10.00 2.6487 982.9 0.01405 176.52 435.14 0.9157 1.8985 2.307 1.509 1.860 626.
216.9 118.0 13.86 122.5 16.96 6.50 –10.00 –9.00 2.7249 977.7 0.01361 178.80 434.84 0.9240 1.8934 2.325 1.537 1.881 617.
216.5 116.1 13.95 121.3 17.18 6.29 –9.00 –8.00 2.8027 972.5 0.01319 181.09 434.51 0.9324 1.8882 2.345 1.566 1.904 609.
216.1 114.1 14.03 120.1 17.42 6.09 –8.00 –7.00 2.8821 967.1 0.01278 183.39 434.17 0.9408 1.8830 2.365 1.597 1.927 600.
215.6 112.2 14.12 118.9 17.66 5.89 –7.00 –6.00 2.9632 961.7 0.01238 185.71 433.79 0.9491 1.8778 2.386 1.629 1.952 591.
215.2 110.3 14.20 117.7 17.91 5.70 –6.00 –5.00 3.0459 956.2 0.01200 188.05 433.38 0.9576 1.8725 2.408 1.663 1.979 582.
214.7 108.4 14.30 116.5 18.17 5.50 –5.00 –4.00 3.1303 950.6 0.01162 190.40 432.95 0.9660 1.8672 2.432 1.699 2.007 573.
214.2 106.6 14.39 115.3 18.44 5.30 –4.00 –3.00 3.2164 945.0 0.01126 192.77 432.48 0.9744 1.8618 2.457 1.737 2.037 564.
213.7 104.8 14.48 114.1 18.73 5.11 –3.00 –2.00 3.3042 939.2 0.01091 195.16 431.99 0.9829 1.8563 2.484 1.777 2.068 555.
213.1 102.9 14.58 112.9 19.03 4.92 –2.00 –1.00 3.3938 933.4 0.01057 197.57 431.46 0.9914 1.8509 2.512 1.819 2.102 546.
212.6 101.2 14.68 111.6 19.34 4.73 –1.00 0.00 3.4851 927.4 0.01024 200.00 430.89 1.0000 1.8453 2.542 1.865 2.138 536.
212.0 99.4 14.79 110.4 19.67 4.54 0.00 1.00 3.5783 921.4 0.00992 202.45 430.29 1.0086 1.8397 2.574 1.913 2.176 527.
211.5 97.6 14.89 109.2 20.02 4.35 1.00 2.00 3.6733 915.2 0.00961 204.93 429.65 1.0172 1.8340 2.609 1.965 2.218 518.
210.9 95.9 15.00 108.0 20.38 4.17 2.00 3.00 3.7701 909.0 0.00931 207.43 428.97 1.0259 1.8282 2.645 2.020 2.262 508.
210.3 94.2 15.12 106.8 20.76 3.99 3.00 4.00 3.8688 902.6 0.00901 209.95 428.25 1.0346 1.8223 2.685 2.080 2.309 499.
209.6 92.5 15.24 105.5 21.17 3.80 4.00 5.00 3.9695 896.0 0.00872 212.50 427.48 1.0434 1.8163 2.727 2.144 2.360 489.
209.0 90.8 15.36 104.3 21.60 3.62 5.00 6.00 4.0720 889.4 0.00845 215.08 426.67 1.0523 1.8102 2.772 2.213 2.416 480.
208.3 89.1 15.49 103.1 22.06 3.45 6.00 7.00 4.1765 882.6 0.00817 217.69 425.81 1.0612 1.8041 2.822 2.289 2.476 470.
207.6 87.5 15.62 101.8 22.54 3.27 7.00 8.00 4.2831 875.6 0.00791 220.34 424.89 1.0702 1.7977 2.875 2.370 2.541 460.
206.9 85.8 15.76 100.6 23.06 3.10 8.00 9.00 4.3916 868.4 0.00765 223.01 423.92 1.0792 1.7913 2.934 2.460 2.612 451.
206.2 84.2 15.91 99.4 23.61 2.93 9.00 10.00 4.5022 861.1 0.00740 225.73 422.88 1.0884 1.7847 2.998 2.558 2.690 441.
205.4 82.6 16.06 98.1 24.21 2.76 10.00 11.00 4.6149 853.6 0.00715 228.49 421.79 1.0976 1.7779 3.068 2.666 2.776 431.
204.6 80.9 16.22 96.9 24.84 2.59 11.00 12.00 4.7297 845.9 0.00691 231.29 420.62 1.1070 1.7710 3.145 2.786 2.871 421.
203.8 79.3 16.39 95.6 25.53 2.42 12.00 13.00 4.8466 837.9 0.00668 234.13 419.37 1.1165 1.7638 3.232 2.919 2.977 411.
203.0 77.7 16.56 94.4 26.27 2.26 13.00 14.00 4.9658 829.7 0.00645 237.03 418.05 1.1261 1.7565 3.328 3.068 3.095 401.
202.1 76.1 16.75 93.1 27.08 2.10 14.00 15.00 5.0871 821.2 0.00622 239.99 416.64 1.1359 1.7489 3.436 3.237 3.228 391.
201.2 74.4 16.95 91.9 27.96 1.95 15.00 16.00 5.2108 812.4 0.00600 243.01 415.12 1.1458 1.7411 3.558 3.429 3.378 381.
200.3 72.8 17.16 90.6 28.93 1.79 16.00 17.00 5.3368 803.3 0.00578 246.10 413.50 1.1559 1.7329 3.698 3.649 3.550 370.
199.3 71.2 17.39 89.4 29.99 1.64 17.00 18.00 5.4651 793.8 0.00557 249.26 411.76 1.1663 1.7244 3.858 3.905 3.748 360.
198.3 69.5 17.64 88.1 31.16 1.49 18.00 19.00 5.5958 783.8 0.00536 252.52 409.89 1.1769 1.7155 4.044 4.204 3.979 349.
197.2 67.8 17.90 86.9 32.47 1.35 19.00 20.00 5.7291 773.4 0.00515 255.87 407.87 1.1877 1.7062 4.264 4.560 4.252 338.
196.1 66.1 18.19 85.7 33.94 1.20 20.00 21.00 5.8648 762.4 0.00494 259.33 405.67 1.1989 1.6964 4.526 4.990 4.578 326.
194.9 64.4 18.50 84.5 35.61 1.06 21.00 22.00 6.0031 750.8 0.00474 262.93 403.26 1.2105 1.6860 4.846 5.519 4.976 314.
193.6 62.7 18.85 83.4 37.52 0.93 22.00 23.00 6.1440 738.4 0.00453 266.68 400.63 1.2225 1.6749 5.248 6.185 5.472 302.
192.3 60.9 19.23 82.4 39.74 0.80 23.00 24.00 6.2877 725.0 0.00433 270.61 397.70 1.2352 1.6629 5.767 7.049 6.107 288.
190.8 59.0 19.66 81.5 42.35 0.67 24.00 25.00 6.4342 710.5 0.00412 274.78 394.43 1.2485 1.6498 6.467 8.212 6.949 274.
189.1 57.0 20.16 80.8 45.51 0.55 25.00 26.00 6.5837 694.5 0.00391 279.26 390.71 1.2627 1.6353 7.460 9.862 8.121 259.
187.2 55.0 20.73 80.5 49.44 0.44 26.00 27.00 6.7361 676.4 0.00369 284.14 386.39 1.2783 1.6189 8.97 12.38 9.87 243.
185.0 52.8 21.42 80.7 54.56 0.33 27.00 28.00 6.8918 655.3 0.00346 289.62 381.20 1.2958 1.5999 11.55 16.69 12.78 225.
182.1 50.3 22.27 81.9 61.73 0.23 28.00 29.00 7.0509 629.4 0.00320 296.07 374.61 1.3163 1.5763 16.95 25.74 18.63 205.
178.2 47.5 23.41 85.2 73.19 0.13 29.00 30.00 7.2137 593.3 0.00290 304.55 365.13 1.3435 1.5433 35.34 55.82 36.66 177.
171.3 43.8 25.17 95.4 98.02 0.05 30.00 30.98c 7.3773 467.6 0.00214 332.25 332.25 1.4336 1.4336 ∞ ∞ ∞ 0.
0.0 — — ∞ ∞ 0.00 30.98 Temperatures are on the ITS-90 scale a = triple point c = critical point 20.36 2001 ASHRAE Fundamentals Handbook (SI) Fig. 15 Pressure-Enthalpy Diagram for Refrigerant 50 (Methane) Note: The reference states for enthalpy and entropy differ from those in the table.
Thermophysical Properties of Refrigerants 20.37 Refrigerant 50 (Methane) Properties of Saturated Liquid and Saturated Vapor Temp., °C Pres-sure, MPa Density, kg/m3 Liquid Volume, m3/kg Vapor Enthalpy, kJ/kg Entropy, kJ/(kg·K) Specific Heat cp, kJ/(kg·K) cp/cv Velocity of Sound, m/s Viscosity, µPa·s Thermal Cond., mW/(m·K) Surface Tension, mN/m Temp., °C Liquid Vapor Liquid Vapor Liquid Vapor Vapor Liquid Vapor Liquid Vapor Liquid Vapor –182.46a 0.01170 451.5 3.9881 –71.82 472.44 –0.7099 5.2911 3.368 2.110 1.341 1539.
249.1 293.0 3.50 273.9 10.20 18.76 –182.46 –180.00 0.01590 448.2 3.0071 –63.53 477.22 –0.6199 5.1853 3.377 2.118 1.343 1516.
252.2 189.5 3.73 208.3 9.15 18.09 –180.00 –175.00 0.02823 441.4 1.7755 –46.58 486.76 –0.4429 4.9910 3.399 2.137 1.348 1470.
258.0 163.9 3.93 202.1 9.78 16.74 –175.00 –170.00 0.04723 434.5 1.1081 –29.49 495.99 –0.2735 4.8208 3.426 2.162 1.355 1422.
263.4 143.4 4.12 195.5 10.42 15.43 –170.00 –165.00 0.07509 427.4 0.72466 –12.24 504.85 –0.1108 4.6704 3.457 2.192 1.365 1373.
268.3 126.8 4.32 188.7 11.09 14.17 –165.00 –161.48b 0.10133 422.4 0.55054 0.00 510.83 0.0000 4.5746 3.481 2.218 1.373 1338.
271.5 116.8 4.46 183.9 11.58 13.30 –161.48 –160.00 0.11429 420.2 0.49291 5.19 513.28 0.0459 4.5363 3.492 2.229 1.377 1323.
272.7 112.9 4.52 181.8 11.79 12.94 –160.00 –155.00 0.16757 412.7 0.34665 22.82 521.22 0.1973 4.4156 3.533 2.274 1.392 1273.
276.5 101.3 4.73 174.8 12.53 11.75 –155.00 –150.00 0.23784 405.0 0.25077 40.69 528.60 0.3440 4.3058 3.580 2.328 1.412 1221.
279.7 91.4 4.95 167.7 13.30 10.60 –150.00 –145.00 0.32817 397.1 0.18583 58.84 535.34 0.4866 4.2049 3.635 2.393 1.436 1168.
282.3 82.8 5.17 160.5 14.13 9.49 –145.00 –140.00 0.44177 388.8 0.14054 77.30 541.38 0.6257 4.1111 3.701 2.472 1.466 1114.
284.3 75.2 5.40 153.4 15.01 8.43 –140.00 –135.00 0.58192 380.2 0.10813 96.13 546.63 0.7618 4.0228 3.780 2.569 1.504 1059.
285.6 68.6 5.63 146.3 15.96 7.42 –135.00 –130.00 0.75201 371.1 0.08440 115.39 550.97 0.8956 3.9384 3.876 2.690 1.552 1002.
286.2 62.7 5.88 139.2 16.99 6.44 –130.00 –125.00 0.95550 361.6 0.06666 135.17 554.29 1.0276 3.8566 3.996 2.842 1.613 943.
286.2 57.3 6.15 132.1 18.13 5.52 –125.00 –120.00 1.1959 351.4 0.05315 155.58 556.43 1.1585 3.7759 4.148 3.038 1.694 882.
285.4 52.4 6.43 125.0 19.39 4.64 –120.00 –118.00 1.3033 347.2 0.04865 163.94 556.91 1.2108 3.7437 4.220 3.133 1.733 858.
284.8 50.5 6.55 122.1 19.95 4.31 –118.00 –116.00 1.4174 342.8 0.04457 172.44 557.16 1.2631 3.7112 4.302 3.240 1.777 832.
284.2 48.7 6.68 119.3 20.53 3.98 –116.00 –114.00 1.5384 338.3 0.04087 181.08 557.15 1.3155 3.6785 4.393 3.362 1.827 807.
283.4 47.0 6.81 116.4 21.15 3.66 –114.00 –112.00 1.6668 333.6 0.03750 189.88 556.87 1.3681 3.6454 4.497 3.501 1.885 780.
282.5 45.3 6.95 113.5 21.81 3.35 –112.00 –110.00 1.8026 328.8 0.03442 198.85 556.28 1.4209 3.6117 4.615 3.662 1.951 754.
281.4 43.6 7.09 110.7 22.52 3.04 –110.00 –108.00 1.9462 323.7 0.03160 208.03 555.37 1.4741 3.5773 4.751 3.848 2.028 726.
280.2 42.0 7.24 107.8 23.28 2.75 –108.00 –106.00 2.0978 318.4 0.02900 217.42 554.09 1.5278 3.5419 4.910 4.068 2.119 698.
278.9 40.4 7.40 104.9 24.12 2.46 –106.00 –104.00 2.2578 312.9 0.02662 227.06 552.39 1.5821 3.5054 5.097 4.331 2.227 670.
277.4 38.8 7.57 102.0 25.05 2.19 –104.00 –102.00 2.4264 307.1 0.02441 236.99 550.24 1.6372 3.4675 5.322 4.650 2.358 641.
275.7 37.2 7.76 99.0 26.09 1.92 –102.00 –100.00 2.6040 301.0 0.02236 247.25 547.56 1.6934 3.4278 5.596 5.044 2.520 610.
273.9 35.6 7.96 96.1 27.28 1.66 –100.00 –95.00 3.0895 283.6 0.01781 274.80 537.86 1.8408 3.3174 6.654 6.608 3.153 529.
268.4 31.7 8.55 88.6 31.27 1.07 –95.00 –90.00 3.6399 261.7 0.01384 306.71 521.52 2.0062 3.1791 9.10 10.37 4.64 437.
261.3 27.5 9.39 81.2 38.68 0.55 –90.00 –85.00 4.2648 227.5 0.00993 349.75 488.81 2.2241 2.9632 21.88 30.57 12.16 320.
250.2 22.3 10.95 77.5 62.99 0.13 –85.00 –82.59c 4.5992 162.7 0.00615 415.59 415.59 2.5624 2.5624 ∞ ∞ ∞ 0.
0.0 — — ∞ ∞ 0.00 –82.59 Temperatures are on the ITS-90 scale a = triple point b = normal boiling point c = critical point Refrigerant 50 (Methane) Properties of Gas at 0.101 325 MPa (one standard atmosphere) Temp., °C Density, kg/m3 Enthalpy, kJ/kg Entropy, kJ/(kg·K) Cp, kJ/ (kg·K) cp/cv Vel. Sound, m/s Visc., µPa·s Therm Cond., mW/(m·K) Temp., °C Density, kg/m3 Enthalpy, kJ/kg Entropy, kJ/(kg·K) Cp, kJ/ (kg·K) cp/cv Vel. Sound, m/s Visc., µPa·s Therm Cond., mW/(m·K) –161.5a 1.8164 510.83 4.5746 2.218 1.373 271.5 4.46 11.58 50.0 0.6058 966.50 6.8565 2.293 1.295 465.1 11.98 37.86 –160.0 1.7899 514.11 4.6037 2.208 1.371 273.5 4.52 11.74 60.0 0.5875 989.57 6.9269 2.321 1.290 471.4 12.29 39.35 –155.0 1.7064 525.08 4.6986 2.183 1.366 280.3 4.71 12.30 70.0 0.5703 1012.92 6.9959 2.350 1.285 477.6 12.60 40.87 –150.0 1.6310 535.95 4.7887 2.165 1.362 286.9 4.90 12.87 80.0 0.5541 1036.57 7.0638 2.380 1.281 483.7 12.90 42.42 –145.0 1.5626 546.74 4.8746 2.151 1.359 293.2 5.10 13.45 90.0 0.5388 1060.52 7.1307 2.411 1.276 489.6 13.20 44.01 –140.0 1.5000 557.47 4.9567 2.141 1.356 299.4 5.29 14.03 100.0 0.5243 1084.79 7.1967 2.443 1.271 495.5 13.49 45.63 –135.0 1.4425 568.15 5.0355 2.132 1.354 305.4 5.48 14.62 110.0 0.5106 1109.39 7.2617 2.476 1.266 501.2 13.79 47.28 –130.0 1.3894 578.79 5.1112 2.125 1.351 311.2 5.67 15.21 120.0 0.4975 1134.32 7.3259 2.510 1.262 506.8 14.08 48.96 –120.0 1.2947 599.99 5.2543 2.116 1.348 322.5 6.06 16.40 130.0 0.4852 1159.60 7.3894 2.545 1.257 512.3 14.36 50.67 –110.0 1.2124 621.11 5.3879 2.109 1.345 333.3 6.44 17.58 140.0 0.4734 1185.22 7.4522 2.580 1.253 517.7 14.64 52.41 –100.0 1.1403 642.18 5.5133 2.106 1.343 343.8 6.81 18.77 150.0 0.4622 1211.21 7.5143 2.616 1.248 523.1 14.92 54.18 –90.0 1.0764 663.23 5.6315 2.104 1.341 353.8 7.19 19.96 160.0 0.4515 1237.55 7.5759 2.652 1.244 528.3 15.20 55.97 –80.0 1.0194 684.27 5.7433 2.105 1.339 363.5 7.56 21.15 170.0 0.4413 1264.25 7.6368 2.689 1.240 533.5 15.47 57.79 –70.0 0.9683 705.33 5.8496 2.107 1.337 372.9 7.92 22.31 180.0 0.4315 1291.32 7.6972 2.726 1.236 538.6 15.74 59.63 –60.0 0.9221 726.42 5.9509 2.111 1.335 382.0 8.29 23.49 190.0 0.4222 1318.76 7.7571 2.763 1.232 543.7 16.01 61.50 –50.0 0.8802 747.57 6.0479 2.118 1.332 390.7 8.64 24.68 200.0 0.4132 1346.57 7.8165 2.800 1.228 548.7 16.28 63.39 –40.0 0.8419 768.78 6.1409 2.126 1.330 399.2 9.00 25.89 210.0 0.4047 1374.76 7.8755 2.837 1.224 553.6 16.54 65.30 –30.0 0.8069 790.10 6.2304 2.137 1.327 407.5 9.35 27.12 220.0 0.3965 1403.32 7.9340 2.875 1.221 558.5 16.80 67.23 –20.0 0.7747 811.53 6.3168 2.149 1.324 415.4 9.69 28.36 230.0 0.3886 1432.26 7.9921 2.912 1.217 563.4 17.06 69.18 –10.0 0.7450 833.09 6.4003 2.164 1.320 423.2 10.03 29.63 240.0 0.3810 1461.57 8.0498 2.950 1.214 568.2 17.31 71.15 0.0 0.7175 854.82 6.4814 2.181 1.316 430.7 10.37 30.93 250.0 0.3737 1491.26 8.1071 2.988 1.211 572.9 17.56 73.14 10.0 0.6919 876.72 6.5601 2.200 1.312 437.9 10.70 32.25 260.0 0.3667 1521.32 8.1640 3.025 1.207 577.6 17.81 75.14 20.0 0.6682 898.82 6.6368 2.221 1.308 445.0 11.02 33.61 270.0 0.3599 1551.76 8.2205 3.062 1.204 582.3 18.06 77.16 30.0 0.6460 921.14 6.7117 2.243 1.304 451.9 11.35 34.99 280.0 0.3534 1582.57 8.2768 3.100 1.201 586.9 18.31 79.20 40.0 0.6252 943.69 6.7849 2.268 1.299 458.6 11.66 36.41 290.0 0.3471 1613.75 8.3326 3.137 1.198 591.5 18.55 81.25 300.0 0.3411 1645.31 8.3882 3.174 1.196 596.0 18.79 83.31 a = saturated vapor at the normal boiling point 20.38 2001 ASHRAE Fundamentals Handbook (SI) Fig. 16 Pressure-Enthalpy Diagram for Refrigerant 170 (Ethane) Note: The reference states for enthalpy and entropy differ from those in the table.
Thermophysical Properties of Refrigerants 20.39 Refrigerant 170 (Ethane) Properties of Saturated Liquid and Saturated Vapor Temp., °C Pres-sure, MPa Density, kg/m3 Liquid Volume, m3/kg Vapor Enthalpy, kJ/kg Entropy, kJ/(kg·K) Specific Heat cp, kJ/(kg·K) cp/cv Velocity of Sound, m/s Viscosity, µPa·s Thermal Cond., mW/(m·K) Surface Tension, mN/m Temp., °C Liquid Vapor Liquid Vapor Liquid Vapor Vapor Liquid Vapor Liquid Vapor Liquid Vapor –175.00 0.00001 643.2 3650.1 –203.34 384.40 –1.4833 4.5049 2.319 1.182 1.305 1950.
188.2 938.4 3.26 249.7 3.35 30.35 –175.00 –170.00 0.00002 637.8 1349.5 –191.67 390.36 –1.3673 4.2752 2.346 1.188 1.302 1917.
192.7 788.2 3.41 245.5 3.64 29.51 –170.00 –165.00 0.00005 632.4 553.41 –179.91 396.38 –1.2560 4.0725 2.355 1.191 1.300 1882.
197.2 672.9 3.55 241.2 3.93 28.67 –165.00 –160.00 0.00013 626.9 248.15 –168.13 402.45 –1.1496 3.8931 2.354 1.192 1.298 1846.
201.5 582.8 3.70 236.7 4.23 27.83 –160.00 –155.00 0.00027 621.4 120.22 –156.38 408.60 –1.0479 3.7339 2.348 1.189 1.296 1809.
205.8 511.0 3.84 232.0 4.54 26.99 –155.00 –150.00 0.00055 615.8 62.310 –144.66 414.80 –0.9508 3.5921 2.341 1.184 1.294 1772.
210.0 453.0 3.99 227.3 4.85 26.15 –150.00 –145.00 0.00103 610.2 34.255 –132.97 421.05 –0.8577 3.4654 2.335 1.177 1.293 1736.
214.2 405.3 4.14 222.5 5.17 25.31 –145.00 –140.00 0.00186 604.6 19.833 –121.30 427.33 –0.7685 3.3520 2.330 1.169 1.293 1699.
218.2 365.6 4.29 217.6 5.50 24.48 –140.00 –135.00 0.00318 599.0 12.018 –109.66 433.63 –0.6826 3.2500 2.328 1.163 1.292 1661.
222.1 332.0 4.44 212.7 5.83 23.64 –135.00 –130.00 0.00521 593.3 7.5823 –98.01 439.93 –0.5998 3.1580 2.329 1.161 1.291 1624.
225.8 303.4 4.59 207.8 6.18 22.81 –130.00 –125.00 0.00823 587.6 4.9578 –86.35 446.20 –0.5198 3.0749 2.332 1.162 1.291 1587.
229.3 278.7 4.74 202.8 6.53 21.98 –125.00 –120.00 0.01259 581.8 3.3464 –74.67 452.42 –0.4423 2.9994 2.338 1.170 1.290 1550.
232.7 257.1 4.90 197.9 6.89 21.15 –120.00 –115.00 0.01867 576.0 2.3237 –62.96 458.58 –0.3671 2.9306 2.346 1.183 1.290 1512.
235.8 238.2 5.05 192.9 7.27 20.33 –115.00 –110.00 0.02696 570.1 1.6549 –51.19 464.66 –0.2940 2.8678 2.357 1.201 1.290 1475.
238.7 221.4 5.21 188.0 7.66 19.51 –110.00 –105.00 0.03799 564.1 1.2057 –39.37 470.64 –0.2227 2.8104 2.369 1.225 1.290 1437.
241.4 206.4 5.36 183.0 8.06 18.69 –105.00 –100.00 0.05236 558.1 0.89645 –27.47 476.52 –0.1531 2.7576 2.384 1.254 1.291 1400.
243.8 192.9 5.52 178.2 8.47 17.88 –100.00 –95.00 0.07075 552.0 0.67885 –15.48 482.28 –0.0851 2.7090 2.401 1.288 1.293 1362.
246.0 180.6 5.68 173.3 8.90 17.07 –95.00 –90.00 0.09388 545.7 0.52261 –3.41 487.91 –0.0185 2.6641 2.421 1.326 1.295 1324.
248.0 169.5 5.84 168.5 9.35 16.26 –90.00 –88.60b 0.10133 544.0 0.48698 0.00 489.47 0.0000 2.6522 2.426 1.337 1.296 1313.
248.6 166.6 5.88 167.1 9.48 16.04 –88.60 –85.00 0.12253 539.4 0.40836 8.78 493.40 0.0469 2.6226 2.442 1.368 1.298 1286.
249.8 159.3 6.00 163.7 9.81 15.46 –85.00 –80.00 0.15753 533.0 0.32340 21.08 498.75 0.1111 2.5841 2.466 1.412 1.303 1247.
251.3 149.9 6.16 159.0 10.30 14.67 –80.00 –78.00 0.17350 530.4 0.29561 26.04 500.85 0.1365 2.5695 2.476 1.431 1.305 1232.
251.8 146.4 6.23 157.1 10.50 14.35 –78.00 –76.00 0.19068 527.7 0.27071 31.02 502.92 0.1617 2.5553 2.486 1.451 1.308 1217.
252.3 143.0 6.30 155.2 10.70 14.04 –76.00 –74.00 0.20914 525.1 0.24835 36.02 504.96 0.1868 2.5415 2.497 1.471 1.310 1201.
252.8 139.6 6.36 153.3 10.91 13.72 –74.00 –72.00 0.22892 522.4 0.22823 41.04 506.97 0.2117 2.5280 2.508 1.491 1.313 1186.
253.2 136.4 6.43 151.5 11.12 13.41 –72.00 –70.00 0.25010 519.7 0.21009 46.09 508.96 0.2364 2.5149 2.520 1.512 1.316 1171.
253.6 133.2 6.50 149.7 11.33 13.10 –70.00 –68.00 0.27272 517.0 0.19369 51.16 510.92 0.2611 2.5021 2.532 1.533 1.319 1155.
253.9 130.2 6.56 147.8 11.55 12.78 –68.00 –66.00 0.29687 514.3 0.17885 56.26 512.85 0.2856 2.4897 2.545 1.555 1.323 1140.
254.2 127.2 6.63 146.0 11.77 12.47 –66.00 –64.00 0.32258 511.5 0.16538 61.39 514.74 0.3099 2.4776 2.558 1.577 1.327 1124.
254.4 124.3 6.70 144.2 12.00 12.16 –64.00 –62.00 0.34994 508.7 0.15314 66.54 516.61 0.3342 2.4657 2.571 1.600 1.331 1108.
254.6 121.5 6.77 142.4 12.23 11.86 –62.00 –60.00 0.37900 505.9 0.14200 71.72 518.44 0.3583 2.4542 2.585 1.624 1.335 1093.
254.8 118.8 6.84 140.6 12.46 11.55 –60.00 –58.00 0.40983 503.1 0.13183 76.93 520.24 0.3824 2.4429 2.600 1.648 1.340 1077.
254.9 116.1 6.91 138.8 12.70 11.24 –58.00 –56.00 0.44250 500.2 0.12255 82.17 522.00 0.4063 2.4318 2.615 1.673 1.345 1061.
254.9 113.5 6.99 137.0 12.95 10.94 –56.00 –54.00 0.47707 497.3 0.11405 87.44 523.73 0.4302 2.4210 2.631 1.698 1.350 1046.
254.9 111.0 7.06 135.2 13.20 10.64 –54.00 –52.00 0.51360 494.4 0.10626 92.74 525.42 0.4539 2.4104 2.647 1.724 1.356 1030.
254.9 108.5 7.13 133.5 13.45 10.33 –52.00 –50.00 0.55216 491.5 0.09912 98.07 527.07 0.4776 2.4000 2.664 1.751 1.362 1014.
254.8 106.1 7.21 131.7 13.71 10.03 –50.00 –48.00 0.59283 488.5 0.09254 103.44 528.68 0.5012 2.3899 2.682 1.778 1.369 998.
254.7 103.7 7.28 130.0 13.98 9.73 –48.00 –46.00 0.63567 485.4 0.08649 108.85 530.25 0.5247 2.3799 2.700 1.807 1.376 982.
254.5 101.4 7.36 128.2 14.25 9.44 –46.00 –44.00 0.68074 482.4 0.08091 114.29 531.78 0.5481 2.3700 2.719 1.836 1.383 966.
254.3 99.1 7.44 126.5 14.53 9.14 –44.00 –42.00 0.72813 479.3 0.07576 119.78 533.26 0.5715 2.3603 2.739 1.867 1.391 950.
254.0 96.9 7.51 124.8 14.82 8.85 –42.00 –40.00 0.77789 476.1 0.07100 125.30 534.70 0.5949 2.3508 2.760 1.898 1.400 934.
253.6 94.7 7.60 123.1 15.11 8.55 –40.00 –38.00 0.83010 473.0 0.06658 130.86 536.08 0.6182 2.3414 2.782 1.931 1.409 918.
253.2 92.6 7.68 121.4 15.41 8.26 –38.00 –36.00 0.88483 469.7 0.06250 136.47 537.42 0.6414 2.3321 2.805 1.964 1.419 902.
252.8 90.5 7.76 119.7 15.72 7.97 –36.00 –34.00 0.94215 466.5 0.05870 142.12 538.70 0.6646 2.3229 2.829 2.000 1.430 885.
252.3 88.5 7.85 118.1 16.04 7.68 –34.00 –32.00 1.0021 463.2 0.05517 147.82 539.93 0.6878 2.3138 2.855 2.036 1.441 869.
251.7 86.5 7.93 116.4 16.37 7.40 –32.00 –30.00 1.0649 459.8 0.05189 153.56 541.10 0.7110 2.3048 2.881 2.075 1.453 852.
251.1 84.5 8.02 114.8 16.71 7.11 –30.00 –28.00 1.1304 456.4 0.04883 159.36 542.20 0.7341 2.2958 2.909 2.115 1.467 836.
250.4 82.6 8.11 113.1 17.05 6.83 –28.00 –26.00 1.1988 452.9 0.04598 165.21 543.25 0.7573 2.2869 2.939 2.158 1.481 819.
249.6 80.7 8.21 111.5 17.41 6.55 –26.00 –24.00 1.2702 449.4 0.04331 171.11 544.22 0.7804 2.2780 2.970 2.202 1.496 802.
248.8 78.9 8.30 109.9 17.78 6.27 –24.00 –22.00 1.3446 445.8 0.04082 177.07 545.13 0.8036 2.2691 3.004 2.250 1.513 785.
247.9 77.0 8.40 108.2 18.17 6.00 –22.00 –20.00 1.4222 442.1 0.03849 183.09 545.96 0.8268 2.2602 3.039 2.300 1.531 768.
247.0 75.2 8.50 106.6 18.57 5.72 –20.00 –18.00 1.5029 438.4 0.03630 189.18 546.71 0.8500 2.2513 3.077 2.353 1.550 751.
246.0 73.5 8.61 105.0 18.98 5.45 –18.00 –16.00 1.5870 434.6 0.03425 195.33 547.38 0.8733 2.2423 3.117 2.410 1.572 734.
244.9 71.7 8.72 103.4 19.41 5.18 –16.00 –14.00 1.6744 430.7 0.03233 201.56 547.95 0.8966 2.2333 3.161 2.471 1.595 716.
243.8 70.0 8.83 101.9 19.86 4.92 –14.00 –12.00 1.7652 426.7 0.03052 207.86 548.43 0.9200 2.2241 3.208 2.537 1.621 699.
242.6 68.3 8.95 100.3 20.33 4.65 –12.00 –10.00 1.8596 422.6 0.02881 214.24 548.81 0.9435 2.2149 3.259 2.608 1.649 681.
241.3 66.6 9.07 98.7 20.83 4.39 –10.00 –8.00 1.9575 418.5 0.02720 220.71 549.07 0.9671 2.2055 3.314 2.686 1.680 663.
239.9 65.0 9.20 97.1 21.35 4.13 –8.00 –6.00 2.0592 414.2 0.02568 227.27 549.22 0.9908 2.1960 3.375 2.771 1.715 645.
238.5 63.3 9.33 95.6 21.89 3.88 –6.00 –4.00 2.1647 409.8 0.02425 233.92 549.24 1.0147 2.1862 3.441 2.864 1.754 626.
236.9 61.7 9.47 94.0 22.48 3.63 –4.00 –2.00 2.2741 405.3 0.02289 240.69 549.12 1.0388 2.1762 3.515 2.967 1.798 608.
235.3 60.1 9.62 92.4 23.09 3.38 –2.00 0.00 2.3875 400.6 0.02161 247.57 548.84 1.0630 2.1660 3.597 3.082 1.847 589.
233.6 58.4 9.77 90.9 23.75 3.13 0.00 5.00 2.6891 388.1 0.01867 265.35 547.38 1.1247 2.1387 3.852 3.440 2.003 540.
229.0 54.4 10.20 87.0 25.62 2.54 5.00 10.00 3.0181 374.2 0.01607 284.17 544.56 1.1887 2.1083 4.217 3.953 2.232 488.
223.8 50.4 10.72 83.1 27.94 1.97 10.00 15.00 3.3763 358.3 0.01373 304.41 539.84 1.2561 2.0732 4.787 4.759 2.598 432.
217.9 46.3 11.35 79.1 30.98 1.43 15.00 20.00 3.7660 339.3 0.01158 326.76 532.31 1.3292 2.0304 5.801 6.226 3.267 371.
211.3 41.9 12.18 75.2 35.30 0.93 20.00 25.00 4.1901 314.8 0.00951 352.66 519.87 1.4124 1.9732 8.108 9.751 4.869 304.
203.6 37.0 13.39 71.9 42.53 0.48 25.00 30.00 4.6535 276.7 0.00725 387.16 494.49 1.5219 1.8760 20.52 29.62 13.66 226.
194.0 30.6 15.72 74.3 61.97 0.11 30.00 32.18c 4.8718 206.6 0.00484 438.26 438.26 1.6868 1.6868 ∞ ∞ ∞ 0.
0.0 — — ∞ ∞ 0.00 32.18 Temperatures are on the ITS-90 scale b = normal boiling point c = critical point 20.40 2001 ASHRAE Fundamentals Handbook (SI) Fig. 17 Pressure-Enthalpy Diagram for Refrigerant 290 (Propane) Note: The reference states for enthalpy and entropy differ from those in the table.
Thermophysical Properties of Refrigerants 20.41 Refrigerant 290 (Propane) Properties of Saturated Liquid and Saturated Vapor Temp., °C Pres-sure, MPa Density, kg/m3 Liquid Volume, m3/kg Vapor Enthalpy, kJ/kg Entropy, kJ/(kg·K) Specific Heat cp, kJ/(kg·K) cp/cv Velocity of Sound, m/s Viscosity, µPa·s Thermal Cond., mW/(m·K) Surface Tension, mN/m Temp., °C Liquid Vapor Liquid Vapor Liquid Vapor Vapor Liquid Vapor Liquid Vapor Liquid Vapor –150.00 0.00001 694.9 4340.8 –123.36 402.32 –0.6872 3.5819 1.959 1.020 1.227 1869.
168.8 1352.
3.55 193.1 3.68 31.84 –150.00 –140.00 0.00003 684.8 869.55 –103.72 412.68 –0.5338 3.3449 1.971 1.052 1.219 1799.
174.9 992.0 3.80 188.0 4.28 30.29 –140.00 –130.00 0.00012 674.8 224.81 –83.94 423.33 –0.3906 3.1534 1.985 1.083 1.211 1731.
180.8 766.4 4.05 182.4 4.90 28.76 –130.00 –120.00 0.00041 664.6 71.174 –64.01 434.26 –0.2560 2.9978 2.001 1.115 1.204 1664.
186.4 614.9 4.31 176.7 5.55 27.24 –120.00 –110.00 0.00116 654.4 26.518 –43.90 445.45 –0.1288 2.8708 2.021 1.148 1.198 1598.
191.7 507.3 4.56 170.7 6.23 25.73 –110.00 –100.00 0.00289 644.1 11.281 –23.58 456.87 –0.0080 2.7670 2.043 1.183 1.193 1532.
196.8 427.5 4.82 164.6 6.94 24.23 –100.00 –90.00 0.00642 633.6 5.3500 –3.02 468.49 0.1075 2.6820 2.069 1.221 1.189 1467.
201.5 365.9 5.08 158.5 7.67 22.74 –90.00 –80.00 0.01300 623.1 2.7761 17.82 480.25 0.2182 2.6125 2.098 1.263 1.186 1402.
205.9 317.0 5.34 152.3 8.43 21.27 –80.00 –70.00 0.02433 612.4 1.5522 38.98 492.14 0.3249 2.5557 2.132 1.308 1.183 1338.
209.9 277.3 5.60 146.1 9.22 19.81 –70.00 –60.00 0.04258 601.4 0.92393 60.50 504.09 0.4282 2.5094 2.169 1.358 1.182 1273.
213.4 244.4 5.85 140.0 10.04 18.36 –60.00 –50.00 0.07043 590.3 0.57957 82.43 516.07 0.5285 2.4718 2.211 1.412 1.183 1209.
216.3 216.8 6.11 134.1 10.88 16.94 –50.00 –40.00 0.11095 578.8 0.37998 104.81 528.03 0.6263 2.4416 2.258 1.472 1.185 1145.
218.7 193.2 6.36 128.2 11.76 15.53 –40.00 –42.08b 0.10129 581.2 0.41358 100.11 525.54 0.6061 2.4473 2.248 1.459 1.185 1158.
218.3 197.8 6.31 129.4 11.57 15.82 –42.08 –38.00 0.12088 576.5 0.35085 109.35 530.41 0.6456 2.4363 2.268 1.484 1.186 1132.
219.1 188.9 6.41 127.0 11.94 15.25 –38.00 –36.00 0.13148 574.1 0.32442 113.90 532.79 0.6648 2.4312 2.278 1.497 1.187 1119.
219.5 184.7 6.47 125.9 12.12 14.98 –36.00 –34.00 0.14279 571.8 0.30039 118.48 535.17 0.6839 2.4264 2.288 1.510 1.188 1106.
219.8 180.7 6.52 124.7 12.30 14.70 –34.00 –32.00 0.15483 569.4 0.27852 123.08 537.54 0.7030 2.4218 2.299 1.523 1.189 1093.
220.1 176.7 6.57 123.6 12.48 14.42 –32.00 –30.00 0.16764 567.0 0.25858 127.70 539.91 0.7220 2.4173 2.310 1.536 1.190 1080.
220.4 172.9 6.62 122.4 12.67 14.15 –30.00 –28.00 0.18125 564.6 0.24036 132.34 542.28 0.7409 2.4131 2.321 1.550 1.191 1067.
220.7 169.2 6.67 121.3 12.86 13.87 –28.00 –26.00 0.19569 562.2 0.22370 137.01 544.63 0.7598 2.4091 2.332 1.564 1.192 1055.
220.9 165.5 6.73 120.2 13.05 13.60 –26.00 –24.00 0.21100 559.8 0.20844 141.70 546.99 0.7785 2.4053 2.344 1.578 1.193 1042.
221.1 162.0 6.78 119.1 13.24 13.32 –24.00 –22.00 0.22720 557.3 0.19444 146.41 549.33 0.7973 2.4016 2.356 1.593 1.195 1029.
221.3 158.6 6.83 117.9 13.44 13.05 –22.00 –20.00 0.24433 554.9 0.18158 151.15 551.67 0.8159 2.3981 2.368 1.607 1.196 1016.
221.4 155.2 6.89 116.8 13.63 12.78 –20.00 –18.00 0.26242 552.4 0.16975 155.91 554.00 0.8346 2.3948 2.380 1.622 1.198 1003.
221.6 152.0 6.94 115.7 13.83 12.51 –18.00 –16.00 0.28151 549.9 0.15885 160.70 556.32 0.8531 2.3916 2.393 1.637 1.199 990.
221.6 148.8 6.99 114.6 14.03 12.24 –16.00 –14.00 0.30163 547.3 0.14880 165.51 558.63 0.8716 2.3886 2.406 1.653 1.201 977.
221.7 145.7 7.05 113.6 14.23 11.97 –14.00 –12.00 0.32281 544.8 0.13952 170.36 560.94 0.8901 2.3857 2.420 1.669 1.203 964.
221.7 142.7 7.10 112.5 14.44 11.71 –12.00 –10.00 0.34510 542.2 0.13093 175.23 563.23 0.9085 2.3830 2.433 1.685 1.206 951.
221.7 139.7 7.16 111.4 14.65 11.44 –10.00 –8.00 0.36852 539.6 0.12299 180.12 565.51 0.9269 2.3804 2.447 1.702 1.208 938.
221.6 136.8 7.22 110.3 14.86 11.18 –8.00 –6.00 0.39312 537.0 0.11562 185.05 567.78 0.9452 2.3779 2.461 1.718 1.210 925.
221.5 134.0 7.27 109.3 15.08 10.91 –6.00 –4.00 0.41892 534.3 0.10879 190.00 570.04 0.9635 2.3755 2.476 1.736 1.213 912.
221.4 131.3 7.33 108.2 15.30 10.65 –4.00 –2.00 0.44597 531.7 0.10244 194.98 572.29 0.9818 2.3733 2.491 1.753 1.216 899.
221.2 128.6 7.39 107.2 15.52 10.39 –2.00 0.00 0.47430 529.0 0.09653 200.00 574.52 1.0000 2.3711 2.507 1.771 1.219 886.
221.0 125.9 7.45 106.1 15.74 10.13 0.00 2.00 0.50394 526.2 0.09103 205.05 576.74 1.0182 2.3691 2.523 1.790 1.222 873.
220.8 123.4 7.51 105.1 15.97 9.87 2.00 4.00 0.53495 523.5 0.08591 210.12 578.94 1.0364 2.3671 2.539 1.809 1.225 859.
220.5 120.8 7.57 104.1 16.21 9.61 4.00 6.00 0.56734 520.7 0.08113 215.23 581.13 1.0545 2.3653 2.556 1.828 1.229 846.
220.2 118.4 7.63 103.1 16.44 9.35 6.00 8.00 0.60117 517.9 0.07666 220.38 583.29 1.0727 2.3635 2.573 1.848 1.232 833.
219.8 116.0 7.69 102.1 16.69 9.10 8.00 10.00 0.63646 515.0 0.07249 225.56 585.44 1.0908 2.3618 2.591 1.869 1.236 820.
219.4 113.6 7.75 101.0 16.93 8.84 10.00 12.00 0.67327 512.1 0.06858 230.77 587.57 1.1089 2.3601 2.609 1.890 1.241 807.
219.0 111.3 7.82 100.1 17.18 8.59 12.00 14.00 0.71162 509.2 0.06492 236.02 589.68 1.1269 2.3586 2.628 1.912 1.245 794.
218.5 109.0 7.88 99.1 17.44 8.34 14.00 16.00 0.75156 506.3 0.06149 241.31 591.77 1.1450 2.3570 2.647 1.934 1.250 780.
218.0 106.8 7.95 98.1 17.70 8.09 16.00 18.00 0.79312 503.3 0.05827 246.63 593.83 1.1631 2.3556 2.667 1.957 1.255 767.
217.4 104.6 8.02 97.1 17.97 7.84 18.00 20.00 0.83635 500.2 0.05524 251.99 595.87 1.1811 2.3542 2.688 1.981 1.261 754.
216.8 102.4 8.09 96.1 18.25 7.59 20.00 22.00 0.88128 497.2 0.05240 257.40 597.88 1.1992 2.3528 2.709 2.006 1.267 741.
216.1 100.3 8.16 95.2 18.53 7.35 22.00 24.00 0.92796 494.0 0.04972 262.84 599.87 1.2173 2.3515 2.731 2.031 1.273 727.
215.4 98.2 8.23 94.2 18.82 7.10 24.00 26.00 0.97643 490.9 0.04720 268.33 601.82 1.2353 2.3501 2.754 2.058 1.280 714.
214.6 96.2 8.31 93.3 19.11 6.86 26.00 28.00 1.0267 487.7 0.04483 273.86 603.75 1.2534 2.3488 2.778 2.085 1.287 700.
213.8 94.2 8.38 92.3 19.42 6.62 28.00 30.00 1.0789 484.4 0.04259 279.43 605.64 1.2715 2.3475 2.803 2.114 1.295 687.
213.0 92.2 8.46 91.4 19.73 6.38 30.00 32.00 1.1330 481.1 0.04047 285.05 607.49 1.2896 2.3463 2.829 2.144 1.303 674.
212.1 90.3 8.54 90.5 20.05 6.14 32.00 34.00 1.1890 477.7 0.03847 290.72 609.31 1.3078 2.3450 2.856 2.176 1.312 660.
211.1 88.3 8.63 89.6 20.38 5.91 34.00 36.00 1.2471 474.3 0.03658 296.44 611.09 1.3259 2.3437 2.884 2.209 1.322 647.
210.1 86.4 8.71 88.7 20.72 5.67 36.00 38.00 1.3072 470.8 0.03479 302.21 612.82 1.3441 2.3424 2.913 2.243 1.333 633.
209.0 84.6 8.80 87.8 21.08 5.44 38.00 40.00 1.3694 467.3 0.03310 308.03 614.51 1.3623 2.3410 2.944 2.280 1.344 619.
207.9 82.7 8.89 86.9 21.44 5.21 40.00 42.00 1.4337 463.6 0.03149 313.91 616.15 1.3806 2.3396 2.976 2.319 1.356 606.
206.7 80.9 8.99 86.0 21.82 4.98 42.00 44.00 1.5002 459.9 0.02996 319.84 617.74 1.3989 2.3382 3.010 2.359 1.369 592.
205.4 79.1 9.09 85.1 22.21 4.75 44.00 46.00 1.5689 456.2 0.02851 325.84 619.27 1.4173 2.3367 3.046 2.403 1.384 578.
204.1 77.3 9.19 84.2 22.61 4.53 46.00 48.00 1.6400 452.3 0.02714 331.89 620.75 1.4357 2.3351 3.083 2.449 1.400 565.
202.8 75.6 9.29 83.3 23.04 4.30 48.00 50.00 1.7133 448.4 0.02583 338.01 622.15 1.4542 2.3335 3.124 2.499 1.417 551.
201.3 73.9 9.40 82.5 23.47 4.08 50.00 55.00 1.9073 438.2 0.02282 353.60 625.36 1.5007 2.3289 3.236 2.640 1.468 516.
197.5 69.6 9.70 80.3 24.66 3.54 55.00 60.00 2.1170 427.2 0.02014 369.68 628.05 1.5479 2.3234 3.372 2.814 1.534 480.
193.2 65.4 10.04 78.2 26.00 3.02 60.00 65.00 2.3433 415.5 0.01775 386.31 630.08 1.5958 2.3167 3.541 3.036 1.623 444.
188.4 61.2 10.42 76.1 27.55 2.51 65.00 70.00 2.5872 402.7 0.01559 403.60 631.30 1.6448 2.3084 3.761 3.333 1.745 407.
183.2 57.1 10.87 74.1 29.37 2.02 70.00 75.00 2.8499 388.6 0.01363 421.73 631.44 1.6954 2.2977 4.067 3.755 1.925 368.
177.4 52.9 11.41 72.0 31.60 1.56 75.00 80.00 3.1327 372.6 0.01182 440.98 630.08 1.7481 2.2836 4.537 4.413 2.212 328.
171.0 48.6 12.09 70.0 34.47 1.12 80.00 85.00 3.4371 353.6 0.01010 461.89 626.46 1.8046 2.2640 5.383 5.601 2.743 284.
163.9 44.0 12.97 68.1 38.49 0.71 85.00 90.00 3.7651 329.1 0.00841 485.83 618.71 1.8683 2.2341 7.475 8.506 4.062 236.
155.7 38.9 14.28 66.8 45.13 0.35 90.00 95.00 4.1206 286.4 0.00638 519.30 598.40 1.9566 2.1714 23.21 29.14 13.52 177.
146.1 31.4 17.03 73.4 64.87 0.06 95.00 96.70c 4.2477 220.5 0.00454 559.09 559.09 2.0630 2.0630 ∞ ∞ ∞ 0.
0.0 — — ∞ ∞ 0.00 96.70 Temperatures have been converted from the IPTS-68 scale of the original formulation to the ITS-90 scale b = normal boiling point c = critical point 20.42 2001 ASHRAE Fundamentals Handbook (SI) Fig. 18 Pressure-Enthalpy Diagram for Refrigerant 600 (n-Butane) Note: The reference states for enthalpy and entropy differ from those in the table.
Thermophysical Properties of Refrigerants 20.43 Refrigerant 600 (n-Butane) Properties of Saturated Liquid and Saturated Vapor Temp., °C Pres-sure, MPa Density, kg/m3 Liquid Volume, m3/kg Vapor Enthalpy, kJ/kg Entropy, kJ/(kg·K) Specific Heat cp, kJ/(kg·K) cp/cv Velocity of Sound, m/s Viscosity, µPa·s Thermal Cond., mW/(m·K) Surface Tension, mN/m Temp., °C Liquid Vapor Liquid Vapor Liquid Vapor Vapor Liquid Vapor Liquid Vapor Liquid Vapor –100.00 0.00017 699.2 149.43 –12.42 450.99 0.0377 2.7143 2.010 1.229 1.132 1513.
167.4 785.7 4.51 170.6 6.33 28.03 –100.00 –95.00 0.00028 694.5 91.091 –2.37 457.16 0.0949 2.6746 2.012 1.245 1.130 1493.
169.6 710.6 4.63 166.8 6.61 27.33 –95.00 –90.00 0.00046 689.8 57.322 7.70 463.40 0.1506 2.6389 2.015 1.260 1.129 1473.
171.8 646.6 4.74 163.2 6.91 26.64 –90.00 –85.00 0.00072 685.1 37.133 17.78 469.70 0.2050 2.6070 2.020 1.276 1.127 1452.
174.0 591.3 4.85 159.8 7.21 25.95 –85.00 –80.00 0.00112 680.4 24.700 27.90 476.07 0.2580 2.5785 2.027 1.293 1.125 1431.
176.1 543.3 4.96 156.5 7.52 25.26 –80.00 –75.00 0.00168 675.6 16.834 38.05 482.51 0.3099 2.5531 2.036 1.310 1.124 1409.
178.1 501.3 5.08 153.3 7.83 24.58 –75.00 –70.00 0.00247 670.8 11.731 48.26 489.01 0.3608 2.5305 2.046 1.327 1.123 1386.
180.1 464.3 5.19 150.3 8.16 23.90 –70.00 –65.00 0.00356 666.0 8.3446 58.52 495.57 0.4107 2.5105 2.058 1.346 1.121 1363.
182.1 431.5 5.31 147.4 8.49 23.22 –65.00 –60.00 0.00502 661.2 6.0489 68.85 502.19 0.4597 2.4929 2.072 1.364 1.120 1339.
184.0 402.3 5.42 144.6 8.83 22.55 –60.00 –55.00 0.00695 656.4 4.4616 79.25 508.87 0.5080 2.4774 2.087 1.384 1.119 1315.
185.8 376.0 5.54 142.0 9.18 21.88 –55.00 –50.00 0.00947 651.5 3.3443 89.73 515.61 0.5554 2.4640 2.104 1.404 1.118 1291.
187.5 352.4 5.66 139.4 9.55 21.22 –50.00 –45.00 0.01271 646.6 2.5441 100.30 522.40 0.6022 2.4524 2.121 1.425 1.118 1266.
189.2 331.0 5.78 136.8 9.92 20.56 –45.00 –40.00 0.01680 641.7 1.9622 110.95 529.24 0.6484 2.4425 2.140 1.446 1.117 1241.
190.8 311.5 5.90 134.4 10.30 19.90 –40.00 –35.00 0.02192 636.7 1.5327 121.70 536.12 0.6940 2.4342 2.159 1.468 1.117 1215.
192.3 293.8 6.02 132.0 10.69 19.25 –35.00 –30.00 0.02823 631.7 1.2114 132.56 543.05 0.7391 2.4274 2.179 1.491 1.116 1189.
193.7 277.5 6.14 129.7 11.09 18.60 –30.00 –25.00 0.03594 626.6 0.96795 143.52 550.02 0.7836 2.4218 2.200 1.515 1.116 1163.
195.1 262.5 6.27 127.5 11.51 17.96 –25.00 –20.00 0.04526 621.5 0.78128 154.58 557.02 0.8277 2.4175 2.222 1.539 1.116 1137.
196.3 248.6 6.39 125.3 11.94 17.33 –20.00 –15.00 0.05641 616.3 0.63659 165.76 564.06 0.8714 2.4143 2.244 1.565 1.117 1111.
197.4 235.8 6.52 123.1 12.38 16.69 –15.00 –10.00 0.06965 611.1 0.52323 177.05 571.12 0.9146 2.4121 2.267 1.591 1.117 1085.
198.4 223.9 6.65 121.0 12.83 16.07 –10.00 –5.00 0.08522 605.9 0.43356 188.46 578.21 0.9575 2.4109 2.291 1.618 1.118 1058.
199.3 212.8 6.78 118.9 13.29 15.44 –5.00 0.00 0.10340 600.5 0.36197 200.00 585.31 1.0000 2.4106 2.316 1.646 1.119 1031.
200.1 202.4 6.91 116.8 13.77 14.83 0.00 –0.53b 0.10132 601.1 0.36890 198.76 584.55 0.9955 2.4106 2.313 1.643 1.119 1034.
200.1 203.5 6.90 117.0 13.72 14.89 –0.53 2.00 0.11147 598.3 0.33744 204.65 588.16 1.0169 2.4107 2.326 1.657 1.120 1021.
200.4 198.5 6.97 116.0 13.97 14.58 2.00 4.00 0.12002 596.2 0.31490 209.32 591.01 1.0338 2.4110 2.336 1.669 1.120 1010.
200.7 194.6 7.02 115.2 14.17 14.34 4.00 6.00 0.12906 594.0 0.29419 214.01 593.86 1.0506 2.4113 2.346 1.681 1.121 999.
200.9 190.9 7.07 114.4 14.37 14.09 6.00 8.00 0.13863 591.8 0.27512 218.72 596.71 1.0673 2.4118 2.357 1.693 1.122 988.
201.1 187.2 7.13 113.6 14.57 13.85 8.00 10.00 0.14874 589.6 0.25754 223.45 599.56 1.0840 2.4124 2.367 1.705 1.123 977.
201.3 183.6 7.18 112.8 14.77 13.61 10.00 12.00 0.15941 587.4 0.24132 228.21 602.42 1.1007 2.4130 2.378 1.717 1.123 967.
201.5 180.1 7.24 112.0 14.98 13.37 12.00 14.00 0.17066 585.2 0.22633 232.98 605.28 1.1173 2.4138 2.388 1.729 1.124 956.
201.7 176.7 7.30 111.2 15.19 13.13 14.00 16.00 0.18251 582.9 0.21246 237.78 608.13 1.1339 2.4147 2.399 1.742 1.125 945.
201.8 173.3 7.35 110.4 15.41 12.89 16.00 18.00 0.19499 580.6 0.19962 242.60 610.99 1.1505 2.4157 2.410 1.755 1.126 934.
201.9 170.1 7.41 109.6 15.62 12.65 18.00 20.00 0.20810 578.4 0.18772 247.44 613.84 1.1670 2.4168 2.421 1.768 1.127 923.
202.0 166.9 7.47 108.9 15.84 12.41 20.00 22.00 0.22188 576.1 0.17668 252.31 616.70 1.1834 2.4180 2.432 1.781 1.128 912.
202.0 163.8 7.53 108.1 16.06 12.17 22.00 24.00 0.23634 573.8 0.16642 257.19 619.55 1.1998 2.4192 2.444 1.794 1.129 901.
202.1 160.7 7.58 107.3 16.28 11.94 24.00 26.00 0.25151 571.4 0.15688 262.11 622.40 1.2162 2.4206 2.455 1.808 1.130 891.
202.1 157.7 7.64 106.5 16.51 11.70 26.00 28.00 0.26741 569.1 0.14800 267.04 625.25 1.2325 2.4220 2.467 1.821 1.132 880.
202.1 154.8 7.70 105.8 16.74 11.47 28.00 30.00 0.28406 566.8 0.13972 272.00 628.10 1.2489 2.4235 2.479 1.835 1.133 869.
202.0 151.9 7.76 105.0 16.97 11.24 30.00 32.00 0.30149 564.4 0.13201 276.98 630.95 1.2651 2.4251 2.491 1.849 1.134 858.
202.0 149.1 7.82 104.2 17.21 11.01 32.00 34.00 0.31971 562.0 0.12480 281.98 633.79 1.2814 2.4267 2.503 1.864 1.136 847.
201.9 146.4 7.88 103.5 17.44 10.78 34.00 36.00 0.33875 559.6 0.11807 287.02 636.63 1.2976 2.4285 2.515 1.878 1.138 836.
201.8 143.7 7.94 102.7 17.69 10.55 36.00 38.00 0.35863 557.1 0.11177 292.07 639.46 1.3138 2.4302 2.528 1.893 1.139 825.
201.6 141.0 8.01 101.9 17.93 10.32 38.00 40.00 0.37937 554.7 0.10588 297.15 642.29 1.3299 2.4321 2.541 1.908 1.141 814.
201.4 138.4 8.07 101.2 18.18 10.09 40.00 42.00 0.40101 552.2 0.10036 302.26 645.12 1.3461 2.4340 2.553 1.924 1.143 803.
201.2 135.9 8.13 100.4 18.43 9.86 42.00 44.00 0.42355 549.7 0.09518 307.39 647.94 1.3622 2.4359 2.567 1.939 1.145 792.
201.0 133.4 8.20 99.7 18.68 9.64 44.00 46.00 0.44703 547.2 0.09032 312.55 650.76 1.3783 2.4379 2.580 1.955 1.147 781.
200.7 130.9 8.26 98.9 18.94 9.41 46.00 48.00 0.47147 544.7 0.08575 317.74 653.57 1.3943 2.4400 2.593 1.971 1.149 770.
200.5 128.5 8.33 98.2 19.20 9.19 48.00 50.00 0.49690 542.1 0.08146 322.95 656.37 1.4104 2.4421 2.607 1.988 1.151 759.
200.1 126.2 8.39 97.4 19.46 8.97 50.00 55.00 0.56492 535.6 0.07181 336.10 663.34 1.4504 2.4475 2.643 2.031 1.158 731.
199.2 120.4 8.56 95.6 20.14 8.42 55.00 60.00 0.63963 528.9 0.06349 349.43 670.25 1.4902 2.4532 2.681 2.076 1.165 703.
198.0 114.9 8.74 93.7 20.84 7.87 60.00 65.00 0.72142 522.0 0.05628 362.95 677.09 1.5301 2.4590 2.720 2.124 1.173 675.
196.7 109.5 8.92 91.9 21.56 7.34 65.00 70.00 0.81068 515.0 0.05000 376.66 683.84 1.5698 2.4649 2.762 2.175 1.183 647.
195.1 104.4 9.11 90.1 22.31 6.81 70.00 75.00 0.90785 507.7 0.04452 390.59 690.50 1.6095 2.4709 2.807 2.230 1.194 619.
193.3 99.5 9.30 88.3 23.09 6.29 75.00 80.00 1.0133 500.1 0.03970 404.72 697.04 1.6492 2.4769 2.856 2.289 1.207 591.
191.2 94.7 9.50 86.5 23.90 5.78 80.00 85.00 1.1276 492.3 0.03546 419.09 703.45 1.6890 2.4829 2.909 2.354 1.222 563.
188.8 90.0 9.72 84.7 24.74 5.28 85.00 90.00 1.2511 484.1 0.03170 433.71 709.69 1.7288 2.4887 2.967 2.426 1.240 534.
186.1 85.5 9.94 83.0 25.61 4.79 90.00 95.00 1.3842 475.6 0.02837 448.59 715.74 1.7688 2.4944 3.031 2.506 1.262 505.
183.1 81.1 10.17 81.3 26.53 4.31 95.00 100.00 1.5276 466.7 0.02539 463.76 721.55 1.8089 2.4997 3.104 2.598 1.288 477.
179.8 76.8 10.42 79.5 27.49 3.84 100.00 105.00 1.6816 457.4 0.02272 479.25 727.09 1.8492 2.5046 3.188 2.705 1.320 448.
176.1 72.5 10.69 77.8 28.50 3.38 105.00 110.00 1.8470 447.4 0.02032 495.11 732.28 1.8899 2.5089 3.286 2.831 1.360 418.
172.1 68.4 10.98 76.2 29.58 2.93 110.00 115.00 2.0242 436.8 0.01814 511.37 737.06 1.9310 2.5124 3.406 2.986 1.412 388.
167.5 64.2 11.30 74.5 30.73 2.50 115.00 120.00 2.2140 425.4 0.01616 528.11 741.31 1.9727 2.5150 3.555 3.183 1.481 358.
162.5 60.1 11.65 72.8 31.99 2.08 120.00 125.00 2.4172 413.0 0.01434 545.43 744.89 2.0153 2.5162 3.751 3.443 1.576 327.
157.0 56.0 12.04 71.0 33.41 1.68 125.00 130.00 2.6346 399.1 0.01266 563.48 747.56 2.0590 2.5156 4.027 3.812 1.716 294.
150.8 51.8 12.50 69.3 35.11 1.30 130.00 135.00 2.8673 383.3 0.01108 582.52 748.96 2.1045 2.5122 4.452 4.386 1.939 261.
143.9 47.5 13.06 67.7 37.36 0.94 135.00 140.00 3.1166 364.5 0.00956 603.05 748.39 2.1528 2.5046 5.221 5.419 2.352 225.
136.2 42.9 13.77 66.5 40.78 0.60 140.00 145.00 3.3844 339.8 0.00802 626.27 744.23 2.2068 2.4889 7.110 7.919 3.368 186.
127.4 37.7 14.81 67.0 47.18 0.31 145.00 150.00 3.6740 297.5 0.00618 657.56 729.69 2.2790 2.4495 19.83 23.68 9.84 141.
117.5 30.4 16.95 74.7 65.81 0.06 150.00 152.01c 3.7960 227.8 0.00439 695.70 695.70 2.3679 2.3679 ∞ ∞ ∞ 0.
0.0 — — ∞ ∞ 0.00 152.01 Temperatures have been converted from the IPTS-68 scale of the original formulation to the ITS-90 scale b = normal boiling point c = critical point 20.44 2001 ASHRAE Fundamentals Handbook (SI) Fig. 19 Pressure-Enthalpy Diagram for Refrigerant 600a (Isobutane) Note: The reference states for enthalpy and entropy differ from those in the table.
Thermophysical Properties of Refrigerants 20.45 Refrigerant 600a (Isobutane) Properties of Saturated Liquid and Saturated Vapor Temp., °C Pres-sure, MPa Density, kg/m3 Liquid Volume, m3/kg Vapor Enthalpy, kJ/kg Entropy, kJ/(kg·K) Specific Heat cp, kJ/(kg·K) cp/cv Velocity of Sound, m/s Viscosity, µPa·s Thermal Cond., mW/(m·K) Surface Tension, mN/m Temp., °C Liquid Vapor Liquid Vapor Liquid Vapor Vapor Liquid Vapor Liquid Vapor Liquid Vapor –100.00 0.00036 683.8 68.449 –7.88 430.08 0.0607 2.5902 1.881 1.131 1.145 1446.
168.3 932.0 4.49 138.3 6.03 25.53 –100.00 –95.00 0.00059 679.0 42.929 1.57 435.75 0.1145 2.5518 1.898 1.152 1.143 1422.
170.5 833.4 4.60 138.3 6.31 24.88 –95.00 –90.00 0.00094 674.1 27.744 11.11 441.50 0.1672 2.5174 1.916 1.172 1.140 1399.
172.6 749.8 4.71 138.0 6.61 24.24 –90.00 –85.00 0.00146 669.2 18.429 20.73 447.34 0.2191 2.4866 1.934 1.192 1.138 1375.
174.7 678.4 4.83 137.5 6.91 23.60 –85.00 –80.00 0.00220 664.3 12.552 30.44 453.25 0.2700 2.4592 1.952 1.213 1.136 1351.
176.7 616.8 4.94 136.7 7.22 22.96 –80.00 –75.00 0.00323 659.4 8.7478 40.25 459.24 0.3202 2.4348 1.970 1.235 1.134 1328.
178.6 563.3 5.06 135.7 7.54 22.32 –75.00 –70.00 0.00465 654.4 6.2265 50.15 465.30 0.3695 2.4132 1.989 1.256 1.132 1304.
180.6 516.5 5.18 134.6 7.87 21.69 –70.00 –65.00 0.00655 649.5 4.5186 60.15 471.43 0.4181 2.3941 2.009 1.279 1.131 1279.
182.4 475.3 5.29 133.2 8.21 21.05 –65.00 –60.00 0.00906 644.5 3.3382 70.25 477.62 0.4660 2.3773 2.028 1.302 1.130 1255.
184.2 438.8 5.41 131.7 8.56 20.42 –60.00 –55.00 0.01233 639.4 2.5070 80.44 483.88 0.5133 2.3627 2.049 1.325 1.128 1230.
185.9 406.3 5.53 130.1 8.93 19.79 –55.00 –50.00 0.01650 634.3 1.9115 90.74 490.19 0.5599 2.3501 2.069 1.349 1.128 1206.
187.5 377.3 5.65 128.3 9.30 19.16 –50.00 –45.00 0.02176 629.2 1.4780 101.15 496.56 0.6060 2.3392 2.090 1.374 1.127 1181.
189.0 351.2 5.78 126.4 9.68 18.53 –45.00 –40.00 0.02831 624.0 1.1577 111.66 502.98 0.6515 2.3300 2.112 1.400 1.126 1155.
190.4 327.6 5.90 124.4 10.08 17.91 –40.00 –35.00 0.03636 618.8 0.91762 122.28 509.45 0.6966 2.3223 2.134 1.426 1.126 1130.
191.8 306.2 6.03 122.4 10.49 17.29 –35.00 –30.00 0.04614 613.6 0.73542 133.02 515.96 0.7411 2.3161 2.156 1.453 1.126 1104.
193.0 286.7 6.15 120.3 10.91 16.67 –30.00 –25.00 0.05791 608.2 0.59546 143.87 522.51 0.7852 2.3111 2.180 1.482 1.126 1079.
194.2 269.0 6.28 118.1 11.34 16.06 –25.00 –20.00 0.07194 602.8 0.48671 154.84 529.09 0.8289 2.3073 2.204 1.511 1.127 1053.
195.2 252.7 6.42 115.9 11.79 15.45 –20.00 –15.00 0.08850 597.4 0.40134 165.94 535.71 0.8722 2.3046 2.228 1.540 1.128 1026.
196.1 237.7 6.55 113.6 12.25 14.84 –15.00 –11.61b 0.10132 593.7 0.35378 173.53 540.22 0.9013 2.3034 2.245 1.561 1.128 1008.
196.6 228.3 6.64 112.1 12.57 14.43 –11.61 –10.00 0.10789 591.9 0.33363 177.16 542.36 0.9151 2.3029 2.253 1.571 1.129 1000.
196.9 223.9 6.69 111.3 12.73 14.23 –10.00 –5.00 0.13042 586.3 0.27944 188.51 549.03 0.9577 2.3022 2.279 1.603 1.130 973.
197.5 211.2 6.83 109.1 13.22 13.63 –5.00 0.00 0.15642 580.6 0.23569 200.00 555.73 1.0000 2.3023 2.306 1.636 1.132 947.
198.0 199.3 6.97 106.8 13.72 13.03 0.00 2.00 0.16786 578.3 0.22057 204.63 558.41 1.0168 2.3026 2.317 1.649 1.133 936.
198.2 194.8 7.03 105.9 13.93 12.79 2.00 4.00 0.17993 576.0 0.20663 209.29 561.09 1.0336 2.3030 2.328 1.663 1.133 925.
198.3 190.5 7.09 105.0 14.14 12.56 4.00 6.00 0.19265 573.6 0.19376 213.97 563.78 1.0504 2.3035 2.340 1.677 1.134 914.
198.4 186.2 7.15 104.0 14.35 12.32 6.00 8.00 0.20605 571.3 0.18186 218.67 566.47 1.0670 2.3041 2.351 1.691 1.135 903.
198.5 182.1 7.20 103.1 14.56 12.08 8.00 10.00 0.22014 568.9 0.17085 223.39 569.16 1.0837 2.3048 2.363 1.705 1.136 893.
198.5 178.1 7.27 102.2 14.78 11.85 10.00 12.00 0.23495 566.6 0.16065 228.14 571.85 1.1003 2.3057 2.374 1.719 1.137 882.
198.6 174.2 7.33 101.3 15.00 11.61 12.00 14.00 0.25050 564.2 0.15118 232.91 574.54 1.1169 2.3066 2.386 1.734 1.139 871.
198.6 170.3 7.39 100.4 15.22 11.38 14.00 16.00 0.26682 561.8 0.14239 237.71 577.23 1.1335 2.3076 2.398 1.748 1.140 860.
198.6 166.6 7.45 99.5 15.45 11.15 16.00 18.00 0.28392 559.3 0.13422 242.53 579.92 1.1500 2.3088 2.411 1.763 1.141 849.
198.5 163.0 7.51 98.7 15.67 10.91 18.00 20.00 0.30184 556.9 0.12661 247.38 582.61 1.1664 2.3100 2.423 1.779 1.143 838.
198.4 159.5 7.58 97.8 15.90 10.68 20.00 22.00 0.32058 554.4 0.11952 252.25 585.30 1.1829 2.3113 2.436 1.794 1.144 827.
198.3 156.1 7.64 96.9 16.13 10.45 22.00 24.00 0.34019 551.9 0.11292 257.15 587.99 1.1993 2.3127 2.449 1.810 1.146 816.
198.2 152.7 7.71 96.0 16.37 10.22 24.00 26.00 0.36068 549.4 0.10675 262.07 590.67 1.2157 2.3141 2.462 1.826 1.148 805.
198.0 149.4 7.77 95.1 16.61 9.99 26.00 28.00 0.38208 546.9 0.10099 267.02 593.35 1.2321 2.3156 2.475 1.842 1.149 794.
197.8 146.3 7.84 94.3 16.85 9.76 28.00 30.00 0.40441 544.3 0.09560 272.00 596.03 1.2484 2.3173 2.489 1.859 1.151 783.
197.6 143.1 7.91 93.4 17.09 9.54 30.00 32.00 0.42770 541.7 0.09055 277.00 598.71 1.2647 2.3189 2.502 1.875 1.153 772.
197.4 140.1 7.98 92.6 17.34 9.31 32.00 34.00 0.45197 539.1 0.08583 282.03 601.38 1.2810 2.3207 2.517 1.892 1.156 761.
197.1 137.1 8.04 91.7 17.58 9.08 34.00 36.00 0.47725 536.5 0.08139 287.09 604.04 1.2972 2.3225 2.531 1.910 1.158 749.
196.7 134.2 8.12 90.9 17.84 8.86 36.00 38.00 0.50356 533.9 0.07723 292.18 606.70 1.3135 2.3243 2.546 1.928 1.160 738.
196.4 131.4 8.19 90.0 18.09 8.64 38.00 40.00 0.53093 531.2 0.07332 297.30 609.36 1.3297 2.3262 2.560 1.946 1.163 727.
196.0 128.6 8.26 89.2 18.35 8.41 40.00 42.00 0.55938 528.5 0.06965 302.44 612.00 1.3459 2.3282 2.576 1.964 1.166 716.
195.5 125.9 8.33 88.4 18.61 8.19 42.00 44.00 0.58894 525.7 0.06619 307.62 614.64 1.3621 2.3302 2.591 1.983 1.168 705.
195.1 123.2 8.41 87.6 18.87 7.97 44.00 46.00 0.61964 523.0 0.06293 312.83 617.27 1.3783 2.3322 2.607 2.003 1.172 694.
194.6 120.6 8.48 86.7 19.14 7.75 46.00 48.00 0.65150 520.2 0.05986 318.07 619.89 1.3945 2.3343 2.624 2.023 1.175 682.
194.0 118.0 8.56 85.9 19.41 7.53 48.00 50.00 0.68455 517.3 0.05696 323.34 622.50 1.4107 2.3364 2.640 2.043 1.178 671.
193.4 115.5 8.64 85.1 19.69 7.32 50.00 55.00 0.77255 510.1 0.05041 336.66 628.97 1.4510 2.3418 2.684 2.096 1.188 643.
191.8 109.5 8.84 83.2 20.39 6.78 55.00 60.00 0.86860 502.6 0.04470 350.20 635.34 1.4914 2.3473 2.731 2.154 1.200 614.
189.9 103.7 9.06 81.2 21.11 6.25 60.00 65.00 0.97312 494.9 0.03972 363.96 641.61 1.5318 2.3528 2.782 2.216 1.213 586.
187.7 98.2 9.28 79.4 21.86 5.72 65.00 70.00 1.0866 486.9 0.03534 377.96 647.73 1.5722 2.3583 2.837 2.285 1.229 557.
185.2 92.9 9.52 77.5 22.64 5.21 70.00 75.00 1.2095 478.5 0.03148 392.23 653.69 1.6127 2.3637 2.898 2.362 1.249 528.
182.4 87.9 9.77 75.7 23.44 4.71 75.00 80.00 1.3424 469.8 0.02805 406.77 659.44 1.6534 2.3688 2.966 2.448 1.272 499.
179.2 83.0 10.04 73.9 24.28 4.21 80.00 85.00 1.4858 460.6 0.02500 421.62 664.94 1.6943 2.3736 3.043 2.548 1.302 469.
175.7 78.3 10.34 72.1 25.16 3.73 85.00 90.00 1.6402 450.9 0.02228 436.81 670.13 1.7355 2.3779 3.132 2.666 1.338 440.
171.8 73.7 10.65 70.4 26.09 3.26 90.00 95.00 1.8063 440.6 0.01983 452.39 674.95 1.7770 2.3815 3.237 2.810 1.386 409.
167.5 69.2 11.00 68.6 27.09 2.80 95.00 100.00 1.9848 429.6 0.01761 468.40 679.28 1.8191 2.3842 3.367 2.989 1.448 378.
162.8 64.8 11.39 66.9 28.17 2.36 100.00 105.00 2.1764 417.6 0.01560 484.92 683.02 1.8619 2.3857 3.531 3.222 1.533 347.
157.5 60.5 11.83 65.2 29.39 1.93 105.00 110.00 2.3820 404.4 0.01375 502.08 685.96 1.9057 2.3856 3.752 3.544 1.655 314.
151.8 56.2 12.35 63.4 30.83 1.53 110.00 115.00 2.6025 389.6 0.01204 520.05 687.84 1.9508 2.3831 4.073 4.020 1.842 280.
145.5 51.8 12.96 61.7 32.68 1.14 115.00 120.00 2.8390 372.4 0.01043 539.15 688.20 1.9981 2.3772 4.600 4.808 2.158 244.
138.6 47.3 13.72 60.3 35.31 0.78 120.00 125.00 3.0928 351.2 0.00887 560.06 686.09 2.0492 2.3657 5.665 6.384 2.805 205.
130.9 42.4 14.75 59.6 39.54 0.46 125.00 130.00 3.3656 321.2 0.00724 584.77 679.00 2.1088 2.3425 9.16 11.29 4.85 162.
122.0 36.5 16.38 60.9 47.81 0.18 130.00 134.70c 3.6400 224.4 0.00446 638.92 638.92 2.2397 2.2397 ∞ ∞ ∞ 0.
0.0 — — ∞ ∞ 0.00 134.70 Temperatures have been converted from the IPTS-68 scale of the original formulation to the ITS-90 scale b = normal boiling point c = critical point 20.46 2001 ASHRAE Fundamentals Handbook (SI) Fig. 20 Pressure-Enthalpy Diagram for Refrigerant 1150 (Ethylene) Note: The reference states for enthalpy and entropy differ from those in the table.
Thermophysical Properties of Refrigerants 20.47 Refrigerant 1150 (Ethylene) Properties of Saturated Liquid and Saturated Vapor Temp., °C Pres-sure, MPa Density, kg/m3 Liquid Volume, m3/kg Vapor Enthalpy, kJ/kg Entropy, kJ/(kg·K) Specific Heat cp, kJ/(kg·K) cp/cv Velocity of Sound, m/s Viscosity, µPa·s Thermal Cond., mW/(m·K) Surface Tension, mN/m Temp., °C Liquid Vapor Liquid Vapor Liquid Vapor Vapor Liquid Vapor Liquid Vapor Liquid Vapor –169.16a 0.00012 654.6 252.64 –158.09 409.42 –1.1789 4.2787 2.429 1.187 1.333 1767.
202.7 1246.
3.51 270.0 5.96 28.14 –169.16 –165.00 0.00025 649.3 129.69 –147.97 414.35 –1.0835 4.1160 2.432 1.188 1.333 1740.
206.7 598.1 1.64 264.9 6.62 27.31 –165.00 –160.00 0.00053 643.0 62.719 –135.81 420.24 –0.9736 3.9408 2.432 1.189 1.333 1707.
211.3 514.5 2.47 258.1 6.59 26.33 –160.00 –155.00 0.00107 636.6 32.557 –123.66 426.12 –0.8685 3.7848 2.429 1.191 1.334 1673.
215.9 448.6 3.13 251.3 6.71 25.35 –155.00 –150.00 0.00203 630.1 17.974 –111.52 431.97 –0.7679 3.6454 2.424 1.194 1.334 1639.
220.3 395.8 3.66 244.5 6.92 24.38 –150.00 –145.00 0.00362 623.6 10.472 –99.41 437.78 –0.6715 3.5204 2.419 1.198 1.335 1604.
224.5 352.7 4.09 237.9 7.18 23.42 –145.00 –140.00 0.00614 617.1 6.3951 –87.33 443.54 –0.5790 3.4080 2.414 1.203 1.336 1569.
228.6 317.2 4.44 231.2 7.47 22.47 –140.00 –135.00 0.01000 610.5 4.0710 –75.27 449.24 –0.4902 3.3065 2.409 1.210 1.337 1534.
232.5 287.5 4.74 224.7 7.78 21.53 –135.00 –130.00 0.01565 603.8 2.6880 –63.22 454.85 –0.4046 3.2145 2.406 1.218 1.339 1498.
236.3 262.3 5.00 218.3 8.09 20.59 –130.00 –125.00 0.02368 597.1 1.8331 –51.19 460.37 –0.3220 3.1310 2.404 1.228 1.341 1463.
239.8 240.7 5.23 212.0 8.39 19.66 –125.00 –120.00 0.03474 590.3 1.2865 –39.16 465.79 –0.2423 3.0548 2.404 1.240 1.344 1427.
243.2 222.0 5.44 205.8 8.70 18.74 –120.00 –115.00 0.04961 583.4 0.92612 –27.12 471.08 –0.1651 2.9850 2.406 1.254 1.348 1390.
246.3 205.7 5.63 199.7 9.00 17.83 –115.00 –110.00 0.06911 576.5 0.68196 –15.06 476.22 –0.0903 2.9210 2.409 1.271 1.353 1354.
249.2 191.3 5.81 193.7 9.30 16.93 –110.00 –105.00 0.09420 569.4 0.51239 –2.97 481.21 –0.0176 2.8619 2.416 1.290 1.358 1316.
251.8 178.5 5.99 187.9 9.61 16.04 –105.00 –103.77b 0.10133 567.7 0.47899 0.00 482.41 0.0000 2.8481 2.418 1.295 1.360 1307.
252.4 175.6 6.03 186.5 9.68 15.83 –103.77 –100.00 0.12585 562.2 0.39198 9.16 486.03 0.0532 2.8073 2.424 1.312 1.365 1279.
254.2 167.0 6.16 182.2 9.92 15.16 –100.00 –98.00 0.14059 559.3 0.35377 14.02 487.90 0.0810 2.7866 2.429 1.321 1.368 1264.
255.1 162.7 6.23 179.9 10.04 14.82 –98.00 –96.00 0.15662 556.4 0.32007 18.90 489.74 0.1085 2.7664 2.434 1.331 1.371 1249.
255.9 158.6 6.30 177.7 10.17 14.47 –96.00 –94.00 0.17402 553.5 0.29026 23.79 491.55 0.1358 2.7468 2.439 1.342 1.375 1233.
256.7 154.7 6.37 175.5 10.30 14.12 –94.00 –92.00 0.19285 550.5 0.26382 28.69 493.32 0.1628 2.7277 2.445 1.353 1.379 1218.
257.4 150.9 6.44 173.3 10.44 13.78 –92.00 –90.00 0.21320 547.5 0.24030 33.60 495.06 0.1896 2.7092 2.451 1.365 1.383 1203.
258.1 147.2 6.51 171.2 10.57 13.44 –90.00 –88.00 0.23514 544.5 0.21933 38.53 496.76 0.2161 2.6911 2.458 1.377 1.387 1187.
258.7 143.6 6.58 169.0 10.71 13.10 –88.00 –86.00 0.25874 541.4 0.20058 43.48 498.43 0.2425 2.6734 2.465 1.391 1.392 1172.
259.3 140.2 6.65 166.9 10.85 12.76 –86.00 –84.00 0.28409 538.4 0.18378 48.44 500.05 0.2686 2.6562 2.473 1.404 1.397 1156.
259.9 136.9 6.72 164.8 11.00 12.42 –84.00 –82.00 0.31127 535.3 0.16869 53.41 501.64 0.2945 2.6394 2.482 1.419 1.402 1140.
260.4 133.7 6.80 162.7 11.15 12.09 –82.00 –80.00 0.34034 532.2 0.15510 58.41 503.18 0.3202 2.6229 2.491 1.434 1.408 1125.
260.8 130.6 6.87 160.7 11.30 11.75 –80.00 –78.00 0.37141 529.0 0.14284 63.43 504.68 0.3457 2.6069 2.501 1.450 1.414 1109.
261.2 127.5 6.95 158.6 11.46 11.42 –78.00 –76.00 0.40454 525.8 0.13176 68.47 506.14 0.3711 2.5911 2.512 1.467 1.420 1093.
261.5 124.6 7.02 156.6 11.62 11.09 –76.00 –74.00 0.43982 522.6 0.12172 73.53 507.55 0.3963 2.5757 2.524 1.484 1.427 1077.
261.8 121.7 7.10 154.6 11.79 10.77 –74.00 –72.00 0.47733 519.3 0.11260 78.61 508.91 0.4214 2.5606 2.536 1.503 1.435 1061.
262.1 118.9 7.18 152.6 11.96 10.44 –72.00 –70.00 0.51716 516.1 0.10431 83.72 510.23 0.4463 2.5457 2.549 1.522 1.443 1044.
262.3 116.2 7.26 150.6 12.14 10.12 –70.00 –68.00 0.55939 512.7 0.09675 88.86 511.49 0.4710 2.5311 2.563 1.543 1.451 1028.
262.4 113.6 7.34 148.6 12.33 9.80 –68.00 –66.00 0.60411 509.4 0.08985 94.03 512.70 0.4957 2.5168 2.578 1.565 1.460 1012.
262.5 111.0 7.42 146.7 12.52 9.48 –66.00 –64.00 0.65141 506.0 0.08354 99.23 513.85 0.5202 2.5026 2.594 1.588 1.470 995.
262.5 108.5 7.51 144.7 12.72 9.16 –64.00 –62.00 0.70136 502.5 0.07776 104.46 514.95 0.5446 2.4887 2.611 1.612 1.480 978.
262.4 106.0 7.60 142.8 12.92 8.85 –62.00 –60.00 0.75406 499.0 0.07246 109.72 515.99 0.5689 2.4749 2.629 1.638 1.491 962.
262.3 103.6 7.68 140.9 13.14 8.53 –60.00 –58.00 0.80960 495.5 0.06758 115.02 516.97 0.5932 2.4614 2.648 1.665 1.503 945.
262.2 101.2 7.78 139.0 13.36 8.23 –58.00 –56.00 0.86807 491.9 0.06310 120.36 517.88 0.6173 2.4479 2.668 1.694 1.516 928.
262.0 98.9 7.87 137.1 13.59 7.92 –56.00 –54.00 0.92955 488.2 0.05896 125.74 518.73 0.6414 2.4346 2.690 1.725 1.529 911.
261.7 96.6 7.96 135.2 13.84 7.61 –54.00 –52.00 0.99414 484.5 0.05514 131.16 519.51 0.6654 2.4214 2.714 1.757 1.544 894.
261.3 94.4 8.06 133.4 14.09 7.31 –52.00 –50.00 1.0619 480.8 0.05161 136.62 520.21 0.6894 2.4083 2.739 1.792 1.560 876.
260.9 92.2 8.16 131.5 14.35 7.01 –50.00 –48.00 1.1330 476.9 0.04834 142.14 520.84 0.7133 2.3953 2.766 1.829 1.577 859.
260.5 90.1 8.27 129.7 14.63 6.71 –48.00 –46.00 1.2075 473.0 0.04530 147.70 521.39 0.7372 2.3824 2.795 1.869 1.596 841.
259.9 87.9 8.38 127.8 14.91 6.42 –46.00 –44.00 1.2854 469.1 0.04249 153.32 521.86 0.7611 2.3694 2.826 1.912 1.616 823.
259.3 85.8 8.49 126.0 15.22 6.13 –44.00 –42.00 1.3669 465.0 0.03987 158.99 522.24 0.7850 2.3565 2.859 1.958 1.638 806.
258.7 83.8 8.60 124.2 15.53 5.84 –42.00 –40.00 1.4521 460.9 0.03743 164.73 522.53 0.8089 2.3436 2.895 2.007 1.662 787.
257.9 81.8 8.72 122.3 15.86 5.55 –40.00 –38.00 1.5410 456.7 0.03515 170.52 522.72 0.8328 2.3306 2.934 2.061 1.688 769.
257.1 79.7 8.84 120.5 16.21 5.27 –38.00 –36.00 1.6339 452.4 0.03303 176.39 522.81 0.8568 2.3176 2.976 2.119 1.717 751.
256.3 77.8 8.97 118.7 16.58 4.99 –36.00 –34.00 1.7307 448.0 0.03105 182.33 522.79 0.8809 2.3045 3.022 2.182 1.749 732.
255.3 75.8 9.10 116.8 16.97 4.71 –34.00 –32.00 1.8315 443.5 0.02919 188.35 522.65 0.9050 2.2913 3.072 2.252 1.784 714.
254.3 73.8 9.24 115.0 17.38 4.44 –32.00 –30.00 1.9366 438.9 0.02745 194.45 522.38 0.9292 2.2779 3.127 2.328 1.823 695.
253.2 71.9 9.39 113.2 17.82 4.17 –30.00 –28.00 2.0459 434.1 0.02581 200.65 521.98 0.9535 2.2643 3.188 2.413 1.866 676.
252.0 70.0 9.54 111.3 18.29 3.90 –28.00 –26.00 2.1596 429.2 0.02427 206.94 521.44 0.9780 2.2505 3.254 2.507 1.914 656.
250.7 68.1 9.69 109.5 18.78 3.64 –26.00 –24.00 2.2779 424.2 0.02283 213.34 520.74 1.0027 2.2365 3.329 2.612 1.969 637.
249.4 66.2 9.86 107.6 19.32 3.38 –24.00 –22.00 2.4008 419.0 0.02146 219.85 519.86 1.0276 2.2221 3.412 2.731 2.031 617.
248.0 64.3 10.04 105.8 19.90 3.13 –22.00 –20.00 2.5284 413.6 0.02017 226.49 518.80 1.0527 2.2074 3.506 2.866 2.101 596.
246.4 62.4 10.22 103.9 20.53 2.88 –20.00 –18.00 2.6610 408.0 0.01895 233.27 517.53 1.0781 2.1922 3.612 3.021 2.182 576.
244.8 60.5 10.42 102.0 21.22 2.63 –18.00 –16.00 2.7985 402.2 0.01780 240.21 516.04 1.1039 2.1765 3.735 3.200 2.277 555.
243.1 58.7 10.63 100.0 21.98 2.39 –16.00 –14.00 2.9412 396.1 0.01670 247.32 514.28 1.1300 2.1602 3.878 3.411 2.387 533.
241.3 56.8 10.86 98.1 22.83 2.15 –14.00 –12.00 3.0893 389.8 0.01565 254.63 512.24 1.1567 2.1431 4.046 3.661 2.520 511.
239.3 54.9 11.10 96.1 23.78 1.92 –12.00 –10.00 3.2428 383.0 0.01466 262.17 509.87 1.1839 2.1252 4.247 3.965 2.680 489.
237.3 52.9 11.36 94.1 24.88 1.70 –10.00 –8.00 3.4019 375.9 0.01370 269.96 507.10 1.2118 2.1062 4.493 4.339 2.877 465.
235.1 51.0 11.65 92.0 26.15 1.48 –8.00 –6.00 3.5669 368.3 0.01278 278.07 503.89 1.2406 2.0859 4.802 4.815 3.127 441.
232.8 49.0 11.98 89.9 27.66 1.26 –6.00 –4.00 3.7379 360.1 0.01189 286.56 500.12 1.2705 2.0640 5.201 5.438 3.452 416.
230.3 46.9 12.34 87.8 29.52 1.06 –4.00 –2.00 3.9152 351.1 0.01103 295.51 495.66 1.3018 2.0400 5.739 6.291 3.894 390.
227.6 44.8 12.75 85.6 31.88 0.86 –2.00 0.00 4.0990 341.2 0.01018 305.06 490.32 1.3350 2.0132 6.507 7.526 4.526 362.
224.6 42.6 13.23 83.5 35.04 0.67 0.00 5.00 4.5896 308.8 0.00802 333.52 470.25 1.4327 1.9243 11.68 16.10 8.72 283.
214.9 36.2 15.01 79.8 52.44 0.25 5.00 9.20c 5.0418 214.2 0.00467 399.43 399.43 1.6614 1.6614 ∞ ∞ ∞ 0.
0.0 — — ∞ ∞ 0.00 9.20 Temperatures are on the ITS-90 scale a = triple point b = normal boiling point c = critical point 20.48 2001 ASHRAE Fundamentals Handbook (SI) Fig. 21 Pressure-Enthalpy Diagram for Refrigerant 1270 (Propylene) Note: The reference states for enthalpy and entropy differ from those in the table.
Thermophysical Properties of Refrigerants 20.49 Refrigerant 1270 (Propylene) Properties of Saturated Liquid and Saturated Vapor Temp., °C Pres-sure, MPa Density, kg/m3 Liquid Volume, m3/kg Vapor Enthalpy, kJ/kg Entropy, kJ/(kg·K) Specific Heat cp, kJ/(kg·K) cp/cv Velocity of Sound, m/s Surface Tension, mN/m Temp., °C Liquid Vapor Liquid Vapor Liquid Vapor Vapor Liquid Vapor –140.00 0.00005 716.4 584.26 –98.40 429.52 –0.5068 3.4585 1.893 1.015 1.242 1754.
180.7 31.76 –140.00 –130.00 0.00018 704.9 156.30 –79.22 439.77 –0.3679 3.2579 1.939 1.039 1.235 1683.
186.9 30.05 –130.00 –120.00 0.00059 693.6 50.921 –59.65 450.23 –0.2357 3.0938 1.974 1.064 1.229 1617.
192.7 28.36 –120.00 –110.00 0.00166 682.2 19.445 –39.76 460.89 –0.1100 2.9589 2.002 1.090 1.223 1554.
198.3 26.69 –110.00 –100.00 0.00403 670.8 8.4527 –19.61 471.72 0.0099 2.8477 2.026 1.118 1.218 1493.
203.5 25.04 –100.00 –90.00 0.00881 659.4 4.0858 0.78 482.67 0.1243 2.7556 2.050 1.149 1.214 1432.
208.4 23.41 –90.00 –80.00 0.01753 647.8 2.1562 21.41 493.71 0.2339 2.6793 2.075 1.183 1.211 1371.
212.9 21.80 –80.00 –70.00 0.03230 636.1 1.2238 42.32 504.78 0.3393 2.6159 2.103 1.220 1.209 1309.
216.9 20.22 –70.00 –60.00 0.05575 624.1 0.73807 63.53 515.82 0.4411 2.5632 2.135 1.262 1.209 1247.
220.4 18.67 –60.00 –50.00 0.09107 611.9 0.46834 85.09 526.78 0.5397 2.5191 2.171 1.309 1.211 1184.
223.4 17.14 –50.00 –48.00 0.09987 609.5 0.42991 89.45 528.95 0.5591 2.5112 2.179 1.319 1.211 1171.
223.9 16.83 –48.00 –47.69b 0.10129 609.1 0.42431 90.12 529.29 0.5621 2.5100 2.180 1.321 1.211 1169.
224.0 16.79 –47.69 –46.00 0.10931 607.0 0.39528 93.82 531.12 0.5783 2.5036 2.187 1.330 1.212 1158.
224.4 16.53 –46.00 –44.00 0.11944 604.5 0.36402 98.21 533.28 0.5975 2.4962 2.195 1.340 1.213 1145.
224.8 16.23 –44.00 –42.00 0.13029 601.9 0.33575 102.62 535.44 0.6166 2.4891 2.203 1.351 1.214 1133.
225.3 15.93 –42.00 –40.00 0.14187 599.4 0.31014 107.05 537.58 0.6356 2.4822 2.212 1.362 1.215 1120.
225.7 15.63 –40.00 –38.00 0.15424 596.9 0.28688 111.49 539.72 0.6545 2.4756 2.221 1.373 1.216 1107.
226.1 15.34 –38.00 –36.00 0.16743 594.3 0.26574 115.95 541.85 0.6733 2.4692 2.230 1.385 1.217 1094.
226.4 15.04 –36.00 –34.00 0.18147 591.7 0.24648 120.43 543.96 0.6920 2.4630 2.239 1.396 1.218 1081.
226.7 14.75 –34.00 –32.00 0.19639 589.1 0.22892 124.94 546.07 0.7106 2.4570 2.249 1.408 1.220 1068.
227.0 14.45 –32.00 –30.00 0.21223 586.5 0.21286 129.46 548.17 0.7292 2.4512 2.259 1.421 1.221 1056.
227.3 14.16 –30.00 –28.00 0.22903 583.9 0.19818 134.00 550.25 0.7477 2.4457 2.269 1.433 1.223 1043.
227.5 13.87 –28.00 –26.00 0.24682 581.2 0.18472 138.56 552.32 0.7661 2.4402 2.279 1.446 1.225 1030.
227.7 13.58 –26.00 –24.00 0.26564 578.5 0.17237 143.14 554.38 0.7844 2.4350 2.289 1.459 1.227 1017.
227.9 13.29 –24.00 –22.00 0.28553 575.8 0.16102 147.75 556.42 0.8027 2.4299 2.300 1.473 1.229 1004.
228.1 13.01 –22.00 –20.00 0.30653 573.1 0.15058 152.37 558.46 0.8209 2.4250 2.311 1.487 1.231 991.
228.2 12.72 –20.00 –18.00 0.32866 570.4 0.14095 157.02 560.47 0.8390 2.4203 2.323 1.501 1.234 978.
228.3 12.44 –18.00 –16.00 0.35198 567.6 0.13207 161.70 562.47 0.8571 2.4157 2.335 1.515 1.236 965.
228.3 12.15 –16.00 –14.00 0.37653 564.8 0.12386 166.39 564.46 0.8751 2.4112 2.347 1.530 1.239 952.
228.3 11.87 –14.00 –12.00 0.40233 562.0 0.11627 171.12 566.42 0.8931 2.4068 2.359 1.546 1.242 939.
228.3 11.59 –12.00 –10.00 0.42944 559.2 0.10924 175.86 568.37 0.9110 2.4026 2.372 1.562 1.245 926.
228.2 11.31 –10.00 –8.00 0.45788 556.3 0.10272 180.64 570.30 0.9289 2.3985 2.385 1.578 1.248 912.
228.1 11.03 –8.00 –6.00 0.48770 553.5 0.09667 185.44 572.21 0.9468 2.3946 2.398 1.594 1.252 899.
228.0 10.76 –6.00 –4.00 0.51895 550.6 0.09105 190.26 574.10 0.9645 2.3907 2.412 1.612 1.256 886.
227.8 10.48 –4.00 –2.00 0.55166 547.6 0.08582 195.12 575.97 0.9823 2.3869 2.426 1.629 1.260 873.
227.6 10.21 –2.00 0.00 0.58588 544.6 0.08094 200.00 577.82 1.0000 2.3832 2.441 1.648 1.264 860.
227.4 9.94 0.00 2.00 0.62163 541.6 0.07640 204.91 579.65 1.0177 2.3796 2.456 1.666 1.268 847.
227.1 9.67 2.00 4.00 0.65898 538.6 0.07216 209.85 581.45 1.0353 2.3761 2.471 1.686 1.273 834.
226.7 9.40 4.00 6.00 0.69795 535.5 0.06819 214.83 583.22 1.0529 2.3726 2.487 1.706 1.278 820.
226.4 9.13 6.00 8.00 0.73860 532.4 0.06449 219.83 584.97 1.0705 2.3693 2.504 1.726 1.284 807.
226.0 8.87 8.00 10.00 0.78096 529.3 0.06102 224.87 586.69 1.0881 2.3660 2.521 1.748 1.289 794.
225.5 8.60 10.00 12.00 0.82508 526.1 0.05777 229.94 588.39 1.1056 2.3627 2.538 1.770 1.296 781.
225.0 8.34 12.00 14.00 0.87100 522.9 0.05472 235.04 590.05 1.1232 2.3595 2.557 1.793 1.302 767.
224.5 8.08 14.00 16.00 0.91877 519.7 0.05185 240.18 591.68 1.1407 2.3563 2.575 1.817 1.309 754.
223.9 7.82 16.00 18.00 0.96843 516.4 0.04916 245.35 593.28 1.1582 2.3532 2.595 1.842 1.316 741.
223.3 7.56 18.00 20.00 1.0200 513.0 0.04663 250.57 594.85 1.1757 2.3501 2.615 1.868 1.324 727.
222.6 7.31 20.00 22.00 1.0736 509.6 0.04425 255.82 596.37 1.1932 2.3470 2.637 1.895 1.333 714.
221.9 7.05 22.00 24.00 1.1292 506.2 0.04201 261.11 597.86 1.2107 2.3439 2.659 1.923 1.342 701.
221.1 6.80 24.00 26.00 1.1869 502.7 0.03990 266.44 599.31 1.2282 2.3409 2.682 1.952 1.351 687.
220.3 6.55 26.00 28.00 1.2467 499.2 0.03790 271.81 600.72 1.2457 2.3378 2.706 1.983 1.362 674.
219.4 6.31 28.00 30.00 1.3086 495.6 0.03601 277.23 602.09 1.2632 2.3348 2.731 2.015 1.373 661.
218.5 6.06 30.00 32.00 1.3728 491.9 0.03423 282.69 603.40 1.2807 2.3317 2.757 2.049 1.385 647.
217.5 5.82 32.00 34.00 1.4392 488.2 0.03254 288.20 604.67 1.2983 2.3286 2.785 2.085 1.398 634.
216.5 5.57 34.00 36.00 1.5080 484.4 0.03095 293.76 605.89 1.3159 2.3255 2.815 2.123 1.412 620.
215.4 5.33 36.00 38.00 1.5791 480.6 0.02943 299.37 607.05 1.3335 2.3223 2.845 2.163 1.427 606.
214.3 5.10 38.00 40.00 1.6526 476.6 0.02800 305.04 608.15 1.3511 2.3191 2.878 2.206 1.443 593.
213.1 4.86 40.00 42.00 1.7287 472.6 0.02664 310.76 609.20 1.3688 2.3158 2.913 2.252 1.460 579.
211.8 4.63 42.00 44.00 1.8072 468.6 0.02534 316.53 610.17 1.3866 2.3124 2.950 2.300 1.480 565.
210.5 4.40 44.00 46.00 1.8883 464.4 0.02411 322.37 611.08 1.4044 2.3089 2.990 2.352 1.501 551.
209.1 4.17 46.00 48.00 1.9721 460.1 0.02294 328.28 611.91 1.4223 2.3054 3.032 2.409 1.524 538.
207.7 3.95 48.00 50.00 2.0585 455.8 0.02182 334.25 612.65 1.4402 2.3017 3.078 2.469 1.549 524.
206.2 3.72 50.00 55.00 2.2868 444.4 0.01925 349.51 614.13 1.4855 2.2919 3.211 2.645 1.624 488.
202.2 3.18 55.00 60.00 2.5333 432.1 0.01696 365.31 614.95 1.5316 2.2809 3.380 2.868 1.723 452.
197.7 2.65 60.00 65.00 2.7990 418.8 0.01489 381.77 614.94 1.5788 2.2683 3.604 3.164 1.859 415.
192.8 2.15 65.00 70.00 3.0853 404.1 0.01302 399.08 613.86 1.6275 2.2534 3.922 3.579 2.056 377.
187.3 1.67 70.00 75.00 3.3935 387.3 0.01129 417.54 611.31 1.6787 2.2352 4.419 4.216 2.365 336.
181.3 1.21 75.00 80.00 3.7253 367.4 0.00967 437.75 606.58 1.7338 2.2118 5.320 5.339 2.921 294.
174.6 0.79 80.00 85.00 4.0827 341.6 0.00808 461.14 598.00 1.7967 2.1788 7.524 7.941 4.231 247.
167.2 0.41 85.00 90.00 4.4687 298.5 0.00629 493.66 579.40 1.8835 2.1196 — — — 195.
159.2 0.10 90.00 92.42c 4.6646 223.4 0.00448 540.41 540.41 2.0097 2.0097 ∞ ∞ ∞ 0.
0.0 0.00 92.42 Temperatures have been converted from the IPTS-68 scale of the original formulation to the ITS-90 scale b = normal boiling point c = critical point 20.50 2001 ASHRAE Fundamentals Handbook (SI) Fig. 22 Pressure-Enthalpy Diagram for Refrigerant 702 (Normal Hydrogen) Thermophysical Properties of Refrigerants 20.51 Refrigerant 702 (Normal Hydrogen) Properties of Saturated Liquid and Saturated Vapor Temp., K Absolute Pressure, MPa Density, kg/m3 Liquid Volume, m3/kg Vapor Enthalpy, kJ/kg Entropy, kJ/(kg·K) Specific Heat cp, kJ/(kg·K) cp/cv Vapor Velocity of Sound, m/s Viscosity, µPa·s Thermal Cond., mW/(m·K) Surface Tension, mN/m Temp., K Liquid Vapor Liquid Vapor Liquid Vapor Liquid Vapor Liquid Vapor Liquid Vapor 13.95a 0.00776 76.90 7.2871 218.1 667.4 14.082 46.224 07.78 10.90 1.669 1362.
304.5 25.5 0.66 76.2 10.37 3.181 13.95 14.00 0.00797 76.86 7.1136 218.6 667.8 14.108 46.147 07.71 10.89 1.672 1360.
305.1 25.3 0.67 76.7 10.43 3.171 14.00 15.00 0.01334 76.02 4.5226 226.3 676.9 14.610 44.697 7.15 10.86 1.701 1318.
316.2 22.2 0.74 84.1 11.62 2.970 15.00 16.00 0.02113 75.12 3.0172 233.4 685.6 15.075 43.411 07.30 11.00 1.719 1272.
325.2 19.8 0.81 90.1 12.68 2.771 16.00 17.00 0.03200 74.18 2.0940 240.9 693.7 15.530 42.260 07.72 11.19 1.739 1226.
333.2 17.8 0.87 94.8 13.68 2.575 17.00 18.00 0.04663 73.20 1.5017 249.0 701.3 15.984 41.220 8.26 11.43 1.764 1185.
340.6 16.2 0.94 98.4 14.67 2.381 18.00 19.00 0.06577 72.18 1.1068 257.8 708.3 16.441 40.270 08.85 11.71 1.795 1147.
347.3 14.8 1.00 101.1 15.68 2.191 19.00 20.00 0.09020 71.11 0.83478 267.3 714.6 16.904 39.393 09.49 12.04 1.833 1111.
353.3 13.6 1.06 103.0 16.72 2.003 20.00 20.39b 0.10132 70.67 0.75195 271.2 716.8 17.086 39.068 9.74 12.20 1.851 1097.
355.4 13.2 1.08 103.6 17.13 1.931 20.39 21.00 0.12072 69.96 0.64193 277.4 720.1 17.374 38.576 10.15 12.46 1.881 1075.
358.6 12.6 1.12 104.2 17.81 1.819 21.00 22.00 0.15816 68.73 0.50178 288.3 724.8 17.852 37.806 10.88 12.96 1.940 1039.
363.2 11.6 1.18 104.9 18.96 1.638 22.00 23.00 0.20336 67.41 0.39766 299.9 728.4 18.340 37.074 11.68 13.58 2.013 1001.
367.2 10.8 1.25 105.0 20.18 1.460 23.00 24.00 0.25717 65.98 0.31878 312.3 731.0 18.840 36.369 12.58 14.35 2.106 960.
370.6 10.1 1.32 104.6 21.50 1.287 24.00 25.00 0.32045 64.43 0.25795 325.8 732.3 19.351 35.683 13.65 15.34 2.224 918.
373.3 09.4 1.39 103.8 22.92 1.117 25.00 26.00 0.39404 62.75 0.21028 340.3 732.2 19.877 35.007 14.94 16.63 2.378 872.
375.4 8.7 1.46 102.5 24.48 0.953 26.00 27.00 0.47879 60.91 0.17233 356.1 730.4 20.421 34.331 16.56 18.36 2.586 824.
376.9 08.1 1.54 100.8 26.23 0.793 27.00 28.00 0.57555 58.87 0.14165 373.5 726.5 20.989 33.642 18.72 20.79 2.877 772.
377.8 07.5 1.63 98.7 28.22 0.639 28.00 29.00 0.68516 56.55 0.11647 392.7 720.2 21.596 32.925 21.85 24.41 3.310 715.
378.2 6.9 1.74 96.0 30.56 0.492 29.00 30.00 0.80844 53.76 0.09540 414.7 710.5 22.267 32.157 27.13 30.32 4.013 650.
378.1 06.4 1.86 92.6 33.46 0.352 30.00 31.00 0.94620 50.17 0.07735 441.4 696.2 23.059 31.298 38.73 41.37 5.321 573.
377.7 05.8 2.04 88.3 37.32 0.222 31.00 32.00 1.0993 44.89 0.06132 477.9 674.4 24.112 30.271 84.77 67.41 8.383 482.
377.6 5.1 2.35 82.3 43.40 0.105 32.00 33.00 1.2684 34.38 0.04665 547.5 640.5 26.097 28.951 — — — — — — — — — 0.011 33.00 33.19c 1.3152 30.11 0.03321 577.2 577.2 26.962 26.962 ∞ ∞ ∞ 00 0.
000.0 — — ∞ ∞ 0.000 33.19 Temperatures are on the IPTS-68 scale a = triple point b = normal boiling point c = critical point Refrigerant 702 (Normal Hydrogen) Properties of Gas at 0.101 325 MPa (one standard atmosphere) Temp., °C Density, kg/m3 Enthalpy, kJ/kg Entropy, kJ/(kg·K) cp, kJ/ (kg·K) cp/cv Vel. Sound, m/s Viscosity, µPa·s Thermal Cond., mW/(m·K) Temp., °C Density, kg/m3 Enthalpy, kJ/kg Entropy, kJ/(kg·K) cp, kJ/ (kg·K) cp/cv Vel. Sound, m/s Viscosity, µPa·s Thermal Cond., mW/(m·K) −252.8a 1.3299 0716.8 39.068 12.20 1.851 0355.4 1.08 017.10 000.0 0.0899 3843.3 69.168 14.20 1.410 1261.1 08.40 172.58 −250.0 1.1366 0749.3 40.564 11.45 1.793 0385.4 1.22 019.29 005.0 0.0883 3914.4 69.425 14.22 1.409 1272.1 08.50 175.07 −245.0 0.9089 805.0 42.744 10.91 1.744 431.8 1.47 23.26 10.0 0.0867 3985.5 69.679 14.24 1.408 1283.0 8.61 177.56 −240.0 0.7609 0859.0 44.508 10.69 1.720 0472.2 1.72 027.06 015.0 0.0852 4056.8 69.928 14.27 1.407 1293.9 08.71 180.05 −235.0 0.6557 0912.1 46.001 10.58 1.706 0508.7 1.95 030.62 020.0 0.0838 4128.2 70.174 14.29 1.406 1304.7 08.81 182.48 −230.0 0.5767 964.8 47.300 10.52 1.696 542.3 2.18 34.00 25.0 0.0824 4199.7 70.416 14.31 1.405 1315.4 8.92 184.88 −225.0 0.5150 1017.4 48.452 10.49 1.688 0573.6 2.40 037.23 030.0 0.0810 4271.3 70.654 14.32 1.405 1326.0 09.02 187.28 −220.0 0.4655 1069.7 49.487 10.47 1.682 0603.0 2.60 040.37 035.0 0.0797 4342.9 70.888 14.34 1.404 1336.6 09.12 189.67 −215.0 0.4247 1122.1 50.429 10.48 1.676 630.6 2.80 43.47 40.0 0.0784 4414.7 71.119 14.36 1.403 1347.1 9.22 192.01 −210.0 0.3906 1174.6 51.294 10.51 1.668 0656.6 2.98 046.55 045.0 0.0772 4486.5 71.347 14.37 1.403 1357.5 09.32 194.32 −200.0 0.3366 1280.1 52.845 10.62 1.650 0704.1 3.34 052.56 050.0 0.0760 4558.3 71.571 14.38 1.402 1367.9 09.42 196.78 −190.0 0.2958 1387.2 54.217 10.81 1.629 746.5 3.67 58.30 55.0 0.0748 4630.3 71.792 14.39 1.402 1378.2 9.52 199.33 −180.0 0.2639 1496.4 55.457 11.05 1.604 0784.7 3.98 064.12 060.0 0.0737 4702.3 72.010 14.40 1.401 1388.4 09.62 201.86 −170.0 0.2382 1608.2 56.597 11.32 1.580 0819.7 4.28 070.43 065.0 0.0726 4774.3 72.224 14.41 1.401 1398.6 09.72 204.37 −160.0 0.2171 1722.8 57.657 11.61 1.556 852.3 4.57 77.02 70.0 0.0716 4846.4 72.436 14.42 1.401 1408.7 9.81 206.88 −150.0 0.1994 1840.3 58.652 11.89 1.534 0883.2 4.85 083.56 075.0 0.0705 4918.5 72.644 14.43 1.400 1418.8 09.91 209.39 −140.0 0.1844 1960.7 59.592 12.17 1.515 0912.7 5.12 090.15 080.0 0.0695 4990.7 72.850 14.44 1.400 1428.8 10.01 211.88 −130.0 0.1715 2083.7 60.483 12.44 1.499 941.2 5.38 96.62 85.0 0.0686 5062.9 73.053 14.44 1.400 1438.7 10.10 214.34 −120.0 0.1603 2209.3 61.331 12.68 1.484 0968.8 5.64 103.03 090.0 0.0676 5135.1 73.254 14.45 1.400 1448.6 10.20 216.72 −110.0 0.1505 2337.3 62.140 12.90 1.471 0995.7 5.89 109.55 095.0 0.0667 5207.4 73.451 14.45 1.399 1458.4 10.29 219.04 −100.0 0.1418 2467.3 62.913 13.10 1.461 1022.1 6.14 115.95 100.0 0.0658 5279.7 73.646 14.46 1.399 1468.2 10.39 221.42 −90.0 0.1340 2599.2 63.654 13.28 1.452 1047.9 6.38 122.20 110.0 0.0641 5424.3 74.029 14.47 1.399 1487.5 10.58 226.21 −80.0 0.1271 2732.8 64.364 13.44 1.444 1073.2 6.62 128.33 120.0 0.0625 5569.0 74.401 14.47 1.399 1506.7 10.77 230.86 −70.0 0.1208 2867.9 65.046 13.58 1.437 1098.0 6.85 134.30 130.0 0.0609 5713.8 74.765 14.48 1.398 1525.6 — — −60.0 0.1152 3004.4 65.702 13.71 1.431 1122.5 7.08 140.15 140.0 0.0594 5858.6 75.120 14.48 1.398 1544.3 — — −50.0 0.1100 3142.0 66.333 13.82 1.426 1146.5 7.31 145.84 150.0 0.0580 6003.4 75.466 14.49 1.398 1562.8 — — −40.0 0.1053 3280.7 66.941 13.91 1.422 1170.1 7.53 151.44 160.0 0.0567 6148.3 75.805 14.49 1.398 1581.0 — — −30.0 0.1010 3420.2 67.527 14.00 1.418 1193.4 7.75 156.92 170.0 0.0554 6293.2 76.135 14.49 1.398 1599.1 — — −20.0 0.0970 3560.6 68.093 14.07 1.415 1216.3 7.97 162.24 180.0 0.0542 6438.1 76.459 14.50 1.398 1616.9 — — −10.0 0.0933 3701.7 68.639 14.14 1.412 1238.8 8.19 167.47 190.0 0.0530 6583.1 76.775 14.50 1.398 1634.6 — — 200.0 0.0519 6728.1 77.085 14.50 1.397 1652.0 — — a = saturated vapor at normal boiling point 20.52 2001 ASHRAE Fundamentals Handbook (SI) Fig. 23 Pressure-Enthalpy Diagram for Refrigerant 702p (Parahydrogen) Thermophysical Properties of Refrigerants 20.53 Refrigerant 702p (Parahydrogen) Properties of Saturated Liquid and Saturated Vapor Temp., K Absolute Pressure, MPa Density, kg/m3 Liquid Volume, m3/kg Vapor Enthalpy, kJ/kg Entropy, kJ/(kg·K) Specific Heat cp, kJ/(kg·K) cp/cv Vapor Velocity of Sound, m/s Viscosity, µPa·s Thermal Cond., mW/(m·K) Surface Tension, mN/m Temp., K Liquid Vapor Liquid Vapor Liquid Vapor Liquid Vapor Liquid Vapor Liquid Vapor 13.80a 0.00705 77.04 7.8437 −307.2 140.1 04.980 37.428 07.72 10.68 1.690 1373.
305.0 26.0 0.65 075.3 10.46 3.124 13.80 14.00 0.00790 76.87 7.1198 −305.7 141.9 05.088 37.115 07.53 10.70 1.692 1364.
307.0 25.3 0.67 076.9 10.65 3.086 14.00 15.00 0.01343 76.01 4.4920 −298.3 150.9 5.588 35.628 7.13 10.84 1.704 1316.
316.4 22.2 0.74 084.1 11.62 2.894 15.00 16.00 0.02152 75.12 2.9640 −291.1 159.5 06.051 34.295 07.30 11.01 1.720 1269.
325.1 19.8 0.81 090.0 12.60 2.705 16.00 17.00 0.03284 74.19 2.0392 −283.5 167.5 06.504 33.107 07.72 11.22 1.741 1224.
333.0 17.8 0.87 094.7 13.59 2.516 17.00 18.00 0.04807 73.22 1.4541 −275.4 175.0 6.958 32.042 8.26 11.46 1.767 1183.
340.3 16.2 0.94 098.3 14.60 2.330 18.00 19.00 0.06796 72.19 1.0683 −266.6 181.8 07.416 31.077 08.85 11.76 1.800 1145.
346.8 14.8 1.00 101.1 15.64 2.146 19.00 20.00 0.09326 71.11 0.80448 −257.2 188.0 07.880 30.193 09.48 12.12 1.840 1109.
352.6 13.6 1.06 103.0 16.71 1.963 20.00 20.28b 0.10132 70.80 0.74656 −254.4 189.5 8.009 29.960 9.66 12.23 1.853 1099.
354.1 13.3 1.07 103.4 17.01 1.913 20.28 20.50 0.10818 70.54 0.70371 −252.2 190.8 08.114 29.776 09.81 12.33 1.864 1091.
355.3 13.1 1.09 103.7 17.25 1.873 20.50 21.00 0.12474 69.96 0.61847 −247.1 193.3 08.350 29.373 10.15 12.56 1.890 1073.
357.8 12.6 1.12 104.2 17.82 1.783 21.00 21.50 0.14305 69.35 0.54589 −241.7 195.7 8.589 28.983 10.51 12.81 1.920 1055.
360.2 12.1 1.15 104.7 18.39 1.693 21.50 22.00 0.16320 68.73 0.48372 −236.2 197.8 08.829 28.604 10.87 13.09 1.952 1036.
362.3 11.6 1.18 104.9 18.98 1.605 22.00 22.50 0.18529 68.08 0.43016 −230.5 199.7 09.071 28.235 11.26 13.39 1.989 1018.
364.3 11.2 1.21 105.0 19.59 1.517 22.50 23.00 0.20942 67.42 0.38378 −224.6 201.3 9.315 27.875 11.67 13.74 2.030 999.
366.2 10.8 1.25 105.0 20.21 1.429 23.00 23.50 0.23570 66.72 0.34340 −218.5 202.6 09.562 27.521 12.10 14.12 2.075 0979.
367.9 10.4 1.28 104.9 20.86 1.342 23.50 24.00 0.26423 66.00 0.30808 −212.1 203.6 09.811 27.174 12.56 14.56 2.127 0959.
369.4 10.1 1.32 104.7 21.53 1.256 24.00 24.50 0.29511 65.25 0.27705 −205.5 204.4 10.064 26.832 13.06 15.04 2.185 939.
370.8 9.7 1.35 104.3 22.23 1.171 24.50 25.00 0.32845 64.47 0.24966 −198.7 204.7 10.320 26.492 13.61 15.60 2.251 0918.
372.1 09.4 1.39 103.8 22.95 1.086 25.00 26.00 0.40291 62.80 0.20378 −184.2 204.4 10.843 25.820 14.87 16.96 2.414 0873.
374.1 08.7 1.46 102.6 24.51 0.920 26.00 27.00 0.48849 60.97 0.16716 −168.4 202.3 11.385 25.145 16.47 18.81 2.635 825.
375.5 8.1 1.54 100.9 26.25 0.757 27.00 28.00 0.58610 58.93 0.13744 −151.2 198.1 11.954 24.454 18.61 21.44 2.948 0773.
376.3 07.5 1.63 098.7 28.23 0.599 28.00 29.00 0.69673 56.60 0.11286 −132.0 191.3 12.560 23.727 21.70 25.45 3.426 0716.
376.4 06.9 1.74 096.0 30.57 0.445 29.00 30.00 0.82143 53.86 0.09208 −110.2 180.6 13.223 22.933 26.73 32.31 4.238 653.
375.9 6.4 1.86 92.6 33.45 0.299 30.00 31.00 0.96149 50.48 0.07387 0−84.4 164.1 13.981 22.011 36.68 46.58 5.916 0582.
374.8 05.8 2.04 088.3 37.31 0.161 31.00 32.00 1.1185 45.81 0.05667 0−51.0 135.8 14.938 20.788 66.67 93.30 11.3570 0501.
373.2 05.1 2.35 082.3 43.38 0.038 32.00 32.94c 1.2838 31.36 0.03189 40.3 40.3 17.615 17.615 ∞ ∞ ∞ 0.
0.0 — — ∞ ∞ 0.000 32.94 Temperatures are on the IPTS-68 scale a = triple point b = normal boiling point c = critical point Refrigerant 702p (Parahydrogen) Properties of Gas at 0.101 325 MPa (one standard atmosphere) Temp., °C Density, kg/m3 Enthalpy, kJ/kg Entropy, kJ/(kg·K) cp, kJ/ (kg·K) cp/cv Vel. Sound, m/s Viscosity, µPa·s Thermal Cond., mW/(m·K) Temp., °C Density, kg/m3 Enthalpy, kJ/kg Entropy, kJ/(kg·K) cp, kJ/ (kg·K) cp/cv Vel. Sound, m/s Viscosity, µPa·s Thermal Cond., mW/(m·K) −252.9a 1.3395 0189.5 29.960 12.23 1.853 0354.1 1.07 17.01 −50.0 0.1100 3030.3 60.270 15.71 1.356 1118.1 07.31 164.51 −250.0 1.1364 0223.4 31.524 11.45 1.793 0385.5 1.22 19.33 −40.0 0.1053 3186.7 60.955 15.56 1.361 1144.9 07.53 168.14 −245.0 0.9088 279.1 33.703 10.91 1.744 431.8 1.47 23.27 −30.0 0.1010 3341.5 61.605 15.41 1.366 1171.1 7.75 171.75 −240.0 0.7608 0333.0 35.467 10.69 1.720 0472.2 1.72 27.03 −20.0 0.0970 3495.0 62.224 15.28 1.370 1196.8 07.97 175.31 −235.0 0.6557 0386.2 36.960 10.58 1.706 0508.7 1.95 30.58 −10.0 0.0933 3647.2 62.814 15.16 1.374 1221.9 08.19 178.87 −230.0 0.5767 438.9 38.260 10.54 1.694 542.0 2.18 33.99 0.0 0.0899 3798.3 63.377 15.06 1.377 1246.5 8.40 182.44 −225.0 0.5150 0491.6 39.415 10.53 1.683 0572.8 2.40 37.34 5.0 0.0883 3873.5 63.650 15.01 1.379 1258.6 08.50 184.23 −220.0 0.4655 0544.3 40.456 10.57 1.672 0601.2 2.60 40.72 10.0 0.0867 3948.5 63.917 14.97 1.381 1270.5 08.61 186.05 −215.0 0.4247 597.3 41.410 10.66 1.656 627.0 2.80 44.21 15.0 0.0852 4023.2 64.179 14.93 1.382 1282.3 8.71 187.89 −210.0 0.3906 0651.0 42.295 10.81 1.637 0650.4 2.98 47.87 20.0 0.0838 4097.8 64.436 14.89 1.383 1294.0 08.81 189.73 −200.0 0.3366 0761.4 43.918 11.31 1.587 0690.5 3.34 55.69 25.0 0.0824 4172.1 64.687 14.86 1.384 1305.6 08.92 191.58 −190.0 0.2958 877.9 45.409 12.01 1.532 723.9 3.67 64.19 30.0 0.0810 4246.4 64.934 14.83 1.386 1317.0 9.02 193.46 −180.0 0.2639 1002.1 46.819 12.84 1.479 0753.5 3.98 73.55 35.0 0.0797 4320.4 65.176 14.80 1.387 1328.3 09.12 195.36 −170.0 0.2382 1134.8 48.171 13.69 1.435 0781.3 4.28 83.79 40.0 0.0784 4394.3 65.414 14.77 1.388 1339.5 09.22 197.29 −160.0 0.2171 1275.8 49.475 14.50 1.401 808.7 4.57 94.32 45.0 0.0772 4468.1 65.648 14.75 1.388 1350.5 9.32 199.25 −150.0 0.1994 1424.3 50.732 15.17 1.376 0836.2 4.85 104.41 50.0 0.0760 4541.8 65.878 14.72 1.389 1361.5 09.42 201.20 −140.0 0.1844 1578.7 51.938 15.69 1.358 0864.2 5.12 113.74 55.0 0.0748 4615.4 66.104 14.70 1.390 1372.3 09.52 203.16 −130.0 0.1715 1737.6 53.088 16.05 1.347 892.4 5.38 122.26 60.0 0.0737 4688.9 66.326 14.69 1.391 1383.1 9.62 205.18 −120.0 0.1603 1899.3 54.180 16.27 1.341 0920.9 5.64 129.99 65.0 0.0726 4762.2 66.545 14.67 1.391 1393.7 09.72 207.24 −110.0 0.1505 2062.6 55.213 16.37 1.338 0949.5 5.89 136.76 70.0 0.0716 4835.6 66.760 14.65 1.392 1404.3 09.81 209.33 −100.0 0.1418 2226.3 56.187 16.36 1.338 978.1 6.14 142.70 75.0 0.0705 4908.8 66.972 14.64 1.392 1414.7 9.91 211.44 −90.0 0.1340 2389.6 57.104 16.29 1.340 1006.7 6.38 147.96 80.0 0.0695 4981.9 67.180 14.63 1.393 1425.1 10.01 213.53 −80.0 0.1271 2551.9 57.967 16.17 1.343 1035.1 6.62 152.62 85.0 0.0686 5055.1 67.386 14.62 1.393 1435.3 10.10 215.59 −70.0 0.1208 2712.9 58.779 16.03 1.347 1063.2 6.85 156.90 90.0 0.0676 5128.1 67.588 14.60 1.394 1445.5 10.20 217.63 −60.0 0.1152 2872.4 59.546 15.87 1.352 1090.8 7.08 160.82 95.0 0.0667 5201.1 67.788 14.60 1.394 1455.6 10.29 219.65 100.0 0.0658 5274.1 67.985 14.59 1.394 1465.6 10.39 221.73 a = saturated vapor at normal boiling point 20.54 2001 ASHRAE Fundamentals Handbook (SI) Fig. 24 Pressure-Enthalpy Diagram for Refrigerant 704 (Helium) Note: The reference states for enthalpy and entropy differ from those in the table.
Thermophysical Properties of Refrigerants 20.55 Refrigerant 704 (Helium) Properties of Saturated Liquid and Saturated Vapor Temp., K Absolute Pressure, MPa Density, kg/m3 Liquid Volume, m3/kg Vapor Enthalpy, kJ/kg Entropy, kJ/(kg·K) Specific Heat cp, kJ/(kg·K) cp/cv Vapor Velocity of Sound, m/s Viscosity, µPa·s Thermal Cond., mW/(m·K) Surface Tension, mN/m Temp., K Liquid Vapor Liquid Vapor Liquid Vapor Liquid Vapor Liquid Vapor Liquid Vapor 2.18a 0.00486 146.24 0.87297 2.34 25.56 1.4040 12.0746 6.318 6.061 1.747 217.
83.2 — — — — 0.388 2.18 2.20 0.00515 146.19 0.83068 2.48 25.66 1.4682 12.0035 5.800 6.076 1.750 217.
83.6 — — — — 0.385 2.20 2.30 0.00653 145.87 0.67831 2.98 26.05 1.6865 11.7174 4.164 6.139 1.763 216.
85.0 — — — — 0.371 2.30 2.40 0.00814 145.46 0.56257 3.35 26.44 1.8414 11.4586 3.217 6.199 1.778 216.
86.3 — — — — 0.356 2.40 2.50 0.01000 144.96 0.47252 3.66 26.81 1.9607 11.2211 2.700 6.258 1.795 216.
87.6 — — — — 0.342 2.50 2.60 0.01213 144.38 0.40113 3.93 27.17 2.0604 11.0011 2.453 6.318 1.813 217.
88.8 — — — — 0.328 2.60 2.70 0.01454 143.72 0.34364 4.18 27.52 2.1502 10.7956 2.372 6.380 1.834 217.
90.0 — — — — 0.314 2.70 2.80 0.01727 143.00 0.29674 4.44 27.86 2.2356 10.6024 2.394 6.446 1.857 217.
91.1 — — — — 0.300 2.80 2.90 0.02032 142.21 0.25805 4.70 28.19 2.3196 10.4198 2.477 6.516 1.882 216.
92.1 — — — — 0.286 2.90 3.00 0.02373 141.34 0.22582 4.97 28.50 2.4039 10.2464 2.598 6.592 1.910 214.
93.0 — — — — 0.272 3.00 3.10 0.02750 140.42 0.19871 5.26 28.79 2.4894 10.0808 2.740 6.676 1.941 213.
93.9 — — — — 0.258 3.10 3.20 0.03166 139.43 0.17574 5.56 29.07 2.5765 9.9222 2.897 6.768 1.976 210.
94.8 — — — — 0.244 3.20 3.30 0.03622 138.38 0.15612 5.88 29.33 2.6653 9.7694 3.062 6.872 2.015 208.
95.5 — — — — 0.231 3.30 3.40 0.04121 137.25 0.13925 6.22 29.57 2.7559 9.6216 3.234 6.989 2.059 205.
96.3 — — — — 0.217 3.40 3.50 0.04664 136.06 0.12466 6.58 29.79 2.8482 9.4781 3.414 7.122 2.108 202.
96.9 — — — — 0.203 3.50 3.60 0.05252 134.80 0.11195 6.96 29.98 2.9422 9.3381 3.603 7.274 2.165 199.
97.5 3.5 1.00 17.9 7.31 0.190 3.60 3.70 0.05888 133.45 0.10081 7.35 30.16 3.0380 9.2007 3.803 7.449 2.229 196.
98.1 3.4 1.04 18.1 7.56 0.177 3.70 3.80 0.06573 132.03 0.09101 7.77 30.31 3.1355 9.0653 4.020 7.654 2.303 193.
98.6 3.4 1.07 18.2 7.82 0.164 3.80 3.90 0.07310 130.51 0.08232 8.21 30.43 3.2351 8.9312 4.257 7.894 2.389 189.
99.1 3.3 1.11 18.4 8.09 0.150 3.90 4.00 0.08100 128.90 0.07459 8.67 30.52 3.3367 8.7975 4.523 8.179 2.491 186.
99.5 3.3 1.15 18.5 8.36 0.138 4.00 4.10 0.08945 127.17 0.06767 9.16 30.57 3.4407 8.6633 4.826 8.521 2.611 182.
99.8 3.2 1.19 18.6 8.66 0.125 4.10 4.20 0.09847 125.32 0.06144 9.68 30.59 3.5475 8.5277 5.179 8.938 2.756 178.
100.1 3.2 1.24 18.6 8.97 0.112 4.20 4.23b 0.10132 124.73 0.05967 9.84 30.59 3.5806 8.4861 5.299 9.083 2.806 176.
100.2 3.2 1.25 18.7 9.06 0.108 4.23 4.30 0.10809 123.33 0.05581 10.22 30.57 3.6577 8.3896 5.600 9.455 2.934 173.
100.4 3.1 1.28 18.7 9.30 0.099 4.30 4.40 0.11832 121.17 0.05067 10.80 30.50 3.7719 8.2476 6.118 10.1100 3.158 169.
100.6 3.1 1.33 18.8 9.66 0.087 4.40 4.50 0.12920 118.81 0.04596 11.42 30.36 3.8912 8.0999 6.776 10.9640 3.448 164.
100.8 3.0 1.38 18.8 10.07 0.075 4.50 4.60 0.14075 116.20 0.04161 12.09 30.16 4.0171 7.9440 7.646 12.117 3.838 159.
100.9 3.0 1.43 18.8 10.54 0.063 4.60 4.70 0.15301 113.27 0.03753 12.83 29.87 4.1517 7.7767 8.863 13.7540 4.388 153.
101.0 2.9 1.48 18.9 11.08 0.051 4.70 4.80 0.16602 109.90 0.03367 13.64 29.45 4.2986 7.5924 10.7000 16.2440 5.224 147.
101.1 2.8 1.55 19.0 11.76 0.040 4.80 4.90 0.17983 105.89 0.02993 14.57 28.87 4.4641 7.3821 13.813 20.464 6.637 140.
101.3 2.7 1.62 19.1 12.63 0.029 4.90 5.00 0.19453 100.83 0.02617 15.69 28.02 4.6615 7.1273 20.2400 29.0940 9.531 133.
101.6 2.6 1.70 19.3 13.86 0.018 5.00 5.10 0.21023 93.53 0.02206 17.20 26.63 4.9283 6.7774 40.7700 55.8660 18.545 124.
102.5 2.5 1.80 19.9 15.77 0.008 5.10 5.20c 0.22746 69.64 0.01436 21.71 21.71 5.7639 5.7639 ∞ ∞ ∞ 0.
0.0 — — ∞ ∞ 0.000 5.20 Temperatures are on the EPT-76 scale a = lower lambda point b = normal boiling point c = critical point Refrigerant 704 (Helium) Properties of Gas at 0.101 325 MPa (one standard atmosphere) Temp., °C Density, kg/m3 Enthalpy, kJ/kg Entropy, kJ/(kg·K) cp, kJ/ (kg·K) cp/cv Vel. Sound, m/s Viscosity, µPa·s Thermal Cond., mW/(m·K) Temp., °C Density, kg/m3 Enthalpy, kJ/kg Entropy, kJ/(kg·K) cp, kJ/ (kg·K) cp/cv Vel. Sound, m/s Viscosity, µPa·s Thermal Cond., mW/(m·K) −268.9a 16.758 30.59 8.4861 9.083 2.806 100.2 1.25 9.05 100.0 0.1307 1953.16 32.7129 5.193 1.667 1137.0 23.15 181.41 −260.0 3.7508 82.19 15.2748 5.327 1.708 213.6 2.72 20.17 120.0 0.1240 2057.02 32.9840 5.193 1.667 1167.0 24.00 188.10 −250.0 2.1049 134.88 18.2572 5.236 1.678 284.4 3.93 28.69 140.0 0.1180 2160.88 33.2417 5.193 1.667 1196.3 24.84 194.69 −240.0 1.4677 187.10 20.1325 5.213 1.671 340.1 4.93 35.90 160.0 0.1126 2264.74 33.4872 5.193 1.667 1224.9 25.67 201.19 −220.0 0.9155 291.20 22.5899 5.200 1.668 430.1 6.60 48.56 180.0 0.1076 2368.60 33.7216 5.193 1.667 1252.9 26.49 207.60 −200.0 0.6655 395.16 24.2500 5.196 1.667 504.3 8.04 59.86 200.0 0.1031 2472.46 33.9459 5.193 1.667 1280.2 27.29 213.93 −180.0 0.5228 499.06 25.5057 5.195 1.667 568.8 9.35 70.30 220.0 0.0989 2576.32 34.1609 5.193 1.667 1307.0 28.09 220.18 −160.0 0.4305 602.95 26.5160 5.194 1.667 626.7 10.39 80.09 240.0 0.0950 2680.18 34.3673 5.193 1.667 1333.2 28.88 226.35 −140.0 0.3659 706.82 27.3614 5.194 1.667 679.7 11.56 89.41 260.0 0.0915 2784.04 34.5659 5.193 1.667 1358.9 29.66 232.46 −120.0 0.3182 810.69 28.0882 5.193 1.666 728.8 12.68 98.32 280.0 0.0882 2887.90 34.7571 5.193 1.667 1384.1 30.43 238.50 −100.0 0.2815 914.56 28.7256 5.193 1.666 774.9 13.75 106.90 300.0 0.0851 2991.76 34.9416 5.193 1.667 1408.9 31.20 244.47 −80.0 0.2524 1018.42 29.2932 5.193 1.667 818.3 14.79 115.20 320.0 0.0822 3095.62 35.1197 5.193 1.667 1433.3 31.96 250.39 −60.0 0.2287 1122.28 29.8049 5.193 1.667 859.6 15.80 123.25 340.0 0.0795 3199.48 35.2919 5.193 1.667 1457.2 32.71 256.24 −40.0 0.2091 1226.14 30.2707 5.193 1.667 898.9 16.79 131.09 360.0 0.0770 3303.34 35.4586 5.193 1.667 1480.8 33.45 262.04 −20.0 0.1926 1330.00 30.6980 5.193 1.667 936.7 17.75 138.73 380.0 0.0747 3407.20 35.6201 5.193 1.667 1504.0 34.19 267.79 0.0 0.1785 1433.87 31.0929 5.193 1.667 972.9 18.70 146.20 400.0 0.0725 3511.06 35.7767 5.193 1.667 1526.8 34.92 273.48 20.0 0.1663 1537.73 31.4599 5.193 1.667 1007.9 19.62 153.50 420.0 0.0704 3614.92 35.9288 5.193 1.667 1549.3 35.65 279.13 40.0 0.1557 1641.58 31.8026 5.193 1.667 1041.6 20.52 160.67 440.0 0.0684 3718.78 36.0765 5.193 1.667 1571.5 36.37 284.73 60.0 0.1464 1745.44 32.1241 5.193 1.667 1074.4 21.41 167.70 460.0 0.0665 3822.64 36.2201 5.193 1.667 1593.4 37.08 290.28 80.0 0.1381 1849.30 32.4269 5.193 1.667 1106.1 22.29 174.61 480.0 0.0648 3926.50 36.3599 5.193 1.667 1615.0 37.79 295.78 500.0 0.0631 4030.36 36.4960 5.193 1.667 1636.3 38.49 301.25 a = saturated vapor at normal boiling point 20.56 2001 ASHRAE Fundamentals Handbook (SI) Fig. 25 Pressure-Enthalpy Diagram for Refrigerant 728 (Nitrogen) Thermophysical Properties of Refrigerants 20.57 Refrigerant 728 (Nitrogen) Properties of Saturated Liquid and Saturated Vapor Temp., K Absolute Pressure, MPa Density, kg/m3 Liquid Volume, m3/kg Vapor Enthalpy, kJ/kg Entropy, kJ/(kg·K) Specific Heat cp, kJ/(kg·K) cp/cv Vapor Velocity of Sound, m/s Viscosity, µPa·s Thermal Cond., mW/(m·K) Surface Tension, mN/m Temp., K Liquid Vapor Liquid Vapor Liquid Vapor Liquid Vapor Liquid Vapor Liquid Vapor 63.15a 0.01252 869.7 1.48190 −150.87 64.52 2.4194 5.8303 2.019 1.187 1.390 1022.0 159.5 285.9 4.17 174.6 5.38 12.24 63.15 64.00 0.01461 866.0 1.28530 −149.15 65.32 2.4464 5.7975 2.017 1.196 1.390 1010.0 160.4 272.8 4.23 172.6 5.47 12.03 64.00 66.00 0.02063 857.4 0.93539 −145.12 67.17 2.5084 5.7248 2.013 1.218 1.393 983.
162.4 245.7 4.37 168.1 5.68 11.55 66.00 68.00 0.02850 848.6 0.69509 −141.09 68.98 2.5684 5.6576 2.013 1.240 1.396 958.
164.4 222.9 4.51 163.6 5.90 11.07 68.00 70.00 0.03857 839.8 0.52632 −137.05 70.74 2.6267 5.5952 2.016 1.262 1.401 933.
166.2 203.4 4.65 159.3 6.12 10.59 70.00 72.00 0.05124 830.9 0.40532 −133.01 72.45 2.6835 5.5370 2.020 1.284 1.407 910.
168.0 186.6 4.79 155.0 6.35 10.12 72.00 74.00 0.06696 822.0 0.31694 −128.95 74.10 2.7388 5.4827 2.027 1.306 1.414 887.
169.6 172.0 4.94 150.7 6.58 9.65 74.00 76.00 0.08616 812.8 0.25127 −124.87 75.69 2.7929 5.4319 2.036 1.327 1.423 864.
171.2 159.1 5.08 146.5 6.82 9.19 76.00 77.35b 0.10132 806.6 0.21639 −122.11 76.73 2.8286 5.3993 2.042 1.341 1.430 850.
172.2 151.2 5.19 143.7 6.99 8.88 77.35 78.00 0.10934 803.6 0.20170 −120.78 77.21 2.8457 5.3841 2.046 1.348 1.433 842.
172.7 147.7 5.23 142.4 7.07 8.73 78.00 80.00 0.13698 794.2 0.16375 −116.65 78.67 2.8975 5.3389 2.058 1.368 1.446 821.
174.0 137.5 5.39 138.3 7.33 8.27 80.00 82.00 0.16961 784.7 0.13431 −112.50 80.04 2.9482 5.2962 2.072 1.389 1.460 799.
175.3 128.3 5.54 134.3 7.59 7.83 82.00 84.00 0.20776 775.0 0.11118 −108.32 81.32 2.9980 5.2557 2.088 1.411 1.476 778.
176.4 120.0 5.70 130.3 7.86 7.38 84.00 86.00 0.25198 765.1 0.09281 −104.10 82.52 3.0469 5.2169 2.105 1.434 1.494 756.
177.4 112.4 5.86 126.3 8.14 6.94 86.00 88.00 0.30281 755.0 0.07806 −99.84 83.61 3.0951 5.1798 2.126 1.458 1.515 735.
178.2 105.5 6.02 122.4 8.44 6.51 88.00 90.00 0.36083 744.6 0.06611 −95.54 84.59 3.1426 5.1441 2.148 1.485 1.540 713.
179.0 99.1 6.19 118.5 8.75 6.09 90.00 92.00 0.42661 734.0 0.05633 −91.19 85.45 3.1894 5.1095 2.174 1.516 1.568 692.
179.6 93.2 6.37 114.7 9.07 5.67 92.00 94.00 0.50074 723.2 0.04827 −86.78 86.19 3.2357 5.0759 2.204 1.551 1.600 670.
180.1 87.7 6.54 110.8 9.40 5.25 94.00 96.00 0.58381 712.0 0.04156 −82.32 86.79 3.2816 5.0431 2.238 1.591 1.637 647.
180.4 82.6 6.73 107.0 9.76 4.84 96.00 98.00 0.67641 700.6 0.03594 −77.78 87.24 3.3270 5.0108 2.277 1.639 1.680 624.
180.6 77.7 6.92 103.2 10.14 4.44 98.00 100.00 0.77917 688.7 0.03120 −73.16 87.52 3.3721 4.9789 2.323 1.695 1.731 601.
180.6 73.2 7.12 99.5 10.54 4.05 100.00 105.00 1.0846 657.0 0.02218 −61.20 87.38 3.4844 4.8995 2.481 1.892 1.903 540.
180.0 62.8 7.66 90.0 11.67 3.10 105.00 110.00 1.4676 621.1 0.01595 −48.41 85.74 3.5978 4.8173 2.745 2.230 2.186 473.
178.4 53.5 8.29 80.5 13.09 2.21 110.00 115.00 1.9393 578.5 0.01144 −34.27 81.92 3.7164 4.7268 3.260 2.898 2.728 397.
175.5 44.6 9.09 70.5 15.07 1.39 115.00 120.00 2.5130 522.3 0.00799 −17.57 74.33 3.8496 4.6155 4.601 4.691 4.134 309.
171.0 35.7 10.230 60.2 18.80 0.66 120.00 125.00 3.2080 424.7 0.00490 6.76 55.64 4.0359 4.4270 — — — — — — — — — 0.08 125.00 126.19c 3.3978 313.1 0.00319 29.83 29.83 4.2154 4.2154 ∞ ∞ ∞ 0.
0.0 — — ∞ ∞ 0.00 126.19 temperatures are on the IPTS-68 scale a = triple point b = normal boiling point c = critical point Refrigerant 728 (Nitrogen) Properties of Gas at 0.101 325 MPa (one standard atmosphere) Temp., °C Density, kg/m3 Enthalpy, kJ/kg Entropy, kJ/(kg·K) cp, kJ/ (kg·K) cp/cv Vel. Sound, m/s Viscosity, µPa·s Thermal Cond., mW/(m·K) Temp., °C Density, kg/m3 Enthalpy, kJ/kg Entropy, kJ/(kg·K) cp, kJ/ (kg·K) cp/cv Vel. Sound, m/s Viscosity, µPa·s Thermal Cond., mW/(m·K) −195.8a 4.6213 76.73 5.3993 1.341 1.430 172.2 5.19 6.99 150.0 0.8065 439.66 7.1966 1.047 1.397 419.0 22.80 33.72 −180.0 3.7557 94.62 5.6104 1.075 1.432 194.1 6.26 8.57 160.0 0.7879 450.13 7.2210 1.048 1.396 423.8 23.19 34.33 −160.0 3.0593 115.92 5.8177 1.059 1.419 215.3 7.59 10.58 170.0 0.7701 460.62 7.2450 1.049 1.396 428.6 23.56 34.93 −140.0 2.5860 137.03 5.9894 1.052 1.412 234.2 8.88 12.53 180.0 0.7531 471.11 7.2684 1.050 1.395 433.3 23.94 35.54 −120.0 2.2415 158.02 6.1363 1.048 1.409 251.6 10.12 14.35 190.0 0.7368 481.62 7.2913 1.051 1.394 438.0 24.31 36.14 −100.0 1.9789 178.95 6.2648 1.046 1.406 267.8 11.29 16.11 200.0 0.7213 492.13 7.3138 1.053 1.394 442.6 24.68 36.74 −90.0 1.8697 189.40 6.3235 1.045 1.405 275.6 11.86 16.97 210.0 0.7063 502.67 7.3358 1.054 1.393 447.1 25.04 37.34 −80.0 1.7719 199.85 6.3790 1.044 1.405 283.1 12.41 17.80 220.0 0.6920 513.21 7.3574 1.055 1.392 451.6 25.40 37.93 −70.0 1.6840 210.28 6.4316 1.043 1.404 290.4 12.95 18.62 230.0 0.6782 523.77 7.3786 1.057 1.391 456.0 25.75 38.53 −60.0 1.6044 220.72 6.4818 1.043 1.404 297.5 13.48 19.42 240.0 0.6650 534.35 7.3994 1.059 1.390 460.4 26.11 39.12 −50.0 1.5320 231.14 6.5296 1.042 1.403 304.4 14.00 20.21 250.0 0.6523 544.94 7.4199 1.060 1.389 464.7 26.45 39.71 −40.0 1.4660 241.56 6.5753 1.042 1.403 311.2 14.51 20.98 260.0 0.6401 555.55 7.4400 1.062 1.389 468.9 26.80 40.30 −30.0 1.4054 251.99 6.6190 1.042 1.403 317.9 15.01 21.73 270.0 0.6283 566.18 7.4597 1.064 1.388 473.2 27.14 40.89 −20.0 1.3496 262.40 6.6610 1.042 1.402 324.4 15.50 22.47 280.0 0.6169 576.83 7.4791 1.066 1.387 477.3 27.48 41.48 −10.0 1.2981 272.82 6.7014 1.042 1.402 330.7 15.98 23.20 290.0 0.6060 587.50 7.4983 1.068 1.386 481.4 27.82 42.07 0.0 1.2504 283.23 6.7402 1.041 1.402 337.0 16.46 23.92 300.0 0.5954 598.18 7.5171 1.070 1.385 485.5 28.15 42.66 10.0 1.2061 293.65 6.7777 1.041 1.402 343.1 16.92 24.63 310.0 0.5852 608.89 7.5356 1.072 1.384 489.6 28.48 43.25 20.0 1.1649 304.06 6.8138 1.041 1.401 349.1 17.38 25.32 320.0 0.5753 619.62 7.5538 1.074 1.383 493.6 28.81 43.83 30.0 1.1263 314.47 6.8487 1.041 1.401 355.0 17.83 26.01 330.0 0.5658 630.36 7.5718 1.076 1.381 497.5 29.14 44.42 40.0 1.0903 324.89 6.8825 1.041 1.401 360.8 18.28 26.69 340.0 0.5565 641.13 7.5895 1.078 1.380 501.4 29.46 45.00 50.0 1.0565 335.30 6.9153 1.042 1.401 366.5 18.72 27.36 350.0 0.5476 651.92 7.6070 1.080 1.379 505.3 29.79 45.59 60.0 1.0247 345.72 6.9470 1.042 1.400 372.1 19.15 28.02 360.0 0.5390 662.74 7.6242 1.082 1.378 509.1 30.11 46.17 70.0 0.9948 356.14 6.9778 1.042 1.400 377.7 19.58 28.68 370.0 0.5306 673.57 7.6412 1.085 1.377 512.9 30.42 46.76 80.0 0.9666 366.56 7.0077 1.042 1.400 383.1 20.00 29.33 380.0 0.5225 684.43 7.6579 1.087 1.376 516.7 30.74 47.34 90.0 0.9399 376.99 7.0369 1.043 1.400 388.5 20.42 29.97 390.0 0.5146 695.31 7.6745 1.089 1.375 520.4 31.05 47.93 100.0 0.9147 387.42 7.0652 1.043 1.399 393.7 20.83 30.61 400.0 0.5069 706.22 7.6908 1.092 1.374 524.1 31.36 48.51 110.0 0.8908 397.85 7.0928 1.044 1.399 398.9 21.23 31.24 420.0 0.4923 728.10 7.7228 1.096 1.371 531.4 31.98 49.68 120.0 0.8681 408.29 7.1197 1.044 1.398 404.0 21.63 31.86 440.0 0.4785 750.08 7.7541 1.101 1.369 538.6 32.58 50.84 130.0 0.8466 418.74 7.1459 1.045 1.398 409.1 22.03 32.48 460.0 0.4655 772.15 7.7846 1.106 1.367 545.6 33.19 52.00 140.0 0.8261 429.19 7.1716 1.046 1.397 414.1 22.42 33.10 480.0 0.4531 794.32 7.8144 1.111 1.365 552.6 33.78 53.16 500.0 0.4414 816.59 7.8436 1.116 1.363 559.4 34.37 54.31 a = saturated vapor at normal boiling point 20.58 2001 ASHRAE Fundamentals Handbook (SI) Fig. 26 Pressure-Enthalpy Diagram for Refrigerant 729 (Air) Thermophysical Properties of Refrigerants 20.59 Refrigerant 729 (Air) Properties of Liquid on the Bubble Line and Vapor on the Dew Line Absolute Pressure, MPa Temperature, K Density, kg/m3 Liquid Volume, m3/kg Vapor Enthalpy, kJ/kg Entropy, kJ/(kg·K) Specific Heat cp, kJ/(kg·K) cp/cv Velocity of Sound, m/s Viscosity, µPa·s Thermal Cond., mW/(m·K) Surface Tension, mN/m Bubble Dew Liquid Vapor Liquid Vapor Liquid Vapor Vapor Liquid Vapor Liquid Vapor Liquid Vapor 0.00625a 59.75 64.00 957.7 2.91977 –162.91 63.49 2.4486 6.1162 1.922 1.025 1.406 1013.
160.
440.3 4.40 174.7 5.62 14.12 0.00800 61.08 65.18 952.2 2.32465 –160.35 64.61 2.4909 6.0635 1.919 1.028 1.407 1004.
161.
399.8 4.49 172.1 5.75 13.80 0.01000 62.33 66.32 946.9 1.88989 –157.95 65.67 2.5298 6.0160 1.917 1.031 1.408 996.
162.
367.2 4.57 169.8 5.87 13.49 0.01500 64.72 68.51 936.8 1.29824 –153.38 67.69 2.6017 5.9306 1.914 1.038 1.410 979.
165.
316.2 4.73 165.3 6.10 12.91 0.02000 66.51 70.18 929.1 0.99507 –149.94 69.20 2.6540 5.8707 1.914 1.044 1.413 966.
167.
285.3 4.85 162.0 6.28 12.48 0.03000 69.20 72.70 917.6 0.68434 –144.79 71.44 2.7298 5.7872 1.915 1.054 1.418 945.
169.
248.0 5.04 157.2 6.55 11.84 0.04000 71.23 74.63 908.7 0.52481 –140.88 73.09 2.7853 5.7286 1.918 1.062 1.422 928.
171.
225.0 5.18 153.5 6.77 11.37 0.05000 72.89 76.20 901.4 0.42717 –137.69 74.41 2.8294 5.6834 1.921 1.070 1.427 914.
172.
208.9 5.30 150.6 6.95 10.98 0.06000 74.30 77.55 895.2 0.36102 –134.97 75.51 2.8662 5.6467 1.924 1.077 1.431 902.
174.
196.7 5.40 148.1 7.10 10.65 0.08000 76.65 79.78 884.7 0.27679 –130.44 77.28 2.9259 5.5891 1.930 1.089 1.439 882.
175.
179.0 5.57 144.1 7.36 10.11 0.10000 78.57 81.61 876.0 0.22518 –126.71 78.67 2.9737 5.5446 1.937 1.100 1.447 865.
177.
166.4 5.71 140.8 7.58 9.67 0.10132b 78.69 81.72 875.5 0.22245 –126.49 78.75 2.9766 5.5420 1.937 1.101 1.448 864.
177.
165.7 5.72 140.6 7.59 9.64 0.15000 82.32 85.20 858.6 0.15460 –119.38 81.22 3.0642 5.4641 1.954 1.126 1.466 830.
179.
145.7 6.00 134.4 8.02 8.82 0.20000 85.22 87.96 844.9 0.11825 –113.66 83.02 3.1317 5.4070 1.970 1.148 1.484 803.
181.
132.5 6.22 129.5 8.37 8.18 0.30000 89.68 92.23 823.0 0.08082 –104.76 85.46 3.2322 5.3262 2.003 1.190 1.520 760.
183.
115.4 6.57 122.2 8.95 7.21 0.40000 93.14 95.55 805.3 0.06152 –97.72 87.06 3.3078 5.2683 2.036 1.230 1.556 725.
184.
104.3 6.86 116.5 9.44 6.48 0.50000 96.01 98.32 790.0 0.04966 –91.78 88.15 3.3693 5.2227 2.070 1.270 1.593 695.
185.
96.1 7.11 111.8 9.87 5.88 0.60000 98.49 100.71 776.4 0.04161 –86.56 88.91 3.4216 5.1847 2.106 1.311 1.631 669.
185.
89.6 7.33 107.7 10.26 5.38 0.80000 102.67 104.74 752.2 0.03132 –77.57 89.74 3.5085 5.1228 2.180 1.398 1.712 623.
186.
79.8 7.72 100.9 10.98 4.55 1.00000 106.14 108.10 730.8 0.02498 –69.83 89.94 3.5800 5.0722 2.263 1.493 1.802 583.
186.
72.4 8.07 95.1 11.65 3.88 1.50000 113.06 114.76 683.2 0.01623 –53.52 88.62 3.7224 4.9703 2.518 1.796 2.088 499.
184.
59.2 8.87 83.5 13.31 2.62 2.00000 118.48 119.94 638.9 0.01163 –39.45 85.36 3.8373 4.8841 2.895 2.258 2.524 427.
182.
49.7 9.64 74.0 15.19 1.70 2.50000 123.02 124.25 593.8 0.00871 –26.22 80.29 3.9402 4.8013 3.533 3.068 3.278 362.
179.
42.0 10.50 65.7 17.82 1.01 3.00000 126.97 127.94 543.3 0.00661 –12.71 72.86 4.0412 4.7123 4.885 4.876 4.917 298.
177.
35.0 11.57 58.9 22.79 0.48 3.78502c 132.62 132.62 302.6 0.00331 38.10 38.10 4.4166 4.4166 — — — — — — — — — 0.00 Temperatures are on the IPTS-68 scale a = triple point b = bubble and dew points at 0.101 325 MPa c = critical point Refrigerant 729 (Air) Properties of Gas at 0.101 325 MPa (one standard atmosphere) Temp., °C Density, kg/m3 Enthalpy, kJ/kg Entropy, kJ/(kg·K) cp, kJ/ (kg·K) cp/cv Vel. Sound, m/s Viscosity, µPa·s Thermal Cond., mW/(m·K) Temp., °C Density, kg/m3 Enthalpy, kJ/kg Entropy, kJ/(kg·K) cp, kJ/ (kg·K) cp/cv Vel. Sound, m/s Viscosity, µPa·s Thermal Cond., mW/(m·K) –191.4a 4.4953 78.75 5.5420 1.101 1.448 176.9 5.72 7.59 150.0 0.8338 424.80 7.2170 1.017 1.394 411.7 23.75 34.30 –180.0 3.8899 91.02 5.6825 1.053 1.434 190.6 6.53 8.76 160.0 0.8145 434.98 7.2408 1.019 1.394 416.4 24.15 34.94 –160.0 3.1651 111.75 5.8842 1.026 1.421 211.7 7.92 10.80 170.0 0.7961 445.18 7.2640 1.020 1.393 421.1 24.55 35.57 –140.0 2.6745 132.15 6.0502 1.016 1.414 230.4 9.26 13.22 180.0 0.7786 455.39 7.2868 1.022 1.392 425.7 24.94 36.20 –120.0 2.3179 152.42 6.1921 1.012 1.410 247.6 10.54 14.62 190.0 0.7617 465.62 7.3091 1.024 1.391 430.2 25.32 36.82 –100.0 2.0463 172.63 6.3161 1.009 1.408 263.5 11.76 16.38 200.0 0.7456 475.86 7.3310 1.025 1.390 434.7 25.70 37.45 –90.0 1.9332 182.71 6.3727 1.008 1.407 271.1 12.35 17.23 210.0 0.7302 486.12 7.3525 1.027 1.389 439.1 26.08 38.07 –80.0 1.8321 192.79 6.4263 1.008 1.406 278.5 12.93 18.07 220.0 0.7154 496.40 7.3735 1.029 1.388 443.4 26.46 38.69 –70.0 1.7411 202.86 6.4771 1.007 1.406 285.7 13.49 18.90 230.0 0.7012 506.70 7.3942 1.031 1.387 447.8 26.83 39.31 –60.0 1.6588 212.93 6.5255 1.007 1.405 292.7 14.04 19.70 240.0 0.6875 517.01 7.4145 1.033 1.386 452.0 27.19 39.93 –50.0 1.5840 222.99 6.5717 1.006 1.405 299.5 14.58 20.49 250.0 0.6743 527.35 7.4345 1.035 1.385 456.2 27.56 40.55 –40.0 1.5156 233.06 6.6158 1.006 1.404 306.2 15.11 21.27 260.0 0.6617 537.71 7.4541 1.037 1.384 460.4 27.92 41.16 –30.0 1.4530 243.11 6.6580 1.006 1.404 312.7 15.63 22.03 270.0 0.6495 548.09 7.4734 1.039 1.383 464.5 28.28 41.78 –20.0 1.3953 253.17 6.6986 1.006 1.404 319.1 16.14 22.78 280.0 0.6378 558.49 7.4923 1.041 1.381 468.6 28.63 42.39 –10.0 1.3421 263.23 6.7375 1.006 1.403 325.4 16.65 23.52 290.0 0.6264 568.91 7.5110 1.043 1.380 472.6 28.98 43.00 0.0 1.2928 273.29 6.7750 1.006 1.403 331.5 17.14 24.23 300.0 0.6155 579.35 7.5294 1.045 1.379 476.6 29.33 43.61 10.0 1.2470 283.35 6.8112 1.006 1.402 337.5 17.63 24.95 310.0 0.6049 589.81 7.5475 1.048 1.378 480.5 29.67 44.22 20.0 1.2043 293.41 6.8461 1.006 1.402 343.4 18.10 25.66 320.0 0.5947 600.30 7.5653 1.050 1.377 484.4 30.02 44.83 30.0 1.1644 303.48 6.8799 1.007 1.402 349.2 18.58 26.36 330.0 0.5849 610.81 7.5829 1.052 1.376 488.3 30.36 45.44 40.0 1.1272 313.55 6.9126 1.007 1.401 354.9 19.04 27.06 340.0 0.5753 621.35 7.6002 1.055 1.375 492.1 30.69 46.05 50.0 1.0922 323.62 6.9443 1.008 1.401 360.5 19.50 27.74 350.0 0.5661 631.90 7.6173 1.057 1.373 495.9 31.03 46.66 60.0 1.0594 333.70 6.9750 1.008 1.400 365.9 19.95 28.42 360.0 0.5572 642.48 7.6341 1.059 1.372 499.6 31.36 47.26 70.0 1.0284 343.79 7.0048 1.009 1.400 371.3 20.39 29.10 370.0 0.5485 653.09 7.6508 1.062 1.371 503.4 31.69 47.87 80.0 0.9993 353.88 7.0338 1.010 1.399 376.7 20.83 29.76 380.0 0.5401 663.72 7.6672 1.064 1.370 507.1 32.02 48.47 90.0 0.9717 363.98 7.0620 1.011 1.399 381.9 21.27 30.42 390.0 0.5320 674.37 7.6833 1.066 1.369 510.7 32.35 49.07 100.0 0.9456 374.09 7.0895 1.011 1.398 387.0 21.69 31.08 400.0 0.5241 685.04 7.6993 1.069 1.368 514.3 32.67 49.67 110.0 0.9209 384.21 7.1162 1.012 1.397 392.1 22.12 31.73 420.0 0.5089 706.47 7.7307 1.074 1.365 521.5 33.31 50.87 120.0 0.8975 394.34 7.1423 1.014 1.397 397.1 22.53 32.38 440.0 0.4947 727.99 7.7613 1.078 1.363 528.5 33.95 52.07 130.0 0.8752 404.48 7.1678 1.015 1.396 402.0 22.95 33.02 460.0 0.4812 749.61 7.7912 1.083 1.361 535.4 34.57 53.26 140.0 0.8540 414.64 7.1927 1.016 1.395 406.9 23.35 33.66 480.0 0.4684 771.32 7.8204 1.088 1.359 542.3 35.19 54.45 500.0 0.4563 793.12 7.8490 1.093 1.357 549.0 35.80 55.64 a = dew point temperature 20.60 2001 ASHRAE Fundamentals Handbook (SI) Fig. 27 Pressure-Enthalpy Diagram for Refrigerant 732 (Oxygen) Thermophysical Properties of Refrigerants 20.61 Refrigerant 732 (Oxygen) Properties of Saturated Liquid and Saturated Vapor Temp., K Absolute Pressure, MPa Density, kg/m3 Liquid Volume, m3/kg Vapor Enthalpy, kJ/kg Entropy, kJ/(kg·K) Specific Heat cp, kJ/(kg·K) cp/cv Vapor Velocity of Sound, m/s Viscosity, µPa·s Thermal Cond., mW/(m·K) Surface Tension, mN/m Temp., K Liquid Vapor Liquid Vapor Liquid Vapor Liquid Vapor Liquid Vapor Liquid Vapor 54.36a 0.00015 1306.1 96.543 −193.61 49.11 2.0887 6.5537 1.673 0.926 1.395 1123.
140.3 982.0 4.01 202.9 05.04 22.68 054.36 55.00 0.00018 1303.5 79.987 −192.55 49.68 2.1083 6.5124 1.672 0.928 1.394 1127.
141.1 928.5 4.06 202.1 05.09 22.50 055.00 60.00 0.00073 1282.0 21.462 −184.19 54.19 2.2537 6.2266 1.673 0.948 1.390 1127.
147.0 630.3 4.42 195.2 5.54 21.12 60.00 65.00 0.00233 1259.7 7.2190 −175.81 58.66 2.3878 5.9950 1.677 0.967 1.387 1102.
152.6 462.0 4.78 188.1 6.00 19.76 065.00 70.00 0.00626 1237.0 2.8925 −167.42 63.09 2.5121 5.8051 1.678 0.978 1.387 1066.
158.1 358.8 5.15 180.9 6.46 18.41 070.00 075.00 0.01455 1213.9 1.3293 −159.02 67.45 2.6279 5.6476 1.679 0.979 1.392 1027.
163.3 290.6 5.53 173.8 6.94 17.09 75.00 080.00 0.03012 1190.5 0.6809 −150.61 71.69 2.7363 5.5151 1.682 0.974 1.402 987.
168.4 242.6 5.91 166.7 7.43 15.78 080.00 085.00 0.05683 1166.6 0.38047 −142.18 75.75 2.8383 5.4021 1.688 0.969 1.417 947.
173.1 207.1 6.30 159.6 7.93 14.49 085.00 090.00 0.09935 1142.1 0.22794 −133.69 79.55 2.9349 5.3042 1.699 0.970 1.436 906.
177.3 179.6 6.69 152.6 8.45 13.22 90.00 090.19b 0.10132 1141.2 0.22386 −133.37 79.69 2.9384 5.3008 1.699 0.971 1.437 904.
177.5 178.7 6.71 152.3 8.47 13.17 090.19 095.00 0.16308 1116.9 0.14450 −125.12 83.04 3.0269 5.2181 1.715 0.982 1.460 864.
181.0 157.7 7.09 145.5 9.00 11.98 095.00 100.00 0.25400 1090.9 0.09592 −116.45 86.16 3.1150 5.1411 1.738 1.006 1.491 822.
184.1 139.5 7.50 138.5 9.58 10.75 100.00 105.00 0.37853 1063.8 0.06612 −107.64 88.85 3.1999 5.0712 1.767 1.046 1.528 779.
186.4 124.2 7.92 131.4 10.19 09.56 105.00 110.00 0.54340 1035.5 0.04699 0−98.64 91.05 3.2821 5.0066 1.807 1.101 1.576 735.
188.1 111.0 8.35 124.3 10.86 08.39 110.00 115.00 0.75559 1005.6 0.03424 −89.42 92.72 3.3623 4.9460 1.858 1.177 1.638 689.
189.1 99.4 8.79 117.1 11.60 7.25 115.00 120.00 1.0223 0973.9 0.02544 0−79.90 93.75 3.4409 4.8881 1.927 1.276 1.721 642.
189.4 89.0 9.26 109.9 12.44 06.14 120.00 125.00 1.3509 0939.7 0.01919 0−70.02 94.06 3.5188 4.8314 2.021 1.411 1.835 592.
189.0 79.5 9.76 102.5 13.42 05.07 125.00 130.00 1.7491 902.5 0.01463 −59.66 93.47 3.5967 4.7746 2.153 1.600 2.000 540.
187.8 70.8 10.31 95.1 14.63 4.04 130.00 135.00 2.2250 0861.0 0.01120 0−48.65 91.74 3.6757 4.7157 2.354 1.886 2.252 484.
185.7 62.5 10.94 87.5 16.21 03.05 135.00 140.00 2.7878 0813.2 0.00856 0−36.70 88.47 3.7577 4.6518 2.691 2.370 2.682 423.
182.8 54.6 11.68 79.8 18.49 02.13 140.00 145.00 3.4478 755.1 0.00646 −23.22 82.83 3.8464 4.5777 3.368 3.369 3.561 355.
178.8 46.5 12.64 72.3 22.28 1.27 145.00 150.00 4.2186 0675.5 0.00465 00−6.67 72.56 3.9512 4.4794 5.464 6.625 6.314 274.
172.8 37.8 14.15 66.2 30.28 00.51 150.00 154.58c 5.0430 0436.1 0.00229 0032.42 32.42 4.1974 4.1974 ∞ ∞ ∞ 0.
0.0 — — ∞ ∞ 0.00 154.58 Temperatures are on the IPTS-68 scale a = triple point b = normal boiling point c = critical point Refrigerant 732 (Oxygen) Properties of Gas at 0.101 325 MPa (one standard atmosphere) Temp., °C Density, kg/m3 Enthalpy, kJ/kg Entropy, kJ/(kg·K) cp, kJ/ (kg·K) cp/cv Vel. Sound, m/s Viscosity, µPa·s Thermal Cond., mW/(m·K) Temp., °C Density, kg/m3 Enthalpy, kJ/kg Entropy, kJ/(kg·K) cp, kJ/ (kg·K) cp/cv Vel. Sound, m/s Viscosity, µPa·s Thermal Cond., mW/(m·K) 0−183.0a 4.4671 079.69 5.3008 0.971 1.437 177.5 06.71 08.47 150.0 0.9215 387.55 6.7299 0.948 1.379 389.3 27.00 34.83 −180.0 4.3120 082.52 5.3317 0.948 1.439 181.1 06.93 08.75 160.0 0.9002 397.05 6.7521 0.951 1.377 393.7 27.47 35.53 −160.0 3.5050 101.23 5.5137 0.931 1.422 201.0 8.46 10.64 170.0 0.8799 406.58 6.7738 0.954 1.375 398.0 27.94 36.23 −140.0 2.9596 119.78 5.6646 0.924 1.414 218.8 09.96 12.52 180.0 0.8605 416.13 6.7951 0.957 1.373 402.2 28.40 36.92 −120.0 2.5640 138.20 5.7936 0.919 1.409 235.2 11.43 14.38 190.0 0.8419 425.73 6.8161 0.961 1.372 406.4 28.86 37.61 −100.0 2.2630 156.56 5.9062 0.916 1.407 250.4 12.85 16.12 200.0 0.8241 435.35 6.8366 0.964 1.370 410.5 29.31 38.30 0−90.0 2.1378 165.72 5.9577 0.916 1.406 257.6 13.54 16.98 210.0 0.8070 445.00 6.8568 0.967 1.368 414.5 29.76 38.99 0−80.0 2.0258 174.87 6.0063 0.915 1.405 264.6 14.21 17.82 220.0 0.7906 454.68 6.8766 0.970 1.367 418.5 30.21 39.67 −70.0 1.9251 184.02 6.0525 0.915 1.404 271.5 14.88 18.65 230.0 0.7749 464.40 6.8962 0.973 1.365 422.5 30.65 40.35 0−60.0 1.8340 193.16 6.0964 0.914 1.404 278.1 15.53 19.46 240.0 0.7598 474.14 6.9153 0.976 1.363 426.4 31.08 41.03 0−50.0 1.7512 202.30 6.1383 0.914 1.403 284.6 16.16 20.26 250.0 0.7453 483.92 6.9342 0.980 1.362 430.3 31.52 41.71 −40.0 1.6756 211.45 6.1784 0.914 1.402 290.9 16.79 21.05 260.0 0.7313 493.74 6.9528 0.983 1.360 434.2 31.95 42.39 0−30.0 1.6063 220.59 6.2168 0.915 1.401 297.1 17.40 21.83 270.0 0.7178 503.58 6.9711 0.986 1.359 438.0 32.37 43.06 0−20.0 1.5425 229.74 6.2537 0.915 1.401 303.1 18.00 22.60 280.0 0.7048 513.45 6.9891 0.989 1.357 441.7 32.79 43.73 −10.0 1.4836 238.90 6.2892 0.916 1.400 309.0 18.60 23.35 290.0 0.6923 523.36 7.0068 0.992 1.355 445.4 33.21 44.40 0000.0 1.4290 248.06 6.3233 0.917 1.399 314.8 19.18 24.10 300.0 0.6802 533.29 7.0243 0.995 1.354 449.1 33.62 45.06 0010.0 1.3784 257.23 6.3563 0.918 1.398 320.5 19.75 24.84 310.0 0.6686 543.26 7.0416 0.998 1.352 452.8 34.03 45.72 20.0 1.3312 266.41 6.3882 0.919 1.397 326.0 20.32 25.58 320.0 0.6573 553.26 7.0586 1.001 1.351 456.4 34.44 46.38 0030.0 1.2871 275.61 6.4190 0.920 1.396 331.4 20.87 26.30 330.0 0.6464 563.28 7.0753 1.004 1.350 460.0 34.84 47.03 0040.0 1.2459 284.82 6.4489 0.922 1.395 336.7 21.42 27.03 340.0 0.6358 573.34 7.0919 1.007 1.348 463.6 35.24 47.68 50.0 1.2072 294.05 6.4779 0.924 1.394 341.9 21.96 27.75 350.0 0.6256 583.43 7.1082 1.010 1.347 467.1 35.64 48.33 0060.0 1.1709 303.29 6.5061 0.926 1.392 347.1 22.49 28.47 360.0 0.6157 593.54 7.1243 1.013 1.345 470.6 36.03 48.98 0070.0 1.1367 312.56 6.5335 0.928 1.391 352.1 23.02 29.19 370.0 0.6062 603.68 7.1402 1.016 1.344 474.1 36.42 49.62 80.0 1.1045 321.85 6.5602 0.930 1.390 357.0 23.54 29.90 380.0 0.5969 613.86 7.1559 1.019 1.343 477.5 36.81 50.26 0090.0 1.0740 331.16 6.5862 0.932 1.388 361.9 24.05 30.61 390.0 0.5879 624.05 7.1714 1.021 1.342 480.9 37.20 50.89 0100.0 1.0452 340.49 6.6116 0.935 1.387 366.6 24.56 31.32 400.0 0.5792 634.28 7.1867 1.024 1.340 484.3 37.58 51.53 110.0 1.0179 349.85 6.6363 0.937 1.385 371.3 25.06 32.02 420.0 0.5624 654.81 7.2167 1.029 1.338 491.0 38.34 52.78 0120.0 0.9919 359.23 6.6605 0.940 1.384 375.9 25.55 32.73 440.0 0.5467 675.45 7.2461 1.034 1.336 497.6 39.09 54.02 0130.0 0.9673 368.64 6.6841 0.943 1.382 380.5 26.04 33.43 460.0 0.5318 696.19 7.2748 1.039 1.334 504.2 39.82 55.25 140.0 0.9439 378.08 6.7072 0.945 1.380 384.9 26.52 34.13 480.0 0.5176 717.02 7.3028 1.044 1.332 510.6 40.55 56.47 500.0 0.5042 737.95 7.3302 1.049 1.330 517.0 41.27 57.67 a = saturated vapor at normal boiling point 20.62 2001 ASHRAE Fundamentals Handbook (SI) Fig. 28 Pressure-Enthalpy Diagram for Refrigerant 740 (Argon) Thermophysical Properties of Refrigerants 20.63 Refrigerant 740 (Argon) Properties of Saturated Liquid and Saturated Vapor Temp., K Absolute Pressure, MPa Density, kg/m3 Liquid Volume, m3/kg Vapor Enthalpy, kJ/kg Entropy, kJ/(kg·K) Specific Heat cp, kJ/(kg·K) cp/cv Vapor Velocity of Sound, m/s Viscosity, µPa·s Thermal Cond., mW/(m·K) Surface Tension, mN/m Temp., K Liquid Vapor Liquid Vapor Liquid Vapor Liquid Vapor Liquid Vapor Liquid Vapor 83.80a 0.06896 1417.2 0.24653 −121.05 42.59 1.3314 3.2841 1.067 — — 853.
208.8 282.7 6.68 135.6 5.25 13.42 083.80 84.00 0.07053 1416.0 0.24148 −120.84 42.65 1.3339 3.2803 1.067 — — 852.
207.4 281.0 6.69 135.3 5.27 13.37 084.00 86.00 0.08820 1404.1 0.19666 −118.70 43.29 1.3591 3.2426 1.074 — — 839.
196.7 264.7 6.86 132.3 5.41 12.86 86.00 87.29b 0.10132 1396.3 0.17312 −117.30 43.69 1.3751 3.2193 1.078 — — 830.
192.3 254.9 6.97 130.3 5.51 12.53 087.29 88.00 0.10910 1392.0 0.16174 −116.53 43.91 1.3838 3.2069 1.081 — 825.
190.4 249.7 7.04 129.3 5.56 12.35 088.00 90.00 0.13362 1379.7 0.13423 −114.35 44.50 1.4081 3.1730 1.089 — — 812.
186.4 235.9 7.21 126.3 5.71 11.85 90.00 92.00 0.16212 1367.2 0.11233 −112.15 45.06 1.4320 3.1408 1.097 — — 798.
183.8 223.2 7.39 123.4 5.86 11.35 092.00 94.00 0.19500 1354.5 0.09473 −109.94 45.59 1.4555 3.1100 1.106 — — 784.
182.2 211.4 7.57 120.5 6.02 10.86 094.00 96.00 0.23266 1341.6 0.08045 −107.70 46.08 1.4788 3.0807 1.116 — — 771.
181.2 200.4 7.75 117.6 6.19 10.37 96.00 98.00 0.27553 1328.4 0.06877 −105.44 46.55 1.5018 3.0526 1.127 — — 757.
180.6 190.2 7.93 114.7 6.36 9.89 098.00 100.00 0.32400 1315.0 0.05914 −103.16 46.97 1.5245 3.0257 1.138 0.617 1.854 742.
180.3 180.7 8.12 111.8 6.53 9.41 100.00 102.00 0.37853 1301.3 0.05114 −100.85 47.35 1.5469 2.9998 1.151 0.644 1.856 728.
180.2 171.8 8.31 109.0 6.71 8.94 102.00 104.00 0.43952 1287.4 0.04445 0−98.51 47.68 1.5691 2.9748 1.165 0.670 1.864 714.
180.3 163.4 8.51 106.2 6.90 8.47 104.00 106.00 0.50743 1273.1 0.03881 0−96.15 47.96 1.5912 2.9507 1.179 0.695 1.879 699.
180.5 155.5 8.71 103.4 7.10 8.01 106.00 108.00 0.58268 1258.6 0.03403 −93.75 48.19 1.6130 2.9272 1.195 0.720 1.900 684.
180.8 148.0 8.92 100.6 7.31 7.56 108.00 110.00 0.66574 1243.7 0.02996 −91.32 48.35 1.6347 2.9044 1.213 0.746 1.926 669.
181.0 141.0 9.13 097.9 7.53 7.11 110.00 112.00 0.75704 1228.5 0.02646 −88.85 48.44 1.6562 2.8821 1.232 0.773 1.959 653.
181.3 134.3 9.35 095.2 7.76 6.66 112.00 114.00 0.85705 1212.9 0.02345 −86.35 48.46 1.6777 2.8602 1.253 0.801 1.997 637.
181.6 127.9 9.57 92.5 8.00 6.23 114.00 116.00 0.96622 1196.9 0.02085 −83.80 48.40 1.6990 2.8387 1.277 0.832 2.043 621.
181.9 121.9 9.80 089.8 8.25 5.80 116.00 118.00 1.0850 1180.4 0.01858 −81.21 48.25 1.7204 2.8174 1.303 0.867 2.096 604.
182.1 116.1 10.05 087.1 8.52 5.37 118.00 120.00 1.2139 1163.4 0.01659 −78.56 48.01 1.7417 2.7964 1.333 0.905 2.158 587.
182.3 110.6 10.30 84.4 8.81 4.95 120.00 125.00 1.5835 1118.4 0.01260 −71.69 46.92 1.7951 2.7440 1.429 1.026 2.363 541.
182.5 097.6 10.98 077.8 9.62 3.95 125.00 130.00 2.0270 1068.5 0.00964 −64.33 45.01 1.8496 2.6907 1.572 1.209 2.676 491.
182.3 085.7 11.77 071.1 10.64 2.99 130.00 135.00 2.5530 1011.5 0.00737 −56.29 41.97 1.9065 2.6344 1.809 1.515 3.195 434.
181.4 74.5 12.74 64.3 12.02 2.10 135.00 140.00 3.1710 0942.4 0.00558 −47.15 37.26 1.9684 2.5713 2.277 2.120 4.201 369.
179.8 063.3 14.00 057.4 14.17 1.28 140.00 145.00 3.8929 0849.1 0.00408 −35.87 29.57 2.0418 2.4931 3.577 3.807 6.895 290.
177.0 051.4 15.92 051.2 18.85 0.57 145.00 150.66c 4.8600 530.9 0.00188 −3.56 −3.56 2.2500 2.2500 ∞ ∞ ∞ 0.
0.0 — — ∞ ∞ 0.00 150.66 Temperatures are on the IPTS-68 scale a = triple point b = normal boiling point c = critical point Refrigerant 740 (Argon) Properties of Gas at 0.101 325 MPa (one standard atmosphere) Temp., °C Density, kg/m3 Enthalpy, kJ/kg Entropy, kJ/(kg·K) cp, kJ/ (kg·K) cp/cv Vel. Sound, m/s Viscos-ity, µPa·s Thermal Cond., mW/(m·K) Temp., °C Density, kg/m3 Enthalpy, kJ/kg Entropy, kJ/(kg·K) cp, kJ/ (kg·K) cp/cv Vel. Sound, m/s Viscos-ity, µPa·s Thermal Cond., mW/(m·K) −185.86a 5.7764 043.69 3.2193 — — 192.3 06.97 5.51 150.0 1.1505 220.08 4.0527 0.521 1.668 383.3 30.06 23.47 −180.0 5.3850 046.63 3.2518 0.554 1.729 177.6 07.45 5.87 160.0 1.1239 225.29 4.0648 0.521 1.668 387.8 30.61 23.90 −160.0 4.3729 057.75 3.3600 0.543 1.689 196.2 09.07 7.14 170.0 1.0985 230.49 4.0767 0.521 1.668 392.2 31.14 24.31 −140.0 3.6931 068.48 3.4474 0.532 1.685 213.9 10.71 8.45 180.0 1.0742 235.70 4.0884 0.521 1.668 396.6 31.67 24.73 −120.0 3.2000 079.06 3.5214 0.527 1.681 229.9 12.33 9.97 190.0 1.0510 240.91 4.0997 0.521 1.668 401.0 32.20 25.14 −100.0 2.8246 089.58 3.5860 0.525 1.677 244.7 13.92 10.93 200.0 1.0288 246.12 4.1109 0.521 1.668 405.3 32.72 25.54 −90.0 2.6684 094.83 3.6155 0.525 1.676 251.8 14.69 11.51 210.0 1.0075 251.32 4.1217 0.521 1.668 409.5 33.23 25.94 −80.0 2.5288 100.08 3.6433 0.524 1.675 258.6 15.46 12.09 220.0 0.9871 256.53 4.1324 0.521 1.667 413.8 33.74 26.34 −70.0 2.4031 105.31 3.6698 0.524 1.674 265.3 16.21 12.67 230.0 0.9674 261.74 4.1429 0.521 1.667 417.9 34.25 26.74 −60.0 2.2894 110.55 3.6949 0.523 1.673 271.8 16.95 13.25 240.0 0.9486 266.94 4.1531 0.521 1.667 422.1 34.75 27.13 −50.0 2.1861 115.78 3.7189 0.523 1.672 278.1 17.67 13.81 250.0 0.9304 272.15 4.1631 0.521 1.667 426.2 35.24 27.51 −40.0 2.0917 121.01 3.7418 0.523 1.672 284.3 18.38 14.37 260.0 0.9130 277.36 4.1730 0.521 1.667 430.2 35.74 27.90 −30.0 2.0052 126.23 3.7638 0.522 1.671 290.4 19.08 14.91 270.0 0.8962 282.56 4.1827 0.521 1.667 434.2 36.22 28.28 −20.0 1.9256 131.45 3.7848 0.522 1.671 296.3 19.77 15.45 280.0 0.8799 287.77 4.1922 0.521 1.667 438.2 36.71 28.66 −10.0 1.8521 136.67 3.8051 0.522 1.670 302.1 20.45 15.98 290.0 0.8643 292.98 4.2015 0.521 1.667 442.1 37.18 29.03 0.0 1.7840 141.89 3.8245 0.522 1.670 307.8 21.12 16.50 300.0 0.8492 298.18 4.2107 0.521 1.667 446.1 37.66 29.40 10.0 1.7207 147.11 3.8433 0.522 1.670 313.4 21.77 17.01 310.0 0.8347 303.39 4.2197 0.521 1.667 449.9 38.13 29.77 20.0 1.6619 152.33 3.8614 0.522 1.670 318.9 22.42 17.51 320.0 0.8206 308.59 4.2285 0.521 1.667 453.8 38.60 30.13 30.0 1.6069 157.54 3.8789 0.522 1.669 324.4 23.05 18.01 330.0 0.8070 313.80 4.2372 0.521 1.667 457.6 39.06 30.49 40.0 1.5554 162.76 3.8958 0.521 1.669 329.7 23.68 18.49 340.0 0.7938 319.00 4.2458 0.521 1.667 461.4 39.52 30.85 50.0 1.5072 167.97 3.9122 0.521 1.669 334.9 24.30 18.98 350.0 0.7811 324.21 4.2542 0.521 1.667 465.1 39.98 31.21 60.0 1.4618 173.19 3.9281 0.521 1.669 340.0 24.91 19.45 360.0 0.7687 329.41 4.2625 0.521 1.667 468.8 40.43 31.56 70.0 1.4191 178.40 3.9435 0.521 1.669 345.1 25.51 19.92 370.0 0.7568 334.62 4.2707 0.521 1.667 472.5 40.88 31.91 80.0 1.3789 183.61 3.9585 0.521 1.669 350.1 26.10 20.38 380.0 0.7452 339.82 4.2787 0.521 1.667 476.2 41.33 32.26 90.0 1.3408 188.82 3.9730 0.521 1.668 355.0 26.69 20.84 390.0 0.7340 345.03 4.2866 0.521 1.667 479.8 41.77 32.61 100.0 1.3048 194.03 3.9872 0.521 1.668 359.9 27.27 21.29 400.0 0.7231 350.24 4.2944 0.520 1.667 483.4 42.21 32.95 110.0 1.2707 199.24 4.0010 0.521 1.668 364.7 27.84 21.74 420.0 0.7022 360.64 4.3096 0.520 1.667 490.5 43.08 33.63 120.0 1.2384 204.45 4.0144 0.521 1.668 369.4 28.40 22.18 440.0 0.6825 371.05 4.3244 0.520 1.667 497.5 43.94 34.30 130.0 1.2076 209.66 4.0275 0.521 1.668 374.1 28.96 22.62 460.0 0.6639 381.46 4.3388 0.520 1.667 504.5 44.78 34.96 140.0 1.1784 214.87 4.0402 0.521 1.668 378.7 29.52 23.05 480.0 0.6462 391.87 4.3528 0.520 1.667 511.3 45.62 35.61 500.0 0.6295 402.28 4.3665 0.520 1.667 518.0 46.45 36.25 a = saturated vapor at normal boiling point 20.64 2001 ASHRAE Fundamentals Handbook (SI) Fig. 29 Enthalpy-Concentration Diagram for Ammonia-Water Solutions Prepared by: Kwang Kim and Keith Herold, Center for Environmental Energy Engineering, University of Maryland at College Park Thermophysical Properties of Refrigerants 20.65 Specific Volume of Saturated Ammonia Solutions, m3/kg Temp., °C Concentration, Ammonia (Mass basis) Temp., °C 0 10 20 30 40 50 60 70 80 90 100 −10 0.00100 0.00103 0.00106 0.00109 0.00114 0.00118 0.00122 0.00128 0.00135 0.00142 0.00151 −10 0 0.00100 0.00103 0.00107 0.00110 0.00114 0.00119 0.00124 0.00130 0.00137 0.00146 0.00156 0 10 0.00100 0.00104 0.00107 0.00111 0.00115 0.00120 0.00125 0.00132 0.00139 0.00149 0.00160 10 20 0.00100 0.00104 0.00108 0.00112 0.00116 0.00121 0.00127 0.00133 0.00142 0.00152 0.00164 20 30 0.00100 0.00105 0.00108 0.00113 0.00117 0.00123 0.00128 0.00135 0.00145 0.00156 0.00168 30 40 0.00101 0.00105 0.00109 0.00114 0.00119 0.00124 0.00130 0.00138 0.00148 0.00159 0.00173 40 50 0.00101 0.00106 0.00110 0.00115 0.00120 0.00125 0.00132 0.00140 0.00151 0.00163 0.00177 50 60 0.00102 0.00106 0.00111 0.00116 0.00121 0.00127 0.00134 0.00143 0.00154 0.00167 0.00183 60 70 0.00102 0.00107 0.00112 0.00117 0.00122 0.00129 0.00136 0.00146 0.00158 0.00172 0.00190 70 80 0.00103 0.00108 0.00113 0.00118 0.00124 0.00130 0.00139 0.00149 0.00162 0.00178 0.00198 80 90 0.00104 0.00109 0.00114 0.00119 0.00125 0.00132 0.00141 0.00153 0.00167 0.00184 0.00208 90 100 0.00104 0.00110 0.00115 0.00121 0.00127 0.00135 0.00145 0.00157 0.00172 0.00191 0.00219 100 Prepared under Research Project No. 271-RP, sponsored by TC 8.3.
Data reference: B.H. Jennings, Ammonia water properties (paper presented at ASHRAE meeting, January 1965).
Refrigerant Temperature (t′ = °C) and Enthalpy (h = kJ/kg) of Lithium Bromide Solutions Temp., (t = °C) Percent LiBr 0 10 20 30 40 45 50 55 60 65 70 20 t′ 20 19.1 17.7 15.0 9.8 5.8 −0.4 −7.7 −15.8 −23.4# −29.3# h 84.0 67.4 52.6 40.4 33.5 33.5 38.9 53.2 78.0 111.0# 145.0# 30 t′ 30.0 29.0 27.5 24.6 19.2 15.0 8.6 1.0 −7.3 −15.2# −21.6# h 125.8 103.3 84.0 68.6 58.3 56.8 60.5 73.5 96.8 128.4# 161.7# 40 t′ 40.0 38.9 37.3 34.3 28.5 24.1 17.5 9.8 1.3 −7.0# −14.0# h 167.6 139.5 115.8 96.0 82.5 79.7 82.2 93.5 115.4 146.0# 178.3# 50 t′ 50.0 48.8 47.2 44.0 37.9 33.3 26.5 18.5 9.9 1.3 −6.3# h 209.3 175.2 147.0 123.4 106.7 102.6 103.8 114.0 134.5 163.5 195.0# 60 t′ 60.0 58.8 57.0 53.6 47.3 42.5 35.5 27.3 18.4 9.5 1.4# h 251.1 211.7 179.1 151.4 131.7 125.8 125.8 134.7 153.7 181.4 211.9# 70 t′ 70.0 68.7 66.8 63.3 56.6 51.6 44.4 36.1 27.0 17.7 9.0# h 293.0 247.7 210.5 178.8 155.7 148.9 148.0 155.6 173.2 199.4 228.8# 80 t′ 80.0 78.6 76.7 73.0 66.0 60.8 53.4 44.8 35.6 26.0 16.7# h 334.9 287.8 243.6 207.3 181.0 172.8 170.0 176.2 192.6 217.2 245.7# 90 t′ 90.0 88.6 86.5 82.6 75.4 70.0 62.3 53.6 44.1 34.2 24.3# h 376.9 321.1 275.6 235.4 206.1 195.8 192.3 197.1 212.2 235.6 262.9# 100 t′ 100.0 98.5 96.3 92.3 84.7 79.1 71.3 62.4 52.7 42.4 32.0 h 419.0 357.6 307.9 263.8 231.0 219.9 214.6 218.2 231.5 253.5 279.7 110 t′ 110.0 108.4 106.2 101.9 94.1 88.3 80.2 71.1 61.3 50.6 39.7 h 461.3 394.3 340.1 292.4 255.9 243.3 236.8 239.1 251.0 271.4 296.3 120 t′ 120.0 118.3 116.0 111.6 103.4 97.5 89.2 79.9 69.8 58.9 47.3 h 503.7 431.0 372.5 320.9 281.0 267.0 259.0 260.0 270.2 289.5 313.4 130 t′ 130.0 128.3 125.8 121.3 112.8 106.7 92.8 88.7 78.4 67.1 55.0 h 546.5 468.4 404.5 349.6 306.2 290.7 281.0 280.4 289.1 306.9 330.2 140 t′ 140.0 138.2 135.7 130.9 122.2 115.8 107.1 97.4 87.0 75.3 62.7 h 589.1 505.6 437.8 377.9 331.3 314.2 303.2 301.1 308.1 324.7 346.9 150 t′ 150.0 148.1 145.5 140.6 131.5 125.0 116.1 106.2 95.5 83.5 70.3 h 632.2 542.7 470.5 406.8 356.6 337.8 325.5 321.6 327.3 342.7 363.6 160 t′ 160.0 158.1 155.3 150.3 140.9 134.2 125.0 115.0 104.1 91.8 78.9 h 675.6 580.8 503.1 435.4 381.9 361.2 347.7 342.2 346.1 360.3 380.1 170 t′ 170.0 168.0 165.2 159.9 150.3 143.3 134.0 123.7 112.7 100.0 85.7 h 719.2 618.9 536.1 464.3 406.8 384.9 369.9 362.9 365.4 378.3 396.0 180 t′ 180.0 177.9 175.0 169.6 159.6 152.5 142.9 132.5 121.2 108.2 93.3 h 763.2 657.1 569.4 493.4 432.1 408.8 392.1 383.4 384.3 395.8 411.3 Extensions of data above 115°C are well above the original data and should be used with care.
Supersaturated solution.
20.66 2001 ASHRAE Fundamentals Handbook (SI) Fig. 30 Enthalpy-Concentration Diagram for Water-Lithium Bromide Solutions Thermophysical Properties of Refrigerants 20.67 Fig. 31 Equilibrium Chart for Aqueous Lithium Bromide Solutions Reprinted by permission of Carier Corp.
20.68 2001 ASHRAE Fundamentals Handbook (SI) REFERENCES Except where noted below, tables for the halocarbon refrigerants and their blends have been calculated using: McLinden, M.O., S.A. Klein, E.W. Lemmon, and A.P. Peskin. 2000a. NIST Standard Reference Database 23: Thermodynamic and transport proper-ties of refrigerants and refrigerant mixtures—REFPROP, Version 6.10.
Standard Reference Data Program, National Institute of Standards and Technology, Gaithersburg, MD.
Tables for the hydrocarbons and inorganic fluids have been cal-culated using: Lemmon, E.W., A.P. Peskin, M.O. McLinden, and D.G. Friend. 2000. NIST Standard Reference Database 12: Thermodynamic and transport proper-ties of pure fluids, Version 5.0. Standard Reference Data Program.
The underlying sources for these computer packages are listed below by fluid and property. The reference listed under “Equation of state” was used for vapor pressure, liquid density, vapor volume, enthalpy, entropy, specific heat, and velocity of sound. R-12 Equation of state Marx, V., A. Pruß, and W. Wagner. 1992. Neue Zustandsgleichungen für R-12, R-22, R-11 und R-113. Beschreibung des thermodynamishchen Zustandsverhaltens bei Temperaturen bis 525 K und Drücken bis 200 MPa. VDI-Fortschritt-Ber 19(57). VDI Verlag, Düsseldorf.
Viscosity Klein, S.A., M.O. McLinden, and A. Laesecke. 1997. An improved extended corresponding states method for estimation of viscosity of pure refrigerants and mixtures. Int. Journal of Refrigeration 20:208-17.
Thermal conductivity McLinden, M.O., S.A. Klein, and R.A. Perkins. 2000b. An extended cor-responding states model for the thermal conductivity of refrigerants and refrigerant mixtures. International Journal of Refrigeration 23:43-63.
Surface tension Okada, M. and K. Watanabe. 1988. Surface tension correlations for several fluorocarbon refrigerants, Heat Transfer—Japanese Research 17:35-52.
R-22 Equation of state Kamei, A., S.W. Beyerlein, and R.T. Jacobsen. 1995. Application of non-linear regression in the development of a wide range formulation for HCFC-22. International Journal of Thermophysics 16(5):1155-64.
Viscosity Klein et al. 1997. op. cit.
Thermal conductivity McLinden et al. 2000b. op. cit.
Surface tension Okada, M. and K. Watanabe. 1988. Surface tension correlations for several fluorocarbon refrigerants. Heat Transfer—Japanese Research 17:35-52.
R-23 Data calculated using computer progams based on Lemmon, E.W., R.T. Jacobsen., S.G. Penoncello, and S.W. Beyerlein.
1994. Computer programs for the calculation of thermodynamic properties of cryogens and other fluids. Advances in Cryogenic Engineering 39:1891-97.
Equation of state Penoncello, S.G., Z. Shan, and R.T. Jacobsen. 2000. A fundamental equa-tion for the calculation of the thermodynamic properties of trifluoro-methane (R-23). ASHRAE Transactions 106(1). Viscosity and thermal conductivity Shan, Z., S.G. Penoncello, and R.T. Jacobsen. 2000. A generalized model for viscosity and thermal conductivity of trifluoromethane (R-23).
ASHRAE Transactions 106(1). Surface tension Penoncello, S.G. 1999. Thermophysical properties of trifluoromethane (R-23). Final report ASHRAE Research Project 997.
R-32 Equation of state Tillner-Roth, R. and A. Yokozeki. 1997. An international standard equa-tion of state for difluoromethane (R-32) for temperatures from the triple point at 136.34 K to 435 K and pressures up to 70 MPa. Journal of Physical and Chemical Reference Data 26:1273-1328.
Viscosity Klein et al. 1997. op. cit.
Thermal conductivity McLinden et al. 2000b. op. cit.
Surface tension Okada, M. and Y. Higashi. 1995. Experimental surface tensions for HFC-32, HCFC-124, HFC-125, HCFC-141b, HCFC-142b, and HFC-152a. International Journal of Thermophysics 16(3):791-800.
R-123 Equation of state Younglove, B.A. and M.O. McLinden. 1994. An international standard equation-of-state formulation of the thermodynamic properties of refriger-ant 123 (2,2-dichloro-1,1,1-trifluoroethane). Journal of Physical and Chemical Reference Data 23(5):731-79.
Viscosity Tanaka, Y. and T. Sotani. 1995. Chapter 2: Transport properties (thermal conductivity and viscosity), R-123. Thermodynamic and Physical Proper-ties. International Institute of Refrigeration, Paris.
Thermal conductivity Laesecke, A., R.A. Perkins, and J.B. Howley. 1996. An improved correla-tion for the thermal conductivity of HCFC-123 (2,2-dichloro-1,1,1-triflu-oroethane). International Journal of Refrigeration 19:231-38.
Surface tension Okada and Higashi. 1995. op. cit.
R-124 Equation of state de Vries, B., R. Tillner-Roth, and H.D. Baehr. 1995. Thermodynamic prop-erties of HCFC-124. 19th International Congress of Refrigeration, Inter-national Institute of Refrigeration, IVa:582-89.
Viscosity Klein et al. 1997. op. cit.
Thermal conductivity McLinden et al. 2000b. op. cit.
Surface tension Okada and Higashi. 1995. op. cit.
R-125 Equation of state Sunaga, H., R. Tillner-Roth, H. Sato, and K. Watanabe. 1998. A thermo-dynamic equation of state for pentafluoroethane (R-125). International Journal of Thermophysics 19:1623-35.
Viscosity Klein et al. 1997. op. cit.
Thermal conductivity McLinden et al. 2000b. op. cit.
Surface tension Okada and Higashi. 1995. op. cit.
R-134a Equation of state Tillner-Roth, R. and H.D. Baehr. 1994. An international standard formula-tion of the thermodynamic properties of 1,1,1,2-tetrafluoroethane (HFC-134a) covering temperatures from 170 K to 455 K at pressures up to 70 MPa. Journal of Physical and Chemical Reference Data 23:657-729.
Viscosity Laesecke, A. 2000. Data reassessment and full surface correlation of the viscosity of HFC-134a (1,1,1,2-tetrafluoroethane). Journal of Physical and Chemical Reference Data (submitted).
Thermal conductivity McLinden et al. 2000b. op. cit.
Surface tension Okada, M. and Y. Higashi. 1994. Surface tension correlation of HFC-134a and HCFC-123. CFCs, The Day After. Proceedings of Joint Meeting of IIR Commissions B1, B2, E1, and E2, 541-48.
R-143a Equation of state Lemmon, E.W. and R.T. Jacobsen. 2001. An international standard formu-lation for the thermodynamic properties of 1,1,1-trifluoroethane (HFC-143a) for temperatures from 161 to 500 K and pressures to 60 MPa.
Journal of Physical and Chemical Reference Data (in press).
Viscosity Klein et al. 1997. op. cit.
Thermal conductivity McLinden et al. 2000b. op. cit.
Thermophysical Properties of Refrigerants 20.69 Surface tension Schmidt, J.W., E. Carrillo-Nava, and M.R. Moldover. 1996. Partially halo-genated hydrocarbons CHFCl-CF3, CF3-CH3, CF3-CHF-CHF2, CF3-CH2-CF3, CHF2-CF2-CH2F, CF3-CH2-CHF2, CF3-O-CHF2: Critical tempera-ture, refractive indices, surface tension and estimates of liquid, vapor and critical densities. Fluid Phase Equilibria 122:187-206.
R-152a Equation of state Outcalt, S.L. and M.O. McLinden. 1996. A modified Benedict-Webb-Rubin equation of state for the thermodynamic properties of R152a (1,1-difluoroethane). Journal of Physical and Chemical Reference Data 25(2):605-36.
Viscosity and thermal conductivity Krauss, R., V.C. Weiss, T.A. Edison, J.V. Sengers, and K. Stephan. 1996.
Transport properties of 1,1-Difluoroethane (R-152a). International Jour-nal of Thermophysics 17:731-57.
Surface tension Okada and Higashi. 1995. op. cit.
R-245fa Equation of state Extended corresponding states model of: Huber, M.L. and J.F. Ely. 1994. A predictive extended corresponding states model for pure and mixed refrigerants including an equation of state for R-134a. International Journal of Refrigeration 17:18-31.
Fitted to the data of: Defibaugh, D.R. and M.R. Moldover. 1997. Compressed and saturated liq-uid densities for 18 halogenated organic compounds. Journal of Chemical and Engineering Data 42:160-68.
Viscosity Klein et al. 1997. op. cit.
Thermal conductivity McLinden et al. 2000b. op. cit.
Surface tension Schmidt et al. 1996. op. cit.
R-404A Equation of state Lemmon, E.W. and R.T. Jacobsen. 1999. Thermodynamic properties of mixtures of R-32, R-125, R-134a, and R-152a. International Journal of Thermophysics 20:1629-38.
Viscosity Klein et al. 1997. op. cit.
Thermal conductivity McLinden et al. 2000b. op. cit.
Surface tension Moldover, M.R. and J.C. Rainwater. 1988. Interfacial tension and vapor-liquid equilibria in the critical region of mixtures. J. Chem. Phys. 88: 7772-80.
R-407C Equation of state Lemmon and Jacobsen. 1999. op. cit.
Viscosity Klein et al. 1997. op. cit.
Thermal conductivity McLinden et al. 2000b. op. cit.
Surface tension Moldover and Rainwater. 1988. op. cit.
R-410A Equation of state Lemmon and Jacobsen. 1999. op. cit.
Viscosity Klein et al. 1997. op. cit.
Thermal conductivity McLinden et al. 2000b. op. cit.
Surface tension Moldover and Rainwater. 1988. op. cit.
R-507A Equation of state Lemmon and Jacobsen. 1999. op. cit.
Viscosity Klein et al. 1997. op. cit.
Thermal conductivity McLinden et al. 2000b. op. cit.
Surface tension Moldover and Rainwater. 1988. op. cit.
R-717 (Ammonia) Equation of state Tillner-Roth, R., F. Harms-Watzenberg, and H.D. Baehr. 1993. Eine neue Fundamentalgleichung für Ammoniak. DKV-Tagungsbericht 20(II): 167-81.
Viscosity Fenghour, A., W.A. Wakeham, V. Vesovic, J.T.R. Watson, J. Millat, and E.
Vogel. 1995. The viscosity of ammonia. Journal of Physical and Chemical Reference Data 24:1649-67.
Thermal conductivity Tufeu, R., D.Y. Ivanov, Y. Garrabos, and B. Le Neindre. 1984. Thermal conductivity of ammonia in a large temperature and pressure range includ-ing the critical region. Ber. Bunsen-Ges. Phys. Chem. 88:422-27.
Surface tension Stairs, R.A. and M.J. Sienko. 1956. Surface tension of ammonia and of solutions of alkalai halides in ammonia. J. Am. Chem. Soc. 78: 920-923.
R-718 (Water/Steam) Data computed using Harvey, A.H., S.A. Klein, and A.P. Peskin. 1999. NIST Standard Refer-ence Database 10. NIST/ASME Steam properties database, Version 2.2.
Standard Reference Data Program.
Equation of state Wagner, W. and A. Pruß. 1999. New international formulation for the ther-modynamic properties of ordinary water substance for general and scien-tific use. Journal of Physical and Chemical Reference Data (to be submitted).
Viscosity and thermal conductivity Kestin, J., J.V. Sengers, B. Kamgar-Parsi, and J.M.H. Levelt Sengers.
1984. Thermophysical properties of fluid H2O. Journal of Physical and Chemical Reference Data 13:175.
Surface tension IAPWS. 1995. Physical chemistry of aqueous systems: Meeting the needs of industry. Proceedings 12th International Conference on the Properties of Water and Steam, Orlando. Begell House, Inc., A139-A142. International Association for the Properties of Steam.
R-744 (Carbon Dioxide) Equation of state Span, R. and W. Wagner. 1996. A new equation of state for carbon dioxide covering the fluid region from the triple-point temperature to 1100 K at pressures up to 800 MPa. Journal of Physical and Chemical Reference Data 26:1509-96.
Viscosity Fenghour et al. 1995. op. cit.
Thermal conductivity Vesovic, V., W.A. Wakeham, G.A. Olchowy, J.V. Sengers, J.T.R. Watson, and J. Millat. 1990. The transport properties of carbon dioxide. Journal of Physical and Chemical Reference Data 19:763-808.
Surface tension Rathjen, W. and J. Straub. 1977. Chapter 18, Temperature dependence of surface tension, coexistence curve, and vapor pressure of CO2, CClF3, CBrF3, and SF6. Heat transfer in boiling. Academic Press, New York.
R-50 (Methane) Equation of state Setzmann, U. and W. Wagner. 1991. A new equation of state and tables of thermodynamic properties for methane covering the range from the melt-ing line to 625 K at pressures to 1000 MPa. Journal of Physical and Chem-ical Reference Data 20:1061-1151.
Viscosity Younglove, B.A. and J.F. Ely. 1987. Thermophysical properties of fluids.
II. Methane, ethane, propane, isobutane and normal butane. Journal of Physical and Chemical Reference Data 16:577-798.
Thermal conductivity Friend, D.G., J.F. Ely, and H. Ingham. 1989. Thermophysical properties of methane. Journal of Physical and Chemical Reference Data 18(2):583-638.
Surface tension Somayajulu, G.R. 1988. A generalized equation for surface tension from the triple point to the critical point. Int. J. of Thermophysics 9:559-66.
R-170 (Ethane) Equation of state, viscosity, and thermal conductivity Friend, D.G., H. Ingham, and J.F. Ely. 1991. Thermophysical properties of ethane. Journal of Physical and Chemical Reference Data 20(2):275-347.
20.70 2001 ASHRAE Fundamentals Handbook (SI) Surface tension Soares, V.A.M., B.d.J.V.S. Almeida, I.A. McLure, and R.A. Higgins. 1986.
Surface tension of pure and mixed simple substances at low temperature.
Fluid Phase Equilibria 32:9-16.
R-290 (Propane) Equation of state and thermal conductivity Younglove and Ely. 1987. op. cit.
Viscosity Vogel, E., C. Kuchenmeister, E. Bich, and A. Laesecke. 1998. Reference correlation of the viscosity of propane. Journal of Physical and Chemical Reference Data 27:947-70.
Thermal conductivity Marsh, K., R. Perkins, and M.L.V. Ramires. 2001. Measurement and cor-relation of the thermal conductivity of propane from 86 to 600 K at pres-sures to 70 MPa. Journal of Chemical and Engineering Data (in press). Surface tension: Baidakov, V.G. and I.I. Sulla. 1985. Surface tension of propane and isob-utane at near-critical temperatures. Russian Journal of Physical Chemistry 59:551-54.
R-600 (n-Butane) Equation of state, viscosity, and thermal conductivity Younglove and Ely. 1987. op. cit.
Surface tension Calado, J.C.G., I.A. McLure, and V.A.M. Soares. 1978. Surface tension for octafluorocyclobutane, n-butane and their mixtures from 233 K to 254 K, and vapour pressure, excess gibbs function and excess volume for the mix-ture at 233 K. Fluid Phase Equilibria 2:199-213.
Coffin, C.C. and O. Maass. 1928. The preparation and physical properties of a-, b- and g-butylene and normal and isobutane. Journal of the American Chemical Society 50:1427-37.
R-600a (Isobutane) Equation of state, viscosity, and thermal conductivity Younglove and Ely. 1987. op. cit.
Surface tension Baidakov and Sulla. 1985. op. cit.
R-1150 (Ethylene) Equation of state Smukala, J., R. Span, and W. Wagner. 1999. A new fundamental equation for the fluid state of ethylene for temperatures from the melting line to 450 K and pressure to 300 MPa. Fortschritt-Berichte VDI, Duesseldorf, Series 3, Number 616.
Viscosity and thermal conductivity Holland, P.M., Eaton, B.E. and Hanley, H.J.M. 1983. A correlation of the viscosity and thermal conductivity data of gaseous and liquid ethylene.
Journal of Physical and Chemical Reference Data 12:917-32.
Surface tension Soares, V.A.M., B.d.J.V.S. Almeida, I.A. McLure, and R.A. Higgins. 1986.
Surface tension of pure and mixed simple substances at low temperature.
Fluid Phase Equilibria 32:9-16.
R-1270 (Propylene) Equation of state Angus, S., B. Armstrong, and K.M. de Reuck. 1980. International thermo-dynamic tables of the fluid state—7: Propylene. Pergammon Press, Oxford, England.
Surface tension Maass, O. and C.H. Wright. 1921. Some physical properties of hydrocar-bons containing two and three carbon atoms. Journal of the American Chemical Society 43:1098-1111.
†R-702 (Hydrogen) Thermodynamic data computed using the ALLPROPS database, Version 4.0: Lemmon, E.W., R.T. Jacobsen, S.G. Penoncello, and S.W. Beyerlein.
1994. Computer programs for the calculation of thermodynamic proper-ties of cryogens and other fluids. Advances in Cryogenic Engineering 39:1891-97.
Transport data computed using the NIST 12 database, Version 3.0: Friend, D.G., R.D. McCarty, and V. Arp. 1992. NIST Thermophysical properties of pure fluids database, Version 3.0. Standard Reference Data Program.
†Indicates data reprinted from the 1997 ASHRAE Handbook.
Equation of state, viscosity, and thermal conductivity McCarty, R.D. 1975. Hydrogen: Technology survey—Thermophysical properties. NASA SP-3089.
Surface tension Liley, P.E. and P.D. Desai. 1993. ASHRAE Thermophysical properties of refrigerants.
†R-702p (Parahydrogen) Thermodynamic data computed using the ALLPROPS database, Version 4.0: Lemmon et al. 1994. op. cit.
Transport data computed using the NIST12 database, Version 3.0: Friend et al. 1992. op. cit.
Equation of state, viscosity, and thermal conductivity Younglove, B.A. 1982. Thermophysical properties of fluids. I. Argon, eth-ylene, parahydrogen, nitrogen, nitrogen trifluoride, and oxygen. Journal of Physical and Chemical Reference Data 11(Supplement No. 1). Surface tension Liley and Desai. 1993. op. cit.
†R-704 (Helium) Thermodynamic data computed using the ALLPROPS database, Version 4.0: Lemmon et al. 1994. op. cit.
Transport data computed using the NIST12 database, Version 3.0: Friend et al. 1992. op. cit.
Equation of state, viscosity, and thermal conductivity Arp, V.D., R.D. McCarty, and D.G. Friend. 1995. Thermophysical proper-ties of helium-4 from 0.8 to 1500 K with pressures to 2000 MPa. NIST Technical Note 1334 (revised).
Surface tension Liley and Desai. 1993. op. cit.
†R-728 (Nitrogen) Thermodynamic data computed using the ALLPROPS database, Version 4.0: Lemmon et al. 1994. op. cit.
Equation of state Jacobsen, R.T., R.B. Stewart, and M. Jahangiri. 1986. Thermodynamic properties of nitrogen from the freezing line to 2000 K at pressures to 1000 MPa. Journal of Physical and Chemical Reference Data 15(2):735-909.
Viscosity and thermal conductivity Ely, J.F. 1997. Correlation contained in the AIRPROPS database, Version 1.0. Surface tension Lemmon, E.W. and S.G. Penoncello. 1994. The surface tension of air and air component mixtures. Advances in Cryogenic Engineering 39:1927-34.
†R-729 (Air) Data computed using the AIRPROPS database, Version 1.0.
Lemmon, E.W. 1997. NIST thermophysical properties of air and air com-ponent mixtures, Version 1.0. Standard Reference Data Program.
Equation of state, viscosity, and thermal conductivity Jacobsen, R.T., S.G. Penoncello, S.W. Beyerlein, D.G. Friend, J.F. Ely, J.C. Rainwater, and W.M. Haynes. 1995. Thermophysical properties of air. National Institute of Standards and Technology, Supplement to NASP Technical Memorandum 1005, NASA Langley Research Center.
Surface tension Lemmon and Penoncello. 1994. op.cit.
†R-732 (Oxygen) Thermodynamic data computed using the ALLPROPS database, Version 4.0: Lemmon et al. 1994. op. cit.
Equation of state Schmidt, R. and W. Wagner. 1985. A new form of the equation of state for pure substances and its application to oxygen. Fluid Phase Equilibria 19:175-200.
Viscosity and thermal conductivity Ely. 1997. op. cit.
Surface tension Lemmon and Penoncello. 1994. op.cit.
†R-740 (Argon) Thermodynamic data computed using the ALLPROPS database, Version 4.0: Lemmon et al. 1994. op. cit.
Equation of state Stewart, R.B. and R.T. Jacobsen. 1989. Thermodynamic properties of argon from the triple point to 1200 K with pressures to 1000 MPa. Journal of Physical and Chemical Reference Data 18(2):639-798.
Viscosity and thermal conductivity Ely. 1997. op. cit.
Surface tension Lemmon and Penoncello. 1994. op.cit.
21.1 CHAPTER 21 PHYSICAL PROPERTIES OF SECONDARY COOLANTS (BRINES) Brines ........................................................................................................................................... 21.1 Inhibited Glycols .......................................................................................................................... 21.4 Halocarbons ............................................................................................................................... 21.12 Nonhalocarbon, Nonaqueous Fluids ......................................................................................... 21.12 N MANY refrigeration applications, heat is transferred to a sec-I ondary coolant, which can be any liquid cooled by the refriger-ant and used to transfer heat without changing state. These liquids are also known as heat transfer fluids, brines, or secondary refrigerants.
Other ASHRAE Handbooks describe various applications for secondary coolants. In the 1998 ASHRAE Handbook—Refrigera-tion, refrigeration systems are discussed in Chapter 4, their uses in food processing are found in Chapters 14 through 28, and ice rinks are discussed in Chapter 34. In the 1999 ASHRAE Handbook— Applications, solar energy use is discussed in Chapter 32, thermal storage in Chapter 33, and snow melting in Chapter 49.
This chapter describes the physical properties of several second-ary coolants and provides information on their use. The chapter also includes information on corrosion protection. Additional informa-tion on corrosion inhibition can be found in Chapter 47 of the 1999 ASHRAE Handbook—Applications and Chapter 4 of the 1998 ASHRAE Handbook—Refrigeration.
BRINES Physical Properties Water solutions of calcium chloride and sodium chloride are the most common refrigeration brines. Tables 1 and 2 list the properties of pure calcium chloride brine and sodium chloride brine. For com-mercial grades, use the formulas in the footnotes to these tables. Fig-ures 1 and 5 give the specific heats for calcium chloride and sodium chloride brines and are used for computation of heat loads with ordi-nary brine (Carrier 1959). Figures 2 and 6 show the ratio of the mass of the solution to that of water, which is commonly used as the mea-sure of salt concentration. Viscosities are given in Figures 3 and 7.
Figures 4 and 8 show thermal conductivity of calcium and sodium brines at varying temperatures and concentrations.
Brine applications in refrigeration are mainly in the industrial machinery field and in skating rinks. Corrosion is the principal problem for calcium chloride brines, especially in ice-making tanks where galvanized iron cans are immersed. The preparation of this chapter is assigned to TC 3.1, Refrigerants and Brines.
Fig. 1 Specific Heat of Calcium Chloride Brines Fig. 2 Density of Calcium Chloride Brines 21.2 2001 ASHRAE Fundamentals Handbook (SI) Table 1 Properties of Pure Calcium Chloridea Brines Pure CaCl2, % by Mass Specific Heat at 15°C, J/(kg·K) Crystallization Starts, °C Density at 16°C, kg/m3 Density at Various Temperatures, kg/m3 CaCl2 Brine −20°C −10°C 0°C 10°C 0 4184 0.0 0.0 999 5 3866 −2.4 52.2 1044 1042 1041 6 3824 −2.9 63.0 1049 1051 1050 7 3757 −3.4 74.2 1059 1060 1059 8 3699 −4.1 85.5 1068 1070 1068 9 3636 −4.7 96.9 1078 1079 1077 10 3577 −5.4 108.6 1087 1088 1086 11 3523 −6.2 120.5 1095 1097 1095 12 3464 −7.1 132.5 1104 1107 1104 13 3414 −8.0 144.8 1113 1116 1114 14 3364 −9.2 157.1 1123 1126 1123 15 3318 −10.3 169.8 1132 1140 1136 1133 16 3259 −11.6 182.6 1141 1150 1145 1142 17 3209 −13.0 195.7 1152 1160 1155 1152 18 3163 −14.5 209.0 1161 1170 1165 1162 19 3121 −16.2 222.7 1171 1179 1175 1172 20 3084 −18.0 236.0 1180 1189 1185 1182 21 3050 −19.9 249.6 1189 22 2996 −22.1 264.3 1201 1214 1210 1206 1202 23 2958 −24.4 278.7 1211 24 2916 −26.8 293.5 1223 1235 1231 1227 1223 25 2882 −29.4 308.2 1232 26 2853 −32.1 323.1 1242 27 2816 −35.1 338.5 1253 28 2782 −38.8 354.0 1264 29 2753 −45.2 369.9 1275 29.87 2741 −55.0 378.8 1289 30 2732 −46.0 358.4 1294 32 2678 −28.6 418.1 1316 34 2636 −15.4 452.0 1339 aMass of Type 1 (77% min.) CaCl2 = (mass of pure CaCl2)/(0.77). Mass of Type 2 (94% min.) CaCl2 = (mass of pure CaCl2)/(0.94).
Table 2 Properties of Pure Sodium Chloridea Brines Pure NaCl, % by Mass Specific Heat at 15°C, J/(kg·K) Crystallization Starts, °C Density at 16°C, kg/m3 Density at Various Temperatures, kg/m3 NaCl Brine −10°C −0°C 10°C 20°C 0 4184 0.0 0.0 1000 5 3925 −2.9 51.7 1035 1038.1 1036.5 1034.0 6 3879 −3.6 62.5 1043 1045.8 1043.9 1041.2 7 3836 −4.3 73.4 1049 1053.7 1051.4 1048.5 8 3795 −5.0 84.6 1057 1061.2 1058.9 1055.8 9 3753 −5.8 95.9 1065 1069.0 1066.4 1063.2 10 3715 −6.6 107.2 1072 1076.8 1074.0 1070.6 11 3678 −7.3 118.8 1080 1084.8 1081.6 1078.1 12 3640 −8.2 130.3 1086 1092.4 1089.6 1085.6 13 3607 −9.1 142.2 1094 1100.3 1097.0 1093.2 14 3573 −10.1 154.3 1102 1108.2 1104.7 1100.8 15 3544 −10.9 166.5 1110 1119.4 1116.2 1112.5 1108.5 16 3515 −11.9 178.9 1118 1127.6 1124.2 1120.4 1116.2 17 3485 −13.0 191.4 1126 1135.8 1132.2 1128.3 1124.0 18 3456 −14.1 204.1 1134 1144.1 1140.3 1136.2 1131.8 19 3427 −15.3 217.0 1142 1153.4 1148.5 1144.3 1139.7 20 3402 −16.5 230.0 1150 1160.7 1156.7 1154.1 1147.7 21 3376 −17.8 243.2 1158 1169.1 1165.0 1160.5 1155.8 22 3356 −19.1 256.6 1166 1177.6 1173.3 1168.7 1163.9 23 3330 −20.6 270.0 1174 1186.1 1181.7 1177.0 1172.0 24 3310 −15.7 283.7 1182 1194.7 1190.1 1185.3 1180.3 25 3289 −8.8 297.5 1190 25.2 0.0 aMass of commercial NaCl required = (mass of pure NaCl required)/(% purity).
bMass of water per unit volume = Brine mass minus NaCl mass.
Physical Properties of Secondary Coolants (Brines) 21.3 Fig. 3 Viscosity of Calcium Chloride Brines Fig. 4 Thermal Conductivity of Calcium Chloride Brines Fig. 5 Specific Heat of Sodium Chloride Brines Fig. 6 Density of Sodium Chloride Brines 21.4 2001 ASHRAE Fundamentals Handbook (SI) Ordinary salt (sodium chloride) is used where contact with cal-cium chloride is intolerable (e.g., the brine fog method of freezing fish and other foods). It is used as a spray in air cooling of unit cool-ers to prevent frost formation on coils. In most refrigerating work, the lower freezing point of calcium chloride solution makes it more convenient to use.
Commercial calcium chloride, available as Type 1 (77% mini-mum) and Type 2 (94% minimum), is marketed in flake, solid, and solution forms; flake form is used most extensively. Commercial sodium chloride is available both in crude (rock salt) and refined grades. Because magnesium salts tend to form sludge, their pres-ence in sodium or calcium chloride is undesirable.
Corrosion Inhibition Brine systems must be treated to control corrosion and deposits.
The standard chromate treatment program is the most effective. Cal-cium chloride brines require a minimum of 1800 mg/kg of sodium chromate with pH 6.5 to 8.5. Sodium chloride brines require a min-imum of 3600 mg/kg of sodium chromate and a pH of 6.5 to 8.5.
Sodium nitrite at 3000 mg/kg in calcium brines or 4000 mg/kg in sodium brines controls pH between 7.0 and 8.5, and it should pro-vide adequate protection. Organic inhibitors are available that may provide adequate protection where neither chromates nor nitrites can be used.
Before using any chromate-based inhibitor package, review fed-eral, state, and local regulations concerning the use and disposal of chromate-containing fluids. If the regulations prove too restrictive, an alternative inhibition system should be considered.
INHIBITED GLYCOLS Ethylene glycol and propylene glycol, inhibited for corrosion control, are used as aqueous freezing point depressants (antifreeze) and heat transfer media. Their chief attributes are their ability to lower the freezing point of water, their low volatility, and their rel-atively low corrosivity when properly inhibited. Inhibited ethylene glycol solutions have better physical properties than propylene gly-col solutions, especially at lower temperatures. However, the less toxic propylene glycol is preferred for applications involving pos-sible human contact or where mandated by regulations.
Physical Properties Ethylene glycol and propylene glycol are colorless, practically odorless liquids that are miscible with water and many organic com-pounds. Table 3 shows properties of the pure materials.
The freezing and boiling points of aqueous solutions of ethylene glycol and propylene glycol are given in Tables 4 and 5. Note that increasing the concentration of ethylene glycol above 60% by mass causes the freezing point of the solution to increase. Propylene glycol solutions above 60% by mass do not have freezing points. Instead of freezing, propylene glycol solutions become a glass (glass being an Fig. 7 Viscosity of Sodium Chloride Brines Fig. 8 Thermal Conductivity of Sodium Chloride Brines (Carrier 1959) Table 3 Physical Properties of Ethylene Glycol and Propylene Glycol Property Ethylene Glycol Propylene Glycol Relative molecular mass 62.07 76.10 Density at 20°C, kg/m3 1113 1036 Boiling point, °C at 101.3 kPa 198 187 at 6.67 kPa 123 116 at 1.33 kPa 89 85 Vapor pressure at 20°C, Pa 6.7 9.3 Freezing point, °C −12.7 Sets to glass below −51°C Viscosity, mPa·s at 0°C 57.4 243 at 20°C 20.9 60.5 at 40°C 9.5 18.0 Refractive index nD at 20°C 1.4319 1.4329 Specific heat at 20°C, kJ/(kg·K) 2.347 2.481 Heat of fusion at −12.7°C, kJ/kg 187 — Heat of vaporization at 101.3 kPa, kJ/kg 846 688 Heat of combustion at 20°C, MJ/kg 19.246 23.969 Physical Properties of Secondary Coolants (Brines) 21.5 amorphous, undercooled liquid of extremely high viscosities that has all the appearances of a solid). On the dilute side of the eutectic, ice forms on freezing; on the concentrated side, solid glycol separates from solution on freezing. The freezing velocity of such solutions is often quite slow; but, in time, they set to a hard, solid mass.
Physical properties (i.e., density, specific heat, thermal conduc-tivity, and viscosity) for aqueous solutions of ethylene glycol can be found in Tables 6 through 9 and Figures 9 through 12; similar data for aqueous solutions of propylene glycol can be found in Tables 10 through 13 and Figures 13 through 16. Densities are for aqueous solutions of industrially inhibited glycols. These densities are somewhat higher than those for pure glycol and water alone.
Typical corrosion inhibitor packages do not significantly affect the other physical properties. The physical properties for the two fluids are similar, with the exception of viscosity. At the same concen-tration, aqueous solutions of propylene glycol are more viscous than solutions of ethylene glycol. This higher viscosity accounts for the majority of the performance difference between the two fluids.
The choice of glycol concentration depends on the type of protec-tion required by the application. If the fluid is being used to prevent equipment damage during idle periods in cold weather, such as win-terizing coils in an HVAC system, 30% ethylene glycol or 35% pro-pylene glycol is sufficient. These concentrations will allow the fluid to freeze. As the fluid freezes, it forms a slush that expands and flows into any available space. Therefore, expansion volume must be included with this type of protection. If the application requires that the fluid remain entirely liquid, a concentration with a freezing point 3°C below the lowest expected temperature should be chosen.
Avoid excessive glycol concentration because it increases initial cost and adversely affects the physical properties of the fluid.
Table 4 Freezing and Boiling Points of Aqueous Solutions of Ethylene Glycol Percent Ethylene Glycol Freezing Point, °C Boiling Point, °C at 100.7 kPa By Mass By Volume 0.0 0.0 0.0 100.0 5.0 4.4 −1.4 100.6 10.0 8.9 −3.2 101.1 15.0 13.6 −5.4 101.7 20.0 18.1 −7.8 102.2 21.0 19.2 −8.4 102.2 22.0 20.1 −8.9 102.2 23.0 21.0 −9.5 102.8 24.0 22.0 −10.2 102.8 25.0 22.9 −10.7 103.3 26.0 23.9 −11.4 103.3 27.0 24.8 −12.0 103.3 28.0 25.8 −12.7 103.9 29.0 26.7 −13.3 103.9 30.0 27.7 −14.1 104.4 31.0 28.7 −14.8 104.4 32.0 29.6 −15.4 104.4 33.0 30.6 −16.2 104.4 34.0 31.6 −17.0 104.4 35.0 32.6 −17.9 105.0 36.0 33.5 −18.6 105.0 37.0 34.5 −19.4 105.0 38.0 35.5 −20.3 105.0 39.0 36.5 −21.3 105.0 40.0 37.5 −22.3 105.6 41.0 38.5 −23.2 105.6 42.0 39.5 −24.3 105.6 43.0 40.5 −25.3 106.1 44.0 41.5 −26.4 106.1 45.0 42.5 −27.5 106.7 46.0 43.5 −28.8 106.7 47.0 44.5 −29.8 106.7 48.0 45.5 −31.1 106.7 49.0 46.6 −32.6 106.7 50.0 47.6 −33.8 107.2 51.0 48.6 −35.1 107.2 52.0 49.6 −36.4 107.2 53.0 50.6 −37.9 107.8 54.0 51.6 −39.3 107.8 55.0 52.7 −41.1 108.3 56.0 53.7 −42.6 108.3 57.0 54.7 −44.2 108.9 58.0 55.7 −45.6 108.9 59.0 56.8 −47.1 109.4 60.0 57.8 −48.3 110.0 65.0 62.8 a 112.8 70.0 68.3 a 116.7 75.0 73.6 a 120.0 80.0 78.9 −46.8 123.9 85.0 84.3 −36.9 133.9 90.0 89.7 −29.8 140.6 95.0 95.0 −19.4 158.3 aFreezing points are below −50°C.
Table 5 Freezing and Boiling Points of Aqueous Solutions of Propylene Glycol Percent Propylene Glycol Freezing Point, °C Boiling Point, °C at 100.7 kPa By Mass By Volume 0.0 0.0 0.0 100.0 5.0 4.8 −1.6 100.0 10.0 9.6 −3.3 100.0 15.0 14.5 −5.1 100.0 20.0 19.4 −7.1 100.6 21.0 20.4 −7.6 100.6 22.0 21.4 −8.0 100.6 23.0 22.4 −8.6 100.6 24.0 23.4 −9.1 100.6 25.0 24.4 −9.6 101.1 26.0 25.3 −10.2 101.1 27.0 26.4 −10.8 101.1 28.0 27.4 −11.4 101.7 29.0 28.4 −12.0 101.7 30.0 29.4 −12.7 102.2 31.0 30.4 −13.4 102.2 32.0 31.4 −14.1 102.2 33.0 32.4 −14.8 102.2 34.0 33.5 −15.6 102.2 35.0 34.4 −16.4 102.8 36.0 35.5 −17.3 102.8 37.0 36.5 −18.2 102.8 38.0 37.5 −19.1 103.3 39.0 38.5 −20.1 103.3 40.0 39.6 −21.1 103.9 41.0 40.6 −22.1 103.9 42.0 41.6 −23.2 103.9 43.0 42.6 −24.3 103.9 44.0 43.7 −25.5 103.9 45.0 44.7 −26.7 104.4 46.0 45.7 −27.9 104.4 47.0 46.8 −29.3 104.4 48.0 47.8 −30.6 105.0 49.0 48.9 −32.1 105.0 50.0 49.9 −33.5 105.6 51.0 50.9 −35.0 105.6 52.0 51.9 −36.6 105.6 53.0 53.0 −38.2 106.1 54.0 54.0 −39.8 106.1 55.0 55.0 −41.6 106.1 56.0 56.0 −43.3 106.1 57.0 57.0 −45.2 106.7 58.0 58.0 −47.1 106.7 59.0 59.0 −49.0 106.7 60.0 60.0 −51.1 107.2 65.0 65.0 a 108.3 70.0 70.0 a 110.0 75.0 75.0 a 113.9 80.0 80.0 a 118.3 85.0 85.0 a 125.0 90.0 90.0 a 132.2 95.0 95.0 a 154.4 aAbove 60% by mass, solutions do not freeze but become a glass.
21.6 2001 ASHRAE Fundamentals Handbook (SI) Table 6 Density of Aqueous Solutions of Ethylene Glycol Concentrations in Volume Percent Ethylene Glycol Temperature, °C 10% 20% 30% 40% 50% 60% 70% 80% 90% –35 1089.94 1104.60 1118.61 1132.11 –30 1089.04 1103.54 1117.38 1130.72 –25 1088.01 1102.36 1116.04 1129.21 1141.87 –20 1071.98 1086.87 1101.06 1114.58 1127.57 1140.07 –15 1070.87 1085.61 1099.64 1112.99 1125.82 1138.14 –10 1054.31 1069.63 1084.22 1098.09 1111.28 1123.94 1136.09 −5 1036.85 1053.11 1068.28 1082.71 1096.43 1109.45 1121.94 1133.91 0 1018.73 1035.67 1051.78 1066.80 1081.08 1094.64 1107.50 1119.82 1131.62 5 1017.57 1034.36 1050.33 1065.21 1079.33 1092.73 1105.43 1117.58 1129.20 10 1016.28 1032.94 1048.76 1063.49 1077.46 1090.70 1103.23 1115.22 1126.67 15 1014.87 1031.39 1047.07 1061.65 1075.46 1088.54 1100.92 1112.73 1124.01 20 1013.34 1029.72 1045.25 1059.68 1073.35 1086.27 1098.48 1110.13 1121.23 25 1011.69 1027.93 1043.32 1057.60 1071.11 1083.87 1095.92 1107.40 1118.32 30 1009.92 1026.02 1041.26 1055.39 1068.75 1081.35 1093.24 1104.55 1115.30 35 1008.02 1023.99 1039.08 1053.07 1066.27 1078.71 1090.43 1101.58 1112.15 40 1006.01 1021.83 1036.78 1050.62 1063.66 1075.95 1087.51 1098.48 1108.89 45 1003.87 1019.55 1034.36 1048.05 1060.94 1073.07 1084.46 1095.27 1105.50 50 1001.61 1017.16 1031.81 1045.35 1058.09 1070.06 1081.30 1091.93 1101.99 55 999.23 1014.64 1029.15 1042.54 1055.13 1066.94 1078.01 1088.48 1098.36 60 996.72 1011.99 1026.36 1039.61 1052.04 1063.69 1074.60 1084.90 1094.60 65 994.10 1009.23 1023.45 1036.55 1048.83 1060.32 1071.06 1081.20 1090.73 70 991.35 1006.35 1020.42 1033.37 1045.49 1056.83 1067.41 1077.37 1086.73 75 988.49 1003.34 1017.27 1030.07 1042.04 1053.22 1063.64 1073.43 1082.61 80 985.50 1000.21 1014.00 1026.65 1038.46 1049.48 1059.74 1069.36 1078.37 85 982.39 996.96 1010.60 1023.10 1034.77 1045.63 1055.72 1065.18 1074.01 90 979.15 993.59 1007.09 1019.44 1030.95 1041.65 1051.58 1060.87 1069.53 95 975.80 990.10 1003.45 1015.65 1027.01 1037.55 1047.32 1056.44 1064.92 100 972.32 986.48 999.69 1011.74 1022.95 1033.33 1042.93 1051.88 1060.20 105 968.73 982.75 995.81 1007.71 1018.76 1028.99 1038.43 1047.21 1055.35 110 965.01 978.89 991.81 1003.56 1014.46 1024.52 1033.80 1042.41 1050.38 115 961.17 974.91 987.68 999.29 1010.03 1019.94 1029.05 1037.50 1045.29 120 957.21 970.81 983.43 994.90 1005.48 1015.23 1024.18 1032.46 1040.08 125 953.12 966.59 979.07 990.38 1000.81 1010.40 1019.19 1027.30 1034.74 Note: Density in kg/m3.
Table 7 Specific Heat of Aqueous Solutions of Ethylene Glycol Concentrations in Volume Percent Ethylene Glycol Temperature, °C 10% 20% 30% 40% 50% 60% 70% 80% 90% –35 3.068 2.844 2.612 2.370 –30 3.088 2.866 2.636 2.397 –25 3.107 2.888 2.660 2.423 2.177 –20 3.334 3.126 2.909 2.685 2.450 2.206 –15 3.351 3.145 2.931 2.709 2.477 2.235 –10 3.560 3.367 3.165 2.953 2.733 2.503 2.264 −5 3.757 3.574 3.384 3.184 2.975 2.757 2.530 2.293 0 3.937 3.769 3.589 3.401 3.203 2.997 2.782 2.556 2.322 5 3.946 3.780 3.603 3.418 3.223 3.018 2.806 2.583 2.351 10 3.954 3.792 3.617 3.435 3.242 3.040 2.830 2.610 2.380 15 3.963 3.803 3.631 3.451 3.261 3.062 2.854 2.636 2.409 20 3.972 3.815 3.645 3.468 3.281 3.084 2.878 2.663 2.438 25 3.981 3.826 3.660 3.485 3.300 3.106 2.903 2.690 2.467 30 3.989 3.838 3.674 3.502 3.319 3.127 2.927 2.716 2.496 35 3.998 3.849 3.688 3.518 3.339 3.149 2.951 2.743 2.525 40 4.007 3.861 3.702 3.535 3.358 3.171 2.975 2.770 2.554 45 4.015 3.872 3.716 3.552 3.377 3.193 3.000 2.796 2.583 50 4.024 3.884 3.730 3.569 3.396 3.215 3.024 2.823 2.612 55 4.033 3.895 3.745 3.585 3.416 3.236 3.048 2.850 2.641 60 4.042 3.907 3.759 3.602 3.435 3.258 3.072 2.876 2.670 65 4.050 3.918 3.773 3.619 3.454 3.280 3.097 2.903 2.699 70 4.059 3.930 3.787 3.636 3.474 3.302 3.121 2.929 2.728 75 4.068 3.941 3.801 3.653 3.493 3.324 3.145 2.956 2.757 80 4.077 3.953 3.816 3.669 3.512 3.345 3.169 2.983 2.786 85 4.085 3.964 3.830 3.686 3.532 3.367 3.193 3.009 2.815 90 4.094 3.976 3.844 3.703 3.551 3.389 3.218 3.036 2.844 95 4.103 3.987 3.858 3.720 3.570 3.411 3.242 3.063 2.873 100 4.112 3.999 3.872 3.736 3.590 3.433 3.266 3.089 2.902 105 4.120 4.010 3.886 3.753 3.609 3.454 3.290 3.116 2.931 110 4.129 4.022 3.901 3.770 3.628 3.476 3.315 3.143 2.960 115 4.138 4.033 3.915 3.787 3.647 3.498 3.339 3.169 2.989 120 4.147 4.045 3.929 3.804 3.667 3.520 3.363 3.196 3.018 125 4.155 4.056 3.943 3.820 3.686 3.542 3.387 3.223 3.047 Note: Specific heat in kJ/(kg·K).
Physical Properties of Secondary Coolants (Brines) 21.7 Table 8 Thermal Conductivity of Aqueous Solutions of Ethylene Glycol Concentrations in Volume Percent Ethylene Glycol Temperature, °C 10% 20% 30% 40% 50% 60% 70% 80% 90% –35 0.328 0.307 0.289 0.274 –30 0.333 0.312 0.293 0.276 –25 0.339 0.316 0.296 0.279 0.263 –20 0.371 0.344 0.321 0.300 0.281 0.265 –15 0.377 0.349 0.325 0.303 0.283 0.266 –10 0.415 0.383 0.354 0.329 0.306 0.286 0.268 −5 0.460 0.422 0.389 0.359 0.333 0.309 0.288 0.269 0 0.511 0.468 0.429 0.395 0.364 0.336 0.312 0.290 0.271 5 0.520 0.476 0.436 0.400 0.368 0.340 0.314 0.292 0.272 10 0.528 0.483 0.442 0.405 0.373 0.343 0.317 0.294 0.274 15 0.537 0.490 0.448 0.410 0.377 0.346 0.320 0.296 0.275 20 0.545 0.497 0.453 0.415 0.380 0.349 0.322 0.298 0.276 25 0.552 0.503 0.459 0.419 0.384 0.352 0.324 0.299 0.278 30 0.559 0.509 0.464 0.424 0.387 0.355 0.327 0.301 0.279 35 0.566 0.515 0.469 0.428 0.391 0.358 0.329 0.303 0.280 40 0.572 0.520 0.473 0.431 0.394 0.360 0.331 0.304 0.281 45 0.577 0.525 0.477 0.435 0.397 0.363 0.332 0.306 0.282 50 0.583 0.529 0.481 0.438 0.399 0.365 0.334 0.307 0.283 55 0.588 0.534 0.485 0.441 0.402 0.367 0.336 0.308 0.284 60 0.592 0.538 0.488 0.444 0.404 0.369 0.337 0.310 0.285 65 0.596 0.541 0.491 0.446 0.406 0.371 0.339 0.311 0.286 70 0.600 0.544 0.494 0.449 0.408 0.372 0.340 0.312 0.287 75 0.603 0.547 0.496 0.451 0.410 0.374 0.341 0.313 0.288 80 0.606 0.549 0.498 0.452 0.411 0.375 0.342 0.314 0.288 85 0.608 0.551 0.500 0.454 0.413 0.376 0.343 0.314 0.289 90 0.610 0.553 0.501 0.455 0.414 0.377 0.344 0.315 0.290 95 0.612 0.555 0.503 0.456 0.415 0.378 0.345 0.316 0.290 100 0.613 0.556 0.504 0.457 0.416 0.379 0.346 0.316 0.291 105 0.614 0.556 0.504 0.458 0.416 0.379 0.346 0.317 0.291 110 0.614 0.557 0.505 0.458 0.417 0.380 0.347 0.317 0.292 115 0.614 0.557 0.505 0.458 0.417 0.380 0.347 0.318 0.292 120 0.613 0.556 0.504 0.458 0.417 0.380 0.347 0.318 0.293 125 0.612 0.555 0.504 0.458 0.417 0.380 0.347 0.318 0.293 Note: Thermal conductivity in W/(m·K).
Table 9 Viscosity of Aqueous Solutions of Ethylene Glycol Concentrations in Volume Percent Ethylene Glycol Temperature, °C 10% 20% 30% 40% 50% 60% 70% 80% 90% –35 66.93 93.44 133.53 191.09 –30 43.98 65.25 96.57 141.02 –25 30.50 46.75 70.38 102.21 196.87 –20 15.75 22.07 34.28 51.94 74.53 128.43 –15 11.74 16.53 25.69 38.88 55.09 87.52 –10 6.19 9.06 12.74 19.62 29.53 41.36 61.85 −5 3.65 5.03 7.18 10.05 15.25 22.76 31.56 45.08 0 2.08 3.02 4.15 5.83 8.09 12.05 17.79 24.44 33.74 5 1.79 2.54 3.48 4.82 6.63 9.66 14.09 19.20 25.84 10 1.56 2.18 2.95 4.04 5.50 7.85 11.31 15.29 20.18 15 1.37 1.89 2.53 3.44 4.63 6.46 9.18 12.33 16.04 20 1.21 1.65 2.20 2.96 3.94 5.38 7.53 10.05 12.95 25 1.08 1.46 1.92 2.57 3.39 4.52 6.24 8.29 10.59 30 0.97 1.30 1.69 2.26 2.94 3.84 5.23 6.90 8.77 35 0.88 1.17 1.50 1.99 2.56 3.29 4.42 5.79 7.34 40 0.80 1.06 1.34 1.77 2.26 2.84 3.76 4.91 6.21 45 0.73 0.96 1.21 1.59 2.00 2.47 3.23 4.19 5.30 50 0.67 0.88 1.09 1.43 1.78 2.16 2.80 3.61 4.56 55 0.62 0.81 0.99 1.29 1.59 1.91 2.43 3.12 3.95 60 0.57 0.74 0.90 1.17 1.43 1.69 2.13 2.72 3.45 65 0.53 0.69 0.83 1.06 1.29 1.51 1.88 2.39 3.03 70 0.50 0.64 0.76 0.97 1.17 1.35 1.67 2.11 2.67 75 0.47 0.59 0.70 0.89 1.07 1.22 1.49 1.87 2.37 80 0.44 0.55 0.65 0.82 0.98 1.10 1.33 1.66 2.12 85 0.41 0.52 0.60 0.76 0.89 1.00 1.20 1.49 1.90 90 0.39 0.49 0.56 0.70 0.82 0.92 1.09 1.34 1.71 95 0.37 0.46 0.52 0.65 0.76 0.84 0.99 1.21 1.54 100 0.35 0.43 0.49 0.60 0.70 0.77 0.90 1.10 1.40 105 0.33 0.40 0.46 0.56 0.65 0.71 0.82 1.00 1.27 110 0.32 0.38 0.43 0.53 0.60 0.66 0.76 0.91 1.16 115 0.30 0.36 0.41 0.49 0.56 0.61 0.70 0.83 1.07 120 0.29 0.34 0.38 0.46 0.53 0.57 0.64 0.77 0.98 125 0.28 0.33 0.36 0.43 0.49 0.53 0.60 0.71 0.90 Note: Viscosity in mPa·s.
21.8 2001 ASHRAE Fundamentals Handbook (SI) Table 10 Density of Aqueous Solutions of an Industrially Inhibited Propylene Glycol Concentrations in Volume Percent Propylene Glycol Temperature, °C 10% 20% 30% 40% 50% 60% 70% 80% 90% –35 1072.92 1079.67 1094.50 1092.46 –30 1071.31 1077.82 1090.85 1088.82 –25 1062.11 1069.58 1075.84 1087.18 1085.15 –20 1060.49 1067.72 1073.74 1083.49 1081.46 –15 1050.43 1058.73 1065.73 1071.51 1079.77 1077.74 –10 1039.42 1048.79 1056.85 1063.61 1069.16 1076.04 1074.00 −5 1027.24 1037.89 1047.02 1054.84 1061.37 1066.69 1072.27 1070.24 0 1013.85 1025.84 1036.24 1045.12 1052.71 1059.00 1064.09 1068.49 1066.46 5 1012.61 1024.32 1034.46 1043.09 1050.44 1056.50 1061.36 1064.68 1062.65 10 1011.24 1022.68 1032.55 1040.94 1048.04 1053.88 1058.51 1060.85 1058.82 15 1009.75 1020.91 1030.51 1038.65 1045.52 1051.13 1055.54 1057.00 1054.96 20 1008.13 1019.01 1028.35 1036.24 1042.87 1048.25 1052.44 1053.12 1051.09 25 1006.40 1016.99 1026.06 1033.70 1040.09 1045.24 1049.22 1049.22 1047.19 30 1004.54 1014.84 1023.64 1031.03 1037.18 1042.11 1045.87 1045.30 1043.26 35 1002.56 1012.56 1021.09 1028.23 1034.15 1038.85 1042.40 1041.35 1039.32 40 1000.46 1010.16 1018.42 1025.30 1030.98 1035.47 1038.81 1037.38 1035.35 45 998.23 1007.64 1015.62 1022.24 1027.69 1031.95 1035.09 1033.39 1031.35 50 995.88 1004.99 1012.69 1019.06 1024.27 1028.32 1031.25 1029.37 1027.34 55 993.41 1002.21 1009.63 1015.75 1020.72 1024.55 1027.28 1025.33 1023.30 60 990.82 999.31 1006.44 1012.30 1017.04 1020.66 1023.19 1021.27 1019.24 65 988.11 996.28 1003.13 1008.73 1013.23 1016.63 1018.97 1017.19 1015.15 70 985.27 993.12 999.69 1005.03 1009.30 1012.49 1014.63 1013.08 1011.04 75 982.31 989.85 996.12 1001.21 1005.24 1008.21 1010.16 1008.95 1006.91 80 979.23 986.44 992.42 997.25 1001.05 1003.81 1005.57 1004.79 1002.76 85 976.03 982.91 988.60 993.17 996.73 999.28 1000.86 1000.62 998.58 90 972.70 979.25 984.65 988.95 992.28 994.63 996.02 996.41 994.38 95 969.25 975.47 980.57 984.61 987.70 989.85 991.06 992.19 990.16 100 965.68 971.56 976.36 980.14 983.00 984.94 985.97 987.94 985.91 105 961.99 967.53 972.03 975.54 978.16 979.90 980.76 983.68 981.64 110 958.17 963.37 967.56 970.81 973.20 974.74 975.42 979.38 977.35 115 954.24 959.09 962.97 965.95 968.11 969.45 969.96 975.07 973.03 120 950.18 954.67 958.26 960.97 962.89 964.03 964.38 970.73 968.69 125 945.99 950.14 953.41 955.86 957.55 958.49 958.67 966.37 964.33 Note: Density in kg/m3.
Table 11 Specific Heat of Aqueous Solutions of Propylene Glycol Concentrations in Volume Percent Propylene Glycol Temperature, °C 10% 20% 30% 40% 50% 60% 70% 80% 90% –35 3.096 2.843 2.572 2.264 –30 3.118 2.868 2.600 2.295 –25 3.358 3.140 2.893 2.627 2.326 –20 3.378 3.162 2.918 2.655 2.356 –15 3.586 3.397 3.184 2.943 2.683 2.387 –10 3.765 3.603 3.416 3.206 2.968 2.710 2.417 −5 3.918 3.779 3.619 3.435 3.228 2.993 2.738 2.448 0 4.042 3.929 3.793 3.636 3.455 3.250 3.018 2.766 2.478 5 4.050 3.940 3.807 3.652 3.474 3.272 3.042 2.793 2.509 10 4.058 3.951 3.820 3.669 3.493 3.295 3.067 2.821 2.539 15 4.067 3.962 3.834 3.685 3.513 3.317 3.092 2.849 2.570 20 4.075 3.973 3.848 3.702 3.532 3.339 3.117 2.876 2.600 25 4.083 3.983 3.862 3.718 3.551 3.361 3.142 2.904 2.631 30 4.091 3.994 3.875 3.735 3.570 3.383 3.167 2.931 2.661 35 4.099 4.005 3.889 3.751 3.590 3.405 3.192 2.959 2.692 40 4.107 4.016 3.903 3.768 3.609 3.427 3.217 2.987 2.723 45 4.115 4.027 3.917 3.784 3.628 3.449 3.242 3.014 2.753 50 4.123 4.038 3.930 3.801 3.648 3.471 3.266 3.042 2.784 55 4.131 4.049 3.944 3.817 3.667 3.493 3.291 3.070 2.814 60 4.139 4.060 3.958 3.834 3.686 3.515 3.316 3.097 2.845 65 4.147 4.071 3.972 3.850 3.706 3.537 3.341 3.125 2.875 70 4.155 4.082 3.985 3.867 3.725 3.559 3.366 3.153 2.906 75 4.163 4.093 3.999 3.883 3.744 3.581 3.391 3.180 2.936 80 4.171 4.104 4.013 3.900 3.763 3.603 3.416 3.208 2.967 85 4.179 4.115 4.027 3.916 3.783 3.625 3.441 3.236 2.997 90 4.187 4.126 4.040 3.933 3.802 3.647 3.465 3.263 3.028 95 4.195 4.136 4.054 3.949 3.821 3.670 3.490 3.291 3.058 100 4.203 4.147 4.068 3.966 3.841 3.692 3.515 3.319 3.089 105 4.211 4.158 4.082 3.982 3.860 3.714 3.540 3.346 3.119 110 4.219 4.169 4.095 3.999 3.879 3.736 3.565 3.374 3.150 115 4.227 4.180 4.109 4.015 3.898 3.758 3.590 3.402 3.181 120 4.235 4.191 4.123 4.032 3.918 3.780 3.615 3.429 3.211 125 4.243 4.202 4.137 4.049 3.937 3.802 3.640 3.457 3.242 Note: Specific heat in kJ/(kg·K).
Physical Properties of Secondary Coolants (Brines) 21.9 Table 12 Thermal Conductivity of Aqueous Solutions of Propylene Glycol Concentrations in Volume Percent Propylene Glycol Temperature, °C 10% 20% 30% 40% 50% 60% 70% 80% 90% –35 0.296 0.275 0.255 0.237 –30 0.300 0.277 0.256 0.237 –25 0.329 0.303 0.278 0.257 0.236 –20 0.334 0.306 0.280 0.257 0.236 –15 0.369 0.338 0.309 0.282 0.258 0.236 –10 0.410 0.375 0.342 0.312 0.284 0.259 0.235 −5 0.456 0.416 0.380 0.346 0.314 0.285 0.259 0.235 0 0.510 0.464 0.423 0.385 0.349 0.317 0.286 0.259 0.234 5 0.518 0.472 0.429 0.389 0.353 0.319 0.288 0.260 0.234 10 0.527 0.479 0.434 0.394 0.356 0.321 0.289 0.260 0.233 15 0.535 0.485 0.440 0.398 0.359 0.323 0.290 0.260 0.233 20 0.543 0.492 0.445 0.402 0.362 0.325 0.291 0.261 0.232 25 0.550 0.498 0.449 0.406 0.365 0.327 0.292 0.261 0.231 30 0.557 0.503 0.454 0.409 0.367 0.329 0.293 0.261 0.231 35 0.563 0.508 0.458 0.412 0.370 0.330 0.293 0.261 0.230 40 0.569 0.513 0.462 0.415 0.372 0.331 0.294 0.261 0.229 45 0.575 0.518 0.466 0.418 0.374 0.333 0.294 0.260 0.229 50 0.580 0.522 0.469 0.420 0.375 0.334 0.295 0.260 0.228 55 0.585 0.526 0.472 0.423 0.377 0.335 0.295 0.260 0.227 60 0.589 0.529 0.475 0.425 0.378 0.335 0.295 0.260 0.227 65 0.593 0.532 0.477 0.426 0.379 0.336 0.295 0.259 0.226 70 0.596 0.535 0.479 0.428 0.380 0.336 0.295 0.259 0.225 75 0.599 0.538 0.481 0.429 0.381 0.337 0.295 0.258 0.224 80 0.602 0.540 0.482 0.430 0.382 0.337 0.295 0.258 0.223 85 0.604 0.541 0.484 0.431 0.382 0.337 0.295 0.257 0.222 90 0.606 0.543 0.484 0.431 0.382 0.337 0.294 0.256 0.221 95 0.607 0.544 0.485 0.432 0.382 0.336 0.294 0.256 0.220 100 0.608 0.544 0.485 0.432 0.382 0.336 0.293 0.255 0.219 105 0.609 0.544 0.485 0.432 0.382 0.335 0.292 0.254 0.218 110 0.609 0.544 0.485 0.431 0.381 0.335 0.292 0.253 0.217 115 0.608 0.544 0.485 0.430 0.380 0.334 0.291 0.252 0.216 120 0.608 0.543 0.484 0.429 0.379 0.333 0.290 0.251 0.215 125 0.606 0.542 0.482 0.428 0.378 0.332 0.288 0.250 0.214 Note: Thermal conductivity in W/(m·K).
Table 13 Viscosity of Aqueous Solutions of Propylene Glycol Concentrations in Volume Percent Propylene Glycol Temperature, °C 10% 20% 30% 40% 50% 60% 70% 80% 90% –35 524.01 916.18 1434.22 3813.29 –30 330.39 551.12 908.47 2071.34 –25 110.59 211.43 340.09 575.92 1176.09 –20 73.03 137.96 215.67 368.77 696.09 –15 33.22 49.70 92.00 140.62 239.86 428.19 –10 11.87 23.27 34.78 62.78 94.23 159.02 272.94 −5 4.98 9.08 16.75 24.99 43.84 64.83 107.64 179.78 0 2.68 4.05 7.08 12.37 18.40 31.32 45.74 74.45 122.03 5 2.23 3.34 5.61 9.35 13.85 22.87 33.04 52.63 85.15 10 1.89 2.79 4.52 7.22 10.65 17.05 24.41 37.99 60.93 15 1.63 2.36 3.69 5.69 8.34 12.96 18.41 28.00 44.62 20 1.42 2.02 3.06 4.57 6.65 10.04 14.15 21.04 33.38 25 1.25 1.74 2.57 3.73 5.39 7.91 11.08 16.10 25.45 30 1.11 1.52 2.18 3.09 4.43 6.34 8.81 12.55 19.76 35 0.99 1.34 1.88 2.60 3.69 5.15 7.12 9.94 15.60 40 0.89 1.18 1.63 2.21 3.11 4.25 5.84 7.99 12.49 45 0.81 1.06 1.43 1.91 2.65 3.55 4.85 6.52 10.15 50 0.73 0.95 1.26 1.66 2.29 3.00 4.08 5.39 8.35 55 0.67 0.86 1.13 1.47 1.99 2.57 3.46 4.51 6.95 60 0.62 0.78 1.01 1.30 1.75 2.22 2.98 3.82 5.85 65 0.57 0.71 0.91 1.17 1.55 1.93 2.58 3.28 4.97 70 0.53 0.66 0.83 1.06 1.38 1.70 2.26 2.83 4.26 75 0.49 0.60 0.76 0.96 1.24 1.51 1.99 2.47 3.69 80 0.46 0.56 0.70 0.88 1.12 1.35 1.77 2.18 3.22 85 0.43 0.52 0.65 0.81 1.02 1.22 1.59 1.94 2.83 90 0.40 0.49 0.61 0.75 0.93 1.10 1.43 1.73 2.50 95 0.38 0.45 0.57 0.70 0.86 1.01 1.30 1.56 2.23 100 0.35 0.43 0.53 0.66 0.79 0.92 1.18 1.42 2.00 105 0.33 0.40 0.50 0.62 0.74 0.85 1.08 1.29 1.80 110 0.32 0.38 0.47 0.59 0.69 0.79 1.00 1.19 1.63 115 0.30 0.36 0.45 0.56 0.64 0.74 0.93 1.09 1.48 120 0.28 0.34 0.43 0.53 0.60 0.69 0.86 1.02 1.35 125 0.27 0.32 0.41 0.51 0.57 0.65 0.80 0.95 1.24 Note: Viscosity in mPa·s.
21.10 2001 ASHRAE Fundamentals Handbook (SI) Fig. 9 Density of Aqueous Solutions of Industrially Inhibited Ethylene Glycol (vol. %) Fig. 10 Specific Heat of Aqueous Solutions of Industrially Inhibited Ethylene Glycol (vol. %) Fig. 11 Thermal Conductivity of Aqueous Solutions of Industrially Inhibited Ethylene Glycol (vol. %) Fig. 12 Viscosity of Aqueous Solutions of Industrially Inhibited Ethylene Glycol (vol. %) Fig. 13 Viscosity of Aqueous Solutions of Industrially Inhibited Ethylene Glycol (vol. %) Fig. 14 Specific Heat of Aqueous Solutions of Industrially Inhibited Propylene Glycol (vol. %) Physical Properties of Secondary Coolants (Brines) 21.11 Additional physical property data is available from suppliers of industrially inhibited ethylene and propylene glycol.
Corrosion Inhibition Commercial ethylene glycol or propylene glycol, when pure, is generally less corrosive than water to common metals used in con-struction. However, aqueous solutions of these glycols assume the corrosivity of the water from which they are prepared and can become increasingly corrosive with use if they are not properly inhibited. Without inhibitors, glycols oxidize into acidic end prod-ucts. The amount of oxidation is influenced by temperature, degree of aeration, and, to some extent, the particular combination of metal components to which the glycol solution is exposed.
Corrosion inhibition can be described by classifying additives as either (1) corrosion inhibitors, or (2) environmental stabilizers and adjusters. Corrosion inhibitors form a surface barrier that protects the metal from attack. These barriers are usually formed by adsorp-tion of the inhibitor by the metal, by reaction of the inhibitor with the metal, or by the incipient reaction product. In most cases, metal sur-faces are covered by films of their oxides that inhibitors reinforce.
Environmental stabilizers or adjusters, while not corrosion inhibitors in the strict sense, decrease corrosion by stabilizing or favorably altering the overall environment. An alkaline buffer such as borax is an example of an environmental stabilizer, since its prime purpose is to maintain an alkaline condition (pH above 7).
Some chelating agents function as stabilizers by removing from the solution certain deleterious ions that accelerate the corrosion pro-cess or mechanism; however, exercise caution in their use because improper combinations of pH and concentration may lead to exces-sive corrosion.
Certain oxidants, such as sodium chromate, should not be used with glycol solutions, because the glycol can oxidize prematurely.
Generally, combinations of the two types of additives, inhibitors, and stabilizers offer the best corrosion resistance in a given system.
Commercial inhibited glycols are available from several suppliers.
Service Considerations Design Considerations. Inhibited glycols can be used at temper-atures as high as 175°C. However, maximum-use temperatures vary from fluid to fluid. Therefore, the manufacturer’s suggested temper-ature-use ranges should be followed. In systems with a high degree of aeration, the bulk fluid temperature should not exceed 82°C; however, temperatures up to 175°C are permissible in a pressurized system if air intake is eliminated. Maximum film temperatures should not exceed 28°C above the bulk temperature. Nitrogen blan-keting minimizes oxidation when the system operates at elevated temperatures for extended periods.
Minimum operating temperatures are typically −23°C for ethylene glycol solutions and −18°C for propylene glycol solutions. Operation below these temperatures is generally impractical, because the vis-cosity of the fluids builds dramatically, thus increasing pumping power requirements and reducing heat transfer film coefficients.
Standard materials can be used with most inhibited glycol solu-tions except galvanized steel, because the galvanizing material, zinc, reacts with a portion of the inhibitor package found in most formulated glycols.
Because the removal of sludge and other contaminants is critical, install suitable filters. If inhibitors are rapidly and completely adsorbed by such contamination, the fluid is ineffective for corrosion inhibition. Consider such adsorption when selecting filters.
Storage and Handling. Inhibited glycol concentrates are stable, relatively noncorrosive materials with high flash points. These flu-ids can be stored in mild steel, stainless steel, or aluminum vessels.
However, aluminum should be used only when the fluid tempera-ture is below 66°C. Corrosion in the vapor space of vessels may be a problem, because the fluid’s inhibitor package cannot reach these surfaces to protect them. To prevent this problem, a coating may be used. Suitable coatings include novolac-based vinyl ester resins, high-bake phenolic resins, polypropylene, and polyvinylidene fluo-ride. To ensure the coating is suitable for a particular application and temperature, the manufacturer should be consulted. Since the chem-ical properties of an inhibited glycol concentrate differ from those of its dilutions, the effect of the concentrate on different containers should be known when selecting storage.
Choose transfer pumps only after considering temperature-vis-cosity data. Centrifugal pumps with electric motor drives are often used. Materials compatible with ethylene or propylene glycol should be used for pump packing material. Mechanical seals are also satisfactory. Welded mild steel transfer piping with a minimum diameter is normally used in conjunction with the piping, although flanged and gasketed joints are also satisfactory.
Preparation Before Application. Before an inhibited glycol is charged into a system, remove residual contaminants such as sludge, rust, brine deposits, and oil so the contained inhibitor func-tions properly. Avoid strong acid cleaners; if they are required, con-sider inhibited acids. Completely remove the cleaning agent before charging with inhibited glycol.
Fig. 15 Thermal Conductivity of Aqueous Solutions of Industrially Inhibited Propylene Glycol (vol. %) Fig. 16 Viscosity of Aqueous Solutions of Industrially Inhibited Propylene Glycol (vol. %) 21.12 2001 ASHRAE Fundamentals Handbook (SI) Dilution Water. Use distilled, deionized, or condensate water, because water from some sources contains elements that reduce the effectiveness of the inhibited formulation. If water of this quality is unavailable, water containing less than 25 mg/kg chloride, less than 25 mg/kg sulfate, and less than 100 mg/kg of total hardness may be used.
Fluid Maintenance. Glycol concentrations can be deter-mined by refractive index, gas chromatography, or Karl Fischer analysis for water (assuming that the concentration of other fluid components, such as inhibitor, is known). Using density to determine glycol concentration is unsatisfactory because (1) density measurements are temperature sensitive, (2) inhibitor concentrations can change density, (3) values for propylene glycol are close to those of water, and (4) propylene glycol val-ues are maximum at 70 to 75% concentration.
A rigorous inhibitor monitoring and maintenance schedule is essential to maintain a glycol solution in relatively noncorro-sive condition for a long period. However, a specific schedule is not always easy to establish, because inhibitor depletion rate depends on the particular conditions of use. Analysis of sam-ples immediately after installation, after two to three months, and after six months should establish the pattern for the sched-ule. Visually inspecting the solution and filter residue can de-tect active corrosion.
Many manufacturers of inhibited glycol-based heat transfer fluids provide analytical service to ensure that their heat trans-fer fluid remains in good condition. This analysis may include some or all of the following: percent of ethylene and/or propy-lene glycol, freezing point, pH, reserve alkalinity, corrosion in-hibitor evaluation, contaminants, total hardness, metal content, and degradation products. If maintenance on the fluid is re-quired, recommendations may be given along with the analysis results.
Properly inhibited and maintained glycol solutions provide bet-ter corrosion protection than brine solutions in most systems. A long, though not indefinite, service life can be expected. Avoid indiscriminate mixing of inhibited formulations. Exercise caution in replacing brine systems with inhibited glycols because brine com-ponents are incompatible with glycol formulations.
HALOCARBONS Many common refrigerants are used as secondary coolants as well as primary refrigerating media. Their favorable properties as heat transfer fluids include low freezing points, low viscosities, non-flammability, and good stability. Chapters 19 and 20 present physi-cal and thermodynamic properties for common refrigerants. Table 14 lists two halocarbon compounds that are commonly used as sec-ondary coolants. Table 15 gives vapor pressure, specific heat, ther-mal conductivity, density, and viscosity values for methylene chloride (R-30). Table 16 gives the same properties for trichloro-ethylene (R-1120).
Table 9 in Chapter 19 summarizes comparative safety charac-teristics for halocarbons. Threshold Limit Values and Biological Exposure Indices (ACGIH) has more information on halocarbon toxicity.
Construction materials and stability factors in halocarbon use are discussed in Chapter 19 of this volume and Chapter 5 of the 1998 ASHRAE Handbook—Refrigeration. Note particularly that methyl-ene chloride and trichloroethylene should not be used in contact with aluminum components.
NONHALOCARBON, NONAQUEOUS FLUIDS In addition to the aforementioned fluids, numerous other second-ary refrigerants are available. These fluids have been used primarily by the chemical processing and pharmaceutical industries. They have been used rarely in the HVAC and allied industries due to their Table 14 Freezing and Boiling Points of Halocarbon Coolants Refrigerant Name Freezing Point, °C Boiling Point, °C 30 Methylene chloride −96.7 39.8 1120 Trichloroethylene −86.1 87.2 Table 15 Properties of Liquid Methylene Chloride (R-30) Temper-ature, °C Vapor Pressure, kPa Specific Heat, kJ/(kg·K) Thermal Conductivity, W/(m·K) Density, kg/m3 Viscosity, mPa·s 60 175 1.24 0.128 1254 0.32 50 137 1.22 0.132 1271 0.34 40 100 1.21 0.136 1289 0.37 30 70.5 1.20 0.140 1307 0.40 20 47.0 1.19 0.144 1325 0.44 10 30.3 1.18 0.147 1342 0.48 0 18.8 1.17 0.150 1359 0.53 −10 11.3 1.16 0.154 1377 0.59 −20 6.7 1.16 0.157 1395 0.66 −30 3.8 1.15 0.160 1412 0.76 −40 2.18 1.15 0.163 1430 0.88 −50 1.22 1.14 0.166 1448 1.05 −60 0.69 1.14 0.169 1465 1.29 −70 0.38 1.14 0.171 1483 1.68 −80 0.21 1.14 0.174 1501 2.50 Table 16 Properties of Liquid Trichloroethylene (R-1120) Temper-ature, °C Vapor Pressure, kPa Specific Heat, kJ/(kg·K) Thermal Conductivity, W/(m·K) Density, kg/m3 Viscosity, mPa·s 60 39.5 0.965 0.107 1391 0.40 50 29.0 0.954 0.109 1409 0.44 40 19.8 0.943 0.112 1426 0.48 30 12.8 0.932 0.115 1444 0.52 20 7.8 0.922 0.118 1462 0.57 10 4.60 0.912 0.120 1480 0.63 0 2.55 0.902 0.123 1498 0.70 −10 1.37 0.892 0.126 1515 0.78 −20 0.70 0.883 0.12l8 1532 0.87 −30 0.36 0.875 0.131 1548 0.99 −40 0.168 0.867 0.134 1565 1.14 −50 0.076 0.860 0.137 1581 1.33 −60 0.033 0.853 0.139 1597 1.60 −70 0.014 0.846 0.142 1612 1.93 −80 0.006 0.840 0.145 1627 2.45 Table 17 Summary of Physical Properties of Polydimethylsiloxane Mixture and d-Limonene Polydimethyl-siloxane Mixture d-Limonene Flash point, °C, closed cup 46.7 46.1 Boiling point, °C 175 154.4 Freezing point, °C −111.1 −96.7 Operational temperature range, °C −73.3 to 260 None published Physical Properties of Secondary Coolants (Brines) 21.13 cost and relative novelty. Before choosing these types of fluids, con-sider electrical classifications, disposal, potential worker exposure, process containment, and other relevant issues.
Tables 17 through 19 contain physical property information on a mixture of dimethylsiloxane polymers of various relative molecu-lar masses (Dow Corning 1989) and d-limonene. Information on d-limonene is limited; it is based on measurements made over small data temperature ranges or simply on standard physical property estimation techniques. The compound is an optically active terpene (molecular formula C10H16) derived as an extract from orange and lemon oils. The “d” indicates that the material is dextrorotatory, which is a physical property of the material that does not affect the transport properties of the material significantly.
The mixture of dimethylsiloxane polymers can be used with most standard construction materials; d-limonene, however, can be quite corrosive, easily autooxidizing at ambient temperatures. This fact should be understood and considered before using d-limonene in a system.
REFERENCES ACGIH. 1998. Threshold limit values and biological exposure indices. Pub-lished annually by the American Conference of Governmental Industrial Hygienists, Cincinnati, OH.
Carrier Air Conditioning Company. 1959. Basic data, Section 17M. Syra-cuse, NY.
Dow Corning USA. 1989. Syltherm heat transfer liquids. Midland, MI.
BIBLIOGRAPHY Born, D.W. 1989. Inhibited glycols for corrosion and freeze protection in water-based heating and cooling systems. Midland, MI.
CCI. Calcium chloride for refrigeration brine. Manual RM-1. Calcium Chlo-ride Institute.
Dow Chemical USA. 1994. Engineering manual for DOWFROST and DOWFROST HD heat transfer fluids. Midland, MI.
Dow Chemical USA. 1996. Engineering manual for Dowtherm SR-1 and Dowtherm 4000 heat transfer fluids. Midland, MI.
Fontana, M.G. 1986. Corrosion engineering. McGraw-Hill, New York.
NACE. 1973. Corrosion inhibitors. National Association of Corrosion Engineers, Houston, TX.
NACE. 1991. NACE corrosion engineer’s reference book. NACE. 2000. Corrosion: Understanding the basics.
Union Carbide Corporation. 1994. Ucartherm heat transfer fluids. South Charleston, WV.
Table 18 Properties of a Polydimethylsiloxane Heat Transfer Fluid Temper-ature, °C Vapor Pressure, kPa Viscosity, mPa·s Density, kg/m3 Heat Capacity, kJ/(kg·K) Thermal Conductivity, W/(m·K) −73 0.00 12.4 924.6 1.410 0.1294 −70 0.00 11.2 922.1 1.418 0.1288 −60 0.00 8.26 913.5 1.443 0.1269 −50 0.00 6.24 905.0 1.469 0.1251 −40 0.00 4.83 896.4 1.495 0.1231 −30 0.00 3.81 887.9 1.520 0.1212 −20 0.00 3.07 879.3 1.546 0.1192 −10 0.01 2.51 870.7 1.572 0.1171 0 0.03 2.09 862.0 1.597 0.1150 10 0.08 1.76 853.3 1.623 0.1129 20 0.16 1.49 844.5 1.649 0.1108 30 0.32 1.29 835.5 1.674 0.1086 40 0.61 1.12 826.5 1.700 0.1064 50 1.09 0.98 817.3 1.726 0.1042 60 1.85 0.86 807.9 1.751 0.1019 70 3.02 0.77 798.4 1.777 0.0996 80 4.76 0.69 788.7 1.803 0.0973 90 7.25 0.62 778.8 1.828 0.0949 100 10.73 0.56 768.7 1.854 0.0925 110 15.45 0.51 758.3 1.880 0.0901 120 21.75 0.47 747.7 1.905 0.0877 130 29.95 0.43 736.8 1.931 0.0852 140 40.45 0.40 725.6 1.957 0.0827 150 53.67 0.37 714.1 1.982 0.0802 160 70.06 0.34 702.3 2.008 0.0777 170 90.10 0.32 690.2 2.033 0.0751 180 114.29 0.30 677.7 2.059 0.0725 190 143.17 0.28 664.8 2.085 0.0699 200 177.27 0.26 651.6 2.110 0.0673 210 217.14 0.25 638.0 2.136 0.0646 220 263.36 0.24 623.9 2.162 0.0620 230 316.47 0.22 609.5 2.187 0.0593 240 377.03 0.21 594.5 2.213 0.0566 250 445.61 0.20 579.1 2.239 0.0538 260 522.74 0.19 563.3 2.264 0.0511 Table 19 Physical Properties of d-Limonene Temper-ature, °C Specific Heat, kJ/(kg ·K) Viscosity, mPa·s Density, kg/m3 Thermal Conductivity, W/(m·K) −73 1.27 3.8 914.3 0.137 −50 1.39 3 897.1 0.133 −25 1.51 2.3 878.3 0.128 0 1.65 1.8 859.2 0.124 25 1.78 1.4 839.8 0.119 50 1.91 1.1 820.1 0.114 75 2.04 0.8 800 0.11 100 2.17 0.7 779.5 0.105 125 2.3 0.5 758.4 0.1 150 2.41 0.4 736.6 0.096 Note: Properties are estimated or based on incomplete data.
22.1 CHAPTER 22 SORBENTS AND DESICCANTS Desiccant Applications ................................................................................................................. 22.1 Desiccant Cycle ............................................................................................................................ 22.1 Types of Desiccants ...................................................................................................................... 22.3 Desiccant Isotherms ..................................................................................................................... 22.5 Desiccant Life ............................................................................................................................... 22.5 Cosorption of Water Vapor and Indoor Air Contaminants .......................................................... 22.6 ORPTION refers to the binding of one substance to another.
SSorbents are materials that have an ability to attract and hold other gases or liquids. They can be used to attract gases or liquids other than water vapor, a characteristic that makes them very useful in chemical separation processes. Desiccants are a subset of sor-bents; they have a particular affinity for water.
Virtually all materials are desiccants; that is, they attract and hold water vapor. Wood, natural fibers, clays, and many synthetic mate-rials attract and release moisture as commercial desiccants do, but they lack the holding capacity. For example, woolen carpet fibers attract up to 23% of their dry mass in water vapor, and nylon can take up almost 6% of its mass in water. In contrast, a commercial desiccant takes up between 10 and 1100% of its dry mass in water vapor, depending on its type and on the moisture available in the environment. Furthermore, commercial desiccants continue to attract moisture even when the surrounding air is quite dry, a char-acteristic that other materials do not share.
All desiccants behave in a similar way—they attract moisture until they reach equilibrium with the surrounding air. Moisture is usually removed from the desiccant by heating it to temperatures between 50 and 260°C and exposing it to a scavenger airstream.
After the desiccant dries, it must be cooled so that it can attract moisture once again. Sorption always generates sensible heat equal to the latent heat of the water vapor taken up by the desiccant plus an additional heat of sorption that varies between 5 and 25% of the latent heat of the water vapor. This heat is transferred to the desic-cant and to the surrounding air.
The process of attracting and holding moisture is described as either adsorption or absorption, depending on whether the desiccant undergoes a chemical change as it takes on moisture. Adsorption does not change the desiccant, except by the addition of the mass of water vapor; it is similar in some ways to a sponge soaking up water.
Absorption, on the other hand, changes the desiccant. An example of an absorbent is table salt, which changes from a solid to a liquid as it absorbs moisture.
DESICCANT APPLICATIONS Desiccants can dry either liquids or gases, including ambient air, and are used in many air-conditioning applications, particularly when • The latent load is large in comparison to the sensible load.
• The cost of energy to regenerate the desiccant is low compared to the cost of energy to dehumidify the air by chilling it below its dew point.
• The moisture control level for the space would require chilling the air to subfreezing dew points if compression refrigeration alone were used to dehumidify the air.
• The temperature control level for the space or process requires continuous delivery of air at subfreezing temperatures.
In any of these situations, the cost of running a vapor compres-sion cooling system can be very high. A desiccant process may offer considerable advantages in energy, initial cost of equipment, and maintenance.
Because desiccants are able to attract and hold more than simply water vapor, they can remove contaminants from airstreams to improve indoor air quality. Desiccants have been used to remove organic vapors and, in special circumstances, to control microbio-logical contaminants (Batelle 1971, Buffalo Testing Laboratory 1974). Hines et al. (1991) have also confirmed the usefulness of desiccants in removing vapors that can degrade indoor air quality.
Desiccant materials are capable of adsorbing hydrocarbon vapors while they are collecting moisture from air. These desiccant cosorp-tion phenomena show promise of improving indoor air quality in typical building HVAC systems.
Desiccants are also used in drying compressed air to low dew points. In this application, moisture can be removed from the desic-cant without heat. Desorption is accomplished using differences in vapor pressures compared to the total pressures of the compressed and ambient pressure airstreams.
Finally, desiccants are used to dry the refrigerant circulating in air-conditioning and refrigeration systems. This reduces corrosion in refrigerant piping and prevents valves and capillaries from becoming clogged with ice crystals. In this application, the desic-cant is not regenerated; it is discarded when it has adsorbed its limit of water vapor.
This chapter discusses the water sorption characteristics of des-iccant materials and explains some of the implications of those char-acteristics in ambient pressure air-conditioning applications.
Information on other applications for desiccants can be found in Chapters 14 and 29 of this volume, Chapters 6, 25, 34, 41, and 46 of the 1998 ASHRAE Handbook—Refrigeration, Chapters 1, 2, 5, 8, 15, 17, 20, 27, and 44 of the 1999 ASHRAE Handbook—Applica-tions, and Chapters 22 and 44 of the 2000 ASHRAE Handbook— Systems and Equipment.
DESICCANT CYCLE Practically speaking, all desiccants function by the same mech-anism—transferring moisture because of a difference between the water vapor pressure at their surface and that of the surrounding air.
When the vapor pressure at the desiccant surface is lower than that of the air, the desiccant attracts moisture. When the surface vapor pressure is higher than that of the surrounding air, the desiccant releases moisture.
Figure 1 shows the relationship between the moisture content of the desiccant and its surface vapor pressure. As the moisture content of the desiccant rises, so does the water vapor pressure at its surface.
At some point, the vapor pressure at the desiccant surface is the The preparation of this chapter is assigned to TC 3.5, Desiccant and Sorp-tion Technology.
22.2 2001 ASHRAE Fundamentals Handbook (SI) same as that of the air—the two are in equilibrium. Then, moisture cannot move in either direction until some external force changes the vapor pressure at the desiccant or in the air.
Figure 2 shows the effect of temperature on the vapor pressure at the desiccant surface. Both higher temperature and increased mois-ture content increase the vapor pressure at the surface. When the surface vapor pressure exceeds that of the surrounding air, moisture leaves the desiccant—a process called reactivation or regenera-tion. After the desiccant is dried (reactivated) by the heat, its vapor pressure remains high, so that it has very little ability to absorb moisture. Cooling the desiccant reduces its surface vapor pressure so that it can absorb moisture once again. The complete cycle is illustrated in Figure 3.
The economics of desiccant operation depend on the energy cost of moving a given material through this cycle. The dehumidifica-tion of air (loading the desiccant with water vapor) generally pro-ceeds without energy input other than fan and pump costs. The major portion of energy is invested in regenerating the desiccant (moving from point 2 to point 3) and cooling the desiccant (point 3 to point 1).
Regeneration energy is equal to the sum of three variables: 1. The heat necessary to raise the desiccant to a temperature high enough to make its surface vapor pressure higher than that of the surrounding air 2. The heat necessary to vaporize the moisture it contains (about 2465 kJ/kg) 3. The small amount of heat from desorption of the water from the desiccant The cooling energy is proportional to (1) the mass of the desic-cant and (2) the difference between its temperature after regenera-tion and the lower temperature that allows the desiccant to remove water from the airstream once again.
The cycle is similar when desiccants are regenerated using pres-sure differences in a compressed air application. The desiccant is saturated in a high-pressure chamber (i.e., that of the compressed air). Then valves open, isolating the compressed air from the mate-rial, and the desiccant is exposed to air at ambient pressure. The vapor pressure of the saturated desiccant is much higher than ambi-ent air at normal pressures; thus the moisture leaves the desiccant for the surrounding air. An alternate desorption strategy uses a small portion of the dried air, returning it to the moist desiccant bed to reabsorb the moisture, then venting the air to the atmosphere at ambient pressures.
Table 1 shows the range of vapor pressures over which the des-iccant must operate in space-conditioning applications. It converts the relative humidity at 21°C to dew point and the corresponding vapor pressure. The greater the difference between the air and des-iccant surface vapor pressures, the greater the ability of the material to absorb moisture from the air at that moisture content.
Fig. 1 Desiccant Water Vapor Pressure as Function of Moisture Content (Harriman 1990) Fig. 2 Desiccant Water Vapor Pressure as Function of Desiccant Moisture Content and Temperature (Harriman 1990) Fig. 3 Desiccant Cycle (Harriman 1990) Table 1 Vapor Pressures of Different Relative Humidities at 21°C Relative Humidity at 21°C, % Dew Point, °C Vapor Pressure, kPa 10 −12.4 0.23 20 −3.6 0.47 30 1.9 0.70 40 6.0 0.94 50 9.3 1.17 60 12.0 1.40 70 14.4 1.64 80 16.5 1.87 90 18.3 2.11 100 20.0 2.34 Sorbents and Desiccants 22.3 The ideal desiccant for a particular application depends on the range of water vapor pressures likely to occur in the air, the tem-perature of the regeneration heat source, and the moisture sorption and desorption characteristics of the desiccant within those con-straints. In commercial practice, however, most desiccants can be made to perform well in a wide variety of operating situations through careful engineering of the mechanical aspects of the dehu-midification system. Some of these hardware issues are discussed in Chapter 22 of the 2000 ASHRAE Handbook—Systems and Equipment.
TYPES OF DESICCANTS Desiccants can be liquids or solids and can hold moisture through absorption or adsorption, as described earlier. Most absor-bents are liquids, and most adsorbents are solids.
Liquid Absorbents Liquid absorption dehumidification can best be illustrated by comparison to the operation of an air washer. When air passes through an air washer, its dew point approaches the temperature of the water supplied to the machine. Air that is more humid is dehumidified and air that is less humid is humidified. In a similar manner, a liquid absorption dehumidifier brings air into contact with a liquid desiccant solution. The liquid has a vapor pressure lower than water at the same temperature, and the air passing over the solution approaches this reduced vapor pressure; it is dehumidified.
The vapor pressure of a liquid absorption solution is directly pro-portional to its temperature and inversely proportional to its con-centration. Figure 4 illustrates the effect of increasing desiccant concentration on the water vapor pressure at its surface. The figure shows the vapor pressure of various solutions of water and triethyl-ene glycol, a common commercial desiccant. As the glycol content of the mixture increases, the vapor pressure of the mixture decreases. This pressure difference allows the glycol solution to absorb moisture from the air whenever the vapor pressure of the air is greater than that of the solution.
From a slightly different perspective, the vapor pressure of a given concentration of absorbent solution approximates the vapor pressure values of a fixed relative humidity line on a psychro-metric chart. Higher solution concentrations give lower equilib-rium relative humidities, which allow the absorbent to dry air to lower levels.
Figure 5 illustrates the effect of temperature on the vapor pres-sure of various solutions of water and lithium chloride (LiCl), another liquid desiccant in common use. A solution that is 25% lithium chloride has a vapor pressure of 1.25 kPa at a temperature of 21°C. If the same 25% solution is heated to 38°C, its vapor pres-sure more than doubles to 3.34 kPa. Expressed another way, the 21°C, 25% solution is in equilibrium with air at a 10.5°C dew point.
The same 25% solution at 38°C is at equilibrium with an airstream at a 26°C dew point. The warmer the desiccant, the less moisture it can attract from the air. In standard practice, the behavior of a liquid desiccant is con-trolled by adjusting its temperature, its concentration, or both. Des-iccant temperature is controlled by simple heaters and coolers.
Concentration is controlled by heating the desiccant to drive mois-ture out into a waste airstream or directly to the ambient.
Commercially available liquid desiccants have an especially high water-holding capacity. Each molecule of LiCl, for example, can hold two water molecules, even in the dry state. Above two water molecules per molecule of LiCl, the desiccant becomes a liq-uid and continues to absorb water. If the solution is in equilibrium with air at 90% rh, approximately 26 water molecules are attached to each molecule of LiCl. This represents a water absorption of more than 1000% on a dry mass basis. Fig. 4 Surface Vapor Pressure of Water-Triethylene Glycol Solutions (Dow 1981) Fig. 5 Surface Vapor Pressure of Water-Lithium Chloride Solutions (Foote Mineral 1988) 22.4 2001 ASHRAE Fundamentals Handbook (SI) As a practical matter, however, the absorption process is limited by the exposed surface area of the desiccant and by the contact time allowed for the reaction. More surface area and more contact time allow the desiccant to approach its theoretical capacity. Commercial desiccant systems stretch these limits by flowing the liquid desic-cant onto an extended surface much like in a cooling tower.
Solid Adsorbents Adsorbents are solid materials with a tremendous internal surface area per unit of mass; a single gram can have more than 4600 m2 of surface area. Structurally, adsorbents resemble a rigid sponge, and the surface of the sponge in turn resembles the ocean coastline of a fjord. This analogy indicates the scale of the differ-ent surfaces in an adsorbent. The fjords can be compared to the capillaries in the adsorbent. The spaces between the grains of sand on the fjord beaches can be compared to the spaces between the individual molecules of the adsorbent, all of which have the capacity to hold water molecules. The bulk of the adsorbed water is contained by condensation into the capillaries, and the majority of the surface area that attracts individual water molecules is in the crystalline structure of the material itself.
Adsorbents attract moisture because of the electrical field at the desiccant surface. The field is not uniform in either force or charge, so specific sites on the desiccant surface attract water molecules that have a net opposite charge. When the complete surface is covered, the adsorbent can hold still more moisture because vapor condenses into the first water layer and fills the capillaries throughout the material. As with liquid absorbents, the ability of an adsorbent to attract moisture depends on the difference in vapor pressure between its surface and the air.
The capacity of solid adsorbents is generally less than the capac-ity of liquid absorbents. For example, a typical molecular sieve adsorbent can hold 17% of its dry mass in water when the air is at 21°C and 20% rh. In contrast, LiCl can hold 130% of its mass at the same temperature and relative humidity. But solid adsorbents have several other favorable characteristics.
For example, molecular sieves continue to adsorb moisture even when they are quite hot, allowing dehumidification of very warm airstreams. Also, several solid adsorbents can be manufactured to precise tolerances, with pore diameters that can be closely con-trolled. This means they can be tailored to adsorb molecules of a specific diameter. Water, for example, has an effective molecular diameter of 3.2 nm. A molecular sieve adsorbent with an average pore diameter of 4.0 nm adsorbs water but has almost no capacity for larger molecules, such as organic solvents. This selective adsorption characteristic is useful in many applications. For exam-ple, several desiccants with different pore sizes can be combined in series to remove first water and then other specific contaminants from an airstream.
Adsorption Behavior. The adsorption behavior of solid adsor-bents depends on (1) total surface area, (2) total volume of capillar-ies, and (3) range of capillary diameters. A large surface area gives the adsorbent a larger capacity at low relative humidities. Large cap-illaries provide a high capacity for condensed water, which gives the adsorbent a higher capacity at high relative humidities. A narrow range of capillary diameters makes an adsorbent more selective in the vapor molecules it can hold.
In designing a desiccant, some trade-offs are necessary. For example, materials with large capillaries necessarily have a smaller surface area per unit of volume than those with smaller capillaries.
As a result, adsorbents are sometimes combined to provide a high adsorption capacity across a wide range of operating conditions.
Figure 6 illustrates this point using three noncommercial silica gel adsorbents prepared for use in laboratory research. Each has a dif-ferent internal structure, but since they are all silicas, they have sim-ilar surface adsorption characteristics. Gel 1 has large capillaries, making its total volume large but its total surface area small. It has a large adsorption capacity at high relative humidities but adsorbs a small amount at low relative humidities.
In contrast, Gel 8 has a capillary volume one-seventh the size of Gel 1, but a total surface area almost twice as large. This gives it a higher capacity at low relative humidities but a lower capacity to hold the moisture that condenses at high relative humidities.
Silica gels and most other adsorbents can be manufactured to provide optimum performance in a specific application, balancing capacity against strength, mass, and other favorable characteristics (Bry-Air 1986).
Types of Solid Adsorbents. General classes of solid adsorbents include • Silica gels • Zeolites • Synthetic zeolites (molecular sieves) • Activated aluminas • Carbons • Synthetic polymers Silica gels are amorphous solid structures formed by condens-ing soluble silicates from solutions of water or other solvents.
They have the advantages of relatively low cost and relative sim-plicity of structural customizing. They are available as large as spherical beads about 5 mm in diameter or as small as grains of a fine powder.
Gel Number Total Surface Area, m2/g Average Capillary Diameter, nm Total Volume of Capillaries, mm3/g 1 315 21 1700 5 575 3.8 490 8 540 2.2 250 Fig. 6 Adsorption and Structural Characteristics of Some Experimental Silica Gels (Oscic and Cooper 1982) Sorbents and Desiccants 22.5 Zeolites are aluminosilicate minerals. They occur in nature and are mined rather than synthesized. Zeolites have a very open crys-talline lattice that allows molecules like water vapor to be held inside the crystal itself like an object in a cage. Particular atoms of an aluminosilicate determine the size of the openings between the “bars” of the cage, which in turn governs the maximum size of the molecule that can be adsorbed into the structure. Synthetic zeolites, also called molecular sieves, are crystalline aluminosilicates manufactured in a thermal process. Controlling the temperature of the process and the composition of the ingredient materials allows close control of the structure and surface charac-teristics of the adsorbent. At a somewhat higher cost, this provides a much more uniform product than naturally occurring zeolites.
Activated aluminas are oxides and hydrides of aluminum that are manufactured in thermal processes. Their structural characteris-tics can be controlled by the gases used to produce them and by the temperature and duration of the thermal process.
Carbons are most frequently used for adsorption of gases other than water vapor because they have a greater affinity for the nonpo-lar molecules typical of organic solvents. Like other adsorbents, carbons have a large internal surface and especially large capillar-ies. This capillary volume gives them a high capacity to adsorb water vapor at relative humidities of 45 to 100%.
Synthetic polymers have potential for use as desiccants as well.
Long molecules, like those found in polystyrenesulfonic acid sodium salt (PSSASS), are twisted together like the strands of string.
Each of the many sodium ions in the long PSSASS molecules has the potential to bind several water molecules, and the spaces between the packed strings can also contain condensed water, giving the polymer a capacity exceeding that of many other solid adsorbents.
DESICCANT ISOTHERMS Figure 7 shows a rough comparison of the sorption characteris-tics of different desiccants. Large variations from these isotherms occur because manufacturers use different methods to optimize the materials for different applications. The suitability of a given desic-cant to a particular application is generally governed as much by the engineering of the mechanical system that presents the material to the airstreams as by the characteristics of the material itself.
Several sources give details of desiccant equipment design and information about desiccant isotherm characteristics. Brunauer (1945) considers five basic isotherm shape types. Each isotherm shape is determined by the dominant sorption mechanisms of the desiccant, which give rise to its specific capacity characteristics at different vapor pressures. Isotherm shape can be important in designing the optimum desiccant for applications where a narrow range of operating conditions can be expected. Collier (1986, 1988) illustrates how an optimum isotherm shape can be used to ensure a maximum coefficient of performance in one particular air-condi-tioning desiccant application. DESICCANT LIFE The useful life of desiccant materials depends largely on the quantity and type of contamination in the airstreams they dry. In commercial equipment, desiccants last from 10 000 to 100 000 h and longer before they need replacement. Normally, two mecha-nisms cause the loss of desiccant capacity: (1) change in desiccant sorption characteristics through chemical reactions with contami-nants and (2) loss of effective surface area through clogging or hydrothermal degradation.
Liquid absorbents are more susceptible to chemical reaction with airstream contaminants other than water vapor than are solid adsorbents. For example, certain sulfur compounds can react with LiCl to form lithium sulfate, which is not a desiccant. If the concen-tration of sulfur compounds in the airstream were below 10 mg/kg and the desiccant were in use 24 h a day, the capacity reduction would be approximately 10 to 20% after three years of operation. If the concentration were 30 mg/kg, this reduction would occur after one year.
Solid adsorbents tend to be less chemically reactive and more sensitive to clogging, a function of the type and quantity of partic-ulate material in the airstream. In some situations, certain adsor-bents are sensitive to hydrothermal stress due to the thermal expansion and contraction of the desiccant material on rapid changes in desiccant moisture content. For example, silica gel that must move between an airstream above 95% rh at low temperatures and a reactivating airstream at high temperatures six times per hour, 24 h a day can partly fracture; this may mean a 10% reduction in capacity over the course of one year. For applications where such capacity reduction is undesirable, thermally stabilized desiccants are used in place of more sensitive materials.
In air-conditioning applications, desiccant equipment is designed to minimize the need for desiccant replacement in much the same way that vapor compression cooling systems are designed to avoid the need for compressor replacement. Unlike filters, desiccants are seldom intended to be frequently replaced during normal service in an air-drying application.
The sources for isotherms presented in the figure include PSSASS: Czanderna (1988) Lithium chloride: Munters Corporation—Cargocaire Division and Kathabar, Inc.
Triethylene glycol: Dow Chemical Corporation Silica gel: Davison Chemical Division of W.R. Grace Co.
Activated carbon: Calgon Corporation Activated alumina: LaRoche Industries Inc.
Molecular sieve: Davison Chemical Division of W.R. Grace Co.
Fig. 7 Sorption Isotherms of Various Desiccants 22.6 2001 ASHRAE Fundamentals Handbook (SI) COSORPTION OF WATER VAPOR AND INDOOR AIR CONTAMINANTS Hines et al. (1991) have confirmed that many desiccant materials can collect common indoor pollutants at the same time they collect water vapor from ambient air. This characteristic promises to become useful in future air-conditioning systems where the quality of indoor air is especially important.
The behavior of different desiccant and vapor mixtures is com-plex, but in general, pollutant sorption reactions can be classified into five categories: • Humidity-neutral sorption • Humidity-reduced sorption • Humidity-enhanced sorption • Humidity-pollutant displacement • Desiccant-catalyzed pollutant conversion Humidity-reduced sorption is illustrated by the behavior of water vapor and chloroform on activated carbon. Sorption is humidity-neutral until relative humidity exceeds 45%, when the uptake of chloroform is reduced. The adsorbed water blocks sites that would otherwise attract and hold chloroform. In contrast, water and carbo-nyl chloride mixtures on activated carbon demonstrate humidity-enhanced sorption. Here, sorption of the pollutant increases at high relative humidities. Hines et al. (1991) attribute this phenomenon to the high water solubility of carbonyl chloride.
REFERENCES Batelle. 1971. Project No. N-0914-5200-1971. Batelle Memorial Institute, Columbus, OH.
Brunauer, S. 1945. The adsorption of gases and vapors, Vol. I. Princeton Uni-versity Press, Princeton, NJ. This information is quoted and expanded in The physical chemistry of surfaces, by Arthur W. Adamson. John Wiley and Sons, New York, 1982.
Bry-Air. 1986. MVB Series engineering data. Bry-Air Inc., Sunbury, OH.
Buffalo Testing Laboratory. 1974. Report No. 65711-1974.
Collier, R.K. Advanced desiccant materials assessment. Research Report 5084-243-1089. Phase I-1986, Phase II-1988. Gas Research Institute, Chicago.
Czanderna, A.W. 1988. Polymers as advanced materials for desiccant appli-cations. Research Report NREL/PR-255-3308. National Renewable Energy Laboratory, Golden, CO.
Dow. 1981. Guide to glycols. Dow Chemical Corporation, Organic Chemi-cals Division, Midland, MI.
Foote Mineral. 1988. Lithium chloride technical data. Bulletin 151. Foote Mineral Corporation, Exton, PA.
Harriman, L.G., III. 1990. The dehumidification handbook, 2nd ed. Munters Cargocaire, Amesbury, MA.
Hines, A.J., T.K. Ghosh, S.K. Loyalka, and R.C. Warder, Jr. 1991. Investi-gation of co-sorption of gases and vapors as a means to enhance indoor air quality. ASHRAE Research Project 475-RP and Gas Research Insti-tute Project GRI-90/0194. Gas Research Institute, Chicago.
Oscic, J. and I.L. Cooper. 1982. Adsorption. John Wiley and Sons, New York.
BIBLIOGRAPHY Adamson, A.W. 1982. The physical chemistry of surfaces. John Wiley and Sons, New York.
Falcone, J.S., Jr., ed. 1982. Soluble silicates. Symposium Series 194. Amer-ican Chemical Society, Washington, D.C.
Ruthven, D.M. 1984. Principles of adsorption and adsorption processes.
John Wiley and Sons, New York.
SUNY Buffalo School of Medicine. Effects of glycol solution on micro-biological growth. Niagrara Blower Report No. 03188.
Valenzuela, D. and A. Myers. 1989. Adsorption equilibrium data handbook.
Simon and Schuster/Prentice-Hall, Englewood Cliffs, NJ.
23.1 CHAPTER 23 THERMAL AND MOISTURE CONTROL IN INSULATED ASSEMBLIES—FUNDAMENTALS Terminology and Symbols ....................................................... 23.1 THERMAL INSULATION ....................................................... 23.2 Basic Materials ....................................................................... 23.2 Physical Structure and Form .................................................. 23.2 Properties ................................................................................ 23.2 HEAT FLOW ........................................................................... 23.4 Factors Affecting Thermal Performance ................................ 23.4 Thermal Transmittance ........................................................... 23.6 Factors Affecting Heat Transfer Across Air Spaces ............................................................................ 23.7 Calculating Overall Thermal Resistance ................................ 23.8 Calculating Interface Temperatures ....................................... 23.8 Heat Flow Calculations .......................................................... 23.8 INSULATION THICKNESS .................................................... 23.9 Economic Thickness ................................................................ 23.9 Economic Thickness: Mechanical Systems ............................. 23.9 Economic Thickness: Building Envelopes ............................ 23.10 MOISTURE IN BUILDINGS ................................................ 23.11 Moisture Problems in Buildings ........................................... 23.11 Properties of Water Vapor in Air .......................................... 23.13 Moisture in Building Materials ............................................. 23.13 Moisture Migration ............................................................... 23.14 Water Vapor Retarders and Airflow Retarders .................... 23.15 Steady-State Design Tools .................................................... 23.17 Mathematical Models ............................................................ 23.19 Preventing Surface Condensation ......................................... 23.20 ROPER DESIGN of space heating, air-conditioning, refrig-Peration, and other industrial systems requires knowledge of thermal insulations and thermal-moisture behavior of building structures. This chapter deals with heat and moisture transfer defi-nitions, fundamentals and properties of thermal insulation materi-als, heat flow calculations, economic thickness of insulation, and the fundamentals of moisture as it relates to building components and systems.
TERMINOLOGY AND SYMBOLS The following heat and moisture transfer definitions and sym-bols are commonly used in the building industry.
Thermal transmission, heat transfer, or rate of heat flow. The flow of heat energy induced by a temperature difference. Heat may be transferred by conduction, convection, radiation, and mass trans-fer. These can occur separately or in combinations, depending on specific circumstances.
Thermal conductivity, k. The time rate of heatflow through a unit area of 1 m thick homogeneous material in a direction perpendicular to isothermal planes, induced by a unit temperature gradient. (ASTM Standard C 168 defines homogeneity.) Units are W/(m·K). Thermal conductivity must be evaluated for a specific mean temperature and moisture content, because in most materials it varies with temperature and moisture content.
For porous materials, heat flows by a combination of modes and may depend on orientation, direction, or both. The measured prop-erty of such materials may be called effective or apparent thermal conductivity. The specific test conditions (i.e., sample thickness, orientation, environment, environmental pressure, surface temper-ature, mean temperature, temperature difference, and moisture con-tent) should be reported with the values. With thermal conductivity, the symbol kapp is used to denote the lack of pure conduction or to indicate that all values reported are apparent.
Thermal resistivity, Ru. The reciprocal of thermal conductivity.
Units are m·K/W.
Thermal conductance, C-factor, C. The time rate of heat flow through a unit area of a body induced by a unit temperature differ-ence between the body surfaces. Units are W/(m2·K).
When the two defined surfaces of mass-type (i.e., nonreflec-tive) thermal insulation have unequal areas, as in the case of radial heat flow through a curved block or through a pipe cover-ing (see Table 2 in Chapter 3), or through materials of nonuni-form thickness, an appropriate mean area and mean thickness must be given. Heat flow formulas involving materials that are not uniform slabs must contain shape factors to account for the area variation involved.
When heat flow occurs by conduction alone, the thermal con-ductance of a material may be obtained by dividing the thermal conductivity of the material by its thickness. When several modes of heat transfer are involved, the apparent or effective thermal conductance may be obtained by dividing the apparent thermal conductivity by the thickness.
Where air circulates within or passes through insulation, as it may with low-density fibrous materials, the effective thermal con-ductance is affected.
Thermal conductances and resistances of the more common building materials and industrial insulations are tabulated in Table 4 in Chapter 25.
Heat transfer film coefficient (or surface coefficient of heat transfer or surface film conductance), h or f. Heat transferred between a surface and a fluid per unit time per unit area driven by a unit temperature difference between the surface and the fluid in con-tact with it, in W/(m2·K).
Surface film resistance. The reciprocal of the heat transfer film coefficient, in m2·K/W. Subscripts i and o often denote inside and outside surface resistances and conductances, respectively.
The surrounding space must be air or other fluids for convection to take place. If the space is evacuated, the heat flow only occurs by radiation.
Thermal resistance R-value, R. Under steady-state conditions, the mean temperature difference between two defined surfaces of material or construction that induces unit heat flow through a unit area, in m2·K/W.
Thermal transmittance, U-factor, U. The rate of heat flow per unit area under steady-state conditions from the fluid on the warm side of a barrier to the fluid on the cold side, per unit temperature difference between the two fluids. It is determined by first evaluat-ing the R-value, including the surface film resistances, and then computing its reciprocal, U, in W/(m2·K). The U-factor is some-times called the overall coefficient of heat transfer. In building The preparation of this chapter is assigned to TC 4.4, Building Materials and Building Envelope Performance.
23.2 2001 ASHRAE Fundamentals Handbook (SI) practice, the heat transfer fluid is air. The temperature of the fluid is obtained by averaging its temperature over a finite region near the surface involved.
Thermal emittance, ε. The ratio of the radiant flux emitted by a body to that emitted by a blackbody at the same temperature and under the same conditions.
Effective emittance of an air space, E. The combined effect of emittances from the boundary surfaces of an air space, where the boundaries are parallel and of a dimension much larger than the dis-tance between them. Table 2 in Chapter 25 lists values of E for var-ious air spaces.
Surface reflectance, ρ. The fraction of the radiant flux falling on a surface that is reflected by it.
Water vapor permeance, M. The rate of water vapor transmis-sion by diffusion per unit area of a body between two specified par-allel surfaces, induced by unit vapor pressure difference between the two surfaces, ng/(s·m2·Pa).
Water vapor permeability, µ. The rate of water vapor transmis-sion by diffusion per unit area of flat material of unit thickness induced by unit vapor pressure difference between two surfaces, under specified temperature and humidity conditions. When perme-ability varies with psychrometric conditions, the spot or specific permeability defines the property at a specific condition in ng/(s·m·Pa), where the vapor pressure difference is in pascals.
Water vapor resistance, Z. The reciprocal of permeance—sig-nifies a resistance to moisture flow, TPa·m2·s/kg.
THERMAL INSULATION Thermal insulations are materials or combinations of materials that, when properly applied, retard the rate of heat by conductive, convective, and/or radiative transfer modes. Thermal insulations can be fibrous, particulate, film or sheet, block or monolithic, open-cell or closed-cell, or composites of these materials that can be chemically or mechanically bound or supported.
By retarding heat flow, thermal insulations can serve one or more of the following functions: • Conserve energy by reducing heat loss or gain of piping, ducts, vessels, equipment, and structures • Control surface temperatures of equipment and structures for personnel protection and comfort • Help control the temperature of a chemical process, a piece of equipment, or a structure • Prevent vapor condensation on surfaces. However, thermal insu-lation may promote moisture condensation and subsequent dam-age in a building envelope, if the insulation application is improperly installed or poorly designed • Reduce temperature fluctuations within an enclosure when heat-ing or cooling is not needed or available • Reduce temperature variations within a conditioned space for increased personal comfort • Provide fire protection Thermal insulation can serve additional functions, although such secondary functions should be consistent with its capabilities and primary purpose. Under certain conditions, insulations may • Add structural strength to a wall, ceiling, or floor section • Provide support for a surface finish • Impede water vapor transmission and air infiltration • Prevent or reduce damage to equipment and structures from exposure to fire and freezing conditions • Reduce noise and vibration • Reduce growth of mold and mildew Thermal insulation is used to control heat flow at all tempera-tures, the limiting value being its survival temperature.
BASIC MATERIALS Thermal insulations normally consist of the following basic materials and composites: • Inorganic, fibrous, or cellular materials such as glass, rock, or slag wool; and calcium silicate, bonded perlite, vermiculite, and ceramic products. In the past, asbestos insulations was applied. But, because asbestos has been shown to be a carcin-ogen, extreme caution must be used if it is encountered.
• Organic fibrous materials such as cellulose, cotton, animal hair, wood, pulp, cane, or synthetic fibers; and organic cellular materials such as cork, foamed rubber, polystyrene, polyure-thane, and other polymers.
• Metallic or metallized organic reflective membranes. These surfaces must face an air-filled, gas-filled, or evacuated space to be effective.
PHYSICAL STRUCTURE AND FORM Mass-type insulation can be cellular, granular, or fibrous solid material. Reflective insulation consists of smooth-surfaced sheets of metal foil or foil-surfaced material separated by air spaces.
The physical forms of industrial and building insulations include the following: Loose-fill insulations consist of fibers, powders, granules, or nodules. They are usually poured or blown into walls or other spaces.
Insulating cement is a loose material that is mixed with water or a suitable binder to obtain plasticity and adhesion. It is troweled or blown wet on a surface and dried in place. Both loose-fill and insu-lating cement are suited for covering irregular spaces.
Flexible and semirigid insulations consist of organic and inor-ganic materials with and without binders and with varying degrees of compressibility and flexibility. These insulations are generally available as blanket, batt, or felt insulation, and in either sheets or rolls. Coverings and facings may be fastened to one or both sides and serve as reinforcing, air or vapor retarders or both, reflective surfaces, or surface finishes. These coverings include combinations of laminated foil, glass, cloth or plastics and paper, wire mesh, or metal lath. Although standard sizes are generally used, thickness and shape of insulation can be any dimension handled conveniently.
Rigid materials are available in rectangular blocks, boards, or sheets, which are preformed during manufacture to standard lengths, widths, and thicknesses. Insulation for pipes and curved surfaces is supplied in sections or segments, with radii of curvature available to suit all standard sizes of pipe and tubing.
Reflective materials are available in sheets and rolls of single layer or multilayer construction and in preformed shapes with inte-gral air spaces.
Formed-in-place insulations are available as liquid compo-nents or expandable pellets that can be poured, frothed, or sprayed in place to form rigid or semirigid foam insulation. Fibrous materi-als mixed with liquid binders can also be sprayed in place, and in some products, the binder is also a foam.
Accessory materials for thermal insulation include mechanical and adhesive fasteners, exterior and interior finishes, vapor and air retarder coatings, jackets and weather coatings, sealants, lagging adhesives, membranes, and flashing compounds. ASTM Standard C 168 defines terms related to thermal insulating materials.
PROPERTIES In addition to low thermal conductivity, the selection of thermal insulation may involve secondary criteria. Characteristics such as Thermal and Moisture Control in Insulated Assemblies—Fundamentals 23.3 resiliency or rigidity, acoustical energy absorption, water vapor per-meability, airflow resistance, fire hazard and fire resistance, ease of application, applied cost, health and safety aspects, or other param-eters may influence the choice among materials that have almost equal thermal performance values.
Thermal Properties Thermal resistance is a measure of the effectiveness of thermal insulation to retard heat flow. A material with a high thermal resis-tance (low thermal conductance) is an effective insulator.
Heat flows through most thermal insulations by a combination of gas and solid conduction, radiation, and convection. Heat transfer through materials or systems is controlled by factors such as length of heat flow paths, temperature, temperature difference characteris-tics of the system, and environmental conditions.
Although heat transmission characteristics are usually deter-mined by measuring thermal conductivity, this property does not strictly apply to thermal insulation. A particular sample of a mate-rial has a unique value of thermal conductivity for a particular set of conditions. This value may not be representative of the material at other conditions and should be called apparent thermal conduc-tivity. For details, refer to ASTM Standards C 168, C 177, C 236, C 335, C 518, C 976, and C 1045.
Reflective insulations impede radiant heat transfer because the surfaces have high reflectance and low emittance values. (Table 1 and Table 2 in Chapter 25 give typical design values.) To be effec-tive, the reflective face of both single and multiple layer reflective insulations must face an air or evacuated space. Multiple layers of reflective materials and smooth and parallel sealed air spaces increase overall thermal resistance. Air exchange and movement must be inhibited.
Mass-type insulation can be combined with reflective surfaces and air spaces to obtain a higher thermal resistance. However, each design must be evaluated because maximum thermal performance depends on such factors as condition of the insulation, shape and form of the construction, the means to avoid air leakage and move-ment, and the condition of the installed reflective surfaces and their aging characteristics.
Design values of effective or apparent thermal conductivity, thermal conductance, and thermal resistance for the most common insulations are listed in Table 4 in Chapter 25. These values have been selected as typical and useful for engineering calculations. The manufacturer or test results of the insulation under appropriate con-ditions can give values for a particular insulation.
Other thermal properties that can be important are specific heat, heat capacity, thermal diffusivity, the coefficient of thermal expan-sion, and the maximum temperature limit. Heat capacity is the product of specific heat and mass. Because the rate of temperature change within an insulation is proportional to its thermal diffusivity for a given thickness, thermal diffusivity becomes important for applications where temperature varies with time. Chapter 3 covers symbols, definitions, and methods of calculation in steady-state heat transfer.
Mechanical Properties Some insulations have sufficient structural strength for load bearing. They are used occasionally to support load-bearing roofs and floors, form self-supporting partitions, or stiffen struc-tural panels. For such applications, one or more of the following properties of an insulation may be important: strength in com-pression, tension, shear, impact, flexure, and resistance to vibra-tion. These mechanical properties vary with basic composition, density, cell size, fiber diameter and orientation, type and amount of binder (if any), and temperature and environmental conditioning.
Health and Safety Most thermal insulations have good resistance to fire, vermin, rot, objectionable odors, and vapors; some are a potential risk to health and safety. These risks can result from direct exposure to these materials and accessories while they are being stored or transported, during or after installation, or as a result of interven-ing or indirect actions or events, such as aging, fire, or physical disturbance. The potential health and safety effects of thermal insulation can be considered in two categories: (1) those related to storage, handling, and installation operations and (2) those that occur after installation (such as aging). Potential hazards during manufacture are not discussed here. Correct handling, installation, and precautionary measures can reduce or eliminate these problems.
Potential health effects range from temporary irritation to seri-ous changes in body functions. The principal concerns are with insulation containing asbestos. Questions have also been raised about man-made fibers. To date, research is inconclusive as to their potential hazard; however, they can be very annoying in installation, and the use of proper safeguards is desirable. Poten-tial traumatic injury can occur from direct contact with materials that are sharp, rough, have protrusions or abrasive surfaces, permit overheating, or transmit electrical energy.
Combustion of insulation materials and accessories may release heat, hazardous gases, fibers, and particulates. Manufacturers’ rec-ommendations and applicable government codes and standards (ASTM Standard C 930) have more details.
Acoustics Some thermal insulations are used as acoustical control materi-als, whether or not thermal performance is a design requirement.
Acoustical efficiency depends on the physical structure of the mate-rial. Materials with open, porous surfaces have sound absorption capability. Those with high density and resilient characteristics can act as vibration insulators; either alone or in combination with other materials, some are effective barriers to sound transmission. Insula-tions for sound conditioning include flexible and semirigid, formed-in-place fibrous materials, and rigid fibrous insulation.
Sound absorption insulations are normally installed on interior surfaces or used as interior surfacing materials. Rigid insulations are fabricated into tile or blocks, edge-treated to facilitate mechan-ical or adhesive application, and prefinished during manufacture.
Some insulation units have a natural porous surface, and others include mechanical perforations to facilitate the entry of sound waves. Still others have a diaphragm or decorative film surfacing attached only to the edges of the units, which allows the sound waves to reach the fibrous backing by diaphragm action.
Flexible, semirigid, and formed-in-place fibrous materials used for sound absorption are available in a variety of thicknesses and densities that determine their sound absorption characteristics.
When density is increased by reducing the thickness of the mate-rial, sound absorption is generally reduced; however, as thickness increases, the influence of density decreases.
Thermal insulations improve sound transmission loss when installed in discontinuous construction. A wall of staggered-stud construction that uses resilient clips or channels on one side of the stud or resilient insulation boards of special manufacture to prevent acoustical coupling mechanically between the surfaces reduces sound transmission. A sound absorption thermal insulation blanket in a wall cavity reduces sound transmission, depending on the type of construction.
In floor construction, resilient channels or separate floor and ceiling joists form a discontinuous construction. Sound-absorbing thermal insulation placed within this construction further reduces sound transmission. Sound-deadening boards underlying finish 23.4 2001 ASHRAE Fundamentals Handbook (SI) flooring absorb impact sounds and improve the airtightness of the construction, thus reducing airborne sound transmission.
Thermal insulation boards can be placed under mechanical equipment to isolate vibration. The imposed loading and natural resonant frequency of materials are critical for proper design.
Because material must deflect properly under load to provide iso-lation, the system should be neither overloaded nor underloaded.
For further information on sound and vibration control, refer to Chapter 46 of the 1999 ASHRAE Handbook—Applications.
Other Properties Other properties of insulating materials that can be important, depending on the application, include density, resilience, resistance to settling, permanence, reuse or salvage value, ease of handling, dimensional uniformity and stability, resistance to chemical action and chemical change, resistance to moisture penetration, ease in fab-rication, application of finishes, and sizes and thicknesses obtainable.
HEAT FLOW FACTORS AFFECTING THERMAL PERFORMANCE A wide variety of physical, environmental, application, and, in some cases, aging factors affect the thermal performance of insulations.
Thermal conductivity k is a property of a homogeneous material.
Building materials, such as lumber, brick, and stone, are usually considered homogeneous. Most thermal insulation and many other building materials are porous and consist of combinations of solid matter with small voids.
For many materials, conduction is not the only mode of heat transfer. Consequently, the term apparent thermal conductivity describes the heat flow properties of materials. Some materials with low thermal conductivities are almost purely conductive (silica opacified aerogel, corkboard, etc.).
The apparent thermal conductivity of insulation varies with form and physical structure, environment, and application conditions.
Form and physical structure vary with the basic material and man-ufacturing process. Typical variations include density, cell size, diameter and arrangement of fibers or particles, degree and extent of bonding materials, transparency to thermal radiation, and the type and pressure of gas within the insulation.
Environment and application conditions include mean tempera-ture, temperature gradient, moisture content, air infiltration, orien-tation, and direction of heat flow. Thermal performance values for insulation materials and systems are usually obtained by standard methods listed in Volume 04.06 of the Annual Book of ASTM Stan-dards. The methods apply mainly to laboratory measurements on dried or conditioned samples at specific mean temperatures and tem-perature gradient conditions. Although the fundamental heat trans-mission characteristics of a material or system can be determined accurately, actual performance in a structure may vary from that indicated in the laboratory. The design of the envelope, its construc-tion, and the materials used may all affect the test procedure. These factors are detailed in ASTM STP 544, STP 660, STP 718, STP 789, STP 879, STP 885, STP 922, STP 1030, and STP 1116.
The effective thermal conductivity of some thermal insulation materials varies with density. Figure 1 illustrates this variation at one mean temperature for a number of materials currently used to insulate building envelopes. For most mass-type insulations, there is a minimum in the respective apparent thermal conductivity versus density. This minimum depends on the type and form of material, temperature, and direction of heat flow. For fibrous materials, the values of density at which the minimum value occurs increase as both the fiber diameter or cell size and the mean temperature Fig. 1 Apparent Thermal Conductivity Versus Density of Several Thermal Insulations Used as Building Insulations Fig. 2 Typical Variation of Apparent Thermal Conductivity with Fiber Diameter and Density Thermal and Moisture Control in Insulated Assemblies—Fundamentals 23.5 increase. These effects are shown in Figure 2 (Lotz 1969) and Fig-ure 3, respectively.
Other factors that affect thermal performance include compac-tion and settling of insulation, air permeability, type and amount of binder used, additives that may influence the bond or contact between fibers or particles, and the type and form of radiation trans-fer inhibitor, if used. In cellular materials, the factors that influence thermal performance and strength properties are the same as those that control the thermal conductivity of the basic structured mate-rial: size and shape of the cells, thickness of the cell walls, gas con-tained in the cells, orientation of the cells, and radiation characteristics of the cell surfaces.
A change in density caused by the degree of compaction of insulation powders affects their effective thermal conductivity.
Insulating concretes made from lightweight aggregates can be produced in a wide range of densities, with corresponding thermal conductances.
Fibrous insulations reach a minimum conductivity when fibers are uniformly spaced and perpendicular to the direction of heat flow. Generally, a decrease in the diameter of the fiber lowers the conductivity for the same density (Figure 2). For cellular insulation, a specific combination of cell size, density, and gas composition produces optimum thermal conductivity.
At temperatures below 200 to 300°C, a large portion of heat transfer across most insulations occurs by conduction through the air or other gas in the insulations (Rowley et al. 1952, Lander 1955, Simons 1955, Verschoor and Greebler 1952). The overall heat trans-fer can be closely approximated by supposition of gas conduction with other mechanisms of heat transfer, each determined separately.
If the gas in the insulation is replaced by another gas with a different thermal conductivity, the apparent thermal conductivity of the insu-lation is changed by an amount approximately equal to the differ-ence in conductivity of the two gases. For example, replacing air with a fluorinated hydrocarbon gas can lower the apparent thermal conductivity of the insulation by as much as 50%.
Fluorocarbon-expanded cellular plastic foams with a high pro-portion (greater than 90%) of closed cells retain the fluorocarbon within the cells for extended periods. As these products are initially produced, they have apparent thermal conductivities of approxi-mately 0.016 W/(m·K) at 24°Cwhen the gas contained has a mean-free-path greater than the dimensions within the cells. However, this value increases with time as atmospheric gases diffuse into the cells and, over a long period of time, the fluorocarbon gas dissolves in the polymer or diffuses out.
The rates of diffusion and increase in apparent thermal conduc-tivity depend on several factors, including permeance of the cell walls to the gases involved, age of the foam, temperature, geometry of the insulation (thickness), and integrity of the surface protection provided. The significance of the surface protection is apparent when foams are encased in gas-impermeable membranes or some water vapor retarders (Brandreth 1986, Tye 1987). For estimating the long-term change in thermal resistance of unfaced rigid closed-cell plastic foams, a test method (ASTM Standard C 1303) has been developed that involves slicing and scaling under controlled labo-ratory conditions. Christian et al. (1995) provides an example of an application of this accelerated aging test method.
Brandreth (1986) and Tye (1988) show that the aging process of polyurethane and polyisocyanurate materials is reasonably well understood analytically and confirmed experimentally. The domi-nant parameters for minimum aging are as follows: • Closed-cell content > 90%, preferably > 95% • Small uniform cell diameter << 1 mm, with a larger proportion of polymer in windows • Small anisotropy in cell structure • High density • Increased thickness • High initial pressure of fluorocarbon blowing agent in cell • Polymer highly resistant to gas diffusion and solubility • Polymer distributed evenly in struts and windows of cells • Low aging temperature Aging may be further reduced, particularly for laminated and spray-applied products, with higher density polymer skins, or by well-adhered facings and coverings with low gas and moisture permeance characteristics. An oxygen diffusion rate of less than 3.5 mm3/(m2·day) for 25 µm thickness of barrier is one criterion used by some industry organizations manufacturing laminated products. The adhesion of any facing must be continuous, and every effort must be made in the manufacturing process to eliminate or minimize the shear plane layer at the foam/substrate interface (Ostrogorsky and Glicksman 1986).
Closed-cell phenolic-type materials and products, which are blown with similar gases, age differently and much more slowly.
The reasons for this are believed to be higher material density, smaller more uniform cell size with a larger proportion of polymer in windows, and a basic polymer more resistant to gas diffusion.
The average distance, or mean free path, that an enclosed gas molecule travels before striking another gas molecule increases as pressure within an insulation decreases. When the mean free path equals the average distance a gas molecule travels before striking a solid part of the insulation, the apparent thermal conductivity of the insulation decreases with decreasing pressure. Correspondingly, for materials such as silica gel and fine carbon black, which have an average pore size smaller than the mean free path of air at atmo-spheric pressure, it is possible to attain thermal conductivity values lower than those for still air (Verschoor and Greebler 1952).
For homogeneous, dense materials, the primary mode of heat transfer is conduction. However, as the temperature increases, the heat transmission by thermal radiation and possible convection becomes a greater part of the total heat transferred. The magnitude of radiation and convection transfer depends on temperature differ-ence, direction of heat flow, the nature of materials involved, and geometric considerations.
Because heat is transferred partly by radiation in low-density insulation, measured apparent thermal conductivity depends on test thickness. The thickness effect increases the apparent thermal con-ductivity measured at installed thickness over that commonly deter-mined at 25 mm (Pelanne 1979). From a thermal resistance standpoint, the effect is small, typically less than 10% even for thin (e.g., 25 mm) low-density (e.g., 5.5 kg/m3) insulation. The effect becomes negligible for typical building applications (e.g., 150 mm insulation with a density of 11 kg/m3).
Fig. 3 Typical Variation of Apparent Thermal Conductivity with Mean Temperature and Density for Fibrous Insulations 23.6 2001 ASHRAE Fundamentals Handbook (SI) Environmental and Application Conditions The apparent conductivity of insulating materials generally in-creases with increasing temperature. Figure 4 shows typical varia-tions with mean temperature. However, some materials, such as fluorocarbon-expanded, closed-cell urethanes, have an inflection in the curve over the temperature range where there is a change of phase of the fluorocarbon from gas to liquid (see Table 10 in Chapter 25).
The apparent thermal conductivity of a sample at one mean tem-perature (average of the two surface temperatures) only applies to the material at the particular thickness tested. Further testing is required to obtain values suitable for all thicknesses. The rate of change of apparent thermal conductivity with temperature and envi-ronmental conditions varies with the type and density of material.
Insulating materials that permit a large percentage of heat trans-fer by radiation, such as low-density fibrous and cellular products, show the greatest change in apparent thermal conductivity with changes of temperature and surrounding surface emittance. The ASTM Standard Test Methods recognize the importance of radia-tion in heat transmission and require that the surfaces of all test apparatus plates be painted or otherwise treated to have a total emit-tance greater than 0.8 at operating temperature.
The effect of temperature alone on structural integrity is not ordi-narily important for most materials in low-temperature insulation applications. Decomposition, excessive linear shrinkage, softening, or some other effects of temperature alone limit the maximum tem-perature for which a material is suited. At extreme temperatures, selecting materials for a specific service becomes more critical and must be based on experience and actual performance data (see Table 4 and Table 10 in Chapter 25).
Convection and air infiltration in or through some insulation sys-tems may increase heat transfer across them. Low-density, loose-fill, large open-cell and fibrous insulations, and poorly designed or installed reflective systems are most susceptible to increased heat transfer by air filtration and convection. The temperature difference across an insulation, as well as the height, thickness, or width of the insulated space, influences the amount of convection. In some cases, natural convection may be inherent in the systems (Wilkes and Rucker 1983, Wilkes and Childs 1992). However, in many cases, the effect of convection in or through insulation can be minimized by careful design of an insulated structure (Donnelly et al. 1976).
Gaps between both board- and batt-type insulation can lower the effectiveness. Board-type insulation may not be perfectly square, may be installed improperly, and may be applied to uneven surfaces. For example, a 4% void area around batt insula-tion can produce a 50% loss in effective thermal resistance of ceil-ing application with R = 3.4 m2·K/W (Verschoor 1977). Similar results have been obtained with different test conditions and for wall configurations (Lewis 1979, Hedlin 1985, Tye et al. 1981).
Preformed joints in board-type insulation allow it to fit together without air gaps. Boards and batts can be installed in two layers, with joints between layers offset and staggered to eliminate gaps.
Chapter 24 has further details on these effects.
The presence of moisture may decrease the thermal performance of the installed insulation. The apparent thermal conductivity of construction materials increases with moisture content. If moisture condenses in the insulation, it may further reduce thermal resis-tance, and, perhaps physically damage the system. The reduction in thermal resistance depends on the material, the moisture content, and its distribution.
More information on the effects of moisture is provided in the section on Effect of Moisture on Heat Flow. Section A3 of the CIBSE Guide (1986) and Chapter 24 cover the thermal properties of building structures affected by moisture.
THERMAL TRANSMITTANCE The method of calculating an overall coefficient of heat trans-mission requires knowledge of (1) the apparent thermal conduc-tivity and thickness of homogeneous components, (2) thermal conductance of nonhomogeneous components, (3) surface conduc-tances of both sides of the construction, and (4) conductance of air spaces in the construction. Procedures for calculating thermal con-ductance and resistance, and definitions of heat transfer terms and symbols are included in this chapter and in Chapter 3. In some con-struction, multidimensional heat flow effects are significant. Paral-lel heat flow paths of different resistances occur in wood frame house construction, for example. In such cases, the hot box method (ASTM Standard C 1363) or a multi-dimensional computer model may be used to determine overall thermal transmittance.
Surface Conductance Surface conductance is the heat transfer to or from the surface by the combined effects of radiation, convection, and conduction. Each of these transport modes can vary independently. Heat transfer by radiation between two surfaces is controlled by the character of the surfaces (emittance and reflectivity), the temperature difference between them, and the solid angle through which they see each other. Heat transfer by convection and conduction is controlled by surface roughness, air movement, and temperature difference between the air and surface. Table 1 illustrates the importance of the effect of temperature of surrounding surfaces on surface heat flux caused by radiation.
In many cases, because the thermal resistance (reciprocal of conductance) of the internal parts of the wall is high compared Fig. 4 Apparent Thermal Conductivity Versus Mean Temper-ature for Various Materials (in Air at Atmospheric Pressure) (Glaser et al. 1967, Pelanne 1977) Thermal and Moisture Control in Insulated Assemblies—Fundamentals 23.7 with the surface resistance, the surface factors are of minor im-portance. However, surface resistances on a window with a single pane of glass constitute almost the entire resistance and are very important. Raber and Hutchinson (1945) analyzed the factors affecting surface conductance and the difference between surface and air temperatures.
The convective part of the surface conductance is markedly affected by the nature of the air movement on the surface, illus-trated by Figure 1 in Chapter 25. On smooth surfaces, surface length (Parmelee and Huebscher 1947) also affects the convection part of conductance; the average value decreases as the surface length increases. Moreover, Parmelee and Aubele (1950) observed that only under certain conditions can outdoors be treated as a blackbody radiating at an effective air temperature. Therefore, selection of surface conductance coefficients for a building becomes a matter of judgment. Surface conductances in Table 1 in Chapter 25 are applicable to ordinary building materials. In special cases, where surface conductances become important factors in the overall rates of heat transfer, more accurate coefficients may be required. Principles and data given in Chapter 3 can be applied in such cases.
FACTORS AFFECTING HEAT TRANSFER ACROSS AIR SPACES Heat transfer across an air space is affected by the nature of the boundary surfaces, the intervening air space, the orientation of the air space, the distance between boundary surfaces, and the direction of heat flow. Air space conductance coefficients repre-sent the total conductance from one surface bounding the air space to the other. The total conductance is the sum of a radia-tion component and a convection and conduction component. In all cases, the spaces are considered airtight with no through air leakage.
The radiation portion of the coefficient is affected by the temperature of the two boundary surfaces and by their respec-tive surface properties. For surfaces that can be considered ideal graybodies, the surface properties can be characterized by emis-sivity. The combined effect of the emittances of the boundary surfaces of an air space is expressed by the effective emittance E of the air space. The radiation component is practically not affected by the thickness of the space, its orientation, the direc-tion of heat flow, or the order of emittance (hot or cold surface).
In contrast the heat transfer by convection and conduction com-bined is affected markedly by orientation of the air space and the direction of heat flow, by the temperature difference across the space, and, in some cases, by the thickness of the space. It is also slightly affected by the mean temperature of its surfaces.
For air spaces in building construction, the radiation and con-vection-conduction components can vary independently of each other.
Table 3 in Chapter 25 lists the thermal resistance values of sealed air spaces of uniform thickness and moderately smooth, plane, parallel surfaces. These data are based on experimental measurements (Robinson et al. 1954). Resistance values for sys-tems with air spaces can be estimated from these results if emis-sivity values are corrected for field conditions. However, the resistance value of some common composite building insulation systems involving mass-type insulation with a reflective surface in conjunction with an air space may be appreciably lower than the estimated value (Palfey 1980). This is true particularly if the air space is not sealed or of uniform thickness. The thermal resistance values for plane air spaces in Table 3 in Chapter 25 represent typical values; for critical applications, the effective-ness of a particular design should be confirmed by actual test data undertaken by using ASTM hot box methods (ASTM Stan-dards C 236 and C 976). This test is especially necessary for constructions combining reflective and nonreflective thermal insulation.
For narrow air spaces, defined as those for which the product of the temperature difference (in kelvins) and the cube of the space thickness (in millimetres) is less than 27 000 for heat flow horizontally or downward, or less than 9000 for heat flow upward, convection is practically suppressed. The conductance for these spaces is the sum of the radiative heat transfer coeffi-cient and that for heat conduction alone through air. The radia-tion and conduction component can be computed by the method shown in the footnote to Table 3 in Chapter 25. Effects of differ-ent mean temperatures, temperature differences, and effective emittances may be found in this table.
To obtain high thermal resistance with reflective insulation, a series of multiple air layers bounded by reflective surfaces is needed. The total resistance equals the sum of the resistance values across each air space. All air layers or spaces must be sealed because air moving between the layers can increase the heat flow. Depend-ing on the type of reflective insulation, one or both sides may have highly reflective surfaces. Except for thick horizontal air spaces with heat flow down, little is gained thermally by the addition of a second highly reflective surface to the same air space. If an air space has only one reflective surface, and conditions are completely dry, the side on which the reflective surface is placed makes no appre-ciable difference in the rate of heat transfer. In typical building sit-uations moisture is present and the reflective facing should always be placed on the warm side of the air space, else trace condensation will at times increase the emittance and reduce the insulation value (see Chapter 25, Table 2). A reflective surface placed on the warm side of an air space may also act as a water vapor retarder if the material and its joints have sufficiently low permeance (see the sec-tion on Vapor Retarder Functions and Properties).
The emittance of a surface is the measure of its ability to emit radiant energy and, for the same temperature and wavelength, is equal to its absorptance (ratio of the radiant energy absorbed by a surface to the total radiant energy falling on it). The ratio of the energy reflected by the surface to that falling on it is the reflectance.
For an opaque surface, reflectance is equal to one minus the emit-tance. This emittance varies with surface type and condition and radiation wavelength.
For reflective insulation used with heating, air-conditioning, and refrigeration applications, the emittance value for long-wave-length (infrared) radiation is important, not the value for the shorter wavelengths of the visible spectrum. Visible brightness is not a true measure of the reflectance for thermal radiation because the reflec-tance for light and for long-wavelength radiation is unrelated.
Table 2 in Chapter 25 lists typical emittance values for reflective surfaces and building materials, and the corresponding emittance factors for air spaces.
Chemical action, dust or oil accumulation, or the presence of condensation or frost can change a reflective surface enough to reduce its reflectance and increase its emittance. Chemical changes include oxidation, corrosion, or tarnishing caused by air, moisture, wet plaster, or the chemical treatment of wood spacing strips or other adjoining structural members. Surface emittance values should be obtained by tests.
Table 1 Variation in Surface Heat Flux for Vertical Surfaces at 26.7°C with Different Temperatures of Surrounding Surface (21.1°C Ambient Still Air; 0.83 Emittance) Surrounding Surface Temperature, °C Surface Heat Flux, W/m2 23.9 21.1 18.3 15.6 10 Convection 20.8 20.8 20.8 20.8 20.8 Radiation 13.9 27.1 40.4 53.6 78.6 Total 34.7 47.9 61.2 74.4 99.4 23.8 2001 ASHRAE Fundamentals Handbook (SI) CALCULATING OVERALL THERMAL RESISTANCE Using the principles of heat flow presented in Chapter 3, calculat-ing heat flow by the overall thermal resistance method is preferred.
The total resistance to heat flow through a flat building construc-tion composed of parallel layers such as a flat ceiling, floor, or wall (or curved surface if the curvature is small) is the numerical sum of the resistances (R-values) of all layers of the construction in series: (1) where R1, R2, …, Rn = individual resistances of the layers R = resistance of construction from inside surface to outside surface However, in buildings, to obtain the overall resistance RT, the film resistances Ri and Ro from Table 1 in Chapter 25 must be added to R: (2) The U-factor (thermal transmittance) is the reciprocal of RT, or (3) Thus, for a wall with air space construction, consisting of two homogeneous materials of conductivities k1 and k2 and thickness x1 and x2, respectively, and separated by an air space of conductance C, the overall resistance is (4) where hi and ho are the heat transfer film coefficients.
Series and Parallel Heat Flow Paths In many installations, components are arranged so that heat flows in parallel paths of different conductances. If no heat flows between lateral paths, heat flow in each path may be calculated using Equations (1) and (2). The average transmittance is then (5) where a, b, …, n are respective fractions of a typical basic area com-posed of several different paths with transmittances Ua, Ub, …, Un.
If heat can flow laterally with little resistance in any continuous layer so that transverse isothermal planes result, total average resis-tance RT (av) is the sum of the resistance of the layers between such planes. Each layer is calculated by the appropriate Equation (1) or modification of Equation (4), using the resistance values. This is a series combination of layers, of which one (or more) provides par-allel paths.
The calculated heat flow, assuming parallel heat flow only, is usually considerably lower than that calculated with the assumption of combined series-parallel heat flow. The actual heat flow is some value between the two calculated values. In the absence of test val-ues for the combination, an intermediate value should be used.
Examination of the construction usually reveals whether a value closer to the higher or lower calculated value should be used. Gen-erally, if the construction contains any highly conductive layer in which lateral conduction is very high compared to heat flow through the wall, a value closer to the series-parallel calculation should be used. If, however, no layer has a high lateral conductance, a value closer to the parallel heat flow calculation should be used. In the case where the presence or absence of high lateral conductance is unclear, use the arithmetic mean of the isothermal-planes and par-allel-path methods. A more precise value of the thermal transmit-tance may be obtained by using an appropriate computer program for two- or three-dimensional heat flow.
CALCULATING INTERFACE TEMPERATURES The temperature at any interface can be calculated, since the temperature drop through any component of the wall is proportional to its resistance. Thus, the temperature drop ∆tn through Rn in Equa-tion (1) is (6) where ti and to are the indoor and outdoor temperatures, respectively.
Hence, the temperature at the interface between Rn and Rn+1 is (7) For building materials having nonuniform or irregular sections such as hollow clay tile or concrete blocks, the R-value of the unit as manufactured should be used.
If the resistances of materials in a wall are highly dependent on temperature, the mean temperature must be known to assign the cor-rect value. In such cases, it is perhaps most convenient to use a trial-and-error procedure for the calculation of the total resistance RT.
First, the mean operating temperature for each layer is estimated, and R-values for the particular materials are selected. The total resistance RT is then calculated as in Equation (4), and the temper-ature at each interface is calculated using Equations (6) and (7). The mean temperature of each component (arithmetic mean of its sur-face temperatures) can then be used to obtain second generation R-values. This procedure can then be repeated until the R-values have been correctly selected for the resulting mean temperatures.
Generally, this can be done in two or three trial calculations.
HEAT FLOW CALCULATIONS Equation (8) is used to calculate heat flow through flat surfaces; Equation (9) is used for cylindrical surfaces (Figure 5).
R R1 R2 R3 R4 … Rn + + + + + = RT Ri R Ro + + = U 1 RT ------= RT 1 hi ----x1 k1 -----1 C ----x2 k2 -----1 ho -----+ + + + = Uav aUa bUb … nUn + + + = tn ∆ Rn ti to – ( ) RT ------------------------= tn n – 1 + ti ∆tm m 1 = n ∑ – = Fig. 5 Heat Flow Through Cylindrical Surfaces Thermal and Moisture Control in Insulated Assemblies—Fundamentals 23.9 (8) (9) where qs = rate of heat transfer per unit area of outer surface of insulation R = surface-to-surface thermal resistance from Equation (1) kn = thermal conductivity of insulation layer n at calculated mean temperature tis = temperature of inner surface tos = temperature of outer surface ri,n = inner radius of insulation layer n ro,n = outer radius of insulation layer n rs = outer radius of outer insulation layer N = total number of insulation layers ln = natural or Naperian logarithm Heat flow per unit length of pipe is the preferred unit. The terms that appear in the denominator of Equation (9) represent the resis-tances to heat flow based on logarithmic mean thickness. The ther-mal resistances for flat surfaces should not be confused with the thermal resistance of a cylindrical insulation where the surface areas involved are never equal.
INSULATION THICKNESS ECONOMIC THICKNESS Economics can be used (1) to select the optimum insulation thickness for a specific insulation, or (2) to evaluate two or more insulation materials for least cost for a given thermal performance.
In either case, economics determine the most cost-effective solution for insulating over a specific period (FEA 1976). This solution can be reached by different techniques, but only one solution exists for a given set of economic variables.
Greater than optimum insulation thickness may also require a greater capital investment for the structure and piping. However, other concerns such as limited energy availability, sustainability, or comfort may alter the results of an economic analysis alone.
Life-cycle costing spreads the initial cost of the insulation over the number of years the insulation is expected to be in service. The life cycle selected affects the economic thickness of the insulation.
Thinner insulation pays back on a short life cycle, and thicker insu-lation pays back over a longer cycle. An insulation system designed to pay for itself with energy savings in a short time, and that stays in service longer, does not necessarily produce the lowest total cost over the service period.
The annualized cost of the installed insulation must also be adjusted for the cost of money, which can be based on a desired rate of return on the insulation investment. Insulation system mainte-nance costs should also be included in the annual cost.
Because fuel cost is likely to change during the depreciation period (the life of the facility or payback period), the average cost should be estimated and this value used rather than current cost.
The total annual cost is the sum of the annual cost of insulation and the annual cost of lost energy. For each incremental increase in insulation thickness, the corresponding change in the total cost is mt = mc + ms where mt is the incremental change in total annual costs, mc is the incremental change in cost of insulation, and ms is the incremental change in cost of lost energy plus any change in the cost of the mechanical equipment needed.
Initially, as insulation is applied, the total cost decreases because the incremental energy savings is greater than the incremental cost of insulation. Additional insulation reduces this cost up to a thickness where the change in the total cost is equal to zero. At this point, mt = mc + ms = 0, and no further reduction can be obtained.
Beyond this thickness, the incremental insulation costs become greater than the additional energy savings derived by adding another increment of insulation. However, if total insulation costs then exceed total insulation benefits, the optimal level of insulation for a given component from a purely economics viewpoint is no insulation at all. This may occur if the startup costs are extremely high, as for installation of insulation in a sealed wall cavity.
ECONOMIC THICKNESS: MECHANICAL SYSTEMS Determining the most profitable thickness of insulation for mechanical systems is difficult. The economics of each plant (including cost of producing energy, cost of insulating, discount rate or cost of capital, and potential for energy loss) indicate the preferred amount of insulation. Various types of equipment and piping also require different economic thicknesses. This analysis is further complicated because future energy cost and the life of the facility and insulation must also be considered. For every plant, these factors dictate different solutions to the economic analysis. Furthermore, additional insulation may lower heat loss or gain, possibly allowing for lower capacity equipment and thus for lower first costs.
Economic Analysis The cost of installed insulation increases with thickness. This incremental cost is for both labor and material. Insulation is often applied in multiple layers (1) because materials are not manufac-tured in single layers of sufficient thickness and, (2) in many cases, to compensate for expansion and contraction. Figure 6 shows installed costs for a multilayer application. The average slope of the curves increases with the number of layers because labor and material costs increase at a more rapid rate as thickness increases.
Because the optimal economic thickness is the lowest total cost of lost energy and installed insulation over the life of the insulation, qs tis tos – R -----------------= qs tis tos – rs ro n , ri n , ⁄ ( ) ln kn ⁄ m 1 = N ∑ --------------------------------------------------------------= Fig. 6 Determination of Economic Thickness of Installed Insulation of Mechanical Equipment 23.10 2001 ASHRAE Fundamentals Handbook (SI) these two costs must be compared in similar terms. Either the annual cost of the insulation must be compared to the average annual cost of lost energy, or the cost of the energy lost each year must be expressed in present dollars and compared with the total cost of the insulation investment. The former method, annualizing the insula-tion cost and comparing this with the average annual cost of lost energy, is easier to compute.
Insulation reduces the size and capital costs of the heating and cooling equipment required for an installation because it lowers energy demand. This capital cost may be annualized by consider-ing the plant depreciation period, cost of money, annual energy output for the plant, and operational expenses.
Figure 6 shows curves of total annual costs of operation, insu-lation costs, and lost energy costs. Point A on the total cost curve corresponds to the economic insulation thickness, which, in this example, is in the double-layer range. Viewing the calculated eco-nomic thickness as a minimum thickness provides a hedge against unforeseen fuel price increases and conserves energy.
ECONOMIC THICKNESS: BUILDING ENVELOPES In buildings such as residences and warehouses, the internal energy gains are insignificant compared with the heat losses and gains through the envelope. For these buildings, the heating and cooling requirements are roughly proportional to the difference between the indoor and outdoor temperature. For commercial, industrial, and institutional buildings, internal heat loads can be sig-nificant, and the heating and cooling requirements are not as directly related to the indoor/outdoor temperature difference. In both types of buildings, solar heat can be an important factor and should be evaluated.
Dominant Heat Loss and Gain Through Envelope Thermal insulation is generally installed in building envelope components (e.g., ceilings, walls, and floors) to reduce space heat-ing and cooling costs on a long-term basis. Additional benefits may include increased occupant comfort, reduced heating and cooling system capacity, and elimination of condensation on wall surfaces in cold climates. When possible these benefits should be consid-ered. The economically optimal insulation thickness (best measured in terms of thermal resistance) in an envelope component minimizes total life-cycle space heating and cooling costs attributable to it.
Total life-cycle costs are the sum of present-value heating and cool-ing costs over the useful lifetime of the insulation plus the installed cost of the insulation and the installed cost of the heating and cool-ing equipment.
Ifthe R-value of the insulation used is continuously variable (e.g., loose-fill insulation in attics), uniformly small increments of insula-tion can be used to determine appropriate optimal thicknesses; or calculus can be used to determine an exactoptimum.If the insulation materials used are available only in discrete levels of thermal resis-tance (R = 2, R= 3, R = 5), the increment used in determining optimal thickness should be based on differences between those levels (R = 2 over R= 0, R= 3 over R = 2, R= 5 over R= 3). Where discrete incre-ments of resistance are used, determining the resistance level for which incremental savings equal incremental costs may not be fea-sible.In such cases, the selection should be leftto the judgmentofthe analyst based on the level of conservation desired. In addition, an increase in insulation may make it possible to reduce the size of the heating and cooling equipment, which becomes a discrete reduction in equipment cost.
If a building envelope component requires structural modifica-tions to accommodate increased insulation thickness, this cost must be included in the installed cost of the additional insulation. Gen-erally, such modifications should only be considered when they are less costly than the use of more efficient (i.e., lower thermal conductivity) but more expensive insulation materials than those ordinarily used.
Typically, the incremental energy savings and insulation costs differ for each building envelope component; therefore, the optimal insulation level differs for each component in the same building.
Less efficient heating plants and higher costs of heating energy necessitate higher optimal insulation levels in each building enve-lope component. Conversely, more efficient heating equipment reduces the optimal insulation level. The effects of climate, cooling energy costs, and cooling equipment efficiency on optimal insula-tion levels are less clear and differ widely, depending on overall building design and operational profile.
Dominant Internal Heat Loads In buildings with dominant internal loads, the energy require-ments vary so widely that no generalizations can be made regard-ing insulation. This contrasts with envelope-dominated structures, in which more insulation reduces energy consumption (Hart 1981).
In internal-load-dominated buildings with both annual heating and cooling loads, higher thermal resistance increases cooling energy consumption while reducing heating energy consumption.
Therefore, the calculation of economically optimum resistance becomes quite complex and involves multiple measure or hourly methods described in Chapter 31. Spielvogel (1974), Burch and Hunt (1978), and Rudoy (1975) give more details.
Figure 7 shows the results of these calculations for a building in Columbus, Ohio, with 26 W/m2 of internal heat gains that operates 24 h per day (Spielvogel 1974). This solution is not the only one possible, but illustrates problems faced by the designer. In this case, thermal resistance increases, the U-factor decreases, annual heating energy decreases, and annual cooling energy increases. The energy optimum exists at Point Y in Figure 7, where the total heating and cooling energy is at a minimum. Because the cost of cooling energy differs from the cost of heating energy, the economic optimum will not be the same as the energy optimum.
These results occur in some localities where there are far more hours per year with outdoor temperatures between 10 and 24°Cthan Fig. 7 Example of Optimal Thermal Resistance for Building with Internal Heat Gains (Adapted from Spielvogel 1974) Thermal and Moisture Control in Insulated Assemblies—Fundamentals 23.11 between 24 and 38°C. At temperatures between 24 and 38°C, low U-factors result in less energy consumption for cooling. However, for temperatures between 10 and 24°C, low U-factors inhibit the flow of internal heat from the building, thereby creating higher cooling loads and higher energy requirements than those in buildings with higher U-factors. What might be saved at outdoor temperatures over 24°C can be more than spent in additional cool-ing energy at temperatures below 24°C. Economizer cycles could offset these excess internal gains with ventilation air, however.
Where the hours of use or the quantity of internal heat gains vary from room to room, the optimum thermal resistance also varies. For example, in a cold climate, a hotel kitchen requires little or no insu-lation, because the internal heat is sufficient to heat the space almost all year. In a meeting room adjacent to the kitchen, substantially more insulation is justified. Thus, the economic thermal resistance of any envelope element, such as a roof, will not be the same throughout the entire building.
This type of analysis must include the level and duration of inter-nal gains and the nature of the energy consumption of the heating and cooling systems. Most buildings need evaluation of walls, roofs, and floors on a room-by-room basis. Computer programs make these more complex analyses possible. Due to the wide diver-sity of building types, internal gains, system types, and operating conditions, no simple rules can establish U-factors for minimum energy consumption.
Effectiveness of Added Insulation The effectiveness of added insulation varies with many factors, including climate, original insulation level, preparation costs, and predicted life, based on payback calculations. Building codes gener-ally balance life-cycle costs between construction, financing, and energy expenditures. Figure 8 shows typical relationships between life-cycle costs and energy consumption. Individual points on the curve represent different combinations of ceiling, wall, and floor insulation in R-values and glazing types (single, double, or triple).
Because life-cycle costs vary not only with construction and energy costs but also with climatic factors, the profiles of this curve vary according to locality. In Figure 8, the optimal condition for this example is attained with R = 5.3 m2·K/W attic insulation, R = 3.4 wall insulation, R = 2 floor insulation, and double glazing. However, the effectiveness of added insulation can be determined only by ana-lyzing actual conditions.
MOISTURE IN BUILDINGS MOISTURE PROBLEMS IN BUILDINGS Moisture control is necessary to avoid moisture-related problems with building energy performance, building maintenance and dura-bility, and human comfort and health. Moisture degradation is the largest factor limiting the useful life of a building and can be visible or invisible. Invisible degradation includes degradation of the ther-mal resistance of building materials and decrease in the strength and stiffness of some materials. Visible moisture degradation may be in the form of (1) mold and mildew, (2) the decay of wood-based mate-rials, (3) spalling of masonry and concrete caused by freeze-thaw cycles, (4) hydration of plastic materials, (5) corrosion of metals, (6) damage due to expansion of materials (e.g., buckling of wood floors), and (7) a decline in visual appearance (e.g., buckling of wood siding or efflorescence of masonry materials, which is the for-mation of a salt crust from the leaching of free alkalies). In addition, high moisture levels can lead to odors and mold spores in indoor air, which can seriously affect occupant health and comfort. Short sum-maries of such moisture conditions and related performance and health issues follow.
Mold, Mildew, Dust Mites, and Human Health Mold and mildew in buildings are offensive, and the spores can cause respiratory problems and other allergic reactions in humans.
Mold and mildew will grow on most surfaces if the relative humid-ity at the surface is above a critical value and the surface tempera-ture is conducive to growth. The longer the relative humidity remains above the critical value, the more likely is visible mold growth; and the higher the humidity or temperature, the shorter is the time needed for germination. The surface relative humidity is a complex function of material moisture content, material properties, and local temperature and humidity conditions. In addition, mold growth depends on the type of surface. Fully recognizing the com-plexity of the issue, the International Energy Agency Annex 14 (1990) nevertheless established a surface humidity criterion for design purposes: The monthly average surface relative humidity should remain below 80%. Others have proposed more stringent criteria, the most stringent requiring that surface relative humidity remain below 70% at all times. Although there still is no agreement on which criterion is most appropriate, mold and mildew can usu-ally be avoided by limiting surface moisture conditions over 80% to short time periods. These criteria should only be relaxed for nonpo-rous surfaces that are regularly cleaned. Hukka and Viitanen (1999) developed a mathematical model for the prediction of a mold growth index. This model was successfully implemented and linked to a hygrothermal model by Karagiozis and Salonvaara (1998).
Most molds grow at temperatures above approximately 5°C. Mois-ture accumulation below 5°Cmay not cause mold and mildew if the material is allowed to dry below the critical moisture content before the temperature rises above 5°C.
Dust mites can trigger allergies and asthma (Burge et al. 1994).
Dust mites thrive at high relative humidities (over 70%) at room temperature, but will not survive sustained relative humidities below 50% (Burge et al. 1994). These relative humidities relate to local conditions in typical places that mites tend to inhabit such as mattresses, carpets, soft furniture, etc.
Paint Failure and Other Appearance Problems Moisture trapped behind paint films may cause failure of the paint. Water or condensation may also cause streaking or staining.
Excessive swings in the moisture content of wood-based panels or boards may cause buckling or warping. Excessive moisture in masonry and concrete may cause efflorescence, a white powdery area or lines, or, when combined with low temperatures, may cause freeze-thaw damage and spalling (chipping).
Fig. 8 Typical Relationship of Life-Cycle Cost to Energy Use 23.12 2001 ASHRAE Fundamentals Handbook (SI) Structural Failures Structural failures due to decay of wood are rare but have occurred (e.g., Merrill and TenWolde 1989). Decay generally requires wood moisture content atfiber saturation (usually about 30%) or higher and temperatures between 10 and 40°C. Wood moisture contents above fiber saturation are only possible in green lumber or by absorption of liquid water from condensation, leaks, ground water, or other satu-rated materials in contact with the wood. To maintain a safety margin, 20% moisture content is sometimes used during field inspections as the maximum allowable moisture level. Because wood moisture con-tent can vary widely with sample location, a local moisture content of 20% or higher may indicate fiber saturation elsewhere. Once estab-lished, decay fungi produce water that enables them to maintain moisture conditions conducive to their own growth.
Rusting or corrosion of nails, nail plates, or other metal building components is also a potential cause of structural failure. Corrosion may occur at high relative humidities near the metal surface or as a result of liquid water from elsewhere. Wood moisture content over 20% encourages corrosion of steel fasteners in wood, especially if the wood is treated with preservatives. In buildings, metal fasteners are often the coldest surfaces, encouraging condensation on and corrosion of the fasteners.
Effect of Moisture on Heat Flow Moisture in the building envelope can significantly degrade the thermal performance of most insulation materials. Bomberg and Shirtliffe (1978), Epstein and Putnam (1977), Hedlin (1977, 1983, 1987, 1988a, 1988b), Jespersen (1960), Joy (1957), Knab et al.
(1980), Kumaran (1989), Kyle and Desjarlais (1994), Langlais et al.
(1983), Paljack (1973), Pedersen (1990), Pedersen et al. (1991), Shapiro and Motakef (1990), Thomas et al. (1983), Tobiasson and Richard (1979), Tobiasson (1987, 1991), and Tye (1987) investi-gated the effect of moisture content on heat flow and showed that the effect depends on the type of insulation material, the moisture content, the temperature of the insulation material, the insulation material’s thermal history, and the building envelope’s interior and exterior environments. The reported relationships between the ther-mal performance of the insulation material and heat flow can vary significantly; the variations are more pronounced in open-cell and fibrous insulations. Rapid vapor transfer through vapor-permeable insulations during testing accounts for these variations. Variations can also be due to the location of water in the insulation layer. Kyle and Desjarlais (1994) estimated that water distribution can account for a difference of up to 25% in heat flow in certain cases.
Moisture can contribute to both sensible and latent heat flow, as well as through mass transfer by diffusion or convection. Evapora-tion on the warm side of the insulation and condensation or adsorp-tion on the cold side add a latent heat component to the heat flow.
Under steady-state conditions, the effect of moisture on thermal resistance may be small. Verschoor (1985) showed that an insulated residential-type stud wall panel with a poor vapor retarder on the warm side accumulated 1.5 kg of moisture per square metre when exposed to conditions continuously below freezing (steady-state) for 31 days. All of the accumulated moisture was located in the 13 mm layer of mineral fiber insulation immediately adjacent to the cold-side sheathing and at the interface between the insulation and the sheathing. In a continuation of the test program, to a total expo-sure period of 60 days, the rate of moisture gain remained constant during the entire period. Subsequently, the same test wall was sub-jected to diurnal outside temperature cycles through the freezing point. With this exposure, most of the accumulated moisture was found in the sheathing and the bottom of the test wall rather than in the coldest insulation layer. Except for the sheathing, there was neg-ligible change in the overall thermal performance of the wall.
The behavior may be different for closed-cell plastic foam insu-lations. Field observations of low-temperature insulated tanks have shown that when no air space exists between the cold surface and the insulation, the expected accumulation of condensed moisture did not occur.
Degradation of thermal resistance is more pronounced when daily reversals in temperature across the insulation drive moisture back and forth through the insulation layer and is exacerbated when the insulation material has a high water vapor permeance. A wetted compact low-slope roof is a good example of these phenomena. Dur-ing the nighttime, moisture migrates upward through the insulation layer and can condense in the upper part of the insulation layer on the underside of the membrane. The following day, the increase in ambi-ent temperature coupled with solar radiation heats the membrane and reverses the vapor pressure difference, evaporating some or all of the condensed water and driving it downward into the roofing system, transporting heat in the process. If sufficient water vapor can be driven downward, it may condense somewhere in the lower portion of the insulation layer, releasing its heat of condensation.
Hedlin (1988b) and Shuman (1980) experimentally showed that, for building envelopes containing permeable fibrous insulations that underwent temperature reversals, the rate of energy transfer increased sharply as the moisture content (MC) increased to approx-imately 1% by volume. The rate of increase in energy transfer diminished rapidly with further increases in moisture content.
Energy transfer for permeable insulation with 1% MC by volume was roughly double that of dry insulation. Pedersen et al. (1991) analytically reproduced Hedlin’s results; he demonstrated the high mobility of moisture in a permeable insulation and showed that latent effects are appreciable for a wide variety of North American climates. The latent effects typically add to the energy load and can increase peak energy demand. The extra load is added to the build-ing load in the warm afternoon, and nearly the same amount of heat is removed in the cool evening.
Certain organic insulations such as wood fiberboard and perlite are hygroscopic and can contain 1% MC by volume if installed at equilibrium conditions; water leakage into the building envelope component is not a prerequisite for significant increases in heat flow.
In studies of hourly temperature and moisture content variations in wood frame wall cavities, Duff (1971) showed measurable levels of daily moisture migration across the cavity. Pedersen et al. (1991) demonstrated the same phenomenon in compact low-slope roofs.
The building envelope is exposed to ever-changing exterior con-ditions; fluctuations in the outdoor air temperature and the amount of solar radiation affect the temperature profile through the building envelope. These temperature changes affect the magnitude and the direction of the vapor pressure gradients. Water vapor transfer rates and direction are constantly changing, adjusting not only to the ther-mal changes but to changes in moisture concentration. However, conditions may exist where the average vapor pressure drive is in one direction, for example, upward in a low-slope roof during win-ter. In this situation, moisture accumulates in the insulation just below the impermeable roof membrane, and, assuming these con-ditions can be maintained for a reasonable length of time, water accumulates or frost forms unless the top layer of the insulation has adequate moisture absorption capacity (e.g., perlite or wood fiber-board). Under similar conditions in the summer, moisture is driven down to the vapor retarder or deck (if the deck is less permeable than the insulation layer). If there are no layers in the system that are less permeable than the insulation, the water vapor simply diffuses into the building interior.
Under conditions where the vapor pressure changes slowly or where the insulation layer has an extremely low water vapor per-meance, little water vapor is transported. Moisture still affects the sensible heat transfer in the building envelope component. Epstein and Putnam (1977) and Larsson et al. (1977) showed a nearly linear increase in sensible energy transfer of approximately 3 to 5% for each volume percent increase in moisture content in cellular plastic insulations. For example, an insulation material with a 5% MC by Thermal and Moisture Control in Insulated Assemblies—Fundamentals 23.13 volume has 15 to 25% greater energy transfer than the dry insulation material. Other field studies by Dechow and Epstein (1978) and Ovstaas et al. (1983) have shown similar results for insulations installed in below-grade applications such as foundation walls.
Moisture Effect on Heat Storage Moisture also affects the thermal storage capacity of certain hygroscopic building materials. At 10% MC, nearly 30% of the heat storage capacity of wood is in the water held in the cell walls. Since the specific heat of wood is a function of its temperature and mois-ture content (but almost independent of density and species), heat storage calculations must include an estimate of the equilibrium in-service moisture content of wood building components.
PROPERTIES OF WATER VAPOR IN AIR Chapter 6 describes the properties of moist air and defines the various terms associated with water vapor in air, such as dew-point temperature, dry-bulb and wet-bulb temperatures, and relative humidity. Chapter 6 also explains the use of the psychrometric chart, as well as the physics of heating and cooling moist air.
MOISTURE IN BUILDING MATERIALS Many common building materials are porous. The pores provide a large internal surface area, which generally has an affinity for water. In some materials, such as wood, moisture may also be adsorbed in the cell wall itself. The amount of water in these hygro-scopic (water-attracting) materials is related to the relative humidity (rh) of the surrounding atmosphere. When relative humidity of the surrounding air rises, hygroscopic materials gain moisture (adsorp-tion), and when the relative humidity drops these materials lose moisture (desorption). The relationship between relative humidity and the moisture content (MC) at a particular temperature can be represented in a graph called a sorption isotherm. Often isotherms obtained by adsorption are not identical to isotherms obtained by desorption, because the material tends to retain moisture when it is drying. This difference between desorption and adsorption iso-therms is called hysteresis. At high relative humidity, small pores become entirely filled with water. The maximum moisture content is reached when all pores are filled with water. But experimentally this can only be achieved in a vacuum, by boiling the material or by keeping the material in contact with water for an extremely long time (i.e., many years). In practice, the maximum moisture content of porous materials is lower. This lower maximum moisture content is sometimes referred to as the capillary saturation moisture con-tent. Figure 9 shows typical sorption curves, giving the equilib-rium moisture content (EMC) as a function of relative humidity.
The EMC increases with relative humidity, especially above 80% rh. The EMC decreases slightly with increasing temperature.
Chapter 22 describes hygroscopic substances and their use as dehumidifying agents. Table 2 in Chapter 11 of the 1999 ASH-RAE Handbook—Applications has data on the moisture content of various common materials in equilibrium with the atmosphere at various relative humidities.
Wood and many other hygroscopic materials change dimensions with variations in moisture content. The moisture content at which cell walls of wood are saturated but no free water exists in cell cav-ities is the fiber saturation point. It represents the upper limit for moisture gain from the air as water vapor. It is also the upper limit of swelling; although more water can be absorbed in cell cavities, additional swelling does not occur. Wood moisture content is expressed as a percentage of its oven-dry mass. The average fiber saturation point for most species is about 30%. The average EMC at 20°C and 45% rh (heating season indoor conditions) is about 8.5%.
At 24°C and 70% rh (summer conditions), the EMC is about 13%.
The resulting dimensional changes in wood are proportional to the change in moisture content, but vary with species and direction of grain. For white oak, which is representative of interior trim mate-rial, 4.5% MC variation causes 2.5% volumetric shrinkage or swell-ing. Longitudinal dimensional change in straight-grained wood is insignificant, but it increases with crossgrain and other irregularities.
Fig. 9 Typical Sorption Isotherms for Wood, Concrete, and Gypsum (Hysteresis Is Ignored) Table 2 Linear and Volumetric Shrinkage Values of Wood, from Green to Oven Dry Moisture Content Wood Species Radial % Tangential % Volumetric % Hardwoods Birch Yellow 7.3 9.5 16.8 Oak Northern red 4.0 8.6 13.7 Southern red 4.7 11.3 16.1 White 5.6 10.5 16.3 Softwoods Cedar Northern white 2.2 4.9 7.2 Western red cedar 2.4 5.0 6.8 Douglas fir Coast 4.8 7.6 12.4 Fir White 3.3 7.0 9.8 Hemlock Western 4.2 7.8 12.4 Pine Eastern white 2.1 6.1 8.2 Longleaf 5.1 7.5 12.2 Ponderosa 3.9 6.2 9.7 Redwood Young growth 2.2 4.9 7.0 Imported wood Lauan 3.8 8.0 — Note: Values expressed as a percentage of the green dimension (FPL 1999).
23.14 2001 ASHRAE Fundamentals Handbook (SI) Accordingly, residential wood trusses with top and bottom chords exposed to different temperature and moisture regimes can show measurable seasonal vertical movements (Plewes 1976).
Thermal expansion of wood is usually outweighed by shrinkage or swelling due to moisture content changes. The linear thermal expansion coefficient for wood across the grain ranges from about 12.6 × 10− 6 per kelvin for light wood species to about 45 × 10− 6 per kelvin for dense wood species, which is small compared to that of many other materials. The thermal expansion coefficient parallel to the grain is between 3.1 × 10− 6 and 4.5 × 10− 6 per kelvin.
Most plant and animal fibers undergo dimensional changes sim-ilar to those noted in wood. Typical values for wood can be found in Table 2; the Wood Handbook (FPL 1999) gives more detailed information. Related changes in other materials are not as well documented. However, different expansion rates caused by temper-ature and moisture changes in different materials used in composite constructions should be considered (Baker 1964, BRS 1974).
Porous materials also absorb water in liquid form when in con-tact with it. Liquid water may be present because of leaks, rain pen-etration, flooding, or surface condensation. Wetting may be so complete that the material reaches the capillary saturation moisture content.
MOISTURE MIGRATION Liquid water and water vapor migrate by a variety of moisture transport mechanisms. The following are some of the most impor-tant mechanisms: • Liquid flow by gravity or air pressure differences • Capillary suction of liquid water in porous building materials • Movement of water vapor by air movement • Water vapor diffusion by vapor pressure differences Although in the past many moisture control strategies focused on control of vapor diffusion through the installation of vapor (diffu-sion) retarders, the other mechanisms, when present, can move far greater amounts of moisture. Therefore, liquid flow, capillary suc-tion, and air movement should be controlled first.
Liquid flow by gravity and air pressure difference is not dis-cussed here, but a short description of the other mechanisms fol-lows. A more comprehensive treatment of moisture transport and storage may be found in Kumaran et al. (1994) and Kuenzel (1998).
Capillary Suction Within small pores of diameter less than approximately 0.1 µm, molecular attraction between the surface of the capillary and the water molecules causes a capillary suction defined by (10) where s = capillary suction σ = surface tension of water r = radius of the capillary θ = contact wetting angle The contact wetting angle is the angle between the water menis-cus and the capillary surface. The smaller this angle is, the larger the capillary suction. In hydrophilic (water-attracting) materials, the contact wetting angle is less than 90°, and in hydrophobic (water-repelling) materials, the angle is between 90° and 180°. Capillary suction is greater in smaller capillaries, so capillary suction moves moisture from larger to smaller capillaries. Although surface ten-sion is a function of temperature (the higher the temperature, the lower the surface tension), this effect is very small.
Isothermal and nonisothermal movement may occur in the liquid phase, or in the vapor phase if the capillaries are not completely filled. The transfer in the vapor phase is by vapor diffusion and is caused by a difference in vapor saturation pressure in the capillaries.
Thompson’s law states that the saturation vapor pressure in equi-librium with the water in a capillary is given by (11) where p″ = saturation vapor pressure in capillary p′ = saturation vapor pressure in ambient air at same temperature as p″ ρ = density of water R = gas constant T = absolute temperature Equation (11) shows that the saturation vapor pressure is lower in smaller capillaries than in larger capillaries, which causes vapor diffusion from larger water-filled capillaries to smaller capillaries. Saturation vapor pressure in the capillaries is lower at lower temperatures, which causes diffusion from higher to lower temperatures.
If the vapor pressure in the ambient air is in equilibrium with the saturation vapor pressure in the capillaries, Equation (11) can be rewritten as (12) where φis the relative humidity of the ambient air.
Capillary flow of water can be expressed as a function of suction pressure gradients follows: (13) where wm = water flux km = water permeability coefficient Air Movement Water vapor movement by air can be represented by w = Wρv (14) where w = water vapor flux (flow per unit area) W = humidity ratio of source air ρ = density of air v = airflow velocity Even small airflows can carry large amounts of water vapor when compared to vapor diffusion. Airflow retarders are designed to inhibit the flow of air and thereby the transport of water vapor into the construction.
Water Vapor Diffusion Water vapor also migrates by diffusion through air and building materials. Although moisture diffusion through air is relatively rapid, distribution of vapor in air is dominated by convection. This results in minimal vapor pressure differentials between connected spaces and rapid flow to condensing surfaces, such as cold glass or dehumidifier coils. Although movement by air convection usually dominates, when present, vapor diffusion can be an important mode of transportation in industrial applications such as cold storage facilities or built-in refrigerators. The control of diffusion also becomes more important with increasingly airtight construction.
The equation used to calculate water vapor diffusion flux through materials is based on a form of Fick’s law: s 2σ θ cos r ------------------= p″ p′ exp s ρRT ----------- p′ exp 2σ θ cos rρRT ------------------ = = s ρRT φ ln = wm km – ds dx ------= Thermal and Moisture Control in Insulated Assemblies—Fundamentals 23.15 (15) where w = water vapor flux (flow rate per unit area) p = water vapor pressure x = distance along flow path µ = water vapor permeability According to Equation (15), water vapor flow by diffusion is proportional to the water vapor pressure gradient and closely paral-lels Fourier’s equation for heat flow. The actual diffusion of vapor through a material is complex. The apparent water vapor permeabil-ity is a function of relative humidity and temperature and may vary spatially due to variations in material properties.
ASTM Standard E 96 describes test methods for the measure-ment of water vapor permeability. The rate of moisture transfer through a material is determined gravimetrically while maintaining a steady temperature and vapor pressure differential across the spec-imen. The standard identifies two tests: a dry cup (0% to 50% rh) and a wet cup (50% to 100% rh). In the wet-cup tests, liquid water movement may occur and the resulting water vapor transport coef-ficients should be treated with care (Kuenzel 1995).
Permeance is usually expressed in ng/(s·m2·Pa), and permeabil-ity in ng/(s·m·Pa). Whereas permeability refers to water vapor flux per unit thickness, the term permeance is used in reference to a material of a specific thickness. For example, a material that is 50 mm thick generally is assumed to have half the permeance of 25 mm thick material, even though permeance of many materials often is not strictly proportional to thickness. In many cases, this assump-tion ignores the effect of moisture content variations within the material, and the effect of cracks or holes in the surface. It is inap-propriate to use the term permeability in reference to inhomoge-neous or composite materials, such as plywood or gypsum board with paper facings. Whenever a building product is made of two dis-similar materials, only water vapor permeance is meaningful for characterizing vapor transfer through that product.
Methods have been developed that allow measurement of water vapor transport with temperature gradients across the spec-imen (Douglas et al. 1992, Krus 1996, Galbraith et al. 1998).
These test methods promise more accurate data on isothermal vapor transfer through materials and will eventually allow better distinction between various transport modes.
Combined Liquid and Vapor Flow It is nearly impossible to experimentally distinguish between liq-uid flows and vapor flows in hygroscopic materials. To distinguish the moisture flow in each phase, nonisothermal tests may be per-formed. In most practical applications in moisture analysis both vapor and liquid flow may be treated as parallel processes. There-fore, moisture flow is often expressed as the sum of two transport equations—one using vapor pressure as a driving potential for vapor flow and the other using either capillary suction or relative humidity as driving potentials for liquid moisture flow. These con-servation equations may be written as follows: (16) where m = moisture content of the building material wv = vapor diffusion flux ww = liquid transport flux Sw = moisture source or sink The symbol ∇ represents the nabla operator, which denotes diver-gence, or the gradients along the three spatial coordinates. The vapor and liquid transport flux densities may be given by (17) (18) where µ = water vapor permeability of building materials s = capillary suction pressure psat = water vapor saturation pressure Dp = liquid transport coefficient related to capillary suction Dφ = liquid transport coefficient related to relative humidity φ = relative humidity Previous methods of calculation used moisture content and tem-perature as driving potentials for combined vapor and liquid move-ment. This approach has the disadvantage that moisture content is discontinuous at interfaces between different materials and, there-fore, cannot be used as a moisture flow potential at those interfaces.
Equation (17) and Equation (18) provide the advantage of continu-ity at material interfaces and maintain physically meaningful mate-rial parameters.
WATER VAPOR RETARDERS AND AIRFLOW RETARDERS Water vapor retarders and airflow retarders combine to control the movement of moisture and air. Although their functions are dif-ferent, a single component may serve both functions. The designer assesses the needs for moisture and air movement control in the building envelope and provides a system that combines the required vapor retarder and airflow retarder properties.
Airflow Retarder Functions and Properties In addition to a vapor retarder, control of moisture requires an effective airflow retarder (also referred to as an air barrier or air infiltration barrier). Without effective control of airflow, vapor retarders are completely ineffective. However, airflow may accel-erate the drying of a wet building component, as described by Kara-giozis and Salonvaara (1999a,b).
A vapor retarder may also be an airflow retarder, that is, an air/vapor retarder. In the past, many designs were based on this, and measures were taken to ensure that the vapor retarder was con-tinuous in order to control airflow through it. This remains a valid approach. Some recent designs treat airflow retarders and vapor retarders as separate entities, but an airflow retarder should not be placed in a location where it can cause moisture to condense if it also has vapor retarder properties. For example, an airflow re-tarder placed on the cold side of a building envelope may cause condensation, particularly if the vapor retarder is ineffective and the airflow retarder is impermeable to moisture.
However, a cold-side air/vapor retarder that also has sufficient insulation may result in a lower potential for condensation by raising the temperature of the surface of the air/vapor retarder, but this requires careful installation and sealing of joints.
Air leakage characteristics of airflow retarders can be determined with ASTM Standard E 283 or ASTM Standard E 1424. Di Lenardo et al. (1995) discuss specific air leakage criteria for airflow retarders in cold climates. These specifications call for maximum permissible air leakage rates between 0.05 and 0.2 L/s per square metre (as mea-sured with an air pressure difference of 75 Pa), depending on the water vapor permeance of the outermost layer of the building enve-lope. The highest permissible air leakage rate of the airflow retarder applies if the permeance of the outermost layer is greater than 570 ng/(s·m2·Pa), and the lowest rate applies if the permeance is less than 60 ng/(s·m2·Pa). Intermediate values are provided as well. The recommendations apply only to heating climates.
The effectiveness of an airflow retarder can be greatly reduced if openings, even small ones, exist in it. Such openings can be w µ – dp dx ------= t ∂ ∂m wv ww + ( ) Sw + ∇ – = wv µ p or wv µ φpsat ( ) ∇ – = ∇ – = ww Dp s or ww Dφ φ ∇ – = ∇ – = 23.16 2001 ASHRAE Fundamentals Handbook (SI) caused by poor design, poor workmanship during application, poorly sealed joints and edges, insufficient coating thickness, improper caulking and flashing, uncompensated thermal expan-sion, mechanical forces, aging, and other forms of degradation.
Common faults or leaks occur at electrical boxes, plumbing pen-etrations, telephone and television wiring, and other unsealed openings in the structure.
A ceiling airflow retarder needs to be continuous at chases for plumbing, ducts, flues, and electrical wiring. In flat roofing, mechan-ical fasteners are sometimes used to adhere the system to the deck.
These often penetrate the airflow retarder, and the resulting hole may allow air and accompanying moisture leakage into the roof.
Because it resists airflow, an airflow retarder must withstand pressures exerted by chimney (stack) effects, wind effects, or both, during construction and over the life of the building. The magnitude of the pressure varies, depending on the type of build-ing and the sequence of construction. At one extreme, single-fam-ily dwellings may be built with the exterior cladding partly or entirely installed and insulation in place before the airflow retarder is added. Chimney effects in such buildings are small even in cold weather, so stresses on the airflow retarder during construction are small.
At the other extreme, in tall buildings, wind and chimney-effect forces are much greater than they are in single-family or other low-rise buildings. A fragile, unprotected sheet material should not be used as an airflow retarder (or vapor retarder) in a tall building because it will probably be torn by the wind before construction is completed.
In summary, an airflow retarder must • Meet air permeability requirements • Be continuous, i.e., – Tight joints in the airflow retarder must be constructible – Effective bonds in the airflow retarder must be made at inter-sections (e.g., wall/roof) – Dimensional changes due to temperature changes or shrinkage must be accommodated without damage to joints or the retarder material • Be strong enough to support the stresses applied to it, i.e., – It must not be ruptured or excessively deformed by air pres-sures due to wind and stack effects – Where an adhesive is used to complete a joint, it must be designed to withstand forces that might gradually peel it away A small penetration across an airflow retarder may seriously affect its performance. A penetration concentrates the air/vapor flow in such a way that large local deposits of water and ice are pos-sible. In this situation calculations of moisture flow and accumula-tion using permeance values are useless when airflow is involved.
In addition, the following properties of an airflow retarder may be important, depending on the application: • Elasticity • Thermal stability • Fire and flammability resistance • Inertness to deteriorating elements • Ease of installation More information on air leakage in buildings may be found in Chapter 26.
Vapor Retarder Functions and Properties A vapor retarder retards water vapor diffusion but does not totally prevent its transmission. The requirements for buildings are entirely different than those for pipes and equipment. Conditions on the inside and outside of buildings vary continually, and air move-ment and ventilation can provide wetting as well as drying at vari-ous times. Moisture entering one side of a wall cavity can be stored and released at a later time, or transmitted immediately out of the cavity through the other side. Requirements for vapor retarders in building components are therefore not extremely stringent.
In contrast, moisture that enters the insulation of cold storage facilities, cold pipes, or equipment is unlikely to escape, except dur-ing periods when the facilities, pipes, or equipment are not in use and are allowed to warm up. Vapor retarders for cold pipe or equip-ment applications must therefore have an extremely low permeance [e.g., less that than 3 ng/(s·m2·Pa), or lower for severe conditions].
However, Korsgaard (1993) demonstrated that water condensing on an insulated cold pipe can be removed continuously by wicking action with a specially designed and installed wick system, as long as pipe temperatures are above freezing.
In HVAC applications, vapor retarders are applied to thermal insulation on tanks, cold pipes, ducts, refrigerated enclosures, and buildings. If conditions are conducive to condensation, water vapor retarders help (1) keep the insulation dry, thereby reducing the cool-ing load; (2) prevent structural damage by rot, corrosion, or the expansion of freezing water; and (3) reduce paint problems on exte-rior wall construction (ASTM Standard C 755).
In addition to vapor permeance, the following properties of vapor retarders are important, depending on the application: • Mechanical strength in tension, shear, impact, and flexure • Adhesion • Elasticity • Thermal stability • Fire and flammability resistance • Inertness to other deteriorating elements • Ease of fabrication, application, and joint sealing Any vapor retarder’s effectiveness depends on its vapor per-meance, installation, and location within the insulated section. The vapor retarder is usually located at or near the surface exposed to the higher water vapor pressure. For residences in heating climates, this is usually the winter-warm side of the insulation.
Under conditions of reversible water vapor flow that can occur during temperature cycling of industrial insulations or of special-purpose buildings, the selection and location of water vapor retard-ers require special study and treatment (Stachelek 1955).
Vapor retarder material is usually a thin sheet or coating. How-ever, a construction of several materials, some perhaps of substan-tial thickness, could also constitute a vapor retarder system.
Water vapor permeances and permeabilities of some vapor retarders and other building materials are given in Table 9 in Chapter 25.
Vapor retarders that allow substantial summer drying while functioning as effective vapor retarders during the heating season are sometimes called “smart” vapor retarders. One type of smart vapor retarder has a low permeance to vapor but is permeable to liq-uid water, allowing the drying of condensed moisture. Korsgaard and Pedersen (1989, 1992) describe such a vapor retarder composed of a synthetic fabric sandwiched between staggered strips of plastic film. The fabric wicks free water from the building envelope, while the plastic film retards vapor flow into it.
Another smart vapor retarder provides low vapor permeance at low relative humidities but much higher permeance at high rela-tive humidity. During the heating season, indoor humidity usually is below 50%, and the permeance of the smart vapor retarder is low. In the summer and even during winter days with high solar heat gains, when the temperature gradient is reversed, moisture moving from the exterior of the wall or roof raises the relative humidity at the vapor retarder. This leads to a higher vapor per-meance of the vapor retarder and the potential for the wall or roof to dry out. One such vapor retarder, a nylon film, is described by Kuenzel (1998). Below 50% rh, the permeance of the film is less than 60 ng/(s·m2·Pa), but it becomes more permeable at above 60% rh, reaching 2070 ng/(s·m2·Pa) at 90% rh.
Thermal and Moisture Control in Insulated Assemblies—Fundamentals 23.17 Vapor retarder material for pipes is usually sheet metal with sol-dered seams; heavy foil with wide, sealed overlaps; plastic pipe; or other very low permeance systems.
Classification of Vapor Retarders Historically, a material or system with a permeance of 60 ng/(s·m2·Pa) or less qualifies as a vapor retarder. More recently, further classification of vapor retarders has been proposed. For example, the Canadian General Standards Board (CGSB) has spec-ified Type I vapor retarders as retarders with a permeance of 15 ng/(s·m2·Pa) or less, and Type II as retarders with a permeance of 45 ng/(s·m2·Pa) or less before aging and 60 ng/(s·m2·Pa) or less after aging.
Water vapor retarders are classified as rigid, flexible, or coating materials. Rigid retarders include reinforced plastics, aluminum, and stainless steel. These retarders usually are mechanically fas-tened in place and are vapor-sealed at the joints.
Flexible retarders include metal foils, laminated foil and treated papers, coated felts and papers, and plastic films or sheets. Such retarders are supplied in roll form or as an integral part of a building material (e.g., insulation). Accessory materials are required for seal-ing joints.
Coating retarders may be semifluid or mastic; paint (arbitrarily called surface coatings); or hot melt, including thermofusible sheet materials. Their basic composition may be asphaltic, resinous, or polymeric, with or without pigments and solvents, as required to meet design conditions. They can be applied by spray, brush, trowel, roller, dip or mop, or in sheet form, depending on the type of coating and surface to which it is applied. Potentially, each of these materials is an airflow retarder; however, to meet airflow retarder specifications, it must satisfy the requirements for strength, conti-nuity, and air permeance.
Designers have many options. For example, the conditions for control of airflow and moisture movement might be achieved using an interior finish, such as drywall, to provide strength and stiffness, along with a low-permeability coating, such as a vapor retarder paint, to provide the required low level of permeance. In this case, edge sealing is needed to establish continuity with adjacent airflow retarder/vapor retarder components.
Other designs may use more than one component. However, (1) any component that qualifies as a vapor retarder usually also impedes airflow, and is thus subject to pressure differences that it must resist, and (2) any component that impedes airflow often also retards vapor movement and may promote condensation or frost formation.
Additional information regarding the control of moisture and air-flow through the use of vapor retarders and airflow retarders may be found in Chapters 24 and 26 and in Construction Specifications Canada (1990).
Several studies found a significant increase in the apparent per-meance of vapor retarders as a result of small holes in the vapor retarder. For example, Seiffert (1970) reports a 100-fold increase in the permeance of aluminum foil when it is 0.014% perforated, and a 4000-fold increase when 0.22% of the surface is perforated. In general, penetrations particularly degrade a vapor retarder’s effec-tiveness if the vapor retarder has a very low permeance (e.g., poly-ethylene or aluminum foil). In addition, perforations may lead to additional air leakage, which further erodes the effectiveness of the vapor retarder.
STEADY-STATE DESIGN TOOLS Traditional methods for moisture design of the exterior building envelope all have severe limitations, and the results are sometimes difficult to interpret. However, these methods are used by design professionals and form the basis for current codes dealing with moisture control and vapor retarders.
The three best-known manual steady-state design tools for eval-uating the probability of condensation within exterior envelopes (exterior walls, roofs, or ceilings) are (1) the dew-point method, (2) the Glaser diagram, and (3) the Kieper diagram. All three methods compare the vapor pressures within the envelope, as calculated by simple vapor diffusion equations, with the saturation pressures, which are based on the calculated temperatures within the envelope.
If the calculated vapor pressure is above the saturation pressure at any point within the envelope, condensation is indicated.
The dew-point method, used in North America, and the Glaser diagram, commonly used in Europe and elsewhere, are almost identical. They differ slightly in the formulation of the vapor diffu-sion equation for flow through a building material and in definition of terms; the main difference lies in the graphical procedures. These methods are often misused, especially when condensation is present.
The Kieper diagram, a variant of the previous two methods, was introduced by Kieper et al. (1976) and described in greater detail by Trethowen (1979) and TenWolde (1983, 1994). As with the dew-point method and the Glaser diagram, the Kieper diagram is based entirely on vapor diffusion theory. The advantages of this method are that (1) the same diagram can be used for different wall configurations, as long as indoor and outdoor conditions are not changed; and (2) the calculation does not need to be repeated if con-densation is indicated.
Both the dew-point method and the Kieper diagram allow sim-ple estimation of the effect of wall or roof cavity ventilation by representing the effect of ventilation on thermal and vapor trans-port through the addition of parallel thermal and vapor diffusion resistances (Trethowen 1979, TenWolde and Carll 1992). The par-allel resistances account for the heat and vapor bypassing the exte-rior material layers with outside ventilation air. The magnitude of the parallel resistances may be determined from the following equations: (19) (20) where Rt,par = parallel thermal resistance, m2·K/W Rv,par = parallel vapor flow resistance, Pa·m2·s/ng S = surface area of the wall or ceiling, m2 = cavity ventilation airflow rate, m3/s = density of air, kh/m3 c = ratio of humidity ratio and vapor pressure, approximately 6.13 g/(kg·kPa) cp = specific heat, J/(kg·K) Although this method of including ventilation only approximates the actual effects of ventilation, it can be a useful tool.
Many people advocate abandoning steady-state design tools because of their severe limitations. Perhaps the greatest limitation is that their focus is restricted to prevention of sustained interstitial condensation. Many building failures (such as mold and mildew, buckling of siding, and paint failure) are not necessarily related to surface condensation.
Conversely, limited condensation can often be tolerated, depend-ing on the materials involved, the temperature conditions, and the speed at which the material dries out. Wetting and drying cycles cannot be accurately analyzed with steady-state tools because these tools neglect moisture storage in the building materials. Another weakness is that these methods exclude all moisture transfer mech-anisms other than vapor diffusion. Results obtained with any of these methods should therefore be considered as approximations and be used with extreme care.
Rt par , S Qρcp -------------= Rv par , S Qρc -----------= Q ρ 23.18 2001 ASHRAE Fundamentals Handbook (SI) The validity and usefulness of any of these methods depend on the judicious selection of boundary conditions and material proper-ties. Specifically, they should only be used to estimate seasonal mean conditions, rather than daily or even weekly mean conditions.
Furthermore, permeances may vary with moisture content and the effect of rain splash, flashing imperfections, leaky or poorly formed joints, weather exposure, and sunshine can have overriding effects.
For those who want to use these simple tools, despite their short-comings, a short description of the dew-point method is presented, with an example of its use. The Glaser diagram closely parallels the dew-point method. A comprehensive description of these methods can be found in TenWolde (1994).
Dew-Point Method The dew-point method is based on a slight modification of dif-fusion Equation (15): (21) where w = water vapor flux through a layer of material, ng/s·m2 µ = water vapor permeability of material, ng/(s·m·Pa) ∆p = vapor pressure difference across the layer, Pa d = thickness of the layer, m The term µ/d represents the permeance of the material. Water vapor resistance Z is defined as the inverse of the permeance. Then Equa-tion (21) can be written as (22) Example 1. The dew-point method is explained and demonstrated for the frame wall construction and materials described in Table 3. Assume 21°C, 40% rh indoors, and − 6.7°C, 50% rh outdoors.
Solution: Step 1. Calculate the temperature drop across each material. The temperature drop is proportional to the R-value as follows: Table 4 lists the resulting temperature drops and resulting temperatures at each surface.
Step 2. Find the saturation vapor pressures corresponding to the surface temperatures (Table 4). These values can be found in Table 2 in Chapter 6, or in other psychrometric tables or charts.
Step 3. Calculate the vapor pressure drops across each material.
These are calculated in much the same way as the temperature drops in step 1.
where p = vapor pressure, Pa Z = vapor diffusion resistance, Pa·m2·s/ng From Table 3, the total resistance of the wall with the vapor retarder is Zwall = 1/9200 + 1/290 + 1/1724 + 1/29 + 1/2010 + 1/57 000 = 0.0391 Pa·m2·s/ng The vapor pressure drop across the wall is calculated from indoor and outdoor relative humidities and the indoor and outdoor saturation vapor pressures (see Table 4).
∆pwall = pindoor −poutdoor = (40/100) 2.496 −(50/100) 0.3701 = 0.8134 kPa w µ ∆p d -------– = w p ∆ Z ------– = Tmaterial ∆ Twall ∆ ------------------------Rmaterial Rwall ---------------------= pmaterial ∆ pwall ∆ ------------------------Zmaterial Zwall ---------------------= Table 3 Approximate Thermal and Vapor Diffusion Properties of Wall in Example 1 Air Film or Material Thermal Resistance R, m2 ·K/W Permeance M, ng/(s·m2·Pa) Diffusion Resistance Z, Pa·m2 ·s/ng Air film (still) 0.12 9200a 0.00011a Gypsum board, painted 0.079 290 0.0035 Insulation 1.9 1700 0.00058 Plywood sheathing 0.11 29 0.0345 Wood siding 0.18b 2010b 0.0005b Air film (wind) 0.03 57000a 0.000017a Total 2.42 Not applicable 0.0392 aApproximate values; permeances of surface air films are very large compared to those of other materials and do not affect results of calculations.
bApproximate values; permeance reflects limited ventilation of back of siding.
Table 4 Calculated Temperature Drops, Surface Temperatures, and Saturation Vapor Pressures in Example 1 Air Film or Material or Surface Temperature Drop, °C Surface Temperature, °C SaturationVapor Pressure at Surface, kPa Indoor air — 21 2.496 Surface air film 1.3 — — Interior wall surface — 19.7 2.299 Gypsum board 0.9 — — Gypsum board/Insulation — 18.8 2.168 Insulation 21.9 — — Insulation/Sheathing — − 3.1 0.4843 Plywood sheathing 1.2 — — Sheathing/Siding — − 4.3 0.442 Wood siding 2.0 — — Exterior wall surface — − 6.3 0.3796 Surface air film 0.3 — — Outdoor air — − 6.6 0.3701 Table 5 Initial and Final Calculations of Vapor Pressure Drops and Surface Vapor Pressures in Example 1 Air Film or Material or Surface Satura-tion Vapor Pressure, kPa Initial Calculation Vapor Pressure, kPa Final Calculation Vapor Pressure, kPa Drop At Surface Drop At Surface Indoor air (40% rh) 2.496 — 0.9986 — 0.9986 Surface air film — 0.0024 — 0.0135 — Interior wall surface 2.299 — 0.9962 — 0.9851 Gypsum board — 0.0716 — 0.4292 — Gyp. board/Insulation 2.168 — 0.9246 — 0.5559 Insulation — 0.0122 — 0.0716 — Insulation/Sheathing 0.4843 — 0.9124 — 0.4843 Plywood sheathing — 0.7169 — 0.2951 — Sheathing/Siding 0.442 — 0.1955 — 0.1892 Wood siding — 0.0101 — 0.0040 — Exterior wall surface 0.3796 — 0.1854 — 0.1852 Surface air film — 0.0003 — 0.00014 — Outdoor air (50% rh) 0.3701 — 0.1851 — 0.1851 Thermal and Moisture Control in Insulated Assemblies—Fundamentals 23.19 As with temperatures, the vapor pressures at the surfaces of each material can easily be determined from the vapor pressure drops. Table 5 lists the results for the example wall in the initial calculation column.
Step 4. Figure 10 shows the saturation and calculated vapor pres-sures. Comparison with saturation pressures reveals that the calculated vapor pressure on the interior surface of the sheathing (0.9125 kPa) is well above the saturation pressure at that location (0.4843 kPa). This indicates condensation, probably on the surface of the sheathing. It does not indicate condensation within the insulation.
If the location of the condensation or the condensation rate is of interest, additional calculations (steps 5 and 6) are necessary.
Step 5. Figure 10 shows that the calculated vapor pressure exceeds the saturation vapor pressure by the greatest amount at the interior sur-face of the plywood sheathing. Therefore, this is the most likely loca-tion for condensation to occur. With condensation at that surface, the vapor pressure should equal the saturation vapor pressure at that loca-tion (see the final calculation column in Table 5).
Step 6. The change of vapor pressure on the plywood sheathing alters all other vapor pressures as well as the vapor flow through the wall. The calculation of vapor pressures is similar to the calculation in step 3, but the wall is now divided into two parts: one part on the inte-rior of the condensation plane (i.e., gypsum board and insulation) and the other part on the exterior (plywood sheathing and wood siding).
The vapor pressure drop over the first part of the wall is ∆p1 = 0.9986 −0.4843 = 0.5143 kPa and that over the second part is ∆p2 = 0.4843 −0.1851 = 0.2992 kPa The vapor diffusion resistances of both parts of the wall are Z1 = 1/9200 + 1/290 + 1/1724 = 0.0041 Pa·m2·s/ng Z2 = 1/29 + 1/2010 + 1/5700 = 0.035 Pa·m2·s/ng The vapor pressure drops across each material can now be calcu-lated from Final calculations of vapor pressure are shown in Table 5. The vapor pressure no longer exceeds the saturation vapor pressure, which means that the condensation plane was chosen correctly. Figure 10 shows the vapor pressure profile (labeled as Vapor pressure, final cal-culation).
Vapor flow is no longer the same throughout the wall—vapor flow into the wall from the indoor air increases as a result of the lower vapor pressure at the plywood surface, while flow from the wall to the outside decreases. The difference between the two flows is the rate of moisture accumulation: wc = ∆p1/Z1 − ∆p2/Z2 =0.5143/0.0041 −0.2992/0.035 = 117 µg/(s·m2) This amounts to about 10 g/day·m2. In our example, the plywood surface is below freezing, and this moisture would probably accumu-late as frost. The moisture content of the plywood would be increased by 1% after about a week of condensation at this rate.
The dew-point method can be summarized as follows: 1. Calculate temperature drops and surface temperatures.
2. Find corresponding saturation vapor pressures.
3. Calculate vapor pressure drops and vapor pressures.
4. Check whether the saturation pressure is above the vapor pressure at all surfaces; if so, no condensation is indicated. Vapor flow through the wall may be determined if desired.
If condensation is indicated, the following steps may be fol-lowed: 1. Select the condensation surface. In most cases it is the surface where the difference between the calculated vapor pressure and the saturation vapor pressure is the highest. Vapor pressure at this surface should be set equal to the saturation vapor pressure.
2. Recalculate the vapor pressures; if any vapor pressures are above saturation, steps 5 and 6 should be repeated with a different choice for the condensation surface.
3. If needed, calculate the rate of condensation.
MATHEMATICAL MODELS The rapid advance in computer technology has made it possible to develop computer models capable of analyzing and predicting the thermal and moisture behavior of building components. A small number are extensions of the steady-state analysis methods de-scribed before, but with the transient models it is possible to pre-dict hourly drying and wetting of building components in a variety of conditions, climates, and design configurations. The information in this section is largely based on a review of the state of the art of heat and moisture transport modeling for buildings, which identi-fied 37 different models of various complexity (Hens 1996). Some of those models do not include moisture transport and are therefore not included here. Most of the models are research tools that are not readily available, but some are available either commercially, free of charge, or through a consultant. Karagiozis (2001) describes the more sophisticated design tools, indicating both the capabilities and the limitations. ASTM Manual 40 covers available design tools and approaches to investigate the performance of building envelope sys-tems in terms of heat and moisture transport.
For many applications, the actual behavior of an assembly under transient climatic conditions must be simulated in order to account for short-term processes like driving rain absorption, summer con-densations, and phase changes. Understanding the application lim-its of each model is an important part of applying mathematical models to develop design guidelines.
The features of a complete moisture analysis model include: 1. Transient heat, air, and moisture transport formulation, incorporating physics of • Vapor transport • Liquid transport • Airflow • Heat and moisture storage/capacity • Condensation and evaporation • Freezing and thawing 2. One- or two-dimensional spatial formulation 3. Material properties as functions of moisture content, relative humidity, or temperature • Thermal properties (density, heat capacity, thermal conductivity) • Moisture properties (porosity, sorption, water retention, vapor permeability, liquid diffusivity) • Airflow properties (air permeability) Fig. 10 Dew-Point Calculation in Example 1 pmaterial ∆ pi ∆ ------------------------Zmaterial Zi ---------------------for i 1 2 , = = 23.20 2001 ASHRAE Fundamentals Handbook (SI) 4. Boundary conditions (generally on an hourly basis) • Incident solar radiation and sky radiation (depending on inclination and orientation) • Wind-driven rain at exterior surfaces (depending on location, and aerodynamics) • Wind speed, orientation, and pressure • Interior and exterior temperature and relative humidity • Interior moisture sources and stratification 5. Surface conditions • Convective heat transfer coefficients • Mass transfer coefficients • Short wave absorptivity • Long wave emissivity • Precipitation absorptivity 6. Building systems and subsystem effects • Water penetration rates through subsystems (joints, cracks, etc) • Air leakage (cracks, joints, e.g., around a window) • Additional sources of moisture While not all these features are required for every analysis, addi-tional features may be required in some applications, such as mois-ture flow through unintentional cracks and intentional openings. To accurately model these cracks and openings, supplementary exper-iments may be used to define their performance under various loads.
In Feature 6, additional laboratory-controlled tests provide perfor-mance data specific to each subsystem (e.g., water entry through a crack) (Straube and Burnett 1997). In most cases, field measure-ment of system and subsystem effects is preferred.
The validation, verification, and benchmarking of hygrothermal models is a formidable task. Little internationally accepted experi-mental data exist to benchmark hygrothermal models. The main dif-ficulty lies in the fact that it is difficult to measure moisture flows and moisture transport potentials even in laboratory-controlled con-ditions. Even a validated model should be verified for each new application. Two-dimensional and three-dimensional hygrothermal models are extremely complex and mostly require the direct exper-tise of the authors.
In most hygrothermal models, moisture content, temperature, relative humidity, and related fluxes are common outputs of simu-lations. Results have to be checked for consistency, accuracy, grid independency, and sensitivity to parameter changes. The results may be used to determine the moisture tolerance of an envelope subjected to various interior and exterior loads. The heat fluxes may be used to determine the thermal performance under the influence of moisture and airflow. Furthermore, the transient out-put data may be used to assess durability and indoor air quality.
Post-processing tools concerning durability (e.g., corrosion, mold growth, freezing and thawing, hygrothermal dilation, and indoor air humidity) are currently being developed. For instance, Kara-giozis and Kuenzel (1999) developed a model to estimate the rate of mold growth and corrosion. It is expected that hygrothermal models will be incorporated in whole-building simulation tools (Karagiozis and Salonvaara 1999b).
Transient models enable an hour-by-hour analysis of moisture conditions in building components and a much more realistic calcu-lation than steady-state diffusion models. However, many are not easy to use and require judgment and expertise on the part of the user. Existing moisture transport models are one- or two-dimen-sional, which requires the user to devise a realistic representation of a three-dimensional building component. Users need to be aware of which transport phenomena and types of boundary conditions are included and which are not. For instance, some models are not able to handle rain wetting of the exterior.
Results from an analysis tend to be very sensitive to the choice of indoor and outdoor conditions, but exact conditions are seldom known. Indoor and outdoor conditions to be used for the purpose of building design have not yet been established, although standards are being developed. Accurate data for all the materials in a compo-nent is difficult to find. Finally, interpretation of the results is not easy; accurate data on what moisture and temperature conditions materials can tolerate are not often available.
PREVENTING SURFACE CONDENSATION Surface condensation occurs when water vapor comes in contact with a surface that has a temperature lower than the dew point of that vapor. The insulation should be sufficiently thick to ensure that the insulation surface temperature always exceeds the dew-point temperature to prevent surface condensation from occurring on the warm side of insulated rooms, pipes, ducts, and equipment.
The concept of a temperature index coefficient τ, also called the condensation resistance factor, is useful for calculating the sur-face temperature: (23) where ts = surface temperature on warm side of insulation tc = temperature on cold side of insulation ta = ambient temperature on warm side of insulation The minimum temperature index coefficient to avoid surface con-densation is (24) where td is the dew point of ambient air.
For flat surfaces and insulation, the temperature index coeffi-cient can be stated as (25) where L = thickness of the insulation k = thermal conductivity of the insulation hi = surface heat transfer coefficient The minimum insulation thickness to avoid surface condensation on a flat surface can be calculated from (26) The minimum insulation thickness for pipe insulation is given by (27) where rs,min = minimum outer radius of the insulation ri = inner radius of the insulation The term to the left of the equal sign in Equation (27) is called equivalent thickness. Equivalent thickness is the thickness of insu-lation on a flat surface required to give the same rate of heat trans-mission per unit area of outer surface of insulation as on a cylinder or pipe. Figure 11 provides a convenient way to convert equivalent thickness to actual thickness.
τ ts tc – ta tc – --------------= τ min td tc – ta tc – --------------= L k ⁄ L k ⁄ 1 h ⁄ + ---------------------------------= Lmin τ min 1 τ min – -------------------- k hi ----= rs min , rs min , ri ------------- ln τ min 1 τ min – -------------------- k hi ----= Thermal and Moisture Control in Insulated Assemblies—Fundamentals 23.21 STANDARDS ASTM. 1997. Standard terminology relating to thermal insulating materials.
Standard C 168-97.
ASTM. 1997. Standard test method for steady-state heat flux measurements and thermal transmission properties by means of the guarded-hot-plate apparatus. Standard C 177-97.
ASTM. 1989. Standard test method for steady-state thermal performance of building assemblies by means of guarded hot box. Standard C 236-89(1993).
ASTM. 1995. Standard test method for steady-state heat transfer properties of horizontal pipe insulation. Standard C 335-95.
ASTM. 1998. Standard test method for steady-state thermal transmission properties by means of the heat flow meter apparatus. Standard C 518-98.
ASTM. 1997. Standard practice for selection of vapor retarders for thermal insulation. Standard C 755-97.
ASTM. 1999. Standard classification of potential health and safety concerns associated with thermal insulation materials and accessories. Standard C 930-99.
ASTM. 1990. Standard test method for thermal performance of building assemblies by means of a calibrated hot box. Standard C 976-90(1996).
ASTM 1997. Standard practice for calculating thermal transmission proper-ties from steady-state heat flux measurements. Standard C 1045-97.
ASTM. 2000. Standard test method for estimating the long-term change in the thermal resistance of unfaced rigid closed cell plastic foams by slicing and scaling under controlled laboratory conditions. Standard C 1303-00.
ASTM 1997. Standard test method for the thermal performance of building assemblies by means of a hot box apparatus. Standard C 1363-97.
ASTM. 1995. Standard test methods for water vapor transmission of mate-rials. Standard E 96-95.
ASTM. 1991. Standard test method for determining the rate of air leakage through exterior windows, curtain walls, and doors under specified pres-sure differences across the specimen. Standard E 283-91.
ASTM. 1991. Standard test method for determining the rate of air leakage through exterior windows, curtain walls, and doors under specified pressure and temperature differences across the specimen. Standard E 1424-91(2000).
ASTM. Annual. Annual book of ASTM standards. Volume 4.06, Thermal insulation and environmental acoustics.
REFERENCES ASHRAE. 1989. Thermal Performance of Exterior Envelopes of Buildings IV.
ASHRAE. 1992. Thermal Performance of Exterior Envelopes of Buildings V.
ASTM. 1974. Heat transmission measurements in thermal insulations. Spe-cial Technical Publication STP 544. American Society for Testing and Materials, West Conshohocken, PA.
ASTM. 1979. Thermal transmission measurements of insulation. Special Technical Publication STP 660.
ASTM. 1980. Thermal insulation performance. Special Technical Publica-tion STP 718.
ASTM. 1983. Thermal insulation materials and systems for energy conser-vation in the 80s. Special Technical Publication STP 789.
ASTM. 1985. Guarded hot plate and heat flow meter methodology. Special Technical Publication STP 879.
ASTM. 1985. Building applications of heat flux transducers. Special Tech-nical Publication STP 885.
ASTM. 1988. Thermal insulation—Materials and systems. Special Techni-cal Publication STP 922.
ASTM. 1990. Insulation materials, testing, and applications. Special Tech-nical Publication STP 1030.
ASTM. 1991. Insulation materials, testing, and applications, Second vol-ume. Special Technical Publication STP 1116.
Baker, M.C. 1964. Thermal and moisture deformation in building materials.
Canadian Building Digest 56. National Research Council of Canada, Ottawa.
Bomberg, M. and C. Shirtliffe. 1978. Influence of moisture and moisture gradients on heat transfer through porous building materials. ASTM Spe-cial Technical Publication STP 660:211-33.
Brandreth, D.A., ed. l986. Advances in foam aging—A topic in energy con-servation series. Caissa Editions, Yorklyn, DE.
BRS. 1974. Estimation of thermal and moisture movements and stresses, Parts 1, 2, and 3. Building Research Station Digest 227, 228, and 229, Watford, England.
Burch, D.M. and C.M. Hunt. 1978. Retrofitting an existing wood frame res-idence to reduce its heating and cooling energy requirements. ASHRAE Transactions 84:176.
Burge, H.A., H.J. Su, and J.D. Spengler. 1994. Moisture, organisms, and health effects. In Moisture control in buildings. ASTM Manual Series: MNL 18, Chapter 6.
Christian, J.E., A. Desjarlais, R. Graves, and T.L. Smith. 1995. Five-year field study confirms the PIMA standard for estimating polyisocyanurate insulation long-term thermal performance. 11th Conference of Roofing Technology, National Roofing Contractors Association, Rosemont, IL.
CIBSE. 1986. Thermal properties of building structures. In CIBSE Guide: Reference Data. The Chartered Institution of Building Services Engi-neers, London, England.
Construction Specifications Canada. 1990. Tek-AID on Air Barrier Sys-tems. 1 St. Clair Street W., Toronto, M4W 1K6, Canada.
Dechow, F.J. and K.A. Epstein. 1978. Laboratory and field investigations of moisture absorption and its effect on thermal performance of various insulations. ASTM Special Technical Publication STP 660:234-60.
Di Lenardo, B., W.C. Brown, W.A. Dalgleish, K. Kumaran, and G.F. Poirier.
1995. Technical guide for air barrier systems for exterior walls of low-rise buildings. Canadian Construction Materials Centre, National Research Council Canada, Ottawa, Ontario.
Fig. 11 Conversion of Equivalent Thickness to Actual Thickness for Pipe Insulation 23.22 2001 ASHRAE Fundamentals Handbook (SI) Donnelly, R.G., V.J. Tennery, D.L. McElroy, T.G. Godfrey, and J.O. Kolb.
1976. Industrial thermal insulation, An assessment. Oak Ridge National Laboratory Report TM-5283 and TM-5515, TID-27120.
Douglas, J.S., T.H. Kuehn, and J.W. Ramsey. 1992. A new moisture perme-ability measurement method and representative test data. ASHRAE Transactions 98(2):513-19.
Duff, J.E. 1971. The effect of air conditioning on the moisture conditions in wood walls. USDA Forest Service Research Paper SE-78. Madison, WI.
Epstein, K.A. and L.E. Putnam. 1977. Performance criteria for the protected membrane roof system. Proceedings of the Symposium on Roofing Tech-nology. National Institute of Standards and Technology, Gaithersburg, MD, and National Roofing Contractors Association, Rosemont, IL.
FEA. 1976. Economic thickness for industrial insulation. GPO No. 41-018-00115-8. U.S. Government Printing Office, Washington, D.C.
FPL. 1999. Wood handbook. FPL-GTR-113, Forest Products Laboratory, Forest Service, U.S. Department of Agriculture.
Galbraith, G.H., R.C. McLean, and J.S. Guo. 1998. Moisture permeability data presented as a mathematical relationship. Building research and information 20 (6):364-72.
Glaser, P.E., I.A. Black, R.S. Lindstrom, F.E. Ruccia, and A.E. Wechsler.
1967. Thermal insulation systems—A survey. NASA SP5027.
Hart, G.H. 1981. Heating the perimeter zone of an office building: An ana-lytical study using the proposed ASHRAE comfort standard (55-74R).
ASHRAE Transactions 87(2):529.
Hedlin, C.P. 1977. Moisture gains by foam plastic roof insulations under controlled temperature gradients. Journal of Cellular Plastics 13(5): 313-19.
Hedlin, C.P. 1983. Effect of moisture on thermal resistances of some insu-lations in a flat roof system under field-type conditions. ASTM Special Technical Publication STP 789:602-25.
Hedlin, C.P. 1985. Effect of insulation joints on heat loss through flat roofs.
ASHRAE Transactions 91(2B):608-22.
Hedlin, C.P. 1987. Seasonal variations in the modes of heat transfer in a moist porous thermal insulation in a flat roof. Journal of Thermal Insu-lation 11:54-66.
Hedlin, C.P. 1988a. Heat flow through a roof insulation having moisture contents between 0 and 1% by volume, in summer. ASHRAE Trans-actions 94(2):1579-94.
Hedlin, C.P. 1988b. Heat transfer in a wet porous thermal insulation in a flat roof. Journal of Thermal Insulation 11:165-88.
Hens, H. 1996. Heat, air and moisture transfer in highly insulated envelope parts, task 1: Modelling. Final Report, Volume 1, International Energy Agency, Annex 24. Catholic University-Leuven, Laboratorium for Building Physics, Leuven, Belgium.
Hukka, A. and H.A. Viitanen. 1999. A mathematical model of mold growth on wooden materials. Wood Science and Technology 300, 475-85.
International Energy Agency. 1990. Guidelines and practice, Volume 2.
International Energy Agency, Annex XIV, Leuven, Belgium.
Jespersen, H.B. 1960. The effect of moisture on insulating materials made of plastic foam, impregnated mineral wool, vermiculite, and concrete. Den-mark Technological Institute.
Joy, F.A. 1957. Thermal effect of moisture on insulation containing mois-ture. In Thermal conductivity measurements and applications of thermal insulations. ASTM Special Technical Publication STP 217.
Karagiozis, A.N. 2001. Advanced hygrothermal modeling for hygrothermal research. In Moisture Analysis Manual, Chapter 10. ASTM Manual 40, American Society for Testing and Materials.
Karagiozis, A.N. and H.M. Kuenzel. 1999. A state-of-the-art moisture engi-neering model. ASHRAE Transactions 105(1).
Karagiozis, A.N. and Kuenzel H.M. 2001. Moisture engineering practice.
ASHRAE Transactions 107(1).
Karagiozis, A.N. and H.M. Salonvaara. 1998. EIFS hygrothermal perfor-mance due to initial construction moisture as a function of air leakage, interior cavity insulation, and climate conditions. Thermal Performance of the Exterior Envelopes of Buildings VII, pp. 179-88. ASHRAE.
Karagiozis, A.N. and H.M. Salonvaara. 1999a. Hygrothermal performance of EIFS-clad walls: Effect of vapor diffusion and air leakage on the dry-ing of construction moisture. ASTM STP 1352, pp 32-51.
Karagiozis, A.N. and H.M. Salonvaara. 1999b. Whole building hygrother-mal performance. Building Physics in the Nordic Countries 2:745-53.
Kieper, G., W. Caemmerer, and A. Wagner. 1976. A new diagram to evaluate the performance of building constructions with a view to water vapor dif-fusion. Presented at CIB Working Group W40 Meeting, Washington, DC.
Knab, L.I., D.R. Jenkins, and R.G. Mathey. 1980. The effect of moisture on the thermal conductance of roofing systems. Building Science Series No.
123. National Institute of Standards and Technology, Gaithersburg, MD.
Korsgaard, V. 1993. Innovative concept to prevent moisture formation and icing of cold pipe insulation. ASHRAE Transactions 99(1):270-73.
Korsgaard, V. and C.R. Pedersen. 1989. Transient moisture distribution in flat roofs with hygro diode vapor retarder. In Thermal Performance of Exterior Envelopes of Buildings IV. ASHRAE.
Korsgaard, V. and C.R. Pedersen.1992. Laboratory and practical experience with a novel water-permeable vapor retarder. In Thermal Performance of Exterior Envelopes of Buildings V, pp. 480-90. ASHRAE.
Krus, M. 1996. Moisture transport and storage coefficients of porous min-eral building materials, theoretical principals and new test methods, IRB-Verlag, Stuttgart, Germany.
Kuenzel, H.M. 1995. Simultaneous heat and moisture transport in building components, one- and two-dimensional calculation using parameters.
Ph.D. thesis. Fraunhofer Institute for Building Physics, Germany.
Kuenzel, H.M. 1998. More moisture load tolerance of construction assem-blies through the application of a smart vapor retarder. In Thermal Per-formance of the Exterior Envelopes of Buildings VII, pp. 129-32.
ASHRAE.
Kumaran, M.K. 1989. Experimental investigation on simultaneous heat and moisture transport through thermal insulation. Proceedings of the (Con-seil International du Batiment-International Building Council) CIB 11th International Conference 2:275-84.
Kumaran, M.K., G.P. Mitalas, and M.T. Bomberg. 1994. Fundamentals of transport and storage of moisture in building materials and components.
In Moisture control in buildings. ASTM Manual Series: MNL 18, Chap-ter 1.
Kyle, D.M. and A.O. Desjarlais. 1994. Assessment of technologies for con-structing self-drying low-slope roofs. Oak Ridge National Laboratory Report ORNL/CON-380. Oak Ridge, TN.
Lander, R.M.1955. Gas is an important factor in the thermal conductivity of most insulating materials. ASHRAE Transactions 61:151.
Langlais, C., M. Hyrien, and S. Klarsfeld. 1983. Influence of moisture on heat transfer through fibrous insulating materials. ASTM Special Tech-nical Publication STP 789:563-81.
Larsson, L.E., J. Ondrus, and B.A. Petersson. 1977. The protected mem-brane roof (PMR)—A study combining field and laboratory tests. Pro-ceedings of the Symposium on Roofing Technology. National Institute of Standards and Technology, Gaithersburg, MD, and National Roofing Contractors Association, Rosemont, IL.
Lewis, J.E. 1979. Thermal evaluation of the effects of gaps between adjacent roof insulation panels. Journal of Thermal Insulation (October): 80-103.
Lotz, W.A. 1969. Facts about thermal insulation. ASHRAE Journal (June): 83-84.
Merrill, J.L. and A. TenWolde. 1989. Overview of moisture-related damage in one group of Wisconsin manufactured homes. ASHRAE Transactions 95(1):405-14.
Ojanen, T., R. Kohonen, and M.K Kumaran. 1994. Modeling heat, air, and moisture transport through building materials and components. In Mois-ture control in buildings. ASTM Manual Series: MNL 18, Chapter 2.
Ostrogorsky, A.G. and L.R. Glicksman. 1986. Laboratory tests of effective-ness of diffusion barriers. Journal of Cellular Plastics 22:303.
Ovstaas, G., S. Smith, W. Strzepek, and G. Titley. 1983. Thermal perfor-mance of various insulations in below-earth-grade perimeter application.
ASTM Special Technical Publication STP 789:435-54.
Palfey, A.J. 1980. Thermal performance of low emittance building sheath-ing. Journal of Thermal Insulation 3.
Paljak, I. 1973. Condensation in slabs of cellular plastics. Materiaux et Con-structions 6(31):53.
Parmelee, G.V. and WW. Aubele. 1950. ASHVE Research Report No.1399.
Heat flow through unshaded glass: Design data for load calculations.
ASHVE Transactions 56:371.
Parmelee, G.V. and R.G. Huebscher. 1947. Forced convection heat transfer from flat surfaces. ASHVE Transactions 53:245.
Pedersen-Rode, C. 1990. Combined heat and moisture transfer in building construction. Report No.214. Thermal Insulation Laboratory, Technical University of Denmark, Lyngby, Denmark.
Pedersen-Rode, C., T.W. Petrie, G.E. Courville, P.W. Childs, and K.E.
Wilkes. 1991. Moisture migration and drying rates for low slope roofs— Preliminary results. Proceedings of the 3rd International Symposium on Roofing Technology. National Roofing Contractors Association, Rose-mont, IL.
Thermal and Moisture Control in Insulated Assemblies—Fundamentals 23.23 Pelanne, C.M. 1977. Heat flow principles in thermal insulation. Journal of Thermal Insulation 1:48.
Pelanne, C.M. 1979. Thermal insulation heat flow measurements: Require-ments for implementation. ASHRAE Journal 21(3):51.
Plewes, W.G. 1976. Upward deflection of wood trusses in winter. Building Research Note 107, National Research Council of Canada, Ottawa.
Raber, B.F. and F.W. Hutchinson. 1945. Radiation corrections for basic con-stants used in the design of all types of heating systems. ASHVE Trans-actions 51:213.
Robinson, H.E., F.J. Powlitch, and R.S. Dill. 1954. The thermal insulating value of airspaces. Housing and Home Finance Agency, Housing Research Paper No. 32, U.S. Government Printing Office, Washington, D.C.
Rowley, F.B., R.C. Jordan, C.E. Lund, and R.M. Lander. 1952. Gas is an important factor in the thermal conductivity of most insulating materials.
ASHVE Transactions 58:15.
Rudoy, W. 1975. Effect of building envelope parameters on annual heating/ cooling load. ASHRAE Journal 17(7):19.
Seiffert, K. 1970. Damp diffusion and buildings. Elsevier, Amsterdam, Netherlands.
Shapiro, A.P. and S. Motakef. 1990. Unsteady heat and mass transfer with phase change in a porous slab: Analytical solutions and experimental results. International Journal of Heat and Mass Transfer 33(1):163-73.
Shuman, E.C. 1980. Field measurement of heat flux through a roof with sat-urated thermal insulation and covered with black and white granules.
ASTM Special Technical Publication STP 718:519-39.
Simons, E. 1955. In-place studies of insulated structures. Refrigerating Engineering 63:40 and 63:128.
Spielvogel, L.G. 1974. More insulation can increase energy consumption.
ASHRAE Journal 16(1):61.
Stachelek, S.J. 1955. Vapor barrier requirements for cold storage structures.
Industrial Refrigeration 34.
Straube, J.F. and E.F.P. Burnett. 1997. Rain control and screened wall sys-tems. 7th Conference on Building Science and Technology, Durability of Buildings-Design, Maintenance, Codes and Practices, pp. 17-37. Uni-versity of Waterloo, Ontario, Canada.
TenWolde, A. 1983. The Kieper and MOISTWALL moisture analysis meth-ods for walls. In Thermal Performance of Exterior Envelopes of Build-ings II, pp. 1033-51. ASHRAE.
TenWolde, A. 1994. Design tools. In Moisture control in buildings. ASTM Manual 18, Chapter 11.
TenWolde A. and C. Carll. 1992. Effect of cavity ventilation on moisture in walls and roofs. In Thermal Performance of the Exterior Envelopes of Buildings V, pp. 555-62. ASHRAE.
Thomas, W.C., G.P. Bal, and R.J. Onega. 1983. Heat and moisture transfer in glass fiber roof-insulating material. ASTM Special Technical Publica-tion STP 789:582-601.
Tobiasson, W. and J. Richard. 1979. Moisture gain and its thermal conse-quences for common roof insulations. Proceedings of the 5th Conference on Roofing Technology, National Institute of Standards and Technology, Gaithersburg, MD, and National Roofing Contractors Association, Rose-mont, IL.
Tobiasson, W., A. Greatorex, and D. Van Pelt. 1987. Wetting of polystyrene and urethane roof insulations in the laboratory and on a protected mem-brane roof. In Thermal Insulation, Materials and Systems. ASTM Spe-cial Technical Publication STP 922:421-30.
Tobiasson, W., A. Greatorex, and D. Van Pelt. 1991. New wetting curves for common insulations. In International Symposium on Roofing Technol-ogy. National Institute of Standards and Technology, Gaithersburg, MD, and National Roofing Contractors Association, Rosemont, IL.
Trechsel, H.R. and M. Bomberg, eds. 1989. Water vapor transmission through materials and systems. ASTM Special Technical Publication STP 1039.
Trethowen, H.A. 1979. The Kieper method for building moisture design.
BRANZ Reprint 12, Building Research Association of New Zealand, New Zealand.
Tye, R.P. 1987. Assessment of foam in-place urethane foam insulation used in buildings. ORNL/Sub-86/56525/1. Oak Ridge National Laboratory, Oak Ridge, TN.
Tye, R.P. 1988. Aging of cellular plastics: A comprehensive bibliography.
Journal of Thermal Insulation 11:196-222.
Tye, R.P. and C.F. Baker. 1987. Development of experimental data on cellu-lar plastic insulation under simulated winter exposure conditions. In Thermal Insulation, Materials and Systems, ASTM Special Technical Publication STP 922:518-37.
Tye, R.P. and A.O. Desjarlais. 1981. Performance characteristics of foam-in-place urea formaldehyde insulation. ORNL/Sub-78/86993/1. Oak Ridge National Laboratory, Oak Ridge, TN.
Verschoor, J.D. 1977. Effectiveness of building insulation applications.
USN/CEL Report No. CR78.006—NTIS No. AD-AO53 452/9ST.
Verschoor J.D. 1985. Measurement of water vapor migration and storage in composite building construction. ASHRAE Transactions 91(2):390-403.
Verschoor, J.D. and P. Greebler. 1952. Heat transfer by gas conductivity and radiation in fibrous insulations. ASME Transactions 74(6):961-68.
Wilkes, K.E. and P.W. Childs. 1992. Thermal performance of fiberglass and cellulose attic insulations. In Thermal Performance of the Exterior Enve-lopes of Buildings V, pp. 357-67. ASHRAE.
Wilkes, K.E. and J.L. Rucker. 1983. Thermal performance of residential attic insulation E. Energy and Buildings 5:263-77.
BIBLIOGRAPHY ASHRAE. 1979. Thermal Performance of Exterior Envelopes of Buildings.
ASHRAE. 1982. Thermal Performance of Exterior Envelopes of Buildings II.
ASHRAE. 1985. Thermal Performance of Exterior Envelopes of Buildings III.
ASTM. 1989. Standard practice for determination of heat gain or loss and the surface temperature of insulated pipe and equipment systems by the use of a computer program. Standard C 680-89(1995). American Society for Testing and Materials, West Conshohocken, PA.
ASTM. 1994. Standard test method for water vapor transmission rate of flexible barrier materials using an infrared detection technique. Standard F 372-94.
Bomberg, M., ed. 1991. Moisture research in North America: 25 Years of building science. Lund University, Lund, Sweden.
ECON-I. 1973. How to determine economic thickness of thermal insulation.
Thermal Insulation Manufacturers Association, Mt. Kisco, NY.
FEA. 1974. Retrofitting existing housing for energy conservation: An eco-nomic analysis. Federal Energy Administration, Washington, D.C.
Hite, S.C. and J.L. Dray. 1948. Research in home humidity control.
Research Series No. 106. Purdue University, Engineering Experiment Station, West Lafayette, IN.
Rose, W.B. and A. TenWolde, eds. 1993. Bugs, mold & rot II: Workshop pro-ceedings. National Institute of Building Sciences, Washington, D.C.
Trechsel, H.R., ed. 1994. Moisture control in buildings. ASTM Manual Series: MNL 18.
24.1 CHAPTER 24 THERMAL AND MOISTURE CONTROL IN INSULATED ASSEMBLIES—APPLICATIONS GENERAL BUILDING INSULATION PRACTICE ................ 24.1 Wood Frame Construction ...................................................... 24.1 Cold-Formed Steel Frame Construction ................................ 24.1 Heavy Steel Frame Construction ............................................ 24.2 Masonry and Concrete Construction ...................................... 24.2 Foundation and Floor Systems ............................................... 24.3 Low-Slope Roof Deck Construction ........................................ 24.3 Insulation Field Performance Characteristics ....................... 24.3 MOISTURE CONTROL IN BUILDINGS ............................... 24.3 Control of Liquid Water Entry ................................................ 24.3 Control of Water Vapor Migration ......................................... 24.4 Moisture Control Options ....................................................... 24.4 Moisture Control Options for Heating Climates .................... 24.4 Moisture Control Options for Mixed Climates ....................... 24.7 Moisture Control Options for Warm, Humid Climates ........... 24.8 Membrane Roof Systems ......................................................... 24.9 Moisture Control in Foundations ......................................... 24.10 Envelope Component Intersections ....................................... 24.12 Moisture Control in Commercial and Institutional Buildings ............................................................................ 24.12 INDUSTRIAL AND COMMERCIAL INSULATION PRACTICE ........................................................................ 24.14 Pipes ...................................................................................... 24.14 Ducts ..................................................................................... 24.16 N THE ORIGINAL planning phase of buildings, the thermal and Imoisture design and long-term performance must be considered.
Installation of adequate insulation and moisture control assemblies during construction can be much more economical than installation later. Proper selection of thermal insulation and moisture control assemblies must be based on • Thermal and moisture properties of the materials • Other properties required by the location of the materials • Space availability • Compatibility of the materials with adjacent materials • Interior and exterior climate • Performance expectations Types of thermal insulation, their properties, economic thick-ness, and principles of moisture control and moisture transport are discussed in Chapters 23 and 25. Insulation in various assemblies that can be used interchangeably for a given construction, as well as specific moisture control options for various climatic regions, are discussed in this chapter. For specific industrial applications of insulated assemblies see the appropriate chapter in other ASHRAE Handbooks. In the 1998 ASHRAE Handbook—Refrigeration, for refrigerators and freezers, see Chapters 47, 48, and 49; for insula-tion systems for refrigerant piping, see Chapter 32 and this chapter; for refrigerated facility design, see Chapters 13 and 39; for trucks, trailers, and containers, see Chapter 29; for marine refrigeration, see Chapter 30; and for environmental test facilities, see Chapter 37. GENERAL BUILDING INSULATION PRACTICE WOOD FRAME CONSTRUCTION Wood framing members and structural panels such as plywood, particleboard, and fiberboard only provide limited resistance to heat flow; therefore, wood frame construction is well suited to applica-tion of both cavity insulation and surface-applied insulation. The most common materials for cavity insulation are glass fiber, mineral fiber, cellulose, and spray-applied foams. For surface applications, a wide variety of sheathing insulations exists. Roof decks of wood, metal, or preformed units may be insulated on top of or below the deck. Attic construction with conventional rafters and ceiling joists or roof trusses can be insulated between framing members with batt, blanket, or loose-fill insulation. In warm climates, radiant barriers, low emissivity surfaces, and reflective insulations further reduce cooling loads. The Radiant Barrier Attic Fact Sheet (DOE 1991) provides information on climatic areas best suited for radiant barrier applications. This document also provides comparative information on the relative performance of these products versus conventional fibrous insulations.
The cavities of cathedral ceiling construction (in which the ceiling insulation and interior finish are parallel to the roof plane) can be insulated using glass fiber, cellulose, rigid foam, or spray-applied insulation. The surface above cathedral ceiling fram-ing may be insulated with insulating panels or structural insulated panels (SIPs). The placement of insulation directly beneath a sloped roof deck in standard flat-ceiling construction, with or without ven-tilation, has been called “cathedralized” construction.
The wall cavities of wood frame construction can be insulated with batt, blanket, and loose-fill or spray-applied insulation. When using insulation materials in wall applications, extra care must be taken during the installation to eliminate voids within the wall cav-ity. When installing loose-fill insulation during retrofit of existing construction, all cavities should be checked prior to installation for obstructions such as fire stop headers and wiring that could prevent complete filling of the cavity. In addition, the material must be installed at the manufacturer’s recommended density to ensure the desired thermal performance. In addition to being properly insulated, the exterior envelope of a building should be constructed to minimize airflow into or through the building envelope. Airflow may degrade the thermal perfor-mance of insulation and cause excessive moisture accumulation in the building envelope. The use and function of airflow retarders are discussed in both in Chapter 23 and in this chapter.
COLD-FORMED STEEL FRAME CONSTRUCTION Conventional light frame construction with cold-formed steel framing has many characteristics in common with light frame wood construction. The greatest differences are the increased thermal conductivity and the dimension and shape characteristics of steel framing members. Barbour et al. (1994) and Tuluca and Gorthala (1999) found that the conductivity and framing member spacing The preparation of this chapter is assigned to TC 4.4, Building Materials and Building Envelope Performance.
24.2 2001 ASHRAE Fundamentals Handbook (SI) have the most significant roles in determining overall R-value (U-factor). They determined that the thickness of cold-formed steel does not play a significant role in the R-value (U-factor). The par-allel path method overestimates the R-values for building assem-blies containing cold-formed steel framing, and the isothermal planes calculation methods underestimates the R-value. For meth-ods to calculate heat flow in framing assemblies using cold-formed steel, see Chapter 25.
The most common cavity insulation materials are glass fiber, mineral fiber, cellulose, and spray-applied foams. Cavity insulation should be full width to cover between the steel framing members; it should not be a nominal width such as used for wood stud construc-tion. Surface applications of continuous insulation sheathing, such as rigid foam board (i.e., extruded or expanded polyisocyanurate or polystyrene), may be applied to the exterior or interior of the assem-bly framing to provide a thermal break. Rigid board insulation applied as a sheathing does not provide structural lateral bracing.
Thermal bridging may be more severe with cold-formed steel-framed trusses than with conventional joist and rafter framing.
Roof trusses have chords and web members that extend through the insulation into an unconditioned attic space and act as a thermal bridge to the bottom chord. A continuous insulating sheathing on the exterior or interior of the roof framing provides a thermal break.
HEAVY STEEL FRAME CONSTRUCTION Buildings and structures with heavy, steel framing supports and exterior metal cladding are usually insulated between the frame and cladding with faced blanket or spray insulation. In heating climates, the facing may serve as a combination vapor retarder and interior finish, which must be protected against physical damage. Other types of insulation can be used by adding framing or furring to the inside of frame or exterior siding. Insulation securements that com-press batt and blanket insulations reduce thermal resistance. This reduction, plus the thermal bridging caused by the screw or bolt penetrating the insulation, may cause condensation during cold weather.
The exterior cladding of structural steel-framed buildings may be (1) custom walls for specific projects, where most construction components are developed on a one-time basis; (2) commercial walls, where standard components are adapted to a particular build-ing design; and (3) industrial-type walls, where standard metal sheets are fluted or ribbed to form field-assembled sandwiches to meet job conditions. The cladding may consist of prefabricated panels or sections that may be classified as one of the following general types: 1. To withstand the elements, single-thickness facings are usually inserted in subframing with the windows to form a veneer over separate backup walls. Thermal insulation is integral with, or added to, the exterior of the backup wall and covers the edge of the floor slab or spandrel beam. In heating climates, this method reduces heat loss and keeps floors warm at their exterior edge.
Insulating the exterior of the framing reduces thermal movement of the building structure.
2. Sandwich construction or adhesive-bonded panels are gener-ally three-ply: exterior skin, core materials, and interior skin.
This type of panel, when manufactured with concrete faces and an insulation core, comes in large sizes and can be attached directly to the exterior of the building frame. When using metal facings, the width of the panels is usually restricted to match that of the standard formed sheet. These panels are installed on the exterior of the building framing. The thermal performance required for a particular installation (including the interfaces between walls, floors, ceiling, doors, and windows) should be checked carefully because adding insulation later is difficult.
3. Mechanically fastened panels generally have a hollow box shape, with the exterior facing nesting over the interior facing.
The cavity can be filled with flexible or semirigid insulation. The edges should vent to the exterior of the building to allow the panel to serve as a rain screen. The panels are normally installed in a subframing system.
4. Industrial metal panels have an exterior facing of standard ribbed, corrugated V-beam materials and an interior facing of proprietary metal pans or standard corrugated or ribbed sheet-ing. Insulation varies from semirigid to rigid, depending on the design of the inner surfacing materials. It provides good thermal and moisture control when installed with tight inner surface construction.
All curtain wall panels with insulation in the cavity, or as a core, should be sufficiently tight at their edges to prevent the entrance of free water or moisture. However, because some moisture may enter the wall from the inside or outside, the wall should be capable of drying out. Panel edges should not be hermetically sealed. For cold climates, the subframing members or wall mullions should be of noncontinuous construction. The exterior to interior path should have a thermal break or insulated mullion cover on the interior or exterior to reduce the hazard of condensation.
MASONRY AND CONCRETE CONSTRUCTION Because concrete masonry unit walls and masonry cavity walls have hollow vertical cavities, insulating materials can be placed within the wall itself. Loose-fill insulation, such as water-repellent perlite and vermiculite, as well as foam inserts and foamed-in-place insulations, are used. This insulation method is more effective with low-density masonry units. Rigid insulation can be placed between the wythes in a cavity wall or veneered construction, and furring on the inside of the wall is still used in many areas.
The thickness and density of concrete masonry wall construc-tion coupled with its interior air cavities moderate the heat trans-mission. The thermal mass or inertia of such heavy construction causes a time lag in heat migration, which may lower the peak gains or losses. Insulation placed on the outside of a concrete masonry wall enhances this time lag effect and helps protect the structure from expansion and contraction caused by temperature extremes.
Precast or poured-in-place solid concrete walls are insulated similarly to solid masonry construction. The design and method of fabricating the panels dictates the type of insulation selected. Insulating concrete forms are manufactured from foam plastic.
Generally they are made in units that resemble concrete masonry units, although they are typically somewhat larger. The individual units, which have interlocking edges, are stacked to the desired height to create a form for the concrete for a wall. Where openings for doors and windows are desired, units are omitted. The top, bot-tom, and sides of the opening are formed and braced with framing lumber to retain the concrete in the same way that an opening would be formed in a conventionally formed concrete wall.
Steel reinforcing and anchor bolts, if required, are placed, the forms are plumbed and braced, and concrete is placed in the form.
To this point, construction of the concrete wall proceeds in much the same way as it would for a conventionally formed concrete wall. However, the foam plastic forming material is left in place to serve as building insulation and to reduce sound transmission through the wall after the concrete has hardened.
Although the concrete contributes to the R-value of the wall, the primary insulation value is obtained from the foam plastic forms.
The R-value for the walls depends on the thickness and type of foam plastic, but the typical R-value for an approximately 200 mm thick wall ranges from 3.5 to 5.3 m2·K/W. The method used to tie the foam plastic panels together to create the form also affects the R-value. Ties to connect the outside panels are usually made of plastic or metal, but some manufactured forms use continuous foam plastic across the width of the wall.
Thermal and Moisture Control in Insulated Assemblies—Applications 24.3 FOUNDATION AND FLOOR SYSTEMS Perimeter foundation insulation may be applied on the outside or inside of foundation walls. Rigid mineral fiber or cellular plastic insulation is commonly used as perimeter exterior insulation. Exte-rior insulation may be applied below grade vertically or may extend as skirting down and outward from the building. These two profiles have comparable heat loss. Insulation applied on the exterior, espe-cially skirting, should resist compressive forces from the soil and from the backfilling process. The thermal performance of vertically applied insulation may be degraded if it remains wet, so drainage at the base of the foundation may be required by manufacturers of some perimeter insulations. Some insulation products are designed to facilitate vertical drainage.
Perimeter insulation usually needs protection from physical dam-age and ultraviolet radiation if it extends above grade level. The exposed section of a foundation that extends from grade up to the top of the foundation can be a source of considerable heat loss. However, in buildings subject to termite infestation, the exposed foundation at grade should remain uncovered to facilitate inspection. Exterior insulation that extends as skirting from a shallow foun-dation can prevent frost heave of the slab without a perimeter foot-ing down to frost depth. The insulation thickness is selected so that even during the coldest weather, the soil beneath the insulation remains above freezing.
A slab foundation may be insulated by vertical and horizontal insulation, usually of rigid foam. Insulation at the perimeter of the building provides more resistance to heat loss than beneath the slab inside the building. Perimeter insulation usually includes vertical (installed either inside or outside the stem wall) and horizontal pan-els. The horizontal panels may extend from the foundation wall inward for a distance of 600 mm or more, or outward as skirting.
In crawl space construction, insulation may be applied either to the perimeter walls or to the floor framing. Rigid foam panels are often installed at the inside of the crawl space walls. The cavities at the band joist may be insulated with batts or custom fitted panels of rigid insulation. In vented crawl space construction, the floor fram-ing may be insulated—commonly with batt or blanket insulation.
The facing is stapled to the underside of the floor framing and the exposed batt faces down. Unless additional measures are taken to keep the insulation in place, such as wire strapping, rigid insulation panels, or a reinforced plastic membrane (like the “belly paper” used in manufactured housing), it may fall down. Another floor insulation method uses double-sided, perforated aluminum foil draped and stapled over floor joists (BRANZ 1983).
For a sag depth of about 100 mm, laboratory R-values had a mean of 1.2 m2·K/W and ranged from 1.0 to 2.5 m2·K/W, depending on the humidity above the floor space. Field measurements have shown R-values up to 2.5 m2·K/W for carpeted floors over good installations with a sag of 100 mm.
Basement insulation may be applied at the interior as well as the exterior. Insulation should not be placed on the interior of a base-ment unless extra measures have been taken to ensure proper drain-age. Insulating concrete forms (see above) are often used for basement wall construction. Control of pests around below-grade insulation, especially termite and insects, is a continuing concern.
Details on foundation insulation may be found in the Building Foundation Design Handbook (Labs et al. 1988).
LOW-SLOPE ROOF DECK CONSTRUCTION Almost all low-slope structural roof deck construction requires thermal insulation to economically maintain the design indoor envi-ronment. For low-slope roofs, insulation should be placed on top of the deck, on the outside of the structure. This location moderates the deck temperature, which reduces thermal movement of the deck and the potential for underdeck condensation. Traditionally, the insula-tion is placed under the roof membrane so that it functions as a base for the built-up roof (BUR) or single-ply membrane. For a stable base, a good bond must be established between the insulation and the roof deck, and between the insulation and the BUR. Ineffective bonds that allow the BUR to move in reaction to stresses from tem-perature changes often cause the roof to fail. Single-ply membrane roofs are frequently installed with ballasted, mechanically fastened or fully adhered systems. Insulation can also be placed above the membrane in a protected or inverted roof system. With this approach, the roof membrane is installed on the deck, where it functions as a waterproofing mem-brane and vapor retarder. Insulation can be placed above and below membranes to function both as the base and protector for the mem-brane. Some insulation can be wetted and then successfully dried; however, while wet, it has a greatly reduced insulation value.
INSULATION FIELD PERFORMANCE CHARACTERISTICS Convection and air infiltration in some insulation systems may increase the heat transfer across them. Low-density loose-fill, large open-cell, and fibrous insulations, and poorly designed or installed reflective systems are most susceptible to increased heat transfer caused by natural and forced convection (air infiltration). A temper-ature differential across the insulation, as well as the height, thickness, or width of the insulated space, influences the amount of convection.
When a membrane with low air permeance is applied to one surface or when the cavity is filled with insulation, natural convection is reduced significantly and apparent thermal conductivities measured by standard test methods apply. The heat loss due to air convection may not be significant for many types of insulation products, such as batts and higher density loose-fill insulations. Convective heat loss potentials should be obtained from insulation manufacturers.
The effectiveness of thermal insulation is seriously impaired when it is installed incorrectly. For example, a 4% void area in wall insulation with an R-value of 1.9 m2·K/W increases heat loss by 15%. A 4% void area in ceiling insulation with an R-value of 3.3 m2·K/W causes a 50% increase in heat loss. Verschoor (1977) found that air interchange around thin wall insulation installed ver-tically with air spaces on both sides increases heat loss by 60%.
Lecompte (1990) found significant losses (up to 300%) as a func-tion of the size and distribution of openings around insulation materials. Other factors, including vibration, temperature cycling, and other mechanical forces, can affect thermal performance by causing settling or other dimensional changes. Chapter 23 gives more information on the effect of moisture on thermal properties of building structures.
MOISTURE CONTROL IN BUILDINGS Not all moisture problems can be avoided at all times. Proper design can help reduce the risk and make a building more tolerant to moisture. The recommendations in this chapter are intended to pro-vide guidance.
Strategies to control moisture accumulation fall into two general categories: (1) minimizing moisture entry into the building enve-lope and (2) removing moisture from the building envelope. Once basic moisture transport mechanisms and specific moisture control practices are understood, roof, wall, and foundation constructions for various climates can be reviewed to determine whether each sig-nificant moisture transport mechanism is controlled. Because it is not possible to prevent moisture migration completely, construction should include drainage, ventilation, removal by capillary suction, or other provisions to carry away unwanted water.
CONTROL OF LIQUID WATER ENTRY Moisture problems in buildings are frequently caused by liquid water entering through leaking roofs or the foundation, or through 24.4 2001 ASHRAE Fundamentals Handbook (SI) the walls due to wind-driven rain or rain splashing. Poor flashing details are often a major cause of water entry into walls and roofs.
Rainwater should be carried away from the foundation through gut-ters, downspouts, and positive grading. A rain screen can minimize penetration of walls due to raindrop momentum, capillarity, gravity, and air pressure difference. The rain screen wall is designed so that the air pressure difference across the exterior rain screen is nearly zero at all times. A rain screen wall contains three components: an airflow retarder system, a pressure-equalization chamber, and a rain screen. The airflow retarder must be able to resist pressures from wind, the stack effect, and mechanical ventilation. The pres-sure-equalization chamber separates the rain screen and the air flow retarder system. It may be an air cavity or may be filled with a self-draining material to prevent water that penetrates the rain screen from reaching the airflow retarder system. The chamber should consist of separate compartments to avoid lateral airflow, especially around corners of the building. Each chamber compart-ment is vented to the outside through the rain screen to provide pres-sure equalization and must be flashed to the outside to drain water that has penetrated the rain screen. The rain screen must contain suf-ficient vents to provide pressure equalization; that is, the airflow resistance of the rain screen must be much lower than that of the air-flow retarder.
CONTROL OF WATER VAPOR MIGRATION Water vapor entry into the building envelope can be limited by airflow retarders and water vapor retarders. As described in Chapter 23, airflow retarders are intended to restrict airflow, and thereby water vapor flow, whereas water vapor retarders are designed to restrict vapor flow by diffusion.
Air Leakage Control Past research demonstrated that air movement is more effective than water vapor diffusion for transporting water vapor within the building envelope. In order to minimize moisture penetration by air leakage, the building envelope should be as airtight as possible. The airflow retarder must also be sufficiently strong and well supported to resist wind loads.
In the past, air leakage in residential buildings provided suffi-cient ventilation, and the air leakage paths rarely led to interstitial condensation. However, in airtight buildings mechanical ventilation must be provided to ensure acceptable air quality and prevent mois-ture and health problems caused by excessive indoor humidity. Ven-tilation or drainage must go to the outside of the airtight layer of construction or it will increase air leakage of the building. To avoid condensation on the airtight layer, either the temperature of the layer must be kept above the dew point by locating it on the warm side of the insulation, or the permeance of the layer must be adequate to permit vapor transmission.
As described in the section on Leakage Distribution in Residen-tial Buildings in Chapter 26, air leakage through the building enve-lope is not confined to doors and windows. Although 6 to 22% of the air leakage occurs at windows and doors, 18 to 50% typically takes place through walls, and 3 to 30% through the ceiling. Leakage often occurs between the sill plate and the foundation, through inte-rior walls, electrical outlets, plumbing penetrations, and cracks at the top and bottom of the exterior walls. More detailed information can be found in Chapter 26.
Not all cracks and openings can be sealed in existing buildings, nor can absolutely tight construction be achieved in new buildings.
However, an effort should be made to provide as tight an enclosure as possible to reduce leakage and minimize potential condensation within the envelope. Such measures also reduce energy loss.
Moisture accumulation in the building envelope can also be min-imized by controlling the dominant direction of airflow. This can be accomplished by operating the building at a small negative or positive air pressure, depending on climate. In cooling climates, the pressure should be positive to prevent the entry of humid outside air into the envelope. In heating climates, the building pressure should be neither strongly negative, which could risk drawing soil gas or combustion products to the indoors, nor strongly positive, which could risk driving moisture into building envelope cavities.
MOISTURE CONTROL OPTIONS Options for moisture control under heating conditions often dif-fer from those under cooling conditions, even though the physical principles of moisture movement are the same. Therefore, the selec-tion of moisture control options depends on whether the local cli-mate is predominantly a heating or cooling climate. The Moisture Control Handbook (Lstiburek and Carmody 1991) recommends a three-step procedure for designing energy-efficient roofs, walls, and foundations with inherent moisture control capabilities: 1. Identify the climate: heating, cooling, or mixed 2. Determine the potential moisture transport mechanisms in each part of the exterior envelope: liquid flow, capillary suction, air movement, and vapor diffusion 3. Select the moisture control strategies: control moisture entry, control liquid moisture accumulation (condensation), or remove moisture by draining, venting, or diffusion The definitions of climate zones are somewhat arbitrary. Lstibu-rek and Carmody (1991) recommend that heating climates be defined as climates with 2200 heating kelvin-days (base 18.3°C) or more. Cooling climates are defined as warm, humid climates where one or both of the following conditions occur: (1) a 19.5°C or higher wet-bulb temperature for 3000 or more hours during the warmest six consecutive months of the year; (2) a 23°C or higher wet-bulb temperature for 1750 or more hours during the warmest six consec-utive months of the year.
Mixed climates are all climates that do not fall under the defini-tions of heating or cooling. Regions with heating climates in North America generally include the northern half of the United States, Alaska, and all of Canada. The climate in southeastern coastal regions of the United States generally can be characterized as cool-ing. However, the local climate should be evaluated to determine whether to design for heating, cooling, or mixed-climate conditions.
MOISTURE CONTROL OPTIONS FOR HEATING CLIMATES Surface Condensation Heating climates are defined as climates with 2200 heating kelvin-days (base 18.3°C) or more. In such climates, occasional win-dow condensation is common in buildings during winter and fall.
Lowering indoor humidity to minimize surface condensation on windows is one approach, but increasing the interior surface tem-perature of the window using multiple glazing, low-emittance glaz-ing, low-conductivity spacers, appropriate selection of window frame, or gas-filled glazing may be more effective. Higher thermal resistance in windows has the added advantages of saving energy, improving occupant comfort, and reducing the possibility of con-densate damage to the interior adjacent to the window (e.g., staining of the wall, rotting of the window sill, and mold growth).
Windows should remain clear most of the time. Some condensa-tion may appear around the window perimeter, but should disappear with a warming trend. This criterion should be used to decide which glazing should be installed to maintain the desired humidity, or whether to reduce the humidity to avoid condensation during the coldest periods.
Local condensation and mildew growth on walls and ceilings is often the result of low inside surface temperatures due to insufficient or faulty insulation. Increasing the thermal insulation or eliminating Thermal and Moisture Control in Insulated Assemblies—Applications 24.5 the voids in the insulation is the obvious remedy. If the problem is due to infiltration of cold air, an attempt should be made to eliminate the air leakage. However, in existing buildings these measures are often difficult or too expensive. In these cases, the only alternatives are lowering the indoor humidity, raising the indoor temperature, or increasing the air circulation near the surface.
Indoor Humidity Control A common cause of moisture problems during the heating sea-son is excessive indoor humidity. This is caused by an improper bal-ance between moisture generation and moisture removal. This balance can be changed by reducing the sources of moisture or by increasing the removal rate, usually by ventilation or dehumidifica-tion. However, it is important to avoid lowering the relative humid-ity too far below the lower comfort limit, which is generally about 25 to 30% rh.
Because water vapor is introduced into the building from various sources, the moisture content of the air in an occupied building without dehumidification is always higher than that of the outdoor air. Christian (1994) provides a detailed discussion of various indi-vidual moisture sources, primarily in residences. The section on Internal Moisture Gains in Chapter 20 of the 2000 ASHRAE Hand-book—Systems and Equipment states that a family of four produces an average of 320 g/h of moisture. TenWolde (1988, 1994) reports production rates between 135 and 330 g/h for one to two adults, with an average of 230 g/h. European sources report rates between 270 and 540 g/h for families without children, and rates between 210 and 950 g/h for families with one to three children (Christian 1994).
These numbers demonstrate that moisture production rates in resi-dences can vary widely. Moisture production rates for various kinds of livestock and plants can be found in Chapter 10.
A residential crawl space or basement can contribute significant amounts of additional water vapor. Trethowen (1994) reported an average moisture release of 0.40 kg/m2·day from moist or wet crawl spaces. Moisture released from building materials that are drying (construction moisture) in a new building can add large amounts of water vapor. Exposed soil surfaces in crawl spaces or cellars should be covered with vapor retarder membranes (see the section on Crawl Spaces).
If indoor humidity is excessive, and source reduction is impos-sible, increasing the ventilation rate should be considered. Ventila-tion may be natural or mechanical, and mechanical ventilation may be exhaust, inlet, or balanced. An air-to-air heat recovery device can be included to reduce the heating energy penalty from an exhaust fan. Other approaches include the use of mechanical dehu-midifiers and insulated vent stacks that extend from the living space through the roof. Short-term ventilation procedures such as occasionally opening a window or door may lower humidity momentarily, but it will rise to its original level soon after the window or door has been closed.
This is due to the evaporation of stored moisture into the indoor air.
When water vapor is released from showers or cooking, much of it is adsorbed by hygroscopic materials (paper, wood, fabrics, etc.) in the building. Some temporary storage in the form of surface con-densation may also occur. This moisture is released more slowly at a later time. Moisture storage effectively dampens the effect of short-term (hourly or daily) changes in moisture release or weather conditions on indoor humidity. Stored moisture also slows the effect of ventilation and dehumidification because this moisture needs to be released and removed before the indoor humidity can be lowered permanently. The evaporation of moisture stored during the sum-mer’s periods of high relative humidities is the cause of high relative humidity and window condensation in early fall.
In cold or cool winter climates, house ventilation can be an effec-tive method for moisture removal. In these climates, a ventilation level of 0.35 air changes per hour (ACH) (as recommended in ASHRAE Standard 62) is generally sufficient to prevent excessive indoor humidity and most window condensation (TenWolde 1994).
Ventilation is primarily required to ensure acceptable indoor air quality. If the recommended minimum ventilation levels are achieved, additional ventilation is probably unnecessary and inef-fective for humidity control. In mild, humid climates, ventilation rates greater than 0.35 ACH may be needed for humidity control, but in such climates other means of moisture removal, such as dehu-midifiers, should be considered.
Some suggest that high indoor humidity is caused by vapor retarders that lock in moisture. However, only a small fraction of the total moisture generated can be removed by vapor diffusion through the building envelope. Most high indoor humidity is due to inade-quate ventilation, inadequate dehumidification and air condition-ing, or an unusually large moisture source in the building. Vapor Retarders and Airflow Retarders Vapor retarders are recommended and often mandated in heating climates, and should be placed on the interior (warm) side of the insulation. Airflow retarders are also necessary, but their placement in the wall or ceiling assembly is probably less critical and still sub-ject to debate. Vapor retarder and airflow retarder functions may be combined in one material. Airflow retarder placement on the exte-rior prevents cold air from penetrating the insulation (wind wash-ing) and therefore improves thermal performance of the building envelope. However, in heating climates, airflow retarders on the exterior should have a high water vapor permeance. Special airflow retarder materials for exterior use with sufficiently high water vapor permeance are commonly available. Exterior airflow retarders do not prevent penetration and circulation of warm indoor air inside the wall or ceiling/roof cavity. Conversely, interior airflow retarders do not prevent wind washing. Interior airflow retarders do not need to have a high water vapor permeance. For additional general guid-ance on the placement and properties of airflow retarders, see Chap-ter 23.
The use of vapor retarders in compact low-slope roofing systems has been a long-standing issue for the roofing industry. Unlike other portions of the building envelope, water intrusion into a low-slope roof due to membrane failure is inevitable. Wet insulation performs below thermal performance levels specified during design. A sur-vey by Kyle and Desjarlais (1994) has indicated that the average energy efficiency of the entire roofing inventory in the United States is reduced by approximately 40% due to moisture contamination.
Powell and Robinson (1971) studied these problems and stated that the “most practical and economical solution to the problem of moisture in insulated flat-roof constructions (is) to provide a design that would have in-service self-drying characteristics.” A self-dry-ing roof uses the local meteorological conditions to create a vapor drive into the building interior. Desjarlais (1995) demonstrated that climates with up to 5000 heating kelvin-days create annually aver-aged downward vapor drives. In a self-drying roof, any leakage into the roofing system is passively driven into the building interior; if the leak is repaired, the roof system will dry. A vapor retarder prevents including the self-drying characteris-tics in the roofing design by placing an impermeable layer between the roof insulation and the building interior. A vapor retarder should only be placed in a roofing system when the amount of wintertime water uptake that the roofing system experiences exceeds the mois-ture limit of the insulation material in the roof. Desjarlais (1995) offers guidelines on how to determine these limits.
Penetrations through the airflow retarder (such as electrical out-lets, light fixtures, or plumbing stacks) should be minimized. Any penetration should be sealed carefully. Special airtight electrical boxes are available. Limited or minor penetrations of the vapor retarder are not of great concern, if an effective airflow retarder is placed elsewhere in the wall or ceiling.
24.6 2001 ASHRAE Fundamentals Handbook (SI) Attics and Cathedral Ceilings Attics and cathedral ceilings are protected from interior moisture in heating climates first by limiting the entry of moisture into roof cavities, and second by ventilating the cavities to minimize accumu-lation of moisture. Entry of interior moisture is limited by designing and constructing effective airflow and vapor retarders between the interior space and roof cavities.
Attics. Ventilation is required in all United States and Canadian Model Building Codes. Studies on roof cavity and attic ventilation aimed at reducing paint peeling were conducted by Rowley et al.
(1939), Britton (1948), and Jordan et al. (1948). These studies con-cluded with recommendations for venting of attics. Britton found that air movement from a wet foundation through chases and wall space could reach the attic and cause moisture damage to roof sheathing. Jordan et al. and later reports demonstrated the use of air barriers as the primary requirement for keeping moisture out of attics. Colder attics were found to require more ventilation to pre-vent frost. A review of the studies (Rose 1992) concluded that sup-port for attic ventilation was at times contradictory and the specific requirement for 1 m2 of vent area for 300 m2 of floor space not resolved by the findings.
The four commonly cited reasons for attic ventilation are (1) pre-venting moisture damage, (2) enhancing the service life of temper-ature-sensitive roofing materials, (3) preventing ice dams, and (4) reducing the cooling load (TenWolde and Rose 1999). In some cases, venting may be inconsistent with the moisture control design approach. If the attic is vented, care should be taken to prevent entry of snow and to prevent airflow that might degrade the thermal per-formance of insulating materials (Hens and Janssens 1999).
Moisture. Vents have been shown to provide effective lowering of moisture levels in roof sheathing for attics constructed with a sin-gle unconditioned space, sloped roof, and a tight ceiling plane (Jor-dan 1948). It is relatively easy and inexpensive to install vents in such an attic without compromising the effectiveness of the ceiling insulation. In heating climates, attic ventilation usually provides a measure of protection from excessive moisture accumulation in the roof sheathing. If indoor humidity is high and humid indoor air leaks into the attic, attic vents by themselves may not prevent mois-ture accumulation.
Moisture control in attics in heating climates depends primarily on (1) maintaining lower indoor humidity levels during cold weather, (2) assuring maintainable airtightness and vapor resistance in the ceiling, and (3) attic ventilation (NRC 1963).
Temperature. A ventilated attic is cooler in the summer than an unventilated attic, and ventilation can reduce the temperature of shingles during daylight hours. Asphalt shingle manufacturers encourage ventilation as a prescriptive practice (ARMA 1997). In one study, the temperature difference due to power or turbine ven-tilation over soffit ventilation led to significant differences in max-imum attic air temperatures, but was not shown be an effective energy conservation method in moderately or heavily insulated ceil-ings (Burch and Treado 1978). It is not clear that attic air tempera-ture reduction is a significant factor in extending the service life of shingles (TenWolde and Rose 1999), since the long term studies on the temperature effects on shingle service life are incomplete.
Ice Dams. Ventilation of roofs, coupled with additional insula-tion and reductions in air exfiltration, reduces ice dam damage dur-ing winter in cold regions (Buska et al 1998). Where heat sources are located in the unconditioned attic space, large amounts of ven-tilation may be needed to prevent ice dams, necessitating mechani-cal attic ventilation (Tobiasson et al. 1994). Such heat sources may include furnaces, air handlers, or ductwork with conductive or con-vective heat losses.
Reducing heat loss into the attic by effective insulation, air leak-age control, and avoidance of heat sources such as uninsulated or leaky heating ducts in the attic, possibly coupled with ventilation, is a positive way of reducing ice dams and moisture damage (Fugler 1999). Damage due to ice damming in roof valleys and eaves can also be prevented by installing a waterproof underlayment of suffi-cient width beneath the shingles.
Other Considerations. Roofs with absorbent claddings, such as wood shingles or cement or clay tiles are subject to solar-driven moisture penetration (Cunningham et al. 1990). Moisture is driven into the roof when it is wetted by rain or dew and subsequently exposed to sunshine. When the moisture source is from the exterior, an impermeable membrane under the shingles or tiles can greatly reduce moisture transfer into the roof; but measures should be taken to prevent water accumulation on the underside of this membrane. Leaks cause another moisture load on roofs. Roof leaks are prop-erly addressed by repair rather than by ventilation.
Cathedral Ceilings. Cathedral ceiling construction is inherently prone to a wider range of conditions than attic construction because this type of construction has isolated air cavities in each rafter bay.
Vented attics perform better than vented cathedral ceilings for the same framing type (Goldberg 1999). Although providing effective ventilation to attics with simple geometries is relatively easy and inexpensive, providing soffit and ridge ventilation to each individ-ual cavity in a cathedral ceiling may be difficult or impractical.
Improperly installed insulation can obstruct the area designed or intended to provide ventilation. (Tobiasson 1994). An airtight ceil-ing plane, a vapor retarder, and foam air chutes between the sheath-ing and the top of the insulation effectively control moisture in cathedral ceilings with fiberglass insulation (Rose 1995). Hens and Jannsens (1999) pointed out that moisture control is assured only if airtightness is effective and can be maintained. They showed that the consequences of air entry and wind washing in insulated cathe-dral ceilings are detrimental, leading to degraded thermal perfor-mance, moisture response and overall durability. TenWolde and Carll (1992) showed that ventilation of roof cavities may cause increased air leakage, and that the net moisture effect depends on whether the principal source of makeup air is from indoors or out-doors.
Goldberg et al. (1999) noted that unvented attics and cathedral ceilings show a better retention of thermal resistance of the fibrous insulation than similar vented assemblies, though this benefit is smaller for attics than for cathedral ceilings. With careful attention to design for air- and vapor-tightness, unvented cathedral ceilings can be expected to perform satisfactorily in cold heating climates.
Operating Practices Details of indoor humidity control are discussed in the section on Indoor Humidity Control. Buildings in heating climates should not be operated at substantial positive indoor air pressures, which drive moist air into the building envelope and increase the potential for moisture accumulation. Large negative pressures should also be avoided if any unsealed combustion equipment is operated in the building. Negative pressure in the basement or in slab-on-grade buildings should also be avoided when there is potential radon leak-age from the soil into the building, unless a subslab depressurization system has been installed.
Other Considerations In heating climates, it is important to design for excessive indoor humidity. If the anticipated indoor humidity will be high, then extra care must be taken in design and construction by using air barriers in conjunction with building pressure regulation.
In general, mechanical equipment should be kept within the con-ditioned space of the building. This approach reduces the number of openings through the building envelope and reduces the energy losses associated with exterior equipment and ductwork. Several design options permit installation of insulation below the roof plane, as in cathedralized construction (Rose 1995).
Thermal and Moisture Control in Insulated Assemblies—Applications 24.7 Example of Residential Wall Construction for Heating Climates Figure 1 shows the cross section of a residential wall for heating climates. Moisture control is handled in the following ways: • Rain. The brick veneer, an air space, and the building paper form an effective rain screen. The air space behind the brick veneer provides a capillary break for any rainwater absorbed by the brick and mortar. Mortar should not breach the air space and touch the building paper, as this would allow rainwater to bypass the capil-lary break. The building paper protects the fiberboard or gypsum from any water penetrating the rain screen • Air movement. The sheathing and building paper serve as an air-flow retarder. Sufficient airtightness can be obtained by airtight installation of the sheathing (i.e., installed vertically with joints over the studs, with sealant or caulk used at the joints). • Vapor diffusion. Vapor diffusion from the inside is inhibited by the polyethylene vapor retarder. MOISTURE CONTROL OPTIONS FOR MIXED CLIMATES Mixed or temperate climates fall neither under the definition of a heating climate, nor under the definition of a hot, humid climate.
Mixed climates may be heating- or cooling-dominated. This zone includes areas with hot and dry climates (e.g., Arizona). Buildings in mixed climates may encounter high interior levels of humidity during winter and high exterior levels of humidity during summer.
Summer cooling or winter heating for comfort in mixed climates does not usually create serious vapor problems in exterior walls and ceilings. The summer outdoor dew point, especially during peak values, may exceed the design dew-point temperature in common use, but it seldom exceeds 24°C for a prolonged period. Condensa-tion within exterior walls exposed to an indoor temperature of 24°C is seldom as serious as winter condensation.
In a study of a wood-sided house in Athens, Georgia, Duff (1956) showed that under summer cooling conditions, temperatures were lower outside than inside long enough to prevent moisture buildup from damaging the structure. This was true regardless of whether or not a low-permeance material was placed near the inner surface.
However, masonry or brick-veneered structures with a low-per-meance vapor retarder (e.g., vinyl wallpaper or polyethylene) near inner surfaces do have a moisture buildup under summer cooling conditions.
Vapor Retarders and Airflow Retarders Airtight construction is recommended in all climates. Airflow retarders provide protection from excessive moisture accumulation in the building envelope during cooling and heating, and may reduce energy consumption. In mixed climates, the need for low-permeance vapor retarders in most types of buildings is less pronounced than in heating climates or in warm, humid climates. However, if a vapor retarder is deemed necessary in a mixed cli-mate zone, its placement presents somewhat of a dilemma. Under cooling conditions, a vapor retarder would normally be located on the outside of the insulation. But under winter conditions, it would be located on the inner side. Use of vapor retarders at both locations is undesirable because it restricts moisture movement into the insu-lation as well as the escape of any moisture. In dwelling construc-tion, the vapor retarder should be placed to protect against the more serious condensation (winter or summer). However, if indoor humidity is kept below 35% (at 21°C) during winter, a vapor retarder on the inside of the insulation is probably not necessary in mixed climates.
The choice and placement of a vapor retarder, airflow retarder, and other materials minimize the potential for condensation while allowing for some drying. For example, if a vapor retarder is installed on the interior, an exterior airflow retarder and/or sheath-ing should have sufficient permeance to allow drying. The corre-sponding situation in cold storage buildings, in which a more serious reversal of vapor flow conditions from winter to summer may occur, requires individual analysis.
Attics and Cathedral Ceilings Venting of attics and cathedral ceilings during winter in a mixed climate has similar benefits and drawbacks as in a heating climate.
Venting may provide benefits for moisture control in attics where effective vents can be installed relatively easily and cheaply, and where the ceiling is tightened against air leakage. Unvented cathe-dral ceilings can provide satisfactory moisture performance in mixed climates when the system (1) is designed to control moisture migration, and (2) contains an airflow retarder that is maintained.
More detailed discussion of ventilation of attics and cathedral ceil-ings can be found in the sections on Attics and Cathedral Ceilings under Moisture Control Options for Heating Climates.
Both vented and unvented construction should be designed and constructed to exclude interior moisture from cathedral ceiling cav-ities. As in heating climates, vents in cathedral ceilings may be less effective and beneficial than vents in attics; therefore, vents should be considered a design option. Ductwork should be kept in the conditioned space of the building in order to improve energy efficiency. In hot, dry climates, energy losses through ductwork located in unconditioned attics is greater than energy losses in attics using cathedral construction, in which the insulated envelope is located at the roof (Rudd 1998). Example of Residential Wall Construction for Mixed Climates Figure 2 shows an example of a residential wall detail for mixed climates, with rigid insulating sheathing serving as a vapor retarder and air retarder. Moisture control is handled in the following ways: • Rain. The combination of siding and airtight foam sheathing serves as a screen system and controls rain penetration. The air cavities behind the siding should be sufficient to act as a capillary break. If the air space is insufficient, the siding may be installed Fig. 1 Example of Residential Wall Construction for Heating Climates Source: Lstiburek and Carmody (1991). Reprinted with permission.
24.8 2001 ASHRAE Fundamentals Handbook (SI) on furring strips to provide the air space. With vinyl or aluminum siding, liquid absorption and capillary moisture transfer are not a concern.
• Air movement. The rigid insulating sheathing can be caulked at the top and bottom plates and at the joints to provide an exterior air flow retarder. Alternatively, caulking of the gypsum board can provide an interior airflow retarder.
• Vapor diffusion. The impermeable rigid insulation acts as a vapor retarder. During cooling periods, vapor diffusion from the outside is impeded at the exterior sheathing surface. During heat-ing periods, vapor diffusion from the inside is inhibited at the interior surface of the foam sheathing. To keep moisture conden-sation to a minimum, this first condensing surface temperature should be elevated through the use of foam sheathing with a high R-value. For mixed climates, the thermal resistance of the insu-lating sheathing in this example should be 1.2 m2·K/W or greater, with R = 2 m2·K/W in the cavity. MOISTURE CONTROL OPTIONS FOR WARM, HUMID CLIMATES Warm, humid cooling climates are defined as climates where one or both of the following conditions occur: (1) a 19.5°C or higher wet-bulb temperature for 3000 or more hours during the warmest six consecutive months of the year; (2) a 23°C or higher wet-bulb temperature for 1500 or more hours during the warmest six consec-utive months of the year. Depending on local experience with mois-ture problems, humid climate design criteria may also be desirable in locations that do not quite meet the foregoing conditions. In warm, humid climates dehumidification by air conditioning or other means is the most practical approach to moisture removal from the conditioned space. The overall latent-cooling load is com-posed of diffusion, ventilation, infiltration, and internally generated latent cooling loads. Because the latent-cooling load on an air con-ditioner in high-humidity climates frequently exceeds the sensible load, a system should be capable of handling the latent load without overcooling. In residential buildings, oversized air conditioners may not alleviate the problem of high humidity due to short cycling.
Approaches to solving this problem include proper sizing of the sys-tem, the use of reheat, or design for variable flow rates.
Airflow Retarders and Water Vapor Retarders Construction should be airtight, as in all other climates. Many moisture and condensation problems in cooling climates have been found to be caused by excessive leakage of outside air into the building envelope. Airflow retarders in cooling climates are best placed on the exterior. Negative pressures of the indoor space should be avoided. Under high-humidity conditions, ambient water vapor diffuses through building materials from the outside into air-conditioned spaces. Exterior surfaces should have lower per-meance than interior surfaces in high-humidity climates. Paints and finishes can provide the necessary permeance, with lower per-meance at the outside surface and higher permeance toward the inside.
Low-permeance paints, vinyl wall paper, or any other similar low-permeance material should not be used on the inside of walls and ceilings in warm, humid cooling climates.
Vapor retarders, if used, should be on the outside of the insula-tion. Then, any water vapor that enters the construction can flow to the inside of the building, where it can be removed by the air con-ditioner instead of accumulating in the wall or roof. Note that this recommendation is the reverse of the recommended practice for cold climates.
Attics and Cathedral Ceilings The commonly stated rules for attic and cathedral ceiling con-struction—ventilation and vapor retarder toward the inside—per-tain to cold climates and not to warm, humid climates with indoor air conditioning. Common sense suggests that venting with rela-tively humid outdoor air means higher levels of moisture in the attic or cathedral ceiling. Higher moisture levels in vented attics in hot, humid climates do not lead to moisture damage in sheathing or framing. However, higher moisture levels in attic cavities may affect chilled surfaces of the ceiling and cold surfaces of mechanical equipment. When cooling ducts are located in the attic space, attic ventilation with humid outdoor air may increase the chance of con-densation on the ducts.
As in all climates, airtight construction is desirable. In warm, humid climates, airtight construction usually reduces the latent load. Insulation and interior finishes should be selected and installed with an understanding that vapor diffusion is primarily inward.
As with other climates, a ventilated attic in a warm, humid cli-mate is noticeably cooler in the summer than an unventilated attic.
Beal and Chandra (1995) found that venting can greatly affect the temperature difference across the ceiling.
Other Considerations To encourage drying, shaded exterior surfaces that do not benefit from the evaporative effects of sun and wind (such as inside cor-ners) should be avoided or minimized. Building components that are prone to thermal bridging (such as exterior cantilevers, columns, foundations, or window and door frames) are of special concern.
Although these solutions may not totally eliminate mold and mil-dew growth, they substantially reduce the potential.
Serious wetting within walls can occur in summer under certain conditions. The National Research Council of Canada tested the walls of huts of brick masonry finished inside with furring, insula-tion, a vapor retarder, and plasterboard. The walls were opened dur-ing a sunny period following rain. Extensive wetting was observed in the insulation, particularly on the back of the vapor retarder. The absorptive brick wall had accumulated substantial quantities of water during the rainfall. Subsequent heating by the sun had driven the moisture as vapor into the wall, where it condensed and caused Fig. 2 Example of Residential Wall Construction for Mixed Climates Source: Lstiburek and Carmody (1991). Reprinted with permission.
Thermal and Moisture Control in Insulated Assemblies—Applications 24.9 serious wetting. The construction had no protection in the form of parging or paper on the inside of the brick. The study showed that walls with exterior coverings capable of absorbing and storing considerable quantities of water during a rain, and providing little resistance to vapor flow into the insulation from outdoors, may experience serious interior wetting by condensation.
No wetting occurred in a similar construction when a saturated sheathing paper was placed between the insulation and the brick.
Thus, a moderate vapor flow resistance, such as that provided by parging or a good sheathing paper on the outside of the insulation, can effectively stop vapor flow in such cases.
Operation and Maintenance Because the potential for damage to a building and its contents is substantial in an air-conditioned building in humid climates, it is more important to properly operate and maintain the building and its air conditioning system in humid climates than it is in others. The chilled water supply temperature and flow should be reliable, and multiple chillers and pumps should be considered to ensure contin-uous uninterrupted dehumidification.
Raising the chilled water supply temperature to conserve energy should not be attempted under these conditions, as this would impair the dehumidification capacity of the air-conditioning system.
Lowering the cooling thermostat setting generally increases the chance for mold and condensation in exterior walls, especially in locations where the cooled air is blown directly towards the wall. Example of Residential Wall Construction for Warm, Humid Climates Figure 3 shows an example of a residential wall detail for warm, humid climates, with rigid insulation serving as a vapor retarder and airflow retarder. Moisture control is handled in the following ways: • Rain. The combination of airtight foam sheathing and siding serves as a rain screen system and controls rain penetration. The air cavities behind the siding should be sufficient to act as a cap-illary break. If the air space is insufficient, the siding may be installed on furring strips to provide the air space. With vinyl or aluminum siding, liquid absorption and capillary moisture trans-fer are not a concern. Wood siding may be backprimed to prevent moisture absorption through the back, and installation of wedges and clips on wood lapped siding should be considered to mini-mize capillary transport between the boards.
• Air movement. The exterior sheathing is the best location for an air seal, using either an adhesive or caulk to fasten the sheathing to the framing members.
• Vapor diffusion. In warm, humid climates, the dominant source of moisture is the outside air, and moisture is typically driven toward the interior. This means that the best location for the vapor retarder is at or near the exterior wall surface. Vapor-permeable paint should be used on the interior gypsum wallboard. MEMBRANE ROOF SYSTEMS Because most membrane roof systems in commercial and insti-tutional construction are highly resistant to vapor leakage, conden-sation must be prevented when insulation is placed between the heated interior and the roof membrane. Wet insulation in low-slope roof construction is difficult to dry. Drainage is likely to be so slow as to be ineffective. Ventilation to the outside is not effective for dry-ing roof insulation, because forces acting to remove the moisture are small. The vents themselves may present a hazard to the insulation by admitting moisture and drifting snow. Also, water leaks can occur where the vents penetrate the roof unless they are properly installed. Finally, vents may allow chimney action to carry air upward through openings in the deck and ceiling. Then as air flows to the outside, further moisture is drawn into the roof with the replacement air and may condense.
A vapor retarder in a conventional flat roof can trap moisture in the roof cavity. The decision whether to use a vapor retarder depends on interior humidity and climate. The absence of a vapor retarder allows vapor to enter a roof during the heating season, but it also facilitates the removal of moisture in warm weather. This may not be acceptable in buildings with high indoor humidity or in extremely cold climates, when a large accumulation of frost or liq-uid condensation results in dripping. Where humidities are lower, or the climate less severe, the roof system may successfully store moisture through the heating season without problems (Baker 1980). The success of this strategy, however, also depends on the airtightness of the roof assembly. More information on this can be found in the section on Self-Drying, Low-Slope Roof Systems.
Regular inspection of the membrane and flashings helps prevent water leakage into the roof. Infrared scanners or capacitance meters can help detect wet insulation, which can be removed or possibly dried out.
Inverted Roof Systems The top layers in protected membrane or inverted roof systems are not waterproof; therefore, insulation is exposed to rainwater. To remain effective, it must be able to resist moisture penetration.
Extruded polystyrene board has been used extensively. Insulation moisture content commonly ranges up to 4 or 5% by volume.
Some insulations are damaged by freezing and thawing, which fracture cell walls and allow water into an otherwise low-permeance material. When free moisture is available, the rate of entry increases rapidly as the temperature gradient increases (Hedlin 1977). Even when insulation is immersed in ponded water, moisture absorption through the edges is less than through the upper and lower surfaces, because the temperature gradient is normal to the roof surface.
Protective measures can reduce moisture gains. Roof slope per-forms much the same function for protected membrane roofs as it does for conventional ones. Covering the bottom surface of the insulation with a low-permeance layer inhibits moisture entry Fig. 3 Example of Residential Wall Construction for Warm, Humid Climates Source: Lstiburek and Carmody (1991). Reprinted with permission.
24.10 2001 ASHRAE Fundamentals Handbook (SI) there. The upper surface should be open to the atmosphere so that water can evaporate freely. If it is trapped against the upper surface (e.g., by paving stones), solar heating may drive the water into the insulation.
Where a high thermal resistance is required, roofs may combine conventional and protected membrane systems when they are applied in two separate lifts. The protected membrane system may be applied to existing conventional roofs to increase the thermal resistance, if the roof structure can support the added weight. This addition keeps the roof membrane warmer, so that the chance of moisture condensation on the underside of the roof membrane is significantly reduced.
Self-Drying, Low-Slope Roof Systems A major cause of roof replacement is excessive accumulation of water in the roofing system. Historically, this accumulation has been minimized by delaying its ingress into the roofing system through the use of improved roofing membranes and periodic planned maintenance. Of course, most roofing systems eventually leak. Without periodic inspection, small leaks in a roofing system containing a vapor retarder or some other low-permeance layer (such as an asphalt mopping) can go undetected for long periods and lead to a major roof system failure. The self-drying roof facilitates the controlled out flow of water vapor into the building interior, pre-venting any long-term accumulation of water in the roof. Although they have not been optimized, the roofing industry has constructed self-drying roofs for many years. A roof installed without a vapor retarder or a low-permeance layer is effectively a self-drying roof.
A self-drying roof should be considered whenever the average yearly vapor drive is into the building interior. Tobiasson and Har-rington (1985) have produced vapor drive maps for the continental United States. Desjarlais (1995) reported that this condition (vapor drive to the interior) is satisfied for climatic regions having less than 5000 heating kelvin-days (18.3°C base).
The self-drying roof system must be carefully designed and include special features. The deck system must be reasonably per-meable to water vapor so that downward drying can be maximized.
The water vapor permeance of the insulation materials must be selected so that the anticipated wintertime wetting is maintained at a level that the insulation materials can tolerate. Water vapor absorptive layer(s) should be included in the roof system so that a major leak into the roof can be controlled without leakage into the building interior. The self-drying roof must not contain a vapor retarder or any layers that are relatively impermeable. A suggested roof design procedure for self-drying roofs has been proposed by Kyle and Desjarlais (1994). The drying rate at the bottom depends on the airflow and the drying potential beneath the roof.
MOISTURE CONTROL IN FOUNDATIONS Grading Many of the problems related to moisture in foundations are due to the failure to discharge rainwater away from the building foundation.
Good construction practice generally requires exposed founda-tion between soil grade and the top of the foundation, as an inspec-tion site for pest control. Traditionally the height of exposed foundation has been 200 mm, although recent codes have reduced this requirement to 150 mm. Because of the likelihood of heat loss through this exposed foundation, covering with an insulating mate-rial is desirable wherever pest inspections are not necessary.
The soil should slope away from the building at a 5% grade for the first 3 m around the building perimeter. That is, the fall from grade to a point 3 m away from the building should be a minimum of 150 mm. The soil should be covered with a cap of relatively impermeable soil, in order to maximize surface flow of water away from the foundation. The soil backfill around the foundation is likely to settle in the first years after construction, requiring correc-tion to achieve the proper grade. If the general slope of the soil on the site is toward the building, then swales or drains should be used to divert surface flow around the building.
Rainwater discharge from the downspouts should be managed to ensure that it does not contribute to saturation of any soil that is in contact with the foundation. The discharge may consist of extend-ers, splash blocks, or designed drainage systems to carry the roof runoff away from the building. On sloped lots, downspouts should discharge on the downhill side where possible.
Footing drains are traditionally installed to ensure against a ris-ing water table; nevertheless, they may assist in draining water that may accumulate directly from surface rains. The footing drains should discharge water to an appropriate discharge site such as a storm sewer, a sump pump, or, if the site permits, daylight. Any footings for basements or crawl spaces, where the interior grade is below the exterior, should be provided with footing drains.
Gutters, downspouts, and below-grade drainage systems require maintenance. Below-grade drainage systems should be designed with cleanouts.
Floor Slab Summer surface condensation may form on concrete floors on grade, especially during the first few years after construction. Car-peting tends to lower the interior slab surface temperature, increas-ing the condensation potential. Dehumidification and ventilation may be sufficient to avoid odors or floor cover bonding problems caused by moisture, which are generally more objectionable than actual damage to the floor covering.
Entry of ground moisture can be further reduced by isolating the slab with the placement of a low-permeability membrane over the soil beneath the slab and by using coarse gravel to break the capil-lary moisture rise. Application of a membrane is difficult, however, because it can be easily damaged during construction.
Sealing of floor slabs and basements against the entry of radon should also be considered. Although soil cover sheets are com-monly referred to as vapor retarders, they can also act as a water-proofing membrane when exposed to liquid water.
Control of the slab surface temperature is important in order to minimize the need for mechanical dehumidification, particularly with solar-oriented designs that emphasize the effects of thermal mass. In localities with severe summer surface condensation prob-lems, low-density concrete should be considered for floor slabs to increase their insulating value, or insulation should be added under the slab.
Crawl Spaces Moisture problems generally occur when improper drainage or grading around the house leads to wet soil or even standing water in the crawl space. Evaporation of moisture then causes high humidity in the crawl space and often in the rest of the building. Sometimes the wet soil leads to high moisture content in wood framing mem-bers in the floor and in the band joist (header joist). Any source of subfloor warmth (heating ducts, furnaces) is likely to seriously increase the evaporation from wet subfloor soil (Trethowen and Middlemass 1988, Trethowen 1988). Providing proper drainage of water away from the foundation is critical for moisture control (ASHRAE 1994). Dewatering techniques, including sump pumps, drain tiles, etc., should be used to keep the soil in the crawl space as dry as possible.
Ground covers that restrict evaporation of water from the soil into the crawl space provide an effective way to prevent moisture and humidity problems. It is important to seal any ducts in the crawl space, to avoid venting clothes dryers into the crawl space, and to repair any leaking water pipes. A minimum clearance of 450 mm.
Thermal and Moisture Control in Insulated Assemblies—Applications 24.11 between the crawl space soil and the underside of any wood framing members is recommended and often required. Good access into and around the crawl space is very important.
Whether or not to ventilate a crawl space has been a controversial issue. Most building codes require installation of vents to provide ventilation with outside air, but a symposium on crawl space design concluded that there is no compelling technical basis for crawl space ventilation requirements (ASHRAE 1994). A distinction must be made between conditioned and unconditioned crawl spaces. Conditioned crawl spaces have insulated perimeter walls and may contain plumbing and heating runs. Conditioned crawl spaces should not be ventilated with outdoor air. If air circulation is desired, indoor air should be used. One way to accomplish this is by exhausting indoor air through the insulated crawl space, which may be done in conjunction with an air-to-air heat exchanger for energy efficiency (Samuelson 1994). Unconditioned crawl spaces have an insulated floor above the crawl space. Ventilation with outside air is permitted but not always necessary. Unvented crawl spaces must have a ground cover, which should cover 100% of the crawl space soil. Ground cover treatments for conditioned and unconditioned crawl spaces are similar.
The ground cover material should have a permeance of no more than 57 ng/(s·m2·Pa) and must be rugged enough to withstand foot and knee traffic. The most commonly used material is 0.15 mm polyethylene. The membrane ground cover may be covered with a thin slab of concrete to prevent entry of rodents. Before laying the membrane, all debris must be removed and the soil leveled. Edges need only be lapped 100 to 150 mm, and no sealing is required. The membrane need not be carried up the face of the wall.
If control of entry of radon or other soil gases is desired, a mini-mum 0.15 mm polyethylene ground cover is recommended. Some have recommended that the ground cover should be weighted down and sealed at the perimeter and overlapped to retard radon entry, but others argue that weighting and sealing may lead to water ponding on top of the ground cover. If radon control is not of primary importance, the ground cover may be cut in several low spots to provide drainage should ponding occur. The primary function of the ground cover (i.e., moisture control or radon control) should govern the decision. Example of Residential Foundation Construction Details Figure 4 shows a cross section detail of a typical residential base-ment for mixed climates. Moisture control has been handled in the following ways: • Rain and groundwater. Rain is carried away by gutters, down-spouts, grading away from the building, and a cap of low-per-meance backfill material. Subgrade drainage prevents water from reaching the foundation wall by use of a drain screen (gravel and footer drain connected to daylight, sump, or storm sewer).
• Liquid moisture transport. A dampproof coating is installed on the exterior of the foundation wall and over the top of the footing to control water entry. Capillary moisture movement into the slab is inhibited by a gravel pad 100 mm thick.
• Air movement. All air leakage openings (i.e., floor slab/wall intersection, rim joist area) are caulked and sealed.
• Vapor diffusion. Dampproofing on the wall and polyethylene under the slab inhibit vapor diffusion into the slab and foundation walls. During heating periods, vapor may diffuse from the interior into the rim joist framing, where it may accumulate. To reduce moisture accumulation, the temperature of the rim joist is raised through the installation of exterior insulation. Figure 5 shows an example of a moisture-controlling residential slab-on-grade foundation detail for warm, humid climates, with insulation laid horizontally beneath the perimeter of the floor. Rigid insulation is also placed in the vertical joint between the wall and the slab. Because the rigid insulation can act as a conduit for insects Fig. 4 Example of Residential Basement Construction for Mixed Climates Source: Lstiburek and Carmody (1991). Reprinted with permission.
Fig. 5 Example of Residential Slab-on-Grade Construction in Warm, Humid Climates Source: Lstiburek and Carmody (1991). Reprinted with permission.
24.12 2001 ASHRAE Fundamentals Handbook (SI) into the building, additional protection such as metal flashings or other treatments may be necessary.
• Rain and groundwater. The bottom of the gravel layer is the grade level adjacent to the perimeter. The ground should be graded to direct water away from the building.
• Liquid moisture transport. The granular layer under the slab provides a capillary break between the soil and the slab. This pad can also be integrated into a subslab ventilation system to provide radon mitigation, if needed. Extension of the vapor diffusion retarder over the top of the foundation wall and appropriate flash-ing for the brick facing serve as a capillary break protecting the above-grade wall from ground moisture.
• Air movement and vapor diffusion. The vapor retarder placed under the slab restricts both moist soil gas entry and vapor diffu-sion through the slab. Ductwork located in slabs increases the risk of ground source moisture entering the conditioned space if groundwater and soil gas are permitted to seep into the ducts.
ENVELOPE COMPONENT INTERSECTIONS A moisture control strategy must consider not only envelope components, but also how these components come together. Com-ponent intersections are especially prone to air leakage and thermal bridging and therefore require special care.
Exterior Wall Corners. Mold and mildew often grow in exte-rior corners during heating periods due to cold surfaces caused by (1) thermal bridges, where structural members penetrate the insula-tion and provide a low-resistance heat flow path; (2) wind washing; (3) increased heat loss due to the fin effect; and (4) poor circulation of indoor air. Figure 6 shows heat loss effects at building corners.
Insulating sheathings and two-stud corners help prevent cold inte-rior surfaces and corner moisture problems.
Wall/Window Intersections. Restricting air-transported mois-ture at all potential openings makes a major contribution to the over-all building tightness. The airflow retarder must be continuous.
Figure 7 shows several details that help form a continuous airflow retarder at a window jamb. Wall/Roof Intersections. Exterior wall/ceiling intersections are other common cool spots during the heating season caused by reduced attic insulation at the eaves and wind washing. High-heel trusses that allow installation of more insulation, wind baffles, and rigid insulation exterior sheathing all help control moisture at these locations. Figure 8 shows the heat loss mechanisms at attic/wall intersections and how to minimize the risk of moisture problems. Wall/Floor Intersections. Air leakage at rim joist assemblies is avoided by making sure the airflow retarder is continuous. Caulking and sealing are necessary at all polyethylene seams, as shown in Figure 4. Floor structural members penetrating the insulation can cause thermal bridging, but the use of insulating sheathing helps minimize this (see Figure 4).
Wall/Foundation Intersections. Concrete footings are fre-quently poured directly in contact with the ground, which occasion-ally becomes damp or wet. Concrete used for most residential foundations has the right degree of porosity to provide capillary suc-tion, which draws water into the footings and then into the founda-tion wall. This water usually evaporates into the inside space undetected. However, the moisture occasionally manifests itself as a ring of dampness visible at the bottom interior surface of a base-ment or crawl space wall. Gravel and capillary breaks installed between the footing and the foundation wall are effective moisture control strategies in these below-grade envelope intersections. Sev-eral techniques to control capillary moisture below grade are shown in Figure 4 and Figure 5.
MOISTURE CONTROL IN COMMERCIAL AND INSTITUTIONAL BUILDINGS Moisture control in commercial and institutional buildings often requires approaches different from those in residential buildings.
Indoor humidity conditions can vary greatly from one building to the next, depending on the use and requirements. The building enve-lope should be designed to perform well with these indoor condi-tions. Special thought should be given to the moisture control features of the HVAC equipment and the building envelope for cer-tain buildings with special indoor humidity conditions or require-ments, such as swimming pools, hospitals, and museums.
Fig. 6 Heat Loss at Building Corners Source: Lstiburek and Carmody (1991). Reprinted with permission.
Thermal and Moisture Control in Insulated Assemblies—Applications 24.13 Materials commonly used in commercial and institutional con-struction tend to be more moisture-tolerant and decay-resistant than those used in residential construction. Airflow retarder systems are often poorly designed and executed. As a result, air leakage through the building envelope is common. Air leakage through hollow con-crete masonry is often greater than in other types of construction.
Upward airflow in the cavities is not always adequately blocked, and parallel random leakage paths are found between gypsum wall-board or other finishes and the block face.
Air can leak through exterior walls where the structural system or services penetrate the air barrier or at joints between dissimilar materials or components. For example, masonry cannot form a tight seal with structural steel columns and beams. To reduce this prob-lem, the structural frame should be inside and separate from the exterior wall. The resulting curtain wall can then incorporate a more continuous air barrier and be protected from the fluctuating weather by insulation applied to the outside. Exterior cladding to control rain penetration is best applied following the weather-tightening system.
The interior wythe (masonry course) and air barrier should be acces-sible for maintenance of the air seal and joints.
The deterioration of exterior structural elements of a building and damage to the interior through condensation from air leakage has an important bearing on the operation and maintenance costs of the building. Improving the airtightness of internal floors and parti-tions, particularly in high-rise buildings, redistributes the pressure differences caused by the stack effect and reduces the pressure dif-ference across the exterior wall on each floor. This approach also improves ventilation and air distribution and reduces the air circu-lation between occupancies on different floors. It also helps control smoke movement in the case of fire and may enable a more equita-ble apportioning of energy charges for space heating between indi-vidual units in apartment buildings.
Leakage in actual buildings often occurs through holes cut acci-dentally or deliberately in a reasonably tight membrane or compo-nent, such as the penetration of services through specified air or vapor retarders or solid components. Other leakage openings in the exterior envelope result from dimensional changes in improperly placed materials or from inadequate sealants or membranes applied to bridge joints or cracks that will eventually open.
Fig. 7 Interior Airflow Retarder Details at Window Jamb Source: Lstiburek and Carmody (1991). Reprinted with permission.
Fig. 8 Heat Loss Effect at Ceiling Edge Source: Lstiburek and Carmody (1991). Reprinted with permission.
24.14 2001 ASHRAE Fundamentals Handbook (SI) INDUSTRIAL AND COMMERCIAL INSULATION PRACTICE For pipe materials, selection, application, and installation, see Chapter 41 of the 2000 ASHRAE Handbook—Systems and Equip-ment. For insulation systems for refrigerant piping, see also Chapter 32 of the 1998 ASHRAE Handbook—Refrigeration.
If the equipment is subject to wide changes in temperature, the insulation installation should be designed to accommodate the asso-ciated dimensional change. If the equipment is operated at below ambient temperature, the application should be designed to resist the accumulation of moisture in the insulation.
PIPES Small pipes are insulated with cylindrical half-sections of insu-lation, with factory-applied jackets that form a hinge-and-lap, or with flexible, closed-cell material. Large pipes can be insulated with flexible material or with curved, flat segmented, or cylindrical half-, third-, or quarter-sections of rigid insulation, particularly where removal for frequent servicing of the pipe is necessary. Fittings (valves, tees, crosses, and elbows) are insulated with preformed fit-ting insulation, fabricated fitting insulation, individual pieces cut from sectional straight pipe insulation, or insulating cements. Fit-ting insulations should always be equal in thermal performance to the pipe insulation.
Securing Methods The method of securement varies with the size of pipe, form and weight of the insulation, and the type of jacketing (i.e., separate or factory-applied). Insulation with certain factory-applied jacketing can be secured on small piping by cementing the overlapping jacket.
Large piping may require supplemental wiring or banding. Insula-tion on large piping requiring separate jacketing is wired or banded in place, and the jacket is cemented, wired, or banded, depending on the type. Insulation with factory-applied metal or PVC jacketing is secured by specific design of the jacket and its joint closure. The flexible closed-cell materials require no jacket for most applications and are applied using specially formulated contact adhesives.
Insulation Finish for Above-Ambient Temperatures Pipe insulation finishes for indoor use are usually governed by location. The finishes are factory-applied jackets designed to meet fire safety requirements. For maximum fire safety, unusual expo-sure conditions, or appearance, factory or separately applied metal or PVC jackets may be used. An outdoor finish protects the insula-tion from the weather. Chemical exposure, mechanical abuse, and appearance are additional considerations.
Pipes that operate at temperatures above 260°C may require two layers of insulation in order to accommodate the large dimension change of pipe and insulation materials.
Insulation Finish for Below-Ambient Temperatures Piping at temperatures below ambient is insulated to control heat gain and prevent condensation of moisture from the ambient air. Because piping is an absolute barrier to the passage of water vapor, the outer surface of the insulation must be covered by an impervious membrane or cover, which also helps protect the pipe against corrosion.
Retarder treatment should be recommended by the insulation manufacturer as established by performance testing. The insulation should be as dry as possible, and therefore should be protected from undue weather exposure.
Vapor seals for straight pipe insulation are generally designed to meet operating temperature, fire safety, and appearance require-ments. Jacketing commonly consists of various combinations of laminates of paper, aluminum foil, plastic film, and glass fiber reinforcing. An important feature of such jacketing is very low permeance in a relatively thin layer, which provides flexibility for ease of cementing and sealing laps and end-joint strips. This type of jacketing is commonly used indoors without additional treat-ment. In some cases of operating temperatures below −20°C, mul-tilayer insulation and jacketing may be used. Flexible closed-cell materials must be carefully cemented to avoid openings in the insulations.
Insulation fittings are usually vapor sealed by applying suitable materials in the field, and may vary with the type of insulation and operating temperature. For temperatures above −12°C, the vapor seal can be a lapped spiral wrap of plastic film adhesive tape or a rel-atively thin coat of vapor-seal mastic. For temperatures below −12°C, common practice is to apply two coats of vapor-seal mastic reinforced with open weave glass or other fabric. The thickness of the mastic increases with decreasing temperature. With long lines of piping, the insulation should be sealed off every 5 or 6 m to limit water penetration if vapor seal damage occurs.
For dual-temperature service, where piping is alternately cold and hot, the vapor-seal finish, including mastics, must withstand pipe movement and exposure temperatures without deterioration.
When flexible closed-cell insulation is used, it should be applied slightly compressed to prevent it from being strained when the pip-ing expands.
Outdoor pipe insulation may be vapor-sealed in the same manner as indoor piping, by applying added weather protection jacketing without damage to the retarder and sealing it to keep out water. In some instances, heavy-duty weather and vapor-seal finish may be used.
Because cold piping frequently operates year-round, a constant vapor drive exists under humid conditions. Even with vapor retarder insulation, jackets, and vapor sealing of joints and fittings, moisture inevitably accumulates in permeable insulations. This moisture not only reduces the thermal resistance of the insulation, it also accel-erates condensation on the jacket surface. Since periodic insulation replacement is the only known solution, the piping installation should be accessible for such replacement and should have a means for draining water. As an alternative, very low-permeance insulat-ing materials (e.g., materials not exceeding 0.6 ng/(s· m2· Pa) by the wet cup method) have been used to extend the life of the system and reduce replacement frequency. The lower the permeance of the insulation material, the longer its life, provided good workmanship is practiced during installation.
Surface Temperature In elevated-temperature applications, the surface temperature of the insulation system should be below that at which personnel com-ing into contact with the surface could be harmed. In below-ambient temperature applications, the surface temperature of the system should be above the dew point to prevent condensation.
Compared to a jacket with a nonreflective surface, a jacket with a reflective surface has a higher surface temperature for hot appli-cations and a lower surface temperature for cold-temperature appli-cations, because the lower emissivity reduces the rate of heat exchange. Therefore, adding a reflective jacket could produce a sur-face temperature capable of burning personnel on hot applications and causing condensation on cold applications. The jacketing mate-rial used also contributes to the relative safety at a given surface temperature. For example, at 80°C, a stainless steel jacket blisters skin more severely than a canvas jacket does. Insulating Pipes to Prevent Freezing If the surrounding air temperature remains sufficiently low for an extended period, insulation cannot prevent freezing of still water or of water flowing at a rate insufficient for the available heat content to offset the heat loss. Insulation can only prolong the time required Thermal and Moisture Control in Insulated Assemblies—Applications 24.15 for water to freeze or prevent freezing if water flow is maintained at a sufficient rate. The first section of Table 1 can be used to estimate the thickness of insulation necessary to prevent freezing of still water in pipes. The second section of Table 1 gives the minimum flow of water at an initial temperature of 5.5°C to prevent the tem-perature of the pipe from reaching 0°C at the end of the exposed length.
To calculate time θ (in hours) required for water to cool to 0°C, the following equation can be used: (1) where θ = time for water to cool to freezing, h ρ = density of water = 1000 kg/m3 cp = specific heat of water = 4200 J/(kg·K) Di = inside diameter of pipe, m Dp = outer diameter of pipe or inner diameter of insulation, m DI = outer diameter of insulation, m RT = Rp + RI + Ra = combined thermal resistance of pipe wall, insulation, and exterior air film per metre of pipe, m·K/W Ra = 1/(haπDI) = resistance between ambient air and outer surface of insulation per metre of pipe, m·K/W ha = air heat transfer coefficient (see Chapter 3 for values) RI = ln(DI/Dp)/(2πkI) = resistance of thermal insulation per metre of pipe, m·K/W Rp = ln(Dp/Di)/(2πkp) = resistance of pipe per metre of pipe, m·K/W (Rp ≈ 0 for metal pipe) kI = thermal conductivity of insulation, W/(m·K) kp = thermal conductivity of pipe material, W/(m·K) (see Table 7 in Chapter 41 of the 2000 ASHRAE Handbook—Systems and Equipment for thermal conductivity of various plastic pipes) ta = ambient air temperature, °C ti = initial water temperature, °C tf = freezing temperature, °C When unusual conditions make it impractical to maintain protec-tion with insulation alone, a hot trace pipe or, preferably, electric resistance heating cable is required along the bottom or top of the water pipe. The heating system then supplies the heat lost through the insulation. The insulation thickness is determined by the cost of the heating system, the insulation, and the heat loss.
Pipe bursting is not an immediate consequence of water pipes reaching freezing temperatures. Clean water and pipes usually supercool several degrees below freezing before any ice is formed.
Then, upon nucleation, dendritic ice forms in the water and the tem-perature rises to freezing. Ice can be formed from water only by the release of the latent heat of fusion (334 kJ/kg) through the pipe insu-lation. With well-insulated pipes, this release of latent heat may be greatly retarded. Pipe bursting in water pipes has been shown (Gor-don 1996) to be a consequence not of ice crystal growth in the pipe, but of elevated fluid pressure within a confined pipe section occluded by a growing ice blockage.
Underground Pipe Insulation Both heated and cooled underground piping systems are insu-lated. Protecting underground insulated piping is more difficult than protecting aboveground piping. Groundwater conditions, including chemical or electrolytic contributions by the soil and the existence of water pressure, require a special design to protect insulated pipes from corrosion. Walk-through tunnels, conduits, or integral protec-tive coverings are generally provided to protect the pipe and insula-tion from water. Examples and general design features of conduits and a description of tunnels can be found in Chapter 11 of the 2000 ASHRAE Handbook—Systems and Equipment.
Temperatures Above Ambient. Piping for heated systems in walk-through tunnels is usually covered with sectional insulation and finished with effective mechanical protection such as metal or waterproofing jackets. The use of walk-through tunnels is declining because of cost.
Conduit systems are generally used for underground insulated piping systems. The most successful application is sectional insula-tion with the conduit sized for drainage and adequate drying of insu-lation on heated piping in the event of accidental flooding. BRAB (1975) gives detailed design criteria for conduit systems. The crite-ria require that (1) all systems provide for draining and insulation drying, (2) the insulation withstand boiling and drying without physical damage and loss of insulating value, and (3) the conduit casing is watertight in the field. The insulation should be a noncon-ductor of electricity, verminproof, and chemically and dimension-ally stable at the operating temperature of the pipe.
BRAB (1964) describes evaluative tests and field investigations, which have shown that calcium silicate is resistant to severe boiling action. Fibrous glass (density of 64 to 110 kg/m3) will not withstand boiling when a conduit becomes flooded, and wet poured-in-place insulations are likely to remain partially wet for their installed life.
The thickness of insulation for underground piping is not deter-mined on the same basis as above ground piping. Chapter 11 of the 2000 ASHRAE Handbook—Systems and Equipment provides details for determining thickness.
Temperatures Below Ambient. Integrally protected, insulated piping buried directly in the ground is commonly used for chilled water. However, since no heat is available to drive out moisture, an absolute protective covering against water and insulation with low permeance and water absorption is extremely important. Cellular glass with proper protection has been widely used for this type of application. The acceptance of plastic foams is increasing, but their long-term performance has not yet been established.
Conduit for chilled water piping requires a different approach than for hot piping. Insulation must have low conductivity, and con-duit design must use this low conductivity and maintain continuing performance. More recent designs use low-conductivity plastic foam insulation with plastic pipe as the internal water-carrying pip-ing and as the external conduit.
Where the temperature difference between the pipe at 5°C and the soil at 13 to 16°C is small, pipe size, length, and flow rate may economically justify direct burial without insulation. However, good piping protection may be required.
Table 1 Estimated Requirements to Prevent Freezing of Water in Pipes Steel Pipe Nominal Diameter, mm Insulation Thickness, mm 50 75 100 50 75 100 Time to Cool Water to Freezing, h Water Mass Flow Rate per Unit Length of Exposed Pipe to Prevent Freezing, g/(s·m) 15 0.27 0.32 0.36 0.23 0.19 0.16 25 0.61 0.75 0.85 0.29 0.23 0.20 40 1.16 1.46 1.69 0.37 0.29 0.24 50 1.67 2.13 2.49 0.44 0.33 00.27 80 2.83 3.71 4.42 0.61 0.43 0.35 100 4.07 5.43 6.54 0.77 0.53 0.42 125 5.45 7.36 8.96 0.97 0.64 0.50 150 6.86 9.37 11.5 1.20 0.76 0.58 200 9.59 13.3 16.5 1.79 1.03 0.76 250 12.6 17.6 22.1 2.54 1.32 0.93 300 15.4 21.7 27.4 3.71 1.69 1.14 Design Conditions: Surrounding air temperature ta = −28°C, initial water temperature ti = 5.5°C, and insulation thermal conductivity kI = 0.043 W/(m· K). Thermal resis-tances of pipe and air film at surface of insulation are ignored. Calculations are for 40ST steel pipe. See Table 2 in Chapter 41 of the 2000 ASHRAE Handbook—Systems and Equipment for actual pipe dimensions.
θ ρcpπ 3600 ------------ Di 2 ----- 2 RT α ( ) ti ta – tf ta – -------------ln = 24.16 2001 ASHRAE Fundamentals Handbook (SI) DUCTS The need for duct insulation is influenced by (1) duct location, whether indoors or outdoors; (2) the effect of heat loss or gain on equipment size and operating cost; (3) the need to prevent conden-sation on low-temperature ducts; (4) the need to control temperature change in long duct lengths; and (5) the need to control noise with interior duct lining.
All ducts exposed to outdoor conditions, as well as those passing through unconditioned spaces, should be insulated. While analyses of temperature change, heat loss or gain, and other factors affecting the economics of thermal insulation are seldom made for residential installations, they are essential for large commercial and industrial projects.
The U-factor for uninsulated sheet metal ducts is affected by air velocity, the emittance of the metal, and the shape of the duct. An ap-proximate value of 5.7 W/(m2·K) may be used. For insulated ducts, U-factors of 1.4 and 0.74 W/(m2·K) represent 25 and 50 mm thick rigid insulation with a thermal conductivity of 0.039 W/(m·K) at 24°C mean temperature. A method for determining heat loss or gain for ducts is given in Chapter 34.
Materials for Ducts, Insulations, and Liners Ducts within buildings can be of insulated sheet metal, fibrous glass, or insulated flexible ducts, all of which provide combined air barrier, thermal insulation, and sound absorption. Ducts embedded in or below floor slabs may be of compressed fiber, ceramic tile, or other rigid materials.
Duct insulations include semirigid boards and flexible blanket types, composed of organic and inorganic materials in fibrous, cel-lular, or bonded particle forms. Insulations for exterior surfaces may have attached vapor barriers or facings, or vapor barriers may be applied separately. When applied to the duct interior as a liner, insu-lation both insulates thermally and absorbs sound. Duct liner insu-lations have sound-permeable coatings or other treatment on the side facing the airstream to withstand air velocities or duct cleaning without deterioration.
Per UL Standard 181, fibrous glass air ducts are tested to 63.5 m/s and are rated at 25 m/s and at a pressure of at least 500 Pa. Pri-mary use is for low-pressure systems tested at 1.5 times the recom-mended static pressure. Maximum recommended air temperature is 120°C. A complete system provides greater decibel attenuation than is usually provided by standard duct liners, with greater reduction in airborne equipment noise and crosstalk. Higher design velocities are also possible.
To satisfy most building codes, duct insulation and fibrous glass duct materials must meet the fire hazard requirements of (1) NFPA Standard 90A, to restrict spread of smoke, heat, and fire through duct systems, and to minimize ignition sources; and (2) NFPA Stan-dard 90B, on supply ducts, controls, clearances, heating panels, return ducts, air filters, and heat pumps. Local code authorities should also be consulted.
Where thermally insulated air-conditioning ducts pass through unconditioned spaces, such as attics, the maximum allowable heat flux should be no greater than that required by NFPA Standard 90A.
Securing Methods Exterior rigid duct insulation can be attached with adhesive, with supplemental preattached pins and clips, or with wiring or banding.
Liners can be attached with adhesive and supplemental pins and clips. Flexible duct wraps do not require attachment except on bot-tom duct panels greater than 600 mm wide. For larger ducts a pin no more than 600 mm on center is sufficient.
Manufacturers provide information on the construction of fibrous glass duct systems. Preformed round duct for straight runs is combined with fittings fabricated from straight duct. Rectangular ducts and fittings are fabricated by grooving, folding, and taping.
Metal accessories such as turning vanes, splitters, and dampers are incorporated into the system. When rectangular ducts exceed pre-determined dimensions for particular static pressures, ductwork must be reinforced. The Sheet Metal and Air Conditioning Contrac-tors National Association’s (SMACNA) Fibrous Glass Duct Con-struction Standards (1992) have further information.
Heating Ducts The effect of duct insulation on residential heating system equip-ment size can be marginal. However, insulation can reduce operat-ing costs significantly, depending on unit costs for heating and the extent of duct exposed to outside conditions. In addition, duct insu-lation maintains the supply air temperature, which may keep the air entering the conditioned space within a more comfortable range.
Vapor retarders are not required on exterior insulation of ducts used only for heating, but they must be provided for ducts used for alternate heating and cooling.
Cooling Ducts Insulation can significantly reduce operating costs and cooling equipment size. The advantage of adequate insulation is especially significant in areas with high dry-bulb and dew-point temperatures.
Ducts for summer air conditioning are insulated with the same materials as heating ducts. Ducts in any unconditioned space should be insulated and have vapor retarders to prevent condensation.
Joints and laps in the vapor retarder must be sealed. Flexible closed-cell insulation does not always need a supplemental vapor retarder, but care must be taken to form vapor-tight seams at joints.
REFERENCES ARMA 1997. Ventilation and moisture control for residential roofing. Tech-nical Bulletin 209-RR-86. Asphalt Roofing Manufacturers Association, Calverton, MD.
ASHRAE. 1994. Recommended practices for controlling moisture in crawl spaces. Technical Data Bulletin 10(3).
ASHRAE. 1999. Ventilation for acceptable indoor air quality. ANSI/ ASHRAE Standard 62-1999.
Baker, M.C. 1980. Roofs, Chapters 6 and 8. Multiscience Publications, Mon-treal, Canada.
Barbour, E.J. Goodrow, J. Kosny, and J.E. Christian. 1994 thermal perfor-mance of steel-framed walls. Prepared for American Iron and Steel Insti-tute by NAHB Research Center.
Beal, D. and S. Chandra. 1995. The measured summer performance of tile roof systems and attic ventilation strategies in hot, humid climates. In: Thermal Performance of the Exterior Envelopes of Buildings VI, pp.
753-760. ASHRAE.
BRAB. 1964. Field evaluation of underground heat distribution systems.
Technical Report 47. Building Research Advisory Board, Federal Con-struction Council, National Academy of Sciences, National Research Council, Washington, D.C.
BRAB. 1975. Criteria for underground distribution. Technical Report 66.
Standing Committee on Mechanical Engineering of the FCC, BRAB, and NRC.
BRANZ. 1983. Insulating suspended timber frame floors. Bulletins 177 and 233. Building Research Association, New Zealand.
Britton, R.R. 1948. Condensation in walls and roofs. Technical Papers No.
1, 2, 3, and 8. Housing and Home Finance Agency, Washington, D.C.
Burch, D.M. and S.J. Treado. 1978. Ventilating residences and their attics for energy conservation—An experimental study. In: Summer attic and whole-house ventilation. Special Publication 548. National Institute of Standards and Technology, Gaithersburg, MD.
Buska, J., W. Tobiasson, and A. Greatorex. 1998. Roof ventilation to prevent problematic icings at eaves. ASHRAE Transactions 104(2).
Christian, J.E. 1994. Moisture sources. In: Moisture Control in Buildings, pp. 176-82. ASTM Manual 18. American Society for Testing and Mate-rials, West Conshohocken, PA.
Cunningham, M.J., G.A. Tsongas, and D. McQuade. 1990. Solar-driven moisture transfer through absorbent roofing materials. ASHRAE Trans-actions 96(2):465-71.
Thermal and Moisture Control in Insulated Assemblies—Applications 24.17 Desjarlais, A.O. 1995. Self-drying roofs: What! No dripping! In: Proceed-ings, Thermal Performance of the Exterior Envelopes of Buildings VI, pp. 763-73. ASHRAE.
DOE. 1991. Radiant barrier attic fact sheet. DOE/CE-0335P. U.S. Dept. of Energy.
Duff, J.E. 1956. The effect of air conditioning of water vapor permeability on temperature and humidity. ASHRAE Transactions 62:437.
Fugler, D. 1999. Conclusions from ten years of Canadian attic research.
ASHRAE Transactions 105(1).
Goldberg, L., P. Huelman, and B. Bridges. 1999. A preliminary experimen-tal assessment of the comparative thermal performance of attics and cathedral ceilings in a cold climate. ASHRAE Transactions 105(1).
Gordon, J. 1996. An investigation into freezing and bursting water pipes in residential construction. Research Report 96-1. University of Illinois Building Research Council.
Hedlin, C.P. 1977. Moisture gains by foam plastic roof insulations under controlled temperature gradients. Journal of Cellular Plastics 13(5).
NRCC 16317.
Hens, H, and A. Janssens. 1999. Heat and moisture response of vented and compact cathedral ceilings: A test house evaluation. ASHRAE Transac-tions 105(1).
Jordan, C.A., E.C. Peck, F.A. Strange, and L.V. Teesdale. 1948. Attic con-densation in tightly built houses. Housing and Home Finance Agency Technical Bulletin No 6.
Korsgaard, V. and C.R. Pedersen. 1992. Laboratory and practical experience with a novel water-permeable vapor retarder. In: Thermal Performance of the Exterior Envelopes of Buildings V, pp. 480-90. ASHRAE.
Kyle, D.M. and A.O. Desjarlais. 1994. Assessment of technologies for con-structing self-drying low-slope roofs. Report ORNL/CON-380. Oak Ridge National Laboratory, Oak Ridge, TN.
Labs, K., J. Carmody, R. Sterling, L. Shen, Y.J. Huang, and D. Parker. 1988.
Building foundation design handbook. Report ORNL/sub/86-72143/1.
Oak Ridge National Laboratory.
Lecompte, J. 1990. Influence of natural convection in an insulated cavity on the thermal performance of a wall. Insulation Materials, Testing and Applications, pp. 397-420. ASTM STP 1030. American Society for Test-ing and Materials, West Conshohocken, PA.
Lstiburek, J. and J. Carmody. 1991. The moisture control handbook—New low-rise, residential construction. ORNL/Sub/89-SD350/1. Martin Marietta Energy Systems, Oak Ridge National Laboratory, TN.
NFPA. 1993. Standard for the installation of air conditioning and ventilating systems. Standard 90A-93, National Fire Protection Association, Quincy, MA.
NFPA. 1993. Standard for the installation of warm air heating and air con-ditioning systems. Standard 90B-93. National Fire Protection Associa-tion, Quincy, MA.
NRC 1963. Humidified buildings. Canadian Building Digest 42.
Powell, F.J. and H.E. Robinson. 1971. The effect of moisture on the heat transfer performance of insulated flat-roof constructions. Building Sci-ence Series 37. National Institute of Standards and Technology, Gaith-ersburg, MD.
Rose, W. 1992. Measured values of temperature and sheathing content in residential attic assemblies. In: Thermal Performance of the Exterior Envelopes of Buildings V, pp. 379-90. ASHRAE.
Rose, W.B. 1995. Attic construction with sheathing-applied insulation.
ASHRAE Transactions 101(2):789-98.
Rowley, F.B., A.B. Algren, and C.E. Lund. 1939. Condensation of moisture and its relation to building construction and operation. ASHVE Transac-tions 45:231-52.
Rudd, A.F. and J.W. Lstiburek. 1998. Vented and sealed attics in hot cli-mates. ASHRAE Transactions 104(2):1199-1210.
Samuelson, I. 1994. Moisture control in crawl spaces. In: Recommended practices for controlling moisture in crawl spaces. ASHRAE Technical Data Bulletin 10(3):pp. 58-64.
Sanders, C. 1996. Final report: Volume 2, Task 2: Environmental conditions.
International Energy Agency Annex 24: Heat, Air and Moisture Transfer Through New and Retrofitted Insulated Envelope Parts. K.U. Leuven, Belgium.
SMACNA. 1992. Fibrous glass duct construction standards, 6th ed. Sheet Metal and Air Conditioning Contractor’s National Association, Chan-tilly, VA.
TenWolde, A. 1988. A mathematical model for indoor humidity in homes during winter. In: Proceedings of symposium on air infiltration, ventila-tion, and moisture transfer, pp. 3-32. BTECC-National Institute for Building Science, Washington D.C.
TenWolde, A. and C. Carll. 1992. Effect of cavity ventilation on moisture in walls and roofs. In: Thermal Performance of the Exterior Envelopes of Buildings V, pp. 555-62. ASHRAE.
TenWolde, A. 1994. Ventilation, humidity, and condensation in manufac-tured houses during winter. ASHRAE Transactions 100(1):103-15.
TenWolde, A. and W. Rose. 1999. Issues related to venting of attics and cathedral ceilings. ASHRAE Transactions 105(1).
Tobiasson, W. and M. Harrington. 1985. Vapor drive maps of the U.S. Ther-mal Performance of the Exterior Envelopes of Buildings III, pp. 663-72.
Tobiasson, W., J. Buska, and A. Greatorex. 1994. Ventilating attics to mini-mize icing at eaves. Energy and Buildings 20:229-34.
Trethowen, H.A. 1988. A survey of subfloor ground evaporation rates. Study Report 13. Building Research Association of New Zealand.
Trethowen, H.A. 1994. Three surveys of subfloor moisture in New Zealand.
ASHRAE Transactions 100(1):1427-38.
Trethowen, H.A. and G. Middlemass. 1988. A survey of moisture damage in southern New Zealand buildings. Study Report 7. Building Research Association of New Zealand.
Tuluca, A. and R. Gorthala. 1999. Thermal performance of cold-formed steel ceiling/roof framing assemblies. ASHRAE Research Project 981.
Verschoor, J.D. 1977. Effectiveness of building insulation applications.
USN/CEL Report No. CR78.006-NTIS No. AD-A053 452/9ST.
BIBLIOGRAPHY IEA. 1991. Vol. 2, Guidelines and practice. Annex XIV, Condensation and Energy. International Energy Agency, Leuven, Belgium.
Trechsel, H.R., ed. 1994. Moisture control in buildings. Manual 18. Amer-ican Society for Testing and Materials, West Conshohocken, PA.
25.1 CHAPTER 25 THERMAL AND WATER VAPOR TRANSMISSION DATA Building Envelopes ....................................................................................................................... 25.1 Calculating Overall Thermal Resistances ................................................................................... 25.2 Mechanical and Industrial Systems ........................................................................................... 25.15 Calculating Heat Flow for Buried Pipelines ............................................................................. 25.20 HIS CHAPTER presents thermal and water vapor transmission Tdata based on steady-state or equilibrium conditions. Chapter 3 covers heat transfer under transient or changing temperature condi-tions. Chapter 23 discusses selection of insulation materials and procedures for determining overall thermal resistances by simpli-fied methods.
BUILDING ENVELOPES Thermal Transmission Data for Building Components The steady-state thermal resistances (R-values) of building com-ponents (walls, floors, windows, roof systems, etc.) can be calcu-lated from the thermal properties of the materials in the component; or the heat flow through the assembled component can be measured directly with laboratory equipment such as the guarded hot box (ASTM Standard C 236) or the calibrated hot box (ASTM Standard C 976).
Tables 1 through 6 list thermal values, which may be used to cal-culate thermal resistances of building walls, floors, and ceilings.
The values shown in these tables were developed under ideal con-ditions. In practice, overall thermal performance can be reduced sig-nificantly by such factors as improper installation and shrinkage, settling, or compression of the insulation (Tye and Desjarlais 1983; Tye 1985, 1986).
Most values in these tables were obtained by accepted ASTM test methods described in ASTM Standards C 177 and C 518 for materials and ASTM Standards C 236 and C 976 for building enve-lope components. Because commercially available materials vary, not all values apply to specific products.
The most accurate method of determining the overall thermal resistance for a combination of building materials assembled as a building envelope component is to test a representative sample by a hot box method. However, all combinations may not be conve-niently or economically tested in this manner. For many simple con-structions, calculated R-values agree reasonably well with values determined by hot box measurement.
The performance of materials fabricated in the field is especially subject to the quality of workmanship during construction and installation. Good workmanship becomes increasingly important as the insulation requirement becomes greater. Therefore, some engi-neers include additional insulation or other safety factors based on experience in their design.
Figure 1 shows how convection affects surface conductance of several materials. Other tests on smooth surfaces show that the aver-age value of the convection part of the surface conductance decreases as the length of the surface increases.
Vapor retarders, which are discussed in Chapters 23 and 24, require special attention. Moisture from condensation or other sources may reduce the thermal resistance of insulation, but the effect of moisture must be determined for each material. For exam-ple, some materials with large air spaces are not affected signifi-cantly if the moisture content is less than 10% by mass, while the effect of moisture on other materials is approximately linear.
Ideal conditions of components and installations are assumed in calculating overall R-values (i.e., insulating materials are of uni-form nominal thickness and thermal resistance, air spaces are of uniform thickness and surface temperature, moisture effects are not involved, and installation details are in accordance with design).
The National Institute of Standards and Technology Building Mate-rials and Structures Report BMS 151 shows that measured values differ from calculated values for certain insulated constructions. For this reason, some engineers decrease the calculated R-values a moderate amount to account for departures of constructions from requirements and practices.
Tables 3 and 2 give values for well-sealed systems constructed with care. Field applications can differ substantially from laboratory test conditions. Air gaps in these insulation systems can seriously degrade thermal performance as a result of air movement due to both natural and forced convection. Sabine et al. (1975) found that the tabular values are not necessarily additive for multiple-layer, low-emittance air spaces, and tests on actual constructions should be conducted to accurately determine thermal resistance values.
The preparation of this chapter is assigned to TC 4.4, Thermal Insulation and Moisture Retarders.
Fig. 1 Surface Conductance for Different Surfaces as Affected by Air Movement 25.2 2001 ASHRAE Fundamentals Handbook (SI) Values for foil insulation products supplied by manufacturers must also be used with caution because they apply only to sys-tems that are identical to the configuration in which the product was tested. In addition, surface oxidation, dust accumulation, condensation, and other factors that change the condition of the low-emittance surface can reduce the thermal effectiveness of these insulation systems (Hooper and Moroz 1952). Deteriora-tion results from contact with several types of solutions, either acidic or basic (e.g., wet cement mortar or the preservatives found in decay-resistant lumber). Polluted environments may cause rapid and severe material degradation. However, site inspections show a predominance of well-preserved installations and only a small number of cases in which rapid and severe dete-rioration has occurred. An extensive review of the reflective building insulation system performance literature is provided by Goss and Miller (1989).
CALCULATING OVERALL THERMAL RESISTANCES Relatively small, highly conductive elements in an insulating layer called thermal bridges can substantially reduce the average thermal resistance of a component. Examples include wood and metal studs in frame walls, concrete webs in concrete masonry walls, and metal ties or other elements in insulated wall panels.
The following examples illustrate the calculation of R-values and U-factors for components containing thermal bridges.
The following conditions are assumed in calculating the design R-values: • Equilibrium or steady-state heat transfer, disregarding effects of thermal storage • Surrounding surfaces at ambient air temperature • Exterior wind velocity of 6.7 m/s (24 km/h) for winter (surface with R = 0.03 m2·K/W) and 3.4 m/s (12 km/h) for summer (surface with R = 0.044 m2·K/W) • Surface emittance of ordinary building materials is 0.90 Wood Frame Walls The average overall R-values and U-factors of wood frame walls can be calculated by assuming either parallel heat flow paths through areas with different thermal resistances or by assuming isothermal planes. Equations (1) through (5) from Chapter 23 are used.
The framing factor or fraction of the building component that is framing depends on the specific type of construction, and it may vary based on local construction practices—even for the same type of construction. For stud walls 400 mm on center (OC), the fraction of insulated cavity may be as low as 0.75, where the fraction of studs, plates, and sills is 0.21 and the fraction of headers is 0.04. For studs 600 mm OC, the respective values are 0.78, 0.18, and 0.04.
Table 1 Surface Conductances and Resistances for Air Position of Surface Direction of Heat Flow Surface Emittance, ε Non-reflective Reflective ε = 0.90 ε = 0.20 ε = 0.05 hi R hi R hi R STILL AIR Horizontal Upward 9.26 0.11 5.17 0.19 4.32 0.23 Sloping—45° Upward 9.09 0.11 5.00 0.20 4.15 0.24 Vertical Horizontal 8.29 0.12 4.20 0.24 3.35 0.30 Sloping—45° Downward 7.50 0.13 3.41 0.29 2.56 0.39 Horizontal Downward 6.13 0.16 2.10 0.48 1.25 0.80 MOVING AIR (Any position) ho R Wind (for winter) Any 34.0 0.030 — — — — 6.7 m/s (24 km/h) Wind (for summer) Any 22.7 0.044 — — — — 3.4 m/s (12 km/h) Notes: 1. Surface conductance hi and ho measured in W/(m2· K); resistance R in m2·K/W.
2. No surface has both an air space resistance value and a surface resistance value.
3. For ventilated attics or spaces above ceilings under summer conditions (heat flow down), see Table 5.
4. Conductances are for surfaces of the stated emittance facing virtual blackbody sur-roundings at the same temperature as the ambient air. Values are based on a surface-air temperature difference of 5.5 K and for surface temperatures of 21°C.
5. See Chapter 3 for more detailed information, especially Tables 5 and 6, and see Fig-ure 1 for additional data.
6. Condensate can have a significant impact on surface emittance (see Table 2).
Table 2 Emittance Values of Various Surfaces and Effective Emittances of Air Spacesa Surface Average Emittance ε Effective Emittance εeff of Air Space One Surface Emittance ε; Other, 0.9 Both Surfaces Emittance ε Aluminum foil, bright 0.05 0.05 0.03 Aluminum foil, with condensate just visible (> 0.5 g/m2) 0.30b 0.29 — Aluminum foil, with condensate clearly visible (> 2.0 g/m2) 0.70b 0.65 — Aluminum sheet 0.12 0.12 0.06 Aluminum coated paper, polished 0.20 0.20 0.11 Steel, galvanized, bright 0.25 0.24 0.15 Aluminum paint 0.50 0.47 0.35 Building materials: wood, paper, masonry, nonmetallic paints 0.90 0.82 0.82 Regular glass 0.84 0.77 0.72 aThese values apply in the 4 to 40 µm range of the electromagnetic spectrum.
bValues are based on data presented by Bassett and Trethowen (1984).
Fig. 2 Insulated Wood Frame Wall (Example 1) Thermal and Water Vapor Transmission Data 25.3 These fractions contain an allowance for multiple studs, plates, sills, extra framing around windows, headers, and band joists. These assumed framing fractions are used in the following example, to illustrate the importance of including the effect of framing in deter-mining the overall thermal conductance of a building. The actual framing fraction should be calculated for each specific construction.
Example 1. Calculate the U-factor of the 38 mm by 90 mm stud wall shown in Figure 2. The studs are at 400 mm OC. There is 90 mm min-eral fiber batt insulation (R = 2.3 m2·K/W) in the stud space. The inside finish is 13 mm gypsum wallboard; the outside is finished with rigid foam insulating sheathing (R = 0.7 m2·K/W) and 13 mm by 200 mm wood bevel lapped siding. The insulated cavity occupies approximately 75% of the transmission area; the studs, plates, and sills occupy 21%; and the headers occupy 4%.
Solution: Obtain the R-values of the various building elements from Tables 1 and 4. Assume R = 7.0 m2·K/W for the wood framing. Also, assume the headers are solid wood, in this case, and group them with the studs, plates, and sills.
Because the U-factor is the reciprocal of R-value, U1 = 0.297 W/(m2·K) and U2 = 0.588 W/(m2·K).
If the wood framing (thermal bridging) is not included, Equation (3) from Chapter 23 may be used to calculate the U-factor of the wall as follows: If the wood framing is accounted for using the parallel-path flow method, the U-factor of the wall is determined using Equation (5) from Chapter 23 as follows: If the wood framing is included using the isothermal planes method, the U-factor of the wall is determined using Equations (2) and (3) from Chapter 23 as follows: For a frame wall with a 600 mm OC stud space, the average overall R-value is 0.25 m2·K/W. Similar calculation procedures may be used to evaluate other wall designs, except those with thermal bridges.
Masonry Walls The average overall R-values of masonry walls can be estimated by assuming a combination of layers in series, one or more of which provides parallel paths. This method is used because heat flows lat-erally through block face shells so that transverse isothermal planes result. Average total resistance RT(av) is the sum of the resistances of the layers between such planes, each layer calculated as shown in Example 2.
Example 2. Calculate the overall thermal resistance and average U-factor of the 194 mm thick insulated concrete block wall shown in Figure 3.
The two-core block has an average web thickness of 25 mm and a face shell thickness of 30 mm. Overall block dimensions are 194 mm by 194 mm by 395 mm. Measured thermal resistances of 1700 kg/m3 concrete and 110 kg/m3 expanded perlite insulation are 0.70 and 20 K·m2/2, respectively.
Solution: The equation used to determine the overall thermal resis-tance of the insulated concrete block wall is derived from Equations (2) and (5) from Chapter 23 and is given below: where RT(av) = overall thermal resistance based on assumption of isothermal planes Ri = thermal resistance of inside air surface film (still air) Ro = thermal resistance of outside air surface film (24 km/h wind) Rf = total thermal resistance of face shells Rc = thermal resistance of cores between face shells Rw = thermal resistance of webs between face shells aw = fraction of total area transverse to heat flow represented by webs of blocks ac = fraction of total area transverse to heat flow represented by cores of blocks From the information given and the data in Table 1, determine the val-ues needed to compute the overall thermal resistance.
Ri = 0.12 Ro = 0.03 Rf = 2 × 0.032 × 0.70 = 0.045 Rc = (0.194 −2 × 0.032)(20) = 2.60 Rw = (0.194 −2 × 0.032)(0.70) = 0.091 aw = 3 × 25/395 = 0.190 ac = 1 −0.190 = 0.810 Using the equation given, the overall thermal resistance and average U-factor are calculated as follows: Based on guarded hot box tests of this wall without mortar joints, Tye and Spinney (1980) measured the average R-value for this insu-lated concrete block wall as 0.551 m2·K/W.
Element R (Insulated Cavity) R (Studs, Plates, and Headers) 1. Outside surface, 24 km/h wind 0.03 0.03 2. Wood bevel lapped siding 0.14 0.14 3. Rigid foam insulating sheathing 0.70 0.70 4. Mineral fiber batt insulation 2.30 — 5. Wood stud — 0.63 6. Gypsum wallboard 0.08 0.08 7. Inside surface, still air 0.12 0.12 3.37 1.70 Uav U1 1 R1 ------0.30 W/ m2 K ⋅ ( ) = = = Uav 0.75 0.297 × ( ) 0.25 0.588 × ( ) + 0.37 W m2 K ⋅ ( ⁄ = = T av ( ) 4.98 1 0.75 2.30 ⁄ ( ) 0.25 0.63 ⁄ ( ) + [ ] ⁄ 0.2 + + = 2.47 K m2 W ⁄ ⋅ = Uav 0.40 W m2 K ⋅ ( ) ⁄ = Fig. 3 Insulated Concrete Block Wall (Example 2) RT av ( ) Ri Rf aw Rw ------ac Rc -----+ 1 – Ro + + + = RT av ( ) 0.12 0.045 0.091 2.60 × ( ) 0.810 0.91 × ( ) 0.190 2.60 × ( ) + ------------------------------------------------------------------------------0.03 + + + = 0.612 K m2/W ⋅ = Uav 1 0.612 ⁄ 1.63 W m2 K ⋅ ( ) ⁄ = = 25.4 2001 ASHRAE Fundamentals Handbook (SI) Table 3 Thermal Resistances of Plane Air Spacesa,b,c, K·m2/W Position of Air Space Direction of Heat Flow Air Space 13 mm Air Spacec 20 mm Air Spacec Mean Temp.d, °C Temp.
Diff.d, °C Effective Emittance εeff d,e Effective Emittance εeff d,e 0.03 0.05 0.2 0.5 0.82 0.03 0.05 0.2 0.5 0.82 Horiz.
Up 32.2 5.6 0.37 0.36 0.27 0.17 0.13 0.41 0.39 0.28 0.18 0.13 10.0 16.7 0.29 0.28 0.23 0.17 0.13 0.30 0.29 0.24 0.17 0.14 10.0 5.6 0.37 0.36 0.28 0.20 0.15 0.40 0.39 0.30 0.20 0.15 − 17.8 11.1 0.30 0.30 0.26 0.20 0.16 0.32 0.32 0.27 0.20 0.16 − 17.8 5.6 0.37 0.36 0.30 0.22 0.18 0.39 0.38 0.31 0.23 0.18 − 45.6 11.1 0.30 0.29 0.26 0.22 0.18 0.31 0.31 0.27 0.22 0.19 − 45.6 5.6 0.36 0.35 0.31 0.25 0.20 0.38 0.37 0.32 0.26 0.21 45° Slope Up 32.2 5.6 0.43 0.41 0.29 0.19 0.13 0.52 0.49 0.33 0.20 0.14 10.0 16.7 0.36 0.35 0.27 0.19 0.15 0.35 0.34 0.27 0.19 0.14 10.0 5.6 0.45 0.43 0.32 0.21 0.16 0.51 0.48 0.35 0.23 0.17 − 17.8 11.1 0.39 0.38 0.31 0.23 0.18 0.37 0.36 0.30 0.23 0.18 − 17.8 5.6 0.46 0.45 0.36 0.25 0.19 0.48 0.46 0.37 0.26 0.20 − 45.6 11.1 0.37 0.36 0.31 0.25 0.21 0.36 0.35 0.31 0.25 0.20 − 45.6 5.6 0.46 0.45 0.38 0.29 0.23 0.45 0.43 0.37 0.29 0.23 Vertical Horiz.
32.2 5.6 0.43 0.41 0.29 0.19 0.14 0.62 0.57 0.37 0.21 0.15 10.0 16.7 0.45 0.43 0.32 0.22 0.16 0.51 0.49 0.35 0.23 0.17 10.0 5.6 0.47 0.45 0.33 0.22 0.16 0.65 0.61 0.41 0.25 0.18 − 17.8 11.1 0.50 0.48 0.38 0.26 0.20 0.55 0.53 0.41 0.28 0.21 − 17.8 5.6 0.52 0.50 0.39 0.27 0.20 0.66 0.63 0.46 0.30 0.22 − 45.6 11.1 0.51 0.50 0.41 0.31 0.24 0.51 0.50 0.42 0.31 0.24 − 45.6 5.6 0.56 0.55 0.45 0.33 0.26 0.65 0.63 0.51 0.36 0.27 45° Slope Down 32.2 5.6 0.44 0.41 0.29 0.19 0.14 0.62 0.58 0.37 0.21 0.15 10.0 16.7 0.46 0.44 0.33 0.22 0.16 0.60 0.57 0.39 0.24 0.17 10.0 5.6 0.47 0.45 0.33 0.22 0.16 0.67 0.63 0.42 0.26 0.18 − 17.8 11.1 0.51 0.49 0.39 0.27 0.20 0.66 0.63 0.46 0.30 0.22 − 17.8 5.6 0.52 0.50 0.39 0.27 0.20 0.73 0.69 0.49 0.32 0.23 − 45.6 11.1 0.56 0.54 0.44 0.33 0.25 0.67 0.64 0.51 0.36 0.28 − 45.6 5.6 0.57 0.56 0.45 0.33 0.26 0.77 0.74 0.57 0.39 0.29 Horiz.
Down 32.2 5.6 0.44 0.41 0.29 0.19 0.14 0.62 0.58 0.37 0.21 0.15 10.0 16.7 0.47 0.45 0.33 0.22 0.16 0.66 0.62 0.42 0.25 0.18 10.0 5.6 0.47 0.45 0.33 0.22 0.16 0.68 0.63 0.42 0.26 0.18 − 17.8 11.1 0.52 0.50 0.39 0.27 0.20 0.74 0.70 0.50 0.32 0.23 − 17.8 5.6 0.52 0.50 0.39 0.27 0.20 0.75 0.71 0.51 0.32 0.23 − 45.6 11.1 0.57 0.55 0.45 0.33 0.26 0.81 0.78 0.59 0.40 0.30 − 45.6 5.6 0.58 0.56 0.46 0.33 0.26 0.83 0.79 0.60 0.40 0.30 Air Space 40 mm Air Spacec 90 mm Air Spacec Horiz.
Up 32.2 5.6 0.45 0.42 0.30 0.19 0.14 0.50 0.47 0.32 0.20 0.14 10.0 16.7 0.33 0.32 0.26 0.18 0.14 0.27 0.35 0.28 0.19 0.15 10.0 5.6 0.44 0.42 0.32 0.21 0.16 0.49 0.47 0.34 0.23 0.16 − 17.8 11.1 0.35 0.34 0.29 0.22 0.17 0.40 0.38 0.32 0.23 0.18 − 17.8 5.6 0.43 0.41 0.33 0.24 0.19 0.48 0.46 0.36 0.26 0.20 − 45.6 11.1 0.34 0.34 0.30 0.24 0.20 0.39 0.38 0.33 0.26 0.21 − 45.6 5.6 0.42 0.41 0.35 0.27 0.22 0.47 0.45 0.38 0.29 0.23 45° Slope Up 32.2 5.6 0.51 0.48 0.33 0.20 0.14 0.56 0.52 0.35 0.21 0.14 10.0 16.7 0.38 0.36 0.28 0.20 0.15 0.40 0.38 0.29 0.20 0.15 10.0 5.6 0.51 0.48 0.35 0.23 0.17 0.55 0.52 0.37 0.24 0.17 − 17.8 11.1 0.40 0.39 0.32 0.24 0.18 0.43 0.41 0.33 0.24 0.19 − 17.8 5.6 0.49 0.47 0.37 0.26 0.20 0.52 0.51 0.39 0.27 0.20 − 45.6 11.1 0.39 0.38 0.33 0.26 0.21 0.41 0.40 0.35 0.27 0.22 − 45.6 5.6 0.48 0.46 0.39 0.30 0.24 0.51 0.49 0.41 0.31 0.24 Vertical Horiz.
32.2 5.6 0.70 0.64 0.40 0.22 0.15 0.65 0.60 0.38 0.22 0.15 10.0 16.7 0.45 0.43 0.32 0.22 0.16 0.47 0.45 0.33 0.22 0.16 10.0 5.6 0.67 0.62 0.42 0.26 0.18 0.64 0.60 0.41 0.25 0.18 − 17.8 11.1 0.49 0.47 0.37 0.26 0.20 0.51 0.49 0.38 0.27 0.20 − 17.8 5.6 0.62 0.59 0.44 0.29 0.22 0.61 0.59 0.44 0.29 0.22 − 45.6 11.1 0.46 0.45 0.38 0.29 0.23 0.50 0.48 0.40 0.30 0.24 − 45.6 5.6 0.58 0.56 0.46 0.34 0.26 0.60 0.58 0.47 0.34 0.26 45° Slope Down 32.2 5.6 0.89 0.80 0.45 0.24 0.16 0.85 0.76 0.44 0.24 0.16 10.0 16.7 0.63 0.59 0.41 0.25 0.18 0.62 0.58 0.40 0.25 0.18 10.0 5.6 0.90 0.82 0.50 0.28 0.19 0.83 0.77 0.48 0.28 0.19 − 17.8 11.1 0.68 0.64 0.47 0.31 0.22 0.67 0.64 0.47 0.31 0.22 − 17.8 5.6 0.87 0.81 0.56 0.34 0.24 0.81 0.76 0.53 0.33 0.24 − 45.6 11.1 0.64 0.62 0.49 0.35 0.27 0.66 0.64 0.51 0.36 0.28 − 45.6 5.6 0.82 0.79 0.60 0.40 0.30 0.79 0.76 0.58 0.40 0.30 Horiz.
Down 32.2 5.6 1.07 0.94 0.49 0.25 0.17 1.77 1.44 0.60 0.28 0.18 10.0 16.7 1.10 0.99 0.56 0.30 0.20 1.69 1.44 0.68 0.33 0.21 10.0 5.6 1.16 1.04 0.58 0.30 0.20 1.96 1.63 0.72 0.34 0.22 − 17.8 11.1 1.24 1.13 0.69 0.39 0.26 1.92 1.68 0.86 0.43 0.29 − 17.8 5.6 1.29 1.17 0.70 0.39 0.27 2.11 1.82 0.89 0.44 0.29 − 45.6 11.1 1.36 1.27 0.84 0.50 0.35 2.05 1.85 1.06 0.57 0.38 − 45.6 5.6 1.42 1.32 0.86 0.51 0.35 2.28 2.03 1.12 0.59 0.39 aSee Chapter 23, section Factors Affecting Heat Transfer Across Air Spaces. Thermal resistance values were determined from the relation, R = 1/C, where C = hc + εeff hr, hc is the conduction-convection coefficient, εeff hr is the radiation coefficient ≈ 0.227εeff [(tm + 273)/100]3, and tm is the mean temperature of the air space. Values for hc were determined from data developed by Robinson et al. (1954). Equations (5) through (7) in Yarbrough (1983) show the data in this table in analytic form. For extrapolation from this table to air spaces less than 12.5 mm (as in insulating window glass), assume hc = 21.8(1 + 0.00274 tm)/l where l is the air space thickness in mm, and hc is heat transfer in W/(m2·K) through the air space only.
bValues are based on data presented by Robinson et al. (1954). (Also see Chapter 3, Tables 3 and 4, and Chapter 38). Values apply for ideal conditions (i.e., air spaces of uniform thickness bounded by plane, smooth, parallel surfaces with no air leakage to or from the space). When accurate values are required, use overall U-factors deter-mined through calibrated hot box (ASTM C 976) or guarded hot box (ASTM C 236) testing. Thermal resistance values for multiple air spaces must be based on careful estimates of mean temperature differences for each air space.
cA single resistance value cannot account for multiple air spaces; each air space requires a separate resistance calculation that applies only for the established bound-ary conditions. Resistances of horizontal spaces with heat flow downward are sub-stantially independent of temperature difference.
dInterpolation is permissible for other values of mean temperature, temperature differ-ence, and effective emittance εeff . Interpolation and moderate extrapolation for air spaces greater than 90 mm are also permissible.
eEffective emittance εeff of the air space is given by 1/εeff = 1/ε1 + 1/ε2 −1, where ε1 and ε2 are the emittances of the surfaces of the air space (see Table 2).
Thermal and Water Vapor Transmission Data 25.5 Table 4 Typical Thermal Properties of Common Building and Insulating Materials—Design Valuesa Description Density, kg/m3 Conductivityb (k), W/(m·K) Conductance (C), W/(m2·K) Resistancec (R) Specific Heat, kJ/(kg·K) 1/k, K·m/W For Thickness Listed (1/C), K·m2/W BUILDING BOARD Asbestos-cement board ..................................................
1900 0.58 — 1.73 — 1.00 Asbestos-cement board ......................................3.2 mm 1900 — 187.4 — 0.005 — Asbestos-cement board ......................................6.4 mm 1900 — 93.7 — 0.011 — Gypsum or plaster board....................................9.5 mm 800 — 17.6 — 0.056 1.09 Gypsum or plaster board..................................12.7 mm 800 — 12.6 — 0.079 — Gypsum or plaster board..................................15.9 mm 800 — 10.1 — 0.099 — Plywood (Douglas fir)d..................................................
540 0.12 — 8.66 — 1.21 Plywood (Douglas fir)........................................6.4 mm 540 — 18.2 — 0.055 — Plywood (Douglas fir)........................................9.5 mm 540 — 12.1 — 0.083 — Plywood (Douglas fir)......................................12.7 mm 540 — 9.1 — 0.11 — Plywood (Douglas fir)......................................15.9 mm 540 — 7.3 — 0.14 — Plywood or wood panels..................................19.0 mm 540 — 6.1 — 0.16 1.21 Vegetable fiber board Sheathing, regular densitye .......................12.7 mm 290 — 4.3 — 0.23 1.30 ..................................................................19.8 mm 290 — 2.8 — 0.36 — Sheathing intermediate densitye................12.7 mm 350 — 5.2 — 0.19 1.30 Nail-base sheathinge .................................12.7 mm 400 — 5.3 — 0.19 1.30 Shingle backer.............................................9.5 mm 290 — 6.0 — 0.17 1.30 Shingle backer.............................................7.9 mm 290 — 7.3 — 0.14 — Sound deadening board.............................12.7 mm 240 — 4.2 — 0.24 1.26 Tile and lay-in panels, plain or acoustic .................
290 0.058 — 17.
— 0.59 ....12.7 mm 290 — 4.5 — 0.22 — ....19.0 mm 290 — 3.0 — 0.33 — Laminated paperboard ..................................
480 0.072 — 13.9 — 1.38 Homogeneous board from repulped paper....
480 0.072 — 13.9 — 1.17 Hardboarde Medium density .......................................................
800 0.105 — 9.50 — 1.30 High density, service-tempered grade and service grade......................................................................
880 0.82 — 8.46 — 1.34 High density, standard-tempered grade....................
1010 0.144 — 6.93 — 1.34 Particleboarde Low density..............................................................
590 0.102 — 9.77 — 1.30 Medium density .......................................................
800 0.135 — 7.35 — 1.30 High density.............................................................
1000 0.170 — 5.90 — 1.30 Underlayment.............................................15.9 mm 640 — 6.9 — 0.14 1.21 Waferboard.....................................................................
590 0.01 — 11.0 — — Wood subfloor..................................................19.0 mm — — 6.0 — 0.17 1.38 BUILDING MEMBRANE Vapor—permeable felt...................................................
— — 94.9 — 0.011 Vapor—seal, 2 layers of mopped 0.73 kg/m2 felt..........
— — 47.4 — 0.21 Vapor—seal, plastic film................................................
— — — — Negl.
FINISH FLOORING MATERIALS Carpet and fibrous pad...................................................
— — 2.73 — 0.37 1.42 Carpet and rubber pad....................................................
— — 4.60 — 0.22 1.38 Cork tile .............................................................3.2 mm — — 20.4 — 0.049 2.01 Terrazzo...............................................................25 mm — — 71.0 — 0.014 0.80 Tile—asphalt, linoleum, vinyl, rubber...........................
— — 113.6 — 0.009 1.26 vinyl asbestos...........................................................
1.01 ceramic.....................................................................
0.80 Wood, hardwood finish.......................................19 mm — 8.35 — 0.12 INSULATING MATERIALS Blanket and Battf,g Mineral fiber, fibrous form processed from rock, slag, or glass approx. 75-100 mm............................................
6.4-32 — 0.52 — 1.94 approx. 90 mm ...................................................
6.4-32 — 0.44 — 2.29 approx. 90 mm ...................................................
19-26 — 0.38 — 2.63 approx. 140-165 mm..........................................
6.4-32 — 0.30 — 3.32 approx. 140 mm .................................................
10-16 — 0.27 — 3.67 approx. 150-190 mm..........................................
6.4-32 — 0.26 — 3.91 approx. 210-250 mm..........................................
6.4-32 — 0.19 — 5.34 approx. 250-330 mm..........................................
6.4-32 — 0.15 — 6.77 Board and Slabs Cellular glass..................................................................
136 0.050 — 19.8 — 0.75 Glass fiber, organic bonded............................................
64-140 0.036 — 27.7 — 0.96 Expanded perlite, organic bonded..................................
16 0.052 — 19.3 — 1.26 Expanded rubber (rigid).................................................
72 0.032 — 31.6 — 1.68 Expanded polystyrene, extruded (smooth skin surface) (CFC-12 exp.) .............................................................
29-56 25.6 2001 ASHRAE Fundamentals Handbook (SI) Expanded polystyrene, extruded (smooth skin surface) (HCFC-142b exp.)h.....................................................
29-56 0.029 — 34.7 — 1.21 Expanded polystyrene, molded beads............................
16 0.037 — 26.7 — — 20 0.036 — 27.7 — — 24 0.035 — 28.9 — — 28 0.035 — 28.9 — — 32 0.033 — 30.2 — — Cellular polyurethane/polyisocyanuratei (CFC-11 exp.) (unfaced).............................................
24 0.023-0.026 — 43.3-38.5 — 1.59 Cellular polyisocyanuratei (CFC-11 exp.) (gas-permeable facers)................................................
24-40 0.023-0.026 — 43.3-38.5 — 0.92 Cellular polyisocyanuratej (CFC-11 exp.) (gas-impermeable facers)............................................
32 0.020 — 48.8 — 0.92 Cellular phenolic (closed cell) (CFC-11, CFC-113 exp.)k 32 0.017 — 56.8 — — Cellular phenolic (open cell)....................................
29-35 0.033 — 30.5 — — Mineral fiber with resin binder ................................
240 0.042 — 23.9 — 0.71 Mineral fiberboard, wet felted Core or roof insulation.............................................
260-270 0.049 — 20.4 — Acoustical tile ..........................................................
290 0.050 — 19.8 — 0.80 Acoustical tile ..........................................................
340 0.053 — 18.7 — Mineral fiberboard, wet molded Acoustical tilel .........................................................
370 0.060 — 16.5 — 0.59 Wood or cane fiberboard Acoustical tilel ...........................................12.7 mm — — 4.5 — 0.22 1.30 Acoustical tilel ...........................................19.0 mm — — 3.0 — 0.33 Interior finish (plank, tile)...................................
240 0.050 — 19.8 — 1.34 Cement fiber slabs (shredded wood with Portland cement binder) ............................................................
400-430 0.072-0.076 — 13.9-13.1 — — Cement fiber slabs (shredded wood with magnesia oxysulfide binder).......................................................
350 0.082 — 12.1 — 1.30 Loose Fill Cellulosic insulation (milled paper or wood pulp) ........
37-51 0.039-0.046 — 25.6-21.7 — 1.38 Perlite, expanded............................................................
32-66 0.039-0.045 — 25.6-22.9 — 1.09 66-120 0.045-0.052 — 22.9-19.4 — — 120-180 0.052-0.060 — 19.4-16.6 — — Mineral fiber (rock, slag, or glass)g approx. 95-130 mm..................................................
9.6-32 — — — 1.94 0.71 approx. 170-220 mm................................................
9.6-32 — — — 3.35 — approx. 190-250 mm................................................
9.6-32 — — — 3.87 — approx. 260-350 mm................................................
9.6-32 — — — 5.28 — Mineral fiber (rock, slag, or glass)g approx. 90 mm (closed sidewall application) ..........
32-56 — — — 2.1-2.5 Vermiculite, exfoliated...................................................
110-130 0.068 — 14.8 — 1.34 64-96 0.063 — 15.7 — — Spray Applied Polyurethane foam .........................................................
24-40 0.023-0.026 — 43.3-38.5 — — Ureaformaldehyde foam ................................................
11-26 0.032-0.040 — 31.5-24.7 — — Cellulosic fiber...............................................................
56-96 0.042-0.049 — 23.9-20.4 — — Glass fiber ......................................................................
56-72 0.038-0.039 — 26.7-25.6 — — Reflective Insulation Reflective material (ε < 0.5) in center of 20 mm cavity forms two 10 mm vertical air spacesm........................
— — 1.76 — 0.57 — METALS (See Chapter 38, Table 3) ROOFING Asbestos-cement shingles..............................................
1900 — 27.0 — 0.037 1.00 Asphalt roll roofing........................................................
1100 — 36.9 — 0.026 1.51 Asphalt shingles.............................................................
1100 — 12.9 — 0.077 1.26 Built-up roofing ..................................................10 mm 1100 — 17.0 — 0.058 1.46 Slate ....................................................................13 mm — — 114 — 0.009 1.26 Wood shingles, plain and plastic film faced ..................
— — 6.0 — 0.166 1.30 PLASTERING MATERIALS Cement plaster, sand aggregate......................................
1860 0.72 — 1.39 — 0.84 Sand aggregate..............................................10 mm — — 75.5 — 0.013 0.84 Sand aggregate..............................................20 mm — — 37.8 — 0.026 0.84 Table 4 Typical Thermal Properties of Common Building and Insulating Materials—Design Valuesa (Continued) Description Density, kg/m3 Conductivityb (k), W/(m·K) Conductance (C), W/(m2·K) Resistancec (R) Specific Heat, kJ/(kg·K) 1/k, K·m/W For Thickness Listed (1/C), K·m2/W Thermal and Water Vapor Transmission Data 25.7 Gypsum plaster: Lightweight aggregate ..................................13 mm 720 — 17.7 — 0.056 — Lightweight aggregate ..................................16 mm 720 — 15.2 — 0.066 — Lightweight aggregate on metal lath.............19 mm — — 12.1 — 0.083 — Perlite aggregate.............................................................
720 0.22 — 4.64 — 1.34 Sand aggregate.........................................................
1680 0.81 — 1.25 — 0.84 Sand aggregate..............................................13 mm 1680 — 63.0 — 0.016 — Sand aggregate..............................................16 mm 1680 — 51.7 — 0.019 — Sand aggregate on metal lath ........................19 mm — — 43.7 — 0.023 — Vermiculite aggregate ..............................................
720 0.24 — 4.09 — — MASONRY MATERIALS Masonry Units Brick, fired clay .............................................................
2400 1.21-1.47 — 0.83-0.68 — — 2240 1.07-1.30 — 0.94-0.77 — — 2080 0.92-1.12 — 1.08-0.89 — — 1920 0.81-0.98 — 1.24-1.02 — 0.79 1760 0.71-0.85 — 1.42-1.18 — — 1600 0.61-0.74 — 1.65-1.36 — — 1440 —0.52-0.62 — 1.93-1.61 — — 1280 0.43-0.53 — 2.31-1.87 — — 1120 0.36-0.45 — 2.77-2.23 — — Clay tile, hollow 1 cell deep ........................................................75 mm — — 7.10 — 0.14 0.88 1 cell deep ......................................................100 mm — — 5.11 — 0.20 — 2 cells deep.....................................................150 mm — — 3.75 — 0.27 — 2 cells deep.....................................................200 mm — — 3.07 — 0.33 — 2 cells deep.....................................................250 mm — — 2.56 — 0.39 — 3 cells deep.....................................................300 mm — — 2.27 — 0.44 — Concrete blocksn, o Limestone aggregate 200 mm, 16.3 kg, 2210 kg/m3 concrete, 2 cores......
— — — — — — Same with perlite filled cores ...............................
— — 2.73 — 0.37 — 300 mm, 25 kg, 2210 kg/m3 concrete, 2 cores.........
— — — — — — Same with perlite filled cores ...............................
— — 1.53 — 0.65 — Normal mass aggregate (sand and gravel) 200 mm 15-16 kg, 2020-2180 kg/m3 concrete, 2 or 3 cores — — 5.1-5.8 — 0.20-0.17 0.92 Same with perlite filled cores ...............................
— — 2.84 — 0.35 — Same with vermiculite filled cores .......................
— — 3.0-4.1 — 0.34-0.24 — 300 mm, 22.7 kg, 2000 kg/m3 concrete, 2 cores......
— — 4.60 — 0.217 0.92 Medium mass aggregate (combinations of normal and low mass aggregate) 200 mm, 12-13 kg, 1550-1790 kg/m3 concrete, 2 or 3 cores ..................
— — 3.3-4.4 — 0.30-0.22 — Same with perlite filled cores ...............................
— — 1.5-2.5 — 0.65-0.41 — Same with vermiculite filled cores .......................
— — 1.70 — 0.58 — Same with molded EPS (beads) filled cores.........
— — 1.82 — 0.56 — Same with molded EPS inserts in cores................
— — 2.10 — 0.47 — Low mass aggregate (expanded shale, clay, slate or slag, pumice) 150 mm 7.3-7.7 kg, 1360-1390 kg/m3 concrete, 2 or 3 cores — — 3.0-3.5 — 0.34-0.29 — Same with perlite filled cores ...............................
— — 1.36 — 0.74 — Same with vermiculite filled cores .......................
— — 1.87 — 0.53 — 200 mm, 8.6-10.0 mm, 1150-1380 kg/m3 concrete, — — 1.8-3.1 — 0.56-0.33 0.88 Same with perlite filled cores ...............................
— — 0.9-1.3 — 1.20-0.77 — Same with vermiculite filled cores .......................
— — 1.1-1.5 — 0.93-0.69 — Same with molded EPS (beads) filled cores.........
— — 1.19 — 0.85 — Same with UF foam filled cores ...........................
— — 1.25 — 0.79 — Same with molded EPS inserts in cores................
— — 1.65 — 0.62 — 300 mm, 14.5-16.3 kg, 1280-1440 kg/m3 concrete, 2 or 3 cores............................................................
— — 2.2-2.5 — 0.46-0.40 — Same with perlite filled cores ...............................
— — 0.6-0.9 — 1.6-1.1 — Same with vermiculite filled cores .......................
— — 0.97 — 1.0 — Stone, lime, or sand Quartzitic and sandstone.........................................
2880 10.4 — 0.10 — — 2560 6.2 — 0.16 — — 2240 3.5 — 0.29 — — 1920 1.9 — 0.53 — 0.79 Calcitic, dolomitic, limestone, marble, and granite ....
2880 4.3 — 0.23 — — 2560 3.2 — 0.32 — — 2240 2.3 — 0.43 — — 1920 1.6 — 0.63 — 0.79 1600 1.1 — 0.90 — — Table 4 Typical Thermal Properties of Common Building and Insulating Materials—Design Valuesa (Continued) Description Density, kg/m3 Conductivityb (k), W/(m·K) Conductance (C), W/(m2·K) Resistancec (R) Specific Heat, kJ/(kg·K) 1/k, K·m/W For Thickness Listed (1/C), K·m2/W 25.8 2001 ASHRAE Fundamentals Handbook (SI) Gypsum partition tile 75 by 300 by 760 mm, solid.....................................
— — 4.50 — 0.222 0.79 75 by 300 by 760 mm, 4 cells..................................
— — 4.20 — 0.238 — 100 by 300 by 760 mm, 3 cells ................................
— — 3.40 — 0.294 — Concreteso Sand and gravel or stone aggregate concretes (concretes with more than 50% quartz or quartzite sand have conductivities in the higher end of the range).............
2400 1.4-2.9 — 0.69-0.35 — — 2240 1.3-2.6 — 0.77-0.39 — 0.8-1.0 2080 1.0-1.9 — 0.99-053 — — Limestone concretes ......................................................
2240 1.60 — 0.62 — — 1920 1.14 — 0.88 — — 1600 0.79 — 1.26 — — Gypsum-fiber concrete (87.5% gypsum, 12.5% wood chips) 816 0.24 — 4.18 — 0.88 Cement/lime, mortar, and stucco ...................................
1920 1.40 — 0.71 — — 1600 0.97 — 1.04 — — 1280 0.65 — 1.54 — — Lightweight aggregate concretes Expanded shale, clay, or slate; expanded slags; cinders; pumice (with density up to 1600 kg/m3); and scoria (sanded concretes have conductivities in the higher end of the range) ........................................
1920 0.9-1.3 — 1.08-0.76 — — 1600 0.68-0.89 — 1.48-1.12 — 0.84 1280 0.48-0.59 — 2.10-1.69 — 0.84 960 0.30-0.36 — 3.30-2.77 — — 640 0.18 — 5.40 — — Perlite, vermiculite, and polystyrene beads ................
800 0.26-0.27 — 3.81-3.68 — 640 0.20-0.22 — 4.92-4.65 — 0.63-0.96 480 0.16 — 6.31 — — 320 0.12 — 8.67 — — Foam concretes ..............................................................
1920 0.75 — 1.32 — — 1600 0.60 — 1.66 — — 1280 0.44 — 2.29 — — 1120 0.36 — 2.77 — — Foam concretes and cellular concretes ..........................
960 0.30 — 3.33 — — 640 0.20 — 4.92 — — 320 0.12 — 8.67 — — SIDING MATERIALS (on flat surface) Shingles Asbestos-cement .........................................................
1900 — 27.0 — 0.037 — Wood, 400 mm, 190 mm exposure .............................
— — 6.53 — 0.15 1.30 Wood, double, 400 mm, 300 mm exposure ................
— — 4.77 — 0.21 1.17 Wood, plus insul. backer board, 8 mm........................
— — 4.03 — 0.25 1.30 Siding Asbestos-cement, 6.4 mm, lapped ..............................
— — 27.0 — 0.037 1.01 Asphalt roll siding.......................................................
— — 36.9 — 0.026 1.47 Asphalt insulating siding (12.7 mm bed.)...................
— — 3.92 — 0.26 1.47 Hardboard siding, 11 mm ...........................................
— — 8.46 — 0.12 1.17 Wood, drop, 20 by 200 mm.........................................
— — 7.21 — 0.14 1.17 Wood, bevel, 13 by 200 mm, lapped...........................
— — 6.98 — 0.14 1.17 Wood, bevel, 19 by 250 mm, lapped...........................
— — 5.40 — 0.18 1.17 Wood, plywood, 9.5 mm, lapped ................................
— — 9.60 — 0.10 1.22 Aluminum, steel, or vinylp, q, over sheathing Hollow-backed.........................................................
— — 9.31 — 0.11 1.22q Insulating-board backed...........................................
9.5 mm nominal .......................................................
— — 3.12 — 0.32 1.34 9.5 mm nominal, foil backed ...................................
— — 1.93 — 0.52 — Architectural (soda-lime float) glass..............................
— — 56.8 — 0.018 0.84 WOODS (12% moisture content)e,r Hardwoods 1.63s Oak..............................................................................
659-749 0.16-0.18 — 6.2-5.5 — Birch............................................................................
682-726 0.167-0.176 — 6.0-5.7 — Maple ..........................................................................
637-704 0.157-0.171 — 6.4-5.8 — Ash..............................................................................
614-670 0.153-0.164 — 6.5-6.1 — Softwoods 1.63s Southern pine ..............................................................
570-659 0.144-0.161 — 6.9-6.2 — Douglas fir-Larch........................................................
536-581 0.137-0.145 — 7.3-6.9 — Southern cypress .........................................................
502-514 0.130-0.132 — 7.7-7.6 — Hem-Fir, Spruce-Pine-Fir ...........................................
392-502 0.107-0.130 — 9.3-7.7 — West coast woods, Cedars...........................................
347-502 0.098-0.130 — 10.3-7.7 — California redwood .....................................................
392-448 0.107-0.118 — 9.4-8.5 — Table 4 Typical Thermal Properties of Common Building and Insulating Materials—Design Valuesa (Continued) Description Density, kg/m3 Conductivityb (k), W/(m·K) Conductance (C), W/(m2·K) Resistancec (R) Specific Heat, kJ/(kg·K) 1/k, K·m/W For Thickness Listed (1/C), K·m2/W Thermal and Water Vapor Transmission Data 25.9 Assuming parallel heat flow only, the calculated resistance is higher than that calculated on the assumption of isothermal planes.
The actual resistance generally is some value between the two cal-culated values. In the absence of test values, examination of the con-struction usually reveals whether a value closer to the higher or lower calculated R-value should be used. Generally, if the construc-tion contains a layer in which lateral conduction is high compared with transmittance through the construction, the calculation with isothermal planes should be used. If the construction has no layer of high lateral conductance, the parallel heat flow calculation should be used.
Hot box tests of insulated and uninsulated masonry walls con-structed with block of conventional configuration show that thermal resistances calculated using the isothermal planes heat flow method agree well with measured values (Van Geem 1985, Valore 1980, Shu et al. 1979). Neglecting horizontal mortar joints in conventional block can result in thermal transmittance values up to 16% lower than actual, depending on the density and thermal properties of the masonry, and 1 to 6% lower, depending on the core insulation mate-rial (Van Geem 1985, McIntyre 1984). For aerated concrete block walls, other solid masonry, and multicore block walls with full mor-tar joints, neglecting mortar joints can cause errors in R-values up to 40% (Valore 1988). Horizontal mortar joints usually found in con-crete block wall construction are neglected in Example 2.
Constructions Containing Metal Curtain and metal stud-wall constructions often include metallic and other thermal bridges, which can significantly reduce the ther-mal resistance. However, the capacity of the adjacent facing mate-rials to transmit heat transversely to the metal is limited, and some contact resistance between all materials in contact limits the reduc-tion. Contact resistances in building structures are only 0.01 to 0.1 K·m2/W—too small to be of concern in many cases. However, the contact resistances of steel framing members may be important.
Also, in many cases (as illustrated in Example 3), the area of metal in contact with the facing greatly exceeds the thickness of the metal, which mitigates the contact reistance effects.
Thermal characteristics for panels of sandwich construction can be computed by combining the thermal resistances of the various layers. R-values for the assembled sections should be determined on a representative sample by using a hot box method. If the sample is a wall section with air cavities on both sides of fibrous insulation, the sample must be of representative height since convective air-flow can contribute significantly to heat flow through the test sec-tion. Computer modeling can also be useful, but all heat transfer mechanisms must be considered.
In Example 3, the metal member is only 0.5 mm thick, but it is in contact with adjacent facings over a 32 mm-wide area. The steel member is 90 mm deep, has a thermal resistance of approximately 0.0019 K·m2/W, and is virtually isothermal. The calculation involves careful selection of the appropriate thickness for the steel member. If the member is assumed to be 0.5 mm thick, the fact that the flange transmits heat to the adjacent facing is ignored, and the heat flow through the steel is underestimated. If the member is assumed to be 32 mm thick, the heat flow through the steel is over-estimated. In Example 3, the steel member behaves in much the same way as a rectangular member 32 mm thick and 90 mm deep Notes for Table 4 a Values are for a mean temperature of 24°C. Representative values for dry materials are intended as design (not specification) values for materials in normal use. Thermal values of insulating materials may differ from design values depending on their in-situ properties (e.g., density and moisture content, orientation, etc.) and variability experienced during manufacture.
For properties of a particular product, use the value supplied by the manu-facturer or by unbiased tests.
b The symbol λ is also used to represent thermal conductivity.
c Resistance values are the reciprocals of C before rounding off C to two decimal places.
d Lewis (1967).
e U.S. Department of Agriculture (1974).
f Does not include paper backing and facing, if any. Where insulation forms a boundary (reflective or otherwise) of an airspace, see Tables 2 and 3 for the insulating value of an airspace with the appropriate effective emittance and temperature conditions of the space.
g Conductivity varies with fiber diameter. (See Chapter 23, Factors Affect-ing Thermal Performance.) Batt, blanket, and loose-fill mineral fiber insu-lations are manufactured to achieve specified R-values, the most common of which are listed in the table. Due to differences in manufacturing pro-cesses and materials, the product thicknesses, densities, and thermal con-ductivities vary over considerable ranges for a specified R-value.
h This material is relatively new and data are based on limited testing.
i For additional information, see Society of Plastics Engineers (SPI) Bulle-tin U108. Values are for aged, unfaced board stock. For change in conduc-tivity with age of expanded polyurethane/polyisocyanurate, see Chapter 23, Factors Affecting Thermal Performance.
j Values are for aged products with gas-impermeable facers on the two major surfaces. An aluminum foil facer of 25 µm thickness or greater is generally considered impermeable to gases. For change in conductivity with age of expanded polyisocyanurate, see Chapter 23, Factors Affecting Thermal Performance, and SPI Bulletin U108.
k Cellular phenolic insulation may no longer be manufactured. The thermal conductivity and resistance values do not represent aged insulation, which may have a higher thermal conductivity and lower thermal resistance.
l Insulating values of acoustical tile vary, depending on density of the board and on type, size, and depth of perforations.
mCavity is framed with 20 mm wood furring strips. Caution should be used in applying this value for other framing materials. The reported value was derived from tests and applies to the reflective path only. The effect of studs or furring strips must be included in determining the overall perfor-mance of the wall.
n Values for fully grouted block may be approximated using values for con-crete with a similar unit density.
o Values for concrete block and concrete are at moisture contents represen-tative of normal use.
p Values for metal or vinyl siding applied over flat surfaces vary widely, depending on amount of ventilation of airspace beneath the siding; whether airspace is reflective or nonreflective; and on thickness, type, and application of insulating backing-board used. Values are averages for use as design guides, and were obtained from several guarded hot box tests (ASTM C 236) or calibrated hot box (ASTM C 976) on hollow-backed types and types made using backing of wood fiber, foamed plastic, and glass fiber. Departures of ±50% or more from these values may occur.
q Vinyl specific heat = 1.0 kJ/(kg·K) r See Adams (1971), MacLean (1941), and Wilkes (1979). The conductivity values listed are for heat transfer across the grain. The thermal conductiv-ity of wood varies linearly with the density, and the density ranges listed are those normally found for the wood species given. If the density of the wood species is not known, use the mean conductivity value. For extrapo-lation to other moisture contents, the following empirical equation devel-oped by Wilkes (1979) may be used: where ρ is density of the moist wood in kg/m3, and M is the moisture con-tent in percent.
s From Wilkes (1979), an empirical equation for the specific heat of moist wood at 24°C is as follows: where ∆cp accounts for the heat of sorption and is denoted by where M is the moisture content in percent by mass.
k 0.7494 4.895 10 3 – × 1.503 + 10 4 – M × ( )ρ 1 0.01M + ------------------------------------------------------------------------------------+ = cp 0.1442 0.299 0.01M + ( ) 1 0.01M + ( ) ----------------------------------------cp ∆ + = cp ∆ M 0.008037 1.325 – 10 4 – M × ( ) = 25.10 2001 ASHRAE Fundamentals Handbook (SI) with a thermal resistance of 0.0019 (32/0.5) = 0.12 K·m2/W does.
The Building Research Association of New Zealand (BRANZ) commonly uses this approximation.
Example 3. Calculate the C-factor of the insulated steel frame wall shown in Figure 4. Assume that the steel member has an R-value of 0.12 K·m2/W and that the framing behaves as though it occupies approxi-mately 8% of the transmission area.
Solution. Obtain the R-values of the various building elements from Table 4.
Therefore, C1 = 0.476; C2 = 3.57 W/(m2·K).
If the steel framing (thermal bridging) is not considered, the C-factor of the wall is calculated using Equation (3) from Chapter 23 as follows: If the steel framing is accounted for using the parallel flow method, the C-factor of the wall is determined using Equation (5) from Chapter 23 as follows: If the steel framing is included using the isothermal planes method, the C-factor of the wall is determined using Equations (2) and (3) from Chapter 23 as follows: For this insulated steel frame wall, Farouk and Larson (1983) mea-sured an average R-value of 1.16 m2·K/W.
In ASHRAE/IESNA Standard 90.1-1989, one method given for determining the thermal resistance of wall assemblies containing metal framing involves using a parallel path correction factor Fc, which is listed in Table 8C-2 of the standard. For 38 mm by 90 mm steel framing, 400 mm OC, Fc = 0.50. Using the correction factor method, an R-value of 1.13 m2·K/W (0.08 + 1.94 × 0.50 + 0.08) is obtained for the wall described in Example 3.
Zone Method of Calculation For structures with widely spaced metal members of substantial cross-sectional area, calculation by the isothermal planes method can result in thermal resistance values that are too low. For these constructions, the zone method can be used. This method involves two separate computations—one for a chosen limited portion, Zone A, containing the highly conductive element; the other for the remaining portion of simpler construction, Zone B. The two com-putations are then combined using the parallel flow method, and the average transmittance per unit overall area is calculated. The basic laws of heat transfer are applied by adding the area conductances CA of elements in parallel, and adding area resistances R/A of ele-ments in series.
The surface shape of Zone A is determined by the metal element.
For a metal beam (see Figure 5), the Zone A surface is a strip of width W that is centered on the beam. For a rod perpendicular to panel surfaces, it is a circle of diameter W. The value of W is calcu-lated from Equation (1), which is empirical. The value of d should not be less than 13 mm for still air.
(1) where m = width or diameter of metal heat path terminal, mm d = distance from panel surface to metal, mm Generally, the value of W should be calculated using Equation (1) for each end of the metal heat path; the larger value, within the limits of the basic area, should be used as illustrated in Example 4.
Example 4. Calculate transmittance of the roof deck shown in Figure 5.
Tee-bars at 600 mm OC support glass fiber form boards, gypsum con-crete, and built-up roofing. Conductivities of components are: steel, 45 W/(m·K); gypsum concrete, 0.24 W/(m·K); and glass fiber form board, 0.036 W/(m·K). Conductance of built-up roofing is 17 W/(m·K).
Element R (Insul.) R (Framing) 1. 13 mm gypsum wallboard 0.08 0.08 2. 90 mm mineral fiber batt insulation 1.94 — 3. Steel framing member — 0.12 4. 13 mm gypsum wallboard 0.08 0.08 R1 = 2.10 R2 = 0.28 Fig. 4 Insulated Steel Frame Wall (Example 3) Cav C1 1 R1 ⁄ 0.476 W/ m2 K ⋅ ( ) = = = Cav 0.92 0.476 × ( ) 0.08 3.57 × ( ) + = 0.724 W m2 K ⋅ ( ) ⁄ = RT av ( ) 1.38 m2 K ⋅ W ⁄ = RT av ( ) 0.08 1 0.92 ( ) 1.94 ( ) 0.08 0.12 ⁄ ( ) + ----------------------------------------------------------------------0.08 + + = 1.037 m2 K W ⁄ ⋅ = Cav 0.96 W/(m2 K) ⋅ = Fig. 5 Gypsum Roof Deck on Bulb Tees (Example 4) W m 2d + = Thermal and Water Vapor Transmission Data 25.11 Solution: The basic area is 0.6 m2 with a tee-bar across the middle.
This area is divided into Zones A and B.
Zone A is determined from Equation (1) as follows: Top side W = m + 2d = 15 + (2 × 40) = 95 mm Bottom side W = m + 2d = 50 + (2 × 13) = 76 mm Using the larger value of W, the area of Zone A is (1.0 × 95/1000) = 0.095 m2. The area of Zone B is 0.600 −0.095 = 0.505 m2.
To determine area transmittance for Zone A, divide the structure within the zone into five sections parallel to the top and bottom sur-faces (Figure 5). The area conductance CA of each section is calculated by adding the area conductances of its metal and nonmetal paths. Area conductances of the sections are converted to area resistances R/A and added to obtain the total resistance of Zone A.
Area transmittance of Zone A = 1/(R/A) = 1/3.59 = 0.279.
For Zone B, the unit resistances are added and then converted to area transmittance, as shown in the following table.
Because unit transmittance = 1/R = 0.927, the total area transmit-tance UA is calculated as follows: Overall R-values of 0.805 and 0.854 m2·K/W have been measured in two guarded hot box tests of a similar construction.
When the steel member represents a relatively large proportion of the total heat flow path, as in Example 4, detailed calculations of resistance in sections 3, 4, and 5 of Zone A are unnecessary; if only the steel member is considered, the final result of Example 4 is the same. However, if the heat flow path represented by the steel mem-ber is small, as for a tie rod, detailed calculations for sections 3, 4, and 5 are necessary. A panel with an internal metallic structure and bonded on one or both sides to a metal skin or covering presents spe-cial problems of lateral heat flow not covered in the zone method.
Modified Zone Method for Metal Stud Walls with Insulated Cavities The modified zone method is similar to the parallel path method and the zone method. All three methods are based on parallel-path calculations. Figure 6 shows the width w of the zone of thermal anomalies around a metal stud. This zone can be assumed to equal the length of the stud flange L (parallel path method), or can be cal-culated as a sum of the length of stud flange and a distance double that from wall surface to metal Σdi (zone method). In the modified zone method the width of the zone depends on the following three parameters: • Ratio between thermal resistivity of sheathing material and cavity insulation • Size (depth) of stud • Thickness of sheathing material The modified zone method is explained in Figure 6 (which can be copied and used as a calculation form). The wall cross section Section Area × Conductance = CA 1 = R CA A Air (outside, 24 km/h) 0.095 × 34 3.23 0.31 No. 1, Roofing 0.095 × 17 1.62 0.62 No. 2, Gypsum concrete 0.095 × 0.24/0.030 0.76 1.32 No. 3, Steel 0.015 × 45/0.015 45 } 0.022 No. 3, Gypsum concrete 0.080 × 0.24/0.015 1.28 No. 4, Steel 0.003 × 45/0.025 5.4 } 0.181 No. 4, Glass fiberboard 0.092 × 0.036/0.025 0.13 No. 5, Steel 0.050 × 45/0.005 450 0.002 Air (inside) 0.095 × 9.26 0.88 1.14 Total R/A = 3.59 Section Resistance, R Air (outside, 24 km/h) 1/34 = 0.029 Roofing 1/17 = 0.059 Gypsum concrete 0.045/0.24 = 0.188 Glass fiberboard 0.025/0.036= 0.694 Air (inside) 1/9.26 = 0.108 Total resistance = 1.078 Zone B = 0.505 × 0.927 = 0.468 Zone A = 0.279 Total area transmittance of basic area = 0.747 Transmittance = 0.747 W/(m2·K) Resistance = 0.80 K·m2/W Fig. 6 Modified Zone Method R-Value Calculation Form for Metal Stud Walls 25.12 2001 ASHRAE Fundamentals Handbook (SI) shown in Figure 6, is divided into two zones: the zone of thermal anomalies around metal stud w and the cavity zone cav. Wall mate-rial layers are grouped into an exterior and interior surface sec-tions—A (sheathing, siding) and B (wallboard)—and interstitial sections I and II (cavity insulation, metal stud flange).
Assuming that the layers or layer of wall materials in wall section A are thicker than those in wall section B, as show by the cross sec-tion in Figure 6, they can be described as follows: (2) where n = number of material layer (of thickness di) between metal stud flange and wall surface for section A m = number of material layer (of thickness dj) for section B Then, the width of the zone of thermal anomalies around the metal stud w can be estimated by (3) where L = stud flange size di = thickness of material layer in section A zf = zone factor, which is shown in Figure 7 (zf = 2 for zone method) Kosny and Christian (1995) verified the accuracy of the modi-fied zone method for over 200 simulated cases of metal frame walls with insulated cavities. For all configurations considered the dis-crepancy between results were within ±2%. Hot box measured R-values for 15 metal stud walls tested by Barbour et al. (1994) were compared with results obtained by Kosny and Christian (1995) and McGowan and Desjarlais (1997). The modified zone method was found to be the most accurate simple method for estimating the clear wall R-value of light-gage steel stud walls with insulated cav-ities. However, this analysis does not apply to construction with metal sheathing. Also, ASHRAE Standard 90.1 may require a differ-ent method of analysis.
Ceilings and Roofs The overall R-value for ceilings of wood frame flat roofs can be calculated using Equations (1) through (5) from Chapter 23. Prop-erties of the materials are found in Tables 1, 3, 2, and 4. The fraction of framing is assumed to be 0.10 for joists at 400 mm OC and 0.07 for joists at 600 mm OC. The calculation procedure is similar to that shown in Example l. Note that if the ceiling contains plane air spaces (see Table 3), the resistance depends on the direction of heat flow, i.e., whether the calculation is for a winter (heat flow up) or summer (heat flow down) condition.
For ceilings of pitched roofs under winter conditions, calcu-late the R-value of the ceiling using the procedure for flat roofs.
Table 5 can be used to determine the effective resistance of the attic space under summer conditions for varying conditions of ventilation air temperature, airflow direction and rates, ceiling resistance, roof or sol-air temperatures, and surface emittances (Joy 1958).
The R-value is the total resistance obtained by adding the ceiling and effective attic resistances. The applicable tempera-ture difference is that difference between room air and sol-air temperatures or between room air and roof temperatures (see Table 5, footnote f ). Table 5 can be used for pitched and flat res-idential roofs over attic spaces. When an attic has a floor, the ceiling resistance should account for the complete ceiling-floor construction.
Windows and Doors Table 4 of Chapter 30 lists U-factors for various fenestration products. Table 6 lists U-factors for exterior wood and steel doors.
All U-factors are approximate, because a significant portion of the resistance of a window or door is contained in the air film resis-tances, and some parameters that may have important effects are not considered. For example, the listed U-factors assume the surface temperatures of surrounding bodies are equal to the ambient air tem-perature. However, the indoor surface of a window or door in an actual installation may be exposed to nearby radiating surfaces, such as radiant heating panels, or opposite walls with much higher or lower temperatures than the indoor air. Air movement across the indoor surface of a window or door, such as that caused by nearby heating and cooling outlet grilles, increases the U-factor; and air movement (wind) across the outdoor surface of a window or door also increases the U-factor.
Uo Concept Uo is the combined thermal transmittance of the respective areas of gross exterior wall, roof or ceiling or both, and floor assemblies.
The Uo equation for a wall is as follows: (4) where Uo = average thermal transmittance of gross wall area Ao = gross area of exterior walls di dj j 1 = m ∑ ≥ i 1 = n ∑ w L zf di i 1 = n ∑ + = Use zf = − 0.5 for walls when total thickness of layer of materials attached to one side of metal frame ≤16 mm and thermal resistivity of sheathing ≤10.4 m·K/W.
Use zf = + 0.5 for walls when total thickness of layer of materials attached to one side of metal frame ≤16 mm and thermal resistivity of sheathing > 10.4 m·K/W .
Find zf in chart above for walls when total thickness of layer of materials attached to one side of metal frame > 16 mm.
Fig. 7 Modified Zone Factor for Calculating R-Value of Metal Stud Walls with Cavity Insulation Uo UwallAwall UwindowAwindow UdoorAdoor + + ( ) Ao ⁄ = Thermal and Water Vapor Transmission Data 25.13 Table 5 Effective Thermal Resistance of Ventilated Atticsa (Summer Condition) NONREFLECTIVE SURFACES Ventilation Air Temperature, °C Sol-Airf Temperature, °C No Ventilationb Natural Ventilation Power Ventilationc Ventilation Rate per Square Metre of Ceiling, L/s 0 0.5d 2.5 5.1 7.6 Ceiling Resistance Re, K·m2/W 1.8 3.5 1.8 3.5 1.8 3.5 1.8 3.5 1.8 3.5 49 0.33 0.33 0.49 0.60 1.11 1.64 1.69 2.82 1.94 3.52 27 60 0.33 0.33 0.49 0.62 1.14 1.76 1.72 2.99 2.11 3.70 71 0.33 0.33 0.49 0.63 1.18 1.94 1.76 3.17 2.29 3.87 49 0.33 0.33 0.44 0.49 0.81 1.18 1.07 1.76 1.21 2.29 32 60 0.33 0.33 0.46 0.55 0.92 1.39 1.34 2.11 1.51 2.64 71 0.33 0.33 0.48 0.60 1.02 1.58 1.50 2.46 1.76 2.99 49 0.33 0.33 0.39 0.40 0.58 0.77 0.70 1.06 0.72 1.21 38 60 0.33 0.33 0.42 0.48 0.74 1.07 1.02 1.53 1.14 1.76 71 0.33 0.33 0.46 0.56 0.88 1.34 1.27 1.94 1.46 2.29 REFLECTIVE SURFACESg 49 1.14 1.14 1.43 1.55 2.29 2.99 2.99 4.40 3.34 5.28 27 60 1.14 1.14 1.44 1.58 2.46 3.17 3.17 4.58 3.52 5.46 71 1.14 1.14 1.46 1.62 2.64 3.17 3.34 4.75 3.70 5.63 49 1.14 1.14 1.32 1.41 1.76 2.29 2.11 2.99 2.29 3.34 32 60 1.14 1.14 1.36 1.46 2.11 2.64 2.46 3.52 2.82 3.87 71 1.14 1.14 1.39 1.51 2.29 2.82 2.82 3.87 3.17 4.40 49 1.14 1.14 1.23 1.30 1.41 1.76 1.50 2.11 1.55 2.11 38 60 1.14 1.14 1.28 1.37 1.76 2.11 1.94 2.64 2.11 2.82 71 1.14 1.14 1.34 1.44 1.94 2.46 2.29 3.17 2.64 3.52 aAlthough the term effective resistance is commonly used when there is attic ven-tilation, this table includes values for situations with no ventilation. The effective resistance of the attic added to the resistance (1/U ) of the ceiling yields the effec-tive resistance of this combination based on sol-air (see Chapter 29) and room temperatures. These values apply to wood frame construction with a roof deck and roofing that has a conductance of 5.7 W/(m2· K).
bThis condition cannot be achieved in the field unless extreme measures are taken to tightly seal the attic.
cBased on air discharging outward from attic.
dWhen attic ventilation meets the requirements stated in Chapter 26, 0.5 L/s per square metre is assumed as the natural summer ventilation rate.
eWhen determining ceiling resistance, do not add the effect of a reflective surface fac-ing the attic, as it is accounted for in the Reflective Surface part of this table.
fRoof surface temperature rather than sol-air temperature (see Chapter 29) can be used if 0.04 is subtracted from the attic resistance shown.
gSurfaces with effective emittance εeff = 0.05 between ceiling joists facing attic space.
Table 6 Transmission Coefficients U for Wood and Steel Doors, W/(m2·K) Nominal Door Thickness, mm Description No Storm Door Wood Storm Doorc Metal Storm Doord Wood Doorsa,b 35 Panel door with 11 mm panelse 3.24 1.87 2.10 35 Hollow core flush door 2.67 1.70 1.82 35 Solid core flush door 2.21 1.48 1.59 45 Panel door with 11 mm panelse 3.07 1.82 2.04 45 Hollow core flush door 2.61 1.65 1.82 45 Panel door with 29 mm panelse 2.21 1.48 1.59 45 Solid core flush door 2.27 — 1.48 57 Solid core flush door 1.53 1.14 1.19 Steel Doorsb 45 Fiberglass or mineral wool core with steel stiffeners, no thermal breakf 3.41 — — 45 Paper honeycomb core without thermal breakf 3.18 — — 45 Solid urethane foam core without thermal breaka 2.27 — — 45 Solid fire rated mineral fiberboard core without thermal breakf 2.16 — — 45 Polystyrene core without thermal break [18 gage (1.31 mm) commercial steel]f 1.99 — — 45 Polyurethane core without thermal break (18 gage commercial steel)f 1.65 — — 45 Polyurethane core without thermal break [24 gage (0.70 mm) residential steel]f 1.65 — — 45 Polyurethane core with thermal break and wood perimeter (24 gage residential steel)f 1.14 — — 45 Solid urethane foam core with thermal breaka 1.14 — 0.91 Note: All U-factors for exterior doors in this table are for doors with no glazing, except for the storm doors which are in addition to the main exterior door. Any glazing area in exterior doors should be included with the appropriate glass type and analyzed as a window (see Chapter 30). Interpolation and moderate extrapolation are permitted for door thicknesses other than those specified.
aValues are based on a nominal 810 mm by 2030 mm door size with no glazing.
bOutside air conditions: 24 km/h wind speed, − 18°C air temperature; inside air condi-tions: natural convection, 21°C air temperature.
cValues for wood storm door are for approximately 50% glass area.
dValues for metal storm door are for any percent glass area.
e55% panel area.
fASTM C 236 hot box data on a nominal 910 mm by 2130 mm door size with no glazing.
25.14 2001 ASHRAE Fundamentals Handbook (SI) Uwall = thermal transmittance of all elements of opaque wall area Awall = opaque wall area Uwindow = thermal transmittance of window area (including frame) Awindow = window area (including frame) Udoor = thermal transmittance of door area Adoor = door area (including frame) Where more than one type of wall, window, or door is used, the UA term for that exposure should be expanded into its subelements, as shown in Equation (3).
(5) Example 5. Calculate Uo for a wall 10 m by 2.4 m, constructed as in Example 1. The wall contains two double-glazed (12.7 mm airspace) fixed windows with wood/vinyl frames. (From Table 4 in Chapter 30, U = 2.98 W/(m2·K).) One window is 1500 mm by 860 mm and the second 900 mm by 760 mm. The wall also contains a 45 mm solid core flush door with a metal storm door 860 mm by 2000 mm (U = 1.42 W/(m2·K) from Table 6).
Solution: The U-factor for the wall was obtained in Example 1. The areas of the different components are Therefore, the combined thermal transmittance for the wall is Slab-on-Grade and Below-Grade Construction Heat transfer through basement walls and floors to the ground depends on the following factors: (1) the difference between the air temperature within the room and that of the ground and outside air, (2) the material of the walls or floor, and (3) the thermal conductivity of the surrounding earth. The latter varies with local conditions and is usually unknown. Because of the great thermal inertia of the sur-rounding soil, ground temperature varies with depth, and there is a substantial time lag between changes in outdoor air temperatures and corresponding changes in ground temperatures. As a result, ground-coupled heat transfer is less amenable to steady-state representation than above-grade building elements. However, several simplified procedures for estimating ground-coupled heat transfer have been developed. These fall into two principal categories: (1) those that reduce the ground heat transfer problem to a closed form solution, and (2) those that use simple regression equations developed from statistically reduced multidimensional transient analyses.
Closed form solutions, including the ASHRAE arc-length pro-cedure discussed in Chapter 28 by Latta and Boileau (1969), gener-ally reduce the problem to one-dimensional, steady-state heat transfer. These procedures use simple, “effective” U-factors or ground temperatures or both. Methods differ in the various param-eters averaged or manipulated to obtain these effective values.
Closed form solutions provide acceptable results in climates that have a single dominant season, because the dominant season per-sists long enough to permit a reasonable approximation of steady-state conditions at shallow depths. The large errors (percentage) that are likely during transition seasons should not seriously affect building design decisions, since these heat flows are relatively insignificant when compared with those of the principal season.
The ASHRAE arc-length procedure is a reliable method for wall heat losses in cold winter climates. Chapter 28 discusses a slab-on-grade floor modeldeveloped by one study. Although both procedures give results comparable to transient computer solutions for cold cli-mates, their results for warmer U.S. climates differ substantially.
Research conducted by Hougten et al. (1942) and Dill et al.
(1945) indicates a heat flow of approximately 6.3 W/m2 through an uninsulated concrete basement floor with a temperature difference of 11 K between the basement floor and the air 150 mm above it. A U-factor of 5.7 W/(m2·K) is sometimes used for concrete basement floors on the ground. For basement walls below grade, the temper-ature difference for winter design conditions is greater than for the floor. Test results indicate that at the midheight of the below-grade portion of the basement wall, the unit area heat loss is approxi-mately twice that of the floor.
For concrete slab floors in contact with the ground at grade level, tests indicate that for small floor areas (equal to that of a 7.5 m by 7.5 m house) the heat loss can be calculated as proportional to the length of exposed edge rather than total area. This amounts to 1.40 W per linear metre of exposed edge per degree Celcius differ-ence between the indoor air temperature and the average outdoor air temperature. This value can be reduced appreciably by installing insulation under the ground slab and along the edge between the floor and abutting walls. In most calculations, if the perimeter loss is calculated accurately, no other floor losses need to be considered.
Chapter 28 contains data for load calculations and heat loss values for below-grade walls and floors at different depths.
The second category of simplified procedures uses transient two-dimensional computer models to generate the ground heat transfer data that are then reduced to compact form by regression analysis (Mitalas 1982, 1983; Shipp 1983). These are the most accurate procedures available, but the database is very expensive to generate. In addition, these methods are limited to the range of cli-mates and constructions specifically examined. Extrapolating beyond the outer bounds of the regression surfaces can produce significant errors.
Apparent Thermal Conductivity of Soil Effective or apparent soil thermal conductivity is difficult to esti-mate precisely and may change substantially in the same soil at dif-ferent times due to changed moisture conditions and the presence of freezing temperatures in the soil. Figure 8 shows the typical appar-ent soil thermal conductivity as a function of moisture content for different general types of soil. The figure is based on data presented in Salomone and Marlowe (1989) using envelopes of thermal behavior coupled with field moisture content ranges for different soil types. In Figure 8, the term well-graded applies to granular soils with good representation of all particle sizes from largest to small-est. The term poorly graded refers to granular soils with either a uni-form gradation, in which most particles are about the same size, or a skip (or gap) gradation, in which particles of one or more interme-diate sizes are not present.
Although thermal conductivity varies greatly over the complete range of possible moisture contents for a soil, this range can be nar-rowed if it is assumed that the moisture contents of most field soils lie between the “wilting point” of the soil (i.e., the moisture content of a soil below which a plant cannot alleviate its wilting symptoms) and the “field capacity” of the soil (i.e., the moisture content of a soil that has been thoroughly wetted and then drained until the drainage rate has become negligibly small). After a prolonged dry spell, the moisture will be near the wilting point, and after a rainy period, the soil will have a moisture content near its field capacity. The mois-ture contents at these limits have been studied by many agricultural researchers, and data for different types of soil are given by Salomone and Marlowe (1989) and Kersten (1949). The shaded UoAo Uwall 1Awall 1 Uwall 2Awall 2 … Uwall mAwall m + + + = Uwindow 1Awindow 1 Uwindow 2Awindow 2 … + + + Uwindow nAwindow n Udoor 1Adoor 1 + + Udoor 2Adoor 2 … Udoor oAdoor o + + + Awindow 1.500 0.860 × ( ) 0.900 0.760 × ( ) + 1.97 m2 = = Adoor 0.860 2.000 × ( ) 1.72 m2 = = Awall 10 2.4 × ( ) 1.97 1.72 + ( ) – 20.31 m2 = = Uo 0.404 20.31 × ( ) 2.90 1.97 × ( ) 1.42 1.72 × ( ) + + 10 2.4 × ----------------------------------------------------------------------------------------------------------------------= 0.68 W m2 K ⋅ ( ) ⁄ = Thermal and Water Vapor Transmission Data 25.15 areas on Figure 8 approximate (1) the full range of moisture con-tents for different soil types and (2) a range between average values of each limit.
Table 7 gives a summary of design values for thermal conduc-tivities of the basic soil classes. Table 8 gives ranges of thermal conductivity for some basic classes of rock. The value chosen depends on whether heat transfer is being calculated for minimum heat loss through the soil, as in a ground heat exchange system, or a maximum value, as in peak winter heat loss calculations for a basement. Hence, a high and a low value are given for each soil class.
As heat flows through the soil, the moisture tends to move away from the source of heat. This moisture migration provides initial mass transport of heat, but it also dries the soil adjacent to the heat source, hence lowering the apparent thermal conductivity in that zone of soil.
Trends typical in a soil when other factors are held constant are: • k increases with moisture content • k increases with increasing dry density of a soil • k decreases with increasing organic content of a soil • k tends to decrease for soils with uniform gradations and rounded soil grains (because the grain-to-grain contacts are reduced) • k of a frozen soil may be higher or lower than that of the same unfrozen soil (because the conductivity of ice is higher than that of water but lower than that of the typical soil grains). Differences in k below moisture contents of 7 to 8% are quite small. At approximately 15% moisture content, differences in k-factors may vary up to 30% from unfrozen values.
When calculating annual energy use, values that represent typi-cal site conditions as they vary during the year should be chosen. In climates where ground freezing is significant, accurate heat transfer simulations should include the effect of the latent heat of fusion of water. The energy released during this phase change significantly retards the progress of the frost front in moist soils.
Water Vapor Transmission Data for Building Components Table 9 gives typical water vapor permeance and permeability values for common building materials. These values can be used to calculate water vapor flow through building components and assemblies using equations in Chapter 23.
MECHANICAL AND INDUSTRIAL SYSTEMS Thermal Transmission Data Table 10 lists the thermal conductivities of various materials used as industrial insulations. These values are functions of the arithmetic mean of the temperatures of the inner and outer surfaces for each insulation.
Heat Loss from Pipes and Flat Surfaces Tables 11A, 11B, and 12 give heat losses from bare steel pipes and flat surfaces and bare copper tubes. These tables were calcu-lated using ASTM Standard C 680. User inputs for the programs described in the standard include operating temperature, ambient temperature, pipe size, insulation type, number of insulation layers, and thickness for each layer. A program option allows the user to input a surface coefficient or surface emittance, surface orientation, and wind speed. The computer uses this information to calculate the heat flow and the surface temperature. The programs calculate the surface coefficients if the user has not already supplied them.
The equations used in ASTM C 680 are (6) where hcv = convection surface coefficient, W/(m2·K) d = diameter for cylinder, mm. For flat surfaces and large cylinders (d > 600 mm), use d = 600 mm.
Tavg = average temperature of air film = (Ta + Ts)/2, K Ta = temperature of ambient air, K Ts = temperature of surface, K ∆T = surface to air temperature difference, K Wind = air speed, km/h C = constant depending on shape and heat flow condition = 11.58 for horizontal cylinders = 14.08 for longer vertical cylinders = 15.89 for vertical plates = 20.40 for horizontal plates, warmer than air, facing upward = 10.15 for horizontal plates, warmer than air, facing downward = 10.15 for horizontal plates, cooler than air, facing upward = 20.40 for horizontal plates, cooler than air, facing downward Table 7 Typical Apparent Thermal Conductivity Values for Soils, W/(m2 · K) Normal Range Recommended Values for Designa Lowb Highc Sands 0.6 to 2.5 0.78 2.25 Silts 0.9 to 2.5 1.64 2.25 Clays 0.9 to 1.6 1.12 1.56 Loams 0.9 to 2.5 0.95 2.25 aReasonable values for use when no site- or soil-specific data are available.
bModerately conservative values for minimum heat loss through soil (e.g., use in soil heat exchanger or earth-contact cooling calculations). Values are from Salomone and Marlowe (1989).
cModerately conservative values for maximum heat loss through soil (e.g., use in peak winter heat loss calculations). Values are from Salomone and Marlowe (1989).
Table 8 Typical Apparent Thermal Conductivity Values for Rocks, W/(m2 · K) Normal Range Pumice, tuff, obsidian 0.5 to 2.2 Basalt 0.5 to 2.6 Shale 0.9 to 4.0 Granite 1.7 to 4.3 Limestone, dolomite, marble 1.2 to 4.3 Quartzose sandstone 1.4 to 7.8 Fig. 8 Trends of Apparent Thermal Conductivity of Moist Soils hcv C 1 d -- 0.2 1 Tavg ----------- 0.181 T 0.266 ∆ ( ) 1 0.7935 Wind ( ) + = 25.16 2001 ASHRAE Fundamentals Handbook (SI) (7) where hrad = radiation surface coefficient, W/(m2·K) ε = surface emittance σ = Stefan-Boltzmann constant = 5.4 × 10− 8 W/(m2·K4) Example 6. Compute the total annual heat loss from 50 m of nominal 50 mm bare steel pipe in service 4000 h per year. The pipe is carrying steam at 70 kPa (gage) and is exposed to an average air temperature of 27°C.
Solution: The pipe temperature is taken as the steam temperature, which is 115.2°C, obtained by interpolation from Steam Tables. By interpolation in Table 11A between and 82 and 138°C, heat loss from a nominal 50 mm pipe is 274 W/m. Total annual heat loss from the entire line is 274 × 50 × 4000 × 3600 = 197 GJ.
In calculating heat flow, Equations (8) and (9) from Chapter 23 generally are used. For dimensions of standard pipe and fitting sizes, refer to the Piping Handbook. For insulation product dimensions, refer to ASTM Standard C 585, or to the insulation manufacturers’ literature.
Table 9 Typical Water Vapor Permeance and Permeability Values for Common Building Materialsa Material Thickness, mm Permeance, ng/(s·m2·Pa) Resistanceh, TPa·m2·s/kg Permeability, ng/(s·m·Pa) Resistance/mh, TPa·m·s/kg Construction Materials Concrete (1:2:4 mix) 4.7 0.21 Brick masonry 100 46f 0.022 Concrete block (cored, limestone aggregate) 200 137f 0.0073 Tile masonry, glazed 100 6.9f 0.14 Asbestos cement board 3 220-458d 0.0017-0.0035 With oil-base finishes 17-29d 0.0035-0.052 Plaster on metal lath 19 860f 0.0012 Plaster on wood lath 630e 0.0016 Plaster on plain gypsum lath (with studs) 1140f 0.00088 Gypsum wall board (plain) 9.5 2860f 0.00035 Gypsum sheathing (asphalt impregnated) 13 29f 0.038 Structural insulating board (sheathing quality) 29-73f 0.038-0.014 Structural insulating board (interior, uncoated) 13 2860-5150f 0.00035-0.00019 Hardboard (standard) 3.2 630f 0.0016 Hardboard (tempered) 3.2 290f 0.0034 Built-up roofing (hot mopped) 0.0 ∞ Wood, sugar pine 0.58-7.8f,b 172.0-131 Plywood (douglas fir, exterior glue) 6.4 40f 0.025 Plywood (douglas fir, interior glue) 6.4 109f 0.0092 Acrylic, glass fiber reinforced sheet 1.4 6.9f 0.145 Polyester, glass fiber reinforced sheet 1.2 2.9f 0.345 Thermal Insulations Air (still) 174f 0.0057 Cellular glass 0.0d ∞ Corkboard 3.0-3.8d 0.33-0.26 14e 0.076 Mineral wool (unprotected) 245e 0.0059 Expanded polyurethane [R = 1.94 W/(m2·K)] board stock 0.58-2.3d 1.72-0.43 Expanded polystyrene—extruded 1.7d 0.57 Expanded polystyrene—bead 2.9-8.4d 0.34-0.12 Phenolic foam (covering removed) 38 0.026 Unicellular synthetic flexible rubber foam 0.029d 34-4.61 Plastic and Metal Foils and Filmsc Aluminum foil 0.025 0.0d ∞ Aluminum foil 0.009 2.9d 0.345 Polyethylene 0.051 9.1d 0.110 2133 Polyethylene 0.1 4.6d 0.217 2133 Polyethylene 0.15 3.4d 0.294 2133 Polyethylene 0.2 2.3d 0.435 2133 Polyethylene 0.25 1.7d 0.588 2133 Polyvinylchloride, unplasticized 0.051 39d 0.026 Polyvinylchloride, plasticized 0.1 46-80d 0.032 Polyester 0.025 42d 0.042 Polyester 0.09 13d 0.075 Polyester 0.19 4.6d 0.22 Cellulose acetate 0.25 263d 0.0035 Cellulose acetate 3.2 18d 0.054 hrad εσ Ta 4 Ts 4 – ( ) Ta Ts – ---------------------------------= Thermal and Water Vapor Transmission Data 25.17 Table 9 Typical Water Vapor Permeance and Permeability Values for Common Building Materialsa (Concluded ) Material Unit Mass, kg/m2 Permeance, ng/(s·m2·Pa) Resistanceh, TPa·m2·s/kg Dry-Cup Wet-Cup Other Dry-Cup Wet-Cup Other Building Paper, Felts, Roofing Papersg Duplex sheet, asphalt laminated, aluminum foil one side 0.42 0.1 10 10 0.1 Saturated and coated roll roofing 3.18 2.9 14 0.34 0.071 Kraft paper and asphalt laminated, reinforced 0.33 17 103 0.059 0.0097 Blanket thermal insulation backup paper, asphalt coated 0.30 23 34-240 0.043 0.029-0.0042 Asphalt-saturated and coated vapor retarder paper 0.42 11-17 34 0.091-0.059 0.029 Asphalt-saturated, but not coated, sheathing paper 0.21 190 1160 0.0053 0.00086 0.73 kg/m2 asphalt felt 0.68 57 320 0.017 0.0031 0.73 kg/m2 tar felt 0.68 230 1040 0.0043 0.00096 Single-kraft, double 0.16 1170 2400 0.00056 0.00042 Liquid-Applied Coating Materials Thickness, Commercial latex paints (dry film thickness)i µm Vapor retarder paint 70 26 0.038 Primer-sealer 30 360 0.0028 Vinyl acetate/acrylic primer 50 424 0.0024 Vinyl-acrylic primer 40 491 0.0020 Semi-gloss vinyl-acrylic enamel 60 378 0.0026 Exterior acrylic house and trim 40 313 0.0032 Paint—2 coats Asphalt paint on plywood 23 0.043 Aluminum varnish on wood 17-29 0.059-0.034 Enamels on smooth plaster 29-86 0.034-0.012 Primers and sealers on interior insulation board 51-20 0.020-.0083 Various primers plus 1 coat flat oil paint on plaster 91-172 0.011-0.0058 Flat paint on interior insulation board 229 0.0044 Water emulsion on interior insulation board 1716-4863 0.00058-0.00021 Unit Mass, Paint-3 coats kg/m2 Exterior paint, white lead and oil on wood siding 17-57 0.0059-0.017 Exterior paint, white lead-zinc oxide and oil on wood 51 0.020 Styrene-butadiene latex coating 0.6 629 0.0016 Polyvinyl acetate latex coating 1.2 315 0.0032 Chlorosulfonated polyethylene mastic 1.1 97 0.010 2.2 3.4 0.29 Asphalt cutback mastic, 1.6 mm, dry 8.0 0.125 4.8 mm, dry 0 ∞ Hot melt asphalt 0.6 29 0.034 1.1 5.7 0.175 aThis table permits comparisons of materials; but in the selection of vapor retarder materials, exact values for permeance or permeability should be obtained from the manufacturer or from laboratory tests. The values shown indicate variations among mean values for materials that are similar but of different density, orientation, lot, or source. The values should not be used as design or specification data. Values from dry-cup and wet-cup methods were usually obtained from investigations using ASTM E 96 and C 355; values shown under others were obtained by two-temperature, special cell, and air velocity methods.
bDepending on construction and direction of vapor flow.
cUsually installed asvaporretarders, although sometimesused asan exterior finish and elsewhere near the cold side, where special considerations are then required for warm side barrier effectiveness.
dDry-cup method.
eWet-cup method.
fOther than dry- or wet-cup method.
gLow permeance sheets used as vapor retarders. High permeance used elsewhere in construction.
hResistance and resistance/mm values have been calculated as the reciprocal of the permeance and permeability values.
iCast at 0.25 mm wet film thickness.
25.18 2001 ASHRAE Fundamentals Handbook (SI) Table 10 Typical Thermal Conductivity for Industrial Insulations at Various Mean Temperatures—Design Valuesa Material Accepted Max.
Temp. for Use, b °C Typical Density, kg/m3 Typical Conductivity k in W/(m·K) at Mean Temperature, °C − 73 − 59 − 46 − 32 − 18 − 4 10 24 38 93 150 200 370 480 BLANKETS AND FELTS ALUMINOSILICATE FIBER 980 64 0.035 0.046 0.078 0.143 0.148 7 to 10 µm diameter fiber 1100 96-128 0.036 0.043 0.069 0.112 0.137 1200 64 0.032 0.042 0.065 0.085 0.107 3 µm diameter fiber MINERAL FIBER (Rock, slag, or glass) 650 96-190 0.037 0.046 0.056 0.078 Blanket, metal reinforced 540 40-96 0.035 0.045 0.058 0.088 180 <12 0.036 0.037 0.040 0.043 0.048 0.052 0.076 Blanket, flexible, fine-fiber 12 0.035 0.036 0.039 0.042 0.046 0.049 0.069 organic bonded 16 0.033 0.035 0.036 0.039 0.042 0.046 0.062 24 0.030 0.032 0.033 0.036 0.039 0.040 0.053 32 0.029 0.030 0.032 0.033 0.036 0.037 0.048 48 0.027 0.029 0.030 0.032 0.033 0.035 0.045 Blanket, flexible, textile fiber, 180 10 0.039 0.040 0.042 0.043 0.045 0.046 0.072 0.098 organic bonded 12 0.037 0.039 0.040 0.042 0.045 0.046 0.069 0.095 16 0.035 0.036 0.037 0.039 0.042 0.045 0.065 0.086 24 0.032 0.033 0.035 0.036 0.039 0.042 0.056 0.073 48 0.029 0.030 0.032 0.033 0.035 0.036 0.046 0.059 Felt, semirigid organic bonded 200 48-130 0.035 0.036 0.037 0.039 0.050 0.063 450 48 0.023 0.024 0.026 0.027 0.029 0.030 0.032 0.033 0.035 0.050 0.079 Laminated and felted without binder 650 120 0.050 0.065 0.086 BLOCKS, BOARDS, AND PIPE INSULATION MAGNESIA 320 176-192 0.050 0.055 0.060 85% CALCIUM SILICATE 650 176-240 0.055 0.059 0.063 0.075 0.089 0.104 980 192-240 0.091 0.107 0.137 CELLULAR GLASS 480 125-131 0.035 0.036 0.037 0.040 0.042 0.043 0.046 0.048 0.049 0.059 0.071 0.101 0.146 DIATOMACEOUS SILICA 870 336-352 0.092 0.098 0.104 1040 368-400 0.101 0.108 0.115 MINERAL FIBER (Glass) Organic bonded, block, and boards 200 48-160 0.023 0.024 0.026 0.027 0.029 0.032 0.035 0.036 0.037 0.048 0.058 Nonpunking binder 540 48-160 0.037 0.045 0.055 0.075 Pipe insulation, slag, or glass 180 48-64 0.029 0.030 0.032 0.033 0.035 0.042 260 48-160 0.039 0.032 0.035 0.036 0.037 0.048 0.058 Inorganic bonded block 540 160-240 0.048 0.055 0.065 0.079 980 240-384 0.046 0.053 0.060 0.075 0.089 0.107 Pipe insulation, slag, or glass 540 160-240 0.048 0.055 0.065 0.079 Resin binder 240 0.033 0.035 0.036 0.037 0.040 0.042 RIGID POLYSTYRENE Extruded (R-12 exp.)(smooth skin sur-face) 80 29-56 0.023 0.023 0.024 0.023 0.024 0.026 0.027 0.029 Molded beads 80 16 0.024 0.027 0.029 0.030 0.032 0.035 0.036 0.037 0.040 20 0.024 0.026 0.027 0.029 0.032 0.033 0.035 0.036 0.040 24 0.023 0.024 0.027 0.029 0.030 0.032 0.033 0.035 0.037 28 0.023 0.024 0.026 0.027 0.029 0.032 0.033 0.035 0.036 RIGID POLYURETHANE/POLYISOCYANURATEc,d Unfaced (R-11 exp.) 120 24-40 0.023 0.025 0.026 0.026 0.026 0.025 0.023 0.025 0.025 RIGID POLYISOCYANURATE Gas-imperm. facers (R-11 exp.) 100 32 0.017 0.019 0.020 0.022 RIGID PHENOLIC Closed cell (R-11, R-113 exp.) 48 0.016 0.017 0.017 0.018 RUBBER, Rigid foamed 70 72 0.029 0.030 0.032 0.033 VEGETABLE AND ANIMAL FIBER Wool felt (pipe insulation) 80 320 0.040 0.043 0.045 0.048 INSULATING CEMENTS MINERAL FIBER (Rock, slag, or glass) With colloidal clay binder 980 380-480 0.071 0.079 0.088 0.105 0.122 With hydraulic setting binder 650 480-640 0.108 0.115 0.122 0.137 LOOSE FILL Cellulose insulation (milled pulverized paper or wood pulp) 40-48 0.037 0.039 0.042 Mineral fiber, slag, rock, or glass 32-80 0.027 0.030 0.033 0.036 0.037 0.040 0.045 Perlite (expanded) 48-80 0.032 0.035 0.036 0.039 0.040 0.043 0.045 0.048 0.050 Silica aerogel 120 0.019 0.020 0.022 0.022 0.023 0.024 0.026 Vermiculite (expanded) 110-130 0.056 0.058 0.060 0.063 0.065 0.068 0.071 64-96 0.049 0.050 0.055 0.058 0.060 0.063 0.066 aRepresentative values for dry materials, which are intended as design (not specification) values for materials in normal use. Insula-tion materials in actual service may have thermal values that vary from design values depending on their in-situ properties (e.g., den-sity and moisture content). For properties of a particular product, use the value supplied by the manufacturer or by unbiased tests.
bThese temperatures are generally accepted as maximum. When operating temperature approaches these limits, follow the manufacturers’ recommendations.
cSome polyurethane foams are formed by means that produce a stable product (with respect to k), but most are blown with refrigerant and will change with time.
dSee Table 4, footnote i.
eSee Table 4, footnote j.
Thermal and Water Vapor Transmission Data 25.19 Examples 7 and 8 illustrate how Equations (8) and (9) from Chapter 23 can be used to determine heat loss from both flat and cylindrical surfaces. Figure 9 shows surface resistance as a function of heat transmission for both flat and cylindrical surfaces. The sur-face emittance is assumed to be 0.85 to 0.90 in still air at 27°C.
Example 7. Compute the heat loss from a boiler wall if the interior insula-tion surface temperature is 600°C and ambient still air temperature is 27°C. The wall is insulated with 115 mm of mineral fiber block and 13 mm of mineral fiber insulating and finishing cement.
Solution: Assume that the mean temperature of the mineral fiber block is 370°C, the mean temperature of the insulating cement is 93°C, and the surface resistance Rs is 0.11 m2·K/W.
From Table 10, k1 = 0.089 and k2 = 0.115. Using Equation (8) from Chapter 23, As a check, from Figure 9, at 378 W/m2, Rs ≅0.10. The mean tempera-ture of the mineral fiber block is and the mean temperature of the insulating cement is From Table 10, at 354°C, k1 = 0.087; at 87°C, k2 = 0.114.
Using these adjusted values to recalculate qs, From Figure 9, at 373 W/m2, Rs = 0.10. The mean temperature of the mineral fiber block is and the mean temperature of the insulating cement is From Table 10, at353°C, k1 = 0.087; at 86°C, k2 = 0.114.
Because Rs, k1, and k2 do not change at these values, qs = 373.1 W/m2.
Example 8. Compute heat loss per square foot of outer surface of insula-tion if pipe temperature is 650ºC and ambient still air temperature is 27°C. The pipe is nominal 150 mm steel pipe, insulated with a nominal 75 mm thick diatomaceous silica as the inner layer and a nominal 50 mm thick calcium silicate as the outer layer.
Solution: From Chapter 41 of the 2000 ASHRAE Handbook—Equip-ment, ro = 84.1 mm. A nominal 75 mm thick diatomaceous silica insu-lation to fit a nominal 150 mm steel pipe is 76.7 mm thick. A nominal 50 mm thick calcium silicate insulation to fit over the 52.8 mm diato-maceous silica is 52.8 mm thick. Therefore, ri = 160.8 mm and rs = 213.6 mm.
Assume that the mean temperature of the diatomaceous silica is 315°C, the mean temperature of the calcium silicate is 120°C and the surface resistance Rs is 0.09. From Table 10, k1 = 0.095; k2 = 0.060. By Equation (9) from Chapter 23, From Figure 9, at 2.43 W/m2, Rs = 0.11. The mean temperature of the diatomaceous silica is and the mean temperature of the calcium silicate is From Table 10, k1 = 0.104; k2 = 0.066. Recalculating, From Figure 9 at 264 W/m2, Rs = 0.11. The mean temperature of the diatomaceous silica is and the mean temperature of the calcium silicate is s 600 27 – 0.115 0.089 ⁄ ( ) 0.013 0.115 ⁄ ( ) 0.11 + + -----------------------------------------------------------------------------------------------------378 W m ⁄ = = 0.115 0.089 ⁄ 1.292; 1.292 2 ⁄ 0.646 = = 600 0.646 1.505 ------------- 600 27 – ( ) – 354° C = 0.013 0.115 ⁄ 0.113; 0.113 2 ⁄ 0.057; 1.292 0.057 – 1.349 = = = 600 1.349 1.505 ------------- 600 27 – ( ) – 87° C = qs 573 0.115 0.087 ⁄ ( ) 0.013 0.114 ⁄ ( ) 0.10 + + -----------------------------------------------------------------------------------------------------= 372.1 W m2 ⁄ = 0.115 0.087 ⁄ 1.322; 1.322 2 ⁄ 0.661 = = 600 0.661 1.536 -------------573 – 353° C = 0.013 0.114 ⁄ 0.1140; 0.1140 2 ⁄ 0.057; 1.322 0.057 + 1.379 = = = 600 1.379 1.536 -------------573 – 86° C = Fig. 9 Surface Resistance as Function of Heat Transmission for Flat Surfaces and Cylindrical Surfaces Greater than 600 mm in Diameter qs 650 27 – 0.2136 160.8 84.1 ⁄ ( ) ln 0.095 ----------------------------------------------------------0.2136 213.6 160.8 ⁄ ( ) ln 0.060 -------------------------------------------------------------0.09 + + ------------------------------------------------------------------------------------------------------------------------------------------------= 623 1.457 1.011 0.09 + + ( ) -------------------------------------------------------243 W m2 ⁄ = = 2136 ln 160.8 84.1 ⁄ ( ) 0.095 ⁄ 1.46; 1.46 2 ⁄ 0.7 = = 50 650 27 – ( )0.73 1.457 1.011 0.11 + + --------------------------------------------------– 474° C = 0.2136 ln 213.6 160.8 ⁄ ( ) 0.060 ----------------------------------------------------------------1.01; 1.01 2 ----------0.51; 1.46 0.51 + 1.97 = = = 650 650 27 – ( )1.97 1.457 1.011 0.11 + + --------------------------------------------------– 174° C = qs 623 0.2136 ln 160.8 84.1 ⁄ ( ) 0.104 -------------------------------------------------------------0.2136 ln 213.6 160.8 ⁄ ( ) 0.066 ----------------------------------------------------------------0.11 + + -----------------------------------------------------------------------------------------------------------------------------------------------------= 264 W m2 ⁄ = 0.2136 ln 160.8 84.1 ⁄ ( ) 0.104 -------------------------------------------------------------1.33; 1.33 2 ----------0.665 = = 650 0.665 2.360 ⁄ 623 ----------------------------------– 474° C = 0.2136 ln 213.6 160.8 ⁄ ( ) 0.066 ----------------------------------------------------------------0.919; 0.919 2 -------------0.46; 1.33 0.46 + 1.79 = = = 650 1.79 2.360 -------------623 – 177° C = 25.20 2001 ASHRAE Fundamentals Handbook (SI) From Table 10, at 474°C, k1 = 0.104; at 177°C, k2 = 0.066. Since Rs, k1, and k2 do not change at 264 W/m2, this value is qs. The heat flow per square metre of the inner surface of the insulation is Because trial-and-error techniques are tedious, the computer programs previously described should be used to estimate heat flows per unit area of flat surfaces or per unit length of piping, and interface temperatures including surface temperatures.
Several methods can be used to determine the most effective thickness of insulation for piping and equipment. Table 13 shows the recommended insulation thicknesses for three different pipe and equipment insulations. Installed cost data can be developed using procedures described by the Federal Energy Administra-tion (1976). Computer programs capable of calculating thickness information are available from several sources. Also, manufac-turers of insulations offer computerized analysis programs for designers and owners to evaluate insulation requirements. For more information on determining economic insulation thickness, see Chapter 23.
Chapters 3 and 23 give guidance concerning process control, personnel protection, condensation control, and economics. For specific information on sizes of commercially available pipe insulation, see ASTM Standard C 585 and consult with the North American Insulation Manufacturers Association (NAIMA) and its member companies.
CALCULATING HEAT FLOW FOR BURIED PIPELINES In calculating heat flow to or from buried pipelines, the ther-mal properties of the soil must be assumed. Table 7 gives the apparent thermal conductivity values of various soil types, and Figure 8 shows the typical trends of apparent soil thermal con-ductivity with moisture content for various soil types. Table 8 provides ranges of apparent thermal conductivity for various types of rock. Kernsten (1949) also discusses thermal properties of soils. Carslaw and Jaeger (1959) give methods for calculating the heat flow taking place between one or more buried cylinders and the surroundings.
Table 11A Heat Loss from Bare Steel Pipe to Still Air at 27°Ca, W/m Nominal Pipe Sizeb, mm Pipe Inside Temperature, °C 82 138 194 249 305 360 416 471 527 582 15 57.0 141.5 252.9 396.2 577.5 804.2 1084.6 1427.7 1843.2 2341.8 20 69.7 173.1 310.0 486.5 710.4 991.0 1338.6 1764.4 2280.9 2901.1 25 85.3 212.2 380.7 598.4 875.4 1223.0 1654.1 2182.9 2824.8 3595.7 32 105.4 262.2 471.3 742.2 1097.6 1522.0 2061.9 2724.8 3530.1 4498.3 40 119.1 296.5 533.5 841.0 1233.7 1728.2 2343.1 3098.8 4017.5 5122.1 50 145.9 363.4 654.8 1034.3 1519.8 2132.4 2895.1 3833.6 4975.3 6349.1 80 207.5 517.8 935.5 1481.7 2182.8 3069.4 4175.4 5537.5 7194.9 9189.4 100 261.0 652.1 1180.3 1872.7 2763.7 3892.5 5302.3 7040.3 9156.5 11703.9 150 372.0 930.9 1689.2 2687.7 3977.8 5613.6 7661.8 10189.6 13269.7 16978.6 200 474.1 1187.5 2158.5 3440.4 5098.6 7207.5 9848.7 13110.2 17084.9 21870.4 250 580.7 1455.7 2649.7 4229.1 6275.4 8880.7 12145.7 16178.5 21093.2 27008.8 300 676.5 1693.6 3078.9 4905.8 7262.6 10246.0 13958.4 18505.7 23993.9 30527.9 400 838.2 2103.6 3837.5 6138.3 9125.6 12933.1 17706.2 23598.9 30772.3 39392.1 500 1030.3 2587.4 4725.0 7566.1 11258.2 15965.5 21865.2 29144.1 37995.6 48617.0 600 1219.8 3064.5 5601.0 8975.6 13363.2 18957.6 25966.0 34605.7 45100.0 57674.1 Table 11B Heat Loss from Flat Surfaces to Still Air at 27°C, W/m Surface Inside Temperature, °C 82 138 194 249 305 360 416 471 527 582 Vertical surface 668.4 1679.3 3065.9 4909.6 7311.8 10388.7 14269.8 19097.8 25028.3 32229.2 Horizontal surface Facing up 739.3 1847.2 3342.5 5303.0 7827.4 11030.0 15042.5 20003.8 26070.3 33409.2 Facing down 578.3 1465.7 2713.4 4408.7 6655.3 9571.0 13286.1 17944.6 23702.2 30727.3 aCalculations from ASTM C680; steel: k = 45.3 W/(m2· K); ε = 0.94.
bLosses per square metre of pipe for pipes larger than 600 mm can be considered the same as losses per square metre for 600 mm pipe.
qo qs rs ro ⁄ ( ) 264 213.6 84.1 ⁄ ( ) 6701 W/m2 = = = Thermal and Water Vapor Transmission Data 25.21 CODES AND STANDARDS ASTM. 1990. Standard practice for inner and outer diameters of rigid ther-mal insulation for nominal sizes of pipe and tubing. Standard C585-90.
American Society for Testing and Materials, West Conshohocken, PA.
ASTM. 1991. Standard test method for steady-state heat flux measurements and thermal transmission properties by means of the heat flow meter apparatus. Standard C 518-91.
ASTM. 1993. Standard test method for steady-state heat flux measurements and thermal transmission properties by means of the guarded-hot-plate apparatus. Standard C 177-85 (Revised 1993).
ASTM. 1993. Standard test method for steady-state thermal performance of building assemblies by means of a guarded hot box. Standard C 236-89 (Revised 1993).
ASTM. 1995. Standard practice for determination of heat gain or loss and the surface temperatures of insulated pipe and equipment systems by the use of a computer program. Standard C 680-89 (Revised 1995).
ASTM. 1996. Standard test method for thermal performance of building assemblies by means of a calibrated hot box. Standard C 976-90 (Revised 1996).
REFERENCES Adams, L. 1971. Supporting cryogenic equipment with wood. Chemical Engineering (May):156-58.
Barbour, E., J. Goodrow, J. Kosny, and J.E. Christian. 1994. Thermal per-formance of steel-framed walls. Prepared for American Iron and Steel Institute by NAHB Research Center.
Bassett, M.R. and H.A. Trethowen. 1984. Effect of condensation on emit-tance of reflective insulation. Journal of Thermal Insulation 8 (Octo-ber):127.
Carslaw, H.S. and J.C. Jaeger. 1959. Conduction of heat in solids. Oxford University Press, Amen House, London, England, 449.
Dill, R.S., W.C. Robinson, and H.E. Robinson. 1945. Measurements of heat losses from slab floors. National Bureau of Standards. Building Materi-als and Structures Report, BMS 103.
Economic thickness for industrial insulation. 1976. GPO No. 41-018-001 15-8, Federal Energy Administration, Washington, D.C.
Farouk, B. and D.C. Larson. 1983. Thermal performance of insulated wall systems with metal studs. Proceedings of the 18th Intersociety Energy Conversion Engineering Conference, Orlando, FL.
Farouki, O.T. 1981. Thermal properties of soil. CRREL Monograph 81-1, United States Army Corps of Engineers Cold Regions Research and Engineering Laboratory, December.
Fishenden, M. 1962. Tables of emissivity of surfaces. International Journal of Heat and Mass Transfer 5:67-76.
Goss, W.P. and R.G. Miller. 1989. Literature review of measurement and pre-diction of reflective building insulation system performance: 1900-1989.
ASHRAE Transactions 95(2).
Hooper, F.C. and W.J. Moroz. 1952. The impact of aging factors on the emis-sivity of reflective insulations. ASTM Bulletin (May):92-95.
Hougten, F.C., S.I. Taimuty, C. Gutberlet, and C.J. Brown. 1942. Heat loss through basement walls and floors. ASHVE Transactions 48:369.
Joy, F.A. 1958. Improving attic space insulating values. ASHAE Transac-tions 64:251.
Kersten, M.S. 1949. Thermal properties of soils. University of Minnesota, Engineering Experiment Station Bulletin 28, June.
Table 12 Heat Loss from Bare Copper Tube to Still Air at 27°Ca, W/m Nominal Tube Size, mm Tube Inside Temperature, °C 120 150 180 210 240 270 300 330 8 6.8 13.6 21.0 29.4 38.3 48.0 58.2 69.1 15 10.6 20.9 32.7 45.6 59.5 74.5 90.4 107.4 20 14.1 28.0 43.6 60.8 79.5 99.6 121.1 144.0 25 17.6 34.8 54.2 75.6 98.8 123.9 150.6 179.2 32 20.9 41.4 64.6 89.9 117.6 147.4 179.4 213.5 40 24.2 47.9 74.6 104.1 136.0 170.5 207.6 247.1 50 30.6 60.4 94.2 131.4 171.8 215.6 262.4 312.7 Dull ε = 0.44 80 42.9 84.7 131.8 184.0 240.7 302.1 368.3 439.1 100 54.8 107.9 168.2 234.7 307.2 385.7 470.3 561.1 150 77.6 152.8 238.0 332.2 435.1 546.7 667.1 796.7 200 99.7 196.1 305.4 426.4 558.6 702.2 857.3 1024.4 250 121.2 238.2 371.0 518.1 678.9 853.8 1042.9 1246.8 300 142.2 279.6 435.3 607.8 796.9 1002.4 1224.9 1464.9 8 5.2 10.4 16.2 22.6 29.3 36.4 43.7 51.4 15 7.9 15.8 24.7 34.3 44.5 55.2 66.4 78.0 20 10.3 20.8 32.5 45.1 58.5 72.7 87.4 102.6 25 12.7 25.5 39.8 55.4 71.8 89.2 107.2 126.1 32 14.9 30.1 46.9 65.2 84.6 105.0 126.5 148.7 40 17.1 34.4 53.8 74.8 97.0 120.4 144.9 170.5 50 21.3 42.9 67.0 93.0 120.8 150.0 180.6 212.5 Bright ε = 0.08 80 29.3 58.8 91.9 127.6 165.7 205.8 247.8 291.7 100 36.8 73.8 115.3 160.3 208.2 258.6 311.5 366.5 150 50.9 102.2 159.5 221.7 288.0 358.0 431.2 507.7 200 64.2 128.9 201.2 279.7 363.5 451.8 544.4 641.2 250 77.1 154.5 241.2 335.4 435.7 541.7 653.0 769.2 300 89.4 179.2 279.9 389.1 505.6 628.7 757.9 893.1 aCalculations from ASTM C680; for copper: k = 401.5 W/(m·K).
25.22 2001 ASHRAE Fundamentals Handbook (SI) Table 13 Recommended Thicknesses for Pipe and Equipment Insulation Nom.
Dia., mm MINERAL FIBER (Fiberglass and Rock Wool) CALCIUM Process Temperature, °C Process Temp., °C 65 120 175 225 300 350 400 450 500 550 65 120 175 Thickness 25 38 50 63 75 88 100 100 113 125 25 38 50 15 Heat loss 8 15 23 32 41 52 63 81 96 110 12 23 33 Surface temp.
22 24 24 26 26 27 28 30 31 31 24 26 27 Thickness 25 38 50 63 88 100 100 113 125 138 25 50 63 25 Heat loss 11 20 29 29 47 59 76 92 110 130 15 25 37 Surface temp.
23 24 26 27 26 27 29 30 31 32 24 24 26 Thickness 25 50 63 75 100 100 100 138 138 150 38 65 75 40 Heat loss 13 21 32 43 52 70 90 99 123 146 16 28 40 Surface temp.
73 74 77 79 79 82 86 84 88 90 73 75 78 Thickness 38 50 75 88 100 100 100 138 150 200 38 63 75 50 Heat loss 12 24 33 45 59 78 101 110 132 161 18 31 45 Surface temp.
22 24 24 25 26 28 31 29 31 33 23 24 26 Thickness 38 63 88 100 100 113 113 150 163 175 50 75 88 80 Heat loss 15 27 37 52 72 90 117 128 148 177 20 36 52 Surface temp.
22 23 24 25 27 28 31 31 31 32 23 24 26 Thickness 38 75 100 100 100 125 138 150 175 188 50 75 100 100 Heat loss 18 28 40 61 85 98 121 146 167 198 24 41 56 Surface temp.
22 23 23 26 28 30 29 31 31 32 21 24 25 Thickness 50 75 100 100 113 125 138 163 188 200 50 88 100 150 Heat loss 20 37 52 78 100 125 153 174 200 236 32 49 72 Surface temp.
22 23 24 26 28 29 31 31 32 33 23 24 26 Thickness 50 88 100 100 125 125 138 175 200 213 63 88 100 200 Heat loss 25 40 82 93 111 149 182 196 225 266 34 60 86 Surface temp.
22 23 24 27 27 30 32 31 32 33 23 24 26 Thickness 50 88 100 100 125 138 138 188 213 225 63 100 100 250 Heat loss 31 48 74 111 131 163 211 217 249 295 39 63 102 Surface temp.
22 23 25 27 28 29 32 31 32 33 23 24 27 Thickness 50 88 100 100 125 138 138 188 213 241 63 100 100 300 Heat loss 35 55 84 126 148 185 239 243 279 318 45 72 116 Surface temp.
22 23 25 28 28 30 33 31 32 33 23 24 27 Thickness 50 88 100 100 125 138 163 188 225 241 63 100 100 350 Heat loss 38 59 90 136 159 198 227 260 285 338 49 78 125 Surface temp.
22 23 25 28 28 30 31 32 32 33 23 24 27 Thickness 63 88 100 100 138 138 175 200 225 250 75 100 100 400 Heat loss 36 65 101 151 164 219 237 273 313 357 48 86 138 Surface temp.
22 23 26 28 28 31 30 31 32 33 22 24 28 Thickness 63 88 100 100 138 138 175 200 225 250 75 100 100 450 Heat loss 39 72 111 166 180 240 259 298 340 388 53 95 153 Surface temp.
22 23 26 28 28 31 31 31 32 33 23 24 28 Thickness 63 88 100 100 138 138 175 200 225 250 75 100 100 500 Heat loss 43 79 121 182 196 261 281 322 368 419 58 104 167 Surface temp.
22 24 26 28 28 31 31 32 32 33 23 25 28 Thickness 63 100 100 100 138 150 188 200 225 250 75 100 100 600 Heat loss 51 83 141 212 228 283 308 371 422 479 68 122 195 Surface temp.
22 23 26 28 28 30 30 32 33 34 23 25 28 Thickness 63 100 100 100 138 163 188 213 250 250 75 100 100 750 Heat loss 62 101 172 258 275 319 368 422 462 568 83 148 237 Surface temp.
22 23 26 29 29 29 31 32 32 34 23 25 28 Thickness 63 100 100 100 138 175 200 225 250 250 63 100 100 900 Heat loss 74 118 203 304 322 350 406 467 534 656 114 174 280 Surface temp.
22 23 26 29 29 29 30 31 32 34 23 25 28 Thickness 50 88 100 113 138 213 241 250 250 250 63 88 100 Flat Heat loss 10 13 19 26 30 26 30 37 45 56 12 19 27 Surface temp.
22 23 25 27 28 27 28 29 32 34 23 25 27 Consult manufacturer’s literature for product temperature limitations. Table is based on typical operating conditions, e.g., 18°Cambient temperature and 12 km/h wind speed, and may not represent actual conditions of use. Units for thickness, heat loss, and surface temperature are in mm, W/m (W/m2 for flat surfaces), and °C, respectively.
Thermal and Water Vapor Transmission Data 25.23 Table 13 Recommended Thicknesses for Pipe and Equipment Insulation (Concluded) Nom.
Dia., mm SILICATE CELLULAR GLASS Process Temperature, °C Process Temperature, °C 225 300 350 400 450 500 550 65 120 175 225 300 350 400 Thickness 63 75 88 100 100 100 100 38 38 50 63 75 88 100 15 Heat loss 40 51 61 72 86 104 123 9 22 33 46 60 75 88 Surface temp.
27 28 28 29 31 33 34 21 24 26 28 28 29 29 Thickness 75 88 100 100 100 100 100 38 50 63 75 88 100 100 25 Heat loss 47 58 69 86 105 125 148 12 24 37 50 65 83 108 Surface temp.
27 28 28 30 32 34 37 22 24 25 26 27 28 31 Thickness 88 100 100 100 100 125 125 38 63 75 100 100 100 100 40 Heat loss 52 65 83 102 123 134 158 14 27 42 54 76 101 132 Surface temp.
27 27 29 31 33 33 34 22 24 25 26 28 31 33 Thickness 88 100 113 125 138 150 150 38 63 75 100 100 100 113 50 Heat loss 59 72 86 102 118 136 160 16 30 45 59 81 109 135 Surface temp.
27 28 29 29 31 31 33 22 23 25 26 28 30 32 Thickness 100 113 125 138 150 150 150 38 75 88 100 100 113 125 80 Heat loss 68 84 101 118 137 68 194 21 34 52 72 101 127 155 Surface temp.
27 28 29 29 31 32 34 23 23 25 26 29 30 32 Thickness 100 113 125 138 150 163 175 50 75 100 100 100 113 125 100 Heat loss 79 97 116 136 158 180 205 21 39 57 84 117 144 178 Surface temp.
27 28 29 31 32 32 33 22 23 24 27 29 31 32 Thickness 100 113 125 138 150 175 200 50 88 100 100 113 138 150 150 Heat loss 101 124 147 171 197 215 235 29 46 71 107 138 164 204 Surface temp.
28 29 31 32 33 33 33 22 23 25 28 29 30 32 Thickness 113 125 125 150 175 200 213 63 88 100 100 125 138 163 200 Heat loss 112 138 176 192 211 234 266 29 56 87 129 155 195 229 Surface temp.
28 29 32 32 32 32 33 22 23 26 28 29 31 32 Thickness 100 125 138 150 188 188 225 63 100 100 100 138 138 175 250 Heat loss 143 161 192 224 234 259 294 36 61 102 153 171 229 254 Surface temp.
29 30 30 32 32 32 33 22 23 26 29 29 31 31 Thickness 100 125 138 175 200 213 241 63 100 100 100 138 138 188 300 Heat loss 163 184 256 227 252 288 317 40 68 116 174 193 259 273 Surface temp.
30 30 32 31 31 32 33 22 23 26 29 29 32 31 Thickness 100 125 138 175 200 225 241 63 100 100 100 138 138 200 350 Heat loss 176 197 233 242 252 296 338 45 76 129 191 211 282 282 Surface temp.
30 31 32 31 31 32 33 22 23 27 29 29 33 31 Thickness 100 138 163 188 200 225 250 63 100 100 100 138 138 200 400 Heat loss 196 203 228 255 295 325 357 51 85 143 213 233 312 310 Surface temp.
31 29 30 31 32 32 33 22 24 27 30 30 33 31 Thickness 100 138 163 188 213 225 250 63 100 100 100 138 138 200 450 Heat loss 216 223 249 278 308 353 387 57 92 158 236 256 342 337 Surface temp.
31 30 31 31 31 32 33 22 24 27 30 30 33 31 Thickness 100 138 163 188 213 241 250 63 100 100 113 138 138 200 500 Heat loss 235 242 270 300 333 366 418 62 101 172 234 278 372 364 Surface temp.
31 30 31 31 32 32 33 22 24 27 29 30 33 31 Thickness 100 138 163 188 213 241 250 63 100 100 125 138 138 200 600 Heat loss 276 282 312 346 382 420 478 73 118 201 250 323 432 419 Surface temp.
31 31 31 31 32 32 34 22 24 27 28 31 34 32 Thickness 100 138 175 200 225 250 250 63 100 100 138 138 138 200 750 Heat loss 335 339 354 393 434 479 566 89 144 244 279 389 521 501 Surface temp.
31 31 31 31 32 32 34 22 24 27 28 31 34 32 Thickness 100 163 188 200 225 250 250 63 100 100 138 138 138 200 900 Heat loss 394 345 390 456 504 554 654 106 169 287 327 456 611 583 Surface temp.
32 29 30 31 32 33 34 23 24 27 28 31 34 32 Thickness 138 163 188 213 241 250 250 63 100 100 138 138 188 213 Flat Heat loss 28 32 35 37 41 47 56 11 16 28 30 42 41 48 Surface temp.
27 28 29 29 31 32 34 23 24 28 29 32 32 34 Consult manufacturer’s literature for product temperature limitations. Table is based on typical operating conditions, e.g., 18°C ambient temperature and 12 km/h wind speed, and may not represent actual conditions of use. Units for thickness, heat loss, and surface temperature are in mm, W/m (W/m2 for flat surfaces), and °C, respectively.
25.24 2001 ASHRAE Fundamentals Handbook (SI) Kosny, J. and J.E. Christian. 1995. Reducing the uncertainties associated with using the ASHRAE zone method for R-value calculations of metal frame walls. ASHRAE Transactions 101(2).
Latta, J.K. and G.G. Boileau. 1969. Heat losses from house basements.
Canadian Building 19(10).
Lewis, W.C. 1967. Thermal conductivity of wood-base fiber and particle panel materials. Forest Products Laboratory, Research Paper FPL 77, June.
MacLean, J.D. 1941. Thermal conductivity of wood. ASHVE Transactions 47:323.
McElroy, D.L., D.W. Yarbrough, and R.S. Graves. 1987. Thickness and den-sity of loose-fill insulations after installation in residential attics. Ther-mal insulation: Materials and systems. F.J. Powell and S.L. Matthews, eds. ASTM STP 922:423-505.
McGowan, A. and A.O. Desjarlais. 1997. An investigation of common ther-mal bridges in walls. ASHRAE Transactions 103(2).
McIntyre, D.A. 1984. The increase in U-value of a wall caused by mortar joints, ECRC/M1843. The Electricity Council Research Centre, Copen-hurst, England, June.
Mitalas, G.P. 1982. Basement heat loss studies at DBR/NRC, NRCC 20416.
Division of Building Research, National Research Council of Canada, September.
Mitalas, G.P. 1983. Calculation of basement heat loss. ASHRAE Transac-tions 89(1B):420.
Prangnell, R.D. 1971. The water vapor resistivity of building materials—A literature survey. Materiaux et Constructions 4:24 (November).
Robinson, H.E., F.J. Powell, and L.A. Cosgrove. 1957. Thermal resistance of airspaces and fibrous insulations bounded by reflective surfaces.
National Bureau of Standards, Building Materials and Structures Report BMS 151.
Robinson, H.E., F.J. Powlitch, and R.S. Dill. 1954. The thermal insulation value of airspaces. Housing and Home Finance Agency, Housing Research Paper No. 32.
Sabine, H.J., M.B. Lacher, D.R. Flynn, and T.L. Quindry. 1975. Acoustical and thermal performance of exterior residential walls, doors and win-dows. NBS Building Science Series 77. National Institute of Standards and Technology, Gaithersburg, MD.
Salomone, L.A. and J.I. Marlowe. 1989. Soil and rock classification accord-ing to thermal conductivity: Design of ground-coupled heat pump sys-tems. EPRI CU-6482, Electric Power Research Institute, August.
Shipp, P.H. 1983. Basement, crawlspace and slab-on-grade thermal perfor-mance. Proceedings of the ASHRAE/DOE Conference, Thermal Perfor-mance of the Exterior Envelopes of Buildings II, ASHRAE SP 38:160-79.
Shu, L.S., A.E. Fiorato, and J.W. Howanski. 1979. Heat transmission coef-ficients of concrete block walls with core insulation. Proceedings of the ASHRAE/DOE-ORNL Conference, Thermal Performance of the Exte-rior Envelopes of Buildings, ASHRAE SP 28:421-35.
Tye, R.P. 1985. Upgrading thermal insulation performance of industrial pro-cesses. Chemical Engineering Progress (February):30-34.
Tye, R.P. 1986. Effects of product variability on thermal performance of thermal insulation. Proceedings of the First Asian Thermal Properties Conference, Beijing, People’s Republic of China.
Tye, R.P. and A.O. Desjarlais. 1983. Factors influencing the thermal perfor-mance of thermal insulations for industrial applications. Thermal insula-tion, materials, and systems for energy conservation in the ’80s. F.A.
Govan, D.M. Greason, and J.D. McAllister, eds. ASTM STP 789:733-48.
Tye, R.P. and S.C. Spinney. 1980. A study of various factors affecting the thermal performance of perlite insulated masonry construction. Dyna-tech Report No. PII-2. Holometrix, Inc. (formerly Dynatech R/D Com-pany), Cambridge, MA.
USDA. 1974. Wood handbook. Wood as an engineering material. Forest Products Laboratory, U.S. Department of Agriculture Handbook No. 72, Tables 3-7 and 4-2, and Figures 3-4 and 3-5.
Valore, R.C. 1980. Calculation of U-values of hollow concrete masonry.
American Concrete Institute, Concrete International 2(2):40-62.
Valore, R.C. 1988. Thermophysical properties of masonry and its constitu-ents, Parts I and II. International Masonry Institute, Washington, D.C.
Valore, R., A. Tuluca, and A. Caputo. 1988. Assessment of the thermal and physical properties of masonry block products (ORNL/Sub/86-22020/1), September.
Van Geem, M.G. 1985. Thermal transmittance of concrete block walls with core insulation. ASHRAE Transactions 91(2).
Wilkes, K.E. 1979. Thermophysical properties data base activities at Owens-Corning Fiberglas. Proceedings of the ASHRAE/DOE-ORNL Conference, Thermal Performance of the Exterior Envelopes of Build-ings, ASHRAE SP 28:662-77.
Yarbrough, E.W. 1983. Assessment of reflective insulations for residential and commercial applications (ORNL/TM-8891), October.
Yellott, J.I. 1965. Thermal and mechanical effects of solar radiation on steel doors. ASHRAE Transactions 71(2):42.
26.1 CHAPTER 26 VENTILATION AND INFILTRATION Basic Concepts and Terminology ........................................... 26.1 Driving Mechanisms for Ventilation and Infiltration .................................................................... 26.5 ASHRAE Standard 62 .............................................................. 26.8 Indoor Air Quality .................................................................. 26.9 Thermal Loads ........................................................................ 26.9 Natural Ventilation ............................................................... 26.10 Residential Air Leakage ........................................................ 26.12 Residential Ventilation .......................................................... 26.16 Residential Ventilation Requirements ................................... 26.18 Simplified Models of Residential Ventilation and Infiltration .................................................................. 26.20 Nonresidential Air Leakage .................................................. 26.24 Nonresidential Ventilation .................................................... 26.26 Tracer Gas Measurements .................................................... 26.26 Symbols ................................................................................. 26.28 ROVIDING a comfortable and healthy indoor environment for Pbuilding occupants is the primary concern of HVAC engineers.
Comfort and indoor air quality (IAQ) depend on many factors, including thermal regulation, control of internal and external sources of pollutants, supply of acceptable air, removal of unaccept-able air, occupants’ activities and preferences, and proper operation and maintenance of building systems. Ventilation and infiltration are only part of the acceptable indoor air quality and thermal com-fort problem. HVAC designers, occupants, and building owners must be aware of and address other factors as well. Choosing appro-priate ventilation and infiltration rates to solve thermal comfort problems and to reduce energy consumption can affect indoor air quality and may be against code, so such procedures should be approached with care and be under the direction of a registered pro-fessional engineer with expertise in HVAC analysis and design.
HVAC design engineers and others concerned with building ven-tilation and indoor air quality should obtain a copy of ASHRAE Standard 62. This standard is reviewed regularly and contains ven-tilation design and evaluation requirements for commercial and res-idential buildings. In the design of a new building or the analysis of an existing building, the version of Standard 62 that has been adopted by the local code authority must be determined. An existing building may be required to meet current code, or it may be grand-fathered under an older code. If a project involves infiltration in res-idences, then ASHRAE Standards 119 and 136 should be consulted.
The last chapter of each year’s ASHRAE Handbook (Chapter 39 of this volume) has a list of current standards.
This chapter focuses on commercial and institutional buildings, where ventilation concerns usually dominate, and on single- and multifamily residences, where infiltration is important. The basic concepts and terminology for both are presented before more advanced analytical and design techniques are given. Ventilation of industrial buildings is covered in Chapter 28 of the 1999 ASHRAE Handbook—Applications. However, many of the fundamental ideas and terminology covered in this chapter can also be applied to industrial buildings.
BASIC CONCEPTS AND TERMINOLOGY Outdoor air that flows through a building is often used to dilute and remove indoor air contaminants. However, the energy required to condition this outdoor air can be a significant portion of the total space-conditioning load. The magnitude of the outdoor airflow into the building must be known for proper sizing of the HVAC equipment and evaluation of energy consumption. For buildings without mechanical cooling and dehumidification, proper ventila-tion and infiltration airflows are important for providing comfort for occupants. ASHRAE Standard 55 specifies conditions under which 80% or more of the occupants in a space will find it ther-mally acceptable. Chapter 8 of this volume also addresses thermal comfort. Additionally, the flow of air into buildings and between zones will affect fires and the movement of smoke. Smoke manage-ment is addressed in Chapter 51 of the 1999 ASHRAE Handbook— Applications.
Ventilation and Infiltration Air exchange of outdoor air with the air already in a building can be divided into two broad classifications: ventilation and infiltra-tion. Ventilation air is air used to provide acceptable indoor air quality. It may be composed of forced or natural ventilation, infiltra-tion, suitably treated recirculated air, transfer air, or an appropriate combination (ASHRAE Standard 62). Ventilation includes the intentional introduction of air from the outside into a building; it is further subdivided into natural ventilation and forced ventilation.
Natural ventilation is the flow of air through open windows, doors, grilles, and other planned building envelope penetrations, and it is driven by natural and/or artificially produced pressure differentials.
Forced ventilation, shown in Figure 1, is the intentional movement of air into and out of a building using fans and intake and exhaust vents; it is also called mechanical ventilation.
Infiltration is the flow of outdoor air into a building through cracks and other unintentional openings and through the normal use of exterior doors for entrance and egress. Infiltration is also known as air leakage into a building. Exfiltration, depicted in Figure 1, is the leakage of indoor air out of a building through similar types of openings. Like natural ventilation, infiltration and exfiltration are driven by natural and/or artificial pressure differences. These forces are discussed in detail in the section on Driving Mechanisms for Ventilation and Infiltration. Transfer air is air that moves from one interior space to another, either intentionally or not.
The preparation of this chapter is assigned to TC 4.3, Ventilation Require-ments and Infiltration.
Fig. 1 Two-Space Building with Forced Ventilation, Infiltration, and Exfiltration 26.2 2001 ASHRAE Fundamentals Handbook (SI) These modes of air exchange differ significantly in how they affect energy consumption, air quality, and thermal comfort, and they can each vary with weather conditions, building operation, and use. Although one mode may be expected to dominate in a particular building, all must be considered for the proper design and operation of an HVAC system.
Modern commercial and institutional buildings are normally required to have forced ventilation and are usually pressurized somewhat to reduce or eliminate infiltration. Forced ventilation has the greatest potential for control of air exchange when the sys-tem is properly designed, installed, and operated; it should pro-vide acceptable indoor air quality and thermal comfort when ASHRAE Standard 62 and Standard 55 requirements are fol-lowed. Forced ventilation equipment and systems are described in Chapters 1, 2, and 9 of the 2000 ASHRAE Handbook—Systems and Equipment.
In commercial and institutional buildings, natural ventilation, such as through operable windows, may not be desirable from the point of view of energy conservation and comfort. In commercial and institutional buildings with mechanical cooling and forced ven-tilation, an air- or water-side economizer cycle may be preferable to operable windows for taking advantage of cool outdoor conditions when interior cooling is required. Infiltration may be significant in commercial and institutional buildings, especially in tall, leaky, or partially pressurized buildings and in lobby areas.
In most of the United States, residential buildings typically rely on infiltration and natural ventilation to meet their ventilation needs. Neither is reliable for ventilation purposes,because they depend on weather conditions, building construction, and mainte-nance. However, natural ventilation, usually through operable win-dows, is more likely to allow occupants to control airborne contaminants and interior air temperature, but it can have a substan-tial energy cost if used while the residence’s heating or cooling equipment is operating.
In place of operable windows, small exhaust fans may be pro-vided for localized venting in residential spaces such as kitchens and bathrooms. Not all local building codes require that the exhaust be vented to the outside. Instead, the code may allow the air to be treated and returned to the space or to be discharged to an attic space. Poor maintenance of these treatment devices can make nonducted vents ineffective for ventilation purposes. Condensa-tion in attics should be avoided. In northern Europe and in Can-ada, some building codes require general forced ventilation in residences, and heat recovery heat exchangers are popular for reducing the energy impact. Residential buildings with low rates of infiltration and natural ventilation require forced ventilation at rates given in ASHRAE Standard 62.
Forced-Air Distribution Systems Figure 2 shows a simple air-handling unit (AHU) or air-handler that conditions air for a building. Air brought back to the air handler from the conditioned space is return air (ra). The portion of the return air that is discharged to the environment is exhaust air (ea), and the part of the return air that is reused is recirculated air (ca). Air brought in intentionally from the envi-ronment is outdoor or outside air (oa). Because outside air may need treatment to be acceptable for use in a building, it should not be called “fresh air.” The outside air and the recirculated air are combined to form mixed air (ma), which is then conditioned and delivered to the thermal zone as supply air (sa). Any portion of the mixed air that intentionally or unintentionally circumvents conditioning is bypass air (ba). Due to the wide variety of air-handling systems, the airflows shown in Figure 2 may not all be present in a particular system as defined here. Also, more com-plex systems may have additional airflows.
Outside Air Fraction The outside airflow being introduced to a building or zone by an air-handling unit can also be described by the outside air fraction Xoa, which is the ratio of the volumetric flow rate of outside air brought in by the air handler to the total supply airflow rate: (1) When expressed as a percentage, the outside air fraction is called the percent outside air. The design outside airflow rate for a building’s or zone’s ventilation system is found through evaluating the require-ments of ASHRAE Standard 62. The supply airflow rate is that required to meet the thermal load. The outside air fraction and per-cent outside air then describe the degree of recirculation, where a low value indicates a high rate of recirculation, and a high value shows little recirculation. Conventional all-air air-handling sys-tems for commercial and institutional buildings have approximately 10 to 40% outside air.
100% outside air means no recirculation of return air through the air-handling system. Instead, all the supply air is treated outside air, also known as makeup air (ka), and all return air is discharged directly to the outside as relief air (la), or exhausted by separate exhaust fans. An air-handling unit that provides 100% outside air to offset air that is exhausted is typically called a makeup air unit (MAU).
When outside air via forced ventilation is used to provide venti-lation air (as is common in commercial and institutional buildings), this outside air is delivered to spaces as all or part of the supply air.
With a variable air volume (VAV) system, the outside air fraction of the supply air may need to be increased when the flow rate of the supply air is reduced to meet a particular thermal load.
Room Air Movement Air movement within spaces affects the diffusion of ventilation air and therefore indoor air quality and comfort. Two distinct flow patterns are commonly used to characterize air movement in rooms: displacement flow and entrainment flow. Displacement flow, shown in Figure 3, is the movement of air within a space in a piston-or plug-type motion. No mixing of the room air occurs in ideal displacement flow, which is desirable for removing pollutants gen-erated within a space. A laminar flow air distribution system that sweeps air across a space may produce displacement flow.
Entrainment flow, shown in Figure 4, is also known as con-ventional mixing. Systems with ceiling-based supply air diffusers and return air grilles are common examples of air distribution sys-tems that produce entrainment flow. Entrainment flow with very poor mixing within the room has been called short-circuiting flow Fig. 2 Simple All-Air Air-Handling Unit with Associated Airflows Xoa Qoa Qsa ---------Qoa Qma ----------Qoa Qoa Qca + -------------------------= = = Ventilation and Infiltration 26.3 because much of the supply air leaves the room without mixing with the room air. There is little evidence that properly designed, installed, and operated air distribution systems exhibit short-cir-cuiting behavior. There is some evidence that poorly designed, installed, or operated systems can exhibit short-circuiting behav-ior, especially ceiling-based systems in the heating mode (Offer-mann and Int-Hout 1989).
Perfect mixing occurs when supply air is instantly and evenly distributed throughout a space. Perfect mixing is also known as complete or uniform mixing; the air may be called well stirred or well mixed. This theoretical performance is approached by entrain-ment flow systems that have good mixing and by displacement flow systems that allow too much mixing (Rock et al. 1995). The outdoor air requirements given in Table 2 of ASHRAE Standard 62 assume delivery of ventilation air with perfect mixing within spaces. For more detailed information on space air diffusion, see Chapter 32.
The supply air that enters a space through a diffuser is also known as primary air. A jet is formed as this primary air leaves the diffuser. Secondary air is the room air entrained into the jet.
Total air is the combination of primary air and secondary air at a specific point in a jet. The term primary air is also used to describe the supply air provided to fan-powered mixing boxes by a central air-handling unit.
Air Exchange Rate The air exchange rate I compares airflow to volume and is (2) where Q = volumetric flow rate of air into space, m3/s V = interior volume of space, m3 The air exchange rate has units of 1/time. When the time unit is hours, the air exchange rate is also called air changes per hour (ACH). The air exchange rate may be defined for several different situations. For example, the air exchange rate for an entire building or zone served by an air-handling unit compares the amount of out-side air brought into the building or zone to the total interior volume.
This nominal air exchange rate IN is (3) where Qoa is the outdoor airflow rate including ventilation and infil-tration. The nominal air exchange rate describes the outside air ven-tilation rate entering a building or zone. It does not describe recirculation or the distribution of the ventilation air to each space within a building or zone.
For a particular space, the space air exchange rate IS compares the supply airflow rate Qsa to the volume of that space: (4) The space air exchange rate for a particular space or zone includes recirculated as well as outside air in the supply air, and it is used fre-quently in the evaluation of supply air diffuser performance and space air mixing.
Time Constants Time constants τ, which have units of time (usually in hours or seconds), are also used to describe ventilation and infiltration. One time constant is the time required for one air change in a building, zone, or space if ideal displacement flow existed. It is the inverse of the air exchange rate: (5) The nominal time constant compares the interior volume of a building or zone to the volumetric outdoor airflow rate: (6) Like the nominal air exchange rate, the nominal time constant does not describe recirculation of air within a building or zone. It also does not characterize the distribution of the outside air to individual spaces within a building or zone.
The space time constant compares the interior volume of a par-ticular space to the total supply airflow rate to that space. The space time constant is the inverse of the space air exchange rate: (7) The space time constant includes the effect of recirculated air, if present, as well as that of outside air introduced to the space through the supply air. If infiltration is significant in a space, then the infil-tration flow rate should be included when determining both the space air exchange rate and the space time constant.
Averaging Time-Varying Ventilation When assessing time-varying ventilation in terms of controlling indoor air quality, the quantity of interest is often the temporal aver-age rather than the peak. The concept of effective ventilation (Sher-man and Wilson 1986, Yuill 1986, and Yuill 1991) describes the Fig. 3 Displacement Flow Within a Space Fig. 4 Entrainment Flow Within a Space I Q V ----= IN Qoa V ---------= IS Qsa V ---------= τ 1 I --V Q ----= = τN V Qoa ---------= τS V Qsa ---------= 26.4 2001 ASHRAE Fundamentals Handbook (SI) proper ventilation rate averaging process. In this concept, the aver-age (effective) rate is the steady-state rate that would yield the same average contaminant concentration over the period of interest in the occupied space as does the actual sequence of time-varying discrete ventilation rates over the same period and in the same space. This effective rate is only equal to the simple arithmetic average rate when the discrete ventilation rates are constant over the period of interest and the contaminant concentration has reached its steady-state value. Simple arithmetic averaging of instantaneous ventila-tion rates or concentrations cannot generally be used to determine these averages due to the nonlinear response of indoor concentra-tions to the ventilation rate variations.
An important constraint in the effective ventilation concept is that the contaminant source strength F must be constant over the period of interest or must be uncorrelated with the ventilation rate.
These conditions are satisfied in many residential and commercial buildings because the emission rates of many contaminants that are controlled by whole-building ventilation vary slowly. Sherman and Wilson (1986) describe how to deal with pollutants that have step-wise constant emission rates. Pollutants such as carbon monoxide, radon, and formaldehyde, whose emission rates can be affected by ventilation, cannot be analyzed with this concept and require more complex analyses.
For constant source-strength pollutants, the relationship between effective air exchange rate, effective ventilation rate, volumetric flow, source strength, average concentration, and time-averaged effective turnover time is given by (8) The time-averaged effective turnover time in Equation (8) represents the characteristic time for the concentration in the occu-pied space to approach steady state over the period of interest. It can be determined from a sequence of discrete instantaneous ventilation air change rates Ii using the following (Sherman and Wilson 1986): (9) for Ii > 0, (10) for Ii = 0, (11) where ∆t = length of each discrete time period τe = time-averaged effective turnover time τe,i = instantaneous turnover time in period i τe,i–1 = instantaneous turnover time in previous period ASHRAE Standard 136 provides a set of factors to assist in cal-culating the annual effective air exchange rate for houses. These factors were determined using Equations (9) through (11).
Age of Air The age of air θage (Sandberg 1981) is the length of time that some quantity of outside air has been in a building, zone, or space.
The “youngest” air is at the point where outside air enters the build-ing by forced or natural ventilation or through infiltration (Grieve 1989). The “oldest” air may be at some location in the building or in the exhaust air. When the characteristics of the air distribution sys-tem are varied, a longer age of air suggests poorer outside air deliv-ery compared to a shortage of air for the same location. The age of air has units of time, usually in seconds or minutes, so it is not a true “efficiency” or “effectiveness” measure. The age of air concept, however, has gained wide acceptance in Europe and is used increas-ingly in North America.
The age of air can be evaluated for existing buildings using tracer gas methods. Using either the decay (step-down) or the growth (step-up) tracer gas method, the zone average or nominal age of air θage,N can be determined by taking concentration measurements in the exhaust air. The local age of air θage,L is evaluated through tracer gas measurements at any desired point in a space, such as at a worker’s desk. Once time-dependent data of tracer gas concentra-tion are available, the age of air can be calculated from (12) where Cin = concentration of tracer gas being injected.
Because evaluation of the age of air requires integration to infi-nite time, an exponential tail is usually added to the known concen-tration data (Farrington et al. 1990).
Air Change Effectiveness Ventilation effectiveness is a description of an air distribution system’s ability to remove internally generated pollutants from a building, zone, or space. Air change effectiveness is a description of an air distribution system’s ability to deliver ventilation air to a building, zone, or space. The HVAC design engineer does not have knowledge or control of actual pollutant sources within buildings, so Table 2 of ASHRAE Standard 62 defines outdoor air require-ments for typical, expected building uses. For most projects, there-fore, the air change effectiveness is of more relevance to HVAC system design than the ventilation effectiveness. Various definitions for air change effectiveness have been proposed. The specific mea-sure that meets the local code requirements must be determined, if any is needed at all.
Air change effectiveness measures εI are nondimensional gages of ventilation air delivery. One common definition of air change effectiveness is the ratio of a time constant to an age of air: (13) The nominal air change effectiveness εI,N shows the effective-ness of outside air delivery to the entire building, zone, or space: (14) where the nominal time constant τN is usually calculated from mea-sured airflow rates.
The local air change effectiveness εI,L shows the effectiveness of outside air delivery to one specific point in a space: (15) where τN is found either through airflow measurements or from tracer gas concentration data. An εI,L value of 1.0 indicates that the air distribution system delivers air equivalent to that of a system with perfectly mixed air in the spaces. A value less than 1.0 shows less than perfect mixing with some degree of stagnation. A value of εI,L greater than 1.0 suggests that a degree of plug or displacement flow is present at that point (Rock 1992).
Currently, the HVAC design engineer must assume that a prop-erly designed, installed, operated, and maintained air distribution Im Q V ----F VC --------1 τe ----= = = τe τe 1 N ---τe i , i 1 = N ∑ = τe i , 1 Ii∆t – ( ) exp – Ii ------------------------------------τe i 1 – , Ii∆t – ( ) exp + = τe i , ∆t τe i 1 – , + = θage Cin C – Cin Co – -------------------- θ d θ 0 = ∞ ∫ = εI τ θage ----------= εI N , τN θage N , ---------------= εI L , τN θage L , --------------= Ventilation and Infiltration 26.5 system provides an air change effectiveness of about 1. Therefore, the Table 2 values of ASHRAE Standard 62 are appropriate for the design of commercial, institutional, and residential buildings when the Ventilation Rate Procedure is used. If the Indoor Air Quality Procedure of Standard 62 is used, then actual pollutant sources and the air change effectiveness must be known for the successful design of HVAC systems that have fixed ventilation airflow rates.
ASHRAE Standard 129 describes a method for measuring air change effectiveness of mechanically vented spaces and buildings with limited air infiltration, exfiltration, and air leakage with sur-rounding indoor spaces.
DRIVING MECHANISMS FOR VENTILATION AND INFILTRATION Natural ventilation and infiltration are driven by pressure differ-ences across the building envelope caused by wind and air density differences due to temperature differences between indoor and out-door air (buoyancy, or the stack effect). Mechanical air-moving sys-tems also induce pressure differences across the envelope due to the operation of appliances, such as combustion devices, leaky forced-air thermal distribution systems, and mechanical ventilation sys-tems. The indoor-outdoor pressure difference at a location depends on the magnitude of these driving mechanisms as well as on the characteristics of the openings in the building envelope (i.e., their locations and the relationship between pressure difference and air-flow for each opening).
Stack Pressure Stack pressure is the hydrostatic pressure caused by the weight of a column of air located inside or outside a building. It can also occur within a flow element such as a duct or chimney that has vertical separation between its inlet and outlet. The hydrostatic pressure in the air depends on density and the height of interest above a refer-ence point.
Air density is a function of local barometric pressure, tempera-ture, and humidity ratio, as described by Chapter 6. As a result, stan-dard conditions should not be used to calculate the density. For example, a building site at 1500 m has an air density that is about 20% less than if the building were at sea level. An air temperature increase from −30 to 20°C causes a similar air density difference.
Combined, these elevation and temperature effects reduce the air density about 45%. Moisture effects on density are generally negli-gible, so the dry air density can be used instead, except in hot, humid climates when the air is hot and close to saturation. For example, sat-urated air at 40°C has a density about 5% greater than that of dry air.
Assuming temperature and barometric pressure are constant over the height of interest, the stack pressure decreases linearly as the separation above the reference point increases. For a single column of air, the stack pressure can be calculated as (16) where ps = stack pressure, Pa pr = stack pressure at reference height, Pa g = gravitational constant, 9.81 m/s2 ρ = indoor or outdoor air density, kg/m3 H = height above reference plane, m For tall buildings or when significant temperature stratification occurs indoors, Equation (16) should be modified to include the density gradient over the height of the building.
Temperature differences between indoors and outdoors cause stack pressure differences that drive airflows across the building envelope. Sherman (1991) showed that any single-zone building can be treated as an equivalent box from the point of view of stack effect, if its leaks follow the power law. The building is then char-acterized by an effective stack height and neutral pressure level (NPL) or leakage distribution (see the section on Neutral Pressure Level). Once calculated, these parameters can be used in physical, single-zone models to estimate infiltration.
Neglecting vertical density gradients, the stack pressure differ-ence for a horizontal leak at any vertical location is given by (17) where To = outdoor temperature, K Ti = indoor temperature, K ρo = outdoor air density, kg/m3 ρi = indoor air density, kg/m3 HNPL = height of neutral pressure level above reference plane without any other driving forces, m Chastain and Colliver (1989) showed that when there is stratifi-cation, the average of the vertical distribution of temperature differ-ences is more appropriate to use in Equation (17) than the localized temperature difference near the opening of interest.
By convention, stack pressure differences are positive when the building is pressurized relative to outdoors, which causes flow out of the building. Therefore, in the absence of other driving forces and assuming no stack effect is within the flow elements themselves, when the indoor air is warmer than outdoors, the base of the build-ing is depressurized and the top is pressurized relative to outdoors; when the indoor air is cooler than outdoors, the reverse is true.
In the absence of other driving forces, the location of the NPL is influenced by leakage distribution over the building exterior and by interior compartmentation. As a result, the NPL is not necessarily located at the mid-height of the building nor is it necessarily unique.
NPL location and leakage distribution are described later in the sec-tion on Combining Driving Forces.
For a penetration through the building envelope for which (1) there is a vertical separation between its inlet and outlet and (2) the air inside the flow element is not at the indoor or outdoor tempera-ture, such as in a chimney, more complex analyses than Equation (17) are required to determine the stack effect at any location on the building envelope.
Wind Pressure When wind impinges on a building, it creates a distribution of static pressures on the building’s exterior surface that depends on the wind direction, wind speed, air density, surface orientation, and sur-rounding conditions. Wind pressures are generally positive with respect to the static pressure in the undisturbed airstream on the windward side of a building and negative on the leeward sides. How-ever, pressures on these sides can be negative or positive, depending on wind angle and building shape. Static pressures over building sur-faces are almost proportional to the velocity pressure of the undis-turbed airstream. The wind pressure or velocity pressure is given by the Bernoulli equation, assuming no height change or pressure losses: (18) where pw = wind surface pressure relative to outdoor static pressure in undisturbed flow, Pa ρ = outside air density, kg/m3 (about 1.2) U = wind speed, m/s Cp = wind surface pressure coefficient, dimensionless ps pr ρgH – = ∆ps ρo ρi – ( )g HNPL H – ( ) = ρo To T – i Ti ---------------- g HNPL H – ( ) = pw CpρU2 2 ------= 26.6 2001 ASHRAE Fundamentals Handbook (SI) Cp is a function of location on the building envelope and wind direc-tion. Chapter 16 provides additional information on the values of Cp. Most pressure coefficient data are for winds normal to building surfaces. Unfortunately, for a real building, this fixed wind direction rarely occurs, and when the wind is not normal to the upwind wall, these pressure coefficients do not apply. A harmonic trigonometric function was developed by Walker and Wilson (1994) to interpolate between the surface average pressure coefficients on a wall that were measured with the wind normal to each of the four building surfaces. This function was developed for low-rise buildings three stories or less in height. For each wall of the building, Cp is given by (19) where Cp(1) = pressure coefficient when wind is at 0° Cp(2) = pressure coefficient when wind is at 180° Cp(3) = pressure coefficient when wind is at 90° Cp(4) = pressure coefficient when wind is at 270° φ = wind angle measured clockwise from the normal to Wall 1 The measured data used to develop the harmonic function from Akins et al. (1979) and Wiren (1985) show that typical values for the pressure coefficients are Cp(1) = 0.6, Cp(2) = −0.3, Cp(3) = Cp(4) = −0.65. Because of the geometry effects on flow around a building, the application of this interpolation function is limited to low-rise buildings that are of rectangular plan form (i.e., not L-shaped) with the longest wall less than three times the length of the shortest wall.
For less regular buildings, simple correlations are inadequate and building-specific pressure coefficients are required. Chapter 16 dis-cusses wind pressures for complex building shapes and for high-rise buildings in more detail.
The wind speed most commonly available for infiltration calcula-tions is the wind speed measured at the local weather station, typically the nearest airport. This wind speed needs to be corrected for reduc-tions due to the difference between the height where the wind speed is measured and the height of the building and reductions due to shelter effects.
The reference wind speed used to determine pressure coeffi-cients is usually the wind speed at the eaves height for low-rise buildings and the building height for high-rise buildings. However, meteorological wind speed measurements are made at a different height (typically 10 m) and at a different location. The difference in terrain between the measurement station and the building under study must also be accounted for. Chapter 16 shows how to calcu-late the effective wind speed UH from the reference wind speed Umet using boundary layer theory and estimates of terrain effects.
In addition to the reduction in wind pressures due to the reduc-tion in wind speed, the effects of local shelter also act to reduce wind pressures. The shielding effects of trees, shrubbery, and other build-ings within several building heights of a particular building produce large-scale turbulence eddies that not only reduce effective wind speed but also alter wind direction. Thus, meteorological wind speed data must be reduced carefully when applied to low buildings. Ventilation rates measured by Wilson and Walker (1991) for a row of houses showed reductions in ventilation rates of up to a fac-tor of three when the wind changed direction from perpendicular to parallel to the row. They recommended estimating wind shelter for winds perpendicular to each side of the building and then using the interpolation function in Equation (20) to find the wind shelter for intermediate wind angles: (20) where s = shelter factor for the particular wind direction φ s(i) = shelter factor when wind is normal to Wall i (i = 1 to 4, for four sides of a building) Although the above method gives a realistic variation of wind shelter effects with wind direction, estimates for the numerical val-ues of wind shelter factor s for each of the four cardinal directions must be provided. Table 11 in the section on Residential Calculation Examples lists typical shelter factors. The wind speed used in Equa-tion (18) is then given by (21) The magnitude of the pressure differences found on the surfaces of buildings varies rapidly with time because of turbulent fluctua-tions in the wind (Grimsrud et al. 1979, Etheridge and Nolan 1979). However, the use of average wind pressures to calculate pressure differences is usually sufficient to calculate average infil-tration values.
Mechanical Systems The operation of mechanical equipment, such as supply or exhaust systems and vented combustion devices, affects pressure differences across the building envelope. The interior static pressure adjusts such that the sum of all airflows through the openings in the building envelope plus equipment-induced airflows balance to zero.
To predict these changes in pressure differences and airflow rates caused by mechanical equipment, the location of each opening in the envelope and the relationship between pressure difference and airflow rate for each opening must be known. The interaction between mechanical ventilation system operation and envelope air-tightness has been discussed for low-rise buildings (Nylund 1980) and for office buildings (Tamura and Wilson 1966, 1967b; Persily and Grot 1985a).
Air exhausted from a building by a whole-building exhaust system must be balanced by increasing the airflow into the build-ing through other openings. As a result, the airflow at some loca-tions changes from outflow to inflow. For supply fans, the situation is reversed and envelope inflows become outflows.
Thus, the effects a mechanical system has on a building must be considered. Depressurization caused by an improperly designed exhaust system can increase the rate of radon entry into a build-ing and interfere with the proper operation of combustion device venting or other exhaust systems. Depressurization can also force moist outdoor air through the building envelope; for example, during the cooling season in hot humid climates, moisture may condense within the building envelope. A similar phenomenon, but in reverse, can occur during the heating season in cold cli-mates if the building is depressurized.
The interaction between mechanical systems and the building envelope also pertains to systems serving zones of buildings. The performance of zone-specific exhaust or pressurization systems is affected by the leakage in zone partitions as well as in exterior walls.
Mechanical systems can also create infiltration-driving forces in single-zone buildings. Specifically, some single-family houses with central forced-air duct systems have multiple supply registers, yet only a central return register. When internal doors are closed in these houses, large positive indoor-outdoor pressure differentials are created for rooms with only supply registers, whereas the room Cp φ ( ) 1 2 --{ Cp 1 ( ) Cp 2 ( ) + [ ] φ cos2 ( ) 1 4 ⁄ = Cp 1 ( ) Cp 2 ( ) – [ ] φ cos ( )3 4 ⁄ + Cp 3 ( ) Cp 4 ( ) + [ ] φ sin2 ( ) 2 + Cp 3 ( ) Cp 4 ( ) – [ ] φ sin } + s 1 2 --s 1 ( ) s 2 ( ) + [ ] φ cos2 s 1 ( ) s 2 ( ) – [ ] φ cos + s 3 ( ) s 4 ( ) + [ ] φ sin2 s 3 ( ) s 4 ( ) – [ ] φ sin + + = U sUH = Ventilation and Infiltration 26.7 with the return duct tends to depressurize relative to outside. This is caused by the resistance of internal door undercuts to flow from the supply register to the return (Modera et al. 1991). The magni-tudes of the indoor-outdoor pressure differentials created have been measured to average 3 to 6 Pa (Modera et al. 1991).
Building envelope airtightness and interzonal airflow resistance can also affect the performance of mechanical systems. The actual airflow rate delivered by these systems, particularly ventilation sys-tems, depends on the pressure they work against. This effect is the same as the interaction of a fan with its associated ductwork, which is discussed in Chapter 34 of this volume and Chapter 18 of the 2000 ASHRAE Handbook—Systems and Equipment. The building enve-lope and its leakage must be considered part of the ductwork in determining the pressure drop of the system.
Duct leakage can cause similar problems. Supply leaks to the outside will tend to depressurize the building; return leaks to the outside will tend to pressurize it.
Combining Driving Forces The pressure differences due to wind pressure, stack pressure, and mechanical systems are considered in combination by adding them together and then determining the airflow rate through each opening due to this total pressure difference. The air flows must be determined in this manner, as opposed to adding the airflow rates due to the separate driving forces, because the airflow rate through each opening is not linearly related to pressure difference. For uniform indoor air temperatures, the total pressure difference across each leak can be written in terms of a reference wind param-eter PU and stack effect parameter PT common to all leaks: (22) (23) where T = air temperature, K.
The pressure difference across each leak (with positive pressures for flow into the building) is then given by (24) where ∆pI = pressure that acts to balance inflows and outflows (including mechanical system flows). Equation (24) can then be applied to every leak for the building with the appropriate values of Cp, s, and H. Thus, each leak is defined by its pressure coefficient, shelter, and height. Where indoor pressures are not uniform, more complex analyses are required.
Neutral Pressure Level The neutral pressure level (NPL) is that location or locations in the building envelope where there is no pressure difference. Internal partitions, stairwells, elevator shafts, utility ducts, chimneys, vents, operable windows, and mechanical supply and exhaust systems complicate the analysis of NPL location. An opening with a large area relative to the total building leakage causes the NPL to shift toward the location of the opening. In particular, chimneys and openings at or above roof height raise the NPL in small buildings.
Exhaust systems increase the height of the NPL; outdoor air supply systems lower it.
Figure 5 qualitatively shows the addition of driving forces for a building with uniform openings above and below mid-height and without significant internal resistance to airflow. The slopes of the pressure lines are a function of the densities of the indoor and out-door air. In Figure 5A, with inside air warmer than outside and pressure differences caused solely by thermal forces, the NPL is at mid-height, with inflow through lower openings and outflow through higher openings. Direction of flow is always from the higher to the lower pressure region. Figure 5B presents qualitative uniform pressure differences caused by wind alone, with opposing effects on the windward and leeward sides. When the temperature difference and wind effects both exist, the pressures due to each are added together to determine the total pressure difference across the building envelope. In Figure 5B, there is no NPL because no locations on the building envelope have zero pressure difference. Figure 5C shows the combination, where the wind force of Figure 5B has just balanced the thermal force of Figure 5A, causing no pressure difference at the top wind-ward or bottom leeward side.
The relative importance of the wind and stack pressures in a building depends on building height, internal resistance to vertical airflow, location and flow resistance characteristics of envelope PU ρo UH 2 2 -------= PT gρo To Ti – Ti ---------------- = ∆p s2CpPU HPT ∆pI + + = Fig. 5 Distribution of Inside and Outside Pressures over Height of Building 26.8 2001 ASHRAE Fundamentals Handbook (SI) openings, local terrain, and the immediate shielding of the building.
The taller the building is and the smaller its internal resistance to air-flow, the stronger the stack effect will be. The more exposed a build-ing is, the more susceptible it will be to wind. For any building, there will be ranges of wind speed and temperature difference for which the building’s infiltration is dominated by the stack effect, the wind, or the driving pressures of both (Sinden 1978a). These building and terrain factors determine, for specific values of temperature differ-ence and wind speed, in which regime the building’s infiltration lies.
The effect of mechanical ventilation on envelope pressure differ-ences is more complex and depends on both the direction of the ven-tilation flow (exhaust or supply) and the differences in these ventilation flows among the zones of the building. If mechanically supplied outdoor air is provided uniformly to each story, the change in the exterior wall pressure difference pattern is uniform. With a nonuniform supply of outdoor air (for example, to one story only), the extent of pressurization varies from story to story and depends on the internal airflow resistance. Pressurizing all levels uniformly has little effect on the pressure differences across floors and vertical shaft enclosures, but pressurizing individual stories increases the pressure drop across these internal separations. Pressurization of the ground level is often used in tall buildings to reduce the stack pres-sures across entries.
Available data on the NPL in various kinds of buildings are lim-ited. The NPL in tall buildings varies from 0.3 to 0.7 of total build-ing height (Tamura and Wilson 1966, 1967a). For houses, especially houses with chimneys, the NPL is usually above mid-height. Oper-ating a combustion heat source with a flue raises the NPL further, sometimes above the ceiling (Shaw and Brown 1982).
Thermal Draft Coefficient Compartmentation of a building also affects the NPL location.
Equation (17) provides a maximum stack pressure difference, given no internal airflow resistance. The sum of the pressure differences across the exterior wall at the bottom and at the top of the building, as calculated by these equations, equals the total theoretical draft for the building. The sum of the actual top and bottom pressure differ-ences, divided by the total theoretical draft pressure difference, equals the thermal draft coefficient. The value of the thermal draft coefficient depends on the airflow resistance of the exterior walls relative to the airflow resistance between floors. For a building without internal partitions, the total theoretical draft is achieved across the exterior walls (Figure 6A), and the thermal draft coeffi-cient equals 1. In a building with airtight separations at each floor, each story acts independently, its own stack effect being unaffected by that of any other floor (Figure 6B). The theoretical draft is min-imized in this case, and each story has an NPL.
Real multistory buildings are neither open inside (Figure 6A), nor airtight between stories (Figure 6B). Vertical air passages, stair-wells, elevators, and other service shafts allow airflow between floors. Figure 6C represents a heated building with uniform open-ings in the exterior wall, through each floor, and into the vertical shaft at each story. Between floors, the slope of the line representing the inside pressure is the same as that shown in Figure 6A, and the discontinuity at each floor (Figure 6B) represents the pressure dif-ference across it. Some of the pressure difference maintains flow through openings in the floors and vertical shafts. As a result, the pressure difference across the exterior wall at any level is less than it would be with no internal flow resistance.
Maintaining airtightness between floors and from floors to vertical shafts is a means of controlling indoor-outdoor pressure differences due to the stack effect and therefore infiltration (Lovaett and Wilson 1994). Good separation is also conducive to the proper operation of mechanical ventilation and smoke man-agement systems. However, care is needed to avoid pressure dif-ferences that could prevent door opening in an emergency.
Tamura and Wilson (1967b) showed that when vertical shaft leakage is at least two times the envelope leakage, the thermal draft coefficient is almost one and the effect of compartmentation is negligible. Measurements of pressure differences in three tall office buildings by Tamura and Wilson (1967a) indicated that the thermal draft coefficient ranged from 0.8 to 0.9 with the ventila-tion systems off.
ASHRAE Standard 62 ASHRAE Standard 62 provides guidance on ventilation and indoor air quality and includes two alternative procedures for determining design ventilation rates. In the Ventilation Rate Pro-cedure, indoor air quality is assumed to be acceptable if (1) the concentrations of six pollutants in the incoming outdoor air meet the U.S. EPA national ambient air quality standards (EPA 1986), and (2) the outdoor air ventilation rates meet or exceed values (which depend on the space type) provided in a table. The minimum outside air ventilation per person for any type of space is 8 L/s. This minimum rate will maintain the indoor CO2 con-centration within 0.07% (700 parts per million) of the outdoor con-centration, assuming a typical CO2 generation rate per occupant Fig. 6 Compartmentation Effect in Buildings Ventilation and Infiltration 26.9 (Janssen 1989). This minimum outside air ventilation rate was based, in part, on research by Berg-Munch et al. (1986), Yaglou et al. (1936), Iwashita et al. (1989), and Cain et al. (1993) that indi-cated that 8 L/s was required to satisfy the odor perceptions of 80% or more of visitors.
The other design approach in ASHRAE Standard 62 is the Indoor Air Quality Procedure. In this procedure, acceptable indoor air quality is achieved through the control of indoor contam-inant concentrations. Such control can be realized through source control, air cleaning, and ventilation. It allows for either or both improved indoor air quality and reduced energy consumption.
Chapter 24 of the 2000 ASHRAE Handbook—Systems and Equip-ment has information on air cleaning.
A combination of source control and local exhaust, as opposed to dilution with ventilation air, is the method of choice in industrial environments. Industrial ventilation is discussed in Chapters 28 and 29 of the 1999 ASHRAE Handbook—Applications and in Industrial Ventilation: A Manual of Recommended Practice (ACGIH 1998).
INDOOR AIR QUALITY Outdoor air requirements for acceptable indoor air quality (IAQ) have long been debated, and different rationales have produced rad-ically different ventilation standards (Grimsrud and Teichman 1989, Janssen 1989, Klauss et al. 1970, Yaglou et al. 1936, Yaglou and Witheridge 1937). Historically, the major considerations have included the amount of outdoor air required to control moisture, car-bon dioxide (CO2), odors, and tobacco smoke generated by occu-pants. These considerations have led to prescriptions of a minimum rate of outdoor air supply per occupant. More recently, the mainte-nance of acceptable indoor concentrations of a variety of additional pollutants that are not generated primarily by occupants has been a major concern. Information on contaminants can be found in Chap-ter 12, and odors are covered in Chapter 13.
Indoor pollutant concentrations depend on the strength of pollut-ant sources and the total rate of pollutant removal. Pollutant sources include the outdoor air; indoor sources such as occupants, furnish-ings, and appliances; and the soil adjacent to the building. Pollutant removal processes include dilution with outside air, local exhaust ventilation, deposition on surfaces, chemical reactions, and air-cleaning processes. If (1) general building ventilation is the only significant pollutant removal process, (2) the indoor air is thor-oughly mixed, and (3) the pollutant source strength and ventilation rate have been stable for a sufficient period; then the steady-state indoor pollutant concentration is given by (25) where Ci = steady-state indoor concentration, µg/m3 Co = outdoor concentration, µg/m3 S = total pollutant source strength, µg/s Qoa = ventilation rate, m3/s Variation in pollutant source strengths (rather than variation in ventilation rate) is considered the largest cause of building-to-building variation in the concentrations of pollutants that are not generated by occupants. Turk et al. (1989) found that a lack of correlation between average indoor respirable particle concentra-tions and whole-building outdoor ventilation rate indicated that source strength, high outdoor concentrations, building volume, and removal processes are important. Because pollutant source strengths are highly variable, maintenance of minimum ventila-tion rates does not ensure acceptable indoor air quality in all situ-ations. The lack of health-based concentration standards for many indoor air pollutants, primarily due to the lack of health data, makes the specification of minimum ventilation rates even more difficult.
In cases of high contaminant source strengths, impractically high rates of ventilation are required to control contaminant levels, and other methods of control are more effective. Removal or reduction of contaminant sources is a very effective means of control. Con-trolling a localized source by means of local exhaust, such as range hoods or bathroom exhaust fans, can also be effective.
Particles can be removed with various types of air filters. Gas-eous contaminants with higher relative molecular mass can be con-trolled with activated carbon or alumina pellets impregnated with a substance such as potassium permanganate. Chapter 24 of the 2000 ASHRAE Handbook—Systems and Equipment has information on air cleaning.
THERMAL LOADS Outdoor air introduced into a building constitutes a large part of the total space-conditioning (heating, cooling, humidification, and dehumidification) load, which is one reason to limit air exchange rates in buildings to the minimum required. Air exchange typically represents 20 to 50% of a building’s thermal load. Chapter 28 and Chapter 29 cover thermal loads in more detail.
Air exchange increases a building’s thermal load in three ways.
First, the incoming air must be heated or cooled from the outdoor air temperature to the indoor or supply air temperature. The rate of energy consumption due to this sensible heating or cooling is given by (26) where qs = sensible heat load, W Q = airflow rate, m3/s ρ = air density, kg/m3 (about 1.2) cp = specific heat of air, J/(kg·K) (about 1000) ∆t = temperature difference between indoors and outdoors, K Second, air exchange modifies the moisture content of the air in a building. The rate of energy consumption associated with these latent loads (neglecting the energy associated with any condensate) is given by (27) where ql = latent heat load, W ∆W = humidity ratio difference between indoors and outdoors, mass water/unit mass dry air (kg/kg) Finally, air exchange can change a building’s thermal load by altering the performance of the envelope insulation system. Air-flow through the insulation can decrease the thermal load due to heat exchange between infiltrating or exfiltrating air and the ther-mal insulation. Conversely, air moving in and out of the insulation from outside can increase the thermal load. Experimental and numerical studies have demonstrated that significant thermal cou-pling can occur between air leakage and insulation layers, thereby modifying the heat transmission in building envelopes. In particu-lar, a number of researchers (Wolf 1966; Berlad et al. 1978; Bankvall 1987; Lecompte 1987) have shown that convective air-flow through air-permeable insulation in an envelope assembly may degrade its effective thermal resistance. This R-value degrada-tion occurs when outside air moves through and/or around the insu-lation within the wall cavity and returns to the outdoors without reaching the conditioned space. A literature review by Powell et al.
(1989) summarized the findings about air movement effects on the effective thermal resistance of porous insulation under various con-ditions. The effect of such airflow on insulation system perfor-mance is difficult to quantify but should be considered. Airflow Ci Co S Qoa ⁄ + = qs Qρcp∆t 1200Q t ∆ = = ql Q W 4775 1.998∆t + [ ] ∆ = 26.10 2001 ASHRAE Fundamentals Handbook (SI) within the insulation system can also decrease the system’s perfor-mance due to moisture condensation in and on the insulation.
Even if air flows only through cracks instead of through the insu-lation, the actual heating/cooling load due to the combined effect of conduction and airflow heat transfer can be lower than the heat-ing/cooling load calculated by Equation (26). This reduction in total heating/cooling load is a consequence of the thermal coupling between conduction and convection heat transfer and is called infil-tration heat recovery (IHR). This effect appears to be significant in building envelopes, based on preliminary laboratory and numerical work under controllable conditions by several investigators.
Using a computer simulation, Kohonen et al. (1987) found that the conduction/infiltration thermal interaction reduced total heat-ing load by 15%. Several experimental studies by Claridge et al.
(1988), Claridge and Bhattacharyya (1990), Liu and Claridge (1992a, 1992b, 1992c, 1995), Timusk et al. (1992), and others using a test cell under both steady-state and dynamic conditions found that the actual energy attributed to air infiltration can be 20 to 80% of the values given by Equation (27). Judkoff et al. (1997) measured heat recovery using a mobile home under steady-state conditions. They found that up to 40% heat recovery occurs during exfiltration through the envelope of the mobile home. Buchanan and Sherman (2000) carried out two- and three-dimensional com-putational fluid dynamics simulations to study the fundamental physics of the IHR process and developed a simple macro-scale mathematical model based on the steady-state one-dimensional convection-diffusion equation to predict a heat recovery factor.
Their results show that the traditional method may overpredict the infiltration energy load.
Infiltration Degree-Days Heating and cooling degree-days are a simple way to character-ize the severity of a particular climate. Heating and cooling degree-day values are based on sensible temperature data, but infiltration loads are both sensible and latent. Infiltration degree days (IDDs) more fully describe a climate and can be used to estimate heat loss or gain due to infiltration in residences (Sherman 1986). Total infil-tration degree-days is the sum of the heating and cooling infiltration degree-days and is calculated from hour-by-hour weather data and base conditions using weather weighted by infiltration rate. The selection of base conditions is an important part of the calculation of the IDDs. ASHRAE Standard 119 lists IDDs for many locations with a particular set of base conditions.
NATURAL VENTILATION Natural ventilation is the flow of outdoor air due to wind and ther-mal pressures through intentional openings in the building’s shell.
Under some circumstances, it can effectively control both tempera-ture and contaminants in mild climates, but it is not considered prac-tical in hot and humid climates or in cold climates. Temperature control by natural ventilation is often the only means of providing cooling when mechanical air conditioning is not available. The arrangement, location, and control of ventilation openings should combine the driving forces of wind and temperature to achieve a desired ventilation rate and good distribution of ventilation air through the building. However, intentional openings cannot always guarantee adequate temperature and humidity control or indoor air quality because of the dependence on natural (wind and stack) effects to drive the flow (Wilson and Walker 1992).
Natural Ventilation Openings Natural ventilation openings include (1) windows, doors, dormer (monitor) openings, and skylights; (2) roof ventilators; (3) stacks connecting to registers; and (4) specially designed inlet or outlet openings.
Windows transmit light and provide ventilation when open.
They may open by sliding vertically or horizontally; by tilting on horizontal pivots at or near the center; or by swinging on pivots at the top, bottom, or side. The type of pivoting used is important for weather protection and affects airflow rate.
Roof ventilators provide a weather-resistant air outlet. Capacity is determined by the ventilator’s location on the roof; the resistance to airflow of the ventilator and its ductwork; the ventilator’s ability to use kinetic wind energy to induce flow by centrifugal or ejector action; and the height of the draft.
Natural-draft or gravity roof ventilators can be stationary, pivot-ing, oscillating, or rotating. Selection criteria include ruggedness, corrosion resistance, stormproofing features, dampers and operat-ing mechanisms, noise, cost, and maintenance. Natural ventilators can be supplemented with power-driven supply fans; the motors need only be energized when the natural exhaust capacity is too low.
Gravity ventilators can have manual dampers or dampers controlled by thermostat or wind velocity.
A natural-draft roof ventilator should be positioned so that it receives the full, unrestricted wind. Turbulence created by sur-rounding obstructions, including higher adjacent buildings, impairs a ventilator’s ejector action. The ventilator inlet should be conical or bell mounted to give a high flow coefficient. The opening area at the inlet should be increased if screens, grilles, or other structural mem-bers cause flow resistance. Building air inlets at lower levels should be larger than the combined throat areas of all roof ventilators.
Stacks or vertical flues should be located where wind can act on them from any direction. Without wind, stack effect alone removes air from the room with the inlets.
Required Flow for Indoor Temperature Control The ventilation airflow rate required to remove a given amount of heat from a building can be calculated from Equations (26) and (27) if the quantity of heat to be removed and the indoor-outdoor temperature difference are known.
Airflow Through Large Intentional Openings The relationship describing the airflow through a large inten-tional opening is based on the Bernoulli equation with steady, incompressible flow. The general form that includes stack, wind, and mechanical ventilation pressures across the opening is (28) where Q = airflow rate, m3/s CD = discharge coefficient for opening, dimensionless A = cross-sectional area of opening, m2 ρ = air density, kg/m3 ∆p = pressure difference across opening, Pa The discharge coefficient CD is a dimensionless number that depends on the geometry of the opening and the Reynolds number of the flow.
Flow Caused by Wind Only Factors due to wind forces that affect the ventilation rate include average speed, prevailing direction, seasonal and daily variation in speed and direction, and local obstructions such as nearby build-ings, hills, trees, and shrubbery. Liddament (1988) reviewed the rel-evance of wind pressure as a driving mechanism. A multiflow path simulation model was developed and used to illustrate the effects of wind on air exchange rate.
Wind speeds may be lower in summer than in winter; direc-tional frequency is also a function of season. Natural ventilation systems are often designed for wind speeds of one-half the sea-sonal average. The following equation shows the rate of air forced Q CDA 2 p ∆ ρ ⁄ = Ventilation and Infiltration 26.11 through ventilation inlet openings by wind or determines the proper size of openings to produce given airflow rates: (29) where Q = airflow rate, m3/s Cv = effectiveness of openings (Cv is assumed to be 0.5 to 0.6 for perpendicular winds and 0.25 to 0.35 for diagonal winds) A = free area of inlet openings, m2 U = wind speed, m/s Inlets should face directly into the prevailing wind. If they are not advantageously placed, flow will be less than that predicted by Equation (29); if the inlets are unusually well placed, flow will be slightly more. Desirable outlet locations are (1) on the leeward side of the building directly opposite the inlet, (2) on the roof, in the low-pressure area caused by a flow discontinuity of the wind, (3) on the side adjacent to the windward face where low-pressure areas occur, (4) in a dormer on the leeward side, (5) in roof ventilators, or (6) by stacks. Chapter 16 gives a general description of the wind pressure distribution on a building. The inlets should be placed in the exterior high-pressure regions; the outlets should be placed in the exterior low-pressure regions.
Flow Caused by Thermal Forces Only If building internal resistance is not significant, the flow caused by stack effect can be expressed by (30) where Q = airflow rate, m3/s CD = discharge coefficient for opening ∆HNPL = height from midpoint of lower opening to NPL, m Ti = indoor temperature, K To = outdoor temperature, K Equation (30) applies when Ti > To. If Ti < To, replace Ti in the denominator with To, and replace (Ti − To) in the numerator with (To − Ti). An average temperature should be used for Ti if there is thermal stratification. If the building has more than one opening, the outlet and inlet areas are considered equal. The discharge coef-ficient CD accounts for all viscous effects such as surface drag and interfacial mixing.
Estimation of ∆HNPL is difficult for naturally ventilated build-ings. If one window or door represents a large fraction (approxi-mately 90%) of the total opening area in the envelope, then the NPL is at the mid-height of that aperture, and ∆HNPL equals one-half the height of the aperture. For this condition, flow through the opening is bidirectional (i.e., air from the warmer side flows through the top of the opening, and air from the colder side flows through the bot-tom). Interfacial mixing occurs across the counterflow interface, and the orifice coefficient can be calculated according to the follow-ing equation (Kiel and Wilson 1986): (31) If enough other openings are available, the airflow through the opening will be unidirectional, and mixing cannot occur. A dis-charge coefficient of CD = 0.65 should then be used. Additional information on stack-driven airflows for natural ventilation can be found in Foster and Down (1987).
Greatest flow per unit area of openings is obtained when inlet and outlet areas are equal; Equations (30) and (31) are based on this equality. Increasing the outlet area over inlet area (or vice versa) increases airflow but not in proportion to the added area. When openings are unequal, use the smaller area in Equation (30) and add the increase as determined from Figure 7.
Natural Ventilation Guidelines Several general guidelines should be observed in designing for natural ventilation. Some of these may conflict with other climate-responsive strategies (such as using orientation and shading devices to minimize solar gain) or other design considerations.
1. In hot, humid climates, use mechanical cooling. If mechanical cooling is not available, air velocities should be maximized in the occupied zones. In hot, arid climates, consider evaporative cooling. Airflow throughout the building should be maximized for structural cooling, particularly at night when the tempera-ture is low.
2. Topography, landscaping, and surrounding buildings should be used to redirect airflow and give maximum exposure to breezes. Vegetation can funnel breezes and avoid wind dams, which reduce the driving pressure differential around the build-ing. Site objects should not obstruct inlet openings.
3. The building should be shaped to expose maximum shell open-ings to breezes.
4. Architectural elements such as wing walls, parapets, and overhangs should be used to promote airflow into the build-ing interior.
5. The long facade of the building and the majority of the door and window openings should be oriented with respect to the prevailing summer breezes. If there is no prevailing direction, openings should be sufficient to provide ventilation regardless of wind direction.
6. Windows should be located in opposing pressure zones. Two openings on opposite sides of a space increase the ventilation flow. Openings on adjacent sides force air to change direction, providing ventilation to a greater area. The benefits of the win-dow arrangement depend on the outlet location relative to the direction of the inlet airstream.
7. If a room has only one external wall, better airflow is achieved with two widely spaced windows.
8. If the openings are at the same level and near the ceiling, much of the flow may bypass the occupied level and be ineffective in diluting contaminants there.
9. Vertical distance between openings is required to take advan-tage of the stack effect; the greater the vertical distance, the greater the ventilation.
Q CvAU = Q CDA 2g HNPL ∆ Ti To – ( ) Ti ⁄ = Fig. 7 Increase in Flow Caused by Excess Area of One Opening over the Other CD 0.40 0.0045 Ti To – + = 26.12 2001 ASHRAE Fundamentals Handbook (SI) 10. Openings in the vicinity of the NPL are least effective for ther-mally induced ventilation. If the building has only one large opening, the NPL tends to move to that level, which reduces the pressure across the opening.
11. Greatest flow per unit area of total opening is obtained by inlet and outlet openings of nearly equal areas. An inlet window smaller than the outlet creates higher inlet velocities. An outlet smaller than the inlet creates lower but more uniform airspeed through the room.
12. Openings with areas much larger than calculated are some-times desirable when anticipating increased occupancy or very hot weather.
13. Horizontal windows are generally better than square or vertical windows. They produce more airflow over a wider range of wind directions and are most beneficial in locations where pre-vailing wind patterns shift.
14. Window openings should be accessible to and operable by occupants.
15. Inlet openings should not be obstructed by indoor partitions.
Partitions can be placed to split and redirect airflow but should not restrict flow between the building’s inlets and outlets.
16. Vertical airshafts or open staircases can be used to increase and take advantage of stack effects. However, enclosed staircases intended for evacuation during a fire should not be used for ventilation.
RESIDENTIAL AIR LEAKAGE Most infiltration in residential buildings in the U.S. is dominated by envelope leakage. However, trends in new construction are towards tighter envelopes such that envelope leakage is reduced in newer housing.
Envelope Leakage Measurement Envelope leakage of a building can be measured with pressur-ization testing (commonly called a blower-door test). Fan pres-surization is relatively quick and inexpensive, and it characterizes building envelope airtightness independent of weather conditions.
In this procedure, a large fan or blower is mounted in a door or window and induces a large and roughly uniform pressure differ-ence across the building shell (ASTM Standard E 779, ASTM Standard E 1827, ISO Standard 9972, and CGSB Standard 149.10.
The airflow required to maintain this pressure difference is then measured. The leakier the building is, the more airflow is neces-sary to induce a specific indoor-outdoor pressure difference. The airflow rate is generally measured at a series of pressure differ-ences ranging from about 10 Pa to 75 Pa.
The results of a pressurization test, therefore, consist of several combinations of pressure difference and airflow rate data. An exam-ple of typical data is shown in Figure 8. These data points charac-terize the air leakage of a building and are generally converted to a single value that serves as a measure of the building’s airtightness.
There are several different measures of airtightness, most of which involve fitting the data to a curve describing the relationship between the airflow Q through an opening in the building envelope and the pressure difference ∆p across it. This relationship is called the leakage function of the opening. The form of the leakage function depends on the geometry of the opening. Background the-oretical material relevant to leakage functions may be found in Hop-kins and Hansford (1974), Etheridge (1977), Kronvall (1980), Chastain et al. (1987), and Walker et al. (1997).
The openings in a building envelope are not uniform in geometry and, generally, the flow never becomes fully developed. Each open-ing in the building envelope can be described by Equation (32), commonly called the power law equation: (32) where c = flow coefficient, m3/(s·Pan) n = pressure exponent, dimensionless Sherman (1992b) showed how the power law can be developed analytically by looking at developing laminar flow in short pipes.
Equation (32) only approximates the relationship between Q and ∆p. Measurements of single cracks (Honma 1975, Krieth et al.
1957) have shown that n can vary if ∆p changes over a wide range.
Additional investigation of pressure/flow data for simple cracks by Chastain et al. (1987) further indicated the importance of ade-quately characterizing the three-dimensional geometry of openings and the entrance and exit effects. Walker et al. (1997) showed that for the arrays of cracks in a building envelope over the range of pressures acting during infiltration, n is constant. A typical value for n is about 0.65. c and n can be determined for a building using fan pressurization techniques.
Airtightness Ratings In some cases, the predicted airflow rate is converted to an equiv-alent or effective air leakage area as follows: (33) where AL = equivalent or effective air leakage area, cm2 Qr = predicted airflow rate at ∆pr (from curve fit to pressurization test data), m3/s ρ = air density, kg/m3 ∆pr = reference pressure difference, Pa CD = discharge coefficient All the openings in the building shell are combined into an over-all opening area and discharge coefficient for the building when the equivalent or effective air leakage area is calculated. Some users of the leakage area approach set CD = 1. Others set CD ≈ 0.6 (i.e., the discharge coefficient for a sharp-edged orifice). The air leakage area Fig. 8 Airflow Rate Versus Pressure Difference Data from Whole-House Pressurization Test Q c p ∆ ( )n = AL 10 000Qr ρ 2 pr ∆ ⁄ CD -------------------------= Ventilation and Infiltration 26.13 of a building is, therefore, the area of an orifice (with an assumed value of CD) that would produce the same amount of leakage as the building envelope at the reference pressure.
An airtightness rating, whether based on an air leakage area or a predicted airflow rate, is generally normalized by some factor to account for building size. Normalization factors include floor area, exterior envelope area, and building volume.
With the wide variety of possible approaches to normalization and reference pressure difference, and the use of the air leakage area concept, many different airtightness ratings are being used. Refer-ence pressure differences include 4, 10, 25, 50, and 75 Pa. Refer-ence pressure differences of 4 and 10 Pa are advocated because they are closer to the pressure differences that actually induce air exchange and, therefore, better model the flow characteristics of the openings. While this may be true, they are outside the range of mea-sured values in the test; therefore, the predicted airflow rates at 4 and 10 Pa are subject to significant uncertainty. The uncertainty in these predicted airflow rates and the implications for quantifying airtightness are discussed in Persily and Grot (1985b), Chastain (1987), and Modera and Wilson (1990). Round robin tests by Mur-phy et al. (1991) to determine the repeatability and reproducibility of fan pressurization devices found that subtle errors in fan calibra-tion or operator technique are greatly exaggerated when extrapolat-ing the pressure versus flow curve out to 4 Pa, with errors as great as ±40%, mainly due to the fan calibration errors at low flow.
Some common airtightness ratings include the effective air leak-age area at 4 Pa assuming CD = 1.0 (Sherman and Grimsrud 1980); the equivalent air leakage area at 10 Pa assuming CD = 0.611 (CGSB Standard 149.10); and the airflow rate at 50 Pa, divided by the build-ing volume to give units of air changes per hour (Blomsterberg and Harrje 1979).
Conversion Between Ratings Air leakage areas at one reference pressure difference can be converted to air leakage areas at an other reference pressure differ-ence according to: (34) where Ar,1 = air leakage area at reference pressure difference ∆pr,1, cm2 Ar,2 = air leakage area at reference pressure difference ∆pr,2, cm2 CD,1 = discharge coefficient used to calculate Ar,1 CD,2 = discharge coefficient used to calculate Ar,2 n = pressure exponent from Equation (32) An air leakage area at one reference pressure difference can be converted to an airflow rate at some other reference pressure differ-ence according to (35) where Qr,2 = airflow rate at reference difference ∆pr,2, m3/s.
The flow coefficient c in Equation (32) may be converted to an air leakage area according to (36) Finally, an air leakage area may be converted to the flow coeffi-cient c in Equation (32) according to (37) Equations (34) through (37) require the assumption of a value of n, unless it is reported with the measurement results. When whole-building pressurization test data are fitted to Equation (32), the value of n generally lies between 0.6 and 0.7. Therefore, using a value of n in this range is reasonable.
Building Air Leakage Data Fan pressurization measures a building property that ideally var-ies little with time and weather conditions. In reality, unless the wind and temperature differences during the measurement period are sufficiently mild, the pressure differences they induce during the test will interfere with the test pressures and cause measurement errors. Persily (1982) and Modera and Wilson (1990) studied the effects of wind speed on pressurization test results. Several experi-mental studies have also shown variations on the order of 20 to 40% over a year in the measured airtightness in homes (Persily 1982, Kim and Shaw 1986, Warren and Webb 1986).
Figure 9 summarizes envelope leakage measured North Ameri-can housing (Sherman and Dickerhoff 1998) and from several Euro-pean and Canadian sources (AIVC 1994). This figure shows the large range of measured envelope tightness but can still be used to illustrate typical and extreme values in the housing stock.
ASHRAE Standard 119 establishes air leakage performance levels for residential buildings. These levels are in terms of the normalized leakage area An: (38) where An = normalized leakage area, dimensionless AL = effective leakage area at 4 Pa (CD = 1.0), cm2 Af = gross floor area (within exterior walls), m2 H = building height, m Ho = reference height of one-story building = 2.5 m Air Leakage of Building Components The fan pressurization procedure discussed in the section on En-velope Leakage Measurement enables the measurement of whole-building air leakage. The location and size of individual openings in building envelopes are extremely important because they influence the air infiltration rate of a building as well as the heat and moisture transfer characteristics of the envelope. Additional test procedures exist for pressure-testing individual building components such as windows, walls, and doors; they are discussed in ASTM Standards E 283 and E 783 for laboratory and field tests, respectively.
Leakage Distribution Dickerhoff et al. (1982) and Harrje and Born (1982) studied the air leakage of individual building components and systems. The fol-lowing points summarize the percentages of whole-building air leakage area associated with various components and systems. The values in parentheses include the range determined for each compo-nent and the mean of the range.
Walls (18 to 50%; 35%). Both interior and exterior walls con-tribute to the leakage of the structure. Leakage between the sill plate and the foundation, cracks below the bottom of the gypsum wall-board, electrical outlets, plumbing penetrations, and leaks into the attic at the top plates of walls all occur.
Ceiling details (3 to 30%; 18%). Leakage across the top ceiling of the heated space is particularly insidious because it reduces the effectiveness of insulation on the attic floor and contributes to infil-Ar 2 , Ar 1 , CD 1 , CD 2 , ----------- pr 2 , ∆ pr 1 , ∆ ------------ n 0.5 – = Qr 2 , CD 1 , Ar 1 , 10 000 ---------------------2 ρ ---pr 1 , ∆ ( )0.5 n – pr 2 , ∆ ( )n = AL 10 000 c CD --------------------ρ 2 ---∆pr n 0.5 – ( ) = c CDAL 10 000 ----------------2 ρ --- ∆pr ( )0.5 n – = An 0.1 AL Af ------ H Ho ------ 0.3 = 26.14 2001 ASHRAE Fundamentals Handbook (SI) tration heat loss. Ceiling leakage also reduces the effectiveness of ceiling insulation in buildings without attics. Recessed lighting, plumbing, and electrical penetrations leading to the attic are some particular areas of concern.
Forced-air heating and/or cooling systems (3 to 28%; 18%).
The location of the heating or cooling equipment, air handler, or ductwork in conditioned or unconditioned spaces; the venting arrangement of a fuel-burning device; and the existence and loca-tion of a combustion air supply all affect leakage. Modera et al.
(1991) and Robison and Lambert (1989), among others, have shown that the variability of leakage in ducts passing through uncondi-tioned spaces is high, the coefficient of variation being on the order of 50%. Field studies have also shown that in-situ repairs can elim-inate one-quarter to two-thirds of the observed leakage (Cummings and Tooley 1989, Cummings et al. 1990, Robison and Lambert 1989, Jump et al. 1996). The 18% contribution of ducts to total leak-age significantly underestimates their impact because during sys-tem operation, the pressure differentials across the duct leaks are approximately ten times higher than typical pressure differences across the envelope leaks (Modera 1989, Modera et al. 1991) and result in large (factors of two to three) changes in ventilation rate (Walker 1999, Walker et al. 1999, Cummings et al. 1990).
Windows and doors (6 to 22%; 15%). More variation in win-dow leakage is seen among window types (e.g., casement versus double-hung) than among new windows of the same type from dif-ferent manufacturers (Weidt et al. 1979). Windows that seal by com-pressing the weather strip (casements, awnings) show significantly lower leakage than windows with sliding seals.
Fireplaces (0 to 30%; 12%). When a fireplace is not in use, poorly fitting dampers allow air to escape. Glass doors reduce excess air while a fire is burning but rarely seal the fireplace struc-ture more tightly than a closed damper does. Chimney caps or fire-place plugs (with signs that warn they are in place) effectively reduce leakage through a cold fireplace.
Vents in conditioned spaces (2 to 12%; 5%). Exhaust vents in conditioned spaces frequently have either no dampers or dampers that do not close properly.
Diffusion through walls (<1%). Diffusion, in comparison to infiltration through holes and other openings in the structure, is not an important flow mechanism. At 5 Pa, the permeability of building materials produces an air exchange rate of less than 0.01 ACH by wall diffusion in a typical house.
Component leakage areas. Table 1 shows effective air leakage areas for a variety of residential building components at 4 Pa with CD assumed equal to 1 (Colliver et al. 1992). The values in the table present results in terms of air leakage area per unit component. Per unit component means per component, per unit surface area, or per unit length of crack or sash, whichever is appropriate. These air leakage areas may be converted to air leakage areas at other refer-ence pressures, airflow rates, or flow coefficients using Equations (34) through (37). Table 1 can be used to estimate the air leakage area of the building if test data are not available. To obtain the build-ing’s total air leakage area, multiply the overall dimensions or num-ber of occurrences of each building component by the appropriate table entry. The sum of the resulting products is the total building air leakage area. Table 2 gives the result of an example calculation of the effective air leakage area of a residence. Each leakage compo-nent is identified in the first column and described in the second.
The length, area, or number of the component is in the third column.
The fourth column contains the air leakage area per unit component, from Table 1, and the fifth contains the total air leakage area asso-ciated with that component. The sum of the terms in the last column is the total air leakage area of the building, in this case 848 cm2.
Klote and Milke (1992) describe a method for estimating airflows through gaps such as those found around doors.
Multifamily Building Leakage Leakage distribution is particularly important in multifamily apartment buildings. These buildings often cannot be treated as sin-gle zones due to the internal resistance between apartments. More-over, the leakage between apartments varies widely, tending to be small in modern construction, and ranging as high as 60% of the total apartment leakage in turn-of-the-century brick walk-up apart-ment buildings (Modera et al. 1991, Diamond et al. 1986). Little information on interzonal leakage has been reported because of the difficulty and expense of these measurements.
Fig. 9 Envelope Leakage Measurements Ventilation and Infiltration 26.15 Table 1 Effective Air Leakage Areas (Low-Rise Residential Applications Only) Units (see note) Best Estimate Mini-mum Maxi-mum Units (see note) Best Estimate Mini-mum Maxi-mum Ceiling Piping/Plumbing/Wiring penetrations General cm2/m2 1.8 0.79 2.8 Uncaulked cm2 ea 6 2 24 Drop cm2/m2 0.19 0.046 0.19 Caulked cm2 ea 2 1 2 Ceiling penetrations Vents Whole-house fans cm2 ea 20 1.6 21 Bathroom with damper closed cm2 ea 10 2.5 20 Recessed lights cm2 ea 10 1.5 21 Bathroom with damper open cm2 ea 20 6.1 22 Ceiling/Flue vent cm2 ea 31 28 31 Dryer with damper cm2 ea 3 2.9 7 Surface-mounted lights cm2 ea 0.82 Dryer without damper cm2 ea 15 12 34 Chimney cm2 ea 29 21 36 Kitchen with damper open cm2 ea 40 14 72 Crawl space Kitchen with damper closed cm2 ea 5 1 7 General (area for exposed wall) cm2/m2 10 8 17 Kitchen with tight gasket cm2 ea 1 200 mm by 400 mm vents cm2 ea 129 Walls (exterior) Door frame Cast-in-place concrete cm2/m2 0.5 0.049 1.8 General cm2 ea 12 2.4 25 Clay brick cavity wall, finished cm2/m2 0.68 0.05 2.3 Masonry, not caulked cm2/m2 5 1.7 5 Precast concrete panel cm2/m2 1.2 0.28 1.65 Masonry, caulked cm2/m2 1 0.3 1 Low-density concrete block, unfinished cm2/m2 3.5 1.3 4 Wood, not caulked cm2/m2 1.7 0.6 1.7 Wood, caulked cm2/m2 0.3 0.1 0.3 Low-density concrete block, painted or stucco cm2/m2 1.1 0.52 1.1 Trim cm2/lmc 1 Jamb cm2/lmc 8 7 10 High-density concrete block, unfinished cm2/m2 0.25 Threshold cm2/lmc 2 1.2 24 Doors Continuous air infiltration barrier cm2/m2 0.15 0.055 0.21 Attic/crawl space, not weatherstripped cm2 ea 30 10 37 Rigid sheathing cm2/m2 0.35 0.29 0.41 Window framing Attic/crawl space, weatherstripped cm2 ea 18 8 18.5 Masonry, uncaulked cm2/m2 6.5 5.7 10.3 Attic fold down, not weatherstripped cm2 ea 44 23 86 Masonry, caulked cm2/m2 1.3 1.1 2.1 Wood, uncaulked cm2/m2 1.7 1.5 2.7 Attic fold down, weatherstripped cm2 ea 22 14 43 Wood, caulked cm2/m2 0.3 0.3 0.5 Attic fold down, with insulated box cm2 ea 4 Windows Attic from unconditioned garage cm2 ea 0 0 0 Awning, not weatherstripped cm2/m2 1.6 0.8 2.4 Double, not weatherstripped cm2/m2 11 7 22 Awning, weatherstripped cm2/m2 0.8 0.4 1.2 Double, weatherstripped cm2/m2 8 3 23 Casement, weatherstripped cm2/lmc 0.24 0.1 3 Elevator (passenger) cm2 ea 0.26 0.14 0.35 Casement, not weatherstripped cm2/lmc 0.28 General, average cm2/lmc 0.31 0.23 0.45 Double horizontal slider, not weatherstripped cm2/lmc 1.1 0.019 3.4 Interior (pocket, on top floor) cm2 ea 14 Interior (stairs) cm2/lmc 0.9 0.25 1.5 Double horizontal slider, wood, weatherstripped cm2/lmc 0.55 0.15 1.72 Mail slot cm2/lmc 4 Sliding exterior glass patio cm2 ea 22 3 60 Double horizontal slider, aluminum, weatherstripped cm2/lmc 0.72 0.58 0.8 Sliding exterior glass patio cm2/m2 5.5 0.6 15 Storm (difference between with and without) cm2 ea 6 3 6.2 Double-hung, not weatherstripped cm2/lmc 2.5 0.86 6.1 Double-hung, weatherstripped cm2/lmc 0.65 0.2 1.9 Single, not weatherstripped cm2 ea 21 12 53 Double-hung with storm, not weatherstripped cm2/lmc 0.97 0.48 1.7 Single, weatherstripped cm2 ea 12 4 27 Vestibule (subtract per each location) cm2 ea 10 Double-hung with storm, weatherstripped cm2/lmc 0.79 0.44 1 Electrical outlets/Switches Double-hung with pressurized track, weatherstripped cm2/lmc 0.48 0.39 0.56 No gaskets cm2 ea 2.5 0.5 6.2 With gaskets cm2 ea 0.15 0.08 3.5 Jalousie cm2/louver 3.38 Furnace Lumped cm2/lms 0.471 0.009 2.06 Sealed (or no) combustion cm2 ea 0 0 0 Single horizontal slider, weatherstripped cm2/lms 0.67 0.2 2.06 Retention head or stack damper cm2 ea 30 20 30 Retention head and stack damper cm2 ea 24 18 30 Single horizontal slider, aluminum cm2/lms 0.8 0.27 2.06 Floors over crawl spaces General cm2/m2 2.2 0.4 4.9 Single horizontal slider, wood cm2/lms 0.44 0.27 0.99 Without ductwork in crawl space cm2/m2 1.98 Single horizontal slider, wood clad cm2/lms 0.64 0.54 0.81 With ductwork in crawl space cm2/m2 2.25 Fireplace Single-hung, weatherstripped cm2/lms 0.87 0.62 1.24 With damper closed cm2/m2 43 10 92 Sill cm2/lmc 0.21 0.139 0.212 With damper open cm2/m2 350 145 380 Storm inside, heat shrink cm2/lms 0.018 0.009 0.018 With glass doors cm2/m2 40 4 40 Storm inside, rigid sheet with magnetic seal cm2/lms 0.12 0.018 0.24 With insert and damper closed cm2/m2 36 26 46 With insert and damper open cm2/m2 65 40 90 Storm inside, flexible sheet with mechanical seal cm2/lms 0.154 0.018 0.833 Gas water heater cm2 ea 20 15 25 Joints Storm inside, rigid sheet with mechanical seal cm2/lms 0.4 0.045 0.833 Ceiling-wall cm2/lmc 1.5 0.16 2.5 Sole plate, floor/wall, uncaulked cm2/lmc 4 0.38 5.6 Storm outside, pressurized track cm2/lmc 0.528 Sole plate, floor/wall, caulked cm2/lmc 0.8 0.075 1.2 Storm outside, 2-track cm2/lmc 1.23 Top plate, band joist cm2/lmc 0.1 0.075 0.38 Storm outside, 3-track cm2/lmc 2.46 Note: Air leakage areas are based on values found in the literature. The effective air leak-age area (in square centimetres) is based on a pressure difference of 4 Pa and CD = 1.
Abbreviations: m2 = gross area in square metres lmc = linear metre of crack ea = each lms = linear metre of sash 26.16 2001 ASHRAE Fundamentals Handbook (SI) Controlling Air Leakage New Buildings. It is much easier to build a tight building than to tighten an existing building. Elmroth and Levin (1983), Eyre and Jennings (1983), Marbek Resource Consultants (1984), and Nelson et al. (1985) provide information and construction details on airtight building design for houses.
A continuous air infiltration retarder is one of the most effective means of reducing air leakage through walls, around window and door frames, and at joints between major building elements. Partic-ular care must be taken to ensure its continuity at all wall, floor, and ceiling joints; at window and door frames; and at all penetrations of the retarder, such as electrical outlets and switches, plumbing con-nections, and utility service penetrations. Joints in the air-vapor retarder must be lapped and sealed. Plastic vapor retarders installed in the ceiling should be tightly sealed with the vapor retarder in the outside walls and should be continuous over the par-tition walls. A seal at the top of the partition walls prevents leakage into the attic; a plate on top of the studs generally gives a poor seal.
The air infiltration retarder can be installed either on the inside of the wall framing, in which case it usually functions as a vapor retarder as well, or on the outside of the wall framing, in which case it should have a permeance rating high enough to permit diffusion of water vapor from the wall. For a discussion of moisture transfer in building envelopes, see Chapter 23 and Chapter 24.
A continuous air infiltration retarder installed on the outside of wall framing can cover many difficult construction details associ-ated with the installation of continuous air-vapor retarders. Interior air-vapor retarders must be lapped and sealed at electrical outlets and switches, at joints between walls and floors and between walls and ceilings, and at plumbing connections penetrating the wall’s interior finish. The exterior air infiltration retarder can cover these problem areas continuously. Joints in the air infiltration retarder should be lapped and sealed or taped. Exterior air infiltration retard-ers are generally made of a material stronger than plastic film and are more likely to withstand damage during construction. Sealing the wall against air leakage at the exterior of the insulation also cuts down on convection currents within the wall cavity, allowing insu-lation to retain more of its effectiveness.
Existing Buildings. The air leakage sites must first be located in order to tighten the envelope of an existing building. As discussed earlier, air leakage in buildings is due not only to windows and doors, but to a wide range of unexpected and unobvious construc-tion defects. Many important leakage sites can be very difficult to find. A variety of techniques developed to locate leakage sites are described in ASTM Standard E 1186 and Charlesworth (1988).
Once leakage sites are located, they can be repaired with mate-rials and techniques appropriate to the size and location of the leak. Harrje et al. (1979), Diamond et al. (1982), and Energy Resource Center (1982) include information on airtightening in existing residential buildings. With these procedures, the air leak-age of residential buildings can be reduced dramatically. Depend-ing on the extent of the tightening effort and the experience of those doing the work, residential buildings can be tightened any-where from 5% to more than 50% (Blomsterberg and Harrje 1979, Harrje and Mills 1980, Jacobson et al. 1986, Verschoor and Col-lins 1986, Giesbrecht and Proskiw 1986). Much less information is available for airtightening large, commercial buildings, but the same general principles apply (Parekh et al. 1991, Persily 1991).
RESIDENTIAL VENTILATION Typical infiltration values in housing in North America vary by a factor of about ten, from tightly constructed housing with seasonal average air exchange rates of about 0.2 air changes per hour (ACH) to loosely constructed housing with air exchange rates as great as 2.0 ACH. Figure 10 and Figure 11 show histograms of infiltration rates measured in two different samples of North American housing (Grimsrud et al. 1982, Grot and Clark 1979). Figure 10 shows the average seasonal infiltration of 312 houses located in different areas in North America. The median infiltration value of this sample is 0.5 ACH. Figure 11 represents measurements in 266 houses located in 16 cities in the United States. The median value of this sample is 0.9 ACH. The group of houses contained in the Figure 10 sample is biased toward new, energy-efficient houses, while the group in Fig-ure 11 represents older, low-income housing in the United States.
Table 2 Example of Calculation of Building Effective Air Leakage Area Based on Component Leakage Areas Component Description Size or Number × AL per unit = AL, cm2 Sills Uncaulked 43.2 m 4.0 cm2/m 173 Electrical outlets 20 0.5 cm2 ea 10 Windows Sliding 13.1 m2 4.0 cm2/m2 75 Framing 13.1 m2 1.7 cm2/m2 Exterior doors Single 5.7 m2 7.7 cm2/m2 54 Framing 5.7 m2 1.7 cm2/m2 Fireplace Without damper 1 350 cm2 ea 350 Penetrations Pipes 7 6.0 cm2 ea 42 Heating ducts Ducts untaped, in basement 1 144 cm2 ea 144 Calculated total building air leakage area Ac = 848 cm2 Fig. 10 Histogram of Infiltration Values—New Construction Fig. 11 Histogram of Infiltration Values— Low-Income Housing Ventilation and Infiltration 26.17 Additional studies have found average values for houses in regional areas. Palmiter and Brown (1989) and Parker et al. (1990) found a heating season average of 0.40 ACH (range: 0.13 to 1.11 ACH) for 134 houses in the Pacific Northwest. In a comparison of 292 houses incorporating energy-efficient features (including mea-sures to reduce air infiltration and provide ventilation heat recovery) with 331 control houses, Parker et al. (1990) found an average of about 0.25 ACH (range: 0.02 to 1.63 ACH) for the energy-efficient houses versus 0.49 (range: 0.05 to 1.63 ACH) for the control. Ek et al. (1990) found an average of 0.5 ACH (range: 0.26 to 1.09) for 93 double-wide manufactured homes also in the Pacific Northwest.
Canadian housing stock has been characterized by Yuill and Comeau (1989) and Riley (1990). While these studies do not repre-sent random samples of North American housing, they indicate the distribution of infiltration rates expected in a group of buildings.
Occupancy influences have not been measured directly and vary widely. Desrochers and Scott (1985) estimated that they add an average of 0.10 to 0.15 ACH to unoccupied values. Kvisgaard and Collet (1990) found that in 16 Danish dwellings, the users on aver-age provided 63% of the total air exchange rate. Ventilation air requirements for houses in the U.S. have tradi-tionally been met on the assumption that the building envelope is leaky enough that infiltration will suffice. Possible difficulties with this approach include low infiltration when natural forces (temperature difference and wind) are weak, unnecessary energy consumption when such forces are strong, drafts in cold climates, lack of control of ventilation rates to meet changing needs, poor humidity control, potential for interstitial condensation from exfiltration in cold climates or infiltration in hot humid climates, and lack of opportunity to recover the energy used to condition the ventilation air. The solution to these concerns is to have a rea-sonably tight building envelope and a properly designed and oper-ated mechanical ventilation system.
ASHRAE Standard 119 and the National Building Code of Can-ada (NRCC 1995) encourage the transition to tighter envelope con-struction. Hamlin (1991) shows a 30% increase in airtightness of tract-built Canadian houses between 1982 and 1989. Also, 82% of the newer houses had natural air exchange rates below 0.3 ACH in March. Yuill (1991) derived a procedure to show the extent to which infiltration contributes toward meeting ventilation air requirements.
As a result, the National Building Code of Canada has requirements for mechanical ventilation capability in all new dwelling units.
ASHRAE Standard 62 gives ventilation air requirements for houses, essentially 0.35 ACH with at least 8 L/s per occupant. Cana-dian Standards Association (CSA) Standard F326 expands the requirements for residential mechanical ventilation systems to cover air distribution within the house, thermal comfort, minimum tem-peratures for equipment and ductwork, system controls, pressuriza-tion and depressurization of the dwelling, installation requirements, and verification of compliance. Verification can be by design or by test, but the total rate of outside air delivery must be measured.
Mechanical ventilation is being used in houses, especially in energy-efficient housing demonstration programs (Riley 1990, Palmiter et al. 1991). Possible systems can be characterized as local or central; exhaust, supply, or balanced; with forced-air or radiant/ hydronic heating/cooling systems; with or without heat recovery; and with continuous operation or controlled by occu-pants, demand (i.e., by pollutant sensing), timers or humidity. Note that not all combinations are viable. Various options are described by Fisk et al. (1984), Hekmat et al. (1986), Sibbitt and Hamlin (1991), Palmiter et al. (1991), Yuill et al. (1991), Holton et al.
(1997), Sherman and Matson (1997), Reardon and Shaw (1997), and Lubliner et al. (1997).
The simplest systems use bathroom and kitchen fans to augment infiltration. Noise, installed capacity, durability under continuous operation, distribution to all rooms (especially bedrooms), envelope moisture, combustion safety, and energy efficiency issues need to be addressed. Many present bath and kitchen fans are ineffective ven-tilators because of poor installation and design. However, properly specified and installed exhaust fans can form part of good whole-house ventilation systems and are so specified in some Canadian building codes.
Some central supply systems use a central air-handling unit blower to induce air from the outdoors and distribute it. However, the blower operates intermittently if thermostatically controlled and provides little ventilation in mild weather. Continuous blower oper-ation increases energy consumption. If the blower operates contin-uously when the heat source is off, the combination of lower mixed air temperature and high air speed can cause cold air drafts. To offset these problems, some systems use electronically commutated blower motors, which allow efficient continuous operation at lower speeds. Some others use a timer to cycle the blower when thermo-static demands are inadequate to cause the blower to operate when needed for ventilation (Rudd 1998).
Central exhaust systems use leakage sites and, in some cases, intentional and controllable openings in the building envelope as the supply. Such systems are suitable for retrofit in existing houses.
Energy can be recovered from the exhaust airstream with a heat pump to supplement domestic hot water and/or space heating.
For new houses with tightly constructed envelopes, balanced ventilation with passive heat recovery (air-to-air heat exchangers or heat recovery ventilators) can be appropriate in some climates.
Fan-induced supply and exhaust air flows at nearly equal rates over a heat exchanger, where heat and sometimes moisture is transferred between the airstreams. This reduces the energy required to condition the ventilation air by typically 60 to 80% (Cutter 1987). It also reduces the thermal comfort problem that occurs when untempered air is introduced directly into the house.
Airflow balance, leakage between streams, biological contamina-tion of wet surfaces, frosting, and first cost are concerns associated with these systems.
The type of ventilation system can be selected based on house leakage class as defined in ASHRAE Standard 119. Balanced air-to-air systems with heat recovery are optimal for tight houses (leakage classes A–C). The leakier the house is, the larger is the contribution from infiltration and the less effective is heat recovery ventilation. Tightening the envelope beyond the level of ASHRAE Standard 119 may be warranted in extreme climates to better use the heat recovery effect (Sherman and Matson 1997). In mild cli-mates, these systems can also effectively be used in leakage classes D–F. Central exhaust systems should not be used for leakage classes A–C unless special provisions are made for air inlets; oth-erwise their operation may depressurize the house enough to cause backdrafting through fossil-fueled appliances. Unbalanced systems (either supply or exhaust) are optimal for leakage classes D–F.
Ventilation systems are normally not needed for leakage classes G–J, but for those cases in which they are, an unbalanced system is usually the best choice.
Residential Ventilation Zones For guidance in the selection of residential ventilation systems, Sherman (1995) developed four climatic zones for the United States. These zones are shown in Figure 12 for the continental United States. Alaska is in Zone 1, and Hawaii is in Zone 4.
Zone 1 includes the severe climates of the northern tier of states.
A Zone 1 residence that meets airtightness and energy conservation standards probably cannot meet its ventilation needs through infil-tration and will require forced (mechanical) ventilation. Zone 2 includes the moderate climates where careful design and construc-tion may allow buildings to simultaneously meet energy standards and ventilation needs through infiltration and mechanical exhaust.
The mild climates in Zone 3 allow residences to meet both ASH-RAE Standards 119 and 62 over a substantial range of airtightness.
Zone 4 residences have relatively small energy penalties associated 26.18 2001 ASHRAE Fundamentals Handbook (SI) with infiltration or ventilation. In this zone, natural ventilation is usually preferred to forced ventilation as a technique to supplement infiltration.
RESIDENTIAL VENTILATION REQUIREMENTS Traditionally, residential ventilation has been provided by natu-ral ventilation and infiltration. Sherman and Matson (1997) showed that most of the older building stock is sufficiently leaky that infil-tration alone can meet the minimum requirements of ASHRAE Standard 62. Houses built or retrofitted to new standards have sub-stantially tighter envelopes and insufficient infiltration to meet ven-tilation standards. In most circumstances, concerns over safety, noise, comfort, air quality, and energy minimize occupant use of operable windows. As a result, these houses require supplemental mechanical ventilation to satisfy these standards.
Simply meeting the minimum residential ventilation rates is not always sufficient to adequately dilute all contaminants. For some buildings, such ventilation may not meet the requirements of indi-viduals with allergies or chemical sensitivities or when there are unusual sources. In these cases, source control or extra ventilation is required to manage the contaminant levels. Therefore, especially in single-family dwellings, occupants must be responsible for intro-ducing, monitoring, and controlling the sources in the indoor envi-ronment, as well as for operating the dwelling unit to meet their individual needs.
Source Control When considering how much whole-house ventilation should be supplied, typical and unusual significant sources of indoor pollution need to be controlled. This can be done either by mitigating the source itself or by using local exhaust to extract the contaminants before they can mix into the indoor environment. Typical sources that should be considered include the following: Clothes Dryers. Clothes dryer exhaust is heavily laden with moisture and laundry by-products. Many moisture problems have been traced to clothes dryers vented indoors. Exhaust from clothes dryers, which is typically about 70 L/s, should be vented directly to the outdoors.
Combustion. Water and carbon dioxide are always emitted dur-ing combustion. Other more dangerous compounds can be emitted as well. All these by products should be vented directly outdoors.
Venting of combustion appliances should meet all applicable codes, but for buildings with naturally aspirated combustion appliances within the pressure boundary, excessive depressurization due to exhaust systems should be avoided. In addition, a depressurization safety test should be considered, such as described in ASTM Stan-dard E 1998 or CGSB Standard 51.71.
Carbon monoxide is one of the most pervasive indoor contam-inants. It can come from virtually any source of combustion, includ-ing automobiles. Because even combustion appliances that meet manufacturers specifications can interact with the building and emit carbon monoxide, at least one carbon monoxide alarm meeting UL Standard 2034 should be installed in each dwelling that has com-bustion appliances (including fireplaces) within the pressure bound-ary or attached garages.
Garages. Garages contain many sources of contaminants. Doors between garages and occupied space should be well sealed (with gaskets or weatherstripping) and possibly be self-closing. Depres-surized sections of HVAC systems, such as air handlers or return ducts, should not be located in garages. If such sections must pass through garages, they should be well sealed.
Particulates. The ventilation system should be designed such that return and outdoor air is filtered before passing through the thermal conditioning components. Pressure drops associated with this filtration should be considered in the design of the air-handling system. Particulate filters or air cleaners should have a minimum efficiency of 60% for 3 mm particles, which is equivalent to a MERV 6 designated filter according to ASHRAE Standard 52.2.
Outdoor Air. Outdoor air may contain unacceptably high levels of pollutants, including ozone, pollen, carbon monoxide, particulate matter, odors, etc. In such cases, it may be impossible to provide acceptable indoor air quality; increased ventilation rates can actu-ally decrease indoor air quality. In areas in which this problem may be anticipated, controls should be provided to allow the occupants to temporarily reduce the ventilation rate. Air cleaning should be considered for sensitive individuals.
Local Exhaust Ventilation The single most important source control mechanism in dwell-ings apart from source elimination is local exhaust. All wet rooms and other spaces designed to allow specific contaminant release should be provided with local exhaust. These spaces include kitch-ens, utility rooms, bathrooms, and toilets. Workshops, recreation rooms, smoking areas, art studios, and hobby rooms may also require local ventilation and/or air cleaning to remove contaminants generated by the activities involved. If unvented combustion appli-ances must be used, then rooms with these appliances should meet the same general ventilation requirements for kitchens because such appliances generate significant amounts of moisture even when burning properly.
Mechanical ventilation is the preferred method of providing local ventilation. Normally, it is designed to operate intermittently to exhaust the contaminated air outside when the contaminant is being produced. However, in many circumstances, a continuous and lower-flow-rate exhaust can work as well.
Continuous Local Mechanical Ventilation. A continuously-operating mechanical exhaust is intended to operate without occu-pant intervention. This exhaust may be part of a balanced mechan-ical ventilation system. The system should be designed to operate during all hours in which the dwelling is occupied. Override control should be provided if needed. The minimum delivered ventilation should be at least that given in Table 3.
Fig. 12 Airtightness Zones for Residences in the United States (Sherman 1995) Table 3 Continuous Exhaust Airflow Rates Application Continuous Flow Notes Kitchen 5 ACH Based on kitchen volume Utility room 10 L/s Not less than 2 room air changes per hour Bathroom 10 L/s Toilet 10 L/s Ventilation and Infiltration 26.19 Intermittent Local Mechanical Ventilation. An intermit-tently operating local mechanical exhaust is intended to be oper-ated as needed by the occupant and should be designed with this intent. Shut-off timers, occupancy controls, multiple speed fans, and switching integral with room lighting are helpful, provided they do not impede occupant control. The minimum airflow rating should be at least that given in Table 4, or as mandated by local codes.
Alternative Local Ventilation. Cleaning of recirculated air can sometimes be substituted for local ventilation, if it can be shown to be effective in removing contaminants of concern. Natural ventila-tion is not generally a suitable method for local ventilation in most climates and spaces. Use of natural ventilation can cause re-entrain-ment problems when air flows into rather than out of the space, and contaminated exfiltrating air reenters the building. In milder cli-mates, it may be acceptable to use natural ventilation, when the con-taminant of concern is related to odor rather than health or safety.
Purpose-designed passive exhaust systems have shown acceptable ventilation in European settings and may be considered in lieu of mechanical systems.
Whole-House Ventilation While control of significant sources of pollution in a dwelling is important, whole-house ventilation may still be needed. Each dwelling should be provided with outdoor air according to Table 5.
The rate is the sum of the Area-Based and Occupancy-Based col-umns. Design occupancy can be based on the number of bedrooms as follows: first bedroom, two persons; each additional bedroom, one person.
Natural whole-house ventilation that relies on occupant oper-ation should not be used to make up any part of the minimum total whole-house ventilation. However, because occupancy and sources vary significantly, the capacity to ventilate above minimum rates can be provided by operable openings such as doors and windows.
Continuous whole-house mechanical ventilation systems are intended to operate without occupant intervention. The ventilation effects of multiple local systems may be combined to meet whole-house requirements.
Intermittent whole-house mechanical ventilation systems are intended to be operated automatically and regularly, but not necessarily continuously. The system may consist of supply, exhaust, or balanced mechanical systems. It should be designed so that it can operate automatically based on a timer. A supplemen-tary control mechanism such as a humidistat or indoor air quality sensor may be used.
The effective ventilation rate of an intermittent system is the product of its delivered capacity, its fractional on-time, and the temporal ventilation effectiveness from Table 6. Systems cycling at least once every 3 h may be assumed to have perfect ventilation effectiveness. Multiple systems may be combined, but the combi-nation should only be supply or exhaust values and not both.
Air Distribution Ventilation air should be provided to each habitable room through mechanical and natural air distribution. If a room does not have a balanced air supply (or inlet) and return (or exhaust), path-ways for transfer air should be provided. These pathways may be door undercuts, transfer ducts with grilles, or simply grilles where ducts are not necessary.
In houses without central air handlers, special provisions to dis-tribute outdoor air may be required. Rooms in which occupants spend many continuous hours, such as bedrooms, may require spe-cial consideration.
Selection Principles for Ventilation Systems Determining the right ventilation approach is part of optimizing the entire building, and it can rarely be done without considering the building envelope, the climate, the needs of the users, and costs. For example, mechanical exhaust systems should not be used in hot, humid climates (because of possible condensation problems) when the building has or is likely to have air conditioning installed. Sim-ilarly, mechanical supply ventilation systems should not be used in very cold climates.
Occupant comfort, energy efficiency, ease of use, service life, first and life-cycle cost, value-added features, and indoor environ-mental quality should be considered when selecting a strategy and system. HVAC (and related) systems can be a potential cause of poor indoor air quality. For example, occupants may not use the ventilation systems as intended if operation results in discomfort (e.g., drafts) or excessive energy use. The resulting lack of ventila-tion might produce poor indoor air quality. Therefore, careful design, operation, and maintenance is necessary to provide opti-mum effectiveness.
All exhaust, supply, or air-handler fans have the potential to change the pressure of the living space relative to the outside. High-volume fans, such as the air handler and some cooking exhaust fans, can cause high levels of depressurization, particularly in tightly constructed homes. Considering these effects is essential in design.
Depressurization of the living space relative to outside may cause backdrafting of combustion appliances and the migration of con-taminants (such as radon or other soil gases, car exhaust, insulation particles, etc.) into the living space. Depressurization can also result in moisture intrusion into building cavities in warm, moist climates, which may cause structural damage and fungal growth. Pressuriza-tion of the living space can cause condensation in building cavities in cold climates, resulting in damage to the structural integrity of the home. Excess pressure can best be prevented by balanced ventila-tion systems and nonleaky duct systems. In addition, adequate path-ways must be available for all return air to the air-handling devices.
Occupant activities and operation of fans that exhaust air from the home (including leaky ducts on air conditioners, furnaces, or heat pumps) may produce depressurization of the structure. Several options to address backdrafting concerns include • Using combustion appliances with isolated (or sealed) combustion systems • Locating combustion appliances in a ventilated room isolated from depressurized zones by well-sealed partitions • Installing supply fans to balance or partially balance the exhaust from the zone • Testing to ensure that depressurization will not be excessive Table 4 Intermittent Exhaust Airflow Rates Application Continuous Flow Notes Kitchen 50 L/s Vented range hood required if less than 5 kitchen air changes per hour Utility room 25 L/s Not less than 2 room air changes per hour Bathroom 25 L/s Toilet 25 L/s Table 5 Total Ventilation Air Requirements Area Based Occupancy Based 0.10 L/s per square metre of floor space 8 L/s per person, based on normal occupancy Table 6 Ventilation Effectiveness for Intermittent Fans Fractional On-Time, f Temporal Ventilation Effectiveness f ≤ 35% 0.33 35% < f ≤ 60% 0.50 60% < f ≤ 80% 0.75 80% < f 1.0 26.20 2001 ASHRAE Fundamentals Handbook (SI) The system must be designed, built, operated, and maintained in a way that discourages the growth of biological contaminants. Typ-ical precautions include sloping condensate drain pans toward the drain, keeping condensate drains free of obstructions, maintaining cooling coils free of dirt and other obstructions, and checking and eliminating any cause of moisture inside ducts.
Selecting a Whole-House Ventilation System Whole-house ventilation can be provided through mechanical or natural means or by a combination thereof. Regardless of the approach chosen, it is necessary to consider where the outdoor air comes from, how it enters the house, how it is distributed, and how it leaves the house. Systems that are uncomfortable, expensive to operate, unsafe, noisy, or in other ways unacceptable to the occu-pants are not likely to be used.
Natural Ventilation. The use of operable windows and other natural ventilation openings is rarely an acceptable means of pro-viding base levels of whole-house ventilation (Wilson and Walker 1992), but when it is, it can be a cost-effective alternative. Operable windows allow the user to control the ventilation by the size of the opening. Windows are present for light and egress and can serve as a ventilation system; most detached dwellings have sufficient oper-able windows for natural ventilation. Environmental factors may inhibit effective occupant use of the windows. During cold periods, open windows can cause thermal discomfort. Noise or security issues may also reduce the desirability of opening windows. Finally, the uncontrolled nature of the natural ventilation may increase energy use in many climates, but it can be energy-efficient in some mild climates.
Mechanical Ventilation. Whole-house systems may run contin-uously or intermittently. Intermittent systems must supply more ventilation air and, thus, may cost more to temper the outside air and to run the fans. The system can consist of supply, exhaust, or a bal-anced combination of the two. Fans that are noisy are likely to be unacceptable to many occupants and will be disabled; noise should be reduced by using quiet fans or by remote mounting. Continuous ventilation requires less power if the system is designed with low-resistance ducting. The electrical energy required for operation should be calculated together with the energy required to temper the outside air introduced through ventilation.
Supply Ventilation. A supply ventilation system usually has a single intake for the ventilation air. This air is distributed through the house either by a dedicated duct system or by the thermal distri-bution system. The outdoor air may be filtered in a supply ventila-tion system to remove pollen and dust.
Envelope leakage, exhaust stacks, and flues provide pathways for exhaust air. Supply ventilation can result in indoor pressuriza-tion. This is generally unacceptable in very cold climates because the exfiltrating air can cause condensation in the building envelope.
However, continuous supply ventilation systems can be designed in conjunction with specific building envelope features to account for this condensation.
Supply ventilation can mitigate radon entry or backdrafting problems and may reduce interstitial moisture problems in hot, humid climates. In temperate or severe climates, the supply air, if delivered directly to rooms without tempering, can cause thermal discomfort or draftiness. A dedicated duct system increases the cost of a supply ventilation system.
Exhaust Ventilation. An exhaust ventilation system usually has a central mechanical exhaust. Air enters the house though envelope leakage, open windows, or designed inlets. Because the air intake is dispersed, thermal discomfort is less than that with supply ventila-tion. An exhaust air heat pump can recover energy, but it may not be economical in many climates. The exhaust fan depressurizes the house, which can aggravate radon or backdrafting problems.
Because of potential interstitial space moisture problems, exhaust ventilation may be unacceptable in hot, moist climates when the indoor air is mechanically cooled.
Although air intake through building leaks reduces the particu-late concentration somewhat, filtration of outdoor air is not gener-ally possible with exhaust ventilation. In unusually tight houses, envelope leakage may be insufficient to provide enough supply air, so designed intakes may be needed.
Balanced Ventilation. In a balanced mechanical ventilation sys-tem, there is usually a mechanical exhaust either centrally located or ducted from locations likely to have high contaminant levels. There is a single outside air intake for the ventilation air, which is then dis-persed through the house. The systems are designed to produce equal supply and exhaust flows. With equal flows, the system cre-ates neither pressurization nor depressurization, so problems associ-ated with house pressures are reduced. It is also possible to use slight positive pressurization to reduce infiltration. Filtration and temper-ing of the incoming air can be accomplished at the central unit.
Most balanced systems feature either sensible heat recovery or total heat recovery. Sensible heat recovery systems provide temper-ing of the incoming air using the exiting air as a source for heat (or cooling). Total heat recovery systems provide both sensible and latent tempering of the incoming air.
Because of its energy recovery properties, a balanced system becomes more attractive as total space-conditioning costs increase, such as in severe climates. The balanced system is initially the most costly of the three systems, but total operating costs may be less, particularly if high-efficiency fan/motor assemblies are used.
Selecting a Local Ventilation System The two most common methods for providing local ventilation to damp rooms are natural ventilation through an operable window and mechanical exhaust fans. These methods have the same design issues as whole-house systems. Balanced or supply-only systems should not be used for local ventilation to reduce spreading of mois-ture and contaminants through the house.
SIMPLIFIED MODELS OF RESIDENTIAL VENTILATION AND INFILTRATION This section describes several calculation procedures, ranging from simple estimation techniques to more physical models. Orme (1999) provides a more thorough review of simplified models. The air exchange rate of a building cannot be reliably deduced from the building’s construction or age or from a simple visual inspection.
Some measurement is necessary, such as a pressurization test of envelope airtightness or a detailed quantification of the leakage sites and their magnitude. The air exchange rate of a building may be cal-culated given (1) the location and leakage function for every open-ing in the building envelope and between major building zones, (2) the wind pressure coefficients over the building envelope, and (3) any mechanical ventilation airflow rates. These inputs are gen-erally unavailable for all except very simple structures or extremely well studied buildings. Therefore, assumptions as to their values must be made. The appropriateness of these assumptions deter-mines the accuracy of predictions of air exchange rates.
Empirical Models These models of residential infiltration are based on statistical fits of infiltration rate data for specific houses. They use pressurization test results to account for house airtightness and take the form of simple relations between infiltration rate, an airtightness rating, and, in most cases, weather conditions. Empirical models account for envelope infiltration only and do not deal with intentional ventila-tion. In one approach, the calculated air exchange rate at 50 Pa based on a pressurization test is simply divided by a constant approxi-mately equal to 20 (Sherman 1987). This technique does not account for the effect of infiltration driving mechanisms on air Ventilation and Infiltration 26.21 exchange. Empirical models that do account for weather effects have been developed by Reeves et al. (1979), Kronvall (1980), and Shaw (1981).
The latter two models account for building air leakage using the values of c and n from Equation (32). The only other inputs required are wind speed and temperature difference. Such empirical models predict long-term (one-week) infiltration rates very well in the houses from which they were developed; they do not, however, work as well in other houses due to the building-specific nature of leakage distribution, wind pressure, and internal partitioning. Pers-ily and Linteris (1983) and Persily (1986) show comparisons between measured and predicted house infiltration rates for these and other models. The average long-term differences between mea-surements and predictions are generally on the order of 40%, although individual predictions can be off by 100% or more.
Multizone Models Multicell models of air exchange treat buildings as a series of interconnected zones and assume that the air within each zone is well mixed. Several such models have been developed by Allard and Herrlin (1989), Etheridge and Alexander (1980), Liddament and Allen (1983), Walton (1984, 1989), Herrlin (1985), and Feustel and Raynor-Hoosen (1990). They are all based on a mass balance for each zone of the building. These mass balances are used to solve for interior static pressures within the building by requiring that the inflows and outflows for each zone balance to zero. The models require the user to input a location and leakage function for every opening in the building envelope and in relevant interior partitions, a value for the wind pressure coefficient Cp at the location of each building envelope leakage site, temperatures for each zone, and any mechanical ventilation airflow rates. Such detailed information is difficult to obtain for a building. Wind pressure coefficient data in the literature, air leakage measurement results from the building or its components, and air leakage data from the literature can be used.
These models not only solve for whole-building and individual zone air exchange rates, but also determine airflow rates and pressure dif-ferences between zones. These interzone airflow rates are useful for predicting pollutant transport within buildings with well mixed zones. Caution should be used when applying them to the prediction of smoke movement patterns in the event of a fire. Multizone mod-els have the advantage of being able to model very complex repre-sentations of buildings by using personal computers; however, determining the correct inputs to these models is difficult. As a result of these uncertainties, multizone models are best used to bound a solution rather than determining an absolute solution.
Monte Carlo simulation coupled with the multizone modeling is a useful technique to determine these bounds, if the probability dis-tribution for the uncertain parameters can be defined.
Single-Zone Models Several procedures have been developed to calculate building air exchange rates that are based on physical models of the building interior as a single zone. These single-zone models are only appro-priate to buildings with no internal resistance to airflow and are therefore inappropriate to large, multizone buildings. Models of this type have been developed by the Institute of Gas Technology (IGT) (Cole et al. 1980), the Building Research Establishment (Warren and Webb 1980), the Lawrence Berkeley National Laboratory (LBNL) (Sherman and Grimsrud 1980), and the University of Alberta (AIM-2) (Walker and Wilson 1998). These models have exhibited average errors on the order of 40% for many measure-ments on groups of houses and can be off by 100% in individual cases (Persily 1986, Walker and Wilson 1998). The following sec-tion on Residential Calculation Examples uses a basic model and an enhanced model (Walker and Wilson 1998, Palmiter and Bond 1994, Hamlin and Pushka 1994, CHBA 1994, Bradley 1993).
The basic model uses the effective air leakage area AL at 4 Pa, which can be obtained from a whole-building pressurization test. If a test value is not available, the data in Table 1 can be used to esti-mate the air leakage area of the building. To obtain the building’s total air leakage area, multiply the overall dimensions or number of occurrences of each building component by the appropriate table entry. The sum of the resulting products is the total building air leak-age area. See Table 2 for an example.
The enhanced model uses pressurization test results to charac-terize house air leakage through the leakage coefficient c and the pressure exponent n. The enhanced model improves on the basic model by using a power law to represent the envelope leakage, including a flue as a separate leakage site, and having separate wind effects for houses with crawl spaces or slab/basement foundations.
For both models, the user must input wind speed, temperature difference, information on distribution of leakage over the building envelope, a wind shelter (or local shielding for the basic model) parameter, and a terrain coefficient. The predictive accuracy of the enhanced model can be very good, typically ±10% when the param-eters are well known for the building in question (Walker and Wil-son 1998, Palmiter and Bond 1994, Sherman and Modera 1986). All these single-zone models are sensitive to the values of the inputs, which are quite difficult to determine.
Superposition of Wind and Stack Effects Simplified physical models of infiltration solve the problem of two natural driving forces (wind and stack) separately and then combine them in a process called superposition. Superposition is necessary because each physical process can affect the internal and external pressures on the structure, which can cause interactions between physical processes that are otherwise independent. An exact solution is impossible because detailed properties of all the building leaks are unknown and because leakage is a nonlinear pro-cess. For this reason, most modelers have developed a simplified superposition process to combine stack and wind effects. Sherman (1992a) compares various superposition procedures and derives a generalized superposition equation that involves some simple leak-age distribution parameters. He shows that the result is always sub-additive. Typically only 35% of the infiltration due to the smaller effect can be added to the larger effect. Depending on the details, that percentage could go as high as 85% or as low as zero. Walker and Wilson (1993) compared several superposition techniques to measured data. Sherman, as well as Walker and Wilson, found quadrature [shown in Equation (39)] to be a robust superposition technique: (39) The following sections discuss how this superposition is com-bined with the calculation of the wind and stack flows to determine the total flow.
Residential Calculation Examples Basic Model. The following calculations are based on the LBNL model (Sherman and Grimsrud 1980), which uses the effective air leakage area at 4 Pa. This leakage area can be obtained from a whole-building pressurization test. Using the effective air leakage area, the airflow rate due to infiltration is calculated according to (40) where Q = airflow rate, m3/s AL = effective air leakage area, cm2 Cs = stack coefficient, (L/s)2/(cm4·K) Q Qs 2 Qw 2 + = Q AL 1000 ------------ Cs t ∆ CwU 2 + = 26.22 2001 ASHRAE Fundamentals Handbook (SI) ∆t = average indoor-outdoor temperature difference for time interval of calculation, K Cw = wind coefficient, (L/s)2/[cm4 · (m/s)2] U = average wind speed measured at local weather station for time interval of calculation, m/s Table 7 presents values of Cs for one-, two-, and three-story houses.
The value of the wind coefficient Cw depends on the local shelter class of the building (described in Table 8) and the building height.
Table 9 presents values of Cw for one-, two-, and three-story houses in shelter classes 1 through 5. In calculating the values in Table 7 and Table 9, the following assumptions were made regarding input to the basic model: • Terrain used for converting meteorological to local wind speeds is that of a rural area with scattered obstacles • R = 0.5 (half of the building leakage in the walls) • X = 0 (equal amounts of leakage in the floor and ceiling) • Heights of one-, two-, and three-story buildings = 2.5, 5.0, and 7.5 m, respectively Example 1. Estimate the infiltration at design conditions for a two-story house in Lincoln, Nebraska. The house has an effective air leakage area of 500 cm2 and a volume of 340 m3, and the predominant wind is per-pendicular to the street (shelter class 3). The indoor air temperature is 20°C.
Solution: The 99% design temperature for Lincoln is −19°C. Assume a design wind speed of 6.7 m/s. From Equation (40), with Cs = 0.000 290 from Table 7 and Cw = 0.000 231 from Table 9, the airflow rate due to infiltration is From Equation (2), the air exchange rate I is equal to Q divided by the building volume: Example 2. Calculate the average infiltration during a one-week period in January for a one-story house in Portland, Oregon. During this period, the average indoor-outdoor temperature difference is 17 K, and the average wind speed is 2.7 m/s. The house has a volume of 255 m3 and an effective air leakage area of 690 cm2, and it is located in an area with buildings and trees within 10 m in most directions (shelter class 4).
Solution: From Equation (40), the airflow rate due to infiltration is The air exchange rate is therefore Example 3. Estimate the average infiltration over the heating season in a two-story house with a volume of 310 m3 and the air leakage area cal-culated in Table 2 (848 cm2). The house is located on a lot with several large trees but no other close buildings (shelter class 3). The average wind speed during the heating season is 3.2 m/s, while the average indoor-outdoor temperature difference is 20 K.
Solution: From Equation (40), the airflow rate due to infiltration is The average air exchange rate is therefore Enhanced Model. This section presents a simple, single-zone approach to calculating air infiltration rates in houses based on the AIM-2 model (Walker and Wilson 1998). The airflow rate due to infiltration is calculated using (41) (42) where Qs = stack airflow rate, m3/s Qw = wind airflow rate, m3/s c = flow coefficient, m3/(s·Pan) Cs = stack coefficient, (Pa/K)n Cw = wind coefficient, (Pa·s2/m2)n s = shelter factor In calculating the tabulated values of Cs, Cw, and s, the following assumptions were made: • Each story is 2.5 m high.
• The flue is 15 cm in diameter and reaches 2 m above the upper ceiling.
• The flue is unsheltered.
• Half of the envelope leakage (not including the flue) is in the walls and one-quarter each is at the floor and ceiling, respectively.
• n = 0.67 Using typical values for terrain factors, house height, and wind speed measurement height, the windspeed multiplier G (given in Table 10) uses a relationship based on equations found in Chapter 16.
Example 4. Estimate the infiltration at design conditions for a two-story slab-on-grade house with a flue in Lincoln, Nebraska. The house has Table 7 Basic Model Stack Coefficient Cs House Height (Stories) One Two Three Stack coefficient 0.000 145 0.000 290 0.000 435 Table 8 Local Shelter Classes Shelter Class Description 1 No obstructions or local shielding 2 Typical shelter for an isolated rural house 3 Typical shelter caused by other buildings across the street from the building under study 4 Typical shelter for urban buildings on larger lots where sheltering obstacles are more than one building height away 5 Typical shelter produced by buildings or other structures that are immediately adjacent (closer than one house height): e.g., neighboring houses on the same side of the street, trees, bushes, etc.
Table 9 Basic Model Wind Coefficient Cw Shelter Class House Height (Stories) One Two Three 1 0.000 319 0.000 420 0.000 494 2 0.000 246 0.000 325 0.000 382 3 0.000 174 0.000 231 0.000 271 4 0.000 104 0.000 137 0.000 161 5 0.000 032 0.000 042 0.000 049 Q 500 1000 ------------ 0.000 290 39 × ( ) 0.000 231 6.72 × ( ) + = 0.0736 m3 s ⁄ 265 m3 h ⁄ = = I 265 m3 h ⁄ ( ) 340 m3 ⁄ = 0.78 h 1 – 0.78 ACH = = Q 690 1000 ------------ 0.000 145 17 × ( ) 0.000 104 2.72 × ( ) + = 0.0392 m3 s ⁄ 141 m3 h ⁄ = = I 141 255 ⁄ 0.55 h 1 – 0.55 ACH = = = Q 848 1000 ------------ 0.000 290 20 × ( ) 0.000 231 3.22 × ( ) + = 0.077 m3 s ⁄ 276 m3 h ⁄ = = I 276 310 ⁄ 0.89 h 1 – 0.89 ACH = = = Qs cCs∆tn = Qw cCw sU ( )2n = Ventilation and Infiltration 26.23 a flow coefficient of c = 0.051 m3/(s·Pan) and a pressure exponent of n = 0.67 (this corresponds to effective leakage area of 500 cm2 at 4 Pa). The building volume is 340 m3. The 97.5% design tempera-ture is –19°C, and the design wind speed is 6.7 m/s.
Solution: For a slab-on-grade two-story house with a flue, Table 12 gives Cs = 0.089 (Pa/K)n and Cw = 0.156 (Pa·s2/m2)n. The house is maintained at 20°C indoors. The building wind speed is determined by taking the design wind speed and multiplying by the wind speed multi-plier G from Table 10: From Table 8, the shelter class for a typical urban house is 4. Table 11 gives the shelter factor for a two-story house with a flue and shelter class 4 as s = 0.64. The stack flow is calculated using Equation (41): The wind flow is calculated using Equation (42): Substituting Qs and Qw into Equation (39) gives Q = 0.059 m3/s = 214 m3/h. From Equation (2), the air exchange rate I is equal to Q divided by the building volume: Example 5. Estimate the average infiltration over a one-week period for a single-story crawl space house in Redmond, Washington. The house has a flow coefficient of c = 0.078 m3/(s·Pan) and a pressure exponent of n = 0.6 (this corresponds to effective leakage area of 690 cm2 at 4 Pa). The building volume is 255 m3. During this period, the average indoor-outdoor temperature difference is 16 K, and the wind speed is 2.7 m/s. The house is electrically heated and has no flue. Solution: For a single-story house with no flue, Cs = 0.054 (Pa/K)n. For a crawl space, Cw = 0.128 (Pa·s2/m2)n. From Table 10, for a one-story house, G = 0.48.
Table 11 gives shelter factor s = 0.50 for a house with no flue and shel-ter class 4. The stack flow is calculated using Equation (41): The wind flow is calculated using Equation (42): Substituting Qs and Qw into Equation (39) gives Q = 0.023 m3/s = 83 m3/h. From Equation (2), the air exchange rate I is equal to Q divided by the building volume: Example 6. Estimate the infiltration for a three-story house in San Fran-cisco, California. The house has a flow coefficient of c = 0.102 m3/(s·Pan) and a pressure exponent of n = 0.67 (this corresponds to effective leakage area of 1000 cm2 at 4 Pa). The building volume is 395 m3. The indoor-outdoor temperature difference is 5 K, and the wind speed is 4.47 m/s. The house has a flue and a crawl space foundation. Solution: For a three-story house with a flue, Cs = 0.107 (Pa/K)n. For a crawl space, Cw = 0.154 (Pa·s2/m2)n. From Table 10, for a three-story house, G = 0.67.
The prevailing wind blows along the row of houses parallel to the street, so the house has a shelter class of 5. Table 11 gives the shelter factor for a three-story house with a flue and shelter class 5 as s = 0.43.
Substituting Qs and Qw in Equation (39) gives Q = 0.039 m3/s = 140 m3/h.
Combining Residential Infiltration and Mechanical Ventilation Significant infiltration and mechanical ventilation often occur simultaneously in residences. The pressure difference from Equa-tion (24) can be used for each building leak, and the flow network (including mechanical ventilation) for the building can be solved to find the flow through all the leaks while accounting for the effect of the mechanical ventilation. However, for simplified models, the natural infiltration and mechanical ventilation are usually deter-mined separately and require a superposition method to combine the flow rates.
Sherman (1992) compares various superposition procedures and derives a generalized superposition equation that involves some simple leakage distribution parameters. He shows that the result is always subadditive. For small unbalanced fans, typically only half the flow contributes to the total, but this fraction can be anywhere between 0% and 100% depending on the leakage distribution.
When the fan flow becomes large, infiltration may be ignored.
Table 10 Enhanced Model Wind Speed Multiplier G House Height (Stories) One Two Three Wind speed multiplier G 0.48 0.59 0.67 Table 11 Enhanced Model Shelter Factor s Shelter Class No Flue One Story with Flue Two Story with Flue Three Story with Flue 1 1.00 1.10 1.07 1.06 2 0.90 1.02 0.98 0.97 3 0.70 0.86 0.81 0.79 4 0.50 0.70 0.64 0.61 5 0.30 0.54 0.47 0.43 Table 12 Enhanced Model Stack and Wind Coefficients One Story Two Story Three Story No Flue With Flue No Flue With Flue No Flue With Flue Cs 0.054 0.069 0.078 0.089 0.098 0.107 Cw for basement/ slab 0.156 0.142 0.170 0.156 0.170 0.167 Cw for crawl space 0.128 0.128 0.142 0.142 0.151 0.154 U GUmet 0.59 6.7 ( ) 3.95 m/s = = = Qs 0.051 ( ) 0.089 ( ) 20 19 – ( ) – [ ]0.67 0.053 m3 s ⁄ = = Qw 0.051 ( ) 0.156 ( ) 0.64 3.95 × ( )1.34 0.027 m3 s ⁄ = = I 214 m3 h ⁄ ( ) 340 m3 ⁄ = 0.63 h 1 – 0.63 ACH = = U GUmet 0.48 2.7 ( ) 1.3 m/s = = = Qs 0.078 ( ) 0.054 ( ) 16 ( )0.6 0.022 m3 s ⁄ = = Qw 0.078 ( ) 0.128 ( ) 0.50 1.3 × ( )1.2 0.006 m3 s ⁄ = = I 83 m3 h ⁄ ( ) 255 m3 ( ) ⁄ = 0.32 h 1 – 0.32 ACH = = U GUmet 0.67 4.47 ( ) 3.0 m/s = = = Qs 0.102 ( ) 0.107 ( ) 5 ( )0.67 0.032 m3 s ⁄ = = Qw 0.102 ( ) 0.154 ( ) 0.43 3.0 × ( )1.34 0.022 m3 s ⁄ = = I 140 m3 h ⁄ ( ) 395 m3 ⁄ = 0.35 h 1 – 0.35 ACH = = 26.24 2001 ASHRAE Fundamentals Handbook (SI) In special cases when the leakage distribution is known and highly skewed, it may be necessary to work through the superposi-tion method in more detail. For example, in a wind-dominated situ-ation, a supply fan will have a much bigger effect than an exhaust fan on changing the total ventilation rate; the same situation holds for houses having high neutral levels in cold climates. For the gen-eral case, when the details are not known or can be assumed to be broad and typical, the following superposition gives good results: (43) NONRESIDENTIAL AIR LEAKAGE Commercial Building Envelope Leakage There is currently only one industry standard for the measurement of envelope leakage in tall buildings: CGSB Standard 149.15. It uti-lizes the building’s own air handlers. ASTM Standard E779 and CGSB Standard 149.10 are intended for small detached buildings such as houses and provide no guidelines for dealing with problems arising in tall buildings, such as stack and wind effects. Tall buildings require refinement and extensions of established procedures because of obstacles to accurate measurement not present in small buildings, including large envelope leakage area, interfloor leakage, vertical shafts, and large wind and stack pressures. Recent work by Bahnfleth et al. (1999) also shows how to deal with some of these issues.
The building envelopes of large commercial buildings are often thought to be quite airtight. The National Association of Architec-tural Metal Manufacturers specifies a maximum leakage per unit of exterior wall area of 300 cm3/(s·m2) at a pressure difference of 75 Pa exclusive of leakage through operable windows. Tamura and Shaw (1976a) found that, assuming a flow exponent n of 0.65 in Equation (32), air leakage measurements in eight Canadian office buildings with sealed windows ranged from 610 to 2440 cm3/ (s·m2). Persily and Grot (1986) ran whole-building pressurization tests in large office buildings that showed that pressurization airflow rate divided by building volume is relatively low compared to that of houses. However, if these airflow rates are normalized by build-ing envelope area instead of by volume, the results indicate enve-lope airtightness levels similar to those in typical American houses.
In a study of eight U.S. office buildings, Persily and Grot (1986) found air leakage ranging from 1080 to 5220 cm3/(s·m2) at 75 Pa.
This means that office building envelopes are leakier than expected.
Typical air leakage values per unit wall area at 75 Pa are 500, 1500, and 3000 cm3/(s·m2) for tight, average, and leaky walls, respec-tively (Tamura and Shaw 1976a).
Grot and Persily (1986) also found that eight recently con-structed office buildings had infiltration rates ranging from 0.1 to 0.6 ACH with no outdoor air intake. The infiltration rates of these buildings exhibited varying degrees of weather dependence, gener-ally much lower than that measured in houses.
Air Leakage Through Internal Partitions In large buildings, the air leakage associated with internal parti-tions becomes very important. Elevator, stair, and service shaft walls; floors; and other interior partitions are the major separations of concern in these buildings. Their leakage characteristics are needed to determine infiltration through exterior walls and airflow patterns within a building. These internal resistances are also impor-tant in the event of a fire to predict smoke movement patterns and evaluate smoke management systems. Table 13 gives air leakage areas (calculated at 75 Pa with CD = 0.65) for different internal partitions of commercial buildings (Klote and Milke (1992)). Figure 13 presents examples of measured air leak-age rates of elevator shaft walls (Tamura and Shaw (1976b)), the type of data used to derive the values in Table 13. Chapter 51 of the 1999 ASHRAE Handbook—Applications should be consulted for perfor-mance models and applications of smoke management systems.
Leakage openings at the top of elevator shafts are equivalent to orifice areas of 0.4 to 1.0 m2. Air leakage rates through stair shaft and elevator doors are shown in Figure 14 as a function of average crack width around the door. The air leakage areas associated with other openings within commercial buildings are also important for air movement calculations. These include interior doors and parti-tions, suspended ceilings in buildings where the space above the ceiling is used in the air distribution system, and other components of the air distribution system.
Air Leakage Through Exterior Doors Door infiltration depends on the type of door, room, and build-ing. In residences and small buildings where doors are used infre-quently, the air exchange associated with a door can be estimated based on air leakage through cracks between the door and the frame.
A frequently opened single door, as in a small retail store, has a much larger amount of airflow than a closed door.
Air Leakage Through Automatic Doors Automatic doors are a major source of air leakage in buildings.
They are normally installed in locations where large numbers of Qcomb Qbal Qunbal 2 Qinfiltration 2 + + = Table 13 Air Leakage Areas for Internal Partitions in Commercial Buildings (at 75 Pa and CD = 0.65) Construction Element Wall Tightness Area Ratio AL/Aw Stairwell walls Tight 0.14 × 10−4 Average 0.11 × 10−3 Loose 0.35 × 10−3 Elevator shaft walls Tight 0.18 × 10−3 Average 0.84 × 10−3 Loose 0.18 × 10−2 AL/Af Floors Average 0.52 × 10−4 AL = air leakage area Aw = wall area Af = floor area Fig. 13 Air Leakage Rates of Elevator Shaft Walls Ventilation and Infiltration 26.25 people use the doors. They stay open longer with each use than man-ual doors. The air leakage through automatic doors can be reduced by the installation of a vestibule. However, pairs of automatic doors on the inside and outside of a vestibule normally have overlapping open periods, even when used by only one person at a time. There-fore, it is important that designers take into account the airflow through automatic doors when calculating the heating and cooling loads in the spaces next to them.
To calculate the average airflow rate through an automatic door, the designer must take into account the area of the door, the pressure difference across it, the discharge coefficient of the door when it is open, and the fraction of time that it is open. Obtaining the discharge coefficient is complicated by the fact that it changes as the door opens and closes.
To simplify this calculation, Figure 15 has been developed (Yuill 1996) to combine the discharge coefficients of doors as they open and close with the fraction of time that doors are open at a particular level of use. This figure presents an overall airflow coefficient as a function of the people using a door per hour. To obtain the average infiltration rate through an automatic door, the user must multiply this coefficient by the opening area of the door and by the square root of the pressure difference between the outdoor air and the indoor air at the location of the door. The pressure difference across a door in a building depends on the wind pressure on the building, the stack effect due to the indoor-outdoor temperature difference, and the effect of the operation of the air-handling system. It also depends on the leakage characteristics of the exterior walls of the building and of internal partitions.
Two simple methods are presented here. The first method uses simplifying assumptions to determine design values for Rp, the square root of the pressure difference across the automatic door, given in Figure 16. The second method requires explicit calculation of envelope pressures. In Figure 16, the airflows shown for ambient temperatures of 27 and 38°C, represented by dotted lines, are outward flows. They inter-cept the vertical axis at a lower point than the other lines because the wind pressure coefficients on the downwind face of the building (where the greatest outward flows will occur) are lower than on the upward face. In many buildings, the pressure in the building is con-trolled by varying the flow rate through the return fan(s) or by con-trolling the relief air dampers. These systems are usually set to maintain a pressure above ambient in the lobby. Subtracting the inte-rior pressure maintained in the lobby from the wind pressure gives the net pressure for estimating airflow through the door.
Method 1. For the first method, the infiltration rate through the automatic door is given by (44) where Q = airflow rate, L/s CA = airflow coefficient from Figure 15, L/(s·m2·Pa0.5) A = area of the door opening, m2 Rp = pressure factor from Figure 16, (Pa)0.5 Method 2. The airflow Q is given by (45) where Q = airflow rate, L/s CA = airflow coefficient from Figure 15, L/(s·m2·Pa0.5) A = area of the door opening, m2 ∆p = pressure difference across the door, Pa To find ∆p, it is necessary to find the pressure differential due to wind and that due to the stack effect. In order to give the largest pos-sible pressure difference across the door, there are no interactions between the two natural pressures: Fig. 14 Air Leakage Rate of Door Versus Average Crack Width Fig. 15 Airflow Coefficient for Automatic Doors Fig. 16 Pressure Factor for Automatic Doors Q CAARp = Q CAA ∆p = 26.26 2001 ASHRAE Fundamentals Handbook (SI) (46) where pU = wind-induced surface pressure relative to static pressure, Pa ∆ps = pressure difference due to stack effect, Pa Example Calculations It is desired to find the maximum possible infiltration through an automatic door located on the ground floor of a 20-story building. The area of the door is 0.91 m × 2.1 m = 1.9 m2. Each floor is 4 m high.
Approximately 300 people per hour pass through the door. The design wind conditions are 6.7 m/s, the indoor temperature is 21°C, and the outdoor temperature is −7°C. The airflow coefficient from Figure 15 (using the line for doors without vestibules) is approximately 300 L/(s·m2·Pa0.5).
Method 1: The pressure factor from Figure 16 is 7.9 Pa0.5. Equation (44) gives the door flow as Method 2: The worst possible case for the wind surface pressure coefficient Cp at any point and in any position on the ground floor of the building is inferred from figures in Chapter 16 to be about 0.75. Using this in Equation (18), together with the specified wind speed, results in pw = 20 Pa. Assume that H is one-half the height of the door (1.1 m). In order to have the maximum pressure across the door, assume the neu-tral pressure plane is located halfway up the building such that Substituting these values into Equation (17) gives ∆ps = −47 Pa.
This is the maximum stack pressure difference given no internal resis-tance to airflow. To find the actual stack pressure difference, it is neces-sary to multiply this by a draft coefficient. We will assume that this coefficient is 0.9, which is the highest value that has been found for tall buildings. Therefore, ∆ps = 0.9(−47 Pa) = –42 Pa. The total pressure is then ∆p = 20 − (−42) = 62 Pa. Substituting into Equation (45), If the building had a vestibule, the airflow coefficient would be read from Figure 15 using the line for doors with vestibules, and it would be approximately 208 L/(s·m2·Pa0.5), reducing the airflow to 3100 L/s into the building.
NONRESIDENTIAL VENTILATION Commercial and institutional building ventilation systems are typically designed to provide a slight pressurization to minimize infiltration. This pressurization is achieved by having the outside or makeup airflow rate higher than the exhaust or relief airflow rate. In these buildings, infiltration is usually neglected except in areas such as lobbies, where infiltration can be important due to doors. As dis-cussed in the section on Driving Mechanisms for Ventilation and Infiltration, wind and the stack effect can also cause significant infiltration and exfiltration. Ventilation airflow rates for commercial and institutional buildings are typically determined using proce-dures in ASHRAE Standard 62, Ventilation for Acceptable Indoor Air Quality. In these procedures for designing forced ventilation systems, no credit is given for infiltration. However, weather-driven pressure differentials may be significant and need to be considered when designing the ventilation system.
ASHRAE Standard 62 includes two procedures for obtaining acceptable indoor air quality: the Ventilation Rate Procedure and the Indoor Air Quality Procedure. The Ventilation Rate Procedure is by far the more commonly used.
Ventilation Rate Procedure In the Ventilation Rate Procedure, the design ventilation rate is determined based on a table of minimum ventilation requirements for different space types. These requirements are expressed as an outdoor airflow rate per occupant or per unit floor area, depending on the space type. These ventilation rates are based on air pollut-ants generated by people, activities, and building materials and furnishings. The HVAC designer faces several challenges in designing an air distribution system to deliver outdoor air to the occupants of a build-ing. The first is to determine whether the outdoor air is acceptable for use and to design a system for cleaning the air if it is not accept-able. A second goal is to design an air intake and distribution system that will deliver the required level of outdoor air to the occupied portions of the building. This outdoor air must be delivered not only at the design conditions, but throughout the year. The task is com-plicated by weather-related variations in indoor-outdoor pressure difference. Other complications include pressure variations due to building components such as exhaust fans or dirty filters, and prob-ably most significantly by supply flow variations associated with the operation of variable air volume (VAV) systems (Mumma and Wong 1990, Janu et al. 1995). After the design level of outdoor air is brought into the building, it must then be delivered to the occu-pants. This issue is related to the discussion in the section on Air Change Effectiveness presented earlier in this chapter.
TRACER GAS MEASUREMENTS The only reliable way to determine the air exchange rate of an existing building is to measure it. Several tracer gas measurement procedures exist (including a standard test method: ASTM Standard E 741), all involving an inert or nonreactive gas used to label the indoor air (Hunt 1980; Sherman et al. 1980; Harrje et al. 1981; Lagus and Persily 1985; Dietz et al. 1986; Charlesworth 1988; Per-sily 1988; Fisk et al. 1989; Lagus 1989; Sherman 1989a, 1989b; Fortmann et al. 1990; Harrje et al. 1990; Persily and Axley 1990; Sherman 1990). The tracer is released into the building in a specified manner, and the concentration of the tracer within the building is monitored and related to the building’s air exchange rate. A variety of tracer gases and associated concentration detection devices have been used. Desirable qualities of a tracer gas are detectability, non-reactivity, nontoxicity, neutral buoyancy, relatively low concentra-tion in ambient air, and low cost (Hunt 1980).
All tracer gas measurement techniques are based on a mass bal-ance of the tracer gas within the building. Assuming the outdoor concentration is zero and the indoor air is well mixed, this total bal-ance takes the following form: (47) where V = volume of space being tested, m3 C(θ) = tracer gas concentration at time θ dC/dθ = time rate of change of concentration, s−1 F(θ) = tracer gas injection rate at time θ, m3/s Q(θ) = airflow rate out of building at time θ, m3/s θ = time, s In Equation (47), density differences between indoor and out-door air are generally ignored for moderate climates; therefore, Q also refers to the airflow rate into the building. While Q is often referred to as the infiltration rate, any measurement includes both mechanical and natural ventilation in addition to infiltration. The ratio of Q to the volume V being tested has units of 1/time (often converted to ACH) and is the air exchange rate I.
Equation (47) is based on the assumptions that (1) no unknown tracer gas sources exist; (2) the airflow out of the building is the ∆p pU ∆ps – = Q 300 1.9 ( )7.9 4500 L/s = = HNPL 1 2 -- 20 stories ( ) 4 m story ------------40 m = = Q 306 1.9 ( ) 62 4580 L/s = = V dC dθ ------- F θ ( ) Q θ ( )C θ ( ) – = Ventilation and Infiltration 26.27 dominant means of removing the tracer gas from the space (i.e., the tracer gas does not react chemically within the space and/or is not adsorbed onto or by interior surfaces), and (3) the tracer gas concen-tration within the building can be represented by a single value (i.e., the tracer gas is uniformly mixed within the space).
Three different tracer gas procedures are used to measure air exchange rates: (1) decay or growth, (2) constant concentration, and (3) constant injection.
Decay or Growth Decay. The simplest tracer gas measurement technique is the decay method (also known as the step-down method). A small amount of tracer gas is injected into the space and is allowed to mix with the interior air. After the injection, F = 0 and then the solution to Equation (47) is (48) where Co is the concentration of the tracer in the space at θ = 0.
Equation (48) is generally used to solve for I by measuring the tracer gas concentration periodically during the decay and fitting the data to the logarithmic form of Equation (48): (49) Like all tracer gas techniques, the decay method has advantages and disadvantages. One advantage is that because logarithms of concentration are taken, only relative concentrations are needed, which can simplify the calibration of the concentration-measuring equipment. Also, the tracer gas injection rate need not be mea-sured, although it must be controlled so that the tracer gas concen-trations are within the range of the concentration-measuring device. The concentration-measuring equipment can be located on site, or building samples can be collected in suitable containers and analyzed elsewhere.
The most serious problem with the decay technique is imperfect mixing of the tracer gas with the interior air, both at initial injection and during the decay. Equations (47) and (48) employ the assump-tion that the tracer gas concentration within the building is uniform.
If the tracer is not well mixed, this assumption is not appropriate and the determination of I will be subject to errors. It is difficult to esti-mate the magnitude of the errors due to poor mixing, and little anal-ysis of this problem has been performed.
Growth. The growth or step-up method is similar to the decay method except that the initial tracer gas concentration is low and is increased during the test.
Constant Concentration In the constant concentration technique, the tracer gas injection rate is adjusted to maintain a constant concentration within the building. If the concentration is truly constant, then Equation (47) reduces to (50) There is less experience with this technique than with the decay procedure, but an increasing number of applications exist (Kumar et al. 1979, Collet 1981, Bohac et al. 1985, Fortmann et al. 1990, Walker and Wilson 1998, Wilson and Walker 1993, Walker and For-est 1995).
Because the tracer gas injection is continuous, no initial mixing period is required. Another advantage is that the tracer gas injection into each zone of the building can be separately controlled; thus, the amount of outdoor air flowing into each zone can be determined.
This procedure is best suited for longer term continuous monitoring of fluctuating infiltration rates. This procedure has the disadvantage of requiring the measurement of absolute tracer concentrations and injection rates. Also, imperfect mixing of the tracer and the interior air causes a delay in the response of the concentration to changes in the injection rate.
Constant Injection In the constant injection procedure, the tracer is injected at a con-stant rate, and the solution to Equation (47) becomes (51) After sufficient time, the transient term reduces to zero, the con-centration attains equilibrium, and Equation (51) reduces to (52) Equation (52) is valid only when air exchange rate I and airflow rate Q are constant; thus, this technique is only appropriate for systems at or near equilibrium. It is particularly useful in spaces with mechanical ventilation or with high air exchange rates. Con-stant injection requires the measurement of absolute concentrations and injection rates.
Dietz et al. (1986) used a special case of the constant injection technique. This technique uses permeation tubes as a tracer gas source. The tubes release the tracer at an ideally constant rate into the building being tested. A sampling tube packed with an adsorbent collects the tracer from the interior air at a constant rate by diffusion.
After a sampling period of one week or more, the sampler is removed and analyzed to determine the average tracer gas concen-tration within the building during the sampling period.
Solving Equation (47) for C and taking the time average gives (53) where < … > denotes time average. (Note that the time average of dC/dθ is assumed to equal zero.) Equation (53) shows that the average tracer concentration and the injection rate F can be used to calculate the average of the inverse airflow rate. The average of the inverse is less than the inverse of the actual average, with the magnitude of this difference depending on the distribution of airflow rates during the measure-ment period. Sherman and Wilson (1986) calculated these dif-ferences to be about 20% for one-month averaging periods.
Differences greater than 30% have been measured when there were large changes in air exchange rate due to occupant airing of houses; errors from 5 to 30% were measured when the variation was due to weather effects (Bohac et al. 1987). Longer averaging periods and large changes in air exchange rates during the measurement periods generally lead to larger differences between the average inverse exchange rate and the inverse of the actual average rate.
Multizone Air Exchange Measurement Equation (47) is based on the assumption of a single, well-mixed enclosure, and the techniques described are for single-zone mea-surements. Airflow between internal zones and between the exterior and individual internal zones has led to the development of multi-zone measurement techniques (Harrje et al. 1985, Sherman and Dickerhoff 1989, Fortmann et al. 1990, Harrje et al. 1990). These techniques are important when considering the transport of pollut-ants from one room of a building to another. For a theoretical devel-opment, see Sinden (1978b). Multizone measurements typically use either multiple tracer gases for the different zones or the constant concentration technique. A proper error analysis is essential in all multizone flow determination (Charlesworth 1988, D’Ottavio et al.
1988).
C θ ( ) Coe Iθ – = C θ ( ) ln Co Iθ – ln = Q θ ( ) F θ ( ) C ⁄ = C θ ( ) F Q ⁄ ( ) 1 e Iθ – – ( ) = Q F C ⁄ = F<1 Q ⁄ > = = 26.28 2001 ASHRAE Fundamentals Handbook (SI) SYMBOLS A = area, m2 or cm2 c = flow coefficient, m3/(s·Pan) cp = specific heat, J/(kg·K) or kJ/(kg·K) C = concentration C = time averaged concentration CA = airflow coefficient for automatic doors, L/(s·m2·Pa0.5) CD = discharge coefficient Cp = pressure coefficient Cs = stack flow coefficient, (L/s)2/(cm4·K) or (Pa/K)n Cv = effectiveness of openings Cw = wind flow coefficient, (L/s)2/[cm4·(m/s)2] or [Pa/(m/s)2]n F = tracer gas injection rate, m3/s F = time-averaged contaminant source strength, m3/s f = fractional on-time g = wind speed reduction factor h = specific enthalpy, kJ/kg H = height, m i = hour of year I = air exchange rate, 1/time Ii = instantaneous air exchange rate, 1/time Im = effective air exchange rate, 1/time IDD = infiltration degree-days, K·day n = pressure exponent N = number of discrete time periods in period of interest p = pressure, Pa P = parameter q = heat rate, W Q = volumetric flow rate, m3/s Q = effective volumetric flow rate, m3/s s = shelter factor S = source strength, µg/s t = relative temperature, °C T = absolute temperature, K U = wind speed, m/s V = volume, m3 W = humidity ratio, kg/kg εI = air change effectiveness φ = wind angle, degrees ρ = air density, kg/m3 θ = time θage = age of air τ = time constant Subscripts b = base ba = bypass air c = calculated ca = recirculated air e = effective ea = exhaust air f = floor i = indoor or time counter for summation (instantaneous) H = building height, eaves or roof ka = makeup air l = latent la = relief air L = leakage or local ma = mixed air met = meteorological station location n = normalized N = nominal NPL = neutral pressure level o = outdoor, initial condition, or reference oa = outside air p = pressure r = reference s = sensible or stack sa = supply air S = space or source w = wind REFERENCES ACGIH. 1998. Industrial ventilation: A manual of recommended practice, 23rd ed. American Conference of Governmental Industrial Hygienists, Lansing, MI.
AIVC. 1994. An analysis and data summary of the AIVC’s numerical data-base. AIVC Technical Note 44.
AIVC. 1999. AIRBASE bibliographic database. International Energy Agency Air Infiltration and Ventilation Centre, Coventry, UK.
Akins, R.E., Peterka, J.A., and Cermak, J.E. 1979. Averaged pressure coef-ficients for rectangular buildings. Vol.1, Proceeding of the Fifth Interna-tional Wind Engineering Conference, Fort Collins, U.S., July, pp. 369-80.
Allard, F. and M. Herrlin. 1989. Wind-induced ventilation. ASHRAE Trans-actions 95(2):722-28.
ASHRAE. 1988. Air leakage performance for detached single-family resi-dential buildings. ANSI/ASHRAE Standard 119-1988 (RA 94).
ASHRAE. 1992. Thermal environmental conditions for human occupancy.
ANSI/ASHRAE Standard 55-92.
ASHRAE. 1997. Measuring air change effectiveness. ANSI/ASHRAE Standard 129-97 ASHRAE. 1999. Method of testing general ventilation air cleaning devices for removal efficiency by particle size. Standard 52.2-1999. ASHRAE. 1999. Ventilation for acceptable indoor air quality. ANSI/ ASHRAE Standard 62-1999.
ASHRAE. 2000. A method of determining air change rates in detached dwellings. ANSI/ASHRAE Standard 136-2000.
ASTM. 1987. Practices for air leakage site detection in building envelopes.
Standard E 1186-87(R 1992). American Society for Testing and Materi-als, West Conshohocken, PA.
ASTM. 1991. Test method for determining the rate of air leakage through exterior windows, curtain walls, and doors under specified pressure dif-ferences across the specimen. Standard E 283-91.
ASTM. 1993. Test method for field measurement of air leakage through installed exterior windows and doors. Standard E 783-93.
ASTM. 1993. Test methods for determining air change in a single zone by means of a tracer gas dilution. Standard E 741-93.
ASTM. 1996. Standard test methods for determining airtightness of build-ings using an orifice blower door. Standard E 1827-96.
ASTM. 1999. Measuring air leakage by the fan pressurization method. Stan-dard E 779-99.
ASTM. 1999. Standard guide for assessing depressurization-induced back-drafting and spillage from vented combustion appliances. ASTM Stan-dard E 1998-99.
Bahnfleth, W.P., G.K. Yuill, and B.W. Lee. 1999. Protocol for field testing of tall buildings to determine envelope air leakage rates. ASHRAE Trans-actions 105(2).
Bankvall, C.G. 1987. Air movements and thermal performance of the build-ing envelope. Thermal Insulation: Materials and Systems. F.J. Powell and S.L. Mathews, eds. ASTM, Philadelphia, pp. 124-31.
Berg-Munch, B., G. Clausen, and P.O. Fanger. 1986. Ventilation require-ments for the control of body odor in spaces occupied by women. Envi-ronmental International 12(1-4):195.
Berlad, A.L., N. Tutu, Y. Yeh, R. Jaung, R. Krajewski, R. Hoppe and F.
Salzano. 1978. Air intrusion effects on the performance of permeable insulation systems. In Thermal insulation performance, D. McElroy and R. Tye, eds. ASTM STP 718, pp. 181-94 (October).
Blomsterberg, A.K. and D.T. Harrje. 1979. Approaches to evaluation of air infiltration energy losses in buildings. ASHRAE Transactions 85(1):797.
Bohac, D., D.T. Harrje, and G.S. Horner. 1987. Field study of constant con-centration and PFT infiltration measurements. Proceedings of the 8th IEA Conference of the Air Infiltration and Ventilation Centre, Uberlin-gen, Germany.
Bohac, D., D.T. Harrje, and L.K. Norford. 1985. Constant concentration infiltration measurement technique: An analysis of its accuracy and field measurements, 176. Proceedings of the ASHRAE-DOE-BTECC Confer-ence on the Thermal Performance of the Exterior Envelopes of Buildings III, Clearwater Beach, FL.
Bradley, B. 1993. Implementation of the AIM-2 infiltration model in HOT-2000. Report for Department of Natural Resources Canada.
Buchanan, C.R. and M. H. Sherman. 2000. A mathematical model for infil-tration heat recovery. Proceedings of the 21st AVIC Annual Conference, Innovations in Ventilation Technology, 26-29 September, Air Infiltration and Ventilation Centre, Coventry, UK. Lawrence Berkeley National Lab-oratory Report LBNL-44294.
Ventilation and Infiltration 26.29 Cain, W.S. et al. 1983. Ventilation requirements in buildings—I. Control of occupancy odor and tobacco smoke odor. Atmos. Environ. 17(6): 1183-97.
CGSB. 1986. Determination of the airtightness of building envelopes by the fan depressurization method. CGSB Standard 149.10-M86. Canadian General Standards Board, Ottawa.
CGSB. 1995. The spillage test: Method to determine the potential for pres-sure-induced spillage from vented, fuel-fired, space heating appliances, water heaters, and fireplaces. CAN/CGSB Standard 51.71-95.
CGSB. 1996. Determination of the overall envelope airtightness of build-ings by the fan pressurization method using the building’s air handling systems. CGSB Standard 149.15-96.
CHBA. 1994. HOT2000 technical manual. Canadian Home Builders Asso-ciation, Ottawa, Canada.
Charlesworth, P.S. 1988. Measurement of air exchange rates. Chapter 2 in Air exchange rate and airtightness measurement techniques—An applications guide. Air Infiltration and Ventilation Centre, Coventry, Great Britain.
Chastain, J.P. 1987. Pressure gradients and the location of the neutral pres-sure axis for low-rise structures under pure stack conditions. Unpub-lished M.S. thesis. University of Kentucky, Lexington.
Chastain, J.P., D.G. Colliver, and P.W. Winner, Jr. 1987. Computation of dis-charge coefficients for laminar flow in rectangular and circular openings.
ASHRAE Transactions 93(2):2259-83.
Chastain, J.P. and D.G. Colliver. 1989. Influence of temperature stratifica-tion on pressure differences resulting from the infiltration stack effect.
ASHRAE Transactions 95(1):256-68.
Claridge, D.E., M. Krarti, and S. Bhattacharyya. 1988. Preliminary mea-surements of the energy impact of infiltration in a test cell. Proceedings of Fifth Annual Symposium on Improving Building Energy Efficiency in Hot and Humid Climates, pp. 308-17.
Claridge, D.E., and S. Bhattacharyya. 1990. The measured impact of infil-tration in a test cell. ASME Journal of Solar Energy Engineering 112: 123-26.
Cole, J.T., T.S. Zawacki, R.H. Elkins, J.W. Zimmer, and R.A. Macriss. 1980.
Application of a generalized model of air infiltration to existing homes.
ASHRAE Transactions 86(2):765.
Collet, P.F. 1981. Continuous measurements of air infiltration in occupied dwellings. Proceedings of the 2nd IEA Conference of the Air Infiltration Centre, p.147. Stockholm, Sweden.
Colliver, D.G., W.E. Murphy, and W. Sun. 1992. Evaluation of the tech-niques for the measurement of air leakage of building components. Final Report of ASHRAE Research Project RP-438. University of Kentucky, Lexington.
CSA. 1991. Residential mechanical ventilation systems. CAN/CSA-F326-M91. Canadian Standards Association, Toronto.
Cummings, J.B. and J.J. Tooley, Jr. 1989. Infiltration and pressure differ-ences induced by forced air systems in Florida residences. ASHRAE Transactions 96(20):551-60.
Cummings, J.B., J.J. Tooley, Jr., and R. Dunsmore. 1990. Impacts of duct leakage on infiltration rates, space conditioning energy use and peak electrical demand in Florida homes. Proceedings of ACEEE Summer Study, Pacific Grove, CA. American Council for an Energy-Efficient Economy, Washington, D.C.
Cutter. 1987. Air-to-air heat exchangers. In Energy design update. Cutter Information Corporation, Arlington, MA.
Desrochers, D. and A.G. Scott. 1985. Residential ventilation rates and indoor radon daughter levels. Transactions of the APCA Specialty Conference, Indoor Air Quality in Cold Climates: Hazards and Abatement Measures, p. 362. Ottawa, Canada.
Diamond, R.C., M.P. Modera, and H.E. Feustel. 1986. Ventilation and occu-pant behavior in two apartment buildings. Lawrence Berkeley National Laboratory Report LBL-21862. Presented at the 7th AIC Conference, Stratford-upon-Avon, UK.
Diamond, R.C., J.B. Dickinson, R.D. Lipschutz, B. O’Regan, and B. Shohl.
1982. The house doctor’s manual. Report PUB 3017. Lawrence Berkeley National Laboratory, Berkeley, CA.
Dickerhoff, D.J., D.T. Grimsrud, and R.D. Lipschutz. 1982. Component leak-age testing in residential buildings. Proceedings of the American Council for an Energy-Efficient Economy, 1982 Summer Study, Santa Cruz, CA.
Report LBL 14735. Lawrence Berkeley National Laboratory, Berkeley, CA.
Dietz, R.N., R.W. Goodrich, E.A. Cote, and R.F. Wieser. 1986. Detailed description and performance of a passive perfluorocarbon tracer system for building ventilation and air exchange measurement. In Measured air leakage of buildings, p. 203. ASTM STP 904. H.R. Trechsel and P.L.
Lagus, eds.
D’Ottavio, T.W., G.I. Senum, and R.N. Dietz. 1988. Error analysis tech-niques for perfluorocarbon tracer derived multizone ventilation rates.
Building and Environment 23(40).
Ek, C.W., S.A. Anisko, and G.O. Gregg. 1990. Air leakage tests of manu-factured housing in the Northwest United States. In Air change rate and airtightness in buildings, pp. 152-64. ASTM STP 1067. M.H.
Sherman, ed.
Elmroth, A. and P. Levin. 1983. Air infiltration control in housing, a guide to international practice. Report D2:1983. Air Infiltration Centre, Swed-ish Council for Building Research, Stockholm.
Energy Resource Center. 1982. How to house doctor. University of Illinois, Chicago.
Etheridge, D.W. 1977. Crack flow equations and scale effect. Building and Environment 12:181.
Etheridge, D.W. and D.K. Alexander. 1980. The British gas multi-cell model for calculating ventilation. ASHRAE Transactions 86(2):808.
Etheridge, D.W. and J.A. Nolan. 1979. Ventilation measurements at model scale in a turbulent flow. Building and Environment 14(1):53.
Eto, J. 1990. The HVAC costs of increased fresh air ventilation rates in office buildings. Proceedings of Indoor Air ’90 4:53-58. International Confer-ence on Indoor Air Quality and Climate, Ottawa, Canada.
Eto, J. and C. Meyer. 1988. The HVAC costs of increased fresh air ventila-tion rates in office buildings. ASHRAE Transactions 94(2):331-45.
Eyre, D. and D. Jennings. 1983. Air-vapour barriers—A general perspective and guidelines for installation. Energy, Mines, and Resources Canada, Ottawa.
Farrington, R., D. Martin, and R. Anderson. 1990. A comparison of dis-placement efficiency, decay time constant, and age of air for isothermal flow in an imperfectly mixed enclosure. Proceedings of the ACEEE 1990 Summer Study on Energy Efficiency in Buildings, pp. 4.35-4.43. Ameri-can Council for an Energy-Efficient Economy, Washington, D.C.
Feustel, H.E. and A. Raynor-Hoosen, eds. 1990. Fundamentals of the mul-tizone air flow model—COMIS. Technical Note 29. Air Infiltration and Ventilation Centre, Coventry, Great Britain.
Fisk, W.J., R.J. Prill, and O. Steppanen. 1989. A multi-tracer technique for studying rates of ventilation, air distribution patterns and air exchange efficiencies. Proceedings of Conference on Building Systems—Room Air and Air Contaminant Distribution, pp. 237-40. ASHRAE, Atlanta.
Fisk, W.J., R.K. Spencer, D.T. Grimsrud, F.J. Offermann, B. Pedersen, and R. Sextro. 1984. Indoor air quality control techniques: A critical review. Report LBL 16493. Lawrence Berkeley National Laboratory, Berkeley, CA.
Fortmann, R.C., N.L. Nagda, and H.E. Rector. 1990. Comparison of meth-ods for the measurement of air change rates and interzonal airflows to two test residences. In Air change rate and airtightness in buildings, pp.
104-18. ASTM STP 1067. M.H. Sherman, ed.
Foster, M.P. and M.J. Down. 1987. Ventilation of livestock buildings by nat-ural convection. Journal of Agricultural Engineering Research 37:1.
Giesbrecht, P. and G. Proskiw. 1986. An evaluation of the effectiveness of air leakage sealing. In Measured air leakage of buildings, p. 312. ASTM STP 904. H.R. Trechsel and P.L. Lagus, eds.
Grieve, P.W. 1989. Measuring ventilation using tracer-gases. Brüel and Kjær, Denmark.
Grimsrud, D.T. and K.Y. Teichman. 1989. The scientific basis of Standard 62-1989. ASHRAE Journal 31(10):51-54.
Grimsrud, D.T., M.H. Sherman, and R.C. Sonderegger. 1982. Calculating infiltration: Implications for a construction quality standard. Proceedings of the ASHRAE-DOE Conference on the Thermal Performance of the Exterior Envelope of Buildings II, p. 422. Las Vegas, NV.
Grimsrud, D.T., M.H. Sherman, R.C. Diamond, P.E. Condon, and A.H.
Rosenfeld. 1979. Infiltration-pressurization correlations: Detailed mea-surements in a California house. ASHRAE Transactions 85(1):851.
Grot, R.A. and R.E. Clark. 1979. Air leakage characteristics and weather-ization techniques for low-income housing. Proceedings of the ASHRAE-DOE Conference on the Thermal Performance of the Exterior Envelopes of Buildings, p. 178. Orlando, FL.
Grot, R.A. and A.K. Persily. 1986. Measured air infiltration and ventilation rates in eight large office buildings. In Measured air leakage of build-ings, p. 151. ASTM STP 904. H.R. Trechsel and P.L. Lagus, eds.
26.30 2001 ASHRAE Fundamentals Handbook (SI) Hamlin, T.L. 1991. Ventilation and airtightness in new, detached Canadian housing. ASHRAE Transactions 97(2):904-10.
Hamlin, T. and W. Pushka. 1994. Predicted and measured air change rates in houses with predictions of occupant IAQ comfort. Proceedings of the 15th AIVC Conference, Buxton, UK, pp. 771-75.
Harrje, D.T. and G.J. Born. 1982. Cataloguing air leakage components in houses. Proceedings of the ACEEE 1982 Summer Study, Santa Cruz, CA. American Council for an Energy-Efficient Economy, Washing-ton, D.C.
Harrje, D.T. and T.A. Mills, Jr. 1980. Air infiltration reduction through ret-rofitting, 89. Building air change rate and infiltration measurements.
ASTM STP 719. C.M. Hunt, J.C. King, and H.R. Trechsel, eds.
Harrje, D.T., G.S. Dutt, and J. Beyea. 1979. Locating and eliminating obscure but major energy losses in residential housing. ASHRAE Trans-actions 85(2):521.
Harrje, D.T., R.A. Grot, and D.T. Grimsrud. 1981. Air infiltration site mea-surement techniques. Proceedings of the 2nd IEA Conference of the Air Infiltration Centre, p. 113. Stockholm, Sweden.
Harrje, D.T., G.S. Dutt, D.L. Bohac, and K.J. Gadsby. 1985. Documenting air movements and infiltration in multicell buildings using various tracer techniques. ASHRAE Transactions 91(2):2012-27.
Harrje, D.T., R.N. Dietz, M. Sherman, D.L. Bohac, T.W. D’Ottavio, and D.J.
Dickerhoff. 1990. Tracer gas measurement systems compared in a mul-tifamily building. In Air change rate and airtightness in buildings, p. 5-12. ASTM STP 1067. M.H. Sherman, ed.
Hekmat, D., H.E. Feustel, and M.P. Modera. 1986. Impacts of ventilation strategies on energy consumption and indoor air quality in single-family residences. Energy and Buildings 9(3):239.
Herrlin, M.K. 1985. MOVECOMP: A static-multicell-airflow-model. ASH-RAE Transactions 91(2B):1989.
Holton, J.K., M.J. Kokayko, and T.R. Beggs. 1997. Comparative ventilation system evaluations. ASHRAE Transactions 103(2):675-92.
Honma, H. 1975. Ventilation of dwellings and its disturbances. Faibo Grafiska, Stockholm.
Hopkins, L.P. and B. Hansford. 1974. Air flow through cracks. Building Ser-vice Engineer 42(September):123.
Hunt, C.M. 1980. Air infiltration: A review of some existing measurement techniques and data. Building air change rate and infiltration mea-surements, p. 3. ASTM STP 719. C.M. Hunt, J.C. King, and H.R.
Trechsel, eds.
ISO. 1995. Thermal insulation—Determination of building air tightness— Fan pressurization method. ISO Standard 9972. International Organiza-tion for Standardization, Geneva.
Iwashita, G., K. Kimura., et al. 1989. Pilot study on addition of old units for perceived air pollution sources. Proceedings of the SHASE Annual Meeting, pp. 3221-324. Society of Heating, Air-Conditioning and Sani-tary Engineers of Japan, Tokyo.
Jacobson, D.I., G.S. Dutt, and R.H. Socolow. 1986. Pressurization testing, infiltration reduction, and energy savings. In Measured air leakage of buildings, p. 265. ASTM STP 904. H.R. Trechsel and P.L. Lagus, eds.
Janssen, J.E. 1989. Ventilation for acceptable indoor air quality. ASHRAE Journal 31(10):40-48.
Janu, G.J., J.D. Wegner, C.G. Nesler. 1995. Outdoor air flow control for VAV systems. ASHRAE Journal 37(4):62-68.
Judkoff, R., J.D. Balcomb, C.E. Handcock, G. Barker, and K. Subbarao.
1997. Side-by-side thermal tests of modular offices: A validation study of the STEM method. Report. NREL, Golden, CO.
Jump, D.A., I.S. Walker, and M.P. Modera. 1996. Field measurements of efficiency and duct retrofit effectiveness in residential forced air distri-bution systems. Proceedings of the 1996 ACEEE Summer Study, pp.1.147-1.156. ACEEE, Washington, D.C.
Kiel, D.E. and D.J. Wilson. 1986. Gravity driven airflows through open doors, 15.1. Proceedings of the 7th IEA Conference of the Air Infiltration Centre, Stratford-upon-Avon, United Kingdom.
Kim, A.K. and C.Y. Shaw. 1986. Seasonal variation in airtightness of two detached houses. In Measured air leakage of buildings, p. 17. ASTM STP 904. H.R. Trechsel and P.L. Lagus, eds.
Klauss, A.K., R.H. Tull, L.M. Rootsd, and J.R. Pfafflino. 1970. History of the changing concepts in ventilation requirements. ASHRAE Journal 12(6):51-55.
Kohonen, R., T. Ojanen, and M. Virtanen. 1987. Thermal coupling of leak-age flows and heating load of buildings. 8th AIVC Conference, Berlin.
Kohonen, R., and T. Ojanen. 1987. Non-steady state coupled diffusion and convection heat and mass transfer in porous media. 5th International Conference on Numerical Methods in Thermal Problems, Montreal.
Kreith, F., and R. Eisenstadt. 1957. Pressure drop and flow characteristics of short capillary tubes at low Reynolds numbers. ASME Transactions, pp.
1070-78.
Kronvall, J. 1980. Correlating pressurization and infiltration rate data— Tests of an heuristic model. Lund Institute of Technology, Division of Building Technology, Lund, Sweden.
Kumar, R., A.D. Ireson, and H.W. Orr. 1979. An automated air infiltration measuring system using SF6 tracer gas in constant concentration and decay methods. ASHRAE Transactions 85(2):385.
Kvisgaard, B. and P.F. Collet. 1990. The user’s influence on air change. In Air change rate and airtightness in buildings, pp. 67-76. ASTM STP 1067. M.H. Sherman, ed.
Lagus, P.L. 1989. Tracer measurement instrumentation suitable for infiltra-tion, air leakage, and air flow pattern characterization. Proceedings of the Conference on Building Systems—Room Air and Air Contaminant Dis-tribution, pp. 97-102. ASHRAE, Atlanta.
Lagus, P. and A.K. Persily. 1985. A review of tracer-gas techniques for mea-suring airflows in buildings. ASHRAE Transactions 91(2B):1075.
Lamming, S., and Salmon, J. 1998. Wind data for design of smoke control systems. ASHRAE Transactions 104(1).
Lecompte, J.G.N. 1987. The influence of natural convection in an insulated cavity on thermal performance of a wall. Insulation Materials Testing and Applications. ASTM, Florida.
Liddament, M.W. 1988. The calculation of wind effect on ventilation.
ASHRAE Transactions 94(2):1645-60.
Liddament, M. and C. Allen. 1983. The validation and comparison of math-ematical models of air infiltration. Technical Note 11. Air Infiltration Centre, Bracknell, Great Britain.
Liu, M. and D.E. Claridge. 1992a. The measured energy impact of infiltra-tion under dynamic conditions. Proceedings of the 8th Symposium on Improving Building Systems in Hot and Humid Climates, Dallas, TX.
Liu, M. and D.E. Claridge. 1992b. The measured energy impact of infiltra-tion in a test cell. Proceedings of the 8th Symposium on ImprovingBuild-ing Systems in Hot and Humid Climates, Dallas, TX.
Liu, M. and D.E. Claridge. 1992c. The energy impact of combined solar radiation/infiltration/conduction effects in walls and attics. Proceedings of Thermal Performance of Exterior Envelopes of Buildings, 5th ASHRAE/DOE/BTECC Conference, Clearwater Beach, FL.
Liu, M. and D.E. Claridge. 1995. Experimental methods for identifying infiltration heat recovery in building. Proceedings of the Thermal Per-formance of Exterior Envelopes of Buildings, 6th ASHRAE/DOE/ BTECC Conference, Clearwater Beach, FL.
Lubliner, M., D.T. Stevens, and B. Davis. 1997. Mechanical ventilation in HVD-code manufactured housing in the Pacific Northwest. ASHRAE Transactions 103(1):693-705.
Marbek Resource Consultants. 1984. Air sealing homes for energy conser-vation. Energy, Mines and Resources Canada, Buildings Energy Tech-nology Transfer Program, Ottawa.
Modera, M.P. 1989. Residential duct system leakage: Magnitude, impacts, and potential for reduction. ASHRAE Transactions. 96(2):561-69.
Modera, M.P. and D.J. Wilson. 1990. The effects of wind on residential building leakage measurements. Air change rate and airtightness in buildings, pp. 132-45. ASTM STP 1067. M.H. Sherman, ed.; Lawrence Berkeley National Laboratory Report LBL-24195.
Modera, M.P., D. Dickerhoff, R. Jansky, and B. Smith. 1991. Improving the energy efficiency of residential air distribution systems in California.
Report LBL-30866. Lawrence Berkeley National Laboratory, Berkeley, CA.
Mumma, S.A., and R.J. Bolin. 1994. Real-time, on-line optimization of VAV system control to minimize the energy consumption rate and to satisfy ASHRAE Standard 62-1989 for all occupied zones. ASHRAE Trans-actions 94(1):168-79.
Mumma, S.A., and Y.M. Wong. 1990. Analytical evaluation of outdoor air-flow rate variation vs. supply airflow rate variation in VAV systems when the outside air damper position is fixed. ASHRAE Transactions 90(1):1197-1208.
Murphy, W.E., D.G. Colliver, and L.R. Piercy. 1991. Repeatability and reproducibility of fan pressurization devices in measuring building air leakage. ASHRAE Transactions 97(2):885-95.
Ventilation and Infiltration 26.31 Nelson, B.D., D.A. Robinson, and G.D. Nelson. 1985. Designing the enve-lope—Guidelines for buildings. Thermal Performance of the Exterior Envelopes of Buildings III. Proceedings of the ASHRAE/DOE/BTECC Conference, Florida. ASHRAE SP 49, pp. 1117-22.
NRCC. 1995. National Building Code of Canada. National Research Coun-cil of Canada, Ottawa.
Nylund, P.O. 1980. Infiltration and ventilation. Report D22:1980. Swedish Council for Building Research, Stockholm.
Offermann, F., and D. Int-Hout. 1989. Ventilation effectiveness measure-ments of three supply/return air configurations. Environment Interna-tional 15(1-6):585-92.
Orme, M. 1999. Applicable models for air infiltration and ventilation calculations. AIVC Technical Note 51. AIVC, Coventry. UK.
Palmiter, L. and I. Brown. 1989. The Northwest Residential Infiltration Sur-vey, description and summary of results. Proceedings of the ASHRAE/ DOE/BTECC/CIBSE Conference—Thermal Performance of the Exte-rior Envelopes of Buildings IV, Florida, pp. 445-57.
Palmiter, L., I.A. Brown, and T.C. Bond. 1991. Measured infiltration and ven-tilation in 472 all-electric homes. ASHRAE Transactions 97(2):979-87.
Palmiter, L. and T. Bond. 1994. Modeled and measured infiltration II—A detailed case study of three homes. EPRI Report TR 102511. Electric Power Research Institute, Palo Alto, CA.
Parekh, A., K. Ruest, and M. Jacobs. 1991. Comparison of airtightness, indoor air quality and power consumption before and after air-sealing of high-rise residential buildings. Proceedings of the 12th AIVC Confer-ence: Air Movement and Ventilation Control within Buildings, Air Infil-tration and Ventilation Centre, Coventry, Great Britain.
Parker, G.B., M. McSorley, and J. Harris. 1990. The Northwest Residential Infiltration Survey: A field study of ventilation in new houses in the Pacific Northwest. In Air change rate and airtightness in buildings, pp.
93-103. ASTM STP 1067. M.H. Sherman, ed.
Persily, A. 1982. Repeatability and accuracy of pressurization testing. Pro-ceedings of the ASHRAE-DOE Conference, Thermal Performance of the Exterior Envelopes of Buildings II, Las Vegas, NV.
Persily, A.K. 1986. Measurements of air infiltration and airtightness in pas-sive solar homes. In Measured air leakage of buildings, p. 46. ASTM STP 904. H.R. Trechsel and P.L. Lagus, eds.
Persily, A.K. 1988. Tracer gas techniques for studying building air exchange. Report NBSIR 88-3708. National Institute of Standards and Technology, Gaithersburg, MD.
Persily, A.K. 1991. Design guidelines for thermal envelope integrity in office buildings. Proceedings of the 12th AIVC Conference: Air Move-ment and Ventilation Control within Buildings, Air Infiltration and Ven-tilation Centre, Coventry, Great Britain.
Persily, A.K. and J. Axley. 1990. Measuring airflow rates with pulse tracer techniques. In Air change rate and airtightness in buildings, pp. 31-51.
ASTM STP 1067. M.H. Sherman, ed.
Persily, A.K. and R.A. Grot. 1985a. The airtightness of office building enve-lopes. Proceedings of the ASHRAE-DOE-BTECC Conference on the Thermal Performance of the Exterior Envelopes of Buildings III, Clear-water Beach, FL. p. 125.
Persily, A.K. and R.A. Grot. 1985b. Accuracy in pressurization data analy-sis. ASHRAE Transactions 91(2B):105.
Persily, A.K. and R.A. Grot. 1986. Pressurization testing of federal build-ings. In Measured air leakage of buildings, p. 184. ASTM STP 904. H.R.
Trechsel and P.L. Lagus, eds.
Persily, A.K. and G.T. Linteris. 1983. A comparison of measured and pre-dicted infiltration rates. ASHRAE Transactions 89(2):183.
Powell, F., M. Krarti, and A. Tuluca. 1989. Air movement influence on the effective thermal resistance of porous insulations: A literature survey.
Journal of Thermal Insulation 12:239-251.
Reardon, J.T. and C.-Y. Shaw. 1997. Evaluation of five simple ventilation strategies suitable for houses without forced-air heating. ASHRAE Trans-actions 103(1):731-744.
Reeves, G., M.F. McBride, and C.F. Sepsy. 1979. Air infiltration model for residences. ASHRAE Transactions 85(1):667.
Riley, M. 1990. Indoor air quality and energy conservation: The R-2000 home program experience. Proceedings of Indoor Air ’90 5:143. Inter-national Conference on Indoor Air Quality and Climate, Ottawa, Canada.
Robison, P.E. and L.A. Lambert. 1989. Field investigation of residential infil-tration and heating duct leakage. ASHRAE Transactions 95(2):542-50.
Rock, B.A. 1992. Characterization of transient pollutant transport, dilution, and removal for the study of indoor air quality. Ph.D. diss., University of Colorado at Boulder. University Microfilms International.
Rock, B.A., M.J. Brandemuehl, and R. Anderson. 1995. Toward a simplified design method for determining the air change effectiveness. ASHRAE Transactions 101(1):217-27.
Rudd, A.F. 1998. Design/sizing methodology and economic evaluation of central-fan-integrated supply ventilation systems. ACEEE 1998 Summer Study on Energy Efficiency in Buildings. 23-28 August, Pacific Grove, California. American Council for an Energy Efficient Economy, Wash-ington, D.C.
Sandberg, M.H. 1981. What is ventilation efficiency? Building and Environ-ment 16:123-35.
Shaw, C.Y. 1981. A correlation between air infiltration and air tightness for a house in a developed residential area. ASHRAE Transactions 87(2):333.
Shaw, C.Y. and W.C. Brown. 1982. Effect of a gas furnace chimney on the air leakage characteristic of a two-story detached house, 12.1. Proceed-ings of the 3rd IEA Conference of the Air Infiltration Centre, London.
Sherman, M.H. 1986. Infiltration degree-days: A statistic for quantifying infiltration-related climate. ASHRAE Transactions 92(2):161-81.
Sherman, M.H. 1987. Estimation of infiltration from leakage and climate indications. Energy and Buildings 10(1):81.
Sherman, M.H. 1989a. Uncertainty in airflow calculations using tracer gas measurements. Building and Environment 24(4):347-54.
Sherman, M.H. 1989b. On the estimation of multizone ventilation rates from tracer gas measurements. Building and Environment 24(4):355-62.
Sherman, M.H. 1990. Tracer gas techniques for measuring ventilation in a single zone. Building and Environment 25(4):365-74.
Sherman, M.H. 1991. Single-zone stack-dominated infiltration modeling.
Proceedings of the 12th Air Infiltration and Ventilation Conference, Ottawa, Canada.
Sherman, M.H. 1992a. Superposition in infiltration modeling. Indoor Air 2:101-14.
Sherman, M.H. 1992b. A power law formulation of laminar flow in short pipes. Journal of Fluids Engineering 114:601-05. LBL Report 29414, Lawrence Berkeley Laboratory, University of California.
Sherman, M.H. 1995. The use of blower door data. Indoor Air 5:215-24.
Sherman, M.H. and Dickerhoff, D.J. 1998. Airtightness of U.S. dwellings.
ASHRAE Transactions 104(2).
Sherman, M.H. and D. Dickerhoff. 1989. Description of the LBL multitracer measurement system. Proceedings of the ASHRAE/DOE/BTECC/ CIBSE Conference—Thermal Performance of the Exterior Envelopes of Buildings IV, pp. 417-32.
Sherman, M.H. and D.T. Grimsrud. 1980. Infiltration-pressurization corre-lation: Simplified physical modeling. ASHRAE Transactions 86(2):778.
Sherman, M.H., D.T. Grimsrud, P.E. Condon, and B.V. Smith. 1980. Air infiltration measurement techniques, 9. Proceedings of the 1st IEA Sym-posium of the Air Infiltration Centre, London. Report LBL 10705.
Lawrence Berkeley National Laboratory, Berkeley, CA.
Sherman, M.H. and N. Matson. 1997. Residential ventilation and energy characteristics. ASHRAE Transactions 103(1):717-30.
Sherman, M.H. and M.P. Modera. 1986. Comparison of measured and pre-dicted infiltration using the LBL infiltration model. In Measured air leakage of buildings, p. 325. ASTM STP 904. H.R. Trechsel and P.L.
Lagus, eds.
Sherman, M.H. and D.J. Wilson. 1986. Relating actual and effective venti-lation in determining indoor air quality. Building and Environment 21(3/4):135.
Sibbitt, B.E., and T. Hamlin. 1991. Meeting Canadian residential ventilation standard requirements with low-cost systems. Canada Mortgage and Housing Corporation, Ottawa.
Sinden, F.W. 1978a. Wind, temperature and natural ventilation—Theoreti-cal considerations. Energy and Buildings 1(3):275.
Sinden, F.W. 1978b. Multi-chamber theory of air infiltration. Building and Environment 13:21-28.
Subbarao, K., J.D. Burch, C.E. Handcock, A. Lekov, and J.D. Balcomb.
1990. Measuring the energy performance of buildings through short-term tests. ACEEE 1990 Summer Study on Energy Efficiency in Buildings 10: 245-252.
Swedish Building Code. 1980. Thermal insulation and air tightness. SBN 1980.
Tamura, G.T. and C.Y. Shaw. 1976a. Studies on exterior wall airtightness and air infiltration of tall buildings. ASHRAE Transactions 82(1):122.
Tamura, G.T. and C.Y. Shaw. 1976b. Air leakage data for the design of ele-vator and stair shaft pressurization system. ASHRAE Transactions 82(2):179.
26.32 2001 ASHRAE Fundamentals Handbook (SI) Tamura, G.T. and A.G. Wilson. 1966. Pressure differences for a nine-story building as a result of chimney effect and ventilation system operation.
ASHRAE Transactions 72(1):180.
Tamura, G.T. and A.G. Wilson. 1967a. Pressure differences caused by chim-ney effect in three high buildings. ASHRAE Transactions 73(2):II.1.1.
Tamura, G.T. and A.G. Wilson. 1967b. Building pressures caused by chim-ney action and mechanical ventilation. ASHRAE Transactions 73(2): II.2.1.
Timusk, J., A.L. Seskus, and K. Linger. 1992. A systems approach to extend the limit of envelope performance. Proceedings of the Thermal Perfor-mance of Exterior Envelopes of Buildings, 6th ASHRAE/DOE/BTECC Conference, Clearwater Beach, FL. Turk, B.T., D.T. Grimsrud, J.T. Brown, K.L. Geisling-Sobotka, J. Harrison, and R.J. Prill. 1989. Commercial building ventilation rates and particle concentrations. ASHRAE Transactions 95(1):422-33.
U.S. Environmental Protection Agency. Code of Federal regulations. Title 40. Part 50 (current national ambient air quality standards) July 1, 1986.
Verschoor, J.D. and J.O. Collins. 1986. Demonstration of air leakage reduc-tion program in navy family housing. In Measured air leakage of build-ings, p. 294. ASTM STP 904. H.R. Trechsel and P.L. Lagus, eds.
Walker, I.S. 1999. Distribution system leakage impacts on apartment build-ing ventilation rates. ASHRAE Transactions105(1):943-50.
Walker, I.S. and T.W. Forest. 1995. Field measurements of ventilation rates in attics. Building and Environment 30(3):333-47, Elsevier Science Ltd., Pergamon Press, U.K.
Walker, I.S. and D.J. Wilson. 1993. Evaluating models for superposition of wind and stack effects in air infiltration. Building and Environment 28(2): 201-10, Pergamon Press.
Walker, I.S. and D.J. Wilson. 1994. Simple methods for improving estimates of natural ventilation rates. Proceedings of the 15th AIVC Conference, Buxton, UK. September.
Walker, I.S. and D.J. Wilson. 1998. Field validation of equations for stack and wind driven air infiltration calculations. International Journal of HVAC&R Research 4(2).
Walker, I.S., D.J. Wilson., and M.H. Sherman. 1997. A comparison of the power law to quadratic formulations for air infiltration calculations.
Energy and Buildings 27(3).
Walker, I., M. Sherman, J. Siegel, D. Wang, C. Buchanan, and M. Modera.
1999. Leakage diagnostics, sealant longevity, sizing and technology transfer in residential thermal distribution systems: Part II. LBNL Report 42691.
Walton, G.N. 1984. A computer algorithm for predicting infiltration and interroom airflows. ASHRAE Transactions 90(1B):601.
Walton, G.N. 1989. Airflow network models for element-based building air-flow modeling. ASHRAE Transactions 95(2):611-20.
Warren, P.R. and B.C. Webb. 1980. The relationship between tracer gas and pressurization techniques in dwellings. Proceedings of the 1st IEA Sym-posium of the Air Infiltration Centre, London.
Warren, P.R. and B.C. Webb. 1986. Ventilation measurements in housing.
CIBSE Symposium, Natural Ventilation by Design. Chartered Institution of Building Services Engineers, London.
Weidt, J.L., J. Weidt, and S. Selkowitz. 1979. Field air leakage of newly installed residential windows. Proceedings of the ASHRAE-DOE Con-ference, Thermal Performance of the Exterior Envelopes of Buildings, p. 149. Orlando, FL.
Wilson, D.J. and I.S. Walker. 1991. Wind shelter effects on a row of houses.
Proceedings of the 12th AIVC Conference, Ottawa, Canada.
Wilson, D.J. and I.S. Walker. 1992. Feasibility of passive ventilation by con-stant area vents to maintain indoor air quality in houses. Proceedings of Indoor Air Quality ’92, ASHRAE/ACGIH/AIHA Conference, San Fran-cisco. October.
Wilson, D.J. and I.S. Walker. 1993. Infiltration data from the Alberta Home Heating Research Facility. Air Infiltration and Ventilation Centre Tech-nical Note 41. AIVC, Coventry, England.
Wiren, B.G. 1984. Wind pressure distributions and ventilation losses for a single-family house as influenced by surrounding buildings—A wind tunnel study. Proceedings of the Air Infiltration Centre Wind Pressure Workshop, pp. 75-101. Brussels, Belgium.
Wolf, S. 1996. A theory of the effects of convective air flow through fibrous thermal insulation. ASHRAE Transactions 72(1):III 2.1-III 2.9.
Yaglou, C.P. and W.N. Witheridge. 1937. Ventilation requirements. ASHVE Transactions 43:423.
Yaglou, C.P., E.C. Riley, and D.I. Coggins. 1936. Ventilation requirements.
ASHVE Transactions 42:133.
Yuill, G.K. 1986. The variation of the effective natural ventilation rate with weather conditions. Renewable Energy Conference ’86. Solar Energy Society of Canada Inc., pp. 70-75.
Yuill, G.K. 1991. The development of a method of determining air change rates in detached dwellings for assessing indoor air quality. ASHRAE Transactions 97(2).
Yuill, G.K. 1996. Impact of high use automatic doors on infiltration. Report to ASHRAE (763-RP), July.
Yuill, G.K. and G.M. Comeau. 1989. Investigation of the indoor air quality, air tightness and air infiltration rates of a random sample of 78 houses in Winnipeg, pp. 122-27. Proceedings of IAQ ’89, The Human Equation— Health and Comfort. ASHRAE, Atlanta.
Yuill, G.K., M.R. Jeanson, and C.P. Wray. 1991. Simulated performance of demand-controlled ventilation systems using carbon dioxide as an occu-pancy indicator. ASHRAE Transactions 97(2):963-68.
BIBLIOGRAPHIC DATABASE AIRBASE, a database of bibliographic references that contains abstracts in English of technical papers covering air infiltration in buildings, has been developed by the Air Infiltration and Ventilation Centre (AIVC 1992). Most of the articles are concerned with the pre-diction, measurement, and reduction of air infiltration and leakage rates. Abstracts are also included for selected papers relating to indoor air quality, occupant behavior, thermal comfort, ventilation efficiency, natural and mechanical ventilation, wind pressure and its influence on infiltration, energy-saving measures, and moisture and condensation.
27.1 CHAPTER 27 CLIMATIC DESIGN INFORMATION Climatic Design Conditions ........................................................................................................... 27.1 Other Sources of Climatic Information.......................................................................................... 27.4 United States Design Conditions.................................................................................................... 27.6 Canadian Design Conditions....................................................................................................... 27.22 World Design Conditions ............................................................................................................. 27.27 Selected Monthly Percentiles of Temperature and Humidity for United States Locations .......... 27.54 HIS CHAPTER provides tables of climatic conditions for 1459 Tlocations in the United States, Canada, and around the world.
These summaries include values of dry-bulb, wet-bulb and dew-point temperature and wind speed with direction at various frequen-cies of occurrence. This information is commonly used for design, sizing, distribution, installation, and marketing of heating, ventilat-ing, air-conditioning, and dehumidification equipment; as well as for other energy-related processes in residential, agricultural, com-mercial, and industrial applications. Sources of other information such as degree-days and typical weather years for energy calcula-tions are also described.
The design information in this chapter was developed largely as part of a research project (ASHRAE 1997c, Colliver et al. 2000).
The information includes design values of dry-bulb with mean coin-cident wet-bulb temperature, design wet-bulb with mean coincident dry-bulb temperature, and design dew-point with mean coincident dry-bulb temperature and corresponding humidity ratio. These data allow the designer to consider various operational peak conditions.
Design values of wind speed (ASHRAE 1994, Lamming and Salmon 1998) allow for the design of smoke management systems in buildings.
The design conditions in this chapter are provided for those loca-tions for which long-term hourly observations were available (at least 12 years of data). Consequently, many U.S. locations listed in previous versions (1993 and before) of this chapter are no longer listed because they lacked long-term data. The number of Canadian and international locations has increased significantly.
Warm-season temperature and humidity conditions correspond to annual percentile values of 0.4, 1.0, and 2.0. Cold-season condi-tions are based on annual percentiles of 99.6 and 99.0. The use of annual percentiles to define the design conditions ensures that they represent the same probability of occurrence anywhere, regardless of the seasonal distribution of extreme temperature and humidity.
In the 1993 and earlier versions of this chapter, seasonal rather than annual percentiles were used to define the design conditions.
As a result, the summer and winter months used for the calculation of design conditions varied depending on location. For instance, summer design conditions for the U.S. were calculated over the four-month period from June through September, whereas Cana-dian summer design conditions were based on only the month of July. The following sections describe how the annual percentiles were selected to yield design conditions that are similar to those previously calculated on a seasonal basis for most of the United States.
For a variety of reasons, such as seasonal variations in solar geometry, occupancy, or use of a building, there is sometimes a need for percentiles of temperature and humidity for specific months.
New monthly tables for some U.S. locations have been added in this edition to meet this need.
CLIMATIC DESIGN CONDITIONS Annual design conditions for the United States appear in Tables 1A and 1B, for Canada in Tables 2A and 2B, and for international locations in Tables 3A and 3B. Figure 1 is a world map showing locations.
Information on station location, period analyzed, heating design conditions, wind, mean annual extreme and standard deviation of minimum and maximum dry-bulb temperature, and mean daily tem-perature range is listed in Tables 1A, 2A, and 3A. Information on the design conditions for cooling and humidity control is provided in Tables 1B, 2B, and 3B.
The information provided in Tables 1A, 2A, and 3A in the indi-cated column numbers is 1. Name of the observing station as it appears in the data set from which it was abstracted, the World Meteorological Organization station number, latitude, longitude, elevation, standard pressure at elevation (see Chapter 6 for the equations used to calculate standard pressure), and the period analyzed 2. Dry-bulb temperature corresponding to 99.6% and 99.0% annual cumulative frequency of occurrence (cold) 3. Wind speed corresponding to 1.0%, 2.5% and 5.0% annual cumulative frequency of occurrence 4. Wind speed corresponding to the 0.4% and 1.0% cumulative frequency of occurrence for the coldest month (lowest aver-age dry-bulb temperature), and the mean coincident dry-bulb temperature 5. Mean wind speed coincident with the 99.6% dry-bulb tem-perature in Column 2 and 0.4% dry-bulb temperature from Column 2 of Tables 1B, 2B, and 3B, and the wind direction most frequently occurring with the 99.6% and 0.4% dry-bulb The preparation of this chapter is assigned to TC 4.2, Weather Information.
Fig. 1 Location of Weather Stations 27.2 2001 ASHRAE Fundamentals Handbook (SI) temperatures (direction is reported in degrees true: 360 repre-sents a north wind, 90 east, etc.) 6. Average of annual extreme maximum and minimum dry bulb temperatures and standard deviations Information provided in Tables 1B, 2B, and 3B includes 1. Station name 2. Dry-bulb temperature corresponding to 0.4%, 1.0%, and 2.0% annual cumulative frequency of occurrence and the mean coin-cident wet-bulb temperature (warm) 3. Wet-bulb temperature corresponding to 0.4%, 1.0%, and 2.0% annual cumulative frequency of occurrence and the mean coin-cident dry bulb temperature 4. Dew-point temperature corresponding to 0.4%, 1.0%, and 2.0% annual cumulative frequency of occurrence and the mean coin-cident dry-bulb temperature and the humidity ratio (calculated for the dew-point temperature at the standard atmospheric pres-sure at the elevation of the station) 5. Mean daily range, which is the mean of the difference between daily maximum and minimum dry-bulb temperatures for the warmest month (highest average dry-bulb temperature) Values of Cumulative Frequency of Occurrence Representing the Design Conditions Values of ambient dry-bulb, dew-point, and wet-bulb tempera-ture and wind speed corresponding to the various annual percentiles represent the value that is exceeded on average by the indicated per-centage of the total number of hours in a year (8760). The 0.4%, 1.0%, 2.0%, and 5.0% values are exceeded on average 35, 88, 175, and 438 h per year, respectively, for the period of record.
The 99.0% and 99.6% (cold) values are defined in the same way but are usually viewed as the values for which the corresponding weather element is less than the design condition for 88 and 35 h, respectively.
Mean coincident values are the average of the indicated weather element occurring concurrently with the corresponding design value.
These design conditions were calculated from the frequency dis-tribution analyzed from data sets observed over several years. Gaps due to missing data were filled as needed, as explained in the section on Calculation of Design Conditions. The design values occur more frequently than the corresponding nominal percentile in some years and less frequently in others.
Data Sources The following three primary sources of observational data sets were used for the calculation of design values: 1. Hourly weather observations from Surface Airways Meteoro-logical and Solar Observing Network (SAMSON) data from NCDC (National Climatic Data Center) for 239 United States observing locations from 1961 through 1990 (NCDC 1991) 2. Hourly observational records in the DATSAV2 format (ASH-RAE 1995b, Plantico and Lott 1995) for 538 United States loca-tions for the period 1982 through 1993, including the 239 SAM-SON locations, and for 860 international locations (most data for international locations consist of observations taken every 3 h) 3. Hourly weather records for the period 1953 through 1993 for 145 Canadian locations from the Canadian Weather Energy and Engineering Data Sets (CWEEDS) produced by Environment Canada (1993b) Two primary periods of record were used in the calculations. The values for the United States SAMSON and Canadian CWEEDS locations are generally based on the period from 1961 through 1993.
DATSAV2 data were used for the 1991 through 1993 period for the SAMSON sites. The values for international locations and the rest of the United States, whose data were analyzed from the DATSAV2 files, are based on the period 1982 through 1993. DATSAV2 is a comprehensive database containing hourly observations for loca-tions around the world collected from global telecommunications circuits. It is quality-controlled and archived by the Air Force Com-bat Climatology Center at Asheville, NC. Tables 1A, 2A, and 3A indicate the period of record used for each location.
In summary, the data source for United States locations with the period identified as “6193” is SAMSON data supplemented with DATSAV2 data for the last 3 years. The source for United States locations with the period “8293” is DATSAV2.
Calculation of Design Conditions Details of the analysis procedures are available in ASHRAE (1997c) and Colliver et al. (2000), including the measures used to ensure that the number and distribution of missing data, both by month and by hour of the day, did not introduce significant biases into the analysis. Generally, the annual cumulative frequency distri-bution was constructed from the relative frequency distributions compiled for each month. Each individual month’s data were included if they met screening criteria for completeness and unbi-ased distribution of missing data. Although the minimum period of record selected for this analysis was 12 years (1982 through 1993), some variation and gaps in observing programs meant that some months’ data were unusable due to incompleteness.
A station’s design conditions were included in this chapter only if there were data from at least 8 months that met the screening cri-teria from the period of record for each month of the year. For instance, there had to be 8 months each for January, February, March, etc. whose data met the completeness screening criteria.
Gaps of up to 5 h were filled. A month’s data were included if the month was at least 85% complete after filling and the difference between the number of day and nighttime observations was less than 60.
Relationship Between Design Conditions and Previously Published Design Temperatures The design conditions in this chapter are calculated on a different basis from the design conditions published in the 1993 and previous editions of this Handbook. Previous design conditions were based on a 4-month summer period and a 3-month winter period in the United States, on the months of July and January in Canada, and on the warmest 4-month period and coldest 3-month period in interna-tional locations. Although generally suitable as design values, the different periods resulted in design temperatures representing dif-ferent annual probabilities of occurrence, depending on the country; and within countries, on the distribution of temperature and humid-ity conditions throughout the year typical of regional climatic zones. The design conditions in this chapter explicitly represent the same annual probability of occurrence in any location, regardless of country or general climatic conditions.
The annual cumulative frequencies of occurrence representing the design dry-bulb temperatures generally correspond to the sea-sonal design temperatures in the following fashion for locations in the mid-latitude, continental locations (characterized by a hot sum-mer and cold winter). The 0.4% annual value is about the same as the 1.0% summer design temperature in the 1993 ASHRAE Hand-book. The 1.0% annual value is about 0.5 K lower than the 2.5% summer design temperature in the 1993 ASHRAE Handbook, and the 2.0% annual condition corresponds approximately to the 5.0% summer design temperature in the 1993 ASHRAE Handbook.
In Canadian continental locations, the 0.4% annual condition is about the same as the 2.5% July design temperature in the 1993 ASHRAE Handbook. In the United States for the Pacific region and southern coastal locations, where the extremes are generally more widely distributed throughout the year, the values in this chapter Climatic Design Information 27.3 represent more extreme conditions than the design temperatures in the 1993 ASHRAE Handbook.
Annual 99.6% and 99.0% design conditions represent a slightly colder condition than the previous cold season design temperatures, although there is considerable variability in this relationship from location to location.
Further details concerning differences between the design con-ditions in this Chapter and the 1993 and previous versions are described in ASHRAE (1997c) and Colliver et al. (2000).
Applicability and Characteristics of the Design Conditions The sets of design values in this chapter represent different psy-chrometric conditions. Design data based on dry-bulb temperature represent peak occurrences of the sensible component of ambient outdoor conditions. Design values based on wet-bulb temperature are related to the enthalpy of the outdoor air. Conditions based on dew point relate to the peaks of the humidity ratio. The designer, engineer, or other user must decide which set(s) of conditions and probability of occurrence apply to the design situation under con-sideration. The addition of the new psychrometric design conditions allows for several viewpoints of operational peak loads. Additional sources of information on the frequency and duration of extremes of temperature and humidity are provided later in the section on Other Sources of Climatic Information. Further information is available from Harriman et al. (1999).
Heating Design Conditions. The 99.6% and 99.0% design con-ditions in Column 2 of Tables 1A, 2A, and 3A are often used in the sizing of heating equipment. In cold spells, dry-bulb temperatures below the design conditions can last for a week or more.
Columns 4 and 5 of Tables 1A, 2A, and 3A provide information useful for estimating peak loads accounting for infiltration. Column 4 provides extreme wind speeds only for the coldest month, with the mean coincident dry-bulb temperature. Column 5 provides the mean wind speed and direction coincident to the corresponding per-centile design dry-bulb temperature.
Cooling and Dehumidification Design Conditions. The 0.4%, 1.0%, and 2.0% dry-bulb temperatures and mean coincident wet-bulb temperatures in Column 2 of Tables 1B, 2B, and 3B often rep-resent conditions on hot, mostly sunny days. These are useful for cooling applications, especially air-conditioning.
Design conditions based on wet-bulb temperature in Column 3 represent extremes of the total sensible plus latent heat of outdoor air. This information is useful for cooling towers, evaporative cool-ers, and fresh air ventilation system design.
The design conditions based on dew-point temperatures in Col-umn 4 of Tables 1B, 2B, and 3B are directly related to extremes of humidity ratio, which represent peak moisture loads from the weather. Extreme dew-point conditions may occur on days with moderate dry-bulb temperatures resulting in high relative humidity.
These values are especially useful for applications involving humidity control, such as desiccant cooling and dehumidification, cooling-based dehumidification, and fresh air ventilation systems.
The values are also used as a check point when analyzing the behav-ior of cooling systems at part-load conditions, particularly when such systems are used for humidity control as a secondary function.
The humidity ratio values in Column 2 correspond to the combi-nation of dew-point temperature and the mean coincident dry-bulb temperature calculated at the standard pressure at the elevation of the location.
Wind. Design wind speeds in Column 3 of Tables 1A, 2A, and 3A have been produced as a result of the requirement for the design of smoke management systems. Annual percentiles of 1.0, 2.5, and 5.0 have been determined appropriate for this application. The val-ues for the United States SAMSON sites and Canadian locations have been taken from ASHRAE (1994), in which adjustments to the standard 10 m anemometer height were made. Wind speed values for other locations are taken from ASHRAE (1997c), in which no adjustment for nonstandard anemometer height is made.
Annual Extreme Temperatures. Column 6 of Tables 1A, 2A, and 3A provides the mean and standard deviation of the annual extreme maximum and minimum dry-bulb temperatures. The prob-ability of occurrence of very extreme conditions can be required for the operational design of equipment to ensure continuous operation and serviceability (regardless of whether the heating or cooling loads are being met). These values were calculated from the extremes of the hourly temperature observations. The true maxi-mum and minimum temperatures for any day generally occur between hourly readings. Thus, the mean maximum and minimum temperatures calculated in this way will be about 0.5 K less extreme than the mean daily extreme temperatures observed with maximum and minimum thermometers.
Return period (or recurrence interval) is defined as the recipro-cal of the annual probability of occurrence. For instance, the 50-year return period maximum dry-bulb temperature has a probabil-ity of occurring or being exceeded of 2.0% (i.e., 1/50) each year.
This statistic does not indicate how often the condition will occur in terms of the number of hours each year (as in the design condi-tions based on percentiles) but describes the probability of the con-dition occurring at all in any year. The following method can be used to estimate the return period (recurrence interval) of extreme temperatures.
Tn = M + IFs (1) where Tn = n-year return period value of extreme dry-bulb temperature to be estimated, years M = mean of the annual extreme maximum or minimum dry-bulb temperatures, °C s = standard deviation of the annual extreme maximum or minimum dry-bulb temperatures, °C I = 1, if maximum dry-bulb temperatures are being considered = –1 if minimum extremes are being considered F = For example, the 50-year return period extreme maximum dry-bulb temperature estimated for Terre Haute, Indiana is 40°C (M = 35.5°C, s = 1.8, n = 50, I = 1). Similarly, the 100-year return period extreme minimum dry-bulb temperature for Winnipeg, Manitoba, is –44°C (M = –36°C, s = 2.6, n = 100, I = –1).
This calculation is based on the assumptions that the annual max-ima and minima are distributed according to the Gumbel (Type 1 Extreme Value) distribution and are fitted with the method of moments (Lowery and Nash 1970). The uncertainty or standard error using this method increases with increasing standard devia-tion, increasing value of return period, and decreasing length of the period of record. It can be significant. For instance, the standard error in the 50-year return period maximum dry-bulb temperature estimated at a location with a 12-year period of record can be 3 K or more. Thus, the uncertainty of return-period values estimated in this way will be greater for locations from the DATSAV2 data sets than those from the longer SAMSON and CWEEDS data sets.
Mean Daily Range. The mean daily range is the mean difference between the daily maximum and minimum temperatures during the hottest month. These values are calculated from the extremes of the hourly temperature observations. The true daily temperature range is generally about 1 K greater, for the same reason as explained in the previous section.
Monthly Tables Selected monthly percentiles of temperature and humidity are provided in Tables 4A and 4B for 239 U.S. locations from the SAMSON data set. These monthly values are derived from the 6 π ------- 0.5772 ln ln n n 1 – ------------ + – 27.4 2001 ASHRAE Fundamentals Handbook (SI) same analysis that results in the design conditions in Tables 1A and 1B. These monthly summaries provide additional information when seasonal variations in solar geometry and intensity, building or facility occupancy, or building use patterns require consider-ation. In particular, these values can be used when determining air-conditioning loads during periods of maximum solar radiation.
Table 4A contains the location name and WMO station number and the 0.4%, 1.0%, and 2.0% value of the wet-bulb temperature and mean coincident dry-bulb temperature for the indicated month and annual period. The percentiles for the annual period are the same as those in Table 1B for the same location and are listed in Table 4A for convenience.
Table 4B contains the location name and WMO station number and the 0.4%, 1.0%, and 2.0% value of the dry-bulb temperature and mean coincident wet-bulb temperature for the indicated month and annual period. The percentiles for the annual period are the same as those in Table 1B for the same location and are listed in Table 4B for convenience.
For a 30-day month, the 0.4%, 1.0%, and 2.0% values of occur-rence represent the value that occurs or is exceeded for a total of 3, 7, or 14 h, respectively, per month on average over the period of record.
Monthly percentile values of dry-bulb or wet-bulb temperature may be higher or lower than the design conditions corresponding to the same nominal percentile, depending on the month and the seasonal distribution of the parameter at that location. Generally, for the hottest or most humid months of the year, the monthly percentile value will exceed the design condition for the same element corre-sponding to the same nominal percentile. For instance, Table 1B shows that the 0.4% design dry-bulb temperature at Mobile, Ala-bama, is 34°C. Table 4B shows that the 0.4 monthly percentile dry-bulb temperature exceeds 34°C for the months of June, July, and August, with values of 35°C, 36°C, and 35°C, respectively.
It is essential to remember that the design conditions are based on an annual analysis. The 0.4% annual value represents a value occur-ring or exceeded 35 h on average every year over the period of record, whereas the 0.4% monthly value occurs or is exceeded 3 h on average for the month that it represents, over the period of record.
Representativeness of Data and Sources of Uncertainty The information in the tables was obtained by direct analysis of the observations from the indicated locations. The design values are provided and used as an estimate of the annual cumulative fre-quency of occurrence of the weather conditions at the recording sta-tion for several years into the future. Several sources of uncertainty affect the accuracy of using the design conditions to represent other locations or periods.
The most important of these factors is spatial representativeness.
Most of the observed data for which design conditions were calcu-lated were collected from airport observing sites, most of which are flat, grassy, open areas, away from buildings and trees or other local influences. Data representing the psychrometric conditions are gen-erally properties of air masses, rather than local features, and tend to vary on regional scales. As a result, a particular value often may rea-sonably represent an area extending several kilometres. However, significant variations can occur with changes in local elevation, across large metropolitan areas, or in the vicinity of large bodies of water. Judgment must always be exercised in assessing the represen-tativeness of the design conditions. It is especially important to note the elevation of locations because design conditions vary signifi-cantly for locations whose elevations differ by a few hundred metres or more. An applied climatologist should be consulted in estimating design conditions for locations not explicitly listed in this chapter.
Weather conditions vary from year to year and, to some extent, from decade to decade due to the inherent variability of climate.
Similarly, values representing design conditions vary depending on the period of record used in the analysis. Thus, due to short-term cli-matic variability, there is always some uncertainty in using the design conditions from one period to represent another period. Typ-ically, the values of design dry-bulb temperature vary less than 1 K from decade to decade, but larger variations can occur. Differing periods used in the analysis can lead to differences in design condi-tions between nearby locations at similar elevations. Design condi-tions may show trends in areas of increasing urbanization or other regions experiencing extensive changes to land use. Longer term climatic change due to human or natural causes may also introduce trends into design conditions, but no conclusive evidence or consen-sus of opinion is available on the rapidity or nature of such changes.
Wind speed is very sensitive to local exposure features such as terrain and surface cover. Wind speed values in Columns 3, 4, and 5 of Tables 1A, 2A, and 3A are often representative of a flat, open exposure, such as at airports. Wind engineering methods, as de-scribed in Chapter 16, can be used to account for exposure dif-ferences between airport and building sites. This is a complex procedure, best undertaken by an experienced applied climatologist or wind engineer having knowledge of the exposure of the observ-ing and building sites and surrounding regions.
OTHER SOURCES OF CLIMATIC INFORMATION Joint Frequency Tables of Psychrometric Conditions Most of the design values in this chapter were developed by a research project (ASHRAE 1997c, and Colliver et al. 2000). The joint frequency tables from this project provide the annual and monthly joint frequency of occurrence of various psychrometric combinations for each location in the tables. The complete set of joint frequency tables is available on CD (ASHRAE 1999) The Engineering Weather Data (NCDC 1999) CD, an update of Air Force Manual 88-29, was compiled by the U.S. Air Force Com-bat Climatology Center. This CD contains several tabular and graphical summaries of temperature, humidity, and wind speed information for hundreds of locations in the United States and around the world. In particular, it contains detailed joint frequency tables of temperature and humidity for each month, binned at 1°F and 3-h local time-of-day intervals.
The International Station Meteorological Climate Summary (ISMCS) is a CD-ROM containing several tables of climatic sum-mary information for over 7000 locations around the world (NCDC 1996). Only several hundred of these locations have observed hourly weather data. A table providing the joint frequency of dry-bulb temperature and wet-bulb temperature depression is provided for the locations with hourly observations. It can be used as an aid in estimating design conditions for locations for which no other information is available.
The monthly frequency distribution of dry-bulb temperatures and mean coincident wet-bulb temperatures is available for 134 Canadian locations from Environment Canada (1983-1987).
Degree-Days and Climatic Normals Degree-day summary information and climatic normals for the United States are available on CD-ROM in The Climatography of the U.S. (NCDC 1992a, 1992b, 1994); and, for Canada, in the Cana-dian Climate Normals, 1961 to 1990 (Environment Canada 1993a).
Typical Year Data Sets Software exists to simulate the annual energy performance of buildings requiring a 1-year data set (8760 h) of weather conditions.
Many data sets in different record formats have been developed to meet this requirement. The data represent a typical year from the viewpoint of weather-induced energy loads on a building. No Climatic Design Information 27.5 explicit effort was made to represent extreme conditions, so these files do not represent design conditions.
Weather Year for Energy Calculations Version 2 (WYEC2) for 52 U.S. and 5 Canadian locations is available from ASHRAE (ASHRAE 1997d). A user manual and software toolkit explains the development, format, and characteristics of this data and provides access and toolkit software.
Typical Meteorological Year Version 2 (TMY2) files are avail-able for 239 U.S. locations from the National Renewable Energy Laboratory (Marion and Urban 1995). These were produced using an objective statistical algorithm to select the most typical month from the long-term record. These files were originally intended for the design of solar energy systems, so solar radiation values were weighted heavily.
Canadian Weather Year for Energy Calculation (CWEC) files for 47 Canadian locations were developed for use with the Canadian National Energy Code, using the TMY algorithm and software (Environment Canada 1993b).
Examples of the use of these files for energy calculations in both residential and commercial buildings, including the differences amongst the files, are available in Crawley (1998) and Huang (1998).
Seasonal Percentiles of Dew-Point Temperature Seasonal percentiles of dew-point temperature are available for 381 U.S. and Canadian locations in ASHRAE (1995a), with their development summarized in Colliver et al. (1995).
Sequences of Extreme Temperature and Humidity Durations Colliver et al. (1998) and ASHRAE (1997b) compiled ex-treme sequences of 1-, 3-, 5-, and 7-day duration for 239 U.S.
(SAMSON data) and 144 Canadian (CWEEDS data) locations based independently on the following five criteria: high dry-bulb temperature, high dew-point temperature, high enthalpy, low dry-bulb temperature, and low wet-bulb depression. For the criteria associated with high values, the sequences are selected according to annual percentiles of 0.4, 1.0, and 2.0. For the criteria corre-sponding to low values, annual percentiles of 99.6, 99.0, and 98.0 are reported. The data included for each hour of a sequence are solar radiation, dry-bulb and dew-point temperature, atmospheric pressure, and wind speed and direction. Accompanying informa-tion allows the user to go back to the source SAMSON and CWEEDS data and obtain sequences with different characteris-tics (i.e., different probability of occurrence, windy conditions, low or high solar radiation, etc.). These extreme sequences are available on CD (ASHRAE 1997a).
These sequences were developed primarily to assist the design of heating or cooling systems having a finite capacity before re-generation is required or systems that rely on thermal mass to limit loads. The information is also useful where information on the hourly weather sequence during extreme episodes is required for design.
Observational Data Sets For detailed designs, custom analysis of the most appropriate long-term weather record is best. National weather services are gen-erally the best source of long-term observational data. The WMO World Data Center A at the National Climate Data Center in Ashe-ville, NC, collects and makes available a significant volume of archived weather observations from around the world.
The SAMSON and CWEEDS data sets provide long-term hourly data, including solar radiation values for the United States and Canada.
Increasingly, information about weather and climate services and data sets, as well as the data sets themselves, is becoming avail-able through the Internet and World Wide Web. Information supple-mentary to this chapter may be posted from time to time on the ASHRAE Technical Committee 4.2 web site, the link to which is available from the ASHRAE web site (www.ashrae.org).
REFERENCES ASHRAE. 1994. Wind data for design of smoke control systems. Research Report RP-816.
ASHRAE. 1995a. Design data for the 1%, 2½%, and 5% occurrences of extreme dew point temperature, with mean coincident dry-bulb temper-ature. Research Report RP-754.
ASHRAE. 1995b. Weather data sets for ASHRAE research. Research Report RP-889.
ASHRAE. 1997a. Design weather sequence viewer 2.1 (CD-ROM).
ASHRAE. 1997b. Sequences of extreme temperature and humidity for design calculations. Research Report RP-828.
ASHRAE. 1997c. Updating the tables of design weather conditions in the ASHRAE Handbook of Fundamentals. Research Report RP-890.
ASHRAE. 1997d. WYEC2 data and toolkit (CD-ROM).
ASHRAE. 1999. Weather data viewer 2.1 (CD-ROM).
Colliver, D.G., H. Zhang, R.S. Gates, and K.T. Priddy. 1995. Determination of the 1%, 2.5%, and 5% occurrences of extreme dew-point temperatures and mean coincident dry-bulb temperatures. ASHRAE Transactions 101(2):265-86.
Colliver D.G., R.S. Gates, H. Zhang, and K.T. Priddy. 1998. Sequences of extreme temperature and humidity for design calculations. ASHRAE Transactions 104(1a):133-44.
Colliver, D.G., R.S. Gates, T.F. Burkes, and H. Zhang. 2000. Development of the design climatic data for the 1997 ASHRAE Handbook—Funda-mentals. ASHRAE Transactions 106(1).
Crawley, D.B. 1998. Which weather data should you use for energy simula-tions of commercial buildings? ASHRAE Transactions 104(2):498-515.
Environment Canada. 1983-1987. Principal station data. PSD 1 to 134.
Atmospheric Environment Service, Downsview, Ontario.
Environment Canada. 1993A. Canadian climate normals (1961-1990).
Environment Canada. 1993b. Canadian weather energy and engineering data sets (CWEEDS files) and Canadian weather for energy calculations (CWEC files) user’s manual.
Harriman, L.G., D.G. Colliver, and H.K. Quinn. 1999. New weather data for energy calculations. ASHRAE Journal 41(3):31-38.
Huang, J. 1998. The impact of different weather data on simulated residen-tial heating and cooling loads. ASHRAE Transactions 104(2):516-27.
Lamming S.D. and J.R. Salmon. 1998. Wind data for design of smoke con-trol systems ASHRAE Transactions 104(1A):742-51.
Lowery, M.D. and J.E. Nash. 1970. A comparison of methods of fitting the double exponential distribution. Journal of Hydrology 10(3):259-75.
Marion, W. and K. Urban. 1995. User’s Manual for TMY2s, typical meteo-rological years, derived from the 1961-1990 National Solar Radiation Data Base. NREL/SP-463-7668, E95004064. National Renewable Energy Laboratory, Golden, CO.
NCDC. 1991. Surface airways meteorological and solar observing network (SAMSON) data set. National Climatic Data Center, Asheville, NC.
NCDC. 1992A. Monthly normals of temperature, precipitation, and heating and cooling degree-days. In Climatography of the U.S. #81.
NCDC. 1992B. Annual degree-days to selected bases (1961-1990). In Cli-matography of the U.S. #81.
NCDC. 1994. U.S. division and station climatic data and normals.
NCDC. 1996. International station meteorological climate summary (ISMCS).
NCDC. 1999. Engineering weather data.
Plantico, M.S. and J.N. Lott. 1995. Foreign weather data servicing at NCDC.
ASHRAE Transactions 101(1):484-90.
27.6 2001 ASHRAE Fundamentals Handbook (SI) Table 1A Heating and Wind Design Conditions—United States Heating Dry Bulb Extreme Wind Speed, m/s Coldest Month MWS/PWD to DB Extr. Annual Daily Lat.
Long.
Elev., m StdP, kPa 0.4% 1% 99.6% 0.4% Mean DB StdD DB Station WMO# Dates 99.6% 99% 1% 2.5% 5% WS MDB WS MDB MWS PWD MWS PWD Max.
Min.
Max. Min.
1a 1b 1c 1d 1e 1f 1g 2a 2b 3a 3b 3c 4a 4b 4c 4d 5a 5b 5c 5d 6a 6b 6c 6d ALABAMA Anniston 722287 33.58 85.85 186 99.11 8293 −7.1 −4.5 7.3 6.3 5.7 7.9 8.3 6.9 7.8 2.5 300 3.3 240 36.7 −12.3 1.8 4.1 Birmingham 722280 33.57 86.75 192 99.04 6193 −7.8 −5.2 8.4 7.5 6.7 8.9 4.8 8.0 5.4 3.3 340 4.0 320 36.7 −12.7 1.8 3.6 Dothan 722268 31.32 85.45 122 99.87 8293 −2.4 −0.2 8.2 7.4 6.5 8.4 7.0 7.8 8.2 4.0 320 3.4 320 37.0 −8.7 0.9 4.0 Huntsville 723230 34.65 86.77 196 98.99 6193 −9.4 −6.6 10.4 9.0 8.0 10.4 4.2 9.3 4.4 4.6 340 4.3 270 36.3 −13.9 1.7 4.2 Mobile 722230 30.68 88.25 67 100.52 6193 −3.1 −1.0 9.7 8.6 7.8 10.1 8.9 9.2 9.1 4.3 360 3.9 320 36.2 −7.6 1.1 3.5 Montgomery 722260 32.30 86.40 62 100.58 6193 −4.7 −2.8 8.8 7.7 6.7 9.1 6.9 8.1 7.4 3.1 360 3.5 270 36.8 −9.6 1.6 3.5 Muscle Shoals/IntlFlorence 723235 34.75 87.62 168 99.32 8293 −8.8 −6.0 8.2 7.3 6.3 8.4 5.5 7.7 5.7 3.9 360 3.1 290 36.9 −14.0 1.7 5.1 Ozark, Fort Rucker 722269 31.28 85.72 91 100.24 8293 −2.5 −0.5 7.1 5.9 5.2 7.6 9.4 6.6 8.6 2.4 340 2.2 300 37.1 −7.8 1.3 3.3 Tuscaloosa 722286 33.22 87.62 52 100.70 8293 −6.7 −4.2 7.7 6.4 5.7 8.1 8.4 7.0 10.3 2.1 360 3.3 240 37.3 −11.9 1.0 3.8 ALASKA Adak, NAS 704540 51.88 176.65 4 101.28 8293 −7.0 −5.1 15.3 13.4 11.9 17.8 0.9 15.1 1.8 2.0 210 4.6 170 19.7 −11.7 1.9 1.6 Anchorage, Elemendorf AFB 702720 61.25 149.80 65 100.55 8293 −25.1 −22.0 7.7 6.2 5.3 8.1 −3.6 6.6 −3.6 1.3 50 3.1 260 25.0 −27.8 1.8 3.6 Anchorage, Fort Richardson 702700 61.27 149.65 115 99.95 8293 −28.2 −25.0 8.3 6.3 5.0 8.9 1.6 6.8 2.2 1.4 50 2.2 270 26.6 −30.7 1.2 3.5 Anchorage, Intl Airport 702730 61.17 150.02 40 100.85 6193 −25.6 −22.5 9.8 8.5 7.5 10.3 −7.7 8.6 −7.8 1.7 10 3.7 290 24.8 −27.8 1.6 4.0 Annette 703980 55.03 131.57 34 100.92 6193 −10.8 −8.4 13.8 11.8 10.2 13.7 5.1 12.4 4.6 4.2 40 3.8 320 27.3 −12.2 2.1 3.0 Barrow 700260 71.30 156.78 4 101.28 6193 −40.3 −38.0 12.5 11.1 9.9 13.4 −16.3 11.5 −18.1 3.3 140 5.3 90 18.4 −42.9 2.6 2.4 Bethel 702190 60.78 161.80 46 100.77 6193 −33.3 −31.1 13.8 12.1 10.7 15.1 −13.3 13.4 −15.0 5.6 20 5.1 360 25.7 −35.3 1.8 3.7 Bettles 701740 66.92 151.52 196 98.99 6193 −45.2 −42.3 8.2 7.3 6.2 8.4 −11.7 7.1 −14.2 0.8 340 3.6 190 29.4 −48.4 2.2 3.2 Big Delta, Ft. Greely 702670 64.00 145.73 391 96.72 6193 −42.8 −39.5 15.0 12.7 11.1 16.8 −17.9 14.6 −16.3 1.4 180 3.8 180 28.6 −44.2 1.8 4.2 Cold Bay 703160 55.20 162.73 31 100.95 6193 −14.3 −12.1 16.8 15.0 13.3 20.4 1.3 17.8 1.1 6.6 340 7.1 140 19.3 −16.7 2.2 2.9 Cordova 702960 60.50 145.50 13 101.17 8293 −20.2 −17.4 9.6 8.3 7.2 10.9 4.2 9.6 3.2 0.4 340 3.7 240 26.2 −23.0 2.8 3.0 Deadhorse 700637 70.20 148.47 17 101.12 8293 −37.6 −36.7 14.1 12.7 11.3 15.3 −18.1 13.5 −21.6 5.3 240 5.4 60 25.3 −46.2 7.9 2.9 Dillingham 703210 59.05 158.52 29 100.98 8293 −28.9 −25.1 11.2 9.9 8.8 12.6 −6.5 10.9 −6.3 2.3 40 4.3 180 23.3 −32.8 1.7 5.2 Fairbanks, Eielson AFB 702650 64.67 147.10 167 99.33 8293 −36.1 −34.8 6.2 5.2 4.4 6.4 −6.2 4.9 −9.0 0.1 150 2.1 290 30.8 −43.6 2.1 4.3 Fairbanks, Intl Airport 702610 64.82 147.87 138 99.68 6193 −44.0 −40.7 7.8 6.7 5.8 7.1 −11.6 5.4 −11.9 0.7 10 3.7 220 30.3 −44.4 2.1 4.3 Galena 702220 64.73 156.93 46 100.77 8293 −36.3 −35.2 8.1 6.9 5.9 8.5 −10.0 7.2 −9.3 0.2 270 2.2 320 28.9 −45.5 1.4 5.8 Gulkana 702710 62.15 145.45 481 95.68 6193 −42.4 −39.3 11.7 10.5 9.3 10.0 −8.6 8.3 −8.1 1.3 360 3.2 180 27.8 −43.4 1.8 4.1 Homer 703410 59.63 151.50 22 101.06 8293 −17.7 −15.5 9.9 8.9 8.1 10.4 −4.4 9.3 −2.6 3.8 30 4.3 270 21.3 −20.5 2.2 3.8 Juneau 703810 58.37 134.58 7 101.24 8293 −15.8 −13.7 12.1 10.2 9.1 13.0 4.1 11.0 3.6 2.1 360 4.1 230 27.0 −18.1 1.4 2.7 Kenai 702590 60.57 151.25 29 100.98 8293 −29.8 −25.5 10.1 9.0 8.2 11.0 −3.9 9.9 −4.5 1.1 30 4.2 270 23.9 −32.9 1.9 4.1 Ketchikan 703950 55.35 131.70 29 100.98 8293 −10.3 −6.7 11.0 9.7 8.7 12.9 5.7 10.8 5.5 2.4 280 4.8 320 25.5 −13.8 1.0 2.9 King Salmon 703260 58.68 156.65 15 101.14 6193 −31.1 −28.4 14.3 12.4 10.7 14.9 2.1 12.6 2.1 3.0 360 5.4 270 25.3 −35.1 1.9 4.0 Kodiak, State USCG Base 703500 57.75 152.50 34 100.92 6193 −13.8 −11.4 15.0 13.2 11.6 15.1 −2.3 13.5 −1.3 8.1 300 4.8 320 24.4 −17.2 2.0 3.4 Kotzebue 701330 66.87 162.63 5 101.26 6193 −37.6 −35.2 15.7 13.8 12.4 16.7 −10.1 14.5 −10.0 3.0 70 5.1 300 23.7 −39.3 2.7 3.6 McGrath 702310 62.97 155.62 103 100.09 6193 −43.8 −40.8 8.1 7.1 6.2 8.0 −5.2 6.4 −11.4 0.6 310 3.3 340 28.2 −46.8 1.8 3.9 Middleton Island 703430 59.43 146.33 14 101.16 8293 −7.6 −6.0 17.7 15.3 13.3 18.9 1.5 16.6 2.5 8.2 330 3.7 260 19.0 −9.5 2.7 3.8 Nenana 702600 64.55 149.08 110 100.01 8293 −46.2 −42.5 7.3 6.3 5.5 8.0 −12.4 6.9 −13.1 1.1 250 3.0 60 30.4 −46.6 2.3 4.0 Nome 702000 64.50 165.43 7 101.24 6193 −34.8 −32.3 13.5 11.9 10.5 13.7 −8.2 12.3 −7.9 1.6 20 5.2 260 24.6 −37.2 2.3 3.5 Northway 702910 62.97 141.93 525 95.18 8293 −36.7 −35.5 6.9 6.0 5.3 6.4 −24.8 5.4 −21.3 0.1 300 3.2 290 28.3 −47.6 1.5 3.3 Port Heiden 703330 56.95 158.62 29 100.98 8293 −21.1 −18.8 17.0 14.2 12.3 16.9 2.4 14.4 1.7 7.4 60 6.9 160 23.1 −23.8 2.4 4.1 Saint Paul Island 703080 57.15 170.22 9 101.22 6193 −18.7 −16.1 18.4 16.3 14.6 20.8 −4.3 18.3 −6.2 8.3 350 6.4 240 14.4 −19.2 2.9 3.8 Sitka 703710 57.07 135.35 20 101.08 8293 −8.9 −6.2 10.4 9.3 8.4 10.9 4.3 9.8 5.1 3.4 70 3.9 230 24.7 −11.5 3.4 2.8 Talkeetna 702510 62.30 150.10 109 100.02 6193 −33.4 −29.7 7.8 7.1 6.2 8.4 −10.7 7.8 −9.5 1.7 50 3.3 200 27.9 −37.0 1.6 4.4 Valdez 702750 61.13 146.35 10 101.20 8293 −15.5 −13.8 10.8 8.7 7.3 12.5 −10.5 10.0 −9.2 6.6 70 4.3 240 24.4 −17.2 2.0 3.4 Yakutat 703610 59.52 139.67 9 101.22 6193 −19.6 −16.4 10.5 8.6 7.2 11.0 2.3 9.3 0.6 0.8 100 3.8 320 24.1 −22.4 2.2 3.9 ARIZONA Flagstaff 723755 35.13 111.67 2137 78.15 6193 −17.0 −13.5 9.4 8.2 7.4 9.2 −1.9 8.0 −1.4 1.5 20 4.2 220 31.6 −23.2 1.4 4.1 Kingman 723700 35.27 113.95 1033 89.52 8293 −5.7 −3.0 11.5 10.2 9.1 10.8 7.6 9.2 6.3 2.4 90 5.8 240 39.5 −9.4 1.0 3.8 Page 723710 36.93 111.45 1304 86.61 8293 −6.7 −4.7 8.5 7.2 6.0 7.2 5.7 5.5 4.6 2.0 300 3.3 360 39.9 −13.5 2.0 6.8 Phoenix, Intl Airport 722780 33.43 112.02 337 97.34 6193 1.2 3.0 8.5 7.1 6.1 7.6 15.1 6.4 14.5 2.4 90 4.2 270 45.4 −1.2 1.2 2.6 Phoenix, Luke AFB 722785 33.53 112.38 332 97.40 8293 1.6 3.3 8.3 6.8 5.7 7.2 14.5 5.6 12.9 1.9 340 3.8 210 46.0 −1.3 1.2 2.1 Prescott 723723 34.65 112.42 1537 84.17 6193 −9.3 −6.9 9.9 8.4 7.4 9.2 5.3 8.0 5.7 2.9 190 4.8 230 36.7 −13.8 1.2 3.4 Safford, Agri Center 722747 32.82 109.68 950 90.42 8293 −6.3 −3.1 7.5 6.2 5.2 6.8 10.2 5.6 8.8 1.6 110 3.0 310 40.9 −11.5 2.1 6.4 Tucson 722740 32.12 110.93 779 92.31 6193 −0.6 1.1 10.9 9.5 8.2 10.6 13.1 9.4 13.1 3.1 140 5.2 300 42.2 −3.9 1.6 2.2 Winslow 723740 35.02 110.73 1488 84.68 8293 −12.4 −9.8 11.5 9.8 8.4 10.5 7.5 8.4 7.2 2.3 140 4.2 250 38.0 −16.0 2.7 3.5 Yuma 722800 32.65 114.60 63 100.57 8293 4.7 6.4 8.7 7.4 6.5 8.8 14.8 7.5 14.3 2.0 30 3.0 280 46.7 −1.6 1.0 6.6 ARKANSAS Blytheville, Eaker AFB 723408 35.97 89.95 78 100.39 8293 −11.1 −7.8 9.9 8.6 7.6 10.4 2.3 9.3 3.2 4.3 10 2.9 240 37.3 −14.4 2.8 5.0 Fayetteville 723445 36.00 94.17 381 96.83 8293 −14.4 −10.6 9.2 8.4 7.9 9.4 6.9 8.6 6.7 3.9 350 4.6 190 37.5 −18.4 1.8 5.1 Fort Smith 723440 35.33 94.37 141 99.64 6193 −10.6 −7.5 8.9 7.9 7.1 9.2 7.9 8.2 5.1 4.0 320 4.2 270 38.8 −14.3 2.2 3.7 Little Rock, AFB 723405 34.92 92.15 95 100.19 6193 −8.9 −6.2 8.8 7.9 7.1 8.8 5.4 8.0 5.5 4.2 360 4.0 200 38.6 −12.2 2.1 3.4 Texarkana 723418 33.45 93.98 119 99.90 8293 −6.5 −4.0 8.4 7.6 6.6 8.9 8.4 8.1 9.1 3.9 50 3.8 190 38.3 −10.4 1.7 4.2 CALIFORNIA Alameda, NAS 745060 37.78 122.32 4 101.28 8293 4.4 5.4 9.3 8.2 7.3 8.9 10.6 7.6 11.3 2.6 120 3.6 300 34.0 −4.1 2.7 7.9 Arcata/Eureka 725945 40.98 124.10 66 100.53 6193 −1.1 0.1 9.2 8.3 7.6 9.2 11.9 8.2 10.8 2.2 90 4.3 320 27.9 −3.2 2.5 1.8 Bakersfield 723840 35.43 119.05 150 99.54 6193 0.2 1.6 8.3 7.1 6.2 8.3 13.6 6.3 11.9 2.1 90 5.4 310 42.4 −2.2 1.3 1.9 Barstow/Daggett 723815 34.85 116.78 587 94.47 6193 −2.1 −0.2 13.2 11.9 10.2 13.2 14.6 11.2 12.3 2.6 270 5.2 290 43.9 −5.7 1.4 2.7 Blue Canyon 725845 39.28 120.72 1611 83.41 8293 −6.0 −4.7 6.8 5.5 4.7 7.3 1.6 6.1 1.7 2.3 70 2.7 290 31.6 −11.8 2.1 10.2 Burbank/Glendale 722880 34.20 118.35 236 98.52 8293 4.0 5.1 8.0 6.4 5.5 8.8 13.4 7.8 13.7 0.9 330 3.5 180 41.0 0.7 1.7 2.5 WMO# = World Meteorological Organization number AFB = Air Force Base NAS = Naval Air Station WSO = Weather Service Office AAF = Army Air Field MCAS = Marine Corps Air Station DB = dry-bulb temperature, °C Lat. = North latitude ° Long. = West longitude ° Elev. = elevation, m StdP = standard pressure at station elevation, kPa WS = wind speed, m/s PWD = prevailing wind direction, ° True Climatic Design Information 27.7 Table 1B Cooling and Dehumidification Design Conditions—United States Station Cooling DB/MWB Evaporation WB/MDB Dehumidification DP/MDB and HR Range of DB 0.4% 1% 2% 0.4% 1% 2% 0.4% 1% 2% DB MWB DB MWB DB MWB WB MDB WB MDB WB MDB DP HR MDB DP HR MDB DP HR MDB 1 2a 2b 2c 2d 2e 2f 3a 3b 3c 3d 3e 3f 4a 4b 4c 4d 4e 4f 4g 4h 4i 5 ALABAMA Anniston 34.8 24.3 33.7 24.3 32.1 23.9 26.3 32.0 25.6 31.1 25.0 30.2 24.9 20.4 29.1 24.2 19.6 28.0 23.7 19.0 27.2 10.9 Birmingham 34.7 23.9 33.4 23.8 32.3 23.5 25.7 31.8 25.1 31.2 24.6 30.3 24.0 19.3 28.6 23.4 18.7 27.9 22.9 18.1 27.4 10.4 Dothan 35.0 24.4 34.1 24.3 33.2 24.2 26.5 32.2 25.9 31.3 25.4 30.5 25.2 20.6 28.4 24.6 19.9 27.9 24.2 19.4 27.5 9.7 Huntsville 34.5 23.8 33.2 23.6 31.9 23.3 25.6 31.6 25.0 30.9 24.4 30.0 23.9 19.3 28.6 23.3 18.6 27.9 22.8 18.0 27.3 10.3 Mobile 34.3 24.7 33.4 24.6 32.5 24.4 26.3 31.6 25.8 31.0 25.5 30.6 25.1 20.3 28.5 24.6 19.8 28.1 24.2 19.3 27.8 9.2 Montgomery 35.1 24.6 34.0 24.4 32.9 24.3 26.3 32.6 25.8 31.8 25.3 31.1 24.6 19.8 29.3 24.1 19.2 28.7 23.7 18.6 28.3 10.4 Muscle Shoals/Florence 35.3 24.2 34.2 24.0 33.1 23.6 25.7 32.2 25.3 31.5 24.8 30.8 24.3 19.6 27.9 23.8 19.0 27.6 23.4 18.6 27.3 11.1 Ozark, Fort Rucker 35.2 25.1 34.3 24.8 33.4 24.5 27.0 32.3 26.3 31.7 25.8 30.9 25.5 20.9 29.4 25.0 20.3 28.7 24.5 19.7 28.3 10.0 Tuscaloosa 35.2 24.8 34.2 24.8 33.2 24.4 26.5 32.5 26.0 31.8 25.5 31.0 25.1 20.3 28.8 24.5 19.6 28.2 24.1 19.1 27.9 10.9 ALASKA Adak, NAS 15.2 12.6 14.0 11.4 13.0 10.3 12.9 14.8 11.7 13.7 10.6 12.5 11.5 8.4 14.6 10.4 7.8 13.1 9.3 7.3 11.9 5.4 Anchorage, Elemendorf AFB 21.9 14.6 20.3 13.9 19.0 13.1 15.7 20.7 14.7 18.9 13.9 17.7 13.7 9.9 16.5 12.9 9.3 16.2 11.9 8.7 15.6 7.0 Anchorage, Fort Richardson 23.4 15.5 21.4 14.4 19.9 13.7 16.1 22.2 15.1 20.3 14.2 18.8 13.5 9.8 18.0 12.2 9.0 16.5 11.7 8.7 16.2 8.6 Anchorage, Intl Airport 21.5 14.7 19.9 13.9 18.5 13.1 15.6 20.6 14.6 18.7 13.8 17.4 13.4 9.7 16.8 12.7 9.2 16.3 11.8 8.7 15.6 7.0 Annette 23.3 16.0 21.1 15.1 19.1 13.9 16.7 22.3 15.6 20.1 14.6 18.2 14.3 10.2 18.3 13.6 9.7 17.0 12.9 9.3 15.9 5.8 Barrow 13.6 10.8 11.2 9.3 9.0 7.6 11.1 13.3 9.3 11.1 7.6 8.9 9.7 7.5 12.1 7.9 6.6 10.5 6.4 6.0 8.8 5.9 Bethel 22.1 14.7 19.9 13.7 18.0 12.8 15.5 20.7 14.4 18.6 13.5 16.9 13.4 9.7 16.4 12.6 9.1 15.3 11.8 8.6 14.7 7.4 Bettles 26.1 15.9 24.1 15.1 22.2 14.3 17.0 24.2 15.9 22.5 15.0 20.8 14.2 10.3 18.9 13.1 9.6 17.9 12.2 9.0 17.1 10.8 Big Delta, Ft. Greely 25.5 14.8 23.6 14.3 21.9 13.5 16.2 23.3 15.1 22.1 14.2 20.4 13.3 10.0 18.6 12.2 9.3 17.1 11.2 8.7 16.2 9.6 Cold Bay 15.4 12.4 14.1 11.6 13.0 11.1 13.0 14.8 12.2 13.6 11.4 12.7 12.2 8.9 13.6 11.4 8.4 12.8 10.7 8.0 12.2 4.1 Cordova 21.3 14.8 19.3 13.8 17.2 13.2 15.6 20.5 14.4 18.4 13.5 16.7 13.4 9.6 17.2 12.2 8.9 15.5 11.6 8.5 14.9 7.5 Deadhorse 18.8 13.8 16.2 12.5 14.3 11.4 14.2 17.9 12.9 16.4 11.4 14.2 12.0 8.7 16.6 10.7 8.0 15.2 9.3 7.3 13.6 7.6 Dillingham 20.5 14.1 18.7 13.3 16.7 12.4 14.9 19.2 13.9 17.6 13.0 16.3 13.3 9.6 16.5 11.9 8.7 15.2 11.1 8.2 14.1 7.3 Fairbanks, Eielson AFB 27.4 16.2 25.6 15.5 23.7 14.9 17.5 25.8 16.4 23.7 15.6 22.3 14.5 10.5 19.1 13.6 9.9 18.9 12.2 9.0 18.9 10.8 Fairbanks, Intl Airport 27.1 15.8 25.2 15.1 23.4 14.6 17.0 24.8 16.1 23.4 15.2 21.6 14.2 10.3 18.2 13.3 9.7 17.7 12.4 9.1 17.3 10.3 Galena 25.4 15.9 23.5 14.9 21.4 14.3 17.0 23.2 16.0 21.5 15.1 20.5 14.5 10.4 19.0 13.6 9.8 18.2 12.4 9.0 17.7 8.5 Gulkana 24.7 14.2 22.7 13.2 20.8 12.5 15.1 22.9 14.0 21.3 13.1 19.7 11.6 9.0 16.8 10.7 8.5 15.7 9.7 7.9 15.1 11.3 Homer 18.1 13.4 16.6 12.9 15.7 12.3 14.1 17.1 13.4 16.1 12.7 15.2 12.5 9.1 14.9 11.7 8.6 14.7 11.1 8.2 14.1 6.6 Juneau 23.1 15.5 20.8 14.7 19.1 13.7 16.1 21.6 15.2 20.0 14.3 18.0 14.0 10.0 17.3 13.4 9.6 16.1 12.8 9.2 15.4 7.7 Kenai 20.1 13.3 18.6 12.7 16.8 12.2 14.3 18.4 13.6 17.0 12.9 15.9 12.8 9.2 15.0 11.8 8.6 14.4 11.2 8.3 14.0 7.4 Ketchikan 21.5 15.8 20.0 15.1 18.8 14.2 16.4 20.4 15.6 19.2 14.9 17.9 14.8 10.5 17.5 14.1 10.1 16.6 13.6 9.7 16.0 5.7 King Salmon 21.4 14.3 19.4 13.4 17.7 12.6 15.2 20.1 14.1 18.2 13.2 16.8 13.1 9.4 16.3 12.2 8.8 15.3 11.3 8.3 14.5 8.6 Kodiak, State USCG Base 20.2 14.3 18.2 13.3 16.7 12.6 15.1 18.8 14.1 17.2 13.2 15.9 13.3 9.6 16.3 12.6 9.1 15.2 11.9 8.7 14.1 6.2 Kotzebue 19.8 15.2 18.0 14.3 16.3 13.2 15.8 19.3 14.6 17.5 13.5 15.9 14.1 10.0 17.6 13.0 9.3 16.3 12.1 8.8 15.2 4.9 McGrath 24.9 15.3 22.9 14.3 21.0 13.6 16.2 23.2 15.2 21.1 14.3 19.5 13.7 9.9 17.3 12.8 9.3 16.7 11.9 8.8 15.9 9.7 Middleton Island 16.8 12.0 15.6 10.8 14.8 10.6 12.8 15.9 12.1 14.8 11.7 14.0 11.1 8.2 13.3 10.6 8.0 13.1 10.1 7.7 12.9 3.2 Nenana 26.4 15.7 24.6 14.9 22.8 14.0 16.7 24.1 15.6 22.8 14.8 21.3 13.7 9.9 18.3 12.8 9.3 18.2 11.5 8.6 17.0 11.8 Nome 20.4 13.7 18.2 12.9 16.2 11.9 14.7 19.1 13.6 17.1 12.6 15.6 12.7 9.1 15.9 11.7 8.6 14.8 10.8 8.0 14.0 6.1 Northway 25.5 14.7 23.6 13.9 21.4 13.3 15.5 24.2 14.6 21.6 13.7 20.4 12.2 9.4 16.4 11.4 8.9 15.9 10.7 8.5 15.1 11.1 Port Heiden 18.0 12.5 16.2 11.3 15.0 10.7 13.1 16.8 12.0 15.3 11.2 14.4 10.8 8.1 14.8 10.0 7.7 14.0 9.3 7.3 12.7 5.4 Saint Paul Island 11.9 10.3 11.1 9.7 10.3 9.2 10.7 11.7 10.1 10.8 9.5 10.2 10.2 7.8 11.1 9.6 7.4 10.4 9.1 7.2 9.8 3.0 Sitka 19.1 15.1 17.8 14.2 16.3 13.7 15.8 18.1 14.9 16.8 14.2 15.8 14.7 10.5 16.5 14.1 10.1 15.7 13.6 9.7 15.1 5.1 Talkeetna 24.7 15.7 22.7 14.6 21.0 14.1 16.6 23.1 15.6 21.3 14.7 19.6 14.1 10.1 17.9 13.2 9.6 16.8 12.4 9.1 15.9 9.1 Valdez 20.5 13.5 18.8 13.0 16.8 12.2 14.2 19.3 13.5 17.6 12.7 16.2 11.8 8.6 14.9 11.4 8.4 14.1 11.0 8.2 13.3 6.8 Yakutat 19.0 13.2 17.1 12.7 15.7 12.4 14.3 16.9 13.7 15.8 13.1 15.1 13.4 9.6 14.6 12.8 9.2 14.1 12.3 8.9 13.7 6.7 ARIZONA Flagstaff 29.5 13.1 28.1 12.9 26.7 12.8 16.3 23.1 15.6 22.6 14.9 22.1 14.3 13.3 18.1 13.4 12.5 17.7 12.6 11.8 17.3 15.3 Kingman 37.2 17.7 36.1 17.2 35.0 16.9 21.4 28.0 19.6 29.6 18.8 30.2 19.3 16.0 24.9 16.4 13.2 23.8 15.0 12.1 24.3 13.8 Page 37.1 16.8 36.0 16.4 34.9 16.1 19.1 29.3 18.3 30.2 17.7 30.2 15.7 13.1 23.5 14.6 12.2 23.2 13.6 11.4 23.4 13.2 Phoenix, Intl Airport 43.2 20.9 42.0 20.9 40.9 20.8 24.2 35.8 23.7 35.4 23.2 35.0 21.5 16.8 27.8 20.6 15.9 28.6 19.6 14.9 29.6 12.8 Phoenix, Luke AFB 43.4 21.8 41.7 21.7 40.6 21.5 25.4 35.9 24.7 36.0 24.0 35.3 23.1 18.6 29.6 21.6 16.9 29.7 20.6 15.9 30.2 14.0 Prescott 34.2 15.7 32.7 15.6 31.4 15.4 19.3 27.2 18.6 26.7 18.0 26.0 17.2 14.9 21.6 16.3 14.0 21.4 15.5 13.3 21.3 14.1 Safford, Agri Center 39.0 18.7 37.2 19.0 36.0 18.5 21.9 31.9 21.4 31.7 20.8 31.3 19.3 15.8 24.9 18.7 15.2 24.6 18.0 14.5 25.0 19.3 Tucson 40.1 18.5 38.8 18.4 37.6 18.4 22.2 31.0 21.7 30.4 21.3 30.1 20.3 16.5 24.3 19.6 15.8 24.5 18.9 15.1 24.9 16.3 Winslow 35.2 15.8 34.0 15.5 32.8 15.2 18.6 26.6 17.8 27.0 17.2 26.8 16.0 13.6 21.5 15.2 13.0 20.8 14.3 12.2 20.7 15.2 Yuma 44.0 22.3 42.6 22.5 41.2 22.1 26.4 35.5 25.7 35.2 25.1 35.1 24.3 19.4 30.7 23.3 18.2 31.2 21.9 16.7 31.5 13.2 ARKANSAS Blytheville, Eaker AFB 36.0 25.7 34.8 25.2 33.7 24.8 27.5 33.3 26.6 32.6 25.8 31.6 25.8 21.3 31.2 25.0 20.3 30.2 24.2 19.3 29.2 10.4 Fayetteville 35.2 24.1 34.0 24.0 32.2 23.6 25.6 32.4 24.9 31.9 24.3 30.6 23.7 19.4 29.4 23.2 18.8 28.7 22.2 17.7 27.4 11.9 Fort Smith 37.0 24.3 35.4 24.3 34.0 24.1 26.0 33.6 25.4 32.8 24.9 32.1 24.0 19.2 29.5 23.4 18.6 28.8 22.9 18.0 28.3 11.9 Little Rock, AFB 36.3 25.1 34.8 24.9 33.6 24.6 26.7 33.3 26.1 32.7 25.6 31.8 24.9 20.2 29.8 24.4 19.6 29.3 23.9 19.0 28.8 10.8 Texarkana 36.1 25.0 34.9 24.8 33.9 24.7 26.6 32.9 26.1 32.5 25.7 31.9 25.0 20.4 29.7 24.5 19.8 29.2 24.1 19.3 28.8 11.4 CALIFORNIA Alameda, NAS 28.5 18.4 26.0 17.6 24.3 17.0 19.3 25.9 18.5 24.7 17.9 23.0 16.9 12.1 20.9 16.2 11.5 20.3 15.7 11.2 19.7 8.2 Arcata/Eureka 21.3 15.5 19.7 15.0 18.6 14.6 16.8 19.7 15.9 18.6 15.3 17.7 15.6 11.2 17.8 14.9 10.7 17.1 14.2 10.2 16.4 8.6 Bakersfield 39.9 21.3 38.5 20.8 37.1 20.3 22.6 36.6 21.8 35.6 21.2 35.1 17.9 13.1 28.9 16.7 12.1 28.6 15.7 11.3 28.4 14.7 Barstow/Daggett 41.8 20.2 40.5 19.7 39.1 19.2 22.4 35.1 21.6 35.1 20.7 35.0 18.8 14.7 27.4 16.9 13.0 29.6 15.1 11.5 29.5 15.4 Blue Canyon 28.8 15.1 27.1 14.1 25.9 13.5 16.4 26.5 15.4 25.5 14.5 24.1 12.0 10.6 21.1 10.9 9.9 21.2 9.9 9.2 20.1 9.2 Burbank/Glendale 36.7 20.4 34.8 20.3 33.3 20.0 23.2 32.0 22.3 31.7 21.5 30.0 20.3 15.4 26.6 19.5 14.7 25.7 18.8 14.0 24.8 13.0 DP = dew-point temperature, °C MWB = mean coincident wet-bulb temperature, °C MDB = mean coincident dry-bulb temperature, °C MWS = mean coincident wind speed, m/s StdD = standard deviation, °C HR = humidity ratio, grams of moisture per kilogram of dry air A = airport ANGB = Air National Guard Base MCAF = Marine Corps Air Facility NAF = Naval Air Facility NAWS = Naval Air Weapons Station RAF = Royal Air Force 27.8 2001 ASHRAE Fundamentals Handbook (SI) Table 1A Heating and Wind Design Conditions—United States Heating Dry Bulb Extreme Wind Speed, m/s Coldest Month MWS/PWD to DB Extr. Annual Daily Lat.
Long.
Elev., m StdP, kPa 0.4% 1% 99.6% 0.4% Mean DB StdD DB Station WMO# Dates 99.6% 99% 1% 2.5% 5% WS MDB WS MDB MWS PWD MWS PWD Max.
Min.
Max. Min.
1a 1b 1c 1d 1e 1f 1g 2a 2b 3a 3b 3c 4a 4b 4c 4d 5a 5b 5c 5d 6a 6b 6c 6d Fairfield, Travis AFB 745160 38.27 121.93 19 101.10 8293 −0.3 1.0 12.4 10.9 9.9 11.7 11.5 9.7 11.2 1.6 20 4.2 240 40.4 −3.1 1.9 2.2 Fresno 723890 36.77 119.72 100 100.13 6193 −1.2 0.1 7.6 6.7 5.9 7.4 11.4 6.1 11.2 1.7 90 3.9 290 41.8 −3.4 1.2 2.1 Lancaster/Palmdale 723816 34.73 118.22 715 93.03 8293 −5.7 −4.3 13.5 12.6 11.3 13.1 8.9 11.4 9.6 1.0 260 6.4 240 41.5 −9.5 1.1 3.3 Lemoore, Reeves NAS 747020 36.33 119.95 72 100.46 8293 −1.2 0.2 8.6 7.2 6.1 9.0 9.3 7.1 10.3 1.6 150 3.0 360 43.1 −8.1 2.3 5.8 Long Beach 722970 33.82 118.15 12 101.18 6193 4.5 5.8 8.5 7.2 6.3 8.5 14.3 7.1 14.4 1.6 300 4.6 270 38.9 1.7 2.5 1.6 Los Angeles 722950 33.93 118.40 32 100.94 6193 6.2 7.4 9.2 7.9 7.1 8.9 13.6 7.4 13.3 2.8 70 4.4 250 35.9 3.5 2.8 1.7 Marysville, Beale AFB 724837 39.13 121.43 34 100.92 8293 −0.4 0.9 9.0 7.6 6.4 10.1 11.6 8.5 11.4 1.3 20 2.2 200 41.2 −3.1 1.8 2.3 Merced, Castle AFB 724810 37.38 120.57 57 100.64 8293 −1.0 0.1 8.1 6.5 5.5 9.2 10.4 7.7 9.4 0.9 110 3.9 320 40.1 −3.3 1.5 2.0 Mount Shasta 725957 41.32 122.32 1080 89.01 8293 −8.9 −6.3 6.2 5.3 4.6 6.3 2.5 5.5 3.0 1.9 60 2.0 180 35.1 −12.5 1.5 3.7 Mountain View, Moffet NAS 745090 37.42 122.05 12 101.18 8293 2.2 3.8 8.4 7.5 6.5 8.7 12.0 7.1 11.1 0.3 140 3.8 330 36.5 −5.1 1.4 6.8 Ontario 722865 34.05 117.60 287 97.92 8293 1.4 3.4 9.9 8.4 7.7 12.7 16.7 9.4 13.7 2.0 10 5.8 240 42.1 −1.4 1.9 1.4 Oxnard, Pt. Mugu NAWS 723910 34.12 119.12 2 101.30 8293 3.8 5.0 10.0 8.4 7.1 11.0 14.1 9.3 14.6 2.3 20 5.2 50 33.8 −4.5 2.8 5.9 Paso Robles 723965 35.67 120.63 255 98.30 8293 −3.2 −1.8 10.0 9.1 8.2 9.5 11.1 8.1 10.3 1.3 110 4.9 300 42.5 −6.3 1.2 2.7 Red Bluff 725910 40.15 122.25 108 100.03 8293 −1.5 −0.1 10.4 9.4 8.4 11.7 11.8 10.1 9.9 2.7 340 4.2 160 43.9 −3.8 1.8 2.1 Riverside, March AFB 722860 33.88 117.27 469 95.82 8293 1.0 2.2 8.2 6.8 5.9 9.8 10.8 8.1 12.9 0.6 210 4.1 300 41.4 −1.9 1.3 1.8 Sacramento, Mather Field 724835 38.55 121.30 29 100.98 8293 −1.3 0.1 9.1 7.4 6.1 10.7 11.4 9.0 10.6 0.7 120 2.5 310 40.8 −3.4 2.8 2.4 Sacramento, McClellan AFB 724836 38.67 121.40 23 101.05 8293 −0.4 1.0 8.8 7.3 6.2 10.1 11.5 8.5 11.1 1.0 340 2.4 220 41.4 −2.7 1.4 2.7 Sacramento, Metro 724839 38.70 121.58 7 101.24 6193 −0.8 0.6 10.0 8.6 7.7 10.1 10.5 8.8 9.7 1.3 340 3.8 220 41.6 −2.9 3.7 1.7 Salinas 724917 36.67 121.60 26 101.01 8293 0.7 1.9 9.5 8.5 7.9 10.4 10.7 9.3 10.6 2.9 130 4.7 310 34.9 −1.6 2.6 1.2 San Bernardino, Norton AFB 722866 34.10 117.23 353 97.16 8293 1.0 2.3 7.4 5.7 4.8 9.3 13.2 7.2 12.7 0.7 50 3.4 250 42.9 −1.6 1.4 1.5 San Diego, Intl Airport 722900 32.73 117.17 9 101.22 6193 6.7 7.9 8.2 7.3 6.6 8.8 15.0 7.3 15.4 1.5 70 4.6 310 34.7 4.1 3.6 2.4 San Diego, Miramar NAS 722930 32.85 117.12 128 99.80 8293 4.0 5.3 5.7 4.9 4.2 6.8 14.9 5.2 15.1 1.2 90 2.8 310 38.7 −2.8 2.2 7.6 San Francisco 724940 37.62 122.38 5 101.26 6193 2.7 3.9 13.0 11.5 10.4 11.9 11.4 9.9 11.2 2.4 160 5.6 300 34.7 0.8 2.4 1.7 San Jose Intl Airport 724945 37.37 121.93 17 101.12 8293 1.6 3.1 9.0 8.2 7.4 8.8 13.1 7.8 13.4 0.4 160 4.4 320 38.2 −2.7 1.7 5.0 Santa Barbara 723925 34.43 119.83 3 101.29 8293 1.2 2.8 9.0 7.7 6.4 8.5 14.2 7.3 14.3 0.6 40 4.4 260 36.2 −2.0 3.7 3.6 Santa Maria 723940 34.90 120.45 73 100.45 6193 0.1 1.4 10.4 9.3 8.3 9.4 14.7 8.0 14.9 1.9 110 4.7 300 35.1 −3.0 2.8 1.6 Stockton 724920 37.90 121.25 8 101.23 8293 −1.1 0.1 9.6 8.4 7.6 10.6 11.1 9.2 9.7 1.8 110 4.7 280 41.1 −3.5 1.7 1.9 Victorville, George AFB 723825 34.58 117.38 876 91.23 8293 −2.7 −1.1 9.8 8.4 7.3 10.0 9.6 8.2 8.4 1.4 160 4.0 180 41.0 −5.9 1.7 3.1 COLORADO Alamosa 724620 37.45 105.87 2299 76.59 6193 −27.4−24.0 11.7 10.3 9.2 10.3 0.4 8.9 −1.1 1.5 190 5.4 240 31.2 −32.8 1.1 4.4 Colorado Springs 724660 38.82 104.72 1881 80.68 6193 −18.7−15.3 12.9 11.0 9.6 12.4 1.7 10.4 0.7 3.2 20 5.4 160 34.8 −23.0 1.1 3.8 Craig 725700 40.50 107.53 1915 80.34 8293 −28.8−24.7 11.6 9.1 7.5 9.7 0.4 7.4 −2.7 1.1 270 3.9 250 33.7 −34.8 1.1 5.9 Denver 724699 39.75 104.87 1625 83.26 8293 −19.7−16.1 10.4 8.8 7.5 11.1 4.1 9.5 4.4 2.7 180 4.1 160 36.3 −23.7 1.3 3.9 Eagle 724675 39.65 106.92 1993 79.56 6193 −25.0−21.7 9.8 8.5 7.7 8.9 0.4 7.8 −0.2 1.2 90 5.0 230 34.1 −30.8 1.8 4.3 Grand Junction 724760 39.12 108.53 1475 84.82 6193 −16.8−13.8 10.0 8.5 7.4 7.6 0.6 6.4 −1.4 2.2 70 4.9 290 37.7 −19.6 1.1 4.7 Limon 724665 39.27 103.67 1635 83.16 8293 −21.2−17.0 12.2 10.4 9.3 11.9 −1.7 10.0 −4.0 3.9 160 5.4 200 35.5 −25.2 1.2 3.6 Pueblo 724640 38.28 104.52 1439 85.19 6193 −18.5−15.0 14.1 12.2 10.5 13.2 6.7 11.4 6.1 2.3 270 5.4 140 38.8 −24.4 1.1 4.3 Trinidad 724645 37.27 104.33 1756 81.93 8293 −18.8−14.7 11.1 9.7 8.5 10.7 5.0 9.2 5.3 2.3 290 4.6 210 36.7 −23.4 1.1 3.8 CONNECTICUT Bridgeport 725040 41.17 73.13 5 101.26 6193 −13.3−10.9 11.8 10.2 9.2 15.2 −1.6 13.3 −1.6 6.1 320 6.0 230 33.6 −16.9 1.6 2.7 Hartford, Brainard Field 725087 41.73 72.65 6 101.25 6193 −16.9−14.3 10.1 8.9 7.9 10.1 −3.8 9.1 −3.1 3.3 320 4.9 250 35.9 −21.0 1.3 3.2 Windsor Locks, Bradley Fld 725080 41.93 72.68 55 100.67 8293 −15.9−13.2 9.5 8.4 7.7 10.0 −1.2 8.9 −1.8 3.3 360 4.7 240 36.0 −20.4 1.1 3.2 DELAWARE Dover, AFB 724088 39.13 75.47 9 101.22 8293 −10.2 −7.5 9.7 8.4 7.4 10.4 2.4 9.4 1.4 3.6 340 3.9 240 36.2 −14.3 1.8 3.4 Wilmington 724089 39.68 75.60 24 101.04 6193 −12.4 −9.9 11.1 9.7 8.6 11.9 −1.9 10.3 −1.3 5.1 290 4.7 240 35.3 −16.2 1.5 3.8 FLORIDA Apalachicola 722200 29.73 85.03 6 101.25 8293 −0.5 1.7 8.3 7.4 6.6 8.5 10.4 7.7 10.8 2.5 360 4.1 220 33.8 −5.1 3.7 4.1 Cape Canaveral, NASA 747946 28.62 80.72 3 101.29 8293 3.1 5.3 8.6 7.6 6.8 9.5 15.5 8.3 15.8 3.6 320 3.5 220 35.6 −1.6 0.8 3.4 Daytona Beach 722056 29.18 81.05 11 101.19 6193 0.9 3.0 9.6 8.5 7.7 9.6 16.1 8.6 15.8 3.2 310 4.8 240 35.3 −2.7 1.1 2.4 Fort Lauderdale/Hollywood 722025 26.07 80.15 7 101.24 8293 7.9 10.2 9.9 8.9 8.1 10.0 20.8 9.0 21.4 4.0 330 4.8 120 36.2 3.9 0.6 3.4 Fort Myers 722106 26.58 81.87 5 101.26 8293 5.7 8.2 8.6 7.9 7.1 8.9 17.9 8.2 18.8 2.8 30 4.0 70 36.2 1.3 0.7 2.6 Gainesville 722146 29.68 82.27 46 100.77 8293 −1.0 0.8 8.3 7.4 6.4 8.4 18.2 7.6 16.8 1.8 300 3.8 270 36.1 −6.3 1.0 4.0 Homestead, AFB 722026 25.48 80.38 2 101.30 8293 8.8 10.9 7.7 6.7 6.0 7.7 21.0 6.8 21.0 2.6 360 3.2 120 34.9 4.9 1.2 3.1 Jacksonville, Cecil Field NAS 722067 30.22 81.88 25 101.03 8293 −0.8 1.0 8.2 7.2 6.4 8.4 16.4 7.4 16.4 1.5 290 3.0 270 37.7 −6.7 1.1 4.9 Jacksonville, Intl Airport 722060 30.50 81.70 9 101.22 6193 −1.7 0.2 9.4 8.2 7.4 9.5 12.3 8.6 12.5 2.7 310 3.9 230 36.5 −5.4 1.2 2.8 Jacksonville, Mayport Naval 722066 30.40 81.42 5 101.26 8293 1.2 3.8 8.5 7.4 6.4 9.3 12.0 8.1 13.0 2.7 310 3.3 270 37.4 −6.6 1.2 7.3 Key West 722010 24.55 81.75 6 101.25 6193 12.6 14.4 10.0 9.0 8.2 10.5 18.3 9.6 18.9 5.5 50 4.2 140 33.0 10.3 0.7 2.2 Melbourne 722040 28.10 80.65 11 101.19 8293 3.5 5.9 9.4 8.5 8.0 10.0 16.4 9.0 16.6 4.1 320 4.9 120 36.2 −1.2 1.0 3.7 Miami, Intl Airport 722020 25.82 80.28 4 101.28 6193 7.6 9.8 10.1 9.0 8.2 9.6 19.9 8.8 20.4 4.5 340 5.0 150 34.4 3.6 1.2 2.8 Miami, New Tamiami A 722029 25.65 80.43 3 101.29 8293 7.1 9.6 9.4 8.5 8.0 9.5 22.5 8.7 22.5 3.6 360 5.0 130 35.0 3.8 1.1 3.5 Milton, Whiting Field NAS 722226 30.72 87.02 61 100.59 8293 −2.3 −0.3 8.1 7.0 6.1 8.4 10.0 7.4 11.0 2.9 340 2.7 330 37.1 −8.4 1.0 4.8 Orlando 722050 28.43 81.32 32 100.94 8293 2.9 5.3 9.0 8.0 7.2 9.4 18.8 8.4 18.2 3.6 330 3.9 290 35.6 −1.7 0.9 3.6 Panama City, Tyndall AFB 747750 30.07 85.58 5 101.26 8293 0.5 2.9 8.2 7.2 6.3 8.4 11.3 7.5 11.3 3.8 360 3.3 240 34.5 −4.5 1.3 3.5 Pensacola, Sherman AFB 722225 30.35 87.32 9 101.22 8293 −2.0 −0.1 10.3 9.1 8.1 11.2 6.3 9.8 8.7 4.1 360 4.5 200 37.6 −9.3 3.5 4.2 Saint Petersburg 722116 27.92 82.68 3 101.29 8293 6.1 8.5 9.4 8.4 7.8 9.9 18.2 8.9 17.3 4.8 10 4.4 230 35.9 1.6 1.0 2.6 Sarasota/Bradenton 722115 27.40 82.55 9 101.22 8293 3.8 6.0 9.9 8.5 7.8 10.2 19.5 9.1 19.2 2.1 40 4.1 270 35.9 −1.9 0.9 6.7 Tallahassee 722140 30.38 84.37 21 101.07 6193 −4.0 −2.1 7.9 7.0 6.2 8.6 11.2 7.7 12.1 1.2 350 3.6 360 36.7 −8.2 1.1 2.6 Tampa, Intl Airport 722110 27.97 82.53 3 101.29 6193 2.2 4.2 8.4 7.5 6.8 9.4 14.8 8.3 15.1 3.5 20 4.4 270 34.9 −1.8 0.7 2.7 Valparaiso, Eglin AFB 722210 30.48 86.53 26 101.01 8293 −1.3 0.6 8.3 7.3 6.3 8.2 9.3 7.2 10.4 2.6 360 3.3 210 35.9 −7.1 1.1 3.4 Vero Beach 722045 27.65 80.42 8 101.23 8293 4.0 6.1 9.1 8.3 7.7 9.3 19.4 8.4 19.6 3.6 310 4.7 240 35.6 −0.6 1.1 3.6 WMO# = World Meteorological Organization number AFB = Air Force Base NAS = Naval Air Station WSO = Weather Service Office AAF = Army Air Field MCAS = Marine Corps Air Station DB = dry-bulb temperature, °C Lat. = North latitude ° Long. = West longitude ° Elev. = elevation, m StdP = standard pressure at station elevation, kPa WS = wind speed, m/s PWD = prevailing wind direction, ° True Climatic Design Information 27.9 Table 1B Cooling and Dehumidification Design Conditions—United States Station Cooling DB/MWB Evaporation WB/MDB Dehumidification DP/MDB and HR Range of DB 0.4% 1% 2% 0.4% 1% 2% 0.4% 1% 2% DB MWB DB MWB DB MWB WB MDB WB MDB WB MDB DP HR MDB DP HR MDB DP HR MDB 1 2a 2b 2c 2d 2e 2f 3a 3b 3c 3d 3e 3f 4a 4b 4c 4d 4e 4f 4g 4h 4i 5 Fairfield, Travis AFB 36.4 19.6 34.7 19.4 32.7 19.0 21.3 33.6 20.3 32.1 19.5 30.9 16.9 12.1 24.7 15.9 11.3 23.5 15.1 10.7 22.9 16.1 Fresno 39.6 21.4 38.1 20.9 36.6 20.4 22.7 36.8 21.9 35.6 21.2 34.6 18.0 13.1 29.7 16.8 12.1 28.6 15.9 11.4 27.7 17.2 Lancaster/Palmdale 38.5 19.1 36.7 18.5 35.4 18.0 21.1 34.2 20.3 32.7 19.4 32.3 16.9 13.1 26.5 15.5 12.0 27.2 14.4 11.2 27.2 15.5 Lemoore, Reeves NAS 39.5 22.1 38.2 21.7 36.5 20.9 23.8 36.3 22.8 35.6 21.9 34.7 19.5 14.4 31.7 18.4 13.4 30.6 16.8 12.1 30.0 18.3 Long Beach 33.2 19.5 30.9 19.2 29.1 19.0 21.9 29.4 21.2 28.0 20.6 26.8 19.6 14.4 24.4 18.9 13.7 24.1 18.3 13.2 23.7 9.3 Los Angeles 29.2 17.7 27.0 17.6 25.4 17.9 21.0 25.8 20.3 24.7 19.8 23.8 19.4 14.2 23.6 18.7 13.6 22.8 18.1 13.1 22.1 6.1 Marysville, Beale AFB 38.4 21.1 36.4 20.4 35.0 19.8 22.4 35.9 21.5 34.8 20.6 33.1 17.2 12.3 29.7 16.1 11.5 27.8 15.3 10.9 27.0 16.6 Merced, Castle AFB 37.2 20.8 36.0 20.7 34.7 20.1 22.3 35.3 21.6 34.0 20.8 33.1 17.7 12.8 27.2 16.4 11.7 29.1 15.6 11.2 27.4 16.8 Mount Shasta 32.9 16.8 30.9 16.1 29.5 15.3 18.0 30.6 17.0 28.7 16.1 28.0 13.4 10.9 23.4 11.7 9.8 22.9 10.6 9.1 21.5 17.8 Mountain View, Moffet NAS 31.2 18.2 28.9 18.2 26.7 17.6 20.0 27.8 19.2 26.6 18.5 25.7 16.7 11.9 23.6 16.0 11.4 23.0 15.4 10.9 22.2 10.0 Ontario 38.7 21.5 36.5 21.0 35.0 20.7 23.7 34.5 23.0 33.4 22.2 32.2 20.9 16.1 26.5 19.9 15.1 26.8 19.1 14.4 25.6 15.4 Oxnard, Pt. Mugu NAWS 28.5 16.6 26.1 17.8 24.8 17.9 21.3 25.1 20.5 24.1 19.7 23.2 20.0 14.7 23.4 19.1 13.9 22.8 18.4 13.3 22.2 8.1 Paso Robles 38.8 19.8 36.7 19.2 34.9 18.5 20.9 36.2 20.0 34.7 19.2 32.8 15.9 11.6 24.2 14.6 10.7 22.8 13.9 10.2 21.6 21.0 Red Bluff 40.7 20.9 38.8 20.3 36.7 19.5 22.5 36.5 21.6 35.0 20.8 33.7 18.4 13.4 27.6 16.9 12.2 26.4 16.0 11.5 25.3 16.4 Riverside, March AFB 38.4 20.0 36.4 19.9 35.1 19.4 22.5 33.6 21.7 32.7 21.0 32.2 19.3 14.9 26.1 18.2 13.9 26.7 16.9 12.8 26.3 16.1 Sacramento, Mather Field 38.5 20.7 36.3 20.0 34.8 19.3 21.6 36.2 20.8 34.4 20.0 33.1 16.1 11.5 26.1 15.4 11.0 25.1 14.7 10.5 24.2 18.7 Sacramento, McClellan AFB 38.8 21.0 36.8 20.5 35.2 19.9 22.3 36.2 21.4 35.0 20.6 33.5 17.0 12.2 28.7 16.2 11.5 27.0 15.4 11.0 26.2 16.5 Sacramento, Metro 37.8 20.8 36.0 20.3 34.2 19.7 22.0 35.7 21.1 34.3 20.3 32.8 16.8 12.0 27.9 15.9 11.3 26.4 15.2 10.8 25.4 18.5 Salinas 28.5 17.0 25.8 16.8 24.1 16.2 18.9 25.4 18.1 23.9 17.3 22.5 16.4 11.7 20.7 15.7 11.2 19.9 15.2 10.8 19.2 10.4 San Bernardino, Norton AFB 39.5 20.9 38.1 21.0 36.2 20.4 23.5 34.7 22.7 34.4 21.9 33.2 20.0 15.3 28.5 19.0 14.4 28.4 18.1 13.6 28.0 17.5 San Diego, Intl Airport 29.4 19.6 27.4 19.4 26.1 19.2 22.5 26.3 21.7 25.6 20.9 24.7 21.2 15.8 25.1 20.2 14.9 24.3 19.4 14.2 23.5 4.9 San Diego, Miramar NAS 33.4 20.6 31.0 19.7 29.4 19.3 22.2 29.6 21.5 28.6 20.8 27.4 19.9 14.8 25.6 19.2 14.2 24.8 18.6 13.7 23.9 9.7 San Francisco 28.4 17.0 25.6 16.4 23.3 15.8 18.0 25.9 17.2 23.8 16.6 22.1 15.2 10.8 19.4 14.6 10.4 18.9 14.1 10.1 18.5 9.3 San Jose Intl Airport 34.1 19.5 31.7 18.9 29.8 18.5 20.9 31.1 20.1 29.4 19.4 28.2 17.0 12.2 24.8 16.3 11.6 24.2 15.7 11.2 23.3 12.4 Santa Barbara 28.6 17.6 26.4 17.8 25.2 17.5 20.5 25.2 19.7 24.5 19.1 23.6 18.9 13.7 23.1 18.1 13.0 21.9 17.1 12.2 21.2 10.0 Santa Maria 29.9 17.2 27.5 16.7 25.6 16.1 19.1 27.2 18.2 25.3 17.5 23.9 15.9 11.4 21.0 15.3 11.0 20.7 14.7 10.5 19.9 10.8 Stockton 38.0 20.8 36.1 20.1 34.7 19.6 21.8 35.5 21.0 34.5 20.2 33.1 16.5 11.8 25.3 15.7 11.2 25.3 15.1 10.7 24.8 16.9 Victorville, George AFB 38.4 18.2 36.7 18.1 35.6 17.7 20.8 31.3 20.1 31.3 19.3 31.1 18.1 14.5 25.5 16.2 12.8 26.0 14.9 11.8 25.4 15.7 COLORADO Alamosa 29.0 12.8 27.8 12.7 26.6 12.4 15.4 24.0 14.8 23.2 14.2 22.7 13.0 12.4 16.7 12.1 11.6 16.5 11.2 11.0 16.9 17.3 Colorado Springs 32.1 14.4 30.6 14.2 29.0 14.2 17.1 25.3 16.4 25.1 15.8 24.6 14.7 13.2 19.1 13.9 12.5 18.7 13.1 11.9 18.5 13.8 Craig 30.9 13.9 29.7 13.4 28.4 13.0 15.8 26.3 14.9 25.7 14.1 25.2 11.9 11.0 19.1 10.9 10.3 18.6 10.0 9.7 17.9 20.2 Denver 33.8 15.3 32.3 15.2 30.8 15.1 18.1 27.2 17.3 26.6 16.7 25.8 15.6 13.7 20.4 14.6 12.8 20.1 13.7 12.1 20.1 14.9 Eagle 31.2 14.7 29.9 14.1 28.4 13.7 16.6 26.8 15.7 25.6 15.0 24.6 13.7 12.5 17.8 12.7 11.7 18.1 11.6 10.8 18.3 20.1 Grand Junction 35.7 16.1 34.4 15.7 33.1 15.4 18.4 28.9 17.7 28.3 17.1 27.9 15.7 13.3 20.9 14.5 12.4 21.7 13.2 11.3 21.9 14.8 Limon 32.3 15.4 31.0 15.3 29.5 15.1 17.9 26.1 17.3 25.8 16.8 25.2 15.8 13.7 19.2 15.1 13.1 18.7 14.5 12.6 18.7 14.9 Pueblo 36.2 16.8 34.6 16.7 33.1 16.6 19.7 28.7 19.0 28.4 18.4 28.2 17.3 14.8 21.7 16.4 14.0 21.7 15.6 13.2 21.6 16.3 Trinidad 33.8 16.1 32.1 15.7 30.8 15.6 18.5 29.0 17.8 28.2 17.1 27.0 15.5 13.7 21.4 14.7 13.0 20.7 13.9 12.3 20.8 15.7 CONNECTICUT Bridgeport 30.2 22.8 28.8 22.1 27.6 21.5 24.3 28.1 23.5 27.1 22.8 26.2 23.2 18.0 26.2 22.4 17.1 25.6 21.7 16.4 24.9 7.8 Hartford, Brainard Field 32.9 23.0 31.2 22.1 29.7 21.2 24.3 30.8 23.4 28.9 22.6 27.5 22.5 17.3 27.2 21.8 16.5 26.3 20.9 15.7 25.6 11.6 Windsor Locks, Bradley Fld 33.2 22.7 31.2 21.8 29.7 21.0 24.2 30.6 23.3 28.9 22.5 27.5 22.2 17.0 27.2 21.5 16.3 26.1 20.8 15.6 25.2 11.6 DELAWARE Dover, AFB 33.7 24.5 31.9 23.9 30.6 23.6 26.2 30.9 25.5 30.0 24.7 29.0 25.0 20.1 28.8 24.3 19.3 27.8 23.6 18.4 27.3 9.0 Wilmington 32.9 23.8 31.5 23.1 30.1 22.6 25.3 30.6 24.6 29.3 23.8 28.5 23.9 18.8 27.9 23.1 17.9 27.1 22.4 17.2 26.6 9.4 FLORIDA Apalachicola 33.2 26.0 32.0 25.8 31.4 25.6 27.1 31.3 26.7 30.8 26.2 30.4 25.9 21.2 29.3 25.5 20.7 29.1 25.1 20.2 28.7 7.4 Cape Canaveral, NASA 33.5 25.5 32.2 25.4 31.6 25.1 26.7 31.1 26.2 30.8 25.9 30.4 25.5 20.7 29.1 25.1 20.2 28.7 24.7 19.7 28.2 8.9 Daytona Beach 33.2 25.0 32.2 24.8 31.3 24.8 26.3 31.1 25.8 30.6 25.6 30.2 25.0 20.1 29.1 24.6 19.6 28.7 24.2 19.1 28.3 8.6 Fort Lauderdale/Hollywood 33.3 25.5 32.2 25.7 31.8 25.4 27.0 31.1 26.6 30.6 26.2 30.3 25.7 21.0 29.2 25.5 20.7 29.0 25.1 20.2 28.8 6.3 Fort Myers 34.6 25.0 33.9 25.0 33.4 25.0 26.9 31.5 26.5 30.9 26.1 30.8 25.7 21.0 28.7 25.3 20.5 28.5 24.9 20.0 28.2 9.4 Gainesville 34.2 25.1 33.5 24.9 32.2 24.7 26.5 31.8 26.1 31.2 25.6 30.6 25.2 20.4 28.9 24.7 19.8 28.4 24.4 19.5 28.0 10.4 Homestead, AFB 33.1 26.1 32.1 26.0 31.6 25.7 27.2 31.4 26.8 31.1 26.4 30.7 26.0 21.4 30.3 25.5 20.7 29.8 25.1 20.2 29.6 6.5 Jacksonville, Cecil Field NAS 35.7 24.5 34.8 24.4 34.0 24.2 26.2 32.6 25.7 32.1 25.2 31.5 24.6 19.7 28.8 24.1 19.1 28.2 23.7 18.6 27.9 11.1 Jacksonville, Intl Airport 34.7 25.2 33.7 25.1 32.7 24.8 26.6 32.2 26.2 31.7 25.7 31.0 25.2 20.3 29.2 24.7 19.7 28.7 24.3 19.2 28.3 9.9 Jacksonville, Mayport Naval 34.8 25.3 33.6 25.4 32.2 25.0 27.1 31.8 26.6 31.4 26.1 31.0 25.7 21.0 29.9 25.2 20.3 29.6 24.8 19.8 29.2 8.5 Key West 32.4 26.1 31.9 26.0 31.4 25.8 27.0 30.8 26.7 30.6 26.4 30.4 25.9 21.3 29.6 25.6 20.8 29.4 25.2 20.4 29.2 4.5 Melbourne 33.8 26.3 32.9 26.2 31.8 26.0 27.7 31.7 27.2 31.1 26.8 30.6 26.6 22.2 30.0 26.1 21.5 29.7 25.6 20.9 29.4 8.5 Miami, Intl Airport 32.8 25.2 32.2 25.1 31.6 24.9 26.4 30.4 26.1 30.3 25.8 29.9 25.4 20.6 28.3 25.0 20.1 28.1 24.7 19.7 28.0 6.3 Miami, New Tamiami A 33.6 25.4 33.0 25.4 32.1 25.2 26.6 31.4 26.2 31.0 26.0 30.7 25.5 20.7 28.5 25.1 20.2 28.3 24.7 19.7 28.1 8.6 Milton, Whiting Field NAS 35.0 25.4 34.1 25.0 33.2 24.7 27.1 32.3 26.5 31.7 25.9 31.0 25.7 21.1 29.9 25.1 20.4 29.2 24.6 19.7 28.7 10.3 Orlando 34.4 24.6 33.7 24.6 33.1 24.4 26.2 31.2 25.9 31.0 25.6 30.6 25.1 20.3 28.2 24.7 19.8 27.8 24.4 19.4 27.4 9.2 Panama City, Tyndall AFB 32.9 26.3 31.8 26.0 31.1 25.9 28.1 31.0 27.5 30.5 27.0 29.9 27.1 22.9 30.1 26.5 22.0 29.4 26.0 21.4 29.1 6.8 Pensacola, Sherman AFB 34.1 25.5 33.3 25.3 32.1 25.3 27.2 31.5 26.6 31.0 26.1 30.9 26.0 21.4 29.6 25.3 20.5 29.4 24.7 19.7 29.3 8.5 Saint Petersburg 34.5 26.6 33.8 26.2 33.1 25.9 28.0 32.4 27.5 31.8 27.1 31.0 26.7 22.3 29.9 26.4 21.9 29.6 26.0 21.4 29.5 7.5 Sarasota/Bradenton 33.7 26.4 33.1 25.9 32.2 26.0 27.7 32.2 27.2 31.6 26.9 31.2 26.3 21.8 30.4 25.9 21.2 30.2 25.6 20.9 29.8 8.8 Tallahassee 34.8 24.7 33.8 24.5 32.9 24.2 26.4 31.6 25.9 31.2 25.6 30.6 25.2 20.3 28.3 24.7 19.7 27.8 24.3 19.3 27.5 10.3 Tampa, Intl Airport 33.6 25.1 32.9 25.1 32.3 24.9 26.7 31.2 26.2 31.2 25.8 30.7 25.3 20.5 29.2 24.9 20.0 28.7 24.5 19.5 28.5 8.3 Valparaiso, Eglin AFB 33.5 25.4 32.2 25.3 31.5 25.0 27.1 31.1 26.5 30.4 26.1 30.1 25.9 21.3 29.4 25.4 20.6 28.8 25.0 20.1 28.4 7.7 Vero Beach 33.4 25.2 32.2 25.5 31.7 25.2 26.5 31.3 26.1 31.0 25.9 30.7 25.1 20.2 29.2 24.8 19.9 28.9 24.6 19.6 28.8 8.7 DP = dew-point temperature, °C MWB = mean coincident wet-bulb temperature, °C MDB = mean coincident dry-bulb temperature, °C MWS = mean coincident wind speed, m/s StdD = standard deviation, °C HR = humidity ratio, grams of moisture per kilogram of dry air A = airport ANGB = Air National Guard Base MCAF = Marine Corps Air Facility NAF = Naval Air Facility NAWS = Naval Air Weapons Station RAF = Royal Air Force 27.10 2001 ASHRAE Fundamentals Handbook (SI) Table 1A Heating and Wind Design Conditions—United States Heating Dry Bulb Extreme Wind Speed, m/s Coldest Month MWS/PWD to DB Extr. Annual Daily Lat.
Long.
Elev., m StdP, kPa 0.4% 1% 99.6% 0.4% Mean DB StdD DB Station WMO# Dates 99.6% 99% 1% 2.5% 5% WS MDB WS MDB MWS PWD MWS PWD Max.
Min.
Max. Min.
1a 1b 1c 1d 1e 1f 1g 2a 2b 3a 3b 3c 4a 4b 4c 4d 5a 5b 5c 5d 6a 6b 6c 6d West Palm Beach 722030 26.68 80.12 6 101.25 6193 5.9 8.1 10.5 9.4 8.5 10.5 20.8 9.5 20.9 4.2 320 5.4 110 34.6 1.7 1.1 2.8 GEORGIA Albany 722160 31.53 84.18 59 100.62 8293 −3.0 −1.2 8.3 7.5 6.7 8.5 9.9 7.9 9.8 1.6 360 3.8 250 37.9 −8.5 1.2 4.0 Athens 723110 33.95 83.32 247 98.39 6193 −6.6 −4.2 8.6 7.6 6.7 8.9 4.6 8.1 4.6 4.2 290 3.9 270 36.8 −11.4 1.9 3.7 Atlanta 722190 33.65 84.42 315 97.60 6193 −7.9 −4.9 9.9 8.7 7.8 10.3 2.5 9.2 2.4 5.5 320 4.1 300 35.3 −12.7 1.9 4.1 Augusta 722180 33.37 81.97 45 100.79 6193 −6.1 −4.1 9.1 7.9 6.9 9.3 7.2 8.3 7.7 2.4 290 4.1 250 37.6 −10.4 2.1 3.1 Brunswick 722137 31.15 81.38 6 101.25 8293 −0.9 0.9 8.2 7.5 7.0 8.5 9.3 7.9 9.2 3.5 350 4.5 250 36.8 −5.8 1.4 4.3 Columbus, Fort Benning 722250 32.33 85.00 71 100.47 8293 −4.8 −2.6 7.1 5.9 5.0 7.8 7.7 6.6 7.5 1.4 320 2.2 240 37.8 −9.9 1.6 3.7 Columbus, Metro Airport 722255 32.52 84.93 121 99.88 6193 −4.8 −2.6 7.7 6.9 6.1 8.1 6.8 7.3 7.6 3.0 310 3.8 310 37.2 −10.1 1.3 3.4 Macon 722170 32.70 83.65 110 100.01 6193 −5.1 −2.8 8.5 7.6 6.7 8.8 7.9 7.9 7.3 3.1 320 4.2 270 37.8 −10.3 1.5 3.6 Marietta, Dobbins AFB 722270 33.92 84.52 326 97.47 8293 −6.2 −3.2 8.1 7.0 6.0 9.0 1.8 8.0 3.4 4.0 340 2.5 300 36.3 −11.1 2.0 3.7 Rome 723200 34.35 85.17 196 98.99 8293 −9.4 −6.0 6.1 5.3 4.5 6.3 5.3 5.6 5.4 2.3 340 2.6 270 36.7 −15.6 2.1 3.9 Savannah 722070 32.13 81.20 15 101.14 6193 −3.5 −1.6 8.8 7.7 6.9 9.5 9.3 8.4 9.6 3.0 270 4.2 270 36.9 −7.7 1.7 3.0 Valdosta, Moody AFB 747810 30.97 83.20 71 100.47 8293 −0.9 1.0 6.9 5.9 5.2 7.3 11.7 6.2 11.3 1.7 360 2.4 300 37.3 −5.9 1.4 4.2 Valdosta, Regional Airport 722166 30.78 83.28 62 100.58 8293 −2.5 −0.8 7.6 6.8 6.1 7.9 12.8 7.1 13.1 1.7 340 3.4 300 37.3 −8.3 1.8 4.3 Waycross 722130 31.25 82.40 46 100.77 8293 −1.7 −0.1 7.0 6.2 5.5 7.1 11.0 6.3 11.2 1.6 250 3.3 240 36.5 −6.2 3.9 4.2 HAWAII Ewa, Barbers Point NAS 911780 21.32 158.07 15 101.14 8293 14.9 16.2 8.9 7.9 7.1 10.0 22.8 8.5 23.8 2.2 40 4.8 60 33.7 1.5 0.9 11.9 Hilo 912850 19.72 155.07 11 101.19 6193 16.3 17.1 8.3 7.3 6.4 9.3 24.5 8.1 24.4 3.2 230 5.4 110 31.3 14.4 0.9 1.0 Honolulu 911820 21.35 157.93 5 101.26 6193 16.0 17.2 10.4 9.5 8.7 10.4 23.5 9.3 23.9 2.3 320 6.6 60 32.5 14.2 1.1 1.2 Kahului 911900 20.90 156.43 20 101.08 6193 14.9 15.9 12.2 11.3 10.6 14.4 24.2 12.7 24.3 2.5 160 8.3 50 33.2 12.4 0.8 2.4 Kaneohe, MCAS 911760 21.45 157.77 3 101.29 8293 19.3 19.9 9.0 8.2 7.4 9.5 23.2 8.5 23.1 3.3 190 4.5 70 31.3 4.4 0.8 16.1 Lihue 911650 21.98 159.35 45 100.79 6193 15.5 16.6 11.7 10.5 9.6 11.1 22.5 10.2 22.7 3.6 270 6.0 60 30.3 13.6 0.8 1.7 Molokai 911860 21.15 157.10 137 99.69 8293 15.3 16.3 10.5 9.9 9.3 10.0 23.6 9.2 23.5 1.7 70 5.9 60 33.4 6.1 2.2 12.2 IDAHO Boise 726810 43.57 116.22 874 91.26 6193 −16.8 −12.6 10.5 9.2 8.0 9.8 2.6 8.5 2.7 2.5 130 4.7 320 39.4 −19.8 1.5 5.1 Burley 725867 42.55 113.77 1265 87.02 8293 −20.4 −16.7 10.3 9.4 8.4 10.4 −1.3 9.7 −2.3 3.0 60 3.6 280 36.5 −23.9 2.2 4.7 Idaho Falls 725785 43.52 112.07 1445 85.13 8293 −24.4 −20.9 12.2 10.4 9.3 12.4 0.1 10.4 −1.5 3.3 360 5.2 180 35.6 −28.9 2.0 5.0 Lewiston 727830 46.38 117.02 438 96.17 8293 −14.2 −9.7 9.1 7.7 6.3 10.9 3.6 8.9 4.5 2.2 280 3.1 310 39.4 −16.3 1.5 5.5 Mountain Home, AFB 726815 43.05 115.87 913 90.83 8293 −17.9 −14.8 10.4 9.2 8.1 10.4 0.3 9.4 −0.3 1.0 90 3.5 350 40.7 −21.1 1.8 4.7 Mullan 727836 47.47 115.80 1011 89.75 8293 −18.5 −13.8 4.6 4.3 3.9 4.7 −7.7 4.2 −6.2 0.8 10 1.8 10 33.1 −21.7 1.1 4.4 Pocatello 725780 42.92 112.60 1365 85.97 6193 −21.6 −17.7 13.1 11.2 10.1 13.4 2.0 12.0 2.1 2.5 50 5.0 250 36.7 −26.0 1.3 5.1 ILLINOIS Belleville, Scott AFB 724338 38.55 89.85 138 99.68 8293 −16.2 −12.4 9.3 8.0 6.9 10.2 0.0 9.1 −0.8 3.2 360 3.1 190 37.5 −19.5 1.7 4.0 Chicago, Meigs Field 725340 41.78 87.75 190 99.06 8293 −20.0 −16.1 10.4 9.6 8.7 11.4 −8.2 10.1 −1.0 5.4 240 5.6 220 36.0 −23.4 1.8 4.5 Chicago, O’Hare Intl A 725300 41.98 87.90 205 98.89 6193 −21.2 −18.1 11.7 10.4 9.2 12.0 −4.6 10.4 −4.9 4.6 270 5.4 230 35.4 −24.6 1.6 3.6 Decatur 725316 39.83 88.87 208 98.85 8293 −19.0 −15.9 10.8 9.8 9.1 12.0 −4.2 10.5 −2.8 5.7 310 5.2 210 37.0 −23.2 3.2 4.0 Glenview, NAS 725306 42.08 87.82 199 98.96 8293 −19.7 −15.7 9.7 8.4 7.4 10.1 −8.4 9.0 −4.0 4.7 250 4.3 240 36.4 −23.6 1.7 4.3 Marseilles 744600 41.37 88.68 225 98.65 8293 −20.4 −17.3 11.4 10.0 8.9 12.6 −7.6 11.2 −6.0 5.2 290 4.5 250 35.4 −24.1 2.2 3.3 Moline/Davenport IA 725440 41.45 90.52 181 99.17 6193 −22.4 −19.3 11.6 10.1 9.0 12.7 −8.7 10.8 −8.0 4.2 290 5.2 200 36.2 −25.7 1.5 3.3 Peoria 725320 40.67 89.68 202 98.92 6193 −21.1 −18.1 11.0 9.7 8.7 11.7 −8.7 10.1 −7.2 4.1 290 4.9 180 35.4 −24.6 1.8 3.4 Quincy 724396 39.95 91.20 234 98.55 8293 −19.9 −16.6 10.8 9.6 8.8 12.5 −5.0 10.7 −5.6 5.4 330 5.2 210 36.2 −23.3 2.0 4.5 Rockford 725430 42.20 89.10 226 98.64 6193 −23.1 −20.1 11.4 10.0 9.1 11.4 −7.6 10.0 −6.9 3.9 290 5.6 200 35.1 −26.6 1.7 3.1 Springfield 724390 39.85 89.67 187 99.10 6193 −19.7 −16.6 11.7 10.4 9.4 12.0 −3.8 10.5 −2.8 4.5 270 5.3 230 36.2 −23.9 1.6 3.1 West Chicago 725305 41.92 88.25 231 98.58 8293 −21.6 −18.0 10.3 9.3 8.4 11.1−10.3 10.1 −6.6 5.1 290 4.8 240 35.6 −25.5 1.8 4.3 INDIANA Evansville 724320 38.05 87.53 118 99.92 6193 −16.2 −12.8 9.8 8.6 7.7 9.9 0.3 8.8 1.3 3.1 320 4.2 240 36.3 −20.2 1.5 4.7 Fort Wayne 725330 41.00 85.20 252 98.33 6193 −19.9 −16.9 11.3 10.0 9.1 12.2 −7.1 10.5 −5.7 4.4 250 5.2 230 34.8 −23.9 2.0 2.9 Indianapolis 724380 39.73 86.27 246 98.40 6193 −19.2 −16.1 10.8 9.5 8.3 11.2 −3.1 9.9 −2.8 3.7 230 4.8 230 34.6 −23.2 1.6 3.8 Lafayette, Purdue Univ 724386 40.42 86.93 185 99.12 8293 −20.5 −16.3 10.0 9.0 8.2 10.7 −3.3 9.7 −2.6 4.2 270 5.3 220 36.2 −24.0 2.1 4.3 Peru, Grissom AFB 725335 40.65 86.15 247 98.39 8293 −19.2 −15.3 10.6 9.3 8.2 12.9 −6.8 10.5 −5.4 4.9 270 4.1 210 35.8 −22.5 2.1 4.1 South Bend 725350 41.70 86.32 236 98.52 6193 −18.9 −16.2 11.3 10.0 9.0 11.6 −5.7 10.3 −4.9 5.7 230 5.3 230 34.7 −23.1 1.8 3.2 Terre Haute 724373 39.45 87.32 178 99.20 8293 −19.2 −15.1 10.1 9.1 8.2 10.4 −0.3 9.5 0.0 3.5 150 4.7 230 35.5 −23.3 1.8 4.4 IOWA Burlington 725455 40.78 91.13 213 98.79 8293 −20.0 −17.4 9.4 8.4 7.7 10.5 −11.0 9.5 −7.7 3.9 310 4.9 200 36.4 −23.5 2.2 3.8 Cedar Rapids 725450 41.88 91.70 265 98.18 8293 −24.0 −20.6 11.2 9.9 9.0 13.1 −11.3 11.6 −9.9 4.6 300 4.7 180 35.5 −26.3 2.0 3.0 Des Moines 725460 41.53 93.65 294 97.84 6193 −22.7 −19.9 12.1 10.6 9.4 12.3 −9.8 10.8 −7.2 4.9 320 5.5 180 36.5 −26.2 1.9 2.8 Fort Dodge 725490 42.55 94.18 355 97.13 8293 −25.0 −21.9 11.9 10.4 9.3 12.9−12.2 11.5 −12.3 4.7 340 5.0 190 35.5 −27.4 2.7 2.7 Lamoni 725466 40.62 93.95 342 97.28 8293 −21.1 −17.7 8.7 7.6 6.9 9.5 −5.0 8.5 −6.9 3.2 320 3.9 210 37.0 −24.3 2.4 3.8 Mason City 725485 43.15 93.33 370 96.96 6193 −26.2 −23.5 12.2 10.4 9.7 13.3−12.6 11.8 −11.3 5.3 300 6.3 200 35.9 −30.4 2.0 6.3 Ottumwa 725465 41.10 92.45 258 98.26 8293 −20.6 −17.9 13.0 11.6 10.4 13.7 −6.5 12.6 −4.5 5.9 320 6.5 200 36.7 −24.2 2.2 3.8 Sioux City 725570 42.40 96.38 336 97.35 6193 −23.8 −21.1 12.9 11.2 10.0 14.0 −9.8 12.3 −8.8 4.7 320 6.4 180 37.2 −27.6 2.0 2.6 Spencer 726500 43.17 95.15 408 96.52 8293 −26.5 −23.8 10.6 9.7 8.8 11.3−10.3 10.1 −10.7 4.6 300 5.4 180 37.3 −29.1 3.5 2.2 Waterloo 725480 42.55 92.40 268 98.15 6193 −25.7 −22.9 12.1 10.8 9.7 12.8−12.1 11.2 −10.4 4.1 300 5.9 180 35.5 −29.1 1.9 3.3 KANSAS Concordia 724580 39.55 97.65 452 96.01 8293 −19.9 −16.0 12.3 11.0 10.0 12.4 0.0 11.1 0.1 6.0 360 7.2 10 40.1 −22.0 2.2 5.2 Dodge City 724510 37.77 99.97 790 92.19 6193 −17.7 −14.7 13.3 11.9 10.6 13.8 −0.7 12.2 −0.2 5.7 10 7.7 200 39.9 −21.1 1.6 3.1 Ft Riley, Marshall AAF 724550 39.05 96.77 325 97.48 8293 −19.0 −14.9 9.4 8.2 7.2 9.1 3.9 7.9 3.0 2.2 350 4.2 180 39.8 −20.7 1.7 5.0 Garden City 724515 37.93 100.72 881 91.18 8293 −19.2 −15.5 13.3 11.8 10.3 13.0 0.1 11.0 1.2 5.3 360 7.2 190 39.9 −22.8 1.5 3.6 Goodland 724650 39.37 101.70 1124 88.53 6193 −19.7 −16.5 14.2 12.3 10.8 13.7 −2.7 11.9 −1.3 5.4 270 5.9 180 38.8 −23.9 1.6 3.7 WMO# = World Meteorological Organization number AFB = Air Force Base NAS = Naval Air Station WSO = Weather Service Office AAF = Army Air Field MCAS = Marine Corps Air Station DB = dry-bulb temperature, °C Lat. = North latitude ° Long. = West longitude ° Elev. = elevation, m StdP = standard pressure at station elevation, kPa WS = wind speed, m/s PWD = prevailing wind direction, ° True Climatic Design Information 27.11 Table 1B Cooling and Dehumidification Design Conditions—United States Station Cooling DB/MWB Evaporation WB/MDB Dehumidification DP/MDB and HR Range of DB 0.4% 1% 2% 0.4% 1% 2% 0.4% 1% 2% DB MWB DB MWB DB MWB WB MDB WB MDB WB MDB DP HR MDB DP HR MDB DP HR MDB 1 2a 2b 2c 2d 2e 2f 3a 3b 3c 3d 3e 3f 4a 4b 4c 4d 4e 4f 4g 4h 4i 5 West Palm Beach 32.9 25.3 32.2 25.3 31.6 25.2 26.4 31.1 26.1 30.9 25.8 30.4 25.2 20.4 28.9 24.8 19.9 28.7 24.5 19.5 28.5 7.3 GEORGIA Albany 35.8 24.6 34.8 24.4 33.9 24.1 26.2 32.5 25.7 31.7 25.3 31.1 24.9 20.1 28.2 24.4 19.5 27.9 24.0 19.0 27.4 11.0 Athens 34.6 23.6 33.3 23.7 32.0 23.2 25.3 31.6 24.7 30.8 24.3 30.1 23.6 19.0 27.8 23.1 18.4 27.4 22.7 17.9 26.9 10.2 Atlanta 33.9 23.8 32.6 23.4 31.3 22.8 25.1 31.2 24.4 30.3 23.9 29.7 23.4 19.0 27.8 22.8 18.3 27.1 22.3 17.7 26.6 9.6 Augusta 35.7 24.4 34.3 24.3 33.2 23.9 25.9 32.8 25.4 31.9 24.9 31.1 24.3 19.3 28.6 23.7 18.6 28.1 23.2 18.1 27.7 11.2 Brunswick 34.1 25.7 32.6 26.1 31.3 25.4 27.1 31.6 26.6 31.0 26.2 30.4 25.7 21.0 29.9 25.4 20.6 29.4 25.0 20.1 28.9 8.0 Columbus, Fort Benning 35.9 24.5 34.7 24.4 33.6 24.3 26.5 32.6 25.9 31.9 25.4 31.3 25.0 20.3 29.2 24.4 19.5 28.5 24.0 19.0 27.9 11.4 Columbus, Metro Airport 35.2 24.2 34.0 23.9 33.0 23.7 25.9 31.9 25.4 31.2 25.0 30.6 24.5 19.8 27.9 24.0 19.2 27.5 23.6 18.6 27.2 10.0 Macon 35.7 24.3 34.4 24.0 33.3 23.8 25.9 32.7 25.4 31.9 25.0 31.1 24.2 19.4 28.4 23.8 18.9 27.9 23.4 18.4 27.5 10.7 Marietta, Dobbins AFB 34.4 23.4 33.0 23.2 31.5 23.1 25.0 31.3 24.5 30.6 24.0 29.9 23.6 19.2 27.6 23.0 18.5 27.1 22.2 17.6 26.1 9.5 Rome 35.5 23.4 34.2 23.5 32.9 23.3 25.5 32.1 25.0 31.5 24.5 30.9 23.9 19.2 28.6 23.4 18.6 28.1 22.9 18.1 28.1 11.5 Savannah 35.0 24.9 33.8 24.6 32.6 24.4 26.3 32.2 25.8 31.5 25.3 30.7 24.8 19.8 28.7 24.3 19.3 28.1 23.9 18.8 27.8 9.7 Valdosta, Moody AFB 35.2 25.1 34.3 24.8 33.5 24.5 26.6 32.9 26.1 31.9 25.6 31.2 25.0 20.3 29.5 24.6 19.8 28.8 24.2 19.3 28.5 9.9 Valdosta, Regional Airport 35.1 25.0 34.2 24.7 33.4 24.4 26.5 32.3 26.0 31.7 25.6 30.9 25.2 20.5 28.6 24.7 19.9 28.0 24.3 19.4 27.6 10.8 Waycross 35.7 24.5 34.7 24.2 33.7 24.0 25.8 32.8 25.4 32.2 25.0 31.5 24.1 19.1 28.9 23.7 18.6 28.5 23.3 18.2 28.2 11.3 HAWAII Ewa, Barbers Point NAS 33.1 22.6 32.0 22.5 31.5 22.5 24.6 29.9 24.1 29.9 23.7 29.6 23.2 18.0 28.1 22.1 16.8 27.8 21.5 16.2 27.5 8.8 Hilo 29.6 23.3 29.1 23.1 28.5 22.9 24.7 27.8 24.2 27.4 23.8 27.2 23.8 18.6 26.3 23.3 18.1 25.9 22.8 17.6 25.7 7.4 Honolulu 31.8 22.9 31.3 22.7 30.7 22.6 24.4 29.1 23.9 28.9 23.5 28.6 23.1 17.8 26.6 22.4 17.2 26.4 21.9 16.6 26.2 6.8 Kahului 31.7 23.3 31.1 23.1 30.4 22.8 24.7 29.6 24.2 29.2 23.8 28.9 23.3 18.1 26.9 22.7 17.4 26.8 22.2 16.9 26.7 8.7 Kaneohe, MCAS 29.9 23.9 29.4 23.6 29.1 23.3 25.4 28.0 24.9 27.9 24.5 27.7 24.7 19.7 27.1 24.1 19.0 27.0 23.5 18.3 26.9 4.1 Lihue 29.7 23.8 29.2 23.6 28.8 23.3 24.9 28.3 24.5 27.8 24.1 27.6 23.9 18.9 26.9 23.4 18.3 26.6 22.9 17.8 26.3 5.3 Molokai 31.1 22.7 30.6 22.6 30.1 22.2 24.4 29.2 23.9 28.5 23.5 28.2 23.2 18.3 26.4 22.7 17.7 26.3 21.9 16.8 26.0 7.4 IDAHO Boise 35.8 17.4 34.2 16.9 32.5 16.4 18.8 32.2 17.9 31.5 17.2 30.4 14.3 11.3 22.3 12.9 10.3 21.8 11.7 9.5 21.8 16.8 Burley 34.2 17.2 32.0 16.5 30.4 16.2 19.2 30.0 18.2 28.8 17.3 28.1 15.6 12.9 23.9 14.4 12.0 22.2 13.3 11.1 22.1 16.1 Idaho Falls 33.2 16.2 31.4 15.7 30.0 15.5 17.9 28.7 17.1 27.7 16.3 27.1 14.7 12.5 21.4 13.6 11.6 20.6 12.0 10.4 20.1 18.9 Lewiston 35.9 18.2 34.1 17.6 32.0 17.0 19.2 32.8 18.4 31.9 17.6 30.3 14.6 10.9 22.4 13.6 10.2 21.5 12.2 9.3 21.5 14.7 Mountain Home, AFB 37.1 17.4 35.3 16.9 33.7 16.3 18.8 32.7 17.9 32.6 17.1 31.8 14.2 11.3 21.4 12.4 10.0 20.5 11.2 9.2 21.6 18.2 Mullan 30.4 16.7 28.9 16.2 26.8 15.6 18.1 27.4 17.2 26.1 16.4 24.9 15.3 12.3 20.4 14.2 11.4 19.9 13.2 10.7 19.1 15.6 Pocatello 33.9 15.9 32.3 15.5 30.7 15.1 17.7 29.0 16.9 28.4 16.1 27.8 14.1 11.9 20.8 12.8 10.9 21.1 11.5 10.0 20.4 17.8 ILLINOIS Belleville, Scott AFB 35.1 25.3 33.9 24.8 32.2 24.6 26.6 33.1 25.8 32.1 25.2 31.1 24.8 20.2 30.6 24.2 19.4 29.7 23.6 18.7 29.0 11.0 Chicago, Meigs Field 33.5 23.6 31.5 22.7 30.1 21.7 25.2 31.1 24.2 29.5 23.3 28.4 23.6 18.9 28.7 22.2 17.3 26.9 21.4 16.4 26.7 8.9 Chicago, O’Hare Intl A 32.8 23.6 31.3 22.8 29.7 21.9 25.1 31.0 24.1 29.5 23.1 28.1 23.3 18.6 28.9 22.4 17.5 27.8 21.4 16.4 26.6 10.9 Decatur 34.5 24.7 32.9 24.1 31.2 23.5 26.1 32.3 25.3 31.4 24.5 30.0 24.5 20.0 30.1 23.7 19.0 29.0 22.9 18.1 28.2 11.1 Glenview, NAS 34.1 23.8 31.9 22.8 30.4 21.8 25.3 32.2 24.3 30.3 23.2 28.7 23.4 18.6 29.5 22.1 17.2 28.0 21.2 16.2 27.3 9.8 Marseilles 33.8 23.6 31.7 23.0 30.2 21.9 25.5 31.6 24.4 30.1 23.5 28.8 23.9 19.3 29.4 22.8 18.0 27.6 21.6 16.7 27.2 10.8 Moline/Davenport IA 33.9 24.3 32.2 23.4 30.7 22.7 25.7 32.0 24.7 30.5 23.9 29.3 23.9 19.2 29.5 23.0 18.1 28.3 22.2 17.2 27.5 11.1 Peoria 33.3 24.3 31.7 23.4 30.2 22.8 25.8 31.6 24.9 30.1 23.9 29.0 24.1 19.5 29.2 23.3 18.5 28.3 22.4 17.6 27.2 10.8 Quincy 34.6 24.4 32.9 24.1 30.9 23.2 25.7 31.7 25.0 31.2 24.2 29.5 24.2 19.7 28.8 23.5 18.8 28.0 22.8 18.0 27.6 10.5 Rockford 32.6 23.5 31.0 22.6 29.6 21.7 25.1 30.7 24.1 29.2 23.1 27.8 23.5 18.8 28.8 22.5 17.7 27.3 21.5 16.6 26.2 11.0 Springfield 34.1 24.5 32.6 24.0 31.2 23.2 26.1 31.9 25.2 30.9 24.4 29.5 24.5 19.9 29.8 23.6 18.9 28.7 22.8 17.9 27.6 10.8 West Chicago 33.0 23.7 31.2 23.1 29.9 22.4 25.7 31.0 24.7 29.7 23.6 28.3 24.2 19.7 29.3 23.3 18.6 28.1 21.9 17.0 26.8 11.0 INDIANA Evansville 34.4 24.7 33.1 24.2 31.9 23.7 26.1 32.3 25.4 31.4 24.7 30.4 24.4 19.6 29.8 23.7 18.8 28.9 23.0 18.0 28.1 11.0 Fort Wayne 32.4 23.2 30.9 22.6 29.6 21.8 24.9 30.2 24.0 29.1 23.1 27.7 23.4 18.7 28.2 22.4 17.7 27.3 21.6 16.7 26.2 11.1 Indianapolis 32.7 24.1 31.3 23.4 30.1 22.7 25.6 31.1 24.8 29.7 24.0 28.6 24.1 19.6 28.7 23.3 18.7 28.0 22.6 17.8 26.9 10.5 Lafayette, Purdue Univ 34.1 23.8 32.2 24.0 30.9 22.6 25.9 31.6 25.0 30.2 24.1 28.9 24.4 19.8 29.5 23.6 18.8 28.3 22.8 17.9 27.5 11.6 Peru, Grissom AFB 33.7 24.1 31.7 23.7 30.3 22.7 26.1 31.6 25.1 29.9 24.1 28.4 24.7 20.3 29.7 23.7 19.1 28.3 22.9 18.2 27.3 10.3 South Bend 32.2 23.0 30.8 22.4 29.3 21.6 24.8 30.2 23.8 28.9 22.9 27.4 23.2 18.5 28.1 22.3 17.5 26.8 21.5 16.6 25.7 10.3 Terre Haute 33.8 24.7 32.1 24.6 30.9 23.7 26.5 31.9 25.6 30.6 24.7 29.4 25.0 20.5 29.9 24.2 19.5 28.9 23.5 18.7 27.9 10.9 IOWA Burlington 34.5 24.4 32.8 24.2 30.9 23.0 25.7 31.9 25.0 31.2 24.2 29.7 24.1 19.5 29.3 23.4 18.7 28.6 22.5 17.7 27.8 10.4 Cedar Rapids 33.9 24.1 31.8 23.3 30.2 22.4 25.5 31.9 24.5 30.2 23.6 28.8 23.9 19.4 28.8 23.1 18.4 28.3 21.9 17.1 26.8 11.1 Des Moines 34.1 24.2 32.3 23.4 30.7 22.7 25.3 31.8 24.6 30.8 23.7 29.6 23.6 19.0 29.4 22.7 18.0 28.4 21.8 17.1 27.4 10.3 Fort Dodge 33.4 23.7 31.3 22.8 29.8 21.8 25.0 31.3 24.0 29.8 23.1 28.4 23.4 19.0 29.0 22.1 17.5 27.9 21.2 16.6 26.3 10.3 Lamoni 35.4 23.2 33.4 23.2 31.4 22.5 25.1 31.7 24.4 30.7 23.7 29.5 23.5 19.1 28.3 22.7 18.2 27.6 21.7 17.1 26.8 10.5 Mason City 32.8 23.4 31.1 22.6 29.5 21.9 25.2 30.8 24.1 29.4 23.1 27.8 23.6 19.3 28.6 22.5 18.0 27.5 21.5 16.9 26.4 11.6 Ottumwa 35.0 23.9 33.2 23.7 31.1 22.9 25.5 32.1 24.7 31.0 24.0 29.8 23.9 19.4 28.9 23.2 18.5 28.4 22.1 17.3 27.3 10.4 Sioux City 34.2 23.8 32.4 23.3 30.8 22.4 25.6 31.7 24.6 30.7 23.7 29.6 23.7 19.3 29.7 22.7 18.2 28.8 21.8 17.1 27.7 11.3 Spencer 33.0 23.7 31.1 22.7 29.7 21.5 24.9 30.9 24.0 29.8 23.0 28.0 23.4 19.1 29.0 22.1 17.6 27.5 21.2 16.7 25.9 11.2 Waterloo 32.9 23.6 31.2 22.7 29.7 21.9 25.1 30.7 24.1 29.6 23.2 28.2 23.4 18.9 28.6 22.4 17.7 27.6 21.5 16.7 26.5 11.1 KANSAS Concordia 37.9 23.0 35.5 22.5 33.7 22.4 25.0 32.1 24.2 31.7 23.5 31.2 23.2 19.0 28.9 22.0 17.6 27.5 21.3 16.9 27.1 12.5 Dodge City 37.8 21.2 36.2 20.8 34.4 20.6 23.2 32.4 22.5 31.9 21.9 31.0 20.8 17.1 26.1 20.1 16.3 25.4 19.4 15.6 25.2 13.5 Ft Riley, Marshall AAF 37.2 23.9 35.4 23.2 33.7 23.2 25.6 32.5 24.8 32.1 24.1 31.3 23.9 19.5 29.8 23.0 18.5 28.6 21.9 17.2 28.0 12.6 Garden City 38.0 20.6 36.0 20.6 34.5 20.6 22.8 31.9 22.2 31.7 21.6 31.3 20.4 16.8 26.1 19.7 16.1 25.6 19.1 15.5 25.1 15.3 Goodland 36.1 18.8 34.3 18.7 32.6 18.5 21.2 29.7 20.6 29.0 19.9 28.8 19.1 15.9 23.5 18.2 15.1 23.0 17.4 14.3 22.8 14.7 DP = dew-point temperature, °C MWB = mean coincident wet-bulb temperature, °C MDB = mean coincident dry-bulb temperature, °C MWS = mean coincident wind speed, m/s StdD = standard deviation, °C HR = humidity ratio, grams of moisture per kilogram of dry air A = airport ANGB = Air National Guard Base MCAF = Marine Corps Air Facility NAF = Naval Air Facility NAWS = Naval Air Weapons Station RAF = Royal Air Force 27.12 2001 ASHRAE Fundamentals Handbook (SI) Table 1A Heating and Wind Design Conditions—United States Heating Dry Bulb Extreme Wind Speed, m/s Coldest Month MWS/PWD to DB Extr. Annual Daily Lat.
Long.
Elev., m StdP, kPa 0.4% 1% 99.6% 0.4% Mean DB StdD DB Station WMO# Dates 99.6% 99% 1% 2.5% 5% WS MDB WS MDB MWS PWD MWS PWD Max.
Min.
Max. Min.
1a 1b 1c 1d 1e 1f 1g 2a 2b 3a 3b 3c 4a 4b 4c 4d 5a 5b 5c 5d 6a 6b 6c 6d Russell 724585 38.87 98.82 568 94.69 8293 −20.1 −16.0 13.0 11.4 10.3 12.9 0.3 11.2 1.5 5.1 10 7.3 190 40.5 −22.1 2.0 4.7 Salina 724586 38.80 97.65 388 96.75 8293 −19.4 −15.4 11.9 10.4 9.7 12.5 0.5 10.6 0.9 4.7 360 6.7 180 41.0 −21.6 1.3 5.5 Topeka 724560 39.07 95.62 270 98.12 6193 −18.8 −15.6 11.2 10.0 8.8 11.3 −2.3 9.9 −1.8 4.1 320 5.5 180 38.0 −22.4 2.1 4.1 Wichita, Airport 724500 37.65 97.43 408 96.52 6193 −16.6 −13.3 12.8 11.3 10.2 12.6 −1.0 11.4 −0.8 5.7 360 7.2 200 40.4 −19.7 1.6 3.5 Wichita, McConnell AFB 724505 37.62 97.27 418 96.40 8293 −16.5 −12.3 11.3 10.1 9.1 11.3 3.3 10.1 2.4 5.1 360 5.4 190 40.7 −18.3 1.5 4.3 KENTUCKY Bowling Green 746716 36.97 86.42 167 99.33 8293 −14.1 −10.2 9.1 8.3 7.6 9.5 4.3 8.5 4.4 2.6 220 4.1 230 36.1 −18.9 1.8 5.7 Covington/Cincinnati Airport 724210 39.05 84.67 267 98.16 6193 −17.5 −14.1 9.9 8.8 7.9 10.9 −1.4 9.6 0.4 4.2 250 4.6 230 35.1 −21.6 1.7 4.7 Fort Campbell, AAF 746710 36.67 87.50 174 99.25 8293 −12.6 −9.4 8.3 7.2 6.2 8.9 4.4 7.8 6.1 1.8 330 2.6 240 36.4 −17.6 1.7 5.3 Fort Knox, Godman AAF 724240 37.90 85.97 230 98.59 8293 −12.8 −9.7 7.8 6.6 5.7 8.2 5.4 7.3 4.0 1.7 290 2.5 270 36.0 −17.8 2.3 4.4 Jackson 724236 37.60 83.32 421 96.37 8293 −14.0 −10.1 7.4 6.4 5.9 8.1 6.6 7.2 4.6 3.2 230 2.8 230 34.4 −19.6 1.5 5.2 Lexington 724220 38.03 84.60 301 97.76 6193 −15.8 −12.4 9.5 8.3 7.5 10.1 3.3 9.0 3.1 3.8 270 4.0 240 34.5 −20.1 1.9 4.6 Louisville 724230 38.18 85.73 149 99.55 6193 −14.5 −11.4 9.7 8.6 7.7 10.0 4.4 8.9 1.1 4.3 290 4.5 250 35.6 −18.3 1.7 4.4 Paducah 724350 37.07 88.77 126 99.82 8293 −13.9 −10.4 9.6 8.4 7.6 9.8 7.3 8.7 5.7 3.6 40 3.9 180 36.6 −18.5 1.6 5.2 LOUISIANA Alexandria, England AFB 747540 31.33 92.55 27 101.00 8293 −3.0 −1.1 7.2 6.0 5.2 7.5 11.9 6.5 9.4 3.0 360 1.5 180 36.4 −6.9 1.2 3.5 Baton Rouge 722317 30.53 91.15 21 101.07 6193 −3.0 −0.9 9.1 8.0 7.1 9.2 8.6 8.3 9.2 3.4 360 3.6 270 35.9 −6.7 1.2 3.0 Bossier City, Barksdale AFB 722485 32.50 93.67 51 100.71 8293 −5.5 −2.8 8.2 7.1 6.2 8.3 9.4 7.3 10.5 3.2 360 2.2 180 37.3 −9.2 1.3 3.7 Lafayette 722405 30.20 91.98 13 101.17 8293 −2.4 −0.2 9.2 8.2 7.3 9.2 12.5 8.3 11.6 4.1 10 3.5 200 36.2 −7.1 0.9 4.5 Lake Charles 722400 30.12 93.22 10 101.20 6193 −1.8 0.1 9.7 8.6 7.7 10.5 9.8 9.3 9.6 4.2 20 3.8 230 35.6 −5.1 1.3 2.6 Leesville, Fort Polk 722390 31.05 93.20 100 100.13 8293 −2.9 −0.9 7.0 6.0 5.2 7.2 10.3 6.1 11.1 2.0 20 1.7 180 36.8 −6.7 1.1 3.3 Monroe 722486 32.52 92.03 24 101.04 8293 −5.7 −2.9 8.5 7.7 6.8 8.8 10.1 8.0 8.5 3.8 10 3.2 230 37.3 −8.6 1.0 4.7 New Orleans, Intl Airport 722310 29.98 90.25 9 101.22 6193 −1.3 0.8 9.4 8.3 7.5 9.4 8.8 8.4 9.2 3.3 340 3.5 360 35.6 −4.8 1.1 2.9 New Orleans, Lakefront A 722315 30.05 90.03 3 101.29 8293 1.8 3.9 9.6 8.6 7.9 9.5 9.6 8.9 9.9 6.4 360 3.8 300 34.4 −6.3 4.5 6.9 Shreveport 722480 32.47 93.82 79 100.38 6193 −5.5 −3.2 9.0 7.9 7.1 9.7 7.8 8.5 8.9 3.9 360 3.8 180 37.4 −8.9 1.7 3.1 MAINE Augusta 726185 44.32 69.80 107 100.05 8293 −19.7 −17.2 10.2 9.3 8.4 11.1 −6.5 10.0 −5.8 4.6 320 5.1 210 33.8 −23.4 1.7 1.9 Bangor 726088 44.80 68.83 59 100.62 8293 −21.5 −19.0 9.7 8.4 7.8 10.7 −7.9 9.4 −6.7 2.9 300 4.6 240 34.4 −26.6 1.6 3.3 Brunswick, NAS 743920 43.88 69.93 23 101.05 8293 −19.1 −16.4 9.0 7.8 6.8 9.5 −2.8 8.3 −3.7 1.6 340 3.8 190 35.5 −24.6 4.4 3.4 Caribou 727120 46.87 68.02 190 99.06 6193 −25.8 −23.2 12.3 10.8 9.6 13.3−10.4 11.8 −11.8 4.4 270 5.6 250 32.2 −30.7 1.6 2.5 Limestone, Loring AFB 727125 46.95 67.88 227 98.63 8293 −25.1 −22.7 10.2 8.9 7.9 11.1 −11.3 9.8 −11.6 3.1 300 3.9 260 32.7 −28.8 1.3 1.6 Portland 726060 43.65 70.32 19 101.10 6193 −19.6 −16.7 10.7 9.3 8.2 10.8 −3.6 9.4 −3.7 3.1 320 5.3 270 33.8 −24.9 2.0 3.1 MARYLAND Camp Springs, Andrews AFB 745940 38.82 76.87 86 100.30 8293 −10.3 −7.6 9.4 8.1 7.2 10.4 −1.0 9.3 0.1 3.2 350 3.8 230 36.9 −15.5 1.6 3.7 Baltimore, BWI Airport 724060 39.18 76.67 47 100.76 6193 −11.6 −9.2 10.8 9.4 8.3 11.2 −0.4 9.8 −0.3 4.5 290 4.9 280 36.2 −15.7 1.6 3.2 Lex Park, Patuxent River 724040 38.28 76.40 12 101.18 8293 −9.0 −6.2 8.8 7.7 6.8 9.8 −0.9 8.4 1.7 3.8 340 3.9 270 36.6 −13.5 1.3 3.4 Salisbury 724045 38.33 75.52 16 101.13 8293 −10.5 −7.7 8.8 8.0 7.2 9.0 1.7 8.3 2.8 2.8 10 4.2 240 36.2 −15.3 1.5 3.2 MASSACHUSETTS Boston 725090 42.37 71.03 9 101.22 6193 −13.7 −11.3 13.1 11.3 10.2 13.5 −1.3 12.2 −2.1 7.5 320 6.2 270 35.4 −17.6 1.5 2.6 East Falmouth, Otis ANGB 725060 41.65 70.52 40 100.85 8293 −11.9 −9.8 11.4 10.0 8.9 11.8 1.0 10.3 0.6 4.2 300 4.5 240 32.2 −15.1 1.4 2.1 Weymouth, S Weymth NAS 725097 42.15 70.93 49 100.74 8293 −14.6 −11.8 8.3 7.3 6.3 8.1 −1.9 7.2 −1.7 3.0 320 3.9 260 36.1 −18.9 2.1 2.1 Worcester 725095 42.27 71.88 308 97.68 6193 −17.7 −15.1 12.0 10.2 8.8 12.9 −5.6 11.4 −6.1 6.1 270 4.3 270 32.4 −21.2 1.1 2.3 MICHIGAN Alpena 726390 45.07 83.57 211 98.82 6193 −21.6 −18.6 9.4 8.3 7.5 9.6 −6.4 8.6 −6.8 2.3 270 5.0 240 34.1 −27.1 1.9 3.3 Detroit, Metro 725370 42.23 83.33 202 98.92 6193 −17.8 −15.1 11.9 10.3 9.2 12.4 −2.5 10.8 −2.8 5.0 240 5.6 230 34.9 −21.7 1.7 3.0 Flint 726370 42.97 83.75 233 98.56 6193 −18.9 −16.3 11.1 9.7 8.8 11.9 −4.4 10.3 −4.9 3.8 230 5.6 230 33.6 −23.3 1.7 2.8 Grand Rapids 726350 42.88 85.52 245 98.42 6193 −17.8 −15.2 11.3 9.9 8.8 11.8 −4.0 10.3 −4.6 3.7 180 5.8 240 33.9 −23.0 1.2 2.9 Hancock 727440 47.17 88.50 329 97.43 6193 −22.6 −19.8 9.5 8.5 7.8 10.2 −7.7 9.1 −8.7 3.7 270 4.2 250 32.4 −26.9 1.6 3.1 Harbor Beach 725386 44.02 82.80 183 99.15 8293 −13.0 −11.1 11.4 9.8 8.4 11.5 −2.8 10.2 −2.9 4.6 220 3.9 230 34.7 −16.6 1.6 2.3 Jackson 725395 42.27 84.47 305 97.71 8293 −19.4 −15.7 9.1 8.3 7.7 10.2 −5.4 9.1 −5.1 3.9 240 5.1 210 34.0 −23.8 1.4 3.1 Lansing 725390 42.77 84.60 266 98.17 6193 −19.6 −16.8 11.7 10.2 9.1 12.5 −5.3 11.2 −4.6 3.6 290 5.7 250 34.4 −24.8 1.6 3.3 Marquette, Sawyer AFB 727435 46.35 87.40 372 96.94 8293 −24.0 −21.3 10.5 9.3 8.2 11.4 −7.9 10.1 −8.2 2.5 280 4.5 210 32.9 −27.5 2.6 2.6 Marquette/Ishpeming A 727430 46.53 87.55 434 96.22 8293 −24.9 −22.1 9.7 8.7 8.1 9.8 −6.6 8.9 −8.8 3.5 270 4.9 230 32.4 −29.9 2.5 2.5 Mount Clemens, ANGB 725377 42.62 82.83 177 99.22 8293 −15.9 −14.0 9.4 8.2 7.3 11.2 −6.0 9.4 −4.3 3.3 280 4.0 230 35.2 −19.4 2.2 1.5 Muskegon 726360 43.17 86.25 193 99.03 6193 −16.2 −13.9 12.1 10.8 9.6 12.4 −4.0 11.2 −3.4 4.6 290 5.4 200 32.0 −20.7 1.5 2.8 Oscoda, Wurtsmith AFB 726395 44.45 83.40 193 99.03 8293 −17.9 −16.1 9.5 8.3 7.4 10.4 −3.6 9.2 −4.3 2.5 220 5.1 200 34.9 −21.6 2.3 2.6 Pellston 727347 45.57 84.80 219 98.72 8293 −22.7 −19.5 11.6 10.1 9.1 12.7 −5.5 10.5 −5.8 1.6 300 6.2 250 33.6 −29.4 1.7 2.7 Saginaw 726379 43.53 84.08 204 98.90 8293 −17.5 −15.6 10.2 9.2 8.4 11.0 −5.7 10.0 −5.0 4.6 260 5.8 240 35.6 −21.3 3.2 2.5 Sault Ste. Marie 727340 46.47 84.37 221 98.70 6193 −24.4 −21.8 10.1 8.9 7.9 10.8 −7.2 9.4 −7.6 2.9 90 4.4 230 31.4 −29.7 1.9 3.0 Seul Choix Point 726399 45.92 85.92 180 99.18 8293 −17.6 −15.8 12.4 10.9 9.6 13.5 −2.7 11.7 −2.7 4.2 300 3.7 200 27.9 −20.7 1.3 3.5 Traverse City 726387 44.73 85.58 190 99.06 6193 −19.6 −16.8 9.6 8.5 7.9 10.3 −5.1 9.2 −5.1 2.9 180 5.7 230 34.5 −24.9 1.6 4.1 MINNESOTA Alexandria 726557 45.87 95.40 434 96.22 8293 −29.0 −26.0 11.1 9.9 9.0 12.5 −11.0 10.8 −13.6 4.4 300 6.2 180 35.5 −32.1 2.0 2.5 Brainerd, Pequot Lakes 727500 46.60 94.32 390 96.73 8293 −31.1 −27.3 5.1 4.4 3.9 5.1−13.3 4.4 −11.9 1.4 320 2.2 190 34.8 −34.3 4.4 3.8 Duluth 727450 46.83 92.18 432 96.24 6193 −29.2 −26.6 11.3 9.9 8.9 11.1 −11.4 9.9 −11.8 4.5 310 5.2 230 32.1 −33.6 1.6 2.6 Hibbing 727455 47.38 92.83 412 96.47 8293 −31.8 −28.8 9.1 8.3 7.7 9.0−10.4 8.3 −10.4 2.8 330 5.0 200 33.1 −36.7 1.4 2.6 International Falls 727470 48.57 93.38 361 97.06 6193 −33.6 −30.8 9.9 8.8 7.9 9.7−12.3 8.7 −13.6 2.6 270 5.0 180 33.3 −38.3 1.9 2.1 Minneapolis-St. Paul 726580 44.88 93.22 255 98.30 6193 −26.5 −23.7 11.1 9.9 8.9 11.2 −11.2 9.9 −10.3 4.2 300 6.3 180 35.8 −30.2 1.9 3.0 Redwood Falls 726556 44.55 95.08 312 97.63 8293 −27.1 −24.4 11.5 9.9 9.0 12.4 −9.8 10.6 −9.7 4.7 280 6.1 180 37.1 −29.9 2.3 2.9 Rochester 726440 43.92 92.50 402 96.59 6193 −27.3 −24.4 13.1 11.8 10.5 14.2 −11.4 12.6 −11.4 5.8 300 6.8 200 34.2 −30.7 2.1 2.9 WMO# = World Meteorological Organization number AFB = Air Force Base NAS = Naval Air Station WSO = Weather Service Office AAF = Army Air Field MCAS = Marine Corps Air Station DB = dry-bulb temperature, °C Lat. = North latitude ° Long. = West longitude ° Elev. = elevation, m StdP = standard pressure at station elevation, kPa WS = wind speed, m/s PWD = prevailing wind direction, ° True Climatic Design Information 27.13 Table 1B Cooling and Dehumidification Design Conditions—United States Station Cooling DB/MWB Evaporation WB/MDB Dehumidification DP/MDB and HR Range of DB 0.4% 1% 2% 0.4% 1% 2% 0.4% 1% 2% DB MWB DB MWB DB MWB WB MDB WB MDB WB MDB DP HR MDB DP HR MDB DP HR MDB 1 2a 2b 2c 2d 2e 2f 3a 3b 3c 3d 3e 3f 4a 4b 4c 4d 4e 4f 4g 4h 4i 5 Russell 37.9 22.1 35.8 22.3 34.2 22.0 24.6 33.0 23.8 32.0 23.0 30.9 22.1 18.0 28.3 21.4 17.2 27.5 20.7 16.5 26.6 13.4 Salina 38.2 23.1 36.2 23.0 34.6 22.9 25.2 33.5 24.5 32.4 23.8 31.6 23.2 18.8 29.3 22.1 17.6 28.4 21.5 16.9 27.6 12.8 Topeka 35.5 24.1 33.8 24.0 32.4 23.7 26.0 32.4 25.3 31.6 24.6 30.9 24.2 19.8 30.3 23.4 18.9 29.3 22.7 18.0 28.5 11.3 Wichita, Airport 37.9 22.6 36.3 22.6 34.5 22.5 24.8 32.6 24.2 32.1 23.6 31.4 22.8 18.4 28.4 22.1 17.6 27.7 21.4 16.9 27.2 12.3 Wichita, McConnell AFB 38.0 23.0 36.0 22.8 34.4 22.7 25.2 33.1 24.5 32.4 23.9 31.6 23.3 19.0 28.8 22.1 17.7 28.5 21.5 17.0 27.7 12.1 KENTUCKY Bowling Green 34.2 24.3 32.9 24.1 31.3 23.6 25.7 31.7 25.1 30.7 24.5 30.0 24.2 19.5 28.7 23.7 18.9 28.0 23.1 18.2 27.4 11.1 Covington/Cincinnati Airport 32.8 23.6 31.4 23.0 30.1 22.4 25.1 30.7 24.3 29.7 23.5 28.4 23.5 18.9 28.6 22.7 18.0 27.4 21.9 17.2 26.4 10.5 Fort Campbell, AAF 35.0 24.8 33.7 24.5 32.1 24.2 26.4 32.3 25.7 31.6 25.1 30.7 24.9 20.4 29.4 24.2 19.5 29.0 23.6 18.8 28.4 10.8 Fort Knox, Godman AAF 34.4 24.5 33.1 23.6 31.4 23.3 25.8 32.2 25.1 31.0 24.4 30.2 24.2 19.7 29.4 23.5 18.8 28.6 22.8 18.0 27.9 10.8 Jackson 32.0 23.2 30.7 22.9 29.7 22.3 24.9 30.4 24.2 29.4 23.5 28.3 23.5 19.3 28.6 22.8 18.5 27.3 21.9 17.4 26.1 10.1 Lexington 32.6 23.2 31.4 22.9 30.2 22.4 24.8 30.7 24.1 29.7 23.4 28.5 23.1 18.5 28.1 22.4 17.7 27.3 21.8 17.1 26.5 10.2 Louisville 33.7 24.6 32.4 24.1 31.2 23.4 25.7 31.9 25.1 30.9 24.3 29.8 24.0 19.2 29.5 23.3 18.4 28.7 22.7 17.8 27.8 10.1 Paducah 35.3 24.9 34.1 24.6 33.1 24.1 26.6 32.8 25.9 32.1 25.3 31.2 25.0 20.4 30.0 24.4 19.7 29.2 23.8 18.9 28.6 11.2 LOUISIANA Alexandria, England AFB 35.0 25.5 34.2 25.3 33.4 25.0 27.1 32.2 26.6 32.2 26.1 31.6 25.7 21.0 29.8 25.1 20.3 29.6 24.6 19.7 29.3 10.2 Baton Rouge 34.2 25.3 33.4 25.1 32.7 24.9 26.7 31.7 26.3 31.3 25.8 30.7 25.4 20.7 28.9 25.0 20.1 28.7 24.6 19.6 28.3 9.3 Bossier City, Barksdale AFB 35.6 25.0 34.6 25.1 33.7 24.9 26.7 32.5 26.1 32.1 25.7 31.6 25.2 20.5 29.1 24.7 19.8 28.5 24.2 19.2 28.2 11.1 Lafayette 34.4 25.7 33.7 25.3 33.0 25.2 26.9 31.9 26.5 31.4 26.1 31.1 25.6 20.9 29.1 25.2 20.4 28.6 24.9 20.0 28.3 9.5 Lake Charles 33.8 25.4 33.0 25.3 32.2 25.2 26.9 31.2 26.6 30.9 26.2 30.6 25.8 21.1 28.9 25.4 20.7 28.7 25.1 20.2 28.4 9.0 Leesville, Fort Polk 35.1 24.8 34.2 24.6 33.5 24.4 26.3 31.6 25.9 31.3 25.5 30.8 25.2 20.6 28.4 24.7 20.0 28.0 24.2 19.4 27.6 10.1 Monroe 35.5 25.5 34.5 25.4 33.7 25.2 27.2 32.9 26.7 32.4 26.2 31.8 25.7 21.0 30.2 25.2 20.4 29.6 24.8 19.9 29.1 10.7 New Orleans, Intl Airport 33.9 26.1 33.1 25.7 32.3 25.6 27.3 31.9 26.8 31.3 26.4 30.7 26.1 21.5 30.2 25.6 20.9 29.4 25.2 20.3 29.0 8.6 New Orleans, Lakefront A 33.9 25.7 33.1 25.3 32.0 25.2 27.1 31.3 26.6 30.7 26.2 30.3 26.0 21.4 29.3 25.5 20.7 28.8 25.1 20.2 28.5 6.6 Shreveport 35.9 24.9 34.8 24.8 33.7 24.6 26.3 32.9 25.8 32.3 25.5 31.8 24.7 19.9 28.8 24.2 19.3 28.4 23.9 18.9 28.2 10.6 MAINE Augusta 30.3 21.5 28.7 20.3 26.9 19.5 22.8 28.1 21.7 26.5 20.8 25.2 21.2 16.1 24.9 20.2 15.1 23.9 19.4 14.3 23.1 10.2 Bangor 30.8 21.5 29.1 20.4 27.1 19.3 23.0 28.2 21.8 27.0 20.7 25.2 21.1 15.9 25.6 20.1 14.9 24.1 19.2 14.1 22.9 11.4 Brunswick, NAS 30.3 21.6 28.7 20.4 26.6 19.4 23.0 28.4 21.9 26.6 20.9 24.8 21.2 15.9 25.4 20.3 15.0 24.3 19.5 14.3 23.4 10.6 Caribou 29.4 20.5 27.6 19.4 26.0 18.8 22.4 26.9 21.2 25.2 20.1 24.2 20.9 16.0 24.6 19.8 14.8 23.7 18.7 13.8 22.3 10.8 Limestone, Loring AFB 28.9 19.9 26.8 18.8 25.3 18.0 21.6 26.1 20.5 24.6 19.5 23.5 20.2 15.3 23.7 19.2 14.4 22.5 18.2 13.5 21.9 10.4 Portland 30.2 21.8 28.4 20.8 26.7 19.8 23.2 28.3 22.1 26.6 21.0 25.1 21.6 16.3 26.1 20.6 15.3 24.6 19.6 14.4 23.3 10.4 MARYLAND Camp Springs, Andrews AFB 34.3 24.1 32.9 23.6 31.1 22.9 25.5 31.3 24.8 30.5 24.1 29.4 24.1 19.2 28.4 23.4 18.4 27.5 22.8 17.7 26.8 10.4 Baltimore, BWI Airport 34.0 23.7 32.6 23.2 31.1 22.5 25.4 31.2 24.6 30.2 23.9 29.3 23.8 18.8 28.1 23.1 17.9 27.3 22.4 17.2 26.6 10.4 Lex Park, Patuxent River NAS 33.9 24.4 32.0 23.9 30.8 23.4 25.9 31.2 25.2 30.5 24.5 29.5 24.5 19.5 29.0 23.8 18.7 28.1 23.1 17.9 27.5 8.8 Salisbury 33.9 25.2 32.1 24.7 30.9 24.0 26.6 31.3 25.7 30.2 25.1 29.4 25.4 20.6 28.8 24.6 19.6 28.0 24.0 18.9 27.2 10.4 MASSACHUSETTS Boston 32.5 22.6 30.7 21.9 28.9 21.1 24.1 30.3 23.2 28.4 22.3 26.9 22.3 17.0 26.7 21.5 16.2 25.9 20.7 15.4 25.4 8.5 East Falmouth, Otis ANGB 29.2 22.1 27.5 22.0 25.9 20.8 23.9 27.1 23.1 25.6 22.3 24.7 23.1 17.9 25.3 22.1 16.9 24.7 21.4 16.1 23.7 8.1 S. Weymouth NAS 33.1 23.0 30.8 22.4 29.3 21.6 24.9 30.5 23.7 28.9 22.8 27.3 23.5 18.4 27.6 22.1 16.9 26.2 21.1 15.8 25.3 10.9 Worcester 29.7 21.6 28.2 20.8 26.7 19.9 23.2 27.6 22.1 26.6 21.2 25.1 21.7 17.0 25.7 20.7 16.0 24.6 19.8 15.0 23.7 9.2 MICHIGAN Alpena 30.8 21.7 28.9 20.4 27.1 19.6 23.1 28.4 21.9 27.1 20.9 25.6 21.4 16.5 26.1 20.3 15.3 24.7 19.2 14.3 23.6 12.7 Detroit, Metro 32.1 22.8 30.6 22.1 29.1 21.3 24.4 29.9 23.4 28.7 22.5 27.4 22.7 17.8 28.2 21.8 16.8 26.5 20.9 15.9 25.7 11.3 Flint 31.3 22.6 29.8 21.8 28.4 20.8 24.1 29.0 23.1 27.9 22.1 26.7 22.6 17.8 27.2 21.5 16.6 25.7 20.6 15.7 24.9 11.4 Grand Rapids 31.8 22.8 30.2 21.8 28.8 20.9 24.3 29.6 23.3 28.1 22.4 27.0 22.8 18.0 27.4 21.8 16.9 26.3 20.9 16.0 25.2 11.5 Hancock 29.7 21.5 28.1 20.4 26.7 19.6 22.9 27.8 21.8 26.4 20.8 25.0 21.3 16.6 25.9 20.3 15.6 24.6 19.3 14.7 23.6 11.4 Harbor Beach 32.0 21.9 30.2 20.7 28.5 20.0 23.5 29.9 22.2 28.1 21.2 26.5 21.1 16.1 28.0 20.1 15.1 26.6 19.2 14.3 25.3 8.0 Jackson 31.3 23.4 30.1 22.8 28.9 21.7 24.9 29.8 23.9 28.4 22.9 27.2 23.6 19.1 28.2 22.2 17.5 27.0 21.4 16.7 25.6 11.3 Lansing 31.9 22.9 30.2 22.0 28.7 21.3 24.4 29.6 23.4 28.2 22.6 27.0 22.9 18.2 27.3 21.9 17.2 26.2 21.1 16.3 25.3 12.1 Marquette, Sawyer AFB 30.0 20.4 28.2 19.9 26.0 18.6 22.5 28.1 21.2 26.0 20.1 23.9 20.7 16.1 25.1 19.7 15.1 23.4 18.8 14.2 22.8 12.3 Marquette/Ishpeming, A 29.7 20.5 27.8 19.6 25.7 18.5 22.1 27.7 20.9 25.7 19.9 24.1 20.4 15.9 25.0 19.4 14.9 23.8 18.4 14.0 22.5 12.3 Mount Clemens, ANGB 32.0 23.4 30.3 22.4 28.9 21.5 24.9 30.3 23.8 28.5 22.8 26.9 23.5 18.7 28.4 22.1 17.1 27.0 21.2 16.2 25.7 10.9 Muskegon 29.7 21.8 28.4 21.1 27.2 20.4 23.6 27.8 22.7 26.7 21.8 25.4 22.3 17.4 26.4 21.3 16.4 25.1 20.5 15.5 24.2 10.1 Oscoda, Wurtsmith AFB 31.6 22.2 29.8 21.7 28.2 20.5 24.0 29.8 22.8 28.2 21.7 26.1 22.1 17.2 26.6 21.0 16.0 25.9 20.1 15.1 24.8 11.9 Pellston 30.8 21.4 29.2 20.6 27.2 19.9 23.1 28.5 22.0 27.3 21.1 25.7 21.3 16.4 25.3 20.4 15.5 24.5 19.6 14.7 24.0 13.3 Saginaw 32.0 23.4 30.3 22.3 28.8 21.3 24.9 30.0 23.7 28.7 22.6 27.1 23.5 18.8 28.1 22.0 17.1 26.5 21.0 16.0 25.5 11.8 Sault Ste. Marie 28.4 20.8 26.6 19.8 24.9 18.7 22.2 26.8 21.0 24.9 19.8 23.6 20.7 15.8 24.7 19.6 14.7 23.4 18.4 13.6 22.1 12.2 Seul Choix Point 25.4 18.9 24.2 18.4 23.1 17.5 21.2 24.2 20.2 22.5 19.1 21.7 20.2 15.2 23.2 19.3 14.4 22.3 18.2 13.4 21.2 7.7 Traverse City 31.7 21.9 29.8 20.9 28.1 20.1 23.4 28.9 22.4 27.7 21.3 26.4 21.7 16.7 26.6 20.6 15.6 25.4 19.6 14.7 24.3 12.2 MINNESOTA Alexandria 31.9 22.5 30.2 21.3 28.6 20.3 24.0 30.0 22.9 28.0 21.7 26.6 22.0 17.6 27.7 21.0 16.5 25.9 20.1 15.6 24.9 10.7 Brainerd, Pequot Lakes 31.1 20.9 29.5 20.0 27.2 19.0 22.5 29.7 21.2 27.7 20.0 25.4 20.1 15.5 27.1 19.1 14.6 24.9 18.1 13.7 23.7 12.0 Duluth 29.1 20.4 27.2 19.3 25.6 18.3 21.9 27.2 20.7 25.4 19.5 24.1 20.2 15.7 25.2 19.0 14.5 23.7 17.8 13.4 22.1 11.2 Hibbing 29.3 21.1 27.1 20.0 25.7 18.8 22.6 27.7 21.4 25.5 20.1 24.1 21.0 16.5 25.7 19.9 15.4 24.4 18.9 14.4 22.6 12.9 International Falls 30.1 20.6 28.2 19.4 26.6 18.7 22.3 27.8 21.1 26.1 19.9 24.7 20.6 16.0 25.4 19.3 14.7 23.9 18.2 13.7 22.7 12.1 Minneapolis-St. Paul 32.8 22.7 31.1 21.9 29.4 21.1 24.4 30.8 23.4 29.1 22.3 27.8 22.5 17.7 28.5 21.4 16.6 27.2 20.4 15.6 26.0 10.6 Redwood Falls 33.5 23.2 31.3 22.5 29.8 21.3 25.2 31.4 24.1 29.4 23.0 27.8 23.7 19.3 28.6 22.2 17.5 27.4 21.2 16.5 26.4 11.5 Rochester 31.1 22.4 29.5 21.8 27.9 20.9 24.2 29.2 23.1 27.7 22.1 26.5 22.7 18.3 27.4 21.6 17.1 26.3 20.5 15.9 24.9 10.9 DP = dew-point temperature, °C MWB = mean coincident wet-bulb temperature, °C MDB = mean coincident dry-bulb temperature, °C MWS = mean coincident wind speed, m/s StdD = standard deviation, °C HR = humidity ratio, grams of moisture per kilogram of dry air A = airport ANGB = Air National Guard Base MCAF = Marine Corps Air Facility NAF = Naval Air Facility NAWS = Naval Air Weapons Station RAF = Royal Air Force 27.14 2001 ASHRAE Fundamentals Handbook (SI) Table 1A Heating and Wind Design Conditions—United States Heating Dry Bulb Extreme Wind Speed, m/s Coldest Month MWS/PWD to DB Extr. Annual Daily Lat.
Long.
Elev., m StdP, kPa 0.4% 1% 99.6% 0.4% Mean DB StdD DB Station WMO# Dates 99.6% 99% 1% 2.5% 5% WS MDB WS MDB MWS PWD MWS PWD Max.
Min.
Max. Min.
1a 1b 1c 1d 1e 1f 1g 2a 2b 3a 3b 3c 4a 4b 4c 4d 5a 5b 5c 5d 6a 6b 6c 6d Saint Cloud 726550 45.55 94.07 312 97.63 6193 −28.7 −25.8 9.8 8.7 7.9 10.1−11.6 9.0 −12.2 3.6 300 5.4 200 34.9 −32.7 1.7 3.1 Tofte 727554 47.58 90.83 241 98.46 8293 −23.1 −21.0 10.6 9.1 7.8 11.2 −9.1 9.9 −7.7 3.7 260 3.4 330 30.1 −28.1 2.5 2.7 MISSISSIPPI Biloxi, Keesler AFB 747686 30.42 88.92 10 101.20 8293 −0.7 1.6 7.5 6.4 5.7 8.0 9.6 7.0 9.9 3.4 360 3.0 210 35.9 −4.9 1.1 4.1 Columbus, AFB 723306 33.65 88.45 67 100.52 8293 −6.5 −4.1 7.9 6.7 5.8 8.3 6.3 7.3 7.8 2.5 360 2.7 240 37.5 −11.1 1.5 3.8 Greenwood 722359 33.50 90.08 47 100.76 8293 −6.9 −4.5 8.4 7.6 6.3 8.5 7.8 8.0 8.3 2.9 360 2.9 180 37.4 −10.6 1.3 4.2 Jackson 722350 32.32 90.08 101 100.12 6193 −6.3 −4.1 9.1 8.0 7.1 9.3 7.1 8.3 7.6 3.2 340 3.6 270 36.7 −10.1 1.5 3.2 McComb 722358 31.18 90.47 126 99.82 8293 −4.9 −2.1 7.5 6.4 5.8 7.6 9.2 6.7 9.2 2.5 350 3.3 230 36.6 −9.2 1.1 4.0 Meridian 722340 32.33 88.75 94 100.20 6193 −6.2 −3.9 8.4 7.5 6.6 8.5 6.2 7.7 7.5 2.7 360 3.6 360 37.2 −10.5 1.6 3.3 Tupelo 723320 34.27 88.77 110 100.01 8293 −7.9 −5.3 8.4 7.5 6.7 8.8 6.6 7.8 6.6 3.2 10 3.3 260 37.1 −12.4 1.6 4.7 MISSOURI Cape Girardeau 723489 37.23 89.57 104 100.08 8293 −14.4 −10.7 9.5 8.5 7.9 10.0 1.7 9.1 2.4 4.2 360 4.3 200 37.8 −18.4 1.6 5.1 Columbia 724450 38.82 92.22 274 98.08 6193 −18.1 −14.9 11.1 9.7 8.7 11.2 −2.6 9.9 −2.2 4.8 310 4.8 200 37.2 −22.2 2.4 3.4 Joplin 723495 37.15 94.50 299 97.78 8293 −15.9 −11.9 10.3 9.3 8.4 10.5 10.0 9.5 8.1 4.4 10 4.9 220 37.8 −19.0 2.2 5.2 Kansas City 724460 39.32 94.72 312 97.63 6193 −18.6 −15.4 11.4 10.1 9.0 11.6 1.2 10.2 0.6 4.2 320 5.7 190 37.9 −21.7 2.3 3.7 Poplar Bluff 723300 36.77 90.47 146 99.58 8293 −13.2 −10.3 7.9 6.5 5.7 7.8 4.5 6.6 3.3 3.0 360 3.0 200 38.3 −16.8 3.8 5.2 Spickard/Trenton 725400 40.25 93.72 270 98.12 8293 −17.4 −14.6 10.4 9.1 8.1 11.0 −1.9 9.6 −0.5 3.5 360 5.0 200 37.7 −20.7 2.9 4.1 Springfield 724400 37.23 93.38 387 96.76 6193 −16.2 −12.8 10.8 9.6 8.7 10.4 1.6 9.6 1.8 4.6 340 4.4 230 36.9 −20.2 1.9 3.6 St. Louis, Intl Airport 724340 38.75 90.37 172 99.28 6193 −16.8 −13.6 11.4 10.0 8.8 11.7 −3.4 10.3 −2.8 5.4 290 4.8 240 37.4 −20.7 1.9 3.4 Warrensburg, Whiteman AFB 724467 38.73 93.55 265 98.18 8293 −17.3 −13.8 9.9 8.6 7.6 10.4 1.1 9.3 0.9 4.0 360 4.0 190 38.1 −20.7 2.2 4.3 MONTANA Billings 726770 45.80 108.53 1088 88.92 6193 −25.1 −21.8 12.3 10.7 9.7 13.3 −3.9 12.0 −1.2 4.2 230 4.4 240 37.3 −28.1 1.6 3.4 Bozeman 726797 45.78 111.15 1364 85.98 8293 −29.0 −24.7 9.2 8.0 6.7 8.9 2.5 7.6 1.2 1.9 140 4.0 360 35.8 −34.0 1.6 4.3 Butte 726785 45.95 112.50 1690 82.60 8293 −30.1 −25.8 10.2 9.2 8.2 9.5 −1.5 8.4 −1.2 1.6 150 5.6 120 33.1 −36.8 1.4 4.4 Cut Bank 727796 48.60 112.37 1175 87.98 6193 −29.2 −26.4 15.2 13.4 12.0 18.0 2.0 15.3 2.4 3.2 320 5.8 270 34.1 −33.1 2.2 3.2 Glasgow 727680 48.22 106.62 700 93.19 6193 −30.2 −27.1 13.1 11.5 10.2 12.5 −7.8 11.0 −9.3 3.6 330 5.9 160 37.3 −33.8 1.8 3.7 Great Falls, Intl Airport 727750 47.48 111.37 1115 88.63 6193 −28.1 −24.8 14.6 13.1 11.6 15.3 3.4 14.0 3.4 3.3 240 5.4 230 36.8 −31.8 1.8 4.1 Great Falls, Malmstrom AFB 727755 47.50 111.18 1075 89.06 8293 −27.4 −23.9 12.7 10.8 9.5 14.8 3.3 13.1 3.3 1.8 240 3.5 260 37.0 −29.8 1.8 4.4 Havre 727770 48.55 109.77 792 92.17 8293 −31.6 −28.4 10.8 9.4 8.4 11.5 1.7 10.2 0.4 2.5 240 3.9 270 38.7 −36.2 2.8 4.5 Helena 727720 46.60 112.00 1188 87.84 6193 −27.5 −23.5 11.1 9.8 8.7 11.2 4.3 9.6 1.7 2.0 290 5.4 280 35.7 −30.8 1.8 4.0 Kalispell 727790 48.30 114.27 906 90.90 6193 −24.2 −19.7 10.5 8.8 7.6 11.2−11.1 9.3 −7.9 2.9 20 4.1 170 34.7 −28.4 1.6 4.8 Lewistown 726776 47.05 109.47 1270 86.97 6193 −27.9 −24.2 11.4 10.1 9.1 13.0 1.4 11.3 1.4 3.3 250 4.7 90 34.8 −31.6 1.9 4.1 Miles City 742300 46.43 105.87 801 92.07 6193 −28.2 −24.7 12.0 10.1 9.0 12.3 −3.8 10.3 −2.8 3.7 290 4.9 140 39.1 −31.7 1.5 3.6 Missoula 727730 46.92 114.08 972 90.18 6193 −22.8 −18.3 9.9 8.6 7.5 9.9 −8.1 8.5 −6.3 2.9 120 4.6 290 36.3 −26.3 1.6 4.6 NEBRASKA Bellevue, Offutt AFB 725540 41.12 95.92 319 97.55 8293 −20.7 −17.2 10.0 8.5 7.4 11.4 −4.8 9.7 −5.2 3.4 330 4.4 190 37.6 −22.7 2.5 3.5 Grand Island 725520 40.97 98.32 566 94.71 6193 −22.2 −19.1 13.3 11.6 10.4 13.0 −6.2 11.5 −7.0 4.8 270 6.7 180 39.0 −25.8 1.8 2.9 Lincoln 725510 40.85 96.75 362 97.05 8293 −21.9 −18.8 11.9 10.3 9.3 12.4 −3.9 10.9 −2.7 4.0 350 6.8 180 39.4 −24.1 3.6 4.5 Norfolk 725560 41.98 97.43 473 95.77 6193 −23.7 −20.7 12.9 11.2 9.9 14.5 −6.8 12.7 −6.3 4.7 340 6.6 190 38.2 −27.7 1.7 3.0 North Platte 725620 41.13 100.68 849 91.53 6193 −23.1 −19.8 13.1 11.2 9.9 12.6 −4.6 10.7 −3.3 3.3 320 5.4 180 38.1 −26.9 1.6 3.7 Omaha, Eppley Airfield 725500 41.30 95.90 299 97.78 6193 −21.8 −18.9 11.4 10.0 8.9 12.0 −6.3 10.4 −8.2 4.6 340 5.5 180 37.9 −25.6 1.8 2.7 Omaha, WSO 725530 41.37 96.02 406 96.54 8293 −22.1 −18.8 10.0 8.8 8.0 11.0 −5.0 9.6 −3.8 4.4 310 4.7 170 36.6 −25.3 2.2 3.6 Scottsbluff 725660 41.87 103.60 1206 87.65 6193 −23.6 −19.4 13.4 11.5 9.9 14.1 1.4 12.2 1.4 3.5 300 5.0 300 38.4 −28.6 1.6 4.4 Sidney 725610 41.10 102.98 1312 86.53 8293 −22.5 −18.2 12.9 10.8 9.6 13.8 0.2 11.4 1.9 4.2 290 5.4 160 38.3 −27.6 2.5 4.8 Valentine 725670 42.87 100.55 792 92.17 8293 −26.5 −22.0 11.9 10.4 9.2 11.6 −4.1 10.1 −2.5 3.8 250 6.6 180 40.2 −29.8 2.4 4.7 NEVADA Elko 725825 40.83 115.78 1565 83.88 6193 −20.6 −17.1 9.5 8.1 6.9 8.8 2.3 7.2 2.7 1.6 70 4.5 230 36.9 −25.1 1.8 4.4 Ely 724860 39.28 114.85 1909 80.40 6193 −21.2 −17.8 12.5 10.8 9.3 11.6 0.4 9.9 −0.9 4.9 190 5.7 230 33.8 −26.1 1.3 4.1 Las Vegas, Intl Airport 723860 36.08 115.17 664 93.60 6193 −2.7 −0.9 13.3 11.5 10.2 11.2 8.9 9.9 9.6 3.3 250 5.5 230 44.1 −6.3 1.2 2.6 Mercury 723870 36.62 116.02 1009 89.78 8293 −4.5 −2.2 11.3 9.8 8.7 11.0 6.7 9.2 5.3 3.4 50 5.5 230 38.8 −7.0 9.8 3.5 North Las Vegas, Nellis AFB 723865 36.23 115.03 570 94.66 8293 −2.4 −0.7 10.8 9.4 8.1 10.1 11.1 8.3 9.6 0.7 20 4.0 210 44.6 −6.1 1.1 2.5 Reno 724880 39.50 119.78 1341 86.22 6193 −13.4 −10.6 11.5 9.9 8.5 11.5 7.5 9.2 6.7 1.4 160 4.6 290 37.2 −17.1 1.2 4.7 Tonopah 724855 38.05 117.08 1654 82.97 6193 −13.7 −10.8 11.3 10.0 9.0 10.5 3.0 9.6 2.3 4.2 340 5.2 180 36.7 −17.4 1.1 3.6 Winnemucca 725830 40.90 117.80 1315 86.49 6193 −17.1 −13.7 10.1 8.7 7.7 9.3 3.8 8.1 3.3 2.2 160 4.9 250 38.4 −22.8 1.3 5.7 NEW HAMPSHIRE Concord 726050 43.20 71.50 105 100.07 6193 −21.9 −18.9 10.0 8.7 7.7 10.0 −6.6 9.0 −6.2 2.0 320 4.6 230 34.9 −27.9 1.6 3.1 Lebanon 726116 43.63 72.30 182 99.16 8293 −21.8 −19.3 7.9 6.9 6.1 8.0 −3.8 7.0 −3.4 1.1 360 4.1 220 34.2 −27.3 1.1 2.9 Mount Washington 726130 44.27 71.30 1910 80.39 8293 −30.8 −28.2 39.4 36.1 32.6 44.4−25.3 41.1 −26.0 32.4 280 9.3 270 18.2 −36.0 1.3 2.3 Portsmouth, Pease AFB 726055 43.08 70.82 31 100.95 8293 −15.5 −12.9 9.4 8.1 7.1 10.0 −3.2 8.9 −2.6 3.4 280 3.7 270 34.2 −18.7 1.2 1.8 NEW JERSEY Atlantic City 724070 39.45 74.57 20 101.08 6193 −13.4 −10.8 11.8 10.4 9.1 12.7 1.9 11.0 0.8 3.9 310 5.1 250 35.7 −17.6 1.6 3.2 Millville 724075 39.37 75.07 25 101.03 8293 −12.2 −9.5 8.6 8.1 7.4 9.0 1.9 8.3 1.6 3.0 300 4.9 240 35.8 −17.6 1.3 4.1 Newark 725020 40.70 74.17 9 101.22 6193 −12.3 −10.0 11.6 10.2 9.1 12.0 −2.2 10.4 −1.6 5.9 260 5.7 230 36.4 −15.6 1.4 2.7 Teterboro 725025 40.85 74.07 3 101.29 8293 −12.0 −9.8 9.2 8.3 7.7 9.5 −1.5 8.5 −1.1 4.8 280 5.2 240 36.0 −16.5 1.4 3.1 Trenton, McGuire AFB 724096 40.02 74.60 41 100.83 8293 −11.8 −9.6 9.8 8.4 7.4 10.4 −0.6 9.4 −0.5 3.6 330 3.4 240 36.2 −16.6 1.2 2.9 NEW MEXICO Alamogordo, Holloman AFB 747320 32.85 106.10 1248 87.20 8293 −6.8 −5.2 9.0 7.5 6.3 8.1 10.2 6.6 8.7 1.2 10 3.5 250 38.8 −10.3 1.6 2.0 Albuquerque 723650 35.05 106.62 1620 83.32 6193 −10.4 −7.6 13.0 11.2 9.6 11.8 1.2 9.8 2.7 3.6 360 4.5 240 37.5 −14.7 1.4 4.1 Carlsbad 722687 32.33 104.27 1004 89.83 8293 −7.4 −5.2 11.3 9.6 8.4 11.1 13.8 9.4 12.0 3.5 340 5.5 150 40.2 −12.6 2.0 3.9 Clayton 723600 36.45 103.15 1515 84.40 8293 −17.1 −12.8 13.6 11.9 10.5 13.4 4.4 11.8 3.9 4.3 40 5.9 200 36.4 −20.3 1.4 4.1 WMO# = World Meteorological Organization number AFB = Air Force Base NAS = Naval Air Station WSO = Weather Service Office AAF = Army Air Field MCAS = Marine Corps Air Station DB = dry-bulb temperature, °C Lat. = North latitude ° Long. = West longitude ° Elev. = elevation, m StdP = standard pressure at station elevation, kPa WS = wind speed, m/s PWD = prevailing wind direction, ° True Climatic Design Information 27.15 Table 1B Cooling and Dehumidification Design Conditions—United States Station Cooling DB/MWB Evaporation WB/MDB Dehumidification DP/MDB and HR Range of DB 0.4% 1% 2% 0.4% 1% 2% 0.4% 1% 2% DB MWB DB MWB DB MWB WB MDB WB MDB WB MDB DP HR MDB DP HR MDB DP HR MDB 1 2a 2b 2c 2d 2e 2f 3a 3b 3c 3d 3e 3f 4a 4b 4c 4d 4e 4f 4g 4h 4i 5 Saint Cloud 32.5 22.3 30.8 21.7 29.3 20.8 24.3 30.3 23.2 29.1 22.1 27.6 22.4 17.8 28.2 21.3 16.5 26.9 20.2 15.5 25.7 11.9 Tofte 26.1 17.5 24.1 16.6 21.9 16.1 19.0 23.2 18.0 22.2 17.0 21.0 17.8 13.1 20.9 16.1 11.8 20.5 15.0 11.0 19.8 7.2 MISSISSIPPI Biloxi, Keesler AFB 33.6 26.1 32.7 25.7 31.6 25.5 27.4 31.4 26.9 30.9 26.4 30.4 26.2 21.6 29.8 25.7 21.0 29.4 25.3 20.5 29.1 7.2 Columbus, AFB 35.6 24.8 34.5 24.6 33.5 24.2 26.4 32.6 25.8 31.9 25.3 31.3 25.0 20.2 29.3 24.3 19.4 28.5 23.9 18.9 27.9 10.7 Greenwood 35.5 25.6 34.6 25.3 33.7 25.0 27.2 32.7 26.7 32.3 26.1 31.7 25.7 21.1 29.8 25.2 20.4 29.5 24.7 19.8 29.1 10.6 Jackson 35.2 24.8 34.1 24.7 33.2 24.6 26.4 32.4 25.9 31.8 25.6 31.2 25.0 20.3 28.7 24.5 19.7 28.4 24.1 19.2 27.9 10.7 McComb 34.5 24.6 33.6 24.5 32.7 24.4 26.1 31.7 25.7 31.0 25.3 30.4 24.8 20.1 28.3 24.4 19.7 27.8 24.1 19.3 27.3 11.0 Meridian 35.3 24.7 34.2 24.5 33.1 24.3 26.3 32.6 25.7 31.9 25.2 31.3 24.7 19.9 28.9 24.1 19.2 28.6 23.6 18.6 28.1 11.3 Tupelo 35.3 24.5 34.3 24.2 33.4 24.1 26.0 31.9 25.5 31.8 25.1 31.2 24.4 19.6 28.4 24.0 19.1 28.1 23.6 18.7 27.5 10.5 MISSOURI Cape Girardeau 35.4 25.2 34.2 24.8 33.0 24.5 26.6 33.2 25.8 32.1 25.3 31.3 24.8 20.1 30.2 24.2 19.4 29.4 23.7 18.8 28.6 11.0 Columbia 34.8 23.8 33.1 23.7 31.7 23.2 25.6 31.8 24.9 31.0 24.1 30.2 23.9 19.5 29.4 23.1 18.5 28.3 22.4 17.7 27.5 11.3 Joplin 35.6 24.1 34.4 23.9 33.0 23.5 25.7 32.5 25.1 31.7 24.5 31.2 24.0 19.6 29.4 23.4 18.9 29.2 22.5 17.8 28.1 11.1 Kansas City 35.5 24.1 33.8 23.9 32.3 23.4 25.7 32.3 25.1 31.7 24.4 30.8 23.9 19.5 29.7 23.2 18.6 29.1 22.5 17.9 28.1 10.4 Poplar Bluff 34.8 24.9 33.6 24.7 32.0 24.3 26.5 32.1 25.8 31.3 25.2 30.5 25.1 20.6 29.3 24.4 19.7 28.6 23.8 19.0 28.0 11.1 Spickard/Trenton 35.7 23.6 34.0 22.9 31.8 22.5 25.6 31.1 24.6 31.3 23.7 30.1 24.2 19.8 28.6 23.0 18.3 28.2 21.6 16.8 27.2 10.9 Springfield 34.8 23.5 33.3 23.5 31.8 23.1 25.3 31.7 24.6 31.1 24.0 30.2 23.5 19.2 29.1 22.8 18.3 28.2 22.2 17.7 27.4 11.6 St. Louis, Intl Airport 35.1 24.6 33.6 24.1 32.2 23.6 26.1 32.3 25.3 31.3 24.6 30.4 24.3 19.7 29.3 23.7 18.9 28.5 22.9 18.1 27.8 10.2 Warrensburg, Whiteman AFB 35.6 24.5 34.1 24.5 32.1 24.0 26.1 32.9 25.4 32.2 24.7 31.3 24.3 19.9 30.2 23.7 19.1 29.3 23.0 18.3 28.6 10.7 MONTANA Billings 34.1 16.9 32.3 16.6 30.4 16.2 18.6 30.1 17.7 29.1 16.9 28.2 14.7 11.9 21.8 13.7 11.1 21.4 12.7 10.5 20.8 14.3 Bozeman 33.0 16.0 30.8 15.6 29.2 15.1 17.6 28.4 16.6 27.6 15.9 27.0 14.2 11.9 20.5 13.1 11.1 19.4 11.7 10.1 19.0 17.6 Butte 30.2 13.8 28.8 13.3 26.9 13.0 15.6 24.5 14.6 24.7 13.9 24.3 12.2 10.9 16.2 11.0 10.0 17.0 10.0 9.4 16.5 17.5 Cut Bank 30.7 15.3 28.8 14.8 26.8 14.2 16.8 27.3 15.8 26.3 14.9 25.2 13.3 11.0 19.5 11.9 10.0 18.2 10.8 9.3 17.4 14.5 Glasgow 34.2 17.6 32.2 17.2 30.2 16.7 19.8 29.4 18.6 28.6 17.7 27.5 16.8 13.0 23.2 15.4 11.9 21.4 14.2 11.0 20.7 14.1 Great Falls, Intl Airport 33.2 16.0 31.3 15.5 29.4 15.1 17.7 28.8 16.8 27.9 15.9 26.9 14.1 11.5 20.6 12.9 10.6 19.6 11.8 9.9 19.1 15.1 Great Falls, Malmstrom AFB 33.7 16.5 31.6 16.1 29.9 15.7 18.4 29.5 17.4 28.1 16.5 27.0 14.8 12.0 21.5 13.7 11.1 20.5 12.3 10.2 20.2 14.6 Havre 34.3 17.1 32.0 16.6 30.2 16.3 19.0 30.4 17.8 28.8 16.9 27.9 15.4 12.0 22.0 14.2 11.1 20.4 13.3 10.5 19.9 15.5 Helena 32.4 15.6 30.7 15.2 28.8 14.8 17.2 27.9 16.3 27.4 15.6 26.6 13.8 11.4 19.9 12.5 10.4 19.1 11.3 9.7 19.1 15.6 Kalispell 31.7 16.7 29.8 16.2 27.9 15.7 18.1 28.3 17.2 27.3 16.3 25.9 14.7 11.7 20.7 13.6 10.9 19.6 12.6 10.2 19.4 16.6 Lewistown 31.9 15.9 29.7 15.6 27.8 15.2 18.0 27.4 16.9 26.6 16.1 25.7 14.7 12.2 21.5 13.5 11.3 20.5 12.4 10.5 19.4 15.7 Miles City 35.9 18.6 34.1 18.2 32.1 17.7 20.7 31.6 19.6 30.2 18.7 29.1 17.3 13.6 24.6 16.1 12.6 23.7 14.9 11.7 22.8 14.4 Missoula 33.0 16.7 31.2 16.2 29.2 15.7 18.1 28.6 17.2 27.9 16.4 27.0 14.7 11.7 20.2 13.5 10.9 20.2 12.6 10.2 18.9 17.4 NEBRASKA Bellevue, Offutt AFB 34.8 24.4 33.0 24.1 31.1 23.2 25.9 31.9 25.1 31.2 24.3 30.0 24.4 20.1 29.6 23.6 19.2 28.6 22.8 18.2 27.7 10.2 Grand Island 35.9 22.3 33.9 21.9 32.1 21.3 24.2 31.7 23.4 31.0 22.6 29.9 22.2 18.1 27.7 21.3 17.1 26.9 20.4 16.1 26.2 12.4 Lincoln 36.3 23.6 34.6 23.2 33.0 22.8 25.5 32.0 24.7 31.5 24.0 30.6 23.8 19.5 29.1 23.0 18.5 28.4 21.9 17.3 27.7 12.4 Norfolk 35.0 23.4 33.3 22.3 31.7 21.9 24.7 32.1 23.9 31.2 23.0 30.0 22.6 18.4 28.6 21.7 17.3 27.7 20.8 16.4 27.1 11.6 North Platte 35.0 20.5 33.2 20.5 31.4 19.9 22.8 30.8 21.9 30.1 21.2 29.3 20.6 16.9 26.8 19.6 15.9 25.8 18.7 15.0 25.1 14.2 Omaha, Eppley Airfield 35.0 24.0 33.2 23.6 31.6 23.0 25.7 32.3 24.8 31.3 23.9 30.2 23.8 19.4 29.7 22.9 18.3 29.0 22.0 17.3 28.0 11.1 Omaha, WSO 34.2 23.7 32.5 23.9 30.8 22.6 25.2 31.7 24.3 30.5 23.6 29.7 23.5 19.2 28.9 22.4 18.0 28.1 21.6 17.1 27.6 9.8 Scottsbluff 35.2 18.2 33.4 17.9 31.7 17.9 20.6 30.3 19.7 29.5 19.0 28.7 17.6 14.6 24.2 16.7 13.8 23.2 15.8 13.0 22.9 16.1 Sidney 35.0 17.3 33.5 17.1 31.4 17.0 19.6 29.1 18.7 28.9 18.0 28.7 16.6 13.9 22.6 15.6 13.0 22.4 14.7 12.3 21.8 15.5 Valentine 36.1 20.2 34.3 19.7 32.1 19.4 22.3 32.2 21.5 31.7 20.6 30.6 19.5 15.7 26.2 18.5 14.7 25.4 17.1 13.4 24.9 14.7 NEVADA Elko 34.9 15.5 33.5 14.9 32.0 14.4 17.2 29.3 16.3 29.1 15.5 28.6 13.9 12.0 19.9 12.2 10.7 19.1 10.4 9.5 19.4 21.3 Ely 31.8 13.6 30.6 13.3 29.3 12.9 15.7 25.6 15.0 25.6 14.2 25.4 12.8 11.7 17.9 11.5 10.7 17.8 10.1 9.7 18.2 19.2 Las Vegas, Intl Airport 42.2 18.9 40.9 18.6 39.6 18.2 21.9 34.8 21.2 33.9 20.6 33.7 18.6 14.5 26.2 17.1 13.2 27.4 15.6 12.0 29.4 13.8 Mercury 39.1 18.2 38.0 17.8 36.5 17.2 20.3 31.2 19.6 31.9 18.9 31.8 17.9 14.5 22.2 15.8 12.7 24.8 14.2 11.4 26.5 14.4 North Las Vegas, Nellis AFB 42.1 19.9 41.0 19.5 39.8 19.0 22.4 34.2 21.7 34.6 21.0 34.2 19.4 15.2 26.1 17.9 13.8 27.9 16.1 12.3 29.1 14.6 Reno 34.8 15.8 33.4 15.4 31.9 14.9 17.2 30.7 16.4 30.2 15.7 29.4 13.1 11.0 20.4 11.4 9.9 20.3 10.0 9.0 20.1 20.7 Tonopah 34.4 14.4 33.1 14.1 31.9 13.7 16.8 28.3 16.0 27.9 15.3 27.4 13.4 11.8 19.3 11.9 10.6 20.2 10.2 9.5 20.6 17.3 Winnemucca 36.0 15.8 34.7 15.4 33.2 14.8 17.4 31.1 16.6 30.7 15.7 30.1 13.4 11.3 20.1 11.5 9.9 19.6 9.8 8.8 19.7 20.8 NEW HAMPSHIRE Concord 32.1 21.7 30.3 21.1 28.8 20.2 23.6 29.4 22.6 27.8 21.7 26.3 21.9 16.8 25.9 21.1 15.9 24.9 20.2 15.0 24.3 13.4 Lebanon 31.3 21.4 29.8 20.6 28.5 19.9 23.2 29.1 22.1 27.7 21.2 26.1 21.2 16.2 26.2 20.5 15.5 24.8 19.7 14.7 23.7 12.8 Mount Washington 15.6 13.1 14.5 12.4 13.5 12.0 14.4 15.0 13.5 13.9 12.5 13.3 14.2 12.8 14.7 13.3 12.0 13.8 12.1 11.1 12.6 4.7 Portsmouth, Pease AFB 31.4 22.5 29.7 21.3 28.2 20.9 24.1 29.0 23.0 27.7 22.0 26.3 22.8 17.6 29.5 21.4 16.1 25.2 20.5 15.2 24.4 10.1 NEW JERSEY Atlantic City 32.9 23.4 31.3 23.0 29.8 22.4 25.1 30.3 24.4 28.9 23.7 27.9 23.8 18.7 27.3 23.1 17.9 26.6 22.4 17.2 26.0 10.1 Millville 33.6 23.9 31.9 23.2 30.6 22.8 25.3 30.8 24.7 29.8 24.1 28.6 24.1 19.1 27.4 23.5 18.4 26.8 23.0 17.8 26.5 10.4 Newark 33.9 23.4 32.2 22.6 30.7 21.9 24.8 30.9 24.2 29.6 23.4 28.3 23.3 18.1 27.4 22.6 17.3 26.9 21.8 16.5 26.5 8.8 Teterboro 33.6 24.2 31.7 23.6 30.3 22.7 25.7 31.3 24.8 30.3 24.0 28.6 24.3 19.2 28.7 23.5 18.3 27.5 22.3 17.0 27.1 10.2 Trenton, McGuire AFB 33.8 24.0 32.0 23.5 30.7 22.8 25.5 31.8 24.7 30.4 23.9 28.9 23.9 18.9 28.6 23.2 18.1 27.5 22.1 16.9 26.8 10.5 NEW MEXICO Alamogordo, Holloman AFB 36.5 17.3 35.3 17.3 34.1 17.3 20.2 30.3 19.7 29.7 19.2 29.2 18.1 15.2 22.3 16.8 14.0 22.3 16.0 13.2 22.6 16.8 Albuquerque 35.3 15.7 33.9 15.6 32.5 15.5 18.5 28.6 18.0 28.0 17.5 27.3 16.2 14.0 20.0 15.4 13.3 20.3 14.6 12.7 20.7 14.1 Carlsbad 38.5 18.3 36.7 18.7 35.5 18.9 22.5 31.0 21.9 30.4 21.3 29.5 20.6 17.3 24.4 20.0 16.6 24.4 19.3 15.9 23.7 14.1 Clayton 34.3 16.6 33.0 16.7 31.3 16.8 19.2 28.7 18.6 28.7 18.1 27.9 16.3 14.0 22.0 15.7 13.4 21.5 15.1 12.9 21.0 14.5 DP = dew-point temperature, °C MWB = mean coincident wet-bulb temperature, °C MDB = mean coincident dry-bulb temperature, °C MWS = mean coincident wind speed, m/s StdD = standard deviation, °C HR = humidity ratio, grams of moisture per kilogram of dry air A = airport ANGB = Air National Guard Base MCAF = Marine Corps Air Facility NAF = Naval Air Facility NAWS = Naval Air Weapons Station RAF = Royal Air Force 27.16 2001 ASHRAE Fundamentals Handbook (SI) Table 1A Heating and Wind Design Conditions—United States Heating Dry Bulb Extreme Wind Speed, m/s Coldest Month MWS/PWD to DB Extr. Annual Daily Lat.
Long.
Elev., m StdP, kPa 0.4% 1% 99.6% 0.4% Mean DB StdD DB Station WMO# Dates 99.6% 99% 1% 2.5% 5% WS MDB WS MDB MWS PWD MWS PWD Max.
Min.
Max. Min.
1a 1b 1c 1d 1e 1f 1g 2a 2b 3a 3b 3c 4a 4b 4c 4d 5a 5b 5c 5d 6a 6b 6c 6d Clovis, Cannon AFB 722686 34.38 103.32 1309 86.56 8293 −12.0 −9.2 11.6 10.1 8.8 11.7 4.5 10.3 3.8 3.4 50 5.0 220 38.3 −14.8 1.3 2.2 Farmington 723658 36.75 108.23 1677 82.74 8293 −13.3 −10.8 10.4 9.5 8.2 9.9 1.9 8.4 1.3 2.5 60 4.6 240 37.0 −18.1 2.1 4.0 Gallup 723627 35.52 108.78 1972 79.77 8293 −18.1 −15.1 10.1 8.8 8.1 8.7 3.8 7.9 2.6 0.6 140 5.0 270 34.3 −24.5 1.3 4.4 Roswell 722680 33.30 104.53 1118 88.60 8293 −9.8 −6.8 9.6 8.4 7.5 9.1 10.5 8.0 9.0 3.4 360 4.9 140 40.5 −14.2 2.6 3.6 Truth Or Consequences 722710 33.23 107.27 1481 84.75 8293 −5.6 −3.2 11.0 9.5 8.2 10.7 6.1 9.3 5.1 3.5 350 4.4 170 38.7 −14.6 1.6 13.2 Tucumcari 723676 35.18 103.60 1239 87.30 6193 −12.6 −9.5 11.1 9.9 9.0 12.5 10.2 10.2 7.2 3.3 50 5.2 230 39.1 −17.1 1.7 4.0 NEW YORK Albany 725180 42.75 73.80 89 100.26 6193 −21.9 −18.8 10.9 9.6 8.6 10.0 −6.6 9.0 −5.8 2.1 300 4.6 230 34.8 −27.6 1.7 3.5 Binghamton 725150 42.22 75.98 497 95.50 6193 −18.9 −16.6 10.6 9.4 8.5 10.9 −6.7 9.6 −7.0 5.6 270 4.8 220 31.7 −22.8 1.7 2.4 Buffalo 725280 42.93 78.73 215 98.77 6193 −16.8 −14.8 13.1 11.4 10.0 15.0 −3.8 13.3 −4.6 5.2 270 5.8 240 32.5 −21.1 1.3 2.9 Central Islip 725035 40.80 73.10 30 100.97 8293 −11.7 −9.6 10.0 9.0 8.2 10.3 0.1 9.4 −0.7 4.6 340 4.7 210 34.6 −16.4 1.8 3.0 Elmira/Corning 725156 42.17 76.90 291 97.88 8293 −18.9 −15.9 9.4 8.3 7.6 10.2 −6.7 9.1 −3.0 2.4 240 5.0 210 35.0 −23.5 2.0 3.4 Glens Falls 725185 43.33 73.62 100 100.13 8293 −23.1 −19.8 8.2 7.3 6.3 8.3 −5.6 7.4 −5.4 1.1 350 4.4 190 34.1 −28.9 1.5 2.6 Massena 726223 44.93 74.85 65 100.55 6193 −26.3 −23.2 9.5 7.9 7.4 10.4 −5.4 9.4 −5.7 1.6 270 4.6 230 33.1 −32.9 1.7 3.3 New York, JFK Airport 744860 40.65 73.78 7 101.24 6193 −11.4 −9.2 12.1 10.5 9.5 13.2 −1.4 11.9 −2.3 7.6 320 5.9 230 35.3 −14.7 1.4 2.6 New York, La Guardia A 725030 40.77 73.90 9 101.22 8293 −10.7 −8.1 12.5 11.0 9.9 13.2 −1.6 12.0 −2.4 7.9 310 5.5 280 36.0 −14.5 1.2 2.4 Newburgh 725038 41.50 74.10 150 99.54 8293 −14.7 −12.0 10.2 9.0 8.2 11.7 −8.6 10.2 −3.6 3.5 260 4.4 230 33.3 −20.1 1.7 3.6 Niagara Falls 725287 43.10 78.95 180 99.18 8293 −15.7 −13.9 11.6 10.0 9.1 13.5 −4.3 12.1 −5.1 5.1 240 5.8 230 32.9 −20.2 1.8 3.4 Plattsburgh, AFB 726225 44.65 73.47 72 100.46 8293 −22.6 −20.2 9.2 8.1 7.1 9.8 −3.0 8.6 −4.7 1.0 350 3.5 260 33.7 −27.4 1.5 2.6 Poughkeepsie 725036 41.63 73.88 51 100.71 8293 −16.7 −14.2 8.1 7.1 6.1 8.4 −3.9 7.8 −3.9 1.2 250 3.9 250 35.3 −22.5 1.7 3.3 Rochester 725290 43.12 77.67 169 99.31 6193 −17.2 −15.0 11.9 10.4 9.3 12.9 −5.3 11.6 −6.1 4.4 230 5.4 250 34.0 −21.4 1.6 2.7 Rome, Griffiss AFB 725196 43.23 75.40 154 99.49 8293 −20.6 −17.4 9.7 8.3 7.2 10.4 −5.3 8.8 −5.5 1.4 330 3.4 260 33.9 −25.9 1.4 2.1 Syracuse 725190 43.12 76.12 124 99.84 6193 −19.6 −16.4 11.4 10.0 9.0 12.7 −6.8 11.1 −5.9 3.3 90 4.8 250 33.7 −24.7 1.7 3.3 Watertown 726227 44.00 76.02 99 100.14 8293 −24.7 −20.9 9.5 8.5 8.0 10.5 −4.3 9.3 −4.0 2.2 80 5.1 240 32.0 −31.5 1.6 3.9 White Plains 725037 41.07 73.70 134 99.73 8293 −13.8 −11.1 8.4 7.6 6.5 8.5 −1.4 8.1 −1.4 5.7 310 3.9 260 34.9 −17.8 1.6 2.5 NORTH CAROLINA Asheville 723150 35.43 82.55 661 93.63 6193 −11.4 −8.8 11.0 9.6 8.5 11.6 −3.3 10.3 −2.0 5.0 340 3.9 340 33.0 −16.3 1.4 3.8 Cape Hatteras 723040 35.27 75.55 3 101.29 6193 −3.6 −1.9 11.4 10.0 8.8 11.9 8.4 10.4 8.5 4.9 340 4.8 230 32.8 −6.8 1.1 2.7 Charlotte 723140 35.22 80.93 234 98.55 6193 −7.6 −5.1 8.8 7.7 6.8 9.1 6.8 8.1 7.1 2.8 50 3.8 240 35.9 −12.0 1.6 3.3 Cherry Point, MCAS 723090 34.90 76.88 9 101.22 8293 −4.7 −2.2 8.3 7.3 6.5 8.7 6.1 7.8 8.8 2.3 10 3.3 240 37.5 −11.0 1.4 4.7 Fayetteville, Fort Bragg 746930 35.13 78.93 74 100.44 8293 −5.5 −3.0 7.6 6.4 5.4 8.4 5.3 7.3 6.4 1.9 10 2.7 240 37.9 −9.7 2.1 3.8 Goldsboro, Johnson AFB 723066 35.33 77.97 33 100.93 8293 −5.6 −3.0 7.5 6.3 5.5 8.2 7.9 6.9 6.9 2.0 270 3.4 260 37.9 −10.2 1.7 4.1 Greensboro 723170 36.08 79.95 270 98.12 6193 −9.7 −7.2 8.6 7.7 6.8 8.9 4.3 8.0 4.2 3.2 290 3.8 230 35.7 −14.2 1.5 2.8 Hickory 723145 35.73 81.38 362 97.05 8293 −7.7 −5.2 7.8 6.5 5.9 8.2 4.9 7.1 5.2 1.9 320 3.9 240 36.0 −13.2 1.8 3.8 Jacksonville, New River MCAF 723096 34.72 77.45 8 101.23 8293 −5.1 −2.6 8.2 7.1 6.3 8.4 9.6 7.4 8.3 2.4 350 3.3 240 37.0 −10.5 1.1 4.9 New Bern 723095 35.07 77.05 6 101.25 8293 −5.4 −2.8 8.1 7.1 6.3 8.3 9.6 7.4 8.4 2.5 10 3.6 240 37.3 −10.5 0.8 4.6 Raleigh/Durham 723060 35.87 78.78 134 99.73 6193 −9.1 −6.4 9.2 8.1 7.1 9.5 5.5 8.4 5.9 3.4 360 4.1 240 35.6 −13.0 1.6 2.9 Wilmington 723013 34.27 77.90 10 101.20 6193 −4.9 −2.9 9.6 8.3 7.5 9.9 10.3 8.8 8.8 3.3 320 4.6 220 36.1 −8.6 1.2 3.2 Winston−Salem 723193 36.13 80.22 296 97.82 8293 −8.0 −5.2 8.4 7.7 6.5 9.3 3.1 8.3 3.6 3.3 290 3.6 240 35.7 −13.1 1.5 3.2 NORTH DAKOTA Bismarck 727640 46.77 100.75 506 95.39 6193 −29.6 −26.6 13.0 11.2 10.0 13.0−10.4 11.3 −8.8 3.0 290 5.9 180 37.9 −34.7 2.0 3.6 Devils Lake 727580 48.10 98.87 443 96.12 8293 −30.6 −28.1 11.4 10.0 8.8 12.1−11.1 10.5−12.0 4.0 300 5.0 10 36.4 −32.7 3.9 2.8 Fargo 727530 46.90 96.80 274 98.08 6193 −29.7 −27.3 13.6 12.0 10.7 14.4−14.2 12.5−13.9 3.6 180 6.4 160 36.6 −32.7 1.9 2.4 Grand Forks, AFB 727575 47.97 97.40 278 98.03 8293 −29.1 −26.6 12.2 10.5 9.4 13.3−12.9 11.6−10.6 3.3 290 5.6 180 36.9 −31.9 2.6 2.7 Minot, AFB 727675 48.42 101.35 508 95.37 8293 −29.2 −26.5 12.5 10.5 9.4 13.6 −7.5 12.2 −8.9 4.5 310 5.3 150 38.1 −31.7 2.1 4.1 Minot, Intl Airport 727676 48.27 101.28 523 95.20 6193 −28.9 −26.6 12.3 10.7 9.8 13.4 −9.8 11.9−10.1 5.3 290 5.8 200 36.9 −31.9 1.7 2.7 Williston 727670 48.18 103.63 581 94.54 8293 −31.0 −27.7 12.2 10.4 9.3 12.6 −3.8 10.6 −6.7 3.6 220 6.4 150 38.2 −34.3 2.5 4.6 OHIO Akron/Canton 725210 40.92 81.43 377 96.88 6193 −17.9 −14.9 10.6 9.3 8.3 11.2 −3.6 10.0 −3.6 5.0 270 4.6 230 33.3 −21.8 1.6 3.9 Cincinnati, Lunken Field 724297 39.10 84.42 147 99.57 8293 −14.9 −11.3 9.2 8.3 7.6 9.7 1.8 8.7 0.8 3.8 260 4.3 210 35.8 −19.7 1.8 5.2 Cleveland 725240 41.42 81.87 245 98.42 6193 −17.4 −14.7 11.5 10.2 9.1 12.0 −2.2 10.5 −2.5 5.5 230 5.3 230 33.9 −21.3 1.6 3.5 Columbus, Intl Airport 724280 40.00 82.88 249 98.37 6193 −17.4 −14.3 10.4 9.1 8.1 10.6 −1.3 9.3 −4.0 4.1 270 4.7 270 34.5 −20.9 1.4 3.9 Columbus, Rickenbckr AFB 724285 39.82 82.93 227 98.63 8293 −16.3 −12.5 9.2 8.0 7.1 10.2 −3.4 9.1 −2.8 3.2 210 3.7 270 35.4 −20.2 2.5 4.8 Dayton, Intl Airport 724290 39.90 84.20 306 97.70 6193 −18.2 −15.2 10.7 9.5 8.5 11.1 −3.4 9.8 −2.4 4.8 270 4.8 240 34.8 −22.2 1.6 3.9 Dayton, Wright-Patterson 745700 39.83 84.05 251 98.35 8293 −17.1 −13.2 9.4 8.2 7.2 10.4 −2.3 9.3 −1.0 3.1 270 3.8 240 35.7 −21.5 1.8 4.3 Findlay 725366 41.02 83.67 247 98.39 8293 −18.9 −15.3 10.2 9.1 8.3 11.1 1.0 9.9 −1.8 4.8 250 5.3 210 34.4 −22.6 2.1 4.4 Mansfield 725246 40.82 82.52 395 96.67 6193 −18.6 −15.8 11.1 10.0 9.0 12.5 −2.5 11.1 −3.3 5.6 240 5.3 240 33.0 −22.2 1.6 3.3 Toledo 725360 41.60 83.80 211 98.82 6193 −19.0 −16.2 10.1 8.8 7.9 11.2 −4.1 9.9 −6.1 4.4 250 5.0 230 35.1 −23.3 1.7 3.0 Youngstown 725250 41.27 80.67 361 97.06 6193 −18.1 −15.4 10.4 9.2 8.4 10.8 −5.4 9.7 −6.1 4.6 230 4.6 230 33.0 −22.0 1.4 3.2 Zanesville 724286 39.95 81.90 274 98.08 8293 −16.8 −12.8 8.7 7.9 7.2 9.3 0.0 8.4 −0.6 3.1 240 4.2 220 34.7 −21.5 2.0 4.7 OKLAHOMA Altus, AFB 723520 34.67 99.27 420 96.38 8293 −10.6 −7.1 10.3 9.3 8.3 10.5 4.2 9.5 5.3 3.9 20 4.4 190 41.5 −13.9 1.9 4.3 Enid, Vance AFB 723535 36.33 97.92 398 96.63 8293 −14.8 −11.0 11.5 10.1 9.0 11.9 3.4 10.3 3.2 5.4 10 4.9 190 40.3 −17.1 2.0 3.7 Lawton, Fort Sill/Post Field 723550 34.65 98.40 362 97.05 8293 −11.3 −7.3 10.6 9.4 8.4 11.8 1.8 10.0 2.4 4.9 10 4.8 170 39.5 −13.4 1.4 4.1 McAlester 723566 34.88 95.78 235 98.53 8293 −12.2 −8.1 8.9 8.1 7.3 9.4 8.5 8.4 7.1 3.8 360 4.1 190 38.8 −15.3 2.2 4.6 Oklahoma City, Tinker AFB 723540 35.42 97.38 394 96.68 8293 −12.1 −8.5 10.7 9.6 8.7 11.0 5.7 9.7 5.5 4.6 10 4.8 190 39.3 −14.3 1.8 3.4 Oklahoma City, W. Rogers A 723530 35.40 97.60 397 96.65 6193 −12.6 −9.6 12.9 11.3 10.3 13.0 0.4 11.5 2.8 6.9 360 5.9 180 39.3 −15.7 1.9 2.7 Tulsa 723560 36.20 95.90 206 98.87 6193 −12.9 −10.1 11.3 10.3 9.2 10.8 7.6 9.7 4.7 4.9 360 5.4 180 39.4 −16.4 2.0 3.1 OREGON Astoria 727910 46.15 123.88 7 101.24 6193 −3.8 −1.8 11.3 9.7 8.5 12.8 10.3 10.9 9.3 3.6 90 5.2 320 30.3 −6.9 2.5 3.4 WMO# = World Meteorological Organization number AFB = Air Force Base NAS = Naval Air Station WSO = Weather Service Office AAF = Army Air Field MCAS = Marine Corps Air Station DB = dry-bulb temperature, °C Lat. = North latitude ° Long. = West longitude ° Elev. = elevation, m StdP = standard pressure at station elevation, kPa WS = wind speed, m/s PWD = prevailing wind direction, ° True Climatic Design Information 27.17 Table 1B Cooling and Dehumidification Design Conditions—United States Station Cooling DB/MWB Evaporation WB/MDB Dehumidification DP/MDB and HR Range of DB 0.4% 1% 2% 0.4% 1% 2% 0.4% 1% 2% DB MWB DB MWB DB MWB WB MDB WB MDB WB MDB DP HR MDB DP HR MDB DP HR MDB 1 2a 2b 2c 2d 2e 2f 3a 3b 3c 3d 3e 3f 4a 4b 4c 4d 4e 4f 4g 4h 4i 5 Clovis, Cannon AFB 35.3 17.9 34.0 18.0 32.6 18.2 21.1 29.1 20.5 28.6 20.0 28.5 19.1 16.3 24.0 18.4 15.6 23.3 17.8 15.0 22.9 13.6 Farmington 34.3 15.8 33.1 15.7 31.5 15.5 18.3 28.6 17.7 28.1 17.1 27.5 15.5 13.5 20.4 14.7 12.8 20.2 13.9 12.2 20.4 16.0 Gallup 31.9 13.8 30.7 13.6 29.6 13.6 16.7 24.7 16.1 24.6 15.6 24.1 14.8 13.4 18.1 14.1 12.8 17.7 13.4 12.2 17.9 17.0 Roswell 36.8 18.2 35.6 18.1 34.5 18.1 21.2 30.3 20.7 30.1 20.2 29.4 19.1 15.9 23.0 18.6 15.4 22.9 18.0 14.8 22.8 13.8 Truth Or Consequences 36.3 16.3 35.1 16.2 33.9 16.1 18.9 29.5 18.4 29.3 17.9 29.0 15.8 13.5 21.7 15.2 12.9 21.7 14.6 12.4 22.1 13.9 Tucumcari 36.4 17.9 35.1 18.1 33.7 17.9 20.8 30.6 20.2 29.7 19.7 28.8 18.5 15.6 22.7 17.8 14.9 22.6 17.2 14.3 22.4 13.8 NEW YORK Albany 32.0 21.8 30.2 21.1 28.7 20.3 23.6 29.3 22.6 27.8 21.8 26.3 21.9 16.8 25.9 21.1 15.9 24.9 20.2 15.1 24.3 13.2 Binghamton 29.4 21.2 27.9 20.4 26.5 19.6 22.6 27.3 21.7 26.0 20.8 24.8 21.2 16.8 25.2 20.3 15.9 24.1 19.5 15.1 23.2 9.7 Buffalo 30.0 21.2 28.7 20.7 27.4 20.0 23.2 27.6 22.3 26.6 21.6 25.5 21.8 16.9 25.8 21.0 16.1 24.9 20.2 15.2 24.1 9.8 Central Islip 31.1 22.6 29.7 22.1 28.3 21.3 24.4 28.6 23.7 27.3 23.1 26.0 23.5 18.4 26.3 22.9 17.7 25.4 21.9 16.6 24.8 8.4 Elmira/Corning 32.1 22.4 30.5 21.4 29.0 20.5 24.0 29.9 23.0 28.0 22.0 26.8 22.1 17.4 27.0 21.3 16.5 25.6 20.5 15.7 24.3 13.4 Glens Falls 31.2 22.6 29.7 21.5 28.2 20.9 24.4 29.3 23.2 27.7 22.1 26.4 23.1 18.1 27.1 21.6 16.5 25.9 20.7 15.5 24.6 12.3 Massena 30.7 22.4 29.1 21.4 27.6 20.5 23.7 28.9 22.7 27.4 21.7 25.9 22.1 16.9 26.8 21.1 15.9 25.4 20.2 15.0 24.5 12.1 New York, JFK Airport 32.5 23.1 30.9 22.4 29.5 21.7 24.6 29.9 23.8 28.8 23.1 27.6 23.1 17.8 26.8 22.4 17.1 26.5 21.7 16.3 25.9 7.7 New York, La Guardia A 33.5 23.5 31.5 22.8 30.2 22.2 25.0 30.8 24.2 29.3 23.6 28.1 23.6 18.4 27.0 23.0 17.8 26.5 21.9 16.6 26.5 8.1 Newburgh 31.1 23.1 29.7 22.1 28.5 21.2 24.7 29.2 23.6 28.1 22.6 26.9 23.5 18.6 27.6 22.0 17.0 26.6 21.0 15.9 25.3 9.5 Niagara Falls 30.7 22.2 29.4 21.5 28.2 20.6 24.1 29.0 23.1 27.2 22.1 26.0 22.8 17.9 27.2 21.6 16.6 25.4 20.8 15.8 24.5 10.5 Plattsburgh, AFB 30.2 21.7 28.6 20.7 26.6 20.0 23.2 28.0 22.2 26.8 21.2 25.3 21.6 16.4 26.1 20.6 15.4 24.7 19.7 14.5 23.8 10.9 Poughkeepsie 33.1 23.7 31.2 22.4 29.5 21.5 24.7 30.6 23.7 29.3 22.8 27.6 23.1 18.0 27.8 21.8 16.6 26.6 21.1 15.9 25.8 12.8 Rochester 31.4 22.6 29.9 21.7 28.4 20.8 24.1 29.2 23.1 27.7 22.1 26.7 22.5 17.6 27.2 21.6 16.5 25.9 20.7 15.6 24.8 11.2 Rome, Griffiss AFB 31.3 21.9 29.8 21.0 28.4 20.3 23.6 29.0 22.6 27.5 21.6 26.3 21.7 16.7 26.7 20.9 15.9 25.5 20.0 15.0 24.2 12.7 Syracuse 31.2 22.4 29.7 21.5 28.3 20.9 23.8 29.2 22.9 28.0 22.0 26.6 22.3 17.2 26.9 21.3 16.2 25.7 20.4 15.3 24.8 11.3 Watertown 29.5 21.7 28.1 21.2 26.4 20.6 23.4 27.7 22.5 26.4 21.6 25.1 21.9 16.8 25.7 21.1 15.9 25.0 20.3 15.2 24.1 11.4 White Plains 31.9 23.3 30.4 22.5 28.9 21.4 24.6 30.2 23.7 28.4 22.9 26.9 23.2 18.3 26.9 22.2 17.2 26.4 21.4 16.3 25.3 10.0 NORTH CAROLINA Asheville 31.0 22.2 29.6 21.7 28.4 21.3 23.6 28.8 22.9 27.8 22.3 26.8 22.2 18.3 25.8 21.5 17.5 25.3 20.9 16.9 24.6 10.8 Cape Hatteras 30.8 25.6 30.1 25.2 29.3 24.8 26.6 29.7 26.1 28.8 25.6 28.4 25.7 21.0 28.5 25.2 20.3 28.1 24.7 19.7 27.5 6.3 Charlotte 34.2 23.4 32.8 23.2 31.6 22.8 24.9 31.2 24.3 30.6 23.8 29.8 23.2 18.5 27.5 22.7 17.9 26.9 22.2 17.4 26.4 9.9 Cherry Point, MCAS 34.8 26.2 33.6 25.5 32.1 25.1 27.2 33.0 26.5 32.1 25.9 31.0 25.6 20.9 30.5 25.0 20.1 30.0 24.5 19.5 29.2 9.2 Fayetteville, Fort Bragg 35.7 24.9 34.5 24.5 33.1 24.1 26.3 32.8 25.7 31.9 25.1 31.1 24.7 19.9 29.1 24.2 19.3 28.6 23.7 18.7 28.1 10.1 Goldsboro, Johnson AFB 35.6 25.2 34.2 24.5 32.9 24.3 26.4 32.8 25.8 31.7 25.2 30.8 24.7 19.8 29.0 24.3 19.3 28.4 23.9 18.8 27.8 10.2 Greensboro 33.4 23.8 32.2 23.3 30.9 22.8 25.1 30.8 24.4 30.1 23.8 29.4 23.5 18.9 27.6 22.9 18.2 27.0 22.3 17.6 26.4 10.3 Hickory 34.2 22.6 32.8 22.5 31.2 22.5 24.6 30.3 24.0 29.7 23.5 29.0 23.4 19.0 26.7 22.8 18.3 26.6 21.8 17.2 25.5 10.9 Jacksonville, New River MCAF 34.6 25.9 33.4 25.4 31.8 24.8 27.1 32.4 26.3 31.6 25.7 30.7 25.5 20.7 30.0 24.9 20.0 29.4 24.4 19.4 28.7 9.5 New Bern 34.6 25.6 33.6 25.3 32.0 24.6 27.0 32.7 26.2 31.7 25.6 30.5 25.4 20.6 29.9 24.8 19.9 29.0 24.3 19.2 28.2 9.5 Raleigh/Durham 33.8 24.4 32.4 23.7 31.2 23.2 25.5 31.3 24.9 30.4 24.3 29.5 24.0 19.2 27.9 23.4 18.5 27.3 22.9 17.9 26.7 10.4 Wilmington 33.9 25.8 32.7 25.3 31.5 24.8 26.8 31.9 26.2 31.0 25.7 30.1 25.6 20.8 29.2 25.0 20.1 28.6 24.5 19.5 28.1 8.7 Winston-Salem 33.6 23.4 31.9 23.1 30.8 22.7 24.8 30.0 24.3 29.8 23.7 29.2 23.6 19.1 27.3 23.0 18.4 26.9 22.0 17.3 26.2 9.8 NORTH DAKOTA Bismarck 34.1 19.9 32.1 19.4 30.1 18.8 22.3 29.9 21.1 29.1 20.1 27.8 19.9 15.5 26.3 18.6 14.3 24.8 17.3 13.2 23.7 14.7 Devils Lake 33.0 20.5 30.6 19.4 28.9 18.7 22.3 30.0 21.0 28.3 19.8 26.4 20.0 15.5 25.6 18.7 14.3 25.1 17.0 12.8 24.0 11.7 Fargo 32.9 21.8 31.0 21.2 29.3 20.3 23.9 30.2 22.8 28.7 21.6 27.4 22.1 17.4 27.6 20.8 16.0 26.7 19.6 14.8 24.9 12.4 Grand Forks, AFB 33.0 21.4 30.9 20.6 29.2 19.8 23.7 29.9 22.4 28.6 21.1 26.9 21.7 16.9 27.3 20.4 15.6 25.6 19.2 14.4 24.3 12.7 Minot, AFB 34.7 20.2 32.1 19.5 30.1 18.9 22.5 30.7 21.2 29.7 20.0 28.0 20.0 15.6 26.4 18.6 14.3 25.3 17.0 12.9 23.9 13.7 Minot, Intl Airport 33.3 19.6 31.1 18.8 29.1 18.2 21.7 29.2 20.5 28.2 19.3 26.4 19.4 15.1 25.4 17.9 13.7 23.9 16.7 12.7 22.7 12.7 Williston 35.4 19.3 33.3 18.7 30.8 18.1 21.6 30.7 20.4 29.9 19.3 28.5 18.9 14.7 25.6 17.1 13.1 24.2 16.0 12.2 23.0 14.3 OHIO Akron/Canton 31.1 22.2 29.6 21.6 28.3 20.9 23.8 29.0 22.9 27.6 22.1 26.6 22.3 17.8 26.7 21.4 16.8 25.7 20.7 16.1 24.9 10.4 Cincinnati, Lunken Field 34.1 23.4 32.2 23.9 31.1 22.6 25.2 31.4 24.6 30.5 24.0 29.1 23.7 18.9 27.8 23.2 18.3 27.1 22.1 17.1 26.9 11.1 Cleveland 31.4 22.9 30.0 22.1 28.6 21.4 24.2 29.7 23.3 28.3 22.4 27.1 22.6 17.8 27.6 21.7 16.9 26.4 20.9 16.0 25.4 10.3 Columbus, Intl Airport 32.4 23.1 31.1 22.7 29.7 21.9 24.7 30.5 23.9 29.2 23.1 27.8 23.0 18.3 27.8 22.3 17.5 27.1 21.5 16.7 26.2 10.7 Columbus, Rickenbckr AFB 33.6 23.6 31.7 22.9 30.3 22.3 25.0 31.2 24.1 30.0 23.3 28.7 23.3 18.6 28.4 22.1 17.2 27.7 21.3 16.4 26.3 11.0 Dayton, Intl Airport 32.4 23.2 31.1 22.6 29.8 21.8 24.7 30.4 23.8 29.2 23.1 27.9 22.9 18.4 28.0 22.2 17.5 26.9 21.4 16.7 26.1 10.7 Dayton, Wright-Paterson 33.6 23.6 31.8 23.1 30.6 22.6 25.4 31.3 24.5 30.2 23.6 28.9 23.9 19.4 28.8 22.9 18.2 28.1 21.8 17.0 27.0 11.0 Findlay 32.2 23.2 30.8 22.3 29.4 21.5 24.7 30.0 23.8 28.5 23.0 27.4 23.5 18.9 27.4 22.1 17.3 26.6 21.4 16.5 25.5 10.5 Mansfield 31.0 22.8 29.6 22.2 28.3 21.4 24.2 29.2 23.4 28.1 22.6 26.8 22.7 18.3 27.1 21.9 17.4 26.3 21.2 16.6 25.4 9.9 Toledo 32.4 22.9 30.8 22.2 29.3 21.5 24.7 29.8 23.8 28.7 22.9 27.3 23.2 18.4 28.0 22.3 17.4 26.7 21.3 16.4 25.7 11.6 Youngstown 30.8 22.2 29.4 21.3 28.1 20.6 23.6 28.9 22.7 27.5 21.8 26.2 22.0 17.4 26.4 21.2 16.5 25.4 20.3 15.7 24.6 11.4 Zanesville 32.3 23.5 31.0 22.9 29.8 21.7 24.6 30.6 23.8 29.2 23.1 28.0 23.1 18.5 27.5 22.0 17.2 26.7 21.4 16.6 25.8 11.5 OKLAHOMA Altus, AFB 39.0 22.8 37.8 22.9 36.2 23.0 25.2 34.0 24.6 33.4 24.1 32.8 23.2 18.9 28.7 22.1 17.7 28.6 21.5 17.0 28.0 13.1 Enid, Vance AFB 38.3 23.2 36.4 23.1 35.0 23.0 25.1 33.5 24.4 33.0 23.8 32.4 22.9 18.5 29.4 21.8 17.3 28.5 21.2 16.6 28.0 12.1 Lawton, Fort Sill/Post Field 37.2 22.7 36.0 23.0 34.8 22.8 25.2 32.5 24.6 32.0 24.1 31.4 23.6 19.3 28.3 22.9 18.4 27.7 21.9 17.3 27.0 11.5 McAlester 36.6 24.5 35.3 24.4 34.0 24.3 26.2 33.1 25.7 32.6 25.2 31.8 24.6 20.2 29.5 24.1 19.6 28.6 23.6 19.0 28.2 12.1 Oklahoma City, Tinker AFB 36.9 23.6 35.6 23.9 34.3 23.6 25.8 33.1 25.1 32.6 24.5 31.9 23.9 19.7 30.6 23.2 18.9 29.4 22.1 17.6 28.5 10.8 Oklahoma City, W. Rogers A 37.3 23.2 35.7 23.1 34.2 22.9 24.9 32.8 24.4 32.4 23.9 31.8 22.8 18.4 28.4 22.3 17.8 27.8 21.7 17.2 27.4 11.7 Tulsa 37.6 24.5 35.9 24.3 34.4 24.1 26.2 33.5 25.6 33.1 25.1 32.4 24.2 19.6 30.6 23.6 18.8 29.6 22.9 18.1 29.1 10.8 OREGON Astoria 24.7 17.6 22.2 16.8 20.5 16.0 18.3 23.7 17.2 21.5 16.4 19.9 16.1 11.5 20.6 15.3 10.9 19.1 14.7 10.5 18.2 7.9 DP = dew-point temperature, °C MWB = mean coincident wet-bulb temperature, °C MDB = mean coincident dry-bulb temperature, °C MWS = mean coincident wind speed, m/s StdD = standard deviation, °C HR = humidity ratio, grams of moisture per kilogram of dry air A = airport ANGB = Air National Guard Base MCAF = Marine Corps Air Facility NAF = Naval Air Facility NAWS = Naval Air Weapons Station RAF = Royal Air Force 27.18 2001 ASHRAE Fundamentals Handbook (SI) Table 1A Heating and Wind Design Conditions—United States Heating Dry Bulb Extreme Wind Speed, m/s Coldest Month MWS/PWD to DB Extr. Annual Daily Lat.
Long.
Elev., m StdP, kPa 0.4% 1% 99.6% 0.4% Mean DB StdD DB Station WMO# Dates 99.6% 99% 1% 2.5% 5% WS MDB WS MDB MWS PWD MWS PWD Max.
Min.
Max. Min.
1a 1b 1c 1d 1e 1f 1g 2a 2b 3a 3b 3c 4a 4b 4c 4d 5a 5b 5c 5d 6a 6b 6c 6d Eugene 726930 44.12 123.22 114 99.96 6193 −6.0 −3.6 9.1 7.9 7.1 9.6 7.9 8.5 7.2 3.5 360 5.2 360 37.0 −9.2 2.1 4.4 Hillsboro 726986 45.53 122.95 62 100.58 8293 −7.0 −4.2 8.5 7.8 6.7 10.2 −3.2 8.4 1.3 3.6 60 3.9 360 37.7 −9.5 2.1 3.5 Klamath Falls 725895 42.15 121.73 1247 87.21 8293 −15.8 −12.5 11.0 9.8 8.4 12.6 3.8 10.2 0.8 2.5 320 3.9 320 36.1 −19.9 2.3 4.8 Meacham 726885 45.52 118.40 1236 87.33 8293 −23.0 −17.7 5.2 4.4 3.8 5.7 0.7 5.0 0.3 0.6 130 2.3 360 33.8 −29.7 2.6 6.8 Medford 725970 42.37 122.87 405 96.55 6193 −6.3 −4.5 8.4 6.9 5.8 8.7 10.7 6.6 10.0 1.2 130 4.2 290 40.2 −9.4 1.9 3.6 North Bend 726917 43.42 124.25 4 101.28 6193 −1.3 0.2 11.1 10.0 9.1 10.1 10.8 8.9 10.1 2.9 140 6.2 340 27.9 −4.6 2.2 3.1 Pendleton 726880 45.68 118.85 456 95.97 6193 −15.9 −11.7 12.3 10.6 9.1 12.0 6.7 10.1 5.7 2.6 140 4.1 310 38.8 −18.1 2.0 6.2 Portland 726980 45.60 122.60 12 101.18 6193 −5.8 −2.9 11.0 9.3 8.1 12.5 2.7 11.1 3.6 5.7 120 5.0 340 37.1 −7.8 2.4 3.3 Redmond 726835 44.25 121.15 938 90.55 6193 −17.2 −12.7 8.9 7.6 7.1 9.1 5.7 8.2 4.9 2.5 320 4.7 340 36.7 −21.5 1.8 5.7 Salem 726940 44.92 123.00 61 100.59 6193 −6.6 −4.1 10.2 8.7 7.5 11.0 8.0 9.6 7.9 2.6 350 4.4 360 37.6 −10.0 1.8 3.7 Sexton Summit 725975 42.62 123.37 1171 88.03 8293 −6.1 −4.4 10.9 9.6 8.7 12.2 2.7 10.9 3.6 4.1 120 2.9 340 31.6 −8.7 2.7 4.7 PENNSYLVANIA Allentown 725170 40.65 75.43 117 99.93 6193 −14.9 −12.3 11.9 10.4 9.2 12.7 −3.3 11.2 −4.6 4.2 270 4.9 240 34.9 −18.7 1.6 2.9 Altoona 725126 40.30 78.32 458 95.94 8293 −15.0 −12.3 9.1 8.2 7.4 10.1 −6.8 9.1 −5.8 4.0 270 3.4 250 33.5 −20.5 2.0 4.4 Bradford 725266 41.80 78.63 653 93.72 6193 −21.2 −18.3 8.5 8.0 7.1 9.6 −5.8 8.4 −6.0 3.1 270 4.2 240 30.6 −26.3 1.6 2.8 Du Bois 725125 41.18 78.90 554 94.84 8293 −17.8 −15.0 9.3 8.3 7.6 10.3 −6.8 9.3 −6.5 5.0 280 4.3 270 32.0 −22.8 1.7 3.9 Erie 725260 42.08 80.18 225 98.65 6193 −16.6 −14.1 11.9 10.7 9.7 12.7 −2.5 11.4 −2.5 6.1 200 5.4 250 32.3 −20.2 1.7 3.6 Harrisburg 725115 40.20 76.77 94 100.20 6193 −13.0 −10.7 10.0 8.7 7.9 10.9 −1.8 9.7 −1.9 3.7 270 4.2 250 36.0 −16.7 1.8 3.2 Philadelphia, Intl Airport 724080 39.88 75.25 9 101.22 6193 −11.9 −9.7 10.9 9.6 8.5 11.7 −0.8 10.1 −1.4 5.2 290 4.8 230 35.7 −15.3 1.6 3.1 Philadelphia, Northeast A 724085 40.08 75.02 37 100.88 8293 −11.7 −9.5 9.3 8.3 7.6 9.8 −0.9 8.6 −1.4 4.5 300 4.4 260 36.2 −15.9 1.4 3.4 Philadelphia, Willow Gr NAS 724086 40.20 75.15 110 100.01 8293 −12.0 −9.8 7.9 6.8 5.9 8.3 −1.0 7.3 −1.2 2.4 300 2.8 250 37.2 −16.9 3.0 3.2 Pittsburgh, Allegheny Co. A 725205 40.35 79.93 382 96.82 8293 −15.3 −11.9 9.2 8.3 7.7 10.2 −4.2 9.2 −4.4 5.0 250 4.8 240 34.2 −20.2 1.7 5.2 Pittsburgh, Intl Airport 725200 40.50 80.22 373 96.92 6193 −16.9 −14.1 10.9 9.5 8.4 11.7 −4.7 10.2 −3.9 4.5 260 4.9 230 33.8 −21.0 1.7 3.7 Wilkes-Barre/Scranton 725130 41.33 75.73 289 97.90 6193 −16.7 −14.2 9.1 8.0 7.3 9.5 −3.1 8.4 −4.2 3.8 230 4.8 220 33.5 −20.6 1.6 2.7 Williamsport 725140 41.25 76.92 160 99.42 6193 −16.6 −13.7 10.3 9.0 8.1 10.7 −4.9 9.5 −3.8 3.4 270 4.6 250 34.4 −21.3 1.7 3.3 RHODE ISLAND Providence 725070 41.73 71.43 19 101.10 6193 −14.8 −12.3 11.9 10.4 9.2 12.1 −0.8 10.4 −0.2 5.1 340 5.7 230 35.1 −18.8 2.1 2.8 SOUTH CAROLINA Beaufort, MCAS 722085 32.48 80.72 12 101.18 8293 −2.3 −0.4 7.9 6.8 6.0 8.4 8.0 7.5 7.4 2.0 300 3.2 270 38.2 −10.8 1.6 4.6 Charleston 722080 32.90 80.03 15 101.14 6193 −3.9 −2.0 9.8 8.6 7.7 9.8 10.8 8.6 10.6 3.3 20 4.5 230 36.4 −7.8 1.3 3.1 Columbia 723100 33.95 81.12 69 100.50 6193 −6.3 −4.2 8.7 7.7 6.8 9.1 8.8 8.1 9.3 2.1 220 4.1 240 37.7 −10.6 1.7 3.1 Florence 723106 34.18 79.72 45 100.79 8293 −5.1 −2.7 8.5 7.8 6.9 8.8 10.5 8.0 9.9 3.0 360 4.3 240 38.0 −9.8 1.5 4.2 Greer/Greenville 723120 34.90 82.22 296 97.82 6193 −7.1 −4.8 8.9 7.9 7.0 9.2 7.3 8.2 6.8 2.7 50 3.9 230 35.9 −11.9 1.4 3.1 Myrtle Beach, AFB 747910 33.68 78.93 8 101.23 8293 −3.8 −1.4 7.9 6.9 6.0 7.9 9.3 6.9 8.2 1.6 360 3.1 290 36.9 −8.6 1.6 4.1 Sumter, Shaw AFB 747900 33.97 80.47 74 100.44 8293 −4.2 −1.9 8.1 7.0 6.1 8.6 8.9 7.5 8.8 2.2 10 3.5 240 37.5 −8.3 1.7 3.4 SOUTH DAKOTA Chamberlain 726530 43.80 99.32 530 95.12 8293 −25.0 −21.8 12.2 10.8 9.5 12.6 −8.0 11.0 −6.8 5.0 270 6.0 190 41.2 −24.5 4.510.2 Huron 726540 44.38 98.22 393 96.69 6193 −27.1 −24.4 13.0 11.3 10.0 12.9−10.2 11.3 −9.3 3.9 290 6.4 180 38.8 −31.6 2.6 3.3 Pierre 726686 44.38 100.28 531 95.11 6193 −25.4 −22.5 12.9 11.2 10.0 14.1 −9.6 12.2 −6.6 4.8 320 6.3 180 41.0 −28.9 2.1 3.2 Rapid City 726620 44.05 103.07 966 90.25 6193 −23.8 −20.6 16.2 13.9 12.1 16.7 −3.6 14.2 −3.3 4.2 350 5.8 160 38.6 −27.3 1.9 3.0 Sioux Falls 726510 43.58 96.73 435 96.21 6193 −26.5 −23.7 12.5 10.9 9.7 13.2 −9.4 11.5 −8.2 3.5 310 6.7 180 37.8 −30.7 2.3 2.7 TENNESSEE Bristol 723183 36.48 82.40 463 95.89 6193 −12.8 −9.8 8.8 7.6 6.6 9.3 1.8 8.3 2.0 2.5 270 3.5 250 33.5 −18.2 1.7 4.2 Chattanooga 723240 35.03 85.20 210 98.83 6193 −9.7 −6.9 8.4 7.5 6.6 8.9 2.8 7.9 3.6 3.2 360 3.6 280 36.2 −14.1 2.0 3.9 Crossville 723265 35.95 85.08 573 94.63 8293 −13.7 −9.7 7.3 6.4 5.9 7.9 0.3 7.0 2.3 1.7 310 3.4 270 33.9 −19.2 2.2 4.8 Jackson 723346 35.60 88.92 132 99.75 8293 −11.0 −7.5 8.8 8.0 7.2 9.2 8.0 8.3 6.7 3.8 360 3.5 240 36.6 −15.6 1.3 4.9 Knoxville 723260 35.82 83.98 299 97.78 6193 −10.5 −7.5 9.2 7.9 6.7 9.5 9.1 8.3 7.3 3.2 50 3.4 250 35.1 −15.8 1.7 4.3 Memphis 723340 35.05 90.00 87 100.28 6193 −8.9 −6.3 9.8 8.6 7.8 10.0 5.3 8.9 5.6 4.4 20 4.1 240 36.9 −12.6 1.6 4.0 Nashville 723270 36.13 86.68 180 99.18 6193 −12.2 −9.0 9.6 8.5 7.6 9.8 7.6 8.8 5.7 3.7 340 4.2 230 36.2 −17.0 1.8 4.3 TEXAS Abilene 722660 32.42 99.68 546 94.94 6193 −8.9 −5.7 11.8 10.8 9.8 11.5 8.7 10.3 7.6 5.2 0 5.0 140 39.1 −12.1 1.6 3.3 Amarillo 723630 35.23 101.70 1099 88.80 6193 −14.4 −11.3 13.4 11.9 10.7 13.4 4.2 12.1 3.1 6.3 20 6.7 200 37.8 −18.2 1.6 3.1 Austin 722540 30.30 97.70 189 99.08 6193 −3.7 −1.3 10.3 9.1 8.0 11.1 5.0 9.7 6.2 5.3 10 4.3 180 38.4 −6.9 1.3 3.3 Beaumont/Port Arthur 722410 29.95 94.02 7 101.24 6193 −1.8 0.2 10.1 9.0 8.1 10.2 10.4 9.2 10.6 4.6 340 4.2 200 35.9 −5.4 1.4 2.5 Beeville, Chase Field NAS 722556 28.37 97.67 58 100.63 8293 −2.1 0.8 9.9 8.8 7.9 10.3 14.3 9.1 11.5 5.7 350 4.1 150 40.1 −5.7 1.4 4.6 Brownsville 722500 25.90 97.43 6 101.25 6193 2.1 4.2 12.1 10.8 9.8 11.4 17.8 10.3 16.4 5.9 330 7.0 160 36.6 −0.4 1.3 2.9 College Station/Bryan 722445 30.58 96.37 98 100.15 8293 −5.8 −1.9 9.2 8.3 7.6 9.2 8.5 8.3 9.3 5.5 350 4.2 170 38.6 −8.2 1.3 4.6 Corpus Christi 722510 27.77 97.50 13 101.17 6193 0.1 2.1 12.5 11.2 10.3 12.1 14.8 10.8 14.4 5.7 360 6.8 140 36.8 −3.7 1.1 2.9 Dallas/Fort Worth, Intl A 722590 32.90 97.03 182 99.16 8293 −8.1 −4.4 11.4 10.3 9.4 11.8 7.7 10.5 8.1 5.6 350 4.6 170 39.4 −10.0 1.7 4.6 Del Rio, Laughlin AFB 722615 29.37 100.78 330 97.42 8293 −2.3 0.1 9.6 8.4 7.4 9.7 8.6 8.3 9.9 3.0 10 4.1 140 40.3 −5.6 1.7 3.2 El Paso 722700 31.80 106.40 1194 87.78 6193 −6.2 −3.9 11.0 9.3 7.9 10.7 10.3 9.2 9.6 2.1 20 3.8 180 40.0 −10.2 1.8 3.4 Fort Worth, Carswell AFB 722595 32.77 97.45 198 98.97 8293 −7.9 −4.2 9.9 8.8 8.0 10.2 6.3 9.1 7.2 5.0 10 3.6 10 39.3 −9.5 1.2 4.5 Fort Worth, Meacham Field 722596 32.82 97.37 216 98.76 6193 −7.2 −4.2 11.9 10.6 9.6 12.2 4.2 10.7 6.5 5.8 350 4.6 180 39.5 −10.2 1.7 3.2 Guadalupe Pass 722620 31.83 104.80 1662 82.89 8293 −10.6 −7.2 22.6 20.3 18.3 22.2 3.9 20.4 2.6 8.7 70 5.9 250 36.4 −12.4 1.6 3.8 Houston, Hobby Airport 722435 29.65 95.28 14 101.16 8293 −1.8 1.0 9.8 8.8 8.1 10.3 11.3 9.4 11.0 5.9 350 3.3 190 36.8 −4.7 1.1 4.5 Houston, Inter Airport 722430 29.97 95.35 33 100.93 6193 −2.6 −0.4 9.1 8.1 7.3 10.0 8.5 8.9 10.9 3.6 340 4.2 180 36.8 −5.6 1.7 3.0 Junction 747400 30.50 99.77 522 95.21 8293 −7.2 −4.8 8.3 7.3 6.5 8.3 11.8 7.1 11.7 2.8 360 4.1 150 39.8 −11.3 1.3 3.7 Killeen, Fort Hood 722576 31.07 97.83 309 97.67 8293 −6.6 −2.8 9.7 8.6 7.7 9.9 9.0 8.7 11.5 4.8 360 3.9 160 38.8 −9.3 1.1 4.8 Kingsville, NAS 722516 27.50 97.82 15 101.14 8293 −0.8 2.0 10.1 9.2 8.3 9.9 16.2 8.9 15.4 4.9 360 4.9 150 38.9 −7.8 1.1 5.6 Laredo 722520 27.55 99.47 155 99.48 8293 −0.1 2.2 10.5 9.7 8.9 10.0 15.0 8.9 16.5 3.9 320 5.8 140 41.3 −2.4 1.2 3.7 WMO# = World Meteorological Organization number AFB = Air Force Base NAS = Naval Air Station WSO = Weather Service Office AAF = Army Air Field MCAS = Marine Corps Air Station DB = dry-bulb temperature, °C Lat. = North latitude ° Long. = West longitude ° Elev. = elevation, m StdP = standard pressure at station elevation, kPa WS = wind speed, m/s PWD = prevailing wind direction, ° True Climatic Design Information 27.19 Table 1B Cooling and Dehumidification Design Conditions—United States Station Cooling DB/MWB Evaporation WB/MDB Dehumidification DP/MDB and HR Range of DB 0.4% 1% 2% 0.4% 1% 2% 0.4% 1% 2% DB MWB DB MWB DB MWB WB MDB WB MDB WB MDB DP HR MDB DP HR MDB DP HR MDB 1 2a 2b 2c 2d 2e 2f 3a 3b 3c 3d 3e 3f 4a 4b 4c 4d 4e 4f 4g 4h 4i 5 Eugene 32.9 19.3 30.6 18.6 28.6 17.9 20.3 30.6 19.3 29.1 18.4 27.4 16.5 11.9 23.3 15.6 11.2 22.5 14.8 10.6 21.4 15.3 Hillsboro 33.5 20.4 30.9 19.4 28.9 18.6 21.4 31.7 20.2 29.8 19.1 27.9 17.8 12.9 26.3 16.3 11.7 23.9 15.5 11.1 22.3 14.8 Klamath Falls 32.9 17.7 30.7 16.9 29.2 16.0 18.7 30.5 17.8 28.9 17.0 27.3 14.7 12.2 23.4 13.8 11.5 22.8 12.8 10.7 21.6 19.0 Meacham 30.3 15.0 28.7 14.4 26.6 13.8 16.0 27.9 15.1 26.9 14.3 25.8 11.1 9.6 19.1 10.2 9.0 18.0 9.2 8.4 17.8 20.6 Medford 36.8 19.3 34.9 18.8 32.9 18.1 20.3 34.5 19.5 32.9 18.6 31.3 15.6 11.6 23.7 14.5 10.8 23.2 13.6 10.2 22.7 18.7 North Bend 21.8 15.8 20.6 15.6 19.6 15.0 16.8 20.3 16.2 19.6 15.6 18.8 15.4 10.9 18.2 14.7 10.4 17.5 14.1 10.0 17.0 7.1 Pendleton 35.9 18.0 33.9 17.3 32.0 16.6 18.8 33.3 17.9 32.0 17.1 30.4 14.0 10.5 21.5 12.7 9.7 20.8 11.5 8.9 21.0 15.1 Portland 32.4 19.5 30.1 18.8 28.1 18.0 20.4 30.8 19.5 28.8 18.6 26.9 16.6 11.8 23.7 15.7 11.2 22.3 15.1 10.7 21.6 12.0 Redmond 33.7 16.5 31.8 15.8 29.9 15.1 17.4 31.2 16.6 30.0 15.7 28.3 12.6 10.2 19.9 11.3 9.3 19.4 10.2 8.6 19.1 19.4 Salem 33.2 19.6 30.8 18.6 28.6 17.9 20.2 31.4 19.3 29.4 18.3 27.4 16.0 11.5 24.1 15.2 10.8 22.9 14.4 10.3 21.6 15.5 Sexton Summit 28.5 15.8 26.4 15.0 24.8 14.2 16.9 26.4 15.9 25.2 15.0 23.3 13.0 10.8 21.3 11.7 9.9 20.2 11.0 9.4 18.9 10.5 PENNSYLVANIA Allentown 32.4 22.6 30.9 22.1 29.6 21.4 24.2 29.9 23.4 28.8 22.7 27.6 22.6 17.5 27.0 21.8 16.7 26.1 21.1 15.9 25.6 10.8 Altoona 31.4 22.2 29.9 21.3 28.6 20.6 23.2 29.2 22.5 28.4 21.7 26.8 21.4 17.0 26.0 20.7 16.2 25.2 20.1 15.6 24.3 10.8 Bradford 28.2 20.4 26.9 19.8 25.6 18.8 22.0 26.3 21.1 25.1 20.2 23.6 20.7 16.6 24.1 19.8 15.8 23.0 19.1 15.0 22.4 11.8 Du Bois 30.0 20.9 28.7 20.3 27.0 19.3 22.5 27.4 21.6 26.3 20.9 25.3 20.9 16.6 24.2 20.3 16.0 23.6 19.7 15.4 22.9 10.8 Erie 29.6 22.0 28.2 21.3 26.9 20.8 23.5 27.7 22.7 26.7 21.8 25.6 22.2 17.4 25.9 21.3 16.4 25.2 20.6 15.6 24.4 8.7 Harrisburg 33.3 23.5 31.7 22.8 30.2 22.2 25.1 30.7 24.2 29.4 23.4 28.2 23.6 18.6 27.8 22.7 17.6 26.8 21.9 16.8 26.1 10.4 Philadelphia, Intl Airport 33.4 23.9 31.9 23.3 30.6 22.6 25.4 31.1 24.7 29.7 23.9 28.7 23.9 18.8 28.2 23.2 18.0 27.3 22.6 17.3 26.6 9.8 Philadelphia, Northeast A 34.1 24.3 32.2 23.5 31.0 22.9 25.7 31.3 24.9 30.7 24.1 29.1 24.3 19.3 28.6 23.5 18.4 27.7 22.5 17.3 27.5 10.6 Philadelphia, Willow Gr NAS 34.0 23.7 32.2 23.2 30.9 22.3 25.3 31.7 24.5 30.3 23.7 29.2 23.6 18.7 28.5 22.9 17.9 27.7 21.7 16.6 27.3 10.8 Pittsburgh, Allegheny Co. A 32.0 22.2 30.7 21.4 29.6 20.9 23.7 29.6 23.1 28.7 22.3 27.4 21.9 17.4 26.3 21.3 16.7 25.6 20.7 16.1 24.9 10.0 Pittsburgh, Intl Airport 31.4 22.1 29.9 21.3 28.7 20.6 23.6 29.2 22.7 27.9 21.9 26.8 21.8 17.3 26.4 21.0 16.4 25.7 20.2 15.6 24.8 10.8 Wilkes-Barre/Scranton 31.0 21.7 29.5 21.2 28.1 20.4 23.4 28.6 22.7 27.4 21.9 26.2 21.9 17.1 25.9 21.2 16.4 25.2 20.4 15.6 24.5 10.4 Williamsport 31.9 22.5 30.3 21.8 28.8 21.0 24.2 29.3 23.3 28.1 22.6 26.8 22.7 17.8 26.6 21.9 16.9 25.7 21.2 16.2 24.8 11.3 RHODE ISLAND Providence 31.8 22.8 30.0 21.8 28.4 21.1 24.3 29.5 23.4 27.5 22.6 26.4 22.9 17.7 26.5 22.1 16.8 25.6 21.3 16.0 25.1 9.7 SOUTH CAROLINA Beaufort, MCAS 35.2 25.6 34.1 25.3 33.1 25.0 26.9 32.2 26.4 31.8 25.9 31.1 25.5 20.7 29.4 25.0 20.1 29.2 24.6 19.6 28.7 9.3 Charleston 34.3 25.6 33.1 25.1 32.0 24.7 26.7 31.9 26.1 31.2 25.6 30.5 25.4 20.7 29.0 24.8 19.8 28.6 24.3 19.2 28.1 9.0 Columbia 35.6 24.2 34.3 23.8 33.1 23.6 25.7 32.3 25.2 31.4 24.7 30.6 24.1 19.2 27.8 23.6 18.6 27.3 23.2 18.1 26.9 11.1 Florence 35.8 24.7 34.5 24.7 33.4 24.3 26.5 32.4 25.8 31.8 25.3 31.1 25.1 20.3 29.2 24.4 19.5 28.3 23.9 18.9 27.7 11.0 Greer/Greenville 33.9 23.3 32.6 23.1 31.3 22.8 24.8 31.1 24.2 30.3 23.8 29.4 23.1 18.5 27.3 22.6 18.0 26.8 22.1 17.4 26.4 10.1 Myrtle Beach, AFB 33.8 26.0 32.2 25.8 31.2 25.4 27.4 31.8 26.7 31.1 26.1 30.3 26.0 21.4 30.8 25.4 20.6 29.9 24.9 20.0 29.1 8.0 Sumter, Shaw AFB 35.1 24.2 33.8 24.1 32.1 23.7 25.8 31.7 25.2 31.0 24.7 30.2 24.4 19.5 28.2 23.9 18.9 27.6 23.4 18.4 27.0 10.3 SOUTH DAKOTA Chamberlain 36.5 22.0 34.5 21.6 32.1 20.9 24.5 32.8 23.4 31.7 22.4 30.3 21.9 17.7 29.0 20.9 16.6 27.7 19.9 15.6 26.9 13.2 Huron 35.1 22.3 33.0 21.8 31.1 21.2 24.5 31.8 23.4 30.3 22.4 29.0 22.4 18.0 28.7 21.3 16.7 27.2 20.3 15.7 26.1 13.4 Pierre 37.1 21.1 34.8 20.8 32.7 20.2 23.3 32.4 22.4 31.4 21.6 30.1 20.8 16.5 27.1 19.8 15.5 26.4 18.8 14.5 25.6 14.2 Rapid City 35.1 18.2 32.9 18.1 30.9 17.7 21.0 29.6 20.1 28.8 19.2 27.7 18.4 14.9 24.6 17.4 14.0 23.8 16.3 13.1 22.9 14.1 Sioux Falls 34.4 22.8 32.4 22.2 30.7 21.4 24.6 31.4 23.6 30.5 22.6 29.1 22.5 18.1 28.7 21.4 17.0 27.7 20.5 16.0 26.5 12.3 TENNESSEE Bristol 31.7 22.3 30.4 22.0 29.3 21.6 23.9 29.6 23.3 28.6 22.6 27.8 22.2 17.9 27.1 21.6 17.2 25.8 21.1 16.6 25.2 10.7 Chattanooga 34.5 23.9 33.1 23.7 31.9 23.3 25.4 31.7 24.8 30.9 24.3 30.0 23.8 19.2 27.8 23.2 18.5 27.5 22.7 17.9 26.9 10.8 Crossville 31.7 22.6 30.6 22.4 29.4 22.1 24.3 29.2 23.6 28.5 23.0 27.5 23.1 19.2 26.7 22.0 17.9 26.1 21.5 17.3 25.5 11.0 Jackson 35.1 25.0 34.0 24.7 33.0 24.4 26.4 33.0 25.7 32.0 25.3 31.2 24.7 20.0 29.7 24.1 19.3 29.2 23.7 18.8 28.7 11.0 Knoxville 33.3 23.4 31.9 23.2 30.7 22.8 24.9 31.0 24.3 30.2 23.8 29.2 23.3 18.7 27.8 22.7 18.1 27.2 22.2 17.5 26.5 10.1 Memphis 35.4 25.3 34.2 25.1 33.2 24.7 26.7 33.1 26.2 32.5 25.6 31.6 25.1 20.4 30.3 24.4 19.6 29.8 23.9 19.0 29.0 9.3 Nashville 34.6 24.2 33.2 23.8 32.0 23.4 25.6 31.9 25.0 31.1 24.4 30.2 23.9 19.2 28.3 23.3 18.5 27.7 22.8 18.0 27.2 10.6 TEXAS Abilene 37.2 21.4 36.0 21.5 34.8 21.4 23.9 31.9 23.4 31.6 22.9 31.1 21.8 17.6 27.1 21.2 17.0 26.6 20.7 16.4 26.2 11.4 Amarillo 35.7 19.2 34.3 19.0 33.1 18.8 21.6 30.2 20.9 29.8 20.5 29.4 19.2 16.0 24.2 18.6 15.3 23.8 18.0 14.8 23.3 12.9 Austin 36.8 23.4 35.8 23.4 34.7 23.4 25.6 31.7 25.1 31.1 24.7 30.4 24.2 19.6 26.9 23.8 19.1 26.7 23.4 18.6 26.6 11.2 Beaumont/Port Arthur 34.4 26.0 33.6 25.9 32.7 25.8 27.4 32.0 26.9 31.5 26.6 31.2 26.3 21.7 29.7 25.8 21.2 29.3 25.4 20.7 28.9 8.8 Beeville, Chase Field NAS 38.1 24.9 36.6 25.2 35.7 25.2 27.7 32.7 27.1 32.6 26.6 32.5 26.5 22.2 29.9 25.8 21.2 29.2 25.3 20.6 28.8 12.0 Brownsville 35.1 25.3 34.3 25.2 33.6 25.2 26.7 31.4 26.3 31.3 26.1 31.1 25.6 20.8 28.3 25.2 20.3 28.2 24.9 20.0 28.0 9.2 College Station/Bryan 36.5 23.9 35.6 24.1 34.7 24.1 26.0 31.8 25.6 31.4 25.3 31.1 24.8 20.1 27.6 24.5 19.7 27.4 24.1 19.2 27.0 11.9 Corpus Christi 34.9 25.5 34.2 25.4 33.4 25.4 27.1 31.9 26.7 31.5 26.3 31.0 25.9 21.2 28.9 25.6 20.9 28.6 25.2 20.4 28.3 9.2 Dallas/Fort Worth, Intl A 37.8 23.6 36.4 23.5 35.3 23.5 25.4 33.2 25.0 32.6 24.6 32.2 23.7 18.9 28.0 23.3 18.5 27.5 22.9 18.0 27.3 11.3 Del Rio, Laughlin AFB 38.3 22.5 36.8 22.6 35.8 22.5 25.4 32.6 24.8 32.1 24.4 31.4 23.8 19.4 27.8 23.2 18.7 27.7 22.3 17.7 27.2 11.6 El Paso 38.1 17.9 36.6 17.8 35.3 17.7 21.1 29.2 20.6 29.1 20.1 29.1 19.3 16.3 22.9 18.5 15.5 22.7 17.7 14.7 23.1 15.6 Fort Worth, Carswell AFB 37.8 23.9 36.3 23.9 35.3 23.9 26.2 33.4 25.7 32.7 25.1 32.3 24.6 20.1 29.3 24.0 19.3 29.0 23.4 18.6 28.7 10.7 Fort Worth, Meacham Field 37.8 23.6 36.6 23.6 35.3 23.6 25.7 32.8 25.2 32.3 24.7 31.9 23.9 19.3 28.4 23.4 18.7 27.9 22.9 18.1 27.6 11.1 Guadalupe Pass 33.5 16.0 31.6 15.8 30.4 15.5 19.1 27.5 18.3 26.6 17.7 26.1 16.7 14.6 21.7 15.7 13.7 21.7 14.9 13.0 21.4 11.6 Houston, Hobby Airport 34.7 25.1 34.0 25.1 33.3 25.0 26.8 31.6 26.4 31.0 26.1 30.8 25.7 21.0 28.7 25.3 20.5 28.2 25.0 20.1 27.9 9.2 Houston, Inter Airport 35.5 24.9 34.4 24.9 33.5 24.9 26.6 32.0 26.2 31.6 25.8 31.1 25.3 20.5 28.4 24.9 20.1 28.3 24.6 19.6 28.2 10.1 Junction 37.5 22.1 36.4 21.7 35.4 21.5 24.2 31.6 23.7 31.3 23.2 30.7 22.6 18.5 26.6 21.6 17.3 26.2 21.2 16.9 26.0 13.8 Killeen, Fort Hood 36.7 23.1 35.8 23.0 34.8 23.1 25.4 32.2 24.9 31.8 24.4 31.1 24.0 19.6 27.2 23.4 18.9 27.1 22.9 18.3 26.7 11.9 Kingsville, NAS 36.2 25.1 35.5 25.4 34.8 25.4 27.2 32.9 26.8 32.6 26.4 32.2 25.9 21.3 29.5 25.4 20.6 29.0 25.0 20.1 28.9 11.0 Laredo 38.9 23.0 38.1 23.3 36.8 23.2 25.9 33.2 25.5 32.6 25.1 31.7 24.4 19.7 27.9 24.1 19.4 27.4 23.7 18.9 27.2 11.8 DP = dew-point temperature, °C MWB = mean coincident wet-bulb temperature, °C MDB = mean coincident dry-bulb temperature, °C MWS = mean coincident wind speed, m/s StdD = standard deviation, °C HR = humidity ratio, grams of moisture per kilogram of dry air A = airport ANGB = Air National Guard Base MCAF = Marine Corps Air Facility NAF = Naval Air Facility NAWS = Naval Air Weapons Station RAF = Royal Air Force 27.20 2001 ASHRAE Fundamentals Handbook (SI) Table 1A Heating and Wind Design Conditions—United States Heating Dry Bulb Extreme Wind Speed, m/s Coldest Month MWS/PWD to DB Extr. Annual Daily Lat.
Long.
Elev., m StdP, kPa 0.4% 1% 99.6% 0.4% Mean DB StdD DB Station WMO# Dates 99.6% 99% 1% 2.5% 5% WS MDB WS MDB MWS PWD MWS PWD Max.
Min.
Max. Min.
1a 1b 1c 1d 1e 1f 1g 2a 2b 3a 3b 3c 4a 4b 4c 4d 5a 5b 5c 5d 6a 6b 6c 6d Lubbock, Intl Airport 722670 33.65 101.82 988 90.01 6193 −11.7 −8.3 13.2 11.4 10.2 13.5 6.3 12.1 6.8 5.3 0 6.2 160 38.9 −15.5 1.4 3.1 Lubbock, Reese AFB 722675 33.60 102.05 1017 89.69 8293 −11.4 −7.8 11.3 9.8 8.6 11.2 9.1 9.8 6.6 4.6 20 4.9 170 39.1 −14.2 1.7 2.7 Lufkin 722446 31.23 94.75 88 100.27 6193 −4.9 −2.6 8.0 7.1 6.3 8.2 6.5 7.4 8.0 2.7 330 3.5 230 37.4 −8.6 1.8 2.9 Marfa 722640 30.37 104.02 1481 84.75 8293 −9.6 −7.3 10.9 9.4 8.2 11.2 6.6 9.6 7.4 2.3 360 4.1 220 36.3 −14.8 1.3 2.8 McAllen 722506 26.18 98.23 33 100.93 8293 1.3 4.2 10.5 9.7 9.0 10.4 20.2 9.4 19.9 4.8 350 6.3 130 40.9 −2.7 2.4 4.5 Midland/Odessa 722650 31.95 102.18 872 91.28 6193 −8.2 −5.4 12.7 11.1 9.9 12.1 9.8 10.4 8.9 4.2 20 5.7 180 39.3 −12.8 1.4 3.8 San Angelo 722630 31.37 100.50 582 94.53 6193 −6.9 −4.4 11.6 10.2 9.2 11.3 11.2 10.0 10.8 4.3 20 4.9 160 39.5 −10.7 1.6 3.4 San Antonio, Intl Airport 722530 29.53 98.47 242 98.45 6193 −3.3 −1.0 9.7 8.6 7.7 10.3 6.3 9.1 6.9 4.4 350 4.4 160 37.7 −7.0 1.6 2.9 San Antonio, Kelly AFB 722535 29.38 98.58 210 98.83 8293 −2.7 −0.2 8.4 7.4 6.5 9.2 10.5 8.1 10.9 3.7 360 3.5 160 39.3 −5.8 1.6 3.6 San Antonio, Randolph AFB 722536 29.53 98.28 232 98.57 8293 −2.9 −0.6 8.4 7.4 6.5 8.8 7.4 7.8 8.7 3.2 340 3.3 150 38.2 −6.5 1.2 3.7 Sanderson 747300 30.17 102.42 865 91.36 8293 −4.9 −2.5 8.4 7.0 5.9 9.0 6.5 7.6 8.7 2.6 360 3.3 120 38.7 −12.8 1.6 4.6 Victoria 722550 28.85 96.92 36 100.89 6193 −1.7 0.7 11.5 10.3 9.2 11.4 9.9 10.2 10.7 5.3 360 5.1 180 37.0 −5.2 1.4 2.9 Waco 722560 31.62 97.22 155 99.48 6193 −5.7 −3.2 11.6 10.4 9.4 13.0 3.1 11.3 5.8 5.9 360 5.1 180 39.8 −9.1 1.6 3.6 Wichita Falls, Sheppard AFB 723510 33.98 98.50 314 97.61 6193 −9.9 −7.1 12.8 11.3 10.2 12.6 5.7 11.2 6.0 5.3 360 5.6 180 41.7 −13.7 1.9 3.7 UTAH Cedar City 724755 37.70 113.10 1714 82.36 6193 −16.5−13.3 11.4 10.0 8.9 10.7 3.6 9.4 4.0 1.7 140 5.3 200 36.1 −21.1 1.3 4.6 Ogden, Hill AFB 725755 41.12 111.97 1459 84.98 8293 −14.7−11.6 9.6 8.4 7.5 10.0 −2.6 8.7 −2.5 4.2 110 2.7 190 35.5 −17.2 1.6 3.5 Salt Lake City 725720 40.78 111.97 1288 86.78 6193 −14.7−11.7 12.0 10.1 8.8 11.9 5.7 9.6 4.2 2.9 160 5.0 340 37.9 −19.2 1.1 3.7 VERMONT Burlington 726170 44.47 73.15 104 100.08 6193 −23.9−21.2 10.4 9.2 8.2 10.8 −1.3 9.5 −2.9 2.9 70 4.9 180 33.8 −28.3 1.5 3.1 Montpelier/Barre 726145 44.20 72.57 355 97.13 8293 −23.1−21.0 9.4 8.3 7.6 9.9 −6.6 8.8 −6.5 1.6 320 4.0 220 32.6 −28.0 2.0 3.3 VIRGINIA Fort Belvoir 724037 38.72 77.18 21 101.07 8293 −11.0 −7.8 8.0 6.4 5.2 8.6 1.9 7.4 1.2 0.9 320 2.5 160 37.8 −16.6 1.3 4.2 Hampton, Langley AFB 745980 37.08 76.37 3 101.29 8293 −6.3 −4.2 9.9 8.6 7.7 9.8 4.8 8.8 4.2 4.3 330 4.1 240 36.1 −10.4 1.8 3.4 Lynchburg 724100 37.33 79.20 286 97.94 6193 −11.1 −8.2 8.5 7.6 6.9 9.2 1.8 8.2 1.8 3.3 360 3.8 230 35.1 −15.2 1.6 3.2 Newport News 723086 37.13 76.50 13 101.17 8293 −7.5 −5.3 8.6 8.0 7.3 8.8 4.7 8.2 4.9 3.5 350 4.5 220 37.1 −11.7 1.3 2.6 Norfolk 723080 36.90 76.20 9 101.22 6193 −6.7 −4.7 11.3 9.9 8.9 11.8 4.3 10.4 4.5 5.4 340 5.2 230 36.2 −9.8 1.6 3.0 Oceana, NAS 723075 36.82 76.03 7 101.24 8293 −5.8 −3.9 9.4 8.3 7.4 9.4 5.4 8.4 5.4 3.4 310 4.0 220 36.5 −10.1 1.0 3.8 Quantico, MCAS 724035 38.50 77.30 4 101.28 8293 −9.0 −6.3 7.4 6.3 5.4 8.3 2.0 6.9 3.6 2.8 340 2.4 230 37.6 −13.6 2.0 3.3 Richmond 724010 37.50 77.33 54 100.68 6193 −10.1 −7.6 8.8 7.9 7.0 9.2 4.2 8.2 4.1 3.1 340 4.4 230 36.4 −14.4 1.4 3.2 Roanoke 724110 37.32 79.97 358 97.10 6193 −11.1 −8.6 10.1 8.8 7.5 12.2 −0.7 10.4 −0.1 4.6 320 4.4 290 35.4 −15.3 1.8 3.1 Sterling 724030 38.95 77.45 98 100.15 6193 −12.8−10.2 9.9 8.3 7.2 11.2 −0.3 9.5 −0.7 2.9 340 4.2 250 36.0 −18.2 1.8 3.9 Washington, R. Reagan A 724050 38.85 77.03 20 101.08 8293 −9.3 −6.5 10.1 8.9 8.1 10.7 1.2 9.5 1.9 5.0 340 4.8 170 37.0 −13.5 1.4 3.8 WASHINGTON Bellingham 727976 48.80 122.53 48 100.75 8293 −9.2 −6.0 10.3 9.1 8.2 12.3 0.8 10.1 1.1 7.4 40 4.0 290 30.5 −11.4 1.7 4.1 Hanford 727840 46.57 119.60 223 98.67 8293 −15.1−11.1 11.2 9.4 8.2 10.8 6.8 8.5 6.7 2.7 20 3.4 20 40.4 −16.9 1.7 5.0 Olympia 727920 46.97 122.90 61 100.59 6193 −7.8 −4.8 9.2 7.9 7.1 9.4 7.0 8.4 7.2 2.1 180 3.8 50 34.6 −12.0 2.2 4.5 Quillayute 727970 47.95 124.55 62 100.58 6193 −5.1 −2.8 14.7 11.9 9.2 18.5 7.1 15.5 7.3 2.9 60 3.8 240 30.8 −7.5 4.7 3.6 Seattle, Intl Airport 727930 47.45 122.30 137 99.69 6193 −4.8 −2.2 9.8 8.6 7.6 10.6 6.9 9.5 6.7 4.4 10 4.5 350 33.4 −7.4 2.0 3.8 Spokane, Fairchild AFB 727855 47.62 117.65 750 92.63 6193 −17.4−13.8 11.9 10.3 9.1 12.7 3.9 11.1 3.4 3.1 50 4.0 240 36.5 −21.4 1.8 4.8 Stampede Pass 727815 47.28 121.33 1209 87.62 8293 −16.1−12.4 9.5 8.3 7.3 12.0 −7.4 10.0 −4.0 5.9 90 3.3 100 29.1 −16.4 1.8 4.0 Tacoma, McChord AFB 742060 47.13 122.48 98 100.15 8293 −7.5 −4.7 8.1 6.9 5.8 9.6 7.1 7.9 7.5 1.0 180 3.1 20 34.5 −10.9 1.5 3.8 Walla Walla 727846 46.10 118.28 367 96.99 8293 −15.6−11.1 9.9 8.6 7.8 10.9 9.7 9.6 8.6 2.7 180 4.1 300 40.3 −17.4 1.8 6.5 Wenatchee 727825 47.40 120.20 379 96.85 8293 −15.9−12.6 9.8 8.6 7.7 7.7 2.0 5.5 −0.3 1.4 100 3.9 280 38.1 −18.7 1.4 4.0 Yakima 727810 46.57 120.53 325 97.48 6193 −15.4−11.8 10.7 9.1 7.6 10.1 8.2 8.3 6.3 2.9 250 3.3 90 38.1 −18.7 1.8 4.7 WEST VIRGINIA Bluefield 724125 37.30 81.20 871 91.29 8293 −14.9−11.3 6.8 6.0 5.4 7.9 1.3 6.5 0.5 2.9 270 2.6 290 31.1 −21.1 2.2 4.7 Charleston 724140 38.37 81.60 299 97.78 6193 −14.4−11.5 8.0 7.0 6.1 9.1 3.3 8.1 0.8 2.9 250 3.5 240 34.4 −18.7 1.6 3.7 Elkins 724170 38.88 79.85 609 94.22 6193 −18.8−15.0 9.0 7.9 7.1 9.8 −0.9 8.6 −1.2 1.6 280 3.7 290 31.3 −24.2 1.6 3.0 Huntington 724250 38.37 82.55 255 98.30 6193 −14.6−11.5 8.3 7.2 6.3 8.8 −0.2 7.7 −0.2 3.4 270 3.6 270 34.2 −19.0 2.8 4.2 Martinsburg 724177 39.40 77.98 170 99.30 8293 −13.1−10.2 9.2 8.2 7.3 10.1 0.6 9.0 1.0 3.3 270 4.0 290 37.0 −19.4 2.2 4.6 Morgantown 724176 39.65 79.92 380 96.84 8293 −15.3−11.7 7.9 6.7 6.0 8.4 0.1 7.8 0.6 2.8 210 3.6 240 33.8 −19.9 2.0 4.8 Parkersburg 724273 39.35 81.43 262 98.22 8293 −15.4−11.5 8.0 7.0 6.3 9.0 0.1 7.9 −1.5 3.1 240 3.7 270 35.0 −19.9 1.7 5.1 WISCONSIN Eau Claire 726435 44.87 91.48 276 98.05 6193 −27.7−24.9 9.6 8.7 7.7 9.5 −9.9 8.8 −10.6 3.1 250 5.8 220 35.2 −31.5 1.8 3.2 Green Bay 726450 44.48 88.13 214 98.78 6193 −24.8−21.9 11.0 9.8 8.8 11.1 −7.3 9.8 −7.6 4.3 270 5.5 200 33.8 −28.1 1.6 3.1 La Crosse 726430 43.87 91.25 202 98.92 6193 −25.6−22.4 10.1 9.0 8.1 10.4−10.5 9.3 −10.7 3.0 310 5.3 180 36.0 −29.3 1.8 3.4 Madison 726410 43.13 89.33 264 98.19 6193 −24.1−21.2 10.7 9.4 8.4 11.2 −8.9 9.9 −8.3 3.6 300 5.4 230 34.7 −27.6 1.8 3.3 Milwaukee 726400 42.95 87.90 211 98.82 6193 −21.7−18.7 12.4 10.9 9.7 12.4 −7.3 10.8 −6.5 5.8 290 6.5 220 34.9 −24.6 1.8 3.7 Wausau 726463 44.93 89.63 366 97.00 8293 −26.2−22.7 8.3 7.4 6.5 8.4 −8.8 7.7 −8.1 3.2 300 4.6 200 33.8 −30.1 1.7 2.6 WYOMING Big Piney 726710 42.57 110.10 2124 78.28 8293 −29.9−25.9 10.5 9.1 7.8 9.9 −4.0 8.4 −5.9 1.5 60 5.1 260 30.3 −35.9 1.5 4.7 Casper 725690 42.92 106.47 1612 83.40 6193 −24.7−20.3 15.0 13.4 12.0 15.4 1.6 14.3 −0.1 4.1 260 5.8 240 35.9 −29.9 1.2 4.7 Cheyenne, Warren AFB 725640 41.15 104.82 1872 80.77 6193 −21.5−17.6 15.1 13.1 11.4 17.1 2.2 14.7 1.3 4.4 290 5.7 290 33.4 −26.3 1.2 4.2 Cody 726700 44.52 109.02 1553 84.01 8293 −25.5−21.8 15.2 12.5 10.4 15.7 1.9 13.4 1.4 2.5 40 4.7 70 34.8 −29.1 2.3 5.2 Gillette 726650 44.35 105.53 1230 87.40 8293 −26.5−21.9 12.7 11.1 9.8 13.5 1.2 12.1 0.7 3.4 260 5.0 140 38.5 −28.7 3.3 5.6 Lander 725760 42.82 108.73 1694 82.56 6193 −25.6−21.6 10.1 8.4 7.0 11.0 3.4 8.4 2.5 1.3 120 4.6 270 34.8 −28.9 1.4 4.3 Rock Springs 725744 41.60 109.07 2060 78.90 6193 −22.6−18.7 12.7 11.2 10.1 14.3 −3.7 13.0 −4.7 3.0 70 5.6 280 32.4 −26.9 1.1 4.4 Sheridan 726660 44.77 106.97 1209 87.62 6193 −25.7−21.9 12.3 10.5 9.1 12.7 −0.1 10.4 −2.6 2.1 280 4.2 120 36.9 −29.7 1.7 3.6 Worland 726665 43.97 107.95 1294 86.72 8293 −30.1−25.0 9.9 8.5 7.3 9.0 −2.2 7.5 −2.2 1.4 210 4.1 220 39.5 −34.3 1.2 5.8 WMO# = World Meteorological Organization number AFB = Air Force Base NAS = Naval Air Station WSO = Weather Service Office AAF = Army Air Field MCAS = Marine Corps Air Station DB = dry-bulb temperature, °C Lat. = North latitude ° Long. = West longitude ° Elev. = elevation, m StdP = standard pressure at station elevation, kPa WS = wind speed, m/s PWD = prevailing wind direction, ° True Climatic Design Information 27.21 Table 1B Cooling and Dehumidification Design Conditions—United States Station Cooling DB/MWB Evaporation WB/MDB Dehumidification DP/MDB and HR Range of DB 0.4% 1% 2% 0.4% 1% 2% 0.4% 1% 2% DB MWB DB MWB DB MWB WB MDB WB MDB WB MDB DP HR MDB DP HR MDB DP HR MDB 1 2a 2b 2c 2d 2e 2f 3a 3b 3c 3d 3e 3f 4a 4b 4c 4d 4e 4f 4g 4h 4i 5 Lubbock, Intl Airport 36.3 19.6 34.9 19.7 33.7 19.6 22.5 30.6 21.9 30.1 21.4 29.7 20.5 17.1 25.0 19.8 16.4 24.6 19.2 15.8 24.2 12.3 Lubbock, Reese AFB 36.4 19.4 35.1 19.4 33.9 19.3 22.7 30.3 22.0 30.0 21.4 29.6 20.7 17.4 25.8 19.8 16.4 25.2 19.1 15.7 24.9 13.2 Lufkin 35.8 24.6 34.7 24.7 33.7 24.6 26.3 32.2 26.0 31.9 25.6 31.6 25.1 20.4 28.6 24.6 19.8 28.3 24.1 19.2 28.0 11.6 Marfa 34.4 16.4 33.2 16.3 31.6 16.4 19.9 27.5 19.3 27.0 18.8 26.7 18.2 15.7 22.2 17.1 14.7 21.7 16.4 14.0 21.4 17.4 McAllen 38.0 24.7 36.7 24.6 35.9 24.7 26.9 32.9 26.4 32.4 26.1 31.6 25.6 20.9 28.5 25.2 20.4 28.0 24.9 20.0 27.7 11.5 Midland/Odessa 37.4 19.6 36.1 19.4 34.8 19.5 22.6 30.3 22.0 30.2 21.5 30.0 20.8 17.2 24.5 20.1 16.4 24.0 19.4 15.8 24.0 13.2 San Angelo 37.5 21.2 36.3 21.2 35.1 21.2 23.7 32.0 23.2 31.4 22.8 30.9 21.7 17.5 26.4 21.1 16.9 25.9 20.7 16.5 25.7 12.4 San Antonio, Intl Airport 36.4 23.0 35.4 23.0 34.4 23.1 25.6 30.7 25.1 30.4 24.7 29.9 24.3 19.9 27.3 23.9 19.3 26.9 23.4 18.8 26.8 10.6 San Antonio, Kelly AFB 37.1 23.2 36.2 23.5 35.3 23.4 26.2 31.5 25.7 31.2 25.2 31.0 25.1 20.7 28.3 24.5 20.0 27.6 24.0 19.4 27.0 11.4 San Antonio, Randolph AFB 36.5 23.4 35.6 23.4 34.7 23.2 25.7 32.4 25.2 31.6 24.7 30.9 24.2 19.7 27.6 23.8 19.2 27.1 23.5 18.8 27.0 12.4 Sanderson 36.2 19.7 35.2 20.0 34.2 20.0 23.2 30.2 22.6 30.0 22.1 30.1 21.2 17.6 26.3 20.6 17.0 25.5 20.0 16.3 25.2 11.5 Victoria 35.2 24.6 34.4 24.7 33.6 24.8 26.4 31.2 26.2 31.1 25.8 30.7 25.4 20.7 28.1 25.1 20.2 27.8 24.7 19.8 27.6 9.7 Waco 38.2 23.9 37.1 23.9 35.9 23.8 25.8 33.7 25.4 33.2 25.0 32.6 24.1 19.3 28.2 23.6 18.7 27.9 23.1 18.2 27.7 12.0 Wichita Falls, Sheppard AFB 39.2 23.1 37.7 23.0 36.4 23.0 25.1 34.0 24.6 33.4 24.1 32.9 22.9 18.4 27.7 22.4 17.7 27.6 21.9 17.2 27.3 13.3 UTAH Cedar City 33.9 14.9 32.6 14.7 31.2 14.3 17.6 26.5 16.8 26.4 16.2 26.3 15.2 13.3 19.7 13.9 12.2 19.8 12.6 11.2 20.2 15.8 Ogden, Hill AFB 33.8 15.9 32.1 15.8 30.7 15.4 18.3 28.4 17.6 27.2 16.8 27.4 15.3 13.0 22.2 14.0 11.9 22.6 12.8 11.0 22.9 12.2 Salt Lake City 35.8 16.7 34.6 16.5 33.2 16.2 18.9 29.6 18.2 29.6 17.5 29.2 15.8 13.1 22.5 14.4 12.0 22.5 13.2 11.0 22.8 15.4 VERMONT Burlington 30.8 21.6 29.1 20.7 27.5 20.0 23.3 28.6 22.2 26.9 21.3 25.8 21.6 16.4 26.2 20.6 15.5 25.1 19.7 14.6 24.0 11.3 Montpelier/Barre 29.6 20.9 28.2 20.1 26.5 19.3 22.5 27.7 21.3 26.5 20.3 25.0 20.6 15.9 25.5 19.7 15.1 24.0 18.8 14.2 22.9 11.7 VIRGINIA Fort Belvoir 35.1 25.3 33.7 24.6 31.9 23.8 26.5 33.2 25.7 31.7 24.9 30.6 24.8 19.9 29.9 24.1 19.0 29.4 23.4 18.2 28.5 11.6 Hampton, Langley AFB 34.4 25.6 33.0 25.1 31.3 24.4 26.6 32.3 25.9 31.5 25.3 30.1 25.1 20.2 29.6 24.5 19.5 28.9 24.0 18.9 28.4 8.3 Lynchburg 33.7 23.6 32.3 23.1 31.1 22.6 24.7 31.2 24.2 30.3 23.6 29.3 23.1 18.4 27.3 22.5 17.8 26.7 21.9 17.2 26.3 10.1 Newport News 34.8 25.3 33.5 24.9 31.8 24.3 26.4 32.8 25.8 31.6 25.2 30.4 24.8 19.9 29.1 24.3 19.3 28.5 23.9 18.8 27.9 10.1 Norfolk 34.0 24.8 32.6 24.2 31.2 23.7 25.8 31.8 25.2 30.8 24.7 29.7 24.3 19.3 28.6 23.8 18.6 27.8 23.2 18.0 27.4 8.5 Oceana, NAS 34.2 25.2 32.9 24.7 31.2 24.0 26.2 31.7 25.6 30.8 25.0 30.1 24.8 19.9 29.2 24.2 19.1 28.5 23.6 18.4 27.9 8.7 Quantico, MCAS 34.4 24.8 33.2 24.4 31.6 23.6 26.2 32.7 25.5 31.7 24.7 30.6 24.4 19.4 30.5 23.7 18.5 29.6 23.0 17.8 28.3 10.3 Richmond 34.5 24.6 33.1 24.1 31.8 23.5 26.0 32.1 25.3 31.1 24.7 30.0 24.4 19.5 28.9 23.8 18.7 28.0 23.2 18.0 27.2 10.6 Roanoke 33.2 22.7 31.8 22.2 30.4 21.6 24.0 30.8 23.4 29.8 22.8 28.7 22.2 17.6 26.7 21.6 16.9 26.1 21.1 16.4 25.6 10.9 Sterling 33.7 23.9 32.2 23.2 30.9 22.6 25.2 31.2 24.5 30.3 23.8 29.2 23.6 18.6 28.2 22.9 17.8 27.4 22.2 17.1 26.6 11.7 Washington, National A 34.8 24.7 33.6 24.2 31.7 23.4 26.0 31.8 25.3 30.9 24.7 30.0 24.6 19.6 28.5 24.0 18.9 28.2 23.4 18.2 27.4 9.2 WASHINGTON Bellingham 26.2 18.5 24.7 17.8 23.1 16.7 19.2 25.8 18.1 23.7 17.1 22.2 16.2 11.6 22.8 15.5 11.1 21.2 14.8 10.6 19.5 9.3 Hanford 37.9 19.4 35.7 18.5 33.9 17.8 20.0 35.7 19.1 34.3 18.3 32.4 14.3 10.4 22.4 13.2 9.7 23.7 11.8 8.8 23.6 14.7 Olympia 30.6 19.3 28.3 18.2 26.3 17.5 19.8 29.5 18.8 27.3 17.8 25.4 16.1 11.5 22.7 15.3 10.9 21.9 14.6 10.4 20.8 14.0 Quillayute 26.4 16.8 23.3 15.9 20.8 15.0 17.7 24.4 16.6 21.9 15.7 19.7 15.3 10.9 18.3 14.7 10.5 17.4 14.1 10.1 16.7 8.6 Seattle, Intl Airport 29.4 18.3 27.4 17.6 25.4 16.8 19.1 28.3 18.1 26.3 17.2 24.5 15.4 11.1 21.6 14.7 10.6 20.5 14.1 10.2 19.8 10.2 Spokane, Fairchild AFB 33.4 16.8 31.5 16.3 29.6 15.7 18.1 30.1 17.2 28.8 16.3 27.5 14.1 11.0 20.2 12.9 10.1 19.9 11.8 9.5 19.4 14.5 Stampede Pass 25.5 13.9 23.6 13.2 21.4 12.5 15.0 23.1 14.1 21.8 13.2 20.3 11.8 10.0 17.3 10.8 9.3 16.0 10.0 8.8 14.4 8.9 Tacoma, McChord AFB 30.0 18.4 28.0 17.4 25.8 16.9 19.2 28.5 18.3 26.5 17.4 24.7 15.7 11.3 21.8 15.0 10.8 20.9 14.3 10.3 20.2 12.5 Walla Walla 36.9 19.0 34.9 18.4 33.1 17.7 20.2 33.6 19.2 32.7 18.4 31.1 15.7 11.7 23.3 14.6 10.8 22.5 13.7 10.2 22.4 15.0 Wenatchee 35.0 19.2 33.1 18.3 30.9 17.3 19.6 32.7 18.8 31.4 18.0 29.7 14.9 11.1 23.8 13.9 10.4 23.8 12.9 9.7 23.3 14.0 Yakima 35.1 18.6 33.2 17.9 31.2 17.2 19.7 32.4 18.7 31.4 17.8 29.9 15.1 11.2 23.8 13.8 10.2 23.2 12.6 9.5 22.4 17.3 WEST VIRGINIA Bluefield 29.2 20.5 28.1 20.3 26.6 19.7 22.2 27.0 21.5 26.1 20.9 25.2 20.8 17.2 24.1 20.1 16.5 23.7 19.6 15.9 23.0 9.1 Charleston 32.5 22.8 31.2 22.5 30.0 21.8 24.6 30.1 23.8 29.2 23.2 28.0 23.0 18.4 27.3 22.3 17.6 26.6 21.7 16.9 25.8 10.6 Elkins 29.6 21.4 28.4 20.9 27.3 20.3 22.9 27.7 22.2 26.7 21.4 25.8 21.4 17.3 25.7 20.7 16.6 24.8 20.1 15.9 23.8 11.7 Huntington 32.7 23.5 31.4 23.0 30.2 22.3 25.0 30.3 24.3 29.6 23.6 28.4 23.5 18.9 27.9 22.8 18.1 26.9 22.1 17.3 26.3 10.6 Martinsburg 34.5 23.1 33.0 22.6 31.1 22.1 24.8 30.7 24.1 30.0 23.4 29.4 23.4 18.6 27.3 22.2 17.2 26.9 21.5 16.5 26.0 12.1 Morgantown 31.8 22.2 30.5 21.8 29.2 21.1 23.9 29.4 23.2 28.3 22.6 27.5 22.2 17.7 25.9 21.6 17.0 25.4 21.0 16.4 24.6 11.3 Parkersburg 32.9 23.1 31.3 22.5 30.1 22.1 24.7 30.5 24.0 29.2 23.3 28.0 23.4 18.8 27.6 22.2 17.4 26.9 21.7 16.9 25.6 10.9 WISCONSIN Eau Claire 32.2 22.9 30.5 21.7 28.9 20.9 24.3 30.1 23.2 28.5 22.2 27.2 22.5 17.8 27.8 21.4 16.6 26.4 20.4 15.6 25.3 11.4 Green Bay 31.2 22.9 29.5 21.9 27.9 21.1 24.2 29.4 23.1 27.9 22.1 26.6 22.6 17.7 27.5 21.5 16.6 26.3 20.4 15.5 25.1 11.5 La Crosse 32.8 23.6 31.1 22.6 29.5 21.8 25.2 30.4 24.1 29.1 23.1 27.7 23.6 18.9 28.2 22.6 17.8 27.0 21.6 16.7 25.8 11.2 Madison 32.1 22.9 30.5 22.1 28.9 21.3 24.4 30.2 23.3 28.6 22.3 27.5 22.7 18.0 28.1 21.6 16.8 26.7 20.6 15.8 25.6 12.2 Milwaukee 31.9 23.3 29.9 22.3 28.3 21.3 24.6 29.9 23.5 28.5 22.4 27.0 23.0 18.2 28.1 21.9 17.0 26.7 20.8 15.9 25.5 9.2 Wausau 31.0 21.8 29.4 20.9 27.9 20.3 23.4 28.3 22.4 27.6 21.4 25.5 21.7 17.1 25.9 20.8 16.2 25.0 20.0 15.4 23.8 10.9 WYOMING Big Piney 28.5 12.0 26.6 11.5 25.3 11.2 13.4 24.0 12.6 23.4 11.9 23.1 10.0 9.9 15.8 8.8 9.1 15.3 7.0 8.1 14.9 18.2 Casper 33.2 14.8 31.7 14.3 30.2 14.2 16.9 27.3 16.1 26.7 15.4 26.1 14.0 12.2 18.7 12.8 11.2 19.0 11.6 10.4 18.5 16.9 Cheyenne, Warren AFB 30.8 14.2 29.3 13.9 27.8 13.8 16.8 24.9 16.1 24.3 15.3 23.7 14.4 12.9 18.7 13.4 12.1 18.3 12.5 11.4 17.9 14.3 Cody 32.9 14.9 30.6 14.4 29.0 13.8 16.3 28.1 15.4 27.4 14.6 26.5 12.5 10.9 21.0 10.9 9.8 18.7 9.8 9.1 18.4 14.1 Gillette 34.5 16.3 32.9 16.0 30.6 15.5 18.3 28.9 17.4 28.5 16.6 27.9 15.1 12.5 22.6 13.8 11.4 20.7 12.3 10.4 20.1 15.9 Lander 32.3 14.8 30.8 14.3 29.2 14.0 16.7 27.4 15.8 26.8 15.1 26.4 13.1 11.5 20.4 11.8 10.6 19.8 10.7 9.8 19.6 14.8 Rock Springs 30.2 12.4 28.9 12.1 27.6 11.8 14.6 23.7 13.8 23.4 13.1 23.2 11.9 11.2 16.4 10.6 10.2 16.3 9.3 9.4 15.9 15.4 Sheridan 34.1 16.8 32.2 16.2 30.2 16.0 18.7 29.2 17.8 28.3 17.0 27.4 15.3 12.6 21.9 14.2 11.7 21.7 13.2 10.9 20.8 16.2 Worland 35.8 17.2 34.1 17.2 32.1 16.4 19.7 30.9 18.7 30.1 17.7 29.1 16.1 13.4 24.0 14.8 12.3 23.8 13.7 11.5 22.7 17.2 DP = dew-point temperature, °C MWB = mean coincident wet-bulb temperature, °C MDB = mean coincident dry-bulb temperature, °C MWS = mean coincident wind speed, m/s StdD = standard deviation, °C HR = humidity ratio, grams of moisture per kilogram of dry air A = airport ANGB = Air National Guard Base MCAF = Marine Corps Air Facility NAF = Naval Air Facility NAWS = Naval Air Weapons Station RAF = Royal Air Force 27.22 2001 ASHRAE Fundamentals Handbook (SI) Table 2A Heating and Wind Design Conditions—Canada Heating Dry Bulb Extreme Wind Speed, m/s Coldest Month MWS/PWD to DB Extr. Annual Daily Lat.
Long.
Elev., m StdP, kPa 0.4% 1% 99.6% 0.4% Mean DB StdD DB Station WMO# Dates 99.6% 99% 1% 2.5% 5% WS MDB WS MDB MWS PWD MWS PWD Max.
Min. Max. Min.
1a 1b 1c 1d 1e 1f 1g 2a 2b 3a 3b 3c 4a 4b 4c 4d 5a 5b 5c 5d 6a 6b 6c 6d ALBERTA Calgary Intl A 718770 51.12 114.02 1084 88.96 6193 −30.0 −27.1 12.3 10.7 9.5 14.1 −1.5 12.4 −2.7 2.9 0 4.7 160 31.8 −33.2 1.5 3.1 Cold Lake A 711200 54.42 110.28 544 94.96 6193 −35.1 −32.2 9.5 8.1 7.1 9.5 −7.8 8.2 −11.8 1.2 270 4.2 180 31.2 −40.0 1.8 3.5 Coronation 718730 52.07 111.45 791 92.18 6193 −33.0 −30.3 11.1 9.5 8.3 12.5 −10.9 10.4 −10.7 3.8 320 4.9 160 33.1 −37.4 1.7 3.5 Edmonton Intl A 711230 53.30 113.58 723 92.94 6193 −33.4 −30.5 10.8 9.2 8.0 10.9 −11.3 9.4 −11.6 2.5 180 3.8 180 30.8 −38.0 1.7 4.5 Fort McMurray A 719320 56.65 111.22 369 96.97 6193 −35.8 −33.9 7.5 6.6 6.0 7.8 −8.7 6.8 −11.9 1.2 90 3.8 250 32.4 −41.0 2.0 2.7 Grande Prairie A 719400 55.18 118.88 669 93.54 6193 −35.4 −32.6 11.8 10.0 8.6 13.0 −0.1 10.8 −2.2 1.3 320 3.5 270 30.6 −40.6 1.5 3.7 Lethbridge A 718740 49.63 112.80 929 90.65 6193 −29.9 −26.9 16.0 14.1 12.4 20.0 3.6 17.6 3.5 2.4 250 5.7 270 34.5 −34.2 1.8 3.6 Medicine Hat A 718720 50.02 110.72 716 93.01 6193 −31.2 −28.4 11.7 10.0 8.8 13.0 2.1 11.1 0.3 2.1 230 4.7 220 36.0 −35.8 2.0 4.0 Peace River A 710680 56.23 117.43 571 94.65 6193 −35.3 −32.9 9.2 8.1 7.4 9.7 −0.9 8.6 −4.6 1.9 0 4.1 270 30.6 −40.9 1.6 3.7 Red Deer A 718780 52.18 113.90 905 90.92 6193 −32.8 −29.6 9.8 8.1 7.5 12.0 −10.8 9.9 −10.4 2.8 200 4.5 180 31.3 −37.2 1.7 3.6 Rocky Mtn. House 719280 52.43 114.92 989 89.99 6193 −31.7 −29.1 8.5 7.2 6.0 8.6 −3.3 7.2 −6.5 1.4 340 3.7 160 30.5 −37.6 1.5 2.8 Vermilion A 53.35 110.83 618 94.12 6193 −34.3 −31.6 10.0 8.6 7.6 9.5 −10.6 8.3 −11.6 1.5 270 4.7 180 32.1 −41.9 2.0 3.7 Whitecourt 719300 54.15 115.78 782 92.28 6193 −34.3 −31.2 7.7 6.8 6.1 8.4 −5.5 7.4 −8.0 1.9 270 3.3 90 30.4 −40.7 1.1 3.0 BRITISH COLUMBIA Abbotsford A 711080 49.03 122.37 58 100.63 6193 −9.3 −6.5 9.1 7.5 6.5 13.0 0.6 11.0 0.9 5.5 90 3.3 220 33.3 −12.5 2.2 3.7 Cape St. James 710310 51.93 131.02 92 100.22 6193 −4.0 −1.7 22.4 20.5 18.0 26.9 4.5 24.1 5.5 9.8 50 5.0 300 20.8 −5.6 1.9 3.1 Castlegar A 718840 49.30 117.63 495 95.52 6693 −15.0 −12.9 8.0 6.9 6.3 9.4 −7.7 8.5 −6.0 3.5 0 3.3 180 36.4 −19.2 1.6 3.9 Comox A 718930 49.72 124.90 24 101.04 6193 −6.2 −4.0 13.1 11.1 9.4 14.0 6.0 12.7 5.8 3.0 290 3.0 340 30.6 −8.5 2.1 2.9 Cranbrook A 718800 49.60 115.78 939 90.54 7093 −25.9 −22.2 9.0 7.9 7.1 8.9 0.7 7.9 0.8 1.0 200 4.6 210 34.2 −29.7 1.5 3.7 Fort Nelson A 719450 58.83 122.58 382 96.82 6193 −36.3 −34.4 7.3 6.3 5.4 6.8 −13.6 5.5 −16.5 0.5 220 2.4 120 30.9 −41.3 1.8 3.5 Fort St. John A 719430 56.23 120.73 695 93.25 6193 −34.4 −31.8 10.9 9.6 8.4 13.1 −4.8 11.2 −5.7 2.9 0 3.8 230 29.7 −37.5 1.6 3.5 Kamloops A 718870 50.70 120.45 346 97.24 6693 −22.1 −18.4 10.4 9.0 7.9 11.2 −3.5 9.6 −2.9 1.9 90 3.4 270 36.7 −25.5 1.5 4.9 Penticton A 718890 49.47 119.60 344 97.26 6193 −15.0 −12.4 10.2 8.8 7.8 12.7 1.1 11.1 1.4 3.7 340 3.8 180 35.4 −17.3 1.5 4.0 Port Hardy A 711090 50.68 127.37 22 101.06 6193 −5.5 −3.5 12.7 10.8 9.2 14.5 3.1 13.0 3.8 3.4 110 4.1 340 24.4 −7.6 1.9 2.7 Prince George A 718960 53.88 122.68 691 93.29 6193 −31.9 −27.8 9.3 7.9 6.8 12.2 0.0 10.4 −4.5 0.9 0 2.7 180 29.4 −38.3 5.1 3.3 Prince Rupert A 718980 54.30 130.43 34 100.92 6393 −13.7 −10.5 12.3 10.2 8.9 13.3 6.4 11.6 5.9 2.5 70 3.6 270 24.1 −16.5 2.5 3.8 Quesnel A 711030 53.03 122.52 545 94.95 6193 −29.8 −25.6 7.8 6.7 6.0 8.3 −8.0 7.4 −6.8 0.3 340 2.4 340 33.1 −34.7 2.2 4.6 Sandspit A 711010 53.25 131.82 6 101.25 6193 −6.3 −4.0 17.1 14.1 12.1 18.8 6.4 16.3 5.5 8.0 320 4.2 270 22.3 −7.8 1.9 2.8 Smithers A 719500 54.82 127.18 523 95.20 6193 −28.2 −24.2 7.7 6.6 5.8 8.2 −5.3 7.2 −7.2 1.3 140 2.8 320 30.9 −32.0 2.2 3.9 Spring Island 714790 50.12 127.93 98 100.15 6193 −1.7 −0.5 18.3 15.6 13.0 19.6 7.5 17.7 7.0 2.6 50 2.8 320 25.5 −3.7 3.4 2.5 Terrace A 719510 54.47 128.58 217 98.75 6193 −19.1 −16.7 11.5 10.1 8.9 14.1 −12.2 12.9 −10.3 8.6 0 3.7 270 31.7 −20.7 2.2 3.2 Tofino A 711060 49.08 125.77 24 101.04 6193 −3.7 −1.9 10.6 9.0 7.9 12.8 7.7 10.8 7.2 2.1 70 3.1 290 27.2 −6.2 2.2 3.1 Vancouver Intl A 718920 49.18 123.17 2 101.30 6193 −7.8 −4.7 10.0 8.3 7.1 11.3 5.2 9.4 5.8 2.7 90 3.3 290 28.0 −10.0 1.6 3.5 Victoria Intl A 717990 48.65 123.43 19 101.10 6193 −5.3 −3.2 8.8 7.3 6.3 10.8 2.5 9.1 3.5 4.5 50 2.8 90 30.3 −7.7 1.8 3.2 Williams Lake A 711040 52.18 122.05 940 90.53 6193 −29.0 −25.5 10.0 8.5 7.4 10.9 −1.7 9.3 −1.2 1.2 320 2.8 140 31.1 −34.1 2.2 4.4 MANITOBA Brandon A 711400 49.92 99.95 409 96.51 6193 −33.7 −31.3 12.0 10.3 9.0 12.6 −16.9 10.9 −16.6 4.1 270 5.5 160 34.6 −37.8 1.7 2.5 Churchill A 719130 58.75 94.07 29 100.98 6193 −37.7 −36.2 15.3 13.3 11.8 16.0 −23.8 13.4 −25.7 6.5 270 5.8 230 29.9 −40.6 2.5 2.2 Dauphin A 718550 51.10 100.05 305 97.71 6193 −33.5 −30.8 12.7 11.1 9.9 13.7 −16.6 12.4 −15.3 4.0 250 5.8 200 34.1 −37.6 1.9 2.4 Portage La Prairie A 718510 49.90 98.27 269 98.13 6193 −31.7 −29.6 11.7 10.1 9.0 12.8 −17.5 11.0 −17.0 3.8 250 5.2 180 35.0 −35.1 1.9 2.3 The Pas A 718670 53.97 101.10 271 98.11 6193 −35.3 −33.3 10.8 9.3 8.3 11.1 −21.1 9.6 −19.0 2.6 290 4.8 160 31.6 −40.0 1.9 2.3 Thompson A 710790 55.80 97.87 218 98.73 6893 −38.9 −36.8 8.9 8.0 7.1 8.3 −20.8 7.4 −21.7 1.4 270 4.5 180 31.6 −44.6 2.1 2.4 Winnipeg Intl A 718520 49.90 97.23 239 98.49 6193 −32.8 −30.6 12.9 11.3 10.1 13.3 −14.9 11.7 −15.0 3.2 320 5.7 180 34.5 −36.2 1.9 2.6 NEW BRUNSWICK Charlo A 717110 47.98 66.33 38 100.87 6793 −25.5 −23.5 10.6 9.3 8.3 12.1 −16.1 10.8 −13.6 5.1 250 4.8 250 31.9 −29.3 1.5 2.5 Chatham A 717170 47.00 65.45 31 100.95 6193 −24.3 −21.7 10.8 9.2 8.1 12.0 −8.9 10.5 −9.1 3.2 270 5.1 230 33.6 −28.8 1.2 2.4 Fredericton A 717000 45.87 66.53 20 101.08 6193 −24.3 −21.5 10.0 8.8 7.7 11.2 −8.3 9.6 −8.0 2.3 270 5.0 230 33.4 −29.6 1.4 3.0 Moncton A 717050 46.12 64.68 71 100.47 6193 −23.3 −20.8 11.7 10.1 8.9 13.2 −7.0 11.6 −7.2 5.6 270 5.8 250 31.5 −27.2 1.1 2.5 Saint John A 716090 45.32 65.88 109 100.02 6193 −22.7 −20.2 11.6 10.0 8.9 14.2 −4.3 12.6 −5.3 4.0 340 5.1 230 29.1 −27.8 2.2 2.6 NEWFOUNDLAND Battle Harbour 718170 52.30 55.83 8 101.23 6193 −25.4 −23.2 18.0 15.6 14.1 21.4 −8.4 18.8 −9.3 8.1 270 7.6 230 25.3 −27.5 2.6 3.3 Bonavista 711960 48.67 53.12 27 101.00 6193 −16.1 −14.0 19.1 17.0 15.1 21.3 −6.3 18.8 −5.3 10.8 280 7.6 230 27.3 −17.9 1.5 3.2 Cartwright 718180 53.70 57.03 14 101.16 6493 −28.0 −25.9 15.8 13.5 11.9 18.0 −7.6 15.7 −7.5 5.5 220 5.2 210 28.7 −30.5 2.3 2.5 Daniels Harbour 711850 50.23 57.58 19 101.10 6693 −21.6 −19.3 18.0 15.6 13.9 20.1 −8.7 17.2 −6.3 5.4 270 6.6 230 23.8 −24.4 1.8 3.6 Deer Lake A 718090 49.22 57.40 22 101.06 6693 −25.1 −21.7 10.6 9.6 8.1 11.4 −6.0 10.0 −7.2 1.1 240 6.1 220 30.2 −30.4 1.3 3.7 Gander Intl A 718030 48.95 54.57 151 99.52 6193 −20.0 −17.8 14.4 12.5 10.9 16.4 −6.5 14.1 −5.8 7.1 270 5.7 230 29.3 −22.1 1.6 3.5 Goose A 718160 53.32 60.37 49 99.03 6193 −30.5 −28.7 11.5 10.0 8.8 13.4 −16.2 11.8 −14.8 4.9 250 5.4 250 31.9 −33.8 2.2 2.0 Hopedale 719000 55.45 60.23 8 101.23 6493 −29.2 −27.5 15.9 13.5 11.8 17.9 −10.6 15.7 −11.9 5.4 250 6.0 250 26.0 −31.6 2.5 3.2 St. John’s A 718010 47.62 52.73 140 99.65 6193 −16.0 −14.1 16.6 14.7 12.9 18.4 −4.7 16.4 −3.7 7.6 290 7.6 250 27.5 −18.6 1.4 2.7 Stephenville A 718150 48.53 58.55 26 101.01 6193 −18.9 −16.3 14.6 12.5 10.3 16.0 −6.3 13.2 −6.3 4.9 50 3.9 250 26.3 −21.4 1.4 3.5 Wabush Lake A 718250 52.93 66.87 551 94.88 6193 −36.2 −34.4 10.0 8.8 7.7 11.1 −20.6 9.5 −19.9 2.1 270 5.2 240 28.0 −41.6 2.1 2.4 NORTHWEST TERRITORIES Cape Parry A 719480 70.17 124.68 17 101.12 6193 −36.7 −36.1 13.8 12.3 11.0 15.1 −24.3 13.0 −24.5 3.3 270 3.8 110 18.8 −41.3 2.7 2.4 Fort Smith A 719340 60.02 111.95 203 98.91 6193 −36.6 −35.4 7.9 7.4 6.5 8.8 −19.5 7.9 −20.6 1.1 150 4.4 180 30.9 −44.6 1.9 3.1 Inuvik UA 719570 68.30 133.48 68 100.51 7393 −41.4 −39.9 7.6 6.7 6.0 8.6 −21.9 7.4 −21.8 0.6 70 3.5 180 28.6 −46.6 1.4 2.8 Norman Wells A 710430 65.28 126.80 74 100.44 6193 −39.7 −37.9 10.8 9.2 7.9 12.9 −20.0 10.8 −21.7 0.8 170 3.7 140 30.5 −45.2 1.7 3.0 Yellowknife A 719360 62.47 114.45 206 98.87 6193 −39.5 −37.9 9.8 8.7 7.8 10.1 −22.0 8.8 −23.0 2.2 50 4.4 160 27.9 −44.1 2.0 2.5 NUNAVUT Baker Lake 719260 64.30 96.08 18 101.11 6393 −40.4 −39.3 16.3 14.2 12.6 18.9 −32.4 16.4 −32.3 5.4 0 4.3 270 25.2 −45.5 2.7 2.4 Cambridge Bay A 719250 69.10 105.12 27 101.00 6193 −39.0 −37.4 15.6 13.7 12.1 16.0 −28.2 13.9 −28.4 4.0 320 5.1 140 19.8 −45.7 2.7 2.1 Chesterfield 719164 63.33 90.72 11 101.19 6393 −37.0 −36.6 14.6 12.9 11.6 15.7 −32.2 13.9 −32.3 6.4 320 5.7 320 22.6 −45.1 7.7 2.7 Coral Harbour A 719150 64.20 83.37 64 100.56 6193 −40.2 −38.9 16.5 14.1 12.1 17.3 −20.6 14.2 −21.6 3.8 340 5.8 270 21.4 −45.0 2.1 2.9 Hall Beach A 710810 68.78 81.25 8 101.23 6193 −40.9 −39.0 14.7 12.8 11.3 15.3 −28.6 13.2 −29.5 4.3 320 4.7 180 17.7 −47.3 2.9 3.0 WMO# = World Meteorological Organization number Elev. = elevation, m PWD = prevailing wind direction, ° True DB = dry-bulb temperature, °C Lat. = North latitude, ° Long. = West longitude, ° StdP = standard pressure at station elevation, kPa WS = wind speed, m/s Climatic Design Information 27.23 Table 2B Cooling and Dehumidification Design Conditions—Canada Station Cooling DB/MWB Evaporation WB/MDB Dehumidification DP/MDB and HR Range of DB 0.4% 1% 2% 0.4% 1% 2% 0.4% 1% 2% DB MWB DB MWB DB MWB WB MDB WB MDB WB MDB DP HR MDB DP HR MDB DP HR MDB 1 2a 2b 2c 2d 2e 2f 3a 3b 3c 3d 3e 3f 4a 4b 4c 4d 4e 4f 4g 4h 4i 5 ALBERTA Calgary Intl A 28.5 15.4 26.4 14.7 24.7 14.1 16.9 25.3 15.9 24.1 15.0 23.0 13.9 11.3 19.6 12.9 10.6 18.4 11.7 9.8 17.8 12.2 Cold Lake A 27.8 17.5 25.8 16.7 24.1 15.7 18.9 25.4 17.8 24.2 16.7 22.4 16.5 12.5 21.9 15.4 11.7 20.5 14.4 10.9 19.2 11.1 Coronation 29.5 16.7 27.5 15.6 25.4 15.2 18.1 26.5 17.1 25.3 16.2 23.9 15.3 12.0 20.7 14.2 11.1 19.5 13.2 10.4 18.7 12.3 Edmonton Intl A 27.6 17.1 25.6 16.5 24.0 15.7 18.9 25.0 17.7 23.9 16.5 22.5 16.4 12.7 22.5 15.2 11.8 20.7 14.1 11.0 19.1 12.1 Fort McMurray A 28.7 17.5 26.5 16.5 24.6 15.5 18.8 26.3 17.7 24.4 16.5 22.6 16.1 12.0 21.5 15.0 11.1 20.1 14.1 10.5 19.0 12.2 Grande Prairie A 27.4 16.4 25.4 15.3 23.6 14.5 17.6 24.8 16.5 23.2 15.4 21.8 15.1 11.6 20.1 14.0 10.8 18.4 13.0 10.1 17.4 11.6 Lethbridge A 30.9 16.3 29.1 15.9 27.1 15.3 18.3 27.3 17.2 26.1 16.2 25.1 15.1 12.0 21.0 14.0 11.2 20.4 12.9 10.4 19.2 13.8 Medicine Hat A 32.2 17.4 30.6 16.9 28.8 16.1 18.7 29.1 17.7 28.1 16.9 26.6 15.4 11.9 21.3 14.3 11.1 20.6 13.2 10.3 20.0 13.9 Peace River A 27.2 16.7 25.3 15.8 23.6 14.9 18.1 24.8 16.8 23.6 15.8 21.8 15.4 11.7 20.8 14.3 10.9 19.4 13.3 10.2 18.3 11.9 Red Deer A 27.9 16.8 25.9 15.9 24.2 15.1 18.2 25.3 17.1 24.1 16.0 22.6 15.4 12.2 21.8 14.3 11.4 20.3 13.3 10.6 19.0 12.7 Rocky Mtn. House 26.9 16.6 25.3 16.0 23.8 15.2 18.0 25.4 17.0 23.8 15.9 22.4 15.5 12.4 20.9 14.4 11.6 19.9 13.5 10.9 18.9 12.5 Vermilion A 28.5 17.6 26.4 16.9 24.7 16.1 19.1 25.6 17.9 24.8 16.9 23.4 16.4 12.6 22.4 15.4 11.8 20.8 14.4 11.0 19.5 12.2 Whitecourt 26.9 16.3 25.1 15.7 23.5 14.9 18.2 24.7 16.9 23.4 15.8 21.6 15.7 12.3 20.7 14.6 11.4 19.3 13.6 10.7 18.0 13.0 BRITISH COLUMBIA Abbotsford A 29.2 19.6 26.9 18.7 25.0 17.7 20.3 28.1 19.1 26.3 18.0 24.4 16.9 12.1 24.7 15.9 11.4 23.0 15.1 10.8 21.0 11.9 Cape St. James 18.0 14.8 16.6 14.2 15.7 13.7 15.5 17.0 14.8 16.0 14.2 15.3 14.8 10.6 15.9 14.2 10.2 15.2 13.7 9.9 14.7 4.2 Castlegar A 33.3 18.0 31.0 17.3 29.0 16.6 19.2 29.6 18.2 28.5 17.2 27.0 15.6 11.8 21.9 14.7 11.1 21.0 13.9 10.5 19.7 15.5 Comox A 26.7 17.2 24.6 16.6 22.9 16.0 18.1 24.6 17.3 23.1 16.5 21.9 15.7 11.2 19.9 15.0 10.7 19.1 14.3 10.2 18.3 9.1 Cranbrook A 31.2 16.2 29.4 15.6 27.3 14.9 17.2 28.1 16.4 26.9 15.6 25.5 13.8 11.0 18.6 12.7 10.3 18.6 11.6 9.5 18.5 13.8 Fort Nelson A 27.6 16.6 25.6 15.7 23.9 14.8 17.9 25.1 16.8 23.5 15.8 22.0 15.4 11.4 19.8 14.3 10.7 18.8 13.3 10.0 18.0 11.7 Fort St. John A 26.3 16.0 24.4 15.0 22.6 14.2 17.2 24.2 16.1 22.4 15.0 20.9 14.7 11.4 19.3 13.7 10.6 18.2 12.5 9.8 17.3 10.4 Kamloops A 33.6 18.1 31.2 17.3 29.2 16.6 19.0 31.3 18.1 29.3 17.2 27.3 15.1 11.2 20.9 14.1 10.5 20.3 13.2 9.9 20.2 13.7 Penticton A 32.1 18.4 30.3 17.7 28.5 17.0 19.4 29.6 18.5 28.4 17.7 27.0 15.7 11.6 22.9 14.7 10.9 22.0 13.8 10.3 21.7 14.7 Port Hardy A 19.9 15.0 18.4 14.3 17.1 13.8 15.8 18.7 15.0 17.5 14.3 16.5 14.5 10.3 16.9 13.9 9.9 16.0 13.3 9.5 15.4 6.9 Prince George A 27.1 15.8 25.4 15.2 23.5 14.2 17.1 25.4 16.0 23.2 15.0 21.7 14.2 11.0 18.6 13.3 10.4 17.8 12.1 9.6 16.9 12.9 Prince Rupert A 18.9 14.4 17.3 13.8 16.3 13.4 15.4 17.9 14.6 16.6 13.9 15.8 14.3 10.2 16.2 13.6 9.8 15.5 13.1 9.4 14.9 5.8 Quesnel A 29.5 16.7 27.2 15.7 25.2 15.1 17.9 26.6 16.8 24.9 15.9 23.4 15.2 11.5 19.2 14.1 10.7 18.4 13.2 10.1 17.3 14.1 Sandspit A 19.7 15.5 18.5 14.8 17.2 14.2 16.2 18.9 15.4 17.7 14.7 16.7 15.1 10.7 16.9 14.4 10.2 16.3 13.8 9.8 15.8 4.8 Smithers A 27.2 16.0 25.0 15.2 22.8 14.3 16.9 25.5 15.9 23.5 14.9 21.4 14.0 10.6 18.3 13.1 10.0 17.6 12.0 9.3 16.8 12.2 Spring Island 20.2 15.3 18.7 14.8 17.2 14.2 16.1 18.8 15.4 17.6 14.8 16.7 15.1 10.8 16.7 14.6 10.5 16.1 14.1 10.2 15.4 4.9 Terrace A 28.2 16.6 25.6 15.8 23.3 14.8 17.5 26.3 16.4 24.2 15.4 22.0 14.3 10.4 18.8 13.4 9.8 17.8 12.6 9.3 17.7 9.5 Tofino A 22.2 16.4 20.1 15.3 18.6 14.5 16.8 21.0 15.8 19.2 15.1 17.6 15.2 10.8 17.3 14.6 10.4 16.6 14.1 10.1 15.8 6.8 Vancouver Intl A 24.6 18.2 23.2 17.6 21.8 16.9 18.8 23.8 18.0 22.4 17.2 21.3 16.6 11.8 21.7 15.9 11.3 20.9 15.3 10.9 19.9 7.8 Victoria Intl A 26.2 17.3 24.1 16.7 22.2 15.9 18.0 25.1 17.1 23.3 16.2 21.6 15.0 10.7 20.3 14.3 10.2 19.3 13.7 9.8 18.3 10.2 Williams Lake A 28.1 14.8 25.9 14.1 23.9 13.4 15.8 25.2 14.9 23.9 14.1 22.3 12.9 10.4 16.9 11.6 9.5 16.2 10.6 8.9 15.8 12.2 MANITOBA Brandon A 30.8 19.6 28.9 19.0 26.8 18.2 21.6 27.7 20.3 26.7 19.1 25.1 19.5 15.0 24.7 18.2 13.8 23.3 16.8 12.6 22.3 13.1 Churchill A 25.0 16.6 22.1 15.3 19.5 14.2 17.5 23.2 16.0 21.2 14.5 18.9 15.1 10.8 20.1 13.5 9.7 18.3 11.8 8.6 16.9 9.3 Dauphin A 30.4 19.6 28.6 18.9 26.6 18.0 21.3 27.6 20.2 26.5 19.0 25.1 19.2 14.5 25.0 18.0 13.4 23.2 16.6 12.3 22.1 12.3 Portage La Prairie A 31.1 20.1 29.2 19.5 27.2 18.4 22.1 28.4 20.8 26.8 19.7 25.6 20.2 15.4 25.6 18.8 14.1 23.8 17.8 13.2 22.8 11.4 The Pas A 28.1 18.6 26.1 17.6 24.5 16.6 19.8 26.1 18.7 24.3 17.7 22.8 17.9 13.3 22.6 16.6 12.2 21.3 15.6 11.4 20.2 10.2 Thompson A 28.1 17.5 26.1 16.6 24.3 15.7 19.1 25.6 17.8 24.1 16.7 22.5 16.6 12.1 21.9 15.3 11.1 20.4 14.2 10.4 19.1 12.8 Winnipeg Intl A 30.8 19.9 29.0 19.6 27.0 18.7 22.1 28.0 20.9 26.8 19.8 25.3 20.2 15.3 25.4 18.9 14.1 23.9 17.9 13.2 22.9 11.4 NEW BRUNSWICK Charlo A 28.2 19.8 25.9 19.1 24.1 18.1 21.3 25.8 20.2 24.2 19.1 22.8 19.8 14.6 23.3 18.9 13.8 22.2 17.9 12.9 21.1 10.2 Chatham A 29.9 20.3 28.1 19.2 26.1 18.3 21.8 27.1 20.7 25.5 19.8 24.0 20.1 14.8 24.1 19.2 14.0 22.9 18.3 13.2 21.9 11.3 Fredericton A 30.0 20.6 28.2 19.7 26.3 18.8 22.1 27.7 21.0 26.0 20.0 24.5 20.2 14.9 25.0 19.3 14.1 23.6 18.4 13.3 22.3 11.5 Moncton A 28.2 20.1 26.4 19.4 24.9 18.5 21.7 26.1 20.6 24.7 19.7 23.4 20.1 14.9 24.1 19.2 14.1 22.9 18.3 13.3 21.8 10.8 Saint John A 25.6 18.4 24.0 17.6 22.2 16.5 20.0 23.8 18.8 22.1 17.8 20.8 18.7 13.7 21.6 17.6 12.8 20.3 16.6 12.0 19.0 9.4 NEWFOUNDLAND Battle Harbour 18.3 14.2 15.8 13.0 14.3 11.9 14.9 17.1 13.4 15.5 12.1 13.9 13.9 9.9 16.2 12.1 8.8 14.5 11.0 8.2 13.0 5.8 Bonavista 23.5 18.3 21.5 17.3 19.9 16.6 19.3 22.1 18.2 20.6 17.1 19.3 18.2 13.1 21.1 17.0 12.2 19.8 16.0 11.4 18.6 6.5 Cartwright 23.9 16.5 21.3 15.1 19.2 14.3 17.2 22.2 15.9 20.2 14.8 18.7 15.2 10.8 19.2 14.0 10.0 18.2 12.8 9.2 17.0 9.7 Daniels Harbour 20.3 17.4 19.1 16.8 18.2 16.1 18.2 19.6 17.4 18.7 16.4 17.7 17.8 12.8 19.2 16.7 11.9 18.1 15.8 11.2 17.3 5.4 Deer Lake A 27.0 18.6 25.2 17.6 23.5 16.8 20.2 24.8 19.1 23.1 18.1 21.9 18.7 13.6 22.5 17.5 12.6 21.8 16.3 11.6 20.3 11.9 Gander Intl A 25.9 18.4 24.2 17.2 22.3 16.6 19.8 23.9 18.8 22.2 17.8 21.1 18.4 13.5 21.9 17.3 12.6 20.9 16.3 11.8 19.7 9.9 Goose A 27.4 17.0 24.9 16.2 22.6 15.3 18.6 24.9 17.4 22.9 16.3 21.5 16.3 11.7 21.0 15.1 10.8 19.8 14.1 10.1 18.4 10.1 Hopedale 21.0 15.0 18.5 13.8 16.2 12.5 15.7 19.9 14.3 17.9 12.9 15.9 13.9 9.9 17.6 12.2 8.9 16.1 10.9 8.1 14.9 7.0 St. John’s A 24.3 18.5 22.8 17.6 21.0 17.1 20.0 22.7 18.9 21.4 17.9 20.2 19.0 14.0 21.7 18.0 13.2 20.6 16.8 12.2 19.5 8.7 Stephenville A 23.1 17.9 21.5 17.5 20.4 17.0 19.4 21.6 18.5 20.6 17.6 19.8 18.6 13.5 20.9 17.6 12.6 20.2 16.5 11.8 19.1 6.9 Wabush Lake A 24.3 15.5 22.0 14.4 20.1 13.7 17.1 21.5 15.9 20.3 14.7 18.7 15.5 11.8 19.0 14.3 10.9 17.4 13.1 10.0 16.4 9.4 NORTHWEST TERRITORIES Cape Parry A 14.6 11.4 12.1 9.8 10.2 8.4 11.6 14.3 9.9 11.9 8.5 10.0 9.8 7.5 13.1 8.5 6.9 11.2 7.0 6.2 9.7 5.4 Fort Smith A 28.0 17.2 25.8 16.2 24.0 15.5 18.4 25.4 17.3 23.8 16.3 22.3 15.9 11.6 20.6 14.8 10.8 19.5 13.8 10.1 18.7 11.9 Inuvik UA 25.6 15.5 23.7 14.7 21.7 13.8 16.5 23.8 15.4 22.3 14.4 20.8 13.3 9.6 19.5 11.9 8.7 18.3 10.9 8.2 17.6 10.2 Norman Wells A 26.9 16.5 25.1 15.8 23.4 15.0 17.7 24.9 16.7 23.4 15.8 21.9 14.9 10.7 19.7 14.0 10.1 19.0 13.1 9.5 18.4 10.3 Yellowknife A 24.9 15.8 23.2 14.7 21.3 13.9 16.7 23.0 15.7 21.5 14.7 20.1 14.2 10.4 18.9 13.2 9.7 18.1 11.9 8.9 17.3 8.0 NUNAVUT Baker Lake 20.6 13.9 18.2 12.8 16.0 11.6 14.7 19.3 13.3 17.3 12.0 15.6 12.5 9.0 17.5 11.0 8.2 14.8 9.9 7.6 13.7 9.1 Cambridge Bay A 15.6 11.7 13.8 10.3 11.9 9.1 12.1 15.1 10.6 13.3 9.4 11.6 10.2 7.8 13.5 8.9 7.1 11.9 7.6 6.5 10.7 6.9 Chesterfield 18.9 12.3 15.8 11.0 13.5 9.7 13.0 18.4 11.1 15.3 9.8 13.1 10.0 7.6 15.4 8.7 7.0 12.5 7.6 6.5 11.2 7.6 Coral Harbour A 17.9 11.9 15.5 10.8 13.5 9.7 12.5 17.0 11.1 15.0 9.9 13.2 10.0 7.7 14.5 8.8 7.1 12.5 7.8 6.6 11.2 8.2 Hall Beach A 13.4 9.8 11.1 8.5 9.3 7.0 10.1 13.0 8.6 11.0 7.1 9.2 8.2 6.8 11.7 6.6 6.0 10.0 5.4 5.6 8.5 5.5 MDB = mean coincident dry-bulb temp., °C MWS = mean coincident wind speed, m/s HR = humidity ratio, grams of moisture per kilogram of dry air MWB = mean coincident wet-bulb temp., °C StdD = standard deviation, °C A = airport DP = dew-point temperature, °C 27.24 2001 ASHRAE Fundamentals Handbook (SI) Table 2A Heating and Wind Design Conditions—Canada Heating Dry Bulb Extreme Wind Speed, m/s Coldest Month MWS/PWD to DB Extr. Annual Daily Lat.
Long.
Elev., m StdP, kPa 0.4% 1% 99.6% 0.4% Mean DB StdD DB Station WMO# Dates 99.6% 99% 1% 2.5% 5% WS MDB WS MDB MWS PWD MWS PWD Max.
Min. Max. Min.
1a 1b 1c 1d 1e 1f 1g 2a 2b 3a 3b 3c 4a 4b 4c 4d 5a 5b 5c 5d 6a 6b 6c 6d Iqaluit A (Frobisher) 719090 63.75 68.55 33 100.93 6193 −39.6 −37.9 14.4 12.5 10.9 17.8 −24.3 15.3 −24.4 1.7 320 5.2 320 20.2 −41.7 2.4 2.6 Resolute A 719240 74.72 94.98 67 100.52 6393 −40.9 −39.7 17.3 15.1 13.3 18.7 −26.0 16.5 −27.7 4.2 320 5.5 110 12.8 −45.0 2.1 2.7 NOVA SCOTIA Greenwood A 713970 44.98 64.92 28 100.99 6193 −18.8 −16.2 13.1 11.3 10.0 15.8 −4.1 13.2 −5.2 3.1 300 6.5 250 31.9 −24.0 1.3 3.0 Halifax Intl A 713950 44.88 63.50 145 99.60 6993 −19.0 −16.7 12.1 10.3 9.1 13.4 −3.1 12.1 −3.6 5.1 320 5.2 200 30.4 −22.4 1.5 2.4 Sable Island 716000 43.93 60.02 4 101.28 6193 −9.8 −8.1 17.0 15.0 13.4 19.3 −1.4 17.4 −1.8 10.6 290 5.7 200 22.8 −12.4 1.4 2.2 Shearwater A 716010 44.63 63.50 51 100.71 6193 −17.0 −15.0 13.1 11.2 9.7 14.5 −4.4 12.7 −4.3 5.2 340 4.5 230 29.5 −20.7 1.8 2.3 Sydney A 717070 46.17 60.05 62 100.58 6193 −18.4 −16.1 13.5 11.8 10.4 15.6 −5.0 13.2 −4.4 5.9 270 6.2 230 30.6 −21.6 1.4 2.5 Truro 713980 45.37 63.27 40 100.85 6193 −22.8 −19.8 10.7 9.2 8.2 13.2 −4.0 11.5 −4.1 2.4 0 4.7 270 30.0 −26.9 1.8 3.2 Yarmouth A 716030 43.83 66.08 43 100.81 6193 −14.0 −12.1 12.5 11.1 10.0 13.5 −3.3 12.5 −2.8 5.3 320 4.4 190 26.2 −17.1 1.5 2.0 ONTARIO Armstrong A 718410 50.30 89.03 351 97.18 6193 −36.3 −34.4 9.6 8.5 7.5 9.4 −19.1 8.2 −19.2 1.2 270 5.5 0 31.2 −44.8 1.8 2.7 Atikokan 717480 48.75 91.62 393 96.69 6793 −35.2 −32.8 7.2 6.3 5.5 7.1 −14.2 6.1 −14.6 0.3 270 3.4 230 31.8 −40.9 2.0 2.1 Big Trout Lake 718480 53.83 89.87 224 98.66 6793 −35.8 −34.4 10.4 9.2 8.2 10.5 −18.1 9.3 −18.9 2.5 290 4.0 200 29.7 −40.9 1.8 2.7 Earlton A 717350 47.70 79.85 243 98.44 6193 −32.6 −29.6 9.6 8.4 7.7 10.6 −10.1 9.3 −10.9 1.8 320 5.2 200 32.6 −38.6 2.1 2.8 Geraldton 718340 49.78 86.93 351 97.18 6893 −35.7 −33.5 9.2 7.9 7.1 9.7 −16.2 8.6 −15.6 0.7 270 4.7 0 30.7 −42.5 2.3 3.7 Gore Bay A 717330 45.88 82.57 193 99.03 6193 −24.3 −21.4 12.0 10.4 9.2 13.1 −5.3 11.4 −5.8 3.1 0 4.8 180 29.7 −29.6 2.5 3.3 Kapuskasing A 718310 49.42 82.47 227 98.63 6193 −34.4 −31.7 8.7 7.6 6.9 9.5 −15.0 8.6 −15.0 1.8 270 4.6 230 32.3 −39.4 1.5 2.8 Kenora A 718500 49.78 94.37 411 96.48 6193 −32.7 −30.2 9.2 8.2 7.5 9.3 −14.0 8.4 −14.7 3.6 320 4.9 180 31.7 −35.8 1.9 2.8 London A 716230 43.03 81.15 278 98.03 6193 −19.3 −16.7 11.2 9.9 8.7 13.2 −6.2 11.4 −6.3 3.8 250 5.1 250 32.2 −23.9 1.9 3.1 Mount Forest 716310 43.98 80.75 415 96.44 6293 −21.5 −19.3 11.0 9.5 8.4 12.4 −7.1 11.0 −7.6 2.7 90 4.4 250 30.4 −26.4 1.2 2.5 Muskoka A 716300 44.97 79.30 282 97.98 6193 −27.0 −24.0 9.4 8.5 7.8 10.3 −6.0 9.0 −5.9 2.9 320 4.1 270 30.8 −34.3 1.2 3.1 North Bay A 717310 46.35 79.43 371 96.95 6193 −27.9 −25.1 8.8 7.7 7.0 10.4 −8.7 9.1 −8.9 2.8 0 4.4 230 30.0 −32.3 1.8 3.0 Ottawa Intl A 716280 45.32 75.67 114 99.96 6193 −24.8 −22.2 10.0 8.8 7.7 11.9 −8.5 10.3 −9.3 3.9 290 4.5 250 33.0 −28.4 1.5 2.8 Sault Ste. Marie A 712600 46.48 84.52 192 99.04 6293 −25.2 −22.3 11.8 10.2 8.8 12.5 −8.0 10.6 −7.9 1.9 90 3.9 220 31.3 −31.9 1.8 3.3 Simcoe 715270 42.85 80.27 241 98.46 6293 −18.6 −16.2 10.6 9.1 8.1 12.5 −4.8 10.6 −4.2 4.6 270 5.0 230 32.5 −23.2 1.6 2.6 Sioux Lookout A 718420 50.12 91.90 390 96.73 6193 −34.5 −31.9 7.5 6.7 6.1 8.4 −16.6 7.5 −16.7 1.9 270 4.0 200 31.9 −39.3 1.8 2.7 Sudbury A 717300 46.62 80.80 348 97.21 6193 −28.4 −25.6 13.2 11.7 10.4 13.4 −10.6 12.1 −10.6 4.6 0 6.2 230 31.8 −32.7 2.2 2.9 Thunder Bay A 717490 48.37 89.32 199 98.96 6193 −30.2 −27.6 10.8 9.2 7.9 11.3 −12.8 9.8 −14.2 3.8 250 5.1 200 32.4 −34.9 2.1 2.6 Timmins A 717390 48.57 81.37 295 97.83 6193 −33.5 −30.5 9.4 8.5 7.6 9.8 −14.3 8.7 −14.1 2.0 180 4.4 250 32.7 −39.3 2.0 2.6 Toronto Intl A 716240 43.67 79.63 173 99.26 6593 −19.9 −17.2 11.6 10.0 8.8 13.0 −5.8 11.4 −5.1 4.1 340 5.3 270 33.1 −24.0 1.6 3.1 Trenton A 716210 44.12 77.53 86 100.30 6193 −22.1 −19.6 11.5 9.9 8.6 13.4 −5.3 11.7 −4.6 2.8 50 5.5 230 31.4 −26.7 1.7 3.1 Wiarton A 716330 44.75 81.10 222 98.69 6193 −20.3 −17.6 11.3 10.0 8.7 11.9 −3.0 10.5 −4.0 3.2 340 5.4 230 30.9 −25.9 1.3 3.9 Windsor A 715380 42.27 82.97 190 99.06 6193 −16.9 −14.6 12.4 10.8 9.4 13.2 −5.4 11.8 −5.7 5.3 230 5.5 250 34.4 −20.5 1.5 3.0 PRINCE EDWARD ISLAND Charlottetown A 717060 46.28 63.13 54 100.68 6193 −21.0 −18.9 10.8 9.9 8.8 15.7 −8.5 13.0 −7.6 5.7 270 5.3 230 29.4 −24.6 1.3 2.6 Summerside A 717020 46.43 63.83 24 101.04 6193 −20.5 −18.4 14.2 12.4 11.0 17.7 −8.0 15.5 −7.9 6.7 270 5.6 200 29.5 −23.6 1.5 2.3 QUEBEC Bagotville A 717270 48.33 71.00 159 99.43 6193 −30.7 −28.4 11.7 10.2 9.1 13.2 −16.0 11.5 −15.2 2.8 270 4.6 270 32.5 −34.5 1.4 2.8 Baie Comeau A 711870 49.13 68.20 22 101.06 6593 −28.4 −25.7 12.1 10.6 9.2 13.2 −10.1 11.3 −10.3 4.7 270 5.7 230 27.8 −33.3 1.8 3.8 Grindstone Island 47.38 61.87 59 100.62 6193 −18.5 −16.3 21.9 19.3 17.2 24.4 −6.5 21.7 −6.2 10.5 290 8.1 250 25.9 −20.4 1.1 3.1 Kuujjuarapik A 719050 55.28 77.77 12 101.18 6193 −36.2 −34.3 12.8 11.1 9.9 11.8 −17.5 10.5 −19.0 3.5 120 6.0 180 29.3 −41.4 2.2 2.6 Kuujuaq A 719060 58.10 68.42 37 100.88 6193 −36.5 −34.8 12.9 10.9 9.5 14.0 −16.9 12.1 −19.0 2.2 230 4.9 180 27.9 −40.5 1.9 2.0 La Grande Riviere A 718270 53.63 77.70 195 99.00 7793 −36.0 −34.3 10.0 8.8 7.7 10.3 −17.6 9.0 −18.8 2.6 270 5.7 240 29.6 −39.2 2.0 1.9 Lake Eon A 714210 51.87 63.28 561 94.76 6193 −34.9 −33.0 10.4 9.2 8.2 10.4 −15.7 9.3 −17.6 2.2 270 4.3 230 26.7 −41.0 1.4 2.2 Mont Joli A 717180 48.60 68.22 52 100.70 6193 −24.5 −22.3 12.6 10.9 9.8 15.8 −13.3 13.3 −12.6 6.5 290 6.3 230 30.6 −27.8 1.7 2.5 Montreal Intl A 716270 45.47 73.75 36 100.89 6193 −24.4 −21.8 10.2 9.0 7.9 13.2 −7.4 11.5 −7.7 3.2 250 5.0 230 32.0 −28.3 1.3 2.5 Montreal Mirabel A 716278 45.68 74.03 82 100.34 7693 −26.9 −24.0 9.2 7.9 6.9 11.1 −10.6 9.5 −11.9 2.6 240 3.8 240 31.2 −31.7 1.0 2.4 Nitchequon 53.20 70.90 536 95.05 6193 −36.3 −35.0 11.3 10.0 9.0 13.0 −21.3 11.2 −20.6 2.5 270 4.7 230 25.6 −43.2 2.0 2.6 Quebec A 717140 46.80 71.38 73 100.45 6193 −26.4 −24.0 10.8 9.4 8.3 13.2 −10.4 11.6 −11.5 4.3 250 5.2 250 31.7 −30.4 1.3 2.7 Riviere Du Loup 717150 47.80 69.55 148 99.56 6693 −25.1 −23.4 9.3 8.3 7.5 10.6 −9.9 9.3 −10.7 4.3 180 5.1 230 29.3 −27.6 1.1 2.6 Roberval A 717280 48.52 72.27 179 99.19 6193 −30.7 −28.5 10.4 9.1 8.1 12.1 −13.1 10.2 −12.0 3.1 270 5.3 220 32.3 −34.3 1.7 2.4 Schefferville A 718280 54.80 66.82 521 95.22 6193 −36.3 −34.7 11.8 10.3 9.2 13.5 −23.3 11.8 −22.8 3.2 320 5.5 270 27.2 −41.4 2.2 2.8 Sept-Iles A 718110 50.22 66.27 55 100.67 6893 −28.6 −26.3 12.1 10.2 9.0 13.2 −8.9 11.1 −10.7 3.7 300 4.9 220 27.7 −32.4 2.4 2.4 Sherbrooke A 716100 45.43 71.68 241 98.46 6393 −29.1 −25.7 8.5 7.4 6.7 9.9 −9.7 8.6 −9.8 2.1 110 4.1 250 31.2 −34.6 1.2 2.8 St. Hubert A 713710 45.52 73.42 27 101.00 6193 −24.4 −21.7 12.1 10.3 9.1 13.5 −7.7 12.2 −7.1 3.1 20 5.6 250 32.6 −28.8 1.4 2.7 Ste. Agathe Des Monts 717200 46.05 74.28 395 96.67 6693 −28.5 −25.9 8.6 7.4 6.5 10.0 −11.0 8.6 −11.7 1.9 290 3.9 270 29.8 −33.2 1.6 2.2 Val d’Or A 717250 48.07 77.78 337 97.34 6193 −33.0 −30.2 9.2 8.1 7.1 9.9 −11.0 8.8 −12.0 2.1 310 4.8 230 31.1 −38.2 1.6 2.7 SASKATCHEWAN Broadview 718610 50.38 102.68 602 94.30 6693 −34.2 −31.7 11.7 10.3 9.1 12.6 −12.7 11.1 −12.9 3.4 290 5.3 160 34.5 −38.1 2.0 3.2 Estevan A 718620 49.22 102.97 581 94.54 6193 −31.8 −29.3 13.2 11.6 10.4 14.3 −11.1 12.7 −11.4 4.5 290 6.4 180 36.3 −35.3 1.8 3.0 Moose Jaw A 718640 50.33 105.55 577 94.58 6193 −32.2 −29.6 13.4 11.7 10.4 15.7 −6.4 13.3 −7.4 3.9 290 5.7 180 36.1 −35.9 1.8 3.0 North Battleford A 718760 52.77 108.25 548 94.91 6193 −34.9 −32.1 11.2 9.9 8.8 11.5 −13.2 10.0 −13.8 1.9 320 5.0 140 33.9 −38.8 1.7 3.0 Prince Albert A 718690 53.22 105.68 428 96.29 6193 −36.5 −34.1 10.2 9.0 7.9 10.5 −16.5 9.2 −16.4 1.4 270 5.1 180 32.9 −41.6 2.1 3.3 Regina A 718630 50.43 104.67 577 94.58 6193 −33.9 −31.1 13.5 11.9 10.1 15.2 −13.1 13.0 −12.5 3.8 270 6.2 180 35.3 −37.8 2.0 2.8 Saskatoon 718660 52.17 106.68 504 95.42 6193 −34.8 −32.1 11.9 10.3 9.1 12.0 −11.4 10.4 −14.2 3.3 290 5.6 180 34.4 −38.5 2.1 3.3 Swift Current A 718700 50.28 107.68 818 91.88 6193 −31.9 −29.4 14.5 12.6 11.0 16.0 −8.4 13.4 −10.0 6.3 270 6.0 180 34.7 −35.9 1.7 3.2 Uranium City A 710760 59.57 108.48 318 97.56 6393 −39.0 −37.0 9.1 7.9 6.9 8.0 −21.7 7.0 −22.5 1.6 70 3.1 230 29.8 −45.1 2.2 1.9 Wynyard 718650 51.77 104.20 561 94.76 6593 −34.1 −31.6 11.8 10.4 9.2 12.2 −10.8 10.7 −11.7 3.6 270 5.6 180 33.5 −37.3 1.9 3.4 Yorkton A 711380 51.27 102.47 498 95.48 6193 −34.6 −32.0 11.2 9.9 8.8 12.1 −14.4 10.3 −16.5 3.0 290 5.3 180 34.1 −38.5 1.8 3.1 YUKON TERRITORY Burwash A 719670 61.37 139.05 806 92.01 6793 −36.6 −35.3 12.4 10.7 9.2 13.1 −0.7 11.1 −0.8 0.2 290 4.1 110 26.3 −48.2 1.9 3.6 Whitehorse A 719640 60.72 135.07 703 93.16 6193 −36.8 −34.7 10.1 9.0 8.0 12.3 −8.6 10.8 −9.5 0.9 340 3.7 140 28.2 −43.7 2.2 3.3 WMO# = World Meteorological Organization number Elev. = elevation, m PWD = prevailing wind direction, ° True DB = dry-bulb temperature, °C Lat. = North latitude, ° Long. = West longitude, ° StdP = standard pressure at station elevation, kPa WS = wind speed, m/s Climatic Design Information 27.25 Table 2B Cooling and Dehumidification Design Conditions—Canada Station Cooling DB/MWB Evaporation WB/MDB Dehumidification DP/MDB and HR Range of DB 0.4% 1% 2% 0.4% 1% 2% 0.4% 1% 2% DB MWB DB MWB DB MWB WB MDB WB MDB WB MDB DP HR MDB DP HR MDB DP HR MDB 1 2a 2b 2c 2d 2e 2f 3a 3b 3c 3d 3e 3f 4a 4b 4c 4d 4e 4f 4g 4h 4i 5 Iqaluit A (Frobisher) 15.7 10.1 13.6 8.9 11.5 8.0 10.6 15.2 9.3 12.8 8.2 11.4 7.8 6.6 11.9 6.6 6.1 10.9 5.6 5.7 9.8 6.9 Resolute A 10.2 7.3 8.6 6.1 6.9 5.1 7.5 10.1 6.2 8.3 5.2 6.9 5.4 5.6 8.4 4.4 5.2 7.4 3.5 4.9 6.2 4.7 NOVA SCOTIA Greenwood A 28.6 20.5 26.8 19.6 25.4 18.7 22.1 26.2 21.2 24.9 20.3 23.8 20.7 15.4 24.2 19.8 14.6 23.4 18.9 13.7 22.4 11.1 Halifax Intl A 26.9 19.7 25.3 18.8 23.8 17.9 21.2 24.9 20.3 23.3 19.3 22.0 20.1 15.1 22.7 19.2 14.2 21.6 18.4 13.5 20.5 9.3 Sable Island 21.1 19.5 20.3 18.8 19.6 18.1 20.0 20.7 19.3 19.9 18.6 19.3 19.7 14.4 20.5 19.0 13.8 19.7 18.3 13.2 19.1 4.6 Shearwater A 25.5 19.1 23.8 18.0 22.1 17.4 20.6 23.5 19.6 22.1 18.7 20.8 19.7 14.5 21.9 18.8 13.7 20.6 18.1 13.1 19.7 7.3 Sydney A 27.1 20.1 25.4 19.3 23.7 18.4 21.5 25.5 20.4 23.8 19.4 22.3 20.2 15.0 23.5 19.2 14.1 22.2 18.3 13.3 21.1 9.6 Truro 26.3 20.3 25.0 19.3 23.7 18.7 21.7 25.0 20.7 23.6 19.8 22.1 20.5 15.2 23.7 19.7 14.5 22.5 19.0 13.9 21.3 10.7 Yarmouth A 22.9 18.6 21.4 18.0 20.3 17.3 19.9 21.7 18.8 20.6 18.0 19.6 19.2 14.0 20.8 18.3 13.3 19.8 17.2 12.3 18.9 7.3 ONTARIO Armstrong A 27.2 18.7 25.8 18.4 24.1 17.3 20.7 25.4 19.3 24.1 18.2 22.5 19.2 14.6 22.6 17.9 13.4 21.8 16.3 12.1 20.5 13.7 Atikokan 28.7 19.5 26.8 18.9 25.2 17.9 21.6 26.4 20.4 24.8 19.3 23.5 20.1 15.5 24.1 19.0 14.5 23.0 17.9 13.5 21.8 12.8 Big Trout Lake 26.0 17.5 24.1 16.9 22.2 16.1 19.2 23.9 18.1 22.1 17.1 21.0 17.6 13.0 21.3 16.4 12.0 20.1 15.4 11.2 19.3 9.1 Earlton A 29.3 20.3 27.2 19.4 25.5 18.4 21.9 26.9 20.7 25.2 19.6 24.1 20.3 15.4 24.6 19.2 14.4 23.2 18.2 13.5 22.0 12.0 Geraldton 27.2 18.9 25.7 18.1 24.1 17.0 20.7 25.0 19.5 23.6 18.4 22.5 19.3 14.7 23.1 18.3 13.8 22.1 16.7 12.4 20.6 12.3 Gore Bay A 26.7 20.0 25.3 19.3 24.0 18.4 21.6 25.1 20.6 23.6 19.7 22.6 20.4 15.4 23.3 19.5 14.6 22.3 18.6 13.8 21.6 9.1 Kapuskasing A 29.0 19.3 26.9 18.5 25.2 17.6 21.1 26.1 19.9 24.7 18.8 23.3 19.4 14.5 23.8 18.3 13.6 22.4 16.9 12.4 21.2 12.5 Kenora A 28.9 19.4 27.0 18.5 25.4 17.8 21.3 26.2 20.2 24.8 19.1 23.7 19.7 15.2 24.0 18.6 14.1 22.9 17.4 13.1 22.0 9.1 London A 29.6 21.9 28.2 21.1 26.7 20.4 23.4 28.0 22.4 26.7 21.5 25.2 21.8 17.0 26.1 21.0 16.2 24.9 20.2 15.4 23.8 11.0 Mount Forest 28.2 21.2 26.5 20.2 25.2 19.2 22.4 26.5 21.3 25.1 20.3 23.7 20.9 16.4 24.8 20.0 15.5 23.5 19.2 14.7 22.5 11.3 Muskoka A 28.7 20.8 26.8 19.8 25.5 19.0 22.3 26.6 21.2 24.9 20.2 23.6 20.9 16.1 24.5 20.0 15.2 23.3 19.0 14.3 22.1 11.5 North Bay A 27.2 19.5 25.6 18.8 24.2 18.0 21.2 25.2 20.2 23.7 19.3 22.5 20.0 15.4 23.2 19.1 14.5 22.1 18.2 13.7 21.2 9.5 Ottawa Intl A 30.1 21.3 28.5 20.5 26.8 19.5 22.8 28.0 21.8 26.4 20.8 25.3 21.1 16.0 25.5 20.2 15.1 24.6 19.2 14.2 23.7 10.3 Sault Ste. Marie A 28.1 20.6 26.1 19.4 24.4 18.3 21.9 26.2 20.7 24.4 19.6 23.2 20.4 15.4 24.3 19.4 14.5 22.9 18.3 13.5 21.6 11.6 Simcoe 29.6 22.0 28.4 21.2 26.7 20.3 23.4 28.1 22.4 26.6 21.5 25.2 21.7 16.8 25.9 20.9 16.0 24.9 20.2 15.3 23.7 10.7 Sioux Lookout A 28.9 19.3 26.9 18.1 25.3 17.4 21.0 26.4 19.8 24.4 18.8 23.1 19.4 14.8 23.2 18.3 13.8 21.9 17.1 12.8 21.1 10.5 Sudbury A 29.0 19.4 27.0 18.6 25.4 17.8 21.2 26.4 20.2 24.8 19.1 23.3 19.7 15.0 23.4 18.7 14.1 22.4 17.7 13.2 21.4 10.6 Thunder Bay A 28.7 19.7 26.5 18.6 24.7 17.6 21.2 26.5 19.8 24.7 18.6 23.0 19.3 14.4 24.4 18.1 13.3 22.2 16.8 12.3 20.9 12.1 Timmins A 29.1 19.2 27.1 18.2 25.3 17.5 21.1 26.4 19.9 24.8 18.8 23.2 19.3 14.6 23.5 18.2 13.6 22.4 17.0 12.6 21.0 12.8 Toronto Intl A 30.3 21.8 28.7 20.9 27.1 20.1 23.3 28.5 22.2 26.9 21.3 25.5 21.6 16.6 26.1 20.7 15.7 25.0 19.8 14.8 23.8 11.2 Trenton A 28.7 21.6 27.1 20.9 25.8 20.1 23.1 27.1 22.1 25.6 21.3 24.6 21.7 16.5 25.5 20.9 15.7 24.5 20.1 14.9 23.5 10.0 Wiarton A 28.0 21.1 26.4 20.3 25.0 19.5 22.3 26.4 21.3 25.0 20.4 23.8 21.0 16.1 24.8 20.1 15.2 23.6 19.2 14.4 22.6 10.0 Windsor A 31.4 22.7 29.9 21.9 28.5 21.2 24.3 29.5 23.3 28.1 22.5 26.9 22.8 17.9 27.7 21.8 16.8 26.1 21.0 16.0 25.1 9.7 PRINCE EDWARD ISLAND Charlottetown A 26.3 20.4 24.9 19.4 23.5 18.5 21.5 25.0 20.5 23.5 19.5 22.4 20.2 15.0 23.6 19.4 14.2 22.4 18.5 13.4 21.4 8.4 Summerside A 26.1 19.9 24.7 19.0 23.5 18.2 21.4 24.6 20.4 23.3 19.4 22.0 20.1 14.8 23.4 19.3 14.1 22.2 18.5 13.4 21.1 8.0 QUEBEC Bagotville A 28.8 19.3 26.7 18.2 24.9 17.4 20.8 26.0 19.7 24.6 18.7 23.2 19.1 14.1 23.1 18.1 13.3 22.1 16.8 12.2 21.0 11.0 Baie Comeau A 23.6 17.1 21.7 16.3 20.3 15.4 18.4 21.8 17.3 20.4 16.3 19.2 16.9 12.1 20.1 15.9 11.3 18.8 15.0 10.7 17.6 9.5 Grindstone Island 22.9 19.1 21.3 18.4 20.2 17.7 20.2 21.9 19.2 20.7 18.3 19.8 19.5 14.3 21.3 18.7 13.6 20.3 17.7 12.8 19.7 4.8 Kuujjuarapik A 24.1 16.1 21.3 14.9 19.0 13.9 17.4 22.1 15.9 20.0 14.4 18.2 15.4 10.9 19.5 13.9 9.9 18.0 12.1 8.8 16.4 8.8 Kuujuaq A 23.2 15.3 20.5 14.0 18.2 13.0 16.4 20.7 14.9 19.1 13.4 17.6 14.4 10.3 18.4 12.9 9.3 17.1 11.1 8.3 15.4 10.4 La Grande Riviere A 25.8 16.5 23.6 15.3 21.4 14.3 18.1 22.9 16.8 21.0 15.6 20.1 16.4 11.9 19.8 15.2 11.0 18.3 13.7 10.0 17.4 11.8 Lake Eon A 23.2 15.6 21.1 14.6 19.5 13.9 17.0 21.0 16.0 19.5 14.9 17.9 15.5 11.8 18.5 14.6 11.1 17.4 13.7 10.5 16.7 9.1 Mont Joli A 26.4 19.6 24.6 18.5 23.2 17.7 20.7 25.0 19.5 23.7 18.4 22.1 19.0 13.9 23.8 17.9 12.9 22.3 16.6 11.9 21.0 9.1 Montreal Intl A 29.5 21.9 28.1 20.9 26.5 20.1 23.1 27.9 22.0 26.5 21.1 25.2 21.3 16.0 26.1 20.4 15.1 24.9 19.6 14.4 23.8 9.8 Montreal Mirabel A 28.8 21.4 27.2 20.3 25.8 19.5 22.5 27.3 21.4 25.8 20.5 24.4 20.9 15.7 25.3 19.9 14.7 24.0 19.1 14.0 22.8 11.2 Nitchequon 22.1 15.7 20.4 14.5 18.8 13.9 17.0 20.1 16.0 18.5 15.0 17.5 15.9 12.1 18.4 14.9 11.3 17.1 13.9 10.6 16.4 7.8 Quebec A 28.7 20.9 26.9 19.9 25.3 18.8 22.5 26.7 21.2 25.2 20.2 23.8 21.0 15.8 24.9 19.9 14.7 23.6 18.7 13.6 22.5 10.6 Riviere Du Loup 25.9 19.9 24.5 19.3 23.3 18.4 21.3 24.5 20.3 23.2 19.2 22.3 20.2 15.2 23.3 19.2 14.2 22.4 18.2 13.3 21.3 8.6 Roberval A 28.5 20.1 26.3 19.1 24.6 18.3 21.6 26.3 20.5 24.8 19.3 23.5 20.0 15.0 24.3 18.9 14.0 23.1 17.8 13.1 22.0 9.9 Schefferville A 23.1 14.6 20.8 13.7 18.8 12.8 16.2 20.5 14.9 19.0 13.8 17.5 14.4 10.9 17.7 13.2 10.1 16.8 11.7 9.1 15.7 8.9 Sept-Iles A 22.3 15.7 20.6 15.2 19.2 14.6 17.5 20.3 16.5 19.2 15.7 18.1 16.3 11.7 18.6 15.4 11.0 17.7 14.5 10.4 16.9 7.7 Sherbrooke A 28.6 20.9 26.8 19.9 25.4 18.9 22.1 27.0 21.0 25.4 20.1 24.0 20.5 15.6 25.0 19.5 14.7 23.6 18.6 13.8 22.5 12.4 St. Hubert A 30.0 21.8 28.5 20.7 26.8 20.0 23.1 28.1 22.1 26.7 21.2 25.4 21.5 16.2 25.8 20.5 15.2 24.8 19.6 14.4 23.9 10.6 Ste. Agathe Des Monts 27.2 19.9 25.7 19.1 24.2 18.2 21.5 25.1 20.4 23.8 19.5 22.8 20.3 15.7 23.7 19.3 14.7 22.4 18.4 13.9 21.3 10.8 Val d’Or A 28.4 19.2 26.4 18.2 24.7 17.3 20.8 26.0 19.7 24.4 18.6 22.7 19.2 14.6 23.0 18.1 13.6 22.3 16.9 12.6 21.1 11.5 SASKATCHEWAN Broadview 30.4 18.5 28.4 17.8 26.3 17.0 20.3 27.3 19.1 26.0 18.0 24.6 18.0 13.9 23.5 16.6 12.7 21.9 15.5 11.8 20.9 13.2 Estevan A 32.3 18.7 30.2 18.3 28.3 17.8 21.1 28.4 19.8 27.1 18.7 26.0 18.9 14.7 24.5 17.2 13.2 22.6 16.1 12.3 21.8 13.2 Moose Jaw A 32.2 17.9 30.3 17.5 28.3 16.8 19.9 28.4 18.7 27.1 17.8 26.0 17.1 13.1 22.7 15.9 12.1 21.4 14.8 11.3 20.5 13.2 North Battleford A 29.8 17.7 27.9 17.0 25.8 16.2 19.2 26.5 18.2 25.6 17.2 24.3 16.6 12.6 21.9 15.5 11.8 20.7 14.5 11.0 19.7 11.7 Prince Albert A 29.1 18.2 27.0 17.5 25.3 16.5 19.7 26.4 18.5 25.2 17.5 23.8 17.1 12.9 23.1 16.0 12.0 21.6 14.9 11.1 20.4 12.1 Regina A 31.5 17.9 29.5 17.5 27.6 16.6 19.9 27.6 18.8 26.8 17.8 25.5 17.1 13.1 23.6 15.9 12.1 21.7 14.9 11.3 20.7 13.1 Saskatoon 30.6 17.7 28.6 17.1 26.5 16.5 19.4 27.2 18.3 26.1 17.3 24.9 16.7 12.6 22.5 15.5 11.7 21.2 14.4 10.9 20.2 12.6 Swift Current A 30.9 17.0 29.0 16.6 26.8 15.9 18.9 27.0 17.7 25.9 16.7 25.1 16.1 12.6 21.6 14.9 11.7 19.9 13.9 10.9 19.4 12.8 Uranium City A 26.0 16.4 24.2 15.3 22.2 14.5 17.7 24.2 16.6 22.3 15.5 20.8 15.2 11.2 19.7 14.2 10.5 18.9 13.3 9.9 17.7 9.5 Wynyard 29.6 17.9 27.4 17.0 25.5 16.6 19.8 26.1 18.5 25.1 17.4 23.8 17.5 13.4 23.3 16.1 12.2 21.0 15.1 11.5 19.8 11.5 Yorkton A 30.0 18.5 28.0 17.5 25.9 16.8 20.1 26.9 18.9 25.4 17.8 24.1 18.0 13.7 23.3 16.4 12.4 21.8 15.3 11.5 20.7 12.2 YUKON TERRITORY Burwash A 23.0 13.6 20.7 12.6 18.8 11.6 14.1 21.5 13.1 19.8 12.1 18.0 10.9 8.9 15.5 9.9 8.3 14.6 9.1 7.9 14.0 11.9 Whitehorse A 25.0 13.8 22.7 12.9 20.6 11.9 14.4 23.0 13.5 21.2 12.6 19.5 11.0 8.9 15.9 10.0 8.3 15.0 9.1 7.8 14.5 11.6 MDB = mean coincident dry-bulb temp., °C MWS = mean coincident wind speed, m/s HR = humidity ratio, grams of moisture per kilogram of dry air MWB = mean coincident wet-bulb temp., °C StdD = standard deviation, °C A = airport DP = dew-point temperature, °C 27.26 2001 ASHRAE Fundamentals Handbook (SI) Table 3A Heating and Wind Design Conditions—World Locations Heating Dry Bulb Extreme Wind Speed, m/s Coldest Month MWS/PWD to DB Extr. Annual Daily Elev., m StdP, kPa 0.4% 1% 99.6% 0.4% Mean DB StdD DB Station WMO# Lat.
Long.
Dates 99.6% 99% 1% 2.5% 5% WS MDB WS MDB MWS PWD MWS PWD Max. Min. Max. Min.
1a 1b 1c 1d 1e 1f 1g 2a 2b 3a 3b 3c 4a 4b 4c 4d 5a 5b 5c 5d 6a 6b 6c 6d ALGERIA Algiers 603900 36.72N 3.25E 25 101.03 8293 2.0 3.1 11.0 9.5 8.3 11.8 12.9 9.8 13.0 0.7 200 5.2 60 40.8 −0.1 2.5 1.4 Annaba 603600 36.83N 7.82E 4 101.28 8293 4.1 5.1 11.2 9.9 8.9 12.1 12.2 10.0 11.5 2.2 220 4.8 240 41.0 1.3 3.2 1.2 Biskra 605250 34.80N 5.73E 87 100.28 8293 5.0 6.2 15.1 13.2 11.6 14.9 13.4 12.9 14.4 3.0 10 4.9 180 44.8 2.3 1.5 1.3 Constantine 604190 36.28N 6.62E 694 93.26 8293 −0.6 0.4 10.5 9.0 7.6 11.8 7.6 9.8 8.0 0.8 320 5.2 240 39.7 −2.7 1.4 0.8 El Golea 605900 30.57N 2.87E 397 96.65 8293 0.3 1.7 11.7 9.9 8.3 11.6 14.6 9.6 13.8 1.8 320 3.7 180 45.2 −2.2 1.3 1.5 Oran 604900 35.63N 0.60W 90 100.25 8293 1.9 3.2 12.8 10.8 9.5 14.6 13.9 11.8 14.2 0.9 200 6.3 240 40.2 −2.1 4.9 6.9 Tebessa 604750 35.48N 8.13E 813 91.93 8293 −1.9 −0.5 11.6 9.9 8.4 13.2 6.4 10.3 9.2 0.6 280 5.0 220 39.4 −3.8 1.7 1.1 ARGENTINA Buenos Aires 875760 34.82S 58.53W 20 101.08 8293 −0.7 1.0 10.4 9.1 8.1 9.8 11.9 8.5 11.5 2.0 270 4.9 270 36.6 −2.9 1.4 1.1 Comodoro Rivadavia 878600 45.78S 67.50W 46 100.77 8293 −1.7 −0.4 19.3 16.5 14.8 19.0 10.4 16.1 10.2 4.7 270 9.8 290 36.0 −3.6 4.0 1.1 Cordoba 873440 31.32S 64.22W 474 95.76 8293 −0.4 1.3 12.2 10.5 9.5 12.6 19.0 10.5 15.8 1.4 270 7.2 20 37.8 −3.9 1.7 1.8 Junin Airport 875480 34.55S 60.95W 81 100.36 8293 −1.3 0.2 11.5 10.1 8.8 11.7 12.8 10.1 12.9 0.6 270 5.2 360 37.4 −3.6 3.3 1.1 Formosa 871620 26.20S 58.23W 60 100.61 8293 4.7 6.6 13.2 11.8 10.2 12.5 19.6 11.7 17.7 2.6 270 7.5 360 39.8 1.2 1.0 1.8 Marcos Juarez 874670 32.70S 62.15W 114 99.96 8293 −1.5 0.2 12.6 10.9 9.9 11.9 13.9 10.5 12.9 1.2 50 4.8 360 38.1 −4.7 1.9 2.1 Mendoza 874180 32.83S 68.78W 704 93.15 8293 −0.9 0.6 9.8 8.0 6.9 8.3 13.3 6.6 11.7 1.3 230 5.1 50 39.0 −3.5 1.5 1.0 Paso De Los Libres 872890 29.68S 57.15W 70 100.49 8293 2.2 4.0 13.7 11.8 10.6 13.7 11.7 12.1 13.9 0.6 180 5.9 360 38.3 0.3 1.1 1.3 Posadas 871780 27.37S 55.97W 125 99.83 8293 4.1 5.9 9.8 8.5 7.5 10.4 21.8 9.2 18.9 2.5 180 5.1 360 38.6 1.3 1.2 2.6 Reconquista 872700 29.18S 59.67W 53 100.69 8293 2.8 4.4 12.2 9.5 9.1 12.0 15.7 9.4 14.3 0.7 200 5.3 50 39.4 −0.1 2.7 1.4 Resistencia 871550 27.45S 59.05W 52 100.70 8293 1.8 3.9 8.8 7.5 6.6 8.3 20.3 7.3 17.8 0.8 50 4.7 20 39.6 −0.7 1.3 1.9 Rio Gallegos 879250 51.62S 69.28W 19 101.10 8293 −8.6 −5.8 23.1 20.7 17.7 17.8 2.8 14.9 3.9 2.7 270 9.1 320 32.1 −11.2 6.9 4.4 Rosario 874800 32.92S 60.78W 25 101.03 8293 −1.0 0.5 13.5 11.7 10.4 12.7 11.9 11.3 11.7 1.0 180 5.7 360 36.5 −3.3 1.6 1.5 Salta Airport 870470 24.85S 65.48W 1216 87.54 8293 −0.9 0.6 8.3 7.1 6.1 9.9 25.5 7.4 17.4 0.9 270 4.6 70 35.9 −3.7 1.7 1.4 San Juan 873110 31.57S 68.42W 598 94.34 8293 −2.0 −0.5 14.3 12.4 10.5 12.7 11.3 10.5 11.1 0.3 360 5.0 180 41.5 −5.1 1.4 1.6 San Miguel De Tucuman 871210 26.85S 65.10W 450 96.03 8293 2.9 4.3 9.8 8.1 6.4 7.9 14.2 6.1 13.4 2.1 360 5.2 90 38.9 0.3 1.9 1.5 ARMENIA Yerevan 377890 40.13N 44.47E 890 91.08 8293 −14.1 −11.7 9.7 7.3 6.2 6.1 4.8 4.4 0.1 0.4 180 2.7 210 38.4 −15.9 3.5 3.8 ASCENSION ISLAND Georgetown 619020 7.97S 14.40W 79 100.38 8293 20.8 21.3 11.5 10.6 10.2 11.3 24.6 10.5 24.4 7.3 90 8.6 120 30.6 18.6 1.0 1.6 AUSTRALIA Adelaide 946720 34.93S 138.52E 4 101.28 8293 4.0 5.2 12.2 10.7 9.6 11.4 11.7 10.1 12.4 1.0 50 5.7 310 39.8 1.8 1.7 1.2 Alice Springs 943260 23.80S 133.90E 541 94.99 8293 1.0 2.3 8.8 7.8 7.0 7.8 18.0 6.8 16.9 0.9 270 3.7 100 42.2 −1.9 1.0 2.0 Brisbane 945780 27.38S 153.10E 5 101.26 8293 6.6 7.8 9.6 8.6 7.7 9.6 15.5 8.3 16.0 1.8 220 5.0 20 35.0 3.8 2.1 1.1 Cairns 942870 16.88S 145.75E 7 101.24 8293 13.2 14.9 8.4 7.5 6.7 7.9 23.0 7.2 22.5 3.4 170 3.6 120 36.2 9.8 1.4 4.9 Canberra 949260 35.30S 149.18E 577 94.58 8293 −3.1 −1.8 10.5 9.3 8.2 10.9 7.9 9.8 8.7 0.0 310 5.3 310 36.1 −6.4 2.2 4.5 Darwin 941200 12.40S 130.87E 30 100.97 8293 17.9 19.0 8.4 7.6 6.9 8.2 26.5 7.4 26.9 3.1 140 5.2 290 36.8 15.4 1.6 1.4 Kalgoorlie/Boulder 946370 30.77S 121.45E 360 97.07 8293 2.0 3.3 9.8 8.7 7.8 10.5 14.8 9.2 14.5 0.5 220 3.9 320 42.0 −0.8 1.9 0.7 Learmonth 943020 22.23S 114.08E 6 101.25 8293 9.4 10.8 11.2 10.2 9.4 10.0 19.9 9.0 19.6 2.1 210 6.1 210 44.2 7.0 1.5 1.6 Perth 946100 31.93S 115.95E 29 100.98 8293 4.8 6.1 10.6 9.5 8.4 10.1 14.4 8.7 14.4 0.3 50 4.3 270 41.5 2.2 1.9 1.2 Port Hedland 943120 20.37S 118.62E 6 101.25 8293 10.7 12.0 10.2 9.1 8.3 10.5 20.2 9.6 21.2 2.3 160 5.4 120 44.0 7.5 1.4 1.2 Sydney Intl Airport 947670 33.95S 151.18E 3 101.29 8293 5.8 6.8 11.3 9.9 8.8 11.1 14.2 9.1 13.4 1.1 320 5.3 300 39.3 3.1 2.9 1.9 Townsville 942940 19.25S 146.75E 6 101.25 8293 9.1 11.1 9.2 8.3 7.5 9.0 22.4 8.0 22.1 0.2 190 4.1 50 38.1 5.9 2.1 1.5 AUSTRIA Aigen/Ennstal (Mil) 111570 47.53N 14.13E 649 93.77 8293 −16.9 −14.1 8.3 7.1 6.1 9.8 3.2 8.6 0.9 0.3 60 3.1 60 31.7 −21.2 1.5 3.5 Graz 112400 47.00N 15.43E 347 97.23 8293 −14.8 −11.2 7.8 6.3 4.9 7.1 1.9 4.9 0.6 0.5 180 3.2 140 32.0 −17.1 1.9 4.5 Innsbruck 111200 47.27N 11.35E 593 94.40 8293 −12.2 −10.2 8.3 6.8 5.5 8.2 5.4 6.5 3.9 0.5 260 3.5 70 32.8 −14.9 1.5 3.8 Klagenfurt 112310 46.65N 14.33E 452 96.01 8293 −15.8 −13.0 6.0 4.8 3.9 4.3 −0.6 3.2 −3.0 1.3 310 2.6 100 32.2 −18.3 2.1 3.9 Linz 110100 48.23N 14.20E 313 97.62 8293 −14.9 −11.2 10.5 9.1 7.7 12.5 3.7 11.1 3.1 2.7 90 3.3 110 33.0 −17.3 2.1 5.5 Salzburg 111500 47.80N 13.00E 450 96.03 8293 −14.1 −11.1 7.8 6.7 5.9 8.9 5.5 7.6 3.6 1.4 130 3.0 330 33.2 −16.4 2.3 4.6 Vienna, Hohe Warte 110350 48.25N 16.37E 200 98.95 8293 −11.1 −8.6 10.1 8.5 7.5 11.9 6.1 10.4 5.7 2.8 240 4.0 140 33.0 −12.8 1.8 3.7 Vienna, Schwechat 110360 48.12N 16.57E 190 99.06 8293 −12.6 −9.9 12.0 10.5 9.4 14.0 6.5 11.9 3.4 2.5 320 5.3 150 33.2 −15.0 1.7 4.0 Zeltweg 111650 47.20N 14.75E 682 93.40 8293 −17.7 −14.8 8.2 6.9 5.6 8.6 2.8 7.3 2.4 0.3 250 3.0 190 30.9 −21.1 1.9 4.2 AZORES Lajes 85090 38.77N 27.10W 55 100.67 8293 8.0 9.1 12.6 10.4 9.2 13.4 13.6 11.7 13.9 1.3 300 3.7 250 28.5 4.9 1.0 2.1 BAHAMAS Nassau 780730 25.05N 77.47W 7 101.24 8293 14.1 15.8 9.3 8.4 7.7 9.5 22.0 8.7 21.9 1.5 300 4.5 130 33.9 10.9 0.7 1.7 BAHRAIN Al-Manamah 411500 26.27N 50.65E 2 101.30 8293 11.0 12.2 11.3 10.3 9.4 11.5 13.5 10.7 14.2 5.8 290 5.0 340 42.9 8.1 1.4 3.1 BELARUS Babruysk (Bobruysk) 269610 53.12N 29.25E 165 99.36 8293 −23.0 −19.1 8.9 7.9 7.1 9.0 −2.8 8.1 −1.0 1.6 210 3.6 200 30.4 −26.4 2.2 11.7 Homyel (Gomel’) 330410 52.45N 31.00E 127 99.81 8293 −21.3 −17.9 7.9 6.9 6.1 7.5 −2.4 6.9 −2.5 1.8 330 3.3 150 30.8 −22.5 2.5 4.4 Hrodna (Grodno) 268250 53.68N 23.83E 135 99.71 8293 −20.3 −17.1 11.7 9.8 8.6 11.0 0.8 9.8 0.0 2.2 270 3.9 180 30.7 −20.5 2.1 5.1 Mahilyow (Mogilev) 268630 53.90N 30.32E 193 99.03 8293 −22.9 −19.5 10.3 9.2 8.1 10.4 −0.3 9.3 −2.0 3.0 30 4.1 200 29.4 −24.0 1.9 4.2 Minsk 268500 53.87N 27.53E 234 98.55 8293 −20.7 −17.6 7.4 6.4 5.9 7.8 −4.8 6.7 −4.4 2.1 300 4.0 70 29.7 −22.2 2.1 4.0 Vitsyebsk (Vitebsk) 266660 55.17N 30.13E 176 99.23 8293 −22.3 −18.7 8.7 7.5 6.5 9.3 −3.2 7.7 −2.3 1.5 30 3.4 210 29.1 −23.8 1.9 3.5 BELGIUM/LUXEMBOURG Antwerp 64500 51.20N 4.47E 14 101.16 8293 −8.7 −6.0 10.6 9.2 8.2 12.3 8.2 10.5 6.6 3.0 50 3.2 90 31.6 −9.4 2.3 4.0 WMO# = World Meteorological Organization number Elev. = elevation, m PWD = prevailing wind direction, ° True DB = dry-bulb temperature, °C Lat. = latitude ° Long. = longitude ° StdP = standard pressure at station elevation, kPa WS = wind speed, m/s Climatic Design Information 27.27 Table 3B Cooling and Dehumidification Design Conditions—World Locations Station Cooling DB/MWB Evaporation WB/MDB Dehumidification DP/MDB and HR Range of DB 0.4% 1% 2% 0.4% 1% 2% 0.4% 1% 2% DB MWB DB MWB DB MWB WB MDB WB MDB WB MDB DP HR MDB DP HR MDB DP HR MDB 1 2a 2b 2c 2d 2e 2f 3a 3b 3c 3d 3e 3f 4a 4b 4c 4d 4e 4f 4g 4h 4i 5 ALGERIA Algiers 35.2 21.7 33.2 21.8 31.7 22.0 25.0 30.3 24.4 29.4 23.8 28.6 23.5 18.4 27.5 23.0 17.8 27.1 22.2 16.9 26.6 11.6 Annaba 34.5 21.8 32.1 22.1 30.5 22.7 25.5 29.1 24.8 28.5 24.2 28.1 24.3 19.2 27.8 23.7 18.5 27.2 23.0 17.8 26.8 9.5 Biskra 42.3 20.4 40.9 20.4 39.6 20.3 22.5 36.3 21.9 36.2 21.5 35.7 18.2 13.2 28.4 17.1 12.3 27.8 16.3 11.7 28.4 11.1 Constantine 37.5 18.9 35.8 19.3 34.0 19.1 21.8 31.3 21.0 30.6 20.3 29.8 19.1 15.1 24.7 18.2 14.3 24.4 17.2 13.4 23.8 15.0 El Golea 42.7 20.4 41.4 20.0 40.1 19.8 22.2 38.6 21.5 38.1 20.8 37.2 16.6 12.4 28.0 15.5 11.5 29.0 14.5 10.8 29.2 14.3 Oran 33.2 20.3 31.4 20.8 30.0 20.8 24.1 29.1 23.5 28.1 23.0 27.3 22.6 17.5 26.4 22.1 17.0 26.3 21.6 16.4 26.0 10.2 Tebessa 37.2 18.2 35.9 18.0 34.4 17.9 20.5 31.6 19.8 30.7 19.2 30.0 17.5 13.8 23.0 16.8 13.2 22.3 16.0 12.6 22.3 15.1 ARGENTINA Buenos Aires 33.9 22.8 32.1 22.3 30.7 21.6 24.7 30.1 23.8 28.9 23.1 28.2 23.2 18.0 26.9 22.4 17.1 26.2 21.7 16.4 25.3 12.0 Comodoro Rivadavia 30.6 16.1 28.7 15.4 26.9 14.7 17.3 27.7 16.4 25.7 15.6 24.7 14.0 10.0 18.8 12.9 9.3 17.8 11.7 8.6 18.5 10.4 Cordoba 34.5 22.6 33.0 22.0 31.7 21.7 25.2 31.1 24.2 30.0 23.3 29.0 23.5 19.4 28.8 22.5 18.2 27.5 21.6 17.2 26.5 11.7 Junin Airport 33.5 22.2 31.9 21.9 30.4 21.5 24.3 29.6 23.4 28.7 22.7 28.1 22.9 17.8 26.5 22.0 16.8 25.8 21.1 15.9 24.9 12.0 Formosa 36.6 24.7 35.4 24.6 34.2 24.4 26.8 32.6 26.4 32.4 25.9 31.7 25.4 20.7 29.7 24.9 20.1 29.0 24.4 19.5 28.4 10.3 Marcos Juarez 35.0 23.2 33.4 23.1 32.0 23.0 25.8 31.3 24.9 30.4 24.1 29.4 24.3 19.5 29.2 23.4 18.4 27.7 22.6 17.6 26.7 12.3 Mendoza 35.4 20.0 34.0 19.4 32.8 19.5 22.7 31.5 21.8 30.3 21.1 29.5 20.0 16.0 27.0 19.1 15.1 26.5 18.2 14.3 26.0 12.3 Paso De Los Libres 35.8 23.6 34.5 23.8 33.1 23.0 26.1 31.7 25.3 30.9 24.7 30.2 24.5 19.6 28.8 23.8 18.8 28.3 23.1 18.0 27.5 11.2 Posadas 35.9 24.5 34.8 24.3 33.8 24.1 26.6 33.0 26.0 32.4 25.5 31.6 24.9 20.3 30.6 24.2 19.4 29.7 23.8 18.9 29.1 10.6 Reconquista 35.5 25.5 34.2 25.2 32.9 24.7 27.3 32.8 26.6 31.9 25.9 30.8 25.9 21.4 30.5 25.2 20.5 29.8 24.6 19.7 28.8 9.9 Resistencia 36.6 24.2 35.2 24.2 34.0 24.2 26.8 32.8 26.2 32.1 25.7 31.1 25.2 20.5 29.9 24.8 20.0 29.3 24.2 19.2 28.5 11.2 Rio Gallegos 24.5 14.1 22.4 13.0 20.9 12.1 15.1 22.8 14.0 21.0 13.0 19.4 11.6 8.5 17.0 10.7 8.0 16.0 9.7 7.5 15.4 10.8 Rosario 34.0 23.1 32.4 22.7 31.1 22.4 25.4 30.4 24.6 29.4 23.8 28.6 24.0 18.9 28.2 23.1 17.9 27.4 22.3 17.0 26.3 11.4 Salta Airport 31.9 18.3 30.4 18.8 29.1 18.7 22.1 27.8 21.5 26.9 21.0 26.2 20.5 17.6 24.5 19.9 17.0 24.2 19.4 16.4 23.5 10.6 San Juan 37.6 20.8 36.1 20.1 34.7 19.9 23.0 33.8 22.2 32.6 21.5 31.7 19.9 15.7 27.9 18.8 14.6 27.5 18.0 13.9 27.0 13.2 San Miguel De Tucuman 35.7 23.6 34.1 23.5 32.8 23.1 26.1 32.3 25.4 31.3 24.7 30.4 24.5 20.6 29.7 23.8 19.7 28.9 23.1 18.9 28.2 9.7 ARMENIA Yerevan 35.6 20.5 34.2 20.4 32.8 19.6 22.2 33.0 21.1 32.5 20.3 31.3 18.4 14.8 29.5 17.1 13.6 27.6 16.1 12.8 27.0 13.6 ASCENSION ISLAND Georgetown 30.0 24.1 29.6 23.9 29.2 23.7 25.0 28.7 24.6 28.4 24.3 28.0 23.9 18.9 27.3 23.5 18.5 26.9 23.1 18.0 26.9 4.3 AUSTRALIA Adelaide 35.2 18.0 33.1 17.8 31.1 17.3 21.0 28.5 20.0 27.4 19.1 26.8 19.0 13.8 23.4 17.7 12.7 22.7 16.3 11.6 22.3 10.8 Alice Springs 40.0 18.1 38.9 17.7 37.5 17.8 23.0 28.5 22.2 28.2 21.5 28.6 21.8 17.6 25.1 20.7 16.4 24.9 19.2 14.9 25.1 13.7 Brisbane 31.2 22.5 30.0 22.4 29.1 22.1 25.1 28.7 24.4 27.9 23.7 27.3 24.1 19.0 27.3 23.2 18.0 26.5 22.6 17.3 26.1 7.6 Cairns 33.0 25.3 32.1 25.1 31.2 24.9 27.1 30.8 26.6 30.5 26.1 29.8 26.1 21.5 29.3 25.6 20.9 28.9 25.1 20.2 28.4 7.3 Canberra 32.5 17.1 30.3 17.1 28.3 16.6 19.7 26.2 18.8 25.6 18.1 24.9 17.8 13.7 21.7 16.7 12.8 21.2 15.8 12.0 20.0 13.3 Darwin 34.0 23.8 33.2 24.2 33.0 24.4 27.7 31.4 27.3 31.0 27.0 30.8 27.0 22.8 30.2 26.2 21.7 29.4 26.1 21.6 29.2 7.2 Kalgoorlie/Boulder 39.0 18.1 37.1 17.7 35.1 17.3 20.9 30.7 20.1 29.7 19.3 29.8 18.6 14.0 23.4 17.2 12.8 21.9 16.2 12.0 21.7 13.6 Learmonth 40.4 21.3 38.8 21.1 37.3 21.2 26.1 31.1 25.6 30.8 25.0 30.4 24.9 20.0 28.7 24.2 19.1 28.3 23.4 18.2 28.6 13.1 Perth 37.2 19.2 35.1 19.0 32.9 18.6 22.0 30.5 21.2 29.7 20.5 28.2 19.6 14.4 24.1 18.7 13.6 23.4 18.0 13.0 23.1 12.5 Port Hedland 40.3 21.4 38.9 21.6 37.6 21.8 27.9 33.1 27.4 32.6 27.0 32.0 26.7 22.3 30.5 26.2 21.6 30.1 25.7 21.0 29.8 10.7 Sydney Intl Airport 32.2 20.0 29.5 19.7 27.9 20.1 23.0 28.0 22.3 26.2 21.7 25.3 21.7 16.4 24.8 21.1 15.8 24.3 20.6 15.3 23.9 6.7 Townsville 33.6 24.2 32.8 24.7 31.9 24.7 27.1 31.0 26.6 30.4 26.2 29.7 26.1 21.5 29.1 25.7 21.0 28.9 25.1 20.2 28.5 6.5 AUSTRIA Aigen/Ennstal (Mil) 27.8 18.7 26.0 17.8 24.3 17.1 19.7 26.3 18.7 24.7 17.7 23.2 17.2 13.3 23.4 16.3 12.5 22.1 15.4 11.8 20.7 11.1 Graz 29.2 20.3 27.9 19.8 26.2 18.6 21.4 27.0 20.5 25.6 19.7 24.9 19.7 15.0 23.8 18.8 14.2 22.7 17.9 13.4 21.8 11.4 Innsbruck 29.2 18.1 27.8 17.5 26.1 16.6 19.1 27.1 18.3 25.5 17.6 23.6 16.9 13.0 20.3 16.1 12.3 19.5 15.2 11.6 19.4 11.4 Klagenfurt 29.4 18.7 27.9 18.4 26.1 17.4 20.1 27.4 19.1 26.3 18.3 24.6 17.5 13.2 22.0 16.8 12.6 21.6 16.0 12.0 20.2 12.8 Linz 29.7 19.1 27.9 18.2 26.1 17.3 20.3 27.4 19.3 25.1 18.5 23.1 18.2 13.6 21.1 17.4 12.9 20.8 16.9 12.5 20.2 10.9 Salzburg 29.8 19.3 27.9 18.5 26.1 17.5 19.9 28.0 19.1 26.9 18.2 24.8 17.2 13.0 22.4 16.2 12.2 21.4 15.8 11.8 21.3 10.5 Vienna, Hohe Warte 30.1 20.1 28.5 19.3 26.8 18.7 21.1 28.4 20.3 26.8 19.6 25.0 18.9 14.0 23.3 18.2 13.4 23.0 17.5 12.8 22.4 9.3 Vienna, Schwechat 30.1 19.4 28.4 18.7 26.9 18.1 20.5 28.6 19.6 26.3 18.9 25.5 18.0 13.2 22.7 17.2 12.6 22.0 16.4 11.9 21.8 10.6 Zeltweg 27.7 19.0 25.9 18.1 24.4 17.3 19.7 26.6 18.8 24.7 17.9 23.4 17.2 13.4 23.3 16.4 12.7 22.0 15.7 12.1 20.8 11.7 AZORES Lajes 26.5 21.9 25.8 21.4 24.9 20.8 22.5 25.4 22.0 24.9 21.5 23.7 21.6 16.4 24.0 21.0 15.8 23.7 20.3 15.1 23.3 6.3 BAHAMAS Nassau 33.0 25.9 32.2 25.7 31.9 25.6 27.2 30.6 26.7 30.4 26.4 30.3 26.2 21.6 28.6 26.0 21.4 28.5 25.4 20.6 28.3 6.9 BAHRAIN Al-Manamah 39.2 25.1 38.2 25.5 37.2 26.2 30.7 34.7 30.2 34.2 29.7 33.8 29.9 27.0 33.7 29.1 25.8 33.4 28.7 25.2 33.4 6.9 BELARUS Babruysk (Bobruysk) 27.8 18.6 26.1 18.4 24.5 17.2 20.1 25.8 19.2 24.4 18.2 23.1 18.3 13.5 23.0 17.2 12.5 21.4 16.3 11.8 20.4 11.5 Homyel (Gomel’) 28.3 18.8 26.7 18.3 25.2 17.6 20.3 25.5 19.4 24.7 18.6 23.5 18.5 13.6 22.5 17.5 12.7 21.8 16.6 12.0 21.2 9.0 Hrodna (Grodno) 27.6 18.7 25.8 18.0 24.1 17.1 20.1 25.9 19.0 23.9 18.0 22.6 18.1 13.2 22.5 17.1 12.4 21.4 16.1 11.6 20.3 10.8 Mahilyow (Mogilev) 26.8 19.0 25.2 18.0 23.7 17.2 19.9 24.8 19.0 24.0 18.0 22.3 18.2 13.4 22.4 17.2 12.6 21.2 16.2 11.8 20.2 9.4 Minsk 27.3 18.3 25.8 18.0 24.1 16.9 19.6 25.4 18.7 24.1 17.8 22.8 17.6 13.0 21.7 16.7 12.2 20.7 15.8 11.5 20.4 9.6 Vitsyebsk (Vitebsk) 26.5 18.5 25.0 17.8 23.5 17.1 19.8 24.8 18.9 23.3 17.9 22.2 18.2 13.4 22.2 17.2 12.6 21.1 16.2 11.8 20.0 8.4 BELGIUM/LUXEMBOURG Antwerp 28.0 20.2 26.3 19.1 24.6 18.4 21.1 26.5 20.1 25.2 19.0 23.4 19.1 13.9 24.0 18.1 13.0 22.6 17.1 12.2 21.4 9.0 MDB = mean coincident dry-bulb temp., °C MWS = mean coincident wind speed, m/s HR = humidity ratio, grams of moisture per kilogram of dry air MWB = mean coincident wet-bulb temp., °C StdD = standard deviation, °C A = airport DP = dew-point temperature, °C 27.28 2001 ASHRAE Fundamentals Handbook (SI) Table 3A Heating and Wind Design Conditions—World Locations Heating Dry Bulb Extreme Wind Speed, m/s Coldest Month MWS/PWD to DB Extr. Annual Daily Elev., m StdP, kPa 0.4% 1% 99.6% 0.4% Mean DB StdD DB Station WMO# Lat.
Long.
Dates 99.6% 99% 1% 2.5% 5% WS MDB WS MDB MWS PWD MWS PWD Max. Min. Max. Min.
1a 1b 1c 1d 1e 1f 1g 2a 2b 3a 3b 3c 4a 4b 4c 4d 5a 5b 5c 5d 6a 6b 6c 6d Brussels 64510 50.90N 4.53E 58 100.63 8293 −9.3 −6.2 12.0 10.4 9.2 13.8 7.7 11.8 7.0 3.1 50 3.6 60 31.5 −9.5 2.1 4.8 Charleroi 64490 50.47N 4.45E 192 99.04 8293 −9.3 −6.2 11.3 10.0 8.8 12.8 6.3 10.7 6.1 4.4 50 3.5 50 31.8 −9.9 2.1 4.4 Florennes 64560 50.23N 4.65E 299 97.78 8293 −10.3 −7.1 10.9 9.4 8.4 12.9 6.4 10.9 4.8 4.1 70 3.6 170 30.7 −10.8 2.0 4.4 Koksijde 64000 51.08N 2.65E 9 101.22 8293 −8.4 −5.8 13.3 11.8 10.4 14.9 7.9 13.1 7.2 4.4 90 4.6 100 30.3 −10.1 1.8 4.3 Luxembourg 65900 49.62N 6.22E 379 96.85 8293 −10.5 −7.9 10.7 9.3 8.2 11.9 5.1 10.2 1.1 4.9 50 3.8 80 31.2 −11.3 2.0 4.0 Oostende 64070 51.20N 2.87E 5 101.26 8293 −7.8 −5.4 15.0 13.2 11.7 18.0 7.8 14.8 7.4 4.9 70 4.6 100 30.0 −9.2 1.8 3.6 Saint Hubert 64760 50.03N 5.40E 557 94.81 8293 −11.8 −8.7 10.0 8.8 7.9 11.5 1.8 10.0 −0.5 5.1 90 3.5 30 28.0 −12.0 1.6 4.5 BENIN Cotonou 653440 6.35N 2.38E 9 101.22 8293 21.7 22.3 8.4 7.7 7.2 9.0 26.5 8.2 26.6 2.1 20 5.5 200 36.5 18.3 2.2 3.7 Parakou 653300 9.35N 2.62E 393 96.69 8293 18.2 19.3 6.4 5.5 4.9 5.5 24.2 5.0 24.4 1.6 40 2.4 40 39.5 12.6 2.8 5.1 BERMUDA Hamilton, Bermuda 780160 32.37N 64.68W 3 101.29 8293 12.9 13.9 13.0 11.4 10.2 13.4 18.3 12.6 18.0 7.2 310 4.8 190 32.1 8.3 0.7 4.2 BOLIVIA Cochabamba 852230 17.45S 66.10W 2531 74.39 8293 1.8 2.9 10.0 8.2 5.8 9.8 18.9 7.6 19.1 0.0 180 2.8 360 31.4 −1.2 1.1 1.3 La Paz 852010 16.52S 68.18W 4014 61.53 8293 −4.0 −3.0 8.5 7.7 6.4 9.6 11.1 8.4 11.0 0.9 330 3.4 60 20.8 −6.0 2.6 1.1 BOSNIA−HERZEGOVINA Banja Luka 132420 44.78N 17.22E 156 99.46 8293 −12.0 −8.9 5.9 4.3 3.4 6.4 11.1 4.5 9.8 1.1 320 2.0 360 36.6 −14.5 2.2 4.2 BRAZIL Arch De Fernando 824000 3.85S 32.42W 56 100.65 8293 22.9 23.3 8.2 7.3 6.6 8.5 26.0 7.8 26.2 3.1 130 4.6 150 36.9 17.0 3.6 8.6 Belem 821930 1.38S 48.48W 16 101.13 8293 22.4 22.8 7.6 6.4 5.5 6.9 28.6 5.7 28.3 0.7 90 2.9 90 35.7 17.3 1.8 7.8 Brasilia 833780 15.87S 47.93W 1061 89.21 8293 9.1 10.8 7.6 6.3 5.4 8.0 21.5 6.5 22.3 0.2 90 3.2 90 34.4 6.7 1.8 3.2 Campinas 837210 23.00S 47.13W 661 93.63 8293 8.2 9.9 10.6 10.0 9.2 10.5 18.3 10.1 17.3 3.7 150 3.3 330 35.3 5.4 1.6 2.5 Campo Grande 836120 20.47S 54.67W 556 94.82 8293 8.0 10.2 11.0 10.0 9.2 12.1 15.6 10.5 16.9 7.3 180 4.8 360 36.7 4.8 1.8 2.1 Caravelas 834970 17.63S 39.25W 4 101.28 8293 16.3 17.3 10.1 8.9 7.9 8.6 23.7 7.8 24.0 0.4 240 5.3 60 35.1 13.7 2.2 1.8 Curitiba 838400 25.52S 49.17W 908 90.88 8293 2.4 4.5 8.2 7.0 5.9 8.9 19.8 7.9 18.4 1.1 270 3.8 300 33.6 −3.6 1.9 6.4 Fortaleza 823980 3.78S 38.53W 25 101.03 8293 22.5 22.9 8.9 8.2 7.6 8.7 29.0 8.2 29.1 2.1 180 6.0 120 36.2 18.5 2.2 5.4 Goiania 834240 16.63S 49.22W 747 92.67 8293 11.5 12.9 8.0 6.6 5.4 7.3 25.2 5.9 25.5 0.3 180 1.8 360 36.8 5.5 0.9 6.0 Maceio 829930 9.52S 35.78W 115 99.95 8293 19.2 19.8 8.0 6.8 6.0 7.7 25.7 6.2 25.7 0.2 180 4.5 100 35.1 14.6 2.2 5.9 Manaus, E.Gomes Airport 821110 3.03S 60.05W 2 101.30 8293 21.1 21.8 5.5 5.0 4.4 5.4 29.2 4.7 29.2 0.3 120 2.2 60 37.8 15.2 0.9 7.8 Manaus, Ponta Pelada 823320 3.15S 59.98W 84 100.32 8293 21.9 22.7 6.6 5.6 5.2 6.9 28.5 5.8 28.5 1.6 30 4.3 90 36.6 17.5 1.1 7.3 Natal 825990 5.92S 35.25W 52 100.70 8293 21.0 21.6 9.3 8.5 7.9 10.0 28.5 9.1 27.9 2.9 150 5.1 100 35.2 16.4 2.6 5.4 Porto Alegre 839710 30.00S 51.18W 3 101.29 8293 4.4 6.1 8.4 7.4 6.4 8.8 11.5 7.5 13.0 0.7 240 3.0 290 37.9 2.8 1.7 2.9 Recife 828990 8.07S 34.85W 19 101.10 8293 21.1 21.8 7.7 6.5 5.9 8.3 26.3 7.5 26.2 2.3 240 5.2 120 36.2 19.0 1.7 2.2 Rio De Janeiro, Galeao 837460 22.82S 43.25W 6 101.25 8293 14.9 15.9 8.5 7.7 6.8 7.6 23.4 6.6 23.0 1.2 320 3.1 50 41.5 10.0 1.4 5.2 Salvador 832480 12.90S 38.33W 6 101.25 8293 20.1 21.0 9.3 8.4 7.6 9.5 24.2 8.8 24.6 0.7 180 5.5 90 34.9 15.0 1.7 7.1 Santarem 822440 2.43S 54.72W 72 100.46 8293 22.2 22.9 8.5 8.0 7.1 8.4 27.2 7.7 27.3 2.6 210 5.4 90 35.8 17.9 1.8 8.9 Sao Paulo 837800 23.62S 46.65W 803 92.04 8293 8.8 9.9 6.9 5.9 5.2 6.4 19.0 5.4 18.0 2.0 160 2.6 330 34.3 5.9 1.5 2.1 Vitoria 836490 20.27S 40.28W 4 101.28 8293 16.3 17.2 9.7 8.4 7.5 8.2 25.0 7.2 24.7 0.6 210 5.1 30 36.9 12.3 1.1 5.7 BULGARIA Botevgrad 156270 42.67N 24.83E 2389 75.73 8293 −21.5 −19.0 34.1 27.8 23.7 34.4 −9.2 28.5 −11.9 13.4 320 3.9 270 19.0 −22.4 1.7 3.4 Burgas 156550 42.48N 27.48E 28 100.99 8293 −8.3 −6.2 13.8 10.7 9.5 13.9 3.0 11.4 0.5 4.2 270 4.9 110 34.5 −10.7 1.7 2.9 Lom 155110 43.82N 23.25E 33 100.93 8293 −10.0 −7.6 12.4 10.3 8.1 13.8 1.5 11.9 4.7 0.8 270 1.9 50 35.5 −12.7 2.6 4.3 Musala 156150 42.18N 23.58E 2927 70.76 8293 −23.4 −20.8 28.2 20.2 16.0 34.3 −7.1 28.3 −8.9 7.5 20 3.3 320 18.6 −24.8 3.8 3.2 Plovdiv 156250 42.13N 24.75E 185 99.12 8293 −9.9 −7.1 13.2 11.8 10.2 14.1 4.8 12.2 5.0 1.1 40 2.6 90 37.0 −14.2 2.3 4.0 Ruse 155350 43.85N 25.95E 45 100.79 8293 −11.3 −8.8 14.1 11.9 9.4 14.2 4.1 12.1 0.5 2.3 50 3.5 270 38.4 −13.9 3.0 2.7 Sofia 156140 42.65N 23.38E 595 94.38 8293 −12.1 −9.9 9.3 7.6 6.4 9.8 0.4 8.2 −0.1 1.1 360 2.4 110 34.3 −16.3 2.7 4.7 Varna 155520 43.20N 27.92E 43 100.81 8293 −8.5 −6.5 14.4 11.5 9.4 17.8 −1.7 13.6 0.3 6.7 360 4.0 90 33.3 −10.8 2.2 2.7 BRUNEI Brunei Intl Airport 963150 4.93N 114.93E 15 101.14 8293 21.4 22.0 7.3 6.3 5.5 8.1 28.0 7.2 27.8 0.0 220 3.7 320 35.4 19.3 1.4 2.6 CANARY ISLANDS Las Palmas 600300 27.93N 15.38W 25 101.03 8293 13.0 13.9 14.2 13.3 12.5 11.9 18.6 10.6 18.5 1.8 320 8.7 30 34.4 10.8 2.4 0.5 Santa Cruz De Tenerife 600250 28.05N 16.57W 72 100.46 8293 13.8 14.1 13.1 11.9 10.9 12.6 20.1 11.3 19.8 3.4 360 8.5 60 38.4 10.3 3.0 3.6 CAPE VERDE Sal Island 85940 16.73N 22.95W 55 100.67 8293 17.1 17.8 12.2 11.3 10.5 12.5 22.2 11.8 22.2 4.6 30 6.5 60 32.7 14.2 1.0 2.4 CHILE Antofagasta 854420 23.43S 70.43W 120 99.89 8293 10.2 11.0 9.9 9.1 8.4 10.1 15.3 9.1 15.1 2.7 90 6.7 190 26.7 6.7 1.5 2.0 Arica 854060 18.33S 70.33W 59 100.62 8293 11.2 12.6 9.0 8.2 7.3 7.6 17.9 7.0 17.8 2.0 90 6.6 210 29.3 7.8 1.6 2.4 Concepcion 856820 36.77S 73.05W 16 101.13 8293 1.8 2.8 13.2 11.5 10.3 16.1 12.2 13.0 11.9 1.8 140 8.5 240 33.2 −1.0 8.6 0.8 Iquique 854180 20.53S 70.18W 52 100.70 8293 12.1 12.9 9.7 8.9 8.2 9.0 17.7 8.1 17.1 2.9 160 6.4 210 28.8 10.0 1.4 1.3 La Serena 854880 29.92S 71.20W 146 99.58 8293 5.8 6.6 8.1 7.3 6.6 8.4 12.7 6.8 13.6 2.8 120 5.8 270 27.0 1.5 6.4 3.5 Puerto Montt 857990 41.42S 73.08W 86 100.30 8293 −2.1 −0.9 12.1 10.2 8.8 13.1 9.4 11.6 8.9 0.7 280 4.4 150 26.9 −4.3 2.1 1.3 Punta Arenas 859340 53.00S 70.85W 37 100.88 8293 −5.0 −3.7 19.9 17.2 14.2 18.2 5.9 13.5 3.1 2.9 290 8.2 270 24.4 −8.2 3.6 2.4 Santiago 855740 33.38S 70.78W 476 95.74 8293 −1.4 −0.1 8.5 7.3 6.3 7.2 9.7 5.3 10.0 0.9 20 5.4 210 38.6 −3.7 5.1 1.3 Temuco 857430 38.75S 72.63W 120 99.89 8293 −1.9 −0.7 10.0 8.3 6.9 13.1 10.4 10.2 11.0 1.0 70 4.5 240 33.0 −4.1 4.7 1.0 CHINA Anda 508540 46.38N 125.32E 150 99.54 8293 −28.6 −26.5 10.7 9.2 7.7 8.0−13.8 6.9 −14.6 1.3 250 5.3 250 33.4 −30.6 1.7 2.4 Andirlangar 518480 37.93N 83.65E 1264 87.03 8293 −18.1 −16.4 6.7 5.5 4.9 4.4 0.0 3.7 −1.7 0.6 200 2.5 40 39.5 −20.5 1.3 1.8 WMO# = World Meteorological Organization number Elev. = elevation, m PWD = prevailing wind direction, ° True DB = dry-bulb temperature, °C Lat. = latitude ° Long. = longitude ° StdP = standard pressure at station elevation, kPa WS = wind speed, m/s Climatic Design Information 27.29 Table 3B Cooling and Dehumidification Design Conditions—World Locations Station Cooling DB/MWB Evaporation WB/MDB Dehumidification DP/MDB and HR Range of DB 0.4% 1% 2% 0.4% 1% 2% 0.4% 1% 2% DB MWB DB MWB DB MWB WB MDB WB MDB WB MDB DP HR MDB DP HR MDB DP HR MDB 1 2a 2b 2c 2d 2e 2f 3a 3b 3c 3d 3e 3f 4a 4b 4c 4d 4e 4f 4g 4h 4i 5 Brussels 27.9 19.6 26.2 18.9 24.5 18.2 20.7 26.5 19.8 24.9 18.7 23.3 18.7 13.6 23.2 17.8 12.9 22.2 16.9 12.1 21.4 9.4 Charleroi 28.0 19.2 26.2 18.6 24.5 17.7 20.4 26.0 19.4 24.7 18.4 23.0 18.3 13.5 23.1 17.4 12.7 21.8 16.6 12.1 20.6 9.3 Florennes 26.9 18.9 25.2 18.3 23.5 17.4 20.2 25.4 19.1 23.9 18.1 22.5 18.4 13.8 22.8 17.3 12.8 21.3 16.4 12.1 20.1 9.0 Koksijde 25.9 19.2 23.9 18.4 22.2 17.6 20.0 24.3 19.0 23.0 18.1 21.4 18.4 13.3 22.5 17.5 12.5 20.9 16.7 11.9 19.9 8.4 Luxembourg 28.0 18.1 26.1 17.4 24.5 16.6 19.5 25.4 18.5 24.2 17.6 22.5 17.5 13.1 21.8 16.6 12.4 20.2 15.7 11.7 19.7 9.5 Oostende 25.2 18.9 23.0 18.2 21.3 17.4 19.7 23.6 18.8 22.2 18.0 20.9 18.2 13.1 21.4 17.5 12.5 20.7 16.7 11.9 19.6 7.8 Saint Hubert 25.2 17.5 23.6 16.9 22.0 15.9 18.7 23.3 17.6 22.0 16.7 20.8 17.0 13.0 20.4 16.0 12.2 19.2 15.2 11.5 18.5 8.1 BENIN Cotonou 32.1 26.3 31.7 26.5 31.4 26.5 27.6 31.1 27.4 30.9 27.2 30.6 26.7 22.3 29.6 26.4 21.9 29.6 26.2 21.6 29.5 4.8 Parakou 36.8 21.6 36.0 21.4 35.2 21.4 25.7 32.8 25.3 32.1 25.0 31.5 23.9 19.7 29.1 23.6 19.3 28.8 23.2 18.9 28.4 11.5 BERMUDA Hamilton, Bermuda 31.2 25.5 30.9 25.4 30.1 25.0 26.8 29.7 26.4 29.3 26.0 29.0 26.1 21.5 28.9 25.4 20.6 28.5 25.0 20.1 28.1 4.6 BOLIVIA Cochabamba 29.1 13.6 28.1 13.5 27.2 13.3 16.2 24.5 15.7 24.4 15.2 23.6 14.1 13.8 18.0 13.2 13.0 16.7 13.0 12.8 16.6 15.0 La Paz 17.3 6.6 16.7 6.4 15.9 6.1 9.2 14.2 8.7 13.8 8.2 13.1 7.2 10.4 10.6 6.8 10.2 10.0 6.2 9.7 9.4 12.6 BOSNIA−HERZEGOVINA Banja Luka 33.1 20.4 31.0 20.3 29.2 19.6 22.3 29.2 21.4 28.4 20.4 27.4 20.1 15.1 26.1 19.0 14.1 24.3 18.2 13.4 23.0 12.7 BRAZIL Arch De Fernando 31.0 25.6 30.3 25.3 30.1 25.2 26.5 29.7 26.2 29.5 26.0 29.3 25.6 21.0 28.7 25.2 20.5 28.5 25.0 20.2 28.4 4.5 Belem 33.1 25.5 32.5 25.3 32.1 25.4 27.0 30.8 26.6 30.7 26.4 30.4 26.1 21.5 28.8 25.9 21.3 28.7 25.2 20.4 28.1 8.2 Brasilia 31.8 18.3 30.9 18.2 30.0 18.5 21.7 26.6 21.3 26.2 21.0 26.1 20.4 17.2 22.6 20.1 16.9 22.3 19.9 16.6 22.1 13.0 Campinas 33.2 22.8 32.2 21.9 31.2 22.0 24.6 29.6 24.0 28.9 23.6 28.3 23.2 19.5 26.0 22.9 19.1 25.6 22.2 18.3 25.0 9.9 Campo Grande 35.0 21.9 34.1 21.9 33.2 21.9 25.1 30.5 24.6 30.2 24.2 29.8 23.9 20.1 26.8 23.2 19.2 26.4 22.8 18.8 26.3 10.2 Caravelas 31.8 25.1 31.2 25.0 30.8 24.9 26.1 30.3 25.8 29.6 25.6 29.2 25.2 20.3 27.4 24.9 20.0 27.2 24.6 19.6 26.9 7.6 Curitiba 30.8 20.3 29.6 20.1 28.5 20.1 22.7 27.2 22.1 26.5 21.6 25.8 21.7 18.3 24.0 21.0 17.5 23.3 20.5 17.0 23.3 9.7 Fortaleza 32.2 25.4 32.1 25.3 31.9 25.3 26.7 30.4 26.5 30.5 26.2 29.9 26.1 21.6 27.3 25.9 21.3 27.4 25.2 20.4 27.4 6.2 Goiania 34.1 19.6 33.2 19.9 32.6 20.0 23.7 29.6 23.2 29.1 22.8 28.7 22.1 18.4 25.5 21.7 17.9 25.0 21.3 17.5 24.5 12.8 Maceio 32.2 24.0 31.8 23.9 31.2 23.8 25.7 29.3 25.3 29.1 25.0 29.0 24.9 20.2 27.2 24.2 19.4 26.6 24.0 19.1 26.4 7.9 Manaus, E.Gomes Airport 35.9 25.4 35.1 25.6 34.6 25.5 28.2 32.1 27.6 31.9 27.1 31.0 27.2 23.0 29.8 26.9 22.6 29.5 26.2 21.6 28.6 11.3 Manaus, Ponta Pelada 34.2 25.2 33.9 25.4 33.1 25.4 27.2 31.4 26.8 31.3 26.6 31.1 26.2 21.8 29.1 26.0 21.6 28.9 25.4 20.8 28.5 7.9 Natal 32.4 25.4 32.1 25.1 31.7 25.0 26.4 30.4 26.1 30.1 25.9 29.9 25.7 21.1 27.8 25.1 20.3 27.6 24.9 20.1 27.5 6.8 Porto Alegre 35.0 24.5 33.5 24.0 32.0 23.3 26.0 31.8 25.2 30.8 24.7 29.8 24.5 19.5 28.5 23.9 18.8 27.6 23.2 18.0 26.8 9.5 Recife 33.1 25.7 32.7 25.6 32.1 25.4 26.6 31.7 26.3 31.2 26.1 30.9 25.2 20.4 28.6 25.1 20.3 28.6 24.9 20.0 28.4 6.3 Rio De Janeiro, Galeao 38.9 26.1 37.1 25.2 35.9 24.9 27.7 35.2 27.0 34.2 26.4 32.6 26.1 21.5 30.1 25.2 20.3 28.9 25.0 20.1 28.9 10.7 Salvador 32.0 25.8 31.2 25.5 31.0 25.4 26.9 30.5 26.5 30.2 26.1 29.7 26.1 21.5 29.3 25.2 20.3 28.9 25.1 20.2 28.7 6.0 Santarem 34.0 25.2 33.2 25.1 33.0 25.3 26.6 31.5 26.4 31.3 26.2 31.0 25.5 20.9 28.1 25.2 20.5 28.0 25.1 20.4 28.0 7.8 Sao Paulo 31.9 20.3 30.9 20.3 29.9 20.4 22.9 27.4 22.3 27.1 21.9 26.8 21.8 18.2 24.9 21.1 17.4 24.1 20.8 17.1 23.8 8.3 Vitoria 34.1 25.2 33.2 25.0 32.4 24.9 26.5 31.2 26.0 30.5 25.7 30.1 25.8 21.1 28.3 25.0 20.1 27.2 24.6 19.6 27.6 8.1 BULGARIA Botevgrad 15.6 9.6 14.1 8.9 12.7 8.1 11.1 14.0 10.2 12.5 9.5 11.7 10.1 10.3 11.8 9.2 9.7 11.0 8.4 9.2 10.2 4.3 Burgas 30.8 22.1 29.1 21.5 28.0 20.6 23.5 27.7 22.6 27.3 21.8 26.3 22.1 16.8 26.2 21.1 15.8 25.2 20.2 14.9 24.4 11.1 Lom 32.3 23.2 30.6 22.2 29.2 21.5 23.9 30.9 23.0 29.7 22.1 28.3 21.6 16.3 28.5 20.7 15.4 27.3 19.8 14.6 26.1 10.6 Musala 13.1 6.9 11.6 6.2 10.3 5.8 8.2 11.4 7.4 10.5 6.6 9.4 6.8 8.8 9.0 5.9 8.3 8.2 5.2 7.9 7.6 5.1 Plovdiv 33.7 21.3 32.1 21.0 30.6 20.5 23.0 31.4 22.0 30.4 21.1 29.0 20.0 15.0 28.3 19.1 14.2 26.8 18.3 13.5 25.3 11.9 Ruse 34.5 22.7 32.6 21.5 31.0 21.0 23.6 32.2 22.5 31.1 21.6 29.8 20.6 15.3 29.2 19.5 14.3 26.9 18.7 13.6 25.6 11.5 Sofia 31.3 18.7 29.5 18.4 27.9 17.7 20.1 28.2 19.2 27.6 18.5 26.4 17.1 13.1 23.5 16.2 12.4 22.5 15.5 11.8 22.0 12.1 Varna 29.6 22.3 28.2 21.9 27.1 20.9 23.6 27.6 22.7 26.8 21.9 25.9 22.3 17.1 26.5 21.3 16.0 25.6 20.4 15.2 24.8 9.6 BRUNEI Brunei Intl Airport 33.6 26.1 33.1 26.3 32.6 26.3 27.7 31.4 27.5 31.2 27.2 30.9 26.9 22.6 30.0 26.5 22.1 29.7 26.2 21.7 29.4 7.8 CANARY ISLANDS Las Palmas 30.2 19.9 28.8 20.1 27.2 20.4 24.7 26.6 23.7 25.8 23.0 25.5 24.0 18.9 25.9 23.0 17.8 25.2 22.1 16.8 24.6 5.5 Santa Cruz De Tenerife 32.9 20.1 30.5 20.1 28.9 20.4 23.8 27.9 23.2 27.2 22.6 26.7 22.2 17.0 26.4 21.9 16.7 26.2 21.1 15.9 25.4 6.9 CAPE VERDE Sal Island 30.1 23.9 29.6 24.0 29.0 23.7 25.7 28.0 25.2 27.6 24.8 27.3 25.0 20.2 27.0 24.4 19.5 26.7 24.0 19.0 26.5 4.9 CHILE Antofagasta 25.0 20.2 24.1 19.2 23.2 18.6 20.7 24.1 20.1 23.2 19.4 22.7 19.3 14.3 22.9 18.9 13.9 22.6 18.1 13.2 21.6 5.8 Arica 28.1 21.6 27.1 20.6 26.2 20.2 22.9 26.3 22.1 25.8 21.1 25.5 21.5 16.3 25.8 20.5 15.3 25.5 19.5 14.3 24.4 6.6 Concepcion 24.3 17.3 23.2 16.7 22.2 16.3 18.5 22.5 17.7 21.7 17.1 21.0 17.0 12.2 19.7 16.1 11.5 18.9 15.7 11.2 18.6 11.2 Iquique 26.8 20.1 26.1 19.7 25.5 19.4 20.7 25.8 20.2 25.4 19.7 24.8 18.7 13.6 24.0 18.1 13.1 23.7 17.6 12.7 23.3 6.5 La Serena 22.4 17.4 21.8 17.1 21.0 16.6 18.0 21.7 17.5 20.9 17.0 20.5 16.5 12.0 19.7 16.0 11.6 19.4 15.5 11.2 18.7 6.6 Puerto Montt 22.7 16.9 21.1 16.1 20.0 15.4 17.7 21.4 16.8 20.1 16.0 19.3 16.2 11.6 19.1 15.3 11.0 18.4 14.4 10.3 17.6 9.8 Punta Arenas 17.8 12.5 16.3 11.4 15.2 10.8 13.2 16.8 12.3 15.3 11.4 14.4 11.3 8.4 14.1 10.5 7.9 13.6 9.7 7.5 12.9 7.2 Santiago 31.9 18.4 30.9 18.2 29.9 18.0 19.9 29.3 19.2 28.8 18.7 28.0 16.1 12.1 24.3 15.4 11.6 23.8 14.9 11.2 23.2 17.5 Temuco 27.4 17.7 25.5 17.1 23.9 16.2 18.7 25.0 17.7 24.3 16.9 23.0 16.2 11.7 21.9 15.1 10.9 20.9 14.2 10.3 19.7 13.7 CHINA Anda 30.8 20.4 29.3 20.2 28.0 19.3 23.7 27.8 22.7 26.4 21.7 25.3 22.4 17.4 26.2 21.4 16.4 25.2 20.6 15.5 23.9 8.3 Andirlangar 36.8 17.2 35.4 16.6 34.0 16.1 18.9 31.4 18.1 31.0 17.4 30.1 15.4 12.8 21.6 14.2 11.8 21.9 13.0 10.9 22.1 14.8 MDB = mean coincident dry-bulb temp., °C MWS = mean coincident wind speed, m/s HR = humidity ratio, grams of moisture per kilogram of dry air MWB = mean coincident wet-bulb temp., °C StdD = standard deviation, °C A = airport DP = dew-point temperature, °C 27.30 2001 ASHRAE Fundamentals Handbook (SI) Table 3A Heating and Wind Design Conditions—World Locations Heating Dry Bulb Extreme Wind Speed, m/s Coldest Month MWS/PWD to DB Extr. Annual Daily Elev., m StdP, kPa 0.4% 1% 99.6% 0.4% Mean DB StdD DB Station WMO# Lat.
Long.
Dates 99.6% 99% 1% 2.5% 5% WS MDB WS MDB MWS PWD MWS PWD Max. Min. Max. Min.
1a 1b 1c 1d 1e 1f 1g 2a 2b 3a 3b 3c 4a 4b 4c 4d 5a 5b 5c 5d 6a 6b 6c 6d Anyang 538980 36.12N 114.37E 76 100.42 8293 −7.9 −6.3 7.3 6.2 5.2 6.4 2.2 5.3 1.7 0.5 290 3.2 180 38.0 −10.8 1.2 2.0 Baoding 546020 38.85N 115.57E 19 101.10 8293 −10.4 −8.9 6.5 5.5 4.7 6.1 0.1 4.9 −2.1 0.7 220 3.0 200 37.5 −12.8 1.9 2.4 Bayan Mod 524950 40.75N 104.50E 1329 86.35 8293 −20.5−18.5 12.3 10.3 8.8 11.5 −6.5 9.4 −5.4 2.5 270 3.4 180 35.5 −23.1 1.1 2.3 Beijing 545110 39.93N 116.28E 55 100.67 8293 −10.4 −9.2 9.2 7.6 6.3 9.4 −1.1 7.9 −2.4 2.0 340 3.1 200 38.1 −13.4 1.9 1.7 Bengbu 582210 32.95N 117.37E 22 101.06 8293 −5.6 −4.2 7.3 6.2 5.3 6.9 3.1 5.8 2.6 1.3 20 3.7 180 37.2 −8.9 1.3 2.3 Changchun 541610 43.90N 125.22E 238 98.50 8293 −24.7−22.7 13.0 10.4 8.7 11.4 −7.7 9.5 −7.3 2.6 270 4.2 250 32.5 −27.3 1.4 2.7 Changsha (576790) 576870 28.23N 112.87E 68 100.51 8293 −1.5 −0.4 7.7 6.5 5.5 7.6 5.3 6.4 4.5 1.5 320 3.1 200 38.1 −4.1 2.1 3.4 Chengdu 562940 30.67N 104.02E 508 95.37 8293 −0.1 1.2 5.1 4.2 3.5 3.8 6.9 3.3 6.6 0.2 20 1.9 200 33.8 −2.6 1.5 1.4 Dalian 546620 38.90N 121.63E 97 100.17 8293 −12.5−10.7 12.6 10.9 9.7 13.0 −10.0 11.5 −7.4 6.5 360 4.1 180 32.8 −14.7 2.2 2.3 Dandong 544970 40.05N 124.33E 14 101.16 8293 −17.0−14.8 9.2 7.8 6.9 10.3 −8.7 8.5 −7.6 2.4 340 2.5 220 31.9 −19.3 1.2 2.9 Datong 534870 40.10N 113.33E 1069 89.12 8293 −20.9−19.2 10.4 8.7 7.4 8.8 −9.5 7.9 −8.1 2.4 360 3.8 160 33.6 −23.9 0.7 1.9 Deqen 564440 28.50N 98.90E 3488 65.87 8293 −7.7 −6.6 11.6 9.8 8.4 14.4 −1.6 11.8 −0.4 1.1 320 5.3 180 24.1 −10.2 3.0 2.6 Dinghai 584770 30.03N 122.12E 37 100.88 8293 −0.7 0.5 9.2 7.7 6.8 8.9 3.9 7.8 4.9 3.3 340 2.9 140 35.2 −2.7 0.9 1.4 Erenhot 530680 43.65N 112.00E 966 90.25 8293 −28.8−26.9 13.0 11.0 9.4 11.2 −13.3 9.2−12.8 2.3 70 5.3 140 35.6 −32.3 1.1 2.7 Fuzhou 588470 26.08N 119.28E 85 100.31 8293 4.2 5.3 8.2 7.1 6.2 7.7 12.4 6.3 11.0 3.0 320 4.4 140 37.9 1.8 0.9 1.7 Golmud 528180 36.42N 94.90E 2809 71.83 8293 −17.6−15.8 8.8 7.5 6.6 7.5 −6.7 6.4 −7.2 1.4 250 2.8 70 30.1 −21.1 1.2 2.4 Guangzhou 592870 23.13N 113.32E 8 101.23 8293 5.3 6.7 6.8 5.7 5.0 6.7 11.7 5.6 11.7 2.7 360 2.3 270 36.4 2.9 0.8 1.5 Guilin 579570 25.33N 110.30E 166 99.35 8293 1.3 2.2 9.2 7.9 6.9 9.7 5.9 8.4 6.0 4.9 20 2.5 20 36.2 −1.0 0.9 2.5 Guiyang 578160 26.58N 106.72E 1074 89.07 8293 −2.2 −1.0 6.7 5.5 4.8 5.6 9.0 4.8 7.3 2.0 40 3.4 160 33.0 −3.1 1.9 0.9 Hami 522030 42.82N 93.52E 739 92.76 8293 −18.0−16.0 7.3 5.7 4.6 4.2 −6.6 3.5 −8.1 1.2 40 1.1 250 39.1 −20.0 1.7 2.3 Hangzhou 584570 30.23N 120.17E 43 100.81 8293 −2.4 −1.1 7.4 6.4 5.5 7.4 4.7 6.3 4.7 1.9 340 3.6 160 37.5 −5.6 1.0 2.1 Harbin 509530 45.75N 126.77E 143 99.62 8293 −29.2−26.8 10.1 8.5 7.3 8.0 −10.4 7.1−13.3 1.1 200 5.0 180 32.8 −32.7 1.3 2.3 Hefei 583210 31.87N 117.23E 36 100.89 8293 −4.4 −2.8 8.3 7.1 6.2 7.4 4.0 6.8 3.7 1.7 340 3.6 180 36.8 −7.4 1.3 3.0 Hohhot 534630 40.82N 111.68E 1065 89.17 8293 −20.2−18.6 8.8 7.4 6.3 7.6 −11.7 6.6−11.6 0.7 360 3.5 180 33.3 −22.4 1.4 1.7 Jinan 548230 36.68N 116.98E 58 100.63 8293 −8.0 −6.4 8.8 7.6 6.7 8.2 0.3 7.1 2.2 2.2 70 4.3 200 36.9 −10.6 1.1 2.3 Jingdezhen 585270 29.30N 117.20E 60 100.61 8293 −2.2 −0.7 6.4 5.4 4.6 6.1 6.2 5.2 5.4 1.7 40 3.2 250 38.1 −5.8 2.0 6.1 Jinzhou 543370 41.13N 121.12E 70 100.49 8293 −16.4−14.8 11.0 9.5 8.2 10.3 −7.9 8.8 −6.8 2.5 360 3.9 200 34.2 −18.3 1.7 2.5 Jixi 509780 45.28N 130.95E 234 98.55 8293 −25.6−23.3 10.3 8.9 7.5 10.4 −16.3 9.3−15.2 2.7 250 3.4 270 33.0 −27.2 1.6 2.3 Kashi 517090 39.47N 75.98E 1291 86.75 8293 −13.6 −11.1 6.8 5.0 3.9 3.4 −1.9 2.8 −2.6 0.6 320 1.8 180 35.5 −15.0 1.8 3.5 Korla 516560 41.75N 86.13E 933 90.61 8293 −14.4−12.0 9.3 7.5 6.3 6.4 −4.2 5.3 −4.7 0.5 70 2.0 40 37.8 −16.6 1.4 4.2 Kowloon 450070 22.33N 114.18E 24 101.04 8293 9.0 10.8 10.0 8.8 8.0 9.1 16.9 8.3 17.1 3.5 330 4.6 250 35.4 7.1 1.2 1.4 Kunming 567780 25.02N 102.68E 1892 80.57 8293 0.1 1.3 8.4 7.2 6.2 8.6 15.4 7.6 15.1 0.7 140 4.9 250 29.8 −2.7 3.3 2.4 Lanzhou 528890 36.05N 103.88E 1518 84.37 8293 −12.2−10.7 4.6 3.7 3.2 2.9 −2.1 2.3 −3.4 0.2 90 2.2 70 34.4 −14.5 1.1 1.9 Lhasa 555910 29.67N 91.13E 3650 64.50 8293 −10.3 −9.1 7.3 6.0 5.0 7.2 4.0 6.0 4.6 1.9 90 2.6 290 29.5 −16.9 4.3 10.8 Liuzhou 590460 24.35N 109.40E 97 100.17 8293 3.1 4.2 6.1 5.2 4.5 5.5 10.0 4.9 8.8 2.5 340 2.6 200 37.1 1.9 1.0 1.0 Longzhou 594170 22.37N 106.75E 129 99.78 8293 6.0 7.3 4.5 3.8 3.3 4.3 16.3 3.5 15.0 0.4 90 2.1 200 37.9 3.6 1.1 1.9 Macau 450110 22.20N 113.53E 59 100.62 8293 6.9 8.3 8.4 7.4 6.6 8.3 9.9 7.6 10.6 6.0 360 3.0 200 34.6 5.3 0.9 1.5 Mudanjiang 540940 44.57N 129.60E 74 100.44 8293 −26.1−24.2 9.7 8.1 6.8 8.8 −15.2 7.7−13.8 1.3 180 3.2 200 32.8 −28.7 1.4 2.4 Nanchang 586060 28.60N 115.92E 50 100.73 8293 −1.0 −0.1 7.1 6.1 5.3 7.4 5.4 6.4 4.9 2.7 20 2.8 220 37.7 −3.3 1.2 2.2 Nanjing 582380 32.00N 118.80E 12 101.18 8293 −5.2 −3.5 7.9 6.9 6.0 7.5 3.3 6.7 2.7 0.8 340 4.0 220 36.5 −8.0 1.4 2.0 Nanning 594310 22.82N 108.35E 73 100.45 8293 5.4 6.4 5.2 4.3 3.5 4.5 16.7 3.9 16.2 0.8 90 1.8 160 37.6 3.6 1.2 1.2 Nenjiang 505570 49.17N 125.23E 243 98.44 8293 −35.4−32.6 12.1 10.1 8.6 8.8 −18.2 7.3−18.5 0.7 20 5.8 160 33.0 −38.0 2.4 2.9 Qingdao 548570 36.07N 120.33E 77 100.40 8293 −7.4 −6.1 13.9 12.0 10.4 15.2 −5.8 12.9 −3.9 7.4 340 4.8 160 32.3 −9.4 1.2 1.7 Qiqihar 507450 47.38N 123.92E 148 99.56 8293 −27.9−25.7 10.2 8.5 7.4 7.9 −15.4 6.6−15.6 1.7 290 4.3 180 33.4 −29.9 2.1 2.7 Shanghai 583670 31.17N 121.43E 7 101.24 8293 −3.1 −1.8 8.7 7.6 6.7 8.7 3.4 7.7 3.8 2.5 290 3.5 200 36.6 −6.0 1.1 1.5 Shantou 593160 23.40N 116.68E 3 101.29 8293 6.7 8.1 7.9 6.7 5.9 7.1 13.8 6.3 13.7 3.0 20 3.6 200 34.9 4.2 1.0 1.8 Shaoguan 590820 24.80N 113.58E 68 100.51 8293 2.5 3.8 6.3 5.3 4.4 6.0 11.3 5.0 11.1 1.1 360 2.2 180 38.2 0.6 1.2 1.3 Shenyang 543420 41.77N 123.43E 43 100.81 8293 −21.0−18.7 9.7 8.1 6.9 8.3 −5.5 7.0 −6.4 1.5 70 4.2 200 33.2 −24.6 1.3 2.8 Shijiazhuang 536980 38.03N 114.42E 81 100.36 8293 −9.8 −8.1 7.3 5.8 4.8 7.2 −1.7 5.1 1.0 1.0 160 2.8 160 38.6 −12.3 1.4 2.4 Taiyuan 537720 37.78N 112.55E 779 92.31 8293 −15.4−13.5 9.5 7.8 6.3 9.1 −2.9 7.0 −1.5 0.8 90 2.9 180 34.6 −19.8 1.1 2.4 Tangshan 545340 39.67N 118.15E 29 100.98 8293 −13.8−12.1 8.9 7.4 6.2 9.1 −2.8 7.4 −2.0 1.1 290 3.1 180 34.5 −16.9 1.0 2.8 Tianjin 545270 39.10N 117.17E 5 101.26 8293 −10.0 −8.5 7.9 6.3 5.3 8.5 −3.7 7.1 −3.8 1.9 340 2.7 250 36.5 −13.6 1.3 2.3 Urumqi 514630 43.78N 87.62E 919 90.76 8293 −23.4−20.6 8.0 5.9 5.0 4.5 −7.4 3.5 −9.4 0.8 180 3.6 320 36.4 −23.7 1.7 3.5 Weifang 548430 36.70N 119.08E 51 100.71 8293 −11.3 −9.6 10.6 9.4 8.1 10.2 −1.3 9.0 −1.6 2.7 290 4.5 200 37.2 −13.6 1.5 1.4 Wenzhou 586590 28.02N 120.67E 7 101.24 8293 0.8 2.0 6.5 5.5 4.9 6.1 7.5 5.3 7.4 1.9 290 3.2 90 36.3 −1.6 1.3 1.0 Wuhan 574940 30.62N 114.13E 23 101.05 8293 −2.8 −1.5 6.6 5.4 4.7 6.0 4.3 5.2 4.1 1.0 20 3.5 220 36.9 −6.2 1.4 3.2 Xi’an 570360 34.30N 108.93E 398 96.63 8293 −6.4 −4.8 6.4 5.3 4.4 5.5 2.0 4.5 1.7 0.4 40 2.2 40 37.4 −10.0 1.2 2.5 Xiamen 591340 24.48N 118.08E 139 99.67 8293 5.9 7.0 9.2 8.1 7.2 8.7 11.3 7.9 11.0 4.4 70 3.7 160 34.9 4.2 0.7 1.4 Xining 528660 36.62N 101.77E 2262 76.94 8293 −15.3−13.7 6.7 5.6 4.8 6.2 −3.9 5.2 −3.7 0.6 290 2.5 140 29.7 −18.4 1.1 1.7 Xuzhou 580270 34.28N 117.15E 42 100.82 8293 −7.0 −5.4 7.3 6.3 5.4 6.5 2.3 5.7 2.1 1.1 270 3.7 160 36.3 −9.9 1.1 2.5 Yaxian 599480 18.23N 109.52E 7 101.24 8293 15.1 16.5 7.4 6.3 5.5 8.2 20.9 6.9 21.8 2.6 20 3.4 180 35.4 13.1 2.1 1.8 Yichang 574610 30.70N 111.30E 134 99.73 8293 −1.2 −0.1 5.1 4.1 3.4 4.4 7.9 3.7 7.3 0.8 110 3.3 140 37.7 −2.5 1.2 1.3 Yichun 507740 47.72N 128.90E 232 98.57 8293 −33.3−30.9 8.4 7.1 6.0 7.0 −17.2 6.0−17.1 0.6 220 3.2 220 32.2 −36.0 1.7 2.2 Yinchuan 536140 38.48N 106.22E 1112 88.66 8293 −18.2−15.6 8.7 6.5 5.2 8.3 −2.6 5.1 −3.4 1.0 20 2.5 20 33.6 −20.6 1.1 2.8 Yingkou 544710 40.67N 122.20E 4 101.28 8293 −17.3−15.6 11.4 10.0 8.6 10.7 −3.5 8.9 −5.0 2.6 20 4.4 200 32.0 −20.0 0.9 3.4 Yining 514310 43.95N 81.33E 663 93.61 8293 −23.1−19.2 7.3 5.6 4.4 5.4 −3.3 4.2 −4.2 0.3 70 1.3 270 36.1 −25.1 1.6 3.7 Yueyang 575840 29.38N 113.08E 52 100.70 8293 −1.2 −0.1 7.4 6.4 5.7 7.2 4.0 6.3 3.8 3.2 20 3.4 250 36.6 −2.7 1.2 1.9 WMO# = World Meteorological Organization number Elev. = elevation, m PWD = prevailing wind direction, ° True DB = dry-bulb temperature, °C Lat. = latitude ° Long. = longitude ° StdP = standard pressure at station elevation, kPa WS = wind speed, m/s Climatic Design Information 27.31 Table 3B Cooling and Dehumidification Design Conditions—World Locations Station Cooling DB/MWB Evaporation WB/MDB Dehumidification DP/MDB and HR Range of DB 0.4% 1% 2% 0.4% 1% 2% 0.4% 1% 2% DB MWB DB MWB DB MWB WB MDB WB MDB WB MDB DP HR MDB DP HR MDB DP HR MDB 1 2a 2b 2c 2d 2e 2f 3a 3b 3c 3d 3e 3f 4a 4b 4c 4d 4e 4f 4g 4h 4i 5 Anyang 34.6 22.6 33.4 23.2 32.2 23.3 27.1 31.1 26.3 30.1 25.6 29.1 26.0 21.6 29.8 25.3 20.6 28.7 24.5 19.7 28.4 8.0 Baoding 34.4 22.0 33.0 22.5 31.7 22.7 26.6 30.3 25.8 29.5 25.0 28.8 25.6 20.9 28.7 24.7 19.8 28.1 23.9 18.8 27.7 8.2 Bayan Mod 32.8 14.9 31.5 14.4 30.0 13.8 17.7 26.7 16.7 25.9 15.8 25.6 14.9 12.5 20.7 13.6 11.4 19.8 12.4 10.5 20.4 11.5 Beijing 34.2 21.9 32.9 21.8 31.5 21.9 26.2 30.5 25.4 29.0 24.7 28.3 25.1 20.3 28.4 24.2 19.2 27.6 23.6 18.5 27.1 8.7 Bengbu 35.2 26.5 33.7 26.0 32.3 25.2 28.0 33.1 27.5 32.1 26.9 31.2 26.7 22.3 30.8 26.2 21.7 30.4 25.7 21.0 29.8 6.6 Changchun 30.1 21.3 28.9 21.0 27.6 20.4 24.4 28.1 23.4 26.7 22.5 25.4 23.3 18.6 27.0 22.4 17.6 25.8 21.4 16.5 24.6 7.5 Changsha (576790) 35.6 26.7 34.6 26.6 33.6 26.2 27.8 32.8 27.4 32.5 27.0 32.0 26.7 22.5 30.3 26.2 21.8 29.8 25.7 21.1 29.6 7.0 Chengdu 31.5 24.7 30.6 24.3 29.6 23.9 26.5 30.0 25.7 28.9 25.1 28.1 25.6 22.2 29.0 24.9 21.2 27.8 24.3 20.5 27.2 6.8 Dalian 30.0 23.1 28.7 22.4 27.6 22.0 25.0 28.0 24.4 26.6 23.7 25.8 24.2 19.3 26.6 23.7 18.8 25.8 23.0 18.0 25.5 5.5 Dandong 29.3 23.9 28.0 22.8 26.9 22.1 25.1 27.8 24.4 26.4 23.7 25.5 24.3 19.3 26.4 23.8 18.7 25.6 23.2 18.0 25.0 6.8 Datong 30.7 17.0 29.2 16.7 27.9 16.5 20.7 26.1 19.8 25.1 19.0 24.2 19.2 15.9 22.9 18.2 14.9 22.2 17.2 14.0 22.2 10.7 Deqen 19.4 11.2 18.2 11.2 17.2 10.9 13.0 16.8 12.5 16.3 12.0 15.8 11.7 13.3 14.3 11.2 12.8 13.7 10.7 12.4 13.2 6.7 Dinghai 32.2 27.2 31.1 26.7 30.1 26.2 27.6 31.5 27.1 30.6 26.6 29.6 26.6 22.3 30.4 26.1 21.6 29.5 25.7 21.1 28.7 5.4 Erenhot 32.7 16.0 31.0 15.5 29.3 15.1 18.8 26.3 18.0 25.6 17.2 24.9 16.7 13.4 21.1 15.6 12.5 20.3 14.5 11.6 20.4 11.7 Fuzhou 35.4 27.0 34.4 26.8 33.3 26.5 27.9 33.9 27.4 33.1 26.9 32.3 26.3 22.0 31.7 25.9 21.4 30.9 25.6 21.1 30.0 7.7 Golmud 26.9 11.3 25.3 10.3 23.8 9.8 12.6 23.8 11.7 22.3 10.8 21.5 8.6 9.8 15.8 7.2 8.9 15.7 6.0 8.2 14.6 10.5 Guangzhou 34.6 26.4 33.9 26.3 33.0 26.2 27.7 31.9 27.4 31.5 27.1 31.1 26.7 22.3 29.5 26.4 21.9 29.1 26.1 21.5 29.0 6.9 Guilin 34.5 25.5 33.5 25.5 32.6 25.3 27.1 31.2 26.7 30.8 26.4 30.5 26.2 22.1 29.0 25.7 21.4 28.5 25.4 21.0 28.3 7.4 Guiyang 30.5 21.3 29.5 21.3 28.6 21.0 22.9 28.0 22.5 27.4 22.1 26.7 21.6 18.6 25.2 21.1 18.0 24.9 20.7 17.5 24.5 7.0 Hami 36.1 19.0 34.6 18.1 33.2 17.8 21.0 32.3 19.9 31.8 19.0 30.5 17.4 13.6 26.4 15.9 12.4 25.1 14.7 11.4 24.1 13.3 Hangzhou 35.8 27.2 34.4 26.6 33.1 26.6 28.2 33.4 27.6 32.8 27.0 31.9 27.0 22.8 31.4 26.2 21.7 30.0 25.9 21.3 29.7 7.5 Harbin 30.4 20.8 29.0 20.0 27.7 20.1 24.1 27.8 23.1 26.3 22.1 25.4 23.0 18.1 26.7 22.0 17.0 25.6 21.0 15.9 24.4 8.5 Hefei 34.8 27.1 33.5 26.8 32.2 26.0 28.0 33.1 27.5 32.5 27.0 31.4 26.7 22.4 30.9 26.2 21.7 30.5 25.7 21.1 29.9 6.1 Hohhot 30.9 17.7 29.4 17.3 28.0 16.7 21.1 26.7 20.0 25.2 19.1 24.4 19.4 16.1 23.9 18.4 15.1 23.0 17.3 14.1 22.1 10.5 Jinan 34.8 22.9 33.5 23.1 32.3 22.9 26.7 31.5 26.1 30.8 25.4 29.6 25.4 20.7 29.5 24.7 19.9 29.0 24.1 19.1 28.3 7.3 Jingdezhen 36.0 26.6 34.7 26.1 33.7 26.0 27.5 33.2 27.1 32.6 26.7 31.9 26.2 21.8 29.7 25.8 21.3 29.5 25.4 20.7 29.3 8.3 Jinzhou 31.1 22.1 29.7 21.4 28.5 21.2 25.2 28.2 24.4 27.2 23.7 26.2 24.3 19.4 27.0 23.5 18.5 26.1 22.8 17.7 25.6 7.5 Jixi 30.3 21.1 28.5 20.3 27.1 19.8 23.5 28.2 22.5 25.9 21.4 25.0 22.1 17.3 26.0 21.1 16.2 25.2 20.1 15.2 24.3 8.4 Kashi 33.5 18.8 32.2 18.1 31.0 17.7 20.8 29.2 19.9 28.9 19.0 28.4 18.2 15.4 25.6 16.9 14.1 24.4 15.8 13.1 24.1 12.3 Korla 34.9 19.2 33.7 18.6 32.4 18.1 21.0 32.0 19.8 30.8 19.0 30.0 17.3 13.9 27.2 16.0 12.7 25.4 14.9 11.9 24.7 10.6 Kowloon 33.2 26.1 32.8 26.1 32.0 26.1 27.6 31.1 27.3 30.7 27.0 30.6 27.0 22.8 29.7 26.2 21.7 29.3 26.1 21.5 29.3 4.5 Kunming 26.6 17.0 25.8 17.2 25.0 17.1 20.1 24.6 19.6 23.8 19.2 23.1 18.7 17.1 22.3 18.4 16.8 21.9 18.1 16.5 21.4 8.5 Lanzhou 31.5 17.7 30.2 17.0 28.8 16.6 19.7 27.4 18.9 26.6 18.1 25.5 17.5 15.1 23.3 16.4 14.1 22.6 15.4 13.2 22.0 11.1 Lhasa 24.8 10.7 23.6 10.8 22.5 10.4 13.1 20.6 12.6 20.0 12.2 19.4 10.7 12.7 15.1 10.2 12.2 14.6 9.8 11.9 14.5 11.7 Liuzhou 35.2 25.5 34.3 25.6 33.6 25.5 27.2 32.2 26.8 31.8 26.5 31.5 26.0 21.6 29.6 25.6 21.1 29.3 25.3 20.7 29.1 7.0 Longzhou 35.8 26.5 34.8 26.4 33.9 26.2 27.9 33.4 27.5 32.7 27.2 32.1 26.7 22.6 30.2 26.3 22.1 29.8 26.0 21.7 29.6 7.6 Macau 33.1 27.3 32.4 27.2 31.7 26.9 28.1 32.0 27.7 31.3 27.4 30.9 27.1 23.0 30.4 26.7 22.4 30.0 26.6 22.3 29.8 4.2 Mudanjiang 30.7 21.5 28.8 20.3 27.4 19.8 23.6 28.7 22.5 26.5 21.5 25.4 22.0 17.2 26.8 21.1 16.2 25.8 20.1 15.2 24.4 8.9 Nanchang 35.7 26.8 34.6 26.7 33.5 26.5 28.1 32.6 27.7 32.2 27.2 31.8 27.0 22.8 30.6 26.5 22.2 30.4 26.1 21.6 30.1 6.9 Nanjing 34.6 27.1 33.2 26.8 31.9 26.0 28.1 32.4 27.6 31.9 27.1 31.1 27.1 22.9 30.8 26.5 22.0 30.3 26.0 21.4 29.8 6.2 Nanning 34.9 26.2 34.1 26.3 33.3 26.1 27.8 32.2 27.4 31.6 27.1 31.2 26.7 22.5 29.9 26.4 22.1 29.6 26.1 21.7 29.3 6.6 Nenjiang 29.8 19.4 28.0 18.7 26.5 18.8 22.4 26.3 21.3 25.2 20.5 24.1 21.1 16.2 24.7 20.1 15.2 23.3 19.1 14.3 22.6 9.5 Qingdao 29.2 23.6 28.2 23.6 27.3 23.4 25.8 27.5 25.3 26.9 24.8 26.4 25.3 20.7 27.0 24.8 20.0 26.5 24.3 19.4 26.0 4.5 Qiqihar 30.7 20.8 29.1 20.3 27.7 20.0 23.7 27.6 22.6 26.8 21.7 25.7 22.5 17.5 26.4 21.3 16.2 25.0 20.3 15.2 24.0 7.9 Shanghai 34.4 27.4 33.1 27.4 31.8 26.7 28.4 33.0 27.7 31.9 27.1 31.0 27.2 23.0 31.0 26.8 22.4 30.6 26.1 21.5 29.6 6.4 Shantou 32.8 26.8 32.0 26.6 31.4 26.4 27.6 31.5 27.2 30.8 27.0 30.3 26.7 22.3 29.6 26.3 21.8 29.2 26.1 21.5 28.9 5.2 Shaoguan 35.7 25.8 34.8 25.8 33.9 25.7 27.1 32.8 26.7 32.2 26.5 31.8 25.8 21.3 29.3 25.5 20.9 29.1 25.2 20.5 28.8 7.6 Shenyang 31.1 23.2 29.9 22.7 28.8 22.0 25.3 29.6 24.5 28.3 23.8 27.1 24.0 19.0 27.9 23.4 18.3 27.1 22.6 17.4 26.3 8.0 Shijiazhuang 34.9 22.5 33.5 22.0 32.1 22.7 26.8 30.8 25.9 29.7 25.1 29.0 25.8 21.3 29.3 24.9 20.2 28.4 24.0 19.1 27.6 8.6 Taiyuan 31.7 20.7 30.3 20.1 29.2 19.9 23.8 28.4 22.8 27.4 22.0 26.5 22.5 18.9 26.3 21.5 17.8 25.6 20.5 16.7 24.9 10.9 Tangshan 32.1 23.2 30.9 22.7 29.9 22.5 26.1 29.6 25.3 28.5 24.6 27.6 25.1 20.3 28.4 24.4 19.4 27.7 23.6 18.5 26.6 7.7 Tianjin 33.5 23.1 32.2 23.0 31.1 22.8 26.6 30.1 25.8 29.5 25.2 28.7 25.5 20.7 29.0 24.8 19.8 28.2 24.1 19.0 27.7 7.2 Urumqi 33.0 16.2 31.4 15.8 30.1 15.4 17.6 28.6 17.0 28.0 16.4 27.2 14.2 11.3 19.1 13.2 10.6 19.5 12.4 10.0 19.9 10.2 Weifang 34.0 23.0 32.5 22.9 31.2 23.0 27.0 31.0 26.1 29.7 25.3 28.5 25.9 21.4 29.3 25.1 20.3 28.3 24.4 19.5 27.5 8.3 Wenzhou 33.8 27.6 32.7 27.2 31.8 26.9 28.3 32.7 27.7 31.8 27.3 30.9 27.1 22.9 31.2 26.7 22.3 30.3 26.3 21.8 29.6 6.5 Wuhan 35.1 27.3 34.0 26.9 32.9 26.5 28.3 32.9 27.9 32.5 27.4 31.8 27.2 23.0 30.8 26.7 22.3 30.7 26.2 21.7 30.4 6.1 Xi’an 34.8 22.7 33.4 22.8 32.0 22.5 25.7 31.4 25.0 30.4 24.2 29.3 24.2 20.1 28.8 23.5 19.2 28.4 22.7 18.3 27.4 8.9 Xiamen 32.7 26.3 32.0 26.1 31.2 26.0 27.2 31.4 26.8 30.7 26.5 30.2 26.1 21.9 29.5 25.8 21.5 28.9 25.6 21.2 28.5 5.9 Xining 27.0 14.0 25.5 13.5 24.1 12.8 15.9 22.3 15.1 22.2 14.4 21.2 13.8 13.0 18.4 12.9 12.3 17.2 12.1 11.6 16.9 10.9 Xuzhou 34.2 25.0 32.8 24.6 31.5 24.1 27.6 32.0 26.9 30.9 26.2 29.8 26.5 22.1 30.1 25.8 21.2 29.3 25.2 20.4 28.8 6.8 Yaxian 32.7 26.9 32.3 26.8 31.9 26.7 28.1 31.7 27.7 31.1 27.5 30.8 27.1 22.9 30.3 26.8 22.4 30.0 26.6 22.2 29.9 5.3 Yichang 35.3 26.2 33.9 25.9 32.6 25.3 27.8 33.1 27.2 32.1 26.7 31.0 26.6 22.5 30.9 26.0 21.7 30.3 25.5 21.1 29.6 7.2 Yichun 29.8 20.5 28.2 19.7 26.7 19.2 22.6 27.1 21.6 25.9 20.7 24.7 21.1 16.2 25.4 20.1 15.2 24.1 19.2 14.4 23.0 10.1 Yinchuan 31.2 19.4 30.0 18.9 28.8 18.3 21.9 28.0 20.9 27.1 20.1 26.1 20.0 16.9 25.4 19.0 15.8 24.4 18.0 14.8 23.7 10.6 Yingkou 30.1 24.2 29.1 23.3 28.2 22.7 25.4 28.7 24.7 27.9 24.1 27.2 24.4 19.4 27.7 23.7 18.5 26.9 23.1 17.9 26.4 6.7 Yining 33.0 19.4 31.6 18.8 30.2 18.3 20.6 30.4 19.8 29.5 19.1 28.3 17.4 13.5 24.6 16.5 12.7 23.2 15.7 12.1 22.8 12.8 Yueyang 34.3 27.3 33.5 26.9 32.7 26.6 28.3 33.0 27.7 32.2 27.2 31.5 27.1 23.0 31.8 26.5 22.2 31.1 26.0 21.5 30.1 5.0 MDB = mean coincident dry-bulb temp., °C MWS = mean coincident wind speed, m/s HR = humidity ratio, grams of moisture per kilogram of dry air MWB = mean coincident wet-bulb temp., °C StdD = standard deviation, °C A = airport DP = dew-point temperature, °C 27.32 2001 ASHRAE Fundamentals Handbook (SI) Table 3A Heating and Wind Design Conditions—World Locations Heating Dry Bulb Extreme Wind Speed, m/s Coldest Month MWS/PWD to DB Extr. Annual Daily Elev., m StdP, kPa 0.4% 1% 99.6% 0.4% Mean DB StdD DB Station WMO# Lat.
Long.
Dates 99.6% 99% 1% 2.5% 5% WS MDB WS MDB MWS PWD MWS PWD Max. Min. Max. Min.
1a 1b 1c 1d 1e 1f 1g 2a 2b 3a 3b 3c 4a 4b 4c 4d 5a 5b 5c 5d 6a 6b 6c 6d Zhangjiakou 544010 40.78N 114.88E 726 92.90 8293 −17.0−15.4 7.7 6.6 5.6 7.3 −9.5 6.4 −8.9 3.1 340 2.5 140 34.8 −19.5 0.6 2.0 Zhanjiang 596580 21.22N 110.40E 28 100.99 8293 7.6 8.9 7.1 5.9 5.2 6.2 12.4 5.4 13.4 3.6 340 3.5 250 36.2 5.7 1.7 2.2 Zhengzhou 570830 34.72N 113.65E 111 100.00 8293 −7.4 −5.9 9.3 7.4 6.2 10.5 4.5 8.6 3.6 1.2 180 3.4 160 37.7 −10.6 1.4 2.3 COLOMBIA Bogota 802220 4.70N 74.13W 2548 74.23 8293 2.2 3.9 9.4 8.0 6.4 10.3 17.6 8.5 17.4 0.2 320 4.5 90 28.0 −0.9 4.7 1.5 COOK ISLANDS Rarotonga Island 918430 21.20S 159.82W 7 101.24 8293 16.8 17.8 11.1 9.8 8.9 11.5 21.9 10.3 22.2 0.5 150 4.9 80 31.5 14.6 1.7 1.4 CROATIA Pula 132090 44.90N 13.92E 63 100.57 8293 −4.1 −2.8 11.5 9.4 7.7 11.8 1.8 9.6 3.5 3.3 20 2.8 270 33.5 −6.2 1.1 2.0 Split 133330 43.53N 16.30E 21 101.07 8293 −1.9 −0.1 10.6 8.4 7.0 10.4 4.9 8.5 6.6 3.9 340 3.7 230 34.6 −7.1 3.9 9.3 Zagreb 131310 45.73N 16.07E 107 100.05 8293 −13.2−10.0 8.5 7.2 5.9 7.7 4.0 6.3 3.9 1.0 240 2.9 230 33.5 −16.5 3.2 4.6 CUBA Guantanamo 783670 19.90N 75.15W 17 101.12 8293 19.2 20.1 10.0 8.9 7.9 9.3 29.2 8.4 29.0 3.5 360 5.2 130 37.6 16.0 2.6 4.6 CYPRUS Akrotiri 176010 34.58N 32.98E 23 101.05 8293 4.6 6.0 11.1 10.0 9.0 12.9 11.4 11.5 12.3 2.3 350 4.3 260 35.2 2.4 1.7 2.4 Larnaca 176090 34.88N 33.63E 2 101.30 8293 3.0 4.6 11.9 10.2 8.9 12.5 12.2 10.8 12.2 3.2 310 5.5 200 36.9 0.8 1.2 1.9 Paphos 176000 34.72N 32.48E 8 101.23 8293 4.0 5.4 10.6 9.2 8.0 13.2 12.9 11.3 13.0 3.9 30 4.0 280 33.5 2.1 1.8 1.9 CZECH REPUBLIC Brno 117230 49.15N 16.70E 246 98.40 8293 −14.4−10.9 10.6 9.2 8.2 11.5 −1.0 9.5 −0.7 3.4 60 4.5 180 32.6 −15.8 1.6 4.0 Cheb 114060 50.08N 12.40E 471 95.79 8293 −15.6−12.4 7.1 6.1 5.3 7.6 2.8 6.4 2.1 1.0 40 2.3 220 32.1 −17.1 2.1 3.5 Ostrava 117820 49.68N 18.12E 256 98.29 8293 −17.1−12.9 10.1 9.1 8.3 11.5 −0.1 10.3 0.6 2.3 20 4.6 190 32.3 −19.6 1.7 5.5 Plzen 114480 49.65N 13.27E 364 97.03 8293 −16.7−12.8 9.4 8.3 7.4 10.7 5.0 9.1 3.5 1.0 20 3.5 120 33.3 −18.2 2.2 5.1 Praded Mountain 117350 50.07N 17.23E 1492 84.64 8293 −19.0−16.4 21.0 18.2 16.1 22.6 −6.9 19.0 −5.4 8.4 20 5.2 180 22.1 −20.2 1.7 4.3 Prague 115180 50.10N 14.28E 366 97.00 8293 −16.1−12.4 12.4 10.4 9.0 13.9 4.0 11.9 2.3 1.9 10 3.5 160 32.8 −18.0 2.0 4.9 Pribyslav 116590 49.58N 15.77E 536 95.05 8293 −16.2−13.0 12.8 11.2 9.8 13.3 1.1 12.1 −0.7 2.1 360 3.9 130 30.4 −18.9 2.7 4.0 DENMARK Alborg 60300 57.10N 9.87E 3 101.29 8293 −13.1 −9.2 13.0 11.4 10.2 14.3 7.0 12.5 5.8 2.6 220 4.7 100 28.0 −14.1 2.2 6.9 Copenhagen 61800 55.63N 12.67E 5 101.26 8293 −11.1 −8.0 13.0 11.6 10.5 13.2 4.3 12.0 3.1 5.1 360 4.7 160 27.5 −10.3 1.8 4.5 Hammerodde 61930 55.30N 14.78E 11 101.19 8293 −6.7 −5.3 19.5 16.7 15.0 20.2 1.1 18.5 1.0 8.9 70 5.2 230 26.7 −5.6 1.9 3.3 Mon Island 61790 54.95N 12.55E 15 101.14 8293 −8.0 −5.7 19.1 15.8 14.3 20.4 2.8 18.2 1.9 6.2 320 4.0 70 25.4 −7.2 2.2 4.3 Odense 61200 55.47N 10.33E 17 101.12 8293 −10.2 −7.7 13.1 11.5 10.2 13.5 5.5 12.2 4.2 3.4 40 4.9 120 29.0 −12.6 2.3 5.3 Skagen 60410 57.77N 10.65E 7 101.24 8293 −9.3 −6.4 18.4 16.0 14.4 18.3 2.0 16.2 3.2 7.4 40 4.6 360 24.5 −8.8 1.9 4.4 Tirstrup 60700 56.30N 10.62E 25 101.03 8293 −13.0 −9.1 12.0 10.5 9.4 12.3 4.6 10.9 3.7 2.4 20 4.8 280 27.8 −14.0 1.9 6.1 ECUADOR Guayaquil 842030 2.15S 79.88W 9 101.22 8293 19.7 19.9 7.3 6.5 6.0 7.7 23.2 7.1 23.1 3.6 210 3.2 40 34.9 10.5 1.3 6.3 Quito 840710 0.15S 78.48W 2812 71.80 8293 7.0 7.9 7.8 6.6 5.9 6.6 17.6 6.0 17.8 0.3 350 4.1 150 28.8 4.7 4.3 1.8 EGYPT Alexandria 623180 31.20N 29.95E 7 101.24 8293 6.8 7.8 10.7 9.2 8.1 13.0 13.6 11.3 14.6 2.1 190 4.3 340 39.0 2.9 1.8 2.1 Cairo 623660 30.13N 31.40E 74 100.44 8293 7.0 8.0 9.5 8.3 7.3 10.3 14.6 8.7 16.4 2.6 210 5.6 350 42.1 3.1 1.6 2.7 Luxor 624050 25.67N 32.70E 88 100.27 8293 4.4 5.7 7.2 6.1 5.2 6.8 17.8 5.8 17.7 0.9 180 2.6 330 46.1 0.9 1.7 1.0 ESTONIA Kopu/Cape Ristna 261150 58.92N 22.07E 9 101.22 8293 −15.1−11.9 13.2 11.1 9.4 12.9 3.2 10.7 2.9 2.3 80 2.7 70 26.4 −14.1 2.3 6.7 Tallinn 260380 59.35N 24.80E 44 100.80 8293 −19.8−16.0 9.2 8.1 7.3 9.8 0.9 8.6 0.0 2.9 140 3.6 40 28.0 −19.6 2.4 4.8 FAEROE ISLANDS Torshavn 60110 62.02N 6.77W 39 100.86 8293 −3.2 −2.3 18.2 15.3 13.7 21.5 5.7 19.2 6.2 5.8 320 4.8 210 18.1 −5.4 1.9 1.4 FIJI Nadi 916800 17.75S 177.45E 18 101.11 8293 16.0 17.1 8.9 7.8 7.0 8.7 25.8 7.7 26.0 1.6 120 5.8 350 34.9 13.1 2.0 3.3 Nausori 916830 18.05S 178.57E 7 101.24 8293 16.9 17.9 9.1 8.2 7.3 8.9 23.8 8.1 23.5 0.3 320 4.7 60 32.9 15.0 1.1 1.0 FINLAND Helsinki 29740 60.32N 24.97E 56 100.65 8293 −23.7−19.5 10.0 8.8 7.9 10.9 1.5 9.7 −0.1 2.4 340 4.8 210 28.4 −24.7 1.7 5.3 Jyvaskyla 29350 62.40N 25.68E 145 99.60 8293 −29.2−24.8 8.9 7.7 6.8 10.2 −2.2 8.4 −3.4 0.7 330 3.8 180 28.5 −30.2 2.4 4.2 Kauhava 29130 63.10N 23.03E 44 100.80 8293 −29.0−25.6 9.4 8.3 7.4 10.4 −0.5 9.3 −0.6 0.8 80 3.8 230 27.6 −29.6 1.3 4.5 Kuopio 29170 63.02N 27.80E 102 100.11 8293 −29.7−25.6 8.4 7.4 6.7 9.3 −1.2 8.3 −2.0 0.6 140 3.3 170 27.5 −29.5 1.4 4.4 Lahti 29650 60.97N 25.63E 84 100.32 8293 −26.3−21.9 6.3 5.4 4.8 6.9 0.8 6.0 −0.3 0.6 350 2.7 150 28.5 −28.1 1.5 4.0 Pello 28440 66.80N 24.00E 84 100.32 8293 −31.4−29.1 6.4 5.6 5.0 6.3 −3.7 5.4 −4.4 0.4 300 3.0 340 27.6 −34.7 2.7 3.0 Pori 29520 61.47N 21.80E 17 101.12 8293 −24.3−20.2 11.1 9.8 8.7 13.2 2.7 11.3 1.7 2.3 90 5.0 140 27.5 −24.8 1.4 4.4 Suomussalmi 28790 64.90N 29.02E 224 98.66 8293 −29.7−27.2 7.4 6.4 5.8 7.9 −1.2 6.9 −3.0 0.5 360 3.1 270 28.2 −32.1 5.6 3.7 Tampere 29440 61.42N 23.58E 112 99.99 8293 −26.2−22.2 8.4 7.5 6.8 9.5 1.2 8.4 0.2 0.8 10 4.0 10 28.8 −27.2 1.4 4.8 Turku 29720 60.52N 22.27E 59 100.62 8293 −23.1−19.6 9.5 8.3 7.4 11.1 0.5 9.4 −0.1 2.7 40 4.0 230 28.4 −23.9 1.2 5.4 FRANCE Bordeaux 75100 44.83N 0.70W 61 100.59 8293 −5.8 −3.0 9.9 8.3 7.1 10.6 10.3 9.0 10.4 1.6 40 3.3 80 35.9 −7.4 1.5 4.1 Clermont-Ferrand 74600 45.78N 3.17E 330 97.42 8293 −9.1 −6.8 10.7 8.9 7.5 11.4 9.4 9.7 9.2 1.4 360 3.4 20 36.2 −11.8 1.6 3.9 Dijon 72800 47.27N 5.08E 227 98.63 8293 −9.8 −6.8 10.2 8.7 7.6 11.0 7.0 9.7 6.0 3.1 20 4.6 170 33.5 −11.3 1.9 4.5 Brest 71100 48.45N 4.42W 103 100.09 8293 −2.8 −1.0 11.8 10.4 9.3 12.7 8.7 11.4 8.0 3.4 120 4.0 40 29.5 −4.4 2.2 2.4 Lyon 74810 45.73N 5.08E 240 98.47 8293 −8.5 −5.1 11.4 9.6 8.2 11.8 6.6 10.0 7.2 1.4 20 5.2 180 34.8 −9.7 1.5 4.1 Marseille 76500 43.45N 5.23E 36 100.89 8293 −3.9 −2.0 16.8 14.4 12.4 17.1 6.7 14.4 6.5 3.8 360 5.6 280 35.1 −5.6 1.8 3.1 Montpellier 76430 43.58N 3.97E 6 101.25 8293 −3.8 −1.8 12.7 11.0 9.6 12.2 9.6 10.6 10.2 3.3 340 5.6 180 35.7 −5.9 1.4 3.0 WMO# = World Meteorological Organization number Elev. = elevation, m PWD = prevailing wind direction, ° True DB = dry-bulb temperature, °C Lat. = latitude ° Long. = longitude ° StdP = standard pressure at station elevation, kPa WS = wind speed, m/s Climatic Design Information 27.33 Table 3B Cooling and Dehumidification Design Conditions—World Locations Station Cooling DB/MWB Evaporation WB/MDB Dehumidification DP/MDB and HR Range of DB 0.4% 1% 2% 0.4% 1% 2% 0.4% 1% 2% DB MWB DB MWB DB MWB WB MDB WB MDB WB MDB DP HR MDB DP HR MDB DP HR MDB 1 2a 2b 2c 2d 2e 2f 3a 3b 3c 3d 3e 3f 4a 4b 4c 4d 4e 4f 4g 4h 4i 5 Zhangjiakou 31.9 18.4 30.5 18.2 29.1 18.0 22.4 27.4 21.5 26.8 20.7 25.9 20.9 17.0 25.3 19.7 15.8 24.6 18.8 14.9 24.1 9.5 Zhanjiang 33.9 26.5 33.1 26.6 32.3 26.6 28.0 31.4 27.7 30.9 27.4 30.6 27.2 23.1 29.8 26.9 22.6 29.5 26.6 22.2 29.4 4.7 Zhengzhou 34.7 23.6 33.4 23.5 32.1 23.5 27.4 31.5 26.6 30.4 25.8 29.4 26.3 22.0 30.1 25.5 21.0 28.9 24.8 20.1 28.4 8.1 COLOMBIA Bogota 21.1 13.3 20.2 13.5 20.0 13.5 15.3 18.9 14.9 18.6 14.6 18.2 14.1 13.8 17.5 13.8 13.5 17.0 13.1 12.9 16.5 11.5 COOK ISLANDS Rarotonga Island 29.7 25.5 29.3 25.3 28.9 25.1 26.5 28.7 26.2 28.4 25.9 28.1 25.9 21.2 28.0 25.6 20.9 27.8 25.2 20.3 27.6 4.6 CROATIA Pula 31.8 21.4 30.2 20.6 29.1 20.2 23.3 27.9 22.5 27.8 21.5 27.1 21.9 16.7 26.0 20.8 15.6 25.1 19.9 14.7 24.1 10.6 Split 32.8 21.2 31.7 20.4 30.2 20.0 22.5 29.9 21.7 29.3 21.1 28.7 20.1 14.8 25.9 19.1 13.9 25.3 18.2 13.1 24.7 10.3 Zagreb 31.1 21.3 29.5 21.0 28.1 20.1 22.5 29.4 21.6 28.1 20.8 26.9 20.2 15.1 25.6 19.2 14.2 24.9 18.5 13.5 23.6 12.3 CUBA Guantanamo 34.2 25.8 34.0 25.7 33.2 25.4 27.6 32.8 27.1 32.6 26.7 32.3 26.1 21.5 31.5 25.6 20.9 31.3 25.1 20.2 30.8 8.5 CYPRUS Akrotiri 32.7 21.7 31.4 22.2 30.3 22.6 25.6 29.2 25.1 28.7 24.6 28.1 24.6 19.6 27.7 24.0 18.9 27.6 23.4 18.2 27.3 7.2 Larnaca 33.8 21.7 32.7 22.1 31.6 22.4 25.7 29.9 25.1 29.5 24.6 29.1 24.2 19.1 28.5 23.8 18.7 28.2 23.1 17.9 27.7 9.9 Paphos 30.9 24.7 30.1 24.6 29.2 24.3 26.2 29.8 25.7 29.2 25.1 28.5 25.1 20.2 29.2 24.6 19.6 28.8 24.0 18.9 28.2 8.6 CZECH REPUBLIC Brno 29.5 19.1 27.7 18.5 26.1 17.6 20.2 27.5 19.3 26.2 18.5 24.3 17.7 13.1 23.2 17.0 12.5 21.9 16.2 11.9 21.3 10.8 Cheb 28.4 18.5 26.6 17.5 24.8 16.7 19.3 26.6 18.3 24.7 17.4 23.2 16.8 12.7 21.8 16.0 12.0 20.2 15.2 11.4 19.5 10.8 Ostrava 29.5 19.3 27.6 18.3 25.8 17.7 20.3 27.6 19.4 26.1 18.5 24.2 17.9 13.3 22.6 17.1 12.6 21.5 16.3 12.0 21.2 11.5 Plzen 29.0 19.4 27.1 18.5 25.3 17.7 20.3 27.5 19.3 25.5 18.4 24.0 17.9 13.4 23.6 17.0 12.7 22.6 16.1 12.0 20.8 11.3 Praded Mountain 18.6 13.0 17.0 12.3 15.6 11.7 14.1 17.3 13.2 15.8 12.3 14.7 13.1 11.3 15.0 12.1 10.6 14.1 11.3 10.0 13.3 5.4 Prague 28.8 18.4 26.8 17.8 25.0 17.1 19.7 26.2 18.7 24.7 17.8 23.4 17.3 12.9 22.1 16.4 12.2 20.6 15.8 11.7 20.3 11.1 Pribyslav 27.0 18.1 25.2 17.5 23.5 16.7 19.1 25.4 18.1 23.5 17.2 22.4 16.8 12.8 21.7 16.0 12.1 20.6 15.2 11.5 19.7 10.3 DENMARK Alborg 25.0 17.1 23.1 16.3 21.5 15.5 18.1 23.7 17.2 21.8 16.2 20.3 16.0 11.4 20.2 15.1 10.7 19.3 14.2 10.1 18.1 8.4 Copenhagen 25.0 17.2 23.2 16.4 21.9 15.8 18.2 23.2 17.4 21.7 16.5 20.4 16.2 11.5 20.0 15.5 11.0 19.4 14.8 10.5 18.9 8.1 Hammerodde 22.8 17.8 21.3 17.4 20.1 16.8 18.8 21.3 18.0 20.5 17.2 19.5 17.9 12.9 20.2 17.0 12.1 19.2 16.2 11.5 18.7 3.9 Mon Island 23.1 18.2 21.6 17.4 20.4 16.7 19.0 21.9 18.1 20.9 17.2 19.7 17.8 12.8 20.8 16.8 12.0 20.0 16.0 11.4 19.1 5.6 Odense 25.8 17.8 24.1 17.0 22.3 16.2 18.8 23.7 17.8 22.9 16.9 21.2 16.9 12.1 21.1 15.9 11.3 20.1 15.0 10.7 19.1 9.6 Skagen 22.1 18.9 20.7 17.9 19.4 17.1 19.5 21.3 18.4 20.3 17.5 18.9 18.7 13.5 20.7 17.6 12.6 19.4 16.8 12.0 18.5 5.3 Tirstrup 25.2 17.5 23.7 16.8 22.0 15.9 18.6 23.5 17.6 21.8 16.6 20.5 17.0 12.2 19.8 16.0 11.4 19.0 15.0 10.7 18.2 9.9 ECUADOR Guayaquil 33.2 24.5 32.9 24.6 32.1 24.4 26.7 31.2 26.2 30.4 25.9 29.6 25.8 21.1 29.3 25.2 20.4 28.2 25.0 20.1 27.9 7.4 Quito 22.0 12.5 21.2 12.3 20.8 12.1 14.6 19.4 14.1 19.0 13.7 18.4 13.0 13.3 16.1 12.2 12.6 15.1 12.1 12.5 14.8 10.2 EGYPT Alexandria 32.5 21.6 30.9 22.9 29.9 23.1 25.0 29.7 24.4 28.8 24.0 28.5 23.5 18.3 27.9 23.0 17.8 27.5 22.5 17.2 27.4 6.3 Cairo 38.0 20.3 36.2 20.5 35.1 20.8 24.1 31.5 23.6 30.4 23.1 30.1 22.2 17.0 26.0 21.9 16.7 25.8 21.2 16.0 25.6 13.3 Luxor 43.1 22.2 42.0 21.9 40.9 21.7 24.0 39.4 23.3 38.7 22.7 38.2 18.9 13.8 33.4 17.8 12.9 32.6 16.9 12.2 32.5 17.0 ESTONIA Kopu/Cape Ristna 22.6 17.5 21.1 17.0 19.8 16.3 18.7 21.6 17.7 20.1 16.9 19.2 17.6 12.6 20.4 16.7 11.9 19.4 15.8 11.2 18.4 4.8 Tallinn 24.9 17.6 23.3 16.9 21.6 16.0 19.0 23.0 17.9 21.9 16.9 20.6 17.4 12.5 20.8 16.3 11.7 20.0 15.2 10.8 18.7 8.2 FAEROE ISLANDS Torshavn 14.3 12.4 13.5 11.8 12.7 11.5 13.0 13.9 12.4 13.1 11.9 12.5 12.6 9.1 13.3 12.0 8.8 12.7 11.5 8.5 12.2 3.0 FIJI Nadi 32.3 25.0 31.7 25.0 31.2 24.8 26.6 30.4 26.2 30.0 25.9 29.7 25.6 20.9 28.5 25.2 20.4 28.2 24.9 20.0 28.0 7.9 Nausori 31.2 25.8 30.6 25.7 30.0 25.4 26.7 29.8 26.3 29.3 26.0 28.9 25.9 21.2 28.2 25.6 20.9 28.0 25.2 20.3 27.7 6.1 FINLAND Helsinki 25.9 17.6 24.1 16.3 22.7 15.9 18.7 23.6 17.6 22.4 16.7 20.8 17.0 12.2 19.6 15.9 11.4 19.2 14.9 10.7 18.4 9.8 Jyvaskyla 25.4 16.8 23.8 16.2 21.9 15.3 18.4 23.0 17.2 22.0 16.1 20.3 16.5 12.0 20.5 15.3 11.1 18.5 14.3 10.3 17.6 9.8 Kauhava 25.0 16.7 23.2 16.1 21.2 15.1 18.2 22.9 17.0 21.3 16.0 20.1 16.5 11.8 20.2 15.3 10.9 18.5 14.3 10.2 17.7 10.0 Kuopio 25.4 16.9 23.7 16.2 21.9 15.7 18.6 23.3 17.4 21.7 16.4 20.5 16.7 12.0 20.7 15.7 11.3 19.1 14.6 10.5 18.1 7.1 Lahti 26.1 17.8 24.5 17.0 22.8 16.2 19.1 24.2 17.9 23.2 16.9 21.6 17.0 12.3 21.2 15.8 11.3 20.0 14.9 10.7 18.8 10.6 Pello 24.2 16.0 22.2 15.4 20.3 14.3 17.4 22.0 16.2 20.4 15.1 18.8 15.5 11.1 19.1 14.4 10.3 17.9 13.3 9.6 16.8 8.8 Pori 24.8 16.9 23.2 15.9 21.5 15.1 18.2 22.8 17.1 21.6 16.1 20.0 16.6 11.8 19.7 15.3 10.9 18.5 14.4 10.3 17.5 9.0 Suomussalmi 24.3 16.1 22.3 15.4 20.4 14.3 17.4 22.6 16.2 20.9 15.2 19.1 15.3 11.2 19.0 14.3 10.4 17.7 13.4 9.8 16.8 8.4 Tampere 26.1 16.8 24.4 16.3 22.8 15.4 18.5 23.7 17.3 22.4 16.2 21.1 16.4 11.8 19.6 15.2 10.9 18.9 14.2 10.2 17.6 10.4 Turku 25.9 17.3 24.1 15.9 22.5 15.3 18.4 23.6 17.4 22.2 16.3 20.4 16.7 12.0 20.6 15.4 11.0 19.0 14.6 10.4 18.1 9.1 FRANCE Bordeaux 32.1 21.2 30.0 20.8 28.1 20.0 23.0 29.6 21.8 27.5 20.9 26.1 21.1 15.9 25.9 20.0 14.8 24.0 19.1 14.0 23.1 11.3 Clermont-Ferrand 31.9 20.9 29.8 20.0 27.9 19.6 22.0 29.5 21.0 28.1 20.0 26.5 19.5 14.8 25.0 18.6 14.0 24.3 17.8 13.3 23.2 12.6 Dijon 30.4 20.2 28.8 19.8 27.0 19.2 21.9 27.6 20.9 26.8 19.9 25.5 20.0 15.1 25.1 18.9 14.1 23.3 18.0 13.3 22.2 11.6 Brest 25.8 18.5 23.5 18.3 21.7 17.2 19.7 24.4 18.6 22.1 17.7 20.1 18.2 13.3 20.6 17.5 12.7 19.5 16.9 12.2 18.9 7.6 Lyon 32.0 20.7 30.0 20.3 28.1 19.6 22.2 29.1 21.1 27.9 20.2 26.5 19.9 15.0 25.7 18.9 14.1 24.1 18.0 13.3 22.9 11.9 Marseille 32.0 21.6 30.7 21.2 29.2 20.6 23.7 28.7 22.8 27.9 21.9 27.0 22.1 16.8 26.2 21.1 15.8 25.8 20.1 14.9 24.9 10.4 Montpellier 32.1 21.7 30.3 21.0 29.0 20.5 24.6 27.8 23.7 27.0 22.8 26.4 23.8 18.7 26.0 22.8 17.5 25.3 21.8 16.5 24.7 10.3 MDB = mean coincident dry-bulb temp., °C MWS = mean coincident wind speed, m/s HR = humidity ratio, grams of moisture per kilogram of dry air MWB = mean coincident wet-bulb temp., °C StdD = standard deviation, °C A = airport DP = dew-point temperature, °C 27.34 2001 ASHRAE Fundamentals Handbook (SI) Table 3A Heating and Wind Design Conditions—World Locations Heating Dry Bulb Extreme Wind Speed, m/s Coldest Month MWS/PWD to DB Extr. Annual Daily Elev., m StdP, kPa 0.4% 1% 99.6% 0.4% Mean DB StdD DB Station WMO# Lat.
Long.
Dates 99.6% 99% 1% 2.5% 5% WS MDB WS MDB MWS PWD MWS PWD Max. Min. Max. Min.
1a 1b 1c 1d 1e 1f 1g 2a 2b 3a 3b 3c 4a 4b 4c 4d 5a 5b 5c 5d 6a 6b 6c 6d Nancy 71800 48.68N 6.22E 217 98.75 8293 −10.2 −8.1 9.4 8.2 7.1 10.3 6.5 9.2 4.1 3.2 60 3.7 220 33.4 −12.2 2.2 3.7 Nantes 72220 47.17N 1.60W 27 101.00 8293 −5.2 −2.8 10.7 9.3 8.1 12.2 11.1 10.6 10.1 3.1 60 4.1 60 33.7 −6.5 1.8 3.7 Nice 76900 43.65N 7.20E 10 101.20 8293 1.6 2.9 11.2 9.4 7.8 10.5 11.9 8.7 11.0 4.7 340 3.6 160 32.2 −0.4 1.5 2.7 Nimes 76450 43.87N 4.40E 62 100.58 8293 −3.3 −1.1 10.4 9.1 7.9 10.1 4.1 8.8 6.3 4.3 20 4.1 40 36.2 −4.7 1.1 3.5 Orleans 72490 47.98N 1.75E 125 99.83 8293 −8.4 −5.3 11.8 10.2 9.0 13.2 10.0 11.6 8.9 3.6 60 3.8 80 33.7 −12.0 1.7 11.3 Paris, De Gaulle 71570 49.02N 2.53E 109 100.02 8293 −7.8 −5.0 12.1 10.3 9.1 14.1 9.7 11.9 7.6 4.6 60 4.1 60 33.1 −9.0 2.2 4.6 Paris, Orly 71490 48.73N 2.40E 96 100.18 8293 −7.1 −4.8 11.2 9.7 8.4 12.8 8.8 10.8 8.5 3.7 20 3.4 100 33.4 −8.2 2.0 4.2 St.-Quentin 70610 49.82N 3.20E 101 100.12 8293 −8.2 −5.6 11.9 10.3 9.1 14.4 8.0 12.3 7.6 5.0 60 3.8 120 31.3 −10.3 2.3 4.3 Strasbourg 71900 48.55N 7.63E 154 99.49 8293 −11.0 −8.2 9.8 8.3 7.2 11.7 8.7 9.5 4.9 2.9 340 3.4 20 34.1 −12.5 1.4 4.5 Toulouse 76300 43.63N 1.37E 153 99.50 8293 −5.8 −3.0 9.8 8.5 7.5 10.0 8.9 8.7 9.1 2.2 280 3.3 140 37.0 −7.2 2.0 4.5 FRENCH POLYNESIA Moruroa Island 919520 21.82S 138.80W 3 101.29 8293 19.6 20.2 13.0 11.7 10.6 14.0 22.4 12.6 21.9 7.0 140 4.7 60 32.0 18.3 2.1 0.5 Papeete, Tahiti 919380 17.55S 149.62W 2 101.30 8293 19.8 20.6 9.7 8.3 7.2 10.0 25.4 8.7 26.1 1.5 120 3.0 260 33.0 18.2 0.4 1.1 GERMANY Aachen 105010 50.78N 6.10E 205 98.89 8293 −10.1 −7.2 10.4 9.1 7.9 11.8 7.9 10.3 6.6 1.9 50 2.5 210 32.3 −9.9 1.9 4.7 Ahlhorn (Ger-AFB) 102180 52.88N 8.23E 56 100.65 8293 −11.8 −9.0 11.2 9.8 8.7 13.4 7.2 11.2 5.7 3.0 90 4.0 10 32.1 −12.8 2.2 5.3 Berlin 103840 52.47N 13.40E 49 100.74 8293 −11.8 −9.2 10.4 9.1 8.1 11.5 6.5 9.5 5.1 3.4 80 3.7 150 33.8 −12.2 2.1 4.9 Bitburg 106100 49.95N 6.57E 374 96.91 8293 −10.9 −8.0 10.2 8.9 7.9 12.0 5.4 10.1 3.5 4.7 60 3.1 10 32.5 −11.5 2.0 3.8 Bremen 102240 53.05N 8.80E 5 101.26 8293 −11.3 −8.7 11.3 9.9 8.8 12.6 6.4 10.9 5.6 3.7 70 4.5 100 32.2 −12.6 3.4 5.3 Bremerhaven 101290 53.53N 8.58E 11 101.19 8293 −9.4 −7.0 13.7 12.2 10.9 15.3 6.3 13.4 5.9 3.6 60 4.3 130 31.1 −9.1 2.2 4.1 Dresden 104880 51.13N 13.78E 226 98.64 8293 −13.3−10.3 9.6 8.3 7.3 10.2 5.3 8.8 4.8 1.9 320 3.0 990 33.6 −14.6 1.6 6.2 Dusseldorf 104000 51.28N 6.78E 44 100.80 8293 −9.9 −6.9 10.4 9.2 8.1 11.8 7.0 10.1 6.4 2.8 60 3.8 130 33.4 −10.8 1.5 4.9 Eggebek (Ger-Navy) 100340 54.63N 9.35E 22 101.06 8293 −11.9 −8.6 12.7 11.3 10.0 14.1 4.5 12.4 3.6 3.2 30 5.0 90 30.0 −13.8 1.7 5.3 Ehrenberg 105440 50.50N 9.95E 925 90.70 8293 −14.8−12.1 15.3 13.5 12.0 16.7 −2.7 14.7 −4.1 6.9 100 5.1 190 28.2 −15.2 1.7 4.7 Frankfurt Am Main 106370 50.05N 8.60E 113 99.97 8293 −11.0 −8.2 10.2 8.8 7.7 11.3 7.2 9.4 5.3 3.3 30 3.9 40 34.1 −12.1 1.7 4.3 Grafenwohr 106870 49.70N 11.95E 415 96.44 8293 −18.9−14.8 6.5 5.5 4.8 7.1 3.8 5.9 2.5 0.6 10 2.1 10 33.5 −21.6 2.3 5.3 Greifswald 101840 54.10N 13.40E 6 101.25 8293 −12.8 −9.4 10.4 9.0 7.9 11.3 5.2 9.5 4.5 1.6 250 3.6 50 31.9 −13.3 2.3 6.1 Hamburg 101470 53.63N 10.00E 16 101.13 8293 −11.6 −8.9 10.2 9.0 8.1 10.6 5.6 9.5 4.2 2.5 60 4.7 90 31.5 −12.5 2.5 5.0 Hannover 103380 52.47N 9.70E 54 100.68 8293 −12.7 −9.8 10.2 8.9 8.0 11.1 6.0 9.6 5.0 2.5 80 4.2 110 32.5 −13.1 2.2 5.6 Heidelberg 107340 49.40N 8.65E 109 100.02 8293 −10.0 −7.1 7.5 6.3 5.3 8.1 6.6 6.9 6.3 1.9 170 2.8 10 35.8 −11.2 1.6 4.8 Hof 106850 50.32N 11.88E 568 94.69 8293 −16.0−13.0 9.7 8.5 7.6 10.4 2.1 9.3 1.1 2.5 140 3.3 150 30.8 −18.0 1.6 3.9 Husum (Ger-AFB) 100260 54.52N 9.15E 37 100.88 8293 −11.2 −8.2 13.1 11.4 10.1 14.4 6.0 12.8 5.4 3.9 50 4.2 90 29.9 −13.2 2.7 5.0 Kap Arkona 100910 54.68N 13.43E 41 100.83 8293 −8.1 −5.9 19.1 17.0 15.2 20.5 2.9 18.6 2.7 6.7 360 5.4 70 27.0 −8.5 2.0 4.6 Kiel/Holtenau (Ger-Navy) 100460 54.38N 10.15E 31 100.95 8293 −9.9 −7.2 11.3 9.9 8.7 12.7 4.8 10.8 3.3 4.0 40 3.8 160 29.7 −12.0 2.7 5.9 Koln 105130 50.87N 7.17E 99 100.14 8293 −11.4 −8.1 9.2 8.1 7.2 11.0 7.4 9.2 6.2 1.9 110 3.5 130 33.4 −13.3 1.7 5.9 Lahr 108050 48.37N 7.83E 156 99.46 8293 −11.4 −8.2 8.7 7.3 6.2 10.0 8.2 8.9 7.2 1.7 20 2.4 120 34.3 −13.5 1.5 5.6 Landsberg (Ger-AFB) 108570 48.07N 10.90E 628 94.00 8293 −14.9−12.1 11.8 9.9 8.3 13.4 5.2 11.6 2.8 1.5 70 2.9 260 32.0 −17.5 1.8 4.3 Leck (Ger-AFB) 100220 54.80N 8.95E 13 101.17 8293 −11.5 −8.2 12.5 11.1 9.9 14.2 5.5 12.6 4.4 2.3 80 4.5 110 29.6 −15.8 2.1 6.1 Leipzig 104690 51.42N 12.23E 133 99.74 8293 −13.4−10.4 12.5 10.8 9.4 13.4 5.5 11.3 4.8 2.8 70 4.0 190 33.6 −14.3 1.8 6.8 Memmingen (Ger-AFB) 109470 47.98N 10.23E 644 93.82 8293 −14.8−12.0 11.2 9.5 8.2 13.1 3.4 11.5 3.4 2.2 50 3.8 220 32.4 −18.2 2.4 4.5 Munich 108660 48.13N 11.70E 529 95.13 8293 −15.4−12.5 11.9 9.6 7.9 12.9 5.3 10.6 4.5 1.6 80 3.6 30 32.5 −18.6 2.0 4.5 Neuburg (Ger-AFB) 108530 48.72N 11.22E 387 96.76 8293 −15.9−12.5 9.4 8.0 6.7 10.2 4.2 9.1 4.3 1.6 60 1.9 200 33.5 −18.5 2.3 6.6 Nordholz (Ger-Navy) 101360 53.77N 8.67E 31 100.95 8293 −10.9 −8.2 13.2 11.7 10.4 15.1 5.8 12.8 5.0 3.8 80 5.0 120 31.3 −11.3 2.5 4.6 Ramstein (US-AFB) 106140 49.43N 7.60E 238 98.50 8293 −11.8 −9.0 8.7 7.5 6.3 9.3 6.6 8.0 6.2 0.9 10 2.4 240 34.0 −13.7 2.1 4.4 Sollingen (Can-AFB) 107220 48.77N 8.10E 128 99.80 8293 −10.8 −8.1 10.0 8.7 7.7 11.5 7.3 10.1 6.5 2.6 30 2.5 10 34.5 −12.4 1.8 4.5 Stuttgart 107380 48.68N 9.22E 419 96.39 8293 −12.7−10.0 9.4 7.9 6.8 10.3 5.0 9.0 4.4 1.9 90 3.0 90 33.2 −15.4 2.2 5.7 GEORGIA Batumi 374840 41.65N 41.63E 6 101.25 8293 −1.7 −0.2 13.5 12.2 10.6 13.7 9.5 12.7 9.8 6.0 130 4.2 300 32.9 −6.3 5.0 6.8 K’ut’aisi (Kutaisi) 373950 42.27N 42.63E 116 99.94 8293 −2.4 −1.1 21.7 18.1 16.0 21.7 7.3 17.8 8.3 3.9 90 9.6 90 36.1 −4.9 1.8 2.3 Sokhumi (Sukhumi) 372600 42.87N 41.13E 13 101.17 8293 −1.5 −0.5 7.9 6.5 5.6 8.4 6.7 7.1 5.4 2.7 50 4.0 320 32.3 −3.8 1.8 2.2 Tbilisi 375490 41.68N 44.95E 467 95.84 8293 −6.0 −4.7 21.7 18.8 16.6 22.6 2.1 20.1 2.1 2.6 320 4.5 180 36.2 −8.7 1.7 2.2 GIBRALTAR North Front 84950 36.15N 5.35W 5 101.26 8293 7.8 8.9 14.6 12.5 11.1 16.0 14.1 14.4 13.9 4.0 270 6.2 200 36.1 4.7 2.6 2.4 GREECE Andravida 166820 37.92N 21.28E 12 101.18 8293 −0.2 1.1 10.0 8.5 7.4 12.0 13.2 10.0 12.8 0.9 130 5.3 350 36.2 −2.8 1.5 1.5 Athens 167160 37.90N 23.73E 15 101.14 8293 1.2 2.9 10.2 9.2 8.4 11.2 8.2 9.8 8.6 3.5 360 6.1 30 37.4 −0.4 2.4 1.7 Elefsis (Hel-AFB) 167180 38.07N 23.55E 31 100.95 8293 0.8 2.1 10.2 9.2 8.3 10.5 9.3 9.9 7.8 2.0 360 5.5 10 39.8 −1.0 2.9 1.8 Iraklion 167540 35.33N 25.18E 39 100.86 8293 5.0 6.8 14.5 12.9 11.2 18.0 11.5 14.8 10.8 6.2 340 4.7 320 36.2 2.7 2.0 2.0 Larisa 166480 39.63N 22.42E 74 100.44 8293 −5.1 −3.5 8.6 7.3 6.0 8.9 7.5 7.6 7.7 0.4 360 3.2 270 40.4 −8.8 2.5 3.0 Preveza 166430 38.95N 20.77E 4 101.28 8293 2.3 3.5 11.7 9.9 8.6 13.8 10.8 10.8 10.2 4.3 40 4.3 250 34.9 0.3 2.3 1.5 Rodhos 167490 36.40N 28.08E 11 101.19 8293 5.0 6.6 10.6 9.7 9.0 12.5 9.3 10.4 9.5 6.0 360 6.4 270 35.3 3.5 2.0 2.0 Soudha 167460 35.48N 24.12E 151 99.52 8293 4.0 5.2 11.7 10.0 8.9 13.4 11.3 11.4 9.2 4.4 20 4.3 300 38.0 2.0 1.5 1.8 Thessaloniki 166220 40.52N 22.97E 4 101.28 8293 −3.8 −2.0 13.1 10.6 8.9 14.2 5.6 12.6 6.0 3.6 110 4.4 180 37.4 −7.2 2.5 1.8 GREENLAND Dundas, Thule Ab 42020 76.53N 68.50W 59 100.62 8293 −37.1−35.9 13.5 10.6 8.4 13.3 −19.1 10.8−18.0 3.0 80 4.0 310 17.2 −38.5 4.2 1.8 Godthab 42500 64.17N 51.75W 70 100.49 8293 −24.5−22.4 20.3 16.1 13.4 18.5 −7.0 14.8 −7.0 5.9 350 4.6 20 17.8 −22.4 2.9 3.4 Kangerlussuaq 42310 67.00N 50.80W 53 100.69 8293 −41.0−39.0 10.0 8.4 7.2 10.3 −5.7 8.4 −6.7 3.8 60 4.9 70 20.5 −40.6 1.4 4.5 WMO# = World Meteorological Organization number Elev. = elevation, m PWD = prevailing wind direction, ° True DB = dry-bulb temperature, °C Lat. = latitude ° Long. = longitude ° StdP = standard pressure at station elevation, kPa WS = wind speed, m/s Climatic Design Information 27.35 Table 3B Cooling and Dehumidification Design Conditions—World Locations Station Cooling DB/MWB Evaporation WB/MDB Dehumidification DP/MDB and HR Range of DB 0.4% 1% 2% 0.4% 1% 2% 0.4% 1% 2% DB MWB DB MWB DB MWB WB MDB WB MDB WB MDB DP HR MDB DP HR MDB DP HR MDB 1 2a 2b 2c 2d 2e 2f 3a 3b 3c 3d 3e 3f 4a 4b 4c 4d 4e 4f 4g 4h 4i 5 Nancy 29.9 20.4 28.0 19.6 26.2 18.6 21.4 27.9 20.3 26.2 19.4 24.7 19.2 14.3 24.6 18.2 13.5 22.9 17.4 12.8 21.4 11.6 Nantes 30.2 20.2 28.2 20.0 26.2 18.8 22.0 27.5 20.9 26.2 19.9 24.6 20.1 14.8 24.4 19.1 13.9 22.9 18.2 13.1 21.3 10.1 Nice 29.1 23.2 28.1 22.8 27.2 22.5 25.3 27.7 24.5 26.9 23.7 26.2 24.4 19.4 27.0 23.8 18.7 26.6 22.8 17.5 25.9 6.8 Nimes 33.2 20.7 31.7 20.4 30.3 20.2 23.0 28.8 22.1 28.1 21.4 27.3 21.4 16.2 24.9 20.5 15.3 24.6 19.5 14.3 23.8 10.9 Orleans 30.1 19.9 28.2 19.1 26.7 18.5 21.7 28.0 20.3 25.9 19.4 24.7 19.6 14.5 25.6 18.2 13.3 22.7 17.3 12.6 21.8 11.8 Paris, De Gaulle 29.8 20.8 27.7 19.9 25.9 19.0 21.9 27.1 20.7 25.9 19.7 24.3 20.0 14.9 24.6 18.9 13.9 23.3 17.9 13.0 22.2 10.4 Paris, Orly 29.9 20.3 28.0 19.4 26.1 18.6 21.4 27.9 20.3 25.9 19.5 24.4 19.2 14.1 23.7 18.2 13.3 22.9 17.4 12.6 21.9 10.2 St.-Quentin 27.9 20.1 26.0 19.4 24.3 18.4 21.3 26.0 20.0 24.8 19.0 23.3 19.7 14.6 23.9 18.4 13.4 22.1 17.4 12.6 21.2 10.5 Strasbourg 30.5 20.9 28.8 20.0 27.0 19.2 21.9 27.9 20.9 26.8 20.0 25.5 19.9 14.9 24.7 18.9 14.0 23.8 18.0 13.2 22.5 11.5 Toulouse 33.0 21.1 31.0 20.7 29.1 20.0 23.0 30.2 21.9 28.0 21.0 27.2 20.9 15.8 26.3 19.9 14.9 25.2 19.0 14.0 23.8 11.9 FRENCH POLYNESIA Moruroa Island 30.5 25.9 30.0 25.6 29.5 25.4 26.8 29.2 26.5 28.9 26.1 28.5 26.2 21.6 28.2 25.7 21.0 27.9 25.4 20.6 27.8 3.8 Papeete, Tahiti 31.8 26.3 31.4 26.0 31.0 25.8 27.0 30.9 26.6 30.5 26.3 30.2 25.8 21.1 30.0 25.5 20.7 29.6 25.1 20.2 29.3 6.1 GERMANY Aachen 28.6 18.9 26.9 18.6 25.2 17.8 20.2 26.8 19.4 25.4 18.5 24.0 18.0 13.3 22.6 17.2 12.6 21.7 16.3 11.9 20.9 8.6 Ahlhorn (Ger-AFB) 28.8 18.5 26.9 18.1 24.9 17.5 20.0 26.8 19.0 25.5 18.0 23.6 17.8 12.9 21.9 16.9 12.1 21.0 16.0 11.4 20.4 10.1 Berlin 29.9 18.8 27.9 18.1 26.1 17.5 20.1 27.0 19.2 25.9 18.3 24.1 17.9 12.9 22.3 16.9 12.1 21.2 15.9 11.4 20.8 9.3 Bitburg 28.8 19.0 26.9 18.6 25.0 17.6 20.0 26.6 19.2 25.4 18.2 24.0 18.0 13.5 22.5 17.0 12.7 21.9 16.0 11.9 21.0 10.3 Bremen 27.9 19.1 26.1 18.1 24.2 17.3 20.0 25.8 19.1 24.6 18.1 22.7 18.1 13.0 21.8 17.1 12.2 21.4 16.1 11.4 20.2 10.0 Bremerhaven 27.1 18.7 25.0 17.9 23.0 17.2 20.1 25.1 19.1 23.1 18.2 22.0 18.5 13.4 21.6 17.5 12.5 21.3 16.6 11.8 20.3 6.5 Dresden 29.7 18.8 27.5 18.2 25.8 17.4 20.1 27.4 19.1 25.3 18.2 24.2 17.6 13.0 22.4 16.8 12.3 21.6 15.9 11.6 20.5 9.8 Dusseldorf 29.6 19.6 27.8 18.6 26.0 17.8 20.5 27.4 19.6 26.0 18.7 24.3 18.2 13.2 22.9 17.4 12.5 22.0 16.5 11.8 21.4 9.7 Eggebek (Ger-Navy) 26.8 18.4 24.8 17.2 22.9 16.5 19.1 25.1 18.1 23.6 17.2 21.7 17.0 12.2 21.3 16.0 11.4 20.2 15.1 10.7 19.3 9.6 Ehrenberg 24.0 16.5 22.2 15.7 20.6 14.9 17.5 22.3 16.5 21.1 15.5 19.5 15.6 12.4 20.0 14.6 11.6 18.3 13.8 11.0 17.1 7.5 Frankfurt Am Main 30.3 19.4 28.5 18.8 26.7 17.9 20.5 27.8 19.6 26.6 18.7 24.8 18.2 13.3 22.6 17.4 12.6 21.5 16.5 11.9 20.9 11.0 Grafenwohr 29.2 18.8 27.8 18.7 25.9 17.7 20.0 27.1 19.1 26.2 18.1 24.7 17.2 12.9 20.3 16.8 12.6 21.4 15.9 11.9 20.7 13.9 Greifswald 27.2 19.0 25.0 18.1 23.2 17.2 19.9 25.4 18.9 23.7 17.9 22.2 18.1 13.0 22.1 17.0 12.1 20.9 16.0 11.4 20.0 9.1 Hamburg 27.8 18.9 25.9 18.0 24.0 17.1 19.9 25.7 18.8 24.3 17.9 22.6 17.8 12.8 22.1 16.9 12.1 21.0 16.0 11.4 20.1 9.3 Hannover 28.8 19.3 26.9 18.4 25.1 17.6 20.3 26.5 19.3 25.3 18.4 23.4 18.3 13.3 22.5 17.3 12.4 21.4 16.3 11.7 20.8 10.4 Heidelberg 32.1 20.3 30.1 19.6 28.2 19.0 21.4 30.0 20.6 28.7 19.7 26.9 19.0 14.0 25.0 18.0 13.1 23.8 17.1 12.4 21.9 11.1 Hof 27.0 17.5 25.0 16.9 23.3 16.1 18.8 25.0 17.8 23.2 16.8 21.9 16.6 12.7 21.3 15.7 11.9 19.7 14.9 11.3 18.7 10.3 Husum (Ger-AFB) 26.1 18.0 24.2 17.6 22.2 16.4 19.5 24.2 18.4 22.9 17.4 21.0 17.8 12.8 21.3 16.8 12.0 19.8 15.9 11.3 19.0 8.6 Kap Arkona 23.1 18.2 21.7 17.7 20.5 16.9 19.1 22.2 18.2 21.1 17.4 20.0 17.7 12.8 20.9 16.9 12.1 19.9 16.1 11.5 19.2 5.1 Kiel/Holtenau (Ger-Navy) 25.8 18.0 24.0 17.0 22.2 16.4 18.7 24.2 17.8 22.7 16.9 21.3 16.8 12.0 21.5 15.9 11.3 20.2 15.0 10.7 19.2 8.6 Koln 29.6 19.2 27.7 18.3 25.9 17.5 20.3 27.1 19.4 25.9 18.6 24.4 18.1 13.2 22.7 17.2 12.4 21.5 16.4 11.8 20.8 11.0 Lahr 30.2 20.7 28.8 20.0 26.9 19.0 21.7 28.5 20.7 27.4 19.8 25.4 19.2 14.2 25.1 18.3 13.4 23.3 17.8 13.0 22.9 11.5 Landsberg (Ger-AFB) 28.2 19.1 26.2 17.7 24.9 17.2 19.4 27.3 18.5 25.2 17.6 23.4 16.8 12.9 22.0 15.9 12.2 21.3 15.0 11.5 20.6 11.2 Leck (Ger-AFB) 26.2 18.2 24.2 17.0 22.2 16.4 19.2 25.0 18.0 23.8 17.0 21.4 17.0 12.1 21.8 16.0 11.4 20.2 15.0 10.7 19.3 9.1 Leipzig 29.7 19.0 27.6 18.4 25.8 17.6 20.2 27.0 19.2 25.7 18.4 24.3 17.8 13.0 22.9 17.0 12.3 21.7 16.1 11.6 21.1 10.3 Memmingen (Ger-AFB) 28.2 19.0 26.8 18.1 25.0 17.0 19.2 27.1 18.4 25.4 17.5 23.6 16.2 12.5 22.9 15.2 11.7 20.9 14.8 11.4 20.7 11.1 Munich 29.0 18.7 27.1 18.0 25.5 17.4 19.6 26.7 18.8 25.6 18.1 24.3 17.1 13.0 22.2 16.4 12.4 21.4 15.7 11.9 20.7 11.2 Neuburg (Ger-AFB) 29.2 19.0 27.2 18.1 25.9 17.6 20.0 27.5 19.1 26.5 18.1 24.7 17.1 12.8 22.9 16.2 12.1 21.8 15.2 11.3 20.7 12.6 Nordholz (Ger-Navy) 27.2 18.3 25.1 17.5 23.1 16.8 19.6 24.8 18.5 23.5 17.6 21.9 17.5 12.6 21.0 16.6 11.9 20.3 15.8 11.3 19.5 8.2 Ramstein (US-AFB) 30.2 19.8 28.2 18.8 26.8 18.2 20.9 28.2 19.7 26.3 18.8 25.4 18.2 13.5 22.1 17.2 12.6 22.4 16.2 11.9 21.6 12.4 Sollingen (Can-AFB) 30.8 20.6 28.8 19.9 27.0 18.9 21.7 28.4 20.8 27.0 19.9 25.3 19.2 14.2 23.9 18.8 13.8 23.4 17.9 13.1 22.5 11.0 Stuttgart 29.1 18.9 27.3 18.3 25.6 17.4 19.9 27.3 19.0 25.7 18.2 24.3 17.3 13.0 23.1 16.5 12.4 21.9 15.8 11.8 21.1 10.8 GEORGIA Batumi 27.7 22.8 26.9 22.2 26.1 21.7 23.8 26.8 23.1 26.0 22.4 25.2 22.8 17.5 26.1 22.1 16.8 25.3 21.4 16.1 24.5 5.8 K’ut’aisi (Kutaisi) 32.1 21.4 30.3 21.2 28.9 21.1 24.2 28.6 23.4 27.2 22.7 26.3 22.9 17.9 26.3 22.2 17.1 25.4 21.6 16.5 25.0 8.3 Sokhumi (Sukhumi) 28.9 22.8 27.8 22.6 26.9 22.2 24.4 27.4 23.7 26.6 23.1 25.9 23.4 18.2 26.2 22.7 17.4 25.5 22.2 16.9 25.0 7.3 Tbilisi 33.5 21.2 31.9 21.2 30.4 20.7 22.9 31.3 22.1 30.2 21.2 28.8 20.2 15.8 27.2 19.4 15.0 26.2 18.6 14.2 25.6 10.2 GIBRALTAR North Front 31.1 20.4 29.2 20.1 27.9 20.1 23.4 26.2 22.9 25.6 22.3 25.0 22.5 17.2 24.6 22.0 16.7 24.6 21.4 16.1 23.7 7.0 GREECE Andravida 32.9 20.9 31.6 21.5 30.3 21.4 24.1 28.9 23.5 28.2 22.8 27.9 22.9 17.7 26.4 22.0 16.7 26.2 21.1 15.8 25.5 11.8 Athens 34.1 20.6 33.0 20.1 31.8 20.1 23.8 29.7 22.9 29.2 22.1 28.5 21.9 16.6 28.2 20.8 15.5 27.5 19.8 14.5 26.7 9.4 Elefsis (Hel-AFB) 36.1 21.1 34.9 20.1 33.2 19.8 23.6 31.3 22.6 30.9 21.6 30.1 21.0 15.7 28.6 19.9 14.7 27.6 18.8 13.7 26.7 10.1 Iraklion 31.2 18.9 29.9 19.5 28.8 19.9 23.2 27.5 22.6 27.3 22.0 26.5 21.8 16.5 26.7 21.0 15.7 26.3 20.1 14.9 25.9 5.9 Larisa 36.0 20.3 34.1 20.1 32.8 19.8 21.9 32.6 21.1 30.9 20.5 30.1 19.1 14.0 23.6 18.2 13.2 23.3 17.6 12.7 23.1 14.0 Preveza 31.2 21.6 29.9 21.6 28.8 22.1 24.4 28.1 23.8 27.6 23.3 27.0 23.2 18.0 26.6 22.7 17.4 26.1 22.1 16.8 25.9 8.0 Rodhos 32.0 21.5 30.8 21.4 29.8 21.3 24.3 27.7 23.8 27.6 23.3 27.3 23.2 18.0 26.1 22.7 17.4 26.1 22.0 16.7 25.9 5.6 Soudha 34.0 19.1 32.2 19.2 30.9 19.0 22.3 27.8 21.6 27.2 21.1 26.8 20.9 15.8 24.1 20.0 15.0 24.0 19.2 14.2 23.9 8.5 Thessaloniki 33.2 21.2 32.1 20.7 30.9 20.5 22.9 30.4 22.2 29.0 21.5 28.9 20.8 15.5 27.0 19.9 14.6 26.3 19.0 13.8 25.4 11.6 GREENLAND Dundas, Thule Ab 12.2 6.6 10.4 6.1 9.1 5.2 7.1 11.4 6.2 10.2 5.4 8.9 4.0 5.1 8.1 3.0 4.7 7.1 2.2 4.5 6.4 4.6 Godthab 14.0 9.4 12.1 8.4 10.5 8.0 10.1 13.0 9.1 11.5 8.2 10.1 8.7 7.0 10.2 7.6 6.5 9.3 6.8 6.2 8.6 5.8 Kangerlussuaq 18.2 10.7 17.1 10.2 15.8 9.5 11.1 17.6 10.4 16.6 9.7 15.1 7.4 6.4 11.3 6.6 6.1 11.3 5.9 5.8 11.1 10.1 MDB = mean coincident dry-bulb temp., °C MWS = mean coincident wind speed, m/s HR = humidity ratio, grams of moisture per kilogram of dry air MWB = mean coincident wet-bulb temp., °C StdD = standard deviation, °C A = airport DP = dew-point temperature, °C 27.36 2001 ASHRAE Fundamentals Handbook (SI) Table 3A Heating and Wind Design Conditions—World Locations Heating Dry Bulb Extreme Wind Speed, m/s Coldest Month MWS/PWD to DB Extr. Annual Daily Elev., m StdP, kPa 0.4% 1% 99.6% 0.4% Mean DB StdD DB Station WMO# Lat.
Long.
Dates 99.6% 99% 1% 2.5% 5% WS MDB WS MDB MWS PWD MWS PWD Max. Min. Max. Min.
1a 1b 1c 1d 1e 1f 1g 2a 2b 3a 3b 3c 4a 4b 4c 4d 5a 5b 5c 5d 6a 6b 6c 6d Narsarsuaq 42700 61.18N 45.42W 26 101.01 8293 −27.8−24.7 20.9 17.4 13.7 23.6 1.0 20.6 1.7 0.9 60 7.5 70 20.0 −26.1 1.4 6.0 GUAM Andersen AFB (Guam) 912180 13.58N 144.93E 185 99.12 8293 23.3 23.7 9.0 7.9 7.2 8.4 25.9 7.9 25.9 3.8 70 3.8 90 32.8 22.0 1.6 0.6 HUNGARY Budapest 128390 47.43N 19.27E 185 99.12 8293 −13.2−10.2 16.1 12.8 10.6 15.6 4.3 12.1 4.2 0.9 170 4.5 200 34.5 −16.5 1.2 4.4 Debrecen 128820 47.48N 21.63E 112 99.99 8293 −14.6−11.6 9.6 8.0 6.7 10.0 1.9 8.3 1.6 1.6 50 2.5 90 33.5 −16.4 1.5 3.7 Nagykanizsa 129250 46.45N 16.98E 141 99.64 8293 −13.5−10.2 8.7 7.3 6.2 8.7 0.3 7.3 1.8 2.1 360 2.7 230 33.0 −17.2 1.5 4.6 Pecs 129420 46.00N 18.23E 203 98.91 8293 −11.2 −9.0 9.7 7.9 6.7 10.3 2.1 8.8 0.6 2.6 320 3.2 270 33.8 −13.5 1.3 3.2 Siofok 129350 46.92N 18.03E 108 100.03 8293 −11.3 −8.5 13.4 11.4 9.5 13.3 −0.7 11.3 4.0 2.0 320 2.2 270 32.8 −13.8 1.7 4.6 Szombathely 128120 47.27N 16.63E 221 98.70 8293 −12.0 −9.5 13.1 10.9 9.0 12.1 −4.0 9.5 −0.2 3.4 270 3.2 180 32.8 −14.1 1.7 3.9 ICELAND Akureyri 40630 65.68N 18.08W 27 101.00 8293 −13.4−11.5 13.4 11.4 9.8 15.3 0.3 13.3 2.9 3.0 160 4.9 180 22.2 −16.5 1.7 1.2 Keflavik 40180 63.97N 22.60W 54 100.68 8293 −8.1 −6.9 18.1 15.3 13.5 20.8 0.8 18.5 0.9 6.3 20 5.0 350 18.7 −11.3 3.6 1.1 Raufarhofn 40770 66.45N 15.95W 10 101.20 8293 −12.2−10.4 16.5 14.4 12.7 19.3 −0.6 17.0 −0.3 6.2 230 6.2 320 20.0 −15.8 2.1 1.7 Reykjavik 40300 64.13N 21.90W 61 100.59 8293 −9.8 −8.1 18.1 15.4 13.7 21.0 2.5 18.6 2.5 4.3 90 5.5 360 18.4 −12.1 1.9 1.4 INDIA Ahmadabad 426470 23.07N 72.63E 55 100.67 8293 11.3 12.8 7.0 5.9 5.1 6.3 23.5 5.3 23.3 1.0 360 3.0 270 43.9 6.3 1.7 3.8 Bangalore 432950 12.97N 77.58E 921 90.74 8293 14.9 15.7 6.0 5.2 4.4 5.0 21.7 4.3 22.1 1.4 90 1.7 90 37.0 12.2 1.1 1.9 Bombay 430030 19.12N 72.85E 14 101.16 8293 16.5 17.6 6.7 6.0 5.3 5.4 26.2 5.0 26.6 0.2 360 3.1 320 38.5 13.4 1.3 1.6 Calcutta 428090 22.65N 88.45E 6 101.25 8293 12.0 13.1 5.6 4.7 3.8 3.3 22.9 3.0 22.6 0.2 360 2.0 180 39.2 10.2 1.1 0.8 Cuddalore 433290 11.77N 79.77E 12 101.18 8293 19.9 20.7 6.2 5.4 4.8 5.8 26.0 5.2 26.5 0.7 320 2.6 250 40.4 17.3 1.5 1.7 Goa/Panaji 431920 15.48N 73.82E 60 100.61 8293 19.6 20.3 7.5 6.4 5.5 5.2 28.6 4.4 28.6 2.2 50 2.7 320 37.2 16.4 1.5 3.0 Hyderabad 431280 17.45N 78.47E 545 94.95 8293 14.5 15.8 9.2 8.3 7.7 5.6 24.5 5.1 25.0 0.4 50 3.7 320 41.6 11.5 1.1 2.0 Jaipur 423480 26.82N 75.80E 390 96.73 8293 6.8 8.2 7.1 5.8 5.0 5.3 17.9 4.4 18.0 0.2 90 3.8 320 43.5 3.9 1.2 1.7 Madras 432790 13.00N 80.18E 16 101.13 8293 19.9 20.5 7.4 6.4 5.7 5.6 26.8 4.8 26.8 0.9 290 3.7 270 41.2 18.3 1.2 1.0 Nagpur 428670 21.10N 79.05E 310 97.66 8293 11.8 13.0 7.6 6.1 5.3 4.9 23.7 3.5 23.3 1.0 360 2.8 320 44.9 9.2 1.3 1.8 New Delhi 421820 28.58N 77.20E 216 98.76 8293 6.6 7.6 7.4 6.3 5.4 6.4 18.9 5.7 18.7 0.7 270 3.3 320 43.4 5.0 1.2 1.1 Poona 430630 18.53N 73.85E 559 94.79 8293 9.8 10.9 5.3 4.4 3.5 3.4 25.9 2.8 25.6 0.0 70 1.5 270 40.4 7.2 0.7 1.5 Sholapur 431170 17.67N 75.90E 479 95.70 8293 16.2 17.2 3.6 3.1 2.5 2.8 23.5 2.4 23.8 0.5 90 0.9 320 42.5 13.1 1.7 1.7 Trivandrum 433710 8.48N 76.95E 64 100.56 8293 22.0 22.6 6.4 5.8 5.1 7.5 28.2 6.5 28.1 1.1 360 2.5 320 37.4 18.3 1.7 2.6 INDIAN OCEAN ISLANDS Diego Garcia Isl.
619670 7.30S 72.40E 3 101.29 8293 23.0 23.8 9.3 8.4 7.7 9.5 26.5 9.0 26.5 4.8 110 3.1 90 35.2 19.9 2.2 6.0 IRELAND Belmullet 39760 54.23N 10.00W 10 101.20 8293 −1.2 0.2 17.6 15.1 13.5 20.1 9.0 17.9 8.6 3.6 90 4.7 180 24.2 −3.1 2.3 2.0 Birr 39650 53.08N 7.88W 72 100.46 8293 −4.2 −2.5 10.4 9.1 8.1 12.5 7.4 11.1 7.4 0.6 90 3.0 150 24.9 −6.9 4.5 2.8 Claremorris 39700 53.72N 8.98W 69 100.50 8293 −3.6 −2.2 13.1 11.4 9.8 15.2 7.5 13.4 7.2 2.7 70 3.6 90 24.8 −6.1 2.4 2.2 Clones 39740 54.18N 7.23W 89 100.26 8293 −3.7 −2.1 12.3 10.5 9.2 13.6 7.2 12.3 7.6 1.8 60 3.1 120 25.3 −6.4 2.3 2.6 Cork 39550 51.85N 8.48W 162 99.39 8293 −1.4 −0.2 15.1 13.3 11.9 17.7 6.6 15.2 7.0 5.7 40 4.2 330 24.1 −3.5 2.0 1.9 Dublin 39690 53.43N 6.25W 85 100.31 8293 −1.6 −0.4 13.8 12.3 10.9 15.6 6.9 13.5 6.8 4.2 250 4.9 230 24.9 −3.6 1.8 1.8 Kilkenny 39600 52.67N 7.27W 64 100.56 8293 −3.7 −2.3 11.9 10.0 8.5 13.1 8.5 11.7 8.4 1.0 360 3.1 180 26.2 −6.6 2.2 2.3 Malin 39800 55.37N 7.33W 25 101.03 8293 −0.3 0.8 20.0 18.1 15.9 22.3 6.1 20.2 6.2 6.2 170 7.2 200 22.4 −2.0 1.2 1.7 Mullingar 39710 53.53N 7.37W 104 100.08 8293 −3.7 −2.4 11.4 9.9 8.8 13.1 6.2 11.5 7.5 1.4 70 3.9 100 25.2 −6.4 2.1 1.9 Rosslare 39570 52.25N 6.33W 25 101.03 8293 0.2 1.2 14.5 13.1 11.8 16.0 5.0 13.9 6.3 6.5 90 5.1 220 22.7 −1.1 1.9 1.3 Shannon 39620 52.70N 8.92W 20 101.08 8293 −2.0 −0.6 13.9 12.0 10.4 16.3 7.2 14.0 7.6 2.9 70 4.1 110 25.7 −4.4 2.4 1.7 Valentia Observatory 39530 51.93N 10.25W 14 101.16 8293 −0.6 0.8 14.9 13.3 11.8 17.1 8.4 15.1 8.7 3.0 60 4.6 270 25.0 −2.8 2.2 1.5 ISRAEL Jerusalem 401840 31.78N 35.22E 754 92.59 8293 0.6 1.6 10.5 9.3 8.3 12.6 5.0 10.5 5.7 2.5 270 4.5 290 36.4 −0.8 3.0 1.3 Lod 401800 32.00N 34.90E 49 100.74 8293 4.2 5.6 10.1 8.7 7.7 11.4 13.3 9.8 12.1 1.7 150 4.8 320 39.3 2.0 1.6 1.6 Ovda (Isr-AFB/Civ) 401980 30.00N 34.83E 432 96.24 8293 2.3 3.7 10.4 8.7 7.8 11.1 9.5 9.2 12.3 2.2 210 4.3 40 40.6 −0.1 0.7 1.3 Tel Aviv-Yafo 401760 32.10N 34.78E 4 101.28 8293 6.4 7.6 12.6 9.9 8.4 13.2 13.4 10.9 13.5 3.1 120 4.1 310 37.2 5.1 3.0 1.4 ITALY Bologna/Borgo (AFB) 161400 44.53N 11.30E 42 100.82 8293 −5.5 −3.9 7.2 5.9 4.9 6.3 4.8 4.9 2.8 0.5 220 2.4 80 36.0 −8.0 1.9 3.3 Brindisi 163200 40.65N 17.95E 10 101.20 8293 2.1 3.9 11.5 9.8 8.5 13.2 10.4 11.6 10.6 3.9 360 4.5 180 37.1 −0.6 2.4 1.3 Catania 164600 37.47N 15.05E 17 101.12 8293 1.8 3.0 10.1 8.5 7.3 11.4 12.0 9.6 12.6 2.3 230 4.6 90 39.3 −0.6 3.0 1.2 Genova 161200 44.42N 8.85E 3 101.29 8293 0.1 2.0 12.1 10.8 9.7 12.6 6.1 11.6 6.7 6.8 40 3.3 50 33.3 −1.6 1.3 2.3 Messina 164200 38.20N 15.55E 51 100.71 8293 6.1 7.3 8.5 7.3 6.2 9.1 13.4 7.8 13.8 2.0 310 2.8 60 36.3 3.2 2.7 2.2 Milan, Linate 160800 45.43N 9.28E 103 100.09 8293 −6.0 −4.1 7.2 5.4 4.3 8.9 7.8 6.3 7.9 0.4 90 2.2 220 34.8 −7.7 3.3 2.8 Milan, Malpensa 160660 45.62N 8.73E 211 98.82 8293 −9.6 −7.9 8.0 5.5 4.2 8.6 6.3 5.1 4.7 0.1 360 1.8 210 34.4 −12.6 1.7 2.6 Naples 162890 40.85N 14.30E 72 100.46 8293 0.0 1.2 10.1 8.2 6.9 11.8 9.7 9.2 9.4 1.8 340 3.4 200 36.4 −2.3 2.1 1.6 Palermo 164050 38.18N 13.10E 34 100.92 8293 6.9 7.9 13.5 11.6 9.9 14.1 13.8 12.7 13.0 4.8 210 5.1 250 37.9 3.6 2.9 2.6 Perugia 161810 43.08N 12.50E 205 98.89 8293 −4.4 −2.9 9.4 8.2 7.1 9.8 7.7 8.6 7.1 0.5 360 2.3 270 35.6 −8.4 1.6 4.0 Pisa 161580 43.68N 10.38E 1 101.31 8293 −3.1 −1.9 10.0 8.4 7.1 10.0 9.5 8.6 8.3 1.6 90 4.0 270 35.3 −6.2 1.1 2.8 Rome 162420 41.80N 12.23E 3 101.29 8293 −0.9 0.1 12.3 10.5 9.2 12.7 8.7 10.3 10.1 3.8 60 5.2 270 34.4 −3.2 2.9 1.7 Ronchi Legionari Ab 161080 45.82N 13.48E 12 101.18 8293 −5.9 −4.1 10.0 8.0 6.2 12.3 3.2 9.1 3.2 1.1 10 2.3 220 35.5 −8.2 1.3 2.3 Torino 160590 45.22N 7.65E 287 97.92 8293 −6.6 −5.0 6.0 4.1 3.1 7.1 8.8 4.4 5.1 0.0 260 0.7 70 33.1 −9.0 1.6 2.2 Venice 161050 45.50N 12.33E 6 101.25 8293 −4.9 −3.1 9.8 7.5 5.9 12.5 2.2 9.4 0.5 1.8 60 2.7 160 33.6 −7.0 2.4 2.2 WMO# = World Meteorological Organization number Elev. = elevation, m PWD = prevailing wind direction, ° True DB = dry-bulb temperature, °C Lat. = latitude ° Long. = longitude ° StdP = standard pressure at station elevation, kPa WS = wind speed, m/s Climatic Design Information 27.37 Table 3B Cooling and Dehumidification Design Conditions—World Locations Station Cooling DB/MWB Evaporation WB/MDB Dehumidification DP/MDB and HR Range of DB 0.4% 1% 2% 0.4% 1% 2% 0.4% 1% 2% DB MWB DB MWB DB MWB WB MDB WB MDB WB MDB DP HR MDB DP HR MDB DP HR MDB 1 2a 2b 2c 2d 2e 2f 3a 3b 3c 3d 3e 3f 4a 4b 4c 4d 4e 4f 4g 4h 4i 5 Narsarsuaq 18.1 9.9 16.7 9.4 15.3 8.6 11.2 15.5 10.4 14.3 9.8 13.7 9.6 7.4 10.6 8.7 7.0 10.1 7.9 6.6 9.7 6.8 GUAM Andersen AFB (Guam) 31.2 26.0 30.7 26.0 30.3 25.8 27.3 29.8 26.9 29.4 26.6 29.2 26.5 22.5 29.0 26.1 22.0 28.5 25.7 21.4 28.2 4.2 HUNGARY Budapest 32.1 20.4 30.2 19.9 28.8 19.0 21.4 30.5 20.6 29.2 19.7 26.9 18.2 13.4 21.6 17.9 13.1 23.4 17.0 12.4 23.5 12.2 Debrecen 31.2 21.7 29.5 21.1 27.9 20.1 22.4 30.1 21.5 28.6 20.6 27.2 19.7 14.6 26.5 18.8 13.8 25.5 18.0 13.1 24.2 11.3 Nagykanizsa 30.6 21.0 29.0 20.6 27.4 19.8 22.0 28.8 21.2 27.5 20.3 26.5 19.9 14.9 25.3 18.9 13.9 23.9 18.1 13.2 23.0 12.6 Pecs 31.3 21.3 29.8 20.8 28.1 20.0 22.2 29.9 21.3 28.9 20.4 27.1 19.4 14.5 26.9 18.5 13.7 26.3 17.6 12.9 24.7 10.5 Siofok 29.8 21.9 28.2 21.3 26.9 20.5 22.9 28.6 21.9 27.3 20.9 26.3 20.8 15.7 27.4 19.8 14.7 26.0 18.8 13.8 24.7 8.1 Szombathely 30.2 20.6 28.4 20.0 26.8 19.4 21.7 28.1 20.8 26.8 19.9 25.7 19.6 14.7 24.7 18.6 13.8 23.7 17.8 13.1 22.5 11.0 ICELAND Akureyri 19.0 13.4 17.4 12.4 15.9 11.5 14.1 18.1 12.8 16.6 11.8 15.4 12.0 8.8 16.0 10.8 8.1 14.7 9.7 7.5 13.3 5.3 Keflavik 14.9 11.1 13.7 10.5 12.8 10.3 12.0 13.6 11.3 12.7 10.8 12.1 11.2 8.3 12.3 10.8 8.1 11.8 10.0 7.7 11.3 4.2 Raufarhofn 15.4 11.5 13.6 10.5 12.3 9.9 12.1 14.7 11.0 13.3 10.1 11.9 10.5 7.9 13.9 9.6 7.4 12.0 8.9 7.1 10.9 4.1 Reykjavik 15.6 11.5 14.2 10.9 13.3 10.4 12.5 14.5 11.7 13.6 11.0 12.9 11.4 8.4 13.4 10.6 8.0 12.5 10.0 7.7 11.7 4.7 INDIA Ahmadabad 42.1 23.5 41.0 23.4 39.7 23.7 28.7 34.6 28.2 33.6 27.8 32.9 27.6 23.7 31.1 27.1 23.0 30.5 26.7 22.4 30.1 12.7 Bangalore 34.4 19.5 33.6 19.4 32.8 19.5 23.4 28.8 22.8 28.0 22.4 27.4 22.2 18.9 25.1 21.6 18.2 24.6 21.2 17.8 24.3 10.7 Bombay 35.0 22.8 34.0 23.3 33.2 24.0 27.7 31.6 27.4 31.3 27.1 30.9 26.7 22.3 30.2 26.4 21.9 29.9 26.1 21.5 29.6 5.2 Calcutta 37.0 25.7 35.9 26.0 35.0 26.3 29.3 34.2 28.9 33.3 28.5 32.7 28.2 24.4 32.2 27.8 23.8 31.6 27.5 23.4 31.1 10.0 Cuddalore 37.4 25.4 36.4 25.5 35.5 25.6 28.7 32.8 28.3 32.4 28.0 32.0 27.7 23.7 31.3 27.3 23.1 31.0 27.0 22.7 30.8 8.2 Goa/Panaji 33.7 25.1 33.2 25.2 32.7 25.1 28.2 31.3 27.6 31.1 27.2 30.6 27.3 23.3 30.5 26.7 22.5 29.8 26.3 21.9 29.3 5.8 Hyderabad 40.3 21.6 39.2 21.5 38.1 21.5 25.2 32.0 24.7 31.2 24.4 30.5 23.7 19.8 27.3 23.3 19.3 26.8 23.0 19.0 26.4 10.5 Jaipur 42.2 20.7 40.8 20.5 39.5 20.8 26.9 31.3 26.5 30.7 26.1 30.3 26.0 22.4 28.7 25.6 21.9 28.3 25.2 21.3 28.0 12.4 Madras 38.1 25.1 37.0 25.2 36.0 25.2 28.3 32.6 27.9 32.0 27.5 31.4 27.3 23.2 30.6 27.0 22.7 30.2 26.6 22.2 29.8 8.1 Nagpur 43.5 21.8 42.2 21.5 41.0 21.3 26.7 32.3 26.2 31.3 25.9 30.6 25.6 21.6 28.6 25.2 21.1 28.2 24.9 20.7 27.9 12.7 New Delhi 41.7 22.0 40.5 22.4 39.2 22.6 28.0 33.2 27.6 32.6 27.2 32.0 26.9 23.2 30.4 26.5 22.6 30.0 26.1 22.1 29.8 12.0 Poona 38.0 19.3 37.0 19.4 36.0 19.2 24.5 29.9 23.9 29.1 23.5 28.3 23.1 19.1 26.6 22.7 18.7 25.9 22.3 18.2 25.5 16.1 Sholapur 40.8 21.9 39.8 21.9 38.8 21.8 26.6 34.0 25.8 33.1 25.1 32.5 24.9 21.2 30.0 24.1 20.2 28.6 23.5 19.4 28.0 11.7 Trivandrum 33.5 25.6 33.0 25.5 32.5 25.4 27.2 31.3 27.0 30.9 26.7 30.6 26.2 21.8 29.2 26.0 21.5 29.0 25.7 21.1 28.8 6.5 INDIAN OCEAN ISLANDS Diego Garcia Isl 32.1 26.6 31.6 26.1 31.1 26.0 28.0 30.3 27.4 30.2 26.9 29.8 27.2 23.0 29.2 26.8 22.4 29.1 26.2 21.6 28.8 5.2 IRELAND Belmullet 21.0 16.9 19.1 16.2 17.7 15.5 17.7 20.0 16.7 18.4 15.9 17.4 16.7 11.9 18.6 15.9 11.3 17.8 15.2 10.8 16.9 4.9 Birr 24.2 17.6 22.2 17.0 20.5 16.2 18.6 22.5 17.6 21.1 16.8 19.7 17.2 12.4 20.3 16.2 11.6 18.8 15.5 11.1 18.6 8.3 Claremorris 22.7 17.8 20.9 16.8 19.2 16.0 18.4 21.6 17.4 20.1 16.5 18.6 17.1 12.3 19.6 16.3 11.7 18.6 15.5 11.1 17.9 7.7 Clones 23.3 17.6 21.4 16.8 19.9 16.0 18.3 22.2 17.4 20.6 16.5 19.2 16.9 12.2 19.7 16.0 11.5 18.8 15.2 10.9 18.3 7.6 Cork 21.7 16.8 20.2 16.2 19.0 15.7 17.7 20.3 17.0 19.2 16.3 18.1 16.9 12.3 18.4 16.1 11.7 17.8 15.5 11.2 17.1 6.7 Dublin 22.0 17.0 20.6 16.3 19.4 15.6 17.9 20.5 17.1 19.7 16.3 18.8 16.8 12.1 19.5 15.9 11.4 18.5 15.1 10.8 17.6 7.0 Kilkenny 24.3 17.7 22.5 16.8 20.9 16.3 18.6 22.5 17.7 21.2 16.9 20.0 17.2 12.4 20.2 16.3 11.7 19.1 15.5 11.1 18.3 8.8 Malin 19.3 15.9 18.1 15.5 17.0 14.8 16.7 18.5 15.9 17.6 15.2 16.7 15.8 11.3 17.5 15.1 10.8 16.8 14.5 10.3 16.2 4.2 Mullingar 23.2 17.5 21.3 16.8 19.7 16.1 18.3 21.7 17.4 20.4 16.6 19.1 17.0 12.3 19.6 16.1 11.6 18.7 15.4 11.1 18.0 8.0 Rosslare 20.0 16.6 18.9 15.9 18.0 15.4 17.3 19.1 16.6 18.2 16.0 17.5 16.5 11.8 18.3 15.9 11.3 17.5 15.2 10.8 16.8 4.9 Shannon 23.8 17.9 21.9 17.0 20.2 16.1 18.6 22.2 17.7 20.9 16.9 19.6 17.1 12.2 20.0 16.3 11.6 19.4 15.6 11.1 18.6 6.7 Valentia Observatory 21.7 17.3 20.0 16.8 18.7 16.1 18.2 20.5 17.5 19.2 16.7 18.2 17.4 12.5 18.9 16.7 11.9 18.5 16.0 11.4 17.8 5.2 ISRAEL Jerusalem 31.6 18.1 30.2 17.7 29.1 17.4 21.3 27.4 20.5 26.3 19.8 25.6 19.6 15.7 23.6 18.7 14.8 22.4 18.1 14.3 21.9 10.2 Lod 34.2 20.5 32.2 22.0 31.2 22.5 25.3 30.3 24.7 29.6 24.2 29.2 23.9 18.9 28.6 23.1 18.0 28.0 22.6 17.4 27.5 9.5 Ovda (Isr-AFB/Civ) 37.6 18.5 36.2 18.2 35.2 18.1 22.8 31.8 21.7 30.7 20.8 29.7 20.1 15.6 26.8 19.0 14.5 25.8 18.0 13.6 24.5 13.9 Tel Aviv-Yafo 31.2 20.6 30.0 23.6 29.3 23.5 25.7 29.1 25.1 28.5 24.6 28.2 24.6 19.6 28.2 24.0 18.9 27.9 23.4 18.2 27.6 5.5 ITALY Bologna/Borgo (AFB) 33.8 23.7 32.2 22.9 31.0 22.4 24.9 31.6 24.1 30.3 23.2 29.4 23.0 17.8 28.2 22.1 16.9 27.4 21.2 15.9 27.0 11.3 Brindisi 32.0 23.0 30.2 23.5 29.1 23.9 26.5 29.0 25.9 28.4 25.1 27.9 25.9 21.3 28.6 25.0 20.1 28.0 24.1 19.0 27.2 7.2 Catania 34.9 22.1 33.0 22.6 31.8 22.5 26.0 29.4 25.3 29.1 24.6 28.5 25.1 20.2 27.9 24.1 19.0 27.3 23.2 18.0 26.9 11.6 Genova 29.8 22.4 28.8 22.4 27.8 22.2 24.7 28.1 24.0 27.3 23.2 26.7 23.6 18.4 27.6 22.9 17.6 27.0 22.0 16.7 26.3 5.8 Messina 31.9 22.5 30.9 22.8 30.0 23.1 26.0 28.9 25.5 28.6 24.9 28.2 25.1 20.3 28.1 24.4 19.5 28.1 23.8 18.8 27.6 5.2 Milan, Linate 31.6 22.8 30.3 22.3 29.2 21.7 24.2 29.7 23.5 28.7 22.6 27.7 22.7 17.6 27.4 21.8 16.7 26.4 21.0 15.9 25.8 10.1 Milan, Malpensa 32.0 23.4 30.8 23.0 29.4 22.5 25.1 30.0 24.1 29.3 23.1 28.3 23.4 18.7 28.6 22.2 17.3 27.1 21.3 16.4 26.3 12.8 Naples 33.2 22.8 31.9 22.6 30.8 22.8 26.0 29.5 25.1 29.1 24.3 28.1 25.0 20.3 28.6 24.0 19.0 27.5 23.0 17.9 26.7 11.0 Palermo 33.2 21.8 31.1 22.8 30.0 23.9 26.6 29.5 26.1 28.9 25.5 28.5 25.9 21.3 29.2 25.1 20.3 28.5 24.5 19.6 27.9 5.3 Perugia 33.2 21.0 32.0 20.7 30.2 20.4 22.9 30.4 22.0 29.1 21.2 28.2 20.6 15.7 26.0 19.8 14.9 25.1 19.0 14.1 24.3 13.9 Pisa 31.9 22.4 30.4 21.8 29.2 21.5 24.5 28.8 23.7 28.1 23.0 27.7 23.1 17.9 26.7 22.2 16.9 26.2 21.4 16.1 25.5 11.8 Rome 30.8 23.3 29.8 23.2 28.9 23.4 26.1 28.6 25.4 27.9 24.6 27.2 25.2 20.3 28.1 24.5 19.5 27.3 23.8 18.7 26.8 9.9 Ronchi Legionari Ab 32.7 22.4 31.1 21.9 29.9 21.4 24.4 28.6 23.5 28.4 22.5 27.9 23.1 17.9 26.9 21.9 16.6 26.0 20.9 15.6 25.6 11.8 Torino 30.8 22.4 29.5 21.9 28.2 21.2 24.0 28.8 23.1 27.7 22.3 26.4 22.5 17.8 25.6 21.8 17.1 25.9 20.9 16.1 25.2 10.5 Venice 30.8 23.3 29.5 22.6 28.2 21.8 25.1 28.4 24.1 27.8 23.1 27.0 24.0 18.9 27.4 22.9 17.6 26.8 21.9 16.6 25.8 9.1 MDB = mean coincident dry-bulb temp., °C MWS = mean coincident wind speed, m/s HR = humidity ratio, grams of moisture per kilogram of dry air MWB = mean coincident wet-bulb temp., °C StdD = standard deviation, °C A = airport DP = dew-point temperature, °C 27.38 2001 ASHRAE Fundamentals Handbook (SI) Table 3A Heating and Wind Design Conditions—World Locations Heating Dry Bulb Extreme Wind Speed, m/s Coldest Month MWS/PWD to DB Extr. Annual Daily Elev., m StdP, kPa 0.4% 1% 99.6% 0.4% Mean DB StdD DB Station WMO# Lat.
Long.
Dates 99.6% 99% 1% 2.5% 5% WS MDB WS MDB MWS PWD MWS PWD Max.
Min.
Max. Min.
1a 1b 1c 1d 1e 1f 1g 2a 2b 3a 3b 3c 4a 4b 4c 4d 5a 5b 5c 5d 6a 6b 6c 6d JAMAICA Kingston 783970 17.93N 76.78W 9 101.22 8293 21.9 22.3 14.9 13.9 12.9 14.8 28.5 13.8 28.7 2.5 330 11.2 110 35.4 20.2 1.3 0.8 Montego Bay 783880 18.50N 77.92W 3 101.29 8293 21.3 21.9 12.8 11.5 10.4 12.6 26.6 11.5 26.8 2.3 140 9.5 70 35.2 15.9 2.0 8.4 JAPAN Aomori 475750 40.82N 140.77E 3 101.29 8293 −7.5 −6.4 9.4 8.4 7.4 9.7 −0.6 8.4 −0.4 3.7 230 4.6 220 33.1 −9.3 1.1 1.7 Asahikawa 474070 43.77N 142.37E 116 99.94 8293 −19.1 −16.4 5.6 5.1 4.4 5.4 −3.0 4.6 −3.7 0.7 80 2.9 270 32.5 −22.5 1.4 2.7 Atsugi 476790 35.45N 139.45E 65 100.55 8293 −1.9 −0.9 10.2 9.0 8.0 9.4 8.2 8.4 7.9 2.0 360 4.6 180 34.8 −4.2 1.5 1.1 Fukuoka 478080 33.58N 130.45E 12 101.18 8293 −1.0 0.0 9.2 8.2 7.3 9.4 4.2 8.6 5.0 3.4 10 5.0 10 35.3 −3.3 1.1 1.3 Hakodate 474300 41.82N 140.75E 36 100.89 8293 −10.3 −8.8 9.1 7.9 6.9 9.3 0.0 7.9 −0.9 2.3 290 3.3 220 29.9 −12.6 1.2 1.5 Hamamatsu 476810 34.75N 137.70E 48 100.75 8293 −1.2 −0.2 9.8 8.8 8.0 9.9 6.8 9.1 6.5 3.7 10 5.6 270 34.2 −3.1 1.5 1.0 Hiroshima 477650 34.40N 132.47E 53 100.69 8293 −1.3 −0.3 9.4 8.2 7.2 8.8 7.2 7.8 6.1 2.8 20 4.4 220 35.0 −2.8 1.8 1.5 Hyakuri (Jasdf) 477150 36.18N 140.42E 35 100.91 8293 −7.0 −5.2 9.7 8.3 7.2 8.5 6.3 7.4 5.5 1.1 250 4.8 130 33.2 −9.8 1.1 1.2 Kadena 479310 26.35N 127.77E 45 100.79 8293 10.1 11.2 10.5 9.2 8.2 10.0 15.9 8.9 16.4 4.3 70 5.2 240 34.6 7.3 1.1 1.4 Kagoshima 478270 31.57N 130.55E 5 101.26 8293 0.3 1.4 7.5 6.4 5.6 6.5 9.5 5.9 9.2 2.5 300 4.0 270 34.5 −1.7 1.1 1.1 Kumamoto 478190 32.82N 130.72E 39 100.86 8293 −2.1 −0.9 6.7 5.6 4.9 6.3 5.5 5.3 6.5 1.4 360 3.6 240 35.6 −4.5 1.2 1.1 Maebashi 476240 36.40N 139.07E 113 99.97 8293 −3.4 −2.2 7.9 6.8 5.9 8.1 6.1 7.2 6.4 3.3 330 3.6 110 36.6 −5.7 1.6 1.7 Maizuru 477500 35.45N 135.32E 22 101.06 8293 −2.2 −1.3 8.4 7.0 5.9 8.2 5.3 6.8 5.1 1.8 240 3.1 20 34.9 −3.9 1.0 1.5 Matsumoto 476180 36.25N 137.97E 611 94.20 8293 −8.9 −7.5 7.9 7.0 6.2 7.4 4.2 7.0 4.6 1.2 30 4.0 170 33.7 −11.6 1.4 1.4 Matsuyama 478870 33.83N 132.78E 34 100.92 8293 −0.6 0.4 6.1 5.3 4.6 6.2 5.7 5.4 6.0 2.0 110 3.4 270 34.1 −2.3 0.9 1.2 Miho (Civ/Jasdf) 477430 35.48N 133.25E 9 101.22 8293 −1.2 −0.8 10.9 9.5 8.4 11.8 1.8 10.4 2.7 6.2 260 5.9 260 34.0 −3.5 1.6 1.2 Miyako Jima Island 479270 24.78N 125.28E 41 100.83 8293 12.3 13.2 11.5 9.9 9.0 10.0 17.9 9.2 17.6 5.6 360 5.8 210 33.3 10.8 0.6 1.2 Morioka 475840 39.70N 141.17E 157 99.45 8293 −9.8 −8.0 8.5 7.4 6.6 8.2 −0.5 7.1 0.3 1.9 140 4.2 190 33.1 −12.4 1.0 1.8 Nagasaki 478170 32.73N 129.87E 35 100.91 8293 0.4 1.4 7.8 6.5 5.7 7.1 9.6 6.1 9.3 2.4 300 2.9 230 34.6 −1.1 1.0 0.9 Nagoya 476350 35.25N 136.93E 17 101.12 8293 −3.0 −1.9 9.6 8.2 7.1 9.7 7.3 8.5 6.7 1.4 350 3.9 10 35.7 −5.1 1.5 1.3 Naha 479300 26.18N 127.65E 8 101.23 8293 11.8 12.8 13.0 11.4 10.2 13.4 14.6 12.2 14.9 7.1 10 5.8 200 32.7 10.5 0.6 1.1 Naze 479090 28.38N 129.50E 7 101.24 8293 9.2 10.1 7.1 6.1 5.3 7.1 13.1 6.3 13.5 3.1 190 3.3 210 34.6 7.4 2.1 0.7 New Tokyo Intl Apt 476860 35.77N 140.38E 44 100.80 8293 −5.1 −3.9 9.7 8.4 7.3 9.3 7.6 8.1 6.1 2.1 330 4.7 10 33.7 −8.3 1.3 2.2 Niigata 476040 37.92N 139.05E 7 101.24 8293 −2.7 −1.7 10.5 9.2 8.1 12.0 3.2 10.2 2.4 5.0 200 4.5 140 35.9 −4.7 2.2 2.1 Nyutabaru (Jasdf) 478540 32.08N 131.45E 82 100.34 8293 −2.0 −0.8 9.5 8.1 7.0 9.8 7.8 8.6 8.7 3.0 270 5.5 230 34.8 −4.6 1.2 1.1 Oita 478150 33.23N 131.62E 13 101.17 8293 −1.0 0.0 7.4 6.3 5.6 7.3 6.4 6.5 6.2 2.9 170 3.9 10 34.4 −3.5 0.9 1.8 Osaka 477710 34.78N 135.45E 15 101.14 8293 −2.0 −1.0 8.4 7.4 6.6 8.0 6.8 7.0 6.8 2.1 10 4.0 10 35.8 −3.6 1.1 1.2 Owase 476630 34.07N 136.20E 27 101.00 8293 −1.0 0.1 7.6 6.3 5.4 7.7 8.0 6.6 8.2 1.3 280 4.1 70 35.2 −3.0 1.4 1.5 Sapporo 474120 43.05N 141.33E 19 101.10 8293 −11.0 −9.5 7.1 6.2 5.4 6.9 −0.2 5.7 −1.7 1.4 130 3.2 150 31.6 −13.6 1.5 2.0 Sendai 475900 38.27N 140.90E 43 100.81 8293 −4.6 −3.5 10.3 8.9 7.7 10.5 2.9 9.2 2.8 2.9 350 3.9 130 32.7 −5.9 2.0 1.3 Shimonoseki 477620 33.95N 130.93E 19 101.10 8293 0.8 1.8 10.8 9.0 7.8 9.5 6.4 8.7 6.6 4.6 330 4.1 100 32.9 −1.0 0.8 1.3 Shizuhama (Jasdf) 476580 34.82N 138.30E 10 101.20 8293 −1.0 0.0 10.8 9.7 8.7 11.3 8.5 10.4 8.2 4.0 10 5.7 260 34.9 −2.8 1.5 1.1 Tokyo, Intl Airport 476710 35.55N 139.78E 8 101.23 8293 −0.8 0.2 12.3 10.8 9.5 11.9 9.9 10.5 7.4 3.3 280 6.4 10 34.4 −2.6 1.4 1.7 Tosashimizu 478980 32.72N 133.02E 33 100.93 8293 1.2 2.5 10.3 8.6 7.4 9.3 12.9 7.7 11.1 3.4 350 3.4 250 31.3 −0.4 1.0 1.2 Wakkanai 474010 45.42N 141.68E 11 101.19 8293 −11.7 −10.3 12.2 10.4 9.2 12.8 −3.6 11.2 −3.7 4.0 190 4.5 240 27.3 −13.0 2.0 2.2 JORDAN Amman 402700 31.98N 35.98E 773 92.38 8293 0.8 1.8 10.3 8.9 7.8 12.1 6.2 9.8 6.0 2.7 90 3.6 290 38.2 −3.3 1.6 6.7 KAZAKHSTAN Almaty (Alma Ata) 368700 43.23N 76.93E 847 91.56 8293 −19.5 −16.1 4.5 3.4 2.8 3.5 −1.9 2.9 −3.4 0.6 360 1.4 330 36.4 −20.2 1.7 3.2 Aqmola (Tselinograd) 351880 51.13N 71.37E 348 97.21 8293 −29.4 −27.0 10.6 9.3 8.2 11.4 −6.7 9.8 −9.4 2.3 270 2.8 230 36.6 −32.4 3.2 3.2 Aqtobe (Aktyubinsk) 352290 50.30N 57.23E 227 98.63 8293 −28.6 −25.6 11.2 9.9 8.3 12.2 −7.1 10.3 −6.9 1.1 190 3.4 60 37.2 −32.0 2.2 2.0 Atyrau (Gur’yev) 357000 47.02N 51.85E −15 101.51 8293 −21.9 −19.3 13.4 11.8 10.0 15.7 −7.5 12.6 −5.5 1.7 90 4.4 140 39.3 −24.6 1.4 2.8 Oral (Ural’sk) 351080 51.25N 51.40E 36 100.89 8293 −27.6 −25.1 12.3 10.3 9.3 14.0 −8.4 12.3 −9.1 2.5 360 4.9 140 37.9 −31.3 2.3 2.7 Pavlodar 360030 52.28N 76.95E 123 99.86 8293 −31.1 −28.5 9.6 8.5 7.5 10.1 −6.2 9.1 −6.7 2.4 60 3.2 230 36.2 −33.0 1.7 3.6 Qaraghandy (Karaganda) 353940 49.80N 73.13E 555 94.83 8293 −28.0 −25.2 10.0 8.7 7.8 11.4 −6.9 9.9 −7.2 2.2 120 3.9 20 35.4 −29.9 1.8 3.8 Qostanay (Kustanay) 289520 53.22N 63.62E 156 99.46 8293 −29.6 −27.1 11.0 9.5 8.5 11.9 −9.8 10.2 −7.2 2.6 180 4.2 90 36.0 −32.4 1.7 2.9 Semey (Semipalatinsk) 361770 50.35N 80.25E 196 98.99 8293 −31.4 −28.8 8.9 7.4 6.4 9.8 −3.0 7.9 −5.2 0.4 90 2.8 90 37.0 −34.3 1.9 3.9 Zhambyl (Dzhambul) 383410 42.85N 71.38E 653 93.72 8293 −20.6 −17.0 12.0 9.7 7.1 12.1 2.5 9.6 2.1 0.8 180 3.9 20 38.8 −21.4 1.7 3.9 KENYA Arissa 637230 0.47S 39.63E 147 99.57 8293 21.1 21.8 13.1 11.5 10.0 13.3 28.3 12.3 28.1 3.1 180 3.6 180 40.6 14.9 4.2 4.6 Kisumu 637080 0.10S 34.75E 1146 88.29 8293 15.8 16.4 9.8 8.4 7.4 8.4 24.7 7.2 26.1 1.6 90 5.7 230 37.8 11.3 6.2 3.9 Lodwar 636120 3.12N 35.62E 515 95.29 8293 20.8 21.8 10.4 9.1 8.3 9.3 28.3 8.3 28.2 1.7 270 5.8 90 41.3 14.3 4.4 5.4 Nairobi 637400 1.32S 36.92E 1624 83.28 8293 9.5 10.8 10.4 9.3 8.4 7.8 21.2 6.6 20.5 2.6 240 6.3 60 32.1 5.2 2.5 2.7 Nakuru 637140 0.27S 36.10E 1901 80.48 8293 8.1 9.0 8.4 7.3 6.0 7.4 21.0 6.3 21.0 0.8 350 3.8 360 34.8 4.6 5.5 2.9 KOREA, NORTH (Democratic People’s Republic of Korea) Anju 470500 39.62N 125.65E 27 101.00 8293 −18.1 −15.7 8.3 7.1 6.1 7.6 −6.6 6.4 −7.4 1.2 140 2.4 230 32.4 −21.9 1.5 4.1 Ch’ongjin 470080 41.78N 129.82E 43 100.81 8293 −14.0 −12.0 6.9 5.4 4.4 6.8 −8.5 5.6 −8.6 2.9 320 1.3 90 31.0 −15.9 1.7 2.8 Changjin 470310 40.37N 127.25E 1081 89.00 8293 −28.2 −26.1 8.9 8.1 7.2 9.1−15.8 8.2 −15.7 0.5 320 2.7 320 29.6 −31.9 2.6 1.9 Haeju 470690 38.03N 125.70E 81 100.36 8293 −12.2 −10.7 9.3 8.0 6.7 8.3 −5.4 7.4 −5.1 3.2 320 3.0 180 32.9 −13.8 2.0 2.5 Hamhung 470410 39.93N 127.55E 38 100.87 8293 −13.9 −12.1 8.2 6.9 5.8 9.4 −6.8 8.0 −5.5 3.2 360 4.0 230 33.8 −16.9 1.6 2.4 Namp’o 470600 38.72N 125.37E 47 100.76 8293 −13.2 −11.5 10.2 8.6 7.4 9.4 −6.1 8.2 −6.0 3.1 320 2.8 270 32.9 −14.7 3.2 2.8 P’yongyang 470580 39.03N 125.78E 38 100.87 8293 −16.2 −14.1 6.4 5.4 4.5 6.5 −7.3 5.6 −7.3 1.1 110 1.8 270 32.9 −18.6 1.3 3.6 Sinuiju 470350 40.10N 124.38E 7 101.24 8293 −16.0 −14.1 7.8 6.6 5.8 8.6 −7.1 7.2 −7.9 2.6 50 2.4 230 33.5 −19.9 2.1 2.8 Wonsan 470550 39.18N 127.43E 36 100.89 8293 −11.0 −9.2 7.0 5.8 5.0 7.1 −3.0 6.2 −3.9 2.8 270 1.6 250 33.7 −13.7 1.6 2.6 WMO# = World Meteorological Organization number Elev. = elevation, m PWD = prevailing wind direction, ° True DB = dry-bulb temperature, °C Lat. = latitude ° Long. = longitude ° StdP = standard pressure at station elevation, kPa WS = wind speed, m/s Climatic Design Information 27.39 Table 3B Cooling and Dehumidification Design Conditions—World Locations Station Cooling DB/MWB Evaporation WB/MDB Dehumidification DP/MDB and HR Range of DB 0.4% 1% 2% 0.4% 1% 2% 0.4% 1% 2% DB MWB DB MWB DB MWB WB MDB WB MDB WB MDB DP HR MDB DP HR MDB DP HR MDB 1 2a 2b 2c 2d 2e 2f 3a 3b 3c 3d 3e 3f 4a 4b 4c 4d 4e 4f 4g 4h 4i 5 JAMAICA Kingston 33.2 25.6 32.9 25.6 32.2 25.4 27.1 31.5 26.6 31.4 26.2 30.9 26.0 21.4 29.5 25.2 20.4 29.3 25.0 20.1 29.2 6.5 Montego Bay 32.2 25.9 32.1 25.9 31.8 25.8 26.9 31.1 26.6 31.1 26.2 30.8 25.8 21.1 29.8 25.2 20.3 29.7 25.0 20.1 29.8 6.3 JAPAN Aomori 29.8 23.5 28.2 22.8 26.6 21.9 24.4 28.8 23.6 27.4 22.8 26.0 23.1 17.9 26.6 22.4 17.1 26.4 21.7 16.4 25.5 7.2 Asahikawa 29.9 22.7 28.0 21.3 26.4 20.2 23.6 28.7 22.7 27.0 21.6 25.1 22.0 16.9 27.5 21.1 16.0 25.6 20.2 15.1 24.6 8.7 Atsugi 32.2 25.0 31.1 24.6 30.0 24.2 26.0 30.7 25.5 28.9 25.0 29.0 25.0 20.2 27.8 24.2 19.3 28.1 23.9 18.9 27.4 6.5 Fukuoka 33.8 24.9 32.7 25.6 31.2 24.9 26.3 31.5 25.9 30.6 25.5 29.8 25.1 20.2 28.5 24.2 19.1 27.6 24.1 19.0 27.9 7.3 Hakodate 27.9 23.0 26.5 22.1 25.1 21.4 23.8 27.0 23.0 25.7 22.1 24.5 22.8 17.6 25.9 22.0 16.7 24.9 21.1 15.8 24.3 6.1 Hamamatsu 32.0 24.7 30.9 24.6 29.9 24.4 26.3 29.8 25.9 28.7 25.5 27.9 25.8 21.2 27.4 25.1 20.3 27.0 24.8 20.0 27.1 6.6 Hiroshima 32.6 25.3 31.6 25.1 30.6 24.8 26.3 30.9 25.8 30.3 25.4 29.5 25.1 20.3 29.0 24.6 19.7 28.3 24.2 19.2 27.9 6.5 Hyakuri (Jasdf) 31.8 25.4 30.1 25.1 28.9 24.6 26.2 30.5 25.7 29.1 25.1 28.2 25.1 20.3 28.1 24.8 19.9 27.9 24.1 19.1 27.0 7.4 Kadena 33.2 27.2 32.9 27.0 32.1 26.8 28.3 32.1 28.0 31.4 27.6 31.1 27.2 23.1 30.7 27.1 23.0 30.4 26.9 22.7 30.1 5.4 Kagoshima 32.7 25.6 32.0 25.4 31.2 25.2 26.6 30.7 26.2 30.1 25.9 29.7 25.6 20.9 28.7 25.2 20.3 28.5 24.9 20.0 28.4 6.2 Kumamoto 33.6 25.4 32.7 25.1 31.6 25.0 26.6 31.0 26.2 30.5 25.7 29.7 25.6 20.9 28.5 25.1 20.3 28.0 24.7 19.8 27.8 7.8 Maebashi 33.5 24.8 32.1 24.2 30.7 23.7 25.6 31.8 25.0 30.7 24.5 29.3 24.0 19.1 28.0 23.5 18.6 28.1 22.9 17.9 27.6 7.5 Maizuru 33.0 25.1 31.8 24.9 30.5 24.5 25.9 31.1 25.4 30.5 24.9 29.5 24.6 19.6 28.5 24.0 18.9 27.8 23.6 18.5 27.4 7.6 Matsumoto 31.9 22.4 30.5 22.0 29.1 21.4 23.3 29.6 22.7 28.6 22.1 27.7 21.7 17.6 26.0 21.0 16.9 25.4 20.4 16.2 25.1 9.4 Matsuyama 32.6 24.8 31.7 24.6 30.8 24.3 25.7 30.8 25.2 30.0 24.9 29.5 24.4 19.4 27.8 24.0 19.0 27.6 23.5 18.4 27.6 6.7 Miho (Civ/Jasdf) 32.2 25.6 31.1 25.2 30.0 24.6 26.1 30.4 25.7 29.5 25.1 28.9 25.1 20.2 28.7 24.2 19.1 27.9 24.0 18.9 27.4 6.4 Miyako Jima Island 32.1 26.8 31.6 26.7 31.1 26.5 27.7 30.7 27.3 30.4 27.1 30.0 27.0 22.8 29.5 26.6 22.3 28.9 26.2 21.7 28.8 4.7 Morioka 30.5 23.8 28.9 23.0 27.5 21.9 24.6 29.0 23.9 27.8 23.2 26.3 23.4 18.5 26.6 22.8 17.9 26.2 22.1 17.1 25.7 7.7 Nagasaki 32.1 25.2 31.1 25.3 30.2 25.2 26.7 29.6 26.2 29.1 25.8 28.8 26.0 21.4 28.8 25.5 20.8 28.1 25.0 20.2 27.7 5.6 Nagoya 33.8 25.2 32.2 24.5 31.1 24.0 26.1 31.0 25.6 30.0 25.1 29.7 25.0 20.1 28.0 24.1 19.0 27.4 23.8 18.7 27.5 7.8 Naha 32.1 26.6 31.2 26.4 31.0 26.4 27.7 30.6 27.3 30.3 27.0 30.2 27.0 22.7 29.9 26.2 21.6 29.3 26.2 21.6 29.2 3.8 Naze 32.5 26.3 31.9 26.2 31.3 26.1 27.2 31.1 26.8 30.7 26.6 30.3 26.1 21.5 29.6 25.7 21.0 29.4 25.5 20.7 29.1 5.4 New Tokyo Intl Apt 31.9 25.5 30.8 25.4 29.2 24.7 26.2 30.5 25.8 29.3 25.2 28.4 25.2 20.4 28.0 24.9 20.1 27.8 24.1 19.1 27.2 7.5 Niigata 32.3 24.8 30.9 24.3 29.7 24.0 25.7 30.6 25.1 29.7 24.5 28.9 24.3 19.3 28.5 23.7 18.5 27.8 23.1 17.9 27.6 6.0 Nyutabaru (Jasdf) 32.1 25.3 30.9 25.3 29.8 25.0 26.3 29.9 26.0 29.2 25.6 28.4 25.8 21.3 27.7 25.1 20.4 27.5 24.9 20.2 27.5 6.1 Oita 32.6 25.4 31.5 25.2 30.4 24.8 26.2 30.8 25.7 30.1 25.3 29.3 25.0 20.1 28.4 24.5 19.5 28.0 24.1 19.0 27.7 6.8 Osaka 34.0 24.8 32.9 24.7 31.8 24.0 26.2 31.3 25.7 30.4 25.3 29.8 25.0 20.1 28.3 24.2 19.2 27.3 24.0 18.9 28.0 8.0 Owase 32.2 24.5 30.7 24.4 29.6 24.3 25.8 29.9 25.3 29.3 24.9 28.7 24.7 19.8 28.1 24.2 19.2 27.6 23.7 18.6 27.1 6.1 Sapporo 29.1 22.7 27.3 21.9 25.7 20.5 23.6 28.0 22.6 26.5 21.6 25.0 22.1 16.8 26.6 21.2 15.9 25.8 20.3 15.0 24.5 6.5 Sendai 30.2 24.1 28.7 23.3 27.4 22.8 25.1 28.6 24.5 27.5 23.8 26.4 24.1 19.1 27.1 23.5 18.4 26.4 22.9 17.7 25.8 5.3 Shimonoseki 31.2 25.2 30.4 24.9 29.6 24.7 26.0 29.9 25.6 29.4 25.2 28.8 24.8 19.9 28.3 24.4 19.4 28.0 24.1 19.0 27.7 4.4 Shizuhama (Jasdf) 32.9 26.1 31.8 25.5 30.2 25.1 26.7 30.9 26.2 30.1 25.9 29.1 26.0 21.4 28.5 25.2 20.4 28.1 24.9 20.0 28.0 6.6 Tokyo, Intl Airport 32.8 25.6 31.2 25.1 30.2 24.8 26.6 31.3 26.1 30.1 25.7 29.1 25.2 20.4 28.6 25.0 20.1 28.6 24.2 19.1 27.8 6.2 Tosashimizu 29.9 25.8 29.2 25.7 28.6 25.5 26.7 29.0 26.4 28.6 26.0 28.1 26.1 21.6 28.4 25.7 21.0 28.1 25.4 20.7 27.8 3.3 Wakkanai 24.8 21.6 23.5 20.8 22.5 20.1 22.2 24.4 21.3 23.2 20.3 22.2 21.4 16.1 23.9 20.5 15.2 22.9 19.6 14.3 22.0 4.2 JORDAN Amman 34.9 18.6 33.2 18.1 31.9 17.8 21.9 28.5 20.9 28.0 20.2 27.4 20.2 16.4 24.4 19.0 15.2 23.2 18.1 14.3 22.5 11.3 KAZAKHSTAN Almaty (Alma Ata) 32.9 18.1 31.4 17.8 29.9 17.2 19.7 29.2 18.9 28.7 18.1 28.1 16.6 13.1 24.7 15.3 12.0 23.0 14.3 11.3 22.9 11.0 Aqmola (Tselinograd) 31.7 17.5 29.6 16.8 27.9 16.4 19.3 27.0 18.4 26.0 17.7 25.6 17.0 12.7 21.5 16.1 11.9 20.5 14.9 11.0 20.4 10.8 Aqtobe (Aktyubinsk) 34.1 19.4 32.0 18.6 30.0 17.7 20.5 30.5 19.7 29.2 18.9 27.6 17.5 12.9 22.9 16.7 12.2 22.2 15.7 11.5 22.3 12.8 Atyrau (Gur’yev) 36.3 20.0 34.5 19.2 32.7 18.9 22.4 30.5 21.3 29.4 20.5 29.0 20.1 14.8 25.7 18.9 13.7 24.2 17.7 12.7 24.2 11.2 Oral (Ural’sk) 33.8 19.5 31.7 18.8 29.8 18.3 21.1 29.9 20.2 28.9 19.5 27.6 18.5 13.4 23.7 17.5 12.6 23.2 16.6 11.9 22.1 12.5 Pavlodar 32.2 18.6 30.4 18.1 28.7 17.2 20.2 27.9 19.5 26.8 18.7 26.2 18.1 13.2 22.5 17.0 12.3 21.8 16.0 11.5 21.5 11.2 Qaraghandy (Karaganda) 31.5 16.1 29.4 15.9 27.7 15.4 18.2 26.1 17.4 25.4 16.7 24.8 16.0 12.2 20.0 14.8 11.2 19.5 13.7 10.5 19.1 11.3 Qostanay (Kustanay) 32.2 18.7 30.2 18.4 28.3 17.8 20.5 28.2 19.7 27.3 18.9 26.1 18.1 13.3 22.9 17.1 12.4 22.5 16.3 11.8 21.9 10.4 Semey (Semipalatinsk) 32.8 18.7 30.7 17.9 29.0 17.5 20.2 28.8 19.5 27.8 18.7 26.6 17.6 12.9 22.5 16.8 12.3 22.2 15.9 11.6 21.7 12.5 Zhambyl (Dzhambul) 35.6 17.6 33.9 17.5 32.5 17.0 19.3 31.4 18.6 31.0 17.9 29.9 15.4 11.8 22.2 14.4 11.1 22.2 13.4 10.4 22.3 13.9 KENYA Arissa 37.2 23.3 36.6 23.2 35.9 23.3 26.0 32.1 25.4 31.7 25.1 31.0 24.5 19.8 28.1 24.1 19.3 27.3 23.7 18.9 26.9 10.6 Kisumu 32.5 18.7 31.5 19.0 30.7 19.2 22.2 28.0 21.7 27.6 21.4 27.2 20.5 17.5 24.7 20.1 17.0 24.2 19.7 16.6 23.8 11.0 Lodwar 37.6 20.6 37.0 20.8 36.5 20.8 24.1 30.4 23.6 30.2 23.2 30.4 22.6 18.4 26.3 22.0 17.8 26.1 21.5 17.2 26.1 11.0 Nairobi 29.0 15.8 28.2 15.8 27.3 15.8 18.7 23.9 18.2 23.2 17.9 22.7 17.4 15.2 19.6 17.1 14.9 19.0 16.8 14.6 18.7 13.5 Nakuru 29.1 14.0 28.2 14.0 27.5 14.0 17.7 23.8 17.2 23.0 16.8 22.5 16.0 14.4 18.9 15.5 13.9 18.6 15.1 13.6 18.3 15.3 KOREA, NORTH (Democratic People’s Republic of Korea) Anju 29.9 23.8 28.6 23.2 27.6 22.6 25.2 28.3 24.5 27.1 23.8 26.4 24.4 19.4 26.9 23.7 18.6 26.2 23.0 17.8 25.7 7.5 Ch’ongjin 27.4 21.9 25.7 21.0 24.5 20.8 23.5 26.1 22.6 24.6 21.7 23.6 22.7 17.5 25.2 21.8 16.5 24.1 21.0 15.7 23.2 5.2 Changjin 25.1 18.5 23.6 17.4 22.2 16.8 20.5 23.6 19.5 21.9 18.6 21.0 19.6 16.4 21.9 18.7 15.5 20.9 17.8 14.6 20.1 9.0 Haeju 30.2 24.0 28.9 22.9 27.7 22.3 25.3 28.7 24.6 27.2 24.0 26.1 24.4 19.5 26.7 23.8 18.8 26.2 23.3 18.3 25.3 6.0 Hamhung 30.7 22.8 29.0 22.0 27.4 21.4 24.8 29.1 24.0 27.6 23.1 25.8 23.6 18.5 27.2 22.9 17.7 26.1 22.2 17.0 25.2 6.7 Namp’o 29.7 24.2 28.6 23.7 27.5 23.0 25.5 28.6 24.8 27.3 24.3 26.4 24.7 19.8 26.9 24.1 19.1 26.3 23.6 18.5 25.8 6.3 P’yongyang 30.7 23.7 29.5 23.1 28.4 22.5 25.3 29.0 24.6 27.9 24.0 26.7 24.3 19.3 27.0 23.7 18.6 26.5 23.1 17.9 26.0 7.6 Sinuiju 30.4 23.7 28.9 22.9 27.7 22.3 25.1 28.3 24.4 27.0 23.8 26.0 24.3 19.3 26.7 23.7 18.5 25.8 23.1 17.9 25.3 7.1 Wonsan 31.0 23.7 29.4 22.4 27.8 21.8 25.3 29.4 24.3 28.0 23.4 26.4 24.1 19.1 27.5 23.2 18.0 26.7 22.3 17.1 26.1 5.2 MDB = mean coincident dry-bulb temp., °C MWS = mean coincident wind speed, m/s HR = humidity ratio, grams of moisture per kilogram of dry air MWB = mean coincident wet-bulb temp., °C StdD = standard deviation, °C A = airport DP = dew-point temperature, °C 27.40 2001 ASHRAE Fundamentals Handbook (SI) Table 3A Heating and Wind Design Conditions—World Locations Heating Dry Bulb Extreme Wind Speed, m/s Coldest Month MWS/PWD to DB Extr. Annual Daily Elev., m StdP, kPa 0.4% 1% 99.6% 0.4% Mean DB StdD DB Station WMO# Lat.
Long.
Dates 99.6% 99% 1% 2.5% 5% WS MDB WS MDB MWS PWD MWS PWD Max. Min. Max. Min.
1a 1b 1c 1d 1e 1f 1g 2a 2b 3a 3b 3c 4a 4b 4c 4d 5a 5b 5c 5d 6a 6b 6c 6d KOREA, SOUTH (Republic of Korea) Cheju 471820 33.50N 126.55E 27 101.00 8293 −1.1 −0.1 12.2 10.7 9.6 12.4 3.0 11.2 4.6 6.2 40 6.3 230 33.8 −3.2 1.2 1.4 Inch’on 471120 37.48N 126.63E 70 100.49 8293 −11.2 −9.5 9.9 8.5 7.3 10.1 −4.6 8.8 −5.3 4.7 320 3.1 230 33.5 −12.8 3.8 2.6 Kangnung 471050 37.75N 128.90E 27 101.00 8293 −8.7 −6.9 8.1 7.0 6.1 8.7 −1.5 7.6 −1.5 5.2 250 2.8 90 34.8 −10.9 1.7 2.3 Kwangju 471560 35.13N 126.92E 72 100.46 8293 −7.1 −5.8 7.7 6.7 5.8 7.5 0.1 6.8 0.4 1.9 20 3.3 250 34.1 −9.2 1.5 1.8 Osan 471220 37.08N 127.03E 12 101.18 8293 −13.9 −11.8 7.7 6.4 5.4 7.3 −2.0 6.2 −1.9 1.3 10 2.6 10 34.8 −16.8 1.2 3.0 Seoul 471100 37.55N 126.80E 19 101.10 8293 −14.1 −12.1 8.6 7.5 6.5 8.3 −4.5 7.2 −3.7 1.2 10 4.2 160 33.5 −16.8 0.9 3.4 Taegu 471430 35.88N 128.62E 61 100.59 8293 −8.2 −6.7 9.1 7.9 7.0 9.8 −0.5 8.6 0.2 3.6 290 3.9 270 35.6 −11.0 1.4 1.7 Taejon 471330 36.30N 127.40E 78 100.39 8293 −11.0 −9.3 6.8 5.8 5.0 5.5 2.6 4.7 1.1 0.3 110 2.8 270 34.9 −13.5 1.9 1.5 Ulsan 471520 35.55N 129.32E 33 100.93 8293 −6.8 −5.4 7.1 6.2 5.4 8.0 −0.3 7.0 −0.5 3.0 320 3.6 140 35.0 −9.3 1.7 1.9 KUWAIT Kuwait 405820 29.22N 47.98E 55 100.67 8293 3.2 5.0 11.5 10.4 9.5 10.5 16.0 9.3 15.3 1.7 300 6.1 340 49.4 0.7 1.3 1.3 KYRGYZSTAN Bishkek (Frunze) 383530 42.85N 74.53E 635 93.93 8293 −22.4 −18.8 9.2 7.7 6.5 8.3 0.0 6.8 0.5 1.2 150 3.4 220 38.4 −24.0 1.2 4.2 Tianshan (Mtn Stn) 369820 41.92N 78.23E 3614 64.80 8293 −32.6 −30.8 9.7 8.5 7.4 9.0−15.8 7.7 −17.4 0.3 360 4.7 210 19.8 −35.6 4.3 2.2 LATVIA Liepaja 264060 56.55N 21.02E 8 101.23 8293 −17.1 −12.9 12.4 10.6 9.5 12.0 3.8 10.4 3.3 3.3 30 3.8 120 28.1 −16.0 1.6 6.1 Riga 264220 56.97N 24.07E 3 101.29 8293 −19.6 −15.5 10.8 9.2 8.2 10.3 2.6 9.2 2.1 2.0 40 4.1 150 29.5 −19.2 2.0 7.3 LIBYA Banghazi 620530 32.08N 20.27E 132 99.75 8293 6.7 7.5 13.5 12.2 10.3 13.1 12.8 10.4 13.7 2.3 90 6.6 350 41.1 3.9 1.0 1.6 Tripoli 620100 32.67N 13.15E 81 100.36 8293 4.1 5.1 10.3 9.4 8.4 9.6 15.0 8.4 14.6 1.7 240 5.6 60 45.5 1.9 1.7 1.1 LIECHTENSTEIN Vaduz 69900 47.13N 9.53E 463 95.89 8293 −11.1 −8.6 10.0 7.6 6.0 9.7 9.9 8.1 9.0 1.2 180 4.5 320 31.7 −13.1 1.1 3.7 LITHUANIA Kaunas 266290 54.88N 23.88E 75 100.43 8293 −19.9 −15.9 10.2 9.1 8.1 10.2 −0.3 9.3 0.2 2.5 70 3.7 180 29.9 −18.7 2.0 4.7 Klaipeda 265090 55.70N 21.15E 10 101.20 8293 −17.4 −13.3 13.7 11.7 10.0 12.8 4.2 10.9 3.9 3.4 70 3.7 140 28.3 −15.6 1.7 5.7 Vilnius 267300 54.63N 25.28E 156 99.46 8293 −20.4 −16.7 11.3 10.1 9.0 11.2 −1.4 10.0 −1.5 2.2 70 4.7 140 30.2 −20.6 1.6 4.3 MACEDONIA Skopje 135860 41.97N 21.65E 239 98.49 8293 −12.4 −9.3 9.0 7.7 6.2 8.3 2.2 6.7 1.1 0.4 50 2.0 270 38.0 −15.8 2.5 5.2 MADEIRA ISLANDS Funchal 85210 32.68N 16.77W 55 100.67 8293 11.9 12.8 13.5 11.9 10.4 15.0 16.3 12.8 16.5 3.6 310 4.9 30 30.7 10.0 2.9 1.0 MALAYSIA George Town 486010 5.30N 100.27E 4 101.28 8293 22.8 22.9 6.5 5.6 5.1 6.0 27.5 5.2 28.4 1.1 350 3.7 270 35.7 21.2 2.2 0.7 Kota Baharu 486150 6.17N 102.28E 5 101.26 8293 21.6 22.2 7.7 6.8 6.1 8.1 27.4 7.4 27.2 0.5 190 4.0 90 35.1 20.1 1.6 0.8 Kuala Lumpur 486470 3.12N 101.55E 22 101.06 8293 21.6 22.0 7.0 6.1 5.3 5.9 29.5 5.1 29.4 0.5 340 3.4 270 36.6 19.9 1.7 1.9 Kuantan 486570 3.78N 103.22E 16 101.13 8293 21.1 21.5 7.1 6.2 5.5 7.3 28.2 6.7 27.9 2.1 350 3.5 230 37.4 14.7 2.9 13.0 Malacca 486650 2.27N 102.25E 9 101.22 8293 22.0 22.4 7.0 6.0 5.2 7.6 29.0 6.8 28.8 1.3 10 3.5 20 36.2 18.8 1.9 3.1 Sitiawan 486200 4.22N 100.70E 8 101.23 8293 21.8 22.3 6.0 5.2 4.5 5.1 28.9 4.4 29.3 0.6 60 3.3 180 37.3 19.0 3.3 2.7 Kuching 964130 1.48N 110.33E 27 101.00 8293 21.8 22.0 5.4 4.7 4.1 5.8 28.0 5.1 27.9 0.9 260 2.2 360 37.3 19.6 2.3 4.1 Miri 964490 4.33N 113.98E 18 101.11 8293 22.4 22.8 8.0 6.7 5.7 7.9 28.0 7.0 28.4 1.1 120 3.9 270 37.0 18.9 4.6 5.7 MALI Bamako 612910 12.53N 7.95W 381 96.83 8293 15.1 16.8 8.9 7.6 6.7 8.2 25.2 7.3 25.0 3.0 40 4.0 80 43.1 9.8 3.4 3.7 MALTA Luqa 165970 35.85N 14.48E 91 100.24 8293 6.8 7.8 11.5 10.2 9.1 12.9 13.2 11.4 13.2 2.6 270 4.1 310 37.3 3.3 2.3 1.7 MARSHALL ISLANDS Kwajalein Atoll 913660 8.73N 167.73E 8 101.23 8293 24.4 24.8 11.1 10.3 9.6 12.4 27.3 11.4 27.5 5.5 70 4.9 70 34.9 15.3 3.9 13.1 MAURITANIA Nouadhibou 614150 20.93N 17.03W 3 101.29 8293 12.9 13.9 14.4 13.4 12.5 13.4 17.2 12.3 17.4 6.3 360 6.3 20 38.3 8.9 1.6 3.5 Nouakchott 614420 18.10N 15.95W 3 101.29 8293 12.8 13.9 10.4 9.5 8.5 11.8 23.7 10.5 23.9 3.8 60 6.3 80 44.8 6.7 0.7 3.7 MEXICO Acapulco 768056 16.77N 99.75W 5 101.26 8293 20.0 20.9 10.2 8.3 7.6 7.7 28.9 6.3 29.1 1.0 320 7.4 200 36.2 15.8 1.5 4.8 Merida 766440 20.98N 89.65W 10 101.20 8293 13.9 15.8 15.1 10.1 8.5 10.0 25.0 8.4 25.0 2.0 90 6.6 140 39.7 8.1 1.2 1.2 Mexico City 766790 19.43N 99.08W 2234 77.21 8293 4.0 5.4 22.6 9.8 8.0 22.8 10.9 9.8 19.1 2.1 90 4.8 360 31.3 0.0 1.2 2.3 Puerto Vallarta (766010) 766014 20.68N 105.25W 6 101.25 8293 14.8 15.6 7.9 6.2 5.4 5.5 25.9 5.3 25.8 0.2 10 7.5 330 34.5 12.4 0.9 0.8 Tampico (765491) 765494 22.28N 97.87W 24 101.04 8293 9.9 11.8 14.5 10.5 9.4 15.1 15.2 12.6 16.6 3.8 270 4.9 90 36.2 6.2 2.4 3.6 Veracruz 766910 19.20N 96.13W 14 101.16 8293 14.0 15.2 20.6 15.2 12.9 20.8 20.9 15.5 20.0 2.0 330 9.6 90 38.4 9.8 2.2 2.5 MICRONESIA Chuuk Intl/Moen Isl 913340 7.47N 151.85E 2 101.30 8293 24.0 24.4 9.2 8.2 7.4 9.4 27.1 8.5 27.6 3.9 100 3.9 40 39.0 13.3 4.4 14.3 MIDWAY ISLAND Midway Island NAF 910660 28.22N 177.37W 4 101.28 8293 14.8 15.4 10.9 9.9 9.1 13.1 19.2 11.7 19.5 4.6 360 4.2 110 31.7 7.5 1.0 11.2 MOLDOVA Chisinau (Kishinev) 338150 47.02N 28.87E 180 99.18 8293 −14.2 −12.0 6.8 5.9 5.2 7.4 −0.3 6.3 −1.9 2.1 300 2.8 200 32.9 −15.4 2.1 3.2 MONGOLIA Ulaanbataar 442920 47.93N 106.98E 1316 86.48 8293 −30.3 −28.6 10.4 9.4 7.6 8.3−17.8 6.6 −17.4 0.8 320 3.7 270 31.4 −32.6 2.9 2.6 Ulaangom 442120 49.97N 92.08E 936 90.57 8293 −40.2 −38.4 7.9 6.0 4.9 3.9−34.0 3.2 −33.8 0.6 180 2.1 50 31.8 −41.6 2.8 2.2 MOROCCO Al Hoceima 601070 35.18N 3.85W 14 101.16 8293 6.9 7.8 10.9 9.5 8.1 10.7 14.5 8.4 14.5 1.3 180 5.2 360 36.4 3.9 5.0 2.3 WMO# = World Meteorological Organization number Elev. = elevation, m PWD = prevailing wind direction, ° True DB = dry-bulb temperature, °C Lat. = latitude ° Long. = longitude ° StdP = standard pressure at station elevation, kPa WS = wind speed, m/s Climatic Design Information 27.41 Table 3B Cooling and Dehumidification Design Conditions—World Locations Station Cooling DB/MWB Evaporation WB/MDB Dehumidification DP/MDB and HR Range of DB 0.4% 1% 2% 0.4% 1% 2% 0.4% 1% 2% DB MWB DB MWB DB MWB WB MDB WB MDB WB MDB DP HR MDB DP HR MDB DP HR MDB 1 2a 2b 2c 2d 2e 2f 3a 3b 3c 3d 3e 3f 4a 4b 4c 4d 4e 4f 4g 4h 4i 5 KOREA, SOUTH (Republic of Korea) Cheju 31.8 25.8 30.2 25.9 29.2 25.7 27.5 30.0 26.8 29.3 26.2 28.8 27.0 22.8 29.1 26.1 21.6 28.4 25.8 21.2 28.2 5.4 Inch’on 30.5 24.4 29.1 23.4 27.8 22.8 25.2 28.8 24.6 27.6 24.0 26.6 24.3 19.4 27.2 23.7 18.7 26.4 23.2 18.1 26.0 5.8 Kangnung 32.5 24.1 30.7 23.3 29.2 22.3 25.3 30.4 24.7 29.4 23.9 27.9 23.9 18.8 27.9 23.2 18.0 27.6 22.6 17.4 26.7 5.8 Kwangju 32.1 25.4 30.9 24.7 29.7 24.0 26.4 30.4 25.7 29.3 25.2 28.5 25.3 20.6 28.4 24.7 19.9 27.7 24.2 19.3 27.4 7.0 Osan 32.2 25.4 31.0 24.7 29.8 24.1 26.6 31.1 25.8 29.7 25.1 28.5 25.8 21.1 29.8 24.8 19.9 28.4 24.0 18.9 27.6 8.0 Seoul 31.8 24.8 30.1 24.0 29.0 23.1 26.5 30.2 25.8 28.1 25.0 26.7 26.0 21.4 28.0 25.1 20.3 27.1 24.2 19.2 26.3 8.0 Taegu 33.6 25.3 32.0 24.3 30.7 23.5 26.3 31.6 25.6 30.6 25.0 29.3 24.9 20.1 29.3 24.2 19.3 28.3 23.6 18.6 27.9 7.3 Taejon 32.5 24.5 31.2 24.0 29.9 23.1 25.9 29.7 25.2 29.0 24.7 28.4 24.9 20.2 27.7 24.3 19.4 27.1 23.7 18.7 26.7 8.0 Ulsan 32.9 25.5 31.4 25.2 29.9 24.3 26.4 31.4 25.8 30.1 25.3 29.0 25.1 20.3 28.8 24.6 19.7 28.4 24.2 19.2 27.9 6.4 KUWAIT Kuwait 47.2 20.6 46.2 20.4 45.2 19.8 28.0 34.7 25.8 33.0 24.1 33.4 26.2 21.8 33.3 23.8 18.8 30.5 21.2 16.0 29.3 15.4 KYRGYZSTAN Bishkek (Frunze) 35.1 19.3 33.7 18.6 32.2 18.1 20.7 32.2 19.9 30.8 19.1 29.6 17.1 13.2 25.1 16.2 12.4 23.4 15.4 11.8 23.1 14.2 Tianshan (Mtn Stn) 13.9 5.5 12.3 4.8 10.8 3.9 6.7 12.1 5.6 10.5 4.8 9.2 4.5 8.2 7.6 3.4 7.6 6.6 2.5 7.1 5.9 11.6 LATVIA Liepaja 24.6 17.9 22.9 16.5 21.2 16.5 19.2 22.4 18.2 21.3 17.2 20.1 18.0 12.9 21.1 16.9 12.1 19.8 16.0 11.4 18.8 5.7 Riga 26.1 18.2 24.3 17.6 22.7 16.6 19.6 23.7 18.7 22.7 17.7 21.4 18.1 13.0 21.8 17.1 12.2 20.5 16.0 11.4 19.9 7.9 LIBYA Banghazi 37.2 22.1 35.2 21.6 33.6 21.3 25.5 31.7 24.6 30.2 24.0 29.2 24.0 19.2 28.1 23.1 18.1 27.0 22.5 17.5 26.5 9.3 Tripoli 41.4 24.3 39.6 23.6 37.7 23.0 27.0 37.2 25.7 34.2 24.7 32.5 24.7 19.9 30.9 23.6 18.6 29.1 22.7 17.6 28.3 13.8 LIECHTENSTEIN Vaduz 28.3 19.2 26.8 18.3 25.3 17.7 20.1 26.8 19.3 25.5 18.5 24.0 17.7 13.4 23.4 17.0 12.8 22.2 16.2 12.2 21.4 9.2 LITHUANIA Kaunas 26.9 19.2 25.2 18.2 23.6 17.1 20.3 25.3 19.2 23.7 18.1 22.1 18.4 13.4 23.0 17.4 12.6 21.9 16.4 11.8 20.2 9.2 Klaipeda 24.9 18.6 23.0 17.5 21.2 16.9 19.6 23.3 18.4 21.7 17.5 20.4 18.1 13.0 21.8 17.1 12.2 20.3 16.2 11.5 19.3 5.4 Vilnius 27.1 18.1 25.3 17.7 23.8 16.7 19.8 25.3 18.7 23.6 17.7 22.2 17.8 13.0 21.9 16.8 12.2 21.0 15.9 11.5 19.9 9.0 MACEDONIA Skopje 35.2 20.2 33.3 19.8 31.8 19.4 21.7 32.3 21.0 31.1 20.1 30.0 18.1 13.4 25.5 17.2 12.6 24.4 16.8 12.3 24.0 15.2 MADEIRA ISLANDS Funchal 27.1 20.3 26.1 20.3 25.2 20.1 22.1 25.4 21.5 24.6 21.0 24.4 21.0 15.8 24.2 20.2 15.0 23.8 19.8 14.6 23.6 4.7 MALAYSIA George Town 32.9 26.0 32.2 25.8 32.0 25.8 27.6 31.3 27.2 30.8 27.0 30.5 26.9 22.6 29.6 26.2 21.6 29.1 26.1 21.5 28.9 7.4 Kota Baharu 32.9 26.2 32.4 26.1 32.0 26.0 27.2 31.2 26.9 31.0 26.6 30.8 26.1 21.5 29.3 25.7 21.0 29.1 25.5 20.7 28.9 7.1 Kuala Lumpur 34.2 25.4 33.8 25.5 33.2 25.5 27.3 32.1 26.9 31.9 26.7 31.5 26.2 21.7 29.4 25.9 21.3 29.0 25.5 20.8 28.7 9.0 Kuantan 33.5 26.0 33.0 25.9 32.5 25.9 27.2 31.7 26.9 31.3 26.6 30.9 26.1 21.5 29.2 25.7 21.0 28.9 25.6 20.9 28.7 8.5 Malacca 33.5 25.3 32.9 25.4 32.4 25.4 27.2 31.1 27.0 30.9 26.7 30.6 26.2 21.6 28.9 26.0 21.4 28.8 25.7 21.0 28.6 8.5 Sitiawan 33.3 26.2 32.9 26.1 32.5 26.1 27.4 32.1 27.1 31.7 26.9 31.3 26.2 21.6 29.7 26.0 21.4 29.6 25.7 21.0 29.2 8.2 Kuching 34.0 26.0 33.2 25.8 32.9 25.8 27.3 32.0 26.8 31.6 26.5 31.3 26.1 21.6 30.0 25.8 21.2 29.4 25.2 20.4 28.5 8.8 Miri 32.2 26.3 31.8 26.3 31.4 26.2 27.6 31.0 27.2 30.6 27.0 30.4 26.6 22.2 29.8 26.2 21.7 29.3 26.1 21.5 29.1 6.6 MALI Bamako 40.0 20.3 39.2 20.3 38.3 20.3 26.2 32.6 25.7 31.9 25.4 31.2 25.0 21.0 28.6 24.2 20.0 27.9 24.0 19.8 27.7 12.3 MALTA Luqa 33.2 21.7 31.3 22.4 30.1 22.2 25.1 28.8 24.5 27.9 24.0 27.7 24.1 19.2 26.8 23.2 18.2 26.4 22.9 17.8 26.2 8.0 MARSHALL ISLANDS Kwajalein Atoll 31.4 26.1 31.2 26.0 30.9 26.0 27.2 30.6 26.9 30.4 26.7 30.2 26.0 21.4 29.8 25.7 21.0 29.5 25.6 20.9 29.3 4.2 MAURITANIA Nouadhibou 33.1 20.6 31.2 20.5 29.8 20.3 24.4 28.4 23.5 27.2 22.7 27.0 23.2 18.0 26.1 22.1 16.8 25.8 21.3 16.0 24.8 8.8 Nouakchott 41.4 21.2 39.7 20.6 37.8 20.4 27.1 31.2 26.6 30.4 26.2 29.9 26.1 21.5 29.1 25.8 21.1 28.8 25.1 20.2 28.5 12.8 MEXICO Acapulco 33.2 26.5 33.1 26.5 32.9 26.5 27.7 32.2 27.3 31.9 27.0 31.7 26.2 21.6 30.4 26.1 21.5 30.2 26.0 21.4 29.8 7.2 Merida 37.8 24.4 36.4 24.5 35.2 24.5 27.0 32.6 26.6 32.3 26.2 31.7 25.7 21.0 29.9 25.2 20.4 28.9 25.0 20.1 28.3 12.5 Mexico City 29.0 13.8 27.9 13.7 26.9 13.5 16.6 23.2 16.1 23.0 15.7 22.0 14.9 14.0 18.4 14.1 13.2 17.8 13.9 13.1 17.3 13.8 Puerto Vallarta (766010) 33.2 27.2 32.9 27.0 32.2 26.7 28.1 32.0 27.6 32.0 27.2 31.4 27.1 22.9 30.9 26.3 21.8 30.3 26.1 21.5 30.1 7.9 Tampico (765491) 33.1 26.9 32.2 26.5 32.0 26.4 28.5 31.5 27.6 31.1 27.1 30.7 27.9 24.0 30.9 26.9 22.6 30.1 26.2 21.7 29.1 6.3 Veracruz 34.2 26.6 33.2 26.7 32.8 26.6 27.7 32.8 27.2 32.1 26.8 31.7 26.2 21.7 29.7 26.1 21.5 29.6 25.8 21.1 29.4 8.3 MICRONESIA Chuuk Intl/Moen Isl 31.2 26.5 31.0 26.4 30.7 26.3 27.2 30.5 27.0 30.3 26.7 30.0 26.2 21.6 29.8 25.9 21.2 29.5 25.7 21.0 29.3 4.1 MIDWAY ISLAND Midway Island NAF 30.7 24.1 30.2 24.0 29.8 23.8 25.1 29.0 24.7 28.9 24.4 28.8 24.0 18.9 27.7 23.6 18.4 27.5 23.2 18.0 27.5 4.5 MOLDOVA Chisinau (Kishinev) 30.2 19.6 28.7 19.1 27.3 18.4 21.1 27.4 20.1 26.6 19.3 25.6 18.9 14.0 24.2 17.9 13.1 23.1 17.0 12.4 22.1 9.1 MONGOLIA Ulaanbataar 27.6 15.4 25.5 14.8 23.7 14.2 17.0 24.0 16.1 23.0 15.2 21.8 14.7 12.3 19.2 13.6 11.4 18.8 12.6 10.7 17.9 9.8 Ulaangom 27.9 16.1 26.3 15.5 24.8 14.9 17.3 25.5 16.4 24.4 15.7 23.0 14.4 11.5 20.2 13.3 10.7 20.2 12.4 10.0 19.1 10.7 MOROCCO Al Hoceima 30.5 22.8 29.1 22.5 27.9 22.5 25.0 27.8 24.4 27.1 23.9 26.6 24.2 19.1 26.9 23.6 18.5 26.3 23.0 17.8 25.6 6.2 MDB = mean coincident dry-bulb temp., °C MWS = mean coincident wind speed, m/s HR = humidity ratio, grams of moisture per kilogram of dry air MWB = mean coincident wet-bulb temp., °C StdD = standard deviation, °C A = airport DP = dew-point temperature, °C 27.42 2001 ASHRAE Fundamentals Handbook (SI) Table 3A Heating and Wind Design Conditions—World Locations Heating Dry Bulb Extreme Wind Speed, m/s Coldest Month MWS/PWD to DB Extr. Annual Daily Elev., m StdP, kPa 0.4% 1% 99.6% 0.4% Mean DB StdD DB Station WMO# Lat.
Long.
Dates 99.6% 99% 1% 2.5% 5% WS MDB WS MDB MWS PWD MWS PWD Max. Min. Max. Min.
1a 1b 1c 1d 1e 1f 1g 2a 2b 3a 3b 3c 4a 4b 4c 4d 5a 5b 5c 5d 6a 6b 6c 6d Casablanca 601550 33.57N 7.67W 62 100.58 8293 5.7 6.7 9.4 8.0 7.1 10.4 14.4 8.4 14.5 2.3 180 3.4 360 35.2 2.8 3.3 1.3 Casablanca/Nouasser 601560 33.37N 7.58W 206 98.87 8293 3.1 4.2 10.2 9.0 8.1 11.5 13.7 9.1 13.7 0.7 160 5.9 340 41.9 1.0 2.2 0.7 Midelt 601950 32.68N 4.73W 1515 84.40 8293 −1.7 −0.6 14.3 12.2 10.4 18.5 7.7 12.4 7.1 2.7 260 4.1 200 35.4 −4.1 0.7 1.2 Ouarzazate 602650 30.93N 6.90W 1140 88.36 8293 0.4 1.5 14.4 12.0 9.9 13.1 12.2 9.9 10.7 0.8 320 5.4 240 39.1 −1.9 0.6 0.9 Oujda 601150 34.78N 1.93W 470 95.80 8293 1.0 2.2 13.3 11.4 9.9 13.7 12.3 12.0 13.0 1.6 240 6.6 360 41.0 −1.5 1.5 1.2 Safi 601850 32.28N 9.23W 45 100.79 8293 5.4 6.4 9.4 8.3 7.5 9.0 14.9 7.9 14.4 3.1 60 5.2 20 40.9 3.0 2.9 1.3 Tanger 601010 35.73N 5.90W 21 101.07 8293 4.8 5.9 19.3 16.7 14.2 18.5 13.1 15.0 14.0 1.8 100 10.6 80 37.2 2.1 2.1 1.8 NETHERLANDS Amsterdam 62400 52.30N 4.77E −2 101.35 8293 −8.3 −6.0 13.8 12.1 10.7 15.5 8.4 13.7 6.7 5.0 70 4.9 70 30.0 −8.9 1.9 4.6 Beek 63800 50.92N 5.78E 116 99.94 8293 −10.0 −7.0 12.1 10.6 9.4 13.3 7.1 11.7 6.1 4.6 60 3.7 40 32.1 −10.6 2.1 4.7 De Bilt 62600 52.10N 5.18E 4 101.28 8293 −9.1 −6.5 8.7 7.6 6.8 9.5 6.6 8.4 5.7 3.1 50 3.7 70 30.7 −10.2 2.0 4.5 Eindhoven 63700 51.45N 5.42E 22 101.06 8293 −9.0 −6.2 11.0 9.5 8.5 12.2 6.8 10.3 6.7 3.3 40 4.1 50 31.8 −10.3 2.0 4.4 Gilze/Rijen 63500 51.57N 4.93E 13 101.17 8293 −9.7 −6.9 10.4 9.1 8.1 12.1 7.8 10.3 5.7 3.8 20 4.3 70 31.4 −10.5 1.8 4.1 Groningen 62800 53.13N 6.58E 4 101.28 8293 −10.1 −7.6 12.4 10.9 9.6 13.8 7.1 12.2 6.5 3.0 50 3.9 100 30.8 −11.7 1.8 4.3 Leeuwarden 62700 53.22N 5.75E 2 101.30 8293 −8.8 −6.7 13.0 11.4 10.0 14.4 5.9 12.9 6.2 3.4 80 4.7 80 29.3 −10.5 1.3 4.1 Rotterdam 63440 51.95N 4.45E −4 101.37 8293 −8.3 −5.9 13.3 11.9 10.6 14.9 6.8 13.1 6.9 4.1 50 4.4 90 30.2 −9.2 1.9 3.7 NETHERLANDS ANTILLES Willemstad 789880 12.20N 68.97W 67 100.52 8293 23.3 23.9 10.4 10.0 9.4 10.4 27.5 9.9 27.4 4.6 100 7.9 80 35.3 21.8 1.8 0.7 NEW CALEDONIA Noumea 915920 22.27S 166.45E 72 100.46 8293 16.1 16.7 12.2 10.8 9.9 11.4 20.2 10.3 20.0 2.9 60 5.3 80 34.0 14.5 1.6 1.1 NEW ZEALAND Auckland 931190 37.02S 174.80E 6 101.25 8293 1.8 2.8 13.7 12.4 11.2 14.5 11.9 12.7 11.9 4.6 240 5.9 20 29.6 1.7 7.6 1.2 Christchurch 937800 43.48S 172.55E 34 100.92 8293 −2.2 −1.2 12.0 10.4 9.4 10.9 8.7 9.4 8.7 0.6 280 7.0 300 33.2 −4.0 6.0 0.7 Taiaroa Head 938960 45.77S 170.73E 76 100.42 8293 3.1 3.7 23.1 20.5 17.8 23.2 6.8 20.6 6.5 8.3 240 7.8 320 25.3 1.6 1.8 0.8 Wellington (934340) 934360 41.33S 174.80E 7 101.24 8293 1.8 2.0 18.7 16.8 14.9 17.9 9.9 15.3 10.1 6.3 10 7.7 360 28.7 1.9 8.0 1.2 NIGER Agadez 610240 16.97N 7.98E 502 95.44 8293 10.3 11.7 14.3 12.2 10.4 15.7 21.5 14.3 21.9 3.1 100 4.8 120 45.2 4.5 2.4 4.4 Niamey 610520 13.48N 2.17E 227 98.63 8293 15.5 16.8 10.1 8.7 7.5 10.5 22.8 9.6 23.2 2.8 40 3.7 40 44.4 11.1 1.7 2.5 NORWAY Bergen 13110 60.30N 5.22E 50 100.73 8293 −9.0 −6.8 11.7 10.2 8.9 13.3 5.1 12.1 4.3 1.5 60 3.5 240 25.7 −11.1 1.5 4.0 Bodo 11520 67.27N 14.37E 13 101.17 8293 −12.8−10.8 16.9 14.7 13.1 19.8 −0.1 17.6 −2.1 8.6 80 5.0 100 24.2 −13.1 2.2 2.6 Oslo/Fornebu 14880 59.90N 10.62E 17 101.12 8293 −18.0−14.9 8.5 7.4 6.5 9.6 4.6 8.4 3.9 0.7 360 3.3 180 29.5 −18.6 3.1 4.8 Oslo/Gardermoen 13840 60.20N 11.08E 204 98.90 8293 −22.0−18.9 8.9 7.8 6.8 9.9 2.1 8.5 1.3 1.1 30 3.3 180 28.0 −23.5 2.5 5.7 Stavanger 14150 58.88N 5.63E 9 101.22 8293 −10.3 −7.9 13.3 11.7 10.4 13.6 3.3 12.1 4.0 1.5 150 5.2 320 26.1 −11.8 1.9 4.1 Svinoy (Lgt-H) 12050 62.33N 5.27E 41 100.83 8293 −2.5 −1.4 23.5 21.1 18.9 25.8 7.6 22.3 5.6 6.2 140 5.6 150 21.2 −4.6 2.3 2.7 Tromso 10250 69.68N 18.92E 10 101.20 8293 −14.2−12.5 13.5 11.9 10.3 15.3 2.2 13.5 1.5 1.1 170 3.4 180 24.1 −15.8 1.4 2.0 Trondheim 12710 63.47N 10.93E 17 101.12 8293 −18.1−14.2 12.3 10.4 8.8 14.4 2.4 12.1 3.4 3.8 120 4.4 260 28.4 −19.1 1.8 4.5 Utsira 14030 59.30N 4.88E 56 100.65 8293 −4.8 −2.8 21.6 19.4 17.5 21.7 3.8 20.1 2.7 6.1 90 5.0 120 22.2 −8.2 1.9 10.1 OMAN Masqat 412560 23.58N 58.28E 15 101.14 8293 16.1 17.0 9.0 7.8 6.8 8.2 23.4 7.2 22.8 2.1 200 5.0 340 46.6 11.0 1.3 4.6 Salalah 413160 17.03N 54.08E 20 101.08 8293 17.4 18.3 9.5 8.3 7.3 12.5 21.1 10.4 22.6 4.4 20 5.2 200 38.4 10.9 2.8 5.6 Thamarit 413140 17.67N 54.03E 445 96.09 8293 9.0 10.8 14.4 13.3 12.2 9.5 22.0 8.4 21.4 3.1 160 4.8 340 43.9 5.8 1.5 1.5 Tur’at Masirah 412880 20.67N 58.90E 19 101.10 8293 17.1 18.4 12.0 10.9 10.1 11.0 19.0 9.6 20.1 6.3 300 5.7 210 41.2 11.8 1.5 5.3 PANAMA Panama 788060 8.92N 79.60W 16 101.13 8293 22.8 22.9 7.5 6.5 5.7 6.8 27.0 5.3 27.8 0.8 10 5.2 10 37.0 19.7 2.5 5.4 Tocumen 787920 9.05N 79.37W 11 101.19 8293 19.8 20.2 7.2 6.3 5.5 5.9 26.9 5.2 27.4 0.2 300 4.3 30 35.7 14.7 1.8 6.3 PARAGUAY Asuncion 862180 25.27S 57.63W 101 100.12 8293 4.9 6.9 10.2 9.0 8.1 11.0 21.4 9.6 20.6 1.0 180 6.2 360 39.6 1.6 2.4 1.5 PERU Arequipa 847520 16.32S 71.55W 2520 74.50 8293 5.3 6.1 11.7 9.4 8.0 13.7 12.3 11.8 12.0 2.8 30 6.3 240 26.3 1.3 1.9 1.9 Cuzco 846860 13.55S 71.98W 3249 67.92 8293 −0.2 0.9 10.8 8.8 6.7 9.9 16.7 7.4 17.1 0.0 90 2.0 330 25.0 −2.5 2.0 1.4 Iquitos 843770 3.75S 73.25W 126 99.82 8293 19.0 20.1 8.6 5.9 4.7 7.3 25.0 5.2 25.5 1.1 170 1.7 330 36.8 11.8 1.8 10.1 Lima 846280 12.00S 77.12W 13 101.17 8293 13.9 14.5 10.6 9.1 8.0 9.4 16.9 8.2 17.3 1.9 170 5.9 170 30.5 9.9 1.2 3.4 Pisco 846910 13.75S 76.28W 7 101.24 8293 11.9 12.8 11.1 9.5 8.3 10.1 18.4 8.4 18.2 0.3 90 5.0 210 31.4 8.3 2.2 3.5 Talara 843900 4.57S 81.25W 90 100.25 8293 15.8 16.0 20.6 18.6 14.7 20.7 18.0 18.7 18.5 9.8 150 6.6 190 34.0 12.8 1.8 4.3 PHILIPPINES Angeles, Clark AFB 983270 15.18N 120.55E 196 98.99 8293 19.8 20.8 6.5 5.4 4.8 5.9 28.4 5.3 27.9 2.2 10 3.1 990 37.4 18.2 1.1 1.2 Baguio 983280 16.42N 120.60E 1501 84.55 8293 11.3 12.3 9.9 6.8 5.3 6.3 18.5 5.4 18.1 1.3 90 1.6 140 33.9 9.6 3.0 0.9 Cebu/Mandaue 986460 10.30N 123.97E 24 101.04 8293 22.9 23.4 8.1 6.8 6.1 8.3 27.5 7.5 27.5 2.3 40 4.0 40 36.6 19.2 1.8 5.7 Olongapo 984260 14.80N 120.27E 17 101.12 8293 21.0 21.9 9.2 8.0 7.0 9.2 29.3 8.3 28.7 1.7 70 4.7 70 37.7 19.3 2.4 1.4 Manila, Aquino Apt 984290 14.52N 121.00E 21 101.07 8293 20.4 21.5 18.2 16.1 14.1 17.8 28.4 15.9 28.5 0.7 90 9.5 90 37.0 6.2 1.0 7.9 POLAND Bialystok 122950 53.10N 23.17E 151 99.52 8293 −20.1−16.2 8.1 7.1 6.1 8.2 −2.5 7.1 −1.2 0.9 310 2.8 180 30.5 −20.5 2.0 6.2 Gdansk 121500 54.38N 18.47E 138 99.68 8293 −17.2−12.9 13.7 11.7 10.1 14.2 4.9 12.1 2.5 0.8 130 5.1 10 30.2 −18.7 2.9 6.1 Katowice 125600 50.23N 19.03E 284 97.96 8293 −15.9−12.6 8.5 7.6 6.8 9.7 4.4 8.7 3.3 1.1 20 3.4 250 31.5 −18.8 1.6 5.3 Kielce 125700 50.82N 20.70E 261 98.23 8293 −18.5−14.7 9.1 7.9 7.0 9.3 −0.5 8.3 0.7 1.9 60 3.2 190 31.1 −20.6 1.6 5.9 Kolobrzeg 121000 54.18N 15.58E 5 101.26 8293 −12.2 −8.7 8.1 7.2 6.4 8.3 3.4 7.4 2.3 2.3 110 2.5 140 32.0 −11.8 2.3 4.5 WMO# = World Meteorological Organization number Elev. = elevation, m PWD = prevailing wind direction, ° True DB = dry-bulb temperature, °C Lat. = latitude ° Long. = longitude ° StdP = standard pressure at station elevation, kPa WS = wind speed, m/s Climatic Design Information 27.43 Table 3B Cooling and Dehumidification Design Conditions—World Locations Station Cooling DB/MWB Evaporation WB/MDB Dehumidification DP/MDB and HR Range of DB 0.4% 1% 2% 0.4% 1% 2% 0.4% 1% 2% DB MWB DB MWB DB MWB WB MDB WB MDB WB MDB DP HR MDB DP HR MDB DP HR MDB 1 2a 2b 2c 2d 2e 2f 3a 3b 3c 3d 3e 3f 4a 4b 4c 4d 4e 4f 4g 4h 4i 5 Casablanca 29.6 22.0 27.3 22.0 26.0 22.0 24.0 26.7 23.3 25.9 22.8 25.2 23.1 18.0 25.3 22.6 17.4 24.8 22.0 16.8 24.3 5.1 Casablanca/Nouasser 35.5 22.0 32.9 21.4 30.5 21.3 23.7 32.1 22.7 30.1 22.0 28.7 21.2 16.3 26.6 20.4 15.5 25.0 20.0 15.1 24.8 11.0 Midelt 33.5 14.6 32.5 14.5 31.4 14.4 16.9 28.2 16.2 27.3 15.7 26.9 13.5 11.6 19.1 12.6 10.9 19.7 11.7 10.3 19.3 13.6 Ouarzazate 37.5 16.7 36.7 16.2 35.9 16.0 18.7 32.2 17.9 31.7 17.2 31.5 14.7 12.0 22.1 13.4 11.0 21.5 12.1 10.1 21.8 13.7 Oujda 36.6 20.9 34.5 20.3 32.6 20.3 23.3 32.2 22.5 30.6 21.8 29.2 21.0 16.6 26.5 20.2 15.8 25.6 19.6 15.2 24.9 13.7 Safi 34.7 21.4 31.6 21.3 29.3 21.0 23.5 30.2 22.7 28.4 22.1 27.0 21.8 16.6 25.8 21.2 15.9 24.8 20.6 15.3 24.1 8.2 Tanger 33.1 21.4 31.5 21.5 30.0 21.2 23.4 29.9 22.7 28.8 22.2 27.4 21.6 16.3 26.1 21.0 15.7 25.3 20.2 14.9 24.9 9.3 NETHERLANDS Amsterdam 26.6 19.0 24.8 18.1 23.1 17.7 20.3 24.8 19.2 23.5 18.4 22.0 18.7 13.5 22.2 17.8 12.8 20.8 17.0 12.1 19.8 8.2 Beek 28.1 19.3 26.3 18.6 24.6 17.9 20.7 26.0 19.7 24.4 18.8 23.1 18.9 13.9 23.2 18.0 13.1 21.8 17.1 12.4 20.9 9.1 De Bilt 27.7 19.0 25.9 18.5 24.0 17.6 20.4 25.9 19.3 24.1 18.3 22.8 18.3 13.2 22.8 17.5 12.5 21.3 16.7 11.9 20.6 8.9 Eindhoven 28.3 19.2 26.6 18.3 24.8 17.7 20.3 26.5 19.3 25.0 18.4 23.4 18.2 13.1 22.7 17.3 12.4 21.0 16.5 11.8 20.3 9.9 Gilze/Rijen 28.0 19.0 26.3 18.2 24.4 17.3 20.2 26.2 19.2 24.3 18.2 22.7 18.2 13.1 22.4 17.3 12.4 20.8 16.5 11.8 20.2 9.6 Groningen 27.1 19.3 25.0 18.3 23.1 17.6 20.6 25.2 19.4 23.3 18.3 21.8 18.8 13.6 23.1 17.9 12.9 21.1 17.0 12.1 20.0 9.7 Leeuwarden 25.9 18.8 23.7 17.8 21.8 17.0 19.7 24.2 18.6 22.3 17.7 20.9 18.0 12.9 21.3 17.1 12.2 20.2 16.3 11.6 19.3 7.6 Rotterdam 26.9 19.6 25.1 18.5 23.4 17.9 20.6 25.4 19.5 23.8 18.6 22.3 18.9 13.7 22.7 18.0 12.9 21.5 17.1 12.2 20.2 8.1 NETHERLANDS ANTILLES Willemstad 32.9 26.4 32.2 26.4 32.0 26.3 27.6 31.6 27.2 31.1 27.0 30.7 26.3 21.9 30.1 26.2 21.8 30.0 26.0 21.5 29.9 5.3 NEW CALEDONIA Noumea 31.1 24.7 30.2 24.5 29.3 24.1 26.0 29.7 25.5 28.8 25.0 28.0 25.1 20.4 28.1 24.6 19.8 27.3 24.1 19.2 26.9 5.2 NEW ZEALAND Auckland 25.2 19.1 24.2 19.1 23.3 18.8 21.2 23.9 20.4 22.9 19.7 22.3 20.2 14.9 22.3 19.4 14.2 21.8 18.7 13.5 21.1 6.3 Christchurch 28.1 16.9 26.1 16.2 24.2 15.5 18.5 25.1 17.6 23.6 16.8 21.6 16.5 11.8 19.4 15.7 11.2 19.3 14.9 10.6 18.4 9.7 Taiaroa Head 20.7 14.1 18.9 13.8 17.7 13.6 16.1 18.3 15.4 17.4 14.9 16.6 15.2 10.9 16.7 14.6 10.5 16.2 14.1 10.1 15.8 4.8 Wellington (934340) 23.1 17.6 21.9 17.4 20.9 16.7 19.0 21.7 18.3 20.7 17.6 19.9 18.0 12.9 20.3 17.2 12.3 19.7 16.5 11.8 19.3 5.4 NIGER Agadez 42.1 19.4 41.4 19.4 40.7 19.1 24.0 33.3 23.5 33.2 23.0 32.9 21.7 17.4 27.4 21.0 16.6 27.5 20.2 15.8 27.9 12.5 Niamey 42.1 21.6 41.2 21.5 40.3 21.2 26.6 35.1 26.1 34.4 25.7 33.8 24.7 20.3 29.3 24.2 19.7 28.9 23.9 19.3 28.9 13.2 NORWAY Bergen 22.6 14.8 20.2 13.8 18.2 12.9 15.9 19.8 15.1 17.9 14.3 17.0 14.9 10.6 16.0 14.0 10.0 15.4 13.1 9.5 14.8 6.4 Bodo 20.9 14.9 18.9 13.7 17.1 12.8 15.4 19.3 14.3 17.5 13.5 16.4 13.9 9.9 16.0 13.0 9.3 15.2 12.1 8.8 14.2 5.0 Oslo/Fornebu 26.5 17.4 24.8 16.4 22.9 15.3 18.4 24.1 17.4 21.9 16.5 20.8 16.6 11.8 19.5 15.8 11.2 18.8 14.8 10.5 18.1 8.8 Oslo/Gardermoen 25.5 15.6 23.7 14.7 21.8 13.7 16.6 22.7 15.6 20.9 14.8 19.7 14.8 10.8 17.1 13.8 10.1 16.8 12.8 9.4 15.9 10.0 Stavanger 22.9 15.2 20.9 14.7 18.9 14.1 16.8 20.6 15.7 18.9 15.1 17.5 15.5 11.0 17.5 14.7 10.5 16.8 13.9 9.9 16.0 6.3 Svinoy (Lgt-H) 17.6 13.7 16.3 13.5 15.3 13.1 14.9 16.6 14.2 15.5 13.5 14.9 14.2 10.2 15.6 13.5 9.7 15.0 12.8 9.3 14.4 2.3 Tromso 20.0 13.8 18.0 13.0 16.2 12.0 14.8 18.6 13.6 17.0 12.7 15.5 13.1 9.4 16.0 12.1 8.8 14.8 11.1 8.2 13.8 6.0 Trondheim 24.0 15.6 21.9 15.2 20.0 14.3 17.4 21.3 16.3 19.9 15.4 18.6 16.0 11.4 18.7 15.0 10.7 17.5 14.0 10.0 16.3 6.9 Utsira 19.2 14.5 17.5 14.2 16.2 13.9 15.7 17.5 15.1 16.6 14.5 15.8 15.1 10.8 16.5 14.5 10.4 15.6 13.9 10.0 14.9 2.9 OMAN Masqat 43.0 22.8 41.8 22.8 40.5 22.8 30.1 34.0 29.5 33.8 29.1 33.6 29.1 25.8 32.8 28.5 24.9 32.6 28.0 24.2 32.4 8.3 Salalah 33.4 21.9 32.7 24.2 32.0 24.7 28.0 31.1 27.6 30.6 27.2 30.4 27.1 22.9 30.1 26.8 22.5 29.9 26.2 21.7 29.5 5.4 Thamarit 42.0 20.4 41.0 20.2 39.9 20.2 26.3 34.2 25.3 32.9 24.6 32.5 24.3 20.3 29.8 23.4 19.2 28.8 22.9 18.6 28.0 14.0 Tur’at Masirah 37.2 23.6 35.7 24.2 34.3 24.9 28.7 32.3 28.0 31.7 27.5 31.1 27.9 24.0 31.0 27.1 22.9 30.1 26.7 22.3 29.8 8.6 PANAMA Panama 34.8 24.7 34.0 25.0 33.2 24.8 27.7 31.8 27.3 31.4 27.0 31.0 26.9 22.6 30.1 26.2 21.7 29.4 26.1 21.5 29.6 8.8 Tocumen 33.8 25.3 33.1 25.2 32.8 25.2 27.2 31.5 26.7 31.1 26.5 31.1 26.1 21.5 29.6 25.8 21.1 29.5 25.2 20.4 28.8 9.7 PARAGUAY Asuncion 36.5 23.9 35.2 24.1 34.2 24.1 26.6 32.9 26.1 32.3 25.7 31.6 25.1 20.5 30.0 24.2 19.4 28.6 24.1 19.2 28.5 10.3 PERU Arequipa 23.9 12.7 23.2 12.0 22.8 11.8 15.2 21.4 14.6 20.7 14.1 20.5 13.0 12.8 17.6 12.2 12.1 16.8 11.8 11.8 16.2 13.0 Cuzco 22.2 11.3 21.8 11.0 20.9 10.8 12.8 19.6 12.2 19.2 11.9 19.0 10.1 11.5 16.1 9.3 10.9 15.3 8.9 10.6 15.2 13.3 Iquitos 34.0 26.9 33.2 26.8 32.9 26.7 27.5 32.6 27.2 32.4 27.0 32.2 26.1 21.8 30.5 25.9 21.6 30.5 25.5 21.0 30.4 9.5 Lima 29.9 24.1 28.8 23.2 27.8 22.6 24.6 28.6 24.0 27.4 23.4 26.6 23.2 18.0 26.8 22.9 17.7 26.8 22.1 16.8 26.3 6.4 Pisco 29.8 24.1 28.3 22.9 27.6 22.4 24.3 28.9 23.6 28.0 22.7 26.7 22.8 17.5 28.2 22.0 16.7 27.0 21.2 15.9 26.1 6.9 Talara 32.0 24.3 31.1 24.0 30.5 23.5 26.0 30.0 25.6 28.7 25.1 28.4 25.1 20.4 28.0 24.7 19.9 27.5 24.1 19.2 27.2 7.9 PHILIPPINES Angeles, Clark AFB 36.0 25.3 34.9 25.0 34.0 25.0 28.0 31.8 27.5 31.6 27.0 30.8 27.1 23.4 30.2 26.8 23.0 30.0 26.1 22.0 29.3 9.8 Baguio 27.7 21.6 26.2 21.1 25.2 20.7 23.2 25.8 22.2 24.8 21.6 24.2 22.5 20.7 25.1 21.4 19.3 24.1 20.7 18.5 23.6 8.2 Cebu/Mandaue 33.8 27.1 33.1 27.0 32.8 26.9 27.8 32.4 27.6 32.3 27.3 31.8 26.4 21.9 30.6 26.2 21.7 30.5 26.1 21.5 30.4 6.9 Olongapo 36.4 25.0 35.7 25.1 34.9 25.3 28.1 32.7 27.6 32.0 27.1 31.9 27.1 22.9 30.9 26.2 21.7 30.0 26.1 21.5 29.8 9.5 Manila, Aquino Apt 35.0 27.0 34.1 26.5 33.4 26.3 28.4 32.8 27.9 32.3 27.5 31.9 27.2 23.0 31.5 26.8 22.5 31.1 26.2 21.7 30.4 8.8 POLAND Bialystok 27.2 19.0 25.5 18.5 23.9 17.5 20.6 25.5 19.3 23.9 18.4 22.8 18.8 13.9 23.3 17.7 12.9 21.6 16.6 12.0 20.4 10.6 Gdansk 26.8 18.6 24.8 17.4 22.9 16.5 19.5 24.9 18.3 22.8 17.2 21.6 17.8 13.0 21.0 16.2 11.7 20.1 15.2 11.0 19.3 9.7 Katowice 28.5 19.5 26.7 18.1 25.0 17.5 20.2 26.8 19.2 25.1 18.3 23.4 18.0 13.4 22.1 17.1 12.6 21.4 16.4 12.1 20.8 10.2 Kielce 28.2 19.2 26.4 18.4 24.6 17.5 20.2 26.3 19.2 24.7 18.4 23.3 18.1 13.4 22.7 17.3 12.8 21.5 16.4 12.0 20.6 11.2 Kolobrzeg 26.4 18.3 23.8 17.3 21.8 17.1 19.4 23.3 18.6 22.5 17.7 21.1 18.0 12.9 21.1 17.1 12.2 20.1 16.2 11.5 19.5 6.7 MDB = mean coincident dry-bulb temp., °C MWS = mean coincident wind speed, m/s HR = humidity ratio, grams of moisture per kilogram of dry air MWB = mean coincident wet-bulb temp., °C StdD = standard deviation, °C A = airport DP = dew-point temperature, °C 27.44 2001 ASHRAE Fundamentals Handbook (SI) Table 3A Heating and Wind Design Conditions—World Locations Heating Dry Bulb Extreme Wind Speed, m/s Coldest Month MWS/PWD to DB Extr. Annual Daily Elev., m StdP, kPa 0.4% 1% 99.6% 0.4% Mean DB StdD DB Station WMO# Lat.
Long.
Dates 99.6% 99% 1% 2.5% 5% WS MDB WS MDB MWS PWD MWS PWD Max. Min. Max. Min.
1a 1b 1c 1d 1e 1f 1g 2a 2b 3a 3b 3c 4a 4b 4c 4d 5a 5b 5c 5d 6a 6b 6c 6d Krakow 125660 50.08N 19.80E 237 98.51 8293 −18.2 −14.4 9.0 8.1 7.3 10.6 4.8 8.9 2.3 1.3 60 2.5 240 31.7 −19.6 1.8 5.2 Lodz 124650 51.73N 19.40E 188 99.09 8293 −16.8 −13.0 10.0 8.8 7.8 10.7 3.4 9.1 0.9 2.3 90 3.6 130 33.2 −17.9 2.2 6.2 Lublin 124950 51.22N 22.40E 240 98.47 8293 −18.8 −14.9 9.2 8.1 7.3 9.4 −1.7 8.4 −1.0 2.2 180 2.3 220 30.9 −18.9 1.4 5.9 Poznan 123300 52.42N 16.83E 92 100.22 8293 −15.9 −11.7 9.8 8.3 7.4 10.4 1.8 8.8 1.4 1.3 90 3.4 220 33.4 −16.3 2.1 6.2 Przemysl 126950 49.80N 22.77E 280 98.01 8293 −17.2 −13.8 10.4 9.3 8.5 12.2 1.2 10.3 0.4 2.4 270 2.7 250 30.2 −19.0 1.0 5.5 Snezka 125100 50.73N 15.73E 1613 83.39 8293 −19.5 −16.6 35.6 30.5 27.7 38.1 −6.9 35.0 −6.7 16.1 340 5.6 200 21.2 −19.6 1.9 5.3 Suwalki 121950 54.13N 22.95E 186 99.11 8293 −20.7 −16.8 11.7 9.3 8.2 11.9 2.0 9.6 −0.9 1.5 20 3.3 300 30.3 −20.7 2.2 4.9 Szczecin 122050 53.40N 14.62E 3 101.29 8293 −13.7 −10.3 9.9 8.6 7.6 10.2 5.3 9.0 3.3 2.0 40 3.9 220 32.4 −14.0 2.4 6.1 Torun 122500 53.03N 18.58E 72 100.46 8293 −17.0 −13.0 7.5 6.5 5.8 7.3 2.2 6.6 1.1 1.9 30 3.1 110 32.9 −17.6 1.6 6.4 Warsaw 123750 52.17N 20.97E 107 100.05 8293 −17.5 −13.4 10.8 9.4 8.4 11.3 −0.4 10.0 0.9 2.1 90 3.9 150 32.7 −17.8 2.1 6.2 Wroclaw 124240 51.10N 16.88E 121 99.88 8293 −16.7 −12.4 9.3 8.1 7.2 10.0 5.7 8.6 3.9 1.8 110 3.5 170 33.3 −17.9 1.9 6.0 PORTUGAL Beja 85620 38.02N 7.87W 247 98.39 8293 2.1 3.4 9.7 8.6 7.8 10.1 12.7 8.7 11.6 3.3 90 4.4 180 40.1 0.1 1.2 1.7 Braganca 85750 41.80N 6.73W 692 93.28 8293 −3.6 −2.4 10.0 8.4 7.3 10.5 3.1 8.8 5.9 1.1 180 3.5 240 36.0 −5.8 1.4 2.0 Coimbra 85490 40.20N 8.42W 140 99.65 8293 1.9 3.2 9.8 7.6 6.4 10.2 12.3 8.3 11.9 1.8 180 2.8 310 38.5 0.0 1.2 1.2 Evora 85570 38.57N 7.90W 321 97.53 8293 2.7 4.0 10.2 9.0 8.2 10.4 10.0 9.0 9.1 4.7 320 3.4 300 38.0 0.8 1.4 1.9 Faro 85540 37.02N 7.97W 4 101.28 8293 4.8 5.9 10.5 9.3 8.3 11.3 14.0 9.7 13.9 2.3 20 4.8 110 36.1 2.1 1.5 2.1 Lisbon 85360 38.78N 9.13W 123 99.86 8293 4.0 5.1 10.2 9.1 8.2 9.9 12.8 8.4 12.9 2.0 50 4.9 330 38.6 1.4 1.7 1.4 Portalegre 85710 39.28N 7.42W 590 94.44 8293 1.3 2.7 10.5 9.0 8.1 10.8 9.1 9.6 8.2 4.7 290 3.7 240 36.8 −1.0 1.5 1.8 Porto 85450 41.23N 8.68W 73 100.45 8293 1.8 2.9 10.6 9.2 8.1 11.6 12.4 10.0 12.2 3.2 90 4.0 330 35.0 −1.0 1.3 1.4 Viana Do Castelo 85430 41.70N 8.80W 18 101.11 8293 0.4 1.5 8.3 7.1 6.0 9.0 12.7 7.6 13.1 0.5 50 2.6 160 36.2 −1.5 1.3 1.0 PUERTO RICO Cieba, Roosevelt Rds 785350 18.25N 65.63W 12 101.18 8293 20.2 21.1 7.8 7.1 6.4 8.1 26.1 7.4 26.3 0.8 330 4.8 80 34.4 18.8 2.3 0.6 San Juan 785260 18.43N 66.00W 19 101.10 8293 20.3 20.8 8.3 7.6 7.1 8.4 27.0 7.7 27.0 1.3 190 5.4 170 34.7 13.6 1.2 12.4 QATAR Ad Dawhah 411700 25.25N 51.57E 10 101.20 8293 10.3 11.6 11.2 9.9 8.8 9.5 18.5 8.5 18.0 3.2 290 6.9 350 46.2 6.1 1.0 3.9 ROMANIA Bucharest 154200 44.50N 26.13E 91 100.24 8293 −13.5 −10.2 9.0 7.8 6.9 8.8 −2.3 7.9 −1.7 1.3 250 2.2 230 36.1 −16.8 2.1 3.6 Cluj-Napoca 151200 46.78N 23.57E 413 96.46 8293 −15.6 −13.5 9.0 7.5 6.1 8.4 −0.2 6.5 −1.3 0.9 270 2.6 140 32.2 −19.8 1.9 3.3 Constanta 154800 44.22N 28.63E 14 101.16 8293 −9.7 −7.1 13.8 11.8 10.3 15.8 −1.6 13.8 1.3 5.2 360 3.7 180 32.8 −11.8 3.0 3.4 Craiova 154500 44.23N 23.87E 195 99.00 8293 −12.3 −9.6 14.2 10.3 8.6 13.7 −1.3 10.1 3.1 2.2 270 1.9 180 36.3 −15.2 2.1 4.7 Galati 153100 45.50N 28.02E 72 100.46 8293 −13.7 −10.9 12.0 9.5 8.6 12.3 −4.0 10.0 −2.7 3.9 20 3.7 230 34.5 −15.6 2.0 3.0 Omul Mountain 152800 45.45N 25.45E 2509 74.60 8293 −25.0 −21.7 39.7 33.6 24.1 40.1 −13.4 39.6 −13.4 14.8 230 2.8 230 22.9 −27.5 7.6 5.8 Satu Mare 150100 47.78N 22.88E 124 99.84 8293 −17.8 −14.4 9.6 8.3 7.1 10.2 0.3 8.7 0.6 0.9 90 2.4 230 34.0 −20.2 2.2 4.3 Timisoara 152470 45.77N 21.25E 88 100.27 8293 −12.7 −9.9 8.7 7.4 6.3 8.0 0.5 6.9 1.0 1.7 360 2.5 200 36.2 −16.5 1.6 4.3 RUSSIA Abakan 298650 53.75N 91.40E 245 98.42 8293 −33.9 −31.1 10.3 9.0 7.6 9.9 −7.6 8.6 −7.9 0.3 350 2.2 50 32.9 −35.4 2.0 3.5 Aldan 310040 58.62N 125.37E 682 93.40 8293 −40.6 −38.1 6.3 5.4 4.8 6.3 −18.9 5.4 −18.3 0.7 200 2.3 180 29.8 −42.8 1.0 3.3 Aleksandrovsk-Sahal 320610 50.90N 142.17E 31 100.95 8293 −27.2 −25.1 13.9 11.9 10.0 15.8 −14.4 12.9 −7.8 3.0 130 5.5 220 26.9 −29.8 2.5 2.7 Anadyr’ 255630 64.78N 177.57E 62 100.58 8293 −38.5 −36.7 21.5 18.0 15.0 23.4 −12.0 21.0 −9.8 6.1 320 4.2 140 22.7 −39.1 3.0 2.7 Apuka 259560 60.45N 169.58E 8 101.23 8293 −27.7 −25.8 17.3 14.9 13.2 19.7 −11.0 17.6 −12.4 6.5 60 5.4 270 20.1 −31.0 2.1 2.7 Arkhangel’sk 225500 64.53N 40.47E 13 101.17 8293 −34.1 −30.3 7.5 6.5 5.8 8.2 −5.4 7.1 −5.2 0.9 130 2.8 140 29.6 −35.7 2.7 4.4 Armavir 370310 44.98N 41.12E 160 99.42 8293 −15.3 −12.5 10.2 8.3 6.8 10.5 −0.8 9.4 0.2 1.0 140 2.5 90 35.9 −20.4 2.5 3.2 Astrakhan’ 348800 46.27N 48.03E 18 101.11 8293 −18.4 −15.7 11.8 9.5 8.5 12.0 −5.7 9.9 −5.6 3.0 270 4.9 90 37.7 −21.6 2.5 5.1 Barnaul 298380 53.40N 83.70E 252 98.33 8293 −29.6 −26.7 12.9 10.4 9.1 14.1 −8.2 11.7 −7.2 3.1 170 4.1 60 33.0 −31.7 3.9 5.1 Blagoveshchensk 315100 50.25N 127.50E 137 99.69 8293 −32.8 −30.5 8.7 7.5 6.2 7.9 −15.6 6.4 −17.2 0.7 310 3.1 180 32.8 −35.2 2.1 2.8 Borzya 309650 50.38N 116.52E 684 93.37 8293 −38.1 −35.7 11.4 9.7 8.3 8.4 −18.3 7.1 −21.2 1.4 80 4.3 150 31.6 −41.0 1.8 2.7 Bratsk 303090 56.07N 101.83E 489 95.59 8293 −35.4 −32.9 9.7 7.8 6.7 8.3 −13.7 6.7 −13.9 1.3 280 3.3 130 30.3 −37.5 2.2 4.4 Bryansk 268980 53.33N 34.23E 217 98.75 8293 −22.1 −19.3 10.4 9.1 8.2 11.1 −1.5 9.3 −4.3 3.0 110 4.5 220 30.0 −24.4 1.7 3.8 Chelyabinsk 286420 55.30N 61.53E 227 98.63 8293 −28.3 −26.2 12.8 10.8 9.3 12.9 −8.6 11.2 −9.4 3.1 340 5.2 190 32.3 −30.4 2.4 3.3 Cherepovets 271130 59.12N 37.93E 131 99.76 8293 −31.5 −27.4 9.8 8.2 6.9 10.1 −3.2 9.0 −4.9 0.9 20 2.9 170 28.8 −35.3 1.7 5.1 Chita 307580 52.02N 113.33E 685 93.36 8293 −35.9 −33.7 10.8 9.5 8.3 9.4 −12.3 7.7 −13.0 0.0 310 3.5 210 32.2 −37.9 1.9 1.9 Dudinka 230740 69.40N 86.17E 19 101.10 8293 −45.4 −42.5 14.4 12.1 10.3 12.9 −18.1 10.6 −18.8 2.5 100 4.4 60 28.0 −48.2 2.7 2.5 Egvekinot 253780 66.35N 179.12W 26 101.01 8293 −36.6 −33.7 15.5 13.4 11.6 18.5 −17.0 16.2 −17.4 0.7 190 3.6 160 24.1 −37.9 5.2 3.7 Groznyy 372350 43.35N 45.68E 162 99.39 8293 −14.8 −12.4 10.5 9.2 8.0 11.2 −2.3 9.6 −1.9 1.3 270 4.2 90 35.6 −19.1 2.3 2.6 Habarovsk/Novy 317350 48.52N 135.17E 72 100.46 8293 −29.8 −28.0 10.2 8.9 8.1 9.4 −16.8 8.4 −18.1 1.7 200 3.8 250 32.1 −32.5 1.7 1.6 Irkutsk 307100 52.27N 104.35E 513 95.31 8293 −33.7 −31.1 10.4 9.3 8.1 9.5 −16.7 8.1 −16.6 1.7 80 3.6 190 30.5 −37.4 1.5 3.8 Izhevsk 284110 56.82N 53.27E 158 99.44 8293 −29.7 −26.6 10.5 9.3 8.0 11.4 −7.9 10.1 −7.2 2.2 100 4.8 160 31.4 −32.8 2.6 2.4 Juzno-Kurilsk 321650 44.02N 145.87E 40 100.85 8293 −12.1 −10.6 15.3 13.1 11.3 16.1 −6.1 14.2 −5.7 6.0 320 3.5 320 24.2 −13.0 2.6 2.7 Juzno-Sahalinsk 321500 46.92N 142.73E 31 100.95 8293 −24.0 −22.2 8.4 7.3 6.3 9.4 −11.4 8.1 −10.8 1.5 360 3.3 180 28.6 −26.9 1.4 2.3 Kaliningrad 267020 54.70N 20.62E 27 101.00 8293 −19.2 −14.4 7.2 6.2 5.4 7.3 2.1 6.3 1.1 1.8 350 2.4 120 30.9 −17.0 2.1 5.4 Kaluga 277030 54.57N 36.37E 201 98.93 8293 −24.5 −21.6 9.3 7.9 7.0 10.3 −6.8 8.2 −5.9 1.5 340 3.8 130 29.2 −30.5 1.4 9.8 Kazan’ 275950 55.78N 49.18E 116 99.94 8293 −27.6 −24.7 12.3 10.5 9.3 13.4 −10.1 12.0 −10.2 4.2 330 4.4 170 31.3 −30.4 1.9 3.0 Kirov 271960 58.65N 49.62E 147 99.57 8293 −32.7 −27.9 9.9 8.7 7.8 10.1 −5.7 9.1 −7.6 3.0 250 3.9 90 30.5 −35.1 1.9 3.8 Kolpashevo 292310 58.30N 82.88E 76 100.42 8293 −38.5 −34.5 8.8 7.4 6.6 8.7 −10.0 7.3 −10.8 1.4 290 3.2 160 30.2 −40.3 1.7 4.6 Krasnodar 349290 45.03N 39.15E 33 100.93 8293 −15.9 −13.0 10.3 9.1 7.9 11.3 5.2 9.5 −1.1 3.0 50 3.6 90 34.7 −20.0 1.3 4.6 Krasnoyarsk 295740 56.00N 92.88E 277 98.04 8293 −33.8 −31.2 10.2 8.5 7.1 11.8 −10.0 9.7 −7.9 1.5 290 2.9 50 32.3 −35.2 3.5 3.9 WMO# = World Meteorological Organization number Elev. = elevation, m PWD = prevailing wind direction, ° True DB = dry-bulb temperature, °C Lat. = latitude ° Long. = longitude ° StdP = standard pressure at station elevation, kPa WS = wind speed, m/s Climatic Design Information 27.45 Table 3B Cooling and Dehumidification Design Conditions—World Locations Station Cooling DB/MWB Evaporation WB/MDB Dehumidification DP/MDB and HR Range of DB 0.4% 1% 2% 0.4% 1% 2% 0.4% 1% 2% DB MWB DB MWB DB MWB WB MDB WB MDB WB MDB DP HR MDB DP HR MDB DP HR MDB 1 2a 2b 2c 2d 2e 2f 3a 3b 3c 3d 3e 3f 4a 4b 4c 4d 4e 4f 4g 4h 4i 5 Krakow 29.2 20.4 27.2 19.3 25.2 18.2 21.2 27.9 20.1 26.0 19.2 24.4 18.9 14.1 24.2 18.0 13.3 22.8 17.1 12.6 21.6 10.9 Lodz 28.7 19.0 26.7 18.2 25.1 17.4 20.1 26.7 19.1 25.2 18.3 23.6 17.9 13.1 22.1 16.9 12.3 21.8 16.1 11.7 20.9 10.4 Lublin 27.6 19.6 25.9 18.9 24.2 17.8 20.5 26.5 19.5 24.8 18.5 23.2 18.4 13.7 23.5 17.5 12.9 22.0 16.7 12.2 21.1 10.0 Poznan 29.2 18.8 27.2 18.0 25.7 17.4 20.2 26.9 19.2 25.5 18.2 23.7 18.0 13.1 22.1 17.0 12.3 21.6 16.0 11.5 20.3 10.9 Przemysl 27.5 19.6 25.9 18.8 24.4 18.0 20.7 26.0 19.7 24.7 18.7 23.2 18.7 14.0 23.6 17.8 13.2 22.0 17.0 12.5 21.2 8.3 Snezka 17.5 12.9 15.7 12.1 14.3 11.3 13.9 16.3 12.7 14.7 11.8 13.9 12.9 11.3 14.7 11.8 10.5 13.7 10.9 9.9 12.8 4.4 Suwalki 26.8 18.7 24.9 18.2 23.3 17.2 20.1 25.1 19.1 23.6 17.9 22.3 18.3 13.5 23.0 17.2 12.6 21.3 16.2 11.8 20.2 10.3 Szczecin 28.5 19.5 26.6 18.8 24.8 18.0 20.8 26.5 19.7 25.0 18.7 23.6 18.9 13.7 23.6 17.8 12.8 22.2 16.8 12.0 21.1 9.4 Torun 28.8 19.3 26.9 18.2 25.2 17.5 20.3 26.4 19.3 25.2 18.4 23.9 18.2 13.2 22.0 17.1 12.3 21.4 16.1 11.5 20.9 10.2 Warsaw 29.0 19.7 27.0 19.0 25.2 17.9 21.0 27.6 19.9 25.3 18.9 24.3 18.9 13.9 24.2 17.9 13.0 22.5 17.0 12.3 21.4 11.0 Wroclaw 29.0 19.5 27.2 18.8 25.5 17.9 20.6 27.2 19.6 25.6 18.7 24.2 18.3 13.4 22.9 17.3 12.5 21.8 16.5 11.9 21.5 10.6 PORTUGAL Beja 37.0 20.9 35.1 20.2 33.3 19.6 21.6 34.6 21.0 33.1 20.3 31.4 18.1 13.4 23.2 17.2 12.7 23.1 16.6 12.2 22.7 16.4 Braganca 33.3 18.4 31.3 17.9 29.6 17.4 19.5 31.1 18.7 29.6 18.0 28.1 15.8 12.2 22.2 15.0 11.6 21.1 14.3 11.1 20.8 13.5 Coimbra 33.9 21.2 31.5 20.6 29.3 20.0 22.3 31.3 21.4 29.7 20.6 28.2 19.3 14.3 26.3 18.6 13.7 24.4 18.0 13.2 23.3 11.9 Evora 35.7 19.9 33.7 19.1 31.8 18.7 20.7 32.5 20.1 30.9 19.4 29.7 17.8 13.3 21.3 17.1 12.7 21.6 16.4 12.1 20.8 13.1 Faro 31.9 20.3 30.1 20.2 29.0 20.3 22.9 27.6 22.2 26.8 21.6 26.4 21.4 16.1 25.1 20.8 15.5 24.8 20.0 14.7 24.2 9.5 Lisbon 34.1 20.7 32.0 20.2 29.9 19.8 22.7 30.8 21.7 28.4 20.9 27.4 20.2 15.1 24.4 19.8 14.7 24.5 18.9 13.9 23.7 10.5 Portalegre 34.6 19.1 32.7 18.5 31.0 17.8 19.9 31.7 19.3 30.5 18.7 29.4 16.7 12.8 20.8 15.9 12.1 20.8 15.2 11.6 20.4 10.8 Porto 30.1 19.4 28.0 19.1 25.9 18.3 20.8 27.2 20.1 25.6 19.4 23.8 19.1 14.0 22.0 18.3 13.3 20.9 18.0 13.1 20.6 9.6 Viana Do Castelo 32.0 21.3 30.0 20.5 27.9 19.7 22.0 30.4 21.2 28.4 20.4 26.5 19.5 14.3 24.6 18.9 13.7 23.1 18.3 13.2 22.3 10.4 PUERTO RICO Cieba, Roosevelt Rds 32.2 25.4 31.9 25.3 31.2 25.0 26.9 30.5 26.5 30.4 26.1 30.1 26.1 21.5 29.4 25.2 20.4 28.9 25.0 20.1 28.9 5.6 San Juan 33.2 25.0 32.2 25.4 31.7 25.4 27.0 30.9 26.6 30.5 26.3 30.3 25.7 21.0 29.3 25.5 20.8 29.1 25.2 20.4 28.8 6.8 QATAR Ad Dawhah 43.0 21.9 41.9 22.1 40.8 22.3 30.5 34.7 29.9 34.1 29.4 33.8 29.4 26.3 33.2 29.0 25.7 33.0 28.2 24.4 33.1 10.8 ROMANIA Bucharest 33.0 22.0 31.2 21.2 29.9 20.7 23.6 30.5 22.6 29.4 21.7 28.3 21.9 16.8 25.7 20.8 15.6 24.8 19.9 14.8 23.7 13.3 Cluj-Napoca 29.2 20.2 27.5 19.5 26.0 18.6 21.4 26.8 20.3 25.5 19.4 24.5 19.8 15.3 24.1 18.6 14.1 22.6 17.7 13.3 21.5 11.4 Constanta 28.5 22.0 27.3 22.0 26.2 21.4 24.0 26.4 23.1 25.9 22.3 25.3 23.2 18.0 25.6 22.2 16.9 25.0 21.4 16.1 24.2 6.8 Craiova 33.2 23.5 31.5 22.6 29.9 21.8 24.8 31.4 23.6 29.9 22.6 28.5 22.9 18.1 28.6 21.7 16.8 26.8 20.7 15.7 25.9 12.2 Galati 31.6 22.0 30.1 21.3 28.7 20.7 23.4 29.1 22.4 28.3 21.5 26.7 21.7 16.5 26.6 20.6 15.4 24.9 19.7 14.5 24.0 11.2 Omul Mountain 14.2 10.1 12.5 9.1 11.1 8.3 11.2 13.3 10.0 11.7 9.0 10.5 10.2 10.6 12.0 9.2 9.9 11.0 8.2 9.2 9.7 6.2 Satu Mare 31.2 21.4 29.5 20.8 27.9 20.0 22.5 29.6 21.5 28.2 20.6 26.7 20.2 15.1 25.6 19.3 14.3 24.6 18.4 13.5 23.4 12.9 Timisoara 33.1 21.0 31.2 20.3 29.5 19.7 21.9 30.4 21.1 29.2 20.4 27.9 19.4 14.3 23.8 18.7 13.7 23.2 17.9 13.0 22.7 12.8 RUSSIA Abakan 29.6 18.0 27.7 17.4 25.8 16.7 19.7 27.1 18.8 25.1 18.0 23.9 17.4 12.8 21.6 16.5 12.1 21.0 15.6 11.4 20.4 10.5 Aldan 27.2 16.3 25.2 15.5 23.3 14.9 17.6 24.9 16.7 23.1 15.8 21.5 15.3 11.8 19.7 14.3 11.0 18.9 13.3 10.3 18.6 10.2 Aleksandrovsk-Sahal 23.2 17.7 21.6 17.2 20.2 16.3 19.2 22.0 18.1 20.5 17.1 19.3 18.1 13.1 20.8 17.0 12.2 19.5 16.1 11.5 18.4 6.3 Anadyr’ 18.5 13.6 16.6 12.7 14.9 11.4 14.1 17.6 13.0 16.0 11.8 14.4 12.2 8.9 15.8 11.2 8.3 14.7 10.2 7.8 13.3 5.4 Apuka 16.0 12.6 14.6 11.9 13.4 11.3 13.3 15.3 12.4 14.0 11.6 13.0 12.2 8.9 13.9 11.5 8.5 13.3 10.7 8.0 12.5 4.7 Arkhangel’sk 26.0 18.5 24.1 17.4 22.2 16.7 19.6 24.2 18.5 23.0 17.3 21.1 17.9 12.9 22.3 16.7 11.9 20.5 15.5 11.0 19.6 9.4 Armavir 32.3 20.8 30.5 20.3 29.0 19.8 22.8 28.6 21.8 27.8 21.0 27.0 20.9 15.9 26.3 19.8 14.8 24.8 18.9 14.0 23.8 12.1 Astrakhan’ 34.3 21.4 32.7 20.8 31.3 20.2 23.4 30.4 22.3 29.2 21.5 28.5 21.4 16.1 26.1 20.3 15.0 25.4 19.3 14.1 24.5 10.8 Barnaul 29.2 18.8 27.4 17.9 25.8 17.4 20.2 26.8 19.3 25.3 18.4 24.1 18.0 13.3 22.7 17.1 12.6 22.0 16.2 11.9 21.0 9.4 Blagoveshchensk 29.9 20.8 28.3 20.3 26.7 19.7 23.0 27.2 22.0 25.7 21.1 24.9 21.7 16.6 25.1 20.7 15.6 24.2 19.8 14.8 23.2 9.2 Borzya 28.3 17.0 26.5 16.8 24.9 16.5 19.7 24.7 18.7 23.8 17.8 22.7 17.8 13.9 22.4 16.9 13.1 21.3 16.0 12.4 20.3 10.6 Bratsk 27.1 17.3 25.2 16.4 23.6 16.1 18.9 24.3 17.9 23.3 17.0 21.9 16.9 12.8 21.1 15.9 12.0 20.4 14.9 11.2 19.7 9.2 Bryansk 27.1 18.9 25.4 18.2 23.9 17.6 20.2 24.7 19.3 23.9 18.4 23.0 18.6 13.8 22.4 17.6 12.9 21.6 16.7 12.2 20.4 8.1 Chelyabinsk 29.4 19.1 27.6 18.5 25.9 17.7 20.4 26.9 19.5 25.7 18.7 24.2 18.2 13.5 23.1 17.4 12.8 22.0 16.5 12.1 21.3 9.2 Cherepovets 26.2 19.4 24.6 18.3 23.0 17.2 20.5 25.0 19.1 23.1 18.1 22.0 18.9 13.9 23.1 17.7 12.9 21.2 16.5 11.9 20.0 10.1 Chita 28.8 18.5 26.8 17.3 25.0 16.3 20.0 26.2 18.7 24.6 17.8 22.9 17.9 14.0 23.1 16.8 13.0 21.2 15.8 12.2 19.8 11.4 Dudinka 24.7 16.5 22.2 15.6 19.7 14.2 17.7 22.6 16.2 20.9 14.8 19.3 15.5 11.0 20.6 13.9 9.9 19.1 12.6 9.1 17.0 8.0 Egvekinot 18.0 11.9 15.8 11.0 14.1 10.1 12.8 16.4 11.5 15.3 10.6 13.4 10.7 8.0 13.9 9.8 7.6 12.2 9.0 7.2 11.2 5.1 Groznyy 32.9 21.4 31.3 20.7 29.8 20.1 22.7 30.6 21.9 28.9 21.1 27.7 20.4 15.4 25.9 19.6 14.6 25.0 19.0 14.1 24.1 10.3 Habarovsk/Novy 30.1 21.2 28.3 20.8 26.7 19.9 22.9 27.2 22.0 26.2 21.1 25.4 21.6 16.4 24.7 20.7 15.5 24.2 19.7 14.5 23.0 9.0 Irkutsk 27.0 17.3 25.3 16.8 23.6 16.1 19.0 24.3 18.1 23.6 17.2 22.0 17.2 13.1 21.1 16.2 12.3 19.9 15.2 11.5 19.2 11.1 Izhevsk 29.3 19.0 27.3 18.5 25.5 17.5 20.6 26.8 19.6 25.5 18.6 23.8 18.4 13.5 23.6 17.4 12.7 22.1 16.6 12.0 21.3 9.6 Juzno-Kurilsk 20.2 18.1 19.1 17.5 18.1 17.0 18.9 19.6 18.1 18.8 17.3 17.9 18.6 13.5 19.3 17.8 12.8 18.4 17.1 12.3 17.7 3.1 Juzno-Sahalinsk 25.3 20.0 23.7 19.2 22.0 18.2 21.3 23.9 20.1 22.3 18.9 21.3 20.4 15.1 22.9 19.3 14.1 21.5 18.0 13.0 20.4 7.8 Kaliningrad 26.9 18.9 25.0 17.7 23.3 16.8 19.9 25.4 18.7 23.4 17.7 22.0 18.0 13.0 22.4 16.9 12.1 21.2 15.9 11.3 20.0 8.5 Kaluga 26.8 19.4 25.2 18.7 23.6 17.4 20.4 25.3 19.4 23.8 18.4 22.6 18.6 13.8 23.6 17.7 13.0 22.1 16.7 12.2 20.6 9.2 Kazan’ 29.3 19.2 27.3 18.8 25.5 17.9 21.0 27.0 19.9 25.0 19.0 24.1 19.0 14.0 23.8 17.9 13.0 22.9 16.9 12.2 21.6 9.2 Kirov 28.0 19.7 25.9 18.4 24.0 17.6 20.8 26.3 19.6 24.5 18.5 22.8 18.8 13.9 24.0 17.7 12.9 22.2 16.7 12.1 21.0 9.6 Kolpashevo 27.9 18.9 26.3 18.1 24.5 17.2 20.7 25.7 19.5 24.3 18.5 22.9 18.9 13.8 23.0 17.8 12.9 22.0 16.7 12.0 20.8 10.0 Krasnodar 32.0 21.7 30.4 21.1 29.0 20.4 23.2 29.4 22.3 28.1 21.5 27.3 21.2 15.9 26.2 20.4 15.1 25.4 19.5 14.3 24.8 11.7 Krasnoyarsk 28.4 18.1 26.6 17.3 24.8 16.6 19.9 26.0 18.8 24.3 17.9 22.9 17.9 13.3 22.1 16.9 12.5 21.1 15.9 11.7 20.3 10.6 MDB = mean coincident dry-bulb temp., °C MWS = mean coincident wind speed, m/s HR = humidity ratio, grams of moisture per kilogram of dry air MWB = mean coincident wet-bulb temp., °C StdD = standard deviation, °C A = airport DP = dew-point temperature, °C 27.46 2001 ASHRAE Fundamentals Handbook (SI) Table 3A Heating and Wind Design Conditions—World Locations Heating Dry Bulb Extreme Wind Speed, m/s Coldest Month MWS/PWD to DB Extr. Annual Daily Elev., m StdP, kPa 0.4% 1% 99.6% 0.4% Mean DB StdD DB Station WMO# Lat.
Long.
Dates 99.6% 99% 1% 2.5% 5% WS MDB WS MDB MWS PWD MWS PWD Max. Min. Max. Min.
1a 1b 1c 1d 1e 1f 1g 2a 2b 3a 3b 3c 4a 4b 4c 4d 5a 5b 5c 5d 6a 6b 6c 6d Kurgan 286610 55.47N 65.40E 79 100.38 8293 −31.6 −28.6 12.5 10.1 8.4 13.4 −7.0 12.1 −6.7 2.4 30 4.3 200 33.6 −34.8 2.1 4.2 Kursk 340090 51.73N 36.27E 210 98.83 8293 −22.7 −19.6 12.0 9.8 8.4 12.9 −7.5 11.3 −4.9 2.3 360 4.2 90 31.2 −25.4 1.3 5.6 Kyakhta 309250 50.37N 106.45E 801 92.07 8293 −29.4 −27.9 8.3 6.4 5.3 5.3 −14.4 4.1 −14.3 0.1 150 2.5 180 32.6 −31.8 1.8 1.9 Magadan 259130 59.58N 150.78E 118 99.92 8293 −28.7 −26.0 10.0 8.8 7.7 10.5 −7.7 9.2 −9.2 0.6 50 3.9 270 23.0 −29.7 3.3 4.5 Magnitogorsk 288380 53.35N 59.08E 382 96.82 8293 −28.5 −25.9 10.3 9.4 7.7 11.0 −14.5 10.1 −12.2 0.6 30 4.2 170 33.9 −31.4 3.0 2.0 Markovo 255510 64.68N 170.42E 33 100.93 8293 −47.5 −45.2 8.0 6.5 5.5 8.2 −12.0 6.7 −13.4 0.2 200 2.2 180 27.4 −50.3 1.6 3.5 Moscow 276120 55.75N 37.63E 156 99.46 8293 −23.1 −20.1 7.8 6.4 5.6 8.3 −3.5 7.3 −5.9 1.5 20 2.0 210 30.1 −25.8 1.8 3.0 Moscow, Vnukovo 275185 55.65N 37.27E 203 98.91 8293 −24.0 −21.1 10.3 9.2 8.2 12.2 −2.9 10.3 −2.7 2.6 10 4.1 240 29.8 −26.7 1.9 3.1 Murmansk 221130 68.97N 33.05E 51 100.71 8293 −28.7 −24.4 12.1 10.7 9.6 13.3 −2.7 11.9 −3.1 4.6 210 4.3 150 26.9 −28.4 2.0 5.7 Nikolayevsk 313690 53.15N 140.70E 68 100.51 8293 −33.1 −30.8 9.8 8.5 7.8 11.9 −8.1 8.4 −16.8 1.6 290 4.7 120 29.3 −35.2 2.1 2.5 Nikolskoe/Beringa 326180 55.20N 165.98E 6 101.25 8293 −11.0 −9.4 17.8 15.4 13.7 19.2 −3.0 17.3 −2.3 6.1 360 4.6 170 17.1 −12.6 2.5 2.2 Nizhniy Novgorod 275530 56.22N 43.82E 82 100.34 8293 −27.2 −23.8 9.3 8.3 7.3 9.9 −7.8 8.6 −4.5 1.8 350 4.1 150 31.6 −30.4 1.8 3.2 Nizhniy Tagil 282400 57.88N 60.07E 258 98.26 8293 −32.2 −28.9 7.4 6.7 5.9 8.3 −16.1 7.0 −17.4 2.4 20 3.6 250 30.9 −34.6 1.9 4.2 Novokuznetsk 298460 53.73N 87.18E 308 97.68 8293 −30.8 −27.4 13.0 10.5 9.2 14.7 −6.8 12.4 −8.0 1.8 340 3.1 150 31.3 −32.2 1.8 4.3 Novosibirsk 296340 55.03N 82.90E 177 99.22 8293 −31.7 −28.4 11.9 10.1 8.7 12.4 −7.1 11.1 −7.7 3.0 220 4.5 180 31.5 −33.7 2.5 4.8 Nyurba 246390 63.28N 118.33E 129 99.78 8293 −52.9 −50.1 7.6 6.4 5.4 6.0 −22.7 5.3 −25.2 0.8 20 2.9 20 33.2 −54.2 5.0 3.3 Olekminsk 249440 60.40N 120.42E 226 98.64 8293 −47.7 −44.7 7.4 6.3 5.4 6.6 −19.3 5.4 −18.8 1.0 90 2.8 90 32.9 −49.0 2.6 3.7 Omsk 286980 54.93N 73.40E 123 99.86 8293 −31.3 −27.9 11.2 9.4 8.2 11.7 −8.5 9.8 −8.8 3.0 210 4.5 120 33.7 −33.4 2.4 3.7 Orel 279060 53.00N 36.03E 203 98.91 8293 −23.5 −19.9 11.8 10.0 8.8 12.5 −5.1 10.3 −4.0 3.0 360 4.5 280 30.7 −25.6 1.5 4.1 Orenburg 351210 51.78N 55.22E 109 100.02 8293 −27.7 −24.8 12.1 10.4 9.2 13.1 −7.4 11.6 −7.0 2.2 260 4.8 160 36.4 −31.1 2.4 2.5 Ozernaja 325940 51.48N 156.48E 29 100.98 8293 −15.3 −13.7 19.7 16.1 13.7 20.3 −2.1 17.1 −3.2 5.0 80 7.1 90 19.2 −18.7 2.8 1.7 Penza 279620 53.13N 45.02E 174 99.25 8293 −25.5 −22.8 12.2 10.2 8.9 13.3 −7.5 11.7 −4.9 2.5 220 4.9 130 33.1 −29.2 2.7 3.1 Perm’ 282250 58.02N 56.30E 172 99.28 8293 −30.9 −27.6 9.2 7.9 7.0 9.6 −8.8 8.2 −8.6 2.5 280 4.5 200 32.0 −33.4 2.3 3.7 Petropavlovsk-Kamca 325400 52.97N 158.75E 24 101.04 8293 −14.8 −13.2 13.3 11.3 9.5 13.6 −4.4 12.0 −5.3 3.7 360 3.7 330 24.2 −18.0 2.0 1.7 Petrozavodsk 228200 61.82N 34.27E 112 99.99 8293 −28.3 −24.1 7.4 6.5 5.9 7.9 −2.9 7.2 −4.2 1.7 90 2.7 290 28.1 −28.4 1.4 5.8 Pskov 262580 57.80N 28.42E 42 100.82 8293 −24.9 −20.1 8.9 7.5 6.5 10.0 −0.7 8.5 −0.9 1.1 150 2.8 130 29.4 −25.5 2.2 5.0 Rostov-Na-Donu 347310 47.25N 39.82E 77 100.40 8293 −16.9 −14.9 13.7 11.9 10.2 15.8 −2.2 14.0 −6.6 4.8 80 5.1 110 34.1 −19.5 1.9 2.7 Rubtsovsk 360340 51.50N 81.22E 215 98.77 8293 −32.2 −29.1 14.0 12.2 10.3 14.6 −6.5 13.3 −4.8 1.8 360 3.6 30 35.3 −34.0 2.0 4.3 Ryazan’ 277310 54.62N 39.72E 170 99.30 8293 −23.5 −20.9 9.7 8.0 6.8 10.3 −4.5 9.1 −4.7 2.8 310 3.3 140 31.1 −26.3 1.4 3.4 Rybinsk 272250 58.00N 38.83E 114 99.96 8293 −28.4 −24.7 9.1 8.1 7.2 9.5 −6.4 8.5 −4.9 2.3 320 3.5 210 29.4 −31.0 1.6 4.4 Samara (Kuybyshev) 289000 53.25N 50.45E 44 100.80 8293 −27.1 −24.7 11.0 9.6 8.1 10.5 −5.9 9.7 −6.5 1.0 320 3.4 110 33.8 −30.4 2.3 2.3 Saratov 341720 51.57N 46.03E 156 99.46 8293 −22.4 −20.3 12.2 10.2 9.0 12.9 −2.0 11.1 −6.0 5.8 300 4.7 70 33.0 −25.1 2.1 2.0 Smolensk 267810 54.75N 32.07E 241 98.46 8293 −22.7 −19.8 7.9 6.7 5.9 7.5 0.6 6.7 −1.4 2.6 280 2.8 230 28.9 −24.6 1.8 3.5 Sochi 371710 43.45N 39.90E 16 101.13 8293 −2.5 −1.2 8.4 7.1 6.3 10.1 6.2 8.3 6.6 3.4 70 3.4 270 30.9 −5.6 1.5 2.1 St Petersburg 260630 59.97N 30.30E 4 101.28 8293 −22.6 −18.9 8.3 7.0 6.2 10.3 0.0 8.4 −0.7 1.2 40 2.6 180 29.0 −24.0 2.0 4.7 Svobodnyy 314450 51.45N 128.12E 197 98.98 8293 −37.3 −34.7 7.7 6.6 6.0 6.9 −18.6 6.1 −19.5 0.6 290 3.2 180 32.4 −40.8 2.0 2.8 Syktyvkar 238040 61.72N 50.83E 119 99.90 8293 −35.5 −31.6 8.6 7.6 6.6 9.0 −7.2 8.0 −7.8 1.6 10 3.0 140 30.3 −37.9 1.8 4.0 Tambov 279470 52.73N 41.47E 139 99.67 8293 −25.1 −22.3 11.8 10.1 9.1 12.9 −4.0 10.3 −2.9 2.1 360 4.2 140 33.5 −28.7 2.8 3.9 Tayshet 295940 55.95N 98.00E 302 97.75 8293 −36.9 −33.7 8.4 7.3 6.3 8.2 −9.0 6.8 −10.0 0.8 330 3.1 40 31.4 −39.5 1.7 3.8 Ufa 287220 54.75N 56.00E 105 100.07 8293 −31.2 −27.7 10.3 9.0 7.9 11.4 −5.2 10.0 −7.3 1.5 140 3.7 50 33.8 −34.2 3.6 3.6 Ulan Ude 308230 51.80N 107.43E 510 95.35 8293 −35.9 −33.8 14.5 12.1 10.2 10.3 −15.7 7.4 −14.3 0.1 50 3.6 20 33.0 −38.5 2.2 2.3 Urup Island 321860 46.20N 150.50E 70 100.49 8293 −11.1 −10.0 20.4 17.7 15.6 22.5 −4.1 20.2 −4.7 10.5 320 4.8 180 21.5 −11.8 2.1 1.8 Ust’ilimsk 301170 58.03N 102.73E 402 96.59 8293 −40.1 −37.4 8.9 7.8 6.9 8.6 −12.4 7.6 −12.4 4.0 300 3.1 170 30.8 −43.2 1.4 6.1 Ust-Kamcatsk 324080 56.22N 162.47E 27 101.00 8293 −30.6 −27.7 13.1 11.0 9.7 15.3 −2.7 12.9 −3.7 2.5 360 3.9 180 24.5 −34.4 4.5 2.5 Vladimir 275320 56.13N 40.38E 170 99.30 8293 −26.6 −23.3 9.8 8.6 7.8 10.3 −5.9 9.3 −5.9 3.0 10 4.1 210 30.1 −29.5 1.7 4.2 Vladivostok 319600 43.12N 131.90E 184 99.13 8293 −22.1 −20.2 15.5 13.7 12.2 14.8 −15.7 13.5 −15.5 9.4 360 4.4 240 29.8 −24.5 1.3 3.1 Volgograd 345600 48.68N 44.35E 145 99.60 8293 −21.2 −18.9 14.4 12.7 11.1 15.1 −6.0 13.1 −7.0 4.1 340 6.6 110 35.4 −24.1 2.1 2.8 Vologda 270370 59.23N 39.87E 131 99.76 8293 −32.6 −28.0 9.1 7.7 6.8 9.9 −8.2 8.2 −6.0 1.8 330 3.2 190 29.1 −35.5 1.9 4.4 Voronezh 341220 51.70N 39.17E 154 99.49 8293 −23.0 −20.4 11.2 9.4 7.9 12.7 −4.7 10.7 −4.1 2.6 20 4.3 160 32.1 −25.8 1.9 4.2 Yakutsk 249590 62.08N 129.75E 103 100.09 8293 −51.9 −50.1 6.5 5.8 5.2 4.9 −37.3 4.1 −38.2 0.7 340 3.2 110 32.0 −52.5 2.4 2.6 Yekaterinburg 284400 56.80N 60.63E 237 98.51 8293 −29.8 −27.2 8.9 7.7 6.8 8.7 −8.5 7.6 −9.8 2.2 280 4.5 300 31.6 −32.6 2.2 3.6 Yelets 279280 52.63N 38.52E 168 99.32 8293 −24.1 −21.2 7.7 6.5 5.8 8.1 −2.8 7.1 −4.1 1.8 10 2.7 140 32.9 −26.4 3.8 4.3 Zyryanka 254000 65.73N 150.90E 43 100.81 8293 −49.0 −46.8 8.2 6.9 5.9 6.4 −28.6 5.3 −29.2 0.3 150 2.7 160 31.4 −50.8 1.7 2.3 SAMOA Pago Pago 917650 14.33S 170.72W 3 101.29 8293 22.0 23.0 11.3 10.3 9.6 11.4 25.5 10.7 25.8 2.4 310 5.1 80 33.6 14.5 1.8 8.9 SAUDI ARABIA Abha 411120 18.23N 42.65E 2084 78.67 8293 5.2 6.8 10.3 9.3 8.4 11.1 15.2 10.0 15.3 0.8 180 5.9 20 33.6 1.5 3.1 1.6 Al Jawf 403610 29.78N 40.10E 684 93.37 8293 0.2 1.8 11.4 9.7 8.4 11.8 9.6 9.7 11.2 2.3 50 3.8 320 43.9 −2.3 2.5 2.0 Al Madinah 404300 24.55N 39.70E 631 93.97 8293 8.8 9.9 9.1 8.0 7.0 8.7 18.3 7.5 19.3 3.0 60 4.1 300 45.8 5.5 1.2 2.2 Al Wajh 404000 26.20N 36.47E 16 101.13 8293 11.7 12.8 12.3 11.1 10.1 11.9 19.8 10.5 20.8 3.0 20 5.7 270 40.3 8.6 1.9 2.6 Ar’ar 403570 30.90N 41.13E 552 94.87 8293 −0.1 1.2 9.8 8.5 7.5 9.3 10.5 8.2 10.9 1.9 270 3.2 240 43.7 −2.3 1.1 1.7 At Ta’if 410360 21.48N 40.55E 1449 85.09 8293 5.6 7.2 10.4 9.4 8.5 10.6 18.3 9.6 18.8 2.4 90 4.4 50 38.2 1.4 1.8 1.6 Az Zahran 404160 26.27N 50.15E 17 101.12 8293 7.0 8.3 11.6 10.1 9.1 9.6 16.1 8.6 15.9 4.1 290 5.9 360 46.7 4.6 1.3 2.1 Ha’il 403940 27.43N 41.68E 1013 89.73 8293 −0.8 1.1 10.3 9.0 7.9 10.4 13.9 9.0 15.4 1.8 180 3.6 180 42.2 −4.1 2.0 2.7 Hafar Al Batin 403730 28.33N 46.17E 355 97.13 8293 2.4 4.0 12.2 10.4 9.3 10.9 13.4 9.8 13.3 2.9 270 4.4 240 47.1 −0.6 1.3 1.0 Jiddah 410240 21.67N 39.15E 12 101.18 8293 14.8 15.9 10.2 9.2 8.3 10.2 25.3 9.3 24.8 2.8 30 5.9 330 45.3 12.0 2.5 1.9 Jizan 411400 16.90N 42.58E 3 101.29 8293 20.1 21.2 9.0 8.0 7.2 7.8 27.6 7.0 27.8 2.4 100 4.6 230 41.5 14.2 2.5 5.7 WMO# = World Meteorological Organization number Elev. = elevation, m PWD = prevailing wind direction, ° True DB = dry-bulb temperature, °C Lat. = latitude ° Long. = longitude ° StdP = standard pressure at station elevation, kPa WS = wind speed, m/s Climatic Design Information 27.47 Table 3B Cooling and Dehumidification Design Conditions—World Locations Station Cooling DB/MWB Evaporation WB/MDB Dehumidification DP/MDB and HR Range of DB 0.4% 1% 2% 0.4% 1% 2% 0.4% 1% 2% DB MWB DB MWB DB MWB WB MDB WB MDB WB MDB DP HR MDB DP HR MDB DP HR MDB 1 2a 2b 2c 2d 2e 2f 3a 3b 3c 3d 3e 3f 4a 4b 4c 4d 4e 4f 4g 4h 4i 5 Kurgan 30.5 19.3 28.5 19.1 26.7 18.3 21.0 28.0 20.2 26.7 19.3 25.1 18.7 13.7 23.8 17.8 12.9 22.8 17.0 12.2 22.1 10.4 Kursk 28.6 19.2 26.9 18.7 25.3 18.2 20.7 25.8 19.8 25.1 18.9 23.8 18.9 14.1 23.4 17.8 13.1 22.4 16.9 12.4 21.4 9.5 Kyakhta 28.2 16.8 26.4 16.2 24.7 15.5 18.5 25.3 17.5 24.2 16.6 22.7 16.2 12.7 20.7 15.1 11.8 20.0 14.1 11.1 19.2 10.0 Magadan 18.2 12.7 16.3 11.9 15.0 11.1 13.7 16.5 12.8 15.1 12.1 14.0 12.4 9.1 14.2 11.8 8.7 13.4 11.1 8.3 13.0 5.4 Magnitogorsk 29.7 18.5 27.7 17.7 26.0 16.9 20.0 26.8 18.9 25.5 18.0 24.5 17.6 13.2 23.0 16.5 12.3 22.0 15.5 11.5 21.0 10.5 Markovo 24.5 15.9 22.4 15.1 20.3 14.0 16.8 22.9 15.6 21.3 14.5 19.3 14.2 10.1 19.2 13.0 9.4 17.4 12.0 8.8 16.5 10.2 Moscow 27.6 19.3 26.0 18.6 24.5 17.8 20.5 25.6 19.5 24.6 18.6 22.9 18.6 13.7 22.9 17.7 12.9 22.1 16.8 12.2 21.2 8.2 Moscow, Vnukovo 27.1 18.8 25.2 18.5 24.0 17.2 20.1 25.2 19.2 24.2 18.2 23.0 18.2 13.4 23.2 17.2 12.6 20.6 16.2 11.8 20.2 9.1 Murmansk 23.6 15.1 21.1 14.0 18.9 13.2 16.2 21.4 15.0 19.9 13.9 18.3 13.9 10.0 18.0 12.6 9.1 17.1 11.6 8.6 15.7 6.8 Nikolayevsk 25.7 19.3 24.1 18.5 22.6 17.6 20.3 24.4 19.2 22.7 18.2 21.9 18.7 13.6 22.6 17.8 12.9 21.5 16.8 12.1 20.2 8.6 Nikolskoe/Beringa 14.0 12.4 12.9 11.8 12.1 11.1 12.7 13.5 11.9 12.7 11.2 12.1 12.2 8.9 13.3 11.5 8.4 12.3 10.7 8.0 11.8 2.5 Nizhniy Novgorod 28.5 19.5 26.8 18.8 25.1 17.8 21.0 26.7 19.9 25.0 18.9 23.6 19.1 14.0 23.8 18.0 13.1 22.2 17.0 12.2 21.6 9.7 Nizhniy Tagil 28.4 18.5 26.6 17.9 24.7 17.2 20.0 25.8 19.0 24.4 18.0 23.1 18.0 13.3 22.2 17.1 12.6 21.4 16.1 11.8 20.0 10.2 Novokuznetsk 28.3 18.0 26.7 17.9 25.0 17.3 20.0 26.2 19.0 24.9 18.1 23.3 17.9 13.3 22.5 16.9 12.5 21.7 16.0 11.8 20.9 9.9 Novosibirsk 28.6 19.0 26.8 18.0 25.2 17.4 20.5 26.0 19.6 24.8 18.6 23.2 18.8 13.9 22.3 17.8 13.0 21.4 16.8 12.2 21.0 9.4 Nyurba 29.0 18.7 26.8 17.8 24.5 16.3 20.2 26.9 18.8 24.8 17.4 22.8 18.0 13.1 22.8 16.6 12.0 21.4 15.1 10.9 19.9 12.7 Olekminsk 29.7 18.5 27.4 17.6 25.1 16.6 19.7 27.2 18.6 25.3 17.5 23.3 17.1 12.5 21.8 16.0 11.7 22.1 15.0 10.9 20.2 11.2 Omsk 30.7 18.9 28.9 18.0 27.2 17.5 20.3 27.7 19.4 26.8 18.5 25.3 17.7 12.9 23.1 16.8 12.2 22.3 15.9 11.5 21.3 10.7 Orel 28.0 18.9 26.4 18.8 24.9 17.8 20.4 25.9 19.5 24.7 18.6 23.6 18.5 13.7 22.8 17.5 12.8 22.1 16.6 12.1 21.2 9.2 Orenburg 33.2 18.9 31.1 18.3 29.3 17.7 20.3 29.4 19.6 28.2 18.9 26.8 17.7 12.9 22.5 16.8 12.1 22.0 15.9 11.4 21.5 11.6 Ozernaja 15.6 13.4 14.2 12.5 13.1 11.5 13.9 15.4 12.8 13.9 12.0 12.8 13.2 9.5 14.7 12.2 8.9 13.4 11.4 8.4 12.7 3.3 Penza 29.7 18.6 27.9 18.0 26.2 17.4 20.6 27.2 19.6 25.4 18.6 24.1 18.3 13.5 23.3 17.5 12.8 22.1 16.6 12.1 21.1 10.4 Perm’ 29.0 19.4 27.1 18.5 25.0 17.9 20.6 26.7 19.6 25.7 18.6 23.8 18.4 13.6 24.1 17.4 12.7 22.4 16.5 12.0 21.0 8.9 Petropavlovsk-Kamca 20.7 15.4 18.9 14.2 17.3 13.2 16.0 20.0 14.7 18.3 13.7 16.7 14.2 10.1 18.0 12.9 9.3 15.9 12.1 8.8 14.8 5.3 Petrozavodsk 24.8 17.9 23.0 16.9 21.3 15.8 19.1 23.2 17.9 21.5 16.7 20.3 17.6 12.8 21.1 16.3 11.7 19.7 15.2 10.9 18.6 7.6 Pskov 26.7 19.1 25.2 17.9 23.6 17.3 20.3 24.7 19.1 23.6 18.1 22.4 18.6 13.5 22.6 17.4 12.5 21.0 16.3 11.6 20.4 9.1 Rostov-Na-Donu 31.4 20.8 29.9 20.2 28.5 19.7 22.3 29.2 21.5 27.6 20.7 26.4 20.1 14.9 25.3 19.3 14.2 24.4 18.6 13.6 24.0 10.3 Rubtsovsk 30.6 19.2 28.9 18.7 27.3 17.9 20.7 27.0 19.9 26.2 19.1 25.0 18.8 14.0 22.7 17.9 13.2 21.8 16.9 12.4 21.4 11.2 Ryazan’ 28.4 19.6 26.6 18.7 25.0 17.8 20.8 26.4 19.8 25.2 18.8 23.6 18.8 13.9 23.6 17.9 13.1 22.2 17.0 12.4 21.5 8.3 Rybinsk 26.6 19.9 24.8 18.6 23.1 17.7 20.6 25.6 19.5 23.4 18.5 22.2 18.7 13.7 23.0 17.8 12.9 22.0 17.0 12.3 20.8 7.2 Samara (Kuybyshev) 31.3 19.8 29.3 19.3 27.6 18.7 22.0 28.8 20.9 27.3 19.8 25.6 19.6 14.4 25.0 18.5 13.4 24.5 17.4 12.5 22.9 11.4 Saratov 30.5 18.7 28.9 18.5 27.4 17.8 20.7 27.7 19.8 26.5 19.0 25.4 18.2 13.4 23.6 17.4 12.7 22.9 16.6 12.0 22.5 8.4 Smolensk 26.0 19.1 24.6 18.1 23.1 17.0 20.0 24.6 19.0 23.3 18.1 22.0 18.3 13.6 22.5 17.4 12.8 21.4 16.4 12.0 20.3 8.4 Sochi 28.1 22.9 27.3 22.7 26.5 22.2 24.1 27.0 23.5 26.5 22.9 25.8 23.2 18.0 26.4 22.5 17.2 25.6 21.8 16.5 25.2 7.5 St Petersburg 26.3 18.4 24.6 17.6 23.0 16.7 19.7 24.4 18.7 23.1 17.7 21.8 17.9 12.9 22.1 16.8 12.0 20.8 15.9 11.3 19.8 7.5 Svobodnyy 29.4 19.6 27.9 19.4 26.2 18.7 22.0 26.4 21.0 25.1 20.1 24.3 20.6 15.6 24.1 19.6 14.7 23.0 18.7 13.9 22.3 10.4 Syktyvkar 28.1 19.1 25.9 18.4 23.8 16.8 20.4 26.4 19.3 24.6 18.1 22.7 18.3 13.4 23.5 17.2 12.5 21.9 16.2 11.7 20.2 9.5 Tambov 29.9 19.5 28.2 18.8 26.5 18.0 21.1 27.7 20.0 25.5 19.1 24.9 18.7 13.8 23.9 17.9 13.1 22.6 17.0 12.3 21.6 10.3 Tayshet 28.2 17.5 26.3 17.4 24.6 16.9 19.6 25.5 18.7 24.2 17.8 22.8 17.6 13.1 22.4 16.7 12.3 21.0 15.7 11.6 20.2 10.7 Ufa 30.3 19.2 28.4 19.0 26.6 18.4 21.2 27.5 20.2 26.6 19.3 25.3 19.0 14.0 24.5 17.9 13.0 22.9 17.0 12.3 22.0 10.6 Ulan Ude 29.3 17.3 27.5 16.9 25.6 16.2 19.3 25.7 18.2 24.7 17.2 23.6 16.9 12.8 22.1 15.9 12.0 21.2 14.9 11.3 19.8 11.5 Urup Island 17.3 15.1 15.5 14.1 14.1 13.1 15.8 16.7 14.4 15.2 13.2 13.9 15.5 11.1 16.2 14.1 10.1 14.6 12.8 9.3 13.4 5.1 Ust’ilimsk 28.0 17.2 25.8 16.5 24.0 15.9 19.0 24.2 18.0 23.4 16.9 22.6 17.2 12.9 21.1 15.9 11.9 20.3 14.7 11.0 19.2 11.2 Ust-Kamcatsk 19.3 14.1 17.2 13.2 15.6 12.3 14.8 17.9 13.7 16.4 12.9 15.0 13.2 9.5 15.8 12.4 9.0 14.8 11.7 8.6 13.9 5.9 Vladimir 27.7 19.5 25.8 18.7 24.1 17.5 20.6 25.9 19.6 24.6 18.6 23.0 18.7 13.8 23.5 17.7 13.0 21.8 16.8 12.2 21.0 8.5 Vladivostok 25.8 20.4 24.0 19.4 22.4 19.0 22.2 24.0 21.2 22.6 20.3 21.5 21.6 16.6 23.0 20.7 15.7 22.0 19.8 14.8 21.0 4.8 Volgograd 32.5 18.6 30.9 18.4 29.3 18.0 20.7 29.1 19.9 27.5 19.1 26.4 18.1 13.3 23.0 17.4 12.7 22.5 16.5 12.0 22.5 10.5 Vologda 26.4 19.1 24.7 18.0 23.1 17.3 20.4 25.3 19.2 23.0 18.1 21.9 18.6 13.7 23.2 17.6 12.8 21.7 16.5 11.9 20.2 9.4 Voronezh 29.4 19.1 27.7 18.2 26.0 17.4 20.5 26.5 19.6 25.6 18.8 24.1 18.4 13.5 23.0 17.5 12.8 22.0 16.7 12.1 21.4 9.7 Yakutsk 29.4 18.7 27.4 18.0 25.4 16.8 20.1 27.5 18.9 25.5 17.8 24.1 17.5 12.7 23.3 16.2 11.7 21.9 15.0 10.8 22.0 11.8 Yekaterinburg 28.7 19.0 27.0 18.3 25.1 17.5 20.5 26.7 19.5 24.9 18.5 23.6 18.4 13.7 23.7 17.5 12.9 21.9 16.5 12.1 21.2 9.5 Yelets 29.1 19.2 27.4 18.7 25.8 17.9 20.7 26.7 19.9 25.2 19.0 24.2 18.7 13.8 23.1 17.9 13.1 22.2 17.1 12.5 21.3 9.5 Zyryanka 28.1 17.4 25.9 16.7 23.4 15.6 18.7 25.9 17.5 24.3 16.3 22.4 15.8 11.3 22.1 14.5 10.4 20.9 13.2 9.5 20.0 9.6 SAMOA Pago Pago 31.3 26.7 31.0 26.7 30.7 26.5 27.6 30.5 27.2 30.2 27.1 30.1 26.6 22.2 29.8 26.3 21.8 29.5 26.1 21.5 29.3 5.2 SAUDI ARABIA Abha 30.5 13.2 29.9 13.4 29.1 13.5 19.6 24.3 19.0 23.9 18.3 23.1 18.2 17.0 22.0 17.2 15.9 21.6 16.7 15.4 21.5 11.9 Al Jawf 40.6 17.0 39.7 16.7 38.6 16.3 18.5 36.5 17.7 35.1 17.3 35.4 14.0 10.8 19.4 12.8 10.0 19.1 11.2 9.0 19.7 14.6 Al Madinah 44.8 18.5 43.2 18.2 42.9 18.0 20.7 36.0 19.9 36.2 19.4 36.4 16.9 13.0 23.1 15.3 11.7 23.7 14.1 10.8 24.3 13.2 Al Wajh 35.0 22.4 33.8 24.3 33.0 25.5 28.6 32.4 28.0 31.6 27.5 31.5 27.8 23.9 31.3 27.0 22.7 31.0 26.2 21.7 30.6 7.3 Ar’ar 41.8 20.2 40.6 19.8 39.5 19.4 23.0 39.2 21.8 37.6 20.7 36.8 16.8 12.8 33.3 15.3 11.6 32.2 14.1 10.7 30.0 14.2 At Ta’if 36.0 18.6 35.2 18.5 34.6 18.5 22.0 31.7 21.2 30.6 20.4 30.4 19.2 16.7 26.8 18.1 15.6 26.8 17.1 14.6 26.3 11.6 Az Zahran 44.0 21.8 42.9 21.7 41.8 22.1 29.9 34.4 29.0 34.3 28.1 33.5 29.0 25.7 32.8 27.9 24.0 32.4 26.9 22.6 32.4 13.3 Ha’il 40.6 18.3 39.8 17.9 38.9 17.6 20.1 36.5 19.2 36.0 18.6 36.0 15.2 12.2 21.5 14.1 11.4 22.3 13.0 10.6 21.6 15.7 Hafar Al Batin 45.2 19.3 44.2 19.0 43.1 18.7 21.4 37.2 20.3 38.0 19.7 37.7 18.0 13.5 23.1 16.3 12.1 22.3 15.1 11.2 21.6 15.3 Jiddah 40.2 22.0 39.0 22.9 38.0 23.6 28.2 34.2 27.9 34.2 27.3 33.5 27.0 22.7 31.5 26.2 21.6 31.5 26.0 21.4 31.2 12.2 Jizan 38.8 28.6 38.0 28.6 37.3 28.4 30.4 36.4 30.0 36.9 29.6 36.2 28.9 25.5 35.8 28.2 24.4 35.3 27.9 24.0 35.0 7.0 MDB = mean coincident dry-bulb temp., °C MWS = mean coincident wind speed, m/s HR = humidity ratio, grams of moisture per kilogram of dry air MWB = mean coincident wet-bulb temp., °C StdD = standard deviation, °C A = airport DP = dew-point temperature, °C 27.48 2001 ASHRAE Fundamentals Handbook (SI) Table 3A Heating and Wind Design Conditions—World Locations Heating Dry Bulb Extreme Wind Speed, m/s Coldest Month MWS/PWD to DB Extr. Annual Daily Elev., m StdP, kPa 0.4% 1% 99.6% 0.4% Mean DB StdD DB Station WMO# Lat.
Long.
Dates 99.6% 99% 1% 2.5% 5% WS MDB WS MDB MWS PWD MWS PWD Max. Min. Max. Min.
1a 1b 1c 1d 1e 1f 1g 2a 2b 3a 3b 3c 4a 4b 4c 4d 5a 5b 5c 5d 6a 6b 6c 6d Khamis Mushayt 411140 18.30N 42.80E 2054 78.96 8293 4.3 6.1 9.2 8.2 7.2 9.7 16.1 8.9 16.1 0.8 150 4.5 30 36.2 1.8 3.8 2.3 Makkah 410300 21.48N 39.83E 310 97.66 8293 15.2 16.8 6.4 5.4 4.8 6.7 25.1 5.6 25.4 1.6 20 3.4 300 47.5 11.7 1.1 2.4 Qasim 404050 26.30N 43.77E 650 93.76 8293 2.7 4.0 9.3 8.1 7.0 8.8 15.8 7.4 13.8 0.8 30 3.7 90 45.9 −0.1 2.1 1.8 Rafha 403620 29.63N 43.48E 447 96.07 8293 0.4 1.9 11.3 9.9 8.8 10.8 12.1 9.7 13.2 1.8 270 4.0 300 46.3 −2.6 1.5 1.5 Riyadh 404380 24.72N 46.72E 612 94.19 8293 5.1 6.8 9.9 8.5 7.5 9.2 15.6 8.0 15.5 1.6 320 4.8 360 46.0 1.9 0.8 1.5 Tabuk 403750 28.37N 36.63E 770 92.41 8293 1.1 2.6 11.0 9.1 7.5 11.0 15.5 9.0 15.4 1.1 110 4.5 270 41.8 −1.3 1.2 1.3 Turayf 403560 31.68N 38.67E 813 91.93 8293 −1.9 −0.1 11.2 9.9 8.8 11.4 7.7 10.0 7.8 2.8 270 4.2 270 40.9 −4.0 1.3 1.9 Yanbu’al Bahr 404390 24.15N 38.07E 1 101.31 8293 10.9 12.1 11.4 10.2 9.2 11.0 22.1 9.8 22.5 1.5 10 7.7 270 45.8 8.6 1.0 1.1 SENEGAL Dakar 616410 14.73N 17.50W 24 101.04 8293 16.2 16.8 10.3 9.4 8.5 10.3 20.8 9.4 20.8 4.5 360 4.3 360 37.9 12.3 2.2 4.3 Saint Louis 616000 16.05N 16.45W 4 101.28 8293 15.3 16.2 10.1 9.0 8.2 10.2 24.6 8.8 24.7 3.2 40 5.0 80 42.4 11.5 1.6 2.0 Tambacounda 616870 13.77N 13.68W 50 100.73 8293 17.0 18.2 7.5 6.5 5.6 7.7 27.9 7.2 27.3 1.5 80 2.8 100 43.1 11.9 1.8 3.8 Ziguinchor 616950 12.55N 16.27W 23 101.05 8293 16.2 17.2 6.3 5.2 4.4 6.8 27.5 5.5 27.2 0.6 40 2.8 60 41.3 12.5 0.5 4.1 SINGAPORE Singapore 486980 1.37N 103.98E 16 101.13 8293 22.8 23.1 8.0 7.2 6.3 8.2 28.5 7.6 28.7 2.0 330 4.7 30 33.9 18.3 1.1 6.7 SLOVAKIA Bratislava 118160 48.20N 17.20E 130 99.77 8293 −13.0 −10.0 9.2 8.0 7.1 10.1 1.4 8.6 2.5 1.5 50 3.5 160 34.3 −15.2 1.6 4.9 Chopok Mountain 119160 48.93N 19.58E 2012 79.38 8293 −21.1 −19.2 23.4 20.7 18.7 27.4 −14.2 24.7 −14.2 13.1 330 4.7 180 17.1 −22.0 1.1 3.6 Kosice 119680 48.70N 21.27E 232 98.57 8293 −13.5 −11.3 12.9 11.0 9.4 13.5 −6.0 11.4 −3.7 3.9 350 3.5 180 31.8 −15.7 1.4 3.7 Lomnicky Stit (Peak) 119300 49.20N 20.22E 2635 73.42 8293 −24.3 −22.3 23.1 19.8 17.1 26.5 −18.1 22.9 −16.5 10.4 310 2.5 180 14.7 −24.9 1.4 3.3 Zilina 118410 49.23N 18.62E 315 97.60 8293 −16.8 −13.6 7.9 6.7 5.6 8.4 2.8 6.8 −0.6 1.7 70 3.2 250 31.7 −19.7 1.2 3.9 SLOVENIA Ljubljana 130140 46.22N 14.48E 385 96.78 8293 −13.0 −10.4 6.2 5.1 4.2 5.5 1.0 4.4 1.4 0.5 290 3.1 130 33.7 −16.2 2.5 3.2 SOUTH AFRICA Bloemfontein 684420 29.10S 26.30E 1348 86.15 8293 −3.5 −2.2 10.6 9.3 8.3 9.2 12.7 8.2 14.2 0.5 220 5.4 270 36.6 −5.6 1.4 1.7 Cape Town 688160 33.98S 18.60E 42 100.82 8293 3.6 4.9 14.5 13.0 11.8 13.7 14.1 12.4 14.2 0.1 40 5.2 170 34.5 1.3 1.6 0.8 Durban 685880 29.97S 30.95E 8 101.23 8293 10.0 11.1 11.9 10.4 9.2 10.6 21.0 9.2 20.5 0.3 340 6.3 30 34.0 7.6 1.2 1.1 Johannesburg 683680 26.13S 28.23E 1700 82.50 8293 1.0 2.8 9.6 8.5 7.6 8.7 12.7 7.9 12.3 4.1 210 4.1 300 31.6 −1.6 1.0 1.7 Marion Island 689940 46.88S 37.87E 22 101.06 8293 −0.9 −0.1 26.7 23.7 21.1 26.3 3.1 23.0 5.9 7.6 200 9.6 290 20.1 −4.6 5.0 2.4 Port Elizabeth 688420 33.98S 25.60E 60 100.61 8293 6.3 7.5 14.6 12.9 11.5 13.7 14.7 12.4 15.5 0.9 270 4.3 290 35.9 3.5 2.1 1.2 Pretoria 682620 25.73S 28.18E 1322 86.42 8293 3.9 5.1 6.4 5.4 4.7 5.8 15.9 5.0 15.2 0.5 220 1.9 270 34.7 1.6 1.6 1.1 SPAIN Barcelona 81810 41.28N 2.07E 6 101.25 8293 0.1 1.6 9.2 7.8 6.9 9.5 10.0 8.2 9.1 3.7 350 4.0 210 32.0 −2.0 1.6 1.9 Granada 84190 37.18N 3.78W 559 94.79 8293 −3.9 −2.4 9.2 8.0 7.0 9.0 9.6 7.4 9.6 0.1 230 5.3 180 39.6 −7.0 0.9 2.8 La Coruna 80010 43.37N 8.42W 67 100.52 8293 3.9 5.2 12.2 10.3 9.1 13.1 11.8 11.5 11.7 2.7 140 3.2 60 30.1 1.6 1.7 1.5 Madrid 82210 40.45N 3.55W 582 94.53 8293 −4.5 −3.1 10.0 8.7 7.5 10.1 8.1 8.4 8.2 0.2 360 3.4 240 38.9 −6.8 1.0 1.7 Malaga 84820 36.67N 4.48W 7 101.24 8293 3.8 4.9 12.1 10.3 9.0 14.4 12.9 12.6 13.1 4.4 320 5.6 320 39.5 0.7 1.8 1.7 Palma 83060 39.55N 2.73E 8 101.23 8293 −0.5 0.8 10.2 9.1 8.1 10.6 12.1 9.3 12.3 0.2 60 4.6 60 37.2 −3.3 1.8 1.2 Salamanca 82020 40.95N 5.50W 795 92.13 8293 −5.2 −4.0 11.9 10.0 8.6 12.6 7.9 10.6 7.2 0.6 80 3.3 300 36.4 −7.9 1.2 2.5 Santander 80230 43.47N 3.82W 65 100.55 8293 2.3 4.0 10.6 8.4 7.0 12.3 10.6 10.2 10.6 2.1 110 3.1 40 33.0 1.2 2.2 1.4 Santiago De Compostela 80420 42.90N 8.43W 367 96.99 8293 −1.2 0.1 9.6 8.3 7.3 10.4 10.3 9.3 9.7 1.4 90 2.9 280 36.2 −4.2 2.3 2.2 Sevilla 83910 37.42N 5.90W 31 100.95 8293 1.2 2.8 9.0 7.9 6.8 9.1 12.9 7.9 12.6 1.1 30 3.4 240 42.9 −1.2 1.5 1.7 Valencia 82840 39.50N 0.47W 62 100.58 8293 0.9 2.2 12.2 10.1 8.4 14.6 14.1 12.0 13.5 1.9 280 5.3 120 37.6 −1.5 2.2 1.5 Zaragoza 81605 41.67N 1.05W 263 98.21 8293 −2.2 −0.9 12.5 10.8 9.6 13.0 7.6 11.4 9.2 2.3 10 3.1 90 38.3 −4.0 5.7 2.6 SWEDEN Goteborg, Landvetter 25260 57.67N 12.30E 169 99.31 8293 −16.2 −12.1 11.4 10.1 9.1 12.2 3.6 10.9 2.7 4.0 40 4.0 310 28.5 −16.4 2.3 5.4 Goteborg, Save 25120 57.78N 11.88E 53 100.69 8293 −16.1 −12.2 12.1 10.7 9.5 12.6 4.8 11.1 3.9 2.1 50 4.1 290 27.9 −16.2 1.7 5.1 Jonkoping 25500 57.77N 14.08E 232 98.57 8293 −20.0 −15.1 11.2 9.9 8.9 12.0 5.1 10.5 3.6 3.1 30 4.5 50 28.5 −21.9 2.6 5.7 Kalmar 26720 56.73N 16.30E 16 101.13 8293 −15.0 −12.0 12.2 10.4 9.5 12.3 5.1 11.4 5.1 2.6 270 4.9 270 29.1 −15.9 2.5 3.9 Karlsborg 25440 58.52N 14.53E 102 100.11 8293 −16.5 −12.9 12.0 10.4 9.1 13.3 3.4 11.5 3.1 4.7 50 2.9 190 27.1 −16.0 2.1 5.6 Karlstad 24180 59.37N 13.47E 55 100.67 8293 −20.6 −17.3 10.0 8.8 8.0 11.6 3.4 10.3 3.4 1.8 350 3.8 200 27.2 −20.5 2.0 5.5 Kiruna 20440 67.82N 20.33E 452 96.01 8293 −30.2 −27.0 11.8 10.2 8.9 13.3 −1.9 11.6 −2.4 2.0 210 4.3 190 24.2 −32.3 1.3 3.0 Malmo 26360 55.55N 13.37E 106 100.06 8293 −13.9 −10.1 13.4 12.0 10.8 14.0 2.8 12.6 2.3 3.8 340 5.2 140 27.8 −13.3 1.9 5.4 Ostersund/Froso 22260 63.18N 14.50E 370 96.96 8293 −25.8 −21.6 11.9 10.2 8.8 15.6 1.0 12.3 0.1 1.2 320 3.0 280 26.5 −27.4 1.7 5.2 Soderhamn 23760 61.27N 17.10E 36 100.89 8293 −21.4 −17.8 9.8 8.4 7.5 10.6 1.2 9.4 −1.4 2.7 290 4.6 130 28.8 −21.4 1.6 4.2 Stockholm, Arlanda 24600 59.65N 17.95E 61 100.59 8293 −18.9 −15.2 10.5 9.3 8.2 12.4 2.6 10.5 2.2 2.0 350 3.7 180 29.1 −18.5 1.9 5.7 Stockholm, Bromma 24640 59.35N 17.95E 11 101.19 8293 −18.3 −15.0 9.4 8.3 7.4 9.1 2.6 8.2 2.5 1.8 320 3.9 200 28.7 −18.4 2.1 5.4 Sundsvall 23660 62.53N 17.45E 10 101.20 8293 −25.5 −22.1 10.5 9.0 7.8 12.7 4.0 10.4 0.0 1.3 310 4.6 140 27.7 −25.5 1.7 5.2 Ungskar 26660 56.03N 15.80E 3 101.29 8293 −11.9 −8.9 18.5 16.6 15.0 19.0 1.3 17.3 2.7 5.1 20 5.4 250 23.2 −10.7 2.4 5.2 Uppsala 24580 59.88N 17.60E 41 100.83 8293 −19.9 −16.2 10.6 9.3 8.3 12.6 1.7 11.0 2.7 2.6 330 4.2 230 28.0 −19.9 1.5 6.6 Visby 25900 57.67N 18.35E 47 100.76 8293 −11.2 −8.9 13.9 12.4 11.2 14.7 0.7 13.5 0.9 5.4 20 5.3 210 27.4 −13.6 2.1 4.7 SWITZERLAND Geneva 67000 46.25N 6.13E 416 96.43 8293 −8.0 −5.2 9.0 7.8 6.7 9.2 3.6 8.2 3.5 2.9 230 3.4 210 33.4 −10.0 1.1 3.8 Interlaken 67340 46.67N 7.88E 580 94.55 8293 −9.7 −7.3 7.1 6.0 5.1 6.5 1.8 5.4 0.2 2.6 190 3.4 280 31.0 −11.6 1.9 4.1 Jungfrau Mountain 67300 46.55N 7.98E 3576 65.12 8293 −26.3 −23.9 21.6 18.1 14.9 21.6 −16.1 19.2 −14.2 7.2 310 5.1 140 9.0 −28.6 3.6 3.8 La Chaux-De-Fonds 66120 47.08N 6.80E 1019 89.67 8293 −14.4 −11.3 8.8 7.6 6.6 9.9 1.8 8.5 0.8 1.2 230 3.2 250 30.0 −18.4 4.8 6.1 Locarno 67620 46.17N 8.88E 198 98.97 8293 −6.3 −4.7 7.2 5.9 4.9 7.0 6.2 5.7 5.3 1.8 90 2.7 240 31.6 −9.1 2.0 2.9 WMO# = World Meteorological Organization number Elev. = elevation, m PWD = prevailing wind direction, ° True DB = dry-bulb temperature, °C Lat. = latitude ° Long. = longitude ° StdP = standard pressure at station elevation, kPa WS = wind speed, m/s Climatic Design Information 27.49 Table 3B Cooling and Dehumidification Design Conditions—World Locations Station Cooling DB/MWB Evaporation WB/MDB Dehumidification DP/MDB and HR Range of DB 0.4% 1% 2% 0.4% 1% 2% 0.4% 1% 2% DB MWB DB MWB DB MWB WB MDB WB MDB WB MDB DP HR MDB DP HR MDB DP HR MDB 1 2a 2b 2c 2d 2e 2f 3a 3b 3c 3d 3e 3f 4a 4b 4c 4d 4e 4f 4g 4h 4i 5 Khamis Mushayt 31.1 13.8 30.6 13.7 30.0 13.4 19.0 24.1 18.2 23.3 17.6 23.1 17.2 15.9 21.8 16.6 15.2 21.5 15.9 14.6 21.0 12.3 Makkah 44.8 24.5 43.8 24.3 42.8 24.0 28.0 39.8 27.2 38.9 26.6 38.3 25.1 21.0 33.9 24.2 19.9 34.3 23.5 19.0 34.3 15.1 Qasim 43.5 19.4 42.8 18.5 41.8 18.0 23.6 35.9 21.2 35.6 19.9 36.8 19.9 15.8 29.1 17.0 13.1 26.4 15.1 11.6 23.7 16.3 Rafha 43.9 20.5 42.8 20.0 41.5 19.4 22.0 40.0 21.2 39.8 20.5 39.5 17.0 12.8 24.1 15.9 11.9 22.3 14.2 10.7 23.8 16.5 Riyadh 44.0 18.0 43.1 17.8 42.2 17.7 20.6 35.6 19.7 36.3 19.0 35.6 16.9 13.0 22.9 15.6 11.9 22.1 14.1 10.8 22.2 14.0 Tabuk 40.1 17.8 39.0 17.4 38.0 17.1 20.0 35.2 19.2 34.9 18.5 34.6 14.8 11.5 24.9 13.4 10.5 24.8 12.8 10.1 24.4 14.8 Turayf 38.9 17.7 37.3 17.0 36.1 16.9 20.2 33.6 19.1 32.8 18.3 32.4 15.8 12.4 25.6 14.2 11.2 24.0 13.2 10.4 23.1 15.2 Yanbu’al Bahr 42.7 24.2 41.2 24.1 39.9 24.2 28.6 35.7 27.8 35.1 27.2 34.6 27.0 22.7 32.2 26.1 21.5 31.4 25.2 20.3 31.1 14.3 SENEGAL Dakar 31.8 23.4 31.0 24.9 30.2 25.1 27.0 30.2 26.6 29.5 26.2 29.0 26.2 21.7 28.8 26.0 21.4 28.6 25.5 20.8 28.4 5.4 Saint Louis 38.1 20.6 36.3 20.3 34.6 20.7 27.9 31.1 27.5 30.6 27.1 30.1 27.1 22.8 29.7 26.7 22.3 29.4 26.3 21.8 29.1 9.0 Tambacounda 40.9 21.3 40.2 21.1 39.3 21.0 27.1 32.5 26.7 31.8 26.3 31.3 25.9 21.4 29.2 25.5 20.8 28.8 25.1 20.3 28.6 12.6 Ziguinchor 38.2 22.4 36.9 21.8 35.7 22.4 28.3 32.5 27.9 32.0 27.5 31.6 27.2 23.0 31.3 26.7 22.3 30.6 26.4 21.9 30.0 15.4 SINGAPORE Singapore 33.0 25.9 32.2 25.9 32.0 25.9 27.2 30.9 27.1 30.7 26.8 30.3 26.2 21.7 28.9 26.2 21.7 28.8 26.1 21.5 28.7 6.3 SLOVAKIA Bratislava 31.8 20.4 30.0 20.0 28.2 19.3 21.6 29.5 20.8 28.6 19.9 26.7 19.0 14.0 25.2 18.1 13.2 24.1 17.2 12.5 23.2 12.3 Chopok Mountain 15.0 11.0 13.6 10.5 12.4 9.8 12.0 13.9 11.3 12.9 10.4 11.9 11.3 10.7 12.4 10.5 10.1 11.9 9.7 9.6 11.1 4.4 Kosice 29.6 20.1 27.9 19.4 26.2 18.6 20.9 28.3 20.1 26.7 19.2 25.1 18.3 13.6 24.5 17.6 13.0 23.2 16.8 12.3 22.4 10.7 Lomnicky Stit (Peak) 11.9 8.1 10.5 7.2 9.3 6.2 9.2 10.9 8.1 9.8 7.2 8.7 8.4 9.5 10.1 7.3 8.8 8.9 6.3 8.2 8.0 4.7 Zilina 29.2 19.5 27.5 18.6 25.7 17.6 20.2 27.6 19.4 26.0 18.4 24.3 17.7 13.2 22.9 16.9 12.5 21.8 16.2 12.0 21.1 12.1 SLOVENIA Ljubljana 30.1 20.0 28.3 19.2 26.9 18.5 20.9 28.4 20.0 26.8 19.2 25.8 18.2 13.7 23.2 17.6 13.2 22.5 16.9 12.6 22.3 12.4 SOUTH AFRICA Bloemfontein 34.0 15.6 32.8 15.4 31.5 15.4 19.2 26.0 18.7 25.9 18.2 25.5 17.5 14.8 20.7 16.9 14.2 20.4 16.1 13.5 20.1 14.6 Cape Town 30.3 19.7 28.6 19.3 27.0 18.6 21.2 27.7 20.5 26.4 19.9 24.9 19.3 14.1 22.7 18.6 13.5 22.4 18.1 13.1 22.2 8.8 Durban 30.3 23.7 29.3 23.7 28.5 23.4 25.5 28.7 24.9 28.1 24.5 27.4 24.5 19.5 27.4 24.0 18.9 27.1 23.5 18.3 26.7 5.5 Johannesburg 29.0 15.6 27.9 15.6 26.9 15.5 18.6 24.9 18.1 24.2 17.6 23.7 16.8 14.8 20.4 16.2 14.2 19.9 15.7 13.7 19.6 10.4 Marion Island 14.0 12.0 12.7 11.3 11.8 10.6 12.6 13.5 11.7 12.4 11.0 11.5 12.2 8.9 13.0 11.4 8.4 12.0 10.7 8.0 11.3 4.5 Port Elizabeth 29.2 18.7 27.3 19.8 26.1 20.0 22.7 25.8 22.1 25.0 21.6 24.3 21.9 16.7 24.2 21.2 16.0 23.6 20.6 15.4 23.3 6.7 Pretoria 31.9 17.6 30.9 17.1 30.0 17.1 20.5 26.8 19.9 26.6 19.4 26.2 18.7 15.9 22.8 18.0 15.2 22.5 17.4 14.6 22.2 9.8 SPAIN Barcelona 29.3 23.4 28.8 23.5 27.9 22.9 25.2 28.1 24.4 27.5 23.6 26.8 24.1 19.0 27.5 23.2 18.0 26.7 22.6 17.3 26.3 8.4 Granada 37.0 19.6 35.3 19.3 34.0 18.9 21.2 32.9 20.5 31.7 19.9 31.2 18.0 13.8 24.7 17.1 13.1 23.8 16.1 12.2 23.5 18.7 La Coruna 25.2 18.6 23.6 18.2 22.3 17.6 19.6 23.8 19.0 22.4 18.4 21.4 18.2 13.2 21.0 17.7 12.8 20.3 17.2 12.4 20.0 5.2 Madrid 36.0 20.7 34.7 20.0 33.1 19.6 21.9 34.4 21.0 33.0 20.1 31.6 17.8 13.7 27.5 16.8 12.9 26.6 15.8 12.0 26.0 16.2 Malaga 34.1 20.2 32.0 19.8 30.1 19.8 23.9 27.5 23.4 27.0 22.7 26.7 22.9 17.6 26.1 22.1 16.8 25.8 21.3 16.0 25.1 9.1 Palma 33.0 23.1 31.4 22.9 30.2 22.9 25.8 29.2 25.0 28.9 24.3 28.5 24.8 19.9 28.4 23.9 18.8 27.6 23.0 17.8 27.0 12.4 Salamanca 33.7 18.5 32.0 17.8 30.2 17.2 19.5 30.9 18.6 29.8 17.8 28.3 15.9 12.4 22.7 15.0 11.7 20.7 14.1 11.1 20.2 15.9 Santander 26.5 19.5 24.7 19.4 23.7 19.0 21.5 24.4 20.7 23.3 20.1 22.8 20.5 15.3 22.7 19.7 14.5 22.4 19.0 13.9 21.9 5.2 Santiago De Compostela 31.2 20.4 29.0 19.5 26.9 18.7 21.5 29.7 20.4 27.6 19.4 25.2 19.0 14.4 24.6 18.1 13.6 22.8 17.2 12.8 21.3 11.8 Sevilla 39.8 23.7 38.0 22.2 36.1 21.7 24.6 36.4 23.3 35.3 22.4 33.3 21.0 15.7 29.1 20.0 14.8 26.7 19.1 13.9 25.5 16.7 Valencia 32.2 21.9 31.0 22.1 30.0 22.1 24.7 29.2 24.1 28.5 23.6 27.8 23.2 18.1 27.1 22.9 17.8 26.9 22.1 16.9 26.4 9.2 Zaragoza 36.0 20.7 34.0 20.3 32.1 19.9 22.4 31.9 21.6 31.5 21.0 29.7 19.2 14.4 24.9 18.8 14.1 24.3 17.9 13.3 25.4 13.4 SWEDEN Goteborg, Landvetter 25.8 16.6 23.9 15.8 22.0 14.8 17.7 23.5 16.7 21.6 15.8 19.9 15.9 11.5 18.7 15.0 10.9 17.7 14.1 10.2 16.7 8.3 Goteborg, Save 25.3 16.8 23.4 16.1 21.6 15.6 18.6 23.0 17.5 21.2 16.6 20.0 17.0 12.2 19.8 16.0 11.4 19.1 15.1 10.8 18.1 7.6 Jonkoping 26.1 16.2 23.9 15.4 22.0 14.6 17.5 23.4 16.6 21.6 15.7 20.2 15.9 11.6 18.3 14.9 10.9 17.5 14.0 10.3 16.7 10.9 Kalmar 26.0 17.4 24.0 16.8 22.4 15.7 18.8 24.0 17.7 22.3 16.7 20.9 17.0 12.2 20.7 15.9 11.3 19.6 14.9 10.6 18.6 10.4 Karlsborg 24.7 17.0 22.8 16.1 21.1 15.4 18.1 23.2 17.1 21.2 16.2 20.1 16.2 11.7 20.1 15.3 11.0 19.1 14.4 10.4 18.2 7.9 Karlstad 25.1 17.1 23.2 16.4 21.5 15.5 18.4 22.6 17.5 21.3 16.6 20.1 17.0 12.2 19.4 16.0 11.4 18.5 15.1 10.8 17.9 8.7 Kiruna 21.0 13.5 19.0 12.4 17.0 11.3 14.2 18.8 13.3 17.4 12.3 16.2 12.4 9.5 15.3 11.3 8.8 14.8 10.2 8.2 13.8 7.6 Malmo 25.0 16.9 23.1 16.3 21.8 15.7 18.7 22.0 17.7 20.9 16.9 19.9 17.7 12.9 20.1 16.8 12.1 18.9 15.8 11.4 18.2 7.9 Ostersund/Froso 23.2 14.6 21.2 13.9 19.3 13.0 15.9 21.7 14.7 19.7 13.7 18.2 13.4 10.0 17.1 12.4 9.4 16.0 11.5 8.8 15.4 7.5 Soderhamn 24.9 16.7 22.9 15.6 21.1 14.9 17.9 22.9 16.8 21.5 15.8 20.0 16.0 11.4 19.4 14.7 10.5 18.6 13.9 9.9 17.6 9.0 Stockholm, Arlanda 26.8 17.1 24.8 16.1 22.8 15.2 18.4 23.6 17.4 21.9 16.4 20.8 17.0 12.2 19.2 15.9 11.4 18.3 14.9 10.7 17.6 9.0 Stockholm, Bromma 26.1 17.1 24.2 16.2 22.5 15.4 18.7 23.2 17.7 22.0 16.7 20.7 17.2 12.3 19.8 16.1 11.5 19.0 15.1 10.7 18.1 8.8 Sundsvall 24.0 16.9 22.1 15.2 20.4 14.6 17.7 21.4 16.5 20.0 15.6 19.0 16.5 11.8 18.8 15.2 10.8 17.5 14.2 10.1 16.6 8.8 Ungskar 21.5 18.3 20.2 17.4 19.2 16.6 18.8 20.8 17.9 19.8 17.1 18.9 18.0 12.9 20.3 17.0 12.1 19.2 16.1 11.4 18.5 3.8 Uppsala 25.3 16.8 23.7 16.2 21.8 15.2 18.3 23.0 17.2 21.8 16.2 20.6 16.4 11.7 20.4 15.2 10.8 19.2 14.2 10.2 18.2 9.3 Visby 24.1 17.1 22.2 16.7 21.0 16.0 18.8 21.8 17.9 21.0 17.0 19.8 17.8 12.8 19.8 16.8 12.0 19.0 16.0 11.4 18.3 7.3 SWITZERLAND Geneva 30.1 19.1 28.5 18.4 26.9 18.0 20.0 27.8 19.4 26.6 18.6 25.2 17.6 13.3 22.2 16.9 12.7 21.6 16.1 12.0 21.1 12.3 Interlaken 27.6 18.6 25.9 18.1 24.3 17.3 19.4 26.4 18.6 24.9 17.7 23.3 17.0 13.0 22.2 16.2 12.4 20.9 15.5 11.8 20.3 9.9 Jungfrau Mountain 6.1 0.7 4.9 0.2 4.0 −0.2 2.7 4.7 1.9 3.6 1.2 2.9 1.7 6.7 3.1 0.8 6.2 2.4 0.1 5.9 1.9 3.8 La Chaux-De-Fonds 25.5 16.4 23.5 16.0 21.9 15.1 17.3 23.7 16.5 22.2 15.7 21.0 15.1 12.1 19.5 14.3 11.5 18.9 13.5 10.9 18.1 9.8 Locarno 29.0 20.9 28.0 20.4 26.9 19.6 22.2 27.6 21.4 26.7 20.6 25.5 20.4 15.4 25.4 19.5 14.6 24.6 18.7 13.9 23.5 9.9 MDB = mean coincident dry-bulb temp., °C MWS = mean coincident wind speed, m/s HR = humidity ratio, grams of moisture per kilogram of dry air MWB = mean coincident wet-bulb temp., °C StdD = standard deviation, °C A = airport DP = dew-point temperature, °C 27.50 2001 ASHRAE Fundamentals Handbook (SI) Table 3A Heating and Wind Design Conditions—World Locations Heating Dry Bulb Extreme Wind Speed, m/s Coldest Month MWS/PWD to DB Extr. Annual Daily Elev., m StdP, kPa 0.4% 1% 99.6% 0.4% Mean DB StdD DB Station WMO# Lat.
Long.
Dates 99.6% 99% 1% 2.5% 5% WS MDB WS MDB MWS PWD MWS PWD Max. Min. Max. Min.
1a 1b 1c 1d 1e 1f 1g 2a 2b 3a 3b 3c 4a 4b 4c 4d 5a 5b 5c 5d 6a 6b 6c 6d Lugano 67700 46.00N 8.97E 276 98.05 8293 −3.7 −2.0 7.9 6.3 5.0 7.5 7.4 5.7 7.4 1.5 360 2.8 190 32.2 −8.1 1.7 5.5 Payerne 66100 46.82N 6.95E 491 95.56 8293 −10.2 −7.2 8.1 6.8 5.8 9.1 8.2 7.9 5.8 1.7 40 3.2 230 32.1 −11.6 1.7 4.2 Saentis (Aut) 66800 47.25N 9.35E 2500 74.68 8293 −19.5 −17.1 18.5 16.0 14.2 19.8 −8.8 18.0 −8.6 6.1 260 2.9 210 17.1 −22.0 1.8 4.6 San Bernardino 67830 46.47N 9.18E 1638 83.13 8293 −14.4 −12.2 10.4 9.3 8.4 11.4 −5.4 10.4 −5.5 1.7 310 3.2 120 23.0 −17.3 1.4 4.7 Zurich 66600 47.38N 8.57E 569 94.67 8293 −10.4 −7.8 8.8 7.2 5.8 10.2 6.6 8.8 5.6 2.6 60 2.2 230 31.7 −11.5 1.5 4.4 SYRIA Damascus 400800 33.42N 36.52E 605 94.27 8293 −4.1 −2.2 11.4 10.1 8.9 11.3 10.2 9.6 9.3 1.3 30 3.3 210 40.8 −7.2 1.6 2.1 TAIWAN Hsinchu 467570 24.82N 120.93E 27 101.00 8293 8.6 9.8 9.7 8.1 7.0 8.4 13.8 7.4 13.0 4.1 40 3.9 270 37.3 6.8 2.4 1.7 Hualien 466990 23.98N 121.60E 19 101.10 8293 11.7 12.8 8.4 6.9 5.8 8.7 17.3 7.2 16.8 2.8 250 3.5 140 35.7 9.0 3.4 1.4 Kaohsiung 467400 22.58N 120.35E 9 101.22 8293 11.2 12.6 9.3 7.8 6.7 8.4 18.9 7.4 18.3 3.4 360 5.8 280 36.3 8.9 2.0 1.4 T’aichung 467510 24.18N 120.65E 112 99.99 8293 7.5 8.9 8.8 7.7 7.1 8.8 16.5 8.1 16.1 2.4 30 4.3 240 36.8 2.8 0.9 2.3 T’ainan (593580) 467410 23.00N 120.22E 14 101.16 8293 10.4 11.6 8.8 7.6 6.8 8.6 16.5 8.0 15.9 4.7 20 4.9 200 36.5 8.2 2.9 1.6 Taipei 466960 25.07N 121.55E 6 101.25 8293 8.8 10.0 9.0 8.0 7.3 8.5 17.2 7.9 17.2 1.8 110 5.1 290 36.7 5.7 1.5 2.1 Taipei Intl Apt 466860 25.08N 121.22E 33 100.93 8293 8.8 9.8 13.1 11.9 11.0 12.4 13.2 11.7 13.7 7.3 40 7.1 260 36.0 6.3 0.8 1.8 TAJIKISTAN Dushanbe 388360 38.55N 68.78E 803 92.04 8293 −7.3 −5.1 5.6 4.4 3.5 5.7 5.2 4.2 5.7 1.1 60 1.5 270 40.7 −10.6 3.5 3.2 Khujand (Leninabad) 385990 40.22N 69.73E 428 96.29 8293 −8.4 −6.3 12.9 11.8 10.4 13.6 −0.4 12.2 0.6 6.1 240 5.0 240 40.7 −12.5 3.2 4.8 THAILAND Bangkok 484560 13.92N 100.60E 12 101.18 8293 18.4 19.9 8.3 7.3 6.4 6.2 27.7 5.3 27.4 1.9 40 4.8 180 38.8 16.3 1.7 1.5 Chiang Mai 483270 18.78N 98.98E 314 97.61 8293 11.9 13.1 7.4 5.9 4.9 5.7 22.3 4.3 23.8 0.4 360 3.1 190 39.5 9.0 0.9 1.0 Chiang Rai 483030 19.92N 99.83E 395 96.67 8293 9.6 11.0 4.3 3.5 3.2 3.9 20.6 3.3 21.8 0.0 20 2.1 180 39.0 6.8 0.9 1.4 Chumphon 485170 10.48N 99.18E 5 101.26 8293 19.1 20.2 7.9 6.8 5.9 8.5 28.1 7.7 28.1 0.0 30 4.0 120 37.7 17.2 1.1 1.3 Hat Yai 485690 6.92N 100.43E 35 100.91 8293 20.9 21.8 8.1 7.1 6.2 8.3 28.5 7.3 28.5 0.2 330 3.3 240 37.7 19.6 1.1 0.8 Phetchabun 483790 16.43N 101.15E 116 99.94 8293 13.6 15.2 4.4 3.9 3.4 4.6 26.1 4.1 26.1 0.1 360 2.6 180 40.2 7.9 0.7 10.6 Phrae 483300 18.17N 100.17E 162 99.39 8293 12.9 14.3 4.3 3.5 3.2 3.3 24.7 3.0 26.3 0.2 30 2.2 240 40.6 9.1 1.1 3.2 Tak 483760 16.88N 99.15E 124 99.84 8293 14.0 15.5 8.1 6.3 5.2 3.6 26.8 3.1 26.8 0.0 270 2.1 270 40.9 11.3 0.7 1.6 TRINIDAD & TOBAGO Port of Spain 789700 10.62N 61.35W 15 101.14 8293 20.1 20.9 8.4 7.8 7.2 9.0 28.9 8.2 28.8 0.1 90 5.2 90 34.4 15.8 1.1 6.5 TUNISIA Bizerte 607140 37.25N 9.80E 3 101.29 8293 3.3 4.6 12.9 11.2 10.0 14.4 12.5 12.2 12.1 0.4 320 5.3 100 41.0 0.8 2.7 0.5 Gabes 607650 33.88N 10.10E 5 101.26 8293 5.8 6.8 9.1 7.4 6.6 10.0 15.1 8.3 14.1 1.9 230 4.6 250 41.9 2.5 2.8 1.9 Gafsa 607450 34.42N 8.82E 314 97.61 8293 2.1 3.5 11.8 10.3 8.8 11.5 11.2 9.5 11.4 2.4 60 3.8 240 42.6 −0.5 1.5 0.9 Kelibia 607200 36.85N 11.08E 30 100.97 8293 5.7 6.6 10.1 8.6 7.5 11.2 13.0 9.6 12.8 2.4 300 3.8 300 35.9 3.4 3.5 1.1 Qairouan (Kairouan) 607350 35.67N 10.10E 68 100.51 8293 4.4 5.5 7.4 6.3 5.5 7.2 12.4 6.0 13.7 1.0 240 3.0 180 44.0 1.4 1.7 1.6 Tunis 607150 36.83N 10.23E 4 101.28 8293 4.9 5.9 11.8 10.4 9.2 12.4 12.6 10.6 13.1 2.8 240 4.9 180 41.7 1.7 2.9 0.7 TURKEY Adana 173500 37.00N 35.42E 66 100.53 8293 −0.2 1.1 8.7 7.5 6.4 9.5 8.9 8.3 9.3 2.5 30 3.5 210 39.4 −3.1 1.0 1.9 Ankara 171280 40.12N 32.98E 949 90.43 8293 −16.9 −13.1 9.2 7.9 6.9 8.7 0.6 7.3 0.8 0.4 20 3.5 270 35.0 −19.0 2.7 5.3 Erzurum 170960 39.92N 41.27E 1758 81.91 8293 −30.7 −27.4 10.5 9.8 8.8 11.7 −4.4 10.0 −5.5 0.0 310 4.3 90 31.1 −33.4 1.8 3.9 Eskisehir 171240 39.78N 30.57E 785 92.24 8293 −11.2 −9.1 8.7 7.8 6.9 8.4 −0.2 7.3 −0.5 1.4 120 3.9 320 35.3 −14.7 1.4 3.5 Istanbul 170600 40.97N 28.82E 37 100.88 8293 −3.2 −1.8 10.4 9.5 9.0 11.8 −0.1 10.1 2.8 6.2 360 5.9 60 34.9 −6.0 2.3 3.5 Izmir/Cigli (Cv/AFB) 172180 38.50N 27.02E 5 101.26 8293 −2.0 −0.8 11.6 10.2 9.2 13.7 14.0 11.7 11.7 2.7 360 6.2 350 38.3 −4.3 2.2 1.0 Malatya 172000 38.43N 38.08E 849 91.53 8293 −12.1 −9.1 10.3 9.2 7.9 11.1 −0.4 9.4 0.2 1.6 210 3.2 60 39.1 −16.1 1.4 3.6 Van 171700 38.45N 43.32E 1661 82.90 8293 −14.7 −13.0 7.2 5.6 5.0 6.9 −0.2 5.5 −0.4 1.9 90 1.4 300 32.0 −16.9 2.6 3.2 TURKMENISTAN Ashgabat (Ashkhabad) 388800 37.97N 58.33E 210 98.83 8293 −6.9 −5.0 9.7 8.4 7.4 8.4 5.7 7.3 4.0 1.8 110 4.3 90 43.1 −9.7 1.4 2.2 Dashhowuz (Tashauz) 383920 41.83N 59.98E 88 100.27 8293 −14.9 −12.1 9.9 8.7 8.0 9.2 0.8 8.3 0.7 2.7 200 4.5 360 42.3 −18.2 1.0 3.3 UNITED KINGDOM & NORTHERN IRELAND Aberdeen/Dyce 30910 57.20N 2.22W 65 100.55 8293 −5.7 −3.0 12.8 11.1 9.9 14.8 4.8 13.0 5.4 1.4 360 4.7 170 25.1 −10.3 1.6 4.8 Aberporth 35020 52.13N 4.57W 134 99.73 8293 −3.1 −1.5 18.3 15.9 14.4 20.8 6.7 18.6 7.5 6.4 90 5.6 130 26.6 −4.6 2.7 2.3 Aughton 33220 53.55N 2.92W 56 100.65 8293 −3.5 −2.1 11.8 10.3 9.2 13.1 6.6 11.2 6.4 3.7 130 3.9 130 27.6 −4.7 2.2 2.3 Aviemore 30630 57.20N 3.83W 220 98.71 8293 −9.4 −6.3 12.8 11.1 9.7 14.6 4.7 13.2 5.0 0.5 360 4.0 200 25.1 −12.9 5.3 4.8 Belfast 39170 54.65N 6.22W 81 100.36 8293 −2.8 −1.4 12.7 11.0 9.7 14.5 5.9 12.8 6.9 1.8 180 3.9 110 25.0 −5.7 2.1 1.9 Birmingham 35340 52.45N 1.73W 99 100.14 8293 −6.2 −4.2 10.2 9.0 8.1 11.7 6.5 10.3 6.1 1.9 70 3.9 100 28.9 −9.2 2.5 4.3 Bournemouth 38620 50.78N 1.83W 11 101.19 8293 −5.4 −3.8 12.0 10.4 9.3 13.3 9.4 11.6 8.6 1.9 20 4.4 90 28.7 −8.0 2.3 2.2 Bristol 37260 51.47N 2.60W 11 101.19 8293 −3.5 −1.7 10.5 9.2 8.0 11.9 8.4 10.2 8.3 3.4 70 3.9 90 29.2 −5.2 2.2 3.0 Camborne 38080 50.22N 5.32W 88 100.27 8293 −1.3 0.2 15.2 13.5 12.3 16.3 6.7 15.1 7.4 5.6 50 5.8 100 24.4 −2.9 2.1 3.0 Cardiff 37150 51.40N 3.35W 67 100.52 8293 −4.0 −2.2 14.0 12.3 10.9 16.9 6.9 14.6 6.6 6.2 60 4.0 60 28.5 −5.6 2.2 2.5 Edinburgh 31600 55.95N 3.35W 41 100.83 8293 −5.9 −3.8 12.5 11.0 9.8 14.7 8.3 13.2 6.8 0.7 250 4.2 250 25.9 −9.1 2.1 3.3 Exeter 38390 50.73N 3.42W 30 100.97 8293 −4.3 −2.7 12.2 10.5 9.3 14.7 8.3 12.7 7.5 3.0 40 4.4 150 27.9 −5.8 2.5 1.9 Finningley 33600 53.48N 1.00W 17 101.12 8293 −5.1 −3.3 12.4 10.7 9.6 13.9 7.3 12.2 8.1 1.8 170 4.4 150 28.9 −7.5 2.4 2.1 Glasgow 31400 55.87N 4.43W 8 101.23 8293 −6.2 −4.2 13.4 11.8 10.3 17.6 7.8 15.0 7.0 1.1 270 4.2 230 26.8 −9.6 1.9 3.1 Hemsby 34960 52.68N 1.68E 14 101.16 8293 −3.0 −1.6 13.3 11.8 10.4 15.3 5.3 13.7 4.4 6.5 70 5.4 230 27.2 −5.2 1.9 2.9 Herstmonceux 38840 50.87N 0.33E 17 101.12 8293 −4.5 −2.7 13.7 12.0 10.4 15.2 8.5 13.3 8.5 3.9 40 4.0 130 26.8 −7.8 5.4 5.0 Jersey/Channel Islands 38950 49.22N 2.20W 84 100.32 8293 −2.5 −0.7 14.7 12.9 11.4 17.6 8.1 14.9 8.0 5.6 100 4.6 70 28.3 −4.2 2.2 3.3 Kirkwall 30170 58.95N 2.90W 21 101.07 8293 −1.7 −0.5 18.2 15.2 13.6 18.4 5.0 16.0 4.0 5.1 270 5.0 120 21.1 −3.6 1.5 1.1 WMO# = World Meteorological Organization number Elev. = elevation, m PWD = prevailing wind direction, ° True DB = dry-bulb temperature, °C Lat. = latitude ° Long. = longitude ° StdP = standard pressure at station elevation, kPa WS = wind speed, m/s Climatic Design Information 27.51 Table 3B Cooling and Dehumidification Design Conditions—World Locations Station Cooling DB/MWB Evaporation WB/MDB Dehumidification DP/MDB and HR Range of DB 0.4% 1% 2% 0.4% 1% 2% 0.4% 1% 2% DB MWB DB MWB DB MWB WB MDB WB MDB WB MDB DP HR MDB DP HR MDB DP HR MDB 1 2a 2b 2c 2d 2e 2f 3a 3b 3c 3d 3e 3f 4a 4b 4c 4d 4e 4f 4g 4h 4i 5 Lugano 29.7 21.4 28.2 20.8 27.0 20.0 22.6 27.9 21.7 26.9 20.9 26.0 20.8 16.0 25.9 19.9 15.1 25.2 19.0 14.3 24.3 9.8 Payerne 28.7 19.4 27.0 18.7 25.4 18.0 20.3 27.0 19.3 25.4 18.5 24.3 17.8 13.6 23.5 17.1 13.0 22.6 16.3 12.3 21.3 11.0 Saentis (Aut) 14.1 8.5 12.6 8.0 11.5 6.8 9.8 12.4 8.9 11.2 8.1 10.4 8.8 9.6 10.5 7.9 9.0 9.6 7.0 8.5 9.0 4.3 San Bernardino 20.7 12.7 19.2 12.4 17.9 11.7 14.7 18.1 13.8 17.5 12.8 16.4 13.3 11.6 16.5 12.3 10.9 15.6 11.3 10.2 14.5 8.3 Zurich 28.1 18.8 26.4 18.1 24.8 17.3 19.7 26.5 18.9 25.0 18.1 23.3 17.5 13.4 21.8 16.8 12.8 21.1 16.1 12.3 20.5 8.9 SYRIA Damascus 38.1 17.9 36.9 17.8 35.8 17.6 20.6 29.6 20.0 29.4 19.4 28.9 18.8 14.7 22.3 17.9 13.8 22.0 16.9 13.0 21.6 18.8 TAIWAN Hsinchu 34.0 27.3 33.3 27.2 32.7 27.0 28.1 32.8 27.7 32.4 27.3 32.0 26.8 22.5 31.6 26.4 22.0 31.2 26.0 21.4 30.7 6.9 Hualien 32.0 26.7 31.5 26.8 31.0 26.6 27.7 31.0 27.3 30.7 27.0 30.5 26.7 22.3 30.4 26.3 21.8 30.2 25.9 21.3 30.0 5.4 Kaohsiung 33.1 26.3 32.4 26.1 32.1 26.1 27.5 31.0 27.2 30.7 26.8 30.5 26.9 22.6 29.3 26.2 21.6 29.1 26.0 21.4 29.1 6.4 T’aichung 34.2 27.7 33.8 27.6 33.0 27.4 28.9 33.1 28.4 32.8 27.9 32.3 27.8 24.2 32.4 27.1 23.2 31.9 26.8 22.7 31.5 8.4 T’ainan (593580) 33.2 27.1 32.7 27.0 32.3 26.8 28.1 31.6 27.7 31.3 27.4 31.1 27.3 23.2 30.0 26.9 22.6 29.7 26.5 22.1 29.6 5.5 Taipei 34.6 26.8 33.9 26.7 33.1 26.6 27.7 32.9 27.4 32.4 27.0 32.0 26.4 21.9 30.1 26.1 21.5 29.9 25.9 21.2 29.8 7.4 Taipei Intl Apt 34.1 26.8 33.2 26.7 32.9 26.6 27.9 32.5 27.5 32.1 27.0 31.7 26.9 22.7 30.9 26.2 21.7 30.1 25.9 21.3 30.0 7.3 TAJIKISTAN Dushanbe 37.1 19.6 36.0 19.3 34.8 19.0 21.7 33.7 20.7 32.8 20.0 32.0 17.7 14.0 28.6 16.5 13.0 27.4 15.5 12.1 26.5 14.2 Khujand (Leninabad) 37.1 19.2 35.8 19.1 34.5 18.8 21.3 33.6 20.5 32.6 19.7 32.2 17.0 12.8 26.2 16.0 12.0 26.2 15.1 11.3 25.5 12.8 THAILAND Bangkok 37.1 26.5 36.1 26.1 35.2 25.7 28.8 34.3 28.1 32.7 27.6 31.9 27.5 23.4 31.3 27.1 22.9 30.6 26.7 22.3 30.3 9.3 Chiang Mai 37.8 22.5 36.8 22.4 35.5 22.6 26.1 31.7 25.7 31.2 25.4 30.8 25.0 20.9 28.1 24.2 19.9 27.8 24.1 19.7 27.4 13.6 Chiang Rai 36.8 22.0 35.6 22.0 34.3 22.6 26.2 31.5 25.9 31.1 25.6 30.7 24.9 21.0 28.7 24.6 20.6 28.4 24.3 20.2 28.0 13.9 Chumphon 35.2 26.2 34.2 26.3 33.5 26.2 27.6 33.2 27.2 32.6 26.9 32.1 26.1 21.5 30.8 25.7 21.0 30.3 25.5 20.7 30.1 9.3 Hat Yai 35.0 24.9 34.2 24.9 33.9 24.9 26.7 31.6 26.6 31.5 26.2 30.9 25.9 21.3 28.5 25.3 20.5 27.9 25.2 20.4 27.8 10.0 Phetchabun 38.3 25.5 37.3 25.4 36.2 25.3 27.7 33.5 27.2 32.9 27.0 32.5 26.2 21.9 30.8 25.9 21.5 30.3 25.6 21.1 29.9 11.6 Phrae 38.6 24.6 37.4 24.8 36.3 24.5 27.3 33.1 27.0 32.5 26.7 32.2 26.0 21.8 30.0 25.6 21.3 29.7 25.3 20.9 29.4 12.3 Tak 39.1 23.2 38.1 23.5 37.1 23.3 26.7 32.2 26.3 31.7 26.1 31.3 25.4 20.9 28.6 25.1 20.5 28.5 24.7 20.0 28.2 10.4 TRINIDAD & TOBAGO Port of Spain 33.0 25.1 32.2 25.0 32.0 24.9 26.6 30.6 26.3 30.3 26.1 30.1 25.7 21.0 28.5 25.2 20.4 27.9 25.1 20.2 27.8 7.9 TUNISIA Bizerte 36.0 22.0 33.6 22.2 31.8 22.0 24.9 29.9 24.3 29.0 23.7 28.6 23.6 18.4 27.1 22.9 17.6 27.0 22.2 16.9 26.4 10.1 Gabes 35.6 21.9 33.1 22.8 31.6 23.0 26.4 30.4 25.8 30.0 25.2 29.6 25.2 20.3 29.5 24.5 19.5 29.2 23.7 18.5 28.7 6.5 Gafsa 40.3 20.0 38.5 20.4 36.8 20.2 22.9 33.2 22.3 32.1 21.7 31.5 20.4 15.7 26.0 19.5 14.8 25.7 18.8 14.1 25.4 13.2 Kelibia 31.5 22.6 30.1 23.0 29.2 23.0 25.5 28.5 24.9 27.8 24.3 27.4 24.6 19.7 27.3 24.0 19.0 27.1 23.3 18.1 26.5 7.3 Qairouan (Kairouan) 40.4 21.8 38.0 21.7 36.2 21.4 24.7 32.3 24.0 31.3 23.3 31.0 23.0 17.9 27.1 22.0 16.8 26.7 21.2 16.0 26.5 14.1 Tunis 36.7 22.6 34.2 23.0 32.9 22.6 25.8 31.0 25.0 30.1 24.2 29.4 24.2 19.1 28.0 23.5 18.3 27.5 22.8 17.5 27.2 12.1 TURKEY Adana 36.1 21.6 34.6 21.8 33.2 22.3 26.0 31.7 25.4 30.5 24.9 29.9 24.5 19.6 27.9 24.0 19.0 28.0 23.4 18.3 27.7 11.0 Ankara 32.0 17.3 30.2 17.1 28.8 16.4 18.6 29.0 17.8 28.1 17.0 27.4 14.8 11.8 23.0 13.9 11.1 22.0 13.0 10.5 21.3 15.8 Erzurum 28.9 16.3 27.5 15.6 26.1 15.1 17.8 26.6 16.8 25.7 15.9 24.7 14.2 12.5 23.4 13.1 11.7 22.4 12.1 10.9 21.2 16.6 Eskisehir 32.2 19.8 30.8 19.4 29.2 18.9 21.4 29.2 20.4 28.5 19.7 28.1 18.8 15.0 26.7 17.6 13.9 25.3 16.7 13.1 24.1 14.4 Istanbul 30.2 21.0 29.1 20.8 28.1 20.3 23.2 27.6 22.5 26.7 21.8 25.8 22.0 16.7 25.3 21.1 15.8 24.7 20.2 15.0 24.4 8.5 Izmir/Cigli (Cv/AFB) 35.8 22.1 34.1 21.6 33.0 21.2 23.3 33.1 22.7 32.1 22.0 31.6 20.1 14.8 28.3 19.2 14.0 27.5 18.8 13.6 27.0 12.8 Malatya 36.3 20.0 35.1 19.6 33.9 19.1 21.1 34.6 20.2 33.7 19.5 33.0 15.9 12.5 30.7 14.8 11.7 29.3 13.9 11.0 28.1 15.2 Van 28.8 19.1 27.6 18.7 26.6 18.2 21.0 27.2 20.0 26.4 19.1 25.7 18.7 16.6 26.6 17.6 15.5 25.8 16.5 14.4 25.1 10.8 TURKMENISTAN Ashgabat (Ashkhabad) 40.1 19.7 38.7 19.6 37.4 19.6 22.9 34.7 22.1 33.5 21.3 32.8 18.9 14.1 29.7 17.9 13.2 29.3 17.0 12.4 28.9 13.4 Dashhowuz (Tashauz) 39.2 23.3 37.4 22.6 35.8 21.9 25.1 36.4 24.1 35.1 23.1 33.9 21.2 16.0 33.5 20.2 15.0 32.3 19.1 14.0 31.1 13.5 UNITED KINGDOM & NORTHERN IRELAND Aberdeen/Dyce 21.7 16.8 20.0 15.8 18.5 14.7 17.5 21.0 16.3 19.4 15.3 17.9 15.8 11.3 19.6 14.9 10.7 18.0 14.0 10.0 16.9 7.2 Aberporth 22.3 16.9 20.2 16.0 18.5 15.4 17.7 20.8 16.8 19.0 16.0 17.9 16.6 12.0 18.6 15.9 11.5 17.8 15.2 11.0 16.9 5.2 Aughton 23.9 17.6 22.0 16.6 20.3 15.8 18.3 22.7 17.4 20.8 16.5 19.5 16.7 12.0 19.8 16.0 11.4 18.9 15.2 10.9 17.9 6.0 Aviemore 23.8 16.2 21.4 15.3 19.4 14.3 17.0 22.1 16.0 20.4 14.9 18.4 14.9 10.9 18.6 14.0 10.2 18.1 13.1 9.6 16.9 8.6 Belfast 22.5 16.7 20.7 15.9 19.2 15.3 17.7 21.1 16.8 19.7 16.0 18.4 16.3 11.7 19.0 15.5 11.1 18.1 14.7 10.5 17.5 7.1 Birmingham 25.7 17.5 23.9 16.5 22.3 16.1 18.5 23.8 17.6 22.4 16.7 20.9 16.7 12.0 20.1 15.9 11.4 19.3 15.0 10.8 18.5 9.4 Bournemouth 25.7 18.2 23.8 17.2 22.3 16.5 19.1 24.3 18.1 22.2 17.3 20.7 17.2 12.3 20.5 16.6 11.8 19.4 16.0 11.4 18.9 10.0 Bristol 26.3 18.2 24.5 17.4 22.8 16.6 19.2 24.6 18.2 22.8 17.3 21.2 17.1 12.2 21.4 16.4 11.7 20.0 15.7 11.2 19.1 7.1 Camborne 21.5 16.6 20.0 16.1 18.8 15.7 17.7 19.7 17.0 18.8 16.5 18.1 17.0 12.3 18.4 16.4 11.8 17.7 15.8 11.3 17.1 4.9 Cardiff 25.0 17.8 23.1 17.2 21.4 16.4 18.8 23.3 17.8 21.5 17.0 20.2 17.2 12.4 20.3 16.5 11.8 19.1 15.8 11.3 18.4 8.2 Edinburgh 22.1 16.3 20.4 15.6 19.0 14.8 17.2 20.8 16.3 19.5 15.5 18.2 15.7 11.2 18.8 14.9 10.6 17.6 14.1 10.1 17.1 8.1 Exeter 25.5 18.2 23.7 17.7 22.2 16.8 19.4 24.3 18.4 22.7 17.6 20.9 17.5 12.6 21.2 16.8 12.0 20.0 16.2 11.6 19.3 8.8 Finningley 25.5 17.7 23.7 17.0 22.0 16.1 18.6 23.9 17.7 22.4 16.8 20.9 16.6 11.8 20.4 15.8 11.2 19.5 15.0 10.7 18.5 9.6 Glasgow 23.7 17.0 21.6 16.0 19.6 15.0 17.7 22.4 16.7 20.5 15.7 18.7 15.9 11.3 19.5 15.0 10.7 18.4 14.2 10.1 17.3 8.1 Hemsby 23.5 18.1 21.8 17.1 20.4 16.5 18.7 22.0 17.8 20.9 17.1 19.6 17.3 12.4 20.3 16.6 11.8 19.2 15.9 11.3 18.4 7.7 Herstmonceux 24.7 18.3 23.2 17.4 21.7 16.7 19.1 23.9 18.2 22.0 17.4 20.5 17.4 12.5 20.9 16.7 11.9 19.5 16.1 11.5 19.0 8.5 Jersey/Channel Islands 24.7 18.1 22.8 17.0 21.1 16.5 18.7 23.4 17.8 21.4 17.2 19.9 17.2 12.4 19.5 16.6 11.9 18.6 16.1 11.6 18.2 6.1 Kirkwall 18.0 14.8 16.5 14.0 15.4 13.3 15.5 17.4 14.5 16.0 13.8 14.9 14.6 10.4 16.4 13.7 9.8 15.2 13.1 9.4 14.3 5.1 MDB = mean coincident dry-bulb temp., °C MWS = mean coincident wind speed, m/s HR = humidity ratio, grams of moisture per kilogram of dry air MWB = mean coincident wet-bulb temp., °C StdD = standard deviation, °C A = airport DP = dew-point temperature, °C 27.52 2001 ASHRAE Fundamentals Handbook (SI) Table 3A Heating and Wind Design Conditions—World Locations Heating Dry Bulb Extreme Wind Speed, m/s Coldest Month MWS/PWD to DB Extr. Annual Daily Elev., m StdP, kPa 0.4% 1% 99.6% 0.4% Mean DB StdD DB Station WMO# Lat.
Long.
Dates 99.6% 99% 1% 2.5% 5% WS MDB WS MDB MWS PWD MWS PWD Max. Min. Max. Min.
1a 1b 1c 1d 1e 1f 1g 2a 2b 3a 3b 3c 4a 4b 4c 4d 5a 5b 5c 5d 6a 6b 6c 6d Lerwick 30050 60.13N 1.18W 84 100.32 8293 −2.2 −1.2 19.2 16.9 15.1 20.1 5.5 18.1 5.7 6.3 350 5.3 180 18.7 −4.4 1.9 1.1 Leuchars 31710 56.38N 2.87W 12 101.18 8293 −4.7 −2.9 14.1 12.4 10.9 16.9 6.4 14.7 6.1 2.2 250 5.8 240 25.6 −7.2 2.6 2.7 London, Gatwick 37760 51.15N 0.18W 62 100.58 8293 −5.6 −3.7 10.2 8.9 7.9 11.5 7.0 10.1 6.3 1.5 80 3.7 70 29.4 −8.7 2.5 3.4 London, Heathrow 37720 51.48N 0.45W 24 101.04 8293 −4.0 −2.3 10.0 8.8 7.9 11.4 8.2 10.0 6.4 2.7 20 4.5 90 30.5 −6.3 2.3 2.3 Lyneham 37400 51.50N 1.98W 156 99.46 8293 −5.6 −3.6 11.7 10.2 9.1 13.1 6.3 11.5 5.0 5.1 30 4.4 70 28.5 −8.0 2.3 3.5 Lynemouth 32620 55.02N 1.42W 30 100.97 8293 −2.3 −0.7 20.2 16.9 14.7 21.2 5.8 19.6 5.0 7.1 190 6.5 260 24.6 −4.4 2.8 2.4 Manchester 33340 53.35N 2.27W 78 100.39 8293 −4.2 −2.7 11.2 10.0 8.9 12.7 5.5 11.2 6.3 2.5 90 3.9 130 28.3 −6.4 2.4 2.1 Nottingham 33540 53.00N 1.25W 117 99.93 8293 −4.9 −3.2 10.7 9.5 8.4 12.4 5.0 11.0 6.2 3.5 20 3.6 210 29.1 −7.1 2.6 3.2 Oban 31140 56.42N 5.47W 4 101.28 8293 −3.2 −1.5 13.2 10.6 9.2 15.2 5.8 13.2 6.4 1.4 180 2.9 180 25.5 −5.5 1.8 2.2 Plymouth 38270 50.35N 4.12W 27 101.00 8293 −1.7 −0.3 15.2 13.4 11.8 17.6 9.5 14.9 8.8 3.7 80 4.5 80 27.0 −3.3 2.2 2.1 Stansted Airport 36830 51.88N 0.23E 106 100.06 8293 −5.2 −3.5 11.1 9.7 8.5 12.5 6.9 10.7 6.2 3.2 30 4.5 130 28.6 −7.5 2.3 3.5 Stornoway 30260 58.22N 6.32W 13 101.17 8293 −1.8 −0.4 16.7 14.5 13.0 18.8 7.2 16.4 6.1 2.8 300 4.4 160 21.3 −4.2 1.8 1.7 Valley 33020 53.25N 4.53W 11 101.19 8293 −2.7 −1.1 17.9 15.3 13.7 18.7 7.9 16.5 8.2 4.5 80 4.2 50 26.7 −4.0 2.3 2.1 Wyton Raf 35660 52.35N 0.12W 41 100.83 8293 −5.3 −3.6 11.9 10.4 9.3 13.2 6.6 11.7 6.4 3.4 40 4.4 100 29.2 −8.0 2.4 4.1 UKRAINE Chernihiv (Chernigov) 331350 51.48N 31.28E 137 99.69 8293 −21.6−18.3 9.6 8.5 7.7 9.9 −4.0 8.8 −5.0 2.4 110 4.1 160 31.8 −23.1 2.4 5.2 Chernivtsi (Chernovtsky) 336580 48.27N 25.97E 240 98.47 8293 −16.6−14.1 12.0 9.7 8.4 12.4 −3.5 10.5 −3.4 3.7 320 3.7 110 31.7 −19.1 1.9 3.4 Dnipropetrovs’k 345040 48.37N 35.08E 142 99.63 8293 −17.6−15.3 12.1 10.2 9.2 12.9 −2.8 11.1 −2.1 4.1 50 4.9 90 34.1 −19.3 1.4 3.5 Donets’k 345190 48.07N 37.77E 226 98.64 8293 −18.6−16.3 13.2 11.7 9.7 16.4 −6.0 14.1 −5.0 2.9 70 4.8 100 32.7 −20.8 2.1 3.0 Kerch 339830 45.40N 36.42E 49 100.74 8293 −11.6 −9.5 12.6 10.9 9.5 15.1 −3.6 12.3 −3.6 5.1 30 5.0 40 32.5 −12.9 2.1 3.1 Kharkiv (Khar’kov) 343000 49.87N 36.13E 152 99.51 8293 −19.2−16.8 10.2 9.0 8.2 12.2 −2.4 10.1 −5.8 2.8 30 4.8 110 32.3 −20.8 1.5 4.3 Kherson 339020 46.67N 32.62E 48 100.75 8293 −15.8−13.2 10.4 9.0 8.0 12.3 −3.0 10.5 −3.2 3.3 270 3.4 80 34.0 −16.9 1.2 2.9 Kirovohrad (Kirovograd) 337110 48.48N 32.25E 172 99.28 8293 −19.0−16.5 9.8 8.6 7.8 10.4 −2.4 9.3 −1.9 2.1 310 4.8 100 32.3 −20.4 1.4 2.9 Kryvyy Rih (Krivoy Rog) 337910 47.93N 33.33E 125 99.83 8293 −18.0−15.6 12.2 10.2 9.0 14.1 −4.3 11.8 −3.2 4.0 50 4.3 90 33.4 −19.6 1.3 3.5 Kyyiv (Kiev) 333450 50.40N 30.45E 168 99.32 8293 −19.0−15.9 9.6 8.2 7.1 9.6 −8.5 8.2 −7.2 3.0 270 2.9 180 30.6 −19.4 1.9 5.1 Luhans’k 345230 48.60N 39.27E 62 100.58 8293 −20.1−17.7 13.3 11.2 9.4 14.3 −8.2 12.6 −5.8 2.9 90 4.5 90 34.0 −22.9 1.9 3.2 Mariupol’ (Zdanov) 347120 47.07N 37.50E 70 100.49 8293 −15.5−13.4 15.6 13.9 12.2 17.1 −3.3 15.9 −5.0 5.3 70 4.4 90 31.4 −17.1 1.3 2.4 Odesa 338370 46.48N 30.63E 35 100.91 8293 −14.1 −11.2 12.2 10.3 9.2 13.2 −2.9 11.4 −2.6 5.0 360 4.4 180 33.2 −15.8 1.7 3.5 Poltava 335060 49.60N 34.55E 159 99.43 8293 −19.4−16.6 10.8 9.3 7.9 12.4 −6.6 10.4 −7.0 2.9 360 3.1 70 31.8 −21.2 1.6 4.6 Rivne (Rovno) 333010 50.58N 26.13E 234 98.55 8293 −19.5−16.0 12.2 10.1 8.5 12.1 −2.4 10.5 −1.5 2.8 270 4.2 130 30.5 −20.7 1.5 5.6 Simferopol’ 339460 45.02N 33.98E 181 99.17 8293 −13.1−10.6 12.2 10.3 9.0 12.4 −1.0 11.1 −2.4 3.8 50 4.7 50 33.6 −15.2 1.6 3.2 Sumy 332750 50.88N 34.78E 174 99.25 8293 −21.8−18.5 10.3 9.3 8.1 10.4 −1.9 9.5 −2.5 2.5 330 3.4 130 31.8 −23.3 2.2 5.2 Uzhhorod (Uzhgorod) 336310 48.63N 22.27E 118 99.92 8293 −14.7−12.0 8.4 7.2 6.2 9.2 −1.1 7.3 −0.1 1.3 100 3.0 170 32.5 −17.8 1.8 3.8 Vinnytsya (Vinnitsa) 335620 49.23N 28.47E 298 97.80 8293 −19.1−16.3 12.4 10.3 8.9 12.3 −4.0 11.1 −4.2 3.5 340 4.2 180 30.4 −20.7 1.6 4.5 Zaporizhzhya 346010 47.80N 35.25E 86 100.30 8293 −17.7−15.1 10.8 9.3 7.8 12.5 −2.3 10.5 −1.9 3.7 360 4.2 220 33.6 −18.8 1.2 3.0 Zhytomyr (Zhitomir) 333250 50.27N 28.63E 227 98.63 8293 −19.8−16.4 10.8 9.3 8.2 10.5 0.3 9.4 −1.3 2.9 90 3.8 190 30.5 −20.4 1.7 4.8 UNITED ARAB EMIRATES Abu Dhabi 412170 24.43N 54.65E 27 101.00 8293 10.9 12.0 9.6 8.5 7.7 9.3 20.7 8.2 20.7 2.0 200 4.2 320 46.5 7.7 0.5 1.4 Dubai 411940 25.25N 55.33E 5 101.26 8293 12.0 13.0 9.4 8.3 7.4 9.8 18.8 8.5 19.9 1.8 170 4.9 270 45.5 9.7 1.4 1.0 Ra’s Al Khaymah 411840 25.62N 55.93E 31 100.95 8293 9.7 11.0 8.1 6.9 6.1 7.7 20.3 6.5 21.5 1.0 210 4.6 320 45.9 6.3 0.8 1.3 Sharjah 411960 25.33N 55.52E 33 100.93 8293 9.2 10.7 8.6 7.5 6.7 8.9 19.8 7.5 20.4 2.1 120 4.5 270 46.2 5.3 1.2 2.3 URUGUAY Colonia Del Sacramento 865600 34.45S 57.83W 23 101.05 8293 3.8 5.0 14.1 11.9 10.3 13.5 10.4 11.6 9.4 4.0 50 3.9 360 35.1 1.3 3.3 1.7 Montevideo 865800 34.83S 56.00W 32 100.94 8293 1.8 3.1 14.5 12.5 10.7 13.5 11.4 12.0 12.8 3.5 330 6.4 360 36.1 −0.4 1.7 1.2 Paso De Los Toros 864600 32.80S 56.52W 75 100.43 8293 1.1 2.5 11.5 10.2 8.6 11.0 12.9 9.9 12.9 1.0 280 4.3 330 37.5 −1.1 1.3 1.2 Rocha 865650 34.48S 54.30W 18 101.11 8293 0.9 2.3 10.6 9.1 8.1 10.4 11.8 9.1 11.5 0.6 310 4.1 360 35.6 −1.5 2.4 1.4 Salto 863600 31.38S 57.95W 34 100.92 8293 1.3 2.9 8.6 7.7 6.4 8.5 14.1 7.6 15.1 0.6 120 3.9 20 39.2 −0.9 1.4 1.5 Treinta Y Tres 865000 33.22S 54.38W 46 100.77 8293 0.4 1.8 8.3 6.5 5.5 9.8 11.0 7.5 11.4 0.9 270 2.7 290 37.2 −1.7 1.3 1.0 UZBEKISTAN Samarqand (Samarkand) 386960 39.70N 67.00E 724 92.92 8293 −11.1 −8.2 11.8 9.8 8.3 9.7 3.2 8.3 2.1 3.6 140 4.5 50 38.5 −13.0 1.3 3.5 Tashkent 384570 41.27N 69.27E 489 95.59 8293 −10.3 −7.8 6.0 5.1 4.4 6.1 7.0 5.1 5.7 0.9 90 1.7 300 40.7 −12.2 1.2 3.0 VANUATU Luganville 915540 15.52S 167.22E 44 100.80 8293 18.8 19.9 8.3 7.4 6.4 8.0 24.8 7.3 25.1 0.7 290 4.0 100 31.6 16.5 0.9 1.7 VENEZUELA Caracas 804150 10.60N 66.98W 48 100.75 8293 20.9 21.6 5.5 4.7 4.1 5.4 27.2 4.8 27.4 0.3 140 2.7 340 36.1 15.2 1.2 7.8 VIETNAM Ho Chi Minh City 489000 10.82N 106.67E 19 101.10 8293 20.0 21.0 17.2 11.4 7.7 11.2 28.2 7.3 27.5 1.9 360 4.1 160 38.5 13.1 2.8 11.4 WAKE ISLAND Wake Island 912450 19.28N 166.65E 4 101.28 8293 21.8 22.4 12.8 11.5 10.5 13.5 23.6 12.4 24.1 6.5 40 6.2 80 34.4 19.7 1.7 1.4 WALLIS & FUTUNA ISLAND Wallis Islands 917530 13.23S 176.17W 27 101.00 8293 22.1 22.8 9.2 8.2 7.5 9.3 26.7 8.5 26.5 1.1 160 4.2 100 32.6 19.8 1.7 1.3 YUGOSLAVIA Belgrade 132720 44.82N 20.28E 99 100.14 8293 −11.5 −8.9 11.1 9.1 7.8 10.2 −0.4 8.9 0.1 2.5 10 2.7 120 36.2 −14.6 2.2 4.6 Palic 130670 46.10N 19.77E 105 100.07 8293 −12.5 −9.6 7.6 6.6 5.5 7.7 3.2 6.7 2.5 1.9 50 2.4 180 35.0 −15.8 2.9 4.2 Podgorica (Titograd) 134620 42.37N 19.25E 33 100.93 8293 −4.1 −2.8 10.9 9.2 7.8 10.5 6.1 9.3 5.6 3.1 360 3.3 180 37.1 −6.8 1.2 2.4 ZIMBABWE Harare 677750 17.92S 31.13E 1503 84.52 8293 7.0 8.0 8.9 7.9 7.1 8.4 15.6 7.4 15.8 2.5 120 4.5 60 32.4 4.3 1.8 1.7 WMO# = World Meteorological Organization number Elev. = elevation, m PWD = prevailing wind direction, ° True DB = dry-bulb temperature, °C Lat. = latitude ° Long. = longitude ° StdP = standard pressure at station elevation, kPa WS = wind speed, m/s Climatic Design Information 27.53 Table 3B Cooling and Dehumidification Design Conditions—World Locations Station Cooling DB/MWB Evaporation WB/MDB Dehumidification DP/MDB and HR Range of DB 0.4% 1% 2% 0.4% 1% 2% 0.4% 1% 2% DB MWB DB MWB DB MWB WB MDB WB MDB WB MDB DP HR MDB DP HR MDB DP HR MDB 1 2a 2b 2c 2d 2e 2f 3a 3b 3c 3d 3e 3f 4a 4b 4c 4d 4e 4f 4g 4h 4i 5 Lerwick 15.8 13.5 14.7 12.8 13.9 12.4 14.1 15.1 13.4 14.2 12.8 13.5 13.6 9.8 14.5 13.0 9.4 13.8 12.4 9.1 13.1 3.8 Leuchars 22.1 16.0 20.4 15.2 18.9 14.4 16.9 20.6 16.0 19.2 15.2 17.9 15.4 10.9 18.3 14.6 10.4 17.2 13.9 9.9 16.6 7.8 London, Gatwick 26.4 18.4 24.7 17.4 23.1 16.8 19.3 25.0 18.3 23.0 17.5 21.5 17.3 12.5 21.0 16.5 11.8 19.9 15.8 11.3 19.2 9.8 London, Heathrow 27.4 18.7 25.7 17.7 24.1 17.2 19.6 26.0 18.7 23.8 17.8 22.4 17.4 12.5 21.3 16.7 11.9 20.7 16.0 11.4 20.0 9.2 Lyneham 25.6 18.0 23.7 16.8 22.0 16.0 18.7 24.1 17.6 22.0 16.7 20.6 16.7 12.1 20.2 16.0 11.6 18.7 15.2 11.0 18.3 8.8 Lynemouth 20.6 16.0 19.3 15.3 18.2 14.7 16.9 19.2 16.2 18.3 15.5 17.5 16.0 11.4 17.8 15.2 10.8 17.1 14.6 10.4 16.7 4.9 Manchester 25.2 17.3 23.1 16.4 21.5 15.6 18.3 23.2 17.4 21.7 16.6 20.3 16.5 11.9 20.0 15.7 11.3 19.2 14.8 10.6 18.3 7.6 Nottingham 25.5 18.0 23.6 17.2 21.9 16.3 19.0 24.0 17.9 22.3 16.9 20.8 17.1 12.4 21.3 16.1 11.6 19.8 15.3 11.0 18.6 8.9 Oban 22.7 16.2 20.7 15.3 18.9 14.6 17.1 21.4 16.1 19.4 15.3 17.9 15.5 11.0 18.3 14.7 10.4 17.5 14.0 10.0 16.8 5.8 Plymouth 23.8 17.3 22.1 16.6 20.6 16.1 18.4 22.3 17.6 20.4 17.0 19.4 17.1 12.2 19.5 16.6 11.9 18.7 16.0 11.4 18.0 6.1 Stansted Airport 25.9 17.7 24.2 17.1 22.7 16.4 19.0 24.4 18.0 22.4 17.1 21.0 17.2 12.4 21.0 16.3 11.7 19.5 15.5 11.1 18.8 9.3 Stornoway 18.3 15.1 16.8 14.2 15.7 13.4 15.7 17.6 14.7 16.1 14.0 15.3 14.7 10.5 16.4 14.1 10.1 15.5 13.4 9.6 14.8 4.8 Valley 23.4 17.3 21.2 16.3 19.4 15.4 17.8 22.1 16.9 20.0 16.1 18.5 16.4 11.7 18.9 15.7 11.2 17.8 15.1 10.7 17.1 5.9 Wyton Raf 26.3 18.0 24.5 17.3 22.8 16.6 19.2 24.5 18.1 22.6 17.2 21.4 17.3 12.4 21.2 16.4 11.7 19.8 15.6 11.1 18.9 9.3 UKRAINE Chernihiv (Chernigov) 28.7 19.8 27.2 19.2 25.6 18.2 21.1 26.8 20.2 25.7 19.2 23.9 19.1 14.1 23.9 18.2 13.3 22.8 17.4 12.7 21.9 10.1 Chernivtsi (Chernovtsky) 28.4 19.8 26.8 18.8 25.4 18.3 20.7 26.4 19.8 25.4 19.0 24.2 18.8 14.0 23.6 17.8 13.1 22.6 17.0 12.5 22.0 9.0 Dnipropetrovs’k 30.7 19.6 29.2 19.1 27.7 18.6 21.3 27.7 20.4 26.6 19.6 25.6 19.2 14.2 23.8 18.3 13.4 23.1 17.4 12.7 22.3 10.4 Donets’k 29.8 19.0 28.4 18.9 26.9 18.1 21.0 27.9 20.1 26.4 19.2 24.6 18.6 13.8 23.6 17.9 13.2 22.8 17.1 12.5 22.1 10.9 Kerch 29.6 20.7 28.4 20.6 27.3 20.1 22.8 27.3 21.9 26.8 21.1 25.8 21.2 15.9 25.7 20.2 15.0 24.8 19.4 14.2 24.0 8.1 Kharkiv (Khar’kov) 29.6 19.2 28.1 18.6 26.6 18.2 20.9 26.7 20.0 25.5 19.2 24.5 18.9 14.0 23.1 18.1 13.3 22.8 17.3 12.6 21.8 9.1 Kherson 31.5 20.1 29.9 19.8 28.5 19.1 21.9 28.6 20.9 27.0 20.2 26.1 19.8 14.6 23.9 18.9 13.8 23.7 18.1 13.1 22.6 11.8 Kirovohrad (Kirovograd) 29.9 18.4 28.4 18.1 26.8 17.6 20.5 26.3 19.5 25.3 18.7 24.4 18.6 13.7 22.8 17.6 12.9 22.0 16.7 12.1 20.9 11.4 Kryvyy Rih (Krivoy Rog) 30.7 19.4 29.3 18.8 27.7 18.3 21.1 27.6 20.2 26.6 19.4 25.7 19.0 14.0 23.8 18.0 13.1 23.0 17.1 12.4 22.0 11.7 Kyyiv (Kiev) 28.2 19.7 26.7 18.9 25.2 18.1 20.8 26.3 19.9 25.0 19.1 23.9 19.0 14.1 23.5 18.1 13.3 22.6 17.2 12.5 21.6 9.2 Luhans’k 31.2 19.2 29.5 18.8 27.9 18.0 21.1 28.2 20.2 27.0 19.4 25.6 18.8 13.7 24.3 17.9 12.9 22.5 17.1 12.3 22.1 10.6 Mariupol’ (Zdanov) 29.0 21.6 27.9 20.9 26.7 20.6 23.2 27.1 22.3 26.3 21.4 25.5 22.0 16.8 25.4 20.8 15.6 24.8 19.9 14.7 24.0 8.4 Odesa 30.1 19.6 28.8 19.4 27.2 19.0 22.2 26.1 21.2 26.0 20.3 25.1 20.9 15.6 24.5 19.7 14.5 23.6 18.8 13.7 22.6 10.2 Poltava 29.4 19.0 27.8 18.7 26.4 18.1 20.8 26.3 19.9 25.6 19.1 24.6 18.9 14.0 23.7 17.9 13.1 22.6 17.0 12.4 21.7 9.8 Rivne (Rovno) 27.8 19.5 26.2 18.7 24.7 17.9 20.6 26.0 19.6 24.9 18.7 23.4 18.7 13.9 23.7 17.7 13.1 22.0 16.8 12.3 20.9 10.3 Simferopol’ 30.6 19.3 29.2 18.9 27.8 18.5 21.0 27.2 20.3 26.3 19.6 25.6 19.2 14.3 23.2 18.3 13.5 22.8 17.4 12.7 22.1 11.3 Sumy 28.8 19.2 27.2 18.8 25.6 18.0 20.9 26.3 19.9 25.0 19.0 23.8 19.0 14.1 23.4 18.0 13.2 22.1 17.2 12.5 21.5 9.5 Uzhhorod (Uzhgorod) 30.1 20.5 28.6 19.8 27.1 19.0 21.4 28.2 20.5 27.4 19.7 25.9 18.9 13.9 25.2 18.0 13.1 24.0 17.2 12.5 22.9 10.4 Vinnytsya (Vinnitsa) 27.8 19.0 26.3 18.4 24.9 17.8 20.2 25.4 19.4 24.4 18.5 23.4 18.4 13.8 22.9 17.5 13.0 21.9 16.7 12.3 20.9 10.1 Zaporizhzhya 30.9 19.7 29.5 19.0 28.0 18.5 21.4 27.5 20.5 26.8 19.7 25.7 19.4 14.3 24.3 18.4 13.4 23.4 17.6 12.7 22.2 11.2 Zhytomyr (Zhitomir) 27.8 19.4 26.3 18.5 24.8 17.8 20.4 25.9 19.5 24.8 18.6 23.4 18.5 13.7 23.2 17.6 13.0 21.9 16.8 12.3 20.9 10.5 UNITED ARAB EMIRATES Abu Dhabi 43.8 23.6 42.5 23.4 41.1 23.8 30.3 35.0 29.7 34.3 29.2 34.1 29.2 26.0 32.9 28.8 25.4 32.8 28.0 24.2 32.5 12.8 Dubai 41.9 23.8 40.7 24.0 39.3 24.5 30.2 34.6 29.7 34.2 29.2 34.1 29.2 25.9 33.0 28.8 25.3 32.8 28.0 24.1 32.7 9.7 Ra’s Al Khaymah 43.7 24.6 42.7 25.0 41.7 25.1 30.1 37.6 29.5 37.1 29.0 36.5 28.4 24.8 34.2 27.9 24.1 34.0 27.1 22.9 33.6 12.6 Sharjah 43.1 24.9 41.9 25.2 40.9 25.2 30.1 37.1 29.4 36.1 28.8 35.6 28.8 25.4 33.5 28.0 24.2 33.0 27.1 22.9 32.7 13.3 URUGUAY Colonia Del Sacramento 31.2 23.4 29.9 22.7 28.7 22.2 24.7 29.2 24.0 28.1 23.2 27.2 23.4 18.2 27.7 22.6 17.4 26.7 22.0 16.7 25.9 8.3 Montevideo 31.9 22.1 30.0 21.6 28.2 21.1 24.2 28.6 23.5 27.1 22.7 26.3 23.1 17.9 26.2 22.2 16.9 25.0 21.8 16.5 24.7 9.3 Paso De Los Toros 34.8 22.5 33.0 22.3 31.3 21.8 24.8 30.8 24.1 29.4 23.3 28.4 23.2 18.1 27.5 22.5 17.4 26.7 21.9 16.7 25.8 11.3 Rocha 31.5 22.6 29.8 22.3 28.3 21.9 24.7 28.8 23.9 27.7 23.1 26.5 23.6 18.5 26.8 22.8 17.6 25.8 22.1 16.8 24.9 10.6 Salto 36.4 23.6 34.8 23.4 33.3 23.1 26.1 32.6 25.2 31.7 24.5 30.8 24.2 19.2 29.7 23.5 18.4 28.1 22.8 17.6 27.3 12.2 Treinta Y Tres 33.4 22.6 31.8 22.2 30.1 21.9 24.9 29.9 24.1 28.9 23.4 27.4 23.6 18.5 26.9 22.8 17.6 26.2 22.1 16.9 25.5 11.6 UZBEKISTAN Samarqand (Samarkand) 35.7 19.7 34.5 19.1 33.3 18.7 21.0 33.4 20.0 31.8 19.2 31.5 16.7 13.0 27.1 15.7 12.2 25.0 14.7 11.4 24.6 13.8 Tashkent 38.0 21.4 36.7 20.1 35.2 20.1 24.0 35.0 22.5 33.5 21.2 32.2 20.2 15.8 31.5 18.5 14.2 28.7 17.1 13.0 28.0 14.9 VANUATU Luganville 30.3 25.3 30.0 25.3 29.6 25.2 26.6 29.2 26.2 28.9 26.0 28.7 25.7 21.1 28.3 25.5 20.8 28.0 25.2 20.4 27.7 5.8 VENEZUELA Caracas 33.2 28.7 32.7 28.5 32.0 28.2 30.2 32.0 29.6 31.6 29.1 31.0 29.9 27.2 31.7 29.1 25.9 31.1 28.7 25.3 30.6 7.0 VIETNAM Ho Chi Minh City 35.1 25.2 34.2 25.2 33.9 25.2 27.2 32.2 26.9 31.9 26.6 31.6 26.1 21.5 29.8 25.7 21.0 29.3 25.2 20.4 28.7 8.2 WAKE ISLAND Wake Island 31.8 26.0 31.4 26.0 31.1 25.8 27.4 30.1 26.9 29.8 26.5 29.7 26.6 22.2 29.2 26.0 21.4 28.9 25.6 20.8 28.7 4.5 WALLIS & FUTUNA ISLAND Wallis Islands 31.1 26.9 30.7 26.7 30.4 26.5 27.5 30.3 27.2 30.0 27.0 29.8 26.7 22.4 29.5 26.3 21.8 29.3 26.1 21.6 29.0 4.7 YUGOSLAVIA Belgrade 33.4 21.8 31.8 21.1 29.9 20.5 22.7 30.3 21.8 29.6 21.1 28.6 20.1 15.0 26.8 19.2 14.1 25.4 18.4 13.4 24.4 12.3 Palic 32.2 20.9 30.5 20.5 29.0 19.6 21.9 30.3 21.1 29.0 20.4 27.7 19.2 14.1 25.2 18.4 13.4 24.3 17.6 12.8 23.8 11.3 Podgorica (Titograd) 35.1 21.8 33.8 21.6 32.2 20.9 23.0 32.5 22.4 31.6 21.7 30.5 20.2 14.9 25.9 19.4 14.2 26.4 18.8 13.7 25.8 11.7 ZIMBABWE Harare 30.1 16.6 29.1 16.3 28.3 16.2 20.0 24.7 19.6 24.2 19.2 23.6 18.9 16.5 21.0 18.4 16.0 20.6 18.1 15.7 20.5 11.7 MDB = mean coincident dry-bulb temp., °C MWS = mean coincident wind speed, m/s HR = humidity ratio, grams of moisture per kilogram of dry air MWB = mean coincident wet-bulb temp., °C StdD = standard deviation, °C A = airport DP = dew-point temperature, °C 27.54 2001 ASHRAE Fundamentals Handbook (SI) Table 4A Design Wet-Bulb—Mean Coincident Dry-Bulb Temperature Location and WMO# % Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Annual WB MDB WB MDB WB MDB WB MDB WB MDB WB MDB WB MDB WB MDB WB MDB WB MDB WB MDB WB MDB WB MDB 1a 1b 2a 2b 3a 3b 4a 4b 5a 5b 6a 6b 7a 7b 8a 8b 9a 9b 10a 10b 11a 11b 12a 12c 13a 13b 14a 14b ALABAMA Birmingham 722280 0.4 17.9 19.8 18.9 21.1 20.5 24.9 22.0 26.4 24.0 29.5 25.7 32.2 26.5 32.9 26.2 31.9 25.2 30.4 23.0 26.8 20.9 23.1 19.8 22.3 25.7 31.8 1 17.1 18.8 18.0 20.4 19.8 23.6 21.4 25.7 23.5 28.9 25.3 31.7 26.1 32.4 25.8 31.7 24.8 30.2 22.3 26.2 20.1 22.3 18.9 20.8 25.1 31.2 2 16.3 17.8 17.1 19.2 19.1 22.9 20.7 25.2 23.0 28.3 24.8 31.1 25.8 32.1 25.5 31.6 24.4 30.0 21.7 25.6 19.4 22.1 17.9 19.9 24.6 30.3 Huntsville 723230 0.4 17.1 18.8 17.9 20.1 19.7 24.0 21.9 26.1 23.7 28.9 25.3 31.6 26.4 33.2 26.2 31.6 25.1 30.8 22.6 26.2 20.1 22.4 18.7 20.8 25.6 31.6 1 16.1 17.9 16.9 19.1 18.9 22.8 21.1 25.9 23.1 28.1 25.0 31.3 26.1 32.7 25.7 31.4 24.7 30.2 21.9 25.7 19.2 21.8 17.7 19.4 25.0 30.9 2 15.2 16.8 16.1 18.2 18.1 21.7 20.3 24.8 22.6 27.6 24.6 30.7 25.8 32.1 25.3 31.2 24.3 29.8 21.3 25.0 18.5 21.0 16.9 18.8 24.4 30.0 Mobile 722230 0.4 21.0 22.5 21.4 23.1 22.2 24.6 23.6 27.2 24.9 29.1 26.7 32.1 26.9 32.6 26.8 32.2 26.3 30.7 24.7 28.1 23.3 25.5 22.4 23.8 26.3 31.6 1 20.4 22.1 20.8 22.4 21.8 24.2 23.2 26.7 24.4 28.8 26.2 31.7 26.6 32.2 26.5 31.7 25.9 30.4 24.3 27.4 22.7 24.6 21.9 23.4 25.8 31.0 2 19.8 21.4 20.2 21.8 21.4 23.6 22.7 26.0 24.1 28.6 25.8 31.2 26.3 31.9 26.3 31.4 25.7 30.2 23.8 27.1 22.1 24.4 21.4 22.9 25.5 30.6 Montgomery 722260 0.4 19.4 22.2 20.1 22.9 21.2 25.2 22.9 27.4 24.6 30.4 26.3 33.3 27.0 33.8 26.9 33.3 25.9 31.7 24.0 28.5 22.1 25.6 21.2 24.1 26.3 32.6 1 18.5 20.8 19.3 21.8 20.7 25.0 22.3 26.7 24.1 29.8 25.8 32.4 26.7 33.1 26.6 32.9 25.6 31.4 23.3 27.1 21.3 24.2 20.5 23.2 25.8 31.8 2 17.7 19.8 18.6 21.1 20.1 23.8 21.7 26.3 23.7 29.2 25.6 32.1 26.3 32.7 26.3 32.3 25.2 31.0 22.8 27.2 20.6 23.6 19.6 22.2 25.3 31.1 ARIZONA Flagstaff 723755 0.4 5.2 12.2 6.3 13.8 7.3 15.9 9.6 20.2 11.8 20.2 14.9 24.8 17.3 23.5 17.3 24.0 15.7 21.9 11.9 17.2 8.3 15.2 5.7 11.6 16.3 23.1 1 4.3 10.1 5.4 12.4 6.6 14.6 8.8 19.1 11.3 20.7 14.2 26.1 16.8 23.3 16.8 23.6 15.1 21.6 10.9 17.6 7.4 14.9 4.8 10.7 15.6 22.6 2 3.6 9.4 4.6 11.1 5.9 14.2 8.2 18.4 10.7 20.3 13.6 26.0 16.4 23.1 16.4 23.2 14.5 21.7 10.3 18.2 6.7 14.3 3.9 9.7 14.9 22.1 Phoenix, Intl Airport 722780 0.4 14.6 18.4 15.3 24.2 16.2 27.9 17.9 34.1 19.7 36.3 23.1 36.6 24.7 36.6 25.1 36.8 24.4 35.9 21.2 29.0 17.3 25.2 14.6 18.7 24.2 35.8 1 13.8 18.0 14.7 22.8 15.6 27.3 17.3 33.2 19.1 35.2 22.2 37.4 24.4 36.6 24.7 35.9 23.9 35.7 20.7 28.4 16.4 22.8 13.8 17.4 23.7 35.4 2 13.2 18.2 14.1 21.1 15.0 26.7 16.7 32.4 18.7 34.7 21.4 38.2 24.1 36.0 24.4 35.4 23.4 34.5 20.1 28.2 15.7 22.8 13.2 18.0 23.2 35.0 Prescott 723723 0.4 8.6 14.7 9.7 18.3 10.3 20.1 11.9 24.4 14.7 25.6 17.4 29.6 20.3 27.0 20.6 28.4 18.8 27.7 15.1 21.2 11.3 19.2 8.9 13.4 19.3 27.2 1 7.8 13.8 8.9 16.3 9.7 19.1 11.3 23.5 14.1 24.9 16.8 30.1 19.6 27.0 20.0 27.4 18.2 26.8 14.4 20.6 10.4 17.7 8.0 13.4 18.6 26.7 2 7.0 13.1 8.1 15.6 9.2 18.7 10.8 23.0 13.6 24.7 16.2 29.9 19.2 27.2 19.5 27.6 17.7 26.6 13.7 21.4 9.8 17.4 7.2 12.6 18.0 26.0 Tucson 722740 0.4 12.8 18.3 13.4 22.1 14.2 27.9 15.9 30.9 18.0 32.9 21.0 33.2 22.7 31.0 23.0 32.5 22.2 31.1 19.3 25.8 15.4 23.3 13.4 17.8 22.2 31.0 1 12.2 18.0 12.8 21.4 13.6 26.4 15.2 29.3 17.5 31.9 20.5 33.4 22.3 31.0 22.7 32.5 21.8 31.1 18.8 24.8 14.7 22.1 12.4 17.5 21.7 30.4 2 11.6 17.9 12.3 20.2 13.0 25.6 14.6 28.9 16.9 30.8 20.0 33.8 22.0 30.8 22.4 31.5 21.4 30.6 18.2 25.6 14.1 21.7 11.9 17.2 21.3 30.1 ARKANSAS Fort Smith 723440 0.4 16.2 19.8 16.5 21.4 19.7 25.6 21.9 28.0 24.5 30.8 26.3 33.3 26.7 34.9 26.8 34.9 25.6 32.2 23.0 27.9 19.9 23.7 17.8 21.7 26.0 33.6 1 14.9 17.9 15.6 19.9 18.9 24.7 21.3 26.6 23.8 30.1 25.9 32.9 26.3 34.2 26.4 34.3 25.1 31.6 22.3 27.4 19.1 23.3 16.9 20.2 25.4 32.8 2 13.5 15.9 14.8 19.1 18.1 23.7 20.6 26.4 23.2 28.9 25.5 32.3 25.9 33.6 26.0 33.7 24.7 31.2 21.5 26.3 18.3 22.2 15.8 18.7 24.9 32.1 Little Rock, AFB 723405 0.4 17.9 20.8 17.9 20.5 20.7 24.9 22.8 27.3 24.8 30.3 26.9 33.4 27.4 34.3 27.4 33.7 26.3 32.3 23.8 28.1 21.0 24.2 19.3 21.6 26.7 33.3 1 17.1 19.1 17.2 20.1 20.0 24.1 22.1 26.2 24.3 29.9 26.5 32.6 27.0 34.1 26.9 33.2 25.8 31.4 23.0 27.2 20.2 23.1 18.3 20.4 26.1 32.7 2 15.9 18.1 16.2 19.2 19.3 23.4 21.5 25.7 23.9 29.4 26.1 31.8 26.7 33.9 26.6 32.8 25.4 30.7 22.3 26.4 19.6 22.2 17.3 19.3 25.6 31.8 CALIFORNIA Arcata/Eureka 725945 0.4 14.3 15.8 14.9 17.3 13.6 15.6 14.5 17.1 15.6 18.7 16.3 19.3 16.8 19.0 17.8 20.7 18.1 22.5 16.7 21.6 15.7 17.7 14.6 15.7 16.8 19.7 1 13.6 15.1 14.1 16.1 13.0 15.1 13.6 16.1 14.9 17.7 15.6 18.4 16.2 18.4 17.2 19.5 17.4 20.8 16.0 19.2 15.0 16.9 13.8 15.2 15.9 18.6 2 13.0 14.5 13.4 15.2 12.4 14.5 12.9 15.3 14.3 16.6 15.1 17.8 15.7 18.1 16.6 18.8 16.7 19.8 15.4 18.6 14.4 16.3 13.2 14.6 15.3 17.7 Bakersfield 723840 0.4 16.2 19.9 16.7 22.1 17.7 25.1 19.1 31.6 20.8 36.2 22.5 38.4 23.6 39.2 23.9 36.5 22.7 35.7 20.4 33.7 17.1 23.2 15.2 18.4 22.6 36.6 1 14.8 18.3 15.9 21.1 16.9 23.7 18.3 30.3 20.1 35.2 21.8 38.3 23.1 38.3 23.3 35.6 22.1 33.1 19.8 32.7 16.3 23.3 14.3 17.4 21.8 35.6 2 13.9 17.4 15.2 20.2 16.2 22.9 17.5 28.9 19.6 33.6 21.2 37.1 22.6 37.3 22.7 35.8 21.6 33.7 19.2 31.0 15.7 22.4 13.4 17.2 21.2 35.1 Barstow/Daggett 723815 0.4 13.3 19.1 13.6 20.4 14.4 25.6 15.8 31.6 18.5 34.9 20.6 40.2 23.3 35.9 23.6 34.6 22.4 31.2 18.3 33.2 15.1 26.4 14.0 20.3 22.4 35.1 1 12.2 17.3 12.9 20.7 13.8 25.1 15.2 30.9 17.8 34.1 20.1 39.7 22.8 35.7 23.0 34.7 21.8 32.7 17.7 32.5 14.2 23.7 12.3 18.8 21.6 35.1 2 11.3 16.7 12.2 19.9 13.3 24.5 14.7 29.8 17.3 32.9 19.6 39.3 22.4 35.6 22.4 35.3 21.1 32.0 17.1 31.3 13.5 22.7 11.3 16.9 20.7 35.0 Fresno 723890 0.4 15.2 17.5 16.3 20.3 17.9 23.9 19.0 30.3 21.1 35.3 22.4 37.6 23.7 38.8 24.1 37.9 22.4 36.4 20.2 33.6 16.6 22.3 15.1 16.7 22.7 36.8 1 13.9 16.9 15.7 19.8 16.9 23.4 18.3 28.6 20.4 34.0 21.9 36.8 23.3 38.0 23.3 36.7 21.8 35.8 19.6 32.2 15.9 21.7 13.7 15.9 21.9 35.6 2 13.0 15.6 15.0 19.4 16.2 22.3 17.5 27.5 19.7 32.9 21.3 36.1 22.8 37.2 22.8 36.3 21.3 34.2 19.0 30.8 15.4 21.4 12.8 15.0 21.2 34.6 Long Beach 722970 0.4 15.8 20.3 16.4 21.6 17.0 23.8 18.3 27.1 19.6 27.9 20.8 28.8 22.1 30.2 23.1 31.6 23.0 31.1 20.8 28.4 18.4 23.7 16.2 18.9 21.9 29.4 1 15.3 19.5 15.9 20.2 16.4 22.2 17.6 25.1 18.9 25.9 20.2 27.9 21.7 29.0 22.4 30.6 22.4 29.6 20.2 26.7 17.8 22.1 15.6 19.3 21.2 28.0 2 14.9 18.7 15.4 19.3 15.8 21.1 16.9 23.6 18.2 23.9 19.7 26.6 21.2 28.2 21.9 29.5 21.9 28.9 19.7 25.8 17.3 21.6 15.1 18.6 20.6 26.8 Los Angeles 722950 0.4 15.8 18.7 16.2 19.6 16.3 20.5 17.7 23.3 18.6 23.8 19.9 25.9 21.3 25.7 22.1 26.9 21.8 28.1 20.2 25.0 17.9 22.1 16.1 18.4 21.0 25.8 1 15.3 18.1 15.7 18.9 15.8 19.8 17.0 21.9 17.9 22.4 19.3 24.4 20.8 25.0 21.6 26.7 21.3 26.9 19.6 24.1 17.4 21.1 15.4 18.0 20.3 24.7 2 14.8 17.5 15.2 18.2 15.3 18.8 16.5 21.1 17.4 21.4 18.8 23.6 20.4 24.5 21.2 25.7 20.8 26.1 19.1 23.2 16.9 20.3 14.9 17.5 19.8 23.8 Sacramento, Metro 724839 0.4 14.6 15.8 16.1 19.2 17.7 22.8 19.7 28.0 21.2 32.6 22.7 36.9 23.3 37.9 23.9 36.6 21.6 35.2 19.6 31.8 17.1 21.6 14.6 16.2 22.0 35.7 1 13.6 15.1 15.2 18.2 16.7 22.0 18.6 26.7 20.3 31.7 21.9 36.2 22.5 36.7 22.4 35.8 21.0 34.2 19.0 30.3 16.3 21.1 13.6 15.4 21.1 34.3 2 12.8 14.4 14.5 17.4 15.8 20.8 17.6 25.3 19.6 30.7 21.2 34.9 21.9 35.7 21.7 35.2 20.5 32.9 18.4 29.1 15.5 19.8 12.8 14.7 20.3 32.8 San Diego, Intl Airport 722900 0.4 16.6 18.7 16.9 19.7 16.6 20.8 18.2 22.8 19.0 22.7 20.9 25.4 22.9 26.7 23.3 27.2 23.9 28.2 20.8 25.0 18.6 21.7 16.8 18.4 22.5 26.3 1 16.1 18.0 16.3 19.2 16.2 19.9 17.6 21.8 18.4 22.4 20.2 24.4 22.3 26.1 22.8 26.7 23.3 27.1 20.3 24.2 18.0 21.3 16.1 18.4 21.7 25.6 2 15.6 17.7 15.9 18.8 15.7 19.2 16.9 21.3 17.9 21.7 19.8 23.9 21.7 25.5 22.4 26.2 22.7 26.6 20.0 23.5 17.5 20.7 15.6 18.1 20.9 24.7 San Francisco 724940 0.4 14.8 16.2 15.2 17.7 15.4 19.6 16.9 24.7 17.4 25.6 18.6 28.6 18.7 28.4 18.6 27.1 19.1 28.6 18.1 26.2 15.9 18.8 15.0 16.0 18.0 25.9 1 14.1 15.2 14.5 16.5 14.7 18.1 15.7 21.8 16.6 23.8 17.7 26.5 18.0 26.2 17.9 25.0 18.3 25.9 17.4 23.8 15.3 18.2 14.4 15.5 17.2 23.8 2 13.4 14.5 13.9 16.1 14.1 17.3 14.9 19.7 15.9 22.6 16.9 24.4 17.4 24.4 17.4 23.6 17.7 24.5 16.9 22.3 15.0 17.8 13.7 14.9 16.6 22.1 Santa Maria 723940 0.4 15.7 18.6 16.6 21.4 16.5 22.8 17.6 27.6 17.3 24.1 17.9 26.6 20.4 23.4 19.7 28.2 20.8 30.3 18.6 28.3 16.7 22.9 15.4 19.3 19.1 27.2 1 15.1 18.3 15.8 21.0 15.9 21.3 16.9 26.2 16.7 23.1 17.4 25.3 19.4 25.8 19.2 27.0 20.0 28.3 18.1 25.9 16.1 20.7 14.7 18.9 18.2 25.3 2 14.5 17.9 15.2 19.3 15.2 20.0 16.2 24.8 16.2 21.9 16.9 24.2 18.6 25.9 18.7 25.9 19.1 26.8 17.6 24.8 15.7 20.4 14.1 19.3 17.5 23.9 COLORADO Alamosa 724620 0.4 3.2 8.8 4.6 11.9 6.3 16.8 8.4 19.4 11.2 21.3 14.8 24.8 16.5 25.1 17.3 24.4 14.6 22.7 10.9 18.4 6.3 14.4 3.1 8.3 15.4 24.0 1 2.0 7.8 3.4 10.5 5.5 15.9 7.8 18.9 10.6 21.2 14.1 24.2 15.8 24.7 16.1 24.1 13.7 22.4 10.2 17.4 5.6 13.7 2.1 7.1 14.8 23.2 2 1.1 6.7 2.7 9.2 4.7 14.7 7.3 17.5 10.1 20.3 13.7 24.4 15.5 24.3 15.6 23.8 13.1 21.5 9.3 17.8 4.9 12.3 1.2 6.5 14.2 22.7 Boulder 724699 0.4 6.7 16.1 7.9 18.8 8.9 21.3 11.8 23.8 14.9 23.5 18.0 27.9 19.4 27.6 18.9 26.9 16.9 26.6 12.4 24.1 9.3 20.6 7.2 18.1 18.1 27.2 1 5.8 14.8 6.8 16.5 8.0 20.0 11.1 22.5 14.2 22.9 17.4 27.4 18.8 27.7 18.4 26.8 16.2 25.0 11.8 23.6 8.6 19.5 6.3 16.2 17.3 26.6 2 4.9 13.6 5.8 15.4 7.3 18.7 10.5 21.4 13.7 22.8 16.9 26.8 18.3 27.8 17.9 26.3 15.6 24.8 11.3 22.9 7.8 18.6 5.4 14.7 16.7 25.8 Colorado Springs 724660 0.4 5.7 15.4 6.8 18.2 7.8 19.9 10.4 21.7 13.6 22.1 16.6 26.3 18.2 26.3 18.0 25.4 16.1 24.8 11.7 24.1 7.8 19.4 5.7 16.1 17.1 25.3 1 4.7 14.5 5.6 16.0 6.8 18.2 9.7 20.4 13.0 21.2 16.1 25.6 17.7 26.2 17.5 25.1 15.4 24.1 11.0 22.8 7.1 18.1 4.9 14.7 16.4 25.1 2 3.8 12.9 4.5 14.3 6.1 17.3 9.1 19.9 12.4 20.9 15.7 25.1 17.3 25.7 17.2 24.8 14.8 23.8 10.4 21.8 6.5 17.1 4.1 13.0 15.8 24.6 Eagle 724675 0.4 3.2 6.4 4.7 10.6 7.4 15.2 9.9 21.3 12.8 24.6 16.3 29.1 18.1 28.8 17.6 28.7 15.2 25.4 11.8 22.2 7.4 15.6 3.9 8.2 16.6 26.8 1 2.4 5.3 3.8 9.1 6.4 14.6 9.2 19.9 12.2 24.0 15.4 27.5 17.3 28.1 16.9 27.0 14.4 25.2 10.9 21.3 6.4 13.8 3.0 6.9 15.7 25.6 2 1.6 4.7 3.1 7.9 5.6 13.7 8.4 18.7 11.7 22.8 14.8 27.3 16.8 27.4 16.4 25.8 13.9 24.6 10.2 19.8 5.6 12.4 2.1 5.5 15.0 24.6 Grand Junction 724760 0.4 4.8 8.9 7.6 13.1 9.9 19.8 11.9 24.4 14.9 26.1 17.6 30.2 19.4 30.4 19.3 30.1 17.2 27.1 13.6 19.8 9.3 16.9 6.1 10.2 18.4 28.9 1 4.0 8.2 6.8 13.1 8.9 17.6 11.3 23.1 14.3 25.7 17.0 30.7 19.0 29.4 18.8 29.3 16.6 26.0 12.9 20.3 8.6 15.1 5.1 8.9 17.7 28.3 2 3.1 6.7 5.9 11.3 8.1 17.1 10.7 22.4 13.7 25.5 16.4 30.7 18.6 29.2 18.4 28.4 16.1 26.1 12.3 20.9 7.9 13.9 4.0 7.6 17.1 27.9 WMO# = World Meteorological Organization number WB = wet-bulb temperature, °C MDB = mean coincident dry-bulb temperature, °C Climatic Design Information 27.55 Table 4B Design Dry-Bulb—Mean Coincident Wet-Bulb Temperature Location and WMO# % Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Annual DB MWB DB MWB DB MWB DB MWB DB MWB DB MWB DB MWB DB MWB DB MWB DB MWB DB MWB DB MWB DB MWB 1a 1b 2a 2b 3a 3b 4a 4b 5a 5b 6a 6b 7a 7b 8a 8b 9a 9b 10a 10b 11a 11b 12a 12c 13a 13b 14a 14b ALABAMA Birmingham 722280 0.4 21.0 16.0 24.2 16.3 27.7 17.3 29.9 19.4 33.3 21.6 35.3 24.0 37.1 24.6 35.8 24.3 34.7 23.3 30.1 20.4 25.7 18.2 22.9 18.2 34.7 23.9 1 19.6 15.3 22.8 15.6 26.6 17.1 29.1 19.0 32.0 21.4 34.4 23.6 35.9 24.3 34.8 23.8 33.7 23.0 29.2 20.2 24.7 17.9 21.9 17.9 33.4 23.8 2 18.6 15.2 21.3 14.9 25.3 16.8 28.3 18.4 30.8 21.5 33.5 23.4 34.9 24.4 34.0 24.0 32.7 23.0 28.2 19.9 23.7 17.3 20.8 17.4 32.3 23.5 Huntsville 723230 0.4 19.8 15.3 22.7 15.9 26.7 16.8 29.9 19.8 32.5 20.8 35.1 23.4 36.5 24.6 35.8 24.1 34.2 23.3 29.6 20.5 25.1 17.5 21.9 17.9 34.5 23.8 1 18.7 15.0 21.3 14.6 25.4 16.8 28.9 19.1 31.4 21.2 34.1 23.2 35.6 24.3 34.9 23.7 33.0 23.3 28.6 19.7 23.9 17.2 20.6 17.3 33.2 23.6 2 17.6 14.2 19.9 14.1 24.2 16.3 27.9 18.3 30.5 21.3 33.3 23.2 34.7 24.4 34.0 23.7 32.1 22.7 27.7 19.1 22.7 16.9 19.4 15.9 31.9 23.3 Mobile 722230 0.4 23.9 19.6 25.1 18.4 27.8 19.4 30.2 21.3 33.2 22.4 35.2 24.7 35.7 25.1 35.2 25.3 34.1 24.2 31.2 22.9 27.6 21.1 24.9 21.3 34.3 24.7 1 23.0 19.3 23.9 18.3 26.8 18.4 29.3 20.8 32.2 22.5 34.5 24.5 35.0 25.3 34.5 25.1 33.4 24.2 30.4 21.9 26.6 20.5 24.1 20.7 33.4 24.6 2 22.0 18.6 23.0 18.4 25.9 18.5 28.5 20.4 31.4 22.1 33.8 24.2 34.4 25.0 33.8 24.9 32.8 24.2 29.7 21.4 25.8 20.2 23.4 20.3 32.5 24.4 Montgomery 722260 0.4 23.3 17.7 25.0 18.2 28.3 19.1 30.1 20.6 33.4 22.4 36.3 24.4 36.6 25.6 36.6 24.7 35.3 23.8 31.6 22.3 27.7 20.1 25.6 19.9 35.1 24.6 1 22.0 16.8 23.8 17.1 27.3 18.4 29.3 20.3 32.4 22.4 35.3 24.3 35.7 25.1 35.6 24.8 34.4 23.7 30.6 21.5 26.6 19.5 24.2 19.2 34.0 24.4 2 20.7 16.4 22.6 16.6 26.4 17.9 28.6 19.9 31.6 22.4 34.4 24.0 34.9 25.0 34.7 24.9 33.6 23.7 29.7 20.8 25.5 18.7 23.0 18.8 32.9 24.3 ARIZONA Flagstaff 723755 0.4 14.2 4.7 16.2 5.8 17.8 6.4 22.4 8.4 26.1 10.1 31.8 13.3 31.3 13.7 29.8 13.9 27.8 13.1 25.2 9.7 19.3 6.8 14.6 4.7 29.5 13.1 1 12.2 3.6 14.3 4.5 16.8 5.7 21.1 7.8 25.1 9.9 30.6 12.7 30.4 13.6 28.8 14.2 26.7 12.6 24.1 9.3 18.0 6.3 13.3 3.8 28.1 12.9 2 10.7 2.9 12.9 3.6 15.8 5.2 20.0 7.3 24.1 9.5 29.6 12.3 29.6 13.3 27.9 14.0 25.9 12.3 22.9 9.3 16.6 5.9 11.8 3.2 26.7 12.8 Phoenix, Intl Airport 722780 0.4 25.6 12.0 29.1 13.0 32.6 14.9 37.1 16.7 40.8 18.2 45.2 20.2 44.7 21.3 43.6 22.4 42.2 21.3 38.3 18.1 30.9 14.5 25.3 11.8 43.2 20.9 1 24.2 11.4 27.7 12.7 31.3 14.3 35.8 16.4 39.7 17.7 44.2 20.0 43.8 21.4 42.8 22.3 41.2 21.1 37.1 18.0 29.8 14.5 24.2 11.5 42.0 20.9 2 23.1 11.1 26.4 12.4 30.0 13.9 34.7 15.9 38.5 17.4 43.3 19.7 43.2 21.4 42.1 22.2 40.3 21.2 36.1 18.0 28.7 13.9 23.0 10.9 40.9 20.8 Prescott 723723 0.4 18.2 7.4 21.2 8.2 23.6 9.1 27.6 10.6 30.8 12.8 36.1 15.3 36.3 16.0 34.4 16.9 32.7 15.9 29.7 12.4 23.0 9.4 17.9 6.9 34.2 15.7 1 16.7 6.8 19.8 7.8 22.2 8.6 26.3 10.5 29.6 12.3 35.2 15.2 35.3 16.1 33.4 17.1 31.7 15.7 28.7 12.2 21.8 9.3 16.7 6.4 32.7 15.6 2 15.3 6.0 18.4 7.1 21.0 8.4 25.1 10.1 28.6 12.1 34.2 14.6 34.4 16.1 32.6 16.8 30.8 15.3 27.4 12.0 20.7 8.7 15.6 5.9 31.4 15.4 Tucson 722740 0.4 25.3 10.7 28.1 12.1 31.0 13.3 35.2 14.5 38.5 16.3 42.7 18.8 41.8 19.1 39.7 20.2 38.6 18.9 35.8 16.2 29.4 13.3 25.0 10.6 40.1 18.5 1 23.7 10.2 26.4 11.3 29.5 12.7 33.8 13.8 37.4 16.1 41.4 18.5 40.8 19.1 38.9 20.3 37.8 18.9 34.6 16.0 28.3 12.7 23.9 10.3 38.8 18.4 2 22.5 9.9 25.2 10.8 28.2 12.1 32.4 13.6 36.2 15.6 40.6 18.2 39.9 18.9 38.1 20.0 36.9 18.6 33.4 15.5 27.3 12.2 22.7 10.1 37.6 18.4 ARKANSAS Fort Smith 723440 0.4 21.1 14.2 24.2 13.9 28.1 17.4 31.0 19.4 32.5 22.5 36.1 24.7 39.3 23.8 39.1 24.4 36.1 23.2 32.1 20.3 26.5 17.5 22.3 16.7 37.0 24.3 1 19.2 13.1 22.4 13.8 26.7 16.6 29.7 19.2 31.8 22.2 35.0 24.4 38.3 24.1 38.0 24.2 34.9 23.4 30.9 20.2 25.3 17.3 21.0 16.1 35.4 24.3 2 17.6 12.3 20.7 13.4 25.4 16.7 28.6 18.7 30.9 21.9 34.2 24.3 37.3 24.4 37.0 24.3 33.9 23.4 29.6 20.0 24.0 16.9 19.4 14.6 34.0 24.1 Little Rock, AFB 723405 0.4 21.8 15.9 24.6 14.4 27.7 18.4 29.9 20.3 32.9 22.8 36.2 24.4 39.0 24.9 38.3 24.7 35.5 24.4 31.5 20.9 26.4 18.8 22.4 18.2 36.3 25.1 1 20.2 15.2 22.5 14.6 26.4 17.8 28.8 19.6 32.1 22.4 35.2 24.6 37.8 25.7 37.2 24.6 34.3 23.9 30.2 20.6 24.9 18.4 21.2 17.1 34.8 24.9 2 18.8 14.6 20.9 14.4 25.1 17.5 27.9 19.6 31.3 22.2 34.4 24.2 36.7 25.6 36.2 25.1 33.4 23.8 29.2 20.5 23.7 17.8 19.8 16.5 33.6 24.6 CALIFORNIA Arcata/Eureka 725945 0.4 18.5 11.1 19.7 12.9 18.6 11.8 18.9 12.7 20.2 14.9 20.9 15.7 20.6 15.4 21.8 16.7 25.3 15.9 24.3 15.2 19.6 13.6 17.3 11.7 21.3 15.5 1 17.0 11.7 17.9 12.3 16.8 12.1 17.4 12.3 18.4 14.4 19.4 15.0 19.6 15.2 20.6 16.6 22.7 15.9 21.9 14.7 18.4 13.4 16.4 12.3 19.7 15.0 2 15.9 11.6 16.8 11.9 15.7 11.5 16.3 12.2 17.2 13.8 18.4 14.6 19.0 15.2 19.7 15.9 21.1 16.0 19.9 14.5 17.4 13.6 15.6 11.9 18.6 14.6 Bakersfield 723840 0.4 22.2 13.8 25.3 14.3 28.1 15.4 33.9 17.6 38.0 19.7 41.3 21.0 41.7 22.3 41.2 22.2 39.4 21.2 36.0 19.6 27.7 14.9 21.6 13.3 39.9 21.3 1 20.5 13.5 23.8 13.7 26.6 14.8 32.1 17.2 36.6 19.4 40.2 21.1 40.8 21.6 40.2 21.7 38.3 21.2 34.4 19.1 26.2 14.7 20.1 12.6 38.5 20.8 2 19.1 12.8 22.6 13.7 25.3 14.8 30.7 16.4 35.3 18.8 39.0 20.7 40.0 21.4 39.3 21.3 37.2 20.6 33.2 18.5 24.8 14.4 19.0 11.9 37.1 20.3 Barstow/Daggett 723815 0.4 22.5 10.7 26.2 12.3 29.7 13.0 34.4 15.3 37.5 17.2 42.5 19.6 43.6 21.1 43.0 20.8 40.2 19.5 36.7 17.5 28.6 14.2 23.1 11.8 41.8 20.2 1 21.3 10.4 24.7 11.3 28.2 12.6 32.9 14.6 36.4 17.0 41.7 19.3 42.7 20.6 42.2 20.8 39.3 19.5 35.4 17.0 27.1 13.0 21.7 11.0 40.5 19.7 2 20.1 9.4 23.6 10.8 26.9 12.2 31.6 14.1 35.3 16.4 40.8 18.9 42.1 20.2 41.3 20.7 38.3 18.9 34.1 16.4 25.7 12.4 20.2 10.1 39.1 19.2 Fresno 723890 0.4 19.4 12.9 23.4 14.7 26.7 16.3 32.8 17.1 37.3 19.8 40.5 21.3 41.4 22.1 40.8 22.3 38.8 21.4 35.3 19.7 26.0 14.7 19.2 12.7 39.6 21.4 1 17.9 12.7 22.2 14.2 25.2 15.4 31.4 17.0 36.1 19.4 39.6 20.9 40.7 21.7 39.9 21.6 37.7 20.9 33.8 18.9 24.6 14.6 17.8 12.1 38.1 20.9 2 16.8 12.3 21.0 13.8 24.0 14.8 29.9 16.4 34.9 18.9 38.6 20.5 39.7 21.6 38.9 21.5 36.6 20.3 32.2 18.1 23.3 14.3 16.6 12.1 36.6 20.4 Long Beach 722970 0.4 27.8 13.5 27.6 14.1 28.4 14.8 31.6 15.3 31.5 17.3 33.5 19.5 33.3 20.5 34.4 21.6 36.4 19.7 35.4 17.3 30.6 14.9 27.0 12.6 33.2 19.5 1 26.2 13.1 26.2 13.5 25.9 14.6 29.1 15.6 28.9 17.7 30.6 19.2 31.7 20.7 32.7 21.4 34.2 20.3 32.9 17.2 28.6 14.6 25.4 12.6 30.9 19.2 2 24.3 12.7 24.6 13.5 24.2 13.9 27.0 15.3 26.6 16.9 28.8 19.0 30.4 20.2 31.3 21.0 32.4 20.3 30.8 16.8 26.9 13.9 23.8 12.3 29.1 19.0 Los Angeles 722950 0.4 26.4 12.9 26.3 13.4 25.4 13.1 27.8 15.2 27.1 16.9 29.5 18.7 28.0 19.8 28.9 20.7 33.1 18.8 32.4 16.4 28.8 14.5 26.1 12.4 29.2 17.7 1 24.8 12.6 24.3 12.8 23.2 13.5 25.1 14.8 24.4 17.1 26.1 18.5 26.7 19.9 27.7 20.9 30.6 19.1 29.9 16.0 26.9 13.7 24.3 12.2 27.0 17.6 2 23.0 11.9 22.8 12.7 21.7 12.7 23.4 14.7 22.8 16.7 24.4 18.1 25.7 19.9 26.7 20.6 28.4 19.4 27.8 15.4 25.4 13.8 22.8 11.7 25.4 17.9 Sacramento, Metro 724839 0.4 17.9 12.1 21.9 14.2 25.0 15.7 30.3 18.2 35.7 19.1 39.4 21.0 40.2 21.9 39.4 21.7 37.9 20.6 34.3 18.9 25.2 15.1 17.8 12.5 37.8 20.8 1 16.5 11.9 20.7 13.9 23.6 15.3 28.9 16.9 34.2 18.9 38.0 20.7 38.8 21.4 38.3 21.2 36.5 20.1 32.8 18.3 23.8 14.8 16.7 12.5 36.0 20.3 2 15.5 11.9 19.4 13.6 22.3 14.8 27.5 16.6 32.9 18.6 36.7 20.2 37.6 21.1 37.1 20.8 35.1 19.5 31.1 17.7 22.2 14.3 15.7 11.8 34.2 19.7 San Diego, Intl Airport 722900 0.4 25.8 12.7 26.1 13.7 25.6 13.8 27.8 14.6 26.8 15.7 29.5 20.0 29.3 21.5 29.6 21.7 32.7 20.4 32.2 16.9 28.1 14.1 24.7 11.9 29.4 19.6 1 24.3 12.4 24.6 13.3 23.8 13.4 25.8 15.1 24.6 17.1 26.7 19.4 28.1 21.2 28.6 21.8 30.8 20.4 29.7 16.7 26.2 13.8 23.4 11.9 27.4 19.4 2 22.7 12.1 22.8 12.8 22.3 13.2 23.9 14.8 23.1 17.0 25.2 19.4 26.9 21.1 27.5 21.6 29.1 20.7 27.6 16.7 24.7 13.4 22.2 11.8 26.1 19.2 San Francisco 724940 0.4 17.8 11.9 20.4 13.4 21.9 13.8 26.2 16.5 28.2 16.5 31.4 17.8 29.9 18.3 28.3 17.8 31.4 17.7 29.5 16.3 22.7 13.8 17.5 12.3 28.4 17.0 1 16.7 11.9 18.9 12.0 20.1 13.1 23.7 14.6 25.9 15.9 28.4 17.1 27.3 17.5 26.4 17.5 29.4 17.2 27.4 15.7 21.2 13.4 16.5 12.4 25.6 16.4 2 15.8 12.4 17.7 12.3 18.8 12.6 21.8 13.9 23.5 15.3 25.8 16.4 25.2 17.0 24.5 17.0 27.3 16.7 25.4 15.8 19.8 13.2 15.7 12.6 23.3 15.8 Santa Maria 723940 0.4 25.5 13.2 26.1 14.0 25.8 14.2 30.6 16.8 26.7 15.5 30.1 17.3 31.2 18.2 29.3 19.1 33.1 18.3 33.0 17.2 28.4 13.9 26.0 13.1 29.9 17.2 1 24.0 12.8 24.5 13.6 24.1 13.7 28.3 16.1 24.6 15.8 26.7 16.6 28.5 18.4 27.7 18.7 30.8 18.6 30.6 15.6 26.9 13.8 24.9 12.3 27.5 16.7 2 22.8 12.0 23.0 13.3 22.6 13.8 26.2 15.0 22.9 15.4 24.9 16.4 26.6 18.0 26.4 18.5 28.6 18.5 28.6 16.0 25.6 13.9 23.6 11.9 25.6 16.1 COLORADO Alamosa 724620 0.4 10.4 2.4 13.4 3.3 18.4 5.6 21.8 7.4 26.2 9.7 30.3 12.8 30.8 12.7 29.5 13.9 27.3 12.1 23.1 8.8 16.6 5.4 10.2 2.0 29.0 12.8 1 8.7 1.4 12.1 2.6 17.3 4.9 20.7 6.8 25.1 9.4 29.5 12.2 30.1 13.3 28.7 13.6 26.5 11.7 22.2 8.4 15.4 5.1 8.8 1.4 27.8 12.7 2 7.1 0.8 10.5 2.3 15.8 4.2 19.7 6.4 24.0 8.8 28.7 12.1 29.3 13.2 27.9 13.4 25.6 11.6 21.3 8.2 14.1 4.4 7.4 0.7 26.6 12.4 Boulder 724699 0.4 17.3 5.6 19.8 7.3 23.1 8.1 26.8 10.6 29.9 12.6 34.7 14.9 35.8 15.6 34.8 15.6 32.5 14.6 27.8 10.8 22.5 8.4 19.1 6.7 33.8 15.3 1 15.6 5.4 17.9 6.1 21.4 7.7 25.4 9.9 28.8 12.4 33.6 14.9 35.1 15.4 33.8 15.2 31.3 14.2 26.7 10.8 21.0 8.0 17.2 5.8 32.3 15.2 2 14.1 4.7 16.2 5.6 19.9 6.8 24.1 9.4 27.6 11.9 32.8 14.9 34.3 15.6 32.9 15.1 30.1 13.8 25.6 10.5 19.6 7.4 15.4 5.2 30.8 15.1 Colorado Springs 724660 0.4 16.6 4.9 18.9 6.3 21.5 7.3 25.3 9.2 28.6 11.1 33.5 13.8 34.3 14.4 32.8 14.6 30.3 13.6 26.8 10.6 21.3 7.2 17.1 5.3 32.1 14.4 1 15.2 4.4 16.9 5.4 19.9 6.3 24.0 8.8 27.4 10.8 32.4 13.7 33.4 14.6 31.8 14.3 29.4 13.3 25.7 10.1 19.2 6.5 15.4 4.6 30.6 14.2 2 13.4 3.4 15.1 4.0 18.2 5.7 22.8 8.3 26.2 10.8 31.2 13.5 32.5 14.7 30.8 14.4 28.4 13.2 24.4 9.6 17.9 6.1 13.9 3.9 29.0 14.2 Eagle 724675 0.4 7.8 2.4 11.7 4.2 17.8 6.0 22.9 8.9 27.4 11.2 32.1 14.4 33.2 15.2 32.4 14.9 30.1 13.1 25.0 10.6 16.8 6.8 9.3 3.1 31.2 14.7 1 6.4 1.6 10.3 3.4 16.2 6.1 21.5 8.2 26.4 11.0 31.1 14.0 32.3 14.9 31.4 14.9 28.9 13.2 23.8 9.7 15.1 6.1 7.8 2.0 29.9 14.1 2 5.3 1.1 8.9 2.6 14.6 5.3 20.3 7.7 25.4 10.7 30.2 13.6 31.6 14.9 30.6 14.8 27.6 12.8 22.6 9.5 13.6 5.0 6.3 1.7 28.4 13.7 Grand Junction 724760 0.4 10.4 3.8 15.9 6.4 21.9 8.6 27.3 10.8 31.3 13.1 36.9 15.9 37.4 16.7 36.4 16.4 33.2 15.2 27.4 11.6 18.9 8.5 11.4 4.9 35.7 16.1 1 8.9 3.2 14.6 5.5 20.6 8.2 26.1 10.2 30.3 12.7 35.9 15.8 36.6 16.5 35.7 16.6 32.2 15.0 26.5 11.2 17.4 7.7 10.1 4.3 34.4 15.7 2 7.7 2.6 13.1 5.0 19.2 7.4 24.8 9.9 29.4 12.3 35.0 14.9 35.9 16.1 34.9 16.2 31.2 14.3 25.4 10.9 16.1 6.9 8.7 3.2 33.1 15.4 WMO# = World Meteorological Organization number DB = dry-bulb temperature, °C MWB = mean coincident wet-bulb temperature, °C 27.56 2001 ASHRAE Fundamentals Handbook (SI) Table 4A Design Wet-Bulb—Mean Coincident Dry-Bulb Temperature Location and WMO# % Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Annual WB MDB WB MDB WB MDB WB MDB WB MDB WB MDB WB MDB WB MDB WB MDB WB MDB WB MDB WB MDB WB MDB 1a 1b 2a 2b 3a 3b 4a 4b 5a 5b 6a 6b 7a 7b 8a 8b 9a 9b 10a 10b 11a 11b 12a 12c 13a 13b 14a 14b Pueblo 724640 0.4 8.1 17.8 9.2 21.6 9.8 22.1 12.9 22.6 16.2 25.8 19.4 29.6 20.5 29.8 20.8 29.4 18.4 27.7 13.9 24.5 9.7 22.6 8.1 18.4 19.7 28.7 1 6.9 16.2 8.2 19.2 9.1 21.3 12.1 23.2 15.6 26.2 18.7 29.6 20.1 29.2 20.2 29.0 17.8 26.8 13.3 26.2 9.1 20.8 7.2 16.9 19.0 28.4 2 6.1 15.3 7.1 17.6 8.6 20.4 11.4 22.4 15.0 25.6 18.2 28.9 19.7 29.0 19.8 28.4 17.3 26.6 12.8 25.7 8.6 19.7 6.2 15.5 18.4 28.2 CONNECTICUT Bridgeport 725040 0.4 10.0 10.9 9.1 10.9 11.2 14.1 16.1 21.6 20.7 24.9 23.2 28.8 25.2 30.1 25.5 29.4 24.4 28.1 20.4 22.3 17.3 19.1 12.4 13.6 24.3 28.1 1 8.6 9.9 7.6 9.2 10.1 12.6 14.9 19.8 19.6 23.4 22.5 27.4 24.7 29.5 25.0 28.2 23.9 27.1 19.8 22.1 16.3 17.7 11.1 12.2 23.5 27.1 2 6.9 7.9 6.3 7.8 9.2 11.6 13.5 17.1 18.5 22.3 21.9 26.3 24.1 28.8 24.6 27.8 23.3 26.0 19.1 21.2 15.5 16.7 10.1 11.2 22.8 26.2 Hartford, Brainard Field 725087 0.4 10.7 12.5 11.6 13.6 14.9 19.0 18.1 26.3 22.0 29.7 24.2 31.2 25.6 33.2 25.4 31.7 24.3 30.8 20.3 23.5 18.0 20.3 13.8 15.1 24.3 30.8 1 8.2 10.6 9.8 12.6 13.3 17.0 16.7 23.1 21.0 28.7 23.5 30.7 25.1 32.3 24.7 30.8 23.6 29.2 19.1 23.3 16.6 19.1 11.7 13.6 23.4 28.9 2 5.9 7.9 7.6 10.2 11.7 15.9 15.6 21.1 20.2 26.3 22.9 29.7 24.6 31.5 24.1 29.6 22.8 27.6 18.3 22.1 15.2 17.3 9.7 11.2 22.6 27.5 DELAWARE Wilmington 724089 0.4 12.9 15.0 14.1 16.7 17.5 22.7 19.3 26.7 23.4 29.2 25.2 31.1 26.5 32.2 26.1 31.7 25.2 30.4 21.5 24.6 19.0 21.6 15.9 17.9 25.3 30.6 1 11.3 13.0 12.7 14.7 15.9 20.7 18.4 25.2 22.3 27.9 24.6 30.3 25.9 31.8 25.6 30.4 24.6 29.4 20.8 23.7 17.5 20.1 14.5 16.4 24.6 29.3 2 9.7 10.9 11.0 13.4 14.4 17.7 17.4 22.7 21.5 26.7 24.0 29.5 25.4 31.2 25.1 29.8 24.1 28.8 20.0 22.6 16.6 18.6 12.9 14.3 23.8 28.5 FLORIDA Daytona Beach 722056 0.4 21.7 25.3 21.7 26.7 22.4 27.4 23.6 28.4 25.1 30.1 26.3 31.6 26.9 32.0 26.8 31.3 26.4 30.3 25.3 28.8 23.8 26.6 22.6 26.2 26.3 31.1 1 21.2 24.8 21.2 25.7 21.9 26.9 22.9 27.9 24.6 29.3 25.9 31.3 26.6 31.6 26.6 31.0 26.1 30.2 25.0 28.7 23.3 26.3 22.0 25.4 25.8 30.6 2 20.7 24.4 20.7 24.9 21.6 26.3 22.5 27.4 24.1 28.8 25.7 30.8 26.3 31.4 26.3 30.9 25.8 29.9 24.6 28.1 22.9 25.9 21.4 24.8 25.6 30.2 Jacksonville, Intl Airport 722060 0.4 21.1 24.4 21.3 25.4 22.3 27.2 23.7 28.7 25.3 30.4 26.7 33.4 27.4 33.7 27.2 32.8 26.6 31.3 25.2 29.1 23.4 26.4 22.0 25.3 26.6 32.2 1 20.5 23.7 20.8 24.4 21.8 26.8 23.0 27.9 24.7 29.7 26.3 32.6 26.9 32.9 26.8 32.3 26.2 31.1 24.8 28.6 22.9 25.8 21.5 24.3 26.2 31.7 2 19.9 23.2 20.2 23.7 21.3 26.1 22.4 27.3 24.2 29.1 25.9 31.9 26.7 32.4 26.5 31.9 25.9 30.7 24.4 27.6 22.4 25.2 20.9 23.7 25.7 31.0 Key West 722010 0.4 24.3 26.8 24.4 27.1 25.0 27.7 25.7 28.9 26.6 30.0 27.4 31.1 27.4 31.6 27.5 31.1 27.2 30.7 26.8 30.2 25.9 28.1 24.8 27.0 27.0 30.8 1 23.9 26.4 24.1 26.6 24.6 27.1 25.3 28.4 26.2 29.4 27.1 30.8 27.0 31.2 27.3 30.8 26.9 30.6 26.4 29.8 25.6 27.6 24.5 26.5 26.7 30.6 2 23.6 25.8 23.7 26.2 24.3 26.8 25.0 28.1 25.8 29.1 26.8 30.5 26.8 30.9 26.9 30.7 26.7 30.4 26.1 29.6 25.2 27.6 24.2 26.2 26.4 30.4 Miami, Intl Airport 722020 0.4 23.5 26.4 23.4 26.8 24.1 27.7 24.9 28.9 25.8 29.1 26.6 30.7 26.8 30.9 26.8 30.8 26.7 30.1 26.2 29.3 25.2 27.9 24.3 27.1 26.4 30.4 1 23.2 26.1 23.0 26.3 23.6 27.2 24.4 28.2 25.3 28.8 26.2 30.3 26.4 30.9 26.7 30.6 26.3 30.1 25.8 29.2 24.9 27.8 23.8 26.5 26.1 30.3 2 22.8 25.8 22.7 26.1 23.4 26.7 24.0 28.1 25.0 28.6 25.8 29.9 26.2 30.8 26.4 30.4 26.2 30.1 25.6 29.2 24.6 27.3 23.4 25.8 25.8 29.9 Tallahassee 722140 0.4 21.5 23.4 21.4 23.3 22.4 25.7 23.6 27.4 25.1 29.6 26.6 32.6 27.2 32.7 26.9 32.2 26.3 30.6 24.9 28.5 22.9 25.3 22.6 24.2 26.4 31.6 1 20.8 22.6 20.8 22.7 21.9 24.8 23.1 26.9 24.6 29.0 26.2 32.1 26.8 32.0 26.6 31.9 26.0 30.4 24.4 27.8 22.5 24.8 22.0 23.6 25.9 31.2 2 20.1 21.7 20.2 22.1 21.4 24.0 22.6 26.3 24.2 28.8 25.8 31.3 26.4 31.6 26.3 31.7 25.7 30.3 24.0 27.8 22.2 24.4 21.3 22.8 25.6 30.6 Tampa, Intl Airport 722110 0.4 22.4 25.7 22.4 26.2 23.4 26.7 24.5 27.9 25.7 30.5 26.7 30.9 27.4 31.7 27.3 31.7 26.6 30.9 25.7 29.9 24.3 27.7 23.4 26.3 26.7 31.2 1 21.8 24.7 21.9 25.3 23.0 26.7 24.0 27.8 25.2 30.0 26.4 31.0 26.9 31.3 26.9 31.5 26.3 31.1 25.4 29.6 23.8 27.2 22.8 25.8 26.2 31.2 2 21.3 24.0 21.5 24.6 22.5 26.4 23.5 27.3 24.7 29.3 26.1 31.2 26.7 31.2 26.6 31.4 26.0 31.0 25.1 29.3 23.4 26.6 22.3 25.0 25.8 30.7 West Palm Beach 722030 0.4 23.1 26.6 23.4 26.9 23.9 28.2 24.6 28.4 25.7 29.5 26.7 30.8 26.9 31.5 26.9 31.6 26.8 30.8 25.9 29.5 25.0 28.1 23.9 27.2 26.4 31.1 1 22.8 26.3 22.9 26.8 23.4 27.4 24.0 28.1 25.3 29.1 26.3 30.7 26.6 31.4 26.7 31.3 26.4 30.6 25.7 29.2 24.6 27.6 23.4 26.7 26.1 30.9 2 22.4 25.6 22.5 26.3 23.0 26.8 23.5 27.8 24.9 28.7 25.9 30.5 26.3 31.3 26.4 31.2 26.2 30.6 25.3 29.1 24.2 27.1 22.9 26.2 25.8 30.4 GEORGIA Athens 723110 0.4 16.9 19.0 17.9 20.2 19.6 23.4 21.2 25.7 23.9 29.1 25.1 32.1 26.2 33.2 25.8 32.3 25.1 30.9 22.9 26.4 20.6 22.2 19.4 20.7 25.3 31.6 1 16.1 17.9 16.9 19.3 18.8 22.2 20.4 25.3 23.2 28.3 24.7 31.6 25.8 32.5 25.6 32.1 24.6 30.1 22.1 25.2 19.8 22.1 18.5 20.2 24.7 30.8 2 15.1 16.6 16.1 18.3 18.1 21.9 19.9 24.7 22.6 27.8 24.3 31.1 25.4 31.9 25.2 31.4 24.2 29.4 21.4 24.7 19.1 21.1 17.3 18.8 24.3 30.1 Atlanta 722190 0.4 16.9 18.7 17.8 20.2 18.9 23.8 21.0 25.3 23.3 28.9 24.9 31.7 26.5 33.6 25.8 31.7 24.7 30.1 22.3 25.6 20.1 22.2 18.9 20.9 25.1 31.2 1 16.2 17.9 17.1 19.2 18.3 22.2 20.1 24.9 22.6 28.2 24.4 30.9 25.8 32.0 25.3 31.4 24.2 29.5 21.6 24.9 19.4 21.9 17.9 19.6 24.4 30.3 2 15.2 16.9 16.2 18.4 17.6 21.8 19.6 23.9 22.1 27.6 24.1 30.3 25.3 31.2 24.9 31.1 23.7 29.3 20.8 24.4 18.8 21.0 16.9 18.7 23.9 29.7 Augusta 722180 0.4 18.4 20.6 19.3 23.1 20.3 25.9 22.2 27.4 24.4 30.8 25.9 33.1 26.9 33.8 26.7 32.9 25.6 31.9 23.6 27.9 21.7 24.8 20.2 22.9 25.9 32.8 1 17.6 20.5 18.3 21.8 19.7 24.8 21.3 26.8 23.5 29.3 25.5 32.3 26.4 33.7 26.3 32.8 25.2 31.3 23.0 26.8 21.1 24.2 19.3 21.7 25.4 31.9 2 16.7 18.8 17.4 20.6 19.0 23.5 20.7 26.1 22.9 29.0 25.1 31.8 26.2 33.2 25.8 32.2 24.8 30.8 22.4 26.6 20.4 23.3 18.4 21.1 24.9 31.1 Columbus, Metro Airport 722255 0.4 19.1 21.2 19.4 21.4 21.0 25.1 22.4 27.3 24.4 29.3 26.3 32.4 26.8 33.9 26.5 32.4 25.6 30.6 23.8 28.1 21.5 24.3 21.0 23.0 25.9 31.9 1 18.2 20.1 18.6 21.3 20.2 24.3 21.8 26.9 23.7 28.8 25.8 31.9 26.3 33.1 26.1 31.9 25.2 30.0 23.1 27.1 20.8 23.3 20.2 22.4 25.4 31.2 2 17.5 19.2 17.9 20.6 19.7 23.5 21.2 25.9 23.3 28.3 25.3 31.4 26.0 32.4 25.8 31.5 24.9 29.6 22.5 26.6 20.3 22.6 19.3 21.4 25.0 30.6 Macon 722170 0.4 18.8 20.8 19.6 22.5 20.8 25.1 22.6 27.7 24.4 30.2 26.0 33.9 26.8 34.2 26.7 33.1 25.6 31.2 23.6 28.1 21.7 24.1 20.8 23.1 25.9 32.7 1 18.0 20.2 18.8 21.7 20.2 24.3 21.8 26.6 23.9 29.6 25.6 33.0 26.3 33.5 26.3 32.8 25.2 30.8 23.1 27.2 20.9 23.7 19.9 22.1 25.4 31.9 2 17.3 19.4 18.1 20.7 19.7 23.5 21.2 26.2 23.3 28.2 25.2 31.9 26.0 33.1 25.9 32.4 24.9 30.6 22.6 26.6 20.4 23.0 19.1 21.3 25.0 31.1 Savannah 722070 0.4 19.6 22.6 20.1 24.4 21.2 26.2 22.9 27.7 24.7 29.5 26.2 32.6 27.2 33.0 26.9 32.8 25.8 30.8 24.7 28.2 22.6 25.5 21.0 23.8 26.3 32.2 1 19.1 22.2 19.4 23.4 20.5 25.5 22.2 27.0 24.2 29.2 25.8 32.1 26.8 32.6 26.5 32.3 25.6 30.7 24.1 27.6 21.9 24.6 20.3 23.2 25.8 31.5 2 18.4 21.4 18.8 21.9 19.9 24.2 21.5 26.4 23.7 28.8 25.5 31.6 26.4 32.4 26.2 31.9 25.2 30.1 23.4 26.6 21.4 23.9 19.6 22.3 25.3 30.7 HAWAII Hilo 912850 0.4 23.0 26.8 23.1 27.1 23.1 26.6 23.6 27.1 24.0 27.8 23.9 27.6 24.7 27.6 25.2 28.2 25.2 28.7 25.2 28.7 24.6 27.7 23.6 26.9 24.7 27.8 1 22.7 26.2 22.7 27.1 22.6 26.1 23.0 26.6 23.6 27.4 23.6 27.3 24.3 27.3 24.7 27.7 24.8 28.3 24.7 28.0 24.2 27.2 23.3 26.7 24.2 27.4 2 22.3 25.9 22.3 26.4 22.3 25.7 22.7 26.1 23.2 27.1 23.3 27.1 24.0 27.2 24.5 27.4 24.5 27.9 24.4 27.6 23.8 26.9 22.9 26.3 23.8 27.2 Honolulu 911820 0.4 23.4 26.1 23.0 26.8 22.9 26.8 22.9 27.2 23.6 28.3 23.6 29.1 24.5 29.5 24.8 29.9 25.1 29.5 25.1 29.1 24.2 28.1 23.8 27.1 24.4 29.1 1 23.0 26.0 22.6 26.3 22.4 26.5 22.6 27.0 23.2 27.9 23.3 29.1 24.1 29.2 24.3 29.8 24.6 29.6 24.7 28.7 23.8 27.9 23.4 26.7 23.9 28.9 2 22.7 25.8 22.2 25.9 22.1 26.3 22.3 26.9 22.9 27.9 22.9 28.8 23.6 28.9 24.0 29.7 24.3 29.4 24.3 28.6 23.5 27.7 23.0 26.4 23.5 28.6 Kahului 911900 0.4 23.3 26.9 23.0 27.4 23.0 27.7 23.1 27.7 24.3 28.8 24.2 29.9 24.7 29.8 25.2 30.1 25.2 30.1 25.2 30.2 24.3 28.4 23.9 27.8 24.7 29.6 1 22.8 26.6 22.6 26.8 22.6 27.1 22.8 27.7 23.8 28.2 23.9 29.6 24.4 29.6 24.7 29.5 24.7 30.0 24.8 29.7 23.9 28.4 23.4 27.4 24.2 29.2 2 22.4 26.1 22.3 26.5 22.2 26.9 22.4 27.4 23.3 27.7 23.5 29.2 24.1 29.3 24.4 29.2 24.4 29.7 24.4 29.2 23.6 28.0 23.0 26.9 23.8 28.9 Lihue 911650 0.4 23.2 25.6 23.2 26.1 23.0 26.4 23.4 26.6 23.8 27.5 24.4 28.6 25.2 28.4 25.5 28.7 25.3 28.8 25.2 28.2 24.4 26.5 23.6 25.8 24.9 28.3 1 22.8 25.6 22.7 25.4 22.7 26.2 22.9 26.2 23.5 27.1 24.0 28.2 24.7 27.9 25.1 28.5 25.1 28.6 24.8 27.8 24.1 26.6 23.3 25.6 24.5 27.8 2 22.4 25.3 22.3 25.2 22.3 25.7 22.7 25.9 23.3 26.7 23.6 27.7 24.4 27.6 24.8 28.2 24.7 28.3 24.5 27.6 23.7 26.4 23.0 25.5 24.1 27.6 IDAHO Boise 726810 0.4 7.2 9.8 9.6 14.5 11.0 19.5 14.1 24.3 16.9 29.1 19.0 32.9 20.2 32.7 19.9 34.1 17.7 29.8 14.1 25.7 10.8 15.5 7.8 11.2 18.8 32.2 1 6.6 9.5 8.4 12.7 10.2 18.0 13.2 22.8 15.9 28.2 18.3 31.8 19.5 32.9 19.2 32.9 17.0 28.9 13.5 24.7 10.1 14.7 6.9 9.8 17.9 31.5 2 6.0 8.7 7.5 11.5 9.4 16.1 12.3 22.2 15.2 27.1 17.7 30.8 18.9 32.6 18.7 31.8 16.3 28.0 12.9 23.1 9.3 13.4 6.2 9.4 17.2 30.4 Pocatello 725780 0.4 5.1 8.4 7.7 12.1 9.1 16.9 12.1 22.5 15.2 26.1 18.2 30.2 19.1 29.7 18.8 28.8 16.3 26.3 12.8 24.4 9.3 14.5 5.9 9.3 17.7 29.0 1 4.3 7.4 6.4 10.7 8.3 15.5 11.2 21.4 14.4 25.5 17.3 29.4 18.5 29.8 18.0 28.8 15.6 26.3 12.2 22.6 8.4 13.4 4.9 8.0 16.9 28.4 2 3.6 6.2 5.4 9.2 7.4 13.8 10.4 20.2 13.7 24.1 16.7 29.0 17.9 29.4 17.5 28.5 15.0 25.9 11.5 21.8 7.6 12.6 4.1 6.9 16.1 27.8 ILLINOIS Chicago, O’Hare Intl Airport 725300 0.4 10.7 12.3 10.6 13.3 16.8 21.3 19.3 25.4 22.7 28.3 24.7 31.1 26.6 33.1 26.3 32.4 24.5 29.4 20.4 26.4 16.6 19.2 14.3 15.8 25.1 31.0 1 8.1 9.5 8.8 11.2 15.4 19.2 18.4 23.7 21.9 27.4 24.1 30.7 25.8 32.2 25.7 31.8 23.8 29.1 19.4 24.6 15.7 18.4 12.6 14.1 24.1 29.5 2 5.8 7.3 7.1 9.3 14.2 17.7 17.6 22.3 21.2 27.2 23.6 29.7 25.3 31.4 25.1 30.7 23.1 28.3 18.5 23.2 14.8 16.9 10.1 11.8 23.1 28.1 WMO# = World Meteorological Organization number WB = wet-bulb temperature, °C MDB = mean coincident dry-bulb temperature,temperature, °C Climatic Design Information 27.57 Table 4B Design Dry-Bulb—Mean Coincident Wet-Bulb Temperature Location and WMO# % Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Annual DB MWB DB MWB DB MWB DB MWB DB MWB DB MWB DB MWB DB MWB DB MWB DB MWB DB MWB DB MWB DB MWB 1a 1b 2a 2b 3a 3b 4a 4b 5a 5b 6a 6b 7a 7b 8a 8b 9a 9b 10a 10b 11a 11b 12a 12c 13a 13b 14a 14b Pueblo 724640 0.4 19.1 7.4 22.2 9.0 25.0 9.2 28.4 11.6 32.8 13.9 38.1 16.5 38.4 17.4 36.9 17.3 34.4 15.9 30.2 12.4 23.8 9.3 19.8 7.4 36.2 16.8 1 17.5 6.2 20.4 7.8 23.5 8.6 27.2 10.9 31.8 13.4 36.9 16.1 37.4 17.4 35.9 17.1 33.4 15.4 29.3 12.2 22.2 8.8 17.9 6.6 34.6 16.7 2 16.1 5.7 18.6 6.7 22.1 8.1 26.1 10.2 30.8 13.2 35.7 15.7 36.6 17.2 35.0 16.9 32.3 15.4 28.1 11.9 20.8 8.1 16.4 5.9 33.1 16.6 CONNECTICUT Bridgeport 725040 0.4 11.5 9.6 11.6 7.1 15.5 9.7 23.8 14.7 26.9 19.4 31.1 21.8 33.1 23.9 31.7 23.4 30.1 23.6 24.3 18.9 19.5 16.1 14.2 11.9 30.2 22.8 1 9.9 7.9 9.9 6.9 14.3 9.4 21.3 13.8 25.4 17.4 29.5 21.3 31.9 23.0 30.7 23.3 28.7 22.3 22.9 17.9 18.4 15.3 12.7 10.6 28.8 22.1 2 8.6 6.4 8.4 5.9 12.8 8.2 19.1 12.1 24.1 17.1 28.2 20.7 30.6 22.7 29.8 23.0 27.4 22.0 22.1 17.7 17.3 14.8 11.6 9.6 27.6 21.5 Hartford, Brainard Field 725087 0.4 13.1 10.6 15.2 11.1 21.9 13.2 28.5 17.1 32.3 20.4 33.9 22.7 35.1 24.4 33.9 24.1 32.4 23.7 26.7 18.2 21.8 15.8 15.7 12.7 32.9 23.0 1 10.6 7.4 12.6 9.2 19.1 12.0 25.6 15.4 30.3 19.6 32.9 21.9 34.1 23.8 32.8 23.4 30.7 22.7 25.1 17.6 19.9 15.3 13.9 10.8 31.2 22.1 2 8.3 5.3 10.7 7.5 16.7 11.0 23.2 13.9 28.6 18.9 31.6 22.0 33.1 23.6 31.8 22.8 29.0 21.7 23.6 16.8 18.1 14.5 11.7 9.0 29.7 21.2 DELAWARE Wilmington 724089 0.4 15.5 11.2 18.2 11.9 24.1 15.9 28.7 18.0 31.8 21.6 33.8 23.3 35.3 24.7 33.9 23.9 32.8 24.1 27.0 19.5 23.0 17.6 18.4 15.3 32.9 23.8 1 13.4 10.5 16.2 11.7 22.1 14.8 26.7 17.8 30.3 21.1 32.7 23.2 34.2 24.3 32.9 24.1 31.6 23.5 25.6 18.9 21.2 17.1 16.6 14.1 31.5 23.1 2 11.7 9.3 14.2 10.4 19.5 13.2 24.4 16.3 29.1 19.9 31.7 22.7 33.3 24.1 32.0 23.7 30.2 22.8 24.3 18.6 19.6 15.4 14.8 12.3 30.1 22.6 FLORIDA Daytona Beach 722056 0.4 27.2 20.5 28.4 20.8 29.8 21.1 31.6 21.6 33.1 22.8 34.4 25.0 34.7 25.4 34.1 25.2 32.5 24.8 30.8 23.8 28.7 22.6 27.5 21.3 33.2 25.0 1 26.4 20.4 27.4 20.3 29.0 20.7 30.6 21.2 32.1 22.6 33.5 24.9 33.9 25.3 33.3 25.6 31.8 24.9 30.1 23.6 27.9 22.2 26.8 20.9 32.2 24.8 2 25.6 19.9 26.1 19.9 28.0 20.6 29.7 21.3 31.2 22.3 32.7 24.8 33.2 25.4 32.6 25.4 31.3 24.9 29.5 23.5 27.3 21.9 26.0 20.7 31.3 24.8 Jacksonville, Intl Airport 722060 0.4 26.7 19.8 27.9 20.2 30.1 20.4 32.0 20.9 34.4 22.4 35.8 25.3 36.2 25.9 35.2 25.6 33.9 25.1 31.8 23.8 28.2 22.0 27.1 20.8 34.7 25.2 1 25.7 19.7 26.8 19.3 29.1 20.4 31.2 20.8 33.2 22.6 35.0 24.9 35.4 25.8 34.6 25.6 33.2 25.0 30.8 23.1 27.5 21.5 26.2 20.2 33.7 25.1 2 24.6 19.0 25.7 18.8 28.1 20.0 30.4 20.8 32.3 22.2 34.2 24.7 34.7 25.5 34.0 25.6 32.6 24.9 29.9 22.9 26.8 21.1 25.3 19.8 32.7 24.8 Key West 722010 0.4 27.8 23.6 27.9 23.6 28.9 24.0 30.2 24.8 31.2 25.3 32.4 26.2 33.0 26.2 33.1 26.1 32.5 26.1 31.3 25.9 29.5 25.1 28.0 23.8 32.4 26.1 1 27.2 23.2 27.5 23.4 28.4 23.8 29.7 24.4 30.8 25.2 32.0 26.0 32.7 26.1 32.8 26.1 32.1 26.0 30.8 25.7 29.1 24.8 27.6 23.4 31.9 26.0 2 26.8 22.8 27.1 23.2 27.9 23.6 29.1 23.9 30.4 25.1 31.7 25.9 32.3 26.0 32.4 26.1 31.8 25.9 30.6 25.5 28.6 24.5 27.1 23.2 31.4 25.8 Miami, Intl Airport 722020 0.4 28.4 22.1 28.9 22.1 30.2 21.8 31.8 22.7 32.3 23.1 33.3 24.9 33.9 25.2 33.6 25.4 32.9 25.4 31.9 24.7 29.8 24.1 28.5 23.0 32.8 25.2 1 27.8 22.1 28.2 21.8 29.4 22.4 30.8 22.7 31.6 23.2 32.6 25.1 33.2 25.3 33.1 25.4 32.4 25.4 31.3 24.6 29.3 23.7 27.9 22.5 32.2 25.1 2 27.1 21.8 27.6 21.6 28.8 22.1 30.1 22.4 30.9 23.1 32.0 24.9 32.7 25.3 32.6 25.3 31.9 25.3 30.7 24.5 28.7 23.3 27.3 22.1 31.6 24.9 Tallahassee 722140 0.4 24.8 19.7 26.2 18.1 29.1 19.1 31.7 20.2 34.0 22.3 36.0 24.9 36.1 25.5 35.7 25.2 34.6 24.3 31.8 22.5 28.1 21.0 26.0 20.3 34.8 24.7 1 23.9 19.2 25.2 18.3 28.1 19.0 30.7 20.4 33.1 22.2 35.1 24.4 35.3 25.3 35.0 25.3 33.9 24.2 31.1 22.3 27.3 20.6 25.1 20.2 33.8 24.5 2 22.9 18.7 24.1 18.1 27.2 18.3 29.8 19.9 32.3 21.8 34.3 24.0 34.7 25.1 34.3 25.1 33.3 24.3 30.3 21.9 26.5 20.1 24.2 19.6 32.9 24.2 Tampa, Intl Airport 722110 0.4 27.3 20.4 28.0 20.9 29.3 21.4 31.1 21.5 33.4 23.2 34.3 24.7 34.1 25.5 34.3 25.5 33.8 24.8 32.2 24.2 30.0 22.6 28.3 21.9 33.6 25.1 1 26.7 20.8 27.3 20.2 28.7 21.2 30.4 21.7 32.7 23.2 33.6 24.6 33.6 25.5 33.8 25.5 33.2 24.9 31.5 23.9 29.2 22.3 27.4 20.9 32.9 25.1 2 25.9 20.3 26.6 20.3 28.1 21.1 29.8 21.4 32.1 22.8 33.1 24.6 33.3 25.4 33.4 25.4 32.8 24.9 30.9 23.6 28.5 22.0 26.7 20.8 32.3 24.9 West Palm Beach 722030 0.4 28.3 22.1 28.9 22.1 30.3 22.3 31.8 22.1 32.3 23.6 33.3 24.9 34.2 25.3 33.7 25.7 32.9 25.3 31.7 24.6 29.7 23.6 28.5 22.5 32.9 25.3 1 27.5 21.8 28.2 21.8 29.4 22.1 30.7 22.2 31.4 23.2 32.6 25.1 33.4 25.3 33.1 25.7 32.4 25.3 31.2 24.4 29.1 23.3 27.9 22.3 32.2 25.3 2 26.8 21.5 27.4 21.7 28.6 22.1 29.7 22.2 30.8 23.2 32.1 25.2 32.8 25.4 32.7 25.7 31.9 25.3 30.6 24.3 28.4 23.2 27.3 22.2 31.6 25.2 GEORGIA Athens 723110 0.4 20.7 14.3 23.3 14.5 26.9 16.8 29.9 18.5 32.7 21.6 35.8 23.3 37.0 24.2 35.6 24.2 33.4 23.7 29.6 20.6 25.4 17.8 22.6 17.6 34.6 23.6 1 19.1 13.7 21.9 14.7 25.8 16.7 29.0 18.2 31.7 21.0 34.7 23.1 36.1 23.9 34.5 23.9 32.6 23.3 28.4 20.1 24.4 17.7 21.1 17.4 33.3 23.7 2 17.9 13.1 20.6 13.8 24.6 15.9 28.0 17.7 30.8 21.0 33.7 23.1 35.2 23.9 33.7 24.1 31.7 22.9 27.4 19.3 23.2 16.7 19.8 16.4 32.0 23.2 Atlanta 722190 0.4 20.3 14.4 22.9 14.6 26.8 16.7 29.5 18.3 31.6 21.6 34.7 23.2 36.4 24.3 35.1 24.1 33.3 23.4 28.8 20.2 25.1 18.0 21.9 17.4 33.9 23.8 1 19.1 14.7 21.7 14.0 25.6 16.5 28.5 17.6 30.7 20.7 33.7 22.8 35.3 24.0 34.1 24.0 32.2 23.0 27.8 19.5 23.9 17.3 20.8 16.8 32.6 23.4 2 17.8 13.8 20.3 13.9 24.4 15.6 27.6 17.3 30.0 20.3 32.9 22.8 34.3 24.1 33.2 23.8 31.3 22.6 26.8 19.1 22.8 16.7 19.6 15.8 31.3 22.8 Augusta 722180 0.4 23.1 16.2 25.4 17.4 28.5 18.2 31.4 19.1 33.9 22.1 37.1 23.9 37.9 24.6 36.7 24.7 34.9 24.1 31.1 21.4 27.7 19.5 24.9 19.1 35.7 24.4 1 21.6 16.0 24.1 16.3 27.6 17.7 30.5 18.8 32.9 21.8 35.8 23.7 36.9 24.5 35.7 24.7 34.0 24.0 30.2 21.3 26.2 18.8 23.6 18.2 34.3 24.3 2 20.1 15.1 22.7 15.4 26.5 17.1 29.6 18.6 32.0 21.3 34.8 23.6 35.9 24.7 34.9 24.7 33.1 23.7 29.2 20.5 25.1 18.2 22.1 17.1 33.2 23.9 Columbus, Metro Airport 722255 0.4 22.7 16.8 24.6 16.3 28.4 18.4 30.8 19.2 32.9 21.8 36.2 23.6 37.2 24.8 36.2 24.4 34.7 23.2 31.3 21.6 26.8 18.9 24.7 19.0 35.2 24.2 1 21.6 16.5 23.4 16.2 27.3 17.7 30.1 19.4 32.1 21.4 35.3 23.8 36.3 24.6 35.2 24.3 33.9 23.3 30.3 20.9 25.8 18.4 23.5 18.9 34.0 23.9 2 20.2 16.0 22.2 15.7 26.3 17.2 29.2 18.9 31.4 21.3 34.6 23.4 35.4 24.5 34.6 24.2 33.1 23.2 29.2 20.6 24.9 18.2 22.4 18.4 33.0 23.7 Macon 722170 0.4 22.9 16.6 25.2 16.9 28.5 18.4 31.4 19.8 34.2 21.5 36.9 24.1 37.8 25.0 36.8 24.8 35.0 23.8 31.6 21.7 27.5 19.9 24.6 19.0 35.7 24.3 1 21.6 16.1 23.9 16.6 27.4 17.8 30.4 19.1 33.3 21.9 36.0 23.8 36.8 24.6 35.7 24.8 34.1 23.7 30.5 21.1 26.2 18.8 23.4 18.3 34.4 24.0 2 20.2 15.7 22.6 15.7 26.3 17.4 29.5 18.9 32.4 21.6 35.1 23.6 35.9 24.6 34.9 24.5 33.3 23.4 29.5 20.6 25.2 18.4 22.3 18.1 33.3 23.8 Savannah 722070 0.4 24.8 18.4 26.4 18.7 29.1 19.2 31.8 19.7 34.0 22.3 36.3 24.5 37.2 25.1 35.7 25.5 34.0 24.4 30.8 22.2 27.7 20.7 25.8 19.9 35.0 24.9 1 23.5 17.8 25.0 17.5 27.8 18.6 30.7 19.4 32.9 22.2 35.3 24.3 36.2 25.3 34.9 25.3 33.2 24.4 30.0 22.0 26.7 20.6 24.8 18.9 33.8 24.6 2 22.2 17.2 23.7 17.2 26.7 18.4 29.6 19.5 31.9 21.8 34.3 24.1 35.2 25.3 34.2 25.3 32.4 24.2 29.1 21.7 25.7 19.7 23.4 18.3 32.6 24.4 HAWAII Hilo 912850 0.4 29.1 21.4 29.3 21.8 28.8 21.7 28.5 22.2 29.4 23.1 29.5 22.7 29.6 23.3 29.8 23.9 30.2 23.9 30.1 23.9 29.2 23.4 28.9 22.6 29.6 23.3 1 28.3 21.4 28.4 21.5 28.2 21.3 27.9 22.0 28.8 22.7 29.0 22.5 29.1 23.2 29.3 23.7 29.7 23.8 29.6 23.7 28.7 23.1 28.2 22.2 29.1 23.1 2 27.6 21.3 27.8 21.3 27.5 21.1 27.3 21.7 28.2 22.3 28.6 22.3 28.6 23.0 29.0 23.4 29.2 23.6 29.1 23.5 28.2 22.8 27.7 21.9 28.5 22.9 Honolulu 911820 0.4 28.4 21.5 28.5 21.2 29.4 21.2 29.6 21.3 30.7 22.0 31.4 22.3 31.9 22.8 32.4 23.1 32.6 23.4 31.9 23.2 30.8 22.7 29.5 22.3 31.8 22.9 1 27.9 21.3 28.1 21.1 28.9 21.1 29.2 21.1 30.2 21.8 31.0 22.1 31.6 22.7 32.0 22.9 32.1 23.2 31.4 23.0 30.2 22.6 28.7 21.8 31.3 22.7 2 27.5 21.1 27.7 21.0 28.4 20.8 28.8 21.0 29.8 21.7 30.7 22.0 31.2 22.5 31.7 22.8 31.7 23.0 31.0 22.8 29.8 22.5 28.3 21.6 30.7 22.6 Kahului 911900 0.4 28.7 22.2 29.0 21.6 30.1 21.7 30.1 21.8 31.1 22.3 31.8 22.7 32.2 23.4 32.4 23.9 32.4 23.8 31.8 23.6 30.8 22.9 29.5 22.7 31.7 23.3 1 28.2 21.6 28.5 21.4 29.3 21.4 29.6 21.8 30.6 22.4 31.2 22.4 31.6 23.4 31.8 23.5 31.9 23.6 31.3 23.4 30.3 22.9 28.9 22.3 31.1 23.1 2 27.7 21.3 28.0 21.2 28.7 21.3 29.0 21.6 30.1 22.2 30.7 22.4 31.0 23.2 31.3 23.3 31.4 23.3 30.9 23.3 29.8 22.6 28.5 22.0 30.4 22.8 Lihue 911650 0.4 27.7 21.6 28.2 21.3 27.9 21.7 28.2 22.4 28.7 22.7 29.6 23.9 29.7 24.1 30.6 24.3 30.2 24.2 29.7 23.8 28.5 22.7 27.8 22.1 29.7 23.8 1 27.1 21.3 27.4 21.1 27.3 21.6 27.6 21.9 28.2 22.5 29.1 23.4 29.4 23.8 30.0 24.2 29.9 24.0 29.4 23.6 28.1 22.6 27.2 22.0 29.2 23.6 2 26.6 21.2 26.8 21.2 26.8 21.4 27.2 21.8 27.9 22.3 28.6 22.9 29.1 23.5 29.6 23.8 29.6 23.8 29.0 23.4 27.8 22.6 26.8 21.9 28.8 23.3 IDAHO Boise 726810 0.4 11.0 7.0 15.7 8.3 21.2 10.1 27.1 13.0 32.7 15.5 37.0 17.2 38.0 18.3 37.8 18.2 33.7 16.2 27.9 13.3 18.1 9.1 12.6 6.9 35.8 17.4 1 9.9 6.2 14.1 7.7 19.4 9.8 25.4 12.0 31.1 15.4 35.6 17.0 36.9 17.9 36.6 17.7 32.3 15.9 26.3 12.8 16.5 8.8 11.1 6.4 34.2 16.9 2 8.9 5.6 12.7 6.9 17.6 8.7 23.7 11.6 29.3 14.6 34.1 16.7 35.9 17.7 35.4 17.4 31.2 15.4 24.6 12.3 15.1 8.4 9.8 5.9 32.5 16.4 Pocatello 725780 0.4 9.6 4.3 13.6 7.1 18.5 8.2 25.3 10.7 29.2 13.5 34.6 16.3 35.6 16.3 35.7 15.9 32.0 14.4 26.9 11.9 17.2 8.2 10.5 4.3 33.9 15.9 1 7.8 3.8 11.8 5.6 16.9 7.6 23.9 10.2 28.1 13.1 33.3 15.9 34.8 16.3 34.6 15.8 30.8 14.2 25.3 11.2 15.7 7.6 8.9 4.3 32.3 15.5 2 6.7 3.2 10.3 4.7 15.2 6.9 22.1 9.9 26.9 12.7 31.9 15.7 34.1 16.1 33.8 15.7 29.6 13.9 23.7 10.8 14.3 6.6 7.6 3.4 30.7 15.1 ILLINOIS Chicago, O’Hare Intl Airport 725300 0.4 12.1 10.4 14.1 9.8 23.2 15.1 28.4 17.7 31.5 20.8 33.9 21.8 35.3 24.5 34.4 24.9 32.4 23.2 28.2 18.8 21.1 15.2 16.3 14.0 32.8 23.6 1 10.0 7.8 11.9 8.6 20.9 14.5 26.5 17.0 30.2 20.5 32.6 22.1 34.2 24.4 33.3 24.2 30.9 22.6 26.5 18.6 19.3 14.8 14.4 13.0 31.3 22.8 2 7.7 5.7 9.8 6.8 18.8 13.1 24.4 16.3 28.9 20.0 31.7 22.0 33.1 24.4 32.2 23.7 29.5 21.6 24.7 17.2 17.7 13.8 12.1 9.7 29.7 21.9 WMO# = World Meteorological Organization number DB = dry-bulb temperature, °C MWB = mean coincident wet-bulb temperature, °C 27.58 2001 ASHRAE Fundamentals Handbook (SI) Table 4A Design Wet-Bulb—Mean Coincident Dry-Bulb Temperature Location and WMO# % Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Annual WB MDB WB MDB WB MDB WB MDB WB MDB WB MDB WB MDB WB MDB WB MDB WB MDB WB MDB WB MDB WB MDB 1a 1b 2a 2b 3a 3b 4a 4b 5a 5b 6a 6b 7a 7b 8a 8b 9a 9b 10a 10b 11a 11b 12a 12c 13a 13b 14a 14b Moline/Davenport Intl Airport 725440 0.4 9.8 11.3 10.2 14.2 16.7 21.8 19.8 26.9 23.0 29.2 25.5 31.9 26.9 33.6 26.9 33.3 25.0 31.0 21.1 27.1 16.4 19.6 13.7 15.8 25.7 32.0 1 7.3 9.5 8.8 12.6 15.5 19.8 18.8 24.9 22.3 28.3 24.8 31.0 26.4 33.0 26.3 32.5 24.2 29.7 20.1 24.6 15.6 18.4 12.0 13.4 24.7 30.5 2 5.3 7.3 7.3 10.1 14.2 17.9 18.0 23.7 21.6 27.6 24.2 30.7 25.9 32.2 25.7 31.7 23.5 28.6 19.2 23.6 14.8 17.1 9.4 11.2 23.9 29.3 Peoria 725320 0.4 11.1 12.2 11.4 14.0 17.4 21.9 20.4 25.6 23.4 28.1 25.7 31.0 27.3 33.1 26.8 32.3 25.2 30.8 20.8 25.2 16.8 19.4 14.7 16.0 25.8 31.6 1 8.9 9.7 10.1 12.7 16.1 20.6 19.3 24.1 22.6 27.4 25.1 30.3 26.6 32.2 26.2 32.1 24.3 29.4 20.0 24.2 16.1 19.1 12.9 13.9 24.9 30.1 2 6.8 7.9 8.8 10.9 14.9 17.7 18.3 22.9 21.8 27.0 24.4 29.7 26.0 32.0 25.6 31.2 23.6 28.2 19.2 22.8 15.2 17.6 10.7 11.9 23.9 29.0 Rockford 725430 0.4 8.2 9.1 8.8 11.8 16.0 20.8 19.3 25.9 22.7 27.9 25.0 31.6 26.5 33.0 26.3 31.9 24.2 29.8 20.4 25.7 15.8 18.6 13.0 14.2 25.1 30.7 1 5.9 7.0 7.3 9.4 14.6 18.1 18.3 24.9 21.7 27.4 24.2 30.4 25.9 32.1 25.7 30.9 23.6 28.8 19.6 23.7 15.0 17.0 11.1 12.1 24.1 29.2 2 4.0 5.3 5.8 7.6 13.2 16.3 17.2 22.2 21.0 27.1 23.6 29.4 25.4 31.1 25.0 30.2 22.9 27.7 18.6 22.4 14.2 16.1 7.9 9.1 23.1 27.8 Springfield 724390 0.4 12.8 14.1 13.2 15.3 18.1 22.7 21.1 26.1 24.1 28.9 25.8 31.7 27.7 32.7 27.1 33.0 25.4 31.1 21.3 25.7 17.9 20.8 15.6 17.1 26.1 31.9 1 10.7 12.1 11.9 14.7 16.8 20.8 19.9 24.9 23.2 27.7 25.2 31.0 26.9 32.4 26.6 32.2 24.7 30.3 20.4 25.0 17.0 19.7 14.1 15.6 25.2 30.9 2 8.8 10.5 10.4 12.9 15.9 19.7 19.1 23.8 22.4 27.4 24.7 30.4 26.4 32.3 26.0 31.7 24.1 29.5 19.7 23.9 16.2 18.8 12.3 13.4 24.4 29.5 INDIANA Evansville 724320 0.4 15.0 17.2 15.4 18.7 19.0 23.7 21.3 26.6 24.3 29.9 26.4 32.7 27.1 33.6 26.9 32.9 25.6 31.2 21.9 26.1 18.8 21.7 16.5 18.9 26.1 32.3 1 13.7 15.7 14.4 16.6 18.0 21.9 20.5 26.0 23.6 29.2 25.8 32.1 26.7 33.6 26.4 32.3 25.1 30.8 21.1 25.2 18.0 20.9 15.3 17.6 25.4 31.4 2 12.3 14.1 13.3 15.4 16.9 20.7 19.7 24.5 23.0 28.1 25.2 31.5 26.3 32.8 25.9 31.9 24.4 30.2 20.2 24.7 17.3 19.9 14.3 16.3 24.7 30.4 Fort Wayne 725330 0.4 11.5 12.7 11.2 13.3 16.7 19.8 19.1 24.3 22.9 27.7 24.8 30.8 26.2 31.9 26.2 31.9 24.6 29.6 20.1 24.5 16.7 19.7 14.9 16.5 24.9 30.2 1 9.4 10.4 9.8 12.4 15.6 19.4 18.3 24.1 22.0 27.4 24.2 30.2 25.6 31.5 25.6 31.0 23.8 28.9 19.2 23.3 15.8 18.2 13.5 14.6 24.0 29.1 2 7.1 8.4 8.3 10.5 14.4 17.9 17.5 22.7 21.3 26.9 23.6 29.3 25.1 30.7 24.8 29.5 23.2 28.1 18.5 22.7 15.2 17.7 11.7 12.7 23.1 27.7 Indianapolis 724380 0.4 14.1 15.9 13.4 15.2 17.4 20.8 19.7 24.7 23.3 28.4 25.7 30.8 26.8 32.3 26.7 32.1 25.0 30.4 21.1 24.8 18.0 20.5 15.4 16.8 25.6 31.1 1 11.7 12.7 12.0 14.1 16.4 20.0 18.9 23.2 22.6 27.7 25.0 30.1 26.2 32.0 26.1 31.4 24.4 29.5 20.2 23.4 17.3 19.5 14.6 16.1 24.8 29.7 2 9.7 10.9 10.6 12.9 15.5 19.3 18.2 22.8 21.9 27.2 24.3 29.6 25.8 31.5 25.6 30.7 23.8 28.8 19.5 22.8 16.4 18.6 13.3 14.8 24.0 28.6 South Bend 725350 0.4 11.4 12.4 10.8 13.6 16.5 20.7 18.9 24.8 22.9 27.6 24.7 30.3 26.2 32.6 25.9 31.3 24.1 29.2 20.2 24.7 16.8 19.1 14.8 16.2 24.8 30.2 1 9.4 10.7 9.2 11.7 15.3 19.7 18.1 23.2 21.8 26.6 24.1 29.9 25.6 31.3 25.4 30.8 23.5 28.0 19.4 23.7 16.0 18.3 13.7 14.9 23.8 28.9 2 6.8 8.1 7.7 10.0 14.1 17.8 17.4 22.1 21.1 26.4 23.5 28.9 25.0 30.6 24.7 29.9 23.0 27.1 18.4 22.2 15.1 17.4 11.6 13.0 22.9 27.4 IOWA Des Moines 725460 0.4 8.4 10.5 10.3 14.3 16.1 21.8 19.6 26.8 22.9 28.4 25.6 31.8 26.8 33.5 26.1 32.9 24.8 30.4 20.8 25.9 16.1 19.1 13.5 15.1 25.3 31.8 1 6.2 8.5 8.4 13.4 14.8 19.0 18.6 24.9 22.1 27.4 25.0 31.0 26.2 32.6 25.6 32.2 24.1 30.1 19.8 24.9 15.2 17.4 10.9 12.6 24.6 30.8 2 4.7 7.8 7.0 11.2 13.4 18.5 17.7 23.9 21.2 26.9 24.4 30.4 25.7 32.1 25.1 31.7 23.5 28.9 18.8 22.9 14.2 16.5 7.7 10.7 23.7 29.6 Mason City 725485 0.4 3.7 6.0 7.1 11.3 14.8 19.5 18.6 26.3 22.2 27.8 25.4 31.4 26.7 32.2 26.3 31.6 24.5 29.8 20.4 23.8 14.9 16.8 10.4 11.4 25.2 30.8 1 2.7 4.3 5.3 8.2 13.0 16.6 17.6 24.1 21.2 26.6 24.6 30.6 26.1 31.6 25.6 31.1 23.4 28.1 19.1 22.7 13.8 15.7 5.9 7.2 24.1 29.4 2 1.9 3.3 3.8 5.8 11.2 14.6 16.3 22.2 20.4 25.9 23.9 29.8 25.4 30.8 24.9 30.5 22.7 27.1 18.0 21.8 12.4 14.7 3.8 5.3 23.1 27.8 Sioux City 725570 0.4 5.9 10.2 9.2 14.6 15.3 23.3 18.9 27.7 22.5 28.9 25.3 32.0 27.2 33.0 26.3 32.8 24.6 31.0 20.1 25.1 14.5 17.9 9.1 11.8 25.6 31.7 1 4.7 8.5 7.7 12.6 13.6 18.9 17.9 25.3 21.7 27.8 24.8 31.4 26.4 32.3 25.8 32.3 23.8 29.9 19.2 24.5 13.3 16.6 6.4 9.8 24.6 30.7 2 3.5 6.8 6.3 10.1 12.3 18.2 16.9 23.8 20.9 27.2 24.2 31.1 25.9 31.9 25.2 31.6 23.2 28.3 17.9 22.6 11.9 15.3 4.9 7.8 23.7 29.6 Waterloo 725480 0.4 5.2 7.6 7.8 11.8 15.4 20.6 18.8 25.8 22.7 27.9 25.2 31.7 26.6 32.8 26.2 31.7 24.6 30.1 20.4 24.8 15.2 17.2 12.1 13.6 25.1 30.7 1 3.7 5.4 5.8 9.2 13.7 18.2 17.7 23.9 21.7 27.1 24.5 30.9 26.0 31.9 25.6 31.1 23.7 28.4 19.3 24.4 14.2 16.4 9.1 10.3 24.1 29.6 2 2.7 4.4 4.5 6.9 12.1 16.1 16.7 22.1 20.9 26.8 23.9 29.8 25.3 31.0 24.9 30.4 22.9 27.5 18.2 23.0 12.8 15.2 5.9 7.4 23.2 28.2 KANSAS Dodge City 724510 0.4 9.6 17.8 11.8 21.3 14.9 21.8 18.4 25.7 21.6 29.3 23.8 32.3 24.1 32.8 24.0 32.1 22.4 30.8 19.0 25.4 15.1 19.2 10.4 16.8 23.2 32.4 1 8.2 16.1 10.5 18.7 13.8 22.4 17.3 24.1 20.8 27.8 23.2 32.4 23.6 32.9 23.4 32.2 21.8 30.3 18.3 24.6 13.8 17.9 9.1 15.4 22.5 31.9 2 7.1 14.2 9.3 17.2 12.9 20.9 16.5 23.7 20.1 27.1 22.7 32.0 23.2 32.7 23.0 32.2 21.3 29.8 17.6 23.7 12.8 17.3 7.9 13.9 21.9 31.0 Goodland 724650 0.4 7.6 16.9 9.8 20.2 11.8 22.0 15.3 23.9 18.9 25.6 21.4 29.6 22.4 30.2 21.9 29.4 20.2 28.4 15.7 22.6 11.3 19.2 8.6 17.4 21.2 29.7 1 6.6 15.1 8.4 17.8 10.7 20.3 14.6 23.4 18.2 24.9 20.9 29.1 21.8 29.8 21.3 29.6 19.6 27.7 14.9 23.3 10.3 18.9 7.4 14.8 20.6 29.0 2 5.6 13.3 7.6 16.3 9.8 19.2 13.8 23.0 17.6 24.6 20.4 28.8 21.4 29.8 21.0 29.4 19.0 27.1 14.1 22.9 9.5 17.6 6.2 14.0 19.9 28.8 Topeka 724560 0.4 12.5 15.4 13.7 17.9 17.7 22.8 21.2 27.6 23.7 29.1 26.3 32.4 27.2 33.3 26.7 33.2 25.4 31.4 22.0 26.1 17.7 21.1 14.5 16.7 26.0 32.4 1 10.1 12.9 12.1 16.7 16.9 22.0 20.4 26.4 22.9 28.6 25.8 31.6 26.7 33.1 26.2 32.7 24.8 31.3 20.9 25.4 16.7 19.9 13.3 15.5 25.3 31.6 2 8.5 12.2 10.4 16.2 15.8 20.9 19.5 24.8 22.2 27.7 25.2 30.8 26.2 32.7 25.8 32.1 24.3 30.5 20.1 24.5 15.8 19.0 11.2 13.7 24.6 30.9 Wichita, Airport 724500 0.4 12.8 14.7 13.6 18.3 18.0 23.4 20.7 26.8 23.4 29.5 25.2 33.6 25.7 33.1 25.7 32.8 24.7 31.4 21.7 26.8 17.6 20.7 14.6 16.4 24.8 32.6 1 10.2 13.1 12.6 16.7 17.0 21.7 19.9 25.5 22.7 28.7 24.7 32.6 25.2 32.8 25.2 32.7 24.1 30.7 20.7 25.2 16.7 19.6 13.0 15.1 24.2 32.1 2 8.7 12.3 11.2 16.9 15.9 20.8 19.3 24.6 22.1 27.7 24.2 31.8 24.8 32.7 24.7 32.5 23.5 29.8 19.8 24.3 15.7 18.7 11.3 13.2 23.6 31.4 KENTUCKY Covington/ Cincinnati OH, Intl Airport 724210 0.4 14.2 16.1 14.3 16.5 17.7 22.1 19.7 24.9 23.1 28.2 25.5 30.5 26.6 32.8 25.9 31.3 24.3 30.0 20.7 24.7 17.9 20.9 15.6 18.3 25.1 30.7 1 13.1 14.8 13.3 15.6 16.6 20.9 18.9 24.3 22.4 27.7 24.8 30.2 26.0 32.0 25.3 30.9 23.8 29.4 20.0 24.3 17.1 19.2 14.8 17.1 24.3 29.7 2 11.3 12.8 12.0 14.1 15.6 19.6 18.2 23.6 21.8 27.0 24.2 29.6 25.4 31.1 24.8 30.4 23.4 28.5 19.3 23.3 16.3 18.5 13.6 15.6 23.5 28.4 Lexington 724220 0.4 14.3 16.7 14.6 17.0 17.8 22.9 19.9 25.6 23.2 28.1 24.7 30.3 26.1 32.1 25.5 31.4 24.4 30.1 21.1 25.1 18.3 22.2 15.8 18.3 24.8 30.7 1 13.2 15.3 13.9 16.4 16.9 21.1 19.1 24.4 22.4 27.7 24.3 29.9 25.6 31.4 25.1 31.3 23.9 29.4 20.3 23.7 17.4 20.3 15.1 17.4 24.1 29.7 2 12.0 13.9 13.0 15.1 16.0 20.2 18.3 23.8 21.9 27.2 23.8 29.5 25.1 31.0 24.6 30.7 23.4 28.4 19.6 23.6 16.6 19.6 14.4 16.5 23.4 28.5 Louisville 724230 0.4 15.0 17.7 15.4 18.0 18.3 23.4 20.4 26.1 23.8 29.2 25.8 31.7 26.9 33.8 26.4 32.7 25.2 30.8 21.8 25.7 18.9 21.7 16.3 18.9 25.7 31.9 1 13.9 15.7 14.5 17.5 17.5 22.0 19.7 25.3 23.2 28.6 25.3 31.2 26.4 33.2 26.1 32.4 24.7 30.3 21.1 24.9 18.1 20.6 15.4 18.1 25.1 30.9 2 12.5 14.6 13.4 15.5 16.6 21.1 19.1 24.6 22.6 27.7 24.7 30.6 25.9 32.2 25.6 32.1 24.2 29.7 20.3 24.6 17.3 20.0 14.6 16.8 24.3 29.8 LOUISANA Baton Rouge 722317 0.4 21.1 23.6 21.7 24.2 22.8 26.6 23.7 28.1 25.7 30.7 26.8 32.6 27.3 32.9 27.4 32.6 26.7 30.9 25.2 28.9 23.4 25.8 22.8 24.4 26.7 31.7 1 20.5 22.8 21.1 23.6 22.2 25.3 23.3 27.7 25.2 29.9 26.6 32.1 26.9 32.2 26.9 31.8 26.3 30.7 24.6 28.3 22.9 25.3 21.9 24.1 26.3 31.3 2 19.9 22.4 20.3 22.6 21.7 24.9 22.9 27.1 24.7 29.1 26.2 31.4 26.7 31.8 26.7 31.4 25.8 30.4 24.0 27.9 22.3 25.0 21.4 23.3 25.8 30.7 Lake Charles 722400 0.4 21.2 23.1 21.6 23.4 22.4 24.9 24.1 27.1 25.7 29.8 27.4 31.3 27.4 32.0 27.4 32.1 26.9 30.8 25.8 29.1 24.0 26.1 22.8 24.3 26.9 31.2 1 20.7 22.3 21.0 22.3 22.0 24.3 23.6 26.7 25.3 29.2 26.9 30.9 27.1 31.6 27.1 31.6 26.6 30.6 25.3 28.6 23.4 25.5 22.3 23.4 26.6 30.9 2 20.2 21.3 20.4 21.4 21.7 23.7 23.3 26.3 24.9 28.6 26.6 30.8 26.8 31.3 26.8 31.3 26.3 30.4 24.8 27.9 22.9 24.9 21.7 23.0 26.2 30.6 New Orleans, Intl Airport 722310 0.4 21.7 23.8 22.4 24.5 23.6 25.9 24.7 27.8 25.8 30.0 27.4 32.4 28.1 33.6 27.9 33.0 27.4 31.7 25.7 29.4 24.1 26.3 23.1 25.0 27.3 31.9 1 21.0 23.4 21.8 23.7 23.0 25.5 24.1 27.6 25.4 29.8 26.9 31.8 27.7 32.9 27.5 32.1 26.9 31.0 25.1 28.9 23.6 25.8 22.6 24.4 26.8 31.3 2 20.4 22.6 21.2 22.9 22.5 25.1 23.6 26.9 25.1 29.4 26.7 31.4 27.3 32.2 27.2 31.6 26.6 30.6 24.6 27.9 23.0 25.1 22.0 23.9 26.4 30.7 Shreveport 722480 0.4 19.4 22.6 19.8 22.6 21.3 25.9 23.5 27.8 25.3 30.7 26.9 33.0 26.8 33.9 26.8 33.6 26.3 32.2 24.7 29.0 22.3 25.4 20.9 23.4 26.3 32.9 1 18.7 21.1 19.1 21.7 20.7 24.9 22.9 27.4 24.8 30.2 26.3 32.3 26.4 33.5 26.6 33.3 25.9 31.8 24.1 28.1 21.6 24.3 20.1 22.6 25.8 32.3 2 18.2 20.5 18.2 21.3 20.2 23.9 22.4 26.5 24.3 29.5 25.9 31.8 26.2 33.3 26.3 32.9 25.6 31.7 23.4 27.2 20.9 23.9 19.3 21.7 25.5 31.8 WMO# = World Meteorological Organization number WB = wet-bulb temperature, °C MDB = mean coincident dry-bulb temperature, °C Climatic Design Information 27.59 Table 4B Design Dry-Bulb—Mean Coincident Wet-Bulb Temperature Location and WMO# % Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Annual DB MWB DB MWB DB MWB DB MWB DB MWB DB MWB DB MWB DB MWB DB MWB DB MWB DB MWB DB MWB DB MWB 1a 1b 2a 2b 3a 3b 4a 4b 5a 5b 6a 6b 7a 7b 8a 8b 9a 9b 10a 10b 11a 11b 12a 12c 13a 13b 14a 14b Moline/Davenport Intl Airport 725440 0.4 11.9 9.4 15.2 10.2 23.8 15.3 29.3 18.1 32.0 21.0 34.8 23.6 36.1 25.5 35.9 25.3 33.3 23.2 28.7 19.9 21.3 14.7 16.0 13.8 33.9 24.3 1 9.9 7.1 12.6 7.9 21.6 13.6 27.5 17.8 30.9 20.8 33.8 23.2 35.1 25.0 34.2 25.3 31.7 23.1 27.2 18.9 19.4 14.8 13.6 11.0 32.2 23.4 2 7.8 5.2 10.5 7.1 19.4 13.0 25.7 16.8 29.7 19.9 32.7 22.8 34.1 24.7 33.1 24.9 30.3 22.4 25.6 18.2 18.0 13.9 11.7 8.9 30.7 22.7 Peoria 725320 0.4 12.5 9.9 15.3 10.1 23.9 15.3 28.4 18.6 31.1 21.2 34.0 23.3 36.1 25.2 35.7 24.9 32.4 23.3 27.9 19.3 21.4 15.6 16.2 14.1 33.3 24.3 1 10.6 8.4 13.1 9.3 22.1 14.6 26.9 17.7 30.2 20.5 32.9 23.1 34.8 25.2 33.8 24.9 31.3 22.9 26.6 18.6 19.8 15.2 14.4 12.3 31.7 23.4 2 8.6 6.2 11.3 8.4 19.9 13.7 25.3 17.1 29.1 20.3 32.0 22.7 33.7 24.8 32.6 24.8 30.1 22.8 25.0 17.7 18.1 14.4 12.5 10.7 30.2 22.8 Rockford 725430 0.4 9.4 8.4 12.3 8.4 22.2 14.8 28.6 18.1 31.1 20.5 33.8 22.6 35.1 24.3 34.4 24.4 31.8 22.6 27.8 19.9 20.0 14.8 14.5 12.8 32.6 23.5 1 7.3 5.6 9.8 6.7 20.1 12.8 26.6 17.1 30.2 20.2 32.6 22.4 33.9 24.6 32.9 24.1 30.6 22.6 26.1 18.3 17.9 14.3 12.2 11.0 31.0 22.6 2 5.7 3.7 7.9 5.2 17.6 12.4 24.4 16.7 29.1 19.7 31.6 22.3 32.8 24.0 31.7 23.8 29.2 21.9 24.5 17.5 16.4 13.3 9.6 7.4 29.6 21.7 Springfield 724390 0.4 14.8 12.1 16.9 12.6 24.6 16.4 29.0 18.6 32.3 21.7 35.2 23.7 36.6 25.6 35.7 25.3 33.6 23.5 29.6 19.7 22.8 15.9 17.3 14.2 34.1 24.5 1 12.6 10.4 15.0 10.8 22.9 15.8 27.7 18.4 31.3 21.2 34.0 23.3 35.3 25.6 34.4 25.1 32.3 23.1 28.1 19.1 21.3 15.8 15.9 13.8 32.6 24.0 2 10.7 8.3 13.3 10.4 21.1 14.7 26.3 17.9 30.2 20.8 33.1 23.1 34.3 25.0 33.3 25.2 31.1 23.0 26.6 18.1 19.7 15.2 13.9 11.5 31.2 23.2 INDIANA Evansville 724320 0.4 17.9 14.2 19.9 13.9 25.8 16.9 29.6 19.2 32.3 22.0 35.5 24.4 36.6 25.6 36.2 25.4 33.7 23.2 29.5 20.0 24.2 16.8 19.3 16.1 34.4 24.7 1 16.1 13.4 17.9 13.1 24.1 16.3 28.6 18.6 31.4 21.5 34.4 24.1 35.3 25.3 34.9 24.9 32.7 23.3 28.3 19.1 22.9 16.4 17.9 14.9 33.1 24.2 2 14.5 11.8 16.6 12.2 22.7 16.0 27.3 18.3 30.6 21.3 33.6 23.5 34.5 25.1 33.9 24.7 31.8 23.7 27.1 18.7 21.5 15.8 16.7 14.0 31.9 23.7 Fort Wayne 725330 0.4 13.0 11.3 14.2 10.3 22.6 15.4 28.1 17.5 31.2 20.8 33.5 21.5 35.0 23.7 33.6 24.2 32.0 22.6 27.6 17.9 21.6 15.5 16.5 14.7 32.4 23.2 1 10.9 9.1 12.0 9.3 20.9 14.8 26.4 17.4 30.1 20.3 32.5 21.9 33.6 23.9 32.4 24.0 30.6 22.5 26.2 18.2 19.7 15.1 14.8 13.1 30.9 22.6 2 8.6 6.9 10.4 8.1 19.0 13.6 24.2 16.4 29.0 20.0 31.6 22.1 32.6 23.9 31.4 23.6 29.4 21.8 24.5 17.2 18.1 14.6 13.1 11.4 29.6 21.8 Indianapolis 724380 0.4 15.8 13.9 17.0 11.8 24.0 16.1 27.8 18.1 31.1 21.3 33.4 22.8 35.3 24.8 33.9 24.6 32.4 23.0 27.9 18.7 22.5 16.2 17.6 14.6 32.7 24.1 1 13.3 11.3 15.1 10.7 22.3 14.9 26.5 17.6 30.2 21.1 32.4 22.4 34.1 24.9 32.8 25.1 31.3 23.2 26.6 18.6 21.2 16.0 16.2 14.1 31.3 23.4 2 11.3 9.2 13.4 10.4 20.7 14.2 24.9 17.2 29.2 20.4 31.6 22.9 33.0 24.8 31.9 24.6 30.2 22.7 25.2 18.1 19.8 15.3 14.9 13.1 30.1 22.7 South Bend 725350 0.4 12.5 11.3 14.3 10.0 22.9 15.0 27.8 17.1 31.2 20.3 33.6 21.8 34.9 24.4 33.8 24.8 31.9 22.4 27.2 18.2 21.3 15.6 16.5 14.8 32.2 23.0 1 10.7 9.1 12.0 9.5 20.8 13.8 25.9 17.1 30.1 20.3 32.4 21.9 33.6 24.1 32.4 23.8 30.1 22.1 25.8 18.0 19.4 14.8 14.9 13.3 30.8 22.4 2 8.5 6.3 10.1 7.3 18.8 13.2 23.7 16.1 28.9 19.6 31.4 21.6 32.4 23.7 31.3 23.4 28.9 21.6 24.2 17.4 17.9 14.6 12.9 11.4 29.3 21.6 IOWA Des Moines 725460 0.4 11.6 6.8 16.1 8.3 24.5 14.4 29.3 18.3 30.8 20.5 34.9 23.8 36.9 24.5 36.6 24.3 33.1 23.3 28.4 19.2 20.4 14.4 15.5 12.4 34.1 24.2 1 9.6 6.3 13.5 7.9 21.8 13.5 27.6 17.3 29.9 20.3 33.3 23.2 35.5 24.7 35.0 24.6 31.5 23.2 26.8 18.4 18.7 13.8 13.2 9.8 32.3 23.4 2 7.7 4.4 11.3 6.9 19.5 12.3 25.6 16.5 28.9 20.1 32.2 23.0 34.4 24.6 33.6 24.1 30.2 22.2 25.3 17.5 17.3 13.3 10.9 7.3 30.7 22.7 Mason City 725485 0.4 6.6 3.3 11.2 6.2 21.3 13.9 28.8 16.8 31.1 19.6 34.6 23.8 35.1 24.8 34.4 24.7 32.0 23.3 27.8 18.1 18.3 13.6 11.4 10.4 32.8 23.4 1 4.7 2.3 8.3 4.8 17.7 11.6 26.6 16.7 29.8 18.8 33.2 22.4 33.9 24.4 33.0 24.3 30.3 21.8 25.8 17.1 16.6 12.4 7.8 5.7 31.1 22.6 2 3.4 1.7 6.1 3.4 15.1 10.8 24.3 15.1 28.4 18.5 31.9 21.9 32.9 23.9 31.8 23.8 28.8 21.2 23.9 16.3 15.1 11.9 5.5 3.6 29.5 21.9 Sioux City 725570 0.4 11.2 5.4 15.4 8.1 24.6 13.5 30.9 17.2 31.9 19.6 35.9 23.0 36.8 24.4 35.7 24.6 33.3 22.1 28.8 18.1 19.6 12.0 12.7 8.1 34.2 23.8 1 8.9 4.2 12.9 7.4 21.9 13.0 28.7 16.9 30.7 20.0 34.4 23.2 35.4 24.5 34.5 24.2 31.7 22.3 27.1 17.3 17.9 12.0 10.5 6.6 32.4 23.3 2 7.1 3.4 10.7 6.2 19.3 11.4 26.6 16.1 29.6 19.5 33.2 22.9 34.3 24.6 33.3 24.0 30.3 21.4 25.5 16.4 16.4 10.8 8.3 4.5 30.8 22.4 Waterloo 725480 0.4 7.9 4.6 11.7 7.4 22.6 13.4 28.3 17.1 31.1 20.7 34.0 23.3 35.4 24.4 35.1 25.2 32.1 22.6 27.7 19.0 18.9 13.6 13.7 12.2 32.9 23.6 1 6.0 3.3 9.3 6.2 19.4 12.4 26.5 16.4 29.9 20.2 32.8 22.8 34.2 24.2 33.0 24.1 30.5 22.8 25.9 18.1 17.1 12.9 10.6 8.8 31.2 22.7 2 4.7 2.5 7.3 4.1 16.8 11.8 24.2 15.5 28.7 19.3 31.8 22.3 33.2 24.1 31.9 23.9 29.1 21.5 24.2 16.8 15.7 12.3 8.0 5.5 29.7 21.9 KANSAS Dodge City 724510 0.4 18.9 8.7 23.8 10.9 27.9 13.0 31.1 15.5 34.1 17.8 38.5 20.7 40.1 21.3 39.3 20.7 36.5 19.9 32.1 15.7 24.6 12.5 19.1 8.9 37.8 21.2 1 16.9 7.4 21.1 9.3 26.1 12.6 29.4 14.6 32.7 18.2 37.1 21.1 39.1 21.0 38.2 21.3 35.3 19.6 30.7 15.6 22.7 11.8 17.1 7.9 36.2 20.8 2 14.8 6.7 18.9 8.5 24.1 11.4 27.9 14.6 31.1 17.8 35.9 20.5 38.3 21.2 37.3 21.2 34.1 19.7 29.1 15.3 20.8 10.7 15.1 7.3 34.4 20.6 Goodland 724650 0.4 18.4 7.1 21.7 9.4 25.9 10.0 29.6 12.8 32.1 15.4 37.6 18.0 38.4 18.8 37.3 18.9 34.8 17.3 30.5 13.2 22.4 9.6 18.7 7.8 36.1 18.8 1 16.0 6.0 19.1 7.8 23.4 9.7 27.8 12.5 30.5 15.2 36.1 18.5 37.3 19.1 36.3 18.8 33.4 17.3 29.0 13.2 21.0 9.7 16.6 6.7 34.3 18.7 2 13.7 5.2 17.2 7.2 21.7 9.0 26.1 12.3 29.2 15.1 34.7 18.1 36.3 19.2 35.2 18.4 32.2 16.9 27.4 12.8 19.3 8.8 14.4 5.6 32.6 18.5 Topeka 724560 0.4 16.8 11.3 20.8 11.2 26.7 15.8 30.2 18.8 31.6 21.4 35.3 24.1 38.5 24.2 37.9 23.8 35.3 22.8 30.7 18.4 23.4 15.7 18.0 12.5 35.5 24.1 1 14.5 9.0 18.8 10.9 24.6 14.7 28.6 18.6 30.6 21.2 34.2 24.1 36.9 24.4 36.4 24.1 34.1 23.0 29.1 18.9 21.7 14.9 16.4 11.5 33.8 24.0 2 12.7 7.9 16.7 10.1 22.8 14.3 27.3 17.9 29.6 20.8 33.2 24.1 35.8 24.5 35.2 24.2 32.8 22.8 27.6 18.6 20.2 14.2 14.9 10.9 32.4 23.7 Wichita, Airport 724500 0.4 17.1 10.0 22.0 11.7 26.6 15.5 30.1 17.9 32.7 21.6 37.9 22.3 41.4 22.3 39.7 22.6 36.9 21.9 31.5 18.1 23.3 15.0 17.7 11.6 37.9 22.6 1 15.1 9.2 19.9 10.7 24.7 15.0 28.4 18.2 31.4 20.9 36.7 22.7 39.8 22.6 38.5 22.6 35.4 22.1 29.8 18.5 21.7 14.6 16.2 11.7 36.3 22.6 2 13.2 7.9 18.0 9.9 22.9 14.4 26.9 17.3 30.2 20.3 35.5 22.7 38.6 22.7 37.5 22.7 34.0 21.8 28.3 18.1 20.3 14.2 14.9 10.3 34.5 22.5 KENTUCKY Covington/ Cincinnati OH, Intl Airport 724210 0.4 16.6 13.3 18.3 12.3 24.7 15.7 28.3 18.3 30.8 21.2 33.4 22.4 35.2 24.4 34.3 23.9 32.3 23.0 27.7 19.2 22.8 16.6 18.9 14.9 32.8 23.6 1 15.1 12.6 16.7 12.3 23.1 15.2 27.1 17.6 29.9 21.1 32.5 22.6 34.0 24.4 33.1 23.7 31.3 22.4 26.7 18.3 21.2 15.2 17.4 14.0 31.4 23.0 2 13.2 10.6 15.2 11.4 21.5 14.8 25.7 16.9 28.9 20.3 31.7 22.8 33.1 24.3 32.2 23.4 30.1 22.2 25.4 17.7 19.8 14.9 16.0 13.4 30.1 22.4 Lexington 724220 0.4 17.7 13.5 19.1 12.2 24.8 15.8 28.0 18.2 30.7 21.2 32.9 22.0 34.9 24.0 34.4 23.7 32.3 22.7 27.7 18.3 23.7 17.5 19.4 15.1 32.6 23.2 1 15.8 12.1 17.8 12.2 23.6 15.4 27.1 17.6 29.8 20.8 32.1 22.3 33.6 23.7 33.2 23.4 31.2 22.3 26.7 18.3 22.3 16.2 18.0 14.1 31.4 22.9 2 14.3 11.7 16.5 11.7 22.1 14.7 25.9 17.4 28.9 20.5 31.4 22.3 32.8 23.8 32.3 23.2 30.3 22.2 25.6 17.9 20.8 15.2 16.8 14.0 30.2 22.4 Louisville 724230 0.4 18.2 13.8 20.3 13.1 25.9 16.9 29.2 18.8 31.1 22.4 33.9 23.8 36.0 25.2 35.4 25.1 33.2 23.7 28.6 20.2 24.1 17.3 19.8 15.2 33.7 24.6 1 16.6 13.1 18.6 12.8 24.4 15.4 28.2 18.3 30.3 21.7 33.0 23.7 34.7 25.0 34.3 25.0 32.2 23.6 27.4 19.3 22.7 16.3 18.4 14.8 32.4 24.1 2 15.1 12.1 17.1 12.3 22.9 15.4 27.1 17.9 29.5 21.6 32.4 23.6 33.9 24.9 33.4 24.6 31.2 23.2 26.5 19.2 21.4 15.9 17.4 14.3 31.2 23.4 LOUISANA Baton Rouge 722317 0.4 25.1 20.1 25.9 19.8 28.7 20.1 30.4 21.8 33.2 23.7 35.1 25.4 35.2 25.6 35.2 25.4 34.1 25.1 31.8 23.0 28.2 21.3 26.0 21.4 34.2 25.3 1 23.9 19.7 24.9 18.9 27.8 20.4 29.7 21.7 32.4 23.3 34.4 25.1 34.6 25.5 34.6 25.5 33.4 24.9 30.9 22.7 27.4 21.1 25.0 20.7 33.4 25.1 2 22.8 18.9 23.8 18.3 26.9 19.7 28.9 21.4 31.7 23.0 33.9 24.7 34.1 25.4 33.9 25.6 32.8 24.7 30.1 22.4 26.6 20.8 24.2 20.3 32.7 24.9 Lake Charles 722400 0.4 23.8 20.1 24.6 18.8 26.7 20.6 29.7 20.7 32.2 23.1 34.0 25.3 34.9 25.4 35.2 25.7 33.9 24.4 31.2 23.8 27.9 21.6 25.1 21.6 33.8 25.4 1 23.0 19.8 23.7 19.1 25.9 20.1 28.7 21.4 31.3 22.8 33.4 25.1 34.3 25.5 34.5 25.6 33.2 24.6 30.6 23.3 27.1 21.7 24.3 21.2 33.0 25.3 2 22.0 18.9 22.8 18.8 25.2 19.6 27.9 21.4 30.7 23.1 32.9 25.2 33.7 25.6 33.9 25.6 32.5 24.9 29.9 22.9 26.3 21.4 23.6 20.9 32.2 25.2 New Orleans, Intl Airport 722310 0.4 25.2 20.8 25.8 20.6 28.0 20.9 29.8 22.5 32.2 23.7 34.5 26.2 35.4 26.7 35.1 26.3 33.6 25.6 31.3 24.4 28.0 22.4 26.1 22.1 33.9 26.1 1 24.1 20.2 24.9 19.8 27.1 21.1 29.1 22.8 31.6 23.6 33.9 25.8 34.7 26.6 34.3 26.0 32.9 25.4 30.3 23.8 27.2 22.3 25.2 21.6 33.1 25.7 2 23.0 19.9 23.9 19.7 26.3 20.9 28.5 22.3 30.9 23.6 33.3 25.4 34.1 26.3 33.7 25.9 32.4 25.4 29.6 23.1 26.4 21.8 24.5 21.1 32.3 25.6 Shreveport 722480 0.4 24.4 18.0 26.2 17.0 28.9 18.7 30.3 20.5 32.6 23.7 35.7 25.4 37.6 25.2 37.6 24.8 35.9 24.6 32.3 21.8 27.6 19.8 24.7 19.4 35.9 24.9 1 22.9 17.4 24.6 16.8 27.7 18.4 29.4 21.2 31.8 23.4 34.9 24.9 36.7 25.1 36.8 24.9 34.8 24.4 31.3 21.6 26.6 19.6 23.6 18.9 34.8 24.8 2 21.4 16.6 23.3 15.9 26.5 18.2 28.7 20.4 31.2 22.9 34.2 24.8 35.9 25.1 36.1 24.8 34.0 24.3 30.3 21.6 25.7 19.5 22.5 18.4 33.7 24.6 WMO# = World Meteorological Organization number DB = dry-bulb temperature, °C MWB = mean coincident wet-bulb temperature, °C 27.60 2001 ASHRAE Fundamentals Handbook (SI) Table 4A Design Wet-Bulb—Mean Coincident Dry-Bulb Temperature Location and WMO# % Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Annual WB MDB WB MDB WB MDB WB MDB WB MDB WB MDB WB MDB WB MDB WB MDB WB MDB WB MDB WB MDB WB MDB 1a 1b 2a 2b 3a 3b 4a 4b 5a 5b 6a 6b 7a 7b 8a 8b 9a 9b 10a 10b 11a 11b 12a 12c 13a 13b 14a 14b MAINE Caribou 727120 0.4 6.2 6.6 6.2 7.3 8.1 11.9 14.1 18.1 19.6 27.6 22.7 27.2 24.1 28.8 23.8 29.7 21.4 24.9 17.9 21.3 14.1 14.9 7.8 7.9 22.4 26.9 1 3.4 4.0 4.7 6.8 6.9 9.4 12.7 15.7 18.3 23.8 21.6 26.3 23.4 28.6 22.6 26.5 20.4 24.1 16.7 20.1 12.5 13.4 5.3 6.3 21.2 25.2 2 2.0 2.9 3.0 4.3 5.7 8.1 11.1 14.8 17.4 23.0 20.7 25.1 22.8 27.8 21.8 25.3 19.5 22.0 15.3 18.0 10.9 12.0 3.7 4.4 20.1 24.2 Portland 726060 0.4 8.3 9.4 7.9 9.3 11.1 14.4 14.9 19.6 20.3 26.2 22.9 29.1 24.7 31.0 24.6 30.1 23.2 28.7 17.9 21.8 14.5 16.1 11.0 11.9 23.2 28.3 1 6.4 7.2 6.6 7.7 9.3 12.3 13.4 18.4 19.1 24.6 22.1 28.0 24.1 30.2 23.7 28.6 22.2 26.8 17.1 20.4 13.6 14.9 9.6 10.7 22.1 26.6 2 4.6 5.8 5.4 6.9 7.9 10.4 12.1 16.7 17.7 22.9 21.3 27.2 23.4 28.8 23.0 27.5 21.2 24.6 16.3 19.3 12.6 13.9 8.1 9.3 21.0 25.1 MARYLAND Glen Burnie/ Baltimore, BWI Airport 724060 0.4 13.6 16.4 15.0 18.2 17.9 24.1 19.6 26.9 23.4 29.4 25.4 31.8 26.6 33.1 26.3 31.6 25.2 30.5 21.7 25.2 19.3 22.7 16.4 18.6 25.4 31.2 1 12.1 14.5 13.5 17.3 16.4 21.6 18.8 25.3 22.7 28.2 24.8 30.8 26.0 32.3 25.7 31.3 24.6 29.9 21.1 24.1 18.0 20.6 15.0 17.4 24.6 30.2 2 10.3 12.5 11.8 14.7 15.2 19.9 17.9 23.8 21.8 27.2 24.3 30.2 25.6 31.7 25.2 30.7 24.1 29.5 20.2 23.3 17.2 19.3 13.4 15.5 23.9 29.3 MASSACHUSETTS Boston 725090 0.4 12.3 13.0 11.9 14.3 14.5 17.8 17.4 26.0 21.4 29.4 24.1 31.1 25.2 32.6 25.3 31.2 24.1 30.7 20.1 23.4 17.8 20.4 14.5 15.9 24.1 30.3 1 10.3 12.2 10.1 12.0 12.9 17.1 16.2 22.5 20.4 26.9 23.2 30.2 24.7 32.1 24.7 30.1 23.3 28.6 19.1 22.3 16.4 18.5 13.2 14.4 23.2 28.4 2 8.1 10.2 8.2 10.4 11.4 14.3 14.9 19.7 19.4 25.1 22.5 29.3 24.2 30.9 24.0 29.1 22.7 26.9 18.2 21.0 15.3 17.8 11.4 13.3 22.3 26.9 Worcester 725095 0.4 10.4 11.5 10.5 11.8 13.1 17.0 16.6 23.6 20.6 26.5 22.9 28.6 24.4 29.5 24.8 29.1 23.3 28.3 18.5 21.2 16.2 18.6 12.7 14.2 23.2 27.6 1 8.1 10.3 8.6 10.2 11.6 15.4 15.1 21.2 19.5 25.4 22.0 27.7 23.8 28.9 23.8 27.8 22.4 26.8 17.6 20.6 15.0 16.5 11.4 12.8 22.1 26.6 2 5.9 7.4 6.7 8.7 10.0 13.3 14.1 18.9 18.4 24.0 21.3 26.7 23.2 28.2 23.1 26.6 21.5 24.9 16.8 20.3 13.8 15.6 9.3 10.9 21.2 25.1 MICHIGAN Alpena 726390 0.4 5.7 6.9 6.1 8.7 12.8 16.2 17.6 26.1 21.1 26.6 23.1 29.9 24.9 30.7 24.4 29.3 22.8 28.4 18.7 23.6 14.5 16.9 10.6 11.7 23.1 28.4 1 4.0 5.3 4.4 6.3 10.7 14.7 16.5 22.7 20.0 25.2 22.2 28.8 24.2 30.3 23.6 28.8 21.9 26.2 17.6 21.1 13.0 14.7 7.4 8.4 21.9 27.1 2 2.7 4.1 3.1 5.1 8.6 12.6 14.9 19.7 19.1 24.4 21.4 27.5 23.4 29.0 23.0 27.5 21.0 24.7 16.3 19.6 11.4 13.3 5.0 6.2 20.9 25.6 Detroit, Metro 725370 0.4 10.4 11.7 9.6 11.9 15.7 19.8 18.9 24.2 22.9 28.3 24.6 30.7 25.8 32.7 25.8 31.9 24.1 29.2 19.9 24.8 16.1 18.4 13.9 15.6 24.4 29.9 1 7.7 9.1 8.1 10.4 14.1 18.0 17.9 23.4 22.1 27.7 23.9 30.0 25.2 31.6 25.0 30.7 23.3 28.4 18.8 23.0 15.1 17.4 12.1 13.6 23.4 28.7 2 5.8 7.0 6.6 8.3 12.7 16.8 16.9 22.5 21.2 27.2 23.2 29.2 24.5 30.3 24.2 29.3 22.6 27.4 17.8 21.7 14.2 16.7 10.0 11.5 22.5 27.4 Flint 726370 0.4 9.1 10.2 8.4 10.3 15.2 18.6 18.6 24.6 22.8 28.2 23.8 29.5 25.9 31.6 25.6 30.7 23.7 29.0 19.6 23.7 16.0 18.5 13.7 15.2 24.1 29.0 1 7.2 8.6 7.1 8.8 13.7 17.2 17.7 23.1 21.8 27.1 23.3 29.5 25.1 30.2 24.7 29.6 22.9 27.3 18.7 22.6 15.0 16.9 11.7 12.6 23.1 27.9 2 4.9 6.7 5.6 7.5 12.0 15.5 16.7 21.9 20.8 25.7 22.7 28.7 24.3 29.4 23.9 28.3 22.2 25.9 17.7 21.4 14.1 16.2 9.3 10.8 22.1 26.7 Grand Rapids 726350 0.4 9.2 10.3 9.1 11.2 15.6 20.1 19.1 25.1 22.3 27.4 24.3 29.8 25.7 31.3 25.6 31.3 23.8 29.0 20.2 24.8 15.8 18.6 13.4 14.8 24.3 29.6 1 7.1 8.1 7.0 8.9 14.2 18.2 18.1 23.5 21.6 27.2 23.7 29.6 25.1 30.7 24.9 30.4 23.1 27.3 19.0 23.1 14.8 16.7 11.5 12.6 23.3 28.1 2 4.6 5.8 5.5 7.2 12.4 15.9 17.1 21.4 20.8 26.8 23.1 28.9 24.4 29.9 24.3 29.3 22.4 26.5 18.0 21.7 14.1 16.0 8.7 10.4 22.4 27.0 Hancock 727440 0.4 5.5 6.2 5.5 7.6 12.4 14.7 17.3 22.8 20.3 26.4 23.1 29.1 24.3 30.1 24.4 29.8 22.4 26.5 18.3 21.9 13.9 15.8 8.6 9.5 22.9 27.8 1 3.4 4.6 4.1 5.5 10.5 12.6 15.9 21.5 19.6 25.9 22.3 27.2 23.7 29.1 23.6 28.4 21.6 24.9 17.3 20.7 12.8 14.8 6.2 7.1 21.8 26.4 2 1.9 2.8 2.7 4.1 8.5 10.8 14.7 18.4 18.7 24.3 21.6 26.6 23.1 27.9 23.0 27.5 20.8 23.7 16.3 19.4 11.7 13.3 3.8 4.6 20.8 25.0 Lansing 725390 0.4 9.6 10.5 9.3 10.8 15.6 18.9 19.3 24.9 22.9 27.3 24.4 29.8 25.8 31.7 25.3 31.1 24.1 28.7 19.8 24.4 16.1 18.5 13.6 14.9 24.4 29.6 1 7.5 8.9 7.6 9.6 14.2 17.6 18.1 23.3 21.8 27.0 23.7 29.3 25.2 30.8 24.7 30.3 23.4 27.6 18.9 22.8 15.0 17.7 11.6 12.7 23.4 28.2 2 5.2 6.7 6.0 7.6 12.7 15.9 17.1 21.6 21.0 26.3 23.1 28.8 24.6 29.7 24.2 29.4 22.8 26.8 18.0 21.7 14.2 16.2 9.1 10.2 22.6 27.0 Muskegon 726360 0.4 7.7 8.6 6.9 9.0 14.5 18.4 17.9 23.1 21.3 25.7 22.9 28.3 24.7 29.3 25.1 29.8 23.2 27.3 19.2 22.8 15.1 16.8 11.7 12.8 23.6 27.8 1 5.8 6.7 5.6 7.2 12.8 15.8 16.8 21.3 20.3 25.2 22.4 27.6 24.1 28.7 24.5 28.8 22.6 26.3 18.3 21.7 14.3 15.9 10.1 11.0 22.7 26.7 2 4.0 4.8 4.3 6.0 11.3 14.8 15.7 19.8 19.6 24.7 21.8 26.7 23.6 27.9 23.9 27.9 21.9 25.2 17.5 20.7 13.4 15.1 7.7 8.8 21.8 25.4 Sault Ste. Marie 727340 0.4 2.9 3.4 3.4 4.2 6.9 8.9 15.6 20.6 20.6 26.6 22.0 27.5 24.2 29.1 23.6 28.3 22.4 26.8 17.3 20.4 12.3 12.8 6.6 7.1 22.2 26.8 1 1.8 2.3 2.3 3.0 5.8 7.7 14.0 18.3 19.3 24.4 21.0 26.7 23.3 28.0 22.8 26.9 21.2 23.9 16.1 19.1 10.9 12.2 4.2 4.7 21.0 24.9 2 1.0 1.5 1.6 2.3 4.8 6.3 12.3 15.4 18.1 23.3 20.1 25.1 22.4 27.4 21.9 25.8 20.1 22.4 15.0 17.6 9.4 10.7 2.8 3.3 19.8 23.6 Traverse City 726387 0.4 5.2 6.2 5.8 8.2 13.4 17.8 18.2 25.0 21.7 27.9 23.7 30.7 24.9 31.8 24.7 30.1 23.4 28.6 19.8 24.2 14.9 17.0 11.4 13.4 23.4 28.9 1 3.7 5.3 4.3 6.2 11.8 15.2 17.2 23.4 20.7 27.5 22.8 29.7 24.2 30.8 23.9 29.1 22.6 27.3 18.7 22.7 13.8 16.1 8.4 10.1 22.4 27.7 2 2.7 4.2 3.1 4.9 10.0 13.7 15.9 21.5 19.7 25.6 21.9 28.5 23.6 29.4 23.3 27.9 21.7 25.4 17.6 21.2 12.6 14.7 5.7 6.6 21.3 26.4 MINNESTOA Duluth 727450 0.4 1.5 3.3 3.2 6.9 7.9 11.8 14.2 19.9 19.1 25.7 22.1 26.9 23.7 29.9 23.1 28.8 21.4 26.1 16.6 20.0 11.1 12.3 3.7 4.4 21.9 27.2 1 0.5 1.9 1.9 4.2 6.2 9.1 12.6 18.6 18.0 24.5 20.9 26.6 23.0 28.8 22.3 27.8 20.3 23.6 15.4 18.6 9.3 10.6 1.5 2.4 20.7 25.4 2 −0.3 0.9 0.9 2.9 4.6 8.1 11.0 16.8 17.1 22.5 19.9 25.6 22.3 27.9 21.6 26.4 19.3 22.5 14.3 18.0 7.3 9.8 0.7 1.5 19.5 24.1 International Falls 727470 0.4 1.2 3.1 3.1 5.8 7.8 12.2 14.9 22.3 20.0 27.9 22.3 28.4 24.2 29.8 23.6 28.9 21.9 25.4 16.4 21.1 10.0 11.2 1.9 2.9 22.3 27.8 1 0.1 1.3 1.8 4.1 6.4 9.5 13.6 20.4 19.1 26.8 21.3 26.8 23.3 29.1 22.8 27.9 20.5 23.9 15.1 19.9 8.2 11.7 0.6 1.5 21.1 26.1 2 −0.8 0.2 0.8 2.6 4.9 8.2 12.2 18.8 18.1 24.3 20.5 26.1 22.6 28.4 22.1 27.2 19.3 23.2 14.1 18.1 6.8 8.9 −0.2 0.8 19.9 24.7 Minneapolis-St. Paul 726580 0.4 3.0 5.3 6.3 10.3 13.7 17.9 18.0 25.0 21.7 27.8 24.7 31.5 26.0 32.2 25.6 32.1 24.0 30.1 19.6 23.1 14.0 16.1 7.9 9.2 24.4 30.8 1 1.9 3.7 4.3 6.9 11.7 16.1 16.9 23.2 20.8 26.7 24.0 30.3 25.3 31.8 24.9 31.3 22.9 28.3 18.6 22.6 12.4 15.1 4.1 5.3 23.4 29.1 2 1.2 2.8 3.1 4.9 10.0 13.7 15.7 21.7 19.9 25.7 23.3 29.4 24.7 31.1 24.1 30.2 22.0 26.7 17.3 21.9 10.9 13.4 2.8 4.3 22.3 27.8 Rochester 726440 0.4 2.9 4.7 6.3 8.5 14.0 18.0 18.1 25.2 21.6 26.6 24.4 29.9 26.3 31.2 25.2 30.6 23.6 27.7 19.2 22.7 14.0 16.2 9.0 10.3 24.2 29.2 1 2.1 3.5 3.9 6.0 12.2 14.8 16.8 22.6 20.7 26.1 23.8 29.1 25.2 30.2 24.6 29.7 22.5 26.7 18.1 21.9 12.8 14.7 5.4 6.6 23.1 27.7 2 1.3 2.5 2.6 4.0 10.1 13.0 15.7 20.9 19.9 24.6 23.0 28.3 24.4 29.2 23.9 29.0 21.7 25.5 16.9 20.9 11.4 13.9 3.1 4.2 22.1 26.5 Saint Cloud 726550 0.4 2.6 5.1 5.6 8.4 12.4 16.6 17.8 24.2 21.3 27.3 24.3 31.1 25.9 31.8 25.3 31.4 23.9 29.3 19.1 21.6 12.7 14.3 4.4 6.5 24.3 30.3 1 1.7 3.7 3.9 6.4 10.6 15.3 16.6 22.1 20.5 27.6 23.6 30.2 25.2 31.2 24.7 30.6 22.8 27.7 18.2 21.9 11.1 12.9 2.8 4.2 23.2 29.1 2 0.9 2.4 2.7 5.1 8.8 12.4 14.9 21.1 19.7 26.4 22.9 28.8 24.6 30.7 24.0 29.7 21.8 26.4 16.9 21.1 9.6 12.7 1.7 2.8 22.1 27.6 MISSISSIPPI Jackson 722350 0.4 19.9 22.9 20.6 23.8 21.8 26.0 23.4 27.9 25.0 30.1 26.7 32.8 27.1 34.1 26.9 32.5 26.2 31.8 24.5 28.4 22.5 25.4 21.4 23.8 26.4 32.4 1 19.2 21.6 19.9 22.7 21.2 25.1 22.9 27.0 24.5 29.4 26.2 32.4 26.8 33.5 26.7 32.4 25.8 31.2 23.8 27.4 21.8 24.2 20.7 22.9 25.9 31.8 2 18.4 20.8 19.1 21.6 20.7 24.4 22.3 26.1 23.9 29.1 25.8 31.8 26.5 32.9 26.4 32.0 25.6 30.8 23.1 26.7 21.1 24.1 20.1 22.2 25.6 31.2 Meridian 722340 0.4 19.6 21.8 20.3 22.9 21.4 25.4 23.1 27.3 24.7 30.4 26.4 32.2 27.1 34.1 26.8 32.4 25.9 31.4 24.0 28.3 21.8 24.2 21.2 23.7 26.3 32.6 1 18.8 20.8 19.6 21.8 20.8 24.5 22.5 26.6 24.2 29.7 25.8 32.0 26.7 33.5 26.4 32.4 25.6 31.1 23.3 27.2 21.2 23.9 20.7 23.1 25.7 31.9 2 18.1 20.2 18.7 21.3 20.3 23.8 21.9 25.9 23.7 28.9 25.5 32.0 26.3 33.1 26.2 32.4 25.2 30.9 22.7 26.1 20.6 23.7 19.8 22.2 25.2 31.3 MISSOURI Columbia 724450 0.4 13.5 16.3 13.8 18.0 17.8 23.3 20.5 26.8 23.9 28.7 25.9 31.8 26.7 33.0 26.4 32.6 24.9 31.1 21.3 26.3 17.7 21.3 15.7 18.3 25.6 31.8 1 11.8 14.1 12.7 16.4 16.8 22.1 19.7 25.9 22.9 27.6 25.4 31.0 26.3 32.6 25.8 32.1 24.3 30.7 20.4 24.4 16.8 20.7 14.3 17.1 24.9 31.0 2 10.0 12.2 11.4 15.5 15.8 21.1 18.9 25.6 22.1 26.9 24.8 30.5 25.8 32.1 25.3 31.6 23.7 29.9 19.7 24.0 15.9 19.3 12.7 15.0 24.1 30.2 Kansas City 724460 0.4 13.2 16.3 13.3 17.8 17.7 22.4 21.1 27.6 23.5 28.4 25.8 31.9 26.7 32.9 26.6 32.5 25.2 32.0 22.1 26.2 17.6 21.5 15.1 17.4 25.7 32.3 1 10.9 13.6 11.8 16.1 16.7 21.9 20.3 26.6 22.6 27.9 25.3 31.4 26.3 32.9 26.1 32.4 24.6 31.1 20.8 26.1 16.9 20.6 13.6 16.2 25.1 31.7 2 9.0 11.8 10.4 15.0 15.8 20.6 19.3 25.1 21.9 27.6 24.9 31.2 25.8 32.6 25.6 32.2 24.1 30.3 20.1 24.3 16.0 19.4 11.8 14.3 24.4 30.8 WMO# = World Meteorological Organization number WB = wet-bulb temperature, °C MDB = mean coincident dry-bulb temperature, °C Climatic Design Information 27.61 Table 4B Design Dry-Bulb—Mean Coincident Wet-Bulb Temperature Location and WMO# % Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Annual DB MWB DB MWB DB MWB DB MWB DB MWB DB MWB DB MWB DB MWB DB MWB DB MWB DB MWB DB MWB DB MWB 1a 1b 2a 2b 3a 3b 4a 4b 5a 5b 6a 6b 7a 7b 8a 8b 9a 9b 10a 10b 11a 11b 12a 12c 13a 13b 14a 14b MAINE Caribou 727120 0.4 6.6 6.3 8.4 5.8 13.1 7.6 21.0 13.0 28.9 18.2 30.8 19.7 31.8 22.2 30.6 22.5 27.4 20.1 22.7 17.3 15.2 13.6 8.4 7.4 29.4 20.5 1 4.4 2.8 6.3 4.9 10.3 5.3 18.5 11.3 26.8 17.0 29.6 19.3 30.7 21.8 29.3 20.9 25.7 19.1 20.5 16.2 13.7 12.1 6.6 5.3 27.6 19.4 2 3.1 1.8 4.5 2.6 8.7 4.9 16.2 10.1 25.1 15.3 28.4 19.1 29.6 21.4 28.2 20.0 24.4 18.2 18.6 14.6 12.4 10.6 4.6 3.4 26.0 18.8 Portland 726060 0.4 9.5 7.7 10.3 7.2 16.0 9.9 22.6 13.4 29.1 18.8 32.0 21.4 32.9 23.7 32.2 22.6 29.8 22.6 24.1 16.7 17.7 12.9 12.4 9.8 30.2 21.8 1 7.9 5.9 8.4 5.8 13.4 8.6 19.5 12.4 26.8 17.7 30.1 21.2 31.6 22.9 30.7 22.4 28.1 21.4 22.1 15.9 15.7 12.4 10.9 9.3 28.4 20.8 2 6.3 3.9 7.1 4.8 11.2 6.9 17.3 10.7 24.4 16.2 28.6 20.2 30.3 21.8 29.5 21.8 26.2 20.2 20.3 15.2 14.4 11.4 9.4 7.9 26.7 19.8 MARYLAND Glen Burnie/ Baltimore, BWI Airport 724060 0.4 16.9 13.5 20.5 12.8 25.7 16.5 30.1 18.3 32.8 21.6 34.6 23.4 36.4 24.5 35.2 24.4 33.9 23.2 28.2 19.4 24.2 18.3 19.8 15.5 34.0 23.7 1 14.7 10.7 17.9 11.7 23.6 14.8 28.1 17.4 31.3 20.4 33.7 23.2 35.3 24.3 34.1 24.0 32.6 23.3 26.7 19.2 22.3 16.3 17.9 14.0 32.6 23.2 2 12.9 9.7 16.1 11.4 21.2 14.3 26.1 16.6 29.9 20.3 32.7 22.7 34.3 24.2 33.1 23.7 31.3 22.7 25.3 18.7 20.8 15.6 16.1 12.8 31.1 22.5 MASSACHUSETTS Boston 725090 0.4 14.2 11.7 14.9 12.2 20.7 12.7 27.1 16.8 31.2 20.0 34.0 22.3 35.3 24.3 33.6 23.6 32.1 23.3 26.1 18.3 21.5 16.6 16.6 13.7 32.5 22.6 1 12.2 9.8 12.8 9.7 17.4 11.9 23.9 15.4 29.1 18.8 32.7 22.1 34.0 23.8 32.6 23.3 30.1 22.3 24.4 17.4 19.9 15.1 14.9 12.9 30.7 21.9 2 10.4 7.9 10.9 7.7 15.1 10.6 21.5 13.5 27.3 17.8 31.2 21.5 32.7 23.2 31.4 22.6 28.4 21.3 22.9 16.8 18.2 14.6 13.5 11.2 28.9 21.1 Worcester 725095 0.4 12.1 9.9 13.0 8.9 18.9 10.9 26.1 15.0 29.1 18.8 30.7 21.8 31.9 22.8 30.9 23.1 29.2 22.7 24.6 16.6 19.4 14.9 14.6 12.3 29.7 21.6 1 10.2 7.8 10.9 8.3 16.7 10.4 23.3 14.4 27.5 18.0 29.6 20.9 30.9 22.7 29.6 22.1 27.7 21.9 23.1 15.9 17.7 13.8 13.2 11.1 28.2 20.8 2 8.1 5.5 8.9 6.3 14.4 9.2 20.9 12.6 26.1 17.3 28.5 20.2 29.9 22.2 28.7 21.7 26.2 20.4 21.6 15.9 16.2 13.2 11.1 9.2 26.7 19.9 MICHIGAN Alpena 726390 0.4 7.4 5.7 8.7 4.9 17.3 12.1 27.6 16.8 29.4 18.6 32.7 21.3 33.9 23.1 31.9 23.1 29.3 21.8 24.5 17.8 17.6 13.1 12.6 10.2 30.8 21.7 1 5.8 3.8 6.8 4.2 14.9 9.9 23.9 15.5 28.0 18.5 31.2 20.7 32.4 22.9 30.7 22.5 27.8 20.8 22.3 16.1 15.6 12.6 8.8 7.0 28.9 20.4 2 4.3 2.6 5.3 2.7 12.5 8.3 21.2 13.8 26.3 17.9 29.6 20.4 31.1 22.1 29.4 21.7 26.2 19.8 20.6 15.1 13.9 11.3 6.3 4.7 27.1 19.6 Detroit, Metro 725370 0.4 11.6 10.2 12.9 9.2 22.0 14.5 27.6 17.5 31.2 20.9 33.2 22.3 34.8 23.7 33.3 24.6 31.3 22.6 26.9 17.8 20.2 14.9 15.8 13.7 32.1 22.8 1 9.4 7.7 10.6 7.7 19.6 13.5 25.7 16.8 29.8 20.5 32.2 22.0 33.4 23.8 32.1 23.7 30.0 22.2 25.1 17.8 18.4 14.5 13.9 11.9 30.6 22.1 2 7.2 5.6 8.6 6.2 17.1 11.6 23.3 15.9 28.5 19.9 31.2 21.9 32.3 23.3 31.1 22.7 28.7 21.8 23.4 16.4 17.0 13.8 11.6 9.6 29.1 21.3 Flint 726370 0.4 10.6 9.3 10.9 7.4 20.8 12.9 27.1 17.4 30.1 20.9 32.8 21.5 33.9 22.9 32.4 24.2 30.9 22.3 26.1 18.3 19.9 14.8 15.6 13.2 31.3 22.6 1 8.9 6.7 9.3 6.8 18.9 12.9 25.1 16.7 28.9 20.3 31.4 21.6 32.6 23.6 31.2 23.3 29.1 21.8 24.3 17.4 18.2 14.3 13.4 11.4 29.8 21.8 2 6.7 5.0 7.6 5.4 16.5 11.6 22.9 15.9 27.7 19.3 30.4 21.4 31.6 23.2 30.2 22.4 27.7 20.9 22.8 16.8 16.7 13.6 10.9 9.1 28.4 20.8 Grand Rapids 726350 0.4 10.3 9.0 11.3 8.1 21.1 14.2 27.1 17.8 30.9 20.5 32.9 21.8 33.6 23.4 33.3 23.2 30.7 22.3 26.2 18.5 19.8 14.3 15.0 12.9 31.8 22.8 1 8.3 6.7 9.3 7.0 19.1 12.8 25.4 17.1 29.5 20.2 31.8 21.6 32.7 23.2 32.1 23.4 29.2 22.1 24.8 18.0 17.8 13.8 12.9 11.1 30.2 21.8 2 6.2 4.3 7.4 5.2 16.9 12.6 23.2 15.7 28.2 19.6 30.9 21.4 31.9 23.4 30.8 22.6 27.8 21.0 23.3 17.3 16.3 13.3 10.3 8.7 28.8 20.9 Hancock 727440 0.4 6.3 5.3 7.6 5.1 16.1 10.7 24.8 16.6 29.5 18.9 31.1 21.1 32.7 22.6 31.3 23.2 27.9 21.2 24.6 16.8 17.1 12.8 10.0 8.8 29.7 21.5 1 4.8 3.2 6.1 3.8 13.4 9.5 22.2 14.8 28.0 17.9 29.7 20.7 31.2 22.2 30.1 22.6 26.4 20.2 22.8 15.9 15.3 12.3 7.5 6.1 28.1 20.4 2 3.1 1.7 4.4 2.2 11.2 7.6 20.2 13.6 26.6 17.2 28.6 20.2 29.9 21.9 29.0 22.0 25.2 19.1 20.9 15.4 13.7 11.1 4.7 3.4 26.7 19.6 Lansing 725390 0.4 11.2 8.8 11.3 8.6 21.2 13.2 26.9 17.9 30.7 20.7 33.2 21.6 34.1 23.5 33.6 24.1 31.1 22.6 26.4 18.1 20.3 14.9 15.0 13.2 31.9 22.9 1 9.2 7.5 9.8 7.4 19.3 12.9 25.4 17.4 29.4 21.2 31.9 21.9 33.1 23.5 32.2 23.7 29.4 22.1 24.9 18.0 18.2 14.5 13.0 11.3 30.2 22.0 2 6.9 5.2 7.7 5.7 17.0 12.1 23.1 16.1 28.0 19.8 30.8 21.7 32.1 23.4 30.9 22.7 28.0 21.4 23.1 16.9 16.6 13.6 10.4 8.7 28.7 21.3 Muskegon 726360 0.4 8.7 7.4 9.4 6.6 19.3 13.0 25.0 17.5 28.8 19.1 30.7 21.1 31.7 23.0 31.4 23.7 28.7 21.8 24.3 18.2 17.8 14.2 12.9 11.4 29.7 21.8 1 6.7 5.6 7.4 5.0 17.3 11.3 22.9 15.6 27.7 18.3 29.7 20.7 30.6 22.6 30.1 23.1 27.5 21.1 23.1 17.4 16.4 13.6 11.2 10.2 28.4 21.1 2 5.2 3.8 5.9 4.1 15.3 10.8 21.2 14.1 26.6 18.4 28.8 20.3 29.8 22.2 29.0 22.6 26.6 21.1 21.8 16.5 15.4 13.0 9.1 7.6 27.2 20.4 Sault Ste. Marie 727340 0.4 3.7 2.7 4.6 2.2 10.2 6.1 22.3 14.5 28.2 19.4 29.7 20.7 31.3 22.3 29.8 21.9 27.2 22.0 22.6 16.7 13.6 10.9 7.5 6.4 28.4 20.8 1 2.4 1.5 3.7 2.0 8.4 4.9 19.6 12.2 26.6 17.9 28.2 19.5 29.9 22.3 28.7 21.2 25.4 20.0 20.1 15.2 12.4 10.4 5.1 3.7 26.6 19.8 2 1.7 0.8 2.8 1.2 6.8 4.0 17.4 10.8 24.9 16.3 26.8 19.2 28.6 21.3 27.6 21.0 23.9 18.9 18.3 14.4 11.1 9.1 3.4 2.6 24.9 18.7 Traverse City 726387 0.4 7.1 4.4 8.3 5.4 19.2 12.4 27.8 17.7 30.9 20.2 33.8 21.4 34.2 22.9 33.2 22.8 29.9 22.2 25.8 18.6 18.7 13.3 13.2 11.6 31.7 21.9 1 5.7 3.4 6.9 3.9 16.8 10.6 25.2 16.2 29.3 19.6 32.1 21.4 33.1 22.9 31.5 22.4 28.6 21.4 24.1 17.6 16.6 12.8 10.0 7.8 29.8 20.9 2 4.4 2.4 5.2 2.6 14.2 9.8 22.4 14.2 27.4 18.1 30.6 20.4 31.9 22.3 30.2 21.6 27.2 20.5 22.4 16.5 15.3 12.4 7.2 5.1 28.1 20.1 MINNESTOA Duluth 727450 0.4 4.0 1.2 6.9 2.9 12.8 6.7 23.1 12.5 28.2 17.3 29.8 20.0 31.8 22.7 31.1 21.4 27.6 19.9 22.5 14.4 14.2 8.7 5.0 3.1 29.1 20.4 1 2.3 0.1 4.7 1.7 10.2 5.5 20.4 11.1 26.4 16.2 28.5 19.4 30.4 21.6 29.6 21.5 25.6 19.3 20.6 13.9 11.8 8.1 2.8 0.7 27.2 19.3 2 1.1 -0.4 3.2 0.5 8.2 4.4 18.2 10.2 25.0 14.8 27.2 18.7 29.2 21.2 28.3 20.3 24.3 17.6 18.8 12.9 10.1 7.2 1.7 0.6 25.6 18.3 International Falls 727470 0.4 3.1 0.9 6.0 2.6 13.1 7.1 24.7 13.5 30.4 18.4 31.4 20.1 32.7 23.0 31.9 21.7 27.8 19.8 23.6 14.7 14.0 9.4 3.7 1.5 30.1 20.6 1 1.5 -0.2 4.6 1.5 10.8 5.6 22.2 11.9 28.7 17.7 30.1 19.8 31.3 22.0 30.3 21.3 26.2 19.3 21.4 14.2 11.4 7.1 1.9 0.4 28.2 19.4 2 0.3 -0.9 3.0 0.4 8.8 4.3 19.9 11.6 26.9 16.5 28.7 18.8 30.1 21.2 29.2 20.4 24.6 17.8 19.3 13.1 9.7 6.8 0.9 -0.3 26.6 18.7 Minneapolis-St. Paul 726580 0.4 5.6 2.5 10.1 6.2 19.4 12.1 27.8 16.2 31.3 18.9 34.2 22.6 35.6 23.7 34.4 23.9 31.7 22.4 26.7 17.3 17.7 12.5 9.6 7.5 32.8 22.7 1 4.2 1.6 7.4 4.1 16.6 10.9 25.3 15.5 29.8 18.9 32.8 22.4 34.3 23.6 32.9 23.3 29.8 21.4 24.6 16.8 15.8 11.6 6.1 3.4 31.1 21.9 2 3.1 1.0 5.6 2.6 14.3 8.9 23.2 14.5 28.3 18.2 31.5 21.6 33.1 22.9 31.8 23.0 28.3 21.1 22.8 16.0 14.1 10.3 4.6 2.6 29.4 21.1 Rochester 726440 0.4 5.2 3.0 9.0 6.2 18.8 12.4 26.8 16.4 29.7 19.1 32.6 21.8 33.8 23.2 32.8 23.4 29.7 22.6 25.7 17.1 16.9 12.6 10.3 9.0 31.1 22.4 1 3.6 1.7 6.3 3.6 15.8 11.1 24.5 16.7 28.4 18.9 31.3 21.6 32.6 23.4 31.2 23.6 28.3 21.2 23.8 16.4 15.4 12.1 6.6 5.2 29.5 21.8 2 2.6 1.2 4.4 2.2 13.7 9.6 22.3 14.6 27.0 18.4 30.2 21.2 31.4 23.2 30.1 22.9 26.8 19.9 22.3 15.8 14.1 11.1 4.3 2.9 27.9 20.9 Saint Cloud 726550 0.4 5.3 2.3 8.8 4.8 18.9 12.7 27.3 15.2 31.2 19.3 34.2 21.7 35.4 23.1 34.0 23.2 31.3 22.7 25.6 16.6 17.1 10.6 6.8 3.8 32.5 22.3 1 4.1 1.4 6.9 3.7 15.7 9.9 25.1 15.2 29.5 19.6 32.5 21.6 34.1 23.2 32.8 22.6 29.7 21.2 24.1 15.6 14.8 9.3 4.8 2.5 30.8 21.7 2 2.8 0.4 5.4 2.3 13.3 8.0 22.9 14.4 28.1 17.9 31.4 20.9 32.7 23.1 31.7 22.5 28.2 20.3 22.5 15.4 12.9 9.1 3.1 1.4 29.3 20.8 MISSISSIPPI Jackson 722350 0.4 24.1 18.1 25.9 18.3 28.7 19.7 30.2 21.1 33.0 22.7 36.1 24.1 36.9 25.8 36.4 25.1 35.5 24.3 31.6 21.6 27.7 20.0 25.1 20.1 35.2 24.8 1 22.7 18.0 24.7 17.8 27.6 18.8 29.5 20.9 32.2 22.2 35.1 24.2 36.0 25.3 35.6 24.9 34.3 24.1 30.7 21.3 26.7 19.8 24.1 19.8 34.1 24.7 2 21.3 17.1 23.3 16.8 26.6 18.2 28.7 20.2 31.5 22.2 34.4 24.3 35.1 25.2 34.9 24.9 33.5 24.1 29.8 20.9 25.8 19.3 23.0 19.1 33.2 24.6 Meridian 722340 0.4 23.6 17.7 25.8 17.7 28.8 19.2 30.4 20.6 33.2 22.6 36.3 24.2 37.4 25.6 36.4 25.0 35.7 23.4 31.6 21.4 27.3 19.5 25.0 19.8 35.3 24.7 1 22.3 16.8 24.3 16.9 27.7 18.4 29.5 20.3 32.3 22.4 35.3 24.2 36.3 25.1 35.6 24.9 34.4 23.3 30.4 20.8 26.3 18.9 23.9 19.2 34.2 24.5 2 21.0 16.8 23.0 16.7 26.5 17.7 28.8 19.8 31.6 22.4 34.4 24.0 35.4 25.2 34.8 24.8 33.6 23.6 29.4 20.5 25.3 18.6 22.7 18.6 33.1 24.3 MISSOURI Columbia 724450 0.4 17.7 12.7 20.2 12.2 26.2 15.9 29.3 18.4 30.8 20.8 34.8 23.3 38.7 23.5 37.2 23.8 34.1 23.1 29.9 18.9 23.8 16.2 18.9 14.8 34.8 23.8 1 15.0 10.3 18.3 11.7 24.5 15.3 28.2 18.2 30.0 20.8 33.3 23.8 36.3 24.3 35.8 23.8 32.8 23.1 28.2 18.8 22.3 15.5 17.5 13.8 33.1 23.7 2 13.2 9.2 16.2 10.4 22.8 14.7 27.0 17.8 29.1 20.7 32.3 23.4 35.1 24.3 34.8 23.6 31.7 22.9 26.7 18.3 20.8 14.9 15.6 11.8 31.7 23.2 Kansas City 724460 0.4 16.5 12.4 19.9 11.4 25.9 15.8 30.2 18.8 31.3 20.9 35.5 23.8 38.3 24.2 37.8 24.4 34.6 23.3 30.5 19.0 23.8 15.9 18.1 13.7 35.5 24.1 1 14.8 9.8 17.8 10.7 24.2 14.8 28.4 18.6 30.5 20.3 34.1 23.7 37.1 24.1 36.5 23.8 33.3 23.2 29.0 19.1 22.0 15.0 16.8 12.7 33.8 23.9 2 12.8 8.3 15.9 9.9 22.5 14.3 26.9 18.1 29.6 20.6 33.1 23.6 35.8 24.3 35.3 23.8 32.2 23.1 27.4 18.4 20.5 14.7 15.2 11.0 32.3 23.4 WMO# = World Meteorological Organization number DB = dry-bulb temperature, °C MWB = mean coincident wet-bulb temperature, °C 27.62 2001 ASHRAE Fundamentals Handbook (SI) Table 4A Design Wet-Bulb—Mean Coincident Dry-Bulb Temperature Location and WMO# % Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Annual WB MDB WB MDB WB MDB WB MDB WB MDB WB MDB WB MDB WB MDB WB MDB WB MDB WB MDB WB MDB WB MDB 1a 1b 2a 2b 3a 3b 4a 4b 5a 5b 6a 6b 7a 7b 8a 8b 9a 9b 10a 10b 11a 11b 12a 12c 13a 13b 14a 14b Springfield 724400 0.4 13.6 16.7 14.9 18.9 18.1 23.2 20.8 26.6 23.4 28.4 25.7 31.4 26.4 32.6 25.8 32.2 24.7 31.1 21.6 26.1 18.1 21.2 15.9 18.3 25.3 31.7 1 12.4 14.4 13.8 17.3 17.3 21.8 20.0 25.6 22.7 27.9 25.1 30.6 25.9 32.2 25.4 32.1 24.2 30.1 20.8 25.3 17.2 20.7 15.0 17.7 24.6 31.1 2 11.3 13.3 12.7 16.4 16.5 20.9 19.4 24.6 22.1 27.1 24.6 29.9 25.5 31.8 25.0 32.1 23.7 29.5 19.9 23.4 16.4 19.7 13.7 16.2 24.0 30.2 St. Louis, Intl Airport 724340 0.4 14.5 17.8 15.0 18.9 18.6 24.3 21.3 26.9 23.7 29.5 26.0 32.2 27.3 33.9 26.5 33.4 25.8 31.4 21.9 25.7 18.6 21.5 16.7 19.0 26.1 32.3 1 12.7 14.4 13.7 16.6 17.8 23.0 20.2 26.1 23.1 28.4 25.4 31.5 26.8 33.5 26.1 32.9 25.1 30.8 21.3 25.0 17.9 21.0 15.4 16.9 25.3 31.3 2 11.1 13.4 12.3 15.7 16.9 21.2 19.5 24.9 22.4 27.9 25.0 30.8 26.3 32.6 25.7 32.4 24.4 29.8 20.6 24.2 17.1 19.6 13.9 15.6 24.6 30.4 MONTANA Billings 726770 0.4 5.8 12.3 7.6 16.3 9.1 19.3 12.7 25.6 15.9 26.8 19.2 30.8 20.2 30.2 19.3 31.1 17.1 29.2 13.1 25.3 9.3 18.2 6.4 13.6 18.6 30.1 1 4.7 10.6 6.2 13.2 8.2 17.9 11.8 23.7 15.2 25.9 18.4 29.8 19.6 30.3 18.6 31.3 16.3 27.9 12.5 24.7 8.3 16.4 5.3 11.8 17.7 29.1 2 3.8 9.3 5.4 12.4 7.4 16.5 11.0 21.8 14.6 25.3 17.8 28.9 18.9 30.3 18.0 30.2 15.6 27.1 11.8 22.6 7.3 14.7 4.4 10.1 16.9 28.2 Cut Bank 727796 0.4 6.3 11.3 6.6 12.3 7.4 15.4 10.9 22.1 14.2 23.8 16.4 26.7 19.1 26.6 17.6 29.9 15.4 26.8 12.1 24.2 8.4 15.6 6.3 10.9 16.8 27.3 1 5.1 9.9 5.3 10.7 6.6 13.4 9.9 20.6 13.4 22.7 15.9 26.6 18.2 25.8 17.2 28.7 14.6 25.8 11.3 22.8 7.3 14.3 5.1 9.8 15.8 26.3 2 4.1 8.4 4.3 9.3 5.6 11.5 9.0 18.6 12.6 21.8 15.3 25.9 17.3 26.0 16.6 28.5 13.8 25.9 10.7 21.2 6.3 12.1 4.2 8.6 14.9 25.2 Glasgow 727680 0.4 3.9 6.9 6.3 11.5 9.4 17.4 13.4 24.6 17.4 28.5 20.5 31.6 22.4 28.9 20.3 30.2 17.4 29.0 13.7 24.7 8.9 15.0 5.3 9.4 19.8 29.4 1 3.0 5.9 5.0 9.6 8.2 15.4 12.3 22.6 16.4 26.9 19.6 29.8 21.4 28.8 19.6 30.1 16.7 26.5 12.8 23.2 7.8 14.1 3.9 7.8 18.6 28.6 2 2.2 4.7 3.9 7.4 7.0 13.6 11.4 21.3 15.6 25.2 18.9 28.9 20.5 28.7 18.9 29.9 16.0 26.3 11.9 21.4 6.6 11.9 2.8 5.8 17.7 27.5 Great Falls, Intl Airport 727750 0.4 6.4 12.4 7.4 14.8 8.9 18.3 12.4 24.2 15.7 25.9 18.4 29.5 19.6 29.3 18.4 30.3 16.1 29.2 12.8 24.9 8.8 16.1 6.9 12.9 17.7 28.8 1 5.4 10.7 6.3 13.1 7.9 16.6 11.5 22.5 14.8 24.9 17.6 28.7 18.7 29.1 17.7 29.1 15.3 27.7 12.2 23.7 8.0 15.3 5.9 11.6 16.8 27.9 2 4.6 9.8 5.4 11.6 7.1 14.7 10.6 20.7 13.9 24.3 16.9 27.8 18.0 28.8 17.2 28.7 14.7 26.9 11.4 22.8 7.3 14.2 5.0 10.4 15.9 26.9 Helena 727720 0.4 5.2 10.0 6.8 13.1 8.5 18.3 12.1 22.5 15.0 25.4 18.2 27.3 18.6 28.4 17.9 28.1 16.3 27.1 12.3 22.3 8.3 15.3 5.8 11.3 17.2 27.9 1 4.4 8.9 5.7 11.1 7.6 16.1 11.1 20.9 14.3 25.1 17.2 27.7 17.9 28.4 17.3 28.0 15.6 26.2 11.7 21.4 7.3 13.9 4.7 9.6 16.3 27.4 2 3.6 7.7 4.8 10.1 6.7 14.2 10.2 19.9 13.6 23.9 16.5 27.1 17.4 28.1 16.8 27.9 14.8 25.6 10.9 20.4 6.4 12.2 3.9 8.3 15.6 26.6 Kalispell 727790 0.4 4.9 6.7 5.6 8.8 8.6 14.8 12.9 22.1 16.3 26.3 18.6 28.3 19.8 29.8 19.2 29.2 16.6 26.5 12.9 20.9 8.9 11.7 5.6 7.2 18.1 28.3 1 3.8 5.5 4.7 7.7 7.4 13.1 11.8 20.3 15.2 24.2 17.9 27.6 19.0 29.6 18.3 28.6 15.8 25.6 12.1 19.7 7.6 10.3 4.4 6.7 17.2 27.3 2 3.0 4.8 3.9 6.4 6.4 11.4 10.8 18.2 14.4 22.9 17.3 26.7 18.4 28.8 17.8 28.4 15.2 24.7 11.3 17.9 6.6 8.9 3.6 5.2 16.3 25.9 Lewistown 726776 0.4 5.9 10.7 7.0 13.3 8.3 17.3 11.6 23.2 15.3 25.3 18.6 27.2 20.1 27.9 18.9 29.9 16.8 27.0 13.5 22.6 8.9 17.9 6.3 12.6 18.0 27.4 1 4.8 9.7 5.8 11.2 7.3 14.8 10.8 20.7 14.4 24.6 17.7 26.7 19.2 27.6 18.2 28.9 15.6 25.7 12.4 22.9 7.8 14.9 5.2 10.8 16.9 26.6 2 3.8 8.2 4.7 9.6 6.4 13.5 10.0 19.4 13.6 22.7 17.1 25.7 18.4 27.5 17.4 28.2 14.7 24.9 11.5 21.4 6.8 13.5 4.3 8.9 16.1 25.7 Miles City 742300 0.4 5.3 9.5 7.6 14.1 10.7 19.1 14.3 25.4 18.3 28.6 21.5 32.1 22.6 32.4 21.2 32.2 18.4 29.1 14.1 25.4 9.8 17.3 5.8 10.7 20.7 31.6 1 4.3 8.4 6.3 11.9 9.6 17.5 13.2 23.9 17.3 27.8 20.7 31.6 21.8 32.5 20.5 31.2 17.7 29.2 13.4 24.1 8.6 15.1 4.7 9.3 19.6 30.2 2 3.3 6.5 5.4 10.6 8.6 15.9 12.4 22.2 16.5 26.8 19.9 30.3 21.2 31.9 19.8 31.1 17.0 27.9 12.7 22.8 7.5 13.4 3.6 7.6 18.7 29.1 Missoula 727730 0.4 4.9 6.9 6.0 10.4 9.1 16.1 13.2 22.1 16.7 26.8 18.9 29.4 19.5 30.6 19.2 29.6 16.6 29.2 12.9 21.9 8.7 12.4 5.4 8.1 18.1 28.6 1 3.8 5.7 5.2 8.6 8.3 14.7 12.2 21.0 15.7 24.8 18.2 28.4 18.8 30.3 18.3 28.3 16.0 27.2 12.3 21.0 7.6 10.9 4.3 6.4 17.2 27.9 2 3.1 4.8 4.4 7.3 7.3 12.9 11.3 19.9 14.8 23.9 17.6 27.7 18.2 28.8 17.7 28.7 15.3 25.4 11.4 19.2 6.7 9.8 3.4 5.2 16.4 27.0 NEBRASKA Grand Island 725520 0.4 7.2 14.3 9.9 17.3 14.7 22.9 18.6 26.2 21.8 28.7 24.6 31.5 25.6 32.8 25.2 31.8 23.2 29.1 18.9 25.4 13.9 17.2 8.6 15.1 24.2 31.7 1 6.0 11.9 8.5 15.5 13.5 20.2 17.6 25.6 20.8 27.3 23.8 32.2 24.9 32.6 24.6 31.6 22.6 29.7 17.9 23.1 12.5 16.6 7.2 12.4 23.4 31.0 2 4.7 9.6 7.4 13.6 12.2 18.9 16.6 24.0 20.1 26.7 23.2 31.2 24.4 32.1 24.0 31.2 21.9 29.3 16.9 22.7 11.4 15.4 5.9 11.5 22.6 29.9 Norfolk 725560 0.4 7.1 12.4 9.3 15.8 14.6 21.0 19.1 29.3 21.7 29.1 24.9 32.1 25.7 33.3 25.6 33.1 23.5 30.7 17.9 24.4 14.2 17.1 7.9 13.9 24.7 32.1 1 5.9 12.1 8.2 14.1 13.4 19.7 17.9 26.9 20.9 27.5 24.4 32.1 25.2 32.8 25.1 32.6 22.9 30.4 16.7 22.7 12.8 15.6 6.4 11.0 23.9 31.2 2 4.5 9.3 6.9 11.9 12.3 17.7 16.8 23.9 20.2 26.4 23.8 31.6 24.7 32.2 24.6 32.0 22.3 29.4 15.9 22.3 11.4 14.9 5.2 9.6 23.0 30.0 North Platte 725620 0.4 6.4 14.3 8.9 17.9 12.2 23.2 16.4 25.9 19.9 26.7 23.4 32.1 24.2 31.1 23.6 31.5 21.5 29.2 16.6 23.7 11.7 19.1 7.7 15.6 22.8 30.8 1 5.3 12.1 7.9 16.3 11.1 20.1 15.6 23.9 19.2 26.3 22.5 31.1 23.6 31.1 23.0 31.1 20.7 28.8 15.6 23.6 10.6 17.6 6.3 13.5 21.9 30.1 2 4.2 10.1 6.8 14.6 10.1 19.3 14.6 23.5 18.4 25.4 21.9 30.2 23.1 30.8 22.4 30.5 20.1 28.1 14.8 23.4 9.4 16.1 5.1 11.9 21.2 29.3 Omaha, Eppley Airfield 725500 0.4 7.5 11.2 10.6 15.3 16.3 22.8 19.8 27.8 23.1 28.6 25.9 32.7 27.3 33.3 26.6 32.5 24.7 30.7 20.9 25.1 15.9 19.1 12.2 13.4 25.7 32.3 1 6.2 10.2 8.8 14.9 15.1 21.0 18.9 26.1 22.3 28.7 25.3 31.9 26.6 32.8 26.0 32.6 24.1 29.9 20.0 25.1 14.9 17.8 9.4 12.7 24.8 31.3 2 4.9 8.3 7.7 12.1 13.7 19.2 17.9 24.2 21.4 27.5 24.6 31.4 25.9 32.7 25.4 32.2 23.5 29.1 19.0 24.3 13.6 16.8 6.9 10.7 23.9 30.2 Scottsbluff 725660 0.4 6.3 13.7 8.4 17.2 10.2 21.8 13.7 24.9 17.3 25.9 20.4 30.3 21.9 31.0 21.5 31.1 18.9 30.2 14.2 24.7 10.2 20.3 7.9 16.4 20.6 30.3 1 5.4 11.9 7.3 15.7 9.2 20.2 13.0 24.3 16.7 25.6 19.8 30.2 21.3 30.8 20.8 30.6 18.3 28.3 13.3 24.8 9.2 18.5 6.5 14.3 19.7 29.5 2 4.5 10.7 6.3 14.3 8.4 18.5 12.3 22.2 16.0 25.2 19.3 29.3 20.8 30.7 20.3 30.1 17.6 27.6 12.6 23.8 8.3 17.3 5.3 12.8 19.0 28.7 NEVADA Elko 725825 0.4 6.2 10.0 8.0 14.2 9.1 18.2 11.3 22.9 14.5 27.0 17.0 31.6 18.5 29.9 18.4 29.3 16.4 28.6 12.8 24.1 9.7 15.7 6.0 10.8 17.2 29.3 1 5.2 8.8 7.1 12.1 8.3 16.7 10.6 21.4 13.7 25.5 16.3 30.5 17.9 29.8 17.8 28.6 15.6 27.6 11.9 24.4 8.6 14.7 5.2 9.2 16.3 29.1 2 4.2 7.7 6.2 11.3 7.6 15.1 9.9 20.3 12.9 24.3 15.6 29.6 17.4 29.9 17.2 27.8 14.9 25.8 11.2 23.1 7.8 13.2 4.4 7.7 15.5 28.6 Ely 724860 0.4 4.9 10.1 6.7 13.1 7.5 15.8 10.1 21.4 12.7 25.2 15.2 27.6 16.8 27.2 16.9 26.3 14.7 24.6 11.1 22.3 8.3 15.3 5.4 11.4 15.7 25.6 1 4.1 9.2 5.6 11.4 6.7 15.3 9.2 20.6 11.9 23.9 14.5 28.2 16.2 26.3 16.3 25.6 14.1 24.5 10.4 22.4 7.2 14.6 4.3 10.2 15.0 25.6 2 3.2 7.7 4.7 10.3 6.1 14.4 8.4 19.4 11.4 22.5 13.9 27.7 15.8 26.1 15.8 25.1 13.5 24.1 9.9 21.9 6.3 13.9 3.3 9.0 14.2 25.4 Las Vegas, Intl Airport 723860 0.4 11.3 16.8 12.8 21.0 13.8 26.1 15.7 31.2 17.9 32.8 20.3 37.5 22.8 35.9 23.0 34.6 21.2 31.7 17.2 27.1 13.8 22.2 11.2 16.9 21.9 34.8 1 10.3 15.9 11.9 19.1 13.0 24.6 14.9 30.4 17.3 31.8 19.6 37.8 22.3 35.7 22.5 34.7 20.6 31.4 16.5 27.5 13.1 21.1 10.2 15.3 21.2 33.9 2 9.4 15.3 11.2 18.8 12.4 23.7 14.3 28.7 16.8 32.2 19.0 37.6 21.9 35.7 22.1 34.6 20.1 31.8 15.8 27.7 12.4 21.1 9.4 14.7 20.6 33.7 Reno 724880 0.4 8.3 13.6 9.7 17.0 9.9 20.4 12.3 24.4 15.1 27.6 17.4 31.3 18.4 30.9 18.3 30.6 16.0 29.3 13.9 26.9 10.5 17.6 8.6 13.4 17.2 30.7 1 7.1 12.0 8.6 15.3 9.1 18.9 11.4 23.4 14.4 26.8 16.7 30.9 17.8 30.7 17.7 30.1 15.5 28.8 13.2 26.6 9.8 17.4 7.6 13.0 16.4 30.2 2 6.2 11.2 7.8 14.5 8.4 17.6 10.7 22.4 13.8 26.6 16.1 30.2 17.3 31.1 17.2 29.7 15.1 29.1 12.5 25.4 9.1 16.6 6.7 11.8 15.7 29.4 Tonopah 724855 0.4 7.0 12.0 7.6 14.1 8.9 19.3 10.6 24.7 13.4 27.2 16.2 31.4 17.9 28.9 17.9 27.5 15.7 25.7 12.4 25.3 9.1 17.3 6.2 12.3 16.8 28.3 1 5.8 11.5 6.8 13.7 8.1 18.1 9.8 23.3 12.8 26.3 15.4 30.9 17.3 29.0 17.4 27.8 15.2 25.7 11.7 24.6 8.3 16.5 5.2 10.4 16.0 27.9 2 4.9 9.7 6.1 13.1 7.3 16.4 9.2 22.2 12.2 25.1 14.7 30.3 16.8 28.7 16.9 27.8 14.6 25.9 11.2 23.8 7.5 15.2 4.6 9.3 15.3 27.4 Winnemucca 725830 0.4 8.3 12.2 9.4 16.0 10.3 20.4 12.1 23.6 15.1 29.7 17.5 33.2 19.0 31.5 18.5 32.4 16.0 27.4 13.3 27.1 10.1 17.8 8.2 12.9 17.4 31.1 1 6.9 11.8 8.4 14.7 9.4 18.8 11.3 23.1 14.4 27.2 16.8 32.6 18.3 31.6 17.8 30.4 15.5 27.1 12.7 26.4 9.4 16.8 7.2 12.0 16.6 30.7 2 6.1 10.4 7.5 13.4 8.6 16.8 10.7 22.1 13.8 27.0 16.1 31.8 17.7 30.8 17.3 29.9 15.0 28.4 12.1 24.9 8.7 15.6 6.3 10.8 15.7 30.1 NEW HAMPSHIRE Concord 726050 0.4 9.1 10.6 8.9 11.1 13.2 17.5 17.5 25.3 21.2 29.4 23.6 29.9 24.8 31.0 24.7 30.8 23.4 29.2 19.1 22.5 16.5 18.8 12.3 13.4 23.6 29.4 1 5.9 8.0 7.2 9.8 11.0 15.6 15.9 22.2 20.2 27.6 22.8 28.7 24.3 30.7 23.9 29.8 22.6 27.9 18.0 21.9 14.9 17.2 9.8 11.9 22.6 27.8 2 3.9 5.8 5.6 7.5 9.6 13.3 14.7 19.8 19.1 26.1 22.2 28.2 23.8 30.1 23.2 28.1 21.8 26.2 17.2 20.7 13.5 15.9 7.2 8.6 21.7 26.3 NEW JERSEY Atlantic City 724070 0.4 13.3 15.1 13.9 16.4 16.7 22.3 19.1 26.6 23.4 29.6 25.0 30.8 26.3 32.1 26.1 31.4 25.1 29.8 21.6 24.6 18.8 21.4 15.6 17.0 25.1 30.3 1 12.2 13.4 12.5 14.5 15.2 19.1 18.1 24.6 22.2 26.3 24.4 30.6 25.7 31.4 25.5 30.4 24.5 29.1 20.8 23.4 17.8 19.6 14.6 15.9 24.4 28.9 2 11.1 12.3 11.3 13.2 13.7 17.0 16.8 21.9 21.3 26.4 23.8 29.9 25.3 30.7 25.0 29.9 23.9 28.3 20.1 22.7 16.9 18.2 13.5 14.7 23.7 27.9 Newark 725020 0.4 13.2 16.2 13.2 15.4 17.0 21.9 18.7 26.9 22.7 30.4 24.7 32.3 26.1 32.9 25.7 32.4 25.1 31.4 21.2 23.8 18.6 21.7 15.4 17.2 24.8 30.9 1 11.0 12.7 11.5 14.4 15.1 19.0 17.7 24.6 21.9 28.7 24.0 30.8 25.5 32.7 25.2 31.1 24.5 29.7 20.4 23.3 17.3 19.4 14.0 15.6 24.2 29.6 2 9.1 11.2 10.1 12.6 13.4 17.1 16.8 22.7 21.1 27.2 23.4 29.9 24.9 31.4 24.7 30.2 23.9 28.8 19.7 22.7 16.4 18.5 12.2 13.6 23.4 28.3 WMO# = World Meteorological Organization number WB = wet-bulb temperature, °C MDB = mean coincident dry-bulb temperature, °C Climatic Design Information 27.63 Table 4B Design Dry-Bulb—Mean Coincident Wet-Bulb Temperature Location and WMO# % Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Annual DB MWB DB MWB DB MWB DB MWB DB MWB DB MWB DB MWB DB MWB DB MWB DB MWB DB MWB DB MWB DB MWB 1a 1b 2a 2b 3a 3b 4a 4b 5a 5b 6a 6b 7a 7b 8a 8b 9a 9b 10a 10b 11a 11b 12a 12c 13a 13b 14a 14b Springfield 724400 0.4 18.4 11.9 21.5 13.1 25.8 16.3 29.0 19.2 30.2 21.4 34.4 23.5 37.5 23.1 37.0 23.3 33.9 22.8 30.2 18.4 23.8 16.8 19.7 13.8 34.8 23.5 1 16.3 11.0 19.6 11.7 24.4 15.7 27.4 18.6 29.4 21.3 33.2 23.3 36.0 23.6 36.0 23.4 32.4 22.4 28.5 18.7 22.5 15.4 18.4 13.7 33.3 23.5 2 14.7 9.6 17.9 11.3 22.9 14.9 26.3 18.0 28.7 20.8 32.2 23.4 34.9 23.9 35.0 23.3 31.4 22.7 27.1 18.6 21.2 15.3 16.9 13.1 31.8 23.1 St. Louis, Intl Airport 724340 0.4 18.8 12.8 21.7 12.9 27.2 16.1 30.4 18.6 31.9 21.6 34.9 24.0 37.9 25.3 37.6 24.9 34.5 24.1 29.9 19.6 24.4 16.1 19.9 15.6 35.1 24.6 1 16.1 10.7 19.3 12.2 25.3 15.9 29.0 18.4 31.1 21.1 34.0 23.9 36.5 25.1 36.0 24.7 33.4 23.8 28.4 19.6 22.9 15.7 18.1 13.4 33.6 24.1 2 14.1 10.1 17.2 10.8 23.5 15.3 27.7 17.9 30.1 20.8 33.3 23.6 35.4 24.8 34.9 24.4 32.2 23.2 26.9 18.7 21.4 15.4 16.4 13.1 32.2 23.6 MONTANA Billings 726770 0.4 13.2 5.3 16.4 7.1 20.6 8.3 26.8 11.9 30.4 14.9 35.2 17.1 36.4 17.6 36.3 17.4 32.6 15.6 27.4 12.6 18.9 8.8 14.4 6.2 34.1 16.9 1 11.4 4.4 14.4 5.8 18.7 7.9 24.7 11.4 28.7 14.3 33.6 16.9 35.3 17.3 34.9 17.0 31.2 15.1 25.3 11.9 17.1 7.9 12.5 5.0 32.3 16.6 2 9.9 3.3 12.9 5.1 16.9 7.2 22.9 10.6 27.3 13.6 32.2 16.7 34.1 17.1 33.9 16.9 29.7 14.8 23.6 11.2 15.4 6.9 11.0 3.9 30.4 16.2 Cut Bank 727796 0.4 12.5 5.1 13.4 5.9 16.5 6.9 23.6 10.3 26.8 12.9 30.4 15.5 32.1 16.0 34.0 16.2 30.6 14.2 25.5 11.5 16.6 7.8 12.1 5.7 30.7 15.3 1 10.7 4.4 11.4 4.7 14.4 6.2 21.3 9.4 25.1 12.5 29.1 15.0 30.9 15.5 32.6 15.8 28.9 14.1 23.6 10.9 14.8 7.1 10.6 4.6 28.8 14.8 2 8.8 3.7 10.0 4.2 12.4 5.1 19.3 8.6 23.6 12.1 27.9 14.6 29.9 15.8 31.4 15.5 27.2 13.2 21.9 10.4 12.8 5.8 8.9 3.7 26.8 14.2 Glasgow 727680 0.4 7.7 3.6 12.2 6.0 18.3 8.6 26.7 12.8 31.4 16.5 35.7 18.7 36.3 18.6 36.9 18.2 32.4 15.9 26.6 12.5 16.4 8.4 10.0 4.8 34.2 17.6 1 6.3 2.7 9.7 4.6 16.4 7.7 24.7 11.7 29.4 15.4 33.8 18.1 35.1 18.2 35.6 17.6 30.7 15.6 24.4 12.3 14.4 7.3 8.2 3.7 32.2 17.2 2 5.1 2.1 7.8 3.6 14.2 6.7 22.7 11.0 27.8 14.8 32.1 17.5 33.9 17.8 34.2 17.3 29.1 15.1 22.4 11.7 12.4 6.2 6.3 2.6 30.2 16.7 Great Falls, Intl Airport 727750 0.4 13.3 5.2 15.8 6.7 19.2 8.4 25.9 11.8 29.1 14.1 33.4 16.1 35.2 16.4 35.9 16.2 32.1 14.7 26.8 12.2 18.1 7.9 13.9 6.1 33.2 16.0 1 11.7 5.1 14.1 5.9 17.3 7.3 23.9 10.9 27.9 13.8 32.1 16.1 34.2 16.2 34.6 16.0 30.7 14.2 25.0 11.6 16.3 7.4 12.7 5.2 31.3 15.5 2 10.4 3.9 12.7 5.1 15.6 6.5 22.1 10.2 26.4 13.3 30.5 16.1 33.1 16.3 33.5 15.8 29.3 14.0 23.4 11.2 14.8 6.8 11.2 4.7 29.4 15.1 Helena 727720 0.4 11.4 4.8 13.7 6.0 19.3 8.2 24.8 11.1 28.9 13.9 33.2 15.5 35.1 15.8 34.8 15.8 31.2 15.0 24.9 11.3 16.1 7.4 12.3 5.4 32.4 15.6 1 9.4 4.2 12.1 5.4 16.9 7.1 22.9 10.6 27.5 13.2 31.8 15.7 33.8 16.1 33.4 15.7 29.7 14.3 23.1 11.1 14.6 7.1 10.4 4.4 30.7 15.2 2 8.1 3.3 10.7 4.4 15.1 6.4 21.2 9.7 26.0 13.1 30.2 15.3 32.7 15.7 32.2 15.4 28.0 13.8 21.4 10.4 12.9 6.1 8.8 3.6 28.8 14.8 Kalispell 727790 0.4 7.0 4.3 9.6 4.9 16.1 8.1 23.9 12.4 28.7 14.8 31.2 17.4 33.6 17.6 34.2 17.3 29.7 15.2 22.5 12.2 12.4 7.8 8.3 5.1 31.7 16.7 1 6.0 3.7 8.2 4.2 13.9 6.8 21.8 11.2 26.7 14.2 30.0 16.7 32.7 17.1 33.1 16.8 28.1 14.9 20.9 11.4 11.0 7.1 6.8 4.0 29.8 16.2 2 5.1 2.7 6.9 3.6 12.2 5.8 19.7 10.2 25.0 13.8 28.7 16.2 31.7 16.9 31.9 16.4 26.6 14.5 19.1 10.7 9.7 6.2 5.7 3.4 27.9 15.7 Lewistown 726776 0.4 12.1 5.5 14.0 6.4 18.1 8.0 24.1 10.9 27.4 13.9 30.6 16.2 33.9 16.8 34.8 16.9 31.3 14.4 26.4 12.7 18.3 8.1 13.4 5.6 31.9 15.9 1 10.3 4.1 11.7 5.3 16.1 6.8 22.3 10.1 26.1 13.2 29.1 16.2 32.7 16.7 33.7 16.3 29.8 14.0 24.7 11.8 16.3 7.3 11.6 4.7 29.7 15.6 2 8.9 3.4 10.3 4.4 14.1 6.0 20.6 9.6 24.6 13.2 27.7 15.9 31.4 16.3 32.4 15.9 28.3 13.8 23.0 10.8 14.5 6.4 10.1 3.7 27.8 15.2 Miles City 742300 0.4 10.6 4.6 15.1 6.7 20.7 9.8 28.1 13.2 32.1 16.4 37.5 18.9 38.4 19.2 37.9 18.3 34.4 16.7 27.5 13.3 18.2 9.3 12.0 5.0 35.9 18.6 1 8.6 4.0 12.8 6.1 18.5 9.1 25.8 12.6 30.6 16.0 35.2 18.8 37.2 19.6 36.8 18.4 32.7 16.4 25.4 13.1 16.1 8.0 9.7 4.3 34.1 18.2 2 6.9 3.2 10.9 5.2 16.6 8.2 24.1 11.8 29.1 15.7 33.8 18.3 36.0 19.1 35.8 18.2 30.9 15.9 23.7 12.3 14.3 7.1 8.1 3.4 32.1 17.7 Missoula 727730 0.4 7.3 4.4 11.3 5.4 17.6 8.7 25.0 12.6 29.2 15.5 33.1 17.2 35.2 17.2 35.4 17.0 30.9 15.6 23.9 12.3 13.6 7.6 8.8 5.2 33.0 16.7 1 6.1 3.6 9.3 4.6 15.7 7.4 23.0 11.6 27.5 14.7 31.9 17.3 34.1 16.9 34.1 16.5 29.4 15.1 22.2 11.6 11.8 6.8 7.1 4.0 31.2 16.2 2 5.2 2.9 7.9 4.0 14.1 7.0 21.1 10.8 25.9 14.1 30.4 16.7 33.2 16.9 33.1 16.4 27.8 14.7 20.3 10.9 10.3 6.5 5.5 3.0 29.2 15.7 NEBRASKA Grand Island 725520 0.4 14.9 6.8 18.9 8.9 26.1 13.6 31.0 16.9 32.8 19.1 37.3 22.3 38.7 22.5 37.3 22.4 34.6 20.6 29.9 16.0 21.2 11.7 16.2 7.6 35.9 22.3 1 12.4 5.4 16.6 8.2 23.2 12.1 28.8 16.0 31.2 18.3 35.9 21.9 37.5 22.6 35.9 22.2 33.1 20.9 28.1 16.2 19.3 10.9 13.7 6.7 33.9 21.9 2 10.1 4.3 14.1 7.1 20.8 11.1 26.7 15.4 29.5 18.5 34.6 21.9 36.3 22.8 34.8 22.4 31.7 20.6 26.6 15.2 17.7 10.2 11.5 5.7 32.1 21.3 Norfolk 725560 0.4 14.5 6.4 17.6 8.8 24.8 13.7 32.3 17.3 31.4 18.9 36.5 22.7 37.5 23.9 36.4 23.4 34.4 21.6 28.6 15.9 20.0 12.1 14.0 8.8 35.0 23.4 1 12.1 5.6 14.9 7.6 21.8 11.9 30.2 17.3 30.2 19.3 35.3 23.1 36.3 23.6 35.2 23.9 33.2 21.1 26.8 15.3 18.1 10.7 11.8 6.1 33.3 22.3 2 9.6 4.1 12.7 6.6 19.3 10.9 28.0 15.8 28.7 18.8 33.9 22.6 35.1 23.7 34.2 23.1 31.9 20.7 25.1 14.5 16.4 9.5 9.7 4.9 31.7 21.9 North Platte 725620 0.4 15.2 6.1 19.4 8.1 24.5 10.9 30.1 14.0 31.4 16.9 36.2 21.4 37.7 20.8 36.8 20.2 34.3 19.3 29.8 14.6 21.9 10.3 16.6 7.3 35.0 20.5 1 12.7 4.9 17.2 7.2 22.6 10.1 28.1 14.6 29.9 16.8 34.5 20.8 36.5 20.9 35.5 20.7 33.0 18.8 28.2 13.9 20.0 9.2 14.4 6.0 33.2 20.5 2 10.6 4.1 15.2 6.8 20.7 9.7 25.9 13.2 28.6 16.7 33.1 20.6 35.3 20.9 34.2 20.6 31.6 18.2 26.6 13.6 17.8 8.8 12.1 5.1 31.4 19.9 Omaha, Eppley Airfield 725500 0.4 12.7 6.5 17.3 9.5 26.1 14.7 30.4 17.5 32.1 20.9 36.2 23.7 37.9 24.1 36.8 24.6 34.0 22.6 29.3 19.1 21.1 13.7 15.1 10.6 35.0 24.0 1 10.7 5.4 14.8 8.6 23.4 13.3 28.6 17.2 30.8 20.4 34.8 23.6 36.5 24.6 35.4 24.3 32.6 22.5 27.7 17.9 19.5 13.2 13.2 8.7 33.2 23.6 2 8.8 4.7 12.7 7.2 21.2 12.9 26.8 16.6 29.7 20.1 33.4 23.3 35.3 24.6 34.2 24.2 31.2 22.1 26.2 17.5 18.1 12.4 11.2 7.2 31.6 23.0 Scottsbluff 725660 0.4 14.9 5.3 18.8 8.1 23.3 9.6 28.5 12.2 31.6 14.7 36.4 17.9 38.2 18.6 36.3 18.1 34.4 17.2 28.9 13.2 21.8 9.7 17.1 6.9 35.2 18.2 1 13.1 4.8 16.3 6.6 21.3 8.6 26.9 11.9 30.2 14.7 34.9 17.7 37.0 18.4 35.1 18.4 32.9 16.4 27.3 12.3 20.1 8.6 15.4 6.1 33.4 17.9 2 11.4 4.0 15.1 6.1 19.5 7.9 25.2 11.6 28.9 14.6 33.6 17.3 35.8 18.5 34.1 18.2 31.4 15.8 25.9 12.0 18.1 8.0 13.3 5.1 31.7 17.9 NEVADA Elko 725825 0.4 11.6 5.5 16.2 6.9 19.9 8.4 25.3 10.4 29.7 13.1 35.4 15.4 37.0 16.8 36.4 16.1 33.3 14.8 28.2 11.6 19.1 8.2 11.9 4.9 34.9 15.5 1 9.9 4.4 14.2 6.3 18.3 7.2 24.0 9.8 28.3 12.4 34.2 15.0 36.0 16.2 35.3 15.4 32.2 14.5 26.7 11.1 17.6 7.3 10.4 4.4 33.5 14.9 2 8.4 3.6 12.5 5.5 16.6 6.9 22.4 9.2 27.1 12.0 33.0 14.7 35.2 15.9 34.5 15.2 31.1 13.7 25.3 10.6 15.7 6.8 9.1 3.7 32.0 14.4 Ely 724860 0.4 11.9 3.8 15.3 5.5 18.2 6.5 23.3 9.1 27.1 12.0 32.2 13.3 33.6 14.2 32.8 13.6 30.2 12.8 25.5 10.1 18.8 6.9 13.3 4.4 31.8 13.6 1 10.1 3.2 13.5 4.4 16.7 6.1 22.1 8.6 25.9 11.2 31.4 13.2 32.8 14.0 32.1 13.7 29.2 12.7 24.5 9.8 17.1 6.3 11.8 3.7 30.6 13.3 2 8.6 2.4 11.8 4.2 15.4 5.5 20.7 8.0 24.8 10.3 30.5 12.8 32.1 13.8 31.3 13.6 28.2 12.3 23.3 9.3 15.7 5.4 9.9 2.9 29.3 12.9 Las Vegas, Intl Airport 723860 0.4 20.4 9.4 24.8 10.9 28.9 12.4 33.8 14.5 38.3 16.4 43.1 18.4 44.0 19.2 43.2 19.6 39.9 18.0 35.4 15.6 26.4 12.5 20.1 8.8 42.2 18.9 1 19.1 8.7 23.2 10.3 27.6 12.2 32.7 14.1 37.1 15.9 42.2 18.2 43.2 19.1 42.2 19.4 39.2 18.1 34.3 15.1 25.2 11.7 18.7 8.7 40.9 18.6 2 17.8 8.3 21.9 9.9 26.4 11.7 31.4 13.5 35.8 15.4 41.3 17.7 42.5 19.1 41.4 19.4 38.3 17.9 32.9 14.6 23.9 11.3 17.6 8.2 39.6 18.2 Reno 724880 0.4 15.3 6.6 19.2 8.6 22.0 9.2 26.8 11.3 31.2 14.2 34.9 15.9 36.9 16.6 36.3 16.2 33.7 14.9 29.4 13.1 20.9 9.6 15.9 6.7 34.8 15.8 1 13.8 6.3 17.6 7.6 20.4 8.4 25.5 10.8 29.7 13.6 33.9 15.6 36.0 16.3 35.4 16.0 32.4 14.4 28.1 12.8 19.6 8.7 14.4 6.7 33.4 15.4 2 12.4 5.3 16.2 6.7 19.0 7.8 24.0 10.4 28.4 12.9 32.9 14.9 35.1 16.1 34.6 15.7 31.4 14.2 26.8 12.1 18.1 8.0 12.9 6.1 31.9 14.9 Tonopah 724855 0.4 14.5 5.8 17.3 6.2 21.0 8.0 26.2 10.0 29.8 12.7 34.9 14.6 36.1 15.0 35.7 14.7 32.3 13.7 28.4 11.4 20.3 7.7 13.7 4.7 34.4 14.4 1 12.8 4.7 16.2 5.8 19.8 7.2 24.6 9.4 28.7 11.9 34.0 13.6 35.4 14.8 34.8 14.8 31.3 13.5 27.2 11.0 18.9 7.1 12.7 4.0 33.1 14.1 2 11.4 3.9 15.1 5.3 18.4 6.8 23.2 8.8 27.7 11.2 33.0 13.3 34.7 14.6 33.9 14.6 30.4 13.1 26.0 10.4 17.6 6.8 11.6 3.8 31.9 13.7 Winnemucca 725830 0.4 14.3 7.2 18.1 8.4 22.1 8.6 27.2 10.9 32.3 14.6 36.5 15.7 38.1 16.8 37.8 16.4 34.1 14.4 29.6 12.6 20.7 9.0 14.8 7.1 36.0 15.8 1 12.6 6.2 16.5 7.3 20.3 8.7 25.7 10.7 30.6 13.6 35.5 15.9 37.2 16.3 36.7 16.1 33.1 14.4 28.1 12.1 18.9 8.4 13.2 6.2 34.7 15.4 2 11.3 5.3 15.1 6.9 18.8 7.9 24.2 10.0 29.3 12.8 34.4 15.4 36.3 16.1 35.7 15.6 32.0 13.9 26.7 11.6 17.5 7.9 11.7 5.6 33.2 14.8 NEW HAMPSHIRE Concord 726050 0.4 11.2 9.2 12.3 8.3 19.7 11.6 27.4 16.0 32.1 18.9 33.3 22.2 34.7 23.2 33.2 23.3 30.9 22.7 26.1 16.7 20.3 15.4 14.2 11.8 32.1 21.7 1 8.6 5.6 10.0 6.9 16.7 10.4 24.4 14.4 30.2 18.9 32.1 21.4 33.4 22.8 31.9 22.3 29.5 21.4 24.1 16.7 18.2 13.7 11.8 9.1 30.3 21.1 2 6.5 3.6 7.9 5.1 14.1 8.6 22.2 12.7 28.4 18.1 30.7 20.8 32.3 22.1 30.9 21.8 28.0 20.6 22.3 15.6 16.4 12.4 9.3 6.5 28.8 20.2 NEW JERSEY Atlantic City 724070 0.4 16.2 11.2 18.6 12.5 24.0 15.1 28.3 17.8 32.3 21.4 34.4 23.1 35.2 24.2 33.9 24.1 32.6 23.8 26.9 19.3 22.7 17.3 18.1 14.0 32.9 23.4 1 14.0 11.1 15.9 11.1 21.0 13.7 26.3 16.6 30.6 20.0 33.0 22.7 34.2 24.3 32.7 23.9 31.1 23.4 25.6 19.2 20.9 16.2 16.6 13.6 31.3 23.0 2 12.7 10.8 14.0 10.5 18.4 12.8 23.8 15.5 28.9 19.9 31.8 22.9 33.2 23.9 31.7 23.6 29.7 22.5 24.3 18.6 19.4 15.7 15.3 13.2 29.8 22.4 Newark 725020 0.4 15.7 11.7 17.4 11.4 24.0 14.9 29.1 17.3 32.7 21.3 34.9 22.8 36.6 24.1 34.9 24.5 33.7 24.2 27.2 19.1 22.8 16.7 18.0 14.1 33.9 23.4 1 13.5 10.3 15.3 11.0 21.4 13.8 26.8 16.3 31.3 20.1 33.8 22.5 35.3 24.1 33.6 23.9 32.2 23.5 25.8 18.1 21.1 16.2 16.2 13.4 32.2 22.6 2 11.4 8.4 13.1 9.2 18.9 12.6 24.5 15.3 29.8 19.8 32.7 21.8 34.2 23.8 32.5 23.4 30.5 22.4 24.7 17.7 19.4 15.0 14.3 11.7 30.7 21.9 WMO# = World Meteorological Organization number DB = dry-bulb temperature, °C MWB = mean coincident wet-bulb temperature, °C 27.64 2001 ASHRAE Fundamentals Handbook (SI) Table 4A Design Wet-Bulb—Mean Coincident Dry-Bulb Temperature Location and WMO# % Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Annual WB MDB WB MDB WB MDB WB MDB WB MDB WB MDB WB MDB WB MDB WB MDB WB MDB WB MDB WB MDB WB MDB 1a 1b 2a 2b 3a 3b 4a 4b 5a 5b 6a 6b 7a 7b 8a 8b 9a 9b 10a 10b 11a 11b 12a 12c 13a 13b 14a 14b NEW MEXICO Albuquerque 723650 0.4 7.3 12.8 8.4 17.6 10.1 21.0 12.2 25.1 15.1 25.9 17.9 28.3 19.1 29.4 19.3 28.9 18.1 27.3 14.8 21.4 10.8 16.2 7.3 12.7 18.5 28.6 1 6.3 12.3 7.7 15.9 9.2 19.9 11.5 24.1 14.4 25.7 17.4 28.2 18.8 29.4 19.0 28.5 17.6 26.4 14.1 20.4 9.9 15.6 6.3 11.9 18.0 28.0 2 5.5 11.3 6.9 14.8 8.6 19.5 10.9 23.2 13.8 25.3 16.9 28.0 18.5 29.3 18.6 28.2 17.2 25.5 13.4 20.6 9.1 15.6 5.7 11.1 17.5 27.3 Tucumcari 723676 0.4 9.7 18.7 11.2 20.5 12.7 23.7 15.5 25.4 18.4 26.4 20.7 31.4 21.8 30.6 21.6 30.7 20.4 28.2 17.1 25.2 13.1 20.7 10.2 17.2 20.8 30.6 1 8.8 18.3 10.2 20.7 11.5 22.6 14.7 23.4 17.7 26.6 20.2 30.2 21.3 30.7 21.2 30.4 19.9 28.2 16.3 23.7 12.3 19.5 9.4 17.4 20.2 29.7 2 7.9 16.4 9.4 19.2 10.7 22.6 13.9 22.6 17.1 25.9 19.7 29.7 20.9 30.9 20.7 30.0 19.5 27.8 15.7 23.2 11.4 20.1 8.6 17.1 19.7 28.8 NEW YORK Albany 725180 0.4 9.2 10.8 8.9 10.8 13.1 17.5 17.5 25.5 21.4 28.7 23.6 30.0 24.9 30.9 24.7 30.9 23.4 29.5 19.1 22.3 16.5 18.7 12.3 13.4 23.6 29.3 1 5.9 8.2 7.2 9.8 11.0 15.4 16.0 22.2 20.5 27.7 22.9 28.8 24.3 30.6 23.9 29.7 22.6 27.9 18.2 22.3 14.9 17.3 9.7 11.7 22.6 27.8 2 3.9 5.7 5.7 7.4 9.6 13.1 14.7 20.0 19.4 26.8 22.2 28.2 23.8 29.8 23.3 28.2 21.8 26.2 17.3 20.8 13.6 15.9 7.3 8.7 21.8 26.3 Binghamton 725150 0.4 9.5 10.6 9.7 10.9 14.2 19.6 16.8 24.4 20.9 26.4 22.7 27.7 23.9 28.9 23.5 28.5 22.8 26.9 17.9 21.2 15.4 17.6 12.3 13.8 22.6 27.3 1 7.4 8.3 8.4 10.1 12.5 16.8 16.1 22.3 20.1 25.3 21.8 26.8 23.3 28.2 22.9 27.9 22.0 26.0 17.2 20.2 14.4 16.7 10.8 12.2 21.7 26.0 2 5.2 6.6 6.7 8.3 11.1 14.7 15.1 20.6 19.3 24.2 21.2 26.3 22.7 27.6 22.3 26.6 21.4 24.9 16.5 19.6 13.2 14.9 8.7 9.8 20.8 24.8 Buffalo 725280 0.4 10.2 12.2 10.3 12.4 14.3 19.4 17.6 24.3 21.3 26.6 23.3 28.6 24.2 29.3 24.2 28.6 23.4 27.4 18.8 22.9 15.7 18.6 13.6 15.2 23.2 27.6 1 7.5 9.8 8.7 10.6 12.7 17.3 16.6 23.3 20.6 26.1 22.4 27.9 23.8 28.8 23.5 27.6 22.7 26.8 18.0 21.7 14.8 17.1 12.1 13.9 22.3 26.6 2 5.9 7.7 7.1 9.3 11.6 15.4 15.5 20.9 19.7 24.5 21.8 26.9 23.4 27.8 22.9 27.1 22.0 25.5 17.3 20.6 13.7 16.2 10.2 12.1 21.6 25.5 Massena 726223 0.4 7.2 8.9 8.2 10.0 12.6 17.3 17.6 22.8 21.8 28.2 24.1 29.3 25.3 31.2 24.9 29.5 23.7 28.2 18.8 23.6 15.3 18.7 11.1 12.2 23.7 28.9 1 5.2 7.1 6.5 8.2 10.8 15.7 16.2 21.6 20.9 27.1 23.4 28.6 24.6 30.1 24.0 29.1 22.9 26.8 18.0 21.8 13.9 16.0 8.9 10.3 22.7 27.4 2 3.8 5.3 5.1 6.8 9.2 12.5 14.8 19.8 19.9 25.2 22.6 27.7 23.9 29.4 23.3 27.8 21.9 25.0 17.0 20.5 12.6 14.8 6.6 7.7 21.7 25.9 New York, John F Kennedy Airport 744860 0.4 11.8 13.4 12.4 14.0 15.1 20.3 18.1 25.3 22.4 28.6 24.3 31.9 25.6 31.8 25.7 31.2 24.6 30.0 20.9 23.7 17.8 20.6 14.2 15.6 24.6 29.9 1 10.3 12.2 10.9 13.3 13.4 17.1 17.0 22.7 21.6 27.4 23.6 30.4 25.1 30.8 25.1 30.2 24.1 29.1 20.2 22.7 16.6 18.5 13.0 14.3 23.8 28.8 2 8.8 10.7 9.5 11.8 11.9 15.3 15.8 21.3 20.6 26.2 22.9 29.2 24.6 30.2 24.6 29.3 23.5 27.7 19.4 22.1 15.7 17.7 11.7 13.0 23.1 27.6 Rochester 725290 0.4 9.7 11.7 10.3 12.4 15.0 20.5 18.4 24.7 22.1 27.8 24.5 29.9 25.5 31.3 25.2 30.6 24.0 28.9 19.3 23.9 16.3 18.9 13.9 16.3 24.1 29.2 1 7.2 9.4 8.7 10.7 13.4 17.5 17.4 23.7 21.4 27.2 23.7 29.1 24.9 30.3 24.3 28.9 23.3 27.9 18.5 22.3 15.1 17.6 11.7 13.8 23.1 27.7 2 5.8 8.2 6.9 8.8 12.0 16.6 16.4 22.2 20.6 26.3 22.9 28.2 24.3 29.7 23.6 27.8 22.6 26.6 17.7 21.7 14.1 16.8 9.8 11.9 22.1 26.7 Syracuse 725190 0.4 9.7 12.5 10.2 12.6 14.9 21.0 18.4 25.4 22.2 28.0 23.8 29.9 25.3 31.0 25.1 30.2 23.9 29.5 19.0 23.8 16.3 19.5 13.4 15.2 23.8 29.2 1 7.1 9.1 8.3 10.9 13.1 18.1 17.3 23.8 21.3 27.4 23.2 29.3 24.6 30.1 24.3 28.6 23.2 27.8 18.3 22.5 15.3 18.1 11.7 13.7 22.9 28.0 2 5.4 7.7 6.7 8.7 11.5 16.1 16.0 21.5 20.3 25.6 22.6 28.3 24.0 29.6 23.6 28.0 22.4 26.6 17.5 21.1 13.9 16.6 9.2 11.7 22.0 26.6 NORTH CAROLINA Asheville 723150 0.4 14.3 16.8 15.1 17.7 16.7 22.2 19.0 24.2 21.8 25.7 23.7 29.5 24.7 30.2 24.2 29.6 23.0 27.9 20.2 23.7 18.0 19.7 16.8 18.1 23.6 28.8 1 13.2 14.9 13.9 16.3 16.1 19.7 18.0 22.9 20.9 25.2 23.1 28.4 24.3 29.6 23.8 29.3 22.6 27.4 19.6 22.2 17.3 19.3 15.7 16.7 22.9 27.8 2 12.2 13.6 12.9 15.4 15.4 18.7 17.3 21.9 20.4 24.9 22.6 27.8 23.8 29.1 23.4 28.5 22.2 26.7 18.9 21.7 16.4 18.7 14.6 16.2 22.3 26.8 Cape Hatteras 723040 0.4 19.0 20.3 18.7 20.8 19.7 22.0 20.8 23.8 24.3 26.9 26.2 29.3 27.4 30.9 27.3 30.4 26.1 28.8 24.5 27.0 21.9 24.0 20.4 22.2 26.6 29.7 1 18.3 19.4 17.8 19.6 19.0 20.9 20.2 23.2 23.6 26.3 25.8 28.8 27.1 30.5 26.9 29.9 25.7 28.7 23.8 26.0 21.4 23.1 19.7 21.3 26.1 28.8 2 17.4 18.6 17.1 18.6 18.4 20.3 19.7 22.5 23.1 25.7 25.4 28.3 26.8 30.1 26.6 29.6 25.3 28.4 23.2 25.1 20.8 22.4 19.0 20.3 25.6 28.4 Charlotte 723140 0.4 16.4 18.2 17.2 20.4 18.6 23.6 20.6 26.9 23.6 29.3 25.1 31.1 25.6 32.1 25.4 31.8 24.6 30.3 22.3 26.5 20.1 22.7 19.0 20.6 24.9 31.2 1 15.5 17.7 16.3 18.9 17.9 22.3 19.8 25.2 22.8 28.2 24.5 30.8 25.2 31.7 25.1 31.4 24.1 29.9 21.6 24.8 19.5 22.1 17.8 19.1 24.3 30.6 2 14.4 16.3 15.3 17.4 17.1 20.9 19.1 24.6 22.2 27.9 24.1 30.4 24.9 31.5 24.8 31.0 23.7 29.4 20.9 24.3 18.7 20.8 16.6 18.2 23.8 29.8 Greensboro 723170 0.4 16.1 18.4 16.7 20.0 18.6 23.2 20.2 25.0 23.5 28.7 25.3 30.7 26.1 32.3 25.7 31.2 24.3 29.9 21.7 25.3 19.9 22.3 18.3 19.9 25.1 30.8 1 14.8 16.9 15.6 17.9 17.8 21.8 19.6 24.1 22.6 27.7 24.6 30.4 25.6 31.6 25.2 30.9 23.8 29.2 21.0 24.7 19.1 21.0 17.1 18.3 24.4 30.1 2 13.5 15.3 14.7 16.6 16.9 20.8 18.9 23.4 22.1 27.1 24.1 30.3 25.2 31.1 24.9 30.7 23.4 28.6 20.3 23.7 18.2 20.8 16.0 17.2 23.8 29.4 Raleigh/Durham 723060 0.4 17.3 19.0 17.8 21.1 19.1 24.6 20.7 25.6 23.8 29.2 25.7 31.1 26.6 32.5 26.2 31.7 24.9 30.2 22.6 25.9 20.3 22.6 19.2 21.3 25.5 31.3 1 16.3 18.2 16.8 19.9 18.4 23.3 19.9 25.8 23.1 28.4 25.1 30.8 26.1 32.2 25.8 31.6 24.5 29.7 21.8 25.2 19.7 21.8 18.1 20.0 24.9 30.4 2 15.3 17.6 15.8 18.2 17.6 21.7 19.3 24.8 22.5 27.8 24.6 30.5 25.6 31.5 25.4 31.2 24.1 29.1 21.3 24.4 18.9 21.1 16.9 19.0 24.3 29.5 Wilmington 723013 0.4 19.7 21.2 20.0 22.8 20.8 24.3 22.7 26.9 24.9 30.2 26.7 32.5 27.8 33.5 27.4 32.7 26.3 30.8 24.7 27.9 22.9 25.1 21.1 22.8 26.8 31.9 1 18.9 20.3 19.1 21.9 20.2 23.6 21.9 26.1 24.3 28.9 26.2 31.7 27.3 33.0 27.1 32.0 25.8 30.1 24.1 26.9 22.2 24.1 20.1 21.8 26.2 31.0 2 18.2 19.8 18.3 20.6 19.6 22.9 21.2 25.3 23.7 27.9 25.8 31.0 26.9 32.2 26.8 31.6 25.6 29.7 23.4 26.0 21.4 23.4 19.2 20.9 25.7 30.1 NORTH DAKOTA Bismarck 727640 0.4 3.7 7.3 6.4 11.3 10.3 18.8 15.1 24.1 20.2 27.8 22.8 30.2 24.6 31.2 22.9 31.6 20.5 29.1 15.3 23.5 10.1 15.5 4.5 8.6 22.3 29.9 1 2.9 5.8 4.9 9.2 8.7 15.7 13.9 23.2 19.0 26.3 21.8 29.9 23.7 30.3 22.2 31.1 19.5 28.2 14.3 22.4 8.5 14.1 3.3 6.2 21.1 29.1 2 2.1 4.8 3.7 6.9 7.3 13.4 12.8 21.0 17.9 25.2 21.1 29.0 22.8 30.2 21.6 30.7 18.6 26.3 13.5 21.0 7.3 12.2 2.3 5.0 20.1 27.8 Fargo 727530 0.4 2.2 4.0 4.8 7.6 10.8 15.7 16.9 25.3 21.2 29.4 24.7 29.8 25.8 31.5 25.1 32.1 23.1 29.4 17.6 22.3 11.3 13.9 4.0 6.5 23.9 30.2 1 1.3 2.7 3.2 5.6 8.9 13.9 15.4 23.6 20.2 27.0 23.6 29.6 25.0 30.7 24.2 31.0 22.0 27.8 16.4 21.0 9.6 12.8 2.0 4.0 22.8 28.7 2 0.6 1.7 2.1 3.9 7.4 11.3 14.1 21.0 19.3 25.7 22.7 28.5 24.2 30.1 23.4 29.9 20.8 25.7 15.4 20.8 8.3 11.2 0.9 2.2 21.6 27.4 Minot, Intl Airport 727676 0.4 3.4 6.1 5.3 9.7 9.3 16.6 14.4 25.6 19.2 27.3 21.8 28.9 24.1 30.6 23.0 31.5 20.2 30.4 15.0 22.5 9.5 14.9 3.8 7.2 21.7 29.2 1 2.7 5.3 4.1 7.5 7.9 14.3 13.4 22.7 18.1 27.5 21.1 28.7 23.1 29.8 22.2 30.2 18.6 26.7 14.1 21.7 7.9 12.7 2.8 5.8 20.5 28.2 2 1.9 4.3 2.9 5.6 6.6 12.1 12.3 21.2 17.1 25.6 20.2 28.0 22.2 28.3 21.3 29.1 17.6 26.3 13.2 20.2 6.8 10.9 1.9 4.6 19.3 26.4 OHIO Akron/Canton 725210 0.4 11.7 13.3 11.9 13.8 15.6 19.8 18.3 23.7 22.4 26.2 24.0 29.7 25.1 30.4 24.9 30.3 23.3 28.5 19.3 23.5 16.5 19.3 14.2 16.2 23.8 29.0 1 10.3 11.7 10.6 12.4 14.5 18.4 17.3 23.4 21.3 25.9 23.4 28.7 24.5 30.2 24.2 29.2 22.8 27.9 18.5 21.7 15.6 18.1 13.2 14.8 22.9 27.6 2 8.5 10.2 9.5 11.3 13.5 17.7 16.6 22.4 20.3 25.6 22.7 27.8 24.0 29.4 23.6 28.2 22.2 26.9 17.7 21.3 14.8 17.0 11.8 13.5 22.1 26.6 Cleveland 725240 0.4 12.3 14.4 12.3 14.3 16.6 21.2 19.1 25.1 22.8 27.3 24.4 30.3 25.5 31.7 25.3 31.1 24.0 29.3 19.9 24.5 17.2 20.1 14.3 16.9 24.2 29.7 1 10.5 12.2 10.9 13.3 15.2 19.8 18.1 24.2 21.8 27.3 23.8 29.5 25.0 30.9 24.6 29.9 23.4 28.6 18.9 23.0 16.1 18.3 13.2 15.3 23.3 28.3 2 8.3 10.2 9.3 11.4 13.9 18.4 17.2 23.1 21.0 26.6 23.1 28.4 24.4 30.2 23.9 28.8 22.8 27.6 18.1 22.0 15.2 17.4 11.8 13.8 22.4 27.1 Columbus, Intl Airport 724280 0.4 13.0 14.9 13.1 15.4 16.7 22.1 19.1 24.7 23.4 29.0 25.1 30.7 25.9 31.6 25.6 31.6 24.3 30.3 20.3 24.6 17.2 20.5 14.8 16.8 24.7 30.5 1 11.6 13.2 12.1 13.9 15.6 20.6 18.3 24.4 22.4 27.2 24.5 30.1 25.2 31.2 25.1 31.2 23.6 28.9 19.5 23.3 16.3 19.3 13.8 15.8 23.9 29.2 2 9.5 11.2 10.7 13.1 14.7 18.9 17.4 23.4 21.6 26.5 23.8 29.3 24.8 30.7 24.5 30.1 22.9 28.2 18.7 22.8 15.4 18.3 12.6 14.5 23.1 27.8 Dayton, Intl Airport 724290 0.4 13.1 14.7 13.2 15.3 16.9 21.6 19.4 24.1 23.1 28.4 24.9 30.6 25.8 31.9 25.8 31.5 24.0 30.0 20.2 23.9 17.6 20.5 15.0 16.7 24.7 30.4 1 11.8 13.1 11.9 13.5 15.7 19.9 18.5 23.3 22.3 26.9 24.2 29.8 25.3 31.6 25.1 30.8 23.5 28.9 19.4 23.3 16.7 19.3 14.0 15.3 23.8 29.2 2 9.7 11.1 10.7 12.9 14.8 18.1 17.7 22.8 21.5 26.6 23.6 29.2 24.8 30.9 24.6 29.9 22.9 27.9 18.7 22.7 15.8 18.2 12.6 14.1 23.1 27.9 Mansfield 725246 0.4 12.4 15.2 12.0 13.5 16.1 20.7 18.1 23.9 22.3 26.8 24.3 29.4 25.6 31.1 25.3 29.9 23.6 28.2 19.6 23.2 17.0 20.2 14.6 16.4 24.2 29.2 1 11.1 12.6 10.8 12.7 14.9 18.7 17.4 23.7 21.2 25.8 23.7 28.8 24.8 30.3 24.7 29.2 23.0 28.0 18.9 22.4 16.1 18.0 13.5 15.2 23.4 28.1 2 9.1 11.3 9.6 11.6 13.7 18.0 16.7 22.4 20.3 24.4 23.2 27.9 24.3 29.5 24.1 28.8 22.4 27.1 18.2 22.0 15.2 17.7 12.2 13.7 22.6 26.8 Toledo 725360 0.4 11.2 12.6 10.9 12.8 16.1 20.2 18.9 24.5 22.9 28.2 24.4 30.8 26.1 32.3 26.3 31.9 24.4 29.4 19.8 24.9 16.6 19.3 14.3 16.1 24.7 29.8 1 9.3 10.7 9.3 11.8 14.6 19.7 17.9 24.0 22.1 27.4 23.9 30.3 25.5 31.2 25.5 30.9 23.7 28.3 18.9 23.2 15.6 17.7 12.8 14.2 23.8 28.7 2 6.8 8.2 7.7 9.7 13.5 17.5 17.2 23.3 21.2 27.1 23.3 28.8 24.8 30.2 24.7 29.3 23.0 27.1 18.1 22.1 14.6 16.9 11.2 12.5 22.9 27.3 Youngstown 725250 0.4 11.4 12.8 11.8 14.5 15.6 19.7 18.0 24.2 22.2 26.9 23.7 29.5 25.1 30.4 24.5 30.2 23.4 28.3 19.2 23.6 16.6 20.0 14.1 16.3 23.6 28.9 1 9.8 11.3 10.3 12.5 14.4 18.6 17.2 24.1 21.1 26.3 23.2 28.4 24.4 29.9 23.8 29.2 22.8 27.6 18.4 21.7 15.6 18.4 13.1 14.7 22.7 27.5 2 7.8 9.7 9.0 11.3 13.3 17.5 16.4 22.9 20.2 25.5 22.4 27.6 23.8 29.4 23.2 28.3 22.2 26.5 17.6 20.8 14.6 16.7 11.7 13.1 21.8 26.2 WMO# = World Meteorological Organization number WB = wet-bulb temperature, °C MDB = mean coincident dry-bulb temperature, °C Climatic Design Information 27.65 Table 4B Design Dry-Bulb—Mean Coincident Wet-Bulb Temperature Location and WMO# % Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Annual DB MWB DB MWB DB MWB DB MWB DB MWB DB MWB DB MWB DB MWB DB MWB DB MWB DB MWB DB MWB DB MWB 1a 1b 2a 2b 3a 3b 4a 4b 5a 5b 6a 6b 7a 7b 8a 8b 9a 9b 10a 10b 11a 11b 12a 12c 13a 13b 14a 14b NEW MEXICO Albuquerque 723650 0.4 15.8 6.3 20.1 7.0 24.2 8.8 28.4 10.9 32.4 12.8 37.6 15.6 37.5 15.9 35.1 16.5 33.3 15.1 28.8 11.9 21.4 8.6 16.1 5.9 35.3 15.7 1 14.4 5.1 18.7 6.8 23.0 8.4 27.2 10.3 31.2 12.3 36.4 14.8 36.4 16.4 34.4 16.5 32.2 15.5 27.5 11.6 20.2 8.4 14.6 5.4 33.9 15.6 2 13.1 4.7 17.3 6.2 21.8 7.9 26.2 9.8 30.2 12.1 35.4 14.6 35.6 16.3 33.6 16.5 31.1 15.3 26.4 11.7 18.9 7.8 13.2 4.6 32.5 15.5 Tucumcari 723676 0.4 21.1 8.3 24.7 10.1 27.5 11.3 30.7 13.1 34.6 15.2 39.1 17.3 38.2 18.8 36.8 18.3 34.5 17.7 31.3 14.4 25.8 11.6 21.2 8.9 36.4 17.9 1 19.6 8.2 22.8 9.4 25.8 10.4 29.8 12.7 33.5 14.8 37.7 17.0 37.2 18.9 35.9 18.7 33.5 17.2 30.2 13.9 24.1 10.8 19.8 8.4 35.1 18.1 2 18.1 7.4 21.1 8.4 24.5 9.8 28.6 12.1 32.5 14.6 36.6 16.7 36.4 18.8 35.2 18.5 32.4 17.2 28.9 13.7 22.6 10.3 18.4 7.8 33.7 17.9 NEW YORK Albany 725180 0.4 11.3 9.1 12.1 8.3 19.6 11.9 27.4 16.0 31.8 19.2 33.3 22.2 34.4 23.4 33.2 23.3 30.9 22.8 26.1 16.7 20.2 15.4 14.1 11.7 32.0 21.8 1 8.6 5.6 10.0 6.9 16.7 10.4 24.4 14.7 30.1 19.1 32.0 21.6 33.2 22.8 31.9 22.3 29.4 21.4 24.2 16.8 18.2 13.8 11.8 9.1 30.2 21.1 2 6.4 3.5 7.9 5.1 14.1 8.6 22.3 12.9 28.3 18.4 30.6 20.8 32.2 22.1 30.8 21.9 27.9 20.6 22.3 15.8 16.3 12.4 9.4 6.7 28.7 20.3 Binghamton 725150 0.4 11.0 8.5 12.4 8.6 20.7 13.8 26.4 16.2 28.6 18.6 30.1 21.1 31.6 21.7 30.5 21.8 28.7 21.2 23.8 16.5 18.5 14.4 14.1 11.9 29.4 21.2 1 8.9 7.4 10.4 8.2 18.1 11.9 24.2 14.9 27.4 18.6 29.0 20.7 30.6 21.9 29.4 21.6 27.5 21.2 22.3 15.7 17.2 12.8 12.3 10.1 27.9 20.4 2 6.9 4.7 8.7 6.1 15.8 10.7 22.1 13.7 26.2 17.9 27.9 20.1 29.7 21.6 28.4 20.9 26.2 20.3 21.1 15.3 15.8 12.5 10.5 8.1 26.5 19.6 Buffalo 725280 0.4 12.2 9.8 13.1 8.8 21.2 13.2 26.0 16.9 28.9 19.6 31.1 20.9 32.2 21.8 31.2 22.9 29.6 22.5 24.9 17.7 19.7 14.2 15.7 12.9 30.0 21.2 1 10.1 7.4 11.4 8.2 18.8 12.5 24.2 15.8 27.8 18.9 30.1 20.7 31.1 21.6 30.1 21.7 28.4 21.4 23.7 16.6 18.3 13.2 14.2 11.9 28.7 20.7 2 8.1 5.2 9.7 7.2 16.2 10.7 22.3 14.3 26.7 18.2 29.1 20.3 30.1 21.4 29.2 21.2 27.1 20.4 22.4 15.9 17.1 12.7 12.3 10.1 27.4 20.0 Massena 726223 0.4 9.9 7.2 10.6 7.7 18.1 11.5 25.6 16.4 30.4 20.4 31.5 22.8 32.9 23.8 31.8 23.7 29.2 22.8 24.5 17.4 19.1 14.3 13.1 10.7 30.7 22.4 1 7.2 5.1 8.6 5.9 15.5 10.7 23.2 15.2 28.7 19.7 30.4 22.3 32.0 23.3 30.6 22.7 27.8 22.1 23.2 17.2 17.1 13.4 10.6 8.9 29.1 21.4 2 5.6 3.4 7.1 5.1 13.3 8.6 21.1 13.1 27.0 18.6 29.3 21.4 31.1 22.6 29.6 22.1 26.5 20.8 21.7 15.7 15.1 12.3 8.2 6.3 27.6 20.5 New York, John F Kennedy Airport 744860 0.4 14.3 10.6 16.1 10.6 21.4 14.3 26.8 16.5 31.3 21.2 33.7 22.8 35.2 24.1 33.4 24.6 32.3 23.5 26.2 19.1 22.0 16.3 16.2 12.4 32.5 23.1 1 12.4 9.8 14.2 10.3 19.1 12.6 24.5 16.2 29.6 20.0 32.4 22.4 34.0 23.7 32.2 23.4 31.0 23.1 24.8 18.2 19.9 15.1 14.9 12.4 30.9 22.4 2 11.0 8.5 12.4 9.2 16.9 11.0 22.6 14.7 28.1 19.4 31.3 21.7 32.9 23.4 31.3 23.2 29.5 21.9 23.4 17.8 18.4 15.0 13.6 11.4 29.5 21.7 Rochester 725290 0.4 12.2 8.7 13.1 9.5 22.4 13.9 27.3 17.6 30.6 20.6 32.2 22.3 34.0 23.7 32.5 23.4 30.9 22.7 26.1 18.1 20.2 14.8 16.1 13.6 31.4 22.6 1 10.1 7.2 11.2 8.3 19.1 12.6 25.4 16.4 29.1 20.1 31.2 22.1 32.8 23.3 31.4 22.8 29.6 22.6 24.7 17.3 18.6 13.9 14.1 11.0 29.9 21.7 2 8.2 5.4 9.4 6.6 16.7 11.3 23.3 15.4 27.9 19.0 30.3 21.7 31.8 23.1 30.2 22.1 28.1 21.7 23.2 16.6 17.3 13.4 12.0 9.4 28.4 20.8 Syracuse 725190 0.4 12.8 9.9 14.0 9.2 22.3 13.9 27.4 17.1 30.4 20.8 32.2 22.6 33.3 23.7 32.5 23.4 30.7 22.7 26.0 17.2 20.7 14.6 15.7 12.2 31.2 22.4 1 9.8 6.3 11.4 8.0 19.4 11.9 25.2 16.4 29.2 19.6 31.1 21.8 32.4 23.3 31.3 22.7 29.3 22.6 24.2 17.3 18.9 14.4 14.0 11.3 29.7 21.5 2 7.7 5.2 9.4 6.3 16.8 11.0 23.2 14.9 27.7 18.9 30.0 21.2 31.4 22.9 30.1 22.1 27.9 21.7 22.7 16.3 17.2 13.4 12.0 8.7 28.3 20.9 NORTH CAROLINA Asheville 723150 0.4 18.1 11.7 20.9 12.3 24.6 14.9 28.2 16.4 29.2 19.6 31.9 22.0 33.2 22.9 32.2 22.9 30.2 21.8 26.4 17.8 22.9 14.9 19.9 14.7 31.0 22.2 1 16.4 11.2 18.9 11.3 23.3 13.6 26.9 15.9 28.1 19.1 30.8 21.6 32.3 22.7 31.2 22.5 29.1 21.4 25.3 17.4 21.7 14.4 18.3 14.4 29.6 21.7 2 15.1 10.5 17.3 11.2 21.9 13.2 25.8 15.4 27.2 18.7 29.8 21.2 31.3 22.4 30.3 22.3 28.3 21.1 24.2 16.7 20.4 14.3 16.9 13.2 28.4 21.3 Cape Hatteras 723040 0.4 20.8 18.1 21.4 17.7 23.2 18.3 25.8 19.7 28.7 23.1 30.8 25.1 32.4 26.6 31.9 26.1 30.7 25.1 28.3 22.9 25.1 21.0 22.4 19.7 30.8 25.6 1 19.9 17.5 20.2 17.1 22.2 18.0 24.7 19.2 27.7 22.1 29.9 24.8 31.6 26.2 31.3 25.7 30.1 24.8 27.2 22.4 24.2 20.2 21.6 19.2 30.1 25.2 2 19.1 17.1 19.1 16.3 21.3 17.3 23.8 18.7 26.8 21.9 29.3 24.4 31.0 25.9 30.7 25.6 29.5 24.4 26.3 22.1 23.3 19.8 20.8 18.4 29.3 24.8 Charlotte 723140 0.4 20.1 13.9 22.9 15.1 26.9 16.1 30.2 17.9 32.3 20.9 34.6 23.1 36.6 23.5 35.4 23.4 33.2 22.8 29.0 19.9 25.4 17.8 21.6 17.7 34.2 23.4 1 18.5 13.4 21.1 13.6 25.4 15.8 29.2 17.7 31.3 21.1 33.6 23.0 35.6 23.6 34.4 23.6 32.4 22.8 28.0 19.2 24.1 17.2 20.4 16.8 32.8 23.2 2 17.3 13.2 19.6 13.1 23.9 15.2 28.1 17.2 30.2 20.7 32.7 22.7 34.7 23.5 33.6 23.6 31.6 22.7 26.9 19.2 22.8 16.7 19.0 15.0 31.6 22.8 Greensboro 723170 0.4 19.3 13.9 22.1 13.6 26.8 16.8 29.9 17.6 31.6 20.9 34.2 23.7 35.6 24.4 34.8 24.1 32.8 22.7 28.4 19.9 24.6 17.6 21.1 17.2 33.4 23.8 1 17.9 13.6 20.0 13.2 24.9 16.1 28.8 17.1 30.6 20.8 33.2 23.6 34.8 23.9 33.7 24.1 31.8 22.6 27.3 19.1 23.2 16.4 19.8 16.2 32.2 23.3 2 16.4 12.2 18.6 12.4 23.3 14.8 27.6 16.9 29.7 20.4 32.3 22.9 33.8 24.1 32.8 24.0 30.8 22.3 26.2 18.3 22.1 15.7 18.2 14.7 30.9 22.8 Raleigh/Durham 723060 0.4 20.8 15.6 23.3 14.9 27.4 17.4 30.7 17.8 32.1 21.6 34.5 24.1 35.9 24.3 34.9 24.7 33.1 23.6 29.1 20.3 25.7 17.9 22.4 17.6 33.8 24.4 1 19.4 14.9 21.5 14.7 25.9 16.6 29.4 18.0 31.1 21.2 33.4 23.8 34.9 24.7 33.9 24.6 32.1 23.3 27.8 20.0 24.2 17.4 21.1 16.9 32.4 23.7 2 18.0 14.2 19.8 13.9 24.4 15.7 28.2 17.4 30.1 20.9 32.5 23.4 34.1 24.6 33.0 24.3 31.1 22.9 26.7 19.2 23.1 17.0 19.8 16.3 31.2 23.2 Wilmington 723013 0.4 23.0 17.9 24.5 18.3 27.6 18.6 30.8 20.0 32.7 22.4 34.9 25.4 35.9 26.0 35.0 26.3 33.3 25.3 29.8 22.6 26.7 21.1 24.3 19.3 33.9 25.8 1 21.6 17.2 23.2 17.8 26.1 18.6 29.6 19.5 31.5 22.3 33.7 25.1 35.0 26.3 34.1 26.3 32.3 24.9 28.7 22.1 25.7 20.9 23.2 18.7 32.7 25.3 2 20.4 17.1 21.9 16.5 24.7 17.9 28.2 19.5 30.4 21.8 32.8 24.9 34.1 26.2 33.4 26.1 31.4 24.6 27.8 21.7 24.8 19.6 22.1 18.3 31.5 24.8 NORTH DAKOTA Bismarck 727640 0.4 7.7 3.4 11.7 6.2 19.6 9.9 27.6 13.6 31.2 17.9 35.1 20.1 37.1 20.8 36.7 20.3 33.6 18.7 26.9 13.9 17.8 9.5 9.4 3.6 34.1 19.9 1 6.1 2.6 9.4 4.7 16.4 8.4 25.4 12.8 29.5 17.1 33.5 19.9 35.4 21.1 35.4 19.7 31.6 17.6 24.8 13.7 15.1 7.8 6.9 2.9 32.1 19.4 2 4.9 2.0 7.2 3.5 13.9 6.7 23.2 11.9 27.8 16.6 31.7 19.3 33.8 20.4 34.1 19.7 29.7 17.0 22.8 12.7 12.8 6.7 5.3 2.1 30.1 18.8 Fargo 727530 0.4 4.1 1.8 7.8 4.4 16.4 9.9 27.8 15.1 31.5 18.9 33.6 22.1 35.3 22.8 35.6 22.6 32.2 20.1 26.1 15.6 15.7 9.6 6.9 4.2 32.9 21.8 1 2.9 1.1 5.7 3.2 13.9 8.9 25.3 14.1 30.2 18.2 32.3 21.2 33.8 22.9 34.2 22.1 30.4 20.1 23.8 14.9 13.8 8.8 4.2 1.8 31.0 21.2 2 1.8 0.4 3.9 2.0 11.6 7.0 22.8 13.1 28.7 17.7 30.9 20.9 32.4 21.9 32.8 22.1 28.8 19.3 21.9 14.2 11.9 7.8 2.5 0.7 29.3 20.3 Minot, Intl Airport 727676 0.4 6.4 3.1 10.0 5.3 17.8 8.3 27.1 13.3 31.4 17.3 34.4 18.7 35.6 21.1 35.8 19.3 32.6 17.7 25.6 13.9 15.9 8.6 7.9 3.5 33.3 19.6 1 5.5 2.5 7.8 3.7 14.9 7.7 24.9 12.7 29.4 16.8 32.4 18.7 34.2 20.1 34.6 19.3 30.6 17.3 23.6 13.2 13.7 7.4 6.2 2.8 31.1 18.8 2 4.4 1.8 6.2 2.7 12.4 6.3 22.4 11.7 27.7 16.1 30.8 18.3 32.8 19.9 33.3 19.7 28.6 16.4 21.7 12.3 11.9 6.1 4.7 1.8 29.1 18.2 OHIO Akron/Canton 725210 0.4 13.7 11.3 14.9 10.2 22.8 14.1 26.8 16.3 29.9 19.4 32.1 21.5 33.3 22.4 32.6 23.3 30.4 22.1 25.5 17.1 20.8 15.3 16.6 13.4 31.1 22.2 1 12.1 10.0 13.4 10.3 20.9 13.3 25.4 16.1 28.7 19.6 31.0 21.7 32.1 22.9 31.2 22.7 29.3 21.9 24.4 17.0 19.3 14.1 15.2 12.4 29.6 21.6 2 10.4 8.2 12.0 9.1 19.0 12.4 23.9 15.9 27.6 19.1 30.1 21.0 31.3 22.7 30.1 22.1 28.2 21.3 23.2 16.4 18.1 13.7 13.9 11.5 28.3 20.9 Cleveland 725240 0.4 14.7 11.6 15.2 10.7 23.6 15.0 27.3 17.9 30.3 20.9 32.4 22.2 33.7 23.8 32.5 24.1 31.2 22.7 26.4 17.8 21.5 15.6 17.5 13.9 31.4 22.9 1 12.6 9.9 13.7 10.6 21.4 14.1 25.9 17.3 29.2 20.4 31.3 22.1 32.7 23.8 31.4 23.4 29.9 22.5 24.9 17.9 20.0 14.9 15.7 12.3 30.0 22.1 2 10.7 8.0 12.0 9.2 19.6 13.3 24.2 16.4 28.2 19.8 30.5 21.9 31.8 23.4 30.3 22.7 28.8 22.0 23.6 16.8 18.4 14.2 14.1 11.7 28.6 21.4 Columbus, Intl Airport 724280 0.4 15.2 12.6 16.6 11.4 24.3 15.0 27.9 17.4 31.3 21.0 33.2 22.0 34.6 24.1 33.7 23.8 32.1 22.8 27.1 18.6 22.2 16.1 17.4 14.0 32.4 23.1 1 13.4 11.2 15.1 11.5 22.7 15.3 26.8 17.2 30.2 20.7 32.3 22.2 33.4 23.9 32.7 23.9 30.9 22.5 26.1 18.0 20.8 14.9 16.2 13.3 31.1 22.7 2 11.6 9.0 13.6 10.4 20.7 13.7 25.2 16.9 29.1 20.2 31.4 22.2 32.6 23.6 31.7 23.4 29.9 22.1 24.8 17.0 19.2 14.6 14.8 12.3 29.7 21.9 Dayton, Intl Airport 724290 0.4 15.1 12.8 16.6 11.4 23.7 15.5 27.6 17.3 30.8 20.9 33.4 21.3 35.1 23.7 33.9 24.6 31.9 22.8 27.0 18.2 22.2 16.3 17.0 13.4 32.4 23.2 1 13.3 11.6 14.8 11.6 22.1 14.4 26.3 17.0 29.9 20.4 32.3 22.3 33.7 23.8 32.7 23.7 30.7 22.2 25.8 18.0 20.7 15.8 15.8 13.3 31.1 22.6 2 11.3 9.0 13.1 9.9 20.3 13.6 24.7 16.8 28.9 20.4 31.4 22.0 32.6 23.7 31.7 23.4 29.7 21.8 24.5 16.9 19.2 14.6 14.6 12.3 29.8 21.8 Mansfield 725246 0.4 14.6 12.3 15.5 9.8 22.9 14.1 26.9 16.9 28.9 20.2 32.1 22.7 33.4 23.8 31.9 23.7 30.4 22.6 25.7 18.2 21.1 15.8 16.8 14.0 31.0 22.8 1 13.1 11.0 13.6 9.6 21.2 13.9 25.6 16.3 27.9 19.2 30.9 21.7 32.3 23.5 30.9 23.2 29.3 22.0 24.3 17.8 20.0 14.7 15.4 13.2 29.6 22.2 2 11.2 8.8 12.2 9.3 19.4 12.8 24.1 15.8 26.9 18.9 29.8 21.7 31.3 23.3 30.1 22.6 28.3 21.5 23.2 17.0 18.5 14.3 14.0 11.7 28.3 21.4 Toledo 725360 0.4 12.8 11.1 13.8 9.9 22.9 14.7 28.2 17.4 31.8 20.8 33.8 21.9 34.7 23.7 33.7 25.1 32.0 22.4 27.3 18.0 21.1 14.8 16.1 14.1 32.4 22.9 1 10.9 8.9 11.8 8.9 20.9 13.7 26.5 17.1 30.4 20.7 32.6 21.7 33.6 23.8 32.4 24.1 30.6 21.9 25.7 17.7 19.3 14.7 14.6 12.4 30.8 22.2 2 8.7 6.8 10.1 7.2 18.6 12.7 24.2 16.1 29.1 19.5 31.6 21.8 32.4 23.8 31.2 23.3 29.2 21.8 24.1 16.7 17.7 13.9 12.7 11.3 29.3 21.5 Youngstown 725250 0.4 13.2 10.8 15.2 10.1 23.0 14.3 27.5 16.7 29.7 19.9 31.7 21.3 33.2 22.7 32.4 23.2 30.6 21.4 25.6 17.5 21.2 15.1 16.8 13.6 30.8 22.2 1 11.4 9.4 13.3 10.0 20.8 13.1 25.9 16.1 28.7 19.4 30.7 21.4 32.0 23.0 30.9 22.7 29.2 21.6 24.4 16.6 19.5 13.8 15.3 12.0 29.4 21.3 2 9.7 7.6 11.5 8.6 18.8 12.6 24.1 15.5 27.7 18.8 29.8 21.2 31.1 22.8 29.9 21.7 28.1 21.2 23.1 16.1 17.9 13.6 13.7 11.5 28.1 20.6 WMO# = World Meteorological Organization number DB = dry-bulb temperature, °C MWB = mean coincident wet-bulb temperature, °C 27.66 2001 ASHRAE Fundamentals Handbook (SI) Table 4A Design Wet-Bulb—Mean Coincident Dry-Bulb Temperature Location and WMO# % Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Annual WB MDB WB MDB WB MDB WB MDB WB MDB WB MDB WB MDB WB MDB WB MDB WB MDB WB MDB WB MDB WB MDB 1a 1b 2a 2b 3a 3b 4a 4b 5a 5b 6a 6b 7a 7b 8a 8b 9a 9b 10a 10b 11a 11b 12a 12c 13a 13b 14a 14b OKLAHOMA Oklahoma City, Will Rogers Airport 723530 0.4 14.9 16.7 15.7 20.3 18.8 23.2 21.3 27.3 24.1 30.6 25.8 32.8 25.6 33.3 25.3 33.7 24.8 32.1 22.7 28.1 19.1 22.1 16.5 18.5 24.9 32.8 1 13.1 15.6 14.7 18.9 17.9 22.9 20.8 26.2 23.4 29.5 25.2 32.5 25.2 33.3 25.0 33.1 24.3 31.6 21.9 26.2 18.1 21.3 15.6 17.8 24.4 32.4 2 11.4 14.7 13.7 18.8 17.2 22.1 20.1 25.1 22.8 28.4 24.8 31.9 24.9 33.2 24.6 32.9 23.9 31.0 20.9 25.5 17.2 20.8 14.4 16.7 23.9 31.8 Tulsa 723560 0.4 15.0 17.4 16.2 20.1 19.7 24.4 22.2 27.6 24.8 30.8 26.9 33.7 27.2 34.9 26.6 34.3 25.9 33.2 23.1 28.2 19.8 22.2 17.5 19.9 26.2 33.5 1 14.1 16.8 15.4 19.4 18.7 23.7 21.4 26.9 24.1 29.9 26.4 32.5 26.8 34.4 26.2 34.2 25.4 32.4 22.4 27.3 18.9 22.2 16.3 18.7 25.6 33.1 2 12.5 14.7 14.4 19.3 17.8 22.1 20.8 26.2 23.6 28.9 25.9 31.9 26.3 33.9 25.8 34.0 24.9 32.0 21.6 26.2 18.1 22.1 15.1 17.5 25.1 32.4 OREGON Astoria 727910 0.4 11.8 12.8 13.1 14.8 12.6 15.3 14.0 18.4 17.3 23.7 18.6 24.3 19.3 25.6 20.4 26.8 19.2 26.1 16.9 22.4 13.9 15.5 13.0 14.2 18.3 23.7 1 11.3 12.2 12.3 14.3 11.8 14.0 13.1 16.6 16.3 21.5 17.4 22.2 18.4 23.8 19.2 24.4 18.4 24.5 15.9 19.9 13.3 14.5 12.1 13.3 17.2 21.5 2 10.8 11.8 11.4 13.4 11.2 13.2 12.4 15.4 15.3 19.4 16.6 20.7 17.7 22.3 18.2 22.8 17.7 22.7 15.3 18.2 12.8 14.1 11.6 12.6 16.4 19.9 Eugene 726930 0.4 13.1 14.7 13.6 15.6 14.4 18.0 16.8 22.9 20.4 27.3 21.1 31.2 21.6 32.3 21.4 33.8 19.9 29.4 17.2 24.6 14.9 16.4 13.4 14.4 20.3 30.6 1 12.3 13.4 12.9 14.6 13.5 17.1 15.7 21.6 19.1 26.3 20.2 29.2 20.8 31.9 20.7 32.4 19.1 28.7 16.5 23.0 14.1 15.8 12.4 13.6 19.3 29.1 2 11.6 12.9 12.1 13.8 12.7 16.0 14.8 20.2 18.1 25.1 19.4 28.2 20.1 31.1 20.1 30.8 18.6 27.8 15.8 21.6 13.3 14.9 11.6 12.7 18.4 27.4 Medford 725970 0.4 10.8 13.3 12.6 17.2 14.1 21.5 16.7 26.7 19.6 31.3 20.8 34.5 21.6 36.0 21.2 36.6 19.7 33.1 16.9 28.9 13.2 17.0 11.4 14.1 20.3 34.5 1 9.9 12.8 11.6 16.7 13.0 19.3 15.6 24.8 18.4 29.0 20.2 33.3 20.9 35.1 20.7 35.7 19.1 32.6 16.3 27.6 12.3 15.3 10.5 12.2 19.5 32.9 2 9.3 11.9 10.8 15.0 12.2 17.6 14.6 23.1 17.4 28.0 19.5 32.2 20.4 34.8 20.1 34.2 18.4 31.3 15.6 26.1 11.7 14.6 9.4 11.6 18.6 31.3 North Bend 726917 0.4 13.9 15.5 14.4 16.9 13.5 16.1 14.1 17.8 16.1 20.9 16.8 20.5 17.1 20.4 17.9 20.7 17.7 22.1 16.7 21.6 15.3 16.9 14.2 15.6 16.8 20.3 1 13.2 14.9 13.6 15.9 12.8 15.1 13.3 17.0 15.2 18.2 16.1 19.8 16.6 20.0 17.3 20.1 17.0 21.1 15.9 19.8 14.7 16.6 13.5 15.0 16.2 19.6 2 12.5 14.0 12.8 14.7 12.2 14.6 12.7 15.7 14.5 17.2 15.5 18.8 16.2 19.4 16.8 19.7 16.4 20.2 15.4 18.7 14.1 15.8 12.9 14.4 15.6 18.8 Pendleton 726880 0.4 10.8 14.6 11.5 15.3 12.3 18.0 15.1 23.3 18.6 29.3 19.4 32.3 20.8 33.4 19.4 34.4 17.8 31.0 15.2 25.4 12.1 16.6 10.8 14.4 18.8 33.3 1 9.4 12.9 10.4 14.1 11.4 17.0 14.1 22.8 17.5 27.6 18.7 31.1 19.8 34.8 18.9 34.0 17.2 29.9 14.4 23.6 11.1 15.4 9.7 13.1 17.9 32.0 2 8.5 11.9 9.5 13.2 10.7 15.7 13.3 21.6 16.4 26.8 18.0 31.1 19.1 34.4 18.4 33.5 16.6 28.9 13.7 22.6 10.2 13.9 8.8 12.4 17.1 30.4 Portland 726980 0.4 12.3 13.9 13.2 15.1 13.5 18.3 16.3 22.4 19.6 29.1 20.7 31.0 21.6 32.9 21.8 33.5 20.2 29.7 17.3 24.8 14.3 16.6 12.7 14.1 20.4 30.8 1 11.3 12.9 12.3 14.7 12.8 17.4 15.2 21.1 18.3 26.3 19.8 29.7 20.9 31.8 21.1 32.2 19.5 28.8 16.4 22.3 13.4 15.3 11.8 13.4 19.5 28.8 2 10.7 12.3 11.4 13.9 12.0 16.1 14.4 19.9 17.3 24.9 19.1 28.3 20.3 31.1 20.4 30.6 18.8 26.9 15.8 20.8 12.6 14.6 10.9 12.5 18.6 26.9 Redmond 726835 0.4 8.8 12.9 9.8 14.9 10.6 18.4 13.2 22.8 16.4 28.3 18.1 30.3 18.6 31.2 18.5 31.3 16.6 30.1 14.2 26.7 10.9 16.3 9.2 12.7 17.4 31.2 1 7.6 11.4 8.8 13.4 9.5 17.5 12.3 22.8 15.4 26.8 17.4 30.4 18.0 31.7 17.9 31.6 15.9 29.3 13.4 24.9 10.0 15.9 8.0 11.9 16.6 30.0 2 6.7 10.3 8.1 12.7 8.7 15.6 11.4 20.8 14.4 24.9 16.6 29.2 17.4 31.4 17.4 31.3 15.4 28.2 12.7 23.3 9.1 13.9 7.0 11.1 15.7 28.3 Salem 726940 0.4 12.5 14.1 13.5 15.6 13.8 17.4 16.3 21.8 19.5 28.8 21.1 31.2 21.7 33.3 21.4 33.9 19.7 29.7 17.3 24.5 14.4 16.2 12.9 14.3 20.2 31.4 1 11.8 13.3 12.7 14.6 13.0 16.8 15.1 21.3 18.4 27.2 20.1 30.1 20.9 32.6 20.7 33.1 19.1 29.1 16.4 23.2 13.6 15.4 11.9 13.3 19.3 29.4 2 11.1 12.6 11.9 13.9 12.3 16.0 14.3 20.1 17.5 25.4 19.3 28.7 20.2 31.6 20.1 31.6 18.5 27.9 15.6 21.4 12.8 14.6 11.3 12.8 18.3 27.4 PENNSYLVANIA Allentown 725170 0.4 12.2 13.8 12.4 13.9 16.6 21.6 18.6 26.5 22.2 28.3 24.1 30.7 25.4 32.1 25.2 30.3 24.2 29.8 20.2 23.0 18.1 20.6 14.7 16.5 24.2 29.9 1 9.5 11.4 10.7 13.1 15.0 18.8 17.6 24.1 21.5 27.7 23.4 29.7 24.9 31.7 24.6 29.8 23.6 29.0 19.6 22.6 17.0 19.1 12.4 13.6 23.4 28.8 2 7.0 8.7 8.6 11.1 13.3 17.1 16.7 23.0 20.7 26.8 22.9 28.8 24.3 30.7 24.0 29.1 23.0 27.8 18.8 22.2 15.8 17.9 10.1 11.6 22.7 27.6 Bradford 725266 0.4 9.3 10.2 9.3 10.4 14.2 18.1 16.4 22.7 20.4 24.6 21.8 26.7 23.3 27.8 23.2 27.4 22.2 26.2 17.8 21.3 15.2 16.8 12.3 13.2 22.0 26.3 1 7.0 7.7 7.8 9.3 12.9 16.2 15.5 21.3 19.4 24.2 21.3 26.1 22.7 27.6 22.3 26.6 21.3 25.1 16.9 19.9 14.3 16.4 10.8 11.7 21.1 25.1 2 5.1 6.4 6.4 7.9 11.5 15.2 14.6 20.3 18.6 23.7 20.8 25.4 22.2 26.7 21.7 25.6 20.6 23.6 16.3 18.9 13.2 14.7 9.3 10.1 20.2 23.6 Erie 725260 0.4 11.3 13.1 11.2 13.2 15.5 21.2 18.1 23.4 22.3 26.6 23.4 28.7 25.1 29.9 24.5 29.1 23.5 27.8 19.0 23.5 16.2 19.1 14.2 16.9 23.5 27.7 1 9.4 11.4 9.8 11.9 14.0 18.2 17.1 22.3 21.1 25.9 22.8 27.8 24.3 28.3 23.8 28.0 22.8 27.2 18.1 22.1 15.3 17.9 12.9 14.8 22.7 26.7 2 7.1 9.2 8.3 10.4 12.7 16.6 16.2 20.9 20.1 24.8 22.2 26.8 23.7 27.8 23.3 27.2 22.1 25.8 17.4 21.1 14.3 16.7 11.2 13.4 21.8 25.6 Harrisburg 725115 0.4 12.0 14.2 12.1 14.3 16.9 22.2 19.3 27.1 23.6 29.1 24.9 30.6 26.4 32.1 26.2 31.3 25.1 30.1 20.8 24.1 18.6 21.0 14.9 16.8 25.1 30.7 1 9.4 11.4 10.3 12.9 15.3 19.8 18.4 25.2 22.6 27.4 24.2 30.0 25.8 31.6 25.6 31.1 24.4 29.1 20.1 22.9 17.4 19.7 12.6 14.9 24.2 29.4 2 7.1 9.0 8.9 12.1 13.9 17.7 17.4 23.8 21.6 26.3 23.7 29.5 25.2 31.1 24.9 30.3 23.7 28.4 19.3 22.1 16.1 18.7 10.3 11.9 23.4 28.2 Philadelphia, Intl Airport 724080 0.4 13.5 14.8 14.1 16.3 17.4 22.1 19.5 27.7 23.3 29.5 25.3 31.3 26.6 33.3 26.3 32.2 25.4 30.8 21.8 24.6 19.3 22.0 15.9 16.9 25.4 31.1 1 12.2 13.7 12.8 14.7 16.1 20.6 18.5 25.6 22.4 28.2 24.7 30.8 26.1 32.2 25.8 31.3 24.8 29.9 21.1 23.7 18.1 20.1 14.7 16.0 24.7 29.7 2 10.2 11.8 11.2 13.4 14.6 18.2 17.5 23.4 21.7 27.7 24.2 30.2 25.6 31.4 25.3 30.4 24.2 28.8 20.2 23.2 17.0 19.1 13.1 14.4 23.9 28.7 Pittsburgh, Intl Airport 725200 0.4 12.3 14.7 12.6 14.8 15.9 20.7 18.2 24.4 21.9 27.4 23.6 29.7 24.9 30.7 24.6 30.8 23.5 28.9 19.3 23.3 16.3 20.0 14.2 17.0 23.6 29.2 1 10.6 12.4 11.1 13.1 14.8 19.4 17.4 23.7 21.2 26.8 23.0 29.0 24.3 30.2 23.9 29.6 22.8 27.7 18.4 22.5 15.6 18.7 13.4 15.9 22.7 27.9 2 8.8 10.7 9.7 12.8 13.7 18.9 16.6 23.0 20.2 26.2 22.4 28.4 23.7 29.4 23.3 28.6 22.1 27.1 17.6 21.7 14.7 17.8 12.1 14.2 21.9 26.8 Wilkes-Barre/ Scranton 725130 0.4 11.2 12.9 11.7 13.3 15.4 20.8 17.7 25.5 21.7 27.4 23.5 28.9 24.7 30.5 24.3 29.8 23.4 28.4 19.4 22.7 16.9 19.2 13.7 15.2 23.4 28.6 1 9.2 10.6 10.2 12.2 14.0 18.2 16.8 23.3 20.7 26.2 22.8 28.3 24.1 29.9 23.7 28.8 22.9 27.3 18.6 21.3 15.8 17.9 11.8 13.3 22.7 27.4 2 6.6 8.4 8.4 10.3 12.5 16.7 15.8 21.8 19.9 25.4 22.3 27.6 23.6 28.9 23.3 28.2 22.2 26.5 17.8 20.9 14.7 17.2 9.8 11.2 21.9 26.2 Williamsport 725140 0.4 9.9 11.8 11.2 12.9 15.8 21.0 18.4 25.8 22.3 27.8 24.1 29.2 25.4 31.4 25.1 30.8 23.9 29.3 19.8 22.4 17.1 19.1 12.6 14.2 24.2 29.3 1 7.4 8.8 9.3 12.3 14.1 18.1 17.6 23.8 21.3 26.9 23.4 28.8 24.9 30.7 24.6 29.9 23.3 28.0 19.2 22.0 15.9 18.1 10.4 11.3 23.3 28.1 2 5.6 7.3 7.4 9.5 12.4 16.6 16.7 21.9 20.5 26.3 22.9 27.9 24.4 29.8 24.0 28.9 22.7 26.8 18.3 21.1 14.7 16.6 8.3 9.9 22.6 26.8 RHODE ISLAND Providence 725070 0.4 12.7 13.3 11.9 13.3 14.1 17.2 17.3 25.2 21.6 28.4 24.1 30.7 25.6 32.1 25.6 31.2 24.3 29.7 20.5 22.6 17.8 20.3 14.5 15.7 24.3 29.5 1 10.7 12.4 10.3 12.1 12.7 15.9 15.9 21.8 20.4 26.9 23.2 29.0 24.9 31.2 24.8 29.4 23.6 27.6 19.5 22.1 16.6 18.6 13.2 14.4 23.4 27.5 2 8.6 9.9 8.8 10.9 11.4 14.4 14.8 19.7 19.4 24.9 22.4 28.2 24.3 30.3 24.2 28.5 22.9 26.1 18.7 21.6 15.5 17.7 11.8 13.3 22.6 26.4 SOUTH CAROLINA Charleston 722080 0.4 19.4 22.4 20.1 24.1 20.8 25.4 22.7 28.1 24.9 29.4 26.8 32.4 27.9 33.7 27.2 32.9 26.3 30.4 24.6 27.6 22.4 25.3 21.0 23.7 26.7 31.9 1 18.7 21.1 19.2 22.6 20.2 24.3 21.8 26.9 24.3 29.3 26.3 31.7 27.3 32.7 26.8 32.4 25.9 30.1 24.1 27.2 21.9 24.3 20.2 22.6 26.1 31.2 2 18.0 20.4 18.4 21.1 19.6 23.6 21.2 25.9 23.8 28.7 25.8 31.0 26.9 32.0 26.4 31.9 25.6 30.0 23.6 26.7 21.4 23.7 19.4 21.6 25.6 30.5 Columbia 723100 0.4 19.2 22.1 19.4 22.9 20.2 25.7 22.2 28.1 24.3 29.9 25.7 32.9 26.6 32.9 26.3 33.1 25.6 31.7 23.6 27.8 21.6 24.6 20.3 22.8 25.7 32.3 1 18.2 20.1 18.4 21.4 19.7 24.7 21.4 27.0 23.6 29.2 25.2 32.0 26.1 32.8 25.9 32.7 25.2 30.9 23.0 26.9 20.8 23.7 19.5 22.2 25.2 31.4 2 17.2 19.4 17.5 20.1 19.1 24.2 20.7 26.1 23.0 28.6 24.8 31.2 25.8 32.3 25.7 32.2 24.8 30.2 22.4 26.1 20.2 23.1 18.5 21.0 24.7 30.6 Greer/Greenville 723120 0.4 16.1 18.5 17.0 19.2 18.4 23.6 20.8 26.4 23.5 28.7 24.8 31.6 25.8 32.5 25.4 31.4 24.4 29.9 22.0 26.2 19.8 22.2 18.4 19.7 24.8 31.1 1 15.1 17.4 15.8 17.9 17.8 22.0 19.8 25.5 22.8 28.1 24.4 31.3 25.3 31.9 25.0 31.2 24.0 29.4 21.2 25.1 19.1 21.4 17.2 18.6 24.2 30.3 2 14.1 16.2 14.8 17.3 17.0 20.4 19.1 24.6 22.2 27.7 24.1 30.4 25.0 31.4 24.6 30.6 23.6 28.7 20.6 24.2 18.3 20.3 16.2 17.9 23.8 29.4 SOUTH DAKOTA Huron 726540 0.4 4.8 8.9 7.2 12.9 13.4 21.6 17.6 26.3 21.7 28.2 24.9 32.5 26.0 32.6 25.3 32.2 23.1 30.9 18.0 23.7 12.2 15.7 6.1 10.2 24.5 31.8 1 3.6 6.7 5.9 10.2 11.6 17.4 16.7 23.8 20.9 27.4 24.1 31.1 25.3 32.4 24.7 31.6 22.3 29.6 16.9 21.8 10.9 14.2 4.6 8.2 23.4 30.3 2 2.6 5.2 4.8 8.0 10.0 15.4 15.7 23.1 20.0 26.2 23.4 30.0 24.8 32.1 24.1 30.7 21.6 28.3 15.9 21.4 9.5 13.9 3.3 6.4 22.4 29.0 Pierre 726686 0.4 5.7 11.2 7.9 15.4 12.1 20.7 16.4 25.9 20.7 28.9 23.9 31.6 25.1 34.6 23.7 33.4 22.1 31.7 16.4 25.1 11.3 16.2 6.3 11.7 23.3 32.4 1 4.5 8.9 6.7 13.1 10.8 19.2 15.5 25.0 19.7 27.5 23.2 31.3 24.3 33.2 23.2 32.5 21.1 29.7 15.6 23.9 9.9 16.1 5.1 9.9 22.4 31.4 2 3.4 7.0 5.6 11.0 9.4 16.4 14.5 23.3 18.9 26.5 22.5 30.7 23.7 32.8 22.7 32.2 20.2 28.9 14.7 22.9 8.7 15.0 4.0 8.1 21.6 30.1 Rapid City 726620 0.4 6.8 14.2 7.9 16.5 10.6 21.3 14.1 25.2 18.3 26.5 21.8 29.6 22.3 30.7 21.8 30.0 18.9 29.4 14.1 24.3 9.9 18.9 7.4 15.5 21.0 29.6 1 5.7 12.7 6.8 14.6 9.5 19.4 13.2 23.7 17.4 25.6 21.0 28.6 21.7 30.8 21.1 29.7 18.2 28.3 13.3 24.8 9.1 18.2 6.1 13.4 20.1 28.8 2 4.7 10.7 5.9 13.1 8.6 17.3 12.4 22.8 16.7 24.8 20.3 28.1 21.2 30.2 20.5 29.7 17.4 27.4 12.7 23.9 8.1 16.1 5.1 11.6 19.2 27.7 WMO# = World Meteorological Organization number WB = wet-bulb temperature, °C MDB = mean coincident dry-bulb temperature, °C Climatic Design Information 27.67 Table 4B Design Dry-Bulb—Mean Coincident Wet-Bulb Temperature Location and WMO# % Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Annual DB MWB DB MWB DB MWB DB MWB DB MWB DB MWB DB MWB DB MWB DB MWB DB MWB DB MWB DB MWB DB MWB 1a 1b 2a 2b 3a 3b 4a 4b 5a 5b 6a 6b 7a 7b 8a 8b 9a 9b 10a 10b 11a 11b 12a 12c 13a 13b 14a 14b OKLAHOMA Oklahoma City, Will Rogers Airport 723530 0.4 19.8 11.1 24.6 13.3 27.7 15.6 30.6 19.1 32.5 21.4 36.1 23.8 39.9 22.8 38.9 23.1 36.3 23.1 32.1 18.7 25.2 16.1 20.6 13.9 37.3 23.2 1 17.9 10.4 22.7 12.7 26.2 15.6 29.2 18.8 31.4 21.7 35.0 23.4 38.8 22.9 38.0 23.1 35.2 22.7 30.4 19.6 23.9 15.4 19.1 12.8 35.7 23.1 2 16.6 10.6 20.9 11.8 24.7 15.1 28.0 17.8 30.4 21.3 34.1 23.5 37.8 23.2 37.2 23.1 34.1 22.5 29.1 19.2 22.6 15.6 17.9 12.9 34.2 22.9 Tulsa 723560 0.4 20.2 12.1 24.2 13.6 28.0 16.4 31.0 19.3 32.1 23.5 36.0 24.6 40.1 24.4 39.4 24.3 36.5 23.8 32.5 19.7 26.1 16.3 21.2 15.0 37.6 24.5 1 18.3 12.0 22.3 12.8 26.4 16.4 29.7 19.2 31.3 22.6 35.0 24.6 38.8 24.6 38.5 24.2 35.1 23.5 30.8 20.8 24.7 16.6 19.8 14.4 35.9 24.3 2 16.8 11.2 20.6 12.3 24.9 16.0 28.5 18.6 30.5 21.7 34.2 24.4 37.8 24.8 37.7 24.3 34.1 23.8 29.5 20.2 23.3 16.6 18.4 14.1 34.4 24.1 OREGON Astoria 727910 0.4 14.3 10.4 17.0 10.8 17.9 10.9 20.4 12.7 24.2 17.2 25.0 18.1 26.6 19.3 27.6 19.7 28.1 18.2 23.4 16.2 16.8 12.4 14.7 12.6 24.7 17.6 1 13.2 10.2 15.3 10.8 16.2 10.4 18.4 12.3 21.9 16.2 22.7 17.1 24.3 18.0 25.3 18.6 26.4 17.7 21.5 15.1 15.6 12.4 13.6 11.7 22.2 16.8 2 12.4 10.2 14.2 10.8 14.7 10.1 16.6 11.8 19.9 15.1 20.9 16.3 22.7 17.5 23.2 18.1 24.2 16.9 19.8 14.4 14.7 12.2 12.8 11.2 20.5 16.0 Eugene 726930 0.4 15.1 12.7 17.1 12.2 19.9 12.8 23.9 16.3 28.8 19.6 32.8 20.2 35.6 20.4 35.9 20.3 33.3 18.4 27.6 16.1 17.8 13.9 15.0 12.7 32.9 19.3 1 13.9 11.8 15.8 11.8 18.5 12.5 22.2 15.4 27.2 18.4 30.8 19.3 34.1 19.8 34.3 19.4 31.7 17.8 25.3 15.4 16.4 13.5 14.0 12.2 30.6 18.6 2 13.0 11.3 14.6 11.5 17.3 12.0 20.7 14.3 25.6 17.6 29.3 18.8 32.8 19.4 32.8 19.4 30.0 17.7 23.3 15.0 15.4 12.8 12.9 11.4 28.6 17.9 Medford 725970 0.4 15.8 8.9 20.0 11.4 23.0 13.0 28.2 15.6 33.4 18.5 36.9 19.8 39.0 19.8 39.7 20.2 36.8 18.6 31.5 16.3 19.1 11.8 15.1 10.4 36.8 19.3 1 14.4 8.7 18.0 10.3 21.4 12.3 26.4 14.7 31.4 17.8 35.3 19.3 38.0 19.9 38.1 19.6 35.3 17.8 29.3 15.5 17.4 11.2 13.6 9.2 34.9 18.8 2 12.9 8.4 16.4 9.9 19.7 11.3 24.8 14.2 29.6 16.8 33.9 18.7 36.8 19.5 36.8 19.2 33.8 17.3 27.3 15.1 16.1 10.4 12.3 8.7 32.9 18.1 North Bend 726917 0.4 16.6 13.1 18.6 13.2 18.0 11.8 20.0 13.6 21.7 15.8 21.6 16.2 21.8 16.2 22.4 16.4 25.7 15.7 23.8 15.8 18.6 13.8 16.7 13.4 21.8 15.8 1 15.5 12.1 17.1 12.3 16.5 11.7 17.8 12.7 19.2 14.4 20.2 15.4 21.1 16.0 21.4 16.3 22.9 16.0 21.3 15.1 17.4 13.5 15.6 12.8 20.6 15.6 2 14.6 12.1 15.8 12.3 15.4 11.4 16.4 12.3 17.9 13.8 19.3 14.9 20.4 15.6 20.9 16.1 21.6 15.7 19.8 14.5 16.6 13.4 14.8 12.4 19.6 15.0 Pendleton 726880 0.4 15.2 9.8 16.9 10.3 19.8 11.4 25.9 14.1 31.8 17.2 35.8 18.1 38.9 19.7 38.3 18.5 33.3 16.9 27.1 14.3 18.2 10.4 15.7 9.7 35.9 18.0 1 14.1 8.9 15.3 9.7 18.4 10.7 23.9 13.8 30.1 16.8 34.6 17.8 37.6 18.8 36.8 18.2 32.1 16.4 25.3 13.8 16.7 10.2 14.1 8.8 33.9 17.3 2 12.8 8.1 14.1 8.9 16.9 9.8 22.2 12.7 28.3 15.8 33.2 17.2 36.3 18.2 35.6 17.9 30.7 16.0 23.8 13.2 15.2 9.5 12.8 8.4 32.0 16.6 Portland 726980 0.4 14.4 11.7 17.1 11.6 20.3 12.2 24.7 15.4 30.3 18.1 32.9 20.0 35.8 20.3 36.3 20.8 32.9 18.2 26.7 16.1 17.4 12.6 14.7 12.6 32.4 19.5 1 13.4 10.8 15.7 10.7 19.0 12.0 22.9 14.6 28.6 17.7 30.9 19.0 33.6 20.3 34.1 20.2 31.4 18.1 24.6 15.2 16.2 12.6 13.5 11.6 30.1 18.8 2 12.6 10.2 14.7 10.7 17.7 11.1 21.3 13.6 26.7 16.9 29.4 18.6 31.9 19.9 32.3 19.4 29.7 17.7 22.8 14.8 15.3 11.9 12.6 10.9 28.1 18.0 Redmond 726835 0.4 14.4 8.2 16.6 8.5 20.7 9.5 26.1 12.3 30.2 15.2 33.6 16.8 36.1 17.4 36.3 17.2 32.8 15.7 28.2 13.2 18.8 9.6 14.3 7.6 33.7 16.5 1 12.5 6.9 15.3 7.7 18.7 8.6 24.1 11.7 28.5 14.3 32.3 16.6 34.9 16.9 34.9 16.8 31.3 15.1 26.7 13.1 16.8 8.9 12.8 7.3 31.8 15.8 2 11.0 6.1 13.8 7.1 17.1 7.9 22.0 10.7 26.7 13.8 30.9 15.8 33.9 16.6 33.6 16.4 29.9 14.6 24.7 12.0 14.9 8.3 11.6 6.3 29.9 15.1 Salem 726940 0.4 14.8 11.4 17.6 11.4 19.9 12.2 24.2 15.5 29.8 18.9 33.9 19.9 36.0 20.4 36.6 20.3 33.7 18.3 27.6 15.6 17.3 13.2 14.9 12.8 33.2 19.6 1 13.8 11.6 15.8 11.4 18.7 12.0 22.4 14.4 28.1 17.8 31.7 19.3 34.4 20.1 34.7 19.7 31.9 17.8 25.2 14.9 16.0 13.1 13.7 11.7 30.8 18.6 2 12.8 10.9 14.8 11.2 17.3 11.4 20.8 13.7 26.4 17.1 30.0 18.7 32.9 19.8 32.9 19.2 30.2 17.4 23.3 14.9 15.1 12.4 12.8 11.0 28.6 17.9 PENNSYLVANIA Allentown 725170 0.4 13.9 11.2 16.3 11.0 23.4 14.8 28.9 17.2 31.4 20.9 33.4 22.7 34.9 23.9 33.5 23.3 32.3 22.8 26.2 18.6 21.9 16.9 16.6 14.3 32.4 22.6 1 11.6 9.1 13.6 9.9 21.0 13.6 26.4 16.7 30.0 19.7 32.3 22.1 33.8 23.3 32.3 22.7 30.9 22.5 24.7 18.1 20.1 15.7 14.3 11.9 30.9 22.1 2 9.3 6.7 11.6 8.7 18.7 12.6 24.1 15.3 28.7 19.1 31.2 21.7 32.8 23.3 31.4 22.7 29.6 21.5 23.4 17.7 18.5 14.9 12.0 9.8 29.6 21.4 Bradford 725266 0.4 10.5 9.3 11.8 7.9 20.0 13.1 25.6 15.0 27.8 18.3 28.9 20.2 30.7 21.6 29.6 21.2 27.2 21.4 23.6 16.1 18.4 13.7 13.5 11.8 28.2 20.4 1 8.6 6.4 10.0 7.1 17.7 11.7 23.8 14.3 26.8 17.8 27.9 19.8 29.5 21.4 28.3 20.9 26.2 20.6 22.2 15.3 16.9 13.0 12.1 10.3 26.9 19.8 2 6.8 4.7 8.6 5.8 15.7 10.9 21.9 13.8 25.6 16.9 27.2 19.7 28.6 21.0 27.3 20.5 25.1 19.6 21.0 14.6 15.6 12.3 10.5 8.7 25.6 18.8 Erie 725260 0.4 13.7 10.9 14.3 10.7 22.6 13.9 25.4 16.6 28.5 19.9 30.8 21.4 31.6 22.8 30.9 23.3 29.8 22.3 25.7 17.6 20.8 15.0 17.3 13.9 29.6 22.0 1 11.8 9.5 12.6 9.2 20.1 13.0 24.0 16.1 27.4 19.4 29.8 21.2 30.3 22.8 29.9 22.7 28.3 22.3 23.9 17.2 19.1 14.0 15.2 12.4 28.2 21.3 2 9.8 6.9 10.8 7.9 18.1 11.8 22.6 15.4 26.3 19.1 28.8 20.8 29.5 22.3 28.8 21.9 27.1 21.1 22.5 16.1 17.6 13.4 13.5 10.8 26.9 20.8 Harrisburg 725115 0.4 15.3 11.3 16.8 10.4 24.2 15.3 29.3 17.8 31.7 22.3 34.2 22.8 36.2 23.6 34.4 23.8 32.9 23.5 26.8 19.6 22.3 16.4 17.3 14.2 33.3 23.5 1 12.1 9.0 14.2 9.6 21.9 13.2 27.5 17.6 30.4 20.2 32.9 22.6 34.8 23.8 33.3 24.3 31.6 22.8 25.3 18.5 20.8 16.4 15.1 11.8 31.7 22.8 2 9.8 6.4 12.1 8.5 19.6 13.0 25.2 16.3 29.1 19.9 31.8 22.5 33.7 23.6 32.2 23.7 29.9 22.6 23.9 17.7 19.3 15.7 12.4 9.1 30.2 22.2 Philadelphia, Intl Airport 724080 0.4 15.6 12.8 17.7 12.2 24.2 15.6 29.7 18.3 32.6 21.9 34.3 23.2 35.9 25.2 34.6 24.6 32.9 24.4 27.3 20.5 23.4 18.1 17.8 15.4 33.4 23.9 1 13.8 11.6 15.8 12.2 22.0 14.7 27.1 17.5 30.8 20.9 33.3 23.3 34.6 24.4 33.4 24.3 31.6 23.3 25.8 19.4 21.3 17.0 16.3 14.2 31.9 23.3 2 12.1 9.9 13.9 11.1 19.6 13.2 24.8 16.1 29.4 20.3 32.2 22.8 33.6 24.4 32.5 23.9 30.4 22.9 24.6 18.7 19.7 15.9 14.8 12.4 30.6 22.6 Pittsburgh, Intl Airport 725200 0.4 15.0 11.5 16.6 10.8 24.1 14.3 27.7 16.7 30.2 19.8 32.5 21.8 33.9 23.1 33.2 23.5 31.0 22.5 25.9 17.2 21.8 15.2 17.6 13.6 31.4 22.1 1 13.1 9.8 14.6 9.9 22.2 13.4 26.4 16.1 29.1 19.7 31.3 21.3 32.7 22.7 31.8 22.7 29.8 21.8 24.9 16.8 20.2 14.2 16.2 12.2 29.9 21.3 2 11.3 8.4 13.0 9.1 20.3 12.6 24.9 15.6 28.0 18.7 30.3 21.1 31.7 22.6 30.6 22.1 28.6 21.0 23.8 16.2 18.8 13.2 14.6 11.7 28.7 20.6 Wilkes-Barre/ Scranton 725130 0.4 13.4 11.1 14.4 10.7 22.3 13.8 28.1 16.6 30.3 19.7 31.8 21.6 33.7 22.3 31.9 22.8 30.2 22.2 25.1 17.4 20.5 15.8 15.7 12.6 31.0 21.7 1 11.0 8.8 12.4 9.9 19.7 12.4 25.8 15.6 29.0 18.8 30.8 21.4 32.4 22.6 30.9 22.2 29.0 21.6 23.8 17.0 18.8 15.0 13.5 11.5 29.5 21.2 2 8.7 6.4 10.8 8.0 17.8 12.1 23.8 14.7 27.8 18.2 29.8 20.9 31.3 22.3 29.9 22.1 27.9 21.1 22.4 16.3 17.5 14.0 11.5 9.3 28.1 20.4 Williamsport 725140 0.4 12.0 9.4 13.9 9.6 23.0 14.2 28.8 17.2 31.8 20.0 32.9 22.2 35.0 23.7 32.9 23.7 31.1 22.6 25.4 18.2 20.3 15.5 14.0 12.4 31.9 22.5 1 9.6 6.8 12.3 9.2 20.6 12.7 26.4 16.4 30.1 19.6 31.7 22.2 33.5 22.9 31.8 23.1 29.7 22.2 24.0 17.2 18.8 15.0 11.9 9.8 30.3 21.8 2 7.4 5.2 10.4 6.7 18.1 11.6 24.5 15.4 28.7 18.9 30.5 21.2 32.3 23.0 30.7 22.6 28.4 21.4 22.5 16.6 17.4 14.0 10.2 7.6 28.8 21.0 RHODE ISLAND Providence 725070 0.4 14.0 12.3 14.7 10.4 19.7 12.4 26.4 16.2 30.7 20.3 32.9 22.4 34.7 24.5 32.9 23.8 31.9 23.3 25.6 17.9 21.2 16.2 16.4 13.1 31.8 22.8 1 12.4 10.4 12.7 10.3 17.3 11.6 23.5 14.9 28.7 19.5 31.6 21.9 33.3 23.7 31.8 23.3 29.8 22.3 24.1 18.0 19.5 15.9 15.1 12.7 30.0 21.8 2 10.5 8.2 11.2 8.1 15.3 10.2 21.4 13.2 26.8 18.3 30.3 21.3 31.9 23.1 30.7 23.0 28.1 21.3 22.8 17.2 17.9 14.9 13.5 11.6 28.4 21.1 SOUTH CAROLINA Charleston 722080 0.4 23.9 18.3 25.6 18.3 28.4 18.4 31.3 20.1 33.3 22.4 35.4 25.0 36.2 25.9 35.1 26.1 33.5 25.0 30.6 22.5 27.4 20.9 24.9 19.7 34.3 25.6 1 22.7 17.2 24.2 17.8 27.1 18.5 30.2 19.6 32.3 22.4 34.4 24.9 35.3 26.2 34.3 25.9 32.7 24.9 29.5 21.8 26.3 20.4 23.9 19.1 33.1 25.1 2 21.3 16.9 22.8 16.5 25.9 18.2 29.0 19.7 31.2 22.3 33.4 24.7 34.5 26.1 33.7 25.8 31.9 24.6 28.6 21.6 25.6 19.6 22.8 18.3 32.0 24.7 Columbia 723100 0.4 23.7 17.2 25.7 17.1 29.1 18.1 31.9 19.2 33.9 21.8 36.6 23.5 38.2 23.9 36.7 24.5 34.6 23.9 30.9 21.7 27.4 20.4 24.8 18.7 35.6 24.2 1 21.8 15.8 24.1 16.4 27.8 17.4 31.1 18.7 33.0 21.2 35.6 23.4 37.1 23.9 35.7 24.6 33.7 23.8 29.8 20.8 26.3 18.9 23.6 18.1 34.3 23.8 2 20.3 15.5 22.6 14.5 26.5 16.9 30.1 18.7 32.2 21.2 34.7 23.3 36.0 24.2 34.9 24.3 32.9 23.7 28.9 20.3 25.2 18.0 22.2 17.8 33.1 23.6 Greer/Greenville 723120 0.4 20.4 13.9 22.5 14.4 26.7 16.3 29.9 18.4 32.4 20.8 34.9 23.2 36.3 23.4 34.9 23.7 32.7 22.8 28.7 19.4 25.1 17.5 21.1 15.7 33.9 23.3 1 18.4 13.2 21.1 13.7 25.3 15.8 28.9 17.7 31.2 20.8 33.8 22.9 35.4 23.5 33.9 23.3 31.7 23.0 27.8 19.3 23.6 16.6 19.9 15.7 32.6 23.1 2 17.1 12.4 19.4 12.1 23.9 15.5 27.9 17.4 30.2 20.6 32.9 22.7 34.4 23.7 33.1 23.6 30.8 22.7 26.8 18.6 22.4 16.0 18.6 14.8 31.3 22.8 SOUTH DAKOTA Huron 726540 0.4 9.4 4.4 13.6 6.7 22.6 12.4 29.2 15.8 30.7 19.6 35.7 22.4 38.0 23.3 37.5 22.2 34.9 20.5 27.8 15.7 18.5 10.3 11.0 6.1 35.1 22.3 1 7.2 3.4 11.1 5.7 19.4 10.4 26.8 15.3 29.3 19.4 34.0 22.7 36.9 22.9 35.9 22.1 32.8 20.4 25.8 15.3 16.3 9.1 8.6 4.4 33.0 21.8 2 5.4 2.4 8.6 4.3 16.3 9.1 24.4 14.3 28.1 19.0 32.7 21.6 35.4 22.9 34.4 21.9 31.1 20.1 23.8 14.6 14.4 8.9 6.6 3.3 31.1 21.2 Pierre 726686 0.4 11.7 5.3 15.9 7.8 22.5 11.8 29.7 14.8 32.4 18.4 37.4 21.4 40.7 22.1 39.5 21.5 36.4 19.1 29.6 14.7 19.8 9.7 12.5 5.8 37.1 21.1 1 9.3 4.3 13.7 6.5 20.0 9.9 27.6 14.2 30.8 17.9 35.5 21.2 38.9 22.1 37.9 20.8 34.6 19.3 27.2 14.4 17.6 9.0 10.3 4.7 34.8 20.8 2 7.4 3.1 11.2 5.4 17.3 8.8 25.3 13.4 29.2 17.6 33.8 20.7 37.6 21.8 36.6 20.8 32.6 18.6 25.2 13.7 15.4 7.9 8.4 3.7 32.7 20.2 Rapid City 726620 0.4 15.2 6.2 17.6 7.1 22.6 10.1 27.9 12.8 30.5 15.8 36.5 17.9 38.3 19.2 37.1 18.3 34.6 16.6 28.9 12.9 20.9 9.3 16.1 6.7 35.1 18.2 1 13.3 5.3 15.7 6.4 20.2 9.0 26.3 12.4 29.0 15.7 33.9 18.0 36.7 18.9 35.9 17.9 32.9 16.2 27.1 12.4 18.9 8.7 14.0 6.0 32.9 18.1 2 11.4 4.4 13.8 5.7 18.2 8.1 24.3 11.6 27.5 15.3 32.0 18.6 35.2 19.1 34.6 18.1 31.4 15.8 25.4 11.9 16.9 7.8 12.0 4.6 30.9 17.7 WMO# = World Meteorological Organization number DB = dry-bulb temperature, °C MWB = mean coincident wet-bulb temperature, °C 27.68 2001 ASHRAE Fundamentals Handbook (SI) Table 4A Design Wet-Bulb—Mean Coincident Dry-Bulb Temperature Location and WMO# % Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Annual WB MDB WB MDB WB MDB WB MDB WB MDB WB MDB WB MDB WB MDB WB MDB WB MDB WB MDB WB MDB WB MDB 1a 1b 2a 2b 3a 3b 4a 4b 5a 5b 6a 6b 7a 7b 8a 8b 9a 9b 10a 10b 11a 11b 12a 12c 13a 13b 14a 14b Sioux Falls 726510 0.4 4.4 8.2 7.4 12.9 14.2 20.4 17.8 25.5 21.4 28.1 24.7 31.6 25.8 32.9 25.5 32.6 23.7 30.5 19.1 24.4 13.3 15.9 7.1 11.4 24.6 31.4 1 3.2 6.6 6.1 10.1 12.5 18.3 16.9 24.1 20.6 27.4 24.1 30.8 25.3 32.4 24.9 31.6 22.8 29.6 17.8 22.4 11.9 14.8 4.7 8.2 23.6 30.5 2 2.2 4.8 4.7 8.3 10.9 16.4 15.8 22.7 19.9 26.4 23.4 30.2 24.8 31.8 24.3 30.7 21.9 28.0 16.7 21.4 10.5 13.9 3.6 6.3 22.6 29.1 TENNESSEE Bristol 723183 0.4 13.9 16.7 15.0 17.5 17.0 22.6 19.4 24.3 22.3 27.2 24.0 30.1 24.8 30.6 24.6 30.4 23.3 28.9 20.4 24.7 17.9 21.4 15.9 19.3 23.9 29.6 1 12.9 15.4 13.9 16.7 16.2 21.2 18.6 23.9 21.6 26.6 23.4 29.3 24.4 30.0 24.2 30.0 22.8 28.7 19.6 23.1 17.1 20.4 14.9 17.7 23.3 28.6 2 11.9 14.3 12.9 15.6 15.4 19.8 17.8 23.3 21.0 26.7 22.9 28.7 24.0 29.7 23.8 29.5 22.3 28.1 18.9 22.5 16.3 19.6 13.9 16.6 22.6 27.8 Chattanooga 723240 0.4 15.9 17.8 16.8 19.2 19.0 24.1 20.7 26.7 23.8 29.0 25.4 31.4 26.7 34.0 26.3 32.6 24.9 30.6 22.4 26.3 19.7 22.2 18.1 20.3 25.4 31.7 1 15.0 16.8 15.8 18.2 18.1 22.7 20.1 25.4 23.2 28.6 25.0 31.3 26.0 32.7 25.7 31.9 24.5 29.9 21.6 24.8 18.8 21.2 17.1 18.9 24.8 30.9 2 13.9 15.7 15.0 17.4 17.3 21.1 19.4 24.6 22.7 28.1 24.6 30.5 25.6 32.1 25.3 31.4 24.1 29.4 20.9 24.2 18.1 20.3 16.1 17.8 24.3 30.0 Knoxville 723260 0.4 15.4 18.3 16.2 19.0 18.7 22.8 20.7 25.4 23.5 28.3 25.2 30.6 26.1 32.6 25.5 31.5 24.4 30.1 21.8 26.3 19.2 22.3 17.9 20.6 24.9 31.0 1 14.5 17.2 15.3 18.1 17.8 22.1 19.9 24.7 22.9 27.6 24.6 30.2 25.6 31.7 25.1 31.2 24.0 29.6 20.9 24.6 18.4 21.2 16.6 18.7 24.3 30.2 2 13.3 15.3 14.4 17.3 16.9 20.8 19.2 24.3 22.3 27.1 24.2 29.7 25.2 31.3 24.7 30.7 23.6 28.9 20.2 23.7 17.6 20.3 15.4 18.1 23.8 29.2 Memphis 723340 0.4 17.7 19.7 18.4 21.2 20.6 24.2 22.4 27.1 24.9 30.2 26.8 32.6 27.7 34.3 27.4 33.4 26.3 32.1 23.8 27.8 20.8 23.3 19.6 21.4 26.7 33.1 1 16.8 18.7 17.3 20.1 19.9 23.2 21.8 26.4 24.4 29.8 26.3 32.3 27.3 34.2 26.9 33.1 25.8 31.6 22.9 26.6 20.1 22.9 18.5 20.3 26.2 32.5 2 15.8 18.3 16.3 19.1 19.1 22.5 21.3 25.4 23.9 29.2 25.8 31.9 26.9 33.7 26.6 32.8 25.3 30.7 22.1 26.2 19.5 22.1 17.4 19.4 25.6 31.6 Nashville 723270 0.4 16.8 18.9 17.2 19.7 19.4 24.1 21.4 25.4 24.1 28.9 25.7 32.1 26.4 32.7 26.1 32.2 25.1 30.9 22.5 26.2 19.9 23.2 18.2 19.8 25.6 31.9 1 15.6 17.3 16.3 18.9 18.6 22.5 20.6 25.6 23.4 28.7 25.2 31.5 26.1 32.4 25.7 31.9 24.6 30.1 21.7 25.8 19.1 21.6 17.1 19.1 25.0 31.1 2 14.5 16.1 15.4 17.6 17.8 21.6 19.9 24.3 22.9 28.2 24.8 30.9 25.7 32.2 25.4 31.6 24.2 29.7 21.0 24.9 18.4 20.7 16.2 18.0 24.4 30.2 TEXAS Abilene 722660 0.4 15.9 19.6 16.2 21.9 19.4 25.8 21.8 28.6 23.7 30.9 25.1 33.9 24.5 32.6 24.1 32.6 23.9 31.3 22.4 28.1 19.4 23.0 16.8 20.2 23.9 31.9 1 14.7 18.3 15.4 20.9 18.5 24.3 21.1 27.7 23.1 30.4 24.5 33.0 24.0 32.3 23.8 32.2 23.4 30.8 21.8 27.2 18.7 22.7 16.1 19.7 23.4 31.6 2 13.5 17.7 14.6 20.1 17.8 23.1 20.4 26.3 22.5 29.5 24.1 31.9 23.6 32.2 23.4 32.1 23.1 30.3 21.3 26.8 18.0 22.1 15.3 18.6 22.9 31.1 Amarillo 723630 0.4 10.2 18.4 11.6 21.3 14.4 20.8 16.9 24.8 19.8 26.6 21.9 30.7 22.4 31.3 22.3 30.1 21.2 28.2 18.3 25.8 14.7 19.3 11.4 16.6 21.6 30.2 1 9.1 16.5 10.4 19.3 13.3 21.5 16.2 23.4 19.3 26.5 21.4 30.2 21.9 30.8 21.8 30.0 20.7 28.4 17.6 24.2 13.4 18.6 10.1 16.3 20.9 29.8 2 7.9 16.2 9.5 18.2 12.3 21.2 15.5 22.7 18.7 25.8 21.1 29.9 21.6 30.6 21.4 29.7 20.2 28.1 16.9 23.7 12.6 17.6 9.0 16.1 20.5 29.4 Austin 722540 0.4 20.1 22.0 19.9 22.3 21.9 26.2 23.7 29.4 25.7 31.1 26.1 32.9 25.8 32.5 25.7 32.2 25.7 30.9 24.8 29.2 22.6 25.6 20.7 22.9 25.6 31.7 1 19.3 21.5 19.2 22.2 21.3 24.9 23.1 28.2 25.2 30.6 25.7 32.4 25.6 32.0 25.3 31.9 25.3 30.3 24.4 28.0 22.0 24.8 20.1 22.2 25.1 31.1 2 18.5 20.8 18.6 21.9 20.8 23.9 22.7 27.2 24.6 29.7 25.4 31.9 25.2 31.6 25.1 31.7 25.1 30.1 24.0 28.2 21.6 24.3 19.6 21.5 24.7 30.4 Beaumont/ Port Arthur 722410 0.4 21.3 23.1 21.4 23.2 22.6 25.4 24.7 27.4 26.2 29.6 27.8 32.3 28.0 32.7 27.9 32.7 27.4 31.7 26.1 29.9 24.2 26.4 23.1 24.2 27.4 32.0 1 20.7 22.4 20.8 22.4 22.1 24.7 24.2 26.8 25.7 29.3 27.4 31.7 27.7 32.4 27.6 32.4 27.1 31.4 25.7 29.1 23.7 26.1 22.3 23.8 26.9 31.5 2 20.2 21.4 20.3 21.8 21.7 24.0 23.8 26.4 25.3 28.9 26.9 31.2 27.4 32.0 27.3 32.1 26.8 31.0 25.2 28.4 23.2 25.3 21.7 23.1 26.6 31.2 Brownsville 722500 0.4 22.4 26.1 22.8 27.3 23.8 29.3 25.4 30.8 26.7 31.7 27.2 32.4 26.8 31.8 26.8 31.5 26.9 31.4 26.3 30.8 24.8 28.2 23.6 26.6 26.7 31.4 1 22.0 25.4 22.2 25.4 23.3 28.3 25.0 30.3 26.3 31.3 26.9 31.9 26.4 31.6 26.5 31.4 26.7 31.3 26.0 30.7 24.5 27.8 23.2 25.9 26.3 31.3 2 21.7 24.7 21.7 24.8 22.9 27.4 24.6 29.4 25.9 30.7 26.7 31.6 26.3 31.6 26.3 31.6 26.3 31.1 25.7 30.2 24.1 27.5 22.8 25.3 26.1 31.1 Corpus Christi 722510 0.4 22.4 25.6 22.3 26.2 24.1 28.4 25.5 30.1 26.9 31.3 27.6 32.6 27.2 33.3 27.3 33.0 27.4 31.8 26.7 30.7 25.1 27.7 23.2 25.7 27.1 31.9 1 21.9 24.3 21.7 24.6 23.3 27.3 25.1 29.2 26.5 30.6 27.3 32.0 26.9 32.7 26.9 32.4 27.1 31.4 26.3 30.1 24.6 27.3 22.6 25.3 26.7 31.5 2 21.3 23.4 21.2 23.6 22.8 26.1 24.6 28.4 26.2 30.1 26.9 31.4 26.7 32.4 26.8 32.1 26.8 31.1 25.9 29.6 24.1 27.0 22.2 24.4 26.3 31.0 El Paso 722700 0.4 10.9 16.1 12.1 19.7 13.0 24.5 16.1 28.9 18.1 30.0 20.7 30.3 21.8 29.4 21.8 30.1 21.0 28.1 18.4 25.5 14.5 20.9 11.8 17.6 21.1 29.2 1 10.1 16.2 11.3 20.2 12.4 24.7 15.1 26.3 17.4 28.7 20.2 30.6 21.4 29.6 21.4 29.4 20.6 27.8 17.7 24.9 13.6 19.8 10.8 15.9 20.6 29.1 2 9.3 15.7 10.6 19.3 11.8 24.3 14.3 25.9 16.8 28.2 19.8 30.3 21.1 29.6 21.1 29.1 20.2 27.6 17.1 24.1 12.8 19.1 10.1 15.8 20.1 29.1 Fort Worth, Meacham Field 722596 0.4 18.0 20.4 18.4 22.2 21.0 25.5 23.2 27.4 25.3 31.3 26.3 33.8 26.3 33.1 25.8 33.9 25.7 31.9 24.0 29.3 21.3 24.3 19.4 21.9 25.7 32.8 1 17.3 19.7 17.7 21.4 20.3 24.8 22.6 27.2 24.6 30.3 25.9 33.2 25.8 32.7 25.6 33.8 25.3 31.2 23.5 27.8 20.8 24.1 18.7 21.3 25.2 32.3 2 16.4 18.8 16.9 20.8 19.8 23.7 22.1 26.7 24.1 29.7 25.5 32.9 25.6 32.6 25.2 33.3 25.0 30.8 23.0 27.2 20.2 23.4 18.0 20.4 24.7 31.9 Houston, Intercontinental AP 722430 0.4 21.3 23.7 21.6 23.8 22.6 25.5 24.6 28.4 25.7 30.5 26.9 32.6 26.9 32.8 26.9 32.6 26.8 31.9 25.7 29.9 23.9 26.8 22.3 24.6 26.6 32.0 1 20.8 23.2 20.9 23.0 22.1 25.0 24.0 28.3 25.4 30.2 26.7 32.4 26.7 32.4 26.7 32.1 26.5 31.3 25.2 29.2 23.4 26.1 21.8 23.6 26.2 31.6 2 20.3 22.6 20.3 22.4 21.7 24.4 23.5 27.3 25.1 29.9 26.3 31.9 26.4 32.1 26.3 31.9 26.2 30.7 24.8 28.5 22.9 25.5 21.2 23.3 25.8 31.1 Lubbock, Intl Airport 722670 0.4 12.1 16.7 13.2 21.5 15.8 20.5 18.2 25.1 21.1 28.7 23.2 31.2 23.4 31.1 23.0 31.5 22.5 29.4 19.8 25.1 16.3 20.2 13.4 17.2 22.5 30.6 1 11.0 17.3 12.3 20.8 14.9 20.6 17.6 24.2 20.5 27.6 22.6 30.7 22.9 30.8 22.7 31.1 21.9 29.2 19.1 24.6 15.3 18.8 12.0 16.5 21.9 30.1 2 9.7 17.7 11.3 18.8 14.1 20.4 17.1 23.4 19.9 26.8 22.2 30.0 22.5 30.6 22.3 30.8 21.5 28.7 18.4 24.1 14.5 18.0 10.9 16.7 21.4 29.7 Lufkin 722446 0.4 20.6 23.4 20.6 22.1 22.2 25.9 24.3 28.4 25.6 30.9 26.8 32.6 26.9 32.8 26.8 32.4 26.8 32.1 25.3 29.4 22.7 25.6 21.9 23.8 26.3 32.2 1 19.9 22.4 19.9 21.9 21.7 25.2 23.6 27.9 25.1 30.1 26.4 32.1 26.6 32.7 26.4 32.3 26.3 31.4 24.7 28.6 22.1 25.2 21.1 23.1 26.0 31.9 2 19.3 21.6 19.2 21.9 21.1 24.2 23.2 27.4 24.7 29.4 26.2 31.8 26.3 32.4 26.2 32.3 25.9 31.1 24.1 28.0 21.7 24.5 20.3 22.5 25.6 31.6 Midland/ Odessa 722650 0.4 13.7 17.5 14.7 21.7 17.1 22.4 19.5 25.1 21.5 29.6 23.4 30.4 23.3 30.9 23.0 30.0 22.4 29.6 20.9 26.4 17.9 21.3 14.6 17.9 22.6 30.3 1 12.6 16.3 13.7 20.1 16.3 22.4 18.9 24.8 21.1 28.8 22.9 30.2 22.9 31.3 22.6 30.1 21.9 29.4 20.3 26.2 16.9 20.8 13.7 17.6 22.0 30.2 2 11.6 17.0 12.8 19.6 15.6 22.6 18.4 24.7 20.5 27.6 22.4 30.0 22.4 30.8 22.3 30.3 21.7 29.2 19.7 25.1 16.1 19.9 12.7 16.3 21.5 30.0 San Angelo 722630 0.4 16.4 20.1 16.4 21.6 19.2 25.5 21.4 28.8 23.4 31.2 24.6 33.1 24.1 32.6 24.0 32.9 23.9 31.1 22.7 28.9 19.9 23.2 17.1 20.7 23.7 32.0 1 15.6 18.9 15.7 20.9 18.6 24.0 20.8 27.8 22.9 30.7 24.1 32.7 23.8 32.4 23.6 32.4 23.4 30.9 22.1 27.7 19.2 23.6 16.4 20.1 23.2 31.4 2 14.4 18.0 15.1 19.9 17.9 23.8 20.3 26.9 22.4 29.8 23.7 32.0 23.4 32.0 23.3 32.1 23.1 30.3 21.6 27.1 18.5 22.9 15.8 19.2 22.8 30.9 San Antonio, Intl Airport 722530 0.4 19.7 21.9 19.9 22.3 21.7 26.2 23.7 29.0 25.6 30.8 26.4 32.0 26.1 31.3 25.6 31.4 25.7 30.3 24.7 28.7 22.5 25.3 20.8 22.9 25.6 30.7 1 19.1 21.6 19.3 22.7 21.2 25.1 23.4 28.5 25.0 30.2 25.9 31.4 25.7 30.6 25.2 30.9 25.2 30.0 24.3 28.3 21.9 24.8 20.2 22.3 25.1 30.4 2 18.6 20.7 18.7 22.1 20.8 24.1 22.9 27.4 24.6 29.6 25.6 31.0 25.3 30.5 25.0 30.6 25.0 29.9 23.9 28.1 21.6 24.4 19.7 21.8 24.7 29.9 Victoria 722550 0.4 21.3 23.9 21.4 23.6 22.7 26.4 24.6 27.8 26.2 30.4 26.9 32.3 26.8 32.1 26.7 31.3 26.8 30.5 25.8 29.8 24.4 26.8 22.7 24.2 26.4 31.2 1 20.8 23.1 20.8 22.8 22.2 25.3 24.1 27.8 25.7 30.3 26.7 31.6 26.4 31.6 26.4 31.4 26.4 30.4 25.4 29.4 23.8 26.8 22.0 23.9 26.2 31.1 2 20.4 22.6 20.3 22.4 21.8 24.7 23.6 27.2 25.3 29.7 26.3 31.1 26.3 31.6 26.2 31.4 26.2 30.4 25.1 28.8 23.4 25.9 21.6 23.4 25.8 30.7 Waco 722560 0.4 18.7 21.3 19.1 23.1 21.4 26.5 23.9 28.4 25.5 32.3 26.7 34.6 26.3 34.9 26.2 34.4 26.2 32.8 24.7 30.1 22.0 26.0 19.8 22.7 25.8 33.7 1 18.0 20.6 18.4 22.6 20.9 25.8 23.3 28.0 25.1 31.4 26.2 33.9 25.8 34.4 25.8 34.3 25.7 32.4 24.2 29.1 21.5 24.9 19.1 21.4 25.4 33.2 2 17.2 19.6 17.7 21.3 20.2 24.2 22.7 27.3 24.6 30.7 25.8 33.4 25.6 34.2 25.6 34.3 25.3 32.0 23.7 28.4 20.9 24.0 18.5 20.8 25.0 32.6 Wichita Falls, Sheppard AFB 723510 0.4 15.9 18.7 17.1 23.2 19.9 25.6 22.5 28.9 24.7 32.2 26.1 34.8 25.8 34.8 25.5 34.5 25.1 33.1 23.6 29.5 19.8 23.8 17.2 20.6 25.1 34.0 1 14.4 17.2 15.9 21.4 19.2 25.2 21.9 27.9 24.0 31.5 25.6 34.4 25.3 34.6 25.1 34.3 24.6 32.3 22.8 27.7 19.1 22.9 16.4 19.4 24.6 33.4 2 13.0 17.4 14.9 20.2 18.3 24.1 21.2 27.2 23.4 30.4 25.1 33.6 24.9 34.4 24.7 34.0 24.3 31.8 22.1 27.4 18.4 22.7 15.4 18.3 24.1 32.9 UTAH Cedar City 724755 0.4 7.1 12.2 8.1 14.8 9.1 17.9 11.1 22.7 13.8 24.8 16.6 29.6 18.9 26.3 18.6 27.2 16.4 26.1 12.8 20.7 9.4 17.3 7.0 13.1 17.6 26.5 1 5.7 11.4 7.3 13.2 8.4 17.2 10.3 21.6 13.3 24.1 15.8 28.8 18.3 26.6 18.0 26.7 15.9 25.6 12.1 20.7 8.7 16.2 5.9 11.8 16.8 26.4 2 4.8 9.7 6.4 13.1 7.7 16.6 9.7 20.8 12.7 24.0 15.1 30.0 17.8 26.4 17.5 26.6 15.4 25.0 11.5 21.1 7.9 15.4 5.1 10.3 16.2 26.3 Salt Lake City 725720 0.4 6.7 10.9 8.5 14.4 10.6 19.4 13.3 24.6 16.5 26.7 18.7 31.6 20.1 30.4 20.3 29.1 17.8 28.3 14.0 24.1 10.7 16.4 6.8 12.0 18.9 29.6 1 5.6 9.3 7.8 12.9 9.7 17.9 12.4 23.3 15.7 25.8 17.9 31.0 19.5 30.5 19.6 28.6 17.2 26.6 13.4 23.7 9.7 15.5 6.1 10.5 18.2 29.6 2 4.7 7.9 7.1 12.3 9.1 16.8 11.9 22.2 15.0 25.3 17.2 30.3 19.0 30.2 19.0 28.8 16.7 27.3 12.8 22.4 9.0 15.1 5.2 9.3 17.5 29.2 WMO# = World Meteorological Organization number WB = wet-bulb temperature, °C MDB = mean coincident dry-bulb temperature, °C Climatic Design Information 27.69 Table 4B Design Dry-Bulb—Mean Coincident Wet-Bulb Temperature Location and WMO# % Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Annual DB MWB DB MWB DB MWB DB MWB DB MWB DB MWB DB MWB DB MWB DB MWB DB MWB DB MWB DB MWB DB MWB 1a 1b 2a 2b 3a 3b 4a 4b 5a 5b 6a 6b 7a 7b 8a 8b 9a 9b 10a 10b 11a 11b 12a 12c 13a 13b 14a 14b Sioux Falls 726510 0.4 9.2 4.0 13.4 7.0 22.9 12.9 29.6 16.1 31.1 19.6 35.3 22.4 37.4 23.6 35.9 22.8 33.1 21.7 27.5 16.3 18.2 10.7 11.3 7.4 34.4 22.8 1 6.8 3.1 10.7 5.5 20.1 11.3 27.3 15.8 29.7 19.2 33.9 22.2 36.1 23.5 34.7 22.8 31.6 21.0 25.6 15.9 16.6 10.1 8.6 4.6 32.4 22.2 2 5.0 1.9 8.7 4.5 17.3 10.2 24.7 14.9 28.2 18.6 32.6 21.7 34.9 23.3 33.6 22.9 30.1 20.7 23.8 15.1 14.9 9.5 6.4 3.3 30.7 21.4 TENNESSEE Bristol 723183 0.4 18.0 12.7 20.3 14.2 24.8 14.9 28.0 17.4 30.1 20.6 32.4 22.2 33.7 22.4 33.5 22.8 31.4 21.7 27.3 17.7 23.8 16.4 20.7 15.5 31.7 22.3 1 16.3 11.5 18.7 12.1 23.3 14.8 26.9 16.7 29.1 19.9 31.4 21.8 32.7 22.7 32.2 23.1 30.6 21.3 26.3 17.8 22.3 15.4 18.6 14.2 30.4 22.0 2 14.9 11.1 17.1 11.2 22.1 14.0 25.9 16.2 28.2 19.7 30.7 21.4 31.8 22.7 31.1 22.7 29.8 21.4 25.4 17.3 20.9 14.7 17.1 13.1 29.3 21.6 Chattanooga 723240 0.4 19.1 14.5 22.4 14.7 26.8 17.2 30.1 18.6 32.4 21.9 34.8 23.3 37.1 24.9 35.9 24.6 33.9 23.5 29.2 20.2 24.7 17.0 21.3 17.4 34.5 23.9 1 17.7 13.5 20.8 14.1 25.4 16.2 29.0 17.9 31.3 21.8 33.9 23.1 35.9 24.6 34.9 24.2 32.8 23.1 28.0 19.4 23.3 16.8 19.8 15.9 33.1 23.7 2 16.5 12.5 19.2 12.9 24.0 15.5 28.1 17.5 30.3 21.2 33.1 23.3 34.9 24.3 34.0 24.1 31.8 22.7 27.1 19.1 22.1 16.3 18.5 15.6 31.9 23.3 Knoxville 723260 0.4 19.1 14.7 21.7 15.7 25.6 16.3 28.9 18.0 31.2 20.8 33.5 22.2 35.4 24.1 34.7 23.8 32.8 22.6 28.0 20.2 24.4 17.5 21.5 16.7 33.3 23.4 1 17.7 13.5 20.1 13.3 24.3 16.0 27.9 17.9 30.2 20.9 32.5 22.7 34.6 24.1 33.6 23.6 31.8 22.5 27.1 19.4 22.9 16.7 19.9 16.2 31.9 23.2 2 16.1 12.3 18.7 12.8 23.1 15.4 26.9 17.7 29.4 20.7 31.8 22.6 33.7 24.1 32.8 23.8 30.9 22.4 26.2 18.8 21.7 16.3 18.1 14.6 30.7 22.8 Memphis 723340 0.4 20.8 16.3 23.6 16.2 27.0 17.7 30.1 19.7 32.8 23.0 35.7 24.5 37.8 26.2 36.6 25.4 35.4 24.3 30.6 21.3 26.1 18.0 22.3 18.5 35.4 25.3 1 19.6 15.7 22.0 15.4 25.9 17.4 29.0 19.3 31.9 22.5 34.8 24.6 36.7 26.0 35.6 25.5 33.8 24.5 29.6 20.7 24.8 18.4 21.1 17.7 34.2 25.1 2 18.4 14.9 20.7 14.8 24.7 17.2 28.1 19.3 31.2 22.4 34.1 24.4 35.7 25.8 34.8 25.4 32.9 24.3 28.7 20.1 23.6 17.8 19.8 16.7 33.2 24.7 Nashville 723270 0.4 19.6 14.9 22.2 13.8 26.8 17.2 29.7 18.6 31.9 22.2 35.2 23.5 36.6 24.6 36.3 24.3 34.2 22.9 29.3 20.1 25.3 17.8 21.2 16.9 34.6 24.2 1 18.3 14.7 20.8 14.4 25.4 16.7 28.7 18.6 31.1 21.9 34.1 23.6 35.7 24.5 35.2 24.2 33.1 23.3 28.5 19.8 23.8 17.2 20.0 15.8 33.2 23.8 2 17.1 13.4 19.6 13.5 24.0 16.2 27.8 18.3 30.2 21.4 33.3 23.1 34.7 24.6 34.2 24.1 32.2 23.0 27.6 19.3 22.6 16.1 18.9 15.4 32.0 23.4 TEXAS Abilene 722660 0.4 24.3 12.9 27.2 13.3 30.7 15.8 33.4 17.1 36.1 19.7 37.9 21.4 38.9 21.4 38.2 21.6 36.4 20.9 32.9 17.8 27.8 15.8 23.7 12.8 37.2 21.4 1 22.6 11.8 25.4 12.6 29.1 15.7 32.2 17.5 34.8 19.5 36.7 21.8 38.0 21.4 37.4 21.8 35.3 21.1 31.7 18.8 26.3 16.1 22.4 13.4 36.0 21.5 2 21.1 11.4 23.7 12.2 27.6 15.2 30.8 17.2 33.6 19.7 35.7 21.9 37.3 21.4 36.8 21.6 34.4 21.1 30.6 19.2 25.1 15.7 21.1 13.2 34.8 21.4 Amarillo 723630 0.4 21.2 9.1 24.6 10.3 27.8 11.4 31.4 13.7 34.5 15.4 38.0 18.2 37.3 19.3 36.4 19.6 34.6 18.1 31.4 15.2 25.2 11.6 21.2 9.3 35.7 19.2 1 19.2 8.1 22.4 9.3 26.4 11.3 29.8 13.7 33.4 14.9 36.5 18.6 36.4 19.3 35.6 19.7 33.5 18.1 30.1 14.7 23.8 11.4 19.6 8.8 34.3 19.0 2 17.4 7.1 20.4 8.6 24.7 10.8 28.6 12.4 32.2 15.3 35.3 18.6 35.7 19.6 34.7 19.6 32.6 18.1 28.9 14.7 22.5 10.9 18.0 7.8 33.1 18.8 Austin 722540 0.4 25.4 15.7 28.2 15.9 30.6 17.4 32.2 19.4 33.9 23.1 36.6 24.2 37.9 22.9 38.4 23.3 36.6 23.2 33.2 22.4 28.9 20.1 25.7 18.1 36.8 23.4 1 24.1 16.4 26.3 16.8 28.9 18.4 30.9 20.2 33.0 22.9 35.6 24.0 37.3 23.2 37.7 23.3 35.7 23.1 32.3 21.8 27.7 19.6 24.4 17.6 35.8 23.4 2 22.9 16.1 24.8 16.1 27.5 17.8 29.9 20.5 32.2 23.1 34.7 23.8 36.7 23.4 37.0 23.3 34.8 23.3 31.4 21.7 26.8 19.4 23.4 17.7 34.7 23.4 Beaumont/ Port Arthur 722410 0.4 23.9 20.2 25.2 18.3 27.1 19.9 30.0 20.7 32.2 24.1 34.6 25.9 35.6 26.3 36.1 26.1 34.6 25.2 31.8 24.3 28.3 21.8 25.4 21.8 34.4 26.0 1 23.1 19.3 24.1 18.8 26.2 19.8 29.0 21.8 31.5 24.1 33.9 25.7 34.9 26.4 35.1 26.1 33.9 25.6 31.1 24.2 27.4 22.4 24.6 21.3 33.6 25.9 2 22.3 19.1 23.2 18.8 25.4 19.5 28.1 21.8 30.8 24.1 33.3 25.7 34.4 26.2 34.4 26.2 33.2 25.7 30.2 23.6 26.7 21.9 23.8 21.1 32.7 25.8 Brownsville 722500 0.4 27.8 21.4 29.1 19.3 31.6 21.3 32.9 24.1 33.6 25.3 35.3 25.9 35.9 25.2 35.8 25.1 35.1 25.2 33.2 25.3 30.7 23.3 28.8 21.7 35.1 25.3 1 26.9 21.2 27.9 20.7 30.3 21.8 31.9 23.7 33.0 25.2 34.7 25.7 35.4 25.1 35.4 25.2 34.6 25.2 32.6 25.1 29.9 23.3 27.9 21.9 34.3 25.2 2 26.1 20.7 26.9 20.4 29.1 21.6 31.1 23.5 32.5 25.2 34.1 25.6 34.9 25.2 35.0 25.2 34.0 25.2 31.9 24.7 29.3 23.1 27.2 21.6 33.6 25.2 Corpus Christi 722510 0.4 27.1 20.4 28.9 19.1 31.1 20.1 32.2 21.7 33.0 25.5 34.7 26.1 35.7 25.7 36.0 25.3 35.1 25.7 33.1 24.8 30.1 23.1 28.2 20.7 34.9 25.5 1 25.9 20.5 27.2 18.8 29.4 19.4 31.1 22.6 32.3 25.3 34.1 25.9 35.3 25.6 35.4 25.4 34.4 25.4 32.2 25.1 29.2 23.1 27.1 21.3 34.2 25.4 2 24.8 20.2 26.0 19.5 28.2 20.1 30.1 23.1 31.6 25.1 33.6 25.8 34.8 25.5 35.0 25.4 33.8 25.6 31.6 24.7 28.6 22.7 26.2 20.9 33.4 25.4 El Paso 722700 0.4 21.5 9.0 25.0 10.6 28.2 11.4 32.4 14.1 35.8 15.9 40.9 17.9 39.6 18.8 37.6 18.6 35.8 17.7 32.1 15.3 25.8 12.3 21.3 9.4 38.1 17.9 1 20.0 8.5 23.6 9.9 27.1 11.5 31.2 13.4 34.9 15.2 39.7 17.7 38.6 18.7 36.8 18.7 35.1 17.9 31.1 15.2 24.7 11.7 20.4 8.9 36.6 17.8 2 18.8 8.0 22.4 9.4 26.1 11.1 30.2 12.9 34.1 14.9 38.8 17.7 37.7 18.6 36.1 18.7 34.2 17.7 30.0 15.2 23.7 11.2 19.2 8.4 35.3 17.7 Fort Worth, Meacham Field 722596 0.4 23.7 14.3 26.3 15.2 29.2 18.2 31.7 20.2 33.8 23.8 37.8 24.6 39.8 23.3 39.2 23.4 36.8 23.6 33.7 20.3 28.2 18.2 24.3 17.0 37.8 23.6 1 22.1 14.2 24.8 15.0 27.7 17.8 30.1 19.9 32.8 23.1 36.3 24.1 38.7 23.4 38.4 23.4 35.9 23.2 32.2 20.8 26.9 18.3 22.9 16.5 36.6 23.6 2 20.7 13.8 23.4 14.5 26.4 17.1 29.0 19.7 31.8 22.3 35.5 23.9 37.8 23.5 37.8 23.6 35.0 23.3 31.1 20.8 25.7 17.7 21.5 16.6 35.3 23.6 Houston, Intercontinental AP 722430 0.4 25.7 20.2 27.2 18.2 28.7 19.5 31.1 22.3 33.0 24.0 35.7 25.3 36.9 25.1 37.2 24.6 35.2 24.6 32.5 23.6 29.3 21.5 26.2 20.6 35.5 24.9 1 24.6 19.2 25.8 18.7 27.7 20.2 30.0 21.8 32.3 24.1 34.9 25.2 36.1 24.8 36.4 24.8 34.4 24.7 31.7 23.3 28.4 21.8 25.4 20.4 34.4 24.9 2 23.6 18.6 24.6 17.9 26.8 19.1 29.1 21.9 31.7 23.8 34.2 24.9 35.4 24.9 35.7 24.8 33.7 24.6 30.9 22.8 27.6 21.5 24.6 20.1 33.5 24.9 Lubbock, Intl Airport 722670 0.4 22.7 9.7 26.1 11.4 29.1 12.2 32.4 14.6 36.1 16.8 38.7 19.4 37.8 20.1 36.6 20.3 34.7 19.3 31.7 16.1 26.4 12.4 22.4 10.2 36.3 19.6 1 20.9 8.9 24.1 10.4 27.8 12.2 31.0 14.5 34.8 16.6 37.5 18.7 36.8 20.2 35.8 20.7 33.8 19.2 30.5 15.7 25.1 12.0 20.9 9.5 34.9 19.7 2 19.3 8.6 22.5 10.1 26.3 11.7 29.9 13.5 33.6 16.3 36.4 18.8 36.0 20.4 35.0 20.6 33.1 19.1 29.4 15.7 23.8 11.8 19.4 8.9 33.7 19.6 Lufkin 722446 0.4 25.1 18.3 25.9 17.1 29.4 19.6 30.6 22.4 32.7 23.4 35.3 24.9 37.2 24.9 38.1 24.5 35.7 24.2 32.8 22.6 28.3 20.2 25.2 19.9 35.8 24.6 1 23.8 18.8 24.7 17.2 27.9 19.3 29.8 22.1 32.1 23.3 34.6 25.1 36.5 24.9 36.9 24.8 34.9 24.6 31.8 22.4 27.1 20.4 24.3 19.7 34.7 24.7 2 22.6 17.8 23.7 16.5 26.7 18.7 29.1 21.4 31.5 23.3 34.0 24.8 35.7 24.8 36.1 24.6 34.2 24.4 30.9 22.2 26.2 19.9 23.5 19.5 33.7 24.6 Midland/Odessa 722650 0.4 23.9 10.7 27.4 12.3 30.6 13.0 33.8 15.0 37.3 17.5 39.1 18.5 38.8 20.1 38.4 20.3 36.6 19.7 32.8 16.8 27.9 13.2 24.2 11.1 37.4 19.6 1 22.3 9.8 25.6 11.2 29.1 12.4 32.7 14.2 36.1 16.2 38.0 19.0 37.8 20.2 37.6 20.2 35.2 19.7 31.6 16.9 26.6 13.7 22.8 10.7 36.1 19.4 2 20.9 9.4 24.1 10.8 27.9 12.2 31.4 14.4 35.0 15.8 37.1 19.2 36.9 20.1 36.7 20.1 34.2 19.4 30.6 16.7 25.2 13.1 21.4 10.3 34.8 19.5 San Angelo 722630 0.4 25.3 11.7 27.9 13.7 31.4 15.6 34.3 17.1 37.4 19.4 38.4 21.7 38.9 21.1 39.1 21.3 36.7 21.2 33.1 20.0 28.5 15.3 24.8 13.1 37.5 21.2 1 23.7 12.1 26.3 13.0 29.7 15.4 33.2 16.8 35.9 19.6 37.2 21.7 38.1 21.3 38.0 21.4 35.7 21.2 31.8 18.7 27.0 16.3 23.3 13.1 36.3 21.2 2 22.1 11.8 24.7 12.8 28.4 15.4 32.1 16.9 34.6 19.6 36.2 21.4 37.5 21.3 37.2 21.3 34.7 21.2 30.7 19.1 25.9 16.3 22.1 12.9 35.1 21.2 San Antonio, Intl Airport 722530 0.4 25.8 16.0 28.3 16.4 31.1 18.1 32.9 20.1 35.2 21.7 36.8 23.6 37.5 22.5 37.9 23.2 36.0 22.6 33.3 22.3 29.1 19.5 25.7 18.1 36.4 23.0 1 24.2 15.9 26.9 15.9 29.4 17.7 31.4 20.3 33.8 22.7 35.9 23.7 36.8 22.8 37.1 22.9 35.2 22.8 32.3 21.7 27.9 19.7 24.7 17.5 35.4 23.0 2 23.1 15.7 25.4 15.7 28.1 17.7 30.4 19.8 32.7 22.6 35.1 23.4 36.2 23.0 36.4 23.0 34.6 23.1 31.4 21.6 26.9 19.4 23.7 17.3 34.4 23.1 Victoria 722550 0.4 25.8 19.3 27.3 17.4 29.4 19.5 31.1 19.6 33.0 24.4 35.2 25.2 36.2 24.3 36.7 24.3 35.3 24.1 32.7 23.4 29.5 22.2 26.6 20.7 35.2 24.6 1 24.8 19.2 25.8 17.3 28.1 18.9 30.2 21.6 32.1 24.4 34.6 25.1 35.6 24.4 35.9 24.4 34.6 24.4 31.8 23.7 28.6 22.2 25.7 20.4 34.4 24.7 2 23.8 18.9 24.7 17.8 27.2 18.8 29.4 21.6 31.4 23.9 33.9 25.1 35.1 24.7 35.3 24.6 33.8 24.5 31.1 23.6 27.7 21.7 25.0 20.2 33.6 24.8 Waco 722560 0.4 24.6 15.8 26.9 16.3 29.8 18.3 32.3 20.1 34.0 24.0 37.6 24.7 39.4 23.9 40.0 24.1 37.3 23.8 33.9 20.7 29.0 19.4 25.0 17.0 38.2 23.9 1 23.1 15.1 25.3 16.1 28.3 18.4 30.8 20.8 33.2 23.6 36.7 24.8 38.8 23.8 39.0 23.9 36.3 23.8 32.9 21.1 27.7 19.3 23.6 17.0 37.1 23.9 2 21.6 15.1 23.9 15.3 27.0 17.9 29.7 20.2 32.4 23.0 35.8 24.4 38.2 23.8 38.3 24.0 35.6 23.7 31.7 21.6 26.6 19.0 22.3 16.6 35.9 23.8 Wichita Falls, Sheppard AFB 723510 0.4 23.1 12.6 26.9 14.6 31.1 16.6 33.1 19.4 35.9 21.5 39.6 23.2 41.5 22.7 40.7 22.8 37.8 22.7 33.7 19.3 27.6 17.1 23.2 14.5 39.2 23.1 1 21.4 12.3 24.9 13.7 29.3 16.3 31.7 19.2 34.6 21.9 37.9 23.3 40.6 22.9 39.7 23.0 36.8 22.7 32.2 20.3 26.2 16.9 21.7 14.0 37.7 23.0 2 19.7 11.3 23.1 12.8 27.3 16.8 30.3 18.7 33.3 21.7 36.9 23.4 39.7 23.1 38.9 23.1 35.8 22.6 31.1 20.0 24.9 16.6 20.3 13.1 36.4 23.0 UTAH Cedar City 724755 0.4 14.4 5.8 17.6 7.1 20.8 8.0 25.0 10.1 29.5 12.6 35.1 14.8 35.8 15.4 34.7 15.2 32.1 14.3 27.4 11.1 20.1 8.4 15.0 5.7 33.9 14.9 1 12.4 5.3 15.9 6.1 19.4 7.4 23.8 9.6 28.1 11.7 34.1 14.7 35.0 15.4 33.9 15.2 31.2 14.3 26.4 10.8 18.9 7.6 13.5 5.0 32.6 14.7 2 10.6 4.1 14.5 5.4 18.2 7.1 22.6 9.2 27.0 11.4 33.0 14.2 34.2 15.0 33.1 15.1 30.2 13.8 25.3 10.2 17.6 7.2 12.1 4.3 31.2 14.3 Salt Lake City 725720 0.4 11.6 5.7 15.9 7.6 20.9 9.6 26.9 12.2 30.7 13.7 36.8 16.3 37.6 17.2 36.7 16.9 33.7 15.7 27.8 12.9 19.4 9.6 13.4 6.2 35.8 16.7 1 10.2 4.9 14.5 6.7 19.6 9.1 25.4 11.8 29.6 13.6 35.8 16.2 36.9 17.1 35.8 16.6 32.5 15.4 26.4 12.3 17.8 8.8 11.8 5.4 34.6 16.5 2 8.6 4.3 13.1 6.5 18.2 8.4 24.0 11.1 28.7 13.6 34.7 16.1 36.2 17.0 35.2 16.7 31.5 15.1 25.0 11.9 16.4 8.2 10.0 4.7 33.2 16.2 WMO# = World Meteorological Organization number DB = dry-bulb temperature, °C MWB = mean coincident wet-bulb temperature, °C 27.70 2001 ASHRAE Fundamentals Handbook (SI) Table 4A Design Wet-Bulb—Mean Coincident Dry-Bulb Temperature Location and WMO# % Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Annual WB MDB WB MDB WB MDB WB MDB WB MDB WB MDB WB MDB WB MDB WB MDB WB MDB WB MDB WB MDB WB MDB 1a 1b 2a 2b 3a 3b 4a 4b 5a 5b 6a 6b 7a 7b 8a 8b 9a 9b 10a 10b 11a 11b 12a 12c 13a 13b 14a 14b VERMONT Burlington 726170 0.4 7.4 8.8 9.1 10.6 13.3 17.2 17.3 24.0 21.3 28.4 23.4 29.4 24.6 30.9 24.4 30.2 23.7 27.6 18.2 22.2 15.4 18.2 12.0 13.3 23.3 28.6 1 5.4 7.1 6.3 7.9 11.3 15.2 16.0 21.9 20.2 25.7 22.7 28.2 23.9 29.9 23.6 28.7 22.7 26.6 17.3 20.9 14.2 17.1 9.4 10.9 22.2 26.9 2 3.4 5.1 4.8 6.7 9.6 13.3 14.6 19.3 19.2 24.9 21.9 27.5 23.4 29.1 22.8 27.2 21.7 24.9 16.5 19.7 12.7 15.1 6.9 8.7 21.3 25.8 VIRGINA Lynchburg 724100 0.4 14.7 17.2 15.9 19.3 18.2 23.3 19.4 24.2 23.7 29.0 24.8 30.7 25.7 32.3 25.3 32.3 24.3 30.8 21.6 25.7 19.2 21.1 17.3 19.5 24.7 31.2 1 13.4 17.5 14.6 17.4 17.3 23.4 18.7 24.9 22.8 27.6 24.3 30.6 25.3 31.8 24.9 31.7 23.9 30.7 20.9 25.2 18.4 21.1 16.3 18.6 24.2 30.3 2 12.0 15.0 13.3 16.4 16.3 21.2 18.1 24.6 22.1 26.8 23.9 29.9 24.9 31.4 24.5 30.9 23.5 29.6 20.1 24.0 17.7 20.3 15.0 17.2 23.6 29.3 Norfolk 723080 0.4 17.5 20.1 18.5 21.3 19.5 24.4 20.6 26.8 24.1 30.1 25.8 32.7 26.8 33.1 26.5 32.1 25.5 30.7 22.8 26.4 20.7 24.0 19.2 21.8 25.8 31.8 1 16.5 18.9 17.4 20.6 18.7 23.8 19.9 25.8 23.2 28.3 25.2 31.9 26.4 32.8 26.2 31.9 25.0 30.7 22.2 26.2 20.1 22.8 18.1 20.2 25.2 30.8 2 15.3 17.9 16.1 18.6 17.8 21.9 19.2 24.9 22.4 27.6 24.8 30.9 26.0 32.3 25.7 31.2 24.6 29.8 21.6 24.8 19.3 22.1 17.2 19.4 24.7 29.7 Richmond 724010 0.4 16.7 18.7 17.1 20.4 19.3 24.7 20.5 26.9 24.3 29.4 26.0 32.8 27.1 33.7 26.7 32.9 25.5 31.6 22.4 26.2 20.6 22.7 18.6 20.7 26.0 32.1 1 15.3 17.8 16.0 19.4 18.4 23.4 19.9 26.2 23.6 29.1 25.4 32.2 26.7 32.9 26.2 32.3 24.9 30.9 21.8 25.4 19.7 21.7 17.3 19.2 25.3 31.1 2 13.8 15.9 14.8 17.6 17.4 21.7 19.3 25.5 22.9 28.2 25.0 31.4 26.2 32.3 25.8 31.8 24.5 29.8 21.2 24.8 18.9 21.3 16.1 18.1 24.7 30.0 Roanoke 724110 0.4 13.5 16.9 14.9 18.4 17.4 23.8 18.8 26.2 22.6 28.6 24.0 30.9 25.1 32.1 24.7 31.2 23.8 30.7 20.7 24.9 18.6 21.1 16.0 18.2 24.0 30.8 1 12.2 16.4 13.6 17.1 16.3 21.6 18.2 25.2 21.8 27.8 23.5 30.2 24.6 31.4 24.3 31.0 23.3 29.7 19.9 23.6 17.6 20.4 15.0 17.8 23.4 29.8 2 11.1 14.1 12.3 16.1 15.4 20.3 17.6 23.7 21.1 26.3 23.0 29.6 24.1 31.0 23.9 30.6 22.8 28.8 19.3 23.3 16.6 19.2 13.7 16.1 22.8 28.7 Sterling 724030 0.4 13.9 17.7 15.1 17.3 17.8 23.5 19.6 25.9 23.7 29.7 25.3 31.2 26.4 33.2 25.9 32.2 25.5 29.6 21.5 25.2 19.3 22.5 16.8 19.0 25.2 31.2 1 11.8 14.3 13.9 16.4 16.6 21.8 18.9 25.8 22.9 28.4 24.8 30.8 25.8 32.4 25.4 31.4 24.7 29.9 20.8 24.4 18.3 21.1 15.4 17.8 24.5 30.3 2 10.3 12.3 12.2 14.9 15.6 20.1 18.1 24.5 22.0 27.2 24.2 30.1 25.3 31.8 25.0 30.7 24.0 29.4 20.1 22.9 17.2 19.8 13.8 15.8 23.8 29.2 WASHINGTON Olympia 727920 0.4 11.9 12.6 12.4 14.4 12.8 15.4 15.2 21.4 18.1 26.7 20.1 30.0 21.1 31.6 21.3 31.9 19.3 28.9 16.4 22.3 13.2 14.8 11.8 13.0 19.8 29.5 1 11.0 11.7 11.7 13.1 12.1 15.0 14.2 19.8 17.3 26.1 19.3 28.4 20.4 30.2 20.6 31.2 18.5 26.3 15.7 20.2 12.5 13.8 10.9 11.8 18.8 27.3 2 10.3 11.1 11.1 12.5 11.3 14.2 13.3 18.7 16.4 24.1 18.6 27.0 19.8 29.6 19.9 29.4 17.9 25.2 15.0 18.1 11.8 12.9 10.3 11.1 17.8 25.4 Quillayute 727970 0.4 11.3 12.3 12.1 14.4 11.5 14.0 13.1 17.9 16.4 24.3 17.6 25.6 18.7 27.9 19.4 28.8 18.4 26.1 15.8 20.2 13.2 14.1 11.7 12.2 17.7 24.4 1 10.5 11.1 11.3 13.0 10.8 12.8 12.4 16.8 15.3 22.4 16.7 23.7 17.9 25.5 18.4 25.9 17.6 24.2 15.1 18.0 12.7 13.3 11.2 11.7 16.6 21.9 2 10.1 10.6 10.4 11.8 10.3 12.1 11.7 15.8 14.4 19.2 15.7 21.6 17.1 23.4 17.7 23.7 16.8 22.6 14.6 16.8 12.2 12.9 10.7 11.2 15.7 19.7 Seattle, Intl Airport 727930 0.4 11.1 12.3 12.0 14.5 12.1 14.9 14.7 20.3 17.6 26.9 19.1 28.9 20.1 30.4 20.6 30.8 18.9 27.0 16.0 21.6 13.1 14.7 11.6 12.8 19.1 28.3 1 10.3 11.7 11.3 13.4 11.3 14.6 13.7 20.2 16.7 25.2 18.4 27.7 19.4 29.5 19.9 29.4 18.2 25.9 15.3 19.7 12.4 13.9 10.7 11.9 18.1 26.3 2 9.8 11.2 10.7 12.8 10.7 13.9 12.8 18.4 15.8 23.2 17.7 26.3 18.9 28.4 19.2 28.2 17.5 24.6 14.7 18.3 11.7 13.1 10.1 11.2 17.2 24.5 Spokane, Fairchild AFB 727855 0.4 7.0 8.8 8.6 10.7 10.4 15.7 14.2 22.8 16.8 28.2 18.6 29.9 19.4 31.7 19.1 31.4 16.9 28.9 14.3 23.4 10.1 12.8 7.8 9.3 18.1 30.1 1 5.8 7.3 7.7 9.9 9.5 13.6 12.8 20.7 15.6 25.9 17.8 28.7 18.7 31.1 18.5 30.9 16.3 27.9 13.3 21.7 9.1 10.9 6.5 7.7 17.2 28.8 2 5.0 6.5 6.8 9.2 8.7 12.6 11.7 18.1 14.8 24.3 17.2 27.7 18.1 30.3 17.9 30.2 15.6 26.8 12.5 19.6 8.3 9.9 5.6 6.8 16.3 27.5 Yakima 727810 0.4 8.3 11.8 9.9 15.2 11.8 18.4 15.1 23.8 17.9 29.6 19.5 33.2 21.5 33.3 20.9 33.0 18.9 30.0 15.8 25.4 11.4 15.8 9.0 12.2 19.7 32.4 1 7.4 10.9 9.1 13.7 11.0 17.1 13.9 21.9 16.9 28.4 18.8 31.7 20.6 32.6 20.2 32.6 18.1 29.4 14.8 23.9 10.6 14.1 7.7 10.5 18.7 31.4 2 6.4 9.6 8.3 12.6 10.1 15.6 12.9 21.1 16.0 26.7 18.2 30.3 19.7 32.7 19.7 32.3 17.4 28.6 13.9 22.1 9.8 13.2 6.6 9.0 17.8 29.9 WEST VIRGINA Charleston 724140 0.4 14.1 17.7 14.6 18.8 17.2 23.5 19.0 25.2 23.2 28.4 24.6 30.3 25.8 32.1 25.3 30.8 24.3 29.5 20.8 24.7 17.9 22.6 15.8 19.7 24.6 30.1 1 13.1 16.8 13.7 17.2 16.3 22.0 18.3 25.6 22.4 27.2 24.1 29.9 25.2 31.2 24.8 30.4 23.7 28.8 20.1 24.3 17.1 21.7 14.9 18.4 23.8 29.2 2 12.1 15.1 12.7 16.4 15.4 20.7 17.7 25.0 21.6 26.7 23.6 29.2 24.7 30.3 24.3 29.7 23.2 28.1 19.3 23.5 16.3 20.3 14.1 17.5 23.2 28.0 Elkins 724170 0.4 12.4 15.0 13.2 16.4 15.8 21.1 17.9 23.8 20.9 25.6 23.0 27.4 24.1 28.9 23.8 28.6 22.8 27.7 19.4 22.9 16.8 20.7 14.9 18.1 22.9 27.7 1 11.5 13.8 12.3 15.2 14.7 19.5 17.0 23.3 20.4 25.1 22.4 27.2 23.7 28.5 23.2 27.8 22.3 26.7 18.6 22.1 15.7 19.6 13.8 17.1 22.2 26.7 2 10.3 13.0 11.3 14.2 13.8 18.6 16.3 22.3 19.8 24.2 21.9 26.8 23.1 27.9 22.7 27.4 21.8 26.0 17.8 21.6 14.9 17.7 12.7 15.4 21.4 25.8 Huntington 724250 0.4 14.4 18.2 14.7 17.7 17.4 23.2 19.7 26.1 23.6 28.7 25.2 30.6 26.2 32.0 26.0 31.8 24.1 30.0 20.8 24.7 18.2 22.2 16.0 19.7 25.0 30.3 1 13.4 16.5 13.8 17.2 16.6 22.1 18.8 25.3 22.8 27.7 24.6 29.9 25.6 31.4 25.3 30.7 23.7 29.4 20.1 24.2 17.3 20.9 15.2 18.5 24.3 29.6 2 12.3 14.7 12.8 15.9 15.6 20.7 18.1 24.8 22.1 27.3 24.1 29.5 25.2 30.7 24.8 30.2 23.3 28.6 19.4 23.4 16.5 20.1 14.2 16.9 23.6 28.4 WISCONSIN Eau Claire 726435 0.4 3.3 5.0 5.9 8.7 13.5 18.7 18.1 24.7 22.1 28.5 24.5 31.4 26.0 32.2 25.6 32.2 23.6 28.8 19.5 22.9 14.0 15.6 8.8 10.7 24.3 30.1 1 2.1 3.8 4.1 6.3 11.5 15.7 17.1 23.1 21.2 27.4 23.8 30.2 25.2 31.7 24.8 30.7 22.7 27.3 18.6 21.9 12.7 14.6 4.6 5.2 23.2 28.5 2 1.3 2.8 3.0 4.9 9.6 13.4 16.0 21.4 20.3 26.1 23.0 29.3 24.5 30.6 24.0 29.4 21.9 25.8 17.4 21.5 11.0 13.4 2.9 4.0 22.2 27.2 Green Bay 726450 0.4 4.1 5.2 4.9 6.2 13.8 16.9 18.6 24.8 22.5 28.1 24.6 31.2 25.8 31.3 25.7 31.2 23.6 28.2 19.7 23.6 14.4 16.3 9.1 10.1 24.2 29.4 1 2.9 4.0 3.4 5.0 11.6 14.6 17.3 22.7 21.6 26.9 23.8 29.7 25.1 30.2 24.7 29.4 22.8 26.8 18.4 22.2 13.2 14.7 6.6 7.5 23.1 27.9 2 1.9 3.1 2.6 4.0 9.6 12.3 16.1 20.3 20.6 25.6 23.1 28.4 24.4 29.3 23.9 29.1 21.8 25.4 17.2 20.1 11.8 13.5 4.1 4.7 22.1 26.6 La Crosse 726430 0.4 4.9 6.6 6.8 9.7 14.4 19.7 18.8 25.7 23.0 29.1 25.3 31.6 27.1 32.4 26.4 32.3 24.7 30.1 20.2 23.9 14.8 16.4 10.2 11.6 25.2 30.4 1 3.3 5.2 5.1 7.4 12.8 16.7 17.7 23.9 21.7 26.4 24.6 30.3 26.3 31.5 25.6 31.8 23.8 28.4 19.2 22.9 13.9 15.7 6.8 8.1 24.1 29.1 2 2.2 3.6 3.8 5.8 11.1 14.8 16.6 21.7 20.8 25.9 23.8 29.4 25.4 30.6 24.9 29.8 22.9 27.0 18.1 21.8 12.3 14.5 4.2 5.3 23.1 27.7 Madison 726410 0.4 6.4 8.1 7.2 10.0 15.1 19.5 18.4 25.4 22.2 27.6 24.8 30.9 26.0 32.2 25.7 31.4 24.0 29.0 19.8 24.9 15.1 17.2 11.4 12.9 24.4 30.2 1 4.4 6.1 5.5 7.9 13.6 17.3 17.4 23.5 21.4 27.2 23.9 30.1 25.2 31.1 24.9 30.3 23.1 27.8 18.9 23.3 14.1 16.1 8.9 9.7 23.3 28.6 2 2.9 4.4 4.1 6.2 12.0 15.9 16.4 21.7 20.6 26.4 23.2 29.1 24.6 30.5 24.2 29.8 22.2 26.4 17.9 21.3 13.1 14.9 6.1 7.3 22.3 27.5 Milwaukee 726400 0.4 7.6 8.8 7.5 9.8 15.4 18.9 18.5 24.7 22.3 27.5 24.4 30.4 26.1 32.2 25.8 31.5 24.4 28.8 20.2 25.6 15.7 17.8 12.4 13.9 24.6 29.9 1 5.3 6.5 5.8 7.7 14.1 17.7 17.5 23.4 21.3 27.0 23.8 29.8 25.4 31.2 25.3 30.6 23.4 28.2 18.9 23.1 14.7 16.7 9.6 11.0 23.5 28.5 2 3.7 4.9 4.4 6.4 12.0 15.8 16.3 21.1 20.4 25.5 23.1 29.1 24.8 30.2 24.6 29.6 22.6 26.2 17.8 21.3 13.6 15.6 7.4 8.6 22.4 27.0 WYOMING Casper 725690 0.4 4.1 8.8 5.9 12.9 8.2 17.4 10.8 22.1 14.0 24.2 17.1 27.8 18.0 28.0 17.9 29.1 15.3 26.9 11.3 22.7 7.7 16.0 4.9 10.8 16.9 27.3 1 3.3 7.6 4.8 10.8 7.0 15.7 10.1 20.8 13.4 22.8 16.6 27.3 17.5 27.9 17.3 28.0 14.6 25.6 10.7 22.1 6.8 15.0 4.1 9.5 16.1 26.7 2 2.6 6.6 4.0 9.4 6.3 14.9 9.4 19.8 12.8 22.6 16.0 26.7 17.0 27.3 16.8 27.2 14.1 25.2 10.2 20.9 5.9 13.7 3.3 8.2 15.4 26.1 Cheyenne, Warren Afb 725640 0.4 4.4 12.2 5.8 14.1 7.1 17.9 10.2 20.1 13.6 21.0 16.8 24.9 18.0 25.7 17.9 25.1 15.6 23.3 11.0 22.1 7.4 16.6 5.1 13.3 16.8 24.9 1 3.6 10.9 4.7 12.5 6.3 16.3 9.6 19.8 12.9 21.2 16.3 25.2 17.4 25.8 17.2 24.8 14.8 22.7 10.2 21.1 6.7 15.9 4.2 11.8 16.1 24.3 2 2.8 9.7 3.8 11.6 5.5 14.9 8.9 18.6 12.3 20.5 15.7 24.8 17.0 25.3 16.7 24.3 14.1 22.3 9.6 20.3 5.8 14.7 3.4 10.6 15.3 23.7 Lander 725760 0.4 3.7 9.2 5.4 11.0 7.4 17.1 10.9 22.4 13.6 23.7 16.8 27.8 18.0 28.6 17.9 28.3 15.2 25.9 11.6 21.7 7.5 15.0 4.6 10.3 16.7 27.4 1 2.8 8.0 4.4 10.1 6.4 15.4 9.9 21.3 12.9 23.3 16.2 27.6 17.4 28.2 17.2 27.9 14.6 26.1 10.8 21.6 6.6 13.9 3.6 9.5 15.8 26.8 2 2.1 7.1 3.4 9.1 5.8 14.2 9.2 19.5 12.3 22.6 15.6 26.4 16.9 27.7 16.6 27.2 13.9 25.3 10.2 20.9 5.7 12.9 2.7 8.3 15.1 26.4 Rock Springs 725744 0.4 2.9 6.4 4.3 8.6 6.1 14.4 8.8 20.3 11.2 20.6 14.1 27.2 16.1 22.7 15.3 22.4 14.1 22.2 9.6 18.1 6.3 12.5 3.6 7.6 14.6 23.7 1 2.2 5.1 3.2 6.6 5.2 13.3 8.1 18.6 10.6 20.9 13.5 25.9 15.3 23.8 14.8 22.3 13.0 20.9 8.9 18.4 5.4 11.4 2.7 6.0 13.8 23.4 2 1.4 4.0 2.4 6.2 4.4 11.7 7.4 17.4 10.1 20.5 12.9 24.9 14.8 24.1 14.4 22.4 12.3 20.9 8.3 18.2 4.6 10.6 1.8 4.9 13.1 23.2 Sheridan 726660 0.4 5.2 11.4 7.2 15.4 9.3 19.3 13.3 23.9 16.3 26.6 19.7 30.3 20.3 28.8 19.1 30.9 17.1 28.5 13.2 25.2 9.3 18.8 6.4 14.3 18.7 29.2 1 4.4 9.8 6.1 12.9 8.4 17.7 12.0 22.4 15.7 25.4 18.9 28.8 19.5 28.8 18.6 30.1 16.1 27.6 12.4 23.8 8.2 17.2 5.4 12.7 17.8 28.3 2 3.7 8.9 5.1 11.1 7.6 16.3 11.2 21.1 15.1 24.2 18.3 28.3 18.9 29.4 18.1 29.7 15.5 27.4 11.7 22.7 7.3 15.7 4.3 10.4 17.0 27.4 WMO# = World Meteorological Organization number WB = wet-bulb temperature, °C MDB = mean coincident dry-bulb temperature, °C Climatic Design Information 27.71 Table 4B Design Dry-Bulb—Mean Coincident Wet-Bulb Temperature Location and WMO# % Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Annual DB MWB DB MWB DB MWB DB MWB DB MWB DB MWB DB MWB DB MWB DB MWB DB MWB DB MWB DB MWB DB MWB 1a 1b 2a 2b 3a 3b 4a 4b 5a 5b 6a 6b 7a 7b 8a 8b 9a 9b 10a 10b 11a 11b 12a 12c 13a 13b 14a 14b VERMONT Burlington 726170 0.4 9.4 7.3 11.3 8.3 18.9 13.6 26.0 16.1 30.4 19.7 32.0 21.4 33.4 23.1 31.8 22.6 28.9 22.6 23.7 17.5 18.8 15.2 13.4 11.5 30.8 21.6 1 7.2 5.4 8.4 5.2 16.1 10.4 23.0 14.0 28.6 18.4 30.7 21.0 32.3 22.8 30.6 21.9 27.6 21.9 22.3 16.5 17.2 13.9 11.7 9.0 29.1 20.7 2 5.4 3.1 6.9 4.7 13.6 9.2 21.0 13.4 26.9 17.7 29.6 20.8 31.1 22.4 29.6 21.4 26.2 20.6 20.9 15.6 15.8 12.4 9.2 6.8 27.5 20.0 VIRGINA Lynchburg 724100 0.4 19.8 13.7 22.3 13.6 26.8 17.0 30.9 17.6 31.2 21.0 33.7 23.6 35.9 23.6 35.1 23.4 33.7 23.2 28.2 20.0 24.8 16.0 21.3 16.6 33.7 23.6 1 17.3 11.9 19.8 12.2 24.9 15.9 29.3 16.7 30.3 20.8 32.8 23.1 34.9 23.7 34.1 23.4 32.3 23.1 27.2 18.9 23.4 15.5 19.5 15.2 32.3 23.1 2 15.5 11.1 18.2 11.4 23.0 14.7 27.9 16.4 29.3 20.3 31.9 22.7 34.1 23.7 33.2 23.3 31.3 22.4 26.2 18.4 21.9 14.9 17.5 14.1 31.1 22.6 Norfolk 723080 0.4 21.4 16.5 23.6 16.7 27.2 18.4 30.3 18.6 32.3 22.1 34.8 24.6 35.9 25.3 35.4 25.1 34.0 24.4 29.2 21.2 26.0 19.3 22.6 17.7 34.0 24.8 1 19.6 15.5 21.6 15.3 25.6 16.8 28.9 18.3 31.1 21.4 33.6 24.2 35.1 25.1 34.2 25.1 32.6 24.1 27.9 20.9 24.7 18.3 21.3 17.2 32.6 24.2 2 18.1 14.8 19.9 15.0 23.8 16.2 27.6 17.9 30.1 20.9 32.8 23.9 34.2 25.1 33.2 24.8 31.3 23.6 26.5 20.4 23.4 17.8 20.1 16.7 31.2 23.7 Richmond 724010 0.4 20.1 15.0 22.8 13.7 27.7 18.0 31.7 18.6 33.1 22.3 35.3 25.0 36.4 25.4 35.7 25.1 34.4 24.0 29.5 20.5 26.1 18.1 21.9 17.1 34.5 24.6 1 18.2 13.8 20.9 14.1 25.9 16.9 30.1 18.3 31.9 21.4 34.2 24.3 35.6 25.1 34.6 25.2 33.2 24.1 28.1 20.2 24.5 17.6 20.7 16.3 33.1 24.1 2 16.8 12.4 19.1 13.5 24.0 15.3 28.6 17.9 30.8 21.0 33.3 23.7 34.8 24.9 33.7 24.8 31.9 23.3 26.8 19.4 23.1 16.7 18.9 14.6 31.8 23.5 Roanoke 724110 0.4 18.8 12.4 21.6 13.1 26.4 15.8 30.0 17.1 31.4 20.5 33.6 22.4 35.9 23.1 34.9 22.8 32.9 22.7 28.0 18.6 24.2 15.6 20.1 14.6 33.2 22.7 1 16.7 10.9 19.5 11.8 24.6 14.7 28.4 16.3 30.3 20.0 32.7 22.3 34.7 22.8 33.5 22.8 31.8 22.4 26.8 17.8 22.6 15.4 18.5 13.7 31.8 22.2 2 15.0 9.8 17.7 10.9 22.8 13.7 27.0 15.9 29.1 19.3 31.8 21.8 33.6 23.0 32.4 22.7 30.6 21.8 25.8 17.5 21.2 14.5 16.9 12.6 30.4 21.6 Sterling 724030 0.4 17.2 12.2 20.3 12.4 25.8 16.1 29.7 17.6 31.8 21.8 34.2 23.6 36.1 24.3 34.9 24.2 33.8 24.0 28.2 19.2 24.2 17.5 20.4 16.4 33.7 23.9 1 15.2 11.1 17.9 11.7 23.8 15.3 28.1 17.6 30.7 21.1 33.2 23.1 34.9 24.3 33.8 23.9 32.4 23.3 26.9 19.1 22.6 16.3 18.2 14.7 32.2 23.2 2 12.9 9.3 16.1 12.2 21.7 14.6 26.4 17.1 29.6 20.4 32.2 23.1 34.0 24.2 32.9 23.6 31.3 22.9 25.5 18.7 21.1 15.6 16.4 13.4 30.9 22.6 WASHINGTON Olympia 727920 0.4 12.9 11.7 16.3 10.3 19.0 11.1 23.8 13.9 28.9 17.2 31.5 18.8 33.4 20.0 33.8 20.3 30.4 18.3 23.6 15.3 15.5 12.4 13.2 11.5 30.6 19.3 1 12.1 10.6 14.7 10.3 17.4 10.3 21.8 13.3 26.9 16.8 29.6 18.7 31.8 19.8 32.2 19.9 28.4 17.5 21.7 14.9 14.3 11.8 11.9 10.6 28.3 18.2 2 11.3 9.9 13.3 10.2 16.1 9.9 20.0 12.8 25.1 15.8 28.1 18.0 30.2 19.4 30.6 19.4 27.1 17.3 19.9 14.2 13.4 11.3 11.2 10.1 26.3 17.5 Quillayute 727970 0.4 13.3 10.1 16.6 10.3 16.9 10.1 21.1 12.3 26.4 15.6 27.9 16.5 29.2 17.4 29.6 19.2 28.3 17.3 22.6 15.3 15.2 10.9 12.6 11.1 26.4 16.8 1 11.9 9.6 14.6 9.8 15.3 9.6 19.0 11.2 23.8 14.4 24.8 16.4 27.0 17.1 27.2 17.7 26.6 16.7 20.3 13.4 14.0 11.8 11.9 10.9 23.3 15.9 2 11.0 9.3 13.0 9.6 13.9 9.2 17.1 11.0 21.0 13.7 22.2 15.7 24.7 17.0 24.9 17.0 24.4 15.9 18.5 13.5 13.2 11.7 11.3 10.4 20.8 15.0 Seattle, Intl Airport 727930 0.4 13.4 9.7 17.2 10.3 18.2 10.4 23.3 12.6 28.1 16.9 30.4 18.4 32.2 19.2 32.6 19.8 29.3 17.5 23.1 15.3 16.3 10.7 13.2 11.3 29.4 18.3 1 12.4 9.6 15.4 9.8 16.9 10.1 21.3 13.1 26.1 15.9 28.7 17.9 30.4 19.0 30.8 19.2 27.7 17.2 21.3 14.7 14.9 11.2 12.3 10.3 27.4 17.6 2 11.6 9.2 14.0 9.4 15.6 9.6 19.6 11.8 24.2 15.6 27.2 17.1 29.1 18.5 29.2 18.6 26.2 16.8 19.7 13.9 14.0 11.1 11.4 9.6 25.4 16.8 Spokane, Fairchild AFB 727855 0.4 9.1 6.9 11.9 7.9 16.9 9.7 24.3 13.4 29.8 15.6 33.1 16.9 35.7 17.8 35.8 17.1 31.6 16.1 24.6 13.3 13.2 8.8 9.6 7.2 33.4 16.8 1 7.8 5.6 10.7 7.0 15.3 8.7 21.8 11.9 28.1 14.9 31.8 16.7 34.6 17.2 34.6 17.1 29.8 15.4 22.8 12.7 11.7 8.6 8.2 5.9 31.5 16.3 2 6.6 4.8 9.6 6.2 13.9 7.8 20.0 11.1 26.3 13.8 30.4 16.3 33.6 16.8 33.4 16.8 28.6 15.0 21.2 11.8 10.6 7.7 7.2 5.3 29.6 15.7 Yakima 727810 0.4 13.2 7.6 15.9 9.3 20.1 10.6 25.8 14.1 32.2 17.2 35.3 18.5 37.6 19.5 37.7 19.4 32.6 17.8 26.6 15.1 16.7 10.7 13.2 8.3 35.1 18.6 1 11.3 7.0 14.7 8.5 18.6 9.9 24.4 12.9 30.4 16.0 33.9 17.6 36.5 19.3 36.2 19.2 31.3 17.2 24.9 14.2 15.4 9.8 11.2 7.1 33.2 17.9 2 10.1 6.2 13.3 7.7 17.2 9.3 22.8 12.2 28.8 15.6 32.6 17.3 35.4 18.6 35.0 18.6 30.1 16.7 23.1 13.6 14.2 9.0 9.5 6.1 31.2 17.2 WEST VIRGINA Charleston 724140 0.4 19.2 13.1 21.2 12.8 26.5 15.1 30.2 17.4 31.3 20.3 33.0 22.2 34.5 23.8 34.3 22.9 32.3 22.2 27.7 18.8 24.8 16.5 21.2 14.8 32.5 22.8 1 17.2 11.7 19.4 12.3 25.1 14.6 28.8 17.1 30.3 19.6 32.2 22.0 33.4 23.3 33.1 22.8 31.1 22.6 26.8 18.7 23.3 15.4 19.3 13.8 31.2 22.5 2 15.7 11.2 17.8 11.5 23.7 13.9 27.8 16.4 29.6 19.8 31.4 22.2 32.6 23.7 32.0 22.8 30.1 22.1 25.7 17.9 22.0 14.7 18.0 13.3 30.0 21.8 Elkins 724170 0.4 16.3 10.5 18.4 12.2 24.2 13.8 27.3 15.3 28.5 19.0 29.9 20.9 31.6 21.7 31.2 21.3 29.2 21.1 25.5 17.1 22.4 14.7 19.1 13.9 29.6 21.4 1 14.8 10.2 16.7 11.4 22.4 12.6 26.2 15.5 27.6 18.5 29.1 20.6 30.6 21.9 30.2 21.4 28.4 21.4 24.5 16.8 21.0 13.8 17.4 13.0 28.4 20.9 2 13.3 9.9 15.4 9.9 21.1 12.6 25.1 14.9 26.8 18.1 28.4 20.5 29.7 22.0 29.2 21.4 27.5 20.9 23.6 16.1 19.7 13.3 15.8 12.0 27.3 20.3 Huntington 724250 0.4 18.8 14.0 20.9 12.9 26.8 15.7 30.1 17.4 31.1 21.3 33.1 22.6 34.9 24.4 34.4 23.8 32.5 22.6 27.8 19.3 24.7 17.1 20.9 15.4 32.7 23.5 1 17.1 12.0 19.4 12.0 25.1 14.6 28.8 17.3 30.2 20.6 32.3 22.5 33.7 23.9 33.3 23.8 31.2 22.7 26.8 18.4 23.1 15.8 19.3 14.2 31.4 23.0 2 15.5 11.2 17.7 11.4 23.6 14.3 27.7 16.8 29.4 20.0 31.6 22.6 32.9 23.9 32.4 23.7 30.2 22.3 25.8 18.1 21.7 14.8 17.8 13.4 30.2 22.3 WISCONSIN Eau Claire 726435 0.4 5.4 2.9 8.8 5.3 19.3 12.6 28.0 17.4 30.9 20.1 33.7 23.1 35.3 24.1 33.8 23.9 30.7 22.6 25.7 17.6 17.0 11.7 10.1 9.1 32.2 22.9 1 4.2 1.9 6.8 3.9 16.5 10.9 25.3 15.2 29.7 19.6 32.2 22.4 33.9 23.8 32.4 23.7 29.1 21.3 24.0 17.1 15.2 12.1 6.0 3.8 30.5 21.7 2 2.9 1.1 5.2 2.5 14.1 9.0 23.3 14.5 28.5 18.3 31.1 21.6 32.6 23.2 31.3 22.8 27.6 20.6 22.3 16.1 13.9 10.6 4.3 2.5 28.9 20.9 Green Bay 726450 0.4 5.5 3.4 6.6 4.3 18.2 13.2 26.7 18.1 29.7 20.7 33.3 22.1 33.6 23.8 32.7 24.4 29.6 22.5 24.9 18.6 17.1 13.2 10.2 8.9 31.2 22.9 1 4.2 2.7 5.4 3.3 15.1 10.9 24.2 16.2 28.6 20.3 31.9 22.2 32.4 23.6 31.3 23.7 28.1 21.7 23.1 17.2 15.3 12.3 7.7 6.6 29.5 21.9 2 3.2 1.8 4.3 2.2 12.8 9.2 21.7 14.8 27.3 19.4 30.5 21.6 31.3 23.2 30.1 22.6 26.8 20.6 21.6 16.3 13.6 11.5 5.3 3.6 27.9 21.1 La Crosse 726430 0.4 6.9 4.2 10.2 6.2 21.2 14.1 28.4 17.2 31.2 20.8 34.3 23.2 35.6 25.2 34.6 25.1 31.8 23.3 26.3 18.3 18.0 13.1 11.4 10.0 32.8 23.6 1 5.3 3.1 8.1 4.6 18.0 11.8 26.3 16.6 30.0 19.9 32.9 22.7 34.2 24.4 33.1 24.3 30.1 21.9 24.7 17.6 16.2 12.6 8.3 6.3 31.1 22.6 2 4.1 2.0 6.4 3.4 15.8 10.1 24.1 15.4 28.6 19.2 31.8 22.4 33.1 24.0 31.8 23.4 28.5 21.5 23.4 16.8 15.0 12.1 5.8 3.6 29.5 21.8 Madison 726410 0.4 8.6 6.4 10.6 6.3 21.1 13.0 27.4 17.3 30.4 19.9 33.2 22.7 34.7 23.7 33.6 24.2 30.8 22.5 26.3 19.1 18.8 13.4 13.0 11.5 32.1 22.9 1 6.3 4.2 8.3 5.1 18.7 12.4 25.6 16.4 29.3 20.0 32.2 22.3 33.4 23.6 32.3 23.5 29.4 21.5 24.7 17.6 17.0 13.3 10.0 8.3 30.5 22.1 2 4.7 2.7 6.6 3.9 16.4 11.2 23.4 15.3 28.2 19.4 31.3 22.1 32.3 23.4 31.2 23.2 28.2 21.3 23.1 16.7 15.2 12.4 7.8 5.7 28.9 21.3 Milwaukee 726400 0.4 9.3 7.2 10.6 6.9 21.3 13.9 27.2 17.4 29.7 20.1 33.4 22.7 34.3 24.1 33.9 24.8 31.0 22.8 26.5 19.3 19.2 14.7 13.9 12.1 31.9 23.3 1 6.9 5.1 8.2 5.3 18.7 13.5 24.6 16.8 28.5 19.7 32.1 22.2 33.1 24.2 32.4 24.0 29.3 22.6 24.4 18.1 17.3 14.2 11.2 9.6 29.9 22.3 2 5.2 3.4 6.8 4.1 16.1 11.8 22.3 14.6 27.2 19.4 30.7 21.7 31.9 23.8 30.9 23.3 27.9 21.3 22.6 16.6 15.7 13.5 8.8 7.3 28.3 21.3 WYOMING Casper 725690 0.4 10.0 3.5 13.9 5.8 18.9 7.4 24.8 9.9 28.4 12.1 34.2 15.0 35.4 15.3 34.3 15.3 31.8 13.6 25.9 10.3 17.4 7.2 12.0 4.3 33.2 14.8 1 8.5 2.5 11.7 4.2 17.1 6.4 23.2 9.2 27.3 11.8 32.9 14.4 34.6 15.0 33.4 14.9 30.5 13.3 24.6 9.9 15.9 6.3 10.2 3.4 31.7 14.3 2 7.4 1.8 10.3 3.6 15.6 5.9 21.7 8.9 26.2 11.7 31.6 14.1 33.6 14.9 32.6 14.6 29.3 12.6 23.0 9.4 14.4 5.4 9.0 2.8 30.2 14.2 Cheyenne, Warren AFB 725640 0.4 13.3 3.7 15.7 4.8 19.4 6.8 23.5 9.0 27.2 11.2 31.5 14.0 33.2 14.9 32.0 14.3 29.6 12.6 25.1 9.9 18.5 6.8 14.6 4.2 30.8 14.2 1 11.8 2.9 13.9 4.1 17.6 5.8 22.3 8.7 25.9 10.8 30.3 13.9 32.2 14.6 31.0 14.1 28.5 12.7 24.0 9.5 17.2 6.0 13.2 3.7 29.3 13.9 2 10.5 2.4 12.3 3.4 15.7 5.0 20.8 8.1 24.6 10.9 29.2 13.8 31.3 14.4 30.1 13.8 27.4 12.3 22.7 8.9 15.8 5.4 11.6 2.9 27.8 13.8 Lander 725760 0.4 10.2 3.0 12.9 4.7 18.3 7.1 24.1 9.6 27.6 11.5 33.3 14.5 34.6 15.3 33.4 15.0 30.4 13.5 25.2 10.5 16.3 6.7 11.4 4.1 32.3 14.8 1 8.7 2.5 11.1 3.8 16.6 5.8 22.7 9.8 26.6 11.4 31.9 14.4 33.6 15.1 32.5 14.9 29.2 12.8 23.6 10.1 14.9 6.2 9.9 3.2 30.8 14.3 2 7.4 1.8 9.7 2.9 14.9 5.5 21.1 8.7 25.4 11.1 30.7 14.1 32.7 14.9 31.7 14.6 28.1 12.4 22.2 9.4 13.3 5.4 8.5 2.6 29.2 14.0 Rock Springs 725744 0.4 7.3 2.6 10.4 3.2 15.9 5.4 21.8 8.3 25.5 9.8 31.1 12.8 32.3 13.0 31.2 12.4 28.4 11.8 22.7 8.3 14.3 5.2 8.6 2.7 30.2 12.4 1 5.8 1.6 8.7 2.6 14.2 4.7 20.7 7.3 24.4 9.6 30.0 12.3 31.3 12.8 30.4 12.4 27.4 11.1 21.6 7.9 12.9 4.7 6.9 1.8 28.9 12.1 2 4.6 0.7 7.3 1.9 12.7 4.1 19.3 6.7 23.3 9.0 29.0 11.9 30.5 12.4 29.6 12.2 26.3 10.7 20.4 7.8 11.7 4.0 5.7 0.9 27.6 11.8 Sheridan 726660 0.4 12.8 4.8 15.4 6.7 20.9 8.6 25.9 12.1 29.6 14.7 35.1 17.7 36.7 17.5 36.1 17.2 33.3 15.3 27.6 12.2 20.2 8.8 15.1 6.1 34.1 16.8 1 10.8 4.1 13.4 5.8 19.1 8.1 24.2 11.4 28.1 14.4 33.0 17.3 35.5 17.1 35.1 16.8 31.7 14.6 25.8 11.6 18.2 8.0 13.1 5.1 32.2 16.2 2 9.3 3.1 11.9 4.7 16.9 7.3 22.6 10.4 26.7 14.0 31.3 16.6 34.2 17.2 33.9 16.4 30.2 14.5 24.0 11.1 15.9 7.2 11.1 4.0 30.2 16.0 WMO# = World Meteorological Organization number DB = dry-bulb temperature, °C MWB = mean coincident wet-bulb temperature, °C 28.1 CHAPTER 28 RESIDENTIAL COOLING AND HEATING LOAD CALCULATIONS Residential Features ............................................................... 28.1 COOLING LOAD ................................................................... 28.1 Load Components ................................................................... 28.1 Load Calculation .................................................................... 28.5 HEATING LOAD .................................................................... 28.6 General Procedure .................................................................. 28.7 Selecting Heating Design Conditions ..................................... 28.7 Estimating Temperatures in Adjacent Unheated Spaces ........ 28.8 Calculating Heat Loss from Crawl Spaces ............................. 28.9 Calculating Transmission Heat Loss .................................... 28.10 Calculating Infiltration Heat Loss ........................................ 28.12 PICKUP LOAD ..................................................................... 28.14 HIS CHAPTER covers the engineering basis of modified T residential load calculation procedures for the nonengineer.
The procedures described in Chapter 29 may be used to calculate a heating or cooling load for residential buildings.
RESIDENTIAL FEATURES With respect to heating and cooling load calculation and equip-ment sizing, the unique features distinguishing residences from other types of buildings are the following: • Unlike many other structures, residences are usually occupied and conditioned 24 h per day, virtually every day of the cooling and heating seasons.
• Residential system loads are primarily imposed by heat loss or gain through structural components and by air leakage or ventilation. Internal loads, particularly those from occupants and lights, are small in comparison to those in commercial or industrial structures.
• Most residences are conditioned as a single zone. Unit capacity cannot be redistributed from one area to another as loads change from hour to hour; however, exceptions do occur.
• Most residential cooling systems use units of relatively small capacity (about 5 to 18 kW cooling, 18 to 32 kW heating).
Because loads are largely affected by outside conditions, and few days each season are design days, the unit operates at only partial load during most of the season; thus, an oversized unit is detrimental to good system performance, especially for cooling in areas of high wet-bulb temperature.
• Dehumidification occurs during cooling unit operation only, and space condition control is usually limited to use of room thermostats (sensible heat-actuated devices).
• Multifamily living units are similar to single-family detached houses, but the living units may not all have surfaces exposed in all directions. This affects load calculation.
Categories of Residences Single-Family Detached. A house in this category usually has exposed walls in four directions, often more than one story, and a roof. The cooling system is a single-zone, unitary system with a sin-gle thermostat. Two-story houses may have a separate cooling sys-tem for each floor. The rooms are reasonably open and generally have a centralized air return. In this configuration, both air and load from rooms are mixed, and a load-leveling effect, which requires a distribution of air to each room that is different from a pure com-mercial system, results. Because the amount of air supplied to each room is based on the load for that room, proper load calculation pro-cedures must be used.
Multifamily Buildings. Unlike single-family detached units, multifamily units by definition do not have exposed surfaces facing in all directions. Rather, each unit has only one or two exposed sur-faces and possibly a roof. Two exposed walls will be at right angles, and both east and west walls will not be exposed in a given living unit. Each living unit has a single unitary cooling system or a single fan-coil unit, and the rooms are relatively open to one another. This configuration does not have the same load-leveling effect as a sin-gle-family detached house, but it is not a commercial building.
Therefore, a specific load calculation procedure is required.
Other Categories. Many buildings do not fall into either of the above categories. Critical to the designation of a single-family de-tached building is the exposure of both east and west walls. There-fore, some multifamily structures should be treated as single-family detached when the exposed surfaces are oriented in a particular way.
Examples include duplexes or apartments with either exposed east, west, and south walls or exposed east, west, and north walls, with or without a roof; and apartments, town houses, or condominiums with only east and west or north and south exposed walls.
COOLING LOAD LOAD COMPONENTS A cooling load calculation determines total sensible cooling load due to heat gain (1) through structural components (walls, floors, and ceilings); (2) through windows; (3) caused by infiltration and ventilation; and (4) due to occupancy. The latent portion of the cool-ing load is evaluated separately. While the entire structure may be considered a single zone, equipment selection and system design should be based on a room-by-room calculation. For proper design of the distribution system, the amount of conditioned air required by each room must be known.
Peak Load Computation To select a properly sized cooling unit, the peak or maximum load (block load) for each zone must be computed. Because this procedure may vary considerably for different types of buildings, each building type has to be considered; the block load for a single-family detached house with one central system is the sum of all the room loads. If the house has a separate system for each zone, each zone block load (i.e., the sum of the loads for all rooms in each zone) is required. When a house is zoned with one central cooling system, the block load must be computed for the complete house as if it were one zone. In multifamily structures, each living unit has a zone load that equals the sum of the room loads. For apartments with separate systems, the block load for each unit establishes the system size.
Apartment buildings with a central cooling system (i.e., a hydronic system with fan-coils in each apartment) require a block load calcu-lation for the complete structure to size the central system; each unit The preparation of this chapter is assigned to TC 4.1, Load Calculation Data and Procedures.
28.2 2001 ASHRAE Fundamentals Handbook (SI) load establishes the size of the fan-coil and air distribution system for each apartment. One of the methods discussed in Chapter 29 may be used to calculate the block load.
Indoor Temperature Swing For hour-by-hour load calculations, allowing for a swing in indoor temperature results in lower peak loads. Because the indoor temperature does swing, such an allowance gives a more reasonable equipment capacity. The tables in this section are based on an assumed indoor temperature swing of no more than 1.5 K on a design day, when the residence is conditioned 24 h per day and the thermostat is set at 24°C.
Cooling Load Due to Heat Gain Through Structure The sensible cooling load due to heat gains through the walls, floor, and ceiling of each room is calculated using appropriate cooling load temperature differences (CLTDs) (Tables 1 and 2) and U-factors for summer conditions. For ceilings under naturally vented attics or beneath vented flat roofs, the combined U-factor for the roof, vented space, and ceiling should be used. The mass of the walls is a variable in Table 2 and is important in calculating energy use, but it is not used in Table 1 because of the averaging technique required to develop the CLTDs. Values in Tables 1 and 2 assume a dark color because color is an unpredictable variable in any residence.
Daily range (outdoor temperature swing on a design day) signif-icantly affects the equivalent temperature difference. Tables 1 and 2 list daily temperature ranges classified as high, medium, and low.
Tables 1, 2, and 3 in Chapter 27 list outdoor daily ranges of dry-bulb temperature for different locations.
Cooling Load Due to Heat Gain Through Windows Direct application of procedures for calculating cooling load due to heat gain for flat glass (discussed in Chapters 29 and 30) results in unrealistically high cooling loads for residential installations.
Window glass load factors (GLFs), modified for single- and mul-tifamily residential cooling load calculations and including solar heat load plus air-to-air conduction, are given in Tables 3 and 4.
Table 5 lists the shading coefficients (SCs) and U-factors used to compile Tables 3 and 4.
In application, the area of each window is multiplied by the appropriate GLF. The effects of permanent outside shading devices should be considered separately in determining the cooling load.
Shaded glass is considered the same as north-facing glass. The shade line factor (SLF) is the ratio of the distance a shadow falls beneath the edge of an overhang to the width of the overhang (Table 6). Therefore, assuming the overhang is at the top of the window, the shade line equals the SLF times the overhang width. The shaded and sunlit glass areas may then be computed separately. The tabulated values are the average of the shade line values for 5 h of maximum solar intensity on each wall orientation shown. Northeast- and northwest-facing windows are not effectively protected by roof overhangs; in most cases, they should not be considered shaded.
Infiltration Natural air leakage in residential structures is less in summer than in winter, largely because wind velocities are lower in most localities. The data in Tables 7 and 8 showing space air changes per hour (ACH) apply to both single- and multifamily housing, Table 1 CLTD Values for Single-Family Detached Residencesa Daily Temperature Rangeb Design Temperature, °C 29 32 35 38 41 43 L M L M H L M H M H M H All walls and doors North 4 2 7 4 2 10 7 4 10 7 10 13 NE and NW 8 5 11 8 5 13 11 8 13 11 13 16 East and West 10 7 13 10 7 16 13 10 16 13 16 18 SE and SW 9 6 12 9 6 14 12 9 14 12 14 17 South 6 3 9 6 3 12 9 6 12 9 12 14 Roofs and ceilings Attic or flat built-up 23 21 26 23 21 28 26 23 28 26 28 31 Floors and ceilings Under conditioned space, over unconditioned room, or over crawl space 5 2 7 5 2 8 7 5 8 7 8 11 Partitions Inside or shaded 5 2 7 5 2 8 7 5 8 7 8 11 aCooling load temperature differences (CLTDs) for single-family detached houses, duplexes, or multifamily, with both east and west exposed walls or only north and south exposed walls, K.
bL denotes low daily range, less than 9 K; M denotes medium daily range, 9 to 14 K; and H denotes high daily range, greater than 14 K.
Table 2 CLTD Values for Multifamily Residencesa Daily Temperature Rangeb Design Temperature, °C 29 32 35 38 41 43 L M L M H L M H M H M H Walls and doorsc Low 8 6 11 9 7 13 12 9 14 12 15 18 N Medium 7 6 10 8 6 13 11 9 14 12 14 17 High 5 3 8 6 4 11 9 7 12 9 12 15 Low 13 9 16 12 9 18 15 12 18 14 17 20 NE Medium 11 8 14 11 9 17 14 12 16 14 16 19 High 9 7 12 9 7 14 12 10 14 12 14 17 Low 18 15 21 18 15 24 21 18 23 21 23 26 E Medium 17 13 19 16 13 22 19 16 22 18 22 24 High 13 10 16 13 10 19 16 13 18 16 18 21 Low 17 15 19 17 14 23 21 17 23 21 23 26 SE Medium 16 12 18 15 12 21 18 15 21 18 21 24 High 12 9 14 12 9 18 15 12 17 15 18 21 Low 14 12 16 14 12 19 17 14 20 18 21 24 S Medium 12 10 14 12 10 17 14 12 17 15 18 21 High 9 6 11 9 7 14 12 9 14 12 15 18 Low 22 20 24 22 19 28 26 22 28 26 29 32 SW Medium 18 16 21 19 16 24 22 19 25 22 26 29 High 13 10 16 13 11 20 17 14 19 17 20 23 Low 24 23 27 25 22 30 28 26 31 29 32 35 W Medium 21 18 23 21 18 26 23 21 27 24 27 31 High 14 12 17 15 13 21 18 15 21 18 21 24 Low 18 17 21 19 17 24 22 19 24 22 25 28 NW Medium 16 14 18 16 13 21 18 16 22 19 22 25 High 11 9 14 11 9 17 14 12 17 14 18 21 Roof and ceiling Attic or flat built-up Light 32 29 36 33 31 39 36 33 39 36 40 43 Flat built-up Medium or heavy 12 10 13 12 10 14 13 12 14 13 14 16 Floors and ceiling Under or over uncondi-tioned space, crawl space 5 2 7 5 2 8 7 5 8 7 8 11 Partitions Inside or shaded 5 2 7 5 2 8 7 5 8 7 8 11 aCooling load temperature differences (CLTDs) for multifamily low-rise or single-family detached if zoned with separate temperature control for each zone, K.
bL denotes low daily range, less than 9 K; M denotes medium daily range, 9 to 14 K; and H denotes high daily range, greater than 14 K.
cLow denotes low-density; medium denotes medium-density; and high denotes high-density construction.
Residential Cooling and Heating Load Calculations 28.3 Table 3 Window Glass Load Factors (GLFs) for Single-Family Detached Residencesa Design Temperature, °C Regular Single Glass Regular Double Glass Heat-Absorbing Double Glass Clear Triple Glass 29 32 35 38 41 43 29 32 35 38 41 43 29 32 35 38 41 43 29 32 35 No inside shading North 107 114 129 148 151 158 95 95 107 117 120 129 63 63 73 79 82 88 85 85 95 NE and NW 199 205 221 237 243 262 173 177 186 196 199 208 114 117 123 132 139 139 158 158 167 East and West 278 284 300 315 322 337 243 246 255 265 268 278 161 161 170 177 186 186 221 221 230 SE and SWb 249 255 271 287 290 309 218 221 230 240 243 252 142 145 155 161 170 170 196 199 205 Southb 167 173 189 205 211 227 145 148 158 167 170 180 98 98 107 114 123 123 132 132 142 Horizontal skylight 492 492 508 524 527 539 432 435 442 451 454 464 284 287 293 300 303 309 391 394 401 Draperies, venetian blinds, translucent roller shades, fully drawn North 57 60 73 85 91 104 50 50 60 69 73 82 41 44 50 57 60 66 47 50 57 NE and NW 101 104 120 132 136 148 91 95 101 110 114 123 76 76 85 91 91 101 88 88 95 East and West 142 145 158 170 173 186 126 129 139 145 148 158 104 104 114 120 120 129 123 123 129 SE and SWb 126 129 145 155 161 173 114 117 123 132 136 145 91 95 101 107 110 117 110 114 120 Southb 85 88 104 117 120 132 76 79 88 98 98 107 63 66 73 79 82 88 73 76 82 Horizontal skylight 246 249 262 271 274 284 224 224 233 240 243 249 183 186 192 199 199 205 218 218 224 Opaque roller shades, fully drawn North 44 47 63 73 79 91 41 44 54 60 63 73 38 38 47 54 54 63 41 41 47 NE and NW 79 82 98 107 114 126 73 76 85 95 95 104 66 69 76 82 85 91 73 73 82 East and West 107 114 126 139 142 155 101 104 114 120 123 132 91 95 101 107 110 117 101 101 110 SE and SWb 98 101 114 126 132 145 91 95 104 110 114 123 82 85 91 98 101 107 91 91 98 Southb 66 69 85 95 101 114 63 63 73 82 85 95 57 60 66 73 76 82 60 63 69 Horizontal skylight 189 192 202 214 218 227 180 180 189 196 199 205 164 164 173 180 180 186 177 180 186 aGlass load factors (GLFs) for single-family detached houses, duplexes, or multifam-ily residences, with both east and west exposed walls or only north and south exposed walls, W/m2.
bCorrect by +30% for latitude of 48° and by −30% for latitude of 32°. Use linear interpolation for latitude from 40 to 48 and from 40 to 32°.
To obtain GLF for other combinations of glass and/or inside shading: GLFa = (SCa/SCt)(GLFt − UtDt) + UaDt, where the subscripts a and t refer to the alternate and table values, respectively. SCt and Ut are given in Table 5. Dt = (ta − 24), where ta = to − (DR/2); to is the outdoor design temperature and DR is the daily range.
Table 4 Window Glass Load Factors (GLFs) for Multifamily Residencesa Design Temperature, °C Regular Single Glass Regular Double Glass Heat-Absorbing Double Glass Clear Triple Glass 29 32 35 38 41 43 29 32 35 38 41 43 29 32 35 38 41 43 29 32 35 No inside shading North 126 139 155 170 183 202 107 114 123 132 139 148 73 76 82 91 95 104 95 101 107 NE 278 281 287 300 306 315 246 249 252 262 265 268 164 164 167 173 173 180 224 224 230 East 429 432 438 448 454 464 378 382 385 394 397 401 249 249 255 262 262 265 344 344 350 SE 407 410 423 438 445 454 344 356 366 375 378 385 227 237 243 249 249 255 312 325 331 Southb 278 287 303 319 331 347 240 246 255 265 271 281 158 164 170 177 183 189 214 221 227 SW 486 501 517 533 549 565 423 432 442 451 457 467 281 287 293 300 306 312 382 388 394 West 549 561 577 593 606 621 476 486 495 505 511 520 315 322 328 334 341 347 432 438 445 NW 388 401 416 432 445 464 337 344 353 363 369 382 224 227 237 243 249 255 303 309 315 Horizontal 785 795 807 823 833 845 688 694 703 713 719 725 454 460 467 473 479 486 624 631 637 Draperies, venetian blinds, translucent roller shades, fully drawn North 66 79 91 104 114 126 57 66 73 82 88 98 47 54 60 66 73 79 54 60 66 NE 136 139 145 158 161 164 123 126 129 139 142 145 104 104 107 114 114 117 123 123 126 East 211 214 221 233 237 240 192 196 199 205 208 211 158 158 161 170 170 173 189 189 192 SE 202 205 218 230 233 243 183 186 192 199 202 208 151 151 158 164 164 170 180 180 186 Southb 142 151 164 177 186 199 126 132 139 148 155 164 104 107 114 123 126 132 120 126 132 SW 249 262 274 287 296 309 221 227 237 246 252 262 180 186 196 202 208 214 214 218 224 West 281 290 303 315 325 337 249 255 265 271 278 287 205 208 218 224 227 237 240 246 252 NW 199 208 221 233 243 255 177 183 192 199 208 214 145 151 158 164 170 177 170 173 180 Horizontal 397 404 416 426 432 445 356 363 369 378 382 391 293 296 303 309 315 322 347 350 356 Opaque roller shades, fully drawn North 54 66 79 91 101 114 47 54 63 73 79 88 44 47 57 63 69 76 47 50 57 NE 104 107 110 123 126 132 98 101 104 114 110 117 91 88 95 101 101 107 101 98 104 East 161 164 167 180 192 205 151 155 158 167 164 173 142 142 145 151 151 155 155 155 158 SE 155 158 167 180 183 192 145 148 155 164 164 173 132 136 142 148 148 155 145 145 151 Southb 110 120 132 145 155 167 101 107 117 126 132 132 91 98 104 110 117 123 101 104 110 SW 192 205 218 230 243 255 180 186 196 205 211 221 164 170 177 183 189 196 177 183 189 West 214 224 237 252 262 274 202 208 214 224 230 240 183 189 196 202 208 214 199 202 208 NW 155 164 177 189 199 211 142 148 158 167 173 183 129 136 142 148 155 161 142 145 151 Horizontal 306 312 322 334 341 350 287 293 300 306 312 322 262 268 274 281 284 290 284 290 293 aGlass load factors (GLFs) for multifamily low-rise or single-family detached resi-dences if zoned with separate temperature control for each zone, W/m2.
bCorrect by +30% for latitude of 48° and by −30% for latitude of 32°. Use linear interpolation for latitude from 40 to 48 and from 40 to 32°.
To obtain GLF for other combinations of glass and/or inside shading: GLFa = (SCa/SCt)(GLFt − UtDt) + UaDt, where the subscripts a and t refer to the alternate and table values, respectively. SCt and Ut are given in Table 5. Dt = (ta − 24), where ta = to − (DR/2); to is the outdoor design temperature and DR is the daily range.
28.4 2001 ASHRAE Fundamentals Handbook (SI) although most of the raw data were for single-family structures (McQuiston 1984). Construction may be defined as follows: Tight. Good multifamily construction with close-fitting doors, windows, and framing is considered tight. New houses with full vapor retarder, no fireplace, well-fitted windows, weather-stripped doors, one story, and less than 140 m2 floor area fall into this category.
Medium. Medium structures include new, two-story frame houses or one-story houses more than 10 years old with average maintenance, a floor area greater than 140 m2, average fit windows and doors, and a fireplace with damper and glass closure. Below-average multifamily construction falls in this category.
Loose. Loose structures are poorly constructed single- and mul-tifamily residences with poorly fitted windows and doors. Examples include houses more than 20 years old, of average maintenance, having a fireplace without damper or glass closure, or having more than an average number of vented appliances. Average manufac-tured homes are in this category.
Ventilation Residential air-conditioning systems may introduce outdoor air, although it is not a code requirement in most localities. Posi-tive ventilation should be considered, however, if the anticipated infiltration is less than about 0.5 ACH. When positive means of introducing outdoor air are used, controls, either manual or auto-matic, should be provided, and an energy recovery device should be considered.
Occupancy Even though occupant density is low, occupancy loads should be estimated. Sensible heat gain per sedentary occupant is assumed to be 67 W. To prevent gross oversizing, the number of occupants should not be overestimated. Recent census studies recommend that the total number of occupants be based on two persons for the first bedroom, plus one person for each additional bedroom. The occu-pancy load should then be distributed equally among the living areas because the maximum load occurs when most of the residents occupy these areas.
Household Appliances Appliance loads are concentrated mainly in the kitchen and laun-dry areas. Based on contemporary living conditions in single-family houses, a sensible load of 470 W should be divided between the kitchen and/or laundry and the adjoining room or rooms. A sensible utility load of 470 W may be added if the laundry room contains con-tinuously operating appliances such as refrigerators and/or freezers.
For multifamily units, the sensible heat gain values should be about 350 W. These values assume that the cooking range and clothes dryer are vented. Further allowances should be considered when unusual lighting intensities, computers, or other equipment is present.
Air Distribution System—Heat Loss/Gain Whenever the air distribution system is outside the conditioned space (i.e., in attics, crawl spaces, or other unconditioned spaces) heat loss or gains to the ducts or pipes must be included in the cal-culated load and should be considered in equipment selection.
Latent Heat Sources The latent cooling load has three main sources: outdoor air, occu-pants, and miscellaneous sources, such as cooking, laundry, and Table 5 Shading Coefficients and U-Factors for Residential Windows Glass Type Inside Shade None Drapery, Venetian Blind, or Translucent Roller Shade Opaque Roller Shade SC U SC U SC U Single 1.00 5.91 0.50 4.60 0.38 4.60 Double 0.88 3.46 0.45 3.12 0.36 3.12 Heat-absorbing 0.58 2.56 0.37 2.50 0.33 2.50 Triple 0.80 2.50 0.44 2.27 0.36 2.27 Note: U is in W/(m2· K).
Table 6 Shade Line Factors (SLFs) Direction Window Faces Latitude, Degrees N 24 32 36 40 44 48 52 East 0.8 0.8 0.8 0.8 0.8 0.8 0.8 SE 1.8 1.6 1.4 1.3 1.1 1.0 0.9 South 9.2 5.0 3.4 2.6 2.1 1.8 1.5 SW 1.8 1.6 1.4 1.3 1.1 1.0 0.9 West 0.8 0.8 0.8 0.8 0.8 0.8 0.8 Note: Shadow length below the overhang equals the shade line factor times the over-hang width. Values are averages for the 5 h of greatest solar intensity on August 1.
Table 7 Winter Air Exchange Rates (ACH) as Function of Airtightness Class Outdoor Design Temperature, °C 10 4 −1 −7 −12 −18 −23 −29 −34 −40 Tight 0.41 0.43 0.45 0.47 0.49 0.51 0.53 0.55 0.57 0.59 Medium 0.69 0.73 0.77 0.81 0.85 0.89 0.93 0.97 1.00 1.05 Loose 1.11 1.15 1.20 1.23 1.27 1.30 1.35 1.40 1.43 1.47 Note: Values are for 6.7 m/s (24 km/h) wind and indoor temperature of 20°C.
Table 8 Summer Air Exchange Rates (ACH) as Function of Airtightness Class Outdoor Design Temperature, °C 29 32 35 38 41 43 Tight 0.33 0.34 0.35 0.36 0.37 0.38 Medium 0.46 0.48 0.50 0.52 0.54 0.56 Loose 0.68 0.70 0.72 0.74 0.76 0.78 Note: Values are for 3.4 m/s (12 km/h) wind and indoor temperature of 24°C.
Fig. 1 Effect of Infiltration on Latent Load Factor Residential Cooling and Heating Load Calculations 28.5 bathing. The miscellaneous latent loads are largely covered by out-door air because most residences have exhaust fans and clothes dry-ers that vent most of the moisture from these sources. This vent air is accounted for in the infiltration calculation. McQuiston (1984) esti-mated latent load factors for typical houses located in geographic regions ranging from very dry to very wet using the transfer function method (Figure 1). A latent factor LF (LF = 1/SHF) of 1.3 or a sen-sible heat factor SHF (SHF = sensible load/total load) of 0.77 matches the performance of typical residential vapor compression cooling systems. Homes in almost all other regions of North America have cooling loads with an SHF greater than 0.77 and latent factors less than 1.3. Figure 1 may be used to estimate the total cooling load by reading LF as a function of the design humidity ratio and airtight-ness. Then qtotal = (LF)qsensible. If the humidity ratio is less than 0.01, set LF = 1.0.
LOAD CALCULATION The cooling load calculation procedures are summarized in Table 9.
Example 1. A single-family detached house (Figure 2) is located in the south central United States at 36°N latitude.
Roof construction. Conventional roof-attic-ceiling combination, vented to remove moisture with 150 mm of fibrous batt insulation and vapor retarder [U = 0.28 W/(m2·K)]. Wall construction. Frame with 100 mm face brick, 90 mm fibrous batt insulation, 19 mm polystyrene sheathing, and 13 mm gypsum wall-board [U = 0.34 W/(m2·K)]. Ceiling height is 2.4 m throughout.
Floor construction. 100 mm concrete slab on grade.
Fenestration. Clear double glass, 3 mm thick, in and out. Assume closed, medium-color venetian blinds. The window glass has a 600 mm overhang at the top.
Doors. Solid core flush with all-glass storm doors [U = 1.82 W/(m2·K)].
Outdoor design conditions. Temperature of 36°C dry bulb with a 13 K daily range and a humidity ratio of 0.0136 kg vapor/kg dry air (23.7°C wet bulb).
U-factors for all external surfaces are based on a 3.4 m/s (12 km/h) wind velocity.
Indoor design conditions. Temperature of 24°C dry bulb and 50% rh.
Occupancy. Four persons, based on two for the master bedroom and one for each additional bedroom. Assign to the living room.
Appliances and lights. Assume 470 W for the kitchen, and assign 50% to the living room. Assume 470 W for the utility room, and assign 25% to the kitchen and 25% to the storage room.
The conditioning equipment is located in the garage, and the con-struction of the house is considered medium.
Find the sensible, latent, and total cooling load; size the cooling unit; and compute the air quantity for each room.
Fig. 2 Floor Plan of Single-Family Detached House Table 9 Summary of Procedures for Residential Cooling Load Calculations Load Source Equation Tables and Notes Glass and window areas q = (GLF)A Glass load factors may be found in Tables 3 and 4 according to window orien-tation, type of glass, type of interior shading, and outdoor design temperature.
The GLF includes effects of both transmission and solar radiation. Glass shaded by overhangs is treated as north glass. Table 6 gives shade line factors.
Doors q = UdA(CLTD) Door CLTD values are in Tables 1 and 2 according to orientation, outdoor design temperature, and design daily temperature range.
Above-grade exterior walls q = UwA(CLTD) Wall CLTD values are in Tables 1 and 2 based on the outdoor design tempera-ture, daily range, and orientation.
Partitions to unconditioned space q = UpA∆t Where ∆t is the temperature difference across the partition.
Ceilings and roofs q = UrA(CLTD) Tables 1 and 2 for CLTD, based on outdoor design temperature and daily range.
Exposed floors q = UfA(CLTD) Tables 1 and 2 for CLTD, based on outdoor design temperature and daily range.
Infiltration q = 1.2Q∆t Air exchange rates are given in Tables 7 and 8.
Q = ACH × (room volume) × 1000/3600 Internal loads— Plan 67 W per person.
Divide occupants evenly among rooms not used as bedrooms. If number of occupants is not known, assume two people for first bedroom and one person for each additional bedroom.
People, appliances, lights The appliance and light load of 470 W is divided between the kitchen and adjoining room and the laundry and adjoining room. Use 350 W for multifam-ily units.
Total loads Total cooling load = LF × (Sum of individual sensible cooling load components) Load factors are from Figure 1 according to outdoor design humidity ratio and airtightness classification.
q =sensible cooling load, W ∆t =design temperature difference between outside and inside air, K A =area of applicable surface, m2 U =U-factors for appropriate construction, W/(m2·K) Q = volumetric airflow rate, L/s ACH = air changes per hour, 1/h GLF = glass load factor, W/m2 CLTD = cooling load temperature difference, K LF = latent load multiplier 28.6 2001 ASHRAE Fundamentals Handbook (SI) Solution: The cooling load must be made on a room-by-room basis to determine the proper distribution of air. The calculations follow the procedure outlined in the section on Load Components.
Walls, roof, windows, and doors. The calculations for the living room and the kitchen, where q = UA(CLTD) for the walls, roof, and door and q = A(GLF) for the windows, are outlined in Table 10. The glass shaded by the overhang is treated as north-facing glass, with the shaded area computed using Table 6.
Internal and infiltration sensible cooling loads. Compute as follows.
For the living room: Infiltration. Using Table 8, Q =ACH (room volume) × 1000/3600 Q =0.5 × 106.9 × 1000/3600 = 14.85 L/s q =1.2Q(to − ti) q =1.2 × 14.85(36 − 24) = 214 W Occupants. Assuming 67 W per person, q =67 × (persons) q =67 × 4 = 268 W Appliances. Assuming that 50% of the kitchen appliance load is picked up in the living room, q =0.5 × (kitchen appliance load) q =0.5 × 470 = 235 W For the kitchen: Infiltration.
Q =0.5 × 54.2 × 1000/3600 = 7.5 L/s q =1.2 × 7.5(36 − 24) = 108 W No occupants.
q =0 Appliances. Assuming 25% of the utility appliance load is picked up in the kitchen, q = (470/2) + (470/4) = 352.5 W For the total sensible cooling load for these two rooms and the cooling load for the remaining rooms, see Table 11. At this point, the sensible cooling load for the house is 5.75 kW. Depending on the design of the air distribution system, heat losses from the supply and return ducts may add to the cooling load. These may be more accu-rately estimated after designing the system; however, to size the cool-ing unit, duct losses should be included initially. If all ducts are in the attic space, a duct loss of l0% of the space sensible cooling load is rea-sonable. For a counterflow system, with ducts below the slab, a 5% loss is more reasonable.
An infiltration rate of 0.5 ACH may not be adequate for good indoor air quality, so some outdoor air should be introduced. This additional cooling load may be estimated in the same way as the infiltration load.
Assume that the entire duct system is in the attic; that is, the total sensible cooling load with a 10% duct loss is 1.1 × 5.75 = 6.33 kW.
Also, assume that additional outdoor air is needed to assure good indoor air quality, so the total infiltration and outdoor ventilation air is 0.75 ACH. This increases the infiltration rate by 50%, or about 0.47 kW. The total sensible cooling load is then increased to 6.80 kW (Table 11).
The total cooling load (sensible plus latent) may be estimated by applying the latent factor (LF) from Figure 1. For a design humidity ratio of 0.0136 kg vapor per kg dry air, LF = 1.15 for a house of medium construction. Hence, the total cooling load equals 1.15 × 6.80 = 7.82 kW.
The load raises the temperature of the cooling air 9 to 12 K as it leaves the rooms. The total design flow from the air conditioner can be estimated by the following equation: (1) where Qtot =total airflow, L/s q =total sensible load, W 1.2 =density times specific heat of cooling air ∆t =temperature difference of air entering and leaving room, K For a temperature difference of 10 K, the total airflow is estimated from Equation (1) as The exact design flow can be determined only after the cooling unit has been selected. Then, the supply air quantities can be computed. Air should be supplied to each room on the basis of the room sensible cool-ing load: where Qrm =airflow to each room, L/s qrm =room sensible cooling load, W Thus, for the example, If the living space in Example 1 were a multifamily unit (assume that the north, south, and east walls are not exposed surfaces), the calculation procedure would be the same, except that Table 2 would have been used for the CLTDs and Table 4 for the GLFs. Assump-tions regarding infiltration, ventilation, and appliance loads are dif-ferent for smaller multifamily units.
HEATING LOAD Calculating a residential heating load involves estimating the maximum (block) heat loss of each room or space to be heated and the simultaneous maximum (block) heat loss for the building, while maintaining a selected indoor air temperature during periods of design outdoor weather conditions. Heat losses are mainly • Transmission losses or heat transferred through the confining walls, glass, ceiling, floor, or other surfaces • Infiltration losses or energy required to warm outdoor air leaking in through cracks and crevices around doors and windows, Table 10 Transmission Cooling Load for Example 1 Item Net Area, m2 GLF, W/m2 U-Factor, W/(m2·K) CLTD, K Cooling Load, kW Reference Living Room West wall 8.4 0.34 14 0.040 Table 1 Partition (garage) 17.5 0.40 7 0.049 Table 1 Roof 44.5 0.28 27 0.336 Table 1 West door 1.9 1.82 14 0.048 Table 1 West glass 3.1 141 0.437 Table 3 Shaded glass 1.2 63 0.076 Table 3 Kitchen East wall 12.4 0.34 14 0.059 Table 1 Roof 22.6 0.28 27 0.171 Table 1 East glass 1.25 141 0.176 Table 3 Shaded glass 1.0 63 0.063 Table 3 Table 11 Summary of Sensible Cooling Load Estimate for Example 1 Room Roof, Walls, and Doors Glass People Appli-ances Infil-tration Total kW (qrm) Room L/s (Qrm) Living room 0.47 0.51 0.27 0.24 0.21 1.70 142 Kitchen 0.23 0.24 0.35 0.11 0.93 77 Utility and storage 0.41 0.35 0.13 0.89 74 Bedroom No. 1 0.17 0.16 0.08 0.41 34 Bedroom No. 2 0.19 0.25 0.08 0.52 43 Master bedroom and bath 0.50 0.24 0.24 0.98 82 Bath 0.16 0.08 0.08 0.32 27 Total 2.13 1.48 0.27 0.94 0.93 5.75 479 Duct loss (10%) 0.58 Outdoor ventilation air 0.47 Total 6.80 kW Qtot 1000q 1.2∆t ---------------= Qtot 1000 5.75 × 1.2 10 × ----------------------------479 L/s = = Qrm Qtot qrm q ⁄ ( ) = Qrm 479 5.75 ⁄ ( )qrm = Residential Cooling and Heating Load Calculations 28.7 through open doors and windows, and through porous building materials GENERAL PROCEDURE To calculate a design heating load, prepare the following infor-mation about building design and weather data at design conditions.
1. Select outdoor design weather conditions: temperature, wind direction, and wind speed. Winter climatic data can be found in Chapter 27, or selected weather conditions and temperatures appropriate for the application may be used. Weather station data may differ significantly from values in Chapter 27.
2. Select the indoor air temperature to be maintained in each space during design weather conditions.
3. Temperatures in adjacent unheated spaces, attached garages, and attics can be estimated at the outdoor ambient temperature.
4. Select or compute heat transfer coefficients for outside walls and glass; for inside walls, nonbasement floors, and ceilings if these are next to unheated spaces; and for the roof if it is next to heated spaces.
5. Determine the net area of outside wall, glass, and roof next to heated spaces, as well as any cold walls, floors, or ceilings next to unheated spaces. These determinations can be made from building plans or from the actual building, using inside dimensions.
6. Compute transmission heat losses for each kind of wall, glass, floor, ceiling, and roof in the building by multiplying the heat transfer coefficient in each case by the area of the surface and the temperature difference between indoor air and outdoor air or adjacent lower temperature spaces.
7. Compute heat losses from basement or grade-level slab floors using the methods in this chapter.
8. Select unit values, and compute the energy associated with infiltration of cold air around outside doors, windows, porous building materials, and other openings. These unit values depend on the kind or width of crack, wind speed, and the temperature difference between indoor and outdoor air. An alternative method is to use air changes (see Chapter 26).
9. When positive ventilation using outdoor air is provided by an air-heating or air-conditioning unit, the energy required to warm the outdoor air to the space temperature must be provided by the unit. The principle for calculation of this load component is identical to that for infiltration. If mechanical exhaust from the space is provided in an amount equal to the outdoor air drawn in by the unit, the unit must also provide for natural infiltration losses. If no mechanical exhaust is used and the outdoor air supply equals or exceeds the amount of natural infiltration that can occur without ventilation, some reduction in infiltration may occur.
10. The sum of the coincidental transmission losses or heat transmitted through the confining walls, floor, ceiling, glass, and other surfaces, plus the energy associated with cold air entering by infiltration or the ventilation air required to replace mechanical exhaust, represents the total heating load.
11. Include the pickup loads that may be required in intermittently heated buildings using night thermostat setback. Pickup loads frequently require an increase in heating equipment capacity to bring the temperature of structure, air, and material contents to the specified temperature. See Figure 9.
12. Use materials and data in Chapters 25, 26, 27, and others as appropriate to the calculations. See Table 12.
SELECTING HEATING DESIGN CONDITIONS The ideal solution to a basic heating system design is a plant with a maximum output capacity equal to the heating load that develops with the most severe local weather conditions. However, this solution is usually uneconomical. Weather records show that severe weather conditions do not repeat annually. If heating sys-tems were designed for maximum weather conditions, excess capacity would exist during most of the system’s operating life. In many cases, an occasional failure of a heating plant to maintain a preselected indoor design temperature during brief periods of severe weather is not critical.
Outdoor Design Temperature Before selecting an outdoor design temperature from Chapter 27, the designer should consider the following for residential buildings: • Is the structure heavy, medium, or light?
• Is the structure insulated?
• Is the structure exposed to high wind?
• Is the load from infiltration or ventilation high?
• Is there more glass area than normal?
• During what part of the day will the structure be used?
• What is the nature of occupancy?
• Will there be long periods of operation at reduced indoor temperature?
• What is the amplitude between local maximum and minimum daily temperatures?
• Are there local conditions that cause significant variation from temperatures reported by the weather service?
• What auxiliary heating devices will be in the building?
Before selecting an outdoor design temperature, the designer must keep in mind that, if the outdoor to indoor design temperature difference is exceeded, the indoor temperature may fall, depending on (1) the thermal mass of the structure and its contents, (2) whether the internal load was included in calculations, (3) the duration of the cold period, and (4) internal heat generated by appliances, etc.
The effect of wind on the heating requirements of any building should be considered because Table 12 Summary of Loads, Equations, and References for Calculating Design Heating Loads Heating Load Equation Reference, Table, Description Roofs, ceilings, walls, glass ➤Chapter 24, Tables 1, 2, and 4 ➤Temperature difference between inside and outside design dry bulbs, Chapter 26. For temperatures in unheated spaces, see Equation (2); for attic temperatures, see Equation (3).
➤Area calculated from plans Walls below grade ➤ ➤ See Table 14.
Use Figure 6 to assist in determining ∆t.
Floors Above grade ➤For crawl space temperatures, see Equation (4).
➤See Table 16.
On grade ➤See Equation (6).
➤Perimeter of slab Below grade ➤ ➤ Use Figure 6 to assist in determining ∆t.
See Table 15.
Infiltration and ventilation air Sensible ➤Volume of outdoor air entering build-ing. See Chapter 25 for estimating methods for infiltration.
Latent ➤Humidity ratio difference, if humidi-fication is to be added q = U A ∆t q = U A ∆t q = U A ∆t q = F2 P ∆t q = U A ∆t qs = 0.018 Q ∆t qt = 80.7 Q ∆W 28.8 2001 ASHRAE Fundamentals Handbook (SI) • Wind movement increases the heat transmission of walls, glass, and roof, affecting poorly insulated walls to a much greater extent than well-insulated walls.
• Wind materially increases the infiltration of cold air through cracks around doors and windows and even through building materials themselves (see Chapter 26).
Theoretically, on a design basis, the most unfavorable combina-tion of temperature and wind speed should be chosen. A building may require more heat on a windy day with a moderately low out-door temperature than on a quiet day with a much lower outdoor temperature. The worst combination of wind and temperature varies by building because wind speed has a greater effect on buildings with relatively high infiltration rates. The building heating load may be calculated for several combinations of temperature and wind speed on record, and the worst combination may be selected; how-ever, except for critical applications, designers generally find such a degree of refinement unnecessary. No correlation has been shown between the design temperatures in Chapter 27 and the simulta-neous maximum wind speed. If a designer prefers the air change method for computing infiltration rates, such correlation is not important. Designers who use the crack method can use a leakage rate at a wind speed of 6.7 m/s (24 km/h), unless local experience has established that another speed is more appropriate. Abnormally high wind speeds may have an effect on infiltration and the U-factor of the building components (see Chapter 23).
Indoor Design Temperature The indoor temperature for comfort heating may vary depending on building use, type of occupancy, or code requirements. Chapter 8 and ASHRAE Standards 55 and 55a define the relationship between temperature and comfort.
ESTIMATING TEMPERATURES IN ADJACENT UNHEATED SPACES Heat loss from heated rooms to unheated rooms or spaces must be based on the estimated or assumed temperature in such unheated spaces. This temperature will be in between the indoor and outdoor temperatures. If the surface area adjacent to the heated room and that exposed to the outdoors are equal and if the heat transfer coefficients are equal, the temperature in the unheated space may be assumed equal to the mean of the indoor and outdoor design temperatures. If, however, the surface areas and coefficients are unequal, the temperature in the unheated space should be esti-mated by (2) where ρcp = density times specific heat of air = 1.2 kJ/(m3·K) for standard air tu = temperature in unheated space, °C ti = indoor design temperature of heated room, °C to = outdoor design temperature, °C A1, A2, A3, etc. = areas of surfaces of unheated space adjacent to heated spaces, m2 Aa, Ab, Ac, etc. = areas of surfaces of unheated space exposed to outdoors, m2 U1, U2, U3, etc. = heat transfer coefficients of surfaces of A1, A2, A3, etc., W/(m2·K) Ua, Ub, Uc, etc. = heat transfer coefficients of surfaces of Aa, Ab, Ac, etc., W/(m2·K) Qo = rate of introduction of outside air into unheated space by infiltration and/or ventilation, L/s Example 2. Calculate the temperature in an unheated space adjacent to a heated room with surface areas (A1, A2, and A3) of 9, 11, and 13 m2 and overall heat transfer coefficients (U1, U2, and U3) of 0.8, 1.1, and 1.4 W/(m2·K), respectively. The surface areas of the unheated space exposed to the outdoors (Aa and Ab) are 9 and 13 m2, respectively, and the corresponding overall heat transfer coefficients are 0.6 and 1.7 W/(m2·K). The sixth surface is on the ground and can be neglected for this example, as can the effect of introduction of outdoor air into the unheated space. Assume ti = 21°C and to = −23°C.
Solution: Substituting into Equation (2), Temperatures in unheated spaces with large glass areas and two or more surfaces exposed to the outdoors (e.g., sleeping porches and sun parlors) are generally assumed to be the same as that of the out-doors.
Attic Temperature An attic is a space having an average distance of 0.3 m or more between a ceiling and the underside of the roof. Estimating attic temperature is a special case of estimating temperature in an adja-cent unheated space and can be done using (3) where ρcp = air density times specific heat = 1.20 kJ/(m3·K) for standard air ta = attic temperature, °C tc = indoor temperature near top floor ceiling, °C to = outdoor temperature, °C Ac = area of ceiling, m2 Ar = area of roof, m2 Aw = area of net vertical attic wall surface, m2 Ag = area of attic glass, m2 Uc = heat transfer coefficient of ceiling, W/(m2·K), based on surface conductance of 12.5 W/(m2·K) (upper surface, see Table 2 in Chapter 25); 12.5 = reciprocal of one-half the air space resistance Ur = heat transfer coefficient of roof, W/(m2·K), based on surface conductance of 12.5 W/(m2·K) (upper surface, see Table 2 in Chapter 25); 12.5 = reciprocal of one-half the air space resistance Uw = heat transfer coefficient of vertical wall surface, W/(m2·K) Ug = heat transfer coefficient of glass, W/(m2·K) Vc = rate of introduction of outside air into the attic space by ventilation per square metre of ceiling area, L/(s·m2) Example 3. Calculate the temperature in an unheated attic assuming tc = 21°C; to = −12°C; Ac = 100 m2; Ar = 120 m2; Aw = 10 m2; Ag = 1.0 m2; Ur = 2.8 W/(m2·K); Uc = 2.3 W/(m2·K); Uw = 1.7 W/(m2·K); Ug = 6.4 W/(m2·K); and Vc = 2.5 L/(s·m2).
Solution: Substituting these values into Equation (3), Equation (3) includes the effect of air interchange that would take place through attic vents or louvers intended to preclude attic condensation. Test data from Joy et al. (1956), Joy (1958), and Rowley et al. (1940) indicate that a reduction in the temperature tu [ti A1U1 A2U2 A3U3 etc.
+ + + ( ) = to(ρcpQo AaUa AbUb AcUc etc.)] + + + + + (A1U1 A2U2 A3U3 etc.
+ + + ÷ ρcpQo AaUa AbUb AcUc etc.) + + + + + tu [21 9 0.8 × 11 1.1 × 13 1.4 × + + ( ) = 23 – ( )(9 0.6 × 13 1.7 × )] + + (9 0.8 × 11 1.1 × 13 1.4 × + + ÷ 9 0.6 × 13 1.7) × + + tu 155 65 ⁄ 2.4°C = = ta AcUctc to ρcpAcVc ArUr AwUw AgUg + + + ( ) + Ac Uc ρcpVc + ( ) ArUr AwUw AgUg + + + ------------------------------------------------------------------------------------------------------------------= ta [ 100 2.3 × 20 × ( ) 12 – ( ) + (1.2 100 × 2.5 × = 120 2.8 × 10 + 1.7 × 1.0 6.4 × )] + + [100(2.3 1.2 2.5) × + ÷ 120 2.8 × 10 1.7 1.0 6.4 × + ] × + + ta 3313 – 889 ⁄ 3.7 – °C = = Residential Cooling and Heating Load Calculations 28.9 difference between attic air and outside air is linear as attic ventila-tion rates increase from 0 to 2.5 L/(s·m2) of the ceiling area. When attic ventilation meets the requirements in Chapter 24, 2.5 L/(s·m2) is the approximate ventilation rate for design conditions.
This reduction in temperature difference affects the overall heat loss of a residence with an insulated ceiling by only 1 or 2%.
Equation (3) does not consider factors such as heat exchange between chimney and attic or solar radiation to and from the roof.
Because of these effects, attic temperatures are frequently higher than values calculated using Equation (3). However, Equation (3) can be used to calculate attic temperature because the resulting error is generally less than that introduced by neglecting the roof and assum-ing that the attic temperature is equal to the outdoor air temperature.
When relatively large louvers are installed (customary in southern regions of the United States), the attic temperature is often assumed to be the average of the indoor and outdoor air temperatures.
For an approximate method of calculating heat losses through attics, the combined ceiling and roof coefficient may be used (see Table 5 in Chapter 25).
CALCULATING HEAT LOSS FROM CRAWL SPACES A crawl space can be considered a half basement. To prevent ground moisture from evaporating and causing a condensation problem, sheets of vapor retarder (e.g., polyethylene film) are used to cover the ground surface (see Chapter 24). Most codes require crawl spaces to be adequately vented all year round. However, vent-ing the crawl space in the heating season causes substantial heat loss through the floor.
The space may be insulated in several ways: the crawl space ceil-ing (floor above the crawl space) can be insulated, or the perimeter wall can be insulated either on the outside or on the inside. If the floor above is insulated, the crawl space vents should be kept open because the temperature of the crawl space is likely to be below the dew point of the indoor space. If the perimeter wall is insulated, the vents should be kept closed in the heating season and open the remainder of the year.
Crawl Space Temperature The crawl space temperature depends on such factors as venting, heating ducts, and the heating plant. When the crawl space is well ventilated, its temperature is close to that of the ambient air temper-ature. When the crawl space vent is closed for the heating season, or if the space is used as a plenum (i.e., part of the forced-air heating system), the crawl space temperature approaches that of the indoor conditioned space. In the former case, the floor above the crawl space, the heating ductwork, and the utility pipes should be insu-lated similarly to the walls and ceiling of a house.
The following steady-state equation can be used to estimate the temperature of a crawl space.
where qf = heat loss through floor into crawl space, W qp = heat loss from crawl space through foundation walls and sill box, W qg = heat loss into ground, W qa = heat loss due to ventilation of crawl space, W Latta and Boileau (1969) estimated the air exchange rate for an uninsulated basement at 0.67 ACH under winter conditions. In more detail, the above equation can be repeated as (4) where ti = indoor air temperature (i.e., air above ceiling of crawl space), °C to = outdoor air temperature, °C tg = ground temperature (constant), °C tc = crawl space temperature, °C Af = area of floor above, m2 Ap = area of perimeter, exposed foundation wall plus sill box, m2 Ag = area of ground below (Af = Ag), m2 Uf = average heat transfer coefficient through floor, W/(m2·K) Ug = average heat transfer coefficient through ground (horizontal air film and 3 m of soil), W/(m2·K) Up = combined heat transfer coefficient of sill box and foundation wall (both above and below grade), W/(m2·K) Vc = volume of crawl space, m3 ρcp = volumetric heat capacity of air = 1.2 kJ/(m3·K) 0.67 = assumed air exchange rate, volumes/hour Example 4. A crawl space of 120 m2 with a 44 m perimeter is considered.
The construction of the perimeter wall is shown in Figure 3. The indoor, outdoor, and the deep-down ground temperatures are 20, −12, and 10°C, respectively. Estimate the heat loss and crawl space tempera-ture with and without insulation. The heat transmission coefficient (U-factor) for each component is indicated in Table 13.
Solution: Three cases are examined.
Case A. This base case is a vented and uninsulated crawl space. The crawl space temperature approaches that of the outdoors, −12°C, and the heat loss is 1.42 × 120[20 − (−12)] = 5450 W.
Case B. The crawl space is vented. The floor above is insulated with an R = 1.94 K·m2/W blanket; no insulation on the perimeter. The temperature of the crawl space approaches that of the outdoors, −12°C.
The heat loss is calculated as Case C. The crawl space is not vented during the heating season.
The floor above is not insulated, but the perimeter wall is insulated with R = 0.95 K·m2/W down to 900 mm below grade.
qf qp qg qa + + = U f Af ti tc – ( ) Up Ap tc to – ( ) Ug Ag tc tg – ( ) + = 0.67ρcpVc tc to – ( ) 3.6 ⁄ + Fig. 3 Uninsulated Crawl Space qf 120 0.432 × 20 12 – ( ) – [ ] 1660 W = = 28.10 2001 ASHRAE Fundamentals Handbook (SI) The crawl space temperature is solved using Equation (4): tc = 10.5°C. The heat loss is 1620 W.
The results show that base case A can potentially lose the most heat. However, when the floor above is insulated, the crawl space must be vented to eliminate any condensation potential, and the heating duct-work and utility pipeline in the crawl space must be adequately insu-lated. When the perimeter is insulated, the vents must be closed during the heating season and opened for the rest of the year; the heating duct-work and utility pipeline do not need insulation.
CALCULATING TRANSMISSION HEAT LOSS Steady-state heat loss by conduction and convection heat transfer through any surface is (5) where q = heat transfer through wall, glass, roof, ceiling, floor, or other exposed surface, W A = area of surface, m2 U = air-to-air heat transfer coefficient, W/(m2·K) ti = indoor air temperature near surface involved, °C to = outdoor air temperature or temperature of adjacent unheated space, °C Example 5. Calculate the transmission loss through a 200 mm brick wall having an area of 14 m2, if the indoor temperature ti is 21°C, and the outdoor temperature to is −23°C.
Solution: The overall heat transfer coefficient U of a plain 200 mm brick wall is 2.33 W/(m2·K). Substituting into Equation (5), Through Ceiling and Roof Transmission heat loss through top floor ceilings, attics, and roofs may be estimated by either of two methods: 1. Substitute in Equation (5) the ceiling area A, the indoor/outdoor temperature difference (ti − to), and the proper U-factor: Flat roofs. Use appropriate coefficients in Equation (3) if side walls extend appreciably above the ceiling or the floor below.
Pitched roofs. Calculate the combined roof and ceiling coeffi-cient as outlined in Chapter 25.
2. For pitched roofs, estimate the attic temperature (based on the indoor and outdoor design temperatures) using Equation (3), and substitute for to in Equation (5), obtaining the value of ta, together with the ceiling area A and the ceiling U-factor. Attic temperatures do not need to be calculated for flat roofs, as the ceiling-roof heat loss can be determined as suggested in Method 1 above.
From the Basement The basement interior is considered conditioned space if a min-imum temperature of 5.5 Κ below indoor design air temperature is maintained over the heating season. In many instances, the house heating plant, water heater, and heating ducts are in the basement, so it remains at or above 10°C.
Heat transmission from the below-grade portion of the basement wall to the ambient air cannot be estimated by simple, one-dimen-sional heat conduction. In fact, field measurement of an uninsulated basement by Latta and Boileau (1969) showed that the isotherms near the wall are not parallel lines but closer to radial lines centered at the intersection of the grade line and the wall. Therefore, heat flow paths approximately follow a concentric circular pattern (Figure 4).
Such heat flow paths are altered when insulation is added to the wall or floor. An extreme case would be no heat loss from the base-ment wall and floor (i.e., infinite insulation applied to the wall and floor). In this case, the isotherms would be horizontal lines parallel to the grade line, and the heat flow would be vertical. When finite insulation or partial insulation is applied to the wall and floor, the heat flow paths take shapes somewhere between the circular and vertical lines (Figure 5).
Ground Temperature. Ground temperatures assumed for esti-mating basement heat losses will differ for basement floors and walls. The temperatures under floors are generally higher than those adjacent to walls. This is discussed further in the section on Base-ment Design Temperatures.
Table 13 Estimated U-Factors for Insulated and Uninsulated Crawl Spaces Component Uninsulated Insulateda W/K per metre of Perimeter W/K per metre of Perimeter 400 mm exposed concrete blocks 1.22 0.316 190 mm sill box 0.324 0.123 1st 300 mm block wall below grade 0.614 0.22 2nd 300 mm block wall below grade 0.381 0.242 3rd 300 mm block wall below grade 0.23 0.173 Total for perimeter wall 2.77 1.07 W/(m2· K) W/(m2· K) Ground 0.437 0.437 Floor above crawl space 1.42 0.432a aPerimeter walls are insulated with R = 0.95 K·m2/W; the floor is insulated with R = 1.94 K·m2/W blanket or batts.
Case Venting Insulation Heat Loss Through Floor Above, W Temperature of Crawl Space, °C A Yes None 5450 −12 B Yes R = 1.94 K·m2/W on floor above 1660 −12 C No R = 0.95 K·m2/W on perimeter wall 1620 10.5 qf 120 1.42 20 tc – ( ) × = qp 44 1.07 × tc 12 – ( ) – [ ] = qg 120 0.437 × tc 10 – ( ) = qa 120 0.9 1.2 × × 0.67 × tc 12 – ( ) – [ ] 3.6 ⁄ = q UA ti to – ( ) = q 14 2.33 21 23 – ( ) – [ ] × 1435 W = = Fig. 4 Heat Flow from Basement Residential Cooling and Heating Load Calculations 28.11 Through Basement Walls Houghten et al. (1942) observed nonuniform heat flux across the basement wall with respect to the depth of the wall because each heat flow path contains a different thermal resistance. For a base-ment wall that has its top portion exposed to ambient air, heat may be conducted vertically through the concrete wall and dissipated to the ambient from the top portion of the wall (Wang 1979, Bligh et al. 1978). Under certain conditions, this vertical heat flux be-comes significant and should not be ignored.
Once the heat paths are known or assumed, a steady-state analysis can calculate the overall heat transmission coefficient for each seg-ment of the basement wall. Referring to Figures 4 and 5, the total thermal resistance for each depth increment of the basement wall can be found by summing the thermal resistances along each heat flow path. Based on these resistances, the heat loss at each depth incre-ment can be estimated for a unit temperature difference between the basement and the average mean winter temperature. Table 14 lists such heat loss values at different depths for an uninsulated and an insulated concrete wall (Latta and Boileau 1969). Also listed are the lengths of the heat flow path through the soil (circular path).
Through Basement Floors The same steady-state design used for the basement wall can be applied to the basement floor, except that the length of the heat flow path is longer (see Figure 4). Thus, the heat loss through the base-ment floor is much smaller than that through the wall. An average value for the heat loss through the basement floor can be multiplied by the floor area to give total heat loss from the floor. Table 15 lists typical values.
Basement Design Temperatures Although internal design temperature is given by basement air temperature, none of the usual external design air temperatures apply because of the heat capacity of the soil. However, ground sur-face temperature fluctuates about a mean value by an amplitude A, which varies with geographic location and surface cover. Therefore, suitable external design temperatures can be obtained by subtract-ing A for the location from the mean winter air temperature ta. Val-ues for ta can be obtained from meteorological records, and A can be estimated from the map in Figure 6. This map is part of one prepared by Chang (1958) giving annual ranges in ground temperature at a depth of 100 mm. Example 6. Consider a basement 8.5 m wide by 9.1 m long sunk 1.8 m below grade, with R = 1.47 K·m2/W insulation applied to the top 0.6 m of the wall below grade. Assume an internal air temperature of 21°C and an external design temperature (ta − A) of 7°C. Solution: Wall (using Table 14) Floor (using Table 15) Total If a basement is completely below grade and unheated, its tem-perature ranges between that in the rooms above and that of the ground. Basement windows lower the basement temperature when it is cold outdoors, and heat given off by the heating plant increases the basement temperature. The exact basement temperature is inde-terminate if the basement is not heated. In general, heat from the heating plant sufficiently warms the air near the basement ceiling to make unnecessary an allowance for floor heat loss from rooms located over the basement.
Table 14 Heat Loss Below Grade in Basement Walls Depth, m Path Length Through Soil, m Heat Loss Coefficient, W/(m2· K)a Un-insulated R = 0.73 m2· K/W R = 1.47 m2· K/W R = 2.20 m2· K/W 0 to 0.3 0.2 2.33 Σb 0.86 3.53 0.53 Σb 0.38 Σb 0.3 to 0.6 0.69 1.26 3.59 0.66 1.52 0.45 0.98 0.36 0.74 0.6 to 0.9 1.18 0.88 4.47 0.53 2.05 0.38 1.36 0.30 1.04 0.9 to 1.2 1.68 0.67 5.14 0.45 2.50 0.34 1.70 0.27 1.31 1.2 to 1.5 2.15 0.54 5.68 0.39 2.89 0.30 2.00 0.25 1.56 1.5 to 1.8 2.64 0.45 6.13 0.34 3.23 0.27 2.27 0.23 1.79 1.8 to 2.1 3.13 0.39 6.52 0.30 3.53 0.25 2.52 0.21 2.00 Source: Latta and Boileau (1969).
aSoil conductivity was assumed to be 1.38 W/(m· K).
bΣ = heat loss to current depth.
Table 15 Heat Loss Through Basement Floors Heat Loss Coefficient, W/(m2·K) Depth of Foundation Wall below Grade, m Shortest Width of House, m 6 7.3 8.5 9.7 1.5 0.18 0.16 0.15 0.13 1.8 0.17 0.15 0.14 0.12 2.1 0.16 0.15 0.13 0.12 Note: ∆t = (ta − A) Fig. 5 Heat Flow Path for Partially Insulated Basement Wall 0.3 m below grade ....................... 0.53 × 0.3 = 0.159 W/(m· K) 0.6 m below grade ....................... 0.45 × 0.3 = 0.135 W/(m· K) 0.9 m below grade ....................... 0.88 × 0.3 = 0.264 W/(m· K) 1.2 m below grade ....................... 0.67 × 0.3 = 0.201 W/(m· K) 1.5 m below grade ....................... 0.54 × 0.3 = 0.162 W/(m· K) 1.8 m below grade ....................... 0.45 × 0.3 = 0.135 W/(m· K) Total per metre length of wall .......................... 1.056 W/(m· K) Basement perimeter................................ 2(8.5 + 9.1) = 35.2 m Total wall heat loss............................1.056 × 35.2 = 37.2 W/K Average heat loss per m2................................... 0.14 W/(m2·K) Floor area 8.5 × 9.1 ..................................................... 77.35 m2 Total floor heat loss ........................... 0.14 × 77.35 = 10.8 W/K Total basement heat loss below grade....37.2 + 10.8 = 48 W/K Design temperature difference .......................21 − (−7) = 28 K Maximum rate of heat loss from below-grade basement..................................48 × 28 = 1344 W 28.12 2001 ASHRAE Fundamentals Handbook (SI) Transient Calculations for Basement Walls The heat loss from basement walls can be estimated more accu-rately with a finite element or finite difference computer program by transient simulations (Wang 1979, Bligh et al. 1978). The solution is in the form of heat loss over time, which can be converted to an average U-factor. This approach also offers the possibility for esti-mating the depth below grade to which insulation is economical.
Direct and indirect evidence of hollow concrete block walls shows that a convective path exists within the blocks vertically along the wall (Harrje et al. 1979). Therefore, insulation should be arranged to reduce this convective heat transfer.
Peony et al. (1979) showed that the dynamic thermal perfor-mance of a masonry wall is better when insulation is placed on the exterior. Moreover, transient simulation showed that insulation is more effective when it is placed on the exterior side of the basement wall. Depending on the exposed portion of the block wall and the temperature difference between indoor and outdoor air, exterior application can be 10 to 20% more efficient than a corresponding interior application. However, such exterior insulation must be installed properly to maintain its integrity.
Calculating Transmission Heat Loss from Floor Slabs Concrete slab floors may be (1) unheated, relying for warmth on heat delivered above floor level by the heating system, or (2) heated, containing heated pipes or ducts that constitute a radiant slab or por-tion of it for complete or partial heating of the house.
The perimeter insulation of a slab-on-grade floor is quite impor-tant for comfort and energy conservation. In unheated slab floors, the floor edge must be insulated in order to keep the floor warm. Down-drafts from windows or exposed walls can create pools of chilly air over considerable areas of the floor. In heated slab floors, the floor edge must be insulated to prevent excessive heat loss from the heat-ing pipe or duct embedded in the floor or from the baseboard heater.
Wang (1979) and Bligh et al. (1978) found that heat loss from an unheated concrete slab floor is mostly through the perimeter rather than through the floor and into the ground. Total heat loss is more nearly proportional to the length of the perimeter than to the area of the floor, and it can be estimated by the following equation for both unheated and heated slab floors: (6) where q = heat loss through perimeter, W F2 = heat loss coefficient per foot of perimeter (see Table 16), W/(m·K) P = perimeter or exposed edge of floor, m ti = indoor temperature, °C (For the heated slab, ti is the weighted average heating duct or pipe temperature.) to = outdoor temperature, °C Vertical “I”-shaped systems are used to insulate slab floor perim-eters. In the “I” system, the insulation is placed vertically next to the exposed slab edge, extending downward below grade, as shown in Figure 7.
Breaks or joints must be avoided when the insulation is installed; otherwise, local thermal bridges can be formed, and the overall effi-ciency of the insulation is reduced.
Transient Calculations for Floor Slabs Figure 8 shows four basic slab-on-grade constructions analyzed with a finite element computer program by Wang (1979). Figures 8A-C represent unheated slabs; Figure 8D can be considered a heated slab. Each was investigated with and without insulation of R-0.95 K·m2/W under three climatic conditions (4130, 2970, and 1640 kelvin days). Table 16 lists the results in terms of heat loss coef-ficient F2, based on kelvin days.
Table 16 shows that the heat loss coefficient F2 is sensitive to both construction and insulation. The reverse loss, or heat loss into the ground and outward through the edges of the slab and founda-tion wall, is significant when heating pipes, heating ducts, or base-board heaters are placed near the slab perimeters. To prevent reverse loss, the designer may find it advantageous to use perimeter insula-tion even in warmer climates. For severe winter regions (above 3300 kelvin days), the insulation value should be increased to R >1.8 K·m2/W.
Figure 8A shows that this construction benefits from the wall insulation between block and brick; the insulation is extended roughly 400 mm below the slab floor. Without this wall insulation, the heat loss coefficient F2 would be close to that of the 100 mm block wall construction (Figure 8B). Table 16 can be used to esti-mate F2 under different kelvin days of heating season weather.
CALCULATING INFILTRATION HEAT LOSS Infiltration of outside air causes both sensible and latent heat loss. The energy required to raise the temperature of outdoor infil-trating air to indoor air temperature is the sensible component. The energy associated with net loss of moisture from the space is the latent component. Infiltration is discussed in detail in Chapter 26.
Fig. 6 Lines of Constant Amplitude q F2P ti to – ( ) = Fig. 7 “I”-Shaped or Vertical Insulation System Residential Cooling and Heating Load Calculations 28.13 Sensible Heat Loss The energy required to warm outdoor air entering by infiltration to the temperature of the room is given by (7) where qs = heat flow required to raise temperature of air leaking into building from to to ti, W cp = specific heat of air, kJ/(kg·K) Q = volumetric flow of outdoor air entering building, L/s ρ = density of air at temperature to, kg/m3 Using standard air [ρ = 1.20 kg/m3 and cp = 1.0 kJ/(kg·K)], Equation (7) reduces to (8) The volumetric flow Q of outdoor air entering depends on wind speed and direction, width of cracks or size of openings, type of openings, and other factors explained in Chapter 26. Two methods used to obtain the quantity of infiltration air are the crack length and the air change. Louvers and doors and the direction they face, as well as any other factors affecting infiltration, may need to be considered.
Latent Heat Loss When moisture must be added to the indoor air to maintain win-ter comfort conditions, the energy needed to evaporate an amount of water equivalent to what is lost by infiltration (latent component of infiltration heat loss) must be determined. This energy may be cal-culated by (9) where ql = heat flow required to increase moisture content of air leakage into building from Wo to Wi, W Q = volumetric flow of outdoor air entering building, L/s ρ = density of air at temperature ti, kg/m3 Wi = humidity ratio of indoor air, g/kg (dry air) Wo = humidity ratio of outdoor air, g/kg (dry air) hfg = latent heat of vapor at ti, kJ/kg If the latent heat of vapor hfg is 2500 kJ/kg, and the air density is 1.2 kg/m3, Equation (7) reduces to (10) Crack Length Method The basis of calculation for the crack method is that the amount of crack used for computing the infiltration heat loss should not be less than one-half the total length of crack in the outside walls of the room. In a building without partitions, air entering through cracks on the windward side must leave through cracks on the leeward side. Therefore, one-half the total crack for each side and end of the building is used for calculation. In a room with one exposed wall, all the crack is used. With two, three, or four exposed walls, either the wall with the crack that will result in the greatest air leakage or at least one-half the total crack is used, whichever is greater.
In residences, total infiltration loss of the house is generally con-sidered equal to the sum of infiltration losses of the various rooms.
But, at any given time, infiltration takes place only on the windward side or sides and not on the leeward. Therefore, for determining total heat requirements of larger buildings, it is more accurate to base total infiltration loss on the wall with the most total crack or on at Fig. 8 Slab-on-Grade Foundation Insulation qs cpQρ ti to – ( ) = qs 1.2Q ti to – ( ) = Table 16 Heat Loss Coefficient F2 of Slab Floor Construction, W/K per metre of Perimeter Construction Insulation Kelvin Days (18°C Base) 1640 K·d/yr 2970 K·d/yr 4130 K·d/yr 200 mm block wall, brick facing Uninsulated R = 0.95 K·m2/W from edge to footer 1.07 1.17 1.24 0.83 0.86 0.97 100 mm block wall, brick facing Uninsulated R = 0.95 from edge to footer 1.38 1.45 1.61 0.81 0.85 0.93 Metal stud wall, stucco Uninsulated R = 0.95 from edge to footer 1.99 2.07 2.32 0.88 0.92 1.00 Poured concrete wall with duct near perimetera Uninsulated R = 0.95 from edge to footer, 910 mm under floor 3.18 3.67 4.72 1.11 1.24 1.56 aWeighted average temperature of the heating duct was assumed at 43°C during the heating season (outdoor air temperature less than 18°C).
ql Qρ Wi Wo – ( )hfg 1000 ----------------------------------------= ql 3.0Q Wi Wo – ( ) = 28.14 2001 ASHRAE Fundamentals Handbook (SI) least half the total crack in the building, whichever is greater. When the crack method rather than Equations (8) and (10) is used for esti-mating leakage, the heat loss in terms of the crack length may be expressed as (11) and (12) where B = air leakage for wind velocity and type of window or door crack involved, L/s per metre of crack L = length of window or door crack to be considered, m Air Change Method Some designers base infiltration on an estimated number of air changes rather than the length of window cracks. The number of air changes given in Chapter 26 should be considered only as a guide.
When calculating infiltration losses by the air change method, Equations (8) and (10) can be used by substituting for Q the volume of the room multiplied by the number of air changes.
Exposure Factors Some designers use empirical exposure factors to increase cal-culated heat loss of rooms or spaces on the side(s) of the building exposed to prevailing winds. However, exposure factors are not needed with the method of calculating heat loss described in this chapter. Instead, they may be (1) regarded as safety factors to allow for additional capacity for rooms or spaces exposed to prevailing winds or (2) used to account for the effects of radiation loss, partic-ularly in the case of multistory buildings. Tall buildings may have severe infiltration heat losses induced by stack effect that require special analysis. Although a 15% exposure allowance is often assumed, the actual allowance, if any, is largely a matter of experi-ence and judgment; no test data are available from which to develop rules for the many conditions encountered.
PICKUP LOAD For intermittently heated buildings and night thermostat setback, additional heat is required to raise the temperature of air, building materials, and material contents of a building to the specified tem-perature. The pickup load, which is the rate at which this additional heat must be supplied, depends on the heat capacity of the structure, its material contents, and the time in which these are to be heated.
Relatively little information on pickup load exists; however, Smith (1941, 1942) addressed pickup loads for buildings heated only occasionally, such as auditoriums and churches. Nelson and MacArthur (1978) studied the relationship between thermostat set-back, furnace capacity, and recovery time. Based on this limited information, the following design guidelines are offered.
Because design outdoor temperatures generally provide a sub-stantial margin for outdoor temperatures typically experienced during operating hours, many engineers make no allowance for this additional heat in most buildings. However, the additional heat should be computed and allowed for, as conditions require.
In the case of intermittently heated buildings, an additional 10% capacity should be provided.
In buildings with setback-type thermostats, the furnace must be oversized to reestablish the space temperature in an acceptable time. The amount of oversizing depends on such factors as the amount of setback, inside-to-outside temperature difference, building construction, and acceptable pickup time. Figure 9 shows this relationship for a particular residence. As a rule for residences, a 5.6 K night setback requires 40% oversizing for acceptable pickup time and minimum energy requirements (Nel-son and MacArthur 1978). For smaller setback, the oversizing can be proportionally less. If daytime as well as night setback is practiced, oversizing of up to 60% is warranted.
REFERENCES ASHRAE. 1992. Thermal environmental conditions for human occupancy.
ANSI/ASHRAE Standard 55-1992.
ASHRAE. 1995. Addendum to ANSI/ASHRAE 55-1992. ANSI/ASHRAE Standard 55a-1995.
Bligh, T.P., P. Shipp, and G. Meixel. 1978. Energy comparisons and where to insulate earth sheltered buildings and basements. Earth covered settle-ments, U.S. Department of Energy Conference, Fort Worth, TX.
Chang, J.H. 1958. Ground temperature. Bluehill Meteorological Observa-tory, Harvard University, Cambridge, MA.
Harrje, D.T., G.S. Dutt, and J. Beyea. 1979. Locating and eliminating obscure but major energy losses in residential housing. ASHRAE Trans-actions 85(2).
Houghten, F.C., S.I. Taimuty, C. Gutberlet, and C.J. Brown. 1942. Heat loss through basement walls and floors. ASHVE Transactions 48:369.
Joy, F.A. 1958. Improving attic space insulating values. Heating, Piping and Air Conditioning 30(1):223.
Joy, F.A., J.J. Zabrony, and S. Bhaduri. 1956. Insulating value of reflective elements in an attic under winter conditions. Pennsylvania State Univer-sity, University Park, PA.
Latta, J.K. and G.G. Boileau. 1969. Heat losses from house basements.
Canadian Building 19(10):39.
McQuiston, F.C. 1984. A study and review of existing data to develop a stan-dard methodology for residential heating and cooling load calculations.
ASHRAE Transactions 90(2A):102-36.
McQuiston, F.C. and J.D. Spitler. 1992. Cooling and heating load calcula-tion manual, 2nd ed. ASHRAE, Atlanta.
Nelson, L.W. and J.W. MacArthur. 1978. Energy savings through thermostat setback. ASHRAE Transactions 84(2):3l9-34.
Peony, B.A., F.J. Powell, and D.M. Burch. 1979. Dynamic thermal perfor-mance of an experimental masonry building. NBS Report 10 664, National Institute of Standards and Technology, Gaithersburg, MD.
Rowley, F.B., A.B. Algren, and C.E. Lund. 1940. Methods of moisture con-trol and their application to building construction. Bulletin No. 17 XLIII(4):28. University of Minnesota Engineering Experiment Station.
Smith, E.G. 1941. Heat requirement of intermittently heated buildings.
Texas A&M Engineering Experiment Station Series No. 62 (November).
College Station, TX.
Smith, E.G. 1942. A method of compiling tables for intermittent heating.
Heating, Piping, and Air Conditioning 14(6):386.
Wang, F.S. 1979. Mathematical modeling and computer simulation of insu-lation systems in below grade applications. ASHRAE/DOE Conference on Thermal Performance of the Exterior Envelopes of Buildings, Orlando, FL.
qs 1.2BL ti to – ( ) = ql 3.0BL Wi Wo – ( ) = Fig. 9 Furnace Operating Times Required to Pick Up Space Temperature Following 2.8 and 5.6 K Night Setback 29.1 CHAPTER 29 NONRESIDENTIAL COOLING AND HEATING LOAD CALCULATION PROCEDURES Cooling Load Principles ......................................................... 29.2 Initial Design Considerations ................................................. 29.2 HEAT SOURCES AND HEAT GAIN CALCULATION CONCEPTS ............................................. 29.3 Time Delay Effect .................................................................... 29.3 People ..................................................................................... 29.3 Lighting ................................................................................... 29.3 Electric Motors ....................................................................... 29.7 Appliances ............................................................................... 29.8 Heat Gain Through Fenestration Areas ............................... 29.13 Heat Gain Through Exterior Surfaces .................................. 29.15 Sol-Air Temperature ............................................................. 29.15 Outdoor Air Temperatures .................................................... 29.16 Heat Gain Through Interior Surfaces ................................... 29.18 Infiltration and Ventilation Heat Gain ................................. 29.18 Latent Heat Gain from Moisture Through Permeable Building Materials .......................................... 29.19 Heat Gain from Miscellaneous Sources ................................ 29.19 HEAT BALANCE METHOD OF COOLING LOAD CALCULATION ..................................................... 29.20 Heat Balance Model Assumptions ........................................ 29.20 Elements of Heat Balance Model .......................................... 29.20 General Zone for Load Calculation ...................................... 29.23 Mathematical Description of Heat Balance Procedure ........ 29.23 Input Required for Heat Balance Procedure ........................ 29.25 RADIANT TIME SERIES (RTS) METHOD .......................... 29.25 RTS Cooling Load Assumptions and Principles .................. 29.26 Overview of the Radiant Time Series Method ....................... 29.26 Radiant Time Series Procedure ............................................ 29.27 COMPARISON WITH PREVIOUS METHODS ................... 29.37 HEATING LOAD PRINCIPLES ........................................... 29.38 HIS CHAPTER presents two load calculation methods that Trepresent a significant departure from those in common use.
The technology involved, however—the principle of calculating a heat balance for a given space—is not new. The first of the two methods is the heat balance (HB) method. The calculation proce-dures and scientific principles are explained in equation format.
These equations are coded in a generic computer program named Hbfort, released with Cooling and Heating Load Calculation Prin-ciples (Pedersen et al. 1998), and linked to a user interface program to allow input and output in either inch-pound or SI units. The source code for these programs has been refined and enhanced under ASHRAE Research Project 987 and can be found in the ASHRAE Load Calculation Toolkit.
The second method is called the radiant time series (RTS) method, which is a simplified method directly related to and derived from the HB calculation procedure. This chapter presents the principles of both procedures rather than a working tool for the cooling load practitioner. The load prediction of an actual, multiple-room building requires a complex computer program. While the procedures described in this chapter are the most reliable means for estimating cooling load for a defined building space, the methods described in earlier editions of the Handbook series are valid for many applications. Each of these earlier proce-dures is, however, a simplification of the heat balance principles, and their use requires experience to deal with atypical situations or unusual circumstances. In fact, any cooling or heating load estimate is no better than the assumptions used to define conditions and parameters such as physical makeup of the various envelope sur-faces, conditions of occupancy and use, and ambient weather con-ditions outside the building. The experience of the practitioner can never be ignored.
The procedures described in this chapter are concerned with a given space or zone in a building. When estimating the loads for a group of spaces such as might be handled by a single air-handling system, the assembled zones must be analyzed to consider (1) the simultaneous effects taking place; (2) any diversification of heat gains for occupants, lighting, or other internal load sources; (3) ventilation; and/or (4) any other unique circumstances. With large buildings that involve more than a single HVAC system, simulta-neous loads and any additional diversities also must be consid-ered. The methods presented in this chapter are expressed as hourly load summaries, reflecting 24 h input schedules and pro-files of the individual load variables. Specific systems and appli-cations may require different profiles.
In comparing the HB and RTS methods against the methods described in earlier versions of this chapter, the primary difference is the directness of approach as opposed to the simplifying tech-niques necessitated by the limited computer capability available in earlier days. The transfer function method (TFM), for example, required many calculation steps. Also, this method was originally designed for energy analysis with emphasis on daily, monthly, and annual energy use and, thus, was more oriented to average hourly cooling loads than peak design loads.
The total equivalent temperature differential method with time averaging (TETD/TA) has been a highly reliable—if subjec-tive—method of load estimating since its initial presentation in the 1967 ASHRAE Handbook—Fundamentals. Originally conceived as a manual method of calculation, it proved suitable only as a com-puter application because of the need to calculate an extended pro-file of hourly heat gain values from which the radiant components had to be averaged over a time perceived to represent the general mass of the building involved. Because this perception of thermal storage characteristics of a given building was almost entirely sub-jective, with little specific information for the user to judge varia-tions, the TETD/TA method’s primary usefulness has always been to the experienced engineer.
The cooling load temperature differential method with solar cooling load factors (CLTD/CLF) was an attempt to simplify the two-step TFM and TETD/TA methods into a single-step technique that allowed proceeding directly from raw data to cooling load with-out the intermediate conversion of radiant heat gain to cooling load.
A series of factors were taken from cooling load calculation results (produced by more sophisticated methods) as “equivalent tempera-ture differences” for use in traditional conduction (q = UA∆t) equa-The preparation of this chapter is assigned to TC 4.1, Load Calculation Data and Procedures.
29.2 2001 ASHRAE Fundamentals Handbook (SI) tions. The results, however, are approximate cooling load values rather than simple heat gain values. The simplifications required for this process limit the applicability of this method. In fact, past ver-sions of this chapter listed several areas of potential use that should be avoided as beyond the range of applicability.
COOLING LOAD PRINCIPLES The variables affecting cooling load calculations are numerous, often difficult to define precisely, and always intricately interre-lated. Many cooling load components vary in magnitude over a wide range during a 24 h period. Because these cyclic changes in load components often are not in phase with each other, each must be analyzed to establish the maximum cooling load for a building or zone. A zoned system (a system of conditioning equipment serving several independent areas, each with its own temperature control) need provide no greater total cooling load capacity than the largest hourly summary of simultaneous zone loads throughout a design day; however, it must handle the peak cooling load for each zone at its individual peak hour. At certain times of the day during the heat-ing or intermediate seasons, some zones may require heating while others require cooling.
Calculation Accuracy A realistic cooling load calculation gives values adequate for acceptable system performance. Variation in the heat transmission coefficients of typical building materials and composite assemblies, the differing motivations and skills of those who construct the build-ing, and the manner in which the building is actually operated are some of the variables that make a precise calculation impossible.
Even if the designer uses reasonable procedures to account for these factors, the calculation can never be more than a good estimate of the actual cooling load.
Heat Flow Rates In air-conditioning design, four related heat flow rates, each of which varies with time, must be differentiated: (1) space heat gain, (2) space cooling load, (3) space heat extraction rate, and (4) cooling coil load.
Space Heat Gain. This instantaneous rate of heat gain is the rate at which heat enters into and/or is generated within a space. Heat gain is classified by (1) the mode in which it enters the space and (2) whether it is a sensible or latent gain.
Mode of entry. The mode of entry includes (1) solar radiation through transparent surfaces; (2) heat conduction through exterior walls and roofs; (3) heat conduction through ceilings, floors, and interior partitions; (4) heat generated in the space by occupants, lights, and appliances; (5) energy transfer as a result of ventilation and infiltration of outdoor air; and (6) miscellaneous heat gains.
Sensible or latent heat. Sensible heat is added directly to the conditioned space by conduction, convection, and/or radiation.
Latent heat gain occurs when moisture is added to the space (e.g., from vapor emitted by occupants and equipment). To maintain a constant humidity ratio, water vapor must condense on the cooling apparatus and be removed at a rate equal to the rate it is added to the space. The amount of energy required to offset the latent heat gain essentially equals the product of the rate of condensation and the latent heat of condensation. In selecting cooling apparatus, it is necessary to distinguish between sensible and latent heat gain.
Every cooling apparatus has a maximum sensible heat removal capacity and a maximum latent heat removal capacity for particular operating conditions.
Space Cooling Load. This is the rate at which heat must be removed from the space to maintain a constant space air tempera-ture. The sum of all space instantaneous heat gains at any given time does not necessarily (or even frequently) equal the cooling load for the space at that same time.
Radiant heat gain. Radiant energy must first be absorbed by the surfaces that enclose the space (walls, floor, and ceiling) and the objects in the space (furniture, etc.). When these surfaces and objects become warmer than the surrounding air, some of their heat is transferred to the air by convection. The composite heat storage capacity of these surfaces and objects determines the rate at which their respective surface temperatures increase for a given radiant input and thus governs the relationship between the radiant portion of heat gain and its corresponding part of the space cooling load (Figure 1). The thermal storage effect is critically important in dif-ferentiating between instantaneous heat gain for a given space and its cooling load at that moment. Predicting the nature and magnitude of this phenomenon in order to estimate a realistic cooling load for a particular combination of circumstances has long been a subject of interest to design engineers. The section on Bibliography lists some of the early work on the subject.
Space Heat Extraction Rate. The rate at which heat is removed from the conditioned space equals the space cooling load only if the room air temperature is held constant. Along with the intermittent operation of the cooling equipment, the control sys-tem characteristics usually permit a minor cyclic variation or swing in room temperature. Therefore, a proper simulation of the control system gives a more realistic value of energy removal over a fixed period than using the values of the space cooling load.
However, this concept is primarily important for estimating energy use over time; it is not needed to calculate design peak cooling load for equipment selection.
Cooling Coil Load. The rate at which energy is removed at the cooling coil that serves one or more conditioned spaces equals the sum of the instantaneous space cooling loads (or space heat extrac-tion rate if it is assumed that the space temperature does not vary) for all the spaces served by the coil, plus any external loads. Such external loads include fan heat gain, duct heat gain, and outdoor air heat and moisture brought into the cooling equipment to satisfy the ventilation requirement.
Cooling Load Estimation in Practice Frequently, a cooling load must be calculated before every parameter in the conditioned space can be properly or completely defined. An example is a cooling load estimate for a new building with many floors of unleased spaces where detailed partition requirements, furnishings, lighting selection, and layout cannot be predefined. Potential tenant modifications once the building is occupied also must be considered. The load estimating process requires proper engineering judgment that includes a thorough understanding of heat balance fundamentals.
INITIAL DESIGN CONSIDERATIONS To calculate a space cooling load, detailed building design infor-mation and weather data at selected design conditions are required.
Generally, the following steps should be followed.
Fig. 1 Origin of Difference Between Magnitude of Instantaneous Heat Gain and Instantaneous Cooling Load Nonresidential Cooling and Heating Load Calculation Procedures 29.3 Data Assembly Building Characteristics. Obtain characteristics of the building.
Building materials, component size, external surface colors, and shape are usually determined from building plans and specifications.
Configuration. Determine building location, orientation, and external shading from building plans and specifications. Shading from adjacent buildings can be determined by a site plan or by vis-iting the proposed site but should be carefully evaluated as to its probable permanence before it is included in the calculation. The possibility of abnormally high ground-reflected solar radiation (i.e., from adjacent water, sand, or parking lots) or solar load from adja-cent reflective buildings should not be overlooked.
Outdoor Design Conditions. Obtain appropriate weather data, and select outdoor design conditions. For outdoor design conditions for a large number of weather stations, see Chapter 27. Note, how-ever, that these values for the design dry-bulb and mean coincident wet-bulb temperatures may vary considerably from data traditionally used in various areas. Use judgment to ensure that results are consis-tent with expectations. Also, consider prevailing wind velocity and the relationship of a project site to the selected weather station.
In recent years, several research projects have greatly expanded the amount of available weather data (Colliver et al. 1995, 1998, 2000). In addition to the conventional dry-bulb with mean coinci-dent wet-bulb, data are now available for wet-bulb and dew-point with mean coincident dry-bulb. The peak load for a space that requires both large quantities of outside air and close control of moisture may occur at peak wet-bulb or peak dew-point conditions when the corresponding dry-bulb temperature is significantly lower than normal design conditions. Indoor Design Conditions. Select indoor design conditions, such as indoor dry-bulb temperature, indoor wet-bulb tempera-ture, and ventilation rate. Include permissible variations and con-trol limits.
Operating Schedules. Obtain a proposed schedule of lighting, occupancy, internal equipment, appliances, and processes that con-tribute to the internal thermal load. Determine the probability that the cooling equipment will be operated continuously or shut off dur-ing unoccupied periods (e.g., nights and/or weekends).
Date and Time. Select the time of day and month to do the cool-ing load calculation. Frequently, several different times of day and several different months must be analyzed to determine the peak load time. The particular day and month are often dictated by peak solar conditions. For southern exposures in north latitudes above 32° having large fenestration areas, the peak space cooling load usu-ally occurs in December or January. To calculate a space cooling load under these conditions, the warmest temperature for the winter months must be known. These data can be found for the United States in Chapter 27, Table 4B.
Additional Considerations The proper design and sizing of all-air or air-and-water central air-conditioning systems require more than calculation of the cool-ing load in the space to be conditioned. The type of air-conditioning system, fan energy, fan location, duct heat loss and gain, duct leak-age, heat extraction lighting systems, and type of return air system all affect system load and component sizing. Adequate system design and component sizing require that system performance be analyzed as a series of psychrometric processes.
HEAT SOURCES AND HEAT GAIN CALCULATION CONCEPTS TIME DELAY EFFECT The energy absorbed by walls, floor, furniture, etc., contributes to space cooling load only after a time lag, with some part of this energy still present and reradiating after the heat sources have been switched off or are no longer present (Figure 2). There is always significant delay between the time of switching on or otherwise activating a heat source and the point when reradi-ated energy equals that being instantaneously stored. This time lag must be considered when calculating cooling load because the load felt by the space can be much lower than the instantaneous heat gain being generated, and the peak load for the space may be affected significantly.
PEOPLE Table 1 gives representative rates at which heat and moisture are given off by human beings in different states of activity. Often these sensible and latent heat gains constitute a large fraction of the total load. Even for short-term occupancy, the extra heat and mois-ture brought in by people may be significant. Chapter 8 should be consulted for detailed information; however, Table 1 summarizes design data representing conditions commonly encountered.
The conversion of sensible heat gain from people to space cool-ing load is affected by the thermal storage characteristics of that space, since some percentage of the sensible load is radiant energy.
Latent heat gains are considered instantaneous.
LIGHTING Because lighting is often the major space load cooling compo-nent, an accurate estimate of the space heat gain it imposes is needed. Calculation of this load component is not straightforward; the rate of cooling load due to lighting at any given moment can be quite different from the heat equivalent of power supplied instanta-neously to those lights.
Instantaneous Heat Gain from Lighting The primary source of heat from lighting comes from light-emit-ting elements, or lamps, although significant additional heat may be generated from associated appurtenances in the light fixtures that house such lamps. Generally, the instantaneous rate of heat gain from electric lighting may be calculated from qel = WFulFsa (1) where qel = heat gain, W W = total light wattage Ful = lighting use factor Fsa = lighting special allowance factor The total light wattage is obtained from the ratings of all lamps installed, both for general illumination and for display use.
The lighting use factor is the ratio of the wattage in use, for the conditions under which the load estimate is being made, to the total Fig. 2 Thermal Storage Effect in Cooling Load from Lights 29.4 2001 ASHRAE Fundamentals Handbook (SI) installed wattage. For commercial applications such as stores, the use factor would generally be unity.
The special allowance factor is for fluorescent fixtures and/or fixtures that are either ventilated or installed so that only part of their heat goes to the conditioned space. For fluorescent or high-intensity discharge fixtures, the special allowance factor accounts primarily for ballast losses. Table 2 shows that the special allowance factor for a two-lamp fluorescent fixture ranges from 0.94 for T8 lamps with an electronic ballast to 1.21 for energy-saver T12 lamps with a standard electromagnetic ballast. High-intensity discharge fixtures, such as metal halide, may have special allowance factors varying from 1.07 to 1.44, depending on the lamp wattage and quan-tity of lamps per fixture, and should be dealt with individually. A wide variety of lamp and ballast combinations is available, and bal-last catalog data provide the overall fixture wattage.
For ventilated or recessed fixtures, manufacturers’ or other data must be sought to establish the fraction of the total wattage that may be expected to enter the conditioned space directly (and subject to time lag effect) versus that which must be picked up by return air or in some other appropriate manner.
Light Heat Components Cooling load caused by lights recessed into ceiling cavities is made up of two components: one part (known as the heat-to-space load) comes from the light heat directly contributing to the space heat gain, and the other is the light heat released into the above-ceil-ing cavity, which (if used as a return air plenum) is mostly picked up by the return air that passes over or through the light fixtures. In such a ceiling return air plenum, this second part of the load (some-times referred to as heat-to-return) never enters the conditioned space. It does, however, add to the overall load and significantly influences the load calculation.
Even though the total cooling load imposed on the cooling coil from these two components remains the same, the larger the fraction of heat output picked up by the return air, the more the space cooling load is reduced. The minimum required airflow rate for the condi-tioned space is decreased as the space cooling load decreases. Sup-ply fan power decreases accordingly, which ultimately results in reduced energy consumption for the system and possibly reduced equipment size as well.
For ordinary design load estimation, the heat gain for each com-ponent may be calculated simply as a fraction of the total lighting load by using judgment to estimate heat-to-space and heat-to-return percentages (Mitalas and Kimura 1971).
Return Air Light Fixtures Two generic types of return air light fixture are available—those that allow and those that do not allow return air to flow through the lamp chamber. The first type is sometimes called a heat-of-light fix-ture. The percentage of light heat released through the plenum side of various ventilated fixtures can be obtained from lighting fixture manufacturers. For representative data, see Nevins et al. (1971).
Even unventilated fixtures lose some heat to plenum spaces; how-ever, most of the heat ultimately enters the conditioned space from a dead-air plenum or is picked up by return air via ceiling return air openings. The percentage of heat to return air ranges from 40 to 60% for heat-to-return ventilated fixtures or 15 to 25% for unventi-lated fixtures.
Plenum Temperatures As heat from lighting is picked up by the return air, the temper-ature differential between the ceiling cavity and the conditioned space causes part of that heat to flow from the ceiling back to the conditioned space. Return air from the conditioned space can be ducted to capture light heat without passing through a ceiling ple-num as such, or the ceiling space can be used as a return air plenum, causing the distribution of light heat to be handled in distinctly dif-ferent ways. Most plenum temperatures do not rise more than 0.5 to 1.5 K above space temperature, thus generating only a relatively small thermal gradient for heat transfer through plenum surfaces but Table 1 Representative Rates at Which Heat and Moisture Are Given Off by Human Beings in Different States of Activity Degree of Activity Total Heat, W Sensible Heat, W Latent Heat, W % Sensible Heat that is Radiantb Adult Male Adjusted, M/Fa Low V High V Seated at theater Theater, matinee 115 95 65 30 Seated at theater, night Theater, night 115 105 70 35 60 27 Seated, very light work Offices, hotels, apartments 130 115 70 45 Moderately active office work Offices, hotels, apartments 140 130 75 55 Standing, light work; walking Department store; retail store 160 130 75 55 58 38 Walking, standing Drug store, bank 160 145 75 70 Sedentary work Restaurantc 145 160 80 80 Light bench work Factory 235 220 80 140 Moderate dancing Dance hall 265 250 90 160 49 35 Walking 4.8 km/h; light machine work Factory 295 295 110 185 Bowlingd Bowling alley 440 425 170 255 Heavy work Factory 440 425 170 255 54 19 Heavy machine work; lifting Factory 470 470 185 285 Athletics Gymnasium 585 525 210 315 Notes: 1. Tabulated values are based on 24°C room dry-bulb temperature. For 27°C room dry bulb, the total heat remains the same, but the sensible heat values should be decreased by approximately 20%, and the latent heat values increased accord-ingly.
2. Also refer to Table 4, Chapter 8, for additional rates of metabolic heat generation.
3. All values are rounded to nearest 5 W.
aAdjusted heat gain is based on normal percentage of men, women, and children for the application listed, with the postulate that the gain from an adult female is 85% of 85% of that for an adult male, and that the gain from a child is 75% of that for an adult male.
bValues approximated from data in Table 6, Chapter 8, where is air velocity with limits shown in that table.
cAdjusted heat gain includes 18 W for food per individual (9 W sensible and 9 W latent).
dFigure one person per alley actually bowling, and all others as sitting (117 W) or standing or walking slowly (231 W).
Nonresidential Cooling and Heating Load Calculation Procedures 29.5 Table 2 Typical Nonincandescent Light Fixtures Description Ballast Watts/Lamp Lamps/Fixture Lamp Watts Fixture Watts Special Allowance Factor Description Ballast Watts/Lamp Lamps/Fixture Lamp Watts Fixture Watts Special Allowance Factor Compact Fluorescent Fixtures Twin, (1) 5 W lamp Mag-Std 5 1 5 9 1.80 Twin, (2) 40 W lamp Mag-Std 40 2 80 85 1.06 Twin, (1) 7 W lamp Mag-Std 7 1 7 10 1.43 Quad, (1) 13 W lamp Electronic 13 1 13 15 1.15 Twin, (1) 9 W lamp Mag-Std 9 1 9 11 1.22 Quad, (1) 26 W lamp Electronic 26 1 26 27 1.04 Quad, (1) 13 W lamp Mag-Std 13 1 13 17 1.31 Quad, (2) 18 W lamp Electronic 18 2 36 38 1.06 Quad, (2) 18 W lamp Mag-Std 18 2 36 45 1.25 Quad, (2) 26 W lamp Electronic 26 2 52 50 0.96 Quad, (2) 22 W lamp Mag-Std 22 2 44 48 1.09 Twin or multi, (2) 32 W lamp Electronic 32 2 64 62 0.97 Quad, (2) 26 W lamp Mag-Std 26 2 52 66 1.27 Fluorescent Fixtures (1) 450 mm, T8 lamp Mag-Std 15 1 15 19 1.27 (4) 1200 mm, T8 lamp Electronic 32 4 128 120 0.94 (1) 450 mm, T12 lamp Mag-Std 15 1 15 19 1.27 (1) 1500 mm, T12 lamp Mag-Std 50 1 50 63 1.26 (2) 450 mm, T8 lamp Mag-Std 15 2 30 36 1.20 (2) 1500 mm, T12 lamp Mag-Std 50 2 100 128 1.28 (2) 450 mm, T12 lamp Mag-Std 15 2 30 36 1.20 (1) 1500 mm, T12 HO lamp Mag-Std 75 1 75 92 1.23 (1) 600 mm, T8 lamp Mag-Std 17 1 17 24 1.41 (2) 1500 mm, T12 HO lamp Mag-Std 75 2 150 168 1.12 (1) 600 mm, T12 lamp Mag-Std 20 1 20 28 1.40 (1) 1500 mm, T12 ES VHO lamp Mag-Std 135 1 135 165 1.22 (2) 600 mm, T12 lamp Mag-Std 20 2 40 56 1.40 (2) 1500 mm, T12 ES VHO lamp Mag-Std 135 2 270 310 1.15 (1) 600 mm, T12 HO lamp Mag-Std 35 1 35 62 1.77 (1) 1500 mm, T12 HO lamp Mag-ES 75 1 75 88 1.17 (2) 600 mm, T12 HO lamp Mag-Std 35 2 70 90 1.29 (2) 1500 mm, T12 HO lamp Mag-ES 75 2 150 176 1.17 (1) 600 mm, T8 lamp Electronic 17 1 17 16 0.94 (1) 1500 mm, T12 lamp Electronic 50 1 50 44 0.88 (2) 600 mm, T8 lamp Electronic 17 2 34 31 0.91 (2) 1500 mm, T12 lamp Electronic 50 2 100 88 0.88 (1) 900 mm, T12 lamp Mag-Std 30 1 30 46 1.53 (1) 1500 mm, T12 HO lamp Electronic 75 1 75 69 0.92 (2) 900 mm, T12 lamp Mag-Std 30 2 60 81 1.35 (2) 1500 mm, T12 HO lamp Electronic 75 2 150 138 0.92 (1) 900 mm, T12 ES lamp Mag-Std 25 1 25 42 1.68 (1) 1500 mm, T8 lamp Electronic 40 1 40 36 0.90 (2) 900 mm, T12 ES lamp Mag-Std 25 2 50 73 1.46 (2) 1500 mm, T8 lamp Electronic 40 2 80 72 0.90 (1) 900 mm, T12 HO lamp Mag-Std 50 1 50 70 1.40 (3) 1500 mm, T8 lamp Electronic 40 3 120 106 0.88 (2) 900 mm, T12 HO lamp Mag-Std 50 2 100 114 1.14 (4) 1500 mm, T8 lamp Electronic 40 4 160 134 0.84 (2) 900 mm, T12 lamp Mag-ES 30 2 60 74 1.23 (1) 1800 mm, T12 lamp Mag-Std 55 1 55 76 1.38 (2) 900 mm, T12 ES lamp Mag-ES 25 2 50 66 1.32 (2) 1800 mm, T12 lamp Mag-Std 55 2 110 122 1.11 (1) 900 mm, T12 lamp Electronic 30 1 30 31 1.03 (3) 1800 mm, T12 lamp Mag-Std 55 3 165 202 1.22 (1) 900 mm, T12 ES lamp Electronic 25 1 25 26 1.04 (4) 1800 mm, T12 lamp Mag-Std 55 4 220 244 1.11 (1) 900 mm, T8 lamp Electronic 25 1 25 24 0.96 (1) 1800 mm, T12 HO lamp Mag-Std 85 1 85 120 1.41 (2) 900 mm, T12 lamp Electronic 30 2 60 58 0.97 (2) 1800 mm, T12 HO lamp Mag-Std 85 2 170 220 1.29 (2) 900 mm, T12 ES lamp Electronic 25 2 50 50 1.00 (1) 1800 mm, T12 VHO lamp Mag-Std 160 1 160 180 1.13 (2) 900 mm, T8 lamp Electronic 25 2 50 46 0.92 (2) 1800 mm, T12 VHO lamp Mag-Std 160 2 320 330 1.03 (2) 900 mm, T8 HO lamp Electronic 25 2 50 50 1.00 (2) 1800 mm, T12 lamp Mag-ES 55 2 110 122 1.11 (2) 900 mm, T8 VHO lamp Electronic 25 2 50 70 1.40 (4) 1800 mm, T12 lamp Mag-ES 55 4 220 244 1.11 (1) 1200 mm, T12 lamp Mag-Std 40 1 40 55 1.38 (2) 1800 mm, T12 HO lamp Mag-ES 85 2 170 194 1.14 (2) 1200 mm, T12 lamp Mag-Std 40 2 80 92 1.15 (4) 1800 mm, T12 HO lamp Mag-ES 85 4 340 388 1.14 (3) 1200 mm, T12 lamp Mag-Std 40 3 120 140 1.17 (1) 1800 mm, T12 lamp Electronic 55 1 55 68 1.24 (4) 1200 mm, T12 lamp Mag-Std 40 4 160 184 1.15 (2) 1800 mm, T12 lamp Electronic 55 2 110 108 0.98 (1) 1200 mm, T12 ES lamp Mag-Std 34 1 34 48 1.41 (3) 1800 mm, T12 lamp Electronic 55 3 165 176 1.07 (2) 1200 mm, T12 ES lamp Mag-Std 34 2 68 82 1.21 (4) 1800 mm, T12 lamp Electronic 55 4 220 216 0.98 (3) 1200 mm, T12 ES lamp Mag-Std 34 3 102 100 0.98 (1) 2400 mm, T12 ES lamp Mag-Std 60 1 60 75 1.25 (4) 1200 mm, T12 ES lamp Mag-Std 34 4 136 164 1.21 (2) 2400 mm, T12 ES lamp Mag-Std 60 2 120 128 1.07 (1) 1200 mm, T12 ES lamp Mag-ES 34 1 34 43 1.26 (3) 2400 mm, T12 ES lamp Mag-Std 60 3 180 203 1.13 (2) 1200 mm, T12 ES lamp Mag-ES 34 2 68 72 1.06 (4) 2400 mm, T12 ES lamp Mag-Std 60 4 240 256 1.07 (3) 1200 mm, T12 ES lamp Mag-ES 34 3 102 115 1.13 (1) 2400 mm, T12 ES HO lamp Mag-Std 95 1 95 112 1.18 (4) 1200 mm, T12 ES lamp Mag-ES 34 4 136 144 1.06 (2) 2400 mm, T12 ES HO lamp Mag-Std 95 2 190 227 1.19 (1) 1200 mm, T8 lamp Mag-ES 32 1 32 35 1.09 (3) 2400 mm, T12 ES HO lamp Mag-Std 95 3 285 380 1.33 (2) 1200 mm, T8 lamp Mag-ES 32 2 64 71 1.11 (4) 2400 mm, T12 ES HO lamp Mag-Std 95 4 380 454 1.19 (3) 1200 mm, T8 lamp Mag-ES 32 3 96 110 1.15 (1) 2400 mm, T12 ES VHO lamp Mag-Std 185 1 185 205 1.11 (4) 1200 mm, T8 lamp Mag-ES 32 4 128 142 1.11 (2) 2400 mm, T12 ES VHO lamp Mag-Std 185 2 370 380 1.03 (1) 1200 mm, T12 ES lamp Electronic 34 1 34 32 0.94 (3) 2400 mm, T12 ES VHO lamp Mag-Std 185 3 555 585 1.05 (2) 1200 mm, T12 ES lamp Electronic 34 2 68 60 0.88 (4) 2400 mm, T12 ES VHO lamp Mag-Std 185 4 740 760 1.03 (3) 1200 mm, T12 ES lamp Electronic 34 3 102 92 0.90 (2) 2400 mm, T12 ES lamp Mag-ES 60 2 120 123 1.03 (4) 1200 mm, T12 ES lamp Electronic 34 4 136 120 0.88 (3) 2400 mm, T12 ES lamp Mag-ES 60 3 180 210 1.17 (1) 1200 mm, T8 lamp Electronic 32 1 32 32 1.00 (4) 2400 mm, T12 ES lamp Mag-ES 60 4 240 246 1.03 (2) 1200 mm, T8 lamp Electronic 32 2 64 60 0.94 (2) 2400 mm, T12 ES HO lamp Mag-ES 95 2 190 207 1.09 (3) 1200 mm, T8 lamp Electronic 32 3 96 93 0.97 (4) 2400 mm, T12 ES HO lamp Mag-ES 95 4 380 414 1.09 29.6 2001 ASHRAE Fundamentals Handbook (SI) a relatively large percentage reduction in space cooling load. (Many engineers believe that a major reason for plenum temperatures not becoming more elevated is due to leakage into the plenum from sup-ply air ducts normally concealed there.) Energy Balance Where the ceiling space is used as a return air plenum, an energy balance requires that the heat picked up from the lights into the return air (1) become a part of the cooling load to the return air (rep-resented by a temperature rise of the return air as it passes through the ceiling space), (2) be partially transferred back into the condi-tioned space through the ceiling material below, and/or (3) may be partially “lost” (from the space) through the floor surfaces above the plenum. In a multistory building, the conditioned space fre-quently gains heat through its floor from a similar plenum below, offsetting the loss just mentioned. The radiant component of heat leaving the ceiling or floor surface of a plenum is normally so small that all such heat transfer is considered convective for calculation purposes.
Figure 3 shows a schematic diagram of a typical return air ple-num. The following equations, using the heat flow directions shown in Figure 3, represent the heat balance of a return air plenum design for a typical interior room in a multifloor building: q1 = UcAc(tp – tr) (2) (1) 2400 mm, T12 ES lamp Electronic 60 1 60 69 1.15 (1) 2400 mm, T8 HO lamp Electronic 59 1 59 68 1.15 (2) 2400 mm, T12 ES lamp Electronic 60 2 120 110 0.92 (1) 2400 mm, T8 VHO lamp Electronic 59 1 59 71 1.20 (3) 2400 mm, T12 ES lamp Electronic 60 3 180 179 0.99 (2) 2400 mm, T8 lamp Electronic 59 2 118 109 0.92 (4) 2400 mm, T12 ES lamp Electronic 60 4 240 220 0.92 (3) 2400 mm, T8 lamp Electronic 59 3 177 167 0.94 (1) 2400 mm, T12 ES HO lamp Electronic 95 1 95 80 0.84 (4) 2400 mm, T8 lamp Electronic 59 4 236 219 0.93 (2) 2400 mm, T12 ES HO lamp Electronic 95 2 190 173 0.91 (2) 2400 mm, T8 HO lamp Electronic 86 2 172 160 0.93 (4) 2400 mm, T12 ES HO lamp Electronic 95 4 380 346 0.91 (4) 2400 mm, T8 HO lamp Electronic 86 4 344 320 0.93 (1) 2400 mm, T8 lamp Electronic 59 1 59 58 0.98 Circular Fluorescent Fixtures Circlite, (1) 20 W lamp Mag-PH 20 1 20 20 1.00 (2) 200 mm circular lamp Mag-RS 22 2 44 52 1.18 Circlite, (1) 22 W lamp Mag-PH 22 1 22 20 0.91 (1) 300 mm circular lamp Mag-RS 32 1 32 31 0.97 Circline, (1) 32 W lamp Mag-PH 32 1 32 40 1.25 (2) 300 mm circular lamp Mag-RS 32 2 64 62 0.97 (1) 150 mm circular lamp Mag-RS 20 1 20 25 1.25 (1) 400 mm circular lamp Mag-Std 40 1 40 35 0.88 (1) 200 mm circular lamp Mag-RS 22 1 22 26 1.18 High-Pressure Sodium Fixtures (1) 35 W lamp HID 35 1 35 46 1.31 (1) 250 W lamp HID 250 1 250 295 1.18 (1) 50 W lamp HID 50 1 50 66 1.32 (1) 310 W lamp HID 310 1 310 365 1.18 (1) 70 W lamp HID 70 1 70 95 1.36 (1) 360 W lamp HID 360 1 360 414 1.15 (1) 100 W lamp HID 100 1 100 138 1.38 (1) 400 W lamp HID 400 1 400 465 1.16 (1) 150 W lamp HID 150 1 150 188 1.25 (1) 1000 W lamp HID 1000 1 1000 1100 1.10 (1) 200 W lamp HID 200 1 200 250 1.25 Metal Halide Fixtures (1) 32 W lamp HID 32 1 32 43 1.34 (1) 250 W lamp HID 250 1 250 295 1.18 (1) 50 W lamp HID 50 1 50 72 1.44 (1) 400 W lamp HID 400 1 400 458 1.15 (1) 70 W lamp HID 70 1 70 95 1.36 (2) 400 W lamp HID 400 2 800 916 1.15 (1) 100 W lamp HID 100 1 100 128 1.28 (1) 750 W lamp HID 750 1 750 850 1.13 (1) 150 W lamp HID 150 1 150 190 1.27 (1) 1000 W lamp HID 1000 1 1000 1080 1.08 (1) 175 W lamp HID 175 1 175 215 1.23 (1) 1500 W lamp HID 1500 1 1500 1610 1.07 Mercury Vapor Fixtures (1) 40 W lamp HID 40 1 40 50 1.25 (1) 250 W lamp HID 250 1 250 290 1.16 (1) 50 W lamp HID 50 1 50 74 1.48 (1) 400 W lamp HID 400 1 400 455 1.14 (1) 75 W lamp HID 75 1 75 93 1.24 (2) 400 W lamp HID 400 2 800 910 1.14 (1) 100 W lamp HID 100 1 100 125 1.25 (1) 700 W lamp HID 700 1 700 780 1.11 (1) 175 W lamp HID 175 1 175 205 1.17 (1) 1000 W lamp HID 1000 1 1000 1075 1.08 Abbreviations: Mag = electromagnetic; ES = energy saver; Std = standard; HID = high-intensity discharge; HO = high output; VHO = very high output; PH = preheat; RS = rapid start Table 2 Typical Nonincandescent Light Fixtures (Concluded) Description Ballast Watts/Lamp Lamps/Fixture Lamp Watts Fixture Watts Special Allowance Factor Description Ballast Watts/Lamp Lamps/Fixture Lamp Watts Fixture Watts Special Allowance Factor Fig. 3 Schematic Diagram of Typical Return Air Plenum Nonresidential Cooling and Heating Load Calculation Procedures 29.7 q2 = UfAf(tp – tfa) (3) q3 = 1.23Q(tp – tr) (4) qlp – q2 – q1 – q3 = 0 (5) (6) where q1 = heat gain to space from plenum through ceiling, W q2 = heat loss from plenum through floor above, W q3 = heat gain “pickup” by return air, W Q = return airflow, L/s qlp = light heat gain to plenum via return air, W qlr = light heat gain to space, W qf = heat gain from plenum below, through floor, W qw = heat gain from exterior wall, W qr = space cooling load, including appropriate treatment of qlr, qf, and/or qw, W tp = plenum temperature, °C tr = space temperature, °C tfa = space temperature of floor above, °C ts = supply air temperature, °C From heat balance Equation (5), tp can be found as the resultant return air temperature or plenum temperature. The results, although rigorous and best solved by computer, are important in determining the cooling load, which affects equipment size selection, future energy consumption, and other factors.
Equations (2) through (6) are simplified to illustrate the heat bal-ance relationship. Heat gain into a return air plenum is not limited to the heat of lights alone. Exterior walls directly exposed to the ceil-ing space will transfer heat directly to or from the return air. For sin-gle-story buildings or the top floor of a multistory building, the roof heat gain or loss enters or leaves the ceiling plenum rather than entering or leaving the conditioned space directly. The supply air quantity calculated by Equation (6) is only for the conditioned space under consideration and is assumed equal to the return air quantity.
The amount of airflow through a return plenum above a condi-tioned space may not be limited to that supplied into the space under consideration; it will, however, have no noticeable effect on plenum temperature if the surplus comes from an adjacent plenum operating under similar conditions. Where special conditions exist, heat bal-ance Equations (2) through (6) must be modified appropriately.
Finally, even though the building’s thermal storage has some effect, the amount of heat entering the return air is small and may be con-sidered as convective for calculation purposes.
ELECTRIC MOTORS Instantaneous heat gain from equipment operated by electric motors within a conditioned space is calculated as qem = (P/EM)FUMFLM (7) where qem = heat equivalent of equipment operation, W P = motor power rating, W EM = motor efficiency, decimal fraction < 1.0 FUM = motor use factor, 1.0 or decimal fraction < 1.0 FLM = motor load factor, 1.0 or decimal fraction < 1.0 The motor use factor may be applied when motor use is known to be intermittent with significant nonuse during all hours of oper-ation (e.g., overhead door operator). For conventional applications, its value would be 1.0.
The motor load factor is the fraction of the rated load being delivered under the conditions of the cooling load estimate. In Equation (7), it is assumed that both the motor and the driven equipment are in the conditioned space. If the motor is outside the space or airstream, qem = PFUMFLM (8) When the motor is inside the conditioned space or airstream but the driven machine is outside, (9) Equation (9) also applies to a fan or pump in the conditioned space that exhausts air or pumps fluid outside that space.
Tables 3A and 3B give average efficiencies and related data representative of typical electric motors, generally derived from the lower efficiencies reported by several manufacturers of open, drip-proof motors. These reports indicate that totally enclosed Q qr q1 + 1.23 tr ts – ( ) -----------------------------= Table 3A Average Efficiencies and Related Data Representative of Typical Electric Motors Motor Name-plate or Rated Horse-power Motor Type Nominal rpm Full Load Motor Effi-ciency, % Location of Motor and Driven Equipment with Respect to Conditioned Space or Airstream A B C (kW) Motor in, Driven Equip-ment in, W Motor out, Driven Equip-ment in, W Motor in, Driven Equip-ment out, W 0.05 (0.04)Shaded pole 1500 35 105 35 70 0.08 (0.06)Shaded pole 1500 35 170 59 110 0.125 (0.09)Shaded pole 1500 35 264 94 173 0.16 (0.12)Shaded pole 1500 35 340 117 223 0.25 (0.19) Split phase 1750 54 346 188 158 0.33 (0.25) Split phase 1750 56 439 246 194 0.50 (0.37) Split phase 1750 60 621 372 249 0.75 (0.56) 3-phase 1750 72 776 557 217 1 (0.75) 3-phase 1750 75 993 747 249 1.5 (1.1) 3-phase 1750 77 1453 1119 334 2 (1.5) 3-phase 1750 79 1887 1491 396 3 (2.2) 3-phase 1750 81 2763 2238 525 5 (3.7) 3-phase 1750 82 4541 3721 817 7.5 (5.6) 3-phase 1750 84 6651 5596 1066 10 (7.5) 3-phase 1750 85 8760 7178 1315 15 (11.2) 3-phase 1750 86 13 009 11 192 1820 20 (14.9) 3-phase 1750 87 17 140 14 913 2230 25 (18.6) 3-phase 1750 88 21 184 18 635 2545 30 (22.4) 3-phase 1750 89 25 110 22 370 2765 40 (30) 3-phase 1750 89 33 401 29 885 3690 50 (37) 3-phase 1750 89 41 900 37 210 4600 60 (45) 3-phase 1750 89 50 395 44 829 5538 75 (56) 3-phase 1750 90 62 115 55 962 6210 100 (75) 3-phase 1750 90 82 918 74 719 8290 125 (93) 3-phase 1750 90 103 430 93 172 10 342 150 (110) 3-phase 1750 91 123 060 111 925 11 075 200 (150) 3-phase 1750 91 163 785 149 135 14 738 250 (190) 3-phase 1750 91 204 805 186 346 18 430 Table 3B Typical Overload Limits with Standard Motors Watts Motor Type 40 to 190 120 to 250 500 to 560 750 and up AC open 1.4 1.35 1.25 1.15 AC TEFCa and DC — 1.0 1.0 1.0 Note: Some shaded pole, capacitor start, and special purpose motors have a service fac-tor varying from 1.0 up to 1.75.
aSome totally enclosed fan-cooled (TEFC) motors have a service factor above 1.0.
qem P 1.0 EM – EM --------------------- FUMFLM = 29.8 2001 ASHRAE Fundamentals Handbook (SI) fan-cooled (TEFC) motors are slightly more efficient. For speeds lower or higher than those listed, efficiencies may be 1 to 3% lower or higher, depending on the manufacturer. Should actual voltages at motors be appreciably higher or lower than rated nameplate voltage, efficiencies in either case will be lower. If electric motor load is an appreciable portion of cooling load, the motor efficiency should be obtained from the manufacturer. Also, depending on design, the maximum efficiency might occur any-where between 75 to 110% of full load; if underloaded or over-loaded, the efficiency could vary from the manufacturer’s listing.
Overloading or Underloading Heat output of a motor is generally proportional to the motor load, within the overload limits. Because of typically high no-load motor current, fixed losses, and other reasons, FLM is generally assumed to be unity, and no adjustment should be made for under-loading or overloading unless the situation is fixed, can be accu-rately established, and the reduced load efficiency data can be obtained from the motor manufacturer.
Radiation and Convection Unless the manufacturer’s technical literature indicates other-wise, the heat gain normally should be equally divided between radi-ant and convective components for the subsequent cooling load calculations.
APPLIANCES In a cooling load estimate, heat gain from all appliances—elec-trical, gas, or steam—should be taken into account. Because of the variety of appliances, applications, schedules, use, and installations, estimates can be very subjective. Often, the only information avail-able about heat gain from equipment is that on its nameplate. Cooking Appliances These appliances include common heat-producing cooking equipment found in conditioned commercial kitchens. Marn (1962) concluded that appliance surfaces contributed most of the heat to commercial kitchens and that when applicances were installed under an effective hood, the cooling load was independent of the fuel or energy used for similar equipment performing the same operations.
Gordon et al. (1994) and Smith et al. (1995) found that gas appli-ances may exhibit slightly higher heat gains than their electric coun-terparts under wall-canopy hoods operated at typical ventilation rates. This is due to the fact that the heat contained in the combus-tion products exhausted from a gas appliance may increase the tem-peratures of the appliance and surrounding surfaces, as well as the hood above the appliance, more than the heat produced by its elec-tric counterpart. These higher temperature surfaces radiate heat to the kitchen, adding moderately to the radiant gain directly associ-ated with the appliance cooking surface.
Marn (1962) confirmed that where the appliances are installed under an effective hood, only radiant gain adds to the cooling load; convected and latent heat from the cooking process and combustion products are exhausted and do not enter the kitchen. Gordon et al.
(1994) and Smith et al. (1995) substantiated these findings.
Sensible Heat Gain for Hooded Cooking Appliances. To establish a heat gain value, nameplate energy input ratings may be used with appropriate usage and radiation factors. Where specific rating data are not available (nameplate missing, equipment not yet purchased, etc.) or as an alternative approach, recommended heat gains listed in Table 5 for a wide variety of commonly encountered equipment items may be used. In estimating the appliance load, probabilities of simultaneous use and operation for different appli-ances located in the same space must be considered.
The radiant heat gain from hooded cooking equipment can range from 15 to 45% of the actual appliance energy consumption (Talbert et al. 1973, Gordon et al. 1994, Smith et al. 1995). This ratio of heat gain to appliance energy consumption may be expressed as a radia-tion factor. It is a function of both appliance type and fuel source.
The radiation factor FR is applied to the average rate of appliance energy consumption, determined by applying usage factor FU to the nameplate or rated energy input. Marn (1962) found that radiant heat temperature rise can be substantially reduced by shielding the fronts of cooking appliances. Although this approach may not always be practical in a commercial kitchen, radiant gains can also be reduced by adding side panels or partial enclosures that are inte-grated with the exhaust hood.
Heat Gain from Meals. For each meal served, the heat trans-ferred to the dining space is approximately 15 W, of which 75% is sensible and 25% is latent.
Heat Gain for Electric and Steam Appliances. The average rate of appliance energy consumption can be estimated from the nameplate or rated energy input qinput by applying a duty cycle or usage factor FU. Thus, the sensible heat gain qsensible for generic types of electric, steam, and gas appliances installed under a hood can be estimated using one of the following equations: qsensible = qinputFUFR (10) or qsensible = qinput FL (11) where FL is defined as the ratio of sensible heat gain to the manu-facturer’s rated energy input.
Table 4 lists usage factors, radiation factors, and load factors based on appliance energy consumption rate for typical electrical, steam, and gas appliances under standby or idle conditions.
Unhooded Equipment. For all cooking appliances not installed under an exhaust hood or directly vent-connected and located in the conditioned area, the heat gain may be estimated as 50% (FU = 0.50) Table 4A Hooded Electric Appliance Usage Factors, Radiation Factors, and Load Factors Appliance Usage Factor FU Radiation Factor FR Load Factor FL = FUFR Elec/Steam Griddle 0.16 0.45 0.07 Fryer 0.06 0.43 0.03 Convection oven 0.42 0.17 0.07 Charbroiler 0.83 0.29 0.24 Open-top range without oven 0.34 0.46 0.16 Hot-top range without oven 0.79 0.47 0.37 with oven 0.59 0.48 0.28 Steam cooker 0.13 0.30 0.04 Sources: Alereza and Breen (1984), Fisher (1998).
Table 4B Hooded Gas Appliance Usage Factors, Radiation Factors, and Load Factors Appliance Usage Factor FU Radiation Factor FR Load Factor FL = FUFR Gas Griddle 0.25 0.25 0.06 Fryer 0.07 0.35 0.02 Convection oven 0.42 0.20 0.08 Charbroiler 0.62 0.18 0.11 Open-top range without oven 0.34 0.17 0.06 Sources: Alereza and Breen (1984), Fisher (1998).
Nonresidential Cooling and Heating Load Calculation Procedures 29.9 or the rated hourly input, regardless of the type of energy or fuel used. On average, 34% of the heat may be assumed to be latent and the remaining 66% sensible. Note that cooking appliances venti-lated by “ductless” hoods should be treated as unhooded appliances from the perspective of estimating heat gain. In other words, all energy consumed by the appliance and all moisture produced by the cooking process is introduced to the kitchen as a sensible or latent cooling load.
Recommended Heat Gain Values. As an alternative proce-dure, Table 5 lists recommended rates of heat gain from typical commercial cooking appliances. The data in the “with hood” col-umns assume installation under a properly designed exhaust hood connected to a mechanical fan exhaust system.
Hospital and Laboratory Equipment Hospital and laboratory equipment items are major sources of heat gain in conditioned spaces. Care must be taken in evaluating the probability and duration of simultaneous usage when many components are concentrated in one area, such as a laboratory, an operating room, etc. Commonly, heat gain from equipment in a laboratory ranges from 50 to 220 W/m2 or, in laboratories with outdoor exposure, as much as four times the heat gain from all other sources combined.
Medical Equipment. It is more difficult to provide generalized heat gain recommendations for medical equipment than for general office equipment because medical equipment is much more varied in type and in application. Some heat gain testing has been done and can be presented, but the equipment included represents only a small sample of the type of equipment that may be encountered.
The data presented for medical equipment in Table 6 are relevant for portable and bench-top equipment. Medical equipment is very specific and can vary greatly from application to application. The data are presented to provide guidance in only the most general sense. For large equipment, such as MRI, engineers must obtain heat gain from the manufacturer.
Laboratory Equipment. Equipment in laboratories is similar to medical equipment in that it will vary significantly from space to space. Chapter 13 of the 1999 ASHRAE Handbook—Applications discusses heat gain from equipment, stating that it may range from 50 to 270 W/m2 in highly automated laboratories. Table 7 lists some values for laboratory equipment, but, as is the case for medical equipment, it is for general guidance only. Wilkins and Cook (1999) also examined laboratory equipment heat gains.
Office Equipment Computers, printers, copiers, calculators, checkwriters, posting machines, etc., can generate 9 to 13 W/m2 for general offices or 18 to 22 W/m2 for purchasing and accounting departments. ASHRAE Research Project 822 developed a method to measure the actual heat gain from equipment in buildings and the radiant/convective percentages (Hosni et al. 1998; Jones et al. 1998). This methodol-ogy was then incorporated into ASHRAE Research Project 1055 and applied to a wide range of equipment (Hosni et al. 1999) as a follow-up to independent research by Wilkins et al. (1991) and Wilkins and McGaffin (1994). Komor (1997) found similar results.
Analysis of measured data showed that results for office equipment could be generalized, but results from laboratory and hospital equipment proved too diverse. The following general guidelines for office equipment are a result of these studies.
Nameplate Versus Measured Energy Use. Nameplate data rarely reflect the actual power consumption of office equipment.
Actual power consumption of such equipment is assumed equal to the total (radiant plus convective) heat gain, but the ratio of such energy to the nameplate value varies widely. ASHRAE Research Project 1055 (Hosni et al. 1999) found that for general office equip-ment with nameplate power consumption of less than 1000 W, the actual ratio of total heat gain to nameplate ranged from 25% to 50%, but when all tested equipment is considered, the range is broader.
Generally, if the nameplate value is the only information known and no actual heat gain data are available for similar equipment, it would be conservative to use 50% of nameplate as heat gain and more nearly correct if 25% of nameplate were used. Much better results can be obtained, however, by considering the heat gain as being pre-dictable based on the type of equipment.
Office equipment is grouped into categories such as computers, monitors, printers, facsimile machines, and copiers, with heat gain results within each group analyzed to establish patterns.
Computers. Based on tests by Hosni et al. (1999) and Wilkins and McGaffin (1994), nameplate values on computers should be ignored when performing cooling load calculations. Table 8 pre-sents typical heat gain values for computers with varying degrees of safety factor.
Monitors. Based on monitors tested by Hosni et al. (1999), heat gain correlates approximately with screen size as qmon = 0.2S – 20 (12) where qmon = heat gain from monitor, W S = nominal screen size, mm Wilkins and McGaffin tested ten monitors (330 to 480 mm), finding the average heat gain value to be 60 W. This testing was done in 1992 when DOS was prevalent and the Windows™ operat-ing system was just being introduced. Monitors displaying Win-dows consumed more power than those displaying DOS. Table 8 tabulates typical values.
Laser Printers. Hosni et al. (1999) found that the power con-sumed by laser printers, and therefore the heat gain, depended largely on the level of throughput for which the printer was designed. It was observed that smaller printers are used more inter-mittently and that larger printers may run continuously for longer periods. Table 9 presents data on laser printers.
These data can be applied by taking the value for continuous operation and then applying an appropriate diversity factor. This would likely be most appropriate for larger open office areas.
Another approach could be to take the value that most closely matches the expected operation of the printer with no diversity. This may be appropriate when considering a single room or small area.
Copiers. Hosni et al. (1999) also tested five copy machines con-sidered to be of two types, desktop and office (freestanding high-volume copiers). Larger machines used in production environments were not addressed. Table 9 summarizes of the results. It was observed that desktop copiers rarely operated continuously but that office copiers frequently operated continuously for periods of an hour or more. Miscellaneous Office Equipment. Table 10 presents data on miscellaneous office equipment such as vending machines and mailing equipment.
Diversity. The ratio of the measured peak electrical load at the equipment panels to the sum of the maximum electrical load of each individual item of equipment is the usage diversity. A small, one- or two-person office containing equipment listed in Tables 8 through 10 can be expected to contribute heat gain to the space at the sum of the appropriate listed values. Progressively larger areas with many equipment items will always experience some degree of usage diversity resulting from whatever percentage of such equipment is not in operation at any given time.
Wilkins and McGaffin (1994) measured diversity in 23 areas within five different buildings totaling over 25 600 m2. Diversity was found to range between 37 and 78%, with the average (normal-ized based on area) being 46%. Figure 4 illustrates the relationship between nameplate, the sum of the peaks, and the actual electrical load with diversity accounted for, based on the average of the total 29.10 2001 ASHRAE Fundamentals Handbook (SI) Table 5 Recommended Rates of Heat Gain From Typical Commercial Cooking Appliances Appliance Size Energy Rate, W Recommended Rate of Heat Gain,a W Without Hood With Hood Rated Standby Sensible Latent Total Sensible Electric, No Hood Required Barbeque (pit), per kilogram of food capacity 36 to 136 kg 88 — 57 31 88 27 Barbeque (pressurized) per kilogram of food capacity 20 kg 210 — 71 35 106 33 Blender, per litre of capacity 1.0 to 3.8 L 480 — 310 160 470 150 Braising pan, per litre of capacity 102 to 133 L 110 — 55 29 84 40 Cabinet (large hot holding) 0.46 to 0.49 m3 2080 — 180 100 280 85 Cabinet (large hot serving) 1.06 to 1.15 m3 2000 — 180 90 270 82 Cabinet (large proofing) 0.45 to 0.48 m3 2030 — 180 90 270 82 Cabinet (small hot holding) 0.09 to 0.18 m3 900 — 80 40 120 37 Cabinet (very hot holding) 0.49 m3 6150 — 550 280 830 250 Can opener 170 — 170 — 170 0 Coffee brewer 12 cup/2 brnrs 1660 — 1100 560 1660 530 Coffee heater, per boiling burner 1 to 2 brnrs 670 — 440 230 670 210 Coffee heater, per warming burner 1 to 2 brnrs 100 — 66 34 100 32 Coffee/hot water boiling urn, per litre of capacity 11 L 120 — 79 41 120 38 Coffee brewing urn (large), per litre of capacity 22 to 38 L 660 — 440 220 660 210 Coffee brewing urn (small), per litre of capacity 10 L 420 — 280 140 420 130 Cutter (large) 460 mm bowl 750 — 750 — 750 0 Cutter (small) 360 mm bowl 370 — 370 — 370 0 Cutter and mixer (large) 28 to 45 L 3730 — 3730 — 3730 0 Dishwasher (hood type, chemical sanitizing), per 100 dishes/h 950 to 2000 dishes/h 380 — 50 110 160 50 Dishwasher (hood type, water sanitizing), per 100 dishes/h 950 to 2000 dishes/h 380 — 56 123 179 56 Dishwasher (conveyor type, chemical sanitizing), per 100 dishes/h5000 to 9000 dishes/h 340 — 41 97 138 44 Dishwasher (conveyor type, water sanitizing), per 100 dishes/h 5000 to 9000 dishes/h 340 — 44 108 152 50 Display case (refrigerated), per cubic metre of interior 0.17 to 1.9 m3 1590 — 640 0 640 0 Dough roller (large) 2 rollers 1610 — 1610 — 1610 0 Dough roller (small) 1 roller 460 — 460 — 460 0 Egg cooker 12 eggs 1800 — 850 570 1420 460 Food processor 2.3 L 520 — 520 — 520 0 Food warmer (infrared bulb), per lamp 1 to 6 bulbs 250 — 250 — 250 250 Food warmer (shelf type), per square metre of surface 0.28 to 0.84 m3 2930 — 2330 600 2930 820 Food warmer (infrared tube), per metre of length 1.0 to 2.1 m 950 — 950 — 950 950 Food warmer (well type), per cubic metre of well 20 to 70 L 37400 — 12400 6360 18760 6000 Freezer (large) 2.07 m3 1340 — 540 — 540 0 Freezer (small) 0.51 m3 810 — 320 — 320 0 Griddle/grill (large), per square metre of cooking surface 0.43 to 1.1 m2 29000 — 1940 1080 3020 1080 Griddle/grill (small), per square metre of cooking surface 0.20 to 0.42 m2 26200 — 1720 970 2690 940 Hot dog broiler 48 to 56 hot dogs 1160 — 100 50 150 48 Hot plate (double burner, high speed) 4900 — 2290 1590 3880 1830 Hot plate (double burner stockpot) 4000 — 1870 1300 3170 1490 Hot plate (single burner, high speed) 2800 — 1310 910 2220 1040 Hot water urn (large), per litre of capacity 53 L 130 — 50 16 66 21 Hot water urn (small), per litre of capacity 7.6 L 230 — 87 30 117 37 Ice maker (large) 100 kg/day 1090 — 2730 — 2730 0 Ice maker (small) 50 kg/day 750 — 1880 — 1880 0 Microwave oven (heavy duty, commercial) 20 L 2630 — 2630 — 2630 0 Microwave oven (residential type) 30 L 600 to 1400 — 600 to 1400 — 600 to 1400 0 Mixer (large), per litre of capacity 77 L 29 — 29 — 29 0 Mixer (small), per litre of capacity 11 to 72 L 15 — 15 — 15 0 Press cooker (hamburger) 300 patties/h 2200 — 1450 750 2200 700 Refrigerator (large), per cubic metre of interior space 0.71 to 2.1 m3 780 — 310 — 310 0 Refrigerator (small) per cubic metre of interior space 0.17 to 0.71 m3 1730 — 690 — 690 0 Rotisserie 300 hamburgers/h 3200 — 2110 1090 3200 1020 Serving cart (hot), per cubic metre of well 50 to 90 L 21200 — 7060 3530 10590 3390 Serving drawer (large) 252 to 336 dinner rolls 1100 — 140 10 150 45 Serving drawer (small) 84 to 168 dinner rolls 800 — 100 10 110 33 Skillet (tilting), per litre of capacity 45 to 125 L 180 — 90 50 140 66 Slicer, per square metre of slicing carriage 0.06 to 0.09 m2 2150 — 2150 — 2150 680 Soup cooker, per litre of well 7 to 11 L 130 — 45 24 69 21 Steam cooker, per cubic metre of compartment 30 to 60 L 214000 — 17000 10900 27900 8120 Steam kettle (large), per litre of capacity 76 to 300 L 95 — 7 5 12 4 Steam kettle (small), per litre of capacity 23 to 45 L 260 — 21 14 35 10 Syrup warmer, per litre of capacity 11 L 87 — 29 16 45 14 Nonresidential Cooling and Heating Load Calculation Procedures 29.11 Toaster (bun toasts on one side only) 1400 buns/h 1500 — 800 710 1510 480 Toaster (large conveyor) 720 slices/h 3200 — 850 750 1600 510 Toaster (small conveyor) 360 slices/h 2100 — 560 490 1050 340 Toaster (large pop-up) 10 slice 5300 — 2810 2490 5300 1700 Toaster (small pop-up) 4 slice 2470 — 1310 1160 2470 790 Waffle iron 0.05 m2 1640 — 700 940 1640 520 Electric, Exhaust Hood Required Broiler (conveyor infrared), per square metre of cooking area 0.19 to 9.5 m2 60800 — — — — 12100 Broiler (single deck infrared), per square metre of broiling area 0.24 to 0.91 m2 34200 — — — — 6780 Charbroiler, per linear metre of cooking surface 0.6 to 2.4 m 10600 8900 — — — 2700 Fryer (deep fat) 15 to 23 kg oil 14000 850 — — — 350 Fryer (pressurized), per kilogram of fat capacity 6 to 15 kg 1010 — — — — 38 Griddle, per linear metre of cooking surface 0.6 to 2.4 m 18800 3000 — — — 1350 Oven (full-szie convection) 12000 5000 — — — 850 Oven (large deck baking with 15.2 m3 decks), per cubic metre of oven spacer 0.43 to 1.3 m3 17300 — — — — 710 Oven (roasting), per cubic metre of oven space 0.22 to 0.66 m3 28300 — — — — 1170 Oven (small convection), per cubic metre of oven space 0.04 to 0.15 m3 107000 — — — — 1520 Oven (small deck baking with 7.7 m3 decks), per cubic metre of oven space 0.22 to 0.66 m3 28700 — — — — 1170 Open range (top), per 2 element section 2 to 10 elements 4100 1350 — — — 620 Range (hot top/fry top), per square metre of cooking surface 0.36 to 0.74 m2 22900 — — — — 8500 Range (oven section), per cubic metre of oven space 0.12 to 0.32 m3 40600 — — — — 1660 Gas, No Hood Required Broiler, per square metre of broiling area 0.25 46600 190b 16800 9030 25830 3840 Cheese melter, per square metre of cooking surface 0.23 to 0.47 32500 190b 11600 3400 15000 2680 Dishwasher (hood type, chemical sanitizing), per 100 dishes/h 950 to 2000 dishes/h 510 190b 150 59 209 67 Dishwasher (hood type, water sanitizing), per 100 dishes/h 950 to 2000 dishes/h 510 190b 170 64 234 73 Dishwasher (conveyor type, chemical sanitizing), per 100 dishes/h 5000 to 9000 dishes/h 400 190b 97 21 118 38 Dishwasher (conveyor type, water sanitizing), per 100 dishes/h 5000 to 9000 dishes/h 400 190b 110 23 133 41 Griddle/grill (large), per square metre of cooking surface 0.43 to 1.1 m2 53600 1040 3600 1930 5530 1450 Griddle/grill (small), per square metre of cooking surface 0.23 to 0.42 m2 45400 1040 3050 1610 4660 1260 Hot plate 2 burners 5630 390b 3430 1020 4450 1000 Oven (pizza), per square metre of hearth 0.59 to 1.2 m2 14900 190b 1970 690 2660 270 Gas, Exhaust Hood Required Braising pan, per litre of capacity 102 to 133 L 3050 190b — — — 750 Broiler, per square metre of broiling area 0.34 to 0.36 m3 68900 1660 — — — 5690 Broiler (large conveyor, infrared), per square metre of cooking area/minute 0.19 to 9.5 m2 162000 6270 — — — 16900 Broiler (standard infrared), per square metre of broiling area 0.22 to 0.87 m2 61300 1660 — — — 5040 Charbroiler (large), per linear metre of cooking area 0.6 to 2.4 m 34600 21000 — — — 3650 Fryer (deep fat) 15 to 23 kg 23500 1640 — — — 560 Oven (bake deck), per cubic metre of oven space 0.15 to 0.46 m3 79400 190b — — — 1450 Griddle, per linear metre of cooling surface 0.6 to 2.4 m 24000 6060 — — — 1540 Oven (full-size convection) 20500 8600 — — — 1670 Oven (pizza), per square metre of oven hearth 0.86 to 2.4 m2 22800 190b — — — 410 Oven (roasting), per cubic metre of oven space 0.26 to 0.79 m3 44500 190b — — — 800 Oven (twin bake deck), per cubic metre of oven space 0.31 to 0.61 m3 45400 190b — — — 810 Range (burners), per 2 burner section 2 to 10 burners 9840 390 — — — 1930 Range (hot top or fry top), per square metre of cooking surface 0.26 to 0.74 m3 37200 1040 — — — 10700 Range (large stock pot) 3 burners 29300 580 — — — 5740 Range (small stock pot) 2 burners 11700 390 — — — 2290 Range top, open burner (per 2 element section) 2 to 6 elements 11700 4000 — — — 640 Steam Compartment steamer, per kilogram of food capacity/h 21 to 204 kg 180 — 14 9 23 7 Dishwasher (hood type, chemical sanitizing), per 100 dishes/h 950 to 2000 dishes/h 920 — 260 110 370 120 Dishwasher (hood type, water sanitizing), per 100 dishes/h 950 to 2000 dishes/h 920 — 290 120 410 130 Dishwasher (conveyor, chemical sanitizing), per 100 dishes/h 5000 to 9000 dishes/h 350 — 41 97 138 44 Dishwasher (conveyor, water sanitizing), per 100 dishes/h 5000 to 9000 dishes/h 350 — 44 108 152 50 Steam kettle, per litre of capacity 12 to 30 L 160 — 12 8 20 6 Sources: Alereza and Breen (1984), Fisher (1998).
aIn some cases, heat gain data are given per unit of capacity. In those cases, the heat gain is calculated by: q = (recommended heat gain per unit of capacity) × (capacity) bStandby input rating is given for entire appliance regardless of size.
Table 5 Recommended Rates of Heat Gain From Typical Commercial Cooking Appliances (Concluded) Appliance Size Energy Rate, W Recommended Rate of Heat Gain,a W Without Hood With Hood Rated Standby Sensible Latent Total Sensible 29.12 2001 ASHRAE Fundamentals Handbook (SI) area tested. Data on actual diversity can be used as a guide, but diversity varies significantly with occupancy. The proper diversity factor for an office of mail order catalog telephone operators is dif-ferent from that for an office of sales representatives who travel regularly.
Heat Gain per Unit Area. Wilkins (1998) and Wilkins and Hosni (2000) summarized recent research on a heat gain per unit area basis. The diversity testing showed that the actual heat gain per unit area, or load factor, ranged from 4.7 to 11.6 W/m2, with an average (normalized based on area) of 8.7 W/m2. Spaces tested Table 6 Recommended Heat Gain from Typical Medical Equipment Equipment Nameplate, W Peak, W Average, W Anesthesia system 250 177 166 Blanket warmer 500 504 221 Blood pressure meter 180 33 29 Blood warmer 360 204 114 ECG/RESP 1440 54 50 Electrosurgery 1000 147 109 Endoscope 1688 605 596 Harmonical scalpel 230 60 59 Hysteroscopic pump 180 35 34 Laser sonics 1200 256 229 Optical microscope 330 65 63 Pulse oximeter 72 21 20 Stress treadmill N/A 198 173 Ultrasound system 1800 1063 1050 Vacuum suction 621 337 302 X-ray system 968 82 X-ray system 1725 534 480 X-ray system 2070 18 Source: Hosni et al. (1999) Table 7 Recommended Heat Gain from Typical Laboratory Equipment Equipment Nameplate, W Peak, W Average, W Analytical balance 7 7 7 Centrifuge 138 89 87 Centrifuge 288 136 132 Centrifuge 5500 1176 730 Electrochemical analyzer 50 45 44 Electrochemical analyzer 100 85 84 Flame photometer 180 107 105 Fluorescent microscope 150 144 143 Fluorescent microscope 200 205 178 Function generator 58 29 29 Incubator 515 461 451 Incubator 600 479 264 Incubator 3125 1335 1222 Orbital shaker 100 16 16 Oscilloscope 72 38 38 Oscilloscope 345 99 97 Rotary evaporator 75 74 73 Rotary evaporator 94 29 28 Spectronics 36 31 31 Spectrophotometer 575 106 104 Spectrophotometer 200 122 121 Spectrophotometer N/A 127 125 Spectro fluorometer 340 405 395 Thermocycler 1840 965 641 Thermocycler N/A 233 198 Tissue culture 475 132 46 Tissue culture 2346 1178 1146 Source: Hosni et al. (1999) Table 8 Recommended Heat Gain from Typical Computer Equipment Continuous, W Energy Saver Mode, W Computersa Average value 55 20 Conservative value 65 25 Highly conservative value 75 30 Monitorsb Small monitor (330 to 380 mm) 55 0 Medium monitor (400 to 460 mm) 70 0 Large monitor (480 to 510 mm) 80 0 Sources: Hosni et al. (1999), Wilkins and McGaffin (1994).
aBased on 386, 486, and Pentium grade.
bTypical values for monitors displaying Windows environment.
Table 9 Recommended Heat Gain from Typical Laser Printers and Copiers Continuous, W 1 page per min., W Idle, W Laser Printers Small desktop 130 75 10 Desktop 215 100 35 Small office 320 160 70 Large office 550 275 125 Copiers Desktop copier 400 85 20 Office copier 1,100 400 300 Source: Hosni et al. (1999).
Table 10 Recommended Heat Gain from Miscellaneous Office Equipment Appliance Maximum Input Rating, W Recommended Rate of Heat Gain, W Mail-processing equipment Folding machine 125 80 Inserting machine, 3,600 to 6,800 pieces/h 600 to 3300 390 to 2150 Labeling machine, 1,500 to 30,000 pieces/h 600 to 6600 390 to 4300 Postage meter 230 150 Vending machines Cigarette 72 72 Cold food/beverage 1150 to 1920 575 to 960 Hot beverage 1725 862 Snack 240 to 275 240 to 275 Other Bar code printer 440 370 Cash registers 60 48 Check processing workstation, 12 pockets 4800 2470 Coffee maker, 10 cups 1500 1050 sens., 450 latent Microfiche reader 85 85 Microfilm reader 520 520 Microfilm reader/printer 1150 1150 Microwave oven, 28 L 600 400 Paper shredder 250 to 3000 200 to 2420 Water cooler, 30 L/h 700 350 Nonresidential Cooling and Heating Load Calculation Procedures 29.13 were fully occupied and highly automated, comprising 21 unique areas in five buildings, with a computer and monitor at every work-station. Table 11 presents a range of load factors with a subjective description of the type of space to which they would apply. Table 12 presents more specific data that can be used to better quantify the amount of equipment in a space and the expected load factor. The medium load density is likely to be appropriate for most standard office spaces. Medium/heavy or heavy load densities may be encountered but can be considered extremely conservative esti-mates even for densely populated and highly automated spaces.
Radiant Convective Split. Hosni et al. (1999) found that the radiant-convective split for equipment was fairly uniform, the most important differentiating feature being whether or not the equip-ment had a cooling fan. Table 13 is a summary of those results.
HEAT GAIN THROUGH FENESTRATION AREAS The primary weather-related variable influencing the cooling load for a building is solar radiation. The effect of solar radiation is more pronounced and immediate in its impact on exposed non-opaque surfaces. The calculation of solar heat gain and conduc-tive heat transfer through various glazing materials and associated mounting frames, with or without interior and/or exterior shading devices, is discussed in Chapter 30. This chapter covers the application of such data to the overall heat gain evaluation and the conversion of the calculated heat gain into a composite cooling load for the conditioned space. Table 14 includes some useful solar equations.
Fenestration Direct Solar, Diffuse Solar, and Conductive Heat Gains For fenestration heat gain, use the following equations: Table 11 Recommended Load Factors for Various Types of Offices Load Density of Office Load Factor, W/m2 Description Light 5.4 Assumes 15.5 m2/workstation (6.5 workstations per 100 m2) with computer and monitor at each plus printer and fax. Computer, monitor, and fax diversity 0.67, printer diversity 0.33.
Medium 10.8 Assumes 11.6 m2/workstation (8.5 workstations per 100 m2) with computer and monitor at each plus printer and fax. Computer, monitor, and fax diversity 0.75, printer diversity 0.50.
Medium/ Heavy 16.1 Assumes 9.3 m2/workstation (11 workstations per 100 m2) with computer and monitor at each plus printer and fax. Computer and monitor diversity 0.75, printer and fax diversity 0.50.
Heavy 21.5 Assumes 7.8 m2/workstation (13 workstations per 100 m2) with computer and monitor at each plus printer and fax. Computer and monitor diversity 1.0, printer and fax diversity 0.50.
Source: Wilkins and McGaffin (1994).
Fig. 4 Office Equipment Load Factor Comparison (Wilkins and McGaffin 1994) Table 12 Cooling Load Estimates for Various Office Load Densities Num-ber Each, W Total, W Diver-sity Load, W Light Load Densitya Computers 6 55 330 0.67 220 Monitors 6 55 330 0.67 220 Laser printer—small desk top 1 130 130 0.33 43 Fax machine 1 15 15 0.67 10 Total Area Load 494 Recommended equipment load factor = 5.4 W/m2 Medium Load Densitya Computers 8 65 520 0.75 390 Monitors 8 70 560 0.75 420 Laser printer—desk 1 215 215 0.5 108 Fax machine 1 15 15 0.75 11 Total Area Load 929 Recommended equipment load factor = 10.8 W/m2 Medium/Heavy Load Densitya Computers 10 65 650 1 650 Monitors 10 70 700 1 700 Laser printer—small office 1 320 320 0.5 160 Facsimile machine 1 30 30 0.5 15 Total Area Load 1525 Recommended equipment load factor = 16.1 W/m2 Heavy Load Densitya Computers 12 75 900 1 900 Monitors 12 80 960 1 960 Laser printer-small office 1 320 320 0.5 160 Facsimile machine 1 30 30 0.5 15 Total Area Load 2035 Recommended equipment load factor = 21.5 W/m2 Source: Wilkins and McGaffin (1994).
a See Table 11 for descriptions of load densities.
Table 13 Summary of Radiant-Convective Split for Office Equipment Device Fan Radiant Convective Computer Yes 10 to 15% 85 to 90% Monitor No 35 to 40% 60 to 65% Computer and monitor – 20 to 30% 70 to 80% Laser printer Yes 10 to 20% 80 to 90% Copier Yes 20 to 25% 75 to 80% Fax machine No 30 to 35% 65 to 70% Source: Hosni et al. (1999).
29.14 2001 ASHRAE Fundamentals Handbook (SI) Direct beam solar heat gain qb: qb = AED SHGC(θ)IAC (13) Diffuse solar heat gain qd: qd = A(Ed + Er)〈SHGC〉D IAC (14) Conductive heat gain qc: qc = UA(Tout – Tin) (15) Total fenestration heat gain Q: Q = qb + qd + qc (16) where A = window area, m2 ED, Ed, and Er = direct, diffuse, and ground-reflected irradiance, calculated using the equations in Table 14 SHGC(θ) = direct solar heat gain coefficient as a function of incident angle θ; may be interpolated between values in Table 13 of Chapter 30 〈SHGC〉D = diffuse solar heat gain coefficient (also referred to as hemispherical SHGC); from Table 13 of Chapter 30 Tin = inside temperature, °C Tout = outside temperature, °C U = overall U-factor, including frame and mounting orientation from Table 4 of Chapter 30, W/(m2·K) IAC = inside shading attenuation coefficient, = 1.0 if no inside shading device If specific window manufacturer’s SHGC and U-factor data are available, those should be used. For fenestration equipped with inside shading (blinds, drapes or shades), IAC is listed in Tables 19, 20, and 22 of Chapter 30. The inside shading attentuation coeffi-cients given are used to calculate both direct and diffuse solar heat gains.
Exterior Shading Nonuniform exterior shading, caused by roof overhangs, side fins, or building projections, requires separate hourly calculations for the externally shaded and unshaded areas of the window in ques-tion, with the inside shading SHGC still used to account for any internal shading devices. The areas, shaded and unshaded, depend on the location of the shadow line on a surface in the plane of the glass. Sun (1968) developed fundamental algorithms for analysis of shade patterns. McQuiston and Spitler (1992) provide graphical data to facilitate shadow line calculation. An alternative, more accurate method described by Todorovic et al. (1993) first calculates cooling loads as if the external shading were absent, then adjusts (reduces) the result to account for the shading effect. This correction applies a “negative cooling load fac-tor,” calculated in much the same way as a conventional cooling load but using the time-varying area of the shaded portion of the glass as the heat gain element. Todorovic (1987) describes the solu-tion of the moving shade line problem in the context of consequent cooling load.
The equations for calculating shade angles [Chapter 30, Equa-tions (107) to (110)] can be used to determine the shape and area of moving shadow falling across a given window from external shad-ing elements during the course of a design day. Thus, a subprofile of heat gain for that window can be created by separating its sunlit and shaded areas for each hour.
Table 14 Solar Equations Solar Angles All angles are in degrees. The solar azimuth φ and the surface azimuth ψ are measured in degrees from south; angles to the east of south are negative, and angles to the west of south are positive. Calculate solar altitude, azimuth, and surface incident angles as follows: Apparent solar time AST, in decimal hours: AST = LST + ET/60 + (LSM – LON)/15 Hour angle H, degrees: H = 15(hours of time from local solar noon) = 15(AST – 12) Solar altitude β: sinβ = cosLcosδcosH + sinLsinδ Solar azimuth φ: cosφ = (sinβsin L – sinδ)/(cosβcosL) Surface-solar azimuth γ: γ = φ – ψ Incident angle θ: cosφ = cosβcosγsinΣ + sinβcosΣ where ET = equation of time, decimal minutes L = latitude LON = local longitude, decimal degrees of arc LSM = local standard time meridian, decimal degrees of arc = 60° for Atlantic Standard Time = 75° for Eastern Standard Time = 90° for Central Standard Time = 105° for Mountain Standard Time = 120° for Pacific Standard Time = 135° for Alaska Standard Time = 150° for Hawaii-Aleutian Standard Time LST = local standard time, decimal hours δ = solar declination, ° ψ = surface azimuth, ° Σ = surface tilt from horizontal, horizontal = 0° Values of ET and δ are given in Table 7 of Chapter 30 for the 21st day of each month.
Direct, Diffuse, and Total Solar Irradiance Direct normal irradiance EDN If β > 0 Otherwise, EDN = 0 Surface direct irradiance ED If cosθ > 0 ED = EDN cosθ Otherwise, ED = 0 Ratio Y of sky diffuse on vertical surface to sky diffuse on horizontal surface If cosθ > –0.2 Y = 0.55 + 0.437 cosθ + 0.313 cos2θ Otherwise, Y = 0.45 Diffuse irradiance Ed Vertical surfaces Ed = CYEDN Surfaces other than vertical Ed = CEDN(1 + cos Σ)/2 Ground-reflected irradiance Er = EDN(C + sin β)ρg(l – cos Σ)/2 Total surface irradiance Et = ED + Ed + Er where A = apparent solar constant B = atmospheric extinction coefficient C = sky diffuse factor CN = clearness number multiplier for clear/dry or hazy/humid locations. See Figure 5 in Chapter 32 of the 1999 ASHRAE Handbook—Applications for CN values.
Ed = diffuse sky irradiance Er = diffuse ground-reflected irradiance ρg = ground reflectivity Values of A, B, and C are given in Table 7 of Chapter 30 for the 21st day of each month. Values of ground reflectivity ρg are given in Table 10 of Chapter 30.
EDN A B β sin ⁄ ( ) exp --------------------------------- CN = Nonresidential Cooling and Heating Load Calculation Procedures 29.15 Temperature Considerations To estimate the conductive heat gain through fenestration at any time, applicable values of the outdoor and indoor dry-bulb temperatures must be used. Chapter 27 gives monthly cooling load design values of outdoor dry-bulb temperatures for many locations. These are generally midafternoon temperatures; for other times, the method described below in the section on Outdoor Air Temperatures can be used to estimate temperatures. Where local microclimatic conditions prevail or data are not included in Chapter 27, local weather stations or the National Oceanic and Atmospheric Administration can supply temperature data. Winter design temperatures should not be used because such data are for heating design rather than coincident conductive heat gain with sunlit glass during the winter months.
HEAT GAIN THROUGH EXTERIOR SURFACES Heat gain through exterior opaque surfaces is derived from the same elements of solar radiation and thermal gradient as that for fenestration areas. It differs primarily as a function of the mass and nature of the wall or roof construction, since those elements affect the rate of conductive heat transfer through the composite assembly to the interior surface.
SOL-AIR TEMPERATURE Sol-air temperature is the temperature of the outdoor air that in the absence of all radiation changes gives the same rate of heat entry into the surface as would the combination of incident solar radia-tion, radiant energy exchange with the sky and other outdoor sur-roundings, and convective heat exchange with the outdoor air.
Heat Flux into Exterior Sunlit Surfaces The heat balance at a sunlit surface gives the heat flux into the surface q/A as (17) where α = absorptance of surface for solar radiation Et = total solar radiation incident on surface, W/(m2·K) ho = coefficient of heat transfer by long-wave radiation and convection at outer surface, W/(m2·K) to = outdoor air temperature, °C ts = surface temperature, °C ε = hemispherical emittance of surface ∆R = difference between long-wave radiation incident on surface from sky and surroundings and radiation emitted by blackbody at outdoor air temperature, W/m2 Assuming the rate of heat transfer can be expressed in terms of the sol-air temperature te, (18) and from Equations (17) and (18), (19) Horizontal Surfaces. For horizontal surfaces that receive long-wave radiation from the sky only, an appropriate value of ∆R is about 63 W/m2, so that if ε = 1 and ho = 17 W/(m2·K), the long-wave correction term is about 4 K (Bliss 1961) and the correction itself thus −4 K.
Vertical Surfaces. Because vertical surfaces receive long-wave radiation from the ground and surrounding buildings as well as from the sky, accurate ∆R values are difficult to determine. When solar radiation intensity is high, surfaces of terrestrial objects usually have a higher temperature than the outdoor air; thus, their long-wave radiation compensates to some extent for the sky’s low emit-tance. Therefore, it is common practice to assume ε∆R = 0 for ver-tical surfaces.
Tabulated Temperature Values The sol-air temperatures in Table 15 have been calculated based on ε∆R/ho values of 4 K for horizontal surfaces and 0 K for vertical surfaces; total solar intensity values used for the calculations were calculated using the equations included in Table 14.
Surface Colors Sol-air temperature values are given for two values of the param-eter α/ho (Table 15); the value of 0.026 is appropriate for a light-colored surface, while 0.052 represents the usual maximum value for this parameter (i.e., for a dark-colored surface or any surface for which the permanent lightness can not reliably be anticipated).
Example 1. Calculating sol-air temperature using solar equations. Cal-culate the sol-air temperature at 3 P.M., Central Daylight Time on July 21 for a vertical light-colored wall surface, facing southwest, located at 40° North latitude and 90° West longitude, with an outdoor temperature of 34.4°C. The clearness number CN is assumed to be 1.0 and ground reflectivity ρg = 0.2.
Solution: Sol-air temperature is calculated using Equation (19). For a light-colored wall, α/ho = 0.026, and for vertical surfaces, ε∆R/ho = 0.
The solar irradiance Et must be determined using the equations in Table 14.
Solar Angles: ψ = southwest orientation = +45°. Σ = surface tilt from horizontal (where horizontal = 0°) = 90° for vertical wall surface. Local solar time (LST) is one hour earlier than Daylight Time, so 3 P.M. DST = 2 P.M.
LST = hour 14. Calculate solar altitude, solar azimuth, surface solar azimuth, and incident angle as follows.
From Table 7 in Chapter 30, solar position data and constants for July 21 are ET = –6.2 min δ = 20.6° A = 1085 W/m2 B = 0.207 C = 0.136 Local standard meridian (LSM) for Central Time Zone = 90°.
Apparent solar time AST AST = LST + ET/60 + (LSM – LON)/15 = 14 + (–6.2/60) + [(90 – 90)/15] = 13.897 Hour angle H, degrees: H = 15(AST – 12) = 15(13.897 – 12) = 28.45° Solar altitude β: sin β = cosLcosδcos H + sin L sin δ = cos(40)cos(20.6)cos(28.45) + sin(40)sin(20.6) = 0.8566 β = sin–1(0.8566) = 58.9° Solar azimuth φ: cosφ = (sinβsin L – sin δ)/(cosβcos L) = [(sin(58.9)sin(40) – sin(20.6)]/[cos(58.9)cos(40)] = 0.502 φ = cos–1(0.502) = 59.9° Surface-solar azimuth γ: γ = φ – ψ = 59.9 – 45 = 14.9° q A ---αEt ho to ts – ( ) ε∆R – + = q A ---ho te ts – ( ) = te to αEt ho ---------ε∆R ho ----------– + = 29.16 2001 ASHRAE Fundamentals Handbook (SI) Incident angle θ: cosθ = cosβcosγsin Σ + sinβcos Σ = cos(58.9)cos(14.9)sin(90) + sin(58.9)cos(90) = 0.499 θ = cos–1(0.499) = 60.1° Direct, Diffuse, and Total Solar Irradiance: Direct normal irradiance EDN EDN = = = 852 W/m2 Surface direct irradiance ED ED = EDN cosθ = (852)cos(60.1) = 425 W/m2 Ratio Y of sky diffuse on vertical surface to sky diffuse on horizon-tal surface Y = 0.55 + 0.437 cosθ + 0.313 cos2θ = 0.55 + 0.437 cos(60.1) + 0.313 cos2(60.1) = 0.846 Diffuse irradiance Ed Vertical surfaces Ed = CYEDN = (0.136)(0.846)(852) = 98 W/m2 Ground-reflected irradiance Er Er = EDN(C + sinβ)ρg(l – cosΣ)/2 = (852)0.136 + sin(58.9)[1 – cos(90)]/2 = 85 W/m2 Total surface irradiance Et Et = ED + Ed + Er = 425 + 98 + 85 = 608 W/m2 Sol-air temperature [from Equation (19)]: te = to + αEt /ho – ε∆R/ho = 34.4 + (0.026)(608) – 0 = 50.2°C This procedure was used to calculate the sol-air temperatures included in Table 15. Due to the tedious solar angle and intensity calculations, use of a simple computer spreadsheet or other com-puter software implementing these calculations can reduce the effort involved. A spreadsheet is illustrated in Table 16, calculating a 24 h sol-air temperature profile for the data of this example. OUTDOOR AIR TEMPERATURES The hourly air temperatures in Column 2, Table 16, are for a location with a design temperature of 35°C and a range of 11.7 K. To compute corresponding temperatures for other locations, select a suitable design temperature from Table 4B of Chapter 27 for the month being calculated and note the outdoor daily range from Table 1B of Chapter 27. For each hour, take the percentage of the daily range indicated in Table 17 of this chapter and subtract it from the design temperature.
Example 2. Air temperature calculation. Calculate the July dry-bulb temperature at 1200 h for Reno, Nevada.
Solution: From Table 1B, Chapter 27, the daily range is 20.7 K, and from Table 4B, Chapter 27, the 1% design dry-bulb temperature for July is 36°C. From Table 17, the percentage of the daily range at 1200 h is 23%. Thus, the dry-bulb temperature at 1200 is 36 − (0.23 × 20.7) = 31.2°C.
Table 15 Sol-Air Temperatures for July 21, 40° North Latitude Local Standard Hour Air Temp. to, °C Light-Colored Surface, α/ho = 0.026 Local Standard Hour Air Temp. to, °C Dark-Colored Surface, α/ho = 0.052 N NE E SE S SW W NW Horiz.
N NE E SE S SW W NW Horiz.
1 24.4 24.4 24.4 24.4 24.4 24.4 24.4 24.4 24.4 20.6 1 24.4 24.4 24.4 24.4 24.4 24.4 24.4 24.4 24.4 20.6 2 24.4 24.4 24.4 24.4 24.4 24.4 24.4 24.4 24.4 20.6 2 24.4 24.4 24.4 24.4 24.4 24.4 24.4 24.4 24.4 20.6 3 23.9 23.9 23.9 23.9 23.9 23.9 23.9 23.9 23.9 20.0 3 23.9 23.9 23.9 23.9 23.9 23.9 23.9 23.9 23.9 20.0 4 23.3 23.3 23.3 23.3 23.3 23.3 23.3 23.3 23.3 19.4 4 23.3 23.3 23.3 23.3 23.3 23.3 23.3 23.3 23.3 19.4 5 23.3 23.3 23.3 23.3 23.3 23.3 23.3 23.3 23.3 19.4 5 23.3 23.3 23.3 23.3 23.3 23.3 23.3 23.3 23.3 19.4 6 23.3 27.7 34.5 35.3 29.7 24.3 24.3 24.3 24.3 23.1 6 23.3 32.1 45.6 47.3 36.1 25.3 25.3 25.3 25.3 23.1 7 23.9 28.4 39.6 43.0 36.2 25.9 25.8 25.8 25.8 29.0 7 23.9 32.9 55.4 62.1 48.5 28.0 27.8 27.8 27.8 29.0 8 25.0 27.9 40.2 45.8 40.7 28.3 27.6 27.6 27.6 35.1 8 25.0 30.7 55.4 66.6 56.4 31.5 30.2 30.2 30.2 35.1 9 26.7 29.9 38.9 46.0 43.7 33.4 29.8 29.8 29.8 41.2 9 26.7 33.2 51.1 65.4 60.7 40.2 33.0 33.0 33.0 41.2 10 28.3 31.9 36.4 44.0 44.7 38.0 31.9 31.9 31.9 46.2 10 28.3 35.4 44.4 59.7 61.1 47.7 35.6 35.4 35.4 46.2 11 30.6 34.4 34.6 41.2 44.6 42.2 35.3 34.4 34.4 50.7 11 30.6 38.2 38.7 51.8 58.7 53.8 40.2 38.2 38.2 50.7 12 32.2 36.1 36.1 36.4 41.8 44.6 42.5 37.0 36.1 53.2 12 32.2 40.0 40.0 40.6 51.3 57.0 53.1 41.9 40.0 53.2 13 33.9 37.7 37.7 37.7 39.7 45.8 47.1 43.4 37.9 54.3 13 33.9 41.5 41.5 41.5 45.5 57.7 60.8 52.8 42.0 54.3 14 34.4 38.1 38.1 38.1 38.2 44.6 50.2 49.2 41.5 52.9 14 34.4 41.7 41.7 41.7 42.0 54.8 66.6 63.9 48.6 52.9 15 35.0 38.4 38.2 38.2 38.2 42.4 51.8 53.7 46.4 50.3 15 35.0 41.7 41.5 41.5 41.5 49.8 69.1 72.5 57.8 50.3 16 34.4 37.4 37.2 37.2 37.2 38.4 50.4 55.2 49.2 45.5 16 34.4 40.4 39.9 39.9 39.9 42.4 66.8 75.9 63.9 45.5 17 33.9 38.1 36.0 36.0 36.0 36.1 46.9 53.7 49.8 40.0 17 33.9 42.2 38.1 38.1 38.1 38.4 60.2 73.5 65.7 40.0 18 32.8 37.5 34.0 34.0 34.0 34.0 40.5 46.9 45.5 33.6 18 32.8 42.3 35.2 35.2 35.2 35.2 48.4 60.9 58.3 33.6 19 30.6 31.0 30.6 30.6 30.6 30.6 31.0 31.5 31.5 26.8 19 30.6 31.5 30.7 30.7 30.7 30.7 31.4 32.5 32.5 26.8 20 29.4 29.4 29.4 29.4 29.4 29.4 29.4 29.4 29.4 25.6 20 29.4 29.4 29.4 29.4 29.4 29.4 29.4 29.4 29.4 25.6 21 28.3 28.3 28.3 28.3 28.3 28.3 28.3 28.3 28.3 24.4 21 28.3 28.3 28.3 28.3 28.3 28.3 28.3 28.3 28.3 24.4 22 27.2 27.2 27.2 27.2 27.2 27.2 27.2 27.2 27.2 23.3 22 27.2 27.2 27.2 27.2 27.2 27.2 27.2 27.2 27.2 23.3 23 26.1 26.1 26.1 26.1 26.1 26.1 26.1 26.1 26.1 22.2 23 26.1 26.1 26.1 26.1 26.1 26.1 26.1 26.1 26.1 22.2 24 25.0 25.0 25.0 25.0 25.0 25.0 25.0 25.0 25.0 21.1 24 25.0 25.0 25.0 25.0 25.0 25.0 25.0 25.0 25.0 21.1 A B β sin ⁄ ( ) exp --------------------------------- CN 1085 0.207 58.9 ( ) sin ⁄ [ ] exp ------------------------------------------------------- 1.0 ( ) Nonresidential Cooling and Heating Load Calculation Procedures 29.17 Data Limitations The outdoor daily range is the difference between the average daily maximum and average daily minimum temperatures during the warmest month. More reliable results could be obtained by determining or estimating the shape of the temperature curve for typical hot days at the building site and considering each month sep-arately. Peak cooling load is often determined by solar heat gain through fenestration; this peak may occur in winter months and/or at a time of day when outside air temperature is not at its peak.
Table 16 Solar Calculations—Sol-Air Temperature for Example 1 GENERAL INPUT: SURFACE INPUT: SOLAR CONSTANTSa TIME ZONE MERIDIANS Time Zone = 3 Azimuth = 45 Month Equation of Time, min.
Declination, degrees A B C Month = 7 Tilt = 90 Longitude = 90 SOL-AIR TEMP. INPUT: W/m2 (Dimensionless Ratios) Latitude = 40 Absorptance = 0.45 1 –11.2 –20 1230 0.142 0.058 Time Zone Standard Meridian Clearness = 1 h (outside) =17.3 2 –13.9 –10.8 1215 0.144 0.060 Ground Refl.
= 0.2 Emittance = 1 3 –7.5 0 1186 0.156 0.071 1 Atlantic 60 ∆R = 0 4 1.1 11.6 1136 0.180 0.097 2 Eastern 75 5 3.3 20 1104 0.196 0.121 3 Central 90 SELECTED DATA: 6 –1.4 23.45 1088 0.205 0.134 4 Mountain 105 For Month = 7 7 –6.2 20.6 1085 0.207 0.136 5 Pacific 120 Equation of Time = –6.2 8 –2.4 12.3 1107 0.201 0.122 6 Alaska 135 Declination = 20.6 9 7.5 0 1151 0.177 0.092 7 Hawaii 150 A = 1085 10 15.4 –10.5 1192 0.160 0.073 B = 0.207 11 13.8 –19.8 1221 0.149 0.063 C = 0.136 12 1.6 –23.45 1233 0.142 0.057 Local Std Time Meridian = 90 DIRECT BEAM SOLAR HEAT GAIN DIFFUSE SOLAR HEAT GAIN Local Standard Hour Apparent Solar Time, hours Hour Angle Solar Altitude Solar Azimuth Direct Normal Irradiance Surface Incident Angle Surface Direct Irradiance Ground Diffuse Y Ratio Sky Diffuse Total Diffuse Irradiance Total Surface Irradiance Outside Temp.
Sol-Air Temp.
1 0.90 166.55 –28.1 –165.7 — 139.3 0.0 0.0 0.45 — 0.0 0.0 24.4 24.4 2 1.90 151.55 –23.8 –150.8 — 151.6 0.0 0.0 0.45 — 0.0 0.0 24.4 24.4 3 2.90 136.55 –17.1 –137.7 — 162.7 0.0 0.0 0.45 — 0.0 0.0 23.9 23.9 4 3.90 121.55 –8.6 –126.2 — 167.8 0.0 0.0 0.45 — 0.0 0.0 23.3 23.3 5 4.90 106.55 1.3 –116.2 0 161.1 0.0 0.0 0.45 0.0 0.0 0.0 23.3 23.3 6 5.90 91.55 11.9 –107.0 399 149.7 0.0 13.7 0.45 24.4 38.1 38.1 23.3 24.3 7 6.90 76.55 23.1 –98.1 641 137.3 0.0 33.9 0.45 39.2 73.1 73.1 23.9 25.8 8 7.90 61.55 34.6 –88.8 754 124.7 0.0 53.0 0.45 46.1 99.1 99.1 25.0 27.6 9 8.90 46.55 46.0 –78.0 814 112.2 0.0 69.6 0.45 49.8 119.4 119.4 26.7 29.8 10 9.90 31.55 56.8 –63.6 847 100.0 0.0 82.5 0.48 55.7 138.2 138.2 28.3 31.9 11 10.90 16.55 66.0 –41.0 865 88.4 24.8 90.8 0.56 66.2 157.0 181.8 30.6 35.3 12 11.90 1.55 70.6 4.4 871 75.4 220.1 94.0 0.68 80.6 174.6 394.7 32.2 42.5 13 12.90 13.45 67.5 34.6 867 67.8 327.0 91.9 0.76 89.6 181.4 508.5 33.9 47.1 14 13.90 28.45 58.9 59.8 852 60.1 425.0 84.6 0.85 98.0 182.6 607.6 34.4 50.2 15 14.90 43.45 48.3 75.4 822 55.0 471.6 72.6 0.90 101.0 173.6 645.2 35.0 51.8 16 15.90 58.45 37.0 86.7 769 53.4 458.4 56.7 0.92 96.4 153.1 611.5 34.4 50.4 17 16.90 73.45 25.5 96.2 671 55.6 379.0 38.0 0.90 81.8 119.8 498.8 33.9 46.9 18 17.90 88.45 14.2 105.1 467 61.1 225.4 17.8 0.83 53.0 70.8 296.1 32.8 40.5 19 18.90 103.45 3.4 114.2 33 69.3 11.8 0.6 0.74 3.4 4.0 15.8 30.6 31.0 20 19.90 118.45 –6.6 124.0 — 79.1 0.0 0.0 0.64 — 0.0 0.0 29.4 29.4 21 20.90 133.45 –15.5 135.2 — 90.2 0.0 0.0 0.55 — 0.0 0.0 28.3 28.3 22 21.90 148.45 –22.6 147.9 — 101.9 0.0 0.0 0.45 — 0.0 0.0 27.2 27.2 23 22.90 163.45 –27.5 162.5 — 114.2 0.0 0.0 0.45 — 0.0 0.0 26.1 26.1 24 23.90 178.45 –29.4 178.3 — 126.7 0.0 0.0 0.45 — 0.0 0.0 25.0 25.0 a See Table 7 in Chapter 30.
Table 17 Percentage of Daily Temperature Range Time, h % Time, h % Time, h % 1 87 9 71 17 10 2 92 10 56 18 21 3 96 11 39 19 34 4 99 12 23 20 47 5 100 13 11 21 58 6 98 14 3 22 68 7 93 15 0 23 76 8 84 16 3 24 82 29.18 2001 ASHRAE Fundamentals Handbook (SI) HEAT GAIN THROUGH INTERIOR SURFACES Whenever a conditioned space is adjacent to a space with a dif-ferent temperature, transfer of heat through the separating physical section must be considered. The heat transfer rate is given by q = UA(tb – ti) (20) where q = heat transfer rate, W U = coefficient of overall heat transfer between adjacent and conditioned space, W/(m2·K) A = area of separating section concerned, m2 tb = average air temperature in adjacent space, °C ti = air temperature in conditioned space, °C Values of U can be obtained from Chapter 25. Temperature tb may differ greatly from ti. The temperature in a kitchen or boiler room, for example, may be as much as 8 to 28 K above the outdoor air temperature. Actual temperatures in adjoining spaces should be measured when possible. Where nothing is known except that the adjacent space is of conventional construction, contains no heat sources, and itself receives no significant solar heat gain, tb – ti may be considered to be the difference between the outdoor air and con-ditioned space design dry-bulb temperatures minus 3 K. In some cases, the air temperature in the adjacent space will correspond to the outdoor air temperature or higher.
Floors For floors directly in contact with the ground or over an under-ground basement that is neither ventilated nor conditioned, heat transfer may be neglected for cooling load estimates.
INFILTRATION AND VENTILATION HEAT GAIN Ventilation Outdoor air must be introduced to ventilate conditioned spaces.
Chapter 26 suggests minimum outdoor air requirements for repre-sentative applications, but the minimum levels are not necessarily adequate for all psychological attitudes and physiological re-sponses. Where maximum economy in space and load is essential, as in submarines or other restricted spaces, as little as 0.5 L/s of out-door air per person can be sufficient, provided that recirculated air is adequately decontaminated (Consolazio and Pecora 1947).
Local codes and ordinances frequently specify ventilation requirements for public places and for industrial installations. For example, minimum requirements for safe practice in hospital oper-ating rooms are given in NFPA Standard 99. Although 100% out-door air is sometimes used in operating rooms, this standard does not require it, and limiting the outdoor air to 6 to 8 changes per hour is finding increasing acceptance. ASHRAE Standard 62 recommends minimum ventilation rates for most common applications. For general applications, such as offices, 10 L/s per person is suggested.
Ventilation air is normally introduced at the air-conditioning apparatus rather than directly into the conditioned space and thus becomes a cooling coil load component instead of a space load com-ponent. Calculations for estimating this heat gain are discussed later in the section on Heat Gain Calculations Using Standard Air Values.
Reducing heat gain from outdoor air by using filtered recircu-lated air in combination with outdoor air should be considered.
Recirculated air can also be treated to control odor (see Chapter 13 of this volume and Chapter 44 of the 1999 ASHRAE Handbook— Applications).
Infiltration The principles of estimating infiltration in buildings, with emphasis on the heating season, are discussed in Chapter 26. For the cooling season, infiltration calculations are usually limited to doors and windows. Air leakage through doors can be estimated using the information in Chapter 26. Table 3 in Chapter 26, adjusted for the average wind velocity in the locality, may be used to compute infiltration for windows. In calculating window infil-tration for an entire structure, the total window area on all sides of the building is not involved since wind does not act on all sides simultaneously. In any case, infiltration from all windows in any two adjacent wall exposures should be included. A knowledge of the prevailing wind direction and velocity is helpful in selecting exposures.
When economically feasible, sufficient outdoor air should be introduced as ventilation air through the air-conditioning equipment to maintain a constant outward escape of air and thus eliminate the infiltration portion of the gain. The pressure maintained must over-come wind pressure through cracks and door openings. When the quantity of outside air introduced through the cooling equipment is not sufficient to maintain the required pressure to eliminate infiltra-tion, the entire infiltration load should be included in the space heat gain calculations.
Standard Air Volumes Because the specific volume of air varies appreciably, calcula-tions are more accurate when made on the basis of air mass instead of volume. However, volume values are often required for selection of coils, fans, ducts, etc., in which cases volume values based on measurement at standard conditions may be used for accurate results. One standard value is 1.2 kg (dry air)/m3 (0.833 m3/kg).
This density corresponds to about 16°C at saturation and 21°C dry air (at 101.325 kPa). Because air usually passes through the coils, fans, ducts, etc., at a density close to standard, the accuracy desired normally requires no correction. When airflow is to be measured at a particular condition or point, such as at a coil entrance or exit, the corresponding specific volume can be read from the psychrometric chart.
Example 3. Standard air calculation. Assume outdoor air at standard conditions is flowing at 10 m3/s. What is the flow rate when the out-door air is at 35°C dry-bulb and 24°C wet-bulb (0.893 m3/kg)?
Solution: The measured rate at that condition should be 10 (0.893/0.833) = 10.7 m3/s.
Heat Gain Calculations Using Standard Air Values Air-conditioning design often requires calculation of the following.
1. Total heat Total heat gain qt corresponding to the change of a given stan-dard flow rate Qs through an enthalpy difference ∆h is qt = 1.2Qs∆h (21) where air density = 1.2 kg/m3.
2. Sensible heat Sensible heat gain qs corresponding to the change of dry-bulb temperature ∆t for given airflow (standard conditions) Qs is qs = 1.2(1.006 + 1.84W )Qs∆t (22) where 1.006 = specific heat of dry air, kJ/(kg·K) W = humidity ratio, kg (water)/kg (air) 1.84 = specific heat of water vapor, kJ/(kg·K) The specific heats are for a range from about −75 to 90°C. When W = 0, the value of 1.20(1.006 + 1.84W) = 1.21; when W = 0.01, the value is 1.23; when W = 0.02, the value is 1.25; and when W = 0.03, Nonresidential Cooling and Heating Load Calculation Procedures 29.19 the value is 1.27. Because a value of W = 0.01 approximates condi-tions found in many air-conditioning problems, the sensible heat change (in W) can normally be found as qs = 1.23Qs∆t (23) 3. Latent heat Latent heat gain ql corresponding to the change of humidity ratio ∆W for given airflow (standard conditions) Qs is ql = 1.20 × 2500Qs∆W = 3010Qs∆W (24) where 2500 is the approximate heat content of 50% rh vapor at 24°C less the heat content of water at 10°C. 50% rh at 24°C is a common design condition for the space, and 10°C is normal condensate tem-perature from cooling and dehumidifying coils.
The constants 1.20, 1.23, and 3010 are useful in air-conditioning calculations at sea level (101.325 kPa) and for normal temperatures and moisture ratios. For other conditions, more precise values should be used. For an altitude of 1500 m (84.556 kPa), appropriate values are 1.00, 1.03, and 2500.
LATENT HEAT GAIN FROM MOISTURE THROUGH PERMEABLE BUILDING MATERIALS The diffusion of moisture through all common building materials is a natural phenomenon that is always present. Chapters 23 and 24 cover the principles and specific methods used to control moisture.
Moisture transfer through walls is often neglected in the usual com-fort air-conditioning application because the actual rate is quite small and the corresponding latent heat gain is insignificant. The permeability and permeance values for various building materials are given in Table 9, Chapter 25. Vapor retarders are frequently installed to keep moisture transfer to a minimum.
Certain industrial applications call for a low moisture content to be maintained in a conditioned space. In such cases, the latent heat gain accompanying moisture transfer through walls may be greater than any other latent heat gain. This gain is computed by qm = MA∆pv(hg – hf) (25) where qm = latent heat gain, W M = permeance of wall assembly, ng/(s·m2·Pa) A = area of wall surface, m2 ∆pv = vapor pressure difference, Pa hg = enthalpy at room conditions, kJ/kg hf = enthalpy of water condensed at cooling coil, kJ/kg = 2500 kJ/kg when room temperature is 24°C and condensate off coil is 10°C HEAT GAIN FROM MISCELLANEOUS SOURCES The calculation of the cooling load is affected by such factors as (1) type of HVAC system, (2) effectiveness of heat exchange sur-faces, (3) fan location, (4) duct heat gain or loss, (5) duct leakage, (6) heat-extraction lighting systems, (7) type of return air system, and (8) sequence of controls. System performance needs to be ana-lyzed as a sequence of individual psychrometric processes. The most straightforward method first defines all known (or desired) state points on a psychrometric chart. Next, the actual entering and leaving dry- and wet-bulb conditions are calculated for such com-ponents as the cooling and/or heating coils (based on zone or space load), the amount of outside air introduced into the system through the equipment, and the amount of heat gain or loss at various points.
This overall process must verify that the space conditions origi-nally sought can actually be met by the designed system by consid-ering all sensible and latent heat changes to the air as it travels from the space conditions through the return air system and equipment back to the conditioned space. If the design is successful (i.e., within the degree of correctness of the various design assumptions), appro-priate equipment components can safely be selected. If not, the designer must judge if the results will be “close enough” to satisfy the needs of the project, or if one or more assumptions and/or design criteria must first be modified and the calculations rerun.
Heat Gain from Fans Fans that circulate air through HVAC systems add energy to the system by one or all of the following processes: • Temperature rise in the airstream from fan inefficiency.
Depending on the equipment, fan efficiencies generally range between 50 and 70%, with an average value of 65%. Thus, some 35% of the energy required by the fan appears as instantaneous heat gain to the air being transported.
• Temperature rise in the airstream as a consequence of air static and velocity pressure. The “useful” 65% of the total fan energy that creates pressure to move air spreads out throughout the entire air transport system in the process of conversion to sensible heat.
Designers commonly assume that the temperature change equivalent of this heat occurs at a single point in the system, depending on fan location as noted below.
• Temperature rise from heat generated by motor and drive inefficiencies. The relatively small gains from fan motors and drives are normally disregarded unless the motor and/or drive are physically located within the conditioned airstream. Equations (7), (8), and (9) may be used to estimate heat gains from typical motors. Belt drive losses are often estimated as 3% of the motor power rating. Conversion to temperature rise is calculated by Equation (26).
The location of each fan relative to other elements (primarily the cooling coil) and the type of system (e.g., single zone, multizone, double-duct, terminal reheat, VAV), along with the concept of equipment control (space temperature alone, space temperature and relative humidity, etc.), must be known before the analysis can be completed. A fan located upstream of the cooling coil (blowthrough supply fan, return air fan, outside air fan) adds the heat equivalent of its inefficiency to the airstream at that point; thus, a slightly elevated entering dry-bulb temperature to the cooling coil results. A fan located downstream of the cooling coil raises the dry-bulb temper-ature of air leaving the cooling coil. This rise can be offset by reduc-ing the cooling coil temperature or, alternatively, by increasing airflow across the cooling coil as long as its impact on space condi-tions is considered.
Duct Heat Gain and Leakage Unless return air duct systems are extensive or subjected to rig-orous conditions, only the heat gained or lost by supply duct sys-tems is significant; it is normally estimated as a percentage of space sensible cooling load (usually about 1%) and applied to the dry-bulb temperature of the air leaving the coil in the form of an equivalent temperature reduction.
Air leakage out of (or into) ductwork can have much greater impact than conventional duct heat gain or loss, but it is normally about the same or less. Outward leakage from supply ducts is a direct loss of cooling and/or dehumidifying capacity and must be offset by increased airflow (sometimes reduced supply air temper-atures) unless it enters the conditioned space directly. Inward leak-age to return ducts causes temperature and/or humidity variations, but these are often ignored under ordinary circumstances due to the low temperature and pressure differentials involved. Chapter 34 has further details on duct sealing and leakage.
A well-designed and installed duct system should not leak more than 1 to 3% of the total system airflow. All HVAC equipment and 29.20 2001 ASHRAE Fundamentals Handbook (SI) volume control units connected into a duct system are usually deliv-ered from manufacturers with allowable leakage not exceeding 1 or 2% of maximum airflow rating. Where duct systems are specified to be sealed and leak tested, both low- and medium-pressure types can be constructed and required to fall within this range. Designers nor-mally assume this loss to approximate 1% of the space load, handled in a similar manner to that for duct heat gain. Latent heat consider-ations are frequently ignored.
Poorly designed or installed duct systems can have leakage rates of 10 to 30%. Leakage from low-pressure lighting troffer connec-tions lacking proper taping and sealing can be 35% or more of the terminal air supply. Improperly sealed high-pressure systems can leak as much as 10% or more from the high-pressure side alone.
Such extremes destroy the validity of any load calculation.
Although leaks do not always affect overall system loads enough to cause problems, they will always adversely impact required supply air quantities. Also, uninsulated supply ductwork running through return air plenums results in high thermal leakage, which reduces the space-cooling capability of the supply air and may cause con-densation during a warm startup.
HEAT BALANCE METHOD OF COOLING LOAD CALCULATION The estimation of cooling load for a space involves calculating a surface-by-surface conductive, convective, and radiative heat bal-ance for each room surface and a convective heat balance for the room air. Sometimes called the exact solution, these principles form the foundation for all methods described in this chapter.
Some of the computations required by this rigorous approach to calculating space cooling load make the use of modern digital com-puters essential. The heat balance procedure is not new. Many energy calculation programs have used it in some form for many years. The first implementation that incorporated all the elements to form a complete method was NBSLD (Kusuda 1967). The heat bal-ance procedure is also implemented in both the BLAST and TARP energy analysis programs (Walton 1983). Prior to the implementa-tion of ASHRAE Research Project 875, the method had never been described completely or in a form applicable to cooling load calcu-lations. The papers resulting from RP-875 describe the heat balance procedure in detail (Pedersen et al. 1997, Liesen and Pedersen 1997, McClellan and Pedersen 1997).
The HB method is codified in the software called Hbfort that accompanies Cooling and Heating Load Calculation Principles (Pedersen et al. 1998).
HEAT BALANCE MODEL ASSUMPTIONS All calculation procedures involve some kind of model. All mod-els require simplifying assumptions and therefore are approximate.
The most fundamental assumption is that the air in the thermal zone can be modeled as well mixed, meaning it has a uniform tempera-ture throughout the zone. ASHRAE Research Project 664 (Fisher et al. 1997) established that this assumption is valid over a wide range of conditions. The next major assumption is that the surfaces of the room (walls, windows, floor, etc.) can be treated as having • Uniform surface temperatures • Uniform long-wave (LW) and short-wave (SW) irradiation • Diffuse radiating surfaces • One-dimensional heat conduction within The resulting formulation is called the heat balance model. It is important to note that the foregoing assumptions, although com-mon, are quite restrictive and set certain limits on the information that can be obtained from the model. ELEMENTS OF HEAT BALANCE MODEL Within the framework of the foregoing assumptions, the heat balance model can be viewed as four distinct processes: 1. Outside face heat balance 2. Wall conduction process 3. Inside face heat balance 4. Air heat balance Figure 5 shows the relationship between these processes for a single opaque surface. The top part of the figure, inside the shaded box, is repeated for each of the surfaces enclosing the zone. The process for transparent surfaces would be similar but would have the absorbed solar component appear in the conduction process block instead of at the outside face. Also, the absorbed component would split into an inward-flowing fraction and an outward-flow-ing fraction. These components would participate in the surface heat balances. Outside Face Heat Balance The heat balance on the outside face of each surface is (26) where q″ αsol = absorbed direct and diffuse solar radiation flux (q/A), W/m2 q″ LWR = net long-wave radiation flux exchange with air and surroundings, W/m2 q″ conv = convective exchange flux with outside air, W/m2 q″ ko = conductive flux (q/A) into wall, W/m2 All terms are positive for net flux to the face except the conduction term, which is traditionally taken to be positive in the direction from outside to inside the wall. Fig. 5 Schematic of Heat Balance Processes in a Zone q″ αsol q″ LWR q″ conv q″ ko – + + 0 = Nonresidential Cooling and Heating Load Calculation Procedures 29.21 Each of the heat flux terms in Equation (26) has been modeled in several ways, and in some formulations the first three terms are combined by using the sol-air temperature.
Wall Conduction Process The wall conduction process has been formulated in more ways than any of the other processes. Among the possible ways to model this process are 1. Numerical finite difference 2. Numerical finite element 3. Transform methods 4. Time series methods This process introduces part of the time dependence inherent in the load calculation process. Figure 6 shows schematically the sur-face temperatures on the inside and outside faces of the wall element and corresponding conductive heat fluxes away from the outside face and toward the inside face. All four quantities are functions of time. The direct formulation of the process has the two temperature functions as input or known quantities and the two heat fluxes as outputs or resultant quantities.
In some models, the surface heat transfer coefficients are included as part of the wall element. Then the temperatures in ques-tion are the inside and outside air temperatures. This is not an acceptable formulation because it hides the heat transfer coeffi-cients and prohibits changing them as airflow conditions change.
Also, it prohibits treating the internal long-wave radiation exchange appropriately.
Since the heat balances on both sides of the element induce both the temperature and heat flux, the solution technique must deal with this simultaneous condition. From a computational standpoint, two methods that have been used widely are a finite difference proce-dure and a method using conduction transfer functions. Because of the computational time advantage, the conduction transfer function formulation has been selected for the heat balance procedure pre-sented here. Inside Face Heat Balance The heart of the heat balance method is the internal heat balance involving the inside faces of the zone surfaces. This heat balance has many heat transfer components, and they are all coupled. Both long-wave (LW) and short-wave (SW) radiation are important, as well as wall conduction and convection to the air. The inside face heat balance for each surface can be written as follows: q″ LWX + q″ SW + q″ LWS + q″ ki + q″ sol + q″ conv = 0 (27) where q″ LWX = net long-wave radiant flux exchange between zone surfaces, W/m2 q″ SW = net short-wave radiation flux to surface from lights, W/m2 q″ LWS = long-wave radiation flux from equipment in zone, W/m2 q″ ki = conductive flux through the wall, W/m2 q″ sol = transmitted solar radiative flux absorbed at surface, W/m2 q″ conv = convective heat flux to zone air, W/m2 These terms are explained in the following paragraphs.
LW Radiation Exchange among Zone Surfaces. The two lim-iting cases for modeling internal LW radiation exchange are 1. Zone air is completely transparent to LW radiation 2. Zone air completely absorbs LW radiation from surfaces in the zone Most heat balance models treat air as completely transparent, and then it does not participate in the LW radiation exchange among the surfaces in the zone. The second model is attractive because it can be formulated simply using a combined radiative and convective heat transfer coefficient from each surface to the zone air and in that way decouples the radiant exchange among surfaces in the zone. How-ever, because the transparent air model allows the radiant exchange and is more realistic, the second model is inferior.
Furniture in a zone increases the amount of surface area that can participate in the radiative and convective heat exchanges. It also adds thermal mass to the zone. These two changes can affect the time response of the zone cooling load.
SW Radiation from Lights. The short-wavelength radiation from lights is usually assumed to be distributed over the surfaces in the zone in some manner. The HB procedure retains this approach but allows the distribution function to be changed. LW Radiation from Internal Sources. The traditional model for this source defines a radiative/convective split for the heat introduced into a zone from equipment. The radiative part is then distributed over the surfaces in the zone in some manner. This is not a completely realistic model, and it departs from the heat bal-ance principles. If it were handled in a true heat balance model, the equipment surfaces would be treated just as other LW radiant sources within the zone. However, since information about the surface temperature of equipment is rarely known, it is reason-able to keep the radiative/convective split concept even though it ignores the true nature of the radiant exchange. ASHRAE Research Project 1055 (Hosni et al. 1999) determined radia-tive/convective splits for many additional equipment types, as listed in Table 13.
Transmitted Solar Heat Gain. The calculation procedure for determining transmitted solar energy through fenestration as de-scribed in Chapter 30 uses the solar heat gain coefficient (SHGC) directly rather than relating it to double-strength glass as is done when using a shading coefficient (SC). The difficulty with this plan is that the SHGC includes both the transmitted solar and the inward-flowing fraction of the solar radiation absorbed in the window. With the heat balance method, this latter part should be added to the con-duction component so that it can be included in the inside face heat balance.
Transmitted solar radiation is also distributed over surfaces in the zone in a prescribed manner. It is possible to calculate the actual position of beam solar radiation, but this involves partial surface irradiation, which is inconsistent with the rest of the zone model that assumes uniform conditions over an entire surface. Using SHGC to Calculate Solar Heat Gain The total solar heat gain through fenestration consists of the directly transmitted solar radiation plus the inward-flowing frac-tion of the solar radiation that is absorbed in the glazing system.
Fig. 6 Schematic of Wall Conduction Process 29.22 2001 ASHRAE Fundamentals Handbook (SI) Both parts contain beam and diffuse contributions. The transmit-ted radiation goes directly onto surfaces within the zone and is accounted for in the surface inside heat balance. The zone heat balance model accommodates the resulting heat fluxes without difficulty. The second part, the inward-flowing fraction of the absorbed solar radiation, gets involved in an interaction with the other surfaces of the enclosure through long-wave radiant exchange and with the zone air through convective heat transfer.
As such, it is dependent both on the geometric and radiative properties of the zone enclosure and the convection characteris-tics inside and outside the zone. The solar heat gain coefficient (SHGC) combines the transmitted solar radiation and the inward-flowing fraction of the absorbed radiation. The SHGC is defined as (28) where τ = solar transmittance of glazing αk = solar absorptance of the kth layer of the glazing system n = number of layers NK = inward-flowing fraction of absorbed radiation in the kth layer Note that Equation (28) is written in a generic way. It can be written for a specific incidence angle and/or radiation wave-length and integrated over the wavelength and/or angle, but the principle is the same in each case. Refer to Chapter 30 for the specific expressions.
Unfortunately, the inward-flowing fraction N interacts with the zone in many ways. This interaction can be expressed as N = f(inside convection coefficient, outside convection coefficient, glazing system overall heat transfer coefficient, zone geometry, zone radiation properties) The only way to model these interactions correctly is to com-bine the window model with the zone heat balance model and solve the two simultaneously. This combination has been done recently in some energy analysis programs but is not generally available in load calculation procedures. In addition, the SHGC used for rating glazing systems is based on specific values of the inside, outside, and overall heat transfer coefficients and does not include any zonal long-wavelength radiation considerations. So, the challenge is to devise a way to use SHGC values within the framework of a heat balance calculation in the most accurate way possible. This will be done in the following paragraphs.
Using SHGC Data. The normal incidence SHGC used to rate and characterize glazing systems is not sufficient for determining solar heat gain for load calculations. These calculations require solar heat gain as a function of the incident solar angle in order to determine the hour-by-hour gain profile. Thus, it is necessary to use angular SHGC values and also diffuse SHGC values. These can be obtained from the Window 4.1 program (LBL 1994). This program does a detailed optical and thermal simulation of a glaz-ing system and, when applied to a single clear layer, produces the information shown in Table 18.
Table 18 shows the parameters as a function of incident solar angle and also the diffuse values. The specific parameters shown are Vtc = transmittance in the visible spectrum Rf and Rb = front and back surface visible reflectances Tsol = solar transmittance [symbol τ in Equations (28), (29), and (30)] Rf and Rb = front and back surface solar reflectances Abs1 = solar absorptance for layer 1, which is the only layer in this case [symbol α in Equations (28), (29), and (30)] SHGC = solar heat gain coefficient at the center of the glazing The parameters used for heat gain calculations are Tsol, Abs, and SHGC. For the specific convective conditions assumed in the Win-dow 4.1 program, the inward-flowing fraction of the absorbed solar can be obtained by rearranging Equation (28) to give: Nkαk = SHGC – τ (29) This quantity, when multiplied by the appropriate incident solar intensity, will provide the amount of absorbed solar radiation that flows inward. In the heat balance formulation for zone loads, this heat flux is combined with that caused by conduction through the glazing and included in the surface heat balance. The outward-flowing fraction of the absorbed solar radiation is used in the heat balance on the outside face of the glazing and is determined from (1 – Nk) αk = αk – Nkαk = αk – (SHGC – τ) (30) If there is more than one layer, the appropriate summation of absorptances must be done.
There is some potential inaccuracy in using the Window 4.1 SHGC values because the inward-flowing fraction part was deter-mined under specific conditions for the inside and outside heat transfer coefficients. However, it is possible to rerun the program with inside and outside coefficients of one’s own choosing. Nor-mally, however, this is not a large effect, and only in highly absorp-tive glazing systems might it cause significant error. SHGC τ Nkαk k=1 n ∑ + = Table 18 Single Layer Glazing Data Produced by Window 4.1 Parameter Incident Angle Diffuse (Hemis.) 0 10 20 30 40 50 60 70 80 90 Vtc 0.901 0.901 0.9 0.897 0.89 0.871 0.824 0.706 0.441 0 0.823 Rf 0.081 0.081 0.082 0.083 0.09 0.108 0.155 0.271 0.536 1 0.146 Rb 0.081 0.081 0.082 0.083 0.09 0.108 0.155 0.271 0.536 1 0.146 Tsol 0.85 0.85 0.848 0.844 0.835 0.814 0.766 0.652 0.399 0 0.77 Rf 0.075 0.074 0.075 0.076 0.082 0.099 0.144 0.255 0.509 1 0.136 Rb 0.075 0.074 0.075 0.076 0.082 0.099 0.144 0.255 0.509 1 0.136 Abs1 0.075 0.076 0.077 0.08 0.083 0.087 0.091 0.093 0.092 0 0.084 SHGC 0.87 0.87 0.868 0.865 0.857 0.837 0.79 0.677 0.423 0 0.792 Source: LBL (1994).
Nonresidential Cooling and Heating Load Calculation Procedures 29.23 For solar heat gain calculations, then, it seems reasonable to use the generic window property data that comes from Window 4.1.
Considering Table 18, the procedure is as follows: 1. Determine the angle of incidence for the glazing. 2. Determine the corresponding SHGC.
3. Evaluate Nkαk using Equation (28).
4. Multiply Tsol by the incident beam radiation intensity to get the transmitted beam solar radiation. 5. Multiply Nkαk by the incident beam radiation intensity to get inward-flowing absorbed heat.
6. Repeat steps 2 through 5 with the diffuse parameters and diffuse radiation. 7. Add the beam and diffuse components of transmitted and inward-flowing absorbed heat. Table 13 in Chapter 30 contains SHGC information for many additional glazing systems. That table is similar to Table 18 but is slightly abbreviated. Again, the information needed for heat gain calculations is Tsol, SHGC, and Abs. The same caution regarding the inside and outside heat transfer coefficients applies to the information in Table 13 in Chapter 30.
Those values were also obtained with specific inside and outside heat transfer coefficients, and the inward-flowing fraction N is dependent upon those values. Convection to Zone Air. The inside convection coefficients pre-sented in past editions of this chapter and used in most load calcu-lation procedures and energy programs are based on very old, natural convection experiments and do not accurately describe the heat transfer coefficients in a mechanically ventilated zone. In pre-vious load calculation procedures, these coefficients were buried in the procedures and could not be changed. A heat balance formula-tion keeps them as working parameters. In this way, research results such as those from ASHRAE Research Project 664 (Fisher 1998) can be incorporated into the procedures. It also permits determining the sensitivity of the load calculation to these parameters. Air Heat Balance In heat balance formulations aimed at determining cooling loads, the capacitance of the air in the zone is neglected and the air heat balance is done as a quasi-steady balance in each time period. Four factors contribute to the air heat balance: qconv + qCE + qIV + qsys = 0 (31) where qconv = convective heat transfer from surfaces, W qCE = convective parts of the internal loads, W qIV = sensible load due to infiltration and ventilation air, W qsys = heat transfer to/from the HVAC system, W The convection from zone surfaces qconv is the sum of all the convective heat transfer quantities from the inside surface heat bal-ance. This comes to the air via the convective heat transfer coeffi-cient on the surfaces.
The convective parts of the internal loads qCE is the compan-ion to q″ LWS , the radiant contribution from internal loads described previously [Equation (27)]. It is added directly to the air heat bal-ance. Such a treatment also violates the tenets of the heat balance approach because surfaces producing the internal loads exchange heat with the zone air through normal convective processes. How-ever, once again, this level of detail is generally not included in the heat balance, so it is included directly into the air heat balance instead.
In keeping with the well-mixed model for the zone air, any air that enters by way of infiltration or ventilation (qIV) is immedi-ately mixed with the zone air. The amount of infiltration air is uncer-tain. Sometimes it is related to the indoor-outdoor temperature difference and wind speed; however it is determined, it is added directly to the air heat balance.
The conditioned air that enters the zone from the HVAC system and provides qsys is also mixed directly with the zone air. GENERAL ZONE FOR LOAD CALCULATION The heat balance procedure is tailored to a single thermal zone, shown in Figure 7. The definition of a thermal zone depends on how the fixed temperature is going to be controlled. If air is circu-lated through an entire building or an entire floor in such a way that it is uniformly well stirred, the entire building or floor could be con-sidered a thermal zone. On the other hand, if each room has a dif-ferent control scheme, each room may need to be considered as a separate thermal zone. The framework needs to be flexible enough to accommodate any zone arrangement, but the heat balance aspect of the procedure also requires that a complete zone be described.
This zone consists of four walls, a roof or ceiling, a floor, and a “thermal mass surface” (described in the section on Input Required for Heat Balance Procedure). Each wall and the roof can include a window (or skylight in the case of the roof). This makes a total of 12 surfaces, any of which may have zero area if it is not present in the zone to be modeled. The heat balance processes for this general zone are formulated for a 24 h steady-periodic condition. The variables of the problem are the inside and outside temperatures of the 12 surfaces plus either the HVAC system energy required to maintain a specified air tem-perature or the air temperature, if the system capacity is specified.
This makes a total of 25 × 24 or 600 variables. While it is possible to set up the problem for a simultaneous solution of these variables, the relatively weak coupling of the problem from one hour to the next permits a double iterative approach. One iteration is through all the surfaces in each hour, and the other iteration is through the 24 h of a day. This procedure automatically reconciles the nonlinear as-pects of the surface radiative exchange and the other heat flux terms.
MATHEMATICAL DESCRIPTION OF HEAT BALANCE PROCEDURE Conduction Process Because it links the outside and inside heat balances, the wall conduction process regulates the time dependence of the cooling load. For the heat balance procedure presented here, the wall con-duction process is formulated using conduction transfer functions (CTFs), which relate conductive heat fluxes to the current and past surface temperatures and the past heat fluxes. The general form for the inside heat flux is Fig. 7 Schematic View of General Heat Balance Zone 29.24 2001 ASHRAE Fundamentals Handbook (SI) (32) For the outside heat flux, the form is (33) where Xj = outside CTF, j = 0,1,…nz Yj = cross CTF, j = 0,1,…nz Zj = inside CTF, j = 0,1,…nz Φj = flux CTF, j = 1,2,…nq θ = time = time step Tsi = inside face temperature, °C Tso = outside face temperature, °C q″ ki = conductive heat flux on inside face, W/m2 q″ ko = conductive heat flux on outside face, W/m2 The subscript following the comma indicates the time period for the quantity in terms of the time step . Also, the first terms in the series have been separated from the rest in order to facilitate solving for the current temperature in the solution scheme.
The two summation limits nz and nq depend on wall construction and depend somewhat on the scheme used for calculating the CTFs.
If nq = 0, the CTFs are generally referred to as response factors, but then theoretically nz is infinite. The values for nz and nq are gener-ally set to minimize the amount of computation. A development of CTFs can be found in Hittle (1981).
Heat Balance Equations The primary variables in the heat balance for the general zone are the 12 inside face temperatures and the 12 outside face temperatures at each of the 24 h, assigning i as the surface index and j as the hour index, or, in the case of CTFs, the sequence index. Then, the primary variables are: Tsoi,j = outside face temperature, i = 1,2,…12; j = 1,2,…24 Tsii,j = inside face temperature, i = 1,2,…12; j = 1,2,…24 In addition, qsysj = cooling load, j = 1,2,…24 Equations (26) and (33) are combined and solved for Tso to pro-duce 12 equations applicable in each time step: (34) where To = outside air temperature hco = outside convection coefficient, introduced by using q″ conv = hco(To – Tso) Equation (34) shows the need for separating Zi,0 because the con-tribution of the current surface temperature to the conductive flux can be collected with the other terms involving that temperature.
Equations (27) and (32) are combined and solved for Tsi to pro-duce the next 12 equations: (35) where Ta = zone air temperature hci = convective heat transfer coefficient on the inside, obtained from q″conv = hci(Ta – Tsi) Note that in Equations (34) and (35), the opposite surface tempera-ture at the current time appears on the right-hand side. The two equations could be solved simultaneously to eliminate those vari-ables. Depending on the order of updating the other terms in the equations, this can have a beneficial effect on solution stability.
The remaining equation comes from the air heat balance, Equa-tion (31). This provides the cooling load qsys at each time step: (36) In Equation (36), the convective heat transfer term is expanded to show the interconnection between the surface temperatures and the cooling load. Overall HB Iterative Solution Procedure The iterative HB procedure consists of a series of initial calcula-tions that proceed sequentially, followed by a double iteration loop as shown in the following steps: 1. Initialize areas, properties, and face temperatures for all surfaces, 24 h.
2. Calculate incident and transmitted solar flux for all surfaces and hours.
3. Distribute transmitted solar energy to all inside faces, 24 h.
4. Calculate internal load quantities for all 24 h.
5. Distribute LW, SW, and convective energy from internal loads to all surfaces for all hours.
6. Calculate infiltration and ventilation loads for all hours.
7. Iterate the heat balance according to the following scheme: 8. Display results.
q″ t ( ) ki ZoTsi θ , – ZjTsi θ jδ – , j=1 nz ∑ – = YoTso θ , YjTso θ jδ – , j=1 nz ∑ Φjq″ ki θ jδ – , j=1 nq ∑ + + + q″ t ( ) ko YoTsi θ , – YjTsi θ jδ – , j=1 nz ∑ – = XoTso θ , XjTso θ jδ – , j=1 nz ∑ Φjq″ ko θ jδ – , j=1 nq ∑ + + + δ δ Tsoi j , Tsii j k – , k=1 nz ∑ Yi k , Tsoi j k – , k=1 nz ∑ Zi k , – Φi k , q″ koi j k – , k=1 nq ∑ – = + q″αsoli j , q″LWRi j , Tsii j , Yi 0 , Tojhcoi j , Zi 0 , hcoi j , + ( ) ⁄ + + + Tsii j , Tsii j , Yi 0 , Tsoi j k – , k–1 nz ∑ Yi k , + = Tsii j k – , k=1 nz ∑ Zi k , – Φi k , q″ kii j k – , k=1 nq ∑ Tajhcij q″LWS + + + q″LWX q″SW q″sol e Zi 0 , hcii j , + ( ) ⁄ + + + qsysj Aihci Tsii j , Taj – ( ) i=1 12 ∑ qCE qIV + + = For Day = 1 to Maxdays For j = 1 to 24 {hours in the day} For SurfaceIter = 1 to MaxIter For i = 1 to 12 {The twelve zone surfaces} Evaluate Equations (34) and (35) Next i Next SurfaceIter Evaluate Equation (36) Next j If not converged, Next Day Nonresidential Cooling and Heating Load Calculation Procedures 29.25 Generally, four or six surface iterations are sufficient to provide convergence. The convergence check on the day iteration should be based on the difference between the inside and the outside conduc-tive heat flux terms qk. A limit such as requiring the difference between all inside and outside flux terms to be less than 1% of either flux works well.
INPUT REQUIRED FOR HEAT BALANCE PROCEDURE Previous methods for calculating cooling loads attempted to sim-plify the procedure by precalculating representative cases and grouping the results with various correlating parameters. This gen-erally tended to reduce the amount of information required to apply the procedure. In the case of the heat balance procedure, no precal-culations are made, so the procedure requires a fairly complete description of the zone.
Global Information. Because the procedure incorporates a solar calculation, some global information is required, including latitude, longitude, time zone, month, day of month, north axis of the zone, and the zone height (floor to floor). Additionally, to take full advan-tage of the flexibility of the method to incorporate, for example, variable outside heat transfer coefficients, such things as wind speed, wind direction, and terrain roughness may be specified. Nor-mally, these variables and others default to some reasonable set of values, but the flexibility remains. Wall Information (Each Wall). Because the walls are involved in three of the fundamental processes (external and internal heat bal-ance and wall conduction), each wall of the zone requires a fairly large set of variables. They include • Facing angle with respect to building north • Tilt (degrees from horizontal) • Area • Solar absorptivity outside • Long-wave emissivity outside • Short-wave absorptivity inside • Long-wave emissivity inside • Exterior boundary temperature condition (solar vs. nonsolar) • External roughness • Layer-by-layer construction information Again, some of these parameters can be defaulted, but they are changeable, and they indicate the more fundamental character of the heat balance method since they are related to true heat transfer processes.
Window Information (Each Window). The situation for win-dows is similar to that for walls, but the windows require some addi-tional information because of their role in the solar load. The necessary parameters include: • Area • Normal solar transmissivity • Normal SHGC • Normal total absorptivity • Long-wave emissivity outside • Long-wave emissivity inside • Surface-to-surface thermal conductance • Reveal (for solar shading) • Overhang width (for solar shading) • Distance from overhang to window (for solar shading) Roof and Floor Details. The roof and floor surfaces are speci-fied similarly to walls. The main difference is that the ground out-side boundary condition will probably be specified more often for a floor.
Thermal Mass Surface Details. An “extra” surface, called a thermal mass surface, can serve several functions. It is included in the radiant heat exchange with the other surfaces in the space but is only exposed to the inside air convective boundary condition. As an example, this surface would be used to account for the movable par-titions in a space. The construction of the partitions is specified layer by layer, similar to specification for walls, and those layers store and release heat via the same conduction mechanism as walls.
As a general definition, the extra thermal mass surface should be sized to represent all of the surfaces in the space that are exposed to the air mass, except the walls, roof, floor, and windows. In the for-mulation, both sides of the thermal mass participate in the exchange. Internal Heat Gain Details. The space can be subjected to sev-eral internal heat sources: people, lights, electrical equipment, and infiltration. In the case of infiltration, the energy is assumed to go immediately into the air heat balance, so it is the least complicated of the heat gains. For the others, several parameters must be speci-fied. These include the following fractions: • Fraction of heat gain that is sensible energy • Fraction of heat gain that is latent energy • Fraction of energy that enters as short-wave radiation • Fraction of energy that enters as long-wave radiation • Fraction of energy that enters the air immediately as convection • Activity level of people • Fraction of energy of lighting heat gain that goes directly to the return air Radiant Distribution Functions. As mentioned previously, the generally accepted assumptions for the heat balance method include specifying the distribution of radiant energy from several sources to the surfaces that enclose the space. This requires a distribution func-tion that specifies the fraction of the total radiant input that is absorbed by each surface. The types of radiation that require distri-bution functions are • Long-wave radiation from equipment and lights • Short-wave radiation from lights • Transmitted solar radiation Other Required Information. Additional flexibility is included in the model so that results of research can be incorporated easily.
This includes the capability to specify such things as • Heat transfer coefficients/convection models • Solar coefficients • Sky models The amount of input information required may seem extensive, but many of the parameters can be set to default values in most rou-tine applications. However, all of the parameters listed can be changed when necessary to fit unusual circumstances or when addi-tional information is obtained.
RADIANT TIME SERIES (RTS) METHOD The radiant time series (RTS) method is a new simplified method for performing design cooling load calculations that is derived from the heat balance (HB) method described above. It effectively replaces all other simplified (non-heat-balance) methods, such as the transfer function method (TFM), the cooling load temperature difference/cooling load factor (CLTD/CLF) method, and the total equivalent temperature difference/time averaging (TETD/TA) method.
The casual observer might well ask why yet another load calcu-lation method is necessary. This method was developed in response to the desire to offer a method that is rigorous, yet does not require iterative calculations, and that quantifies each component contribu-tion to the total cooling load. In addition, it is desirable for the user to be able to inspect and compare the coefficients for different con-struction and zone types in a form illustrating their relative impact 29.26 2001 ASHRAE Fundamentals Handbook (SI) on the result. These characteristics of the RTS method make it easier to apply engineering judgment during the cooling load calculation process.
The RTS method is suitable for peak design load calculations, but it should not be used for annual energy simulations due to its inherent limiting assumptions. The RTS method, while simple in concept, involves too many calculations to be used practically as a manual method, although it can easily be implemented in a simple computerized spreadsheet, as illustrated in the examples. For a man-ual cooling load calculation method, refer to the CLTD/CLF method included in the 1997 ASHRAE Handbook—Fundamentals.
RTS COOLING LOAD ASSUMPTIONS AND PRINCIPLES Design cooling loads are based on the assumption of steady-periodic conditions (i.e., the design day’s weather, occupancy, and heat gain conditions are identical to those for preceding days such that the loads repeat on an identical 24 h cyclical basis). Thus, the heat gain for a particular component at a particular hour is the same as 24 h prior, which is the same as 48 h prior, etc. This assumption is the basis for the RTS derivation from the HB method.
Cooling load calculations must address two time-delay effects inherent in building heat transfer processes: (1) delay of conductive heat gain through opaque massive exterior surfaces (walls, roofs, or floors) and (2) delay of radiative heat gain conversion to cooling loads.
Exterior walls and roofs conduct heat due to temperature differ-ences between outdoor and indoor air. In addition, solar energy on exterior surfaces is absorbed, then transferred by conduction to the building interior. Due to the mass and thermal capacity of the wall or roof construction materials, there is a substantial time delay in heat input at the exterior surface becoming heat gain at the interior surface.
As described earlier in the section on Cooling Load Principles, most heat sources transfer energy to a room by a combination of convection and radiation. The convection part of heat gain immedi-ately becomes cooling load. The radiation part must first be absorbed by the finishes and mass of the interior room surfaces and becomes cooling load only when it is later transferred by convection from those surfaces to the room air. Thus, radiant heat gains become cooling loads over a delayed period of time.
OVERVIEW OF THE RADIANT TIME SERIES METHOD Figure 8 gives an overview of the radiant time series method. In the calculation of solar radiation, transmitted solar heat gain through windows, sol-air temperature, and infiltration, the RTS method is exactly the same as previous simplified methods (TFM and TETD/TA). Important areas that are different include the com-putation of conductive heat gain, the splitting of all heat gains into radiant and convective portions, and the conversion of radiant heat gains into cooling loads. Fig. 8 Overview of Radiant Time Series Method Nonresidential Cooling and Heating Load Calculation Procedures 29.27 The RTS method accounts for both conduction time delay and radiant time delay effects by multiplying hourly heat gains by 24 h time series. The time series multiplication, in effect, distributes heat gains over time. Series coefficients, which are called radiant time factors and conduction time factors, are derived using the heat balance method. Radiant time factors reflect the percentage of an earlier radiant heat gain that becomes cooling load during the cur-rent hour. Likewise, conduction time factors reflect the percentage of an earlier heat gain at the exterior of a wall or roof that becomes heat gain at the inside during the current hour. By definition, each radiant or conduction time series must total 100%.
These series can be used to easily compare the time-delay impact of one construction versus another. This ability to compare choices is of particular benefit in the design process, when all construction details may not have been decided. Comparison can illustrate the magnitude of difference between the choices, allowing the engineer to apply judgment and make more informed assumptions in estimat-ing the load.
Figure 9 illustrates CTS values for three walls with similar U-fac-tors but with light to heavy construction. Figure 10 illustrates CTS for three walls with similar construction but with different amounts of insulation, thus with significantly different U-factors. Figure 11 illus-trates RTS values for zones varying from light to heavy construction.
RADIANT TIME SERIES PROCEDURE The general procedure for calculating cooling load for each load component (lights, people, walls, roofs, windows, appliances, etc.) with RTS is as follows: 1. Calculate 24 h profile of component heat gain for design day (for conduction, first account for conduction time delay by applying conduction time series).
2. Split heat gains into radiant and convective parts (see Table 19 for radiant and convective fractions).
3. Apply appropriate radiant time series to radiant part of heat gains to account for time delay in conversion to cooling load.
4. Sum convective part of heat gain and delayed radiant part of heat gain to determine cooling load for each hour for each cooling load component.
After calculating cooling loads for each component for each hour, sum those to determine the total cooling load for each hour and select the hour with the peak load for design of the air-conditioning system. This process should be repeated for multiple design months to determine the month when the peak load occurs, especially with windows on southern exposures (northern exposure in southern lat-itudes), which can result in higher peak room cooling loads in win-ter months than in summer.
Conductive Heat Gain Using Conduction Time Series In the RTS method, conduction through exterior walls and roofs is calculated using conduction time series (CTS). Wall and roof con-ductive heat input at the exterior is defined by the familiar conduc-tion equation as Fig. 9 CTS for Light to Heavy Walls Fig. 10 CTS for Walls with Similar Mass and Increasing Insulation Fig. 11 RTS for Light to Heavy Construction Table 19 Convective and Radiant Percentages of Total Sensible Heat Gain Heat Gain Source Radiant Heat, % Convective Heat, % Transmitted solar, no inside shade 100 0 Window solar, with inside shade 63 37 Absorbed (by fenestration) solar 63 37 Fluorescent lights, suspended, unvented 67 33 Fluorescent lights, recessed, vented to return air 59 41 Fluorescent lights, recessed, vented to return air and supply air 19 81 Incandescent lights 80 20 People See Table 1 Conduction, exterior walls 63 37 Conduction, exterior roofs 84 16 Infiltration and ventilation 0 100 Machinery and appliances (see Table 13) 20 to 80 80 to 20 Sources: Pedersen et al. (1998), Hosni et al. (1999).
29.28 2001 ASHRAE Fundamentals Handbook (SI) qi,θ−n = UA(te,θ−n – trc) (37) where qi,θ−n = conductive heat input for the surface n hours ago, W U = overall heat transfer coefficient for the surface, W/(m2·K) A = surface area, m2 te,θ−n = sol-air temperature n hours ago, °C trc = presumed constant room air temperature, °C Conductive heat gain through walls or roofs can be calculated using conductive heat inputs for the current hours and past 23 h and conduction time series: qθ = c0qi,θ + c1qi,θ−1 + c2qi,θ−2 + c3qi,θ−3 + … + c23qi,θ−23 (38) where qθ = hourly conductive heat gain for the surface, W qi,θ = heat input for the current hour, W qi,θ−n = heat input n hours ago, W c0, c1, etc. = conduction time factors Conduction time factors for representative wall and roof types are included in Tables 20 and 21. Those values were derived by first calculating conduction transfer functions for each example wall and roof construction. The assumption of steady-periodic heat input conditions for design load calculations allowed the conduction transfer functions to be reformulated into periodic response factors Table 20 Wall Conduction Time Series (CTS) Wall Number = CURTAIN WALLS STUD WALLS EIFS BRICK WALLS 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 U-factor, W/(m2·K) 0.428 0.429 0.428 0.419 0.417 0.406 0.413 0.668 0.305 0.524 0.571 0.377 0.283 0.581 0.348 0.628 0.702 0.514 0.581 0.389 Total R 2.3 2.3 2.3 2.4 2.4 2.5 2.4 1.5 3.3 1.9 1.7 2.7 3.5 1.7 2.9 1.6 1.4 1.9 1.7 2.6 Mass, kg/m2 31.0 20.9 80.0 25.5 84.6 25.6 66.7 36.6 38.3 130.9 214.1 214.7 215.8 290.6 304.0 371.7 391.5 469.3 892.2 665.1 Thermal Capacity, W/(m2·K) 8.5 5.4 19.0 7.0 20.5 9.3 17.1 10.2 10.6 33.2 49.2 49.3 49.7 66.6 70.6 89.1 86.7 108.0 218.4 161.5 Hour Conduction Time Factors, % 0 18 25 8 19 6 7 5 11 2 1 0 0 0 1 2 2 1 3 4 3 1 58 57 45 59 42 44 41 50 25 2 5 4 1 1 2 2 1 3 4 3 2 20 15 32 18 33 32 34 26 31 6 14 13 7 2 2 2 3 3 4 3 3 4 3 11 3 13 12 13 9 20 9 17 17 12 5 3 4 6 3 4 4 4 0 0 3 1 4 4 4 3 11 9 15 15 13 8 5 5 7 3 4 4 5 0 0 1 0 1 1 2 1 5 9 12 12 13 9 6 6 8 4 4 4 6 0 0 0 0 1 0 1 0 3 8 9 9 11 9 7 6 8 4 4 5 7 0 0 0 0 0 0 0 0 2 7 7 7 9 9 7 7 8 5 4 5 8 0 0 0 0 0 0 0 0 1 6 5 5 7 8 7 7 8 5 4 5 9 0 0 0 0 0 0 0 0 0 6 4 4 6 7 7 6 7 5 4 5 10 0 0 0 0 0 0 0 0 0 5 3 3 5 7 6 6 6 5 4 5 11 0 0 0 0 0 0 0 0 0 5 2 2 4 6 6 6 6 5 5 5 12 0 0 0 0 0 0 0 0 0 4 2 2 3 5 5 5 5 5 5 5 13 0 0 0 0 0 0 0 0 0 4 1 2 2 4 5 5 4 5 5 5 14 0 0 0 0 0 0 0 0 0 3 1 2 2 4 5 5 4 5 5 5 15 0 0 0 0 0 0 0 0 0 3 1 1 1 3 4 4 3 5 4 4 16 0 0 0 0 0 0 0 0 0 3 1 1 1 3 4 4 3 5 4 4 17 0 0 0 0 0 0 0 0 0 2 1 1 1 2 3 4 3 4 4 4 18 0 0 0 0 0 0 0 0 0 2 0 0 1 2 3 3 2 4 4 4 19 0 0 0 0 0 0 0 0 0 2 0 0 1 2 3 3 2 4 4 4 20 0 0 0 0 0 0 0 0 0 2 0 0 0 1 3 3 2 4 4 4 21 0 0 0 0 0 0 0 0 0 1 0 0 0 1 2 2 1 4 4 4 22 0 0 0 0 0 0 0 0 0 1 0 0 0 1 2 2 1 4 4 3 23 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 3 4 3 Total Percentage 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 Layer ID from outside to inside (see Table 22) F01 F01 F01 F01 F01 F01 F01 F01 F01 F01 F01 F01 F01 F01 F01 F01 F01 F01 F01 F01 F09 F08 F10 F08 F10 F11 F07 F06 F06 F06 M01 M01 M01 M01 M01 M01 M01 M01 M01 M01 F04 F04 F04 G03 G03 G02 G03 I01 I01 I01 F04 F04 F04 F04 F04 F04 F04 F04 F04 F04 I02 I02 I02 I04 I04 I04 I04 G03 G03 G03 I01 G03 I01 I01 M03 I01 I01 I01 I01 M15 F04 F04 F04 G01 G01 G04 G01 F04 I04 M03 G03 I04 G03 M03 I04 M05 M01 M13 M16 I04 G01 G01 G01 F02 F02 F02 F02 G01 G01 F04 F04 G01 I04 F02 G01 G01 F02 F04 F04 G01 F02 F02 F02 0 0 0 0 F02 F02 G01 G01 F02 G01 0 F02 F02 0 G01 G01 F02 0 0 0 0 0 0 0 0 0 F02 F02 0 F02 0 0 0 0 F02 F02 0 Wall Number Descriptions 1. Spandrel glass, insulation board, gyp board 2. Metal wall panel, insulation board, gyp board 3. 25 mm stone, insulation board, gyp board 4. Metal wall panel, sheathing, batt insulation, gyp board 5. 25 mm stone, sheathing, batt insulation, gyp board 6. Wood siding, sheathing, batt insulation, 13 mm wood 7. 25 mm stucco, sheathing, batt insulation, gyp board 8. EIFS finish, insulation board, sheathing, gyp board 9. EIFS finish, insulation board, sheathing, batt insulation, gyp board 10. EIFS finish, insulation board, sheathing, 200 mm LW CMU, gyp board 11. Brick, insulation board, sheathing, gyp board 12. Brick, sheathing, batt insulation, gyp board 13. Brick, insulation board, sheathing, batt insulation, gyp board 14. Brick, insulation board, 200 mm LW CMU 15. Brick, 200 mm LW CMU, batt insulation, gyp board 16. Brick, insulation board, 200 mm HW CMU, gyp board 17. Brick, insulation board, brick 18. Brick, insulation board, 200 mm LW concrete, gyp board 19. Brick, insulation board, 300 mm HW concrete, gyp board 20. Brick, 200 mm HW concrete, batt insulation, gyp board Nonresidential Cooling and Heating Load Calculation Procedures 29.29 as demonstrated by Spitler and Fisher (1999a). The periodic re-sponse factors were further simplified by dividing the 24 periodic response factors by the respective overall wall or roof U-factor to form the conduction time series (CTS). The conduction time factors can then be used in Equation (38) and provide a means for compar-ison of time delay characteristics between different wall and roof constructions. Construction material data used in the calculations for walls and roofs included in Tables 20 and 21 are listed in Table 22.
Heat gains calculated for walls or roofs using periodic response factors (and thus CTS) are identical to those calculated using con-duction transfer functions for the steady periodic conditions assumed in design cooling load calculations. The methodology for calculating periodic response factors from conduction transfer func-tions was originally developed as part of ASHRAE Research Project 875 (Spitler et al. 1997, Spitler and Fisher 1999b). Example 4. Wall heat gain using conduction time series. Using the data from Example 1 and Table 15, calculate the heat gain at 3 P.M. Central Daylight Time on July 21 through 9.3 m2 of a wall composed of light-colored 100 mm brick, 50 mm of insulation (R = 1.76 m2·K/W) and 200 mm lightweight concrete block. An air space is included between the brick and insulation. Overall U-factor of the wall is 0.386 W/ (m2·K). Inside room temperature is 23.9°C. Solution: Conductive heat gain is calculated using Equations (37) and (38). First, calculate the 24 h heat input profile using Equation (37) and the sol-air temperatures from Table 15 for a southwest-facing wall with light exterior color: qi1 = (0.386)(9.3)(24.4 – 23.9) = 2.0 W qi,2 = (0.386)(9.3)(24.4 – 23.9) = 2.0 qi,3 = (0.386)(9.3)(23.9 – 23.9) = 0 qi,4 = (0.386)(9.3)(23.3 – 23.9) = –2.0 qi,5 = (0.386)(9.3)(23.3 – 23.9) = –2.0 qi,6 = (0.386)(9.3)(24.3 – 23.9) = 1.6 Table 20 Wall Conduction Time Series (CTS) (Concluded) Wall Number = CONCRETE BLOCK WALL PRECAST AND CAST-IN-PLACE CONCRETE WALLS 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 U-factor, W/(m2·K) 0.383 0.335 0.414 1.056 0.834 0.689 0.673 0.418 0.434 0.650 0.387 0.467 0.434 0.266 3.122 Total R 2.6 3.0 2.4 0.9 1.2 1.5 1.5 2.4 2.3 1.5 2.6 2.1 2.3 3.8 0.3 Mass, kg/m2 108.8 108.8 224.3 94.3 107.1 168.9 143.9 144.6 262.5 291.8 274.7 488.1 469.9 698.9 683.2 Thermal Capacity, W/(m2·K) 27.4 27.4 57.0 23.1 26.9 42.0 34.5 34.6 61.3 68.9 64.9 122.7 118.3 175.7 171.0 Hour Conduction Time Factors, % 0 0 1 0 1 1 0 1 2 1 3 1 2 1 0 1 1 2 11 3 1 10 8 1 2 2 3 2 2 2 2 11 2 8 21 12 2 20 18 3 3 3 4 5 3 4 8 21 3 12 20 16 5 18 18 6 5 6 5 8 3 7 12 20 4 12 15 15 7 14 14 8 6 7 6 9 5 8 12 15 5 11 10 12 9 10 11 9 6 8 6 9 5 8 11 10 6 9 7 10 9 7 8 9 6 8 6 8 6 8 9 7 7 8 5 8 8 5 6 9 6 7 5 7 6 8 8 5 8 7 3 6 8 4 4 8 6 7 5 6 6 7 7 3 9 6 2 4 7 3 3 7 6 6 5 6 6 6 6 2 10 5 2 3 6 2 2 7 5 6 5 5 6 6 5 2 11 4 1 3 6 2 2 6 5 5 5 5 5 5 4 1 12 3 1 2 5 1 2 5 5 5 4 4 5 4 3 1 13 2 1 2 4 1 1 4 5 4 4 4 5 4 2 1 14 2 0 1 4 1 1 4 4 4 4 3 4 4 2 0 15 2 0 1 3 1 1 3 4 3 4 3 4 3 2 0 16 1 0 1 3 0 1 2 4 3 4 3 4 3 1 0 17 1 0 1 2 0 0 2 3 3 4 2 4 3 1 0 18 1 0 0 2 0 0 1 3 2 4 2 4 2 1 0 19 1 0 0 2 0 0 1 3 2 3 2 3 2 1 0 20 1 0 0 2 0 0 1 3 2 3 2 3 2 1 0 21 1 0 0 2 0 0 1 3 2 3 2 3 1 1 0 22 1 0 0 1 0 0 1 3 2 3 1 3 1 1 0 23 0 0 0 1 0 0 1 2 2 2 1 3 1 0 0 Total Percentage 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 Layer ID from outside to inside (see Table 22) F01 F01 F01 F01 F01 F01 F01 F01 F01 F01 F01 F01 F01 F01 F01 F07 M08 M08 M09 M11 M11 M11 F06 M13 F06 M15 M16 M16 F07 M08 M05 F02 F04 F04 I01 I04 I02 I01 I04 I02 I04 I05 F02 M05 F02 I04 — G01 G01 F04 G01 M11 M13 G01 M15 G01 G01 — I04 — G01 — F02 F02 G01 F02 F02 G01 F02 G01 F02 F02 — G01 — F02 — — — F02 — — F02 — F02 — — — F02 — Wall Number Descriptions 21. 200 mm LW CMU, batt insulation, gyp board 22. 200 mm LW CMU with fill insulation, batt insulation, gyp board 23. 25 mm stucco, 200 mm HW CMU, batt insulation, gyp board 24. 200 mm LW CMU with fill insulation 25. 200 mm LW CMU with fill insulation, gyp board 26. 300 mm LW CMU with fill insulation, gyp board 27. 100 mm LW concrete, board insulation, gyp board 28. 100 mm LW concrete, batt insulation, gyp board 29. 100 mm LW concrete, board insulation, 100 mm LW concrete 30. EIFS finish, insulation board, 200 mm LW concrete, gyp board 31. 200 mm LW concrete, batt insulation, gyp board 32. EIFS finish, insulation board, 200 mm HW concrete, gyp board 33. 200 mm HW concrete, batt insulation, gyp board 34. 300 mm HW concrete, batt insulation, gyp board 35. 300 mm HW concrete 29.30 2001 ASHRAE Fundamentals Handbook (SI) qi,7 = (0.386)(9.3)(25.8 – 23.9) = 6.8 qi,8 = (0.386)(9.3)(27.6 – 23.9) = 13.2 qi,9 = (0.386)(9.3)(29.8 – 23.9) = 21.1 qi,10 = (0.386)(9.3)(31.9 – 23.9) = 28.8 qi,11 = (0.386)(9.3)(35.3 – 23.9) = 40.9 qi,12 = (0.386)(9.3)(42.5 – 23.9) = 66.7 qi,13 = (0.386)(9.3)(47.1 – 23.9) = 83.3 qi,14 = (0.386)(9.3)(50.2 – 23.9) = 94.6 qi,15 = (0.386)(9.3)(51.8 – 23.9) = 100.1 qi,16 = (0.386)(9.3)(50.4 – 23.9) = 94.9 qi,17 = (0.386)(9.3)(46.9 – 23.9) = 82.4 qi,18 = (0.386)(9.3)(40.5 – 23.9) = 59.5 qi,19 = (0.386)(9.3)(31.0 – 23.9) = 25.4 qi,20 = (0.386)(9.3)(29.4 – 23.9) = 19.9 qi,21 = (0.386)(9.3)(28.3 – 23.9) = 15.9 qi,22 = (0.386)(9.3)(27.2 – 23.9) = 12.0 qi,23 = (0.386)(9.3)(26.1 – 23.9) = 8.0 qi,24 = (0.386)(9.3)(25.0 – 23.9) = 4.0 Total 24 h heat input = 779.1 W These data are used with conduction time series to calculate the wall heat gain. From Table 20, the most similar wall construction is wall number 14. This is a brick and block wall that has similar mass and thermal capacity. Using Equation (38), the conduction time factors for wall 14 can be used in conjunction with the 24 h heat input profile calculated above to determine the wall heat gain at 3 P.M. Central Day-light Time (which is actually 2 P.M. or hour 14 local standard time): Table 21 Roof Conduction Time Series (CTS) SLOPED FRAME ROOFS WOOD DECK METAL DECK ROOFS CONCRETE ROOFS Roof Number 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 U-factor, W/(m2·K) 0.249 0.227 0.255 0.235 0.239 0.231 0.393 0.329 0.452 0.370 0.323 0.206 0.297 0.304 0.296 0.288 0.315 0.313 0.239 Total R 4.0 4.4 3.9 4.2 4.2 4.3 2.5 3.0 2.2 2.7 3.1 4.9 3.4 3.3 3.4 3.5 3.2 3.2 4.2 Mass, kg/m2 26.7 21.0 14.0 34.7 55.5 34.9 48.9 55.9 23.9 30.9 25.0 27.2 57.6 149.2 214.3 279.3 360.7 474.5 362.3 Thermal Capacity, W/(m2·K) 7.3 4.6 3.5 12.9 20.2 13.0 21.0 22.1 7.8 8.9 8.1 8.9 15.7 37.6 52.8 67.9 92.8 121.3 91.8 Hour Conduction Time Factors, % 0 6 10 27 1 1 1 0 1 18 4 8 1 0 1 2 2 2 3 1 1 45 57 62 17 17 12 7 3 61 41 53 23 10 2 2 2 2 3 2 2 33 27 10 31 34 25 18 8 18 35 30 38 22 8 3 3 5 3 6 3 11 5 1 24 25 22 18 10 3 14 7 22 20 11 6 4 6 5 8 4 3 1 0 14 13 15 15 10 0 4 2 10 14 11 7 5 7 6 8 5 1 0 0 7 6 10 11 9 0 1 0 4 10 10 8 6 7 6 8 6 1 0 0 4 3 6 8 8 0 1 0 2 7 9 8 6 6 6 7 7 0 0 0 2 1 4 6 7 0 0 0 0 5 7 7 6 6 6 7 8 0 0 0 0 0 2 5 6 0 0 0 0 4 6 7 6 6 6 6 9 0 0 0 0 0 1 3 5 0 0 0 0 3 5 6 6 5 5 5 10 0 0 0 0 0 1 3 5 0 0 0 0 2 5 5 6 5 5 5 11 0 0 0 0 0 1 2 4 0 0 0 0 1 4 5 5 5 5 5 12 0 0 0 0 0 0 1 4 0 0 0 0 1 3 5 5 4 5 4 13 0 0 0 0 0 0 1 3 0 0 0 0 1 3 4 5 4 4 4 14 0 0 0 0 0 0 1 3 0 0 0 0 0 3 4 4 4 4 3 15 0 0 0 0 0 0 1 3 0 0 0 0 0 2 3 4 4 4 3 16 0 0 0 0 0 0 0 2 0 0 0 0 0 2 3 4 3 4 3 17 0 0 0 0 0 0 0 2 0 0 0 0 0 2 3 4 3 4 3 18 0 0 0 0 0 0 0 2 0 0 0 0 0 1 3 3 3 3 2 19 0 0 0 0 0 0 0 2 0 0 0 0 0 1 2 3 3 3 2 20 0 0 0 0 0 0 0 1 0 0 0 0 0 1 2 3 3 3 2 21 0 0 0 0 0 0 0 1 0 0 0 0 0 1 2 3 3 3 2 22 0 0 0 0 0 0 0 1 0 0 0 0 0 1 2 3 2 2 2 23 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 2 2 2 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 Layer ID from outside to inside (see Table 22) F01 F01 F01 F01 F01 F01 F01 F01 F01 F01 F01 F01 F01 F01 F01 F01 F01 F01 F01 F08 F08 F08 F12 F14 F15 F13 F13 F13 F13 F13 F13 M17 F13 F13 F13 F13 F13 F13 G03 G03 G03 G05 G05 G05 G03 G03 G03 G03 G03 G03 F13 G03 G03 G03 G03 G03 M14 F05 F05 F05 F05 F05 F05 I02 I02 I02 I02 I03 I02 G03 I03 I03 I03 I03 I03 F05 I05 I05 I05 I05 I05 I05 G06 G06 F08 F08 F08 I03 I03 M11 M12 M13 M14 M15 I05 G01 F05 F03 F05 F05 F05 F03 F05 F03 F05 F03 F08 F08 F03 F03 F03 F03 F03 F16 F03 F16 — G01 G01 G01 — F16 — F16 — — F03 — — — — — F03 — F03 — F03 F03 F03 — F03 — F03 — — — — — — — — — Roof Number Descriptions 1. Metal roof, batt insulation, gyp board 2. Metal roof, batt insulation, suspended acoustical ceiling 3. Metal roof, batt insulation 4. Asphalt shingles, wood sheathing, batt insulation, gyp board 5. Slate or tile, wood sheathing, batt insulation, gyp board 6. Wood shingles, wood sheathing, batt insulation, gyp board 7. Membrane, sheathing, insulation board, wood deck 8. Membrane, sheathing, insulation board, wood deck, suspended acoustical ceiling 9. Membrane, sheathing, insulation board, metal deck 10. Membrane, sheathing, insulation board, metal deck, suspended acoustical ceiling 11. Membrane, sheathing, insulation board, metal deck 12. Membrane, sheathing, plus insulation boards, metal deck 13. 50 mm concrete roof ballast, membrane, sheathing, insulation board, metal deck 14. Membrane, sheathing, insulation board, 100 mm LW concrete 15. Membrane, sheathing, insulation board, 150 mm LW concrete 16. Membrane, sheathing, insulation board, 200 mm LW concrete 17. Membrane, sheathing, insulation board, 150 mm HW concrete 18. Membrane, sheathing, insulation board, 200 mm HW concrete 19. Membrane, 150 mm HW concrete, batt insulation, suspended acoustical ceiling Nonresidential Cooling and Heating Load Calculation Procedures 29.31 Table 22 Thermal Properties and Code Numbers of Layers Used in Wall and Roof Descriptions for Tables 20 and 21 Layer ID Description Thickness, mm Conductivity, W/(m·K) Density, kg/m3 Specific Heat, kJ/(kg·K) Resistance, m2·K/W R Mass, kg/m2 Thermal Capacity, W·h/(m2·K) Notes F01 Outside surface resistance — — — — 0.04 0.04 — — 1 F02 Inside vertical surface resistance — — — — 0.12 0.12 — — 2 F03 Inside horizontal surface resistance — — — — 0.16 0.16 — — 3 F04 Wall air space resistance — — — — 0.15 0.15 — — 4 F05 Ceiling air space resistance — — — — 0.18 0.18 — — 5 F06 EIFS finish 9.5 0.72 1856 0.84 — 0.01 17.7 4.12 6 F07 25 mm stucco 25.4 0.72 1856 0.84 — 0.04 47.2 10.98 6 F08 Metal surface 0.8 45.28 7824 0.50 — 0.00 6.0 0.83 7 F09 Opaque spandrel glass 6.4 0.99 2528 0.88 — 0.01 16.1 3.39 8 F10 25 mm stone 25.4 3.17 2560 0.79 — 0.01 65.1 14.39 9 F11 Wood siding 12.7 0.09 592 1.17 — 0.14 7.5 2.45 10 F12 Asphalt shingles 3.2 0.04 1120 1.26 — 0.08 3.6 1.24 F13 Built-up roofing 9.5 0.16 1120 1.46 — 0.06 10.7 4.35 F14 Slate or tile 12.7 1.59 1920 1.26 — 0.01 24.4 8.52 F15 Wood shingles 6.4 0.04 592 1.30 — 0.17 3.8 1.36 F16 Acoustic tile 19.1 0.06 368 0.59 — 0.31 7.0 1.14 11 F17 Carpet 12.7 0.06 288 1.38 — 0.22 3.7 1.41 12 F18 Terrazzo 25.4 1.80 2560 0.79 — 0.01 65.1 14.39 13 G01 16 mm gyp board 15.9 0.16 800 1.09 — 0.10 12.7 3.85 G02 16 mm plywood 15.9 0.12 544 1.21 — 0.14 8.6 2.92 G03 13 mm fiberboard sheathing 12.7 0.07 400 1.30 — 0.19 5.1 1.83 14 G04 13 mm wood 12.7 0.15 608 1.63 — 0.08 7.7 3.51 15 G05 25 mm wood 25.4 0.15 608 1.63 — 0.17 15.5 7.01 15 G06 50 mm wood 50.8 0.15 608 1.63 — 0.33 30.9 14.03 15 G07 100 mm wood 101.6 0.15 608 1.63 — 0.66 61.8 28.05 15 I01 25 mm insulation board 25.4 0.03 43 1.21 — 0.88 1.1 0.37 16 I02 50 mm insulation board 50.8 0.03 43 1.21 — 1.76 2.2 0.74 16 I03 75 mm insulation board 76.2 0.03 43 1.21 — 2.64 3.3 1.11 16 I04 89 mm batt insulation 89.4 0.05 19 0.96 — 1.94 1.7 0.46 17 I05 154 mm batt insulation 154.4 0.05 19 0.96 — 3.34 3.0 0.79 17 I06 244 mm batt insulation 243.8 0.05 19 0.96 — 5.28 4.7 1.25 17 M01 100 mm brick 101.6 0.89 1920 0.79 — 0.11 195.2 43.16 18 M02 150 mm LW concrete block 152.4 0.49 512 0.88 — 0.31 78.1 19.08 19 M03 200 mm LW concrete block 203.2 0.50 464 0.88 — 0.41 94.3 23.06 20 M04 300 mm LW concrete block 304.8 0.71 512 0.88 — 0.43 156.2 38.16 21 M05 200 mm concrete block 203.2 1.11 800 0.92 — 0.18 162.7 41.65 22 M06 300 mm concrete block 304.8 1.40 800 0.92 — 0.22 244.0 62.47 23 M07 150 mm LW concrete block (filled) 152.4 0.29 512 0.88 — 0.53 78.1 19.08 24 M08 200 mm LW concrete block (filled) 203.2 0.26 464 0.88 — 0.78 94.3 23.06 25 M09 300 mm LW concrete block (filled) 304.8 0.29 512 0.88 — 1.04 156.2 38.16 26 M10 200 mm concrete block (filled) 203.2 0.72 800 0.92 — 0.28 162.7 41.65 27 M11 100 mm lightweight concrete 101.6 0.53 1280 0.84 — 0.19 130.1 30.29 M12 150 mm lightweight concrete 152.4 0.53 1280 0.84 — 0.29 195.2 45.43 M13 200 mm lightweight concrete 203.2 0.53 1280 0.84 — 0.38 260.3 60.58 M14 150 mm heavyweight concrete 152.4 1.95 2240 0.90 — 0.08 341.6 85.47 M15 200 mm heavyweight concrete 203.2 1.95 2240 0.90 — 0.10 455.5 113.96 M16 300 mm heavyweight concrete 304.8 1.95 2240 0.90 — 0.16 683.2 170.94 M17 50 mm LW concrete roof ballast 50.8 0.19 640 0.84 — 0.27 32.5 7.57 28 Notes: The following notes give sources for the data in this table.
1. Chapter 25, Table 1 for 3.4 m/s wind 2. Chapter 25, Table 1 for still air, horizontal heat flow 3. Chapter 25, Table 1 for still air, downward heat flow 4. Chapter 25, Table 3 for 40 mm space, 32.2°C, horizontal heat flow, 0.82 emittance 5. Chapter 25, Table 3 for 90 mm space, 32.2°C, downward heat flow, 0.82 emittance 6. EIFS finish layers approximated by Chapter 25, Table 4 for 10 mm cement plaster, sand aggregate 7. Chapter 38, Table 3 for steel (mild), 22 gage 8. Chapter 25, Table 4 for architectural glass 9. Chapter 25, Table 4 for marble and granite 10. Chapter 25, Table 4, density assumed same as Southern pine 11. Chapter 25, Table 4 for mineral fiberboard, wet molded, acoustical tile 12. Chapter 25, Table 4 for carpet and rubber pad, density assumed same as fiberboard 13. Chapter 25, Table 4, density assumed same as stone 14. Chapter 25, Table 4 for nail-base sheathing 15. Chapter 25, Table 4 for Southern pine 16. Chapter 25, Table 4 for expanded polystyrene 17. Chapter 25, Table 4 for glass fiber batt, specific heat per glass fiber board 18. Chapter 25, Table 4 for clay fired brick 19. Chapter 25, Table 4, 7.3 kg block, 200 mm × 400 mm face 20. Chapter 25, Table 4, 8.6 kg block, 200 mm × 400 mm face 21. Chapter 25, Table 4, 14.5 kg block, 200 mm × 400 mm face 22. Chapter 25, Table 4, 15 kg normal weight block, 200 mm × 400 mm face 23. Chapter 25, Table 4, 22.7 kg normal weight block, 200 mm × 400 mm face 24. Chapter 25, Table 4, 7.3 kg block, vermiculite fill 25. Chapter 25, Table 4, 8.6 kg block, 200 mm × 400 mm face, vermiculite fill 26. Chapter 25, Table 4, 14.5 kg block, 200 mm × 400 mm face, vermiculite fill 27. Chapter 25, Table 4, 15 kg normal weight block, 200 mm × 400 mm face, vermic-ulite fill 28. Chapter 25, Table 4 for 640 kg/m3 LW concrete 29.32 2001 ASHRAE Fundamentals Handbook (SI) q14 = c0qi,14 + c1qi,13 + c2qi,12 + c3qi,11 + … + c23qi,15 = (0.01)(94.6) + (0.01)(83.3) + (0.02)(66.7) + (0.05)(40.9) + (0.08)(28.8) + (0.09)(21.1) + (0.09)(13.2) + (0.09)(6.8) + (0.08)(1.6) + (0.07) (–2) + (0.07)(–2) + (0.06)(0) + (0.05)(2) + (0.04)(2) +(0.04)(4) + (0.03)(8) + (0.03)(12) + (0.02)(15.9) + (0.02)(19.9) + (0.02)(25.4) + (0.01)(59.4) + (0.01)(82.4) + (0.01)(94.9) + (0.0)(93.1) = 15.5 W Due to the tedious calculations involved, use of a simple computer spreadsheet or other computer software implementing these calculations can reduce the effort involved. A spreadsheet is illustrated in Table 23, calculating a 24 h heat gain profile for the data of this example.
Calculating Cooling Load Using RTS The instantaneous cooling load is defined as the rate at which heat energy is convected to the zone air at a given point in time. The computation of cooling load is complicated by the radiant exchange between surfaces, furniture, partitions, and other mass in the zone.
Most heat gain sources transfer energy by both convection and radi-ation. Radiative heat transfer introduces to the process a time depen-dency that is not easily quantified. Radiation is absorbed by the thermal masses in the zone and then later transferred by convection into the space. This process creates a time lag and dampening effect.
The convection portion of heat gains, on the other hand, is assumed to immediately become cooling load in the hour in which that heat gain occurs.
Heat balance procedures calculate the radiant exchange between surfaces based on their surface temperatures and emissivities, but they typically rely on estimated “radiative-convective splits” to determine the contribution of internal loads, including people, light-ing, appliances, and equipment, to the radiant exchange. The radiant time series procedure further simplifies the heat balance procedure by also relying on an estimated radiative-convective split of wall and roof conductive heat gain instead of simultaneously solving for the instantaneous convective and radiative heat transfer from each surface, as is done in the heat balance procedure.
Thus, the cooling load for each load component (lights, peo-ple, walls, roofs, windows, appliances, etc.) for a particular hour is the sum of the convective portion of the heat gain for that hour plus the time-delayed portion of radiant heat gains for that hour and the previous 23 h. Table 19 contains recommendations for splitting each of the heat gain components into convective and radiant portions.
The radiant time series method converts the radiant portion of hourly heat gains to hourly cooling loads using radiant time factors, the coefficients of the radiant time series. Radiant time factors are used to calculate the cooling load for the current hour on the basis of current and past heat gains. The radiant time series for a particular zone gives the time-dependent response of the zone to a single pulse of radiant energy. The series shows the portion of the radiant pulse that is convected to the zone air for each hour. Thus, r0 represents the fraction of the radiant pulse convected to the zone air in the cur-rent hour r1 in the previous hour, and so on. The radiant time series thus generated is used to convert the radiant portion of hourly heat gains to hourly cooling loads according to the following equation: Qr,θ = r0qr,θ + r1qr,θ−1 + r2qr,θ−2 + r3qr,θ−3 + … + r23qr,θ−23 (39) where Qr,θ = radiant cooling load (Qr) for the current hour (θ), W qr,θ = radiant heat gain for the current hour, W qr,θ−n = radiant heat gain n hours ago, W r0, r1, etc. = radiant time factors The radiant cooling load for the current hour, which is calculated using RTS and Equation (39), is added to the convective portion to determine the total cooling load for that component for that hour.
Radiant time factors are generated by a heat balance based pro-cedure. A separate series of radiant time factors is theoretically required for each unique zone and for each unique radiant energy distribution function assumption. For most common design appli-cations, RTS variation depends primarily on the overall massive-ness of the construction and the thermal responsiveness of the surfaces the radiant heat gains strike. One of the goals in developing RTS was to provide a simplified method that was based directly on the heat balance method; thus, it was deemed desirable to generate the RTS coefficients directly from a heat balance. To this end, a heat balance computer program was developed. This program, called Hbfort, which may be used to gen-erate RTS coefficients, is included as part of Cooling and Heating Load Calculation Principles (Pedersen et al. 1998). The RTS pro-cedure is described by Spitler et al. (1997). The procedure for gen-erating RTS coefficients may be thought of as analogous to the custom weighting factor generation procedure used by DOE 2.1 (Kerrisk et al. 1981; Sowell 1988a, 1988b). In both cases, a zone model is pulsed with a heat gain. In the case of DOE 2.1, the result-ing loads are used to estimate the best values of the transfer function method weighting factors to most closely match the load profile. In the case of the procedure described here, a unit periodic heat gain pulse is used to generate loads for a 24 h period. As long as the heat gain pulse is a unit pulse, the resulting loads are equivalent to the RTS coefficients.
Two different series of radiant time factors are utilized–one for direct transmitted solar heat gain (radiant energy assumed to be dis-tributed to the floor and furnishings only) and one for all other types of heat gains (radiant energy assumed to be uniformly distributed on all internal surfaces). Nonsolar RTS apply to radiant heat gains from people, lights, appliances, walls, roofs, and floors. Also, for diffuse Table 23 Wall Heat Gain for Example 4 Wall Area = 9.3 m2 CTS Table 20 Wall Number: 14 Inside Temperature = 23.9°C Wall U-Factor = 0.386 Hour Sol-Air Temp., °C Inside Temp., °C Heat Input, W CTS Heat Gain, W 1 24.4 23.9 2.0 1% 47.8 2 24.4 23.9 2.0 1% 43.8 3 23.9 23.9 — 2% 39.5 4 23.3 23.9 2.0 5% 35.2 5 23.3 23.9 2.0 8% 30.8 6 24.3 23.9 1.6 9% 26.7 7 25.8 23.9 6.8 9% 23.0 8 27.6 23.9 13.2 9% 19.6 9 29.8 23.9 21.1 8% 16.7 10 31.9 23.9 28.8 7% 14.5 11 35.3 23.9 40.9 7% 13.0 12 42.5 23.9 66.7 6% 12.8 13 47.1 23.9 83.3 5% 13.5 14 50.2 23.9 94.6 4% 15.5 15 51.8 23.9 100.1 4% 19.4 16 50.4 23.9 94.9 3% 25.0 17 46.9 23.9 82.4 3% 31.7 18 40.5 23.9 59.5 2% 38.9 19 31.0 23.9 25.4 2% 45.6 20 29.4 23.9 19.9 2% 51.0 21 28.3 23.9 15.9 1% 54.4 22 27.2 23.9 12.0 1% 55.3 23 26.1 23.9 8.0 1% 54.0 24 25.0 23.9 4.0 0% 51.3 779.1 100% 779.1 Nonresidential Cooling and Heating Load Calculation Procedures 29.33 Table 24 Representative Nonsolar RTS Values for Light to Heavy Construction % Glass Interior Zones Light Medium Heavy Light Medium Heavy With Carpet No Carpet With Carpet No Carpet With Carpet No Carpet With Carpet No Carpet With Carpet No Carpet With Carpet No Carpet 10% 50% 90% 10% 50% 90% 10% 50% 90% 10% 50% 90% 10% 50% 90% 10% 50% 90% Hour Radiant Time Factor, % 0 47 50 53 41 43 46 46 49 52 31 33 35 34 38 42 22 25 28 46 40 46 31 33 21 1 19 18 17 20 19 19 18 17 16 17 16 15 9 9 9 10 9 9 19 20 18 17 9 9 2 11 10 9 12 11 11 10 9 8 11 10 10 6 6 5 6 6 6 11 12 10 11 6 6 3 6 6 5 8 7 7 6 5 5 8 7 7 4 4 4 5 5 5 6 8 6 8 5 5 4 4 4 3 5 5 5 4 3 3 6 5 5 4 4 4 5 5 4 4 5 3 6 4 5 5 3 3 2 4 3 3 2 2 2 4 4 4 4 3 3 4 4 4 3 4 2 4 4 4 6 2 2 2 3 3 2 2 2 2 4 3 3 3 3 3 4 4 4 2 3 2 4 3 4 7 2 1 1 2 2 2 1 1 1 3 3 3 3 3 3 4 4 4 2 2 1 3 3 4 8 1 1 1 1 1 1 1 1 1 3 2 2 3 3 3 4 3 3 1 1 1 3 3 4 9 1 1 1 1 1 1 1 1 1 2 2 2 3 3 2 3 3 3 1 1 1 2 3 3 10 1 1 1 1 1 1 1 1 1 2 2 2 3 2 2 3 3 3 1 1 1 2 3 3 11 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 3 3 3 1 1 1 2 2 3 12 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 3 3 3 1 1 1 1 2 3 13 1 1 1 0 1 0 1 1 1 1 1 1 2 2 2 3 3 2 1 1 1 1 2 3 14 0 0 1 0 1 0 1 1 1 1 1 1 2 2 2 3 2 2 1 0 1 1 2 3 15 0 0 1 0 0 0 1 1 1 1 1 1 2 2 2 2 2 2 0 0 1 1 2 3 16 0 0 0 0 0 0 1 1 1 1 1 1 2 2 2 2 2 2 0 0 1 1 2 3 17 0 0 0 0 0 0 1 1 1 1 1 1 2 2 2 2 2 2 0 0 1 1 2 2 18 0 0 0 0 0 0 1 1 1 1 1 1 2 2 1 2 2 2 0 0 1 1 2 2 19 0 0 0 0 0 0 0 1 0 0 1 1 2 2 1 2 2 2 0 0 1 0 2 2 20 0 0 0 0 0 0 0 0 0 0 1 1 2 1 1 2 2 2 0 0 0 0 2 2 21 0 0 0 0 0 0 0 0 0 0 1 1 2 1 1 2 2 2 0 0 0 0 2 2 22 0 0 0 0 0 0 0 0 0 0 1 0 1 1 1 2 2 2 0 0 0 0 1 2 23 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 2 2 1 0 0 0 0 1 2 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 Table 25 Representative Solar RTS Values for Light to Heavy Construction % Glass Light Medium Heavy With Carpet No Carpet With Carpet No Carpet With Carpet No Carpet 10% 50% 90% 10% 50% 90% 10% 50% 90% 10% 50% 90% 10% 50% 90% 10% 50% 90% Hour Radiant Time Factor, % 0 53 55 56 44 45 46 52 54 55 28 29 29 47 49 51 26 27 28 1 17 17 17 19 20 20 16 16 15 15 15 15 11 12 12 12 13 13 2 9 9 9 11 11 11 8 8 8 10 10 10 6 6 6 7 7 7 3 5 5 5 7 7 7 5 4 4 7 7 7 4 4 3 5 5 5 4 3 3 3 5 5 5 3 3 3 6 6 6 3 3 3 4 4 4 5 2 2 2 3 3 3 2 2 2 5 5 5 2 2 2 4 4 4 6 2 2 2 3 2 2 2 1 1 4 4 4 2 2 2 3 3 3 7 1 1 1 2 2 2 1 1 1 4 3 3 2 2 2 3 3 3 8 1 1 1 1 1 1 1 1 1 3 3 3 2 2 2 3 3 3 9 1 1 1 1 1 1 1 1 1 3 3 3 2 2 2 3 3 3 10 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 3 3 3 11 1 1 1 1 1 1 1 1 1 2 2 2 2 2 1 3 3 2 12 1 1 1 1 1 0 1 1 1 2 2 2 2 1 1 2 2 2 13 1 1 0 1 0 0 1 1 1 2 2 2 2 1 1 2 2 2 14 1 0 0 0 0 0 1 1 1 1 1 1 2 1 1 2 2 2 15 1 0 0 0 0 0 1 1 1 1 1 1 1 1 1 2 2 2 16 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 2 2 2 17 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 2 2 2 18 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 2 2 2 19 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 2 2 2 20 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 2 2 2 21 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 2 2 2 22 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 2 1 1 23 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 2 1 1 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 29.34 2001 ASHRAE Fundamentals Handbook (SI) solar heat gain and direct solar heat gain from fenestration with inside shading (blinds, drapes, etc.), the nonsolar RTS should be used. Radiation from those sources is assumed to be more uniformly distributed onto all room surfaces. Effect of beam solar radiation distribution assumptions is addressed by Hittle (1999).
Representative RTS data for light, medium, and heavyweight constructions are provided in Tables 24 and 25. Those were calcu-lated using the Hbfort computer program (Pedersen et al. 1998) with zone characteristics listed in Table 26. Customized RTS values may be calculated using the HB method where the zone is not reasonably similar to these typical zones or where more precision is desired.
ASHRAE Research Project 942 compared HB and RTS results over a wide range of zone types and input variables (Spitler et al.
1998, Rees et al. 2000). In general, total cooling loads calculated using the RTS method closely agreed with or were slightly higher than those of the HB method with the same inputs. The project examined more than 5000 test cases of varying zone parameters.
The dominating variable was overall thermal mass, and results were grouped into lightweight, U.S. medium-weight, U.K. medium-weight, and heavyweight construction. Best agreement between RTS and HB results was obtained for light- and medium-weight construction. Greater differences occurred in heavyweight cases, with RTS generally predicting slightly higher peak cooling loads than HB. Greater differences also were observed in zones with extremely high internal radiant loads and large glazing areas or with a very lightweight exterior envelope. In this case, heat balance cal-culations predict that some of the internal radiant load will be trans-mitted to the outdoor environment and never becomes cooling load within the space. The RTS methodology does not account for energy being transferred out of the space to the environment and thus pre-dicted higher cooling loads.
Example 5. Wall cooling load using radiant time series. Using the data from Example 4, calculate the cooling load at 3 P.M. Central Daylight Time on July 21 through 9.3 m2 of a wall composed of 100 mm brick, 50 mm of insulation (R = 1.76 m2·K/W), and 200 mm lightweight con-crete block in an office building of average commercial construction. Solution: Total cooling load for the wall is calculated by summing the convective and radiant portions. The convective portion is simply the wall heat gain for the hour being calculated, from Example 4, times the convective fraction for walls from Table 19: Qc = (15.8)(37%) = 5.85 W Table 26 RTS Representative Zone Construction for Tables 24 and 25 Construction Class Exterior Wall Roof/Ceiling Partitions Floor Furnishings Light steel siding, 50 mm insulation, air space, 19 mm gyp 100 mm LW concrete, ceiling air space, acoustic tile 19 mm gyp, air space, 19 mm gyp acoustic tile, ceiling air space, 100 mm LW concrete 25 mm wood @ 50% of floor area Medium 100 mm face brick, 50 mm insulation, air space, 19 mm gyp 100 mm HW concrete, ceiling air space, acoustic tile 19 mm gyp, air space, 19 mm gyp acoustic tile, ceiling air space, 100 mm HW concrete 25 mm wood @ 50% of floor area Heavy 100 mm face brick, 200 mm HW concrete air space, 50 mm insulation, 19 mm gyp 200 mm HW concrete, ceiling air space, acoustic tile 19 mm gyp, 200 mm HW concrete block, 19 mm gyp acoustic tile, ceiling air space, 200 mm HW concrete 25 mm wood @ 50% of floor area Table 27 Wall Heat Gain and Cooling Load for Example 5 Wall Area = 9.3 m2 CTS Table 20 RTS medium-weight construction, 50% glass, with carpet Inside Temp. = 23.9°C Wall Number: 14 Wall U-factor = 0.386 Hour Sol-Air Temp Inside Temp Heat Input CTS Heat Gain Split Heat Gain Nonsolar RTS Radiant Cooling Load Total Cooling Load Convective % Radiant % 37% 63% 1 24.4 23.9 2.0 1% 47.8 17.7 30.1 49% 29.0 46.7 2 24.4 23.9 2.0 1% 43.8 16.2 27.6 17% 27.4 43.6 3 23.9 23.9 — 2% 39.5 14.6 24.9 9% 25.6 40.2 4 23.3 23.9 2.0 5% 35.2 13.0 22.2 5% 23.7 36.7 5 23.3 23.9 2.0 8% 30.8 11.4 19.4 3% 21.6 33.0 6 24.3 23.9 1.6 9% 26.7 9.9 16.8 2% 19.6 29.5 7 25.8 23.9 6.8 9% 23.0 8.5 14.5 2% 17.7 26.2 8 27.6 23.9 13.2 9% 19.6 7.3 12.4 1% 15.9 23.2 9 29.8 23.9 21.1 8% 16.7 6.2 10.5 1% 14.3 20.5 10 31.9 23.9 28.8 7% 14.5 5.4 9.1 1% 13.0 18.3 11 35.3 23.9 40.9 7% 13.0 4.8 8.2 1% 11.9 16.7 12 42.5 23.9 66.7 6% 12.8 4.7 8.0 1% 11.3 16.0 13 47.1 23.9 83.3 5% 13.5 5.0 8.5 1% 11.2 16.2 14 50.2 23.9 94.6 4% 15.5 5.8 9.8 1% 11.6 17.4 15 51.8 23.9 100.1 4% 19.4 7.2 12.3 1% 12.8 20.0 16 50.4 23.9 94.9 3% 25.0 9.2 15.7 1% 14.8 24.1 17 46.9 23.9 82.4 3% 31.7 11.7 19.9 1% 17.5 29.2 18 40.5 23.9 59.5 2% 38.9 14.4 24.5 1% 20.7 35.1 19 31.0 23.9 25.4 2% 45.6 16.9 28.7 1% 23.9 40.8 20 29.4 23.9 19.9 2% 51.0 18.9 32.2 1% 26.9 45.8 21 28.3 23.9 15.9 1% 54.4 20.1 34.3 0% 29.2 49.3 22 27.2 23.9 12.0 1% 55.3 20.4 34.8 0% 30.4 50.9 23 26.1 23.9 8.0 1% 54.0 20.0 34.0 0% 30.7 50.7 24 25.0 23.9 4.0 0% 51.3 19.0 32.3 0% 30.1 49.1 779.1 100% 779.1 288.3 490.8 100% 490.8 779.1 Nonresidential Cooling and Heating Load Calculation Procedures 29.35 The radiant portion of the cooling load is calculated using conduc-tive heat gains for the current and past 23 h, the radiant fraction for walls from Table 19 (63%), and radiant time series from Table 24, in accordance with Equation (39). From Table 24, select the RTS for medium-weight construction, assuming 50% glass and carpeted floors as is often found in modern office construction. Take the wall heat gains from Table 23. Thus, the radiant cooling load for the wall is Qr,14 = r0(0.63)qi,14 + r1(0.63)qi,13 + r2(0.63)qi,12 + r3(0.63)qi,11 + …. + r23(0.63)qi,15 = (0.49)(0.63)(15.5) + (0.17)(0.63)(13.5) + (0.09)(0.63)(12.8) + (0.05)(0.63)(13) + (0.03)(0.63)(14.5) + (0.02)(0.63)(16.7) + (0.02)(0.63)(19.6) + (0.01)(0.63)(23.0) + (0.01)(0.63)(26.7) + (0.01)(0.63)(30.8) + (0.01)(0.63)(35.2) + (0.01)(0.63)(39.5) + (0.01)(0.63)(43.8) + (0.01)(0.63)(47.8) + (0.01)(0.63)(51.3) + (0.01)(0.63)(54.0) + (0.01)(0.63)(55.3) + (0.01)(0.63)(54.4) + (0.01)(0.63)(51.0) + (0.01)(0.63)(45.6) + (0.00)(0.63)(38.9) + (0.00)(0.63)(31.7) + (0.00)(0.63)(25.0) + (0.00)(0.63)(19.4) = 11.6 W The total wall cooling load at the designated hour is thus Qwall = Qc + Qr14 = 5.8 + 11.6 = 17.4 W Again, due to the tedious calculations involved, use of a simple computer spreadsheet or other computer software implementing these calculations can reduce the effort involved. The spreadsheet illustrated in Table 23 is expanded in Table 27 to include splitting the heat gain into convective and radiant portions, applying RTS to the radiant por-tion, and totaling the convective and radiant loads to determine a 24 hour cooling load profile for the data of this example. Figure 12 shows wall heat input, heat gain, and total cooling load versus time for the 24 h calculated.
Example 6. Window cooling load using radiant time series. Calculate the cooling load for a 1.858 m2 double-glazed bronze low-e window at 3 P.M. Central Daylight Time on July 21, at the location and for the con-ditions defined in Example 1, in an office building of average commer-cial construction. Use SHGC data for glass type 17f from Table 13 in Chapter 30. Use window U-factor = 3.18. Solution: To determine the window cooling load, first calculate the 24 h heat gain profile for the window, then split those heat gains into radiant and convective portions, apply the appropriate RTS to the radiant por-tion, then sum the convective and radiant cooling load components to determine total window cooling load at the designated time. The win-dow heat gain components are calculated using Equations (13) through (15).
From Example 1, at hour 14 standard time (3 P.M. Central Daylight Time): ED = 425 W/m2 Ed = 98 W/m2 Er = 85 W/m2 θ = 60.1° From Chapter 30, Table 13, for glass type 17f, SHGC(θ) = SHGC(60.1) = 0.349 (interpolated) 〈SHGC〉D = 0.38 Therefore, window heat gain components for hour 14 are qb14 = AED SHGC(θ) = (1.858)(425)(0.349) = 276 W qd14 = A(Ed + Er) 〈SHGC〉D = (1.858)(98 + 85)(0.38) = 129 W qc14 = UA(tout – tin) = (3.18)(1.858)(34.4 – 23.9) = 62.4 This procedure is repeated to determine these values for a 24 h heat gain profile, as illustrated in Table 28.
Total cooling load for the window is calculated by summing the convective and radiant portions. For windows with inside shading (blinds, drapes, etc.), the direct beam, diffuse, and conductive heat gains may be summed and treated together in calculating cooling loads.
However, in this example, the window does not have inside shading, and the direct beam solar heat gain should be treated separately from the diffuse and conductive heat gains. The direct beam heat gain, with-out inside shading, is treated as 100% radiant, and solar RTS factors from Table 25 are used to convert the beam heat gains to cooling loads.
The diffuse and conductive heat gains can be totaled and split into radi-ant and convection portions according to Table 19 percentages, and nonsolar RTS factors from Table 24 are used to convert the radiant por-tion to cooling load. The solar beam cooling load is calculated using heat gains for the current hour and past 23 h and radiant time series from Table 25, in accordance with Equation (39). From Table 25, select the solar RTS for medium-weight construction, assuming 50% glass and carpeted floors as is often found in modern office construction. Using Table 28 values for direct solar heat gain, the radiant cooling load for the window direct beam solar component is Qb,14 = r0qb,14 + r1qb,13 + r2qb,12 + r3qb,11 + … + r23qb,15 = (0.54)(276) + (0.16)(175) + (0.08)(82) + (0.04)(1) + (0.03)(0) + (0.02)(0) + (0.01)(0) + (0.01)(0) + (0.01)(0) + (0.01)(0) + (0.01)(0) + (0.01)(0) + (0.01)(0) + (0.01)(0) + (0.01)(0) + (0.01)(0) + (0.01)(0) + (0.01)(0) + (0.01)(0) + (0.00)(6) + (0.00)(143) + (0.00)(262) + (0.00)(326) + (0.00)(329) = 184 W This process is repeated for other hours; results are listed in Table 29.
For diffuse and conductive heat gains, the radiant fraction accord-ing to Table 19 is 63%. The radiant portion is processed using nonsolar RTS coefficients from Table 24. The results are listed in Table29.
The total window cooling load at the designated hour is thus Qwindow = Qb + Qdiff + cond = 184 + 170 = 354 W Again, due to the tedious calculations involved, use of a simple computer spreadsheet or other computer software implementing these calculations can reduce the effort involved. The spreadsheet illustrated in Table 28 is expanded in Table 29 to include splitting the heat gain into convective and radiant portions, applying RTS to the radiant por-tion, and totaling the convective and radiant loads to determine a 24 h cooling load profile for the data of this example.
Example 7. Internal cooling load using radiant time series. Calculate the cooling load at 10 A.M. for a 100 m2 office space due to suspended fluorescent lights with a total load of 2000 W, in an office building of average commercial construction. The lights are 50% on at 7 A.M., 100% on from 8 A.M. to 6 P.M., then off the remainder of the time.
Solution: To determine the lighting cooling load, first calculate the 24 h heat gain profile for the lighting load, then split those heat gains into radiant and convective portions, apply the appropriate RTS to the radiant portion, and sum the convective and radiant cooling load com-ponents to determine total cooling load at the designated time. The lighting heat gain profile, based on the schedule indicated, is q1 = (2000 W)(0%) = 0 q2 = (2000 W)(0%) = 0 q3 = (2000 W)(0%) = 0 q4 = (2000 W)(0%) = 0 q5 = (2000 W)(0%) = 0 q6 = (2000 W)(0%) = 0 Fig. 12 Wall Heat Input, Heat Gain for Example 5 and Cooling Load Using CTS and RTS 29.36 2001 ASHRAE Fundamentals Handbook (SI) q7 = (2000 W)(50%) = 1000 q8 = (2000 W)(100%) = 2000 q9 = (2000 W)(100%) = 2000 q10 = (2000 W)(100%) = 2000 q11 = (2000 W)(100%) = 2000 q12 = (2000 W)(100%) = 2000 q13 = (2000 W)(100%) = 2000 q14 = (2000 W)(100%) = 2000 q15 = (2000 W)(100%) = 2000 q16 = (2000 W)(100%) = 2000 q17 = (2000 W)(100%) = 2000 q18 = (2000 W)(100%) = 2000 q19 = (2000 W)(0%) = 0 q20 = (2000 W)(0%) = 0 q21 = (2000 W)(0%) = 0 q22 = (2000 W)(0%) = 0 q23 = (2000 W)(0%) = 0 q24 = (2000 W)(0%) = 0 The convective portion is simply the lighting heat gain for the hour being calculated times the convective fraction for unvented fluorescent lighting from Table 19: Qc,10 = (2000)(33%) = 660 W Table 28 Solar Calculations—Solar Heat Gain Using SHGC—for Example 6 Time Zone = 3 Surface: Month Equa-tion of Time, min.
Decli-nation degrees A B C Time Zone Std. Merid-ian W/m2 (Dimensionless Ratios) Month = 7 Azimuth 45 1 Atlantic 60 Longitude = 90 Tilt 90 1 –11.2 –20 1230 0.142 0.058 2 Eastern 75 Latitude = 40 2 –13.9 –10.8 1215 0.144 0.060 3 Central 90 Clearness = 1 Fenestration 3 –7.5 0 1186 0.156 0.071 4 Mountain 105 Ground Refl. = 0.2 Area1.858 m2 4 1.1 11.6 1136 0.180 0.097 5 Pacific 120 Room Temp. =23.9 U 0.56 5 3.3 20 1104 0.196 0.121 6 Alaska 135 IAC 1.0 7 Hawaii 150 Angle SHGC: 6 –1.4 23.45 1088 0.205 0.134 For Month = 7 0 0.450 7 –6.2 20.6 1085 0.207 0.136 Equation of Time = –6.2 40 0.420 8 –2.4 12.3 1107 0.201 0.122 Declination = 20.6 50 0.400 9 7.5 0 1151 0.177 0.092 A = 344 60 0.350 10 15.4 –10.5 1192 0.160 0.073 B = 0.207 70 0.270 11 13.8 –19.8 1221 0.149 0.063 C = 0.136 80 0.140 12 1.6 –23.45 1233 0.142 0.057 Local Standard Time Meridian = 90 90 — Hemis: 0.380 Direct Beam Solar Heat Gain Diffuse Solar Heat Gain Conduction Total Local Standard Hour Apparent Solar Time, hours Hour Angle Solar Altitude Solar Azimuth Direct Normal Irradiance Surface Incident Angle Surface Direct, W/m2 Direct SHGC Direct Solar Heat Gain, W Ground Diffuse Y Ratio Sky Diffuse Total Diffuse, W/m2 Hemis. SHGC Diffuse Solar Heat Gain, W Outside Temp.
Conductive Heat Gain, W Window Heat Gain, W 1 0.90 166.55 –28.1 –165.7 — 139.3 0.0 — 0.0 0.0 0.45 0.00 0.0 0.380 0.0 24.4 3.3 3 2 1.90 151.55 –23.8 –150.8 — 151.6 0.0 — 0.0 0.0 0.45 0.00 0.0 0.380 0.0 24.4 3.3 3 3 2.90 136.55 –17.1 –137.7 — 162.7 0.0 — 0.0 0.0 0.45 0.00 0.0 0.380 0.0 23.9 0.0 0 4 3.90 121.55 –8.6 –126.2 — 167.8 0.0 — 0.0 0.0 0.45 0.00 0.0 0.380 0.0 23.3 3.3 3 5 4.90 106.55 1.3 –116.2 0 161.1 0.0 — 0.0 0.0 0.45 0.01 0.0 0.380 0.0 23.3 3.3 3 6 5.90 91.55 11.9 –107.0 399 149.7 0.0 — 0.0 13.7 0.45 24.40 38.1 0.380 26.9 23.3 3.3 3 7 6.90 76.55 23.1 –98.1 641 137.3 0.0 — 0.0 33.9 0.45 39.21 73.1 0.380 51.6 23.9 0.0 52 8 7.90 61.55 34.6 –88.8 754 124.7 0.0 — 0.0 53.0 0.45 46.11 99.1 0.380 70.0 25.0 6.6 77 9 8.90 46.55 46.0 –78.0 814 112.2 0.0 — 0.0 69.6 0.45 49.80 119.4 0.380 84.3 26.7 16.4 101 10 9.90 31.55 56.8 –63.6 847 100.0 0.0 — 0.0 82.5 0.48 55.70 138.2 0.380 97.5 28.3 26.3 124 11 10.90 16.55 66.0 –41.0 865 88.4 24.8 0.023 1.1 90.8 0.56 66.21 157.0 0.380 110.8 30.6 39.4 151 12 11.90 1.55 70.6 4.4 871 75.4 220.1 0.200 81.9 94.0 0.68 80.61 174.6 0.380 123.3 32.2 49.2 254 13 12.90 13.45 67.5 34.6 867 67.8 327.0 0.287 174.6 91.9 0.76 89.55 181.4 0.380 128.1 33.9 59.1 362 14 13.90 28.45 58.9 59.8 852 60.1 425.0 0.349 275.9 84.6 0.85 98.02 182.6 0.380 128.9 34.4 62.4 467 15 14.90 43.45 48.3 75.4 822 55.0 471.6 0.375 328.6 72.6 0.90 101.05 173.6 0.380 122.6 35.0 65.6 517 16 15.90 58.45 37.0 86.7 769 53.4 458.4 0.383 326.2 56.7 0.92 96.40 153.1 0.380 108.1 34.4 62.4 497 17 16.90 73.45 25.5 96.2 671 55.6 379.0 0.372 262.0 38.0 0.90 81.82 119.8 0.380 84.6 33.9 59.1 406 18 17.90 88.45 14.2 105.1 467 61.1 225.4 0.341 142.7 17.8 0.83 52.96 70.8 0.380 50.0 32.8 52.5 245 19 18.90 103.45 3.4 114.2 33 69.3 11.8 0.276 6.0 0.6 0.74 3.36 4.0 0.380 2.8 30.6 39.4 48 20 19.90 118.45 –6.6 124.0 — 79.1 0.0 0.151 0.0 0.0 0.64 0.00 0.0 0.380 0.0 29.4 32.8 33 21 20.90 133.45 –15.5 135.2 — 90.2 0.0 — 0.0 0.0 0.55 0.00 0.0 0.380 0.0 28.3 26.3 26 22 21.90 148.45 –22.6 147.9 — 101.9 0.0 — 0.0 0.0 0.45 0.00 0.0 0.380 0.0 27.2 19.7 20 23 22.90 163.45 –27.5 162.5 — 114.2 0.0 — 0.0 0.0 0.45 0.00 0.0 0.380 0.0 26.1 13.1 13 24 23.90 178.45 –29.4 178.3 — 126.7 0.0 — 0.0 0.0 0.45 0.00 0.0 0.380 0.0 25.0 6.6 7 Nonresidential Cooling and Heating Load Calculation Procedures 29.37 The radiant portion of the cooling load is calculated using lighting heat gains for the current hour and past 23 h, the radiant fraction from Table 19 (67%), and radiant time series from Table 24, in accordance with Equation (39). From Table 24, select the RTS for medium-weight construction, assuming 50% glass and carpeted floors as is often found in modern office construction. Thus, the radiant cooling load for light-ing is Qr,10 = r0(0.67)q10 + r1(0.67)q9 + r2(0.67)q8 + r3(0.67)q7 + …. + r23(0.67)q11 = (0.49)(0.67)(2000) + (0.17)(0.67)(2000) + (0.09)(0.67)(2000) + (0.05)(0.67)(1000) + (0.03)(0.67)(0) + (0.02)(0.67)(0) + (0.02)(0.67)(0) + (0.01)(0.67)(0) + (0.01)(0.67)(0) + (0.01)(0.67)(0) + (0.01)(0.67)(0) + (0.01)(0.67)(0) + (0.01)(0.67)(0) + (0.01)(0.67)(0) + (0.01)(0.67)(0) + (0.01)(0.67)(0) + (0.01)(0.67)(2000) + (0.01)(0.67)(2000) + (0.01)(0.67)(2000) + (0.01)(0.67)(2000) + (0.00)(0.67)(2000) + (0.00)(0.67)(2000) + (0.00)(0.67)(2000) + (0.00)(0.67)(2000) = 1092 W The total lighting cooling load at the designated hour is thus Qwall = Qc,10 + Qr,10 = 660 + 1092 = 1752 W COMPARISON WITH PREVIOUS METHODS Some users of this chapter during the past eight editions may have been disappointed that it did not contain (nor was it accom-panied by) fully developed multiple-room, multiple-system cal-culation computer programs for immediate use by HVAC designers. Nor have the several attempts to create and present hand-calculation procedures been acceptable in practice, as they were simply incapable of dealing realistically with contemporary demands. As ASHRAE evolves in the 21st century, the issue of practical load calculation procedures clearly demands computer operation.
Who should provide the computer applications for this pur-pose? While it could be argued that ASHRAE itself is the best source of experience and talent to generate such tools, the nature of ASHRAE as a technical society prevents this type of activity.
Instead, ASHRAE has provided technically reliable mechanisms for both the heat balance and the radiant time series procedures to be accurately implemented, as the most scientifically correct methodologies available, anticipating that various qualified pri-vate enterprises will take these frameworks and complete their development into practical applications for HVAC designers.
The user may question what benefits may be expected now that the TFM, TETD/TA, and CLTD/CLF procedures presented in earlier chapter versions have been superseded (not invalidated or discredited). The primary benefit will be improved accuracy, with reduced dependency upon purely subjective input (such as determining a proper time-averaging period for TETD/TA, or ascertaining appropriate safety factors to add to the “rounded off” TFM results). As a generic example, the space sensible cool-ing load for the traditional little ASHRAE store building (used for example purposes since the 1940s) was calculated by means of the heat balance procedure and independently calculated by application of the radiant time series procedure, with each set of results plotted as one of the load profile curves of Figure 13. Also plotted on this chart are the corresponding curves produced by Table 29 Fenestration Cooling Load for Example 6 Direct Beam Solar Cooling Load Diffuse Solar and Conduction Cooling Load Total Hour Direct Solar Heat Gain, W Convect., % Rad., % Solar RTS Coefficients from Table 25 Direct Solar Radiant Cooling Load Direct Solar Total Cooling Load Diffuse Solar Heat Gain Cond. Heat Gain, Btu/h Tot Diff. & Cond., Btu/h Convect., % Rad., % Nonsolar RTS Coefficients from Table 24 Diff. & Cond. Radiant Cooling Load Diff. & Cond. Total Cooling Load Total Window Cooling Load 0% 100% 37% 63% 1 0 — — 54% 16 16 — 3.28299 3 1.21 2.07 49% 15 16 32 2 0 — — 16% 16 16 — 3 3 1.21 2.07 17% 14 15 31 3 0 — — 8% 16 16 — 0 0 0.00 0.00 9% 12 12 28 4 0 — — 4% 16 16 — –3 3 1.21 2.07 5% 10 9 25 5 0 — — 3% 16 16 0 –3 3 1.21 2.06 3% 9 8 24 6 0 — — 2% 16 16 27 –3 24 8.73 14.86 2% 16 25 41 7 0 — — 1% 15 15 52 0 52 19.10 32.52 2% 26 46 61 8 0 — — 1% 13 13 70 7 77 28.33 48.24 1% 26 66 79 9 0 — — 1% 11 11 84 16 101 37.26 63.45 1% 49 86 97 10 0 — — 1% 7 7 98 26 124 45.81 78.00 1% 60 106 113 11 1 — 4 1% 5 5 111 39 150 55.59 94.65 1% 73 128 133 12 82 — 1 1% 46 46 123 49 173 63.83 108.68 1% 84 148 194 13 175 — 82 1% 108 108 128 59 187 69.26 117.92 1% 94 163 271 14 276 — 175 1% 184 184 129 62 191 70.78 120.51 1% 100 170 354 15 329 — 276 1% 239 239 123 66 188 69.65 118.59 1% 102 172 410 16 326 — 329 1% 260 260 108 62 170 63.07 107.39 1% 98 161 422 17 262 — 326 1% 238 238 85 59 144 53.16 90.52 1% 90 143 381 18 143 — 262 1% 171 171 50 53 102 37.92 64.57 1% 74 112 283 19 6 — 143 1% 78 78 3 39 42 15.62 26.60 1% 50 66 144 20 0 — — 0% 45 45 — 33 33 12.15 20.68 1% 39 51 95 21 0 — — 0% 29 29 — 26 26 9.72 16.54 0% 31 41 70 22 0 — — 0% 22 22 — 20 20 7.29 12.41 0% 26 33 55 23 0 — — 0% 18 18 — 1 13 4.86 8.27 0% 22 26 44 24 0 — — 0% 16 16 — 7 7 2.43 4.14 0% 18 20 36 1599 — 1599 100% 1599 1599 1,190 634 1823 675 1149 100% 1149 1823 3422 29.38 2001 ASHRAE Fundamentals Handbook (SI) the TFM and TETD/TA methodologies in the 1997 edition of this chapter. The user may draw his or her own conclusions from this chart.
HEATING LOAD PRINCIPLES Techniques for estimating design heating load for commercial, institutional, and industrial applications are essentially the same as for those estimating design cooling loads for such uses, except that (1) temperatures outside the conditioned spaces are generally lower than the space temperatures maintained; (2) credit for solar heat gains or for internal heat gains is not included; and (3) the thermal storage effect of building structure or content is ignored.
Heat losses (negative heat gains) are thus considered to be instan-taneous, heat transfer essentially conductive, and latent heat treated only as a function of replacing space humidity lost to the exterior environment.
This simplified approach is justified because it evaluates “worst case” conditions that can reasonably occur during a heating season.
The worst case is the load that must be met under design interior and exterior conditions, including infiltration and/or ventilation, but with no solar effect (at night or on cloudy winter days) and before the periodic presence of people, lights, and appliances has an offset-ting effect.
Safety Factors and Load Allowances. Before mechanical cooling became a usual procedure, buildings included much less insulation, large operable windows, and generally more infiltra-tion-prone assemblies than the energy-efficient and much tighter buildings typical of today. Allowances of 10 to 20% of the net calculated heating load for piping losses to unheated spaces and 10 to 20% more for a warm-up load were common practice, along with occasional other safety factors reflecting the experi-ence and/or concern of the individual designer. Such measures are rarely used today, with the uncompensated net heating load normally considered as having an adequate margin for error.
Armstrong et al. (1992a, 1992b) provide a design method to deal with warm-up and cool-down load.
Cooling Needs During Noncooling Months. Perimeter spaces exposed to high solar heat gain often justify mechanical cooling during sunlit portions of traditional heating months, as do com-pletely interior spaces with significant internal heat gain. Such spaces can also have significant heating loads during nonsunlit hours or after periods of nonoccupancy when adjacent spaces have cooled below interior design temperatures. The loads involved can be estimated conventionally to offset or to compensate for them and prevent overheating, but they have no direct relationship to design heating loads for the spaces in question.
Other Considerations. Calculation of design heating load esti-mates for this general category of applications has essentially become a subset of the more involved and complex estimation of cooling loads for such spaces. Chapter 31 discusses using the heat-ing load estimate to predict or analyze energy consumption over time. Special provisions to deal with special problems are covered in the 1999 ASHRAE Handbook—Applications and the 2000 ASHRAE Handbook—Systems and Equipment.
REFERENCES Alereza, T. and J.P. Breen, III. 1984. Estimates of recommended heat gain due to commercial appliances and equipment. ASHRAE Transactions 90(2A):25-58.
ASHRAE. 1999. Ventilation for acceptable indoor air quality. ASHRAE Standard 62-1999.
Armstrong, P.R., C.E. Hancock III, and J.E. Seem. 1992a. Commercial building temperature recovery—Part I: Design procedure based on a step response model. ASHRAE Transactions 98(1):381-96.
Armstrong, P.R., C.E. Hancock III, and J.E. Seem. 1992b. Commercial building temperature recovery—Part II: Experiments to verify the step response model. ASHRAE Transactions 98(1):397-410.
Bliss, R.J.V. 1961. Atmospheric radiation near the surface of the ground.
Solar Energy 5(3):103.
Consolazio, W. and L.J. Pecora. 1947. Minimal replenishment air required for living spaces. ASHVE Standard 53:127.
Colliver, D.G., H. Zhang, R.S. Gates, and K.T. Priddy. 1995. Determination of the 1%, 2.5%, and 5% occurences of extreme dew-point temperatures and mean coincident dry-bulb temperatures. ASHRAE Transactions 101(2):265-86.
Colliver, D.G., R.S. Gates, H. Zhang, and K.T. Priddy. 1998. Sequences of extreme temperature and humidity for design calculations. ASHRAE Transactions 104(1A):133-44.
Colliver, D.G., R.S. Gates, T.F. Burke, and H. Zhang. 2000. Development of the design climatic data for the 1997 ASHRAE Handbook—Fundamen-tals. ASHRAE Transactions 106(1):3.14.
Fisher, D.E. and C.O. Pedersen. 1997. Convective heat transfer in building energy and thermal load calculations. ASHRAE Transactions 103(2): 137-48.
Fisher, D.R. 1998. New recommended heat gains for commercial cooking equipment. ASHRAE Transactions 104(2):953-60.
Gordon, E.B., D.J. Horton, and F.A. Parvin. 1994. Development and appli-cation of a standard test method for the performance of exhaust hoods with commercial cooking appliances. ASHRAE Transactions 100(2): 988-99.
Hittle, D.C. and C.O. Pedersen. 1981. Calculating building heating loads using the frequency of multi-layered slabs. ASHRAE Transactions 87(2):545-68.
Hittle, D.C. 1999. The effect of beam solar radiation on peak cooling loads.
ASHRAE Transactions 105(2):510-13.
Hosni, M.H., B.W. Jones, and J.M. Snipes. 1998. Total heat gain and the split between radiant and convective heat gain from office and laboratory equipment in buildings. ASHRAE Transactions 104(1A):356-65.
Hosni, M.H., B.W. Jones, and H. Xu. 1999. Experimental results for heat gain and radiant/convective split from equipment in buildings. ASHRAE Transactions 105(2):527-39.
Jones, B.W., M.H. Hosni, and J.M. Snipes. 1998. Measurements of radiant heat gain from office equipment using a scanning radiometer. ASHRAE Transactions 104(1B):1775-83.
Kerrisk, J.F., N.M. Schnurr, J.E. Moore, and B.D. Hunn. 1981. The custom weighting-factor method for thermal load calculations in the DOE-2 computer program. ASHRAE Transactions 87(2):569-84.
Komor, P. 1997. Space cooling demands from office plug loads. ASHRAE Journal 39(12):41-44.
Kusuda, T. 1967. NBSLD, The computer program for heating and cooling loads for buildings. Bss 69 and NBSIR 74-574. National Bureau of Stan-dards.
LBL. 1994. Window 4.1: Program description. Lawrence Berkeley Labora-tory. Liesen, R.J. and C.O. Pedersen. 1997. An evaluation of inside surface heat balance models for cooling load calculations. ASHRAE Transactions 103(2):485-502.
McClellan, T.M. and C.O. Pedersen. 1997. The sensitivity of cooling load calculations to outside heat balance models. ASHRAE Transactions 103(2):469-84.
Fig. 13 Cooling Load Profile Comparison Nonresidential Cooling and Heating Load Calculation Procedures 29.39 McQuiston, F.C. and J.D. Spitler. 1992. Cooling and heating load calcula-tion manual. 2nd ed. ASHRAE.
Marn, W.L. 1962. Commercial gas kitchen ventilation studies. Research Bulletin No. 90 (March). Gas Association Laboratories, Cleveland, OH.
Mitalas, G.P. 1972. Transfer function method of calculating cooling loads, heat extraction rate, and space temperature. ASHRAE Transactions 14(12):52.
Mitalas, G.P. and K. Kimura. 1971. A calorimeter to determine cooling load caused by lights. ASHRAE Transactions 77(2)65.
Nevins, R.G., H.E. Straub, and H.D. Ball. 1971. Thermal analysis of heat removal troffers. ASHRAE Transactions 77(2):58-72.
NFPA. 1999. Standard for health care facilities. Standard 99-99. National Fire Protection Association, Quincy, MA.
Pedersen, C.O., D.E. Fisher, and R.J. Liesen. 1997. A heat balance based cooling load calculation procedure. ASHRAE Transactions 103(2):459-68.
Pedersen, C.O., D.E. Fisher, J.D. Spitler, and R.J. Liesen. 1998. Cooling and heating load calculation principles. ASHRAE, Atlanta.
Rees, S.J., J.D. Spitler, M.G. Davies, and P. Haves. 2000. Qualitative com-parison of North American and UK cooling load calculation models.
International Journal of Heating, Ventilating, Air-Conditioning and Refrigerating Research 6(1):75-99.
Smith, V.A., R.T. Swierczyna, C.N. Claar. 1995. Application and enhance-ment of the standard test method for the performance of commercial kitchen ventilation systems. ASHRAE Transactions 101(2).
Sowell, E.F. 1988a. Cross-check and modification of the DOE-2 program for calculation of zone weighting factors. ASHRAE Transactions 94(2):737-53.
Sowell, E.F. 1988b. Load calculations for 200,640 zones. ASHRAE Trans-actions 94(2):716-36.
Spitler, J.D. and D.E. Fisher. 1999a. Development of periodic response fac-tors for use with the radiant time series method. ASHRAE Transactions 105(2).
Spitler, J.D. and D.E. Fisher. 1999b. On the relationship between the radiant time series and transfer function methods for design cooling load calcu-lations. International Journal of Heating, Ventilating, Air-Conditioning and Refrigerating Research 5(2).
Spitler, J.D., D.E. Fisher, and C.O. Pedersen. 1997. The radiant time series cooling load calculation procedure. ASHRAE Transactions 103(2).
Spitler, J.D., S.J. Rees, and P. Haves. 1998. Quantitive comparison of North American and UK cooling load calculation procedures—Part 1: method-ology, Part II: results. ASHRAE Transactions 104(2):36-46, 47-61.
Sun, T.-Y. 1968. Shadow area equations for window overhangs and side-fins and their application in computer calculation. ASHRAE Transactions 74(1):I-1.1 to I-1.9.
Talbert, S.G., L.J. Canigan, and J.A. Eibling. 1973. An experimental study of ventilation requirements of commercial electric kitchens. ASHRAE Transactions 79(1):34.
Todorovic, B., L. Marjanovic, and D. Kovacevic. 1993. Comparison of dif-ferent calculation procedures for cooling load from solar radiation through a window. ASHRAE Transactions 99(2):559-64.
Walton, G. 1983. Thermal analysis research program reference manual.
National Bureau of Standards.
Wilkins, C.K. 1998. Electronic equipment heat gains in buildings. ASHRAE Transactions 104(1B):1784-89.
Wilkins, C.K., R. Kosonen, and T. Laine. 1991. An analysis of office equip-ment load factors. ASHRAE Journal 33(9):38-44.
Wilkins, C.K. and N. McGaffin. 1994. Measuring computer equipment loads in office buildings. ASHRAE Journal 36(8):21-24.
Wilkins, CK. and M.R. Cook. 1999. Cooling loads in laboratories. ASHRAE Transactions 105(1):744-49.
Wilkins, C.K. and M.H. Hosni. 2000. Heat gain from office equipment.
ASHRAE Journal 42(6):33-44.
BIBLIOGRAPHY Alford, J.S., J.E. Ryan, and F.O. Urban. 1939. Effect of heat storage and vari-ation in outdoor temperature and solar intensity on heat transfer through walls. ASHVE Transactions 45:387.
American Gas Association. 1948. A comparison of gas and electric use for commercial cooking. Cleveland, OH.
American Gas Association. 1950. Gas and electric consumption in two col-lege cafeterias. Cleveland, OH.
ASHRAE. 1975. Procedure for determining heating and cooling loads for computerized energy calculations, algorithms for building heat transfer subroutines.
ASHRAE. 1979. Cooling and heating load calculation manual.
BLAST Support Office. 1991. BLAST User Reference. Urbana-Champaign.
University of Illinois.
Brisken, W.R. and G.E. Reque. 1956. Thermal circuit and analog computer methods, thermal response. ASHAE Transactions 62:391.
Buchberg, H. 1955. Electric analog prediction of the thermal behavior of an inhabitable enclosure. ASHAE Transactions 61:339-386.
Buchberg, H. 1958. Cooling load from thermal network solutions ASHAE Standard 64:111.
Buffington, D.E. 1975. Heat gain by conduction through exterior walls and roofs—transmission matrix method. ASHRAE Transactions 81(2):89.
Burch, D.M., B.A. Peavy, and F.J. Powell. 1974. Experimental validation of the NBS load and indoor temperature prediction model. ASHRAE Trans-actions 80(2):291.
Burch, D.M., J.E. Seem, G.N. Walton, and B.A. Licitra. 1992. Dynamic evaluation of thermal bridges in a typical office building. ASHRAE Transactions 98:291-304.
Butler, R. 1984. The computation of heat flows through multi-layer slabs.
Building and Environment 19(3):197-206.
Ceylan, H.T., and G.E. Myers. 1985. Application of response-coefficient method to heat-conduction transients. ASHRAE Transactions 91:30-39.
Chiles, D.C. and E.F. Sowell. 1984. A counter-intuitive effect of mass on zone cooling load response. ASHRAE Transactions 91(2A):201-208.
Chorpening, B.T. 1997. The sensitivity of cooling load calculations to win-dow solar transmission models. ASHRAE Transactions 103(1).
Clarke, J.A. 1985. Energy simulation in building design. Adam Hilger Ltd., Boston.
Davies, M.G. 1996. A time-domain estimation of wall conduction transfer function coefficients. ASHRAE Transactions 102(1).
DeAlbuquerque, A.J. 1972. Equipment loads in laboratories. ASHRAE Jour-nal 14(10):59.
Falconer, D.R., E.F. Sowell, J.D. Spitler, and B.B. Todorovic. 1993. Elec-tronic tables for the ASHRAE load calculation manual. ASHRAE Trans-actions 99(1):193-200.
Harris, S. M. and F.C. McQuiston. 1988. A study to categorize walls and roofs on the basis of thermal response. ASHRAE Transactions 94(2):688-714.
Headrick, J.B. and D.P. Jordan. 1969. Analog computer simulation of heat gain through a flat composite roof section. ASHRAE Transactions 75(2):21.
Hittle, D.C. 1981. Calculating building heating and cooling loads using the frequency response of multilayered slabs, Ph.D. thesis, Department of Mechanical and Industrial Engineering, University of Illinois, Urbana, IL.
Hittle, D.C. and R. Bishop. 1983. An improved root-finding procedure for use in calculating transient heat flow through multilayered slabs. Inter-national Journal of Heat and Mass Transfer 26:1685-93.
Houghton, D.G., C. Gutherlet, and A.J. Wahl. 1935. ASHVE Research Report No. 1001—Cooling requirements of single rooms in a modern office building. ASHVE Transactions 41:53.
Kimura and Stephenson. 1968. Theoretical study of cooling loads caused by lights. ASHRAE Transactions 74(2):189-97.
Kusuda, T. 1969. Thermal response factors for multilayer structures of var-ious heat conduction systems. ASHRAE Transactions 75(1):246 Leopold, C.S. 1947. The mechanism of heat transfer, panel cooling, heat storage. Refrigerating Engineering 7:33.
Leopold, C.S. 1948. Hydraulic analogue for the solution of problems of ther-mal storage, radiation, convection, and conduction. ASHVE Transac-tions 54:3-9.
Livermore, J.N. 1943. Study of actual vs predicted cooling load on an air conditioning system. ASHVE Transactions 49:287.
Mackey, C.O. and N.R. Gay, 1949. Heat gains are not cooling loads. ASHVE Transactions 55:413.
Mackey, C.O. and N.R. Gay. 1952. Cooling load from sunlit glass. ASHVE Transactions 58:321.
Mackey, C.O. and N.R. Gay. 1954. Cooling load from sunlit glass and wall.
ASHVE Transactions 60:469.
Mackey, C.O. and L.T. Wright, Jr. 1944. Periodic heat flow—homogeneous walls or roofs. ASHVE Transactions 50:293.
Mackey, C.O. and L.T. Wright, Jr. 1946. Periodic heat flow—composite walls or roofs. ASHVE Transactions 52:283.
29.40 2001 ASHRAE Fundamentals Handbook (SI) Mast, W.D. 1972. Comparison between measured and calculated hour heat-ing and cooling loads for an instrumented building. ASHRAE Symposium Bulletin No. 72-2.
McBridge, M.F., C.D. Jones, W.D. Mast, and C.F. Sepsey. 1975. Field vali-dation test of the hourly load program developed from the ASHRAE algorithms. ASHRAE Transactions 1(1):291.
Mitalas, G.P. 1968. Calculations of transient heat flow through walls and roofs. ASHRAE Transactions 74(2):182-88.
Mitalas, G.P. 1969. An experimental check on the weighting factor method of calculating room cooling load. ASHRAE Transactions 75(2):22.
Mitalas, G.P. 1973. Calculating cooling load caused by lights. ASHRAE Transactions 75(6):7.
Mitalas, G.P. and D.G. Stephenson. 1967. Room thermal response factors.
ASHRAE Transactions 73(2): III.2.1.
Mitalas, G.P. and J.G. Arsenault. 1970. Fortran IV program to calculate Z-transfer functions for the calculation of transient heat transfer through walls and roofs. Use of Computers For Environmental Engineering Related to Buildings, 633-68. National Bureau of Standards, Gaithers-burg, MD.
Mitalas, G.P. 1978. Comments on the Z-transfer function method for calcu-lating heat transfer in buildings. ASHRAE Transactions, 84(1):667-74.
Nottage, H.B. and G.V. Parmelee. 1954. Circuit analysis applied to load esti-mating. ASHVE Transactions 60:59.
Nottage, H.B. and G.V. Parmelee. 1955. Circuit analysis applied to load esti-mating. ASHAE Transactions 61(2):125.
Ouyang, K. and F. Haghighat. 1991. A procedure for calculating thermal response factors of multi-layer walls-state space method. Building and Environment 26(2):173-77.
Parmelee, G.V., P. Vance, and A.N. Cherny. 1957. Analysis of an air condi-tioning thermal circuit by an electronic differential analyzer. ASHAE Transactions 63:129.
Paschkis, V. 1942. Periodic heat flow in building walls determined by elec-tric analog method. ASHVE Transactions 48:75.
Peavy, B.A. 1978. Determination and verification of thermal response fac-tors for thermal conduction applications. Peavy, B.A. 1978. A note on response factors and conduction transfer func-tions. ASHRAE Transactions 84(1):688-90.
Peavy, B.A., F.J. Powell, and D.M. Burch. 1975. Dynamic thermal perfor-mance of an experimental masonry building. NBS Building Science Series 45 (July).
Romine, T.B., Jr. 1992. Cooling load calculation, Art or science? ASHRAE Journal, 34(1), p. 14.
Rudoy, W. 1979. Don’t turn the tables. ASHRAE Journal 21(7):62.
Rudoy, W. and F. Duran. 1975. Development of an improved cooling load calculation method. ASHRAE Transactions 81(2):19-69.
Seem, J.E., S.A. Klein, W.A. Beckman, and J.W. Mitchell. 1989. Transfer functions for efficient calculation of multidimensional transient heat transfer. Journal of Heat Transfer 111:5-12.
Sowell, E.F. and D.C. Chiles. 1984a. Characterization of zone dynamic response for CLF/CLTD tables. ASHRAE Transactions 91(2A):162-78.
Sowell, E.F. and D.C. Chiles. 1984b. Zone descriptions and response char-acterization for CLF/CLTD calculations. ASHRAE Transactions 91(2A): 179-200.
Spitler, J.D. 1996. Annotated guide to building load calculation models and algorithms. ASHRAE, Atlanta.
Spitler, J.D., F.C. McQuiston, and K.L. Lindsey. 1993. The CLTD/SCL/CLF cooling load calculation method. ASHRAE Transactions 99(1).
Spitler, J.D. and F.C. McQuiston. 1993. Development of a revised cooling and heating calculation manual. ASHRAE Transactions 99(1).
Stephenson, D.G. 1962. Method of determining non-steady-state heat flow through walls and roofs at buildings. The Journal of the Institution of Heating and Ventilating Engineers 30:5.
Stephenson, D.G. and G.P. Mitalas. 1967. Cooling load calculation by ther-mal response factor method. ASHRAE Transactions 73(2):III.1.1.
Stephenson, D.G. and G.P. Mitalas. 1971. Calculation of heat transfer func-tions for multi-layer slabs. ASHRAE Transactions 77(2):117-26.
Stewart, J.P. 1948. Solar heat gain through walls and roofs for cooling load calculations. ASHVE Transactions 54:361.
Sun, T.-Y. 1968. Computer evaluation of the shadow area on a window cast by the adjacent building, ASHRAE Journal, September 1968.
Todorovic, B. 1987. The effect of the changing shade line on the cooling load calculations. ASHRAE videotape “Practical applications for cool-ing load calculations.” Todorovic B. 1982. Cooling load from solar radiation through partially shaded windows, taking heat storage effect into account. ASHRAE Transactions 88(2):924-937.
Todorovic, B. 1984. Distribution of solar energy following its transmittal through window panes. ASHRAE Transactions 90(1B):806-15.
Todorovic, B. 1989. Heat storage in building structure and its effect on cool-ing load; Heat and mass transfer in building materials and structure.
Hemisphere publishing, New York, 603-14.
Todorovic, B. and D. Curcija. 1984. Calculative procedure for estimating cooling loads influenced by window shading, using negative cooling load method. ASHRAE Transactions 2:662.
Vild, D.J. 1964. Solar heat gain factors and shading coefficients. ASHRAE Journal 6(10):47.
Wilkins, C.K. and N. McGaffin. 1994. Measuring computer equipment loads in office buildings. ASHRAE Journal 36(8):21-24.
York, D.A. and C.C. Cappiello. 1981. DOE-2 engineers manual (Version 2.1A), Lawrence Berkeley Laboratory and Los Alamos National Labo-ratory.
30.1 CHAPTER 30 FENESTRATION Fenestration Components ....................................................... 30.1 Determining Fenestration Energy Flow ................................. 30.3 U-FACTOR (THERMAL TRANSMITTANCE) ....................... 30.4 Determining Fenestration U-Factors ..................................... 30.4 Indoor and Outdoor Surface Heat Transfer Coefficients ....... 30.5 Representative U-Factors for Fenestration Products ............. 30.7 Representative U-Factors for Doors .................................... 30.11 Examples ............................................................................... 30.12 SOLAR HEAT GAIN AND VISIBLE TRANSMITTANCE ..... 30.13 Determining Incident Solar Flux .......................................... 30.13 Optical Properties ................................................................. 30.17 Solar-Optical Properties of Glazing ..................................... 30.18 Solar Heat Gain Coefficient ................................................. 30.36 Calculation of Solar Heat Gain ............................................ 30.41 SHADING DEVICES AND FENESTRATION ATTACHMENTS .................................. 30.42 Exterior Shading ................................................................... 30.44 Indoor and Between-Glass Shading Devices on Simple Fenestrations ......................................................... 30.47 Completely Shaded Glazings ................................................ 30.49 VISUAL AND THERMAL CONTROLS ................................ 30.54 AIR LEAKAGE ...................................................................... 30.55 DAYLIGHTING ..................................................................... 30.56 Daylight Prediction ............................................................... 30.56 Light Transmittance and Daylight Use ................................. 30.58 SELECTING FENESTRATION ............................................. 30.59 Annual Energy Performance ................................................. 30.59 Condensation Resistance ...................................................... 30.60 Occupant Comfort and Acceptance ...................................... 30.62 Durability .............................................................................. 30.64 Codes and Standards ............................................................ 30.64 Symbols ................................................................................. 30.65 ENESTRATION is an architectural term that refers to the ar-Frangement, proportion, and design of window, skylight, and door systems within a building. Fenestration components include glazing material, either glass or plastic; framing, mullions, muntins, dividers, and opaque door slabs; external shading devices; internal shading devices; and integral (between-glass) shading systems. For our pur-poses, fenestration and fenestration systems will refer to the basic assemblies and components of exterior window, skylight, and door systems within the building envelope. Fenestration can serve as a physical and/or visual connection to the outdoors, as well as a means to admit solar radiation. The solar radiation provides natural lighting, referred to as daylighting, and heat gain to a space. Fenestration can be fixed or operable, and operable units can allow natural ventilation to a space and egress in low-rise buildings.
Fenestration affects building energy use through four basic mechanisms—thermal heat transfer, solar heat gain, air leakage, and daylighting. The energy impacts of fenestration can be mini-mized by (1) using daylight to offset lighting requirements, (2) using appropriate glazings and shading strategies to control solar heat gain to supplement heating through passive solar gain and min-imize cooling requirements, (3) using appropriate glazing to mini-mize conductive heat loss, and (4) specifying low air leakage fenestration products. In addition, natural ventilation strategies can reduce energy use for cooling and fresh air requirements.
Today designers, builders, energy codes, and energy-efficiency incentive programs [such as Energy Star (www.energystar.gov) and the LEED Green Building Program (www.usgbc.org)] are asking more and more from fenestration systems. Window, skylight, and door manufacturers are responding with new and improved prod-ucts to meet those demands. With the advent of computer simulation software, designing to improve thermal performance of fenestration products has become much easier. Through participation in rating and certification programs (such as those of the National Fenestra-tion Rating Council) that require the use of this software, fenestra-tion manufacturers can take credit for these improvements through certified ratings that are credible to designers, builders, and code officials. A designer should consider architectural requirements, thermal performance, economic criteria, and human comfort when selecting fenestration. Typically, a wide range of fenestration prod-ucts are available that meet the specifications for a project. Refining the specifications to improve the energy performance and enhance a living or work space can result in lower energy costs, increased productivity, and improved thermal and visual comfort. Carmody et al. (1996) and CEA (1995) provide guidance for carrying out these requirements.
FENESTRATION COMPONENTS Fenestration consists of glazing, framing, and in some cases shading devices and insect screens. Glazing The glazing unit may have single glazing or multiple glazing.
The most common glazing material is glass, although plastic is also used. The glass or plastic may be clear, tinted, coated, lami-nated, patterned, or obscured. Clear glass transmits more than 80% of the incident solar radiation and more than 75% of the visi-ble light. Tinted glass is available in many colors, all of which dif-fer in the amount of solar radiation and visible light they transmit and absorb. Coatings on glass affect the transmission of solar radi-ation, and visible light may affect the absorptance of room temper-ature radiation. Some coatings are highly reflective (such as mirrors), while others are designed to have a very low reflectance.
Some coatings result in a visible light transmittance that is as much as 1.4 times higher than the solar heat gain coefficient (desirable for good daylighting while minimizing cooling loads). Laminated glass is made of two panes of glass adhered together. The inter-layer between the two panes of glass is typically plastic and may be clear, tinted, or coated. Patterned glass is a durable ceramic frit applied to a glass surface in a decorative pattern. Obscured glass is translucent and is typically used in privacy applications.
Insulating Glazing Units Insulating glazing units (IGUs) are hermetically sealed, multi-ple-pane assemblies consisting of two or more glazing layers held and bonded at their perimeter by a spacer bar typically containing a desiccant material. The desiccated spacer is surrounded on at least two sides by a sealant that adheres the glass to the spacer. Figure 1 shows the construction of a typical IGU.
The preparation of this chapter is assigned to TC 4.5, Fenestration.
30.2 2001 ASHRAE Fundamentals Handbook (SI) Glazing. Common types of glass used in IGUs are clear, tinted, and low emissivity (low-e). Due to its energy efficiency, daylight-ing, and comfort benefits, low-e coated glass is now used in more than 30% of all the fenestration products installed in the United States. Tinted and reflective glazing can also be used to reduce solar heat gain through fenestration products. Low-e coatings can also be applied to thin plastic films for use in IGUs. There are two types of low-e coating: high-solar-gain and low-solar-gain. The first of these primarily reduces heat conduction through the glazing system and is intended for cold climates. The second, for hot climates, reduces solar heat gain by blocking admission of the infrared por-tion of the solar spectrum. There are two ways of achieving low-solar-gain low-e performance. The first is with a special multilayer solar infrared reflecting coating. The second is with a solar infrared absorbing outer glass. To protect the inner glazing and the building interior from the absorbed heat from this outer glass, a cold-climate-type low-e coating is also used to reduce conduction of heat from the outer pane to the inner one. In addition, argon and krypton gas are used in lieu of air in the gap between the panes in combination with low-e glazing to further reduce energy transfer. Some manu-facturers construct IGUs with one or more suspended, low-e coated plastic films between the glass panes and with a spacer that has bet-ter insulating properties and a dual sealant that improves the seal around the gas spaces.
Spacer. The spacer serves to separate the panes of glass and to provide the surface for primary and secondary sealant adhesion.
Several types of spacers are used in IGU construction today. Each type provides different heat transfer properties depending on the spacer material and geometry. Heat transfer at the edge of the IGU is greater than at the center of the IGU due to greater heat flow through the spacer system.
Spacer systems have been developed to minimize the heat flow at the edge of the IGU. These spacer systems are referred to as warm edge spacers. In IGU construction, warm edge spacer designs reduce edge heat transfer by substituting materials that have lower thermal conductivity than aluminum (e.g., stainless steel, galva-nized steel, tin plated steel, polymers, or foamed silicone). Tradi-tional spacers are often made of aluminum. Fusing or bending the corners of the spacer minimizes moisture and hydrocarbon vapor transmission into the air space through the corners. Desiccants such as molecular sieve or silica gel are also used to absorb moisture that was initially trapped in the IGU during assembly or gradually diffuses through the seals after construction.
Sealant(s). Several different sealant configurations are being used successfully in modern IGU construction. In all sealant con-figurations, the primary seal minimizes moisture and hydrocarbon transmission. In dual-seal construction, the secondary seal provides structural integrity between the lites of the IGU. A secondary seal ensures long-term adhesion and greater resistance to solvents, oils, and short-term water immersion. In typical dual-seal construction, the primary seal is made of compressed polyisobutylene (PIB), and the secondary seal is made of silicone, polysulfide, or polyurethane.
Single-seal construction depends on a single sealant to provide adhesion of the glass to the spacer as well as minimizing moisture and hydrocarbon transmission. Single-seal construction is generally more cost-efficient than dual-seal systems. A third type of sealant used in IGU construction takes advantage of advanced cross-linking polymers that provide both low moisture transmission and equiva-lent structural properties to dual-seal systems. These sealants are typically referred to as dual seal equivalent (DSE) materials.
Desiccants. Typical desiccants include molecular sieve, silica gel, or a matrix of both materials. Desiccants are used to absorb moisture that was initially trapped in the IGU during assembly or that gradually diffused through the seals after construction.
Gas Fill. The hermetically sealed space between glass panes in an IGU is most often filled with air. In some cases, argon and kryp-ton gas are used in lieu of air in the space between the panes to fur-ther reduce the energy transfer.
Fig. 1 Insulating Glazing Unit (IGU) Construction Detail Fig. 2 Types of Residential Windows Fenestration 30.3 Framing The three main categories of window framing materials are wood, metal, and polymers. Wood has good structural integrity and insulating value but low resistance to weather, moisture, warpage, and organic degradation (from mold and insects). Metal is durable and has excellent structural characteristics, but it has very poor thermal performance. The metal of choice in windows is almost exclusively aluminum, due to ease of manufacture, low cost, and low mass, but aluminum has a thermal conductivity roughly 1000 times that of wood or polymers. The poor thermal performance of metal-frame windows can be improved with a thermal break (a nonmetal component that separates the metal frame exposed to the outside from the surfaces exposed to the inside). Polymer frames are made of extruded vinyl or poltruded fiberglass (glass-reinforced polyester). Their thermal and structural performance is similar to that of wood, although vinyl frames for large windows must be reinforced.
Manufacturers sometimes combine these materials as clad units (e.g., vinyl-clad aluminum, aluminum-clad wood, vinyl-clad wood) to increase durability, improve thermal performance, or improve aesthetics. In addition, curtain wall systems for commercial build-ings may be structurally glazed, and the exterior “framing” is sim-ply rubber gaskets or silicone.
Residential windows can be categorized by operator type, as shown in Figure 2. Traditionally there are several basic window types: casements, fixed picture windows, horizontal and vertical sliders, pivoting, awning, or projecting windows, dual acting win-dows, and special applications such as skylights and greenhouse or garden window inserts. The glazing system can be mounted either directly in the frame (a direct-glazed or direct-set window, which is not operable) or in a sash that moves in the frame (for an operating window). In operable windows, a weather-sealing system between the frame and sash reduces air and water leakage.
Shading Shading devices are available in a wide range of products that differ greatly in their appearance and energy performance. Shading devices include interior and exterior blinds, integral blinds, interior and exterior screens, shutters, draperies, and roller shades. Shading devices on the exterior of the glazing reduce solar heat gain more effectively than interior devices. However, interior devices are eas-ier to operate and adjust. Some products help insulate the indoors from the outdoors, while others redirect incoming solar radiation to minimize visual and thermal discomfort. Overhangs and vegetation can be effective shading too.
DETERMINING FENESTRATION ENERGY FLOW Energy flows through fenestration via (1) conductive and con-vective heat transfer caused by the temperature difference between outdoor and indoor air, (2) net long-wave (above 2500 nm) radiative exchange between the fenestration and its surrounding and between glazing layers, and (3) short-wave (below 2500 nm) solar radiation incident on the fenestration product, either directly from the sun or reflected from the ground or adjacent objects. Simplified calcula-tions are based on the observation that the temperatures of the sky, ground, and surrounding objects (and hence their radiant emission) correlate with the exterior air temperature. The radiative inter-changes are then approximated by assuming that all the radiating surfaces (including the sky) are at the same temperature as the out-door air. With this assumption, the basic equation for the instanta-neous energy flow Q through a fenestration is (1) where Q = instantaneous energy flow, W U = overall coefficient of heat transfer (U-factor), W/(m2·K) tin = interior air temperature, °C tout = exterior air temperature, °C Apf = total projected area of fenestration, m2 SHGC = solar heat gain coefficient, nondimensional Et = incident total irradiance, W/m2 The quantities U and SHGC are instantaneous performance indi-ces. The principal justification for Equation (1) is its simplicity, achieved by collecting all the linked radiative, conductive, and con-vective energy transfer processes into U and SHGC. These quanti-ties vary because (1) convective heat transfer rates vary as fractional powers of temperature differences or free-stream speeds, (2) varia-tions in temperature due to the weather or climate are small on the absolute temperature scale (°R) that controls radiative heat transfer rates, (3) fenestration systems always involve at least two thermal resistances in series, and (4) solar heat gain coefficients depend on solar incident angle and spectral distribution.
In the discussion of this chapter, Q is divided into two parts: Q = Qth + Qsol (2) where Qth = instantaneous energy flow due to indoor-outdoor temperature difference (thermal energy flow) Qsol = instantaneous energy flow due to solar radiation (solar energy flow) The section on U-Factor (Thermal Transmittance) deals with Qth, while the section on Solar Heat Gain and Visible Transmittance discusses Qsol. In the latter section, both the effects of direct solar radiation and those of solar radiation scattered by the sky or ground are included.
Equation (1) presents a fenestration as it might appear on a build-ing plan: a featureless, planar object filling an opening in the build-ing envelope. Real fenestrations, however, are composite three-dimensional objects that may consist of frames, sashes, mullions, and other structural elements, as well as glazing systems. The latter in turn may contain structural spacers as well as glazing layers.
There may in addition be shading elements, either as separate attachments or integrated into the glazing system.
The heat transfer through such an assembly of elements is calcu-lated by dividing the fenestration area into parts, each of which has an energy flow that is more simply calculated than the total: (3) where qv = energy flux (energy flow per unit area) of the vth part Av = area of the vth part This subdivision is applied to each of the terms in Equation (2) separately; for example, the thermal heat transfer through glazings and frames is frequently different, so that it is useful to make the fol-lowing separation: (4) where the subscript f refers to the frame, and g refers to the glazing (both for thermal energy flow). Similarly, solar radiation will have a different effect on the frame and the glazed area of a fenestration (since the former is generally opaque), so that (5) Q UApf tout tin – ( ) SHGC ( )ApfEt + = Q Avqv v ∑ = Qth Afqf Agqg + = Qsol Aopqop Asqs + = 30.4 2001 ASHRAE Fundamentals Handbook (SI) where the subscript op refers to the (opaque) frame (for solar energy flow), and s refers to the (solar-transmitting) glazing. This division into frame and glazing areas can be and usually is different for the solar and thermal energy flows. Subdivisions of this sort, when Equation (3) is compared with Equation (1), effectively make the overall U-factor and solar heat gain coefficient area-averaged quan-tities. This area averaging is described explicitly in the appropriate sections below. Note that in more complicated fenestrations, where the glazing portion may contain opaque shading elements, the opaque portion is that part that can never under any conditions admit solar radiation in any form other than heat. A window with a closed, perfectly opaque blind would not be considered an opaque element because sometimes the blind may be open. A section of curtain wall consisting of wall or frame elements with an exterior cover of glass (for uniform appearance) would be an opaque element in spite of its transparent covering.
A second type of subdivision occurs when, for a given part of the fenestration system, energy flow is driven by physical processes that are more complicated than those assumed in Equation (1). For example, the heat transfer through a glazing consists of a “contact” (i.e., glass-to-air) part and a radiative part, and the latter (qR) may depend on radiant temperatures that are different from the air tem-peratures in Equation (1): (6) U-FACTOR (THERMAL TRANSMITTANCE) In the absence of sunlight, air infiltration, and moisture conden-sation, the first term in Equation (1) represents the rate of thermal heat transfer through a fenestration system. Most fenestration sys-tems consist of transparent multipane glazing units and opaque ele-ments comprising the sash and frame (hereafter called frame). The glazing unit’s heat transfer paths include a one-dimensional center-of-glass contribution and a two-dimensional edge contribution. The frame contribution is primarily two-dimensional.
Consequently, the total rate of heat transfer through a fenestra-tion system can be calculated knowing the separate heat transfer contributions of the center glass, edge glass, and frame. (When present, glazing dividers, such as decorative grilles and muntins, also affect heat transfer, and their contribution must be considered.) The overall U-factor is estimated using area-weighted U-factors for each contribution by (7) where the subscripts cg, eg, and f refer to the center-of-glass, edge-of-glass, and frame, respectively. Apf is the area of the fenestration product’s rough opening in the wall or roof less installation clear-ances. When a fenestration product has glazed surfaces in only one direction (typical windows), the sum of the areas equals the pro-jected area. Skylights, greenhouse/garden windows, bay/bow win-dows, etc., because they extend beyond the plane of the wall/roof, have greater surface area for heat loss than a window with a similar glazing option and frame material; consequently, U-factors for such products are expected to be greater.
DETERMINING FENESTRATION U-FACTORS Center-of-Glass U-Factor Heat flow across the central glazed portion of a multipane unit must consider both convective and radiative transfer in the gas space. Convective heat transfer is estimated based on high-aspect-ratio, natural convection correlations for vertical and inclined air layers (El Sherbiny et al. 1982, Shewen 1986, Wright 1996). Radi-ative heat transfer (ignoring gas absorption) is quantified using a more fundamental approach. Computational methods solving the combined heat transfer problem have been devised (Rubin 1982a, 1982b, Hollands and Wright 1982).
Especially for single glass, U-factors depend strongly on indoor and outdoor film coefficients. The U-factor for single glass is (8) where ho, hi = outdoor and indoor respective glass surface heat transfer coefficients, W/(m2·K) L = glass thickness, mm k = thermal conductivity, W/(m·K) Values for Ucg at standard indoor and outdoor conditions depend on such glazing construction features as the number of glazing lights, the gas-space dimensions, the orientation relative to vertical, the emissivity of each surface, and the composition of the fill gas.
Several computer programs can be used to estimate glazing unit heat transfer for a wide range of glazing construction (Arasteh et al.
1994, Finlayson and Arasteh 1993, Wright 1995c). The National Fenestration Rating Council calls for WINDOW 4.1 (LBL 1994) as a standard calculation method for the center glazing. In Canada, the VISION program (Wright 1995b) is used to determine center-glaz-ing properties for the Canadian Standards Association (CSA Stan-dard A440.2). Figure 3 shows the effect of gas space width on Ucg for vertical double- and triple-paned glazing units. U-factors are plotted for air, argon, and krypton fill gases and for high (uncoated) and low (coated) values of surface emissivity. Gas space widths greater than 13 mm have no significant effect on Ucg, but greater glazing unit thicknesses decrease Uo since the length of the shortest heat flow path through the frame increases. A low-emissivity coating com-bined with krypton gas fill offers significant potential for reducing heat transfer in narrow gap-width glazing units.
Edge-of-Glass U-Factor Insulating glazing units usually have continuous spacer members around the glass perimeter to separate the glazing and provide an edge seal. Aluminum spacers greatly increase conductive heat transfer between the contacted inner and outer glazing, thereby degrading the thermal performance of the glazing unit locally. The edge-of-glass area is typically taken to be a band 65 mm wide around the sightline. The width of this area is determined from the extent of two-dimensional heat transfer effects in current computer models, which are based on conduction-only analysis. In reality, due to convective and radiative effects, this area may extend beyond 65 mm (Beck et al. 1995, Curcija and Goss 1994, Wright and Sul-livan 1995b), and it will depend on the type of insulating glazing unit and its thickness.
In low-conductivity frames, the heat flow at the edge-of-glass and frame area is through the spacer, and so the type of spacer has a greater impact on the edge-of-glass and frame U-factor. In metal frames, the edge-of-glass and frame U-factor varies little with the type of spacer (metal or insulating) because there is a significant heat flow through the highly conductive frame near the edge-of-glass area.
Frame U-Factor Fenestration frame elements consist of all structural members exclusive of the glazing units and include sash, jamb, head, and sill members; meeting rails and stiles; mullions; and other glazing dividers. Estimating the rate of heat transfer through the frame is q qC qR + = Uo UcgAcg UegAeg UfAf + + Apf ------------------------------------------------------------= U 1 1 ho ⁄ 1 hi ⁄ L 1000k ⁄ + + -----------------------------------------------------------= Fenestration 30.5 complicated by (1) the variety of fenestration products and frame configurations, (2) the different combinations of materials used for frames, (3) the different sizes available, and, to a lesser extent, (4) the glazing unit width and spacer type. Internal dividers or grilles have little impact on the fenestration U-factor, provided there is at least a 3 mm gap between the divider and each panel of glass.
Computer simulations found that frame heat loss in most fenes-tration is controlled by a single component or controlling resistance, and only changes in this component significantly affect frame heat loss (EEL 1990). For example, the frame U-factor for thermally bro-ken aluminum fenestration products is largely controlled by the depth of the thermal break material in the heat flow direction. For aluminum frames without a thermal break, the inside film coeffi-cient provides most of the resistance to heat flow. For vinyl- and wood-framed fenestrations, the controlling resistance is the shortest distance between the inside and outside surfaces, which usually depends on the thickness of the sealed glazing unit.
Carpenter and McGowan (1993) experimentally validated frame U-factors for a variety of fixed and operable fenestration product types, sizes, and materials using computer modeling techniques.
Table 1 lists frame U-factors for a variety of frame and spacer mate-rials and glazing unit thicknesses. Frame and edge U-factors are normally determined by two-dimensional computer simulation. The National Fenestration Rating Council requires that frame and edge U-factors be determined using the THERM (Arasteh et al. 2000) and FRAME (EEL 1995) computer programs. The Canadian Stan-dards Association requires that frame and edge U-factors be deter-mined using FRAME.
Curtain Wall Construction A curtain wall is an exterior building wall that carries no roof or floor loads and consists entirely or principally of glass and other sur-facing materials supported by a framework. A curtain wall typically has a metal frame. To improve the thermal performance of standard metal frames, manufacturers provide both traditional thermal breaks as well as thermally improved products. The traditional ther-mal break type is poured and debridged, where urethane is poured into a metal U-channel in the frame and then the bottom of the chan-nel is removed by machine. For this system to work well, there must be a thermal break between the interior and the exterior for all of the frame components, including those in any operable sash. Skip debridging (incomplete pour and debridging used for increased structural strength) can significantly degrade the U-factor. Bolts that penetrate the thermal break also degrade performance, but to a lesser degree. Griffith et al. (1998) showed that stainless steel bolts spaced 300 mm on center increased the frame U-factor by 18%. The paper also concluded that, in general, the isothermal planes method referenced in Chapter 25 provides a conservative approach to deter-mining U-factors.
Thermally improved curtain wall products are a more recent development. In these products, most of the metal frame tends to be located on the interior with only a metal cap exposed on the exterior.
Plastic spacers isolate the glazing assembly from both the metal cap on the exterior and the metal frame on the interior. These products can have significantly better thermal performance than standard metal frames, but it is important to minimize the number and area of the bolts that penetrate from exterior to interior.
INDOOR AND OUTDOOR SURFACE HEAT TRANSFER COEFFICIENTS Part of the overall thermal resistance of a fenestration system is due to the convective and radiative heat transfer between the exposed surfaces and the environment. Surface heat transfer coeffi-cients ho and hi at the outer and inner glazing surfaces, respectively, combine the effects of radiation and convection.
The wind speed and orientation of the building are important in determining ho. This relationship has long been studied, and many correlations have been proposed for ho as a function of wind speed.
However, no universal relationship has been accepted, and limited field measurements at low wind speeds by Klems (1989) show sig-nificant difference with values used by others.
Convective heat transfer coefficients are usually determined at standard temperature and air velocity conditions on each side.
Wind speed can vary from less than 0.2 m/s for calm weather, free convection conditions, to over 29 m/s for storm conditions. A standard value of 29 W/(m2·K) corresponding to a 6.7 m/s wind is often used to represent winter design conditions. At near-zero wind speed, ho varies with outside air and surface temperature, orientation to vertical, and air moisture content. At low wind speeds, the overall surface heat transfer coefficient can be as low as 6.8 W/(m2·K) (Yazdanian and Klems 1993).
For natural convection at the inner surface of a vertical fenestra-tion product, the inner surface coefficient hi depends on the indoor air and glass surface temperatures and on the emissivity of the glass inner surface. Table 2 shows the variation of hi for winter (ti = 21°C) Fig. 3 Center of Glass U-Factor for Vertical Double- and Triple-Pane Glazing Units 30.6 2001 ASHRAE Fundamentals Handbook (SI) and summer (ti = 24°C) design conditions, for a range of glass types and heights. Designers often use hi = 8.3 W/(m2·K), which corre-sponds to ti = 21°C, glass temperature of –9°C, and uncoated glass with eg = 0.84. For summer conditions, the same value [hi = 8.3 W/(m2·K)] is normally used, and it corresponds approximately to glass temperature of 35°C, ti = 24°C, and eg = 0.84. For winter con-ditions, this most closely approximates single glazing with clear glass that is 600 mm tall, but it overestimates the value as the glaz-ing unit conductance decreases and height increases. For summer conditions, this value approximates all types of glass that are 600 mm tall but, again, is less accurate as the glass height increases.
If the room surface of the glass has a low-e coating, the hi values are about halved at both winter and summer conditions.
Heat transfer between the glazing surface and its environment is driven not only by the local air temperatures but also by the radiant temperatures to which the surface is exposed. The radiant tempera-ture of the indoor environment is generally assumed to be equal to the indoor air temperature. While this is a safe assumption where a small fenestration product is exposed to a large room with surface temperatures equal to the air temperature, it is not valid in rooms where the fenestration product is exposed to other large areas of glazing surfaces (e.g., greenhouse, atrium) or to other cooled or heated surfaces (Parmelee and Huebscher 1947).
The radiant temperature of the outdoor environment is frequently assumed to be equal to the outdoor air temperature. This assumption may be in error, since additional radiative heat loss occurs between a fenestration and the clear sky (Duffie and Beckman 1980). There-fore, for clear-sky conditions, some effective outdoor temperature to,e should replace to in Equation (1). For methods for determining to,e, see, for example, work by AGSL (1992). Note that a fully cloudy sky is assumed in ASHRAE design conditions.
The air space in an insulating glass panel made up of glass with no reflective coating on the air space surfaces has a coefficient hs of 7.4 W/(m2·K). When a reflective coating is applied to an air space surface, hs can be selected from Table 3 by first calculating the effective air space emissivity es,e by Equation (9): Table 1 Representative Fenestration Frame U-Factors in W/(m2·K)—Vertical Orientation Frame Material Type of Spacer Product Type/Number of Glazing Layers Operable Fixed Garden Window Plant-Assembled Skylight Curtain Walle Sloped/Overhead Glazinge Singleb Doublec Tripled Singleb Doublec Tripled Singleb Doublec Singleb Doublec Tripled Singlef Doubleg Tripleh Singlef Doubleg Tripleh Aluminum without thermal break All 13.51 12.89 12.49 10.90 10.22 9.88 10.67 10.39 44.57 39.86 39.01 17.09 16.81 16.07 17.32 17.03 16.30 Aluminum with thermal breaka Metal 6.81 5.22 4.71 7.49 6.42 6.30 39.46 28.67 26.01 10.22 9.94 9.37 10.33 9.99 9.43 Insulated n/a 5.00 4.37 n/a 5.91 5.79 n/a 26.97 23.39 n/a 9.26 8.57 n/a 9.31 8.63 Aluminum-clad wood/ reinforced vinyl Metal 3.41 3.29 2.90 3.12 2.90 2.73 27.60 22.31 20.78 Insulated n/a 3.12 2.73 n/a 2.73 2.50 n/a 21.29 19.48 Wood /vinyl Metal 3.12 2.90 2.73 3.12 2.73 2.38 5.11 4.83 14.20 11.81 10.11 Insulated n/a 2.78 2.27 n/a 2.38 1.99 n/a 4.71 n/a 11.47 9.71 Insulated fiberglass/ vinyl Metal 2.10 1.87 1.82 2.10 1.87 1.82 Insulated n/a 1.82 1.48 n/a 1.82 1.48 Structural glazing Metal 10.22 7.21 5.91 10.33 7.27 5.96 Insulated n/a 5.79 4.26 n/a 5.79 4.26 Note: This table should only be used as an estimating tool for the early phases of design.
aDepends strongly on width of thermal break. Value given is for 9.5 mm.
bSingle glazing corresponds to individual glazing unit thickness of 3 mm (nominal).
cDouble glazing corresponds to individual glazing unit thickness of 19 mm (nominal).
dTriple glazing corresponds to individual glazing unit thickness of 34.9 mm (nominal).
eGlass thickness in curtainwall and sloped/overhead glazing is 6.4 mm.
fSingle glazing corresponds to individual glazing unit thickness of 6.4 mm (nominal).
gDouble glazing corresponds to individual glazing unit thickness of 25.4 mm (nominal).
hTriple glazing corresponds to individual glazing unit thickness of 44.4 mm (nominal).
n/a Not applicable Table 2 Indoor Surface Heat Transfer Coefficient hi in W/(m2·K)—Vertical Orientation (Still Air Conditions) Glazing ID Glazing Type Glazing Height, m Winter Conditions Summer Conditions Glass Temp., °C Temp. Diff., °C hi, W/(m2·K) Glass Temp., °C Temp. Diff., °C hi, W/(m2·K) 1 Single glazing 0.6 −9 30 8.04 33 9 4.12 1.2 −9 30 7.42 33 9 3.66 1.8 −9 30 7.10 33 9 3.43 5 Double glazing with 12.7 mm air space 0.6 7 14 7.72 35 11 4.28 1.2 7 14 7.21 35 11 3.80 1.8 7 14 6.95 35 11 3.55 23 Double glazing with e = 0.1 on surface 2 and 12.7 mm argon space 0.6 13 8 7.44 34 10 4.20 1.2 13 8 7.00 34 10 3.73 1.8 13 8 6.77 34 10 3.49 43 Triple glazing with e = 0.1 on surfaces 2 and 5 and 12.7 mm argon spaces 0.6 17 4 7.09 40 16 4.61 1.2 17 4 6.72 40 16 4.08 1.8 17 4 6.53 40 16 3.81 Notes: Glazing ID refers to fenestration assemblies in Table 4.
Winter conditions: room air temperature ti = 21°C, outdoor air temperature to = −18°C, no solar radiation Summer conditions: room air temperature ti = 24°C, outdoor air temperature to = 32°C, direct solar irradiance ED = 748 W/m2 hi = hic + hiR = 1.46(∆T/L)0.25 + eΓ(T4 g – T4 i )/∆T where ∆T = Tg – Ti, K; L = glazing height, m; Tg = glass temperature, K Fenestration 30.7 (9) where eo and ei are the hemispherical emissivities of the two air space surfaces. Hemispherical emissivity of ordinary uncoated glass is 0.84 over a wavelength range of 0.4 to 40 µm.
REPRESENTATIVE U-FACTORS FOR FENESTRATION PRODUCTS Table 4 lists computed U-factors for a variety of generic fenes-tration products. The table is based on ASHRAE-sponsored research involving laboratory testing and computer simulation of various fenestration products. In the past, test data were used to pro-vide more accurate results for specific products (Hogan 1988).
Computer simulations (with validation by testing) are now accepted as the standard method for accurate product-specific U-factor deter-mination. The simulation methodologies are specified in the National Fenestration Rating Council’s NFRC 100 or Canadian Standards Association (CSA) Standard A440.2 and are based on algorithms published in ISO Standard 15099. The International Energy Conservation Code (ICC 2000) and various state energy codes in the United States, the National Energy Code in Canada, and ASHRAE Standards 90.1 and 90.2 all cite these standards. Fenes-tration needs to be rated in accordance with the NFRC or CSA stan-dards for code compliance. The use of Table 4 should be limited to that of an estimating tool for the early phases of design.
Values in Table 4 are listed at winter design conditions for verti-cal installation and for skylights and other sloped installations with glazing surfaces that are sloped 20° from the horizontal. Data are based on center-of-glass and edge-of-glass component U-factors and assume that there are no dividers. However, they apply only to the specific design conditions described in the footnotes in the table, and they are typically used only to determine peak load conditions for sizing heating equipment. While these U-factors have been determined for winter conditions, they can also be used to estimate heat gain during peak cooling conditions, since conductive gain, which is one of several variables, is usually a small portion of the total heat gain for fenestration in direct sunlight. Glazing designs and framing materials may be compared in choosing a fenestration system that needs a specific winter design U-factor.
Table 4 lists 48 glazing types. (A subset of these types is included in Table 13, which lists solar heat gain coefficients and visible light transmittance.) The multiple glazing categories are appropriate for sealed glazing units and the addition of storm sash to other glazing units. No distinction is made between flat and domed units such as skylights. For acrylic domes, use an average gas-space width to determine the U-factor. Note that garden window and sloped/pyra-mid/barrel vault skylight U-factors are approximately twice those of other similar products. While this is partially due to the difference in slope in the case of the sloped/pyramid/barrel vault skylights, it is largely because these products project out from the surface of the wall or roof. For instance, the skylight surface area, which includes the curb, can vary from 13 to 240% greater than the rough opening area, depending on the size and mounting method. Unless otherwise noted, all multiple-glazed units are filled with dry air. Argon units are assumed to be filled with 90% argon (Elmahdy and Yusuf 1995).
U-factors for CO2-filled units are similar to argon fills. For spaces up to 13 mm, argon/SF6 (sulfur hexafluoride) mixtures up to 70% SF6 are generally the same as argon fills. The use of krypton gas can provide U-factors lower than those for argon for glazing spaces less than 13 mm.
Table 4 provides data for six values of hemispherical emissivity and for 6.4 and 12.7 mm gas space widths. The emissivity of various low-e glasses varies considerably between manufacturers and pro-cesses. When the emissivity is between the listed values, interpola-tion may be used. When manufacturers’ data are not available for Table 3 Air Space Coefficients for Horizontal Heat Flow Air Space Thick-ness, mm Air Space Temp., °C Air Temp. Diff., °C Air Space Coefficient hs, W/(m2·K) Effective Emissivity es, e 0.82 0.72 0.40 0.20 0.10 0.05 13 −15 5 5.0 4.6 3.3 2.6 2.2 2.0 15 5.1 4.7 3.5 2.7 2.3 2.1 30 5.7 5.3 4.0 3.2 2.8 2.7 40 6.0 5.6 4.3 3.6 3.2 3.0 50 6.3 5.9 4.6 3.8 3.4 3.2 0 5 5.7 5.2 3.7 2.8 2.3 2.1 15 5.7 5.3 3.8 2.9 2.4 2.2 30 6.1 5.7 4.2 3.3 2.8 2.6 40 6.4 6.0 4.5 3.5 3.1 2.8 50 6.7 6.2 4.7 3.8 3.3 3.1 10 5 6.1 5.6 4.0 3.0 2.4 2.2 15 6.2 5.7 4.0 3.0 2.5 2.2 30 6.5 6.0 4.3 3.3 2.8 2.5 40 6.8 6.2 4.6 3.5 3.0 2.8 50 7.0 6.5 4.8 3.8 3.3 3.0 30 5 7.2 6.6 4.6 3.3 2.7 2.4 15 7.3 6.6 4.6 3.3 2.7 2.4 30 7.4 6.8 4.7 3.5 2.8 2.5 40 7.6 6.9 4.9 3.6 3.0 2.7 50 7.8 7.2 5.1 3.9 3.2 2.9 50 5 8.4 7.7 5.2 3.7 2.9 2.5 15 8.5 7.7 5.2 3.7 2.9 2.6 30 8.5 7.8 5.3 3.8 3.0 2.6 40 8.6 7.9 5.4 3.9 3.1 2.7 50 8.8 8.0 5.5 4.0 3.2 2.8 10 −15 5 5.5 5.1 3.9 3.1 2.7 2.5 30 5.7 5.3 4.0 3.2 2.9 2.7 50 6.1 5.7 4.4 3.6 3.2 3.1 0 5 6.2 5.7 4.3 3.3 2.9 2.6 30 6.3 5.8 4.4 3.4 3.0 2.7 50 6.6 6.1 4.6 3.7 3.2 3.0 10 5 6.7 6.2 4.6 3.5 3.0 2.8 30 6.8 6.3 4.6 3.6 3.1 2.8 50 7.0 6.5 4.8 3.8 3.2 3.0 30 5 7.8 7.2 5.2 3.9 3.3 3.0 30 7.9 7.2 5.2 4.0 3.3 3.0 50 8.0 7.3 5.3 4.0 3.4 3.1 50 5 9.1 8.3 5.9 4.3 3.6 3.2 30 9.1 8.4 5.9 4.4 3.6 3.2 50 9.2 8.4 6.0 4.4 3.6 3.3 7 −15 <50 6.5 6.1 4.9 4.1 3.7 3.5 0 <50 7.3 6.8 5.3 4.4 3.9 3.7 10 <50 7.8 7.3 5.6 4.6 4.1 3.8 30 <50 9.0 8.4 6.3 5.1 4.4 4.1 50 <50 10.3 9.5 7.1 5.6 4.8 4.4 6 −15 <50 7.1 6.7 5.4 4.6 4.2 4.0 0 <50 7.9 7.4 5.9 5.0 4.5 4.3 10 <50 8.4 7.9 6.2 5.2 4.7 4.4 30 <50 9.6 9.0 7.0 5.7 5.1 4.7 50 <50 11.0 10.2 7.8 6.2 5.5 5.1 5 −15 <50 7.8 7.4 6.2 5.4 5.0 4.8 0 <50 8.7 8.2 6.7 5.8 5.3 5.1 10 <50 9.2 8.7 7.1 6.0 5.5 5.2 30 <50 10.5 9.9 7.8 6.6 5.9 5.6 50 <50 11.9 11.2 8.7 7.2 6.4 6.0 es e , 1 1 eo ⁄ 1 ei ⁄ 1 – + --------------------------------------= 30.8 2001 ASHRAE Fundamentals Handbook (SI) Table 4 U-Factors for Various Fenestration Products in W/(m2·K) Product Type Glass Only Vertical Installation Operable (including sliding and swinging glass doors) Fixed Frame Type ID Glazing Type Center of Glass Edge of Glass Aluminum Without Thermal Break Aluminum With Thermal Break Reinforced Vinyl/ Aluminum Clad Wood Wood/ Vinyl Insulated Fiberglass/ Vinyl Aluminum Without Thermal Break Aluminum With Thermal Break Reinforced Vinyl/ Aluminum Clad Wood Wood/ Vinyl Insulated Fiberglass/ Vinyl Single Glazing 1 3.2 mm glass 5.91 5.91 7.24 6.12 5.14 5.05 4.61 6.42 6.07 5.55 5.55 5.35 2 6.4 mm acrylic/polycarbonate 5.00 5.00 6.49 5.43 4.51 4.42 4.01 5.60 5.25 4.75 4.75 4.58 3 3.2 mm acrylic/polycarbonate 5.45 5.45 6.87 5.77 4.82 4.73 4.31 6.01 5.66 5.15 5.15 4.97 Double Glazing 4 6.4 mm air space 3.12 3.63 4.93 3.70 3.25 3.13 2.77 3.94 3.56 3.19 3.17 3.04 5 12.7 mm air space 2.73 3.36 4.62 3.42 3.00 2.87 2.53 3.61 3.22 2.86 2.84 2.72 6 6.4 mm argon space 2.90 3.48 4.75 3.54 3.11 2.98 2.63 3.75 3.37 3.00 2.98 2.85 7 12.7 mm argon space 2.56 3.24 4.49 3.30 2.89 2.76 2.42 3.47 3.08 2.73 2.70 2.58 Double Glazing, e = 0.60 on surface 2 or 3 8 6.4 mm air space 2.95 3.52 4.80 3.58 3.14 3.02 2.67 3.80 3.41 3.05 3.03 2.90 9 12.7 mm air space 2.50 3.20 4.45 3.26 2.85 2.73 2.39 3.42 3.03 2.68 2.66 2.54 10 6.4 mm argon space 2.67 3.32 4.58 3.38 2.96 2.84 2.49 3.56 3.17 2.82 2.80 2.67 11 12.7 mm argon space 2.33 3.08 4.31 3.13 2.74 2.62 2.28 3.28 2.89 2.54 2.52 2.40 Double Glazing, e = 0.40 on surface 2 or 3 12 6.4 mm air space 2.78 3.40 4.66 3.46 3.03 2.91 2.56 3.66 3.27 2.91 2.89 2.76 13 12.7 mm air space 2.27 3.04 4.27 3.09 2.70 2.58 2.25 3.23 2.84 2.49 2.47 2.35 14 6.4 mm argon space 2.44 3.16 4.40 3.21 2.81 2.69 2.35 3.37 2.98 2.63 2.61 2.49 15 12.7 mm argon space 2.04 2.88 4.09 2.93 2.55 2.43 2.10 3.04 2.65 2.31 2.29 2.17 Double Glazing, e = 0.20 on surface 2 or 3 16 6.4 mm air space 2.56 3.24 4.49 3.30 2.89 2.76 2.42 3.47 3.08 2.73 2.70 2.58 17 12.7 mm air space 1.99 2.83 4.05 2.89 2.52 2.39 2.07 2.99 2.60 2.26 2.24 2.13 18 6.4 mm argon space 2.16 2.96 4.18 3.01 2.63 2.51 2.17 3.13 2.74 2.40 2.38 2.26 19 12.7 mm argon space 1.70 2.62 3.83 2.68 2.33 2.21 1.89 2.75 2.36 2.03 2.01 1.90 Double Glazing, e = 0.10 on surface 2 or 3 20 6.4 mm air space 2.39 3.12 4.36 3.17 2.78 2.65 2.32 3.32 2.93 2.59 2.56 2.45 21 12.7 mm air space 1.82 2.71 3.92 2.77 2.41 2.28 1.96 2.84 2.45 2.12 2.10 1.99 22 6.4 mm argon space 1.99 2.83 4.05 2.89 2.52 2.39 2.07 2.99 2.60 2.26 2.24 2.13 23 12.7 mm argon space 1.53 2.49 3.70 2.56 2.22 2.10 1.79 2.60 2.21 1.89 1.86 1.76 Double Glazing, e = 0.05 on surface 2 or 3 24 6.4 mm air space 2.33 3.08 4.31 3.13 2.74 2.62 2.28 3.28 2.89 2.54 2.52 2.40 25 12.7 mm air space 1.70 2.62 3.83 2.68 2.33 2.21 1.89 2.75 2.36 2.03 2.01 1.90 26 6.4 mm argon space 1.87 2.75 3.96 2.81 2.44 2.32 2.00 2.89 2.50 2.17 2.15 2.03 27 12.7 mm argon space 1.42 2.41 3.61 2.48 2.15 2.02 1.71 2.50 2.11 1.79 1.77 1.67 Triple Glazing 28 6.4 mm air space 2.16 2.96 4.11 2.89 2.51 2.45 2.16 3.10 2.73 2.38 2.33 2.25 29 12.7 mm air space 1.76 2.67 3.80 2.60 2.25 2.19 1.91 2.76 2.39 2.05 2.01 1.93 30 6.4 mm argon space 1.93 2.79 3.94 2.73 2.36 2.30 2.01 2.90 2.54 2.19 2.15 2.07 31 12.7 mm argon space 1.65 2.58 3.71 2.52 2.17 2.12 1.84 2.66 2.30 1.96 1.91 1.84 Triple Glazing, e = 0.20 on surface 2,3,4, or 5 32 6.4 mm air space 1.87 2.75 3.89 2.69 2.32 2.27 1.98 2.86 2.49 2.15 2.10 2.03 33 12.7 mm air space 1.42 2.41 3.54 2.36 2.02 1.97 1.70 2.47 2.10 1.77 1.73 1.66 34 6.4 mm argon space 1.59 2.54 3.67 2.48 2.13 2.08 1.80 2.61 2.25 1.91 1.87 1.80 35 12.7 mm argon space 1.25 2.28 3.40 2.23 1.91 1.86 1.59 2.32 1.96 1.63 1.59 1.52 Triple Glazing, e = 0.20 on surfaces 2 or 3 and 4 or 5 36 6.4 mm air space 1.65 2.58 3.71 2.52 2.17 2.12 1.84 2.66 2.30 1.96 1.91 1.84 37 12.7 mm air space 1.14 2.19 3.31 2.15 1.84 1.78 1.52 2.23 1.86 1.54 1.49 1.43 38 6.4 mm argon space 1.31 2.32 3.45 2.27 1.95 1.90 1.62 2.37 2.01 1.68 1.63 1.56 39 12.7 mm argon space 0.97 2.05 3.18 2.03 1.72 1.67 1.41 2.08 1.71 1.39 1.35 1.29 Triple Glazing, e = 0.10 on surfaces 2 or 3 and 4 or 5 40 6.4 mm air space 1.53 2.49 3.63 2.44 2.10 2.05 1.77 2.57 2.20 1.86 1.82 1.75 41 12.7 mm air space 1.02 2.10 3.22 2.07 1.76 1.71 1.45 2.13 1.76 1.44 1.40 1.33 42 6.4 mm argon space 1.19 2.23 3.36 2.19 1.87 1.82 1.55 2.27 1.91 1.58 1.54 1.47 43 12.7 mm argon space 0.80 1.92 3.05 1.90 1.61 1.56 1.30 1.93 1.57 1.25 1.21 1.15 Quadruple Glazing, e = 0.10 on surfaces 2 or 3 and 4 or 5 44 6.4 mm air spaces 1.25 2.28 3.40 2.23 1.91 1.86 1.59 2.32 1.96 1.63 1.59 1.52 45 12.7 mm air spaces 0.85 1.96 3.09 1.94 1.65 1.60 1.34 1.98 1.62 1.30 1.26 1.19 46 6.4 mm argon spaces 0.97 2.05 3.18 2.03 1.72 1.67 1.41 2.08 1.71 1.39 1.35 1.29 47 12.7 mm argon spaces 0.68 1.83 2.96 1.82 1.54 1.48 1.23 1.84 1.47 1.16 1.11 1.05 48 6.4 mm krypton spaces 0.68 1.83 2.96 1.82 1.54 1.48 1.23 1.84 1.47 1.16 1.11 1.05 Notes: 1. All heat transmission coefficients in this table include film resistances and are based on winter conditions of −18°C outdoor air temperature and 21°C indoor air tempera-ture, with 6.7 m/s outdoor air velocity and zero solar flux. With the exception of single glazing, small changes in the indoor and outdoor temperatures will not significantly affect overall U-factors. The coefficients are for vertical position except skylight val-ues, which are for 20° from horizontal with heat flow up.
2. Glazing layer surfaces are numbered from the outdoor to the indoor. Double, triple, and quadruple refer to the number of glazing panels. All data are based on 3 mm glass, unless otherwise noted. Thermal conductivities are: 0.917 W/(m·K) for glass, and 0.19 W/(m·K) for acrylic and polycarbonate.
3. Standard spacers are metal. Edge-of-glass effects assumed to extend over the 65 mm band around perimeter of each glazing unit.
Fenestration 30.9 Table 4 U-Factors for Various Fenestration Products in W/(m2·K) (Concluded) Vertical Installation Sloped Installation ID Garden Windows Curtain Wall Glass Only (Skylights) Manufactured Skylight Site-Assembled Sloped/Overhead Glazing Aluminum Without Thermal Break Wood/ Vinyl Aluminum Without Thermal Break Aluminum With Thermal Break Structural Glazing Center of Glass Edge of Glass Aluminum Without Thermal Break Aluminum With Thermal Break Reinforced Vinyl/ Aluminum Clad Wood Wood/ Vinyl Aluminum Without Thermal Break Aluminum With Thermal Break Structural Glazing 14.76 13.13 6.93 6.30 6.30 6.76 6.76 11.24 10.73 9.96 8.34 7.73 7.09 7.09 1 13.23 11.71 6.11 5.48 5.48 5.85 5.85 10.33 9.82 9.07 7.45 6.90 6.26 6.26 2 14.00 12.42 6.52 5.89 5.89 6.30 6.30 10.79 10.27 9.52 7.89 7.31 6.67 6.67 3 10.30 9.16 4.47 3.84 3.59 3.29 3.75 7.44 6.32 5.94 4.79 4.64 3.99 3.74 4 9.72 8.68 4.14 3.51 3.26 3.24 3.71 7.39 6.27 5.90 4.74 4.59 3.95 3.70 5 9.97 8.88 4.28 3.65 3.40 3.01 3.56 7.19 6.06 5.70 4.54 4.40 3.75 3.50 6 9.47 8.47 3.99 3.36 3.11 3.01 3.56 7.19 6.06 5.70 4.54 4.40 3.75 3.50 7 10.05 8.95 4.33 3.70 3.45 3.07 3.60 7.24 6.11 5.75 4.59 4.45 3.80 3.55 8 9.38 8.40 3.94 3.31 3.06 3.01 3.56 7.19 6.06 5.70 4.54 4.40 3.75 3.50 9 9.63 8.61 4.09 3.46 3.21 2.78 3.40 6.98 5.86 5.49 4.34 4.20 3.56 3.31 10 9.13 8.19 3.80 3.17 2.92 2.78 3.40 6.98 5.86 5.49 4.34 4.20 3.56 3.31 11 9.80 8.75 4.18 3.55 3.30 2.90 3.48 7.09 5.96 5.59 4.44 4.30 3.66 3.41 12 9.05 8.12 3.75 3.12 2.87 2.84 3.44 7.03 5.91 5.54 4.39 4.25 3.61 3.36 13 9.30 8.33 3.89 3.26 3.01 2.50 3.20 6.73 5.60 5.24 4.09 3.96 3.32 3.07 14 8.71 7.83 3.55 2.92 2.67 2.61 3.28 6.83 5.70 5.34 4.19 4.06 3.41 3.16 15 9.47 8.47 3.99 3.36 3.11 2.61 3.28 6.83 5.70 5.34 4.19 4.06 3.41 3.16 16 8.62 7.76 3.50 2.87 2.63 2.61 3.28 6.83 5.70 5.34 4.19 4.06 3.41 3.16 17 8.88 7.98 3.65 3.02 2.77 2.22 3.00 6.47 5.34 4.99 3.84 3.72 3.07 2.83 18 8.19 7.40 3.26 2.63 2.38 2.27 3.04 6.52 5.39 5.04 3.89 3.77 3.12 2.87 19 9.21 8.26 3.84 3.22 2.97 2.50 3.20 6.73 5.60 5.24 4.09 3.96 3.32 3.07 20 8.36 7.55 3.36 2.73 2.48 2.50 3.20 6.73 5.60 5.24 4.09 3.96 3.32 3.07 21 8.62 7.76 3.50 2.87 2.63 2.04 2.88 6.31 5.18 4.84 3.69 3.57 2.93 2.68 22 7.94 7.18 3.11 2.48 2.23 2.16 2.96 6.41 5.29 4.94 3.79 3.67 3.03 2.78 23 9.13 8.19 3.80 3.17 2.92 2.39 3.12 6.62 5.50 5.14 3.99 3.87 3.22 2.97 24 8.19 7.40 3.26 2.63 2.38 2.44 3.16 6.67 5.55 5.19 4.04 3.91 3.27 3.02 25 8.45 7.62 3.41 2.78 2.53 1.93 2.79 6.21 5.08 4.73 3.58 3.48 2.83 2.58 26 7.76 7.04 3.01 2.39 2.14 2.04 2.88 6.31 5.18 4.84 3.69 3.57 2.93 2.68 27 see see 3.58 2.97 2.65 2.22 3.00 6.38 5.07 4.77 3.63 3.65 3.02 2.71 28 note note 3.24 2.63 2.31 2.04 2.88 6.22 4.92 4.62 3.48 3.51 2.88 2.56 29 7 7 3.39 2.77 2.46 1.99 2.83 6.17 4.86 4.56 3.43 3.46 2.83 2.51 30 3.14 2.53 2.21 1.87 2.75 6.07 4.76 4.46 3.33 3.36 2.73 2.41 31 see see 3.34 2.73 2.41 1.93 2.79 6.12 4.81 4.51 3.38 3.41 2.78 2.46 32 note note 2.95 2.33 2.02 1.76 2.67 5.96 4.65 4.36 3.22 3.26 2.63 2.32 33 7 7 3.09 2.48 2.16 1.59 2.54 5.81 4.50 4.21 3.07 3.11 2.49 2.17 34 2.80 2.19 1.87 1.53 2.49 5.75 4.44 4.15 3.02 3.07 2.44 2.12 35 see see 3.14 2.53 2.21 1.65 2.58 5.86 4.55 4.26 3.12 3.16 2.53 2.22 36 note note 2.70 2.09 1.77 1.53 2.49 5.75 4.44 4.15 3.02 3.07 2.44 2.12 37 7 7 2.85 2.24 1.92 1.36 2.36 5.60 4.29 4.00 2.86 2.92 2.29 1.97 38 2.55 1.94 1.62 1.25 2.28 5.49 4.18 3.90 2.76 2.82 2.19 1.87 39 see see 3.05 2.43 2.11 1.53 2.49 5.75 4.44 4.15 3.02 3.07 2.44 2.12 40 note note 2.60 1.99 1.67 1.42 2.41 5.65 4.34 4.05 2.91 2.97 2.34 2.02 41 7 7 2.75 2.14 1.82 1.19 2.23 5.44 4.13 3.84 2.71 2.77 2.14 1.82 42 2.40 1.79 1.47 1.14 2.19 5.38 4.07 3.79 2.66 2.72 2.09 1.78 43 2.80 2.19 1.87 1.25 2.28 5.49 4.18 3.90 2.76 2.82 2.19 1.87 44 see see 2.45 1.84 1.52 1.08 2.14 5.33 4.02 3.74 2.60 2.67 2.04 1.73 45 note note 2.55 1.94 1.62 1.02 2.10 5.28 3.97 3.69 2.55 2.62 1.99 1.68 46 7 7 2.31 1.69 1.38 0.91 2.01 5.17 3.86 3.59 2.45 2.52 1.90 1.58 47 2.31 1.69 1.38 0.74 1.87 5.01 3.70 3.43 2.29 2.38 1.75 1.43 48 4. Product sizes are described in Figure 4, and frame U-fac-tors are from Table 1.
5. Use U = 3.40 W/(m2·K) for glass block with mortar but without reinforcing or framing.
6. The use of this table should be limited to that of an estimat-ing tool for the early phases of design.
7. Values for triple- and quadruple-glazed garden windows are not listed as these are not common products.
8. Minor differences exist between the data in this table and U-factors determined using NFRC 100-91 because the data in this table are generated using modified heat transfer correlations for glazing cavities (Wright 1996) and indoor fenestration sur-faces (Curcija and Goss 1995b).
30.10 2001 ASHRAE Fundamentals Handbook (SI) low-e glass, assume that glass with a pyrolytic (hard) coating has an emissivity of 0.40 and that glass with a sputtered (soft) coating has an emissivity as low as 0.36. Tinted glass does not change the winter U-factor. Also, some reflective glass may have an emissivity less than 0.84. Values listed are for insulating glass units using alumi-num edge spacers. If an insulated or nonmetallic spacer is used, the U-factors are approximately 0.17 W/(m2·K) lower.
Fenestration product types are subdivided first by vertical ver-sus sloped installation and then into two general categories— manufactured and site-assembled. “Manufactured” is intended to represent products delivered as a complete unit to the site. These products are typically installed in low-rise residential and small commercial/institutional/industrial buildings. For vertical sliders, horizontal sliders, casement, awning, pivoted, and dual-action win-dows, and for sliding and swinging glass doors, use the operable category. For picture windows, use the fixed category. For prod-ucts that project out from the surface of the wall, use the garden window category. For skylights, use the sloped skylight category.
“Site-assembled” is intended to represent products where frame extrusions are assembled on site into a fenestration product and then glazing is added on site. These products are typically installed in high-rise residential and larger commercial/institutional/industrial buildings. Curtain walls are typically made up of vision (transparent portion) and spandrel (opaque portion) panels. Table 4 contains rep-resentative U-factors for the vision panel (including mullions) for these assemblies. The spandrel portion of curtain walls usually con-sists of a metal pan filled with insulation and covered with a sheet of glass or other weatherproof covering. Although the U-factor in the center of the spandrel panel can be quite low, the metal pan is a ther-mal bridge, significantly increasing the U-factor of the assembly.
Two-dimensional simulation validated by testing of a curtain wall having an aluminum frame with a thermal break found that the U-factor for the edge of the spandrel panel (the 65 mm band around the perimeter adjacent to the frame) was 40% of the way toward the U-factor of the frame. The U-factor was 0.34 W/(m2·K) for the cen-ter of the spandrel, 2.56 for the edge of the spandrel, and 6.02 for the frame (Carpenter and Elmahdy 1994). Two-dimensional heat trans-fer analysis or physical testing is recommended to determine the U-factor of spandrel panels. Use the sloped/overhead glazing cate-gory for sloped glazing panels comparable to curtain walls.
Physical testing of double-glazed units showed U-factors of 5.7 W/(m2·K) for a thermally broken aluminum pyramidal skylight and 7.4 W/(m2·K) for an aluminum-frame half-round barrel vault (both normalized to a rough opening of 2.4 m by 2.4 m). Until more conclusive results are available, U-factors for these systems can be estimated by multiplying the “site-assembled sloped/overhead glaz-ing” values in Table 4 by the ratio of total product surface area (including curbs) to rough opening area. These ratios range from 1.2 to 2.0 for low-slope skylights, 1.4 to 2.1 for pyramid assemblies sloped at 45°, and 1.7 to 2.9 for semicircular barrel vault assemblies.
An example calculation is provided in Example 4.
The U-factors in Table 4 are based on the definitions of the six product types, frame sizes, and proportion of frame to glass area shown in Figure 4. Four of the products are manufactured fenestra-tion products. The operable category glazing units are 1.35 m2 in area, and the overall size corresponds to a 900 mm by 1500 mm fen-estration product. The fixed (nonoperable) category is about 1.44 m2 in area, and the overall size corresponds to a 1200 mm by 1200 mm window. The garden window category is 1.35 m2 in projected area (3.15 m2 in surface area) and 1500 mm wide by 900 mm high by 380 mm deep. The manufactured skylight category is a nominal 0.72 m2 in area, corresponding to a 600 mm by 1200 mm skylight. The nom-inal dimensions of a roof-mounted skylight correspond to centerline spacing of roof framing members; consequently, the rough opening dimensions are 570 mm by 1180 mm. The curtain wall and sloped/overhead glazing categories are a nominal 1.44 m2 in area, representing repeating 1200 mm by 1200 mm panels. The nominal dimensions correspond to centerline spacing of the head and sill and vertical mullions.
Six frame types are listed (although not all for any one category) in order of improving thermal performance. The most conservative assumption is to use the frame category of aluminum frame without Frame Material Frame Width, mm Operable Fixed Garden Window Skylight Curtain Wall Sloped/Overhead Glazing Aluminum without thermal break 38 33 44 18 57 57 Aluminum with thermal break 53 33 n/a 18 57 57 Aluminum-clad wood/reinforcing vinyl 71 41 n/a 23 n/a n/a Wood/vinyl 71 41 44 23 n/a n/a Insulated fiberglass/vinyl 79 46 n/a n/a n/a n/a Structural glazing n/a n/a n/a n/a 57 64 Fig. 4 Standard Fenestration Units Fenestration 30.11 a thermal break (although there are products on the market that have higher U-factors). The aluminum frame with a thermal break is for frames having at least a 10 mm thermal break between the inside and outside for all members including both the frame and the operable sash, if applicable. (Products are available with significantly wider thermal breaks, which achieve considerable improvement.) The re-inforced vinyl/aluminum clad wood category represents vinyl-frame products, such as sliding glass doors or large windows that have extensive metal reinforcing within the frame and wood products with extensive metal, usually on the exterior surface of the frame. Both of these factors provide short circuits, which degrade the thermal per-formance of the frame material. The wood/vinyl frame is meant to represent the improved thermal performance that is possible if the thermal short circuits from the previous frame category do not exist.
Insulated fiberglass/vinyl represents fiberglass or vinyl frames that do not have metal reinforcing and whose frame cavities are filled with insulation. For several site-assembled product types, there is a structural glazing frame category that is intended to represent prod-ucts where sheets of glass are butt-glazed to each other using a seal-ant only, and none of the framing members is exposed to the exterior.
For glazing with a steel frame, use aluminum frame values. For alu-minum window with wood trim or vinyl cladding, use the values for aluminum. Frame type refers to the primary unit. Thus, when storm sash is added over another fenestration product, use the values given for the nonstorm product.
To estimate the overall U-factor of a fenestration product that differs significantly from the assumptions given in Table 4 and/or Figure 4, first determine the area that is frame/sash, center-of-glass, and edge-of-glass (based on a 65 mm band around the perimeter of each glazing unit). Next, determine the appropriate component U-factors. These can be taken either from the standard values listed in italics in Table 4 for glass, from the values in Table 1 for frames, or from some other source such as test data or computed factors.
Finally, multiply the area and the component U-factors, sum these products, and then divide by the rough opening in the building enve-lope where this product will fit to obtain the overall U-factor Uo.
Table 5 provides approximate data to convert the overall U-fac-tor at one wind condition to a U-factor at another. REPRESENTATIVE U-FACTORS FOR DOORS Doors are often an overlooked component in the thermal integ-rity of the building envelope. Although swinging and revolving doors represent a small portion of the shell in residential, commer-cial, and institutional buildings, their U-factor is usually many times higher than that of the walls or ceilings. In some storage and industrial buildings, loading bay doors (overhead doors) represent a significant area of high heat loss. Table 6 contains representative U-factors for swinging, overhead, and revolving doors determined through computer simulation (Carpenter and Hogan 1996). These are generic values, and product-specific values determined in accordance with standards should be used whenever available.
NFRC 100, Section B, and CSA Standard A453 give procedures for evaluating the performance of swinging doors. Overhead doors are often evaluated in accordance with National Association of Garage Door Manufacturers (NAGDM) Standard 105. Where these standards are cited in codes, they must be used for compliance.
Swinging doors can be divided into two categories: slab and stile-and-rail. A stile-and-rail door is a swinging door with a full-glass insert supported by horizontal rails and vertical stiles. The stiles and rails are typically either solid wood members or extruded aluminum or vinyl, as shown in Figure 5. Most residential doors are slab type with either solid wood, steel, or fiberglass skin over foam insulation in a wood frame with aluminum sill. The edges of the Table 5 Glazing U-Factors for Various Wind Speeds Wind Speed, km/h 24 12 0 U-Factor, W/(m2· K) 0.5 0.46 0.42 1.0 0.92 0.85 1.5 1.33 1.27 2.0 1.74 1.69 2.5 2.15 2.12 3.0 2.56 2.54 3.5 2.98 2.96 4.0 3.39 3.38 4.5 3.80 3.81 5.0 4.21 4.23 5.5 4.62 4.65 6.0 5.03 5.08 6.5 5.95 5.50 Table 6 U-Factors of Doors in W/(m2·K) Door Type No Glazing Single Glazing Double Glazing with 12.7 mm Air Space Double Glazing with e = 0.10, 12.7 mm Argon SWINGING DOORS (Rough Opening—970 mm × 2080 mm) Slab Doors Wood slab in wood framea 2.61 6% glazing (560 × 200 lite) — 2.73 2.61 2.50 25% glazing (560 × 910 lite) — 3.29 2.61 2.38 45% glazing (560 × 1620 lite) — 3.92 2.61 2.21 More than 50% glazing Use Table 5 (operable) Insulated steel slab with wood edge in wood framea 0.91 6% glazing (560 × 200 lite) — 1.19 1.08 1.02 25% glazing (560 × 910 lite) — 2.21 1.48 1.31 45% glazing (560 × 1630 lite) — 3.29 1.99 1.48 More than 50% glazing Use Table 5 (operable) Foam insulated steel slab with metal edge in steel frameb 2.10 6% glazing (560 × 200 lite) — 2.50 2.33 2.21 25% glazing (560 × 910 lite) — 3.12 2.73 2.50 45% glazing (560 × 1630 lite) — 4.03 3.18 2.73 More than 50% glazing Use Table 5 (operable) Cardboard honeycomb slab with metal edge in steel frame 3.46 Stile- and -Rail Doors Sliding glass doors/ French doors Use Table 5 (operable) Site-Assembled Stile- and -Rail Doors Aluminum in aluminum frame — 7.49 5.28 4.49 Aluminum in aluminum frame with thermal break — 6.42 4.20 3.58 REVOLVING DOORS (Rough Opening—2080 mm × 2130 mm) Aluminum in aluminum frame Open — 7.49 — — Closed — 3.69 — — SECTIONAL OVERHEAD DOORS (Nominal—3050 mm × 3050 mm) Uninsulated steel (nominal U = 6.53) 6.53 — — — Insulated steel (nominal U = 0.62) 1.36 — — — Insulated steel with thermal break (nominal U = 0.45) 0.74 — — — Note: All dimensions are in millimetres.
a Thermally broken sill [add 0.17 W/(m2·K) for non-thermally broken sill] b Non-thermally broken sill c Nominal U-factors are through the center of the insulated panel before consideration of thermal bridges around the edges of the door sections and due to the frame.
30.12 2001 ASHRAE Fundamentals Handbook (SI) steel skin door are normally wood to provide a thermal break. In commercial construction, doors are either steel skin over foam insu-lation in a steel frame (i.e., utility doors) or a full glass door made up of aluminum stiles, rails, and frame (i.e., entrance doors). The most important factors affecting door U-factor are material construction, glass size, and glass type. Frame depth, slab width, and number of panels have a minor effect on door performance. Sidelites and dou-ble doors have U-factors similar to a single door of the same con-struction. For wood slab doors in a wood frame, the glazing area does not have much effect on the U-factor. For the insulated steel slab in a wood frame, however, glazing area has a strong effect on U-factor. Typical commercial insulated slab doors have a U-factor approximately twice that of residential insulated doors, the prime reason being thermal bridging of the slab edge and the steel frame.
Stile-and-rail doors, even if thermally broken, have U-factors 50% higher than a full-glass commercial steel slab door.
There are three generic types of overhead doors: roll-up, uninsu-lated sectional, and insulated sectional. Metal roll-up doors consist of small metal plates of approximately 65 mm in height that “roll up” around a metal rod to open. Sectional doors consist of a series of 600 mm high sections that travel in a track to open. There is a wide range in the design of insulated overhead doors. Factors affect-ing heat transfer include width of insulation, thermal break design (if any), and design of interior skin. For the uninsulated sectional door, there is very little difference between the center value and the total value: essentially the value of single glazing. The center of the insulated door has low U-factors, but thermal bridging at the door and section edges significantly increases the total U-factor. For doors without thermally broken edges, the total value is 2.5 to 3.3 times greater than the center value. The addition of a good thermal break design reduces this increase to a 1.6 multiplier.
Many commercial buildings use revolving entrance doors. Most of these doors are of similar design: single glazing in an aluminum frame without thermal break. The door, however, can be in two posi-tions: closed (X shaped as viewed from above) or open (+ shaped).
At nighttime, these doors are locked in the X position, effectively creating a double-glazed system. During the daytime, the door revolves and is often left positioned so that there is only one glazing between the inside and outside (+ position). U-factors are given in Table 6 for both positions.
EXAMPLES Example 1. Estimate the U-factor for a manufactured fixed fenestration product with a reinforced vinyl frame and double-glazing with a sput-ter-type low-e coating (e = 0.10). The gap is 13 mm wide and argon-filled, and the spacer is metal.
Solution: Locate the glazing system type in the first column of Table 4 (ID = 23), then find the appropriate product type (fixed) and frame type (reinforced vinyl). The U-factor listed (in the tenth column of U-fac-tors) is 1.89 W/(m2·K).
Example 2. Estimate a representative U-factor for a wood-framed, 970 mm by 2080 mm swinging French door with eight 280 mm by 400 mm panes (true divided panels), each consisting of clear dou-ble-glazing with a 6.5 mm air space and a metal spacer.
Solution: Without more detailed information, assume that the dividers have the same U-factor as the frame and that the divider edge has the same U-factor as the edge-of-glass. Calculate the center-of-glass, edge-of-glass, and frame areas: Select the center-of-glass, edge-of-glass, and frame U-factors. These component U-factors are 3.29 and 3.63 W/(m2·K) (from Table 4, glaz-ing ID = 4, U-factor columns 1 and 2) and 2.90 W/(m2·K) (from Table 1, wood frame, metal spacer, operable, double-glazing), respectively.
From Equation (7), Example 3. Estimate the overall average U-factor for a multifloor curtain wall assembly that is part vision glass and part opaque spandrel. The typical floor-to-floor height is 3.6 m, and the building module is 1.2 m as reflected in the spacing of the mullions both horizontally and verti-cally. For a representative section 1.2 m wide and 3.6 m tall, one of the modules is glazed and the other two are opaque. The mullions are alu-minum frame with a thermal break 80 mm wide and centered on the module. The IGU is double glazing with a pyrolytic low-e coating (e = 0.40) and has a 13 mm gap filled with air and a metal spacer. The span-drel panel has a metal pan backed by R = 3.5 m2·K/W insulation and no intermediate reinforcing members. Solution: It is necessary to calculate the U-factor for the glazed module and for the opaque spandrel modules and then to do an area-weighted average to determine the average U-factor for the overall curtain wall assembly.
First, calculate the overall U-factor for the glazed module. Calcu-late the center-of-glass, edge-of-glass, and frame areas. The glazed area is 1120 mm by 1120 mm (1200 mm module, 1200 mm of mullions on each edge).
Select the center-of-glass, edge-of-glass, and frame U-factors. These component U-factors are 2.27 and 3.04 W/(m2·K) (from Table 4, ID = 13, columns 1 and 2) and 9.94 W/(m2·K) (from Table 1, aluminum frame with a thermal break, metal spacer, curtain wall, double glazing), respectively. From Equation (7), Then, calculate the overall U-factor for the two opaque spandrel mod-ules. The center-of-spandrel, edge-of spandrel, and frame areas are the same as the glazed module. The frame U-factor is the same. Cal-culate the center-of-spandrel U-factor. In this particular case, the R-value of the insulation does not need to be rated as there are no intermediate framing members penetrating it and providing thermal short circuits. When the resistance of the insulation (3.5 m2·K/W) is Fig. 5 Details of Stile-and-Rail Door Acg 8 280 130 – ( ) 400 130 – ( ) [ ] 106 ⁄ 0.324 m2 = = Aeg 8 280 400 × ( ) 106 ⁄ 0.324 – 0.572 m2 = = Af 970 2080 × ( ) 106 ⁄ 8 280 400 × ( ) 106 ⁄ – 1.122 m2 = = Uo 3.12 0.324 × ( ) 3.63 0.572 × ( ) 2.90 1.122 × ( ) + + 0.97 2.08 × ( ) ----------------------------------------------------------------------------------------------------------------------= 3.14 W/(m2 K) ⋅ = Acg 1120 130 – ( ) 1120 130 – ( ) 106 ⁄ 0.9801 m2 = = Aeg 1120 1120 × ( ) 106 ⁄ 0.9801 – 0.2743 m2 = = Af 1200 1200 × ( ) 106 ⁄ 1120 1120 × ( ) – [ ] 106 ⁄ 0.1856 m2 = = Uglazing module 2.27 0.9801 × ( ) 3.04 0.2743 × ( ) 9.94 0.1856 × ( ) + + 1.2 1.2 × ( ) -------------------------------------------------------------------------------------------------------------------------------= 3.41 W/(m2 K) ⋅ = Fenestration 30.13 added to the exterior air film resistance of 0.03 m2·K/W and the inte-rior air film resistance of 0.12 m2·K/W (from Table 1, Chapter 25), the total resistance is 3.65 m2·K/W and the U-factor is 1/3.65 = 0.274 W/(m2·K). The edge-of-spandrel U-factor is 40% of the way to the frame U-factor, which is 0.274 + [0.40(9.94 – 0.274)] = 4.14 W/(m2·K).
Finally, calculate the overall average U-factor for the curtain wall assembly, including the one module of vision glass and the two mod-ules of opaque spandrel.
Note that even with double glazing having a low-e coating and with R-20 in the opaque areas, this curtain wall with metal pans only has an overall R-value of approximately 0.38 m2·K/W.
Example 4. Estimate the U-factor for a semicircular barrel vault that is 6 m wide (3 m tall) and 10 m long mounted on a 150 mm curb. The barrel vault has an aluminum frame without a thermal break. The glazing is dou-ble with a 13 mm gap width filled with air and a low-e coating (e = 0.20).
Solution: An approximation can be made by multiplying the U-factor for a site-assembled sloped/overhead glazing product having the same frame and glazing features by the ratio of the surface area (including the curb) of the barrel vault to the rough opening area in the roof that the barrel vault fits over. First, determine the surface area (including the curb) of the barrel vault: Area of the curved portion of the barrel vault = (π × diameter/2) × length = (3.14 × 6/2) × 10 = 94.25 m2 Area of the two ends of the barrel vault = 2π(radius2)/2 = πr2 = 3.14 × 32 = 28.27 m2 Area of the curb = perimeter × curb height = (6 + 10 + 6 + 10) × 0.150 = 4.8 m2 Total surface area of the barrel vault = 94.25 + 28.27 + 4.8 = 127.3 m2 Second, determine the rough opening area in the roof that the barrel vault fits over: = length × width = 6 × 10 = 60 m2 Third, determine the ratio of the surface area to the rough opening area: = 127.3/60 = 2.12 Fourth, determine the U-factor from Table 4 of a site-assembled sloped/overhead glazing product having the same frame and glazing features. The U-factor is 4.06 W/(m2·K) (ID = 17, 12th column on the second page of Table 4).
Fifth, determine the estimated U-factor of the barrel vault.
Ubarrel vault = Usloped overhead glazing × surface area/rough opening for the barrel vault = 4.06 × 2.12 = 8.61 W/(m2·K) SOLAR HEAT GAIN AND VISIBLE TRANSMITTANCE Fenestration solar heat gain has two components. First is directly transmitted solar radiation. The quantity of radiation enter-ing the fenestration directly is governed by the solar transmittance of the glazing system. Multiplying the incident irradiance by the glazing area and its solar transmittance yields the solar heat enter-ing the fenestration directly. The second component is the absorbed solar radiation, radiation that is removed from the main beam and absorbed in the glazing and framing materials of the window, some of which is subsequently conducted to the interior of the building.
DETERMINING INCIDENT SOLAR FLUX Solar Radiation The flux of solar radiation on a surface normal (perpendicular) to the sun’s rays above the earth’s atmosphere at the mean earth-sun distance of 149.5 × 106 km (Allen 1973) is defined as the solar con-stant Esc. The currently accepted value is 1367 W/m2 (Iqbal 1983).
Because the earth’s orbit is slightly elliptical, the extraterrestrial radiant flux Eo varies from a maximum of 1413 W/m2 on January 3, when the earth is closest to the sun (aphelion), to a minimum of 1332 W/m2 on July 4, when the earth-sun distance reaches its max-imum (perihelion).
The earth’s orbital velocity also varies throughout the year, so apparent solar time, as determined by a solar time sundial, varies somewhat from the mean time kept by a clock running at a uni-form rate. This variation, called the equation of time, is given in Table 7. The conversion between local standard time and solar time involves two steps. First the equation of time is added to the local standard time, and then a longitude correction is added. This longitude correction is four minutes of time per degree difference between the local (site) longitude and the longitude of the local standard meridian for that time zone. Standard meridians are found every 15° from 0° at Greenwich, England (Greenwich Merid-ian). In the United States and Canada, these values are 60° for Atlantic Standard Time, 75° for Eastern Standard Time, 90° for Central Standard Time, 105° for Mountain Standard Time, 120° for Pacific Standard Time, 135° for Alaska Standard Time, and 150° for Hawaii-Aleutian Standard Time.
Equation (10) relates apparent solar time (AST) to local standard time (LST) as follows: AST = LST + ET/60 + (LSM – LON)/15 (10) where AST = apparent solar time, decimal hours LST = local solar time, decimal hours ET = equation of time, decimal minutes LSM = local standard time meridian, decimal ° of arc LON = local longitude, decimal ° of arc Because the earth’s equatorial plane is tilted at an angle of 23.45° to the orbital plane, the solar declination δ (the angle between the Uopaque spandrel module 0.274 0.9801 × ( ) 4.14 0.2743 × ( ) 9.94 0.1856 × ( ) + + 1.2 1.2 × ( ) ----------------------------------------------------------------------------------------------------------------------------------= 2.26 W/(m2 K) ⋅ = Ucurtain wall 3.41 1.2 1.2 × ( ) × [ ] 2.26 2 × 1.2 1.2 × ( ) × [ ] + 3 1.2 1.2 × ( ) × ---------------------------------------------------------------------------------------------------------------= 2.64 W/(m2 K) ⋅ = Table 7 Extraterrestrial Solar Irradiance and Related Data Eo, W/m2 Equation of Time, min Declina-tion, degrees A B C W/m2 (Dimensionless Ratios) Jan 1416 −11.2 −20.0 1230 0.142 0.058 Feb 1401 −13.9 −10.8 1215 0.144 0.060 Mar 1381 −7.5 0.0 1186 0.156 0.071 Apr 1356 1.1 11.6 1136 0.180 0.097 May 1336 3.3 20.0 1104 0.196 0.121 June 1336 −1.4 23.45 1088 0.205 0.134 July 1336 −6.2 20.6 1085 0.207 0.136 Aug 1338 −2.4 12.3 1107 0.201 0.122 Sep 1359 7.5 0.0 1151 0.177 0.092 Oct 1380 15.4 −10.5 1192 0.160 0.073 Nov 1405 13.8 −19.8 1221 0.149 0.063 Dec 1417 1.6 −23.45 1233 0.142 0.057 Note: Data are for 21st day of each month during the base year of 1964.
30.14 2001 ASHRAE Fundamentals Handbook (SI) earth-sun line and the equatorial plane) varies through out the year, as shown in Figure 6, Table 7, and Equation (11). This variation causes the changing seasons with their unequal periods of daylight and darkness. The following equation can be used to estimate the declination from the day of year η, but it is more accurate to look up the actual declination in an astronomical or nautical almanac for the actual year and date in question.
δ = 23.45 sin {[360(284 + η)]/365} (11) The spectral distribution of solar radiation beyond the earth’s atmosphere (Figure 7) resembles the radiant energy emitted by a blackbody at about 6000 K. The peak solar spectral irradiance of 2130 W/(m2·K) is reached at 0.451 µm (451 nm) in the green por-tion of the visible spectrum.
In passing through the earth’s atmosphere, the sun’s radiation is reflected, scattered, and absorbed by dust, gas molecules, ozone, water vapor, and water droplets (fog and clouds). The extent of this depletion at any given time is determined by atmospheric composi-tion and length of the atmospheric path traversed by the sun’s rays.
This length is expressed in terms of the air mass m, which is the ratio of the mass of atmosphere in the actual earth-sun path to the mass that would exist if the sun were directly overhead at sea level (m = 1.0). For most purposes, the air mass at any time equals the cosecant of the solar altitude multiplied by the ratio of the existing barometric pressure to standard pressure. Beyond the atmosphere, m = 0.
Most ultraviolet solar radiation is absorbed by the ozone in the upper atmosphere, while part of the radiation in the short-wave por-tion of the spectrum is scattered by air molecules, imparting the blue color to the sky. Water vapor in the lower atmosphere causes the characteristic absorption bands observed in the solar spectrum at sea level (Figure 7). For a solar altitude β of 41.8° (air mass m = 1.5), the total solar direct beam flux on a clear day at sea level can be divided into spectral regions as follows. Less than 3% of the total is in the ultraviolet, 47% is in the visible region, and the remaining 50% is in the infrared (ASTM Standard E 891). The maximum spectral irra-diance occurs at 0.61 µm, and little solar energy (less than 5% of the spectrum) exists at wavelengths beyond 2.1 µm.
It is interesting to see what fraction of the total solar irradiance lies in the visible part of the spectrum. Since the limits of the visible portion vary from observer to observer (and because the eye is not very sensitive to radiation at the spectral limits of vision), the frac-tions of total irradiance and illuminance found between different spectral limits at the edge of the visible portion of the spectrum can be calculated. The results are shown in Table 8 for the ASTM air mass m = 1.5 terrestrial spectrum shown in Figure 8.
The solar spectral distribution shown in Figure 7 for m = 0 is the World Radiation Center’s 1985 standard extraterrestrial spectrum for a solar constant of 1367 Wm2 (Wehrli 1985). The one for m = 1.5 in Figure 8 is from ASTM Standard E 891. This latter takes no account of monthly variations in irradiance caused by changes in the Fig. 6 Motion of Earth around Sun Fig. 7 Terrestrial and Extraterrestrial Solar Spectral Irradiances Table 8 Portions of Total Solar Spectral Irradiance Contained in Portions of Visible Spectrum Wavelength, nm Percent Irradiance Percent Illuminance Start End 370 770 54.4 100.0 380 760 52.2 100.0 390 750 50.2 99.9 400 740 47.4 99.9 410 730 44.9 99.8 420 720 41.9 99.8 430 710 39.5 99.8 440 700 36.7 99.8 450 690 35.3 99.5 460 680 31.1 99.1 Note: The integrated total irradiance = 950 W/m2 and illuminance = 100 klx.
Fig. 8 Comparison of Standard Air Mass m = 1.5 Solar Spectrum with Direct Beam Spectra Through Atmospheres Characteristic of southwest in winter (SWWINT) and southeastern U.S. in summer (SESUMM) for two solar altitude angles (McCluney 1996) Fenestration 30.15 earth-sun distance and by variations in the atmosphere’s constituent particulates and gases.
When variations in atmospheric constituents and air mass are considered, the solar spectral distribution is seen to vary, as illus-trated in Figure 8, for two different atmospheric conditions and for two solar altitude angles, and in Figure 9 for both direct and diffuse radiation components and a low sun angle. It is clear that the spec-tral distribution for low sun angle beam radiation is significantly shifted toward longer wavelengths. This shift can be seen visually as a reddening of the sun near to the horizon. Clear sky diffuse radi-ation is generally shifted toward the blue end of the spectrum.
Upon passage through the atmosphere, extraterrestrial solar radi-ation is reduced in magnitude due to absorption by atmospheric gases and particulates. The strength of this absorption varies with wavelength, and the terrestrial solar spectrum exhibits definite “dips” in regions of strong absorption, called absorption bands.
The most prominent atmospheric gases contributing to this effect are listed below: • Ozone. Strongest absorption in the ultraviolet, some in the visible. Concentration variable.
• H2O. Strongest absorption in near and far IR. Highly variable.
• CO2. Strongest absorption in near and far IR. Slightly variable.
• O2, CH4, N2O, CFCs. Strongest absorption mostly in the IR. Concentration almost constant.
• NO2. Strongest absorption in the visible. Highly variable in polluted areas.
The effect of aerosols and other particulates on terrestrial solar radiation can be significant. Diffuse sky radiation is solar beam radi-ation that has been multiply scattered out of the direct beam and downward through the atmosphere to the earth’s surface. This scat-tering is produced by 30 different atmospheric molecules (of which the above are the most significant optically) and by larger particles of different types, including aerosols of water, dust, smoke, and par-ticulates of other kinds.
More information on atmospheric optics can be found in Chapter 44 of the Optical Society of America’s Handbook of Optics (Bass 1995) and in Iqbal (1983).
Glazing systems exhibiting strong spectral selectivity (strong changes in their optical properties over the solar spectrum) will selectively pass more or less radiation in different parts of the spec-trum. This effect can cause substantial changes in the solar heat gain coefficient of the glazing system when the shape of the solar spec-trum shifts appreciably. This in turn can cause errors in solar heat gain predictions when the actual solar radiation on a fenestration system has a spectrum that is different from the standard spectrum used to determine the solar heat gain coefficient of that system (McCluney 1996). These errors are typically 5 to 10% but can be substantially greater in special cases.
Some short-wavelength radiation scattered by air molecules, dust, and other particulates in the atmosphere reaches the earth in the form of diffuse sky radiation Ed. Since this diffuse radiation comes from all parts of the sky, its irradiance is difficult to predict and varies as moisture and particulate content and sun angle change throughout any given day. For completely overcast condi-tions, the diffuse component accounts for all solar radiant heat gain of fenestrations.
The total short-wavelength irradiance Et reaching a terrestrial surface is the sum of the direct solar radiation ED, the diffuse sky radiation Ed, and the solar radiation Er reflected from surrounding surfaces. The irradiance on the fenestration aperture of the direct beam component ED is the product of the direct normal irradiation EDN and the cosine of the angle of incidence θ between the incom-ing solar rays and a line normal (perpendicular) to the surface: (12) A method for computing all the factors on the right side of Equa-tion (12) is presented in the sections on Direct Normal Irradiance and Diffuse and Ground-Reflected Radiation. Perez et al. (1986), Gueymard (1987), Solar Energy (1988), and Gueymard (1993) give more detailed models, which separate the diffuse sky radiation into different components. Gueymard (1995) provides a comprehensive spectrally based model for calculating the spectral and broadband totals of all three terms in Equation (12), for cloudless sky condi-tions. The Gueymard model allows user input of the concentrations of a variety of atmospheric constituents, including particulates.
The importance of the diffuse component is illustrated in Figure 9, which shows that at low sun angles the diffuse component con-tains more radiant flux than the direct beam component, even on a clear day, and that the spectral distributions of the two components are quite different. Although the total irradiances are relatively modest for both of these components, they are not insignificant for annual energy performance calculations.
Vertical windows receive considerable quantities of diffuse sky radiation over the course of a year. The diffuse component is an important part of solar radiant heat gain.
Determining Solar Angle The sun’s position in the sky is conveniently expressed in terms of the solar altitude β above the horizontal and the solar azimuth φ measured from the south (Figure 10). These angles, in turn, depend on the local latitude L; the solar declination δ, which is a function of the date [Table 7 or Equation (11)]; and the apparent solar time, expressed as the hour angle H, where H = 15(AST – 12) (13) Equations (14) and (15) relate β and φ to the three angles just mentioned: (14) (15) Figure 10 shows the solar position angles and incident angles for horizontal and vertical surfaces. Line OQ leads to the sun, the north-south line is NOS, and the east-west line is EOW. Line OV is perpendicular to the horizontal plane in which the solar azimuth φ (angle HOS) and the surface azimuth Ψ (angle POS) are located.
Angle HOP is the surface solar azimuth γ, defined as γ = φ − ψ (16) The solar azimuth φ is positive for afternoon hours and negative for morning hours. Likewise, surfaces that face west have a positive Fig. 9 Comparison of Direct and Diffuse Solar Spectra for Low Solar Altitude Angle Et EDN θ cos Ed Er + + = β sin L δ H cos cos cos L δ sin sin + = φ cos β L sin sin δ sin – β L cos cos ---------------------------------------= 30.16 2001 ASHRAE Fundamentals Handbook (SI) surface azimuth ψ; those facing east have a negative surface azi-muth (Table 9). If γ is greater than 90°or less than −90°, the surface is in the shade. Table 9 gives values in degrees for the surface azi-muth ψ, applicable to the orientations of interest. The angle of incidence θ for any surface is defined as the angle between the incoming solar rays and a line normal to that surface.
For the horizontal surface shown in Figure 10, the incident angle θH is QOV; for the vertical surface, the incident angle θV is QOP.
For any surface, the incident angle θ is related to β, γ, and the tilt angle of the surface Σ by cosθ = cosβ cosγsinΣ + sinβcosΣ (17) where Σ = tilt angle of surface from horizontal.
When the surface is horizontal, Σ = 0° and cosθH = sinβ (18) For a vertical surface, Σ = 90° and cosθV = cosβcosγ (19) Direct Normal Irradiance At the earth’s surface on a clear day, direct normal irradiation, or solar irradiance EDN, is represented by (20) where A = apparent solar irradiation at air mass m = 0 (Table 7) B = atmospheric extinction coefficient (Table 7) Values of A and B vary during the year because of seasonal changes in the dust and water vapor content of the atmosphere and because of the changing earth-sun distance. Equation (20) does not give the maximum value of EDN that can occur in each month but yields values that are representative of conditions on cloudless days for a relatively dry and clear atmosphere. For very clear atmo-spheres, EDN can be 15% higher than indicated by Equation (20), using values of A and B in Table 7. For locations where clear, dry skies predominate (e.g., at high elevations) or, conversely, where hazy and humid conditions are fre-quent, values found by using Equation (20) and Table 7 should be multiplied by the Clearness Numbers in Threlkeld and Jordan (1958), reproduced as Figure 5 in Chapter 32 of the 1999 ASHRAE Handbook—Applications. This broadband model should only be used for determining fenestration solar gain when the glazing sys-tem is not strongly spectrally selective and when its angular selec-tivity closely matches that of single-pane glass.
Diffuse and Ground-Reflected Radiation The following equations can be used to generate Ed and Er, where all angles are in degrees. The solar azimuth φ and the surface azimuth ψ are measured in degrees from south; angles to the east of south are negative, and angles to the west of south are positive. Val-ues of A, B, and C are given in Table 7 for the 21st day of each month. Values for other dates can be obtained by interpolation.
The ratio Y of sky diffuse radiation on a vertical surface to sky diffuse radiation on a horizontal surface is given by (21) Diffuse irradiance Ed is given by Ed = CYEDN for vertical surfaces (22) for surfaces other than vertical (23) Ground-reflected irradiance Er is given by for surfaces at all orientations (24) where ρg is ground reflectivity, often taken to be 0.2 for typical mix-ture of ground surfaces. For other surfaces, Table 10 lists angle-dependent solar reflectances.
Example 5. Find the solar azimuth and altitude at 3:00 P.M. central daylight savings time on June 21 in St. Louis, MO.
Solution: Central daylight savings time of 3:00 P.M. re-expressed in decimal hours at local standard time is LST = 14.0. The latitude and lon-gitude of St. Louis, MO, are 38.6°N and 90.2°W, respectively. The local standard meridian of the central time zone is 90°W. The equation of time (Table 7) is –1.4 min. From Equation (10), apparent solar time (AST) = 14.0 − 1.4/60 + (90 − 90.2)/15 = 13.9633. From Equation (13), H = 15 × (13.9633 − 12) = 29.45°. Table 7 gives the solar declination δ on June 21 as 23.45°.
Thus, by Equation (14), sin β = cos(38.6°) cos(23.45°) cos(29.45°) + sin(38.6°) sin(23.45°) = 0.873 β = 60.76° Using Equation (15), φ = 67.4° Table 9 Surface Orientations and Azimuths, Measured from South Orientation N NE E SE S SW W NW Surface azimuth ψ 180° −135° −90° −45° 0 45° 90° 135° Fig. 10 Solar Angles for Vertical and Horizontal Surfaces EDN A B β sin ⁄ ( ) exp ---------------------------------= Table 10 Solar Reflectances of Foreground Surfaces Foreground Surface Incident Angle 20° 30° 40° 50° 60° 70° New concrete 0.31 0.31 0.32 0.32 0.33 0.34 Old concrete 0.22 0.22 0.22 0.23 0.23 0.25 Bright green grass 0.21 0.22 0.23 0.25 0.28 0.31 Crushed rock 0.20 0.20 0.20 0.20 0.20 0.20 Bitumen and gravel roof 0.14 014 0.14 0.14 0.14 0.14 Bituminous parking lot 0.09 0.09 0.10 0.10 0.11 0.12 Adapted from Threlkeld (1962) Y 0.55 0.437 θ cos 0.313cos2θ + + = for cosθ 0.2 – > Y 0.45 = for cosθ 0.2 – ≤ Ed CEDN 1 Σ cos + 2 ---------------------= Er = EDN C β sin + ( )ρg 1 Σ cos – 2 ---------------------φ cos 60.76 ( ) sin 38.6 ( ) 23.45 ( ) sin – sin 60.76 ( ) 38.6 ( ) cos cos ----------------------------------------------------------------------------------0.384 = = Fenestration 30.17 Example 6. For the conditions of Example 5, find the incident angle at a window facing west.
Solution: From Example 5, φ = 67.4°. From Table 9, ψ = 90°. From Equation (16), γ = 67.4° − 90° = –22.6°.
A negative surface solar azimuth γ indicates that the sun is south of the normal to the surface. Thus, using Equation (19), cosθ = cos(60.76°) cos(–22.6°) = 0.451 θ = 63.2° Example 7. Find the direct, diffuse, and ground-reflected components of the solar irradiation on the window in Example 6.
Solution: From Example 5, sin β = 0.873, and from Table 7, A = 1088 W/m2 and B = 0.205. Therefore, from Equation (20), From Table 7, C = 0.134. From Equation (21), Y = 0.55 + 0.437 × 0.451 + 0.313 × (0.451)2 = 0.811. From Equation (22), From Equation (24), assuming a ground reflectivity ρg of 0.2, Thermal Infrared Radiation Any material above a temperature of absolute zero emits elec-tromagnetic radiation. The rate of emission depends upon the tem-perature of the material and can be expressed in a simple equation, called the Stefan-Boltzmann law: (25) where Eb = hemispherical total emissive power of black body, W/m2 T = temperature, K σ = Stefan-Boltzmann constant, W/(m2·K4) The emissivity e of a surface is the ratio of the emission of ther-mal radiant flux from the surface to the flux that would be emitted by a blackbody emitter at the same temperature. Given the temper-ature and emissivity of a surface, the emitted irradiance spectrum can be computed from where E is the emitted irradiance in units of flux per unit area, e is the emissivity of the surface, σ is the Stefan-Boltzmann constant, and T is the absolute temperature of the surface.
The maximum value of hemispherical emissivity e for any material is 1.0, in which case the surface emits the theoretical max-imum amount of radiation possible. In this case, the surface is called a blackbody, and the radiation emitted by the surface is called blackbody radiation.
Occasionally, a related quantity called normal emissivity is used.
The relationship between them is illustrated in Figure 11. The spectral distribution of blackbody radiation is illustrated in Figure 12 for tem-peratures ranging from 300 K (room temperature) to 20 000 K.
OPTICAL PROPERTIES Solar radiation (including both direct rays from the sun and dif-fuse rays from the sky, clouds, and surrounding objects) incident on a fenestration system is partly transmitted and partly reflected by the glazings of that system. An additional fraction is absorbed within the glazings and/or the coatings on their surfaces. The fraction of incident flux that is reflected is called the reflectance R, the fraction absorbed is called the absorptance A, and the fraction transmitted is the transmittance T. The sum of the transmittance T, absorptance A, and reflectance R of a glazing layer is unity: (26) However, this is complicated by the fact that radiation incident on a surface can have nonconstant distributions over the directions of incidence and over the wavelength (or frequency) scale. Thus, when measuring one or more of the optical properties, the wave-length distribution and direction of incident (and emerging) radia-tion must be specified.
Angular Dependence The concept of solid angle is needed to understand angular dependence. A solid angle is defined and enclosed by all rays join-ing a point to a closed curve. For a closed curve on a sphere of radius R, the solid angle ϖ is the ratio of the projected area A on the sphere to the square of R. A sphere has a solid angle of 4π steradians (4π sr); a hemisphere has a 2π sr solid angle.
Radiation incident on a point in a surface comes to that point from many directions in some solid angle. For a cone of half angle α, the solid angle defined by the circular top and point bottom of that cone is given by ϖ = 2π (1 − cos α) (27) EDN 1088 exp 0.205 0.873 ⁄ ( ) [ ] ⁄ 860.3 W m2 ⁄ = = EDN θ ( ) cos 860.3 0.451 388 = × W m2 ⁄ = Ed 0.134 ( ) 0.811 ( ) 860.3 ( ) 93.5 W m2 ⁄ = = Er 860.3 ( ) 0.134 0.873 + ( ) 0.2 ( ) 2 ⁄ 86.6 W m2 ⁄ = = Eb σT 4 = E eσT 4 = Fig. 11 Illustration of Difference Between Normal and Hemispherical Emissivity Fig. 12 Spectral Distributions of Blackbody Radiation at Different Source Temperatures T R A + + 1 = 30.18 2001 ASHRAE Fundamentals Handbook (SI) In measuring transmittance or reflectance, a sample is illumi-nated over a specified solid angle. The reflected or transmitted flux is then collected within another solid angle. The size of a conical solid angle and the direction of its axis need to be specified to obtain meaningful results. A conical solid angle is bounded by a right cir-cular cone.
ASTM Standards E 903, E 1084, E 971, and E 972, as well as NFRC 300, which refers to ASTM Standard E 971, refer to conical-hemispherical measurements of optical properties. The reason is that for most thermal or HVAC design calculations, only the total flux transmitted into a hemispherical solid angle, due to the direct solar beam incident in a small conical solid angle, is of interest. All transmitted solar irradiance is considered heat gain in these applica-tions, regardless of the directional distribution of the transmitted radiation.
For complex fenestrations (those with nonspecular components), for many daylighting applications, and for some passive solar space-heating applications, the directional distribution of transmit-ted radiation is of interest. In such cases, it is important to know what is called the biconical transmittance and reflectance of fen-estration systems, defined for conical solid angles of incidence and emergence (see Figure 13).
Biconical optical properties are needed (1) to treat diffuse sky as well as direct beam radiation, (2) to handle the directional distribu-tion of the flux entering a room through a window, and (3) to calcu-late the angle-dependent optical and solar gain properties of multiple-pane window systems and complex glazing systems, including those with integral or attached shading devices.
Spectral Dependence Frequency and wavelength are related through λf = c (28) where λ = wavelength, m f = frequency, Hz c = speed of light = 3.0 × 108 m/s in air at standard atmospheric pressure The wavelength dependence of radiometric quantities is denoted with a subscript λ attached to the optical quantity, thus φλ, Eλ, and Lλ. The wavelength dependency of optical properties is denoted by the functional notation, thus α(λ), ρ(λ), τ(λ), and e(λ).
All solar radiant flux becomes heat when it is absorbed by materials such as glazings, window frames, and room surfaces.
This includes radiation in the UV, visible, and IR portions of the spectrum. Recall from Equation (26) that the sum of the transmit-tance, reflectance, and absorptance is 1.0.
There is an additional relationship among the optical properties that is of interest and importance. It is called Kirchhoff’s law (McCluney 1994a): (29) where θ and φ are angles defining the directional dependence of the spectral absorptance A(λ) and the spectral emissivity e(λ). A con-sequence of Equations (26) and (29) is that for opaque materials a good absorber is a good emitter and a poor reflector, and vice versa, but only on a wavelength-by-wavelength basis or over a defined wavelength interval. A surface appearing to be an excellent reflec-tor in the visible portion of the spectrum may have a high emissivity (and low reflectance) over most of the infrared spectrum, or vice versa.
Source Spectra. Radiation incident on a surface has a distribu-tion not only over direction within some solid angle but also over a range of wavelengths. The latter distribution is called a spectrum.
For terrestrial applications, it is only after the extraterrestrial solar spectrum has been modified by passage through the atmosphere that it is of interest (Figures 8 and 9).
SOLAR-OPTICAL PROPERTIES OF GLAZING The solar-optical properties of a glazing are the wavelength-inte-grated (or total) transmittance, reflectance, and absorptance of the glazing to incident solar radiation. If the spectral optical properties [T(λ), R(λ), A(λ)] of the glazing and the spectral irradiance E(λ) incident on the glazing are known, the solar optical properties can be calculated from (30) where p stands for transmittance, reflectance, or absorptance, and the limits of integration are those values of wavelength outside of which the solar spectral irradiance is negligibly small.
If E(λ) in Equation (30) is a standard solar spectral irradiance distribution, then the optical property resulting from this equation is called the solar optical property. If the spectral properties p(λ) are available only at a set of discrete wavelengths λk (e.g., measured data), then the following discrete sum approximation of Equation (30) can be calculated: (31) where is the spectral weight for the wavelength λk and where ∆λk = 1/2(λk+1 – λk–1).
Unless explicitly stated otherwise, solar-optical properties used in this chapter are calculated using the tabulated ASTM Standard E 891 solar spectrum and Equation (31).
Many window glazings do not have strong spectral selectivity over the solar spectrum, so their spectral optical properties can be Fig. 13 Geometry for Definition of Biconical Transmittance and Reflectance A λ θ φ , , ( ) e λ θ φ , , ( ) = p p λ ( )E λ ( ) λ d λmin λmax ∫ E λ ( ) λ d λmin λmax ∫ -------------------------------------------= p wkp λk ( ) k=1 M ∑ = wk E λk ( ) E λk ( ) λk ∆ k=1 M ∑ --------------------------------= Fenestration 30.19 considered constant, even if the source spectrum changes substan-tially. In these cases, the transmitted spectral irradiance can be determined by multiplying the incident irradiance by the solar transmittance.
Figure 14 shows the spectral transmittance at normal incidence of typical architectural glasses. The approximate transmittance of total incident solar radiation through clear float glass at an incident angle of 0° ranges from 86% for 2.4 mm thick glass to 84% for 3.2 mm thick glass to 78% for 6.4 mm thick glass. Actual transmit-tance varies with the amount of iron or other absorbers in the glass.
Low iron content glass has a relatively constant spectral transmit-tance over the entire solar spectrum.
Figure 15 shows the normal incidence spectral transmittances of several common commercially available glazings. Figure 16 shows the normal incidence spectral transmittances and exterior reflec-tances of a variety of additional coated and tinted glasses, indicating the strong spectral selectivity that is now available from some glass and window manufacturers.
Angular Dependence of Glazing Optical Properties As Figure 17 shows, the optical properties of a single sheet of clear glass depend on the angle of incidence. This variation is small for incident angles below 40° but becomes significant at larger angles.
Fig. 14 Spectral Transmittance for Typical Architectural Glass Fig. 15 Spectral Transmittances of Commercially Available Glazings (McCluney 1993) Fig. 16 Spectral Transmittances and Reflectances of Strongly Spectrally Selective Commercially Available Glazings (McCluney 1996) Fig. 17 Transmittance and Reflectance of Plane, Parallel, Glass Plate refractive index n = 1.55, thickness t = 3.2 mm, absorptivity α = 0.01/m Fig. 18 Variations with Incident Angle of Solar-Optical Properties for (A) Double-Strength Sheet Glass, (B) Clear Plate Glass, and (C) Heat-Absorbing Plate Glass 30.20 2001 ASHRAE Fundamentals Handbook (SI) Figure 18 compares the properties of glasses of different thick-ness and composition. As the incident angle increases from zero, transmittance diminishes, reflectance increases, and absorptance first increases because of the lengthened optical path and then decreases as more incident radiation is reflected. While the shapes of the property curves are superficially similar, note that both the magnitude of the transmittance at normal incidence and the angle at which the transmittance changes significantly vary with glass type and thickness. The three curves all have slightly different shapes.
For coated glasses or for multiple-pane glazing systems, this differ-ence is more pronounced. One cannot assume that all glazings or glazing systems have a universal angular dependence. This is one of the inadequacies of the shading coefficient methodology for deter-mining solar heat gain that led to its elimination from this edition.
In North America, peak summertime solar gains occur with east-and west-facing vertical windows at angles of incidence ranging from about 25 to 55°. The peak solar gain for horizontal glazings occurs typically at small angles of incidence. For north- and south-facing vertical glazings, peak summertime solar gains occur at angles of incidence greater than about 40° (McCluney 1994b).
Angles of incidence important for annual energy performance calculations range from 5° to over 80° for east- and west-facing ver-tical windows and for horizontal glazings. This range is only slightly diminished for south-facing windows. For north-facing windows, the direct beam solar gains are small and their angles of incidence range from 62 to 86° (McCluney 1994b).
Optical Properties of Single Glazing Layers The optical properties of a single layer of glazing material are outlined in Figure 19. The layer has a thickness d and is charac-terized by a surface reflectivity and transmissivity, ρ and τ, for each of the two surfaces (denoted f and b in the figure) and an absorptivity, α, which is a volumetric property of the material (assumed of uniform composition). In general, τ and ρ are charac-teristics of the interface between the material and the adjacent medium; they may in principle be different for the two surfaces (e.g., for a coated surface, or where a material layer is adjacent to another material rather than air). All three properties, transmissiv-ity, reflectivity, and absorptivity, depend on the wavelength of the incident radiation, and τ and ρ also depend on the incident angle θ of the radiation incident on the layer.
The transmittance T and the front reflectance Rf of a layer (as opposed to a surface) contain the effects of multiple reflections between the two surfaces of the layer, as indicated in Figure 19, as well as the effects of absorption during the passage through the layer (one or more times), due to the volume absorptivity α. The same is true of the back reflectance Rb, which is the reflectance of the layer for radiation incident on back side b and which is not illustrated in the figure. For non-normal incidence, surface reflectances are in general different for the two possible polarizations of light, conven-tionally denoted s (TE) and p (TM). We distinguish these below by a subscript µ (= s,p) on the surface reflectance. [A pre-subscript is used where there is a possibility of confusion with later notational additions.] The transmittance and reflectances are given by (32) (33) (34) where ζ is the angle at which radiation incident at angle θ propa-gates within the glazing layer (the refracted angle). Since sunlight is unpolarized, the transmittance and reflectances of an isolated glaz-ing layer are then calculated from (35) (36) (37) The transmittance is the same for incident radiation incident (of a given polarization) on either surface, as can be seen from the symmetry of Equation (32) in the indices f and b. Front and back reflectances, however, may differ. The angular and wavelength de-pendence of these quantities is emphasized in the equations through explicit function reference [e.g., T(θ,λ)]. This dependence will not always be made explicit (in the interest of brevity of equa-tions) but should not be forgotten.
The transmittance and reflectances are the basic measurable quantities for an isolated glazing layer in air. Measurements on glaz-ing layers are typically made at normal incidence, and the properties at other angles must be inferred from these measurements. A sys-tematic compilation of these measured properties for most glazings manufactured in the United States is maintained by the National Fenestration Rating Council, Silver Spring, MD, and is available on the World Wide Web at or at [see also LBL (1994) and NFRC (2000a)].
It follows from conservation of energy that the average absorp-tance of the layer must be defined as the fraction of the incident radi-ation that is neither transmitted nor reflected by the layer. Note that when the layer surfaces have different properties, this results in dif-ferent absorptances for front and back incidence. It also produces an angular dependence in the layer absorptance a that is not present in the absorptivity α: (38a) (38b) and, since these are linear relations, Fig. 19 Optical Properties of a Single Glazing Layer T θ λ , ( ) µ τµ f θ λ , ( )τµ b θ λ , ( )e α λ ( )d ζ cos ---------------– 1 ρµ f θ λ , ( )ρµ b θ λ , ( )e 2α λ ( )d ζ cos -------------------– – ------------------------------------------------------------------------= Rf µ θ λ , ( ) ρµ f θ λ , ( ) ρµ b θ λ , ( ) T θ λ , ( ) µ e α λ ( )d ζ cos ---------------– + = Rb µ θ λ , ( ) ρµ b θ λ , ( ) ρµ f θ λ , ( ) T θ λ , ( ) µ e α λ ( )d ζ cos ---------------– + = T θ λ , ( ) 1 2 -- T θ λ , ( ) s T θ λ , ( ) p + [ ] = Rf θ λ , ( ) 1 2 -- Rf θ λ , ( ) s Rf θ λ , ( ) p + [ ] = Rb θ λ , ( ) 1 2 -- Rb θ λ , ( ) s Rb θ λ , ( ) p + [ ] = af µ θ λ , ( ) 1 T θ λ , ( ) µ – Rf µ θ λ , ( ) – = ab µ θ λ , ( ) 1 T θ λ , ( ) µ – Rb µ θ λ , ( ) – = Fenestration 30.21 (38c) (38d) for the unpolarized quantities. A lowercase symbol is used here in anticipation of the discussion of multilayer glazing systems below.
Uncoated Glazings For uncoated glazings, the interface reflectivities ρ of the two sur-faces are the same and may be determined from the Fresnel equations: (39a) (39b) where ζ is the refracted angle and may be calculated from Snell’s law and the real part of the refractive index n relative to air: (40) At normal incidence, the two polarizations are indistinguishable, and Equation (39) reduces to (41) and since (42) for any surface, a measurement of the spectral transmittance and re-flectance at normal incidence may be used with Equations (32) and (33) to determine the refractive index and absorptance as a function of wavelength. Note that n is also wavelength-dependent, although for most glazing materials the dependence is weak over the solar spectrum. Once these quantities are known, the equations may be used to calculate the properties at all angles.
ASHRAE “Standard” Glass In the discussion of single glazing, and historically in the context of the solar heat gain factor and shading coefficient methodology, ASHRAE has used as a calculation standard the properties of “one-eighth-inch, clear double-strength glass” (DSG). The wavelength-averaged properties of this standard glazing are calculated from the following equations: (43) (44) where the coefficients (ts)n and (as)n are given in Table 11. The hemispherical average quantities may be calculated by averaging Equations (43) and (44), which yields (45) (46) Determining the Properties of Uncoated Glazing Layers from Normal Incidence Measurements For uncoated glazings, the front and back transmissivities, re-flectivities, transmittances, reflectances, and isolated-layer absorp-tances are equal, the two polarizations are indistinguishable, and at normal incidence Equations (32) and (33) become (47) (48) while Equation (42) becomes (49) These three equations can be solved to yield ρ(0, λ) and α(λ): (50) where (51) (52) The real part of the refractive index (relative to air) is then cal-culated by solving Equation (41): (53) Thus, given spectroscopic measurements of the transmittance and reflectance at normal incidence of an uncoated glazing layer, one can determine the basic parameters needed for a complete cal-culation of the optical properties of that layer.
Example 8. Construct an approximate model of the optical properties of a single layer of uncoated 3 mm clear glass, suitable for use in calcula-tions involving selective glazings that have different properties in the visible and NIR regions. Use this model to calculate the properties under the conditions of Example 5.
Solution: The spectral transmittance and reflectance of clear 3 mm glass are shown in Figure 20. [The source of these data is “generic” clear glass in the NFRC (2000b) spectral data library.] It can be seen af θ λ , ( ) 1 T θ λ , ( ) – Rf θ λ , ( ) – = ab θ λ , ( ) 1 T θ λ , ( ) – Rb θ λ , ( ) – = ρs θ λ , ( ) θ ζ – ( ) sin θ ζ + ( ) sin ------------------------- 2 = ρp θ λ , ( ) θ ζ – ( ) tan θ ζ + ( ) tan ------------------------- 2 = θ sin n ζ and sin ζ arc θ sin n ----------- sin = = ρ 0 λ , ( ) n λ ( ) 1 – [ ]2 n λ ( ) 1 + [ ]2 ---------------------------- and n λ ( ) 1 ρ 0 λ , ( ) + 1 ρ 0 λ , ( ) – --------------------------------= = τµ θ λ , ( ) ρµ θ λ , ( ) + 1 = TDSG θ ( ) ts ( )n θ n cos n= 0 5 ∑ = aDSG f θ ( ) aDSG b θ ( ) as ( )n θ n cos n= 0 5 ∑ = = TDSG 〈 〉D 2 ts ( )n n 2 + ------------n= 0 5 ∑ = Table 11 Coefficients for Double-Strength Glass (DSG) for Calculation of Transmittance and Absorptance n (as)n (ts)n 0 0.01154 −0.00885 1 0.77674 2.71235 2 −3.94657 −0.62062 3 8.57881 −7.07329 4 −8.38135 9.75995 5 3.01188 −3.89922 aDSG f 〈 〉D aDSG b 〈 〉 = D 2 as ( )n n 2 + -------------n= 0 5 ∑ = T 0 λ , ( ) τ 0 λ , ( ) [ ]2e α λ ( )d – 1 ρ 0 λ , ( ) [ ]2e 2α λ ( )d – – -------------------------------------------------------= R 0 λ , ( ) ρ 0 λ , ( ) 1 T 0 λ , ( )e α λ ( )d – + [ ] = τ 0 λ , ( ) ρ 0 λ , ( ) + 1 = ρ 0 λ , ( ) P P2 4 2 R 0 λ , ( ) – [ ]R 0 λ , ( ) – – 2 2 R 0 λ , ( ) – [ ] --------------------------------------------------------------------------------= P T 0 λ , ( ) [ ]2 R 0 λ , ( ) [ ]2 – 2R 0 λ , ( ) + = α λ ( ) 1 d --R 0 λ , ( ) ρ 0 λ , ( ) – ρ 0 λ , ( )T 0 λ , ( ) -----------------------------------------ln – = n λ ( ) 1 ρ 0 λ , ( ) + 1 ρ 0 λ , ( ) – -------------------------------= 30.22 2001 ASHRAE Fundamentals Handbook (SI) that the variation with wavelength is not very great, and although a sin-gle average transmittance and reflectance would be adequate, for use with spectrally selective glazings it is better to construct the “2-band” model also shown in the figure, which characterizes the properties sep-arately for the visible region (320 to 780 nm wavelength) and the NIR region (780 to 2500 nm). About 99% of the ASTM solar spectrum lies within these two regions. The corresponding normal incidence proper-ties are T(0,vis) = 0.876, R(0,vis) = 0.081, T(0,NIR) = 0.791, and R(0,NIR) = 0.069.
From Equation (51), one then calculates P(vis) = 1.922 and P(NIR) = 1.758. Equation (50) yields surface reflectances of ρ(0,vis) = 0.44 and ρ(0,NIR) = 0.041, and Equation (53) gives refractive indices for the two wavelength regions of n(vis) = 1.530 and n(NIR) = 1.507. From Equa-tion (52), we obtain the values α(vis) = 0.0149 and α(NIR) = 0.0508.
These are the parameters necessary for calculating the optical properties.
For the conditions of Example 5, the incident angle is found in Example 6 to be 63.2°. At this incident angle, Equation (40) gives refracted angles of ζ(vis) = 35.7° and ζ(NIR) = 36.3°. For polarization s, Equation (39a) gives ρs(63.2°,vis) = 0.218, ρs(63.2°,NIR) = 0.210, and for polarization p, Equation (39b) gives ρp(63.2°,vis) = 0.0066, and ρp(63.2°,NIR) = 0.0072. Putting these values into Equations (32) through (34) gives for the transmittances and reflectances for each polarization: sT(63.2°,vis) = 0.604, sR(63.2°,vis) = 0.343, sT(63.2°,NIR) = 0.533, sR(63.2°,NIR) = 0.303, pT(63.2°,vis) = 0.934, pR(63.2°,vis) = 0.012, pT(63.2°,NIR) = 0.816, and pR(63.2°,NIR) = 0.012. Note that the reflected radiation is almost completely polarized. The unpolarized quantities are then calculated from Equations (35) and (36): T(63.2°,vis) = 1/2(0.604 + 0.934) = 0.769 and R(63.2°,vis) = 1/2 (0.343 + 0.012) = 0.178, and, similarly, T(63.2°,NIR) = 0.674 and R(63.2°,NIR) = 0.157.
Coated Glazings While in principle the equations in the preceding sections could be used to calculate the properties of coated glazings, this is not currently practical. To obtain the necessary basic information about the structure of complex coatings would require spectropho-tometric measurements at angles other than normal incidence. The instruments for making such measurements are not widely avail-able nor are there yet standardized procedures for making the mea-surements and extracting coating properties from them.
Until such capabilities are available, the following approximate procedure should be used to model coated glazings (Finlayson and Arasteh 1993). Coated glazing properties should vary from these estimates by no more than ±20% at 60° incidence (Rubin et al.
1999).
The spectral transmittance and reflectances at any incident angle are approximated from those at normal incidence by (54) (55) where (56) and (57) The constants in Equations (56) and (57) are selected from the entries in Table 12 appropriate to the value of the spectrally aver-aged transmittance at normal incidence, T(0). Since Equations (54) and (55) make wavelength-independent modifications to the prop-erties at normal incidence, the spectral averaging may be done first, yielding (58) (59) Example 9. Construct an approximate model of the optical properties of a single layer of 3 mm clear glass with a selective low-emissivity coating on surface 2. Use this model to calculate the properties under the condi-tions of Example 5.
Solution: The spectral transmittance and reflectances of selective 3 mm glass at normal incidence are shown in Figure 21. [The source of these data is “EE-170-3.CIG” glass in the NFRC (2000b) spectral data library.] The 2-band model shown in the figure has the following nor-mal-incidence spectral properties: T(0,vis) = 0.657, Rf(0,vis) = 0.132, Rb(0,vis) = 0. 122, T(0,NIR) = 0.088, Rf(0,NIR) = 0.614, and Rb(0,NIR) = 0.830. It can be seen from Figure 21 that this model crudely repre-sents the difference in glazing optical properties for the two spectral regions.
For the crude optical model available for coated glass, the angular dependence enters only through TREF and RREF. Following the method of Example 11 to average the spectral properties, one finds that the spec-tral average transmittance at normal incidence for this glazing, T(0), is 0.379. Since this is less than 0.645, constants for computing these two functions using Equations (56) and (57) are taken from the lower two rows of Table 12. At an incident angle of 63.2° (from Example 6), these functions are TREF(63.2°) = 0.810 and RREF(63.2°) = 0.081. The trans-mittance and reflectances are then calculated for each wavelength region using Equations (58) and (59): Table 12 Polynomial Coefficients for Calculation of Reference Angular Functions for Coated Glazings Condition m 0 1 2 3 4 T(0) > 0.645 TA −0.0015 3.355 −3.840 1.460 0.0288 RA 0.999 −0.563 2.043 −2.532 1.054 T(0) ≤ 0.645 TA −0.002 2.813 −2.341 −0.05725 0.599 RA 0.997 −1.868 6.513 −7.862 3.225 Fig. 20 Transmittance and Reflectance of Clear 3 mm Glass Approximated by a 2-Band Model of Spectrally Weighted Transmittance and Reflectance T θ λ , ( ) T 0 λ , ( )TREF θ ( ) = Ri θ λ , ( ) Ri 0 λ , ( ) 1 RREF θ ( ) – [ ] RREF θ ( ) i f b , = ( ) + = TREF θ ( ) TA ( )m θ m cos m= 0 4 ∑ = RREF θ ( ) RA ( )m θ TREF θ ( ) – m cos m= 0 4 ∑ = T θ ( ) T 0 ( ) TREF θ ( ) [ ] = Ri θ ( ) Ri 0 ( ) 1 RREF θ ( ) – [ ] RREF θ ( ) i f b , = ( ) + = Fenestration 30.23 T(63.2°,vis) = 0.532, Rf(63.2°,vis) = 0.202, Rb(63.2°,vis) = 0.193, T(63.2°,NIR) = 0.071, Rf(63.2°,NIR) = 0.646, and Rb(63.2°,NIR) = 0.844 Optical Properties of Multiple-Layer Glazing Systems For the optical properties of glazing systems consisting of mul-tiple glazing layers, interreflections may occur between layers, which means that the effect of a particular layer on the overall prop-erties may depend on its position within the assembly as well as on the transmittance and reflectances of the particular layer. We must therefore expand the glazing layer considerations above to apply to the overall properties of systems and subsystems of glazing layers.
The notation for doing this is illustrated in Figure 22.
In the following section, polarization indices are omitted from the equations. In principle, these equations should be used to make separate calculations of the optical properties of the glazing system for each of the two polarizations, s and p, and the results should be averaged using Equations (35) through (37). This should be done if a highly accurate result is desired (and data are available). However, since accurate data on coated glazings are lacking and polarization effects are relatively small, the polarization-averaged quantities will be used for each layer. This is the approximation used in commonly available computer calculations such as Wright (1995b) and LBL (1994).
The position of each layer in a multilayer glazing system consist-ing of L layers is characterized by its layer number n as shown in Figure 22. (By convention, layer 1 is the layer closest to the sun.) The individual layer transmittances, reflectances, and absorptances for the nth layer (the properties of the layer when isolated in air) are denoted by adding a single subscript to the property symbol: Tn, Rf n, Rb n, af n and ab n. The properties of a subsystem consisting of the lay-ers n through m, inclusive (n and m may be written as capital letters N and M to avoid confusion and emphasize their role in specifying a subsystem), are denoted by symbols with two indices for transmit-tances and reflectances and by script capital letters for absorptances (A): Tn(θ,λ) = isolated-layer transmittance of the nth layer (in an L-layer system) TN,M(θ,λ) = transmittances of the subsystem consisting of layers N through M (in an L-layer system) and similarly for reflectances, while a f n(θ,λ) = isolated-layer front absorptance for the nth layer (in an L-layer system) A f n:(N,M)(θ,λ) = actual front absorptance for the nth layer in the subsystem consisting of layers N through M (in an L-layer system); the fraction of the radiation incident on layer N that is absorbed in layer n, including effects of multiple reflections from layers N through M Ab n:(N,M)(θ,λ) = actual back absorptance for the nth layer in the subsystem consisting of layers N through M (in an L-layer system); the fraction of the (backward-going) radiation incident on layer M that is absorbed in layer n, including effects of multiple reflections from layers N through M. A quantity such as T1,L(θ,λ) refers to the total overall system property. Note that A f n:(n,n)(θ,λ) = a f n and Ab n:(n,n)(θ,λ) = ab n. (A sub-system consisting of one layer is the same as an isolated layer.) The properties of any subsystem can be calculated by use of the following recursion relations and proceeding from left to right in Figure 22 (Finlayson and Arasteh 1993): (60) (61) (62) where it is always the case that m ≥ n, and a subsystem consisting of one layer is the same as an isolated layer [e.g., Tn,n(θ,λ) ≡ Tn(θ,λ)].
These equations allow one to build up the properties of the L-layer system by beginning with the isolated properties of the first layer and successively adding additional layers. The absorptance of the nth layer in the system is then calculated from Fig. 21 Transmittance and Reflectance at Normal Incidence of a Selective Low-e Glass Approximated by a 2-Band Model of Spectrally Weighted Transmittance and Reflectance Fig. 22 Multilayer Glazings Considered as Systems and Subsystems Tn m 1 + , θ λ , ( ) Tn m , θ λ , ( )Tm 1 + θ λ , ( ) 1 Rn m , b θ λ , ( )Rm 1 + f θ λ , ( ) – -----------------------------------------------------------------= Rn m 1 + , f θ λ , ( ) Rn m , f θ λ , ( ) Tn m , θ λ , ( ) [ ]2Rm 1 + b θ λ , ( ) 1 Rn m , b θ λ , ( )Rm 1 + f θ λ , ( ) – -----------------------------------------------------------------+ = Rn m 1 + , b θ λ , ( ) Rm 1 + b θ λ , ( ) Tm 1 + θ λ , ( ) [ ]2Rn m , b θ λ , ( ) 1 Rn m , b θ λ , ( )Rm 1 + f θ λ , ( ) – -----------------------------------------------------------------+ = 30.24 2001 ASHRAE Fundamentals Handbook (SI) (63) or (64) Note that in Equations (63) and (64), combinations of subscripts will arise for the first and last layer absorptances that refer to non-existent layers (the layer before the first or after the last). Rather than write separate equations for these special cases, we define the nonexistent layers to have transmittance of one and reflectance of zero: (65) Example 10. Calculate the properties of a double-glazed unit having the coated glass of Example 9 as an outer glass and the clear glass of Example 8 as an inner glass, under the conditions of Example 5.
Solution: From Example 9, the optical properties of the outer glazing, which is layer 1, at 63.2° (Example 6) are T1(63.2°,vis) = 0.532, (63.2°,vis) = 0.202, and (63.2°,vis) = 0.193 for the visible region and T1(63.2°,NIR) = 0.071, (63.2°,NIR) = 0.646, and (63.2°,NIR) = 0.844 for the NIR region. From Example 8, the corresponding (unpo-larized) values for the inner glazing (layer 2) are T2(63.2°,vis) = 0.769 and (63.2°,vis) = (63.2°,vis) = 0.178 for the visible region, and T2(63.2°,NIR) = 0.674 and (63.2°,NIR) = (63.2°,NIR) = 0.157 for the NIR region. Inserting these values into Equation (60) gives the following for the glazing system’s overall transmittance: and Similarly, Equation (61) gives (63.2°,vis) = 0.202 + (0.532)2 × (0.178)/[1 – (0.193)(0.178)] = 0.254 and (63.2°,NIR) = 0.446 for the front reflectances in the two spectral regions, and Equation (62) gives the back reflectances, (63.2°,vis) = 0.178 + (0.769)2 × (0.193)/[1 – (0.193)(0.178)] = 0.296 and (63.2°,NIR) = 0.600. In order to cal-culate the layer absorptances in the glazing system, we must first calcu-late the isolated-layer absorptances for both glazings using Equations (38c) and (38d): We then use Equations (63) and (65) to calculate the layer front absorp-tances in the glazing system: and Equations (64) and (65) to calculate the back absorptances: (63.2°,vis) = 0.219, (63.2°,vis) = 0.061, (63.2°,NIR) = 0.066, and (63.2°,NIR) = 0.279.
Spectral Averaging of Glazing Properties For calculating fenestration heat transfer, it is generally suffi-cient to use spectrally averaged glazing system properties as defined in Equation (30) or (31). While for special combinations of climate and location it may be desirable to carry out this averaging using variant solar spectra as weighting functions in these equations, as discussed in the section on Solar Radiation under Determining Inci-dent Solar Flux, it is seldom either feasible or necessary to carry out heat transfer calculations using a detailed, time-dependent solar spectrum and the spectral glazing properties.
For multiple-layer glazing systems, the spectral averaging of Equation (30) or (31) should in general be applied to the system spectral properties at each angle [Equations (60) through (64)].
Since all glazing layer properties are to some extent both angle and wavelength dependent, and since these equations are nonlinear in the glazing properties, this is the only procedure that is valid in prin-ciple. In general, when a glazing property is used without a specific function reference to wavelength, it assumes that this averaging procedure has been carried out. For example, for a multiple-layer glazing, (66) where Tn,m(θ, λ) has been obtained from the individual layer prop-erties and the application of Equation (60). The analogous defini-tions apply for , , , and and the other wavelength-dependent quantities appearing in the An: 1 L , ( ) f θ λ , ( ) T1 n 1 – , θ λ , ( )an f θ λ , ( ) 1 R1 n 1 – , b θ λ , ( )Rn L , f θ λ , ( ) – -------------------------------------------------------------------= T1 n , θ λ , ( )Rn 1 L , + f θ λ , ( )an b θ λ , ( ) 1 R1 n , b θ λ , ( )Rn 1 + L , f θ λ , ( ) – -------------------------------------------------------------------------------+ An: 1 L , ( ) b θ λ , ( ) Tn 1 L , + θ λ , ( )an b θ λ , ( ) 1 R1 n , b θ λ , ( )Rn 1 + L , f θ λ , ( ) – --------------------------------------------------------------------= Tn L , θ λ , ( )R1 n 1 – , f θ λ , ( )an f θ λ , ( ) 1 R1 n , f θ λ , ( )Rn 1 + L , b θ λ , ( ) – ------------------------------------------------------------------------------+ T1 0 , TL 1 L , + 1 = = RL 1 L , + f RL 1 L , + b R1 0 , f R1 0 , b 0 = = = = R1 f R1 b R1 f R1 b R2 f R2 b R2 f R2 b T1 2 , 63.2°,vis ( ) T1 63.2°,vis ( )T2 63.2°,vis ( ) 1 R1 b 63.2°,vis ( )R2 f 63.2°,vis ( ) – ------------------------------------------------------------------------------= 0.532 ( ) 0.769 ( ) 1 0.193 ( ) 0.178 ( ) – ----------------------------------------------0.424 = = T1 2 , 63.2°,NIR ( ) T1 63.2°,NIR ( )T2 63.2°,NIR ( ) 1 R1 b 63.2°,NIR ( )R2 f 63.2°,NIR ( ) – ------------------------------------------------------------------------------------= 0.071 ( ) 0.674 ( ) 1 0.844 ( ) 0.157 ( ) – ----------------------------------------------0.055 = = R1 2 , f R1 2 , f R1 2 , b R1 2 , b a1 f 63.2°,vis ( ) 1 0.532 – 0.202 – 0.266 = = a1 b 63.2°,vis ( ) 0.275 = a1 f 63.2° NIR , ( ) 0.283 = a1 b 63.2° NIR , ( ) 0.085 = a2 f 63.2°,vis ( ) a2 b 63.2°,vis ( ) 0.053 = = a2 f 63.2° NIR , ( ) a2 b 63.2° NIR , ( ) 0.165 = = A1: 1 2 , ( ) f 63.2°,vis ( ) a1 f 63.2°,vis ( ) = T1 63.2°,vis ( )R2 f 63.2°,vis ( )a1 b 63.2°,vis ( ) 1 R1 b 63.2°,vis ( )R2 f 63.2°,vis ( ) – ------------------------------------------------------------------------------------------------------+ 0.266 0.532 ( ) 0.178 ( ) 0.275 ( ) 1 0.193 ( ) 0.178 ( ) – --------------------------------------------------------+ 0.293 = = A2: 1 2 , ( ) f 63.2° vis , ( ) T1 63.2°,vis ( )a2 b 63.2°,vis ( ) 1 R1 b 63.2°,vis ( )R2 f 63.2°,vis ( ) – ------------------------------------------------------------------------------= 0.532 ( ) 0.053 ( ) 1 0.193 ( ) 0.178 ( ) – ----------------------------------------------0.029 = = A1: 1 2 , ( ) f 63.2°,NIR ( ) 0.285 = A2:1 2 , f 63.2°,NIR ( ) 0.014 = A1: 1 2 , ( ) b A2: 1 2 , ( ) b A1: 1 2 , ( ) b A2: 1 2 , ( ) b Tn m , θ ( ) ESTD λ ( )Tn m , θ λ , ( ) λ d λmin λmax ∫ ESTD λ ( ) λ d λmin λmax ∫ -------------------------------------------------------------------= Rn m , f θ ( ) Rn m , b θ ( ) An: 1 L , ( ) f θ ( ) An: 1 L , ( ) b θ ( ) Fenestration 30.25 section on Optical Properties of Multiple-Layer Glazing Systems.
Unless otherwise stated, the standard spectrum, ESTD(λ), used in the calculation is the ASTM Standard E 891 spectrum.
However, as explained in this section, many glazings have properties that vary little over the wavelength region spanned by the solar spectrum. For systems containing only these glazings, much computational labor can be saved by calculating the wave-length-averaged properties (as a function of incident angle) for each isolated glazing layer and using these properties in Equa-tions (60) through (64). While this procedure is less accurate, it may prove adequate for systems involving clear and nearly clear glazings. In using it, one should, however, examine the spectral properties of the glazings to ensure that the potential errors in the approximation are understood. One should particularly note Fig-ures 15, 16, 23, and 24. While clear architectural glazings are generally engineered to have constant spectral properties in the visible region to avoid objectionable coloration, this may not apply to the substantial part of the solar spectrum outside the vis-ible region.
The values of T1,m(θ), , and are tabulated for a number of common single (m = 1), double (m = 2), and tri-ple (m = 3) glazing systems in Table 13, and these may be used in combination with the wavelength-averaged versions of Equations (63), (64), (38c), and (38d). One obtains the wavelength-aver-aged equation by simply dropping the references to wavelength in the equations. At the level of accuracy implied by the use of Table 13, it will be sufficient to use the wavelength-averaged properties listed in the table for layers or subsystems in Equa-tions (63) and (64) to calculate the layer absorptances in the sys-tem. For example, if one were interested in a double-glazed system with a tinted and a clear glass, the properties listed in Table 13 for single-glazed clear and single-glazed tinted glass could be used for the layer properties in calculating the in-system layer absorptances. A problem arises for some low-e coated glaz-ings, which are not listed in Table 13 as single glazings because they are never utilized that way (due to the fragility of the coat-ing). In this case, one must use the formulas or consult the NFRC (2000b) data library and its associated (free) computer software to obtain the layer properties.
Even when some glazings have a significant spectral depen-dence, it may be possible to simplify the calculation by evaluating Equation (31) over only a small number of selected wavelengths.
For example, a common situation is a glazing system consisting primarily of clear glazing layers but containing one “selective” or “hot climate” low-e glazing layer (see Figure 24 and associated text). This glazing layer will have very different properties in the visible and near-infrared parts of the solar spectrum. One could, therefore, calculate its spectral average separately over the two spectral regions and utilize the center wavelengths of the two regions in Equation (31)—a “two-band” model. If the other glaz-ings in the system have properties that can be considered constant within each of the two spectral regions, a model of this type will generally produce adequate results.
Example 11. Calculate the spectral average properties of the glazings in Examples 8, 9, and 10. How do the results using a 2-band model compare with those for a single-band model? How do the results obtained from the formulas compare with those obtained by linearly interpolating the values in Table 13?
Solution: One must first calculate the spectral weights to be applied to the 2-band model. This is done by integrating the ASTM spectrum (numerically) to determine the fraction of energy contained in the vis-ible (320 to 780 nm) and NIR (780 to 2500 nm) bands. Energy out-side of either band is neglected (about 1.4% of the solar spectrum).
These weights are wvis = 0.512 and wNIR = 0.488. The spectral aver-age properties are then calculated using Equation (31). For the selec-tive glazing of Example 9, the spectral average properties at 63.2° are T1(63.2°) = (0.512)(0.532) + (0.488)(0.071) = 0.307, (63.2°) = 0.419, (63.2°) = 0.511, (63.2°) = 0.274, and (63.2°) = 0.182.
For the clear glazing of Example 8, the spectral average properties are T2(63.2°) = 0.723, (63.2°) = (63.2°) = 0.168, and (63.2°) = (63.2°) = 0.107. For the double-glazed system of Example 10, the spectral average properties are T1,2(63.2°) = 0.244, (63.2°) = 0.446, (63.2°) = 0.444, (63.2°) = 0.289, (63.2°) = 0.144, (63.2°) = 0.022, and (63.2°) = 0.167.
To obtain the results for a single-band model, we repeat the calcula-tions of Example 10 using the spectral average quantities for the two glazing layers instead of the separate values for the visible and NIR bands. When we do this, we obtain the spectral average properties T1,2 (63.2°) = 0.243, (63.2°) = 0.436, (63.2°) = 0.460, (63.2°) = 0.285, (63.2°) = 0.144, (63.2°) = 0.037, and (63.2°) = 0.153. As can be seen, the results for the two meth-ods are not identical but are relatively close together for this glazing system.
The exact glazing system used in this example is not listed in Table 13 (as will frequently be the case in practice), so we must locate the glaz-ing system that is closest in properties. The emissivity of the selective coating on surface 2 of the outer glazing is 0.05, and the transmittance and reflectances for the insulating glass unit at normal incidence are T(0°) = 0.331, Rf(0°) = 0.388, and Rb(0°) = 0.396. The most similar unit listed in Table 13 is ID 25a, which has T(0°) = 0.37, Rf(0°) = 0.35, and Rb(0°) = 0.39 (i.e., slightly lower front reflectance and therefore higher transmittance).
Near the required angle of incidence, the table values listed are T(60°) = 0.29, Rf(60°) = 0.40, Rb(60°) = 0.43 and T(70°) = 0.22, Rf(70°) = 0.47, and Rb(70°) = 0.50. (The subscripts 1,2 are understood from the system description.) Linear interpolation in angle yields the values [T1,2 (63.2°)]25a = 0.27, [ (63.2°)]25a = 0.42, and [ (63.2°)]25a = 0.45.
Since the normal incidence transmittance of ID 25a in the table was about 0.04 higher than that of the glazing system in which we are interested (and the front reflectance correspondingly 0.04 lower), we might correct the above calculation by multiplying by the ratio of the normal incidence value for the desired glazing to that of table entry 25a. When we do this, we obtain the values [T1,2(63.2°)]table = 0.27 × 0.331/0.37 = 0.24, [ (63.2°)]table = 0.42 × 0.388/0.35 = 0.47, and [ (63.2°)]table = 0.45 × 0.396/0.39 = 0.46. These values are quite close to the ones obtained above from use of the exact formulas.
Angular Averaging of Glazing Properties As stressed in the previous sections, the solar-optical properties of glazings depend on the incident angle of the radiation passing through the glazing. It is relatively simple to account for this depen-dence in the case of beam solar radiation, since at any particular time the radiation can be considered to be incident from a single direction, and from this direction the incident angle can be calcu-lated as described in the section on Determining Solar Angle under Determining Solar Incident Flux. However, for the diffuse solar and ground-reflected radiation, the situation is more complicated. In principle, the energy flow through the glazing should be a sum over the individual energy flows resulting from incident radiation from each direction; for example, the radiant flux directly transmitted through an L-layer glazing due to sky and ground radiation would be (67) where λ denotes solar wavelengths and hem stands for hemispheri-cal. Esky, ground(θ, φ, λ) is the spectral radiance of the portion of the sky viewed in the direction (θ, φ) from the glazing, and dϖ is the infinitesimal element of solid angle corresponding to this direction.
While such calculations can be carried out for specific sky condi-tions using detailed sky data or models, such as those cited in the text following Equation (12) in the section on Solar Radiation, the labor of such a calculation will be worthwhile only for very specific purposes. More generally, a drastic simplifying assumption, alluded to in Equation (12), is made. Both the sky and ground radiation are assumed to be ideally diffuse (i.e., to have a sky radiance that is R1 m , f θ ( ) R1 m , b θ ( ) R1 f R1 b a1 f a1 b R2 f R2 b a2 f a2 b R1 2 , f R1 2 , b A1: 1 2 , ( ) f A1: 1 2 , ( ) b A2: 1 2 , ( ) f A2: 1 2 , ( ) b R1 2 , f R1 2 , b A1: 1 2 , ( ) f A1: 1 2 , ( ) b A2: 1 2 , ( ) f A2: 1 2 , ( ) b R1 2 , f R1 2 , b R1 2 , f R1 2 , b qtrans Esky ground , θ φ λ , , ( )T1 L , θ λ , ( ) θ cos ( ) ϖ d λ d ∫ hem ∫ λ ∫ = 30.26 2001 ASHRAE Fundamentals Handbook (SI) Table 13 Visible Transmittance (Tv), Solar Heat Gain Coefficient (SHGC), Solar Transmittance (T), Front Reflectance (Rf), Back Reflectance (Rb), and Layer Absorptances ( ) for Glazing and Window Systems ID Glazing System Center Glazing Tv Center-of-Glazing Properties Total Window SHGC at Normal Incidence Total Window Tv at Normal Incidence Incidence Angles Aluminum Other Frames Aluminum Other Frames Glass Thick., mm Normal 0.00 40.00 50.00 60.00 70.00 80.00 Hemis., Diffuse Operable Fixed Operable Fixed Operable Fixed Operable Fixed Uncoated Single Glazing 1a 3 CLR 0.90 SHGC 0.86 0.84 0.82 0.78 0.67 0.42 0.78 0.75 0.78 0.64 0.75 0.77 0.80 0.66 0.78 T 0.83 0.82 0.80 0.75 0.64 0.39 0.75 R f 0.08 0.08 0.10 0.14 0.25 0.51 0.14 Rb 0.08 0.08 0.10 0.14 0.25 0.51 0.14 Af 1 0.09 0.10 0.10 0.11 0.11 0.11 0.10 1b 6 CLR 0.88 SHGC 0.81 0.80 0.78 0.73 0.62 0.39 0.73 0.71 0.74 0.60 0.71 0.75 0.79 0.64 0.77 T 0.88 0.87 0.85 0.80 0.69 0.43 0.80 R f 0.08 0.09 0.11 0.15 0.27 0.53 0.14 Rb 0.08 0.09 0.11 0.15 0.27 0.53 0.14 Af 1 0.16 0.17 0.18 0.19 0.19 0.17 0.17 1c 3 BRZ 0.68 SHGC 0.73 0.71 0.68 0.64 0.55 0.34 0.65 0.64 0.67 0.54 0.64 0.58 0.61 0.50 0.59 T 0.65 0.62 0.59 0.55 0.46 0.27 0.56 R f 0.06 0.07 0.08 0.12 0.22 0.45 0.12 Rb 0.06 0.07 0.08 0.12 0.22 0.45 0.12 Af 1 0.29 0.31 0.32 0.33 0.33 0.29 0.31 1d 6 BRZ 0.54 SHGC 0.62 0.59 0.57 0.53 0.45 0.29 0.54 0.54 0.56 0.46 0.54 0.45 0.48 0.39 0.47 T 0.49 0.45 0.43 0.39 0.32 0.18 0.41 R f 0.05 0.06 0.07 0.11 0.19 0.42 0.10 Rb 0.05 0.68 0.66 0.62 0.53 0.33 0.10 Af 1 0.46 0.49 0.50 0.51 0.49 0.41 0.48 1e 3 GRN 0.82 SHGC 0.70 0.68 0.66 0.62 0.53 0.33 0.63 0.62 0.64 0.52 0.61 0.70 0.73 0.60 0.71 T 0.61 0.58 0.56 0.52 0.43 0.25 0.53 R f 0.06 0.07 0.08 0.12 0.21 0.45 0.11 Rb 0.06 0.07 0.08 0.12 0.21 0.45 0.11 Af 1 0.33 0.35 0.36 0.37 0.36 0.31 0.35 1f 6 GRN 0.76 SHGC 0.60 0.58 0.56 0.52 0.45 0.29 0.54 0.53 0.55 0.45 0.53 0.65 0.68 0.55 0.66 T 0.47 0.44 0.42 0.38 0.32 0.18 0.40 R f 0.05 0.06 0.07 0.11 0.20 0.42 0.10 Rb 0.05 0.06 0.07 0.11 0.20 0.42 0.10 Af 1 0.47 0.50 0.51 0.51 0.49 0.40 0.49 1g 3 GRY 0.62 SHGC 0.70 0.68 0.66 0.61 0.53 0.33 0.63 0.62 0.64 0.52 0.61 0.52 0.55 0.45 0.54 T 0.61 0.58 0.56 0.51 0.42 0.24 0.53 R f 0.06 0.07 0.08 0.12 0.21 0.44 0.11 Rb 0.06 0.07 0.08 0.12 0.21 0.44 0.11 Af 1 0.33 0.36 0.37 0.37 0.37 0.32 0.35 1h 6 GRY 0.46 SHGC 0.59 0.57 0.55 0.51 0.44 0.28 0.52 0.53 0.54 0.44 0.52 0.39 0.41 0.34 0.40 T 0.46 0.42 0.40 0.36 0.29 0.16 0.38 R f 0.05 0.06 0.07 0.10 0.19 0.41 0.10 Rb 0.05 0.06 0.07 0.10 0.19 0.41 0.10 Af 1 0.49 0.52 0.54 0.54 0.52 0.43 0.51 1i 6 BLUGRN 0.75 SHGC 0.62 0.59 0.57 0.54 0.46 0.30 0.55 0.55 0.57 0.46 0.54 0.64 0.67 0.55 0.65 T 0.49 0.46 0.44 0.40 0.33 0.19 0.42 R f 0.06 0.06 0.07 0.11 0.20 0.43 0.11 Rb 0.06 0.06 0.07 0.11 0.20 0.43 0.11 Af 1 0.45 0.48 0.49 0.49 0.47 0.38 0.48 Reflective Single Glazing 1j 6 SS on CLR 8% 0.08 SHGC 0.19 0.19 0.19 0.18 0.16 0.10 0.18 0.18 0.18 0.15 0.17 0.07 0.07 0.06 0.07 T 0.06 0.06 0.06 0.05 0.04 0.03 0.05 R f 0.33 0.34 0.35 0.37 0.44 0.61 0.36 Rb 0.50 0.50 0.51 0.53 0.58 0.71 0.52 Af 1 0.61 0.61 0.60 0.58 0.52 0.37 0.57 1k 6 SS on CLR 14% 0.14 SHGC 0.25 0.25 0.24 0.23 0.20 0.13 0.23 0.23 0.24 0.19 0.22 0.12 0.12 0.10 0.12 T 0.11 0.10 0.10 0.09 0.07 0.04 0.09 R f 0.26 0.27 0.28 0.31 0.38 0.57 0.30 Rb 0.44 0.44 0.45 0.47 0.52 0.67 0.46 Af 1 0.63 0.63 0.62 0.60 0.55 0.39 0.60 An f Fenestration 30.27 1l 6 SS on CLR 20% 0.20 SHGC 0.31 0.30 0.30 0.28 0.24 0.16 0.28 0.28 0.29 0.24 0.27 0.17 0.18 0.15 0.17 T 0.15 0.15 0.14 0.13 0.11 0.06 0.13 R f 0.21 0.22 0.23 0.26 0.34 0.54 0.25 Rb 0.38 0.38 0.39 0.41 0.48 0.64 0.41 Af 1 0.64 0.64 0.63 0.61 0.56 0.40 0.60 1m 6 SS on GRN 14% 0.12 SHGC 0.25 0.25 0.24 0.23 0.21 0.14 0.23 0.23 0.24 0.19 0.22 0.10 0.11 0.09 0.10 T 0.06 0.06 0.06 0.06 0.04 0.03 0.06 R f 0.14 0.14 0.16 0.19 0.27 0.49 0.18 Rb 0.44 0.44 0.45 0.47 0.52 0.67 0.46 Af 1 0.80 0.80 0.78 0.76 0.68 0.48 0.75 1n 6 TI on CLR 20% 0.20 SHGC 0.29 0.29 0.28 0.27 0.23 0.15 0.27 0.27 0.27 0.22 0.26 0.17 0.18 0.15 0.17 T 0.14 0.13 0.13 0.12 0.09 0.06 0.12 R f 0.22 0.22 0.24 0.26 0.34 0.54 0.26 Rb 0.40 0.40 0.42 0.44 0.50 0.65 0.43 Af 1 0.65 0.65 0.64 0.62 0.57 0.40 0.62 1o 6 TI on CLR 30% 0.30 SHGC 0.39 0.38 0.37 0.35 0.30 0.20 0.35 0.35 0.36 0.30 0.34 0.26 0.27 0.22 0.26 T 0.23 0.22 0.21 0.19 0.16 0.09 0.20 R f 0.15 0.15 0.17 0.20 0.28 0.50 0.19 Rb 0.32 0.33 0.34 0.36 0.43 0.60 0.36 Af 1 0.63 0.65 0.64 0.62 0.57 0.40 0.62 Uncoated Double Glazing 5a 3 CLR CLR 0.81 SHGC 0.76 0.74 0.71 0.64 0.50 0.26 0.66 0.67 0.69 0.56 0.66 0.69 0.72 0.59 0.70 T 0.70 0.68 0.65 0.58 0.44 0.21 0.60 R f 0.13 0.14 0.16 0.23 0.36 0.61 0.21 Rb 0.13 0.14 0.16 0.23 0.36 0.61 0.21 Af 1 0.10 0.11 0.11 0.12 0.13 0.13 0.11 Af 2 0.07 0.08 0.08 0.08 0.07 0.05 0.07 5b 6 CLR CLR 0.78 SHGC 0.70 0.67 0.64 0.58 0.45 0.23 0.60 0.61 0.63 0.52 0.61 0.66 0.69 0.57 0.68 T 0.61 0.58 0.55 0.48 0.36 0.17 0.51 R f 0.11 0.12 0.15 0.20 0.33 0.57 0.18 Rb 0.11 0.12 0.15 0.20 0.33 0.57 0.18 Af 1 0.17 0.18 0.19 0.20 0.21 0.20 0.19 Af 2 0.11 0.12 0.12 0.12 0.10 0.07 0.11 5c 3 BRZ CLR 0.62 SHGC 0.62 0.60 0.57 0.51 0.39 0.20 0.53 0.55 0.57 0.46 0.54 0.53 0.55 0.45 0.54 T 0.55 0.51 0.48 0.42 0.31 0.14 0.45 R f 0.09 0.10 0.12 0.16 0.27 0.49 0.15 Rb 0.12 0.13 0.15 0.21 0.35 0.59 0.19 Af 1 0.30 0.33 0.34 0.36 0.37 0.34 0.33 Af 2 0.06 0.06 0.06 0.06 0.05 0.03 0.06 5d 6 BRZ CLR 0.47 SHGC 0.49 0.46 0.44 0.39 0.31 0.17 0.41 0.44 0.46 0.37 0.43 0.40 0.42 0.35 0.41 T 0.38 0.35 0.32 0.27 0.20 0.08 0.30 R f 0.07 0.08 0.09 0.13 0.22 0.44 0.12 Rb 0.10 0.11 0.13 0.19 0.31 0.55 0.17 Af 1 0.48 0.51 0.52 0.53 0.53 0.45 0.50 Af 2 0.07 0.07 0.07 0.07 0.06 0.04 0.07 5e 3 GRN CLR 0.75 SHGC 0.60 0.57 0.54 0.49 0.38 0.20 0.51 0.53 0.55 0.45 0.53 0.63 0.66 0.54 0.65 T 0.52 0.49 0.46 0.40 0.30 0.13 0.43 R f 0.09 0.10 0.12 0.16 0.27 0.50 0.15 Rb 0.12 0.13 0.15 0.21 0.35 0.60 0.19 Af 1 0.34 0.37 0.38 0.39 0.39 0.35 0.37 Af 2 0.05 0.05 0.05 0.04 0.04 0.03 0.04 5f 6 GRN CLR 0.68 SHGC 0.49 0.46 0.44 0.39 0.31 0.17 0.41 0.43 0.45 0.37 0.43 0.57 0.60 0.49 0.59 T 0.39 0.36 0.33 0.29 0.21 0.09 0.31 R f 0.08 0.08 0.10 0.14 0.23 0.45 0.13 Rb 0.10 0.11 0.13 0.19 0.31 0.55 0.17 Af 1 0.49 0.51 0.05 0.53 0.52 0.43 0.50 Af 2 0.05 0.05 0.05 0.05 0.04 0.03 0.05 Table 13 Visible Transmittance (Tv), Solar Heat Gain Coefficient (SHGC), Solar Transmittance (T), Front Reflectance (Rf), Back Reflectance (Rb), and Layer Absorptances ( ) for Glazing and Window Systems (Continued) ID Glazing System Center Glazing Tv Center-of-Glazing Properties Total Window SHGC at Normal Incidence Total Window Tv at Normal Incidence Incidence Angles Aluminum Other Frames Aluminum Other Frames Glass Thick., mm Normal 0.00 40.00 50.00 60.00 70.00 80.00 Hemis., Diffuse Operable Fixed Operable Fixed Operable Fixed Operable Fixed An f 30.28 2001 ASHRAE Fundamentals Handbook (SI) 5g 3 GRY CLR 0.56 SHGC 0.60 0.57 0.54 0.48 0.37 0.20 0.51 0.53 0.55 0.45 0.52 0.48 0.50 0.41 0.49 T 0.51 0.48 0.45 0.39 0.29 0.12 0.42 R f 0.09 0.09 0.11 0.16 0.26 0.48 0.14 Rb 0.12 0.13 0.15 0.21 0.34 0.59 0.19 Af 1 0.34 0.37 0.39 0.40 0.41 0.37 0.37 Af 2 0.05 0.06 0.06 0.xx 0.xx 0.xx 0.xx 5h 6 GRY CLR 0.41 SHGC 0.47 0.44 0.42 0.37 0.29 0.16 0.39 0.42 0.43 0.35 0.41 0.35 0.37 0.30 0.36 T 0.36 0.32 0.29 0.25 0.18 0.07 0.28 R f 0.07 0.07 0.08 0.12 0.21 0.43 0.12 Rb 0.10 0.11 0.13 0.18 0.31 0.55 0.17 Af 1 0.51 0.54 0.56 0.57 0.56 0.47 0.53 Af 2 0.07 0.07 0.07 0.06 0.05 0.03 0.06 5i 6 BLUGRN CLR 0.67 SHGC 0.50 0.47 0.45 0.40 0.32 0.17 0.43 0.45 0.46 0.38 0.44 0.57 0.60 0.49 0.58 T 0.40 0.37 0.34 0.30 0.22 0.10 0.32 R f 0.08 0.08 0.10 0.14 0.24 0.46 0.13 Rb 0.11 0.11 0.14 0.19 0.31 0.55 0.17 Af 1 0.47 0.49 0.50 0.51 0.50 0.42 0.48 Af 2 0.06 0.06 0.06 0.05 0.04 0.03 0.05 5j 6 HI-P GRN CLR 0.59 SHGC 0.39 0.37 0.35 0.31 0.25 0.14 0.33 0.35 0.36 0.30 0.34 0.50 0.53 0.43 0.51 T 0.28 0.26 0.24 0.20 0.15 0.06 0.22 R f 0.06 0.07 0.08 0.12 0.21 0.43 0.11 Rb 0.10 0.11 0.13 0.19 0.31 0.55 0.17 Af 1 0.62 0.65 0.65 0.65 0.62 0.50 0.63 Af 2 0.03 0.03 0.03 0.03 0.02 0.01 0.03 Reflective Double Glazing 5k 6 SS on CLR 8%, CLR 0.07 SHGC 0.13 0.12 0.12 0.11 0.10 0.06 0.11 0.13 0.13 0.11 0.12 0.06 0.06 0.05 0.06 T 0.05 0.05 0.04 0.04 0.03 0.01 0.04 R f 0.33 0.34 0.35 0.37 0.44 0.61 0.37 Rb 0.38 0.37 0.38 0.40 0.46 0.61 0.40 Af 1 0.61 0.61 0.60 0.58 0.53 0.37 0.56 Af 2 0.01 0.01 0.01 0.01 0.01 0.01 0.01 5l 6 SS on CLR 14%, CLR 0.13 SHGC 0.17 0.17 0.16 0.15 0.13 0.08 0.16 0.17 0.17 0.13 0.15 0.11 0.12 0.09 0.11 T 0.08 0.08 0.08 0.07 0.05 0.02 0.07 R f 0.26 0.27 0.28 0.31 0.38 0.57 0.30 Rb 0.34 0.33 0.34 0.37 0.44 0.60 0.36 Af 1 0.63 0.64 0.64 0.63 0.61 0.56 0.60 Af 2 0.02 0.02 0.02 0.02 0.02 0.02 0.02 5m 6 SS on CLR 20%, CLR 0.18 SHGC 0.22 0.21 0.21 0.19 0.16 0.09 0.20 0.21 0.21 0.17 0.20 0.15 0.16 0.13 0.16 T 0.12 0.11 0.11 0.09 0.07 0.03 0.10 R f 0.21 0.22 0.23 0.26 0.34 0.54 0.25 Rb 0.30 0.30 0.31 0.34 0.41 0.59 0.33 Af 1 0.64 0.64 0.63 0.62 0.57 0.41 0.61 Af 2 0.03 0.03 0.03 0.03 0.02 0.02 0.03 5n 6 SS on GRN 14%, CLR 0.11 SHGC 0.16 0.16 0.15 0.14 0.12 0.08 0.14 0.16 0.16 0.13 0.14 0.09 0.10 0.08 0.10 T 0.05 0.05 0.05 0.04 0.03 0.01 0.04 R f 0.14 0.14 0.16 0.19 0.27 0.49 0.18 Rb 0.34 0.33 0.34 0.37 0.44 0.60 0.36 Af 1 0.80 0.80 0.79 0.76 0.69 0.49 0.76 Af 2 0.01 0.01 0.01 0.01 0.01 0.01 0.01 5o 6 TI on CLR 20%, CLR 0.18 SHGC 0.21 0.20 0.19 0.18 0.15 0.09 0.18 0.20 0.20 0.16 0.19 0.15 0.16 0.13 0.16 T 0.11 0.10 0.10 0.08 0.06 0.03 0.09 R f 0.22 0.22 0.24 0.27 0.34 0.54 0.26 Rb 0.32 0.31 0.32 0.35 0.42 0.59 0.35 Af 1 0.65 0.66 0.65 0.63 0.58 0.41 0.62 Af 2 0.02 0.02 0.02 0.02 0.02 0.01 0.02 5p 6 TI on CLR 30%, CLR 0.27 SHGC 0.29 0.28 0.27 0.25 0.20 0.12 0.25 0.27 0.27 0.22 0.26 0.23 0.24 0.20 0.23 T 0.18 0.17 0.16 0.14 0.10 0.05 0.15 R f 0.15 0.15 0.17 0.20 0.29 0.51 0.19 Rb 0.27 0.27 0.28 0.31 0.40 0.58 0.31 Af 1 0.64 0.64 0.63 0.62 0.58 0.43 0.61 Af 2 0.04 0.04 0.04 0.04 0.03 0.02 0.04 Table 13 Visible Transmittance (Tv), Solar Heat Gain Coefficient (SHGC), Solar Transmittance (T), Front Reflectance (Rf), Back Reflectance (Rb), and Layer Absorptances ( ) for Glazing and Window Systems (Continued) ID Glazing System Center Glazing Tv Center-of-Glazing Properties Total Window SHGC at Normal Incidence Total Window Tv at Normal Incidence Incidence Angles Aluminum Other Frames Aluminum Other Frames Glass Thick., mm Normal 0.00 40.00 50.00 60.00 70.00 80.00 Hemis., Diffuse Operable Fixed Operable Fixed Operable Fixed Operable Fixed An f Fenestration 30.29 Low-e Double Glazing, e = 0.2 on surface 2 17a 3 LE CLR 0.76 SHGC 0.65 0.64 0.61 0.56 0.43 0.23 0.57 0.57 0.59 0.49 0.57 0.65 0.68 0.55 0.66 T 0.59 0.56 0.54 0.48 0.36 0.18 0.50 R f 0.15 0.16 0.18 0.24 0.37 0.61 0.22 Rb 0.17 0.18 0.20 0.26 0.38 0.61 0.24 Af 1 0.20 0.21 0.21 0.21 0.20 0.16 0.20 Af 2 0.07 0.07 0.08 0.08 0.07 0.05 0.07 17b 6 LE CLR 0.73 SHGC 0.60 0.59 0.57 0.51 0.40 0.21 0.53 0.53 0.55 0.45 0.53 0.62 0.65 0.53 0.64 T 0.51 0.48 0.46 0.41 0.30 0.14 0.43 R f 0.14 0.15 0.17 0.22 0.35 0.59 0.21 Rb 0.15 0.16 0.18 0.23 0.35 0.57 0.22 Af 1 0.26 0.26 0.26 0.26 0.25 0.19 0.25 Af 2 0.10 0.11 0.11 0.11 0.10 0.07 0.10 Low-e Double Glazing, e = 0.2 on surface 3 17c 3 CLR LE 0.76 SHGC 0.70 0.68 0.65 0.59 0.46 0.24 0.61 0.62 0.64 0.52 0.61 0.65 0.68 0.55 0.66 T 0.59 0.56 0.54 0.48 0.36 0.18 0.50 R f 0.17 0.18 0.20 0.26 0.38 0.61 0.24 Rb 0.15 0.16 0.18 0.24 0.37 0.61 0.22 Af 1 0.11 0.12 0.13 0.13 0.14 0.15 0.12 Af 2 0.14 0.14 0.14 0.13 0.11 0.07 0.13 17d 6 CLR LE 0.73 SHGC 0.65 0.63 0.60 0.54 0.42 0.21 0.56 0.57 0.59 0.49 0.57 0.62 0.65 0.53 0.64 T 0.51 0.48 0.46 0.41 0.30 0.14 0.43 R f 0.15 0.16 0.18 0.23 0.35 0.57 0.22 Rb 0.14 0.15 0.17 0.22 0.35 0.59 0.21 Af 1 0.17 0.19 0.20 0.21 0.22 0.22 0.19 Af 2 0.17 0.17 0.17 0.15 0.13 0.07 0.16 17e 3 BRZ LE 0.58 SHGC 0.57 0.54 0.51 0.46 0.35 0.18 0.48 0.51 0.52 0.43 0.50 0.49 0.52 0.42 0.50 T 0.46 0.43 0.41 0.36 0.26 0.12 0.38 R f 0.12 0.12 0.14 0.18 0.28 0.50 0.17 Rb 0.14 0.15 0.17 0.23 0.35 0.60 0.21 Af 1 0.31 0.34 0.35 0.37 0.38 0.35 0.34 Af 2 0.11 0.11 0.10 0.10 0.08 0.04 0.10 17f 6 BRZ LE 0.45 SHGC 0.45 0.42 0.40 0.35 0.27 0.14 0.38 0.40 0.42 0.34 0.40 0.38 0.40 0.33 0.39 T 0.33 0.30 0.28 0.24 0.17 0.07 0.26 R f 0.09 0.09 0.10 0.14 0.23 0.44 0.13 Rb 0.13 0.14 0.16 0.21 0.34 0.58 0.20 Af 1 0.48 0.51 0.52 0.54 0.53 0.45 0.50 Af 2 0.11 0.11 0.10 0.09 0.07 0.04 0.09 17g 3 GRN LE 0.70 SHGC 0.55 0.52 0.50 0.44 0.34 0.17 0.46 0.49 0.50 0.41 0.48 0.60 0.62 0.51 0.61 T 0.44 0.41 0.38 0.33 0.24 0.11 0.36 R f 0.11 0.11 0.13 0.17 0.27 0.48 0.16 Rb 0.14 0.15 0.17 0.23 0.35 0.60 0.21 Af 1 0.35 0.38 0.39 0.41 0.42 0.37 0.38 Af 2 0.11 0.10 0.10 0.09 0.07 0.04 0.09 17h 6 GRN LE 0.62 SHGC 0.42 0.40 0.38 0.34 0.26 0.13 0.36 0.38 0.39 0.32 0.37 0.53 0.55 0.45 0.54 T 0.29 0.26 0.24 0.21 0.15 0.06 0.23 R f 0.08 0.08 0.09 0.13 0.22 0.43 0.13 Rb 0.13 0.14 0.16 0.21 0.34 0.58 0.20 Af 1 0.56 0.59 0.61 0.61 0.59 0.48 0.58 Af 2 0.09 0.09 0.09 0.08 0.08 0.04 0.08 17i 3 GRY LE 0.51 SHGC 0.53 0.50 0.48 0.42 0.33 0.17 0.45 0.47 0.49 0.40 0.47 0.43 0.45 0.37 0.44 T 0.43 0.40 0.38 0.33 0.24 0.11 0.35 R f 0.11 0.11 0.13 0.17 0.27 0.48 0.16 Rb 0.14 0.15 0.17 0.22 0.35 0.60 0.21 Af 1 0.58 0.60 0.61 0.61 0.59 0.48 0.59 Af 2 0.08 0.08 0.08 0.08 0.07 0.04 0.08 Table 13 Visible Transmittance (Tv), Solar Heat Gain Coefficient (SHGC), Solar Transmittance (T), Front Reflectance (Rf), Back Reflectance (Rb), and Layer Absorptances ( ) for Glazing and Window Systems (Continued) ID Glazing System Center Glazing Tv Center-of-Glazing Properties Total Window SHGC at Normal Incidence Total Window Tv at Normal Incidence Incidence Angles Aluminum Other Frames Aluminum Other Frames Glass Thick., mm Normal 0.00 40.00 50.00 60.00 70.00 80.00 Hemis., Diffuse Operable Fixed Operable Fixed Operable Fixed Operable Fixed An f 30.30 2001 ASHRAE Fundamentals Handbook (SI) 17j 6 GRY LE 0.37 SHGC 0.39 0.37 0.35 0.31 0.24 0.13 0.33 0.35 0.36 0.30 0.34 0.31 0.33 0.27 0.32 T 0.27 0.25 0.23 0.20 0.14 0.06 0.21 R f 0.09 0.09 0.11 0.14 0.23 0.44 0.14 Rb 0.13 0.14 0.16 0.22 0.34 0.58 0.20 Af 1 0.55 0.58 0.59 0.59 0.58 0.48 0.56 Af 2 0.09 0.09 0.08 0.07 0.06 0.03 0.08 17k 6 BLUGRN LE 0.62 SHGC 0.45 0.42 0.40 0.36 0.28 0.14 0.38 0.40 0.42 0.34 0.40 0.53 0.55 0.45 0.54 T 0.22 0.20 0.18 0.15 0.11 0.07 0.17 R f 0.07 0.07 0.08 0.12 0.20 0.44 0.11 Rb 0.13 0.14 0.16 0.21 0.34 0.58 0.20 Af 1 0.48 0.51 0.53 0.54 0.54 0.45 0.51 Af 2 0.11 0.10 0.10 0.09 0.07 0.03 0.09 17l 6 HI-P GRN LE 0.55 0.241 0.34 0.32 0.30 0.27 0.21 0.11 0.29 0.31 0.32 0.26 0.30 0.47 0.49 0.40 0.48 T 0.22 0.19 0.18 0.15 0.10 0.04 0.17 R f 0.07 0.07 0.08 0.11 0.20 0.41 0.11 Rb 0.13 0.14 0.16 0.21 0.33 0.58 0.20 Af 1 — — — — — — — Af 2 — — — — — — — Low-e Double Glazing, e = 0.1 on surface 2 21a 3 LE CLR 0.75 SHGC 0.54 0.52 0.49 0.44 0.34 0.18 0.46 0.48 0.50 0.41 0.47 0.64 0.67 0.55 0.65 T 0.48 0.45 0.43 0.37 0.27 0.13 0.40 R f 0.24 0.24 0.26 0.29 0.38 0.58 0.28 Rb 0.27 0.27 0.28 0.32 0.42 0.62 0.31 Af 1 0.23 0.25 0.26 0.28 0.30 0.26 0.26 Af 2 0.06 0.06 0.06 0.06 0.05 0.04 0.06 21b 6 LE CLR 0.72 SHGC 0.51 0.49 0.47 0.42 0.32 0.17 0.44 0.45 0.47 0.38 0.45 0.61 0.64 0.53 0.63 T 0.42 0.40 0.37 0.32 0.24 0.11 0.35 R f 0.20 0.20 0.22 0.26 0.34 0.55 0.25 Rb 0.24 0.24 0.25 0.29 0.38 0.58 0.28 Af 1 0.30 0.32 0.32 0.33 0.35 0.29 0.32 Af 2 0.08 0.09 0.09 0.09 0.08 0.05 0.08 21l 6 HI-P GRN W/LE CLR 0.57 SHGC 0.31 0.30 0.29 0.26 0.21 0.12 0.27 0.28 0.29 0.24 0.27 0.48 0.51 0.42 0.50 T 0.22 0.21 0.19 0.17 0.12 0.06 0.18 R f 0.07 0.07 0.09 0.13 0.22 0.46 0.12 Rb 0.23 0.23 0.24 0.28 0.37 0.57 0.27 Af 1 0.67 0.68 0.67 0.66 0.62 0.46 0.65 Af 2 0.04 0.05 0.05 0.05 0.04 0.03 0.04 Low-e Double Glazing, e = 0.1 on surface 3 21c 3 CLR LE 0.75 SHGC 0.60 0.58 0.56 0.51 0.40 0.22 0.52 0.53 0.55 0.45 0.53 0.64 0.67 0.55 0.65 T 0.48 0.45 0.43 0.37 0.27 0.13 0.40 R f 0.26 0.27 0.28 0.32 0.42 0.62 0.31 Rb 0.24 0.24 0.26 0.29 0.38 0.58 0.28 Af 1 0.12 0.13 0.14 0.14 0.15 0.15 0.13 Af 2 0.14 0.15 0.15 0.16 0.16 0.10 0.15 21d 6 CLR LE 0.72 SHGC 0.56 0.55 0.52 0.48 0.38 0.20 0.49 0.50 0.51 0.42 0.49 0.61 0.64 0.53 0.63 T 0.42 0.40 0.37 0.32 0.24 0.11 0.35 R f 0.24 0.24 0.25 0.29 0.38 0.58 0.28 Rb 0.20 0.20 0.22 0.26 0.34 0.55 0.25 Af 1 0.19 0.20 0.21 0.22 0.23 0.22 0.21 Af 2 0.16 0.17 0.17 0.17 0.16 0.10 0.16 21e 3 BRZ LE 0.57 SHGC 0.48 0.46 0.44 0.40 0.31 0.17 0.42 0.43 0.44 0.36 0.42 0.48 0.51 0.42 0.50 T 0.37 0.34 0.32 0.27 0.20 0.08 0.30 R f 0.18 0.17 0.19 0.22 0.30 0.50 0.21 Rb 0.23 0.23 0.25 0.29 0.37 0.57 0.28 Af 1 0.34 0.37 0.38 0.39 0.39 0.35 0.37 Af 2 0.11 0.12 0.12 0.12 0.11 0.07 0.11 Table 13 Visible Transmittance (Tv), Solar Heat Gain Coefficient (SHGC), Solar Transmittance (T), Front Reflectance (Rf), Back Reflectance (Rb), and Layer Absorptances ( ) for Glazing and Window Systems (Continued) ID Glazing System Center Glazing Tv Center-of-Glazing Properties Total Window SHGC at Normal Incidence Total Window Tv at Normal Incidence Incidence Angles Aluminum Other Frames Aluminum Other Frames Glass Thick., mm Normal 0.00 40.00 50.00 60.00 70.00 80.00 Hemis., Diffuse Operable Fixed Operable Fixed Operable Fixed Operable Fixed An f Fenestration 30.31 21f 6 BRZ LE 0.45 SHGC 0.39 0.37 0.35 0.31 0.24 0.13 0.33 0.35 0.36 0.30 0.34 0.38 0.40 0.33 0.39 T 0.27 0.24 0.22 0.19 0.13 0.05 0.21 R f 0.12 0.12 0.13 0.16 0.24 0.44 0.16 Rb 0.19 0.20 0.22 0.25 0.34 0.55 0.24 Af 1 0.51 0.54 0.55 0.56 0.55 0.46 0.53 Af 2 0.10 0.10 0.10 0.10 0.09 0.05 0.10 21g 3 GRN LE 0.68 SHGC 0.46 0.44 0.42 0.38 0.30 0.16 0.40 0.41 0.42 0.34 0.40 0.58 0.61 0.50 0.59 T 0.36 0.32 0.30 0.26 0.18 0.08 0.28 R f 0.17 0.16 0.17 0.20 0.29 0.48 0.20 Rb 0.23 0.23 0.25 0.29 0.37 0.57 0.27 Af 1 0.38 0.41 0.42 0.43 0.43 0.38 0.40 Af 2 0.10 0.11 0.11 0.11 0.10 0.06 0.10 21h 6 GRN LE 0.61 SHGC 0.36 0.33 0.31 0.28 0.22 0.12 0.30 0.32 0.33 0.27 0.31 0.52 0.54 0.44 0.53 T 0.24 0.21 0.19 0.16 0.11 0.05 0.18 R f 0.11 0.10 0.11 0.14 0.22 0.43 0.14 Rb 0.19 0.20 0.22 0.25 0.34 0.55 0.24 Af 1 0.56 0.59 0.61 0.61 0.59 0.48 0.58 Af 2 0.09 0.09 0.09 0.08 0.08 0.04 0.08 21i 3 GRY LE 0.52 SHGC 0.46 0.44 0.42 0.38 0.30 0.16 0.39 0.41 0.42 0.35 0.40 0.44 0.46 0.38 0.45 T 0.35 0.32 0.30 0.25 0.18 0.08 0.28 R f 0.16 0.16 0.17 0.20 0.28 0.48 0.20 Rb 0.23 0.23 0.25 0.29 0.37 0.57 0.27 Af 1 0.39 0.42 0.43 0.44 0.44 0.38 0.41 Af 2 0.10 0.11 0.11 0.11 0.10 0.06 0.10 21j 6 GRY LE 0.37 SHGC 0.34 0.32 0.30 0.27 0.21 0.12 0.28 0.31 0.32 0.26 0.30 0.31 0.33 0.27 0.32 T 0.23 0.20 0.18 0.15 0.11 0.04 0.17 R f 0.11 0.11 0.12 0.15 0.23 0.44 0.15 Rb 0.20 0.20 0.22 0.25 0.34 0.55 0.24 Af 1 0.58 0.60 0.61 0.61 0.59 0.48 0.59 Af 2 0.08 0.08 0.08 0.08 0.07 0.04 0.08 21k 6 BLUGRN LE 0.62 SHGC 0.39 0.37 0.34 0.31 0.24 0.13 0.33 0.35 0.36 0.30 0.34 0.53 0.55 0.45 0.54 T 0.28 0.25 0.23 0.20 0.14 0.06 0.22 R f 0.12 0.12 0.13 0.16 0.24 0.44 0.16 Rb 0.23 0.23 0.25 0.28 0.37 0.57 0.27 Af 1 0.51 0.54 0.56 0.56 0.55 0.46 0.53 Af 2 0.08 0.09 0.08 0.08 0.08 0.05 0.08 Low-e Double Glazing, e = 0.05 on surface 2 25a 3 LE CLR 0.72 SHGC 0.41 0.40 0.38 0.34 0.27 0.14 0.36 0.37 0.38 0.31 0.36 0.61 0.64 0.53 0.63 T 0.37 0.35 0.33 0.29 0.22 0.11 0.31 R f 0.35 0.36 0.37 0.40 0.47 0.64 0.39 Rb 0.39 0.39 0.40 0.43 0.50 0.66 0.42 Af 1 0.24 0.26 0.26 0.27 0.28 0.23 0.26 Af 2 0.04 0.04 0.04 0.04 0.03 0.03 0.04 25b 6 LE CLR 0.70 SHGC 0.37 0.36 0.34 0.31 0.24 0.13 0.32 0.34 0.34 0.28 0.33 0.60 0.62 0.51 0.61 T 0.30 0.28 0.27 0.23 0.17 0.08 0.25 R f 0.30 0.30 0.32 0.35 0.42 0.60 0.34 Rb 0.35 0.35 0.35 0.38 0.44 0.60 0.37 Af 1 Af 2 25c 6 BRZ W/LE CLR 0.42 SHGC 0.26 0.25 0.24 0.22 0.18 0.10 0.23 0.24 0.25 0.20 0.23 0.36 0.37 0.31 0.37 T 0.18 0.17 0.16 0.14 0.10 0.05 0.15 R f 0.15 0.16 0.17 0.21 0.29 0.51 0.20 Rb 0.34 0.34 0.35 0.37 0.44 0.60 0.37 Af 1 0.63 0.63 0.63 0.61 0.57 0.42 0.60 Af 2 0.04 0.04 0.04 0.04 0.03 0.03 0.04 Table 13 Visible Transmittance (Tv), Solar Heat Gain Coefficient (SHGC), Solar Transmittance (T), Front Reflectance (Rf), Back Reflectance (Rb), and Layer Absorptances ( ) for Glazing and Window Systems (Continued) ID Glazing System Center Glazing Tv Center-of-Glazing Properties Total Window SHGC at Normal Incidence Total Window Tv at Normal Incidence Incidence Angles Aluminum Other Frames Aluminum Other Frames Glass Thick., mm Normal 0.00 40.00 50.00 60.00 70.00 80.00 Hemis., Diffuse Operable Fixed Operable Fixed Operable Fixed Operable Fixed An f 30.32 2001 ASHRAE Fundamentals Handbook (SI) 25d 6 GRN W/LE CLR 0.60 SHGC 0.31 0.30 0.28 0.26 0.21 0.12 0.27 0.28 0.29 0.23 0.27 0.51 0.53 0.44 0.52 T 0.22 0.21 0.20 0.17 0.13 0.06 0.18 R f 0.10 0.10 0.12 0.16 0.25 0.48 0.15 Rb 0.35 0.34 0.35 0.37 0.44 0.60 0.37 Af 1 0.64 0.64 0.64 0.63 0.59 0.43 0.62 Af 2 0.05 0.05 0.05 0.05 0.04 0.03 0.05 25e 6 GRY W/LE CLR 0.35 SHGC 0.24 0.23 0.22 0.20 0.16 0.09 0.21 0.23 0.23 0.19 0.21 0.30 0.31 0.26 0.30 T 0.16 0.15 0.14 0.12 0.09 0.04 0.13 R f 0.12 0.13 0.15 0.18 0.26 0.49 0.17 Rb 0.34 0.34 0.35 0.37 0.44 0.60 0.37 Af 1 0.69 0.69 0.68 0.67 0.62 0.45 0.66 Af 2 0.03 0.03 0.03 0.03 0.03 0.02 0.03 25f 6 BLUE W/LE CLR 0.45 SHGC 0.27 0.26 0.25 0.23 0.18 0.11 0.24 0.25 0.26 0.21 0.24 0.38 0.40 0.33 0.39 T 0.19 0.18 0.17 0.15 0.11 0.05 0.16 R f 0.12 0.12 0.14 0.17 0.26 0.49 0.16 Rb 0.34 0.34 0.35 0.37 0.44 0.60 0.37 Af 1 0.66 0.66 0.65 0.64 0.60 0.44 0.63 Af 2 0.04 0.04 0.04 0.04 0.04 0.03 0.04 25g 6 HI-P GRN W/LE CLR 0.53 SHGC 0.27 0.26 0.25 0.23 0.18 0.11 0.23 0.25 0.26 0.21 0.24 0.45 0.47 0.39 0.46 T 0.18 0.17 0.16 0.14 0.10 0.05 0.15 R f 0.07 0.07 0.09 0.13 0.22 0.46 0.12 Rb 0.35 0.34 0.35 0.38 0.44 0.60 0.37 Af 1 0.71 0.72 0.71 0.69 0.64 0.47 0.68 Af 2 0.04 0.04 0.04 0.04 0.03 0.02 0.04 Triple Glazing 29a 3 CLR CLR CLR 0.74 SHGC 0.68 0.65 0.62 0.54 0.39 0.18 0.57 0.60 0.62 0.51 0.59 0.63 0.66 0.54 0.64 T 0.60 0.57 0.53 0.45 0.31 0.12 0.49 R f 0.17 0.18 0.21 0.28 0.42 0.65 0.25 Rb 0.17 0.18 0.21 0.28 0.42 0.65 0.25 Af 1 0.10 0.11 0.12 0.13 0.14 0.14 0.12 Af 2 0.08 0.08 0.09 0.09 0.08 0.07 0.08 Af 3 0.06 0.06 0.06 0.06 0.05 0.03 0.06 29b 6 CLR CLR CLR 0.70 SHGC 0.61 0.58 0.55 0.48 0.35 0.16 0.51 0.54 0.56 0.46 0.53 0.60 0.62 0.51 0.61 T 0.49 0.45 0.42 0.35 0.24 0.09 0.39 R f 0.14 0.15 0.18 0.24 0.37 0.59 0.22 Rb 0.14 0.15 0.18 0.24 0.37 0.59 0.22 Af 1 0.17 0.19 0.20 0.21 0.22 0.21 0.19 Af 2 0.12 0.13 0.13 0.13 0.12 0.08 0.12 Af 3 0.08 0.08 0.08 0.08 0.06 0.03 0.08 29c 6 HI-P GRN CLR CLR 0.53 SHGC 0.34 0.31 0.29 0.25 0.19 0.10 0.28 0.31 0.32 0.26 0.30 0.45 0.47 0.39 0.46 T 0.20 0.17 0.15 0.12 0.07 0.02 0.15 R f 0.06 0.07 0.08 0.11 0.20 0.41 0.11 Rb 0.13 0.14 0.16 0.22 0.35 0.57 0.20 Af 1 0.64 0.67 0.68 0.68 0.66 0.53 0.65 Af 2 0.06 0.06 0.05 0.05 0.05 0.03 0.05 Af 3 0.04 0.04 0.04 0.03 0.02 0.01 0.04 Triple Glazing, e = 0.2 on surface 2 32a 3 LE CLR CLR 0.68 SHGC 0.60 0.58 0.55 0.48 0.35 0.17 0.51 0.53 0.55 0.45 0.53 0.58 0.61 0.50 0.59 T 0.50 0.47 0.44 0.38 0.26 0.10 0.41 R f 0.17 0.19 0.21 0.27 0.41 0.64 0.25 Rb 0.19 0.20 0.22 0.29 0.42 0.63 0.26 Af 1 0.20 0.20 0.20 0.21 0.21 0.17 0.20 Af 2 0.08 0.08 0.08 0.09 0.08 0.07 0.08 Af 3 0.06 0.06 0.06 0.06 0.05 0.03 0.06 Table 13 Visible Transmittance (Tv), Solar Heat Gain Coefficient (SHGC), Solar Transmittance (T), Front Reflectance (Rf), Back Reflectance (Rb), and Layer Absorptances ( ) for Glazing and Window Systems (Continued) ID Glazing System Center Glazing Tv Center-of-Glazing Properties Total Window SHGC at Normal Incidence Total Window Tv at Normal Incidence Incidence Angles Aluminum Other Frames Aluminum Other Frames Glass Thick., mm Normal 0.00 40.00 50.00 60.00 70.00 80.00 Hemis., Diffuse Operable Fixed Operable Fixed Operable Fixed Operable Fixed An f Fenestration 30.33 32b 6 LE CLR CLR 0.64 SHGC 0.53 0.50 0.47 0.41 0.29 0.14 0.44 0.47 0.49 0.40 0.47 0.54 0.57 0.47 0.56 T 0.39 0.36 0.33 0.27 0.17 0.06 0.30 R f 0.14 0.15 0.17 0.21 0.31 0.53 0.20 Rb 0.16 0.16 0.19 0.24 0.36 0.57 0.22 Af 1 0.28 0.31 0.31 0.34 0.37 0.31 0.31 Af 2 0.11 0.11 0.11 0.11 0.10 0.08 0.11 Af 3 0.08 0.08 0.08 0.07 0.05 0.03 0.07 Triple Glazing, e = 0.2 on surface 5 32c 3 CLR CLR LE 0.68 SHGC 0.62 0.60 0.57 0.49 0.36 0.16 0.52 0.55 0.57 0.46 0.54 0.58 0.61 0.50 0.59 T 0.50 0.47 0.44 0.38 0.26 0.10 0.41 R f 0.19 0.20 0.22 0.29 0.42 0.63 0.26 Rb 0.18 0.19 0.21 0.27 0.41 0.64 0.25 Af 1 0.11 0.12 0.13 0.14 0.15 0.15 0.13 Af 2 0.09 0.10 0.10 0.10 0.10 0.08 0.10 Af 3 0.11 0.11 0.11 0.10 0.08 0.04 0.10 32d 6 CLR CLR LE 0.64 SHGC 0.56 0.53 0.50 0.44 0.32 0.15 0.47 0.50 0.51 0.42 0.49 0.54 0.57 0.47 0.56 T 0.39 0.36 0.33 0.27 0.17 0.06 0.30 R f 0.16 0.16 0.19 0.24 0.36 0.57 0.22 Rb 0.14 0.15 0.17 0.21 0.31 0.53 0.20 Af 1 0.17 0.19 0.20 0.21 0.22 0.22 0.19 Af 2 0.13 0.14 0.14 0.14 0.13 0.10 0.13 Af 3 0.15 0.16 0.15 0.14 0.12 0.05 0.14 Triple Glazing, e = 0.1 on surface 2 and 5 40a 3 LE CLR LE 0.62 SHGC 0.41 0.39 0.37 0.32 0.24 0.12 0.34 0.37 0.38 0.31 0.36 0.53 0.55 0.45 0.54 T 0.29 0.26 0.24 0.20 0.13 0.05 0.23 R f 0.30 0.30 0.31 0.34 0.41 0.59 0.33 Rb 0.30 0.30 0.31 0.34 0.41 0.59 0.33 Af 1 0.25 0.27 0.28 0.30 0.32 0.27 0.28 Af 2 0.07 0.08 0.08 0.08 0.07 0.06 0.07 Af 3 0.08 0.09 0.09 0.09 0.07 0.04 0.08 40b 6 LE CLR LE 0.59 SHGC 0.36 0.34 0.32 0.28 0.21 0.10 0.30 0.33 0.34 0.27 0.32 0.50 0.53 0.43 0.51 T 0.24 0.21 0.19 0.16 0.10 0.03 0.18 R f 0.34 0.34 0.35 0.38 0.44 0.61 0.37 Rb 0.23 0.23 0.25 0.28 0.36 0.56 0.27 Af 1 0.24 0.25 0.26 0.28 0.30 0.25 0.26 Af 2 0.10 0.11 0.11 0.11 0.10 0.07 0.10 Af 3 0.09 0.09 0.09 0.08 0.07 0.03 0.08 Triple Glazing, e = 0.05 on surface 2 and 4 40c 3 LE LE CLR 0.58 SHGC 0.27 0.25 0.24 0.21 0.16 0.08 0.23 0.25 0.25 0.21 0.24 0.49 0.52 0.42 0.50 T 0.18 0.17 0.16 0.13 0.08 0.03 0.14 R f 0.41 0.41 0.42 0.44 0.50 0.65 0.44 Rb 0.46 0.45 0.46 0.48 0.53 0.68 0.47 Af 1 0.27 0.28 0.28 0.29 0.30 0.24 0.28 Af 2 0.12 0.12 0.12 0.12 0.11 0.07 0.12 Af 3 0.02 0.02 0.02 0.02 0.01 0.01 0.02 40d 6 LE LE CLR 0.55 SHGC 0.26 0.25 0.23 0.21 0.16 0.08 0.22 0.24 0.25 0.20 0.23 0.47 0.49 0.40 0.48 T 0.15 0.14 0.12 0.10 0.07 0.02 0.12 R f 0.33 0.33 0.34 0.37 0.43 0.60 0.36 Rb 0.39 0.38 0.38 0.40 0.46 0.61 0.40 Af 1 0.34 0.36 0.36 0.37 0.36 0.28 0.35 Af 2 0.15 0.15 0.15 0.14 0.12 0.08 0.14 Af 3 0.03 0.03 0.03 0.03 0.02 0.01 0.03 KEY: CLR = clear, BRZ = bronze, GRN = green, GRY = gray, BLUGRN = bluegreen, SS = stainless steel reflective coating, TI = titanium reflective coating Reflective coating decsriptors include percent visible transmittance as x%.
HI-P GRN = high performance green tinted glass, LE = low-emissivity coating Tv = visible transmittance, T = solar transmittance, SHGC = solar heat gain coefficient, and H. = hemispherical SHGC ID #’ s refer to U-Factors in Table 5.
Table 13 Visible Transmittance (Tv), Solar Heat Gain Coefficient (SHGC), Solar Transmittance (T), Front Reflectance (Rf), Back Reflectance (Rb), and Layer Absorptances ( ) for Glazing and Window Systems (Continued) ID Glazing System Center Glazing Tv Center-of-Glazing Properties Total Window SHGC at Normal Incidence Total Window Tv at Normal Incidence Incidence Angles Aluminum Other Frames Aluminum Other Frames Glass Thick., mm Normal 0.00 40.00 50.00 60.00 70.00 80.00 Hemis., Diffuse Operable Fixed Operable Fixed Operable Fixed Operable Fixed An f 30.34 2001 ASHRAE Fundamentals Handbook (SI) independent of direction). In addition, the spectral dependence is assumed to be the same as for beam solar radiation. These simplifi-cations are implicit in the use of the quantities Ed and Er in Equation (12). To calculate these quantities from one of the detailed sky or ground models mentioned in that section (or from detailed data), rather than from the simpler equations given elsewhere in this chap-ter, one would use the following equations: (68) (69) Since these are assumed to be direction-independent quantities, and since for vertical glazings the sky and ground subtend equal solid angles, in most cases the fact that these quantities originate from distinct solid angle regions is ignored. More careful consi-deration must be given to these quantities for tilted glazings or for direction-dependent shading (such as overhangs and venetian blinds) when high accuracy in the results is desired.
The optical properties to be used with these quantities are then the hemispherical averages of the corresponding angle-dependent properties. For example, for an L-layer unshaded glazing, the aver-age transmittance, front reflectance, and layer-specific overall front absorptance would be (70) (71) (72) and similarly for the back reflectance and layer-specific absorptan-ces. The simplified expression at the end of each equation follows from the fact that the quantities being averaged are independent of the angle φ.
Example 12. Calculate the hemispherical average properties for the two glazings and the glazing system in Examples 8, 9, and 10, and compare the results with Table 13.
Solution: Equations (70) through (72) are used, and the integration is done numerically, using an angular grid of 10° intervals extending from 0° to 90°. To carry out this calculation, all of the properties in the three examples must be recalculated at each angle. A single-band model rather than a 2-band is used to reduce the calculational labor. Values of T1,2(θ) used in calculating are shown in Table 14, together with the angular weighting functions used in the numerical integration.
The angular weighting functions are the values, at each angle, of 2(cosθ)(sinθ)∆θ at that angle, where ∆θ is the grid interval (10°) expressed in radians.
The resulting values are = 0.335, = 0.401, = 0.496, = 0.264, and = 0.169 for the selective glazing of Example 9; = 0.761, = = 0.137, and = = 0.101 for the clear glazing of Example 8; = 0.277, = 0.405, = 0.442, and = 0.271, = 0.138, = 0.036, and = 0.141 for the double-glazing system of Example 10. The corresponding values listed for system 25a in Table 13 are = 0.31, = 0.39, and = 0.42.
When we correct them as in Example 11 for the differences between the glazing system of interest and ID 25a at normal incidence, we obtain = 0.31 × 0.331/0.37 = 0.28, = 0.39 × 0.388/0.35 = 0.43, and = 0.42 × 0.396/0.39 = 0.43. These are reasonably close to the values we obtained from the calculation using the more detailed equations.
Spectrally Selective Glazing The spectral range from 350 nm to over 50 µm contains radiation from both the sun and sky incident upon fenestration systems as well as the longer wavelength thermal radiation. Thermal radia-tion is emitted by warm bodies both outside and inside the building.
Figure 23 shows the human eye spectral response, the solar spec-trum for an air mass m = 1.5, and a room temperature blackbody radiation spectrum. The blackbody radiation is scaled to compare with the solar spectrum. Spectral selectivity was defined previously as strong changes in the optical properties of a glazing system over the spectrum. Figure 23 illustrates the basic concept in terms of the spectral reflectance and Figure 24, the spectral transmittance of ideal low solar gain glazings. One can see in this figure the human eye spectral response (called the human photopic visibility func-tion), an air mass 1.5 solar spectrum, and a room temperature (24°C) blackbody radiation spectrum. The latter has been scaled up to bet-ter compare it with the shape of the solar spectrum. What is clear Ed Esky θ φ λ , , ( ) θ cos ( ) ϖ d λ d ∫ sky ∫ λ ∫ = Er Eground θ φ λ , , ( ) θ cos ( ) ϖ d λ d ∫ ground ∫ λ ∫ = T1 L , 〈 〉D T1 L , θ ( ) θ cos ( ) ∫ ϖ d hem ∫ θ cos ( ) ϖ d ∫ hem ∫ --------------------------------------------------------= 2 T1 L , θ ( ) θ cos ( ) θ sin ( ) θ d 0 π 2 ⁄ ∫ = R1 L , f 〈 〉D R1 L , f θ ( ) θ cos ( ) ϖ d ∫ hem ∫ θ cos ( ) ϖ d ∫ hem ∫ ---------------------------------------------------------= 2 R1 L , f θ ( ) θ cos ( ) θ sin ( ) θ d 0 π 2 ⁄ ∫ = An: 1 L , ( ) f 〈 〉D An: 1 L , ( ) f θ ( ) θ cos ( ) ϖ d ∫ hem ∫ θ cos ( ) ϖ d ∫ hem ∫ -----------------------------------------------------------------= 2 An: 1 L , ( ) f θ ( ) θ cos ( ) θ sin ( ) θ d 0 π 2 ⁄ ∫ = Table 14 Angular Weighting Function and Double Glazing System (Example 10) Transmittance Calculated for the Angular Mesh Used to Calculate Hemispherical Average Transmittance θº Angular Weight T1,2(θ) 0 0.000 0.331 10 0.060 0.329 20 0.113 0.325 30 0.153 0.318 40 0.174 0.309 50 0.174 0.294 60 0.153 0.260 70 0.113 0.194 80 0.060 0.094 90 0.000 0.000 T1 2 , 〈 〉D T1 〈 〉D R1 f 〈 〉D R1 b 〈 〉D a1 f 〈 〉D a1 b 〈 〉D T2 〈 〉D R2 f 〈 〉D R2 b 〈 〉D a2 f 〈 〉D a2 b 〈 〉D T1 2 , 〈 〉D R1 2 , f 〈 〉D R1 2 , b 〈 〉D A1: 1 2 , ( ) f 〈 〉D A1: 1 2 , ( ) b 〈 〉D A2: 1 2 , ( ) f 〈 〉D A2: 1 2 , ( ) b 〈 〉D T1 2 , 〈 〉D [ ]25a R1 2 , f 〈 〉D [ ]25a R1 2 , b 〈 〉D [ ]25a T1 2 , 〈 〉D [ ]table R1 2 , f 〈 〉D [ ]table R1 2 , b 〈 〉D [ ]table Fenestration 30.35 from this diagram is the separation of the solar spectrum from the emission spectrum characteristic of an interior room of a multiple-pane glazing system.
Almost all architectural glass is opaque to the long-wave radia-tion emitted by surfaces at temperatures below about 1200°C. This characteristic produces the greenhouse effect, by which solar radi-ation passing through a window is partially retained inside by the following mechanism. Radiation absorbed by surfaces within the room is emitted as long-wavelength radiation, and it cannot escape directly through the glass since it is opaque to all radiation beyond 4.5 µm. Instead, the radiation from the room surfaces falling on the glass is absorbed and re-emitted to both sides as determined by sev-eral parameters, such as the inside and outside film heat transfer coefficients, the surface emissivities, and other glazing properties.
A good reflector in the long-wavelength infrared portion of the spectrum can be a poor reflector and a good transmitter in the solar portion. Because of the conservation of energy (T + R + A = 1.0), a high reflectance in the long-wavelength infrared portion of the spectrum means a low transmittance and absorptance. Because of Kirchhoff’s law [Equation (29)], a low absorptance means a low emissivity as well. This is the principle of operation of the high-solar-gain (or cold-climate) low-e coating on window glass. Such a coating has high transmittance over the entire solar spectrum, producing high solar heat gain while being highly reflective to the long-wavelength infrared radiation emitted by the interior surfaces, reflecting this radiation back inward. The low-e in low-e coating refers to a low emissivity over the long-wavelength portion of the spectrum.
Another characteristic of glazing is that at a given wavelength, or over a defined range of wavelengths, the transmittance is the same in both directions. A glazing with a coating or consisting of multiple glazing layers will have one solar transmittance value, but it will have two reflectance values—one for radiation approaching each side of the system.
Figure 24 shows a hypothetical glazing system with improved performance for hot climates. In this case, the sharp reflectance edge that the ideal cold-climate low-e coating exhibited just past the end of the solar spectrum in Figure 23 is shifted closer to the edge of the visible portion of the spectrum, thereby increasing the solar near-infrared (NIR) reflectance of the glazing. This results, as seen in Figure 24, in a drop in the hot-climate transmittance to the right of the visible portion of the spectrum. The effect is to reflect the near-infrared portion of the solar spectrum outside, reducing solar gain, while still admitting visible light in the wavelength region below about 800 nm. This hot-climate, solar-gain-rejecting coating also exhibits low emissivity over the long-wavelength spectrum, and is therefore also properly termed a low-e coating. To distinguish the cold- from the hot-climate version, a glazing with this type of spectral response is often termed selective low-e. This is somewhat of a misnomer, because both hot- and cold-climate glazings are spectrally selective. Another terminology is high-solar-gain low-e glazing system for cold climates, contrasted with low-solar-gain low-e glazing system for hot climates.
The reduced infrared transmittance for the hot-climate glazing in Figure 24 is ideally achieved by high reflectance and low absorp-tance (meaning also low emissivity). It can also be accomplished with high infrared absorptance, if the flow of the absorbed solar radiation to the interior of the building can be reduced, introducing a second approach to the construction of a hot-climate, low-solar-gain glazing system. In this case, the outer pane of a multiple-pane glazing system is made to have good visible transmittance but high absorptance over the solar infrared spectrum. To protect the interior of the building from the heat of this absorbed radiation, additional glazings, gas spaces, and cold-climate type, low-e coatings are added.
By this means, radiation, conduction, and convection of heat from the hot outer pane to the interior ones and to the interior of the building are reduced because of the coating, the insulating gas space, and the additional panes. Such a glazing system for hot cli-mates is insulated primarily not to protect the building from conduc-tive heat losses in winter but to protect the interior from the solar radiant heat absorbed by the hot outer pane in summer. Several man-ufacturers offer this kind of nonreflecting spectrally selective glaz-ing system for commercial buildings having large cooling loads. (If this glazing system is placed in a flip window, one that rotates open for cleaning and can be closed with either side facing outward, then flipping it over in winter can make it an effective trap for passive solar heating of the interior (McCluney and Jindra 2000). Such a system will have two different SHGC values, depending upon which way the window is flipped.
Figure 24 shows that glazings intended for hot climates should have (1) high transmittance over the visible portion of the spectrum to let daylight in for both illumination and view and (2) low trans-mittance over all other portions of the spectrum to reduce solar heat gain. In contrast, glazings intended for very cold climates should have high transmittance over the whole solar spectrum, from 380 nm Fig. 23 Solar Spectrum, Human Eye Response Spectrum, Scaled Blackbody Radiation Spectrum, and Idealized Glazing Reflectance Spectrum Fig. 24 Demonstration of Two Spectrally Selective Glazing Concepts, Showing Ideal Spectral Transmittances for Glazings Intended for Hot and Cold Climates 30.36 2001 ASHRAE Fundamentals Handbook (SI) to over 3500 nm, for maximum admission of solar radiant heat gain and light. In addition, glazings for cold climates should have low transmittance over the long-wavelength portion of the spectrum in order to block the radiant heat emitted by the relatively warm interior surfaces of buildings, preventing its escape to the outside.
Extreme spectral selectivity in glazing systems in the visible por-tion of the spectrum can produce an unwanted color shift in the transmitted light. The color of the transmitted light and its color-ren-dering properties should be considered in the design.
SOLAR HEAT GAIN COEFFICIENT Fenestration solar heat gain has two components. First is the directly transmitted solar radiation. The quantity of radiation enter-ing the fenestration directly is governed by the solar transmittance of the glazing system. Multiplying the incident irradiance by the glazing area and its solar transmittance yields the solar heat entering the fenestration directly. The second component is the inward-flow-ing portion of the absorbed solar radiation, radiation that is absorbed in the glazing and framing materials of the window and is subse-quently conducted, convected, and radiated to the interior of the building.
The solar heat gain coefficient values presented in this chapter are based on a standard spectral irradiance distribution for air mass 1.5. This spectrum is recommended by the National Fenestration Rating Council (NFRC) for the purpose of rating fenestrations for instantaneous energy performance using defined environmental and incident irradiance conditions. The NFRC standard spectrum can be found in ASTM Standard E 891. For real solar gain calculation sit-uations, and with glazing systems exhibiting spectral selectivity over the solar spectrum, the incident solar spectrum used should be altered from the representative one contained in this standard in order to be more realistic for the atmospheric conditions at the loca-tion and time of the calculation. Gueymard (1987a, 1987b, 1993a, 1993b, and 1995) has developed an algorithm to guide a computer-ized methodology for calculating realistic solar spectra that are sen-sitive to changes in the atmospheric constituents. The ASTM standard spectrum is different from Parry Moon’s (Moon 1940) air mass 2 spectrum used by the glazing industry in the past. Both are different from solar spectral distributions incident upon fenestrations with different atmospheric conditions and for different sun angles. These differences will have little impact on the SHGC of glazing systems containing only glazings with relatively flat spectral transmittances, that is, glazings that are not strongly spectrally selective. Glazings that do exhibit strong spectral selec-tivity (such as those shown in Figures 15 and 16), however, can have different SHGC values. The visible transmittances are less sensitive to solar spectral changes. However, for glazings with very strong spectral selectivity in the visible portion of the spectrum, such as those exhibiting strong color, the visible transmittance also can be sensitive to the shape of the incident spectrum.
Absorbed solar radiation, including ultraviolet, visible, and infrared radiation from the sun and sky, is turned into heat inside the absorbing material. In a window, the glazing system temperature rises as a result to some approximately equilibrium value at which the energy gains from absorbed radiation are balanced by equal losses. The absorbed solar radiation is dissipated through the mech-anisms of conduction, convection, and radiation. Some heat goes outside the building, and the remainder goes inside, adding to the directly transmitted solar radiation. The magnitude of what is called the inward-flowing fraction N of the absorbed radiation depends on the nature of the air boundary layers adjacent to both sides of the glazing, including any gas between the panes of a multiple-pane glazing system (Ni is often used to distinguish the inward-flowing fraction from the outward-flowing fraction, No. However, since only the inward-flowing fraction is used here, the subscript i is dropped for clarity.).
The concept of the solar heat gain coefficient is best illustrated for the case of a single glass pane in direct sunlight. Let ED be the direct solar irradiance incident upon a single pane of glass, T be its solar transmittance, A be the solar absorptance, and N be the inward-flowing fraction of the absorbed radiation. In this case, the total solar gain (heat flow per unit area) qb that enters the space due to the incident solar radiation is (73) in units of energy flux per unit area, W/m2. Multiple glazings are discussed in the section on Calculation of Solar Heat Gain Coefficient.
The quantity in parenthesis in Equation (73) is called the solar heat gain coefficient or SHGC. It is the fraction of incident irradi-ance that enters through the glazing and becomes heat gain. It includes both the directly transmitted portion and the absorbed and re-emitted portion: (74) The SHGC is needed to determine the solar radiant heat gain through a window’s glazing system. The SHGC for certain defined conditions of spectrum and incident angle θ should be included along with U-factor and other instantaneous performance properties in any manufacturer’s description of a window’s energy perfor-mance. Since the optical properties A and T vary with the angle of incidence [defined as the angle between the rays incident on the glazing and the normal (perpendicular) to the glazing], according to Equation (74), the solar heat gain coefficient is also a function of angle of incidence. Once the incident irradiance and SHGC are known for a given angle of incidence, the solar gain (from direct beam radiation) can be computed with the following equation: (75) Optical properties also vary with wavelength. The quantities A and T are spectral averages, as described by Equation (30).
Calculation of Solar Heat Gain Coefficient In the most general way, the solar heat gain q and the solar heat gain coefficient SHGC are defined as angle-dependent and spec-trally dependent properties: (76) where the definition of Equation (74) implies an angle- and wave-length-dependent solar heat gain coefficient: (77) This can in turn be used to define a wavelength-averaged solar heat gain coefficient: (78) where ED(λ) = incident solar spectral irradiance T(θ,λ) = spectral transmittance of the glazing system A(θ,λ) = total spectral absorptance of the glazing system qb ED T NA + ( ) = SHGC T NA + = qb EDSHGC = q θ ( ) ED λ ( ) T θ λ , ( ) NA θ λ , ( ) + [ ] λ d λ ∫ = ED λ ( )SHGC θ λ , ( ) λ d λ ∫ = SHGC θ λ , ( ) T θ λ , ( ) NA θ λ , ( ) + = SHGC θ ( ) ED λ ( ) T θ λ , ( ) NA θ λ , ( ) + [ ] λ d λ ∫ ED λ ( ) λ d λ ∫ ------------------------------------------------------------------------------= Fenestration 30.37 Starting with this edition of the Handbook, Equations (76) through (78) indicate the preferred way of determining the solar gain of glazing systems and calculating the solar heat gain coeffi-cient. At least two computer programs are available to assist in the calculation (Arasteh et al. 1994, AGSL 1992). This approach has been adopted by the National Fenestration Rating Council in NFRC 200 for the rating, certification, and labeling of windows for energy performance and by the Canadian Standards Association (CSA Standard A440.2). The method is valid for strongly spectrally selec-tive glazing systems as well as for nonselective ones. In these pro-grams, the overall system optical properties at a given incident angle are calculated for each wavelength and the results averaged following Equation (31). The ASTM Standard E 891 spectrum is used in the averaging. The wavelength-averaged properties (at a given incident angle) can then be used in Equation (76).
When a glazing system is not strongly spectrally selective, the solar-weighted spectral broadband values of the optical properties can be used, and the integral over wavelength shown in Equations (76) and (78) is not needed. In this case, and for a general unshaded glazing system, which may consist of several glazing layers, each glazing layer will have its own individual inward-flowing fraction of the absorbed radiation for that layer. Let the glazings be num-bered from the outside inward, and k be the glazing index. Then, the SHGC is given by (79) where = front transmittance of the glazing system L = number of glazing layers = absorptance of layer k Nk = inward-flowing fraction for layer k The wavelength-independent quantities in this equation are ob-tained by the averaging procedure of Equation (66). Conversely, the wavelength-dependent form of Equation (79) is obtained by adding the wavelength dependence to the indicated angle dependence; the Nk and the layer decomposition are independent of wavelength.
The solar heat gain coefficient is a combination of the solar-optical properties discussed in the section on Solar Optical Proper-ties of Glazings and a new set of quantities Nk, which account for the fact that of the energy absorbed in a given layer, the fraction that reaches the interior space will depend on the location of the layer. The Nk are thermal in origin; they depend on the heat trans-fer properties of the assembly rather than on its optical properties. The inward-flowing fractions can be calculated from simplified heat transfer models, using the following equation: (80) where U = U-factor of the glazing ho,k = effective heat transfer coefficient between the exterior environment and the kth glazing layer The effective heat transfer coefficient can be calculated in a one-dimensional model as the reciprocal of the sum of the thermal resistances of all elements between layer k and the exterior (see Chapter 3). For example, for single glazing, (81) where U is the U-factor of the glazing and ho is the exterior heat transfer coefficient (see Chapter 3). For double glazing, the two inward-flowing fractions are (82) where the numeric subscripts obey the usual layer numbering con-vention (1 is the exterior glazing).
In more complicated multilayer glazing systems, it is advisable to perform a detailed heat transfer analysis of the system to deter-mine the values of Nk, since the effective heat transfer coefficients and U depend (weakly) on the glazing layer temperatures and other environmental conditions (e.g., Finlayson and Arasteh 1993, LBL 1994, and Wright 1995c).
Diffuse Radiation For incident radiation that is diffuse, the hemispherical average solar heat gain coefficient must be used. This may be calculated either by a direct hemispherical averaging of Equation (79) using the analog of Equation (70) as follows: (83) or equivalently by simply using the hemispherically averaged solar-optical properties in Equation (79): (84) In any case, Nk is unaffected in the averaging process, since it does not depend on the incident angle or the wavelength. Consider-ations of the solar spectrum discussed previously in the sections on Spectral Averaging of Glazing Properties and Angular Averaging of Glazing Properties also apply to the solar heat gain coefficient.
Solar Gain Through Frame and Other Opaque Elements Figure 25 illustrates the mechanisms by which a window pro-vides solar gain. It is assumed that all of the directly transmitted solar radiation is absorbed at indoor surfaces, where it is converted to heat. Solar gain also enters a building through opaque elements such as the frame and any mullion or dividers that are part of the fen-estration system because a portion of the solar energy absorbed at the surfaces of these elements is redirected to the indoor side by heat transfer.
The solar heat gain coefficient of the fenestration system can be calculated while accounting for solar gain through the opaque ele-ments by area-weighting the solar heat gain coefficients of the glaz-ing, frame, and M divider elements. Thus, (85) SHGC θ ( ) T1 L , f θ ( ) NkAk: 1 L , ( ) f θ ( ) k=1 L ∑ + = T1 L , f θ ( ) Ak: 1 L , ( ) f Nk U ho k , ---------= N U ho -----= N1 U ho -----, = N2 ho U + ho ----------------= SHGC θ ( ) 〈 〉D SHGC θ ( ) θ cos ( ) ϖ d ∫ hem ∫ θ cos ( ) ϖ d ∫ hem ∫ -------------------------------------------------------------= 2 SHGC θ ( ) θ cos ( ) θ d 0 π 2 ⁄ ∫ = SHGC 〈 〉D T1 L , f 〈 〉D Nk Ak: 1 L , ( ) f 〈 〉D k=1 L ∑ + = SHGC SHGCgAg SHGCfAf AiSHGCi i=1 M ∑ + + Ag Af Ai i=1 M ∑ + + --------------------------------------------------------------------------------------------------= 30.38 2001 ASHRAE Fundamentals Handbook (SI) where SHGCg, SHGCf , and SHGCi are the solar heat gain coeffi-cients of the glazed area, frame, and ith divider, respectively. Ag, Af, and Ai are the corresponding projected areas.
In some cases, it is useful to have an overall SHGC for the opaque elements only, which is defined by (86) where SHGCf can be estimated (Wright 1995a) using (87) where is the solar absorptivity of the outdoor surface of the frame, Uf is the frame U-factor, and hf is the heat transfer coefficient (radiative plus convective) between the frame and the outdoor envi-ronment. The projected-to-surface area ratio (Af/Asurf) corrects for the fact that Uf is based on projected area Af and hf is based on the exposed outdoor frame surface area Asurf. SHGCi can be calculated in the same way: (88) The outdoor side heat transfer coefficients hf and hi can be esti-mated using ASHRAE (1996): (89) where hco is the convective heat transfer coefficient between the frame (or divider) surface and the outdoor environment, ef is the emissivity (long-wave) of the outdoor frame (or divider) surface, Tout is the outdoor absolute temperature, and σ is the Stefan-Boltz-mann constant.
Shading Coefficient—A Historical Perspective Before modern complex windows were marketed in quantity, the determination of fenestration solar gain was substantially simpler.
Frame and edge effects were largely ignored, and attention focused on the glazing, which was typically made up of single-pane clear or tinted glass. ASHRAE provided a method for calculating the inci-dent solar beam irradiance for any direction of incidence and tabu-lated the resulting solar gains through single-pane clear glass in units of flux per unit area in tables of what were called solar heat gain factors (SHGFs). These factors, having units of W/m2, are to be distinguished from the solar heat gain coefficient, which is dimensionless. Solar heat gain factors provide the total solar radiant heat gain through a standard single-glazing system, including both the directly transmitted radiant component and the inward flowing fraction of the radiation absorbed in the glazing system.
The engineer’s job is relatively easy with this method. One first figures out the angle of direct beam incidence on the glass for a typ-ical peak solar gain date and time. This is done by (1) using equa-tions provided in the section on Determining Solar Angle under Determining Incident Solar Flux or (2) by looking up the data for a latitude close to that of the building being designed for a chosen glazing orientation and for a given time of day expected to produce peak solar gain, in Tables 16 through 22 in Chapter 29 of the 1997 ASHRAE Handbook—Fundamentals. These tables provide the solar heat gain factor for clear single-pane glass (the so-called standard reference glazing, having T = 0.86, R = 0.08, and a = 0.06 at normal incidence) under these conditions.
The next step is simply a matter of multiplying the solar heat gain factor by the area of the glazing, producing the solar gain expected, in W. This provides directly the solar gain for single-pane clear glass.
In order to deal with windows having shades or tinted glass, the concept of a shading coefficient was introduced. The idea behind the shading coefficient was to find a multiplicative factor for the shaded glass that allowed the engineer to correct the previously determined solar gain number through clear glass to the proper value for the window with a shade or tint. This could be done because the commonly used shading systems (interior shades and drapes) were assumed not to have strongly varying angular proper-ties, leaving the single glazing as the only determinant of angular dependence. Spectral variation was not considered. This concept was later extended to tinted glazings, which often do not violate the basic underlying assumptions, and to glass blocks and single-glazed windows with interior venetian blinds, for which the application is more dubious.
The shading coefficient was defined to be the ratio of the solar heat gain coefficient of a glazing system at a particular angle of inci-dence and incident solar spectrum to that for standard reference glazing at the same angle of incidence and spectral distribution: (90) For single-pane clear and many single-pane tinted glazings, this ratio remains constant as the solar spectral shape varies and as the angle of incidence varies. Thus, a single number can be used to con-vert from the reference SHGC to the SHGC for the tinted glazing at the angle of incidence selected. Due to its lack of sensitivity to angle of incidence, the same SC value works for beam radiation at any angle of incidence, as well as for diffuse radiation. This meant that the complicated calculation of solar heat gain transmission through glass could be done once and tabulated, a quantity appearing in pre-vious editions of this Handbook as the SHGF. For other systems, the SHGF obtained from the table could simply be multiplied by the SC, a great simplification. The value of the SC for standard refer-ence glass is 1.0, but the SHGC for this glass is 0.87 at normal inci-Fig. 25 Components of Solar Radiant Heat Gain with Double-Pane Window, Including Both Frame and Glazing Contributions SHGCop SHGCfAf AiSHGCi i=1 M ∑ + Aop -----------------------------------------------------------= Aop Af = Ai i=1 M ∑ + SHGCf αf s Uf hf ----- Af Asurf ------------ = αf s SHGCi αi s Ui hi ----- Ai Asurf i , --------------- = hf or hi hco 4σefTout 3 + = SC SHGC θ ( )test SHGC θ ( )ref --------------------------------= Fenestration 30.39 dence, using ASHRAE standard summer conditions, and for the standard ASTM solar spectrum. This was useful for dealing with glazings of different thicknesses (and was later extended to tinted glazings). Because the principal property causing angular depen-dence in single glazing is reflectance, angular properties do not depend strongly on thickness, so that the SC defined by Equation (90) remains approximately constant. While tinted glazings, having different spectral transmission from the reference glazing, have a very different SHGC, this difference does not depend strongly on angle, so again SC is approximately constant. Since it is easiest to determine the transmittance of a glazing at normal incidence, it became standard practice to calculate the SC for clear and tinted sin-gle glazing from the following relation: (91) This equation applies only to the glazing portion of single-pane tinted or clear windows. It does not include frame effects. It may still be used to determine the SC of commercially available, single, uncoated glazing products from the solar heat gain coefficient (or vice versa) published by the manufacturer.
Solar Heat Gain Coefficient, Visible Transmittance, and Spectrally Averaged Solar-Optical Property Values Table 13 lists the visible transmittance, solar transmittance, front and back reflectance, and solar heat gain coefficients for common glazing and window systems. The window systems include win-dows with aluminum or metal frames and windows with other frames that have a lower conductivity (e.g., thermally broken alu-minum, wood, vinyl, and fiberglass). As can be seen in Table 13, the total window solar heat gain coefficient varies with the type of oper-ator, size of the fenestration product, and type of frame. The glazing Tv, Tsol, Rf, Rb, and SHGC values have been calculated using manufacturers’ spectral data following methods described in the section on Solar-Optical Properties of Glazing (Finlayson et al.
1994, Wright 1995a). The glazing values are given for 3 mm and 6 mm glass and will vary with glass thickness and glass manufacturer.
The values shown are average values and may vary by ±0.05. It is recommended that actual values be determined using the methods described in this chapter with detailed spectral data from NFRC (1994). The front reflectance is the reflectance of the unit to the out-side, and the back reflectance is the reflectance to the room side.
The visible transmittances are center-glazing values at normal incidence. A rule of thumb is to select a glazing unit whose visi-ble transmittance is greater than its solar heat gain coefficient, especially if daylighting strategies will be used in the building.
For maximum light with minimum solar gain, there are fenestra-tion products available having visible transmittance that is 1.4 times their SHGC.
The solar heat gain coefficients are center-glazing values and total window values. The center-glazing solar heat gain coefficients are given at normal incidence (0°) and at 40°, 50°, 60°, 70°, and 80° incidence angles. For angles other than those listed, straight-line interpolation can be used between the two closest angles for which values are shown.
The solar transmittances and front and back reflectances are also center-glazing values and are given at normal incidence (0°) and at 40°, 50°, 60°, 70°, and 80° incidence angles. The solar absorptances can be calculated from these values using Equation (26). The effec-tive inward-flowing fraction of absorbed radiation for the entire sys-tem (not layer-specific values) can be determined from Equation (74) by inserting the solar transmittance and corresponding SHGC.
The total window solar heat gain coefficients in Table 13 assume normal incidence. The operable and fixed window sizes in Table 4 were used. To calculate the frame area, the frame heights shown in Figure 4 for aluminum and aluminum-clad wood/wood/vinyl were used. The frame area for the aluminum windows is 15% for the operable size and 11% for the fixed size. The frame area for other frames is 27% for the operable size and 13% for the fixed size. The ratio of projected frame area to frame surface area is assumed to be 1.0, based on Wright (1995a).
The frame solar heat gain coefficients used to determine the total window solar heat gain coefficients are calculated according to the section on solar heat gain coefficients for frames and other nonglaz-ing elements in this chapter. The frame U-factors are taken from Table 1. The frame absorptance is assumed to be 0.5. The outside film coefficient is 22.2 W/(m2·K), corresponding to a wind speed of 3.4 m/s. For the aluminum window, the frame solar heat gain coef-ficient is 0.14 for the operable window and 0.11 for the fixed win-dow. For the other frames, the frame solar heat gain coefficient was found to vary between 0.02 and 0.07 for the various lower conduc-tivity frame types. A frame solar heat gain coefficient of 0.04 is used for the operable window, and 0.03 is used for the fixed window.
These values correspond directly to the aluminum-clad wood/rein-forced vinyl frames.
For energy calculations on a daylit building, the visible transmit-tance for the entire window should be used. The visible transmittance of a window can be calculated by multiplying the fraction of glazing area by the center-glazing visible transmittance (see Example 13).
The U-factor of a window listed in Table 13 can be found in Table 4. The ID number for each entry in Table 13 refers to an ID number in Table 4. For a particular glazing system in Table 13, the corresponding glazing system should be the glazing system with that ID number or following that ID number in Table 4. Remember that while the gap width and gas fill have a negligible impact on the solar heat gain coefficient and other optical properties, they are important factors when determining U-factors.
Example 13. Estimate the overall visible light transmittance for an operable wood casement window with clear, uncoated 6 mm double glazing.
Solution: From Table 13, ID 5b, the center-glazing visible light trans-mittance is 0.78. The operable window has 27% frame area with a wood frame. The overall visible light transmittance is Tv = 0.27(0) + 0.73(0.78) = 0.57 Passive Solar Gain Energy analysis of a fenestration product should include the value of passive solar gain through the product in winter. As described in Chapter 32 of the 1999 ASHRAE Handbook—Applica-tions, the magnitude of this energy gain depends on such variables as latitude and orientation. In some cases, properly designed and operated fenestration allows more energy into the building over a heating season than it loses, thus making it energy-contributing rather than energy-consuming. Excessive solar gain during the cool-ing season must be controlled, however.
Direct beam admission to occupied spaces can often produce severe localized glare and overheating conditions. Judicious use of shades and other fenestration control strategies, as well as place-ment and orientation of workstations and furniture, can alleviate these problems in most cases.
Solar Gain Rejection and Internal Load Dominated Buildings For some buildings in certain climates, preventing solar gain is more important from an energy perspective than improving thermal insulation using multiple panes of glazing. For example, internal load dominated buildings in cool, clear climates can have substantial daytime solar and internal heat gains. These gains can be rejected by conduction through the building envelope and/or forced ventilation through the HVAC system. Preventing excessive solar gain through the fenestration systems of such buildings is very important.
SC SHGC 0.87 ----------------= 30.40 2001 ASHRAE Fundamentals Handbook (SI) Airflow Windows If properly managed, airflow between panes of a double-glazed window can improve fenestration performance. In normal use, a venetian blind is located between the glazing layers. Ventilation air from the room enters the double-glazed cavity, flows over the blind, and is, in some designs, exhausted from the building or returned through the ducts to the central HVAC system. In cold weather, the window acts as a heat exchanger when sunlit so that the inner glass temperature nearly equals the room air temperature and improves thermal comfort.
The apparent conductance across the inner glazing is very low, but this is misleading since additional heat is lost to the outdoors from the moving airstream in the window cavity. During sunny win-ter days, the blind acts as a solar air collector; heat removed by the moving air can be used elsewhere in the building. In the summer, the window can have a very low shading coefficient if the blinds are appropriately placed since the majority of solar gains are removed from the window. These systems can control window heat transfer under many different operating conditions. Sodergren and Bostrom (1971) and Brandle and Boehm (1982) give details on airflow or exhaust windows.
Skylights Skylight solar heat gain strongly depends on the configuration of the space below or adjacent to (i.e., in sloped applications) the skylight formed by the skylight curb and any associated light well.
Five aspects must be considered (1) the transmittance and absorptance of the skylight unit, (2) the transmitted solar flux that reaches the aperture of the light well, (3) whether that aperture is covered by a diffuser, (4) the transmitted solar flux that strikes the walls of the light well, and (5) the reflectance of the walls of the light well. Data for flat skylights, which may be considered as sloped glazings, are found in Tables 4 and 13.
Domed Skylights. Solar and total heat gains for domed skylights can be determined by the same procedure used for windows. Table 15 gives SHGCs for plastic domed skylights at normal incidence.
Manufacturers’ literature has further details. Given the poorly defined incident angle conditions for domed skylights, it is best to use these values without correction for incident angle, together with the correct (angle-dependent) value of incident solar irradiance.
Results should be considered approximate. In the absence of other data, these values may also be used to make estimates for skylights on slanted roofs.
Glass Block Walls Glass block can be used for light transmission through exterior walls when optical clarity for view is not needed or wanted. Table 16 describes a variety of glass block patterns and gives solar heat gain coefficients to be applied to the solar irradiances so that approximate instantaneous solar heat gains can be calculated.
Convection and low-temperature radiative heat gain for all hol-low glass block panels fall within a narrow range. Differences in SHGCs are largely the result of differences in the transmittance of the glass blocks for solar radiation. Solar heat gain coefficients for any particular glass block pattern vary depending on orientation and time of day. The SHGC for western exposures in the morning (in the Table 15 Solar Heat Gain Coefficients for Domed Horizontal Skylights Dome Light Diffuser (Translucent) Curb Solar Heat Gain Coefficient Height, in.
Width-to-Height Ratio Clear Yes 0 ∞ 0.53 τ = 0.86 τ = 0.58 9 5 0.50 18 2.5 0.44 Clear 0 ∞ 0.86 τ = 0.86 None 9 5 0.77 18 2.5 0.70 Translucent 0 ∞ 0.50 τ = 0.52 None 18 2.5 0.40 Translucent 0 ∞ 0.30 τ = 0.27 None 9 5 0.26 18 2.5 0.24 Table 16 Shading Coefficients and U-Factors for Standard Hollow Glass Block Wall Panels Type of Glass Blocka Description of Glass Block Solar Heat Gain Coefficient U-Factor,c W/(m2·K) in Sun in Shadeb Type I Glass colorless or aqua A, D: Smooth B, C: Smooth or wide ribs, or flutes horizontal or vertical, or shallow configuration 0.65 0.40 2.9 E: None Type IA Same as Type I except ceramic enamel on A 0.27 0.20 2.9 Type II Same as Type I except glass fiber screen partition E 0.44 0.34 2.7 Type III Glass colorless or aqua A, D: Narrow vertical ribs or flutes.
B, C: Horizontal light-diffusing prisms, or horizontal light-directing prisms 0.33 0.27 2.7 E: Glass fiber screen Type IIIA Same as Type III except E: Glass fiber screen with green ceramic spray coating 0.25 0.18 2.7 or glass fiber screen and gray glass or glass fiber screen with light-selecting prisms Type IV Same as Type I except reflective oxide coating on A 0.16 0.12 2.9 aAll values are for 200 by 200 by 100 mm block, set in light-colored mortar. For 300 by 300 by 100 mm block, increase coefficients by 15%, and for 150 by 150 by 100 mm block reduce coefficents by 15%.
bFor NE, E, and SE panels in shade, add 50% to the values listed for panels in the shade.
cValues shown are the same for all size block.
Fenestration 30.41 shade) is depressed because of the heat storage within the block, whereas the SHGC for eastern exposures in the afternoon (in the shade) is elevated as the stored heat is dissipated. Time lag effects from heat storage are estimated by using solar gains and air-to-air temperature differences for one hour earlier than the time for which the load calculation is made.
Calorimeter tests of Type 1A glass block showed little difference in solar heat gains between glass block with either black or white ceramic enamel on the exterior of the block. Because white and black ceramic enamel surfaces represent the two extremes for reflecting or absorbing solar energy, glass block with enamel sur-faces of other colors should have solar heat gain coefficients between these values. Since glass blocks are good examples of strongly angularly selective fenestrations, all cautions in the section on Angular Dependence of Glazing Optical Properties apply here.
Plastic Materials for Glazing Generally, the factors outlined for glass apply also to glazing materials such as acrylic, polycarbonate, polystyrene, or other plas-tic panels. If the solar transmittance, absorptance, and reflectance are known, an SHGC and a shading coefficient can be calculated in the same way as for glass. These properties can be obtained from the manufacturer or be determined by simple laboratory tests. The National Fenestration Rating Council has developed standards for testing the optical properties of glazing (NFRC 301, NFRC 300).
In selecting plastic panels for glazing, possible deterioration from the sun, expansion and contraction because of temperature extremes, and possible damage from abrasion are concerns.
CALCULATION OF SOLAR HEAT GAIN As indicated in the section on Determining Fenestration Energy Flow, the solar energy flow through a fenestration may be divided into two parts, opaque and glazing portions, qop and qs, respectively, as given in Equation (5).
The glazing solar energy flux can be split into that due to incident beam radiation (b) and that due to incident diffuse radiation (d), which includes both diffuse sky radiation and radiation scattered (reflected) from the ground: (92) The net heat balance that would occur for a sunlit glazing if there were no diffuse radiation is shown in Figure 26. This net heat balance does not include any of the heat flows contained in Qth in Equation (1) (i.e., those resulting from inside-outside tem-perature differences). The heat balance due to sunlight is pictured as an effect superimposed on the thermal effect, with, for exam-ple, glazing temperatures somewhat elevated over their value without the sunlight. This superposition picture cannot be carried too far, however, because the heat flows indicated in Figure 26 as resulting from convection and radiation depend in part on pro-cesses that are nonlinear with respect to temperature [e.g., Equa-tion (25)], so that in reality the two effects cannot be separated. To calculate them, one would need the actual glazing (and other) temperatures, not simply the incremental temperature rise due to the sunlight.
One can see from Figure 26 that the glazing solar energy flow consists of two parts: (93) where qbt = glazing solar energy flux due to transmitted incident beam radiation qba = glazing solar energy flux due to inward heat flow of absorbed beam radiation by convection and radiation However, there is always also diffuse incident radiation, so that the glazing heat flux is in reality given by (94) where qt = glazing solar energy flux due to all transmitted incident solar radiation qa = glazing solar energy flux due to inward flow of absorbed incident solar radiation (by heat transfer processes) It is convenient to separate the solar energy fluxes due to the two types of incident radiation for purposes of calculation. We can do this by making a definition analogous to Equation (93) for the dif-fuse solar heat flux: (95) In order to calculate solar energy fluxes, one must first calculate the incident angle θ from the local standard time and the longitude using Equations (10) and (13) through (17). The direct normal solar irradiance EDN is then calculated from Equation (20), the diffuse sky irradiance Ed from Equation (22) or (23), and the ground-reflected radiation Er from Equation (24). Note that the latter two are assumed to be ideally diffuse radiation. The total incident irradiance Et may be calculated from Equation (12).
Because of the distinctions between beam and diffuse incident radiation, one must determine the overall solar heat gain coefficient SHGCo, if desired, by first calculating Qsol and then solving Equa-tions (2) and (1).
Opaque Fenestration Elements The opaque portion solar energy flux is calculated from (96) where SHGCop is obtained from Equation (86).
Glazing Systems The following discussion of glazing system solar energy fluxes applies to unshaded glazings. For shaded glazings, see the section on Shading Devices and Fenestration Attachments.
The glazing solar energy flux due to incident beam radiation is calculated from (97) qs qb qd + = qb qbt qba + = Fig. 26 Instantaneous Heat Balance for Sunlit Glazing Material qs qt qa + = qd qdt qda + = qop EDN θ cos Ed Er + + ( )SHGCop = qb EDN θSHGC θ ( ) cos = 30.42 2001 ASHRAE Fundamentals Handbook (SI) where the beam solar heat gain coefficient is given by Equation (79). If instead one desires the solar radiant and heat fluxes sepa-rately, one can calculate the glazing transmitted solar flux (which is solar radiation traveling in the incident direction) from (98) and the inward-flowing absorbed solar flux (which is heat) from (99) Calculation of and in these equations is described in the section on Optical Properties of Multiple-Layer Glazing Systems, and determination of Nk is discussed in the section on Calculation of Solar Heat Gain Coefficient.
The glazing solar energy flux due to diffuse incident radiation is calculated from (100) where the hemispherically averaged solar heat gain coefficient is calculated from Equation (83). The solar radiant and heat fluxes can be separately calculated from (101) which is diffusely distributed solar radiation (note that effects of finite glazing size and thickness are neglected), and (102) The hemispherically averaged optical properties are calculated using Equations (70) and (72).
Example 14. Calculate the solar energy flux through the glazing system of Example 10 under the conditions of Example 5 using (a) Table 13 and (b) the equations for optical properties.
Solution: (a) The exact glazing system used in this example is not listed in Table 13 (as will frequently be the case in practice), so we must locate the glazing system that is closest in properties. The emissivity of the selec-tive coating on surface 2 of the outer glazing is 0.05, and the transmis-sion and reflectances for the insulating glass unit at normal incidence are T(0°) = 0.331, Rf(0°) = 0.388, and Rb(0°) = 0.396. The most similar unit is ID 25a, which has T(0°) = 0.37, Rf(0°) = 0.35, and Rb(0°) = 0.39 (i.e., slightly lower front reflectance and therefore higher transmit-tance). Near the required angle of incidence, the table values listed for the solar heat gain coefficient are [SHGC(60°)]25a = 0.34 and [SHGC(70°)]25a = 0.27, and from the diffuse column of the table we obtain = 0.36. Linear interpolation in angle yields the value [SHGC(63.2°)]25a = 0.34 + (0.27 – 0.34)/(70° – 60°) × (63.2° – 60°) = 0.32. This value can then be approximately corrected for the (small) difference between the glazing of interest and ID 25a in Table 13 as follows. We first note that entry 25a has a transmittance that is 0.04 higher than the glazing in question and a front reflectance that is 0.04 lower. This means that the absorptance of the two glazings is the same to within the accuracy of Table 13. Referring to Equation (79), we see that with the absorptance held constant, a change in the transmit-tance of the glazing produces an additive correction to the SHGC that is equal to the difference in transmittance. Referring to Examples 11 and 12 where the transmittances for this case are calculated, we make the correction: and Note that these values are very close to those calculated below in part (b) of the solution. The solar energy flux is then calculated from Equa-tions (92), (97), and (100), using the incident solar fluxes calculated in Example 7: (b) We use the previously calculated optical properties of the glazing to determine the SHGC. To do this we need the inward-flowing fractions N1 and N2 for the two glazing layers. The simplest way to obtain these is to do a heat transfer analysis of the glazing and determine the glazing temperatures under the appropriate conditions. We assume an exterior temperature to of 31.7°C, a wind speed of 3.4 m/s, and an indoor tem-perature ti of 23.9°C. Using methods presented in the section on Deter-mining Fenestration U-Factors, the heat transfer analysis yields a central glazing U of 1.22 W/(m2· K) and glazing layer temperatures of t1 = 31.2°C and t2 = 25.3°C. Since the temperature differences will be proportional to the thermal resistances (which are inversely propor-tional to the effective heat transfer coefficients), N1 = (t1 – to)/(ti – to) = 0.064 and N2 = (t2 – to)/(ti – to) = 0.821. We can then calculate the solar heat gain coefficient for beam radiation incident at 63.2° (from Exam-ples 5 and 6), using the results from Example 11, from Equation (79): and for diffuse radiation, using the results from Example 12, from Equation (84): The solar energy flux is then calculated from Equations (92), (97), and (100), using the incident solar fluxes calculated in Example 7: SHADING DEVICES AND FENESTRATION ATTACHMENTS Fenestrations with shading devices and attachments have a degree of optical and geometric complexity far greater than that of fenestrations with unshaded glazings. ASHRAE has sponsored the development of a method for determining the SHGC of glazing sys-tems of arbitrary complexity. However, there are as yet insufficient data to make this method easy to use as a design tool. For the large number of shading systems for which there are no adequate data, it is still necessary to use the simplified solar heat gain coefficient data to estimate solar gain. In the following section, a detailed calcula-tion method is described, and it should be used for those cases where the necessary data are available.
qbt EDN θT1 L , f θ ( ) cos = qba EDN θ Nk k=1 L ∑ Ak: 1 L , ( ) f θ ( ) cos = T1 L , f θ ( ) Ak: 1 L , ( ) f θ ( ) qd Ed Er + ( ) SHGC 〈 〉D = qdt Ed Er + ( ) T1 L , θ ( ) 〈 〉D = qda Ed Er + ( ) Nn n=1 L ∑ An:1 L , f 〈 〉D = SHGC 〈 〉D [ ]25a SHGC 63.2° ( ) SHGC 63.2° ( ) [ ]25a = T1 2 , 0° ( ) T1 2 , 0° ( ) [ ]25a – { } T1 2 , 63.2° ( ) T1 2 , 0° ( ) -----------------------------× + 0.32 0.33 0.37 – ( ) + 0.24 0.33 ----------× 0.32 0.3 – 0.29 = = = SHGC 〈 〉D SHGC 〈 〉D [ ]25a = T1 2 , 0° ( ) T1 2 , 0° ( ) [ ]25a – { } T1 2 , 〈 〉D T1 2 , 0° ( ) ---------------------× + 0.36 0.33 0.37 – ( ) + 0.25 0.33 ----------× 0.36 0.3 – 0.33 = = = qs qt qa + EDN 63.2° ( )SHGC 63.2° ( ) cos Ed Er + ( ) SHGC 〈 〉D + = = 388.0 ( ) 0.29 ( ) 93.5 86.6 + ( ) 0.33 ( ) 172 W/(m2 ·K) = + = SHGC 63.2° ( ) T1 2 , 63.2° ( ) N1A1: 1 2 , ( ) f 63.2° ( ) N2A2: 1 2 , ( ) f 63.2° ( ) + + = 0.244 0.064 ( ) 0.289 ( ) 0.821 ( ) 0.022 ( ) + + 0.281 = = SHGC 〈 〉D T1 2 , 〈 〉D N1 A1: 1 2 , ( ) f 〈 〉D N2 A2: 1 2 , ( ) f 〈 〉D + + = 0.277 0.064 ( ) 0.271 ( ) 0.821 ( ) 0.036 ( ) + + 0.324 = = qs qt qa + EDN 63.2° ( )SHGC 63.2° ( ) cos Ed Er + ( ) SHGC 〈 〉D + = = 388.0 ( ) 0.281 ( ) 93.5 86.6 + ( ) 0.324 ( ) 167.4 W/(m2 ·K) = + = Fenestration 30.43 Generalized Calculation Procedure for Shading Devices and Fenestration Attachments A complex fenestration system is one that contains one or more nonspecular optical elements in the glazed area of the window. A nonspecular optical element is one for which light (or short-wave infrared radiation) incident on the element from a single spatial direction does not emerge traveling in a single transmitted direction and/or a single reflected direction. Examples of nonspecular ele-ments are shades, drapes, blinds, honeycombs, figured glass, ground glass, and other diffusers, lenses, prisms, and holographic glazings.
These systems may have a more complicated angle dependence than do the specular glazings discussed previously. The optical properties and solar heat gain coefficient may now depend on both angles defining the incident direction rather than simply on the angle θ between the incident direction and the surface normal. In addition, radiation incident from a given direction is not necessarily reflected or transmitted in a unique direction but may be distributed over a variety of directions. One must therefore introduce the con-cept of the directional-hemispherical transmittance (reflectance) [i.e., the fraction of radiation incident from a given direction that is transmitted (reflected) into the complete outgoing hemisphere].
With these extensions, the equation for the solar heat gain coef-ficient of a system with L layers becomes (103) where θ = incident angle relative to normal layer φ = azimuthal angle (in plane of layer, about normal) = directional-hemispherical front transmittance of system = directional absorptance of ith layer in system Nk = inward-flowing fraction of absorbed energy for ith layer in system The usual approach to complex fenestration solar heat gain has been either direct measurement or calculation of SHGC(θ, φ) at some specified incident direction, usually either normal incidence or θ = 30°, φ = 0°. Such values, with SHGC re-expressed as shading coefficient, are given for a variety of systems in Tables 18 through 20 and 22. Although in the past it has been assumed that the result would be the same for all incident directions, this is not generally a true assumption, and users of these tables should be aware that they apply only to incident directions of 30° or less and may not include important azimuthal angle dependence. While the values in these tables may give an approximate “first guess” in the absence of better information, for accurate and reliable data, one should rely on a measurement or calculation for the incident directions of interest unless there is independent reason for believing that incident angle or azimuthal dependencies are unimportant.
Solar-Thermal Separation It is possible to use Equation (103) to calculate the solar heat gain coefficient for a system from separate determinations of the system transmittance , the layer absorptances , and the inward-flowing fractions Nk. This has been termed solar-thermal separation, since the processes determining the Nk are thermal in nature, while those determining and (θ,φ) are solar-optical. Table 17 gives calorimetrically determined values of Nk for a number of generic glazing/shading systems (Klems and Kelley 1996). These are independent of the solar-optical properties of the particular system.
The transmittances and layer absorptances in Equation (103) may be determined by a variety of different methods, ranging from calculation to overall system measurement. The computation pro-cedure has advantages, for example, when one wishes to compare the performance of differing glazings or shading system colors in the same general configuration. Optical data are sometimes more readily available or more economically obtained than overall calo-rimetric measurements of SHGC. For example, is rather simply measurable by an optical technique using an integrating sphere. The layer absorptances present a more diffi-cult problem of determination, since the data easily available are likely to be the directional absorptance of an isolated layer, , whereas is the in-system layer absorptance (i.e., it includes the contributions of absorbed radiation multiply reflected back to the ith layer from all the other layers in the system).
Isolated layer absorptances must be corrected for this effect.
Calculating System Transmittance and Absorptances from Layer Properties The key feature of nonspecular elements is that they produce dis-tributions of outgoing radiation (in the solar-optical spectral region) in the transmitted and/or reflected hemisphere, even for incident radiation from a single direction. This means that they are charac-terized by bidirectional transmittance (BTDF) and reflectance (BRDF) distribution functions that give the outgoing radiance (energy flux per unit area per unit solid angle) as a fraction of the incident irradiance (energy flux per unit area): (104) where θo, φo = angles specifying outgoing direction θi, φi = angles specifying incident direction Table 17 Measured Layer Inward-Flowing Fractions Nk for Typical Fenestration System Bind Angle Below Horizontal Inner Shading Layer Inner Glass Between-Pane Shading Outer Glass Exterior Shading Layer Single glazing with interior shade 0.80 ± 0.08 0.08 ± 0.06 Single glazing with interior venetian blind −45° 0.69 ± 0.05 0.24 ± 0.09 30° 0.83 ± 0.08 0.21 ± 0.07 Closed 0.72 ± 0.07 0.14 ± 0.05 Single glazing with exterior venetian blind 45° 0.46 ± 0.12 0.04 ± 0.01 Double glazing with interior shade 0.85 ± 0.10 0.52 ± 0.12 0.28 ± 0.06 Double glazing with interior venetian blind 45° 0.86 ± 0.06 0.69 ± 0.14 0.21 ± 0.09 Double glazing with between-pane blind 45° 0.69 ± 0.14 0.45 ± 0.06 0.34 ± 0.10 −45° 0.76 ± 0.10 0.40 ± 0.07 0.27 ± 0.14 Low-e double glazing with between-pane blind 35° 0.46 ± 0.12 0.38 ± 0.05 0.32 ± 0.11 Double glazing with exterior venetian blind 45° 0.73 ± 0.13 0.28 ± 0.12 0.03 ± 0.02 SHGC θ φ , ( ) T1 L , fH θ φ , ( ) NkAk: 1 L , ( ) f θ φ , ( ) k=1 L ∑ + = T1 L , fH θ φ , ( ) Ai: 1 L , ( ) f θ φ , ( ) T1 L , fH θ φ , ( ) Ak: 1 L , ( ) f θ φ , ( ) T1 L , fH θ φ , ( ) Ak: 1 L , ( ) f T1 L , fH θ φ , ( ) Ak: 1 L , ( ) f θ φ , ( ) ak f θ φ , ( ) Ak: 1 L , ( ) f θ φ , ( ) ak f θ φ , ( ) I θo φo; θi φi , , ( ) τ θo φo; θi φi , , ( )E = J θo φo; θi φi , , ( ) ρ θo φo; θi φi , , ( )E = 30.44 2001 ASHRAE Fundamentals Handbook (SI) E = incident irradiance I, J = transmitted, reflected radiance In a real nonspecular element, these quantities are also functions of the location in space at which the radiation is incident on the fen-estration, as can be seen by visualizing a venetian blind. However, this level of detail is only useful if one wishes to find the detailed spatial images of the outgoing radiation patterns. For determining the solar heat gain, it is sufficient to consider the process as spatially averaged over the nonspecular device so that it can be considered as a thin uniform layer with only angular dependence (Klems 1994a).
By dividing the (transmission or reflection) hemisphere into a grid of solid angle sections, the bidirectional property functions can be approximated as matrices (Klems 1994b).
There are many standard commercial computer programs exe-cutable on personal computers that can perform the matrix calcula-tions, including many popular spreadsheet programs. Use of this bi-angular grid to characterize a nonspecular layer with azimuthal dependence requires handling matrices that are 145 × 145 elements, and for this level of complexity a special-purpose computer pro-gram for handling the large amount of data involved is probably desirable.
Figure 27 shows the results of such a calculation for an interior buff-colored blind (slat reflectance 62%) in combination with sealed double glazing (3 mm glass panes). The calculation used bidirectional transmittance and reflectance measurements averaged over a 200 mm square section of the blind, with 25 mm slats. These measurements were used to construct layer property matrices with the clear glazing properties taken from published literature (Rubin 1985). The SHGC was calculated using Equation (103), solar-optical transmittances and absorptances calculated by the matrix method of Klems (1994b), and inward-flowing fractions from Table 17 (Klems and Kelley 1996, Klems and Warner 1995).
Simplified Calculation Procedure The detailed methodology has been shown to agree with mea-surement (Klems et al. 1996) and should be used whenever accurate calculations are necessary and the data needed are available. Once the necessary data are available for a shading layer, Klems and Warner (1997) have shown how to calculate the performance for any combination of that shading layer and unshaded glazings. How-ever, the data necessary to use this exact calculation method are not readily available for many shading systems and are difficult to obtain by measurement for some (e.g., venetian blinds). It will therefore be necessary to use a simpler and less accurate method for most design purposes.
Two simplified approaches are described below. For some sim-ple glazing systems (single glazing and clear double glazing) at solar incident angles below 30°, measurements of the solar heat gain coefficient exist for a number of shading configurations. These are presented in tables, and the first approach, described in the sections on Exterior Shading and Indoor and Between Glass Shading Devices on Simple Fenestrations, may be used. For situations not covered by these tables, the second approach, consisting of the fol-lowing simplifying approximation together with the section on Completely Shaded Glazings, should be used.
First, one must determine the fraction (if any) of the incident radiation that passes through the fenestration essentially without encountering the shading element. This is termed the unshaded fraction Fu. For example, an overhang might shadow only part of a fenestration, a venetian blind or louver might under some condi-tions permit radiation to pass through it without encountering any of the slats, and for a drapery of open weave, some radiation passes through the gaps between threads and is effectively specularly transmitted (as opposed to radiation that is diffusely scattered by the threads). For this fraction of the incident radiation, the fenestration is essentially an unshaded one. The unshaded fraction is determined from geometric considerations, either those described below for exterior shading or those described for draperies. Once the unshaded fraction has been determined, the glazed area AG is divided into two equivalent glazings, an unshaded one of area FuAG and a completely shaded one of area (1 – Fu)AG. Heat gain through the unshaded equivalent glazing is calculated using the methods given in the section on Calculation of Solar Heat Gain, in the sub-section on Glazing Systems, for unshaded glazings. Heat gain through the completely shaded equivalent glazing is calculated by the methods given in the section on Completely Shaded Glazings, which assume that the shading layer is a uniform diffuse reflector or transmitter. A uniform diffuse reflector or transmitter is one for which a given incident irradiance produces an outgoing reflected or transmitted radiance that is the same in all directions and is indepen-dent of the incident direction.
EXTERIOR SHADING The most effective way to reduce the solar load on fenestration is to intercept direct radiation from the sun before it reaches the glass.
Fenestration products fully shaded from the outside reduce solar heat gain as much as 80%. In one way or another, fenestration can be shaded by roof overhangs, vertical and horizontal architectural projections, awnings, heavily proportioned exterior louvers, insect or shading screens, patterned screens having a weave designed for sunlight interception, sun screens of narrow fixed louvers, or a vari-ety of vegetative shades including trees, hedges, and trellis vines. In all exterior shading structures, the air must move freely to carry away heat absorbed by the shading and glazing materials. See man-ufacturers’ instructions regarding proper installation for achieving the expected performance by providing suitable free convection ventilation between the shading and glazing. Also, consider the geometry of the structures relative to changing sun position to deter-mine the times and quantities of direct sunlight penetration.
Detailed discussions of the effectiveness of various outside shading devices are given in Pennington (1968), Yellott (1972), and Ewing and Yellott (1976).
The general effect of exterior shading is to attenuate the solar radiation. Some of the beam radiation may pass through or around the shading, and this is accounted for by the unshaded fraction Fu Fig. 27 Contour Plot of Beam SHGC for Double Glazed Window with Interior Venetian Blind with Slats Tilted at 45° Fenestration 30.45 defined above. Some fraction of the remainder of the solar radiation will be transmitted by the shading system and will be incident on the glazing, but in the form of diffuse radiation. The actual angular dis-tribution of this radiation may be quite complex, but for simplicity it is treated here as uniformly diffuse radiation, and it is considered to compose a fraction EAC, the exterior solar attenuation coeffi-cient, of the total incident radiation interacting with the exterior shading system. The solar energy flux through the glazing with exterior shading is then given by (105) The quantities SHGC(θ) and in Equation (105) are the beam and diffuse solar heat gain coefficients of the unshaded glaz-ing system. The second of the two terms on the right-hand side of the equation is the flux through the completely shaded glazing and is discussed in more detail in Equation (114) and the associated text, which makes clear the physical basis of the EAC.
Louvers and Sunshades The ability of horizontal panels or louvers to intercept the direct component of solar radiation depends on their geometry and the profile or shadow-line angle Ω (Figure 28), defined as the angular difference between a horizontal plane and a plane tilted about a hor-izontal axis in the plane of the fenestration until it includes the sun.
The profile angle can be calculated by tan Ω = tan β/cos γ (106) For slat-type sunshades, the transmitted solar radiation consists of straight-through and transmitted-through components. When the profile angle Ω is above the cutoff angle (see Figure 29), straight-through transmission of direct radiation is completely eliminated, but the transmitted diffuse and the reflected-through components remain. Their magnitude depends largely on the reflectance of the sunshade surfaces and of exterior objects.
Narrow horizontal louvers, fabricated in conventional width-to-spacing ratios and framed as window screens, retain their shading characteristics, while gaining in effective transparency (view) by eliminating the coarse striation pattern of wide louvers. Table 18 gives values of Fu and EAC for several types of louvered sun screens. Commercially available sun screens completely exclude direct solar radiation when the profile angle exceeds approximately 26° (Groups 1 and 2) or 40° (Groups 3 and 4). Group designations are defined in the footnote to Table 18.
Roof Overhangs: Horizontal and Vertical Projections In the northern hemisphere, horizontal projections can consider-ably reduce solar heat gain on south, southeast, and southwest expo-sures during late spring, summer, and early fall. On east and west exposures during the entire year, and on south exposures in winter, the solar altitude is generally so low that to be effective, horizontal projections must be excessively long.
The shadow width SW and shadow height SH (Figure 30) pro-duced by the vertical and horizontal projections (PV and PH), respectively, can be calculated using the surface solar azimuth γ and the horizontal profile angle Ω determined by Equation (106).
SW = PV tan γ (107) SH = PH tan Ω (108) Note: When the surface solar azimuth γ is greater than 90° and less than 270°, the fenestration product is completely in the shade; thus, Sw = W + Rw and ASL = 0.
The sunlit ASL and shaded ASH areas of the fenestration product are variable during the day and can be calculated for each moment using the following relations (see Figure 30): ASL = [W − (SW − RW)][H − (SH − RH)] (109) Table 18 Unshaded Fractions (Fu) and Exterior Solar Attenuation Coefficients (EAC) for Louvered Sun Screens Profile Angle Group 1 Group 2 Group 3 Group 4 Transmittance Fu EAC Transmittance Fu EAC Transmittance Fu EAC Transmittance Fu EAC 10° 0.23 0.20 0.15 0.25 0.13 0.02 0.4 0.33 0.18 0.48 0.29 0.3 20° 0.06 0.02 0.15 0.14 0.03 0.02 0.32 0.24 0.18 0.39 0.2 0.3 30° 0.04 0.00 0.15 0.12 0.01 0.02 0.21 0.13 0.18 0.28 0.08 0.3 ≥ 40° 0.04 0.00 0.15 0.11 0.00 0.02 0.07 0.00 0.18 0.2 0.00 0.3 Group 1: Black, width over spacing ratio 1.15/1; 1.1 mm between louvers. Group 2: Light color; high reflectance, otherwise same as Group 1. Group 3: Black or dark color; w/s ratio 0.85/1; 1.5 mm between louvers. Group 4: Light color or unpainted aluminum; high reflectance; otherwise same as Group 3. U-factor = 4.83 W/(m2·K) for all groups when used with single glazing.
q FuEDN θSHGC θ ( ) cos = 1 Fu – ( )EDN θ cos Ed Er + + [ ]EAC SHGC 〈 〉D + SHGC 〈 〉D Fig. 28 Profile Angle for South-Facing Slat-Type Sunshades Fig. 29 Geometry of Slat-Type Sunshades 30.46 2001 ASHRAE Fundamentals Handbook (SI) ASH = A − ASL (110) where A is total fenestration product area. McCluney (1990) de-scribed an algorithm for computer use to calculate the unshaded frac-tion of a window equipped with overhangs, awnings, or side fins.
Example 15. A window in the southwest wall of a building at 40°N lati-tude is 870 mm wide and 1480 mm high. The depth of the horizontal and vertical projections is 150 mm, and they are located 75 mm beyond the edges of the window.
(a) Find the sunlit and shaded area of the window at 3:00 P.M. on July 21. (b) Find the depth of the projections necessary to fully shade the win-dow just described.
Solution: (a) The wall azimuth ψ for a southwest wall is +45° (Table 9). The solar azimuth φ can be calculated using Equation (15). At 3:00 P.M., H = 0.25 × 180 = 45°; from Table 7, for July 21, δ = 20.6°.
Find the solar altitude β using Equation (14): sin β = cos(40)cos(20.6)cos(45) + sin(40)sin(20.6) β = 47.2° Find the solar azimuth φ using Equation (15): cos φ = [sin(47.2)sin(40) – sin(20.6)]/[cos(47.2)cos(40)] φ = 76.7° Thus, from Equation (16), γ = 76.7 − 45 = 31.7°.
Using Equation (107), the width of the vertical projection shadow is SW = 150 |tan(31.7)| = 93 mm Using Equation (106), the profile angle for the horizontal projection is tan Ω = tan(47.2)/cos(31.7) Ω = 51.8° Using Equation (108), the height of the horizontal projection shadow is SH = 150 tan(51.8) = 191 mm Using Equations (109) and (110), the sunlit and shaded area of the window are now ASL = [870 − (93 − 75)] [1480 − (191 − 75)]/106 = 1.162 m2 ASH = (870 × 1480)/106 – 1.162 = 0.128 m2 (b) The shadow length necessary to fully shade the given window SH(fs) and SW(fs) from the horizontal and vertical projection are given by (see Figure 30) SH(fs) = 1480 + 75 = 1555 mm SW(fs) = 850 + 75 = 945 mm Thus, using Equations (107) and (108), PV(fs) = 1555φcot(31.7) = 1224 mm PH(fs) = 945|cot(51.8)| = 1530 mm For this example, because both horizontal and vertical projections do not need to fully shade the window, a horizontal projection of 1224 mm is satisfactory. Also, to accurately analyze the influence of external projections, an hour-by-hour calculation must be performed over the periods of the year for which shadowing is desired.
Example 16. Suppose that the glazing of Examples 10 and 14 has a black exterior louver shade with a width-to-spacing ratio of 1.15/1. Find the solar energy flux through the system if it is on a west-facing orientation under the conditions of Example 5.
Solution: In Example 5, the solar altitude was found to be β = 60.76° and the solar azimuth was found to be φ = 67.4°. The incident angle was found in Example 6 to be θ = 63.2°. Referring to Table 9, we see that ψ = 90° for a west-facing window, and so γ = φ – ψ = 67.4° – 90° = –22.6°. We can discard the sign of γ as irrelevant to the calculation of profile angle, since it merely determines whether the sun lies to the south or the north of the normal to the glazing. From Equation (106), we find that the profile angle is Ω = arctan(tanβ/tanγ) = arctan[tan(60.76°)/tan(22.6°)] = 76.9°.
From the footnote to Table 18, we see that the louver system corre-sponds to Group 1, and from the table, we see that for profile angles greater than or equal to 40° we should use Fu = 0 and EAC = 0.15. In Example 14, the (unshaded) glazing system was determined to have SHGC(63.2°) = 0.29 and = 0.33, respectively, using Table 13. In Example 7, the incident solar fluxes for these conditions were calculated to be EDNcos(θ) = 388 W/m2, Ed = 93.5 W/m2, and Er = 86.6 W/m2. Using Equation (105), we calculate Equations for Computer Calculations of External Shadowing of Inclined Surfaces Incident angle: θ = cos−1 (cos β cos γ sin Σ + sin β cos Σ) Vertical surface: θV = cos−1 (cos β cos γ) Horizontal surface: θH = cos−1 (sin β) Vertical projection profile angle: For θ > 90° ASL = 0 and ASH = A Vertical surface: ∆V = tan−1 (γ) for 90° > γ > 270° Horizontal surface: Fig. 30 Vertical and Horizontal Projections and Related Profile Angles for Vertical Surface Containing Fenestration SHGC 〈 〉D q 0 ( ) 388 ( ) 0.29 ( ) = 1 0 – ( ) 388 ( ) 93.5 86.6 + + [ ] 0.15 ( ) 0.33 ( ) 28 W/m2 = + ∆ γ β cos sin θ cos ----------------------- θ ; 1 – 90° < tan = ∆H γ sin β tan ----------- for all γ 1 – tan = Fenestration 30.47 Horizontal projection profile: Angle: Vertical surface: Horizontal surface: Length of shadow from vertical projection: SW = PV tan ∆ Length of shadow from horizontal projection: SH = PH tan Ω Sunlit and shaded areas of the fenestration product are deter-mined using Equations (109) and (110).
where φ = solar azimuth β = solar altitude γ = surface solar azimuth Σ = surface tilt angle PV = vertical projection depth PH = horizontal projection depth W = fenestration product width H = fenestration product height RW = width of opaque surface between fenestration product and vertical projection RH = height of opaque surface between fenestration product and horizontal projection A = total projected area of the fenestration product θ = angle of incidence Ω= horizontal projection profile angle ∆= vertical projection profile angle INDOOR AND BETWEEN-GLASS SHADING DEVICES ON SIMPLE FENESTRATIONS Venetian Blinds and Roller Shades Most fenestration has some type of internal shading to provide privacy and aesthetic effects, as well as to give varying degrees of sunshine control (Ozisik and Schutrum 1960). This section provides a method of calculating the approximate SHGC for such fenestra-tions for a selection of simple and somewhat common shading ele-ments and glazing systems. The information presented here is based on measurements that predate many modern developments in fen-estration systems. For this reason, desired systems may not be rep-resented in the tables, or it may be difficult to determine whether the desired conditions match those of the table. If information is not available in this section or greater accuracy or detail is desired, the reader should refer to the subsequent section on Completely Shaded Glazings, where calculation formulas are presented. However, the present state of data availability on shading devices does not permit great accuracy in the calculation of heat flow through shaded fenes-trations. The user should proceed with caution.
As discussed in the text and equations associated with Equation (129), the details of solar transmission through a glazing with inte-rior shading are quite complex. However, if one is interested only in the approximate total heat flux through the fenestration, rather than in the detailed separation into transmitted and absorbed (and ther-mally retransmitted) fluxes and the distribution of the absorbed energy among the fenestration layers, it will frequently be the case that measurements made on a fenestration under one set of condi-tions can be extrapolated to other fenestrations and conditions to give an adequate answer. In this case, the heat flux is represented by (111) where the quantity in the brackets represents the heat flow through the unshaded glazing system [compare Equations (97) and (100)] and the constant interior solar attenuation coefficient IAC represents the fraction of that heat flow that enters the room, some energy having been excluded by the shading. The solar heat gain coefficients in Equation (111) may be obtained from Table 13 or calculated by Equations (79) and (84) for unshaded glazing.
Table 19 gives values of IAC (derived from measurements) for a variety of glazing and shading combinations.
It will be noticed that the IAC bears a certain similarity to the shading coefficient. There is, however, an important difference: we must calculate the solar heat flux through the unshaded glazing at the appropriate angle before applying the IAC. With the shading coefficient, only the angular dependence of single glazing was included (through the now-discarded SHGF). The effectiveness of any internal shading device depends on its ability to reflect incom-ing solar radiation back through the fenestration before it can be absorbed and converted into heat within the building. Table 21 lists approximate values of solar-optical properties for the typical indoor shading devices described in Tables 19 and 20.
For shading between the panes of a double-glazing system, an equation similar to Equation (111) can be used, but in this case one defines the unshaded glazing system to be the portion of the glazing that is exterior to the shading and defines a new coefficient, the between-pane solar attenuation coefficient (BAC), to describe the effect of the shading system together with the portion of the glazing that is interior to it: (112) Values of BAC for those systems for which measurements are available are given in Table 20. For other systems, one should use the detailed equations of the section on Completely Shaded Glazings.
Example 17. Find the approximate SHGC and solar heat flux for a heat-absorbing double glazing with a light-colored interior venetian blind under the conditions of Example 5. The heat-absorbing double glazing has a visible transmittance of 0.41.
Solution: Referring to Table 19, we see that an interior, light venetian blind, heat-absorbing double glazing has an IAC value of 0.66. For the conditions of Example 5, the incident angle was found in Example 6 to be θ = 63.2°, and in Example 7 the incident solar fluxes for these conditions were calculated to be EDNcosθ = 388 W/m2, Ed = 93.5 W/m2, and Er = 86.6 W/m2. Referring to Table 13, we see that the closest glazing system corresponds to ID 5h, 6 mm gray glass exterior, clear interior, which has SHGC(60°) = 0.37, SHGC(70°) = 0.29, and 〈SHGC〉D = 0.39. Interpo-lating in incident angle between the two table values gives SHGC(63.2°) = 0.34. Using Equation (111), we find the solar heat flux q = [388 × 0.34 + (93.5 + 86.6) × 0.39]0.66 = 133 W/m2.
There is some ambiguity about the meaning of the SHGC in this case. One could pick out the terms in Equation (111) corresponding to either the effective beam or diffuse SHGC for the overall system, but, given the fact that the IAC values in Table 19 were measured in out-door calorimeters so that the incident radiation consisted of both beam and diffuse radiation, it is most reasonable to calculate an effective SHGC for these conditions by dividing the estimated solar heat flux by the total incident intensity: Example 18. Estimate the solar heat flux for the glazing and conditions of Example 17, but assume that the venetian blind is between the glazing panes.
Solution: Referring to Table 20, we see that a light venetian blind between the panes of a heat-absorbing double-glazing system has a BAC value of 0.28. We use this value in Equation (112), but in contrast to Example 17, we need the SHGC values of the exterior glazing layer rather than the full glazing system. In Example 17, we used ID 5h, Ω β Σ β γ Σ cos cos cos – sin sin θ cos ------------------------------------------------------------------ for θ 1 – 90° < tan = ΩV β tan γ cos ----------- for 90° γ 270° > > 1 – tan = ΩH γ cos β tan ----------- for 90° γ 270° < < 1 – tan = q EDN θ cos SHGC θ ( ) Ed Er + ( ) SHGC 〈 〉D + [ ]IAC = q EDN θSHGC θ ( ) cos Ed Er + ( ) SHGC 〈 〉D + [ ]extBAC = SHGCeff 133 388 93.5 86.6 + + ------------------------------------------0.23 = = 30.48 2001 ASHRAE Fundamentals Handbook (SI) Table 19 Interior Solar Attenuation Coefficients (IAC) for Single or Double Glazings Shaded by Interior Venetian Blinds or Roller Shades Glazing Systema Nominal Thicknessb Each Pane, mm Glazing Solar Transmittanceb Glazing SHGC IAC Venetian Blinds Roller Shades Outer Pane Single or Inner Pane Medium Light Opaque Dark Opaque White Translucent Light Single Glazing Systems Clear, residential 3 0.87 to 0.80 0.86 0.75d 0.68d 0.82 0.40 0.40 Clear, commercial 6 to 13 0.80 to 0.71 0.82 Clear, pattern 3 to 13 0.87 to 0.79 Heat absorbing, pattern 3 0.59 Tinted 5, 5.5 0.74, 0.71 Above glazings, automated blindse 0.86 0.64 0.59 Above glazings, tightly closed vertical blinds 0.85 0.30 0.26 Heat absorbingf 6 0.46 0.59 0.84 0.78 0.66 0.44 0.47 Heat absorbing, pattern 6 Tinted 3, 6 0.59, 0.45 Heat absorbing or pattern 0.44 to 0.30 0.59 0.79 0.76 0.59 0.41 0.47 Heat absorbing 10 0.34 Heat absorbing or pattern 0.29 to 0.15 0.24 0.37 0.99 0.94 0.85 0.66 0.73 Reflective coated glass 0.26 to 0.52 0.83 0.75 Double Glazing Systemsg Clear double, residential 3 0.87 0.87 0.76 0.71d 0.66d 0.81 0.40 0.46 Clear double, commercial 6 0.80 0.80 0.70 Heat absorbing doublef 6 0.46 0.8 0.47 0.72 0.66 0.74 0.41 0.55 Reflective double 0.17 to 0.35 0.90 0.86 Other Glazings (Approximate) 0.83 0.77 0.74 0.45 0.52 ± Range of Variationh 0.15 0.17 0.16 0.21 0.21 a Systems listed in the same table block have the same IAC.
b Values or ranges given for identification of appropriate IAC value; where paired, solar transmittances and thicknesses correspond. SHGC is for unshaded glazing at normal incidence.
c Typical thickness for residential glass.
d From measurements by Van Dyke and Konen (1982) for 45° open vene-tian blinds, 35° solar incidence, and 35° profile angle.
e Use these values only when operation is automated for exclusion of beam solar (as opposed to day-light maximization). Also applies to tightly closed horizontal blinds.
f Refers to gray, bronze and green tinted heat-absorbing glass (on exterior pane in double glazing) g Applies either to factory-fabricated insulating glazing units or to prime windows plus storm windows.
h The listed approximate IAC value may be higher or lower by this amount, due to glazing/shading interactions and variations in the shading properties (e.g., manufacturing tolerances).
Table 20 Between-Glass Solar Attenuation Coefficients (BAC) for Double Glazing with Between-Glass Shading Type of Glass Nominal Thickness, Each Pane Solar Transmittancea Description of Air Space Type of Shading Venetian Blinds Louvered Sun Screen Outer Pane Inner Pane Light Medium Clear out, Clear in 2.4, 3 mm 0.87 0.87 Shade in contact with glass or shade separated from glass by air space.
0.33 0.36 0.43 Clear out, Clear in 6 mm 0.80 0.80 Shade in contact with glass-voids filled with plastic.
— — 0.49 Heat-absorbingb out, Clear in Shade in contact with glass or shade separated from glass by air space.
0.28 0.30 0.37 6 mm 0.46 0.80 Shade in contact with glass-voids filled with plastic.
— — 0.41 aRefer to manufacturers’ literature for exact values.
bRefers to gray, bronze and green tinted heat-absorbing glass.
Table 21 Properties of Representative Indoor Shading Devices Shown in Tables 19 and 20 Indoor Shade Solar-Optical Properties (Normal Incidence) Transmittance Reflectance Absorptance Venetian blindsa (ratio of slat width to slat spacing 1.2, slat angle 45°) Light colored slat 0.05 0.55 0.40 Medium colored slat 0.05 0.35 0.60 Vertical blinds White louvers 0.00 0.77 0.23 Roller shades Light shades (translucent) 0.25 0.60 0.15 White shade (opaque) 0.00 0.65 0.35 Dark colored shade (opaque) 0.00 0.20 0.80 aValues in this table and Tables 19 and 20 are based on horizontal venetian blinds. However, tests show that these values can be used for vertical blinds with good accuracy.
Fenestration 30.49 which Table 13 shows had an exterior glazing that is 6 mm gray glass.
We find that ID 1h is the entry for clear 6 mm gray glass, for which SHGC(60°) = 0.51, SHGC(70°) = 0.44, and 〈SHGC〉D = 0.52. Interpo-lating the beam SHGC values in incident angle gives SHGC(63.2°) = 0.49. We then use Equation (112) to calculate This indicates that having the blind between the glazings yields slightly better solar heat gain rejection than an interior placement.
Draperies Draperies reduce heating and cooling loads, depending on the type and the use by the occupant. Rudoy and Duran (1975) found annual reductions between 5 and 20%. An approximate model for determining the SHGC of free-standing vertical interior shades was developed by McCluney and Mills (1993). Solar heat gain may be estimated by using Equation (111) and the IAC values listed in Table 22. The solar optical properties of drapery fabrics can be deter-mined accurately by laboratory tests (Yellott 1963), and manufac-turers can usually supply solar transmittance and reflectance values for their products. In addition to these properties, the open-ness factor (ratio of the open area between the fibers to the total area of the fabric) is a useful property that can be measured exactly (Keyes 1967, Pennington and Moore 1967). It can also be esti-mated by inspection, since the human eye can readily distinguish between tightly woven fabrics that permit little direct radiation to pass between the fibers and loosely woven fabrics that allow the sun’s rays to pass freely. Drapery fabrics can be classified in terms of their solar-optical properties as having specific values of fabric transmittance and reflectance. Fabric reflectance is the major factor in determining the ability of a fabric to reduce solar heat gain. Based on their appearance, draperies can also be classified by yarn color as dark, medium, and light and by weave as closed, semiopen, and open.
The apparent color of a fabric is determined by the reflectance of the yarn itself. The figure in Table 22 shows yarn reflectance. Fig-ure 31 classifies drapery fabrics into nine types, rated by openness and yarn reflectances.
Figure 31 guides in the use of Table 22 for a fabric-glass combi-nation when the solar-optical properties are unknown. Whenever possible, fabric reflectance and transmittance values should be obtained from the manufacturer, which permits more accurate solar heat flux estimates to be made. Visual estimations of openness and yarn reflectance, interpreted through Table 22, are valuable in judging the effectiveness of drapes for (1) protection from exces-sive radiant energy from either sunlight or sun-heated glass, (2) brightness control, (3) providing either outward view or privacy, and (4) sound control.
Table 22 applies to glass and a single drape hung with 100% full-ness (drapery width is twice the width of the draped area). If the drapery is hung flat, like a fenestration product shade, a different IAC applies; with a low transmittance and high reflectance, the IAC is appreciably lower. As an extreme example, a flat opaque drapery having an aluminized or similar coating with reflectance of 0.80, in combination with 6 mm clear glass, has an SHGC of 0.18, compared with 0.32 for this material in draped form. Pennington and Moore (1967) explain the effect of folding drapery materials to provide 100% fullness and describe a method for calculating SHGC when materials are used flat.
Example 19. A drape with 100% fullness, having a fabric transmittance of 0.20 and a fabric reflectance of 0.40, is used with 6 mm glass. What IAC should be used?
Solution: Locating the point on the ideogram in Table 22 correspond-ing to fabric reflectance 0.40 and transmittance 0.20, we find that it is close to the diagonal line labeled “F”. In column F of the table, we find a value of 0.58 on the line corresponding to 6 mm clear glass. This is the desired IAC value.
Example 20. Determine the fabric designator for a fabric having an open-ness factor of 0.10 and a yarn reflectance of 0.60.
Solution: On the figure in Table 22, these lines intersect in the area of Designator IIL. Refer also to Figure 30. Fabric is semiopen and light in color. Additional information: probable fabric reflectance is 0.50, and fabric transmittance is 0.35.
COMPLETELY SHADED GLAZINGS This section presents approximate models (Klems 2001) for cal-culating the heat flow through glazing systems with shading for all cases not covered by the sections on Exterior Shading and Indoor and Between-Glass Shading Devices on Simple Fenestrations (i.e., glazing or shading systems not covered by the tables or with inci-dent angles above 30°). With the exception of exterior fins and over-hangs, all shading is modeled as a planar, ideally diffuse layer parallel to the glass layers, and the models do not apply if there is more than one shading layer. The effects of specular transmission or partial shading are assumed to have been separated out using the method described above under Simplified Calculation Procedure.
The shading layer is denoted by the subscript S. To apply these models, one or more of the following quantities must be known for the shading (the number depends on the position of the shading in the glazing system): = front directional-hemispherical transmittance of the shading layer = back directional-hemispherical transmittance of the shading layer = front directional-hemispherical reflectance of the shading layer = back directional-hemispherical reflectance of the shading layer q 388 0.49 × 93.5 86.6 + ( ) + 0.52 × [ ]0.41 116 W/m2 = = Fig. 31 Designation of Drapery Fabrics TS fH TS bH RS fH RS bH 30.50 2001 ASHRAE Fundamentals Handbook (SI) Table 22 Interior Solar Attenuation Coefficients for Single and Insulating Glass with Draperies Glazing Glass Trans-mission Glazing SHGC (No Drapes) IAC A B C D E F G H I J Single glass 3 mm clear 0.86 0.87 0.87 0.82 0.74 0.69 0.64 0.59 0.53 0.48 0.42 0.37 6 mm clear 0.80 0.83 0.84 0.79 0.74 0.68 0.63 0.58 0.53 0.47 0.42 0.37 13 mm clear 0.71 0.77 0.84 0.80 0.75 0.69 0.64 0.59 0.55 0.49 0.44 0.40 6 mm heat absorbing 0.46 0.58 0.85 0.81 0.78 0.73 0.69 0.66 0.61 0.57 0.54 0.49 13 mm heat absorbing 0.24 0.44 0.86 0.84 0.80 0.78 0.76 0.72 0.68 0.66 0.64 0.60 Reflective coated — 0.52 0.95 0.90 0.85 0.82 0.77 0.72 0.68 0.63 0.60 0.55 — 0.44 0.92 0.88 0.84 0.82 0.78 0.76 0.72 0.68 0.66 0.62 — 0.35 0.90 0.88 0.85 0.83 0.80 0.75 0.73 0.70 0.68 0.65 — 0.26 0.83 0.80 0.80 0.77 0.77 0.77 0.73 0.70 0.70 0.67 Insulating glass, 6 mm air space (3 mm out and 3 mm in) 0.76 0.77 0.84 0.80 0.73 0.71 0.64 0.60 0.54 0.51 0.43 0.40 Insulating glass 13 mm air space Clear out and clear in 0.64 0.72 0.80 0.75 0.70 0.67 0.63 0.58 0.54 0.51 0.45 0.42 Heat absorbing out and clear in 0.37 0.48 0.89 0.85 0.82 0.78 0.75 0.71 0.67 0.64 0.60 0.58 Reflective coated — 0.35 0.95 0.93 0.93 0.90 0.85 0.80 0.78 0.73 0.70 0.70 — 0.26 0.97 0.93 0.90 0.90 0.87 0.87 0.83 0.83 0.80 0.80 — 0.17 0.95 0.95 0.90 0.90 0.85 0.85 0.80 0.80 0.75 0.75 Interior Solar Attenuation (IAC) Notes: 1. Interior attenuation coefficients are for draped fabrics.
2. Other properties are for fabrics in flat orientation.
3. Use fabric reflectance and transmittance to obtain accu-rate IAC values.
4. Use openness and yarn reflec-tance or openness and fabric reflectance to obtain the vari-ous environmental characteris-tics, or to obtain approximate IAC values.
Classification of Fabrics I = Open weave II = Semiopen weave III = Closed weave D = Dark color M = Medium color L = Light color To obtain fabric designator (IIIL, IM, etc.). Using either (1) fabric transmittance and fabric reflectance coordinates, or (2) openness and yarn reflectance coordinates, find a point on the chart and note the des-ignator for that area. If properties are not known, the classification may be approximated by eye as described in the note in Figure 31. To obtain interior attenuation (IAC). (1) Locate drapery fabric as a point using its known properties, or approximate using its fabric classification designator. For accuracy, use fabric transmittance and fabric reflectance; (2) follow diagonal IAC lines to lettered columns in the table. Find IAC value in selected column on line corresponding to glazing used. For example, IAC is 0.4 for 6 mm clear single glass with IIIL drapery (Column H).
Fenestration 30.51 Note: Unlike specular transmittance, front and back directional-hemispherical transmittances are not necessarily equal.
To match the assumptions of the model, the specular component should be excluded from the transmittance but included in the reflectances. Since the shading layer is assumed to be an ideal dif-fuser, the above quantities are the same for all incident directions. In addition, the notation 〈〉D will be applied to various optical prop-erties of specular glazings. This notation means the hemispherical average of the quantity over all incident directions and is the quan-tity labeled “diffuse” in Table 13. For example, for a particular glass layer j, 〈Tj〉D would be the hemispherical average transmittance of the glass layer, which is also the transmittance for ideally diffuse incident radiation. Another addition to the notation is needed for discussing absorp-tion in this section. In the treatment of unshaded multiple glazings, the notation was used to denote the actual front absorption of the nth layer of an L-layer system. Since only one system was under consideration, no ambiguity could arise from the fact that the nota-tion does not specifically reference the number of layers in the sys-tem. Here it will be necessary to discuss multi-layer systems that are subdivided into one or more subsystems. Since the actual absorp-tion depends on the other layers in the system because of reflections from those layers and because it is referenced to the incident inten-sity on the front (or back) surface of the system, rather than on the specific layer, it is necessary to include the particular subsystem in the notation. The following notation will be used. We assume that the overall system has L layers that are numbered sequentially from outside to inside, as before. Now is the actual front absorption of layer n in subsystem of layers M through N, and similarly for back absorption. In this notation, the front absorptance of layer n in the overall system, which previously was called , would now be denoted . Note that for the subsystem (M, N) the front absorptance is the fraction of the radia-tion incident on the front of layer M that is absorbed in layer n, while the back absorptance is the fraction of the radiation inci-dent on the back side of layer N that is absorbed in layer n.
In all cases, we assume that there is a specular glazing of L layers, labeled as described in the section on Optical Properties of Multi-ple-Layer Glazing Systems in the discussion of unshaded glazing optical properties. The shading layer, denoted S, is not counted in this labeling. Differing locations of the shading layer are discussed in the following sections.
Exterior Shading by Overhangs, Fins, etc.
In this case, we neglect (1) reduction of the sky view by the shad-ing and (2) reflection of diffuse radiation by the face of the shading seen by the glazing. It will usually be the case that this results in an overestimate of the heat flow, but exceptions are possible (e.g., highly reflective ground and light-colored overhang). Here the effect of the shading is simply to eliminate the direct beam radia-tion, and one is left with an unshaded glazing with diffuse radiation incident; the relevant optical properties are the hemispherical aver-ages of those of the unshaded glazing system.
(113) The designer should be aware that because the diffuse solar radi-ation is small compared to that of the direct beam, reflection of direct beam radiation from other surfaces visible to the glazing can result in significant additional heat flow (e.g., beam radiation falling on the “wrong” side of a fin or overhang intended to shade an adja-cent window).
Exterior Shading Layer Examples of planar exterior shading layers include exterior lou-vers or venetian blinds. It is assumed that air has free flow between the exterior shading system and the glazing system. Under these conditions, as Table 17 shows, the fraction of the energy absorbed in the shading that flows inward is small. If this energy flow is neglected, then the whole effect of the shading layer is to modify the intensity of the (diffuse) radiation incident on the unshaded glazing, and the heat flow is (114) where 〈SHGC〉D is for the unshaded glazing system. To compute the amount of energy absorbed in the shading layer, one must estimate the isolated-layer absorptances of the shading layer from (115) When attached to the glazing, and using S = 0 to fit into the labeling scheme, the shading layer front absorptance is then (116) where refers to the unshaded glazing system. may either be obtained from Table 13 or calculated as described previously in the section on Solar-Optical Properties of Glazings. The addition to the heat flow in Equation (114) is then (117) where the inward-flowing fraction NS for the shading layer is taken from the appropriate line of Table 17.
Between-Glass Shading Layer One must first identify the position of the shading layer within the L-layer glazing system, remembering that the labeling of glaz-ing layers did not count the shading layer S. We count the layers, beginning with the one in contact with the exterior air, until we reach the layer next to the shading layer. Let the number of that layer be K. We now relabel the shading layer by setting S = K + 1 and rela-bel all subsequent glazing layers, making the one to the interior of S layer K + 2 (or S + 1), and continuing until we reach glazing layer L, which becomes L + 1 in the new labeling scheme. We now have an L + 1 layer system that consists of three parts: (1) an exterior glazing subsystem of S – 1 layers, (2) the shading layer S, and (3) an interior glazing subsystem, consisting of layers S + 1 to L + 1, a total of (L + 1) − S or L − K layers in all. Both the exterior glazing and interior glazing subsystems are assemblies of unshaded, specular glazing layers, and their optical properties are calculated by the methods described in the section on Optical Properties of Multiple-Layer Glazing Systems for unshaded glazings. The exterior and interior glazing subsystems will often be either a single or double glazing, and, if they are sufficiently common, their properties may then be found in Table 13.
For calculational purposes, it is convenient to consider two other subsystems, one consisting of the shading layer together with the exterior glazing, an S-layer subsystem consisting of layers 1 through S, and the other consisting of the shading layer together with the interior glazing, consisting of layers S through L + 1, a total of L – S + 2 or L – K + 1 layers. For the former subsystem, we will need the directional-hemispherical transmittance: An f θ ( ) An: M N , ( ) f θ ( ) An f θ ( ) An: 1 L , ( ) f θ ( ) An: M N , ( ) f An: M N , ( ) b q Ed Er + ( ) SHGC 〈 〉D = q 1 Fu – ( )EDN θ cos Ed Er + + [ ] = TS fH 1 RS bH R1 L , f 〈 〉D – --------------------------------------- SHGC 〈 〉D aS f 1 TS fH – RS fH – = aS b 1 TS bH – RS bH – = AS: S L , ( ) f aS f aS bTS fH R1 L , f 〈 〉D 1 RS bH R1 L , f 〈 〉D – ---------------------------------------+ = R1 L , f R1 L , f q ∆ 1 Fu – ( )EDN θ cos Ed Er + + [ ]AS fNS = 30.52 2001 ASHRAE Fundamentals Handbook (SI) (118) and the hemispherical back reflectance (which is the same for all incident directions): (119) For the latter subsystem, we need the hemispherical front reflectance: (120) The directional-hemispherical transmittance of the total system is then (121) The absorptances for the kth layer are for k < S (122) for k = S (123) for k > S (124) The hemispherical average of the appropriate above equation then gives .
For convenience, we split the heat flow up into three parts: that due to transmitted direct beam radiation (subscript B), that due to transmitted diffuse beam and ground-reflected radiation (subscript d), and that due to energy absorbed in the various layers and flowing inward by the normal mechanisms of heat transfer (subscript a): (125) We note that qB and qd consist of diffusely distributed radiation in the solar wavelength range, while qa is a heat flux. The quantities in Equation (125) are given by (126) (127) (128) The values of the layer-specific inward-flowing fractions Nk are chosen from the appropriate lines of Table 17. All types of shading should be considered equivalent when using Table 17 with this sim-plified method; accordingly, corresponding entries in the lines for differing blind tilts should be averaged in the case of double glazing.
For systems with more than two glazing layers, for which no information exists in Table 17, the following estimate should be used. If the glazing system contains no low-e coating, then informa-tion for double glazing with between-pane shading should be aver-aged; if the system contains one or more low-e coatings, then the line corresponding to low-e double glazing should be chosen. For k < S, use for Nk the value in Table 17 corresponding to the outer glass; for k = S, use the value for the shading layer; and for k > S, use the value corresponding to the inner glass.
Interior Shading Layer For interior shading, we can consider the shading layer S to form the (L + 1)th layer of an (L + 1) layer system. The total directional-hemispherical transmittance of this system is given by (129) and the total diffuse transmittance of the system is the hemispherical average of this quantity is given by (130) In discussing layer absorptance, we need to consider a layer both in its role in the L-layer unshaded system and in the (L + 1)-layer system formed by adding the shading layer. The L-layer system is an unshaded one, so all quantities pertaining to that system can be cal-culated from Equations (60) through (65), (79), and (84) for the optical properties of multilayer, unshaded glazings above. If it is a sufficiently common system, its properties may be found in Table 13, and the table values may be linearly interpolated to the desired incident angle; see also the previous section on Spectral Averaging of Glazing Properties and Examples 11, 12, and 13. The addition of the shading layer causes an additional contribution to the absorp-tance in each layer of the L-layer system due to the backward inci-dence of diffuse radiation reflected by the shade: T1 S , fH θ ( ) T1 S 1 – , θ ( )TS fH 1 RS fH R1 S 1 – , b 〈 〉D – ---------------------------------------------= R1 S , bH RS bH R1 S 1 – , b θ ( ) 〈 〉DTS bHTS fH 1 RS fH R1 S 1 – , b 〈 〉D – ------------------------------------------------------+ = RS L 1 + , fH RS fH RS 1 L 1 + , + f 〈 〉DTS bHTS fH 1 RS bH RS 1 L 1 + , + f 〈 〉D – -----------------------------------------------------+ = T1 L 1 + , fH θ ( ) T1 S , fH θ ( )TS 1 + L 1 + , fH 1 R1 S , bH RS 1 L 1 + , + f 〈 〉D – ------------------------------------------------------= T1 S 1 – , θ ( )TS fHTS 1 + L 1 + , fH 1 RS fH R1 S 1 – , b 〈 〉D – [ ] 1 R1 S , bH RS 1 L 1 + , + f 〈 〉D – [ ] -------------------------------------------------------------------------------------------------------------= Ak: 1 L 1 + , ( ) f θ ( ) Ak: 1 S 1 – , ( ) f θ ( ) = Ak: 1 S 1 – , ( ) b 〈 〉D T1 S , fH θ ( ) 1 RS L 1 + , fH R1 S 1 – , b 〈 〉D – ------------------------------------------------------+ Ak: 1 L 1 + , ( ) f θ ( ) aS fT1 S 1 – , θ ( ) 1 RS L 1 + , fH R1 S 1 – , b 〈 〉D – ------------------------------------------------------= aS bT1 S , fH θ ( ) RS 1 L 1 + , + fH 〈 〉D 1 R1 S , bH RS 1 L 1 + , + fH 〈 〉D – -----------------------------------------------------------+ Ak: 1 L 1 + , ( ) f θ ( ) Ak: S 1 + L 1 + , ( ) f 〈 〉D T1 S , fH θ ( ) 1 R1 S , bH RS 1 L 1 + , + fH 〈 〉D – ------------------------------------------------------= Ak: 1 L 1 + , ( ) f 〈 〉D q qB qd qa + + = qB EDN θ ( )T1 L 1 + , fH θ ( ) cos = qd Ed Er + ( ) T1 L 1 + , fH 〈 〉D = qa EDN θ ( ) Ak: 1 L 1 + , ( ) f D θ ( )Nk k=1 L+1 ∑ cos = Ed Er + ( ) Ak: 1 L 1 + , ( ) f 〈 〉DNk k=1 L+1 ∑ + T1 L 1 + , fH θ ( ) T1 L , θ ( )TS fH 1 RS fH R1 L , b 〈 〉D – --------------------------------------= T1 L 1 + , fH 〈 〉D T1 L , θ ( ) 〈 〉DTS fH 1 RS fH R1 L , b 〈 〉D – --------------------------------------= Fenestration 30.53 (131) and the hemispherical average of this quantity over incident direc-tion is given by (132) The absorptance in the shade layer is (133) with, again, a hemispherical average given by (134) The heat flow is calculated from Equations (125) through (128).
The inward-flowing fractions Nk in Equation (128) are chosen from Table 17 when possible. The most thermally resistive glazing system with interior shading in this table is double glazing. For higher resistance systems (low-e double glazing, triple glazing, or higher, etc.), there are three possible procedures: 1. Calculate the Nk (k < S) for the unshaded glazing, and for the shading layer (k = S), take the value from the Inner Shading Layer column of Table 17 for the system that is closest to the desired one.
2. Treat the shading layer as an additional (uncoated, unless the shading layer has low emissivity on the exterior side) glazing and calculate the Nk for the resulting (L + 1)-layer glazing.
3. Take N1, N2, and the value for the interior shading layer from the table entries for double glazing with interior shade, and take Nk = N2 for all subsequent glazing layers. Using as an alternative assumption NL = N2 and Nk = N1 for all k < L will give an estimate of the range of error to be expected. Method (1) will produce values of Nk for k ≤ L that tend to be too high, and a value of NS (S = L + 1) that may be too low. Method (2) will produce values of Nk for k ≤ L that tend to be too low and a value of NS that also tends to be too low (since some heat that transfers outward by convection from layer S would be carried to the interior with an unsealed air space). By comparing the resulting solar energy fluxes using each of the three methods, one can obtain an estimate of the sensitivity of the results to uncertainty in calculating Nk.
Example 21. Calculate the solar energy flux through the glazing system of Example 10 with a closed interior translucent light shade, under the conditions of Example 5.
Solution: From Table 21, we take the properties of a translucent light shade to be , , and . We assume that the reflectance is diffuse and that the transmittance is independent of incident angle. By using method (1) above for determining the inward-flowing fractions, we can take the values of N1 = 0.064 and N2 = 0.821 calculated in Example 14, and we take NS = 0.85 from the entry in Table 17 for double glazing with interior shade. We use the property values for an incident angle θ of 63.2° (from Example 6) that were cal-culated for the unshaded glazing in Example 11: T1,2(63.2°) = 0.244, = 0.446, = 0.444, = 0.289, = 0.144, = 0.022, = 0.167. We use the hemispherical average properties calculated in Example 12: = 0.277, = 0.405, = 0.442, = 0.271, = 0.138, = 0.036, and = 0.141. In the shaded fenestration, the shade comprises layer 3 (i.e., S = 3); the total transmittance at an incident angle of 63.2° is calculated from Equation (129).
The layer absorptances are calculated from Equation (131): and Equation (133): From this we calculate the solar heat gain coefficient, using Equation (79), The corresponding hemispherical average properties are calculated from Equations (130), (131), and (134): Using these values, one calculates the diffuse solar heat gain coefficient from Equation (84): Ak: 1 L 1 + , ( ) f θ ( ) = Ak: 1 L , ( ) f θ ( ) Ak: 1 L , ( ) b 〈 〉D T1 L , θ ( )RS fH 1 RS fH R1 L , b 〈 〉D – --------------------------------------+ Ak: 1 L 1 + , ( ) f 〈 〉D = Ak: 1 L , ( ) f 〈 〉D Ak: 1 L , ( ) b 〈 〉D T1 L , 〈 〉DRS fH 1 RS fH R1 L , b 〈 〉D – --------------------------------------+ AS: 1 L 1 + , ( ) f θ ( ) AL 1: 1 L 1 + , ( ) + f θ ( ) ≡ T1 L , θ ( )aS f 1 RS fH R1 L , b 〈 〉D – --------------------------------------= AL 1: 1 L 1 + , ( ) + f 〈 〉D T1 L , 〈 〉DaS f 1 RS fH R1 L , b 〈 〉D – --------------------------------------= TS fH 0.25 = RS fH 0.60 = aS f 0.15 = R1 2 , f 63.2° ( ) R1 2 , b 63.2° ( ) A1: 1 2 , ( ) f 63.2° ( ) A1: 1 2 , ( ) b 63.2° ( ) A2: 1 2 , ( ) f 63.2° ( ) A2: 1 2 , ( ) b 63.2° ( ) T1 2 , 〈 〉D R1 2 , f 〈 〉D R1 2 , b 〈 〉D A1: 1 2 , ( ) f 〈 〉D A1: 1 2 , ( ) b 〈 〉D A2: 1 2 , ( ) f 〈 〉D A2: 1 2 , ( ) b 〈 〉D T1 S , fH 63.2° ( ) T1 2 , 63.2° ( )TS fH 1 RS fH R1 2 , b 〈 〉D – -------------------------------------0.244 ( ) 0.25 ( ) 1 0.60 ( ) 0.442 ( ) – -------------------------------------------0.083 = = = A1: 1 S , ( ) f 63.2° ( ) A1 1 2 , ( ) ; f 63.2° ( ) A1: 1 2 , ( ) b 〈 〉D T1 2 , 63.2° ( )RS fH 1 RS fH R1 2 , b 〈 〉D – --------------------------------------+ = 0.289 0.138 ( ) 0.244 ( ) 0.60 ( ) 1 0.60 ( ) 0.442 ( ) – -------------------------------------------+ 0.316 = = A2: 1 S , ( ) f 63.2° ( ) A2: 1 2 , ( ) f 63.2° ( ) A2: 1 2 , ( ) b 〈 〉D T1 2 , 63.2° ( )RS fH 1 RS fH R1 2 , b 〈 〉D – -------------------------------------+ = 0.022 0.141 ( ) 0.244 ( ) 0.60 ( ) 1 0.60 ( ) 0.442 ( ) – -------------------------------------------+ 0.050 = = AS: 1 S , ( ) f 63.2° ( ) T1 2 , 63.2° ( )aS f 1 RS fH R1 2 , b 〈 〉D – -------------------------------------0.244 ( ) 0.15 ( ) 1 0.60 ( ) 0.442 ( ) – -------------------------------------------0.050 = = = SHGC 63.2° ( ) T1 S , f 63.2° ( ) NkAk: 1 2 , ( ) f 63.2° ( ) k=1 S =3 ( ) ∑ + = T1 S , f 63.2° ( ) N1A1 1 S , ( ) ; f 63.2° ( ) + = N2A2: 1 S , ( ) f 63.2° ( ) NSAS: 1 S , ( ) f 63.2° ( ) 0.083 ( ) = + + 0.064 ( ) 0.316 ( ) 0.821 ( ) 0.050 ( ) 0.85 ( ) 0.050 ( ) 0.187 = + + + T1 S , fH 〈 〉D T1 2 , 〈 〉DT1 S , fH 1 RS fH R1 2 , b 〈 〉D – -------------------------------------0.277 ( ) 0.25 ( ) 1 0.60 ( ) 0.442 ( ) – -------------------------------------------0.094 = = = A1: 1 S , ( ) f 〈 〉D A1: 1 2 , ( ) f 〈 〉D A1: 1 2 , ( ) b 〈 〉D + T1 2 , 〈 〉DRS fH 1 RS fH R1 2 , b 〈 〉D – -------------------------------------= 0.271 0.138 ( ) 0.277 ( ) 0.25 ( ) 1 0.60 ( ) 0.442 ( ) – -------------------------------------------+ 0.302 = = A2: 1 S , ( ) f 〈 〉D A2: 1 2 , ( ) f 〈 〉D A2: 1 2 , ( ) b 〈 〉D + T1 2 , 〈 〉DRS fH 1 RS fH R1 2 , b 〈 〉D – -------------------------------------= 0.036 0.141 ( ) 0.277 ( ) 0.60 ( ) 1 0.60 ( ) 0.442 ( ) – -------------------------------------------+ 0.068 = = AS: 1 S , ( ) f 〈 〉D T1 2 , 〈 〉DaS f 1 RS fH R1 2 , b 〈 〉D – -------------------------------------0.277 ( ) 0.15 ( ) 1 0.60 ( ) 0.442 ( ) – -------------------------------------------0.057 = = = 30.54 2001 ASHRAE Fundamentals Handbook (SI) In Example 7, the incident solar irradiances are calculated: EDNcos (63.2°) = 388 W/m2, Edcos(63.2°) = 93.5 W/m2, and Ercos(63.2°) = 86.6 W/m2.
From these values, the solar energy flux is calculated from Equations (92), (97), and (100): VISUAL AND THERMAL CONTROLS The ideal fenestration system permits optimum light, heat, ven-tilation, and visibility; minimizes moisture and sound transfer between the exterior and the interior; and produces a satisfactory physiological and psychological environment. The controls of an optimum system will react to varying climatological and occupant demands. Fixed controls may have operations or cost advantages or both but do not react to physical and psychological variations.
Variable controls are, therefore, more effective in energy conserva-tion and environmental satisfaction.
Operational Effectiveness of Shading Devices Shading devices vary in their operational effectiveness. Some devices such as overhangs, light shelves, and tinted glazings do not require operation, have long life expectancies, and do not degrade significantly over their effective life. Other types of shading devices, especially operable interior shades, may have reduced effectiveness due to less than optimal operation and degradation of effectiveness over time. It is important to evaluate operational effec-tiveness when considering the actual heat rejection potential of shading devices.
The performance of shading devices for the reduction of peak cooling loads and annual energy use should account for operational effectiveness or reliability in actual operation. Passive devices, such as architectural elements and glazing tinting, are considered 100% effective in operation. Glazing coatings and adherent films may degrade over time. Shade screens are removable and may be assumed to be operated seasonally, but in any given population of users, some will remain in place all year long and some will not be installed or removed at optimum times. Automated shading devices controlled for optimum thermal operation are considered more effective than manual devices, but controls require ongoing mainte-nance, and some occupants may object to the lack of personal control with totally automated devices. Automated shading devices may also be operated for nonthermal purposes such as glare and daylighting optimization, and this may reduce thermal effective-ness. Manually operated devices will be subject to wide variation in use effectiveness, and this diversity in effective use should be con-sidered when evaluating performance.
Indoor Shading Devices While the thermal comfort of occupants within the glazed space may be paramount to the HVAC designer, other factors (see Table 22 and Figure 31) that should be considered, some of which may be more important to the user, include the following: Radiant Energy Protection. Unshaded fenestration products become sources of radiant heat by transmitting short-wave solar radiation and by emitting long-wave radiation to dissipate some of the absorbed solar energy. In winter, glass temperatures usually fall below room air temperature, which may produce thermal discom-fort to occupants near the fenestration. In summer, individuals seated near the unshaded fenestration product may experience dis-comfort from both direct solar rays and long-wave radiation emitted by sun-heated glass. In winter, loss of heat by radiation to cold glass can also cause discomfort. Tightly woven, highly reflective drapes minimize such discomfort; drapes with high openness factors are less effective because they permit short-wave and long-wave radia-tion to pass more freely. Light-colored shading devices with maxi-mum total surface usually provide the best protection since they absorb less heat and tend to lose heat readily by convection to the conditioned air.
Outward Vision. Outward vision is normally desirable in both business and living spaces. Open-weave, dark-colored fabrics of uniform pattern permit maximum outward vision, while uneven pat-tern weaves reduce the ability to see out. A semiopen weave modi-fies the view without completely obscuring the outdoors. Tightly woven fabrics block off outward vision completely.
Privacy. Venetian blinds, either vertical or horizontal, can be adjusted and, when completely closed, afford full privacy. When draperies are closed, the degree of privacy is determined by their color and tightness of weave and the source of the principal illumi-nation. To obscure the view so completely that not even shadows or silhouettes can be detected, fully opaque materials are used. Gener-ally, the more brightly lit side of a partially shaded glazing is the most visible from the opposite side, making the interior fairly pri-vate in daytime, but not at night.
Brightness Control. Visual comfort is essential in many occu-pied areas, and freedom from glare is an important factor in per-forming tasks. Discomfort glare is produced by uneven brightnesses in occupied spaces, with areas or spots that are much brighter than SHGC 〈 〉D T1 S , f 〈 〉D Nk Ak: 1 2 , ( ) f 〈 〉D k=1 S =3 ( ) ∑ + = T1 S , f 〈 〉D N1 A1: 1 S , ( ) f 〈 〉D N2 A2: 1 S , ( ) f 〈 〉D NS AS: 1 S , ( ) f 〈 〉D + + + = 0.094 0.064 ( ) 0.302 ( ) 0.821 ( ) 0.068 ( ) 0.85 ( ) 0.057 ( ) + + + = 0.218 = qs qb qd + EDN 63.2° ( ) cos [ ]SHCG 63.2° ( ) Ed Er + ( ) SHGC 〈 〉D + = = 388 [ ] 0.243 ( ) 93.5 86.6 + ( ) 0.218 ( ) + 133W/m2 = = Table 23 Summary of Environmental Control Capabilities of Draperies Item Designator (Table 22 and Figure 31) ID IM IL IID IIM IIL IIID IIIM IIIL 1. Protection from direct solar radiation and long-wave radiation to or from window areas Fair Fair Fair Fair Good Good Fair Good Good 2. Effectiveness in allowing outward vision through fenestration Good Good Fair Fair Fair Some None None None 3. Effectiveness in attaining privacy (limiting inward vision from outside) None None Poora Poor Fair Faira Goodb Goodb Goodb Gooda Gooda 4. Protection against excessive brightness and glare from sunshine and external objects Mild Mild Mildc Good Good Goodc Good Good Goodc Poorc Poorc Poorc 5. Effectiveness in modifying unattractive or distracting view out of window Little Little Some Some Good Good Blocks Blocks Blocks aGood when bright illumination is on the viewing side.
bTo obscure view completely, material must be completely opaque.
cPoor rating applies to white fabric in direct sunlight. Use off-white color to avoid excessive transmitted light.
Fenestration 30.55 surrounding surfaces. Windows themselves, when they look out onto bright skies or brightly reflecting surfaces, can be glare sources if care is not taken to keep surround brightnesses comparable. A maximum brightness ratio of about 3 to 1 is sometimes quoted.
Moderation of this ratio can be achieved through the use of interior furnishings and wall coverings, which on average have moderately high diffuse reflectances and access to admitted daylight. Con-versely, dark interior surfaces, and those shaded from daylight illu-mination, will accentuate the brightness difference between the window and its surroundings. Interior surface brightness can also be elevated by ample use of interior electric lighting, but this can have adverse consequences for the building’s energy use. In general, larger window apertures admit more sunlight, increasing interior brightnesses without affecting the perceived brightness of the win-dow, all other things being equal.
An important guideline is the dictum that direct sunlight must not strike the eye, and reflected sunlight from bright or shiny surfaces is equally disturbing and even disabling. A tightly woven white fabric with high solar transmittance attains such brilliance when illumi-nated by direct sunshine that, by contrast with its surroundings, it creates excessive glare. Off-white colors should be used so their sur-face brightness is not too great. Venetian blinds permit considerable light to enter by interreflection between slats. When two shading devices are used, the one on the inside (away from the fenestration product) should be darker and more open. With this arrangement, the inside device can be used to control brightness for the other shading devices and, when used alone, to reduce brightness while still permitting some view of the outside.
View Modification. When the view is unattractive or distract-ing, draperies modify the view to some degree, depending on the fabric weave and color (summarized in Table 23), but the fenestra-tion product remains as an effective connection to the outside.
Sound Control. Indoor shading devices, particularly draperies, can absorb some of the sounds originating within the room but have little or no effect in preventing outdoor sounds from entering. For excessive internally generated sound, the usual remedy is to apply acoustical treatment to the ceiling and other room surfaces. While these materials can be effective in controlling sound, they are often located on the two horizontal surfaces (ceiling and floor) and leave the opposing vertical surfaces of glass and bare wall to reflect sound. The noise reduction coefficient (NRC = average absorptance coefficient at four frequencies) for venetian blinds is about 0.10, compared to 0.02 for glass and 0.03 for plaster. For drapery fabrics at 100% fullness, NRC ranges from 0.10 to 0.65, depending on the tightness of weave. Class III (tightly woven) fabrics have NRC val-ues of 0.35 to 0.65. Figure 32 shows the relationship between NRC and openness factor for fabrics of normal weight.
Example 22. To select a drapery fabric, consider the five environmental factors listed in Table 23. Choose a fabric designator that has suitable performance for all the factors important to the case being considered.
If this is not possible, make compromises resulting in an acceptable designator. Determine from Table 23 if the IAC for the chosen designa-tor is satisfactory. Specific cases follow: 1. Where modification of a distracting view is necessary but a degree of outward vision is needed and an IAC of 0.50 with 6 mm gray glass is satisfactory (see Table 23, Item 5), select IIM or IIL; from Item 2, select IIM. The IAC for IIM on Table 23 is approximately 0.46, there-fore satisfactory.
2. Where protection from radiation is paramount and minimum IAC is necessary (see Table 23, Item 1), select a closed weave, IIIM or IIIL.
Since the IAC for IIIL is lowest (see Table 23), choose IIIL.
3. When good outward vision is desired, together with some reduction in brightness, choose IM or ID (Table 23).
Double Drapery Double draperies (two sets of drapery covering the same area) have a light, open weave on the fenestration product side for out-ward vision and daylight when desired and a heavy, closed weave or opaque drapery on the room side to block out sunlight and provide privacy when desired. When properly selected and used, double draperies can provide a reduced U-factor and a lowered IAC.
The reduced U-factor results principally from adding a semi-closed air space to the barrier. A U-factor of about 1.80 W/(m2·K) is achieved using double draperies with single glass, and about 1.17 W/(m2·K) with insulating glass.
To most effectively reduce solar heat gain, the drapery exposed to sunlight should have high reflectance and low transmittance. The light, open-weave drapery should be opened when the heavy drap-ery is closed to prevent entry of sunlight.
Properly used double draperies give (1) extreme flexibility of vision and light intensity, (2) a lowered U-factor and IAC, and (3) an improved comfort condition, since the room-side drapery is more nearly at room temperature. Table 23 gives characteristics of indi-vidual draperies. For large areas, the IAC should be calculated in detail to determine the cooling load.
AIR LEAKAGE Infiltration Through Fenestration Air infiltration through fenestration products affects occupant comfort and energy consumption. Infiltration should not be con-fused with ventilation. Infiltration is the uncontrolled inward leak-age of air caused by pressure effects of wind or differences in air density, such as the stack effect. While fenestration products can be operated to intentionally provide natural ventilation and increase comfort, infiltration should be reasonably minimized to avoid unpleasant accompanying problems. If additional air is required, controlled ventilation is preferable to infiltration. Mechanical ven-tilation provides air in a comfortable manner and when desired. For infiltration, however, the peak supply is more likely to occur as an Fig. 32 Noise Reduction Coefficient Versus Openness Factor for Draperies 30.56 2001 ASHRAE Fundamentals Handbook (SI) uncomfortable draft and when least desired, such as during a storm or the coldest weather.
ASHRAE/IESNA Standard 90.1, ASHRAE’s energy standard for all buildings other than low-rise residential buildings, estab-lishes an air leakage maximum of XX per square metre of gross fen-estration product area (XX/m2 for swinging entrance doors and revolving doors). This air leakage is as determined in accordance with NFRC 400 and ASTM Standard E 283 and allows direct com-parison of all fenestration products: operable and fixed, windows and doors. Most manufactured fenestration products achieve these reason-able standards of maximum air infiltration. However, products that do not completely seal, such as jalousie windows or doors, are not likely to do so and are most appropriate for installation in uncondi-tioned spaces.
For products achieving this infiltration standard, the energy con-sumption due to infiltration is likely to be significantly less than the energy associated with U-factor and solar heat gain coefficient.
Also, while overall air infiltration is a significant component in determining a building’s heating and cooling loads, the infiltration through fenestration products meeting the above standard is gener-ally likely to be a small portion of that total. Chapter 26 presents cal-culation procedures for air infiltration.
Indoor Air Movement Because supply air grilles are frequently located directly below fenestration products, air sweeps the interior glass surface. Heated supply air should be directed away from the glass to prevent large temperature differences between the center and edges of the glass.
These thermal effects must be considered, particularly when annealed glass is used and air is forced over the glass surface during the heating season. Direct flow of heated air over the glass surface can increase the heat transfer coefficient and the temperature differ-ence, causing a substantial increase in heat loss; it may also lead to thermally induced stress and risk of glass breakage.
Systems designed predominantly for cooling lower the glass temperature and rapidly pick up the cooling load. Both tend to improve the comfort condition of the space. However, the air-con-ditioned space has an increased net heat gain caused by (1) increase in the solar heat gain coefficient (SHGC) due to delivery of a larger portion of the absorbed heat to the indoor space, (2) increase in the fenestration U-factor because of the greater convection effect at the indoor surface, and (3) increase in the air-to-air temperature differ-ence since supply air rather than room air is in contact with the indoor glass surface. The principal increase in heat gain with clear glass is the result of the higher U-factor and the greater air-to-air temperature difference.
DAYLIGHTING DAYLIGHT PREDICTION Daylighting is the illumination of building interiors with sunlight and sky light and is known to affect visual performance, lighting quality, health, human performance, and energy efficiency. In many European countries with predominantly cloudy skies, there are codes regulating minimum window size, minimum daylight factor, and window position in order to provide views to all occupants and to create a minimum interior brightness level. For practical reasons, daylighting provides backup interior illumination in the event of power outages. Daylighting may have some positive or negative health effects on the skin, eyes, hormone secretion, and mood. Its temporal variation, intensity, spectral content, and diurnal and tem-poral variation may be used to combat jet lag, sick building syn-drome, and other health problems.
In terms of energy efficiency, daylighting can provide substan-tial whole-building energy reductions in nonresidential buildings through the use of electric lighting controls. Daylight admission can displace the need for electric lighting at the perimeter zone with vertical windows and at the core zone with skylights. Light-ing and its associated cooling energy use constitute 30 to 40% of a nonresidential building’s energy use. Energy use reductions can be achieved, perhaps less reliably, in residential buildings with manual or automated switching of electric lights on and off to match space occupancy. For internal load-dominated buildings, daylight admission must be balanced against solar heat admission to achieve optimum energy efficiency. Since the heat gains from solar radiation typically define peak load conditions, daylighting is also a very effective method of decreasing peak demand. The use of daylighting can not only decrease annual operating costs through energy efficiency but may also reduce capital cost due to mechanical downsizing.
For conventional sidelit nonresidential buildings, three funda-mental relationships for daylight optimization are given as a func-tion of (1) glazing properties and (2) window area or the window-to-wall area ratio (WWR), which is defined as the ratio of the transparent glazing area to the exterior floor-to-floor wall area: 1. Annual cooling energy use (including fan energy use) increases linearly with solar radiation admission, as indicated by the product of SHGC and WWR, but is affected by decreases in electric lighting heat gains.
2. Annual lighting energy use decreases exponentially/asymptoti-cally with daylight admission, as indicated by the product of Tv and WWR.
3. Annual heating energy use (including fan energy use) increases linearly with decreased lighting heat gains. Figure 33 illustrates the first two relationships for a prototypical nonresidential building. A similar relationship can be demonstrated with skylights. To determine the fenestration design that achieves an optimum balance between daylight admission and solar rejection requires iterative calculations where the glazing area and/or glazing solar-optical properties are varied parametrically. For each case, the fol-lowing general steps should be taken for each hour over a year: 1. Interior Daylight Illuminance. Determine the building charac-teristics, configuration, outdoor design conditions, and operat-ing schedules as described in the section on Initial Design Considerations in Chapter 29. These include building orienta-tion, exterior obstructions, ground reflectance, etc. Determine the depth from the window wall for each electric lighting zone.
Typical sidelighting windows can effectively daylight the perim-eter zone to a depth of 1.5 times the head height of the window.
In private offices, one dimming zone is typically cost-effective, while in open plan offices, two zones are cost-effective.
Select a typical task location within each of the lighting zones.
Determine interior daylight illuminance due to all window and skylight sources at these locations. Interior illuminance may be determined using computer simulation tools or physical scale models. Comprehensive explanations of simple and computer-based tools are available (IEA 1999). The majority of these tools can model simple box geometry with noncomplex fenestration systems. Some advanced simulation tools, such as Radiance (Ward 1990) and Adeline (Erhorn and Dirksmöller 2000), are capable of modeling complex geometry and fenestration sys-tems with adequate bidirectional solar-optical data, but this capability is not routine. 2. Lighting Energy Use. Determine the type of lamps, ballasts, and control system to be used in the perimeter zones. Determine whether the lamp is capable of being dimmed or switched. For example, fluorescent lamps can be dimmed, while metal halides cannot be switched or dimmed. Cold, outdoor applications of some lamps may prevent switching. For electronic dimming bal-Fenestration 30.57 lasts, obtain dimming power and light output characteristics.
Obtain control specifications to determine how the system will respond to available light; dead-band ranges, response times, and commissioning will affect the sensitivity and accuracy of the system. The type of switching (on-off, bilevel, multilevel, and continuous dimming controls) will be dictated by both the type of lamp and space use. Determine the task illuminance design set point for each zone.
Determine the percentage electric lighting power reduction Fdaylight [see Equation (135)] that will result with automatic day-light controls, and apply to the installed wattage. Simplified methods for calculating lighting power reductions based on task illuminance levels are given in Robbins (1986). More sophisti-cated programs (Choi and Mistrick 1999) model commercially available photosensor dimming control systems (typically located in the ceiling above the work plane task) more rigor-ously; the spectral and bidirectional response of the photosensor to incident flux is used to determine voltage output, which is then used by the ballast controller algorithm to determine the lighting power reduction. Response delays and commissioning set points will further affect this predicted output. Lights may also be switched manually, but there are no modeling prediction tools for manual switching. Field tests (Jennings et al. 1999) indicate that with bilevel switching, 45% of the lighting zone-hours were at less than full power lighting, with 28% at only one-third of full lighting output levels. Manual switching occurred less in public spaces. Occupancy and other types of switching may occur as well and should be accounted for as a confounding effect with any daylighting controls.
3. Mechanical Energy Use. Determine mechanical energy use due to the fenestration loads and reduced electric lighting heat gains.
Fenestration heat gains and losses may be computed using the section on Determining Fenestration Energy Flow. Instanta-neous lighting heat gains qel, described by Equation (1) in Chap-ter 29, must be multiplied by the power reduction factor Fdaylight.
Mechanical loads and energy use may then be determined as described in Chapter 29. There have been many studies investi-gating the magnitude of change in heating and cooling energy use associated with reductions of lighting energy use in nonres-idential buildings, as will be realized with daylighting controls.
In a DOE-2.1E simulation study (Sezgen and Koomey 2000), the greatest savings were generated in hospitals, large offices, and large hotels; for every $1.00 saved through lighting energy effi-ciency, additional savings as a result of reduced HVAC were $0.26, $0.16, and $0.14, respectively. These results emphasize the need to include HVAC effects when assessing the impact of daylighting. Simplified design tools are available that enable one to conduct such parametric runs for preliminary analysis. For vertical windows, the BEEM program can be used to determine the relative impacts of daylighting on energy use, peak demand, and costs for a given window and lighting control system (Rundquist 1991). Skylighting tools based on regressions using DOE-2 data or simplified DOE-2 procedures are also available (AAMA 1988, Heschong et al. 1998). More comprehensive building energy prediction tools combined with daylighting algorithms, such as DOE-2.1E (Winkelmann 1983), implement hour-by-hour calculations using existing weather data and enable one to evaluate glare, visual comfort, and quality of light as well.
In the United States, a general rule has been that the fenestration area should be at least 20% of the floor area. In Europe, a similar rule was based on a minimum illumination value on the normal work plane from a standard overcast sky condition. In general, it is more energy-efficient to use larger window areas to elevate interior surface brightnesses as a glare reduction strategy than to increase interior electric lighting levels. As window area increases, interior brightness increases while window brightness remains the same. Of course there are mitigating considerations, such as the higher cost of larger windows and increased heat transfers through larger win-dows. The latter problem can be mitigated through the use of insu-lating multiple-pane windows and special coatings to reduce solar gain without serious loss of light transmission, as discussed in the section on Selecting Fenestration.
The secondary visual benefit of fenestration is the amount and quality of light it produces in the work environment. One general rule determined the need for auxiliary electric light by assuming that daylight was adequate for a depth of two-and-one-half times the height of the fenestration product into the room based on a normal sill height. To prevent excessive glare, all fenestration should have sun controls. Variable and removable controls are often more effec-tive in daylight than fixed controls.
For more accurate evaluation of daylight distribution within a space, several prediction tools, such as the Recommended Practice of Daylighting (IESNA 1999), are available. This practice shows a simple way of calculating the daylight distribution on the work plane from windows and skylights with and without controls. Many other daylight prediction tools calculate illuminance from radiant flux transfer or ray tracing. Any or all of the various daylight prediction tools can be used to compare the relative value of daylight distribution from alternative fenestration systems, but ultimately the designer must evaluate costs and benefits to choose between alternative designs. This may be based on energy use or, more properly, on overall costs and ben-efits to the client. Also, the negative possibility of total loss of pro-ductivity from an electric brown-out in a space with no natural ventilation or daylight may be as important as the benefits of many energy-saving schemes.
Fig. 33 Window-to-Wall Ratio Versus Annual Electricity Use (kWh/m2·floor·year) 30.58 2001 ASHRAE Fundamentals Handbook (SI) LIGHT TRANSMITTANCE AND DAYLIGHT USE When daylight is to be the primary lighting system, the minimum expected daylight in the building must be calculated for the building performance cycle and integrated into the lighting calculations.
IESNA (1999) gives daylight design and calculation procedures. In some glazing applications, such as artists’ studios and showrooms, maximum transmittance may be required for adequate daylighting of the interior. Regular clear glass, produced by float, plate, or sheet process, may be the logical choice.
When daylight is a supplementary light source, the electric light-ing can be designed independently of the daylight system. However, adequate switching must be included in the electric distribution to substitute available daylight for electric lighting by automatic or prescribed manual control whenever possible and practical. Photo-sensitive controls automatically adjust shading devices to provide uniform illumination and reduce energy consumption. Manual con-trol is less effective.
Buildings with large areas of glass usually have insulating glass units with clear, tinted, or reflective coatings. The tinted and reflect-ing units reduce the brightness contrast between fenestration prod-ucts and other room surfaces and provide a relatively glare-free environment for most daylight conditions.
Tables 13 and 24 list typical solar energy transmittances and day-light transmittances for various glass types. Manufacturers’ litera-ture has more appropriate values.
The color of glass chosen for a building depends largely on where and how it is used. For commercial building lobbies, show-room fenestration products, and other areas where maximum visi-bility from exterior to interior is required, regular clear glass is generally best. Clear glass with a low-e coating is also suitable for these locations, including for retail storefronts, as it only decreases the light transmittance by 10% or so. For other glass areas, a tinted glass may best complement the interior colors. Bronze, gray, and reflective-film glasses also give some privacy to building occupants during daylight hours. Patterned, etched, or sandblasted glass that diffuses lighting is available. In warm climates, a tinted outer glass in an insulated double-pane system can have solar heat gain rejec-tion benefits, while providing good color rendering illumination of the interior without apparent color.
The primary purpose of a fenestration product is not just to save energy but to provide a view of the exterior. One sees out of a fen-estration product by virtue of the light from the outside that comes through that fenestration product into the occupant’s eyes. The light from outside is valuable not only for views of the outdoors but for providing daylight illumination of the interior.
Some buildings are designed especially to use the daylight com-ing through fenestration systems in displacing electric lighting and its attendant energy costs. Using daylighting to displace electric lighting benefits the energy bill directly through reduced direct con-sumption by the lighting systems involved and indirectly through reduced electrically produced heat gain that may have to be removed by the air-conditioning system.
The light-transmitting properties of fenestration systems are therefore of great importance, not only for permitting views of the outdoors but also for admitting daylight to reduce electric lighting.
It is conceivable that one could design a fenestration product with excellent solar heat gain performance for hot climates (meaning a very low solar heat gain coefficient) but very poor view and daylight illumination performance. If this problem is bad enough, it can cause occupants to turn on electric lights indoors during the day-time, which adds to the electric bill and possibly causes problems of thermal discomfort as well.
The light-transmitting property of a fenestration product is called the visible transmittance Tv. It is similar to the solar-weighted solar transmittance, except that an additional weighting function is needed, in this case to account for the spectral response of the human eye.
In most applications, it is important to have a high visible trans-mittance. In northern climates, a good solar heat gain is also impor-tant for offsetting wintertime heating costs. In southern climates, a low solar heat gain is good for offsetting summertime cooling costs.
In the latter situation, it is difficult to have both a high visible trans-mittance and a low solar heat gain coefficient. Figures 34 and 35 show a plot of visible transmittance versus SHGC for a number of glazing systems covering a range of spectral selectivities. The data are for normal incidence and a single, ASTM standard solar spectral distribution.
A rule of thumb is to select a glazing unit having a visible trans-mittance greater than its shading solar heat gain coefficient, espe-cially if daylighting strategies will be used in the building. For maximum light with minimum solar gain, there are fenestration prod-ucts available having a visible transmittance 1.4 times the SHGC.
Three different zones are delineated on Figure 35. In the neutral zone, it is possible to have colorless glazing systems, meaning glaz-ings with approximately uniform transmittance over the visible spectrum. Of course, one can have glazings in this zone with color, but this is not necessary. In the color zone, the only way to achieve higher visible transmittance for a given level of solar heat gain coef-ficient is by stripping off some of the red and blue wavelengths at the edges of the V-λ function with a spectrally selective glazing transmittance, imparting color to the transmitted radiation (or by Table 24 Daylight Transmittance for Various Types of Glass Type of Glass Visible Transmittance Tv 3 mm regular sheet or float glass 0.86 to 0.91 3 mm gray sheet 0.31 to 0.71 5 mm gray sheet 0.61 5.5 mm gray sheet 0.14 to 0.56 6 mm gray sheet 0.52 6 mm green/float glass 0.75 6 mm gray plate glass 0.44 6 mm bronze plate glass 0.49 13 mm gray plate glass 0.21 13 mm bronze plate glass 0.25 Coated glasses (single, laminated, insulating) 0.07 to 0.50 Fig. 34 Visible Transmittance Versus SHGC for Several Glazings with Different Spectral Selectivities Fenestration 30.59 otherwise altering the spectral transmittance and hence the color over the visible portion of the spectrum). In the forbidden zone, no combination of visible transmittance and solar heat gain coefficient is possible for normal incidence and for the solar spectral distribu-tion used. (Changing the solar spectral distribution used to calculate Tv and SHGC will shift the transition curves somewhat. A low solar altitude angle, direct-beam spectrum will move the curves to the left on the plot in Figure 35.) It can be seen that the glazings that trans-mit more solar radiant heat than light cluster on the lower portion of the plot.
The Tv versus SHGC chart can be a useful tool for illustrating the degree of spectral selectivity attained by a glazing system. These concepts lead to an index of spectral selectivity that can be useful.
It is called the light-to-solar-gain ratio, or LSG, defined as (135) Some characteristic values for Tv, SHGC, and LSG are given in Table 25 for several different glazings, using the ASTM standard spectral distribution at normal incidence to calculate the values.
The LSG can be useful in spotting errors in the calculation of the SHGC. Values of SHGC that lie outside reasonable ranges can be spotted fairly quickly and used to identify possible problems in cal-culations or measurements. In general, it is very difficult and there-fore unlikely to have a useful glazing system for buildings with an LSG value greater than 2.0. Values near 0 should be particularly sus-pect, since they indicate a glazing that transmits considerably more heat than light and would be unlikely candidates for general use.
Generally, a high value of LSG is desired for residential buildings in hot climates, to maximize daylight admission with minimal solar heat gain. This is also true for internal load-dominated nonresiden-tial buildings in many climates, since solar gain rejection is often desired for such buildings, even in cool or cold climates. An LSG value somewhat below 1.0 would be appropriate in cold climates for residential buildings and nonresidential buildings without strong internal cooling loads.
SELECTING FENESTRATION Since fenestration systems provide so many functions and because environmental conditions and user needs vary widely, it is difficult to make a completely optimal selection of a fenestration system. Aesthetic and cost considerations are perhaps the most important to residential users, with visual and comfort performance also being of interest. Considering annual energy costs, peak load consequences, and acoustic characteristics, the choice is seldom optimal. The HVAC system designer, fortunately, has a more restricted range of interests, mainly dealing with the energy conse-quences of a particular fenestration selection. This section therefore focuses on fenestration energy performance determination.
ANNUAL ENERGY PERFORMANCE Instantaneous energy performance indices (U-factor, solar heat gain coefficient, air leakage, etc.) are typically used to compare fenestration systems under a fixed set of conditions. However, the absolute and relative effect of these indices on a building’s heat-ing and cooling load can fluctuate as environmental conditions change. As a result, these indices alone are not good indicators of the annual energy performance (energy savings/costs) attributable to the fenestration. Furthermore, fenestration annual energy per-formance is difficult to quantify in and of itself because of the many dynamic responses that occur between the fenestration sys-tem and the total environment in which it is installed. The four basic mechanisms of fenestration energy performance that were each addressed previously in the chapter—thermal heat transfer, solar heat gains, air leakage, and daylighting—should all be taken into account but are not independent of many other parameters that influence fenestration annual energy performance. As a result, the annual energy performance of fenestration systems can be accurately determined only when a large number of variables are considered. Building type and orientation, climate (weather, temperature, wind speed), microclimate (shading from adjacent buildings, trees, terrain), occupant usage patterns, and certain HVAC parameters can significantly affect the annual energy impacts of fenestration systems. For these reasons, the most effective means of establishing fen-estration annual energy performance is through detailed, dynamic, hourly computer simulations for the specific building and climate of interest. Since the instantaneous performance of the fenestration will often vary by differing magnitudes as climatic conditions change, the most accurate simulation results are obtained when these variances are accounted for in a building energy simulation computer program. After constructing the building energy simula-tion model following the procedures defined in Chapters 28 and 29 (for residential and commercial construction, respectively), specific changes to the fenestration system can be modeled, and the annual energy performance changes attributable to fenestration can be quantified. These analytical techniques do not consider issues of performance durability for the various instantaneous indices and should only be used as an initial annual energy performance indica-tor (Mathis and Garries 1995). Fig. 35 Visible Transmittance Versus SHGC at Various Spectral Selectivities (McCluney 1996) LSG Tv SHGC ----------------= Table 25 Spectral Selectivity of Several Glazings Glazing Tv SHGC LSG Reflective blue-green 0.33 0.38 0.87 Film on clear glass 0.19 0.22 0.86 Green tinted, medium 0.75 0.69 1.09 Green low-e 0.71 0.49 1.45 Sun-control low-e + green 0.36 0.23 1.56 Super low-e + clear 0.71 0.40 1.77 Super low-e + green 0.60 0.30 2.00 30.60 2001 ASHRAE Fundamentals Handbook (SI) Simplified Techniques for Rough Estimates of Fenestration Annual Energy Performance While dynamic hourly modeling is certainly the most accurate technique for determining fenestration annual energy performance, it is not readily available to many decision makers and end users of fenestration products simply because it may not be practical or cost-effective. Under these circumstances, it may be useful to assess the relative importance of, or balance the trade-off between, the known instantaneous performance indices of U-factor, SHGC, air leakage, and Tv for any given fenestration system when considering heating, cooling, and lighting loads for many different building types and cli-mates. Mitchell et al. (1999) and Huang et al. (1999) describe per-sonal computer programs that are being developed to run this simplified analysis for residential windows.
Broad generalizations can be made for some classifications of building types and climates. For instance, with large commercial buildings, which require substantial cooling energy use because of high internal loads, significant thermal mass, or high orientation dependency, the primary objective may be to place the most empha-sis on low SHGC to reduce the cooling load. Also, an evaluation of commercial fenestration annual energy use can take into account the trade-off between artificial lighting and the natural daylighting ben-efits associated with a particular fenestration system. Contrary to this, electric lighting loads in low-rise detached residential build-ings are typically very small in comparison to the heating and cool-ing loads because of high envelope-dependent energy use, egress requirements, and occupant usage patterns, and therefore the energy influence of daylighting may be neglected altogether. Yet, despite these generalizations, the problem still exists of balancing and assessing the impact of each of the remaining parameters to estab-lish the seasonal or annual energy performance for cases in which detailed computer modeling is not performed.
Realizing the need for characterization of fenestration annual energy performance, scientists in many different countries have been working over the last several years to develop simplified annual energy performance indices for fenestration. These simpli-fied techniques typically involve using the instantaneous fenestra-tion performance indices to quantify building- and climate-independent scalars of annual or seasonal energy performance for rating purposes. Many of these performance indices have value in that they can be relatively independent of building type, climate, distribution of products, orientation, and other items needed for hourly dynamic building energy analyses. These normalized, sca-lar-based approaches are also limited in accuracy for the same rea-sons. A further limitation with the simplified techniques is that they do not have broad applicability to varied building types (commer-cial versus residential buildings, for example). The usefulness of these scalar-based approaches can be increased when limiting the comparison to a single building type. Currently, the simplified tech-niques for characterizing fenestration annual energy performance are applicable only to fenestration systems for detached residential buildings and are not appropriate for use with multifamily residen-tial or commercial building fenestration systems.
Simplified Residential Annual Energy Performance Ratings Annual energy performance ratings can provide a simple means of product comparisons for consumers. Such ratings have been derived with many assumptions, usually to suit local climatic conditions.
The Canadian Standards Association (CSA Standard A440.2) developed a simplified energy rating applicable to residential heating in the Canadian climate, which has been adopted in the 1995 National Energy Code for Houses. The standard also pro-vides for specific energy ratings to compare products by orienta-tion and climate.
In the United States, where heating and cooling are both signifi-cant, the NFRC is developing a rating system that includes both effects (Crooks et al. 1995, Arasteh et al. 2000).
CONDENSATION RESISTANCE Water vapor condenses in a film on fenestration surfaces that are at temperatures below the dew-point temperature of the inside air. If the surface temperature is below freezing, frost forms. Sometimes, condensation occurs first, and ice from the condensed water forms when temperatures drop below freezing. Condensation frequently occurs on single glazing and on aluminum frames without a thermal break. The edge-seal creates a thermal bridge at the perimeter of the IGU.
The circulation of fill gas due to temperature differences in the IGU cavity contributes to the condensation problem at the bottom of the indoor glazing (Wright and Sullivan 1995a, 1995b; Curcija and Goss 1994, 1995a). In winter, fill gas near the indoor glazing is warmed and flows up, while gas near the outdoor glazing is cooled and flows down. The descending gas becomes progressively colder until it reaches the bottom of the cavity. There, the gas turns and flows to the indoor glazing, resulting in higher heat transfer rates at the bottom. Thus, the bottom edge of the indoor glazing is cooled both by edge-seal conduction and by fill-gas convection. The com-bined effect of these two heat transfer mechanisms is shown in Fig-ure 36. The surface isotherms show a wider band of cold glass at the bottom of the window. Typical condensation patterns match these isotherms. The vertical indoor surface temperature profile also shows the effect of edge-seal conduction and that the minimum indoor surface temperature is near the bottom edge of the glass.
Condensation to the fenestration and surrounding structures can cause extensive structural, aesthetic, and health problems. Specific examples include peeling of paint, rotting of wood, saturation of insulation, and mold growth. Ice can render doors and windows inoperable and prevent egress during an emergency.
Fig. 36 Temperature Distribution on Indoor Surfaces of Insulating Glazing Unit Fenestration 30.61 Energy-efficient housing has been accompanied by reduced ven-tilation. The resulting increase in indoor humidity has contributed to the condensation problem. However, the solution does not lie in the reduction of humidity levels to a minimum. Relative humidity below 20% and above 70% can increase health risks and reduce comfort. Generally, a minimum of 30% rh should be maintained, and 40% to 50% is more desirable (Sterling et al. 1985).
Minimum indoor surface temperatures can be quantified in a variety of ways. Sullivan et al. (1996), Griffith et al. (1996), Elmahdy (1996), Zhao et al. (1996), and de Abreu et al. (1996) dem-onstrated good agreement between detailed two-dimensional numerical simulation and surface temperature measurements using thermographs. Wright and Sullivan (1995c), and Curcija et al.
(1996) developed simplified simulation models to predict conden-sation resistance. Estimates of center-glass and bottom-edge surface temperatures that can be expected for two different glazing systems exposed to a range of outdoor temperature are shown in Figure 36.
Both glazing systems include insulating foam edge seals. High-per-formance glazing systems (e.g., low-e/argon and insulated spacers) permit significantly higher indoor humidity levels.
Current measures of condensation resistance of a fenestration system are the condensation index (CI) as defined by NFRC (2000a), the condensation resistance factor (CRF) as defined by AAMA (1988), or the temperature index (I), as defined in CSA Standards A440 and A440.1. The condensation index is a measure of condensation potential that is based on both area and temperature weighting and is expressed as a minimum of center-of-glazing, edge-of-glazing, and frame CIs. The novelty of this index lies in the fact that it is determined using computer simulation tools unless the overall thermal performance cannot be validated with testing. In the case that thermal performance cannot be validated, a testing option for determining CI is used.
Computer simulation is done for characteristic two-dimensional cross sections in much the same way that U-factors are determined.
The basic difference between U-factor and CI simulations is that more advanced models are used for CI calculations. This is neces-sary because temperatures are intrinsically local quantities, as opposed to U-factors, which are average quantities, and it is neces-sary to provide better models for convective heat transfer in glazing cavities and convective and radiative heat transfer on indoor fenes-tration boundaries. The most general expression of the formula for calculating frame, center-of-glazing, and edge-of-glazing CI is given by the following equation: (136) where i = frame, center-of-glazing, or edge-of-glazing section j = 30%, 50%, and 70% relative humidity tdpp = tdp + 0.3 K tdp = dew-point temperature, °C + = positive values only The other two standards define the values by a single dimension-less number as (137) where th and tc are the warm and cold side temperatures, respec-tively. Figure 38 can be used to determine the acceptable range of CRF/I for a specific climatic zone.
The two standards differ in the methods used to determine tem-perature. The CSA test procedure is based on thermocouple mea-surements at the coldest location on the frame plus three locations on the glass, each X mm above the bottom sightline. The AAMA procedure specifies two separate factors: one for the frame (CRFF), which uses weighted frame temperature obtained from surface tem-perature measurements at predetermined and roving locations on the frame, and one for the IGU (CRFG), which uses the average of six temperatures measured at predetermined locations near the top, middle, and bottom of the glazed area.
Inside details can significantly alter the potential for condensa-tion on window surfaces. Items such as venetian blinds, roll blinds, insect screens, and drapes increase the thermal resistance between the indoor space and the window and lower the temperature of the window surfaces. These window treatments do not prevent migra-tion of moisture, so they can cause increased condensation. Figure 39 shows different situations that affect the potential for condensa-tion. Note that window reveal plays an important role. If the window is placed near the outside of the wall, the increase in the outdoor film coefficient and decrease in the indoor film coefficient cause colder window surfaces. This effect is more pronounced near the corners of the recess where the indoor film coefficient is locally suppressed because air movement is restricted. Also, blinds should be placed at least 100 mm from the plane of the wall to allow some natural con-vection between the window and the blind.
Air leakage, especially in operable sections of fenestration, is another important cause of low surface temperature. Leakage near the edge-of-glass sections can further increase the potential for condensation. However, the drier outdoor air decreases the rela-tive humidity near the leakage sites and, in some cases, offsets the Fig. 37 Minimum Indoor Surface Temperatures Before Condensation Occurs CI 1 – 1 3 --tdpp,j ti – ( )+Ai i ∑ tdpp,j to – ( )A -----------------------------------------j=1 3 ∑ 1 3 ⁄ 100 × = CRF or I t tc – th tc – --------------= 30.62 2001 ASHRAE Fundamentals Handbook (SI) undesirable effect of the lower surface temperatures. The net effect of air leakage cannot readily be determined experimentally or with simulation.
OCCUPANT COMFORT AND ACCEPTANCE Human thermal comfort is an immediate sensation that reflects building occupants’ perceived response to many physical factors.
Unlike much building design that is based primarily on long-term energy and economic considerations, comfort-related design focuses on, and must take heed of, short-term responses of the body’s physiology to its surroundings.
Windows influence thermal comfort through a combination of three mechanisms: long-wave radiation exchange, absorption of solar radiation, and convective draft effects (Figure 40). An under-standing of these phenomena is important to help designers evalu-ate the benefits of improved windows and create comfortable buildings. Although it is well understood that high-performance windows can reduce building energy consumption, a better under-standing of their impact on comfort might lead to further savings.
For example, Hawthorne and Reilly (2000) suggest that significant energy consumption is caused by the standard practice of using perimeter duct distribution in houses to mitigate potential discom-fort caused by windows. They found that perimeter heating is often not necessary when high-performance windows are installed and that heating energy savings of 10 to 15% could result from install-ing a simpler, less expensive duct system. Better windows can allow thermostat settings to be lowered with no loss of comfort.
Another simulation study (Lyons et al. 2000) examined the relative magnitudes of a residential window’s physical influences under a wide variety of winter and summer climates, glazing parameters, and clothing levels. They found that • Long-wave, thermal radiation influences of the window dominate unless direct sun strikes the occupant • Direct solar load has a major influence on perceptions of comfort • For most residential-size windows, draft effects are generally small With all but highly insulating windows, the inside surface temper-ature of the window is heavily influenced by exterior conditions, and this temperature can significantly affect the radiant heat exchange between an occupant and the environment. If this heat exchange moves outside the acceptable range, discomfort will result. Mean radiant temperature (MRT) is commonly used to simplify the char-acterization of the radiant environment. On a cold day, the inside sur-face temperature can easily drop below −9°C for a clear single-pane window and below 4°C for a clear, double-pane window. If the occu-pant is sitting sufficiently near the window, MRT could drop to 13°C for the single-pane case and 17°C for the double-pane case. Based on ASHRAE Standard 55, even the use of the clear double-pane win-dow could result in discomfort. [This example assumes an outdoor air temperature of −18°C, indoor air temperature of 22°C, nonwin-dow surface temperatures of 22°C, occupant-window view factor of 0.3, 0.9 clo (standard winter indoor clothing), and activity level of 1 met.] In addition to the MRT effect, a cold inside glass surface can induce a downward draft that increases air movement, contributing to further discomfort. If direct solar radiation strikes the glazing or occupant, the situation is much more complex.
In winter, the warming effect of sunlight on skin and clothing is often welcome, depending on the compounding effect of other fac-tors such as air temperature. Windows also absorb and transmit a significant amount of solar radiation. Because of such absorption, a solar-heated window may improve MRT for a nearby person. The premise of passive solar design is that occupants will welcome, or at least tolerate, solar gain in exchange for savings on heating energy.
However, it is desirable that the onset of discomfort be able to be Fig. 38 Minimum Condensation Resistance Requirements (th = 20°C) Fig. 39 Location of Fenestration Product Reveals and Blinds/Drapes and Their Effect on Condensation Resistance Fenestration 30.63 predicted; otherwise, the energy-saving design may be defeated if occupants draw shades to prevent overheating.
In summer, solar-heated glass may become uncomfortably hot and, in commercial premises, actually devalue rented space near windows. The inside surface of body-tinted, heat-absorbing glass can routinely reach temperatures above 50°C in summer conditions, raising MRT by as much as 8 K. This can be ameliorated with the addition of a second pane of glass on the inside. Transmitted radia-tion often causes discomfort if it falls directly on the occupant. A person sitting near a window in direct solar radiation can experience heat gain equivalent to a 11 K rise in MRT [Arens et al. (1986)].
Similarly, in residential applications, the perceived need for solar control is affected both by the contribution of window surfaces to MRT and by overheating due to direct solar load.
Advances in window technology, especially high-performance glazings, mean that the designer has a choice of potential glazing systems. On the basis of annual energy performance for heating, cooling, and lighting, these alternatives may give similar outcomes.
However, because they represent different combinations of U-fac-tor, SHGC, and inside glass surface temperature, their comfort out-comes may differ considerably. Research continues to develop tools that will help designers evaluate such difficult trade-offs. In the meantime, several general rules of thumb may be followed: • In heating-dominated climates, windows with the lowest U-factor tend to give the best comfort outcomes. However, there is likely to be a trade-off between the twin goals of maximizing instanta-neous comfort and minimizing annual energy consumption.
• In cooling-dominated climates or for orientations where cooling loads are of concern, windows with the lowest rise in surface tem-perature for a given SHGCtend to give the best comfort outcomes.
Sound Reduction Proper acoustical treatment of exterior walls can decrease noise levels in certain areas. The airtightness of a wall is the primary fac-tor to consider in reducing sound transmission from the exterior.
Once walls and fenestration products are tight, the choice of glass and draperies becomes important. Draperies do not prevent sound from coming through the fenestration; they act as an absorber for sound that does penetrate. Table 26 lists average sound transmission losses for various types of glass. These averages apply for the fre-quency range of 125 to 4000 Hz and were determined by tests based on ASTM Standard E 90. Strength and Safety In addition to its thermal, visual, and aesthetic functions, glass for building exteriors must also perform well structurally. Wind loads are specified in most building codes, and these requirements may be adequate for many structures. However, detailed wind tun-nel tests should be run for tall or unusually shaped buildings and for buildings where the surroundings create unusual wind patterns. The strength of annealed, heat-strengthened, tempered, laminated, and insulated glass is given in ASTM Standard E 1300.
Thermal expansion and contraction of glass can result in break-age of ordinary annealed glass. This expansion and contraction can be caused by solar radiation onto partly shaded glass, by heat traps from drop ceilings and tight-fitting drapes, or by HVAC ducts incor-rectly directed toward the glazing. High-performance tinted and reflective glasses with low-e coatings are usually more vulnerable to thermal stress breakage than clear glass. Heat treating (heat strengthening or fully tempering) the glass resists thermal stress breakage. Heat-strengthened glass, although not a safety glass, is usually preferred to tempered (safety) glass because it typically has less distortion and is much less likely to have spontaneous breakage.
Spontaneous breakage can occur on very rare occasions in tempered glass. The glass manufacturer or fabricator should be consulted for information on thermal stress performance.
Building codes may require glass in certain positions to per-form with certain breakage characteristics, which can be satisfied by tempered, laminated, or wired glass. In this case, glass should meet Federal Standard 16 CFR 1201 or other appropriate break-age performance requirements.
Life-Cycle Costs Alternative building shells should be compared to ensure satis-factory energy use and total energy budget compliance, if required.
ASHRAE Standards 90.1 and 90.2 should be used as a starting point. A life-cycle cost model should be developed for each system considered. See Chapter 35 of the 1999 ASHRAE Handbook— Applications.
Table 26 Sound Transmittance Loss for Various Types of Glass Type of Glass Sound Transmittance Loss, dB 3 mm double-strength sheet glass 24 6 mm plate or float glass 27 13 mm plate glass 32 19 mm. plate glass 35 25 mm plate glass 36 6 mm. laminated glass (11 mm plastic interlayer) 30 25 mm insulating glass 32 13 mm laminated glass (11 mm plastic interlayer) 34 Insulating glass, 150 mm air space, 6 mm plate or float glass 40 Fig. 40 Fenestration Impacts on Thermal Comfort: Long-Wave Radiation, Solar Radiation, Convective Draft 30.64 2001 ASHRAE Fundamentals Handbook (SI) DURABILITY The service life and long-term performance of fenestration sys-tems depend on the durability of all the components that make up the system. Representative samples of IGUs are usually tested (for seal durability) according to test methods to ensure the integrity of the seal. Failure of IGUs is usually indicated by loss of adhesion of sealant to the glass; as a result, fogging occurs inside the glazing cavity.
In the case of argon-filled units, the seal failure means the loss of argon and, hence, degradation in the thermal characteristics of the unit. Extensive work was done at the National Research Council of Canada to study the durability of IGUs filled with argon gas (Elmahdy and Yusuf 1995). The results indicated that, under normal conditions, argon loss due to diffusion through the sealant is very small. However, when cracks or pinholes exist in the sealant, most of the argon gas escapes from the unit, which implies that the imple-mentation of stringent quality control procedures is essential for the production of durable IGUs.
The degradation of organic materials and other chemical compo-nents in the IGUs, as a result of exposure to ultraviolet radiation, is also among the factors affecting the durability and service life of fenestration systems. The use of low-e coating on glass tends to enhance the appearance of chemical deposits on the glass surface.
Also, the insertion of muntin bars in the glazing cavities may result in excessive rate of unit failure during the ultraviolet volatile (fog-ging) test unless strict quality assurance processes are implemented.
The current ASTM (United States) and CGSB (Canada) durability standards are being reviewed to reflect the emergence of new tech-nologies in the fenestration industry.
Insulating glass products have been studied in a 15-year correla-tion study by the Sealed Insulating Glass Manufacturers Associa-tion (SIGMA). During this study, it was found that long-term performance and durability of insulating glass correlated well with the test level to which such a unit’s construction had been manufac-tured with regard to the ASTM Standard E 773 test method and ASTM Standard E 774 specification for sealed insulating glass. The units showing the highest percentage of resistance to seal failure were those that were tested in conformance with the ASTM Stan-dard E 774 Class CBA standard. Units that did not qualify to the A level showed a definite correlation to a higher percentage of failure.
During the field correlation studies, it was found that units glazed in compliance with the SIGMA recommendations perform for longer periods than units not constructed properly, having deficiencies in the glazing system, or not meeting the ASTM requirements.
The durability of fenestration systems is also dependent on the durability of other system components such as the weatherstripping, gaskets, glazing tapes, air seals, and hardware. The wear and tear of these elements with time and use may result in excessive air and water leakage, which will affect the overall performance and the service life of the system. Excessive water leakage may result in damage to the fenestration product, especially the edge seal, as well as the wall section where the product is mounted. Excessive air leakage may lead to frost buildup and condensation on the fenestra-tion surfaces.
Studies conducted at the National Research Council of Canada (Elmahdy 1995) and elsewhere (Patenaude 1995) showed that when windows are tested at high pressure and temperature differentials, they experience air leakage rates which exceed those determined at 75 Pa and zero temperature differential (these conditions are used in rating the window air leakage in U.S. and Canadian standards). In other studies (CANMET 1991, 1993), the effect of pressure and motion cycling on windows resulted in excessive degradation in almost all the window performance factors, particularly the conden-sation resistance, ease of operation, air leakage, and water leakage.
In order to predict long-term performance, the unit construction for insulating glass should be subjected to a test and certification program such as ASTM Standard E 774 Class CBA level and the requirements of SIGMA or CGSB Standard 12.8 certified by the Insulating Glass Manufacturers Association of Canada (IGMAC) or equivalent.
In addition to affecting the fenestration performance factors mentioned above, durability may also affect long-term energy performance.
CODES AND STANDARDS National Fenestration Rating Council (NFRC) The National Fenestration Rating Council (NFRC) was formed in 1989 to respond to a need for fair, accurate, and credible ratings for fenestration products. NFRC has adopted rating procedures for U-factor (NFRC 100), solar heat gain coefficient and visible trans-mittance (NFRC 200), optical properties (NFRC 300), emissivities (NFRC 301), and air leakage (NFRC 400). To provide certified rat-ings, manufacturers follow the requirements in the NFRC Product Certification Program (PCP) which involves working with labora-tories accredited to the NFRC Laboratory Accreditation Program (LAP) and independent certification and inspection agencies accredited through the NFRC Certification Agency Program (CAP).
NFRC 100 was the first of the NFRC rating procedures approved and thus the first NFRC procedure adopted into energy codes in the United States. NFRC 100 requires the use of a combination of state-of-the-art computer simulations and improved thermal testing to determine U-factors for the whole product. The next step is product certification. NFRC has a series of checks and balances to ensure that the rating system is accurately and uniformly employed. Prod-ucts and their ratings are authorized for certification by an NFRC-licensed independent certification and inspection agency (IA).
Finally, two labels are required: the temporary label, which contains the product ratings, and a permanent label, which allows tracking back to the IA and information in the NFRC Product Directory. In addition to informing the buyer, the temporary label provides the building inspector with the information necessary to verify energy code compliance. The permanent label provides access to energy rating information for a future owner, property manager, building inspector, lending agency, or building energy rating organization. This process has a number of noteworthy features that make it superior to previous fenestration energy rating systems and correct past problems: • The procedures provide a means for manufacturers to take credit for all the nuances and refinement in their product design and a common basis for others to compare product claims.
• The involvement of independent laboratories and the IA provides architects, engineers, designers, contractors, consumers, building officials, and utility representatives with greater confidence that the information is unbiased.
• Requiring simulation and testing provides an automatic check on accuracy. This also remedies a shortcoming of previous energy code requirements that relied on testing alone, which allowed manufacturers to perform several tests and then use the best one for code purposes.
• The certification process indicates that the manufacturer is con-sistently producing the product that was rated. This corrects a past concern that manufacturers were able to make an exceptionally high quality sample and obtain a good rating in a test but not con-sistently produce that product.
• There is now a readily visible temporary label that can be used by the building inspector to quickly verify compliance with the energy code.
• There is now a permanent label that enables future access to energy rating information.
Fenestration 30.65 While the NFRC program is similar for other fenestration char-acteristics, there are differences worth pointing out. The solar heat gain coefficient and visible transmittance ratings (NFRC 200), which have been referenced in several codes, are based on simula-tion alone. Optical properties (NFRC 300) and emissivity (NFRC 301) are based on measurements by the manufacturer, with indepen-dent verification. The air leakage ratings (NFRC 400) are based on testing alone. For site-assembled fenestration products (such as cur-tain walls and window walls), there is an NFRC label certificate that fulfills the labeling requirements and serves the certification pur-pose. There must be a separate NFRC label certificate for each “individual product” in a particular project.
United States Energy Policy Act (EPAct) In the United States, the 1992 Energy Policy Act (EPAct) required the development of national fenestration energy rating sys-tems and specified NFRC as the preferred developer. (The U.S.
Department of Energy was to establish procedures if the NFRC did not.) While this recognition provided an impetus for NFRC to develop the desired procedures and programs, the EPAct sections on energy codes have been a key factor in their implementation. EPAct set energy code baselines for state energy codes. The ICC 2000 International Energy Conservation Code (IECC) and ASHRAE/IESNA Standard 90.1-1999, Energy Standard for Buildings Except Low-Rise Residential Buildings are the current successors to the versions cited in the 1992 legislation. The majority of states have adopted the predecessors to the 2000 IECC (including the 1998 IECC and the CABO 1995 Model Energy Code) and to ASHRAE/IESNA Standard 90.1-1999 (i.e., ASH-RAE/IESNA Standard 90.1-1989) into their codes either directly or by reference when adopting a building code published by one of the three national code organizations in the United States. The ICC 2000 International Building Code (the U.S. model building code jointly developed by ICBO, BOCA, and SBCCI) references the 2000 International Energy Conservation Code.
The ICC 2000 International Energy Conservation Code The ICC 2000 International Energy Conservation Code (IECC) references NFRC 100 for U-factor (as did the 1998 IECC and the 1995 Model Energy Code) and NFRC 200 for solar heat gain coef-ficient (SHGC) (as did the 1998 IECC). Section 102.3, which applies to all occupancies, requires U-factors of fenestration prod-ucts (windows, doors, and skylights) to be determined in accor-dance with NFRC 100 by an accredited independent laboratory and labeled and certified by the manufacturer. While the language does not specify NFRC accreditation, it both requires the use of the NFRC rating procedure by an independent entity and requires label-ing and certification.
ASHRAE/IESNA Standard 90.1-1999 In 1999, ASHRAE and IESNA published a comprehensive update to the 1989 version of Standard 90.1. The fenestration rating, labeling, and certification criteria are in Sections 5.2.2 and 5.2.3.
U-factors are to be determined in accordance with NFRC 100, solar heat gain coefficient in accordance with NFRC 200, visible trans-mittance in accordance with NFRC 300, and air leakage in accor-dance with NFRC 400.
For further information on U.S. energy codes, the Building Codes Assistance Project (BCAP) publishes a bimonthly summary entitled “Status of State Energy Codes,” which provides informa-tion on current codes and pending legislation. For additional infor-mation, contact BCAP at 1200 18th Street NW, Suite 900, Washington DC 20036; voice: 202-530-2200; fax: 202-331-9588; e-mail: bcap@ase.org; website: bcap/update. html.
Canadian Standards Association (CSA) In Canada, the Canadian Standards Association (CSA) promul-gates fenestration energy rating standards. CSA Standard A440.2 addresses most fenestration products, and CSA Standard A453 addresses doors. These are companion standards to NFRC 100.
NFRC and CSA have established a Thermal Harmonization Task Force to attempt to harmonize their fenestration energy rating standards.
SYMBOLS a = absorptance in a layer, considered as an isolated layer A = apparent solar constant A = total projected area of the fenestration product AST = apparent solar time B = atmospheric extinction coefficient C = sky diffuse factor e = hemispherical emissivity EDN = direct normal irradiance ED = direct irradiance Ed = diffuse sky irradiance Er = diffuse ground reflected irradiance Et = total irradiance ET = equation of time h = surface heat transfer coefficient H = fenestration product height H = hour angle k = thermal conductivity L = latitude LON = longitude LSM = local standard meridian LST = local standard time n = refractive index PV = vertical projection depth PH = horizontal projection depth q = instantaneous energy flux Q = instantaneous energy flow R = reflectance of a layer or collection of layers (system or subsystem) RH = height of opaque surface between fenestration product and horizontal projection RW = width of opaque surface between fenestration product and vertical projection SHGC = solar heat gain coefficient t = relative temperature T = absolute temperature T = transmittance of a layer or collection of layers (system or subsystem) U = overall coefficient of heat transfer W = fenestration product width Y = ratio of vertical/horizontal sky diffuse α = material absorptivity ß = solar altitude δ = declination ∆= vertical projection profile angle φ = solar azimuth γ = surface solar azimuth η = day of year λ = wavelength θ = incident angle ρg = ground reflectance ϖ = solid angle Ω= horizontal projection profile angle ξ = refractive angle ψ = surface azimuth Σ = surface tilt A = absorptance in a layer or a collection of layers (system or subsystem) REFERENCES AAMA. 1987. Skylight handbook: Design guidelines. American Architec-tural Manufacturers Association, Schamberg, IL.
AAMA. 1988. Voluntary test method for thermal transmittance and conden-sation resistance of windows, doors and glazed wall sections. Publication No. AAMA 1503.1-88. 30.66 2001 ASHRAE Fundamentals Handbook (SI) AGSL. 1992. Vision3, Glazing system thermal analysis—User manual, Department of Mechanical Engineering, University of Waterloo, Water-loo, Ontario, Canada, August.
Allen, C.W. 1973. Astrophysical quantities. The Athlone Press, University of London, London.
Arasteh, D. 1989. An analysis of edge heat transfer in residential windows.
Proceedings of ASHRAE/DOE/BTECC Conference, Thermal Perfor-mance of the Exterior Envelopes of Buildings IV, Orlando, FL, 376-87.
Arasteh, D. et al. 1994. WINDOW 4.1: A PC program for analyzing window thermal performance in accordance with standard NFRC procedures.
Publication LBL-35298, Lawrence Berkeley Laboratory, Energy & Environment Division, Berkeley, CA, 1993.
Arasteh, D. et al. 1995. Recent technical improvements to the WINDOW computer program. Proceedings of Window Innovations Conference ’95, Toronto, Canada.
Arasteh, D., J. Huang, R. Mitchell, B. Clear, and C. Kohler. 2000. A database of window annual energy use in typical North American single family houses. ASHRAE Transactions 106(2).
Arens, E.A., R. Gonzalez, and L. Berglund. 1986. Thermal comfort under an extended range of environmental conditions. ASHRAE Transactions 92(1).
ASHRAE. 1988. Method of measuring solar-optical properties of materials.
ANSI/ASHRAE Standard 74-1988. ASHRAE. 1996. Standard method for determining and expressing the heat transfer and total optical properties of fenestration products. Draft ASHRAE Standard, February.
ASHRAE. 1999. Energy standard for buildings except low-rise residential buildings. ASHRAE/IESNA Standard 90.1-1999. ASTM E. 1981. Recommended practice for laboratory measurements of air borne sound transmission loss of building partitions. Standard E 90-81.
American Society for Testing and Materials, West Conshohocken, PA.
ASTM. 1982. Standard test method for solar absorptance, reflectance, and transmittance of materials using integrating spheres. Standard E 90-82. ASTM. 1986. Standard test method for solar transmittance (terrestrial) of sheet materials using sunlight. Standard E 1084-86.
ASTM. 1987. Standard tables for terrestrial direct normal solar spectral irra-diance for air mass 1.5. Standard E 891-87 (now part of ASTM Standard G 159-98).
ASTM. 1987. Standard tables for terrestrial solar spectral irradiance at air mass 1.5 for a 37° tilted surface. Standard E 892-87 (now part of ASTM Standard G 159-98). ASTM. 1988. Standard practice for calculation of photometric transmittance and reflectance of materials to solar radiation. Standard E 971-88.
ASTM. 1988. Standard test method for solar photometric transmittance of sheet materials using sunlight. Standard E 972-88.
ASTM. 1988. Standard test method for seal durability of sealed insulated glass units. Standard E 773-88.
ASTM. 1991. Standard test method for determining rate of air leakage through exterior windows, curtain walls, and doors under specified pres-sure differences across the specimen. Standard E 283-91.
ASTM. 1992. Standard specification for sealed insulated glass units. Stan-dard E 774-92.
ASTM. 1994. Standard practice for determining the minimum thickness and type of glass required to resist a specific load. Standard E 1300-94.
Bass, M., ed. 1995. Handbook of optics, Vol. I —Fundamentals, techniques, and design, 2nd ed. Optical Society of America, McGraw-Hill, New York.
Beck, F.A., B.T. Griffith, D, Turler, and D. Arasteh. 1995. Using infrared thermography for the creation of a window surface temperature database to validate computer heat transfer models. Proceedings of Window Inno-vations Conference ’95, Toronto, Canada.
Bird, R.E. and R.L. Hulstrom. 1982. Terrestrial solar spectral data sets.
SERI/TR-642-1149. Solar Energy Research Institute.
Bliss, R.W. 1961. Atmospheric radiation near the surface of the ground.
Solar Energy 5(3):103.
Brandle, K. and R.F. Boehm. 1982. Air flow windows: Performance and applications. Proceedings of Thermal Performance of the Exterior Enve-lopes of Buildings II, ASHRAE/DOE Conference, ASHRAE SP38, December.
CANMET. 1991. A study of the long term performance of operating and fixed windows subjected to pressure cycling. Catalogue No. M91-7/214-1993E. Efficiency and Alternative Energy Technology Branch, CAN-MET, Ottawa, Ontario, Canada.
CANMET. 1993. Long term performance of operating windows subjected to motion cycling. Catalogue No. M91-7/235-1993E. Efficiency and Alter-native Energy Technology Branch, CANMET, Ottawa, Ontario, Canada.
Carpenter, S. and A. Elmahdy. 1994. Thermal performance of complex fen-estration systems. ASHRAE Transactions. Carpenter, S. and J. Hogan. 1996. Recommended U-factors for swinging, overhead and revolving doors. ASHRAE Transactions.
Carpenter, S. and A. McGowan. 1993. Effect of framing systems on the ther-mal performance of windows. ASHRAE Transactions 99(1).
CEA. 1995. Energy-efficient residential and commercial windows reference guide. Canadian Electricity Association, Montreal, PQ.
Choi, A.-S. and R.G. Mistrick. 1999. Analysis of daylight responsive dim-ming system performance. Building and Environment 34:231-43.
Crooks, B.P. et al. 1995. NFRC efforts to develop a residential fenestration annual energy rating methodology. Proceedings of Window Innovations Conference ’95, Toronto, Canada.
CSA. 1990. Windows. Publication CAN/CSA-A440-90. Canadian Stan-dards Association, Etobicoke, Ontario.
CSA. 1991. Windows/User selection guide to CSA standards. CAN/CSA A440.1-M90.
CSA. 1993. Energy performance evaluation of windows and sliding glass doors. CAN/CSA A440.2-93.
CSA. 1995. Energy performance evaluation of swinging doors. A453-95.
Curcija, D. and W.P. Goss. 1994. Two-dimensional finite element model of heat transfer in complete fenestration systems. ASHRAE Transactions 100(2).
Curcija, D. and W.P. Goss. 1995a. Three-dimensional finite element model of heat transfer in complete fenestration systems. Window Innovations Conference ’95, Toronto, Canada, June.
Curcija, D. and W.P. Goss. 1995b. New correlations for convective heat transfer coefficient on indoor fenestration surfaces—Compilation of more recent work. Thermal Performance of the Exterior Envelopes of Buildings VI. ASHRAE.
Curcija, D., W.P. Goss, J.P. Power, and Y. Zhao. 1996. “Variable-h” model for improved prediction of surface temperatures in fenestration systems.
Technical Report, University of Massachusetts at Amherst.
de Abreu, P., R.A. Fraser, H.F. Sullivan, and J.L. Wright. 1996. A study of insulated glazing unit surface temperature profiles using two-dimen-sional computer simulation. ASHRAE Transactions 102(2).
Duffie, J.A. and W.A. Beckman. 1980. Solar engineering of thermal pro-cesses. John Wiley & Sons, New York.
Elmahdy, H. 1996. Surface temperature measurement of insulating glass units using infrared thermography. ASHRAE Transactions 102(2).
Elmahdy, A.H. and S.A. Yusuf. 1995. Determination of argon concentration and assessment of the durability of high performance insulating glass units filled with argon gas. ASHRAE Transactions 101(2).
Elmahdy, A.H. 1995. Air leakage characteristics of windows subjected to simultaneous temperature and pressure differentials. Proceedings Win-dow Innovation Conference ’95, CANMET.
El Sherbiny, S.M., et al. 1982. Heat transfer by natural convection across vertical and inclined air layers. Journal of Heat Transfer 104:96-102.
EEL. 1990. FRAME/VISION window performance modelling and sensitiv-ity analysis. Institute for Research in Construction, National Research Council of Canada, Ottawa.
EEL. 1995. The FRAMEplus Toolkit for heat transfer assessment of build-ing components. Enermodal Engineering, Kitchener, ON, and Denver, CO.
Solar Energy. 1988. Erratum 40:175. Erhorn, H. and M. Dirksmöller, eds. 2000. Documentation of the software package ADELINE 3. Fraunhofer-Institut für Bauphysik, Stuttgart.
Ewing, W.B. and J.I. Yellott. 1976. Energy conservation through the use of exterior shading of fenestration. ASHRAE Transactions 82(1):703-33.
Finlayson, E.U. and D. Arasteh. 1993. WINDOW 4.0: Documentation of calculation procedures. Publication LBL-33943/UC-350, Lawrence Berkeley Laboratory, Energy & Environment Division, Berkeley, CA.
Federal Standard 16 CFR 1201. Safety standard for architectural glazing materials.
Galanis, N. and R. Chatiguy. 1986. A critical review of the ASHRAE solar radiation model. ASHRAE Transactions 92(1).
Gates, D.M. 1966. Spectral distribution of solar radiation at the earth’s sur-face. Science 151(2):3710.
Griffith, B.T., D. Turler, and D. Arasteh. 1996. Surface temperature of insu-lated glazing units: Infrared thermography laboratory measurements.
ASHRAE Transactions 102(2).
Fenestration 30.67 Griffith, B., E. Finlayson, M. Yazdanian, and D. Arasteh. 1998. The signif-icance of bolts in the thermal performance of curtain-wall frames for glazed facades. ASHRAE Transactions 105(1). Gueymard, C.A. 1987a. An anisotropic solar irradiance model for tilted sur-faces and its comparison with selected engineering algorithms. Solar Energy 38:367-86. Erratum, Solar Energy 40:175 (1988).
Gueymard, C. 1987b. Solar Energy 38:367-86. Gueymard, C.A. 1993a. Critical analysis and performance assessment of clear sky solar irradiance models using theoretical and measured data.
Solar Energy.
Gueymard, C. 1993b. Development and performance assessment of a clear sky spectral radiation model, Proceedings, 22nd Annual Conference, Solar ’93, Washington, D.C., American Solar Energy Society, 433-38.
Gueymard, C. 1995. A simple model of the atmospheric radiative transfer of sunshine: Algorithms and performance assessment. FSEC-PF-270-95, Florida Solar Energy Center, Cocoa, FL, November.
Hawthorne W. and M.S. Reilly. 2000. The impact of glazing selection on residential duct design and comfort. ASHRAE Transactions 106(1).
Heschong, L., D. Mahone, F. Rubinstein, and J. McHugh. 1998. Skylighting guidelines. The Heschong Mahone Group, Sacramento, CA.
Hickey, J.R. et al. 1981. Observations of the solar constant and its variations: Emphasis on Nimbus 7 results. Symposium on the Solar Constant and the Special Distribution of Solar Irradiance, IAMAP 3rd Scientific Assem-bly, Hamburg, Germany.
Hogan, J.F. 1988. A summary of tested glazing U-values and the case for an industry wide testing program. ASHRAE Transactions 94(2).
Hollands, K.G.T. and J.L. Wright. 1982. Heat loss coefficients and effective τα products for flat plate collectors with diathermous covers. Solar Energy 30:211-16.
Huang, Y.J., J.C. Bull, and R.L. Ritschard. 1987. Technical documentation for a residential energy use database developed in support of ASHRAE Special Project 53. Lawrence Berkeley Laboratory Report LBL-24306.
Huang, J., R. Mitchell, D. Arasteh, and S. Selkowitz. 1999. Residential fen-estration performance analysis using RESFEN 3.1. Thermal Perfor-mance of the Exterior Envelopes of Building VII, ASHRAE.
ICC. 2000. International energy conservation code. International Code Council, Falls Church, VA.
IEA. 1999. Daylighting simulation: Methods, algorithms, and resources. A Report of IEA SHC Task 21/ECBCS Annex 29, December. Available at or LBNL Report 44296, Lawrence Berkeley National Laboratory.
IESNA. 1999. IESNA recommended practice of daylighting. IES RP-5-99.
Illuminating Engineering Society of North America, New York.
Iqbal, M. 1983. An introduction to solar radiation. Academic Press, Toronto.
ISO. 2000. Thermal performance of windows, doors and shading devices— Detailed calculations. ISO 15099: Final Draft International Standard (FDIS). International Organization for Standardization, Geneva.
Jennings, J.D., F.M. Rubinstein, D. DiBartolomeo, and S. Blanc. 1999. Com-parison of control options in private offices in an advanced lighting con-trols testbed. LBNL-43096, Lawrence Berkeley National Laboratory.
Keyes, M.W. 1967. Analysis and rating of drapery materials used for indoor shading. ASHRAE Transactions 73(1):8.4.1.
Klems, J.H. 1989. U-values, solar heat gain, and thermal performance: Recent studies using the MoWitt. ASHRAE Transactions 95(1).
Klems, J.H. 1994a. A new method for predicting the solar heat gain of com-plex fenestration systems: I. Overview and derivation of the matrix layer calculation. ASHRAE Transactions 100(1):1065-72.
Klems, J.H. 1994b. A new method for predicting the solar heat gain of com-plex fenestration systems: II. Detailed description of the matrix layer cal-culation. ASHRAE Transactions 100(1):1073-86.
Klems, J.H. 2001. Solar heat gain through fenestration systems containing shading: Procedures for estimating performance from minimal data.
LBNL-46682. Windows and Daylighting Group, Lawrence Berkeley National Laboratory, Berkeley, CA.
Klems, J.H. and G.O. Kelley. 1996. Calorimetric measurements of inward-flowing fraction for complex glazing and shading systems. ASHRAE Transactions 102(1):947-54.
Klems, J.H. and J.L. Warner. 1995. Measurement of bi-directional optical properties of complex shading devices. ASHRAE Transactions 101(1): 791-801.
Klems, J.H. and J.L. Warner. 1997. Solar heat gain coefficient of complex fenestrations with a venetian blind for differing slat tilt angles. ASHRAE Transactions 103(1):1026-34.
Klems, J.H., J.L. Warner, and G.O. Kelley. 1996. A comparison between cal-culated and measured SHGC for complex fenestration systems. ASHRAE Transactions 102(1):931-39.
LBL. 1994. WINDOW 4.1: A PC program for analyzing window thermal performance of fenestration products. LBL-35298. Windows and Day-lighting Group, Lawrence Berkeley Laboratory, Berkeley, CA.
LBL. 1996. THERM 1.0: A PC program for analyzing the two-dimensional heat transfer through building products. LBL-37371. Windows and Day-lighting Group.
Lyons, P.R.A., D.K. Arasteh, and C. Huizenga. 2000. Window performance for human thermal comfort. ASHRAE Transactions 106(1).
Mathis, R.C. and R. Garries. 1995. Instant, annual life: A discussion on the current practice and evolution of fenestration energy performance rating.
Proceedings of Window Innovations Conference ’95, Toronto, Canada.
McCabe, M.E., et al. 1986. U-value measurements for windows and mov-able insulations from hot box tests in two commercial laboratories.
ASHRAE Transactions 92(1).
McCluney, R. 1987. Determining solar radiant heat gain of fenestration sys-tems. Passive Solar Journal 4(4):439-87.
McCluney, R. 1990. Awning shading algorithm update. ASHRAE Trans-actions 96(1).
McCluney, R. 1991. The death of the shading coefficient? ASHRAE Journal (3):36-45.
McCluney, R. 1993. Sensitivity of optical properties and solar gain of spec-trally selective glazing systems to changes in solar spectrum. Solar ’93, 22nd American Solar Energy Society Conference, Washington, D.C. McCluney, R. 1994a. Introduction to radiometry and photometry. Artech House, Boston.
McCluney, R. 1994b. Angle of incidence and diffuse radiation influences on glazing system solar gain. Proceedings Solar ’94 Conference, American Solar Energy Society, San Jose, CA.
McCluney, R. 1996. Sensitivity of fenestration solar gain to source spectrum and angle of incidence. ASHRAE Transactions 102(2).
McCluney, R. and C. Gueymard. 1992. SUNSPEC 1.0, Operating manual.
FSEC-SW-3-92. Florida Solar Energy Center, Cocoa, FL.
McCluney, R. and L.R. Mills. 1993. Effect of interior shade on window solar gain. ASHRAE Transactions 99(2). Mitchell, R., J. Huang, D. Arasteh, R. Sullivan, and S. Phillip. 1999. RES-FEN 3.1: A PC program for calculating the heating and cooling energy use of windows in residential buildings—Program description. LBNL-40682 Rev. BS-371. Lawrence Berkeley National Laboratory.
Moon, P. 1940. Proposed standard solar radiation curves for engineering use. Journal of the Franklin Institute 11:583.
NAGDM. 1992. Test method for thermal transmittance and air infiltration of garage doors. Standard 105-1992. National Association of Garage Door Manufacturers, Chicago.
NFRC. 1993. NFRC 301-93: Standard test method for emissivity of specular surfaces using spectrometric measurements. National Fenestration Rat-ing Council, Silver Spring, MD.
NFRC. 1994. NFRC 300-94: Procedures for determining solar optical prop-erties of simple fenestration products.
NFRC. 1995. NFRC 200-95: Procedure for determining fenestration prod-uct solar heat gain coefficients at normal incidence.
NFRC. 1995. NFRC 400-95: Procedure for determining fenestration prod-uct air leakage.
NFRC. 1997. NFRC 100-97: Procedure for determining fenestration prod-uct U-factors.
NFRC. 1997. NFRC 100-97 Attachment A: NFRC test procedure for mea-suring the steady-state thermal transmittance of fenestration systems.
NFRC. 1999. NFRC 100-99 Section B: Procedure for determining door sys-tem product thermal properties (Currently limited to U-values).
NFRC. 2000a. Draft NFRC 500: Calculational method and test procedure to determine the condensation index of fenestration products.
NFRC. 2000b. Spectral data library available through www.nfrc.org/soft-ware.html.
Ozisik, N. and L.F. Schutrum. 1960. Solar heat gain factors for windows with drapes. ASHRAE Transactions 66:228.
Parmelee, G.V. and W.W. Aubele. 1952. Radiant energy emission of atmo-sphere and ground. ASHRAE Transactions 58:85.
Parmelee, G.V. and R.G. Huebscher. 1947. Forced convection heat transfer from flat surfaces. ASHVE Transactions 245-84.
Patenaude, A. 1995. Air infiltration rate of windows under temperature and pressure differentials. Proceedings Window Innovation Conference ’95.
CANMET.
30.68 2001 ASHRAE Fundamentals Handbook (SI) Pennington, C.W. 1968. How louvered sun screens cut cooling, heating loads. Heating, Piping, and Air Conditioning (December).
Pennington, C.W. et al. 1964. Experimental analysis of solar heat gain through insulating glass with indoor shading. ASHRAE Journal 2:27.
Pennington, C.W. and G.L. Moore. 1967. Measurement and application of solar properties of drapery shading materials. ASHRAE Transactions 73(1):8.3.1.
Pennington, C.W., C. Morrison, and R. Pena. 1973. Effect of inner surface air velocity and temperatures upon heat loss and gain through insulating glass. ASHRAE Transactions 79(2).
Perez, R. et al. 1986. An anisotropic hourly diffuse radiation model for slop-ing surfaces—Description, performance validation, and site dependency evaluation. Solar Energy 36:481-98.
Reilly, M.S. 1994. Spacer effects on edge-of-glass and frame heat transfer.
ASHRAE Transactions 94:31-33.
Reilly, M.S., F.C. Winkelmann, D.K. Arasteh, and W.L. Carroll. 1992. Mod-eling windows in DOE-2.1E. Energy and Buildings 22(1995):59-66.
Robbins, C.L. 1986. Daylighting: Design and analysis. Van Nostrand Rein-hold, New York.
Rubin, M. 1982a. Solar optical properties of windows. Energy Research 6:122-33.
Rubin, M. 1982b. Calculating heat transfer through windows. Energy Research 6:341-49.
Rubin, M. 1984. Optical constants and bulk optical properties of soda lime silica glasses for windows. Report No. 13572, Applied Sciences Divi-sion, Lawrence Berkeley Laboratory, Berkeley, CA.
Rubin, M. 1985. Optical properties of soda lime silica glasses. Solar Energy Materials 12:275-88.
Rubin, M., R. Powles, and K. von Rottkay. 1999. Models for the angle-dependent optical properties of coated glazing materials. Solar Energy 66(4):267-76.
Rudoy, W. and F. Duran. 1975. Effect of building envelope parameters on annual heating/cooling load. ASHRAE Journal (7):19. Rundquist, R.A. 1991. Calculation procedure for daylighting and fenestra-tion effects on energy and peak demand. ASHRAE Transactions 97(2).
Selkowitz, S.E. 1979. Thermal performance of insulating windows. ASH-RAE Transactions 85(2):669-85.
Sezgen, O. and J.G. Koomey. 1998. Interactions between lighting and space conditioning energy use in U.S. commercial buildings. LBNL-39795, Lawrence Berkeley National Laboratory.
Shewen, E.C. 1986. A Peltier-effect technique for natural convection heat flux measurement applied to the rectangular open cavity. Ph.D. thesis, Department of Mechanical Engineering, University of Waterloo, Ontario.
Smith, W.A. and C.W. Pennington. 1964. Shading coefficients for glass block panels. ASHRAE Journal 5(12):31.
Sodergren, D. and T. Bostrom. 1971. Ventilating with the exhaust air win-dow. ASHRAE Journal 13(4):51.
Sterling, E.M., A. Arundel, and T.D. Sterling. 1985. Criteria for human exposure in occupied buildings. ASHRAE Transactions 91(1).
Sullivan, H.F., J.L. Wright, and R.A. Fraser. 1996. Overview of a project to determine the surface temperatures of insulated glazing units: Thermo-graphic measurement and 2-D simulation. ASHRAE Transactions 102(2).
Sullivan, H.F. and J.L. Wright. 1987. Recent improvements and sensitivity of the VISION glazing system thermal analysis program. Proceedings of the 12th Passive Solar Conference, ASES/SESCI, Portland, OR, 145-49.
TSC. 1996. WinSARC: Solar angles and radiation calculation for MS Win-dows. User’s Manual. Tait Solar Co., Tempe, AZ.
Threlkeld, J.L. and R.C. Jordan. 1958. Direct solar radiation available on clear days. ASHRAE Transactions 64:45.
Vild, D.J. 1964. Solar heat gain factors and shading coefficients. ASHRAE Journal 10:47.
Ward, G.W. 1990. Visualization. Lighting Design + Application 20(6):4-20.
Wehrli. 1985. Extraterrestrial solar spectrum. Publication No. 615, Phys-ikalisch-Metrologisches Observatorium and World Radiation Data Cen-ter, Davos, Switzerland.
Winkelmann, F.C. 1983. Daylighting calculation in DOE-2. LBNL-11353.
Lawrence Berkeley National Laboratory.
Wright, J.L. 1995a. Summary and comparison of methods to calculate solar heat gain. ASHRAE Transactions 101(1).
Wright, J.L. 1995b. VISION4 glazing system thermal analysis: User man-ual. Advanced Glazing System Laboratory, University of Waterloo, August.
Wright, J.L. 1995c. VISION4 glazing system thermal analysis: Reference manual. Advanced Glazing System Laboratory, University of Waterloo, May.
Wright, J.L. 1996. A correlation to quantify convective heat transfer between window glazings. ASHRAE Transactions 102(1):940-46.
Wright, J.L. and H.F. Sullivan. 1995a. A 2-D numerical model for natural convection in a vertical, rectangular window cavity. ASHRAE Trans-actions 100(2).
Wright, J.L. and H.F. Sullivan. 1995b. A 2-D numerical model for glazing system thermal analysis. ASHRAE Transactions 101(1).
Wright, J.L. and H.F. Sullivan. 1995c. A simplified method for the numeri-cal condensation resistance analysis of windows. Window Innovations ’95, Toronto, Canada, June.
Yazdanian, M. and J.H. Klems. 1993. Measurement of the exterior convec-tive film coefficient for windows in low-rise buildings. ASHRAE Trans-actions 100(1).
Yellott, J.I. 1963. Selective reflectance—A new approach to solar heat con-trol. ASHRAE Transactions 69:418.
Yellott, J.I. 1966. Shading coefficients and sun-control capability of single glazing. ASHRAE Transactions 72(1):72.
Yellott, J.I. 1972. Effect of louvered sun screens upon fenestration heat loss.
ASHRAE Transactions 78(l):199-204.
Zhao, Y., D. Curcija, and W.P. Goss. 1996. Condensation resistance valida-tion project—Detailed computer simulations using finite element meth-ods. ASHRAE Transactions 102(2).
BIBLIOGRAPHY Brambley, M.R. and S.S. Penner. 1979. Fenestration devices for energy conservation I. Energy savings during the cooling season. Energy (February).
Burkhardt, W.C. 1975. Solar optical properties of gray and brown solar con-trol series transparent acrylic sheet. ASHRAE Transactions 81(1): 384-97.
Burkhardt, W.C. 1976. Acrylic plastic glazing; properties, characteristics and engineering data. ASHRAE Transactions 82(1):683. Collins, B.L. 1975. Windows and people: A literature survey, psychological reaction with and without windows. National Bureau of Standards, Building Science Series 70. Energy, Mines and Resources Canada. 1987. Enermodal Engineering Ltd.— The effect of frame design on window heat loss phase 1. Report prepared for Renewable Energy Branch, Ottawa.
Johnson, B. 1985. Heat transfer through windows. Swedish Council for Building Research, Stockholm.
Lawrence Berkeley Laboratory. 1987. WINDOW 3.0 users guide. Windows and Daylighting Group, Lawrence Berkeley Laboratory, Berkeley, CA.
Iqbal, M. 1983. An introduction to solar radiation, p. 389. Academic Press, New York.
McCluney, W.R. 1968. Radiometry and photometry. American Journal of Physics 36:977-79. Meyer-Arendt, J.R. 1968. Radiometry and photometry: Units and conver-sion factors. Applied Optics 7:2081-84. Moore, G.L. and C.W. Pennington. 1967. Measurement and application of solar properties of drapery shading materials. ASHRAE Transactions 73(1):3.1-15.
Nicodemus, F.E. 1963. Radiance. American Journal of Physics 31:368.
Nicodemus, F.E. et al. 1976-84. Self-study manual on optical radiation measurements, Part I—Concepts. NBS Technical Notes 910-1, 910-2, 910-3, and 910-4. National Bureau of Standards (now National Insti-tute of Standards and Technology, Gaithersburg, MD). Nicodemus, F.E. et al. 1977. Geometrical considerations and nomenclature for reflectance. NBS Monograph 160, NBS (now NIST), October.
Pennington, C.W. and D.E. McDuffie, Jr. 1970. Effect of inner surface air velocity and temperature upon heat gain and loss through glass fenestra-tion. ASHRAE Transactions 76:190.
Pierpoint, W. and J. Hopkins. 1982. The derivation of a new area source equation. Presented at the Annual Conference of the Illuminating Engi-neering Society, Atlanta, August.
Threlkeld, J.L. 1962. Thermal environmental engineering, p. 321. Prentice Hall, New York.
Van Dyke, R.L. and T.P. Konen. 1982. Energy conservation through inte-rior shading of windows: An analysis, test and evaluation of reflective venetian blinds. LBL-14369. Lawrence Berkeley Laboratory, March.
Yellott, J.I. 1965. Drapery fabrics and their effectiveness in sun control.
ASHRAE Transactions 71(l):260-72.
31.1 CHAPTER 31 ENERGY ESTIMATING AND MODELING METHODS GENERAL CONSIDERATIONS ............................................. 31.1 Forward and Inverse Models .................................................. 31.1 Characteristics of Models ....................................................... 31.2 Choosing an Analysis Method ................................................ 31.3 COMPONENT MODELING AND LOADS ............................ 31.4 Calculating Space Sensible Loads .......................................... 31.4 Ground Heat Transfer ............................................................ 31.7 Secondary System Components ............................................... 31.9 Primary System Components ................................................ 31.13 SYSTEM MODELING ........................................................... 31.16 Overall Modeling Strategies ................................................. 31.16 Degree-Day and Bin Methods ............................................... 31.17 Correlation Methods ............................................................. 31.22 Simulating Secondary and Primary Systems ........................ 31.22 Modeling of System Controls ................................................ 31.22 Integration of System Models ................................................ 31.22 INVERSE MODELING ......................................................... 31.24 Categories of Inverse Methods ............................................. 31.24 Types of Inverse Models ........................................................ 31.25 Examples of Inverse Methods ............................................... 31.29 GENERAL CONSIDERATIONS HE ENERGY requirements and fuel consumption of HVAC Tsystems have a direct impact on the cost of operating a building and an indirect impact on the environment. This chapter discusses methods for estimating energy use for two purposes: modeling for the design of buildings and HVAC systems and associated design optimization (forward modeling); and modeling the energy use of existing buildings for establishing baselines and calculating retrofit savings (inverse modeling).
FORWARD AND INVERSE MODELS A mathematical model is a description of the behavior of a sys-tem. It is made up of three components (Beck and Arnold 1977): 1. Input variables (statisticians call these regressor variables while physicists refer to them as forcing variables), which act on the system. Note that there are two types of such variables: controllable by the experimenter, and such uncontrollable variables as climate.
2. System structure and parameters/properties, which provide the necessary physical description of the system (for example, thermal mass or mechanical properties of the elements).
3. Output (or response or dependent) variables, which describe the reaction of the system to the input variables. Energy use is often a response variable.
The science of mathematical modeling as applied to physical systems involves determining the third component of a system when the other two components are given or specified. We can broadly differentiate between two distinct categories of modeling, the choice of which is dictated essentially by the objective or pur-pose behind the investigation (Rabl 1988).
Forward or Classical Approach. The objective is to predict the output variables of a specified model with known structure and known parameters when subject to specified input variables. In order to ensure accuracy of prediction, the models have tended to become increasingly complex, especially with the advent of cheap and powerful computing power. This approach presumes detailed knowledge not only of the various natural phenomena affecting sys-tem behavior but also of the magnitude of various interactions (e.g., effective thermal mass, heat and mass transfer coefficients, etc.).
The main advantage of this approach is that the system need not be physically built in order to predict its behavior. Thus, this approach is ideal in the preliminary design and analysis stage and is most often employed as such.
Forward modeling as applied to building energy use begins with a physical description of the building system or component of inter-est. For example, we define the building geometry, geographical location, physical characteristics (such as wall material and thick-ness), type of equipment and operating schedules, type of HVAC system, building operating schedules, plant equipment, etc. The peak and average energy use of such a building can then be pre-dicted or simulated by the forward simulation model. The primary benefits of this method are that it is based on sound engineering principles usually taught in colleges and universities and conse-quently has gained widespread acceptance by the design and pro-fessional community. Major government-developed simulation codes, such as BLAST, DOE-2, and EnergyPlus, are based on for-ward simulation models. Figure 1 is a flow chart that illustrates the ordering of the analysis that is typically performed by a building energy simulation program.
Inverse or Data-Driven Approach. In this case, the input and output variables are known and measured, and the objective is to determine a mathematical description of the system and to estimate the system parameters. In contrast to the forward approach, the inverse approach is relevant to the case when the system has already been built and actual performance data are available for model development and/or identification. Two types of perfor-mance data can be used: nonintrusive and intrusive. Intrusive data are gathered under conditions of certain predetermined or planned experiments on the system in order to elicit system response under a wider range of system performance than would have occurred under normal system operation. Such performance data allow for more accurate model specification and identification. When con-straints on system operation do not permit such tests to be per-formed, the model must be identified from nonintrusive data obtained under normal operation.
The inverse modeling approach often allows identification of system models that are not only simpler to use but that are more accurate predictors of future system performance than forward models. The inverse approach arises in many fields, such as physics, biology, engineering, and economics. Although several mono-graphs, textbooks, and even specialized technical journals are avail-able in this area, the approach has not been widely adopted in energy-related curricula and has yet to diffuse in a significant and pervasive manner (as has the forward approach) into the building professional community.
The preparation of this chapter is assigned to TC 4.7, Energy Calculations.
31.2 2001 ASHRAE Fundamentals Handbook (SI) CHARACTERISTICS OF MODELS Forward Models Although the procedures for estimating energy requirements vary considerably in their degree of complexity, they all have three common elements: the calculation of (1) space load, (2) secondary equipment load, and (3) primary equipment energy requirements.
Here, secondary refers to equipment that distributes the heating, cooling, or ventilating medium to conditioned spaces, while pri-mary refers to central plant equipment that converts fuel or electric energy to heating or cooling effect. A major distinction is made between steady-state methods (based on degree-days or tempera-ture bins) and dynamic methods (e.g., based on transfer functions).
The first step in calculating energy requirements is to determine the space load, which is the amount of energy that must be added to or extracted from a space to maintain thermal comfort. The simplest procedures assume that the energy required to maintain comfort is only a function of the outdoor dry-bulb temperature. More detailed methods consider solar effects, internal gains, heat storage in the walls and interiors, and the effects of wind on both building enve-lope heat transfer and infiltration. Chapters 28 and 29 discuss load calculation in detail.
While energy calculations are similar to the heating and cooling load calculations used to size equipment, they are not the same.
Energy calculations are based on average use and typical weather conditions rather than on maximum use and worst-case weather.
Currently, the most sophisticated procedures are based on hourly pro-files for climatic conditions and operational characteristics for a num-ber of typical days of the year or on 8760 h of operation per year.
The second step translates the space load to a load on the sec-ondary equipment. This can be a simple estimate of duct or piping losses or gains or a complex hour-by-hour simulation of an air sys-tem, such as variable air volume with outdoor-air cooling. This step must include the calculation of all forms of energy required by the secondary system (i.e., electrical energy to operate fans and/or pumps, as well as energy in the form of heated or chilled water).
The third step calculates the fuel and energy required by the pri-mary equipment to meet these loads and the peak demand on the utility system. It considers equipment efficiencies and part-load char-acteristics. It is often necessary to keep track of the different forms of energy, such as electrical, natural gas, or oil. In some cases, where cal-culations are required to assure compliance with codes or standards, these energies must be converted to source energy or resource con-sumed, as opposed to energy delivered to the building boundary.
Often, energy calculations lead to an economic analysis to estab-lish the cost-effectiveness of conservation measures (ASHRAE Standard 90.1). Thus, thorough energy analysis provides intermedi-ate data, such as time of energy usage and maximum demand, so that utility charges can be accurately estimated. Although not part of the energy calculations, estimated capital equipment costs should be included in such an analysis.
Complex and often unexpected interactions can occur between the systems or between various modes of heat transfer. For example, radiant heating panels impact the space loads by raising the mean radiant temperature in the space (Howell and Suryanarayana 1990).
As a result, the air temperature can be lowered while maintaining comfort. Compared to a conventional heated air system, radiant panels create a greater temperature difference from the inside sur-face to the outside air. Thus, conduction losses through the walls and roof increase because the inside surface temperatures are greater. At the same time, the heating load due to infiltration or ventilation decreases because of the reduced indoor air to outdoor air tempera-ture difference. The infiltration rate may also decrease because the reduced air temperature difference reduces the stack effect.
Inverse Models The inverse model has to meet requirements very different from the forward model. The inverse model can only contain a relatively small number of parameters because of the limited and often repet-itive information contained in the performance data. (For example, building operation from one day to the next is fairly repetitive.) The inverse model is thus a much simpler model that contains fewer terms representative of aggregated or macroscopic parameters (e.g., overall building heat loss coefficient and time constants). Since the model parameters are deduced from actual building performance, it is much more likely to accurately capture the as-built system per-formance, thus allowing more accurate prediction of future system behavior under certain specific circumstances. Performance data collection and model formulation need to be appropriately tailored for the specific circumstance, which often requires a higher level of skill and expertise of the user. In general, inverse models are less flexible than forward models in evaluating energy implications of different design and operational alternatives, and so they are not a substitute in this regard.
To better understand the uses of inverse models, consider some of the questions that a building professional may ask about an exist-ing building whose energy consumption is known (Rabl 1988): • How does the consumption compare with design predictions (and, in case of discrepancies, are they due to anomalous weather, to unintended building operation, to improper operation, or to other causes)?
• How would the consumption change if the thermostat settings, ventilation rates, or indoor lighting levels were changed?
• How much energy could be saved by retrofits to the building shell, changes to air handler operation from CV to VAV, or changes in the various control settings?
• If the retrofits are implemented, can one verify that the savings are due to the retrofit and not to other causes (e.g., the weather)?
• How can one detect faults in HVAC equipment and optimize con-trol and operation?
Weather Library Dry-bulb temp.
Wet-bulb temp.
Cloud factor Wind speed Pressure Building Description System Description Location Design data Construction data Thermal zones Internal loads Usage profiles Infiltration System types and sizes Supply and return fans Control and schedules Outside air requirements Plant Description Equipment types and sizes Performance characteristics Auxiliary equipment Load assignment Fuel types Economic Data Economic factors Project life First cost Maintenance cost Hourly zone heating and cooling loads Peak heating & cooling loads Hourly equipment loads by system Fuel demand & consumption Life-cycle cost LOADS ANALYSIS SYSTEMS ANALYSIS PLANT ANALYSIS ECONOMIC ANALYSIS Fig. 1 Flow Chart for Building Energy Simulation Program (Ayres and Stamper 1995) Energy Estimating and Modeling Methods 31.3 All the above questions are better addressed by the inverse approach. The forward approach could also be used, for example, by going back to the blueprints of the building and of the HVAC sys-tem, and repeating the analysis performed at the design stage while using actual building schedules and operating modes. This, how-ever, is tedious and labor-intensive, and materials and equipment often perform differently under as-built conditions than as speci-fied. Tuning the forward simulation model is often awkward and labor intensive, although it is still an option (as adopted in the cali-brated inverse approach).
CHOOSING AN ANALYSIS METHOD The most important step in selecting an energy analysis method is to match the method capabilities with project requirements. The method must be capable of evaluating all design options with suffi-cient accuracy to make correct choices. The following factors apply generally (Sonderegger 1985): • Accuracy. The method should be sufficiently accurate to allow correct choices. Because of the many parameters involved in energy estimation, absolutely accurate energy prediction is not possible (Waltz 1992).
• Sensitivity. The method should be sensitive to the design options being considered. The difference in energy use between two choices should be accurate.
• Versatility. The method should allow the analysis of all options under consideration. When different methods must be used to consider different options, an accurate estimate of the differential energy use cannot be made.
• Speed and cost. The total time (gathering data, preparing input, calculations, and analysis of output) to make an analysis should be appropriate to the potential benefits gained. With greater speed, more options can be considered in a given time. The cost of analysis is largely determined by the total time of analysis.
• Reproducibility. The method should not allow so many vaguely defined choices that different analysts would get completely dif-ferent results (Corson 1992).
• Ease of use. This impacts both on the economics of analysis (speed) and the reproducibility of results.
ASHRAE Standard 140, Method of Test for the Evaluation of Building Energy Analysis Computer Programs, has been developed to identify and diagnose differences in predictions that may possibly be caused by algorithmic differences, modeling limitations, or cod-ing errors.
Selecting Energy Analysis Computer Programs The selection of a building energy analysis program depends on its application, the number of times it will be used, the experience of the user, and the hardware available to run it. The first criterion is the capability of the program to deal with the application. For exam-ple, if the effect of a shading device is to be analyzed on a building that will also be shaded by other buildings part of the time, the capa-bility of analyzing detached shading is an absolute requirement, regardless of any other factors.
Because almost all manual methods are now implemented on a computer, the selection of an energy analysis method is the selection of a computer program. Today, all well-known programs run on microcomputers. However, the cost of the computer facilities and the software itself are typically a small part of running a building energy analysis. The major costs are the cost of learning to use the program and the cost of using it. Major issues that influence the cost of learn-ing a program include (1) complexity of the input procedures, (2) quality of the user’s manual, and (3) availability of a good support system to answer questions. As the user becomes more experienced, the cost of learning becomes less important, but the need to obtain and enter a complex set of input data will continue to consume the time of even an experienced user until such data are readily available in electronic form compatible with simulation programs.
The complexity of input is largely influenced by the availability of default values for the input variables. Default values can be used as a simple set of input data when detail is not needed or when the building design is very conventional, but additional complexity can be supplied when needed. Secondary defaults, which can be sup-plied by the user, are also useful in the same way. Some programs allow the user to specify a level of detail. Then the program requests only the information appropriate to that level of detail, using default values for all others.
The quality of the output is another factor to consider. Reports should be easy to read and uncluttered. The titles and headings should be unambiguous. Units should be stated explicitly. The user’s manual should explain the meanings of the data presented.
Graphic output can be very helpful. In most cases, simple summa-ries of overall results are the most useful, but very detailed output is needed for certain studies and also for debugging program input during the early stages of an analysis.
Before a final decision is made, manuals for the most suitable programs should be obtained and reviewed, and, if possible, dem-onstration versions of the programs should be obtained and run.
During this last part of the selection process, support from the soft-ware supplier should be tested. The availability of training should be considered when choosing a more complex program.
The availability of weather data and the availability of a weather data processing subroutine or program are major features of a program. Some programs include subroutine or supplementary programs that allow the user to create a weather file for any site for which weather data is available. Programs that do not have this capability must have weather files for various sites created by the program supplier. In that case, the available weather data and the terms on which the supplier will create new weather data files must be checked.
Auxiliary capabilities, such as economic analysis and design calculations, are a final concern in selecting a program. An eco-nomic analysis may include only the ability to calculate annual energy bills from utility rates, or it might extend to calculations or even to life-cycle cost optimization. An integrated program may save time because some input data will have been entered already for other purposes.
The results of computer calculations should be accepted with caution, as the software vendor does not accept responsibility for the correctness of calculations or the use of the program. A manual calculation should be run to develop a good understanding of the underlying physical processes and building behavior. In addition, the user should (1) review the computer program documentation to determine what calculation procedures are used, (2) compare the results with manual calculations and measured data, and (3) conduct sample tests to confirm that the program delivers acceptable results.
Tools for Energy Analysis The most accurate methods for calculating building energy con-sumption are the most costly because of their intense computational requirements and the needed expertise on the part of the designer or analyst. Simulation programs that assemble component models into system models and then exercise those models with weather and occupancy data are preferred by experts for determining energy use in buildings. This chapter provides descriptions of methods to model heating and cooling loads, including heat flow through build-ing foundations, and methods to model the performance of second-ary and primary HVAC equipment. The chapter continues with information about system-level modeling.
Often, energy consumption at a system or whole-building level must be estimated quickly to study trends, compare systems, or study building effects such as envelope characteristics. For these purposes, simpler methods may be used. Degree-day and bin methods are two 31.4 2001 ASHRAE Fundamentals Handbook (SI) simpler approaches described in the chapter. Finally, the chapter addresses inverse models in detail.
Table 1 classifies methods for analyzing building energy use as forward or inverse and steady-state or dynamic. The U.S. Depart-ment of Energy maintains an up-to-date listing of building energy software with links to other sites that describe energy modeling tools: www.eren.doe.gov/buildings/tools_directory/ COMPONENT MODELING AND LOADS CALCULATING SPACE SENSIBLE LOADS Calculating instantaneous space sensible load is a key step in any building energy simulation. The heat balance method and the weighting factor method are two methods used for these calculations. A third method, the thermal network method, while not widely used, shows promise.
The instantaneous space sensible load is the rate of heat flow into the space air mass. This quantity, sometimes called the cool-ing load, differs from heat gain in that heat gain usually contains a radiative component that passes through the air and is absorbed by other bounding surfaces. Instantaneous space sensible load is entirely convective; even loads from internal equipment, lights, and occupants enter the air by convection from the surface of such objects or by convection from room surfaces that have absorbed the radiant component of energy emitted from these sources. How-ever, some adjustment must be made when radiant cooling and heating systems are evaluated because some of the space load is offset directly by radiant transfer without convective transfer to the air mass.
For equilibrium, the instantaneous space sensible load must match the heat removal rate of the conditioning equipment. Any imbalance in these rates changes the energy stored in the air mass.
Customarily, however, the thermal mass (heat capacity) of the air itself is ignored in an analysis, so that the air is always assumed to be in thermal equilibrium. Under these assumptions, the instanta-neous space sensible load and the rate of heat removal are equal in magnitude and opposite in sign.
Table 1 Classification of Analysis Methods For Building Energy Use Method Forward Inverse Comments Empirical or Black-Box Calibrated Simulation Physical or Gray-Box Steady-State Methods Simple linear regression (Kissock et al. 1998, Ruch and Claridge 1991) — X — — One dependent parameter, one independent parameter. May have slope and y-intercept.
Multiple linear regression (Dhar 1995, Dhar et al. 1998, 1999a,b, Katipamula et al. 1998, Sonderegger 1998) — X — — One dependent parameter, multiple independent parameters.
Modified degree-day method X — — — Based on fixed reference temperature of 18.3°C.
Variable-base degree-day method, or 3-P change point models (Fels 1986, Reddy et al. 1997, Sonderegger 1998) X X — X Variable base reference temperatures.
Change point models: 4-P, 5-P (Fels 1986, Kissock et al. 1998) — X — X Uses daily or monthly utility billing data and average period temperatures.
ASHRAE bin method and inverse bin method (Thamilseran and Haberl 1995) X X — — Hours in temperature bin times load for that bin.
ASHRAE TC 4.7 modified bin method (Knebel 1983) X — — — Modified bin method with cooling load factors.
Multistep parameter identification (Reddy et al. 1999) — — — X Uses daily data to determine overall heat loss and ventilation of large buildings.
Dynamic Methods Thermal network (Sonderegger 1977, Rabl 1988, Reddy 1989) X — — X Uses equivalent thermal parameters (inverse mode).
Response factors (Stephenson and Mitalas 1967, Mitalas and Stephenson 1967, Mitalas 1968, Kusuda 1969) X — — — Tabulated or as used in simulation programs.
Fourier analysis (Shurcliff 1984, Subbarao 1988) X — X X Frequency domain analysis convertible to time domain.
ARMA model (Subbarao 1986, Rabl 1988, Reddy 1989) — — — X Autoregressive Moving Average model.
PSTAR (Subbarao 1988) X — X X Combination of ARMA and Fourier series, includes loads in time domain.
Modal analysis (Bacot et al. 1984, Rabl 1988) X — — X Building described by diagonalized differential equation using nodes.
Differential equation (Rabl 1988) — — — X Analytical linear differential equation.
Computer simulation: DOE-2, BLAST (Norford et al. 1994, Haberl and Bou-Saada 1998, Manke et al. 1996) X — X — Hourly simulation programs with system models.
Computer emulation (HVACSIM+, TRNSYS) (Clark 1985, Klein et al. 1994) X — — — Subhourly simulation programs.
Artificial neural networks (Kreider and Wang 1991, Kreider and Haberl 1994) — X — — Connectionist models.
Energy Estimating and Modeling Methods 31.5 The weighting factor method and the heat balance method use conduction transfer functions (or their equivalents) to calculate transmission heat gain or loss. The principal difference is in the methods used to calculate the subsequent internal heat transfers to the room. Experience with both methods has indicated largely the same results, provided the weighting factors are determined for the specific building under analysis.
Heat Balance Method The heat balance method for calculating net space sensible loads is more fundamental than the weighting factor method. Its develop-ment relies on the first law of thermodynamics (conservation of energy) and the principles of matrix algebra. Because it requires fewer assumptions than the weighting factor method, it is also more flexible. However, the heat balance method requires more calcula-tions at each point in the simulation process, using more computer time. The weighting factors used in the weighting factor method are determined with a heat balance procedure. Although not necessary, linearization is commonly used to simplify the radiative transfer formulation.
The heat balance method allows the net instantaneous sensible heating and/or cooling load to be calculated on the space air mass.
Generally, a heat balance equation is written for each enclosing sur-face, plus one equation for room air. This set of equations can then be solved for the unknown surface and air temperatures. Once these temperatures are known, they can be used to calculate the convec-tive heat flow to or from the space air mass. The heat balance method is developed in Chapter 29 for use in design cooling load calculations, so a fuller description is omitted here. However, one fundamental difference is that the heat balance procedure described in Chapter 29 is aimed at obtaining the design cooling load for a fixed zone air temperature. For building energy analysis purposes, it is preferable to know the actual heat extraction rate.
This may be determined by recasting Equation (36) of Chapter 29 so that the system heat transfer is determined simultaneously with the zone air temperature. The system heat transfer is the rate at which heat is transferred to the space by the system. Although this can be done by simultaneously modeling the zone and the system (Taylor et al. 1990, 1991), it is convenient to make a simple, piece wise-linear representation of the system known as a control profile.
This usually takes the form (1) where = system heat transfer at time step j, W a, b = coefficients that apply over a certain range of zone air temperatures = zone air temperature at time step j, °C The system heat transfer may be considered positive when heating is provided to the space and negative when cooling is pro-vided. It is equal in magnitude but opposite in sign to the zone cool-ing load, as definted in Chapter 29, when the zone air temperature is fixed.
Substituting Equation (1) into Equation (36) of Chapter 29 and solving for the zone air temperature, (2) where = zone air temperature at time step j, °C N = number of zone surfaces Ai = area of ith surface, m2 hci = convection coefficient for ith surface = surface temperature for ith surface at time step j, ??
ρ = density, kg/m3 c = specific heat of air, J/kg ·K V = volumetric flow rate of air, m3/s = outdoor air temperature at time step j, °C = ventilation air temperature at time step j, °C = sum of the convective portions of all internal heat gains at time step j, W The zone air heat balance equation [Equation (2)] must be solved simultaneously with the interior and exterior surface heat balance equations [Equations (35) and (34)] described in Chapter 29. Also, it is necessary to determine the correct temperature range to use the proper set of a and b coefficients. This may be done iteratively.
Once the zone air temperature is found, the actual system heat trans-fer rate may be found directly from Equation (1).
Beyond the treatment of the system heat transfer, there are a few other considerations that may be important in building energy anal-ysis programs. These include simulations over periods as long as a year, treatment of radiant cooling and heating systems, treatment of interzone heat transfer, modeling of convection heat transfer, and modeling of radiation heat transfer.
The heat balance method presented in Chapter 29 assumes the use of a single design day. When utilized in a building energy anal-ysis program, it is most commonly used with a year’s worth of design weather data. In this case, the first day of the year is usually simulated several times until a steady-periodic response is obtained.
Then, each day is simulated sequentially, and, where needed, histor-ical data for surface temperatures and heat fluxes from the previous day are utilized.
When radiant cooling and heating systems areevaluated, the radi-ant source should be identified as a room surface. The calculation procedure considers the radiant source in the heat balance analysis.
Therefore, the heat balance method is preferred over the weighting factor method for evaluating radiant systems. Strand and Pedersen (1997) have described the implementation of heat-source conduc-tion transfer functions, which may be used for modeling of radiant panels, into a heat balance-based building simulation program.
In principle, the heat balance method described above extends directly to multiple spaces, with heat transfer between zones. In this case, some surface temperatures will appear in the surface heat bal-ance equations for two different zones. In practice, however, the size of the coefficient array required for solving the simultaneous equations becomes prohibitively large, and the solution time exces-sive. For this reason, many programs solve only one space at a time and assume that the adjacent space temperatures are either the same as the space in question or some assigned, constant value. Other approaches may remove this limitation (Walton 1980).
Relatively simple exterior and interior convection models may be used for design cooling load calculation procedures. However, it may be desirable to use more sophisticated exterior convection models (Cooper and Tree 1973, Fracastoro et al. 1982, Melo and Hammond 1991, Walton 1983, Yazdanian and Klems 1993) that incorporate the effects of wind speed, wind direction, surface orien-tation, etc. Likewise, a number of more detailed interior convection correlations have been published for use in buildings (Alamdari and Hammond 1982, 1983; Altmayer et al. 1983; Bauman et al. 1983; Bohn et al. 1984; Chandra and Kerestecioglu 1984; Khalifa and Marshall 1990; Spitler et al. 1991; Walton 1983).
Also, more detailed models of exterior long-wave radiation transfer [e.g., Cole (1976) and Walton (1983)] and interior long-wave radiation transfer have been implemented in detailed building qsysj a btaj + = qsysj taj qsysj taj a Aihcitsii j , i=1 N ∑ ρcVinfiljtoj ρcVventjtvj qc int , j + + + + b – Aihci i=1 N ∑ ρcVinfilj ρcVventj + + + ----------------------------------------------------------------------------------------------------------------------------= taj tsii j , toj tvj qc intj , 31.6 2001 ASHRAE Fundamentals Handbook (SI) simulation programs. See Carroll (1980), Davies (1988), Kamal and Novak (1991), Steinman et al. (1989), and Walton (1980) for further discussion of interior radiation heat transfer models.
Weighting Factor Method The weighting factor method of calculating instantaneous space sensible load represents a compromise between simpler methods, such as a steady-state calculation, that ignore the ability of building mass to store energy, and more complex methods, such as complete energy balance calculations. With this method, space heat gains at constant space temperature are determined from a physical descrip-tion of the building, ambient weather conditions, and internal load profiles. Along with the characteristics and availability of heating and cooling systems for the building, space heat gains are used to calculate air temperatures and heat extraction rates. This discussion is in terms of heat gains, cooling loads, and heat extraction rates.
Heat losses, heating loads, and heat addition rates are merely differ-ent terms for the same quantities, depending on the direction of the heat flow.
The weighting factors represent Z-transfer functions (York and Cappiello 1982, Kerrisk et al. 1981). The Z-transform is a method for solving differential equations with discrete data. Two groups of weighting factors are used: heat gain and air temperature.
Heat gain weighting factors represent transfer functions that relate space cooling load to instantaneous heat gains. A set of weighting factors is calculated for each group of heat sources that differ significantly in (1) the relative amounts of energy appearing as convection to the air versus radiation, and (2) in the distribution of radiant energy intensities on different surfaces.
Air temperature weighting factors represent a transfer func-tion that relates room air temperature to the net energy load of the room. The weighting factors for a particular heat source are deter-mined by introducing a unit pulse of energy from that source into the room’s network. The network is a set of equations that represents a heat balance for the room. At each time step (1 h intervals), includ-ing the initial introduction, the energy flow to the room air repre-sents the amount of the pulse that becomes a cooling load. Thus, a long sequence of cooling loads can be generated from which the weighting factors are calculated. Similarly, a unit pulse change in room air temperature can be used to produce a sequence of cooling loads.
A two-step process is used to determine the air temperature and heat extraction rate of a room or building zone for a given set of con-ditions. First, the room air temperature is assumed to be fixed at some reference value. This reference temperature is usually chosen as the average air temperature expected for the room over the sim-ulation period. Instantaneous heat gains are calculated based on this constant air temperature. Various types of heat gains are considered.
Some, such as solar energy entering through windows or energy from lighting, people, or equipment, are independent of the refer-ence temperature. Others, such as conduction through walls, depend directly on the reference temperature.
A space sensible cooling load for the room, defined as the rate at which energy must be removed from the room to maintain the reference value of the air temperature, is calculated for each type of instantaneous heat gain. The cooling load generally differs from the instantaneous heat gain because some energy from the heat gain is absorbed by walls or furniture and stored for later release to the air. At an hour θ, the calculation uses present and past values of the instantaneous heat gain (qθ, qθ –1), past values of the cooling load (Qθ–1, Qθ–2, ...), and the heat gain weighting factors (v0, v1, v2, ..., w1, w2, ...) for the type of heat gain under consideration. Thus, for each type of heat gain qθ, the cooling load Qθ is calculated as (3) The heat gain weighting factors are a set of parameters that quan-titatively determine how much of the energy entering a room is stored and how rapidly the stored energy is released during later hours. Mathematically, the weighting factors are parameters in a Z-transfer function relating the heat gain to the cooling load.
These weighting factors differ for different heat gain sources because the relative amounts of convective and radiative energy leaving various sources differ and because the distribution of radi-ative energy can differ. The heat gain weighting factors also differ for different rooms because room construction influences the amount of incoming energy stored by walls or furniture and the rate at which it is released. Sowell (1988) showed the effects of 14 zone design parameters on zone dynamic response. After the first step, the cooling loads from various heat gains are added to give a total cooling load for the room.
In the second step, the total cooling load is used—along with information on the HVAC system attached to the room and a set of air temperature weighting factors—to calculate the actual heat extraction rate and air temperature. The actual heat extraction rate differs from the cooling load (1) because, in practice, the air tem-perature can vary from the reference value used to calculate the cooling load or (2) because of HVAC system characteristics. The deviation of the air temperature tθ from the reference value at hour θ is calculated as (4) where ERθ is the energy removal rate of the HVAC system at hour θ, and g0, g1, g2, ..., P1, P2, ... are the air temperature weighting fac-tors, which incorporate information about the room, particularly the thermal coupling between the air and the storage capacity of the massive elements.
Tables of values of weighting factors for typical building rooms are presented in the table below. One of the three groups of weight-ing factors, for light, medium, and heavy construction rooms, can be used to approximate the behavior of any room. Some automated simulation techniques allow weighting factors to be calculated spe-cifically for the building under consideration. This option improves the accuracy of the calculated results, particularly for a building with an unconventional design. McQuiston and Spitler (1992) pro-vided electronic tables of weighting factors for a large number of parametrically defined zones.
Two assumptions are made in the weighting factor method.
First, the processes modeled are linear. This assumption is neces-sary because heat gains from various sources are calculated inde-pendently and summed to obtain the overall result (i.e., the superposition principle is used). Therefore, nonlinear processes such as radiation or natural convection must be approximated lin-early. This assumption does not represent a significant limitation because these processes can be linearly approximated with suffi-cient accuracy for most calculations. The second assumption is that system properties influencing the weighting factors are con-stant (i.e., they are not functions of time). This assumption is nec-essary because only one set of weighting factors is used during the entire simulation period. This assumption can limit the use of weighting factors in situations where important room properties vary during the calculation. Two examples are the distribution of Qθ v0qθ v1qθ 1 – … w1Qθ 1 – – w2Qθ 2… – – + + = Normalized Coefficients of Space Air Transfer Functions Room Envelope Construction g0 g1 g2 p0 p1 W/(m2· K) Dimensionless Light +9.54 − 9.82 +0.28 1.0 − 0.82 Medium +10.28 − 10.73 +0.45 1.0 − 0.87 Heavy +10.50 − 11.07 +0.57 1.0 − 0.93 tθ 1 g0 ⁄ Qθ ERθ – ( ) P1 Qθ 1 – ERθ 1 – – ( ) + [ + = P2 Qθ 2 – ERθ 2 – – ( ) … g1tθ 1 – – g2tθ 2 – – … ] – + + Energy Estimating and Modeling Methods 31.7 solar radiation incident on the interior walls of a room, which can vary hourly, and inside surface heat transfer coefficients.
When the weighting factor method is used, a combined radia-tive-convective heat transfer coefficient is used as the inside sur-face heat transfer coefficient. This value is assumed constant even though in a real room (1) the radiant heat transferred from a sur-face depends on the temperature of other room surfaces (not on the room air temperature) and (2) the combined heat transfer coeffi-cient is not constant. Under these circumstances, an average value of the property must be used to determine the weighting factors.
Cumali et al. (1979) have investigated extensions to the weighting factor method to eliminate this limitation.
Thermal Network Methods Although implementations of the thermal network method vary, they all have in common the discretization of the building into a net-work of nodes, with interconnecting paths through which energy flows. In many respects, thermal network models may be consid-ered a refinement of the heat balance method. Where the heat bal-ance model generally uses one node for zone air, the thermal network method might use multiple nodes. For each heat transfer element (wall, roof, floor, etc.), the heat balance model generally has one interior surface node and one exterior surface node; the ther-mal network model may include additional nodes. Heat balance models generally use simple methods for distributing radiation from lights; thermal network models may model the lamp, ballast, and luminaire housing separately. Furthermore, thermal network mod-els depend on a heat balance at each node to determine the node temperature and the energy flow between all connected nodes. The energy flows may include conduction, convection, short-wave radi-ation, and long-wave radiation.
For any mode of energy flow, a range of techniques may be used to model the energy flow between two nodes. Taking conduction heat transfer as an example, the simplest thermal network model would be a resistance-capacitance network (Sowell 1990). By refin-ing the network discretization, the models become what are com-monly thought of as finite difference or finite volume models (Clarke 1985, Lewis and Alexander 1990, Walton 1993).
Thermal network models generally use a set of algebraic and dif-ferential equations. In most implementations, the solution proce-dure is separated from the models so that, in theory, different solvers might be used to perform the simulation. In contrast, most heat bal-ance programs and weighting factor programs interweave the solu-tion technique with the models. Various solution techniques have been used in conjunction with thermal network models. Examples include graph theory combined with Newton-Raphson and predic-tor-corrector ordinary differential equation integration (Buhl et al.
1990) and the use of Euler explicit integration combined with sparse matrix techniques (Walton 1993).
Of the three zone models discussed, thermal network models are the most flexible and have the greatest potential for high accuracy.
As a trade-off, they also require the most computation time, and, in current implementations, they require more user effort to take advantage of the flexibility.
GROUND HEAT TRANSFER The thermal performance of building foundations, including guidelines for placement of insulation, is described in Chapter 24 of this volume and Chapter 42 of the 1999 ASHRAE Handbook— Applications. Chapter 28 provides information needed to calcu-late transmission heat losses through slab foundations and through basement walls and floors. These calculations are appro-priate for design loads but are not intended for estimating annual energy usage. This section provides simplified calculation meth-ods suitable for energy estimates over time periods of arbitrary length.
The thermal performance of building foundations has been largely ignored. It is estimated that in the early 1970s, only 10% of the total energy use of a typical U.S. home was attributed to the heat transfer from its foundation (Labs et al. 1988). Since then, the thermal performance of above-grade building elements has improved significantly, and the contribution of ground-cou-pled heat transfer to total energy use in a typical U.S. home has increased. Shipp and Broderick (1983) estimated that the heat transfer from an uninsulated basement in Columbus, Ohio, can represent up to 67% of the total building envelope heating load.
Earth-contact heat transfer, rated at 1 to 3 EJ of energy annu-ally in U.S. buildings, has an impact similar to infiltration on annual heating and cooling loads in residential buildings (Clar-idge 1988a). Adding insulation to building foundations is esti-mated to save up to 0.5 EJ of annual energy use in the U.S. (Labs et al. 1988).
Simplified Calculation Method for Slab Foundations and for Basements A simplified design tool for calculating heat loss for slabs and basements is presented by modifying the design tool for slab-on-grade floors developed by Krarti and Chuangchid (1999). The pro-posed design tool is easy to use and requires straightforward input parameters with continuously variable values, including foundation size, insulation R-values, soil thermal properties, and indoor and outdoor temperatures. The simplified method provides a set of equations that are suitable for estimating the design, seasonal, and annual total heat loss of a slab or a basement as a function of a wide range of variables.
When the indoor temperature of the building is maintained con-stant, the ground-coupled heat transfer q(θ) varies with time accord-ing to the following equation: (5) where qmean = annual-mean heat loss/gain, W qamp = heat loss/gain amplitude, W φ = phase lag between total slab heat loss/gain and soil surface temperature, s ω = annual angular frequency (ω= 1.992 × 10− 7 rad/s) θ = time, s Equation (5) is convenient and flexible because it can be used to calculate the foundation heat loss/gain not only at any time but also at design conditions and for any time period (such as a heating sea-son or 1 year). In particular, the design heat loss/gain load Qdes for a slab foundation is obtained as follows: (6) The parameters qmean and qamp are functions of such variables as building dimensions, soil properties, and insulation R-values.
Expressions developed by nondimensional analysis allow the cal-culation of qmean and qamp.
The soil conductivity is normalized to form four parameters— Uo, G, H, and D: (7) where ks = soil thermal conductivity, W/(m·K) P = slab perimeter, m A = slab area, m2 For mean calculations, (8) q θ ( ) qmean qamp ωθ φ + ( ) sin + = qdes qmean qamp + = Uo ks A P ⁄ ( )eff b , -------------------------------= A P ⁄ ( )eff b , mean , 1 beff + 0.4 – e + Hb – ( ) [ ] A P ⁄ ( )b = 31.8 2001 ASHRAE Fundamentals Handbook (SI) For annual calculations, (9) where (10) (11) B = basement depth, m (0 m for slab) (12) where Req = equivalent thermal resistance for the entire slab, m2·K/W αs = soil thermal diffusivity, m2/s For uniform insulation configurations (placed horizontally be-neath the slab floor), (13) where Rf = thermal resistance of the floor, m2·K/W Ri = thermal resistance of insulation, m2·K/W Forpartialinsulation configurations (both horizontaland vertical), (14) where c = insulation length of slab, m.
(15) (16) The effective heat-transfer coefficients for mean heat flow Ueff,mean and heat-flow amplitude Ueff,amp, W/m2·K, are (17) (18) where the dimensionless coefficients m and a depend on the insula-tion placement configurations and are provided in Table 2.
The annual-mean slab foundation and basement heat loss/gain can now be defined as (19) where ta = annual average ambient dry-bulb temperature, °C tr = annual average indoor dry-bulb temperature, °C The heat loss/gain amplitude for slab foundations and basements is (20) where tamp = annual amplitude ambient temperature, K.
This simplified model for slab-foundation and basement heat flows provides accurate predictions when A/P is larger than 0.5 m.
To illustrate the use of the simplified models, two examples are pre-sented: one for a slab-on-grade floor for a building insulated with uniform horizontal insulation, and one for a basement structure insulated with uniform insulation.
Example 1. Calculation for Slab Foundations. Determine the annual mean and annual amplitude of total slab heat loss for the slab founda-tion illustrated in Figure 2. The building is located in Denver, Colorado.
Solution: Step 1. Provide the required input data.
Dimensions Slab width = 10.0 m Slab length = 15.0 m Ratio of slab area to slab perimeter, A/P = 3.0 m 102 mm thick reinforced concrete slab, thermal resistance Rf = 0.5 m2·K/W A P ⁄ ( )eff b , amp , 1 beff + e Hb – ( ) A P ⁄ ( )b = Hb A P ⁄ ( )b ksReq ----------------------= beff B A P ⁄ ( )b ----------------------= G ksReq ω αs -----= Req Rf Ri + = Req Rf 1 c A P ⁄ --------------Ri Ri Rf + ( ) --------------------- – ----------------------------------------------------------= H A P ⁄ ( )eff b , ksReq -------------------------------= D 1 H + ( ) 1 1 H ----+ H ln = Table 2 Coefficients m and a for Slab-Foundation Heat Transfer Calculations Insulation Placement m a Uniform—Horizontal 0.40 0.25 Partial—Horizontal 0.34 0.20 Vertical 0.28 0.13 Ueff mean , mUoD = Ueff amp , aUoD0.16G 0.6 – = qmean Ueff mean , A ta tr – ( ) = Qa Ueff a , Atamp = Fig. 2 Slab Foundation for Example 1 Energy Estimating and Modeling Methods 31.9 Soil Thermal Properties Soil thermal conductivity ks = 1.21 W/m·K Soil density ρ = 700 kg/m3 Soil thermal diffusivity αs = 5.975 × 10− 7 m2/s Insulation Uniform insulation R-value Ri = 3.52 m2·K/W Temperatures Indoor temperature tr = 20°C Annual average ambient temperature ta = 6.3°C Annual amplitude ambient temperature tamp = 20 K Annual angular frequency ω= 1.992 × 10− 7 rad/s Step 2. Calculate qmean and qamp values.
The various normalized parameters are first calculated using Equa-tions (7) through (18). Then, the annual mean and amplitude of the foundation slab heat loss/gain are determined using Equations (19) and (20).
Therefore, and Example 2. Calculation for Basements. Determine the annual mean and annual amplitude of total basement heat loss for a building located in Denver, Colorado.
Solution: Step 1. Provide the required input data.
Dimensions Basement width = 10.0 m Basement length = 15.0 m Basement wall height B= 1.5 m Basement slab and wall total area = 225.0 m2 Ratio of slab and wall area to slab and wall perimeter, (A/P)b = 3.629 m 102 mm thick reinforced concrete slab, thermal resistance Rf = 0.5 m2·K/W Soil Thermal Properties Soil thermal conductivity ks = 1.21 W/m·K Soil thermal diffusivity αs = 4.47 × 10− 7 m2/s Insulation Uniform insulation R-value Ri = 1.152 m2·K/W Temperatures Indoor temperature, tr = 22° C Annual average ambient temperature, ta = 10°C Annual amplitude ambient temperature, tamp = 12.7 K Annual angular frequency, ω= 1.992 × 10− 7 rad/s Step 2. Calculate qmean and qamp values.
The normalized parameters are first calculated using Equations (7) through (18). Then the annual mean and amplitude of the basement heat loss are determined using Equations (19) and (20).
Therefore, and Table 3 compares results of the simplified method presented here and the more exact interzone temperature profile estimation (ITPE) (Krarti 1994a, 1994b; Krarti et al. 1988a, 1988b).
SECONDARY SYSTEM COMPONENTS Secondary HVAC systems generally include all elements of the overall building energy system between a central heating and cool-ing plant and the building zones. The precise definition depends heavily on the building design. A secondary system typically includes air-handling equipment, air distribution systems with the Uo ks A P ⁄ ( ) -------------------1.21 3.0 ----------0.4033 = = = H A P ⁄ ( ) ksReq -------------------3.0 1.21 0.5 3.52 + ( ) ----------------------------------------0.6168 = = = D 1 H + ( ) 1 1 H ----+ H ln 1.0748 = = G ksReq ω αs -----1.21 0.5 3.52 + ( ) 1.992 7 – × 10 5.975 7 – × 10 ----------------------------2.8086 = = = qmean Ueff mean , A tr ta – ( ) 0.4 0.4033 × 1.0748 × 150 20 6.3 – ( ) × × = = 356 W = qamp Ueff amp , Atamp 0.25 0.4033 × 1.07480.16 × 2.8086 0.6 – × = = 150 × 20 × 165W = Table 3 Example 2 Heat Loss per Unit Area for the Simplified and ITPE Methods Method Mean (qmean), W Amplitude (qamp), W Simplified 699 208 ITPE solution 658 212 Hb A P ⁄ ( )b ksReq ---------------------3.629 1.21 0.5 1.152 + ( ) -------------------------------------------1.8155 = = = beff B A P ⁄ ( )b ---------------------1.5 3.629 -------------0.4133 = = = A P ⁄ )eff b mean , , 1 0.4133 + 0.4 – e 1.8155 – + ( ) [ ] 3.629 × 3.2731 = = A P ⁄ )eff b amp , , 1 0.4133 + e 1.8155 – [ ] 3.629 × 3.8731 = = Uo mean , ks A P ⁄ ( )eff b mean , , ---------------------------------------------1.21 3.2731 ----------------0.3697 = = = Uo amp , ks A P ⁄ ( )eff b amp , , ------------------------------------------1.21 3.8731 ----------------0.3124 = = = Hmean A P ⁄ ( )eff b mean , , ksReq ---------------------------------------------3.2731 1.21 0.5 1.152 + ( ) -------------------------------------------1.6374 = = = Hamp A P ⁄ ( )eff b amp , , ksReq ------------------------------------------3.8731 1.21 0.5 1.152 + ( ) -------------------------------------------1.9376 = = = Dmean 1 Hmean + ( ) 1 1 Hmean ---------------+ Hmean ln 1.7503 = = Damp 1 Hamp + ( ) 1 1 Hamp -------------+ Hamp ln 1.8839 = = G ksReq ω αs -----1.21 0.5 1.152 + ( ) 1.992 7 – × 10 4.47 7 – × 10 ----------------------------1.3344 = = = qmean Ueff mean , A ta tr – ( ) = 0.4 0.3697 × 1.7503 × 225 22 10 – ( ) × × 699 W = = qamp Ueff amp , Atamp 0.25 0.3124 1.88390.16 × × = = 1.3344 0.6 – 225 12.7 × × × 208 W = 31.10 2001 ASHRAE Fundamentals Handbook (SI) associated ductwork, dampers, fans, and heating, cooling, and humidity conditioning equipment. Secondary systems also include the liquid distribution systems between the central plant and the zone and air-handling equipment, including piping, valves, and pumps.
While the exact design of secondary systems varies dramatically among buildings, they are composed of a relatively small set of generic HVAC components. These components include distribution components (e.g., pumps/fans, pipes/ducts, valves/dampers, head-ers/plenums, fittings) and heat and mass transfer components (e.g., heating coils, cooling and dehumidifying coils, liquid heat exchang-ers, air heat exchangers, evaporative coolers, steam injectors). Most secondary systems can be described by simply connecting these components to form the complete system.
Energy estimation through computer simulation often mimics the modular construction of secondary systems by using modular simulation elements [e.g. the ASHRAE HVAC 2 Toolkit (Brande-muehl 1993, Brandemuehl and Gabel 1994), the simulation pro-gram TRNSYS (Klein et al. 1994), and Annex 10 activities of the International Energy Agency]. To the extent that the secondary sys-tem consumes energy and transfers energy between the building and central plant, an energy analysis can be performed by characterizing the energy consumption of the individual components and the energy transferred among system components. In fact, few of the secondary components consume energy directly, except fans, pumps, furnaces, direct expansion air-conditioning package units with gas-fired heaters, and inline heaters. In this chapter, secondary components are divided into two categories: distribution compo-nents and heat and mass transfer components.
Fans, Pumps, and Distribution Systems The distribution system of an HVAC system affects energy con-sumption in two ways. First, fans and pumps consume electrical energy directly, based on the flow and pressures under which the device operates. Ducts and dampers, or pipes and valves, and the system control strategies affect the flow and pressures at the fan or pump. Second, thermal energy is often transferred to (or from) the fluid due to heat transfer through pipes and ducts and due to the electrical input to fans and pumps. The analysis of system compo-nents should, therefore, account for both direct electrical energy consumption and thermal energy transfer.
Fan and pump performance are discussed in Chapters 18 and 39 of the 2000 ASHRAE Handbook—Systems and Equipment. In addi-tion, Chapter 34 of this handbook covers pressure loss calculations for airflow in ducts and duct fittings. Chapter 35 presents a similar discussion for fluid flow in pipes. While these chapters do not spe-cifically focus on energy estimation, energy use is governed by the same performance characteristics and engineering relationships.
Strictly speaking, performance calculations of the fan and air distri-bution systems in a building require a detailed pressure balance on the entire network. For example, in an air distribution system, the airflow through the fan depends on its physical characteristics, the operating speed, and the pressure differential across the fan. The pressure drop through the duct system depends on the duct design, the position of all dampers, and the airflow through the fan. The interaction between the fan and duct system results in a set of cou-pled, nonlinear algebraic equations. Models and subroutines for performing these calculations are available in the ASHRAE HVAC 2 Toolkit (Brandemuehl 1993).
While a detailed analysis of a distribution system requires flow and pressure balancing among the components, nearly all commer-cially available energy analysis methods approximate the effect of the interactions with part-load performance curves. This procedure eliminates the need to calculate pressure drop through the distribu-tion system at off-design conditions. The part-load curves are often expressed in terms of a power input ratio as a function of the part-load ratio, defined as the ratio of part-load flow to design flow: (21) where PIR = power input ratio W = fan motor power at part load, W Wfull = fan motor power at full load or design, W Q = fan airflow rate at part load, m3/s Qfull = fan airflow rate at full load or design, m3/s fplr = regression function, typically polynomial The exact shape of the part-load curve depends on the effect of flow control on the pressure and fan efficiency and may be calculated using a detailed analysis or measured field data. Figure 3 shows the relationship for three typical fan control strategies, as represented in a simulation program (York and Cappiello 1982). Within the simu-lation program, the curves are represented by polynomial regression equations. Models and subroutines for performing these calculations are also available in the ASHRAE HVAC 2 Toolkit (Brandemuehl 1993).
Figure 4 shows an example of a similar curve for the part-load operation of a fan system in a monitored building (Brandemuehl and Bradford 1999). In this particular case, the fan system represents ten separate air handlers, each with supply and return fans, operating with variable-speed fan control to maintain a set duct static pres-sure. Notice that, although the shape of the curve is similar to the variable-speed curve of Figure 3, the measured data for this partic-ular system exhibit a more linear relationship between power and flow.
PIR W Wfull ------------fplr Q Qfull ----------- = = Fig. 3 Part-Load Curves for Typical Fan Operating Strategies Fig. 4 Fan Part-Load Curve Obtained from Measured Field Data under ASHRAE 823-RP Energy Estimating and Modeling Methods 31.11 Heat transferred to the airstream due to fan operation increases the temperature of the air. While the shaft power of the fan has a direct effect on the heat transfer, motor inefficiencies also heat the air if the motor is mounted inside the airstream. For pumps, this con-tribution is typically assumed to be zero.
The following equation provides a convenient and general model to calculate the heat transferred to the fluid: (22) where qfluid = heat transferred to the fluid, W fm,loss = fraction of the motor heat loss transferred to the fluid stream, dimensionless (= 1 if fan mounted in airstream, = 0 if fan mounted outside airstream) W = fan motor power, W η m = motor efficiency Heat and Mass Transfer Components Secondary HVAC systems comprise such heat and mass transfer components as steam-based air-heating coils, chilled water cooling and dehumidifying coils, shell-and-tube liquid heat exchangers, air-to-air heat exchangers, evaporative coolers, and steam injectors.
While these components do not consume energy directly, their ther-mal performance dictates the interactions between the building loads and the energy-consuming primary components (e.g., chillers, boilers). In particular, the performance of the secondary com-ponents determines the entering fluid conditions for primary com-ponents, which in turn determine the energy efficiencies of the primary equipment. Accurate energy calculations cannot be per-formed without appropriate models of the system heat and mass transfer components.
For example, the load on a chiller is typically described as the sum of zone sensible and latent loads, plus any heat gain from ducts, plenums, fans, pumps, and piping. However, the energy consump-tion of the chiller is determined not only by the load but also by the return chilled water temperature and flow rate. The return water condition is determined by the cooling coil performance and the part-load operating strategy of the air and water distribution system.
The cooling coil might typically be controlled to maintain a constant leaving air temperature by modulating the water flow through the coil. In such a scenario, the cooling coil model must be able to cal-culate the leaving air humidity, leaving water temperature, and leav-ing water flow rate given the cooling coil design characteristics, entering air temperature and humidity, entering airflow, and enter-ing water temperature.
Virtually all building energy simulation programs include, and require, models of heat and mass transfer components. In general, these models are relatively simple. While a coil designer might use a detailed tube-by-tube analysis of conduction and convection heat transfer and condensation on fin surfaces to develop an optimal combination of fin and tube geometry, an energy analyst is more interested in determining the changes in leaving fluid states as oper-ating conditions vary during the year. In addition, the energy analyst is likely to have limited design data on the equipment and, therefore, requires a model with very few parameters that depend on equip-ment geometry and detailed design characteristics.
A typical approach to modeling heat and mass transfer compo-nents for energy calculations is based on an effectiveness-NTU heat exchanger model (Kays and London 1984). The effective-ness-NTU model is described in most heat transfer textbooks and is briefly discussed in Chapter 3. It is particularly appropriate for describing the leaving fluid conditions when the entering fluid con-ditions and equipment design characteristics are known. In addition, this model requires only a single parameter to describe the charac-teristics of the exchanger—the overall transfer coefficient UA— which can be determined from limited design performance data.
Because the classical effectiveness methods were developed for sensible heat exchangers, they are used to perform energy calcula-tions for a variety of sensible heat exchangers in HVAC systems.
For typical finned-tube air-heating coils, the crossflow configura-tion with both fluid streams unmixed is most appropriate. The same configuration typically applies to air-to-air heat exchangers. For liq-uid-to-liquid exchangers, tube-in-tube equipment can be modeled as parallel or counterflow, depending on flow directions; and shell-and-tube equipment can be modeled as either counter- or crossflow, depending on the extent of baffling and the number of tube passes.
The energy analyst must determine the UA to describe the oper-ations of a specific heat exchanger. There are typically two ap-proaches to determine this important parameter: direct calculation and from manufacturers’ data. Given detailed information about the materials, geometry, and construction of the heat exchanger, it is possible to apply fundamental heat transfer principles to calculate the overall heat transfer coefficient. An alternative to direct calcu-lation, and the method most appropriate for energy estimation, is to use manufacturers’ performance data or direct measurements of in-stalled performance. In reporting the design performance of a heat exchanger, a manufacturer typically gives the rate of heat transfer under various operating conditions, with the operating conditions described in terms of entering fluid flow rates and temperatures.
The effectiveness and UA can be calculated from the given heat transfer rate and entering fluid conditions.
Example 3. An energy analyst seeks to perform an evaluation of a hot water heating system that includes a hot water heating coil. The energy analysis program uses an effectiveness-NTU model of the coil and requires the UA of the coil as an input parameter. While detailed infor-mation on the coil geometry and heat transfer surfaces is not available, the manufacturer states that the 1-row hot water heating coil will deliver 240 kW of heat under the following design conditions: Design Performance Entering water temperature thi = 80°C Water mass flow rate = 5.0 kg/s Entering air temperature tci = 20°C Air mass flow rate = 8.0 kg/s Design heat transfer q = 240 kW Solution: The solution proceeds by first determining the heat exchanger UA from the design data, and then using the UA to predict performance at the off-design conditions. The effectiveness-NTU relationships are used for both steps. The key assumption is that the UA is constant for both operating conditions. This example uses the nomenclature as described in Chapter 3 in the section on Overall Heat Transfer.
a) An examination of the flow rates and fluid specific heats allows calculation of the capacity rates Ch and Cc at design conditions and the capacity rate ratio Z.
b) The effectiveness can be directly calculated from the heat trans-fer definition.
c) The effectiveness-NTU relationships for a crossflow heat exchanger with both fluids unmixed allows calculation of the effectiveness in terms of the capacity rate ratio Z and the NTU [the relationships are available from most heat transfer textbooks and specifically can be found in Kays and London (1984)].
qfluid η m 1 η m – ( )fm loss , + [ ]W = Ch m · cp ( )h 5.0 ( ) 4.195 ( ) 20.97 kW/K = = = Cc m · cp ( )c 8.0 ( ) 1.007 ( ) 8.05 kW/K = = = Cmax Ch = Cmin Cc = Z Cmin Cmax ------------0.384 = = ε tco tci – ( ) thi tci – ( ) ----------------------q Cc ⁄ thi tci – ( ) ----------------------240 8.05 ⁄ 80 20 – ( ) --------------------------0.497 = = = = 31.12 2001 ASHRAE Fundamentals Handbook (SI) Given the effectiveness and capacity rate ratio, the NTU can be determined to be NTU = 0.804.
d) The heat transfer UA is then determined from the definition of the NTU.
Application to Cooling and Dehumidifying Coils The analysis of air-cooling and dehumidifying coils requires coupled, nonlinear heat and mass transfer relationships. These relationships form the basis for all HVAC components with mois-ture transfer, including cooling coils, cooling towers, air washers, and evaporative coolers. While the complex heat and mass transfer theory that is presented in many textbooks is often required for cooling coil design, simpler models based on effectiveness con-cepts are usually more appropriate for energy estimation. For example, the bypass factor is a form of effectiveness in the approach of the leaving air temperature to the apparatus dew-point, or coil surface, temperature.
While the effectiveness-NTU method is typically developed and applied in the analysis of sensible heat exchangers, it can also be used to analyze other types of exchangers such as cooling and dehu-midifying coils that couple heat and mass transfer. By redefining the state variables, the capacity rates, and the overall exchange coeffi-cient of these enthalpy exchangers, the effectiveness concept may be used to calculate heat transfer rates and leaving fluid states. For sensible heat exchangers, the state variable is temperature, the capacity is the product of mass flow and fluid specific heat, and the overall transfer coefficient is the conventional overall heat transfer coefficient. For cooling and dehumidifying coils, the state variable becomes moist air enthalpy, the capacity has units of mass flow, and the overall heat transfer coefficient is modified to reflect enthalpy exchange. This approach forms the basis for models by Threlkeld (1970), Elmahdy and Mitalas (1977), Braun (1988), and Brande-muehl (1993). The same principles also underlie the coil model described in Chapter 21 of the 2000 ASHRAE Handbook—Systems and Equipment.
The effectiveness model is based on the observation that, for a given set of entering air and liquid conditions, the heat and mass transfer are bounded by thermodynamic maximum values. Figure 5 shows the limits for leaving air states on a psychrometric chart. Spe-cifically, the leaving chilled water temperature cannot be warmer than the entering air temperature and the leaving air temperature and humidity cannot be lower than the conditions of saturated moist air at the temperature of the entering chilled water.
Figure 5 also shows that the performance of a cooling coil requires the evaluation of two different effectivenesses to identify the leaving air temperature and humidity. An overall effectiveness can be used to describe the approach of the leaving air enthalpy to the minimum possible value. An air-side effectiveness, related to the coil bypass factor, describes the approach of the leaving air tem-perature to the effective wet coil surface temperature.
The effectiveness analysis is accomplished for wet coils by establishing a common state variable for both the moist air and liq-uid streams. As implied by the lower limit of the entering chilled water temperature, this common state variable is the moist air en-thalpy. In other words, all liquid and coil temperatures are trans-formed to the enthalpy of saturated moist air at the liquid or coil temperature. Changes in liquid temperature can similarly be ex-pressed in terms of changes in saturated moist air enthalpy through a saturation specific heat cp,sat defined by the following: (23) Using the definition of Equation (23), the basic effectiveness relationships discussed in Chapter 3 can be written as (24) (25) (26) (27) (28) where q = heat transfer from air to water, kW C = fluid capacity, kg/s = dry air mass flow rate, kg/s = liquid mass flow rate, kg/s cp,l = liquid specific heat, kJ/(kg·K) cp,sat = saturation specific heat, defined by Equation (23), kJ/(kg·K) ha = enthalpy of moist air, kJ/kg hl,sat = enthalpy of saturated moist air at the temperature of the liquid, kJ/kg The cooling coil effectiveness of Equation (25) is defined, then, as the ratio of moist air enthalpies in Figure 5. As in the case of sen-sible heat exchangers, the effectiveness is also a function of the physical coil characteristics and can be obtained by modeling the coil as a counterflow heat exchanger. However, since the heat trans-fer calculations are performed based on enthalpies, the overall transfer coefficient must be based on enthalpy potential rather than temperature potential. The enthalpy-based heat transfer coefficient UAh is related to the conventional temperature-based coefficient by the specific heat: (29) A similar analysis can be performed to evaluate the air-side effectiveness, which identifies the leaving air temperature. While the overall enthalpy-based effectiveness is based on an overall heat transfer coefficient between the chilled water and the air, the air-side effectiveness is based on a heat transfer coefficient between the coil surface and the air.
As with sensible heat exchangers, the overall heat transfer coef-ficients UA can be determined either from direct calculation from coil properties or from manufacturers’ performance data. While a sensible heat exchanger is modeled with a single effectiveness and UA CminNTU 8.05 ( ) 0.804 ( ) 6.47 kW/K = = = cp sat , hl sat , ∆ tl ∆ ------------------= Fig. 5 Psychrometric Schematic of Cooling Coil Processes Ca ha ent , ha lvg , – ( ) Cl hl sat lvg , , hl sat ent , , – ( = = q εCmin ha ent , hl sat ent , , – ( ) = Ca m · a = Cl m · cp ( )l cp sat , ----------------= Cmin min Ca Cl , ( ) = m · a m · l q UA t ∆ UAh h ∆ = = UAh UA t ∆ h ∆ --------------UA cp --------= = Energy Estimating and Modeling Methods 31.13 can be described by a single parameter UA, a wet cooling and dehu-midifying coil requires two parameters to describe the two effec-tivenesses shown in Figure 5. These two parameters are the internal and external UAs—one describes the heat transfer between the chilled water and the air-side surface through the pipe wall and the other between the surface and the moist air. These two parameters can be determined from the sensible and latent capacity of a cooling coil at a single rating condition. A significant advantage of the effectiveness-NTU method is that the component can be described with as little as one measured data point or one manufacturer’s design calculation.
PRIMARY SYSTEM COMPONENTS Primary HVAC systems consume energy and deliver heating and cooling to a building, usually through secondary systems. Pri-mary equipment generally includes chillers, boilers, cooling tow-ers, cogeneration equipment, and plant-level thermal storage equipment. In particular, primary equipment generally represents the major energy-consuming equipment of a building, so accurate characterization of building energy use relies on accurate model-ing of primary equipment energy consumption.
Modeling Strategies The energy consumption characteristics of primary equipment generally depend on equipment design, load conditions, environ-mental conditions, and equipment control strategies. For example, chiller performance depends on the basic equipment design features (e.g., heat exchange surfaces, compressor design), the temperatures and flow through the condenser and evaporator, and the methods for controlling the chiller at different loads and operating conditions (e.g., inlet guide vane control on centrifugal chillers to maintain leaving chilled water temperature set point). In general, these vari-ables that dictate energy consumption vary constantly and require calculations on an hourly basis.
Regression Models. While many secondary components (e.g., heat exchangers, valves) are readily described by fundamental engi-neering principles, the complex nature of most primary equipment has discouraged the use of first-principle models for energy calcu-lations. Instead, the energy consumption characteristics of primary equipment have traditionally been modeled using simple equations developed by regression analysis of manufacturers’ published design data. Because published data are often available only for full-load design conditions, additional correction functions are used to correct the full-load data to part-load conditions. The functional form of the regression equations and correction functions takes many forms, including exponentials, Fourier series, and, most of the time, second- or third-order polynomials. The selection of an appro-priate functional form depends on the behavior of the equipment. In some cases, energy consumption is calculated using direct interpo-lation from tables of data. However, this method often requires excessive data input and computer memory.
The typical approach to modeling primary equipment in energy simulation programs is to assume the following functional form for equipment power consumption: (30) (31) where P = equipment power, kW PIR = energy input ratio PIRnom = energy input ratio under nominal full-load conditions Load = power delivered to the load, kW Cavail = available equipment capacity, kW Cnom = nominal equipment capacity, kW f1 = function relating full-load power at off-design conditions (ta, tb, ...) to full-load power at design conditions f2 = fraction full-load power function, relating part-load power to full-load power f3 = function relating available capacity at off-design conditions (ta, tb, ...) to nominal capacity ta, tb = various operating temperatures that affect power PLR = part-load ratio The part-load ratio is defined as the ratio of the load to the avail-able equipment capacity at given off-design operating conditions.
Like the power, the available, or full-load, capacity will be a func-tion of operating conditions.
The particular forms of the off-design functions f1 and f3 depend on the specific type of primary equipment. For example, for fossil-fuel boilers, full-load capacity and power (or fuel use) can be affected by the thermal losses to ambient temperature. However, these off-design functions are typically considered to be unity in most building simulation programs. For chillers, both capacity and power are affected by the condenser and evaporator temperatures.
These two temperatures are often characterized in terms of their sec-ondary fluids. For direct expansion air-cooled chillers, the operating temperatures are typically the wet-bulb temperature of the air enter-ing the evaporator and the dry-bulb temperature of the air entering the condenser. For liquid chillers, the temperatures are usually the leaving chilled water temperature and the entering condenser water temperature.
As an example, consider the performance of a direct expansion (DX) packaged single-zone rooftop unit. The nominal rated per-formance of these units is typically given for an outdoor air tem-perature of 35°C and evaporator entering coil conditions of 26.7°C dry-bulb temperature and 19.4°C wet-bulb temperature. However, the performance changes as the outdoor temperature and entering coil conditions vary. To account for these effects, the DOE-2.1E simulation program expresses the off-design functions f1 and f3 with biquadratic functions of the outdoor dry-bulb temperature and the coil entering wet-bulb temperature.
(32) (33) The constants in Equations (32) and (33) are given in Table 4.
The fraction full-load power function f2 represents the change in equipment efficiency at part-load conditions and depends heavily on the control strategies used to match load and capacity. Figure 6 shows several possible shapes of these functional relationships.
(Notice that these curves are similar to the fan part-load curves of Figure 3.) Curve 1 represents equipment with constant efficiency, independent of load. Curve 2 represents equipment that is most effi-cient in the middle of its operating range. Curve 3 represents equip-ment that is most efficient at full load. Note that these types of curves apply to both boilers and chillers.
P PIR Load × = PIR PIRnomf1 ta tb … , , ( )f2 PLR ( ) = Cavail Cnomf3 ta tb … , , ( ) = PLR Load Cavail ---------------= Table 4 Correlation Coefficients for Off-Design Relationships Cor-rela-tion 0 1 2 3 4 5 f1 − 1.063931 0.0306584 0.0001269 0.0154213 0.0000497 0.0002096 f3 0.8740302 0.0011416 0.0001711− 0.002957 0.0000102 0.0000592 f1 twb ent , toa , ( ) a0 a1twb ent , a2twb ent , 2 a3toa a4toa 2 a5twb ent , + + + + + toa = f3 twb ent , toa , ( ) c0 c1twb ent , c2twb ent , 2 c3toa c4toa 2 c5twb ent , + + + + + toa = 31.14 2001 ASHRAE Fundamentals Handbook (SI) First-Principle Models. As with the secondary components, engineering first principles can also be used to develop models of primary equipment. Lebrun et al. (1999), Gordon and Ng (1994, 1995), Gordon et al. (1995), and others have sought to develop such models in which unknown model parameters are extracted from measured or published manufacturers’ data.
The energy analyst is often faced with choosing the appropriate model for the job. For example, a complex boiler model is not appropriate if the boiler in question operates at virtually constant efficiency. By similar arguments, a regression-based model might be appropriate when the user has a full dataset of reliable in-situ measurements of the plant. However, first-principle physical mod-els generally have several advantages over pure regression models: • Physical models allow confident extrapolation outside the range of available data.
• Regression is still required to obtain values for unknown physical parameters. However, the values of these parameters usually have physical significance. The engineer can capitalize on this signif-icance to estimate default parameter values, diagnose errors in data analysis through checks for realistic parameter values, and even evaluate potential performance improvements.
• The number of unknown parameters is generally much smaller than the number of unknown coefficients in the typical regression model. For example, the standard ARI compressor model re-quires as many as 30 coefficients, 10 coefficients each for the regressions of capacity, power, and refrigerant flow. By compar-ison, a physical compressor model may have as few as four or five unknown parameters. As a result, the physical models require fewer measured data.
• Data on part-load operation of chillers and boilers are notoriously difficult to obtain. Part-load corrections often represent the great-est uncertainty in the regression models, while causing the great-est effect on annual energy predictions. By comparison, physical models of full-load operation often allow direct extension to part-load operation with little additional required data.
While physical models of primary HVAC equipment are gener-ally based on fundamental engineering analysis and found in many HVAC textbooks, the models described here are specifically based on the work of Bourdouxhe et al. (1994a, 1994b, 1994c) in the development of the ASHRAE HVAC 1 Toolkit (Lebrun et al. 1999).
The behavior of each elementary component is characterized by a limited number of physical parameters, such as heat exchanger heat transfer area or centrifugal compressor impeller blade angle. Values of these parameters are identified, or tuned, based on regression fits of overall performance compared to measured or published data.
While physical models are based on physical characteristics, the values obtained through a regression analysis of manufacturers’ data are not necessarily representative of the actual measured val-ues. Strictly speaking, the parameter values are regression coeffi-cients with somewhat fictitious values, identified to minimize the error in overall system performance. In other words, errors in the fundamental models of the equipment are offset by over or under-estimation of the parameter values.
Boiler Model The thermal model of a water boiler operating in a steady-state regime is shown in Figure 7. It consists of an adiabatic combustion chamber and two heat exchangers. These three components interact through the following fluids: air, fuel, combustion gas, water, and a fictitious fluid representing the environment.
The adiabatic combustion chamber and the two heat exchangers can be modeled using classical thermodynamic principles. For example, the boiler heat exchangers can be described using effec-tiveness-NTU models. That is, knowing the flow rates and entering fluid temperatures, the heat transfer is calculated using a single overall heat transfer coefficient UA. For the water/environment heat exchanger, the environment is considered to be an isothermal reser-voir and can be modeled as a fictitious fluid with infinite capaci-tance. The values of the overall heat transfer coefficients of the two heat exchangers are selected, or tuned to manufacturer’s data, to represent the behavior of a particular boiler.
Most boilers are equipped with combustion control systems that vary fuel input to satisfy changing heating loads. The model of Fig-ure 7 can reflect the physical effects of varying fuel, fluid flow, and stand-by losses to realistically show the degradation of performance at part-load operation. In addition, single-stage boilers with on-off control and two-stage boilers with two distinct firing rates can be modeled as “quasi-static” systems, where the cyclic operation is represented by a linear combination of two steady-state regimes.
Vapor Compression Chiller Models Figure 8 shows a schematic of a vapor compression chiller. In this case, the components include two heat exchangers, an expan-sion valve, and a compressor with a motor and transmission. The components of a chiller are linked through the refrigerant. For energy estimating, a simplified approach is sufficient to represent the refrigerant as a “perfect” fluid with fictitious property values.
That is, refrigerant liquid is modeled as incompressible, and vapor properties are described by ideal gas laws with effective average values of property parameters, such as specific heat.
Condenser and Evaporator Modeling. Both condensers and evaporators are modeled as classical heat exchangers. The two heat exchangers are each assumed to have a constant overall heat transfer coefficient. In addition, the models used in chiller systems suffer from one additional assumption—the refrigerant fluid is assumed to be isothermal for both heat exchangers, which effec-tively ignores the superheated and subcooled regions of the heat exchanger. The assumption of an isothermal refrigerant is partic-ularly crude for the condenser, which sees very high refrigerant temperatures from the compressor discharge. The effect of the Fig. 6 Possible Part-Load Power Curves Fig. 7 Boiler Modeled with Elementary Components Energy Estimating and Modeling Methods 31.15 assumption is to significantly underestimate the mean temperature difference between refrigerant and water in the heat exchanger.
Fortunately, this systematic error is compensated by a significant overestimate of the corresponding heat transfer coefficient.
General Compressor Modeling. The modeling of real com-pressors requires the description of many thermomechanical losses within the compressor. Such losses could include heat loss, fluid friction, throttling losses in valves, or motor and transmission inef-ficiencies. While some of these losses can be modeled within the compressor, others are too complex or unknown to describe in a model for energy calculations.
The general approach used here for compressor modeling is described in the Figure 9. The compressor is described by two dis-tinct internal elements: an idealized internal compressor and a motor-transmission element to account for unknown losses. Sche-matically, the motor-transmission subsystem represents an ineffi-ciency of energy conversion. The losses from these inefficiencies are assumed to heat the fluid prior to compression. Mathematically, it can be modeled by the following linear relationship: (34) where W = electrical power for a hermetic or semihermetic compressor, or shaft power for an open compressor Win = idealized internal compressor power (depends on type of compressor) Wlo = constant electromechanical loss α = proportional power loss factor Wlo and α are empirical parameters determined by performing a regression analysis on manufacturers’ data. Other parameters are also required to model Win, the idealized internal compressor power, depending on the type of compressor.
The following sections describe different modeling techniques for reciprocating, screw, and centrifugal compressors. Detailed modeling techniques are available in the ASHRAE HVAC 1 Toolkit (Lebrun et al. 1999) and associated references.
Modeling the Reciprocating Compressor. The conceptual schematic for a reciprocating compressor, for use with the general model, is shown in Figure 10. The refrigerant enters the compressor at state 1 and is heated to state 1a by the thermomechanical losses of the motor-transmission model in Figure 9. The refrigerant under-goes an isentropic compression process to state 2s, followed by a throttling process to the compressor discharge at state 2. The throt-tling valve is a simplified approach to model the known losses within the compressor due to pressure drops across the suction and discharge valves. Perhaps a more accurate model would include pressure losses at both the inlet and outlet of the compressor, but analysis of compressor data reveals that this simpler model is ade-quate for modeling of typical reciprocating compressors. In fact, many compressors can be adequately modeled with no throttling valve at all.
The refrigerant flow rate through the system must be determined to predict chiller and compressor performance. In general, volumet-ric flow depends on the pressure difference across the compressor.
The compressor refrigerant flow rate is a decreasing function of the pressure ratio due to the re-expansion of the vapor in the clearance volume. With the refrigerant vapor modeled as an ideal gas, the vol-umetric flow rate is given by the following: (35) where V = volumetric flow rate Vs = swept volumetric flow rate (geometric displacement of the compressor) Cf = clearance factor = Vclearance/Vs Pex/Psuc = cylinder pressure ratio γ = specific heat ratio Vs and Cf must be identified using data for the actual reciprocating compressor.
While the models discussed apply to full-load operation, Equa-tion (35) is also valid at part-load conditions. However, the internal power use can be different at part load depending on the particular strategy for capacity modulation, such as on-off cycling, cylinder Fig. 8 Chiller Model Using Elementary Components W Wlo 1 α + ( )Win + = Fig. 9 General Schematic of Compressor Fig. 10 Schematic of Reciprocating Compressor Model V Vs 1 Cf Cf pex psuc ---------- 1 γ ⁄ – + = 31.16 2001 ASHRAE Fundamentals Handbook (SI) unloading, hot-gas bypass, or variable-speed motor. In most cases, simple physical models can be developed to describe these meth-ods, which generally vary the swept volumetric rate. Additional thermomechanical losses can also be modeled flow but often involve additional parameters. For example, the effect of cylinder unloading can be modeled by the following relationship: (36) where Nc = number of cylinders in use Nc,FL = number of cylinders in use in full-load regime Wpump = internal power of the compressor when all the cylinders are unloaded (“pumping” power) Win,FL = full-load power The variable Wpump characterizes the part-load regime of the recip-rocating compressor, and it is assumed to be constant throughout the entire part-load range.
In summary, a realistic physical model of a reciprocating com-pressor, covering both full-load and part-load operations, can be developed based on six parameters: the constant and proportional loss terms of the motor-transmission model Wlo and α, the swept volumetric flow rate Vs of the compressor cylinders, the cylinder clearance volume factor Cf , the fictitious exhaust valve flow area Aex, and the zero-load pumping power of the unloaded compressor Wpump. The entire chiller can then be modeled with two additional parameters for the overall heat transfer coefficients of the con-denser and evaporator.
Modeling of Other Compressors and Chillers. From a mod-eling perspective, the thermodynamic processes of a screw com-pressor are similar to those of a reciprocating compressor.
Physically, the screw compressor transports an initial volumetric flow rate of refrigerant vapor to a higher pressure and density by squeezing it into a smaller space. A realistic physical model of a variable-volume-ratio, twin-screw compressor, covering both full-load and part-load operations, can be developed based on five parameters: the constant and proportional loss terms of the motor-transmission model of Equation (34), the swept volumetric flow rate of the compressor screw, the internal leakage area, and a pumped pressure differential for diverted flow at part-load (Lebrun et al. 1999). The entire chiller can then be modeled with two addi-tional parameters for the overall heat transfer coefficients of the condenser and evaporator.
An idealized internal model of a centrifugal compressor, to be used in conjunction with Equation (34) and Figure 9, can be based on an ideal analysis of a single-stage compressor composed of an isentropic impeller and isentropic diffuser. In addition to the ther-momechanical loss parameters of Equation (34), only three addi-tional parameters are required by this centrifugal compressor model: the peripheral speed of the impeller, the inclination of the vanes at the impeller exhaust, and the impeller exhaust area.
The refrigerant cycle of an absorption chiller is the same as for a vapor compression cycle, except for the absorption-generation sub-system in place of the compressor. The absorption-generation sub-system includes an absorber, a steam-fired generator, a recovery heat exchanger, a pump, and a control valve. All components except the pump and control valve can be modeled as heat exchangers.
Cooling Tower Model A cooling tower is used in primary systems to reject heat from the chiller condenser. The controls typically control the tower fans and pumps to maintain a desired water temperature entering the condenser. Like cooling and dehumidifying coils in secondary sys-tems, the performance of a cooling tower has a strong influence on the energy consumption of the chiller. In addition, tower fans con-sume electrical energy directly.
Fundamentally, a cooling tower is a direct contact heat and mass exchanger. Equations describing the fundamental processes are given in Chapter 5 and in many HVAC textbooks. Chapter 36 of the 2000 ASHRAE Handbook—Systems and Equipment describes the specific performance of cooling towers. In addition, cooling tower performance subroutines are available in Lebrun et al. (1999) and Klein et al. (1994).
For energy calculations, cooling tower performance is typically described in terms of the outdoor wet-bulb temperature, the temper-ature drop of the water flowing through the tower (the range), and the difference between the leaving water temperature and the air wet-bulb temperature (the approach). While simple models assume constant range and approach, more sophisticated models use rating performance data to relate leaving water temperature to the outdoor wet-bulb temperature, water flow, and airflow. Simple cooling tower models, such as those based on a single overall transfer coef-ficient that can be directly inferred from a single tower rating point, are often appropriate for energy calculations.
SYSTEM MODELING OVERALL MODELING STRATEGIES In developing a simulation model for building energy prediction, two basic issues must be considered—(1) modeling of components or subsystems and (2) the overall modeling strategy. Modeling of components, which was discussed in the section on Component Modeling and Loads, results in sets of equations describing the indi-vidual components. The overall modeling strategy refers to the sequence and procedures used to solve these equations. The accu-racy of results and the computer resources required to achieve these results depend on the modeling strategy.
In most building energy programs, the load models are executed for every space for every hour of the simulation period. (Practically all models use 1 h as the time step, which excludes any information on phenomena occurring in a shorter time span.) The load model is followed by running models for every secondary system, one at a time, for every hour of the simulation. Finally, the plant simulation model is executed again for the entire period. Each sequential exe-cution processes the fixed output of the preceding step.
This procedure is illustrated in Figure 11. The solid lines repre-sent data passed from one model to the next. The dashed lines rep-resent information, usually provided by the user, about one model passed to the preceding model. For example, the system information consists of a piecewise-linear function of zone temperature that gives the system capacity.
Because of this loads-systems-plants sequence, certain phenom-ena cannot be modeled precisely. For example, if the heat balance method for computing loads is used, and some component in the system simulation model cannot meet the load, the program can only report the current load. In actuality, the space temperature should readjust until the load matches the equipment capacity, but this cannot be modeled because the loads have been precalculated and fixed. If the weighting factor method is used for loads, this problem is partially overcome, because loads are continually read-justed during the system simulation. However, the weighting factor Win Win FL , 1 Nc Nc F , L ---------------– Wpump + = Fig. 11 Overall Modeling Strategy Energy Estimating and Modeling Methods 31.17 technique is based on linear mathematics, and wide departures of room temperatures from those used during execution of the load program can introduce errors.
A similar problem arises in plant simulation. For example, in an actual building, as the load on the central plant varies the supply chilled water temperature also varies. This variation in turn affects the capacity of the secondary system equipment. In an actual build-ing, when the central plant becomes overloaded, space temperatures should rise to reduce the load. However, in most energy estimating programs, this condition cannot occur; thus, only the overload con-dition can be reported. These are some of the penalties associated with decoupling of the load, system, and plant models.
An alternative strategy, in which all calculations are performed at each time step, is conceivable. Here the load, system, and plant equations are solved simultaneously at each time interval. With this strategy, unmet loads and imbalances cannot occur; conditions at the plant are immediately reflected to the secondary system and then to the load model, forcing them to readjust to the instantaneous conditions throughout the building. The results of this modeling strategy are superior to those currently available, although the mag-nitude and importance of the improvement are uncertain.
The principal disadvantage of the alternative approach, and the reason that it has not been widely used, is that it demands more com-puting resources. However, programs that, to one degree or another, implement simultaneous solution of the loads, system, and plant models have been developed by Clarke (1985), Park et al. (1985), Klein et al. (1994), and Metcalfe et al. (1995).
An economic model, as shown in Figure 11, calculates energy costs (and sometimes capital costs) based on the estimated required input energy. Thus, the simulation model calculates energy usage and cost for any given input weather and internal loads. By apply-ing this model (i.e., determining output for given inputs) at each hour (or other suitable interval), the hour-by-hour energy consump-tion and cost can be determined. Maintaining running sums of these quantities yields monthly or annual energy usage and costs.
These models only compare design alternatives; a large number of uncontrolled and unknown factors usually rule out such models for accurate prediction of utility bills. For example, Miller (1980) found that the dynamics of control of components may have at least minor effects on predicted energy use. The section on Bibliography lists several models, which are also described in Walton (1983) and York and Cappiello (1981). Generally, the load models tend to be the most complex and time-consuming, while the central plant model is the least complex.
Because the detailed models are computationally intensive, sev-eral simplified methods have been developed. These methods include the degree-day method, the bin method, and correlation methods, and they are presented in the next two sections.
DEGREE-DAY AND BIN METHODS Degree-day methods are the simplest methods for energy anal-ysis and are appropriate if the building use and the efficiency of the HVAC equipment are constant. Where efficiency or conditions of use vary with outdoor temperature, the consumption can be calcu-lated for different values of the outdoor temperature and multiplied by the corresponding number of hours; this approach is used in var-ious bin methods. When the indoor temperature is allowed to fluc-tuate or when interior gains vary, models other than simple steady-state models must be used.
Even in an age when computers can easily calculate the energy consumption of a building, the concepts of degree-days and balance point temperature remain valuable tools. The severity of a climate can be characterized concisely in terms of degree-days. Also, the degree-day method and its generalizations can provide a simple estimate of annual loads, which can be accurate if the indoor tem-perature and internal gains are relatively constant and if the heating or cooling systems operate for a complete season. For these reasons, basic steady-state methods continue to be important.
Balance Point Temperature The balance point temperature tbal of a building is defined as that value of the outdoor temperature to at which, for the specified value of the interior temperature ti, the total heat loss qgain is equal to the heat gain from sun, occupants, lights, and so forth.
(37) where Ktot is the total heat loss coefficient of the building in W/K.
For any of the steady-state methods described in this section, heat gains must be the average for the period in question, not for the peak values. In particular, solar radiation must be based on aver-ages, not peak values. The balance point temperature is therefore obtained as (38) Heating is needed only when to drops below tbal. The rate of energy consumption of the heating system is (39) where η h is the efficiency of the heating system, also designated on an annual basis as the annual fuel use efficiency (AFUE), θ is time, and the plus sign above the bracket indicates that only positive val-ues are to be counted. If tbal, Ktot, and η h are constant, the annual heating consumption can be written as an integral: (40) This integral of the temperature difference conveniently summa-rizes the effect of outdoor temperatures on a building. In practice, it is approximated by summing averages over short time intervals (daily or hourly); the result is termed degree-days or degree-hours.
Annual Degree-Day Method Annual Degree-Days. If daily average values of outdoor tem-perature are used for evaluating the integral, the degree-days for heating DDh(tbal) are obtained as (41) with dimensions of K·days. Here the summation is to extend over the entire year or over the heating season. It is a function of tbal, reflecting the roles of ti, heat gain, and loss coefficient. The balance point temperature tbal is also known as the base of the degree-days.
In terms of degree-days, the annual heating consumption is (42) Heating degree-days or degree-hours for a balance point temper-ature of 18.3°C have been widely tabulated based on the observa-tion that this has represented average conditions in typical buildings in the past. The 18.3°C base is assumed whenever tbal is not indi-cated explicitly. The extension of degree-day data to different bases is discussed later.
qgain Ktot ti tbal – ( ) = tbal ti qgain Ktot ------------– = qh Ktot η h --------- tbal to θ ( ) – [ ]+ = Qh yr , Ktot η h ---------tbal to θ ( ) – [ ]+ θ d ∫ = DDh tbal ( ) 1 day ( ) tbal to – ( )+ days ∑ = Qh yr , Ktot η h ---------DDh tbal ( ) = 31.18 2001 ASHRAE Fundamentals Handbook (SI) Cooling degree-days can be calculated using an equation analo-gous to Equation (41) for heating degree-days as (43) While the definition of the balance point temperature is the same as that for heating, in a given building its numerical value for cooling is generally different from that for heating because qi, Ktot, and ti can be different. According to Claridge et al. (1987), tbal can include both solar and internal gains as well as losses to the ground.
Calculating cooling energy consumption using degree-days is more difficult than heating. For cooling, the equation analogous to Equation (42) is (44) for a building whose Ktot does not change. That assumption is gen-erally acceptable during the heating season, when windows are closed and the air exchange rate is fairly constant. However, during the intermediate or cooling season, heat gains can be eliminated, and the onset of mechanical cooling can be postponed by opening windows or increasing the ventilation. (In buildings with mechani-cal ventilation, this is called the economizer mode.) Mechanical air conditioning is needed only when the outdoor temperature extends beyond the threshold tmax. This threshold is given by an equation analogous to Equation (38), with the replacement of the closed win-dow heat transmission coefficient Ktot by its value Kmax for open windows: (45) Kmax varies considerably with wind speed, but a constant value can be assumed for simple cases. The resulting sensible cooling load is shown schematically in Figure 12 as a function of to. The solid line is the load with open windows or increased ventilation; the dashed line shows the load if Ktot were kept constant. The annual cooling load for this mode can be calculated by breaking the area under the solid line into a rectangle and a triangle, or (46) where DDc(tmax) are the cooling degree-days for base tmax, and Nmax is the number of days during the season when to rises above tmax.
This is merely a schematic model of air conditioning. In practice, heat gains and ventilation rates vary, as does the occupant behavior in using the windows and the air conditioner. Also, in commercial buildings with economizers, the extra fan energy for increased ven-tilation must be added to the calculations. Finally, air-conditioning systems are often turned off during unoccupied periods. Therefore, cooling degree-hours better represent the period when equipment is operating than cooling degree-days because degree-days assume uninterrupted equipment operation as long as there is a cooling load.
Latent loads can form an appreciable part of a building’s cooling load. The degree-day method can be used to estimate the latent load during the cooling season on a monthly basis by adding the follow-ing term to Equation (46): (47) where qlatent = monthly latent cooling load, kW = monthly infiltration (total airflow), kg/s hfg = heat of vaporization of water, kJ/kg Wo = outdoor humidity ratio (monthly averaged) Wi = indoor humidity ratio (monthly averaged) The degree-day method assumes that tbal is constant, which is not well satisfied in practice. Solar gains are zero at night, and internal gains tend to be highest during the evening. The pattern for a typical house is shown in Figure 13. As long as to always stays below tbal, the variations average out without changing the consumption. But for the situation in Figure 13, to rises above tbal from shortly after 10:00 A.M. to 10:00 P.M.; the consequences for energy consumption depend on the thermal inertia and on the control of the HVAC sys-tem. If this building had low inertia and if temperature control were critical, heating would be needed at night and cooling during the day. In practice, this effect is reduced by thermal inertia and by the dead band of the thermostat, which allows ti to float.
The closer to is to tbal, the greater the uncertainty. If the occupants keep the windows closed during mild weather, ti will rise above the set point. If they open the windows, the potential benefit of heat gains is reduced. In either case, the true values of tbal become uncer-tain. Therefore, the degree-day method, like any steady-state method, is unreliable for estimating the consumption during mild weather. In fact, the consumption becomes most sensitive to occu-pant behavior and cannot be predicted with certainty.
Despite these problems, the degree-day method (using an appro-priate base temperature) can give remarkably accurate results for the annual heating energy of single-zone buildings dominated by DDc tbal ( ) 1 day ( ) to tbal – ( )+ days ∑ = Qc yr , Ktot η h ---------DDc tbal ( ) = tmax ti qgain Kmax ------------– = Fig. 12 Cooling Load as Function of Outdoor Temperature to Qc Ktot DDc tmax ( ) tmax tbal – ( )Nmax + [ ] = Fig. 13 Variation of Balance Point Temperature and Internal Gains for a Typical House (Nisson and Dutt 1985) qlatent m · hfg Wo Wi – ( ) = m · Energy Estimating and Modeling Methods 31.19 losses through the walls and roof and/or ventilation. Typical build-ings have time constants that are about 1 day, and a building’s ther-mal inertia essentially averages over the diurnal variations, especially if ti is allowed to float. Furthermore, the energy consump-tion in mild weather is small; hence, a relatively large error here has only a small effect on the total for the season.
Variable-Base Annual Degree-Days. The calculation of Qh from degree-days DDh(tbal) depends on the value of tbal. This value varies widely from one building to another because of widely differing personal preferences for thermostat settings and setbacks and because of different building characteristics. In response to the fuel crises of the 1970s, heat transmission coeffi-cients have been reduced, and thermostat setback has become common. At the same time, the energy use by appliances has increased. These trends all reduce tbal (Fels and Goldberg 1986).
Hence, in general, degree days with the traditional base 18.3°C are not to be used.
Figure 14A shows how the heating degree-days vary with tbal for a particular site, in this case, New York. The plot is obtained by evaluating Equation (41) with data for the number of hours per year during which to is within 2.8 K temperature intervals centered at 25°C, 22.2°C, 19.4°C, 16.6°C, …, − 13.9°C. The data for the num-ber of hours in each interval, or bin, are included as labels in this plot. Analogous curves, without these labels, are shown in Figure 14B for three other sites: Houston, Washington, and Denver. If the annual average of to is known, the cooling degree-days to any base below 22 ±1.4°C can also be found.
Seasonal Efficiency. The seasonal efficiency η h of heating equipment depends on such factors as steady-state efficiency, siz-ing, cycling effects, and energy conservation devices. Sometimes it is much lower than and other times it is comparable to steady-state efficiency. Alereza and Kusuda (1982) developed expressions to estimate the seasonal efficiency for a variety of furnaces, if infor-mation on rated input and output is available. These expressions correlate seasonal efficiency with variables determined by using the equipment simulation capabilities of a large hourly simulation pro-gram and typical equipment performance curves supplied by the National Institute of Standards and Technology (NIST): (48) where η ss = steady-state efficiency (rated output/input) αD = fraction of heat loss from ducts CFpl = part-load correction factor The dimensionless term CFpl is a characteristic of the part-load effi-ciency of the heating equipment, which may be calculated as follows: Gas Forced-Air Furnaces With pilot CFpl = 0.6328 + 0.5738(RLC) − 0.3323(RLC)2 With intermittent ignition CFpl = 0.7791 + 0.1983(RLC) −0.0711(RLC)2 With intermittent ignition and loose stack damper CFpl = 0.9276 + 0.0732(RLC) − 0.0284(RLC)2 Oil Furnaces Without Stack Damper CFpl = 0.7092 + 0.6515(RLC) −0.4711(RLC)2 Resistance Electric Furnaces CFpl = 1.0 These equations are based on many annual simulations for the equipment. The dimensionless ratio RLC of building design load to the capacity (rated output) of the equipment is defined as follows: where BLC = building loss coefficient, W/K tod = outside design temperature, °C CHT = capacity (rated output) of heating equipment, W The building loss coefficient BLC can be defined as design-day heat loss/(tbal −tod). The design-day heat loss includes both infiltra-tion and ground losses. Duct losses as a percentage of the design-day heat loss are added via the factor (1 + αD). RLC assumes values in the range 0 to 1.0, appropriate for typical cases when the heating equipment is oversized. Seasonal efficiency is also discussed by Chi and Kelly (1978), Parker et al. (1980), and Mitchell (1983).
Monthly Degree-Days Many formulas have been proposed for estimating the degree-days relative to an arbitrary base when detailed data are not avail-able. The basic idea is to assume a typical probability distribution of Fig. 14 Annual Heating Days DDh(tbal) as Function of Balance Temperature tbal η η ssCFpl 1 αD + ------------------= RLC BLC CHT ------------ tbal tod – ( ) 1 αD + ( ) = 31.20 2001 ASHRAE Fundamentals Handbook (SI) temperature data, characterized by its average and by its standard deviation σ . Erbs et al. (1983) developed a model that needs as input only the average for each month of the year. The standard devi-ations σ m for each month are then estimated from the correlation (49) This is a dimensional equation with t and σin °C; σ yr is the standard deviation of the monthly average temperatures about the annual average : (50) To obtain a simple expression for the degree-days, a normalized temperature variable φis defined as (51) where N = number of days in the month (N has units of day/month and φ has units of ). While temperature distributions can be different from month to month and location to location, most of this variability can be accounted for by the average and the stan-dard deviation of . Being centered around and scaled by σ m, the quantity φ eliminates these effects. In terms of φ, the monthly heating degree-days for any location are well approximated by (52) where a = 1.698 .
For nine locations spanning most climatic zones of the United States, Erbs et al. (1983) verified that the annual heating degree days can be estimated with a maximum error of 175 K·days if Equation (52) is used for each month. For cooling degree-days, the largest error is 150 K·days. Such errors are quite acceptable, representing less than 5% of the total.
Table 5 lists monthly heating degree-days for New York City, using the model of Erbs et al. (1983), given monthly averages of to as reproduced in column 2 of Table 5. The degree-days are based on a balance temperature of 15.6°C. Column 2 lists the given values of monthly average outdoor temperature, and N is the number of days in the month. Intermediate quantities are shown in columns 4 and 5, and to, yr and σ yr are shown at the bottom. Column 6 shows the monthly and annual results.
Table 6 contains degree-day data for several sites and monthly averaged outdoor temperatures needed for the algorithm. More complete tabulations of the latter are contained in Cinquemani et al.
(1978) and in local climatological data summaries available from the National Climatic Data Center, Asheville, NC (NOAA 1973 and www.ncdc.noaa.gov). Monthly degree-day data at various bases, as well as other climatic information for 209 U.S. and 14 Canadian cit-ies, may be found in Appendix 3 to Balcomb et al. (1982).
Bin Method For many applications, the degree-day method should not be used, even with the variable-base method, because the heat loss coefficient Ktot, the efficiency η h of the HVAC system, or the bal-ance point temperature tbal may not be sufficiently constant. The efficiency of a heat pump, for example, varies strongly with out-door temperature; or the efficiency of the HVAC equipment may be affected indirectly by to when the efficiency varies with the load, a common situation for boilers and chillers. Furthermore, in most commercial buildings, the occupancy has a pronounced pat-tern, which affects heat gain, indoor temperature, and ventilation rate.
In such cases, a steady-state calculation can yield good results for the annual energy consumption if different temperature intervals and time periods are evaluated separately. This approach is known as the bin method because the consumption is calculated for several values of the outdoor temperature to and multiplied by the number of hours Nbin in the temperature interval (bin) centered around that temperature: to to σ m 3.54 0.0290 t o – 0.0644σ yr + = t o yr , σ yr 1 12 ------t o t o yr , – ( ) 2 1 12 ∑ = φ t bal t o – σ m N ---------------------= month day ⁄ t o t o DDh tbal ( ) σ mN1.5 φ 2 --ln e aθ – eaθ + ( ) 2a -----------------------------------+ = day month ⁄ Table 5 Degree-Day Calculation from Monthly Averaged Data Month N, day/mo.
DDh(tbal), K·days January 0.1 31 2.03 1.32 463 February 0.8 28 2.01 1.34 399 March 5.1 31 1.89 0.95 312 April 11.2 30 1.71 0.41 133 May 16.8 31 1.55 − 0.21 31 June 22.0 30 1.40 − 0.92 3 July 24.8 31 1.32 − 1.33 1 August 23.8 31 1.34 − 1.18 1 September 20.2 30 1.45 − 0.66 7 October 14.8 31 1.60 0.02 59 November 8.6 30 1.79 0.66 202 December 1.9 31 1.98 1.19 406 to,yr 12.51 Sum 2018 σ yr 8.80 Note: Use Equation (52) to calculate DDh(tbal).
to ° C , σ m ° C , φ mo.
day ⁄ , Table 6 Degree-Day and Monthly Average Temperatures for Various Locations Site Variable-Base Heating Degree-Day, K·daysa Monthly Average Outdoor Temperature , °Cb 18.3 15.6 12.8 10.0 7.2 Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Los Angeles, CA 692 290 88 14 0 12.5 13.1 13.6 14.9 16.6 18.1 20.3 20.8 20.4 18.4 15.8 13.8 Denver, CO 3342 2624 2001 1474 1029 − 1.2 0.4 2.8 8.6 13.9 18.9 22.8 22.0 17.1 11.1 4.1 0.3 Miami, FL 114 30 4 0 0 19.6 19.9 21.8 23.9 25.6 27.2 27.9 28.3 27.6 25.4 22.3 20.2 Chicago, IL 3404 2751 2173 1666 1233 − 4.3 − 2.6 2.7 9.9 15.6 21.4 23.7 23.2 18.8 13.0 4.7 − 1.9 Albuquerque, NM 2384 1797 1294 865 535 1.8 4.4 7.7 13.2 18.5 23.7 25.9 24.8 21.2 14.6 6.9 2.3 New York, NY 2727 2104 1559 1100 728 0.1 0.8 5.1 11.2 16.8 22.0 24.8 23.8 20.2 14.8 8.6 1.9 Bismarck, ND 5024 4253 3569 2959 2430 − 13.2 − 10.3 − 3.8 6.1 12.4 17.7 21.6 20.7 14.2 8.2 − 1.7 − 9.1 Nashville, TN 2053 1532 1091 743 473 3.5 5.0 9.3 15.6 20.3 24.8 26.4 25.8 22.2 16.1 9.1 4.7 Dallas/Ft. Worth, TX 1272 858 527 292 139 7.4 9.7 13.2 19.1 23.2 27.6 29.8 29.9 25.7 20.0 13.3 9.0 Seattle, WA 2626 1816 1162 663 334 3.4 5.7 6.7 9.3 12.7 15.4 18.1 17.7 15.3 11.2 7.0 4.7 aSource: NOAA (1973).
bSource: Cinquemani et al. (1978).
t o Energy Estimating and Modeling Methods 31.21 (53) The superscript plus sign indicates that only positive values are counted; no heating is needed when to is above tbal. Equation (53) is evaluated for each bin, and the total consumption is the sum of the Qbin over all bins.
In the United States, the necessary weather data are available in ASHRAE (1995) and USAF (1978). The bins are usually in 2.8 K increments (when derived from 5°F bins) and are often collected in three daily 8 h shifts. Mean coincident wet-bulb temperature data (for each dry-bulb bin) are used to calculate latent cooling loads from infiltration and ventilation. The bin method considers both occupied and unoccupied building conditions and gives credit for internal loads by adjusting the balance point. For example, a calcu-lation could be performed for 5°C outdoors (representing all occur-rences from 3.6 to 6.4°C) and with building operation during the midnight to 0800 shift (5°C outdoors, representing all occurrences from 4°C). Because there are 23 2.8 K bins between –23 and 40.4°C and 3 8 h shifts, 69 separate operating points are calculated.
For many applications, the number of calculations can be reduced.
A residential heat pump (heating mode), for example, could be cal-culated for just the bins below 18.3°C without the three-shift breakdown. The data included in Table 7 are samples of annual totals for a few sites, but ASHRAE (1995) and USAF (1978) include monthly data and data further separated into time intervals during the day.
Sample Annual Bin Data Site Bin 39/ 41 36/ 38 33/ 35 30/ 32 27/ 29 24/ 26 21/ 23 18/ 20 15/ 17 12/ 14 9/ 11 6/ 8 3/ 5 0/ 2 –3/ –1 –6/ –4 –9/ –7 –12/ –10 –15/ –13 –18/ –16 –21/ –19 Chicago, IL 74 176 431 512 960 660 591 780 510 770 686 1671 380 304 125 66 49 11 4 Dallas/Ft. Worth, TX 4 170 322 511 922 1100 1077 750 803 870 581 728 418 464 37 3 Denver, CO 81 217 406 390 570 726 712 902 809 783 750 1467 446 216 106 85 52 44 8 Los Angeles, CA 4 10 9 16 56 194 1016 1874 2280 2208 843 227 23 Miami, FL 14 648 2147 2581 1852 734 390 202 100 76 14 2 Nashville, TN 4 82 366 717 756 1291 831 693 801 670 858 639 793 141 89 29 Seattle, WA 10 88 139 330 497 898 1653 1392 1844 1127 715 40 26 1 Table 7 Calculation of Annual Heating Energy Consumption for Example 4 Climate House Heat Pump Supplemental A B C D E F G H I J K L M N Temp.
Bin, °C Temp.
Diff., tbal – tbin Weather Data Bin, h Heat Loss Rate, kW Heat Pump Integrated Heating Capacity, kW Cycling Capacity Adjust-ment Factora Adjusted Heat Pump Capacity, kWb Rated Electric Input, kW Opera-ting Time Fractionc Heat Pump Supplied Heating, kWhd Seasonal Heat Pump Electric Consump-tion, KWhe Space Load, kWhf Supple-mental Heating Required, kWhg Total Electric Energy Consump-tion, kWhh 16 1.8 693 0.70 12.80 0.764 9.78 3.74 0.072 488 187 485 — 187 13 4.8 801 1.87 12.01 0.789 9.48 3.63 0.197 1 496 573 1 497 — 573 10 7.8 670 3.04 11.22 0.818 9.18 3.52 0.331 2 036 781 2 037 — 781 7 10.8 858 4.21 9.80 0.857 8.40 3.40 0.501 3 611 1 462 3 612 — 1 462 4 13.8 639 5.38 8.49 0.908 7.71 3.18 0.698 3 439 1 418 3 438 — 1 418 1 16.8 793 6.55 7.98 0.955 7.62 3.10 0.860 5 196 2 114 5 195 — 2 114 –2 19.8 141 7.72 7.47 1.000 7.47 3.02 1.000 1 053 426 1 089 36 462 –5 22.8 89 8.89 6.95 1.000 6.95 2.93 1.000 618 261 791 173 434 –8 25.8 29 10.06 6.48 1.000 6.48 2.85 1.000 188 83 292 104 187 –11 28.8 0 11.23 5.69 1.000 — — — — — — — — Totals: 18 125 7 305 18 436 313 7 618 aCycling Capacity Adjustment Factor = 1 −Cd(1 −x), where Cd = degradation coefficient (default = 0.25 unless part load factor is known) and x = building heat loss per unit capacity at temperature bin. Cycling capacity = 1 at the balance point and below. The cycling capacity adjustment factor should be 1.0 at all temperature bins if the manufacturer includes cycling effects in the heat pump capacity (Column E) and associated electrical input (Column H).
bColumn G = Column E × Column F cOperating Time Factor equals smaller of 1 or Column D/Column G dColumn J = Column I × Column G × Column C eColumn K = Column I × Column H × Column C fColumn L = Column C × Column D gColumn M = Column L – Column J hColumn N = Column K + Column M Qbin Nbin Ktot η h --------- tbal to – [ ]+ = Fig. 15 Heat Pump Capacity and Building Load 31.22 2001 ASHRAE Fundamentals Handbook (SI) Equipment performance may vary with load. For heat pumps, the U.S. Department of Energy has adopted test procedures to deter-mine the effect of dynamic operations. The bin method uses these results for a specific heat pump to adjust the integrated capacity for the effect of part-load operation. Figure 15 compares the adjusted heat pump capacity to the building heat loss in Example 4 below.
This type of curve must be developed for each model heat pump as applied to an individual profile. The heat pump cycles on and off above the balance point temperature to meet the house load, while supplemental heat is required at lower temperatures. This cycling can reduce performance, depending on the part-load factor at a given temperature. The cycling capacity adjustment factors used in this example to account for cycling degradation can be calculated from the equation in footnote a of Table 8.
Frosting and the necessary defrost cycle can reduce perfor-mance over steady-state conditions that do not include frosting.
The effects of frosting and defrosting are already integrated into many (but not all) manufacturers’ published performance data.
Example 4 assumes that the manufacturer’s data already accounts for the frosting/defrosting losses (as indicated by the characteristic notch of the capacity curve in Figure 15) and shows how to adjust an integrated performance curve for cycling losses.
Example 4. Estimate the energy requirements for a residence in Nashville, Tenn., with a design heat loss of 11 700 W at 30°C design temperature difference. The inside design temperature is 21°C. It is estimated that internal heat gains average 1250 W. Weather data in terms of hours for each bin are shown in Table 7. Assume the selection of a 10.5 kW heat pump having the characteristics given in Columns E and H of Table 8 and in Figure 15.
Solution: The design heat loss is based on no internal heat generation.
The heat pump system energy input is the net heat requirement of the space (i.e., the envelope loss minus the internal heat generation). The net heat loss per degree and the heating/cooling balance temperature may be computed: Ktot = HL/∆t = 11 700/30 = 390 W/K From Equation (38), tbal = 21 – (1250/390) = 17.8°C Table 8 is then computed, resulting in 7618 kWh.
The modified bin method (Knebel 1983) extends the basic bin method to account for weekday/weekend and partial-day occu-pancy effects, to calculate net building loads (conduction, infiltra-tion, internal loads, and solar loads) at four temperatures, rather than interpolate from design values, and to better describe secondary and primary equipment performance.
CORRELATION METHODS One way to simplify energy analyses is to correlate energy requirements to various inputs. Typically, the result of a correlation is a simple equation that may be used in a calculator or small com-puter program or to develop a graph that provides a quick insight into the energy requirements. Examples of correlation methods are in ASHRAE Standard 90.1, which includes several empirical equa-tions that may be used to predict energy consumption by many types of buildings.
The accuracy of correlation methods depends on the size and accuracy of the database and the statistical means used to develop the correlation. A database generated from measured data can lead to accurate correlations (Lachal et al. 1992). The key to the proper use of a correlation is that the case being studied matches the cases used in developing the database. Inputs to the correlation (the inde-pendent variables) indicate the factors that are considered to have significant impact on energy consumption. A correlation is invalid either when an input parameter is used beyond its valid range (cor-responding to extrapolation rather than interpolation) or when some important feature of the building/system is not included in the avail-able inputs to the correlation.
SIMULATING SECONDARY AND PRIMARY SYSTEMS Traditionally, most energy analysis programs include a set of preprogrammed models that represent various systems, such as variable air volume, terminal reheat, multizone, etc. In this scheme, the equations for each system are arranged so they can be solved sequentially. If this is not possible, then the smallest number of equations that must be solved simultaneously is solved using an appropriate technique. Furthermore, individual equations may vary from hour to hour in the simulation, depending on controls and operating conditions. For example, a dry coil uses different equa-tions than a wet coil.
The primary disadvantage of this scheme is that it is relatively inflexible—in order to modify a system, the program source code may have to be modified and recompiled. Alternative strategies (Klein et al. 1994, Park et al. 1985) have viewed the system as a series of components (e.g. fan, coil, pump, duct, pipe, damper, thermostat) that may be organized in a component library. Users of the program specify the connections between the components.
The program then resolves the specification of components and connections into a set of simultaneous equations.
A refinement of component-based modeling is known as equa-tion-based modeling (Sowell and Moshier 1995, Buhl et al. 1993).
In this scheme, the models do not follow well-defined rules for a solution, and input and output variables are not predetermined.
MODELING OF SYSTEM CONTROLS From a mathematical viewpoint, controls represent equations that must be satisfied at each point during the simulation. For exam-ple, the room thermostat can be represented as a function relating heating and cooling delivery to space temperature. Similarly, cool-ing coil reset controls can be modeled as a relationship between out-side or zone temperature and coil discharge temperature. An accurate secondary system model must ensure that all controls are properly represented and that the governing equations are satisfied at each simulation time step. This often creates a need for iteration or, alternately, for use of values from an earlier solution point.
The controls on space temperature affect the interaction between loads calculations and the secondary system simulation. A realistic model might require a dead band in space temperature in which no heating or cooling is called for; within this range, the true space sen-sible load is zero, and the true space temperature must be adjusted accordingly. If the thermostat has proportional control between zero and full capacity, the space temperature will rise in proportion to the load during cooling and fall similarly during heating. Capacity to heat or cool also varies with space temperature after the control device has reached its maximum because capacity is proportional to the difference between supply and space temperatures. Failure to properly model these phenomena results in overestimating required energy.
INTEGRATION OF SYSTEM MODELS Energy calculations for secondary systems involve construction of the complete system from the set of HVAC components. For example, a VAV system is a single-path system that controls zone temperature by modulating the airflow while maintaining a constant supply air temperature. VAV terminal units, located at each zone, adjust the quantity of air reaching each zone depending on its load requirements. Reheat coils may be included to provide required heating for perimeter zones.
This VAV system simulation consists of a central air-handling unit and a VAV terminal unit with reheat coil located at each zone, Energy Estimating and Modeling Methods 31.23 as shown in Figure 16. The central air-handling unit includes a fan, a cooling coil, a preheat coil, and an outside air economizer. The supply air leaving the air-handling unit is controlled to a fixed set point. The VAV terminal unit at each zone varies the airflow to meet the cooling load. As the zone cooling load decreases, the VAV ter-minal unit decreases the zone airflow until the unit reaches its min-imum position. If the cooling load continues to decrease, the reheat coil will be activated to meet the zone load. As the supply air vol-ume leaving the unit decreases, the fan power consumption will also be reduced. A variable-speed drive is used to control the supply fan.
The simulation is based on system characteristics and zone design requirements. For each zone, the inputs include the sensible and latent loads, the zone set point temperature, and the minimum zone supply air mass flow. System characteristics include the supply air temperature set point, the entering water temperature of the reheat, preheat and cooling coils, the minimum mass flow of outside air, and the economizer temperature/enthalpy set point for minimum airflow.
The algorithm for performing the calculations for this VAV sys-tem is shown in Figure 17. The algorithm directs sequential calcu-lations of system performance. Calculations proceed from the zones forward along the return air path to the cooling coil inlet and back through the supply air path to the cooling coil discharge.
Moving back along the supply air path, the fan entering air tem-perature is calculated by setting fan outlet air temperature to the sys-tem design supply air temperature. The known fan inlet air temperature is then used as both the cooling coil and preheat coil discharge air temperature set point. Moving forward along the return air path, the cooling coil entering air temperature can be determined by sequentially moving through the economizer cycle and the preheat coil.
Unlike temperature, the humidity ratio at any point in a system cannot be explicitly determined due to the dependence of the cool-ing coil performance on the mixed air humidity ratio. The latent load defines the difference between zone humidity and supply air humidity. However, the humidity ratio of the supply air depends on the humidity ratio entering the coil, which in turn depends on that of the return air. This calculation must be performed either by solving simultaneous equations or, as in this case, by an iterative process.
Assuming a trial value for the humidity ratio at the cooling coil discharge (e.g., 13°C, 90% rh), the humidity ratio at all other points throughout the system can be calculated. With known cooling coil inlet air conditions and a design discharge air temperature, the inverted cooling coil subroutine iterates on the coil fluid mass flow to converge on the discharge air temperature with the discharge air humidity ratio as an output. The cooling coil discharge air humidity ratio is then compared to the previous discharge humidity ratio. This iterative process continues calculating through the loop several Fig. 16 Schematic of Variable Air Volume System with Reheat BEGIN LOOP Calculate zone related design requirements • Calculate required supply airflow to meet zone load • Sum actual zone mass airflow rate • Sum zone latent loads IF zone equals last zone THEN Exit Loop END LOOP • Calculate system return air temperature from zone temps • Assume an initial cooling coil leaving air humidity ratio BEGIN LOOP Iterate on cooling coil leaving air humidity ratio • Calculate return air humidity ratio from latent loads • Calculate supply fan power consumption and entering fan air temperature • Calculate mixed air temperature and humidity ratio using an economizer cycle IF mixed air temperature is less than design supply air temperature THEN • Calculate preheat coil load ELSE • Calculate cooling coil load and leaving air humidity ratio ENDIF IF cooling coil leaving air humidity ratio converged THEN Exit Loop END LOOP BEGIN LOOP Calculate the zone reheat coil loads IF zone supply air temperature is greater than system design supply air temperature THEN • Calculate reheat coil load (Subroutine: COILINV/HCDET) ENDIF • Sum reheat coil loads for all zones IF zone equals last zone THEN Exit Loop END LOOP Fig. 17 Algorithm for Calculating Performance of VAV with System Reheat 31.24 2001 ASHRAE Fundamentals Handbook (SI) times until the values of the cooling coil discharge air humidity ratio stabilize within a specified tolerance.
This basic algorithm for simulation of a VAV system might be used in conjunction with a heat balance type of load calculation. For a weighting factor approach, it would have to be modified to allow zone temperatures to vary and consequently zone loads to be read-justed. It should also be enhanced to allow for possible limits on reheat temperature and/or cooling coil limits, zone humidity limits, outside air control (economizers), and/or heat-recovery devices, zone exhaust, return air fan, heat gain in the return air path because of lights, the presence of baseboard-type heaters, and more realistic control profiles. Most current building energy programs incorporate these and other features as user options, as well as algorithms for other types of systems.
INVERSE MODELING CATEGORIES OF INVERSE METHODS Inverse methods for energy-use estimation in buildings and related HVAC&R equipment can be classified into three broad categories. These approaches are widely disparate in data require-ments, in time and effort needed to develop the associated mod-els, in user skill, and in the sophistication and reliability that they provide.
Empirical or “Black-Box” Approach With this approach, a simple or multi-variate regression model is identified between measured energy use and the various influential parameters (climatic variables, building occupancy). The form of the regression models can be either purely statistical or loosely based on some basic engineering formulation of energy use in the building. In any case, the identified model coefficients are such that no (or very little) physical meaning can be assigned to them. This approach can be used with any time scale—monthly, daily, hourly or subhourly—provided appropriate data are available. Single-vari-ate, multivariate, change point, Fourier series, and artificial neural network (ANN) models fall under this category, as noted in Table 1.
Model identification is relatively straightforward, usually requires little effort, and can be used in several diverse types of cir-cumstances. The empirical approach is thus the most widely used inverse approach. Although more sophisticated regression tech-niques such as maximum likelihood and two-stage regression schemes can be used for model identification, least-squares regres-sion is most common. The purely statistical approach is usually ade-quate for evaluating demand-side management (DSM) programs to identify simple and conventional energy conservation measures in an actual building (lighting retrofits, air handler retrofits such as CV to VAV retrofits) and for baseline model development in energy conservation measurement and verification (M&V) projects (Fels 1986; MacDonald and Wasserman 1989; Ruch and Claridge 1991; Reddy et al. 1997; Claridge 1998b; Kissock et al. 1998; Katipamula et al. 1998; Dhar 1995; Dhar et al. 1998, 1999a, 1999b; Miller and Seem 1991; Kreider and Wang 1991; Krarti et al. 1998). It is also appropriate for modeling equipment such as pumps and fans, and even more elaborate equipment such as chillers and boilers, pro-vided the necessary performance data are available (Phelan et al.
1996, Braun 1992, Englander and Norford 1992, Lorenzetti and Norford 1993). Although this approach allows detection or flagging of equipment or system faults, it is usually of limited value for diag-nosis and on-line control (with ANN as a possible exception).
Calibrated Simulation Approach With this approach, one uses an existing building simulation computer program and “tunes” or calibrates the various physical inputs to the program so that observed energy use matches closely with that predicted by the simulation program. Once that is achieved, more reliable predictions can be made than with the sta-tistical approaches. The calibrated simulation approach is ad-vocated in cases where only whole-building metering is available and M&V calls for estimating the energy savings of individual ret-rofits. Practitioners of this approach have largely tended to adopt existing and widely used forward simulation programs such as DOE-2 to subsequently perform the calibration with performance data. Hourly subaggregated monitored energy data (most compati-ble with the time step adopted by most building energy simulation programs) allow the development of the most accurate calibrated model, but analysts are usually forced to work with less data. Tun-ing can be done with monthly data or data that span only a few weeks or months over the year, but the resulting model is very likely to be increasingly less accurate with decrease in performance data.
The main reservations with the widespread use of calibrated simulation are that it is labor-intensive, requires a high level of user skill and knowledge in both simulation and practical build-ing operation, is time-consuming, and is often dependent on the person doing the calibration. Several practical difficulties prevent achieving a calibrated simulation or a simulation that nearly reflects the actual building performance, including (1) the mea-surement and adaptation of weather data for use by the simula-tion programs (e.g., converting global horizontal solar into beam and diffuse solar radiation), (2) the choice of methods used to calibrate the model, and (3) the choice of methods used to mea-sure the required input parameters for the simulation (i.e., the weight of the building, infiltration coefficients, and shading coef-ficients). Truly “calibrated” models have been achieved in only a few applications because they require a very large number of input parameters, a high degree of expertise, and enormous amounts of computing time, patience, and financial resources.
Bronson et al. (1992), Haberl and Bou-Saada (1998), Kaplan et al. (1990), Corson (1992), Bou-Saada and Haberl (1995a, 1995b), Manke et al. (1996), and Norford et al. (1994) provide examples of different methods used to calibrate simulation mod-els.
In recent years, Katipamula and Claridge (1993) and Liu and Claridge (1998) have suggested that models simpler than those used in the detailed simulation programs such as DOE-2 could also serve the intended purpose while allowing model calibration to be done much faster. Typically, the building is divided into two zones: an exterior or perimeter zone and an interior or a core zone. The core zone is assumed to be insulated from the envelope heat losses/gains, while the solar heat gains, infiltration heat loss/gain, and conduction gains/losses from the roof are taken to appear as loads on the external zone only. Given the internal load schedule, the building description, the type of HVAC system, and the climatic parameters, the HVAC system loads can be esti-mated for each hour of the day and for as many days of the year as needed by the simplified systems model. Since there are fewer parameters to vary, the calibration process is much faster. There-fore, these models have a significant advantage over general-pur-pose models in buildings where the HVAC systems can be adequately modeled. These studies, based on the ASHRAE Sim-plified Energy Analysis Procedure (Knebel 1983), illustrate the applicability of this method both to baseline model development for M&V purposes and as a diagnostic tool for identifying poten-tial operational problems and for estimating potential savings from optimized operating parameters.
Gray-Box Approach With this approach, a physical model is first formulated to represent the structure or physical configuration of the building or HVAC&R equipment or system, and then important parame-ters representative of certain key and aggregated physical param-eters and characteristics are identified by a statistical analysis Energy Estimating and Modeling Methods 31.25 (Rabl and Riahle 1992). This approach requires a high level of user expertise both in setting up the appropriate modeling equa-tions and in the estimation of these parameters. Often an intru-sive experimental protocol is necessary for proper parameter estimation that also requires a certain amount of user expertise.
This approach has great potential, especially for fault detection and diagnosis (FDD) and on-line control, but its applicability to whole-building energy use is limited. Examples of parameter estimation studies as applied to building energy use are Son-deregger (1977), Hammersten (1984), Subbarao (1988), Rabl (1988), Reddy (1989), Andersen and Brandemuehl (1992), Braun (1990), Reddy et al. (1999), Gordon and Ng (1995), and Guyon and Palomo (1999).
TYPES OF INVERSE MODELS We distinguish between two types of models: steady-state and dynamic. Steady-state models are those that do not consider such effects such as thermal mass or capacitance that cause short-term temperature transients. Generally these models are appro-priate for monthly, weekly, or daily data and are often used for baseline model development. Dynamic models capture effects such as building warm-up or cool-down periods and peak loads and are appropriate for building load control, FDD, and equip-ment control. A simple criterion to determine whether a model is steady-state or dynamic is to look for the presence of time-lagged variables, either in the response or regressor variables. Steady-state models do not contain time-lagged variables.
Steady-State Models There are several types of steady-state models used for both building and equipment energy use. They are single-variate, mul-tivariate, polynomial, and physical models.
Single-Variate Models. Single-variate models (i.e., models with one regressor variable only) are perhaps the most widely used. They formulate energy use in a building as a function of one driving force that impacts building energy use. An important aspect in identifying statistical models of baseline energy use is the choice of the functional form and the independent (or regres-sor) variables. Extensive studies (Fels 1986, Kissock et al. 1993, Katipamula et al. 1994, Reddy et al. 1997) have clearly indicated that the outdoor dry-bulb temperature is the most important regressor variable, especially at monthly time scales but also at daily time scales.
The simplest steady-state inverse model is one developed by regressing monthly utility consumption data against average bill-ing-period temperatures. The model must identify the balance point temperatures (or change points) at which energy use switches from weather-dependent to weather-independent behav-ior. In its simplest form, the 18.3°C degree-day model is a change-point model that has a fixed change point at 18.3°C.
Other examples include three- and five-parameter Princeton Scorekeeping Methods (PRISM) based on the variable-base degree-day concept (Fels 1986). An allied modeling approach for commercial buildings is the four-parameter (4-P) model devel-oped by Ruch and Claridge (1991), which is based on the monthly mean temperature (and not degree-days). Table 9 shows the appropriate model functional forms. The three parameters are a weather-independent base-level use, a change point, and a tem-perature-dependent energy use, characterized as a slope of a line that is determined by regression. The four parameters include a change point, a slope above the change point, a slope below the change point, and the energy use associated with the change point. An inverse bin method has also been proposed to handle more than four change points (Thamilseran and Haberl 1995).
Figure 18 shows several types of steady-state, single-variate inverse models. Figure 18A shows a simple one-parameter, or constant, model, and Table 9 gives the equivalent notation for calculating the constant energy use using this model. Figure 18B shows a steady-state two-parameter (2-P) model where b0 is the y-axis intercept and b1 is the slope of the regression line for posi-tive values of x, where x represents the ambient air temperature.
The 2-P model represents cases when either heating or cooling is always required.
Figure 18C shows a three-parameter, change-point model.
This model is typical of natural gas energy use in a single-family residence that uses gas for space heating and domestic water heating. In the notation of Table 9 for the three-parameter model, b0 represents the baseline energy use and b1 is the slope of the regression line for values of ambient temperature less than the change point b2. In this type of notation, the superscript plus sign indicates that only positive values of the parenthetical expression are considered. Figure 18D shows a three-parameter model for cooling energy use, and Table 9 provides the appropriate analytic expression.
Figure 18E and Figure 18F illustrate four-parameter models for heating and cooling, respectively. The appropriate expressions for calculating the heating and cooling energy consumption using a four-parameter model are found in Table 9: b0 represents the base-line energy exactly at the change point b3, and b1 and b2 are the lower and upper region regression slopes for ambient air tempera-ture below and above the change point b3. Figure 18G illustrates a 5-P model (Fels 1986). Such a model is useful for modeling build-ings that are electrically heated and cooled. The 5-P model has two change points and a base level consumption value.
The advantage of these steady-state inverse models is that their use can be easily automated and applied to large numbers of build-ings where monthly utility billing data and average daily tempera-tures for the billing period are available. Steady-state single-variate inverse models have also been applied with success to daily data (Kissock et al. 1998). In such a case, the variable-base degree-day method and the monthly mean temperature models described earlier for utility billing data analysis become identical in their functional form. Single-variate models can also be applied to daily data to compensate for differences such as weekday and weekend use by separating the data accordingly and identifying models for each period separately.
Table 8 Single-Variate Models Applied to Utility Billing Data Model Type Independent Variable(s) Form Examples One-parameter or constant (1-P) None E = b0 Non-weather-sensitive demand Two-parameter (2-P) Temperature E = b0 +b1(T ) Three-parameter (3-P) Degree-days/ Temperature E = b0 + b1(DDBT) E = b0 + b1(b2 −T )+ E = b0 + b1(T −b2)+ Seasonal weather-sensitive use (fuel in winter, electricity in summer for cooling) Four-parameter change point (4-P) Temperature E = b0 + b1(b3 −T )+ −b2(T −b3)+ E = b0 – b1(b3 −T )+ + b2(T −b3)+ Energy use in commercial buildings Five-parameter (5-P) Degree-days/ Monthly mean temperature E = b0 −b1(DDTH) + b2(DDTC) E = b0 + b1(b3 −T )+ + b2(T – b4)+ Heating and cooling supplied by same meter Note: DD denotes degree days and T is the monthly mean daily outdoor dry-bulb temperature.
31.26 2001 ASHRAE Fundamentals Handbook (SI) Fig. 18 Steady-State, Single-Variate Models Appropriate for Modeling Energy Use in Residential and Commercial Buildings Energy Estimating and Modeling Methods 31.27 Disadvantages of steady-state single-variate inverse models include insensitivity to dynamic effects (e.g., thermal mass), insen-sitivity to variables other than temperature (e.g., humidity and solar gain), and inappropriateness for some buildings (e.g., buildings with strong on/off schedule-dependent loads or buildings with multiple change points). Moreover, the use of a single-variable, 3-P model such as the PRISM model (Fels 1986) has a physical basis only when energy use above a base level is linearly proportional to degree-days.
This is a good approximation in the case of heating energy use in res-idential buildings where the heating load never exceeds the capacity of the heating system. However, commercial buildings, in general, have higher internal heat generation with simultaneous heating and cooling energy use and are strongly influenced by HVAC system type and control strategy. This makes energy use in commercial buildings less strongly influenced by outdoor air temperature alone.
Therefore, it is not surprising that blind use of single-variate models has had mixed success at modeling energy use in commercial build-ings (MacDonald and Wasserman 1989).
Change-point regression models work best with heating data from buildings with systems that have little or no part-load non-linearities (i.e., systems that become less efficient as they begin to cycle on-off with part loads). In general, change-point regression models do not predict cooling loads as well because outdoor humid-ity has a large influence on latent loads on the cooling coil. Other factors that decrease the accuracy of change-point models include solar effects, thermal lags, and on-off HVAC schedules. Four-parameter models exhibit a better statistical fit than three-parameter models in buildings with continuous, year-round cooling or heating (e.g., grocery stores and office buildings with high internal loads).
However, every model should be checked to ensure that the regres-sion is not falsely indicating an unreasonable relationship.
A major advantage of using a steady-state inverse model to eval-uate the effectiveness of energy conservation retrofits lies in its abil-ity to factor out year-to-year weather variations by using a normalized annual consumption (NAC) (Fels 1986). Basically, the annual energy conservation savings can be calculated by comparing the difference obtained by multiplying the preretrofit and postretro-fit parameters by the weather conditions for the average year. Typ-ically, 10 to 20 years of average daily weather data from a nearby weather service site are used to calculate 365 days of average weather conditions, which are then used to calculate the average preretrofit and postretrofit conditions.
Utilities and government agencies have found it advantageous to prescreen many buildings against test regression models. Such inverse models can be used to develop comparative figures of merit for buildings in a similar standard industrial code (SIC) classifica-tion. In such applications, a minimum goodness of fit is usually established that determines whether the monthly utility billing data are well fitted by the one-, two-, three-, four- or five-parameter model being tested. Comparative figures of merit can then be deter-mined by dividing the parameters by the conditioned floor area to yield average daily energy use per unit area of conditioned space.
For example, an area-normalized comparison of base-level param-eters across residential buildings would be used to analyze weather-independent energy use. Such information can be used by energy auditors to focus their efforts on those systems needing assistance (Haberl and Komor 1990a, 1990b).
Multivariate Models. Two types of steady-state, multivariate models have been reported: 1. Standard multiple-linear or change-point regression models, where the set of data observations is treated without retaining the time series nature of the data (Katipamula et al. 1998).
2. Fourier series models that retain the time-series nature of the building energy use data and capture the diurnal and seasonal cycles according to which buildings are operated (Seem and Braun 1991; Dhar 1995; Dhar et al. 1998, 1999a, 1999b).
These models are a logical extension to single-variate models, provided that the choice of the variables to be included and their functional forms are based on the engineering principles on which HVAC systems and other systems in commercial buildings oper-ate. The goal of modeling energy use by the multivariate approach is to characterize building energy use with a few readily available and reliable input variables. These input variables should be selected with care. The model should contain variables not affected by the retrofit and likely to change (for example, climatic vari-ables) from preretrofit to postretrofit periods. Other less obvious variables, such as changes in operating hours, in base load, and in occupancy levels, should be included in the model if these are not energy conservation measures (ECMs) but variables that may change during the postretrofit period.
Environmental variables that meet the above criteria for mod-eling heating and cooling energy use include outdoor air dry-bulb temperature, solar radiation, and outdoor specific humidity. Some of these parameters are difficult to estimate or measure in an actual building and hence are not good candidates for regressor vari-ables. Further, some of the variables vary little. Although their effect on energy use may be important, an inverse model will implicitly lump their effect into the parameter that represents con-stant load. In commercial buildings, internally generated loads, such as the heat given off by people, lights, and electrical equip-ment, also impact heating and cooling energy use. Such internal loads are difficult to measure in their entirety given the ambiguous nature of occupant loads and latent loads. However, monitored electricity used by internal lights and equipment is a good surro-gate for total internal sensible loads (Reddy et al. 1999). For exam-ple, when the building is fully occupied, it is also likely to be experiencing high internal electric loads, and vice versa.
The effect of environmental variables is important for such buildings as offices but may be less so for mixed-use buildings (e.g., hotels and hospitals) and buildings such as retail buildings, schools, and assembly buildings. Differences in HVAC system behavior dur-ing occupied and unoccupied periods can be modeled by a dummy or indicator variable (Draper and Smith 1981). For some office buildings, there seems to be little need to include such a dummy variable, but its inclusion in the general functional form will provide added flexibility.
Several standard statistical tests exist for evaluating the good-ness-of-fit of the model and the degree of influence that each of the independent variables exerts on the response variable (Draper and Smith 1981, Neter et al. 1989). Although energy use is in fact dependent on several variables, there are strong practical incentives for identifying the simplest model that results in acceptable accu-racy. Multivariate models require more metering and are unusable if even one of the variables becomes unavailable. In addition, some of the regressor variables may be linearly correlated. This condition, called multicollinearity, can result in large uncertainty in the esti-mates of the regression coefficients (i.e., unintended error) and can also lead to poorer model prediction accuracy compared to a model where the regressors are not linearly correlated.
Several authors recommend using principal component anal-ysis (PCA) to overcome multicollinearity effects. PCA was one of the strongest analysis methods in the ASHRAE Predictor Shoot-out I and II contests (Kreider and Haberl 1994, Haberl and Thamilseran 1996). Analysis of multiyear monitored daily energy use in a grocery store found a clear superiority of PCA over mul-tivariate regression models (Ruch et al. 1993), but this conclusion is unproven for commercial building energy use in general. A more general evaluation by Reddy and Claridge (1994) of both analysis techniques using synthetic data from four different geo-graphic locations in the U.S. found that injudicious use of PCA may exacerbate rather than overcome problems associated with multicollinearity. Draper and Smith (1981) also caution against indiscriminate use of PCA.
31.28 2001 ASHRAE Fundamentals Handbook (SI) The functional basis of air-side heating and cooling use in vari-ous HVAC system types has been addressed by Reddy et al. (1995) and subsequently applied to monitored data in commercial build-ings (Katipamula et al. 1994, 1998). Because the quadratic and cross-product terms of the engineering equations are not usually picked up by the multivariate models, one is often left with models for energy use that are strictly linear.
In addition to To, internal electric equipment and lighting load Eint, solar loads qsol, and latent effects via the outdoor dew-point tem-perature Tdp are candidate regressor variables. In commercial build-ings, a major portion of the latent load is due to fresh air ventilation.
However, this load appears only when the outdoor air dew-point temperature exceeds the cooling coil temperature. Hence, the term (Tdp −Ts)+ (where the + sign indicates that the term is to be set to zero if negative, and Ts is the mean surface temperature of the cooling coil, typically about 11 to 13°C) is a more realistic descriptor of the latent loads than is Tdp alone. The use of (Tdp −Ts)+ as a regressor in the model is a simplification that seems to yield good accuracy.
Therefore, a multivariate linear regression model with an engi-neering basis has the following structure: (54) Based on the above discussion, . Introducing indicator vari-able terminology (Draper and Smith 1981), Equation (54) becomes identical to (55) where the indicator variable I is introduced to handle the change in slope of the energy use due to To. The variable I is set equal to 1 for To values to the right of the change point (i.e., for high To range) and set equal to 0 for low To values. As with the single-variate seg-mented models (i.e., 3-P and 4-P models), a search method is used in order to determine the change point that minimizes the total sum of squares of residuals (Fels 1986, Kissock et al. 1993).
Another finding from the Katipamula et al. (1994) study was that the model given by Equation (55), appropriate for VAV systems, could be simplified for constant volume HVAC systems: (56) Note that instead of using (Tdp−Ts)+, one could equally use the absolute humidity potential (W0 −Ws)+, where W0 is the outdoor absolute humidity, and Ws, the absolute humidity level at the dew point of the cooling coil, is typically about 0.009 kg/kg. A final aspect to be kept in mind is that the term should be omitted from the regressor variable set when regressing heating energy use because there are no latent loads on a heating coil.
The above multivariate models have been found to be very accu-rate for daily time scales and slightly less so for hourly time scales.
This is because changes in the way the building is operated during the day and the night lead to different relative effects of the various regressors on energy use, which cannot be accurately modeled by one single hourly model. Breaking up the energy use data into hourly bins corresponding to each hour of the day and then identi-fying 24 individual hourly models leads to appreciably greater accu-racy (Katipamula et al. 1994).
Polynomial Models. Historically, polynomial models have been widely used as pure statistical models to model the behavior of such equipment as pumps, fans, and chillers (Stoecker and Jones 1982).
The theoretical aspects of calculating pump performance are well understood and documented. Pump capacity and efficiency are cal-culated from measurements of pump pressure, flow rate, and pump electrical power input. Phelan et al. (1996) have studied the predic-tive ability of linear and quadratic models for electricity consumed by pumps and water mass flow rate and concluded that quadratic models are superior to linear models. For fans, Phelan et al. (1996) have studied the predictive ability of linear and quadratic polyno-mial single-variate models of fan electricity consumption as a func-tion of the supply air mass flow rate and concluded that, although quadratic models are superior in terms of predicting energy use, the linear model seems to be the better overall predictor of both energy and demand (i.e., maximum monthly power consumed by the fan).
This is a noteworthy conclusion given that a third-order polynomial is warranted analytically as well as from monitored field data pre-sented by previous authors (e.g., Englander and Norford 1992, Lorenzetti and Norford 1993).
Polynomial models have been used to correlate chiller (or evap-orator) thermal cooling capacity or load Qevap and the electrical power consumed by the chiller (or compressor) Ecomp with the rel-evant number of influential physical parameters. For example, based on the functional form of the DOE-2 building simulation soft-ware (York and Cappiello 1982), models for part-load performance of energy equipment and plant, Ecomp can be modeled as the follow-ing triquadratic polynomial: (57) In this model, there are 11 model parameters to identify. How-ever, since all of them are unlikely to be statistically significant, a step-wise regression to the sample data set yields the optimal set of parameters to retain in a given model. Other authors, such as Braun (1992), have used slightly different polynomial forms.
Physical Models. In contrast to polynomial models, which have no physical basis (merely a convenient statistical one), physical models are based on fundamental thermodynamic or heat transfer considerations. These types of models are usually associated with the parameter estimation approach. Often such models are preferred because they generally have fewer parameters. Furthermore, their mathematical formulation can be traced to actual physical princi-ples that govern the performance of the building or equipment.
Hence the model coefficients tend to be more robust, leading to sounder model predictions. Only a few studies have used steady-state physical models for parameter estimation relating to commer-cial building energy use (for example, Reddy et al. 1999). Unlike in single-family residences, it is difficult to perform elaborately planned experiments in large buildings and obtain representative values of indoor fluctuations.
A physical model of a chiller has been proposed by Gordon and Ng (1994, 1995) and Gordon et al. (1995). It is a simple, analytical, universal model for chiller performance based on thermodynamic considerations and linearization of heat losses. The model predicts the dependence of chiller COP [defined as the ratio of chiller (or evaporator) thermal cooling capacity Qevap divided by the electrical power consumed by the chiller (or compressor) Ecomp] with certain key (and easily measurable) parameters such as the fluid (water or refrigerant) return temperature from the condenser , the fluid temperature leaving the evaporator (or the chilled water supply tem-perature to the building) , and the thermal cooling capacity of the evaporator. The complete Gordon-Ng model is a three-parame-ter model that takes the following form: (58) Qbldg β0 β1 To β3 – ( )– β2 To β3 – ( )+ β4 Tdp β6 – ( )– + + + = β + 5 Tdp β6 – ( )+ β7qsol β8Eint + + β4 0 = Qbldg a bTo cI dITo eTdp + f qsol gEint + + + + + + = Qbldg a bTo eTdp + f qsol gEint + + + + = Tdp + Ecomp a bQevap cTcond in dTevap out eQevap 2 + + + + = fTcond in 2 gTevap out 2 hQevapTcond in iTevap out Qevap + + + + jTcond in Tevap out kQevapTcond in Tevap out + + Tcond in Tevap out 1 COP -----------1 Tcond in Tevap out -------------– + Qevap A0 – A1Tcond in A2 Tcond in Tevap out -------------– + = Energy Estimating and Modeling Methods 31.29 The three parameters are identified by multiple linear regression.
Results of applying such models to field monitored data are fully described by the Gordon et al. papers as well as by Phelan et al.
(1996) and Haberl et al. (1997).
Dynamic Models In general, steady-state inverse models are used with monthly and daily data containing one or more independent variables.
Dynamic inverse models are usually used with hourly or subhourly data in cases where the thermal mass of a building is sufficiently sig-nificant to delay the heat gains or losses. Dynamic models tradition-ally required the solution of a set of differential equations. The disadvantages of dynamic inverse models include their complexity and the need for more detailed measurements to tune the model.
Unlike steady-state inverse models, dynamic inverse models usu-ally require a high degree of user interaction and knowledge of the building or system being modeled.
Several residential energy studies using dynamic inverse models based on parameter estimation approaches have been reported, most of them involving intrusive data gathering. Rabl (1988) has classi-fied the various types of dynamic inverse models used for whole-building energy use and drawn attention to the common underlying features of these models. There are essentially four different types of model formulations: thermal network models, time series models, differential equation models, and modal models. They all qualify as parameter-estimation approaches. Table 1 lists several pertinent studies in each category. A few studies (Hammersten 1984, Rabl 1988, Reddy 1989) have evaluated these different approaches with the same data set. A number of papers have reported results of applying such different techniques such as ther-mal network models and ARMA models to residential and commer-cial building energy use (see Table 1). Examples of dynamic inverse models for commercial building are found in Rabl (1988), Andersen and Brandemuehl (1992), and Braun (1990).
Dynamic inverse models based on pure statistical approaches have also been reported. Two examples are machine learning (Miller and Seem 1991) and artificial neural networks (Miller and Seem 1991, Kreider and Wang 1991, Kreider and Haberl 1994).
Neural networks are considered to be intuitive because they learn by example rather than by following programmed rules. The ability to “learn” is one of the key aspects of neural networks. A neural net-work consists of one input layer (which can contain one or more inputs), one or more hidden layers, and an output or target layer.
One challenge of this technology is to construct a net with sufficient complexity to learn accurately without imposing excessive compu-tational time.
The weights of a net are initiated with small random numbers.
Then, the weights are adjusted iteratively or “trained” so that the application of a set of inputs produces the desired set of outputs.
Usually a network is trained with a training data set that consists of many input-output pairs. Artificial neural networks have been trained by a wide variety of methods (McClelland and Rumelhart 1988, Wasserman 1989). One such training method is called back propagation.
Neural networks have been useful in predicting energy use in commercial buildings for such reasons as • Prediction of what a properly operating building should be doing compared to actual operation. If there is a difference, it can be used in an expert system to produce early diagnoses of building operation problems.
• Prediction of what a building, prior to an energy retrofit, would have consumed under present conditions. When compared to the measured consumption of the retrofitted building, the difference represents a good estimate of the energy savings due to the retro-fit. This represents one of the few ways that actual energy savings can be determined after the preretrofit building configuration has ceased to exist.
EXAMPLES OF INVERSE METHODS Modeling of Utility Bill Data The following example (taken from Sonderegger 1998) illus-trates a utility bill analysis. Assume that values of the utility bills over an entire year have been measured. To obtain the equation coefficients through regression, the utility bills must be normalized by the length of the time interval between utility bills. This is equiv-alent to expressing all utility bills, degree-days, and other indepen-dent variables by their daily averages.
An appropriate software program is used in which one assumes values for heating and cooling balance points; from these, the cor-responding heating and cooling degree-days for each utility bill period are determined. Repeated regression is done till the regres-sion equation represents the best fit to the meter data. The model coefficients are then assumed to be tuned. Some computer programs allow direct determination of these optimal model parameters with-out the user’s manual tuning of the parameters.
A widely used statistic to gage the goodness-of-fit of the model is the coefficient of determination R2. A value of R2 = 1 indicates a perfect correlation between actual data and the regression equa-tion; a value of R2 = 0 indicates no correlation. For purposes of tun-ing for a performance contract, as a rule of thumb the value of R2 should never be less than 0.75.
When more than one independent variable is included in the regression, the value of R2 is no longer sufficient to determine the goodness-of-fit. The standard error of the estimate of the coeffi-cients becomes the more important determinant. The smaller the standard error compared to the coefficient’s magnitude, the more reliable the coefficient estimate. To facilitate the significance of individual coefficients, the so-called t-statistics, or simply t-values, are used. These are simply the ratio of the coefficient estimate divided by the standard error of the estimate.
The coefficient of each variable included in the regression has a t-statistic. For a coefficient to be statistically meaningful, the abso-lute value of its t-statistic must be at least 2.0. Another way of stat-ing this is that under no circumstances should a variable be included in a regression if the standard error of its coefficient estimate is greater than half the magnitude of the coefficient. The latter is true even when including a variable that increases the R2. Generally speaking, including more variables in a regression results in a higher R2, but the significance of most individual coefficients will likely decrease.
Figure 19 illustrates how well a regression fit captures measured baseline energy use in a hospital building. Cooling degree-days are found to be a significant variable, with the best fit for a base tem-perature of 12.2°C.
There are good reasons why individual utility bills may be unsuitable to develop a baseline and should be excluded from the regression. For example, a bill may be atypically high because of a one-time equipment malfunction that was subsequently repaired.
However, it is often tempting to look for reasons to exclude bills that fall far from “the line” and not question those that are close to it. For example, bills for periods containing vacations or production shut-downs may look anomalously low, but excluding them from the regression would result in a chronic overestimate of the future base-line during the same period.
Neural Network Models Figure 20 shows results for a single neural network typical of several hundred networks constructed for an academic engineer-ing center located in central Texas. The cooling load is created by solar gains, internal gains, outdoor air sensible heat, and outdoor 31.30 2001 ASHRAE Fundamentals Handbook (SI) air humidity loads. The neural network is used to predict the pre-retrofit energy consumption for comparison with measured con-sumption of the retrofitted building. Six months of preretrofit data were available with which to train a network. The solid lines show the known building consumption data while the dashed lines show the neural network predictions. This figure shows that a neural network trained for one period (September 1989) can predict energy consumption well into the future (here, January 1990).
The network used for this prediction had two hidden layers. The input layer contained eight neurons that receive eight different types of input data as listed below. The output layer consisted of one neu-ron that gave the output datum (chilled water consumption). Each training fact (i.e., training data set), therefore, contained eight input data (independent variables) and one pattern datum (dependent variable). The eight hourly input data used in each hour’s data vec-tor were selected on physical bases (Kreider and Rabl 1994) and were as follows: • Hour number (0 to 2300) • Ambient dry-bulb temperature • Horizontal insolation • Humidity ratio • Wind speed • Weekday/weekend binary flag (0, 1) • Past hour’s chilled water consumption • Second past hour’s chilled water consumption These measured independent variables were able to predict the chilled water use to an RMS error of less than 4% (JCEM 1992).
The choice of an optimal network’s configuration for a given problem remains an art. The number of hidden neurons and layers must be sufficient to meet the requirement of the given applica-tion. However, if too many neurons and layers are used, the net-work tends to memorize data rather than learning, that is, finding the underlying patterns in the data. Further, choosing an exces-sively large number of hidden layers significantly increases the required training time for certain learning algorithms. Anstett and Kreider (1993), Kreider and Wang (1991), Wang and Kreider (1992), and Krarti et al. (1998) report additional case studies for commercial buildings.
Closing Remarks Steady-state and dynamic inverse models can be used with energy management and control systems to predict energy use (Kreider and Haberl 1994). Hourly or daily comparisons of mea-sured energy use against predicted energy use can be used to determine whether systems are being left on unnecessarily or are in need of maintenance. Combinations of predicted energy use and a knowledge-based system can indicate above-normal energy use and diagnose the possible cause of the malfunction if suffi-cient historical information has been previously gathered (Haberl and Claridge 1987). Hourly systems that use artificial neural net-works have also been constructed (Kreider and Wang 1991).
Table 10 presents a decision diagram for selecting a forward or inverse model where use of the model, degree of difficulty in understanding and applying the model, time scale for the data used by the model, calculation time, and input variables used by the models are the criteria used to choose a particular model.
Fig. 19 Variable-Base Degree-Day Model Identification Using Electricity Utility Bills at a Hospital (Sonderegger 1998) Fig. 20 Neural Network Prediction of Whole Building, Hourly Chilled Water Consumption for a Commercial Building Energy Estimating and Modeling Methods 31.31 REFERENCES Alamdari, F. and G.P. Hammond. 1982. Time-dependent convective heat transfer in warm-air heated rooms. Energy Conservation in the Built Environment: Proceedings CIB W67 Third International Symposium.
pp. 209-20. Dublin, Ireland.
Alamdari, F. and G.P. Hammond. 1983. Improved data correlations for buoy-ancy-driven convection in rooms. Building Services Engineering Research and Technology 4(3):106-12.
Alereza, T. and T. Kusuda. 1982. Development of equipment seasonal per-formance models for simplified energy analysis methods. ASHRAE Transactions 88(2):249-62.
Altmayer, E.F., A.J. Gadgil, F.S. Bauman, and R.C. Kammerud. 1983. Cor-relations for convective heat transfer from room surfaces. ASHRAE Transactions 89(2A):61-77.
Andersen, I. and M.J. Brandemuehl. 1992. Heat storage in building thermal mass: A parametric study. ASHRAE Transactions 98(1):910-18.
Anstett, M. and J.F. Kreider. 1993. Application of artificial neural networks to commercial building energy use prediction. ASHRAE Transactions 99(1):505-17.
ASHRAE. 1995. Bin and degree hour weather data for simplified energy calculations.
ASHRAE. 1999. Energy standard for buildings except low-rise residential buildings. ANSI/ASHRAE/IESNA Standard 90.1-1999.
Ayres, M.J. and E. Stamper. 1995. Historical development of building energy calculations. ASHRAE Transactions 101(1):47-55.
Bacot, P., A. Neveu, and J. Sicard. 1984. Analyse modale des phenomenes thermiques en regime variable dans le batiment. Revue Generale de Thermique 267:189.
Balcomb, J.D., R.W. Jones, R.D. McFarland, and W.O. Wray. 1982. Expand-ing the SLR method. Passive Solar Journal 1(2).
Bauman, F., A. Gadgil, R. Kammerud, E. Altmayer, and M. Nansteel. 1983.
Convective heat transfer in buildings: Recent research results. ASHRAE Transactions 89(1A):215-32.
Beck, J.V. and K.J. Arnold, 1977. Parametric estimation in engineering and science. John Wiley and Sons, New York.
Bohn, M.S., A.T. Kirkpatrick, and D.A. Olson. 1984. Experimental study of three-dimensional natural convection high-Rayleigh number. Journal of Heat Transfer 106:339-45.
Bou-Saada, T. and J. Haberl. 1995a. A weather-day typing procedure for dis-aggregating hourly end-use loads in an electrically heated and cooled building from whole-building hourly data. Proceedings 30th IECEC, pp.
349-56.
Bou-Saada, T. and J. Haberl. 1995b. An improved procedure for developing calibrated hourly simulation models. Proceedings Building Simulation ’95. IBPSA. Madison, Wisc.
Bourdouxhe, J.P., M. Grodent, J. Lebrun, and C. Saavedra. 1994a. A toolkit for primary HVAC system energy calculation—Part 1: Boiler model.
ASHRAE Transactions 100(2):759-73.
Bourdouxhe, J.P., M. Grodent, J. Lebrun, C. Saavedra, and K. Silva. 1994b.
A toolkit for primary HVAC system energy calculation—Part 2: Recip-rocating chiller models. ASHRAE Transactions 100(2):774-86.
Bourdouxhe, J.P., M. Grodent, and C. Silva. 1994c. Cooling tower model developed in a toolkit for primary HVAC system energy calculation— Part 1: Model description and validation using catalog data. Proceedings Fourth International Conference on System Simulation in Buildings.
Brandemuehl, M.J. 1993. HVAC 2 toolkit: Algorithms and subroutines for secondary HVAC system energy calculations. ASHRAE.
Brandemuehl, M.J. and S. Gabel. 1994. Development of a toolkit for second-ary HVAC system energy calculations. ASHRAE Transactions 100(1):21-32.
Brandemuehl, M.J. and J.D. Bradford. 1999. Optimal supervisory control of cooling plants without storage. Final Report RP-823. ASHRAE.
Braun, J.E. 1988. Methodologies for the design and control of chilled water systems. Ph.D. thesis, University of Wisconsin-Madison.
Braun, J.E. 1990. Reducing energy costs and peak electrical demand through optimal control of building thermal mass. ASHRAE Transactions 96(2): 876-88.
Braun, J.E. 1992. A comparison of chiller-priority, storage-priority, and optimal control of an ice-storage system. ASHRAE Transactions 98(1): 893-902.
Bronson, D., S. Hinchey, J. Haberl, and D. O’Neal. 1992. A procedure for calibrating the DOE-2 simulation program to non-weather dependent loads. ASHRAE Transactions 98(1):636-52.
Buhl, W.F., A.E. Erdem, J.M. Nataf, F.C. Winkelmann, M.A. Moshier, and E.F. Sowell. 1990. The US EKS: Advances in the SPANK-based energy kernel system. Proceedings Third International Conference on System Simulation in Buildings, pp. 107-50.
Buhl, W.F., A.E. Erdem, F.C. Winkelmann, and E.F. Sowell. 1993. Recent improvements in SPARK: Strong component decomposition, multi-valued objects and graphical interface. Proceedings of Building Simu-lation ’93. IBPSA, pp. 283-390.
Carroll, J.A. 1980. An “MRT method” of computing radiant energy exchange in rooms. Systems Simulation and Economic Analysis, pp. 343-48. San Diego.
Chandra, S. and A.A. Kerestecioglu. 1984. Heat transfer in naturally venti-lated rooms: Data from full-scale measurements. ASHRAE Transactions 90(1B):211-24.
Chi, J. and G.E. Kelly. 1978. A method for estimating the seasonal perfor-mance of residential gas and oil-fired heating systems. ASHRAE Trans-actions 84(1):405.
Cinquemani, V., J.R. Owenby, and R.G. Baldwin. 1978. Input data for solar systems. U.S. Department of Energy Report No. E(49-26)1041.
Clark, D.R. 1985. HVACSIM+ building systems and equipment simulation program: Reference manual. NBSIR 84-2996, U.S. Department of Com-merce, Washington, D.C.
Clarke, J.A. 1985. Energy simulation in building design. Adam Hilger Ltd., Boston.
Claridge, D. 1988a. Design methods for earth-contact heat transfer. Progress in Solar Energy, K. Boer, ed. American Solar Energy Society, Boulder, CO.
Claridge, D. 1998b. A perspective on methods for analysis of measured energy data from commercial buildings, ASME Journal of Solar Energy Engineering 120:150.
Table 9 Capabilities of Different Forward and Inverse Modeling Methods Methods Usagea Difficulty Time Scaleb Calc. Time Variablesc Accuracy Simple linear regression ES Simple D, M Very fast T Low Multiple linear regression D, ES Simple D, M Fast T, H, S, W, t Medium ASHRAE bin method and inverse bin method ES Moderate H Fast T Medium Change point models D, ES Simple H, D, M Fast T Medium ASHRAE TC 4.7 modified bin method ES, DE Moderate H Medium T, S, tm Medium Artificial Neural Networks D, ES, C Complex S, H Fast T, H, S, W, t, tm High Thermal network D, ES, C Complex S, H Fast T, S, tm High Fourier series analysis D, ES, C Moderate S, H Medium T, H, S, W, t, tm High ARMA model D, ES, C Moderate S, H Medium T, H, S, W, t, tm High Modal analysis D, ES, C Complex S, H Medium T, H, S, W, t, tm High Differential equation D, ES, C Complex S, H Fast T, H, S, W, t, tm High Computer simulation (component-based) D, ES, C, DE Very complex S, H Slow T, H, S, W, t, tm Medium Computer simulation (fixed schematic) D, ES, DE Very complex H Slow T, H, S, W, t, tm Medium Computer emulation D, C Very complex S, H Very slow T, H, S, W, t, tm High Notes: aUsage shown includes diagnostics (D), energy savings calculations (ES), design (DE), and control (C).
bTime scales shown are hourly (H), daily (D), monthly (M), and subhourly (S).
cVariables include temperature (T), humidity (H), solar (S), wind (W), time (t), and thermal mass (tm).
31.32 2001 ASHRAE Fundamentals Handbook (SI) Claridge, D.E., M. Krarti, and M. Bida. 1987. A validation study of variable-base degree-day cooling calculations. ASHRAE Transactions 93(2):90-104.
Cole, R.J. 1976. The longwave radiation incident upon the external surface of buildings. The Building Services Engineer 44:195-206.
Cooper, K.W. and D.R. Tree. 1973. A re-evaluation of the average convec-tion coefficient for flow past a wall. ASHRAE Transactions 79:48-51.
Corson, G.C. 1992. Input-output sensitivity of building energy simulations.
ASHRAE Transactions 98(1):618.
Cumali, Z., A.O. Sezgen, et al. 1979. Extensions of methods used in analyz-ing building thermal loads. Proceedings Thermal Performance of the Exterior Envelopes of Buildings, pp. 411-20.
Davies, M.G. 1988. Design models to handle radiative and convective exchange in a room. ASHRAE Transactions 94(2):173-95.
Dhar, A. 1995. Development of Fourier series and artificial neural network approaches to model hourly energy use in commercial buildings. Ph.D.
thesis, ME Dept., Texas A&M University.
Dhar, A., T.A. Reddy, and D.E. Claridge. 1998. Modeling hourly energy use in commercial buildings with Fourier series functional forms. Journal of Solar Energy Engineering 120:217.
Dhar, A., T.A. Reddy, and D.E. Claridge. 1999a. A Fourier series model to predict hourly heating and cooling energy use in commercial buildings with outdoor temperature as the only weather variable. Journal of Solar Energy Engineering 121:47-53.
Dhar, A., T.A. Reddy, and D.E. Claridge. 1999b. Generalization of the Fou-rier series approach to model hourly energy use in commercial buildings.
Journal of Solar Energy Engineering 121:54-62.
Draper, N. and H. Smith. 1981. Applied regression analysis, 2nd ed. John Wiley and Sons, New York.
Elmahdy, A.H. and G.P. Mitalas. 1977. A simple model for cooling and dehumidifying coils for use in calculating energy requirements for build-ings. ASHRAE Transactions 83(2):103-17.
Englander, S.L. and L.K. Norford. 1992. Saving fan energy in VAV sys-tems—Part 1: Analysis of a variable-speed-drive retrofit. ASHRAE Transactions 98(1):3-18.
Erbs, D.G., S.A. Klein, and W.A. Beckman. 1983. Estimation of degree-days and ambient temperature bin data from monthly-average temperatures.
ASHRAE Journal 25(6):60.
Fels, M., ed. 1986. Measuring energy savings: The scorekeeping approach.
Energy and Buildings 9.
Fels, M. and M. Goldberg, 1986. Refraction of PRISM results in compo-nents of saved energy. Energy and Buildings 9:169.
Fracastoro, G., M. Masoero, and M. Cali. 1982. Surface heat transfer in building components. Proceedings Thermal Performance of the Exterior Envelopes of Buildings II, pp. 180-203.
Gordon, J.M. and K.C. Ng. 1994. Thermodynamic modeling of reciprocat-ing chillers. Journal of Applied Physics 75(6):2769-74.
Gordon, J.M and K.C. Ng. 1995. Predictive and diagnostic aspects of a uni-versal thermodynamic model for chillers. International Journal of Heat and Mass Transfer 38(5):807-18.
Gordon, J.M., K.C. Ng, and H.T. Chua. 1995. Centrifugal chillers: Thermo-dynamic modeling and a case study. International Journal of Refrigera-tion 18(4):253-57.
Guyon, G. and E. Palomo. 1999. Validation of two French building energy programs—Part 2: Parameter estimation method applied to empirical validation. ASHRAE Transactions 105(2):709-20.
Haberl, J.S. and T.E. Bou-Saada. 1998. Procedures for calibrating hourly simulation models to measured building energy and environmental data.
ASME Journal of Solar Energy Engineering 120(August):193.
Haberl, J.S. and D.E. Claridge. 1987. An expert system for building energy consumption analysis: Prototype results.
ASHRAE Transactions 93(1):979-98.
Haberl, J. and P. Komor. 1990a. Improving commercial building energy audits: How annual and monthly consumption data can help. ASHRAE Journal 32(8):26-33.
Haberl, J. and P. Komor. 1990b. Improving commercial building energy audits: How daily and hourly data can help. ASHRAE Journal 32(9):26-36.
Haberl, J.S. and S. Thamilseran. 1996. The great energy predictor shootout II: Measuring retrofit savings and overview and discussion of results.
ASHRAE Transactions 102(2):419-35.
Haberl, J.S., T.A. Reddy, I.E. Figuero, and M. Medina. 1997. Overview of LoanSTAR chiller monitoring—Analysis of in-situ chiller diagnostics using ASHRAE RP-827 test method. Paper presented at Cool Sense National Integrated Chiller Retrofit Forum, Presidio, San Francisco, September.
Hammersten, S. 1984. Estimation of energy balances for houses. National Swedish Institute for Building Research, M84.18, December.
Howell, R.H. and S. Suryanarayana. 1990. Sizing of radiant heating sys-tems: Part I and Part II. ASHRAE Transactions 96(1):652-65.
JCEM. 1992. Final report: Artificial neural networks applied to LoanSTAR data. Joint Center for Energy Management Report TR/92/15.
Kamal, S. and P. Novak. 1991. Dynamic analysis of heat transfer in build-ings with special emphasis on radiation. Energy and Buildings 17(3): 231-41.
Kaplan, M., J. McFerran, J. Jansen, and R. Pratt. 1990. Reconciliation of a DOE2.1C model with monitored end-use data from a small office build-ing. ASHRAE Transactions 96(1):981.
Katipamula, S. and D.E. Claridge. 1993. Use of simplified systems model to measure retrofit energy savings. Transactions of the ASME Journal of Solar Energy Engineering 115(May):57-68.
Katipamula, S., T.A. Reddy, and D.E. Claridge. 1994. Development and application of regression models to predict cooling energy consumption in large commercial buildings. Proceedings of the 1994 ASME/JSME/ JSES International Solar Energy Conference, San Francisco, March 27-30, p. 307.
Katipamula, S., T.A. Reddy, and D.E. Claridge. 1998. Multivariate regres-sion modeling. ASME Journal of Solar Energy Engineering 120 (August):176.
Kays, W.M. and A.L. London. 1984. Compact heat exchangers, 3rd ed.
McGraw-Hill, New York.
Kerrisk, J.F., N.M. Schnurr, et al. 1981. The custom weighting-factor method for thermal load calculation in the DOE-2 computer program.
ASHRAE Transactions 87(2):569-84.
Khalifa, A.J.N. and R.H. Marshall. 1990. Validation of heat transfer coeffi-cients on interior building surfaces using a real-sized indoor test cell.
International Journal of Heat and Mass Transfer 33(10):2219-36.
Kissock, J.K., T.A. Reddy, J.S. Haberl, and D.E. Claridge. 1993. E-model: A new tool for analyzing building energy use data. Proceedings Intl. Indust.
Energy Tech. Conf., Texas A&M University.
Kissock, J.K., T.A. Reddy, and D.E. Claridge. 1998. Ambient temperature regression analysis for estimating retrofit savings in commercial build-ings. ASME Journal of Solar Energy Engineering 120:168.
Klein, S.A., W.A. Beckman, et al. 1994. TRNSYS: A transient simulation program. Engineering Experiment Station Report 38-14, University of Wisconsin-Madison.
Knebel, D.E. 1983. Simplified energy analysis using the modified bin method. ASHRAE.
Krarti, M. 1994a. Time varying heat transfer from slab-on-grade floors with vertical insulation. Building and Environment 29(1):55-61.
Krarti, M. 1994b. Time varying heat transfer from horizontally insulated slab-on-grade floors. Building and Environment 29(1):63-71.
Krarti, M. and P. Chuangchid. 1999. Cooler floor heat gain for refrigerated structures, Final Report, ASHRAE Project 953-TRP.
Krarti, M., D.E. Claridge, and J. Kreider. 1988a. The ITPE technique applied to steady-state ground-coupling problems. International Journal of Heat and Mass Transfer 31:1885-98.
Krarti, M., D.E. Claridge, and J. Kreider. 1988b. ITPE method applications to time-varying two-dimensional ground-coupling problems. Interna-tional Journal of Heat and Mass Transfer 31:1899-1911.
Krarti, M., J.F. Kreider, D. Cohen, and P. Curtiss. 1998. Estimation of energy savings for building retrofits using neural networks. ASME Journal of Solar Energy Engineering 120:211.
Kreider, J.F. and J. Haberl. 1994. Predicting hourly building energy usage: The great predictor shootout—Overview and discussion of results. ASH-RAE Transactions 100(2):1104-18.
Kreider, J.F. and A. Rabl. 1994. Heating and cooling of buildings. McGraw-Hill, New York.
Kreider, J.F. and X.A. Wang. 1991. Artificial neural networks demonstration for automated generation of energy use predictors for commercial build-ings. ASHRAE Transactions 97(1):775-79.
Kusuda, T. 1969. Thermal response factors for multi-layer structures of var-ious heat conduction systems. ASHRAE Transactions 75(1):246-71.
Energy Estimating and Modeling Methods 31.33 Labs, K., J. Carmody, R. Sterling, L. Shen, Y. Huang, and D. Parker. 1988.
Building foundation design handbook, ORNL Report Sub/86-72143/1.
Oak Ridge National Laboratory, Oak Ridge, TN.
Lachal, B., W.U. Weber, and O. Guisan. 1992. Simplified methods for the thermal analysis of multifamily and administrative buildings. ASHRAE Transactions 98.
Lebrun, J., J.-P. Bourdouxhe, and M. Grodent. 1999. HVAC 1 toolkit: A tool-kit for primary HVAC system energy calculation. ASHRAE Lewis, P.T. and D.K. Alexander. 1990. HTB2: A flexible model for dynamic building simulation. Building and Environment. pp. 7-16.
Liu, M. and D.E. Claridge. 1998. Use of calibrated HVAC system models to optimize system operation. ASME Journal of Solar Energy Engineering 120:131.
Lorenzetti, D.M. and L.K. Norford. 1993. Pressure reset control of variable air volume ventilation systems. Proceedings ASME International Solar Energy Conference, p. 445, April, Washington, D.C.
MacDonald, J.M. and D.M. Wasserman. 1989. Investigation of metered data analysis methods for commercial and related buildings. Oak Ridge National Laboratory Report ORNL/CON-279.
Manke, J.M., D.C. Hittle, and C.E. Hancock. 1996. Calibrating building energy analysis models using short-term data. Proceedings ASME Inter-national Solar Energy Conference, p.369, San Antonio, TX.
McClelland, J.L. and D.E. Rumelhart. 1988. Exploration in parallel dis-tributed processing. MIT Press, Cambridge, MA.
McQuiston, F.C. and J. D. Spitler. 1992. Cooling and Heating Load Calcu-lation Manual. ASHRAE Melo, C. and G.P. Hammond. 1991. Modeling and assessing the sensitivity of external convection from building facades in heat and mass transfer in building materials and structures. pp. 683-95. Hemisphere Publishing, New York.
Metcalfe, R.R., R.D. Taylor, C.O. Pederson, R.J. Liesen, and D.E. Fisher.
1995. Incorporating a modular system simulation program into a large energy analysis program: The linking of IBLAST and HVACSIM+. Pro-ceedings of Building Simulation ’95.
Miller, D.E. 1980. The impact of HVAC process dynamics on energy use.
ASHRAE Transactions 86(2):535-56.
Miller, R. and J. Seem. 1991. Comparison of artificial neural networks with traditional methods of predicting return from night setback. ASHRAE Transactions 97(2):500-08.
Mitalas, G.P. and D.G. Stephenson. 1967. Room thermal response factors.
ASHRAE Transactions 73(1):III.2.1-III.2.10.
Mitalas, G.P. 1968. Calculations of transient heat flow through walls and roofs. ASHRAE Transactions 74(2):182-88.
Mitchell, J.W. 1983. Energy engineering. John Wiley and Sons, New York.
NOAA. 1973. Degree-days to selected bases. U.S. National Climatic Data Center, Asheville, NC.
Neter, J., W. Wasseran, and M. Kutner. 1989. Applied linear regression mod-els, 2nd ed. Richard C. Irwin, Homewood, IL.
Nisson, J.D.N. and G. Dutt. 1985. The superinsulated home book. John Wiley and Sons, New York.
Norford, L.K., R.H. Socolow, E.S. Hsieh, and G.V. Spadaro. 1994. Two-to-one discrepancy between measured and predicted performance of a low-energy office building: Insights from a reconciliation based on the DOE-2 model. Energy and Buildings 21:121.
Park, C., D.R. Clark, and G.E. Kelly. 1985. An overview of HVACSIM+, a dynamic building/HVAC control systems simulation program. Proceed-ings First Building Energy Simulation Conference.
Parker, W.H., G.E. Kelly, and D. Didion. 1980. A method for testing, rating, and estimating the heating seasonal performance of heat pumps. National Bureau of Standards, NBSIR 80-2002, April.
Phelan, J., M.J. Brandemuehl, and M. Krarti. 1996. Final Report ASHRAE Project RP-827: Methodology development to measure in-situ chiller, fan, and pump performance. JCEM Report No. JCEM/TR/96-3, Univer-sity of Colorado at Boulder.
Rabl, A. 1988. Parameter estimation in buildings: Methods for dynamic analysis of measured energy use. Journal of Solar Energy Engineering 110:52-66.
Rabl, A. and A. Riahle. 1992. Energy signature model for commercial build-ings: Test with measured data and interpretation. Energy and Buildings 19:143-54.
Reddy, T. 1989. Application of dynamic building inverse models to three occupied residences monitored non-intrusively. Proceedings Thermal Performance of Exterior Envelopes of Buildings IV, ASHRAE/DOE/ BTECC/CIBSE.
Reddy, T. and D. Claridge. 1994. Using synthetic data to evaluate multiple regression and principle component analyses for statistical modeling of daily building energy consumption. Energy and Buildings 24:35-44.
Reddy, T.A., S. Katipamula, J.K. Kissock, and D.E. Claridge. 1995. The functional basis of steady-state thermal energy use in air-side HVAC equipment. Journal of Solar Energy Engineering 117:31-39.
Reddy, T.A., N.F. Saman, D.E. Claridge, J.S. Haberl, W.D. Turner, and A. Chalifoux. 1997. Baselining methodology for facility level monthly energy use—Part 1: Theoretical aspects. ASHRAE Transactions 103(2): 336-47.
Reddy, T.A., S. Deng, and D.E. Claridge. 1999. Development of an inverse method to estimate overall building and ventilation parameters of large commercial buildings. Journal of Solar Energy Engineering 121:47.
Ruch, D. and D. Claridge. 1991. A four parameter change-point model for predicting energy consumption in commercial buildings. Proceedings ASME International Solar Energy Conference, 433-40.
Ruch, D., L. Chen, J. Haberl, and D. Claridge. 1993. A change-point prin-ciple component analysis (CP/CAP) method for predicting energy use in commercial buildings: The PCA model. Journal of Solar Energy Engi-neering 115:77-84.
Seem, J.E. and J.E. Braun. 1991. Adaptive methods for real-time forecasting of building electricity demand. ASHRAE Transactions 97(1):710.
Shipp, P.H. and T.B. Broderick. 1983. Analysis and comparison of annual heating loads for various basement wall insulation strategies using tran-sient and steady-state models. Thermal Insulation, Materials, and Sys-tems for Energy Conservation in the 80’s. ASTM STP 789, F.A. Govan, D.M. Greason, and J.D. McAllister, eds. American Society for Testing and Materials, West Conshohocken, PA.
Shurcliff, W.A. 1984. Frequency method of analyzing a building’s dynamic thermal performance. Cambridge, MA.
Sonderegger, R.C. 1977. Dynamic models of house heating based on equiv-alent thermal parameters. Ph.D. thesis, Center for Energy and Environ-mental Studies Report No. 57. Princeton University, Princeton, NJ.
Sonderegger, R.C. 1985. Thermal modeling of buildings as a design tool.
Proceedings of CHMA 2000, Vol. 1.
Sonderegger, R.C. 1998. Baseline equation for utility bill analysis using both weather and non-weather related variables. ASHRAE Transactions 104(2):859-70.
Sowell, E.F. 1988. Classification of 200,640 parametric zones for cooling load calculations. ASHRAE Transactions 94(2):716-36.
Sowell, E.F. 1990. Lights: A numerical lighting/HVAC test cell. ASHRAE Transactions 96(2):780-86.
Sowell, E.F. and M.A. Moshier. 1995. HVAC component model libraries for equation-based solvers. Proceedings of Building Simulation ’95. IBPSA, Madison, WI.
Spitler, J.D., C.O. Pedersen, and D.E. Fisher. 1991. Interior convective heat transfer in buildings with large ventilative flow rates. ASHRAE Trans-actions 97(1):505-15.
Steinman, M., L.N. Kalisperis, and L.H. Summers. 1989. The MRT-correc-tion method: A new method of radiant heat exchange. ASHRAE Trans-actions 95(1):1015-27.
Stephenson, D.G. and G.P. Mitalas. 1967. Cooling load calculations by ther-mal response factor method. ASHRAE Transactions 73(1):III.1.1-III.1.7.
Stoecker, W.F. and J.W. Jones. 1982. Refrigeration and air conditioning, 2nd ed. McGraw-Hill, New York.
Strand, R.K. and C.O. Pedersen. 1997. Implementation of a radiant heating and cooling model into an integrated building energy analysis program.
ASHRAE Transactions 103(1):949-58.
Subbarao, K. 1986. Thermal parameters for single and multi-zone buildings and their determination from performance data. SERI Report SERI/TR-253-2617. Solar Energy Research Institute, Golden, CO.
Subbarao, K. 1988. PSTAR—Primary and secondary terms analysis and re-normalization: A unified approach to building energy simulations and short-term monitoring. SERI/TR-253-3175.
Taylor, R.D., C.O. Pedersen, and L. Lawrie. 1990. Simultaneous simulation of buildings and mechanical systems in heat balance based energy anal-ysis programs. Proceedings Third International Conference on System Simulation in Buildings. Liege, Belgium.
Taylor, R.D. et al. 1991. Impact of simultaneous simulation of buildings and mechanical systems in heat balance based energy analysis programs on system response and control. Proceedings Building Simulation ’91.
Sophia Antipolis, Nice France: IBPSA.
31.34 2001 ASHRAE Fundamentals Handbook (SI) Thamilseran, S. and J. Haberl. 1995. A bin method for calculating energy conservation retrofit savings in commercial buildings. Proceedings 1995 ASME/JSME/JSES Intl. Solar Energy Conference, pp. 111-24.
Threlkeld, J.L. 1970. Thermal environmental engineering, 2nd ed. Prentice-Hall, Englewood Cliffs, NJ.
USAF. 1978. Engineering weather data. Department of the Air Force Man-ual AFM 88-29. U.S. Government Printing Office, Washington, D.C.
Walton, G.N. 1980. A new algorithm for radiant interchange in room loads calculations. ASHRAE Transactions 86(2):190-208.
Walton, G.N. 1983. Thermal analysis research program reference manual.
NBSIR 83-2655. National Institute of Standards and Technology, Gaith-ersburg, MD.
Walton, G.N. 1993. Computer programs for simulation of lighting/HVAC interactions. NISTIR 5322. NIST.
Waltz, J.P. 1992. Practical experience in achieving high levels of accuracy in energy simulations of existing buildings. ASHRAE Transactions 98(1): 606-17.
Wang, X.A. and J.F. Kreider.1992. Improved artificial neural networks for commercial building energy use prediction. Journal of Solar Energy Engineering.
Wasserman, P.D. 1989. Neural computing, theory and practice. Van Nos-trand Reinhold, New York.
Yazdanian, M. and J. Klems. 1993. Measurement of the exterior convective film coefficient for windows in low-rise buildings. ASHRAE Trans-actions 100(1):1087-96.
York, D.A. and C.C. Cappiello, eds. 1982. DOE-2 engineers manual.
Lawrence Berkeley Laboratory Report LBL-11353 (LA-8520-M, DE83004575). National Technical Information Services, Springfield, VA.
BIBLIOGRAPHY NEMVP. 1996. U.S. DOE North-America energy monitoring and verifica-tion protocol. U.S. Department of Energy, Washington, D.C.
Shavit, G. 1995. Short-time-step analysis and simulation of homes and buildings during the last 100 years. ASHRAE Transactions 101(1):856-68.
Sowell, E.F. and G.N. Walton. 1980. Efficient computation of zone loads.
ASHRAE Transactions 86(1):49-72.
Sowell, E.F. and D.C. Hittle. 1995. Evolution of building energy simulation methodology. ASHRAE Transactions 101(1):850-55.
Spitler, J.D. 1996. Annotated guide to load calculation models and algo-rithms. ASHRAE.
Subbarao, K., J. Burch, and C.E. Hancock. 1990. How to accurately measure the load coefficient of a residential building. Journal of Solar Energy Engineering.
U.S. Army. 1979. BLAST, The building loads analysis and system thermo-dynamics program—Users manual. U.S. Army Construction Engineer-ing Research Laboratory Report E-153.
Yuill, G.K. 1990. An annotated guide to models and algorithms for energy calculations relating to HVAC equipment. ASHRAE.
32.1 CHAPTER 32 SPACE AIR DIFFUSION Terminology ............................................................................ 32.1 PRINCIPLES OF JET BEHAVIOR ........................................ 32.1 Air Jet Classification .............................................................. 32.1 Multiple Jets ............................................................................ 32.6 Airflow in Occupied Zone ....................................................... 32.6 ROOM AIR DIFFUSION METHODS .................................... 32.7 Mixing Systems ....................................................................... 32.7 Displacement Ventilation ...................................................... 32.11 Unidirectional Airflow Ventilation ....................................... 32.11 Underfloor Air Distribution and Task/Ambient Conditioning ................................................... 32.11 METHODS OF EVALUATION ............................................. 32.12 Air Diffusion Performance Index (ADPI) ............................. 32.13 SYSTEM DESIGN ................................................................. 32.14 Design Considerations .......................................................... 32.14 Design Procedure ................................................................. 32.15 Outlet Location and Selection ............................................... 32.15 Return Air Design for Optimum Performance ...................... 32.17 Ceiling-Mounted Air Diffuser Systems ................................. 32.17 TERMINOLOGY SPECT ratio. Ratio of length to width of an opening or Acore of a grille.
Axial flow jet. Stream of air whose motion is approximately symmetrical along a line, although some spreading and drop or rise can occur from diffusion and buoyancy effects.
Coefficient of discharge. Ratio of area at vena contracta to area of opening.
Cold air. General term used for supply air at 1.5 to 4.5°C.
Core area. Total plane area of that portion of a grille, included within lines tangent to the outer edges of the outer openings, through which air can pass.
Damper. Device used to vary the volume of air passing through a confined cross section by varying the cross-sectional area.
Diffuser. Outlet discharging supply air in various directions and planes.
Diffusion. Distribution of air within a space by an outlet dis-charging supply air in various directions and planes.
Draft. Undesired local cooling of a body caused by low temper-ature and movement of air.
Drop. Vertical distance that the lower edge of a horizontally pro-jected airstream drops between the outlet and the end of its throw.
Effective area. Net area of an outlet or inlet device through which air can pass; equal to the free area times the coefficient of dis-charge.
Entrainment. Movement of room air into the jet caused by the airstream discharged from the outlet (secondary air motion).
Entrainment ratio. Total air divided by the air discharged from the outlet.
Envelope. Outer boundary of an airstream moving at a percepti-ble velocity.
Exhaust opening or inlet. Any opening through which air is removed from a space.
Free area. Total minimum area of the openings in the air outlet or inlet through which air can pass.
Grille. A covering for any area through which air passes.
Induction. See Entrainment.
Isothermal jet. Air jet with the same temperature as the sur-rounding air.
Lower zone. Room volume below the stratification level created by displacement ventilation.
Nonisothermal jet. Air jet with an initial temperature different from the surrounding air.
Outlet velocity. Average velocity of air emerging from the out-let, measured in the plane of the opening.
Primary air. Air delivered to the outlet by the supply duct.
Radius of diffusion. Horizontal axial distance an airstream travels after leaving an air outlet before the maximum stream velocity is reduced to a specified terminal level (e.g., 0.25, 0.5, 0.75, or 1.0 m/s).
Register. Grille equipped with a damper or control valve.
Spread. Divergence of the airstream in a horizontal and/or ver-tical plane after it leaves the outlet.
Stagnant zone. Area characterized by low air motion and strat-ification. This does not imply poor air quality.
Supply opening or outlet. Any opening through which supply air is delivered into a ventilated space being heated, cooled, humid-ified, or dehumidified. Supply outlets are classified according to their location in a room as sidewall, ceiling, baseboard, or floor out-lets. However, because numerous designs exist, they are more accu-rately described by their construction features. (See Chapter 17 of the 2000 ASHRAE Handbook—Systems and Equipment.) Temperature differential. Temperature difference between pri-mary and room air.
Terminal velocity. Maximum airstream velocity at the end of the throw.
Throw. Horizontal or vertical axial distance an airstream travels after leaving an air outlet before the maximum stream velocity is reduced to a specified terminal velocity (e.g., 0.25, 0.5, 0.75, or 1.0 m/s), defined by ASHRAE Standard 70.
Total air. Mixture of discharged air and entrained air.
Upper zone. Room volume above the stratification level created by displacement ventilation.
Vane. Thin plate in the opening of a grille.
Vane ratio. Ratio of the depth of a vane to the space between two adjacent vanes.
Vena contracta. Smallest area of a fluid stream leaving an orifice.
PRINCIPLES OF JET BEHAVIOR AIR JET CLASSIFICATION As a rule, air supplied into rooms through the various types of outlets (e.g., grilles, ceiling diffusers, perforated panels) is distrib-uted by turbulent air jets. These air jets are the primary factor affect-ing room air motion; Baturin (1972), Christianson (1989), and Murakami (1992) have further information on the relationship between the air jet and the occupied zone. If the air jet is not obstructed by walls, ceiling, or other obstructions, it is considered a free jet. If the air jet is attached to a surface, it is an attached air jet.
The preparation of this chapter is assigned to TC 5.3, Room Air Distribution.
32.2 2001 ASHRAE Fundamentals Handbook (SI) Characteristics of the air jet in a room might be influenced by reverse flows created by the same jet entraining the ambient air.
This air jet is called a confined jet. If the temperature of the sup-plied air is equal to the temperature of the ambient room air, the air jet is called an isothermal jet. A jet with an initial temperature dif-ferent from the temperature of the ambient air is called a noniso-thermal jet. The air temperature differential between supplied and ambient room air generates thermal forces in jets, affecting (1) the trajectory of the jet, (2) the location at which the jet attaches and separates from the ceiling/floor, and (3) the throw of the jet. The sig-nificance of such effects depends on the ratio between the thermal buoyancy of the air and inertial forces (characterized by the Archimedes number Ar).
Depending on diffuser type, air jets can be classified as follows: • Compact air jets are formed by cylindrical tubes, nozzles, square or rectangular openings with a small aspect ratio (unshaded or shaded by perforated plates), grilles, etc. Compact air jets are three-dimensional and axisymmetric at least at some distance from the diffuser opening. The maximum velocity in the cross section of the compact jet is on the axis.
• Linear air jets are formed by slots or rectangular openings with a large aspect ratio. The jet flows are approximately two-dimensional. Air velocity is symmetric in the plane at which air velocities in the cross section are maximum. At some distance from the diffuser, linear air jets tend to transform into compact jets.
• Radial air jets are formed by ceiling cylindrical air diffusers with flat disks or multidiffusers that direct the air horizontally in all directions.
• Conical air jets are formed by cone-type or regulated multidiffuser ceiling-mounted air distribution devices. They have an axis of symmetry. The air flows parallel to the conical surface (the angle at the top of the cone is 120°) with the maximum velocities in the cross section perpendicular to the axis.
• Incomplete radial air jets are formed by outlets with grilles having diverging vanes and a forced angle of expansion. At a distance, this jet tends to transform into a compact one.
• Swirling air jets are formed by diffusers with vortex-forming devices. These devices create rotation, which has, in addition to the axial component of velocity vectors, tangential and radial ones. Depending on the type of air diffuser, swirling jets can be compact, conical, or radial.
Isothermal Jets The shape of jets at a short distance from the outlet face is very similar whether the outlet is round, rectangular, grille-like, or a per-forated panel. The jet discharged from a round opening forms an expanding cone; jets from rectangular outlets rapidly pass from a rectangular to an elliptical cross-sectional shape and then to a circu-lar shape, at a rate depending primarily on the aspect ratio and jet width. Even for wide-angle grilles and annular outlets, the similar-ities permit the same performance analysis for both.
For many conditions of jet discharge, it is possible to analyze jet performance and determine (1) the angle of divergence of the jet boundary, (2) the velocity patterns along the jet axis, (3) the veloc-ity profile at any cross section in the zone of maximum engineer-ing importance, and (4) the entrainment ratios in the same zone (Tuve 1953).
Using the data in this section, the following must be considered: 1. Because the method of finding the jet velocities is based on several approximations, the two recommended equations [Equations (3) and (4)] must be used cautiously for extreme axial and radial distances.
2. The characteristics of the low-velocity regions of ventilating jets are not well understood, and the effects at various Reynolds numbers are not fully known for axial or radial jets.
3. The quantitative treatment of the forces governing room air diffusion problems is limited, and nonisothermal conditions involving buoyant forces are more difficult to predict.
4. Most investigations have addressed free jets, whereas airstreams in practical room air diffusion are not free streams but are influenced by walls, ceilings, floors, and other obstructions.
Angle of Divergence. The angle of divergence is well defined near the outlet face, but the boundary contours are billowy and eas-ily affected by external influences. Near the outlet, as in the room, air movement has local eddies, vortices, and surges. The internal forces governing this air motion are extremely delicate (Nottage et al. 1952b).
Measured angles of divergence (spread) for discharge into large open spaces usually range from 20 to 24° with an average of 22°.
Coalescing jets for closely spaced multiple outlets expand at smaller angles, averaging 18°, and jets discharging into relatively small spaces show even smaller angles of expansion (McElroy 1943). In cases where the outlet area is small compared to the dimensions of the space normal to the jet, the jet may be considered free as long as (1) where X = distance from face of outlet, m AR = cross-sectional area of confined space normal to jet, m2 Jet Expansion Zones. The full length of an air jet (compact, lin-ear, radial, or conical), in terms of the maximum or centerline veloc-ity and temperature differential at the cross section, can be divided into four zones: Zone 1. A core zone; a short zone, extending about four diame-ters or widths from the outlet face, in which the maximum velocity (temperature) of the airstream remains practically unchanged.
Zone 2. A transition zone, the length of which depends on the type of outlet, aspect ratio of the outlet, initial airflow turbulence, and so forth.
Zone 3. A zone of fully established turbulent flow that may be 25 to 100 equivalent air outlet diameters (widths for slot-type air dif-fusers) long.
Zone 4. A zone of diffuser jet degradation, where the maximum air velocity and temperature decreases rapidly. The distance to this zone and its length depend on the velocities and turbulence charac-teristics of the ambient air. In a few diameters or widths, the air velocity becomes less than 0.25 m/s. The characteristics of this zone are still not well understood.
Zone 3 is of major engineering importance because, in most cases, the diffuser jet enters the occupied area within this zone.
Centerline Velocities in Zones 1 and 2. In Zone 1, the ratio Vx/Vo is constant and equal to the ratio of the center velocity of the jet at the start of expansion to the average velocity. The ratio Vx/Vo varies from approximately 1.0 for rounded entrance nozzles to about 1.2 for straight pipe discharges; it has much higher values for diverging discharge outlets.
Experimental evidence indicates that in Zone 2, (2) where Vx = centerline velocity at distance X from outlet, m/s Vo = Vc/CdRfa = average initial velocity at discharge from open-ended duct or across contracted stream at vena contracta of orifice or multiple-opening outlet, m/s Vc = nominal velocity of discharge based on core area, m/s Cd = discharge coefficient (usually between 0.65 and 0.90) Rfa = ratio of free area to gross (core) area X 1.5 AR ≤ Vx Vo ------ 1.13KHo X ----------------------= Space Air Diffusion 32.3 Ho = width of jet at outlet or at vena contracta, m K = centerline velocity constant depending on outlet type and discharge pattern (see Table 1) X = distance from outlet to measurement of centerline velocity Vx, m The aspect ratio (Tuve 1953) and turbulence (Nottage et al.
1952b) primarily affect the centerline velocities in Zones 1 and 2.
Aspect ratio has little effect on the terminal zone of the jet when Ho is greater than 100 mm. This is particularly true of nonisothermal jets. When Ho is very small, the induced air can penetrate the core of the jet, thus reducing the centerline velocities. The difference in per-formance between the radial outlet with a small Ho and the axial out-let with a large Ho shows the importance of the thickness of the jets. When air is discharged from relatively large perforated panels, the constant velocity core formed by the coalescence of the individ-ual jets extends a considerable distance from the panel face. In Zone 1, when the ratio is less than 5, the following equation should be used for estimating centerline velocities (Koestel et al. 1949): (3) Centerline Velocity in Zone 3. In Zone 3, maximum or center-line velocities of straight flow isothermal jets can be determined with accuracy from the following equations: (4) (5) (6) where K = proportionality constant Do = effective or equivalent diameter of stream at discharge from open-end duct or at contracted section, m Ao = core area Ac or duct area, m2 Ac = measured gross (core) area of outlet, m2 Q = discharge from outlet, m3/s Because Ao equals the effective area of the stream, the flow area for commercial registers and diffusers, according to ASHRAE Standard 70, can be used in Equation (4) with the appropriate value of K.
Equation (4) is nondimensional and requires only that consistent units be used. Values of K are listed in Table 2 (Tuve 1953, Koestel et al. 1950).
In multiple-opening outlets and annular ring outlets, the streams coalesce into a solid jet before actual jet expansion takes place. This coalescence affects the proportionality constant K and accounts for some divergence in reported values for similar outlets.
For perforated panels of relatively large size, the values of K given in Table 2 apply only when the ratio is larger than 5 (see the section on Centerline Velocities in Zones 1 and 2).
Low-velocity test results, in the range Vx < 0.75 m/s, indicate that normal values of K should be reduced about 20% for Vx = 0.25 m/s, as used later in Equation (9) for throw.
Determining Centerline Velocities. To correlate data from all four zones, centerline velocity ratios are plotted against distance from the outlet in Figure 2.
The airflow patterns of diffusers are related to the throw K-fac-tors and to the throw distance. In general, diffusers that have a cir-cular airflow pattern have a shorter throw than those with a directional or crossflow pattern. During cooling, the circular pattern has a tendency to curl back from the end of the throw toward the dif-fuser. This action reduces the drop and ensures that the cool air remains near the ceiling. Cross-flow airflow patterns have a longer throw, and the indi-vidual side jets react in a manner similar to jets from sidewall grilles. The airflow jets with this type of pattern have a longer throw and the airflow does not roll back to the diffuser at the end of the throw. The airflow continues to move away from the dif-fuser at low velocities.
The following example illustrates the use of Table 1 and Fig-ure 2.
Example 1. A 300 mm by 450 mm high sidewall grille with a 280 mm by 430 mm core area is selected. From Table 1, K = 5 for Zone 3. If the airflow is 0.3 m3/s, what is the throw to 0.25 m/s, 0.5 m/s, and 0.75 m/s? The grille has 80% free area.
Table 1 Recommended Values for Centerline Velocity Constant K for Commerical Supply Outlets Outlet Type Discharge Pattern Area A Ka High sidewall grilles (Figure 1A) 0° deflection b Core 5.0 Wide deflection Core 3.7 High sidewall linear (Figure 1B) Core less than 100 mm high c Core 3.9 Core more than 100 mm high Core 4.4 Low sidewall (Figure 1C) Up and on wall, no spread Core 4.4 Wide spread c Core 2.6 Baseboard (Figure 1C) Up and on wall, no spread Duct 3.9 Wide spread e Duct 1.8 Floor (Figure 1C) No spread c Core 4.1 Wide spread Core 1.4 Ceiling circular directional (Fig. 1D) 360° horizontal d Duct 1.0 Four-way—little spread Duct 3.3 Ceiling linear (Figure 1E) One-way—horizontal along ceiling c Core 4.8 a These values are representative for commercial outlets with discharge patterns as shown in Figure 1.
b Free area is about 80% of core area.
c Free area is about 50% of core area.
d Cone free area is greater than duct area.
e Face free area is greater than duct area. X Ac ⁄ Distance from Panel Panel Area ⁄ ( ) Vx 1.2Vo CdRfa = Vx Vo ------KDo X -----------1.13K Ao X ---------------------------= = Vx 1.13KVo Ao X ----------------------------------1.13KQ X Ao -------------------= = Vx 1.13KQ X AcCdRfa -------------------------------= Table 2 Recommended Values of Centerline Velocity Constant for Standard Openings Type of Outlet K Vo = 2.5 to 5 m/s Vo = 10 to 50 m/s Free openings Round or square 5.0 6.2 Rectangular, large aspect ratio (<40) 4.3 5.3 Annular slots, axial or radiala — — Grilles and grids Free area 40% or more 4.1 5.0 Perforated panels Free area 3 to 5% 2.7 3.3 Free area 10 to 20% 3.5 4.3 aFor radial slots, use X/H instead of . H is height or width of slot.
Note: K is an index of loss in axial kinetic energy. Interpolate as required. Departures from maximum value indicate losses in Zones 1 and 2 when compared with the jet from a rounded-entrance, circular nozzle.
X A ⁄ X Ac ⁄ 32.4 2001 ASHRAE Fundamentals Handbook (SI) Fig. 1 Airflow Patterns of Different Diffusers Space Air Diffusion 32.5 Solution: From Equation (5): Solving for 0.25 m/s throw, But according to Figure 2, 0.25 m/s is in Zone 4, which is typically 20% less than calculated in Equation (4), or X = 19.5 × 0.80 = 15.6 m Solving for 0.50 m/s throw, Solving for 0.75 m/s throw, Throw. Equation (6) can be transposed to determine the throw X of an outlet if the discharge volume and the centerline velocity are known: (7) Or, if Z = , (8) The maximum throw TV is usually defined as the distance from the outlet face to where the centerline velocity is 0.25 m/s. There-fore, for VT = 0.25 m/s, (9) Any other terminal centerline velocity could be inserted in Equa-tion (9) for VT.
Velocity Profiles of Jets. In Zone 3 of both axial and radial jets, the velocity distribution may be expressed by a single curve (Figure 3) in terms of dimensionless coordinates; this same curve can be used as a good approximation for adjacent portions of Zones 2 and 4. Temperature and density differences have little effect on cross-sectional velocity profiles.
Velocity distribution in Zone 3 can be expressed by the Gauss error function or probability curve, which is approximated by the following equation: (10) where r = radial distance of point under consideration from centerline of jet r0.5V = radial distance in same cross-sectional plane from axis to point where velocity is one-half centerline velocity (i.e., V = 0.5Vx ) Vx = centerline velocity in same cross-sectional plane V = actual velocity at point being considered Experiments show that the conical angle for r0.5V is approxi-mately one-half the total angle of divergence of a jet. The velocity profile curve for one-half of a straight-flow turbulent jet (the other half being a symmetrical duplicate) is shown in Figure 3. For mul-tiple-opening outlets, such as grilles or perforated panels, the veloc-ity profiles are similar, but the angles of divergence are smaller.
Entrainment Ratios. The following are equations for the entrainment of circular jets and of jets from long slots.
For third-zone expansion of circular jets, (11) By substituting from Equation (4), (12) Fig. 2 Chart for Determining Centerline Velocities of Axial and Radial Jets X 1.13KQ Vx Ao -------------------1.13 5 0.3 × × Vx 280 430 × 106 ⁄ -----------------------------------------------4.885 Vx -------------= = = X 4.885 0.25 -------------19.5 m = = X 4.885 0.50 -------------9.8 m = = X 4.885 0.75 -------------6.5 m = = X 1.13K Vx -------------- Q AcCdRfa ---------------------------= CdRfa X 1.13K Vx -------------- Q Z Ac ---------------= Fig. 3 Cross-Sectional Velocity Profiles for Straight-Flow Turbulent Jets TV X 1.13K 0.25 -------------- Q Z Ac ---------------= = r r0.5V ----------- 2 3.3 Vx V -----log = Qx Qo ------2X 1.13K Ao ---------------------------= Qx Qo ------2 Vo Vx ------= 32.6 2001 ASHRAE Fundamentals Handbook (SI) For a continuous slot with active sections up to 3 m and separated by 0.6 m, (13) or, substituting from Equation (2), (14) where Qx = total volumetric flow rate at distance X from face of outlet, m3/s Qo = discharge from outlet, m3/s X = distance from face of outlet, m K = proportionality constant Ao = core area Ac or duct area, m2 Ho = width of slot, m The entrainment ratio Qx/Qo is important in determining total air movement at a given distance from an outlet. For a given outlet, the entrainment ratio is proportional to the distance X [Equation (11)] or to the square root of the distance X [Equation (13)] from the outlet.
Equations (12) and (14) show that, for a fixed centerline velocity Vx, the entrainment ratio is proportional to the outlet velocity. Equations (12) and (14) also show that, at a given centerline and outlet veloc-ity, a circular jet has greater entrainment and total air movement than a long slot. Comparing Equations (11) and (13), the long slot should have a greater rate of entrainment. The entrainment ratio at a given distance is less with a large K than with a small K.
Isothermal Radial Flow Jets In a radial jet, as with an axial jet, the cross-sectional area at any distance from the outlet varies as the square of this distance. Cen-terline velocity gradients and cross-sectional velocity profiles are similar to those of Zone 3 of axial jets, and the angles of divergence are about the same.
A jet from a ceiling plaque has the same form as half of a free radial jet. The jet is wider and longer than a free jet, with the maxi-mum velocity close to the surface. Koestel (1957) provides an equa-tion for radial flow outlets.
Nonisothermal Jets When the temperature of introduced air is different from the room air temperature, the behavior of the diffuser air jet is affected by the thermal buoyancy due to air density difference. The trajec-tory of a nonisothermal jet introduced horizontally is determined by the Archimedes number (Baturin 1972): (15) where g = gravitational acceleration rate, m/s2 Lo = length scale of diffuser outlet equal to hydraulic diameter of outlet, m to = initial temperature of jet, °C ts = temperature of surrounding air, °C Vo = initial air velocity of jet, m/s Ts = room air temperature, K The paths assumed by horizontally projected heated and chilled jets influenced by buoyant forces are significant in heating and cool-ing with wall outlets. Koestel’s equation (1955) describes the behavior of these jets.
Helander and Jakowatz (1948), Helander et al. (1953, 1954, 1957), Yen et al. (1956), and Knaak (1957) developed equations for outlet characteristics that affect the downthrow of heated air.
Koestel (1954, 1955) developed equations for temperatures and velocities in heated and chilled jets. Li et al. (1993, 1995) and Kirkpatrick and Elleson (1996) provide additional information on nonisothermal jets.
Surface Jets (Wall and Ceiling) Jets discharging parallel to a surface with one edge of the outlet coinciding with the surface take the form of one-half of an axial jet discharging from an outlet twice as large, similar to radial jets from ceiling plaques. Entrainment takes place almost exclusively along the surface of a half cone, and the maximum velocity remains close to the surface (Tuve 1953).
Values of K are approximately those for a free jet multiplied by ; that is, the normal maximum of 6.2 for K for free jets becomes 8.8 for a similar jet discharged parallel to and adjacent to a surface.
When a jet is discharged parallel to but at some distance from a solid surface (wall, ceiling, or floor), its expansion in the direction of the surface is reduced, and entrained air must be obtained by recirculation from the jet instead of from ambient air (McElroy 1943, Nottage et al. 1952a, Zhang et al. 1990). The restriction to entrainment caused by the solid surface induces the Coanda effect, which makes the jet attach to a surface a short distance after it leaves the diffuser outlet. The jet then remains attached to the surface for some distance before separating from the surface again.
In nonisothermal cases, the trajectory of the jet is determined by the balance between the thermal buoyancy and the Coanda effect, which depends on the jet momentum and the distance between the jet exit and the solid surface. The behavior of such nonisothermal surface jets has been studied by Kirkpatrick et al. (1991), Wilson et al. (1970), Oakes (1987), and Zhang et al. (1990), each addressing different factors. A more systematic study of these jets in room ven-tilation flows is needed to provide reliable guidelines for designing air diffusion systems.
MULTIPLE JETS Twin parallel air jets act independently until they interfere. The point of interference and its distance from the outlets varies with the distance between the outlets. From the outlets to the point of inter-ference, the maximum velocity, as for a single jet, is on the center-line of each jet. After interference, the velocity on a line midway between and parallel to the two jet centerlines increases until it equals the jet centerline velocity. From this point, a maximum velocity of the combined jet stream is on the midway line, and the profile seems to emanate from a single outlet of twice the area of one of the two outlets.
Koestel and Austin (1956) determined the spacing between out-lets for noninterference between the jets. For a K value of 6.5, the outlets should be placed three to eight diameters apart, with Vo val-ues from 2.5 to 7.5 m/s.
AIRFLOW IN OCCUPIED ZONE Mixing Systems. Laboratory experiments on jets usually involve recirculated air with negligible resistance to flow on the return path of the jet air. Experiments in mine tunnels of small cross-sectional areas, where the return flow of jet air to outlets meets con-siderable resistance, show that expansion of the jet terminates abruptly at a distance that is independent of discharge velocity and is only slightly affected by the size of the outlet. These distances are determined primarily by the size and length of the return path. In a long tunnel with a cross section of 1.5 m by 1.8 m, a jet may not travel more than 7.5 m; in a tunnel with a relatively large section (7.5 m by 18 m), the jet may travel more than 75 m. McElroy (1943) provides data on this phase of jet expansion.
Zhang et al. (1990) found that, for a given heat load and room air supply rate, air velocity in the occupied zone increases when the Qx Qo ------ 2 1.13K ---------------- X Ho ------= Qx Qo ------ 2 Vo Vx ------= Ar gLo to ts – ( ) Vo 2Ts ----------------------------= 2 Space Air Diffusion 32.7 outlet discharge velocity increases. Therefore, the design supply air velocity should be high enough to maintain the jet traveling in the desired direction in order to ensure good mixing before it reaches the occupied zone. Excessively high outlet air velocity would induce high air velocity in the occupied zone and result in thermal discomfort.
Turbulence Production and Transport. The air turbulence within a room is mainly produced at the diffuser jet region by interaction of the supply air with the room air and with the solid surfaces (walls or ceiling) in the vicinity. It is then transported to other parts of the room, including the occupied zone (Zhang et al.
1992). Meanwhile, the turbulence is also damped by viscous effect. Air in the occupied zone usually contains very small amounts of turbulent kinetic energy compared to that in the jet region. Because turbulence may cause thermal discomfort (Fanger et al. 1989), air diffusion systems should be designed so that the primary mixing between the introduced air and the room air occurs away from occupied regions.
ROOM AIR DIFFUSION METHODS Room air diffusion systems can be classified as mixing, displace-ment, unidirectional, underfloor, and task ambient conditioning.
MIXING SYSTEMS In mixing systems, conditioned air is normally discharged from air outlets at velocities much greater than those acceptable in the occupied zone. Conditioned air temperature may be above, below, or equal to the air temperature in the occupied zone, depending on the heating/cooling load. The diffuser jets mix with the ambient room air by entrainment, which reduces the air velocity and equal-izes the air temperature. The occupied zone is ventilated either by the decayed air jet directly or by the reverse flow created by the jets.
Mixing air distribution creates relatively uniform air velocity, tem-perature, humidity, and air quality conditions in the occupied zone.
Outlet Classification and Performance Straub et al. (1956) and Straub and Chen (1957) classified outlets into five groups: Group A. Outlets mounted in or near the ceiling that discharge air horizontally.
Group B. Outlets mounted in or near the floor that discharge air ver-tically in a nonspreading jet.
Group C. Outlets mounted in or near the floor that discharge air vertically in a spreading jet.
Group D. Outlets mounted in or near the floor that discharge air horizontally.
Group E. Outlets mounted in or near the ceiling that project pri-mary air vertically.
Analysis of outlet performance was based on primary air pattern, total air pattern, stagnant air layer, natural convection currents, return air pattern, and room air motion. Figures 4 through 8 show the room air motion characteristics of the five outlet groups; exterior walls are depicted by heavy lines. The principles of air diffusion emphasized by these figures are as follows: 1. The primary air (shown by dark envelopes in Figures 4 through 8) from the outlet down to a velocity of about 0.75 m/s can be treated analytically. The heating or cooling load has a strong effect on the characteristics of the primary air.
2. The total air, shown by gray envelopes in Figures 4 through 8, is influenced by the primary air and is of relatively high velocity (but less than 0.75 m/s). The total air is also influenced by the environment and drops during cooling or rises during heating; it is not subject to precise analytical treatment.
3. Natural convection currents form a stagnant zone from the ceiling down during cooling, and from the floor up during heating. This zone forms below the terminal point of the total air during heating and above the terminal point during cooling.
Because this zone results from natural convection currents, the air velocities within it are usually low (approximately 0.1 m/s), and the air stratifies in layers of increasing temperatures. The concept of a stagnant zone is important in properly applying and selecting outlets because it considers the natural convection currents from warm and cold surfaces and internal loads.
4. A return inlet affects the room air motion only within its immediate vicinity. The intake should be located in the stagnant zone to return the warmest room air during cooling or the coolest room air during heating. The importance of the location depends on the relative size of the stagnant zone, which depends on the type of outlet.
5. The general room air motion (shown by white areas in Figures 4 through 8) is a gentle drift toward the total air. Room conditions are maintained by the entrainment of the room air into the total airstream. The room air motion between the stagnant zone and the total air is relatively slow and uniform. The highest air motion occurs in and near the total airstreams.
Group A Outlets. This group includes high sidewall grilles, sidewall diffusers, ceiling diffusers, linear ceiling diffusers, and similar outlets. High sidewall grilles and ceiling diffusers are illus-trated in Figure 4.
The primary air envelopes (isovels) show a horizontal, two-jet pattern for the high sidewall and a 360° diffusion pattern for the ceil-ing outlet. Although variation of vane settings might cause a dis-charge in one, two, or three jets in the case of the sidewall outlet, or have a smaller diffusion angle for the ceiling outlet, the general effect in each is the same.
During cooling, the total air drops into the occupied zone at a dis-tance from the outlet that depends on air quantity, supply velocity, temperature differential between supply and room air, deflection setting, ceiling effect, and type of loading within the space. Analy-tical methods of relating some of these factors are presented in the section on Principles of Jet Behavior.
The cooling diagram for the high sidewall outlet shows an over-throw condition, which causes the total air to drop along the oppo-site wall and flow slowly for some distance across the floor.
Velocities of about 0.5 to 0.75 m/s may be found near the wall but will dissipate within about 100 mm of the wall.
The cooling diagram for the ceiling outlet shows that the total air movement is counteracted by the rising natural convection currents on the heated wall, and, therefore, drops before reaching the wall. On the other hand, the total air reaches the inside wall and descends for some distance along it. With this type of outlet, temperature variations in the room are minimized, with minimal stagnant volume. The maximum velocity and the maximum tem-perature variation occur in and near the total air envelope; there-fore, the drop region becomes important because it is an area with high effective draft temperature θ [see Equation (16)]. Con-sequently, how far the air drops before velocities and tempera-tures reach acceptable limits must be known.
Because these outlets discharge horizontally near the ceiling, the warmest air in the room is mixed immediately with the cool primary air far above the occupied zone. Therefore, the outlets are capable of handling relatively large quantities of air at large temperature dif-ferentials.
During heating, warm supply air introduced at the ceiling can cause stratification in the space if there is insufficient induction of room air at the outlet. Selecting diffusers properly, limiting the room supply temperature differential, and maintaining air supply rates at a level high enough to ensure air mixing by induction provide ade-quate air diffusion and minimize stratification.
32.8 2001 ASHRAE Fundamentals Handbook (SI) Several building codes and ASHRAE Standard 90.1 require suf-ficient insulation in exterior walls, so most perimeter spaces can be heated effectively by ceiling air diffusion systems. Interior spaces, which generally have only cooling demand conditions, seldom require long-term heating and are seldom a design problem.
Flow rate and velocity for both heating and cooling are the same for the outlets shown in Figure 4. The heating diagram for the side-wall unit shows that, under these conditions, the total air does not descend along the wall. Consequently, higher velocities might be beneficial in eliminating the stagnant zone, since high velocity causes some warm air to reach floor level and counteract stratifica-tion of the stagnant region.
The heating diagram for the ceiling outlet shows the effect of the natural convection currents that produce a larger throw toward the cold exposed wall. The velocity of the total air toward the exposed wall complements the natural convection currents. However, the warm total air loses its downward momentum at its terminal point, and buoyancy forces cause it to rise toward the primary air.
Although these forces are complementary, the heating effect of the total air replaces the cool natural convection currents with warm total air.
Group B Outlets. This group includes floor registers, baseboard units, low sidewall units, linear-type grilles in the floor or window-sill, and similar outlets. Figure 5 illustrates a floor outlet adjacent to an inside wall. Because these outlets have no deflecting vanes, the primary air is discharged in a single, vertical jet. When the total air strikes the ceil-ing, it fans out in all directions and, during cooling, follows the ceil-ing for some distance before dropping toward the occupied zone.
During heating, the total airflow follows the ceiling across the room, then descends partway down the exterior wall.
The cooling diagram shows that a stagnant zone forms outside the total air region above its terminal point. Below the stagnant zone, air temperature is uniform, effecting complete cooling. Also, the space below the terminal point of the total air is cooled satisfac-torily. For example, if total airflow is projected upward for 2.4 m, the region from this level down to the floor will be cooled satisfac-torily. This, however, does not apply to an extremely large space.
Judgment to determine the acceptable size of the space outside the total air is needed. A distance of 4.5 to 6 m between the drop region and the exposed wall is a conservative design value.
A comparison of Figures 4 and 5 for heating shows that the stag-nant region is smaller for Group B outlets than for Group A outlets Fig. 4 Air Motion Characteristics of Group A Outlets (Straub et al. 1956) Space Air Diffusion 32.9 Fig. 5 Air Motion Characteristics of Group B Outlets (Straub et al. 1956) Fig. 6 Air Motion Characteristics of Group C Outlets (Straub et al. 1956) 32.10 2001 ASHRAE Fundamentals Handbook (SI) because the air entrained in the immediate vicinity of the outlet is taken mainly from the stagnant region, which is the coolest air in the room. This results in greater temperature equalization and less buoyancy in the total air than would occur with Group A outlets.
While the temperature gradients for both outlet groups are about the same, the stagnant layer for Group B is lower than that for Group A.
Group C Outlets. This group includes floor diffusers, sidewall diffusers, linear-type diffusers, and other outlets installed in the floor or windowsill (Figure 6).
Although Group C outlets are related to Group B outlets, they are characterized by wide-spreading jets and diffusing action. Total air and room air characteristics are similar to those of Group B, although the stagnant zone formed is larger during cooling and smaller during heating. Diffusion of the primary air usually causes the total air to fold back on the primary and total air during cooling, instead of fol-lowing the ceiling. This diffusing action of the outlets makes it more difficult to project the cool air, but it also provides a greater area for induction of room air. This action is beneficial during heating because the induced air comes from the lower regions of the room.
Group D Outlets. This group includes baseboard and low side-wall registers and similar outlets (Figure 7) that discharge the pri-mary air in single or multiple jets. During cooling, because the air is discharged horizontally across the floor, the total air remains near the floor, and a large stagnant zone forms in the entire upper region of the room.
During heating, the total air rises toward the ceiling because of the buoyant effect of warm air. The temperature variations are uni-form, except in the total air region.
Group E Outlets. This group includes ceiling diffusers, linear-type grilles, sidewall diffusers and grilles, and similar outlets mounted or designed for vertical downward air projection. Figure 8 shows the heating and cooling diagrams for such a ceiling diffuser.
During cooling, the total air projects to and follows the floor, producing a stagnant region near the ceiling. During heating, the total airflow reaches the floor and folds back toward the ceiling. If projected air does not reach the floor, a stagnant zone results.
Factors Affecting Outlet Performance Vanes. Vanes affect grille performance if their depth is at least equal to the distance between the vanes (vane ratio ≥ 1). If the vane ratio is less than unity, effective control by the vanes of the airstream discharged from the grille is impossible. Increasing the vane ratio Fig. 7 Air Motion Characteristics of Group D Outlets (Straub et al. 1956) Fig. 8 Air Motion Characteristics of Group E Outlets (Straub et al. 1956) Space Air Diffusion 32.11 above two has little or no effect, so vane ratios should be between one and two.
A grille discharging air uniformly forward (vanes in straight position) has a spread of 14 to 24°, depending on the type of outlet, the duct approach, and the discharge velocity. Turning the vanes influences the direction and throw of the discharged airstream.
A grille with diverging vanes (vertical vanes with uniformly increasing angular deflection from the centerline to a maximum at each end of 45°) has a spread of about 60° and reduces the throw considerably. With increasing divergence, the quantity of air dis-charged by a grille for a given upstream total pressure decreases.
A grille with converging vanes (vertical vanes with uniformly decreasing angular deflection from the centerline) has a slightly higher throw than a grille with straight vanes, but the spread is approximately the same for both settings. The airstream converges slightly for a short distance in front of the outlet and then spreads more rapidly than air discharged from a grille with straight vanes.
In addition to vertical vanes that normally spread the air horizon-tally, horizontal vanes may spread the air vertically. However, spreading the air vertically risks hitting beams or other obstructions or blowing primary air into the occupied zone at excessive veloci-ties. On the other hand, vertical deflection may increase adherence to the ceiling and reduce the drop.
Beamed Ceilings and Obstructions. In spaces with exposed beams, the outlets should be located below the bottom of the lowest beam level, preferably low enough to employ an upward or arched air path. The air path should be arched sufficiently to miss the beams and prevent the primary or induced airstream from striking furniture and obstacles and producing objectionable drafts (Wilson 1970).
Obstructions influence airflow patterns and can reduce air distribu-tion efficiency. Obstructions can reduce jet throw, increase air veloc-ities in portions of the occupied zone, and create stagnant zones.
Variable Air Volume (VAV) Systems. The design of air distri-bution systems is usually based on the full load (heating/cooling).
When only a partial load exists, VAV systems reduce the supply airflow, which in turn reduces the air velocity at the outlet. There-fore, the different operation modes of the system (airflow and ini-tial temperature difference) should be considered in designing a VAV system air distribution.
DISPLACEMENT VENTILATION In displacement ventilation, conditioned air with a temperature slightly lower than the desired room air temperature in the occupied zone is supplied from air outlets at low air velocities (0.5 m/s or less). The outlets are located at or near the floor level, and the supply air is introduced directly to the occupied zone. Returns through which the warm room air is exhausted from the room are located at or close to the ceiling. The supply air is spread over the floor and then rises as it is heated by the heat sources in the occupied zone.
Heat sources (e.g., person, computer) in the occupied zone create upward convective flows in the form of thermal plumes. These ther-mal plumes tend to remove heat and contaminants within the plume from the occupied zone (Figure 9). The air volume in the plumes increases as they rise because the plumes entrain ambient air. A stratification level exists where the airflow rate in the plumes equals the supply airflow rate. Two dis-tinct zones are thus formed within the room: one lower zone below the stratification level and with no recirculation flow (close to dis-placement flow), and one upper zone, with recirculation flow (Fig-ure 9). The height of the lower zone depends on the supply airflow rate and the characteristics of heat sources and their distribution across the floor area. In a properly designed displacement ventila-tion system, the upper boundary of the lower zone is above the occu-pied zone so that the occupied zone can be ventilated effectively.
For this type of system to function properly, a stable vertically strat-ified temperature field is essential.
In contrast to mixing ventilation, displacement ventilation is designed to minimize mixing of air within the occupied zone. The objective of the displacement ventilation is to create conditions close to supply air conditions in the occupied zone. This type of ventilation was originally used in industrial buildings as an effec-tive method for removing contaminants in the occupied zone. It is now also used for ventilating and cooling office buildings. How-ever, local discomfort due to draft and vertical temperature gradi-ent may be critical (Melikov and Nielsen 1989). Sandberg and Blomqvist (1989) suggest that the maximum convective cooling load in office buildings with displacement ventilation not exceed about 25 W/m2 so that the maximum vertical temperature gradient in the occupied zone will not be larger than 3 K. This is equivalent to 5 L/(s·m2) at a maximum cooling differential of 4 K. Kegel and Schulz (1989) and Svensson (1989) suggest somewhat higher cooling load limits of 30 to 40 W/m2.
One way of increasing the cooling capacity of displacement ven-tilation systems is to recirculate some of the room air in the occu-pied zone through an induction circuit; that is, the room air is induced into the supply air and is mixed before discharge through the low-velocity air terminal device into the room. This reduces the room air temperature gradient for a given cooling load, thus allow-ing a cooling load limit of up to 50 W/m2 (Jackman 1991).
Air diffusers with a large outlet area are used to supply air at low velocity. Displacement ventilation has been compared with conven-tional mixing ventilation (Svensson 1989, Seppanen et al. 1989, Stymne et al. 1991). Design guidelines for displacement ventilation can be found in Scaret (1985), Jackman and Appleby (1990), Jack-man (1991), and Shilkrot and Zhivov (1992).
UNIDIRECTIONAL AIRFLOW VENTILATION In this type of ventilation, air is either (1) supplied from the ceil-ing and exhausted through the floor, or vice versa; or (2) supplied through the wall and exhausted through returns at the opposite wall.
The outlets are uniformly distributed over the ceiling, floor, or wall to provide a low-turbulence “plug”-type flow across the entire room. This type of system is mainly used for clean room ventilation, in which the main objective is to remove contaminant particles from the room. Details about clean room ventilation are given in Chapter 15 of the 1999 ASHRAE Handbook—Applications. Unidirectional flow ventilation is also used in other areas, such as computer rooms and paint booths.
UNDERFLOOR AIR DISTRIBUTION AND TASK/AMBIENT CONDITIONING Underfloor air distribution systems are installed with a raised floor through which conditioned air is delivered to the space through floor grilles or as part of the workstation furniture and partitions. Sometimes called localized ventilation, these systems Fig. 9 Schematic of Displacement Ventilation 32.12 2001 ASHRAE Fundamentals Handbook (SI) supply air to local areas that are often near building occupants or other specific locations in the space. In comparison to conventional ceiling-based air diffusion, underfloor air distribution systems gen-erally have a larger number of supply diffusers directly in the occu-pied zone of the building (e.g., in floors, desks, workstation partitions, or theater seats). Air typically returns at or close to ceil-ing level, so that localized systems benefit from the same overall upward movement of air in the room as displacement ventilation systems. In cooling applications, this air movement efficiently removes heat and contaminant sources from the room. Underfloor air distribution differs from displacement ventilation in that (1) it generally uses higher supply volumes, which enable higher cooling loads to be met; and (2) it supplies air at a higher velocity through smaller diffusers. Because air is delivered directly to the occupied zone, the supply air temperature is usually warmer (above about 17.5°C) than that maintained for conventional ceiling distribution in order to avoid occupant discomfort due to drafts.
Bauman et al. (1999), Hanzawa and Nagasawa (1990), Houghton (1995), Loudermilk (1999), McCarry (1995, 1998), Shute (1992, 1995), Sodec and Craig (1990), Spoormaker (1990), and Tanabe and Kimura (1996) describe the results of laboratory studies, case studies of actual installations, field experiments of system perfor-mance, and present design guidelines.
A well-designed underfloor air distribution system also requires less energy and is more flexible in providing and maintaining build-ing services than traditional overhead systems. Extremely low oper-ational static pressures in the underfloor air supply plenum can reduces central fan energy use. Thermal storage in the exposed structural mass in the underfloor plenum (e.g., concrete slab) can save energy and reduce peak cooling loads. The use of raised floor-ing provides maximum flexibility and significantly lowers costs associated with reconfiguring building services, particularly in buildings having high churn rates. (Churn rate is defined as the annual percentage of workers and their associated work spaces in a building that are reconfigured or undergo significant changes.) Fig-ure 10 shows a schematic diagram of an underfloor air distribution system.
Task/Ambient Conditioning (TAC) Task/ambient conditioning (TAC) is most commonly installed with underfloor air distribution (Arens et al. 1991; Bauman et al.
1991, 1993, 1995, 1998; Bauman and Arens 1996; Faulkner et al.
1993, 1999; Fisk et al. 1991; Matsunawa et al. 1995; Tsuzuki et al.
1999). TAC gives individuals some control over their local environ-ment without adversely affecting that of nearby occupants. Typi-cally, the occupant can control the speed, direction, and, in some cases, temperature of the incoming air supply. TAC systems have been most frequently installed in open-plan offices in which they provide supply air and, in some cases, radiant heating directly into workstations. Figure 11 shows an underfloor TAC system with a local (personal HVAC) diffuser located in the partition in front of the office worker (Matsunawa et al. 1995).
As further evidence of the benefits of providing personal control, field research has found that building occupants who have no indi-vidual control capabilities are twice as sensitive to changes in tem-perature as occupants who do have individual thermal control (de Dear and Brager 1999, Bauman et al. 1998). METHODS OF EVALUATION Standards for Satisfactory Conditions The object of air diffusion in warm-air heating, ventilating, and air-conditioning is to create the proper combination of temperature, humidity, and air motion in the occupied zone of the conditioned room—from the floor to 1.8 m above floor level (Miller 1989). The effective draft temperature combines the effects of air temperature, air motion, and relative humidity in terms of their physiological effects on a human body. Variation from accepted standard limits (ASHRAE Standard 55) may cause occupant discomfort. Lack of uniform conditions within the space or excessive fluctuation of conditions in the same part of the space also produces discomfort.
Discomfort can arise due to any of the following conditions: • Excessive air motion (draft) • Excessive room air temperature variations (horizontal, vertical, or both) • Failure to deliver or distribute air according to the load requirements at different locations • Overly rapid fluctuation of room temperature Draft. Koestel and Tuve (1955) and Reinmann et al. (1959) stud-ied the effect of air motion on comfort and defined draft as any localized feeling of coolness or warmth of any portion of the body due to both air movement and air temperature, with humidity and radiation considered constant. The warmth or coolness of a draft was measured above or below a controlled room condition of 24°C dry-bulb at the center of the room, 0.75 m above the floor, with air moving at about 0.15 m/s.
To define the effective draft temperature θ (the difference in temperature between any point in the occupied zone and the control condition), the investigators used the following equation proposed by Rydberg and Norback (1949) and modified by Straub in discus-sion of a paper by Koestel and Tuve (1955): Fig. 10 Underfloor Air Distribution System Fig. 11 Underfloor TAC and Personal HVAC System (Matsunawa et al. 1995) Space Air Diffusion 32.13 (16) where θ = effective draft temperature, K tx = local airstream dry-bulb temperature, °C tc = average (control) room dry-bulb temperature, °C Vx = local airstream centerline velocity, m/s Equation (16) accounts for the feeling of coolness produced by air motion and is used to establish the neutral line in Figure 12. In summer, the local airstream temperature tx is below the control tem-perature tc. Hence, both temperature and velocity terms are negative when velocity Vx is greater than 0.15 m/s, and they both add to the feeling of coolness. If, in winter, tx is above tc, any air velocity above 0.15 m/s subtracts from the feeling of warmth produced by tx.
Therefore, it is usually possible to have zero difference in effective temperature between location x and the control point in winter, but not in summer.
Houghten et al. (1938) presented data that make it possible to interpret statistically the percentage of room occupants that will object to a given draft condition. Figure 12 presents the data in the form used by Koestel and Tuve (1955). The data show that a person tolerates higher velocities and lower temperatures at ankle level than at neck level. Because of this, conditions in the zone extending from approximately 0.75 to 1.5 m above the floor are more critical than conditions nearer the floor.
Air Velocity. Room air velocities less than 0.25 m/s are generally preferred; however, Figure 12 shows that even higher velocities may be acceptable to some occupants. ASHRAE Standard 55 rec-ommends elevated air speeds at elevated air temperatures. No min-imum air speeds are recommended for comfort, although air speeds below 0.1 m/s are usually imperceptible.
Temperature Gradient. Figure 12 also shows that up to 20% of occupants will not accept an ankle-to-sitting-level gradient of about 2 K. Poorly designed or operated systems in a heating mode can create this condition, which emphasizes the importance of proper selection and operation of perimeter systems. The section on Outlet Classification and Performance describes possible regions of high room air velocities caused by various outlets; the section on Outlet Location and Selection describes how to evaluate acceptable air diffusion.
AIR DIFFUSION PERFORMANCE INDEX (ADPI) A high percentage of people are comfortable in sedentary (office) occupations where the effective draft temperature θ, as defined in Equation (16), is between −1.5 and +1 K and the air velocity is less than 0.35 m/s. If several measurements of air veloc-ity and air temperature are made throughout the occupied zone of an office, the ADPI is the percentage of locations where measurements were taken that meet these specifications for effective draft temper-ature and air velocity. If the ADPI is maximum (approaching 100%), the most desirable conditions are achieved (Miller and Nev-ins 1969, 1970, 1972, 1974; Miller 1971; Miller and Nash 1971; Nevins and Ward 1968; Nevins and Miller 1972).
The ADPI is based only on air velocity and the effective draft temperature (a combination of local temperature variations from the room average) and is not directly related to the dry-bulb temperature or relative humidity. These and similar effects, such as mean radiant temperature, must be accounted for separately according to ASHRAE Standard 55.
The ADPI is for cooling mode conditions; a measurement tech-nique is specified in ASHRAE Standard 113. Heating conditions can be evaluated using ASHRAE Standard 55 guidelines or ISO Standard 7730. The ADPI technique uses isothermal throw data determined under ASHRAE Standard 70 (see Table 4) to predict what will happen under cooling conditions.
ADPI Selection Guide Jet Throw. The throw of a jet is the distance from the outlet to a point where the maximum velocity in the stream cross section has been reduced to a selected terminal velocity. To estimate ADPI, ter-minal velocity VT was selected for all diffusers as 0.25 m/s, except in the case of ceiling slot diffusers, where it was selected as 0.5 m/s.
Each manufacturer gives data for the throw of a jet from various dif-fusers for isothermal conditions and without a boundary wall inter-fering with the jet.
The throw distance of a jet is denoted by TV, where subscript V indicates the terminal velocity for which the throw is given. The characteristic room length L is the distance from the diffuser to the nearest boundary wall in the principle horizontal direction of the air-flow. However, where air injected into the room does not impinge on a wall surface but collides with air from a neighboring diffuser, the characteristic length is one-half the distance between diffusers plus the distance the mixed jet travels downward to reach the occu-pied zone. Table 3 summarizes definitions of characteristic length for various diffusers.
The midplane between diffusers also can be considered the module line when diffusers serve equal modules throughout a θ tx tc – ( ) 8 Vx 0.15 – ( ) – = Fig. 12 Percentage of Occupants Objecting to Drafts in Air-Conditioned Room 32.14 2001 ASHRAE Fundamentals Handbook (SI) space, and a characteristic length consideration can be based on module dimension d.
Load Considerations. The recommendations in Table 4 cover cooling loads of up to 250 W per square metre of floor surface. The loading is distributed uniformly over the floor up to about 22 W/m2, lighting contributes about 30 W/m2, and the remainder is supplied by a concentrated load against one wall that simulates a business machine or a large sun-loaded window. Over this range of data, the maximum ADPI condition is lower for the highest loads; however, the optimum design condition changes only slightly with the load.
Design Conditions. The quantity of air must be known from other design specifications. If it is not known, the solution must be obtained by trial and error.
The devices for which data were obtained are (1) high sidewall grilles, (2) cone-type circular ceiling diffusers, (3) sill grilles, (4) two- and four-slot ceiling diffusers, (5) light troffer diffusers, and (6) square-faced perforated and louvered ceiling diffusers. Table 2 summarizes the results of the recommendations on values of TV/L by giving the value of TV/L at which the ADPI is a maximum for various loads, as well as a range of values of TV/L for which ADPI is above a minimum specified value.
SYSTEM DESIGN DESIGN CONSIDERATIONS Noise The noise generated by diffusers transmits to the occupied space directly and cannot be attenuated. Therefore, the diffusion system design should meet the sound level criteria specified in Chapter 46 of the 1999 ASHRAE Handbook—Applications.
Duct Approaches to Diffuser Outlets The manner in which the airstream approaches the diffuser outlet is important. For correct air diffusion, the velocity of the airstream must be as uniform as possible over the entire cross-sectional area of the connecting duct and must be perpendicular to the outlet face.
Effects of improper duct approach generally cannot be corrected by the diffuser.
If the system is designed carefully, a wall grille installed at the end of a horizontal duct and a ceiling outlet at the end of a vertical duct receive the air perpendicularly and at uniform velocity over the entire duct cross section. However, few outlets are installed in this way. Most sidewall outlets are installed either at the end of vertical ducts or in the side of horizontal ducts, and most ceiling outlets are attached either directly to the bottom of horizontal ducts or to special vertical takeoff ducts that connect the outlet with the horizontal duct.
In all these cases, special devices for directing and equalizing the air-flow are necessary for proper direction and diffusion of the air.
The influence of the duct approach on outlet performance has been investigated for vertical stack heads with plain openings (Nel-son et al. 1940) or equipped with grilles (Nelson et al. 1942) and side outlets on horizontal ducts (Nelson and Smedberg 1943). In tests conducted with the stack heads, splitters or guide vanes in the elbows at the top of the vertical stacks are needed, regardless of the shape of the elbows (rounded, square, or expanding). Cushion chambers at the top of the stack heads are not beneficial. Figure 13 shows the direction of flow, diffusion, and velocity (measured 300 mm from the opening) of the air for various stack heads tested, expanding from a 350 mm by 150 mm stack to a 350 mm by 230 mm opening, without a grille. The air velocity for each was 2.5 m/s in the stack below the elbow, but the direction of flow and the diffusion pattern indicate performance obtained with nonexpanding elbows of similar shapes for velocities from 1 to 2 m/s. In tests conducted with 75 mm by 250 mm, 100 mm by 230 mm, and 150 mm by 150 mm side outlets in a 150 mm by 510 mm hor-izontal duct at duct velocities of 1 to 7 m/s in the horizontal duct sec-tion, multiple curved deflectors produced the best flow characteristics. Vertical guide strips in the outlet were not as effec-tive as curved deflectors. A single scoop-type deflector at the outlet did not improve the flow pattern obtained from a plain outlet and, therefore, was not desirable.
Return and Exhaust Openings Selection. The selection of return and exhaust openings depends on (1) velocity in the occupied zone near the openings, (2) permis-sible pressure drop through the openings, and (3) noise.
Velocity. Airflow patterns and room air movement are not influ-enced by the location of the return and exhaust outlets beyond a dis-tance of one characteristic length of the return or exhaust opening (e.g., square root of the opening area). Air handled by the opening approaches the opening from all directions, and its velocity decreases rapidly as the distance from the opening increases. There-fore, drafty conditions rarely occur near return openings. Table 5 shows recommended return opening face velocities.
Table 3 Characteristic Room Length for Several Diffusers Diffuser Type Characteristic Length L High sidewall grille Distance to wall perpendicular to jet Circular ceiling diffuser Distance to closest wall or intersecting air jet Sill grille Length of room in direction of jet flow Ceiling slot diffuser Distance to wall or midplane between outlets Light troffer diffusers Distance to midplane between outlets plus distance from ceiling to top of occupied zone Perforated, louvered ceiling diffusers Distance to wall or midplane between outlets Table 4 Air Diffusion Performance Index (ADPI) Selection Guide Terminal Device Room Load, W/m2 T0.25/L for Maximum ADPI Maximum ADPI For ADPI Greater than Range of T0.25/L High sidewall grilles 250 1.8 68 — — 190 1.8 72 70 1.5–2.2 125 1.6 78 70 1.2–2.3 65 1.5 85 80 1.0–1.9 Circular ceiling diffusers 250 0.8 76 70 0.7–1.3 190 0.8 83 80 0.7–1.2 125 0.8 88 80 0.5–1.5 65 0.8 93 90 0.7–1.3 Sill grille, straight vanes 250 1.7 61 60 1.5–1.7 190 1.7 72 70 1.4–1.7 125 1.3 86 80 1.2–1.8 65 0.9 95 90 0.8–1.3 Sill grille, spread vanes 250 0.7 94 90 0.6–1.5 190 0.7 94 80 0.6–1.7 125 0.7 94 — — 65 0.7 94 — — Ceiling slot diffusers (for T100/L) 250 0.3 85 80 0.3–0.7 190 0.3 88 80 0.3–0.8 125 0.3 91 80 0.3–1.1 65 0.3 92 80 0.3–1.5 Light troffer diffusers 190 2.5 86 80 <3.8 125 1.0 92 90 <3.0 65 1.0 95 90 <4.5 Perforated, louvered ceil-ing diffusers 35–160 2.0 96 90 1.4–2.7 80 1.0–3.4 Space Air Diffusion 32.15 Permissible pressure drop. Permissible pressure drop depends on the choice of the designer. Proper pressure drop allowances should be made for control or directive devices.
Noise. The problem of noise in return openings is the same as that in supply outlets. In computing room noise levels resulting from the operation of an air-conditioning system, the return opening must be included as part of the total grille area.
Location. The openings should be located to minimize short-circuiting of supply air. If air is supplied by the jets attached to the ceiling, exhaust openings should be located between the jets or at the side of the room away from the supply air jets. In rooms with vertical temperature stratification, such as foundries, computer rooms, theaters, bars, kitchens, dining rooms, and club rooms, exhaust openings should be located near the ceiling to collect warm air, odors, and fumes.
For industrial rooms with gas release, selection of exhaust open-ing locations depends on the density of the released gases and their temperature; locations should be specified for each application.
Exhaust outlets located in walls and doors, depending on their elevation, have the characteristics of either floor or ceiling returns.
In large buildings with many small rooms, return air may be brought through door grilles or door undercuts into the corridors and then to a common return or exhaust. If the pressure drop through door returns is excessive, the air diffusion to the room may be seriously unbalanced by opening or closing the doors. Outward leakage through doors or windows cannot be counted on for dependable results.
System Balancing Ducts and diffusers in a system should be sized so that the supply of air is distributed properly. However, for flexibility, use standard sizes and allow for future redistribution; the system as designed may not be self-balancing. Chapter 36 of the 1999 ASHRAE Handbook— Applications describes the procedures used to balance air distribu-tion systems.
DESIGN PROCEDURE 1. Determine the air volumetric flow requirements based on load and room size. For VAV systems, evaluation should include the range of flow rates from minimum occupied to design load.
2. Select the tentative diffuser type and location within room.
3. Determine the room’s characteristic length L (Table 3).
4. Select the recommended TV/L ratio from Table 4.
5. Calculate the throw distance TV by multiplying the recom-mended TV/L ratio from Table 4 by the room length L.
6. Locate appropriate outlet size from the manufacturer’s catalog.
7. Ensure that this outlet meets other imposed specifications, such as for noise and for static pressure.
Example 2.
Specifications: Room size 6 m by 4 m with 2.5 m ceiling Loading Uniform, 30 W/m2 or 720 W Air volumetric flow 5 × 10−3 m3/(s·m2) or 0.12 m3/s for the one outlet Device High sidewall grille, located at center of 4 m endwall, 230 mm from ceiling Calculations: Characteristic length L = 6 m (length of room: Table 3) Recommended TV/L = 1.5 (Table 4) Throw to 0.25 m/s T50 = 1.5 × 6 = 9 m Refer to the manufacturer’s catalog for a size that gives this isother-mal throw to 0.25 m/s. One manufacturer recommends the following sizes, when vanes are straight, discharging 0.12 m3/s: 400 mm by 100 mm, 300 mm by 125 mm, or 250 mm by 150 mm.
OUTLET LOCATION AND SELECTION No criteria have been established for choosing among the six types of outlets to obtain an optimum ADPI. All outlets tested, when used according to these recommendations, can have ADPI values that are satisfactory (greater than 90% for loads less than 130 W/m2).
The design of an air distribution and air diffusion system is influ-enced by the same factors that influence the design of an air-condi-tioning plant—building use, size, and construction type. Location and selection of the supply outlets is further influenced by the inte-rior design of the building, local sources of heat gain or loss, and outlet performance and design.
Local sources of heat gain or loss promote convection currents or cause stratification; they may, therefore, determine both the type and location of the supply outlets. Outlets should be located to neu-tralize any undesirable convection currents set up by a concentrated load. If a concentrated heat source is located at the occupancy level of the room, the heating effect can be counteracted (1) by directing cool air toward the heat source or (2) by locating an exhaust or return grille adjacent to the heat source. The second method is more economical for cooling applications, since heat is withdrawn at its source rather than dissipated into the conditioned space. Where lighting loads are heavy (50 W/m2) and ceilings relatively high (above 4.5 m), outlets should be located below the lighting load, and the stratified warm air should be removed by an exhaust or return fan. An exhaust fan is recommended if the wet-bulb temperature of the air is above that of the outdoors; a return fan is recommended if the wet-bulb temperature is below this temperature. These methods reduce the requirements for supply air. Enclosed lights are more Table 5 Recommended Return Inlet Face Velocities Inlet Location Velocity Across Gross Area, m/s Above occupied zone > 4 Within occupied zone, not near seats 3 to 4 Within occupied zone, near seats 2 to 3 Door or wall louvers 1 to 1.5 Through undercut area of doors 1 to 1.5 Fig. 13 Outlet Velocity and Air Direction Diagrams for Stack Heads with Expanding Outlets 32.16 2001 ASHRAE Fundamentals Handbook (SI) economical than exposed lights because a considerable portion of the energy is radiant.
Based on the analysis of the outlet performance tests conducted by Straub et al. (1956) and Straub and Chen (1957), the following are selection considerations for outlets in Groups A through E.
Group A Outlets Outlets mounted in or near the ceiling with horizontal air dis-charge should not be used with temperature differentials exceeding 15 K during heating. Researchers have recommended that temper-ature differentials not exceed 8 K during heating (Hart and Int-Hout 1980, Lorch and Straub 1983). Consequently, such outlets should be used for heating buildings located in regions where winter heating is only a minor problem and, in northern latitudes, solely for interior spaces. However, these outlets are particularly suited for cooling and can be used with high airflow rates and large temperature dif-ferentials. They are usually selected for their cooling characteris-tics.
The performance of these outlets is affected by various factors.
Vane deflection settings reduce throw and drop by changing air from a single straight jet to a wide-spreading or fanned-out jet.
Accordingly, a sidewall outlet with 0° deflection has a longer throw and a greater drop than a ceiling diffuser with a single 360° angle of deflection. Sidewall grilles and similar outlets with other deflection settings may have performance characteristics between these two extremes.
Wide deflection settings also cause a ceiling effect, which increases the throw and decreases the drop. To prevent smudging, the total air should be directed away from the ceiling, but this is rarely practicable, except for very high ceilings. For optimum air diffusion in areas with normal ceilings, total air should scrub the ceiling surface.
Drop increases and throw decreases with larger temperature dif-ferentials. For constant temperature differential, airflow rate affects drop more than velocity. Therefore, to avoid drop, several small out-lets may be better in a room than one large outlet.
With the data in the section on Principles of Jet Behavior, the throw may be selected for a portion of the distance between the out-let and wall or, preferably, for the entire distance. For outlets in opposite walls, the throw should be one-half the distance between the walls. Following these recommendations, the air drops before striking the opposite wall or the opposing airstream. To counteract specific sources of heat gain or to provide higher air motion in rooms with high ceilings, it may be necessary to select a longer throw. In no case should the drop exceed the distance from the outlet to the 1.8 m level.
To maintain maximum ventilation effectiveness with ceiling dif-fusers, throws should be kept as long as possible. With VAV designs, some overthrow at maximum design volumes will be desir-able—the highest induction can be maintained at reduced flows.
Adequate induction by a ceiling-mounted diffuser prevents short-circuiting of unmixed supply air between supply outlet and ceiling-mounted returns.
Group B Outlets In selecting these outlets, it is important to provide enough throw to project the air high enough for proper cooling in the occupied zone. An increase in supply air velocity improves air diffusion dur-ing both heating and cooling. Also, a terminal velocity of about 0.75 m/s is found at the same distance from the floor during both heating and cooling. Therefore, outlets should be selected from the data given in the section on Principles of Jet Behavior, with throw based on a terminal velocity of 0.75 m/s.
With outlets installed near the exposed wall, the primary air is drawn toward the wall, resulting in a wall effect similar to the ceil-ing effect for ceiling outlets. This scrubbing of the wall increases heat gain or loss. To reduce scrubbing, outlets should be installed some distance from the wall, or the supply air should be deflected at an angle away from the wall. However, to prevent the air from drop-ping into the occupied zone before it reaches maximum projection the distance should not be too large nor the angle too wide. A dis-tance of 150 mm and an angle of 15° is satisfactory.
These outlets do not counteract natural convection currents unless sufficient outlets are installed around the perimeter of the space — preferably in locations of greatest heat gain or loss (under windows). The effect of drapes and blinds must be considered with outlets installed near windows. If installed correctly, outlets of this type handle large airflow rates with uniform air motion and temper-atures.
Group C Outlets These outlets can be used for heating, even with severe heat load conditions. Higher supply velocities produce better room air diffu-sion than lower velocities, but velocity is not critical in selecting these units for heating.
To achieve the required projection for cooling, the outlets should be used with temperature differentials of less than 8 K. With higher temperature differentials, supply air velocity is not sufficient to project the total air up to the desired level.
The outlets have been used successfully for residential heating, but they may also offer a solution for applications where heating requirements are severe and cooling requirements are moderate. For throw, refer to the section on Principles of Jet Behavior.
Group D Outlets These outlets direct high-velocity total air into the occupied zone, and, therefore, are not recommended for comfort, particularly for summer cooling. For heating, outlet velocities should not be higher than 1.5 m/s, so that air velocities in the occupied zone will not be excessive. These outlets have been applied successfully to process installations where controlled air velocities are desired.
Group E Outlets The different throws shown in the heating and cooling diagrams for these outlets become critical in selecting and applying the out-lets. Because the total air enters the occupied zone for both cooling and heating, outlets are used for either cooling or heating, but sel-dom for both.
During cooling, temperature differential, supply air velocity, and airflow rate have considerable influence on projection. Therefore, low values of each should be selected.
During heating, it is important to select the correct supply air velocity to project the warm air into the occupied zone. Temperature differential is also critical because a small temperature differential reduces variation of the throw during the cyclic fluctuation of the supply air temperature. Vane setting for deflection is as important here as it is for Group B and C outlets.
Investigations by Nevins and Ward (1968) and Miller and Nevins (1969) in full-scale interior test rooms indicate that air temperatures and velocities throughout a room cooled by a ventilating ceiling are a linear function of room load (heat load per unit area) and are not affected significantly by variations in ceiling type, total air temper-ature differential, or air volumetric flow rate. Higher room loading produces wider room air temperature variations and higher veloci-ties, which decrease performance.
These studies also found no appreciable difference in the perfor-mance of air-diffusing ceilings and circular ceiling diffusers for lower room loads (65 W/m2). For higher room loads (250 W/m2), an air-diffusing ceiling system has only slightly larger vertical temper-ature variations and slightly lower room air velocities than a ceiling diffuser system.
Space Air Diffusion 32.17 When the ventilating ceiling is used at exterior exposures, the additional load at the perimeter must be considered. During heat-ing operation, the designer must provide for the cold wall effect, as with any ceiling supply diffusion system. The sound generated by the air supply device must also be considered in total system analysis to ensure that room sound levels do not exceed the design criteria.
RETURN AIR DESIGN FOR OPTIMUM PERFORMANCE An HVAC system operating in the cooling mode performs best when generated heat is removed at its source rather than distrib-uted throughout the conditioned space. Heat from solar and mis-cellaneous loads such as machinery and floor or desk-mounted lamps is difficult to remove at the source. However, return air flowing over ceiling-mounted lighting fixtures keeps most of that heat from being distributed into the conditioned space. In addition to increasing HVAC system efficiency, return air lighting fixtures improve light output and extend the life of the lamps. The manu-facturers of fixtures, ceiling grids, and grilles give performance information (airflow rate, pressure drop, and heat removal rate) of their product. Ball et al. (1971) found that the heat removal perfor-mance of return air fixtures covers a narrow range.
With a suspended ceiling, low operating static pressure across the ceiling must be maintained. Failure to do so can result in return air being forced around the edges of the ceiling panels or, in some cases, through the ceiling panels. The result is often a soiled ceiling and a mechanical system that is choked for return air. To avoid this, the static pressure difference across the ceiling should be as low as possible. If necessary, slotted tees or grilles can be used with return air fixtures to obtain the specified pres-sure drop. A maximum pressure drop of 5 to 7.5 Pa is acceptable under most conditions.
At the typical air supply rates found in office interior zone spaces [usually less than 7.5 L/(s·m2)] and with adequate induction at the supply diffusers, the location of the return diffuser has no effect on air patterns in the space. For most office spaces, it is only necessary that sufficient return outlets be provided to maintain inlet velocities within recommendations (see Table 5).
In spaces expected to operate in a cooling mode most of the time, returning the warmest air in the space can effectively reduce energy costs and increase circulation in the space. This is especially true in climates where economizer systems operate for long periods during the year. In spaces having very high ceilings, with atriums, sky-lights, or large vertical glass surfaces, and where the highest areas are unoccupied, air stratification may be used as an energy-saving measure by locating returns near the occupied zone.
CEILING-MOUNTED AIR DIFFUSER SYSTEMS For the best thermal comfort conditions and highest ventilation effectiveness in an occupied space (i.e., office or retail store), the entire system performance of air diffusers should be considered.
This is particularly true for open spaces, where airstreams from dif-fusers may interact with each other, and for perimeter spaces, where airstreams from diffusers interact with hot or cold perimeter walls.
While throw data for individual diffusers are used in system design, an air diffuser system should maintain a high quality of air diffusion in the occupied space with low temperature variation, good air mix-ing, and no objectionable drafts in the occupied space (typically 150 mm to 1.8 m above the floor).
Adequate ventilation requires that the selected diffusers effec-tively mix (by entrainment) the total air in the room with the sup-plied conditioned air, which is assumed to contain adequate ventilation air.
Interior Spaces An interior space is conditioned exclusively for cooling loads, except after unoccupied periods when the space may have cooled to below a comfortable temperature. Tests by Miller and Nevins (1970), Miller and Nash (1971), Miller (1979), and Hart and Int-Hout (1981) suggest that the air diffusion performance index (ADPI) can be improved by moving diffusers closer together (i.e., specifying more diffusers for a given space and air quantity) and by limiting the value of the supply air/room air temperature difference.
In a given system of diffusers, these studies found an optimum oper-ating range of air volumetric flow rates at a given thermal load. The operating load varies with diffuser design, ceiling height, thermal load, and diffuser orientation. This information can be obtained by constructing a mock-up representing the proposed building space, with several alternatives tested for ADPI values, in accordance with ASHRAE Standard 113. Usually, the diffuser manufacturer has per-formed these tests and can provide the best choice of design options for a particular building. For a VAV system, the diffuser spacing selection should not be based on maximum or design air volumes but rather on the air volume range in which the system is expected to operate most of the time. For VAV applications, Miller (1979) recommends that the designer consider the expected variation in the outlet air volume to ensure that ADPI values remain above a spec-ified minimum.
Perimeter Spaces All-air mechanical systems that handle both heating and cooling thermal loads are commonly used in modern office buildings instead of baseboards for heating and forced air for cooling. State energy codes (most based on ASHRAE Standard 90 series) require that commercial buildings have exterior walls that meet minimum thermal performance criteria for a particular location. Typically, walls of new buildings have design heat losses as low as 200 to 300 W per linear metre of wall.
A successful all-air heating/cooling mechanical system requires the designer to consider several design variables that have been the subject of research by Hart and Int-Hout (1980), Lorch and Straub (1983), and Rousseau (1983). The most important design variables include • Supply air/room air temperature difference • Diffuser type and design • Design heating and cooling loads • Supply air volumetric flow rates • Distance between diffusers and perimeter wall • Direction of air throw (toward wall, away from wall, or both) • Ceiling height • Desired air diffusion performance criteria The diffuser manufacturer is best able to recommend the use of equipment.
For an office environment in cooling mode, the design goal should be an ADPI greater than 80. The ADPI should not be used as a measure of performance for heating conditions. In both cases, ASHRAE Standard 55 recommends that the maximum temperature gradient (the difference in temperature between any two points) should not exceed 3 K. Linear diffusers placed parallel to the perimeter wall perform well. For year-round operation, linear diffusers with two-way throw (i.e., both toward and away from the perimeter wall) work best.
Lorch and Straub (1983) reported optimum performance with a dif-fuser that throws warm air toward the perimeter wall under heating load conditions and chilled air in both directions under cooling load conditions. All researchers found less than optimum performance with high discharge temperatures (greater than 8 K above ambient), both with one-way throw of air away from a cold wall and with one-way throw of chilled air toward the perimeter wall. Under heating 32.18 2001 ASHRAE Fundamentals Handbook (SI) load conditions, the supply air temperature must be limited to avoid excessive thermal stratification. To resolve any uncertainty about performance, a mockup should be constructed with provisions for a cold wall; several variations of the design should be tested so that the best diffuser wall spacing and supply air volumes can be selected. The ADPI, room temperature gradients, or both, measured in accordance with ASHRAE Standard 113, can help gage system performance.
The following principles provide the best air diffusion quality and minimum energy use: • For cooling load conditions, return air should exhaust from a location that takes advantage of any thermal stratification design.
In many cases, this should be a high point in order to take advantage of rising warm air. Cooling supply air should be introduced as close to the heat sources as possible. Alternately, stratification designs may condition only part of the total space. In these cases, conditioned air is supplied and exhausted as close to the occupants as possible. In either case, comfort zone temperature gradients should be maintained within 3 K.
• For heating load conditions, thermal stratification should be discouraged. Heat should be introduced at points low in the large space. Ceiling-mounted fans may reduce stratification.
REFERENCES Arens, E.A., F. Bauman, L. Johnston, and H. Zhang. 1991. Testing of local-ized ventilation systems in a new controlled environment chamber.
Indoor Air 3:263-81.
ASHRAE. 1990. Method of testing for room air diffusion. ANSI/ASHRAE Standard 113-1990.
ASHRAE. 1991. Method of testing for rating the performance of air outlets and inlets. ANSI/ASHRAE Standard 70-1991.
ASHRAE. 1992. Thermal environmental conditions for human occupancy.
ANSI/ASHRAE Standard 55-1992.
Ball, H.D., R.G. Nevins, and H.E. Straub. 1971. Thermal analysis of heat removal troffers. ASHRAE Transactions 77(2).
Baturin, V.V. 1972. Fundamentals of industrial ventilation, 3rd England ed.
Translated by O.M. Blunn. Pergamon Press, New York.
Bauman, F. and E. Arens. 1996. Task/ambient conditioning systems: Engi-neering and application guidelines. Center for Environmental Design Research, University of California, Berkeley.
Bauman, F.S., E.A. Arens, S. Tanabe, H. Zhang, and A. Baharlo. 1995. Test-ing and optimizing the performance of a floor-based task conditioning system. Energy and Buildings 22(3):173-86.
Bauman, F.S., T.G. Carter, A.V. Baughman, and E.A. Arens. 1998. Field study of the impact of a desktop task/ambient conditioning system in office buildings. ASHRAE Transactions 104(1).
Bauman, F.S., L. Johnston, H. Zhang, and E. Arens. 1991. Performance test-ing of a floor-based, occupant-controlled office ventilation system.
ASHRAE Transactions 97(1).
Bauman, F., P. Pecora, and T. Webster. 1999. How low can you go? Air flow performance of low-height underfloor plenums. Center for the Built Environment, University of California, Berkeley.
Bauman, F.S., H. Zhang, E. Arens, and C. Benton. 1993. Localized comfort control with a desktop task conditioning system: Laboratory and field measurements. ASHRAE Transactions 99(2).
Christianson, L.L., ed. 1989. Building systems: Room air and air contami-nant distribution. ASHRAE, Atlanta.
de Dear, R. and G.S. Brager. 1999. Developing an adaptive model of thermal comfort and preference. ASHRAE Transactions 104 (1).
Fanger, P.O., A.K. Melikov, H. Hanzawa, and J. Ring. 1988. Air turbulence and sensation of draft. Energy and Buildings 12:21-39.
Faulkner, D., W.J. Fisk, and D.P. Sullivan. 1993. Indoor air flow and pollut-ant removal in a room with desktop ventilation. ASHRAE Transactions 99(2).
Faulkner, D., W.J. Fisk, D.P. Sullivan, and D.P. Wyon. 1999. Ventilation effi-ciencies of task/ambient conditioning systems with desk-mounted air supplies. Proceedings Indoor Air ’99, Edinburgh, Scotland, 8-13 August.
Fisk, W.J., D. Faulkner, D. Pih, P. McNeel, F. Bauman, and E. Arens. 1991.
Indoor air flow and pollutant removal in a room with task ventilation.
Indoor Air 3:247-62.
Hanzawa, H., and Y. Nagasawa. 1990. Thermal comfort with underfloor air-conditioning systems. ASHRAE Transactions 96(2).
Hart, G.H. and D. Int-Hout. 1980. The performance of a continuous linear diffuser in the perimeter zone of an office environment. ASHRAE Trans-actions 86(2).
Hart, G.H. and D. Int-Hout. 1981. The performance of a continuous linear diffuser in the interior zone of an open office environment. ASHRAE Transactions 87(2).
Helander, L. and C.V. Jakowatz. 1948. Downward projection of heated air.
ASHVE Transactions 54:71.
Helander, L., S.M. Yen, and R.E. Crank. 1953. Maximum downward travel of heated jets from standard long radius ASME nozzles. ASHVE Trans-actions 59:241.
Helander, L., S.M. Yen, and L.B. Knee. 1954. Characteristics of downward jets of heated air from a vertical delivery discharge unit heater. ASHVE Transactions 60:359.
Helander, L., S.M. Yen, and W. Tripp. 1957. Outlet characteristics that affect the downthrow of heated air jets. ASHAE Transactions 63:255.
Houghten, F.C., C. Gutberlet, and E. Witkowski. 1938. Draft temperatures and velocities in relation to skin temperatures and feelings of warmth.
ASHVE Transactions 44:289.
Houghton, D. 1995. Turning air conditioning on its head: Underfloor air dis-tribution offers flexibility, comfort, and efficiency. E Source TU-95-8. E Source, Inc., Boulder, CO.
ISO. 1994. Moderate thermal environments—Determination of the PMV and PPD indices and specification of the conditions for thermal comfort.
ISO Standard 7730-1994. International Organization for Standardiza-tion, Geneva.
Jackman, P.J. 1991. Displacement ventilation. CIBSE National Conference.
Chartered Institution of Building Services Engineers, London.
Jackman, P.J., and P.A. Appleby. 1990. Displacement flow ventilation.
BSRIA Project Report. The Building Services Research and Information Association, Old Bracknell Lane, Berkshire, RG12 4AH, UK.
Kegel, B. and U.W. Schulz. 1989. Displacement ventilation for office build-ings. Proceedings of the 10th AIVC Conference, Helsinki. Air Infiltration and Ventilation Center, University of Warwick Science Park, Barclays Venture Centre, Sir William Lyons Road, Coventry CV4 7EZ, UK.
Kirkpatrick, A. and J. Elleson. 1996. Design guide for cold air distribution systems. ASHRAE, Atlanta.
Kirkpatrick, A., T. Malmstrom, P. Miller, and V. Hassani. 1991. Use of low temperature air for cooling of buildings. Proceedings Building Simulation.
Knaak, R. 1957. Velocities and temperatures on axis of downward heated jet from 4-inch long-radius ASME nozzle. ASHAE Transactions 63:527.
Koestel, A. 1954. Computing temperatures and velocities in vertical jets of hot or cold air. ASHVE Transactions 60:385.
Koestel, A. 1955. Paths of horizontally projected heated and chilled air jets.
ASHAE Transactions 61:213.
Koestel, A. 1957. Jet velocities from radial flow outlets. ASHAE Transac-tions 63:505.
Koestel, A. and J.B. Austin, Jr. 1956. Air velocities in two parallel ventilat-ing jets. ASHAE Transactions 62:425.
Koestel, A. and G.L. Tuve. 1955. Performance and evaluation of room air distribution systems. ASHRAE Transactions 61:533.
Koestel, A., P. Hermann, and G.L. Tuve. 1949. Air streams from perforated panels. ASHVE Transactions 55:283.
Koestel, A., P. Hermann, and G.L. Tuve. 1950. Comparative study of venti-lating jets from various types of outlets. ASHVE Transactions 56:459.
Li, Z., L.L. Christianson, and J.S. Zhang. 1995. Separation distances of nonisothermal air jets. Research Triangle Park, NC.
Li, Z., J.S. Zhang, A.M. Zhivov, and L.L. Christianson. 1993. Characteris-tics of diffuser air jets and airflow in the occupied regions of mechani-cally ventilated rooms: A literature review. ASHRAE Transactions 99(1):1119-27.
Lorch, F.A. and H.E. Straub. 1983. Performance of overhead slot diffusers with simulated heating and cooling conditions. ASHRAE Transactions 89(1).
Loudermilk, K. 1999. Underfloor air distribution solutions for open office applications. ASHRAE Transactions 105(1).
Matsunawa, K., H. Iizuka, and S. Tanabe. 1995. Development and applica-tion of an underfloor air-conditioning system with improved outlets for a “smart” building in Tokyo. ASHRAE Transactions 101(2):887.
McCarry, B.T. 1995. Underfloor air distribution systems: Benefits and when to use the system in building design. ASHRAE Transactions 101(2).
Space Air Diffusion 32.19 McCarry, B.T. 1998. Innovative underfloor system. ASHRAE Journal 40 (3).
McElroy, G.E. 1943. Air flow at discharge of fan-pipe lines in mines. U.S.
Bureau of Mines Report of Investigations, 19.
Melikov, A.K. and J.B. Nielsen. 1989. Local thermal discomfort due to draft and vertical temperature difference in rooms with displacement ventila-tion. ASHRAE Transactions 95(2):1050-57.
Miller, P.L. 1971. Room air distribution performance of four selected out-lets. ASHRAE Transactions 77(2):194.
Miller, P.L. 1979. Design of room air diffusion systems using the air diffu-sion performance index (ADPI). ASHRAE Journal 10:85.
Miller, P.L. 1989. Descriptive methods. In Building systems: Room air and air contaminant distribution, L.L. Christianson, ed. ASHRAE, Atlanta.
Miller, P.L. and R.T. Nash. 1971. A further analysis of room air distribution performance. ASHRAE Transactions 77(2):205.
Miller, P.L. and R.G. Nevins. 1969. Room air distribution with an air distrib-uting ceiling—Part II. ASHRAE Transactions 75:118.
Miller, P.L. and R.G. Nevins. 1970. Room air distribution performance of ventilating ceilings and cone-type circular ceiling diffusers. ASHRAE Transactions 76(1):186.
Miller, P.L. and R.G. Nevins. 1972. An analysis of the performance of room air distribution systems. ASHRAE Transactions 78(2):191.
Murakami, S. 1992. New scales for ventilation efficiency and their applica-tion based on numerical simulation of room airflow. International Sym-posium on Room Air Convection and Ventilation Effectiveness. Nelson, D.W. and G.E. Smedberg. 1943. Performance of side outlets on hor-izontal ducts. ASHVE Transactions 49:58.
Nelson, D.W., H. Krans, and A.F. Tuthill. 1940. The performance of stack heads. ASHVE Transactions 46:205.
Nelson, D.W., D.H. Lamb, and G.E. Smedberg. 1942. Performance of stack heads equipped with grilles. ASHVE Transactions 48:279.
Nevins, R.G. and P.L. Miller. 1972. Analysis, evaluation and comparison of room air distribution performance. ASHRAE Transactions 78(2):235.
Nevins, R.G. and E.D. Ward. 1968. Room air distribution with an air distrib-uting ceiling. ASHRAE Transactions 74:VI.2.1.
Nottage, H.B., J.G. Slaby, and W.P. Gojsza. 1952a. Isothermal ventilation jet fundamentals. ASHVE Transactions 58:107.
Nottage, H.B., J.G. Slaby, and W.P. Gojsza. 1952b. Outlet turbulence inten-sity as a factor in isothermal-jet flow. ASHVE Transactions 58:343.
Oakes, W.C. 1987. Experimental investigation of Coanda jet. M.S. thesis, Michigan State University, East Lansing, MI.
Reinmann, J.J., A. Koestel, and G.L. Tuve. 1959. Evaluation of three room air distribution systems for summer cooling. ASHRAE Transactions 65:717.
Rousseau, W.H. 1983. Perimeter air diffusion performance index tests for heating with a ceiling slot diffuser. ASHRAE Transactions 89(1).
Rydberg, J. and P. Norback. 1949. Air distribution and draft. ASHVE Trans-actions 55:225.
Sandberg, M. and C. Blomqvist. 1989. Displacement ventilation in office rooms. ASHRAE Transactions 95(2):1041-49.
Scaret, E. 1985. Ventilation by displacement Characterization and design implications. Elsevier Science Publishers, New York.
Seppanen, O.A., W.J. Fisk., J. Eto, and D.T. Grimsrud. 1989. Comparison of conventional mixing and displacement air-conditioning and ventilating systems in U.S. commercial buildings. ASHRAE Transactions 95(2): 1028-40.
Shilkrot, E., and A. Zhivov. 1992. Room ventilation with designed vertical air temperature stratification. ROOMVENT-’92, Proceedings 3rd Inter-national Conference on Engineering Aero- and Thermodynamics of Ven-tilated Rooms.
Shute, R.W. 1992. Integrating access floor plenums for HVAC air distribu-tion. ASHRAE Journal 34(10).
Shute, R.W. 1995. Integrated access floor HVAC: Lessons learned. ASHRAE Transactions 101(2).
Sodec, F., and R. Craig. 1990. The underfloor air supply system—The Euro-pean experience. ASHRAE Transactions 96(2).
Spoormaker, H.J. 1990. Low-pressure underfloor HVAC system. ASHRAE Transactions 96(2).
Straub, H.E. and M.M. Chen. 1957. Distribution of air within a room for year-round air conditioning—Part II. University of Illinois Engineering Experiment Station Bulletin No. 442.
Straub, H.E., S.F. Gilman, and S. Konzo. 1956. Distribution of air within a room for year-round air conditioning—Part I. University of Illinois Engi-neering Experiment Station Bulletin No. 435.
Stymne, H., M. Sandberg, and M. Mattsson. 1991. Dispersion pattern of contaminants in a displacement ventilation room—Implications for demand control. Proceedings, 12th Air Movement and Ventilation Con-trol Within Buildings, Ottawa. AIVC Conference.
Svensson, A.G.L. 1989. Nordic experiences of displacement ventilation sys-tems. ASHRAE Transactions 95(2):1013-17.
Tanabe, S. and K. Kimura. 1996. Comparisons of ventilation performance and thermal comfort among displacement, underfloor and ceiling based air distribution systems by experiments in a real sized office chamber.
Proceedings, 5th International Conference on Air Distribution in Rooms, ROOMVENT ’96.
Tuve, G.L. 1953. Air velocities in ventilating jets. ASHVE Transactions 59:261.
Tsuzuki, K., E.A. Arens, F.S. Bauman, and D.P. Wyon. 1999. Individual thermal comfort control with desk-mounted and floor-mounted task/ ambient conditioning (TAC) systems. Proceedings Indoor Air ’99, Edin-burgh, Scotland, 8-13 August.
Wilson, J.D., M.L. Esmay, and S. Persson. 1970. Wall-jet velocity and tem-perature profiles resulting from a ventilation inlet. ASAE Transactions.
Yen, S.M., L. Helander, and L.B. Knee. 1956. Characteristics of downward jets from a vertical discharge unit heater. ASHAE Transactions 62:123.
Zhang, J.S., L.L. Christianson, and G.L. Riskowski. 1990. Regional airflow characteristics in a mechanically ventilated room under nonisothermal conditions. ASHRAE Transactions 96(1):751-59.
Zhang, J.S., L.L. Christianson, G.J. Wu, and G.L. Riskowski. 1992. Detailed measurements of room air distribution for evaluating numerical simula-tion models. ASHRAE Transactions 98(1):58-65.
BIBLIOGRAPHY Int-Hout, D. 1981. Measurement of room air diffusion in actual office envi-ronments to predict occupant thermal comfort. ASHRAE Transactions 87(2).
Poz, M.Y. 1991. Theoretical investigation and practical applications of nonisothrmal jets for the rooms ventilating. Current East/West HVAC Developments. IEI/CIBSE/ABOK Joint Conference.
Zhivov, A. 1990. Variable-air volume ventilation systems for industrial buildings. ASHRAE Transactions 96(2).
33.1 CHAPTER 33 HVAC COMPUTATIONAL FLUID DYNAMICS Theory ..................................................................................... 33.1 Finite Volume Formulations ................................................... 33.3 Finite Element Formulations .................................................. 33.3 Preprocessors ......................................................................... 33.3 Post Processors ....................................................................... 33.3 Energy Equation ..................................................................... 33.3 Multispecies Flows................................................................... 33.3 Viscosity Models ..................................................................... 33.4 CFD Applications ................................................................... 33.4 Symbols .................................................................................... 33.4 HE APPLICATION of computational fluid dynamics (CFD) to Troom air motion, as summarized by Haghighat et al. (1992), began with investigations into room air flow (Neilsen 1974) and natural convection in enclosed cavities (Catton 1978, Ostrach 1982, Markatos and Pericleous 1984, Lin and Nansteel 1987, Hadjisopho-cleous et al. 1988, Gadgil et al. 1984, Chen et al. 1990b). Lemaire (1987) applied the CHAMPHxN code (Pun and Spalding 1976) coupled with radiation to predict air movement and heat transfer in a room heated by a radiator.
Convective heat and mass tranfer has been analyzed using a commercial code by Holmes (1982), Markatos (1983), Jones and Sullivan (1985), and Chen and Van der Kooi (1988). Murakami et al. (1988), Horstman (1988), and Chen et al. (1990a) developed numerical models of ventilation with contaminant transport.
Partitioned models (multiple zones) have also been developed to predict room air motion. Natural convection was investigated by Chang et al. (1982) and Kelkar and Patankar (1985). Contaminant transport models were used in the ventilation models of Haghighat et al. (1989, 1990).
Theory The basis for ventilation computational fluid dynamics analyses are the incompressible Navier-Stokes equations. These equations describe the motion of a viscous Newtonian fluid. For example, the following represent these equations in two dimensions: (1) (2) (3) Because these equations are complex, only a few exact solutions have been obtained for very simple flow conditions. The typical ventilation system is well beyond the realm of an exact solution.
Forms of the Navier-Stokes equations have been developed using numerical methods that divide the flow field into finite volumes or elements. The elements typically assume uniform properties throughout and exchange pressure, momentum, and viscous dissi-pation information with one another. The solution is obtained iteratively either by time step or by a steady flow updating of each element. The new value of pressure or velocity of an element at each iteration may be incorporated into the overall solution gradually using relaxation methods, which help sta-bilize the solution, giving time for information to travel between the elements and allowing each to assert its influence on the entire flow field. An example of this method is the finite difference approxima-tion of the stream function-vorticity formulation of the Navier-Stokes equations in two dimensions: Example. A two-dimensional model of a ventilated room (Figure 1) is modeled using the stream function-vorticity formulation of the Navier-Stokes equations. The room is 3 m by 3 m. Air enters at 0.5 m/s through a 0.3 m wide opening on the left wall near the ceiling and leaves through a 0.3 m wide opening near the floor. The grid representing the room consists of 41 × 41 elements.
Solution: The stream function-vorticity formulation of the Navier-Stokes equations begins with the vorticity transport equation: (4) The stream function is related to the velocity: (5) Vorticity is analogous to rotation of the fluid: (6) Most CFD models have some sort of turbulence model to account for eddy viscosity when the Reynolds number exceeds about 2000. The simplest of the turbulence models uses the Prandtl mixing length: (7) The preparation of this chapter is assigned to TC 4.10, Indoor Environmen-tal Modeling.
∂u ∂t ------u∂u ∂x ------v∂u ∂y ------+ + 1 ρ ---X 1 ρ --- ∂p ∂x ------– v x2 2 ∂ ∂u y2 2 ∂ ∂u + + = ∂v ∂t -----u∂v ∂x -----v∂v ∂y -----+ + 1 ρ ---X 1 ρ --- ∂p ∂y ------– v x2 2 ∂ ∂v y2 2 ∂ ∂v + + = ∂u ∂x ------∂v ∂y -----+ 0 = Fig. 1 Example Two-Dimensional Model u∂ω ∂x -------v∂ω ∂y -------+ v ∂2ω ∂x2 ----------∂2ω ∂y2 ----------+ = u ∂ψ ∂y -------, = v ∂ψ ∂x -------– = ω ∂v ∂x ------∂u ∂y ------+ ∇ – 2ψ = = vt κLCµ 1 4 ⁄ K 1 2 ⁄ = 33.2 2001 ASHRAE Fundamentals Handbook (SI) In recirculating flows, the length to the wall is difficult to measure and is flow-dependent. Equation (5) defines the length scale by the position of streamlines relative to the two boundary values of the stream function: (8) The turbulent kinetic energy K is a function of V ′, the magni-tude of the fluctuating component of the velocity due to turbulence.
V ′ can be assumed to be about 10% of V, the local average velocity, for typical room air models: (9) Equations (7), (8), and (9) are combined to give a simple turbulence model based on the stream function alone: (10) Finite difference equations of the following form are used to approximate Equations (4) and (6): (11) (12) The equations are rearranged so that the current value of the center cell or element is based on that of the surrounding cells. In addition, the vorticity transport Equation (4) is expanded for second-order accuracy as done by Dennis and Hudson (1978): (13) (14) L ψ V ---- or L ψ ψB – V --------------------= whichever is smaller = Fig. 2 Stream Function-Vorticity Solution for Ventilated Room (50 000 Iterations) K 3 2 -- V ′ ( )2 3 2 -- 0.1V ( )2 = = vt κ ψ Cµ 1 4 ⁄ 3 2 -- 0.1 ( ) = or vt κ ψ ψB – Cµ 1 4 ⁄ 3 2 -- 0.1 ( ) = whichever is smaller ∂f ∂s -----fn 1 + fn 1 – – 2 S ∆ ----------------------------≈ ∂2f ∂s2 --------fn 1 + 2fn fn 1 – + – S2 ∆ ------------------------------------------≈ ωi j , 1 xui 1 j , + ∆ 2vi j , ---------------------– x2ui 1 j , + 2 ∆ 8vi j , 2 -----------------------+ ωi 1 j , + 1 xui 1 j , – ∆ 2vi j , --------------------x2ui 1 j , – 2 ∆ 8vi j , 2 -----------------------+ + + ωi 1 j , – … + … 1 xvi j , 1 + ∆ 2vi j , ---------------------– x2vi j , 1 + 2 ∆ 8vi j , 2 -----------------------+ ωi j , 1 + + 1 xvi j , 1 – ∆ 2vi j , ---------------------x2vi j , 1 – 2 ∆ 8vi j , 2 -----------------------+ + ωi j , 1 – + 4 x2 ∆ ui j , 2 vi j , 2 + ( ) 4vi j , 2 ------------------------------------+ -----------------------------------------------------------------------------------------------------= ψi j , ψi 1 j , + ψi 1 j , – ψi j , 1 + ψi j , 1 – ∆x2ωi j , + + + + 4 ------------------------------------------------------------------------------------------------------------= HVAC Computational Fluid Dynamics 33.3 The subscripted velocities shown in Equation (13) are obtained from Equation (5) in the form of Equation (11). For example, the v velocity at location i,j – 1 is or (15) Boundary conditions for vorticity are defined by the apparent rota-tion rate of the nearby fluid passing by: (16) Relaxation parameters are applied to the vorticity and to the viscos-ity. The parameter δ for the vorticity is typically equal to 0.03, so that 3% of the new value and 97% of the old value is used. The viscosity is unstable, so a smaller relaxation parameter is required; 0.01 was used in this example. To illustrate this procedure, a new value of vorticity is calculated using Equation (13) or (16) (depending on the node type).
The current value of vorticity is obtained by adding most of the old value to a fraction of the new value: (17) The inlets and outlets have a fixed, uniform velocity. The stream function is set at each of the three nodes between the walls. Since the wall below the inlet has ψ = 0 and the wall above has ψ = 1.667, the stream function is divided evenly between the inlet nodes: ψ1,38 = 0.25(1.667) = 0.417; ψ1, 39 = 0.5(1.667) = 0.834; ψ1, 40 = 0.75(1.667) = 1.25. The same distribution is applied to the outlet nodes as shown in Figure 1. In a real room, the velocity would not be uniform across the outlet, but for an illustrative example, this simplification is appropriate.
The results of this analysis method are shown in Figure 2.
These results show the overall flow pattern of the room. Ventila-tion air travels along the ceiling, then attaches to the right wall before exiting. This generates a clockwise rotational flow where the eddy viscosity approaches two hundred times the molecular viscos-ity. At this point, the engineer would make a finer grid and run the model again to see if the solution has reached grid independence.
Finite Volume Formulations Commercially available CFD programs are generally used to solve specific applications. Code development requires frequent validation and benchmarking and is best done by companies that specialize in that task. Most commercially available codes are based on finite volume formulations such as the SIMPLE algorithm (Patankar 1988). The finite volume method is distinguished from the finite element method in that the properties and flow conditions of the fluid are assumed uniform throughout the elemental volume and the exchange of momentum, energy, and pressure occurs at the faces.
Finite Element Formulations Less common than the finite volume formulations, finite element formulations are usually available when the flow solver is packaged commercially with a stress analysis code. The finite element method may be differentiated from the finite volume method in the way the properties and flow conditions are point values and infor-mation travels between nodes, through the elements (Baker 1983).
Until recently, the finite element method had the advantage of geometry adaptability.
Preprocessors The first step in the CFD process is building the grid that math-ematically represents the physical model. The preprocessor con-structs the grid from such data as room dimensions, diffuser and duct dimensions, and internal features such as furniture or partitions that may affect flow. The preprocessor usually includes a computer aided design (CAD) package to allow the analyst to construct the required geometry. A good preprocessor usually includes a transla-tor to other CAD packages so that geometry data is interchangeable.
The grid has an exact correspondence to the geometry; open areas, walls, diffusers, etc., are represented by the elements or vol-umes of computational fluid, boundaries, and blocked elements.
The grid may be structured, where the element index or location is described by coordinates of i(x), j(y), and k(z). Or, the grid may be unstructured, in which the elements are arbitrarily arranged and defined by associative identification.
Post Processors After the flow solver has converged on the solution, the data is presented through the post processor. The typical post processor displays contour diagrams of equation variables such as pressure, temperature, velocity, turbulence, and contaminant concentration.
Post processors are available that provide elaborate three-dimen-sional contour and vector plots with color scaling and animated out-put for time-dependent solutions.
Energy Equation When the CFD application involves the exchange of heat, the energy equation is used. The energy equation is similar in structure to the Navier-Stokes equation: (18) The energy equation accommodates several HVAC situations.
For example, heat in a room may be lost or gained through walls, windows, heaters, air conditioners, or occupants. Natural convec-tion may dominate the flow pattern during heating or cooling and affect the comfort of the occupants, causing drafts or reduced air change effectiveness.
The energy equation is also important in compressible flow (high velocity). When the Mach number exceeds 0.2, the effects of com-pressibility begin to appear. The fluid density and velocity are deter-mined in part by the energy equation. The CFD applications that require these high velocities are usually components (valves, duct-ing, ejectors, etc.).
A special class of problems requires the solution of the conduc-tion equation (Laplace’s equation). This type of analysis is called conjugate heat transfer. An example would be a room with an insulated wall of known thermal conductivity, where the solver determines both the temperature profile across the wall and the flow in the room.
Multispecies Flows Often, a system with a mixture of gases must be evaluated. For example, a vent hood in a laboratory may be used to evacuate toxic gases from the room. A CFD model can provide an accurate assess-ment of the hood performance under a wide range of conditions and help to optimize the design for energy consumption.
If the concentration of the second gas is low, its viscous and momentum effects may be ignored. These types of problems may be solved using the single-species equation with the second species being convected along. The diffusion term is handled separately. If the concentration of the second gas is high enough to affect the properties of the mixture, such as density or viscosity, then the fully coupled form of the transport equations must be used.
Another class of multispecies flow problems involves the change of phase of one or more components. The humidity of a room with an evaporative water source would be an example of this.
vi j 1 – , ψi 1 j 1 – , + ( ) ψi 1 j 1 – , – ( ) – 2 x ∆ -------------------------------------------------------------– = vi j , ψi 1 j , + ( ) ψi 1 j , – ( ) – 2 x ∆ -----------------------------------------------– = ωi j , wall ( ) ψadj ψi j wall ( ) , – x2 ∆ -------------------------------------– = ωi j , current ( ) δωi j , new ( ) 1 δ – ( )ωi j , old ( ) + = ρc u∂T ∂x ------v∂T ∂y ------+ k ∂2T ∂x2 ---------∂2T ∂y2 ---------+ = 33.4 2001 ASHRAE Fundamentals Handbook (SI) Chemically reacting flows are often analyzed with CFD. Com-bustion models have been used extensively. These flows are com-plicated by the interaction of the various species, the phase changes, and the changes in properties and heat of formation of the combus-tion products.
Viscosity Models In the classical Navier-Stokes equations, the viscosity is assumed constant. For low Reynolds numbers (i.e., Re < 2500), the laminar form is used. When higher Reynolds numbers are encountered, a means to adjust the viscosity (the eddy viscosity) is required. These types of equations are frequently called turbulence models because the basis for eddy viscosity is the turbulence level. Several turbu-lence models are available, but the most accepted and benchmarked model is the k-ε model. Chen (1995) provides some background for the k-ε application to room airflow. The standard k-ε model is a two-equation model. The first equation describes the turbulent kinetic energy transport and generation rate: (19) Turbulent kinetic energy [Equation (9)] represents the fluctuating component of velocity.
The second equation in the k-ε model is the dissipation rate: (20) where Equation (20) is solved in conjunction with the Equation (19) to obtain the local eddy viscosity: (21) This two-equation model has a counterpart for high Reynolds number flows, but most room air motion studies use a lower Rey-nolds number model.
The near wall conditions for the turbulence model are often defined with wall functions that are based on the classical turbulent boundary layer. They may have two layers (the laminar sublayer and the turbulent layer) or they may have three layers (a laminar sublayer, a buffer layer, and a fully turbulent outer layer). The two-layer wall function is shown as follows: (22) (23) where and (24) Direct numerical simulation (DNS) is an active area of study of turbulence modeling. The basis for DNS turbulence modeling is that there is actually no turbulence model at all. Rather, the motion of the individual eddies within the bulk flow are modeled them-selves. This model requires an extremely small grid size in which an eddy cannot exist due to molecular viscosity. Due to the huge demand on computing resources; only small flow fields, about shoe box size at the most, have been simulated using DNS. Since DNS is really not yet practical for room air simulations, a compromise between it and conventional turbulence models has been developed. This method, large eddy simulation (LES), uses an eddy viscosity for the subgrid scale turbulence, but fully captures the time-dependent behavior of the larger energy bearing eddies.
The application to room airflow is just beginning (Emmerich and McGrattan 1998).
CFD Applications Currently, CFD has been applied to room air motion. The room might represent an auditorium, an aircraft cabin, or an automobile interior. Useful parameters such as velocity and temperature distri-bution, air change effectiveness, and humidity are predicted for the comfort of the occupant.
External flow models have used CFD to predict building infiltra-tion, heat transfer rates for heating and cooling loads, and wind loads for structural design.
Finally, CFD is a useful tool for other internal flows, especially when nonstandard components are analyzed. Complex manifolds can be flow-balanced in one step, and important data such as pres-sure drop, aero/hydrodynamic loads, and heat transfer rates are available with the currently obtainable tools.
SYMBOLS adj = subscript denoting adjacent node c = specific heat, J/(kg·K) based on units of ρ Cε1 = constant = 1.45 Cε2 = constant = 1.92 Cµ = constant = 0.09 f = field variable such as velocity f2 = coefficient = 1 – 0.3exp(–Ret2 ), where Ret is local Reynolds number = ρK2/µε fµ = coefficient = exp[–3.4/(1 + Ret/50)2] i = subscript index in x direction j = subscript index in y direction k = thermal conductivity, W/(m·K) K = kinetic energy of turbulence, m2/s2 L = turbulence length scale, m n = general subscript index P = pressure, Pa s = distance, m T = temperature, K u = fluid velocity in x direction, m/s uτ = friction velocity, m/s v = fluid velocity in y direction, m/s V = local velocity magnitude, m/s V ′ = turbulent velocity fluctuation, m/s x = horizontal distance, m ∆x = node spacing, m y = vertical distance, m ρu∂k ∂x -----ρv∂k ∂y -----+ ∂ ∂x -----µ µt σk -----+ ∂k ∂x -----y ∂ ∂ µ µt σk -----+ ∂k ∂y -----+ = µt 2 ∂u ∂x ------ 2 2 ∂v ∂y ----- 2 ∂u ∂y ------∂v ∂x -----+ 2 + + + ρε – 2µ ∂ k ∂x ------------ 2 ∂ k ∂x ------------ 2 + – ρu∂ε ∂x -----ρv∂ε ∂y -----+ ∂ ∂x -----µ µt σε -----+ ∂ε ∂x -----∂ ∂y -----µ µt σε -----+ ∂ε ∂y -----+ = Cε1µt ε k -- 2 ∂u ∂x ------ 2 2 ∂v ∂y ----- 2 ∂u ∂y ------∂v ∂x -----+ 2 + + + Cε2f2ρε2 k -----E + – E 2 µt ρ ----∂2u ∂x2 -------- 2 ∂2v ∂x2 -------- 2 + ≅ µt Cµfµρk2 ε -----= u uτ -----y+ for 0 y+ < 30 ≤ = u uτ -----1 κ ---ln Ey+ ( ) for 30 y+ < 300 ≤ = y+ ρuτy µ -----------= µ∂u ∂y ------wall τw ρuτ 2 = = HVAC Computational Fluid Dynamics 33.5 y+ = distance from wall in the universal velocity profile, m t = time, s δ = relaxation parameter ε = dissipation, m2/s3 κ = von Karman constant = 0.41 µ = dynamic viscosity, kg/(m·s) µt = effective viscosity, kg/(m·s) ν = kinematic viscosity, m2/s νt = effective viscosity, m2/s ρ = density, kg/m3 σk = constant = 1.0 σε = constant = 1.3 ψ = stream function, m2/s ψB = boundary stream function, m2/s ω = vorticity, 1/s REFERENCES Baker, A.J. 1983. Finite element computational fluid mechanics. Hemi-sphere Publishing, New York.
Catton, I. 1978. Natural convection in enclosures. Heat Transfer, 6.
Chang, L.C., J.R. Lloyd, and K.T. Yang. 1982. A finite difference study of natural convection in complex enclosures. Proceedings of the 7th Inter-national Heat Transfer Conference 2:183-88.
Chen, Q. and J. Van der Kooi. 1988. ACCURACY—A program for com-bined problems of energy analysis, indoor airflow, and air quality.
ASHRAE Transactions 94(2):196-214.
Chen, Q. 1995. Comparison of different k-ε models for indoor air flow com-putations. Numerical Heat Transfer, Part B Fundamentals 28:353-69.
Chen, Q., A. Moser, and A. Huber. 1990a. Prediction of buoyant, turbulent flow by a low-Reynolds-number k-ε model. ASHRAE Transactions 96(1):564-73.
Chen, Q., A. Moser, and P. Suter. 1990b. Indoor air quality and thermal com-fort under six kinds of air diffusion. ASHRAE Transactions 96(1).
Dennis, S.C.R. and J.D. Hudson. 1978. A difference method for solving the Navier-Stokes equations. Numerical Methods in Laminar and Turbulent Flow, pp. 69-80. Taylor Morgan Brebbia. Emmerich, S.J. and K.B. McGrattan. 1998. Application of a large eddy sim-ulation model to study room airflow. ASHRAE Transactions 104(1).
Gadgil, A., F. Bauman, E. Altmayer, and R.C. Kammerund. 1984. Verifica-tion of a natural simulation technique for natural convection. ASME Journal of Solar Energy Engineering 106:366-69.
Hadjisophocleous, G.V., A.C.M. Sousa, and J.E.S. Venart. 1988. Prediction of transient natural convection in enclosures of arbitrary geometry using a non orthogonal numerical model. Numerical Heat Transfer 13:373.
Haghighat, F., J.C.Y. Wang, and Z. Jiang. 1989. Natural convection and air-flow pattern in a partitioned room with turbulent flow. ASHRAE Trans-actions 95(2):600-10.
Haghighat, F., J.C.Y. Wang, and Z. Jiang. 1990. Development of a three-dimensional numerical model to investigate the airflow and age distribu-tion in a multizone enclosure. Proceedings Fifth International Confer-ence on Indoor Air Quality and Climate: Indoor Air ’90 4:183-88.
International Conference on Indoor Air Quality and Climate, 2344 Had-dinton Crescent, Ottawa, Ontario K1H 8J4, Canada.
Haghighat, F., J.C.Y. Wang, Z. Jiang, and F. Allard. 1992. Air movement in buildings using computational fluid dynamics. Transactions of the ASME Journal of Solar Energy Engineering 114:84-92.
Holmes, M.J. 1982. The application of fluid mechanics simulation program PHOENICS to a few typical HVAC problems. Ove Arup and Partners, London.
Horstman, R.H. 1988. Predicting velocity and contamination distribution in ventilated volumes using Navier-Stokes equations. ASHRAE Confer-ence, IAQ 88, pp. 209-30.
Jones, P. and P.O. Sullivan. 1985. Modeling of air flow patterns in large sin-gle volume space. SERC Workshop: Development in Building Simula-tion Programs, Loughborough University.
Kelkar, K.M. and S.V. Patankar. 1985. Numerical prediction of natural con-vection in partitioned enclosures. Numerical Heat Transfer, pp. 63-71. Lemaire, A.D. 1987. The numerical simulation of the air movement and heat transfer in a heated room resp. a ventilated atrium. Proceedings of the International Conference on Air Distribution in Ventilated Space, ROOMVENT-87. NORSKVVS, Oslo.
Lin, D.S. and N.W. Nansteel. 1987. Natural convection heat transfer in a square enclosure containing water near its density maximum. Interna-tional Journal of Heat and Mass Transfer 30(11):2319-29.
Markatos, N.C. 1983. The computer analysis of building ventilation and heating problems. Passive and Low Energy Architecture, pp. 667-75.
Markatos, N.C. and K.A. Pericleous.1984. Laminar and turbulent natural convection in an enclosed cavity. International Journal of Heat and Mass Transfer 27(5):755-72.
Murakami, S., S. Kota, and Y. Suyama. 1988. Numerical and experimental study on turbulent diffusion fields in convection flow type clean rooms.
ASHRAE Transactions 94(2):469-93.
Nielsen, P.V. 1974. Flow in air-conditioned rooms. Ph.D. thesis. Technical University of Denmark, Copenhagen.
Ostrach, S. 1982. Natural convection heat transfer in cavities and cells. Pro-ceedings of the International Heat Transfer Conference. Hemisphere, Washington, D.C., pp. 365-79.
Patankar, S.V. 1988. Elliptic systems: Finite-difference method 1. Handbook of numerical heat transfer, Chap. 6, pp. 215-40. John Wiley and Sons, New York.
Pun, W.M., and D.B. Spalding. 1976. A general computer program for two-dimensional elliptic flows. HTS/76/2. Imperial College, London.
BIBLIOGRAPHY Cebeci, T. 1988. Parabolic systems: Finite-difference method II. Handbook of numerical heat transfer. John Wiley and Sons, New York.
Chen, Q. and Z. Jiang. 1992. Significant questions in predicting room air motion. ASHRAE Transactions 98(1):929-39.
Lin, C.H., T. Han, and C.A. Koromilas. 1992. Effect of HVAC design parameters on passenger thermal comfort. SAE No. 920264. Society of Automotive Engineers, Warrendale, PA.
34.1 CHAPTER 34 DUCT DESIGN BERNOULLI EQUATION ...................................................... 34.1 Head and Pressure .................................................................. 34.2 SYSTEM ANALYSIS ................................................................ 34.2 Pressure Changes in System ................................................... 34.6 FLUID RESISTANCE ............................................................. 34.7 Friction Losses ........................................................................ 34.7 Dynamic Losses ...................................................................... 34.8 Ductwork Sectional Losses ................................................... 34.12 FAN-SYSTEM INTERFACE ................................................. 34.12 DUCT SYSTEM DESIGN ...................................................... 34.14 Design Considerations .......................................................... 34.14 Duct Design Methods ............................................................ 34.18 HVAC Duct Design Procedures ............................................ 34.20 Industrial Exhaust System Duct Design ....................................................................... 34.22 FITTING LOSS COEFFICIENTS ......................................... 34.29 OMMERCIAL, industrial, and residential air duct system Cdesign must consider (1) space availability, (2) space air diffu-sion, (3) noise levels, (4) duct leakage, (5) duct heat gains and losses, (6) balancing, (7) fire and smoke control, (8) initial invest-ment cost, and (9) system operating cost.
Deficiencies in duct design can result in systems that operate incorrectly or are expensive to own and operate. Poor air distribu-tion can cause discomfort, loss of productivity and even adverse health effects; lack of sound attenuators may permit objectionable noise levels. Poorly designed ductwork can result in unbalanced systems. Faulty duct construction or lack of duct sealing produces inadequate airflow rates at the terminals. Proper duct insulation eliminates the problem caused by excessive heat gain or loss.
In this chapter, system design and the calculation of a system’s frictional and dynamic resistance to airflow are considered. Chap-ter 16 of the 2000 ASHRAE Handbook—Systems and Equipment examines duct construction and presents construction standards for residential, commercial, and industrial heating, ventilating, air-conditioning, and exhaust systems.
BERNOULLI EQUATION The Bernoulli equation can be developed by equating the forces on an element of a stream tube in a frictionless fluid flow to the rate of momentum change. On integrating this relationship for steady flow, the following expression (Osborne 1966) results: (1) where v = streamline (local) velocity, m/s P = absolute pressure, Pa (N/m2) ρ = density, kg/m3 g = acceleration due to gravity, m/s2 z = elevation, m Assuming constant fluid density within the system, Equation (1) reduces to (2) Although Equation (2) was derived for steady, ideal frictionless flow along a stream tube, it can be extended to analyze flow through ducts in real systems. In terms of pressure, the relationship for fluid resistance between two sections is (3) where V = average duct velocity, m/s ∆pt,1-2 = total pressure loss due to friction and dynamic losses between sections 1 and 2, Pa In Equation (3), V (section average velocity) replaces v (streamline velocity) because experimentally determined loss coefficients allow for errors in calculating ρv2/2 (velocity pressure) across streamlines.
On the left side of Equation (3), add and subtract pz1; on the right side, add and subtract pz2, where pz1 and pz2 are the values of atmo-spheric air at heights z1 and z2. Thus, (4) The atmospheric pressure at any elevation ( pz1 and pz2) expressed in terms of the atmospheric pressure pa at the same datum elevation is given by (5) (6) Substituting Equations (5) and (6) into Equation (4) and simpli-fying yields the total pressure change between sections 1 and 2.
Assume no change in temperature between sections 1 and 2 (no heat exchanger within the section); therefore, ρ1 = ρ2. When a heat exchanger is located within the section, the average of the inlet and outlet temperatures is generally used. Let ρ = ρ1 = ρ2. (P1 − pz1) and (P2 − pz2) are gage pressures at elevations z1 and z2. (7a) (7b) The preparation of this chapter is assigned to TC 5.2, Duct Design.
v2 2 -----P d ρ ------∫ gz + + constant, N m ⋅ kg ⁄ = v2 2 -----P ρ ---gz + + constant, N m ⋅ kg ⁄ = ρ1V1 2 2 ------------P1 gρ1z1 + + ρ2V2 2 2 ------------P2 gρ2z2 pt 1-2 , ∆ + + + = ρ1V1 2 2 ------------P1 pz1 pz1 – ( ) gρ1z1 + + + ρ2V2 2 2 ------------P2 + = pz2 pz2 – ( ) gρ2z2 pt 1-2 , ∆ + + + pz1 pa gρaz1 – = pz2 pa gρaz2 – = pt 1-2 , ∆ ps 1 , ρV1 2 2 ---------+ ps 2 , ρV2 2 2 ---------+ – = g ρa ρ – ( ) z2 z1 – ( ) + pt 1-2 , ∆ pt ∆ pse ∆ + = 34.2 2001 ASHRAE Fundamentals Handbook (SI) (7c) where ps,1 = static pressure, gage at elevation z1, Pa ps,2 = static pressure, gage at elevation z2, Pa V1 = average velocity at section 1, m/s V2 = average velocity at section 2, m/s ρa = density of ambient air, kg/m3 ρ = density of air or gas within duct, kg/m3 ∆pse = thermal gravity effect, Pa ∆pt = total pressure change between sections 1 and 2, Pa ∆pt,1−2 = total pressure loss due to friction and dynamic losses between sections 1 and 2, Pa HEAD AND PRESSURE The terms head and pressure are often used interchangeably; however, head is the height of a fluid column supported by fluid flow, while pressure is the normal force per unit area. For liquids, it is convenient to measure the head in terms of the flowing fluid. With a gas or air, however, it is customary to measure pressure on a col-umn of liquid.
Static Pressure The term p/ρg is static head; p is static pressure.
Velocity Pressure The term V2/2g refers to velocity head, and the term ρV2/2 refers to velocity pressure. Although velocity head is independent of fluid density, velocity pressure, calculated by Equation (8), is not.
(8) where pv = velocity pressure, Pa V = fluid mean velocity, m/s For air at standard conditions (1.204 kg/m3), Equation (8) becomes (9) Velocity is calculated by Equation (10) or (11).
(10) where Q = airflow rate, L/s A = cross-sectional area of duct, mm2 (11) where A = cross-sectional area of duct, m2.
Total Pressure Total pressure is the sum of static pressure and velocity pressure: (12) or (13) where pt = total pressure, Pa ps = static pressure, Pa Pressure Measurement The range, precision, and limitations of instruments for mea-suring pressure and velocity are discussed in Chapter 14. The manometer is a simple and useful means for measuring partial vacuum and low pressure. Static, velocity, and total pressures in a duct system relative to atmospheric pressure are measured with a pitot tube connected to a manometer. Pitot tube construction and locations for traversing round and rectangular ducts are presented in Chapter 14.
SYSTEM ANALYSIS The total pressure change due to friction, fittings, equipment, and net thermal gravity effect (stack effect) for each section of a duct system is calculated by the following equation: (14) where = net total pressure change for i-section, Pa = pressure loss due to friction for i-section, Pa = total pressure loss due to j-fittings, including fan system effect (FSE), for i-section, Pa = pressure loss due to k-equipment for i-section, Pa = thermal gravity effect due to r-stacks for i-section, Pa m = number of fittings within i-section n = number of equipment within i-section λ = number of stacks within i-section nup = number of duct sections upstream of fan (exhaust/return air subsystems) ndn = number of duct sections downstream of fan (supply air subsystems) From Equation (7), the thermal gravity effect for each nonhori-zontal duct with a density other than that of ambient air is deter-mined by the following equation: (15) where ∆pse =thermal gravity effect, Pa z1 and z2 =elevation from datum in direction of airflow (Figure 1), m ρa =density of ambient air, kg/m3 ρ =density of air or gas within duct, kg/m3 Example 1. For Figure 1, calculate the thermal gravity effect for two cases: (a) air cooled to −34°C, and (b) air heated to 540°C. The density of air at −34°C and 540°C is 1.477 kg/m3 and 0.434 kg/m3, respectively. The density of the ambient air is 1.204 kg/m3. Stack height is 15 m.
Solution: (a) For ρ > ρa (Figure 1A), (b) For ρ < ρa (Figure 1B), pt ∆ pt 1-2 , ∆ pse ∆ – = pv ρV 2 2 ⁄ = pv 0.602V 2 = V 1000Q A ⁄ = V 0.001Q A ⁄ = pt ps ρV2 2 ⁄ + = pt ps pv + = pti ∆ pfi ∆ pij ∆ j=1 m ∑ pik ∆ k=1 n ∑ pseir ∆ r=1 λ ∑ – + + = for i 1 2 … nup ndn + , , , = pti ∆ pfi ∆ pij ∆ pik ∆ pseir ∆ pse ∆ 9.81 ρa ρ – ( ) z2 z1 – ( ) = pse ∆ 9.81 ρa ρ – ( )z = pse ∆ 9.81 1.204 1.477 – ( )15 = 40 Pa – = pse ∆ 9.81 1.204 0.434 – ( )15 = +113 Pa = Duct Design 34.3 Example 2. Calculate the thermal gravity effect for the two-stack system shown in Figure 2, where the air is 120°C and the stack heights are 15 and 30 m. The density of 120°C air is 0.898 kg/m3; ambient air is 1.204 kg/m3.
Solution: For the system shown in Figure 3, the direction of air movement created by the thermal gravity effect depends on the initiating force. The initiating force could be fans, wind, opening and closing doors, and turning equipment on and off. If for any reason air starts to enter the left stack (Figure 3A), it creates a buoyancy effect in the right stack. On the other hand, if flow starts to enter the right stack (Figure 3B), it creates a buoyancy effect in the left stack. In both cases the produced thermal gravity effect is stable and depends on the stack height and magnitude of heating. The starting direction of flow is important when using natural convection for ventilation.
To determine the fan total pressure requirement for a system, use the following equation: (16) where Fup and Fdn=sets of duct sections upstream and downstream of a fan Pt = fan total pressure, Pa ε = symbol that ties duct sections into system paths from the exhaust/return air terminals to the supply terminals Figure 4 illustrates the use of Equation (16). This system has three supply and two return terminals consisting of nine sections con-nected in six paths: 1-3-4-9-7-5, 1-3-4-9-7-6, 1-3-4-9-8, 2-4-9-7-5, 2-4-9-7-6, and 2-4-9-8. Sections 1 and 3 are unequal area; thus, they are assigned separate numbers in accordance with the rules for pse ∆ 9.81 ρa ρ – ( ) z2 z1 – ( ) = 9.81 1.204 0.898 – ( ) 30 15 – ( ) = 45 Pa = Fig. 1 Thermal Gravity Effect for Example 1 Fig. 2 Multiple Stacks for Example 2 Fig. 3 Multiple Stack Analysis Pt pti ∆ iεFup ∑ pti ∆ iεFdn ∑ for i + 1 2 … nup ndn + , , , = = 34.4 2001 ASHRAE Fundamentals Handbook (SI) identifying sections (see Step 4 in the section on HVAC Duct Design Procedures). To determine the fan pressure requirement, the following six equations, derived from Equation (16), are applied.
These equations must be satisfied to attain pressure balancing for design airflow. Relying entirely on dampers is not economical and may create objectionable flow-generated noise.
(17) Fig. 4 Illustrative 6-Path, 9-Section System Pt p1 ∆ p3 ∆ p4 ∆ p9 ∆ p7 ∆ p5 ∆ + + + + + = Pt p1 ∆ p3 ∆ p4 ∆ p9 ∆ p7 ∆ p6 ∆ + + + + + = Pt p1 ∆ p3 ∆ p4 ∆ p9 ∆ p8 ∆ + + + + = Pt p2 ∆ p4 ∆ p9 ∆ p7 ∆ p5 ∆ + + + + = Pt p2 ∆ p4 ∆ p9 ∆ p7 ∆ p6 ∆ + + + + = Pt p2 ∆ p4 ∆ p9 ∆ p8 ∆ + + + = Fig. 5 Single Stack with Fan for Examples 3 and 4 Duct Design 34.5 Example 3. For Figures 5A and 5C, calculate the thermal gravity effect and fan total pressure required when the air is cooled to −34°C. The heat exchanger and ductwork (section 1 to 2) total pressure losses are 170 and 70 Pa respectively. The density of −34°C air is 1.477 kg/m3; ambient air is 1.204 kg/m3. Elevations are 21 m and 3 m as noted in the solutions below.
Solution. (a) For Figure 5A (downward flow), (b) For Figure 5C (upward flow), Example 4. For Figures 5B and 5D, calculate the thermal gravity effect and fan total pressure required when the air is heated to 120°C. The heat exchanger and ductwork (section 1 to 2) total pressure losses are 170 and 70 Pa respectively. The density of 120°C air is 0.898 kg/m3; ambient air is 1.204 kg/m3. Elevations are 21 m and 3 m as noted in the solutions below.
Solution: (a) For Figure 5B (downward flow), (b) For Figure 5D (upward flow), Example 5. Calculate the thermal gravity effect for each section of the sys-tem shown in Figure 6 and the net thermal gravity effect of the system.
The density of ambient air is 1.204 kg/m3, and the lengths are as fol-lows: z1 = 15 m, z2 = 27 m, z4 = 30 m, z5 = 8 m, and z9 = 60 m. The pressure required at section 3 is −25 Pa (−2.7 mm of water). Write the equation to determine the fan total pressure requirement.
Fig. 6 Triple Stack System for Example 5 pse ∆ 9.81 ρa ρ – ( ) z2 z1 – ( ) = 9.81 1.204 1.477 – ( ) 3 21 – ( ) = 48 Pa = Pt pt,3-2 ∆ pse ∆ – = 170 70 + ( ) 48 ( ) – = 192 Pa = pse ∆ 9.81 ρa ρ – ( ) z2 z1 – ( ) = 9.81 1.204 1.477 – ( ) 21 3 – ( ) = 48 – Pa = Pt pt,3-2 ∆ pse ∆ – = 170 70 + ( ) 48 – ( ) – = 288 Pa = pse ∆ 9.81 ρa ρ – ( ) z2 z1 – ( ) = 9.81 1.204 0.898 – ( ) 3 21 – ( ) = 54 Pa – = Pt pt,3-2 ∆ pse ∆ – = 170 70 + ( ) 54 – ( ) – = 294 Pa = pse ∆ 9.81 ρa ρ – ( ) z2 z1 – ( ) = 9.81 1.204 0.898 – ( ) 21 3 – ( ) = 54 Pa = Pt pt,3-2 ∆ pse ∆ – = 170 70 + ( ) 54 ( ) – = 186 Pa = 34.6 2001 ASHRAE Fundamentals Handbook (SI) Solution: The following table summarizes the thermal gravity effect for each section of the system as calculated by Equation (15). The net thermal gravity effect for the system is 118 Pa. To select a fan, use the following equation: PRESSURE CHANGES IN SYSTEM Figure 7 shows total and static pressure changes in a fan/duct system consisting of a fan with both supply and return air duct-work. Also shown are the total and static pressure gradients refer-enced to atmospheric pressure.
For all constant-area sections, the total and static pressure losses are equal. At the diverging transitions, velocity pressure decreases, absolute total pressure decreases, and absolute static pressure can increase. The static pressure increase at these sections is known as static regain.
At the converging transitions, velocity pressure increases in the direction of airflow, and the absolute total and absolute static pres-sures decrease.
At the exit, the total pressure loss depends on the shape of the fit-ting and the flow characteristics. Exit loss coefficients Co can be greater than, less than, or equal to one. The total and static pressure grade lines for the various coefficients are shown in Figure 3. Note that for a loss coefficient less than one, static pressure upstream of the exit is less than atmospheric pressure (negative). The static pres-sure just upstream of the discharge fitting can be calculated by sub-tracting the upstream velocity pressure from the upstream total pressure.
At section 1, the total pressure loss depends on the shape of the entry. The total pressure immediately downstream of the entrance equals the difference between the upstream pressure, which is zero (atmospheric pressure), and the loss through the fitting. The static pressure of the ambient air is zero; several diameters downstream, static pressure is negative, equal to the sum of the total pressure (negative) and the velocity pressure (always positive).
System resistance to airflow is noted by the total pressure grade line in Figure 7. Sections 3 and 4 include fan system effect pressure losses. To obtain the fan static pressure requirement for fan selection where the fan total pressure is known, use (18) where Ps = fan static pressure, Pa Pt = fan total pressure, Pa pv,o = fan outlet velocity pressure, Pa Path (x-x′) Temp., °C ρ, kg/m3 ∆z (zx′ − zx), m ∆ρ (ρa − ρx−x′), kg/m3 ∆pse, Pa [Eq. (15)] 1-2 815 0.324 (27 − 15) +0.880 +104 3-4 540 0.434 0 +0.770 0 4-5 540 0.434 (8 − 30) +0.770 −166 6-7 120 0.898 0 +0.306 0 8-9 120 0.898 (60 − 0) +0.306 +180 Net Thermal Gravity Effect 118 Pt 25 pt 1-7 , ∆ pt 8-9 , ∆ pse ∆ – + + = 25 pt 1-7 , ∆ pt 8-9 , ∆ 118 – + + = pt 1-7 , ∆ pt 8-9 , ∆ 93 – + = Fig. 7 Pressure Changes During Flow in Ducts Ps Pt pv o , – = Duct Design 34.7 FLUID RESISTANCE Duct system losses are the irreversible transformation of mechanical energy into heat. The two types of losses are (1) friction losses and (2) dynamic losses.
FRICTION LOSSES Friction losses are due to fluid viscosity and are a result of momentum exchange between molecules in laminar flow and between individual particles of adjacent fluid layers moving at dif-ferent velocities in turbulent flow. Friction losses occur along the entire duct length.
Darcy, Colebrook, and Altshul-Tsal Equations For fluid flow in conduits, friction loss can be calculated by the Darcy equation: (19) where ∆pf = friction losses in terms of total pressure, Pa f = friction factor, dimensionless L = duct length, m Dh = hydraulic diameter [Equation (24)], mm V = velocity, m/s ρ = density, kg/m3 Within the region of laminar flow (Reynolds numbers less than 2000), the friction factor is a function of Reynolds number only.
For completely turbulent flow, the friction factor depends on Reynolds number, duct surface roughness, and internal protuber-ances such as joints. Between the bounding limits of hydraulically smooth behavior and fully rough behavior, is a transitional rough-ness zone where the friction factor depends on both roughness and Reynolds number. In this transitionally rough, turbulent zone the friction factor f is calculated by Colebrook’s equation (Colebrook 1938-39). Colebrook’s transition curve merges asymptotically into the curves representing laminar and completely turbulent flow.
Because Colebrook’s equation cannot be solved explicitly for f, use iterative techniques (Behls 1971).
(20) where ε = material absolute roughness factor, mm Re = Reynolds number A simplified formula for calculating friction factor, developed by Altshul (Altshul et al. 1975) and modified by Tsal, is (21) Friction factors obtained from the Altshul-Tsal equation are within 1.6% of those obtained by Colebrook’s equation.
Reynolds number (Re) may be calculated by using the following equation.
(22) where ν = kinematic viscosity, m2/s.
For standard air, Re can be calculated by (23) Roughness Factors The roughness factors ε listed in Table 1 are recommended for use with the Colebrook or Altshul-Tsal equation [Equations (20) and (21), respectively]. These values include not only material, but also duct construction, joint type, and joint spacing (Griggs and Khodabakhsh-Sharifabad 1992). Roughness factors for other materials are presented in Idelchik et al. (1994). Idelchik summa-rizes roughness factors for 80 materials including metal tubes; con-duits made from concrete and cement; and wood, plywood, and glass tubes.
Swim (1978) conducted tests on duct liners of varying densities, surface treatments, transverse joints (workmanship), and methods of attachment to sheet metal ducts. As a result of these tests, Swim recommends for design 4.6 mm for spray-coated liners and 1.5 mm for liners with a facing material cemented onto the air side. In both cases, the roughness factor includes the resistance offered by mechanical fasteners and assumes good joints. Liners cut too short result in (1) loss of thermal performance, (2) possible condensation problems, (3) potential damage to the liner (erosion of the blanket or tearing away from the duct surface), and (4) the collection of dirt and debris and the initiation of biological problems. Liner density does not significantly influence flow resistance.
Manufacturers’ data indicate that the absolute roughness for fully extended nonmetallic flexible ducts ranges from 1.1 to 4.6 mm. For fully extended flexible metallic ducts, absolute rough-ness ranges from 0.1 to 2.1 mm. This range covers flexible duct with the supporting wire exposed to flow or covered by the material. Fig-ure 8 provides a pressure drop correction factor for straight flexible duct when less than fully extended.
pf ∆ 1000fL Dh ----------------- ρV2 2 ---------= 1 f -------2 log – ε 3.7Dh --------------2.51 Re f --------------+ = f ′ 0.11 ε Dh ------68 Re ------+ 0.25 = If f ′ 0.018: f f ′ = ≥ If f ′ 0.018: f 0.85f ′ 0.0028 + = < Re DhV 1000ν ---------------= Table 1 Duct Roughness Factors Duct Material Roughness Category Absolute Roughness ε, mm Uncoated carbon steel, clean (Moody 1944) (0.05 mm) Smooth 0.03 PVC plastic pipe (Swim 1982) (0.01 to 0.05 mm) Aluminum (Hutchinson 1953) (0.04 to 0.06 mm) Galvanized steel, longitudinal seams, 1200 mm joints (Griggs et al. 1987) (0.05 to 0.10 mm) Medium smooth 0.09 Galvanized steel, continuously rolled, spiral seams, 3000 mm joints (Jones 1979) (0.06 to 0.12 mm) Galvanized steel, spiral seam with 1, 2, and 3 ribs, 3600 mm joints (Griggs et al. 1987) (0.09 to 0.12 mm) Galvanized steel, longitudinal seams, 760 mm joints (Wright 1945) (0.15 mm) Average 0.15 Fibrous glass duct, rigid Medium rough 0.9 Fibrous glass duct liner, air side with facing material (Swim 1978) (1.5 mm) Fibrous glass duct liner, air side spray coated (Swim 1978) (4.5 mm) Rough 3.0 Flexible duct, metallic (1.2 to 2.1 mm when fully extended) Flexible duct, all types of fabric and wire (1.0 to 4.6 mm when fully extended) Concrete (Moody 1944) (1.3 to 3.0 mm) Re 66.4DhV = 34.8 2001 ASHRAE Fundamentals Handbook (SI) Friction Chart Fluid resistance caused by friction in round ducts can be deter-mined by the friction chart (Figure 9). This chart is based on stan-dard air flowing through round galvanized ducts with beaded slip couplings on 1220 mm centers, equivalent to an absolute roughness of 0.09 mm. Changes in barometric pressure, temperature, and humidity affect air density, air viscosity, and Reynolds number. No corrections to Figure 9 are needed for (1) duct materials with a medium smooth roughness factor, (2) temperature variations in the order of ±15 K from 20°C, (3) elevations to 500 m, and (4) duct pressures from −5 to +5 kPa relative to the ambient pressure. These individual variations in temperature, elevation, and duct pressure result in duct losses within ±5% of the standard air friction chart.
For duct materials other than those categorized as medium smooth in Table 1, and for variations in temperature, barometric pressure (elevation), and duct pressures (outside the range listed), calculate the friction loss in a duct by the Altshul-Tsal and Darcy equations [Equations (21) and (19), respectively].
Noncircular Ducts A momentum analysis can relate average wall shear stress to pressure drop per unit length for fully developed turbulent flow in a passage of arbitrary shape but uniform longitudinal cross-sectional area. This analysis leads to the definition of hydraulic diameter: (24) where Dh = hydraulic diameter, mm A = duct area, mm2 P = perimeter of cross section, mm While the hydraulic diameter is often used to correlate noncircular data, exact solutions for laminar flow in noncircular passages show that such practice causes some inconsistencies. No exact solutions exist for turbulent flow. Tests over a limited range of turbulent flow indicated that fluid resistance is the same for equal lengths of duct for equal mean velocities of flow if the ducts have the same ratio of cross-sectional area to perimeter. From a series of experiments using round, square, and rectangular ducts having essentially the same hydraulic diameter, Huebscher (1948) found that each, for most purposes, had the same flow resistance at equal mean veloci-ties. Tests by Griggs and Khodabakhsh-Sharifabad (1992) also indi-cated that experimental rectangular duct data for airflow over the range typical of HVAC systems can be correlated satisfactorily using Equation (20) together with hydraulic diameter, particularly when a realistic experimental uncertainty is accepted. These tests support using hydraulic diameter to correlate noncircular duct data.
Rectangular Ducts. Huebscher (1948) developed the relation-ship between rectangular and round ducts that is used to determine size equivalency based on equal flow, resistance, and length. This relationship, Equation (25), is the basis for Table 2.
(25) where De = circular equivalent of rectangular duct for equal length, fluid resistance, and airflow, mm a = length one side of duct, mm b = length adjacent side of duct, mm To determine equivalent round duct diameter, use Table 2. Equa-tions (21) or (20) and (19) must be used to determine pressure loss.
Flat Oval Ducts. To convert round ducts to spiral flat oval sizes, use Table 3. Table 3 is based on Equation (26) (Heyt and Diaz 1975), the circular equivalent of a flat oval duct for equal airflow, resis-tance, and length. Equations (21) or (20) and (19) must be used to determine friction loss.
(26) where AR is the cross-sectional area of flat oval duct defined as (27) and the perimeter P is calculated by (28) where P = perimeter of flat oval duct, mm A = major axis of flat oval duct, mm a = minor axis of flat oval duct, mm DYNAMIC LOSSES Dynamic losses result from flow disturbances caused by duct-mounted equipment and fittings that change the airflow path’s direction and/or area. These fittings include entries, exits, elbows, transitions, and junctions. Idelchik et al. (1994) discuss parameters affecting fluid resistance of fittings and presents local loss coeffi-cients in three forms: tables, curves, and equations.
Local Loss Coefficients The dimensionless coefficient C is used for fluid resistance, because this coefficient has the same value in dynamically similar streams (i.e., streams with geometrically similar stretches, equal Reynolds numbers, and equal values of other criteria necessary for dynamic similarity). The fluid resistance coefficient represents the ratio of total pressure loss to velocity pressure at the referenced cross section: (29) where C = local loss coefficient, dimensionless ∆pj = total pressure loss, Pa ρ = density, kg/m3 V = velocity, m/s pv = velocity pressure, Pa Fig. 8 Correction Factor for Unextended Flexible Duct Dh 4A P ⁄ = De 1.30 ab ( )0.625 a b + ( )0.250 --------------------------------= De 1.55AR0.625 P0.250 -----------------------------= AR πa2 4 ⁄ ( ) a A a – ( ) + = P πa 2 A a – ( ) + = C pj ∆ ρV 2 2 ⁄ ( ) ---------------------pj ∆ pv --------= = Duct Design 34.9 Fig. 9 Friction Chart for Round Duct (ρ = 1.20 kg/m3 and ε = 0.09 mm) 34.10 2001 ASHRAE Fundamentals Handbook (SI) Table 2 Circular Equivalents of Rectangular Duct for Equal Friction and Capacitya Lgth Adj.b Length One Side of Rectangular Duct (a), mm 100 125 150 175 200 225 250 275 300 350 400 450 500 550 600 650 700 750 800 900 Circular Duct Diameter, mm 100 109 125 122 137 150 133 150 164 175 143 161 177 191 200 152 172 189 204 219 225 161 181 200 216 232 246 250 169 190 210 228 244 259 273 275 176 199 220 238 256 272 287 301 300 183 207 229 248 266 283 299 314 328 350 195 222 245 267 286 305 322 339 354 383 400 207 235 260 283 305 325 343 361 378 409 437 450 217 247 274 299 321 343 363 382 400 433 464 492 500 227 258 287 313 337 360 381 401 420 455 488 518 547 550 236 269 299 326 352 375 398 419 439 477 511 543 573 601 600 245 279 310 339 365 390 414 436 457 496 533 567 598 628 656 650 253 289 321 351 378 404 429 452 474 515 553 589 622 653 683 711 700 261 298 331 362 391 418 443 467 490 533 573 610 644 677 708 737 765 750 268 306 341 373 402 430 457 482 506 550 592 630 666 700 732 763 792 820 800 275 314 350 383 414 442 470 496 520 567 609 649 687 722 755 787 818 847 875 900 289 330 367 402 435 465 494 522 548 597 643 686 726 763 799 833 866 897 927 984 1000 301 344 384 420 454 486 517 546 574 626 674 719 762 802 840 876 911 944 976 1037 1100 313 358 399 437 473 506 538 569 598 652 703 751 795 838 878 916 953 988 1022 1086 1200 324 370 413 453 490 525 558 590 620 677 731 780 827 872 914 954 993 1030 1066 1133 1300 334 382 426 468 506 543 577 610 642 701 757 808 857 904 948 990 1031 1069 1107 1177 1400 344 394 439 482 522 559 595 629 662 724 781 835 886 934 980 1024 1066 1107 1146 1220 1500 353 404 452 495 536 575 612 648 681 745 805 860 913 963 1011 1057 1100 1143 1183 1260 1600 362 415 463 508 551 591 629 665 700 766 827 885 939 991 1041 1088 1133 1177 1219 1298 1700 371 425 475 521 564 605 644 682 718 785 849 908 964 1018 1069 1118 1164 1209 1253 1335 1800 379 434 485 533 577 619 660 698 735 804 869 930 988 1043 1096 1146 1195 1241 1286 1371 1900 387 444 496 544 590 663 674 713 751 823 889 952 1012 1068 1122 1174 1224 1271 1318 1405 2000 395 453 506 555 602 646 688 728 767 840 908 973 1034 1092 1147 1200 1252 1301 1348 1438 2100 402 461 516 566 614 659 702 743 782 857 927 993 1055 1115 1172 1226 1279 1329 1378 1470 2200 410 470 525 577 625 671 715 757 797 874 945 1013 1076 1137 1195 1251 1305 1356 1406 1501 2300 417 478 534 587 636 683 728 771 812 890 963 1031 1097 1159 1218 1275 1330 1383 1434 1532 2400 424 486 543 597 647 695 740 784 826 905 980 1050 1116 1180 1241 1299 1355 1409 1461 1561 2500 430 494 552 606 658 706 753 797 840 920 996 1068 1136 1200 1262 1322 1379 1434 1488 1589 2600 437 501 560 616 668 717 764 810 853 935 1012 1085 1154 1220 1283 1344 1402 1459 1513 1617 2700 443 509 569 625 678 728 776 822 866 950 1028 1102 1173 1240 1304 1366 1425 1483 1538 1644 2800 450 516 577 634 688 738 787 834 879 964 1043 1119 1190 1259 1324 1387 1447 1506 1562 1670 2900 456 523 585 643 697 749 798 845 891 977 1058 1135 1208 1277 1344 1408 1469 1529 1586 1696 Lgth Adj.b Length One Side of Rectangular Duct (a), mm 1000 1100 1200 1300 1400 1500 1600 1700 1800 1900 2000 2100 2200 2300 2400 2500 2600 2700 2800 2900 Circular Duct Diameter, mm 1000 1093 1100 1146 1202 1200 1196 1256 1312 1300 1244 1306 1365 1421 1400 1289 1354 1416 1475 1530 1500 1332 1400 1464 1526 1584 1640 1600 1373 1444 1511 1574 1635 1693 1749 1700 1413 1486 1555 1621 1684 1745 1803 1858 1800 1451 1527 1598 1667 1732 1794 1854 1912 1968 1900 1488 1566 1640 1710 1778 1842 1904 1964 2021 2077 2000 1523 1604 1680 1753 1822 1889 1952 2014 2073 2131 2186 2100 1558 1640 1719 1793 1865 1933 1999 2063 2124 2183 2240 2296 2200 1591 1676 1756 1833 1906 1977 2044 2110 2173 2233 2292 2350 2405 2300 1623 1710 1793 1871 1947 2019 2088 2155 2220 2283 2343 2402 2459 2514 2400 1655 1744 1828 1909 1986 2060 2131 2200 2266 2330 2393 2453 2511 2568 2624 2500 1685 1776 1862 1945 2024 2100 2173 2243 2311 2377 2441 2502 2562 2621 2678 2733 2600 1715 1808 1896 1980 2061 2139 2213 2285 2355 2422 2487 2551 2612 2672 2730 2787 2842 2700 1744 1839 1929 2015 2097 2177 2253 2327 2398 2466 2533 2598 2661 2722 2782 2840 2896 2952 2800 1772 1869 1961 2048 2133 2214 2292 2367 2439 2510 2578 2644 2708 2771 2832 2891 2949 3006 3061 2900 1800 1898 1992 2081 2167 2250 2329 2406 2480 2552 2621 2689 2755 2819 2881 2941 3001 3058 3115 3170 aTable based on De = 1.30(ab)0.625/(a + b)0.25.
bLength adjacent side of rectangular duct (b), mm.
Duct Design 34.11 Dynamic losses occur along a duct length and cannot be sepa-rated from friction losses. For ease of calculation, dynamic losses are assumed to be concentrated at a section (local) and to exclude friction. Frictional losses must be considered only for relatively long fittings. Generally, fitting friction losses are accounted for by measuring duct lengths from the centerline of one fitting to that of the next fitting. For fittings closely coupled (less than six hydraulic diameters apart), the flow pattern enter-ing subsequent fittings differs from the flow pattern used to determine loss coefficients. Adequate data for these situations are unavailable.
For all fittings, except junctions, calculate the total pressure loss ∆pj at a section by (30) where the subscript o is the cross section at which the velocity pres-sure is referenced. The dynamic loss is based on the actual velocity in the duct, not the velocity in an equivalent noncircular duct. For the cross section to reference a fitting loss coefficient, refer to Step 4 in the section on HVAC Duct Design Procedures. Where necessary (unequal area fittings), convert a loss coefficient from section o to section i using Equation (31), where V is the velocity at the respec-tive sections.
(31) For converging and diverging flow junctions, total pressure losses through the straight (main) section are calculated as (32) For total pressure losses through the branch section, (33) where pv,c is the velocity pressure at the common section c, and Cc,s and Cc,b are losses for the straight (main) and branch flow paths, respectively, each referenced to the velocity pressure at section c.
To convert junction local loss coefficients referenced to straight and branch velocity pressures, use the following equation: (34) where Ci = local loss coefficient referenced to section being calculated (see subscripts), dimensionless Cc,i = straight (Cc,s) or branch (Cc,b) local loss coefficient referenced to dynamic pressure at common section, dimensionless Vi = velocity at section to which Ci is being referenced, m/s Vc = velocity at common section, m/s Subscripts: b = branch s = straight (main) section c = common section The junction of two parallel streams moving at different veloci-ties is characterized by turbulent mixing of the streams, accompa-nied by pressure losses. In the course of this mixing, an exchange of momentum takes place between the particles moving at different velocities, finally resulting in the equalization of the velocity distri-butions in the common stream. The jet with higher velocity loses a part of its kinetic energy by transmitting it to the slower moving jet.
The loss in total pressure before and after mixing is always large and positive for the higher velocity jet and increases with an increase in the amount of energy transmitted to the lower velocity jet. Conse-quently, the local loss coefficient, defined by Equation (29), will always be positive. The energy stored in the lower velocity jet increases as a result of mixing. The loss in total pressure and the local loss coefficient can, therefore, also have negative values for the lower velocity jet (Idelchik et al. 1994).
Table 3 Equivalent Spiral Flat Oval Duct Dimensions Circular Duct Diameter, mm Minor Axis (a), mm 70 100 125 150 175 200 250 275 300 325 350 375 400 450 500 550 600 Major Axis (A), mm 125 205 140 265 180 160 360 235 190 180 475 300 235 200 200 380 290 245 215 224 490 375 305 — 240 250 475 385 325 290 280 485 410 360 — 285 315 635 525 — — 345 325 355 840 — 580 460 425 395 375 400 1115 — 760 — 530 490 460 435 450 1490 — 995 — 675 — 570 535 505 500 1275 — 845 — 700 655 615 580 560 1680 — 1085 — 890 820 765 720 630 1425 — 1150 1050 970 905 810 710 1505 1370 1260 1165 1025 800 1800 1645 1515 1315 1170 1065 900 2165 1985 1705 1500 1350 1000 2170 1895 1690 1120 2455 2170 1950 1250 2795 2495 pj ∆ Copv o , = Ci Co Vi Vo ⁄ ( )2 ----------------------= pj ∆ Cc s , pv c , = pj ∆ Cc b , pv c , = Ci Cc i , Vi Vc ⁄ ( )2 ----------------------= 34.12 2001 ASHRAE Fundamentals Handbook (SI) Duct Fitting Database A duct fitting database, developed by ASHRAE (1994), which includes 228 round and rectangular fittings with the provision to include flat oval fittings, is available from ASHRAE in electronic form with the capability to be linked to duct design programs.
The fittings are numbered (coded) as shown in Table 4. Entries and converging junctions are only in the exhaust/return portion of systems. Exits and diverging junctions are only in supply systems.
Equal-area elbows, obstructions, and duct-mounted equipment are common to both supply and exhaust systems. Transitions and unequal-area elbows can be either supply or exhaust fittings. Fitting ED5-1 (see the section on Fitting Loss Coefficients) is an Exhaust fitting with a round shape (Diameter). The number 5 indicates that the fitting is a junction, and 1 is its sequential number. Fittings SR3-1 and ER3-1 are Supply and Exhaust fittings, respectively. The R indicates that the fitting is Rectangular, and the 3 identifies the fit-ting as an elbow. Note that the cross-sectional areas at sections 0 and 1 are not equal (see the section on Fitting Loss Coefficients). Other-wise, the elbow would be a Common fitting such as CR3-6. Addi-tional fittings are reproduced in the section on Fitting Loss Coefficients to support the example design problems (see Table 12 for Example 8; see Table 14 for Example 9).
DUCTWORK SECTIONAL LOSSES Darcy-Weisbach Equation Total pressure loss in a duct section is calculated by combining Equations (19) and (29) in terms of ∆p, where ΣC is the summation of local loss coefficients within the duct section. Each fitting loss coefficient must be referenced to that section’s velocity pressure.
(35) FAN-SYSTEM INTERFACE Fan Inlet and Outlet Conditions Fan performance data measured in the field may show lower per-formance capacity than manufacturers’ ratings. The most common causes of deficient performance of the fan/system combination are improper outlet connections, nonuniform inlet flow, and swirl at the fan inlet. These conditions alter the aerodynamic characteristics of the fan so that its full flow potential is not realized. One bad con-nection can reduce fan performance far below its rating. No data have been published that account for the effects of fan inlet and out-let flexible vibration connectors.
Normally, a fan is tested with open inlets and a section of straight duct attached to the outlet (ASHRAE Standard 51). This setup results in uniform flow into the fan and efficient static pressure recovery on the fan outlet. If good inlet and outlet conditions are not provided in the actual installation, the performance of the fan suf-fers. To select and apply the fan properly, these effects must be con-sidered, and the pressure requirements of the fan, as calculated by standard duct design procedures, must be increased.
Figure 10 illustrates deficient fan/system performance. The system pressure losses have been determined accurately, and a fan has been selected for operation at Point 1. However, no allowance has been made for the effect of system connections to the fan on fan perfor-mance. To compensate, a fan system effect must be added to the calcu-lated system pressure losses to determine the actual system curve. The point of intersection between the fan performance curve and the actual system curve is Point 4. The actual flow volume is, therefore, deficient by the difference from 1 to 4. To achieve design flow volume, a fan system effect pressure loss equal to the pressure difference between Points 1 and 2 should be added to the calculated system pressure losses, and the fan should be selected to operate at Point 2.
Fan System Effect Coefficients The system effect concept was formulated by Farquhar (1973) and Meyer (1973); the magnitudes of the system effect, called system effect factors, were determined experimentally in the labo-ratory of the Air Movement and Control Association (AMCA) (Brown 1973, Clarke et al. 1978) and published in their Publication 201 (AMCA 1990a). The system effect factors, converted to local loss coefficients, are in the Duct Fitting Database (ASHRAE 1994) for both centrifugal and axial fans. Fan system effect coefficients are only an approximation. Fans of different types and even fans of the same type, but supplied by different manufacturers, do not necessar-ily react to a system in the same way. Therefore, judgment based on experience must be applied to any design.
Fan Outlet Conditions. Fans intended primarily for duct sys-tems are usually tested with an outlet duct in place (ASHRAE Stan-dard 51). Figure 11 shows the changes in velocity profiles at various distances from the fan outlet. For 100% recovery, the duct, includ-ing transition, must meet the requirements for 100% effective duct length [Le (Figure 11)], which is calculated as follows: For Vo > 13 m/s, (36) Table 4 Duct Fitting Codes Fitting Function Geometry Category Sequential Number S: Supply D: round (Diameter) 1. Entries 1,2,3...n 2. Exits E: Exhaust/Return R: Rectangular 3. Elbows 4. Transitions C: Common (supply and return) F: Flat oval 5. Junctions 6. Obstructions 7. Fan and system interactions 8. Duct-mounted equipment 9. Dampers 10. Hoods p ∆ 1000fL Dh -----------------ΣC + ρV 2 2 --------- = Fig. 10 Deficient System Performance with System Effect Ignored Le Vo Ao 4500 ------------------= Duct Design 34.13 For Vo ≤ 13 m/s, (37) where Vo = duct velocity, m/s Le = effective duct length, m Ao = duct area, mm2 As illustrated by Fitting SR7-1 in the section on Fitting Loss Coefficients, centrifugal fans should not abruptly discharge to the atmosphere. A diffuser design should be selected from Fitting SR7-2 (see the section on Fitting Loss Coefficients) or SR7-3 (see ASHRAE 1994).
Fan Inlet Conditions. For rated performance, air must enter the fan uniformly over the inlet area in an axial direction without pre-rotation. Nonuniform flow into the inlet is the most common cause of reduced fan performance. Such inlet conditions are not equiva-lent to a simple increase in the system resistance; therefore, they cannot be treated as a percentage decrease in the flow and pressure from the fan. A poor inlet condition results in an entirely new fan performance. An elbow at the fan inlet, for example Fitting ED7-2 (see the section on Fitting Loss Coefficients), causes turbulence and uneven flow into the fan impeller. The losses due to the fan system effect can be eliminated by including an adequate length of straight duct between the elbow and the fan inlet.
The ideal inlet condition allows air to enter axially and uniformly without spin. A spin in the same direction as the impeller rotation reduces the pressure-volume curve by an amount dependent on the intensity of the vortex. A counterrotating vortex at the inlet slightly increases the pressure-volume curve, but the power is increased substantially.
Inlet spin may arise from a great variety of approach conditions, and sometimes the cause is not obvious. Inlet spin can be avoided by providing an adequate length of straight duct between the elbow and the fan inlet. Figure 12 illustrates some common duct connec-tions that cause inlet spin and includes recommendations for cor-recting spin.
Fans within plenums and cabinets or next to walls should be located so that air may flow unobstructed into the inlets. Fan per-formance is reduced if the space between the fan inlet and the enclosure is too restrictive. The system effect coefficients for fans in an enclosure or adjacent to walls are listed under Fitting ED7-1 (see the section on Fitting Loss Coefficients). The manner in which the airstream enters an enclosure in relation to the fan inlets also affects fan performance. Plenum or enclosure inlets or walls that are not symmetrical with the fan inlets cause uneven flow and/or inlet spin.
Testing, Adjusting, and Balancing Considerations Fan system effects (FSEs) are not only to be used in conjunction with the system resistance characteristics in the fan selection pro-cess, but are also applied in the calculations of the results of testing, adjusting, and balancing (TAB) field tests to allow direct compari-son to design calculations and/or fan performance data. Fan inlet swirl and the effect on system performance of poor fan inlet and out-let ductwork connections cannot be measured directly. Poor inlet flow patterns affect fan performance within the impeller wheel Fig. 11 Establishment of Uniform Velocity Profile in Straight Fan Outlet Duct (Adapted by permission from AMCA Publication 201) Le Ao 350 ------------= 34.14 2001 ASHRAE Fundamentals Handbook (SI) (centrifugal fan) or wheel rotor impeller (axial fan), while the fan outlet system effect is flow instability and turbulence within the fan discharge ductwork.
The static pressure at the fan inlet and the static pressure at the fan outlet may be measured directly in some systems. In most cases, static pressure measurements for use in determining fan total (or static) pressure will not be made directly at the fan inlet and outlet, but at locations a relatively short distance from the fan inlet and downstream from the fan outlet. To calculate fan total pressure for this case from field measurements, use Equation (38), where ∆px-y is the summation of calculated total pressure losses between the fan inlet and outlet sections noted. Plane 3 is used to determine airflow rate. If necessary, use Equation (18) to calculate fan static pressure knowing fan total pressure. For locating measurement planes and calculation procedures, consult AMCA Publication 203 (AMCA 1990b).
(38) where Pt = fan total pressure, Pa ps = static pressure, Pa pv = velocity pressure, Pa FSE = fan system effect, Pa ∆px-y = summarization of total pressure losses between planes x and y, Pa Subscripts (numerical subscripts same as used by AMCA Publication 203): 1 = fan inlet 2 = fan outlet 3 = plane of airflow measurement 4 = plane of static pressure measurement upstream of fan 5 = plane of static pressure measurement downstream of fan sw = swirl DUCT SYSTEM DESIGN DESIGN CONSIDERATIONS Space Pressure Relationships Space pressure is determined by fan location and duct system arrangement. For example, a supply fan that pumps air into a space increases space pressure; an exhaust fan reduces space pressure. If both supply and exhaust fans are used, space pressure depends on the relative capacity of the fans. Space pressure is positive if supply exceeds exhaust and negative if exhaust exceeds supply (Osborne 1966). System pressure variations due to wind can be minimized or eliminated by careful selection of intake air and exhaust vent loca-tions (Chapter 16).
Fire and Smoke Management Because duct systems can convey smoke, hot gases, and fire from one area to another and can accelerate a fire within the system, fire protection is an essential part of air-conditioning and ventilation system design. Generally, fire safety codes require compliance with the standards of national organizations. NFPA Standard 90A exam-ines fire safety requirements for (1) ducts, connectors, and appurte-nances; (2) plenums and corridors; (3) air outlets, air inlets, and fresh air intakes; (4) air filters; (5) fans; (6) electric wiring and equipment; (7) air-cooling and -heating equipment; (8) building construction, including protection of penetrations; and (9) controls, including smoke control.
Fire safety codes often refer to the testing and labeling practices of nationally recognized laboratories, such as Factory Mutual and Underwriters Laboratories (UL). The Building Materials Directory compiled by UL lists fire and smoke dampers that have been tested and meet the requirements of UL Standards 555 and 555S. This directory also summarizes maximum allowable sizes for individual dampers and assemblies of these dampers. Fire dampers are 1.5 h or 3 h fire-rated. Smoke dampers are classified by (1) temperature deg-radation [ambient air or high temperature (120°C minimum)] and (2) leakage at 250 Pa and 1000 Pa pressure difference (2 kPa and 3 kPa classification optional). Smoke dampers are tested under con-ditions of maximum airflow. UL’s Fire Resistance Directory lists the fire resistance of floor/roof and ceiling assemblies with and without ceiling fire dampers.
For a more detailed presentation of fire protection, see Chapter 51 of the 1999 ASHRAE Handbook—Applications and the NFPA Fire Protection Handbook.
Duct Insulation In all new construction (except low-rise residential buildings), air-handling ducts and plenums installed as part of an HVAC air distribution system should be thermally insulated in accordance with Section 6.2.4.2 of ASHRAE Standard 90.1. Duct insulation for new low-rise residential buildings should be in compliance with ASHRAE Standard 90.2. Existing buildings should meet the requirements of ASHRAE Standard 100. The insulation thick-nesses in these standards are minimum values. Economic and ther-mal considerations may justify higher insulation levels. Additional insulation, vapor retarders, or both may be required to limit vapor transmission and condensation.
Duct heat gains or losses must be known for the calculation of supply air quantities, supply air temperatures, and coil loads (see Chapter 29 of this volume and Chapter 2 of the 2000 ASHRAE Hand-book—Systems and Equipment). To estimate duct heat transfer and entering or leaving air temperatures, use the following equations: (39) Fig. 12 Inlet Duct Connections Causing Inlet Spin and Corrections for Inlet Spin (Adapted by permission from AMCA Publication 201) Pt ps 5 , pv 5 , + ( ) p2-5 ∆ FSE2 + + = ps 4 , pv 4 , + ( ) p4-1 ∆ FSE1 FSE1 sw , + + + + ql UPL 1000 ------------ te tl + 2 ------------- ta – = Duct Design 34.15 (40) (41) where ql = heat loss/gain through duct walls, W (negative for heat gain) U = overall heat transfer coefficient of duct wall, W/(m2·K) P = perimeter of bare or insulated duct, mm L = duct length, m te = temperature of air entering duct, °C tl = temperature of air leaving duct, °C ta = temperature of air surrounding duct, °C y = 2.0AVρcp/UPL for rectangular ducts = 0.5DVρcp/UL for round ducts A = cross-sectional area of duct, mm2 V = average velocity, m/s ρ = density of air, kg/m3 cp = specific heat of air, kJ/(kg·K) D = diameter of duct, mm Use Figure 13A to determine U-factors for insulated and uninsu-lated ducts. Lauvray (1978) has shown the effects of (1) compress-ing insulation wrapped externally on sheet metal ducts and (2) insulated flexible ducts with air-porous liners. For a 50 mm thick, 12 kg/m3 fibrous glass blanket compressed 50% during installation, the heat transfer rate increases approximately 20% (see Figure 13A). Pervious flexible duct liners also influence heat transfer significantly (see Figure 13B). At 12.7 m/s, the pervious liner U-factor is 1.87 W/(m2·K); for an impervious liner, U = 1.08 W/(m2·K).
Example 6. A 20 m length of 600 mm by 900 mm uninsulated sheet metal duct, freely suspended, conveys heated air through a space maintained above freezing at 5°C. Based on heat loss calculations for the heated zone, 8100 L/s of standard air [cp = 1.006 kJ/(kg·K)] at a supply air temperature of 50°C is required. The duct is connected directly to the heated zone. Determine the temperature of the air entering the duct and the duct heat loss.
Solution: Calculate duct velocity using Equation (10): Calculate entering air temperature using Equation (40): Calculate duct heat loss using Equation (39): Example 7. Same as Example 6, except the duct is insulated externally with 50 mm thick fibrous glass with a density of 12 kg/m3. The insula-tion is wrapped with 0% compression.
Solution: All values except U remain the same as in Example 6. From Figure 13A, U = 0.83 W/(m2· K) at 15 m/s. Therefore, Insulating this duct reduces heat loss to 20% of the uninsulated value.
Duct System Leakage Leakage in all unsealed ducts varies considerably with the fab-ricating machinery used, the methods for assembly, and installation workmanship. For sealed ducts, a wide variety of sealing methods and products exists. Sealed and unsealed duct leakage tests (AISI/SMACNA 1972, ASHRAE/SMACNA/TIMA 1985, Swim and Griggs 1995) have confirmed that longitudinal seam, transverse joint, and assembled duct leakage can be represented by Equation (42) and that for the same construction, leakage is not significantly different in the negative and positive modes. A range of leakage rates te tl y 1 + ( ) 2ta – y 1 – ( ) ----------------------------------= tl te y 1 – ( ) 2ta + y 1 + ( ) ----------------------------------= V 1000 8100 L/s × 600 mm 900 mm × ----------------------------------------------15 m/s = = U 4.16 W m2 K ⋅ ( ) from Figure 13A ( ) ⁄ = P 2 600 900 + ( ) 3000 mm = = y 2.0 600 mm ( ) 900 mm ( ) 15 m/s ( ) 1.204 kg/m3 ( ) 1.006 ( ) 4.16 W m2 K ⋅ ( ) ⁄ 3000 mm × 20 m × -------------------------------------------------------------------------------------------------------------------------------------78.6 = = te 50°C 78.6 1 + ( ) 2 5°C × ( ) – 78.6 1 – ( ) ---------------------------------------------------------------------51.2°C = = ql 4.16 W m2 K ⋅ ( ) ⁄ 3000 mm × 20 m × 1000 -------------------------------------------------------------------------------------------= 51.2°C 50°C + 2 -------------------------------------5°C – × 11 400 W = Fig. 13 Duct Heat Transfer Coefficients y 394 = te 50.2°C = ql 2250 W = 34.16 2001 ASHRAE Fundamentals Handbook (SI) for longitudinal seams commonly used in the construction of metal ducts is presented in Table 5. Longitudinal seam leakage for unsealed or unwelded metal ducts is about 10 to 15% of total duct leakage.
(42) where Q = duct leakage rate, L/s per m2 C = constant reflecting area characteristics of leakage path ∆ps = static pressure differential from duct interior to exterior, Pa N = exponent relating turbulent or laminar flow in leakage path Analysis of the AISI/ASHRAE/SMACNA/TIMA data resulted in the categorization of duct systems into leakage classes CL based on Equation (43), where the exponent N is assumed to be 0.65. A selected series of leakage classes based on Equation (43) is shown in Figure 14.
(43) where Q = leakage rate, L/s per m2 (surface area) CL = leakage class, mL/s·m2 at 1 Pa Table 6 is a forecast of the leakage class attainable for commonly used duct construction and sealing practices. Connections of ducts to grilles, diffusers, and registers are not represented in the test data.
Leakage classes listed are for a specific duct type, not a system with a variety of duct types, access doors, and other duct-mounted equip-ment. The designer is responsible for assigning acceptable system leakage rates. It is recommended that this be accomplished by using Table 7 as a guideline to specify a ductwork leakage class or by specifying a duct seal level as recommended by Table 8. The designer should take into account attainable leakage rates by duct type and the fact that casings of volume-controlling air terminal units may leak 1 to 2% of their maximum flow. The effects of such leakage should be anticipated, if allowed, and the ductwork should not be expected to compensate for equipment leakage. When a sys-tem leakage class is specified by a designer, it is a performance spec-ification that should not be compromised by prescriptive sealing. A portion of a system may exceed its leakage class if the aggregate system leakage meets the allowable rate. Table 9 can be used to esti-mate the system percent leakage based on the system design leakage class and system duct surface area. Table 9 is predicated on assess-ment at an average of upstream and downstream pressures because use of the highest pressure alone could indicate an artificially high rate. When several duct pressure classifications occur in a system, ductwork in each pressure class should be evaluated independently to arrive at an aggregate leakage for the system.
Leakage tests should be conducted in compliance with SMACNA’s HVAC Air Duct Leakage Test Manual (1985) to verify the intent of the designer and the workmanship of the the installing contractor. Leakage tests used to confirm leakage class should be conducted at the pressure class for which the duct is constructed.
Leakage testing is also addressed in ASHRAE Standard 90.1.
Limited performance standards for metal duct sealants and tapes exist. For guidance in their selection and use refer to SMACNA’s HVAC Duct Construction Standards (1995). Fibrous glass ducts and their closure systems are covered by UL Standards 181and 181A.
Table 5 Unsealed Longitudinal Seam Leakage, Metal Ducts Type of Duct/Seam Leakage, L/s per metre Seam Lengtha Range Average Rectangular Pittsburgh lock 0.015 to 0.87 0.25 Button punch snaplock 0.015 to 0.25 0.10 Round Snaplock 0.06 to 0.22 0.17 Grooved 0.17 to 0.28 0.19 aLeakage rate is at 250 Pa static pressure.
Q C ps N ∆ = Fig. 14 Duct Leakage Classifications CL 1000Q ps 0.65 ∆ -----------------= Table 6 Duct Leakage Classificationa Duct Type Sealedb,c Unsealedc Predicted Leakage Class CL Leakage Rate, L/(s·m2) at 250 Pa Predicted Leakage Class CL Leakage Rate, L/(s·m2) at 250 Pa Metal (flexible excluded) Round and flat oval 4 0.14 42 1.5 (8 to 99) (0.3 to 3.6) Rectangular ≤ 500 Pa (both positive and negative pressures) 17 0.62 68 2.5 (17 to 155) (0.6 to 5.6) > 500 and ≤ 2500 Pa (both positive and negative pressures) 8 0.29 68 2.5 (17 to 155) (0.6 to 5.6) Flexible Metal, Aluminum 11 0.40 42 1.5 (17 to 76) (0.6 to 2.8) Nonmetal 17 0.62 30 1.5 (6 to 76) (0.2 to 2.8) Fibrous glass Round 4 0.14 na na Rectangular 8 0.29 na na aThe leakage classes listed in this table are averages based on tests conducted by AISI/ SMACNA (1972), ASHRAE/SMACNA/TIMA (1985), and Swim and Griggs (1995).
bThe leakage classes listed in the sealed category are based on the assumptions that for metal ducts, all transverse joints, seams, and openings in the duct wall are sealed at pressures over 750 Pa, that transverse joints and longitudinal seams are sealed at 500 and 750 Pa, and that transverse joints are sealed below 500 Pa. Lower leakage classes are obtained by careful selection of joints and sealing methods.
cLeakage classes assigned anticipate about 0.82 joints per metre of duct. For systems with a high fitting to straight duct ratio, greater leakage occurs in both the sealed and unsealed conditions.
Duct Design 34.17 For fibrous glass duct construction standards consult NAIMA (1997) and SMACNA (1992). Flexible duct performance and instal-lation standards are covered by UL 181, UL 181B and ADC (1996).
Soldered or welded duct construction is necessary where sealants are not suitable. Sealants used on exterior ducts must be resistant to weather, temperature cycles, sunlight, and ozone.
Shaft and compartment pressure changes affect duct leakage and are important to health and safety in the design and operation of con-taminant and smoke control systems. Shafts should not be used for supply, return, and/or exhaust air without accounting for their leakage rates. Airflow around buildings, building component leakage, and the distribution of inside and outside pressures over the height of a build-ing, including shafts, are discussed in Chapters 16 and 26. Smoke management system design is covered in Chapter 51 of the 1999 ASHRAE Handbook—Applications and in Klote and Milke (1992).
System Component Design Velocities Table 10 summarizes face velocities for HVAC components in built-up systems. In most cases, the values are abstracted from per-tinent chapters in the 2000 ASHRAE Handbook—Systems and Equipment; final selection of the components should be based on data in these chapters or from manufacturers.
Louvers require special treatment since the blade shapes, angles, and spacing cause significant variations in louver-free area and per-formance (pressure drop and water penetration). Selection and anal-ysis should be based on test data obtained in accordance with AMCA Standard 500-L (1999). This standard presents both pressure drop and water penetration test procedures and a uniform method for cal-culating the free area of a louver. Tests are conducted on a 1220 mm square louver with the frame mounted flush in the wall. For the water penetration tests, the rainfall is 100 mm/h, no wind, and the water flow down the wall is 0.05 L/s per linear metre of louver width.
Use Figure 15 for preliminary sizing of air intake and exhaust louvers. For air quantities greater than 3300 L/s per louver, the air intake gross louver openings are based on 2 m/s; for exhaust lou-vers, 2.5 m/s is used for air quantities of 2400 L/s per louver and greater. For air quantities less than these, refer to Figure 15. These criteria are presented on a per louver basis (i.e., each louver in a bank of louvers) to include each louver frame. Representative pro-duction-run louvers were used in establishing Figure 15, and all data used in that analysis are based on AMCA standard tests. For louvers larger than 1.5 m2, the free areas are greater than 45%, while for lou-vers less than 1.5 m2, the free areas are less than 45%. Unless spe-cific louver data are analyzed, no louver should have a face area less than 0.4 m2. If debris collection on the screen of an intake louver is possible, or if louvers are located at grade with adjacent pedestrian traffic, louver face velocity should not exceed 0.5 m/s.
System and Duct Noise The major sources of noise from air-conditioning systems are diffusers, grilles, fans, ducts, fittings, and vibrations. Chapter 46 of the 1999 ASHRAE Handbook—Applications discusses sound con-trol for each of these sources. Sound control for terminal devices Table 7 Recommended Ductwork Leakage Class by Duct Type Duct Type Leakage Class Leakage Rate, L/(s·m2) at 250 Pa Metal Round 4 0.14 Flat oval 4 0.14 Rectangular 8 0.29 Flexible 8 0.29 Fibrous glass Round 4 0.14 Rectangular 8 0.29 Table 8A Recommended Duct Seal Levelsa Duct Location Duct Type Supply Exhaust Return ≤ 500 Pa > 500 Pa Outdoors A A A A Unconditioned spaces B A B B Conditioned spaces (concealed ductwork) C B B C Conditioned spaces (exposed ductwork) A A B B aSee Table 8B for definition of seal level.
Table 8B Duct Seal Levels Seal Level Sealing Requirementsa A All transverse joints, longitudinal seams, and duct wall penetrations B All transverse joints and longitudinal seams C Transverse joints only aTransverse joints are connections of two duct or fitting elements oriented perpendic-ular to flow. Longitudinal seams are joints oriented in the direction of airflow. Duct wall penetrations are openings made by screws, non-self-sealing fasteners, pipe, tub-ing, rods, and wire. Round and flat oval spiral lock seams need not be sealed prior to assembly, but may be coated after assembly to reduce leakage. All other connections are considered transverse joints, including but not limited to spin-ins, taps and other branch connections, access door frames, and duct connections to equipment.
Table 9 Leakage as Percentage of Airflowa,b Leakage Class System L/s per m2 Duct Surfacec Static Pressure, Pa 125 250 500 750 1000 1500 68 10 15 24 38 49 59 77 12.7 12 19 30 39 47 62 15 10 16 25 33 39 51 20 7.7 12 19 25 30 38 25 6.1 9.6 15 20 24 31 34 10 7.7 12 19 25 30 38 12.7 6.1 9.6 15 20 24 31 15 5.1 8.0 13 16 20 26 20 3.8 6.0 9.4 12 15 19 25 3.1 4.8 7.5 9.8 12 15 17 10 3.8 6 9.4 12 15 19 12.7 3.1 4.8 7.5 9.8 12 15 15 2.6 4.0 6.3 8.2 9.8 13 20 1.9 3.0 4.7 6.1 7.4 9.6 25 1.5 2.4 3.8 4.9 5.9 7.7 8 10 1.9 3 4.7 6.1 7.4 9.6 12.7 1.5 2.4 3.8 4.9 5.9 7.7 15 1.3 2.0 3.1 4.1 4.9 6.4 20 1.0 1.5 2.4 3.1 3.7 4.8 25 0.8 1.2 1.9 2.4 3.0 3.8 4 10 1.0 1.5 2.4 3.1 3.7 4.8 12.7 0.8 1.2 1.9 2.4 3.0 3.8 15 0.6 1.0 1.6 2.0 2.5 3.2 20 0.5 0.8 1.3 1.6 2.0 2.6 25 0.4 0.6 0.9 1.2 1.5 1.9 aAdapted with permission from HVAC Air Duct Leakage Test Manual (SMACNA 1985, Appendix A).
bPercentage applies to the airflow entering a section of duct operating at an assumed pressure equal to the average of the upstream and downstream pressures.
cThe ratios in this column are typical of fan volumetric flow rate divided by total sys-tem surface. Portions of the systems may vary from these averages.
34.18 2001 ASHRAE Fundamentals Handbook (SI) consists of selecting devices that meet the design goal under all operating conditions and installing them properly so that no addi-tional sound is generated. The sound power output of a fan is deter-mined by the type of fan, airflow, and pressure. Sound control in the duct system requires proper duct layout, sizing, and provision for installing duct attenuators, if required. The noise generated by a sys-tem increases with both duct velocity and system pressure. Chapter 46 of the 1999 ASHRAE Handbook—Applications presents meth-ods for calculating required sound attenuation.
Testing and Balancing Each air duct system should be tested, adjusted, and balanced.
Detailed procedures are given in Chapter 36 of the 1999 ASHRAE Handbook—Applications. To properly determine fan total (or static) pressure from field measurements taking into account fan system effect, refer to the section on Fan-System Interface. Equation (38) allows direct comparison of system resistance to design calculations and/or fan performance data. It is important that the system effect magnitudes be known prior to testing. If necessary, use Equation (18) to calculate fan static pressure knowing fan total pressure [Equation (38)]. For TAB calculation procedures of numerous fan/system configurations encountered in the field, refer to AMCA Publication 203 (AMCA 1990b).
DUCT DESIGN METHODS Duct design methods for HVAC systems and for exhaust systems conveying vapors, gases, and smoke are the equal friction method, the static regain method, and the T-method. The section on Indus-trial Exhaust System Duct Design presents the design criteria and procedures for exhaust systems conveying particulates. Equal fric-tion and static regain are nonoptimizing methods, while the T-method is a practical optimization method introduced by Tsal et al. (1988).
To ensure that system designs are acoustically acceptable, noise generation should be analyzed and sound attenuators and/or acous-tically lined duct provided where necessary. Dampers must be installed throughout systems designed by equal friction, static regain, and the T-method because inaccuracies are introduced into these design methods by duct size round-off and the effect of close-coupled fittings on the total pressure loss calculations.
Equal Friction Method In the equal friction method, ducts are sized for a constant pres-sure loss per unit length. The shaded area of the friction chart (Fig-ure 9) is the suggested range of friction rate and air velocity. When energy cost is high and installed ductwork cost is low, a low friction rate design is more economical. For low energy cost and high duct cost, a higher friction rate is more economical. After initial sizing, calculate the total pressure loss for all duct sections, and then resize sections to balance pressure losses at each junction.
Static Regain Method The objective of the static regain method is to obtain the same static pressure at diverging flow junctions by changing downstream duct sizes. This design objective can be developed by rearranging Equation (7a) and setting ps,2 equal to ps,1 (neglecting thermal grav-ity effect term). Thus, Table 10 Typical Design Velocities for HVAC Components Duct Element Face Velocity, m/s LOUVERSa Intake 3300 L/s and greater 2 Less than 3300 L/s See Figure 15 Exhaust 2400 L/s and greater 2.5 Less than 2400 L/s See Figure 15 FILTERSb Panel filters Viscous impingement 1 to 4 Dry-type, extended-surface Flat (low efficiency) Duct velocity Pleated media (intermediate efficiency) Up to 3.8 HEPA 1.3 Renewable media filters Moving-curtain viscous impingement 2.5 Moving-curtain dry media 1 Electronic air cleaners Ionizing type 0.8 to 1.8 HEATING COILSc Steam and hot water 2.5 to 5 1 min., 8 max.
Electric Open wire Refer to mfg. data Finned tubular Refer to mfg. data DEHUMIDIFYING COILSd 2 to 3 AIR WASHERSe Spray type 1.5 to 3.0 Cell type Refer to mfg. data High-velocity spray type 6 to 9 aBased on assumptions presented in text.
bAbstracted from Chapter 24, 2000 ASHRAE Handbook—Systems and Equipment.
cAbstracted from Chapter 23, 2000 ASHRAE Handbook—Systems and Equipment.
dAbstracted from Chapter 21, 2000 ASHRAE Handbook—Systems and Equipment.
eAbstracted from Chapter 19, 2000 ASHRAE Handbook—Systems and Equipment.
Fig. 15 Criteria for Louver Sizing Parameters Used to Establish Figure Intake Louver Exhaust Louver Minimum free area (1220 mm square test section), % 45 45 Water penetration, µL/(m2·s) Negligible (less than 0.6) na Maximum static pressure drop, Pa 35 60 Duct Design 34.19 (44) and (45) where is the total pressure loss from upstream of junction 1 to upstream of junction 2, or the terminal of section 2. The immedi-ate downstream duct size that satisfies Equation (45) is determined by iteration.
To start the design of a system, a maximum velocity is selected for the root section (duct section upstream and/or downstream of a fan). In Figure 17, section 6 is the root for the return air subsystem.
Section 19 is the root for the supply air subsystem. The shaded area on the friction chart (Figure 9) is the suggested range of air velocity. When energy cost is high and installed ductwork cost is low, a lower initial velocity is more economical. For low energy cost and high duct cost, a higher velocity is more economical. All other sections, except terminal sections, are sized iteratively by Equation (45). In Figure 17, terminal sections are 1, 2, 4, 7, 8, 11, 12, 15, and 16. Knowing the terminal static pressure requirements, Equation (45) is used to calculate the duct size of terminal sec-tions. If the terminal is an exit fitting rather than a register, diffuser, or terminal box, the static pressure at the exit of the terminal sec-tion is zero.
The classical static regain method (Carrier Corporation 1960, Chun-Lun 1983) is based on Equation (46), where R is the static pressure regain factor, and ∆pr is the static pressure regain between junctions.
(46) Typically R-values ranging from 0.5 to 0.95 have been used. Tsal and Behls (1988) show that this uncertainty exists because the split-ting of mass at junctions and the dynamic (fitting) losses between junctions are ignored. The classical static regain method using an R-value should not be used because R is not predictable.
T-Method Optimization T-method optimization (Tsal et al. 1988) is a dynamic program-ming procedure based on the tee-staging idea used by Bellman (1957), except that phase level vector tracing is eliminated by opti-mizing locally at each stage. This modification reduces the number of calculations but requires iteration.
Optimization Basis. The objective function, Equation (47), includes both initial system cost and the present worth of energy.
Hours of operation, annual escalation and interest rates, and amor-tization period are also required for optimization.
(47) where E = present worth owning and operating cost Ep = first year energy cost Es = initial cost PWEF = present worth escalation factor (Smith 1968), dimensionless Energy cost is determined by (48) where Qf = fan airflow rate, L/s Ec = unit energy cost, cost/kWh Ed = energy demand cost, cost/kW T = system operating time, h/year Pt = fan total pressure, Pa ηf = fan total efficiency, decimal ηe = motor-drive efficiency, decimal Energy cost depends on both applicable energy rates Ec and demand cost Ed. Since the difference in fan pressure between an optimized and a nonoptimized system is a small part of demand, it is usually neglected. Initial cost includes ducts and HVAC equip-ment, which is primarily the central handling unit. The cost of duct systems is given by the following equations: (49) (50) where Sd = unit ductwork cost/m2 (including material and labor) H = duct height, mm W = duct width, mm L = duct length, m The cost of space required by ducts and equipment is another important factor of duct optimization. Including this cost reduces the size of ducts, thereby increasing energy consumption. Because the space available for ductwork is usually not used for anything else, its cost is ignored.
Both electrical energy rates and ductwork costs vary widely, by a factor of up to eight times for industrial users (DOE). Black iron rectangular ductwork can cost about 3.9 times that of spiral duct-work (Wendes 1989). Combining these ratios yields a factor of 30 to 1 based on locale and type of ductwork. Therefore, a great potential exists for reducing duct system life-cycle cost due to energy and ductwork cost variations.
The following constraints are necessary for duct optimization (Tsal and Adler 1987): • Continuity. For each node, the flow in equals the flow out.
• Pressure balancing. The total pressure loss in each path must equal the fan total pressure; or, in effect, at any junction, the total pressure loss for all paths is the same.
• Nominal duct size. Ducts are constructed in discrete, nominal sizes. Each diameter of a round duct or height and width of a rectangular duct is rounded to the nearest increment, usually 25 or 50 mm. If a lower nominal size is selected, the initial cost decreases, but the pressure loss increases and may exceed the fan pressure. If the higher nominal size is selected, the opposite is true—the initial cost increases, but the section pressure loss decreases. However, this lower pressure at one section may allow smaller ducts to be selected for sections that follow. Therefore, optimization must consider size rounding.
• Air velocity restriction. The maximum allowable velocity is an acoustic limitation (ductwork regenerated noise).
• Construction restriction. Architectural limits may restrict duct sizes. If air velocity or construction constraints are violated during an iteration, a duct size must be calculated. The pressure loss calculated for this preselected duct size is considered a fixed loss.
Calculation Procedure. The T-method comprises the following major procedures: ps 1 , ps 2 , – pt 1-2 , ∆ ρV1 2 2 ---------ρV2 2 2 ---------– – = pt 1-2 , ∆ ρV1 2 2 ---------ρV2 2 2 ---------– = pt 1-2 , ∆ pr ∆ R ρV1 2 2 ---------ρV2 2 2 ---------– = E Ep PWEF ( ) Es + = Ep Qf Ed EcT + ( ) 106ηf ηe --------------------------- Pt = Round Es SdπDL 1000 ⁄ = Rec gular tan Es 2Sd H W + ( )L 1000 ⁄ = 34.20 2001 ASHRAE Fundamentals Handbook (SI) • System condensing. This procedure condenses a multiple-section duct system into a single imaginary duct section with identical hydraulic characteristics and the same owning cost as the entire system. By Equation (1.41) in Tsal et al. (1988), two or more converging or diverging sections and the common section at a junction can be replaced by one condensed section. By applying this equation from junction to junction in the direction to the root section (fan), the entire supply and return systems can be condensed into one section (a single resistance).
• Fan selection. From the condensed system, the ideal optimum fan total pressure Pt opt is calculated and used to select a fan. If a fan with a different pressure is selected, its pressure P opt is considered optimum.
• System expansion. The expansion process distributes the available fan pressure P opt throughout the system. Unlike the condensing procedure, the expansion procedure starts at the root section and continues in the direction of the terminals.
Economic Analysis. Tsal et al. (1988) describe the calculation procedure and include an economic analysis of the T-method.
T-Method Simulation T-method simulation, also developed by Tsal et al. (1990), deter-mines the flow in each duct section of an existing system with a known operating fan performance curve. The simulation version of the T-method converges very efficiently. Usually three iterations are sufficient to obtain a solution with a high degree of accuracy.
Calculation procedure. The simulation version of the T-method includes the following major procedures: • System condensing. This procedure condenses a branched tee system into a single imaginary duct section with identical hydraulic characteristics. Two or more converging or diverging sections and the common section at a junction can be replaced by one condensed section [by Equation (18) in Tsal et al. (1990)]. By applying this equation from junction to junction in the direction to the root section (fan), the entire system, including supply and return subsystems, can be condensed into one imaginary section (a single resistance).
• Fan operating point. This step determines the system flow and pressure by locating the intersection of the fan performance and system curves, where the system curve is represented by the imaginary section from the last step.
• System expansion. Knowing system flow and pressure, the previously condensed imaginary duct section is expanded into the original system with flow distributed in accordance with the ratio of pressure losses calculated in the system condensing step.
Simulation Applications. The need for duct system simulation appears in many HVAC problems. In addition to the following con-cerns that can be clarified by simulation, the T-method is an excel-lent design tool for simulating the flow distribution within a system with various modes of operation.
• Flow distribution in a variable air volume (VAV) system due to terminal box flow diversity • Airflow redistribution due to HVAC system additions and/or modifications • System airflow analysis for partially occupied buildings • Necessity to replace fans and/or motors when retrofitting an air distribution system • Multiple-fan system operating condition when one or more fans shut down • Pressure differences between adjacent confined spaces within a nuclear facility when a design basis accident (DBA) occurs (Farajian et al. 1992) • Smoke management system performance during a fire, when certain fire/smoke dampers close and others remain open HVAC DUCT DESIGN PROCEDURES The general procedure for HVAC system duct design is as follows: 1. Study the building plans, and arrange the supply and return outlets to provide proper distribution of air within each space.
Adjust calculated air quantities for duct heat gains or losses and duct leakage. Also, adjust the supply, return, and/or exhaust air quantities to meet space pressurization requirements.
2. Select outlet sizes from manufacturers’ data (see Chapter 32).
3. Sketch the duct system, connecting supply outlets and return intakes with the air-handling units/air conditioners. Space allocated for supply and return ducts often dictates system layout and ductwork shape. Use round ducts whenever feasible and avoid close-coupled fittings.
4. Divide the system into sections and number each section. A duct system should be divided at all points where flow, size, or shape changes. Assign fittings to the section toward the supply and return (or exhaust) terminals. The following examples are for the fittings identified for Example 6 (Figure 16), and system section numbers assigned (Figure 17). For converging flow fitting 3, assign the straight-through flow to section 1 (toward terminal 1), and the branch to section 2 (toward terminal 4). For diverging flow fitting 24, assign the straight-through flow to section 13 (toward terminals 26 and 29) and the branch to section 10 (toward terminals 43 and 44). For transition fitting 11, assign the fitting to upstream section 4 [toward terminal 9 (intake louver)].
For fitting 20, assign the unequal area elbow to downstream section 9 (toward diffusers 43 and 44). The fan outlet diffuser, fitting 42, is assigned to section 19 (again, toward the supply duct terminals).
5. Size ducts by the selected design method. Calculate system total pressure loss; then select the fan (refer to Chapter 18 of the 2000 ASHRAE Handbook—Systems and Equipment).
6. Lay out the system in detail. If duct routing and fittings vary significantly from the original design, recalculate the pressure losses. Reselect the fan if necessary.
7. Resize duct sections to approximately balance pressures at each junction.
8. Analyze the design for objectionable noise levels, and specify sound attenuators as necessary. Refer to the section on System and Duct Noise.
Example 8. For the system illustrated by Figures 16 and 17, size the duct-work by the equal friction method, and pressure balance the system by changing duct sizes (use 10 mm increments). Determine the system resistance and total pressure unbalance at the junctions. The airflow quantities are actual values adjusted for heat gains or losses, and duct-work is sealed (assume no leakage), galvanized steel ducts with trans-verse joints on 1200 mm centers (ε = 0.09 mm). Air is at 1.204 kg/m3 density.
Because the primary purpose of Figure 16 is to illustrate calculation procedures, its duct layout is not typical of any real duct system. The layout includes fittings from the local loss coefficient tables, with emphasis on converging and diverging tees and various types of entries and discharges. The supply system is constructed of rectangular duct-work; the return system, round ductwork.
Solution: See Figure 17 for section numbers assigned to the system.
The duct sections are sized within the suggested range of friction rate shown on the friction chart (Figure 9). Tables 11 and 12 give the total pressure loss calculations and the supporting summary of loss coeffi-cients by sections. The straight duct friction factor and pressure loss were calculated by Equations (19) and (20). The fitting loss coefficients are from the Duct Fitting Database (ASHRAE 1994). Loss coefficients were calculated automatically by the database program (not by manual interpolation). The pressure loss values in Table 11 for the diffusers (fittings 43 and 44), the louver (fitting 9), and the air-measuring station (fitting 46) are manufacturers’ data.
Duct Design 34.21 Fig. 16 Schematic for Example 8 34.22 2001 ASHRAE Fundamentals Handbook (SI) The pressure unbalance at the junctions may be noted by referring to Figure 18, the total pressure grade line for the system. The system resistance Pt is 679 Pa. Noise levels and the need for duct silencers were not evaluated. To calculate the fan static pressure, use Equation (18): where 119 Pa is the fan outlet velocity pressure.
INDUSTRIAL EXHAUST SYSTEM DUCT DESIGN Chapter 29 of the 1999 ASHRAE Handbook—Applications dis-cusses design criteria, including hood design, for industrial exhaust systems. Exhaust systems conveying vapors, gases, and smoke can be designed by equal friction, or T-method. Systems conveying par-ticulates are designed by the constant velocity method at duct veloc-ities adequate to convey particles to the system air cleaner. For contaminant transport velocities, see Table 5 in Chapter 29 of the 1999 ASHRAE Handbook—Applications.
Two pressure-balancing methods can be considered when designing industrial exhaust systems. One method uses balancing devices (e.g., dampers, blast gates) to obtain design airflow through each hood. The other approach balances systems by adding resis-tance to ductwork sections (i.e., changing duct size, selecting differ-ent fittings, and increasing airflow). This self-balancing method is preferred, especially for systems conveying abrasive materials.
Where potentially explosive or radioactive materials are conveyed, the prebalanced system is mandatory because contaminants could accumulate at the balancing devices. To balance systems by increas-ing airflow, use Equation (51), which assumes that all ductwork has the same diameter and that fitting loss coefficients, including main and branch tee coefficients, are constant.
(51) where Qc = airflow rate required to increase Pl to Ph, L/s Qd = total airflow rate through low-resistance duct run, L/s Ph = absolute value of pressure loss in high-resistance ductwork section(s), Pa Pl = absolute value of pressure loss in low-resistance ductwork section(s), Pa For systems conveying particulates, use elbows with a large cen-terline radius-to-diameter ratio (r/D), greater than 1.5 whenever possible. If r/D is 1.5 or less, abrasion in dust-handling systems can reduce the life of elbows. Elbows are often made of seven or more gores, especially in large diameters. For converging flow fittings, a 30° entry angle is recommended to minimize energy losses and abrasion in dust-handling systems. For the entry loss coefficients of hoods and equipment for specific operations, refer to Chapter 29 of the 1999 ASHRAE Handbook—Applications and to ACGIH (1998).
Fig. 17 System Schematic with Section Numbers for Example 8 Fig. 18 Total Pressure Grade Line for Example 8 Ps 679 119 – 560 Pa = = Qc Qd Ph Pl ⁄ ( )0.5 = Duct Design 34.23 Example 9. For the metalworking exhaust system in Figures 19 and 20, size the ductwork and calculate the fan static pressure requirement for an industrial exhaust designed to convey granular materials. Pressure balance the system by changing duct sizes and adjusting airflow rates.
The minimum particulate transport velocity for the chipping and grind-ing table ducts (sections 1 and 5, Figure 20) is 20 m/s. For the ducts associated with the grinder wheels (sections 2, 3, 4, and 5), the mini-mum duct velocity is 23 m/s. Ductwork is galvanized steel, with the absolute roughness being 0.09 mm. Assume standard air and use ISO diameter sizes, given in the following table: The building is one story, and the design wind velocity is 9 m/s. For the stack, use Design J shown in Figure 13 in Chapter 16 for complete Fig. 19 Metalworking Exhaust System for Example 9 Standard Circular Duct Diameters (ISO 1983) 63 180 500 71 200 560 80 224 630 90 250 710 100 280 800 112 315 900 125 355 1000 140 400 1120 160 450 1250 Note: Dimensions listed are in millimetres.
Fig. 20 System Schematic with Section Numbers for Example 9 34.24 2001 ASHRAE Fundamentals Handbook (SI) Table 11 Total Pressure Loss Calculations by Sections for Example 8 Duct SectionaFitting No.b Duct Element Airflow, L/s Duct Size (Equivalent Round) Velocity, m/s Velocity Pressure, Pa Duct Length,c m Summary of Fitting Loss Coefficientsd Duct Pressure Loss,e Pa/m Total Pressure Loss, Pa Section Pressure Loss, Pa 1 — Duct 700 300 mm φ 9.9 — 4.6 — 3.5 16 — Fittings 700 — 9.9 59 — 0.32 — 19 35 2 — Duct 250 200 mm φ 8.0 — 18.3 — 3.8 70 — Fittings 250 — 8.0 38 — −0.34 — −13 57 3 — Duct 950 300 mm φ 13.4 — 6.1 — 6.2 38 — Fittings 950 — 13.4 109 — 0.60 — 65 103 4 — Duct 950 600 mm × 600 mm (656) 2.6 — 1.5 — 0.1 0 — Fittings 950 — 2.6 4 — 1.09 — 4 9 Louver 950 600 mm × 600 mm — — — — — 25f 29 5 — Duct 950 380 mm φ 8.4 — 18.3 — 1.9 35 — Fittings 950 — 8.4 42 — 1.61 — 68 103 6 — Duct 1900 450 mm φ 11.9 — 9.1 — 3.0 27 — Fittings 1900 — 11.9 86 — 0.87 — 75 102 7 — Duct 275 250 mm × 250 mm (273) 4.4 — 4.3 — 0.9 4 — Fittings 275 — 4.4 12 — 0.26 — 3 43 Diffuser 275 250 mm × 250 mm — — — — — 25f 32 8 — Duct 275 250 mm × 250 mm (273) 4.4 — 1.2 — 0.9 1 — Fittings 275 — 4.4 12 — 1.25 — 15 44 Diffuser 275 250 mm × 250 mm — — — — — 25f 41 9 — Duct 550 500 mm × 250 mm (381) 4.4 — 7.6 — 0.6 5 — Fittings 550 — 4.4 12 — 1.67 — 20 25 10 — Duct 550 400 mm × 250 mm (343) 5.5 — 13.7 — 1.1 15 — Fittings 550 — 5.5 18 — 2.69 — 48 63 11 — Duct 475 250 mm × 250 mm (273) 7.6 — 3.0 — 2.6 8 — Fittings 475 — 7.6 35 — 1.68 — 59 67 12 — Duct 475 250 mm × 250 mm (273) 7.6 — 6.7 — 2.6 17 — Fittings 475 — 7.6 35 — 1.44 — 50 67 13 — Duct 950 350 mm × 250 mm (322) 10.9 — 10.7 — 4.2 45 — Fittings 950 — 10.9 71 — 0.17 — 12 57 14 — Duct 1500 660 mm × 250 mm (414) 9.1 — 4.6 — 2.2 10 — Fittings 1500 — 9.1 50 — 0.13 — 7 17 15 — Duct 200 200 mm × 150 mm (189) 6.7 — 12.2 — 3.2 39 — Fittings 200 — 6.7 27 — 0.57 — 15 54 16 — Duct 200 200 mm × 150 mm (189) 6.7 — 6.1 — 3.2 20 — Fittings 200 — 6.7 27 — 1.74 — 47 67 17 — Duct 400 250 mm × 150 mm (210) 10.7 — 4.2 — 6.9 29 — Fittings 400 — 10.7 69 — 0.73 — 50 79 18 — Duct 1900 800 mm × 250 mm (470) 9.5 — 7.0 — 2.3 16 — Fittings 1900 — 9.5 54 — 2.93 — 158 174 19 — Duct 1900 800 mm × 450 mm (649) 5.3 — 3.7 — 0.5 2 — Fittings 1900 — 5.3 17 — 4.71 — 80 46 Air-measur-ing station 1900 — — — — — — 15f 97 aSee Figure 17.
bSee Figure 16.
cDuct lengths are to fitting centerlines.
dSee Table 12.
eDuct pressure based on a 0.09 mm absolute roughness factor.
fPressure drop based on manufacturers’ data.
Table 12 Loss Coefficient Summary by Sections for Example 8 Duct Section Fitting Number Type of Fitting ASHRAE Fitting No.a Parameters Loss Coefficient 1 1 Entry ED1-3 r/D = 0.2 0.03 2 Damper CD9-1 θ = 0° 0.19 3 Wye (30°), main ED5-1 As/Ac = 1.0, Ab/Ac = 0.444, Qs/Qc = 0.75 0.10 (Cs) Summation of Section 1 loss coefficients............................................................................................................................................ 0.32 2 4 Entry ED1-1 L = 0, t = 1.61 mm (16 gage) 0.50 4 Screen CD6-1 n = 0.70, A1/Ao = 1 0.58 5 Elbow CD3-6 60°, r/D = 1.5, pleated 0.27 6 Damper CD9-1 θ = 0° 0.19 3 Wye (30°), branch ED5-1 As/Ac = 1.0, Ab/Ac = 0.444, Qb/Qc = 0.25 −1.88 (Cb) Summation of Section 2 loss coefficients............................................................................................................................................ −0.34 3 7 Damper CD9-1 θ = 0° 0.19 8 Wye (45°), main ED5-2 As/Ac = 0.445, Ab/Ac = 0.713, Qs/Qc = 0.5 0.41 (Cs) Summation of Section 3 loss coefficients............................................................................................................................................ 0.60 aDuct Fitting Database (ASHRAE 1994) data for fittings reprinted in the section on Fitting Loss Coefficients.
Duct Design 34.25 Table 12 Loss Coefficient Summary by Sections for Example 8 (Concluded) Duct Section Fitting Number Type of Fitting ASHRAE Fitting No.a Parameters Loss Coefficient 4 10 Damper CR9-4 θ = 0°, 5 blades (opposed), L/R = 1.25 0.52 11 Transition ER4-3 L = 750 mm, Ao/A1 = 3.17, θ = 17° 0.57 Summation of Section 4 loss coefficients 1.09 5 12 Elbow CD3-17 45°, mitered 0.34 13 Damper CD9-1 θ = 0° 0.19 8 Wye (45°), branch ED5-2 Qb/Qc = 0.5, As/Ac = 0.445, Ab/Ac = 0.713 1.08 (Cb) Summation of Section 5 loss coefficients 1.61 6 14 Fire damper CD9-3 Curtain type, Type C 0.12 15 Elbow CD3-9 90°, 5 gore, r/D =1.5 0.15 — Fan and system interaction ED7-2 90° elbow, 5 gore, r/D = 1.5, L = 900 mm 0.60 Summation of Section 6 loss coefficients 0.87 7 16 Elbow CR3-3 90°, r/W = 0.70, 1 splitter vane 0.14 17 Damper CR9-1 θ = 0°, H/W = 1.0 0.08 19 Tee, main SR5-13 Qs/Qc = 0.5, As/Ac = 0.50 0.04 (Cs) Summation of Section 7 loss coefficients 0.26 8 19 Tee, branch SR5-13 Qb/Qc = 0.5, Ab/Ac = 0.50 0.73 (Cb) 18 Damper CR9-4 θ = 0°, 3 blades (opposed), L/R = 0.75 0.52 Summation of Section 8 loss coefficients 1.25 9 20 Elbow SR3-1 90°, mitered, H/W1 = 0.625, W o /W1 = 1.25 1.67 Summation of Section 9 loss coefficients 1.67 10 21 Damper CR9-1 θ = 0°, H/W = 0.625 0.08 22 Elbow CR3-10 90°, single-thickness vanes, design 2 0.12 23 Elbow CR3-6 θ = 90°, mitered, H/W = 0.625 1.25 24 Tee, branch SR5-1 r/Wb = 1.0, Qb/Qc = 0.367, As/Ac = 0.530, Ab/Ac = 0.606 1.24 (Cb) Summation of Section 10 loss coefficients 2.69 11 25 Damper CR9-1 θ = 0°, H/W = 1.0 0.08 26 Exit SR2-1 H/W = 1.0, Re = 125 400 1.00 27 Wye, dovetail SR5-14 r/Wc = 1.5, Qb1/Qc = 0.5, Ab1/Ac = 0.714 0.60 (Cb) Summation of Section 11 loss coefficients 1.68 12 28 Damper CR9-1 θ = 0°, H/W = 1.0 0.08 29 Exit SR2-5 θ = 19°, A1/Ao = 3.24, Re = 130 000 0.76 27 Wye, dovetail SR5-14 r/Wc = 1.5, Qb2/Qc = 0.5, Ab2/Ac = 0.714 0.60 (Cb) Summation of Section 12 loss coefficients 1.44 13 30 Damper CR9-1 θ = 0°, H/W = 0.71 0.08 24 Tee, main SR5-1 r/Wb = 1.0, Qs/Qc = 0.633, As/Ac = 0.530, Ab/Ac = 0.606 0.09 (Cs) Summation of Section 13 loss coefficients 0.17 14 31 Damper CR9-1 θ = 0°, H/W = 0.38 0.08 32 Tee, main SR5-13 Qs/Qc = 0.79, As/Ac = 0.825 0.05 (Cs) Summation of Section 14 loss coefficients 0.13 15 48 Elbow CR3-1 θ = 90°, r/W = 1.5, H/W = 0.75 0.19 33 Exit SR2-6 L = 500 mm, Dh = 187 0.27 34 Damper CR9-1 θ = 0°, H/W = 0.75 0.08 35 Tee, main SR5-1 r/Wb = 1.0, Qs/Qc = 0.5, As/Ac = 0.80, Ab/Ac = 0.80 0.03 (Cs) Summation of Section 15 loss coefficients 0.57 16 36 Exit SR2-3 θ = 20°, A1/Ao = 2.0, Re = 75 000 0.63 36 Screen CR6-1 n = 0.8, A1/Ao = 2.0 0.08 37 Damper CR9-1 θ = 0°, H/W = 0.75 0.08 35 Tee, branch SR5-1 r/Wb = 1.0, Qb/Qc = 0.5, As/Ac = 0.80, Ab/Ac = 0.80 0.95 (Cb) Summation of Section 16 loss coefficients 1.74 17 38 Damper CR9-1 θ = 0°, H/W = 0.6 0.08 32 Tee, branch SR5-13 Qb/Qc = 0.21, Ab/Ac = 0.187 0.65 (Cb) Summation of Section 17 loss coefficients 0.73 18 39 Obstruction, pipe CR6-4 Re = 15 000, y = 0, d = 25 mm, Sm/Ao = 0.1, y/H = 0 0.17 40 Transition SR4-1 θ = 25°, Ao/A1 = 0.556, L = 450 mm 0.04 41 Elbows, Z-shaped CR3-17 L = 1000 mm, L/W = 4.0, H/W = 3.2, Re = 240 000 2.53 45 Fire damper CR9-6 Curtain type, Type B 0.19 Summation of Section 18 loss coefficients 2.93 19 42 Diffuser, fan SR7-17 θ1 = 28°, L = 1000 mm, Ao/A1 = 2.67, C1 = 0.59 4.19 (Co) 47 Damper CR9-4 θ = 0°, 8 blades (opposed), L/R = 1.44 0.52 Summation of Section 19 loss coefficients 4.71 aDuct Fitting Database (ASHRAE 1994) data for fittings reprinted in the section on Fitting Loss Coefficients.
34.26 2001 ASHRAE Fundamentals Handbook (SI) rain protection. The stack height, determined by calculations from Chapter 16, is 4.9 m above the roof. This stack height is based on mini-mized stack downwash; therefore, the stack discharge velocity must exceed 1.5 times the design wind velocity.
Solution: For the contaminated ducts upstream of the collector, initial duct sizes and transport velocities are summarized below. The 22.8 m/s velocity in section 4 is acceptable because the transport velocity is not significantly lower than 23 m/s. For the next available duct size (160 mm diameter), the duct velocity is 28.8 m/s, significantly higher than 23 m/s. The following tabulation summarizes design calculations up through the junction after sections 1 and 4.
For the initial design, Design 1, the imbalance between section 1 and section 2 (or 3) is 383 Pa, with section 1 requiring additional resis-tance. Decreasing section 1 duct diameter by ISO sizes results in the least imbalance, 88 Pa, when the duct diameter is 200 mm (Design 3).
Because section 1 requires additional resistance, estimate the new air-flow rate using Equation (51): At 900 L/s flow in section 1, 130 Pa imbalance remains at the junction of sections 1 and 4. By trial-and-error solution, balance is attained when the flow in section 1 is 860 L/s. The duct between the collector and the fan inlet is 355 mm round to match the fan inlet (340 mm diam-eter). To minimize downwash, the stack discharge velocity must exceed 13.5 m/s, 1.5 times the design wind velocity (9 m/s) as stated in the problem definition. Therefore, the stack is 355 mm round, and the stack discharge velocity is 14.5 m/s.
Table 13 summarizes the system losses by sections. The straight duct friction factor and pressure loss were calculated by Equations (19) and (20). Table 14 lists fitting loss coefficients and input parameters neces-sary to determine the loss coefficients. The fitting loss coefficients are from the Duct Fitting Database (ASHRAE 1994). The fitting loss coef-ficient tables are included in the section on Fitting Loss Coefficients for illustration but can not be obtained exactly by manual interpolation since the coefficients were calculated by the duct fitting database algorithms (more significant figures). For a pressure grade line of the system, see Figure 21. The fan total pressure, calculated by Equation (16), is 1992 Pa.
To calculate the fan static pressure, use Equation (18): where 192 Pa is the fan outlet velocity pressure. The fan airflow rate is 1440 L/s, and its outlet area is 0.081 m2 (260 mm by 310 mm). There-fore, the fan outlet velocity is 17.9 m/s.
The hood suction for the chipping and grinding table hood is 560 Pa, calculated by Equation (18) from Chapter 29 of the 1999 ASHRAE Hand-book—Applications [HS = (1 + 0.25) (451) = 560 Pa, where 0.25 is the Duct Section Design Airflow, L/s Transport Velocity, m/s Duct Diameter, mm Duct Velocity, m/s 1 850 20 224 21.6 2,3 290 each 23 125 23.6 4 580 23 180 22.8 5 1430 23 280 23.2 Design No.
D1, mm ∆p1, Pa ∆p2+4, Pa Imbalance, ∆p1 − ∆p2+4 1 224 411 794 −383 2 200 762 850 −88 3 180 1320 712 +609 Q1 = 850 L/s Q2 = 290 L/s; D2 = 125 mm dia.
Q3 = 290 L/s; D3 = 125 mm dia.
Q4 = 850 L/s; D4 = 180 mm dia.
Qc 1 , 850 850 762 ⁄ ( )0.5 900 L/s = = Fig. 21 Total Pressure Grade Line for Example 9 Ps 1992 192 – 1800 Pa = = Table 13 Total Pressure Loss Calculations by Sections for Example 9 Duct Sectiona Duct Element Airflow, L/s Duct Size Velocity, m/s Velocity Pressure, Pa Duct Length,b m Summary of Fitting Loss Coefficientsc Duct Pressure Loss, Pa/md Total Pressure Loss, Pa Section Pressure Loss, Pa 1 Duct 860 200 mm φ 27.4 — 7.0 — 40 280 Fittings 860 — 27.4 451 — 1.12 — 505 785 2,3 Duct 290 125 mm φ 23.6 — 2.7 — 54 146 Fittings 290 — 23.6 336 — 1.06 — 356 502 4 Duct 580 180 mm φ 22.8 — 3.84 — 32 123 Fittings 580 — 22.8 313 — 0.51 — 160 283 5 Duct 1440 280 mm φ 23.4 — 2.7 — 20 54 Fittings 1440 — 23.4 329 — 0.22 — 72 126 — Collector,e fabric 1440 — — — — — — 750 750 6 Duct 1440 355 mm φ 14.5 — 3.7 — 6 22 Fittings 1440 — 14.5 127 — 0.00 — 0 22 7 Duct 1440 355 mm φ 14.5 — 8.5 — 6 51 Fittings 1440 — 14.5 127 — 2.03 — 258 309 aSee Figure 20.
bDuct lengths are to fitting centerlines.
cSee Table 14.
dDuct pressure based on a 0.09 mm absolute roughness factor.
eCollector manufacturers set the fabric bag cleaning mechanism to actuate at a pressure difference of 750 Pa between the inlet and outlet plenums. The pressure difference across the clean media is approximately 400 Pa.
Duct Design 34.27 hood entry loss coefficient Co, and 451 Pa is the duct velocity pressure Pv a few diameters downstream from the hood]. Similarly, the hood suction for each of the grinder wheels is 470 Pa: where 0.4 is the hood entry loss coefficient, and 336 Pa is the duct velocity pressure.
REFERENCES ACGIH. 1998. Industrial ventilation: A manual of recommended practice, 23rd ed. American Conference of Governmental Industrial Hygienists, Lansing, MI.
ADC. 1996. Flexible duct performance and installation standards, 3rd ed.
Air Diffusion Council.
AISI/SMACNA. 1972. Measurement and analysis of leakage rates from seams and joints of air handling systems.
Altshul, A.D., L.C. Zhivotovckiy, and L.P. Ivanov. 1987. Hydraulics and aerodynamics. Stroisdat Publishing House, Moscow.
AMCA. 1999. Laboratory method of testing louvers for rating. Standard 500-L. Air Movement and Control Association, Arlington Heights, IL.
AMCA. 1990a. Fans and systems. Publication 201.
AMCA. 1990b. Field performance measurement of fan systems. Publica-tion 203.
ASHRAE. 1993. Energy-efficient design of new low-rise residential build-ings. ASHRAE Standard 90.2-1993.
ASHRAE. 1994. Duct fitting database.
ASHRAE. 1995. Energy conservation in existing buildings. ASHRAE/ IESNA Standard 100-1995. Addendum 1-1996.
ASHRAE. 1999. Energy standard for buildings except low-rise residential buildings. ASHRAE/IESNA Standard 90.1-1999.
ASHRAE. 1999. Laboratory methods for testing fans for aerodynamic per-formance rating. ANSI/ASHRAE Standard 51-1999. Also ANSI/ AMCA 210-99.
ASHRAE/SMACNA/TIMA. 1985. Investigation of duct leakage. ASH-RAE Research Project 308.
Behls, H.F. 1971. Computerized calculation of duct friction. Building Sci-ence Series 39, p. 363. National Institute of Standards and Technology, Gaithersburg, MD.
Bellman, R.E. 1957. Dynamic programming. Princeton University Press, New York.
Brown, R.B. 1973. Experimental determinations of fan system effect fac-tors. In Fans and systems, ASHRAE Symposium Bulletin LO-73-1, Lou-isville, KY (June).
Carrier Corporation. 1960. Air duct design. Chapter 2 in System design man-ual, Part 2: Air distribution. pp.17-63. Syracuse, NY.
Chun-Lun, S. 1983. Simplified static-regain duct design procedure. ASH-RAE Transactions 89(2A):78.
Clarke, M.S., J.T. Barnhart, F.J. Bubsey, and E. Neitzel. 1978. The effects of system connections on fan performance. ASHRAE Transactions 84(2): 227-63.
Colebrook, C.F. 1938-39. Turbulent flow in pipes, with particular reference to the transition region between the smooth and rough pipe laws. Journal of the Institution of Civil Engineers 11:133. London.
DOE. Electrical sales and revenue, latest edition. Department of Energy, Washington, D.C. (To purchase, call Energy Information Administration 202-512-1800.) Farajian, T., G. Grewal, and R.J. Tsal. 1992. Post-accident air leakage anal-ysis in a nuclear facility via T-method airflow simulation. 22nd DOE/NRC Nuclear Air Cleaning and Treatment Conference, Denver, CO, October.
Farquhar, H.F. 1973. System effect values for fans. In Fans and systems, ASHRAE Symposium Bulletin LO-73-1, Louisville, KY (June).
Griggs, E.I. and F. Khodabakhsh-Sharifabad. 1992. Flow characteristics in rectangular ducts. ASHRAE Transactions 98(1).
Griggs, E.I., W.B. Swim, and G.H. Henderson. 1987. Resistance to flow of round galvanized ducts. ASHRAE Transactions 93(1):3-16.
Heyt, J.W. and M.J. Diaz. 1975. Pressure drop in flat-oval spiral air duct.
ASHRAE Transactions 81(2):221-32.
Huebscher, R.G. 1948. Friction equivalents for round, square and rectangu-lar ducts. ASHVE Transactions 54:101-18.
Hutchinson, F.W. 1953. Friction losses in round aluminum ducts. ASHVE Transactions 59:127-38.
Table 14 Loss Coefficient Summary by Sections for Example 9 Duct Section Fitting Number Type of Fitting ASHRAE Fitting No.a Parameters Loss Coefficient 1 1 Hoodb — Hood face area: 0.9 m by 1.2 m 0.25 2 Elbow CD3-10 90°, 7 gore, r/D = 2.5 0.11 4 Capped wye (45°), with 45° elbow ED5-6 Ab/Ac = 1 0.64 (Cb) 5 Wye (30°), main ED5-1 Qs/Qc = 0.60, As/Ac = 0.510, Ab/Ac = 0.413 0.12 (Cs) Summation of Section 1 loss coefficients .................................................................................................................................................. 1.12 2,3 6 Hoodc — Type hood: For double wheels, dia. = 560 mm each, 0.40 wheel width = 100 mm each; type takeoff: tapered 7 Elbow CD3-12 90°, 3 gore, r/D = 1.5 0.34 8 Symmetrical wye (60°) ED5-9 Qb/Qc = 0.5, Ab/Ac = 0.482 0.32 (Cb) Summation of Sections 2 and 3 loss coefficients....................................................................................................................................... 1.06 4 9 Elbow CD3-10 90°, 7 gore, r/D = 2.5 0.11 10 Elbow CD3-13 60°, 3 gore, r/D = 1.5 0.19 5 Wye (30°), branch ED5-1 Qb/Qc = 0.40, As/Ac = 0.510, Ab/Ac = 0.413 0.21 (Cb) Summation of Section 4 loss coefficients .................................................................................................................................................. 0.51 5 11 Exit, conical diffuser to collector ED2-1 L = 600 mm, L/Do = 2.14, A1/Ao ≈ 16 0.22 Summation of Section 5 loss coefficients .................................................................................................................................................. 0.22 6 12 Entry, bellmouth from collector ER2-1 r/D1 = 0.20 0.00 (C1) Summation of Section 6 loss coefficients .................................................................................................................................................. 0.00 7 13 Diffuser, fan outletd SR7-17 Fan outlet size: 260 mm by 310 mm, 0.39 (Co) Ao/A1 = 1.563 (assume 355 mm by 355 mm outlet rather than 355 mm round), L = 460 mm 14 Capped wye (45°), with 45° elbow ED5-6 Ab/Ac = 1 0.64 (Cb) 15 Stackhead SD2-6 De/D = 1 1.0 Summation of Section 7 loss coefficients .................................................................................................................................................. 2.03 aDuct Fitting Database (ASHRAE 1994) data for fittings reprinted in the section on Fitting Loss Coefficients.
bFrom Industrial Ventilation (ACGIH 1998, Figure VS-80-19).
cFrom Industrial Ventilation (ACGIH 1998, Figure VS-80-11).
dFan specified: Industrial exhauster for granular materials: 530 mm wheel diameter, 340 mm inlet diameter, 260 mm by 310 mm outlet, 6 kW motor.
HS2 3 , 1 0.4 + ( ) 336 ( ) 470 Pa = = 34.28 2001 ASHRAE Fundamentals Handbook (SI) Idelchik, I.E., M.O. Steinberg, G.R. Malyavskaya, and O.G. Martynenko.
1994. Handbook of hydraulic resistance, 3rd ed. CRC Press/Begell House, Boca Raton, Ann Arbor, London, Tokyo.
ISO. 1983. Air distribution—Straight circular sheet metal ducts with a lock type spiral seam and straight rectangular sheet metal ducts—Dimen-sions. Standard 7807:1983. International Organization for Standardiza-tion, Geneva.
Jones, C.D. 1979. Friction factor and roughness of United Sheet Metal Com-pany spiral duct. United Sheet Metal, Division of United McGill Corp., Westerville, OH (August). Based on data contained in Friction loss tests, United Sheet Metal Company Spiral Duct, Ohio State University Engi-neering Experiment Station, File No. T-1011, September, 1958.
Klote, J.H. and J. Milke. 1992. Design of smoke management systems.
ASHRAE, Atlanta.
Lauvray, T.L. 1978. Experimental heat transmission coefficients for operat-ing air duct systems. ASHRAE Journal (June):69.
Meyer, M.L. 1973. A new concept: The fan system effect factor. In Fans and systems, ASHRAE Symposium Bulletin LO-73-1, Louisville, KY (June).
Moody, L.F. 1944. Friction factors for pipe flow. ASME Transactions 66:671.
NAIMA. 1997. Fibrous glass duct construction standards, 3rd ed. North American Insulation Manufacturers Association.
NFPA. Fire protection handbook, latest ed. National Fire Protection Asso-ciation, Quincy, MA.
NFPA. Installation of air conditioning and ventilating systems. ANSI/NFPA Standard 90A, latest ed.
Osborne, W.C. 1966. Fans. Pergamon Press Ltd., London.
SMACNA. 1985. HVAC air duct leakage manual. Sheet Metal and Air Con-ditioning Contractors’ National Association, Chantilly, VA.
SMACNA. 1992. Fibrous glass duct construction standards, 6th ed.
SMACNA. 1995. HVAC duct construction standards, metal and flexible, 2nd ed.
Smith, G.W. 1968. Engineering economy: Analysis of capital expenditures.
The Iowa State University Press, Ames, IA.
Swim, W.B. 1978. Flow losses in rectangular ducts lined with fiberglass.
ASHRAE Transactions 84(2):216.
Swim, W.B. 1982. Friction factor and roughness for airflow in plastic pipe.
ASHRAE Transactions 88(1):269.
Swim, W.B. and E.I. Griggs. 1995. Duct leakage measurement and analysis.
ASHRAE Transactions 101(1).
Tsal, R.J. and M.S. Adler. 1987. Evaluation of numerical methods for duct-work and pipeline optimization. ASHRAE Transactions 93(1):17-34.
Tsal, R.J. and H.F. Behls. 1988. Fallacy of the static regain duct design method. ASHRAE Transactions 94(2):76-89.
Tsal, R.J., H.F. Behls, and R. Mangel. 1988. T-method duct design, Part I: Optimization theory; Part II: Calculation procedure and economic anal-ysis. ASHRAE Transactions 94(2):90-111.
Tsal, R.J., H.F. Behls, and R. Mangel. 1990. T-method duct design, Part III: Simulation. ASHRAE Transactions 96(2).
UL. Published annually. Building materials directory. Underwriters Labora-tories, Northbrook, IL.
UL. Published annually. Fire resistance directory.
UL. Factory-made air ducts and air connectors. UL Standard 181, latest ed.
UL. Closure systems for use with rigid air ducts and air connectors. UL Stan-dard 181A, latest ed.
UL. Closure systems for use with rigid air ducts and air connectors. UL Stan-dard 181B, latest ed.
UL. Fire dampers. Standard UL 555, latest ed.
UL. Smoke dampers. Standard 555S, latest ed.
Wendes, H.C. 1989. Sheet metal estimating. Wendes Mechanical Consulting Services, Elk Grove Village, IL.
Wright, D.K., Jr. 1945. A new friction chart for round ducts. ASHVE Trans-actions 51:303-16.
BIBLIOGRAPHY SMACNA. 1987. Duct research destroys design myths. Videotape (VHS).
Sheet Metal and Air Conditioning Contractors’ National Association, Chantilly, VA.
Duct Design 34.29 FITTING LOSS COEFFICIENTS Fittings to support Examples 8 and 9 and some of the more common fittings are reprinted here.
For the complete fitting database see the Duct Fitting Database (ASHRAE 1994).
ROUND FITTINGS CD3-1 Elbow, Die Stamped, 90 Degree, r/D = 1.5 D, mm 75 100 125 150 180 200 230 250 Co 0.30 0.21 0.16 0.14 0.12 0.11 0.11 0.11 CD3-3 Elbow, Die Stamped, 45 Degree, r/D = 1.5 D, mm 75 100 125 150 180 200 230 250 Co 0.18 0.13 0.10 0.08 0.07 0.07 0.07 0.07 CD3-5 Elbow, Pleated, 90 Degree, r/D = 1.5 D, mm 100 150 200 250 300 350 400 Co 0.57 0.43 0.34 0.28 0.26 0.25 0.25 CD3-6 Elbow, Pleated, 60 Degree, r/D = 1.5 D, mm 100 150 200 250 300 350 400 Co 0.45 0.34 0.27 0.23 0.20 0.19 0.19 34.30 2001 ASHRAE Fundamentals Handbook (SI) CD3-7 Elbow, Pleated, 45 Degree, r/D = 1.5 D, mm 100 150 200 250 300 350 400 Co 0.34 0.26 0.21 0.17 0.16 0.15 0.15 CD3-9 Elbow, 5 Gore, 90 Degree, r/D = 1.5 D, mm 75 150 230 300 380 450 530 600 690 750 1500 Co 0.51 0.28 0.21 0.18 0.16 0.15 0.14 0.13 0.12 0.12 0.12 CD3-10 Elbow, 7 Gore, 90 Degree, r/D = 2.5 D, mm 75 150 230 300 380 450 690 1500 Co 0.16 0.12 0.10 0.08 0.07 0.06 0.05 0.03 CD3-12 Elbow, 3 Gore, 90 Degree, r/D = 0.75 to 2.0 r/D 0.75 1.00 1.50 2.00 Co 0.54 0.42 0.34 0.33 CD3-13 Elbow, 3 Gore, 60 Degree, r/D = 1.5 D, mm 75 150 230 300 380 450 530 600 690 750 1500 Co 0.40 0.21 0.16 0.14 0.12 0.12 0.11 0.10 0.09 0.09 0.09 Duct Design 34.31 CD3-14 Elbow, 3 Gore, 45 Degree, r/D = 1.5 D, mm 75 150 230 300 380 450 530 600 690 750 1500 Co 0.31 0.17 0.13 0.11 0.11 0.09 0.08 0.08 0.07 0.07 0.07 CD3-17 Elbow, Mitered, 45 Degree D, mm 75 150 230 300 380 450 530 600 690 1500 Co 0.34 0.34 0.34 0.34 0.34 0.34 0.34 0.34 0.34 0.34 CD6-1 Screen (Only) A1/Ao Co Values n 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.90 1.00 0.2 155.00 102.50 75.00 55.00 41.25 31.50 24.25 18.75 14.50 11.00 8.00 3.50 0.00 0.3 68.89 45.56 33.33 24.44 18.33 14.00 10.78 8.33 6.44 4.89 3.56 1.56 0.00 0.4 38.75 25.63 18.75 13.75 10.31 7.88 6.06 4.69 3.63 2.75 2.00 0.88 0.00 0.5 24.80 16.40 12.00 8.80 6.60 5.04 3.88 3.00 2.32 1.76 1.28 0.56 0.00 0.6 17.22 11.39 8.33 6.11 4.58 3.50 2.69 2.08 1.61 1.22 0.89 0.39 0.00 0.7 12.65 8.37 6.12 4.49 3.37 2.57 1.98 1.53 1.18 0.90 0.65 0.29 0.00 0.8 9.69 6.40 4.69 3.44 2.58 1.97 1.52 1.17 0.91 0.69 0.50 0.22 0.00 0.9 7.65 5.06 3.70 2.72 2.04 1.56 1.20 0.93 0.72 0.54 0.40 0.17 0.00 1.0 6.20 4.10 3.00 2.20 1.65 1.26 0.97 0.75 0.58 0.44 0.32 0.14 0.00 1.2 4.31 2.85 2.08 1.53 1.15 0.88 0.67 0.52 0.40 0.31 0.22 0.10 0.00 1.4 3.16 2.09 1.53 1.12 0.84 0.64 0.49 0.38 0.30 0.22 0.16 0.07 0.00 1.6 2.42 1.60 1.17 0.86 0.64 0.49 0.38 0.29 0.23 0.17 0.13 0.05 0.00 1.8 1.91 1.27 0.93 0.68 0.51 0.39 0.30 0.23 0.18 0.14 0.10 0.04 0.00 2.0 1.55 1.03 0.75 0.55 0.41 0.32 0.24 0.19 0.15 0.11 0.08 0.04 0.00 2.5 0.99 0.66 0.48 0.35 0.26 0.20 0.16 0.12 0.09 0.07 0.05 0.02 0.00 3.0 0.69 0.46 0.33 0.24 0.18 0.14 0.11 0.08 0.06 0.05 0.04 0.02 0.00 4.0 0.39 0.26 0.19 0.14 0.10 0.08 0.06 0.05 0.04 0.03 0.02 0.01 0.00 6.0 0.17 0.11 0.08 0.06 0.05 0.04 0.03 0.02 0.02 0.01 0.01 0.00 0.00 34.32 2001 ASHRAE Fundamentals Handbook (SI) CD6-4 Round Duct, Depressed to Avoid an Obstruction Co = 0.24 CD9-1 Damper, Butterfly D/Do Co Values θ 0 10 20 30 40 50 60 70 75 80 85 90 0.5 0.19 0.27 0.37 0.49 0.61 0.74 0.86 0.96 0.99 1.02 1.04 1.04 0.6 0.19 0.32 0.48 0.69 0.94 1.21 1.48 1.72 1.82 1.89 1.93 2.00 0.7 0.19 0.37 0.64 1.01 1.51 2.12 2.81 3.46 3.73 3.94 4.08 6.00 0.8 0.19 0.45 0.87 1.55 2.60 4.13 6.14 8.38 9.40 10.30 10.80 15.00 0.9 0.19 0.54 1.22 2.51 4.97 9.57 17.80 30.50 38.00 45.00 50.10 100.00 1.0 0.19 0.67 1.76 4.38 11.20 32.00 113.00 619.00 2010.00 10350.00 99999.00 99999.00 CD9-3 Fire Damper, Curtain Type, Type C Co = 0.12 ED1-1 Duct Mounted in Wall t /D Co Values L /D 0.00 0.002 0.01 0.05 0.10 0.20 0.30 0.50 10.00 0.00 0.50 0.57 0.68 0.80 0.86 0.92 0.97 1.00 1.00 0.02 0.50 0.51 0.52 0.55 0.60 0.66 0.69 0.72 0.72 0.05 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 10.00 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 ED1-3 Bellmouth, with Wall r /D 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.08 0.10 0.12 0.16 0.20 10.00 Co 0.50 0.44 0.37 0.31 0.26 0.22 0.20 0.15 0.12 0.09 0.06 0.03 0.03 Duct Design 34.33 ED2-1 Conical Diffuser, Round to Plenum, Exhaust/Return Systems A1/Ao Co Values L /Do 0.5 1.0 2.0 3.0 4.0 5.0 6.0 8.0 10.0 12.0 14.0 1.5 0.03 0.02 0.03 0.03 0.04 0.05 0.06 0.08 0.10 0.11 0.13 2.0 0.08 0.06 0.04 0.04 0.04 0.05 0.05 0.06 0.08 0.09 0.10 2.5 0.13 0.09 0.06 0.06 0.06 0.06 0.06 0.06 0.07 0.08 0.09 3.0 0.17 0.12 0.09 0.07 0.07 0.06 0.06 0.07 0.07 0.08 0.08 4.0 0.23 0.17 0.12 0.10 0.09 0.08 0.08 0.08 0.08 0.08 0.08 6.0 0.30 0.22 0.16 0.13 0.12 0.10 0.10 0.09 0.09 0.09 0.08 8.0 0.34 0.26 0.18 0.15 0.13 0.12 0.11 0.10 0.09 0.09 0.09 10.0 0.36 0.28 0.20 0.16 0.14 0.13 0.12 0.11 0.10 0.09 0.09 14.0 0.39 0.30 0.22 0.18 0.16 0.14 0.13 0.12 0.10 0.10 0.10 20.0 0.41 0.32 0.24 0.20 0.17 0.15 0.14 0.12 0.11 0.11 0.10 A1/Ao Optimum Angle θ 0.5 1.0 2.0 3.0 4.0 5.0 6.0 8.0 10.0 12.0 14.0 1.5 34 20 13 9 7 6 4 3 2 2 2 2.0 42 28 17 12 10 9 8 6 5 4 3 2.5 50 32 20 15 12 11 10 8 7 6 5 3.0 54 34 22 17 14 12 11 10 8 8 6 4.0 58 40 26 20 16 14 13 12 10 10 9 6.0 62 42 28 22 19 16 15 12 11 10 9 8.0 64 44 30 24 20 18 16 13 12 11 10 10.0 66 46 30 24 22 19 17 14 12 11 10 14.0 66 48 32 26 22 19 17 14 13 11 11 20.0 68 48 32 26 22 20 18 15 13 12 11 ED4-1 Transition, Round to Round, Exhaust/Return Systems Ao/A1 Co Values θ 10 15 20 30 45 60 90 120 150 180 0.06 0.21 0.29 0.38 0.60 0.84 0.88 0.88 0.88 0.88 0.88 0.10 0.21 0.28 0.38 0.59 0.76 0.80 0.83 0.84 0.83 0.83 0.25 0.16 0.22 0.30 0.46 0.61 0.68 0.64 0.63 0.62 0.62 0.50 0.11 0.13 0.19 0.32 0.33 0.33 0.32 0.31 0.30 0.30 1.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 2.00 0.20 0.20 0.20 0.20 0.22 0.24 0.48 0.72 0.96 1.04 4.00 0.80 0.64 0.64 0.64 0.88 1.12 2.72 4.32 5.60 6.56 6.00 1.80 1.44 1.44 1.44 1.98 2.52 6.48 10.10 13.00 15.10 10.00 5.00 5.00 5.00 5.00 6.50 8.00 19.00 29.00 37.00 43.00 ED4-2 Transition, Round to Rectangular, Exhaust/Return Systems Ao/A1 Co Values θ 10 15 20 30 45 60 90 120 150 180 0.06 0.30 0.54 0.53 0.65 0.77 0.88 0.95 0.98 0.98 0.93 0.10 0.30 0.50 0.53 0.64 0.75 0.84 0.89 0.91 0.91 0.88 0.25 0.25 0.36 0.45 0.52 0.58 0.62 0.64 0.64 0.64 0.64 0.50 0.15 0.21 0.25 0.30 0.33 0.33 0.33 0.32 0.31 0.30 1.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 2.00 0.24 0.28 0.26 0.20 0.22 0.24 0.49 0.73 0.97 1.04 4.00 0.89 0.78 0.79 0.70 0.88 1.12 2.72 4.33 5.62 6.58 6.00 1.89 1.67 1.59 1.49 1.98 2.52 6.51 10.14 13.05 15.14 10.00 5.09 5.32 5.15 5.05 6.50 8.05 19.06 29.07 37.08 43.05 34.34 2001 ASHRAE Fundamentals Handbook (SI) ED5-1 Wye, 30 Degree, Converging As/Ac Ab/Ac Cb Values Qb /Qc 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.2 0.2 −24.17 −3.78 −0.60 0.30 0.64 0.77 0.83 0.88 0.98 0.3 −55.88 −9.77 −2.57 −0.50 0.25 0.55 0.67 0.70 0.71 0.4 −99.93 −17.94 −5.13 −1.45 −0.11 0.42 0.62 0.68 0.68 0.5 −156.51 −28.40 −8.37 −2.62 −0.52 0.30 0.62 0.71 0.69 0.6 −225.62 −41.13 −12.30 −4.01 −0.99 0.20 0.66 0.78 0.75 0.7 −307.26 −56.14 −16.90 −5.61 −1.51 0.11 0.73 0.90 0.86 0.8 −401.44 −73.44 −22.18 −7.44 −2.08 0.04 0.84 1.06 1.01 0.9 −508.15 −93.02 −28.15 −9.49 −2.71 −0.03 0.99 1.27 1.20 1.0 −627.39 −114.89 −34.80 −11.77 −3.39 −0.08 1.18 1.52 1.43 0.3 0.2 −13.97 −1.77 0.08 0.59 0.77 0.84 0.88 0.92 1.06 0.3 −33.06 −5.33 −1.09 0.10 0.51 0.66 0.71 0.72 0.74 0.4 −59.43 −10.08 −2.52 −0.41 0.32 0.59 0.67 0.68 0.66 0.5 −93.24 −16.11 −4.30 −1.00 0.14 0.56 0.69 0.70 0.66 0.6 −134.51 −23.45 −6.44 −1.68 −0.03 0.57 0.76 0.77 0.70 0.7 −183.25 −32.08 −8.93 −2.45 −0.21 0.61 0.87 0.88 0.79 0.8 −239.47 −42.01 −11.77 −3.32 −0.38 0.69 1.02 1.03 0.91 0.9 −303.16 −53.25 −14.97 −4.27 −0.56 0.80 1.21 1.23 1.07 1.0 −374.32 −65.79 −18.53 −5.32 −0.73 0.94 1.45 1.47 1.27 0.4 0.2 −9.20 −0.85 0.39 0.71 0.82 0.87 0.90 0.94 1.09 0.3 −22.31 −3.24 −0.38 0.39 0.64 0.73 0.76 0.78 0.85 0.4 −40.52 −6.48 −1.37 0.02 0.48 0.64 0.67 0.66 0.65 0.5 −63.71 −10.50 −2.50 −0.33 0.40 0.63 0.69 0.67 0.63 0.6 −92.00 −15.37 −3.84 −0.71 0.33 0.67 0.75 0.71 0.65 0.7 −125.40 −21.08 −5.40 −1.13 0.28 0.75 0.85 0.80 0.70 0.8 −163.90 −27.65 −7.16 −1.59 0.25 0.86 1.00 0.93 0.80 0.9 −207.52 −35.07 −9.14 −2.09 0.25 1.02 1.18 1.10 0.93 1.0 −256.25 −43.35 −11.33 −2.63 0.26 1.21 1.42 1.31 1.09 0.5 0.2 −6.62 −0.36 0.54 0.77 0.85 0.88 0.90 0.95 1.11 0.3 −16.42 −2.11 −0.01 0.54 0.72 0.78 0.80 0.83 0.96 0.4 −30.26 −4.59 −0.79 0.22 0.54 0.64 0.66 0.64 0.64 0.5 −47.68 −7.55 −1.61 −0.02 0.48 0.63 0.65 0.62 0.59 0.6 −68.93 −11.13 −2.56 −0.28 0.45 0.67 0.69 0.65 0.58 0.7 −94.00 −15.31 −3.65 −0.55 0.44 0.74 0.77 0.71 0.61 0.8 −122.90 −20.12 −4.88 −0.83 0.46 0.85 0.90 0.81 0.68 0.9 −155.63 −25.54 −6.25 −1.12 0.51 1.00 1.06 0.94 0.77 1.0 −192.18 −31.58 −7.77 −1.43 0.59 1.19 1.26 1.12 0.90 0.6 0.2 −5.12 −0.10 0.62 0.79 0.85 0.87 0.90 0.95 1.11 0.3 −13.00 −1.49 0.18 0.61 0.75 0.79 0.82 0.86 1.02 0.4 −24.31 −3.55 −0.50 0.30 0.55 0.62 0.63 0.62 0.63 0.5 −38.41 −5.94 −1.16 0.09 0.48 0.59 0.60 0.57 0.55 0.6 −55.58 −8.80 −1.92 −0.12 0.45 0.61 0.62 0.57 0.52 0.7 −75.83 −12.16 −2.79 −0.33 0.44 0.66 0.67 0.60 0.52 0.8 −99.17 −16.00 −3.76 −0.54 0.46 0.74 0.76 0.67 0.56 0.9 −125.60 −20.33 −4.83 −0.76 0.51 0.86 0.88 0.77 0.62 1.0 −155.12 −25.14 −6.02 −0.99 0.58 1.02 1.04 0.90 0.71 0.7 0.2 −4.24 0.05 0.65 0.80 0.85 0.87 0.89 0.94 1.12 0.3 −11.00 −1.15 0.27 0.63 0.75 0.79 0.82 0.87 1.06 0.4 −20.82 −3.00 −0.38 0.31 0.52 0.59 0.60 0.59 0.61 0.5 −32.99 −5.09 −0.98 0.10 0.43 0.53 0.54 0.52 0.51 0.6 −47.78 −7.58 −1.67 −0.11 0.38 0.52 0.53 0.49 0.45 0.7 −65.22 −10.50 −2.44 −0.32 0.34 0.53 0.54 0.49 0.43 0.8 −85.32 −13.83 −3.30 −0.53 0.33 0.58 0.59 0.52 0.43 0.9 −108.07 −17.58 −4.26 −0.75 0.34 0.66 0.67 0.58 0.46 1.0 −133.48 −21.76 −5.30 −0.97 0.38 0.76 0.78 0.67 0.51 Duct Design 34.35 ED5-1 Wye, 30 Degree, Converging (Continued) As/Ac Ab/Ac Cb Values (Concluded) Qb/Qc 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.8 0.2 −3.75 0.11 0.65 0.79 0.84 0.86 0.88 0.94 1.12 0.3 −9.88 −0.99 0.29 0.63 0.74 0.78 0.81 0.87 1.09 0.4 −18.88 −2.75 −0.36 0.28 0.48 0.55 0.56 0.57 0.61 0.5 −29.98 −4.71 −0.96 0.04 0.36 0.46 0.47 0.46 0.47 0.6 −43.46 −7.05 −1.64 −0.20 0.26 0.41 0.43 0.41 0.39 0.7 −59.34 −9.77 −2.40 −0.44 0.19 0.38 0.41 0.38 0.34 0.8 −77.64 −12.88 −3.26 −0.69 0.13 0.38 0.42 0.37 0.31 0.9 −98.35 −16.38 −4.20 −0.95 0.09 0.40 0.45 0.39 0.30 1.0 −121.48 −20.27 −5.24 −1.23 0.06 0.45 0.51 0.43 0.31 0.9 0.2 −3.52 0.12 0.64 0.78 0.82 0.85 0.88 0.93 1.12 0.3 −9.34 −0.95 0.28 0.60 0.71 0.76 0.80 0.87 1.10 0.4 −17.96 −2.70 −0.40 0.22 0.43 0.50 0.53 0.54 0.60 0.5 −28.58 −4.65 −1.05 −0.07 0.26 0.37 0.40 0.41 0.42 0.6 −41.45 −6.97 −1.77 −0.35 0.12 0.28 0.32 0.32 0.32 0.7 −56.61 −9.66 −2.58 −0.65 0.00 0.21 0.27 0.26 0.24 0.8 −74.08 −12.74 −3.49 −0.97 −0.12 0.16 0.23 0.22 0.18 0.9 −93.84 −16.21 −4.50 −1.30 −0.23 0.13 0.21 0.19 0.14 1.0 −115.92 −20.06 −5.61 −1.66 −0.34 0.11 0.21 0.18 0.11 1.0 0.2 −3.48 0.10 0.62 0.76 0.81 0.84 0.87 0.92 1.11 0.3 −9.22 −1.00 0.23 0.56 0.68 0.74 0.78 0.86 1.11 0.4 −17.76 −2.79 −0.50 0.14 0.37 0.45 0.49 0.52 0.60 0.5 −28.31 −4.82 −1.21 −0.20 0.15 0.28 0.33 0.35 0.38 0.6 −41.06 −7.21 −2.01 −0.55 −0.04 0.15 0.22 0.23 0.25 0.7 −56.09 −9.99 −2.91 −0.92 −0.23 0.03 0.12 0.14 0.15 0.8 −73.39 −13.17 −3.92 −1.32 −0.41 −0.07 0.04 0.06 0.06 0.9 −92.98 −16.75 −5.04 −1.75 −0.60 −0.17 −0.03 −0.01 −0.02 1.0 −114.85 −20.74 −6.28 −2.21 −0.79 −0.26 −0.09 −0.07 −0.09 As/Ac Ab/Ac Cs Values Qs/Qc 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.2 0.2 −16.02 −3.15 −0.80 0.04 0.45 0.69 0.86 0.99 1.10 0.3 −11.65 −1.94 −0.26 0.32 0.60 0.77 0.90 1.01 1.10 0.4 −8.56 −1.20 0.05 0.47 0.68 0.82 0.92 1.02 1.11 0.5 −6.41 −0.71 0.25 0.57 0.73 0.84 0.93 1.02 1.11 0.6 −4.85 −0.36 0.38 0.63 0.76 0.86 0.94 1.02 1.11 0.7 −3.68 −0.10 0.48 0.68 0.79 0.87 0.95 1.03 1.11 0.8 −2.77 0.10 0.56 0.71 0.81 0.88 0.95 1.03 1.11 0.9 −2.04 0.26 0.62 0.74 0.82 0.89 0.95 1.03 1.11 1.0 −1.45 0.38 0.66 0.76 0.83 0.89 0.96 1.03 1.11 0.3 0.2 −36.37 −7.59 −2.48 −0.79 −0.06 0.29 0.47 0.57 0.61 0.3 −26.79 −5.07 −1.42 −0.27 0.21 0.42 0.53 0.59 0.61 0.4 −19.94 −3.49 −0.80 0.02 0.35 0.49 0.56 0.60 0.62 0.5 −15.18 −2.44 −0.41 0.20 0.43 0.54 0.58 0.61 0.62 0.6 −11.73 −1.70 −0.13 0.32 0.49 0.56 0.60 0.61 0.62 0.7 −9.13 −1.14 0.07 0.41 0.53 0.58 0.60 0.61 0.62 0.8 −7.11 −0.72 0.23 0.48 0.57 0.60 0.61 0.62 0.62 0.9 −5.49 −0.38 0.35 0.53 0.59 0.61 0.62 0.62 0.62 1.0 −4.17 −0.11 0.45 0.58 0.61 0.62 0.62 0.62 0.62 0.4 0.2 −64.82 −13.76 −4.74 −1.81 −0.59 −0.02 0.24 0.36 0.39 0.3 −47.92 −9.38 −2.93 −0.94 −0.16 0.19 0.34 0.39 0.40 0.4 −35.81 −6.62 −1.88 −0.46 0.07 0.30 0.38 0.41 0.40 0.5 −27.39 −4.78 −1.20 −0.16 0.22 0.36 0.41 0.42 0.41 0.6 −21.28 −3.48 −0.73 0.04 0.31 0.41 0.43 0.43 0.41 0.7 −16.68 −2.51 −0.38 0.20 0.38 0.44 0.45 0.43 0.41 0.8 −13.10 −1.77 −0.12 0.31 0.44 0.46 0.46 0.44 0.41 0.9 −10.24 −1.18 0.09 0.40 0.48 0.48 0.46 0.44 0.41 1.0 −7.90 −0.69 0.26 0.47 0.51 0.50 0.47 0.44 0.41 34.36 2001 ASHRAE Fundamentals Handbook (SI) ED5-1 Wye, 30 Degree, Converging (Concluded) As/Ac Ab/Ac Cs Values (Concluded) Qs/Qc 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.5 0.2 −101.39 −21.64 −7.61 −3.07 −1.19 −0.34 0.05 0.22 0.26 0.3 −75.05 −14.87 −4.83 −1.75 −0.54 −0.03 0.19 0.26 0.27 0.4 −56.18 −10.59 −3.21 −1.02 −0.20 0.13 0.26 0.29 0.27 0.5 −43.04 −7.74 −2.16 −0.56 0.02 0.23 0.30 0.30 0.27 0.6 −33.51 −5.72 −1.43 −0.24 0.16 0.30 0.33 0.31 0.28 0.7 −26.34 −4.22 −0.90 −0.01 0.27 0.35 0.35 0.32 0.28 0.8 −20.75 −3.06 −0.49 0.16 0.35 0.39 0.37 0.33 0.28 0.9 −16.29 −2.14 −0.17 0.30 0.41 0.41 0.38 0.33 0.28 1.0 −12.64 −1.39 0.10 0.41 0.46 0.44 0.39 0.33 0.28 0.6 0.2 −146.06 −31.26 −11.09 −4.56 −1.89 −0.68 −0.12 0.10 0.16 0.3 −108.19 −21.55 −7.12 −2.69 −0.97 −0.24 0.07 0.17 0.17 0.4 −81.04 −15.40 −4.80 −1.65 −0.48 −0.01 0.17 0.20 0.18 0.5 −62.13 −11.31 −3.30 −0.99 −0.17 0.13 0.22 0.22 0.18 0.6 −48.43 −8.41 −2.25 −0.54 0.03 0.22 0.26 0.24 0.18 0.7 −38.10 −6.25 −1.49 −0.22 0.18 0.29 0.29 0.25 0.19 0.8 −30.07 −4.59 −0.90 0.03 0.30 0.34 0.31 0.25 0.19 0.9 −23.64 −3.27 −0.44 0.23 0.39 0.38 0.33 0.26 0.19 1.0 −18.39 −2.20 −0.06 0.39 0.46 0.42 0.34 0.27 0.19 0.7 0.2 −198.85 −42.62 −15.17 −6.31 −2.68 −1.04 −0.29 0.01 0.08 0.3 −147.33 −29.41 −9.78 −3.77 −1.44 −0.45 −0.04 0.10 0.10 0.4 −110.40 −21.07 −6.64 −2.36 −0.77 −0.14 0.09 0.15 0.11 0.5 −84.67 −15.50 −4.60 −1.48 −0.36 0.05 0.17 0.17 0.11 0.6 −66.02 −11.56 −3.19 −0.86 −0.08 0.18 0.23 0.19 0.12 0.7 −51.97 −8.63 −2.15 −0.42 0.12 0.27 0.27 0.20 0.12 0.8 −41.04 −6.37 −1.35 −0.08 0.27 0.34 0.29 0.21 0.12 0.9 −32.30 −4.58 −0.72 0.19 0.39 0.39 0.32 0.22 0.12 1.0 −25.16 −3.12 −0.21 0.40 0.49 0.43 0.33 0.23 0.13 0.8 0.2 −259.75 −55.70 −19.86 −8.29 −3.56 −1.43 −0.46 −0.06 0.03 0.3 −192.48 −38.47 −12.84 −4.99 −1.95 −0.66 −0.12 0.05 0.05 0.4 −144.25 −27.58 −8.74 −3.16 −1.09 −0.26 0.05 0.11 0.06 0.5 −110.65 −20.32 −6.08 −2.00 −0.55 −0.01 0.15 0.15 0.07 0.6 −86.30 −15.17 −4.24 −1.20 −0.19 0.15 0.22 0.17 0.08 0.7 −67.95 −11.34 −2.88 −0.62 0.08 0.27 0.27 0.19 0.08 0.8 −53.67 −8.40 −1.84 −0.18 0.28 0.36 0.30 0.20 0.08 0.9 −42.26 −6.05 −1.02 0.16 0.44 0.43 0.33 0.21 0.08 1.0 −32.93 −4.15 −0.35 0.44 0.56 0.49 0.36 0.22 0.09 0.9 0.2 −328.76 −70.51 −25.16 −10.53 −4.54 −1.84 −0.62 −0.12 0.00 0.3 −243.63 −48.72 −16.28 −6.35 −2.50 −0.87 −0.20 0.03 0.03 0.4 −182.60 −34.94 −11.09 −4.03 −1.41 −0.37 0.02 0.10 0.04 0.5 −140.07 −25.75 −7.74 −2.57 −0.74 −0.06 0.15 0.14 0.05 0.6 −109.25 −19.24 −5.40 −1.56 −0.28 0.15 0.23 0.17 0.05 0.7 −86.04 −14.40 −3.68 −0.83 0.06 0.30 0.30 0.20 0.06 0.8 −67.96 −10.66 −2.37 −0.27 0.31 0.41 0.34 0.21 0.06 0.9 −53.52 −7.70 −1.33 0.17 0.51 0.50 0.38 0.22 0.06 1.0 −41.71 −5.29 −0.49 0.52 0.67 0.57 0.41 0.23 0.07 1.0 0.2 −405.88 −87.06 −31.07 −13.01 −5.62 −2.29 −0.77 −0.16 −0.02 0.3 −300.78 −60.15 −20.11 −7.85 −3.10 −1.09 −0.26 0.02 0.02 0.4 −225.44 −43.14 −13.70 −4.99 −1.76 −0.47 0.01 0.11 0.04 0.5 −172.93 −31.80 −9.56 −3.18 −0.92 −0.09 0.17 0.17 0.05 0.6 −134.89 −23.76 −6.68 −1.94 −0.35 0.17 0.28 0.20 0.06 0.7 −106.23 −17.78 −4.56 −1.04 0.06 0.36 0.35 0.23 0.06 0.8 −83.92 −13.18 −2.93 −0.35 0.37 0.50 0.41 0.25 0.06 0.9 −66.08 −9.52 −1.65 0.19 0.62 0.61 0.46 0.26 0.07 1.0 −51.51 −6.54 −0.61 0.63 0.81 0.70 0.49 0.28 0.07 Duct Design 34.37 ED5-2 Wye, 45 Degree, Converging As/Ac Ab/Ac Cb Values Qb/Qc 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.2 0.2 −25.19 −3.97 −0.64 0.32 0.67 0.82 0.90 0.96 1.08 0.3 −58.03 −10.14 −2.63 −0.45 0.36 0.69 0.84 0.93 1.08 0.4 −104.08 −18.80 −5.40 −1.51 −0.07 0.52 0.77 0.88 1.01 0.5 −163.36 −29.97 −8.97 −2.87 −0.62 0.29 0.67 0.80 0.84 0.6 −235.59 −43.47 −13.22 −4.44 −1.20 0.12 0.65 0.83 0.85 0.7 −320.90 −59.38 −18.21 −6.25 −1.84 −0.04 0.68 0.91 0.93 0.8 −419.32 −77.73 −23.95 −8.33 −2.56 −0.22 0.72 1.02 1.02 0.9 −530.86 −98.50 −30.44 −10.66 −3.36 −0.40 0.79 1.16 1.14 1.0 −655.51 −121.72 −37.68 −13.26 −4.25 −0.59 0.87 1.33 1.28 0.3 0.2 −14.27 −1.77 0.13 0.66 0.85 0.93 0.97 1.03 1.21 0.3 −33.62 −5.28 −0.95 0.27 0.70 0.87 0.94 1.01 1.19 0.4 −60.85 −10.26 −2.48 −0.30 0.47 0.77 0.88 0.93 1.04 0.5 −95.87 −16.64 −4.44 −1.00 0.21 0.66 0.82 0.84 0.84 0.6 −138.38 −24.26 −6.68 −1.73 0.01 0.66 0.88 0.91 0.88 0.7 −188.60 −33.25 −9.32 −2.58 −0.20 0.68 0.98 1.02 0.95 0.8 −246.54 −43.60 −12.34 −3.54 −0.43 0.72 1.11 1.15 1.03 0.9 −312.21 −55.33 −15.76 −4.61 −0.68 0.78 1.26 1.31 1.13 1.0 −385.59 −68.43 −19.56 −5.79 −0.94 0.86 1.45 1.49 1.24 0.4 0.2 −8.77 −0.64 0.54 0.85 0.95 0.99 1.03 1.09 1.31 0.3 −21.41 −2.85 −0.10 0.63 0.87 0.96 1.00 1.06 1.26 0.4 −39.30 −6.02 −1.05 0.28 0.72 0.87 0.91 0.92 1.00 0.5 −62.10 −9.96 −2.16 −0.06 0.63 0.85 0.90 0.88 0.86 0.6 −89.77 −14.65 −3.42 −0.38 0.61 0.93 0.99 0.95 0.90 0.7 −122.46 −20.19 −4.88 −0.74 0.61 1.04 1.12 1.06 0.95 0.8 −160.18 −26.56 −6.55 −1.15 0.62 1.18 1.29 1.19 1.01 0.9 −202.93 −33.77 −8.44 −1.60 0.64 1.36 1.48 1.35 1.07 1.0 −250.70 −41.83 −10.54 −2.09 0.68 1.56 1.71 1.53 1.15 0.5 0.2 −5.45 0.04 0.79 0.97 1.02 1.04 1.07 1.14 1.39 0.3 −14.10 −1.39 0.40 0.84 0.97 1.00 1.02 1.07 1.28 0.4 −26.48 −3.53 −0.24 0.59 0.83 0.89 0.88 0.85 0.86 0.5 −41.84 −5.96 −0.80 0.51 0.88 0.97 0.95 0.90 0.87 0.6 −60.61 −8.90 −1.46 0.43 0.97 1.09 1.06 0.97 0.90 0.7 −82.80 −12.36 −2.22 0.35 1.09 1.25 1.20 1.08 0.93 0.8 −108.39 −16.35 −3.09 0.27 1.24 1.45 1.38 1.20 0.96 0.9 −137.41 −20.86 −4.07 0.19 1.42 1.68 1.59 1.35 0.99 1.0 −169.84 −25.90 −5.15 0.11 1.63 1.95 1.83 1.52 1.02 0.6 0.2 −5.54 −0.08 0.70 0.91 0.98 1.01 1.05 1.14 1.42 0.3 −14.48 −1.75 0.13 0.64 0.81 0.88 0.92 0.98 1.19 0.4 −27.10 −4.14 −0.68 0.26 0.57 0.68 0.71 0.72 0.76 0.5 −42.84 −6.91 −1.50 −0.02 0.47 0.64 0.68 0.69 0.70 0.6 −62.07 −10.28 −2.48 −0.34 0.37 0.61 0.67 0.66 0.63 0.7 −84.79 −14.26 −3.62 −0.71 0.27 0.59 0.67 0.63 0.54 0.8 −111.02 −18.84 −4.92 −1.12 0.16 0.58 0.67 0.61 0.44 0.9 −140.76 −24.03 −6.40 −1.57 0.04 0.58 0.68 0.59 0.31 1.0 −174.01 −29.83 −8.04 −2.07 −0.08 0.58 0.70 0.56 0.15 0.7 0.2 −3.96 0.25 0.83 0.97 1.01 1.04 1.08 1.17 1.47 0.3 −11.07 −1.10 0.34 0.71 0.83 0.87 0.90 0.95 1.13 0.4 −20.92 −2.92 −0.27 0.43 0.65 0.72 0.73 0.73 0.77 0.5 −33.20 −5.01 −0.85 0.24 0.59 0.69 0.71 0.69 0.70 0.6 −48.21 −7.55 −1.55 0.03 0.53 0.68 0.69 0.65 0.61 0.7 −65.95 −10.56 −2.37 −0.20 0.48 0.68 0.69 0.62 0.49 0.8 −86.42 −14.01 −3.30 −0.46 0.43 0.68 0.69 0.58 0.35 0.9 −109.65 −17.93 −4.35 −0.75 0.38 0.70 0.70 0.53 0.18 1.0 −135.63 −22.32 −5.53 −1.07 0.33 0.72 0.71 0.48 −0.03 34.38 2001 ASHRAE Fundamentals Handbook (SI) ED5-2 Wye, 45 Degree, Converging (Continued) As/Ac Ab/Ac Cb Values (Concluded) Qb/Qc 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.8 0.2 −2.78 0.50 0.91 1.01 1.03 1.05 1.09 1.18 1.49 0.3 −8.58 −0.65 0.47 0.74 0.82 0.85 0.86 0.89 1.02 0.4 −16.29 −2.00 0.05 0.56 0.71 0.75 0.74 0.74 0.78 0.5 −25.98 −3.59 −0.37 0.44 0.68 0.73 0.72 0.69 0.69 0.6 −37.82 −5.52 −0.87 0.31 0.65 0.72 0.70 0.64 0.58 0.7 −51.83 −7.79 −1.44 0.17 0.63 0.73 0.69 0.59 0.43 0.8 −68.01 −10.42 −2.10 0.01 0.62 0.75 0.69 0.53 0.25 0.9 −86.37 −13.39 −2.84 −0.16 0.61 0.77 0.68 0.47 0.03 1.0 −106.91 −16.73 −3.68 −0.35 0.61 0.79 0.68 0.38 −0.25 0.9 0.2 −1.87 0.68 0.98 1.03 1.05 1.06 1.09 1.18 1.49 0.3 −6.70 −0.33 0.54 0.74 0.79 0.80 0.80 0.81 0.87 0.4 −12.69 −1.29 0.29 0.66 0.76 0.77 0.75 0.74 0.78 0.5 −20.37 −2.48 0.00 0.59 0.74 0.75 0.72 0.69 0.67 0.6 −29.77 −3.94 −0.34 0.52 0.73 0.75 0.70 0.63 0.54 0.7 −40.89 −5.66 −0.73 0.45 0.74 0.76 0.68 0.56 0.36 0.8 −53.74 −7.64 −1.18 0.37 0.76 0.78 0.67 0.48 0.13 0.9 −68.32 −9.89 −1.69 0.28 0.77 0.80 0.65 0.38 −0.15 1.0 −84.66 −12.42 −2.27 0.18 0.80 0.83 0.62 0.26 −0.49 1.0 0.2 −1.17 0.81 1.02 1.05 1.05 1.06 1.09 1.18 1.48 0.3 −5.09 −0.02 0.64 0.78 0.81 0.81 0.80 0.80 0.86 0.4 −9.81 −0.72 0.48 0.74 0.79 0.78 0.76 0.74 0.77 0.5 −15.89 −1.61 0.29 0.71 0.79 0.77 0.72 0.68 0.65 0.6 −23.34 −2.69 0.07 0.68 0.80 0.77 0.69 0.60 0.49 0.7 −32.15 −3.96 −0.18 0.66 0.82 0.78 0.67 0.51 0.27 0.8 −42.35 −5.44 −0.47 0.64 0.85 0.79 0.63 0.41 0.00 0.9 −53.94 −7.12 −0.80 0.61 0.88 0.81 0.60 0.28 −0.34 1.0 −66.93 −9.01 −1.17 0.58 0.92 0.82 0.55 0.13 −0.75 As/Ac Ab/Ac Cs Values Qs/Qc 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.2 0.2 −10.16 −2.08 −0.43 0.24 0.62 0.88 1.10 1.29 1.46 0.3 −7.83 −1.20 0.03 0.50 0.77 0.97 1.14 1.30 1.46 0.4 −5.62 −0.59 0.30 0.65 0.85 1.01 1.16 1.31 1.46 0.5 −3.96 −0.18 0.48 0.74 0.90 1.04 1.18 1.32 1.47 0.6 −2.71 0.12 0.60 0.80 0.94 1.06 1.19 1.32 1.47 0.7 −1.75 0.34 0.70 0.85 0.96 1.07 1.19 1.32 1.47 0.8 −0.99 0.52 0.77 0.88 0.98 1.08 1.20 1.32 1.47 0.9 −0.38 0.66 0.82 0.91 0.99 1.09 1.20 1.33 1.47 1.0 0.13 0.77 0.87 0.93 1.00 1.10 1.20 1.33 1.47 0.3 0.2 −23.33 −5.14 −1.67 −0.44 0.12 0.42 0.58 0.67 0.72 0.3 −18.44 −3.44 −0.84 0.00 0.36 0.54 0.64 0.69 0.73 0.4 −13.64 −2.22 −0.34 0.25 0.49 0.60 0.67 0.70 0.73 0.5 −10.00 −1.37 0.00 0.41 0.57 0.64 0.69 0.71 0.73 0.6 −7.26 −0.75 0.24 0.52 0.62 0.67 0.70 0.72 0.73 0.7 −5.15 −0.29 0.41 0.60 0.66 0.69 0.71 0.72 0.73 0.8 −3.48 0.07 0.55 0.66 0.69 0.70 0.71 0.72 0.73 0.9 −2.14 0.36 0.65 0.71 0.72 0.72 0.72 0.72 0.73 1.0 −1.03 0.60 0.74 0.75 0.73 0.73 0.72 0.72 0.73 0.4 0.2 −42.17 −9.48 −3.34 −1.23 −0.31 0.12 0.33 0.42 0.44 0.3 −33.68 −6.60 −1.98 −0.53 0.05 0.31 0.41 0.45 0.45 0.4 −25.24 −4.51 −1.13 −0.13 0.25 0.40 0.46 0.47 0.45 0.5 −18.83 −3.04 −0.57 0.13 0.37 0.46 0.48 0.48 0.46 0.6 −13.99 −1.97 −0.17 0.31 0.46 0.50 0.50 0.48 0.46 0.7 −10.27 −1.17 0.12 0.44 0.52 0.53 0.51 0.49 0.46 0.8 −7.32 −0.54 0.35 0.54 0.57 0.55 0.52 0.49 0.46 0.9 −4.94 −0.04 0.53 0.62 0.61 0.57 0.53 0.49 0.46 1.0 −2.98 0.37 0.68 0.68 0.64 0.58 0.54 0.50 0.46 0.5 0.2 −66.95 −15.18 −5.49 −2.21 −0.81 −0.16 0.14 0.26 0.28 0.3 −53.80 −10.77 −3.45 −1.17 −0.27 0.11 0.26 0.30 0.29 0.4 −40.66 −7.54 −2.16 −0.57 0.02 0.25 0.32 0.33 0.30 0.5 −30.68 −5.27 −1.30 −0.18 0.21 0.33 0.36 0.34 0.30 0.6 −23.15 −3.62 −0.69 0.09 0.33 0.39 0.38 0.35 0.30 0.7 −17.34 −2.38 −0.24 0.29 0.42 0.43 0.40 0.35 0.30 0.8 −12.75 −1.41 0.11 0.44 0.49 0.47 0.41 0.36 0.30 0.9 −9.04 −0.64 0.39 0.56 0.55 0.49 0.43 0.36 0.30 1.0 −5.99 0.00 0.61 0.65 0.59 0.51 0.43 0.36 0.30 Duct Design 34.39 ED5-2 Wye, 45 Degree, Converging (Concluded) As/Ac Ab/Ac Cs Values (Concluded) Qs/Qc 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.6 0.2 −97.90 −22.29 −8.18 −3.41 −1.39 −0.46 −0.03 0.13 0.16 0.3 −79.03 −15.99 −5.28 −1.94 −0.64 −0.09 0.13 0.19 0.17 0.4 −60.15 −11.37 −3.44 −1.09 −0.23 0.10 0.21 0.22 0.18 0.5 −45.80 −8.13 −2.22 −0.55 0.03 0.22 0.26 0.24 0.18 0.6 −34.97 −5.77 −1.35 −0.17 0.20 0.30 0.30 0.25 0.18 0.7 −26.62 −3.98 −0.71 0.11 0.33 0.36 0.32 0.26 0.19 0.8 −20.02 −2.59 −0.21 0.33 0.43 0.41 0.34 0.26 0.19 0.9 −14.68 −1.48 0.18 0.49 0.51 0.44 0.35 0.27 0.19 1.0 −10.29 −0.57 0.51 0.63 0.57 0.47 0.37 0.27 0.19 0.7 0.2 −135.28 −30.88 −11.42 −4.85 −2.08 −0.80 −0.21 0.02 0.06 0.3 −109.64 −22.35 −7.50 −2.88 −1.07 −0.31 0.00 0.09 0.07 0.4 −83.96 −16.08 −5.02 −1.73 −0.52 −0.05 0.11 0.13 0.08 0.5 −64.44 −11.67 −3.36 −0.99 −0.17 0.11 0.18 0.15 0.09 0.6 −49.71 −8.47 −2.19 −0.48 0.06 0.22 0.22 0.17 0.09 0.7 −38.35 −6.04 −1.31 −0.10 0.24 0.30 0.26 0.18 0.09 0.8 −29.37 −4.16 −0.64 0.18 0.37 0.36 0.28 0.19 0.09 0.9 −22.12 −2.65 −0.10 0.41 0.47 0.40 0.30 0.19 0.09 1.0 −16.14 −1.41 0.33 0.60 0.55 0.44 0.32 0.20 0.09 0.8 0.2 −179.32 −41.01 −15.25 −6.55 −2.88 −1.19 −0.41 −0.10 −0.04 0.3 −145.86 −29.89 −10.14 −3.99 −1.58 −0.55 −0.13 0.00 −0.02 0.4 −112.34 −21.71 −6.91 −2.50 −0.86 −0.22 0.01 0.05 −0.01 0.5 −86.85 −15.96 −4.75 −1.54 −0.41 −0.01 0.10 0.08 0.00 0.6 −67.62 −11.78 −3.22 −0.87 −0.10 0.13 0.16 0.10 0.00 0.7 −52.79 −8.62 −2.08 −0.38 0.12 0.23 0.20 0.11 0.00 0.8 −41.06 −6.16 −1.20 0.00 0.29 0.31 0.23 0.12 0.01 0.9 −31.59 −4.19 −0.51 0.29 0.43 0.37 0.26 0.13 0.01 1.0 −23.78 −2.58 0.06 0.53 0.54 0.42 0.28 0.14 0.01 0.9 0.2 −230.27 −52.75 −19.69 −8.53 −3.81 −1.63 −0.63 −0.22 −0.13 0.3 −187.95 −38.69 −13.24 −5.29 −2.16 −0.83 −0.28 −0.10 −0.10 0.4 −145.53 −28.34 −9.15 −3.41 −1.26 −0.41 −0.10 −0.04 −0.09 0.5 −113.27 −21.07 −6.42 −2.19 −0.69 −0.15 0.01 0.00 −0.09 0.6 −88.94 −15.78 −4.48 −1.35 −0.30 0.03 0.09 0.03 −0.08 0.7 −70.16 −11.78 −3.04 −0.73 −0.02 0.16 0.14 0.04 −0.08 0.8 −55.33 −8.67 −1.93 −0.25 0.20 0.26 0.18 0.06 −0.07 0.9 −43.33 −6.18 −1.05 0.12 0.37 0.33 0.21 0.07 −0.07 1.0 −33.46 −4.14 −0.34 0.42 0.50 0.39 0.24 0.08 −0.07 1.0 0.2 −288.39 −66.15 −24.77 −10.80 −4.88 −2.14 −0.87 −0.35 −0.22 0.3 −236.14 −48.79 −16.81 −6.80 −2.85 −1.15 −0.44 −0.20 −0.19 0.4 −183.77 −36.02 −11.76 −4.47 −1.73 −0.63 −0.22 −0.12 −0.18 0.5 −143.95 −27.05 −8.39 −2.98 −1.03 −0.31 −0.08 −0.08 −0.17 0.6 −113.91 −20.52 −6.00 −1.93 −0.55 −0.09 0.01 −0.04 −0.16 0.7 −90.73 −15.58 −4.23 −1.17 −0.20 0.07 0.08 −0.02 −0.16 0.8 −72.41 −11.74 −2.86 −0.58 0.06 0.19 0.13 −0.01 −0.16 0.9 −57.61 −8.66 −1.77 −0.12 0.27 0.28 0.16 0.01 −0.15 1.0 −45.42 −6.15 −0.88 0.25 0.44 0.36 0.20 0.02 −0.15 34.40 2001 ASHRAE Fundamentals Handbook (SI) ED5-3 Tee, Dc < or = 250 mm, Converging CbValues Qb/Qc As/Ac Ab/Ac 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.2 0.2 –24.56 –3.63 –0.36 0.59 0.93 1.08 1.14 1.19 1.27 0.3 –56.72 –9.54 –2.15 –0.01 0.78 1.10 1.23 1.30 1.39 0.4 –101.83 –17.86 –4.68 –0.87 0.52 1.09 1.32 1.41 1.48 0.5 –159.91 –28.59 –7.98 –2.02 0.17 1.05 1.40 1.51 1.51 0.6 –230.83 –41.68 –11.98 –3.39 –0.24 1.03 1.53 1.66 1.61 0.7 –314.56 –57.10 –16.68 –4.98 –0.69 1.04 1.71 1.90 1.82 0.8 –411.18 –74.90 –22.10 –6.82 –1.21 1.04 1.92 2.16 2.05 0.9 –520.69 –95.08 –28.25 –8.90 –1.81 1.04 2.15 2.45 2.31 1.0 –643.09 –117.63 –35.12 –11.24 –2.47 1.04 2.41 2.78 2.58 0.3 0.2 –14.05 –1.55 0.36 0.89 1.08 1.16 1.19 1.23 1.34 0.3 –33.18 –4.91 –0.58 0.64 1.07 1.23 1.30 1.34 1.44 0.4 –60.09 –9.68 –1.94 0.24 1.00 1.29 1.39 1.42 1.47 0.5 –94.80 –15.89 –3.74 –0.33 0.87 1.32 1.46 1.46 1.38 0.6 –136.97 –23.33 –5.84 –0.92 0.81 1.45 1.65 1.66 1.53 0.7 –186.81 –32.14 –8.32 –1.62 0.74 1.61 1.88 1.88 1.70 0.8 –244.33 –42.30 –11.19 –2.43 0.65 1.78 2.14 2.13 1.88 0.9 –309.54 –53.82 –14.44 –3.35 0.54 1.98 2.42 2.41 2.08 1.0 –382.43 –66.70 –18.08 –4.39 0.42 2.19 2.74 2.72 2.29 0.4 0.2 –8.95 –0.54 0.71 1.04 1.15 1.20 1.22 1.26 1.40 0.3 –21.82 –2.70 0.16 0.94 1.19 1.29 1.32 1.35 1.47 0.4 –39.99 –5.81 –0.67 0.73 1.19 1.35 1.39 1.39 1.41 0.5 –63.37 –9.82 –1.75 0.45 1.18 1.42 1.47 1.43 1.32 0.6 –91.72 –14.59 –2.97 0.20 1.26 1.60 1.67 1.60 1.43 0.7 –125.23 –20.24 –4.41 –0.10 1.34 1.81 1.90 1.81 1.56 0.8 –163.91 –26.77 –6.09 –0.45 1.43 2.04 2.16 2.03 1.69 0.9 –207.76 –34.17 –7.99 –0.85 1.53 2.30 2.44 2.28 1.82 1.0 –256.79 –42.45 –10.12 –1.30 1.63 2.58 2.75 2.54 1.95 0.5 0.2 –6.03 0.04 0.91 1.13 1.20 1.22 1.24 1.29 1.44 0.3 –15.35 –1.46 0.56 1.09 1.25 1.30 1.32 1.35 1.46 0.4 –28.59 –3.67 –0.01 0.96 1.26 1.34 1.35 1.32 1.29 0.5 –45.45 –6.42 –0.66 0.85 1.33 1.45 1.45 1.38 1.24 0.6 –65.92 –9.70 –1.41 0.78 1.46 1.64 1.63 1.53 1.32 0.7 –90.12 –13.58 –2.29 0.69 1.61 1.86 1.85 1.70 1.39 0.8 –118.07 –18.07 –3.32 0.57 1.78 2.11 2.09 1.89 1.46 0.9 –149.75 –23.18 –4.49 0.43 1.96 2.38 2.35 2.09 1.53 1.0 –185.19 –28.89 –5.81 0.27 2.16 2.67 2.63 2.31 1.57 0.6 0.2 –4.20 0.39 1.03 1.18 1.22 1.24 1.26 1.30 1.47 0.3 –11.33 –0.72 0.79 1.16 1.27 1.30 1.31 1.33 1.43 0.4 –21.57 –2.42 0.35 1.05 1.25 1.29 1.27 1.22 1.12 0.5 –34.29 –4.35 –0.03 1.07 1.38 1.44 1.41 1.32 1.16 0.6 –49.85 –6.73 –0.50 1.08 1.54 1.63 1.57 1.45 1.19 0.7 –68.26 –9.55 –1.06 1.09 1.71 1.83 1.76 1.58 1.21 0.8 –89.52 –12.81 –1.72 1.10 1.91 2.07 1.97 1.73 1.22 0.9 –113.64 –16.52 –2.47 1.10 2.12 2.32 2.20 1.88 1.21 1.0 –140.62 –20.68 –3.33 1.09 2.35 2.60 2.44 2.03 1.16 Duct Design 34.41 As/Ac Ab/Ac CbValues (Concluded) Qb/Qc 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.7 0.2 –3.00 0.62 1.10 1.21 1.23 1.24 1.26 1.31 1.49 0.3 –8.74 –0.27 0.91 1.19 1.26 1.27 1.27 1.28 1.36 0.4 –16.90 –1.59 0.58 1.11 1.25 1.27 1.24 1.18 1.06 0.5 –26.99 –3.06 0.33 1.17 1.38 1.41 1.36 1.26 1.06 0.6 –39.35 –4.86 0.02 1.22 1.54 1.57 1.50 1.35 1.05 0.7 –53.97 –7.01 –0.35 1.29 1.72 1.76 1.65 1.45 1.02 0.8 –70.87 –9.50 –0.79 1.35 1.91 1.97 1.82 1.54 0.96 0.9 –90.04 –12.34 –1.31 1.41 2.12 2.19 2.00 1.64 0.86 1.0 –111.50 –15.53 –1.89 1.46 2.34 2.43 2.19 1.73 0.72 0.8 0.2 –2.20 0.76 1.14 1.22 1.24 1.24 1.26 1.31 1.49 0.3 –7.04 –0.01 0.95 1.18 1.23 1.23 1.23 1.22 1.27 0.4 –13.77 –1.06 0.71 1.13 1.24 1.24 1.20 1.13 1.00 0.5 –22.11 –2.24 0.54 1.20 1.36 1.36 1.30 1.19 0.97 0.6 –32.33 –3.69 0.31 1.27 1.50 1.50 1.41 1.25 0.90 0.7 –44.42 –5.41 0.04 1.34 1.66 1.65 1.53 1.30 0.81 0.8 –58.40 –7.42 –0.29 1.42 1.83 1.83 1.65 1.35 0.67 0.9 –74.28 –9.72 –0.67 1.49 2.01 2.01 1.78 1.38 0.49 1.0 –92.06 –12.30 –1.12 1.56 2.21 2.20 1.92 1.40 0.24 0.9 0.2 –1.67 0.85 1.16 1.22 1.23 1.24 1.25 1.30 1.48 0.3 –5.95 0.12 0.95 1.14 1.18 1.18 1.16 1.15 1.14 0.4 –11.68 –0.74 0.77 1.12 1.20 1.20 1.16 1.08 0.93 0.5 –18.85 –1.74 0.63 1.18 1.31 1.30 1.23 1.11 0.86 0.6 –27.63 –2.98 0.44 1.24 1.42 1.41 1.31 1.13 0.75 0.7 –38.04 –4.45 0.21 1.30 1.55 1.53 1.39 1.14 0.58 0.8 –50.07 –6.17 –0.07 1.36 1.69 1.66 1.47 1.13 0.37 0.9 –63.75 –8.14 –0.40 1.42 1.83 1.79 1.54 1.11 0.09 1.0 –79.08 –10.36 –0.79 1.46 1.98 1.92 1.61 1.06 –0.26 1.0 0.2 –1.33 0.89 1.16 1.21 1.22 1.22 1.24 1.29 1.46 0.3 –5.30 0.15 0.90 1.08 1.11 1.11 1.09 1.06 0.99 0.4 –10.31 –0.57 0.78 1.09 1.16 1.15 1.11 1.03 0.86 0.5 –16.71 –1.47 0.64 1.13 1.24 1.22 1.15 1.03 0.74 0.6 –24.56 –2.59 0.46 1.17 1.32 1.30 1.20 1.01 0.57 0.7 –33.87 –3.93 0.23 1.20 1.41 1.38 1.24 0.97 0.34 0.8 –44.64 –5.49 –0.05 1.22 1.51 1.46 1.27 0.91 0.05 0.9 –56.89 –7.29 –0.38 1.24 1.59 1.54 1.28 0.82 –0.33 1.0 –70.62 –9.32 –0.77 1.24 1.68 1.61 1.28 0.69 –0.80 ED5–3 Tee, Dc < or = 250 mm, Converging CsValues Qs/Qc As/Ac Ab/Ac 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.2 0.2 18.11 3.42 1.62 1.11 0.90 0.80 0.74 0.70 0.68 0.3 12.67 2.79 1.45 1.04 0.87 0.78 0.73 0.70 0.68 0.4 9.98 2.47 1.36 1.01 0.85 0.77 0.72 0.69 0.67 0.5 8.39 2.27 1.30 0.98 0.84 0.76 0.72 0.69 0.67 0.6 7.34 2.13 1.26 0.96 0.83 0.76 0.72 0.69 0.67 0.7 6.61 2.02 1.22 0.95 0.82 0.75 0.71 0.69 0.67 0.8 6.08 1.94 1.19 0.93 0.81 0.75 0.71 0.68 0.67 0.9 5.68 1.87 1.17 0.92 0.80 0.74 0.70 0.68 0.66 1.0 4.55 1.61 1.05 0.86 0.76 0.71 0.68 0.66 0.65 0.3 0.2 44.33 7.19 2.80 1.57 1.08 0.84 0.71 0.63 0.57 0.3 29.24 5.46 2.33 1.40 1.00 0.80 0.69 0.62 0.57 0.4 21.88 4.59 2.09 1.30 0.96 0.78 0.67 0.61 0.56 0.5 17.62 4.06 1.93 1.24 0.92 0.76 0.66 0.60 0.56 0.6 14.90 3.71 1.82 1.19 0.90 0.74 0.65 0.59 0.55 0.7 13.06 3.45 1.74 1.15 0.88 0.73 0.64 0.59 0.55 0.8 11.78 3.26 1.67 1.12 0.86 0.72 0.63 0.58 0.54 0.9 9.02 2.64 1.41 0.97 0.77 0.66 0.59 0.54 0.51 1.0 8.36 2.52 1.36 0.95 0.75 0.65 0.58 0.54 0.51 ED5-3 Tee, Dc < or = 250 mm, Converging (Continued) 34.42 2001 ASHRAE Fundamentals Handbook (SI) As/Ac Ab/Ac CsValues (Concluded) Qs/Qc 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.4 0.2 78.99 12.25 4.42 2.26 1.39 0.97 0.74 0.60 0.50 0.3 50.14 8.96 3.54 1.92 1.24 0.90 0.70 0.57 0.49 0.4 36.26 7.32 3.08 1.74 1.16 0.85 0.67 0.56 0.48 0.5 28.38 6.35 2.80 1.63 1.10 0.82 0.65 0.54 0.47 0.6 23.50 5.72 2.61 1.54 1.05 0.79 0.63 0.53 0.46 0.7 20.32 5.27 2.46 1.47 1.02 0.77 0.62 0.52 0.45 0.8 14.94 4.13 1.98 1.21 0.85 0.65 0.53 0.46 0.40 0.9 13.55 3.88 1.89 1.16 0.82 0.63 0.52 0.45 0.39 1.0 12.66 3.69 1.80 1.12 0.79 0.62 0.51 0.44 0.39 0.5 0.2 114.73 17.76 6.27 3.07 1.79 1.16 0.81 0.60 0.46 0.3 70.56 12.71 4.92 2.56 1.56 1.05 0.75 0.56 0.44 0.4 49.68 10.24 4.23 2.29 1.43 0.98 0.71 0.54 0.42 0.5 38.12 8.81 3.81 2.11 1.34 0.93 0.68 0.52 0.41 0.6 31.23 7.90 3.53 1.99 1.27 0.88 0.65 0.50 0.39 0.7 21.87 6.00 2.75 1.57 1.01 0.71 0.52 0.40 0.32 0.8 19.30 5.57 2.59 1.49 0.96 0.67 0.50 0.38 0.30 0.9 17.84 5.27 2.46 1.42 0.92 0.65 0.48 0.37 0.29 1.0 17.16 5.05 2.36 1.36 0.88 0.62 0.46 0.35 0.28 0.6 0.2 142.32 22.64 8.06 3.91 2.23 1.39 0.92 0.63 0.44 0.3 84.89 16.05 6.28 3.24 1.92 1.23 0.83 0.58 0.41 0.4 58.43 12.90 5.39 2.88 1.75 1.14 0.78 0.55 0.39 0.5 44.34 11.13 4.86 2.66 1.63 1.07 0.74 0.52 0.37 0.6 29.06 8.20 3.69 2.04 1.25 0.81 0.55 0.38 0.26 0.7 24.71 7.51 3.44 1.91 1.18 0.77 0.52 0.35 0.24 0.8 22.56 7.06 3.26 1.81 1.11 0.72 0.48 0.33 0.22 0.9 21.89 6.78 3.12 1.73 1.06 0.68 0.45 0.30 0.20 1.0 22.24 6.61 3.00 1.65 1.00 0.65 0.43 0.28 0.18 0.7 0.2 152.32 25.82 9.48 4.66 2.65 1.63 1.04 0.68 0.44 0.3 87.85 18.38 7.46 3.88 2.29 1.44 0.94 0.62 0.40 0.4 59.34 14.92 6.47 3.48 2.09 1.33 0.87 0.58 0.37 0.5 35.18 10.56 4.78 2.60 1.55 0.97 0.62 0.38 0.22 0.6 28.26 9.51 4.41 2.42 1.45 0.90 0.57 0.35 0.19 0.7 25.45 8.91 4.16 2.28 1.36 0.85 0.53 0.32 0.17 0.8 25.21 8.60 3.99 2.18 1.29 0.79 0.49 0.28 0.14 0.9 26.68 8.48 3.86 2.08 1.22 0.74 0.45 0.25 0.12 1.0 29.34 8.49 3.77 2.01 1.16 0.70 0.41 0.22 0.10 0.8 0.2 136.74 26.38 10.30 5.22 3.01 1.85 1.17 0.74 0.45 0.3 75.52 19.20 8.32 4.45 2.64 1.66 1.06 0.67 0.41 0.4 37.55 12.79 5.92 3.23 1.91 1.17 0.72 0.42 0.21 0.5 27.25 11.28 5.41 2.98 1.77 1.08 0.66 0.37 0.18 0.6 24.23 10.57 5.10 2.81 1.66 1.01 0.60 0.33 0.14 0.7 25.36 10.32 4.91 2.69 1.57 0.94 0.55 0.29 0.11 0.8 29.09 10.37 4.80 2.59 1.50 0.88 0.50 0.25 0.08 0.9 34.55 10.60 4.74 2.50 1.42 0.82 0.46 0.21 0.05 1.0 41.23 10.98 4.71 2.43 1.36 0.77 0.41 0.18 0.01 0.9 0.2 90.70 23.73 10.34 5.54 3.28 2.05 1.30 0.81 0.47 0.3 29.93 14.20 6.95 3.86 2.30 1.41 0.85 0.48 0.22 0.4 16.27 12.21 6.28 3.55 2.12 1.29 0.77 0.42 0.18 0.5 14.80 11.58 5.96 3.35 1.99 1.20 0.70 0.37 0.14 0.6 19.43 11.62 5.81 3.23 1.89 1.13 0.64 0.32 0.10 0.7 27.55 12.06 5.77 3.14 1.81 1.06 0.59 0.27 0.06 0.8 37.84 12.73 5.79 3.07 1.74 0.99 0.53 0.23 0.02 0.9 49.59 13.57 5.85 3.01 1.67 0.93 0.48 0.18 –0.02 1.0 62.35 14.52 5.94 2.97 1.61 0.87 0.42 0.14 –0.06 1.0 0.2 –6.40 12.70 7.32 4.31 2.64 1.64 1.00 0.56 0.25 0.3 –17.35 10.90 6.66 3.97 2.44 1.51 0.90 0.49 0.20 0.4 –11.05 11.02 6.50 3.82 2.32 1.41 0.83 0.43 0.15 0.5 2.15 11.91 6.54 3.74 2.23 1.33 0.76 0.38 0.10 0.6 18.80 13.18 6.67 3.70 2.16 1.26 0.70 0.32 0.06 0.7 37.42 14.67 6.86 3.68 2.09 1.19 0.63 0.26 0.01 0.8 57.27 16.30 7.09 3.67 2.03 1.12 0.57 0.21 –0.04 0.9 77.95 18.02 7.35 3.66 1.97 1.06 0.51 0.15 –0.09 1.0 99.20 19.80 7.61 3.67 1.92 1.00 0.45 0.10 –0.14 ED5–3 Tee, Dc < or = 250 mm, Converging (Continued) Duct Design 34.43 ED5–3 Tee, Dc > 250 mm, Converging (Continued) As/Ac Ab/Ac CbValues Qb/Qc 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.2 0.2 –26.08 –4.19 –0.70 0.33 0.71 0.87 0.93 0.95 0.93 0.3 –59.71 –10.53 –2.72 –0.43 0.43 0.78 0.91 0.95 0.91 0.4 –106.78 –19.39 –5.53 –1.46 0.05 0.67 0.91 0.97 0.91 0.5 –167.36 –30.77 –9.12 –2.78 –0.42 0.55 0.93 1.02 0.93 0.6 –241.50 –44.68 –13.50 –4.37 –0.97 0.42 0.96 1.10 0.98 0.7 –329.25 –61.15 –18.68 –6.25 –1.62 0.27 1.02 1.21 1.06 0.8 –430.67 –80.18 –24.67 –8.42 –2.37 0.10 1.09 1.35 1.17 0.9 –545.81 –101.78 –31.47 –10.89 –3.22 –0.08 1.17 1.52 1.31 1.0 –674.72 –125.98 –39.08 –13.64 –4.17 –0.28 1.28 1.72 1.48 0.3 0.2 –15.50 –2.16 –0.04 0.58 0.81 0.90 0.93 0.94 0.91 0.3 –35.76 –5.90 –1.20 0.16 0.66 0.85 0.92 0.92 0.88 0.4 –64.09 –11.09 –2.78 –0.38 0.48 0.82 0.93 0.94 0.86 0.5 –100.54 –17.73 –4.78 –1.06 0.29 0.80 0.97 0.98 0.87 0.6 –145.16 –25.85 –7.21 –1.86 0.06 0.80 1.05 1.05 0.90 0.7 –198.01 –35.46 –10.08 –2.81 –0.19 0.82 1.15 1.16 0.96 0.8 –259.13 –46.56 –13.39 –3.89 –0.47 0.85 1.28 1.30 1.05 0.9 –328.59 –59.18 –17.15 –5.11 –0.78 0.89 1.44 1.47 1.17 1.0 –406.44 –73.33 –21.37 –6.48 –1.12 0.94 1.63 1.68 1.32 0.4 0.2 –10.31 –1.18 0.26 0.69 0.84 0.91 0.93 0.93 0.90 0.3 –23.96 –3.65 –0.48 0.43 0.75 0.88 0.91 0.91 0.86 0.4 –42.98 –7.03 –1.46 0.11 0.67 0.87 0.93 0.91 0.84 0.5 –67.44 –11.35 –2.69 –0.26 0.59 0.90 0.97 0.94 0.84 0.6 –97.39 –16.60 –4.17 –0.69 0.52 0.95 1.06 1.01 0.87 0.7 –132.88 –22.81 –5.91 –1.17 0.46 1.03 1.17 1.11 0.92 0.8 –173.96 –29.99 –7.90 –1.73 0.40 1.15 1.33 1.24 1.00 0.9 –220.69 –38.15 –10.16 –2.35 0.35 1.29 1.51 1.40 1.11 1.0 –273.12 –47.31 –12.70 –3.04 0.29 1.45 1.74 1.61 1.26 0.5 0.2 –7.26 –0.62 0.43 0.75 0.86 0.91 0.93 0.93 0.90 0.3 –16.99 –2.35 –0.07 0.57 0.80 0.89 0.91 0.90 0.87 0.4 –30.49 –4.67 –0.72 0.38 0.76 0.89 0.92 0.90 0.85 0.5 –47.82 –7.61 –1.50 0.19 0.75 0.93 0.97 0.93 0.85 0.6 –69.03 –11.17 –2.42 –0.03 0.76 1.01 1.05 0.98 0.88 0.7 –94.17 –15.37 –3.49 –0.26 0.80 1.13 1.17 1.07 0.93 0.8 –123.30 –20.22 –4.71 –0.50 0.87 1.29 1.33 1.20 1.02 0.9 –156.48 –25.73 –6.09 –0.77 0.96 1.48 1.53 1.36 1.13 1.0 –193.74 –31.92 –7.63 –1.07 1.06 1.71 1.77 1.56 1.28 0.6 0.2 –5.28 –0.27 0.54 0.78 0.88 0.91 0.93 0.93 0.91 0.3 –12.43 –1.51 0.18 0.66 0.83 0.89 0.91 0.91 0.89 0.4 –22.29 –3.15 –0.25 0.55 0.82 0.90 0.92 0.91 0.88 0.5 –34.92 –5.19 –0.74 0.46 0.84 0.95 0.96 0.93 0.89 0.6 –50.35 –7.64 –1.30 0.38 0.91 1.05 1.04 0.98 0.93 0.7 –68.66 –10.52 –1.94 0.32 1.01 1.18 1.16 1.07 0.99 0.8 –89.89 –13.83 –2.65 0.26 1.15 1.36 1.33 1.19 1.08 0.9 –114.09 –17.61 –3.46 0.22 1.32 1.59 1.53 1.35 1.21 1.0 –141.33 –21.84 –4.35 0.18 1.54 1.85 1.77 1.54 1.37 0.7 0.2 –3.90 –0.03 0.61 0.81 0.89 0.92 0.94 0.94 0.93 0.3 –9.25 –0.94 0.35 0.72 0.85 0.90 0.92 0.92 0.92 0.4 –16.54 –2.10 0.07 0.66 0.85 0.91 0.93 0.92 0.92 0.5 –25.85 –3.51 –0.22 0.64 0.90 0.97 0.97 0.94 0.95 0.6 –37.21 –5.18 –0.54 0.65 1.00 1.07 1.05 1.00 1.00 0.7 –50.68 –7.13 –0.87 0.70 1.14 1.22 1.17 1.08 1.08 0.8 –66.31 –9.37 –1.24 0.78 1.33 1.41 1.33 1.21 1.20 0.9 –84.17 –11.92 –1.64 0.89 1.56 1.65 1.53 1.36 1.34 1.0 –104.29 –14.78 –2.09 1.03 1.84 1.94 1.78 1.56 1.52 0.8 0.2 –2.90 0.15 0.67 0.83 0.90 0.93 0.94 0.95 0.96 0.3 –6.91 –0.53 0.47 0.76 0.87 0.91 0.93 0.94 0.96 0.4 –12.31 –1.34 0.30 0.74 0.88 0.93 0.94 0.95 0.98 0.5 –19.16 –2.29 0.15 0.77 0.94 0.99 0.98 0.98 1.03 0.6 –27.50 –3.39 0.01 0.84 1.06 1.09 1.06 1.03 1.11 0.7 –37.38 –4.66 –0.11 0.97 1.23 1.24 1.18 1.12 1.21 0.8 –48.87 –6.11 –0.22 1.15 1.46 1.45 1.35 1.25 1.35 0.9 –62.01 –7.75 –0.33 1.37 1.73 1.70 1.55 1.41 1.52 1.0 –76.85 –9.59 –0.44 1.63 2.06 2.00 1.80 1.61 1.73 34.44 2001 ASHRAE Fundamentals Handbook (SI) As/Ac Ab/Ac CbValues (Concluded) Qb/Qc 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.9 0.2 –2.14 0.28 0.71 0.85 0.91 0.94 0.96 0.97 0.99 0.3 –5.14 –0.21 0.57 0.80 0.88 0.92 0.95 0.97 1.02 0.4 –9.09 –0.76 0.47 0.80 0.91 0.94 0.96 0.98 1.06 0.5 –14.06 –1.36 0.42 0.86 0.98 1.01 1.01 1.02 1.14 0.6 –20.08 –2.04 0.42 0.99 1.11 1.12 1.09 1.09 1.24 0.7 –27.21 –2.79 0.47 1.17 1.30 1.27 1.21 1.19 1.38 0.8 –35.50 –3.63 0.55 1.42 1.55 1.49 1.38 1.32 1.55 0.9 –45.01 –4.57 0.66 1.72 1.86 1.75 1.59 1.49 1.75 1.0 –55.79 –5.64 0.80 2.08 2.22 2.06 1.84 1.69 1.99 1.0 0.2 –1.54 0.39 0.74 0.87 0.92 0.95 0.97 0.99 1.03 0.3 –3.75 0.03 0.64 0.83 0.90 0.94 0.97 1.00 1.08 0.4 –6.57 –0.32 0.61 0.85 0.93 0.97 0.99 1.03 1.16 0.5 –10.05 –0.65 0.64 0.94 1.02 1.03 1.04 1.08 1.26 0.6 –14.24 –0.98 0.74 1.10 1.16 1.15 1.13 1.16 1.40 0.7 –19.20 –1.32 0.91 1.33 1.37 1.31 1.26 1.27 1.57 0.8 –24.98 –1.69 1.14 1.63 1.63 1.53 1.43 1.41 1.78 0.9 –31.62 –2.10 1.42 2.00 1.96 1.80 1.64 1.59 2.02 1.0 –39.19 –2.55 1.76 2.43 2.35 2.12 1.90 1.81 2.30 ED5–3 Tee, Dc > 250 mm, Converging CsValues Qs/Qc As/Ac Ab/Ac 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.2 0.2 20.43 3.28 1.45 0.98 0.81 0.73 0.69 0.66 0.64 0.3 12.53 2.40 1.22 0.90 0.77 0.71 0.68 0.66 0.64 0.4 8.78 1.98 1.12 0.86 0.76 0.70 0.67 0.66 0.64 0.5 6.69 1.75 1.06 0.84 0.75 0.70 0.67 0.65 0.64 0.6 5.43 1.61 1.02 0.83 0.74 0.70 0.67 0.65 0.64 0.7 4.64 1.52 1.00 0.82 0.74 0.70 0.67 0.65 0.64 0.8 4.15 1.47 0.98 0.81 0.74 0.69 0.67 0.65 0.64 0.9 3.86 1.43 0.97 0.81 0.74 0.69 0.67 0.65 0.64 1.0 3.71 1.42 0.97 0.81 0.73 0.69 0.67 0.65 0.64 0.3 0.2 51.24 7.11 2.49 1.33 0.90 0.70 0.60 0.54 0.50 0.3 29.57 4.70 1.87 1.10 0.80 0.66 0.58 0.53 0.50 0.4 19.40 3.57 1.58 1.00 0.76 0.64 0.57 0.52 0.50 0.5 13.84 2.96 1.42 0.94 0.73 0.62 0.56 0.52 0.50 0.6 10.58 2.59 1.32 0.90 0.72 0.62 0.56 0.52 0.49 0.7 8.64 2.38 1.27 0.88 0.71 0.61 0.56 0.52 0.49 0.8 7.52 2.25 1.23 0.87 0.70 0.61 0.56 0.52 0.49 0.9 6.95 2.19 1.22 0.87 0.70 0.61 0.56 0.52 0.49 1.0 6.76 2.17 1.21 0.86 0.70 0.61 0.55 0.52 0.49 0.4 0.2 90.30 12.10 3.91 1.85 1.08 0.74 0.55 0.45 0.38 0.3 49.68 7.59 2.74 1.42 0.90 0.65 0.51 0.43 0.37 0.4 30.96 5.51 2.21 1.23 0.82 0.61 0.49 0.42 0.37 0.5 21.00 4.40 1.92 1.13 0.78 0.59 0.48 0.42 0.37 0.6 15.43 3.78 1.76 1.07 0.75 0.58 0.48 0.41 0.37 0.7 12.36 3.44 1.67 1.04 0.74 0.57 0.48 0.41 0.37 0.8 10.86 3.27 1.63 1.02 0.73 0.57 0.47 0.41 0.37 0.9 10.40 3.22 1.61 1.01 0.73 0.57 0.47 0.41 0.37 1.0 10.67 3.25 1.62 1.02 0.73 0.57 0.47 0.41 0.37 0.5 0.2 126.36 16.99 5.39 2.42 1.32 0.81 0.54 0.38 0.28 0.3 65.94 10.28 3.65 1.79 1.05 0.68 0.48 0.35 0.27 0.4 38.84 7.27 2.87 1.51 0.93 0.63 0.45 0.34 0.27 0.5 25.07 5.74 2.47 1.37 0.87 0.60 0.44 0.33 0.26 0.6 17.98 4.95 2.27 1.29 0.84 0.58 0.43 0.33 0.26 0.7 14.69 4.58 2.17 1.26 0.82 0.58 0.43 0.33 0.26 0.8 13.78 4.48 2.15 1.25 0.82 0.57 0.43 0.33 0.26 0.9 14.45 4.56 2.17 1.26 0.82 0.58 0.43 0.33 0.26 1.0 16.24 4.76 2.22 1.28 0.83 0.58 0.43 0.33 0.26 ED5–3 Tee, Dc > 250 mm, Converging (Continued) Duct Design 34.45 As/Ac Ab/Ac CsValues (Concluded) Qs/Qc 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.6 0.2 146.22 20.32 6.54 2.92 1.54 0.89 0.54 0.33 0.20 0.3 70.93 11.95 4.37 2.13 1.20 0.73 0.46 0.30 0.18 0.4 38.66 8.37 3.44 1.80 1.06 0.67 0.43 0.28 0.18 0.5 23.61 6.70 3.00 1.64 0.99 0.64 0.42 0.28 0.18 0.6 17.17 5.98 2.82 1.57 0.97 0.62 0.41 0.27 0.18 0.7 15.64 5.81 2.77 1.56 0.96 0.62 0.41 0.27 0.18 0.8 17.19 5.98 2.82 1.57 0.97 0.62 0.41 0.27 0.18 0.9 20.79 6.38 2.92 1.61 0.98 0.63 0.42 0.27 0.18 1.0 25.82 6.94 3.07 1.66 1.00 0.64 0.42 0.28 0.18 0.7 0.2 137.78 20.74 7.01 3.21 1.70 0.96 0.54 0.29 0.13 0.3 58.74 11.96 4.73 2.39 1.34 0.79 0.47 0.26 0.12 0.4 27.78 8.52 3.84 2.06 1.21 0.73 0.44 0.24 0.11 0.5 16.04 7.21 3.50 1.94 1.15 0.71 0.43 0.24 0.11 0.6 13.91 6.97 3.44 1.92 1.14 0.70 0.42 0.24 0.11 0.7 17.28 7.35 3.54 1.95 1.16 0.71 0.43 0.24 0.11 0.8 24.08 8.10 3.73 2.02 1.19 0.72 0.43 0.24 0.11 0.9 33.17 9.11 3.99 2.12 1.23 0.74 0.44 0.25 0.11 1.0 43.86 10.30 4.30 2.23 1.28 0.76 0.45 0.25 0.11 0.8 0.2 92.97 17.35 6.57 3.21 1.75 0.99 0.55 0.27 0.08 0.3 26.98 10.02 4.67 2.52 1.46 0.86 0.48 0.24 0.07 0.4 6.75 7.77 4.09 2.31 1.37 0.81 0.46 0.23 0.06 0.5 4.83 7.56 4.03 2.29 1.36 0.81 0.46 0.23 0.06 0.6 12.05 8.36 4.24 2.37 1.39 0.83 0.47 0.23 0.07 0.7 24.51 9.75 4.60 2.49 1.45 0.85 0.48 0.24 0.07 0.8 40.23 11.49 5.05 2.66 1.52 0.88 0.50 0.24 0.07 0.9 58.13 13.48 5.57 2.85 1.60 0.92 0.51 0.25 0.07 1.0 77.56 15.64 6.13 3.05 1.68 0.96 0.53 0.26 0.08 0.9 0.2 10.77 10.05 5.20 2.91 1.70 0.99 0.55 0.25 0.04 0.3 –21.27 6.49 4.28 2.57 1.56 0.93 0.52 0.24 0.04 0.4 –19.11 6.73 4.34 2.60 1.57 0.93 0.52 0.24 0.04 0.5 –3.28 8.49 4.80 2.76 1.64 0.97 0.54 0.24 0.04 0.6 19.39 11.01 5.45 3.00 1.74 1.01 0.56 0.25 0.04 0.7 45.97 13.96 6.21 3.27 1.86 1.07 0.58 0.27 0.05 0.8 74.99 17.18 7.05 3.58 1.98 1.13 0.61 0.28 0.05 0.9 105.64 20.59 7.93 3.89 2.12 1.19 0.64 0.29 0.06 1.0 137.43 24.12 8.85 4.23 2.26 1.26 0.67 0.31 0.06 1.0 0.2 –99.78 –0.17 3.15 2.40 1.58 0.98 0.56 0.25 0.02 0.3 –75.42 2.54 3.85 2.65 1.69 1.03 0.58 0.26 0.03 0.4 –38.31 6.66 4.92 3.04 1.86 1.11 0.62 0.28 0.03 0.5 3.90 11.35 6.14 3.48 2.04 1.20 0.66 0.29 0.04 0.6 48.66 16.32 7.43 3.94 2.24 1.29 0.70 0.31 0.04 0.7 94.88 21.46 8.76 4.43 2.45 1.38 0.75 0.33 0.05 0.8 142.01 26.70 10.12 4.92 2.66 1.48 0.79 0.35 0.06 0.9 189.74 32.00 11.49 5.41 2.87 1.58 0.84 0.37 0.07 1.0 237.90 37.35 12.88 5.92 3.08 1.68 0.88 0.39 0.07 ED5–3 Tee, Dc > 250 mm, Converging (Continued) 34.46 2001 ASHRAE Fundamentals Handbook (SI) ED5-6 Capped Wye, Branch with 45-Degree Elbow, Branch 90 Degrees to Main, Converging, r/Db = 1.5 Ab/Ac 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Cb 1.26 1.07 0.94 0.86 0.81 0.76 0.71 0.67 0.64 0.64 ED5-9 Symmetrical Wye, 60 Degree, Db1 ≥ Db2, Converging Ab1/Ac Ab2/Ac Cb1 Values Qb1/Qc 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.2 0.2 −11.95 −1.89 −0.09 0.41 0.62 0.74 0.80 0.80 0.79 0.3 −11.95 −1.89 −0.09 0.41 0.62 0.74 0.80 0.80 0.79 0.3 0.2 −45.45 −9.39 −2.44 −0.41 0.33 0.68 0.89 1.03 1.13 0.3 −16.88 −2.92 −0.09 0.59 0.86 1.02 1.09 1.10 1.08 0.4 0.2 −72.04 −14.00 −4.26 −1.24 −0.10 0.33 0.50 0.57 0.63 0.3 −52.95 −9.91 −2.86 −0.69 0.07 0.30 0.40 0.49 0.62 0.4 −28.86 −6.22 −2.15 −0.57 0.19 0.55 0.72 0.79 0.85 0.5 0.2 −126.04 −23.80 −7.44 −2.64 −0.85 −0.13 0.16 0.26 0.28 0.3 −91.07 −16.91 −5.16 −1.73 −0.46 0.04 0.23 0.29 0.28 0.4 −56.41 −10.07 −2.90 −0.82 −0.07 0.21 0.30 0.31 0.29 0.5 −30.58 −5.23 −1.06 0.00 0.32 0.43 0.47 0.47 0.41 0.6 0.2 −209.81 −39.31 −12.13 −4.35 −1.54 −0.40 0.06 0.22 0.23 0.3 −147.43 −27.69 −8.75 −3.20 −1.13 −0.29 0.05 0.17 0.18 0.4 −85.06 −16.07 −5.38 −2.04 −0.71 −0.17 0.04 0.12 0.13 0.5 −58.22 −11.03 −3.84 −1.49 −0.50 −0.09 0.07 0.11 0.12 0.6 −40.57 −7.86 −2.60 −0.99 −0.26 0.00 0.14 0.21 0.25 0.7 0.2 −291.57 −54.52 −17.03 −6.21 −2.27 −0.68 −0.04 0.19 0.21 0.3 −197.37 −38.02 −12.54 −4.92 −2.01 −0.76 −0.22 0.01 0.08 0.4 −102.97 −21.41 −8.05 −3.64 −1.75 −0.84 −0.40 −0.17 −0.05 0.5 −65.15 −14.75 −6.16 −3.07 −1.61 −0.85 −0.44 −0.22 −0.09 0.6 −48.24 −11.70 −4.97 −2.59 −1.40 −0.76 −0.37 −0.15 −0.03 0.7 −73.02 −16.68 −6.90 −3.29 −1.61 −0.80 −0.29 0.02 0.22 0.8 0.2 −373.33 −69.73 −21.93 −8.08 −3.00 −0.95 −0.13 0.15 0.20 0.3 −247.31 −48.35 −16.32 −6.65 −2.89 −1.24 −0.49 −0.15 −0.02 0.4 −120.88 −26.76 −10.71 −5.24 −2.78 −1.52 −0.84 −0.45 −0.24 Duct Design 34.47 Ab1/Ac Ab2/Ac Cb1 Values (Concluded) Qb1/Qc 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.5 −72.08 −18.46 −8.48 −4.65 −2.71 −1.61 −0.95 −0.55 −0.31 0.6 −55.91 −15.54 −7.35 −4.20 −2.54 −1.53 −0.89 −0.51 −0.30 0.7 −80.68 −20.52 −9.27 −4.90 −2.75 −1.56 −0.80 −0.34 −0.06 0.8 −105.46 −25.49 −11.19 −5.59 −2.96 −1.60 −0.72 −0.18 0.19 0.9 0.2 −479.24 −89.56 −28.39 −10.59 −4.04 −1.41 −0.36 0.01 0.09 0.3 −305.31 −61.27 −21.50 −9.28 −4.39 −2.16 −1.07 −0.54 −0.29 0.4 −131.17 −32.88 −14.60 −7.98 −4.74 −2.91 −1.79 −1.10 −0.68 0.5 −67.90 −22.76 −12.17 −7.53 −4.89 −3.19 −2.05 −1.30 −0.81 0.6 −68.95 −23.08 −12.11 −7.45 −4.84 −3.15 −2.01 −1.26 −0.79 0.7 −90.48 −27.35 −13.58 −7.95 −4.97 −3.16 −1.96 −1.17 −0.65 0.8 −112.02 −31.63 −15.05 −8.44 −5.11 −3.18 −1.90 −1.07 −0.51 0.9 −130.32 −35.19 −16.07 −8.70 −5.18 −3.19 −1.88 −1.08 −0.53 1.0 0.2 −585.16 −109.39 −34.85 −13.11 −5.09 −1.86 −0.59 −0.13 −0.01 0.3 −363.31 −74.20 −26.68 −11.91 −5.90 −3.08 −1.66 −0.94 −0.56 0.4 −141.46 −39.00 −18.50 −10.71 −6.71 −4.29 −2.74 −1.74 −1.12 0.5 −63.71 −27.06 −15.85 −10.41 −7.07 −4.77 −3.16 −2.05 −1.31 0.6 −81.99 −30.62 −16.87 −10.70 −7.13 −4.77 −3.13 −2.02 −1.28 0.7 −100.28 −34.19 −17.89 −11.00 −7.19 −4.76 −3.11 −1.99 −1.24 0.8 −118.58 −37.76 −18.91 −11.29 −7.26 −4.76 −3.09 −1.96 −1.20 0.9 −136.88 −41.32 −19.93 −11.55 −7.32 −4.77 −3.07 −1.98 −1.23 1.0 −155.18 −44.89 −20.95 −11.80 −7.39 −4.78 −3.05 −1.99 −1.25 Ab1/Ac Ab2/Ac Cb2 Values Qb2/Qc 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.2 0.2 −11.95 −1.89 −0.09 0.41 0.62 0.74 0.80 0.80 0.79 0.3 −11.95 −1.89 −0.09 0.41 0.62 0.74 0.80 0.80 0.79 0.3 0.2 −8.24 −1.18 0.05 0.42 0.61 0.73 0.78 0.77 0.76 0.3 −16.88 −2.92 −0.09 0.59 0.86 1.02 1.09 1.10 1.08 0.4 0.2 −6.95 −1.00 0.16 0.53 0.67 0.71 0.72 0.72 0.71 0.3 −16.21 −2.90 −0.44 0.40 0.79 0.98 1.05 1.06 1.05 0.4 −28.86 −6.22 −2.15 −0.57 0.19 0.55 0.72 0.79 0.85 0.5 0.2 −4.82 −0.01 0.56 0.71 0.82 0.89 0.92 0.90 0.89 0.3 −12.27 −1.17 0.44 0.88 1.11 1.25 1.29 1.25 1.23 0.4 −20.76 −2.93 −0.21 0.48 0.73 0.84 0.88 0.87 0.82 0.5 −30.58 −5.23 −1.06 0.00 0.32 0.43 0.47 0.47 0.41 0.6 0.2 −3.68 0.07 0.77 0.98 1.06 1.08 1.08 1.06 1.04 0.3 −9.06 −0.55 0.86 1.27 1.42 1.48 1.49 1.46 1.42 0.4 −17.62 −2.12 0.06 0.60 0.83 0.95 0.98 0.95 0.91 0.5 −28.00 −4.26 −0.99 −0.16 0.20 0.39 0.45 0.41 0.38 0.6 −40.57 −7.86 −2.60 −0.99 −0.26 0.00 0.14 0.21 0.25 0.7 0.2 −5.44 −0.40 0.55 0.86 0.98 1.02 1.04 1.03 1.02 0.3 −9.36 −0.77 0.73 1.20 1.39 1.47 1.49 1.47 1.44 0.4 −19.57 −3.09 −0.44 0.36 0.71 0.89 0.97 0.98 0.97 0.5 −31.88 −6.02 −1.90 −0.63 −0.05 0.26 0.40 0.44 0.46 0.6 −46.44 −9.82 −3.47 −1.41 −0.48 −0.04 0.21 0.36 0.45 0.7 −73.02 −16.68 −6.90 −3.29 −1.61 −0.80 −0.29 0.02 0.22 0.8 0.2 −7.21 −0.87 0.33 0.73 0.90 0.97 1.00 1.00 0.99 0.3 −9.67 −0.99 0.60 1.13 1.36 1.45 1.49 1.48 1.46 0.4 −21.53 −4.06 −0.93 0.11 0.59 0.83 0.96 1.01 1.03 0.5 −35.77 −7.77 −2.82 −1.09 −0.29 0.13 0.35 0.48 0.55 0.6 −52.32 −11.78 −4.34 −1.83 −0.70 −0.09 0.28 0.51 0.65 0.7 −78.89 −18.64 −7.76 −3.71 −1.83 −0.85 −0.22 0.16 0.42 0.8 −105.46 −25.49 −11.19 −5.59 −2.96 −1.60 −0.72 −0.18 0.19 0.9 0.2 −4.98 −0.34 0.54 0.85 0.97 1.03 1.04 1.03 1.01 0.3 −9.97 −1.21 0.48 1.06 1.32 1.44 1.49 1.49 1.48 0.4 −23.54 −4.98 −1.39 −0.12 0.47 0.78 0.95 1.04 1.09 0.5 −40.14 −9.57 −3.69 −1.56 −0.55 −0.01 0.31 0.51 0.63 0.6 −58.25 −14.28 −5.64 −2.53 −1.08 −0.30 0.18 0.49 0.70 0.7 −84.09 −21.02 −8.91 −4.38 −2.22 −1.04 −0.31 0.15 0.46 0.8 −109.92 −27.77 −12.18 −6.22 −3.35 −1.79 −0.81 −0.19 0.23 0.9 −130.32 −35.19 −16.07 −8.70 −5.18 −3.19 −1.88 −1.08 −0.53 1.0 0.2 −2.75 0.19 0.76 0.96 1.05 1.08 1.08 1.06 1.04 0.3 −10.28 −1.43 0.35 0.99 1.29 1.43 1.49 1.50 1.50 0.4 −25.56 −5.89 −1.86 −0.36 0.35 0.72 0.93 1.07 1.15 0.5 −44.52 −11.37 −4.56 −2.02 −0.81 −0.14 0.27 0.54 0.72 0.6 −64.19 −16.77 −6.94 −3.24 −1.47 −0.50 0.09 0.48 0.74 0.7 −89.28 −23.41 −10.05 −5.05 −2.61 −1.24 −0.40 0.14 0.50 0.8 −114.38 −30.04 −13.16 −6.86 −3.75 −1.97 −0.89 −0.20 0.27 0.9 −134.78 −37.47 −17.06 −9.33 −5.57 −3.38 −1.97 −1.09 −0.49 1.0 −155.18 −44.89 −20.95 −11.80 −7.39 −4.78 −3.05 −1.99 −1.25 ED5-9 Symmetrical Wye, 60 Degree, Db1 ≥ Db2, Converging (Continued) 34.48 2001 ASHRAE Fundamentals Handbook (SI) ED7-1 Centrifugal Fan Located in Plenum or Cabinet L/Do 0.30 0.40 0.50 0.75 Co 0.80 0.53 0.40 0.22 ED7-2 Fan Inlet, Centrifugal, SWSI, with 4 Gore Elbow r/Do Co Values L/Do 0.0 2.0 5.0 10.0 0.50 1.80 1.00 0.53 0.53 0.75 1.40 0.80 0.40 0.40 1.00 1.20 0.67 0.33 0.33 1.50 1.10 0.60 0.33 0.33 2.00 1.00 0.53 0.33 0.33 3.00 0.67 0.40 0.22 0.22 SD1-1 Bellmouth, Plenum to Round, Supply Air Systems r/Do 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.08 0.10 0.12 0.16 0.20 10.00 Co 0.50 0.44 0.36 0.31 0.26 0.22 0.20 0.15 0.12 0.09 0.06 0.03 0.03 SD1-2 Conical Bellmouth/Sudden Contraction, Plenum to Round, Supply Air Systems Ao/A1 L/Do Co Values θ 0 10 20 30 45 60 90 120 150 180 0.10 0.025 0.46 0.43 0.42 0.40 0.38 0.37 0.38 0.40 0.43 0.46 0.050 0.46 0.42 0.38 0.33 0.30 0.28 0.31 0.36 0.41 0.46 0.075 0.46 0.39 0.32 0.28 0.23 0.21 0.26 0.32 0.39 0.46 0.100 0.46 0.36 0.30 0.23 0.19 0.17 0.23 0.30 0.38 0.46 0.150 0.46 0.34 0.25 0.18 0.15 0.14 0.21 0.29 0.37 0.46 0.300 0.46 0.31 0.22 0.16 0.13 0.13 0.20 0.28 0.37 0.46 0.600 0.46 0.25 0.17 0.12 0.10 0.11 0.19 0.27 0.36 0.46 0.20 0.025 0.42 0.40 0.38 0.36 0.34 0.34 0.35 0.37 0.39 0.42 0.050 0.42 0.38 0.35 0.30 0.27 0.25 0.29 0.33 0.37 0.42 0.075 0.42 0.36 0.30 0.25 0.21 0.19 0.24 0.30 0.36 0.42 0.100 0.42 0.33 0.27 0.21 0.18 0.15 0.21 0.27 0.35 0.42 0.150 0.42 0.31 0.23 0.17 0.13 0.13 0.19 0.26 0.34 0.42 0.300 0.42 0.28 0.20 0.15 0.12 0.12 0.18 0.26 0.34 0.42 0.600 0.42 0.23 0.15 0.11 0.10 0.10 0.17 0.25 0.33 0.42 Duct Design 34.49 SD1-2 Conical Bellmouth/Sudden Contraction, Plenum to Round, Supply Air Systems (Continued) Ao/A1 L/Do Co Values (Concluded) θ 0 10 20 30 45 60 90 120 150 180 0.40 0.025 0.34 0.32 0.31 0.29 0.28 0.27 0.28 0.30 0.32 0.34 0.050 0.34 0.31 0.28 0.25 0.22 0.20 0.23 0.26 0.30 0.34 0.075 0.34 0.29 0.24 0.20 0.17 0.16 0.19 0.24 0.29 0.34 0.100 0.34 0.27 0.22 0.17 0.14 0.12 0.17 0.22 0.28 0.34 0.150 0.34 0.25 0.18 0.14 0.11 0.10 0.15 0.21 0.27 0.34 0.300 0.34 0.23 0.16 0.12 0.10 0.10 0.15 0.21 0.27 0.34 0.600 0.34 0.18 0.12 0.09 0.08 0.08 0.14 0.20 0.27 0.34 0.60 0.025 0.25 0.24 0.23 0.22 0.20 0.20 0.21 0.22 0.23 0.25 0.050 0.25 0.23 0.21 0.18 0.16 0.15 0.17 0.19 0.22 0.25 0.075 0.25 0.21 0.18 0.15 0.13 0.12 0.14 0.18 0.21 0.25 0.100 0.25 0.20 0.16 0.13 0.11 0.09 0.12 0.16 0.21 0.25 0.150 0.25 0.19 0.14 0.10 0.08 0.08 0.11 0.16 0.20 0.25 0.300 0.25 0.17 0.12 0.09 0.07 0.07 0.11 0.15 0.20 0.25 0.600 0.25 0.14 0.09 0.07 0.06 0.06 0.10 0.15 0.20 0.25 0.80 0.025 0.15 0.14 0.13 0.13 0.12 0.12 0.12 0.13 0.14 0.15 0.050 0.15 0.13 0.12 0.11 0.10 0.09 0.10 0.12 0.13 0.15 0.075 0.15 0.13 0.10 0.09 0.08 0.07 0.08 0.10 0.13 0.15 0.100 0.15 0.12 0.10 0.07 0.06 0.05 0.07 0.10 0.12 0.15 0.150 0.15 0.11 0.08 0.06 0.05 0.04 0.07 0.09 0.12 0.15 0.300 0.15 0.10 0.07 0.05 0.04 0.04 0.07 0.09 0.12 0.15 0.600 0.15 0.08 0.05 0.04 0.03 0.04 0.06 0.09 0.12 0.15 0.90 0.025 0.09 0.08 0.08 0.08 0.07 0.07 0.07 0.08 0.08 0.09 0.050 0.09 0.08 0.07 0.06 0.06 0.05 0.06 0.07 0.08 0.09 0.075 0.09 0.07 0.06 0.05 0.04 0.04 0.05 0.06 0.08 0.09 0.100 0.09 0.07 0.06 0.04 0.04 0.03 0.04 0.06 0.07 0.09 0.150 0.09 0.07 0.05 0.04 0.03 0.03 0.04 0.06 0.07 0.09 0.300 0.09 0.06 0.04 0.03 0.03 0.02 0.04 0.05 0.07 0.09 0.600 0.09 0.05 0.03 0.02 0.02 0.02 0.04 0.05 0.07 0.09 SD2-6 Stackhead De/D 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Co 129 41.02 16.80 8.10 4.37 2.56 1.60 1.00 34.50 2001 ASHRAE Fundamentals Handbook (SI) SD4-1 Transition, Round to Round, Supply Air Systems Ao/A1 Co Values θ 10 15 20 30 45 60 90 120 150 180 0.10 0.05 0.05 0.05 0.05 0.07 0.08 0.19 0.29 0.37 0.43 0.17 0.05 0.04 0.04 0.04 0.06 0.07 0.18 0.28 0.36 0.42 0.25 0.05 0.04 0.04 0.04 0.06 0.07 0.17 0.27 0.35 0.41 0.50 0.05 0.05 0.05 0.05 0.06 0.06 0.12 0.18 0.24 0.26 1.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 2.00 0.44 0.52 0.76 1.28 1.32 1.32 1.28 1.24 1.20 1.20 4.00 2.56 3.52 4.80 7.36 9.76 10.88 10.24 10.08 9.92 9.92 10.00 21.00 28.00 38.00 59.00 76.00 80.00 83.00 84.00 83.00 83.00 16.00 53.76 74.24 97.28 153.60 215.04 225.28 225.28 225.28 225.28 225.28 SD4-2 Transition, Rectangular to Round, Supply Air Systems Ao/A1 Co Values θ 10 15 20 30 45 60 90 120 150 180 0.10 0.05 0.05 0.05 0.05 0.07 0.08 0.19 0.29 0.37 0.43 0.17 0.05 0.05 0.04 0.04 0.06 0.07 0.18 0.28 0.36 0.42 0.25 0.06 0.05 0.05 0.04 0.06 0.07 0.17 0.27 0.35 0.41 0.50 0.06 0.07 0.07 0.05 0.06 0.06 0.12 0.18 0.24 0.26 1.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 2.00 0.60 0.84 1.00 1.20 1.32 1.32 1.32 1.28 1.24 1.20 4.00 4.00 5.76 7.20 8.32 9.28 9.92 10.24 10.24 10.24 10.24 10.00 30.00 50.00 53.00 64.00 75.00 84.00 89.00 91.00 91.00 88.00 16.00 76.80 138.24 135.68 166.40 197.12 225.28 243.20 250.88 250.88 238.08 SD5-1 Wye, 45 Degree, Diverging Ab/Ac Cb Values Qb/Qc 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.1 0.38 0.39 0.48 0.2 2.25 0.38 0.31 0.39 0.46 0.48 0.45 0.3 6.29 1.02 0.38 0.30 0.33 0.39 0.44 0.48 0.48 0.4 12.41 2.25 0.74 0.38 0.30 0.31 0.35 0.39 0.43 0.5 20.58 4.01 1.37 0.62 0.38 0.30 0.30 0.32 0.36 0.6 30.78 6.29 2.25 1.02 0.56 0.38 0.31 0.30 0.31 0.7 43.02 9.10 3.36 1.57 0.85 0.52 0.38 0.31 0.30 0.8 57.29 12.41 4.71 2.25 1.22 0.74 0.50 0.38 0.32 0.9 73.59 16.24 6.29 3.06 1.69 1.02 0.67 0.48 0.38 As/Ac Cs Values Qs/Qc 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.1 0.13 0.16 0.2 0.20 0.13 0.15 0.16 0.28 0.3 0.90 0.13 0.13 0.14 0.15 0.16 0.20 0.4 2.88 0.20 0.14 0.13 0.14 0.15 0.15 0.16 0.34 0.5 6.25 0.37 0.17 0.14 0.13 0.14 0.14 0.15 0.15 0.6 11.88 0.90 0.20 0.13 0.14 0.13 0.14 0.14 0.15 0.7 18.62 1.71 0.33 0.18 0.16 0.14 0.13 0.15 0.14 0.8 26.88 2.88 0.50 0.20 0.15 0.14 0.13 0.13 0.14 0.9 36.45 4.46 0.90 0.30 0.19 0.16 0.15 0.14 0.13 Duct Design 34.51 SD5-9 Tee, Diverging Ab/Ac Cb Values Qb/Qc 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.1 1.20 0.62 0.80 1.28 1.99 2.92 4.07 5.44 7.02 0.2 4.10 1.20 0.72 0.62 0.66 0.80 1.01 1.28 1.60 0.3 8.99 2.40 1.20 0.81 0.66 0.62 0.64 0.70 0.80 0.4 15.89 4.10 1.94 1.20 0.88 0.72 0.64 0.62 0.63 0.5 24.80 6.29 2.91 1.74 1.20 0.92 0.77 0.68 0.63 0.6 35.73 8.99 4.10 2.40 1.62 1.20 0.96 0.81 0.72 0.7 48.67 12.19 5.51 3.19 2.12 1.55 1.20 0.99 0.85 0.8 63.63 15.89 7.14 4.10 2.70 1.94 1.49 1.20 1.01 0.9 80.60 20.10 8.99 5.13 3.36 2.40 1.83 1.46 1.20 As/Ac Cs Values Qs/Qc 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.1 0.13 0.16 0.2 0.20 0.13 0.15 0.16 0.28 0.3 0.90 0.13 0.13 0.14 0.15 0.16 0.20 0.4 2.88 0.20 0.14 0.13 0.14 0.15 0.15 0.16 0.34 0.5 6.25 0.37 0.17 0.14 0.13 0.14 0.14 0.15 0.15 0.6 11.88 0.90 0.20 0.13 0.14 0.13 0.14 0.14 0.15 0.7 18.62 1.71 0.33 0.18 0.16 0.14 0.13 0.15 0.14 0.8 26.88 2.88 0.50 0.20 0.15 0.14 0.13 0.13 0.14 0.9 36.45 4.46 0.90 0.30 0.19 0.16 0.15 0.14 0.13 SD5-10 Tee, Conical Branch Tapered into Body, Diverging Ab/Ac Cb Values Qb/Qc 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.1 0.65 0.24 0.2 2.98 0.65 0.33 0.24 0.18 0.3 7.36 1.56 0.65 0.39 0.29 0.24 0.20 0.4 13.78 2.98 1.20 0.65 0.43 0.33 0.27 0.24 0.21 0.5 22.24 4.92 1.98 1.04 0.65 0.47 0.36 0.30 0.26 0.6 32.73 7.36 2.98 1.56 0.96 0.65 0.49 0.39 0.33 0.7 45.26 10.32 4.21 2.21 1.34 0.90 0.65 0.51 0.42 0.8 59.82 13.78 5.67 2.98 1.80 1.20 0.86 0.65 0.52 0.9 76.41 17.75 7.36 3.88 2.35 1.56 1.11 0.83 0.65 As/Ac Cs Values Qs/Qc 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.1 0.13 0.16 0.2 0.20 0.13 0.15 0.16 0.28 0.3 0.90 0.13 0.13 0.14 0.15 0.16 0.20 0.4 2.88 0.20 0.14 0.13 0.14 0.15 0.15 0.16 0.34 0.5 6.25 0.37 0.17 0.14 0.13 0.14 0.14 0.15 0.15 0.6 11.88 0.90 0.20 0.13 0.14 0.13 0.14 0.14 0.15 0.7 18.62 1.71 0.33 0.18 0.16 0.14 0.13 0.15 0.14 0.8 26.88 2.88 0.50 0.20 0.15 0.14 0.13 0.13 0.14 0.9 36.45 4.46 0.90 0.30 0.19 0.16 0.15 0.14 0.13 34.52 2001 ASHRAE Fundamentals Handbook (SI) SD5-24 Cross, Diverging As/Ac Ab1/Ac Cb1 Values Qb1/Qc 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.20 0.1 2.07 2.08 1.62 1.30 1.08 0.93 0.81 0.72 0.64 0.2 2.07 2.31 2.08 1.83 1.62 1.44 1.30 1.18 0.3 2.07 2.34 2.24 2.08 1.91 1.76 1.62 0.4 0.90 2.07 2.32 2.31 2.21 2.08 1.95 0.5 1.28 2.07 2.30 2.33 2.27 2.18 0.6 1.48 2.07 2.29 2.34 2.31 0.7 0.55 1.60 2.07 2.27 2.33 0.8 0.90 1.68 2.07 2.25 0.9 1.12 1.74 2.07 0.35 0.1 3.25 3.11 2.69 2.32 2.03 1.80 1.61 1.46 0.2 2.44 3.25 3.28 3.11 2.90 2.69 2.49 0.3 1.69 2.88 3.25 3.31 3.23 3.11 0.4 1.12 2.44 3.02 3.25 3.31 0.5 0.69 2.04 2.73 3.09 0.6 0.37 1.69 2.44 0.7 0.11 1.38 0.8 0.9 0.55 0.1 1.50 1.56 1.38 1.20 1.06 0.94 0.84 0.77 0.2 0.89 1.50 1.60 1.56 1.47 1.38 1.28 0.3 0.38 1.20 1.50 1.59 1.59 1.56 0.4 0.00 0.89 1.31 1.50 1.58 0.5 0.61 1.09 1.36 0.6 0.38 0.89 0.7 0.17 0.8 0.9 0.80 0.1 1.20 0.62 0.80 1.28 1.99 2.92 4.07 5.44 7.02 0.2 4.10 1.20 0.72 0.62 0.66 0.80 1.01 1.28 1.60 0.3 8.99 2.40 1.20 0.81 0.66 0.62 0.64 0.70 0.80 0.4 15.89 4.10 1.94 1.20 0.88 0.72 0.64 0.62 0.63 0.5 24.80 6.29 2.91 1.74 1.20 0.92 0.77 0.68 0.63 0.6 35.73 8.99 4.10 2.40 1.62 1.20 0.96 0.81 0.72 0.7 48.67 12.19 5.51 3.19 2.12 1.55 1.20 0.99 0.85 0.8 63.63 15.89 7.14 4.10 2.70 1.94 1.49 1.20 1.01 0.9 80.60 20.10 8.99 5.13 3.36 2.40 1.83 1.46 1.20 1.00 0.1 1.20 0.62 0.80 1.28 1.99 2.92 4.07 5.44 7.02 0.2 4.10 1.20 0.72 0.62 0.66 0.80 1.01 1.28 1.60 0.3 8.99 2.40 1.20 0.81 0.66 0.62 0.64 0.70 0.80 0.4 15.89 4.10 1.94 1.20 0.88 0.72 0.64 0.62 0.63 0.5 24.80 6.29 2.91 1.74 1.20 0.92 0.77 0.68 0.63 0.6 35.73 8.99 4.10 2.40 1.62 1.20 0.96 0.81 0.72 0.7 48.67 12.19 5.51 3.19 2.12 1.55 1.20 0.99 0.85 0.8 63.63 15.89 7.14 4.10 2.70 1.94 1.49 1.20 1.01 0.9 80.60 20.10 8.99 5.13 3.36 2.40 1.83 1.46 1.20 As/Ac Cs Values Qs/Qc 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.1 0.13 0.16 0.2 0.20 0.13 0.15 0.16 0.28 0.3 0.90 0.13 0.13 0.14 0.15 0.16 0.20 0.4 2.88 0.20 0.14 0.13 0.14 0.15 0.15 0.16 0.34 0.5 6.25 0.37 0.17 0.14 0.13 0.14 0.14 0.15 0.15 0.6 11.88 0.90 0.20 0.13 0.14 0.13 0.14 0.14 0.15 0.7 18.62 1.71 0.33 0.18 0.16 0.14 0.13 0.15 0.14 0.8 26.88 2.88 0.50 0.20 0.15 0.14 0.13 0.13 0.14 0.9 36.45 4.46 0.90 0.30 0.19 0.16 0.15 0.14 0.13 For the other branch, subscripts 1 and 2 change places.
Duct Design 34.53 SD5-25 Cross, Conical Branches Tapered into Body, Diverging As/Ac Ab1/Ac Cb1 Values Qb1/Qc 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.20 0.1 2.07 2.08 1.62 1.30 1.08 0.93 0.81 0.72 0.64 0.2 2.07 2.31 2.08 1.83 1.62 1.44 1.30 1.18 0.3 2.07 2.34 2.24 2.08 1.91 1.76 1.62 0.4 0.90 2.07 2.32 2.31 2.21 2.08 1.95 0.5 1.28 2.07 2.30 2.33 2.27 2.18 0.6 1.48 2.07 2.29 2.34 2.31 0.7 0.55 1.60 2.07 2.27 2.33 0.8 0.90 1.68 2.07 2.25 0.9 1.12 1.74 2.07 0.35 0.1 3.25 3.11 2.69 2.32 2.03 1.80 1.61 1.46 0.2 2.44 3.25 3.28 3.11 2.90 2.69 2.49 0.3 1.69 2.88 3.25 3.31 3.23 3.11 0.4 1.12 2.44 3.02 3.25 3.31 0.5 0.69 2.04 2.73 3.09 0.6 0.37 1.69 2.44 0.7 0.11 1.38 0.8 0.9 0.55 0.1 1.50 1.56 1.38 1.20 1.06 0.94 0.84 0.77 0.2 0.89 1.50 1.60 1.56 1.47 1.38 1.28 0.3 0.38 1.20 1.50 1.59 1.59 1.56 0.4 0.00 0.89 1.31 1.50 1.58 0.5 0.61 1.09 1.36 0.6 0.38 0.89 0.7 0.17 0.8 0.9 0.80 0.1 0.65 0.24 0.2 2.98 0.65 0.33 0.24 0.18 0.3 7.36 1.56 0.65 0.39 0.29 0.24 0.20 0.4 13.78 2.98 1.20 0.65 0.43 0.33 0.27 0.24 0.21 0.5 22.24 4.92 1.98 1.04 0.65 0.47 0.36 0.30 0.26 0.6 32.73 7.36 2.98 1.56 0.96 0.65 0.49 0.39 0.33 0.7 45.26 10.32 4.21 2.21 1.34 0.90 0.65 0.51 0.42 0.8 59.82 13.78 5.67 2.98 1.80 1.20 0.86 0.65 0.52 0.9 76.41 17.75 7.36 3.88 2.35 1.56 1.11 0.83 0.65 1.00 0.1 0.65 0.24 0.2 2.98 0.65 0.33 0.24 0.18 0.3 7.36 1.56 0.65 0.39 0.29 0.24 0.20 0.4 13.78 2.98 1.20 0.65 0.43 0.33 0.27 0.24 0.21 0.5 22.24 4.92 1.98 1.04 0.65 0.47 0.36 0.30 0.26 0.6 32.73 7.36 2.98 1.56 0.96 0.65 0.49 0.39 0.33 0.7 45.26 10.32 4.21 2.21 1.34 0.90 0.65 0.51 0.42 0.8 59.82 13.78 5.67 2.98 1.80 1.20 0.86 0.65 0.52 0.9 76.41 17.75 7.36 3.88 2.35 1.56 1.11 0.83 0.65 As/Ac Cs Values Qs/Qc 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.1 0.13 0.16 0.2 0.20 0.13 0.15 0.16 0.28 0.3 0.90 0.13 0.13 0.14 0.15 0.16 0.20 0.4 2.88 0.20 0.14 0.13 0.14 0.15 0.15 0.16 0.34 0.5 6.25 0.37 0.17 0.14 0.13 0.14 0.14 0.15 0.15 0.6 11.88 0.90 0.20 0.13 0.14 0.13 0.14 0.14 0.15 0.7 18.62 1.71 0.33 0.18 0.16 0.14 0.13 0.15 0.14 0.8 26.88 2.88 0.50 0.20 0.15 0.14 0.13 0.13 0.14 0.9 36.45 4.46 0.90 0.30 0.19 0.16 0.15 0.14 0.13 For the other branch, subscripts 1 and 2 change places 34.54 2001 ASHRAE Fundamentals Handbook (SI) RECTANGULAR FITTINGS CR3-1 Elbow, Smooth Radius, Without Vanes Cp Values H/W r/W 0.25 0.50 0.75 1.00 1.50 2.00 3.00 4.00 5.00 6.00 8.00 0.50 1.53 1.38 1.29 1.18 1.06 1.00 1.00 1.06 1.12 1.16 1.18 0.75 0.57 0.52 0.48 0.44 0.40 0.39 0.39 0.40 0.42 0.43 0.44 1.00 0.27 0.25 0.23 0.21 0.19 0.18 0.18 0.19 0.20 0.21 0.21 1.50 0.22 0.20 0.19 0.17 0.15 0.14 0.14 0.15 0.16 0.17 0.17 2.00 0.20 0.18 0.16 0.15 0.14 0.13 0.13 0.14 0.14 0.15 0.15 Angle Factor K θ 0 20 30 45 60 75 90 110 130 150 180 K 0.00 0.31 0.45 0.60 0.78 0.90 1.00 1.13 1.20 1.28 1.40 CR3-3 Elbow, Smooth Radius, One Splitter Vane r /W Cp Values H/W 0.25 0.50 1.00 1.50 2.00 3.00 4.00 5.00 6.00 7.00 8.00 0.55 0.52 0.40 0.43 0.49 0.55 0.66 0.75 0.84 0.93 1.01 1.09 0.60 0.36 0.27 0.25 0.28 0.30 0.35 0.39 0.42 0.46 0.49 0.52 0.65 0.28 0.21 0.18 0.19 0.20 0.22 0.25 0.26 0.28 0.30 0.32 0.70 0.22 0.16 0.14 0.14 0.15 0.16 0.17 0.18 0.19 0.20 0.21 0.75 0.18 0.13 0.11 0.11 0.11 0.12 0.13 0.14 0.14 0.15 0.15 0.80 0.15 0.11 0.09 0.09 0.09 0.09 0.10 0.10 0.11 0.11 0.12 0.85 0.13 0.09 0.08 0.07 0.07 0.08 0.08 0.08 0.08 0.09 0.09 0.90 0.11 0.08 0.07 0.06 0.06 0.06 0.06 0.07 0.07 0.07 0.07 0.95 0.10 0.07 0.06 0.05 0.05 0.05 0.05 0.05 0.06 0.06 0.06 1.00 0.09 0.06 0.05 0.05 0.04 0.04 0.04 0.05 0.05 0.05 0.05 Angle Factor K θ 0 30 45 60 90 K 0.00 0.45 0.60 0.78 1.00 Curve Ratio CR r /W 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00 CR 0.218 0.302 0.361 0.408 0.447 0.480 0.509 0.535 0.557 0.577 Throat Radius/Width Ratio (R /W) r /W 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00 R/W 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 CR3-6 Elbow, Mitered θ Co Values H/W 0.25 0.50 0.75 1.00 1.50 2.00 3.00 4.00 5.00 6.00 8.00 20 0.08 0.08 0.08 0.07 0.07 0.07 0.06 0.06 0.05 0.05 0.05 30 0.18 0.17 0.17 0.16 0.15 0.15 0.13 0.13 0.12 0.12 0.11 45 0.38 0.37 0.36 0.34 0.33 0.31 0.28 0.27 0.26 0.25 0.24 60 0.60 0.59 0.57 0.55 0.52 0.49 0.46 0.43 0.41 0.39 0.38 75 0.89 0.87 0.84 0.81 0.77 0.73 0.67 0.63 0.61 0.58 0.57 90 1.30 1.27 1.23 1.18 1.13 1.07 0.98 0.92 0.89 0.85 0.83 Duct Design 34.55 CR3-9 Elbow, Mitered, 90 Degree, Single-Thickness Vanes (Design 1) Co = 0.11 CR3-10 Elbow, Mitered, 90 Degree, Single-Thickness Vanes (Design 2) Co = 0.12 CR3-12 Elbow, Mitered, 90 Degree, Single-Thickness Vanes (Design 4) Co = 0.33 CR3-14 Elbow, Mitered, 90 Degree, Double-Thickness Vanes (Design 1) Co = 0.38 r = 50 mm s = 40 mm 34.56 2001 ASHRAE Fundamentals Handbook (SI) CR3-15 Elbow, Mitered, 90 Degree, Double-Thickness Vanes (Design 2) Co = 0.25 CR3-16 Elbow, Mitered, 90 Degree, Double-Thickness Vanes (Design 3) Co = 0.41 CR3-17 Elbow, Z-Shaped H/W Cp Values L/W 0.0 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 4.0 8.0 10.0 100.0 0.25 0.00 0.68 0.99 1.77 2.89 3.97 4.41 4.60 4.64 4.60 3.39 3.03 2.70 2.53 0.50 0.00 0.66 0.96 1.72 2.81 3.86 4.29 4.47 4.52 4.47 3.30 2.94 2.62 2.46 0.75 0.00 0.64 0.94 1.67 2.74 3.75 4.17 4.35 4.39 4.35 3.20 2.86 2.55 2.39 1.00 0.00 0.62 0.90 1.61 2.63 3.61 4.01 4.18 4.22 4.18 3.08 2.75 2.45 2.30 1.50 0.00 0.59 0.86 1.53 2.50 3.43 3.81 3.97 4.01 3.97 2.93 2.61 2.33 2.19 2.00 0.00 0.56 0.81 1.45 2.37 3.25 3.61 3.76 3.80 3.76 2.77 2.48 2.21 2.07 3.00 0.00 0.51 0.75 1.34 2.18 3.00 3.33 3.47 3.50 3.47 2.56 2.28 2.03 1.91 4.00 0.00 0.48 0.70 1.26 2.05 2.82 3.13 3.26 3.29 3.26 2.40 2.15 1.91 1.79 6.00 0.00 0.45 0.65 1.16 1.89 2.60 2.89 3.01 3.04 3.01 2.22 1.98 1.76 1.66 8.00 0.00 0.43 0.63 1.13 1.84 2.53 2.81 2.93 2.95 2.93 2.16 1.93 1.72 1.61 Reynolds Number Correction Factor Kr Re/1000 10 20 30 40 60 80 100 140 500 Kr 1.40 1.26 1.19 1.14 1.09 1.06 1.04 1.00 1.00 r = 50 mm s = 60 mm r = 110 mm s = 80 mm Duct Design 34.57 ’ CR6-1 Screen (Only) A1/Ao Co Values n 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.90 1.00 0.2 155.00 102.50 75.00 55.00 41.25 31.50 24.25 18.75 14.50 11.00 8.00 3.50 0.00 0.3 68.89 45.56 33.33 24.44 18.33 14.00 10.78 8.33 6.44 4.89 3.56 1.56 0.00 0.4 38.75 25.63 18.75 13.75 10.31 7.88 6.06 4.69 3.63 2.75 2.00 0.88 0.00 0.5 24.80 16.40 12.00 8.80 6.60 5.04 3.88 3.00 2.32 1.76 1.28 0.56 0.00 0.6 17.22 11.39 8.33 6.11 4.58 3.50 2.69 2.08 1.61 1.22 0.89 0.39 0.00 0.7 12.65 8.37 6.12 4.49 3.37 2.57 1.98 1.53 1.18 0.90 0.65 0.29 0.00 0.8 9.69 6.40 4.69 3.44 2.58 1.97 1.52 1.17 0.91 0.69 0.50 0.22 0.00 0.9 7.65 5.06 3.70 2.72 2.04 1.56 1.20 0.93 0.72 0.54 0.40 0.17 0.00 1.0 6.20 4.10 3.00 2.20 1.65 1.26 0.97 0.75 0.58 0.44 0.32 0.14 0.00 1.2 4.31 2.85 2.08 1.53 1.15 0.88 0.67 0.36 0.40 0.31 0.22 0.10 0.00 1.4 3.16 2.09 1.53 1.12 0.84 0.64 0.49 0.38 0.30 0.22 0.16 0.07 0.00 1.6 2.42 1.60 1.17 0.86 0.64 0.49 0.38 0.29 0.23 0.17 0.13 0.05 0.00 1.8 1.91 1.27 0.93 0.68 0.51 0.39 0.30 0.23 0.18 0.14 0.10 0.04 0.00 2.0 1.55 1.03 0.75 0.55 0.41 0.32 0.24 0.19 0.15 0.11 0.08 0.04 0.00 2.5 0.99 0.66 0.48 0.35 0.26 0.20 0.16 0.12 0.09 0.07 0.05 0.02 0.00 3.0 0.69 0.46 0.33 0.24 0.18 0.14 0.11 0.08 0.06 0.05 0.04 0.02 0.00 4.0 0.39 0.26 0.19 0.14 0.10 0.08 0.06 0.05 0.04 0.03 0.02 0.01 0.00 6.0 0.17 0.11 0.08 0.06 0.05 0.04 0.03 0.02 0.02 0.01 0.01 0.00 0.00 CR6-4 Obstruction, Smooth Cylinder in Rectangular Duct y/H Re/1000 Co Values y/H Re/1000 Co Values Sm /Ao Sm /Ao 0.00 0.05 0.10 0.15 0.20 0.00 0.05 0.10 0.15 0.20 0.00 0.1 0.00 0.10 0.21 0.35 0.47 400 0.00 0.04 0.10 0.16 0.21 0.5 0.00 0.08 0.17 0.28 0.38 500 0.00 0.03 0.07 0.12 0.16 200 0.00 0.08 0.17 0.28 0.38 600 0.00 0.02 0.04 0.06 0.09 300 0.00 0.07 0.16 0.26 0.35 1000 0.00 0.02 0.04 0.07 0.09 400 0.00 0.05 0.11 0.19 0.25 500 0.00 0.04 0.09 0.14 0.19 0.25 0.1 0.00 0.08 0.17 0.28 0.38 600 0.00 0.02 0.05 0.07 0.10 0.5 0.00 0.06 0.14 0.22 0.30 1000 0.00 0.02 0.05 0.08 0.11 200 0.00 0.06 0.14 0.22 0.30 300 0.00 0.06 0.12 0.20 0.28 0.05 0.1 0.00 0.10 0.21 0.34 0.46 400 0.00 0.04 0.09 0.15 0.20 0.5 0.00 0.08 0.17 0.27 0.37 500 0.00 0.03 0.07 0.11 0.15 200 0.00 0.08 0.17 0.27 0.37 600 0.00 0.02 0.04 0.06 0.08 300 0.00 0.07 0.15 0.25 0.34 1000 0.00 0.02 0.04 0.06 0.09 400 0.00 0.05 0.11 0.18 0.24 500 0.00 0.04 0.08 0.13 0.18 0.30 0.1 0.00 0.07 0.16 0.26 0.35 600 0.00 0.02 0.04 0.07 0.10 0.5 0.00 0.06 0.13 0.21 0.28 1000 0.00 0.02 0.05 0.08 0.11 200 0.00 0.06 0.13 0.21 0.28 300 0.00 0.05 0.12 0.19 0.26 0.10 0.1 0.00 0.09 0.20 0.32 0.44 400 0.00 0.04 0.08 0.14 0.19 0.5 0.00 0.07 0.16 0.26 0.35 500 0.00 0.03 0.06 0.10 0.14 200 0.00 0.07 0.16 0.26 0.35 600 0.00 0.02 0.03 0.05 0.07 300 0.00 0.07 0.15 0.24 0.32 1000 0.00 0.02 0.04 0.06 0.08 400 0.00 0.05 0.11 0.17 0.23 500 0.00 0.04 0.08 0.13 0.18 0.35 0.1 0.00 0.07 0.14 0.23 0.32 600 0.00 0.02 0.04 0.07 0.09 0.5 0.00 0.05 0.11 0.19 0.25 1000 0.00 0.02 0.05 0.08 0.10 200 0.00 0.05 0.11 0.19 0.25 300 0.00 0.05 0.11 0.17 0.23 0.15 0.1 0.00 0.09 0.19 0.31 0.42 400 0.00 0.04 0.08 0.12 0.17 0.5 0.00 0.07 0.15 0.25 0.34 500 0.00 0.03 0.06 0.09 0.13 200 0.00 0.07 0.15 0.25 0.34 600 0.00 0.01 0.03 0.05 0.07 300 0.00 0.06 0.14 0.23 0.31 1000 0.00 0.02 0.03 0.05 0.07 400 0.00 0.05 0.10 0.17 0.22 500 0.00 0.04 0.08 0.12 0.17 0.40 0.1 0.00 0.06 0.13 0.20 0.28 600 0.00 0.02 0.04 0.07 0.09 0.5 0.00 0.05 0.10 0.16 0.22 1000 0.00 0.02 0.04 0.07 0.10 200 0.00 0.05 0.10 0.16 0.22 300 0.00 0.04 0.09 0.15 0.20 0.20 0.1 0.00 0.08 0.18 0.29 0.40 400 0.00 0.03 0.07 0.11 0.15 0.5 0.00 0.07 0.14 0.24 0.32 500 0.00 0.02 0.05 0.08 0.11 200 0.00 0.07 0.14 0.24 0.32 600 0.00 0.01 0.03 0.04 0.06 300 0.00 0.06 0.13 0.22 0.29 1000 0.00 0.01 0.03 0.05 0.06 34.58 2001 ASHRAE Fundamentals Handbook (SI) CR9-1 Damper, Butterfly H/W Co Values θ 0 10 20 30 40 50 60 65 70 90 0.12 0.04 0.30 1.10 3.00 8.00 23.00 60.00 100.00 190.00 99999 0.25 0.08 0.33 1.18 3.30 9.00 26.00 70.00 128.00 210.00 99999 1.00 0.08 0.33 1.18 3.30 9.00 26.00 70.00 128.00 210.00 99999 2.00 0.13 0.35 1.25 3.60 10.00 29.00 80.00 155.00 230.00 99999 CR9-3 Damper, Parallel Blades L/R Co Values θ 0 10 20 30 40 50 60 70 80 0.3 0.52 0.79 1.49 2.20 4.95 8.73 14.15 32.11 122.06 0.4 0.52 0.84 1.56 2.25 5.03 9.00 16.00 37.73 156.58 0.5 0.52 0.88 1.62 2.35 5.11 9.52 18.88 44.79 187.85 0.6 0.52 0.92 1.66 2.45 5.20 9.77 21.75 53.78 288.89 0.8 0.52 0.96 1.69 2.55 5.30 10.03 22.80 65.46 295.22 1.0 0.52 1.00 1.76 2.66 5.40 10.53 23.84 73.23 361.00 1.5 0.52 1.08 1.83 2.78 5.44 11.21 27.56 97.41 495.31 CR9-4 Damper, Opposed Blades L/R Co Values θ 0 10 20 30 40 50 60 70 80 0.3 0.52 0.79 1.91 3.77 8.55 19.46 70.12 295.21 807.23 0.4 0.52 0.85 2.07 4.61 10.42 26.73 92.90 346.25 926.34 0.5 0.52 0.93 2.25 5.44 12.29 33.99 118.91 393.36 1045.44 0.6 0.52 1.00 2.46 5.99 14.15 41.26 143.69 440.25 1163.09 0.8 0.52 1.08 2.66 6.96 18.18 56.47 193.92 520.27 1324.85 1.0 0.52 1.17 2.91 7.31 20.25 71.68 245.45 576.00 1521.00 1.5 0.52 1.38 3.16 9.51 27.56 107.41 361.00 717.05 1804.40 CR9-6 Fire Damper, Curtain Type, Type B Co = 0.19 Duct Design 34.59 ER2-1 Bellmouth, Plenum to Round, Exhaust/Return Systems Ao/A1 Co Values r/D1 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.08 0.10 0.12 0.16 0.20 10.00 1.5 0.22 0.20 0.15 0.14 0.12 0.10 0.09 0.07 0.05 0.04 0.03 0.01 0.01 2.0 0.13 0.11 0.08 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.02 0.01 0.01 2.5 0.08 0.07 0.05 0.05 0.04 0.04 0.03 0.02 0.02 0.01 0.01 0.00 0.00 3.0 0.06 0.05 0.04 0.03 0.03 0.02 0.02 0.02 0.01 0.01 0.01 0.00 0.00 4.0 0.03 0.03 0.02 0.02 0.02 0.01 0.01 0.01 0.01 0.01 0.00 0.00 0.00 8.0 0.01 0.01 0.01 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 ER3-1 Elbow, 90 Degree, Variable Inlet/Outlet Areas, Exhaust/Return Systems H/Wo Co Values W1/Wo 0.6 0.8 1.0 1.2 1.4 1.6 2.0 0.25 1.76 1.43 1.24 1.14 1.09 1.06 1.06 1.00 1.70 1.36 1.15 1.02 0.95 0.90 0.84 4.00 1.46 1.10 0.90 0.81 0.76 0.72 0.66 100.00 1.50 1.04 0.79 0.69 0.63 0.60 0.55 ER4-1 Transition, Rectangular, Two Sides Parallel, Symmetrical, Exhaust/Return Systems Ao/A1 Co Values θ 10 15 20 30 45 60 90 120 150 180 0.06 0.26 0.27 0.40 0.56 0.71 0.86 1.00 0.99 0.98 0.98 0.10 0.24 0.26 0.36 0.53 0.69 0.82 0.93 0.93 0.92 0.91 0.25 0.17 0.19 0.22 0.42 0.60 0.68 0.70 0.69 0.67 0.66 0.50 0.14 0.13 0.15 0.24 0.35 0.37 0.38 0.37 0.36 0.35 1.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 2.00 0.23 0.20 0.20 0.20 0.24 0.28 0.54 0.78 1.02 1.09 4.00 0.81 0.64 0.64 0.64 0.88 1.12 2.78 4.38 5.65 6.60 6.00 1.82 1.44 1.44 1.44 1.98 2.53 6.56 10.20 13.00 15.20 10.00 5.03 5.00 5.00 5.00 6.50 8.02 19.10 29.10 37.10 43.10 ER4-3 Transition, Rectangular to Round, Exhaust/Return Systems Ao /A1 Co Values θ 10 15 20 30 45 60 90 120 150 180 0.06 0.30 0.54 0.53 0.65 0.77 0.88 0.95 0.98 0.98 0.93 0.10 0.30 0.50 0.53 0.64 0.75 0.84 0.89 0.91 0.91 0.88 0.25 0.25 0.36 0.45 0.52 0.58 0.62 0.64 0.64 0.64 0.64 0.50 0.15 0.21 0.25 0.30 0.33 0.33 0.33 0.32 0.31 0.30 1.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 2.00 0.24 0.28 0.26 0.20 0.22 0.24 0.49 0.73 0.97 1.04 4.00 0.89 0.78 0.79 0.70 0.88 1.12 2.72 4.33 5.62 6.58 6.00 1.89 1.67 1.59 1.49 1.98 2.52 6.51 10.14 13.05 15.14 10.00 5.09 5.32 5.15 5.05 6.50 8.05 19.06 29.07 37.08 43.05 A0/A1 < or > 1 34.60 2001 ASHRAE Fundamentals Handbook (SI) ER5-2 Tee, Round Tap to Rectangular Main, Converging Qb/Qc 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Cb −12.25 −1.31 0.64 0.94 1.27 1.43 1.40 1.45 1.52 1.49 Qs/Qc 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Cs 2.15 11.91 6.54 3.74 2.23 1.33 0.76 0.38 0.10 ER5-3 Tee, 45 Degree Entry Branch, Converging Qb/Qc 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Cb -18.00 -3.25 -0.64 0.53 0.76 0.79 0.93 0.79 0.90 0.91 Qs/Qc 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Cs 2.15 11.91 6.54 3.74 2.23 1.33 0.76 0.38 0.10 ER5-4 Wye, Symmetrical, Dovetail, Qb/Qc = 0.5, Converging Ab/Ac 0.5 1.0 Cb 0.23 0.28 Branches are identical, Qb1 = Qb2 = Qb, and Cb1 = Cb2 = Cb ER7-1 Fan Inlet, Centrifugal, SWSI, 90 Degree Smooth Radius Elbow (Square) r/H Co Values L/H 0.0 2.0 5.0 10.0 0.50 2.50 1.60 0.80 0.80 0.75 2.00 1.20 0.67 0.67 1.00 1.20 0.67 0.33 0.33 1.50 1.00 0.57 0.30 0.30 2.00 0.80 0.47 0.26 0.26 AS = AC Ab/AC = 0.5 L = 0.25 W, 75 mm min.
AS = AC Ab/AC = 0.5 r/Wc = 1.5 Qb1/Qc = Qb2/Qc = 0.5 Wb1 = Wb2 = Wb Duct Design 34.61 SR1-1 Conical Bellmouth/Sudden Contraction, Plenum to Rectangular, Supply Air Systems Ao/A1 Co Values θ L/Dh 0 10 20 30 45 60 90 120 150 180 0.10 0.025 0.46 0.43 0.42 0.40 0.38 0.37 0.38 0.40 0.43 0.46 0.050 0.46 0.42 0.38 0.33 0.30 0.28 0.31 0.36 0.41 0.46 0.075 0.46 0.39 0.32 0.28 0.23 0.21 0.26 0.32 0.39 0.46 0.100 0.46 0.36 0.30 0.23 0.19 0.17 0.23 0.30 0.38 0.46 0.150 0.46 0.34 0.25 0.18 0.15 0.14 0.21 0.29 0.37 0.46 0.300 0.46 0.31 0.22 0.16 0.13 0.13 0.20 0.28 0.37 0.46 0.600 0.46 0.25 0.17 0.12 0.10 0.11 0.19 0.27 0.36 0.46 0.20 0.025 0.42 0.40 0.38 0.36 0.34 0.34 0.35 0.37 0.39 0.42 0.050 0.42 0.38 0.35 0.30 0.27 0.25 0.29 0.33 0.37 0.42 0.075 0.42 0.36 0.30 0.25 0.21 0.19 0.24 0.30 0.36 0.42 0.100 0.42 0.33 0.27 0.21 0.18 0.15 0.21 0.27 0.35 0.42 0.150 0.42 0.31 0.23 0.17 0.13 0.13 0.19 0.26 0.34 0.42 0.300 0.42 0.28 0.20 0.15 0.12 0.12 0.18 0.26 0.34 0.42 0.600 0.42 0.23 0.15 0.11 0.10 0.10 0.17 0.25 0.33 0.42 0.40 0.025 0.34 0.32 0.31 0.29 0.28 0.27 0.28 0.30 0.32 0.34 0.050 0.34 0.31 0.28 0.25 0.22 0.20 0.23 0.26 0.30 0.34 0.075 0.34 0.29 0.24 0.20 0.17 0.16 0.19 0.24 0.29 0.34 0.100 0.34 0.27 0.22 0.17 0.14 0.12 0.17 0.22 0.28 0.34 0.150 0.34 0.25 0.18 0.14 0.11 0.10 0.15 0.21 0.27 0.34 0.300 0.34 0.23 0.16 0.12 0.10 0.10 0.15 0.21 0.27 0.34 0.600 0.34 0.18 0.12 0.09 0.08 0.08 0.14 0.20 0.27 0.34 0.60 0.025 0.25 0.24 0.23 0.22 0.20 0.20 0.21 0.22 0.23 0.25 0.050 0.25 0.23 0.21 0.18 0.16 0.15 0.17 0.19 0.22 0.25 0.075 0.25 0.21 0.18 0.15 0.13 0.12 0.14 0.18 0.21 0.25 0.100 0.25 0.20 0.16 0.13 0.11 0.09 0.12 0.16 0.21 0.25 0.150 0.25 0.19 0.14 0.10 0.08 0.08 0.11 0.16 0.20 0.25 0.300 0.25 0.17 0.12 0.09 0.07 0.07 0.11 0.15 0.20 0.25 0.600 0.25 0.14 0.09 0.07 0.06 0.06 0.10 0.15 0.20 0.25 0.80 0.025 0.15 0.14 0.13 0.13 0.12 0.12 0.12 0.13 0.14 0.15 0.050 0.15 0.13 0.12 0.11 0.10 0.09 0.10 0.12 0.13 0.15 0.075 0.15 0.13 0.10 0.09 0.08 0.07 0.08 0.10 0.13 0.15 0.100 0.15 0.12 0.10 0.07 0.06 0.05 0.07 0.10 0.12 0.15 0.150 0.15 0.11 0.08 0.06 0.05 0.04 0.07 0.09 0.12 0.15 0.300 0.15 0.10 0.07 0.05 0.04 0.04 0.07 0.09 0.12 0.15 0.600 0.15 0.08 0.05 0.04 0.03 0.04 0.06 0.09 0.12 0.15 0.90 0.025 0.09 0.08 0.08 0.08 0.07 0.07 0.07 0.08 0.08 0.09 0.050 0.09 0.08 0.07 0.06 0.06 0.05 0.06 0.07 0.08 0.09 0.075 0.09 0.07 0.06 0.05 0.04 0.04 0.05 0.06 0.08 0.09 0.100 0.09 0.07 0.06 0.04 0.04 0.03 0.04 0.06 0.07 0.09 0.150 0.09 0.07 0.05 0.04 0.03 0.03 0.04 0.06 0.07 0.09 0.300 0.09 0.06 0.04 0.03 0.03 0.02 0.04 0.05 0.07 0.09 0.600 0.09 0.05 0.03 0.02 0.02 0.02 0.04 0.05 0.07 0.09 SR2-1 Abrupt Exit H/W 0.1 0.2 0.9 1.0 1.1 4.0 5.0 10.0 Co 1.55 1.55 1.55 2.00 1.55 1.55 1.55 1.55 Co = 1.0 Note: Table is LAMINAR flow; Co = 1.0 is TURBULENT flow.
Dh = θ is larger of θ1 and θ2 2 H0W0 H0 W0 + ( ) ---------------------------34.62 2001 ASHRAE Fundamentals Handbook (SI) SR2-3 Plain Diffuser (Two Sides Parallel), Free Discharge A1/Ao Re/1000 Co Values θ 8 10 14 20 30 45 60 90 120 1 50 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 100 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 200 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 400 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 2000 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 2 50 0.50 0.51 0.56 0.63 0.80 0.96 1.04 1.09 1.09 100 0.48 0.50 0.56 0.63 0.80 0.96 1.04 1.09 1.09 200 0.44 0.47 0.53 0.63 0.74 0.93 1.02 1.08 1.08 400 0.40 0.42 0.50 0.62 0.74 0.93 1.02 1.08 1.08 2000 0.40 0.42 0.50 0.62 0.74 0.93 1.02 1.08 1.08 4 50 0.34 0.38 0.48 0.63 0.76 0.91 1.03 1.07 1.07 100 0.31 0.36 0.45 0.59 0.72 0.88 1.02 1.07 1.07 200 0.26 0.31 0.41 0.53 0.67 0.83 0.96 1.06 1.06 400 0.22 0.27 0.39 0.53 0.67 0.83 0.96 1.06 1.06 2000 0.22 0.27 0.39 0.53 0.67 0.83 0.96 1.06 1.06 6 50 0.32 0.34 0.41 0.56 0.70 0.84 0.96 1.08 1.08 100 0.27 0.30 0.41 0.56 0.70 0.84 0.96 1.08 1.08 200 0.24 0.27 0.36 0.52 0.67 0.81 0.94 1.06 1.06 400 0.20 0.24 0.36 0.52 0.67 0.81 0.94 1.06 1.06 2000 0.18 0.24 0.34 0.50 0.67 0.81 0.94 1.05 1.05 SR2-5 Pyramidal Diffuser, Free Discharge A1/Ao Re/1000 Co Values θ 8 10 14 20 30 45 60 90 120 1 50 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 100 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 200 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 400 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 2000 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 2 50 0.65 0.68 0.74 0.82 0.92 1.05 1.10 1.08 1.08 100 0.61 0.66 0.73 0.81 0.90 1.04 1.09 1.08 1.08 200 0.57 0.61 0.70 0.79 0.89 1.04 1.09 1.08 1.08 400 0.50 0.56 0.64 0.76 0.88 1.02 1.07 1.08 1.08 2000 0.50 0.56 0.64 0.76 0.88 1.02 1.07 1.08 1.08 4 50 0.53 0.60 0.69 0.78 0.90 1.02 1.07 1.09 1.09 100 0.49 0.55 0.66 0.78 0.90 1.02 1.07 1.09 1.09 200 0.42 0.50 0.62 0.74 0.87 1.00 1.06 1.08 1.08 400 0.36 0.44 0.56 0.70 0.84 0.99 1.06 1.08 1.08 2000 0.36 0.44 0.56 0.70 0.84 0.99 1.06 1.08 1.08 6 50 0.50 0.57 0.66 0.77 0.91 1.02 1.07 1.08 1.08 100 0.47 0.54 0.63 0.76 0.98 1.02 1.07 1.08 1.08 200 0.42 0.48 0.60 0.73 0.88 1.00 1.06 1.08 1.08 400 0.34 0.44 0.56 0.73 0.86 0.98 1.06 1.08 1.08 2000 0.34 0.44 0.56 0.73 0.86 0.98 1.06 1.08 1.08 10 50 0.45 0.53 0.64 0.74 0.85 0.97 1.10 1.12 1.12 100 0.40 0.48 0.62 0.73 0.85 0.97 1.10 1.12 1.12 200 0.34 0.44 0.56 0.69 0.82 0.95 1.10 1.11 1.11 400 0.28 0.40 0.55 0.67 0.80 0.93 1.09 1.11 1.11 2000 0.28 0.40 0.55 0.67 0.80 0.93 1.09 1.11 1.11 SR2-6 Pyramidal Diffuser, with Wall L/Dh 0.5 1.0 2.0 3.0 4.0 5.0 6.0 8.0 10.0 12.0 14.0 Co 0.49 0.40 0.30 0.26 0.23 0.21 0.19 0.17 0.16 0.15 0.14 θ 26 19 13 11 9 8 7 6 6 5 5 θ is the optimum angle.
Duct Design 34.63 SR3-1 Elbow, 90 Degree, Variable Inlet/Outlet Areas, Supply Air Systems H/W1 Co Values W o/W1 0.6 0.8 1.0 1.2 1.4 1.6 2.0 0.25 0.63 0.92 1.24 1.64 2.14 2.71 4.24 1.00 0.61 0.87 1.15 1.47 1.86 2.30 3.36 4.00 0.53 0.70 0.90 1.17 1.49 1.84 2.64 100.00 0.54 0.67 0.79 0.99 1.23 1.54 2.20 SR4-1 Transition, Rectangular, Two Sides Parallel, Symmetrical, Supply Air Systems Ao/A1 Co Values θ 10 15 20 30 45 60 90 120 150 180 0.10 0.05 0.05 0.05 0.05 0.07 0.08 0.19 0.29 0.37 0.43 0.17 0.05 0.04 0.04 0.04 0.05 0.07 0.18 0.28 0.36 0.42 0.25 0.05 0.04 0.04 0.04 0.06 0.07 0.17 0.27 0.35 0.41 0.50 0.06 0.05 0.05 0.05 0.06 0.07 0.14 0.20 0.26 0.27 1.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 1.00 2.00 0.56 0.52 0.60 0.96 1.40 1.48 1.52 1.48 1.44 1.40 4.00 2.72 3.04 3.52 6.72 9.60 10.88 11.20 11.04 10.72 10.56 10.00 24.00 26.00 36.00 53.00 69.00 82.00 93.00 93.00 92.00 91.00 16.00 66.56 69.12 102.40 143.36 181.76 220.16 256.00 253.44 250.88 250.88 SR4-3 Transition, Round to Rectangular, Supply Air Systems Ao/A1 CoValues θ 10 15 20 30 45 60 90 120 150 180 0.10 0.05 0.05 0.05 0.05 0.07 0.08 0.19 0.29 0.37 0.43 0.17 0.05 0.05 0.05 0.04 0.06 0.07 0.18 0.28 0.36 0.42 0.25 0.06 0.05 0.05 0.04 0.06 0.07 0.17 0.27 0.35 0.41 0.50 0.06 0.07 0.07 0.05 0.06 0.06 0.12 0.18 0.24 0.26 1.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 2.00 0.60 0.84 1.00 1.20 1.32 1.32 1.32 1.28 1.24 1.20 4.00 4.00 5.76 7.20 8.32 9.28 9.92 10.24 10.24 10.24 10.24 10.00 30.00 50.00 53.00 64.00 75.00 84.00 89.00 91.00 91.00 88.00 16.00 76.80 138.24 135.68 166.40 197.12 225.28 243.20 250.88 250.88 238.08 A0/A1 < or > 1 θ is larger of θ1 and θ2 34.64 2001 ASHRAE Fundamentals Handbook (SI) SR5-1 Smooth Wye of Type As + Ab ≥ Ac, Branch 90° to Main, Diverging As/Ac Ab/Ac Cb Values Qb/Qc 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.50 0.25 3.44 0.70 0.30 0.20 0.17 0.16 0.16 0.17 0.18 0.50 11.00 2.37 1.06 0.64 0.52 0.47 0.47 0.47 0.48 1.00 60.00 13.00 4.78 2.06 0.96 0.47 0.31 0.27 0.26 0.75 0.25 2.19 0.55 0.35 0.31 0.33 0.35 0.36 0.37 0.39 0.50 13.00 2.50 0.89 0.47 0.34 0.31 0.32 0.36 0.43 1.00 70.00 15.00 5.67 2.62 1.36 0.78 0.53 0.41 0.36 1.00 0.25 3.44 0.78 0.42 0.33 0.30 0.31 0.40 0.42 0.46 0.50 15.50 3.00 1.11 0.62 0.48 0.42 0.40 0.42 0.46 1.00 67.00 13.75 5.11 2.31 1.28 0.81 0.59 0.47 0.46 As/Ac Ab/Ac Cs Values Qs /Qc 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.50 0.25 8.75 1.62 0.50 0.17 0.05 0.00 −0.02 −0.02 0.00 0.50 7.50 1.12 0.25 0.06 0.05 0.09 0.14 0.19 0.22 1.00 5.00 0.62 0.17 0.08 0.08 0.09 0.12 0.15 0.19 0.75 0.25 19.13 3.38 1.00 0.28 0.05 −0.02 −0.02 0.00 0.06 0.50 20.81 3.23 0.75 0.14 −0.02 −0.05 −0.05 −0.02 0.03 1.00 16.88 2.81 0.63 0.11 −0.02 −0.05 0.01 0.00 0.07 1.00 0.25 46.00 9.50 3.22 1.31 0.52 0.14 −0.02 −0.05 −0.01 0.50 35.00 6.75 2.11 0.75 0.24 0.00 −0.10 −0.09 −0.04 1.00 38.00 7.50 2.44 0.81 0.24 −0.03 −0.08 −0.06 −0.02 SR5-3 Wye of the Type As + Ab > Ac, As = Ac, 45 Degree, Diverging Ab/Ac Cb Values Qb/Qc 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.1 0.60 0.52 0.57 0.58 0.64 0.67 0.70 0.71 0.73 0.2 2.24 0.56 0.44 0.45 0.51 0.54 0.58 0.60 0.62 0.3 5.94 1.08 0.52 0.41 0.44 0.46 0.49 0.52 0.54 0.4 10.56 1.88 0.71 0.43 0.35 0.31 0.31 0.32 0.34 0.5 17.75 3.25 1.14 0.59 0.40 0.31 0.30 0.30 0.31 0.6 26.64 5.04 1.76 0.83 0.50 0.36 0.32 0.30 0.30 0.7 37.73 7.23 2.56 1.16 0.67 0.44 0.35 0.31 0.30 0.8 49.92 9.92 3.48 1.60 0.87 0.55 0.42 0.35 0.32 Qs/Qc 0.1 0.2 0.3 0.4 0.5 0.6 0.8 1.0 Cs 32.00 6.50 2.22 0.87 0.40 0.17 0.03 0.00 SR5-5 Tee of the Type As + Ab > Ac, As = Ac Diverging Ab/Ac Cb Values Qb/Qc 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.1 2.06 1.20 0.99 0.87 0.88 0.87 0.87 0.86 0.86 0.2 5.16 1.92 1.28 1.03 0.99 0.94 0.92 0.90 0.89 0.3 10.26 3.13 1.78 1.28 1.16 1.06 1.01 0.97 0.94 0.4 15.84 4.36 2.24 1.48 1.11 0.88 0.80 0.75 0.72 0.5 24.25 6.31 3.03 1.89 1.35 1.03 0.91 0.84 0.78 0.6 34.56 8.73 4.04 2.41 1.64 1.22 1.04 0.94 0.87 0.7 46.55 11.51 5.17 3.00 2.00 1.44 1.20 1.06 0.96 0.8 60.80 14.72 6.54 3.72 2.41 1.69 1.38 1.20 1.07 Qs/Qc 0.1 0.2 0.3 0.4 0.5 0.6 0.8 1.0 Cs 32.00 6.50 2.22 0.87 0.40 0.17 0.03 0.00 Duct Design 34.65 SR5-11 Tee, Rectangular Main to Round Tap, Diverging Ab/Ac Cb Values Qb/Qc 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.1 1.58 0.94 0.83 0.79 0.77 0.76 0.76 0.76 0.75 0.2 4.20 1.58 1.10 0.94 0.87 0.83 0.80 0.79 0.78 0.3 8.63 2.67 1.58 1.20 1.03 0.94 0.88 0.85 0.83 0.4 14.85 4.20 2.25 1.58 1.27 1.10 1.00 0.94 0.90 0.5 22.87 6.19 3.13 2.07 1.58 1.32 1.16 1.06 0.99 0.6 32.68 8.63 4.20 2.67 1.96 1.58 1.35 1.20 1.10 0.7 44.30 11.51 5.48 3.38 2.41 1.89 1.58 1.38 1.24 0.8 57.71 14.85 6.95 4.20 2.94 2.25 1.84 1.58 1.40 0.9 72.92 18.63 8.63 5.14 3.53 2.67 2.14 1.81 1.58 As/Ac Cs Values Qs/Qc 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.1 0.04 0.2 0.98 0.04 0.3 3.48 0.31 0.04 0.4 7.55 0.98 0.18 0.04 0.5 13.18 2.03 0.49 0.13 0.04 0.6 20.38 3.48 0.98 0.31 0.10 0.04 0.7 29.15 5.32 1.64 0.60 0.23 0.09 0.04 0.8 39.48 7.55 2.47 0.98 0.42 0.18 0.08 0.04 0.9 51.37 10.17 3.48 1.46 0.67 0.31 0.15 0.07 0.04 SR5-13 Tee, 45 Degree Entry Branch, Diverging Ab/Ac Cb Values Qb/Qc 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.1 0.73 0.34 0.32 0.34 0.35 0.37 0.38 0.39 0.40 0.2 3.10 0.73 0.41 0.34 0.32 0.32 0.33 0.34 0.35 0.3 7.59 1.65 0.73 0.47 0.37 0.34 0.32 0.32 0.32 0.4 14.20 3.10 1.28 0.73 0.51 0.41 0.36 0.34 0.32 0.5 22.92 5.08 2.07 1.12 0.73 0.54 0.44 0.38 0.35 0.6 33.76 7.59 3.10 1.65 1.03 0.73 0.56 0.47 0.41 0.7 46.71 10.63 4.36 2.31 1.42 0.98 0.73 0.58 0.49 0.8 61.79 14.20 5.86 3.10 1.90 1.28 0.94 0.73 0.60 0.9 78.98 18.29 7.59 4.02 2.46 1.65 1.19 0.91 0.73 As/Ac Cs Values Qs/Qc 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.1 0.04 0.2 0.98 0.04 0.3 3.48 0.31 0.04 0.4 7.55 0.98 0.18 0.04 0.5 13.18 2.03 0.49 0.13 0.04 0.6 20.38 3.48 0.98 0.31 0.10 0.04 0.7 29.15 5.32 1.64 0.60 0.23 0.09 0.04 0.8 39.48 7.55 2.47 0.98 0.42 0.18 0.08 0.04 0.9 51.37 10.17 3.48 1.46 0.67 0.31 0.15 0.07 0.04 34.66 2001 ASHRAE Fundamentals Handbook (SI) SR5-14 Wye, Symmetrical, Dovetail, Qb/Qc = 0.5, Diverging Ab/Ac 0.5 1.0 Cb 0.30 1.00 Branches are identical: Qb1 = Qb2 = Qb, and Cb1 = Cb2 = Cb SR7-1 Fan, Centrifugal, Without Outlet Diffuser, Free Discharge Ab/Ao 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Co 2.00 2.00 1.00 0.80 0.47 0.22 0.00 SR7-2 Plane Asymmetric Diffuser at Centrifugal Fan Outlet, Free Discharge θ Co Values A1/Ao 1.5 2.0 2.5 3.0 3.5 4.0 10 0.51 0.34 0.25 0.21 0.18 0.17 15 0.54 0.36 0.27 0.24 0.22 0.20 20 0.55 0.38 0.31 0.27 0.25 0.24 25 0.59 0.43 0.37 0.35 0.33 0.33 30 0.63 0.50 0.46 0.44 0.43 0.42 35 0.65 0.56 0.53 0.52 0.51 0.50 SR7-5 Fan Outlet, Centrifugal, SWSI, with Elbow (Position A) Ab/Ao Co Values L/Le 0.00 0.12 0.25 0.50 1.00 10.00 0.4 3.20 2.50 1.80 0.80 0.00 0.00 0.5 2.20 1.80 1.20 0.53 0.00 0.00 0.6 1.60 1.40 0.80 0.40 0.00 0.00 0.7 1.00 0.80 0.53 0.26 0.00 0.00 0.8 0.80 0.67 0.47 0.18 0.00 0.00 0.9 0.53 0.47 0.33 0.18 0.00 0.00 1.0 0.53 0.47 0.33 0.18 0.00 0.00 Duct Design 34.67 SR7-6 Fan Outlet, Centrifugal, SWSI, with Elbow (Position B) Ab/Ao Co Values L/Le 0.00 0.12 0.25 0.50 1.00 10.00 0.4 3.80 3.20 2.20 1.00 0.00 0.00 0.5 2.90 2.20 1.60 0.67 0.00 0.00 0.6 2.00 1.60 1.20 0.53 0.00 0.00 0.7 1.40 1.00 0.67 0.33 0.00 0.00 0.8 1.00 0.80 0.53 0.26 0.00 0.00 0.9 0.80 0.67 0.47 0.18 0.00 0.00 1.0 0.67 0.53 0.40 0.18 0.00 0.00 SR7-7 Fan Outlet, Centrifugal, SWSI, with Elbow (Position C) Ab/Ao Co Values L/Le 0.00 0.12 0.25 0.50 1.00 10.00 0.4 5.50 4.50 3.20 1.60 0.00 0.00 0.5 3.80 3.20 2.20 1.00 0.00 0.00 0.6 2.90 2.50 1.60 0.80 0.00 0.00 0.7 2.00 1.60 1.00 0.53 0.00 0.00 0.8 1.40 1.20 0.80 0.33 0.00 0.00 0.9 1.20 0.80 0.67 0.26 0.00 0.00 1.0 1.00 0.80 0.53 0.26 0.00 0.00 SR7-8 Fan Outlet, Centrifugal, SWSI, with Elbow (Position D) Ab/Ao Co Values L/Le 0.00 0.12 0.25 0.50 1.00 10.00 0.4 5.50 4.50 3.20 1.60 0.00 0.00 0.5 3.80 3.20 2.20 1.00 0.00 0.00 0.6 2.90 2.50 1.60 0.80 0.00 0.00 0.7 2.00 1.60 1.00 0.53 0.00 0.00 0.8 1.40 1.20 0.80 0.33 0.00 0.00 0.9 1.20 0.80 0.67 0.26 0.00 0.00 1.0 1.00 0.80 0.53 0.26 0.00 0.00 SR7-9 Fan Outlet, Centrifugal, DWDI, with Elbow (Position A) Ab/Ao Co L/Le 0.00 0.12 0.25 0.50 1.00 10.00 0.4 3.20 2.50 1.80 0.80 0.00 0.00 0.5 2.20 1.80 1.20 0.53 0.00 0.00 0.6 1.60 1.40 0.80 0.40 0.00 0.00 0.7 1.00 0.80 0.53 0.26 0.00 0.00 0.8 0.80 0.67 0.47 0.18 0.00 0.00 0.9 0.53 0.47 0.33 0.18 0.00 0.00 1.0 0.53 0.47 0.33 0.18 0.00 0.00 34.68 2001 ASHRAE Fundamentals Handbook (SI) SR7-10 Fan Outlet, Centrifugal, DWDI, with Elbow (Position B) Ab/Ao Co L/Le 0.00 0.12 0.25 0.50 1.00 10.00 0.4 4.80 4.00 2.90 1.30 0.00 0.00 0.5 3.60 2.90 2.00 0.84 0.00 0.00 0.6 2.50 2.00 1.50 0.66 0.00 0.00 0.7 1.80 1.30 0.84 0.41 0.00 0.00 0.8 1.25 1.00 0.66 0.33 0.00 0.00 0.9 1.00 0.84 0.59 0.23 0.00 0.00 1.0 0.84 0.66 0.50 0.23 0.00 0.00 SR7-11 Fan Outlet, Centrifugal, DWDI, with Elbow (Position C) Ab/Ao Co L/Le 0.00 0.12 0.25 0.50 1.00 10.00 0.4 5.50 4.50 3.20 1.60 0.00 0.00 0.5 3.80 3.20 2.20 1.00 0.00 0.00 0.6 2.90 2.50 1.60 0.80 0.00 0.00 0.7 2.00 1.60 1.00 0.53 0.00 0.00 0.8 1.40 1.20 0.80 0.33 0.00 0.00 0.9 1.20 0.80 0.67 0.26 0.00 0.00 1.0 1.00 0.80 0.53 0.26 0.00 0.00 SR7-12 Fan Outlet, Centrifugal, DWDI, with Elbow (Position D) Ab/Ao Co L/Le 0.00 0.12 0.25 0.50 1.00 10.00 0.4 4.70 3.80 2.70 1.40 0.00 0.00 0.5 3.20 2.70 1.90 0.85 0.00 0.00 0.6 2.50 2.10 1.40 0.68 0.00 0.00 0.7 1.70 1.40 0.85 0.45 0.00 0.00 0.8 1.20 1.00 0.68 0.26 0.00 0.00 0.9 1.00 0.68 0.57 0.22 0.00 0.00 1.0 0.85 0.68 0.45 0.22 0.00 0.00 SR7-17 Pyramidal Diffuser at Centrifugal Fan Outlet with Ductwork θ C1 Values Ao/A1 1.5 2.0 2.5 3.0 3.5 4.0 10 0.10 0.18 0.21 0.23 0.24 0.25 15 0.23 0.33 0.38 0.40 0.42 0.44 20 0.31 0.43 0.48 0.53 0.56 0.58 25 0.36 0.49 0.55 0.58 0.62 0.64 30 0.42 0.53 0.59 0.64 0.67 0.69 35.1 CHAPTER 35 PIPE SIZING Pressure Drop Equations ........................................................ 35.1 WATER PIPING ..................................................................... 35.3 Flow Rate Limitations ............................................................. 35.3 Hydronic System Piping .......................................................... 35.5 Service Water Piping .............................................................. 35.7 STEAM PIPING .................................................................... 35.10 Low-Pressure Steam Piping .................................................. 35.11 High-Pressure Steam Piping ................................................. 35.13 Steam Condensate Systems ................................................... 35.14 GAS PIPING ......................................................................... 35.17 FUEL OIL PIPING ............................................................... 35.18 HIS CHAPTER includes tables and charts to size piping for Tvarious fluid flow systems. Further details on specific piping systems can be found in appropriate chapters of the ASHRAE Handbook series.
Two related but distinct concerns emerge when designing a fluid flow system: sizing the pipe and determining the flow-pressure rela-tionship. The two are often confused because they can use the same equations and design tools. Nevertheless, they should be deter-mined separately.
The emphasis in this chapter is on the problem of sizing the pipe, and to this end design charts and tables for specific fluids are pre-sented in addition to the equations that describe the flow of fluids in pipes. Once a system has been sized, it should be analyzed with more detailed methods of calculation to determine the pump pres-sure required to achieve the desired flow. Computerized methods are well suited to handling the details of calculating losses around an extensive system.
PRESSURE DROP EQUATIONS Darcy-Weisbach Equation Pressure drop caused by fluid friction in fully developed flows of all “well-behaved” (Newtonian) fluids is described by the Darcy-Weisbach equation: (1) where ∆p = pressure drop, Pa f = friction factor, dimensionless (from Moody chart, Figure 13 in Chapter 2) L = length of pipe, m D = internal diameter of pipe, m ρ = fluid density at mean temperature, kg/m3 V = average velocity, m/s This equation is often presented in specific energy form as (2) where ∆h = energy loss, m g = acceleration of gravity, m/s2 In this form, the density of the fluid does not appear explicitly (although it is in the Reynolds number, which influences f ).
The friction factor f is a function of pipe roughness ε, inside diameter D, and parameter Re, the Reynolds number: (3) where Re = Reynolds number, dimensionless ε = absolute roughness of pipe wall, m µ = dynamic viscosity of fluid, Pa·s The friction factor is frequently presented on a Moody chart (Figure 13 in Chapter 2) giving f as a function of Re with ε/D as a parameter.
A useful fit of smooth and rough pipe data for the usual turbulent flow regime is the Colebrook equation: (4) Another form of Equation (4) appears in Chapter 2, but the two are equivalent. Equation (4) is more useful in showing behavior at limiting cases—as ε/D approaches 0 (smooth limit), the 18.7/Re term dominates; at high ε/D and Re (fully rough limit), the 2ε/D term dominates.
Equation (4) is implicit in f; that is, f appears on both sides, so a value for f is usually obtained iteratively.
Hazen-Williams Equation A less widely used alternative to the Darcy-Weisbach formula-tion for calculating pressure drop is the Hazen-Williams equation, which is expressed as (5) or (6) where C = roughness factor.
Typical values of C are 150 for plastic pipe and copper tubing, 140 for new steel pipe, down to 100 and below for badly corroded or very rough pipe. Valve and Fitting Losses Valves and fittings cause pressure losses greater than those caused by the pipe alone. One formulation expresses losses as (7) where K = geometry- and size-dependent loss coefficient (Tables 1 through 5).
The preparation of this chapter is assigned to TC 6.1, Hydronic and Steam Equipment and Systems.
p ∆ f L D ---- ρV 2 2 --------- = h ∆ p ∆ ρg ------f L D ---- V 2 2g ------ = = Re DVρ µ ⁄ = 1 f -------1.74 2 2ε D -----18.7 Re f ----------------+ log – = f p ∆ 6.819L V C --- 1.852 1 D ---- 1.167 ρg ( ) = h ∆ 6.819L V C --- 1.852 1 D ---- 1.167 = p ∆ Kρ V 2 2 ------ or h ∆ K V 2 2g ------ = = 35.2 2001 ASHRAE Fundamentals Handbook (SI) Table 1 K Factors—Threaded Pipe Fittings Nominal Pipe Dia., mm 90° Standard Elbow 90° Long-Radius Elbow 45° Elbow Return Bend Tee-Line Tee-Branch Globe Valve Gate Valve Angle Valve Swing Check Valve Bell Mouth Inlet Square Inlet Projected Inlet 10 2.5 — 0.38 2.5 0.90 2.7 20 0.40 — 8.0 0.05 0.5 1.0 15 2.1 — 0.37 2.1 0.90 2.4 14 0.33 — 5.5 0.05 0.5 1.0 20 1.7 0.92 0.35 1.7 0.90 2.1 10 0.28 6.1 3.7 0.05 0.5 1.0 25 1.5 0.78 0.34 1.5 0.90 1.8 9 0.24 4.6 3.0 0.05 0.5 1.0 32 1.3 0.65 0.33 1.3 0.90 1.7 8.5 0.22 3.6 2.7 0.05 0.5 1.0 40 1.2 0.54 0.32 1.2 0.90 1.6 8 0.19 2.9 2.5 0.05 0.5 1.0 50 1.0 0.42 0.31 1.0 0.90 1.4 7 0.17 2.1 2.3 0.05 0.5 1.0 65 0.85 0.35 0.30 0.85 0.90 1.3 6.5 0.16 1.6 2.2 0.05 0.5 1.0 80 0.80 0.31 0.29 0.80 0.90 1.2 6 0.14 1.3 2.1 0.05 0.5 1.0 100 0.70 0.24 0.28 0.70 0.90 1.1 5.7 0.12 1.0 2.0 0.05 0.5 1.0 Source: Engineering Data Book (Hydraulic Institute 1979).
Table 2 K Factors—Flanged Welded Pipe Fittings Nominal Pipe Dia., mm 90° Standard Elbow 90° Long-Radius Elbow 45° Long-Radius Elbow Return Bend Standard Return Bend Long-Radius Tee-Line Tee-Branch Globe Valve Gate Valve Angle Valve Swing Check Valve 25 0.43 0.41 0.22 0.43 0.43 0.26 1.0 13 — 4.8 2.0 32 0.41 0.37 0.22 0.41 0.38 0.25 0.95 12 — 3.7 2.0 40 0.40 0.35 0.21 0.40 0.35 0.23 0.90 10 — 3.0 2.0 50 0.38 0.30 0.20 0.38 0.30 0.20 0.84 9 0.34 2.5 2.0 65 0.35 0.28 0.19 0.35 0.27 0.18 0.79 8 0.27 2.3 2.0 80 0.34 0.25 0.18 0.34 0.25 0.17 0.76 7 0.22 2.2 2.0 100 0.31 0.22 0.18 0.31 0.22 0.15 0.70 6.5 0.16 2.1 2.0 150 0.29 0.18 0.17 0.29 0.18 0.12 0.62 6 0.10 2.1 2.0 200 0.27 0.16 0.17 0.27 0.15 0.10 0.58 5.7 0.08 2.1 2.0 250 0.25 0.14 0.16 0.25 0.14 0.09 0.53 5.7 0.06 2.1 2.0 300 0.24 0.13 0.16 0.24 0.13 0.08 0.50 5.7 0.05 2.1 2.0 Source: Engineering Data Book (Hydraulic Institute 1979).
Table 3 Approximate Range of Variation for K Factors 90° Elbow Regular threaded ±20% above 50 mm Tee Threaded, line or branch ±25% ±40% below 50 mm Flanged, line or branch ±35% Long-radius threaded ±25% Globe valve Threaded ±25% Regular flanged ±35% Flanged ±25% Long-radius flanged ±30% Gate valve Threaded ±25% 45° Elbow Regular threaded ±10% Flanged ±50% Long-radius flanged ±10% Angle valve Threaded ±20% Return bend (180°) Regular threaded Regular flanged Long-radius flanged ±25% ±35% ±30% Flanged ±50% Check valve Threaded ±50% Flanged +200% −80% Source: Engineering Data Book (Hydraulic Institute 1979).
Table 4 Comparison of K Factors from Rahmeyer (1999a) with Previous Reference Data for Elbows Previous Ref.a Values Rahmeyer Values 0.6 m/s 1.2 m/s 2.4 m/s 50 mm standard elbow (threaded) 0.60 to 1.0 (1.0) 0.54 0.6 0.68 100 mm long-radius elbow (welded) 0.22 to 0.24 (0.22) 0.28 0.26 0.24 100 mm standard elbow (welded) 0.31 to 0.34 0.40 0.37 0.34 50 mm reducing elbow (50 × 40) (threaded) – 0.76 0.81 0.87 50 mm expanding elbow (40 × 50) (threaded) – 0.65 0.59 0.54 100 mm reducing elbow (100 × 80) (welded) – 0.72 0.57 0.45 100 mm expanding elbow (80 × 100) (welded) – 0.28 0.28 0.27 50 mm reducer (50 × 40) (threaded) – 0.99 0.53 0.28 50 mm expansion (40 × 50) (threaded) – 0.20 0.16 0.13 100 mm reducer (100 × 80) (welded) 0.22 0.40 0.23 0.14 100 mm expansion (80 × 100) (welded) – 0.13 0.11 0.11 a Previous references are Freeman (1941), Crane Co. (1976), and Hydraulic Institute (1979). Numbers in ( ) are from Table 1 and Table 2, above.
Pipe Sizing 35.3 Example 1. Determine the pressure drop for 15°C water flowing at 1 m/s through a nominal 25 mm, 90° threaded elbow.
Solution: From Table 1, the K for a 25 mm, 90° threaded elbow is 1.5.
∆p = 1.5 × 12/2 = 750 Pa The loss coefficient for valves appears in another form as Av, a dimensional coefficient expressing the flow through a valve at a specified pressure drop. (8) where Q = volumetric flow, m3/s Av = valve coefficient, m3/s at ∆p = 1 Pa ∆p = pressure drop, Pa ρ = density of fluid ≈ 1000 kg/m3 for water at temperatures below 120°C Example 2. Determine the volumetric flow through a valve with Av = 0.00024 for an allowable pressure drop of 35 kPa.
Solution: Q = 0.00024 = 0.0014 m3/s = 1.4 L/s Alternative formulations express fitting losses in terms of equiv-alent lengths of straight pipe (Table 6, Table 7, and Figure 4). Pres-sure loss data for fittings are also presented in Idelchik (1986).
Equation (7) and the data in Table 1 and Table 2 are based on the assumption that separated flow in the fitting causes the K factors to be independent of Reynolds number. In reality, the K factor for most pipe fittings varies with Reynolds number. Tests by Rahmeyer (1999a,b) (sponsored as ASHRAE Research Project 968) on 50 mm threaded and 100 mm welded fittings demonstrate the vari-ation and are shown in Table 4 and Table 5. The study also presents K factors of diverting and mixing flows that range between full through flow and full branch flow. It also examined the variation in K factors caused by variations in geometry among manufacturers and by surface defects in individual fittings.
Hegberg (1995) and Rahmeyer (1999a,b) discuss the origins of the data shown in Table 4 and Table 5. The Hydraulic Institute (1979) data appear to have come from Freeman (1941), work that was actually performed in 1895. The work of Giesecke (1926) and Giesecke and Badgett (1931, 1932a,b) may not be representative of present-day fittings.
Calculating Pressure Losses The most common engineering design flow loss calculation selects a pipe size for the desired total flow rate and available or allowable pressure drop.
Because either formulation of fitting losses requires a known diameter, pipe size must be selected before calculating the detailed influence of fittings. A frequently used rule of thumb assumes that the design length of pipe is 50 to 100% longer than actual to account for fitting losses. After a pipe diameter has been selected on this basis, the influence of each fitting can be evaluated.
WATER PIPING FLOW RATE LIMITATIONS Stewart and Dona (1987) surveyed the literature relating to water flow rate limitations. Noise, erosion, and installation and operating costs all limit the maximum and minimum velocities in piping sys-tems. If piping sizes are too small, noise levels, erosion levels, and pumping costs can be unfavorable; if piping sizes are too large, installation costs are excessive. Therefore, pipe sizes are chosen to minimize initial cost while avoiding the undesirable effects of high velocities. Table 5 Comparison of K Factors from Rahmeyer (1999b) with Previous Reference Data for Tees Previous Ref.a Values Rahmeyer Values 0.6 m/s 1.2 m/s 2.4 m/s 50 mm tee 100% branch (threaded) 1.20 to 1.80 (1.4) – 0.93 – 50 mm tee 0% branch (flow through) (threaded) 0.50 to 0.90 (0.90) – 0.19 – 50 mm tee 100% mix (threaded) – – 1.19 – 100 mm tee 100% branch (welded) 0.70 to 1.02 (0.70) – – 0.57 100 mm tee 0% branch (flow through) (welded) 0.15 to 0.34 (0.15) – – 0.06 100 mm tee 100% mix (welded) – – – 0.49 50 mm reducing tee 100% branch (threaded) – – 3.75 – 50 mm reducing tee 0% branch (threaded) – – 2.55 – 50 mm reducing tee 100% mix (threaded) – – 3.36 – 100 mm reducing tee 100% branch (welded) – – – 3.26 100 mm reducing tee 0% branch (welded) – – – 0.04 100 mm reducing tee 100% mix (welded) – – – 0.75 a Previous references are Freeman (1941), Crane Co. (1976), and Hydraulic Institute (1979). Numbers in ( ) are from Table 1 and Table 2, above.
Q Av p ∆ ρ ⁄ = 35 000 1000 ⁄ Table 6 Water Velocities Based on Type of Service Type of Service Velocity, fps Reference General service 1.2 to 3.0 a, b, c City water 0.9 to 2.1 a, b 0.6 to 1.5 c Boiler feed 1.8 to 4.6 a, c Pump suction and drain lines 1.2 to 2.1 a, b aCrane Co. (1976).
bCarrier (1960).
cGrinnell Company (1951).
Table 7 Maximum Water Velocity to Minimize Erosion Normal Operation, h/yr Water Velocity, m/s 1500 4.6 2000 4.4 3000 4.0 4000 3.7 6000 3.0 Source: Carrier (1960).
35.4 2001 ASHRAE Fundamentals Handbook (SI) A variety of upper limits of water velocity and/or pressure drop in piping and piping systems is used. One recommendation places a velocity limit of 1.2 m/s for 50 mm pipe and smaller, and a pressure drop limit of 400 Pa/m for piping over 50 mm. Other guidelines are based on the type of service (Table 6) or the annual operating hours (Table 7). These limitations are imposed either to control the levels of pipe and valve noise, erosion, and water hammer pressure or for economic reasons. Carrier (1960) recommends that the velocity not exceed 4.6 m/s in any case.
Noise Generation Velocity-dependent noise in piping and piping systems results from any or all of four sources: turbulence, cavitation, release of entrained air, and water hammer. In investigations of flow-related noise, Marseille (1965), Ball and Webster (1976), and Rogers (1953, 1954, 1956) reported that velocities on the order of 3 to 5 m/s lie within the range of allowable noise levels for residential and commercial buildings. The experiments showed considerable vari-ation in the noise levels obtained for a specified velocity. Generally, systems with longer pipe and with more numerous fittings and valves were noisier. In addition, sound measurements were taken under widely differing conditions; for example, some tests used plastic-covered pipe, while others did not. Thus, no detailed corre-lations relating sound level to flow velocity in generalized systems are available.
The noise generated by fluid flow in a pipe increases sharply if cavitation or the release of entrained air occurs. Usually the combi-nation of a high water velocity with a change in flow direction or a decrease in the cross section of a pipe causing a sudden pressure drop is necessary to cause cavitation. Ball and Webster (1976) found that at their maximum velocity of 13 m/s, cavitation did not occur in straight pipe; using the apparatus with two elbows, cold water velocities up to 6.5 m/s caused no cavitation. Cavitation did occur in orifices of 1:8 area ratio (orifice flow area is one-eighth of pipe flow area) at 1.5 m/s and in 1:4 area ratio orifices at 3 m/s (Rogers 1954).
Some data are available for predicting hydrodynamic (liquid) noise generated by control valves. The International Society for Measurement and Control compiled prediction correlations in an effort to develop control valves for reduced noise levels (ISA 1985).
The correlation to predict hydrodynamic noise from control valves is (9) where SL = sound level, dB Av = valve coefficient, m3/(s· ) Q = flow rate, m3/s ∆p = pressure drop across valve, Pa t = downstream pipe wall thickness, mm Air entrained in water usually has a higher partial pressure than the water. Even when flow rates are small enough to avoid cavitation, the release of entrained air may create noise. Every effort should be made to vent the piping system or otherwise remove entrained air.
Erosion Erosion in piping systems is caused by water bubbles, sand, or other solid matter impinging on the inner surface of the pipe. Gen-erally, at velocities lower than 30 m/s, erosion is not significant as long as there is no cavitation. When solid matter is entrained in the fluid at high velocities, erosion occurs rapidly, especially in bends.
Thus, high velocities should not be used in systems where sand or other solids are present or where slurries are transported.
Allowances for Aging With age, the internal surfaces of pipes become increasingly rough, which reduces the available flow with a fixed pressure sup-ply. However, designing with excessive age allowances may result in oversized piping. Age-related decreases in capacity depend on the type of water, type of pipe material, temperature of water, and type of system (open or closed) and include • Sliming (biological growth or deposited soil on the pipe walls), which occurs mainly in unchlorinated, raw water systems.
• Caking of calcareous salts, which occurs in hard water (i.e., water bearing calcium salts) and increases with water temperature.
• Corrosion (incrustations of ferrous and ferric hydroxide on the pipe walls), which occurs in metal pipe in soft water. Because oxygen is necessary for corrosion to take place, significantly more corrosion takes place in open systems.
Allowances for expected decreases in capacity are sometimes treated as a specific amount (percentage). Dawson and Bowman (1933) added an allowance of 15% friction loss to new pipe (equiv-alent to an 8% decrease in capacity). The HDR Design Guide (1981) increased the friction loss by 15 to 20% for closed piping systems and 75 to 90% for open systems. Carrier (1960) indicates a factor of approximately 1.75 between friction factors for closed and open systems.
Obrecht and Pourbaix (1967) differentiated between the corro-sive potential of different metals in potable water systems and con-cluded that iron is the most severely attacked, then galvanized steel, lead, copper, and finally copper alloys (i.e., brass). Hunter (1941) and Freeman (1941) showed the same trend. After four years of cold and hot water use, copper pipe had a capacity loss of 25 to 65%.
Aged ferrous pipe has a capacity loss of 40 to 80%. Smith (1983) recommended increasing the design discharge by 1.55 for uncoated cast iron, 1.08 for iron and steel, and 1.06 for cement or concrete.
The Plastic Pipe Institute (1971) found that corrosion is not a problem in plastic pipe; the capacity of plastic pipe in Europe and the United States remains essentially the same after 30 years in use.
Extensive age-related flow data are available for use with the Hazen-Williams empirical equation. Difficulties arise in its applica-tion, however, because the original Hazen-Williams roughness coefficients are valid only for the specific pipe diameters, water velocities, and water viscosities used in the original experiments.
Thus, when the Cs are extended to different diameters, velocities, and/or water viscosities, errors of up to about 50% in pipe capacity can occur (Williams and Hazen 1933, Sanks 1978).
Water Hammer When any moving fluid (not just water) is abruptly stopped, as when a valve closes suddenly, large pressures can develop. While detailed analysis requires knowledge of the elastic properties of the pipe and the flow-time history, the limiting case of rigid pipe and instantaneous closure is simple to calculate. Under these conditions, (10) where ∆ph = pressure rise caused by water hammer, Pa ρ = fluid density, kg/m3 cs = velocity of sound in fluid, m/s V = fluid flow velocity, m/s The cs for water is 1439 m/s, although the elasticity of the pipe reduces the effective value. Example 3. What is the maximum pressure rise if water flowing at 3 m/s is stopped instantaneously? Solution: SL 10logAv 20log p ∆ 30logt – 76.6 + + = Pa ∆ph ρcsV = ∆ph 1000 1439 × 3 × 4.32 MPa = = Pipe Sizing 35.5 Other Considerations Not discussed in detail in this chapter, but of potentially great importance, are a number of physical and chemical considerations: pipe and fitting design, materials, and joining methods must be appropriate for working pressures and temperatures encountered, as well as being suitably resistant to chemical attack by the fluid.
Other Piping Materials and Fluids For fluids not included in this chapter or for piping materials of different dimensions, manufacturers’ literature frequently supplies pressure drop charts. The Darcy-Weisbach equation, with the Moody chart or the Colebrook equation, can be used as an alterna-tive to pressure drop charts or tables.
HYDRONIC SYSTEM PIPING The Darcy-Weisbach equation with friction factors from the Moody chart or Colebrook equation (or, alternatively, the Hazen-Williams equation) is fundamental to calculating pressure drop in hot and chilled water piping; however, charts calculated from these equa-tions (such as Figures 1, 2, and 3) provide easy determination of pres-sure drops for specific fluids and pipe standards. In addition, tables of pressure drops can be found in Hydraulic Institute (1979) and Crane Co. (1976).
The Reynolds numbers represented on the charts in Figures 1, 2, and 3 are all in the turbulent flow regime. For smaller pipes and/or lower velocities, the Reynolds number may fall into the laminar regime, in which the Colebrook friction factors are no longer valid.
Most tables and charts for water are calculated for properties at 15°C. Using these for hot water introduces some error, although the answers are conservative (i.e., cold water calculations overstate the pressure drop for hot water). Using 15°C water charts for 90°C water should not result in errors in ∆p exceeding 20%.
Range of Usage of Pressure Drop Charts General Design Range. The general range of pipe friction loss used for design of hydronic systems is between 100 and 400 Pa/m of pipe. A value of 250 Pa/m represents the mean to which most sys-tems are designed. Wider ranges may be used in specific designs if certain precautions are taken.
Piping Noise. Closed-loop hydronic system piping is generally sized below certain arbitrary upper limits, such as a velocity limit of 1.2 m/s for 50 mm pipe and under, and a pressure drop limit of 400 Pa/m for piping over 50 mm in diameter. Velocities in excess of 1.2 m/s can be used in piping of larger size. This limitation is generally accepted, although it is based on relatively inconclusive experience with noise in piping. Water velocity noise is not caused by water but by free air, sharp pressure drops, turbulence, or a combination of these, which in turn cause cavitation or flashing of water into steam.
Therefore, higher velocities may be used if proper precautions are taken to eliminate air and turbulence.
Air Separation Air in hydronic systems is usually undesirable because it causes flow noise, allows oxygen to react with piping materials, and some-times even prevents flow in parts of a system. Air may enter a sys-tem at an air-water interface in an open system or in an expansion tank in a closed system, or it may be brought in dissolved in makeup water. Most hydronic systems use air separation devices to remove air. The solubility of air in water increases with pressure and decreases with temperature; thus, separation of air from water is best achieved at the point of lowest pressure and/or highest temper-ature in a system. For more information, see Chapter 12 of the 2000 ASHRAE Handbook—Systems and Equipment.
In the absence of venting, air can be entrained in the water and carried to separation units at flow velocities of 0.5 to 0.6 m/s or more in pipe 50 mm and under. Minimum velocities of 0.6 m/s are therefore recommended. For pipe sizes 50 mm and over, minimum velocities corresponding to a pressure loss of 75 Pa are normally used. Maintenance of minimum velocities is particularly important in the upper floors of high-rise buildings where the air tends to come out of solution because of reduced pressures. Higher velocities should be used in downcomer return mains feeding into air separa-tion units located in the basement.
Example 4. Determine the pipe size for a circuit requiring 1.25 L/s flow.
Solution: Enter Figure 1 at 1.25 L/s, read up to pipe size within nor-mal design range (100 to 400 Pa/m), and select 40 mm. Velocity is 1 m/s and pressure loss is 300 Pa/m.
Valve and Fitting Pressure Drop Valves and fittings can be listed in elbow equivalents, with an elbow being equivalent to a length of straight pipe. Table 8 lists equivalent lengths of 90° elbows; Table 9 lists elbow equivalents for valves and fittings for iron and copper.
Example 5. Determine equivalent length of pipe for a 100 mm open gate valve at a flow velocity of approximately 1.33 m/s.
Solution: From Table 8, at 1.33 m/s, each elbow is equivalent to 3.2 m of 100 mm pipe. From Table 9, the gate valve is equivalent to 0.5 elbows. The actual equivalent pipe length (added to measured circuit length for pressure drop determination) will be 3.2 × 0.5, or 1.6 m of 100 mm pipe.
Tee Fitting Pressure Drop. Pressure drop through pipe tees varies with flow through the branch. Figure 4 illustrates pressure drops for nominal 25 mm tees of equal inlet and outlet sizes and for the flow patterns illustrated. Idelchik (1986) also presents data for threaded tees.
Table 8 Equivalent Length in Metres of Pipe for 90° Elbows Velocity, m/s Pipe Size, mm 15 20 25 32 40 50 65 90 100 125 150 200 250 300 0.33 0.4 0.5 0.7 0.9 1.1 1.4 1.6 2.0 2.6 3.2 3.7 4.7 5.7 6.8 0.67 0.4 0.6 0.8 1.0 1.2 1.5 1.8 2.3 2.9 3.6 4.2 5.3 6.3 7.6 1.00 0.5 0.6 0.8 1.1 1.3 1.6 1.9 2.5 3.1 3.8 4.5 5.6 6.8 8.0 1.33 0.5 0.6 0.8 1.1 1.3 1.7 2.0 2.5 3.2 4.0 4.6 5.8 7.1 8.4 1.67 0.5 0.7 0.9 1.2 1.4 1.8 2.1 2.6 3.4 4.1 4.8 6.0 7.4 8.8 2.00 0.5 0.7 0.9 1.2 1.4 1.8 2.2 2.7 3.5 4.3 5.0 6.2 7.6 9.0 2.35 0.5 0.7 0.9 1.2 1.5 1.9 2.2 2.8 3.6 4.4 5.1 6.4 7.8 9.2 2.67 0.5 0.7 0.9 1.3 1.5 1.9 2.3 2.8 3.6 4.5 5.2 6.5 8.0 9.4 3.00 0.5 0.7 0.9 1.3 1.5 1.9 2.3 2.9 3.7 4.5 5.3 6.7 8.1 9.6 3.33 0.5 0.8 0.9 1.3 1.5 1.9 2.4 3.0 3.8 4.6 5.4 6.8 8.2 9.8 35.6 2001 ASHRAE Fundamentals Handbook (SI) Fig. 1 Friction Loss for Water in Commercial Steel Pipe (Schedule 40) Fig. 2 Friction Loss for Water in Copper Tubing (Types K, L, M) Fig. 3 Friction Loss for Water in Plastic Pipe (Schedule 80) Pipe Sizing 35.7 Different investigators present tee loss data in different forms, and it is sometimes difficult to reconcile results from several sources. As an estimate of the upper limit to tee losses, a pressure or head loss coefficient of 1.0 may be assumed for entering and leaving flows (i.e., ∆p = 1.0ρVin 2 /2 + 1.0ρVout 2 /2).
Example 6. Determine the pressure or energy losses for a 25 mm (all openings) threaded pipe tee flowing 25% to the side branch, 75% through. The entering flow is 1 L/s (1.79 m/s).
Solution: From Figure 4, bottom curve, the number of equivalent elbows for the through-flow is 0.15 elbows; the through-flow is 0.75 L/s (1.34 m/s); and the pressure loss is based on the exit flow rate. Table 8 gives the equivalent length of a 25 mm elbow at 1.33 m/s as 0.8 m.
Using Equations (1) and (2) with friction factor f = 0.0263 and diameter D = 26.6 mm, ∆p = (0.15)(0.0263)(0.8/0.0266)(1000)(1.342)/2 = 0.107 kPa pressure drop, or ∆h = (0.15)(0.0263)(0.8/0.0266)(1.342)/[(2)(9.8)] = 0.0109 m loss From Figure 4, top curve, the number of equivalent elbows for the branch flow of 25% is 13 elbows; the branch flow is 0.25 L/s (0.45 m/s); and the pressure loss is based on the exit flow rate. Interpolating from Table 8 gives the equivalent of a 25 mm elbow at 0.45 m/s as 0.75 m. Using Equations (1) and (2) with friction factor f = 0.0334 and diameter = 26.6 mm, ∆p = (13)(0.0334)(0.75/0.0266)(1000)(0.452)/(2) = 1.24 kPa pressure drop, or ∆h = (13)(0.0334)(0.75/0.0266)(0.452)/[(2)(9.8)] SERVICE WATER PIPING Sizing of service water piping differs from sizing of process lines in that design flows in service water piping are determined by the probability of simultaneous operation of a multiplicity of individual loads such as water closets, urinals, lavatories, sinks, and showers.
The full flow characteristics of each load device are readily obtained from manufacturers; however, service water piping sized to handle all load devices simultaneously would be seriously oversized. Thus, a major issue in sizing service water piping is to determine the diversity of the loads.
The procedure shown in this chapter uses the work of R.B.
Hunter for estimating diversity (Hunter 1940, 1941). The present-day plumbing designer is usually constrained by building or plumb-ing codes, which specify the individual and collective loads to be used for pipe sizing. Frequently used codes (including the BOCA National Plumbing Code, Standard Plumbing Code, Uniform Plumbing Code, and National Standard Plumbing Code) contain procedures quite similar to those shown here. The designer must be aware of the applicable code for the location being considered.
Table 9 Iron and Copper Elbow Equivalentsa Fitting Iron Pipe Copper Tubing Elbow, 90° 1.0 1.0 Elbow, 45° 0.7 0.7 Elbow, 90° long-radius 0.5 0.5 Elbow, welded, 90° 0.5 0.5 Reduced coupling 0.4 0.4 Open return bend 1.0 1.0 Angle radiator valve 2.0 3.0 Radiator or convector 3.0 4.0 Boiler or heater 3.0 4.0 Open gate valve 0.5 0.7 Open globe valve 12.0 17.0 Source: Giesecke (1926) and Giesecke and Badgett (1931, 1932a).
aSee Table 8 for equivalent length of one elbow.
Table 10 Proper Flow and Pressure Required During Flow for Different Fixtures Fixture Flow Pressure, kPa (gage)a Flow, L/s Ordinary basin faucet 55 0.2 Self-closing basin faucet 85 0.2 Sink faucet—10 mm 70 0.3 Sink faucet—15 mm 35 0.3 Dishwasher 105 to 175 —b Bathtub faucet 35 0.4 Laundry tube cock—8 mm 35 0.3 Shower 85 0.2 to 0.6 Ball cock for closet 105 0.2 Flush valve for closet 70 to 140 1.0 to 2.5c Flush valve for urinal 105 1.0 Garden hose, 15 m, and sill cock 210 0.3 aFlow pressure is the pressure in the pipe at the entrance to the particular fixture considered.
bVaries; see manufacturers’ data.
cWide range due to variation in design and type of flush valve closets.
Fig. 4 Elbow Equivalents of Tees at Various Flow Conditions (Giesecke and Badgett 1931, 1932b) Notes: 1. Chart is based on straight tees (i.e., branches A, B, and C are the same size).
2. Pressure loss in desired circuit is obtained by selecting the proper curve according to illustrations, determining the flow at the circled branch, and multiplying the pressure loss for the same size elbow at the flow rate in the circled branch by the equivalent elbows indicated.
3. When the size of an outlet is reduced, the equivalent elbows shown in the chart do not apply. Therefore, the maximum loss for any circuit for any flow will not exceed 2 elbow equivalents at the maximum flow occurring in any branch of the tee.
4. Top curve is average of 4 curves, one for each circuit shown.
35.8 2001 ASHRAE Fundamentals Handbook (SI) Federal mandates are forcing plumbing fixture manufacturers to reduce design flows to many types of fixtures, but these may not yet be included in locally adopted codes. Also, the designer must be aware of special considerations; for example, toilet usage at sports arenas will probably have much less diversity than the codes allow and thus may require larger supply piping than the minimum spec-ified by the codes.
Table 10 gives the rate of flow desirable for many common fix-tures and the average pressure necessary to give this rate of flow.
The pressure varies with fixture design.
In estimating the load, the rate of flow is frequently computed in fixture units, which are relative indicators of flow. Table 11 gives the demand weights in terms of fixture units for different plumbing fixtures under several conditions of service, and Figure 5 gives the estimated demand in gallons per minute corresponding to any total number of fixture units. Figures 6 and 7 provide more accurate esti-mates at the lower end of the scale.
The estimated demand load for fixtures used intermittently on any supply pipe can be obtained by multiplying the number of each kind of fixture supplied through that pipe by its weight from Table 11, adding the products, and then referring to the appropriate curve of Figure 5, 6, or 7 to find the demand corresponding to the total fixture units. In using this method, note that the demand for fixture or supply outlets other than those listed in the table of fixture units is not yet included in the estimate. The demands for outlets (e.g., hose connec-tions and air-conditioning apparatus) that are likely to impose con-tinuous demand during heavy use of the weighted fixtures should be estimated separately and added to demand for fixtures used intermit-tently to estimate total demand.
The Hunter curves in Figures 5, 6, and 7 are based on use patterns in residential buildings and can be erroneous for other usages such as sports arenas. Williams (1976) discusses the Hunter assumptions and presents an analysis using alternative assumptions.
Table 11 Demand Weights of Fixtures in Fixture Unitsa Fixture or Groupb Occupancy Type of Supply Control Weight in Fixture Unitsc Water closet Public Flush valve 10 Water closet Public Flush tank 5 Pedestal urinal Public Flush valve 10 Stall or wall urinal Public Flush valve 5 Stall or wall urinal Public Flush tank 3 Lavatory Public Faucet 2 Bathtub Public Faucet 4 Shower head Public Mixing valve 4 Service sink Office, etc.
Faucet 3 Kitchen sink Hotel or restaurant Faucet 4 Water closet Private Flush valve 6 Water closet Private Flush tank 3 Lavatory Private Faucet 1 Bathtub Private Faucet 2 Shower head Private Mixing valve 2 Bathroom group Private Flush valve for closet 8 Bathroom group Private Flush tank for closet 6 Separate shower Private Mixing valve 2 Kitchen sink Private Faucet 2 Laundry trays (1 to 3) Private Faucet 3 Combination fixture Private Faucet 3 Source: Hunter (1941).
aFor supply outlets likely to impose continuous demands, estimate continuous supply separately, and add to total demand for fixtures.
bFor fixtures not listed, weights may be assumed by comparing the fixture to a listed one using water in similar quantities and at similar rates.
cThe given weights are for total demand. For fixtures with both hot and cold water sup-plies, the weights for maximum separate demands can be assumed to be 75% of the listed demand for the supply.
Fig. 5 Demand Versus Fixture Units, Mixed System, High Part of Curve (Hunter 1941) Fig. 6 Estimate Curves for Demand Load (Hunter 1941) Fig. 7 Section of Figure 6 on Enlarged Scale Pipe Sizing 35.9 So far, the information presented shows the design rate of flow to be determined in any particular section of piping. The next step is to determine the size of piping. As water flows through a pipe, the pres-sure continually decreases along the pipe due to loss of energy from friction. The problem is then to ascertain the minimum pressure in the street main and the minimum pressure required to operate the top-most fixture. (A pressure of 100 kPa may be ample for most flush valves, but reference should be made to the manufacturers’ require-ments. Some fixtures require a pressure up to 175 kPa. A minimum of 55 kPa should be allowed for other fixtures.) The pressure differential overcomes pressure losses in the distributing system and the differ-ence in elevation between the water main and the highest fixture.
The pressure loss (in kPa) resulting from the difference in eleva-tion between the street main and the highest fixture can be obtained by multiplying the difference in elevation in metres by the conver-sion factor 9.8.
Pressure losses in the distributing system consist of pressure losses in the piping itself, plus the pressure losses in the pipe fit-tings, valves, and the water meter, if any. Approximate design pressure losses and flow limits for disk-type meters for various rates of flow are given in Figure 8. Water authorities in many local-ities require compound meters for greater accuracy with varying flow; consult the local utility. Design data for compound meters differ from the data in Figure 8. Manufacturers give data on exact pressure losses and capacities.
Figure 9 shows the variation of pressure loss with rate of flow for various faucets and cocks. The water demand for hose bibbs or other large-demand fixtures taken off the building main frequently results in inadequate water supply to the upper floor of a building.
This condition can be prevented by sizing the distribution system so that the pressure drops from the street main to all fixtures are the same. An ample building main (not less than 25 mm where possi-ble) should be maintained until all branches to hose bibbs have been connected. Where the street main pressure is excessive and a pressure reducing valve is used to prevent water hammer or exces-sive pressure at the fixtures, the hose bibbs should be connected ahead of the reducing valve.
The principles involved in sizing upfeed and downfeed systems are the same. In the downfeed system, however, the difference in elevation between the overhead supply mains and the fixtures pro-vides the pressure required to overcome pipe friction. Because friction pressure loss and height pressure loss are not additive, as in an upfeed system, smaller pipes may be used with a downfeed system.
Plastic Pipe The maximum safe water velocity in a thermoplastic piping sys-tem under most operating conditions is typically 1.5 m/s; however, higher velocities can be used in cases where the operating charac-teristics of valves and pumps are known so that sudden changes in flow velocity can be controlled. The total pressure in the system at any time (operating pressure plus surge of water hammer) should not exceed 150% of the pressure rating of the system.
Procedure for Sizing Cold Water Systems The recommended procedure for sizing piping systems is out-lined below.
1. Sketch the main lines, risers, and branches, and indicate the fixtures to be served. Indicate the rate of flow of each fixture.
2. Using Table 11, compute the demand weights of the fixtures in fixture units.
3. Determine the total demand in fixture units and, using Figure 5, 6, or 7, find the expected demand.
4. Determine the equivalent length of pipe in the main lines, risers, and branches. Because the sizes of the pipes are not known, the exact equivalent length of various fittings cannot be determined. Add the equivalent lengths, starting at the street main and proceeding along the service line, the main line of the building, and up the riser to the top fixture of the group served.
5. Determine the average minimum pressure in the street main and the minimum pressure required for the operation of the topmost fixture, which should be 50 to 175 kPa above atmospheric.
6. Calculate the approximate design value of the average pressure drop per unit length of pipe in equivalent length determined in step 4.
Fig. 8 Pressure Losses in Disk-Type Water Meters Fig. 9 Variation of Pressure Loss with Flow Rate for Various Faucets and Cocks A. 1/2 in. laundry bibb (old style) B. Laundry compression faucet C-1. 1/2 in. compression sink faucet (mfr. 1) C-2. 1/2 in. compression sink faucet (mfr. 2) D. Combination compression bathtub faucets (both open) E. Combination compression sink faucet F. Basin faucet G. Spring self-closing faucet H. Slow self-closing faucet (Dashed lines indicate recommended extrapolation) 35.10 2001 ASHRAE Fundamentals Handbook (SI) (11) where ∆p = average pressure loss per metre of equivalent length of pipe, kPa ps = pressure in street main, kPa pf = minimum pressure required to operate topmost fixture, kPa pm = pressure drop through water meter, kPa H = height of highest fixture above street main, m L = equivalent length determined in step 4, m If the system is downfeed supply from a gravity tank, height of water in the tank, converted to kPa by multiplying by 9.8, replaces the street main pressure, and the term 9.8H is added instead of subtracted in calculating ∆p. In this case, H is the ver-tical distance of the fixture below the bottom of the tank.
7. From the expected rate of flow determined in step 3 and the value of ∆p calculated in step 6, choose the sizes of pipe from Figure 1, 2, or 3.
Example 7. Assume a minimum street main pressure of 375 kPa; a height of topmost fixture (a urinal with flush valve) above street main of 15 m; an equivalent pipe length from water main to highest fixture of 30 m; a total load on the system of 50 fixture units; and that the water closets are flush valve operated. Find the required size of supply main.
Solution: From Figure 7, the estimated peak demand is 3.2 L/s. From Table 10, the minimum pressure required to operate the topmost fixture is 105 kPa. For a trial computation, choose the 40 mm meter. From Fig-ure 8, the pressure drop through a 40 mm disk-type meter for a flow of 3.2 L/s is 45 kPa.
The pressure drop available for overcoming friction in pipes and fittings is 375 − 9.8 × 15 − 105 − 45 = 78 kPa.
At this point, estimate the equivalent pipe length of the fittings on the direct line from the street main to the highest fixture. The exact equivalent length of the various fittings cannot be determined since the pipe sizes of the building main, riser, and branch leading to the highest fixture are not yet known, but a first approximation is necessary to ten-tatively select pipe sizes. If the computed pipe sizes differ from those used in determining the equivalent length of pipe fittings, a recalcula-tion using the computed pipe sizes for the fittings will be necessary. For this example, assume that the total equivalent length of the pipe fittings is 15 m.
The permissible pressure loss per metre of equivalent pipe is 78/(30 + 15) = 1.7 kPa/m. A 40 mm building main is adequate.
The sizing of the branches of the building main, the risers, and the fixture branches follows these principles. For example, assume that one of the branches of the building main carries the cold water supply for 3 water closets, 2 bathtubs, and 3 lavatories. Using the permissible pres-sure loss of 1.7 kPa/m, the size of branch (determined from Table 11 and Figures 1 and 7) is found to be 40 mm. Items included in the com-putation of pipe size are as follows: Table 12 is a guide to minimum pipe sizing where flush valves are used.
Velocities exceeding 3 m/s cause undesirable noise in the piping system. This usually governs the size of larger pipes in the system, while in small pipe sizes, the friction loss usually governs the selection because the velocity is low compared to friction loss.
Velocity is the governing factor in downfeed systems, where fric-tion loss is usually neglected. Velocity in branches leading to pump suctions should not exceed 1.5 m/s.
If the street pressure is too low to adequately supply upper-floor fixtures, the pressure must be increased. Constant or variable speed booster pumps, alone or in conjunction with gravity supply tanks, or hydropneumatic systems may be used.
Flow control valves for individual fixtures under varying pres-sure conditions automatically adjust the flow at the fixture to a pre-determined quantity. These valves allow the designer to (1) limit the flow at the individual outlet to the minimum suitable for the pur-pose, (2) hold the total demand for the system more closely to the required minimum, and (3) design the piping system as accurately as is practicable for the requirements.
STEAM PIPING Pressure losses in steam piping for flows of dry or nearly dry steam are governed by Equations (1) through (7) in the section on Pressure Drop Equations. This section incorporates these principles with other information specific to steam systems.
Pipe Sizes Required pipe sizes for a given load in steam heating depend on the following factors: • The initial pressure and the total pressure drop that can be allowed between the source of supply and the end of the return system • The maximum velocity of steam allowable for quiet and dependable operation of the system, taking into consideration the direction of condensate flow • The equivalent length of the run from the boiler or source of steam supply to the farthest heating unit Initial Pressure and Pressure Drop. Table 13 lists pressure drops commonly used with corresponding initial steam pressures for sizing steam piping.
Several factors, such as initial pressure and pressure required at the end of the line, should be considered, but it is most important that (1) the total pressure drop does not exceed the initial gage pres-Fixtures, No. and Type Fixture Units (Table 11 and Note c) Demand (Figure 7) Pipe Size (Figure 1) 3 flush valves 3 × 6 = 18 2 bathtubs 0.75 × 2 × 2 = 3 3 lavatories 0.75 × 3 × 1 = 2.25 Total = 23.25 2.4 L/s 40 mm p ∆ ps 9.8H – pf pm – – ( ) L ⁄ = Table 12 Allowable Number of 25 mm Flush Valves Served by Various Sizes of Water Pipea Pipe Size, mm No. of 25 mm Flush Valves 32 1 40 2-4 50 5-12 65 13-25 75 26-40 100 41-100 aTwo 20 mm flush valves are assumed equal to one 25 mm flush valve but can be served by a 25 mm pipe. Water pipe sizing must consider demand factor, available pressure, and length of run.
Table 13 Pressure Drops Used for Sizing Steam Pipea Initial Steam Pressure, kPab Pressure Drop, Pa/m Total Pressure Drop in Steam Supply Piping, kPa Vacuum return 30 to 60 7 to 14 101 7 0.4 108 30 0.4 to 1.7 115 30 3.5 135 60 10 170 115 20 205 225 30 310 450 35 to 70 445 450 to 1100 70 to 105 790 450 to 1100 105 to 170 1140 450 to 2300 170 to 210 aEquipment, control valves, and so forth must be selected based on delivered pressures.
bSubtract 101 to convert to pressure above atmospheric.
Pipe Sizing 35.11 sure of the system (and in practice it should never exceed one-half the initial gage pressure); (2) the pressure drop is not great enough to cause excessive velocities; (3) a constant initial pressure is main-tained, except on systems specially designed for varying initial pressures (e.g., subatmospheric pressure), which normally operate under controlled partial vacuums; and (4) for gravity return sys-tems, the pressure drop to the heating units does not exceed the water column available for removing condensate (i.e., the height above the boiler water line of the lowest point on the steam main, on the heating units, or on the dry return).
Maximum Velocity. For quiet operation, steam velocity should be 40 to 60 m/s, with a maximum of 75 m/s. The lower the veloc-ity, the quieter the system. When the condensate must flow against the steam, even in limited quantity, the velocity of the steam must not exceed limits above which the disturbance between the steam and the counterflowing water may (1) produce objectionable sound, such as water hammer, or (2) result in the retention of water in certain parts of the system until the steam flow is reduced suffi-ciently to permit the water to pass. The velocity at which these dis-turbances take place is a function of (1) pipe size; (2) the pitch of the pipe if it runs horizontally; (3) the quantity of condensate flow-ing against the steam; and (4) the freedom of the piping from water pockets that, under certain conditions, act as a restriction in pipe size. Table 14 lists maximum capacities for various size steam lines.
Equivalent Length of Run. All tables for the flow of steam in pipes based on pressure drop must allow for pipe friction, as well as for the resistance of fittings and valves. These resistances are gen-erally stated in terms of straight pipe; that is, a certain fitting pro-duces a drop in pressure equivalent to the stated number of feet of straight run of the same size of pipe. Table 15 gives the number of feet of straight pipe usually allowed for the more common types of fittings and valves. In all pipe sizing tables in this chapter, the length of run refers to the equivalent length of run as distinguished from the actual length of pipe. A common sizing method is to assume the length of run and to check this assumption after pipes are sized. For this purpose, the length of run is usually assumed to be double the actual length of pipe.
Example 8. Using Table 15, determine the equivalent length of pipe for the run illustrated.
Measured length =40.0 m 100 mm gate valve = 0.6 m Four 100 mm elbows =10.8 m Two 100 mm tees = 11.0 m Equivalent =62.4 m Sizing Charts Figure 10 is the basic chart for determining the flow rate and velocity of steam in Schedule 40 pipe for various values of pressure drop per unit length, based on saturated steam at standard pressure (101.325 kPa). Using the multiplier chart (Figure 11), Figure 10 can be used at all saturation pressures between 101 and 1500 kPa (see Example 10).
LOW-PRESSURE STEAM PIPING Values in Table 16 (taken from Figure 10) provide a more rapid means of selecting pipe sizes for the various pressure drops listed and for systems operated at 25 and 85 kPa (gage). The flow rates shown for 25 kPa can be used for saturated pressures from 7 to 41 kPa, and those shown for 85 kPa can be used for saturated pressures from 55 to 110 kPa with an error not exceeding 8%.
Table 14 Comparative Capacity of Steam Lines at Various Pitches for Steam and Condensate Flowing in Opposite Directions Pitch of Pipe, mm/m Nominal Pipe Diameter, mm 20 25 32 40 50 Capacity Maximum Velocity Capacity Maximum Velocity Capacity Maximum Velocity Capacity Maximum Velocity Capacity Maximum Velocity 20 0.4 2.4 0.9 2.7 1.5 3.4 2.5 3.7 5.4 4.6 40 0.5 3.4 1.1 3.7 2 4.3 3.3 4.9 6.8 5.5 80 0.7 4.0 1.5 4.6 2.5 5.2 4.2 5.8 8.7 7.3 120 0.8 4.3 1.6 5.2 3.1 6.1 4.7 6.7 10.5 8.2 170 0.9 4.9 1.9 5.8 3.4 6.7 5.3 7.3 11.7 9.1 250 1.0 5.2 2.2 6.7 3.9 7.6 5.9 7.9 12.5 9.8 350 1.2 6.7 2.4 7.3 4.2 7.9 6.4 8.5 12.9 9.8 420 1.3 6.7 2.6 7.6 4.9 9.4 7.5 10.1 14.5 10.1 Source: Laschober et al. (1966).
Capacity in g/s; velocity in m/s.
Table 15 Equivalent Length of Fittings to Be Added to Pipe Run Nominal Pipe Diameter, mm Length to Be Added to Run, m Standard Elbow Side Outside Teeb Gate Valvea Globe Valvea Angle Valvea 15 0.4 0.9 0.1 4 2 20 0.5 1.2 0.1 5 3 25 0.7 1.5 0.1 7 4 32 0.9 1.8 0.2 9 5 40 1.1 2.1 0.2 10 6 50 1.3 2.4 0.3 14 7 65 1.5 3.4 0.3 16 8 80 1.9 4.0 0.4 20 10 100 2.7 5.5 0.6 28 14 125 3.3 6.7 0.7 34 17 150 4.0 8.2 0.9 41 20 200 5.2 11 1.1 55 28 250 6.4 14 1.4 70 34 300 8.2 16 1.7 82 40 350 9.1 19 1.9 94 46 aValve in full-open position.
bValues apply only to a tee used to divert the flow in the main to the last riser.
35.12 2001 ASHRAE Fundamentals Handbook (SI) Fig. 10 Flow Rate and Velocity of Steam in Schedule 40 Pipe at Saturation Pressure of 0 psig Notes: Based on Moody Friction Factor where flow of condensate does not inhibit the flow of steam.
See Figure 11 for obtaining flow rates and velocities of all saturation pressures between 0 and 200 psig; see also Examples 9 and 10.
Pipe Sizing 35.13 Both Figure 10 and Table 16 can be used where the flow of con-densate does not inhibit the flow of steam. Columns B and C of Table 17 are used in cases where steam and condensate flow in opposite directions, as in risers or runouts that are not dripped.
Columns D, E, and F are for one-pipe systems and include risers, radiator valves and vertical connections, and radiator and riser runout sizes, all of which are based on the critical velocity of the steam to permit the counterflow of condensate without noise.
Return piping can be sized using Table 18, in which pipe capac-ities for wet, dry, and vacuum return lines are shown for several val-ues of pressure drop per metre of equivalent length. Example 9. What pressure drop should be used for the steam piping of a system if the measured length of the longest run is 150 m, and the ini-tial pressure must not exceed 14 kPa above atmospheric?
Solution: It is assumed, if the measured length of the longest run is 150 m, that when the allowance for fittings is added, the equivalent length of run does not exceed 300 m. Then, with the pressure drop not over one-half of the initial pressure, the drop could be 7 kPa or less.
With a pressure drop of 7 kPa and a length of run of 300 m, the drop would be 23 Pa/m; if the total drop were 3.5 kPa, the drop would be 12 Pa/m. In both cases, the pipe could be sized for a desired capacity according to Figure 10.
On completion of the sizing, the drop could be checked by taking the longest line and actually calculating the equivalent length of run from the pipe sizes determined. If the calculated drop is less than that assumed, the pipe size is adequate; if it is more, an unusual number of fittings is probably involved, and either the lines must be straightened, or the next larger pipe size must be tried.
HIGH-PRESSURE STEAM PIPING Many heating systems for large industrial buildings use high-pressure steam [100 to 1000 kPa (gage)]. These systems usually have unit heaters or large built-up fan units with blast heating coils.
Temperatures are controlled by a modulating or throttling thermo-static valve or by face or bypass dampers controlled by the room air temperature, fan inlet, or fan outlet.
Fig. 11 Velocity Multiplier Chart for Figure 10 Table 16 Flow Rate of Steam in Schedule 40 Pipe Nominal Pipe Size, mm Pressure Drop, Pa/m 14 Pa/m 28 Pa/m 58 Pa/m 113 Pa/m 170 Pa/m 225 Pa/m 450 Pa/m Sat. Press., kPa Sat. Press., kPa Sat. Press., kPa Sat. Press., kPa Sat. Press., kPa Sat. Press., kPa Sat. Press., kPa 25 85 25 85 25 85 25 85 25 85 25 85 25 85 20 1.1 1.4 1.8 2.0 2.5 3.0 3.7 4.4 4.5 5.4 5.3 6.3 7.6 9.2 25 2.1 2.6 3.3 3.9 4.7 5.8 6.8 8.3 8.6 10 10 12 14 17 32 4.5 5.7 6.7 8.3 9.8 12 14 17 18 21 20 25 29 35 40 7.1 8.8 11 13 15 19 22 26 27 33 31 38 45 54 50 14 17 20 24 29 36 42 52 53 64 60 74 89 107 65 22 27 33 39 48 58 68 83 86 103 98 120 145 173 80 40 48 59 69 83 102 121 146 150 180 174 210 246 302 90 58 69 84 101 125 153 178 214 219 265 252 305 372 435 100 81 101 120 146 178 213 249 302 309 378 363 436 529 617 125 151 180 212 265 307 378 450 536 552 662 643 769 945 1 080 150 242 290 355 422 499 611 718 857 882 1 080 1 060 1 260 1 500 1 790 200 491 605 702 882 1 020 1 260 1 440 1 800 1 830 2 230 2 080 2 580 3 020 3 720 250 907 1110 1 290 1 590 1 890 2 290 2 650 3 280 3 300 4 030 3 780 4 660 5 380 6 550 300 1 440 1 730 2 080 2 460 2 950 3 580 4 160 5 040 5 170 6 240 6 050 7 250 8 540 10 200 Notes: 1. Flow rate is in g/s at initial saturation pressures of 25 and 85 kPa (gage). Flow is based on Moody friction factor, where the flow of condensate does not inhibit the flow of steam.
2. The flow rates at 25 kPa cover saturated pressure from 7 to 41 kPa, and the rates at 85 kPa cover saturated pressure from 55 to 110 kPa with an error not exceeding 8%.
3. The steam velocities corresponding to the flow rates given in this table can be found from Figures 10 and 11.
35.14 2001 ASHRAE Fundamentals Handbook (SI) Use of Basic and Velocity Multiplier Charts Example 10. Given a flow rate of 0.85 kg/s, an initial steam pressure of 800 kPa, and a pressure drop of 2.5 kPa/m, find the size of Schedule 40 pipe required and the velocity of steam in the pipe.
Solution: The following steps are illustrated by the broken line on Fig-ures 10 and 11.
1. Enter Figure 10 at a flow rate of 0.85 kg/s, and move vertically to the horizontal line at 800 kPa. 2. Follow along inclined multiplier line (upward and to the left) to horizontal 101 kPa line. The equivalent mass flow at 101 kPa is about 0.30 kg/s.
3. Follow the 0.30 kg/s line vertically until it intersects the horizontal line at 2500 Pa/m pressure drop. Nominal pipe size is 65 mm. The equivalent steam velocity at 101 kPa is about 165 m/s.
4. To find the steam velocity at 800 kPa, locate the value of 165 m/s on the ordinate of the velocity multiplier chart (Figure 11) at 101 kPa.
5. Move along the inclined multiplier line (downward and to the right) until it intersects the vertical 800 kPa pressure line. The velocity is about 65 m/s.
Note: Steps 1 through 5 would be rearranged or reversed if different data were given. STEAM CONDENSATE SYSTEMS The majority of steam systems used in heating applications are two-pipe systems, in which the two pipes are the “steam” pipe and the “condensate” pipe. This discussion is limited to the sizing of the condensate lines in two-pipe systems.
Two-Pipe Systems When steam is used for heating a liquid to 102°C or less (e.g., in domestic water heat exchangers, domestic heating water con-verters, or air-heating coils), the devices are usually provided with a steam control valve. As the control valve throttles, the absolute Table 17 Steam Pipe Capacities for Low-Pressure Systems Nominal Pipe Size, mm Capacity, g/s Two-Pipe System One-Pipe Systems Condensate Flowing Against Steam Supply Risers Upfeed Radiator Valves and Vertical Connections Radiator and Riser Runouts Vertical Horizontal A Ba Cb Dc E Fb 20 1.0 0.9 0.8 — 0.9 25 1.8 1.8 1.4 0.9 0.9 32 3.9 3.4 2.5 2.0 2.0 40 6.0 5.3 4.8 2.9 2.0 50 12 11 9.1 5.3 2.9 65 20 17 14 — 5.3 80 36 25 25 — 8.2 90 49 36 36 — 15 100 64 54 48 — 23 125 132 99 — — 35 150 227 176 — — 69 200 472 378 — — — 250 882 718 — — — 300 1450 1200 — — — 400 2770 2390 — — — Notes: 1. For one- or two-pipe systems in which condensate flows against the steam flow.
2. Steam at average pressure of 7 kPa (gage) is used as a basis of calculating capacities.
aDo not use Column B for pressure drops of less than 13 Pa per metre of equivalent run. Use Figure 10 or Table 15 instead.
bPitch of horizontal runouts to risers and radiators should be not less than 40 mm/m.
Where this pitch cannot be obtained, runouts over 2.5 m in length should be one pipe size larger than that called for in this table.
cDo not use Column D for pressure drops of less than 9 Pa per metre of equivalent run, except on sizes 80 mm and over. Use Figure 10 or Table 15 instead.
Table 18 Return Main and Riser Capacities for Low-Pressure Systems, g/s Pipe Size, mm 7 Pa/m 9 Pa/m 14 Pa/m 28 Pa/m 57 Pa/m 113 Pa/m Wet Dry Vac.
Wet Dry Vac.
Wet Dry Vac.
Wet Dry Vac.
Wet Dry Vac.
Wet Dry Vac.
G H I J K L M N O P Q R S T U V W X Y Return Main 20 — — — — — 5 — — 13 — — 18 — — 25 — — 36 25 16 8 — 18 9 18 22 10 22 32 13 31 44 14 44 — — 62 32 27 16 — 31 19 31 38 21 38 54 27 54 76 30 76 — — 107 40 43 26 — 50 30 49 60 33 60 85 43 85 120 48 120 — — 169 50 88 59 — 102 67 103 126 72 126 176 93 179 252 104 252 — — 357 65 149 96 — 199 109 171 212 120 212 296 155 300 422 171 422 — — 596 80 237 184 — 268 197 275 338 221 338 473 284 479 674 315 674 — — 953 90 347 248 — 416 277 410 504 315 504 693 407 716 1010 451 1010 — — 1424 100 489 369 — 577 422 567 693 473 693 977 609 984 1390 678 1390 — — 1953 125 — — — — — 993 — — 1220 — — 1730 — — 2440 — — 3440 150 — — — — — 1590 — — 1950 — — 2770 — — 3910 — — 5519 Riser 20 — 6 — — 6 18 — 6 22 — 6 31 — 6 44 — — 62 25 — 14 — — 14 31 — 14 38 — 14 54 — 14 76 — — 107 32 — 31 — — 31 49 — 31 60 — 31 85 — 31 120 — — 169 40 — 47 — — 47 103 — 47 126 — 47 179 — 47 252 — — 357 50 — 95 — — 95 171 — 95 212 — 95 300 — 95 422 — — 596 65 — — — — — 275 — — 338 — — 479 — — 674 — — 953 80 — — — — — 410 — — 504 — — 716 — — 1010 — — 1424 90 — — — — — 564 — — 693 — — 984 — — 1390 — — 1953 100 — — — — — 993 — — 1220 — — 1730 — — 2440 — — 3440 125 — — — — — 1590 — — 1950 — — 2772 — — 3910 — — 5519 Pipe Sizing 35.15 pressure in the load device decreases, removing all pressure moti-vation for flow in the condensate return system. In order to ensure the flow of steam condensate from the load device through the trap and into the return system, it is necessary to provide a vac-uum breaker on the device ahead of the trap. This ensures a mini-mum pressure at the trap inlet of atmospheric pressure plus whatever liquid leg the designer has provided. Then, to ensure flow through the trap, it is necessary to design the condensate sys-tem so that it will never have a pressure above atmospheric in the condensate return line. Vented (Open) Return Systems. To achieve this pressure requirement, the condensate return line is usually vented to the atmosphere (1) near the point of entrance of the flow streams from the load traps, (2) in proximity to all connections from drip traps, and (3) at transfer pumps or feedwater receivers.
With this design, the only motivation for flow in the return sys-tem is gravity. Return lines that are below the liquid level in the downstream receiver or boiler and are thus filled with liquid are called wet returns; those above the liquid level have both liquid and gas in the pipes and are called dry returns.
The dry return lines in a vented return system have flowing liquid in the bottom of the line and gas or vapor in the top (Figure 12A).
The liquid is the condensate, and the gas may be steam, air, or a mix-ture of the two. The flow phenomenon for these dry return systems is open channel flow, which is best described by the Manning equation: (12) where Q = volumetric flow rate, m3/s A = cross-sectional area of conduit, m2 r = hydraulic radius of conduit, m n = coefficient of roughness (usually 0.012) S = slope of conduit, m/m Table 19 is a solution to Equation (12) that shows pipe size capacities for steel pipes with various pitches. Recommended practice is to size vertical lines by the maximum pitch shown, although they would actually have a capacity far in excess of that shown. As the pitch increases, hydraulic jump that could fill the pipe and other transient effects that could cause water hammer should be avoided. Flow values in Table 19 are calculated for Schedule 40 steel pipe, with a factor of safety of 3.0, and can be used for copper pipes of the same nominal pipe size.
The flow characteristics of wet return lines (Figure 12B) are best described by the Darcy-Weisbach equation [Equation (1)]. The motivation for flow is the fluid pressure difference between the entering section of the flooded line and the leaving section. It is common practice, in addition to providing for the fluid pressure dif-ferential, to slope the return in the direction of flow to a collection point such as a dirt leg in order to clear the line of sediment or solids.
Table 20 is a solution to Equation (1) that shows pipe size capacity for steel pipes with various available fluid pressures. Table 20 can also be used for copper tubing of equal nominal pipe size.
Nonvented (Closed) Return Systems. For those systems in which there is a continual steam pressure difference between the point where the condensate enters the line and the point where it leaves (Figure 12C), Table 18 or Table 21, as applicable, can be used for sizing the condensate lines. Although these tables express con-densate capacity without slope, common practice is to slope the Fig. 12 Types of Condensate Return Systems Table 19 Vented Dry Condensate Return for Gravity Flow Based on Manning Equation Nominal Diameter, mm Condensate Flow, g/sa,b Condensate Line Slope 0.5% 1% 2% 4% 15 5 7 10 13 20 10 14 20 29 25 19 27 39 54 32 40 57 80 113 40 60 85 121 171 50 117 166 235 332 65 189 267 377 534 80 337 476 674 953 100 695 983 1390 1970 125 1270 1800 2540 3590 150 2070 2930 4150 5860 a Flow is in g/s of 82°C water for Schedule 40 steel pipes.
b Flow was calculated from Equation (12) and rounded.
Q 1.00Ar2 3 ⁄ S 1 2 ⁄ n ------------------------------------= 35.16 2001 ASHRAE Fundamentals Handbook (SI) lines in the direction of flow to a collection point similar to wet returns to clear the lines of sediment or solids.
When saturated condensate at pressures above the return system pressure enters the return (condensate) mains, some of the liquid flashes to steam. This occurs typically at drip traps into a vented return system or at load traps leaving process load devices that are not valve-controlled and typically have no subcooling. If the return main is vented, the vent lines will relieve any excessive pressure and prevent a back pressure phenomenon that could restrict the flow through traps from valved loads; the pipe sizing would be as described above for vented dry returns. If the return line is not vented, the flash steam results in a pressure rise at that point and the piping could be sized as described above for closed returns, and in accordance with Table 18 or Table 21, as applicable.
The passage of the fluid through the steam trap is a throttling or constant enthalpy process. The resulting fluid on the downstream side of the trap can be a mixture of saturated liquid and vapor. Thus, in nonvented returns, it is important to understand the condition of the fluid when it enters the return line from the trap.
Table 20 Vented Wet Condensate Return for Gravity Flow Based on Darcy-Weisbach Equation Nominal Diameter, mm Condensate Flow, g/sa,b Condensate Pressure, Pa/m 50 100 150 200 250 300 350 400 15 13 19 24 28 32 35 38 41 20 28 41 51 60 68 74 81 87 25 54 79 98 114 129 142 154 165 32 114 165 204 238 267 294 318 341 40 172 248 308 358 402 442 479 513 50 334 482 597 694 779 857 928 994 65 536 773 956 1110 1 250 1 370 1 480 1 590 80 954 1 370 1 700 1 970 2 210 2 430 2 630 2 810 100 1 960 2 810 3 470 4 030 4 520 4 960 5 370 5 750 125 3 560 5 100 6 290 7 290 8 180 8 980 9 720 10 400 150 5 770 8 270 10 200 11 800 13 200 14 500 15 700 16 800 a Flow is in g/s of 82°C water for Schedule 40 steel pipes.
b Flow was calculated from Equation (1) and rounded.
Table 21 Flow Rate for Dry-Closed Returns Pipe Dia. D, mm Supply Pressure = 35 kPa Return Pressure = 0 kPa Supply Pressure = 100 kPa Return Pressure = 0 kPa Supply Pressure = 210 kPa Return Pressure = 0 kPa Supply Pressure = 340 kPa Return Pressure = 0 kPa ∆p/L, Pa/m 15 60 240 15 60 240 15 60 240 15 60 240 Flow Rate, g/s 15 30 66 139 12 26 57 8 16 35 5 12 25 20 64 141 302 26 57 120 16 35 74 11 25 53 25 126 271 572 50 108 229 32 67 141 23 48 101 32 265 567 1 200 106 227 479 66 140 295 47 101 212 40 399 854 1 790 160 343 718 98 210 442 71 151 318 50 786 1 680 a 315 670 a 194 412 a 140 296 a 65 1 260 2 680 a 508 1 070 a 312 662 a 224 476 a 80 2 270 4 790 a 907 1 920 a 559 1 180 a 402 848 a 100 4 690 9 830 a 1 880 3 940 a 1 160 2 420 a 839 1 740 a 150 13 900 a a 5 580 a a 3 440 a a 2 470 a a 200 28 800 a a 11 600 a a 7 110 a a 5 100 a a Pipe Dia. D, mm Supply Pressure = 690 kPa Return Pressure = 0 kPa Supply Pressure = 1030 kPa Return Pressure = 0 kPa Supply Pressure = 690 kPa Return Pressure = 100 kPa Supply Pressure = 1030 kPa Return Pressure = 100 kPa ∆p/L, Pa/m 15 60 240 15 60 240 15 60 240 15 60 240 Flow Rate, g/s 15 4 8 17 3 6 14 7 15 33 5 12 25 20 8 17 37 6 14 29 15 33 71 12 25 53 25 15 33 69 13 26 57 30 63 134 23 49 101 32 32 68 142 25 55 117 63 134 277 48 101 212 40 48 102 214 39 83 176 95 202 418 72 152 315 50 95 200 a 77 164 a 185 391 813 141 296 617 65 151 321 a 123 265 a 299 630 1 300 227 476 983 80 272 573 a 222 467 a 533 1 120 a 403 845 a 100 562 1180 a 459 961 a 1 100 2 290 a 834 1 740 a 150 1 660 a a 1 360 a a 3 260 6 750 a 2 470 5 120 a 200 3 450 a a 2 820 a a 6 730 13 900 a 5 100 10 500 a aFor these sizes and pressure losses, the velocity is above 35 m/s. Select another combination of size and pressure loss.
Pipe Sizing 35.17 The condition of the condensate downstream of the trap can be expressed by the quality x, defined as (13) where mv = mass of saturated vapor in condensate ml = mass of saturated liquid in condensate Likewise, the volume fraction Vc of the vapor in the condensate is expressed as (14) where Vv = volume of saturated vapor in condensate Vl = volume of saturated liquid in condensate The quality and the volume fraction of the condensate down-stream of the trap can be estimated from Equations (13) and (14), respectively.
(15) (16) where h1 = enthalpy of liquid condensate entering trap evaluated at supply pressure for saturated condensate or at saturation pressure corresponding to temperature of subcooled liquid condensate hf2 = enthalpy of saturated liquid at return or downstream pressure of trap hg2 = enthalpy of saturated vapor at return or downstream pressure of trap vf2 = specific volume of saturated liquid at return or downstream pressure of trap vg2 = specific volume of saturated vapor at return or downstream pressure of trap Table 22 presents some values for quality and volume fraction for typical supply and return pressures in heating and ventilating sys-tems. Note that the percent of vapor on a mass basis x is small, while the percent of vapor on a volume basis Vc is very large. This indicates that the return pipe cross section is predominantly occupied by vapor. Figure 13 is a working chart to determine the quality of the condensate entering the return line from the trap for various combi-nations of supply and return pressures. If the liquid is subcooled entering the trap, the saturation pressure corresponding to the liquid temperature should be used for the supply or upstream pressure.
One-Pipe Systems Gravity one-pipe air vent systems in which steam and conden-sate flow in the same pipe, frequently in opposite directions, are considered obsolete and are no longer being installed. Chapter 33 of the 1993 ASHRAE Handbook—Fundamentals or earlier ASH-RAE Handbooks include descriptions of and design information for one-pipe systems.
GAS PIPING Piping for gas appliances should be of adequate size and installed so that it provides a supply of gas sufficient to meet the maximum demand without undue loss of pressure between the point of supply (the meter) and the appliance. The size of gas pipe required depends on (1) maximum gas consumption to be pro-vided, (2) length of pipe and number of fittings, (3) allowable pres-sure loss from the outlet of the meter to the appliance, and (4) specific gravity of the gas.
Insufficient gas flow from excessive pressure losses in gas sup-ply lines can cause inefficient operation of gas-fired appliances and sometimes create hazardous operations. Gas-fired appliances are normally equipped with a data plate giving information on maxi-mum gas flow requirements or Btu input as well as inlet gas pres-sure requirements. The gas utility in the area of installation can give the gas pressure available at the utility’s gas meter. Using the infor-mation, the required size of gas piping can be calculated for satis-factory operation of the appliance(s).
Table 24 gives pipe capacities for gas flow for up to 60 m of pipe based on a specific gravity of 0.60. Capacities for pressures less than 10 kPa may also be determined by the following equation from NFPA/IAS National Fuel Gas Code: (17) where Q = flow rate at 15°C and 101 kPa, L/s Table 22 Flash Steam from Steam Trap on Pressure Drop Supply Pressure, kPa (gage) Return Pressure, kPa (gage) x, Fraction Vapor, Mass Basis Vc, Fraction Vapor, Volume Basis 35 0 0.016 0.962 103 0 0.040 0.985 207 0 0.065 0.991 345 0 0.090 0.994 690 0 0.133 0.996 1030 0 0.164 0.997 690 103 0.096 0.989 1030 103 0.128 0.992 x mv ml mv + ------------------= Vc Vv Vl Vv + -----------------= x h1 hf2 – hg2 hf2 – -------------------= Vc xvg2 vf2 1 x – ( ) xvg2 + ---------------------------------------= Table 23 Estimated Return Line Pressures Pressure Drop, Pa/m Pressure in Return Line, Pa (gage) 200 kPa (gage) Supply 1000 kPa (gage) Supply 30 3.5 9 60 7 18 120 14 35 180 21 52 240 28 70 480 — 138 Fig. 13 Working Chart for Determining Percentage of Flash Steam (Quality) Q 0.0001d2.623 p ∆ CL ⁄ ( )0.541 = 35.18 2001 ASHRAE Fundamentals Handbook (SI) d = inside diameter of pipe, mm ∆p = pressure drop, Pa C = factor for viscosity, density, and temperature = 0.00223(t + 273)s0.848µ0.152 t = temperature, °C s = ratio of density of gas to density of air at 15°C and 101 kPa µ = viscosity of gas, µPa·s (12 for natural gas, 8 for propane) L = pipe length, m Gas service in buildings is generally delivered in the “low-pres-sure” range of 1.7 kPa (gage). The maximum pressure drop allow-able in piping systems at this pressure is generally 125 Pa but is subject to regulation by local building, plumbing, and gas appliance codes (see also the NFPA/IAS National Fuel Gas Code).
Where large quantities of gas are required or where long lengths of pipe are used (e.g., in industrial buildings), low-pressure limitations result in large pipe sizes. Local codes may allow and local gas companies may deliver gas at higher pressures [e.g., 15, 35, or 70 kPa (gage)]. Under these conditions, an allowable pres-sure drop of 10% of the initial pressure is used, and pipe sizes can be reduced significantly. Gas pressure regulators at the appliance must be specified to accommodate higher inlet pressures.
NFPA/IAS (1992) provides information on pipe sizing for various inlet pressures and pressure drops at higher pressures.
More complete information on gas piping can be found in the Gas Engineers’ Handbook (1970).
FUEL OIL PIPING The pipe used to convey fuel oil to oil-fired appliances must be large enough to maintain low pump suction pressure and, in the case of circulating loop systems, to prevent overpressure at the burner oil pump inlet. Pipe materials must be compatible with the fuel and must be carefully assembled to eliminate all leaks. Leaks in suction lines cause pumping problems that result in unreliable burner oper-ation. Leaks in pressurized lines create fire hazards. Cast-iron or aluminum fittings and pipe are unacceptable. Pipe joint compounds must be selected carefully.
Oil pump suction lines should be sized so that at maximum suc-tion line flow conditions, the maximum vacuum will not exceed 34 kPa for distillate grade fuels and 50 kPa for residual oils. Oil supply lines to burner oil pumps should not be pressurized by circulating loop systems or aboveground oil storage tanks to more than 34 kPa, or pump shaft seals may fail. A typical oil circulating loop system is shown in Figure 14.
In assembling long fuel pipe lines, care should be taken to avoid air pockets. On overhead circulating loops, the line should vent air at all high points. Oil supply loops for one or more burners should Table 24 Maximum Capacity of Gas Pipe in Litres per Second Nominal Iron Pipe Size, mm Internal Diameter, mm Length of Pipe, m 5 10 15 20 25 30 35 40 45 50 55 60 8 9.25 0.19 0.13 0.11 0.09 0.08 0.07 0.07 0.06 0.06 0.06 0.05 0.05 10 12.52 0.43 0.29 0.24 0.20 0.18 0.16 0.15 0.14 0.13 0.12 0.12 0.11 15 15.80 0.79 0.54 0.44 0.37 0.33 0.30 0.28 0.26 0.24 0.23 0.22 0.21 20 20.93 1.65 1.13 0.91 0.78 0.69 0.63 0.58 0.54 0.50 0.47 0.45 0.43 25 26.14 2.95 2.03 1.63 1.40 1.24 1.12 1.03 0.96 0.90 0.85 0.81 0.77 32 35.05 6.4 4.4 3.5 3.0 2.7 2.4 2.2 2.1 1.9 1.8 1.7 1.7 40 40.89 9.6 6.6 5.3 4.5 4.0 3.6 3.3 3.1 2.9 2.8 2.6 2.5 50 52.50 18.4 12.7 10.2 8.7 7.7 7.0 6.4 6.0 5.6 5.3 5.0 4.8 65 62.71 29.3 20.2 16.2 13.9 12.3 11.1 10.2 9.5 8.9 8.4 8.0 7.7 80 77.93 51.9 35.7 28.6 24.5 21.7 19.7 18.1 16.8 15.8 14.9 14.2 13.5 100 102.26 105.8 72.7 58.4 50.0 44.3 40.1 36.9 34.4 32.2 30.4 28.9 27.6 Note: Capacity is in litres per second at gas pressures of 3.5 kPa (gage) or less and a pressure drop of 75 kPa; density = 0.735 kg/m3.
Copyright by the American Gas Association and the National Fire Protection Asso-ciation. Used by permission of the copyright holder.
Fig. 14 Typical Oil Circulating Loop Pipe Sizing 35.19 be the continuous circulation type, with excess fuel returned to the storage tank. Dead-ended pressurized loops can be used, but air or vapor venting is more problematic.
Where valves are used, select ball or gate valves. Globe valves are not recommended because of their high pressure drop charac-teristics. Oil lines should be tested after installation, particularly if they are buried, enclosed, or otherwise inaccessible. Failure to perform this test is a frequent cause of later operating difficulties. A suction line can be hydrostatically tested at 1.5 times its maximum operat-ing pressure or at a vacuum of not less than 70 kPa. Pressure or vac-uum tests should continue for at least 60 min. If there is no noticeable drop in the initial test pressure, the lines can be consid-ered tight.
Pipe Sizes for Heavy Oil Table 25 and Table 26 give recommended pipe sizes for handling No. 5 and No. 6 oils (residual grades) and No. 1 and No. 2 oils (dis-tillate grades), respectively. Storage tanks and piping and pumping facilities for delivering the oil from the tank to the burner are important considerations in the design of an industrial oil-burning system.
The construction and location of the tank and oil piping are usu-ally subject to local regulations and National Fire Protection Asso-ciation (NFPA) Standards 30 and 31.
REFERENCES Ball, E.F. and C.J.D. Webster. 1976. Some measurements of water-flow noise in copper and ABS pipes with various flow velocities. The Building Services Engineer 44(2):33.
BOCA. 1992. BOCA National plumbing code, 9th ed. Building Officials and Code Administrators International, Country Club Hills, IL.
Carrier. 1960. Piping design. In System design manual. Carrier Air Condi-tioning Company, Syracuse, NY.
Crane Co. 1976. Flow of fluids through valves, fittings and pipe. Technical Paper No. 410. Crane Company, New York.
Dawson, F.M. and J.S. Bowman. 1933. Interior water supply piping for res-idential buildings. University of Wisconsin Experiment Station, No. 77.
Freeman, J.R. 1941. Experiments upon the flow of water in pipes. American Society of Mechanical Engineers, New York.
Gas engineers’ handbook. 1970. The Industrial Press, New York.
Giesecke, F.E. 1926. Friction of water elbows. ASHVE Transactions 32:303.
Giesecke, F.E. and W.H. Badgett. 1931. Friction heads in one-inch standard cast-iron tees. ASHVE Transactions 37:395.
Giesecke, F.E. and W.H. Badgett. 1932a. Loss of head in copper pipe and fit-tings. ASHVE Transactions 38:529.
Giesecke, F.E. and W.H. Badgett. 1932b. Supplementary friction heads in one-inch cast-iron tees. ASHVE Transactions 38:111.
Grinnell Company. 1951. Piping design and engineering. Grinnell Com-pany, Cranston, RI.
HDR design guide. 1981. Hennington, Durham and Richardson, Omaha, NE.
Hegberg, R.A. 1995. Where did the k-factors for pressure loss in fittings come from? ASHRAE Transactions 101(1): 1264-78.
Howell, R.H. 1985. Evaluation of sizing methods for steam condensate sys-tems. ASHRAE Transactions 91(1).
Hunter, R.B. 1940. Methods of estimating loads in plumbing systems. NBS Report BMS 65. National Institute of Standards and Technology, Gaith-ersburg, MD.
Hunter, R.B. 1941. Water distributing systems for buildings. NBS Report BMS 79. National Institute of Standards and Technology.
Hydraulic Institute. 1979. Engineering data book. Hydraulic Institute, Par-sippany, NJ.
IAPMO. 1994. Uniform plumbing code. International Association of Plumbing and Mechanical Officials, Walnut, CA.
Idelchik, I.E. 1986. Handbook of hydraulic resistance. Hemisphere Publish-ing Corporation, New York.
ISA. 1985. Flow equations for sizing control valves. ANSI/ISA Standard S75.01-85. International Society for Measurement and Control, Research Triangle Park, NC.
Laschober, R.R., G.Y. Anderson, and D.G. Barbee. 1966. Counterflow of steam and condensate in slightly pitched pipes. ASHRAE Transactions 72(1):157.
Marseille, B. 1965. Noise transmission in piping. Heating and Ventilating Engineering (June):674.
NAPHCC. 1996. National standard plumbing code. National Association of Plumbing-Heating-Cooling Contractors, Falls Church, VA.
NFPA. 1992. Installation of oil burning equipment. ANSI/NFPA Standard 31-92. National Fire Protection Association, Quincy, MA.
NFPA. 1993. Flammable and combustible liquids code. ANSI/NFPA Stan-dard 30-93.
NFPA/IAS. 1992. National fuel gas code. ANSI/NFPA Standard 54-92.
National Fire Protection Association, Quincy, MA. ANSI/IAS Standard Z223.1-92. American Gas Association, Arlington, VA.
Obrecht, M.F. and M. Pourbaix. 1967. Corrosion of metals in potable water systems. AWWA 59:977. American Water Works Association, Denver, CO.
Plastic Pipe Institute. 1971. Water flow characteristics of thermoplastic pipe. Plastic Pipe Institute, New York.
Rahmeyer, W.J. 1999a. Pressure loss coefficients of threaded and forged weld pipe fittings for ells, reducing ells, and pipe reducers. ASHRAE Transactions 105(2):334-54.
Rahmeyer, W.J. 1999b. Pressure loss coefficients of pipe fittings for threaded and forged weld pipe tees. ASHRAE Transactions 105(2):355-85.
Rogers, W.L. 1953. Experimental approaches to the study of noise and noise transmission in piping systems. ASHVE Transactions 59:347-60.
Rogers, W.L. 1954. Sound-pressure levels and frequencies produced by flow of water through pipe and fittings. ASHRAE Transactions 60:411-30.
Rogers, W.L. 1956. Noise production and damping in water piping. ASHAE Transactions 62:39.
Table 25 Recommended Nominal Size for Fuel Oil Suction Lines from Tank to Pump (Residual Grades No. 5 and No. 6) Pumping Rate, L/h Length of Run in Metres at Maximum Suction Lift of 4.5 kPa 10 20 30 40 50 60 70 80 90 100 50 40 40 40 50 50 50 65 65 65 80 100 40 40 50 50 65 65 65 65 80 80 200 40 50 50 50 65 65 65 80 80 80 300 50 50 65 65 65 80 80 80 80 80 400 50 50 65 65 80 80 80 80 80 100 500 50 65 65 65 80 80 80 80 100 100 600 65 65 65 80 80 80 100 100 100 100 700 65 65 65 80 80 100 100 100 100 100 800 65 65 80 80 100 100 100 100 100 100 Notes: 1. Sizes (in millimetres) are nominal.
2. Pipe sizes smaller than 25 mm ISO are not recommended for use with residual grade fuel oils.
3. Lines conveying fuel oil from pump discharge port to burners and tank return may be reduced by one or two sizes, depending on piping length and pressure losses.
Table 26 Recommended Nominal Size for Fuel Oil Suction Lines from Tank to Pump (Distillate Grades No. 1 and No. 2) Pumping Rate, L/h Length of Run in Metres at Maximum Suction Lift of 9.0 kPa 10 20 30 40 50 60 70 80 90 100 50 15 15 15 15 15 20 20 20 25 25 100 15 15 15 15 20 20 20 20 25 25 200 15 20 20 20 20 20 25 25 25 25 300 15 20 20 20 20 25 25 25 25 32 400 20 20 20 20 25 25 25 25 32 32 500 20 25 25 25 25 25 32 32 32 32 600 20 25 25 25 25 32 32 32 32 50 700 20 25 25 25 25 32 32 32 50 50 800 20 25 25 25 32 32 32 32 50 50 Note: Sizes (in millimetres) are nominal.
35.20 2001 ASHRAE Fundamentals Handbook (SI) Sanks, R.L. 1978. Water treatment plant design for the practicing engineer.
Ann Arbor Science Publishers, Ann Arbor, MI.
SBCCI. 1994. Standard plumbing code. Southern Building Code Congress International, Birmingham, AL.
Smith, T. 1983. Reducing corrosion in heating plants with special reference to design considerations. Anti-Corrosion Methods and Materials 30 (October):4.
Stewart, W.E. and C.L. Dona. 1987. Water flow rate limitations. ASHRAE Transactions 93(2):811-25.
Williams, G.J. 1976. The Hunter curves revisited. Heating/Piping/Air Con-ditioning (November):67.
Williams, G.S. and A. Hazen. 1933. Hydraulic tables. John Wiley and Sons, New York.
36.1 CHAPTER 36 ABBREVIATIONS AND SYMBOLS Abbreviations for Text, Drawings, and Computer Programs ............................................................ 36.1 Letter Symbols ......................................................................... 36.1 Dimensionless Numbers............................................................ 36.4 Mathematical Symbols .............................................................. 36.4 Subscripts................................................................................. 36.5 Graphical Symbols for Drawings ............................................. 36.5 Piping System Identification ................................................. 36.10 HIS CHAPTER contains information about abbreviations and Tsymbols for heating, ventilating, air-conditioning, and refriger-ating (HVAC&R) engineers.
Abbreviations are shortened forms of names and expressions used in text, drawings, and computer programs. This chapter dis-cusses conventional English language abbreviations that may be different in other languages. A letter symbol represents a quantity or a unit, not its name, and is independent of language. Because of this, use of a letter symbol is preferred over abbreviations for unit or quantity terms. Letter symbols necessary for individual chapters are defined in the chapters where they occur.
Abbreviations are never used for mathematical signs, such as the equality sign (=) or division sign (/), except in computer program-ming, where the abbreviation functions as a letter symbol. Mathe-matical operations are performed only with symbols. Abbreviations should be used only where necessary to save time and space; avoid their usage in documents circulated in foreign countries.
Graphical symbols in this chapter of piping, ductwork, fit-tings, and in-line accessories can be used on scale drawings and diagrams.
Identifying piping by legend and color promotes greater safety and lessens the chance of error in emergencies. Piping identification is now required throughout the United States by the Occupational Safety and Health Administration (OSHA) for some industries and by many federal, state, and local codes.
ABBREVIATIONS FOR TEXT, DRAWINGS, AND COMPUTER PROGRAMS Table 1 gives some abbreviations, as well as others commonly found on mechanical drawings and abbreviations (symbols) used in computer programming. Abbreviations specific to a single subject are defined in the chapters in which they appear. Additional abbre-viations used on drawings can be found in the section on Graphical Symbols for Drawings.
The abbreviations (symbols) used for computer programming for the HVAC&R industries have been developed by ASHRAE Technical Committee 1.5, Computer Applications. These symbols identify computer variables, subprograms, subroutines, and func-tions commonly applied in the industry. Using these symbols enhances comprehension of the program listings and provides a clearly defined nomenclature in applicable computer programs.
Certain programming languages differentiate between real num-bers (numbers with decimals) and integers (numbers without deci-mals) by reserving certain initial letters of a variable for integer numbers. Many of the symbols listed in this chapter begin with these letters and, in order to make them real numbers, must be pre-fixed with a noninteger letter. Some symbols have two or more options listed. The longest abbreviation is preferred and should be used if possible. However, it is sometimes necessary to shorten the symbol to further identify the variable. For instance, the area of a wall cannot be defined as WALLAREA because some computer languages restrict the num-ber of letters in a variable name. Therefore, a shorter variable symbol is applied, and WALLAREA becomes WALLA or WAREA.
Many advanced computer programming languages such as Basic, C, and C++ do not have the limitations of older computer lan-guage compilers. It is good programming practice to include the complete name of each variable and to define any abbreviations in the comments section at the beginning of each module of code.
Abbreviations should be used to help clarify the variables in an equation and not to obscure the readability of the code.
In Table 1, the same symbol is sometimes used for different terms. This liberty is taken because it is unlikely that the two terms would be used in the same program. If such were the case, one of the terms would require a suffix or prefix to differentiate it from the other.
LETTER SYMBOLS Letter symbols include symbols for physical quantities (quantity symbols) and symbols for the units in which these quantities are measured (unit symbols). Quantity symbols, such as I for electric current, are listed in this chapter and are printed in italic type. A unit symbol is a letter or group of letters such as mm for millimetre or a special sign such as ° for degrees and is printed in Roman type. Sub-scripts and superscripts are governed by the same principles. Letter symbols are restricted mainly to the English and Greek alphabets.
Quantity symbols may be used in mathematical expressions in any way consistent with good mathematical usage. The product of two quantities, a and b, is indicated by ab. The quotient is a/b, or ab−1. To avoid misinterpretation, parentheses must be used if more than one slash (/) is employed in an algebraic term; for example, (a/b)/c or a/(b/c) is correct, but not a/b/c.
Subscripts and superscripts, or several of them separated by commas, may be attached to a single basic letter (kernel), but not to other subscripts or superscripts. A symbol that has been modified by a superscript should be enclosed in parentheses before an exponent is added (Xa)3. Symbols can also have alphanumeric marks such as ′ (prime), + (plus), and (asterisk).
More detailed information on the general principles of letter symbol standardization are in standards listed at the end of this chapter. The letter symbols, in general, follow these standards, which are out of print: Y10.3M Letter Symbols for Mechanics and Time-Related Phenomena Y10.4-82 Letter Symbols for Heat and Thermodynamics Other symbols chosen by an author for a physical magnitude not appearing in any standard list should be ones that do not already have different meanings in the field of the text.
The preparation of this chapter is assigned to TC 1.6, Terminology.
36.2 2001 ASHRAE Fundamentals Handbook (SI) Table 1 Abbreviations for Text, Drawings, and Computer Programs Term Text Drawings Program above finished floor — AFF — absolute abs ABS ABS accumulat(e, -or) acc ACCUM ACCUM air condition(-ing, -ed) — AIR COND — air-conditioning unit(s) — ACU ACU air-handling unit — AHU AHU alteration altrn ALTRN — alternating current ac AC AC altitude alt ALT ALT ambient amb AMB AMB American National Standards Institute1 ANSI ANSI — American wire gage AWG AWG — ampere amp AMP AMP, AMPS angle — — ANG angle of incidence — — ANGI apparatus dew point adp ADP ADP approximate approx.
APPROX — area — — A atmosphere atm ATM — average avg AVG AVG azimuth az AZ AZ azimuth, solar — — SAZ azimuth, wall — — WAZ barometer(-tric) baro BARO — bill of material b/m BOM — boiling point bp BP BP Brown & Sharpe wire gage B&S B&S — Celsius °C °C °C center to center c to c C TO C — circuit ckt CKT CKT clockwise cw CW — coefficient coeff.
COEF COEF coefficient, valve flow Cv Cv CV coil — — COIL compressor cprsr CMPR CMPR condens(-er, -ing, -ation) cond COND COND conductance — — C conductivity cndct CNDCT K conductors, number of (3) 3/c 3/c — contact factor — — CF cooling load clg load CLG LOAD CLOAD counterclockwise ccw CCW — cubic centimeter cm3 CC CC cubic metre m3 CU M CU M decibel dB DB DB degree deg. or ° DEG or ° DEG density dens DENS RHO depth or deep dp DP DPTH dew-point temperature dpt DPT DPT diameter dia.
DIA DIA diameter, inside ID ID ID diameter, outside OD OD OD difference or delta diff., ∆ DIFF D, DELTA diffuse radiation — DFRAD direct current dc DC DC direct radiation dir radn DIR RADN DIRAD dry — DRY dry-bulb temperature dbt DBT DB, DBT effectiveness — EFT effective temperature2 ET ET ET efficiency eff EFF EFF efficiency, fin — FEFF efficiency, surface — SEFF electromotive force emf EMF — elevation elev.
EL ELEV entering entr ENT ENT entering water temperature EWT EWT EWT entering air temperature EAT EAT EAT enthalpy — — H entropy — — S equivalent direct radiation edr EDR — evaporat(-e, -ing, -ed, -or) evap EVAP EVAP expansion exp EXP XPAN face area fa FA FA face to face f to f F to F — face velocity fvel FVEL FV factor, correction — — CFAC, CFACT factor, friction — — FFACT, FF fan — — FAN film coefficient,3 inside — — FI, HI film coefficient,3 outside — — FO, HO flow rate, air — — QAR, QAIR flow rate, fluid — — QFL flow rate, gas — — QGA, QGAS freezing point fp FP FP frequency Hz HZ — gage or gauge ga GA GA, GAGE gram g g G gravitational constant g G G greatest temp difference GTD GTD GTD heat — — HT heater — — HTR heat gain HG HG HG, HEATG heat gain, latent LHG LHG HGL heat gain, sensible SHG SHG HGS heat loss — — HL, HEATL heat transfer — — Q heat transfer coefficient U U U height hgt HGT HGT, HT high-pressure steam hps HPS HPS high-temperature hot water hthw HTHW HTHW horsepower hp HP HP hour(s) h h HR humidity, relative rh RH RH humidity ratio W W W incident angle — — INANG indicated kilowatt IkW IkW — International Pipe Std IPS IPS — iron pipe size ips IPS — joule J J J kelvin K K K kilograms kg kg KG kilojoules kJ kJ KJ kilometres per hour km/h km/h KPH kilopascals kPa kPa KPA kilowatt kW kW KW kilowatt hour kWh kWh KWH latent heat LH LH LH, LHEAT least mean temp. difference4 LMTD LMTD LMTD least temp. difference4 LTD LTD LTD leaving air temperature lat LAT LAT leaving water temperature lwt LWT LWT length lg LG LG, L liquid liq LIQ LIQ litre L L L litres per second L/s L/s LPS logarithm (natural) ln LN LN Term Text Drawings Program Abbreviations and Symbols 36.3 logarithm to base 10 log LOG LOG low-pressure steam lps LPS LPS low-temp. hot water lthw LTHW LTHW Mach number Mach MACH — mass flow rate mfr MFR MFR maximum max.
MAX MAX mean effective temp.
MET MET MET mean temp. difference MTD MTD MTD medium-pressure steam mps MPS MPS medium-temp. hot water mthw MTHW MTHW mercury Hg HG HG metre m m M metres per second m/s m/s M/S millilitres per second mL/s mL/s MLPS mL/s standard mL/sS mL/sS MLPSS minimum min.
MIN MIN noise criteria NC NC — normally open n o N O — normally closed n c N C — not applicable na N/A — not in contract n i c N I C — not to scale — N T S — number no.
NO N, NO number of circuits — — NC number of tubes — — NT outside air oa OA OA parts per million ppm PPM PPM Pascal Pa Pa PA Pa (absolute) Pa (abs) Pa A PAA Pa (gage) Pa (gage) Pa G PAG percent % % PCT phase (electrical) ph PH — pipe — — PIPE pressure — PRESS PRES, P pressure, barometric baro pr BARO PR BP critical pressure — — CRIP pressure, dynamic (velocity) vp VP VP pressure drop or difference PD PD PD, DELTP pressure, static sp SP SP pressure, vapor vap pr VAP PR VAP primary pri PRI PRIM radian — — RAD radiat(-e, -or) — RAD — radiation — RADN RAD radius — — R receiver rcvr RCVR REC recirculate recirc.
RECIRC RCIR, RECIR refrigerant (12, 22, etc.) R-12, R-22 R12, R22 R12, R22 relative humidity rh RH RH resist(-ance, -ivity, -or) res RES RES, OHMS return air ra RA RA revolutions rev REV REV revolutions per minute rpm RPM RPM revolutions per second rps RPS RPS roughness rgh RGH RGH, E safety factor sf SF SF saturation sat.
SAT SAT Saybolt seconds Furol ssf SSF SSF Saybolt seconds Universal ssu SSU SSU sea level sl SL SE second s s SEC sensible heat SH SH SH sensible heat gain SHG SHG SHG sensible heat ratio SHR SHR SHR shading coefficient — — SC solar — — SOL Term Text Drawings Program specification spec SPEC — specific gravity SG SG — specific heat sp ht SP HT C sp ht at constant pressure cp cp CP sp ht at constant volume cv cv CV specific volume sp vol SP VOL V, CVOL square sq.
SQ SQ standard std STD STD standard time meridian — — STM static pressure SP SP SP suction suct.
SUCT SUCT, SUC summ(-er, -ary, -ation) — — SUM supply sply SPLY SUP, SPLY supply air sa SA SA surface — — SUR, S surface, dry — — SURD surface, wet — — SURW system — — SYS tabulat(-e, -ion) tab TAB TAB tee — — TEE temperature temp.
TEMP T, TEMP temperature difference TD, ∆t TD TD, TDIF temperature entering TE TE TE, TENT temperature leaving TL TL TL, TLEA thermal conductivity k K K thermal expansion coeff.
— — TXPC thermal resistance R R RES, R thermocouple tc TC TC, TCPL thermostat T STAT T STAT T STAT thick(-ness) thkns THKNS THK total — — TOT total heat tot ht TOT HT — transmissivity — — TAU U-factor — — U unit — — UNIT vacuum vac VAC VAC valve v V VLV vapor proof vap prf VAP PRF — variable var VAR VAR variable air volume VAV VAV VAV velocity vel.
VEL VEL, V velocity, wind w vel.
W VEL W VEL ventilation, vent vent VENT VENT vertical vert.
VERT VERT viscosity visc VISC MU, VISC volt V V E, VOLTS volt ampere VA VA VA volume vol.
VOL VOL volumetric flow rate — — VFR wall — — W, WAL water — — WTR watt W W WAT, W wet bulb wb WB WB wet-bulb temperature wbt WBT WBT width — — WI wind — — WD wind direction wdir WDIR WDIR wind pressure wpr WPR WP, WPRES year yr YR YR zone z Z Z, ZN 1Abbreviations of most proper names use capital letters in both text and drawings.
2The asterisk () is used with ET, effective temperature, as in Chapter 8 of this volume.
3These are surface heat transfer coefficients.
4Letter L also used for Logarithm of these temperature differences in computer programming.
Term Text Drawings Program 36.4 2001 ASHRAE Fundamentals Handbook (SI) LETTER SYMBOLS DIMENSIONLESS NUMBERS MATHEMATICAL SYMBOLS Symbol Description of Item Typical Units a acoustic velocity m/s A area m2 b breadth or width m B barometric pressure kPa c concentration kg/m3 c specific heat kJ/(kg·K) cp specific heat at constant pressure kJ/(kg·K) cv specific heat at constant volume kJ/(kg·K) C coefficient — C fluid capacity rate W/K C thermal conductance W/(m2·K) CL loss coefficient — CP coefficient of performance — d prefix meaning differential — d or D diameter m De or Dh equivalent or hydraulic diameter m Dv mass diffusivity mm2/s e base of natural logarithms — E energy kJ E electrical potential V f film conductance (alternate for h) W/(m2·K) f frequency Hz fD friction factor, Darcy-Weisbach formulation — fF friction factor, Fanning formulation — F force N Fij angle factor (radiation) — g gravitational acceleration m/s2 G mass velocity kg/(s·m2) h heat transfer coefficient W/(m2·K) h hydraulic head m h specific enthalpy kJ/kg ha enthalpy of dry air kJ/kg hD mass transfer coefficient m/s hs enthalpy of moist air at saturation kJ/kg H total enthalpy kJ I electric current A k thermal conductivity W/(m·K) k (or γ) ratio of specific heats, cp/cv — K proportionality constant — KD mass transfer coefficient kg/(h·m2) l or L length m Lp sound pressure dB Lw sound power dB m or M mass kg M relative molecular mass kg/kg mol n or N number in general — N rate of rotation rad/s p or P pressure kPa pa partial pressure of dry air kPa ps partial pressure of water vapor in moist air kPa pw vapor pressure of water in saturated moist air kPa P power kW q time rate of heat transfer W Q total heat transfer kJ Q volumetric flow rate L/s r radius m r or R thermal resistance m2·K/W R gas constant J/(kg·K) s specific entropy kJ/(kg·K) S total entropy kJ/K t temperature °C ∆tm or ∆Tm mean temperature difference K T absolute temperature K u specific internal energy kJ/kg U total internal energy kJ U overall heat transfer coefficient W/(m2·K) v specific volume m3/kg V total volume m3 V linear velocity m/s w mass rate of flow g/s W weight N W humidity ratio of moist air (dry air basis) g/kg W work J Ws humidity ratio of moist air at saturation (dry air basis) g/kg x mole fraction — x quality, mass fraction of vapor — x,y,z lengths along principal coordinate axes m Z figure of merit — α absolute Seebeck coefficient V/K α absorptivity, absorptance radiation — α linear coefficient of thermal expansion 1/K α thermal diffusivity m2/s β volume coefficient of thermal expansion 1/K γ (or k) ratio of specific heats, cp/cv — γ specific weight N/m3 ∆ difference between values — ε emissivity, emittance (radiation) — θ time s, h η efficiency or effectiveness — λ wavelength nm µ degree of saturation — µ dynamic viscosity mPa·s ν kinematic viscosity m2/s ρ density kg/m3 ρ reflectivity, reflectance (radiation) — ρ volume resistivity Ω·m σ Stefan-Boltzmann constant W/(m2·K4) σ surface tension N/m τ stress N/m2 τ time s τ transmissivity, transmittance (radiation) — φ relative humidity — Fo Fourier number ατ/L2 Gr Grashof number L3ρ2βg(∆t)/µ2 Gz Graetz number wcp/kL jD Colburn mass transfer Sh/ReSc1/3 jH Colburn heat transfer Nu/RePr1/3 Le Lewis number α/Dv M Mach number V/a Nu Nusselt number hD/k Pe Peclet number GDcp/k Pr Prandtl number cpµ/k Re Reynolds number ρVD/µ Sc Schmidt number µ/ρDv Sh Sherwood number hDL/Dv St Stanton number h/Gcp Str Strouhal number fd/V equal to = not equal to ≠ approximately equal to ≈ greater than > less than < greater than or equal to ≥ less than or equal to ≤ plus + minus − plus or minus ± a multiplied by b ab, a·b, a × b a divided by b , a/b, ab−1 ratio of the circumference of a circle to its diameter π Symbol Description of Item Typical Units a b --Abbreviations and Symbols 36.5 SUBSCRIPTS GRAPHICAL SYMBOLS FOR DRAWINGS a raised to the power n an square root of a infinity ∞ percent % summation of Σ natural log ln logarithm to base 10 log These are to be affixed to the appropriate symbols. Several sub-scripts may be used together to denote combinations of various states, points, or paths. Often the subscript indicates that a particular property is to be kept constant in a process.
a,b,...
referring to different phases, states or physical conditions of a substance, or to different substances a air a ambient b barometric (pressure) c referring to critical state or critical value c convection db dry bulb dp dew point e base of natural logarithms f referring to saturated liquid f film fg referring to evaporation or condensation F friction g referring to saturated vapor h referring to change of phase in evaporation H water vapor i referring to saturated solid i internal if referring to change of phase in melting ig referring to change of phase in sublimation k kinetic L latent m mean value M molar basis o referring to initial or standard states or conditions p referring to constant pressure conditions or processes p potential r refrigerant r radiant or radiation s referring to moist air at saturation s sensible s referring to isentropic conditions or processes s static (pressure) s surface t total (pressure) T referring to isothermal conditions or processes v referring to constant volume conditions or processes v vapor v velocity (pressure) w wall w water wb wet bulb 1,2,...
different points in a process, or different instants of time Graphical symbols have been extracted from ASME Standard Y32.2.3 and ASME Standard Y32.2.4. Some of these symbols have been modified, and others have been added to reflect current prac-tice. Symbols and quotations are used with permission of the pub-lisher, the American Society of Mechanical Engineers.
Piping Heating High-pressure steam HPS Medium-pressure steam MPS Low-pressure steam LPS a , a0.5 High-pressure condensate HPC Medium-pressure condensate MPC Low-pressure condensate LPC Boiler blowdown BBD Pumped condensate PC Vacuum pump discharge VPD Makeup water MU Atmospheric vent ATV Fuel oil discharge FOD Fuel oil gage FOG Fuel oil suction FOS Fuel oil return FOR Fuel oil tank vent FOV Low-temperature hot water supply HWS Medium-temperature hot water supply MTWS High-temperature hot water supply HTWS Low-temperature hot water return HWR Medium-temperature hot water return MTWR High-temperature hot water return HTWR Compressed air A Vacuum (air) VAC Existing piping (NAME)E Pipe to be removed XX (NAME) XX Air Conditioning and Refrigeration Refrigerant discharge RD Refrigerant suction RS Brine supply B Brine return BR Condenser water supply C Condenser water return CR Chilled water supply CWS Chilled water return CWR Fill line FILL Humidification line H Drain D Hot/chilled water supply HCS Hot/chilled water return HCR Refrigerant liquid RL Heat pump water supply HPWS Heat pump water return HPWR Plumbing Sanitary drain above floor or grade SAN Sanitary drain below floor or grade SAN Storm drain above floor or grade ST Storm drain below floor or grade ST Condensate drain above floor or grade CD Condensate drain below floor or grade CD Vent – – – – – – – – – – – Cold water Hot water Hot water return Gas G G Acid waste ACID Drinking water supply DWS Drinking water return DWR Vacuum (air) VAC Compressed air A Chemical supply pipesa (NAME) Floor drain Funnel drain, open 36.6 2001 ASHRAE Fundamentals Handbook (SI) Fire Safety Devicesb Signal Initiating Detectors Heat (thermal) Gas Smoke Flame Valves Valves for Selective Actuators Air line Ball Butterfly Diaphragm Gate Gate, angle Globe Globe, angle Plug valve Three way Valves Actuators Manual Non-rising sun Outside stem & yoke Lever Gear Electric Motor Solenoid Pneumatic Motor Diaphragm Valves, Special Duty Check, swing gate Check, spring Control, electric-pneumatic Control, pneumatic-electric Hose end drain Lock shield Needle Pressure reducing (number and specify) Quick opening a See section on Piping Identification in this chapter.
b Refer to Standard for Fire Safety Symbols, 1999 Edition (NFPA Standard 170).
Quick closing, fusible link Relief (R) or safety (S) Solenoid Square head cock Unclassified (number and specify) Fittings The following fittings are shown with screwed connections. The sym-bol for the body of a fitting is the same for all types of connections, unless otherwise specified. The types of connections are often specified for a range of pipe sizes, but are shown with the fitting symbol where required. For example, an elbow would be: Flanged Threaded Belt & Spigot Weldeda Soldered Solvent Cement Fitting Symbol Bushing Cap Connection, bottom Connection, top Coupling (joint) Cross Elbow, 90° Elbow, 45° Elbow, turned up Elbow, turned down Elbow, reducing (show sizes) Elbow, base Elbow, long radius Elbow, double branch Elbow, side outlet, outlet up Elbow, side outlet, outlet Down Lateral Reducer, concentric Reducer, eccentric straight invert Reducer, eccentric straight crown Tee Tee, outlet up Tee, outlet down Tee, reducing (show sizes) Tee, side outlet, outlet up a Includes fusion, specify type.
Abbreviations and Symbols 36.7 Tee, side outlet, outlet down Tee, single sweep Union, screwed Union, flanged Piping Specialties Air vent, automatic Air vent, manual Air separator Alignment guide Anchor, intermediate Anchor, main Ball joint Expansion joint Expansion loop Flexible connector Flowmeter, orifice Flowmeter, venturi Flow switch Hanger rod Hanger spring Heat exchanger, liquid Heat transfer surface (indicate type) Pitch of pipe, rise (R) drop (D) Pressure gage and cock Pressure switch Pump (indicate use) Pump suction diffuser Spool piece, flanged Strainer Strainer, blow off Strainer, duplex Tank (indicate use) Thermometer Thermometer well, only Thermostat, electric Thermostat, pneumatic Thermostat, self-contained Traps, steam (indicate type) Unit heater (indicate type) Air Moving Devices and Components Fans (indicate use)a Axial flow Centrifugal Propeller Roof ventilator, intake Roof ventilator, exhaust Roof ventilator, louvered Ductworkb Direction of flow Duct size, first figure is side down Duct section, positive pressure, first figure is top Duct section, negative pressure Change of elevation rise (R) drop (D) Access doors, vertical or horizontal Acoustical lining (insulation) Cowl, (gooseneck) and flashing Flexible connection Flexible duct Sound attenuator Terminal unit, mixing Terminal unit, reheat Terminal unit, variable volume Transitiona Turning vanes Detectors, fire and/or smoke Dampers Backdraft damper Pneumatic operated damper a Units of measurement are not shown herein, but should be shown on drawings. The first of the two dimensions on ducts indicates the side of the duct showing; on duct sections, the top; on grilles and registers, the horizontal edge.
b Adapted from SMACNA, Symbols for Ventilation and Air Conditioning Figure 4.2.
HVAC Duct System Design.
36.8 2001 ASHRAE Fundamentals Handbook (SI) Electric operated damper Fire Damper and sleeve (provide access door) Vertical position Horizontal position Manual volume Manual Splitter Smoke damper (provide access door) Standard branch, supply or return, no splitter Duct, electric heater Grilles, Register and Diffusersb Exhaust grille or register Supply grille or register Grille or register, ceiling Heat stop for fire rated ceiling Louver and screen Louver, door, or wall Door grille Undercut door Ceiling diffuser, rectangular Ceiling diffuser, round Diffuser, linear Diffuser and light fixture combination Transfer grille assembly Refrigeration Compressors Centrifugal a Indicate flat on bottom or top (FOB or FOT), if applicable.
b Show volumetric flow rate at each device.
Reciprocating Rotary Rotary screw Condensers Air cooled Evaporative Water cooled, (specify type) Condensing Units Air cooleda Water cooleda Condenser-Evaporator (Cascade System) Cooling Towers Cooling tower Spray pond Evaporatorsb Finned coil Forced convection Immersion cooling unit Plate coil Pipe coila Liquid Chillers (Chillers only) Direct expansionb Floodedb Tank, closed Tank, open Chilling Units Absorption a L = Liquid being cooled, RL = Refrigerant liquid, RS = Refrigerant suction.
b Specify manifolding.
Abbreviations and Symbols 36.9 Centrifugal Reciprocating Rotary screw Controls Refrigerant Controls Capillary tube Expansion valve, hand Expansion valve, automatic Expansion valve, thermostatic Float valve, high side Float valve, low side Thermal bulb Solenoid valve Constant pressure valve, suction Evaporator pressure regulating valve, thermostatic, throttling Evaporator pressure regulating valve, thermostatic, snap-action Evaporator pressure regulating valve, throttling-type, evaporator side Compressor suction valve, pressure-limiting, throttling-type, compressor side Thermosuction valve Snap-action valve Refrigerant reversing valve Temperature or Temperature-Actuated Electrical or Flow Controls Thermostat, self-contained Thermostat, Remote Bulb a Frequently used diagrammatically as evaporator and/or condenser with label indicat-ing name and type.
b L = Liquid being cooled, RL = Refrigerant liquid, RS = Refrigerant suction.
Pressure or Pressure-Actuated Electrical or Flow Controls Pressure switch Pressure switch, dual (high-low) Pressure switch, differential oil pressure Valve, automatic reducing Valve, automatic bypass Valve, pressure-reducing Valve, condenser water regulating Auxiliary Equipment Refrigerant Filter Strainer Filter and drier Scale trap Drier Vibration absorber Heat exchanger Oil separator Sight glass Fusible plug Rupture disk Receiver, high pressure, horizontal Receiver, high-pressure, vertical Receiver, low-pressure Intercooler Intercooler/desuperheater 36.10 2001 ASHRAE Fundamentals Handbook (SI) PIPING SYSTEM IDENTIFICATION The material in piping systems is identified to promote greater safety and lessen the chances of error, confusion, or inaction in times of emergency. Primary identification should be by means of a lettered legend naming the material conveyed by the piping. In addi-tion to, but not instead of lettered identification, color can be used to identify the hazards or use of the material. The data in this section have been extracted from ASME Stan-dard A13.1.
Definitions Piping Systems. Piping systems include pipes of any kind, fit-tings, valves, and pipe coverings. Supports, brackets, and other accessories are not included. Pipes are defined as conduits for the transport of gases, liquids, semiliquids, or fine particulate dust.
Materials Inherently Hazardous to Life and Property. There are four categories of hazardous materials: • Flammable or explosive materials that are easily ignited, including materials known as fire producers or explosives • Chemically active or toxic materials that are corrosive or are in themselves toxic or productive of poisonous gases • Materials at extreme temperatures or pressures that, when released from the piping, cause a sudden outburst with the potential for inflicting injury or property damage by burns, impingement, or flashing to vapor state • Radioactive materials that emit ionizing radiation Materials of Inherently Low Hazard. All materials that are not hazardous by nature, and are near enough to ambient pressure and temperature that people working on systems carrying these materi-als run little risk through their release.
Fire Quenching Materials. This classification includes sprin-kler systems and other piped fire fighting or fire protection equip-ment. This includes water (for fire fighting), chemical foam, CO2, Halon, and so forth.
Method of Identification Legend. The legend is the primary and explicit identification of content. Positive identification of the content of the piping system is by lettered legend giving the name of the contents, in full or abbre-viated form, as shown in Table 2. Arrows should be used to indicate the direction of flow. Use the legend to identify contents exactly and to provide temperature, pressure, and other details necessary to identify the hazard.
The legend shall be brief, informative, pointed, and simple.
Legends should be applied close to valves and adjacent to changes in direction, branches, and where pipes pass through walls or floors, and as frequently as needed along straight runs to provide clear and positive identification. Identification may be applied by stenciling, tape, or markers (see Figure 1). The number and loca-tion of identification markers on a particular piping system is based on judgment.
Color. Colors listed in Table 3 are used to identify the charac-teristic properties of the contents. Color can be shown on or con-tiguous to the piping by any physical means, but it should be used in combination with a legend. Color can be used in continuous total length coverage or in intermittent displays.
Energy Recovery Equipment Condenser, double bundle Air to Air Energy Recovery Rotary heat wheel Coil loop Heat pipe Fixed plate Plate fin, crossflow Power Sources Motor, electric (number for identification of description in specifications) Engine (indicate fuel) Gas turbine Steam turbine Steam turbine, condensing Electrical Equipmenta Symbols for electrical equipment shown on mechanical drawings are usu-ally geometric figures with an appropriate name or abbreviation, with details described in the specifications. The following are some common examples.b Motor control Disconnect switch, unfused Disconnect switch, fused Time clock Automatic filter panel Lighting panel Power panel a See ARI Standard 130 for preferred symbols of common electrical parts.
b Number each symbol if more than one; see ASME Standard Y32.4.
Table 2 Examples of Legends HOT WATER AIR 700 kPa H.P. RETURN STEAM 700 kPa (gage) Abbreviations and Symbols 36.11 Visibility. Pipe markings should be highly visible. If pipe lines are above the normal line of vision, the lettering is placed below the horizontal centerline of the pipe (Figure 1).
Type and Size of Letters. Provide the maximum contrast between color field and legend (Table 3). Table 4 shows the size of letters recommended. Use of standard size letters of 1/2 in. or larger is recommended. For identifying materials in pipes of less than 3/4 in. in diameter and for valve and fitting identification, use a perma-nently legible tag.
Unusual or Extreme Situations. When the piping layout occurs in or creates an area of limited accessibility or is extremely com-plex, other identification techniques may be required. While a cer-tain amount of imagination may be needed, the designer should always clearly identify the hazard and use the recommended color and legend guidelines.
CODES AND STANDARDS ARI. 1982. Graphic electrical symbols for air-conditioning and refrigeration equipment. Standard 130.
ASME. 1996. Scheme for the identification of piping systems. Standard A13.1.
ASME. 1988. Glossary of terms concerning letter symbols. Standard Y10.1. ASME. 1984. Letter symbols and abbreviations for quantities used in acous-tics. Standard Y10.11.
ASME. 1987. Letter symbols for illuminating engineering. Y10.18.
ASME. 1999. Abbreviations and acronyms. Standard Y14.38-1999 (Revi-sion and redesignation of ASME Y1.1-1989).
ASME. 1999. Graphical symbols for pipe fittings, valves, and piping. Stan-dard Y32.2.3.
ASME. 1998. Graphical symbols for heating, ventilating, and air condition-ing. Standard Y32.2.4.
ASME. 1999. Graphic symbols for plumbing fixtures for diagrams used in architecture and building construction. Standard Y32.4.
IEEE. 1998. American national standard letter symbols for units of measure-ment. IEEE Standard 260.1-1993.
IEEE. 1993. Mathematical signs and symbols for use in physical science and technology. Standard 260.3-1993.
IEEE. 1996. Letter symbols and abbreviations used in acoustics. Standard 260.4-1996.
NEMA. 1998. Safety color code. Standard Z535.1.
NFPA. 1999. Standard for fire safety symbols, 1999 Edition. Standard 170.
Fig. 1 Visibility of Pipe Markings Table 3 Classification of Hazardous Materials and Designation of Colorsa Classification Color Field Colors of Letters for Legend Materials Inherently Hazardous Flammable or explosive Yellow Black Chemically active or toxic Yellow Black Extreme temperatures or pressures Yellow Black Radioactiveb Purple Yellow Materials of Inherently Low Hazard Liquid or liquid admixturec Green Black Gas or gaseous admixture Blue White Fire Quenching Materials Water, foam, CO2, Halon, etc.
Red White aWhen the color scheme above is used, the colors should be as recommended in the latest revision of NEMA Standard Z535.1.
bPreviously specified radioactive markers using yellow or purple are acceptable if already installed and/or until existing supplies are depleted, subject to applicable federal regulations.
cMarkers with black letters on a green color field are acceptable if already installed and/or until existing supplies are depleted.
Table 4 Size of Legend Letters Outside Diameter of Pipe or Covering, mm Length of Color Field A, mm Size of Letters B, mm.
20 to 32 200 13 40 to 50 200 20 65 to 150 300 32 200 to 250 600 65 over 250 800 90 37.1 CHAPTER 37 UNITS AND CONVERSIONS Table 1 Conversions to SI Units Multiply By To Obtain Multiply By To Obtain acre.................................................................
0.4047 ha in2...................................................................
645.2 mm2 atmosphere (standard) 101.325 kPa in3 (volume) ...................................................
16.4 mL bar .................................................................. 100 kPa in3/min (SCIM)..............................................
0.273 mL/s barrel (42 U.S. gal, petroleum) ......................
159.0 L in3 (section modulus).....................................
16390 mm3 ........................
0.1590 m3 in4 (section moment)......................................
416 200 mm4 Btu (International Table)................................
1.055 kJ km/h ...............................................................
0.2778 m/s Btu/ft2.............................................................
11.36 kJ/m2 kWh................................................................
3.60 MJ Btu/ft3.............................................................
37.3 kJ/m3 kW/1000 cfm .................................................
2.12 kJ/m3 Btu/gal............................................................
279 kJ/m3 kilopond (kg force) ........................................
9.81 N Btu·ft/h·ft2· °F ...............................................
1.731 W/(m·K) kip (1000 lbf) .................................................
4.45 kN Btu·in/h·ft2· °F (thermal conductivity, k) ......
0.1442 W/(m·K) kip/in2 (ksi) ....................................................
6.895 MPa Btu/h...............................................................
0.2931 W litre.................................................................
0.001 m3 Btu/h·ft2.........................................................
3.155 W/m2 met .................................................................
58.15 W/m2 Btu/h·ft2· °F (overall heat transfer coefficient, U)...........
micron (µm) of mercury (60°F).....................
133 mPa 5.678 W/(m2·K) mile ................................................................
1.609 km Btu/lb ............................................................. 2.326 kJ/kg mile, nautical.................................................. 1.852 km Btu/lb· °F (specific heat, cp)...........................
4.184 kJ/(kg·K) mph ................................................................
1.609 km/h bushel.............................................................
0.03524 m3 ................................................................
0.447 m/s calorie, gram ..................................................
4.184 J millibar...........................................................
0.100 kPa calorie, kilogram (kilocalorie) .......................
4.184 kJ mm of mercury (60°F)...................................
0.133 kPa centipoise (dynamic viscosity, µ) .................. 1.00 mPa·s mm of water (60°F) .......................................
9.80 Pa centistokes (kinematic viscosity, ν) ............... 1.00 mm2/s ounce (mass, avoirdupois) .............................
28.35 g clo...................................................................
0.155 m2·K/W ounce (force or thrust) ...................................
0.278 N dyne/cm2 ........................................................ 0.100 Pa ounce (liquid, U.S.)........................................
29.6 mL EDR hot water (150 Btu/h)............................
44.0 W ounce inch (torque, moment).........................
7.06 mN·m EDR steam (240 Btu/h)..................................
70.3 W ounce (avoirdupois) per gallon ......................
7.49 kg/m3 EER................................................................
0.293 COP perm (permeance) ..........................................
57.45 ng/(s·m2·Pa) ft..................................................................... 0.3048 m perm inch (permeability) ...............................
1.46 ng/(s·m·Pa) ..................................................................... 304.8 mm pint (liquid, U.S.)...........................................
473 mL ft/min, fpm ..................................................... 0.00508 m/s pound ft/s, fps ........................................................... 0.3048 m/s lb (mass)......................................................
0.4536 kg ft of water.......................................................
2.99 kPa .......................................................
453.6 g ft of water per 100 ft pipe ..............................
0.0981 kPa/m lbf (force or thrust) ......................................
4.448 N ft2 ...................................................................
0.09290 m2 lb/ft (uniform load)......................................
1.49 kg/m ft2·h· °F/Btu (thermal resistance, R) ..............
0.176 m2·K/W lbm/ft·h (dynamic viscosity, µ) ...................
0.4134 mPa·s ft2/s (kinematic viscosity, ν) ..........................
92900 mm2/s lbm/ft·s (dynamic viscosity, µ)....................
1490 mPa·s ft3 ...................................................................
28.32 L lbf·s/ft2 (dynamic viscosity, µ)....................
47.88 Pa·s ...................................................................
0.02832 m3 lb/h...............................................................
0.126 g/s ft3/min, cfm....................................................
0.4719 L/s lb/min ..........................................................
0.00756 kg/s ft3/s, cfs ..........................................................
28.32 L/s lb/h [steam at 212°F (100°C)].....................
0.2843 kW ft·lbf (torque or moment)...............................
1.356 N·m lbf/ft2 ...........................................................
47.9 Pa ft·lbf (work) ...................................................
1.356 J lb/ft2.............................................................
4.88 kg/m2 ft·lbf /lb (specific energy) ..............................
2.99 J/kg lb/ft3 (density, ρ)..........................................
16.0 kg/m3 ft·lbf /min (power)..........................................
0.0226 W lb/gallon.......................................................
120 kg/m3 footcandle.......................................................
10.76 lx ppm (by mass)................................................
1.00 mg/kg gallon (U.S., 231 in3)...................................
3.7854 L psi...................................................................
6.895 kPa gph .................................................................
1.05 mL/s quad (1015 Btu) ..............................................
1.055 EJ gpm ................................................................
0.0631 L/s quart (liquid, U.S.).........................................
0.9463 L gpm/ft2 ...........................................................
0.6791 L/(s·m2) square (100 ft2) ..............................................
9.29 m2 gpm/ton refrigeration .....................................
0.0179 mL/J tablespoon (approximately) ...........................
15 mL grain (1/7000 lb) ............................................
0.0648 g teaspoon (approximately) ..............................
5 mL gr/gal..............................................................
17.1 g/m3 therm (U.S.) ...................................................
105.5 MJ gr/lb................................................................
0.143 g/kg ton, long (2240 lb) .........................................
1.016 Mg horsepower (boiler) (33 470 Btu/h) ...............
9.81 kW ton, short (2000 lb) ........................................
0.907 Mg; t (tonne) horsepower (550 ft·lbf/s)...............................
0.7457 kW ton, refrigeration (12 000 Btu/h)....................
3.517 kW inch................................................................. 25.4 mm torr (1 mm Hg at 0°C)....................................
133 Pa in. of mercury (60°F) .....................................
3.37 kPa watt per square foot .......................................
10.76 W/m2 in. of water (60°F)..........................................
249 Pa yd ...................................................................
0.9144 m in/100 ft, thermal expansion ..........................
0.833 mm/m yd2..................................................................
0.8361 m2 in·lbf (torque or moment) ..............................
113 mN·m yd3..................................................................
0.7646 m3 To Obtain By Divide To Obtain By Divide The preparation of this chapter is assigned to TC 1.6, Terminology.
Conversion factor is exact.
Notes: Units are U.S. values unless noted otherwise.
Litre is a special name for the cubic decimetre. 1 L = 1 dm3 and 1 mL = 1 cm3.
37.2 2001 ASHRAE Fundamentals Handbook (SI) When making conversions, remember that a converted value is no more precise than the original value. For many applications, rounding off the converted value to the same number of significant figures as those in the original value provides sufficient accuracy.
Caution: The conversion values in Table 1 are rounded to three or four significant figures, which is sufficiently accurate for most applications. See ANSI Standard SI-10 (available from ASTM or IEEE) for additional conversions with more significant figures.
Table 2 Conversion Factors Pressure psi in. of water (60°F) in. Hg (32°F) atmosphere mm Hg (32°F) bar kgf/cm2 pascal 1 = 27.708 = 2.0360 = 0.068046 = 51.715 = 0.068948 = 0.07030696 = 6894.8 0.036091 1 0.073483 2.4559 × 10−3 1.8665 2.4884 × 10−3 2.537 × 10−3 248.84 0.491154 13.609 1 0.033421 25.400 0.033864 0.034532 3386.4 14.6960 407.19 29.921 1 760.0 1.01325 1.03323 1.01325 × 105 0.0193368 0.53578 0.03937 1.31579 × 10−3 1 1.3332 × 10−3 1.3595 × 10−3 133.32 14.5038 401.86 29.530 0.98692 750.062 1 1.01972 105 14.223 394.1 28.959 0.96784 735.559 0.980665 1 9.80665 × 104 1.45038 × 10−4 4.0186 × 10−3 2.953 × 10−4 9.8692 × 10−6 7.50 × 10−3 10−5 1.01972 × 10−5 1 Mass lb (avoir.) grain ounce (avoir.) kg 1 = 7000 = 16 = 0.45359 1.4286 × 10−4 1 2.2857 × 10−3 6.4800 × 10−5 0.06250 437.5 1 0.028350 2.20462 1.5432 × 104 35.274 1 Volume cubic inch cubic foot gallon litre cubic metre (m3) 1 = 5.787 × 10−4 = 4.329 × 10−3 = 0.0163871 = 1.63871 × 10−5 1728 1 7.48052 28.317 0.028317 231.0 0.13368 1 3.7854 0.0037854 61.02374 0.035315 0.264173 1 0.001 6.102374 × 104 35.315 264.173 1000 1 Energy Btu ft·lbf calorie (cal) joule (J) = watt-second (W·s) watt-hour (W·h) Note: MBtu, which is 1000 Btu, is confusing and is not used in the Handbook.
1 = 778.17 = 251.9958 = 1055.056 = 0.293071 1.2851 × 10−3 1 0.32383 1.355818 3.76616 × 10−4 3.9683 × 10−3 3.08803 1 4.1868 1.163 × 10−3 9.4782 × 10−4 0.73756 0.23885 1 2.7778 × 10−4 3.41214 2655.22 859.85 3600 1 Density lb/ft3 lb/gal g/cm3 kg/m3 1 = 0.133680 = 0.016018 = 16.018463 7.48055 1 0.119827 119.827 62.4280 8.34538 1 1000 0.0624280 0.008345 0.001 1 Specific Volume ft3/lb gal/lb cm3/g m3/kg 1 = 7.48055 = 62.4280 = 0.0624280 0.133680 1 8.34538 0.008345 0.016018 0.119827 1 0.001 16.018463 119.827 1000 1 Viscosity (absolute) 1 poise = 1 dyne-sec/cm2 = 0.1 Pa·s = 1 g/(cm·s) poise lbf·s/ft2 lbf·h/ft2 kg/(m·s) = N·s/m2 lbm/ft·s 1 = 2.0885 × 10−3 = 5.8014 × 10−7 = 0.1 = 0.0671955 478.8026 1 2.7778 × 10−4 47.88026 32.17405 1.72369 × 106 3600 1 1.72369 × 105 1.15827 × 105 10 0.020885 5.8014 × 10−6 1 0.0671955 14.8819 0.031081 8.6336 × 10−6 1.4882 1 Temperature Temperature Temperature Interval Scale K °C °R °F K °C °R °F Kelvin x K = x x − 273.15 1.8x 1.8x − 459.67 1 K = 1 1 9/5 = 1.8 9/5 = 1.8 Celsius x°C = x + 273.15 x 1.8x + 491.67 1.8x + 32 1°C = 1 1 9/5 = 1.8 9/5 = 1.8 Rankine x°R = x/1.8 (x − 491.67)/1.8 x x − 459.67 1°R = 5/9 5/9 1 1 Fahrenheit x°F = (x + 459.67)/1.8 (x − 32)/1.8 x + 459.67 x 1°F = 5/9 5/9 1 1 Notes: Conversions with are exact.
The Btu and calorie are based on the International Table. All temperature conversions and factors are exact.
The term centigrade is obsolete and should not be used.
38.1 CHAPTER 38 PHYSICAL PROPERTIES OF MATERIALS ALUES in the following tables are in consistent units to Vassist the engineer looking for approximate values. For data on refrigerants, see Chapter 19; for secondary coolants, see Chapter 21. Chapter 25 gives more information on the values for materials used in building construction and insulation. Many properties vary with temperature, material density, and composi-tion. The references document the source of the values and pro-vide more detail or values for materials not listed here. Table 1 Properties of Vapor Material Relative Molecular Mass Normal Boiling Point, °C Critical Temperature, °C Critical Pressure, kPa Density, kg/m3 Specific Heat, J/(kg ·K) Thermal Conductivity, W/(m ·K) Viscosity, µPa·s Alcohol, Ethyl 46.07a 78.6a 243.2b 6 394b 1520j 0.013a 14.2j (289) Alcohol, Methyl 32.04a 65.0a 240.1b 7 977b 1350j 0.0301r 14.8j (272) Ammonia 17.03a −33.2a 132.6b 11 300b 7.72b 2200aa 0.0221b 9.30aa Argon 39.948a −185.9 −122.5 4 860b 1.785b 523c 0.016a 21.0a Acetylene 26.04a −83.7a 36.1b 6 280b 1.17b 1580a 0.0187b 9.34a Benzene 78.11a 80.2a 289.6d 4 924d 2.68e (80) 1300e (80) 0.0071e 7.0a Bromine 159.82a 58.8a 58.8d 10 340d 6.1f (59) 230f (100) 0.0061a 17a Butane 58.12a −0.5a 152.1d 3 797d 2.69g 1580aa 0.014a 7.0a Carbon dioxide 44.01a −78.5a 31.1d 7 384d 1.97g 840g 0.015a 14h Carbon disulfide 76.13h 46.3h 278.9h 7 212h 599.0p (27) Carbon monoxide 28.01a −191.5a −140.3d 3 500d 1.25d 1100f 0.0230a 17a Carbon tetrachloride 153.84g 76.6h 283.3h 4 560h 862q (27) 16.0j Chlorine 70.91a −34.7a 144.1d 7 710d 3.22d 490a 0.0080a 12a Chloroform 119.39h 61.8h 263.4h 5 470h 528j 0.014r 16j Ethyl chloride 64.52h 12.4h 187.3h 5 270h 2.872b 1780r 0.00872j 16.0q Ethylene 28.03h −103.7h 10.0h 5 120h 1.25b 1470aa 0.0176aa 9.60aa Ethyl ether 74.12h 34.7h 192.7h 3 610h 2470h (35) 11.3q Fluorine 38.00h −187.0h −129.2h 5 580h 1.637b 812j 0.0254j 37j Helium 4.0026a −269.0i −267.9h 229i 0.178i 5192aa 0.142aa 19.0aa Hydrogen 2.0159a −253.1i −240.0i 1 316i 0.0900i 14 200j 0.168aa 8.40aa Hydrogen chloride 36.461a −84.9a 51.4d 8 260d 1.640b 800j 0.0131j 13.3j Hydrogen sulfide 34.080a −60.8a 100.4d 9 012d 1.54b 996j 0.0130j 11.6j Heptane (m) 100.21a 98.5a 266.8b 2 720b 3.4k 1990j 0.0185j 7.00j Hexane (m) 86.18a 66.9a 234.8d 3 030d 3.4k 1880j 0.0168j 7.52j Isobutane 58.12f −11.6 135.1j 3 648j 2.47s (21) 1570aa 0.014aa 6.94aa Methane 16.04a −164.0a −81.8j 4 641b 0.718b 2180aa 0.0310aa 10.3aa Methyl chloride 50.49a −24.3a 143.2j 6 678b 2.307b 770aa 0.0093aa 10.1aa Naphthalene 128.19a 218.0 469.1j 3 972j 1310q (25) Neon 20.183a −247.0a −228.8j 2 698j 1030aa 0.0464aa 30.0aa Nitric oxide 30.01a −152.0a −92.9j 6 546j 996j 29.4j Nitrogen 28.01a −195.8a −146.9j 3 394b 1040j 0.0240aa 16.6aa Nitrous oxide 44.01a −88.5a 36.4j 7 235j 850j 0.01731j (26.8) 22.4j Nitrogen tetroxide 92.02a 158.3j 10 133j 842p (27) 0.0401r (55) Oxygen 31.9977 −183.0a −118.6 5 043 913j 0.0244aa 19.1aa n-Pentane 72.53a 36.1 196.7j 3 375j 1680a (27) 0.0152j (26.8) 11.7j Phenol 74.11b 181.4b 418.9b 6 130b 2.6k 1400k 0.017k 12k Propane 44.09g −42.1g 96.7 4 248 2.02g 1571j (4.5) 0.015j 7.40j Propylene 42.08b −47.7l 91.8l 4 622l 1.92l 1460aa 0.014aa 8.06aa Sulfur dioxide 64.06b −10.0b 156.9b 7 874b 2.93b 607l 0.0085j 11.6j Water vapor 18.02b 100.0m 374.0 22 064 0.598m 2050aa 0.0247m 12.1aa Data source unknown. Notes: 1. Properties at 101.325 kPa and 0°C, or the saturation temperature if higher than 0°C, unless otherwise noted in parentheses.
2. Superscript letters indicate data source from the section on References.
38.2 2001 ASHRAE Fundamentals Handbook (SI) Table 2 Properties of Liquids Name or Description Normal Boiling Point, °C at 101.325 kPa Enthalpy of Vaporization , kJ/kg Specific Heat, cp Viscosity Enthalpy of Fusion, kJ/kg Density Thermal Conductivity Vapor Pressure Freezing Point, °C J/ (kg·K) Temp., °C µPa·s Temp., °C kg/m3 Temp., °C W/ (m·K) Temp., °C kPa Temp., °C Acetic acid 118.6a 405.0b 2180b 26 to 95 1 222f 20 195b 1049a 20 0.17b 20 53.3a 99 16.7a Acetone 56.3a 532.4b 2150b 3 to 23 331f 20 98.0b 791a 20 0.1761b 30 53.3a 40 −95.4a Allyl alcohol 97.1a 684.1b 2740b 21 to 96 1 363f 20 853.9a 20 0.180b 25 to 30 53.3a 80 −129.0a n-Amyl alcohol 138.2i 503.1b 4 004f 23 112b 817.9f 15 0.16b 30 13.3a 86 −79.0a Ammonia −33.2a 1357b 4601b 0 266f −33 322.40b 696.8b −45 0.50b −15 to 30 53.3a −45 −77.8a Alcohol-ethyl 78.6a 854.8b 2840b 0 to 98 1 194f 20 108b 789.2a 20 0.182b 20 13.3a 35 117.3a Alcohol-methyl 65.0a 1100b 2510b 15 to 20 592.8f 20 99.3a 791.3a 20 0.215b 20 13.3a 21 −97.8a Aniline 184.4a 434.0b 2140b 8 to 82 4 467.0f 20 114b 1021a 20 0.173b −2 to 20 1.3a 69 −6.2a Benzene 80.2a 394.0h 1720h 20 653a 20 126h 879d 20 0.147h 20 10d 20 5.9a Bromine 58.8a 185d 448f 20 988a 20 66.30d 3119f 20 0.122a 25 22.0d 20 −7.2a n-Butyl alcohol 117.6a 591.5h 2350f 20 2950f 20 125b 811a 20 0.15h 20 0.7d 20 −90.2a n-Butyric acid 163.6a 504.7h 2150f 20 1 540a 20 126a 964a 20 0.16h 12 0.09d 20 −6.2a Calcium chloride brine (20% by mass) 3110i 20 2 000i 20 1180i 20 0.574i 20 −16.2i Carbon disulfide 46.3a 346.1h 1000i 20 360a 20 57.70d 1260d 20 0.16b 30 39.3d 20 −111.2a Carbon tetrachloride 76.7a 195h 842f 20 967a 20 29.80d 1590d 20 0.11j 20 12d 20 −22.8a Chloroform 61.3v 247v 980v 20 562v 20 1489v 20 0.13v 20 21.3v 20 −63.3v n-Decane 174.1b 2000b 20 202b 730b 20 0.15b 20 0.17b 20 −29.8b Ethyl ether 34.5v 351v 2260v 20 230v 20 98.60v 714.6v 20 0.14b 20 58.7v 20 −116.3v Ethyl acetate 77.2v 427.5v 1950v 20 451v 20 119b 838v 20 0.175b 20 9.6b 20 −82.4v Ethyl chloride 12.4j 385.9f (20) 1540f 0 69.04a 897.8a 20 0.310f 1 53.3y 12 −136.4a Ethyl iodide 72.3a 191f (71) 1540f 0 9.90f 20 1935.8a 20 0.370f 30 13.3y 18 −108.0 Ethylene bromide 131.6a 231f (99) 729f 20 28.7f 20 57.73a 2179.3a 20 1.3y 19 9.6a Ethylene chloride 83.6a 365.8f (153) 1260f 20 14.0f 20 88.43a 1235a 20 8.0y 18 −35.4a Ethylene glycol 198.1a 800.1f (344) 181.10a 1109a 20 0.173f 20 0.1y 53 −10.8a Formic acid 99.8a 502.0f (216) 2200f 20 29.7f 20 276.54a 1219a 20 0.180a −2 5.3y 23 7.4a Glycerin (glycerol) 179.9 17 800f 20 1261a 20 0.195a 20 0.1a 51 18.9a Heptane 97.5a 321f 2220j 20 409a 20 140b 684a 20 0.128j 20 4.73y 20 −92.2a Hexane 65.9a 337f 2250j 20 320d 20 150b 658a 20 0.125j 20 16.00y 20 −96.2a Hydrogen chloride −85.9a 444f 54.9f 1190d b.p.
−115.8a Isobutyl alcohol 107.1a 579f 486f 20 3 910f 20 801f 20 0.14f 20 1.3y 20 −109.0a Kerosene 204 to 293b 2000n 20 2 480b 20 820a 20 0.15n 20 Linseed oil 42 900b 20 920d 20 −24.9a Methyl acetate 56.1a 412f 1950f 20 389f 20 971a 20 0.16f 20 22.64y 20 −99.2†a Methyl iodide 41.6a 192f 500f 20 2270a 20 42.7y 20 −67.5a Naphthalene 209.8a 316f 1680f m.p.
901b m.p.
151b 976y m.p.
0.291b 20 79.3a Nitric acid 85.1v 628v 1700v 20 910k 20 166v 1512v 20 0.28v 20 0.236v 20 −42.7v Nitrobenzene 209.9b 330b 1450b 20 2 150b 20 93.69v 1200b 20 1.7b 20 0.001b 20 4.8b Octane 124.8b 306.3b 2100b 20 562b 20 180.70b 703b 20 0.15b 20 0.056b 20 −57.5b Petroleum 230 to 384w 2000 to 3000w 20 7900 to 1.2×106w 20 640 to 1000w 20 n-Pentane 35.1a 357.3h 2330h 20 226d 20 117h 626a 20 0.11h 20 56.7d 20 −130.8a Propionic acid 140.2a 413.6f 1980h 20 1 102a 20 992a 20 0.173 12 0.4d 20 −21.8a Sodium chloride brine 20% by mass 103.9a 3110x 20 1 570x 20 1150x 20 0.583x 20 0.076x 20 −17.4x 10% by mass 100.9a 3620x 20 1 180x 20 1070x 20 0.593x 20 0.087x 20 −7.4x Sodium hydroxide and water (15% by mass) 100.7v 3610b 20 1150b 20 −22.0b Sulfuric acid and water 100% by mass 286.8v 1400b 20 22 000b 20 1833v 20 0.001b 20 9.6b 95% by mass 300.9v 1460v 20 21 000v 20 1836v 20 0.001v 20 −29.2v 90% by mass 259.1v 1600v 20 25 000v 20 1816v 20 0.38b 20 0.001v 20 −10.5v Toluene (C6H5CH3) 108.9b 363b 1690v 20 587v 20 71.90b 867b 20 0.16b 20 0.12b 20 −96.0b Turpentine 148.9a 286v 1700b 20 546b 20 863b 20 0.13b 20 Water 100.0 2257m 4180m 20 988m 20 333.8b 998.20m 20 0.602m 20 2.34 20 −1.0m Xylene [C6H4(CH3)2] Ortho 142.9b 347b 1720b 20 831b 20 128b 881b 20 1.6b 20 0.0260b 20 −26.2b Meta 137.9b 342b 1670b 20 628b 20 109b 867b 0 1.6b 20 0.0290b 20 −48.2b Para 136.9b 340b 1640b 20 670b 20 161b 862b 20 0.0300b 20 11.9b Zinc sulfate and water 10% by mass 3700b 20 1 570a 20 1110r 20 0.583a 20 −2.3a 1% by mass 3300b 20 1 100a 20 1010r 20 0.598a 20 −1.2a Data source unknown.
†Approximate solidification temperature.
Notes: Superscript letters indicate data source from the section on References.
m.p. = melting point b.p. = boiling point Physical Properties of Materials 38.3 Table 3 Properties of Solids Material Description Specific Heat, J/(kg·K) Density, kg/m3 Thermal Conductivity, W/(m ·K) Emissivity Ratio Surface Condition Aluminum (alloy 1100) 896b 2 740u 221u 0.09n Commercial sheet 0.20n Heavily oxidized Aluminum bronze (76% Cu, 22% Zn, 2% Al) 400n 8 280u 100u Asbestos: Fiber 1050b 2 400u 0.170u Insulation 800t 580b 0.16b 0.93b “Paper” Ashes, wood 800t 640b 0.071b (50) Asphalt 920b 2 110b 0.74b Bakelite 1500b 1 300u 17u Bell metal 360t (50) Bismuth tin 170 65.0 Brick, building 800b 1 970u 0.7b 0.93 Brass: Red (85% Cu, 15% Zn) 400u 8 780u 150u 0.030b Highly polished Yellow (65% Cu, 35% Zn) 400u 8 310u 120u 0.033b Highly polished Bronze 435t 8 490t 29d (0) Cadmium 230a 8 650f 92.9b 0.02d Carbon (gas retort) 710a 0.35b (−17) 0.81a Cardboard 0.07b Cellulose 1300b 54t 0.057t Cement (Portland clinker) 670b 1 920i 0.029i Chalk 900t 2 290t 0.83 0.34 About 120°C Charcoal (wood) 840t 240a 0.05a (200) Chrome brick 710b 3 200b 1.2b Clay 920b 1 000t Coal 1000b 1 400t 0.17f (0) Coal tars 1500b (40) 1 200b 0.1b Coke (petroleum, powdered) 1500b (400) 990b 0.95b (400) Concrete (stone) 653b (200) 2 300b 0.93b Copper (electrolytic) 390u 8 910u 393u 0.072n commercial, shiny Cork (granulated) 2030t 86t 0.048t (−5) Cotton (fiber) 1340u 1 500u 0.042u Cryolite (AlF3·3NaF) 1060b 2 900b Diamond 616b 2 420t 47t Earth (dry and packed) 1 500t 0.064 0.41 Felt 330b 0.05b Fireclay brick 829b (100) 1 790t 1b (200) 0.75n At 1000°C Fluorspar (CaF2) 880b 3 190v 1.1v German silver (nickel silver) 400u 8 730u 33u 0.135n Polished Glass: Crown (soda-lime) 750b 2 470u 1.0t (93) 0.94n Smooth Flint (lead) 490b 4 280u 1.4r Heat-resistant 840b 2 230t 1.0t (93) “Wool” 657b 52.0t 0.038t Gold 131u 19 350u 297t 0.02n Highly polished Graphite: Powder 691 0.183 Impervious 670u 1 870u 130u 0.75n Gypsum 1080b 1 200b 0.43b 0.903b On a smooth plate Hemp (fiber) 1352.3u 1 500u Ice: 0°C 2040t 921b 2.24b 0.95 −20°C 1950t 2.44 Iron: Cast 500v (100) 7 210t 47.7b (54) 0.435b Freshly turned Wrought 7 700b 60.4b 0.94b Dull, oxidized Lead 129u 11 300u 34.8u 0.28n Gray, oxidized Leather (sole) 1 000b 0.16b Limestone 909b 1 650b 0.93b 0.36 to 0.90 At 63 to 193°C Linen 0.09b Litharge (lead monoxide) 230b 7 850b Magnesia: Powdered 980b (100) 796b 0.61b (47) Light carbonate 210b 0.059b Magnesite brick 930b (100) 2 530b 3.8b (204) Magnesium 1000b 1 730u 160u 0.55n Oxidized Marble 880b 2 600b 2.6b 0.931b Light gray, polished Nickel, polished 440u 8 890u 59.5u 0.045n Electroplated Paints: White lacquer 0.80n White enamel 0.91n On rough plate Black lacquer 0.80n Black shellac 1 000u 0.26u 0.91n “Matte” finish Flat black lacquer 0.96n Aluminum lacquer 0.39n On rough plate Data source unknown.
Notes: 1. Values are for room temperature unless otherwise noted in parentheses.
2. Superscript letters indicate data source from the section on References.
38.4 2001 ASHRAE Fundamentals Handbook (SI) REFERENCES aHandbook of chemistry and physics, 63rd ed. 1982-83. Chemical Rubber Publishing Co., Cleveland, OH.
bPerry, R.H. Chemical engineers’ handbook, 2nd ed., 1941, 5th ed., 1973.
McGraw-Hill, New York.
cTables of thermodynamic and transport properties of air, argon, carbon dioxide, carbon monoxide, hydrogen, nitrogen, oxygen and steam. 1960.
Pergamon Press, Elmsford, NY.
dAmerican Institute of Physics handbook, 3rd ed. 1972. McGraw-Hill, New York.
eOrganick and Studhalter. 1948. Thermodynamic properties of benzene.
Chemical Engineering Progress (November):847.
fLange. 1972. Handbook of chemistry, rev. 12th ed. McGraw-Hill, New York.
gASHRAE. 1969. Thermodynamic properties of refrigerants.
hReid and Sherwood. 1969. The properties of gases and liquids, 2nd ed.
McGraw-Hill, New York.
iChapter 19, 1993 ASHRAE Handbook—Fundamentals.
jT.P.R.C. data book. 1966. Thermophysical Properties Research Center, W.
Lafayette, IN.
kEstimated.
lCanjar, L.N., M. Goldman, and H. Marchman. 1951. Thermodynamic prop-erties of propylene. Industrial and Engineering Chemistry (May):1183.
mASME steam tables. 1967. American Society of Mechanical Engineers, New York.
nMcAdams, W.H. 1954. Heat transmission, 3rd ed. McGraw-Hill, New York.
oStull, D.R. 1947. Vapor pressure of pure substances (organic compounds).
Industrial and Engineering Chemistry (April):517.
pJANAF thermochemical tables. 1965. PB 168 370. National Technical Information Service, Springfield, VA.
qPhysical properties of chemical compounds. 1955-61. American Chemical Society, Washington, D.C.
rInternational critical tables of numerical data. 1928. National Research Council of USA, McGraw-Hill, New York.
sMatheson gas data book, 4th ed. 1966. Matheson Company, Inc., East Rutherford, NJ.
tBaumeister and Marks. 1967. Standard handbook for mechanical engi-neers. McGraw-Hill, New York.
uMiner and Seastone. Handbook of engineering materials. John Wiley and Sons, New York.
vKirk and Othmer. 1966. Encyclopedia of chemical technology. Interscience Division, John Wiley and Sons, New York.
wGouse and Stevens. 1960. Chemical technology of petroleum, 3rd ed.
McGraw-Hill, New York.
xSaline water conversion engineering data book. 1955. M.W. Kellogg Co.
for U.S. Department of Interior.
yTimmermans, J. Physicochemical constants of pure organic compounds, 2nd ed. American Elsevier, New York.
zWood handbook. 1955. Handbook No. 72. Forest Products Laboratory, U.S.
Department of Agriculture.
aaASHRAE. 1976. Thermophysical properties of refrigerants.
bbLane, G. ed. 1986. Solar heat storage: Latent heat materials, Vol II—Tech-nology. CRC Press, Chicago.
Paper 1300 930b 0.13b 0.92b Pasted on tinned plate Paraffin 1670bb 749bb 0.24b (0) Plaster 2 110b 0.74b (75) 0.91b Rough Platinum 130u 21 470u 69.0u 0.054b Polished Porcelain 750 260u 2.2u 0.92b Glazed Pyrites (copper) 549b 4 200b Pyrites (iron) 569b (69) 4 970v Rock Salt 917u 2 180u Rubber, vulcanized: Soft 2000 1 100t 0.1t 0.86b Rough Hard 1 190t 0.16t 0.95b Glossy Sand 800b 1 520b 0.33b Sawdust 190b 0.05b Silica 1320b 2 240v 1.4t (93) Silver 235u 10 500u 424u 0.02n Polished and at 227°C Snow: Freshly fallen 100y 0.598t At 0°C 500t 2.2t Steel (mild) 500b 7 830b 45.3b 0.12n Cleaned Stone (quarried) 800b 1 500t Tar: Pitch 2500v 1 100u 0.88v Bituminous 1 200t 0.71u Tin 233u 7 290u 64.9u 0.06h Bright and at 50°C Tungsten 130u 19 400u 201u 0.032n Filament at 27°C Wood: Hardwoods— 1900/2700b 370/1100z 0.11/0.255z Ash, white 690z 0.172z Elm, American 580z 0.153z Hickory 800z Mahogany 550u 0.13u Maple, sugar 720z 0.187z Oak, white 2390b 750z 0.176z 0.90n Planed Walnut, black 630z Softwoods— See Table 4, Chapter 24 350/740z 0.11/0.16z Fir, white 430z 0.12z Pine, white 430z 0.11z Spruce 420z 0.11z Wool: Fiber 1360u 1 300u Fabric 110/330u 0.036/0.063u Zinc: Cast 390u 7 130u 110u 0.05n Polished Hot-rolled 390b 7 130b 110b Galvanizing 0.23n Fairly bright Data source unknown.
Notes: 1. Values are for room temperature unless otherwise noted in parentheses.
2. Superscript letters indicate data source from the section on References.
Table 3 Properties of Solids (Continued) Material Description Specific Heat, J/(kg·K) Density, kg/m3 Thermal Conductivity, W/(m ·K) Emissivity Ratio Surface Condition |
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Complete Guide To Weak Acid pH and Ka Calculations
Table of Contents
Introduction
How To Find pH Of a Weak Acid When Ka is Given
How To Find Ka of a Weak Acid When pH is given
How to Calculate % ionization of a weak acid
Weak acids react with water to produce hydronium ion,H3O+,( hydrogen ions (H+) is the concise version of hydronium ion) and conjugate base.
General Equation : HA (aq) <--> H+(aq) + A-(aq)
or HA(aq)+H2O(l) <---> H3O+(aq) + A-(aq)
Weak acids do not dissociate completely. Instead, the acid ionizes only partially in water. Acid ionization constant (equilibrium constant for acids) is Ka.
Ka =[H+][A-]/[HA]
Acid ionization constant describes the extent to which acid ionizes. The higher the Ka, the more acid ionizes and the stronger it is.
How to find pH of a weak acid:
Identify that you have a weak acid (If a question asks for Ka or gives Ka, most likely you are dealing with a weak acid).
Write out the equation for acid dissociation:
HA(aq) <---> H+(aq) + A-(aq)
Create an ICE chart and plug in the initial concentrations.
Write out the Ka expression.
Plug in equilibrium values into the Ka expression and solve for X.
X =[H+]
pH = -log[H+]
Example: What is the pH of 0.10 M nicotinic acid, HC6H4NO2, at 25°C? Ka of nicotinic acid is 1.410^-5
Let's go over the steps:
Identify that you have a weak acid.
We are given a small Ka value of nicotinic acid which implies that it is a weak acid. It is also not one of the strong acid that we memorized.
Write out the equation for acid dissociation: HA (aq) < --> H+(aq) + A-(aq).
Create an ICE chart and plug in the initial concentrations.
Write out the Ka expression.
Plug in equilibrium values into the Ka expression and solve for X. X =[H+].
To make calculations easier and not use the quadratic formula, we assume x is much smaller than 0.10 and cross it out. After we calculate the x value, we need to check the assumption by doing (x/0.10)100% and if it is less than 5% our assumption works. Please note: different books and professors handle this assumption different. Please refer to your professor's notes.
pH = -log[H+]
How to find Ka of a weak acid when pH is given
Write out the equation for acid dissociation: HA(aq) < --> H+(aq) + A-(aq)
Create an ICE chart and plug in the initial concentrations
Determine [H+] from pH: [H+]= 10^-pH
[H+] = x . Plug in the concentration into the equilibrium values instead of x
Write out the Ka expression
Plug in equilibrium values into the Ka expression and solve for Ka
Example: Nicotinic acid (niacin) is a monoprotic acid with the formula HC6H4NO2. A solution that is 0.012 M in nicotinic acid has a pH of 3.39 at 25°C. What is the acid-ionization constant, Ka, for this acid at 25°C?
How to Calculate % ionization of a weak acid
Identify that you have a weak acid (If a question asks for Ka or gives Ka, most likely you are dealing with a weak acid
Write out the equation for acid dissociation:
HA(aq) <--> H+(aq) + A-(aq)
Create an ICE chart and plug in the initial concentrations
Write out the Ka expression
Plug in equilibrium values into the Ka expression and solve for X
% ionization = (X/[HA]initial)100%
Example: Calculate the percent ionization of a 0.20M benzoic acid (HC7H5O2) solution at equilibrium at 25°C if Ka of benzoic acid is 6.310^-5.
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10949 | https://www.math.ucdavis.edu/~webfiles/dissertations/202103_Leroux_dissertation.pdf | Halving point configurations; techniques from algebraic and convex geometry By BRETT ELLIOTT LEROUX DISSERTATION Submitted in partial satisfaction of the requirements for the degree of DOCTOR OF PHILOSOPHY in MATHEMATICS in the OFFICE OF GRADUATE STUDIES of the UNIVERSITY OF CALIFORNIA DAVIS Approved: Luis Rademacher, Chair Eric Babson Jes´ us De Loera Committee in Charge 2021 i © Brett Elliott Leroux, 2021. All rights reserved.
Contents Abstract iv Acknowledgments vi Chapter 1.
Introduction 1 Overview of Chapter 2: Preliminaries 3 Overview of Chapter 3: The k-set problem for algebraic set systems 3 Overview of Chapter 4: Generally k-neighborly embedded manifolds and varieties 4 Overview of Chapter 5: Improved bounds for the expected number of k-sets 5 Overview of Chapter 6: Translations of a fixed convex body in the plane 6 Chapter 2.
Preliminaries 7 2.1.
Discrete geometry and convex geometry 7 2.2.
k-sets and k-facets 8 2.3.
Mass partitions 9 2.4.
Set systems and VC-theory 12 2.5.
Classical real algebraic geometry 13 Chapter 3.
The k-set problem for algebraic set systems 16 3.1.
Introduction 16 3.2.
Preliminaries 18 3.3.
Counting k-facets via maps 21 Chapter 4.
Generally k-neighborly embedded manifolds and varieties 29 4.1.
Introduction 29 4.2.
Generally neighborly embeddings 30 4.3.
Generally neighborly manifolds 32 ii 4.4.
Generally k-neighborly algebraic varieties 33 4.5.
Weakly k-neighborly sets 34 4.6.
Weakly k-neighborly varieties 36 4.7.
Additional evidence 41 4.8.
Neighborly embeddings 42 Chapter 5.
Improved bounds for the expected number of k-sets 44 5.1.
Expected number of k-edges 44 5.2.
Bounding the expected number of k-edges 47 5.3.
On the number of k-edges via the polynomial method 51 Chapter 6.
Translations of a fixed convex body in the plane 59 6.1.
Introduction 60 6.2.
Upper bound for TC-k-sets, probabilistic, k proportional to n 61 6.3.
Lower bound for TC-k-sets, deterministic, k proportional to n 64 6.4.
Bounds on the growth function 66 6.5.
Lower bound for TC-k-sets, probabilistic, some k proportional to n 68 Chapter 7.
Open questions 72 7.1.
The real degree of algebraic varieties 72 7.2.
Conjecture on generally k-neighborly embeddings/manifolds 73 7.3.
Lower bounds for the algebraic k-set problem 73 7.4.
Bound on the number of points of intersection of Z(f) and the k-edge graph 73 7.5.
Polynomial partitioning for k-sets in higher dimensions 74 Bibliography 75 iii Abstract A halving line of a set of points is a line that divides the set of points into two equal parts. The halving lines problem asks: What is the maximum number of distinct halving lines that a set of n points can have? The focus of this dissertation is on results either about or inspired by the halving lines problem and its variations and generalizations. We start out by generalizing the halving lines problem in the most natural way: Given a family of curves or surfaces and a set of points, we want to know how many ways there are to divide the set of points into two equal parts using one of the curves or surfaces in the given family. And we would also like to know what the maximum number of halving curves or surfaces that a set of n points can have is. This type of problem leads us to ask several new questions which are relevant to discrete geometry, convex geometry, as well as real algebraic geometry. Some of our main contributions are as follows: • We study a variation on the halving lines problem when the family of separating curves or surfaces is a parametric family of algebraic curves or surfaces. In some cases, we are able to exactly count the number of halving curves. An example when we obtain an exact count is for the conic sections. These results are similar to a result of Ardila on halving circles.
• The concept of neighborliness is crucial for several of our results. Neighborly polytopes are important to the theory of convex polytopes because of their appearance in the upper bound theorem of McMullen. The moment curve is the standard way to construct neigh-borly polytopes. We define generally neighborly manifolds and algebraic varieties. These objects can be seen as higher-dimensional analogues of the moment curve.
• We study the random version of the original k-set problem in the plane and establish an improved upper bound for the expected number of k-sets. We also investigate how it may be possible to improve our bound using the continuous version of the polynomial partitioning theorem of Guth and Katz. This motivates a question about the points of intersection of an algebraic curve and the k-edge graph of a set of points.
• Another variation on the random version of the k-set problem is introduced and essentially solved: We obtain nearly tight bounds for the expected number of ways one can enclose k points from a random set of points using a translation of a fixed strictly convex body in iv the plane. The motivation is to show that a technique for counting k-sets due to B´ ar´ any and Steiger is nearly tight for a natural variation on the k-set problem.
A theme throughout this work is the investigation of questions whose answers help us under-stand the limits of an argument or proof technique. Most of the ideas presented here also appeared in papers coauthored with Luis Rademacher.
v Acknowledgments First, I would like to thank Luis Rademacher for working with me for the past 3.5 years. I am grateful for his ability to foster math research spanning disparate areas. And I am grateful for his willingness to entertain and stimulate ideas, even those which appear far-fetched.
Many thanks to Jes´ us De Loera who has been a source of advice and inspiration throughout my time in graduate school. I would also like to thank Eric Babson for serving on my thesis and qualifying exam committee and Anne Schilling and Nina Amenta for serving on my qualifying exam committee. Also, I am extremely grateful for the people who have been a part of the ADM, CCACAOO and MADDD seminars at Davis over the last several years and for all the wonderful talks I have been able to see.
Outside of the Davis, I am particularly grateful for the math events I attended at MSRI and UCB. In particular, I would like to thank Adam Sheffer and Joshua Zahl for hosting the Polynomial Method summer school at MSRI during the summer of 2019.
I also want to thank Luis Rademacher (again) and the National Science Foundation for the financial support I have received. This work was supported by the National Science Foundation under Grants CCF-2006994, CCF-1657939, CCF-1422830 and CCF-1934568.
Finally, I would like to thank my friends and family; for their support and love, for the lovely things we’ve seen together, and for asking me questions about math, and even occasionally asking good questions and being truly interested in the answers I gave.
vi CHAPTER 1 Introduction This dissertation is about dividing finite sets of points into two equal parts. The main goal is to understand, in various situations, the number of distinct ways that a set of points can be divided into two equal parts. In order to better understand this sort of problem, we ask some new questions which are relevant to discrete geometry, convex geometry, and real algebraic geometry and which go beyond the basic question of equal division of finite point sets.
The ideas presented here begin with halving lines. Given a set of points in the plane, a halving line is a line that divides the set of points into two equal parts. The halving lines problem asks: What is the maximum number of distinct halving lines that a set of n points can have as a function of n? The halving lines problem is now usually referred to as the k-set problem where a k-set of a set of n points in the plane is a subset of size k which can be separated from the remaining points by a line. The k-set problem asks one to determine the maximum number of k-sets that a set of n points can have. Of course, the halving lines problem is the k = n/2 case of the more general k-set problem.
Figure 1.1. A set of 6 points with all halving lines drawn. This configuration of points has the maximum number of halving lines over all configurations of size 6.
1 The question of determining the maximum number of k-sets for point sets in the plane was first raised by A. Simmons in unpublished work. Straus, also in unpublished work, gave a construction showing an Ω(n log n) lower bound for the k = n/2 case. Lov´ asz [Lov71] published the first paper on k-sets, establishing an O(n3/2) upper bound. See also the paper [ELSS73] of Erd˝ os, Lov´ asz, Simmons, and Straus. The main challenge is that even for the basic k-set problem on the plane, the asymptotics of the maximum number of k-sets is not well understood despite decades of effort.
See Chapter 3 for an overview of the best known bounds.
In the late 1960s and early 1970s when the k-set problem was first studied, the field of computa-tional geometry barely existed. Today, however, computational geometry is one of the main reasons for studying k-sets because they have important applications to geometric algorithms. For example, there are strong connections to order-k Voronoi diagrams [Der82], halfspace range search [CP86], convex hulls and k-hulls [CSY87].
Figure 1.2. A set of 6 points with all 2-edges drawn. (see Definition 2.2.4).
Many of the results proven here are about generalizations of or variations on the original halving lines problem: Given a family of curves or (hyper)surfaces, and a set of points in Rd, the aim is to understand how many combinatorially distinct ways there are to divide the set of points into two equal parts using one of the surfaces in the given family. A generalization of this question asks, for any fixed integer k, how many ways there are to separate k of the points from the remaining points.
The subsets of size k which can be separated from the remaining points by one of the surfaces in the chosen family are called F-k-sets where the notation F specifies the family of surfaces, i.e., the set system, see Chapter 2. The original k-set problem is the case when the family of surfaces is the 2 family of all lines in the plane. As in the case of the original k-set problem, the most interesting and difficult question in this context is how many F-k-sets can there be? More precisely, for any integer n, what is the maximum number of F-k-sets that a set of n points can have as a function of n and k? For most families of surfaces, the halving case, i.e., the case when k is equal to n/2, is the most important case to consider. And if one can solve the problem in this case, the problem has essentially been solved. However, most of our results will be stated for arbitrary values of k, as the generalization is usually not difficult to obtain.
As outlined in the rest of this introduction, our quest to better understand the halving lines problem and its variations leads us to use diverse techniques from convex and discrete geometry as well as real algebraic geometry.
Our use of these techniques leads us to ask, and in some cases answer, several new questions about algebraic or convex sets in Rd. In this way, all of the questions/results in this dissertation are either about or inspired by k-sets.
Below is an outline of the content of each chapter of this document. Chapters 3 and 4 are based on the paper [LR20] written jointly with Luis Rademacher. Chapters 5 and 6 are based on the paper [LR21] which was also written jointly with Luis Rademacher.
Overview of Chapter 2: Preliminaries This chapter reviews some of the most important existing definitions and theorems that are used throughout this dissertation. The definitions and theorems that we review come from the fields of discrete and convex geometry, combinatorics and probability, classical algebraic geometry, and mass partition problems.
Overview of Chapter 3: The k-set problem for algebraic set systems This chapter initiates the study of the k-set problem for more general set systems. A set system is a pair (X, R), where X is a set (called the universe) and R is a family of subsets of X. The subsets are called ranges. For the original k-set problem in the plane, the universe is R2 and the family of ranges is all halfplanes. We focus on algebraic set systems, i.e., set systems whose universe is Rd and whose ranges are described by polynomial inequalities. This means that the separating surfaces are algebraic surfaces. Thus, the problems studied in this section are about the number of ways that one can separate k points from a given set of n points using an algebraic surface from a 3 chosen family. One of our most surprising results is when the family of surfaces is all conic sections in the plane. In this case, we prove a formula which exactly counts, given any set of n points in general position and any k, the number of conic sections which separate k of the points from the remaining points (Theorem 3.3.8). We also study the k-set problem for other algebraic set systems, and when we cannot count k-sets exactly, we at least establish upper bounds.
Overview of Chapter 4: Generally k-neighborly embedded manifolds and varieties The proof of the result mentioned above about conic sections uses a very surprising property of the degree two Veronese map of the plane (Definition 3.2.2). This map has the property that it maps every generic set of points to a neighborly set of points (see Definition 2.1.2 for the definition of neighborly). We call embeddings with this property generally neighborly embeddings or generally k-neighborly embeddings for certain k (Definition 4.2.2). The image of a generally k-neighborly embedding of Rd is called a generally k-neighborly d-manifold.
Recall that the moment curve M is the image of the map ϕ : R 7→Rp where ϕ(t) = (t, t2, . . . , tp).
Generally neighborly d-manifolds should be seen as higher dimensional generalizations of the mo-ment curve for the following reason. The most important property of the moment curve is that any configuration of points on M is neighborly. To find a higher-dimensional version of the moment curve, one would want to find some map ϕ : Rd →Rp (with d ≥2) with the property that every set of points on ϕ(Rd) is neighborly. However, it was shown in [KW08] that no such map exists.
Therefore, the best that we can ask for is that every generic set of points on ϕ(Rd) is neighborly.
When this is the case, ϕ(Rd) is called a generally neighborly d-manifold. When d = 2, generally neighborly d-manifolds do exist. An example is the image of the degree 2 Veronese map of the plane and more examples are given in Chapter 4. However, we conjecture in Conjecture 4.2.5 that for d ≥3, generally neighborly d-manifolds do not exist.
This conjecture is part of the following more general question which we leave open: What is the minimal dimension of the ambient space in which a generally k-neighborly d-dimensional manifold can exist? This question appears as Problem 4.3.2 in Chapter 4.
4 Because we are unable to resolve the main question we have about generally k-neighborly man-ifolds, we study a closely related question about generally k-neighborly algebraic varieties (Defi-nition 4.4.1). Again, the main question is what is the minimal dimension of the ambient space in which a generally k-neighborly d-dimensional algebraic variety can exist? We show that the minimal dimension is 2k + d −1 (Theorem 4.4.3). The proof uses another neighborliness property called weakly neighborly (see Definition 4.5.1). It turns out that all generally k-neighborly vari-eties and manifolds are also weakly k-neighborly and using this property makes it easier to prove Theorem 4.4.3.
The questions in this section are in part inspired by a question asked by Micha Perles in 1982 and studied by Kalai and Wigderson [KW08]. Perles’ question is about neighborly embeddings which we briefly discuss at the end of Chapter 4.
Overview of Chapter 5: Improved bounds for the expected number of k-sets Chapter 5 returns to the original k-set problem in the plane, except that we study the random version of the problem. Instead of trying to determine the maximum number of k-sets that a set of n points can have, we study the expected number of k-sets of a set of n points chosen from some probability distribution on the plane. Our main contribution is an improved upper bound on the expected number of k-sets when the distribution is any probability distribution on the plane such that the measure of every line is 0. In this case, we show that the expected number of k-sets is O(n5/4) (Theorem 5.1.2). The assumption that every line has measure 0 is a very minor restriction on the distribution. This is the first result of this type for probability distributions at this level of generality. Our result is interesting because our O(n5/4) bound is significantly better than the current best known bound on the maximum number of k-sets for deterministic sets of points, which is O(n4/3) [Dey97]. Furthermore, we have some reason to believe that the random version of the k-set problem that we study may be more or less equivalent to the original k-set problem. See Section 5.1 for a discussion of why we believe this may be true.
The proof of our bound on the expected number of k-edges begins by using vertical lines to partition the plane into open vertical strips of equal probability. We investigate how it may be possible to improve our bound by partitioning the plane using the continuous version of the 5 polynomial partitioning theorem of Guth and Katz rather than the basic partition by vertical lines.
This motivates a question about the points of intersection of an algebraic curve and the k-edge graph of a set of points (Question 5.3.1).
Overview of Chapter 6: Translations of a fixed convex body in the plane In this chapter we study the k-set problem for set systems for which the set of ranges consists of all translations of some strictly convex body. That is, for a convex body C ⊂R2 and a set S of n points, we define a TC-k-set of S to be a subset T of S of size k such that there exists a translation of C which contains T in its interior and contains no other points from S. For any strictly convex body C we determine bounds for the expected number of TC-k-sets which are tight up to logarithmic factors (Theorem 6.2.9 and Theorem 6.5.3). The lower bound uses the uniform convergence theorem of Vapnik and Chervonenkis [VC71]. And the upper bound uses a technique due to B´ ar´ any and Steiger [BS94].
6 CHAPTER 2 Preliminaries The point of this chapter is to collect some necessary definitions as well as state some theorems which hopefully will give the reader an idea of the types of techniques that we use.
2.1. Discrete geometry and convex geometry Of all the classic results in discrete geometry, the most important one for our considerations is Radon’s theorem Theorem 2.1.1 (Radon’ s theorem). Any set of d+2 points in Rd can be partitioned into two sets A, B so that the convex hulls of A and B have a common point.
In fact, the theorem we use is a stronger version of Radon’s theorem (Lemma 4.5.4) for the case when the d + 2 points are in general position.
Radon’s theorem is relevant to Chapter 4 because of its connection to neighborly point config-urations and polytopes.
By a polytope, we mean a convex polytope. See [Zie95] for background on polytopes.
Definition 2.1.2. A polytope is k-neighborly if any set of k or fewer vertices forms a face. A d-dimensional polytope is neighborly if it is ⌊d/2⌋-neighborly.
If we are talking about point sets instead of polytopes, we will say that a set of points is k-neighborly (respectively, neighborly) if it is the vertex set of a k-neighborly (respectively, neighborly) polytope.
One way of producing neighborly point sets is by choosing a finite set of points on the moment curve M := {(t, t2, . . . , td) : t ∈R} ⊂Rd. The moment curve is the standard example of an order d curve which is a curve that is intersected by any hyperplane in at most d points. Any finite set of distinct points on an order d curve is neighborly [MSRB71,Stu87].
7 The connection between Radon’s theorem and neighborliness is that it is a simple consequence of Radon’s theorem that if a d-dimensional polytope is (⌊d/2⌋+ 1)-neighborly then it is a simplex.
Our proof of a crucial lemma in Chapter 4 (Lemma 4.6.2) uses a very similar idea, just in a slightly more technical context.
From the area of convex geometry, a classic and important result we will use is about separation of convex sets in Rd.
Two sets Q, R ⊆Rp can be weakly separated if there exist a non-zero a ∈Rp and t ∈R such that Q ⊆{x ∈Rp : a · x ≤t} and R ⊆{x ∈Rp : a · x ≥t}.
We say that the hyperplane {x ∈Rp : a · x = t} weakly separates Q from R. This separation is said to be proper if Q and R are not both contained in {x ∈Rp : a · x = t}.
Now we can state the so-called separating hyperplane theorem: Theorem 2.1.3 (Theorem 1.3.8 in [Sch14]). Let Q, R ⊂Rd be non-empty convex sets. Then Q and R can be properly separated if and only if relint Q ∩relint R = ∅.
Another important notion of separation is called strict separation.
Definition 2.1.4. Two sets Q, R ⊆Rp can be strictly separated if there exist a non-zero a ∈Rp and t ∈R such that Q ⊆{x ∈Rp : a · x < t} and R ⊆{x ∈Rp : a · x > t}. We say that the hyperplane {x ∈Rp : a · x = t} strictly separates Q from R.
2.2. k-sets and k-facets Definition 2.2.1. Let S be a set of points in Rd. A k-set of S is a subset A ⊂S of size k that can be strictly separated from S \ A by a hyperplane.
Definition 2.2.2. We use ak(S) to denote the number of k-sets of the set S ⊂Rd and ad(k, n) for the maximum number of k-sets that a set of n points in Rd can have.
When studying the k-set problem, one usually only considers point sets which are in general linear position.
8 Definition 2.2.3. A set of at least d + 1 points in Rd is in general linear position if no d + 1 (and thus, fewer) points are affinely dependent.
This reduction is justified by the observation that the maximum number of k-sets is attained by a set of points in general linear position (see for example [Wag08]). For point sets in general linear position, one can study the closely related concept of k-facets.
Definition 2.2.4. Let S be a finite set of points in general linear position in Rd and let ∆be a subset of d points from S. The subset ∆along with some orientation of the hyperplane aff∆is a k-facet of S if the open halfspace on the positive side of aff∆contains exactly k points from S. In R2, k-facets are also known as k-edges.
Definition 2.2.5. We use ek(S) to denote the number of k-facets of the set S ⊂Rd and ed(k, n) for the maximum number of k-facets that a set of n points in general linear position in Rd can have.
It seems unlikely that one would be able to determine ed(k, n) or ad(k, n) precisely, so instead efforts have focused on finding the asymptotic behavior of these functions.
The k-set problem asks one to determine the asymptotic behavior of ad(k, n). And the k-facet problem asks one to determine the asymptotic behavior of ed(k, n).
If one is only concerned with the asymptotics, then it suffices to study either k-sets or k-facets since for fixed d and n →∞, ad(k, n) and ed(k, n) have the same asymptotic behavior [Wag08].
2.3. Mass partitions Mass partition theorems describe how a given set of points or collection of measures on Rd is partitioned after dividing Rd into a number of disjoint (usually open) sets.
Probably the most famous theorem of this sort is the ham sandwich theorem: Theorem 2.3.1 (Ham sandwich theorem [ST42]). Given d absolutely continuous1 measures µ1, . . . , µd on Rd, there exists a hyperplane H so that µi(H+) = µi(H−) for all 1 ≤i ≤d. (H+ denotes the open halfspace on the positive side of H and H−is the open halfspace on the negative side of H.) 1The absolutely continuous measures are precisely those which have a density.
9 Part of the reason this theorem is interesting is that its proof is via the Borsuk-Ulam theorem from algebraic topology.
The ham sandwich theorem is relevant to our considerations because it is how one proves polynomial partitioning-type theorems which we make use of in Chapter 5.
The polynomial partitioning theorem of [GK15] has recently been used to solve a number of problems in discrete and combinatorial geometry [Gut16a].
It has also been used to give alternative proofs of some known results, see [KMS12]. Perhaps the most commonly used version of the polynomial partitioning theorem is the following, which we refer to as the discrete version; Theorem 2.3.2 (Discrete polynomial partitioning [GK15]). Let S ⊂Rd be a set of n points. Then for each r ≤n there is a non-zero polynomial f ∈R[x1, . . . , xd] of degree Od(r) such that Rd \Z(f) is the union of a family O of rd pairwise disjoint open sets such that each O ∈O contains at most n/rd points of S. (The notation Od means that the constants in the bounds depend only on d.) In Chapter 5, we are focused on bounding the expected number of k-edges of a sample of points from a distribution on the plane (Theorem 5.3.2). The idea of the proof of this theorem is to use a divide-and-conquer approach which first partitions the plane into a number of cells of equal probability. We then consider separately the expected number of k-edges which cross from one cell to another and the expected number of k-edges that do not cross the boundary of the partition.
Therefore, we need a partitioning theorem which applies to probability distributions rather than finite point sets. This type of result, which we refer to as the continuous version of the polynomial partitioning theorem, has been used to establish improved bounds for the restriction problem in harmonic analysis, see [Gut16b] as a starting point.
Theorem 2.3.3 (Continuous polynomial partitioning [Gut16b]). Let W ∈L1(Rd) with W ≥0.
Then for each r, there is a non-zero polynomial f ∈R[x1, . . . , xd] of degree at most r such that Rd \Z(f) is the union of a family O of Θd(rd) pairwise disjoint open sets such that for all O ∈O, the integrals R O W are equal.
The open sets O in the above theorems are called the cells of the partition.
Using the density of the non-singular polynomials in the space of all polynomials of fixed degree in d variables (see Figs. 2.1a and 2.1b), it is possible to obtain, as a corollary of the continuous 10 (a) Singular cubic (b) Non-singular cubic Figure 2.1. An example of a singular degree 3 algebraic curve and a non-singular degree 3 algebraic curve which are “close” in the space of all degree 3 algebraic curves.
polynomial partitioning theorem, a version where all the irreducible components of the dividing surface Z(f) are non-singular varieties.
Theorem 2.3.4 (Non-singular continuous polynomial partitioning [Gut16b]). Let W ∈L1(Rd) with W ≥0. Then for each r, there is a non-zero polynomial f ∈R[x1, . . . , xd] of degree at most r such that Rd \Z(f) is the union of a family O of Θd(rd) pairwise disjoint open sets such that for all O ∈O, the integrals R O W are within a factor of two of each other. Furthermore, all irreducible components of Z(f) are non-singular.
Notice that the above theorems do not allow us to partition an arbitrary probability distribution, they only apply to those for which there exists a density W. This is the reason why our proof in Section 5.3 only applies to distributions which have a density.
As far as we know, it is an open problem to prove a polynomial partitioning theorem similar to Theorem 2.3.3 for arbitrary probability distributions. The proof of Theorem 2.3.3 relies on the ham sandwich theorem stated above which only applies to distributions which have a density. In order to extend the polynomial partitioning theorem to arbitrary measures, one could attempt to use the more general form of the ham sandwich theorem and repeat the proof of Theorem 2.3.3. However, since an arbitrary distribution may be concentrated in a hyperplane or even a point, one would not be able to guarantee that the integrals R Oi W are all equal, but only that all integrals are at most some fixed value.
11 2.4. Set systems and VC-theory Definition 2.4.1. A set system is a pair (X, R), where X is a set (called the universe) and R is a family of subsets of X The elements of R are called ranges).
We will use the language of set systems in Chapter 3 when we define k-sets for set systems other than halfspaces.
Set systems are also the starting point for VC-theory. The idea of VC-theory is to come up with a way to measure the complexity of a geometric object or family of geometric objects.
Our main use of VC-theory will be of the uniform convergence theorem of Vapnik and Cher-vonenkis [VC71]. We state this theorem below as Theorem 2.4.6. The idea of this theorem is as follows. Say (X, R) is some set system and P is some probability distribution on X. For the uni-form convergence theorem to hold, we also need to assume that the VC-dimension (Definition 2.4.2) of the set system (X, R) is finite. Let X1, . . . , Xm be a sample of m iid points from P. For each R ∈R the relative frequency of R with respect to the sample X1, . . . , Xm is the quotient |{Xi:Xi∈R}| m .
The uniform convergence theorem then says that the relative frequencies of the events R converge uniformly to their probabilities. Another way of saying this is that for any ϵ > 0, the probability that the maximum difference over all R ∈R of the relative frequency of R and the probability of R is less than ϵ converges to 0 as the size of the sample tends to infinity.
First we define VC-dimension. For a set system (X, R) and a subset Y ⊂X, we say that a subset A ⊆Y is induced by a range R ∈R if A = Y ∩R.
For a subset Y ⊂X, we say that Y is shattered by (X, R) if |{R ∩Y : R ∈R}| = 2|Y |. In other words, Y is shattered by the set system (X, R) if all subsets of Y can be induced by intersecting Y with one of the ranges in R.
Definition 2.4.2. The VC-dimension of set system (X, R) is the maximum size of a subset of X that is shattered by (X, R).
Example 2.4.3 (Halfspaces). Consider the set system (X, R) where X = Rd and R is all closed halfspaces. The VC-dimension this set system is d + 1. This follows from Radon’s theorem.
Definition 2.4.4 ( [VC71]). The growth function πR(n) of set system (X, R) is the maximum number of subsets of a set Y ⊆X of size n that can be induced by ranges in R. So the growth 12 function is given by πR(n) = max A⊆X,|A|=n {A ∩R : R ∈R} .
Theorem 2.4.5 ([VC71]). The VC-dimension gives a bound on the growth function: If the VC-dimension of the set system (X, R) is d, then πR(n) ≤ d X i=0 n i .
Theorem 2.4.6 (uniform convergence [VC71]). Suppose that (X, R) is a set system and P is a probability distribution on X such that every R ∈R is measurable according to P and also so that the function supR∈R(·) as defined below is a random variable (i.e., is measurable). For any 0 < ϵ < 1, and {X1, . . . , Xm} a sample of m ≥2/ϵ2 points from P, P sup R∈R P(R) −|{Xi : Xi ∈R}| m < ϵ !
≥1 −4πR(2m)e−ϵ2m/8.
In Chapter 6 we will also need to use the notion of the dual set system.
Definition 2.4.7 (dual set system). The dual set system of set system (X, R) is the set system (R, X∗) where X∗is the family of sets of the form {R ∈R : x ∈R} for x ∈X.
The dual growth function of (X, R) is the growth function of its dual set system.
2.5. Classical real algebraic geometry Given polynomials f1, f2, . . . , fr ∈R[x1, . . . , xp], the affine algebraic variety defined by the fi’s is the set V (C) = {x ∈Cp : f1(x) = 0, . . . , fr(x) = 0}.
Algebraic varieties are most well behaved when the field k is algebraically closed, for example k = C. Our applications, however, require us to study algebraic varieties over the real numbers.
In other words, for a given variety V (C) ⊂Cp, we are mainly interested in the subset V (R) := V (C) ∩Rp of real points. We refer to V (R) as an affine real algebraic variety. Unless otherwise stated, by an algebraic variety or simply variety we mean an affine real algebraic variety.
13 Since we are mainly interested in the set of real points, we will write V for V (R) and will write V (C) to indicate when the complex points are also considered. See [BCR98, Har95] for more definitions from real algebraic geometry.
Definition 2.5.1. A variety V ⊂Rp (resp. V (C) ⊂Cp) is non-degenerate if it is not contained in any hyperplane in Rp (resp. Cp).
One of the most useful properties of varieties is that there is simple and canonical way to decompose them into basic components.
Definition 2.5.2. An algebraic variety is irreducible if it cannot be written as the union of two proper algebraic subvarieties.
Any variety V has a unique decomposition into irreducible components. That is, there is a unique way to write V = V1 ∪. . . ∪Vm where each Vi is an irreducible subvariety and no Vi is contained in any Vj for j ̸= i. In such a decomposition, the Vi are called the irreducible components of V .
We will use facts about the smooth and singular points of a variety. Suppose V (C) is a variety and the ideal of V (C) is generated by the polynomials f1, . . . , fr. The smooth points of V (C) are those points where the Jacobian matrix of the fi’s has maximal rank. A singular point of a variety is a point that is not smooth.
Definition 2.5.3. We use Vsm to denote the set of smooth points of an algebraic variety V , Vsing is the set of singular points.
Given a real variety V ⊂Rp, let VC denote the smallest complex variety which contains V .
It is well known that VC is unique and furthermore that there is a bijection between irreducible components of V and irreducible components of VC, see [Whi57]. Note that V (C) is not always equal to VC. However, it is a useful fact that whenever V contains a smooth real point, V is Zariski 14 dense in V (C) and so V (C) = VC, see [BCR98, Section 2.8].2 We will always assume that our varieties contain smooth real points.
By the dimension of a real algebraic variety V we will mean the dimension of V (C). There is another notion of dimension for real algebraic varieties.
Definition 2.5.4. The real dimension of a real algebraic variety V is the maximal integer d such that there is a homeomorphism of [0, 1]d into some subset of V .
The real dimension of V does not always equal the dimension of V . However, the real dimension of V is never more than the dimension of V (see [BCR98, Proposition 2.8.14]) and if V contains a smooth real point, then these dimensions do agree. This is because around any smooth real point of a d-dimensional variety V ⊂Rp, there is a neighborhood which is a smooth d-dimensional submanifold of Rp [BCR98, Proposition 3.3.11].
In one of the proofs in Section 4.6 we will also consider projective varieties, see [Har95] for background on projective algebraic geometry. We use Pp(K) to denote p-dimensional projective space over the field K = C or K = R.
2There is a potential confusion here about the existence of smooth points in real varieties.
When working with varieties, if you start with a variety V ⊂Rd and then take V to be defined by the polynomials I(V ), then one can prove that every real variety contains smooth real points and thus V (C) = VC always. This is not the approach we take. (Recall that we defined real varieties as the set of real points of a complex variety.) For this reason, in our results later on about real varieties we always add the assumption that they contain smooth real points. But this assumption is only necessary because of the way we defined real varieties.
15 CHAPTER 3 The k-set problem for algebraic set systems In this chapter we study a variation on the k-set/k-facet problem with hyperplanes replaced by algebraic surfaces. In stark contrast to the original k-set/k-facet problem, there are some natural families of algebraic curves for which the number of k-facets can be counted exactly. For example, we show that the number of halving conic sections for any set of 2n + 5 points in general position in the plane is 2 n+2 2 2. Additionally, we give a simple argument which improves the best known bound on the number of k-sets/k-facets for point sets in convex position.
3.1. Introduction It is natural to ask questions similar to the k-set problem but for families of surfaces different from all hyperplanes.
These sorts of questions have been studied in [Der82, Ard04, BHP08, CFSS14, CFSS20]. Ardila’s paper [Ard04] shows that for any set of 2n + 1 points in general position in the plane and any 0 ≤k ≤2n −2, the number of circles that go through 3 points and have k points on one side is exactly 2(k + 1)(2n −k −1). We call this phenomenon exact counting: when for a family of curves (or surfaces), there exists an integer d such that given any generic set of n points and any k, the number of curves which pass through d points and have k points on one side depends only on n and k and not on the points. A result essentially equivalent to Ardila’s was proven earlier in [Der82] by counting vertices of certain Voronoi diagrams. Chevallier et al.
extended the result to convex pseudo-circles in [CFSS20].
Borrowing the language of set theory/computational geometry/learning theory, one can think of the k-set problem as being formulated over a set system (also known as a hypergraph, hypothesis class or range space), namely a universe and a family of subsets of the universe.
In the k-set problem the universe is Rd and the family of subsets is all halfspaces. This paper takes a step towards the understanding of the k-set problem for general set systems. We focus on set systems induced by maps in the following way: given a map ϕ : Rd →Rp, the set system induced by ϕ 16 has universe Rd and family of subsets {ϕ−1(H) : H is a closed halfspace in Rp}. Moreover, most of our results involve maps ϕ with components that are polynomials, so that the separating surfaces in the resulting set system are algebraic surfaces. One of our main examples is the Veronese map (Definition 3.2.2) which induces separators that are algebraic surfaces of degree at most m. The Veronese map is also known as the feature map of the polynomial kernel in machine learning [STC04].
Our contributions: • Exact count. We show that the exact count phenomenon of [Der82,Ard04] (for halving circles) holds for other natural set systems: conic sections (Theorem 3.3.8) and homoge-neous polynomials of fixed even degree on the plane (Theorem 3.3.9).
We prove this by establishing a remarkable property of the corresponding maps: generic point sets are mapped to point sets that form the vertices of a neighborly polytope (Theorem 3.3.4 and Theorem 3.3.5, see Section 3.2.5 for background). This is then combined with the known fact that the number of k-facets of a neighborly point set is given by a formula that depends only on the dimension, on k, and on the number of points [Cla87,CS89], [Wag08, Propo-sition 4.1].
• Convex position bound. We show an improved upper bound on the number of k-sets/k-facets for points in convex position (Theorem 3.3.12). While our argument is simple, we are not aware of any known bounds for the convex case better than the general case in dimension higher than three (the convex case is well understood in two and three dimensions).
• Degree of neighborliness. We study the degree of neighborliness of point sets mapped by a ϕ with components “all monomials of degree at most m” or “all monomials of degree exactly m” (Theorem 3.3.10 and Theorem 3.3.11). In particular, for even m, point sets are mapped into point sets in convex position and the convex position bound gives an improved bound on the number of k-facets.
Outline of the Chapter. Section 3.2 reviews some preliminary material and introduces our gen-eralization of the k-facet problem to set systems which are induced by maps. In Section 3.3 we 17 count k-facets for set systems induced by maps. This amounts to studying k-facets of point sets of the form ϕ(S) where ϕ : Rd →Rp is some map and S ⊂Rd is a finite set of points.
3.2. Preliminaries 3.2.1. Generic properties and general position. After defining k-sets and k-facets for set systems other than halfspaces, we will need to use various notions of general position different from general linear position.
A generic property (of point sets) is one that holds for all but a relatively small number of atypical point sets.
The point sets which satisfy a generic property are said to be in general position. We use the terms “generic point set” and “point set in general position” interchangeably.
In algebraic geometry a generic property is one that holds for a dense and open set. In other fields, a generic property holds almost everywhere. For concreteness in some of our statements we set “generic property” to mean a property that holds in an open and dense set, but this choice is not always crucial.
Now we state more explicitly what it means for a point configuration to be generic.
The collection of all configurations of n-point sets in Rd can be identified with Rdn. If Gn ⊂Rdn is the collection of all generic point configurations of size n, then Gn should be dense and open in Rdn for all n. See [Mat02, Section 1.1] for more on the meaning of general position in discrete geometry.
3.2.2. Set systems induced by maps. Recall that a set system is a pair (X, F) where X is a ground set (or universe) and F is a collection of subsets of X.
We will restrict our attention to set systems which are induced by maps. Suppose we have a map ϕ : Rd →Rp, that is, a map of Rd into some (usually higher dimensional) space. Any such map induces a set system on the ground set Rd in the following way. Let Fϕ consist of all regions R ⊂Rd of the form ϕ−1(H) where H is a closed halfspace in Rp. We say that R is induced by the halfspace H and we say that the set system (Rd, Fϕ) is induced by ϕ. Many interesting set systems are induced by maps.
3.2.3. Set systems induced by Veronese-type maps.
18 Definition 3.2.1. A polynomial map is a map Rd →Rp defined by x 7→(f1(x), . . . , fp(x)) where f1, . . . , fp are polynomials.
Here we introduce our primary examples of polynomial maps and set systems.
Definition 3.2.2. The degree m Veronese map of Rd is the map V d m : Rd →R(d+m m )−1 which maps (x1, . . . , xd) to the vector (in some order) of all non-constant monomials of degree at most m in the d variables x1, . . . , xd.
Definition 3.2.3. The degree m homogeneous Veronese map of Rd is HV d m : Rd →R(d+m−1 m ) which maps (x1, . . . , xd) to the vector (in some order) of all monomials of degree m in the d variables x1, . . . , xd.
We will use the notation (Rd, Pd m) for the set system induced by the degree m Veronese map of Rd and (Rd, Hd m) for the set system induced by the degree m homogeneous Veronese map of Rd.
As a concrete example, consider the degree 2 Veronese map of R2. This is the map V 2 2 : R2 →R5 where V 2 2 (x, y) = (x2, xy, y2, x, y). The set system induced by this map is (R2, P2 2). Its subsets consist of all regions of the plane determined by some conic section.
3.2.4. On k-sets and k-facets for set systems induced by maps. The natural notion of a k-set of a set system is a range that contains exactly k points. It is often more convenient to work with k-facets, but it is not clear how to define them for set systems whose ranges lack a well-defined boundary. This motivates our restriction to set systems induced by maps, as their ranges have a well-defined boundary and interior.
Definition 3.2.4. Given a set system (X, Fϕ) induced by a map ϕ : X →Rp and a finite set S ⊂X, an Fϕ-k-set of S is a subset A ⊂S of size k such that ϕ(A) can be strictly separated from ϕ(S \ A) by a hyperplane.
Definition 3.2.5. Given a set system (X, Fϕ) induced by a map ϕ : X →Rp and a finite set S ⊂X such that ϕ(S) is in general linear position, an Fϕ-k-facet of S is a subset P of p points 19 from S, along with some orientation of the hyperplane affϕ(P), such that the subset ϕ(P) along with the chosen orientation of affϕ(P) is a k-facet of ϕ(S).
Observe that counting Fϕ-k-sets/facets simply amounts to counting k-sets/facets of point sets of the form ϕ(S). Therefore, upper bounds for ed(k, n) and ad(k, n) immediately imply non-trivial upper bounds on the number of Fϕ-k-sets/facets that a set of points may have: Proposition 3.2.6. Given a set system (X, Fϕ) induced by a map ϕ : X →Rp, and a finite subset S ⊂X such that ϕ(S) is in general linear position, the number of Fϕ-k-facets of S is at most ep(k, n).
For Fϕ-k-sets, we do not need to assume that ϕ(S) is in general linear position since k-sets are defined for any point set whether or not it is in general linear position.
Proposition 3.2.7. Given a set system (X, Fϕ) induced by a map ϕ : X →Rp, the number of Fϕ-k-sets that a set of n points in X may have is at most ap(k, n).
Proof. In the case when ϕ is not injective, ϕ(S) may need to be considered as a multiset.
Therefore, we start with the observation that ap(k, n) is the maximum number of k-sets even for point sets which have repeated points, i.e., multisets of points. To see this, observe that perturbing a set of points can only increase the number of k-sets [Wag08]. Therefore, if we start out with a multiset, we can perturb it slightly to create a set (in general linear position) with the same number of points and at least as many k-sets. Now, the number of Fϕ-k-sets of S is equal to the number of k-sets of ϕ(S), which is at most ap(k, n).
□ 3.2.5. Neighborly polytopes. For a set of n points in convex position in the plane, the number of k-facets is precisely n for all values of k. In R3, a similar result is true: the number of k-facets for a set S of n points in general position which form the vertex set of a 3-polytope is 2(k + 1)n −4 k+2 2 , see [Wag08]. There is no such result in dimension d ≥4, i.e., convex position does not force a point set in Rd (d ≥4) to have a specific number of k-facets. In fact, the k-set/k-facet problem for point sets in convex position in R4 is only slightly better understood than the problem for arbitrary point sets, see Theorem 3.3.12.
20 However, if we assume that our point set is not only in convex position but is also neighborly, then e(d) k (S) is determined precisely by |S| and k. See Definition 2.1.2 for the definition of neighborly point sets and [Wag08,Zie95] for a more thorough introduction to neighborly polytopes.
Proposition 3.2.8 ( [Cla87, CS89], [Wag08, Proposition 4.1]). Let S be a neighborly set of n points in general linear position in Rd. Then ek(S) = 2 k+⌈d/2⌉−1 ⌈d/2⌉−1 n−k−⌈d/2⌉ ⌈d/2⌉−1 if d is odd k+d/2−1 d/2−1 n−k−d/2 d/2 + k+d/2 d/2 n−k−d/2−1 d/2−1 if d is even.
3.3. Counting k-facets via maps In this section we count Fϕ-k-facets when Fϕ is a set system induced by a map. When the map ϕ has certain properties, we can say more about the number of Fϕ-k-facets.
3.3.1. Counting k-facets exactly. It turns out that the maps associated to several families of polynomials we have discussed have the surprising property that they map generic point sets into the set of vertices of a neighborly polytope. Given such a map ϕ, we are able to exactly count the number of Fϕ-k-facets for point sets in general position.
Before stating the new results, we recall a result of [Der82,Ard04] which served as motivation.
A halving circle of a point set of size 2n + 1 is a circle which has 3 points on its boundary and n −1 points on either side. In the following theorem general position means that no three points are collinear and no four are concyclic.
Theorem 3.3.1 ( [Der82, Ard04]). Any set S of 2n + 1 points in general position in the plane has exactly n2 halving circles. More generally, for any 0 ≤k ≤2n −2, the number of circles that have 3 points of S on their boundary and k points on one side is exactly 2(k + 1)(2n −k −1)1.
The proof of Theorem 3.3.1 in [Ard04] is by a continuous motion argument.
However, as noted there, it is possible to give a shorter proof using the method of maps as follows. The set system of all circles in the plane can be described as (R2, FC) where C : R2 →R3 is the map C(x, y) = (x, y, x2 + y2). Since C maps generic point sets into convex position (on the surface of 1This formula counts each halving circle twice, once for each orientation.
21 a paraboloid), Theorem 3.3.1 follows from an application of the formula in Proposition 3.2.8 since any 3-polytope is neighborly.
Figure 3.1. A set of 7 points with all halving conic sections drawn.
Now we define halving polynomials for other families of polynomials. Informally, for a finite set S ⊆R2, a halving conic section of S is a conic section inequality having 5 points on its boundary and half of the remaining points of S in its interior. Unlike Theorem 3.3.1 on the circle problem above, we count halving conic sections twice, once for each orientation. This is to be consistent with the standard definition of k-facets (Definition 3.2.5). More precisely, Definition 3.3.2. For a set S of 2n + 5 points in R2, a halving conic section is an FV -n-facet of S where V := V 2 2 is the degree 2 Veronese map of R2.
Definition 3.3.3. For a set S of 2n + m + 1 points in R2, a halving homogeneous polynomial of degree m of S is an H2 m-n-facet of S.
The halving case is a particular case of the more general problem of counting k-facets. It is generally believed that the maximum number of k-facets is maximized by the halving case, so it is considered the most important. However, we state our results counting H2 m-k-facets and FV -k-facets for arbitrary values of k.
22 Theorem 3.3.4. Assume a finite set of points S ⊆R2 is in general linear position. Then the image of S by the degree 2 Veronese map V 2 2 is neighborly.
Proof. First we verify that V 2 2 (S) is the set of vertices of a polytope. There is a bijection between conic sections passing through points of S and hyperplanes passing through the images of those points by V 2 2 . Therefore, for every point v ∈S we need to find a conic section inequality passing through v and with all other points on one side. We can use an inequality of the form (x −a)2 + (y −b)2 ≤r and adjust the constants a, b, r so that (x −a)2 + (y −b)2 ≤r defines a circle that contains v on its boundary and has radius small enough so that no other point in S is in the circle. Now we show that V 2 2 (S) is neighborly. For every 2 points v1, v2 of S we need to find a conic section inequality passing through those points and with all other points on one side. One way to accomplish this is to use the line ax + by = c through v1 and v2. Then (ax + by −c)2 ≤0 is the required conic section inequality.
□ In terms of the terminology defined in Section 4.2, the above result says that V 2 2 is a “generally neighborly embedding”. The same result holds for the even degree homogeneous Veronese map of the plane.
Theorem 3.3.5. Assume m is even and S ⊆R2 is in general position, meaning that no two points of S lie on a common line through the origin. Then the image of S by HV 2 m is neighborly.
Proof. The proof is similar to that of Theorem 3.3.4. For any set {v1, . . . , vk} of k ≤m/2 points of S we need to find a degree m homogeneous polynomial inequality which passes through all the vi and has all other points of S on one side. Let vk+1, . . . , vm/2 be points in the plane that belong to no line passing through the origin and a point of S. For each i, 1 ≤i ≤m/2, let aix + biy = 0 be the line through the origin and vi. Then Qm/2 i=1 (aix + biy)2 ≤0 is a polynomial inequality with the required properties.
□ The next results require us to strengthen our general position assumptions from Theorem 3.3.4 and Theorem 3.3.5.
Definition 3.3.6. A set S ⊂R2 is in general position with respect to conics if S is in general linear position and V 2 2 (S) is in general linear position.
23 Definition 3.3.7. A set S ⊂R2 is in general position with respect to degree m homogeneous polynomials if no two points in S lie on a common line through the origin and HV 2 m(S) is in general linear position.
Theorem 3.3.8. Any set of n points of R2 in general position with respect to conics has exactly 2 k+2 2 n−k−3 2 FV -k-facets where V := V 2 2 is the degree 2 Veronese map of the plane.
Proof. Let S ⊂R2 be a set of n points in general position with respect to conics. There is a bijection between conic sections passing through 5 points of S and hyperplanes passing through 5 points of V (S). Furthermore, there is a bijection between FV -k-facets of S and k-facets of V (S).
By Theorem 3.3.4, V (S) is neighborly. Also, since S is in general position with respect to conics, V (S) is in general linear position. Therefore the number of FV -k-facets of S is given by the formula from Proposition 3.2.8.
□ Theorem 3.3.9. Assume m is even. Any set of 2n + m + 1 points of R2 in general position with respect to degree m homogeneous polynomials has exactly 2 k+m/2 m/2 n−k−m/2−1 m/2 H2 m-k-facets.
Proof. Let S ⊂R2 be a set of 2n + m + 1 points in general position with respect to degree m homogeneous polynomials. As in the last proof, there is a bijection between H2 m-k-facets of S and k-facets of HV 2 m(S). By Theorem 3.3.5, HV 2 m(S) is neighborly. Also, since S is in general position with respect to degree m homogeneous polynomials, HV 2 m(S) is in general linear position.
Since HV 2 m : R2 →Rm+1, the formula from Proposition 3.2.8 in the case d = m + 1 completes the proof.
□ 3.3.2. Lifting the moment curve. We say that a set system (X, F) has the exact count property if the number of F-k-sets for any set of n points in general position depends only on n and k and not on the configuration of points. We can generate many more set systems with the exact counting property by a lifting of the moment curve.
Let f : Rd →R be a function such that {(x, x′) ∈R2d : f(x) ̸= f(x′)} is open and dense and let g : Rd →R be any function. We will say that a set S = {si}i∈[n] of n points in Rd is in general position if Q i,j∈[n],i̸=j f(si) −f(sj) ̸= 0. Note that this is a reasonable definition of general position since if we are considering n-point sets, then point sets (in Rnd) in general position 24 are open and dense in Rnd. Assume that m ≥2 is even. The map ϕ : Rd →Rm+1 given by (3.1) ϕ(x) = f(x), f(x) 2, . . . , f(x) m, g(x) satisfies that, for any set S of n points in general position in Rd, ϕ(S) is neighborly.2 To see this, note that, since m is even, ⌊m+1 2 ⌋= m/2. The projection of ϕ(S) to the first m coordinates is m/2-neighborly since it is a set of n distinct points on the moment curve in Rm. We claim that this implies that ϕ(S) is m/2-neighborly as well: Let π(ϕ(S)) denote the projection to the first m coordinates. The points of ϕ(S) all project to distinct vertices of π(ϕ(S)). By neighborliness of π(ϕ(S)), every subset F of at most m/2 vertices of π(ϕ(S)) forms a face. For any such face, there is a supporting hyperplane H. The preimage π−1(H) of H under the projection π is a hyperplane with normal having last coordinate 0. Moreover, π−1(H) is the supporting hyperplane for a face of ϕ(S) formed by the lifted vertices π−1(F). This shows that ϕ(S) is neighborly.
Given any admissible choice of functions f, g, the map above induces a set system (Rd, Fϕ) with the exact count property.
3.3.3. Improved bounds for Pd m-k-facets and Hd m-k-facets. Results like Theorem 3.3.8 and Theorem 3.3.9 are not possible for any of the other polynomial set systems we have discussed.
However, some progress can be made.
Recall that in Proposition 3.2.6 we proved a non-trivial upper bound for the number of F-k-facets where (X, F) is any set system induced by a map. In this section we show how to improve this result for the set system (Rd, Pd m) for all values of m and d and for the set system (Rd, Hd m) for m even.
We show that the maps which induce (Rd, Pd m) and (Rd, Hd 2m), although not neighborly, still map into convex position with a high degree of neighborliness. Since these maps come up often in many fields, the following results may be useful in other contexts.
Theorem 3.3.10. For a finite set S of points in Rd, V d m(S) is the set of vertices of an ℓ-polytope where ℓ≤ m+d m −1. If V d m/2(S) is in general linear position and m ≥2 is even, then V d m(S) is a m/2+d m/2 −1 -neighborly ℓ-polytope.
2This is saying that ϕ is generally neighborly, using the terminology of Section 4.2. However, note that ϕ may not be an embedding of Rd. Furthermore, if d > 2, any map ϕ constructed as in Eq. (3.1) cannot be an embedding of Rd.
25 Proof. For v ∈S, choose coefficients ai, R so that {x ∈Rm : Pm i=1(xi −ai)2 ≤R} is a ball with v on its boundary and with radius small enough so that no points of S \ v are inside. By the Veronese map V d m, this ball corresponds to a hyperplane in R(m+d m )−1 containing v and with all other points of S on one side. This shows that V d m(v) is a vertex of conv(V d m(S)). For the second claim, let T ⊂S, |T| ≤ m/2+d m/2 −1. Let p(x) = 1 be a degree m/2 polynomial passing through each point of T and no points of S \ T. To show that such a polynomial exists, recall we are assuming that V d m/2(S) is in general position. Therefore, for any |T| points in V d m/2(S) there is a hyperplane passing through precisely those |T| points. This hyperplane corresponds to a degree m/2 polynomial passing through each point of T and no points of S \ T. Then p(x) −1 2 = 0 is a polynomial surface which corresponds to a hyperplane in R(m+d m )−1 which supports conv(T) as a face of conv(S).
□ Theorem 3.3.11. Assume m ≥2 is even. For a finite set S of points in Rd, HV d m(S) is the set of vertices of an ℓ-polytope where ℓ≤ m+d−1 m . If HV d m/2(S) is in general position, meaning no hyperplane through the origin in the image space of HV d m/2 contains more than m/2+d−1 m/2 −1 points of HV d m/2(S), then HV d m(S) is a m/2+d−1 m/2 −1 -neighborly ℓ-polytope.
Proof. For v ∈S, let H = {x ∈Rd : a · x = 0} be a plane through the origin which contains v and contains no other point of S. Then (a · x)m ≤0 is a degree m homogeneous polynomial inequality which, by the homogeneous Veronese map HV d m, corresponds to a hyperplane in R(m+d−1 m ) containing v and with all other points of S on one side. This shows that HV d m(v) is a vertex of conv(HV d m(S)). For the second claim, let T ⊂S, |T| ≤ m/2+d−1 m/2 −1 . Let p(x) = 0 be a degree m/2 homogeneous polynomial passing through each point of T and no points of S\T. To show that such a polynomial exists, recall we are assuming that HV d m/2(S) is in general position. Therefore, for any |T| points in HV d m/2(S) there is a hyperplane passing through the origin and precisely those |T| points. And this hyperplane corresponds to a degree m/2 homogeneous polynomial passing through each point of T and no points of S \ T. Then (p(x))2 = 0 is a degree m homogeneous polynomial surface which corresponds to a hyperplane in R(m+d−1 m ) which supports conv(T) as a face of conv(S).
□ 26 These k-neighborliness results are of interest to us because convex position is a special case of the k-set/facet problem for which we can improve the best known upper bound: Theorem 3.3.12. For a set S of n points in convex position in Rd, ek(S) ≤(n/d)ed−1(k, n −1).
Proof. Let v ∈S. Choose a hyperplane H containing v and with all other points of S on one side of it. Choose another hyperplane H′ parallel to H and with all points of S between H and H′.
Let S′ be the stereographic projection (using v as the “pole”) of S \ v onto H′. We claim that the number of k-facets of S containing v is equal to the number of k-facets of S′ (as a subset of H′, a (d −1)-dimensional subspace).
Assume that conv(v1, . . . , vd−1, v) is a k-facet of S. We claim that for each s ∈S, the stere-ographic projection s′ is on the positive side of aff(v′ 1, . . . , v′ d−1) if and only if s is on the positive side of aff(v1, . . . , vd−1, v). This is seen to be true by observing that aff(s, v) does not intersect aff(v1, . . . , vd−1, v) anywhere other than the point v. This shows that conv(v′ 1, . . . , v′ d−1) is a k-facet of S′. For the converse, assume that conv(v′ 1, . . . , v′ d−1) is a k-facet of S′. Then conv(v1, . . . , vd−1, v) is a k-facet of S for the same reason as above.
Since S′ lies in a hyperplane, it can have at most ed−1(k, n −1) k-facets.
Performing this projection on each point of S and noticing that every k-facet is counted d times shows the desired result.
□ As far as we know, the best known bound for k-facets of n-point sets in R4 in convex position is the same as for general point sets, which is O(n2k2−2/45) [MSSW06]. In R3 the best known bound for general point sets is O(nk2−1/2) [SST01a,SST01b]. This combined with Theorem 3.3.12 gives a bound of O(n2k2−1/2) for the number of k-facets of point sets in convex position in R4.
An argument similar to the proof of Theorem 3.3.12 shows the following generalization (we state it without proof): Proposition 3.3.13. For a set S of n points in m-neighborly position in Rd, ek(S) ≤ n m ed−m(k, n −m) d m .
In dimensions higher than four, the best known bound for k-facets is ed(k, n) = O(nd−ϵd) where ϵd = (4d −3)−d [ABFK92]. Because of the fast decay of the constant ϵd, Proposition 3.3.13 gives 27 an improvement in the best known upper bound which depends on the degree of neighborliness of the point set in question.
Proposition 3.3.13 can be used to improve the bounds for Pd m-k-facets and Hd 2m-k-facets as follows. Recall that the set system (Rd, Pd m) is induced by the map V d m and (Rd, Hd 2m) is induced by the map HV d 2m. Therefore Theorem 3.3.10 and Theorem 3.3.11 along with Proposition 3.3.13 give an improvement in the bound.
28 CHAPTER 4 Generally k-neighborly embedded manifolds and varieties To understand the limits of our argument that provides exact counting of k-facets we introduce a class of maps we call generally neighborly embeddings (Definition 4.2.2), which map generic point sets into neighborly position. The goal is to understand under what conditions the exact count phenomenon can occur for arbitrary set systems induced by a map. This goal leads us to ask new questions about convexity properties of embedded manifolds and algebraic varieties which are interesting beyond their connection to k-facets.
4.1. Introduction The crucial observation that allows us to exactly count F-k-facets for the conic sections and even degree homogeneous polynomials on the plane is that the maps which induce these set systems map generic point sets to neighborly point sets. In this chapter we define a generally neighborly embedding to be an embedding that maps generic point sets to neighborly point sets (Definition 4.2.2). The moment curve map is an example of a generally neighborly embedding of R1.
Moving up one dimension, the degree 2 Veronese map of the plane V 2 2 shows that a generally neighborly embedding of the plane exists (by Theorem 3.3.4). We conjecture (Conjecture 4.2.5) that generally neighborly embeddings of Rd do not exist for d > 2.
In order to provide support for this conjecture, in Section 4.4 we prove a closely related result about algebraic varieties.
More evidence that the conjecture may be true is provided in Section 4.7.
Apart from its relation to the F-k-facets, the problem of determining the existence of generally neighborly embeddings is interesting in its own right. Our definition of generally neighborly em-beddings is similar to and inspired by Micha Perles’ definition of neighborly embeddings which we discuss in Section 4.8.
Our contributions: 29 • Limits of the neighborliness argument. We study the limits of the neighborliness argument above that provides exact counting. We show that, for maps whose image is a variety, the argument only works for points on the plane. We proceed as follows: For the argument to work, one needs the map ϕ : Rd →Rp to map a generic set of points into a k-neighborly set of points for certain k. When ϕ is an embedding, we call the image M := ϕ(Rd) a generally k-neighborly d-manifold (Definition 4.3.1). We study the minimal dimension p so that M is a generally k-neighborly d-manifold and show that p ≤2k+d−1 (Theorem 4.2.4). For the same question with manifolds replaced by algebraic varieties, we show that p = 2k + d −1 (Theorem 4.4.3). This line of work relates to a problem of M. Perles on k-neighborly embeddings (see Section 4.8).
• Weakly neighborly point sets. We leverage weakly k-neighborly point sets (Defini-tion 4.5.1), a notion that is better behaved for our purposes than generally k-neighborly and Perles’ k-neighborly maps (Proposition 4.5.3).
In particular, we study weakly k-neighborly algebraic varieties, and resolve the question of the minimal p so that Rp con-tains a weakly k-neighborly d-dimensional algebraic variety. We show that the minimal dimension is 2k + d −1 (Theorem 4.6.1).
4.2. Generally neighborly embeddings Definition 4.2.1 (embedding). An embedding is a map which is a homeomorphism onto its image.
Definition 4.2.2 (generally k-neighborly embedding). Let ϕ : Rd →Rp be an embedding. For each n ∈N, let Gn ⊂Rdn consist of all configurations of n points in Rd which are mapped to k-neighborly sets by ϕ. Then ϕ is generally k-neighborly if Gn contains a set that is open and dense in Rdn for all n. A generally ⌊p 2⌋-neighborly embedding is called a generally neighborly embedding.
We choose “open and dense” in Definition 4.2.2 for concreteness and readability. For part of our discussion (in particular, Problem 4.2.3 below), it may be reasonable to substitute it by an alternative version of a generic property as discussed in Section 3.2.1.
Observe that the even degree homogeneous Veronese map of the plane, i.e. HV 2 m, is not an embedding because it is not injective. Since the homogeneous Veronese map is one of our prime examples throughout we need to justify why we are now only talking about embeddings.
The 30 reason is that all of the polynomial maps we have considered have the property that they are an embedding of some open subset of Euclidean space. Thus, for the purposes of this section it suffices to assume that our maps are embeddings.
The main question concerning generally k-neighborly embeddings is: Problem 4.2.3. What is the smallest dimension p := pg(k, d) of the image space for which a generally k-neighborly embedding ϕ : Rd →Rp exists?
Theorem 4.2.4. There exists a generally k-neighborly embedding of Rd into R2k+d−1 and so pg(k, n) ≤2k + d −1.
Proof. Consider the embedding ϕ : Rd →R2k+d−1 defined by ϕ(x1, x2, . . . , xd) = (x1, x2 1, x3 1, . . . , x2k 1 , x2, . . . , xd).
For each n ∈N, let Gn ⊂Rdn consist of all configurations of n points in Rd such that no two points in the configuration have the same x1-coordinate. One can verify that Gn is open and dense in Rdn. For any n, let S ∈Gn be some configuration of n points in Gn.
To show that ϕ(S) is k-neighborly, let v1, . . . , vk be k points from S. Consider in the domain of the embedding, Rd, the surface (4.1) k Y i=1 (x1 −vi1)2 = 0.
By expanding Eq. (4.1) we see that this surface corresponds via ϕ to a hyperplane H in R2k+d−1.
Note that v1, . . . , vk satisfy Eq. (4.1) and all other points in S satisfy Qk i=1(x1 −vi1)2 > 0. Using ϕ, we get that H is a face-defining hyperplane that makes ϕ(v1), . . . , ϕ(vk) a face of conv(ϕ(S)).
Therefore, ϕ(S) is k-neighborly. This shows that pg(k, d) ≤2k + d −1.
□ We believe that the bound in the above theorem is actually tight.
Conjecture 4.2.5. pg(k, d) = 2k + d −1.
Observe that for d ≥3, if ϕ : Rd →Rp is a generally k-neighborly embedding then, according to Conjecture 4.2.5, p ≥2k + 2.
This means the conjecture implies that generally neighborly embeddings of Rd do not exist for d ≥3.
31 In the context of the k-set problem, this would mean that set systems like the conic sections do not exist in dimension d ≥3. More precisely, it would imply that, for d ≥3, there is no set system (Rd, F) induced by an embedding (of Rd) for which F-k-facets can be counted by using Proposition 3.2.8.
4.3. Generally neighborly manifolds Here we define generally k-neighborly manifolds which are, for our purposes, equivalent to generally k-neighborly embeddings. The sense in which they are equivalent is made precise below.
Definition 4.3.1. A manifold M ⊂Rp is generally k-neighborly if the set Gn ⊂Mn of configura-tions of n points on M which are k-neighborly contains a set that is open and dense in Mn for all n. A generally ⌊p 2⌋-neighborly manifold is called a generally neighborly manifold.
We ask the same question for manifolds as we did for embeddings: Problem 4.3.2. What is the smallest dimension p of the ambient space in which a generally k-neighborly d-manifold M ⊂Rp exists?
Observe that an open subset of a generally k-neighborly d-manifold is still generally k-neighborly.
Therefore, in the context of Problem 4.3.2 it suffices to assume that the manifold M is (globally) homeomorphic to Rd, that is M = ϕ(Rd) for some embedding ϕ. This observation, along with the following proposition, shows that Problem 4.3.2 is equivalent to Problem 4.2.3.
Proposition 4.3.3. An embedding ϕ : Rd →Rp is generally k-neighborly if and only if M := ϕ(Rd) is a generally k-neighborly d-manifold.
Proof. If ϕ : Rd →Rp is generally k-neighborly, then for each n the set of configurations of n points which are mapped by ϕ to k-neighborly point sets contains a set O that is open and dense in Rdn. Let N ⊂Mn be the set of k-neighborly configurations of n-points in Mn. The set N contains (ϕ×· · ·×ϕ)(O) which is open and dense since ϕ×· · ·×ϕ is a homeomorphism. Therefore M := ϕ(Rd) is a generally k-neighborly d-manifold. The other direction is similar.
□ 32 4.4. Generally k-neighborly algebraic varieties Definition 4.4.1. Let V ⊂Rp be an irreducible real algebraic variety with a smooth real point.
For each n, let Gn ⊂V n sm consist of all configurations of n points on Vsm which are k-neighborly.
Then V is generally k-neighborly if Gn contains a set that is open and dense in V n sm for all n. It is generally neighborly if it is generally ⌊p/2⌋-neighborly.
We make one clarifying remark regarding the above definition. One could replace Vsm every-where in the above definition with V . However, only requiring the property to hold for the smooth points strengthens our results below and does not change the proofs.
Another reason for only considering the smooth points is the following. Loosely speaking, a generally k-neighborly alge-braic variety V is supposed to be a variety such that every generic configuration of points on V is k-neighborly. A generic configuration of points should never contain non-smooth points, so the non-smooth points should be ignored when defining generally k-neighborly algebraic varieties.
The question we are dealing with in this section is the following.
Problem 4.4.2. What is the smallest dimension p := pg,V (k, d) of the ambient space in which a generally k-neighborly d-dimensional algebraic variety V ⊂Rp exists?
Observe that the image of the map ϕ in Theorem 4.2.4 is a d-dimensional generally k-neighborly variety in R2k+d−1. This shows that pg,V (k, d) ≤2k + d −1.
We will prove the following result which completely resolves Problem 4.4.2.
Theorem 4.4.3. Let V ⊂Rp be a generally k-neighborly d-dimensional algebraic variety1. Then p ≥2k + d −1.
Theorem 4.4.3 combined with Theorem 4.2.4 show that pg,V (k, d) = 2k + d −1. In order to prove Theorem 4.4.3 we will first establish a connection between generally k-neighborly varieties and weakly k-neighborly sets. Weak neighborliness is a more usable property that holds for all subsets of points, not just those satisfying a general position assumption. This connection is established in Section 4.5. The proof of Theorem 4.4.3 is then completed in Section 4.6.
Below we list some more examples of generally k-neighborly algebraic varieties.
1Note that “generally k-neighborly” requires V to be irreducible and contain a smooth real point.
33 Example 4.4.4. The image of the degree 2 Veronese map of the plane is a generally 2-neighborly 2-dimensional algebraic variety in R5.
Example 4.4.5. The image of the map ϕ from the proof of Theorem 4.2.4 is a generally k-neighborly d-dimensional algebraic variety in R2k+d−1.
Example 4.4.6. The moment curve is a generally neighborly 1-dimensional algebraic variety. The same is true of any order d curve which is also an algebraic variety.
Example 4.4.7. Theorem 4.4.3 shows that generally neighborly d-dimensional algebraic varieties do not exist for d ≥3.
4.5. Weakly k-neighborly sets It turns out that all generally k-neighborly algebraic varieties satisfy a weaker neighborliness property that holds for all subsets of points (not just those satisfying a general position assump-tion). We call this property weakly k-neighborly (Definition 4.5.1). In the proof of Theorem 4.4.3, we only need to use the fact that Vsm is weakly k-neighborly.
In this section we prove some lemmas concerning weakly k-neighborly sets and the relationship between generally k-neighborly manifolds/varieties and weakly k-neighborly sets.
Definition 4.5.1. A set S ⊆Rp is weakly k-neighborly if for any set T of k points from S, there exists a closed halfspace H with boundary bd(H) such that S ⊂H and T ⊂bd(H).
We will now show that a generally k-neighborly algebraic variety or manifold is weakly k-neighborly. In order to do so, we first show that every finite subset of such a variety or manifold is weakly k-neighborly. We actually state and prove a stronger result (Lemma 4.5.2) that only uses as assumption the “dense” part of “open and dense” in the definition of generally k-neighborly.
We then establish a compactness property of weakly k-neighborly sets. The property is that an arbitrary subset of Rp is weakly k-neighborly if and only if every finite subset is weakly k-neighborly.
These two results together establish that generally k-neighborly algebraic varieties and manifolds are weakly k-neighborly.
34 Lemma 4.5.2. Let M ⊂Rp be a manifold or the set of smooth points of an algebraic variety. If the set N ⊂Mn of configurations of n points on M which are k-neighborly is dense in Mn for all n, then every finite set of points on M is weakly k-neighborly.
Proof. Assume not, so that there exists some finite set S ⊂M and a set T of k points from S such that no closed halfspace contains S and contains T on its boundary.
This means that aff(T) ∩relint conv(S \ T) ̸= ∅(from the separating hyperplane theorem [Sch14, Theorem 1.3.8]).
We can pick |S| small open balls in Rp as follows. For each t ∈T, let Bt be a ball centered at t and for each s ∈S \ T, let As be a ball centered at s. The radii of the balls can be chosen small enough so that any collection of points consisting of one point from each Bt and one point from each As has the property that the affine hull of the points from the Bt intersects the relative interior of the convex hull of the points from the As. This means that any such configuration of points is not k-neighborly. Therefore, the subset of Rp|S| of configurations of points of size |S| on M which are not k-neighborly contains Q t∈T (Bt ∩M) × Q s∈S\T (As ∩M) which is an open subset of M|S|. Therefore, the set of configurations of |S| points on M which are k-neighborly is not dense in M|S|.
□ Proposition 4.5.3 (compactness). Let S ⊆Rp be a (possibly infinite) set. Then for any k ≥1 we have that S is weakly k-neighborly if and only if every finite subset of S is weakly k-neighborly.
Proof. Fix k ≥1. The “only if” direction is clear. We will now prove the “if” direction. Let T ⊆S be a set of k points. Let U = {U ⊆S : U ⊇T and U is finite}. For U ⊆S such that U ⊇T, we will define N(U) ⊆Sp−1 to be the set of unit outer normals to possible halfspaces H such that T ⊆bd(H) and U ⊆H. More precisely, let N(U) = {a ∈Sp−1 : (∀x, y ∈T)a · x = a · y and (∀x ∈T)(∀y ∈U)a · x ≥a · y}.
Clearly N(U) is closed. Let V ⊆U be any finite subfamily. Then ∩U∈VN(U) = N(∪U∈VU) ̸= ∅ by assumption. We have established that {N(U)}U∈U is a family of closed sets with the finite intersection property in compact space Sp−1. This implies ∩U∈UN(U) ̸= ∅. We also have N(S) = N(∪U∈UU) = ∩U∈UN(U) ̸= ∅. That is, there is a halfspace H such that T ⊆bd(H) and S ⊆H.
As T ⊆S was arbitrary, this completes the proof.
□ 35 We require two more lemmas concerning intersections and weak separation of convex sets in Rp. (See Section 2.1 for the definitions of weak and proper separation of sets in Rd.) Lemma 4.5.4 (Radon-type theorem). Let P be a set of p+2 points in Rp in general linear position.
Then there is a partition Q, R of P into two non-empty sets so that relint conv Q∩relint conv R ̸= ∅.
Proof. Let P = {q1, . . . , qp+2}. P is affinely dependent and therefore there exist λ1, . . . , λp+2 such that Pp+2 i=1 λiqi = 0, Pp+2 i=1 λi = 0 and at least one λi is non-zero. Because of the general position assumption, all λi are non-zero.
Let I = {i : λi > 0}, J = {i : λi < 0}.
Both I and J are non-empty. By dividing λis by P i∈I λi we can assume without loss of generality that P i∈I λi = −P i∈J λi = 1. Let Q = {qi : i ∈I}, R = {qi : i ∈J}. Let q = P i∈I λiqi ∈relint conv Q, r = −P i∈J λiqi ∈relint conv R. We have q = r, which completes the proof.
□ Lemma 4.5.5. Let Q, R ⊆Rp be disjoint sets. Suppose aff(Q ∪R) = Rp and relint conv Q ∩ relint conv R ̸= ∅. Then Q, R cannot be weakly separated.
Proof. Assume Q, R can be weakly separated. If the separation is not proper then aff(Q∪R) ̸= Rp. If the separation is proper, then by the separating hyperplane theorem [Sch14, Theorem 1.3.8], relint conv Q ∩relint conv R = ∅.
□ 4.6. Weakly k-neighborly varieties In the previous section we established the connection between generally k-neighborly vari-eties/manifolds and weakly k-neighborly sets.
In this section, we use this connection to prove Theorem 4.4.3. Given an (real) algebraic variety V of dimension d such that Vsm is weakly k-neighborly, we prove a sharp lower bound on the dimension of the ambient space.
Theorem 4.6.1. Assume V ⊂Rp is a non-degenerate d-dimensional irreducible real algebraic variety with a smooth real point. If V \ U is weakly k-neighborly for some proper closed subvariety U, then p ≥2k + d −1.
Before proving Theorem 4.6.1 we prove the special case of algebraic curves and then generalize to higher dimensional varieties.
36 Lemma 4.6.2. Assume C ⊂Rp is a non-degenerate irreducible real algebraic curve with a smooth real point. If C \ U is weakly k-neighborly for some proper closed subvariety U, then p ≥2k.
Proof. First we will show that one can find arbitrarily large point sets in general linear position on C \ U. In order to accomplish this, first observe that C \ U is non-degenerate. Indeed, if it were the case that C \ U is contained in a hyperplane H, then we would have C = (C ∩H) ∪U which is impossible since C is irreducible.
Now assume that S is a set of j points in general linear position on C \U. If it were not possible to find another point s such that S ∪{s} is in general linear position, it would have to be the case that C \ U is contained in the union of all hyperplanes spanned by p points from the set S. Let H be the collection of all hyperplanes spanned by p points in S. Note that H is finite. We have C = [ H∈H (C \ U) ∩H ∪U.
Since C \ U is non-degenerate the above formula would be a representation of C as the union of proper subvarieties. This is impossible since C is irreducible. Therefore we know that C \ U is not contained in the union of all hyperplanes spanned by points in S and so we can always find s so that S ∪{s} is in general linear position. It follows that we can find arbitrarily large point sets in general linear position on C \ U.
Let P be a set of p + 2 points in general linear position on C \ U. By Lemmas 4.5.4 and 4.5.5, there is a partition of P into non-empty sets Q and R so that Q and R cannot be weakly separated.
However, because C \U is weakly k-neighborly, we know that for any set T of k points on C \U there exists a closed halfspace which contains C \U and contains T in its boundary. In other words, any such T can be weakly separated from C \ U. Therefore, it must be that k < min(|Q|, |R|), i.e. k ≤min(|Q|, |R|) −1. Now since min(|Q|, |R|) ≤⌊p+2 2 ⌋, we have that k ≤⌊p+2 2 ⌋−1 and so p ≥2k.
□ The idea of the proof for higher dimensional varieties is to take successive hyperplane sections in order to reduce to the case of curves. So we first need to establish the following lemma concerning hyperplane sections of varieties.
37 Lemma 4.6.3. Assume that V ⊂Rp is a non-degenerate d-dimensional (d ≥2) irreducible variety with a smooth real point and that U is some proper closed subvariety of V . Then for any given open ball B in conv(V \ U) there exists a hyperplane H such that H ∩B ̸= ∅and V ∩H is a non-degenerate, irreducible, (d −1)-dimensional variety with a smooth real point which is not contained in U.
Proof. We identify the set of hyperplanes in Rp with a proper subset of Pp(R). This identifi-cation works as follows. We identify a hyperplane a0 + a1x1 + · · · + apxp = 0 in Rp with the point with homogeneous coordinates (a0, a1, . . . , ap) in Pp(R). Therefore, the set of hyperplanes in Rp is identified with Pp(R) \ {(a0, a1, a2, . . . , ap) : a1 = a2 = · · · = ap = 0}.
Recall from Section 2.5 that around any smooth real point of V there is a neighborhood which is a smooth d-dimensional submanifold of Rp. Let T ⊂Pp(R) be the set of hyperplanes H for which there exists a smooth point of V \U and a neighborhood of that smooth point which has nonempty transversal intersection with H. We know that T is open and that for any H ∈T, H ∩V contains an open subset which is a smooth (d −1)-dimensional submanifold of Rp [GP10, Sections 1.5 and 1.6]. Let H be any hyperplane in T. Clearly the dimension of H ∩V is at most d−1. We claim that H ∩V is a variety of dimension precisely d−1 and that it contains a smooth real point. Recall from Section 2.5 that if a variety has real dimension d then it has dimension at least d. Therefore, the dimension of H ∩V is d−1. Now we show that H ∩V contains a smooth real point. Indeed assume not, so that H ∩V (C) contains no smooth real points.2 This means that H ∩V is contained in the set of singular points of the (d−1)-dimensional variety H ∩V (C). By [BCR98, Proposition 3.3.14], the set of singular points of H ∩V (C) is a variety of dimension at most d −2. Since H ∩V has real dimension d −1 and is contained in a variety of dimension at most d −2 this is a contradiction. So H ∩V contains smooth real points.
Let I be the subset of T consisting of hyperplanes H such that H ∩V is irreducible and non-degenerate. We will show that I is open and dense in the standard topology in T. Let V (C) ⊂Pp(C) be the projective closure of V (C). This means that V (C) = V (C) ∩{x0 ̸= 0}. Because V contains a smooth real point, V is Zariski dense in V (C), that is, V (C) is the smallest complex variety 2By a minor abuse of notation, H ∩V (C) is the complex variety defined by the polynomials defining V along with the polynomial defining H. Similarly, H ∩V is the real variety defined by the polynomials defining V along with the polynomial defining H. So H ∩V is the real part of H ∩V (C).
38 containing V , see [BCR98, Section 2.8]. Therefore, since V is irreducible, by [Whi57, Lemma 7], V (C) is irreducible. Because V (C) is irreducible, it is then a standard fact that V (C) is irreducible.
By projective duality, we can identify the set of all projective hyperplanes with real coefficients in Pp(C) with Pp(R). Let I′ ⊂Pp(R) consist of all projective hyperplanes with real coefficients that have irreducible and non-degenerate intersection with V (C). By [Har95, Theorem 18.10], I′ is Zariski open and dense.3 We will now show that the fact that I′ is Zariski open and dense implies that I is also Zariski open and dense in T. Given a hyperplane H defined by a0 + a1x1 + · · · + apxp = 0 in Rp, there is a corresponding projective hyperplane H defined by a0x0 + a1x1 + · · · + apxp = 0 which is the homogenization of H. Let H be a hyperplane in T and assume that the homogenization H is in I′. We claim that this implies that H ∈I. To establish this claim, we need to verify that H ∩V is irreducible and nondegenerate. Observe that H ∩V (C) = H ∩V (C) ∩{x0 ̸= 0} i.e., H ∩V (C) is an open subset of H ∩V (C). It is a standard fact that a nonempty open subset of an irreducible space is irreducible and dense. Therefore H ∩V (C) is irreducible. Now, since H ∩V (C) is Zariski dense in H ∩V (C), if H ∩V (C) were contained in a hyperplane, H ∩V (C) would be as well. So H ∩V (C) is nondegenerate. We have shown that H ∩V (C) is non-degenerate and irreducible. We claim that because the section H ∩V (C) contains smooth real points, then the non-degeneracy and irreducibility of H ∩V (C) implies non-degeneracy and irreducibility of H ∩V . This relies on the fact mentioned above that if an irreducible variety W contains a smooth real point, then the set of real points is Zariski dense in W(C). This means that H ∩V is Zariski dense in H ∩V (C).
Now by [Whi57, Lemma 7], irreducibility of H ∩V (C) implies that H ∩V is irreducible. Finally, observe that because H ∩V is Zariski dense in H ∩V (C), if H ∩V is degenerate, H ∩V (C) must be as well. Thus H ∩V is non-degenerate.
Recall that T is open (in the standard metric topology) and that I′ is Zariski open and dense in Pp(R) which implies that I′ is open and dense in the standard topology. We established that I contains I′ ∩T. Therefore, we have shown that I is open and dense in the standard topology in T.
3Theorem 18.10 in [Har95] says that the set of hyperplanes which intersect V (C) in a non-degenerate irreducible variety is the complement of a proper subvariety of Pp(C). The intersection of a proper subvariety of Pp(C) with Pp(R) is a proper subvariety of Pp(R). So I′ is Zariski open dense in Pp(R).
39 Let O ⊂Pp(R) be the set of hyperplanes which intersect B. The set O is open (in the standard topology).
We claim that all of this means that I ∩T ∩O is non-empty. To establish this, we need to verify that T ∩O is non-empty. To find H ∈T ∩O, let v be a smooth point of V \ U and let F be a (p −2)-flat having non-empty transversal intersection with a neighborhood of v. Such a flat exists because V has real dimension at least 2. For any point p in B, the hyperplane aff(F ∪p) is in T ∩O.
Therefore, T ∩O is open and non-empty.
Since I is open and dense in T, it follows that I ∩T ∩O ̸= ∅. We claim that any H ∈I ∩T ∩O completes the proof. To show this, it remains to show that for any H ∈I ∩T ∩O, H ∩V has a smooth real point which is not in U. We already know that H ∩V has smooth real points and that H ∩V is not contained in U. So assume for a contradiction that all the smooth real points are contained in U. Then letting S denote the singular points of H ∩V , we have that H ∩V = (H ∩V ∩U) ∪S is the decomposition of H ∩V as the union of two proper closed subvarieties, a contradiction to irreducibility of H ∩V .
□ The lemma established above allows us to generalize Lemma 4.6.2 to higher dimensional vari-eties.
Proof of Theorem 4.6.1. First we show that by making repeated applications of Lemma 4.6.3, we can inductively construct a (p−d+1)-flat L that intersects int conv(V \U) and such that L∩V is a non-degenerate irreducible algebraic curve with a smooth real point which is not contained in U. Say we have some flat F that intersects int conv(V \U) and such that F ∩V is a non-degenerate irreducible d′-dimensional (d′ ≥2) variety with a smooth real point not contained in U. Since F ∩V is not contained in U, F ∩U is a proper subvariety of F ∩V . Let B be an open ball in F that is contained in int conv(V \ U). By Lemma 4.6.3, there exists a hyperplane H in F that intersects B and such that H ∩V is a non-degenerate irreducible (d′ −1)-dimensional variety with a smooth real point not contained in F ∩U and hence not contained in U. Since B is contained in int conv(V \U), H intersects int conv(V \ U). We can repeat this process until we obtain a (p −d + 1)-flat L that intersects int conv(V \ U) and such that C := L ∩V is a non-degenerate irreducible algebraic curve with a smooth real point not contained in U.
40 Now we will show that C \ U is weakly k-neighborly in L, meaning4 that C \ U is weakly k-neighborly as a subset of its affine hull L.
Because V \U is weakly k-neighborly, we know that for any set T of k points on C\U there exists a closed halfspace H (in Rp) which contains C\U and contains T in bd(H). In other words, any such T can be weakly separated from C\U. Notice that we are talking about weak separation in Rp, while we are really interested in weak separation in L. We claim that our assumption that L intersects the interior of the convex hull of V \ U allows us to pass from weakly separating hyperplanes in Rp to weakly separating hyperplanes in L. Indeed, the fact that L intersects int conv(V \ U) means that any closed halfspace H satisfying V \ U ⊂H cannot contain L in its boundary. Therefore bd(H) ∩L is a proper hyperplane in L. To summarize, given a set T of k points on C \ U, by the assumption that any finite set on V \ U is weakly k-neighborly, there exists a hyperplane bd(H) that weakly separates T from V \ U. This and the way we chose L allows us to conclude that bd(H) ∩L is a hyperplane in L that weakly separates T from C \ U. Therefore, C \ U is weakly k-neighborly in L. Since C is not contained in U, C ∩U is a proper subvariety. By Lemma 4.6.2, p −d + 1 ≥2k.
□ We can now prove the lower bound for generally k-neighborly varieties.
Proof of Theorem 4.4.3. We can without loss of generality assume that V is non-degenerate since otherwise we could consider V in aff(V ). By Lemma 4.5.2, every finite set of points on Vsm is weakly k-neighborly. Therefore by Proposition 4.5.3, Vsm is weakly k-neighborly. Since Vsing is a proper closed subvariety (see [BCR98, Proposition 3.3.14]), by Theorem 4.6.1, p ≥2k + d −1.
□ 4.7. Additional evidence In this section, we give some additional comments on the validity of Conjecture 4.2.5. Although we could not resolve the conjecture, the result on algebraic varieties is evidence that it is likely true. In the following we provide more evidence for the conjecture by showing that any manifold violating the conjecture would have to have a property that appears fairly restrictive to us.
4Note that a subset of Rp whose affine hull is a proper subset of Rp is automatically weakly k-neighborly according to Definition 4.5.1, so the requirement here is that halfspace H in that definition is a halfspace of L.
41 Proposition 4.7.1. If a set M ⊂Rp is weakly k-neighborly then for any set S of 2k points in general linear position in M, aff(S) ∩M is contained in the union of all hyperplanes supporting facets of the simplex conv S.
Proof. Assume not, so that S is a set of 2k points in M such that aff(S)∩M is not contained in the union of all hyperplanes supporting facets of conv(S). Then identifying aff(S) with R2k−1, we can find a set S′ of 2k + 1 points in general position in R2k−1 which are weakly k-neighborly.
However, by Lemmas 4.5.4 and 4.5.5, there is a partition Q, R of S′ into two non-empty sets so that Q, R cannot be weakly separated. Since min(Q, R) ≤⌊2k+1 2 ⌋= k, this is a contradiction to weakly k-neighborliness of S′.
□ Proposition 4.7.1 is inspired by the following illustrative example: the possibility of a non-degenerate 2-neighborly curve in R3. By non-degeneracy, pick 4 affinely independent points on the curve. Then by Proposition 4.7.1 the curve would have to be contained in the union of the 4 hyperplanes defined by any 3 of those points.
Assume M ⊂Rp is a weakly k-neighborly embedded d-manifold and S any set of 2k points in general position on M. Notice that if p < 2k + d −1, we would expect aff(S) to intersect M in a manifold of dimension 1 or greater. However, the previous Proposition implies that M ∩affS is contained in a finite number of hyperplanes which is not true of most embedded 1-manifolds.
One approach to proving Conjecture 4.2.5 appears to be showing that, in fact, there is no non-degenerate d-manifold in R2k+d−2 satisfying the conclusion of Proposition 4.7.1.
4.8. Neighborly embeddings Our definition of generally k-neighborly embeddings is similar to the concept of k-neighborly embeddings introduced by Perles in 1982 and studied by Kalai and Wigderson [KW08].
An embedding of a d-dimensional manifold M into Rp is k-neighborly if for every k points on the embedding of M there is a hyperplane H that contains the k points and such that all remaining points of the embedded manifold are on the (strictly) positive side of H.
Requiring that an embedding be k-neighborly is clearly stronger than requiring that it be generally k-neighborly. That is, if an embedding is k-neighborly then it is generally k-neighborly.
However, the reverse implication is certainly not true.
For example, the degree two Veronese 42 embedding V 2 2 (x, y) = (x, y, x2, xy, y2) is only a 1-neighborly embedding while it is a generally 2-neighborly embedding.
In 1982 Perles posed the following problem concerning neighborly embeddings.
Problem 4.8.1. What is the smallest dimension p(k, d) of the ambient space in which a k-neighborly d-dimensional manifold exists?
As in the case of generally neighborly embeddings, for the purposes of this question, it suffices to assume that M = Rd. Kalai and Wigderson proved Theorem 4.8.2 ( [KW08]). k(d + 1) ≤p(k, d) ≤2k(k −1)d.
Improving the bounds in Theorem 4.8.2 appears to be difficult compared to the case of generally k-neighborly embeddings where we were able to conjecture a precise formula for pg(k, d).
Comparing the two definitions, k-neighborly embeddings appear to be the more natural and fundamental class of embeddings to investigate. However, there may be some applications for which the notion of generally k-neighborly embeddings is more appropriate. For example, the authors of [KW08] were interested in neighborly embeddings in part because they may lead us to important examples of k-neighborly polytopes. In particular, by picking points on the embedded manifold, one may produce k-neighborly nonsimplicial polytopes perhaps with other interesting properties.
Since both types of embeddings produce k-neighborly polytopes, our version may be more useful in this context as it is less restrictive.
43 CHAPTER 5 Improved bounds for the expected number of k-sets For a probability distribution P on Rd, we study EP (k, n), the expected number of k-facets of a sample of n random points from P. When P is a distribution on R2 such that the measure of every line is 0, we show that EP (k, n) = O(n5/4) when k = ⌊cn⌋for any fixed c ∈(0, 1). Our argument is based on a technique by B´ ar´ any and Steiger.
We study how it may be possible to improve this bound using the continuous version of the polynomial partitioning theorem. This motivates a question concerning the points of intersection of an algebraic curve and the k-edge graph of a set of points.
5.1. Expected number of k-edges Recall that e2(k, n) denotes the maximum number of k-edges of any set of n points in the plane.
It is widely believed that the true value of e2(k, n) is closer to the best known lower bound than to the best known upper bound. Indeed, Erd˝ os et al. conjectured in [ELSS73] that e2(k, n) = O(n1+ϵ) for any ϵ > 0. Some support for this conjecture is provided by the results one can obtain for the probabilistic version of the k-facet problem.
B´ ar´ any and Steiger initiated the study of the probabilistic version of the k-facet problem [BS94].
The problem was also studied in [Cla04]. Given a probability distribution P on Rd, what is the expected number EP (k, n) of k-facets of X, a sample of n independent random points from P?
Recall that the k-facet problem is only defined for point sets in general position. For this reason, in all of our results concerning EP (k, n), we restrict our attention to distributions P such that the measure of every hyperplane is 0. This is the minimal assumption on distributions P which guarantees that a sample of points from P is in general position with probability 1.
We refer to the original k-facet problem as the deterministic version. By the probabilistic version of the k-facet problem, we mean the question of the value of EP (k, n).
44 Part of the reason for the interest in the probabilistic version of the k-facet problem is that, as noted in [Wag08, Section 4.2], there is some reason to believe that the probabilistic ver-sion may be more or less the same as the original: Circa publication of B´ ar´ any and Steiger’s paper [BS94], the best known lower bound for the deterministic k-facet problem in R2 was ed( n−2 2 , n) = Ω(n log n) [ELSS73]. Using the construction in [ELSS73], B´ ar´ any and Steiger con-struct a probability distribution P with EP ( n−2 2 , n) = Ω(n log n). The Ω(n log n) lower bound for the deterministic k-facet problem was improved to ed( n−2 2 , n) = neΩ(√log n) in [T´ ot01]. As noted in [Wag08], it is possible to use the construction in [T´ ot01] to construct a distribution P ′ with EP ′( n−2 2 , n) = neΩ(√log n). See [Wag08] for some more details.
Upper bounds for the probabilistic k-facet problem which improve the upper bounds for the deterministic version have only been established in special cases. B´ ar´ any and Steiger obtained tight upper bounds for several families of distributions using an integral formula for EP (k, n). We also use the formula in the proofs of our main results so we describe it here. Let X1, . . . , Xd be d random points drawn from P. The assumption that the measure of every hyperplane is zero means that aff(X1, . . . , Xd) is a hyperplane with probability 1. We use aff(X1, . . . , Xd)+ to denote the open half space on the positive side of aff(X1, . . . , Xd) and aff(X1, . . . , Xd)−to denote the open half space on the negative side of aff(X1, . . . , Xd). Define G(t) = P P(aff(X1, . . . , Xd)+) ≤t .
Then given a sample X of n random points from P, the expected number of k-facets of X is EP (k, n) = X F∈(X d) P aff(F)+ or aff(F)−contains exactly k points of X = 2 n d n −d k Z 1 0 tk(1 −t)n−d−kdG(t).
(5.1) B´ ar´ any and Steiger used Eq. (5.1) to show that EP (k, n) = O(nd−1) if P is spherically symmetric.
Also, if P is the uniform distribution on a convex body in R2, they show that EP (k, n) = O(n).
Finally, we remark that Eq. (5.1) can be used to immediately obtain an upper bound for the expected number of n−d 2 -facets of a sample of n points from any1 probability distribution on Rd.
1Recall we assume that hyperplanes have measure 0.
45 The bound is much weaker than the bounds obtained by B´ ar´ any and Steiger for special cases of the distribution, but it is still non-trivial.
Theorem 5.1.1. If P is a Borel probability distribution on Rd such that the measure of any hyperplane is zero, then EP ( n−d 2 , n) = O(nd−1/2).
Proof. From Eq. (5.1) and Stirling’s approximation, EP n −d 2 , n = 2 n d n −d n−d 2 Z 1 0 tk(1 −t)n−d−kdG(t) ≤2 n d n −d n−d 2 1 2n−d = O(nd−1/2).
(5.2) □ In Section 5.2 we use the weak bound in the proof of Theorem 5.1.1 combined with a partition of the plane to show an improved bound for the expected number of k-edges: Theorem 5.1.2. Let P be a Borel probability distribution on R2 such that the measure of every line is zero. Then for any 1 ≤k ≤n−1, EP (k, n) = O n7/4 k1/4(n−k)1/4 (where the constants in big-O are universal).
The proof of Theorem 5.1.2 is in Section 5.2.2. In Section 5.2.1, we review the definition of the k-edge graph (Definition 5.2.1) and state some results needed for the proof of Theorem 5.1.2. In particular, we explain how the k-edge graph can be decomposed into convex chains. The proof of Theorem 5.1.2 also uses a divide and conquer approach where the plane is partitioned into cells by vertical lines.
Section 5.3 outlines how it may be possible to improve the bound in Theorem 5.1.2 by par-titioning the plane with algebraic curves rather than vertical lines.
The existence of algebraic curves which partition the plane in a useful way is a consequence of the continuous version of the polynomial partitioning theorem of [Gut16b] (Theorem 2.3.4). Whether or not using algebraic curves rather than lines leads to an improvement in the bound depends on a question we leave open (Question 5.3.1) which asks for a bound on the maximum number of times that an algebraic 46 curve of degree r can intersect the k-edge graph of a set of n points. We show that this quantity is O(nr2) but, as far as the authors know, it may be possible to improve this bound, see Section 7.4.
5.2. Bounding the expected number of k-edges In this section we review some necessary facts about k-edges and prove Theorem 5.1.2.
5.2.1. Convex/concave chains. Here we recall the “convex chains” technique of [AACS98] which was used in [Dey97] to establish the O(n4/3) bound for planar k-edges. First we need to define the k-edge graph.
Let S be a set of n points in general position in the plane and choose some (x, y) coordinate system. With this choice, we assume without loss of generality that no line spanned by two points in S is vertical. Let Ek be the set of line segments connecting two points x, y ∈S such that there are exactly k points from S in the halfplane below aff(x, y). Therefore, Ek is a subset of the set of all k-edges of S. Throughout, we assume without loss of generality that Ek contains at least half the total number of k-edges. Indeed, we could repeat the analysis after rotating the plane 180 degrees. The line segments Ek define the k-edge graph: Definition 5.2.1. Let S be a set of n points in general position in the plane and assume that no line spanned by two points in S is vertical. For any 0 ≤k ≤n −2, the (geometric) graph Gk = (S, Ek) is called the k-edge graph of S.
The convex chains technique decomposes the k-edge graph Gk of a point set S into the union of a bounded number of convex chains. Each convex chain is the graph of a convex piece-wise linear function defined on some interval of the x-axis. Each chain is formed by some subset of the k-edges in Gk = (S, Ek). A simpler version of the proof of the O(n4/3) bound was established in [HP11] by observing that the k-edge graph can simultaneously be decomposed into the union of concave chains. We use the known fact [Lov71] that the convex/concave chain decompositions imply that a vertical line can intersect the k-edge graph at most min(k −1, n −k −1) times.
Lemma 5.2.2 (Convex/concave chains [HP11, Lemma 9.10] ). Let Gk = (S, Ek) be the k-edge graph of a set S of n points in the plane. The graph Gk can be decomposed into the union of k −1 47 (piece-wise linear) convex chains. Similarly, the graph can be decomposed into the union of n−k+1 (piece-wise linear) concave chains.
These decompositions can be used to show that the total number of crossings in Gk is O(n2), see [Dey97] or [HP11].
5.2.2. Proof of Theorem 5.1.2. We are now ready to prove our bound on the expected number of k-edges. The idea of the proof is to use vertical lines to divide the plane into a number of regions of equal probability. We then bound separately the expected number of k-edges that intersect one of the vertical lines and the expected number of k-edges that do not intersect any of the lines. We remark that a partition of the plane using vertical lines was also used by Lov´ asz in [Lov71] to establish the O(n3/2) bound for the deterministic k-set problem.
Proof of Theorem 5.1.2. For any fixed n, let m := m(n) be an integer whose value will be chosen later and let L be the set consisting of m vertical lines which partition R2 \ S ℓ∈L ℓ into m + 1 open cells such that the measure according to P of each cell is equal to 1/(m + 1). Such a set of lines exists because the measure (according to P) of every line is 0. In order to show how to construct this set of lines, it suffices to show that given any finite measure µ on R2 such that the measure of every line is 0, there exists a vertical line which divides the measure into two equal parts. Indeed, the function f defined by f(t) = µ((x, y) ∈R2 : x ≤t) is a continuous function of t.
The fact that f is continuous is a consequence of the assumption that the measure of every vertical line is 0. Furthermore, as t →−∞, f(t) →0 and as t →∞, f(t) →µ(R2). Therefore, the claim follows from the intermediate value theorem.
Let X = {X1, . . . , Xn} be a sample of n iid points from P. Observe that the probability that two points in X span a vertical line is 0 so we do not need to consider this case. Also, since the measure of every line is 0, X is in general position with probability 1 so we can also ignore the case when X is not in general position. Therefore, we can analyze the k-edges of X using the k-edge graph Gk of X.
We bound the expected number of k-edges in Gk by considering two different types of k-edges separately. First we bound the expected number of k-edges formed by two points in different cells of the partition. Then we bound the expected number of k-edges formed by two points in the same 48 cell of the partition. That is, the expected number of k-edges in Gk is equal to E(number of k-edges in Gk formed by two points in different cells) + E(number of k-edges in Gk formed by two points in the same cell).
(5.3) If conv(X1, X2) is a k-edge formed by two points X1, X2 in different cells, then conv(X1, X2) intersects at least one line in L. So to bound the expected number of k-edges in Gk formed by two points in different cells, it suffices to bound the expected number of k-edges in Gk that intersect a line in L. By Lemma 5.2.2, each line in L intersects at most min(k −1, n −k −1) k-edges in Gk.
Therefore, the first term in Eq. (5.3) is at most m · min(k −1, n −k −1) Now we bound the second term. Recall that the measure according to P of each cell is equal to 1/(m + 1). Therefore, for any fixed i ̸= j, the probability that Xi and Xj are in the same cell is 1/(m + 1) ≤1/m.
We can bound the second term in Eq. (5.3) by E(number of k-edges in Gk formed by two points in the same cell) ≤ X (Xi,Xj)∈(X 2) P (Xi, Xj) is a k-edge AND Xi, Xj are in the same cell = X (Xi,Xj)∈(X 2) P (Xi, Xj) is a k-edge Xi, Xj are in the same cell · P(Xi, Xj are in the same cell) = n 2 · P (X1, X2) is a k-edge X1, X2 are in the same cell · P(X1, X2 are in the same cell) ≤n2 m · P (X1, X2) is a k-edge X1, X2 are in the same cell .
(5.4) 49 Set T := P aff(X1, X2)+ and G(t) = P(T ≤t | X1, X2 are in the same cell). We then have that P (X1, X2) is a k-edge X1, X2 are in the same cell = E P (X1, X2) is a k-edge X1, X2, (X1, X2 are in the same cell) X1, X2 are in the same cell = 2 n −2 k E T k(1 −T)n−2−k X1, X2 are in the same cell = 2 n −2 k Z 1 0 tk(1 −t)n−2−kdG(t) ≤2 n −2 k · k n −2 k · n −2 −k n −2 n−2−k ≤ √n −2 √ k √ n −2 −k .
(5.5) The last inequality follows from Stirling-type upper and lower bounds for factorials, see for example [Hum40]. Therefore, we have that Eq. (5.3) is at most m · min(k −1, n −k −1) + n2 m √n −2 √ k √ n −2 −k and choosing m = Θ n5/4 k1/4(n−k)1/4√ min(k,n−k) makes the above quantity O n7/4 k1/4(n−k)1/4 . Since we could repeat the argument after rotating the plane 180 degrees, the same bound applies to the expected number of k-edges.
□ When k = ⌊cn⌋for some c ∈(0, 1), Theorem 5.1.2 shows that the expected number of k-edges of a sample of n points is O(n5/4). An artifact of the proof is that when k grows much slower than n, the bound we obtain on the expected number of k-edges is worse. For example, when k is a constant and n grows, the theorem only tells us that the expected number of k-edges is O(n3/2).
However, we can combine Theorem 5.1.2 with the best known bound for the deterministic k-edge problem to obtain a bound on the expected number of k-edges that gives a uniform bound for all values of k. Indeed if P is any distribution on R2 such that the measure of every line is zero, then by Theorem 5.1.2 and the main result of [Dey97], the expected number of k-edges of a sample of n points from P is at most min(C1nk1/3, C2n7/4 k1/4(n−2−k)1/4 ) for some constants C1, C2. This quantity is O(n9/7) when k ≤(n −2)/2.
50 5.3. On the number of k-edges via the polynomial method In this section we give another proof of Theorem 5.1.2. The new proof partitions the plane using algebraic curves instead of vertical lines. Given a distribution on R2 which has a density, we use the continuous polynomial partitioning theorem of [Gut16b] (Theorem 2.3.4) to obtain an algebraic curve which divides the plane into a number of cells of equal probability. The rest of the proof is nearly the same as the proof in Section 5.2.2. The reason this alternative proof is interesting is because it motivates the following open question which, if resolved, may lead to an improvement to the bound in Theorem 5.1.2.
Question 5.3.1. What is the maximum (finite) number of times that an irreducible non-singular2 degree r algebraic curve can intersect the k-edge graph of a set of n points in the plane?
It is clear that the quantity in Question 5.3.1 is Ω(nr), and we have some reason to believe that it may be Θ(nr).The best bound we are able to prove is O(nr2) (Lemma 5.3.8). This bound is good enough to reprove Theorem 5.1.2 using polynomial partitioning in the case where the distribution has a density (Theorem 5.3.2). Any improvement to our O(nr2) bound would lead to an improvement in the bound in Theorem 5.3.2: When k is proportional to n, the bound in Theorem 5.3.2 is O(n5/4). An O(nr) bound on the quantity in Question 5.3.1 would allow one to improve the bound in Theorem 5.3.2 from O(n5/4) to O(n7/6) when k is proportional to n.
An O(nr) bound on the quantity in Question 5.3.1 would also have an interesting application to the deterministic k-set problem: It would give another proof of Dey’s O(nk1/3) bound [Dey97, Dey98] on the maximum number of k-edges of a set of n points in the plane in the case where k is proportional to n. The idea of the proof is as follows. Given any set S of n points in general position in the plane, for some r to be chosen later, use the discrete polynomial partitioning theorem (Theorem 2.3.2) to find a degree O(r) polynomial f such that R2 \Z(f) is the union of r2 pairwise disjoint open sets (called cells) each of which contains at most n/r2 points of S. The O(nr) bound on the quantity in Question 5.3.1 implies that the number of k-edges formed by two points of S which are both in different cells of the partition is O(nr). Also, the number of k-edges formed by two points which are both in the same cell is at most r2 · n/r2 2 = O( n2 r2 ). Since we can assume 2One could consider the same question for possibly singular curves, but, for our purposes, it suffices to consider non-singular curves.
51 that S is in general position with respect to degree O(r) algebraic curves, we can assume that the number of points contained in Z(f) is O(r2) and so the number of k-edges formed by two points both of which are in Z(f) is O(r4). Finally, it is not hard to show that the number of k-edges formed by two points where one point is in Z(f) and the other is not and the interior of the k-edge does not intersect Z(f) is O(n). Indeed, the only way this can happen if one point is in Z(f) and the other is in one of the two cells which have the part of Z(f) that contains the first point on their boundary. So there is a O(r2) · 2 · n/r2 bound. Moreover, the number of k-edges formed by two points where one point is in Z(f) and the other is not and the interior of the k-edge does intersect Z(f) is O(nr), again by the O(nr) bound on Question 5.3.1. Choosing r = Θ(n1/3) shows that the total number of k-edges is O(n4/3). See Section 7.4 for a discussion of why we believe the quantity in Question 5.3.1 may be Θ(nr).
There is one technical issue introduced by our application of the polynomial partitioning the-orem to the probabilistic version of the k-edge problem: It can only be applied to distributions which have a density. For this reason, the theorem we prove in this section is slightly weaker than Theorem 5.1.2 because it only applies to distributions which have a density. Although the following theorem is simply a restatement of Theorem 5.1.2 with the added assumption that the distribution has a density, we give a formal statement of the theorem for the sake of readability: Theorem 5.3.2. Let P be a probability distribution on R2 which has a density. Then for any 1 ≤k ≤n −1, EP (k, n) = O n7/4 k1/4(n−k)1/4 (where the constants in big-O are universal).
Before proving Theorem 5.3.2, in Section 5.3.1 we establishes some necessary lemmas concerning algebraic curves.
We make one remark on the requirement that the probability distribution P in Theorem 5.3.2 has a density. As mentioned earlier, in [BS94], B´ ar´ any and Steiger construct a probability distri-bution P with EP ( n−2 2 , n) = Ω(n log n). The distribution P does not have a density. However, if mi is any decreasing sequence whose limit is zero, a slight modification to the previously mentioned construction allows B´ ar´ any and Steiger to construct a probability distribution P ′ which has a den-sity and with EP ′( n−2 2 , n) = Ω(mnn log n). In particular, EP ′( n−2 2 , n) can still be super-linear even if P ′ has a density. This shows that the class of distributions which have a density is an important 52 class of distributions to investigate in the context of the probabilistic k-facet problem. Although the details haven’t been worked out as far as we know, it is also probably not hard to construct a distribution P which has a density and with EP (k, n) = Ω(mnneΩ(√log n)) 5.3.1. Counting intersection points of an algebraic curve and the k-edge graph. Any polynomial f ∈R[x1, x2] defines an algebraic curve Z(f) := {x ∈R2 : f(x) = 0}.
Our use of the polynomial partitioning technique requires us to bound the number of times the k-edge graph Gk of a set of n points can intersect a degree r algebraic curve Z(f), i.e., we must give some answer to Question 5.3.1. Note that the number of points of intersection of Gk and Z(f) could be infinite if Z(f) contained one of the lines spanned by a k-edge in Gk. However, for our purposes, it suffices to bound the number of intersection points in the case when it is finite.
In order to establish our O(nr2) bound on the quantity in Question 5.3.1, we first show how to partition an irreducible algebraic curve into the union of O(r2) convex and concave pieces (Proposition 5.3.6). Combining the convex/concave chains decomposition of the k-edge graph Gk with the partition of a degree r algebraic curve into O(r2) convex and concave pieces allows us to show that a degree r algebraic curve intersects the k-edge graph of a set of n points at most O(nr2) times assuming the number of intersections is finite (Lemma 5.3.8).
First, we show how to partition an irreducible curve Z(f) into the union of a finite number of points and a finite number of convex/concave x-monotone connected curves.
Definition 5.3.3. A connected curve C ⊂R2 is x-monotone if every vertical line intersects it in at most one point.
Definition 5.3.4. An x-monotone curve C is convex (respectively, concave) if for every three points (x1, y1), (x2, y2), (x3, y3) ∈C with x1 < x2 < x3, the point (x2, y2) is below (respectively, above) or on the line joining (x1, y1) and (x3, y3).
In order to break Z(f) into convex/concave pieces, we need to use the inflection points of Z(f).
Definition 5.3.5 ( [Kir92]). A non-singular point (a, b) of an algebraic curve Z(f) is an inflection point if the Hessian curve Hf(x, y) := f2 y fxx −2fxfyfxy + f2 xfyy is equal to zero at (a, b). (The notation fx denotes the partial derivative with respect to x.) 53 Proposition 5.3.6. A non-singular irreducible curve Z(f) ⊂R2 of degree r that is not a vertical line can be partitioned into the union of at most 4r2 points and at most 6r2 x-monotone curves where each x-monotone curve is either convex or concave.
Proof. If Z(f) is a non-vertical line, the conclusion is clearly true. So assume that Z(f) is not a line. Let F = Z(f)∩Z(fy). We know that f depends on y and not just x because otherwise Z(f) would be a vertical line. This means that fy is not identically zero. Now because f is irreducible and the degree of fy is less than the degree of f, the polynomials f and fy cannot have a common factor. Therefore, by B´ ezout’s theorem, |F| ≤r(r −1).
Let I be the set of inflection points of Z(f).
An irreducible curve of degree r ≥2 has at most 3r(r −2) inflection points [Kir92, Proposition 3.33] so |I| ≤3r(r −2). Let C be the set of connected components of Z(f) \ (I ∪F). Because of the removal of the points in F, every curve in C is x-monotone. To show this, let C ∈C. Assume there exists two points a, b ∈C that have the same x-coordinate. Since C is not a vertical line, there must be a point x ∈C that is between a, b and such that x is not on the line through a, b. Therefore, between a and b, the curve must travel in the positive x direction and then in the negative x-direction, meaning there exists a point on the curve between a and b where fy = 0, a contradiction.
Now we show that, because of the removal of the inflection points I, every curve in C is either a convex x-monotone curve or a concave x-monotone curve. Each curve in C is the graph of a function defined in an interval. We claim that, for each curve in C, the second derivative of the associated function exists everywhere and is never zero. Let C ∈C. Since C does not contain a point where fy = 0, using the implicit function theorem, for each (u, v) ∈C, there exists a smooth function φ : (u −ϵ, u + ϵ) →R that gives a local parameterization of the curve near (u, v) [Rut00, Theorem 4.22]. Now a simple calculation shows that if φ′′(x) is equal to 0 at x, then the Hessian curve f2 y fxx −2fxfyfxy + f2 xfyy equals zero at (x, φ(x)). Indeed, we have parameterized Z(f) near a given point in C by x 7→(x, φ(x)). Differentiating f(x, φ(x)) = 0 gives fx + φ′(x)fy = 0 and so φ′(x) = −fx fy . Differentiating again gives φ′′(x)fy + (1, φ′(x)) fxx fxy fxy fyy 1 φ′(x) = 0 and now rearranging and using the fact that φ′(x) = −fx fy and the fact that fy ̸= 0 shows that if φ′′(x) = 0 then the Hessian curve is zero. The inflection points of Z(f) are precisely the points where the 54 Hessian curve is zero. Therefore, since all inflection points were removed, the second derivative of the function whose graph is C is never zero. This means that the function is either strictly convex of strictly concave, and so C is either a convex or concave x-monotone curve.
Now we determine how many distinct curves C can contain. The number of connected compo-nents of Z(f) is at most 2r2 by either [Mil64, Theorem 2] or [She20, Theorem 2.7]. We removed at most r(r −1) + 3r(r −2) points from Z(f). Because Z(f) is non-singular, it has no points of self-intersection.
Therefore, each point which is removed increases the number of connected components of Z(f) \ (I ∪F) by at most 1. Therefore, the number of connected components of Z(f) \ (I ∪F) is at most 2r2 + r(r −1) + 3r(r −2) ≤6r2.
□ The decomposition into convex/concave pieces is useful because of the following fact: Lemma 5.3.7. Let C be an x-monotone convex curve and D an x-monotone concave curve. If the number of points of intersection of C and D is finite, then it is at most 2.
Proof. Assume that C and D intersect in three points (x1, y1), (x2, y2), (x3, y3). Observe that the three points (x1, y1), (x2, y2), (x3, y3) must be contained in a line ℓand we may assume that x1 < x2 < x3. We claim that C and D must both contain the line segment connecting the three points. Indeed, assume that C does not contain the line segment connecting (x1, y1) and (x2, y2).
Then there must be a point (x0, y0) ∈C with x1 < x0 < x2 and (x0, y0) strictly below the line connecting (x1, y1) and (x2, y2). But then the point (x2, y2) is above the line connecting (x0, y0) and (x3, y3), a contradiction to convexity of C. The argument for the other cases is similar.
□ Now we can establish the bound on the number of intersection points between Z(f) and Gk.
Lemma 5.3.8. Let S ⊂R2 be a set of points in general position, Gk = (S, Ek) the k-edge graph of S, and f ∈R[x1, x2] a degree r polynomial such that all irreducible components of Z(f) are non-singular and S ∩Z(f) = ∅. If the number of points of intersection of Z(f) and Gk is finite, then it is at most 13nr2.
Proof. First assume that Z(f) is irreducible. If Z(f) is a line, then it follows from Lemma 5.2.2 that the number of intersection points is at most max(k −1, n −k + 1) ≤13n and we are done.
So assume that Z(f) is not a line. First, we need to decompose the curve Z(f) into the union of 55 convex and concave pieces. By Proposition 5.3.6, Z(f) can be partitioned into the union of 6r2 convex/concave x-monotone curves and at most 4r2 points Let A be the set of convex x-monotone curves, B the set of concave x-monotone curves, and N the set of points in the partition.
By Lemma 5.2.2, Gk can be decomposed into the union of k −1 convex chains C1, . . . , Ck−1, or n −k + 1 concave chains D1, . . . , Dn−k+1.
Recall that we are assuming that no line spanned by two points from S is vertical. Therefore, the convex chains C1, . . . , Ck−1, and the concave chains D1, . . . , Dn−k+1 never contain two points on a vertical line. Furthermore, we claim that any convex or concave chain Ci or Dj intersects Z(f) in only finitely many points. Indeed, if one of these chains intersected Z(f) in infinitely many points, one of the line segments in the chain would have to intersect Z(f) in infinitely many points.
Recall the fact that if a degree r algebraic curve intersects a line in more than r points, then the curve must contain the line. Since we are assuming that Z(f) does not contain any of the points in S, this is not possible. Thus, we can apply Lemma 5.3.7 and the concave chain decomposition of Gk to show that the number of k-edges in Gk that intersect Z(f) at a point contained in one of the convex x-monotone curves in A is at most 2(n −k + 1)6r2. Similarly, the number of k-edges in Gk that intersect Z(f) at a point contained in one of the concave x-monotone curves in B is at most 2(k −1)6r2. Additionally, there are 4r2 points in the set N ⊂Z(f) which are not contained in any of the convex/concave x-monotone curves. Therefore, the total number of intersections is at most 2(n −k + 1)6r2 + 2(k −1)6r2 + 4r2 ≤13nr2.
If Z(f) is not irreducible, then say Z(f) is the union of m irreducible components f1, f2, . . . , fm of degrees r1, r2, . . . , rm. By the above, the number of intersection points of Z(fi) and Gk is at most 13nr2 i . So the total number of intersection points of Z(f) and Gk is at most Pm i=1 13nr2 i ≤ 13nr2.
□ 5.3.2. Proof of Theorem 5.3.2.
Proof. Let W be the density of P. For any fixed n, we use Theorem 2.3.4 applied to W to find a degree r := r(n) (to be chosen later) polynomial f which divides R2 \Z(f) into a family O of Θ(r2) pairwise disjoint open sets such that for all O ∈O, the integrals R O W are within a factor of 2 of each other. Furthermore, all irreducible components of Z(f) are non-singular.
56 Let X = {X1, . . . , Xn} be a sample of n points from P. Observe that the probability that two points in X span a vertical line is 0 so we do not need to consider this case. Also, since P has a density, the measure of every line is 0. This means that X is in general position with probability 1 so we may assume this as well. Therefore, we can analyze the k-edges of X using the k-edge graph Gk of X. Also, since the Lebesgue measure of Z(f) is zero, X ∩Z(f) = ∅with probability 1 so we assume this as well.
We compute the expected number of k-edges in Gk by considering two different types of k-edges separately. First we bound the expected number of k-edges formed by two points in different cells of the partition. Then we bound the expected number of k-edges formed by two points in the same cell of the partition. That is, the expected number of k-edges in Gk is equal to E(number of k-edges in Gk formed by two points in different cells) + E(number of k-edges in Gk formed by two points in the same cell).
(5.6) If conv(X1, X2) is a k-edge formed by two points X1, X2 in different cells, then conv(X1, X2) intersects Z(f).
So to bound the expected number of k-edges in Gk formed by two points in different cells, it suffices to bound the expected number of k-edges that intersect Z(f). We claim that the number of points of intersection between Gk and Z(f) is finite.
This is true because otherwise some k-edge would have to intersect Z(f) infinitely many times. If a degree r algebraic curve intersects a line more than r times it must contain that line. If this were true, then Z(f) would have to contain the two points of X forming the line, but we are assuming that X ∩Z(f) = ∅.
Therefore, the number of points of intersection between Gk and Z(f) is finite and so we can apply Lemma 5.3.8 to show that the first term in Eq. (5.6) is at most 13nr2.
Now we bound the second term. Recall that Z(f) divides R2 \Z(f) into a family O of Θ(r2) pairwise disjoint open sets (called cells) such that for all cells O ∈O, the integrals R O W are within a factor of 2 of each other. Therefore, for any fixed i ̸= j, the probability that Xi and Xj are in the same cell is at most C/r2 for a universal constant C > 0.
Now, the second term in Eq. (5.6) can be bounded using nearly the same argument which we used to bound the second term in Eq. (5.3) in the proof of Theorem 5.1.2. This argument is given in Eqs. (5.4) and (5.5) in the proof of Theorem 5.1.2. The only change is that for any 57 Xi, Xj ∈X, the probability P(X1, X2 are in the same cell) is now upper bounded by C/r2 instead of 1/m. Therefore, we have that Eq. (5.6) is at most 13nr2 + C n2 r2 √n −2 √ k √ n −2 −k and choosing r2 = Θ n3/4 k1/4(n−k)1/4 gives O n7/4 k1/4(n−2−k)1/4 . Since we could repeat the argument after rotating the plane 180 degrees, the same bound applies to the expected number of k-edges.
□ 58 CHAPTER 6 Translations of a fixed convex body in the plane We study a variation on the k-set problem for the set system whose set of ranges consists of all translations of some strictly convex body in the plane. The motivation is to show that the technique by B´ ar´ any and Steiger is tight for a natural family of set systems. For any such set system, we determine bounds for the expected number of k-sets which are tight up to logarithmic factors.
In particular, we show that the two-step argument in the proof of Theorem 5.1.1 (Eq. (5.1) and the general upper bound of the integrand in Eq. (5.2)) is not as loose as it seems, if one applies it to the k-set problem on a set system other than half-planes (generalized in a natural way).
More precisely, let C ⊆R2 be the interior of a fixed convex body. We consider the set system of translations of C and study the expected number of k-sets and k-edges. In addition, we show some deterministic bounds to put our probabilistic bounds in context.
For the case where C is strictly convex, we show: • A relation between k-sets and k-edges that allows one to derive upper bounds on the number of k-sets from upper bounds on the number of k-edges (Lemma 6.2.5).
• For certain natural distributions, the expected number of k-sets and k-edges for a random set of n points and some k proportional to n is Θ∗(n3/2) (where ∗means that polyloga-rithmic factors are ignored) (Theorem 6.2.9 and Theorem 6.5.3). The upper bound uses the B´ ar´ any and Steiger technique, while the lower bound uses the uniform convergence theorem of Vapnik and Chervonenkis [VC71].
• The growth function is O(n2) (Proposition 6.4.1).
For the case where C has C2 boundary, we show that the maximum number of k-sets of n points with k proportional to n is Ω(n2) (Theorem 6.3.2).
59 Some of the assumptions above are chosen for readability, the actual theorems have weaker assumptions in some cases.
6.1. Introduction In this chapter we study a natural variation of the k-set problem for translations of a fixed convex set on the plane, namely, the number of ways in which one can enclose k points out of a given finite set of points by a translation of a convex set so that its boundary strictly separates them from the rest. We will show nearly matching upper and lower bounds on the expected number of ways.
For our lower bound, one of our tools will be the uniform convergence theorem of Vapnik and Chervonenkis [VC71]. This introduces a minor technical complication: their theorem is about abstract set systems without regard to whether sets have a boundary, while the standard k-set problem for lines on the plane ask for strict separation by a line and therefore the natural choice for our generalization is to ask for strict separation by a curve.
Similarly, the other side of our argument, our upper bound, is a variation on the two-step argument in the proof of Theorem 5.1.1 (Eq. (5.1) and the general upper bound of the integrand in Eq. (5.2)), which uses k-edges and therefore also uses the boundary curve in a fundamental way.
A convex body is a compact convex set with non-empty interior. A set C is strictly convex if for all x, y ∈C with x ̸= y and for all λ ∈(0, 1) we have λx + (1 −λ)y ∈int C. A set system is a pair (X, R), where X is a set and R is a family of subsets of X. The elements of R are called ranges. For a set C ⊆R2, let (R2, TC) be the set system of translations of C (that is, TC is the family of translation of C). We are interested in translations of convex sets and it will be notationally convenient to set C to be the interior of a fixed convex body. So, for this section, C will be restricted (at least) to be the interior of a convex body. In this case, when we say that a point lies on the boundary of a range, the point does not lie in the range.
Definition 6.1.1. For a finite subset S ⊆R2, a TC-k-set of S is a subset T ⊆S of k points such that for some Q ∈TC, S ∩bd Q = ∅and T = S ∩Q.
60 6.2. Upper bound for TC-k-sets, probabilistic, k proportional to n This section establishes our upper bound on the expected number of TC-k-sets of a set of n iid points when C is the interior a strictly convex body and k is proportional to n. So for Section 6.2, let C ⊆R2 be the interior of a strictly convex body.
Definition 6.2.1. A set of points in R2 is in general position relative to C if no three points lie on the boundary of some range in TC (i.e., some translation of C).
Lemma 6.2.2. Let (p, q) be a pair of distinct points in R2. Then there are at most two ranges x + C satisfying p, q ∈bd(x + C).
Proof. Up to a rotation we can assume that r := q −p is vertical. Suppose for a contradiction that there are three ranges x + C satisfying p, q ∈bd(x + C). This implies there are three points p1, p2, p3 such that p1, p2, p3, p1 + r, p2 + r, p3 + r ∈bd C. Let f(x1) denote the length of segment “C intersected with the vertical line at x1 ∈R.” Function f is positive in a non-empty interval (a, b), is strictly concave in [a, b] and takes value ∥r∥at three points in [a, b]. But there is no such function so this is a contradiction.
□ From the lemma we conclude: Corollary 6.2.3. Let V ⊆(R2)2 be the set of pairs of distinct points that can appear on the boundary of some range. Then there exists a continuous onto function C : V →TC.
Proof. Every pair in V can appear on the boundary of one or two ranges.
When a pair appears on the boundary of exactly one range, map both orderings of the pair to that range.
When a pair (p, q) appears on the boundary of two ranges, let C1 and C2 be the two translations of C containing p and q on their boundaries. Let C(p, q) be the unique translation that solves maxi∈{1,2} area(Ci∩aff(p, q)+) (where aff(p, q)+ = {r : −(q−p)x(r−p)y +(q−p)y(r−p)x > 0}).
□ From now on we let C(·, ·) denote the function given by Corollary 6.2.3 (with a slight abuse of notation).
61 Definition 6.2.4. For a set of points S in general position relative to C, let a (oriented) TC-k-edge be an ordered pair of points (p, q) ∈V with p, q ∈S (p ̸= q) such that C(p, q) contains k points of S.
We now show a bound relating TC-k-sets and TC-k-edges, which follows from a variation of known continuous deformation arguments [HP11, Lemma 5.15], [Mat02, Chapter 11]. For a finite set of points S in general position relative to C, let ek(S) be the number of TC-k-edges of S and let ak(S) be the number of TC-k-sets of S.
Lemma 6.2.5. For a finite set of points S in general position relative to C and k ≥2 we have ak(S) ≤4 ek−2(S) + ek−1(S) + ek(S) .
Proof. To prove the claim we will construct an injective function f from TC-k-sets of S to a labelled extension of the set of TC-k-edges. The function is defined as follows: Let Q ⊆S be a TC-k-set induced by some range C0. Translate C0 in the x direction until some point p ∈S lies on its boundary to obtain range C′, then translate C′ while keeping p on its boundary (letting p slide along the boundary) until another point q ∈S lies on the boundary to obtain a range C′′ (there may be more than one choice here, pick arbitrarily). We have that cl C′′ contains Q and between zero and two other points from S. From the general position assumption, bd C′′ contains exactly two points from S. Swap points p, q if needed so that C′′ = C(p, q). Pick labels lp, lq ∈{IN, OUT} for p and q according to whether they are in Q. This completes the definition of an f from TC-k-sets of S to S2 × {IN, OUT}2, namely f(Q) = (p, q, lp, lq).
We now show that f is injective. Let Q, Q′ be two TC-k-sets of S induced by ranges C0, C′ 0, respectively, and so that f(Q) = f(Q′) = (p, q, lp, lq). By definition of f we have that Q is equal to S ∩C(p, q) with p and q added according to the labels. But then by definition of f we have that Q′ is also equal to that set and therefore equal to Q. This establishes that f is injective.
To conclude, the image of f contains only pairs (p, q) that are TC-r-edges of S for r ∈{k − 2, k −1, k}. The claim follows.
□ Assumption 6.2.6. Given C, probability distribution P on R2 is such that P bd(x + C) = 0 for all x ∈R2.
62 (In particular the assumption on P holds if P has a density.) Proposition 6.2.7. Let P be a Borel probability distribution satisfying Assumption 6.2.6. Let Y, Z be a pair of iid points, each according to P. Then Y ̸= Z a.s.
Proof. Fix a point b on the boundary of C (so that the origin is on the boundary of −b + C).
Note that P(Y = Z) = E P(Y = Z | Z) ≤E P Y ∈bd(Z −b + C) Z = 0.
□ Proposition 6.2.8. Let P be a Borel probability distribution satisfying Assumption 6.2.6. Let X be a random set of n iid points, each according to P. Then X is in general position relative to C a.s.
Proof. It is enough to prove the claim for n = 3. Let Y, Z, W be three iid random points according to P. By Proposition 6.2.7, Y ̸= Z a.s. Then P (∃a)Y, Z, W ∈bd(a + C) = P Y ̸= Z, (∃a)Y, Z, W ∈bd(a + C) = E P (∃a)Y, Z, W ∈bd(a + C) Y, Z Y ̸= Z = E P W ∈bd C(Y, Z) or W ∈bd C(Z, Y ) Y, Z Y ̸= Z = 0.
□ Theorem 6.2.9 (TC-k-set/edge upper bound, probabilistic, k proportional to n). Let c ∈(0, 1).
Let P be a Borel probability distribution satisfying Assumption 6.2.6. Let X be a random set of n iid points, each according to P. Let An (resp. En) be the number of TC-k-sets (resp. TC-k-edges) of X for k = ⌊cn⌋. Then E(En) ≤O(n3/2) and E(An) ≤O(n3/2) (where the constants in big-O depend only on c).
63 Proof. Let C(p, q) and V be as in Corollary 6.2.3. From Proposition 6.2.8, X is in general position relative to C a.s.
Let X = {X1, . . . , Xn}. Let T = P C(X1, X2) with the additional convention that C(p, q) = ∅ if (p, q) / ∈V . In this way T is defined whenever X1 ̸= X2, that is, a.s. by Proposition 6.2.7. Let G(t) = P(T ≤t) for t ∈R.
Using a variation of Eq. (5.1) and the argument in the proof of Theorem 5.1.2 we get: P (X1, X2) is a TC-k-edge of X = P |C(X1, X2) ∩(X \ {X1, X2})| = k = E P |C(X1, X2) ∩(X \ {X1, X2})| = k X1, X2 = n −2 k E T k(1 −T)n−2−k = n −2 k Z 1 0 tk(1 −t)n−2−kdG(t) and E(En) = n(n −1) n −2 k Z 1 0 tk(1 −t)n−2−kdG(t) ≤n2 n −2 k k n −2 kn −2 −k n −2 n−2−k ≤n2 √n −2 √ k √ n −2 −k ≤O(n3/2).
This proves the first inequality. From this, the second inequality is immediate using Lemma 6.2.5.
□ 6.3. Lower bound for TC-k-sets, deterministic, k proportional to n In this section we show a lower bound on the maximum number of TC-k-sets of a broad family of set systems of the form (R2, TC) for k proportional to n. We illustrate the main idea of the argument in Proposition 6.3.1 for the case where C is a unit square. In Theorem 6.3.2, we then use 64 the argument for the case where C is the interior of a convex body with C2 boundary (actually, slightly more general than that).
While the sets of points in the following results may not be in general position, this is not a true weakness of the results. The reason is that, like in the case of the standard k-set problem for lines, the number of TC-k-sets of a given set of points cannot decrease by applying any sufficiently small perturbation to the points, because any range inducing a TC-k-set must by definition contain no point on its boundary. Thus, the maximum number of TC-k-sets among set in general position is no smaller than the number of TC-k-sets among unrestricted sets of points.
Proposition 6.3.1 (idea of lower bound for maximum, the cross). Let C be the open unit square.
Let 0 < c < c′ < 1. Then for cn ≤k ≤c′n we have max|S|=n ak(S) = Ω(n2) (where the constants in Ωdepend only on c and c′).
Proof. We show the case where n is a multiple of 4 and k = n/2, the rest is similar. Consider a set of points equally spaced on the x and y axes forming a cross. Say, for λ = 4/n, let S = λ Z2 ∩(x-axis ∪y-axis) ∩[−n/4, n/4]2 \ (0, 0) . Then |S| = n and the claim follows.
□ For the next result we assume that the boundary of C is well approximated by its unique tangent line at certain points (locally of class C2). See [Gru07, Section 5.1, subsection “Second-Order Differentiability”] for basic facts about differentiability of the boundary of a convex body.
Theorem 6.3.2 (lower bound for maximum, the cross). Assume that C ⊆R2 is the interior of a convex body such that there exist linearly independent unit vectors u, v ∈R2 and points a, b, c, d ∈ bd C such that bd C is C2 in a neighborhood of a, b, c, d with outer normals u, v, −u, −v, resp. Let 0 < c < c′ < 1. Then for cn ≤k ≤c′n we have max|S|=n ak(S) = Ω(n2) (where the constants in Ω depend only on c, c′ and C).
Proof. We show the case where n is a multiple of 8 and k = n/2, the rest follows easily. Up to an invertible linear transformation, we can assume u = e1 and v = e2, without loss of generality.
Let U = {e1, e2, −e1, −e2} and for p ∈U let v(p) be a locally C2 point on the boundary of C and having outer normal p.
For p ∈U and some t > 0, consider the segment of length 2t perpendicular to the boundary of C at v(p) and centered at v(p), namely s(p) := conv{v(p) −tp, v(p) + tp}. Finally, consider a 65 (one-dimensional) grid g(p) of n/4 equally spaced points on segment s(p). Let our set of points be S = ∪p∈Ug(p). By construction |S ∩C| = n/2. Let ϵ := 2t/(n/4 −1) be the gap between consecutive points in each segment.
We will now show that we can choose t > 0 small enough so that there are Ω(n2) small translation of C that induce different subsets of S, each containing n/2 points. The idea of the argument is to translate C independently in the vertical and horizontal direction, to pick Ω(n) different subsets of n/4 points among the pair of vertical segments and similarly for the horizontal segments. The translations, notated p + C and parameterized by p, form the following grid around the origin: G := {(kϵ, lϵ) ∈R2 : k, l ∈Z ∩[−n/8, n/8]}. By Taylor’s theorem and compactness there exist constants α > 0, tM > 0 (that depend only on C) such that the boundary of C has a C2 parametrization y = φ(x) in a neighborhood of (x0, y0) := v(−e2) such that |φ(x)−y0| ≤α(x−x0)2 for x ∈[x0 −tM, x0 + tM] (and similarly for v(e1), v(−e1), v(e2)). In particular, |φ(x) −y0| ≤αt2 M.
We choose t > 0 small enough so that C contains the same subset of g(−e2) when translated distance less than or equal to t in the horizontal direction. Note that this also ensures that the boundaries of those translations contain no point of g(−e2). The nearest point from g(−e2) to the line y = y0 is at distance ϵ/2 = t n/4−1 > 4t/n so it is enough to have αt2 ≤4t/n, that is, we set t = min{tM, 4 αn}.
With these choices, every p ∈G induces a different TC-k-set of S with k = n/2 and therefore an/2(S) ≥|G| ≥n2/16.
□ To understand the scope of Theorem 6.3.2, note that the condition on C is satisfied when C is the interior of a convex body with C2 boundary. Also, no triangle satisfies the assumptions of Theorem 6.3.2.
6.4. Bounds on the growth function To put our results on the number of TC-k-sets and TC-k-edges in context, we state some basic bounds on the growth function of set system (R2, TC), namely the maximum number of subsets of a set of n point in R2 induced by translations of C. For simplicity some of our bounds have extra assumptions on C that may not be necessary.
66 The growth function [VC71] of set system (R2, TC) is given by n 7→ max S⊆R2,|S|=n {S ∩(x + C) : x ∈R2} .
Proposition 6.4.1. Let C ⊆R2 be the interior of a strictly convex body. The growth function of (R2, TC) is at most n2 −n + 2.
Proof. For the proof we use the notions of a dual set system and dual growth function.
The first step is to notice that (R2, TC) corresponds to the dual set system of (R2, T−C): a range x −C ∈T−C is corresponds to point x ∈R2 and a point y ∈R2 corresponds to range y + C, together with the equivalence “y ∈x −C is equivalent to x ∈y + C”. In this way, the growth function of (R2, TC) is the dual growth function of (R2, T−C).
The second step is to bound the dual growth function of (R2, T−C). Its value at n is bounded by the number of connected components of the complement of n translations of bd(−C), or, equiv-alently, bd(C). Adding n translations of C one by one, this number of connected components is 2 for n = 1 and, using the fact that two translations of bd(C) intersect in at most two points, it increases by at most 2(k−1) when the kth translation is added. Therefore the number of connected components is at most n2 −n + 2.
□ For clarity we state the following summarizing result: Theorem 6.4.2. Let C ⊆R2 be the interior of a strictly convex body with C2 boundary. The growth function of (R2, TC) is Θ(n2) (where the constants in Θ depend only on C).
Proof. Immediate from Proposition 6.4.1 and Theorem 6.3.2.
□ In order to prove our results in Section 6.5 in more generality, we state here a weaker bound on the growth function with weaker assumptions on C.
Theorem 6.4.3. Let C ⊆R2 be the interior of a convex body. The VC-dimension of (R2, TC) is at most 3. The growth function of (R2, TC) is at most n 0 + n 1 + n 2 + n 3 ≤(en/3)3.
Proof. A special case of a result in [NT10] establishes that the VC-dimension of (R2, TC) is at most 3 when C ⊆R2 is a convex body. The bound extends to our case (interior of a convex 67 body) using the observation that if translations of the interior a convex body C shatter a given finite set of points then translations of a scaled down cl C also shatter the same set. The rest follows from the Sauer-Shelah lemma.
□ The VC-dimension bound is tight: C equal to any fixed triangle is a tight example.
6.5. Lower bound for TC-k-sets, probabilistic, some k proportional to n In this section we show, for some k proportional to n, an Ω∗(n3/2) lower bound for the expected number of TC-k-sets for a random sample of n points from the uniform distribution in a set A ⊆R2 sufficiently large to contain translations of C. The restriction to a subset A is necessary as there is no uniform distribution in R2. Our argument uses crucially the fact that translations of C that are contained in A have the same probability under the uniform distribution in A. The minor technical complications introduced by the fact that A is bounded could be avoided by considering a similar set system of translations of a disk (say) on the surface of the two-dimensional sphere (or translations of a shape on the flat torus) with the uniform distribution (a version not studied in this paper).
The idea of the proof is the following: First show that for a random sample X of n points in A, with high probability the number of induced subsets by translations of C contained in A is Ω∗(n2) (Lemma 6.5.2). Then, by VC’s uniform convergence theorem, with high probability each translation of C contained in A contains cn + O∗(√n) points from X for some c. Therefore, by the pigeonhole principle there are Ω∗(n3/2) induced subsets of X that contain exactly the same number of points, that is, X has Ω∗(n3/2) TC-k-sets for some k.
We start by showing that if two translations of C are close then they have a large intersection.
Lemma 6.5.1. Let C ⊆R2 be the interior of a convex body that contains a unit ball. If ∥x∥≤1, then area C ∩(x + C) ≤(1 −∥x∥/2) area(C).
Proof. Consider the function f(x) = area C ∩(x + C) . It is logconcave (by the Pr´ ekopa-Leindler inequality and the fact that f(x) = 1C(x) ∗1−C(x)). Also, f(0) = area(C) ≥π. In other words, log f(x) is concave and, while it is not differentiable at x = 0, we can use directional 68 derivatives and tangent rays at x = 0 to upper bound it by a function of the form x 7→log f(0) + c∥x∥, where c < 0 is an upper bound on the one-sided directional derivative.
We calculate a suitable c now. The one-sided directional derivative at 0 along unit vector v ∈R2 is Df(0)(v) = −2 length(projection of C onto line perpendicular to v) ≤−4 (from the analysis of the movement of chords of C parallel to v: as a chord moves by distance ∆t, it contributes 2∆t and the family of chords is parameterized by values in projection of C onto line perpendicular to v). Thus, log f(0) = log(area C) and D(log f)(0)(v) = Df(0)(v)/f(0) ≤−4/π ≤ −1 (i.e.
we can take c = −1) and these estimates with concavity of log f(x) give log f(x) ≤ log area(C) −∥x∥. That is, f(x) ≤area(C)e−∥x∥. We use the inequality e−t ≤1 −t(1 −1/e) for t ∈[0, 1] to conclude that if ∥x∥≤1, then f(x) ≤area(C) 1−∥x∥(1−1/e) . The claim follows.
□ We show now that the number of ranges induced by translations of C on certain random sets of n points is Ω∗(n2). Because this is meant to be used in the context of TC-k-sets, we show a slightly stronger bound for ranges (induced by translations) that do not contain points on their boundaries.
Lemma 6.5.2 (lower bound on number of ranges, probabilistic). Let C ⊆R2 be the interior of a convex body. Let A ⊆R2 be a compact set such that 2C ⊆A.1 Let X be a set of n iid uniformly random points in A. Let t > 0. Then there exists a constant c6.5.2 > 0 that depends only on A, C and t such that with probability at least 1 −nt, {X ∩(x + C) : x + C ⊆A, X ∩bd(x + C) = ∅} ≥c6.5.2 n log n 2 .
Proof. Let X = {X1, . . . , Xn}. Let B be the ball with center 0 and radius 1. Without loss of generality (up to scaling and translation), B ⊆C.
We will first construct a packing of n2/(c log n)2 translations of C with centers in B with area of pairwise symmetric difference at least about log(n)/n, for some c > 0 to be determined later.
Let G be an n/(c log n)-by-n/(c log n) grid of points with gap (c log n)/n between adjacent rows and columns of points and contained in B. Every pair of points in G is then at distance at least (c log n)/n, and therefore for all x, y ∈G with 0 < ∥x −y∥≤1 we have area (x + C)∆(y + C) = 1The assumption that 2C ⊆A guarantees that the translations of C from our grid G in the proof are contained in A so that the probability computation goes through.
69 2 area(C)−area C ∩((y −x)+C) ≥area(C)∥y −x∥≥area(C)(c log n)/n (using Lemma 6.5.1).
The bound extends to all x, y ∈G with x ̸= y by monotonicity.
We will now show that with probability at least 1−o(1) each x+C with x ∈G induces a different range on X. It is enough to show that for all x, y ∈G with x ̸= y we have (x+C)∆(y+C) ∩X ̸= ∅.
Setting c = (t + 2) area(A)/ area(C) , the probability of this event for some x, y is P (∀i ∈[n])Xi / ∈(x + C)∆(y + C) ≤ 1 −area (x + C)∆(y + C) area(A) !n ≤ 1 −area(C) area(A) (c log n) n n ≤e−(t+2) log n = 1/nt+2.
Thus, with probability at least 1 −n2/nt+2 = 1 −1/nt our event holds for all pairs x, y.
Finally, X ∩∪x∈G bd(x + C) = ∅a.s. The claim follows.
□ We now state and prove our probabilistic lower bound for TC-k-sets for some k proportional to n: Theorem 6.5.3. Let C ⊆R2 be the interior of a convex body. Let A ⊆R2 be a compact set such that 2C ⊆A. Let X be a set of n iid uniformly random points in A. Let a′ k(X) := {X ∩(x + C) : x + C ⊆A, |X ∩(x + C)| = k, X ∩bd(x + C) = ∅} (that is, a′ k(X) is the number of TC-k-sets of X induced by translations of C contained in A). Let p = area(C)/ area(A). Then there exists a function k(n) such that E ak(n)(X) ≥E a′ k(n)(X) ≥ Ω(n3/2/(log n)5/2) and |k(n) −pn| ≤O(√n log n) (where function k(n) and the constants in O, Ω depend only on A and C).
Proof. Let (A, R) be the set system where R is the family of translations of C contained in A. From Theorem 6.4.3 we have that the growth function s(n) of (A, R) satisfies s(n) ≤n3.
70 Fix n. For S ⊆A, let ˆ P(S) = |X ∩S|/n. From Theorem 2.4.6 (VC’s uniform convergence theorem)2 we have, for n ≥2/ϵ2, (6.1) P sup S∈R | ˆ P(S) −p| > ϵ ≤4s(2n)e−ϵ2n/8.
Set ϵ = 4 q 3 log 2n n so that the rhs is at most 1/n3. If we denote by G the complement of the event in (6.1), we have P(G) = 1 −o(1).
Let RX = {X ∩S : S ∈R, X ∩bd S = ∅}. From Lemma 6.5.2 with t = 1 and notation f(n) = c6.5.2(n/ log n)2, we get (6.2) P |RX| ≥f(n) ≥1 −o(1).
Let H denote the event in (6.2).
To conclude, we will show that there is a value k(n) that is independent of X and makes E a′ k(X) large. We have P(X ∈G ∩H) = 1 −o(1). Also for X ∈G ∩H we have X k∈[pn−nϵ,pn+nϵ] a′ k(X) = |RX| ≥f(n).
Therefore E X k∈[pn−nϵ,pn+nϵ] a′ k(X) ≥f(n) P(X ∈G ∩H).
Reordering, P k∈[pn−nϵ,pn+nϵ] E a′ k(X) ≥f(n) P(X ∈G ∩H). Thus, there exists k(n) ∈[pn − nϵ, pn+nϵ] such that E a′ k(X) ≥f(n) P(X ∈G∩H)/(2nϵ+1). That is (using nϵ = O(√n log n)), E a′ k(X) ≥Ω n3/2/(log n)5/2 .
For the bound on E ak(X) , from the definitions we have ak(X) ≥a′ k(X).
□ 2In order to apply VC’s uniform convergence theorem, we need to verify that the function supS∈R(·) as defined in Eq. (6.1) is measurable, i.e., that it is a random variable. This can be verified by observing that R is a permissible class of subsets of A. See [Pol84, Appendix C] for the definition of permissible classes and a proof of the measurability of suprema in this context. One can see that the class R is permissible by indexing it by translation and verifying that the requirements for permissibility are met.
71 CHAPTER 7 Open questions 7.1. The real degree of algebraic varieties The degree of complex algebraic varieties is an extremely well-studied and useful definition. Of course, this definition of degree also applies to real algebraic varieties. The problem is that the degree does not always capture the geometry of a given real variety. For example, the degree of the variety V (x4 −y) is 4. But if you just look at the real points, you would say that the degree should be 2. Here we suggest a new definition of degree for real algebraic varieties (which depends only on the real points) and suggest how it may be related to the property of a real variety being weakly k-neighborly. We remark that a notion of degree for real varieties was suggested in [NR09].
However, the notion of degree given there does not line up with the definition we suggest here.
Definition 7.1.1. Let V ⊂Rp be a d-dimensional real algebraic variety which contains smooth real points. The real degree of V is the maximum integer D so that there exists an open subset O of the set of all (p −d)-flats in Rp with the property that for every flat L in O, |L ∩V (R)| = D.
We say that a d-dimensional variety V ⊂Rp is a variety of minimal real degree if the real degree of V is p −d + 1. Observe that the parametric real algebraic curves of minimal real degree are precisely the curves (called p-order curves or curves of order p) which are studied by Sturmfels in [Stu87] because of their connection to neighborly polytopes.
We need to make one more definition which is inspired by Section 6.5: Definition 7.1.2. A d-dimensional algebraic variety V ⊂Rp is maximally weakly neighborly if V is weakly k-neighborly and p = 2k + d −1.
It is a result of Cordovil and Duchet that a (parameterized) real algebraic curve is maximally weakly neighborly if and only if it is a curve of minimal real degree [CD00, Proposition 3.6]. This equivalence might hold for higher dimensional varieties as well. That is, it might be true that 72 a real variety (possibly with some assumptions on the variety) is maximally weakly neighborly if and only if it is a variety of minimal real degree. The same equivalence may also be true with real varieties replaced by parametric surfaces and hypersurfaces if one extends the definitions of maximally weakly neighborly and minimal real degree to arbitrary parametric (hyper)surfaces in Rd.
7.2. Conjecture on generally k-neighborly embeddings/manifolds Conjecture 4.2.5 remains open. Of course resolving the conjecture in full generality would be ideal, but it may be worthwhile to focus on other special cases instead. For example, one might be able to prove the conjecture for analytic manifolds or for smooth manifolds using tangency properties.
7.3. Lower bounds for the algebraic k-set problem In analogy with the original k-set problem we ask: Is there a polynomial map of the plane into some higher dimensional space which induces a natural set system (R2, F) for which the maximum number of F-k-sets for a set of n points is nℓeΩ(√log n) and o(nℓ+1) for some integer ℓ≥2? Or just Ω(nℓlog n) and o(nℓ+1)? A candidate that we do not fully understand in this context is the map (x, y) 7→(x, y, xy) (or, equivalently up to a linear transformation, (x, y) 7→(x, y, x2 −y2)).
7.4. Bound on the number of points of intersection of Z(f) and the k-edge graph Can the answer to Question 5.3.1 given in Lemma 5.3.8 be improved? We believe it may be possible to improve the bound to O(nr) for the following reason: The k-edge graph Gk of a set of n points behaves somewhat like a degree n algebraic curve when it comes to intersecting it with a line. In particular, a degree n algebraic curve and the k-edge graph of a set of n points both have the property that a line can intersect them at most n times unless the line intersects them infinitely many times. One might expect this phenomenon to also hold for arbitrary algebraic curves, not just lines. If this is the case, the bound on the number of intersection points between a degree r algebraic curve and the k-edge graph of a set of n points should be O(nr) as in the case of the intersection of a degree r algebraic curve and a degree n algebraic curve.
73 7.5. Polynomial partitioning for k-sets in higher dimensions The polynomial partitioning theorem becomes more powerful in higher dimensions. It may be possible to apply it to the k-set problem in dimensions higher than 2. The issue is that there is no analogue of the convex chains decomposition in dimension higher than 2, so this would likely require the discovery of a new property of k-sets or k-facets.
74 Bibliography [AACS98] P. K. Agarwal, B. Aronov, T. M. Chan, and M. Sharir, On levels in arrangements of lines, segments, planes, and triangles, Discrete Comput. Geom. 19 (1998), no. 3, Special Issue, 315–331, Dedicated to the memory of Paul Erd˝ os.
[ABFK92] N. Alon, I. B´ ar´ any, Z. F¨ uredi, and D. J. Kleitman, Point selections and weak ϵ-nets for convex hulls, Combin. Probab. Comput. 1 (1992), no. 3, 189–200.
[Ard04] F. Ardila, The number of halving circles, The American Mathematical Monthly 111 (2004), no. 7, 586– 591.
[BCR98] J. Bochnak, M. Coste, and M.-F. Roy, Real algebraic geometry, Springer, 1998.
[BHP08] S. Buzaglo, R. Holzman, and R. Pinchasi, On s-intersecting curves and related problems, Proceedings of the 24th ACM Symposium on Computational Geometry, College Park, MD, USA, June 9-11, 2008 (M. Teillaud, ed.), ACM, 2008, pp. 79–84.
[BS94] I. B´ ar´ any and W. Steiger, On the expected number of k-sets, Discrete Comput. Geom. 11 (1994), no. 3, 243–263.
[CD00] R. Cordovil and P. Duchet, Cyclic polytopes and oriented matroids, Eur. J. Comb. 21 (2000), no. 1, 49–64.
[CFSS14] N. Chevallier, A. Fruchard, D. Schmitt, and J. Spehner, Separation by convex pseudo-circles, 30th Annual Symposium on Computational Geometry, SOCG’14, Kyoto, Japan, June 08 - 11, 2014 (S. Cheng and O. Devillers, eds.), ACM, 2014, p. 444.
[CFSS20] , Separation by convex pseudo-circles, Discrete & Computational Geometry (2020).
[Cla87] K. L. Clarkson, New applications of random sampling in computational geometry, Discrete & Computa-tional Geometry 2 (1987), 195–222.
[Cla04] K. L. Clarkson, On the expected number of k-sets of coordinate-wise independent points, https:// kenclarkson.org/cwi_ksets/p.pdf, 2004, Manuscript.
[CP86] B. Chazelle and F. P. Preparata, Halfspace range search: An algorithmic application of k-sets, Discret.
Comput. Geom. 1 (1986), 83–93.
[CS89] K. L. Clarkson and P. W. Shor, Application of random sampling in computational geometry, II, Discrete & Computational Geometry 4 (1989), 387–421.
[CSY87] R. Cole, M. Sharir, and C.-K. Yap, On k-hulls and related problems, SIAM J. Comput. 16 (1987), no. 1, 61–77.
75 [Der82] Der-Tsai Lee, On k-nearest neighbor Voronoi diagrams in the plane, IEEE Transactions on Computers C-31 (1982), no. 6, 478–487.
[Dey97] T. K. Dey, Improved bounds on planar k-sets and k-levels, Proceedings 38th Annual Symposium on Foundations of Computer Science (1997), 156–161.
[Dey98] , Improved bounds for planar k-sets and related problems, Discrete & Computational Geometry 19 (1998), no. 3, 373–382.
[ELSS73] P. Erd˝ os, L. Lov´ asz, A. Simmons, and E. G. Straus, Dissection graphs of planar point sets, A survey of combinatorial theory (Proc. Internat. Sympos., Colorado State Univ., Fort Collins, Colo., 1971), 1973, pp. 139–149.
[GK15] L. Guth and N. H. Katz, On the Erd˝ os distinct distances problem in the plane, Ann. of Math. (2) 181 (2015), no. 1, 155–190.
[GP10] V. Guillemin and A. Pollack, Differential topology, AMS Chelsea Publishing, Providence, RI, 2010, Reprint of the 1974 original.
[Gru07] P. M. Gruber, Convex and discrete geometry, Grundlehren der Mathematischen Wissenschaften [Funda-mental Principles of Mathematical Sciences], vol. 336, Springer, Berlin, 2007.
[Gut16a] L. Guth, Polynomial methods in combinatorics, University Lecture Series, vol. 64, American Mathematical Society, Providence, RI, 2016.
[Gut16b] , A restriction estimate using polynomial partitioning, J. Amer. Math. Soc. 29 (2016), no. 2, 371–413.
[Har95] J. Harris, Algebraic geometry, Graduate Texts in Mathematics, vol. 133, Springer-Verlag, New York, 1995, A first course, Corrected reprint of the 1992 original.
[HP11] S. Har-Peled, Geometric approximation algorithms, Mathematical Surveys and Monographs, vol. 173, American Mathematical Society, Providence, RI, 2011.
[Hum40] P. M. Hummel, A note on Stirling’s formula, Amer. Math. Monthly 47 (1940), no. 2, 97–99.
[Kir92] F. Kirwan, Complex algebraic curves, London Mathematical Society Student Texts, vol. 23, Cambridge University Press, Cambridge, 1992.
[KMS12] H. Kaplan, J. Matouˇ sek, and M. Sharir, Simple proofs of classical theorems in discrete geometry via the Guth-Katz polynomial partitioning technique, Discrete Comput. Geom. 48 (2012), no. 3, 499–517.
[KW08] G. Kalai and A. Wigderson, Neighborly embedded manifolds, Discrete & Computational Geometry 40 (2008), no. 3, 319–324.
[Lov71] L. Lov´ asz, On the number of halving lines, Ann. Univ. Sci. Budapest, Etovos, Sect. Math. 14 (1971), 107–108.
[LR20] B.
Leroux and L.
Rademacher, Algebraic k-sets and generally neighborly embeddings, 2020, arXiv:1912.03875.
76 [LR21] , Improved bounds for the expected number of k-sets, 2021, arXiv:2106.04782.
[Mat02] J. Matouˇ sek, Lectures on discrete geometry, Graduate texts in mathematics, vol. 212, Springer, 2002.
[Mil64] J. Milnor, On the Betti numbers of real varieties, Proc. Amer. Math. Soc. 15 (1964), 275–280.
[MSRB71] P. McMullen, G. C. Shephard, J. E. Reeve, and A. Ball, Convex polytopes, vol. 3, CUP Archive, 1971.
[MSSW06] J. Matouˇ sek, M. Sharir, S. Smorodinsky, and U. Wagner, k-sets in four dimensions, Discrete Comput.
Geom. 35 (2006), no. 2, 177–191.
[NR09] J. Nie and K. Ranestad, Algebraic degree of polynomial optimization, SIAM J. Optim. 20 (2009), no. 1, 485–502.
[NT10] M. Nasz´ odi and S. Taschuk, On the transversal number and vc-dimension of families of positive homothets of a convex body, Discrete Mathematics 310 (2010), no. 1, 77–82.
[Pol84] D. Pollard, Convergence of stochastic processes, Springer Series in Statistics, Springer-Verlag, New York, 1984.
[Rut00] J. W. Rutter, Geometry of curves, Chapman & Hall/CRC Mathematics, Chapman & Hall/CRC, Boca Raton, FL, 2000.
[Sch14] R. Schneider, Convex bodies: the Brunn-Minkowski theory, second expanded ed., Encyclopedia of Math-ematics and its Applications, vol. 151, Cambridge University Press, Cambridge, 2014.
[She20] A. Sheffer, Polynomial methods and incidence theory, 000book.pdf, 2020, Accessed: 2021-02-05.
[SST01a] M. Sharir, S. Smorodinsky, and G. Tardos, An improved bound for k-sets in three dimensions, Discrete Comput. Geom., vol. 26, Springer, 2001, pp. 195–204, ACM Symposium on Computational Geometry (Hong Kong, 2000).
[SST01b] M. Sharir, S. Smorodinsky, and G. Tardos, An improved bound for k-sets in three dimensions, Discret.
Comput. Geom. 26 (2001), no. 2, 195–204.
[ST42] A. H. Stone and J. W. Tukey, Generalized “sandwich” theorems, Duke Math. J. 9 (1942), 356–359.
[STC04] J. Shawe-Taylor and N. Cristianini, Kernel methods for pattern analysis, Cambridge University Press, New York, NY, USA, 2004.
[Stu87] B. Sturmfels, Cyclic polytopes and d-order curves, Geometriae Dedicata 24 (1987), no. 1, 103–107.
[T´ ot01] G. T´ oth, Point sets with many k-sets, Discrete & Computational Geometry 26 (2001), no. 2, 187–194.
[VC71] V. Vapnik and A. Chervonenkis, On the uniform convergence of relative frequencies of events to their probabilities, Theory of Probability and its Applications 16 (1971), 264 – 280.
[Wag08] U. Wagner, k-sets and k-facets, Surveys on discrete and computational geometry, Contemp. Math., vol.
453, Amer. Math. Soc., Providence, RI, 2008, pp. 443–513.
[Whi57] H. Whitney, Elementary structure of real algebraic varieties, Ann. of Math. (2) (1957), 545–556.
[Zie95] G. M. Ziegler, Lectures on polytopes, Springer-Verlag, New York, 1995.
77 |
10950 | https://jsmith26.weebly.com/uploads/8/7/2/8/87281618/8.8_rsg.pdf | NC Math 1 Unit 8 Quadratic Functions 8.8 Developed by CHCCS and WCPSS adapted from Illustrative Mathematics READY Topic: Interpreting Quadratic Expressions and Functions. 1. A ball thrown vertically upward at an initial velocity of 𝑣0 feet/second rises a distance of d feet in t seconds, given by 𝑑= 6 + 𝑣0𝑡−16𝑡2. a. Write what this equation would look like if the ball is thrown at 88 feet per second. b. Write what this equation would look like if the ball is thrown to rise a distance of 20 feet in 2 seconds. Can you solve for this missing velocity in this case to find how fast the ball was thrown? 2. The expression −4.9𝑡2 + 17𝑡+ 0.6 describes the height in meters of a basketball 𝑡 seconds after it has been thrown vertically into the air. Interpret the coefficients of each term of this expression in the context of the situation. 3. Three equivalent equations for 𝑓(𝑥) are shown below. 𝑓(𝑥) = −2𝑥2 + 24𝑥−54 𝑓(𝑥) = −2(𝑥−3)(𝑥−9) 𝑓(𝑥) = −2(𝑥−6)2 + 18 a. Which form currently reveals the x-intercepts of the function? b. Circle all values of 𝑥 for which 𝑓(𝑥) = 0. −54 −18 −9 −6 −3 0 3 6 9 18 54 4. The expression −4𝑥2 + 8𝑥+ 12 represents the height in feet of an apple thrown from a person on a ladder near an apple tree to a basket on the ground, where 𝑥 is time in seconds. Write an equivalent expression for this situation that is in factored form. What information can you find easily from each form? NC Math 1 Unit 8 Quadratic Functions 8.8 Developed by CHCCS and WCPSS adapted from Illustrative Mathematics SET Topic: Analyzing and Practicing Projectile Motion problems. 5. A baseball is “popped” straight up by a batter with an initial velocity of 64 ft/sec. The height of the ball above ground is given by a function where t is time in seconds after the ball leaves the bat and h(t) is the height in feet above the ground. The batter hit the ball at an original height of 3 feet off of the ground, and the acceleration due to gravity is -16ft/se𝑐2. a. Write the function that will model this situation: ____ b. Make a table of values for 0 seconds – 4 seconds. c. What is the maximum height that the baseball will reach? d. At what time will the baseball be back down to its original height? 6. A bottle rocket that is originally on the ground is launched vertically with an initial velocity of 128 feet per second. The acceleration due to gravity is -16 feet per second squared. The formula gives the height, h(t), of the rocket after t seconds. a. Write the function that will model this situation: ________ b. How long is the rocket in the air? c. A balloonist sees the rocket go by 4 seconds after it leaves the ground. How high is the balloonist from the ground when he sees the rocket? d. A helicopter is 600 feet in the air. The helicopter sees the rocket approaching the helicopter. Is it possible for the rocket to hit the helicopter if the pilot remains at an altitude of 600 feet? Explain. e. The rocket will pass a sight-seer on a ledge 240 feet high. When does the sight-seer see the rocket pass on its way up? When does the sight-seer see the rocket pass on its way down? 7. Jenny is practicing her diving off of a spring board. She follows the function: ℎ(𝑡) = −𝑡2 + 6𝑡+ 7 where ℎ(𝑡) is her height from the pool over (t) time in seconds. a. How long will it take for Jenny to hit the water? b. What is her maximum height and when will she reach it? c. How high is Jenny at the start (how high is the diving board?) d. What is ℎ(8) and what does it represent in context of this problem? NC Math 1 Unit 8 Quadratic Functions 8.8 Developed by CHCCS and WCPSS Go! Topic: Other Applications of Quadratics. 8. A rectangle has a length that is 2 units longer than the width. If the width is increased by 4 units and the length increased by 3 units, write two equivalent expressions for the area of this new rectangle. 9. A vacant rectangular lot is being turned into a community vegetable garden with a uniform path around it. The area of the lot is represented by the equation 4𝑥2 + 40𝑥= 44 where 𝑥 is the width of the path in meters. Find the width of the path surrounding the garden. 10. The area of a trapezoid is found using the formula 𝐴= 1 2 ℎ(𝑏1 + 𝑏2), where 𝐴 is the area, ℎ is the height, and 𝑏1 and 𝑏2 are the lengths of the bases. Consider the trapezoid below. Write two equivalent expressions for the area of the trapezoid. 11. A town council plans to build a public parking lot. The outline below represents the proposed shape of the parking lot. Write an expression for the area, in square yards, of this proposed parking lot. |
10951 | http://math.uchicago.edu/~may/REU2021/REUPapers/Jin,Alexa.pdf | CANTOR SETS IN TOPOLOGY, ANALYSIS, AND FINANCIAL MARKETS ZIQIAN (ALEXA) JIN Abstract. Cantor sets provide intriguing examples, counterexamples, and illustrations of a variety of concepts in different mathematical fields.
This paper explores the applications of Cantor sets – the Cantor ternary set in particular – to the areas of topology, measure theory, analysis and the real world. Several notable topics discussed include homeomorphism, dimensions, Cantor functions, and the fractal market hypothesis.
Contents 1.
Introduction 1 1.1.
The standard Cantor set 1 1.2.
Ternary representation 2 1.3.
Generalization of the standard Cantor set 3 2.
Topological properties 4 2.1.
Basic topological properties 4 2.2.
Nowhere dense and totally disconnected 5 2.3.
Homeoporphism 5 3.
Applications in measure theory 7 3.1.
The measure of Cantor sets 7 3.2.
Dimension 8 4.
Applications in analysis 10 4.1.
Cantor functions 10 4.2.
Volterra’s Function 11 5.
Applications in the real world: fractal phenomena 12 5.1.
Fractal Geometry in Nature 12 5.2.
Mandelbrot and the Fractal Market 13 5.3.
The fractal market hypothesis 14 Acknowledgments 15 References 15 1. Introduction 1.1. The standard Cantor set. Discovered in 1874 by Henry John Stephen Smith and introduced by German mathematician Georg Cantor in 1883, the Cantor ternary set, also called the standard Cantor set, C is created by iteratively delet-ing the open middle third from a set of line segments. One starts by deleting the Date: DEADLINES: Draft AUGUST 14 and Final version AUGUST 28, 2021.
1 2 ZIQIAN (ALEXA) JIN open middle third 1 3, 2 3 from the unit interval [0, 1], leaving two line segments: 0, 1 3 ∪ 2 3, 1 . Next, the open middle third of each of these remaining segments is deleted, leaving four line segments: 0, 1 9 ∪ 2 9, 1 3 ∪ 2 3, 7 9 ∪ 8 9, 1 . This process is continued ad infinitum, where the nth set is Cn = Cn−1 3 ∪ 2 3 + Cn−1 3 for n ≥1, and C0 = [0, 1]. The standard Cantor set is formed by the intersection of all the Cn, i.e. C = ∞ T n=0 Cn.
Figure 1. Creation of the standard Cantor set 1.2. Ternary representation.
Theorem 1. The standard Cantor set consists of exactly those numbers in [0, 1] that can be written in base 3 without 1’s.
Note that this is sometimes given as an alternative, though perhaps less intuitive, definition of the standard Cantor set.
Proof. We begin by showing that the standard Cantor set only consists of such elements: Write all numbers from the unit interval in base 3. For instance, the number 0.25 in base 10 is written in base 3 as 0.020202... . If a number can be written using only 0 and 2, consider the latter representation of it. For instance, the number 0.1 in base three is the same as 0.022222... in base three. It is easy to check that .1 is in the standard Cantor set, which is why we consider it represented as .022222... .
We proceed by induction, starting with the base case: those in C1 are either smaller than 1/3 or bigger than 2/3. Therefore, their first decimal digits are either 0 or 2.
Now, the inductive step: suppose all elements in Cn can be written in base 3 with only 0 and 2 in the first n decimal places. Then since Cn+1 = Cn 3 ∪ 2 3 + Cn 3 for n ≥1, we deduce that all elements in Cn+1 can be written in base 3 with only 0 and 2 on the first n + 1 decimal places. Specifically, their 2 to (n + 1) digits are copied from the 1 to n digits of the corresponding elements in Cn, and the first CANTOR SETS IN TOPOLOGY, ANALYSIS, AND FINANCIAL MARKETS 3 digit is either 0 or 2 depending on whether they belong to the left or right portion of the Cantor set.
Supposing that there is any 1 that appears in the ternary representation of an element in the Cantor set, it would have to appear in a certain place, say the kth digit. This is impossible by the above induction, which gives that there is no digit where a 1 can appear.
Now for the other direction, we want to show by induction that all the numbers in [0, 1] that can be written in base-3 without 1’s are in the standard Cantor set.
For the base case, we know that all the numbers in [0, 1] that are written base 3 with either 0 or 2 in the first decimal digit are in C1, since all the numbers in [0, 1/3] have a 0 as the first digit of their ternary representation (writing 1/3 = .0222...), and all the numbers in [2/3, 1] have a 2 as the first digit (writing 1 = .222).
Now, suppose all numbers that can be written in base 3 with only 0 and 2 on the first n decimal places are in Cn. Now, we take out from these elements those that have a 1 in the n + 1th decimal place. then we see all the remaining numbers are in Cn+1 Cn+1 = Cn 3 ∪ 2 3 + Cn 3 for n ≥1. If a number has no 1 in its ternary, there will be no Ck which it doesn’t appear in, so it will be in C, the intersection of all Ck.
□ Theorem 2. The standard Cantor set is uncountable One of the most notable properties of the standard Cantor set is that it contains uncountably many points. This is despite C being very small in that it is the result of an intersection of countably many decreasing sets, a notion we will make more precise later.
Proof. First, consider every element of C in its ternary representation. Supposing the standard Cantor set is countable, then there is a listing of all the elements of C, for instance: 0.200200222..., 0.002202002..., 0.202220002..., 0.002020200..., ...
Now, construct a new element by swapping the nth decimal digit of the nth element in the list from either 0 to 2 or 2 to 0. In this case the new element is 0.0202..., which is different from any existing element in the list. However, by the above theorem, this new number must also be in C, which contradicts that we had listed all elements of C. Therefore, it is impossible to find a complete enumeration of all the elements in the standard Cantor set, so it must be uncountable. Note that this is very similar to the common diagonalization argument which shows that R is uncountable.
□ 1.3. Generalization of the standard Cantor set. The word ”ternary” in the standard Cantor set meant that the open middle 1/3 of each interval was being removed during each step of the construction. However, we could have selected a different ratio to remove at each step, say 1/2 or 1/4. In fact, we could even vary 4 ZIQIAN (ALEXA) JIN the ratio we remove in each step, not remove the same amount in each ”branch” of the set on a given step, or not remove intervals from the exact center of the previous intervals. Then, as before, we get a decreasing sequence of sets Ck, and define the generalized Cantor set to be their intersection, C. The figure below provides an example of such a sequence of removed intervals. To obtain a set that is similar enough to our prototypical ternary Cantor set, we have one additional restriction: C cannot contain any intervals.
Although Cantor sets can be defined even more generally, we will limit our discussion in this paper to bounded subsets of R.
Figure 2. Example of the first four iterations of a generalized Cantor set Two notable variants of the standard Cantor set are the ”fat” Cantor sets and the ”thin” Cantor sets, which we will dive into later in the paper.
2. Topological properties Now, we prove several topological properties of the standard Cantor set. At the end of this section, we will show that all Cantor sets, as we have defined them, are homeomorphic to each other, which implies that all Cantor set possess these topological properties.
2.1. Basic topological properties.
Theorem 3. The standard Cantor set is closed.
Proof. The Cantor set is an intersection of countably many sets, each of which is a finite union of closed intervals, so closed itself. Thus, the standard Cantor set is the intersection of countably many closed sets, which implies it is closed.
□ Theorem 4. The standard Cantor set is compact.
Proof. We already have that the standard Cantor set is closed. Furthermore, the set has an upper bound of 1 and a lower bound of 0. Hence, the Cantor set is closed and bounded, and by the Heine-Borel theorem, which states that a subset of R is compact if it is closed and bounded, it follows that the standard Cantor set is compact.
□ Definition 2.1. The set S is perfect if S = S′, where S′ denotes the set of all limit points of S. CANTOR SETS IN TOPOLOGY, ANALYSIS, AND FINANCIAL MARKETS 5 Since a closed set contains all of its limit points, this definition is equivalent to every point of a closed set being a limit point. We already have that the Cantor set is closed, so we only need to show every point of C is a limit point.
Theorem 5. Every point in the standard Cantor set is a limit point.
Proof. Given any point x in the Cantor set, there are 2 situations for its ternary representation: Case 1: There are a finite number of digits. Suppose the number of digits is n.
Approach x through the sequence x + 2/3n+1, x + 2/3n+2, x + 2/3n+3,..., where all the terms are also elements in the standard Cantor set. Therefore, x is a limit point of C.
Case 2: There are infinite digits.
Suppose the point is written in base-3 as 0.x1x2x3x4.... As we have shown, each xk is either 0 or 2. Now, x through the sequence 0.x1, 0.x1x2, 0.x1x2x3, 0.x1x2x3x4,... (where xk is the kth digit). All the terms in the sequence that converges to x are also elements in the standard Cantor set, since xk is either 0 or 2. As a result, x is a limit point of the standard Cantor set □ 2.2. Nowhere dense and totally disconnected.
Definition 2.2. A set is nowhere dense if its closure has empty interior. This definition expresses the idea that a set is not ”tightly clustered” in any location.
Theorem 6. The standard Cantor set is nowhere dense.
Proof. We have shown the Cantor set is closed, so its closure is itself. Suppose a subset of the Cantor set is dense, then the subset contains at least one interval, denoted as A = [a, b] where a ≥0 and b ≤1 (a, b ∈R, a < b). From theorem 2.2, we know that written in base 3, both a and b are composed of only 0 and 2, given that the two points are both in the Cantor set. Now, locate the first from left decimal digit that a and b differ from each other and replace it with a 1. For example, if a=0.20202... and b=0.20222... in base, then let c = 0.2021... in base 3 where the remaining digits don’t really matter. Since the first 3 digits of a, b, and c are the same, we are able to argue that a < c < b and that c is not in the ternary Cantor set since it contains the digit 1. A contradiction is therefore achieved, and the Cantor set is nowhere dense.
□ Definition 2.3 (Totally Disconnected). A totally disconnected space is a topo-logical space that has no non-trivial connected subsets. In other words, the only connected components in any totally disconnected space X are the one-point sets.
Theorem 7. The standard Cantor set is totally disconnected.
Proof. This proceeds from what we established in the previous proof. Specifically, any two elements of the standard Cantor set are separated by at least one point not in C. If this is the case, no two distinct points can be part of the same connected component, so the set is totally disconnected.
□ 2.3. Homeoporphism.
6 ZIQIAN (ALEXA) JIN Definition 2.4. A homeomorphism is a continuous bijection between topological spaces that has a continuous inverse . Homeomorphism is an important concept in topology, since it expresses a notion of topological equivalence. Thus, two sets which are homeomorphic share many topological properties.
Theorem 8. All Cantor sets are homeomorphic to each other.
Proof. Given two Cantor sets C and C′ on the unit interval, suppose they are constructed by the intersection of C0, C1, C2... and C′ 0, C′ 1, C′ 2.... let f0 be the linear map bijection from C0 to C′ 0, both of which are entire intervals, sending endpoint to endpoint. f0(x) is continuous within its domain. Similarly, as shown in figure 3, let f1 be the combination of linear map from the left interval of C1 to left interval C′ 1, and likewise for the right intervals..., and let fk analogue for the kth sets. All these maps are continuous, because they are continuous on disjoint closed intervals.
Figure 3. Examples of the maps Now, define gk as the restriction of fk that maps only C to Ck. Since the domain C is a subset of all Ck for k ∈N, we derive that gk is continuous for all k ∈N. We want to show that these gk converge uniformly to some map g. The range of g will have to be the intersection of all C′ k, so C′.
Denote Mk as the one among 2k intervals of C′ k with the maximum length.
Since the Cantor sets are always nowhere dense, we deduce that limk→∞Mk = 0.
Specifically, if the value of limk→∞Mk = 0 is positive, then a subset of the Cantor set contains at least one interval, contradicting that its closure has empty interior.
Now denote Nk as the supremum of |gk −gm| on the entire unit interval for any m > k. Therefore, given ϵ > 0, there always exists K such that for all k > K, |gk −gm| ≤Nk ≤Mk < ϵ for all x ∈C and m > k. The sequence is Cauchy in the uniform norm, so it uniformly converges to the desired function g.
CANTOR SETS IN TOPOLOGY, ANALYSIS, AND FINANCIAL MARKETS 7 We have successfully shown that gn converge uniformly to g. g is, therefore, a continuous map from C to C′. By the same token, we are able to construct a continuous map from C′ to C by simply reversing the positions of C and C′ and keeping all other aspects of our argument the same.
As a result, in order for there to be a continuous bijection between C and C′, we only need to prove that the map from C to C′ is bijective. In fact, each element in C and C′ can be considered as an infinite sequence of L and R, where L stands for choosing the left interval and R stands for choosing the right interval in a given iteration. A bijective map between a point in C and one in C′ can be established if they have identical L/R sequences, but it is easy to see that gk gives these identical sequences for the first k L/R choices, so g itself gives the desired mapping for all of the infinitely many L/R choices.
Therefore, we have proven that Cantor sets C and C′ are homeomorphic to each other. Since C and C′ can be any arbitrary Cantor sets, we deduce that all the Cantor sets are homeomorphic to each other.
□ 3. Applications in measure theory 3.1. The measure of Cantor sets.
3.1.1. Definition of Lebesgue measure. Now that we have discussed the topological properties of Cantor sets, it is fundamental question also to ask how ”big” they are.
This idea is trivial for finitely many disjoint intervals – just add up the lengths – yet in the infinite case is somewhat more complicated. The concept of the Lebesgue measure, one particularly useful type of measure in mathematics, is basically the total length of the shortest possible intervals that encapsulate a given subset. A full discussion of this measure is beyond the scope of this paper, but it suffices to note that it gives a more rigorous notion of size to sets.
The Lebesgue measure on R satisfy the following properties: 1. m(A) ≥0 2. m(∅) = 0 3. m([a, b]) = b −a 4. It is countably additive. Namely, for all countable collections {Ek}∞ k=1 of pairwise disjoint sets in Σ, m ∞ [ k=1 Ek !
= ∞ X k=1 m (Ek) As an immediate consequence of properties 1 and 4, if A ⊆B then m(A) ≤m(B).
It is easy to check that this consequence along with property 3 implies that points have measure 0, and in fact, countable sets also have measure 0.
We can also calculate the measure of the standard Cantor set.
3.1.2. The standard Cantor set. Since we remove the middle 1/3 of each remaining interval in each iteration, the Lebesgue measure of Cn is (2/3)n (2n intervals each of length 3−n). Each Cn contains C, so the measure of C is no larger than that of any Cn. Taking the limit of it as n goes to infinity gives us zero, which is a fairly counterintuitive result: countable sets all have measure zero, but the Cantor set gives an example of a set that is uncountable and also measure zero.
Now, generalizing the standard Cantor set can lead to even more counter-intuitive results. We begin with a theorem.
8 ZIQIAN (ALEXA) JIN Theorem 9. There exists a nowhere dense set with positive measure.
This theorem can be illustrated by the following category of Cantor sets.
3.1.3. Fat Cantor sets. Instead of removing a constant portion of the original set in each iteration, fat Cantor sets are created by removing progressively smaller portions of the original set in each step such that the ratio of what is being removed to the interval it is being removed from goes to 0 as n goes to infinity.
Ex: remove the middle (1/k)n of Cn−1, where k > 3.
Figure 4. Example of a fat Cantor set Unlike the standard ternary cantor sets, these fat Cantor sets have a positive measure, which is odd because they are nowhere dense and don’t contain even one interval. Take the example mentioned earlier that removes the middle intervals of lengths (1/k)n from Cn−1, k > 3.
The Lebesgue measure of the removed intervals = 1/k + (1/k)2 ∗2 + (1/k)3 ∗4 + ...
= 1/2 ∗(2/k + (2/k)2 + (2/k)3 + ...) = 1/2 ∗2/k ∗1/(1 −2/k) = (1/k) ∗(k/k −2) = 1/(k −2).
Therefore, the Lebesgue measure of the corresponding fat Cantor set is (k−3)/(k− 2) An example of the fat Cantor set is the Smith–Volterra–Cantor set (SVC): k = 4 in this case, and its Lebesgue measure is 1/2.
3.2. Dimension.
3.2.1. Definition. In mathematics, the notion of fractional dimension, an intrinsic property of a set, is an extension of the idea that a line is one-dimensional, a plane is two-dimensional, and space is three-dimensional. First, let us explore one way to approach how the dimensions of, say, a line segment and a rectangle are defined.
A line segment has dimension 1, because as we stretch it to twice its origi-nal length, its ’substance’–length–doubles as well. In the case of a rectangle, if we stretch all sides to twice their original scales, its substance–namely the area– quadruples. Taking the logarithm of 4 over 2 gives us 2.
Put in an equation, we can write that S1/S2 = SD where S1 is the new substance, S2 is the old substance, S stands for the stretch, and D is the dimension. The dimension is the exponent by which the size changes when scaled by a certain amount.
As in the two examples above, you might expect that only integer dimensions are taken. As will be shown below, however, the dimension of mathematical objects are not necessarily integers and can take on many arbitrary values.
CANTOR SETS IN TOPOLOGY, ANALYSIS, AND FINANCIAL MARKETS 9 Figure 5. dimensions of line segments and squares 3.2.2. Dimension of Cantor sets.
Theorem 10. The dimension of the standard Cantor set is equal to log(2)/log(3) = 0.631.
Proof. Because of its self-similar nature, the left third of the object is the exact replica of it as a whole and is of 1/3 of its original scale. So, the standard Cantor set is just the union of two smaller identical Cantor sets. When you scale by a factor of 3, the ’substance’ of the set doubles, because you now have two copies identical to the original. Hence, if the dimension is d, then we may write 3d = 2, giving d = log2/log3 □ Cantor sets serve as some of the easiest examples of objects of non-integer dimen-sion, and in fact, one can slightly modify the construction of the standard Cantor set to achieve a dimension equal to any value in (0, 1).
Theorem 11. For any α ∈(0, 1), there is a set of dimension α Proof. Let’s start with the unit interval as usual. If we take out the middle 1/k of each existing interval in each iteration, then the left (1 −1/k)/2 of the set is the exact replica of it as a whole and is of 1/2 of its original scale. Then: dim = log(2)/log(2/(1 −1/k)) = log(2)/(log(2) −log(1 −1/k)) The dimension approaches 0 as k approaches 1, and it approaches 1 as k ap-proaches infinity. Since k can be any number greater than 1 in R, and the expres-sion for dimension is continuous, we know that there are sets of any dimension from 0 to 1.
□ In fact, we can actually achieve a dimension of 0 using a Cantor set, even though all of the above sets have positive dimension.
Theorem 12. There exists an uncountable set of dimension zero This theorem can be illustrated by another type of generalized Cantor sets called the thin Cantor sets. First let’s have the definition.
10 ZIQIAN (ALEXA) JIN 3.2.3. The Thin Cantor set. The thin Cantor set is created, contrary to the fat counterparts, by removing progressively larger portion of the original set in each step. An example would be removing the middle (1-1/n) of Cn−2 in each iteration.
Similar to the standard Cantor set, the measure of thin Cantor sets is zero. Now, we want to show that the dimension of thin Cantor sets is also zero Proof. Suppose we remove the middle (1-1/n) of Cn−2, denoting Mn = 1 −1/n.
As n goes to infinity, Mn goes to 1. If we have any Cantor set used in the previous theorem, for n sufficiently large, more of each interval is being deleted in the thin Cantor set after step n than the constant ratio being deleted. Thus, the dimension of a thin Cantor set cannot be larger than the dimension of any of the middle 1/k sets, which means the dimension has to be 0, since it is not larger than any number in (0, 1).
□ In a similar manner, it can be argued that fat Cantor sets have dimension 1.
4. Applications in analysis 4.1. Cantor functions. The Cantor function is an example of a function that is continuous, even uniformly continuous, but fails to satisfy the stronger definition of absolute continuity.
It is a notorious counterexample in analysis, because it challenges our intuitions about continuity, derivative, and measure.
The standard Cantor function, or the Cantor ternary function c : [0,1] →[0,1] is defined as follows: 1. Express x in base 3.
2. If x contains a 1, replace every digit strictly after the first 1 by 0.
3. Replace any remaining 2s before the 1 with 1s.
4. Interpret the result in binary.
The result is a ladder-like non-decreasing function that exhibits a point symmetry across (1/2, 1/2).
To find the values of individual points in the domain, for example, let x = 0.25.
Then x = 0.020202... in base 3. Since there is no 1 in the ternary representation, CANTOR SETS IN TOPOLOGY, ANALYSIS, AND FINANCIAL MARKETS 11 replacing all 2s with 1s gives us x’ = 0.010101... Now, we convert it back from base 2 to base 10. x’ = 1/4 + 1/16 + 1/64 +... = 1/4/(1-1/4) = 1/3 4.1.1. Continuity.
Theorem 13. The Cantor function is uniformly continuous.
Proof. Given ϵ > 0, suppose n is the smallest integer such that ϵ > 1/2n. Let δ = 1/3n. Then given x0 ∈D = [0, 1], we have that for any x ∈(x0 −δ, xo + δ), |c(x) −c(x0)| < 1/2n = ϵ. Therefore, the Cantor function is uniformly continuous.
□ 4.1.2. Absolutely Continuous.
Definition 4.1 (Absolute Continuity). Let I be an interval in the real line R.
A function f: I to R is absolutely continuous on I if for every positive number ϵ there is a positive number δ such that whenever a finite sequence of pairwise disjoint sub-intervals (xk, yk) of I with xk < yk ∈I satisfies P k(yk −xk) < δ then P k |f(yk) −f(xk)| < ε Proof. Pick ϵ < 1. Since the standard Cantor set has measure zero, for every δ > 0, we can find a collection of intervals (xk, yk) that cover the points in the standard Cantor set such that P |xk −yk| < δ However, since the Cantor function only changes on the Cantor set, we have P |c(xk) −c(yk)| = 1, contradicting that ϵ < 1.
The standard Cantor function is therefore not absolutely continuous.
□ 4.1.3. Differentiation. The Cantor function has zero derivative on CC (the com-plement of the Cantor Set on the interval (0, 1)) and is not differentiable on the ternary Cantor set Proof. The first half of the theorem is trivial, since the Cantor function is constant on the open set CC. The second half can be proved as follows.
given x ∈C: Case 1: x has a finite number of digits in base 3. Denote that number as n. Let h = 2/3k where k > n. Then limk→∞h = 0. So limh→0(c(h + x) −c(x))/h = limk→∞(c(h + x) −c(x))/h = (1/2k)/(2/3k) = limk→∞3k/2k+1 = ∞. Similarly, it can be proved that the limit does not exist when h is negative.
Case 2: x has an infinite number of digits in base 3. Let y be the first k digits of x, then y is also in C. Then limx→y(c(x)−c(y))/x−y = limk→∞(c(x)−c(y))/x−y > limk→∞(1/2k+1)/(2/3k+1) = limk→∞3k+1/2k+2 = ∞ □ 4.2. Volterra’s Function. The Cantor function shows us some of the limits of the fundamental theorem of calculus that relates differentiation and integration. Going a step further, through one of the fat Cantor sets, Italian mathematician Vito Volterra (1860-1940) constructed a function that is differentiable with bounded derivative, but whose derivative is not integrable. More details on the construction can be found in , but here we provide an overview.
Volterra defined the function F : [0, 1] →R as follows, where C is the Smith-Volterra Cantor set defined on page 8: F(x) = fa,b(x), if x ∈(a, b) for some interval(a, b) ⊂[0, 1]\C 0, if x ∈C 12 ZIQIAN (ALEXA) JIN Figure 6. Volterra’s Function Here, we would like to pick our endpoints a and b so they match exactly those intervals removed in the construction of the SVC set. Thus, each ”hole” in that set has exactly one function placed in it.
Now, we need to specify the function we are filling the gaps with. fa,b : [a, b] →R is defined such that: 1. fa,b(x) = 0, for x = a or x = b 2. fa,b(x) = (x −a)2 sin 1 x−a , for a < x ≤x1, where x1 is the largest number less than or equal to a+b 2 for which (x −a)2 sin 1 x−a has maximum value.
3. fa,b(x) = (x1 −a)2 sin 1 x1−a = (b −x2)2 sin 1 b−x2 , if x1 ≤x ≤x2.
4. fa,b(x) = (b −x)2 sin 1 b−x , if x2 ≤x < b, where x2 is the smallest number greater than or equal to a+b 2 for which (b −x)2 sin 1 b−x has maximum value.
The basic idea about the Volterra function is we use a special property of the function g(x) = x2 sin(1/x). Namely, if we require that g(0) = 0, this function is differentiable everywhere, but its derivative 2x sin(1/x)−cos(1/x) is not continuous at 0. With the above construction, we force, near every point in the SVC, the derivative to be discontinuous.
For the endpoints of the removed intervals, we basically have a copy of the function g around them, so clearly F ′ is discontinuous there. There is a slight subtlety here, because not every point in the SVC is an endpoint, but endpoints are dense in Cantor sets, so the function will still fail to be continuous on all of the Smith-Volterra Cantor set, if we consider approaching any point through successively closer endpoints.
Thus, we have that F ′ fails to be continuous on C, a set of positive measure, so by the Riemann-Lebesgue lemma, it is not Riemann integrable. This is why we needed to specifically use a Cantor set of positive measure.
5. Applications in the real world: fractal phenomena Because of its self-similar nature, the standard Cantor set is the prototype of a fractal. In fact, a well established mathematical branch, fractal geometry is widely applied to study patterns and phenomena in various aspects of our lives, and in this paper, we picked three examples with the closest relationships to the Cantor sets.
5.1. Fractal Geometry in Nature. Among the numerous fractal structures ob-served in nature–spirals, tree branches, snow flakes– Saturn’s rings have a special relationship to the Cantor sets.
Note the different sizes in the gaps of Saturn’s rings in Figure 7, which look like the intervals removed from a Cantor set. The figure on the right consists of the product of fat Cantor set and a circle. The fat Cantor set has positive measure, CANTOR SETS IN TOPOLOGY, ANALYSIS, AND FINANCIAL MARKETS 13 Figure 7.
Left: Saturn’s rings (NASA). Right: a product of a Cantor set and a circle. so it’s product with a circle should have positive area. Thus, the fat Cantor set specifically provides an interesting comparison with Saturn, because if the rings’ cross section were a different Cantor set of zero area, the rings would have almost zero area to reflect light and so would be almost invisible. 5.2. Mandelbrot and the Fractal Market. Compared to unambiguous self-similar patterns in art and nature, the applications of Cantor sets and fractals to the financial world come in a more subtle way. Mathematician Benoit Mandelbrot (1987) once compared markets to turbulent seas in his “Ten Heresies of Finance,” where he argues that :” the very heart of finance is fractal.” In discussing the applications of fractals to analyzing markets, he states that the simplest fractals scale the same way in all directions, hence are called self-similar. If the fractals scale in many different ways at different points–the exact reality of the markets ...
their mathematical properties become intricate and powerful.” Figure 8. Bitcoin Price, for example, in the past 18 hours, with an estimated Hurst Coefficient of around 0.4-0.5 The comparisons with nature lead to idea that financial markets are similar to the behavior of various natural phenomena in the world. The history of shifts from classical to modern views on the market modeling were outlined, visualizing some 14 ZIQIAN (ALEXA) JIN Figure 9. Turbulent sea misconduct of classical approach to market modeling and providing examples of utilizing the fractional approach.
5.3. The fractal market hypothesis.
5.3.1. An alternative to EMH. For decades, the Efficient Market Hypothesis has been the dominant foundation for the modeling of financial markets. It states that stocks always trade at their fair value on exchanges, making it impossible for in-vestors to purchase undervalued stocks or sell stocks for inflated prices. The core idea of the Efficient Market Hypothesis lie in the observation that stock prices ex-hibit random walks, which can be modeled by something called geometric Brownian motion. Modeling with geometric Brownian motion suggests that the percentage change of a stock price in a given future time interval is completely independent of its previous prices. Furthermore, the distribution of the percentage changes after a given time has passed t should be normally distributed, with variance proportional to t.
The model of geometric Brownian motion is useful, but not perfect. For instance, one can modify it by adding a ”drift” term to capture the reality that stock prices tend to increase over time. A more core issue, though, is the idea of fat tails, which reflect the disproportionate influence of rare events on the economy. The reality of fat tails has laid the foundation of a new theory – the “Fractal Market Hypothesis”.
One of the central arguments in the fractal marker hypothesis is that the fre-quency distribution of returns looks the same at different investment horizons, which is the total length of time that an investor expects to hold a security or a portfolio. The longer-term horizons are based more upon fundamental information, and shorter-term investors base their views on more technical information. As long as the market maintains this fractal structure, with no characteristic time scale, the market remains stable because each investment horizon provides liquidity to the others.
As a result, the geometric Brownian motion, as a stochastic process to model stock movements in EMH according to the Black–Scholes model, can be potentially replaced by the fractional Brownian motion with a special parameter Hurst coeffi-cient ”H”. For self-similar time series, H is directly related to fractal dimension, D, CANTOR SETS IN TOPOLOGY, ANALYSIS, AND FINANCIAL MARKETS 15 where 1 < D < 2, such that D = 2 −H. Increments are independent only when H = 1/2, for H > 1/2, increments are positively correlated and for H < 1/2 they are negatively correlated. The values of the Hurst exponent vary between 0 and 1, with higher values indicating a smoother trend, less volatility, and less roughness.
Figure 10. Patterns corresponding to different H coefficients An intriguing point where Cantor sets come into play is when we observe the level set of a one dimensional fractional Brownian motion mentioned above. Assuming in case of figure 8 that the H index of bitcoin price is around 0.5 (0.495 ± 0.102) according to , the chance of the stock price going up or down is close to random.
Therefore, any typical level set (e.g. the red line in figure 7) is a closed, perfect set resulting from the properties of Brownian motions (both fractional and geometric) . Furthermore, the level sets will almost surely not contain any intervals, meaning with the above properties that they must be Cantor sets.
Acknowledgments It is a pleasure to thank my mentor Jake Fiedler for introducing to me the unfamiliar topic of measure theory, discussing and editing my paper, and giving me readings that I found really interesting. I would also like to thank Daniil Rudenko for his engaging lectures on group theory, combinatorics and projective geometry.
Lastly, I want to thank Peter May for organizing the REU program - it has been a great experience.
References Benoit B. Mandelbrot (Author), Richard L. Hudson. The Misbehavior of Markets A Fractal View of Risk, Ruin, and Reward. Profile Books. 2010.
16 ZIQIAN (ALEXA) JIN Walter Rudin, Principles of Mathematical Analysis. McGraw-Hill. 1976.
Edgar E. Peters, Fractal Market Analysis: Applying Chaos Theory to Investment and Eco-nomics. Wiley. 2009.
Juan Carlos Ponce-Campuzanoa, Miguel ´ Angel Maldonado-Aguilar. Vito Volterra’s construc-tion of a nonconstant function with a bounded, non-Riemann integrable derivative. Taylor Francis. 2015.
L. DECREUSEFOND and A.S. ¨ UST¨ UNEL. Stochastic Analysis of the Fractional Brownian Motion. 1997.
Mariusz Tarnopolski. Modeling the price of Bitcoin with geometric fractional Brownian motion: a Monte Carlo approach. 2017. |
10952 | https://www.youtube.com/watch?v=2-HvWEI2Uvw | Finding the Area of Triangles In square Units
Becky Seres
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Finding the area of rectangle, parallelogram and triangles measured in square units.
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hi great six welcome today to today's lesson we're going to be continuing with area and practicing finding the area of a variety of different shapes that we've learned so far I wanted to start with this example today because it looks a little bit different than some of the other examples that we've done so far as you can see we've got a rectangle here and we have a parallelogram over here you'll notice that the dimensions are not labeled but they are actually in square units okay so we're going to have to find the area just by Counting our square units today so we're going to determine which shape has the greater area do we think it is a rectangle or do we think it is the parallels are a I'm still going to proof so I'll start with a rectangle and I'm going to start with my formula area equals base times height okay you can also do length times width but I like to use base times height so my area is so my base is one two three units okay so it is three times my height I'm going to count one two three four five six seven eight so it is three times eight area equals 24 we don't know the measurement so we're just gonna write units squared okay now over to the parallelogram again my formula is the same for parallelogram as it is for rectangle very equals base times height so let's count my base one two three three okay now when finding the height you want to cut through the parallelogram just like you do for the triangle so I'm going to draw a line that cuts through so I can see what my height is here there's my height okay notice how this height is not the same as the side length you don't want to take the side length as the height okay so my height is if you count it one two three four five six seven it is also eight so they have the same area 24 units squared even though they look a little bit different their area is exactly the same okay next example you're going to come across this one we won't answer the whole thing's you're going to come across it when you do your knowledge hook admission you were asked which triangles all have an area of eight square units so you're going to need to find the area of all of these different triangles okay so we'll do one together so let's do a so when finding the area of triangles what is the formula that we use hopefully you're thinking area equals base times height divided by two we have to remember that we need to divide by two so I've written mind here like that so now I'm going to fill in my numbers okay so let's have a look what is my base one two three four it has four square units across okay now my height I'm gonna get out my line tool because I find that this is helpful oops and I'm going to cut through my triangle so I can see the height and I'm going to count the squares so that would be one two three four okay let's see so area equals 16 I think then we don't have the measurement so we're just going to use units okay so you have a chart to fill out oh wait I didn't do it right I forgot to divide by two 16 divided by two is area equals eight units squared so I can include a as one of the triangles that has an area of square unit so you will continue this question so we can put a equals eight minutes or don't forget to the bribe by two I just about good it's an easy mistake to make okay guys let me know if you have any questions underneath the assignment I am here to help you today okay bye bye |
10953 | https://billcookmath.com/papers/2021-10-Synthetic_Division.pdf | The Electronic Journal of Mathematics and Technology, Volume 14, Number 1, ISSN 1933-2823 Synthetic Division: Connecting with Other Mathematical Ideas Katrina Palmer, palmerk@appstate.edu Jaehee Post, postjk@appstate.edu Michael Boss´ e, bossemj@appstate.edu William Bauldry, bauldrywc@appstate.edu William Cook, cookwj@appstate.edu Mathematical Sciences Department Appalachian State University 28608 USA Abstract Synthetic division, as developed by Ruffini in 1804, was limited to division of a polynomial by a linear polynomial factor in the form 𝑥−𝑐. Connected to Ruffini’s method, in the early 1800’s, Horner developed techniques for finding roots and determining the derivatives of polynomials.
Additionally, Horner expanded Ruffini’s method of synthetic division so that a polynomial could be divided by polynomials of higher degree than just 1. Some high school and college students have used synthetic division to divide by a linear polynomial factor. However, few students may know why and how this process works. The increasing use of computers and calculators with algebraic operating systems may further hide the beauty of synthetic division. This paper: (A) describes how synthetic division can be used in the contexts of integer, rational, real, and complex divisors; (B) investigates Horner’s method of synthetic division to divide by polynomials of any degree; and (C) makes connections with the Remainder Theorem and the Zero Product Property. This paper provides the reader with students investigations and an applet for performing polynomial division.
1 Introduction & Leading Questions We introduce this discussion of synthetic division through some investigatory questions. (Answers to these questions appear below.1) Consider the polynomial 𝑃(𝑥) = 2𝑥6 + 3𝑥5 −𝑥4 −3𝑥3 −5𝑥2 −6𝑥−2.
1. What are all possible rational roots for 𝑃(𝑥)? (Hint: consider the Rational Roots Theorem.) 2. What are all the possible real roots for 𝑃(𝑥)?
The Electronic Journal of Mathematics and Technology, Volume 14, Number 1, ISSN 1933-2823 3. Is it possible for 𝑃(𝑥) to have complex roots? Why or why not?
4. What can you deduce from the following synthetic division?
2 −3 4 −5 6 2 ↓ 4 2 12 14 2 1 6 7 20 5. What can you deduce from the following synthetic division?
2 3 −1 −3 −5 −6 −2 𝑖 ↓ 2𝑖 −2 + 3𝑖 −3 −3𝑖 3 −6𝑖 6 −2𝑖 2 2 3 + 2𝑖 −3 + 3𝑖 −6 −3𝑖 −2 −6𝑖 −2𝑖 0 (Web Example #2 on method.html.) 6. Look at the entire multi-step synthetic division. What does it tell us?
2 3 −1 −3 −5 −6 −2 𝑖 ↓ 2𝑖 −2 + 3𝑖 −3 −3𝑖 3 −6𝑖 6 −2𝑖 2 2 3 + 2𝑖 −3 + 3𝑖 −6 −3𝑖 −2 −6𝑖 −2𝑖 0 −𝑖 ↓ −2𝑖 −3𝑖 3𝑖 6𝑖 2𝑖 2 3 −3 −6 −2 0 √ 2 ↓ 2 √ 2 4 + 3 √ 2 6 + √ 2 2 2 3 + 2 √ 2 1 + 3 √ 2 √ 2 0 − √ 2 ↓ −2 √ 2 −3 √ 2 − √ 2 2 3 1 0 −1 2 ↓ −1 −1 2 2 0 −1 ↓ −2 2 0 1 Solutions to the Introductory Questions 1. The rational root theorem (source) says that the only possible rational roots are 𝑥= ±factors of constant term factors of leading term .
Thus the possible rational roots are 𝑥= −1, −2, 1, 2.
2. To determine the possible real roots, you could use Descartes rule of signs (source). Because 𝑃(𝑥) is a sixth degree polynomial, we know that there are either 6,4,2, or 0 real roots since complex roots come in pairs. Graph 𝑃(𝑥) with Desmos.
3. 𝑃(𝑥) may have complex roots because the Fundamental Theorem of Algebra gives 6 roots and we know there are at most 4 rational roots, therefore there must be at least 2 real roots or a complex conjugate pair of roots.
4. The synthetic division demonstrates that 𝑥−2 is not a factor of 𝑃(𝑥) and the division 2𝑥4−3𝑥3+4𝑥2−5𝑥+6 𝑥−2 produces a remainder of 20 𝑥−2. Additionally it reveals that 𝑃(2) = 20.
5. The synthetic division shows that 𝑥−𝑖is a factor of 𝑃(𝑥) because the remainder term is 0, and thus 𝑃(𝑖) = 0.
6. The entire multi-step synthetic division shows that 𝑃(𝑥) = 2(𝑥+ 1)(𝑥+ 1 2)(𝑥+ √ 2)(𝑥− √ 2)(𝑥+ 𝑖)(𝑥−𝑖).
2 The Electronic Journal of Mathematics and Technology, Volume 14, Number 1, ISSN 1933-2823 These initial questions lead to both observations and additional questions. First, most high school and college students have only used synthetic division to divide by a linear factor involving an integer (e.g., dividing by 𝑥−2). In questions 4 and 5, we see that synthetic division can be used for integer, rational, irrational, real and even complex roots.
Second, since in questions 5 and 6 each line of synthetic division results with a remainder of zero (we will later consider the Remainder Theorem), we know that 𝑃(𝑥) = 2𝑥6+3𝑥5−𝑥4−3𝑥3−5𝑥2−6𝑥− 2 = 2(𝑥+1)(𝑥+ 1 2)(𝑥+ √ 2)(𝑥− √ 2)(𝑥+𝑖)(𝑥−𝑖) which can be written as 2(𝑥2+1)(𝑥2−2)(𝑥+ 1 2)(𝑥+1).
Before moving on, let us consider one more possibly unexpected example:2 𝑥3 + 𝑥2 −𝑥+ 15 𝑥2 −2𝑥+ 5 = 𝑥3 + 𝑥2 −𝑥+ 15 𝑥−(1 −2𝑖) 𝑥−(1 + 2𝑖) 1 1 −1 15 1 −2𝑖 ↓ 1 −2𝑖 −2 −6𝑖 −15 1 2 −2𝑖 −3 −6𝑖 0 1 + 2𝑖 ↓ 1 + 2𝑖 3 + 6𝑖 1 3 0 Now we can see that synthetic division can be performed with complex roots. But we should still note that the complex number 1 + 2𝑖is a single complex coefficient and not two coefficients.
With these observations now in place, in the following sections we can investigate how synthetic division works and whether synthetic division is limited to dividing by linear factors.
2 Investigating and Extending Synthetic Division Students are often introduced to synthetic division as shorthand notation for dividing a polynomial by a linear factor . Figure 1 shows an example of long division (2𝑥4 −3𝑥3 + 4𝑥2 −5𝑥+ 6 divided by 𝑥−2) next to its associated form of synthetic division.3 Comparing the two forms illuminates from whence the values in the synthetic division evolve.
2Web Examples #3, #3A, and #3B on
html 3Web Example #4 on 3 The Electronic Journal of Mathematics and Technology, Volume 14, Number 1, ISSN 1933-2823 2𝑥3 +1𝑥2 +6𝑥 +7 𝑅20 𝑥−2 2𝑥4 −3𝑥3 +4𝑥2 −5𝑥 +6 −(2𝑥4 −4𝑥3) ↓ ↓ ↓ +1𝑥3 +4𝑥2 ↓ ↓ −(𝑥3 −2𝑥2) ↓ ↓ 6𝑥2 −5𝑥 ↓ −(6𝑥2 −12𝑥) ↓ 7𝑥 +6 −(7𝑥 −14) 20 2 −3 4 −5 6 2 ↓ 4 2 12 14 2 1 6 7 20 = 𝑓(2) (a) long division (b) synthetic division Figure 1: Side-by-side example showing both long division and synthetic division Thus, as we will later see through the Remainder Theorem, 𝑓(2) = 20, and 𝑓(𝑥) = (2𝑥3 + 𝑥2 + 6𝑥+ 7) · (𝑥−2) + 20.
Below are two figures. One demonstrates a looping algorithm which takes place in the process of long division. The other demonstrates an analogous looping procedure in the process of synthetic division. With some observation, one can see that these looping techniques share much in common.
𝑥+ 3 𝑥−1 𝑥2 + 2𝑥+ 1 −(𝑥2 − 𝑥) 0 + 3𝑥+ 1 −(3𝑥−3) 0 + 4 1 1 2 1 1 3 1 3 4 Figure 2: Looping algorithms for long division and synthetic division.
Possibly to the surprise of many students, this method can also be used to divide by linear factors whose leading coefficient is other than 1. For example,4 let us consider 3𝑥3 +2𝑥−4 divided by 2𝑥−3.
First, notice that we can write the factor 2𝑥−3 into the form 𝑥−3 2. Let us first perform synthetic division by 𝑥−3 2, since this seems to be in the more familiar form with a leading coefficient of 1.
3 0 2 −4 3 2 ↓ 9 2 27 4 105 8 3 9 2 35 4 73 8 4Web Example #5 on 4 The Electronic Journal of Mathematics and Technology, Volume 14, Number 1, ISSN 1933-2823 Let us now do the synthetic division in a slightly modified form by 2𝑥−3.
3 0 2 −4 2 ↓ 9 2 3 ↓ ↓ 27 4 3 2 9 4 35 8 73 8 The details of this process are explained in Section 2.1.
STOP. Let’s be certain to understand the work in the last example; this technique leads to foun-dational ideas for the remainder of this paper. First, notice that division by 𝑥−3 2 and 2𝑥−3 produced the same results. Second, notice that in all previous examples of synthetic division prior to division by 2𝑥−3, only the constant term is on the outside left. In the last example with division by 2𝑥−3, we have two terms on the outside left.
The question which now naturally arises is, ”If we can have either one or two terms on the outside left of the synthetic division, might we be able to have more?” And, if more, is there a limit? And, if we have 𝑎on the outside left, that means we are dividing by 𝑥−𝑎. If we have 𝑎and 𝑏on the outside left, we are dividing by 𝑎𝑥−𝑏. But what does it mean if we have 𝑎, 𝑏, and 𝑐on the outside left, what does this mean that we are dividing by: 𝑎𝑥2 + 𝑏𝑥+ 𝑐, 𝑎𝑥2 −𝑏𝑥+ 𝑐, 𝑎𝑥2 −𝑏𝑥−𝑐, or some other form?
Excuse me? Did we just imply that we can perform synthetic division by a linear or quadratic polynomial, and, by implication, possibly any real polynomial of any degree? Let us examine one example of polynomial division by a sparse5 quartic polynomial. This example uses multiple coeffi-cients of 0 and 1 in the divisor to make the process more transparent and illuminating. We encourage the reader to carefully examine this example to determine from where values in the synthetic division evolve. Consider 7𝑥6 + 6𝑥5 + 4𝑥3 −1 𝑥4 + 1 , leading to the following synthetic division.6 7 6 0 4 0 0 −1 1 ↓ 0 0 0 −7 0 ↓ ↓ 0 0 0 −6 0 ↓ ↓ ↓ 0 0 0 0 0 ↓ ↓ ↓ ↓ ↓ ↓ ↓ −1 ↓ ↓ ↓ ↓ ↓ ↓ ↓ 7 6 0 4 −7 −6 −1 2.1 Expanding the Process in Order to Understand the Compressed Process In the following example (Figure 2.1), each step in the expanded synthetic division technique is taken one at a time. While this makes the example quite lengthy, it should further develop the notion of the technique being used. Also, within the example, products and sums are shown to better demonstrate 5a polynomial with a relatively high number of zero coefficients 6Web Example #6 on 5 The Electronic Journal of Mathematics and Technology, Volume 14, Number 1, ISSN 1933-2823 the process. Our example7 is: 6𝑥5 −5𝑥4 + 4𝑥3 −3𝑥2 + 2𝑥−1 2𝑥2 −𝑥+ 3 .
First set up the synthetic division by putting the coefficients of the divisor across the top (in blue) and the coefficients of the dividend vertically (in red). If the numerator is degree 𝑛and the denominator is degree 𝑚, then, including the column to the left for the divisor, there will be 𝑛+ 1 columns. Including the top row for the dividend and the bottom row for the solution or quotient, there will be 𝑚+ 2 rows. Additionally, the vertical line separating the quotient from the remainder will be after the (𝑛−𝑚+ 1)𝑡ℎnumber across the top. Notice that the leading coefficient of the dividend remains the same (that is 2) while the signs of the non-leading coefficients are changed (to 1 and -3). When you bring down the 6, the leading coefficient of the dividend, you divide by the leading coefficient of the divisor, 2. Next, multiply the 3 by the non-leading coefficients of the dividend.
Then add down the next column and divide by the dividend’s leading coefficient again. Continue this process for a total of 4 times, and thus, 6𝑥5 −5𝑥4 + 4𝑥3 −3𝑥2 + 2𝑥−1 2𝑥2 −1𝑥+ 3 = 3𝑥3 −1𝑥2 −3𝑥−3 2 + 19 2 𝑥+ 7 2 2𝑥2 −1𝑥+ 3.
6 −5 4 −3 2 −1 2 ↓ 1 ↓ −3 ↓ ↓ 6 2 = 3 6 −5 4 −3 2 −1 2 ↓ (3)(1) (3)(−3) 1 ↓ −3 ↓ ↓ 3 (a) Set up & Step 1 (b) Multiply 3 by 1 and -3 6 −5 4 −3 2 −1 2 ↓ 3 −9 1 ↓ ↓ −3 ↓ ↓ ↓ ↓ 3 −5+3 2 = −1 6 −5 4 −3 2 −1 2 ↓ 3 −9 1 ↓ ↓ (−1)(1) (−1)(−3) −3 ↓ ↓ ↓ ↓ 3 −1 (c) Add down column 2 and divide by 2 (d) Multiply -1 by 1 and -3 7Web Example #7 on 6 The Electronic Journal of Mathematics and Technology, Volume 14, Number 1, ISSN 1933-2823 6 −5 4 −3 2 −1 2 ↓ 3 −9 1 ↓ ↓ −1 3 −3 ↓ ↓ ↓ ↓ ↓ ↓ 3 −1 4−9−1 2 = −3 6 −5 4 −3 2 −1 2 ↓ 3 −9 1 ↓ ↓ −1 3 −3 ↓ ↓ ↓ (−3)(1) (−3)(−3) ↓ ↓ ↓ 3 −1 −3 (e) Add down column 3 and divide by 2 (f) Multiply -3 by 1 and -3 6 −5 4 −3 2 −1 2 ↓ 3 −9 1 ↓ ↓ −1 3 −3 ↓ ↓ ↓ −3 9 ↓ ↓ ↓ ↓ 3 −1 −3 −3+3−3 2 = −3 2 6 −5 4 −3 2 −1 2 ↓ 3 −9 1 ↓ ↓ −1 3 −3 ↓ ↓ ↓ −3 9 ↓ ↓ ↓ ↓ (−3 2)(1) (−3 2)(−3) 3 −1 −3 −3 2 (g) Add down column 4 and divide by 2 (h) Multiply −3 2 by 1 and -3 6 −5 4 −3 2 −1 2 ↓ 3 −9 1 ↓ ↓ −1 3 −3 ↓ ↓ ↓ −3 9 ↓ ↓ ↓ ↓ −3 2 9 2 3 −1 −3 −3 2 6 −5 4 −3 2 −1 2 ↓ 3 −9 1 ↓ ↓ −1 3 −3 ↓ ↓ ↓ −3 9 ↓ ↓ ↓ ↓ −3 2 9 2 3 −1 −3 −3 2 19 2 7 2 (i) Simplify row 5 (j) Add down columns 5 & 6 Figure 3: Detailed example of synthetic division with a nonlinear dividend Before formalizing this process in the next section, we provide one more example for the reader to investigate. We recommend that you try this initially on your own and then look at the pattern and solution in the figure to ensure that you did the process correctly and got the right answer. Try:8 𝑥4 + 𝑥2 −𝑥+ 5 𝑥2 + 2𝑥+ 3 .
1 0 1 −1 5 1 ↓ −2 −3 −2 ↓ ↓ 4 6 −3 ↓ ↓ ↓ −4 −6 1 −2 2 1 −1 Additional examples can be seen in . and In the appendix, we provide problems for student investigations regarding synthetic division.
These problems are developed to both deepen the student’s understanding of synthetic division and to help them better understand the material in this paper.
8Web Example #8 on 7 The Electronic Journal of Mathematics and Technology, Volume 14, Number 1, ISSN 1933-2823 3 Horner’s Method of Synthetic Division and an Application Without specifically naming such, we have already investigated Horner’s Method of synthetic divi-sion. Notably, beyond traditional synthetic division, Horner’s Method allows division by polynomials of any degree. However, Horner’s Method has a number of applications such as root finding [4, 6, 7] and determining derivatives [5, 2] . We will consider only polynomial division here.
For any polynomial 𝑃(𝑥), there are a calculable number of operations necessary to evaluate 𝑃(𝑥) at some value 𝑟, of 𝑃(𝑟). Let us first cleverly factor a given polynomial using Horner’s Form. We can see that 𝑞(𝑥) = 𝑥5 −2𝑥4 −𝑥3 + 2𝑥2 −2𝑥+ 4 (1) = ((((𝑥−2) · 𝑥−1) · 𝑥+ 2) · 𝑥−2) · 𝑥+ 4 To compute 𝑞(𝑥) for a specific 𝑥using the first form of 𝑞(𝑥) requires 5 additions with 13 multiplica-tions for a total of 18 operations. Horner’s factored form requires 5 additions and 4 multiplications for a total of 9 operations, significantly reducing the number of operations.
In general, for a fifth degree polynomial: 𝑝(𝑥) = 𝑎5𝑥5 + 𝑎4𝑥4 + 𝑎3𝑥3 + 𝑎2𝑥2 + 𝑎1𝑥+ 𝑎0 = ((((𝑎5𝑥+ 𝑎4) · 𝑥+ 𝑎3) · 𝑥+ 𝑎2) · 𝑥+ 𝑎1) · 𝑥+ 𝑎0 Evaluating 𝑝(𝑟) using Horner’s Form is 𝑝(𝑟) = ((((𝑎5𝑟+ 𝑎4) · 𝑟+ 𝑎3) · 𝑟+ 𝑎2) · 𝑟+ 𝑎1) · 𝑟+ 𝑎0 We can express Horner’s form schematically by 𝑝 𝑎5 𝑎4 𝑎3 𝑎2 𝑎1 𝑎0 𝑟 ↓ 𝑟· 𝑎5 𝑟· 𝐴4 𝑟· 𝐴3 𝑟· 𝐴2 𝑟· 𝐴1 𝑎5 𝐴4 = 𝑟· 𝑎5 + 𝑎4 𝐴3 = 𝑟· 𝐴4 + 𝑎3 𝐴2 = 𝑟· 𝐴3 + 𝑎2 𝐴1 = 𝑟· 𝐴2 + 𝑎1 𝐴0 = 𝑟· 𝐴1 + 𝑎0 Then 𝐴0 = 𝑝(𝑟) For example, evaluating 𝑞from (1) above at 𝑟= −1 is 𝑞 1 −2 −1 2 −2 4 −1 ↓ −1 +3 −2 0 +2 1 −3 +2 0 −2 𝐴0 = 6 Thus, 𝑞(−1) = 6.
Therefore, Horner’s techniques surpass simply performing synthetic division.
In the appendix, we provide problems for student investigations regarding Horner’s Method.
These problems are developed to both deepen the student’s understanding of Horner’s Method and to help them better understand the material in this paper.
4 Remainder Theorem & Extension (nonlinear factors) One of the possible beautiful connections that the reader could have realized from the previous student investigations is in respect to the wonderful Remainder Theorem . (A more detailed, axiomatic, and 8 The Electronic Journal of Mathematics and Technology, Volume 14, Number 1, ISSN 1933-2823 mathematically precise development of the Remainder Theorem is provided in a footnote.9 Readers can investigate the mathematics and style of this footnote and glean much regarding the beauty and intricacy of the mathematics being considered in this section of the paper.) This theorem states that when you divide a polynomial 𝑃(𝑥) by a linear polynomial, you produce another polynomial 𝑄(𝑥) and a remainder 𝑅(a constant polynomial). This is written as 𝑃(𝑥) 𝑥−𝑐= 𝑄(𝑥) + 𝑅 𝑥−𝑐 OR 𝑃(𝑥) = 𝑄(𝑥)(𝑥−𝑐) + 𝑅.
In a number of ways, the Remainder Theorem has been a central connecting fiber to almost all which has preceded. Indeed, the remainder 𝑅is a deeply meaningful notion. When 𝑅= 0, it indicates that 𝑥−𝑐is a factor of 𝑃(𝑥). This can be seen as 𝑃(𝑥) 𝑥−𝑐= 𝑄(𝑥) + 0 𝑥−𝑐= 𝑄(𝑥) + 0 = 𝑄(𝑥) OR 𝑃(𝑥) = 𝑄(𝑥)(𝑥−𝑐).
9 Theorem 1 (Division Algorithm) Let 𝑛, 𝑑∈Z where 𝑑> 0. Then there exist unique integers 𝑞, the quotient, and 𝑟, the remainder, such that 𝑛= 𝑞· 𝑑+ 𝑟 where 0 ≤𝑟< 𝑑. And 𝑑is a factor of 𝑛if and only if 𝑟= 0 The division algorithm extends to polynomials.
Definition 2 (Polynomials over a Field) Let 𝑘be a field. For example: 𝑘could be Q, R, or C. Then 𝑘[𝑥] = 𝑝(𝑥) 𝑝is a polynomial in 𝑥with coefficients in 𝑘 .
Theorem 3 (Division Algorithm for Polynomials) Let 𝑘be a field. Let 𝑓(𝑥) and 𝑑(𝑥) be in 𝑘[𝑥] with 𝑑(𝑥) ≠0. Then there are unique polynomials 𝑞(𝑥), the quotient, and 𝑟(𝑥), the remainder, in 𝑘[𝑥] with 𝑓(𝑥) = 𝑞(𝑥) · 𝑑(𝑥) + 𝑟(𝑥) where either 𝑟(𝑥) = 0 or deg(𝑟) < deg(𝑑).
Note that the degree of the zero polynomial is undefined.
Corollary 4 (Remainder Theorem) Let 𝑘be a field with 𝑓(𝑥) ∈𝑘[𝑥] and 𝑎∈𝑘. Then there is a (unique) 𝑞(𝑥) ∈𝑘[𝑥] such that 𝑓(𝑥) = 𝑞(𝑥) · (𝑥−𝑎) + 𝑓(𝑎).
Corollary 5 (Factor Theorem) Let 𝑘be a field with 𝑓(𝑥) ∈𝑘[𝑥] and 𝑎∈𝑘. Then 𝑎is a root of 𝑓iff (𝑥−𝑎) is a factor of 𝑓(𝑥); i.e., there is a 𝑞(𝑥) ∈𝑘[𝑥] such that 𝑓(𝑥) = 𝑞(𝑥) · (𝑥−𝑎).
9 The Electronic Journal of Mathematics and Technology, Volume 14, Number 1, ISSN 1933-2823 This alone is meaningful, and we saw this repeatedly in earlier examples of synthetic division, where the last term in the synthetic division 𝑃(𝑥) ÷ (𝑥−𝑐) was 0. From that, we knew that 𝑥−𝑐was a factor of 𝑃(𝑥).
However, what does the remainder tell us when 𝑅≠0? Well, it tells us at least two things. First, 𝑥−𝑐is not a factor of 𝑃(𝑥). Second, and this is the same whether or not 𝑅= 0, we always have 𝑅= 𝑃(𝑐), or 𝑅is the 𝑦-value of 𝑃(𝑥) when 𝑥= 𝑐. In other words, 𝑃(𝑐) = 𝑄(𝑐)(𝑐−𝑐) + 𝑅= 𝑅.
This now brings us to a wonderful intersection of some previously posed ideas. Remember that Horner’s Method of synthetic division allowed us to divide a real polynomial by any real polynomial of lower degree, let’s say 𝐷(𝑥), and not simply by a linear polynomial in the form 𝑥−𝑐. So, we need tools which allow 𝐷(𝑥) to be any polynomial.
To our rescue is the Fundamental Theorem of Algebra, which states that every real polynomial can be factored over the reals into a product of linear and irreducible quadratic factors. We can now consider an extension of the Remainder Theorem: Let 𝐼(𝑥) be a real irreducible quadratic polynomial, then 𝑃(𝑥) 𝐼(𝑥) = 𝑄(𝑥) + 𝑅(𝑥) 𝐼(𝑥) OR 𝑃(𝑥) = 𝑄(𝑥)𝐼(𝑥) + 𝑅(𝑥) For example,10 let 𝑃(𝑥) = 𝑥4 −𝑥2 −2𝑥−1 and the irreducible quadratic 𝐼(𝑥) = 𝑥2 + 𝑥+ 1, then 𝑃(𝑥) 𝐼(𝑥) = 𝑥4 −𝑥2 −2𝑥−1 𝑥2 + 𝑥+ 1 = 𝑥2 −𝑥−1 + 0 𝑥2 + 𝑥+ 1.
Therefore, since 𝑅(𝑥) = 0, 𝐼(𝑥) is a factor of 𝑃(𝑥).
If we alter this example slightly11 and let 𝐺(𝑥) = 𝑥4 + 𝑥2 −𝑥+ 5 and the irreducible quadratic 𝐼(𝑥) = 𝑥2 + 𝑥+ 1, then 𝐺(𝑥) 𝐼(𝑥) = 𝑥4 + 𝑥2 −𝑥+ 5 𝑥2 + 𝑥+ 1 = 𝑥2 −𝑥+ 1 + −𝑥+ 4 𝑥2 + 𝑥+ 1.
In this case, since 𝑅(𝑥) is not identically 0, 𝐼(𝑥) is not a factor of 𝐺(𝑥).
Previously, we saw that 𝑃(𝑐) = 𝑅when dividing by the linear expression 𝑥−𝑐. Dividing by a quadratic 𝑞(𝑥) gives two possibilities when 𝑞has real roots. If 𝑞(𝑥) has roots 𝑥= 𝑟1 and 𝑟2, then 𝑅= 𝑅(𝑥), and we (still) have 𝑃(𝑟1) = 𝑅(𝑟1) and 𝑃(𝑟2) = 𝑅(𝑟2). When the quadratic 𝑞(𝑥) has no real roots, i.e., is irreducible over the reals, then what would happen if we evaluated at a complex root 𝑟= 𝑎± 𝑏𝑖of 𝑞(𝑥)? That is, is there a relation between 𝑃(𝑎± 𝑏𝑖) and 𝑅(𝑎± 𝑏𝑖)? Using the Remainder Theorem easily answers this question.
In the appendix, we provide problems for student investigations regarding the Remainder The-orem. These problems are developed to both deepen the student’s understanding of the Remainder Theorem and to help them better understand the material in this paper.
10Web Example #9 on 11Web Example #10 on 10 The Electronic Journal of Mathematics and Technology, Volume 14, Number 1, ISSN 1933-2823 5 Conclusion Explaining, expanding upon, and connecting mathematical ideas is central to student learning. In this paper we have explained the process of synthetic division commonly seen in high school. We have investigated why it works and how it connects to polynomial long division. Commonly seen synthetic division was then extended to help students see that synthetic division could consider linear factors in the form 𝑥−𝑐, where 𝑐is an integer, rational, irrational, and even complex number. We then expanded upon synthetic division by considering Horner’s method for dividing by a polynomial of any degree. We then connected Horner’s method of synthetic division to the Remainder Theorem and the Zero Product Property. This paper also purposely provided two distinct types of instructional aids for the reader: (A) an applet for student experimentation and observation of results and (B) sections of student investigations to accompany major topics. Altogether, we hope that the style and technique of this paper engages the reader and invites the reader to consider both synthetic division and other mathematical topics more deeply.
References Florian Cajori, ”Horner’s method of approximation anticipated by Ruffini.” Bulletin of the Amer-ican Mathematical Society. 17 (1911), no. 8, 389–444.
Horner’s Method, General Report (Assoc. for the Improvement of Geometrical Teaching) 15 (1889), 59–68.
Irwin Hoffman and Larry Kauvar, “Polynomial Synthetic Division,” Math. Teacher 63 (1970), no. 5, 429–431.
William. G. Horner, “A New Method of Solving Numerical Equations of all Orders by Continuous Approximation,” Phil. Trans. Royal Society of London 109 (1819), 308–335.
Dan Kalman, “Differentiation and Synthetic Division,” Two-Year College Math. J. 10 (1979), no. 1, 37–37.
Alex Pathan and Tony Collyer, “The Wonder of Horner’s Method,” Math. Gazette 87 (2003), no. 509, 230–242.
J. J. Price, “Algorithms for Evaluation of Polynomials,” College Math. J 21 (1990), no. 5, 404– 405.
Kurt W. Reimann, “Synthetic Division for Nonlinear Factors,” Math. Teacher 73 (1980), no. 3, 231–233.
Michael Weiss, “Factor and Remainder Theorems: An Appreciation,” Math. Teacher 110 (2016), no. 2, 153–156.
11 The Electronic Journal of Mathematics and Technology, Volume 14, Number 1, ISSN 1933-2823 Appendix This appendix contains problems for students to investigate regarding Synthetic Division, Horner’s Form, and the Remainder Theorem. These questions are excellent for classroom discussions or to gain a deeper understanding of this mathematics in this article. In order to investigate ideas associated with polynomial long division and synthetic division, we have developed and provided an online applet ( Enjoy.
Student Investigations on Synthetic Division 1. Create your own real polynomial 𝑃(𝑥) of degree greater than or equal to 5. Perform synthetic division on 𝑃(𝑥) by each of the following: 𝑥−2; 𝑥+ 3; 𝑥−1 2; 𝑥− √ 2; 𝑥−(2 −3𝑖); 2𝑥−3; and 1 2𝑥+ 2. Check each result using the applet.
2. Reexamine the looping structures in Figure 2. Define the looping structure for the example of the extended polynomial division 𝑥4 + 𝑥2 −𝑥+ 5 𝑥2 + 2𝑥+ 3 provided above.
3. Create your own real polynomial 𝑄(𝑥) with degree of 2 or 3. Perform the extended synthetic division 𝑃(𝑥) ÷ 𝑄(𝑥). Check your result using the applet.
4. Repeat the previous problem with a new polynomial 𝑄(𝑥). Check your result using the applet.
5. Experiment with the applet ( method.html) and see what else you can do in respect to polynomial and synthetic division.
Student Investigations Regarding Horner’s Form 1. Create your own cubic polynomial 𝑃(𝑥). Evaluate 𝑃(3). Factor 𝑃(𝑥) into Horner’s Form and evaluate 𝑃(3). Then perform the synthetic division 𝑃(𝑥) ÷ (𝑥−3). Compare all of your result and compare the amount of work which was necessary for each.
2. Create your own real seventh degree polynomial 𝑃(𝑥).
Evaluate 𝑃(3).
Factor 𝑃(𝑥) into Horner’s Form and evaluate 𝑃(3). Then perform the synthetic division 𝑃(𝑥) ÷ (𝑥−3). Compare all of your result and compare the amount of work which was necessary for each.
3. In your own words, describe every step in the process of Horner’s Method of synthetic division of 𝑛3𝑥3 + 𝑛2𝑥2 + 𝑛1𝑥+ 𝑛0 𝑑2𝑥2 + 𝑑1𝑥+ 𝑑0 . (Hint. You may wish to return to your answer for problem 3 in the previous section.) 4. Analyze all of the previous ideas in this paper. Synthesize these ideas into new mathematical connections that have not yet been presented.
12 The Electronic Journal of Mathematics and Technology, Volume 14, Number 1, ISSN 1933-2823 Student Investigations Regarding the Remainder Theorem 1. We previously defined the Remainder Theorem as: 𝑃(𝑥) 𝑥−𝑐= 𝑄(𝑥) + 𝑅 𝑥−𝑐OR 𝑃(𝑥) = 𝑄(𝑥)(𝑥− 𝑐) + 𝑅. Explain this theorem in your own words through an example of polynomials which you have worked out. Then, explain the theorem again limiting your explanation to using 𝑃(𝑥), 𝑄(𝑥), (𝑥−𝑐), and 𝑅.
2. The Zero Product Property states that for all Complex numbers (including Reals), if 𝑎· 𝑏= 0, then 𝑎= 0 or 𝑏= 0. This theorem has been sneakily at work throughout this entire paper.
Explain why this theorem is important in respect to the Remainder Theorem.
3. The Zero Product Property from the previous question tells us very much about the possible values of 𝑎and 𝑏. However, if 𝑎· 𝑏= 3, what do we know about the possible values of 𝑎and 𝑏?
4. The footnote in the section regarding the Remainder Theorem contains much mathematical notation and symbolism. Define/explain each of the following: (a) Z, Q, R, and C; (b) unique, polynomial, field, root of 𝑓; (c) 𝑘[𝑥] = 𝑝(𝑥) 𝑝is a polynomial in 𝑥with coefficients in 𝑘 ; (d) deg(𝑟) < deg(𝑑).
5. Create three examples of irreducible quadratic polynomials with integer coefficients.
6. Select some integral values for 𝑎, 𝑏, and 𝑘> 0. (Note that only 𝑘needs to be greater than zero.). Expand (𝑎𝑥+ 𝑏)2 + 𝑘. Determine the roots of this quadratic. Change the values of 𝑎, 𝑏, and 𝑘, expand the quadratic, and determine its roots. Do this one more time. What do you notice about the roots of these quadratics?
7. Choose any values for 𝑎, 𝑏, and 𝑐such that 𝑏2 < 4𝑎𝑐. Write these in the form 𝑎𝑥2 + 𝑏𝑥+ 𝑐. Determine the roots of this quadratic. Change the values of 𝑎, 𝑏, and 𝑐(maintaining the relationship 𝑏2 < 4𝑎𝑐) and determine the roots of this quadratic. Do this one more time. What do you notice about the roots of these quadratics?
8. Extend the analysis above of substituting the roots of a quadratic in 𝑃(𝑟𝑖) and relating to 𝑅(𝑟𝑖) to any degree divisor.
9. Let us tinker for a moment with the Remainder Theorem. If we begin with 𝑃(𝑥) = 𝑄(𝑥)(𝑥− 𝑐) + 𝑅, we can then write 𝑃(𝑥) = 𝑄(𝑥)(𝑥−𝑐) + 𝑃(𝑐). With simple algebra, we can rearrange this to 𝑃(𝑥) −𝑃(𝑐) 𝑥−𝑐 = 𝑄(𝑥).
In this context, many students have seen 𝑃(𝑥) −𝑃(𝑐) 𝑥−𝑐 as the limit of 𝑃(𝑥) as 𝑥approaches 𝑐and 𝑄(𝑐) as the slope of the line tangent to 𝑃(𝑥) at 𝑥= 𝑐. Explain how these ideas are connected.
13 |
10954 | https://www.youtube.com/watch?v=9PsC63B_Diw | dy/dx = y² - 2xy / x² - xy | Homogeneous Differential Equation @EAG
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dy/dx = y² - 2xy / x² - xy | Homogeneous Differential Equation @EAG
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okay next problem differential equations solve this differential equation Dy by DX = y² - 2x y x² - x y so Y 2 X Y 1 1 x 2 x 1 y 1 1 + 1 2 okay so degre homogeneous differential equation okay so four clearly it is hom clearly it is homogenous so homogeneous differential equation put y = VX is the substitution okay [Music] y homogeneous differential equation homogeneous now we have toate theate with respect to X so Dy by DX U so UV rule V into u u v d DX so D so one implies v v + x DV by DX that is equal to right side y+ VX y² v² x² - 2 x y and VX by x² is x² - x into y means VX okay next V + x DV by DX so numerator x v² - 2 V by x² X into X x² X into X x² so denominator x² so x² 1us V cancel the like terms in both numerator and denominator next right side X DV by DX is equal to v² - 2 V by 1 - vus V okay so left side V right side minus V now take LCM on right side right side LCM v² - 2 V - V into 1 - V - V into - v- into minus + v² so total 2 v² - 2 V - V - 3 V by 1 - V which is equal to x d by DX okay now variable separable VAR taking where [Music] separable 1 - V by 2 v² - 3B into DB that is equal to DX by X next take integration on [Music] integration so VAR setion this so integral 1 - V by 2 v² - 3v DV is equal to integral DX by X okay left side integral 1 - V by 2 v² - 3 DV so integ first of all we have to take partial fractions for this function 1 - V 2 V 3 Frac so let us 1 - V by 2 v² - 3 V [Music] so linear pols a by V plus b by 2 V - 3 so LCM 1 - V 2us 3 plus b v okay so next sortable Val so put V = to0 so so left side 1- V 1al A into V aus3 A into -3 plus into 0 so - 3 A = 1 a is = -1 by 3 okay next [Music] putal so V is = 3x2 so 1 - V and 3x2 that is equal to a into 2 into 3x 2 - 3 plus b into V and a 3 by 2 so 2 2 cancel 3 - 3 next continuation okay continuation 2 1 2 - 3 - 1 by 2 left side -1 by 2 a into 0 + B into 3x 2 okay so - 1 by 2 = B into 3x2 2 2 cancel 3 the left side B is also - 1 [Music] [Music] by3 fractions [Music] so integral a by V so a 1 by 3 - 1 by 3 by V + B by 2 V - 3 B is also - 1 by 3 - 1x3 by 2v - 3 DV whole DV that is equal to integral DX by X means log X Plus log C okay continuation so - 1 by3 outside this - 1 by3 integral 1 by V DV that means log V so - 1 by3 outside inte 1 by 2us 3 d 1 concept so [Music] log 2 vus 3 [Music] V right side log X Plus log C so- 13 log V so so 1 by that is equal to [Music] log log CX log m + log log M so-3 right so left log m+ log n log MN so log V into root of 2 V - 3 that is equal to- 1 right side it become - 3 log CX okay log V into < TK of 2 V - 3 equal to e log CX power - 3 V into s < TK of 2 V - 3 is equal to C power - 3 into x^ - 3 so x- [Music] 3 so X Cub into V is into Ro otk of 2 V - 3 = C let c d = cus 3 so x- 3 left side say x Cub now put v = y by X so yal VX y byx so so x v y by X into root of 2 into y by x - 3 that is equal to C D okay so X can the x² x² into y = root of 2 into y by x - 3 is = Das okay therefore x² Y into square root of 2 so x² - 3 x² that is equal to C D so therefore X Y into root of 2 x y - 3x² is = c d is the general solution okay which is the general [Music] solution |
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Concepts of Physics explains the different theories and concepts in an easy-to-understand way, making it popular among students. H. C. Verma s book is an all-inclusive theoretical and conceptual guide that covers a vast range of topics. It is an ideal book not only for pre-college students but also for those appearing for competitive exams in the field of engineering and medicine. It is a book that aims to comprehensively guide the students in various aspects of physics thus, making them fully equipped to answer any type of questions that may come in the exams. Volume 2 of the Concepts of Physics starts with a chapter on Heat and Temperature. It moves on to explain the Kinetic Theory of Gases. The concepts of Calorimetry, Law of Thermodynamics, Heat Transfer, Specific Heat Capacities of Gases are discussed in the subsequent chapters. There are separate chapters on Gauss s Law, Electric Field and Potential, and Capacitors. The book also features chapters and detailed explanations on Electric Current in Conductors, Thermal and Chemical effects of Electric Current, and Magnetic Field. The author also elucidates on the concept of Permanent Magnets, Electromagnetic Induction and Electromagnetic Waves. He further stresses on the Magnetic Field due to a current, Magnetic Properties of Matter, Alternating Current and Electric Current through Gases. In the last chapters of the book, the author details out on Bohr s Model and Physics of the Atom, Photoelectric Effect and Wave Particle Duality, X-rays, The Nucleus, Theory of Relativity and, Semiconductors and Semiconductor Devices. Each chapter has been intricately dealt with in the book. It is an ideal study book for all kinds of competitive exams like IIT-JEE and different types of state level exams for engineering and medicine.
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verma is considered as one of the most important book in physics for students whoare preparing for iit jee and other competitive exams.the theory related questions are very intersting in verma i was thankful to flipkart for selling this book for cheap price less than the market value --By pavan nath on Dec 21, 2011
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After going through the first part my eagerness to go through the second one made me to purchase this book . this book is just awesome it clears all fundamentals and basics just like a good teacher , it has everything packed inside the 450 pages. I recommend it to all as its necessary for your jee preparation as well as for strength your "concepts of physics. --By amit das on Apr 23, 2012
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Harish Chandra Verma is an Indian Nuclear Experimental Physicist, educationalist and author. He has authored popular books such as Dominion Status VS Complete Independence, Industrial Families in India, Foundation Science: Physics For Class 9, etc. After procuring his doctorate from the Indian Institute of Technology in Kanpur, he went on to work as a reader and lecturer at Patna University. As a physicist, his chief areas of interest are in Materials Applications, Condensed Matter and also on Earth Science related issues like extinction boundaries and meteorites. Apart from his research in areas of physics involving meteorites and Earth Science, he also gives lectures on topics like India's heritage and cultural values.
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10956 | https://www.albert.io/blog/parametric-equation-for-circle-and-more-ap-precalculus-review/ | Skip to content
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Introduction
Understanding motion through curves and lines is a fundamental part of math, especially in precalculus. Parametric equations step in to make this concept more digestible. They express paths through equations that depend on an independent variable, wonderfully illustrating motion in both curved and linear paths. Here, we’ll dive into the parametric equation for a circle and the parametric equation of a line.
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What Are Parametric Equations?
Parametric equations are unique because they express a set of quantities through functions of one or more parameters (independent variables). Unlike regular equations that relate x and y directly, parametric equations involve another variable, typically t.
Example: Comparing Equations
Traditional Circle Equation: x^2 + y^2 = r^2
Parametric Circle Equation: (x(t), y(t)) = (r \cdot \cos t, r \cdot \sin t)
Notice how the parametric version uses a parameter, t, to independently define positions of x and y.
Parametric Equation for Circle
Let’s work through a circle’s parametric equation using the concept of the unit circle. A unit circle has a radius of 1 and is centered at the origin, (0, 0).
Key Features: Center at (0, 0) and radius 1.
Parametric Formula: (x(t), y(t)) = (\cos t, \sin t)
Domain: 0 \leq t \leq 2\pi
Example 1: Motion Around a Circle
Problem: Describe a point’s position moving counterclockwise around the unit circle from t = 0 to t = \pi/2.
Substitute Values:
At t = 0, x(0) = \cos(0) = 1, y(0) = \sin(0) = 0
At t = \pi/2, x\left(\frac{\pi}{2}\right) = \cos\left(\frac{\pi}{2}\right) = 0, y\left(\frac{\pi}{2}\right) = \sin\left(\frac{\pi}{2}\right) = 1
The point moves from (1, 0) to (0, 1) along the circle’s path.
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Transformations of the Parametric Function
Circles can be moved around the plane and resized by tweaking their parametric equations.
Transformations: Adjust the center to (h, k) and modify the radius to r.
New Parametric Form: (x(t), y(t)) = (h + r \cdot \cos t, k + r \cdot \sin t)
Example 2: Transforming the Circle
Problem: Find the parametric equations for a circle centered at (2, 3) with a radius of 4.
Apply Transformations:
Center (h, k) = (2, 3)
Radius r = 4
New Parametric Equations: (x(t), y(t)) = (2 + 4 \cdot \cos t, 3 + 4 \cdot \sin t)
Parametric Equations for Line Segments
A line can also be expressed using parametric equations. Instead of motion around, consider linear motion.
Definition: Expressed as functions of a parameter for x(t) and y(t).
Example: Line from (x_1, y_1) to (x_2, y_2). The image to the right shows the line from (-1,1) to (1,4)
Example 3: Parametric Equation of a Line Segment
Problem: Create a parametric equation for a line segment from (1, 2) to (5, 6).
Determine Changes:
Change in x = 5 - 1 = 4
Change in y = 6 - 2 = 4
Write Parametric Equations:
Start point: (1, 2)
x(t) = 1 + 4t, \quad y(t) = 2 + 4t, where 0 \leq t \leq 1
The final parametric equation is (1+4t, 2+4t) where 0 \leq t \leq 1.
Quick Reference Vocabulary
| | |
--- |
| Term | Definition |
| Parametric Equation | An equation that expresses a set of quantities as functions of one or more independent variables, called parameters. |
| Unit Circle | A circle with a radius of 1, centered at the origin (0, 0). |
| Transformations | Changes applied to a geometric figure, affecting position, size, or orientation. |
| Linear Path | A straight line described in the coordinate plane. |
| Radius | The distance from the center of a circle to any point on its circumference. |
Conclusion
Understanding parametric equations unlocks the fascinating world of describing motion in both circles and lines. These equations render curvy paths into manageable data points through parameters. Whether illustrating a circle’s geometric beauty or a linear path’s straightforwardness, parametric equations shine in capturing motion’s elegance in precalculus.
Sharpen Your Skills for AP® Precalculus
Are you preparing for the AP® Precalculus exam? We’ve got you covered! Try our review articles designed to help you confidently tackle real-world math problems. You’ll find everything you need to succeed, from quick tips to detailed strategies. Start exploring now!
4.2 Parametric Functions Modeling Planar Motion
4.3 Parametric Functions and Rates of Change
4.5 Implicitly Defined Functions
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10957 | https://arxiv.org/abs/2107.05572 | [2107.05572] The Integer Sequence Transform $a \mapsto b$ where $b_n$ is the Number of Real Roots of the Polynomial $a_0 + a_1x + a_2x^2 + \cdots + a_nx^n$
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arXiv:2107.05572 (math)
[Submitted on 7 Jul 2021 (v1), last revised 13 Aug 2021 (this version, v4)]
Title:The Integer Sequence Transform a \mapsto b where b_n is the Number of Real Roots of the Polynomial a_0 + a_1x + a_2x^2 + \cdots + a_nx^n
Authors:W. Edwin Clark, Mark Shattuck
View a PDF of the paper titled The Integer Sequence Transform $a \mapsto b$ where $b_n$ is the Number of Real Roots of the Polynomial $a_0 + a_1x + a_2x^2 + \cdots + a_nx^n$, by W. Edwin Clark and Mark Shattuck
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Abstract:We discuss the integer sequence transform a \mapsto bwhere b_nis the number of real roots of the polynomial a_0 + a_1x + a_2x^2 + \cdots + a_nx^n. It is shown that several sequences agive the trivial sequence b = (0,1,0,1, 0,1,\ldots), i.e., {b_n = n \bmod 2}, among them the Catalan numbers, central binomial coefficients, n!and \binom{n+k}{n}for a fixed k. We also look at some sequences afor which bis more interesting such as a_n = (n+1)^kfor k \geq 3. Further, general procedures are given for constructing real sequences a_nfor which b_nis either always maximal or minimal.
Subjects:Combinatorics (math.CO)
MSC classes:140B99 (Primary) 12D10 (Secondary)
Cite as:arXiv:2107.05572 [math.CO]
(or arXiv:2107.05572v4 [math.CO] for this version)
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From: W. Edwin Clark [view email]
[v1] Wed, 7 Jul 2021 16:19:56 UTC (8 KB)
[v2] Tue, 13 Jul 2021 01:29:25 UTC (8 KB)
[v3] Sat, 24 Jul 2021 00:13:21 UTC (12 KB)
[v4] Fri, 13 Aug 2021 12:42:36 UTC (17 KB)
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10958 | https://www.teacherspayteachers.com/Product/Break-Apart-Decomposing-for-Addition-Mental-Math-Addition-Strategy-Unit-5766645 | Break Apart (Decomposing) for Addition - Mental Math Addition Strategy Unit
Description
This unit is part of the Mental Math Strategy Collection.
Breaking apart an addend by place value is a powerful mental math strategy for adding numbers with two or more digits. Although this is similar to left-to-right addition, some students prefer it because only one addend is decomposed by place value, rather than both of them.
For example, to solve 43+35, we could first decompose the 35 into 30 and 5. We start by adding 43+30 to make 73, then the remaining 5 to make 78.
This unit includes:
My Math Fact Philosophy
My resources are created with this philosophy in mind:
•Math should be taught using the Concrete-Representational-Abstract model.
•UNDERSTANDING math facts is more important than memorizing math facts. Conceptual understanding is the key to math fact fluency.
•Students must be able to visualize the math in order to really understand it.
•True math fact fluency is more than just speed and accuracy. It also includes flexibility, which is essential to true fluency.
•One of the best ways to build flexibility is by making connections and forming relationships between facts.
Thank you for your interest in my resources,
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www.ShelleyGrayTeaching.com
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10959 | https://chem.libretexts.org/Bookshelves/Biological_Chemistry/Supplemental_Modules_(Biological_Chemistry)/Carbohydrates/Case_Studies/Starch_and_Iodine | Skip to main content
Starch and Iodine
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: Jul 4, 2022
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Blood Glucose Test
Sugar and Teeth
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( \newcommand{\kernel}{\mathrm{null}\,})
Plants store glucose as the polysaccharide starch; the cereal grains (wheat, rice, corn, oats, barley) as well as tubers such as potatoes are also rich in starch. Starch can be separated into two fractions--amylose and amylopectin. Natural starches are mixtures of amylose (10-20%) and amylopectin (80-90%).
Introduction
Amylose forms a colloidal dispersion in hot water whereas amylopectin is completely insoluble. The structure of amylose consists of long polymer chains of glucose units connected by an alpha acetal linkage. Starch - Amylose shows a very small portion of an amylose chain. All of the monomer units are alpha -D-glucose, and all the alpha acetal links connect C #1 of one glucose and to C #4 of the next glucose. As a result of the bond angles in the α acetal linkage, amylose actually forms a spiral much like a coiled spring. See the graphic below, which show four views in turning from a the side to an end view.
Chemical Test for Starch or Iodine
Amylose in starch is responsible for the formation of a deep blue color in the presence of iodine. The iodine molecule slips inside of the amylose coil. Iodine - KI Reagent: Iodine is not very soluble in water, therefore the iodine reagent is made by dissolving iodine in water in the presence of potassium iodide. This makes a linear triiodide ion complex with is soluble that slips into the coil of the starch causing an intense blue-black color.
Starch Test: Add Iodine-KI reagent to a solution or directly on a potato or other materials such as bread, crackers, or flour. A blue-black color results if starch is present. If starch amylose is not present, then the color will stay orange or yellow. Starch amylopectin does not give the color, nor does cellulose, nor do disaccharides such as sucrose in sugar.
Iodine Test: When following the changes in some inorganic oxidation reduction reactions, iodine may be used as an indicator to follow the changes of iodide ion and iodine element. Soluble starch solution is added. Only iodine element in the presence of iodide ion will give the characteristic blue black color. Neither iodine element alone nor iodide ions alone will give the color result. This phenomenon is used in the iodine clock demonstration.
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Charles Ophardt, Professor Emeritus, Elmhurst College; Virtual Chembook
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10960 | https://arxiv.org/abs/1906.03477 | [1906.03477] On a theorem of Sárközy for difference sets and shifted primes
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arXiv:1906.03477 (math)
[Submitted on 8 Jun 2019 (v1), last revised 4 Mar 2020 (this version, v2)]
Title:On a theorem of Sárközy for difference sets and shifted primes
Authors:Ruoyi Wang
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Abstract:We show that if the difference of two elements of a set $A \subseteq [N]$ is never one less than a prime number, then $|A| = O (N \exp (-c (\log N)^{1/3}))$ for some absolute constant $c>0$.
Comments:Title changed. Referee comments incorporated
Subjects:Classical Analysis and ODEs (math.CA); Number Theory (math.NT)
Cite as:arXiv:1906.03477 [math.CA]
(or arXiv:1906.03477v2 [math.CA] for this version)
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Journal reference:Journal of Number Theory, Volume 211, June 2020, Pages 220-234
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[v1] Sat, 8 Jun 2019 15:33:00 UTC (10 KB)
[v2] Wed, 4 Mar 2020 00:05:49 UTC (10 KB)
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10961 | https://flexbooks.ck12.org/cbook/ck-12-middle-school-math-concepts-grade-8/section/6.8/related/lesson/angle-measures-in-given-quadrilaterals-msm6/ | Angle Measures in Given Quadrilaterals | CK-12 Foundation
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Back To Understanding the Angle Measures of QuadrilateralsBack
6.8
Angle Measures in Given Quadrilaterals
Written by:Jen Kershaw, M.ed |Kimberly Hopkins
Fact-checked by:The CK-12 Editorial Team
Last Modified: Sep 01, 2025
[Figure 1]
Ivonne is a farmer. She plants a rhombus-shaped section of corn each year. This year, she has decided to check the angles at each vertex of the corn. She determines that one angle is 110 degrees and the another angle is 70 degrees. Her daughter tells her that she can determine the other angle measures without walking around to them. What are the other angle measures?
In this concept, you will learn how to work with the angle measures of a quadrilateral.
Measuring Angles in Quadrilaterals
Let's look at a square.
[Figure 2]
A square has four right angles. Each right angle is 90∘. You can add up the sum of the interior angles of a square and see how this is related to all quadrilaterals.
90+90+90+90=360∘
The sum of the interior angles of all quadrilaterals is 360∘.
You can use this information to find the measure of missing angles.
[Figure 3]
Write an equation using the variable and given measurements and figure out the measure of the missing angle.
80+75+105+x=360 260+x=360 360−260=x 100=x
The missing angle is equal to 100∘.
Use this information to help you when figuring out missing angle measures in different quadrilaterals.
Also, you can draw specific quadrilaterals using a ruler and a protractor. Use the protractor to be sure that your work is accurate. This is especially important when drawing squares or rectangles or any figure with a right angle.
Start by using a protractor to draw in each of the four right angles. By using a ruler and a protractor, your lines will be straight and we will be able to determine that we have drawn the square correctly.
Drawing it freehand may seem easier, but it does not assure accuracy! The best way to be sure that your work is accurate is to use a protractor and a ruler.
[Figure 4]
Here is the first angle of a square. Now turn the protractor upside down and draw the other angle.
[Figure 5]
Here is the final figure.
[Figure 6]
Examples
Example 1
Earlier, you were given a problem about Ivonne and her rhombus-shaped field of corn.
Ivonne knows that one of the angles of the field is 110 degrees and a different angle is 80 degrees. What are the other angle measures?
First, note the properties of angles in a rhombus.
Opposite sides are congruent
Next, list all of the angle values.
80, 80, 110, and 110
Then, state the other angle measures.
80 and 110
The answer is 80 degrees and 110 degrees.
Example 2
A quadrilateral has the following angle measures: 130, 80 and 95. What is the measure of the missing angle?
First, write an equation.
130+80+95+x=360
Then, solve for the missing angle.
55
The answer is 55 degrees.
Example 3
If one angle of a rectangle is 90 degrees, what are the measures of the other three angles?
First, remember the angle rules for a rectangle.
All of the angles are 90 degrees
Then, state the measures of the other angles,
90 degrees
The answer is 90 degrees.
Example 4
A quadrilateral has the following angle measures: 105, 90 and 88. What is the measure of the missing angle?
First, write an equation.
105+90+88+x=360
Then, solve for the missing angle.
77
The answer is 77 degrees.
Example 5
A parallelogram has two congruent angles that are both 85 degrees. The other two angles are congruent. What is the measure of each missing angle?
First, write an equation.
85+85+x+x=360
Next, simplify the equation.
170+2 x=360
Then, solve for the missing angle.
2 x=190
x=95
The answer is 95 degrees.
Review
Answer each of the following questions about quadrilaterals.
True or false. A quadrilateral will always have only four sides.
The interior angles of a quadrilateral add up to be ___ degrees.
A square will have four _____ degree angles.
A rectangle will have four _____ degree angles.
True or false. A rhombus will also always have four right angles.
If the sum of three of the angles of a quadrilateral is equal to 300∘, it means that the measure of the missing angle is ______.
What is the value of x?
7.
[Figure 7]
8.
[Figure 8]
9.
[Figure 9]
What are all four angles of this rectangle equal to?
[Figure 10]
How many degrees are in a triangle?
Write an equation to show how the angles of the two triangles are equal to 360 degrees.
Identify the following figures.
[Figure 11]
[Figure 12]
Review (Answers)
Click HERE to see the answer key or go to the Table of Contents and click on the Answer Key under the 'Other Versions' option.
Resources
Image Attributions
Back to Angle Measures in Given Quadrilaterals
| Image | Reference | Attributions |
---
| | [Figure 1] | Credit:fishhawk Source: |
| | [Figure 2] | Credit:Dr Abdelkader Dendane Source:CK-12 Properties of Parallelograms License:CC BY-NC |
| | [Figure 3] | Credit:Dr Abdelkader Dendane Source:CK-12 Properties of Parallelograms License:CC BY-NC |
| | [Figure 4] | License:CC BY-NC |
| | [Figure 5] | License:CC BY-NC |
| | [Figure 6] | Source: License:CC BY-NC |
| | [Figure 7] | Credit:Dr Abdelkader Dendane Source:CK-12 Properties of Parallelograms License:CC BY-NC |
| | [Figure 8] | Credit:Dr Abdelkader Dendane Source:CK-12 Properties of Parallelograms License:CC BY-NC |
| | [Figure 9] | Credit:Dr Abdelkader Dendane Source:CK-12 Properties of Parallelograms License:CC BY-NC |
| | [Figure 10] | License:CC BY-NC |
| | [Figure 11] | License:CC BY-NC |
| | [Figure 12] | License:CC BY-NC |
| | [Figure 13] | License:CC BY-NC |
| | [Figure 14] | License:CC BY-NC |
| | [Figure 15] | Credit:Hans Holbein the Elder Source: License:CC BY-NC |
| | [Figure 16] | License:CC BY-NC |
| | [Figure 17] | Credit:By Scientif38 (Own work) [CC0], via Wikimedia Commons Source: |
| | [Figure 18] | Source:CK-12 Foundation |
| | [Figure 19] | License:CC BY-NC |
| | [Figure 20] | License:CC BY-NC |
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10962 | https://www.mometrix.com/academy/incenter-circumcenter-orthocenter-and-centroid/ | Centroid, Incenter, Circumcenter, and Orthocenter (Video)
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Centroid, Incenter, Circumcenter, and Orthocenter
Centroid, Incenter, Circumcenter, and Orthocenter (Video)
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Center of a Triangle Practice Questions
[x] Transcript - [x] Practice
Where is the center of a triangle? How do you find it? It’s not as easy as finding the center of a circle or a rectangle and for a very good reason—there are as many as four different centers to a triangle, depending on how we try to find it! They are the incenter, centroid, circumcenter, and orthocenter.
Today, we’ll look at how to find each one.
Incenter
Let’s start with the incenter. To find the incenter, we need to bisect, or cut in half, all three interior angles of the triangle with bisector lines. Let’s take a look at a triangle with the angle measures given.
The angle on the left is 50 degrees, so we’ll draw a line through it such that it’s broken into two 25-degree angles. We’ll do the same for the 60-degree angle on the right, yielding two 30-degree angles and the 70-degree angle on the top, creating two 35-degree angles, like this:
The point where the three angle bisector lines meet is the incenter.
Centroid
But what if we don’t cut the angles in half, but instead draw a line between each vertex and the midpoint of the line segment on the other side of the triangle? Let’s take a look at another triangle but this time we can see the lengths of the sides instead of the angle measures.
Let’s start by drawing a line between the angle on the left in a way that will cut the opposite side in half. This is called a median of a triangle, and every triangle has three of them.
As we can see, the opposite side that measures 10 meters has been split into two five-meter segments by our median.
Now we need to draw the other two medians:
Centroid
Now that we’ve drawn all three medians, we can see where they intersect. This point is the centroid of the triangle and is our second type of triangle center.
Now that we’ve divided the angles in half to find the incenter and the sides in half to find the centroid, what other methods can we devise to find the other two centers? Remember, there’s four!
Let’s try a variation of the last one. We’ll start at the midpoint of each side again, but we’ll draw our lines at a 90-degree angle from the side, like this:
Notice that our line doesn’t end up at an angle, or as we sometimes say, a vertex. It cuts through another side. That’s totally fine! Let’s do the same thing with the other two sides:
As we can see, all of our sides have perpendicular bisectors and all three of our bisectors meet at a point. This point is called the circumcenter of the triangle.
Orthocenter
Only one center left! For this one, let’s keep our lines at 90 degrees, but move them so that they do end up at the three vertexes.
When we do this, we’re finding the altitudes of a triangle. You might remember altitude because we need it to find the area of a triangle. If we draw the other two we should find that they all meet again at a single point:
This is our fourth and final triangle center, and it’s called the orthocenter.
So, do you think you can remember them all? Pause this video and try to match up the name of the center with the method for finding it:
Thanks for watching, and happy studying!
“Triangle Incenter, Description and Properties – Math Open Reference.”
Center of a Triangle Practice Questions
Question #1:
The center of the triangle below has been determined by constructing a line from each vertex to the opposite side in order to form a 90-degree angle with that side. This location is known as the _____.
Incenter
Centroid
Circumcenter
Orthocenter
[x] Show Answer
Answer:
The orthocenter of a triangle is determined by connecting a line from the vertex to the opposite side, so that a 90-degree angle is formed.
[x] Hide Answer
Question #2:
The center of a triangle can be determined by drawing perpendicular bisectors through the midpoint of each side length of a triangle. The center point where all three lines intersect is known as the ____.
Incenter
Centroid
Circumcenter
Orthocenter
[x] Show Answer
Answer:
The circumcenter of a triangle is located by drawing three perpendicular bisectors from the midpoint of each side length. The intersection point of all three lines is considered the circumcenter.
[x] Hide Answer
Question #3:
Determine if the statement is true or false:
The circumcenter is located at the intersection point of three angle bisectors.
True
False
[x] Show Answer
Answer:
The circumcenter is located by creating three perpendicular bisectors. The point where all three lines intersect is the circumcenter.
[x] Hide Answer
Question #4:
A city planner is designing a triangular park. She plans to plant an oak tree in the center of the triangle. She decides to calculate the midpoint of each side length by measuring the total distance of each side and then dividing by two. From here, she draws a line from each midpoint to the opposite vertex. The point where all three lines intersect is where she plans to plant the tree.
Which method did the city planner use to determine the center of the triangular park?
Incenter
Centroid
Circumcenter
Orthocenter
[x] Show Answer
Answer:
The centroid is determined by connecting a line from the midpoint of each side length, to the opposite vertex. This is the method that the city planner used to determine where to plant the Oak tree.
[x] Hide Answer
Question #5:
Morgan is building a luxury A-frame cabin in the woods. She wants to determine the “center” of the triangular face of the cabin so that she can fit the pieces of glass in properly. Morgan wants to determine the middle of the triangle by bisecting each interior angle. She plans to create three angle bisectors that extend to the opposite side length of the triangle. Morgan will consider the point where all three lines intersect as the “center” of the triangle.
Which method has Morgan used to determine the center of the triangle?
Incenter
Centroid
Circumcenter
Orthocenter
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Answer:
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10963 | https://byjus.com/chemistry/metalbandtheory/ | What is Band Theory of Metals?
Metals conduct electricity with the help of valence electrons present in them.
The atomic orbitals of the metals having the same energy combine to form molecular orbitals which are close in energy to each other to form a band.
In case, the band is partially filled or it overlaps with another higher energy unoccupied conduction band, electrons can easily flow under an applied electric field showing high conductivity.
Table of Contents
Band Theory of Metals Example
Let us take the example of sodium.
Sodium has an atomic configuration of 1s2, 2s2, 2p6, 3s1. It has one unpaired electron in the 3s orbital. The 3s valence atomic orbital of sodium overlaps with another such orbital of the same energy to form molecular orbitals. The atomic orbitals continue to combine in some fashion forming a band. The energy spread of this band is calculated as the difference in energy between the most strongly bound “bonding orbital” and the highest energy “anti bonding orbital”.
In other cases when the gap between the valence band and the conduction band (next higher unoccupied band) is quite high, electrons fail to jump from valence band to the conduction band. Such compounds show very less or no conductivity.
For example glass. When the gap between the valence band and conduction band is small, some electrons may jump from valence band to conduction band and thus show some conductivity. Such substances are known as semiconductors. For example silicon, germanium.
It is evident from the figure, in case of metal, there is no separation between the bands. This helps the incited electrons to easily move from one orbital to another and hence metals are good conductors of electricity. In the case of semiconductors, there is a small gap between the valence band and the conduction band.
Hence, only a small fraction of electrons (having sufficient energy) jumps when incited. However, we can increase the conductivity of such substances by increasing the temperature or doping. In insulators, the difference between the valence band and conduction band is very high. Hence, no conductivity is shown by such substances.
Recommended Videos
Frequently Asked Questions – FAQs
Q1
What is conduction band theory?
The conduction band is the band of electron orbitals that electrons can jump up into from the valence band when excited. When the electrons are in these orbitals, they have enough energy to move freely in the material. This movement of electrons creates an electric current.
Q2
How does band theory explain metallic character of lithium?
Electrons can be fed into one end of a metal wire and removed from the other end without causing any obvious change in the physical and chemical properties of the metal. … If two lithium atoms are brought together, the 1s core electrons remain essentially unchanged since there is virtually no overlap between them.
Q3
What is band theory of semiconductors?
According to the band theory, semiconductors will actually act as insulators at absolute zero. Above this temperature and yet still staying below the melting point of the solid, the metal would act as a semiconductor. Semiconductors are classified by the fully occupied valence band and unoccupied conduction band.
Q4
What is the band gap of conductor?
In a conductor there are no band gaps between the valence and conduction bands. In some metals the conduction and valence bands partially overlap. This means that electrons can move freely between the valence band and the conduction band.
Q5
What is free electron theory of metals?
The treatment of a metal as containing a gas of electrons completely free to move within it. The theory was originally proposed in 1900 to describe and correlate the electrical and thermal properties of metals.
For a detailed discussion on band theory, please visit BYJU’S.
Frequently Asked Questions – FAQs
What is conduction band theory?
The conduction band is the band of electron orbitals that electrons can jump up into from the valence band when excited. When the electrons are in these orbitals, they have enough energy to move freely in the material. This movement of electrons creates an electric current.
How does band theory explain metallic character of lithium?
Electrons can be fed into one end of a metal wire and removed from the other end without causing any obvious change in the physical and chemical properties of the metal. … If two lithium atoms are brought together, the 1s core electrons remain essentially unchanged since there is virtually no overlap between them.
What is band theory of semiconductors?
According to the band theory, semiconductors will actually act as insulators at absolute zero. Above this temperature and yet still staying below the melting point of the solid, the metal would act as a semiconductor. Semiconductors are classified by the fully occupied valence band and unoccupied conduction band.
What is the band gap of conductor?
In a conductor there are no band gaps between the valence and conduction bands. In some metals the conduction and valence bands partially overlap. This means that electrons can move freely between the valence band and the conduction band.
What is free electron theory of metals?
The treatment of a metal as containing a gas of electrons completely free to move within it. The theory was originally proposed in 1900 to describe and correlate the electrical and thermal properties of metals.
For a detailed discussion on band theory, please visit BYJU’S.
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10964 | https://www.youtube.com/watch?v=1kv0gjTHsYY | Euler's number e=lim(1+1/n)^n, Base of Natural Logarithm, most important limit existance proof
Math Physics Engineering
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Posted: 20 Feb 2022
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In this video, we prove the existence of the limit that defines the natural basis for the logarithm. This number was discovered by Leonard Euler and it is called after him and denoted by e. e is one of the most important numbers in mathematics it is just as important as pi. We prove that the limit
e=lim(1+1/n)^n as n tends to infinity exist show a desmos visualization, and say a few words about the properties of e. We also mention Eulers Magical formula e^pi i+1 =0.
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Transcript:
we will prove that this sequence converges and the limit of the sequence exists and this limit is one of the most important limits in all of mathematics this is the limit which leads to the basis of the natural logarithm so first of all let's prove that this limit exists and so this first step of the proof will be to show that this sequence is bounded and we can actually show that for any n that is greater or equal than two this sequence for every n is bounded between two and three uh and okay so n is greater than two because here in case of one then this is just equal to two it doesn't really matter the important thing is that this sequence is bounded from above so let's see how do we do it so in the proof that this is just declaration of the of what we want to prove and here we start uh with this is what we want to prove that this is at least two uh and well at this stage it's not justified but we'll justify it in a moment so now let us use the binomial formula to compute this expression okay and so what we have in using the binomial formula is that okay 1 to the power of n is 1 and then n choose 1 is n times 1 over n and similarly we're going to have here n choose k n to the power 1 over n to the power of k and then 1 to the power of n minus k so we're going to have the sum and now let us rewrite this sum so again since as we said this is n and this is 1 over n then this product is just 1. so we get that this equals for any n that is at least 2 is 1 plus 1 which is at least 2 and those terms are positive so this is already justified and now we will make some computational efforts to show that this is bounded above uh by three so um let's proceed with our computation so we will have to use the formula for the binomial coefficient and now we arrange this expression so basically what we're using that n choose k the general term here it's n factorial divided by uh n minus k factorial and divided by k factorial so um in one of the lectures in my videos i showed that this is in fact the binomial coefficient uh if we take n factorial and divide on the top and divided by n minus k factorial at the bottom this is the expression on top and then we have to divide this by k factorial and this is and times 1 over n to the power of k this is the general term and we sum those terms of those form up to 1 to 1 over n to the power of it and now uh what we're going to do is we'll make some rearrangement because c we have here n to the power of k and here we actually have k terms right because this is n is like n minus 0 and times n minus 1 and up to n minus k minus 1 so here we have actually exactly k terms because from 0 to k minus 1 they're exactly k numbers and now this n one of n's here would be reduced with this n and another n we could have n over n which is 1 minus 1 over n and similarly we'll separate this fraction and just divide by n each such expression so here for example we're going to have 1 minus k minus 1 over n so rearranging this what we have is that we have this sum is actually sum of the following form and notice this pattern that we almost we have here is 1 over k factorial this is the general term and times some coefficient so let's think of this sum as the sum of 1 over k factorial and the factorials appear for nth power from uh 1 up to n all the factorials and all the factorials have this have some coefficient here right and this is 1 over 1 factorial if you want to see this pattern and this one is another extra one on the side so to say and now how this coefficient looks well it's product of numbers one minus one over n and here n is at least two so those numbers are all positive numbers that are strictly smaller than one but as n gets bigger then each term here is closer to one right so if we think of it as like not letting end be very very big or ten to infinity then this term tans almost tends to one we need to be careful here because there are infinitely many terms and we don't want to go into this complication but let's just think of it that this coefficient is strictly positive but no more than 1 here and this is the coefficient of 1 over k factorial i know i'm a bit dwelling on this but this will simplify matters a bit later so uh yeah so this is what we have and now since we proved this product is n is at least 2 and each term here is strictly smaller than 1 and those are positive terms then each coefficient of this one over k factorial so we have this one it is this one and this is one over one factorial and then all those uh factorials that we see here one over two factorial up to one over n factorial here it comes with a coefficient that is strictly smaller than one so if we replace it we remove this coefficient and replace this by one then this sum is of course bigger than the sum right strictly bigger and now we're going to use another identity and uh so here we'll justify it in just one moment basically we are going to uh replace this sum of one over n factorial by this sum of the geometric series and why would we do that well because uh the formula for the geometric series are is uh for the geometric sequence for its sum is quite simple and we have a formula for it so we can compute this and we have a bound from above so using the formula basically what i'm saying here is that each term here if we're starting from um we'll see it in a moment uh so suppose that we have the sum then for this sum as k runs from 1 up to n then the formula for the geometric sequence says that this is this is just the sum this is something known and we've proved it in the previous lecture the formula for the geometric sequence using telescoping series so now we have that this sum is that so we have one minus something positive so what we have here on top is strictly smaller than one and here we have one minus one half which is just two so we have something that is smaller than one times two and this is so all of this is smaller than two and plus one it's strictly smaller than three and now let us justify this replacement well basically uh what we used is that two to the power of n minus one is smaller than n factorial and okay let's see for n equals 1 then it's 2 to the power of 0 it's no not bigger than 1 right and for uh for n which is 2 we have here 2 and this is 1 times 2 it's 2. so it's no bigger than 2 right and then for n equals 3 it also holds it's 2 to the power of 2 okay which is 4 and here it's 1 times 2 times 3 factorial is 6. okay and from this point on we don't from from uh and which is bigger than three we no longer need to check this y well because if n is at least four then what we could say here is that we can split this four into two here and 2 here and then this product of n factorial for n which is at least 4 is product of numbers all of which are at least 2 right so we replace this one by 2 and replace these 4 by 2 and then we have product of numbers which are um we have like uh two times two and all the other numbers are bigger than uh than two here in the product so the product is going to be bigger than the power of two 2 at the power of n minus 1 so of course we have this equality it's a inequality it's kind of obvious and we can also prove it by induction so now what follows is that if to the power of minus one is smaller than a factorial then uh dividing by taking one over we have this bond right so if each term in the sum replaced by one over to the problem n minus one we bound the sum by this sum this one stays this is this one is this one and then this sum can be replaced uh by this sum and we'll have a bound from above and this sum we have already computed therefore the sequence for every n is not bigger than three and so it is bounded above so now let us prove that the sequence is monotone and therefore by the monotone convergence theorem we will see that the sequence converges okay that's i just said this uh so to see this we will compare the formula that we got for the ants element and for the n plus four for uh first element and uh n plus one element i'll see uh that this one is bigger okay so let's do it so we have computed that the formula for the nth element of the sequence is the is is the sum of this form where we have uh this one say other side and then we have 1 over k factorial with this coefficient which is strictly smaller than 1 and we have all these uh factorials from 1 up to n okay and so how would how would the formula for a n plus 1 look well we would have all the same uh coefficients of the factorials of one of the factorial reciprocals so to say so this one is this one and this one is one over one factorial that's the same and the coefficient here so here we're going to have um n plus one over n uh plus one and so this number is strictly bigger this coefficient of 1 over 2 factorial is strictly bigger than this coefficient y because 1 over n plus 1 is smaller than 1 over n and therefore 1 minus this will be bigger and similarly for the general term 1 over k factorial well we just have that all these numbers here are smaller and y minus this is bigger so the coefficient of the reciprocal of k factorial here is bigger than the corresponding coefficient here and this holds for all the terms plus another term that we have here is 1 over n plus 1 factorial which does not appear in the sum but this is a positive term so each each each element that appears here appears here with a bigger coefficient each reciprocal factorial appears here in this sum with a bigger coefficient plus another positive term so of course a n plus 1 is bigger or yeah not not smaller than a a n and therefore that yeah i just stated this in words and therefore we have yeah so we see that this term is strictly smaller than this term and all the reciprocals of factorials appear here and therefore this contains an extra and also this contains an extra positive term so which is positive so this um so a n plus 1 is strictly bigger than a n or bigger or equal for every n which proves that the sequence converges right so this proves that a n is smaller equal to n plus one for every n which is at least one which means that the sequence is monotonically increasing bounded above by three and thus by the monotone convergence uh theorem this sequence converges and let us denote this limit to which it converges by the number e now e is a very important number in all of mathematics and in science in general and it is e after the mathematician learner who discovered this number uh as he was was trying to he computed the derivative of the logarithm and he found that this is actually the natural basis so if you take the derivative we will see it in the future of log in the basis of a of x this function f of x then basically the only basis the only value for a for which the derivative of this log a to x is one over x is this natural basis and so this number is called after euler and uh well it's its value can be calculated approximately uh we can use the sequence by the way there's a even more efficient way to compute this and its numerical value approximate numerical value is given by this of course its goal goes indefinitely um yeah so they say it's after euler and okay this number is super important and it's just as important i i would say as pi and also euler has discovered if this number isn't enough euler has also discovered another formula which is absolutely magical and marvelous and probably one of the most beautiful formulas in all of mathematics it's beyond the scope of the current course but i will just mention it look at this formula it connects the five of the most important numbers in all of mathematics it's the basis of the natural logarithm e it's of course pi it's the identity it's the identity additive identity element in the field of the real numbers or the complex numbers it's the multiplicative identity and uh the square root of minus one uh the the complex number so we will not go uh uh into it what it exactly means but this marvelous connection is just amazing it shows just how beautiful mathematics is and moreover just one remark uh about e is that this number we will prove in the course later that this number is actually irrational so it goes indefinitely and there is no repetitive pattern here and there is no pattern that repeats itself otherwise it would be rational so it's rational and and we will prove it in this course and with more efforts it's also possible to use techniques from calculus to prove that this number is transcendental which means that there is no polynomial with integer coefficients such that e is the root of this polynomial so for example if you look at the square root of 2 for example then this number is a rational but it's a root of the polynomial x squared minus 2 which has integer coefficients so now let us see what we have uh seen in this computation we have seen i want to get back to this point so we have seen that the nth term in the sequence here right this is the ants element of the sequence it is smaller than this sum but actually it turns out that this uh nth element of the sequence this is a n and the sum conver they converge to the same limit and we will see that the more efficient way of computing e with the taylor series will prove it in the future is to compute this sum and so just let us let's just see in decimals how it looks so here we have a list of numbers from 1 up to 10 000. this is our sequence a n and the values of a n and a n are plotted here in green and well in decimals the constant e is known and this is this constant and this is this limit to which both sequences converge and b n is the sequence defined by this sum of the reciprocal factorials i k runs from uh from zero to n and remember that the factorial of zero is one and we see that uh this sequence uh bn of course bigger we computed that the sequence is bigger but they converge to the same limit it's just it's not a proof but i'm showing this to you just visually to see how then here quite rapidly they uh they're close to this number and this sequence bn converges really really really fast to do this value to the limit okay so this is it's about e but of course we'll get back to this number in the future |
10965 | https://byjus.com/cbse-notes/cbse-class-11-physics-notes-chapter-15-waves/ | What are the types of waves Doppler Effect Frequently Asked Questions on CBSE Class 11 Physics Notes Chapter 15 Waves
According to the CBSE Syllabus 2023-24, this chapter has been renumbered as Chapter 14.
What are the types of waves?
We can witness four types of waves. They are:
Mechanical Waves- can exist in material media and follow Newton’s laws.
Transverse waves are referred to those whose particles oscillate in a perpendicular motion of the direction of propagation of the wave
Longitudinal waves are referred to those whose particles oscillate along the way of the propagation of the wave
When the waves move from one point of the medium to another is called a progressive wave
For more information on Waves, watch the below videos
Also See: Wave Motion
Wavelength of a wave
In the case of a progressive wave, the distance between two points in the same phase at that particular time period is known as the wavelength of a wave. The distance is twice the number of two consecutive nodes and antinodes.
Time Period of oscillation
When an element of a medium takes time to move through one complete oscillation then it is called a time period.
Principle of superposition of waves
In a medium when multiple waves transverse simultaneously, the displacement is the algebraic sum of the displacements due to each wave. This phenomenon is referred to as the principle of superposition.
Standing waves
When two identical waves moving in opposite directions interfere, it results in a standing wave. These waves are characterized by zero displacement locations which are fixed and are called nodes and locations of maximum displacements called antinodes.
Related link: Travelling wave
Doppler Effect
The change in the frequency of a wave when the source or the observer or both are moving relative to the medium. This phenomenon is used in different scientific aspects such as planetary science wherein astronomers depend on this effect to identify planets exterior to the solar system. Doppler Effect is an increase (or decrease) in the frequency of sound, light, or other waves as the source and observer move towards (or away from) each other.
| |
| Also Access |
| NCERT Solutions for Class 11 Physics Chapter 15 |
| NCERT Exemplar for Class 11 Physics Chapter 15 |
For more information on Standing Wave and Doppler Effect, watch the below videos
Important Questions
A string of mass 3 kg experiences a tension of 300 N. The stretched string’s length is 20 m. Calculate the time taken by the disturbance to reach the other end when one end is struck with a transverse jerk.
From a tower of height 400 m, a stone is dropped from the top into the water of the pond nearby. Calculate when the splash is heard at the top and the speed of sound in air is 340m/s (g=9.8 m/s2)
The length of the steel wire is 14 m and the mass is 3 kg. Find out the tension in the wire where the speed of the transverse wave on the wire equals the speed of sound in air at a temperature of 30 degrees= 353 m/s
Also Read:
| | |
--- |
| Doppler Effect | Wavelength of Light |
Frequently Asked Questions on CBSE Class 11 Physics Notes Chapter 15 Waves
Q1
What is a travelling wave?
A wave in which the positions of maximum and minimum amplitude travel through the medium is known as a travelling wave.
Q2
What is meant by the superposition of wave?
The superposition principle states that when two or more waves overlap in space, the resultant disturbance is equal to the algebraic sum of the individual disturbances.
Q3
What is meant by time period?
A time period is the time taken for one complete cycle of vibration to pass a given point.
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10966 | https://www.youtube.com/watch?v=qGTYSAeLTOE | Introduction to rates | Ratios, rates, and percentages | 6th grade | Khan Academy
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Learn how rates and how they're related to ratios.
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Transcript:
[Voiceover] What I want
to explore in this video is the notion of a "rate." So, let's look at some examples of rates that you've probably encountered
in your everyday life. So, if you're driving in
your car down the road, and you're looking at the speedometer, you might see that it
says that you are going 35 M-P-H, where the M-P-H stands for 35 miles per, per hour. Well, what's that saying? That's saying, well,
every hour, how many miles are you going if you were to
stay at that current rate. So, it could be a measure of speed. How much distance are you
covering per unit time? And, most typically, when
people talk about rates, that's what they're talking about. They're talking about
how much of something that is happening per unit time. And, it doesn't have to be
even distance per unit time, you might have a, you
might have your hourly rate for someone who is doing
some type of a job. They might say that they're making, they're making $10, so they're making $10. And, actually, let me
write the dollars out so the units become a
little bit more obvious, 10 dollars, dollars per hour, dollars per, dollars per hour. And so, once again,
this is how much money. It's not talking about distance anymore. How much money is being
earned per unit time? And, so, even though rates
are often associated with how much something is
happening per some unit time, and it could be miles per hour, or it could be meters per second, or, in this case, it could be a wage, it could be dollars per hour. Rates don't have to be
just in those terms. In fact, you might say, "All right, "I have a dessert that I really enjoy, "but I'm very conscientious "about, about the number of
calories that I consume." And, you might, you
might see something like, there are 200 calories, calories per serving, per serving. And, so, this is telling
us the number of calories per a serving. And they'll tell us what a serving is. A serving might be a cup or
eight ounces or whatever else. And, so, I could say, "Okay,
look, if I have two servings, "then I'm gonna have 400 calories. "Same way, if I work two hours,
I'm gonna have 20 dollars. "If I, or if I go two hours, "I'm gonna go 70 miles." So, rates give you a sense. It's like, how fast is
something happening? Or how much of one thing is happening for every time something else happens? Now, I can write rates so they look an awful lot like a ratio. And, these words are,
actually, very related, 'cause you see that even
how they're written. R-A-T, R-A-T. Their roots are coming
from the exact same idea. In fact, this rate over
here, 35 miles per hour, it could come from, "Hey, I just, I just went
35 miles in one hour, "what's the ratio?" So, the ratio of miles to hours. And, then, you could
say, "Well, I went 35, "the ratio miles to hours "was 35 to one." Or it could have been, maybe it was 70 to two or something like that. But, that could have been reduced to 35 to one. So, as a ratio, you would typically see
it written like this... Or maybe see it written like,
see it written like this... And, sometimes, you might
even see it written like this, 35 miles to one hour. But, now it's starting
to resemble more of the special case of a ratio,
which we call a "rate." Because, this is the same thing as 35. Instead of writing it
out "miles per hour," you'll often see it written like this, miles per, miles per hour. So, these are very, very related ideas. If you find the ratio between
calories and servings, well, then, you're going
to be able to write, you're going to be able
to express it as a rate and vice versa. Now, why do we care about rates? Well, especially if we're
thinking about things like speed, without rates, it would
be hard to quantify how fast things are happening. Otherwise, we'd be in a
world where we're saying, "Hey, I'm faster than you," or "She's faster than me." But we wouldn't be able to quantify exactly how fast they are. But with rates, we can
say, "Hey, that person ran "a hundred meters in 10 seconds, "they run 10 meters per second." We can quantify exactly how
fast that thing is happening, the rate at which it is happening. Here, instead of saying,
"Hey, a cup of that "is gonna give you, is
gonna give you more energy, or, maybe, contribute more to your weight than a cup of that, and making
these relative comparisons, here, you can actually, you
can actually quantify things. And when we study rate, we're gonna study rate
a lot in mathematics. It's gonna be essential in algebra when we look at the rate of change of a line, how far it moves
in the vertical direction relative to the horizontal direction. We're gonna call that "slope". And you can even imagine
the slope of a hill as how fast is it climbing for
as much as you move forward. But we're also gonna study rates in detail when we go to calculus. In fact, the whole basis
of differential calculus, that you might see later in
high school and early college, is all about measuring instantaneous rate. How fast is something going right now? So, rates are really, really interesting, really, really important. And, I would guess that, if you just look around your life, even over the next few hours, you're going to encounter many, many, many rates. |
10967 | https://oap.unige.ch/journals/sdk/article/view/1271 | Did atmospheric thermal tides cause a daylength locking in the Precambrian? A review on recent results | Sedimentologika
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Vol. 2 No. 1 (2024)/
Publications
Did atmospheric thermal tides cause a daylength locking in the Precambrian? A review on recent results
Authors
Jacques Laskar IMCCE, CNRS, Observatoire de Paris, PSL University, Sorbonne Université, 77 Av. Denfert-Rochereau, 75014, Paris, France
Mohammad Farhat IMCCE, CNRS, Observatoire de Paris, PSL University, Sorbonne Université, 77 Av. Denfert-Rochereau, 75014, Paris, France
Margriet L. Lantink Department of Geoscience, University of Wisconsin-Madison, Madison, WI 53706, Unites States
Pierre Auclair-Desrotour IMCCE, CNRS, Observatoire de Paris, PSL University, Sorbonne Université, 77 Av. Denfert-Rochereau, 75014, Paris, France
Gwenaël Boué IMCCE, CNRS, Observatoire de Paris, PSL University, Sorbonne Université, 77 Av. Denfert-Rochereau, 75014, Paris, France
Matthias Sinnesael IMCCE, CNRS, Observatoire de Paris, PSL University, Sorbonne Université, 77 Av. Denfert-Rochereau, 75014, Paris, France
DOI:
Keywords:
Milankovitch cycles, Thermal atmospheric tides, Earth-Moon distance, Tidal friction, Precambrian Earth, LOD resonant locking
Abstract
After the initial suggestion by Zahnle and Walker (1987) that the torque accelerating the spin rate of the Earth and produced by the heating of the atmosphere by the Sun could counteract the braking luni-solar gravitational torque in the Precambrian, several authors have recently revisited this hypothesis. In these studies, it is argued that the geological evidence of the past spin state of the Earth plays in favor of this atmospheric tidal locking of the length of the day (LOD). In the present review of the recent literature, we show that the drawn conclusions critically depend on LOD estimates based on stromatolite growth band data of Panella at 1.88 and 2.0 Ga which are subject to large uncertainties. When only the most robust cyclostatigraphic estimates of the LOD are retained, the LOD locking hypothesis is not supported. Moreover, our consideration of the published General Circulation Model numerical simulations and of a new analytical model for the thermal atmospheric tides suggest that the atmospheric tidal resonance, which is the crucial ingredient for the LOD locking in the Precambrian, was never of sufficiently large amplitude to allow for this tidal LOD lock.
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Lay summary
Due to tidal interaction between the Earth and the Moon, dissipation occurs, slowing Earth's rotation while causing the Moon to move away from the Earth at a present rate of 3.8 cm per year. Laser reflectors left by Apollo astronauts on the Moon's surface enable extremely precise measurements of this recession. Meanwhile, rock samples brought back from the same Apollo missions helped estimate the Moon's age at 4.25 Ga. Until recently, no physical model could account for Moon's history from its formation near the Earth to its present position. This gap was filled by a scenario fitting both the age of the Moon and its measured present recession (Farhat et al., 2022). Despite agreement with geological evidence, alternate propositions arose last year. One such hypothesis, a revival of ideas proposed by Zahnle and Walker (1987), suggests Earth’s spin stalled in the Precambrian due to a competition between gravitational and thermal atmospheric tides, decelerating and accelerating Earth’s spin, respectively. Thermal atmospheric tides are currently only a small fraction of the complete tidal pull on Earth, but in the past, they may have increased due to a resonance. To clarify this situation, we provide here an extensive discussion of recent literature and conclude that thermal tides were likely never strong enough to balance gravitational tides.
1. Introduction
Since the work of George Darwin, it is known that the body tides exerted by the Sun and Moon on Earth slow down the spin of the Earth and make the Moon recede away from the Earth (Figure 1) (Darwin, 1879; MacDonald, 1964; Goldreich, 1966; Kaula, 1964; Mignard, 1979, 1980; Hut, 1981; Touma & Wisdom, 1994; Néron De Surgy & Laskar, 1997). More elaborate tidal models take into account the oceanic tides, which also slow down the rotation of the Earth and let the Moon go away from the Earth (Webb, 1982; Green et al., 2017; Tyler, 2021; Daher et al., 2021), but none of these tidal models could fit both the present tidal recession of the Moon of 3.83±0.008 cm/yr (Williams & Boggs, 2016) and the age of the Moon of 4.425±0.025 Ga (e.g., Maurice et al., 2020). Elaborated along the lines of Webb (1982), the recent semi-analytical model of Farhat et al. (2022) provides a coherent scenario for the Earth-Moon tidal evolution, with an excellent fit to the present recession rate and the age of the Moon that were included as constraints on the model. It starts with a global ocean in the ancient eons, and then switches to a hemispheric ocean model similar to Webb (1982), but which follows the continental evolution in the most recent 1 Ga. In order to avoid possible circular arguments, no geological proxies of the Earth-Moon distance evolution were used for the elaboration of the model. Despite this, the new model is in good agreement with such geological estimates of the past Earth-Moon distance, and in particular with the recent values obtained by cyclostratigraphic methods (Figure 2). Nevertheless, following the original suggestion of Zahnle and Walker (1987), two recent papers propose that the spin of the Earth was trapped in a resonance between the thermal atmospheric torque and the solid and oceanic torque during the Precambrian Era (Mitchell & Kirscher, 2023; Wu et al., 2023). They also suggest that this stalling of the Earth’s spin rotation speed is correlated with the so-called boring billion (e.g., Holland, 2006). Both studies advocate that this locking of Precambrian day-length is supported by geological observations. However, such a scenario is incompatible with the Earth-Moon evolution presented in Farhat et al. (2022). In this review, which is aimed to the stratigraphic community, we will explicit the differences in the modelling approaches and discuss the use of geological data that can be compared with the models.
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Figure 1.In the Darwin model, the Moon produces a tidal bulge on the Earth, but as the Earth is not totally elastic, this bulge is driven by the fast rotation of the Earth slightly off from the Moon direction (angle δ). This results in a braking torque that slows down the spin of the Earth. By conservation of the angular momentum in the Earth-Moon system, the Moon moves away from the Earth.
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Figure 2.Evolution of the length of day (LOD) in the past following the model of Farhat et al. (2022). The nominal LOD values are in purple, with the uncertainty indicated by the blue lines (adapted from Figure 5 of Farhat et al., 2022). The circles represent the available cyclostratigraphic data with their uncertainty, when available, from various sources: references within Farhat et al. (2022) in blue, Zhou et al. (2022) in red, and Zeeden et al. (2023) in orange. Tidal rhythmites values are represented by yellow squares, with references from Farhat et al. (2022) with the addition of a data point at 3.2 Ga from the Moodies Group (Eulenfeld & Heubeck, 2023).
1.1. Atmospheric thermal tides
Atmospheric thermal tides have been recognized since the 18th century (see Wilkes, 1949). Due to the heating of the atmosphere by the Sun, the atmosphere locally expands and the pressure decreases at the subsolar point, which induces a redistribution of the mass of the atmosphere, with two main components: a diurnal component, which at equilibrium is opposite to the subsolar point, and a semi-diurnal component, orthogonal to the Sun direction (Figure 3). As for the solid tides, since the Earth spin rotation is faster than its orbital motion around the Sun, the Earth rotation drags these atmospheric bulges with a positive offset from their equilibrium position. At present, the gravitational attraction of the Sun on these bulges induces an accelerating torque, opposite to that of the solid and ocean tides (Chapman & Lindzen, 1970; Goldreich & Soter, 1966; Gold & Soter, 1969; Ingersoll & Dobrovolskis, 1978; Dobrovolskis & Ingersoll, 1980; Correia & Laskar, 2001, 2003; Correia et al., 2003; Leconte et al., 2015; Auclair-Desrotour et al., 2017, 2019).
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Figure 3.Thermal atmospheric tides. Due to the heating of the Sun at the subsolar point, a redistribution of the atmosphere creates two main components: a diurnal component (in light blue), and a semi diurnal component (in blue) that are offset from their equilibrium position by an angle δ because of the Earth’s fast rotation speed (ω). They create an accelerating torque on the Earth's spin motion.
1.2. A possible lock of the length of the day
At present, the thermal atmospheric tidal torque is a small part (∼6.4%) of the solid and oceanic tidal friction (Volland, 1990; Farhat et al., 2024), but there are two elements that can change this ratio in favor of the atmospheric thermal tides. First, the atmospheric torque amplitude is dependent on the spin rate of the Earth. As for the oceanic tides, there exists a known resonance of the planetary Lamb wave (Bretherton, 1969), that we name here the Lamb resonance (Farhat et al., 2024), occurring for a faster spin value of the Earth, where the atmospheric torque is largely increased (Lindzen & Blake, 1972; Zahnle & Walker, 1987; Bartlett & Stevenson, 2016). In addition, the oceanic tidal friction is at present close to a resonance, but its value was smaller in the past, for a faster rotation spin value (e.g., Farhat et al., 2022). These elements led Zahnle and Walker (1987) to propose that at some time in the Precambrian, the accelerating thermotidal torque counter-acted the braking luni-solar gravitational tidal torque, which led to a lock of the length of the day (LOD) at about for an extended period of more than 1 Ga. This hypothesis, recently revisited by Bartlett and Stevenson (2016), was probably motivated by the difficulty to fit the existing geological indicators of the past Earth-Moon distance and LOD with more simple models (e.g., Williams, 2000; Hinnov, 2018; Laskar, 2020).
Both in Zahnle and Walker (1987) and Bartlett and Stevenson (2016), a large part of the observational constraints was provided from the Precambrian estimates of the LOD resulting from the analysis of stromatolite deposits (Pannella, 1972a, b) (Figure 4).
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Figure 4.LOD locking due to thermal atmospheric tides resonance for the scenarios of Zahnle and Walker (1987) (in red) and Bartlett and Stevenson (2016) (in grey). The geological indicators that are plotted are limited to pre-2016 published results, adapted from Laskar (2020) and Williams (2000) (see references therein). The dotted red line is the LOD provided by equation 41 of Laskar et al. (2004) with a simple Darwin tidal model. The dotted black line is an empirical fit using a simplified tidal model adjusted to the geological data (Walker & Zahnle, 1986). The stromatolite data points at 1.88 and 2.0 Ga are from Pannella (1972a, b).
2. Geological archives for Precambrian LOD estimates
In his review of geological archives for LOD estimates, Williams (2000) mentioned several varieties of bio-archives (bivalves, corals, brachiopods) which we will not consider here, as they were not present in the Precambrian. For these, one can also consult Rosenberg et al. (1975) or Lambeck (1980). We will concentrate here on the stromatolites (2.1), the tidal rhythmites (2.2), and the promising cyclostratigraphic records (2.3).
2.1. Stromatolites
The model of Bartlett and Stevenson (2016) relies heavily on the adjustment to stromatolite data (Pannella, 1972a, b), although the authors themselves warn the reader, and we can quote from Bartlett and Stevenson (2016, p.7): “However, these data, particularly the early stromatolite data ( Pannella, 1972 ), should not be taken too seriously ( Zahnle & Walker, 1987 ) . Paleontologists Scrutton ( 197 8) and Hofmann ( 1973 ) also found these data to be unreliable and unsuitable for precise quantitative analysis.” These data have also been used in a crucial manner in the recent studies (Mitchell & Kirscher, 2023; Wu et al., 2023; Bao et al., 2022).
Stromatolites are layered organo-sedimentary structures formed by the capture of sediments by microbial organisms, typically formed in shallow marine and lacustrine environments. They are some of the oldest known forms of life, and are an important archive for studying Precambrian paleoenvironments. Studies of recent analogues have shown that daily rhythms of biological growth and binding of sediment can be preserved in stromatolite layering, as the microorganisms (algae) respond actively to daylight (e.g., Logan et al., 1964; Monty, 1967; Davies, 1970; Gebelein, 1969). In addition, environmental fluctuations influence rates of growth and the supply of material, i.e. layering thickness, including tidal and seasonal variations (Gebelein & Hoffman, 1968; Pannella, 1976, 1972b). As such, investigations have been made of diurnal growth layers interpreted from fossil stromatolites that can be structured in larger tidal and seasonal banding; these observations have in turn been used to estimate past LOD and the length of the lunar month. The studies of Precambrian stromatolites by Pannella (1972a, b) are well-known examples that are often used in long-term reconstructions of the past Earth-Moon system (e.g., Bartlett & Stevenson, 2016; Bao et al., 2022; Mitchell & Kirscher, 2023; Wu et al., 2023). The three oldest, most critical LOD data that are often referenced are based on stromatolite sequences from the upper Paleoproterozoic Gunflint Formation (Fralick et al., 2002) and age-correlative Biwabik Formation (Lake Superior region), and the Great Slave Supergroup (Northwestern Territories). However, as emphasized in many studies (e.g., Hofmann, 1973; Scrutton, 1978; Lambeck, 1980), and by Pannella himself in the original publications, these stromatolite-based estimates are rarely, if ever, suitable for precise quantitative interpretation. The formation of daily laminae depends on a fine environmental balance that can easily be disturbed. Stromatolite growth patterns are therefore rarely considered to be complete due to periods of non-deposition and/or post-depositional erosion by for instance storms. For this reason, counts or sequences of laminae should generally be interpreted as minimum estimates, and not as most likely values (Pannella, 1972a, b; Lambeck, 1980). Another source of uncertainty arises from ambiguities associated with determining the exact number of daily laminae per lunar monthly or annual bundle, which is often characterized by a low reproducibility rate.
A lack of internal self-consistency can be noted in the reported growth rhythm hierarchies of the Paleoproterozoic stromatolites studied by Pannella (1972a, b), highlighting the discontinuous nature of the sequences and leaving room for alternative interpretations. More specifically, the counts of the number of diurnal laminae per larger seasonal growth bands yielded a significantly lower interpreted number of days per year than is implied by the observed number of daily increments between lesser growth marks that were believed to indicate the synodic month, obtained from the same sequences. For example, in stromatolites from the ca. 1.88-Ga Gunflint Formation, a maximum of 39 and a mean of 33.4 small-scale laminae were observed per medium-scale growth band interpreted as the synodic month. This ratio implies more than 532 days/year when multiplied by the maximum of 28 observed (and interpreted) fortnightly cycles/year, compared to only 448 diurnal increments that were directly counted within a full seasonal band (Pannella, 1972a). However, for specimens from the time-equivalent Biwabik Formation, Mohr (1975) arrived at a very different growth rhythm hierarchy and conflicting interpretation, suggesting a synodic month of less than 26 days. For the Biwabik Formation, Pannella (1972b) himself counted only 310 diurnal laminae/year directly, while obtaining 442 days/year through extrapolation of interpreted monthly patterns.
Lastly, past LOD estimates based on stromatolites are typically challenged by a lack of (sufficiently precise) independent temporal control on sedimentation/growth rates as well as the absolute age of deposition. This is a challenge that is, however, not unique to the stromatolite archive (tidal rhythmites, for instance, feature similar challenges). Of specific concern here is the uncertainty in the age of the 'Great Slave Supergroup' stromatolite studied by Pannella (1972b). In the literature, this datum point is usually placed around 2 Ga, following the compilation figure of Williams (2000) and given a lack of stratigraphic details provided in (Pannella, 1972b). However, this sample most likely originates from the Pethei Group (Pope & Grotzinger, 2000; Hoffman, 2023, pers. comm.), the current best age constraints of which indicate a significantly younger age between 1.889 and 1.867 Ma (Hoffman et al., 2023). In conclusion, while stromatolites can provide useful insights in past tidal dynamics, they should typically not be considered as providing accurate or precise numerical values for past LOD reconstructions.
2.2. Tidal rhythmites
Tidal rhythmites are laminae deposits related to semi-diurnal or diurnal tidal cycles that can occur in estuaries or deltas. Silty and muddy sediment is carried by ebb tidal currents. These currents transport the sediment in suspension through the main ebb channel to deeper offshore water, where it settles and forms graded layers. During slack water between tides, muddy caps can be deposited on the sandy laminae. The amount of sediment carried by ebb tidal currents, its grain size, and the effectiveness of tides in transporting and depositing sediment, are directly related to the tidal amplitude (Williams, 2000). In an ideal scenario, the analysis of the time series of the thickness of these laminae should allow recovery of all tidal periodicities, such as:
The lunar day, which is the interval between two passages of the Moon at the meridian.
The lunar synodic month, which separates two full Moons, and is thus recovered by the recognition of spring tides of large amplitude, occurring at syzygy, when the Moon and the Earth have same longitude as seen from the Sun.
The tropical year, separating two passages of the Earth at the spring equinox (also called vernal equinox), possibly determined as the time of maximal tidal amplitude in the year.
Finally, if the record is sufficiently long, the nodal period of the Moon (18.6 yr at present) (Walker & Zahnle, 1986).
Although very promising, this method has its drawbacks. The locations where high-quality sequences of such laminae formations can be observed are rare. Moreover, they often lead to divergent analyses when the deposits are analyzed by different groups. The Weeli Wolli Formation in Western Australia, dated 2.45 Ga, was interpreted by Walker and Zahnle (1986) on the basis of the lunar nodal cycle, and led to a reconstructed Earth-Moon distance of Earth radius , while the analysis of (Williams, 1989,1990 resulted in a much larger value of , with the analysis of laminae couplets, grouped in synodic fortnightly increments, and annual cycles. Although the results of these two studies are still consistent within uncertainty (given the very large error bars), these estimates are based on two fundamentally different and mutually incompatible interpretations of the same layering patterns. In the same way, the analysis of the Elatina Formation in South Australia, dated 620 Ma, led Williams (1989, 1990) to determine an Earth-Moon distance of while Sonett and Chan (1998) derived from their analysis of the same sequence. Sonett and Chan (1998) also re-analyzed their previous determination of the Earth-Moon distance for the Big Cottonwood Formation in Utah, at , and found a value corresponding to an Earth-Moon distance of , while their previous determination of the same sequence was (Sonett et al., 1996). We note that the new determinations from Sonett and Chan (1998) are in agreement with the new model of Farhat et al. (2022) (Figure 2). However, given the difficulties associated with interpreting Earth-Moon parameters from these tidal sequences, additional independent studies should be required to further verify the determination of Sonett and Chan (1998).
2.3. Cyclostratigraphy
Due to the gravitational interactions of the Earth with the other planets, the orbital plane of the Earth is moving in a complex motion composed of a slow rotation (precession of the node) and a composition of nearly periodic motions that make the inclination of the Earth orbital plane oscillate (e.g., Laskar, 2020). This induces an oscillation of the tilt of the Earth (or obliquity, ε) of present amplitude 1.3 degrees around its averaged value. The variation of insolation on the surface of the Earth depends also on the precession of perihelion, and on the variation of eccentricity, which are dominated by the so-called long eccentricity term of 405 kyr period and the short eccentricities of main periods 95 kyr and 124 kyr. Finally, the pull of the Moon and Sun on the equatorial bulge of the Earth induces a slow precessional motion of its spin axis at 50.475838 arcsec/yr that corresponds to a period of about 26 kyr (Laskar, 2020). Neglecting the eccentricity of the Earth and Lunar orbit, and the inclination of the Moon, the lunisolar precession frequency can be expressed as , where is the obliquity and where the precession constant is equation 4.14 from Laskar (2020):
(1) α=3/2 G [(m⊙/a 3⊙)+(m M/a3M)] (E d0/γ² 0)γ
where Gis the gravitational constant, ⊙ refers to the Sun, and M to the Moon, m and a are the masses and semimajor axes, γ the Earth's spin angular velocity, γ0 its present value, and Ed0 the dynamical ellipticity at present. As a⊙ can be considered as constant, the precession constant α thus depends mostly on the evolution of aM and γ, which evolve with time under tidal dissipation (highlighted in bold in equation (1)).
The resulting changes in insolation drive climatic changes on Earth (astronomical climate forcing) that can be recorded in the Earth's sedimentary archive. These sediments can be studied today (e.g., Gradstein et al., 2004, 2012, 2020; Montenari, 2018) and inform us on past astronomical changes.
Over very long timescales, beyond 60 Ma, the planetary orbital motions can no longer be reconstructed with accuracy (Laskar et al., 2011a, b), but for the Earth-Moon evolution, the tidal dissipation will dominate, and a reconstruction of the past evolution of the Earth-Moon distance can still be achieved. The variation of and will induce a change in the precession period that can be imprinted in the sedimentary record (Berger et al., 1992; Meyers & Malinverno, 2018). In a reverse way, the determination of the precession frequency from the sedimentary record, and the use of a dynamical model that will link the semi-major axis to the angular spin velocity of the Earth can allow retrieval of both and . This determination requires a time-scale for the sedimentary record, which can be provided either by absolute radiometric age dating, or by the use of the 405 kyr eccentricity period as a metronome for stratigraphic cycles (see Laskar, 2020 and references therein). In recent years, this technique for determining the past state of the Earth-Moon system has made much progress, and many groups have obtained converging results using various methods for the determination of the precession frequency (e.g., Meyers & Malinverno, 2018; Lantink et al., 2022; Zeeden et al., 2023). Moreover, these data are in good agreement with the tidal model of Farhat et al. (2022) (see Figure 6 of Farhat et al., 2022, and Figure 5 of Zeeden et al., 2023).
Another crucial advantage of cyclostratigraphy, compared to, for example, stromatolites and tidal rhythmites, is the greater potential of independent age control by, for example, radioisotopic geochronology and integrated stratigraphic approaches. Using an integrated stratigraphic approach it is also possible to verify interpretations in time-equivalent sections that should have the same time-dependent astronomical signatures (e.g., Olsen et al., 2019; Sinnesael et al., 2019). The coherence of these sets of data leads us to consider that they are the most robust among the geological proxies for the determination of the past precession frequency of the Earth and determination of Earth-Moon system parameters. It should nevertheless be stressed that these parameters do not have the same status in cyclostratigraphic studies. The only primary quantity determined directly with cyclostratigraphy is the Earth luni-solar precession frequency. The other parameters of the Earth-Moon system (Earth-Moon distance and LOD) are retrieved only through a dynamical model that should not be limited to the Earth-Moon system. Indeed, one should consider the exchange of the angular momentum in the Sun-Earth-Moon system, and also the variation of this angular momentum through various mechanisms, the largest one probably being atmospheric thermal tides, although this latest can be neglected in a first approximation.
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Figure 5.Comparison of the data and fit of Mitchell and Kirscher (2023) (black and red lines) with the model of Farhat et al. (2022) (purple curve with uncertainty in blue). As in Figure 2, LOD (in hours) is plotted against age (in Ma). Tidal rhythmites are yellow squares, and cyclostratigraphic data are color circles (light blue with references in Farhat et al. (2022), red from Zhou et al. (2022), and orange from Zeeden et al. (2023)). The stromatolite data from Pannella (1972a, b) are highlighted with a dark green circle while the cyclostratigraphic data from Grotzinger (1986) is circled in light green. Adapted from Figure 2 of Mitchell and Kirscher (2023).
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Figure 6.Comparison of the data and fit of Wu et al. (2023) (in black) with the model of Farhat et al. (2022) (purple curve with uncertainty in blue). As in Figure 2, tidal rhythmites are yellow squares, and cyclostratigraphic data are color circles (light blue with references in Farhat et al. (2022), red from Zhou et al. (2022) and orange from Zeeden et al. (2023)). The stromatolite data from (Pannella, 1972a, b) are highlighted with a dark green circle. Note that these data points seem to be misplaced in the Wu et al. (2023) figure reproduced here (see note 1). Adapted from Figure 3 of Wu et al. (2023).
3. Discussion of the recently published results
The solution of Farhat et al. (2022) is in agreement with the most recent determinations of tidal rhythmites (Sonett & Chan, 1998) and with the recent cyclostratigraphic data (Meyers & Malinverno, 2018; Lantink et al., 2022; Zhou et al., 2022; Zeeden et al., 2023) (Figure 2). One could thus think that the fate of the thermal tides locking hypothesis was settled. However, the recent publication of the two papers (Mitchell & Kirscher, 2023; Wu et al., 2023) in major journals requires some additional discussion to clarify the situation.
3.1. Mitchell and Kirscher (2023)
In their compilation of geological constraints of the Precambrian length of the day, Mitchell and Kirscher (2023) have included most of the available data. Their analysis is purely empirical. They search for the best linear fit, made by pieces over sequences of data. One could wonder on the status of their fit, which is not continuous, as a piecewise linear model would be. The goal is thus not to find an empirical model for the Earth-Moon evolution, but to search for the best fitted trends in the LOD over extended periods. From these fits, they conclude that a LOD lock probably occurred between 1 Ga and 2 Ga. When comparing to the results of Farhat et al. (2022) (Figure 5), one can observe that the stromatolites data at 1.88 Ga and 2 Ga from Pannella (1972a, b) are essential for the conclusions of Mitchell and Kirscher (2023). If these data, which are questionable as we discuss in section 2.1, are not taken into account, the fit will no longer lead to this locked value of the LOD between 1 and 2 Ga.
By contrast, the cyclostratigraphic point of Grotzinger (1986) is nearly exactly on the Farhat et al. (2022) curve (Figure 5). It should be noted, however, that the datum point of Grotzinger (1986) was not originally given in that paper but was derived by Mitchell and Kirscher (2023). Grotzinger (1986) only proposed that there is eustatic sea-level cyclicity within the Milankovitch frequency band recorded in platform carbonates from the Rocknest Formation, at a scale of 1 – 15 m and possibly of 75 – 100 m or 75 – 200 m. Mitchell & Kirscher (2023) then assumed that a 10 m cycle represents climatic precession and 87.5 m is related to short eccentricity. However, this interpretation is poorly constrained. Due to the large uncertainty of this data point, we should consider this as a simple coincidence, until some new quantitative analysis in the spirit of Meyers and Malinverno (2018) is performed on the same 1.89 Ga Rocknest Formation sample (Mitchell & Kirscher, 2023).
3.2. Wu et al. (2023)
In the recent work of Wu et al. (2023), the authors presented a new analytical model of thermal tides to address the resonant locking hypothesis. The model's free parameters were constrained such that the resulting thermotidal torque drives an LOD history that best fits their compilation of LOD geological proxies. As previously, in Figure 6, we compare Wu et al. (2023) (black curve) with Farhat et al. (2022) (purple curve). Here again, one can see that the model of Wu et al. (2023) relies heavily on the stromatolite data of Pannella (1972a, b) to establish the resonance locking of the LOD 1. Moreover, the Wu et al. (2023) curve misses entirely the new cyclostratigraphic determinations of the Earth-Moon state at 2.46 Ga obtained by Lantink et al. (2022) in Joffre Gorge, Australia.
Wu et al. (2023) elaborated a physical model to support their claims. Moreover, the authors performed a suite of GCM (General Circulation Model) numerical simulations, using the LMD-G (Hourdin et al., 2006) and PlaSim (Fraedrich et al., 2005) GCMs, to infer the Earth's paleo-temperature evolution that is required to generate the constrained history of the thermotidal torque. We dedicate the rest of this section to discuss the details behind the model adopted in Wu et al. (2023) and its predictions.
1 Note that these data points from Pannella (Pannella, 1972a, b) occur at a different age position in the figures 1 – 3 of Wu et al. (2023), compare to previous LOD com-pilations (e.g., Williams, 2000; Bartlett and Stevenson, 2016; Mitchell and Kirscher, 2023). In particular, we note that the two closely spaced points at ca. 1.63 Ga most likely represent Pannella’s analyses of the time-correlative Gunflint (1972a) and Bi-wabik (1972b) Formations dated at ca. 1.88 Ga (Fralick et al., 2002). However, used literature data were not provided in Wu et al. (2023) to verify this observation.
3.2.1. The modeled gravitational tides: artificial resonances?
The dynamical evolution of the Earth's rotational motion in Wu et al. (2023) is driven by the luni-solar gravitational tidal torque and the solar thermotidal torque. For the former, the authors used the tidal history of Webb (1982), where Laplace's Tidal Equations (the equations describing the tidal response of a shallow fluid layer; LTEs hereafter) were solved semi-analytically over a hemispherical equatorial ocean on the surface of the Earth. While the work of Webb (1982) was seminal in coupling LTEs with the dynamical evolution of the Earth-Moon system, the modeled history of the lunar orbit in Webb (1982) yielded a lunar formation epoch that is incompatible with the geologically constrained lunar age (see Figure 3 of Webb, 1982).
To efficiently remedy the latter discrepancy, Wu et al (2023) tweak the tidal dissipation history of Webb (1982) by normalizing it with a constant factor (Q1), such that the resultant orbital history of the Moon features its proper temporal origin (see their equation 1 and S1). As a byproduct of this modeling choice, the authors have modified the spectrum of oceanic normal modes in such a way that tidal resonances are characterized with artificial amplitudes(see Green et al., 2017; Daher et al., 2021; Farhat et al., 2022).
Though the authors focus on modeling thermal tides, gravitational tides remain the dominant driver of the Earth's rotational evolution, providing the background of the tidal torque upon which the thermotidal counterpart would significantly contribute only in the vicinity of the Lamb resonance. As such, since the authors are constraining the history of the total torque to fit a compilation of geological LOD proxies, an artificially modeled spectrum of gravitational tidal dissipation may yield an artificial spectrum of thermal tides. Namely, the resultant thermotidal history could be characterized by either an artificial timing of the Lamb resonance occurrence, an artificial amplitude of the Lamb resonance, or both.
3.2.2. Atmospheric thermal tides: model limitations
For the thermotidal contribution to the Earth's rotational history evolution, Wu et al (2023) develop a simplified analytical model of thermal tides that is used to compute the thermotidal torque (Equations S28-S29 therein). The model is parameterized by a number of free parameters (16 in total) that are constrained such that the resulting thermotidal torque, added to the gravitational tidal torque, would drive an LOD history that fits the compilation of LOD geological proxies (see Figure 6).
The developed model essentially resembles a band-pass filter, similar to that developed in Bartlett and Stevenson (2016). It ignores the Coriolis force, which may be significant in the case of a fast rotator like the Earth, along with the vertical velocity of tidal waves. The model also assumes an isothermal structure of the atmosphere. This choice is common in the literature of atmospheric dynamics as it simplifies the mathematical framework of the rather complex theory (e.g., Chapman & Lindzen, 1970; Lindzen & Blake, 1972; Auclair-Desrotour et al., 2019). However, for the Earth, atmospheric temperature measurements (e.g., Figures 2.1-2.3 of Pierrehumbert, 2010) show that the massive troposphere (~80% of atmospheric mass) controlling the tidal mass redistribution is characterized by a negative temperature gradient. The latter is in fact closer to an idealised adiabatic profile than it is to an idealised isothermal profile. These modeling choices could deliver inaccuracies in the determination of the resonant period (Farhat et al., 2024). However, this was somewhat compensated by the authors in modeling the resonant period as a free parameter that is constrained by the geological data.
The other essential quantity of interest is the amplitude of the thermotidal torque when the resonance is encountered. The latter is dependent on several variables, of which the least constrained in the case of the Earth is the rate of energy dissipation by the atmosphere. Namely, as the atmosphere is heated by the shortwave incident solar flux and the infrared emission from the ground, it dissipates energy via multiple pathways including radiative cooling and frictional interactions with the surface. As it is difficult to properly model these mechanisms in the analytical theory, energy dissipation is usually modeled by a free parameter (the parameter Q th in the work of Wu et al., 2023). This unconstrained parameter predominantly controls the amplitude and the spectral width of the resonant thermotidal torque and, consequently, the lifetime of the Lamb resonance and whether it was sufficient to counteract the gravitational tide.
Dissipative radiative transfer and atmospheric cooling, however, can be properly accommodated in GCM simulations. To that end, Wu et al (2023) presents, to date, the first study that uses GCMs to simulate the Lamb resonance specifically for the Earth. Their results, using the two aforementioned GCMs, estimate the dissipation parameter to be Q th ≃ 100, which would render the maximum amplitude of the torque insufficient for the LOD locking. For the LOD evolution, however, the authors used values of that are one order of magnitude larger (Q th ≃ 100) such that the thermotidal torque would be sufficient to counteract the gravitational counterpart. Consequently, the used thermotidal torque is amplified by a factor of ~30 relative to its present value 2. The author's reasoning lies in the need for such a large thermotidal torque so that the LOD proxies, specifically the stromatolites in (Pannella, 1972a, b), can be explained. This brings us back to Section 2.1 in questioning the reliability of this data set as a robust constraint for informing dynamical models, especially when present with evidence from GCMs to the contrary.
² Which means that the amplitude of the surface pressure anomaly is amplified by a factor of ~60 as can be inferred from Farhat et al. (2024).
3.2.3. The asymmetry of the Lamb resonance
An interesting signature of the GCM simulations of the Lamb resonance in Wu et al. (2023) lies in the spectrum of the thermotidal torque shown in their Figure S4. We reproduce this spectrum in Figure 7. The GCM spectrum, shown by the black dots, features an asymmetry in the peaks of the Lamb resonance whereby the two peaks of the torque around the resonance do not share the same amplitude. Namely, the accelerating part of the torque has an amplitude that is almost half that of the decelerating part. The former part is required to occur with a sufficient amplitude such that it counteracts the decelerating gravitational torque, but it appears from this spectrum to be reduced. The authors, however, ignored this signature present in the GCM simulations in favor of the spectrally symmetric Lamb resonance obtained from their analytical model, which is shown by the black curve in Figure 7. Note that the change from an accelerative to a decelerative tidal torque depends on the orientation of the tidally generated atmospheric mass redistribution, and consequently of the tidal bulge, relative to the subsolar point (see Figures 3 and 4 of Farhat et al., 2024 for further detail).
In the recent work of Farhat et al. (2024), the authors propose that such an asymmetry can be obtained if one accounts for the thermal inertia budget in the ground and the lowermost atmospheric layer. Namely, due to the thermal inertia in these layers, the infrared heating of the atmosphere by the ground becomes asynchronous with the incident solar flux. This delayed ground response is shown to be responsible for maneuvering the atmospheric tidal bulge in such a way that creates an amplitude asymmetry between the two peaks. In Figure 7, we show by the red curve how the model of Farhat et al. (2024) can properly explain the spectral asymmetry of the GCM-produced spectrum when taking into account the thermal inertia effects. It is important to note that the reduction of the positive peak of the torque goes hand in hand with the relative contribution of the ground in heating the atmosphere. Namely, the more abundant the greenhouse gases are in the atmosphere, which is predicted for the Precambrian from various geological proxies (see e.g., Catling & Zahnle, 2020), the more the atmosphere would be prone to infrared thermotidal heating, and consequently the more the accelerating thermotidal torque would be reduced.
3.2.4. The temperature problem
One naturally wonders how the discussed modeling limitations carry over to the model predictions. The resulting timing of the Lamb resonance occurrence requires a mean Earth temperature in the Proterozoic of , computed by the authors using the PlaSim GCM (see Figure 7 of Wu et al., 2023). Though a warm climatic interval fits evidence on a Proterozoic glacial gap (e.g., Hoffman et al., 2017), such extreme temperatures are in contrast with geochemical analysis using phosphates (e.g., Blake et al., 2010), geological carbon cycle models (e.g., Sleep & Zahnle, 2001; Krissansen-Totton et al., 2018), numerical results of 3D GCMs (e.g., Charnay et al., 2020), and the fact that solar luminosity was 10-25% lower during the Precambrian (e.g., Gough, 1981). Such extreme temperature estimates would also require elevated amounts of partial pressure of , reaching 200 mbar. This exceeds inferred estimates from various geochemical proxies (see the review by Catling & Zahnle, 2020, and references therein). More importantly, however, this temperature increase will enhance the asynchronous thermotidal heating of the atmosphere by the ground in the infrared, as we describe in Section 3.2.2. The latter would significantly attenuate the peak of the tidal torque near resonance, rendering it insufficient for the LOD locking. The adopted model in Wu et al. (2023) did not account for this feedback effect. Moreover, atmospheric dissipation is also enhanced with the increased temperature, as discussed in Farhat et al. (2024), which has an additional effect of attenuating the resonant amplitude of the torque.
↗
Figure 7.The spectrum of the thermotidal torque as a function of the length of day (LOD) adapted from Figure S4 of Wu et al. (2023). The black dots are simulated using the PlaSim GCM, while the solid black curve shows the prediction of the analytical model of Wu et al (2023). The red curve shows a fit to the GCM results using the analytical model of Farhat et al. (2024), where the physical effects of the delayed thermal response of the ground were taken into account. These effects proved to induce the notable amplitude asymmetry between the peaks around the Lamb resonance. Namely, the accelerating peak of the tidal torque is reduced in amplitude relative to the decelerating peak.
3.3. Bao et al. (2022)
While they do not invoke a thermal tides LOD trapping, Bao et al (2022) try also to reconciliate the LOD history with the Pannella (1972a, b) data. This time, they propose that between 2 Ga and 1.5 Ga, a sudden growth of the Earth core led to a reduction of the spin rate of the Earth. In Figure 8, we have compared their solution with Farhat et al. (2022) (purple curve). In this case again, the scenario depends crucially on the stromatolite data of Pannella (1972a, b). If these data are removed, there is no longer the necessity to search for some peculiar scenario, and it can be recognized that the model of Farhat et al. (2022) fits most of the reliable geological data in a satisfactory manner.
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Figure 8.Comparison of the data and fit of Bao et al. (2022) (in solid blue) with the model of Farhat et al. (2022) (purple curve with uncertainty in green). As in Figure 2, tidal rhythmites are yellow squares, and cyclostratigraphic data are color circles (light blue with references in Farhat et al., 2022, red from Zhou et al., 2022, and orange from Zeeden et al., 2023). The stromatolite data from Pannella (1972a, b) are highlighted with a dark green circle. Adapted from Figure 11 of Bao et al. (2022).
3.4. Farhat et al. (2024)
In their recent work, Farhat et al (2024) have revisited the atmospheric thermal tides computations for rocky planets and in particular for the Earth. They have constructed an ab initio model of thermal tides on rocky planets with a neutrally stratified atmosphere 3. This feature is a major change with respect to previous models, where closed-form solutions are usually obtained assuming that the atmosphere is isothermal, which is a less realistic assumption for the Earth troposphere (Lindzen & McKenzie, 1967; Chapman & Lindzen, 1970; Lindzen & Blake, 1972; Auclair-Desrotour et al., 2019; Wu et al., 2023). Although both atmospheric structures provide appreciable mathematical simplifications, neutral stratification appears to better capture the negative temperature gradient that characterizes the troposphere of the Earth, which contains most of the atmospheric mass. As the stability of stratification with respect to convection determines the strength of the Archimedean force exerted on fluid particles in the vertical direction, the neutral stratification approximation annihilates the buoyancy effects in the tidal response. The upward travelling internal gravity waves are thus filtered out from the solution, leaving only the horizontal compressibility forces responsible for the propagation of the Lamb wave. Another major change with respect to previous models (Lindzen & McKenzie, 1967; Chapman & Lindzen, 1970; Lindzen & Blake, 1972; Ingersoll & Dobrovolskis, 1978; Dobrovolskis & Ingersoll, 1980; Auclair-Desrotour et al., 2019; Wu et al., 2023) is the consideration of heat absorption near the ground level and heat exchange between the atmosphere and ground that takes into account the thermal diffusive processes in the planetary surface layer. This model allows to obtain a closed-form solution for the frequency-dependent atmospheric tidal torque, and is in agreement with simulations using GCMs, both for Earth-like and Venus-like planets. Specifically, when applied to the Earth, their model predicts a resonant rotational period of 22.8 hr, which is in agreement with a recent analysis of pressure data on global scales (Sakazaki & Hamilton, 2020)4, and the GCM prediction of Wu et al (2023). As such, the model predicts the occurrence of the Lamb resonance not in the Precambrian, but in the Phanerozoic, with an amplitude that is insufficient to counteract the luni-solar gravitational tidal torque. This does not exclude the occurrence of the crossing of the resonance, but as the luni-solar gravitational tidal torque remains larger than the thermotidal torque, no LOD trapping can occur. The crossing of the resonance then results only in a small change of the Earth's rotational deceleration: the spin deceleration rate is slightly increased before the resonance, and then reduced to roughly its previous value after the crossing of the resonance.
The Farhat et al. (2024) model depends on two parameters (σ 0, α A), which are the cooling frequency and opacity parameters, respectively. The frequency σ 0 is the inverse of the timescale associated with energy dissipation, which is assumed to result from radiative cooling in the model. The higher σ 0, the more efficient is energy dissipation. The frequency σ 0 is thus tightly associated to the amplitude of the Lamb resonance, and thus related to the parameter Q th appearing in Wu et al. (2023). The opacity parameter α A quantifies the fraction of incident Solar flux that is transferred to the atmosphere in the thermal tidal forcing. Consequently, this parameter takes its values between 0 (no tidal forcing) and 1 (maximum tidal forcing). Other model parameters are related to the atmospheric gas mixture and surface temperature. For the thermotidal torque to cancel the gravitational torque in the Precambrian (and thus the LOD locking to occur in the Precambrian), the (σ 0, α A) pair needs to be below the associated black solid curve of Figure 9.
The observation of the present thermal atmospheric response and the constraint on the cooling frequency σ 0 deduced from the cooling timescales estimated by Leconte et al. (2015), impose the (σ0, αA)pair to be inside the intersection of the two shaded regions, which allows for a narrow parametric area where the Precambrian LOD lock conditions are met. However, Farhat et al. (2024) show that the crossing of the resonance, within the limitations of their analytical model, most probably occurred in the late Paleozoic/early Mesozoic, and not in the Precambrian. In this case, the curve to consider is the dashed line, and the parametric analysis precludes LOD locking (Figure 9).
3 A neutrally stratified atmosphere is an atmosphere where the temperature and pressure dependance with height is such that the Archimedian restoring forces are cancelled.
4 This value is in agreement with the 11.38±0.16 hr semi-diurnal period obtained by analyzing the spectrum of normal modes using pressure data on global scales (see Table 1 of Sakazaki and Hamilton, 2020, first symmetric gravity mode of wav-enumber k = -2).
↗
Figure 9.Amplitude of the Lamb resonance with respect to the two parameters σ0 and αA of the Farhat et al. (2024) model. In order for the thermotidal response to cancel the gravitational counterpart in the Precambrian the parameters (σ0, αA) need to be below the solid black line. The dashed line defines the same threshold needed for the late Paleozoic/early Mesozoic (350-250 Ma). The horizontal shaded area corresponds to typical values of the radiative cooling rate σ0. The other shaded area defines the region of parameter space corresponding to the presently observed semi-diurnal tidal bulge. Adapted from Farhat et al. (2024).
4. Conclusions
The famous astronomer Carl Sagan (1934-1996) used to say that extraordinary claims require extraordinary evidence, which is another version of the Occam's razor in science, stating that simpler explanations should be preferred to more complicated ones, in absence of strong arguments. Adapted to the present problem of the evolution of the Earth-Moon distance and LOD, it can be expressed as: Do we need a LOD lock by thermal tides to explain the evolution of the Earth-Moon system over its age? Is there strong evidence for a LOD lock in the Precambrian?
The answer to the first question is clearly negative, as the Farhat et al. (2022) model provides a coherent scenario for the tidal history of the Earth-Moon system, without the need of a resonant atmospheric tidal lock.
We have seen also how all papers advocating for a LOD lock by thermal tides (Bartlett and Stevenson, 2016; Mitchell and Kirscher, 2023; Wu et al., 2023), or the alternate scenario of a growing Earth's core (Bao et al., 2022) rely critically on the stromatolite LOD estimates of Pannella (1972a, b).
However, as emphasised by Pannella himself, and by several authors that studied these data afterwards, the validity of the stromatolite-based LOD estimates derived from the Paleoproterozoic Gunflint-Biwabik Formations and Great Slave Supergroup should be questioned (see section 2.1). There is thus at present no reliable geological evidence to support these alternate scenarios. Moreover, Mitchell and Kirscher (2023) presented a cyclostratigraphy-based datum from cyclicities in the 1.9-Ga Rocknest Formation (Grotzinger, 1986) which is not compatible with the stromatolite data of Pannella. In addition, the solution of Wu et al. (2023) complies with the questionable stromatolite data of Pannella (1972a, b) but not with the more reliable cyclostratigraphic data of Lantink et al. (2022).
The crucial importance of the stromatolite data at 1.88 Ga and 2.0 Ga used in previous models is an important motivation for the search for alternate estimates of the LOD in this time interval or more generally in the interval of 1.5 Ga to 2.0 Ga. A preference should be given to high resolution cyclostratigraphic data, in the spirit of Meyers and Malinverno (2018). In particular, it would be very useful to re-analyze the cyclostratigraphic data at 1.9 Ga of Grotzinger (1986). More generally, we would like here to emphasize the importance of taking dating (age) uncertainty into consideration when fitting variables through any type of empirical estimate derived from the geological records.
The two recent analytical, semi-analytical, and numerical studies of Wu et al. (2023) and Farhat et al. (2024), although providing opposite conclusions, have improved our understanding of the possibility of atmospheric thermotidal daylength locking in the Precambrian. The problem addressed in Wu et al. (2023) can be summarized as follows: two parameterized spectra of tidal torques, one gravitational and one thermal, are combined, and the parameters of the two counterparts are constrained such that the combined torque drives an LOD evolution that fits a compilation of geological proxies. Much can be appreciated in that work, especially in highlighting the significance of Earth-Moon angular momentum depletion via thermal tides, simulating the Lamb resonance for the Precambrian Earth using GCMs, and establishing a correlation between the resonant period, temperature evolution, and atmospheric compositional variations. Moreover, in the limited sense, the adopted analytical models of Wu et al. (2023), laying down the used spectra of the torques, appear to capture the fundamental dynamical behavior of oceanic and atmospheric tides. However, a closer look at the hierarchy of modeling assumptions in the two models and the constraints imposed by the geological proxies reveal that the story is much more nuanced. In short, stringent constraints on the LOD history were imposed by a subset of quantitatively questionable proxies, as we discuss in Section 2.1. The latter were combined with a spectrum of oceanic tides that does not physically describe the tidal response of the Earth’s paleo-oceans. As such, the modeled atmosphere was constrained to encounter the Lamb resonance with an unrealistic amplitude for the torque and in a whistle-stop fashion such that the stromatolite records in the Proterozoic can be explained.
Using a neutrally stratified analytical model that is more adapted to the Earth’s atmosphere than the usual isothermal model, and taking into account heat diffusion mechanisms in the vicinity of the ground interface, Farhat et al. (2024) conclude that the amplitude of the Lamb resonance is not sufficient for the thermotidal torque to counteract the luni-solar gravitational tidal torque when the crossing of the resonance occurs.
Interestingly, the numerical GCM simulations of Wu et al. (2023) allow to strengthen the analytical model of Farhat et al (2024) by probing the asymmetry of the Lamb resonance (Figure 7). These two studies should provide the basis of future improved models for atmospheric thermal tides, but for now, we should conclude, both by the consideration of the geological evidence and by the comparison of theoretical models, that there are no clear arguments supporting the hypothesis that LOD locking occurred in the past history of the Earth.
Acknowledgements
This project has been supported by the French Agence Nationale de la Recherche (AstroMeso ANR-19-CE31-0002-01) and by the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation program (Advanced Grant AstroGeo-885250). MLL acknowledges funding from the HeisingSimons grant no. 2021 – 2797.
Figures credits. Figure 3 from Mitchell and Kirscher (2023), Figures 3 and S4 from Wu et al. (2023), and Figure 11 from Bao et al. (2022) were adapted according to CC-BY licence (
Conflict of interest
The authors declare having no conflict of interest.
Data availability
Data are available in the quoted references or on the AstroGeo website (www.astrogeo.eu).
Authors contribution
JL designed the study and made the first draft. JL, MF, ML, PAD, GB and MS contributed to the discussion of the work and writing of the paper.
References
Auclair-Desrotour, P., Laskar, J., & Mathis, S. (2017) Atmospheric tides in Earth-like planets. Astronomy & Astrophysics, 603, A107.
Auclair-Desrotour, P., Leconte, J., & Mergny, C. (2019) Generic frequency dependence for the atmospheric tidal torque of terrestrial planets. Astronomy & Astrophysics, 624, A17.
Bao, X., Zhao, H., Zhang, S., Li, X., Tan, W., Li, C., Wu, H., Li, H., & Yang, T. (2022) Length of day at c. 1.1 Ga based on cyclostratigraphic analyses of the Nanfen Formation in the North China craton, and its geodynamic implications. Journal of the Geological Society, 180(1), jgs2022-022.
Bartlett, B.C., & Stevenson, D.J. (2016) Analysis of a Precambrian resonance-stabilized day length. Geophysical Research Letters, 43(11), 5716-5724.
Berger, A., Loutre, M.F., & Laskar, J. (1992) Stability of the Astronomical Frequencies Over the Earth’s History for Paleoclimate Studies. Science, 255(5044), 560-566.
Blake, R.E., Chang, S.J., & Lepland, A. (2010) Phosphate oxygen isotopic evidence for a temperate and biologically active Archaean Ocean. Nature, 464(7291), 1029-1032.
Bretherton, F.P. (1969) Lamb waves in a nearly isothermal atmosphere. Quarterly Journal of the Royal Meteorological Society, 95, 754-757.
Catling, D.C., & Zahnle, K.J. (2020) The Archean atmosphere. Science Advances, 6(9), eaax1420.
Chapman, S., & Lindzen, R.S. (1970) Atmospheric tides: thermal and gravitational, volume 15. Springer Science & Business Media.
Charnay, B., Wolf, E.T., Marty, B., & Forget, F. (2020) Is the faint young sun problem for Earth solved? Space Science Reviews, 216,1-29.
Correia, A., & Laskar, J. (2001) The four final rotation states of Venus. Nature, 411(6839), 767-770.
Correia, A.C., & Laskar, J. (2003) Long-term evolution of the spin of Venus: lI. numerical simulations. Icarus, 163(1), 24-45.
Correia, A.C., Laskar, J., & de Surgy, O.N. (2003) Long-term evolution of the spin of Venus: I. theory. Icarus, 163(1), 1-23.
Daher, H., Arbic, B. K., Williams, J. G., Ansong, J. K., Boggs, D. H., Müller, M., Schindelegger, M., Austermann, J., Cornuelle, B. D., Crawford, E. B., Fringer, O. B., Lau, H. C. P., Lock, S. J., Maloof, A. C., Menemenlis, D., Mitrovica, J. X., Green, J. A. M., & Huber, M. (2021). Long-Term Earth-Moon Evolution With High-Level Orbit and Ocean Tide Models. Journal of Geophysical Research: Planets, 126(12), e2021JE006875.
Darwin, G.H. (1879) On the Precession of a Viscous Spheroid, and on the Remote History of the Earth. Philosophical Transactions of the Royal Society of London Series I, 170, 447-538.
Davies, G.R. (1970) Algal-Laminated Sediments, Gladstone Embayment, Shark Bay, Western Australia. In B.W. Logan, G.R. Davies, J.F. Read, & D.E. Cebulski, editors, Carbonate Sedimentation and Environments, Shark Bay, Western Australia, volume 13, page 0. American Association of Petroleum Geologists.
Dobrovolskis, A.R., & Ingersoll, A.P. (1980) Atmospheric tides and the rotation of Venus I. tidal theory and the balance of torques. Icarus, 41(1),1-17.
Eulenfeld, T., & Heubeck, C. (2023) Constraints on Moon’s Orbit 3.2 Billion Years Ago From Tidal Bundle Data. Journal of Geophysical Research: Planets, 128(1), e2022JE007466.
Farhat, M., Auclair-Desrotour, P., Boué, G., Deitrick, R., & Laskar, J. (2024) Thermal tides in neutrally stratified atmospheres) Revisiting the Earth’s precambrian rotational equilibrium. 11946. Astronomy & Astrophysics, 684, A49.
Farhat, M., Auclair-Desrotour, P., Boué, G., & Laskar, J. (2022) The resonant tidal evolution of the Earth-Moon distance. Astronomy & Astrophysics, 665, L1.
Fraedrich, K., Jansen, H., Kirk, E., Luksch, U., & Lunkeit, F. (2005) The planet simulator) Towards a user friendly model. Meteorologische Zeitschrift, 14(3), 299-304.
Fralick, P., Davis, D.W., & Kissin, S.A. (2002) The age of the Gunflint Formation, Ontario, Canada) single zircon U/Pb age determinations from reworked volcanic ash. Canadian Journal of Earth Sciences, 39(7), 1085-1091.
Gebelein, C., & Hoffman, P. (1968) Intertidal stromatolites from Cape Sable, Florida. Geol. Soc. Am. Spec. Pap, 121, 109.
Gebelein, C.D. (1969) Distribution, morphology, and accretion rate of recent subtidal algal stromatolites, Bermuda. Journal of Sedimentary Research, 39(1), 49-69.
Gold, T., & Soter, S. (1969) Atmospheric tides and the resonant rotation of Venus. Icarus, 11(3), 356-366.
Goldreich, P. (1966) History of the lunar orbit. Reviews of Geophysics, 4(4), 411-439.
Goldreich, P., & Soter, S. (1966) Q in the solar system. Icarus, 5(1-6), 375-389.
Gough, D. (1981) Solar interior structure and luminosity variations. In Physics of Solar Variations) Proceedings of the 14th ESLAB Symposium held in Scheveningen, The Netherlands, 16-19 September, 1980, 21-34. Springer.
Gradstein, F., Ogg, J., Schmitz, K., & Ogg, G. (2012) The Geologic Time Scale 2012, volume 2. Elsevier.
Gradstein, F.M., Ogg, J.G., Schmitz, M.D., & Ogg, G.M., editors (2020) Geological Time Scale 2020. Elsevier.
Gradstein, F.M., Ogg, J.G., & Smith, A.G. (2004) A Geologic Time Scale 2004. Cambridge University Press.
Green, J., Huber, M., Waltham, D., Buzan, J., & Wells, M. (2017) Explicitly modelled deep-time tidal dissipation and its implication for lunar history. Earth and Planetary Science Letters, 461:46-53.
Grotzinger, J.P. (1986) Cyclicity and paleoenvironmental dynamics, Rocknest platform, northwest Canada. GSA Bulletin, 97(10), 12081231.
Hinnov, L.A. (2018) Chapter 3 - Cyclostratigraphy and astrochronology in 2018. In M. Montenari, editor, Stratigraphy & Timescales, volume 3 of Cyclostratigraphy and Astrochronology, pages 1-80. Academic Press.
Hoffman, P. F., Abbot, D. S., Ashkenazy, Y., Benn, D. I., Brocks, J. J., Cohen, P. A., Cox, G. M., Creveling, J. R., Donnadieu, Y., Erwin, D. H., Fairchild, I. J., Ferreira, D., Goodman, J. C., Halverson, G. P., Jansen, M. F., Le Hir, G., Love, G. D., Macdonald, F. A., Maloof, A. C., Partin, C. A., Ramstein, G., Rose, B. E. J., Rose, C. V., Sadler, P. M., Tziperman, E., Voigt, A., & Warren, S. G. (2017). Snowball Earth climate dynamics and Cryogenian geology-geobiology. Science Advances, 3(11), e1600983.
Hoffman, P.F., Macdonald, F.A., Bowring, S.A., Ramezani, J., Buchwaldt, R., Hildebrand, R.S., & Whalen, J.B. (2023) Crustal eduction and slab-failure magmatism in an Orosirian (2.05-1.80Ga) postcollisional cratonic foredeep) geochronology of Seton volcanics and Compton laccoliths, Tu Cho (Great Slave Lake), NWT, Canada. Canadian Journal of Earth Sciences, 60(10), 1359-1384.
Hofmann, H.J. (1973) Stromatolites) Characteristics and utility. Earth-Science Reviews, 9(4), 339-373.
Holland, H.D. (2006) The oxygenation of the atmosphere and oceans. Philosophical Transactions of the Royal Society B) Biological Sciences.
Hourdin, F., Musat, I., Bony, S., Braconnot, P., Codron, F., Dufresne, J.-L., Fairhead, L., Filiberti, M.-A., Friedlingstein, P., Grandpeix, J.-Y., Krinner, G., LeVan, P., Li, Z.-X., & Lott, F. (2006). The LMDZ4 general circulation model: climate performance and sensitivity to parametrized physics with emphasis on tropical convection. Climate Dynamics, 27(7), 787-813.
Hut, P. (1981) Tidal evolution in close binary systems. Astronomy and Astrophysics, 99:126-140.
Ingersoll, A.P., & Dobrovolskis, A.R. (1978) Venus’ rotation and atmospheric tides. Nature, 275(5675), 37-38.
Kaula, W.M. (1964) Tidal Dissipation by Solid Friction and the Resulting Orbital Evolution. Reviews of Geophysics and Space Physics, 2, 661-685.
Krissansen-Totton, J., Arney, G.N., & Catling, D.C. (2018) Constraining the climate and ocean ph of the early Earth with a geological carbon cycle model. Proceedings of the National Academy of Sciences, 115(16), 4105-4110.
Lambeck, K. (1980) The Earth’s variable rotation: geophysical causes and consequences. Cambridge University Press.
Lantink, M.L., Davies, J.H., Ovtcharova, M., & Hilgen, F.J. (2022) Milankovitch cycles in banded iron formations constrain the Earth-Moon system 2.46 billion years ago. Proceedings of the National Academy of Sciences, 119(40), e2117146119.
Laskar, J. (2020) Astrochronology. In F.M. Gradstein, J.G. Ogg, M.B. Schmitz, & G.M. Ogg, editors, Geologic Time Scale 2020, pages 139-158. Elsevier.
Laskar, J., Fienga, A., Gastineau, M., & Manche, H. (2011a) La2010: a new orbital solution for the long-term motion of the Earth. Astronomy and Astrophysics, 532, 89.
Laskar, J., Gastineau, M., Delisle, J., Farrés, A., & Fienga, A. (2011b) Strong chaos induced by close encounters with ceres and vesta. Astronomy and Astrophysics, 532, L4.
Laskar, J., Robutel, P., Joutel, F., Gastineau, M., Correia, A., & Levrard, B. (2004) A long-term numerical solution for the insolation quantities of the Earth. Astronomy & Astrophysics, 428(1), 261-285.
Leconte, J., Wu, H., Menou, K., & Murray, N. (2015) Asynchronous rotation of Earth-mass planets in the habitable zone of lower-mass stars. Science, 347(6222), 632-635.
Lindzen, R.S., & Blake, D. (1972) Lamb waves in the presence of realistic distributions of temperature and dissipation. Journal of Geophysical Research, 77(12), 2166-2176.
Lindzen, R. S., & McKenzie, D. J. (1967). Tidal theory with Newtonian cooling. Pure and Applied Geophysics, 66(1), 90-96.
Logan, B.W., Rezak, R., & Ginsburg, R.N. (1964) Classification and environmental significance of algal stromatolites. The Journal of Geology, 72(1), 68-83.
MacDonald, G.J.F. (1964) Tidal Friction. Reviews of Geophysics and Space Physics, 2, 467-541.
Maurice, M., Tosi, N., Schwinger, S., Breuer, D., & Kleine, T. (2020) A long-lived magma ocean on a young Moon. Science advances, 6(28), eaba8949.
Meyers, S.R., & Malinverno, A. (2018) Proterozoic Milankovitch cycles and the history of the solar system. Proceedings of the National Academy of Sciences, 115(25), 6363-6368.
Mignard, F. (1979) The Evolution of the Lunar Orbit Revisited, I. The Moon and the Planets, 20, 301-315.
Mignard, F. (1980) The Evolution of the Lunar Orbit Revisited, II. The Moon and the Planets, 23(2):185-201.
Mitchell, R., & Kirscher, U. (2023) Mid-Proterozoic day length stalled by tidal resonance. Nature Geosciences, 16, 567-569.
Mohr, R.E. (1975) Measured periodicities of the biwabik (Precambrian) stromatolites and their geophysical significance. In G.D. Rosenberg, and S.K. Runcorn, editors, Growth Rhythms and the History of the Earth’s Rotation, pages 43-56. John Wiley & Sons Ltd, London, New York.
Montenari, M., editor (2018) Cyclostratigraphy and Astrochronology. Academic Press, 1st edition.
Monty, L. (1967) Distribution and structure of recent stromatolitic algal mats, eastern Andros Island, Bahamas. Ann Soc Geol Belgigue, 1966, 55-100.
Néron De Surgy, O., & Laskar, J. (1997) On the long term evolution of the spin of the Earth. Astronomy and Astrophysics, 318(3), 975989.
Olsen, P. E., Laskar, J., Kent, D. V., Kinney, S. T., Reynolds, D. J., Sha, J., & Whiteside, J. H. (2019). Mapping Solar System chaos with the Geological Orrery. Proceedings of the National Academy of Sciences, 116(22), 10664-10673.
Pannella, G. (1972a) Paleontological evidence on the Earth’s rotational history since early precambrian. Astrophysics and Space Science, 16, 212-237.
Pannella, G. (1972b) Precambrian stromatolites as paleontological clocks. Int. Geol. Congr. Rep. Sess., 24th, pages 50-57.
Pannella, G. (1976). Chapter 12.1 Geophysical Inferences from Stromatolite Lamination. In M. R. Walter (Ed.), Developments in Sedimentology (Vol. 20, pp. 673-685): Elsevier.
Pierrehumbert, R.T. (2010) Principles of Planetary Climate. Cambridge University Press.
Pope, M., & Grotzinger, J.P. (2000) Carbonate Sedimentation and Diagenesis in the Evolving Precambrian World, chapter Controls on fabric development and morphology of tufas and stromatolites, Uppermost Pethei Group (1.8 Ga), Great Slave Lake, Northwest Canada, pages 103-121. Society for Sedimentary Geology.
Rosenberg, G.D., & Runcorn, S.K. (1975) Growth Rhythms and the History of the Earth’s Rotation. John Wiley & Sons Ltd, London, New York.
Sakazaki, T., & Hamilton, K. (2020) An array of ringing global free modes discovered in tropical surface pressure data. Journal of the Atmospheric Sciences, 77(7), 2519-2539.
Scrutton, C. T. (1978). Periodic Growth Features in Fossil Organisms and the Length of the Day and Month. In P. Brosche & J. Sündermann (Eds.), Tidal Friction and the Earth’s Rotation (pp. 154-196). Berlin, Heidelberg: Springer Berlin Heidelberg.
Sinnesael, M., De Vleeschouwer, D., Zeeden, C., Batenburg, S. J., Da Silva, A.-C., de Winter, N. J., Dinarès-Turell, J., Drury, A. J., Gambacorta, G., Hilgen, F. J., Hinnov, L. A., Hudson, A. J. L., Kemp, D. B., Lantink, M. L., Laurin, J., Li, M., Liebrand, D., Ma, C., Meyers, S. R., Monkenbusch, J., Montanari, A., Nohl, T., Pälike, H., Pas, D., Ruhl, M., Thibault, N., Vahlenkamp, M., Valero, L., Wouters, S., Wu, H., & Claeys, P. (2019). The Cyclostratigraphy Intercomparison Project (CIP): consistency, merits and pitfalls. Earth-Science Reviews, 199, 102965.
Sleep, N.H., & Zahnle, K. (2001) Carbon dioxide cycling and implications for climate on ancient Earth. Journal of Geophysical Research: Planets, 106(E1), 1373-1399.
Sonett, C., & Chan, M.A. (1998) Neoproterozoic Earth-Moon dynamics: Rework of the 900 ma big cottonwood canyon tidal laminae. Geophysical Research Letters, 25(4), 539-542.
Sonett, C.P., Kvale, E.P., Zakharian, A., Chan, M.A., & Demko, T.M. (1996) Late Proterozoic and Paleozoic tides, retreat of the Moon, and rotation of the Earth. Science, 273(5271), 100-104.
Touma, J., & Wisdom, J. (1994) Evolution of the Earth-Moon system. The Astronomical Journal, 108, 1943-1961.
Tyler, R.H. (2021) On the tidal history and future of the Earth-Moon orbital system. The Planetary Science Journal, 2(2), 70.
Volland, H. (1990) Atmospheric Effects on the Earth’s Rotation. In P. Brosche and J. Sündermann (Eds), Earth’s Rotation from Eons to Days, pages 127-140. Springer, Berlin, Heidelberg.
Walker, J.C., & Zahnle, K.J. (1986) Lunar nodal tide and distance to the Moon during the Precambrian. Nature, 320(6063), 600-602.
Webb, D. (1982) Tides and the evolution of the Earth—Moon system. Geophysical Journal International, 70(1), 261-271. 1365-246x.1982.tb06404.x
Wilkes, M.V. (1949) Oscillations of the Earth’s atmosphere.
Williams, G.E. (1989) Tidal rhythmites: geochronometers for the ancient Earth-Moon system. Episodes Journal of International Geoscience, 12(3), 162-171.
Williams, G.E. (1990) Tidal rhythmites: key to the history of the Earth’s rotation and the Lunar orbit. Journal of Physics of the Earth, 38(6), 475-491.
Williams, G.E. (2000) Geological constraints on the Precambrian history of Earth’s rotation and the Moon’s orbit. Reviews of Geophysics, 38(1), 37-59.
Williams, J.G., & Boggs, D.H. (2016) Secular tidal changes in Lunar orbit and Earth rotation. Celestial Mechanics and Dynamical Astronomy, 126(1), 89-129.
Wu, H., Murray, N., Menou, K., Lee, C., & Leconte, J. (2023) Why the day is 24 hours long: The history of Earth’s atmospheric thermal tide, composition, and mean temperature. Science Advances, 9(27), eadd2499.
Zahnle, K., & Walker, J.C. (1987) A constant daylength during the Precambrian era? Precambrian Research, 37(2), 95-105.
Zeeden, C., Laskar, J., De Vleeschouwer, D., Pas, D., & Da Silva, A.C. (2023) Earth’s rotation and Earth-Moon distance in the Devonian derived from multiple geological records. Earth and Planetary Science Letters, 621, 118348.
Zhou, M., Wu, H., Hinnov, L.A., Fang, Q., Zhang, S., Yang, T., & Shi, M. (2022) Empirical reconstruction of Earth-Moon and solar system dynamical parameters for the past 2.5 billion years from cyclostratigraphy. Geophysical Research Letters, 49(16).
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Lay summary1. Introduction1.1. Atmospheric thermal tides1.2. A possible lock of the length of the day2. Geological archives for Precambrian LOD estimates2.1. Stromatolites2.2. Tidal rhythmites2.3. Cyclostratigraphy3. Discussion of the recently published results3.1. Mitchell and Kirscher (2023)3.2. Wu et al. (2023)3.3. Bao et al. (2022)3.4. Farhat et al. (2024)4. ConclusionsAcknowledgementsConflict of interestData availabilityAuthors contributionReferences
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2024-04-09 — Updated on 2024-04-25
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Laskar, J., Farhat, M., Lantink, M. L., Auclair-Desrotour, P., Boué, G., & Sinnesael, M. (2024). Did atmospheric thermal tides cause a daylength locking in the Precambrian? A review on recent results. Sedimentologika, 2(1).
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Some rights reserved 2024 Jacques Laskar, Mohammad Farhat, Margriet L. Lantink, Pierre Auclair-Desrotour, Gwenaël Boué, Matthias Sinnesael
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10968 | https://pubmed.ncbi.nlm.nih.gov/12439522/ | Noninvasive diagnosis by Doppler ultrasonography of fetal anemia resulting from parvovirus infection - PubMed
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. 2002 Nov;187(5):1290-3.
doi: 10.1067/mob.2002.128024.
Noninvasive diagnosis by Doppler ultrasonography of fetal anemia resulting from parvovirus infection
Erich Cosmi1,Giancarlo Mari,Loredana Delle Chiaie,Laura Detti,Masashi Akiyama,June Murphy,Theodor Stefos,James E Ferguson 2nd,David Hunter,Chaur-Dong Hsu,Alfred Abuhamad,Ray Bahado-Singh
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1 Department of Obstetrics and Gynecology, University of Cincinnati, Ohio 45267, USA.
PMID: 12439522
DOI: 10.1067/mob.2002.128024
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Noninvasive diagnosis by Doppler ultrasonography of fetal anemia resulting from parvovirus infection
Erich Cosmi et al. Am J Obstet Gynecol.2002 Nov.
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. 2002 Nov;187(5):1290-3.
doi: 10.1067/mob.2002.128024.
Authors
Erich Cosmi1,Giancarlo Mari,Loredana Delle Chiaie,Laura Detti,Masashi Akiyama,June Murphy,Theodor Stefos,James E Ferguson 2nd,David Hunter,Chaur-Dong Hsu,Alfred Abuhamad,Ray Bahado-Singh
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1 Department of Obstetrics and Gynecology, University of Cincinnati, Ohio 45267, USA.
PMID: 12439522
DOI: 10.1067/mob.2002.128024
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Objective: The purpose of this study was to evaluate the feasibility of the middle cerebral artery peak systolic velocity for the detection of fetal anemia in pregnancies that are complicated by parvovirus B19 infection.
Study design: Doppler measurements of the middle cerebral artery peak systolic velocity were performed weekly in 32 fetuses at risk for anemia because of maternal parvovirus infection documented by the presence of serum immunoglobulin M antibody. The values of the middle cerebral artery peak systolic velocity and hemoglobin were expressed as multiples of the median. These values were plotted on reference ranges that had been established previously. A cordocentesis was performed either because of fetal ascites or when the middle cerebral artery peak systolic velocity values suggested anemia (middle cerebral artery peak systolic velocity, >1.50 multiples of the median).
Results: Gestational age at study entry ranged from 15.1 to 37 weeks. There were 17 fetuses with middle cerebral artery peak systolic velocity of >1.50 MoM (group 1). Sixteen cordocenteses were performed in these fetuses. All 16 fetuses were anemic (15 severely and 1 mildly). Thirteen fetuses had signs of hydrops (12 with severe and 1 with mild anemia). Group 2 included 15 fetuses with the middle cerebral artery peak systolic velocity values <1.50 MoM. Two cordocenteses were performed. One fetus was mildly anemic; the second fetus was not anemic. The remaining 13 fetuses of this group did not have any complications and were not anemic at birth. The sensitivity of the middle cerebral artery peak systolic velocity (>1.50 MoM) for the prediction of anemia because of parvovirus infection was 94.1%; the specificity was 93.3 %; the positive and negative predictive values were 94.1% and 93.3%, respectively.
Conclusion: Fetal anemia caused by parvovirus infection can be detected noninvasively by Doppler ultrasonography on the basis of an increase in the peak velocity of systolic blood flow in the middle cerebral artery.
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Anemia / diagnosis
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Find period of the following functions <...
Find period of the following functions
(i) f(x)=sin x 2+cos x 3
(ii) f(x)={x}+sin x, where {.} denotes fractional part function
(iii) f(x)=4 cos x.cos 3 x+2
(iv) f(x)=sin 3 x 2−cos x 3−tan 2 x 3
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Know where you stand among peers with ALLEN's JEE Enthusiast Online Test Series Period of sin x 2 is 4 π while period of cos x 3 is 6 π. Hence period of sin x 2+cos x 3 is 12 π {L.C.M of 4 and 6 is 12}
(ii) Period of sin x=2 π
Period of {x}=1
but L.C.M. of 2 π and 1 is not possible as their ratio is irrational number it is aperiodic.
(iii) f(x)=4 cos x.cos 3 x+2
period of f(x) is L.C.M of (2 π,2 π 3)=2 π
but 2 π may or may not be fundamental periodic but fundamental period =2 π n where n ε N. Hence cross checking for n=1,2,3,....... we find π to be fundamental period f(π+x)=4(−cos x)(−cos 3 x)+2=f(x)
(iv) Period of f(x) is L.C.M of 2 π 3/2,2 π 1/3,π 2/3= L.CM. of 4 π 3,6 π,3 π 2=12 π
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2013 AMC 12B Problems/Problem 9
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1 Problem
2 Solution
3 Video Solution by OmegaLearn
4 Video Solution
5 See also
Problem
What is the sum of the exponents of the prime factors of the square root of the largest perfect square that divides ?
Solution
Looking at the prime numbers under , we see that there are factors of , factors of , and factors of . All greater primes are represented once or none in , so they cannot be part of the square. Since we are looking for a perfect square, the exponents on its prime factors must be even, so we can only use of the factors of . Therefore, the prime factorization of the square is . To find the square root of this, we halve the exponents, leaving . The sum of the exponents is
Video Solution by OmegaLearn
~ pi_is_3.14
Video Solution
~someone
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2013 AMC 12B (Problems • Answer Key • Resources)
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10971 | https://www.ck12.org/flexi/cbse-math/complex-numbers/what-are-the-key-properties-of-the-argument-of-a-complex-number/ | What are the key properties of the argument of a complex number? - Characteristics & Formula | CK-12 Foundation
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What are the key properties of the argument of a complex number?
Flexi Says:
The argument of a complex number is a measure of the angle it makes with the positive real axis in the complex plane. It is denoted by θ or a r g(z) where z is the complex number. Here are some key properties of the argument of a complex number:
Range of Argument:
The principal value of the argument of a complex number is always in the interval(−π,π].
Addition of Arguments:
If z 1 and z 2 are two complex numbers, then the argument of their product is the sum of their arguments, i.e., a r g(z 1 z 2)=a r g(z 1)+a r g(z 2).
Subtraction of Arguments:
If z 1 and z 2 are two complex numbers, then the argument of the quotient of the two complex numbers is the difference of their arguments, i.e., a r g(z 1 z 2)=a r g(z 1)−a r g(z 2).
Argument of Conjugate:
If z is a complex number and z¯ is its conjugate, then a r g(z¯)=−a r g(z).
Argument of Reciprocal:
If z is a complex number and 1 z is its reciprocal, then a r g(1 z)=−a r g(z).
Argument of Power:
If z is a complex number and n is a positive integer, then a r g(z n)=n⋅a r g(z).
Remember, these properties hold true when the arguments are in their principal values.
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A Thorough Review Of Process-mapping In Quality Control Inspection
Quality control inspection means far more than dozens of checklists. It’s also about communication, collaboration, and the team being in tune. Here below you’ll find the best practices on mapping inspection processes, and ways to improve them with the help of a quality control inspection software.
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Quality control and inspection difference
Firstly, let’s define what Quality control and Quality control inspection mean. Quality Control is a process that ensures that the manufactured product, provided service or delivered project adheres to the quality criteria and meets customer requirements. Quality control inspection is an activity intended to monitor and test the ongoing state of works, identify and report non-compliance and verify whether the work is executed inline with the applicable specifications.
To help teams streamline the inspection, its process should be automated.
How to automate the quality control inspection process?
The key figures of quality control inspection are usually made up of two teams: Field team (Site Engineers) working on a particular construction site, and Quality team, who firstly review field team’s findings (Quality Managers) and then approve taken findings by responsible parties (Project Managers and Head Engineer).
Let’s map the flowchart for the mentioned groups:
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Here are the steps:
There are four main steps that allow these two groups to ensure the quality of work performed at every step of the project.
Site Engineers: Add details
Site Engineers: Complete Quality Control Checklist
Quality Team: Complete Verification Checklist
Quality Team: Sign off Verification Checklist
Depending on your organization’s structure, some responsibilities in the onboarding process may be shared by various personnel. The contributors are responsible for creating the onboarding checklist template. Senior management or HR can be tasked with making sure that each part of the process is properly assigned and completed.
Quality control inspection software in action
Quality control inspection software has to meet the requirements of both field teams and office crew. Here are the main ones.
Collaborating on QC inspection checklists
The start point of inspection paperwork is collecting raw data. Fluix mobile app allows Site engineers to fill out quality control inspection forms, add measurements, comments, photos with timestamps and geolocation, and finally sign the completed form. All that is possible on an iPad or iPhone, even without connectivity.Since multiple people may work on the same quality control inspection checklist for a longer time, it is usually important for them to have access to the same document in progress. In Fluix app team members get a shared work environment where they can collaborate on the same forms and assign them to each other. In this way working in shifts is no longer a problem since the checklist can be assigned to the team member if it’s not completed before the shift is over.Finally, Fluix mobile app allows engineers to jump between the documents under different inspections and projects, within one account, making paperwork even easier.
Slashing the review process time
The very moment a Site engineer completes the quality control inspection checklist, it is reassigned to the Quality team for review. If the checklist requires rework, the Quality manager can easily return it back to the Site engineers, with comments on what should be corrected. Once all remarks are fixed, Site engineer sends the checklist for another review. The whole review and approval cycle can be also done on mobile devices, right on site. The configured approval workflow allows teams to exchange checklists and comments in an efficient way, without clarification emails and calls.
Perks for Project Leads
One of the most annoying things for the Project lead is to waste time on administration work. This is why it’s crucial to choose a quality control inspection program that offers extensive capabilities and doesn’t require time for their support. Here are some:
Easily add new engineers to work on a project by adding them to right groups
Remove engineers from groups who no longer work on the project and reassign documents in progress to the right engineer
Track progress on the checklists in progress from Document Status section on the backend
Provide granular permissions to those engineers who need extra access or visibility to documents in progress, data storage and reporting
Set naming convention to the templates of quality control checklists to have all the documents named according to the same pattern automatically, which makes it fast to locate them
Store or act on data?
When the review is done, the signed quality control checklist usually flies to the office email and archive folder for safekeeping. Fluix offers both its internal cloud storage or integration with familiar storage such as Google Drive, Dropbox, OneDrive, Box and others.Along with archiving of the completed files Fluix offers to look at data as insights. Data reporting and analysis capabilities allow you to prepare filtered reports and export them in Excel, or third party visualization platforms for further sharing with site teams, managers and customers.
To cover the key aspects of the process, we recommend using these checklists:
Advice from practice
Quality control inspection is an integral part of any construction project, manufacturing, and other spheres where quality is crucial for people safety, high quality of performed work, and consequently reduced costs on maintenance and repairs of the produced or built asset. That is why it’s absolutely crucial to choose the right system that will automate the whole process, no matter how complex it is, the system that is flexible and scalable, the tool that is stable and easy to use.
Ready, Set, And Build Your Perfect Quality Control Inspection Workflow
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10973 | https://chem.libretexts.org/Courses/Portland_Community_College/CH104%3A_Allied_Health_Chemisty_I_(2nd_Edition)/07%3A_Molecules_Covalent_Bonding_and_the_Nomenclature_of_Binary_Covalent_Compounds/7.08%3A_Polarity_of_Bonds | 7.8: Polarity of Bonds - Chemistry LibreTexts
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7: Molecules, Covalent Bonding, and the Nomenclature of Binary Covalent Compounds
CH104: Allied Health Chemisty I (2nd Edition)
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7.8: Polarity of Bonds
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Contents
1. Home
2. Campus Bookshelves
3. Portland Community College
4. CH104: Allied Health Chemisty I (2nd Edition)
5. 7: Molecules, Covalent Bonding, and the Nomenclature of Binary Covalent Compounds
6. 7.8: Polarity of Bonds
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CH104: Allied Health Chemisty I (2nd Edition)
Front Matter
1: Introduction to Chemistry and the Scientific Method
2: Measurement and Significant Figures
3: Dimensional Anlaysis and Density
4: Classifiation of Matter- Properties and Changes
5: The Nuclei of Atoms
6: Ions, Ionic Bonding, and the Nomenclature of Ionic Compounds
7: Molecules, Covalent Bonding, and the Nomenclature of Binary Covalent Compounds
8: Counting Atoms, Ions, and Molecules
9: An Introduction to Chemical Reactions
10: Mass Relations in Chemical Reactions
11: Energy and Chemical Reactions
Back Matter
7.8: Polarity of Bonds
Last updated Aug 10, 2023
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7.7: Molecular Geometry- VSEPR
7.9: Polarity of Molecules
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Table of contents
1. Learning Objectives
2. Electronegativity and Bond Polarity
1. Example 7.8.1
2. Exercise 7.8.1
Exercises
Answers
Learning Objectives
Given a table of electronegativity values, students will be able to determine if a covalent bond is polar or nonpolar.
Electronegativity and Bond Polarity
Although we defined covalent bonding as electron sharing, the electrons in a covalent bond are not always shared equally by the two bonded atoms. Unless the bond connects two atoms of the same element, as in H 2, there will always be one atom that attracts the electrons in the bond more strongly than the other atom does, as in HCl, shown in Figure 7.8.1. A covalent bond that has an equal sharing of electrons (Figure 7.8.1a) is called a nonpolar covalent bond. A covalent bond that has an unequal sharing of electrons, as in Figure 7.8.1b, is called a polar covalent bond.
Figure 7.8.1 Polar versus Nonpolar Covalent Bonds. (a) The electrons in the covalent bond are equally shared by both hydrogen atoms. This is a nonpolar covalent bond. (b) The chlorine atom attracts the electrons in the bond more than the hydrogen atom does, leading to an imbalance in the electron distribution. This is a polar covalent bond.
The distribution of electron density in a polar bond is uneven. It is greater around the atom that attracts the electrons more than the other. For example, the electrons in the H–Cl bond of a hydrogen chloride molecule spend more time near the chlorine atom than near the hydrogen atom. Note that the shaded area around Cl in Figure 7.8.1b is much larger than it is around H.
This imbalance in electron density results in a buildup of partial negative charge (designated as δ−) on one side of the bond (Cl) and a partial positive charge (designated δ+) on the other side of the bond (H). This is seen in Figure 7.8.2a. The separation of charge in a polar covalent bond results in an electric dipole (two poles), represented by the arrow in Figure 7.8.2b. The direction of the arrow is pointed toward the δ− end while the + tail of the arrow indicates the δ+ end of the bond.
Figure 7.8.2: (a) Unequal sharing of the bonding pair of electrons between H and Cl leads to partial positive charge on the H atom and partial negative charge on the Cl. Symbols δ+ and δ– indicate the polarity of the H–Cl bond. (b) The dipole is represented by an arrow with a cross at the tail. The cross is near the δ+ end and the arrowhead coincides with the δ–.
Any covalent bond between atoms of different elements is a polar bond, but the degree of polarity varies widely. Some bonds between different elements are only minimally polar, while others are strongly polar. Ionic bonds can be considered the ultimate in polarity, with electrons being transferred rather than shared. To judge the relative polarity of a covalent bond, chemists use electronegativity, which is a relative measure of how strongly an atom attracts electrons when it forms a covalent bond. There are various numerical scales for rating electronegativity. Figure 7.8.3 shows one of the most popular—the Pauling scale.
Figure 7.8.3 The electronegativity values derived by Pauling follow predictable periodic trends with the higher electronegativities toward the upper right of the periodic table. Fluorine has the highest value (4.0).
The polarity of a covalent bond can be judged by determining the difference in the electronegativities of the two atoms making the bond. The greater the difference in electronegativities, the greater the imbalance of electron sharing in the bond. Although there are no hard and fast rules, the general rule is if the difference in electronegativities isless than about 0.4,the bond is considered nonpolar;if the difference is greater than 0.4,the bond is consideredpolar.If the difference in electronegativities is large enough (generallygreater than about 1.8),the resulting compound is consideredionicrather than covalent.An electronegativity difference of zero, of course,indicates a nonpolar covalent bond.
Figure 7.8.4: As the electronegativity difference increases between two atoms, the bond becomes more ionic.
Example 7.8.1
Describe the electronegativity difference between each pair of atoms and the resulting polarity (or bond type).
C and H
H and H
Na and Cl
O and H
Solution
Carbon has an electronegativity of 2.5, while the value for hydrogen is 2.1. The difference is 0.4, which is rather small. The C–H bond is therefore considered nonpolar.
Both hydrogen atoms have the same electronegativity value—2.1. The difference is zero, so the bond is nonpolar.
Sodium’s electronegativity is 0.9, while chlorine’s is 3.0. The difference is 2.1, which is rather high, and so sodium and chlorine form an ionic compound.
With 2.1 for hydrogen and 3.5 for oxygen, the electronegativity difference is 1.4. We would expect a very polar bond. The sharing of electrons between O and H is unequal with the electrons more strongly drawn towards O.
Exercise 7.8.1
Describe the electronegativity (EN) difference between each pair of atoms and the resulting polarity (or bond type).
C and O
K and Br
N and N
Cs and F
Answer a:
The EN difference is 1.0 , hence polar. The sharing of electrons between C and O is unequal with the electrons more strongly drawn towards O.
Answer b:
The EN difference is greater than 1.8, hence ionic.
Answer c:
Identical atoms have zero EN difference, hence nonpolar.
Answer d:
The EN difference is greater than 1.8, hence ionic.
Exercises
Using Figure 7.8.3, determine which atom in each pair has the higher electronegativity.
H or C
O or Br
Na or Rb
I or Cl
Using Figure 7.8.3, determine which atom in each pair has the lower electronegativity.
Mg or O
S or F
Al or Ga
O or I
Will the electrons be shared equally or unequally across each covalent bond? If unequally, to which atom are the electrons more strongly drawn?
a C–O bond
an F–F bond
an S–N bond
an I–Cl bond
Will the electrons be shared equally or unequally across each covalent bond? If unequally, to which atom are the electrons more strongly drawn?
a C–C bond
a S–Cl bond
an O–H bond
an H–H bond
Arrange the following bonds from least polar to most polar: H-F, H-N, H-O, H-C
Arrange the following bonds from least polar to most polar: C-F, C-N, C-O, C-C
Answers
Using Figure 7.8.3, determine which atom in each pair has the higher electronegativity.
C
O
Na
Cl
Using Figure 7.8.3, determine which atom in each pair has the lower electronegativity.
Mg
S
Al
I
Will the electrons be shared equally or unequally across each covalent bond? If unequally, to which atom are the electrons more strongly drawn?
unequally toward the O
equally
unequally toward the N
unequally toward the Cl
Will the electrons be shared equally or unequally across each covalent bond? If unequally, to which atom are the electrons more strongly drawn?
equally
unequally toward the Cl
unequally toward the O
equally
The electronegativity difference increases from 0.4; 0.9; 1.4; 1.9. Hence, the least to most polar: H-C, H-N, H-O, H-F
The electronegativity difference increases from 0; 0.5; 1.0; 1.5. Hence, the least to most polar: C-C, C-N, C-O, C-F
7.8: Polarity of Bonds is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by LibreTexts.
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10975 | https://www.stonybrook.edu/laser/_dragan/JDraganOpticalPumping.pdf | Optical Pumping and Magnetic Resonance James Dragan Lab Partner: Stefan Evans Physics Department, Stony Brook University, Stony Brook, NY 11794.
(Dated: October 4, 2013) We optically pump electrons in Rb85 to the 52S1/2 , F=2, mf=+2 state and in Rb87 to the 52S1/2, F=3, mf=+3 using the D1 line λ = 795 nm, emitted by a Rb Lamp, which becomes σ+ polarized due to a linear polarizer and a quarter-wave plate. We verify the pumping using an RF signal and stimulated emission to populate the next lower mf level and observe the transmitted light read by a photodiode. We show that this transition is achieved in both isotopes. Using the absorption frequency we determine the Earths Magnetic field BEarth, along the quantization axis z. The presence of this field splits the degenerate states in the hyperfine structure in absence of an external B field, by using theoretical values for gF . We go onto determining the effects of power broadening the absorption dip. Lastly we measure Land˜ eg-factor, gF in both isotopes from measuring an applied B field from Maxwells coils and the corresponding resonance frequency.
1.
INTRODUCTION Optical Pumping is a method first developed by Alfred Kastler1 (who was awarded the Nobel Prize in 1966), that has become a widely used technique in experimental physics ever since. Optical pumping is a process in which electromagnetic radiation is used to pump electrons into a well-defined quantum state. The process by which this happens is dependent on the atomic structure of the sample and the properties of the radiation. The utilization of polarization and the selection rules for m leads to being able to pump the electrons into a dark state where there are no magnetic sublevels, mf, to excite to. This can be used even when there is an mf level to excite to and then the experimenter has a well-defined two-level system. The first thing one must account for is the atomic structure of the atomic sample.
2 1.1.
Fine Structure When an electron undergoes orbital motion, there is an associated orbital magnetic moment defined as ⃗ µ = I ⃗ A where I = −e(ω/2π) and ⃗ A = π⃗ r2.
Rewriting ω in terms of the angular momentum L we find that |⃗ µ| = −eℏ 2m ˆ l = −eℏ 2m p L(L + 1) , (1) where p L(L + 1) are the eigenvalues of the ˆ L operator and eℏ 2m = µB = 9.27 × 10−24JT −1 is the Bohr magneton.
In the presence of an external field, the magnetic moment will undergo a precession due to the cross product of the two terms, resulting n a torque vector. Defining the fields axis along the quantization axis ⃗ Lz one finds that the frequency of precession is the Larmor frequency: ωL = γB0 .
(2) Here γ = µB/ℏis the gyromagnetic ratio. As stated we have defined the quantization axis to be ⃗ lz. It is important to notice that |Lz| < |⃗ L|. This means that the magnetic moment will never completely align with the field and thus it will always precess as described above.
Our next step is to account for the spin of the electron by which S=1/2 and ms = -1/2, 1/2. It is shown that the resutling spin magnetic moment is3 ⃗ µs = −gs e 2m ⃗ S .
(3) The Land´ e g-factor for spin, gs, was theorized by Dirac to be 2, and then shown through Quantum Electrodynamics to be equal to 2.0023. As an aside, the motivation to measure gF in this experiment is due to the fact that this number is disputed and gF depends upon this value.
Next we must change our reference frame to the electron, e−, which sees a nucleus precessing around itself. Because the nucleus is charged, for reasons discussed in later sections, the rotating charge produces a magnetic field. Using the Biot-Savart Law we find ⃗ Bl = Ze2µ0 4πmr3⃗ l .
(4) 3 The electrons spin magnetic moment interacts with this field through the relation HFS = Ze2µ0 8πm2r2 (S · L) (5) where (S · L) is given by the relation J2 = L2 + 2L · S + S2 .
(6) Rearranging the terms, we find L · S = 1 2(J2 + L2 + S2) = ℏ2 2 [J(J + 1) −L(L + 1) −S(S + 1)] .
(7) The full expression for the fine structure correction to the Hamiltonian is given by3 HFS = Egα2 5 1 J + 1/2 .
(8) If the atom had no angular momentum from the nucleus then this would hold enough information to fully describe the energy levels. In the case of this experiment we must take into account the spin of the nucleus.
1.2.
Hyperfine Structure Due to the spin of the nucleus, I = 3/2 in 87Rb and I = 5/2 in 85Rb, I and J couple to give our grand angular momentum quantum number F = |⃗ F| = |⃗ I + ⃗ J|. Looking at the nucleus’ magnetic moment we see ⃗ µI = +gI µN ℏ ⃗ I , (9) where gI is the Land´ e g-factor for the nucleus’ spin, |⃗ I| = ℏ p I(I + 1), and µN = eℏ 2M = µB 1836.
The hyperfine perturbation to the Hamiltonian is given as follows: HHFS = −⃗ µI · ⃗ BJ (10) 4 where ⃗ BJ is given by ⃗ BJ = ⃗ J p J(J + 1) .
(11) Thus HHFS = gIµN ℏ Bj 1 p J(J + 1) (⃗ I · ⃗ J) .
(12) We solve for (⃗ I· ⃗ J) in the same procedure as Eq (1.6) by defining the total atomic angular momentum number ⃗ F = ⃗ I + ⃗ J. Where |⃗ F| = ℏ p F(F + 1) and F has values F = |I −J|, .., |I + J| of integer steps. For each hyperfine level, there are 2F+1 magnetic sublevels, mF . Solving for (⃗ I · ⃗ J) we find (I · J) = 1 2[F(F + 1) −I(I + 1) −J(J + 1)] (13) This gives us our full Hamiltonian for the hyperfine structure: HHFS = gIµNBJ 2 p J(J + 1) [F(F + 1) −I(I + 1) −J(J + 1)] .
(14) This is the full form for our hyperfine structure which we can now use to map out the energy levels of Rubidium.
1.3.
Energy Levels of 85Rb and 87Rb In this experiment we use naturally occurring Rubidium which comes in two isotopes, 85Rb (72% abundance and nuclear spin quantum number I=5/2) and 87Rb (28% abundance and nuclear spin quantum number I=3/2)2.
Alkali Atoms are defined by having a positively charged core with a single valence electron, in our case occupying the 5s orbital. The electron shell of orbitals [(1s2)(2s2)(2p6)(3s2)(3p6)(4s2)(3d10)(4p6)] are all filled making this Rb+ core spherically symmetric with a total angular momentum of Lc=0, spin orbital angular momentum of Sc=0.
The LS-coupled angular momentum quantum number ⃗ Jc=|Jc|=| ⃗ Lc + ⃗ Sc|=0 where J is defined quantum mechanically to take values of increasing integers between |L −S| to |L + S|.
Because the ion core does not contribute any total angular momentum, all the momentum quantum numbers come from the valence electrons. Considering the first electronic ground state, 5 with ionization energy 4.177eV or 296.81nm, in Rubidium (5s)2S1/2 we see that n=5, l=0, S=1/2, L=0 and therefore J = 1/2. Since J takes on only one value there is no fine structure in the (5s)2S1/2 state. For the first excited state, (5p)2P ,however there is fine structure splitting. We find that n=5, l =1, S = 1/2, L=1 and therefore J = 1/2, 3/2 which gives two fine structure levels of (5p)2P1/2 and (5p)2P3/2. The spin orbit coupling energy term is given by Eq. (8).
Using this equation we find that the (5p)2P1/2 is lower than the (5p)2P3/2 state. The energy to couple (5s)2S1/2 to (5p)2P1/2 is given as λ = 795 nm which is referred to as the D1 line. The energy to couple (5s)2S1/2 to (5p)2P3/2 is given as λ = 780 nm which is referred to as D2 line2,3.
In this experiment, both the D1 and D2 line are produced by the Rb Lamp, but we filter out the D2 line so that only the D1 line is incident on the atoms. It should also be noted that the lifetime of these excited states are extremely small, ≈10−8s which is instantaneous with respect to how fast the photodiode can detect changes in the input power.
If we account for the nucleus, and its spin quantum number then we find even finer splittings in energy spectra. Looking at the 85Rb isotope, I = 5/2, we find splittings in both fine structures.
In the ground state 52S1/2 the degeneracy is split into two hyperfine levels F = 2, F =3. For the 52P1/2 state we find F = 2, 3 whereas for the 52P3/2 state F = 1, 2, 3, 4. Looking at the 87Rb isotope with I = 3/2 we find that the ground state 52S1/2 is split to F = 1 and F = 2. In the two excited states we find that 52P1/2 is split into F = 1 and F = 2, while the 52P3/2 state has F = 0, 1, 2, 3.
As stated for each hyperfine F level, there are 2F + 1 mF sublevels if the degeneracy is lifted.
For F = 1 there are 3 mF levels corresponding to mF = -1, 0, +1. For F = 2, mF = -2, -1, 0, +1, +2. For F = 3, mF = -3, -2, -1, 0, +1, +2, +3 and for F = 4, mF = -4, -3, -2, -1, 0, +1, +2, +3, +4. A diagram of the energy structure for each isotope in the 52S1/2 and 52P1/2 state is shown.
These are the corresponding ground state and excited state excited from the D1 line.
6 D1 Line 795 nm F = 2 F = 1 F = 1 F = 2 .8 GHz 6.8 GHz -2 -1 0 +1 +2 -2 -1 0 +1 +2 -1 0 +1 -1 0 +1 D1 Line 795 nm F = 3 F = 2 F = 2 F = 3 -3 -2 -1 0 +1 +2 +3 -3 -2 -1 0 +1 +2 +3 -2 -1 0 +1 +2 -2 -1 0 +1 +2 361.58 MHz 3.035 GHz FIG. 1: The energy diagram for the D1 line for 85Rb and 87Rb is shown along with the corresponding energy spacing.
We find that in the presence of no external field, the hyperfine level F is degenerate. If a field is applied, then there is an energy spacing between mF porportional to the strength of the field. The number of these levels correspond to 2F+1.
In this experiment we use frequencies in the kHz range to make transitions between mF levels once the degeneracy is split. It is clear from the diagram above that we know transitions are made in the same hyperfine F level based on the large energy separation between F levels. We now have enough information to present the concept of optical pumping.
1.4.
Optical Pumping In this experiment the incident light on the Rb cell is σ+ polarized. Due to the selection rules we find that no transition can occur unless ∆m = 0, ±1 6. These solutions correspond to the three distinct polarization types which are π, σ−and σ+ with π referring to linear polarization, σ−referring to left-hand circularly polarized and σ+ referring to right-hand circularly polarized 7 light with respect to the direction of propagation. This experiment utilizes σ+ polarized light to send atoms to the highest mF level, which becomes a dark state, as it has nowhere to excite to after (since we block the D2 line). After a transition is made to an excited state, m′ F = mF + 1, it will decay rapidly and spontaneously. It is important to recall that the lifetime is on the order of ×10−8s. When spontaneous decay occurs, the direction of emission is uniform in all directions and polarization of the emitted light is arbitary, in that it can make a transition of ∆mF = 0, ±1.
This is indicated by the stripped line in the figure below. -2 -1 0 +1 +2 F = 2 F = 1 F = 1 F = 2 FIG. 2: Here the effects of σ+ polarized light incident on an atom are shown. Since the light has angular momentum the selection rules tell us that for σ+ light, ∆m = 1 in a transition. We see this results in electrons going to the right.
This diagram shows optical pumping to the F = 2, mF = +2 level in 87Rb. The same case is true in 85Rb except we optically pump to F = 3, mF = +3. Once the electron has undergone stimulated absorption, it will excite to a higher energy level. When it undergoes spontaneous emission the polarization of the light is random and therefore it can make any of the three transitions indicated by a dashed line. After many lifetimes of absorption and decay/emission we find all the electrons in the F = 2, mF = +2 state. This is the dark state. In the case of σ−light we can pump electrons to the left, to the F = 2, mF = −2 in this case.
8 If the electron undergoes stimulated emission then the polarization it encounters will be σ− sending it back to the original mF level it started at. It quickly becomes clear that to effectively optically pump, the incident light on the atoms must be present over at least ten lifetimes. In our case, if we take the lifetime of the excited state to be 10−8 then in a full second we have ≈108 cycles of absorption and decay or emission. Therefore, practically after only one second, the chances of all the electrons being optically pumped is very high. The efficiency is limited due to other factors that are noted in the Procedure section. At time t = 0 we assume that all the mF states are equally populated due to thermal excitations, kbT ≈1012s >> GHz. It must be noted that a static external field must be present to split the degeneracy of the mf levels, otherwise we could not optically pump. Experimentally the magnetic field from the Earth, Bearth, is enough to split the degeneracy. Now let’s look more in depth at how static fields perturbe the system.
1.5.
Interaction with Static External Fields This experiment utilizes a static magnetic field to split the degeneracy of the mF levels. A similar phenomena occurs when there is a static electric field present but for the purposes of this experiment those such effects will not be discussed. As mentioned, each hyperfine level consists of 2F+1 degenerate sublevels, mF . In the presence of an external magnetic field the degeneracy is broken.
HB = µB ℏ(gsSz + gLLz + gIIz) · Bz (15) Eq. (15) is the Hamiltonian describing the interaction with the magnetic field along the atomic quantization axis. We see that the B field interacts with the magnetic dipole moments of the electron spin, electron orbit, and the nuclear spin. gS, gL, gI are the electron spin, electron orbital, and nuclear ”g-factors”. Here, gL = 1 −me/mnucleus , which we approximate to 1. The exact value of gs has been measured to high precision to be 2.00231930436153(53) which we will approximate to 2 in this lab. To calculate gJ we look at the magnetic moment associated with J: (⃗ µS)J = [(⃗ µL)J + (⃗ µS)J] · J |J| (16) where 9 (⃗ µL)J = µB 2 ⃗ L · ⃗ J |J| (⃗ µS)J = −2µB 2 ⃗ S · ⃗ J |J| .
(17) Solving for ⃗ µJ we find, ⃗ µJ = −µB 2 3 2 + S(S + 1) −L(L + 1) J(J + 1) ⃗ J , (18) which gives us, gJ ≃3 2 + S(S + 1) −L(L + 1) J(J + 1) .
(19) The g-factor to consider is gF . Again we look at the magnetic moment associated with ⃗ F. Solving in the same fashion as above we find, ⃗ µF = −gJ F(F + 1) + J(J + 1) −I(I + 1) 2F(F + 1) µB ℏ ⃗ F , (20) which gives us a value of gF where we neglect the nuclear term because it is a correction of .1%.
We repeat this approximation below for the final form the Hamiltonian for the same reason. gF ≃gj 1 + J(J + 1) −I(I + 1) F(F + 1) .
(21) Writing the Hamiltonian for the hyperfine structure’s interaction with an external magnetic field we find, HHFS = A(⃗ I · ⃗ J) −⃗ µJ · ⃗ B0 −⃗ µI · ⃗ B0 ≃A(⃗ I · ⃗ J) −⃗ µJ · ⃗ B0 , (22) where A(⃗ I · ⃗ J) is the term describing the internal state and −⃗ µJ · ⃗ B0 describes the interaction with the external field. A is the magnetic dipole constant [Hz], dependent on the fine structure state, and has values given in References .
1.5.1.
Zeeman Effect In the weak field limit where J is a good quantum number, | −⃗ µJ · ⃗ B0| ≪|A(⃗ I · ⃗ J)|. We see that ⃗ F precesses around ⃗ B0. In this limit, the external field acts as a perturbation on the hyperfine structure. Thus the energy term can be solved to find, 10 ∆E|FmF > = hν = µBgF mF Bz , (23) where h = 6.62606957(29) × 10−34Js = Plancks constant, µB = 9.27400968(20) × 10−24JT −1 = Bohr Magneton, Bz is the magnetic field in the axis of quantization, mF = change in sublevel number and gF is the Land´ e g-factor for the hyperfine level. This equation describes the energy separation between magnetic sublevels mF , for a given hyperfine structure F in the weak-field limit.
This regime of splitting is called the Zeeman effect. We see that if we use the theoretical values for gF given by Equation (21) one can solve for the external B field.
Bz = ∆E µBgF mF = hν µBgF mF (24) We use this to solve for the Earth’s magnetic field, Bearth by finding the center frequency, ν and using a theoretical value for gF , where mF = 1.
When we rearrange the terms, we find gF = hν µBmF Bz .
(25) This equation can be used to give precise measurements of gF by plotting the absorption fre-quency ν, which corresponds to ∆E, versus the applied magnetic field, Bz.
The slope of the resulting line is porportional to gF . This is the procedure we used to measure the grand angular momentum g-factor, gF .
1.5.2.
Strong Fields Considering the strong field limit where | −⃗ µJ · ⃗ B0| ≫|A(⃗ I · ⃗ J)| we find that the interaction term dominates the hyperfine energies, and thus the hyperfine Hamiltonian is a perturbation on the strong-field eigenstates |J mJ I mI⟩.
∆E = gjmjµBB0 + AmImJ (26) This expression gives the energy separation in the strong field regime, between mI states for a given mJ value, where AmImJ is a small correction that arises since ⃗ I and ⃗ J both precess around ⃗ B0 due to the I-J coupling. The energy shift in this regime is called the Paschen-Back effect.
11 Dealing with the intermediate case requires one to diagonalizable HHFS + HB which becomes more difficult to compute. In this experiment we do not take measurements in this regime but discussions regarding it can be found in References , and .
1.6.
Spin Resonance Classically we know that a magnetic moment will precess around a static magnetic field at the Lamour frequency, wL = γB0. Changing our reference frame into the rotating frame of the magnetic moment we find d ⃗ J dt = γ ⃗ J × ⃗ B0 + ω γ = ⃗ B0,eff , (27) which we discover to be zero if ⃗ ω = −γ ⃗ B0. The gyromagnetic ratio, γ = gµB/ℏin the quantum mechanical treatment. Now lets add an additional rotating magnetic field ⃗ B1 = B1(cos(ωt)⃗ ex − sin(ωt)⃗ ey) which is analogous to the RF signal in this experiment.
If ω = ωL the magnetic moment only experiences ⃗ B1(t) and thus precesses around ⃗ B1(t), which is static in rotating frame, with ωR = γB1 which is our Rabi frequency.
If ω ̸= ωL then, ⃗ Beff = ⃗ B1⃗ erot + ⃗ B0 −ω/γ ⃗ ez | ⃗ Beff| = r B2 1 + ⃗ B0 −ω/γ 2 .
(28) We find the two terms precess around ⃗ Beff at a rate, ΩR = γBeff = q ω2 R + (ωL −ω)2 = q ω2 R + δ2 .
(29) In the analogy of a two-state system, if ω = ωL it is possible to achieve full inversion.
If ω ̸= ωL then we have a case where the state may not invert. Further investigation shows we can use principles describing the time evolution of a two-state system to completely describe how the RF signal excites transitions between mF sublevels.
We already know the interaction term is written H = −⃗ ˜ µ · ⃗ B. Here ⃗ ˜ µ is written as follows: ⃗ ˜ µ = γℏ 2 ⃗ ˜ σ (30) 12 with ⃗ ˆ σ = (ˆ σx, ˆ σy, ˆ σz) (31) are just Pauli’s matrices. Plugging this in to H we find, H = −ℏ 2γB0ˆ σz −ℏ 2γB1(cos(ωt)ˆ σx + sin(ωt)ˆ σy) = −ℏ 2ωLˆ σz −ℏ 2ω1(cos(ωt)ˆ σx + sin(ωt)ˆ σy) (32) Plugging in for ˆ σx, ˆ σy, ˆ σz we find, iℏ ˙ ˜ ag ˙ ˜ ae = −ℏ 2 ωL ω1e−iωt ω1eiωt ωL · ˜ ag ˜ ae (33) , which is the time evolution of a two-state system under a time dependent perturbation found from the Shro¨ edinger equation. Imparting the rotating-wave transformation with rotation operator ˆ Dz(φ) = e−i ˆ Szφ/ℏon ψ we get, |ψ⟩= ˜ age−iφ/2 |g⟩+ ˜ aee−iφ/2 |e⟩= cg |g⟩+ ce |e⟩ (34) where φ = ωt. Making the substitution δ = ω −ωL we find, iℏ ˙ ˜ cg ˙ ˜ ce = −ℏ 2 −δ ω1 ω1 δ · ˜ cg ˜ ce (35) .
This coupled differential equation is analogous to an optically driven 2-level atom. Recall we have a well defined Rabi frequency ωR = γB1 with a detuning δ = ω −γB0. The two coefficients describe the time evolution of the system as it falls to the ground state cg = 1, ce = 0 at time t = π/ωR and is excited back to ce = 1, cg = 0 at time t = 2π/ωR. By imparting the following initial conditions we find that the solution to this differential equation is as follows: cg(0) ce(0) = 0 1 (36) These are chosen because of experimental reasons. We start at the highest mF sublevel and can only excite down to the ”ground state” which is just the next lower mF level. Thus, 13 ˜ cg ˜ ce = cos( 1 2ΩRt) −i δ ΩR sin( 1 2ΩRt) i ωR ΩR sin( 1 2ΩRt) , ΩR ≡ q ω2 R + δ2 .
(37) It is more convienent to write the states as a function of their probability density, Pg or Pe which are defined below.
Pg = ˜ ag ˜ a∗ g = cgc∗ g = 1 −Pe (38) Pe = cec∗ e = ωR ΩR 2 sin2(1 2ΩRt) .
(39) Here, we see if Pg = 1 then Pe = 0 and the system is in the ground state and the same goes with Pe = 1, the system is in the excited state. We can also tell from this formula that if δ = ω −ωL = 0 then ωR/ΩR = 1 resulting in a maximum amplitude of 1 (probability cannot be greater than 1).
Otherwise if δ ̸= 0 our amplitude is reduced, so is the probability of the state making a transition.
Additionally, if ΩR = ωR the frequency of oscillation is minimized unless ΩR > ωR which leads to a higher frequency.
1.7.
Power and Doppler Broadening It is important to note certain aspects of the experiment that can affect the lineshape of the absorption dip. One key factor is Doppler Broadening which describes the change in frequency that an atom traveling with velocity v may encounter. Doppler Broadening causes the absorption dip to take a Gaussian functional form instead of the natural Lorenztian. This is shown below ω ′ = ω ± kv , (40) where ω is the angular frequency of the radiation, k = 2π λ is the wavenumber of the radiation, and v is the velocity of the atom. We choose to add the terms for the atom traveling towards the direction of propagation of the radiation, and a minus sign for the atom traveling away from the direction of propagation. If one solves for ω ′ using f = 200kHz and v = 100m/s , one finds that the change in frequency observed is ≃mHz.
This should hardly produce any effects on the absorption lineshape, which is naturally Lorentzian. Some of the preliminary data has shown 14 Gaussian functions to fit better than a Lorentzian but we believe that is due to some other effects.
Since we are primarily concerned with the center frequency, and not the overal lineshape, our goal is to use a fitting function resulting in the most accurate determination of this value. Regardless the broadening of the lineshape needs to be addressed.
The other effect that can vary the absorption lineshape is power broadening. When the radi-ation power hits a saturation point with respect to a two-level system going through stimulated absorption/emission then the absorption dips haven been shown to broaden. Although the affect of power broadening will not change the functional form of the lineshape, it will broaden the edges of the absorption dip. We study the effects of changing amplitudes and present our information in Section 3.1.
2.
PROCEDURE 2.1.
Apparatus In the experimental setup a Harrison Lab DC Supply feeds 16V to a Rb Lamp, which emits light over a range of frequencies but consists of predominately the D1 line λ1 = 794.76 nm and the D2 line λ2 = 780.02 nm. The light is focused by a lens and then passes through an interference filter (based offthe principles of a Fabry-Perot) which blocks out the D2 line light. This light also passes through a linear polarizer followed by a quarter-wave plate to circularly polarize the D1 light. The light is now σ+ or right hand circularly polarized. The selection rules tell us that for σ+ polarized light the only stimulated transitions between states that can be made are for ∆m = +1 . The light passes through a Rb vapor cell (which is heated to about 50◦C) and excites electrons from the first electronic ground state of Rubidium 5s2S1/2 to 5p2P1/2 via the D1 line. After many lifetimes of absorption and emission (spontaneous or stimulated) the electrons are pumped to the highest mF level of the 5s2S1/2 state; in 85Rb F = 3 mF = +3 and in 87Rb F = 2 mF = +2. This is due to reasons explained in Section 1.4. Because we have blocked the D2 line, the electrons are now in a dark state, meaning they cannot excite to a higher energy state. That is, based upon the polarization of the light, σ+, there is no mF = +3 state to excite to for 87Rb and no mF = +4 state to excite to in 85Rb since we have blocked the D2 line. If it were present then excitation would occur because these sublevels exist in the 5p2P1/2 state.
It should be noted that not 100% of the electrons are optically pumped. Various relaxation 15 processes including collisions with the walls of the cell and collisions with other atoms, cause deexcitation. In our cell, an Argon buffer gas acts to prevent most atoms from hitting the wall but instead Rb atoms collide with the Argon atoms. The cost of having the Argon gas present is that when a collision occurs, the energy levels of Rb are smeared out, or pressure broadened. This is accounted for by heating the Rb Cell to an optimum temperature where the resonant dips are not widened due to this effect. The light that passes through the Rb cell is focused onto a Silicon photodiode, powered by a 9V battery, which outputs a voltage to an Agilent 34401A Digitial Multimeter. The reading on the multimeter is fed to a computer and read using LabVIEW. To efficiently optically pump, it is necessary to excite the state a number of lifetimes, in the order of ten, for the electrons to be pumped to the desired states. This means that in approximately .2 ns the electrons are optically pumped to a dark state at which time no absorption of the light occurs and the photodiode reads a steady voltage. Measurements in this experiment are attributed by applying an RF signal through Helmholtz coils that surrond the Rb cell, perpindicular to the Maxwell coils. The reason for this is based upon the principles of Section 1.6 where the field produced by the Maxwell coils is denoted by B0 and the field from the Helmholtz coils is B1. The Helmholtz coils consist of two loops of an unknown number of turns of 0.0812(5) cm diameter copper wire. The coils are of radius r, separated by their radius (center to center) as required in a Helmholtz coil configuration. A HP 3330B Synthesizer is used to generate RF frequencies in the kHz range. The RF radiation is used for stimulated emission to drive the electron to the next lower mF level in the hyperfine structure; F = 2 mF = + 1 for 87Rb and F = 3 mF = +2 for 85Rb. In this new state, the electrons can now excite to the 5p2P1/2 state via the D1 line. Due to the short lifetime, the electrons fall back to either mF level and the process repeats only if the transition is fully saturated. This means that the cycle repeats roughly 109 times in a second. As a result enough energy is absorbed by the atoms, from the light field, to be detected by the photodiode. As a means of detecting resonance, as we start the frequency sweep a certain range of frequencies will drive this transition and thus absorb energy which will show as an absorption dip read by the photodiode. As discussed, Section 1.7, this lineshape is affected by various process’.
The RF synthesizer has two outputs. One is input to a PRD Model 7805 Amplifier and the other output is connected to an Agilent 34401A Digital Multimeter. The PRD Amplifier sends an output signal to the Helmholtz coils which drives the transitions between magnetic sublevels. In Section 3.1, we study the gain from the amplifier as a function of input frequency. The Agilent Digital Multimeter is connected to a computer which also is read on LabVIEW in terms of a 16 voltage sweep. If one records the center frequency, step size, and number of steps in the frequency sweep it is possible to scale the voltage sweep to a frequency sweep since the relationship is linear.
Recording the frequency sweep in LabVIEW and the voltage read by the photodiode one has all the necessary components to observe resonance corresponding to absorption dips.
Figure 3 shows a diagram of the experimental set up. We now have enough information to understand the data collection process. Now lets look into Maxwell Coils.
2.2.
Maxwell Coils This experiment utilizes a Maxwell coil configuration to produce a homogeneous magnetic field through its center. The field produced is more uniform than that produced by Helmholtz coils.
Fig. 3 shows the relationships between the center coil’s radius R and the number of turns N to the radius of the outside coils, the distance from the center coil, and the number of turns. We find that if a current runs through the wires in the coils a magnetic field is produced which runs through the center of each coil. The direction of the magnetic field lines are parallel or antiparallel to the propagation of the D1 line light depending on the direction the current runs through the wires as described by the right hand rule. As stated the Maxwell coil produces a uniform field through its center which is essential so that each Rb atom ’see’ the same magnetic field otherwise the inaccuracy in our measurements would be large, recall Avagadros number. The coils are wound with 0.132(5) cm diameter copper wire. In the center coil ’B’ there are 14 layers of wire, with 11 turns in odd-numbered layers and 10 turns in even numbered layers with the exception of layer 14 which has 5 turns making a total of 142 turns. The diameter of coil ’B’ is 78.4(5) which is measured to the innermost layer. At a distance of 26.2(2) cm in both directions (from center to center) are coils ’A’ and ’C’. Each coil has 11 turns in odd numbered layers and 10 turns in even numbered layers again with the exception of the outermost layer, 11, which has 5 turns. Each small coil has an inner diameter of 59.1(5) cm.. In Section 3.3 we use the Biot-Savart Law to determine the magnetic field produced by the Maxwell coil as a function of current through the wires. The current is supplied by a HP 6011A DC Power Supply. We measure the current in series using a Keithley 199 Trims Digital Multimeter.
17 Rb Lamp Maxwell Coils Collimating Lens Polariza(on Op(cs D1 Filter Collimating Lens Photodiode RF Coils Rb Cell Experimental Apparatus N turns Radius = R Radius = Radius = turns turns R 3 7 R 3 7 R 4 7 R 4 7 49 64 R 49 64 R FIG. 3: This is a diagram of the experimental apparatus, which is approximately .745m (long not including the base). It is set at an angle of approximately 30◦to the floor. A Rb lamp emits a range of wavelengths consisting of primarly the D1 and D2 line. It passes through a collimator, D1 filter to block the D2 light, a linear polarizer and a quarter-wave plate all of which generates circularly polarized light. The light is incident on the Rb cell and the transmitted light is read onto a photodiode. When absorption occurs the transmitted power decreases which is detected by the photodiode. The setup for the Maxwell Coils are shown including their dimensions. The RF coils which provide the kHz frequency to make transitions between m-sublevels are in a Helmholtz coil configuration which means they are seperated by their radius with an equal amount of turns.
18 3.
DATA AND ANALYSIS Once resonance has been observed, the experiment allows plenty of room to measure various phenomena. We first measured the effects of power broadening in the 87Rb isotope. We used this to determine an optimum amplitude range providing us with a clean absorption dip making the fitting more precise. Afterward we measurd Bearth by determining the center frequency of a transition and by using theoretical values of gF in each isotope from references ,. Once these values are known we can then calculate Bearth from Eq. (24). Afterward we apply a current through the Maxwell coils to produce a magnetic field to change the energy separation between magnetic sublevels in a controlled manner. These effects are studied in the weak and strong field regimes.
In both these cases, and as seen with the Earth’s magnetic field calculation, we use GNUplot to fit a Gaussian function, utilizing the method of least squares fit to determine the center frequency of the absorption dip. The fitting function has the form: f(x) = ae−(x−b)2 2c2 + d .
(41) Naturally the lineshape should have a Lorentzian form, but various experimental effects produce an absorption dip that takes a Gaussian form. Our measurements are concerned with the value of the center frequency which is why we chose to use a Gaussian fitting function.
3.1.
Power Broadening As discussed, one factor contributing to a widened lineshape is power broadening. Here we measure the lineshapes in 87Rb for various output amplitudes from the HP 3330B Synthesizer.
Recall the output signal from the synthesizer is connected to a PRD Model 7805 (RF) Amplifier.
The first thing we did was to measure the frequency dependence on the amplified signal for a fixed amplitude on the synthesizer. Due to the nature of sweeping the frequency, it is important to determine this relation. We find that the amplified signal responds differently for various input frequencies. This is probably due to the internal circuitry of the RF amplifier.
From the figure we verify that the RF amplifiers gain is dependent on input frequency. We are are also able to determine a saturation point, where the RF output reads overdrive, in which the gain is too high for the amplifier. It also provides reason behind why we have to change the synthesizer amplitude to -10dBm when measuring the energy separation in the strong field regime.
19 This is due to the fact that there is no gain from the amplifier for -35dBm input at high frequencies, shown below.
FIG. 4: Here a HP 3330B Synthesizer’s output signal is input to a PRD Model 7805 Amplifier. We find the amplification from the PRD is dependent on the input frequency. Here two different amplitudes are set on the synthesizer while the frequency is stepped. While the PRD notes a 47dBm gain we find a saturation point around 41dBm.
The next step was to sweep over a range of amplitudes on the synthesizer. The goal was to obtain an optimum ratio between signal strength and broadness of the lineshape. We also wanted to resolve apparent asymmetries in the lineshape. In this case we used the 86Rb isotope as our test signal. The frequency sweep was centered at 304.75 kHz with a 1000 steps of 10 Hz at 30ms/step.
Below is a plot showing the various lineshapes for different amplitudes set by the sythesizer.
20 FIG. 5: The absorption dip for 87Rb is shown here for various amplitudes set on the frequency synthesizer.
We see that the wave shape goes from having a slight asymmetry to symmetric with decreasing amplitude.
This graph is also a good indicator of one form of error we have in our data collection methods which is a DC drift of the baseline voltage. Based on the results of the graph, we found that a optimum value for the amplitude set by the synthesizer is between -36dBm and -40dBm. Note that in the strong magnetic field regime, when the RF frequency is high, we need to use a higher power to resolve the absorption which we choose to be -10dBm. This is explained by Fig.5 which shows the frequency dependence of the RF amplifier.
Based upon these results we find that the optimum range of amplitude, before passing through the amplifier, which is -47dBm to -32dBm. The asymmetries are corrected in the applied field as the amplitude is lowered. As an aside, other reasons for the asymmetries in the lineshape are due to pressure broadening from collisions of atoms in the Rb Cell. As stated, a buffer gas of Argon is added to reduce this effect but it is still present. This is our motivation for choosing our primary amplitude to be -35 dBm. We did not choose -40dBm, as this graph would indicate, because of the frequency dependence on the RF amplifier. At -35dBm, the range at which we could resolve dips in terms of frequency was larger than -40dBm. We see in later sections that we have to increase 21 the amplitude to resolve the dips of each transition at higher input frequencies.
3.2.
Measuring the Earth’s B Field When there is no current in the Maxwell Coils, the presence of the Earth’s magnetic field is enough to break the degeneracy of the hyperfine structure. We have shown that in this regime, Zeeman effect, it is possible to find use theoretical values for gF to find Bearth. There is also the presence of ambient magnetic fields from other experiments in this laboratory but we neglect them because they are much smaller than the magnitude of the Earths field. Here we use Eq. (24) to determine the B field, using theoretical values for gF : for 85Rb F=3 gF = 1/2 2 and for 87Rb F=2 gF = 1/3 3.
Changes in this value may vary because: the field itself is moving, alignment with the apparatus, and ambient magnetic fields present in the laboratory. All these factors contribute to the variability in Bearth.
FIG. 6: The Earth’s magnetic field strength along the quantization axis, z, is shown here. We used the absorption frequencies in both isotopes to give comparable measurements over a period of two weeks.
22 If we take the average of the B field found from the 85Rb absorption resonance we find Bearth(avg) = 43.6498(50)µT and for the 87Rb isotope we find that Bearth(avg) = 43.6309(37)µT.
While these values are not within error range of each other, they only differ by .04% giving us reason to trust these calculations. This also implies that we are correctly identifying which ab-sorption dip corresponds to which isotope. Otherwise the calculated values would be incorrect by an amount proportional to the corresponding g-factor. Once we have measured this value, we can split the separation between mF levels even further by applying an external field. In doing so, we also find an external field strength such that it cancels the Earth’s magnetic field which is indicated by observing no absorption dips over a frequency sweep. This is because there is no breaking of the degeneracy in the mF levels and also gives us another measurement of the Earths field to compare to the calculations done here.
3.3.
The Magnetic Field Produced By Maxwell Coils The magnetic field produced by a loop of N turns, radius r, and current I is described by the Biot-Savart Law.
⃗ B = Nµ0I 4π I C d⃗ l × ⃗ r |⃗ r|3 (42) Solving this equation for the magnetic field at a distance z above the center of the coil we find, ⃗ B = Nµ0I 2 · r2 (r)3/2 ,r = r2 + z2 (43) Using propagation of error we find that ∆⃗ B is, ∆⃗ B = ⃗ B ∆I I + 2∆r r + 3∆r r (44) The full expression of ⃗ B is ⃗ B + ∆⃗ B = Nµ0I 2 · r2 (r)3/2 1 ± ∆I I + 2∆r r + 3∆r r (45) We used a Matlab script to sum the individual contribution each layer in each coil has to the magnetic field at the Rb cell which is at the center of coil ’B’ as a function of the applied current.
23 We also considered the error in these calculations. The error increases as B increases because we are multiplying a constant percent error by an increasing B field. The results of our calculations are shown below.
FIG. 7: Using a Matlab script, we were able to determine the magnetic field at the center of a Maxwell coil as a function of the current. By determining the slope of the line (T / A) we can then relate what the magnetic field is for a given current.
We see from the graph that the slope is ∆B/∆I = 4.1725 × 10−05 ± 2.2986 × 10−9 [T/A] . This allows us to calculate the magnetic field B for any given current I by multiplying the slope by a given current. We use this to scale a given current to its corresponding magnetic field accordingly in the following sections.
3.4.
Weak Field Measurements Continuing the concepts from the above sections we know that when an applied magnetic field is small, such that it acts on a perturbation to the hyperfine structure, the energy separation between magnetic sublevels in the hyperfine structure is given by, ∆E|FmF > = hν = µBgF mF Bz (46) 24 Thus as we vary the current through the coils, I, we also vary B linearly as described in the previous section. By sweeping through a range of current we are able to observe what the new value of ∆E by measuring the center frequency of the absorption dip. Below is an example showing the absorption dip and the corresponding fitting function for a given current in the Maxwell Coils.
FIG. 8: A 185.841 ± 0.005294 kHz separation between mF = +3 →+2 absorption dip for 85Rbis shown here. A current of 200.1 mA in the Maxwell coil produces an external field of magnitude 83.4915(22) µT (not subtracting the Earths field). One must include the Earths field to give the total B field, which causes this given frequency gap, the atoms see in the Rb cell. We see that the center frequency is slightly asymmetric which is the reason why we zoom in on the center frequency. In doing so we can measure our center frequency with good precision.
25 FIG. 9: A 279.561 ± 0.004082 kHz separation between mF = +2 →+1 absorption dip for 87Rb is shown here. A current of 200.4 mA in the Maxwell coil produces an external field of magnitude 83.6169(22) µT (not subtracting the Earths field). One must include the Earths field to give the total B field, which causes this given frequency gap that the atoms see in the Rb cell. This graph showcases the accuracy of the fitting which is reflected in the reduced χ2 value.
The figures below showcases that the signal observed when the applied B field through the coils is equal and opposite the magnitude of the Earth’s field. We were able to do this by making the polarity of the current running through the Maxwell coil such that it produced a magnetic field in the direction (directed floor to ceiling) opposite that of the Earth’s field (coming from the ceiling towards the floor). In doing so there is no splitting of the degeneracy of the magnetic sublevels in the hyperfine structure which means no transitions can be made between mF states.
26 FIG. 10: In 85Rb we found that we could cancel the Earth’s magnetic field in a range from Bext = 41.8919µT →47.98 µT. In this region the signal to noise was so low we could not resolve any absorption dips.
27 FIG. 11: In 87Rb we found that we could cancel the Earth’s magnetic field in a range from Bext = 41.8919µT →48.1632µT. Here is a plot of a frequency sweep within this range. We see the character-isitics of noise, because if we were to repeat the same plot for the same parameters it would look different.
We use this procedure to determine the center frequency as we sweep the magnetic field (scaled from the current) in the Zeeman effect region. In doing so we can calculate the Earth’s field, when the two isotopes intercept, and find a value for gF which is given by the slope of the line. The results are as shown.
28 FIG. 12: Here we increased the magnetic field produced by the Maxwell coils to cancel the Earths field. We plot this as a function of the center frequency of absorption. As we expect, as Bext is increased the energy spacing and therefore the frequency between two mF levels decreases. At a certain point the two fields are of equal magnitude but in opposite directions so they cancel to produce zero magnetic field. Experimentally it was hard to resolve any absorption dips in a certain range of Bext which is why there is a range of no data points. As Bext is increased beyond the magnitude of Bearth we see that the frequency increases which is as expected.
For reasons indicated by the figure it was difficult to resolve the energy spacing when Bext ≃ Bearth. In order to fit an accurate slope to each of these lines for purposes of calculating gF and the Earths field (intersection of the two lines) we negate all the frequency values before we cancel the Earths field. This should give us the linear relationship shown below. While the graph is completely non-physical (negative frequency) components shown are. It should be noted that we are able to determine gF from this graph without substracting what the Earths B field from each Bext value which would give Btot because the slope of the line is unchanged. We have shown that gF is given by, 29 gF = hν µBmF Bz (47) Plotting these two lines and using the y = α ∗x + b to find their slopes, α ∝gF ,we find the graphs below.
FIG. 13: Although we are plotting negative frequencies (non-physical) we can determine real variables resulting from the fitting functions. We find that the slopes of each line are proportional to the g-factors for the corresponding hyperfine level F. Additionally, the point of intersection between the two lines can be used to determine the Earth’s magnetic field.
Below is a table indicating the fitting parameters for each isotope. We use these values to give obtain final measurements of gF and Bearth.
slope (Hz / T) = α y-intercept (Hz) Reduced χ2 85Rb 4, 906.33 × 106 ± 1, 011 × 105 −216, 823 ± 4, 472 0.00824549 87Rb 7, 311.98 × 106 ± 1, 482 × 105 −323, 253 ± 6, 559 0.00797794 30 Using Eq. (47) to include propagation of errors we find gF = hα85Rb,87Rb µBmF 1 ± ∆h h + ∆µB µB + ∆α85Rb,87Rb α85Rb,87Rb (48) where α85Rb,87Rb is the slope corresponding to each isotope shown in the table above. In the 85Rb isotope for F = 3 we measure the g-factor to be gF = 0.350(72). Comparing this to the theoretical value given in Reference our measurement is different by 5.16%. In the 87Rb isotope for F = 2, the measured g-factor is gF = 0.52(15). Comparing this to the theoretical value given in Reference our measurement is different by .44%.
Looking back to the fitting functions we can set the two equations of a line equal to each other and solve for the corresponding B value. We find that at the intersection of the two fits, Bz = Bearth = 44.2417 ± 3.61575µT. Here the error is probably due to not being able to resolve any resonance frequencies near Bearth = Bcoils. Other contributions could be ambient magnetic fields that exist in the laboratory as well as placement of many of the metal cabinets and desks.
The error does fall within the range of the values of Bearth measured in Section 3.2.
3.5.
Strong Field Measurements As discussed in Section 1.5.2 if | −⃗ µJ · ⃗ B0| ≫|A(⃗ I · ⃗ J)| then the hyperfine energies act as a perturbation to the interaction term. In this regime J and I are good quantum numbers. There becomes a point where the magnetic field is strong enough, as well as the RF power, for us to split the degeneracy such that we can resolve the transition for each ∆mI state for a given mJ value.
Theoretical evidence for this is supported in References and . Experimental evidence for this is shown in Figures (16 - 21) and Figures (24 - 30). In the Appendix (Section 7) we present the changes in the absorption dip as we increased the applied magnetic field from the Maxwell coil.
In this case we reversed the polarity of the current such that the magnetic field from the Maxwell coil Bext adds to the Earth’s field Bearth which we have calculated to be 44.2417 ± 3.61575µT in the section above. In the Appendix we show the splittings in this region for both isotopes.
As shown that we begin to resolve the transitions between each magnetic sublevel ∆mI = −1.
In the case of 87Rb we believe we observed two photon transitions where ∆mI = −2. At this strong field, the splitting between mI levels is large and equivalently our RF power is high too.
Therefore once an electron deexcites to a lower mI level instead of being optically pumped to the 5p2P1/2 state it can undergo another magnetic resonance transition to the next lower mI level 31 making mI(final) = mI(initial) −2 in this simplified three level system.
For 85Rb I = 5/2, mJ = +1/2, −1/2 and mI = −5/2, −3/2, −1/2, +1/2, +3/2, +5/2 which corresponds to a total of 5 possible transitions. For mJ = +1/2 the states where mI > 0 have a higher energy then states where mI < 0. This phenomena is opposite when mJ = 1/2, mI < 0 states have higher energy than mI > 0 states. In the case of 87Rb I = 3/2, mJ = +1/2, −1/2 and mI = −3/2, −1/2, +1/2, +3/2 making a total of 3 possible transitions. We observe the same phenomena where for mJ = 1/2 the states when mI > 0 have a greater energy than mI < 0 states. The opposite is true for mJ = −1/2.
For the 85Rb isotope, I = 5/2, we obtain results we would expect in terms of the number of absorption dips.
These corresponding to the possible values of mI = +5/2, +3/2, +1/2, −1/2, −3/2, −5/2 making a total of five possible transitions. As shown in the graph we expect the +5/2 →+3/2 transition to be the strongest signal, at the lowest energy, with each lower transition decreasing in signal strength but increasing in energy separation. We also observe some unusual phenomena shown by the presence of a sixth absorption dip. This transition could possibly be from mJ = 1/2, mI = −5/2 to mJ = −1/2, mI = −5/2. There is room to be skeptical of this explanation by looking at Figure 4 in Reference . We expect the energy gap between mI = −5/2 in mJ = +1/2 and mJ = −1/2 to increase as the total magnetic field increase but in our measurements the energy separation between dips trends to be even. Even if we were at such a strong field where the energy separation between mI levels increases linearly, the energy gap between mI = −5/2 for mJ = +1/2 →mJ = −1/2 should be much larger than the gap between mI = −3/2 →−5/2 for mJ = +1/2. We find the energy separation to be nearly even.
For the 87Rb isotope, I = 3/2, we obtain similar results. The number of expected absorption dips (3) is present. This corresponds to the change in mI : +3/2 →+1/2, +1/2 →−1/2, −1/2 →−3/2.
Again since we optically pump to the mI = +3/2 state we expect the +3/2 →+1/2 to be the strongest transition but also occur at a lower energy, Figure 4 in Reference , which is observed experimentally. As with the other isotope we observe extra dips that we do not expect. We cannot explain the second dip seen in Fig. (22). As we increase the B field and observe additional dips the same reasoning can be applied as discussed above. This would mean that transitions are being made between mI = −3/2 for mJ = +1/2 to mJ = −1/2 states. Again we are very skeptical of these results for the same reasons as discussed above by comparing our results to Reference .
In addition, we observe what we believe to be mutliphoton transitions seen to first appear in Fig. (28) and shown completely in Fig. (29). There is no other explanation for these dips to arise and the fact that the growth in between two resonance dips corresponding to two different transitions gives indications that this is the case. It is unusual is that these dips disappear in the 32 next graph which corresponds to a higher magnetic field. A way to resolve this would be if the separation between the two transitions 1 →2, 2 →3 becomes too large for the present RF power to make excitations between 1 →3.
4.
CONCLUSION This experiment demonstrates the application of optical pumping used to prepare states for magnetic resonance. We have shown that electrons can be pumped to a dark state but utilizing σ+ polarization. Spin resonance, induced by a radio frequency signal, can stimulate transitions between magnetic sublevels which can be summarized in the picture of a two-level system. My measuring the frequency at which these transitions occur the values of physical constants such as the Lande´ e g-factor and the Earth’s magnetic field can be obtained.
5.
ACKNOWLEDGEMENTS I would like to thank Dr. Schneble and Ludwig Krinner for their assistance in teaching us the theoretical concepts behind this experiment and for providing multiple references used in this write-up. I would like to thank Dr. Metcalf and John Elgin for helping me with questions pertaining to the experiment. I would also like to thank Mehdi Namaz for his help in the laboratory.
6.
REFERENCES 1. Bloom, Arnold L. Optical Pumping. Scientific American, Inc. 1960.
2. PHY 445 Optical Pumping. 2013 3. Schneble, Dominik. Ultracold Atomic Physics. PHY 565 Quantum Electronics Spring 2012.
4. Steck, Daniel Adam. Rubidium 85 D Line Data. Oregon Center for Optics and Department of Phyiscs, University of Oregon.
5. Steck, Daniel Adam. Rubidium 87 D Line Data. Oregon Center for Optics and Department of Phyiscs, University of Oregon.
6. Griffiths, David. J. Introduction to Quantum Mechanics. Pearson Education, Inc. 2005.
7. NIST Reference on Constants, Units, and Uncertainty.
National Institute for Standards and Technology. June 2011 8. Milonni, Peter W. & Eberly, Joseph H. Laser Physics. Wiley 2010.
33 7.
APPENDIX Here we show the transformation of the absorption dip as we increase the applied field in the strong field regime.
The following graphs show the effects of increasing the current, a larger B field, causes a higher energy separation between magnetic sublevels.
All these graphs are for the strong field regime where the hyperfine structure is a perturbation to the applied field. It is shown that we begin to resolve the transitions between each magnetic sublevel ∆mI = −1.
The following graphs are for the 85Rb isotope for an increasing current. The corresponding B field is noted in the caption which includes Bearth using the value calculated in Section 3.4.
FIG. 14: For B = 231.796 ± 3.6157µT we have an energy spacing of approximately 1072.2 kHz.
34 FIG. 15: For B = 271.463 ± 3.6157µT we have a main dip around 1459 kHz. Here we begin to see multiple dips seperating themselves from main dip. This corresponds to the separation between mI states beginning to be resolvable.
35 FIG. 16: For B = 398.946 ± 3.6157µT the main dip resides around 1842 kHz. The splittings between lower mI states are shown to be approximately 1845.8, 1848, and 1855 kHz respectively. Each weaker dip at a higher energy indicates electrons going from one lower mI state to the one below it.
36 FIG. 17: For B = 458.808 ± 3.6157µT the presence of transitions between mI states are apparent. The center frequencies for each dip are located at approximately 2322, 2326 2329, 2332.5, and 2337 kHz. We also notice an unexpected dip at 2341 kHz.
37 FIG. 18: For B = 544.524 ± 3.6157µT we find our transitions occur at approximately 2512.5, 2517, 2522, 2526, 2529 kHz with the unexpected dip at 2534 kHz respectively.
38 FIG. 19: For B = 587.209 ± 3.6157µT we observe several clean transitions. These transitions are centered at approximately 2709, 2714, 2718.5, 2725, 2730, and 2733 kHz respectively. As we suspect the energy gap betwen the same mI states is increasing for increaseing B-field. The presence of the extra sixth dip, here at 2735 kHz is still evident.
39 FIG. 20: For B = 628.1 ± 3.6157µT these transitions corresponding to m ′ I = mI −1 down to the lowest mI level for that specific hyperfine structure. It makes sense that the weakest signal is the least probable corresponding to m ′ I = −mI not considering the weakest dip here which is unexpected.
40 FIG. 21: For B = 671.076 ± 3.6157µT we continue to see several transitions between mI states as expected.
Additionally the energy spacing between them increases as well. Here the dips are centered at 3195, 3202, 3207.5, 3213.5, 3221, and 3227.5 kHz approximately. It is reassuring the number of dips expected is consistent in our measurements once they were all resolved. This, of course, does not include the dip at 3227.5 kHz which we have possible suspicions for.
The following graphs are for the 87Rb isotope for an increasing current. Recall, in 87Rb I = 3/2 and therefore mI = −3/2, −1/2, +1/2, +3/2. So once this is resolved we should see a total of 4 possible transitions. The corresponding B field is noted in the caption.
41 FIG. 22: For B = 253.993 ± 3.6157µT we have our characteristic absorption dip for ∆mI = −1. The side dip is a result of some unkown phenomena probably related to the apparatus and not the atomic structure.
As we cannot resolve over mI transitions I do not believe this is a mutliphoton transition, ∆mI < 1 42 FIG. 23: For B = 336.859 ± 3.6157µT we begin to see the dip become asymmetric as the other transitions begin to seperate from the main dip.
43 FIG. 24: For B = 419.892 ± 3.6157µT it is clear that other mI transitions have a large enough energy gap to begin to differentiate themselves.
44 FIG. 25: For B = 461.617±3.6157µT the signal strength from these other transitions increase and showcase their center frequencies which are 3201.5, 3205, 3207.5 3210 kHz approximately. As expected we have three transitions present, neglecting the two unknown dips to the right of them all.
45 FIG. 26: For B = 628.392 ± 3.6157µT the transitions now have their own linewidth because the magnetic field is strong enough to split these state enough in terms of energy to resolve their full shape.
46 FIG. 27: For B = 670.158 ± 3.6157µT the lineshape for each of these transitions has become really clean.
The presence of the abnormal fourth and fifth dip to the right is still present and has been for each step in B.
47 FIG. 28: For B = 711.883 ± 3.6157µT we begin to see the arisal of multi-photon transitions where ∆mI > 1 meaning that an electron can deexcite from mI = +3/2 to 0. It makes sense that their dips arise in between the two since the frequency should be roughly the average of the two.
48 FIG. 29: For B = 752.983 ± 3.6157µT the multiphoton transitions have become very prominent and well defined. I cite the arisal of these extra dips as multiphoton transitions because of their center frequency and the number of them 49 FIG. 30: For B = 837.225 ± 3.6157µT we need to sweep over such a large range the absorption dips begin to appear thin. The multiphoton transitions that were present in the previous graph are not now which is very interesting. This could be to the power of the RF not being strong enough to resolve the transitions. |
10976 | https://pdfs.semanticscholar.org/c66e/30afdee7ea98588928168142010590773c3e.pdf | Journal of Mechanical Engineering and Sciences (JMES) ISSN (Print): 2289-4659; e-ISSN: 2231-8380; Volume 9, pp. 1538-1555, December 2015 © Universiti Malaysia Pahang, Malaysia DOI: 1538 Research progresses and future directions on pool boiling heat transfer M. Kumar, V. Bhutani and P. Khatak Mechanical Engineering Department, Guru Jambheshwar University of Science & Technology, Hisar (India)-125001 Email: vijay.bhutani.24@gmail.com Phone: +911662263564; Fax: +911662276025 ABSTRACT This paper reviews the previous work carried on pool boiling heat transfer during heating of various liquids and commodities categorized as refrigerants and dielectric fluids, pure liquids, nanofluids, hydrocarbons and additive mixtures, as well as natural and synthetic colloidal solutions. Nucleate pool boiling is an efficient and effective method of boiling because high heat fluxes are possible with moderate temperature differences. It is characterized by the growth of bubbles on a heated surface. It occurs during boiling of liquids for excess temperature ranging from 5 to 30 °C in various processes related to high vaporization of liquid for specific purposes like sugarcane juice heating for jaggery making, milk heating for khoa making, steam generation, cooling of electronic equipments, refrigeration and etcetera. In this review paper, pool boiling method during heating of liquids for specific purpose is depicted. It is inferred that enhancement in pool boiling heat transfer is a challenging and complex task. Also, recent research and use of various correlations for natural convection pool boiling is reviewed. Keywords: Pool boiling; nucleate boiling; Rohsenow correlation; heating of liquids; pool boiling correlations. INTRODUCTION Boiling is an effective and efficient mode of heat transfer which is used for the transfer of heat for various heating purposes . Many researches have been done since the 1930’s to analyze the boiling process and its characteristics [2, 3]. Rohsenow (1952) proposed correlations for heat flux and heat transfer coefficient for various liquids rosenow. Mostinski, Kutateladze, Labantsov, Kruzhilin, Cooper, Gorenflo, Stephan & Abdelsalam and many other eminent researchers proposed various pool boiling heat transfer correlations for different liquids . Although a lot of research is being carried out on the mechanism of pool boiling, it is not yet accurately understood. Nonlinear mutual interaction between numbers of sub-processes makes the boiling phenomena more complex to understand . The different boiling regimes based on the excess temperature are nucleate boiling, transition boiling, and film boiling [7, 8]. Nucleate boiling is characterized by the growth of bubbles on a heated surface. The bubbles rise from discrete points on a surface, whose temperature is slightly above the liquid’s saturation temperature . Transition boiling (or unstable boiling)s occurs at surface temperature between the maximum attainable temperature in nucleate boiling and the minimum attainable temperature in film boiling. When the heating surface temperature becomes significantly hotter (above 100 °C), film boiling takes place, where Kumar et al. / Journal of Mechanical Engineering and Sciences 9(2015) 1538-1555 1539 a thin layer of vapor is formed that acts as insulation and results in reduced heat transfer [10, 11]. Critical heat flux leads to a drastic rise in heater surface temperature [9, 12, 13]. It is used in various heat exchange systems and in cooling of high-energy–density electronic components [14-16]. Rohsenow proposed the following correlation for pool boiling heat transfer. 3 2 / 1 " )] Pr /( [ ] / ) ( [ n l fg sf pl v l fg l h C T C g h q (1) In this review paper, the potential of pool boiling method for heating of liquids for specific purposes has been depicted by considering various researches conducted on pool boiling by many eminent researchers worldwide. Also, recent research and use of various correlations for natural convection pool boiling is reviewed. Various applications, advantages, disadvantages, and future research directions of pool boiling are also depicted. Earlier reviews on boiling are summarized in Table 1. Table 1. Reviews on pool boiling Authors Year Remarks 1998 Advancement in predicting boiling heat fluxes 2004 Pool boiling heat transfer under reduced gravity. 2005 Influence of lubricants on HT of the refrigerants 2008 Pool boiling heat transfer to HC and NH4 in refrigeration 2011 Boiling HT performance of refrigerants mixtures 2011 Boiling heat transfer enhancement with nanofluids 2011 Statistical analysis of anomalous HT of nanofluids 2011 Progress on nucleate boiling of nanofluids 2011 Critical heat flux enhancement for nanofluids 2012 Pool boiling experiment of multi-component mixtures 2012 Numerical simulation of pool boiling fundamentals 2013 Fundamental issues of critical heat flux RESEARCH CONDUCTED ON POOL BOILING Nucleate pool boiling is a significant method of heat transfer at high rate and thus is studied by various researchers. Various liquids and commodities categorized as refrigerants and dielectric fluids, pure liquids, nanofluids, hydrocarbons and additive mixtures, and natural and synthetic colloidal solutions used for carrying out the research on pool boiling for different purposes have been reviewed in the following sections. Research progresses and future directions on pool boiling heat transfer 1540 Refrigerants and Dielectric Fluids The heat transfer coefficients with the method of regression analysis in natural convection boiling for water, hydrocarbons, cryogenic fluids, and refrigerants were predicted [28, 29]. In addition, the effect of microgravity on pool boiling of Freon 12 and Freon 113 were studied . A correlation was also formulated for the calculation of heat flux density of refrigerants R-113 and R-114 in pool boiling at atmospheric pressure condition . The effects of lubricant mass fraction, viscosity, and miscibility on the pool boiling heat transfer performance of 134a/lubricant mixture with the regression method for 12 different mixtures were reported . A numerical simulation model for heat transfer during boiling of FC-72 based on a numerical macro-layer model was presented . Nucleate pool boiling experiments were performed for pure R-11 with a constant wall temperature condition . Nucleate pool boiling of R-11 on cylindrical copper surfaces at reduced pressure was investigated . The Rohsenow correlation was applied to nucleate boiling of halocarbon refrigerants over cylindrical surfaces and a correlation was developed . The relationship between the flow behavior induced by ultrasonic vibration along with the consequent heat transfer enhancement in natural convection and pool boiling regimes for FC-72 was presented by . The influence of thermo-physical properties on pool boiling heat transfer performance of refrigerants within the evaporator of a refrigeration system was experimentally investigated . The influence of uniform DC electric field on nucleate boiling heat transfer of n-pentane, R-113, and R-123 on a horizontal copper surface was experimentally studied . The pool boiling data for mixtures of R-22/ R-124 on plain tubes at reduced pressures was reported . The pool boiling heat transfer of FC-72 at different pressures on a plain plate heater (15×15 mm2) was studied . The pool boiling heat transfer performance of ammonia within the evaporator of a refrigeration unit with the use of existing correlations was assessed experimentally . A correlation for heat transfer coefficient in the nucleate region based on the Buckingham π theorem for Geva-T and low finned tube was estimated for five liquids (R-113, n-pentane, ethanol, water, and R-11) boiling at atmospheric pressure . The effect of surface roughness on nucleate pool boiling of refrigerant R-113 on horizontal circular copper heating surfaces was experimentally investigated . The effects of surface material (copper, brass, and aluminium) on nucleate boiling heat transfer of R-113 were reported . The experimental investigation of nucleate pool boiling of R-134a and R-123 on enhanced smooth tubes of shell type heat exchangers was presented . The boiling of distilled water, ethanol, R-113, and R-123 on heating surfaces covered with copper fibrous capillary porous structures used in heat pipes with porosity (40%, 70%, and 85%) was investigated experimentally and theoretically . The effect of surface roughness on nucleate boiling heat transfer was studied . Boiling performance of aqueous ammonium chloride as an additive using a nichrome wire heater was experimentally studied . The nucleate boiling heat transfer of gas saturated FC-72 on micro pin finned surface under microgravity was investigated . The guidelines for the design of boiling test for FC-72 dielectric fluid on thin horizontal substrates having large number of artificial nucleation sites were presented . The pool boiling curve of R-14 under 0.1 MPa pressure was experimentally studied . Efficient boiling of the refrigerants is very necessary for the effective refrigeration system. Various researches are being carried out worldwide to explore different methods to enhance the boiling performance of refrigerants. Nucleate boiling under reduced gravity, varying pressures, and different boiling surface roughness has been used to Kumar et al. / Journal of Mechanical Engineering and Sciences 9(2015) 1538-1555 1541 enhance boiling performance. Modified surfaces and increased nucleation sites on the boiling surface are reported to be effective technique to enhance the boiling characteristics of refrigerants. Pure Liquids Pure Liquids are not contaminated with other substances. Some of the pure liquids are pure water, ethanol, benzene, and etcetera. The boiling heat transfer data employed to water and electric heating methods used by various researchers were compared and discussed . A numerical simulation model for heat transfer during boiling of water based on a numerical macro-layer model was presented . The analysis of a sequence of temperature fields obtained from a nucleate pool boiling experiment was investigated . Nucleate pool boiling of distilled water, benzene, and toluene from a horizontally laid plain stainless steel heating tube at atmospheric and sub-atmospheric pressures was experimentally studied . The lateral merger of bubbles during nucleate pool boiling of water was numerically studied . Saturated pool boiling curve for water on a temperature controlled thin copper strip using Couple Map Lattice method in non-linear spatio-temporal chaos dynamics was reproduced . Experiments were conducted to investigate the efficiency of two distinctly different heat transfer enhancement methods using a thin vessel coating and an enhanced insulation structure for external reactor vessel cooling under severe accidental conditions . Nucleate pool boiling characteristics during pool boiling of sub-cooled water on very small wires were studied . The nucleate boiling of saturated water at high heat fluxes was numerically studied . Nucleate pool boiling heat transfer coefficient for several pure liquids on a horizontal rod heater at atmospheric pressure was experimentally measured . The boiling performance characteristics at atmospheric and sub-atmospheric pressures were experimentally investigated . An experimental estimate of the heat flux for pool boiling of water and methanol at atmospheric pressure in a beaker with varying voltage using Rohsenow correlation with regression analysis was presented . A pool boiling experiment with demineralized water on rough surfaces of the tubes gave almost double heat transfer coefficient . The investigation of nucleate boiling phenomena for distilled water at saturated as well as sub-cooled conditions was presented . The potential of the acoustic emission in detection of bubbles to point out the transition zones during boiling process was examined . Heat transfer characteristics of water through pool boiling over flat stainless steel plate heater using Stephen and Abdelsalam correlation (1980) by optimizing values of power index and coefficient was presented . The effect of design parameters on pool boiling heat transfer for water on sintered tube surfaces was experimentally studied . The pool boiling heat transfer performance of de-ionized water on horizontal plates sintered with copper fiber of various geometries under atmospheric pressure was experimentally investigated . Heat transfer characteristics of water through pool boiling over horizontal stainless steel tube heater upto CHF were studied . The pool boiling heat transfer enhancement by adding environment-friendly surfactants to pure water was experimentally described . A study on heat transfer during pool boiling of water at atmospheric pressure over enhanced cylindrical micro-channel test surfaces was carried out . A 2-D numerical simulation on nucleate boiling with help of VOSET method was presented . Pool boiling experiments with synthetic diamond and silicon carbide heaters using water as the boiling liquid under atmospheric pressure was presented . A model predicting the changes in bubble diameter during pool boiling of distilled water using neural networks in modeling with complicated nonlinear relations was presented Research progresses and future directions on pool boiling heat transfer 1542 . Critical heat flux triggering mechanism and dynamic behavior of dry areas in a horizontal pool boiling of saturated water on a transparent indium tin oxide heating surface was observed . The boiling heat transfer behavior of distilled water on horizontal heating surface under atmospheric and sub atmospheric pressure was studied . The effect of microlayer evaporation on heat transfer characteristics for water and ethanol by measuring microlayer thickness formed under a growing bubble was presented . The pool boiling heat transfer characteristics of water on a stainless steel heater was experimentally analyzed . The surface wettability and bubble dynamics during pool boiling of one-component fluids was investigated . A heat transfer enhancement method during cooling of microelectronic elements by the application of ultrasonic fields using wires of different diameters in a pool of subcooled water was presented . A relation for nucleate boiling heat transfer of water through the solid-liquid interface using experimental data was derived and was compared to the existing correlations . A model was proposed to describe the accurate behavior of bubble departure during saturated pool boiling of pure water and ethanol under atmospheric pressure conditions . The pool boiling heat transfer characteristics of water using different treated heating surfaces was studied . Many industrial and commercial processes involve boiling of pure liquids like water, ethanol, Benzene, distilled water, and etcetera. Steam generations, cooling of electronic equipments and others. are the processes where the evaporation of pure liquids is done to absorb high amount of heat generated. Many researchers have discussed various modifications in heating surfaces and use of different pure liquids to enhance pool boiling performance by improving boiling characteristics, i.e. heat flux, critical heat flux, heat transfer coefficient, bubble growth and their departure. Performance of various heaters has been experimentally analyzed to achieve better heat transfer. Nanofluids The nanofluids are engineered colloidal suspension of nanoparticles in a base fluid. The nanoparticles are made of metals, oxides, carbides, and etcetera [81-83]. The boiling heat transfer characteristics of different alumina nano-particle concentrations with water on a horizontal flat smooth surface were studied . Pool boiling heat transfer using nanofluids (γ-alumina nanoparticles, 10-50 nm) was experimentally investigated . . Pool boiling CHF enhancement in nanofluids by forming a porous layer of nanoparticles on the heater surface was presented . Pool boiling heat transfer of ZrO2 based aqueous nanofluids at low volume fraction of solid dispersion was observed . Decreased heat transfer during pool boiling of diluted suspensions of sphere-shaped titania and alumina particles suspended in ethylene glycol-water mixtures was reported . The mechanism of surface coating during nucleate boiling of nanofluids was experimentally explored . A pool boiling heat transfer model for nanofluids based on fractal distribution of nanoparticles and nucleation sites on boiling surfaces was presented . The heat transfer characteristics of CuO nanofluids for low concentrations and at/above atmospheric pressures were experimentally studied and enhanced critical heat flux was observed [91-94]. The pool boiling heat transfer under heating surface with various interlaced wettability using nano-silica particles as the coating element was investigated . A theoretical correlation for pool boiling of TiO2-water nanofluid solution on a stainless steel flat heating surface was developed [96-99]. A correlation for predicting heat transfer coefficient for nucleate pool boiling of TiO2-water nanofluids at Kumar et al. / Journal of Mechanical Engineering and Sciences 9(2015) 1538-1555 1543 several low concentrations using two horizontal circular plate heaters having different surface roughness was presented . The pool boiling heat transfer for saturated water over nanoparticle modified aluminium surfaces having different surface wettability were investigated . During pool boiling experiments on ZnO nanoparticles concentrations with water at atmospheric pressure on an electrically heated Ni-Cr wire, 70 to 80% enhancement in critical heat flux for pure water was reported . Enhancement in pool boiling heat transfer was observed by creating one-dimensionally grown alumina nano porous surface . An empirical correlation was developed to predict the heat flux for nucleate pool boiling of nanofluids . The effect of nanorod length on pool boiling heat transfer for water was experimentally studied . The influence of nanoparticles on the pool boiling heat transfer in open-celled foams at atmospheric pressure was investigated . Heat transfer characteristics during pool boiling of nanofluids on cylindrical surface were investigated and it was observed that the heat transfer coefficient depends upon nanoparticle concentration and boiling pressure . Various nanofluids are being used as additives in pure liquids or as surface coating on the heating surfaces to enhance the heat transfer characteristics. From the literature, it has been observed that low concentrations of nanoparticles in pure liquids cause enhancement in boiling performance, whereas reduction in heat transfer is observed at high concentrations of nanoparticles. Thus, determining the optimal concentration of nanoparticles becomes a challenging task, which significantly affects the pool boiling heat transfer. Hydrocarbons and Additive Mixtures Hydrocarbon, an organic compound consisting of hydrogen and carbon, is the primary energy source for current civilization and is mainly classified into saturated hydrocarbons (Alkanes) and unsaturated hydrocarbons (Alkenes and alkynes). Additive mixtures of two or more liquids are being used in various heat transfer applications to obtain desired properties. An estimation method to predict the heat transfer of nucleate pool boiling in binary mixtures using Colburn analogy was described . An experimental study to determine the effects of binary diffusion and surface tension on the pool boiling heat transfer of dilute aqueous solution of ethylene glycol was presented . The boiling curves obtained for various concentrations of water with cationic surfactant were compared . The heat transfer coefficients of different mixtures were reported lower than those obtained for pure components constituting the mixture for a given heat flux . The influence of thermo-physical properties on pool boiling heat transfer of hydrocarbons (propane and i-butane) within the evaporator of a refrigeration system (Figure 1) was experimentally investigated and compared to data available in the literature . The heat transfer and boiling temperature of different concentration levels of sugar-water solution was studied . Nucleate boiling heat transfer coefficients of mixtures of water-monoethanolamine and water-diethanolamine on a horizontal heating rod at atmospheric pressure were experimentally measured . . Schematic diagram of Gorenflo pool boiling apparatus is shown in Figure 2. Heat transfer coefficient for the nucleate pool boiling of methanol, distilled water, and their mixtures on a plain as well as copper-coated stainless steel tubes at atmospheric and sub-atmospheric pressures were measured . Experiments were conducted to enhance pool boiling heat transfer by adding ammonium chloride as a surfactant in pure water . The heat transfer in saturated nucleate pool boiling of the water/lithium bromide mixture on a uniformly heated vertical cylinder at a pressure of 2 bar was reported . Nucleate pool boiling heat transfer Research progresses and future directions on pool boiling heat transfer 1544 coefficient of ternary mixtures of ethanol, monoethylene glycol, and diethyleneglycol as a new coolant with higher heat transfer coefficient was investigated . An experimental study on n-pentane nucleate boiling at atmospheric pressure and saturation temperature for different gap sizes (Figure 3) in a confined space was conducted . Figure 1. Pool boiling setup . Figure 2. Schematic diagram of Gorenflo pool boiling apparatus . Pool boiling heat transfer in water/glycerol binary solution on a horizontal rod heater for various concentrations at atmospheric pressure was studied . Various correlations for predicting the pool boiling heat transfer coefficient of FK-649 at various saturation conditions were compared to replace engineered fluids . Pool boiling investigation of PF-5060 under reduced gravity and a pressure of 600 mbar was presented . The bubble departure diameters during saturated pool boiling of various binary mixtures under atmospheric pressure conditions were reported . The boiling heat transfer coefficient for nicrome wire immersed in saturated water with and without various concentrations of 2-Ethyl 1-Hexanol as an additive was evaluated . The prediction of pool boiling heat transfer coefficient for multi-component system using Kumar et al. / Journal of Mechanical Engineering and Sciences 9(2015) 1538-1555 1545 artificial neural network method was reported . From the literature, it has been concluded that various binary and ternary mixtures are used to enhance the heat transfer performance. Higher heat transfer coefficients are obtained with the suitable mixtures of hydrocarbons and other commercial liquids as compared to pure liquids. Figure 3. Pool boiling under closed conditions [105, 106]. Natural and Synthetic Colloidal Solutions A colloidal solution is a mixture of a colloid microscopically dispersed throughout another substance. Particles in colloidal solution are smaller and do not settle, distinguishing it from suspension. The major natural colloidal solutions widely used in food processing are milk and sugar cane juice. Milk is an emulsified colloid of liquid butterfat globules dispersed with a water-based solution. Sugar cane juice is an extremely complex liquid medium containing many organic and inorganic constituents in soluble and colloidal form. The pool boiling of sugar cane juice in an aluminium pot heated by an electric hot plate using regression analysis by applying Rohsenow correlation was studied . Pool boiling of milk under open and closed conditions (Figure 4) in aluminum and stainless steel pots has been studied using Rohsenow correlation with the help of linear regression analysis for different heat inputs [125-128]. The convective heat transfer coefficient and heat flux were reported to increase with the increased heat input. The average values of fluid-surface constant Csf for Rohsenow pool boiling correlation during khoa making in an aluminum and stainless steel pot were evaluated as 3 10 8815 .
7 and 3 10 4772 .
9 respectively . From the above literature, it has been observed that pool boiling behavior of natural colloidal solutions, namely milk and sugarcane juice, was studied by applying Rohsenow correlation. Heat transfer coefficient during pool boiling was reported to increase with increased heat inputs. Heat transfer performance was observed to be dependent on proper selection of pot material and heating conditions. In synthetic colloidal solutions, the heat transfer and CHF were found to be dependent on the solution’s concentration. POOL BOILING APPLICATIONS AND ITS CORRELATIONS Pool boiling is an adequate technique for many heat transfer applications because of high heat transfer and high heat flux at moderate temperatures. Some of the applications of Research progresses and future directions on pool boiling heat transfer 1546 pool boiling are: purification of water by boiling, steam generation by rapid evaporation for various industrial purposes , refrigeration system and evaporation of various liquids like refrigerants for industrial purposes, petroleum oil refineries, cooling of electronic equipments [131, 132], processing of milk for khoa making [125-129], processing of sugarcane juice for jaggery making , fluid handling and control system, cooling of nuclear reactor system , heat transfer and optimal system design , as well as impulse drying of paper web in paper industry , and others. Major existing correlations for prediction of pool boiling heat transfer coefficient are listed in Table 2. Table 2. Pool boiling correlations. Researchers Correlations Applications 3 2 / 1 " ] Pr /( [ ] / ) ( [ n l fg sf pl v l fg l h C T C g h q PL, R, HC, & CS 73 .
1 35 .
0 371 .
0 297 .
0 674 .
0 ) / ( 23 .
0 E D C B A d k h b l W, O, R, & C p c F p q h 69 .
0 7 .
0 ") ( 00417 .
0 PL, R, & HC 67 .
0 " 33 .
0 2 67 .
0 ) ( 10 1 075 .
0 q T k h s l v l v PL, R, & HC 3 / 1 4 2 " ] )} /( ){ / ( 9 37 .
3 [ M q C h l k E h pl fg l NF, W, & HC l h P C T k gT q h l k h v fg pl s v l v l s fg l 7 .
0 " 082 .
0 HC, W, & BM 67 .
0 5 .
0 55 .
0 ln 4343 .
0 12 .
0 ) " ( Pr) ln 4343 .
0 ( Pr 55 q M h p R R, PL, & LM 133 .
0 ) / ( ) / " ( po p f o p o R R q q F h h R & HC 4 .
0 67 .
0 5 .
0 Pr Re 7 .
9 p c F p Nu NF, PL, & R 5 .
0 25 .
0 2 )] / ( 1 [ ] / ) ( [ ) 24 / ( " l v v v l fg g h q W & NF PL = Pure liquids, R = Refrigerants, HC = Hydrocarbon, CS = Colloidal Solutions, W = Water, O = Organic fluids, C = Cryogenic fluids, NF = Nanofluids, BM = Binary mixtures, LM = Liquid mixture CONCLUSIONS Pool boiling is an effective and efficient method of heat transfer to liquids. Efficient boiling of the refrigerants is very necessary for an effective refrigeration system. Nucleate boiling under reduced gravity, varying pressures and different boiling surface roughness have been used to enhance boiling performance. Modified surfaces and increased nucleation sites on the boiling surface are found to be effective techniques to enhance the Kumar et al. / Journal of Mechanical Engineering and Sciences 9(2015) 1538-1555 1547 boiling characteristics of refrigerants. Various modifications in heating surfaces and use of pure liquids are found to enhance pool boiling performance by improving boiling characteristics, i.e. heat flux, critical heat flux, heat transfer coefficient, bubble growth and departure, and etcetera. Nanofluids are being used as additives in pure liquids or as surface coating on the heating surfaces to enhance the heat transfer characteristics. Boiling performance enhancement in pure liquids has been observed at low concentrations of nanoparticles. Various binary and ternary mixtures are also used to enhance the heat transfer performance. Higher heat transfer coefficients are obtained with the suitable mixtures of hydrocarbons and other commercial liquids. A correct and significant correlation for heat flux and heat transfer coefficient calculation is still a challenge to be explored. A concerted effort is required to search the optimum mathematical modeling and numerical analysis techniques to predict the inter-relative behavior of the sub-processes involved in boiling. Nanoparticles coatings of different layer thickness applied to the heater surface can be analyzed optimally to enhance the pool boiling heat transfer. Use of acoustic fields, electromagnetic fields, and ultrasonic vibrations can be optimized for the heat transfer enhancement for various liquids. Analysis and design of irregular geometrical corrugation on the heating surfaces can be performed to achieve heat transfer enhancement. ACKNOWLEDGEMENTS The authors would like to thanks to Guru Jambheshwar University of Science & Technology, India for assistance and support. REFERENCES Kumar M, Prakash O, Samsher. Review paper on methods of enhancement of pool boiling heat transfer. 14th ISME International Conference on Mechanical Engineering in Knowledge Age. Delhi. 2005; p. 905-8. Westwater J. Boiling heat transfer. American Scientist. 1959;47:427-46. Han CY, Griffith P. The mechanism of heat transfer in nucleate pool boiling. Cambridge, Mass.: MIT Division of Sponsored Research,; 1962. Rohsenow W. A method of correlating heat-transfer data for surface boiling of liquids. Technical report (Massachusetts Institute of Technology),. 1952;74:969-76. Bejan A, Kraus AD. Heat transfer handbook: John Wiley & Sons; 2003. Fazel SA, Roumana S. Pool boiling heat transfer to pure liquids. WSEAS Conference, USA2010. Dhir V. Boiling heat transfer. Annual Review of Fluid Mechanics. 1998;30:365-401. Gu J, Lu M. The new assumption for calculating heat transfer in pool boiling. Heat and mass transfer. 1999;35:295-7. Boiling heat transfer on external surfaces: Wolverine Tube Inc. Engineering thermal Innovation; 2006. PK N. Heat and mass transfer: Tata Mcgraw Hill Education Private Limited,; 2011. Brumfield LA. Pool boiling enhancement via micro ratchets [MS thesis]: Louisiana State University; 2011. Research progresses and future directions on pool boiling heat transfer 1548 Kandlikar SG. Critical heat flux in subcooled flow boiling-an assessment of current understanding and future directions for research. Multiphase Science and Technology. 2001;13:207-32. Katto Y. Critical heat flux. International Journal of Multiphase Flow. 1994;20:53-90. Bhaumik S, Agarwal V, Gupta S. A generalized correlation of nucleate pool boiling of liquids. Indian Journal of Chemical Technology. 2004;11:719-25. Sriraman SR. Pool boiling on nano-finned surfaces [Thesis Report ]: Texas A&M University; 2007. Nawi MRM, Mamat AMI, Ismail H. Numerical heat transfer analysis of waste heat exchanger for exhaust gas energy recovery. Journal of Mechanical Engineering and Sciences. 2015;8:1498-506. Jungho K. Review of reduced gravity boiling heat transfer: US research. Japan Society of Microgravity Application,. 2003;20:264-71. Eckhard B, Groll A. A critical review of the influence of lubricants on the heat transfer and pressure drop of refrigerants-Part II: Lubricant influence on condensation and pressure drop. HVAC&R Research. 2005;11:511-26. Kotthoff S, Gorenflo D. Pool boiling heat transfer to hydrocarbons and ammonia: A state-of-the-art review. International Journal of Refrigeration. 2008;31:573-602. Saidur R, Kazi S, Hossain M, Rahman M, Mohammed H. A review on the performance of nanoparticles suspended with refrigerants and lubricating oils in refrigeration systems. Renewable and Sustainable Energy Reviews. 2011;15:310-23. Barber J, Brutin D, Tadrist L. A review on boiling heat transfer enhancement with nanofluids. Nanoscale Research Letters. 2011;6:280. Sergis A, Hardalupas Y. Anomalous heat transfer modes of nanofluids: a review based on statistical analysis. Nanoscale Research Letters. 2011;6:1-37. Chung J, Chen T, Maroo S. A review of recent progress on nano/micro scale nucleate boiling fundamentals. Frontiers in Heat and Mass Transfer. 2011;2:1-19. Ahn HS, Kim MH. A review on critical heat flux enhancement with nanofluids and surface modification. Journal of Heat transfer. 2012;134:1-13. Das S, Bhaumik S. Nucleate pool boiling heat transfer in multicomponent mixtures on surfaces: an review. International Journal of Mechanical and Industrial Engineering. 2012;2:98-106. Kunugi T. Brief review of latest direct numerical simulation on pool and film boiling. Nuclear Engineering and Technology. 2012;44:847-54. Cheng L. Fundamental issues of critical heat flux phenomena during flow boiling in microscale-channels and nucleate pool boiling in confined spaces. Heat Transfer Engineering. 2013;34:1016-43. Stephan K, Abdelsalam M. Heat-transfer correlations for natural convection boiling. International Journal of Heat and Mass Transfer. 1980;23:73-87. Yusof TM, Arshad AM, Suziyana MD, Chui LG, Basrawi MF. Experimental study of a domestic refrigerator with POE-Al2O3 nanolubricant. International Journal of Automotive and Mechanical Engineering. 2015;11:2243-52. Zell M, Straub J, Vogel B. Pool boiling under microgravity. Physico Chemical Hydrodynamics. 1989;11:812-23. Kedzierski M. Refrigerant/lubricant mixture boiling heat transfer research at NIST. Sixteenth National Convention of Mechanical Engineers and All India Kumar et al. / Journal of Mechanical Engineering and Sciences 9(2015) 1538-1555 1549 Seminar on Future Trends in Mechanical Engineering, Research and Development, Dept of Mech & Ind Eng, UOR, Roorkee2000. He Y, Shoji M, Maruyama S. Numerical study of high heat flux pool boiling heat transfer. International Journal of Heat and Mass Transfer. 2001;44:2357-73. Lee HC, Oh BD, Bae SW, Kim MH, Lee JY, Song IS. Partial nucleate boiling on the microscale heater maintaining constant wall temperature. Journal of Nuclear Science and Technology. 2003;40:768-74. Silva EF, Stelute EB, Jabardo JMS. The effect of the surface condition on nucleate pool boiling of refrigerant R-11 on cylindrical copper surfaces. Silva. 2002;77:0224.0. Jabardo J, Silva E, Ribatski G, de Barros SF. Evaluation of the Rohsenow correlation through experimental pool boiling of halocarbon refrigerants on cylindrical surfaces. Journal of the Brazilian Society of Mechanical Sciences and Engineering. 2004;26:218-30. Gorenflo D, Chandra U, Kotthoff S, Luke A. Influence of thermophysical properties on pool boiling heat transfer of refrigerants. International Journal of Refrigeration. 2004;27:492-502. Zaghdoudi M, Lallemand M. Pool boiling heat transfer enhancement by means of high DC electric field. Arabian Journal for Science and Engineering. 2005;30:189-212. Chiou CB, Lu DC, Wang CC. Investigations of pool boiling heat transfer of binary refrigerant mixtures. Heat Transfer Engineering. 2007;18:61-72. Zhao JF, Yan N, Li J, Li ZD, Hu WR, H O. Pool boiling heat transfer in microgravity. Journal of the Japan Society of Microgravity Application. 2008;25:257-60. Sathyabhama A, Hegde RN. Prediction of nucleate pool boiling heat transfer coefficient. Thermal Science. 2010;14:353-64. Tarrad AH. A correlation for the prediction of nucleate pool boiling performance of pure liquids from enhanced tubes. Jordan Journal of Mechanical and Industrial Engineering. 2011;5:139-44. Hosseini R, Gholaminejad A, Jahandar H. Roughness effects on nucleate pool boiling of R-113 on horizontal circular copper surfaces. World Academy of Science, Engineering and Technology. 2011;55:580-5. Hosseini R, Gholaminejad A, Nabil M, Samadinia MH. Concerning the effect of surface material on nucleate boiling heat transfer of R-113. Journal of Electronics Cooling and Thermal Control. 2011;1:22-7. Gorgy E, Eckels S. Local heat transfer coefficient for pool boiling of R-134a and R-123 on smooth and enhanced tubes. International Journal of Heat and Mass Transfer. 2012;55:3021-8. Wojcik T, Poniewski M. Heating surface thermal stabilization for pool boiling on porous coverings. Journal of Physics: Conference Series: IOP Publishing; 2012. p. 012138. McHale JP, Garimella SV. Nucleate boiling from smooth and rough surfaces–Part 2: analysis of surface roughness effects on nucleate boiling. Experimental Thermal and Fluid Science. 2013;44:439-55. Gajghate SS, Aacharya AR, Pise AT, Jadhav GS. Experimental study of heat transfer enhancement in pool boiling by using 2-Ethyl 1-Hexanol an additive. Applied Mechanics and Materials: Trans Tech Publ; 2014. p. 1601-6. Research progresses and future directions on pool boiling heat transfer 1550 Zhang Y, Wei J, Xue Y, Kong X, Zhao J. Bubble dynamics in nucleate pool boiling on micro-pin-finned surfaces in microgravity. Applied Thermal Engineering. 2014;70:172-82. Nayak SK, Pattanaik BP. Experimental Investigation on Performance and Emission Characteristics of a Diesel Engine Fuelled with Mahua Biodiesel Using Additive. Energy Procedia. 2014;54:569-79. Zhao C, Gong M, Ding L, Zou X, Chen G, Wu J. An experimental investigation on the entire pool boiling curve of R14 under 0.1 MPa pressure. International Journal of Refrigeration. 2014;41:164-70. Darabi J, Ohadi M, Fanni M, Dessiatoun S, Kedzierski M. Effect of heating boundary conditions on pool boiling experiments. HVAC&R Research. 1999;5:283-96. McSharry PE, Ellepola JH, von Hardenberg J, Smith LA, Kenning DB, Judd K. Spatio-temporal analysis of nucleate pool boiling: identification of nucleation sites using non-orthogonal empirical functions. International Journal of Heat and Mass Transfer. 2002;45:237-53. Mukherjee A, Dhir V. Study of lateral merger of vapor bubbles during nucleate pool boiling. Journal of Heat Transfer, ASME. 2004;126:1023-39. Ghoshdastidar P, Kabelac S, Mohanty A. Numerical modelling of atmospheric pool boiling by the coupled map lattice method. Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science. 2004;218:195-205. Yang J, Cheung F, Rempe J, Suh K, Kim S. Correlations of nucleate boiling heat transfer and critical heat flux for external reactor vessel cooling. ASME 2005 Summer Heat Transfer Conference collocated with the ASME 2005 Pacific Rim Technical Conference and Exhibition on Integration and Packaging of MEMS, NEMS, and Electronic Systems: American Society of Mechanical Engineers; 2005. p. 99-108. Lu F, Peng X, Bourouga B. Nucleate boiling modes of subcooled water on fine wires submerged in a pool. Experimental Heat Transfer. 2006;19:95-111. Son G, Dhir VK. Numerical simulation of nucleate boiling on a horizontal surface at high heat fluxes. International Journal of Heat and Mass transfer. 2008;51:2566-82. Sloan A, Penley S, Wirtz R. Sub-atmospheric pressure pool boiling of water on a screen-laminate enhanced surface. IEEE Semiconductor Thermal Measurement and Management Symposium, 2009; p. 246-53. Islam MS, Haque KT, Saha SC. An experimental investigation of pool boiling at atmospheric pressure. Daffodil International University Journal of Science and Technology. 2011;6:80-6. Sarbu I, Valea ES. Correlations for enhanced boiling heat transfer on modified surfaces tubes. International Journal of Energy and Environment. 2011;5:444-51. Duan X, Buongiorno J, McKrell T. Integrated particle imaging velocimetry and infrared thermometry for high resolution measurement of sub-cooled nucleate pool boiling. The 14th International Topical Meeting on Nuclear Reactor Thermal hydraulics NURETH-14 Toronto Ontario Canada2011. Carmi R, Bussiba A, Widenfeld G, Aharon Y, Alon I, Hochbaum I. Detection of transient zones during water boiling by acoustic emission. Acoustic Emission. 2011;29:89-97. Kumar et al. / Journal of Mechanical Engineering and Sciences 9(2015) 1538-1555 1551 Manikandaprabu B, Mohankumar S, Kathiravan R. Pool boiling of water over a stainless steel flat plate heater. International Journal of Instrumentation, Control and Automation,.1:2231-1890. Wen M-Y, Ho C-Y, Jang K-J. An optimal parametric design to improve pool boiling heat transfer of sintered surfaces. Journal of Engineering and Technology Research. 2012;4:49-56. Qu Z, Li D, Huang J, Xu Z, Liu X, Tao W. Experimental investigations of pool boiling heat transfer on horizontal plate sintered with metallic fiber felt. International Journal of Green Energy. 2012;9:22-38. Muruganantham R, Vignesh G, Vignesh R, Madhan P, Kathiravan R. Pool boiling characteristics of water over a horizontal stainless steel tube heater. International Journal of Mechanical and Production Engineering. 2013;1:23-9. Dikici B, Eno E. Environmentally friendly surfactant additives for nucleate pool boiling enhancement. 41st Annual NATAS Conference: Bowling Green, KY 2014. Mehta JS, Kandlikar SG. Pool boiling heat transfer enhancement over cylindrical tubes with water at atmospheric pressure, Part I: Experimental results for circumferential rectangular open microchannels. International Journal of Heat and Mass Transfer. 2013;64:1205-15. Ling K, Li Z-Y, Tao W-Q. A direct numerical simulation for nucleate boiling by the VOSET method. Numerical Heat Transfer, Part A: Applications: An International Journal of Computation and Methodolog. 2014;65:949-71. Kapitz M, Becker A, Wiesche Sad. Influence of thermophysical wall properties during pool boiling on large diamond and SiC heaters. Heat Transfer Engineering. 2014;35:472-81. Rashidabad VN, Manteghian M, Masoumi M, Mousavian S, Ashouri D. Application of neural networks to predict changing the diameters of bubbles in pool boiling distilled water. International Journal of Chemical, Materials Science and Engineering. 2014;8:16-8. Chu IC, No HC, Song CH, Euh DJ. Observation of critical heat flux mechanism in horizontal pool boiling of saturated water. Nuclear Engineering and Design. 2014;279:189-99. Kumar V, Goel R. Heat transfer from horizontal surface to the pool of distilled water at atmospheric and sub atmospheric pressure. International Journal of Engineering and Management Technology. 2014;2:55-66. Utaka Y, Kashiwabara Y, Ozaki M, Chen Z. Heat transfer characteristics based on microlayer structure in nucleate pool boiling for water and ethanol. International Journal of Heat and Mass Transfer. 2014;68:479-88. Hariharan M, Kathiravan R, Kumar SS, Arunprasath M. Heat transfer analysis of pool boiling in stainless steel flat plate heater with water. International Journal of Scientific and Engineering Research. 2014;5:695-701. Xu X, Qian T. Single-bubble dynamics in pool boiling of one-component fluids. Physical Review E. 2014;89:063002. Hetsroni G, Moldavsky L, Fichman M, Pogrebnyak E, Mosyak A. Ultrasonic enhancement of subcooled pool boiling of freely oscillated wires. International Journal of Multiphase Flow. 2014;67:13-21. Chouarfa F, Benhamza MH, Bendjaballah M. New model of heat transfer in the process of nucleate boiling in a pool: prediction and assessment. Journal of Engineering Physics and Thermophysics. 2014;87:743-52. Research progresses and future directions on pool boiling heat transfer 1552 Hamzekhani S, Falahieh MM, Akbari A. Bubble departure diameter in nucleate pool boiling at saturation: Pure liquids and binary mixtures. International Journal of Refrigeration. 2014;46:50-8. Das S, Kumar SD, Bhowmik S. Enhancement of nucleate pool boiling heat transfer on silicon oxide thin film surface. Procedia Engineering. 2014;90:530-7. Zakaria I, Michael Z, Mohamed WANW, Mamat AMI, Azmi WH, Mamat R, et al. A review of nanofluid adoption in polymer electrolyte membrane (PEM) fuel cells as an alternative coolant. Journal of Mechanical Engineering and Sciences. 2015;8:1351-66. Hussein AM, Sharma KV, Bakar RA, Kadirgama K. Heat transfer enhancement with nanofluids – A Review. Journal of Mechanical Engineering and Sciences. 2013;4:452-61. Azmi WH, Sharma KV, Mamat R, Anuar S. Nanofluid properties for forced convection heat transfer: An overview. Journal of Mechanical Engineering and Sciences. 2013;4:397-408. Bang IC, Chang SH. Boiling heat transfer performance and phenomena of Al 2 O 3–water nano-fluids from a plain surface in a pool. International Journal of Heat and Mass Transfer. 2005;48:2407-19. Wen D, Ding Y. Experimental investigation into the pool boiling heat transfer of aqueous based γ-alumina nanofluids. Journal of Nanoparticle Research. 2005;7:265-74. Kim S, Bang IC, Buongiomo J, Hu L. Study of pool boiling and critical heat flux enhancement in nanofluids. Bulletin of the Polish Academy of Sciences-Technical Sciences. 2007;55:211-6. Chopkar M, Das A, Manna I, Das P. Pool boiling heat transfer characteristics of ZrO 2-water nanofluids from a flat surface in a pool. Heat and Mass Transfer. 2008;44:999-1004. Witharana S, Uanreroro A, Chen H, Ding Y. Pool boiling of titania-water-ethylene glycol nanofluids. In: UK-China Particle Technology Forum. 2009;2:1-3. Phan HT, Caney N, Marty P, Colasson S, Gavillet J. Surface coating with nanofluids: the effects on pool boiling heat transfer. Nanoscale and Microscale Thermophysical Engineering. 2010;14:229-44. Xiao B, Jiang G, Chen L. A fractal study for nucleate pool boiling heat transfer of nanofluids. Science China Physics, Mechanics and Astronomy. 2010;53:30-7. Hegde R, Rao S, Reddy R. Investigations on boiling-induced nanoparticle coating, transient characteristics and effect of pressure in pool boiling heat transfer on a cylindrical surface. Experimental Heat Transfer. 2012;25:323-40. Syam Sundar L, Sharma KV. Laminar Convective heat transfer and friction factor of al2o3 nanofluid in circular tube fitted with twisted tape inserts. International Journal of Automotive and Mechanical Engineering. 2011;3:265-78. Hussein AM, Bakar RA, Kadirgama K, Sharma KV. Experimental measurements of nanofluids thermal properties. International Journal of Automotive and Mechanical Engineering. 2013;7:850-63. Abdolbaqi MK, Azwadi CSN, Mamat R, Azmi WH, Najafi GN. Nanofluids heat transfer enhancement through straight channel under turbulent flow. International Journal of Automotive and Mechanical Engineering. 2015;11:2294-305. Hsu CC, Su TW, Chen PH. Pool boiling of nanoparticle-modified surface with interlaced wettability. Nanoscale Research Letters. 2012;7:1-7. Kumar et al. / Journal of Mechanical Engineering and Sciences 9(2015) 1538-1555 1553 Pal S, Bhaumik S. Development of theoretical correlation for prediction of boiling heat transfer using TiO2-water nanofluid. Energy and Biotechnology. 2012;33:34-8. Syam Sundar L, Sharma KV. An experimental study on heat transfer and friction factor of Al2O3 nanofluid. Journal of Mechanical Engineering and Sciences. 2011;1:99-112. Rao GS, Sharma KV, Chary SP, Bakar RA, Rahman MM, Kadirgama K, et al. Experimental study on heat transfer coefficient and friction factor of al2o3 nanofluid in a packed bed column. Journal of Mechanical Engineering and Sciences. 2011;1:1-15. Mahendran M, Lee GC, Sharma KV, Shahrani A. Performance of evacuated tube solar collector using water-based titanium oxide nanofluid. Journal of Mechanical Engineering and Sciences. 2012;3:301-10. Suriyawong A, Dalkilic A, Wongwises S. Nucleate pool boiling heat transfer correlation for TiO2-water nanofluids. Journal of ASTM International. 2012;9:1-12. Yang CT, Yu CC. Pool boiling of micro-/nanoparticles modified aluminum surface. Advances in Materials Science and Engineering. 2013;2013:1-6. Hiswankar SC, Kshirsagar JM. Determination of critical heat flux in pool boiling using ZnO nanofluids. International Journal of Engineering Research and Technology: ESRSA Publications; 2013. p. 2091-5. Zhang BJ, Kim KJ. Nucleate pool boiling heat transfer augmentation on hydrophobic self-assembly mono-layered alumina nano-porous surfaces. International Journal of Heat and Mass Transfer. 2014;73:551-61. Das S, Bhaumik S. A correlation for the prediction of heat flux for nucleate pool boiling heat transfer of nanofluid. Arabian Journal for Science and Engineering. 2014;39:4997-5006. Demir E, Izci T, Alagoz AS, Karabacak T, Koşar A. Effect of silicon nanorod length on horizontal nanostructured plates in pool boiling heat transfer with water. International Journal of Thermal Sciences. 2014;82:111-21. Xu Z, Zhao C. Influences of nanoparticles on pool boiling heat transfer in porous metals. Applied Thermal Engineering. 2014;65:34-41. Naphon P, Thongjing C. Pool boiling heat transfer characteristics of refrigerant-nanoparticle mixtures. International Communications in Heat and Mass Transfer. 2014;52:84-9. Junjie G, HJ B. Estimation of pool boiling heat transfer coefficients of ethanol-water mixtures. Journal of Chemical Industry and Engineering. 1998;7:95-103. Kandlikar S, Alves L. Effects of surface tension and binary diffusion on pool boiling of dilute solutions: an experimental assessment. Journal of Heat Transfer. 1999;121:488-93. Hetsroni G, Zakin J, Lin Z, Mosyak A, Pancallo E, Rozenblit R. The effect of surfactants on bubble growth, wall thermal patterns and heat transfer in pool boiling. International Journal of Heat and Mass Transfer. 2001;44:485-97. Rao GV, Balakrishnan A. Heat transfer in nucleate pool boiling of multicomponent mixtures. Experimental Thermal and Fluid Science. 2004;29:87-103. Ozdemir M, Pehlivan H. Prediction of the boiling temperature and heat flux in sugar–water solutions under pool-boiling conditions. Heat and Mass Transfer. 2008;44:827-33. Research progresses and future directions on pool boiling heat transfer 1554 Fazel SA, Safekordi A, Jamialahmadi M. Pool boiling heat transfer in water-amines solutions. IJE Transactions A: Basics. 2008;21:113-30. Alam M, Agarwal V. Pool boiling of liquids & their mixtures on enhanced surfaces at sub-atmospheric pressures. International Conference on Chemical & Process Engineering Rome, Italy: Citeseer; 2009. Acharya A, Pise A, Momin I. Ammonium chloride as surfactant for heat transfer enhancement in pool boiling. International Journal of Engineering and Technology. 2011; 3(3): 323-6. Sathyabhama A, Krishnan V. Pool boiling heat transfer to water/lithium bromide mixture. International Conference on Challenges and Opportunities in Mechanical Engineering2012. p. 548-52. Sarafraz M, Peyghambarzadeh S. Nucleate pool boiling heat transfer to Al2O3-water and TiO2-water nanofluids on horizontal smooth tubes with dissimilar homogeneous materials. Chemical and Biochemical Engineering Quarterly. 2012;26:199-206. Cardoso EM, Passos JC. Nucleate boiling of n-pentane in a horizontal confined space. Heat Transfer Engineering. 2013;34:470-8. Fazel SA, Sarafraz M, Shamsabadi AA, Peyghambarzadeh S. Pool boiling heat transfer in diluted water/glycerol binary solutions. Heat Transfer Engineering. 2013;34:828-37. Forrest EC, Hu L-W, Buongiorno J, McKrell TJ. Pool boiling heat transfer performance of a dielectric fluid with low global warming potential. Heat Transfer Engineering. 2013;34:1262-77. Fischer S, Herbert S, Slomski E, Stephan P. Local heat flux investigation during pool boiling single bubble cycles under reduced gravity. Heat Transfer Engineering. 2014;35:482-91. Gajghate S, Acharya A, Pise A. Experimental study of aqueous ammonium chloride in pool boiling heat transfer. Experimental Heat Transfer. 2014;27:113-23. Hakeem M, Kamil M, Asif M. Prediction of pool boiling heat transfer coefficients in pool boiling of liquids using artificial neural network. Journal of Scientific and Industrial Research. 2014;73:536-40. Tiwari G, Prakash O, Kumar S. Evaluation of convective heat and mass transfer for pool boiling of sugarcane juice. Energy Conversion and Management. 2004;45:171-9. Kumar M, Kasana K, Kumar S, Prakash O. Evaluation of convective heat transfer coefficient for pool boiling of milk under closed condition. S-JPSET. 2010;1:91-9. Kumar M, Kasana K, Kumar S, Prakash O. Pool boiling of milk in a stainless steel pot under closed condition. International Journal of Current Research. 2011;3:94-9. Kumar M, Prakash O, Kasana K. An experimental study on the pool boiling heat transfer coefficient of milk. Facta Universitatis, Series Mechanical Engineering 2011;9:61-70. Kumar M, Prakash O, Kasana K. An experimental study on pool boiling of milk. Heat Transfer—Asian Research. 2011;40:159-70. Kumar M, Kasana K, Kumar S, Prakash O. Experimental evaluation of constants for the Rohsenow pool boiling correlation for khoa. SJPSET. 2011;2:21-6. Kumar et al. / Journal of Mechanical Engineering and Sciences 9(2015) 1538-1555 1555 Filipczak G, Troniewski L, Witczak S. Pool boiling of liquid-liquid multiphase systems, evaporation, condensation and heat transfer. Evaporation, Condensation and Heat transfer. Eedited by Amimul Ahsan, Intech Publisher; 2011. Wei J, Yu B, Wang H. Heat transfer mechanisms in vapor mushroom region of saturated nucleate pool boiling. International Journal of Heat and Fluid Flow. 2003;24:210-22. Beitel G. Boiling heat-transfer processes and their application in the cooling of high heat flux devices. Arnold Engineering Development Center; 1993. Yasuo O, Tomoaki K. Numerical study on subcooled pool boiling. Progress in Nuclear Science and Technology. 2011;2:125-9. Golobic I, Pavlovic E, Von Hardenberg J, Berry M, Nelson R, Kenning D, et al. Comparison of a mechanistic model for nucleate boiling with experimental spatio-temporal data. Chemical Engineering Research and Design. 2004;82:1-10. Rudemiller GR. A fundamental study of boiling heat transfer mechanisms related to impulse drying. The Institute of Paper Science and Technology Atlanta Georgia1989. Mostinski I. Application of the rule of corresponding states for calculation of heat transfer and critical heat flux. Teploenergetika. 1963;4:66. Labuntsov D. Heat transfer problems with nucleate boiling of liquids. Thermal Engginering. 1973;19:21-8. Kutateladze SS. Heat transfer and hydrodynamic resistance: Handbook: Energoatomizdat Publishing House; 1990. Boyko-Kruzhilin. International Journal of Heat and Mass Transfer. 1967:10-361. Cooper M. Heat flow rates in saturated nucleate pool boiling: A wide ranging examination using reduced properties. Advances in Heat Transfer. 1984;16:157-239. D. G. Pool boiling. VDI Heat Atlas. 1993. Cornwell K, Houston S. Nucleate pool boiling on horizontal tubes: a convection-based correlation. International Journal of Heat and Mass Transfer. 1994;37:303-9. Zuber N. Hydrodynamic aspects of boiling heat transfer California. USA: Ramo-Wooldridge Corporations, Los Angeles; 1959. |
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Three squares in a triangle
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In a triangle, three identical squares of side lengths 2.8 share a common vertex and are each touching two sides of the triangle. If one of the angles in the triangle is 75 degrees and is opposed to a side of 10.8, then what’s the area of the triangle?
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edited May 31, 2020 at 16:02
Rand al'Thor♦
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1 To the close-voters: you might want to check how tricky this puzzle is, and how many neat "aha" steps are involved in the solution, before thinking it's a straightforward textbook problem rather than an olympiad-style geometry puzzle.Rand al'Thor –Rand al'Thor♦ 2020-05-31 16:03:35 +00:00 Commented May 31, 2020 at 16:03
@Rand I'm not so familiar with PSE mores, but in general on SE sites closure of a question depends on the question and not on its answers.msh210 –msh210 2020-05-31 16:09:45 +00:00 Commented May 31, 2020 at 16:09
@msh210 You're right, but see the PSE policy on maths problems vs puzzles. To a SME it's quite obvious that this question is not a straightforward calculation problem and is going to involve some tricks, even if the exact details of the solution itself are far from obvious.Rand al'Thor –Rand al'Thor♦ 2020-05-31 16:11:59 +00:00 Commented May 31, 2020 at 16:11
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Let's label all the angles:
Note that we have: a+h+i=180,n+p+q=90,
a+h+i=180,n+p+q=90,
a+b+c=90,f+h+j=90,g+i+k=90,
a+b+c=90,f+h+j=90,g+i+k=90,
2 d+p=180,2 e+q=180,2 m+n=180,
2 d+p=180,2 e+q=180,2 m+n=180,
b+d=90,c+e=90,d+f=90,e+g=90,j+m=90,k+m=90,
b+d=90,c+e=90,d+f=90,e+g=90,j+m=90,k+m=90,
therefore
b=f b=f, c=g c=g, j=k j=k.
We're given that one of the angles of the big triangle is 75 degrees, so let's say a=75 a=75, which means
b+c=15 b+c=15, i.e. f+g=15 f+g=15, but also h+i=105 h+i=105, so j+k=2(90)−(f+g)−(h+i)=60 j+k=2(90)−(f+g)−(h+i)=60. Also j=k j=k, so these are 30 30 giving m=60 m=60 and n=60 n=60. (This diagram is not to scale for sure!) Therefore we have an equilateral triangle, and the middle part of the 10.8 10.8 side has length 2.8 2.8. Also p+q=30 p+q=30 and d+e=165 d+e=165.
Now draw diagonals of the squares to make mini-triangles in the h h and i i corners. We have
j=k=30 j=k=30, so the angle at the j j / k k vertex of each mini-triangle is 75 75 degrees. That means the two little triangles are similar to the big one, so f+45=i f+45=i and g+45=h g+45=h. Equivalently b+45=i b+45=i and c+45=h c+45=h, so we have another little similar triangle from drawing the diagonal of the third square to make a mini-triangle in the a a corner.
Now we have the following:
where X+Y=8 X+Y=8 and (from comparing the two similar triangles) X Y=(2.8√2)2=15.68 X Y=(2.8 2–√)2=15.68. So the numbers X X and Y Y are the roots of the quadratic equation t 2−8 t+15.68=0 t 2−8 t+15.68=0, which means they are 8±√64−4(15.68)2=8±√1.28 2=4±√0.32=4±0.4√2.
8±64−4(15.68)−−−−−−−−−−−√2=8±1.28−−−−√2=4±0.32−−−−√=4±0.4 2–√.
Now we can use the SAS formula for area to find the area of one of those small triangles:
1 2 a b sin C=1 2(2.8√2)(4±0.4√2)sin(75)=(5.6√2±1.12)(1+√3 2√2)=(2.8±0.28√2)(1+√3)
1 2 a b sin C=1 2(2.8 2–√)(4±0.4 2–√)sin(75)=(5.6 2–√±1.12)(1+3–√2 2–√)=(2.8±0.28 2–√)(1+3–√)
Now we can use the cosine rule to find the third side of one of those small triangles:
=√(2.8√2)2+(4±0.4√2)2−2(2.8√2)(4±0.4√2)cos(75)=√15.68+(16±3.2√2+0.32)−(22.4√2±4.48)(√6−√2 4)=√32±3.2√2−(11.2±1.12√2)(√3−1)=√(10±√2)(3.2−1.12(√3−1))
That third side is the one which corresponds to the side 10.8 on the big triangle, so the ratio of their areas is
10.8(10±√2)(3.2−1.12(√3−1)),
and the area of the big triangle
=10.8 2(10±√2)(3.2−1.12(√3−1))(2.8±0.28√2)(1+√3)=116.64(0.28)(1+√3)3.2−1.12(√3−1)=32.6592(1+√3)4.32−1.12√3=32.6592(1+√3)(4.32+1.12√3)18.6624−3(1.2544)=32.6592(7.68+5.44√3)14.8992=250.822656+177.666048√3 14.8992=31.352832+22.208256√3 1.8624=16.3296+11.5668√3 0.97.
I do wonder if I've missed a less calculation-heavy method though ... been doing all this by hand except for the last few steps where I reached for a calculator, although the above solution is still exact.
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edited May 31, 2020 at 17:58
answered May 31, 2020 at 16:02
Rand al'Thor♦Rand al'Thor
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I think you should be able to use the sum of the squares of the two "third sides" and the sum of the areas of the two small triangles to get to the result with fewer roots user39583 –user39583 2020-05-31 16:59:38 +00:00 Commented May 31, 2020 at 16:59
@user39583 Well, I've now got an exact figure for the answer at least, but it's not especially pretty.Rand al'Thor –Rand al'Thor♦ 2020-05-31 17:19:42 +00:00 Commented May 31, 2020 at 17:19
@user39583 Oops, good catch. I did some factorisation tricks to simplify the final answer a bit too.Rand al'Thor –Rand al'Thor♦ 2020-05-31 17:58:28 +00:00 Commented May 31, 2020 at 17:58
Looks good now!user39583 –user39583 2020-05-31 18:02:01 +00:00 Commented May 31, 2020 at 18:02
The answer is around 37.49.Display maths –Display maths 2020-05-31 18:34:59 +00:00 Commented May 31, 2020 at 18:34
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10979 | https://www.xconvert.com/unit-converter/meters-of-water-@-4%C2%B0c-to-inches-of-mercury | meters of water @ 4°C to Inches of mercury | Convert mH2O To inHg Online - XConvert
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meters of water @ 4°C (mH 2 O) to Inches of mercury (inHg) conversion
Pressure
From meters of water @ 4°C
mH 2 O
To Inches of mercury
inHg
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meters of water @ 4°C to Inches of mercury conversion table
| meters of water @ 4°C (mH 2 O) | Inches of mercury (inHg) |
--- |
| 0 | 0 |
| 1 | 2.895901839792 |
| 2 | 5.7918036795839 |
| 3 | 8.6877055193759 |
| 4 | 11.583607359168 |
| 5 | 14.47950919896 |
| 6 | 17.375411038752 |
| 7 | 20.271312878544 |
| 8 | 23.167214718336 |
| 9 | 26.063116558128 |
| 10 | 28.95901839792 |
| 20 | 57.918036795839 |
| 30 | 86.877055193759 |
| 40 | 115.83607359168 |
| 50 | 144.7950919896 |
| 60 | 173.75411038752 |
| 70 | 202.71312878544 |
| 80 | 231.67214718336 |
| 90 | 260.63116558128 |
| 100 | 289.5901839792 |
| 1000 | 2895.901839792 |
How to convert meters of water @ 4°c to inches of mercury?
To understand how to convert between meters of water and inches of mercury, it's essential to first grasp the fundamental principle of pressure conversion. This involves understanding the densities of the fluids and the relationship between pressure, density, and height.
Understanding Pressure Conversion
Pressure exerted by a fluid column is given by the formula:
P=ρ g h P = \rho g h
Where:
P P is the pressure,
ρ\rho is the density of the fluid,
g g is the acceleration due to gravity (approximately 9.81 m/s 2 9.81 m/s^2),
h h is the height of the fluid column.
To convert between different units of pressure (in this case, meters of water and inches of mercury), we equate the pressures exerted by both fluids and solve for the desired height.
Conversion Formulas and Constants
Density of water at 4°C (ρ w a t e r\rho_{water}): approximately 1000 k g/m 3 1000 kg/m^3
Density of mercury (ρ m e r c u r y\rho_{mercury}): approximately 13560 k g/m 3 13560 kg/m^3
Since P w a t e r=P m e r c u r y P_{water} = P_{mercury}, we have:
ρ w a t e r⋅g⋅h w a t e r=ρ m e r c u r y⋅g⋅h m e r c u r y\rho_{water} \cdot g \cdot h_{water} = \rho_{mercury} \cdot g \cdot h_{mercury}
We can cancel out g from both sides
ρ w a t e r⋅h w a t e r=ρ m e r c u r y⋅h m e r c u r y\rho_{water} \cdot h_{water} = \rho_{mercury} \cdot h_{mercury}
From this, we derive the conversion formulas:
Meters of Water to Inches of Mercury:
h m e r c u r y=h w a t e r⋅ρ w a t e r ρ m e r c u r y h_{mercury} = h_{water} \cdot \frac{\rho_{water}}{\rho_{mercury}}
Since we need the answer in inches, we must first convert meters to inches (1 meter=39.37 inches 1 \text{ meter} = 39.37 \text{ inches}).
Inches of Mercury to Meters of Water:
h w a t e r=h m e r c u r y⋅ρ m e r c u r y ρ w a t e r h_{water} = h_{mercury} \cdot \frac{\rho_{mercury}}{\rho_{water}}
Since we need the answer in meters, we must convert inches to meters (1 inch=0.0254 meters 1 \text{ inch} = 0.0254 \text{ meters}).
Step-by-Step Conversions
Converting 1 Meter of Water to Inches of Mercury
Plug in the values: h m e r c u r y=1 meter⋅1000 k g/m 3 13560 k g/m 3=0.07375 meters h_{mercury} = 1 \text{ meter} \cdot \frac{1000 kg/m^3}{13560 kg/m^3} = 0.07375 \text{ meters}
Convert meters to inches: 0.07375 meters⋅39.37 inches 1 meter=2.903 inches of mercury 0.07375 \text{ meters} \cdot \frac{39.37 \text{ inches}}{1 \text{ meter}} = 2.903 \text{ inches of mercury}
Therefore, 1 meter of water at 4°C is approximately equal to 2.903 inches of mercury.
Converting 1 Inch of Mercury to Meters of Water
Plug in the values: h w a t e r=1 inch⋅13560 k g/m 3 1000 k g/m 3=13.56 inches h_{water} = 1 \text{ inch} \cdot \frac{13560 kg/m^3}{1000 kg/m^3} = 13.56 \text{ inches}
Convert inches to meters: 13.56 inches⋅0.0254 meters 1 inch=0.3444 meters of water 13.56 \text{ inches} \cdot \frac{0.0254 \text{ meters}}{1 \text{ inch}} = 0.3444 \text{ meters of water}
Therefore, 1 inch of mercury is approximately equal to 0.3444 meters of water.
Real-World Examples
Medical Devices: Blood pressure is often measured in millimeters of mercury (mmHg) in medical contexts. Converting to meters of water can help in calibrating or understanding the pressure readings in different units.
HVAC Systems: Pressure in air conditioning systems might be measured in inches of water column. Converting to other units like inches of mercury or pascals can be necessary for system diagnostics.
Weather Monitoring: Atmospheric pressure, crucial in weather forecasting, can be expressed in various units, including inches of mercury.
Industrial Processes: In industries dealing with fluid dynamics, pressure measurements in tanks or pipes might be in meters of water or similar units, necessitating conversions for compatibility with instruments calibrated in other units.
Historical Context
Evangelista Torricelli, an Italian physicist and mathematician, is best known for his invention of the mercury barometer in 1643. Torricelli's work demonstrated that air had weight and produced a measurable pressure, revolutionizing the understanding of atmospheric phenomena. His invention not only provided a new way to measure pressure but also laid the groundwork for future developments in physics and meteorology.
See below section for step by step unit conversion with formulas and explanations. Please refer to the table below for a list of all the Inches of mercury to other unit conversions.
What is meters of water @ 4°c?
The following sections will provide a comprehensive understanding of meters of water at 4°C as a unit of pressure.
Understanding Meters of Water @ 4°C
Meters of water (mH2O) at 4°C is a unit of pressure that represents the pressure exerted by a column of water one meter high at a temperature of 4 degrees Celsius. This temperature is specified because the density of water is at its maximum at approximately 4°C (39.2°F). Since pressure is directly proportional to density, specifying the temperature makes the unit more precise.
Formation of the Unit
The pressure at the bottom of a column of fluid is given by:
P=ρ⋅g⋅h P = \rho \cdot g \cdot h
Where:
P P is the pressure.
ρ\rho is the density of the fluid.
g g is the acceleration due to gravity (approximately 9.80665 m/s 2 9.80665 \, m/s^2).
h h is the height of the fluid column.
For meters of water at 4°C:
h=1 m h = 1 \, m
ρ=1000 k g/m 3\rho = 1000 \, kg/m^3 (approximately, at 4°C)
g=9.80665 m/s 2 g = 9.80665 \, m/s^2
Therefore, 1 meter of water at 4°C is equal to:
P=(1000 k g/m 3)⋅(9.80665 m/s 2)⋅(1 m)=9806.65 P a P = (1000 \, kg/m^3) \cdot (9.80665 \, m/s^2) \cdot (1 \, m) = 9806.65 \, Pa
Where P a Pa is Pascal, the SI unit of pressure.
Connection to Hydrostatics and Blaise Pascal
The concept of pressure exerted by a fluid column is a fundamental principle of hydrostatics. While no specific law is uniquely tied to "meters of water," the underlying principles are closely associated with Blaise Pascal. Pascal's Law states that pressure applied to a confined fluid is transmitted equally in all directions throughout the fluid. This principle directly relates to how the weight of a water column creates pressure at any point within that column. To learn more about Pascal's Law, visit Britannica's article on Pascal's Principle.
Real-World Examples
Water Tank Levels: Municipal water systems often use meters of water to indicate the water level in storage tanks. Knowing the water level (expressed as pressure head) allows operators to manage water distribution effectively.
Diving Depth: While divers often use meters of seawater (which has a slightly higher density than fresh water), meters of water can illustrate the pressure increase with depth. Each additional meter of depth increases the pressure by approximately 9800 Pa.
Well Water Levels: The static water level in a well can be expressed in meters of water. This indicates the pressure available from the aquifer.
Pressure Sensors: Some pressure sensors and transducers, especially those used in hydraulic or water management systems, directly display pressure readings in meters of water. For example, a sensor might indicate that a pipe has a pressure equivalent to 10 meters of water (approximately 98 kPa).
What is Inches of mercury?
The "inches of mercury" (inHg) is a unit of pressure commonly used in the United States. It's based on the height of a column of mercury that the given pressure will support. This unit is frequently used in aviation, meteorology, and vacuum applications.
Definition and Formation
Inches of mercury is a manometric unit of pressure. It represents the pressure exerted by a one-inch column of mercury at a standard temperature (usually 0°C or 32°F) under standard gravity.
The basic principle is that atmospheric pressure can support a certain height of a mercury column in a barometer. Higher atmospheric pressure corresponds to a higher mercury column, and vice versa. Therefore, the height of this column, measured in inches, serves as a direct indication of the pressure.
Formula and Conversion
Here's how inches of mercury relates to other pressure units:
1 inHg = 3386.39 Pascals (Pa)
1 inHg = 33.8639 millibars (mbar)
1 inHg = 25.4 millimeters of mercury (mmHg)
1 inHg ≈ 0.0334211 atmosphere (atm)
1 inHg ≈ 0.491154 pounds per square inch (psi)
Historical Context: Evangelista Torricelli
The concept of measuring pressure using a column of liquid is closely linked to Evangelista Torricelli, an Italian physicist and mathematician. In 1643, Torricelli invented the mercury barometer, demonstrating that atmospheric pressure could support a column of mercury. His experiments led to the understanding of vacuum and the quantification of atmospheric pressure. Britannica - Evangelista Torricelli has a good intro about him.
Real-World Applications and Examples
Aviation: Aircraft altimeters use inches of mercury to indicate altitude. Pilots set their altimeters to a local pressure reading (inHg) to ensure accurate altitude readings. Standard sea level pressure is 29.92 inHg.
Meteorology: Weather reports often include atmospheric pressure readings in inches of mercury. These readings are used to track weather patterns and predict changes in weather conditions. For example, a rising barometer (increasing inHg) often indicates improving weather, while a falling barometer suggests worsening weather.
Vacuum Systems: In various industrial and scientific applications, inches of mercury is used to measure vacuum levels. For example, vacuum pumps might be rated by the amount of vacuum they can create, expressed in inches of mercury. Higher vacuum levels (i.e., more negative readings) are crucial in processes like freeze-drying and semiconductor manufacturing. For example, common home vacuum cleaners operate in a range of 50 to 80 inHg.
Medical Equipment: Some medical devices, such as sphygmomanometers (blood pressure monitors), historically used mmHg (millimeters of mercury), a related unit. While digital devices are common now, the underlying principle remains tied to pressure measurement.
Interesting Facts
Standard Atmospheric Pressure: Standard atmospheric pressure at sea level is approximately 29.92 inches of mercury (inHg). This value is often used as a reference point for various measurements and calculations.
Altitude Dependence: Atmospheric pressure decreases with altitude. As you ascend, the weight of the air above you decreases, resulting in lower pressure readings in inches of mercury.
Temperature Effects: While "inches of mercury" typically refers to a standardized temperature, variations in temperature can slightly affect the density of mercury and, consequently, the pressure reading.
Complete meters of water @ 4°C conversion table
Enter # of meters of water @ 4°C
| Convert 1 mH 2 O to other units | Result |
--- |
| meters of water @ 4°C to pascals (mH 2 O to Pa) | 9806.65 |
| meters of water @ 4°C to kilopascals (mH 2 O to kPa) | 9.80665 |
| meters of water @ 4°C to megapascals (mH 2 O to MPa) | 0.00980665 |
| meters of water @ 4°C to hectopascals (mH 2 O to hPa) | 98.0665 |
| meters of water @ 4°C to millibar (mH 2 O to mbar) | 98.0665 |
| meters of water @ 4°C to bar (mH 2 O to bar) | 0.0980665 |
| meters of water @ 4°C to torr (mH 2 O to torr) | 73.555924006908 |
| meters of water @ 4°C to millimeters of mercury (mH 2 O to mmHg) | 73.556127270818 |
| meters of water @ 4°C to pounds per square inch (mH 2 O to psi) | 1.4223337722212 |
| meters of water @ 4°C to kilopound per square inch (mH 2 O to ksi) | 0.001422333772221 |
| meters of water @ 4°C to Inches of mercury (mH 2 O to inHg) | 2.895901839792 |
Pressure conversions
meters of water @ 4°C to pascals (mH 2 O to Pa)
meters of water @ 4°C to kilopascals (mH 2 O to kPa)
meters of water @ 4°C to megapascals (mH 2 O to MPa)
meters of water @ 4°C to hectopascals (mH 2 O to hPa)
meters of water @ 4°C to millibar (mH 2 O to mbar)
meters of water @ 4°C to bar (mH 2 O to bar)
meters of water @ 4°C to torr (mH 2 O to torr)
meters of water @ 4°C to millimeters of mercury (mH 2 O to mmHg)
meters of water @ 4°C to pounds per square inch (mH 2 O to psi)
meters of water @ 4°C to kilopound per square inch (mH 2 O to ksi)
meters of water @ 4°C to Inches of mercury (mH 2 O to inHg) |
10980 | https://www.tutorialspoint.com/ratios_and_unit_rates/writing_ratios_using_different_notations.htm | Writing Ratios Using Different Notations
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Writing Ratios Using Different Notations
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Definition
A ratio tells us how much there is of a quantity as compared to another.
The ratio of A to B is read as A is to B and written as A:B
The numbers A and B are called terms of the ratio with A being called the antecedent and B being called the consequent.
The order in a ratio is important. The ratio A:B is not same as B:A
Notation
The ratio of A to B is read as A is to B and A is to B is the word notation of the ratio A:B
The number notation of the ratio A to B is A:B; A ratio is written with a colon between the two quantities that are being compared. For example, the ratio of 2 to 5 is written as 2:5.
A ratio can also be written in fraction notation with a horizontal bar separating the two quantities, for example the ratio 3:7 is written as 3 7 3 7.
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Example 1
Write the word notation, number notation and fraction notation for the following ratio −
There are 2 boys and 3 girls. The ratio of boys to girl is
Solution
Step 1:
The word notation for the ratio of boys to girls is 2 is to 3
Step 2:
The number notation for the ratio of boys to girls is 2:3
Step 3:
The fraction notation for the ratio of boys to girls is 2 3 2 3
Step 4:
Ratio of boys to girls is 2:3 and the ratio of girls to boys is 3:2
Example 2
There are 3 apples and 4 oranges; Find the ratio of oranges to all fruits
Solution
Step 1:
There are 3 apples and 4 oranges. The total number of fruits = 3 + 4 = 7
Step 2:
So the ratio of oranges to all fruits is 4:7
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10981 | https://www.youtube.com/watch?v=ukEtad_aml4 | Graphing Inverse Functions
The Organic Chemistry Tutor
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Posted: 12 Feb 2018
This precalculus video tutorial explains how to graph inverse functions by reflecting the function across the line y = x and by switching the x and y coordinates and plotting the points using a data table. This video contains plenty of examples and practice problems on graphing inverse functions.
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Transcript:
in this lesson we're going to focus on graphing the inverse function so let's say if we have the function f of x and it looks like this now let's say the inverse function is g of x how can we graph the inverse function how can we draw a rough sketch well first you need to draw the line y equals x the inverse function is a reflection of f of x across that line so it's going to look something like let me do that again something like that so that's going to be g of x or the inverse function of f it's symmetric about the line y equals x let's try some more examples so let's say this is f of x it looks like this and let's say here we have the line y equals x go ahead and draw the inverse function of f so it's going to look something like that let's call that g of x so that's a simple way to draw the inverse function just using a rough sketch here's another example that you could try and let's draw the line y equals x first so let's say this is f and looks like that go ahead and draw the inverse function g of x so notice that f started on the x-axis the negative x-axis so g is going to start from the negative y-axis and it's going to just go up and then draw in a way that it reflects across the line y equals x so that's going to be the inverse function g of x now sometimes you may need to graph the inverse function using points so let me use a smaller graph so let's say if we have the first point is going to be at negative three zero the next one is negative one one and let's draw the the line y equals one the line y equals one will have the point one one 2 2 3 3 4 4 negative 1 negative 1 negative 2 negative 2 and so forth it helps if you plot that first now the next point is going to be 1 1 and then 3 2 which is here so that's going to be the graph of f of x using those points go ahead and graph the inverse function g of x so first i'm going to make a table an x y table and this is going to be for f of x so the first point was negative 3 0 the next one was negative 1 1 and then it was one one and then three comma two now the inverse function which we're calling g of x will have these points all you need to do is switch x and y so the first one is going to be zero negative three then one negative one one one and two three so plot them in order let's start with zero negative three that's right here and this was on the x-axis now this point is on the y-axis now the next point one negative one that should be here and then connect the first point with the second point and then we have the point one one so we need to connect these two and then the next point is 2 comma 3 which is here and you can see that the blue line is a reflection of the red line across the line y equals x and so the blue line is g of x notice that the distance between the red line and the line y equals x is equal to the distance between the blue line and the line y equals x they have to be equally distant from the line let's work on another example so let's say we have the point negative three actually start with negative five one and then negative three two and then negative two five and then it's going to be zero one and then 2 negative 4 and then 4 negative 4 and 5 0. so let's say this is the graph of f of x go ahead and graph the inverse function which we'll call g of x so first let's plot the points of the line y equals x so 1 comma 1 2 2 three three four four zero zero and so forth now let's make a table by the way feel free to pause the video if you want to try it so this is going to be for f of x and the first point we said was negative 5 one and then the next point is negative three two you gotta be careful with these steps because if you get the wrong point if you write it wrong then that's gonna mess up the inverse function and its graph so just be careful with these steps now after that the next point we have this one right here that's it has an x value of negative 2 and a y value of 5. and then the next one is zero one and then it was two negative four and then four negative four and then finally the last one is five comma zero now let's switch these points for the inverse function so it's going to be one negative five two negative three five negative 2 1 0 negative 4 2 negative 4 4 and 0 5 in that order so don't forget to plot it one step at a time connecting each point as you go so let's plot one negative five that's going to be over here let me put that in blue and then two negative three so that's in this region and connect the first point to the second point now the next one is five negative two which is here let's label the points let's call this point a b c d e f g so this is going to be a prime as you can see a and a prime they're equally distant from the line y equals x and b and b prime they're equidistant from the line y equals x so that tells you that you're on the right track so this is b prime right here now the next point after five negative two that's going to be 1 0 which is here and then it's going to be negative 4 2. and then negative four four and then zero five so this is c prime d prime e prime f prime and then g prime so you could see the symmetry if you compare the red line and the blue line you can see that they're symmetric about the line y equals x and so that's the graphical relationship between a function and its inverse function and that's it for this video so now you know how to graph an inverse function using drawing a rough sketch or using points you |
10982 | https://en.wikipedia.org/wiki/Weighing_matrix | Weighing matrix - Wikipedia
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1 Properties
2 ApplicationsToggle Applications subsection
2.1 Experimental design
2.2 Optical measurement
2.3 Orthogonal designs
3 Examples
4 Equivalence
5 Existence
6 References
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From Wikipedia, the free encyclopedia
Mathematical weight device
Not to be confused with Weight matrix.
Weighing matrices are so called because of their use in optimally measuring the individual weights of multiple objects.
In mathematics, a weighing matrix of order n{\displaystyle n} and weight w{\displaystyle w} is a matrixW{\displaystyle W} with entries from the set {0,1,−1}{\displaystyle {0,1,-1}} such that:
W W T=w I n{\displaystyle WW^{\mathsf {T}}=wI_{n}}
Where W T{\displaystyle W^{\mathsf {T}}} is the transpose of W{\displaystyle W} and I n{\displaystyle I_{n}} is the identity matrix of order n{\displaystyle n}. The weight w{\displaystyle w} is also called the degree of the matrix. For convenience, a weighing matrix of order n{\displaystyle n} and weight w{\displaystyle w} is often denoted by W(n,w){\displaystyle W(n,w)}.
Weighing matrices are so called because of their use in optimally measuring the individual weights of multiple objects. When the weighing device is a balance scale, the statistical variance of the measurement can be minimized by weighing multiple objects at once, including some objects in the opposite pan of the scale where they subtract from the measurement.
Properties
[edit]
Some properties are immediate from the definition. If W{\displaystyle W} is a W(n,w){\displaystyle W(n,w)}, then:
The rows of W{\displaystyle W} are pairwise orthogonal. Similarly, the columns are pairwise orthogonal.
Each row and each column of W{\displaystyle W} has exactly w{\displaystyle w} non-zero elements.
W T W=w I{\displaystyle W^{\mathsf {T}}W=wI}, since the definition means that W−1=w−1 W T{\displaystyle W^{-1}=w^{-1}W^{\mathsf {T}}}, where W−1{\displaystyle W^{-1}} is the inverse of W{\displaystyle W}.
det W=±w n/2{\displaystyle \det W=\pm w^{n/2}} where det W{\displaystyle \det W} is the determinant of W{\displaystyle W}.
A weighing matrix is a generalization of a Hadamard matrix, which does not allow zero entries. As two special cases, a W(n,n){\displaystyle W(n,n)} is a Hadamard matrix and a W(n,n−1){\displaystyle W(n,n-1)} is equivalent to a conference matrix.
Applications
[edit]
Experimental design
[edit]
See also: Design of experiments §Example
Weighing matrices take their name from the problem of measuring the weight of multiple objects. If a measuring device has a statistical variance of σ 2{\displaystyle \sigma ^{2}}, then measuring the weights of N{\displaystyle N} objects and subtracting the (equally imprecise) tare weight will result in a final measurement with a variance of 2 σ 2{\displaystyle 2\sigma ^{2}}. It is possible to increase the accuracy of the estimated weights by measuring different subsets of the objects, especially when using a balance scale where objects can be put on the opposite measuring pan where they subtract their weight from the measurement.
An order n{\displaystyle n} matrix W{\displaystyle W} can be used to represent the placement of n{\displaystyle n} objects—including the tare weight—in n{\displaystyle n} trials. Suppose the left pan of the balance scale adds to the measurement and the right pan subtracts from the measurement. Each element of this matrix w i j{\displaystyle w_{ij}} will have:
w i j={0 if on the i th trial the j th object was not measured 1 if on the i th trial the j th object was placed in the left pan−1 if on the i th trial the j th object was placed in the right pan{\displaystyle w_{ij}={\begin{cases}0&{\text{if on the }}i{\text{th trial the }}j{\text{th object was not measured}}\1&{\text{if on the }}i{\text{th trial the }}j{\text{th object was placed in the left pan}}\-1&{\text{if on the }}i{\text{th trial the }}j{\text{th object was placed in the right pan }}\\end{cases}}}
Let x{\displaystyle \mathbf {x} } be a column vector of the measurements of each of the n{\displaystyle n} trials, let e{\displaystyle \mathbf {e} } be the errors to these measurements each independent and identically distributed with variance σ 2{\displaystyle \sigma ^{2}}, and let y{\displaystyle \mathbf {y} } be a column vector of the true weights of each of the n{\displaystyle n} objects. Then we have:
x=W y+e{\displaystyle \mathbf {x} =W\mathbf {y} +\mathbf {e} }
Assuming that W{\displaystyle W} is non-singular, we can use the method of least-squares to calculate an estimate of the true weights:
y=(W T W)−1 W x{\displaystyle \mathbf {y} =(W^{T}W)^{-1}W\mathbf {x} }
The variance of the estimated y{\displaystyle \mathbf {y} } vector cannot be lower than σ 2/n{\displaystyle \sigma ^{2}/n}, and will be minimum if and only ifW{\displaystyle W} is a weighing matrix.
Optical measurement
[edit]
An optical mask (3) based on a weighing matrix is used in the measurement of the spectrum of incoming light (4). Depending on the corresponding element of the matrix, the light is either absorbed, or passed to one of two intensity detectors (1,2).
Weighing matrices appear in the engineering of spectrometers, image scanners, and optical multiplexing systems. The design of these instruments involve an optical mask and two detectors that measure the intensity of light. The mask can either transmit light to the first detector, absorb it, or reflect it toward the second detector. The measurement of the second detector is subtracted from the first, and so these three cases correspond to weighing matrix elements of 1, 0, and −1 respectively. As this is essentially the same measurement problem as in the previous section, the usefulness of weighing matrices also applies.
Orthogonal designs
[edit]
"Orthogonal design" redirects here. For the principle of database design, see Principle of orthogonal design.
An orthogonal design of order n{\displaystyle n} and type (s 1,…,s u){\displaystyle (s_{1},\dots ,s_{u})} where s i{\displaystyle s_{i}} are positive integers, is an n×n{\displaystyle n\times n} matrix whose entries are in the set {0,±x 1,…,±x u}{\displaystyle {0,\pm x_{1},\dots ,\pm x_{u}}}, where x i{\displaystyle x_{i}} are commuting variables. Additionally, an orthogonal design must satisfy:
X X T=∑i=0 u s i x i 2{\displaystyle XX^{T}=\sum {i=0}^{u}s{i}x_{i}^{2}}
This constraint is also equivalent to the rows of X{\displaystyle X} being orthogonal and each row having exactly s i{\displaystyle s_{i}} occurrences of x i{\displaystyle x_{i}}. An orthogonal design can be denoted as O D(n;s 1,…,s u){\displaystyle \mathrm {OD} (n;s_{1},\dots ,s_{u})}. An orthogonal design of one variable is a weighing matrix, and so the two fields of study are connected. Because of this connection, new orthogonal designs can be discovered by way of weighing matrices.
Examples
[edit]
Note that when weighing matrices are displayed, the symbol −{\displaystyle -} is used to represent −1. Here are some examples:
This is a W(2,2){\displaystyle W(2,2)}:
(1 1 1−){\displaystyle {\begin{pmatrix}1&1\1&-\end{pmatrix}}}
This is a W(4,3){\displaystyle W(4,3)}:
(1 1 1 0 1−0 1 1 0−−0 1−1){\displaystyle {\begin{pmatrix}1&1&1&0\1&-&0&1\1&0&-&-\0&1&-&1\end{pmatrix}}}
This is a W(7,4){\displaystyle W(7,4)}:
(1 1 1 1 0 0 0 1−0 0 1 1 0 1 0−0−0 1 1 0 0−0−−0 1−0 0 1−0 1 0−1 0 1 0 0 1−−1 0){\displaystyle {\begin{pmatrix}1&1&1&1&0&0&0\1&-&0&0&1&1&0\1&0&-&0&-&0&1\1&0&0&-&0&-&-\0&1&-&0&0&1&-\0&1&0&-&1&0&1\0&0&1&-&-&1&0\end{pmatrix}}}
Another W(7,4){\displaystyle W(7,4)}:
(−1 1 0 1 0 0 0−1 1 0 1 0 0 0−1 1 0 1 1 0 0−1 1 0 0 1 0 0−1 1 1 0 1 0 0−1 1 1 0 1 0 0−){\displaystyle {\begin{pmatrix}-&1&1&0&1&0&0\0&-&1&1&0&1&0\0&0&-&1&1&0&1\1&0&0&-&1&1&0\0&1&0&0&-&1&1\1&0&1&0&0&-&1\1&1&0&1&0&0&-\end{pmatrix}}}
Which is circulant, i.e. each row is a cyclic shift of the previous row. Such a matrix is called a C W(n,k){\displaystyle CW(n,k)} and is determined by its first row. Circulant weighing matrices are of special interest since their algebraic structure makes them easier for classification. Indeed, we know that a circulant weighing matrix of order n{\displaystyle n} and weight k{\displaystyle k} must be of square weight. So, weights 1,4,9,16,...{\displaystyle 1,4,9,16,...} are permissible and weights k≤25{\displaystyle k\leq 25} have been completely classified. Two special (and actually, extreme) cases of circulant weighing matrices are (A) circulant Hadamard matrices which are conjectured not to exist unless their order is less than 5. This conjecture, the circulant Hadamard conjecture first raised by Ryser, is known to be true for many orders but is still open. (B) C W(n,k){\displaystyle CW(n,k)} of weight k=s 2{\displaystyle k=s^{2}} and minimal order n{\displaystyle n} exist if s{\displaystyle s} is a prime power and such a circulant weighing matrix can be obtained by signing the complement of a finite projective plane. Since all C W(n,k){\displaystyle CW(n,k)} for k≤25{\displaystyle k\leq 25} have been classified, the first open case is C W(105,36){\displaystyle CW(105,36)}. The first open case for a general weighing matrix (certainly not a circulant) is W(35,25){\displaystyle W(35,25)}.
Equivalence
[edit]
Two weighing matrices are considered to be equivalent if one can be obtained from the other by a series of permutations and negations of the rows and columns of the matrix. The classification of weighing matrices is complete for cases where w≤5{\displaystyle w\leq 5} as well as all cases where n≤15{\displaystyle n\leq 15}. However, very little has been done beyond this with exception to classifying circulant weighing matrices.
Existence
[edit]
One major open question about weighing matrices is their existence: for which values of n{\displaystyle n} and w{\displaystyle w} does there exist a W(n,w){\displaystyle W(n,w)}? The following conjectures have been proposed about the existence of W(n,w){\displaystyle W(n,w)}:
If n≡2(mod 4){\displaystyle n\equiv 2{\pmod {4}}} then there exists a W(n,w){\displaystyle W(n,w)} if and only if w<n−1{\displaystyle w<n-1} is the sum of two integer squares.
If n≡0(mod 4){\displaystyle n\equiv 0{\pmod {4}}} then there exists a W(n,w){\displaystyle W(n,w)} for each w<n{\displaystyle w<n}.
If n≡4(mod 8){\displaystyle n\equiv 4{\pmod {8}}} then there exists an orthogonal design O D(n;1,1){\displaystyle \mathrm {OD} (n;1,1)} for all k<n{\displaystyle k<n} where k{\displaystyle k} is the sum of three integer squares.
If n≡0(mod 8){\displaystyle n\equiv 0{\pmod {8}}} then there exists an orthogonal design O D(n;1,k){\displaystyle \mathrm {OD} (n;1,k)} for all k<n{\displaystyle k<n}.
If n≡2(mod 4){\displaystyle n\equiv 2{\pmod {4}}} then there exists an orthogonal design O D(n;1,k){\displaystyle \mathrm {OD} (n;1,k)} for all k<n−1{\displaystyle k<n-1} such that k=a 2{\displaystyle k=a^{2}}, a{\displaystyle a} an integer.
Although the last three conjectures are statements on orthogonal designs, it has been shown that the existence of an orthogonal design O D(n;s 1,…,s u){\displaystyle \mathrm {OD} (n;s_{1},\dots ,s_{u})} is equivalent to the existence of X 1,…,X u{\displaystyle X_{1},\dots ,X_{u}} weighing matrices of order n{\displaystyle n} where X i{\displaystyle X_{i}} has weight s i{\displaystyle s_{i}}.
An equally important but often overlooked question about weighing matrices is their enumeration: for a given n{\displaystyle n} and w{\displaystyle w}, how many W(n,w){\displaystyle W(n,w)}'s are there?
References
[edit]
^ abRaghavarao, Damaraju (1960). "Some Aspects of Weighing Designs". The Annals of Mathematical Statistics. 31 (4). Institute of Mathematical Statistics: 878–884. doi:10.1214/aoms/1177705664. ISSN0003-4851.
^ abSeberry, Jennifer (2017). "Some Algebraic and Combinatorial Non-existence Results". Orthogonal Designs. Cham: Springer International Publishing. pp.7–17. doi:10.1007/978-3-319-59032-5_2. ISBN978-3-319-59031-8.
^ abcGeramita, Anthony V.; Pullman, Norman J.; Wallis, Jennifer S. (1974). "Families of weighing matrices". Bulletin of the Australian Mathematical Society. 10 (1). Cambridge University Press (CUP): 119–122. doi:10.1017/s0004972700040703. ISSN0004-9727. S2CID122560830.
^ abRaghavarao, Damaraju (1971). "Weighing Designs". Constructions and combinatorial problems in design of experiments. New York: Wiley. pp.305–308. ISBN978-0471704850.
^ abKoukouvinos, Christos; Seberry, Jennifer (1997). "Weighing matrices and their applications". Journal of Statistical Planning and Inference. 62 (1). Elsevier BV: 91–101. doi:10.1016/s0378-3758(96)00172-3. ISSN0378-3758. S2CID122205953.
^ abcSloane, Neil J. A.; Harwit, Martin (1976-01-01). "Masks for Hadamard transform optics, and weighing designs". Applied Optics. 15 (1). The Optical Society: 107–114. Bibcode:1976ApOpt..15..107S. doi:10.1364/ao.15.000107. ISSN0003-6935. PMID20155192.
^ abcdGeramita, Anthony V.; Seberry, Jennifer (1974). "Orthogonal designs III: weighing matrices". Utilitas Mathematica.
^Charles J. Colbourn (1996). "Orthogonal Designs". In Colbourn, Charles J. (ed.). CRC Handbook of Combinatorial Designs (1 ed.). Boca Raton: CRC Press. p.400. doi:10.1201/9781003040897. ISBN9781003040897.
^Kotsireas, Ilias; Koukouvinos, Christos; Seberry, Jennifer (2008). "New orthogonal designs from weighing matrices". Australasian Journal of Combinatorics. 40: 99–104.
^Arasu, K. T.; Gordon, Daniel M.; Zhang, Yiran (2021). "New nonexistence results on circulant weighing matrices". Cryptography and Communications. 13 (5): 775–789. arXiv:1908.08447v3. doi:10.1007/s12095-021-00492-0. MR4322521.
^Harada, Masaaki; Munemasa, Akihiro (2012). "On the classification of weighing matrices and self-orthogonal codes". J. Combin. Designs. 20: 40–57. arXiv:1011.5382. doi:10.1002/jcd.20295. S2CID1004492.
^Ang, Miin Huey; Arasu, K.T.; Lun Ma, Siu; Strassler, Yoseph (2008). "Study of proper circulant weighing matrices with weight 9". Discrete Mathematics. 308 (13): 2802–2809. doi:10.1016/j.disc.2004.12.029.
^Arasu, K.T.; Hin Leung, Ka; Lun Ma, Siu; Nabavi, Ali; Ray-Chaudhuri, D.K. (2006). "Determination of all possible orders of weight 16 circulant weighing matrices". Finite Fields and Their Applications. 12 (4): 498–538. doi:10.1016/j.ffa.2005.06.009.
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Extended Euclidean Algorithm¶
While the Euclidean algorithm calculates only the greatest common divisor (GCD) of two integers a$a$ and b$b$ , the extended version also finds a way to represent GCD in terms of a$a$ and b$b$ , i.e. coefficients x$x$ and y$y$ for which:
a⋅x+b⋅y=gcd(a,b)
$$a \cdot x + b \cdot y = \gcd(a, b)$$
It's important to note that by Bézout's identity we can always find such a representation. For instance, gcd(55,80)=5$\gcd(55, 80) = 5$ , therefore we can represent 5$5$ as a linear combination with the terms 55$55$ and 80$80$ : 55⋅3+80⋅(−2)=5$55 \cdot 3 + 80 \cdot (-2) = 5$
A more general form of that problem is discussed in the article about Linear Diophantine Equations.
It will build upon this algorithm.
Algorithm¶
We will denote the GCD of a$a$ and b$b$ with g$g$ in this section.
The changes to the original algorithm are very simple.
If we recall the algorithm, we can see that the algorithm ends with b=0$b = 0$ and a=g$a = g$ .
For these parameters we can easily find coefficients, namely g⋅1+0⋅0=g$g \cdot 1 + 0 \cdot 0 = g$ .
Starting from these coefficients (x,y)=(1,0)$(x, y) = (1, 0)$ , we can go backwards up the recursive calls.
All we need to do is to figure out how the coefficients x$x$ and y$y$ change during the transition from (a,b)$(a, b)$ to (b,amodb)$(b, a \bmod b)$ .
Let us assume we found the coefficients (x1,y1)$(x_1, y_1)$ for (b,amodb)$(b, a \bmod b)$ :
b⋅x1+(amodb)⋅y1=g
$$b \cdot x_1 + (a \bmod b) \cdot y_1 = g$$
and we want to find the pair (x,y)$(x, y)$ for (a,b)$(a, b)$ :
a⋅x+b⋅y=g
$$ a \cdot x + b \cdot y = g$$
We can represent amodb$a \bmod b$ as:
amodb=a−⌊ab⌋⋅b
$$ a \bmod b = a - \left\lfloor \frac{a}{b} \right\rfloor \cdot b$$
Substituting this expression in the coefficient equation of (x1,y1)$(x_1, y_1)$ gives:
g=b⋅x1+(amodb)⋅y1=b⋅x1+(a−⌊ab⌋⋅b)⋅y1
$$ g = b \cdot x_1 + (a \bmod b) \cdot y_1 = b \cdot x_1 + \left(a - \left\lfloor \frac{a}{b} \right\rfloor \cdot b \right) \cdot y_1$$
and after rearranging the terms:
g=a⋅y1+b⋅(x1−y1⋅⌊ab⌋)
$$g = a \cdot y_1 + b \cdot \left( x_1 - y_1 \cdot \left\lfloor \frac{a}{b} \right\rfloor \right)$$
We found the values of x$x$ and y$y$ :
{x=y1y=x1−y1⋅⌊ab⌋
$$\begin{cases}
x = y_1 \
y = x_1 - y_1 \cdot \left\lfloor \frac{a}{b} \right\rfloor
\end{cases} $$
Implementation¶
```
int gcd(int a, int b, int& x, int& y) {
if (b == 0) {
x = 1;
y = 0;
return a;
}
int x1, y1;
int d = gcd(b, a % b, x1, y1);
x = y1;
y = x1 - y1 (a / b);
return d;
}
```
The recursive function above returns the GCD and the values of coefficients to x and y (which are passed by reference to the function).
This implementation of extended Euclidean algorithm produces correct results for negative integers as well.
Iterative version¶
It's also possible to write the Extended Euclidean algorithm in an iterative way.
Because it avoids recursion, the code will run a little bit faster than the recursive one.
```
int gcd(int a, int b, int& x, int& y) {
x = 1, y = 0;
int x1 = 0, y1 = 1, a1 = a, b1 = b;
while (b1) {
int q = a1 / b1;
tie(x, x1) = make_tuple(x1, x - q x1);
tie(y, y1) = make_tuple(y1, y - q y1);
tie(a1, b1) = make_tuple(b1, a1 - q b1);
}
return a1;
}
```
If you look closely at the variables a1 and b1, you can notice that they take exactly the same values as in the iterative version of the normal Euclidean algorithm. So the algorithm will at least compute the correct GCD.
To see why the algorithm computes the correct coefficients, consider that the following invariants hold at any given time (before the while loop begins and at the end of each iteration):
x⋅a+y⋅b=a1
$$x \cdot a + y \cdot b = a_1$$
x1⋅a+y1⋅b=b1
$$x_1 \cdot a + y_1 \cdot b = b_1$$
Let the values at the end of an iteration be denoted by a prime ( ′$'$ ), and assume q=a1b1$q = \frac{a_1}{b_1}$ . From the Euclidean algorithm, we have:
a1′=b1
$$a_1' = b_1$$
b1′=a1−q⋅b1
$$b_1' = a_1 - q \cdot b_1$$
For the first invariant to hold, the following should be true:
x′⋅a+y′⋅b=a1′=b1
$$x' \cdot a + y' \cdot b = a_1' = b_1$$
x′⋅a+y′⋅b=x1⋅a+y1⋅b
$$x' \cdot a + y' \cdot b = x_1 \cdot a + y_1 \cdot b$$
Similarly for the second invariant, the following should hold:
x1′⋅a+y1′⋅b=a1−q⋅b1
$$x_1' \cdot a + y_1' \cdot b = a_1 - q \cdot b_1$$
x1′⋅a+y1′⋅b=(x−q⋅x1)⋅a+(y−q⋅y1)⋅b
$$x_1' \cdot a + y_1' \cdot b = (x - q \cdot x_1) \cdot a + (y - q \cdot y_1) \cdot b$$
By comparing the coefficients of a$a$ and b$b$ , the update equations for each variable can be derived, ensuring that the invariants are maintained throughout the algorithm.
At the end we know that a1$a_1$ contains the GCD, so x⋅a+y⋅b=g$x \cdot a + y \cdot b = g$ .
Which means that we have found the required coefficients.
You can even optimize the code more, and remove the variable a1$a_1$ and b1$b_1$ from the code, and just reuse a$a$ and b$b$ .
However if you do so, you lose the ability to argue about the invariants.
Practice Problems¶
UVA - 10104 - Euclid Problem
GYM - (J) Once Upon A Time
UVA - 12775 - Gift Dilemma
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10984 | https://www.cuemath.com/numbers/mixed-number-to-improper-fraction/ | LearnPracticeDownload
Mixed Number to Improper Fraction
In order to convert a mixed number to an improper fraction, we need to multiply the whole number with the denominator and then add this product with the numerator. This forms the new numerator of the improper fraction while the denominator remains the same. The mixed number to improper fraction conversion can be done easily with the help of a few steps discussed on this page.
| | |
--- |
| 1. | Converting Mixed Number to Improper Fraction |
| 2. | How to Add Mixed Numbers to Improper Fractions? |
| 3. | FAQs on Mixed Number to Improper Fraction |
Converting Mixed Number to Improper Fraction
Before learning how to convert a mixed number to an improper fraction, let us quickly go through the definition of mixed numbers and improper fractions. A mixed fraction is one whose value is always greater than 1 and it has a whole number part and a proper fraction. A mixed number example is (3\dfrac{2}{5}) is a mixed number. An improper fraction is one in which the numerator is always greater than or equal to the denominator. Some examples of improper fractions are 4/3, 7/3, 11/5, etc.
Let us understand the method of converting a mixed number to an improper fraction with the help of an example. Let us convert the mixed fraction (7\dfrac{1}{5}) to an improper traction using the following steps:
Step 1: Multiply the denominator with the whole number part. Here, 5 × 7 = 35.
Step 2: Add the numerator to the product obtained in step 1. So, we get, 35 + 1= 36.
Step 3: Write the value obtained in step 2 over the denominator. This will be the new numerator while the denominator will remain the same. So, (7\dfrac{1}{5}) = 36/5.
This is how we convert a mixed number to an improper fraction. Let us understand this with another example.
Example: Convert the mixed number to an improper fraction: (2\dfrac{3}{4})
Solution: We can convert the mixed number to an improper fraction by using the following steps.
Step 1: Let us multiply the denominator with the whole number part. Here, we will multiply 4 by 2, that is, 4 × 2 = 8.
Step 2: Now, we will add this product to the numerator. This will be 8 + 3 = 11.
Step 3: So, 11 will be the new numerator while the denominator (4) will remain the same. This means, (2\dfrac{3}{4}) = 11/4.
The other way to understand this process is the addition of the whole number part and the fractional part. For example, in the same example (7\dfrac{1}{5}), let us add the whole number (7) and the fraction (1/5). We get 7 + 1/5 = 7/1 + 1/5 = (35 + 1)/5 = 36/5. Therefore, this is another way to get the improper fraction from a mixed number.
How to Add Mixed Numbers to Improper Fractions?
In order to add mixed numbers to improper fractions, we first need to convert the mixed number to an improper fraction and then add them using the usual method of addition of fractions. If the given fractions are like fractions, then the addition can be done easily. However, if they are unlike fractions then they need to be converted to like fractions and then added. Let us understand this with the help of an example.
Example 1: Add (3\dfrac{2}{5}) + 14/5.
Solution: We will convert (3\dfrac{2}{5}) to an improper fraction which will be, 17/5. Now 17/5 + 14/5 = 31/5 = (6\dfrac{1}{5}). Therefore, the sum is (6\dfrac{1}{5})
In case of unlike fractions, we need to find the Least Common Multiple (LCM) of the denominators and then convert them to like fractions. After this they can be added in the usual way.
☛ Related Topics
Improper Fraction to Mixed Number
Mixed Fraction to Decimal
Types of Fractions
Equivalent Fractions
Addition and Subtraction of Fractions
Mixed Number to Improper Fraction Examples
Example 1: Convert (5\dfrac{2}{3}) to an improper fraction.
Solution: In this question, a mixed number is given to us and we need to convert it into an improper fraction. Let us follow the steps given below to convert the mixed number to improper fraction:
Step 1: Multiply 3 by 5 ⇒ 3 × 5 = 15.
Step 2: Add 2 to 15 ⇒ 2 + 15 = 17.
Step 3: Write 17 over 3. So, 17/3 is the answer.
Therefore, (5\dfrac{2}{3}) = 17/3.
2. Example 2: What improper fraction is equal to the mixed number (4\dfrac{7}{9})?
Solution: To convert the given mixed number to an improper fraction, let us first multiply the denominator with the whole number. This means, 9 × 4 = 36. Then, let us add this product to the numerator, which is, 36 + 7 = 43. So, this will be the new numerator and the improper fraction will be = 43/9
Therefore, (4\dfrac{7}{9}) = 43/9.
3. Example 3: Which improper fraction is equal to the mixed number (6\dfrac{4}{7})?
Solution: To convert the given mixed number to an improper fraction, let us follow the steps given below:
Step 1: Multiply 7 by 6 ⇒ 7 × 6 = 42.
Step 2: Add 4 to 42 ⇒ 4 + 42 = 46.
Step 3: Write 46 over 7. So, 46/7 is the answer.
Therefore, (6\dfrac{4}{7}) = 46/7.
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Practice Questions on Mixed Number to Improper Fraction
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FAQs on Mixed Number to Improper Fraction
What is the First Step in Changing a Mixed Number to an Improper Fraction?
The first step in changing a mixed number to an improper fraction is the multiplication of the whole number and the denominator of the given mixed number. Then, we add the numerator to the product.
What are the Steps to Convert a Mixed Number to an Improper Fraction?
The steps to convert a mixed fraction to improper fraction are given below. Let us convert (3\dfrac{3}{7}) into an improper fraction.
Step 1: Find the product of the whole number and the denominator of the given mixed fraction. Here, 3 × 7 = 21.
Step 2: Add the product to the numerator. So, 21 + 3 = 24.
Step 3: Write that value over the denominator to express the answer. So, 24/7 is the improper fraction.
How to Change a Mixed Number to an Improper Fraction?
A mixed number consists of two parts - a whole number and a proper fraction. To change a mixed number to an improper fraction, we multiply the whole number by the denominator and then add this product with the numerator. This number becomes the numerator of the improper fraction and the denominator remains the same. For example, let us change the given mixed number to an improper fraction: (3\dfrac{1}{2}). We will first multiply the denominator (2) by the whole number (3) and the product is 2 × 3 = 6. To this product, we will add 1 which is the numerator. This will make it 6 + 1 = 7. So, 7 will become the numerator of the improper fraction and 2 will be the denominator. Therefore, (3\dfrac{1}{2}) is converted to an improper fraction and is written as 7/2.
How to Multiply a Mixed Number to an Improper Fraction?
To multiply a mixed number with an improper fraction, we first need to change the mixed fraction into an improper fraction. After this, we can multiply both the fractions in the usual way. For example, to multiply (2\dfrac{3}{4}) to 7/5, the first step is to convert (2\dfrac{3}{4}) to an improper fraction, which will be 11/4. Now, we can multiply 11/4 and 7/5 which is 11/4 × 7/5 = 77/20 = (3\dfrac{17}{20}).
How to Add Mixed Numbers to Improper Fractions?
In order to add mixed numbers to improper fractions, we first need to convert the mixed number to an improper fraction and then add them using the usual method of addition of fractions. For example, let us add (5\dfrac{3}{4}) + 15/4. We will convert (5\dfrac{3}{4}) to an improper fraction which will be, 23/4. Now 23/4 + 15/4 = 38/4 = 19/2 = (9\dfrac{1}{2}). Therefore, the sum is (9\dfrac{1}{2})
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10985 | https://dictionary.cambridge.org/us/thesaurus/thriving | Cambridge Dictionary +Plus
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Synonyms and antonyms of thriving in English
Thesaurus > achieving a lot, becoming popular, or making a lot of money > thriving
These are words and phrases related to thriving. Click on any word or phrase to go to its thesaurus page. Or,
go to the definition of thriving.
ACHIEVING A LOT, BECOMING POPULAR, OR MAKING A LOT OF MONEY
She had a thriving career.
Synonyms and examples
successful
She's one of the most successful athletes in the franchise's history.
flourishing
Our little town was a flourishing seaport in the 1600s.
triumphant
It was a triumphant victory for the struggling Chicago White Sox.
booming
Business is absolutely booming this month.
off the charts
mainly US
Her career is off the charts!
be flying high
They were flying high after their annual sales numbers tripled.
be riding high
The team was riding high after seven straight wins.
high-flying
She was a high-flying international banker.
to have arrived
His last four albums went platinum - I'd say he's arrived.
have the world at your feet
informal
A fan favorite, and now with two Grammys to her name, the pop star has the world at her feet.
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unsuccessful
She tried a solo voyage around the world, but was ultimately unsuccessful.
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succeed
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If a person or organization succeeds, they achieve something that they have been aiming for. A person or thing that achieves a lot or becomes popular or wealthy can be called a success. A successful person, group, or organization has achieved a lot and become popular or made a lot of money, and something that is successful has achieved the results hoped for.
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thriving | American Thesaurus
thriving
adjective
These are words and phrases related to thriving. Click on any word or phrase to go to its thesaurus page. Or,
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The thriving tomato plants are plentiful this year.
Synonyms
growing fast
blooming
blossoming
flowering
luxuriant
rank
lush
Antonyms
languishing
dying
withering
fading
The pizza parlor did a thriving business. One brother is poor, but the other is thriving.
Synonyms
flourishing
prospering
prosperous
successful
succeeding
busy
vigorous
wealthy
rich
well-to-do
well-off
on Easy Street
Slang
in clover
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failing
bankrupt
unsuccessful
poor
moneyless
penniless
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impoverished
badly off
indigent
broke
Synonyms for thriving from Random House Roget's College Thesaurus, Revised and Updated Edition © 2000 Random House, Inc.
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thriving
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10986 | https://blogs.glowscotland.org.uk/nl/public/ca-mathematics/uploads/sites/20843/2019/03/Implicit-Parametric-Logarithmic-Differentiation.pdf | 1 | P a g e Unit 1: Methods in Algebra and Calculus (H7X2 77) Differentiation 2 – Implicit and Parametric Functions, Logarithmic Differentiation Explicit function: 𝑦 is expressed explicitly as a function of 𝑥 i.e. 𝑦 is the subject of the formula e.g. 𝑦= 2𝑥+ 3 Implicit function: 𝑦 is not expressed explicitly as a function of 𝑥 i.e. 𝑦 is not the subject of the formula e.g. 𝑦−2𝑥= 3 Examples: Given 3𝑥2 + 7𝑥𝑦+ 9𝑦2 = 6, find 𝑑𝑦 𝑑𝑥 Differentiate both sides with respect to 𝑥 𝑑 𝑑𝑥(3𝑥2 + 7𝑥𝑦+ 9𝑦2) = 𝑑 𝑑𝑥(6) 𝑑 𝑑𝑥(3𝑥2) + 𝑑 𝑑𝑥(7𝑥𝑦) + 𝑑 𝑑𝑥(9𝑦2) = 𝑑 𝑑𝑥(6) 6𝑥+ 7 𝑑 𝑑𝑥(𝑥𝑦) + 9 𝑑 𝑑𝑥(𝑦2) = 0 The derivative of 𝑦 is 𝑑𝑦 𝑑𝑥 Use the chain rule for 𝑑 𝑑𝑥(𝑦2): 𝑑 𝑑𝑥(𝑦)2 = 2𝑦× 𝑑𝑦 𝑑𝑥 use the product rule for 𝑑 𝑑𝑥(𝑥𝑦) – 𝑢= 𝑥⇒𝑢′ = 1 𝑣= 𝑦⇒𝑣′ = 𝑑𝑦 𝑑𝑥 𝑑 𝑑𝑥(𝑥𝑦) = 𝑦+ 𝑥𝑑𝑦 𝑑𝑥 This gives: 6𝑥+ 7 (𝑦+ 𝑥 𝑑𝑦 𝑑𝑥) + 9 (2𝑦 𝑑𝑦 𝑑𝑥) = 0 6𝑥+ 7𝑦+ 7𝑥𝑑𝑦 𝑑𝑥+ 18𝑦𝑑𝑦 𝑑𝑥= 0 Now make 𝑑𝑦 𝑑𝑥 the subject: 7𝑥 𝑑𝑦 𝑑𝑥+ 18𝑦 𝑑𝑦 𝑑𝑥= −6𝑥−7𝑦 Take 𝑑𝑦 𝑑𝑥 out as common factor: 𝑑𝑦 𝑑𝑥(7𝑥+ 18𝑦) = −6𝑥−7𝑦 𝑑𝑦 𝑑𝑥= −6𝑥−7𝑦 7𝑥+ 18𝑦 NB – Can’t differentiate 𝑦 with respect to 𝑥 if no 𝑥′𝑠 in the function - 𝑑 𝑑𝑥𝑦= 𝑑𝑦 𝑑𝑥, 𝑑 𝑑𝑥𝑦2 = 2𝑦 𝑑𝑦 𝑑𝑥, 𝑑 𝑑𝑥4𝑦3 = 12𝑦2 𝑑𝑦 𝑑𝑥, etc. - Treat the 𝑦 as a bracket, derivative of outside × inside Bk2 P36 Ex4A Q1 & 3 2 | P a g e Second derivative - 𝑓′′(𝑥) or 𝑑2𝑦 𝑑𝑥2 The 2nd derivative of a function can be used to determine the nature of an SP without the need for a Nature Table: In general: 𝑑2𝑦 𝑑𝑥2 > 0 then min TP 𝑑2𝑦 𝑑𝑥2 < 0 then max TP 𝑑2𝑦 𝑑𝑥2 = 0 then PI but is it or ?? Find the nature of the SPs for the curve 𝑦= 2𝑥3 −2𝑥2 −2𝑥 𝑑𝑦 𝑑𝑥= 6𝑥2 −4𝑥−2 = 2(3𝑥+ 1)(𝑥−1) = 0 so 𝑆𝑃𝑠 @ 𝑥= − 1 3 𝑜𝑟 𝑥= 1 𝑑2𝑦 𝑑𝑥2 = 12𝑥−4 𝑥= − 1 3 𝑔𝑖𝑣𝑒𝑠 𝑑2𝑦 𝑑𝑥2 = −8 ⇒max 𝑇𝑃 𝑥= 1 𝑔𝑖𝑣𝑒𝑠 𝑑2𝑦 𝑑𝑥2 = 8 ⇒min 𝑇𝑃 Second derivative of Implicit Functions Find 𝑑𝑦 𝑑𝑥 and 𝑑2𝑦 𝑑𝑥2 in terms of 𝑥 and 𝑦 only for 𝑥2 + 2𝑥𝑦= 1 Differentiate both sides: 𝑑 𝑑𝑥(𝑥2 + 2𝑥𝑦) = 𝑑 𝑑𝑥(1) 2𝑥+ 2 (𝑥𝑑𝑦 𝑑𝑥+ 𝑦) = 0 2𝑥+ 2𝑥𝑑𝑦 𝑑𝑥+ 2𝑦= 0 2𝑥𝑑𝑦 𝑑𝑥= −2𝑥−2𝑦 𝑑𝑦 𝑑𝑥= −𝑥−𝑦 𝑥 Use the quotient rule and substitute for 𝑑𝑦 𝑑𝑥 where needed: 𝑢= −𝑥−𝑦⇒𝑢′ = −1 − 𝑑𝑦 𝑑𝑥= −1 + 𝑥+𝑦 𝑥= 𝑦 𝑥 𝑣= 𝑥 ⇒𝑣′ = 1 𝑑2𝑦 𝑑𝑥2 = 𝑢′𝑣−𝑢𝑣′ 𝑣2 = 𝑦 𝑥𝑥−(−𝑥−𝑦) 𝑥2 = 𝑦+ 𝑥+ 𝑦 𝑥2 = 2𝑦+ 𝑥 𝑥2 P38 Ex5 Q1 - 1st Column Q3, 5, 7, 9 3 | P a g e Using the Logarithmic Function in Differentiation Find 𝑑𝑦 𝑑𝑥 if 𝑦= 4𝑥 Take ln of both sides: ln 𝑦= ln 4𝑥 ln 𝑦= 𝑥ln 4 Differentiate both sides: 𝑑 𝑑𝑥(ln 𝑦) = 𝑑 𝑑𝑥(𝑥ln 4) 1 𝑦 𝑑𝑦 𝑑𝑥= ln 4 Make 𝑑𝑦 𝑑𝑥 the subject: 𝑑𝑦 𝑑𝑥= 𝑦ln 4 Substitute for 𝑦: 𝑑𝑦 𝑑𝑥= 4𝑥ln 4 𝑦= 𝑥𝑥, Find 𝑑𝑦 𝑑𝑥 ln 𝑦= ln 𝑥𝑥 so ln 𝑦= 𝑥ln 𝑥 𝑢= 𝑥⇒𝑢′ = 1 𝑣= ln 𝑥⇒𝑣′ = 1 𝑥 1 𝑦 𝑑𝑦 𝑑𝑥= ln 𝑥+ 1 so 𝑑𝑦 𝑑𝑥= 𝑦(ln 𝑥+ 1) so 𝑑𝑦 𝑑𝑥= 𝑥𝑥(ln 𝑥+ 1) Find 𝑑𝑦 𝑑𝑥 if 𝑦= 𝑥2√7𝑥−3 1+𝑥 [would normally need product/quotient rule] ln 𝑦= ln ( 𝑥2√7𝑥−3 1+𝑥 ) ln 𝑦= ln(𝑥2) + ln(√7𝑥−3) −ln(1 + 𝑥) ln 𝑦= 2 ln(𝑥) + 1 2 ln(7𝑥−3) −ln(1 + 𝑥) 1 𝑦 𝑑𝑦 𝑑𝑥= 2 𝑥+ 1 2 × 1 7𝑥−3 × 7 − 1 1 + 𝑥 1 𝑦 𝑑𝑦 𝑑𝑥= 2 𝑥+ 7 2(7𝑥−3) − 1 1 + 𝑥 4 | P a g e 1 𝑦 𝑑𝑦 𝑑𝑥= 2 × 2(7𝑥−3)(1 + 𝑥) + 7𝑥(1 + 𝑥) −2𝑥(7𝑥−3) 2𝑥(7𝑥−3)(1 + 𝑥) 1 𝑦 𝑑𝑦 𝑑𝑥= 4(7𝑥2 + 4𝑥−3) + 7𝑥+ 7𝑥2 −14𝑥2 + 6𝑥 2𝑥(7𝑥−3)(1 + 𝑥) 1 𝑦 𝑑𝑦 𝑑𝑥= 21𝑥2 + 29𝑥−12 2𝑥(7𝑥−3)(1 + 𝑥) 𝑑𝑦 𝑑𝑥= 𝑦× 21𝑥2 + 29𝑥−12 2𝑥(7𝑥−3)(1 + 𝑥) 𝑑𝑦 𝑑𝑥= 𝑥2(7𝑥−3) 1 2 1 + 𝑥 × 21𝑥2 + 29𝑥−12 2𝑥(7𝑥−3)(1 + 𝑥) 𝑑𝑦 𝑑𝑥= 𝑥 1 + 𝑥× 21𝑥2 + 29𝑥−12 2(7𝑥−3) 1 2(1 + 𝑥) 𝑑𝑦 𝑑𝑥= 𝑥(21𝑥2 + 29𝑥−12) 2(7𝑥−3) 1 2(1 + 𝑥)2 P40 Ex6 Q1&2 1st column Q3, 4, 5, 8 5 | P a g e Parametric Equations We are most familiar with Cartesian equations where 𝑥 and 𝑦 are linked e.g. 𝑦= 𝑥2 + 4 Sometimes it is more convenient to involve a third variable, 𝑡 or 𝑠 or 𝜃, and express both 𝑥 and 𝑦 in terms of this variable. Equations 𝑥= 𝑥(𝑡) and𝑦= 𝑦(𝑡) are referred to as parametric equations and 𝑡 is referred to as the parameter. We can convert from parametric equations to Cartesian (or rectangular) equations by eliminating 𝑡, 𝑠 or 𝜃. This equation is called the corresponding constraint equation 𝑥= 2𝑡−1 and 𝑦= 1 −𝑡2 Change the subject of one equation to 𝑡: 2𝑡= 𝑥+ 1 ⇒𝑡= 𝑥+1 2 Substitute for 𝑡 in the other equation: 𝑦= 1 −( 𝑥+1 2 ) 2 = 1 − 𝑥2+2𝑥+1 4 Tidy up: 𝑦= 4 4 − 𝑥2+2𝑥+1 4 = 4−𝑥2+2𝑥+1 4 = 3−𝑥2+2𝑥 4 4𝑦= 3 −𝑥2 + 2𝑥 𝑥= 2 sin 2𝜃 and 𝑦= cos 𝜃 Square and add to eliminate sin/cos: 𝑥2 + 𝑦2 = sin2 2𝜃+ cos2 𝜃 𝑥2 + 𝑦2 = (2 sin 𝜃cos 𝜃)2 + cos2 𝜃 𝑥2 + 𝑦2 = 4 sin2 𝜃cos2 𝜃+ cos2 𝜃 𝑥2 + 𝑦2 = 4(1 −cos2 𝜃) cos2 𝜃+ cos2 𝜃 Since 𝑦= cos 𝜃 then 𝑥2 = 4(1 −𝑦2)𝑦2 + 𝑦2 −𝑦2 𝑥2 = 4𝑦2 −4𝑦4 P42 Ex7A All 6 | P a g e 1st Derivative of Parametric Equations We use the Chain Rule to differentiate Parametric Equations: 𝑑𝑦 𝑑𝑥= 𝑑𝑦 𝑑𝑡× 𝑑𝑡 𝑑𝑥 where 𝑑𝑡 𝑑𝑥= 1 𝑑𝑥 𝑑𝑡 i.e. invert 𝑑𝑥 𝑑𝑡 to get 𝑑𝑡 𝑑𝑥 Find 𝑑𝑦 𝑑𝑥 when 𝑥= 4 + 4𝑡 and 𝑦= 3 −3𝑡2 𝑑𝑥 𝑑𝑡= 4 ⇒ 𝑑𝑡 𝑑𝑥= 1 4 𝑑𝑦 𝑑𝑡= −6𝑡 𝑑𝑦 𝑑𝑥= 𝑑𝑦 𝑑𝑡× 𝑑𝑡 𝑑𝑥= −6𝑡× 1 4 = −3𝑡 2 Find a formula for the gradient of the tangent to the curve whose points are given by 𝑥= 𝑎(𝑡−sin 𝑡) and 𝑦= 𝑎(1 −cos 𝑡) 𝑑𝑥 𝑑𝑡= 𝑎−𝑎cos 𝑡⇒ 𝑑𝑡 𝑑𝑥= 1 𝑎−𝑎cos 𝑡 𝑑𝑦 𝑑𝑡= 𝑎sin 𝑡 𝑑𝑦 𝑑𝑥= 𝑑𝑦 𝑑𝑡× 𝑑𝑡 𝑑𝑥= 𝑎sin 𝑡× 1 𝑎−𝑎cos 𝑡= 𝑎sin 𝑡 𝑎−𝑎cos 𝑡= sin 𝑡 1 −cos 𝑡 Find the coordinates of the points on the curve, 𝑥= 1 −𝑡2 and 𝑦= 𝑡3 + 𝑡 at which the gradient = 2. 𝑑𝑥 𝑑𝑡= −2𝑡⇒ 𝑑𝑡 𝑑𝑥= − 1 2𝑡 𝑑𝑦 𝑑𝑡= 3𝑡2 + 1 𝑑𝑦 𝑑𝑥= 𝑑𝑦 𝑑𝑡× 𝑑𝑡 𝑑𝑥= (3𝑡2 + 1) × −1 2𝑡= − 3𝑡2 + 1 2𝑡 𝑑𝑦 𝑑𝑥= 2 ⇒− 3𝑡2+1 2𝑡= 2 so 3𝑡2 + 4𝑡+ 1 = 0 (3𝑡+ 1)(𝑡+ 1) = 0 ⇒𝑡= −1, 𝑡= −1 3 For 𝑡= − 1 3 the 𝑥= 8 9 and 𝑦= − 10 27 so ( 8 9 , − 10 27) For 𝑡= −1 the 𝑥= 0 and 𝑦= −2 so (0, −2) 7 | P a g e 2nd Derivative of Parametric Equations: We make use of the Chain Rule again: 𝑑2𝑦 𝑑𝑥2 = 𝑑 𝑑𝑥( 𝑑𝑦 𝑑𝑥) = 𝑑 𝑑𝑡( 𝑑𝑦 𝑑𝑥) × 𝑑𝑡 𝑑𝑥 Find 𝑑2𝑦 𝑑𝑥2 when 𝑥= 𝑎𝑡2 and 𝑦= 2𝑎𝑡 𝑑𝑥 𝑑𝑡= 2𝑎𝑡⇒ 𝑑𝑡 𝑑𝑥= 1 2𝑎𝑡 𝑑𝑦 𝑑𝑡= 2𝑎 𝑑𝑦 𝑑𝑥= 𝑑𝑦 𝑑𝑡× 𝑑𝑡 𝑑𝑥= 2𝑎× 1 2𝑎𝑡= 1 𝑡 𝑑2𝑦 𝑑𝑥2 = 𝑑 𝑑𝑡(𝑑𝑦 𝑑𝑥) × 𝑑𝑡 𝑑𝑥= 𝑑 𝑑𝑡(1 𝑡) × 1 2𝑎𝑡= −1 𝑡2 × 1 2𝑎𝑡= − 1 2𝑎𝑡3 Find 𝑑2𝑦 𝑑𝑥2 when 𝑥= tan 𝜃 and 𝑦= sin 2𝜃 𝑑𝑥 𝑑𝜃= sec2 𝜃⇒ 𝑑𝜃 𝑑𝑥= 1 sec2 𝜃= cos2 𝜃 𝑑𝑦 𝑑𝜃= 2 cos 2𝜃 𝑑𝑦 𝑑𝑥= 𝑑𝑦 𝑑𝜃× 𝑑𝜃 𝑑𝑥= 2 cos 2𝜃× cos2 𝜃= 2 cos 2𝜃cos2 𝜃 𝑑2𝑦 𝑑𝑥2 = 𝑑 𝑑𝜃(𝑑𝑦 𝑑𝑥) × 𝑑𝜃 𝑑𝑥= 𝑑 𝑑𝜃(2 cos 2𝜃cos2 𝜃) × cos2 𝜃 Using the product/chain rule: 𝑑2𝑦 𝑑𝑥2 = cos2 𝜃(−4 sin 2𝜃× cos2 𝜃+ 2 cos 2𝜃× 2 cos 𝜃× −sin 𝜃) 𝑑2𝑦 𝑑𝑥2 = cos2 𝜃(−4 sin 2𝜃cos2 𝜃−2 cos 2𝜃sin 2𝜃) 𝑑2𝑦 𝑑𝑥2 = −2 sin 2𝜃cos2 𝜃(2 cos2 𝜃+ cos 2𝜃) 𝑑2𝑦 𝑑𝑥2 = −2 sin 2𝜃cos2 𝜃(2 cos2 𝜃+ 2 cos2 𝜃−1) 𝑑2𝑦 𝑑𝑥2 = −2 sin 2𝜃cos2 𝜃(4 cos2 𝜃−1) Bk 2 P44 Ex8A Q1a, 1d, 2a ,2d, 3, 5 8 | P a g e Velocity and Acceleration for parametric functions (Motion in a plane): Reminder: 𝑣= 𝑑𝑠 𝑑𝑡 and 𝑎= 𝑑𝑣 𝑑𝑡= 𝑑2𝑠 𝑑𝑡2 For parametric functions: 𝑆𝑝𝑒𝑒𝑑= |𝑣| = √( 𝑑𝑥 𝑑𝑡) 2 + ( 𝑑𝑦 𝑑𝑡) 2 The direction is given by an angle: tan 𝜃= 𝑦′(𝑡) 𝑥′(𝑡) 𝐴𝑐𝑐𝑒𝑙𝑒𝑟𝑎𝑡𝑖𝑜𝑛= |𝑎| = √( 𝑑2𝑥 𝑑𝑡2) 2 + ( 𝑑2𝑦 𝑑𝑡2) 2 tan 𝜃= 𝑦′′(𝑡) 𝑥′′(𝑡) as above for direction The motion of a particle is modelled by the equations 𝑥= 5𝑡 and 𝑦= 5√3𝑡−5𝑡2. 𝑥 is the horizontal displacement, 𝑦 is the vertical displacement and 𝑡 is the time. Find the position and speed of the particle as well as its direction of motion after 1 second. At 𝑡= 1 𝑥= 5 × 1 = 5 𝑦= 5√3 × 1 −5(1)2 = 5√3 −5 Position is (5,5√3 −5) 𝑑𝑥 𝑑𝑡= 5 𝑑𝑦 𝑑𝑡= 5√3 −10𝑡 At 𝑡= 1 𝑑𝑥 𝑑𝑡= 5 𝑑𝑦 𝑑𝑡= 5√3 −10(1) = 5√3 −10 𝑆𝑝𝑒𝑒𝑑= |𝑣| = √( 𝑑𝑥 𝑑𝑡) 2 + ( 𝑑𝑦 𝑑𝑡) 2 = √(5)2 + (5√3 −10) 2 = 5.2𝑚𝑠−1 Direction: tan 𝜃= 𝑦′(1) 𝑥′(1) = 5√3−10 5 = −0.2679 𝜃= 165° or 𝜃= 345° Since 𝑥′(𝑡) > 0 and 𝑦′(𝑡) < 0 then 𝜃= 345° is the direction 𝑥′(𝑡) < 0 𝑦′(𝑡) > 0 𝑥′(𝑡) > 0 𝑦′(𝑡) > 0 𝑥′(𝑡) < 0 𝑦′(𝑡) < 0 𝑥′(𝑡) > 0 𝑦′(𝑡) < 0 Bk2 P50 Ex1 Odd numbers |
10987 | https://www.holisticdentalarts.com/blog/articles-page-1/understanding-oral-galvanism-in-dental-implants-a-simplified-overview-4 | Meet The Doctor Dr. Zraiqat
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Understanding Oral Galvanism in Dental Implants: A Simplified Overview
Dental Implants & Oral Galvanism
Introduction to Oral Galvanism
Oral galvanism refers to a condition where small electric currents are generated inside the mouth due to the presence of different metals. These metals, found in dental implants and other dental materials like fillings or crowns, can interact with each other and saliva, creating an electric current. This phenomenon has yet to be well known by the general public. Still, dental professionals have studied it to understand its effects, especially in patients with metal implants like titanium.
What is Oral Galvanism?
Oral galvanism occurs when different metals in the mouth, such as titanium, gold, copper, mercury, or metal alloys, interact with saliva and produce an electric current. This process resembles a battery, where different materials exchange electrons and create electricity. In the case of oral galvanism, electricity is not decisive, but it can still cause discomfort or other symptoms in some individuals.
How Do Dental Implants Cause Oral Galvanism?
Many dental implants, including titanium implants, are made of metals or metal alloys. When these metals come into contact with saliva, a good conductor of electricity, it creates the potential for galvanic currents. The implant fixture (the part that connects to the bone), often made of titanium, may have a different electron charge than the metals used in crowns (the visible part of the implant), such as chrome or nickel. The difference in electric potential between these metals can cause a small electric current to flow through the mouth.
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Symptoms of Oral Galvanism
Oral galvanism can cause various symptoms, though they are not always easy to identify. Some common signs of oral galvanism include:
Metallic taste: People with oral galvanism often report a metallic taste in their mouth, which can be due to the presence of different metals and the current generated between them.
Tingling or burning sensation: Some patients experience tingling or burning sensations in their mouth or tongue caused by the low electricity flow in the area.
Increased saliva production: Salivation may increase as the body reacts to metal in the mouth, further encouraging the generation of electric current.
Headaches or facial pain: People sometimes report headaches or pain in the face or jaw. This could be linked to the electrical disturbances caused by the metals.
Other sensory disturbances: Some individuals may feel sensitivity to touch or temperature, which can be linked to the electrical activity caused by the galvanic reaction.
Diagnosing Oral Galvanism
Diagnosing oral galvanism can be tricky because the symptoms may overlap with other conditions. However, dentists use several methods to test for the presence of galvanic currents in the mouth:
Electrical Testing: Dentists can use special instruments to measure the electric current between metals in the mouth. If a significant current is found, it could suggest the presence of oral galvanism.
Medical History: Dentists also take a thorough medical history to rule out other potential causes for the symptoms.
Elimination Method: In some cases, the dentist may replace or remove metal restorations (such as crowns or fillings) to see if the symptoms improve, indicating that galvanism is likely the cause.
Galvanic Corrosion in Dental Implants
A primary concern related to oral galvanism is the potential for galvanic corrosion. This occurs when two metals in a moist environment, such as the mouth, begin to rust or break down. When metals corrode, it can weaken dental implants or other metal restorations. In the worst-case scenario, this could lead to the need to replace the dental work or cause discomfort due to the breakdown of the materials.
Oral Galvanism and Precancerous Lesions
One of the more concerning issues related to oral galvanism is its possible link to oral precancerous lesions. Research has suggested that the electric currents generated in the mouth could contribute to cellular changes in the surrounding tissues. While this is still an area of ongoing study, some scientists believe that constant exposure to low-level electricity could increase the risk of developing oral cancer, especially when combined with other risk factors such as smoking or excessive alcohol consumption.
How Can Oral Galvanism Be Managed?
Managing oral galvanism usually involves reducing the number of different metals in the mouth. Dentists may suggest replacing metal fillings or crowns with materials that do not conduct electricity, such as ceramics or composite resins. Additionally, they may recommend using metals that are less likely to cause a galvanic reaction with each other, such as replacing older mercury amalgam fillings with newer materials.
For patients with dental implants, particularly titanium implants, the dentist may need to monitor any symptoms closely and replace any adjacent metal dental work contributing to the galvanic reaction. Dentists sometimes suggest switching to all-ceramic crowns or other non-metallic alternatives to reduce the risk of oral galvanism.
What Can Patients Do to Prevent Oral Galvanism?
Patients can take several steps to minimize their risk of developing oral galvanism. These include:
Regular Dental Checkups: Seeing a dentist regularly ensures that any potential issues with metal restorations are detected early and can be addressed before they become problematic.
Choosing Biocompatible Materials: When doing dental work, patients can ask their dentist about using materials less likely to cause galvanic reactions, such as ceramic or composite restorations.
Avoiding Multiple Metals: Limiting the number of different metals in the mouth can reduce the risk of oral galvanism. Choosing a single type of metal for all dental work may help prevent galvanic currents from forming.
Monitoring Symptoms: If patients begin to experience symptoms like a metallic taste, tingling, or other unusual sensations, they should inform their dentist so that they can be evaluated for oral galvanism.
Conclusion
Oral galvanism is a condition caused by the interaction of different metals in the mouth, generating small electric currents. While uncommon, it can cause discomfort and contribute to other dental problems if left unchecked. Patients with metal dental implants, fillings, or crowns should be aware of the potential for oral galvanism and discuss any unusual symptoms with their dentist. With proper management and biocompatible materials, oral galvanism can often be minimized or prevented entirely.
in Articles
How to Choose the Right Holistic Dentist in Orange County, CA |
10988 | https://terra-docs.s3.us-east-2.amazonaws.com/IJHSR/Articles/volume4-issue4/2022_44_p40_Cete.pdf | © 2022 Terra Science and Education 40 RESEARCH ARTICLE DOI: 10.36838/v4i4.8 Divisibility Algorithm For Number 12 Efe Çete, Fatma Aykaç, Funda Y. Topal American Collegiate Institute, Göztepe, İnönü Cd. No:476, 35290 Konak/İzmir ABSTRACT: Divisibility, which can mean dividing an integer by another integer without a remainder, also includes finding the remainder in division. In teaching mathematics; odd, even, prime, etc. states of numbers are important for integers and related subjects, it plays a role in determining many of a number’s properties. The divisibility rules of certain numbers are known to students. However, the rules of some numbers are determined by the factors of that number. For example, in the divisibility rule by 12 states, “If the number is divisible by both 3 and 4, it is also divisible by 12.” Our aim in this paper is to express the divisibility rule by 12 differently and independently from its multipliers (3 and 4) and to create a completely new algorithm suitable in order to achieve our goal. For our purpose, firstly, it was desired to create a systematic order with numbers that are a whole multiple of 12. Then, a new divisibility algorithm was developed by working with division including remainders. It has been proven that the algorithm works flawlessly in studies using multiple different sized numbers and their resulting mathematical explanation and proofs.
KEYWORDS: Divisibility, digits, remainders, twelve, division with & without remainders. � Introduction “Divisibility” refers to dividing an integer by an integer without a remainder.¹ Divisibility is a subject that contains the information necessary to estimate the remainder in divi sion with remainders, as equally as it is about division without remainders. In the teaching of this subject, the rules are given to the students in the beginning of their studies. Teaching continues by being reinforced with examples. However, in order to improve students' mathematical skills and provide more permanent learning, students can be given numbers that are divisible by an integer without a remainder, and they can be asked to examine whether there is a certain rule between them.¹,² Thus, the logic of divisibility memorized as a set of rules can be grasped. Yet this would not stay as permanent knowledge without clear explanations rather than straightfor ward rules, which our paper won’t be doing. When studies on divisibility are examined, there are not many studies on the direct divisibility rules. Based on foreign sources, studies on divisibility rules with prime numbers lead the way.³,⁴ The reason for this is that divisibility by a non-prime number is explained by divisibility to all prime factors of the input.⁵ Even though the majority of divisibility rules are prime factor-based, there are some well-known direct rules as well. Certain numbers (such as 2, 3, 4, 5, 7, 8, 9, 10, 11, and 13) have their own divisibility rules that are used by everyone.
For example, A number: • To be divisible by 2 without a remainder, it must be even.
• To be divisible by 3 without a remainder, the sum of the digits of the number must be a multiple of 3.
• To be divisible by 4 without a remainder, the number in the last two digits of the number must be "00" or a number divisible by 4 without a remainder.
Of course, all these divisibility rules have a proof, a premise. However, even though students know these rules by heart, most of them have no idea why and how these apply.¹ Different activities can be developed to increase students’ mathematical reasoning skills of ministry topics and to enable them to produce solution algorithms. In this research, we have examined the number 12, whose rule is accepted such as: "Numbers that are divisible by both 3 and 4 are divisible by 12,” and is among the numbers that don’t have their own divisibility rules.
Purpose: During the preparation of the project, our first goal was to answer the question: "Can we find a divisibility rule for any number other than the numbers whose rule is known and used by everyone?". In line with the answers we gave to the questions asked, we aim to fill a gap in this field and to the number “12”. Just as it was recommended in its teaching, it was started by considering the numbers that are multiples of 12 and their common features.
The study was planned and implemented in order to determine the numbers divisible by 12 and to create an algorithm for finding the remainder and if it is divisible or not.
� Method In this project, an algorithm was created to calculate the divisibility of a number by 12 and what the remainder of dividing the number by 12 is.
This divisibility algorithm, which was created on the basis of existing divisibility rules, includes situations such as four operations, steps, and number values. While creating the algorithm, in the first place, a study was carried out on numbers that are a whole multiple of 12 accommodating the “1k-2k” structure of 12, and with the data obtained in "division by 12 without a remainder", it was aimed to reach the conclusion of "division by 12 with a remainder". Continuing on, the answer to the question: "what will be the remainder as a result of dividing a number by 12" was attempted to be achieved. An ijhighschoolresearch.org 41 algorithm has been developed by taking into account the place values of the numbers whose divisibility is investigated through numerous trials. While applying the algorithm, two similar ways are followed depending on whether the number is odd or even.
If the number is “even”: a. The number in the one’s digit of the number is separated from the numbers in the other digits.
E.g; A three-digit ABC number is separated from the C number on ones-digit from the three-digit ABC number to obtain the two-digit number AB and the number C. (AB and C) b. Then, half of the separated number in the ones digit (C/2) is subtracted from the remaining digits (AB two-digit num ber) (AB - C/2).
c. The number obtained after these operations is divided by 12.
d. If there is a remainder, it is "multiplied by 2 and subtracted from the smallest multiple of 12 bigger than the multiplication result".
Thus, the remainder of the division of the first number by 12 is calculated. If the remainder is "0", the number is divisible by 12 (without a remainder).
If the number is “odd”: a. The number in the one’s digit of the number is separated from the numbers in the other digits.
E.g; A three-digit ABC number is separated from the C number on ones-digit from the three-digit ABC number to obtain the two-digit number AB and the number C. (AB and C) b. Then, “half of 1 more than 1 [(C+1)/2]” of the number in the ones digit is subtracted from the remaining digits (AB two-digit number) [AB-(C+1)/2].
c. The number obtained after these operations is divided by 12.
d. If there is a remainder, it is "multiplied by 2, subtracted from the smallest multiple of 12 bigger than the multiplication result, and subtracted by 1", respectively.
Thus, the remainder of the division of the first number by 12 is calculated.
These steps can be repeated one after the other according to the number of digits of the number we want to calculate its divisibility by 12. It is important to consider the “odd”ness and “even”ness of the remaining numbers after each repetition. A much clearer explanation of the prediscussed topics with ex amples can be found in sections: 3.1, 3.2, 3.3, 5 and appendices.
Findings: First of all, we can see that the algorithm works without errors by experimenting with smaller numbers of 3-4 digits. Below are examples where the algorithm is applied only once. Then, while working with larger numbers with multiple steps, the steps of the application are explained according to the changing even and odd situations.
Application of the algorithm on an even number once: First of all, we can see that the algorithm works without errors byexperimenting with smaller numbers of 3-4 digits. Below are examples where the algorithm is applied only once. Then, while working with larger numbers with multiple steps, the steps of the application are explained according to the changing even and odd situations. Application of the algorithm on an even number once: I. The number in the ones digit of the number is separated from the other digits.
→ For the number 3146, the number is divided into two as 314 and 6 II. Half of the number left is subtracted from the remaining number.
314 III. The result obtained as a result of subtraction is divided by 12. (314–3=311) IV. The remainder obtained as a result of division is multi plied by 2. 11 x 2 = 22 V. The result of the multiplication is subtracted from the smallest positive integer of 12 bigger than the result of the multiplication itself.
24 - 22 = Result of the Operation: The remainder of the division of 3146 by 12 P.S: IV. If the number found as a result of the opera tion is a multiple of 12, we can say that the number is divisible by 12 without doing V. Operation.
Application of the algorithm on an odd number once: I. The number in the ones digit of the number is separated from the other digits. For the number 1035, the number is divided into two as 103 and 5.
II. Half of 1 more than the number left is subtracted from the separated digit.
III. The result obtained as a result of the subtraction is di vided by 12. (103–3=100) IV. The remainder obtained as a result of the division is mul tiplied by 2. 4x2=8 V. The result of the multiplication is subtracted from the smallest positive integer of 12 bigger than the result of the multiplication itself.
12-8=4 VI. In the second operation, "+1" is subtracted from the number obtained as a result of the previous step.
VII. Result of Operation: The remainder of the division of 1035 by DOI: 10.36838/v4i4.8 ijhighschoolresearch.org 42 Application of the algorithm on any number multiple times: I. The number in the ones digit of the number is separated from the other digits. For the number 689814, the number is divided into two as 68981 and 4.
II.Half of the number left is subtracted from the remaining number. (If the reserved number is odd; half of 1 more than the number is subtracted.) Here the newly formed number is odd. In the next step, half of the number will be subtracted. In this way, single or double cases will be taken into account.
Operations 1 and 2 are repeated sequentially until a 2-digit number is obtained. (Let's call this operation a step reduction operation.) (We put a "-" sign in order to remind the steps with odd numbers in order not to make mistakes.) As a result of repeated operations, the remainder is found from the division of the 2-digit number by 12.
To make the operations more understandable, we designated the remainder we found as the number c.
IV. Multiply number c by 2.
12 – 4 = 8 V. Subtract the result of the IV.th operation from the smallest multiple of 12 bigger than 2c.
12 - 8 = 4 The 4th and 5th operations are repeated in reverse (step 4-3-2-1) as the number of steps in the step reduction process.
For the number in step 4 → 4 x 2 = 8 12 - 8 = 4 For the number in step 3 → 4 x 2 = 8 12 - 8 = 4 For the number in step 2(-) → 4 x 2 = 8 12 - 8 (-1) = 3 (Here 1 is subtracted because the number in Step 2 is odd.) For the number in step 1 → 3 x 2 = 6 The number found after these operations gives the remainder of the first number (689814) divided by 12.
Result of Operation: The remainder of the division of 689814 by 12 is As you can see, techniques used for different situations are given in the algorithm in the "Method" section. Based on whether the number is odd or even, the divisibility of numbers by 12 and their remaining states are revealed as a result of simple four operations. You can examine Appendix-1 for an example of a number that is exactly divisible by 12, Appendix-2 for the example where the algorithm is applied more than once and all numbers are odd while applying, and Appendix-3 for the example where the algorithm is applied more than once, and all numbers are even while applying. Mathematical basis of the algorithm: For clarity, let's take a 3-digit number only. Let this number be ABC. If C is even: C is separated from the ABC number. If we write this mathematically; ABC= 100A+10B+C= 10.(AB)+C The number obtained by subtracting C and dividing by 10 will be AB. Then half of C is subtracted from this number AB.
If we write the equivalent of this expression: number is obtained. Let's call this last number x. According to the algorithm, this number x was divided by 12 and the remainder was found. Now let's substitute 12k+m for the number x. The goal is to find m, the remainder. This number m would be multiplied by 2 and subtracted from 12 to find the remainder from the division of ABC by 12 (just like in the 4th and 5th steps of the algorithm). Therefore, the remainder should be 12 - 2m.
Let x be this equation. Since the main purpose is the divisibility of ABC by 12, let's write the equation of ABC.
With the product of insides and outsides method: is found. Let's examine this number now. It is clear that 120k is a multiple of 12 here. Since the initially accepted number C is even, it can be seen that C itself contains a factor of 2 and 6C is a multiple of 12. The only imprecise term here is 10m. To ensure this, let's write 10m=12m-2m and rewrite the equation: The bolded part in the new equation is the remainder of ABC divided by 12. However, since the remainder cannot DOI: 10.36838/v4i4.8 ijhighschoolresearch.org 43 DOI: 10.36838/v4i4.8 be negative, 12 must be added until it becomes positive. So for m<12, the remainder of ABC dividing by 12 is 12-2m. This shows that the algorithm is working correctly. Thus, the conditions for the divisibility of even numbers by 12 are proved.
If C is odd: According to the algorithm, half of 1 more of C is taken and the operations are repeated. Let C+1=D. As the number is odd (ABC), the divisibility of the even number ABD by 12 is checked. Thus, if the remainder from the division of the ABD by 12 is y, it is obvious that the remainder of the division of the ABC by 12 will be y-1.
� Conclusion An algorithm has been created in order to perform a di visibility rule by 12 in a different way other than dividing by known and prime factors. Many examples have been studied to show and control how this algorithm we have created works in odd or even numbers. In order to increase clarity, examples of different situations are included in our study in the upcom ing sections. In order to find the mathematical basis of our algorithm, which gives error-free results in all cases, a proof is presented with the help of the number values, variables, and division-divisibility information.
As a result, since the algorithm developed allows us to obtain smaller numbers by decrementing the digits in very extensive numbers, it has introduced a brand-new method and a distinct rule for divisibility by 12.
Appendices: Appendix-1 I. The number in the ones digit of the number is separated from the other digits.
For the number 288, the number is divided into two as 28 and 8.
II.Half of the number left is subtracted from the remain ing number.
III. The result obtained as a result of subtraction is divid ed by 12. (28-4=24) � Results ● If the remainder in the division operation is "0" as a result of the 3 sequential operations; It means the number is divisible by 12.
● Thus, we can say that the number 288 is divisible by 12 (without a remainder).
Appendix-2 The algorithm is applied more than once, and all numbers are even while applying: I. The number in the ones digit of the number is separated from the other digits.
For the number 378784, the number is divided into two as 37878 and 4.
II. Half of the number left is subtracted from the remain ing number.
Operations 1 and 2 are repeated sequentially until a 2-digit number is obtained.
III. As a result of repeated operations, the remainder is found from the division by 12.
To make the operations more understandable, we called the remainder we found as the number "a".
IV. Multiply a with 2.
10x2=20 V. Subtract the result of the IVth step from the smallest multiple of 12 bigger than a2.
24-20 = 4 The 4th and 5th operations are repeated as many times as the number of steps in the IInd step.
● The number found after these operations gives the re mainder of the first number discussed by dividing it by 12.
● The remainder of the division of 378784 by Appendix-3 ijhighschoolresearch.org 44 DOI: 10.36838/v4i4.8 The algorithm is applied more than once, and all numbers are odd while applying: I. The number in the ones digit of the number is separated from the other digits.
For the number 267673, the number is divided into two as 26767 and 3.
II. Half of 1 more than the number left is subtracted from the remaining number.
Operations 1 and 2 are repeated sequentially until a 2-digit number is obtained.
III. As a result of repeated operations, the remainder is found from the division by 12.
To make the operations more understandable, we called the remainder we found as the number b.
IV. Multiply b with 2.
11x2 = 22 V. Subtract the result of the IVth step from the smallest multiple of 12 bigger than b2 and subtract by 1.
24-22 = 2, 2 - 1 = 1 The 4th and 5th operations are repeated as many times as the number of steps in the IIrd step.
● The number found after these operations gives the re mainder of the first number discussed divided by 12.
● The number found after these operations gives the re mainder of the first number discussed divided by 12.
● The remainder of the division of 267673 by � Acknowledgements Mehmet Gökhan Akbaş, Antalya Bahçeşehir Collage Par korman Campus, Math Educator, gökhan.akbas@bahcesehir.
k12.tr � References 1. Altun, M. (2014). Eğitim fakülteleri ve matematik öğretmenleri için ortaöğretimde matematik öğretimi. Bursa: Aktüel Alfa Akademi.
2. Baki, A. (2008). Kuramdan uygulamaya matematik eğitimi. Ankara: Harf Eğitim Yayıncılıg 3. Smith, F. (1971). Divisibility rules for the first fifteen primes. The Arithmetic Teacher, 18(2), 85-87. 4. Bezuszka, S. J. (1985). A test for divisibility by primes. The Arith metic Teacher, 33(2), 36- 38, 5. Peretti, A. (2005). Some notes on divisibility rules. Department of Economics. The University of Verona, workingpapers/wp2015n19.pdf. ijhighschoolresearch.org |
10989 | https://www.khanacademy.org/math/precalculus/x9e81a4f98389efdf:matrices/x9e81a4f98389efdf:intro-to-matrix-inverses/v/inverse-matrix-introduction | Inverse matrix introduction (video) | Khan Academy
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Precalculus
Course: Precalculus>Unit 7
Lesson 13: Introduction to matrix inverses
Inverse matrix introduction
Invertible matrices and determinants
Invertible matrices and transformations
Inverse matrices and matrix equations
Determine invertible matrices
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Introduction to matrix inverses
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Inverse matrix introduction
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The inverse of a square matrix is another matrix (of the same dimensions), where the multiplication (or composition) of the two matrices results in the identity matrix. This is analogous to inverse functions (if we think of matrices as functions) or reciprocal numbers (if we think of matrices as special numbers). Fascinating!Created by Sal Khan.
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Bogdan Dancu 4 years ago Posted 4 years ago. Direct link to Bogdan Dancu's post “Sal claims that f(f^-1(x)...” more Sal claims that f(f^-1(x)) = x but I'm not sure I understand the reasoning.
Any help would be appreciated! :)
I did take an example (f(x) = 2x) and saw that this is indeed, true.
However, I don't "see" why this works.
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KLaudano 4 years ago Posted 4 years ago. Direct link to KLaudano's post “Let f and f^-1 be inverse...” more Let f and f^-1 be inverse functions. Now, suppose f^-1(x) = y. This should mean that f(y) = x because f is the inverse of f^-1. If we substitute f^-1(x) for y, we get f(f^-1(x)) = x.
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Sachin Guy a year ago Posted a year ago. Direct link to Sachin Guy's post “Suppose Matrix A and B: ...” more Suppose Matrix A and B:
So, Is A ° B the same as A B? Is transformation composition the same as matrix multiplication?
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andreas 3 months ago Posted 3 months ago. Direct link to andreas's post “Yes. I find it easier for...” more Yes. I find it easier for any matrix multiplication to think about transformation. Those are the same thing
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Charlie Omega 2 years ago Posted 2 years ago. Direct link to Charlie Omega's post “Sal has a way of making s...” more Sal has a way of making square brackets that looks like the upper left angle has a minus sign.
or it's just me?
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Frenshell Ceblano 10 days ago Posted 10 days ago. Direct link to Frenshell Ceblano's post “Why don't we compose a ma...” more Why don't we compose a matrix and its inverse?
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SULAGNA NANDI 2 years ago Posted 2 years ago. Direct link to SULAGNA NANDI's post “With regular functions, w...” more With regular functions, we compose a function and its inverse to get x. f^-1(f(x)) = x.
How come, with matrices, we multiply a matrix and its inverse? Why don't we compose a matrix and its inverse?
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deramirez3 a year ago Posted a year ago. Direct link to deramirez3's post “Does this mean that the c...” more Does this mean that the conmutative property holds for the multiplication between the matrix and its inverse? That is, that there is an exception here?
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KLaudano a year ago Posted a year ago. Direct link to KLaudano's post “In general, matrix multip...” more In general, matrix multiplication is not commutative, but there are specific cases where it can be commutative. Multiplication between a matrix and its inverse is one case where it is commutative.
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iamgoingtomars2 2 years ago Posted 2 years ago. Direct link to iamgoingtomars2's post “What would be the inverse...” more What would be the inverse of a zero matrix?
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KLaudano 2 years ago Posted 2 years ago. Direct link to KLaudano's post “The inverse of a zero mat...” more The inverse of a zero matrix does not exist
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Aashay Joglekar 3 years ago Posted 3 years ago. Direct link to Aashay Joglekar's post “How can I find the invers...” more How can I find the inverse of a 33 matrix?
Aashay Joglekar
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TotallyNotAFurryÒwÓ 2 years ago Posted 2 years ago. Direct link to TotallyNotAFurryÒwÓ's post “To find the inverse of a ...” more To find the inverse of a 3x3 matrix, you can use the following steps:
Write down the 3x3 matrix you want to invert and label it as A.
Write down the identity matrix of the same size as A, and label it as I.
For example, if A is a 3x3 matrix, then I would be a 3x3 matrix with 1's on the diagonal and 0's everywhere else.
Combine A and I to form an augmented matrix [A|I].
Use elementary row operations to transform [A|I] into the reduced row echelon form [I|A^-1].
The left half of the matrix should be the identity matrix, and the right half should be the inverse of A.
If you can't get [A|I] to reduce to [I|A^-1], then the matrix A is not invertible.
It's important to note that finding the inverse of a matrix is not always possible. A matrix is invertible if and only if its determinant is nonzero. If the determinant is zero, then the matrix is said to be singular and does not have an inverse.
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Video transcript
[Instructor] We know that when we're just multiplying regular numbers we have the notion of a reciprocal. For example, if I were to take two and I were to multiply it by its reciprocal, it would be equal to one. Or if I were to just take a and a is not equal to zero and I were to multiply it by its reciprocal for any a, that is not equal to zero this will also be equal to one. And this is a number that if I multiply times anything I am just going to get that original number. So that's interesting, put in the back of our minds you learned this many, many years ago. Now we also have something that comes out of our knowledge of functions. We know that if there's some function let's call it f(x) that goes from some set, we call that our domain to some other set we call that our range, that in many cases, not all cases so this is the function f that goes from x to f(x). That in many cases, but not always the case there's another function that can take us back. And we call that other function, the inverse of f. So that if you apply the inverse of f to f(x) you're going to get back to where you were. You're going to get back to x. And we also know that it goes the other way around. For example, if you did f of f inverse of x, that too will get us back to x. So the natural question is is there an analog for an inverse of a function, or for reciprocal when we're multiplying when we think about matrices. So let's play with a few ideas. So let's imagine a matrix as a transformation, which we have already talked about it. When we think about matrices as transformations they really are functions. There are functions that are taking one point in a certain dimensional space let's say in the coordinate plane, to another point it transforms a vector to another vector. For example, let's imagine something that does a clockwise 90 degree rotation. And we know how to construct that transformation matrix which really is a function. What it does is, in our transformation matrix we want to say, what do we do with the one zero unit vector? And what also do we do with the zero one unit vector when you do that transformation? Well, if you're doing a 90 degree clockwise turn, then the one zero unit vector is going to go right over here. And so that's going to be turned into the zero negative one vector. So I'll write that right there. And then the zero one vector is going to be turned into the one zero vector. So let me write it down. This is 90 degrees clockwise and then we can think about what 90 degree counter-clockwise would look like you're going counterclockwise your original one zero vector right over here is going to go over here. It's going to become the zero one vector. So we will write that right over here. And then the zero one vector will then become this vector if you're doing a 90 degree counterclockwise rotation it's going to become the negative one zero vector negative one, zero vector. So in theory these two transformations should undo each other. If I do a transformation that first gets 90 degrees clockwise, and then I apply a transformation that's 90 degrees counter-clockwise I should get back to where we began. Now let's see what happens when we compose these two transformations and we know how to do that. We've already talked about it. We essentially multiply these two matrices. If you were to multiply zero, negative one, one, zero times zero, negative one, one, zero. What do we get? Well, let's see these, this top left this is composing two, two by two matrices is equivalent to multiplying them we've seen that in other videos. And so first we will look at this row and this column and that's going to be zero times zero plus one times one. So that is going to be one. They're going to look at this row and this column. So zero times negative one plus one times zero is just going to be zero. And then we're going to multiply this row times each of those columns. So negative one times zero is zero plus zero times one is zero and then negative one times negative one is one plus zero times zero is one. And look what happened when we took the composition of these two matrices that should undo each other we see that it does. It turns into the identity transformation or the identity matrix. We know that this matrix right over here as a transformation it's just going to map everything onto themselves. Now, this is really interesting because if we view these two by two transformation matrices as functions, we've just shown that if we call this say our first function then can call this it's inverse. And actually we use that same language when we talk about matrices. If we call this as being equal to A we would call this as being equal to A inverse. So if I were to take matrix A and I were to multiply that times its inverse I should get the identity matrix, which is right over here. And here I'm speaking in generalities I'm not even just talking about the two by two case. That should be the three by three case the four by four case so on and so forth. And we also know, that I could have defined this bottom one as A and the top one as A inverse. And so the other way should be true as well. A inverse times A should also be equivalent to the identity matrix. And so that's completely analogous to what we saw in these function examples between a function and its inverse because the other day, as we said an end by end matrix can be viewed as a transformation can be viewed as a function. And we also see that it has analogs to just how we think about multiplication. 'Cause here we could do this multiplication as a composition of transformations but we also can just view this as matrix multiplication. And so if we take a matrix and we multiply it by its inverse, that's analogous to taking a number and multiplying by its reciprocal and we get the equivalent of what in the number world would just be one, but in the matrix world is the identity matrix. 'Cause the identity matrix has this nice property that if I were to take the identity matrix and I were to multiply at times any matrix you're gonna get the original matrix again which is what we saw at least within the analog that we saw in the regular number world.
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10990 | https://www.youtube.com/watch?v=bgcAOKgz9Nc | Parametric equation of the parabola
Heather Whitehead
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Posted: 23 Aug 2017
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Transcript:
Introduction in this video I wanna have a look at parametric equations of parabolas so in Mathematica we're looking out lockers we looked at the four standard forms of the progress we had compare up down left and right and the equations that represent each of those so standard puzzlers again are the ones that have this vertex at the origin so this time we want to have a look at parametric representations all those four standard practice so if we have a look first with the one were most familiar with at a content up problem we know that it has equation x squared equals 4a y in Cartesian form but we can also represent it in with parametric equations x equals 218 y equals 80 squared and T is just our parameter again a is our focal length of our of our problem so that means that this is the X&Y coordinates so the point 2 AP a T squared is a point that lies on that problem now we can prove of those before we do that before we prove it this is on your formula sheet so it tells you that this problem is represented by these parametric equations it doesn't have the other ones but you can work them out from you so we can prove that this is the parametric representation of this by rearranging these so if we take X equal to 80 we'll call that equation 1 we'll call Y is ay T squared equation 2 so from Equation 1 if we rearrange that we could have T equals x over 2 8 if we sub that into 2 we have y equals 8 outside of x over 2a squared which is equal to a times x squared over 4a squared because when we're squaring it with us where everything that's in there now that a will cancel out with one of the AC on here and we'll just have x squared on for 8 that's still equal to Y if we rearrange that we have 4a y equals x squared which is our Cartesian form so summarize that x squared equals 4a Y Summary in a Cartesian form can be represented by X equal to 80 y equals 80 squared in parametric form and that's our concave up forever we can also look at it for our concave down and our concave right and alcohol okay left as well so if we look at each of those we know that a concave down cruller has formed x squared equals minus 4a one parametric equations will be the same here but our Y is going be negative concave right we've got Y squared is 4x so then our X and our Y I've just switched around here so our X is 80 squared and our Y is 280 and for our concave list we've got that minus 4a X so our negatives come back in front of that AET squared so they are the parametric equations for our for standard progress all right let's have a Example look at a few examples so our first one says given the parabola x equals 40 y from 2 T squared find its Cartesian coordinate sorry its Cartesian equation and the point from a problem when T equals plus and minus 2 so for the first part finest Cartesian equation we're going to call this equation 1 and this equation 2 I'm going to rearrange them to eliminate T so from 1 so this is PI from 1 we could rewrite it as T equals x on 4 and then if we sub that into two things lagging in to sub that into 2 we have y equals 2 outside of X on 4 squared okay going this we'd have 2 times x squared on 16 if we simplify our 2 and I 16 divide them both by 2 and we end up with an 8 in the denominator so we have y equals x squared on 8 or we could write that as x squared equals 8y so this is the standard form that we're more familiar with for Part B we want to find the points on the parabola when 10 equals plus or minus 2 so let's have a look at people positive 2 first then we can substitute this into our X and our Y and we have x equals 4 times 2 which is 8 and we have y equals 2 times 2 squared which would be 2 times 4 which would be 8 as well so at t equals true the point is 8 8 then we can look at T equals -2 so we'll substitute that in as well we're going to have 4 - - 4x which will give us a minus 8 and for y we're going to have 2 x minus 2 squared now minus 2 squared so this is a positive 4 so 4 times 2 will give us 8 so then our point is a minus 8 8 for our second example we are asked to find the coordinates of the focus and the equation of the directrix for the parabola x equals 12 t y equals minus 6t squared so it's true method you could use to solve this problem one is you could rewrite these and translate it into a Cartesian equation and then once it's in that standard Cartesian form you can find the focal length and then you can draw a sketch and you can find out your focus under directrix a more direct way to do things though is to recognize the general parametric form that this is in so if we have a look at those and compare them to the four different forms standard forms of a problem we can recognize that this is in the form X equal to 18 y equals minus a T squared so that's going to be a concave down parabola with a vertex at the origin we can also then compare these to this and we can see that our a is going to have to be equal to 6 a there and r6u and instead of that works in our first equation as well because 2 times 6 does it give us that 12 so then we know that we've got if we draw get roughly we've got a concave down parabola with its vertex at the origin we know that our focal length is 6 so our focus is going to be here which would be at minus 6 so our focus is going to have coordinates 0 minus 6 and the directrix has to go along here again the focal length here has to be 6 so our directrix will have equation y equals 6 for our third example is given us the equation of a parabola in Cartesian form and it wants us to write it in parametric form instead so to do that we're going to recognize that this is in the form x squared equals 4a Y and we can use that to help us find what a is so if 4a is equal to 32 because that's the coefficient of 1 then a is going to be equal to 8 now from there we can also recognize that because this is a concave up parabola it's going to satisfy at the parametric equations of the form x equal to 18 y equals AE T squared and now we know what I use our equations are going to be x equals to a will be 2 times 8 which would be 16 T and Y is going to equal 8 T squared so this is the parametric equations that represent this problem our last example is fairly similar except this time is but Y squared equals 12 X that means a concave right parabola so we know that it's in the form at the moment y squared equals 4a X which means that our for a is equal to 12 so a is equal to 3 and we also know that when we translate it into parametric form it's going to be in the form x equals a tu squared and y equals 2a T so that means for us we're going to have x equals 3t squared and Y is equal to 2 times 3 so 6t so that our solution then that's rewriting that Cartesian equation into parametric form instead so having a look at the parametric equation of standard parabolas |
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Shock Wave Interactions and Propagation
15.4.5.2 THE FREE MOLECULAR REGIME
For large Knudsen numbers the presence of the droplet does not disturb the surrounding gas. According to Schaaf and Chambré (1961), the free molecular regime starts at Kn ≃ 7, in which the Knudsen number is defined as
(15.4.64)
The expressions for the free molecular flow regime can be combined with those for the continuum regime. Details are given by Gyarmathy (1982). Somewhat simplified expressions are obtained for fυ ≪ fg and Td ≃ T∞:
(15.4.65)
(15.4.66)
(15.4.67)
(15.4.68)
The parameters Bi depend on the accommodation coefficients αi. We follow here the viewpoint of Mozurkewich (1986) that their values are close to unity. Using numerical data from Smolders and Van Dongen (1992) for water/nitrogen, typical values at 295 K are BH = 0.49, BM = 0.62 and BF = 1.48.
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Chapter
Shock Wave Interactions and Propagation
2001, Handbook of Shock WavesMARINUS E.H. VAN DONGEN
Transfer of Momentum, Mass, and Energy from Gas to Droplets; Dilute Condensable Component
15.4.5.1
: Continuum Regime
15.4.5.2
: The Free Molecular Regime
15.4.5.3
: The Transition Regime
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Book2001, Handbook of Shock WavesMARINUS E.H. VAN DONGEN
Chapter
Transport and Deposition of Aerosol Particles
2.2 Noncontinuum Considerations
In the following discussion, frequent mention will be made of the continuum and free molecular regimes. These terms are used here to distinguish between the two limiting cases characterizing the nature of the particle/gas interaction.a In the continuum limit (large particles or high gas pressures), the gas surrounding the particle appears as a continuous fluid and traditional continuum fluid dynamics apply—such as the Navier–Stokes equations for fluid motion. In the free molecular limit (small particles or low gas pressures), however, the discrete nature of the gas becomes important and individual molecule/particle collisions must be considered. Discrimination between these two regimes is made by comparing the particle diameter to the gas mean free path (which is defined as the average distance a molecule travels between collisions with other gas molecules); a dimensionless parameter known as the Knudsen number is commonly used for these comparisons:
(2.1)
: λ gas mean free path (cm) = μ/(ϕρ)
: dp particle diameter (cm)
: μ gas viscosity (g cm−1 s−1)
: ρ gas density (g/cm3) = PM/RT (for ideal gas)
: mean thermal velocity of the gas molecules (cm/s) = (8RT/πM)1/2
: R universal gas constant (8.31451 × 107 g cm2 s−2 K−1 mol−1)
: M gas molecular weight (g/mol)
: P gas pressure (dyne/cm2 or g cm−1 s−2) (1 atm = 760 torr = 1.01325 × 106 dyne/cm2)
: T gas temperature (K)
Also, ϕ is a dimensionless parameter that depends on the kinetic-theory model used to define the gas mean free path: in this work the value ϕ = 0.491 has been adopted.1 At atmospheric pressures, the mean free path is typically less than 0.1 μm. Gas mean free path is inversely proportional to pressure at constant temperature; for example, the mean free path in air is 0.674 μm at 76 torr, 6.74 μm at 7.6 torr, and 67.4 μm at 760 mtorr at 296 K. Thus, for low-pressure applications, the Knudsen number for submicron particles can be large.
A large Knudsen number (say > 10) corresponds to the free-molecular regime, while a small Knudsen number (say < 0.1) corresponds to the continuum regime. Typically, verified theoretical expressions are available in the literature for the forces acting on particles in both the continuum and free-molecular limits. Unfortunately, theoretical force expressions are difficult to formulate in the transition regime that lies between the continuum and free-molecular regimes (particle size of the order of the mean free path, Kn = 1). Instead, interpolating or correlating functions are used which go to both the continuum and free-molecular expressions in the limit and match experimental data (if available) in between.
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Book2008, Developments in Surface Contamination and Cleaning (Second Edition)Daniel J. Rader, Anthony S. Geller
Review article
A review of transport mechanisms and models for unconventional tight shale gas reservoir systems
2021, International Journal of Heat and Mass TransferSuleiman Akilu, ... Zheng Sun
5 Flow regime for gas flows in tight reservoir system
Shale as an ultra-tight rock is described by tiny multi-sized pore structures. The transport mechanisms in the shale matrix pores are thus different. Whether the flow mechanisms in the narrow channels are dominated by gas molecules interactions with the pore walls or molecule-molecule, the degree of rarefaction of the gas molecules must be determined for various operating conditions. Knudsen number, (Kn) denotes the ratio of the molecular mean free path λ (m) to the characteristics length (Lc) of the physical system is usually used to classify flow regimes as follows :
(13)
The characteristics length or macroscopic length scale corresponds to the flow path which is the pore diameter in shales based on the previous studies [88,104]. One may define the free path as the molecular motion length of gas during two successive collisions and its strength (either long or short) depends on the flow conditions. Mean free path of gas molecules, can be determined using different forms of equations. The most commonly used formula of the mean free path is given by [51,86]
(14)
and the other expression is accordingly
(15)
where, the mean free path is yin which kB (J K−1) is the Boltzmann constant (1.3805 × 10−23), T (K) the temperature, p (Pa) is pressure, and δ (m) is the collision diameter of the gas atoms or molecules, equals 0.40 nm for methane, M is (g mol−1). However, Eqs. (14) and (15) are valid for ideal hydrocarbon gases [106,107]. Real gases do not act ideally as evidence suggests; only at pressures lower than 0.1 MPa would a real gas come close to behaving like an ideal gas. Michel et al. presented a formula for the mean free path of molecules including the compressibility factor, Z from the equation of states to account for the real gas effects:
(16)
Roy et al. proposed a mean free path equation which appears to be a reformed version of Eq. (15) with compressibility factor as follows
(17)
Molecular concentration and molecular diameter of gas are fundamentally related. while Eq. (16) corrects molecular concentration parameter, it does not integrate any correction for effective molecular diameter as reflected by the real gas viscosity model of Chapman-Enskog [108,110]. An alternative model for mean free path equation was put forward with a different form of compression correction as
(18)
This model is considered to be more appropriate [28,84,111]. Zhang et al. carried out a comprehensive analysis of the appropriateness of mean free path expressions in the literature given by Eqs. (14), (15), (16), (17), and (18) over a wide range of pressures at 323.15 K. All the model values are found to be nearly the same when the pressure is less than 6 MPa, and then the values deviated at higher pressures above 30 MPa. However, Eq. (16) had the highest error with the pressure increase followed by Eqs. (14) and (17). Logically, we can expect larger discrepancy in model prediction values for Eq. (14) since λ << δ when p is high. The Eqs. (15) and (18) exhibit accurate values among all the models and were recommended for calculating mean free path for ideal gas and real gas respectively.
According to Knudsen classification, four flow regimes existed in the nanopores include continuum flow (Kn<0.001), slip flow (0.001<Kn<0.1), transition or Knudsen flow (0.1<Kn<10), and free molecular (Kn>10) [109,112,113]. Gas molecules in the pore adhere to the initial equilibrium conditions of the reservoir. When Kn<0.001, the gas flow continuously without slippage. In that case, more collisions between gas molecules occur compared with that of gas molecules and pore surfaces. Eventually, the velocity of gas close to the pore surface is considered to be zero (non-slip) . As the Knudsen number shifts to the next level 0.001<Kn<0.1, mean free path length decreases and gas molecules experience a higher frequency of collision with pore surface resulting in slippage phenomenon [89,114]. Here the velocity of the gas molecules away from the pore wall is not zero, thus the continuum assumptions are still valid. At high Knudsen number (0.1<Kn<10), the average mean free path and pore diameters are of the same order of magnitude since rarefaction becomes more pronounced as expected. Molecule-molecule collisions and molecule-wall collisions are of equal importance in this flow regime however the continuum assumption is no longer relevant [111,115]. At a very high Knudsen number (Kn>10), the gas flow in pores is free-molecular in which the collisions between gas molecules and pore wall are dominant compared to collision amongst the fluid molecules. The Knudsen number range under the shale reservoir conditions is specified in the range of 0.0001 and 6 for pressure between 0.1 and 100 MPa in which the diameter of the shale matrix pore diameter is less than 1 000 nm and temperature ranging from 273 K to 433 K, as shown in Fig. 7 . Considerable attention has been paid to modeling gas flows in the demarcated region that is peculiar to shale gas reservoirs. All four main transport patterns including the continuum flow, slip flow, transition flow, and free-molecule flow are depicted. We can see that transition and Knudsen diffusion (free molecular) flow regimes are mostly observed in nanoscale pores, the slip flow is primarily observed in microscale pores, and the Darcy (continuum) flow regime is mainly observed in microscale pores/fractures and macroscale fractures [61,100,116]. Table 1 provides the flow regimes classifications for the transport in porous media and governing equations applicable to each flow regime zone for various reservoir conditions [29,117].
Table 1. Flow regime classifications and applicable equations Agarwal et al. , Ziarani and Aguilera .
| Molecular model | Boltzmann equation (BE) | | | Collisionless BE |
--- ---
| Continuum model | Euler/Navier Stokes/Burnett equation | | DSMC | MD |
| Knudsen number | (0, 0.001) | (0.001, 0.1) | (0.1, 10) | > 10 |
| Flow regime | Free molecular | Transitional | Slip flow | Continuum |
5.1 Modeling approaches
Two main approaches are commonly used by researchers in the modeling of shale gas flows, namely the continuum and molecular approaches . In the continuum approach, macroscopic fluid properties are solved as a function of the spatial coordinate (independent properties) . While in the molecular approach, fluid is treated as a dense swarm of quantum particles in which the position, inertia, and state of the individual particles can be solved either deterministically or stochastically [75,109,119,120]. Several mathematical models have been developed from both molecular and continuum methods to describe the flow in the pore system. The ultimate goal for each modeling approach is the same for the description of velocity profiles and the pressure distribution of the fluid. Given a single-phase flow, the appropriate modeling approach must be chosen in accordance with the Knudsen number.
Generally, hydrodynamic flow equations by Euler and Navier–Stokes (NS) are used to describe the continuum flow regime (Kn<0.001) where the variations in pore pressure are very significant . In the slip regime (0.001<Kn<0.1), the NS equations remain valid, but some adjustments must be made to capture the non-zero gas velocity in the boundary near the pore surface, which is the Klinkenberg effect . In retrospect, Navier-Stokes equations are solved simply with boundary conditions for slip flows. Maxwell presented a first-order velocity slip expression for gas flow in microscale conduit as follows
(19)
where, C1 is the slip coefficient; which equals unity from Maxwell's derivation, σv is the tangential momentum accommodation coefficient (TMAC), us is the gas velocity; uw is the velocity of the solid wall; (∂u/∂n)w is the velocity gradient along the normal surface. The TMAC is the parameter signifying the extent of momentum exchanges between gas molecules and the wall surface. It accounts for momentum reductions of gas molecules collisions against the wall. That is whether gas molecules would reflect diffusively from the surface or not and varies in the range of 0.2< σv<1.2 [123-125]. The value of the TMAC depends on gas type, temperature, pressure, and wall surface smoothness [18,126]. For an ideal or perfectly smooth surface, gas molecules reflected specularly from the walls leading to the perfect slip condition (σv → 0), hence the tangential momentum of the molecules is conserved . But extremely rough surfaces have a randomly diffusive reflection of molecules; caused in large part by the divergence between the angle of incidence and reflection. Thus the average tangential momentum of the molecules becomes lost (σv = 1) . For real surface, molecules should reflect specularly and reflect diffusively so that only a small fraction of the momentum of incident molecules is lost to the wall. Another first-order slip boundary condition in the form of Maxwell's model was derived from the Boltzmann equation as :
(20)
where, is the relaxation time, k is the Boltzmann constant, T is the temperature in Kelvin, and m is the mass of the molecule. The first-order slip model yields accurate predictions at low Knudsen number Kn< 0.1. However, reports have established that the first-order slip models are not reliable for higher Knudsen number Kn > 0.1 . To improve accuracy and extend the applicability of the Navier-Stokes to higher Knudsen numbers, several attempts have been made to develop second-order slip boundary conditions. Deissler proposed the high-order slip boundary conditions as follows:
(21)
Beskok and Karniadakis define second-order slip boundary condition based on tangential momentum flux in a non-dimensional form as Maxwell's:
(22)
Wu and Boggy presented the second-order slip boundary condition using a Taylor series expansion as follows
(23)
From the available studies, no second-order slip boundary conditions are physically similar. In Eq. (21), the slip velocity accommodation coefficient term directly affects the first-order term while the effect is common to both first and second-order terms in Eq. (23). This raises the question of which type of slip model is correct. A more common system of second-order boundary condition is presented in the form of
(24)
where, C1 and C2 are first-order and second-order slip coefficient, respectively. For example, Lockerby et al. developed a second-order slip boundary condition based on Maxwell's general equation and Burnett constitutive expressions. The value of the second-order slip coefficients in their study depends on the Prandtl number and specific heat capacity of the gas which was given in a range from 0.145–0.19. If C2 → 0, then the slip velocity reduces to the first-order approximation with a similar form as Maxwell's general condition in Eq. (19). The accuracy of slip coefficients C1 and C2 determine the validity of the second-order boundary model for rarefied gas. It is shown that the NS equations can be derived from the Boltzmann equation using the Chapman-Enskog expansion . The slip coefficients from various studies are summarized in Table 2. As one can see from this table, the existing values of the second-order coefficient differ. The signs of the second-order coefficients are also different, suggesting that the slip velocity can be predicted either higher or lower. These discrepancies were considered as factors contributing to the lack of consistent results for gas flows in the slip regime. Aubert and Colin found better predictions of mass flow rate for second-order models when compared with the first-order slip model predictions which suggest that the second-order coefficient enhances the velocity of the gas at the wall surface. The precision of the second-order coefficient is key to extending the NS equations into the transition flow regime . However, the accuracy of the second-order coefficients is still a problem, let alone the comprehensive slip-flow correction model . The slip coefficients can be obtained based on a theoretical framework from kinetic theory , DSMC (direct simulation Monte Carlo) method , Lattice Boltzmann Method (LBM) [134,135] or derived experimentally [126,136-138].
Table 2. Summary of first and second order slip velocity coefficients in the literature.
| Ref | C1 | C2 |
---
| Maxwell | 1 | 0 |
| Deissler | 1 | 9/8 |
| Beskok and Karniadakis | 1 | −1/2 |
| Wu and Bogy | 2/3 | 1/4 |
| Lockerby et al. | 1 | 0.145–0.19 |
In the transition flow regime (0.1<Kn<10), the first and second-order treatment of the slip-flow boundary conditions along with the NS equations may not be accurate because of the increasing rarefaction effects. An alternative approach to extend the slip-flow models into the transition regime is the use of higher-order hydrodynamic equations. Several efforts have been made in that respect following attempts by Burnett and Woods [139-141]. The Burnett-type first higher-order hydrodynamic formulations are special equations based on modifications of the shear stress and heat flux terms capable of approximating flows in the transition regime . Burnett-type models however can suffer from non-linearity and inconsistency and failed to predict rarefied flows very correctly . Direct molecular simulation is considered a better way of solving high Knudsen layer flows rather than linear constitutive relations.
For higher Knudsen flows where rarefied gas behavior exists in the free molecular regime (Kn>10), the flow is usually solved employing collisionless Boltzmann equation. This model is a particle-based kinetic approach and properly captures the Knudsen diffusion over the free flow regime [29,57]. However, directly solving the Boltzmann equation is computationally expensive due to the complexity of the molecular collision term. Simplification of the Boltzmann equation is necessary in order to generate simpler analytical solutions to the problem. One alternative approach for rarefied gas flows is the direct simulation Monte Carlo (DSMC), formally introduced by Bird . Built on Newton technique, the platform can straightforwardly mimic and execute all mechanistic and dynamical behavior of gas flows. The utilization of stochastic algorithms resolves the collision probabilities and scattering distributions easily. It ensures most of the particle trajectories computations are dealt with, as a result, the exorbitant cost penalty is reduced significantly unlike in the molecular dynamic-type simulation approach. Thus, DSMC offers faster descriptions of non-equilibrium flows and has been useful for modeling small-scale interactions on a scale of several nanoseconds in pores . The validity covers the full range of flow regimes from continuum through free molecular . Some researchers claimed that the time penalty cost with respect to modeling low-speed flow in slip- and transition-flow regimes are exorbitantly high . Besides there is a belief that solutions of high Knudsen number gas flow problems by Boltzmann equation and DSMC method are comparable, please see .
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Journal2021, International Journal of Heat and Mass TransferSuleiman Akilu, ... Zheng Sun
Review article
A review of current progress in multiscale simulations for fluid flow and heat transfer problems: The frameworks, coupling techniques and future perspectives
2019, International Journal of Heat and Mass TransferZi-Xiang Tong, ... Wen-Quan Tao
2.3 Particle-based mesoscopic methods
In molecular dynamics, the position, velocity and interaction of every atom or molecule are known. However, the above excessively detailed information is not necessary if the interest of research is only the macroscopic behavior of the system. Then, the models in more coarse-grained scale are preferred. In the coarse-grained models, computational particle can be used to represent a group of molecules. By constituting proper evolution mechanisms of the particles, the macroscopic behavior can be recovered without the detailed information of every molecule. When dissipative force and random perturbation are added to the conservative force between particles to mimic the hydrodynamic behavior, we get the dissipative particle dynamics (DPD). When the motion of particles are discrete in discrete velocity, spatial and temporal space and the collision rules are simplified, we get the lattice gas automata (LGA) and the lattice Boltzmann method (LBM). When the collision between particles are represented by probabilistic sampling, we get the direct simulation Monte Carlo (DSMC) method. These are the typical examples of the particle based methods . Comparing with the macroscopic governing equations and constitutive laws which are based on the continuum assumption, these particle-based methods are still in the discrete level with smaller spatial and temporal scales. On the other hand, the coarse-grained particles have much larger scales than the microscopic molecules. Therefore, they can be regarded as mesoscopic methods. Since Boltzmann equation is the basis of both DSMC and LBM, in the following sections a brief introduction to the Boltzmann equation is given firstly and the introductions to these mesoscopic methods are followed.
2.3.1 Boltzmann equation
The Boltzmann equation can be written as [13,26,27]:
(9)
It describes the evolution of distribution functions f(x, c, t). Here we assume that f is the mass density distribution instead of the number density distribution. The mass of all the molecules at time t, located in the volume element [x, x + dx] and with velocities in the range [c, c + dc], equals f(x, c, t)dxdc. Similarly to Eq. (8) the macroscopic variables are the moments of f with respect to c and C = c − u . The F in Eq. (9) is an external body force, so the left-hand side of Boltzmann equation is the evolution of f in a phase-space consisted of spatial and velocity coordinates. The right-hand-side is an integral term representing the contributions from two-body collisions. Here two molecules with velocities c and c1 collide and the after-collision velocities are and . The corresponding velocity distribution functions are f(x,c,t), , and . The g is the relative velocity |c-c1|, dΩ is the element of solid angle and σ is the differential cross-section which is determined by the potential between molecules. In the equilibrium state the collision term should vanish, and one can derive the Maxwell equilibrium distribution function .
It can be seen that the Boltzmann equation is a complex nonlinear integro-differential equation which is hard to solve. The Chapman-Enskog method and Grad’s moment method are two typical methods for solving the Boltzmann equation [13,26,28]. There are three characteristics of the solution of Boltzmann equation that are important to the coupling with macroscopic and mesoscopic methods.
Firstly, in the absence of external body force, the Boltzmann equation can be rescaled into :
(10)
Here the represents dimensionless variables. The Strouhal’s number is , where Lc and tc are the characteristic length and time of the flow. The important Knudsen’s number Kn is defined as the ratio between mean free path λ and Lc:
(11)
It can be seen in Eq. (10) that Kn determines the importance of the collision term on the gas flow. Therefore Kn measures the degree of rarefaction of gases. It also measures the relative importance of the collision between molecules and the collision between molecule and boundary . The Kn < 0.001 is the continuum flow regime in which the macroscopic Navier-Stokes (N-S) equations and non-slip boundary conditions can be used. The 0.001 < Kn < 0.1 is the slip-flow regime in which the N-S equations are still valid for the bulk flow but the boundary slip should be considered. The 0.1 < Kn < 10 is the transition regime. The continuum assumption is no longer valid and the Boltzmann equation should be solved for the flow. When Kn > 10, the collision between molecules can be neglected and it is the free-molecular regime. Therefore, the Kn is an index to show when the continuum models are not valid and the mesoscopic/microscopic models and multiscale models are needed.
Secondly, the Chapman-Enskog method and Grad’s moment method generate the relations between velocity distribution functions and the macroscopic variables, which are important for the information transfer between different numerical methods. In Chapman-Enskog method, the distribution function f is expanded in power series of a parameter ε which is comparable to Kn:
(12)
The f (1) and f (2) are the deviations from the equilibrium distribution function and they can be represented by macroscopic variables and their gradients. The Euler equations, N-S equations and Burnett equations can be recovered with increasing orders of the approximation. In Grad’s moment method, the distribution function is expanded into series of tensorial Hermite polynomials as [28,29]:
(13)
Here the are functions of the molecular peculiar velocity C, and the coefficients are functions of macroscopic variables. By this expansion the Nth-order moments of the Nth-order polynomial distribution function are the same as those of the original distribution function. It can be seen that Eqs. (12) and (13) generate the relations between mesoscopic distribution functions and the macroscopic variables. Since the velocity distribution function is intrinsically related to the velocities of microscopic molecules, they can be a bridge between the methods in different scales.
Thirdly, the complex collision term can be simplified while the basic properties are still maintained. A well-known simplification is the BGK model , in which the collision is replaced by a relaxation process to the equilibrium distribution function with a relaxation time τ:
(14)
This simple collision term has the same summational invariants with the original collision term and holds the H-theorem. The hydrodynamic equations can be also recovered from Eq. (14). It gains special interests in the mesoscopic methods because it is closely related to the LBM.
2.3.2 Direct simulation Monte Carlo
Boltzmann equation suggests that if the collision term on the right-hand-side is correctly modeled, one can reproduce the hydrodynamics behavior without the detailed simulation of every molecules. Therefore, in DSMC a statistical sampling of the collision is proposed to simplify the calculation of collision between every pair of molecules.
In DSMC the computational particles which represent groups of molecules are simulated. The computational domain is divided into cells and the macroscopic values are calculated in each cell similarly to the MD. The particles experience advection and collision processes in each time step Δtm. In order to decouple the molecular motion with the collision, the Δtm should be small in comparison with the local mean collision time . Thus, the particles moves viΔtm according to their velocity vi in the advection process, and then collide with each other according to the statistical sampling.
The collision step is manipulated cell by cell. In each cell random pairs of particles are selected firstly and the probability for the occurring of collision between each pair is . Here the differential cross-section σ is a function of relative speed g and is determined by the specific molecule model and the potential that are used. (gσ)max is the maximum value of gσ in the cell. As for the number of the samplings of collision in each cell, Bird introduced the “time-counter” (TC) method and later the “no-time-counter” (NTC) method for the DSMC.
The limitation of DSMC is that its spatial and temporal scales must be resolved to the mean free path and mean collision time, and each cell should at least contain 20 particles . Therefore, although the DSMC is an efficient method for rarefied gas flows, it becomes compute-intensive for the continuum flow regime when Kn is small.
2.3.3 Lattice gas automata and lattice Boltzmann method
In DSMC the collision between computational particles is statistically sampled, but the particles still moves in continuous velocity and spatial space. In the LGA model, the computational particles can only stay on the lattice nodes and jump to the neighbor nodes in a time step. The particles have a discrete set of velocity that is linked with the lattice vectors. For example, in the original HPP LGA model the regular square lattice is used [33,34]. The lattice velocities are ci = c[sin(iπ/2), cos(iπ/2)], i = 1–4, as drawn in Fig. 1. If the space and time steps are Δx and Δt, the lattice speed is c = Δx/Δt, so the particles can move from one node to a neighbor node in a time step. At each node there are four cells corresponding to the four velocities and neighbor nodes. Each cell can only be empty or occupied by one particle, so it is a Boolean variable. In a more symmetric FHP model , the hexagonal lattice is used. Each node has six cells with lattice velocities ci = c[sin(iπ/3), cos(iπ/3)], i = 1–6.
Then, similar to the DSMC, the evolution of LGA can be divided into two processes: collision and propagation. In the propagation process the after-collision occupations move to the neighbor nodes corresponding to the directions of the velocities. In the collision process the occupation states of the cells at a node are changed according to the collision rule, which is based on the conservation of mass, momentum and energy. As shown in Fig. 1, the collision rule for HPP model is simple. When two particles with opposite velocity enter a node, rotate both particles by 90°. However, for FHP model the collision rule is more complex because each node contains 26 states and totally 212 possible state transition should be considered in the implementation. Finally, the macroscopic values in LGA are also calculated by ensemble average in each node.
The LGA has many advantages such as the absolute stability, easy boundary condition implementation and intrinsic parallelism. However, it also suffers many problems including statistical noise, violation of the Galilean invariance, exponential complexity of collision operator and spurious invariants. The remedies of these diseases lead to the dawn of the LBM [36–38]. The therapies include continuous distributions instead of Boolean variables , linearized collision operators [40,41] and Boltzmann distribution instead of Fermi-Dirac distribution. Finally, the milestone of this transformation is a simple collision operator with BGK approximation , and the most widely used single-relaxation-time LBM (also called LBGK model) is constructed [42–46].
Qian et al. proposed a family of LBGK model with b velocities on d-dimensional simple cubic lattice (called DdQb model). The evolution equation without external force is
(15)
Here fi is the discrete distribution function and the equilibrium distribution is given by:
(16)
in which wi are the weighting factors and cs is the lattice sound speed. The macroscopic variables can be therefore calculated from the moments of fi:
(17)
in which τ is the deviatoric stress tensor of LBM. The LBGK model has an equation of state . With the Chapman-Enskog expansion procedure, it can be shown that the LBGK model can reproduce N-S equation in the nearly-incompressible limit , and the relation between kinematic viscosity ν and τ is . Taking the most common D2Q9 model as an example, the discrete velocities are c0 = c(0,0), c1∼4 = c(±1,0), c(0,±1) and c5∼8 = c(±1,±1). The weighting factors are w0 = 4/9, w1∼4 = 1/9 and w5∼8 = 1/36 and the lattice sound speed is .
Although the LBM is historically evolved from LGA, quickly it is found that the LBM can be derived from continuum Boltzmann BGK Eq. (14) directly [47–50], and finally the systematic procedures for building higher-order LBM from continuum Boltzmann BGK equation by Hermite expansions and Gauss–Hermite quadrature are proposed [51,52]. Therefore, the LBM has a rigorous kinetic theory basis and inherits several advantages of LGA as a particle-based method, such as the simple implementation of complex boundary condition and parallel computing. It is also an explicit time marching scheme and does not need to solve systems of discrete equations simultaneously. In the past two decades, both the theories and applications of LBM have been extensively expanded, and it has been developed into a powerful numerical method for a wide range of problems. Several recent review papers about LBM can be suggested to the readers [53–57].
Meanwhile, the LBM still has drawbacks comparing with the continuum method. As a natural-born dynamic scheme, LBM is not a method of choice for steady state problems [53,57]. Its efficiency is further limited by small Δt, because the nearly-incompressible limitation requires a large lattice sound speed [58,59]. However, LBM is more efficient than continuum methods for unsteady problems when the time steps of LBM and continuum methods are the same, or when it is used in complex geometries [58–60]. Thus a multiscale method combining LBM with continuum methods is promising.
2.3.4 Dissipative particle dynamics and smoothed dissipative particle dynamics
The other modification of MD is the DPD method, in which the friction effects are added to the interactions to mimic the hydrodynamic behavior. In the DPD method the motion of particle is also described by Newton’s law, but a dissipative force and a random force are added besides the conservative interaction force [61–64]. The motion equations are:
(18)
(19)
Here the is still the conservative force between particles, and a soft repulsion force is usually adopted, in which and . Here rij = ri − rj and . The cut-off distance is assumed to be unit. The dissipative force and random force are give by . Here vij = vi − vj. wDand wR are weight functions and γ and σ are coefficients. ζij is a Gaussian white-noise term.
It can be seen that the dissipative force is proportional to the momentum differences and tends to relax the relative motion, while the random force keeps the system in motion. The symmetry of force terms conserves the system’s momentum. Since the force terms are only related to the relative position and relative velocity, DPD is Galilean invariant . Also, the fluctuation in mesoscale is naturally included in DPD, so DPD is a promising method for the simulation of mesoscopic complex molecules and fluid systems. It has been applied in a variety of research fields such as colloids, polymers, blood flows, and liquids with interfaces [64–66].
However, the original DPD has some disadvantages including the difficulties in applying arbitrary equations of state and specifying the transport coefficients directly. It is also absent of a physical scale. The smoothed dissipative particle dynamics (SDPD) proposed by Español and Revenga can prevent these drawbacks . In contrast with DPD which is built bottom-up from the interaction of particles, the SDPD is constructed top-down from Lagrangian description of Navier-Stokes equations. In SDPD each particle can be regarded as a moving thermodynamic subsystem with mass, entropy and volume. The volume is calculated from weight functions which include a smoothing length h that control the size of the particle and the scale of the simulation [67–69]. The thermodynamics are also introduced by choosing an internal energy function for the particles. Finally, by casting the model in GENERIC framework, the DPD-like thermal fluctuations can be also introduced into the model .
Since the SDPD is constructed from a top-down perspective, both the equations of state and transport coefficients can be specified as inputs of the model. Also, the model maintains the important mesoscopic fluctuation feature. By defining a volume for each particle, a physical scale h is specified. The effects of thermal fluctuation increase with the decreasing of the scale. Therefore, the SDPD solves the drawbacks of the DPD. This top-down approach leads to the flexibility to extend the method to multicomponent problems . The specified scale also facilitates the multiscale method based on SDPD [68,69], which will be discussed in Section 4.2.2.
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Journal2019, International Journal of Heat and Mass TransferZi-Xiang Tong, ... Wen-Quan Tao
Chapter
Pore-Scale Modeling and Simulation in Shale Gas Formations
2019, Petrophysical Characterization and Fluids Transport in Unconventional ReservoirsYang Ning, ... Guan Qin
1.1 Confinement Effect
Gas molecules in an extremely confined space encounter solid surfaces frequently and the mean free path of gas molecules becomes a similar order of magnitude with the characteristic length. The Navier-Stokes equations may not be adequate to describe the flow behavior in shale nanopores due to the failure of capturing the Knudsen layer. The Knudsen layer, shown in Fig. 3, is a kinetic layer that is always formed when gas flows over a solid surface. In an unconfined space, the Knudsen layer is usually negligible because the thickness of a Knudsen layer is much smaller than the characteristic length. However, in nanoscaled shale pores, it has the same order of magnitude as the mean free path of gas molecules and may take up a large portion of the total pore space. Within the Knudsen layer, since the collisions of gas molecules are not sufficient to reach the quasithermodynamic-equilibrium condition, the standard Navier-Stokes equations fail to describe gas slippage. In Fig. 3, it can be seen that the actual slip velocity us is smaller than the slip velocity obtained from the Navier-Stokes equations with a first-order slip boundary condition. In fact, there are many approaches that can successfully capture the flow behavior within the Knudsen layer, but we focus on the effective mean free path approach with higher-order slip boundary conditions in this section because they are common.
1.1.1 Effective mean free path/effective viscosity
Gas flow behavior in shale nanopores is significantly different from the gas flow in the bulk region due to the confinement effect. The mean free path of gas molecules becomes smaller in nanopores because gas molecules collide with solid surfaces more frequently. Therefore, an effective mean free path is introduced to include the confinement effect, which can be written as a function of the Knudsen number:
(1)
where λ∞ is the mean free path in an unconfined system defined as , in which kB is the Boltzmann constant, T is temperature, p is pressure, and d is the diameter of a gas molecule. For simplicity, let us take a gas system between two parallel plates as an example, Ψ can be derived through the probability distribution function of the free path of the gas molecules [11, 12] as
(2)
where the parallel plates are located at z = 0 and z = H. The function ∅ is defined as ∅(y) = 1 + (y − 1)e−y − y2Ei(y), where Ei(y) = ∫1∞t− 1e−ytdt. Fig. 4 shows the nondimensionalized effective mean free path (λe/λ∞) against the system location for Knudsen numbers between 0.01 and 10 that cover from the free molecular regime to the slip flow regime. One should note that λe ≈ λ∞ for the continuum regime (Kn < 0.001). For Kn ≤ 0.1, λe is close to λ∞ in the central region, but it becomes smaller near the walls. λe becomes even smaller as the Knudsen number increases, indicating that the confinement effect becomes more pronounced throughout the entire channel. Note that the exact expression of Ψ is rather complicated because it requires a numerical integration and depends on the location, and it could become impossible for complex pore geometries. To address this complexity, Guo et al. approximated Eq. (2) with
(3)
In Eq. (3), Ψ becomes a function that solely depends on Kn and it can be easily applied to practical problems. Eq. (3) has been shown to fit Eq. (2) quite well over a wide range of Kn .
According to the kinetic theory of gases, the gas viscosity is linearly dependent on the mean free path of gas molecules. Thus, we can also write the effective viscosity of gases in a confined system as
(4)
where μ∞ is the bulk viscosity in an unconfined system, and Ψ(Kn) has the same form with Eq. (3).
1.1.2 Slip boundary condition
As illustrated in Fig. 3, the actual velocity within the Knudsen layer is smaller than the macroscopic slip velocity extrapolated linearly from the Navier-Stokes solutions. In addition, Karniadakis et al. concluded that the first-order slip boundary condition should only be used for gas flows with Kn ≤ 0.1. As Kn increases, the higher-order slip boundary condition must be included in the Navier-Stokes equations in order to capture the correct slip velocity. The second-order slip boundary condition is written as
(5)
where A1 and A2 are the first-order and second-order slip coefficients, respectively. The values of the slip coefficients by different researchers can be found in . For example, Loyalka et al. [15, 16] derived by modifying the Maxwell's argument in the kinetic theory, where σ is the tangential momentum accommodation coefficient that describes the tangential momentum exchange of gas molecules with a solid surface.
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Book2019, Petrophysical Characterization and Fluids Transport in Unconventional ReservoirsYang Ning, ... Guan Qin
Chapter
Vacuum Science and Technology
2002, Materials Science of Thin Films (Second Edition)Milton Ohring
2.3 GAS TRANSPORT AND PUMPING
2.3.1 GAS FLOW REGIMES
In order to better design, modify, or appreciate reduced pressure systems, it is essential to understand concepts of gas flow (Refs. 4–6). An incomplete understanding of gas-flow limitations frequently results in less efficient system performance as well as increased expense. For example, a cheap piece of tubing having the same length and diameter dimensions as the diffusion pump to which it is attached will cut the pumping speed of the latter to approximately one-half of its rated value. In addition to the effectively higher pump cost, a continuing legacy of such a combination will be the longer pumping times required to reach a given level of vacuum each time the system is operated.
Whenever there is a net directed movement of gas in a system under the influence of attached pumps, gas flow is said to occur. Under such conditions the gas experiences a pressure drop. The previous discussion on kinetic theory of gases essentially assumed an isolated sealed system. Although gas molecules certainly move, and with high velocity at that, there is no net gas flow and no pressure gradients in such a system. When gas does flow, however, it is appropriate to distinguish between different regimes of flow. These are dependent on the geometry of the system involved as well as the pressure, temperature, and type of gas under consideration. At one extreme we have free molecular flow, which occurs at low gas densities. The chambers of high-vacuum evaporators and analytical equipment, such as Auger electron spectrometers and electron microscopes, operate within the molecular-flow regime. Here the mean distance between molecular collisions is large compared to the dimensions of the system. Kinetic theory provides an accurate picture of molecular motion under such conditions. At higher pressures, however, the mean distance between successive molecular collisions is reduced, and they predominate relative to molecule–chamber wall collisions. In this case the so-called viscous-flow regime is operative. An important example of such flow occurs in atmospheric chemical vapor-deposition reactors. Compared to molecular flow, viscous flow is quite complex. At low gas velocities the flow is laminar where layered, parallel gas flow lines may be imagined. Under these conditions the laminar flow velocity is zero at the walls of a tube, but it increases to a maximum at the tube axis. For higher flow velocities the gas layers are no longer parallel but swirl and are influenced by any obstacles in the way. In this turbulent flow range, cavities of lower pressure develop between layers. More will be said about viscous flow in Section 6.4.2.
Criteria for distinguishing between the flow regimes are based on the magnitude of the Knudsen number, Kn, which is defined by the ratio of the gas mean free path to a characteristic dimension of the system (e.g., chamber or pipe diameter, Dp), i.e., Kn = λmfp/Dp. Thus, for
(2-12a)
(2-12b)
(2-12c)
Through the use of Eq. 2-5, these limits in air can be alternately expressed by DpP < 5 × 10−3 cm-torr for molecular flow, and DpP > 5 × 10−1 cm-torr in the case of viscous flow. Figure 2-3 serves to map the dominant flow regimes on this basis. The reader should be aware that flow mechanisms may differ in various parts of the same system. Thus, whereas molecular flow will occur in the high vacuum chamber, the gas may flow viscously in the piping near the exhaust pumps.
2.3.2 CONDUCTANCE
Let us reconsider the molecular flow of gas through an orifice of area A that now separates two large chambers maintained at low pressures, P1 and P2. From a phenomenological standpoint, a flow driven by the pressure difference is expected, i.e.,
(2-13)
Here Q is defined as the gas throughput with units of pressure × volume/s (e.g., torr-liters/s). The constant of proportionality C is known as the conductance and has units of liters/s. Alternately, viewing flow through the orifice in terms of kinetic theory we note that the molecular impingements in each of the two opposing directions do not interfere with each other. Therefore, the net gas flow at the orifice plane is given by the difference or (Φ1 − Φ2)A
Through the use of Eq. 2-10 it is easily shown that the conductance of the orifice is
for air at 298 K. The reader will undoubtedly note in the choice of terms the analogy to electrical circuits. If P1 − P2 is associated with electrical pressure or voltage difference, Q may be viewed as a current. Conductances of other components where the gas flow is in the molecular regime can be similarly calculated or measured. Results for a number of important geometric shapes are given in Fig. 2-4 (Ref. 7). It should be noted that conductance is simply a function of the geometry for a specific gas at a given temperature. This is not true of viscous flow where conductance also depends on pressure. As an example, consider air flow at 298 K through a pipe of diameter D and length L connecting the vacuum chamber to the discharge pump. The molecular flow conductance is 12.2D3/L. However, for viscous flow in the same pipe the conductance is equal to 184(P2 + P1)D4/2L with P in torr. Let us assume D = 2.5 cm, L = 100 cm, and (P2 + P1)/2 = 380 torr, which is the average between atmospheric and high vacuum pressure. Substitution and evaluation gives 1.9 liters/s and 27,300 liters/s for the molecular and viscous flow conductances, respectively. This great disparity in conductance means that the geometry of vacuum components becomes increasingly important as the system pressure decreases.
When conductances are joined in series, the system conductance Csys is given by
(2-14)
Clearly Csys is lower than that of any individual conductance. When conductances are connected in parallel,
(2-15)
As an example (Ref. 8), consider the conductance of the cold trap assembly of Fig. 2-5 which isolates a vacuum system above from the pump below. Contributions to the total conductance come from:
Therefore,
upon evaluation. Strictly speaking, C3 and C4 should be multiplied by a correction factor of 1.27, which would have the effect of increasing CTotal to 51.1 liters/s. As we shall soon see, it is always desirable to have as large a conductance as possible. Clearly, the overall conductance is severely limited in this case by that of the annular region between the concentric pipes.
2.3.3 PUMPING SPEED
Pumping is the process of removing gas molecules from the system through the action of pumps. The pumping speed, S, is defined as the volume of gas passing the plane of the inlet port per unit time when the pressure at the pump inlet is P. Thus,
(2-16)
and while the throughput Q can be measured at any plane in the system, P and S refer to quantities measured at the pump inlet.
Although conductance and pumping speed have the same units and may even be equivalent numerically, they have different physical meanings. Conductance implies a component of a given geometry across which a pressure differential exists. Pumping speed refers to a given plane that may be considered to be a pump for preceding portions of the system. To apply these ideas consider a pipe of conductance C connecting a chamber at pressure P to a pump at pressure Pp as shown in Fig. 2-6a. Therefore, Q = C(P − Pp). Elimination of Q through the use of Eq. 2-16 yields
(2-17)
where Sp is the intrinsic speed at the pump inlet (Sp = Q/Pp) and S is the effective pumping speed at the base of the chamber. The latter never exceeds Sp or C and is, in fact, limited by the smaller of these quantities. If, for example, C = Sp in magnitude, then S = Sp/2 and the effective pumping speed is half the rated value for the pump. The lesson, therefore, is to keep conductances large by making ducts between the pump and chamber as short and wide as possible.
Real pumps outgas or release gas into the system as shown in Fig. 2-6b. Account may be taken of this by including an oppositely directed extra throughput term Qp such that
(2-18)
When Q = 0 the ultimate pressure of the pump, P0, is reached and Qp = SpP0. The effective pumping speed is then
(2-19)
and falls to zero as the system pressure reaches the ultimate pump pressure.
An issue of importance in vacuum systems is the time required to achieve a given pressure. The pump-down time may be calculated by noting that the throughput may be defined as the time (t) derivative of the product of volume and pressure, i.e., Q = −d(VP)/dt = −VdP/dt. Employing Eq. 2-18, we write
where Qp includes pump as well as chamber outgassing. Upon integration
(2-20)
where it is assumed that initially P = Pi. During pump-down the pressure thus exponentially decays to P0 with a time constant given by V/Sp. At high pressures where viscous flow is involved, Sp is a function of P and therefore Eq. 2-20 is not strictly applicable in such cases.
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Book2002, Materials Science of Thin Films (Second Edition)Milton Ohring
Review article
Special Issue: Multiphase Flow Research in China
2010, International Journal of Multiphase FlowMingzhou Yu, Jianzhong Lin
In order to apply the TEMOM to entire size regime, it is necessary to combine moment equations in the free molecule regime with equations in the continuum regime by harmonic mean solution or Dahneke’s solution (Otto et al., 1999). Following this solution, Yu and Lin (2009a) studied agglomerate coagulation due to Brownian coagulation in the entire size regime. They found the TEMOM model disposed by Dahneke’s solution (TEMOM-Dahneke) is more accurate than by harmonic mean solution (TEMOM-Harmonic) through comparing their results with the reference sectional model (SM) for different fractal dimensions. In the transition regime, the TEMOM-Dahneke gives more accurate results than the quadrature method of moments with three nodes (QMOM3). The mass fractal dimension was found to play an important role in determining the decay of agglomerate number and the spectrum of agglomerate size distribution. They also found the self-preserving size distribution (SPSD) theory and linear decay law for agglomerate number are only applicable to be in the free molecular regime and continuum plus near-continuum regime, but not perfectly in the transition regime.
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Journal2010, International Journal of Multiphase FlowMingzhou Yu, Jianzhong Lin
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Materials, Preparation, and Properties
2011, Comprehensive Semiconductor Science and TechnologyP. Frigeri, ... S. Franchi
3.12.3 Fundamentals of MBE
MBE is an epitaxial growth technique based on the interaction of species adsorbed from molecular beams of thermal energy on a heated crystalline substrate under UHV conditions (Cho and Arthur, 1975; Chang and Ploog, 1985; Parker, 1985; Arthur, 2002; Joyce and Joyce, 2004). The relevance of (1) UHV conditions, (2) molecular, instead of viscous, regime of growth environment, and (3) heated crystalline substrates is discussed in this section and in Section 3.12.5.1.4. The UHV conditions are required to minimize the incorporation of contaminants at the growth surface and in the epitaxial layer.
As for surface contamination that may affect the growth kinetics and the incorporation of unintentional species, to evaluate the vacuum requirements we recall that the kinetic theory of gases gives the number of atoms impinging on a unit surface area in the unit time at the background pressure P is Φ = P/(2πmkBT)1/2 molecules cm−2 s−1, where m is the atomic mass, kB is the Boltzmann’s constant, and T is the absolute temperature of the gas (Herman and Sitter, 1989). This means, for example, that in a CO background pressure of 10−6 Torr, the number of CO molecules impinging at 25 °C on the substrate surface is 3.8 × 1014 molecules cm−2 s−1. Since the surface density of atoms on a (100) surface of GaAs is approximately 6.3 × 1014 atoms cm−2 s−1, it is immediate to understand that, unless the background pressure is reduced to UHV values, the surface will be covered with a monolayer (ML) of residual gas in few seconds, in the case of unity sticking coefficient.
Another constraint to the UHV requirement arises from the necessity to grow high-purity materials. Since MBE growth rates are usually on the order of 1 ML s−1, to obtain impurity levels lower than one part per million as required by most of the applications, the background pressure should be as low as 10−12 Torr. Considering that not all residual impurities react with (and then stick to) semiconductor surfaces, one impurity per million can be effectively achieved in epitaxial layers with background pressure on the order of 10−11 Torr. As shown in Section 3.12.4, this result is obtained through the suitable choice of components and procedures aimed at reducing the background pressure of residual gases in the MBE chamber.
Molecular beams are generated by the so-called effusion cells by evaporating or sublimating high-purity materials contained in radiatively heated crucibles (Section 3.12.4). By assuming thermal equilibrium between the liquid (or solid) and the vapor into the cell, that is, by assuming that the cell aperture is small as compared to the exposed surface of the evaporating (or sublimating) material (Knudsen, 1909), the flux per unit surface and time of the molecules or atoms impinging on a substrate placed at a distance d from the crucible aperture and perpendicular to the beam is given by Φbeam = (AP/π d2) (NA/(2πMkBT))1/2, where P, NA, and M are the pressure in the cell, the Avogadro’s number, and the molecular weight of the element, respectively; A is the cell aperture area; and T is the absolute temperature (Herman and Sitter, 1989). The effusion cells that satisfy near-equilibrium conditions, called Knudsen cells, generate fluxes corresponding to growth rates of few tenths of nanometer per minute (Cho, 1995), too low for practical applications; therefore, the effectively used cells are constituted by large-aperture crucibles able to generate fluxes corresponding to growth rates in the range of few tenths of nanometer per second.
The flux generated by real cells depends on a number of parameters of the cells that describe the details of its geometry and on how it is located with respect to the substrate (Herman and Sitter, 1989); moreover, the thermal equilibrium of the constituent phases is generally not reached; therefore, the temperature dependence of the flux is experimentally measured and fitted by Φbeam ∝ exp(−ΔE/kBT), where ΔE depends on the source material.
Inside a MBE growth chamber, due to the relative positions of cells and substrates, different values of the fluxes impinge in different positions of substrates. In order to prevent such nonuniformity, the substrate is usually azimuthally rotated during growth (Wasilewski et al., 1991). However, by stopping the rotation of the substrate, it is possible to take advantage of such a nonuniformity to grow layers with thicknesses or compositions or doping levels continuously graded along a diameter of the substrate. This procedure has been used to study parameters that critically depend on the amount of deposited material (coverage); an example is the critical coverage for the 2D to 3D transition in the growth of self-assembled quantum-dots (QDs) (Colocci et al., 1997).
Another very interesting feature of MBE is that growth occurs in a molecular regime instead of that in a viscous one, typical of VPE and MOVPE. The kinetic theory of gases states that the mean free path between collisions of atoms or molecules in a gas is given by L = kBT/(21/2 πPD2), where D is the atomic or molecular diameter of the species (Herman and Sitter, 1989); for operating pressure (with beams switched on) P of 10−6–10−4 Torr, typical of MBE or MOMBE (Section 3.12.4), the mean free path is 5–0.05 m, respectively. Therefore, atoms and molecules produced by effusion cells may reach the substrate, <0.2 m apart, by nearly collision-free paths; this has two implications: (1) beams can be switched on and off by mechanical shutters that have actuation times of tenths of second, the time during which only tenths of ML are grown and (2) no boundary layer, in which the mass transport would take place only by diffusion, is formed in front of the substrate (Razeghi, 1994); boundary layers actually act as sinks or reservoirs of species after the switching on and off of said species, respectively. It is interesting to note that the availability of shutters and the absence of boundary layers, both due to the molecular regime of growth, are instrumental to the achievement of interfaces sharp on an atomic scale, if no hindrance is set by kinetic growth mechanisms (Section 3.12.5).
An aspect of crucial importance in the MBE process is the control of alloys’ composition. Growth rate measurements, obtained from RHEED intensity oscillations (Section 3.12.5.1.2), can be used to determine the composition of III–III–V solid solutions, as long as the sticking coefficients of both group-III species are equal. In this case, the mole fraction xi,j of group-III atoms i and j in the III–III–V alloy is given by the ratio xi,j = Φi,j/(Φi + Φj), where Φi and Φj are the beam fluxes of i and j atoms, respectively, and Φi,j is Φi or Φj. Φi and Φj are related to the measured growth rates Gi and Gj of the binary III–V and III′–V compounds by Gi,j = Φi,j/Ni,j. Ni and Nj are the numbers per unit volume of i and j cations in the binary compound, which are determined by their strain status, as thoroughly treated by Bocchi et al. (1999)). Here, the authors studied the composition of GaAlSb grown on GaSb and compared the results obtained by means of RHEED, Rutherford backscattering spectrometry (RBS), and High-resolution X-ray diffraction (HRXRD).
As for III–V–V′ compounds, the control of composition is more critical owing to the significant differences between the incorporation probabilities of group-V elements, which strongly depends on experimental growth conditions, such as growth temperature and group-III to group-V flux ratio, as, for example, reported by Foxon et al. (1980) for GaInAsP and InAsP alloys grown on GaAs substrate. Bosacchi et al. (1999) studied the composition of fully relaxed GaSbAs grown on GaAs; the results obtained by combined photoluminescence (PL) and RBS measurements were compared with the Ga, Sb, and As fluxes deduced by the analysis of RHEED intensity oscillations under Ga-, Sb-, and As-rich growth conditions. The reported results demonstrate a critical dependence of Sb incorporation, and then of GaSbAs composition, on the flux of each alloy constituent.
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Reference work2011, Comprehensive Semiconductor Science and TechnologyP. Frigeri, ... S. Franchi
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Flame aerosol synthesis of nanostructured materials and functional devices: Processing, modeling, and diagnostics
2016, Progress in Energy and Combustion ScienceShuiqing Li, ... Stephen D. Tse
7.3 Atomic level approach: molecular dynamic simulation
Atomic-level molecular dynamic simulation has been widely attempted to quantitatively characterize the collision–coalescence and other physical/chemical processes of nanoparticles, like heterogeneous catalytic reaction of functional nanoparticles. In this work, the collision and coalescence of two approaching nanoparticles are intensively reviewed as follows.
7.3.1 MD simulations of nanoparticle collision/coagulation
For example, the effect of long-range interactions on particle collision time is a long-existing problem in aerosol science. When the interaction between particles is on the order of the particle radius or longer, coagulation rates between particles are enhanced by increasing the capture radius of individual particles [572–576] and aggregates . An enhancement factor Wenh was proposed by Fuchs to account for long-range interaction for the collision kernel, i.e.
(7.7)
where β is the collision kernel including the effects of long-range interaction; and β0 is that only for Brownian coagulation given in Eqn. (7.5). Ouyang et al. reviewed a variety of expressions for the enhancement factor for both continuum and free-molecular regimes, for both charged and neutral particles . Apart from these two types of long-range interactions, dipole–dipole interaction on enhancing the collision kernel should not be ignored, as nanoparticles can have permanent dipoles due to asymmetry of the crystalline structure or asymmetrical distribution of ions at their surface. These dipoles have been verified by experiments on CdSe and ZnSe and more recently by MD simulation on TiO2 .
Initially, Li and co-workers compared van der Waals (vdW) and dipole–dipole forces between two TiO2 nanoparticles using MD, as shown in Fig. 25(a). The attractive dipole–dipole force is much larger than the vdW force by several orders of magnitude at long distance. Thermal fluctuations of surface ions, especially at high temperatures, can lead to increasing fluctuation of the instantaneous dipole direction and finally to a decrease in magnitude of the time-averaged dipole moment. They further applied MD to quantify the role of dipole–dipole interaction by calculating the enhancement factor Wenh during the Brownian collision of two neutral TiO2 nanoparticles in the free molecular regime . Critical capture radii rf were examined at four characteristic dipole-moment directions/orientations. As shown in Fig. 25(b), dipole–dipole interactions significantly augment the enhancement factor compared with vdW interactions. As the temperature increases, the enhancement factors regress to the values of vdW interactions because of the reduction of the particle dipole moments as a result of thermal fluctuations.
7.3.2 MD simulations of nanoparticle sintering
Upon collision of two particles, sintering (or coalescence) dominates the growth of the ‘merged’ particle and is responsible for determining its final morphology and dimensions. In physics, sintering of nanoparticles is a thermal process that involves mass transport of atoms at the nanoscale. Thus MD simulation is frequently applied to investigate the mass transport routes of sintering. Usually, classic sintering theory of micro-sized ceramic particles cannot be extended to nanoparticles because several different features exist during sintering at the nanoscale.
First, a high portion of surface atoms comprising the nanoparticles change the melting point and lead to size-dependent crystalline-core-amorphous-shell structures of nanoparticles. Various statistical values from MD simulation have been proposed to describe structures and properties of nanoparticles, as summarized in Table 6. Potential energy and Lindemann index have been used to describe the crystallization state of nanoparticles. As presented in Fig. 26, critical diameters for the phase transition of grain structures for different temperatures are predicted by MD. At small particle diameters, the melting point depends not only on the particle temperature (like for bulk material) but also on the particle size. From a thermodynamic consideration, an inverse linear relationship between the difference in temperature and the particle diameter has been proposed, i.e.,
Table 6. Summary of statistical values in MD simulation.
| Statistical value | Content | Expression |
---
| Temperature | Average kinetic energy | , f is the degree of freedom. |
| Potential energy | Melting point | , Uij is the potential energy between every two atoms i and j |
| Lindemann index | Melting point | , rij is the distance between atoms i and j, indicates an average over time. |
| Local lattice orientation | Local crystallization | X component of the normal vector of plane constructed by three neighboring Ti atoms around each O atom as detailed in Zhang et al. |
| Coordination number | Crystallization | , ni is the number of atoms around atom i within the length of crystal lattice l. |
| X-ray diffraction | Crystallization | , b = 2sinθ/λ where 2θ is the scattering angle and λ is the wavelength of the incident radiation taken to be 0.15418 nm, rij is the distance between atoms i and j, fi(b) and fj(b) is the scattering factors for atoms i and j. |
| Distribution function | (Local) crystallization | , where r is the radial distance, is the assemble average of the coordination number of atom j around atom i in a spherical shell with a radius of r and a thickness of dr. represents total radial-distribution function where ci, cj are the concentration of atoms i and j |
| Surface area(m2) | Sintering level | Meyer's method |
| Normalized surface area | Sintering level | a = (a − a0)/(a0 − af), where a0 is the initial surface, af is the initial surface |
| Gyration radius | Sintering level | , M is the total mass of particles and rc is the center-of-mass position of particles. |
| Ratio of moment of inertia | Sintering level | , Iz is the moment of inertial with respect to the Z axis which go through the centers of two particles while Ixy is the moment of inertial with respect to the axis along the direction of collision. |
| Shrinkage | Sintering level | , d1 and d2 are the diameters of the sintering particles and dcom is the distance between their centers of mass at that distance |
| Mean square displacement | Atom mobility | , is the mean square displacement of atoms in time t and d is the dimensionality of the system |
(7.8)
where Tm0 and Tmp are the melting temperatures of the bulk material and the nanoparticle, respectively; and dp is the particle diameter. Moreover, the average coordination number and local lattice orientation reveal that an amorphous shell of 0.4–0.6 nm surrounds a crystalline core, as visualized in Fig. 26(a). Such a crystalline-core-amorphous-shell structure exists due to the highly distorted surface of the nanoparticles. When the nanoparticle size is smaller than a critical diameter, the influence of the amorphous shell penetrates the entire particle, forming a different amorphous grain structure, as illustrated in Fig. 26(b). The critical transition diameters from amorphous to crystalline-core-amorphous-shell structures for TiO2 nanoparticles at different temperatures are shown by the red circles of Fig. 26, based on MD results. As such, different size-dependent structures of nanoparticles can lead to different coalescence processes; and nucleation or recrystallization may be involved during the growth of nanoparticles, which is significantly different from that for conventional sintering.
Second, different sintering mechanisms dominate for different kinds of nanoparticles. Surface diffusion and grain boundary diffusion were observed to be the main mass transport routes for particles with crystal structures, while viscous diffusion was dominant for amorphous particles. In the MD studies, several statistical values like surface area, gyration radius, ratio of moment of inertia, and shrinkage have been proposed to directly characterize the coalescence of the two nanoparticles into a single larger one. A typical sintering process between two 3 nm crystalline-core-amorphous-shell nanoparticles at 1573 K is characterized by temperature and normalized surface area, as shown in Fig. 27. The coalescence (sintering) process can be divided into four stages: (i) before contacting, the two particles rotate with a small angle to adjust their lattice planes (shown in snapshots A and B and regarded as the oriented attachment mechanism, which has also been discovered in the sintering processes of Au [586,587] and Cu particles); (ii) a sintering neck forms between two particles and grows rapidly, accompanied by rapid surface reduction and temperature increase (shown in snapshot C), which is controlled by surface diffusion; (iii) a grain boundary forms at the center of the two crystal cores, with further surface reduction (shown in snapshot D), which is controlled by both grain boundary diffusion and surface diffusion; and finally (iv) the particle shape transforms from ellipsoid to sphere. The sintering process between two amorphous particles is relatively simple, as its coalescence behaves similar to a fusion process between liquid droplets, controlled mainly by viscous diffusion. It should be noted that in all sintering processes, the high ratio of surface energy to total energy results in non-negligible heat release and temperature rise (with positive feedback) during sintering .
In the phenomenological Koch–Friedlander model,
(7.9)
where a is the surface area during sintering; af is the surface area after final coalescence; and τ is the characteristic sintering time. The aforementioned complicated mechanisms present difficulties in deriving a theoretical sintering time, except for some typical cases like the viscous-controlled sintering time of amorphous particles, which can be expressed as τ = ηdp/σ or the solid-state sintering time, which can be expressed by τ = 3kbTvp/(64σDvm) . Molecular simulation then provides an empirical equation after fitting the surface variation line.
Third, sintering can induce nucleation and recrystallization of nanoparticles. In MD simulation, Li and co-workers discovered a unique phase in the sintering of two 3 nm anatase nanoparticles , and Korparde et al. detected phase transformation in multi-particle sintering with multiple phases . Thermodynamically, the free energy of a particle at nanoscale is not only dependent on temperature but on particle size as well. Naicker et al. and Zhang et al. quantified the effect of surface energy on the stability of TiO2 nanoparticles. Phase transition may occur when particle size increases during sintering. For example, for TiO2 particles, anatase phase is less stable than rutile phase for larger particles [481,595]; and amorphous particles may transform into crystalline-core-amorphous-shell nanoparticles during particle growth. Therefore, compared with the geometrical change at the last stage, crystal structure change can lead to significant temperature increase as shown in Fig. 27. Zhou and Fichthorn used MD simulations to probe kinetics of the transformation in individual anatase nanocrystals as well as in nanocrystal aggregates . However, the details of sintering-induced phase transition in physics is still unclear in terms of whether sintering facilitates recrystallization and nucleation kinetically. Further investigation, particularly for phase transition of mixed metal oxides, is needed.
Different from collision–coalescence of nanoparticles in flame synthesis, sintering or thermal deactivation of catalyst supported immobile nanoparticles can be attributed to Ostwald ripening [597,598], which involves inter-particle transport of mobile species with larger particles growing at the expense of smaller particles due to differences in surface energy. Ostwald ripening is a near-equilibrium process of aging, redistribution, or coarsening of matter in various areas [599,600]. It usually takes several hours to transpire for immobile solid particles, and thus its characterization is beyond the current ability of MD simulations. Ostwald ripening in this area has only been investigated by MD simulation in cases of liquid argon clusters in the high-pressure vapor phase .
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10992 | https://www.expii.com/t/pascals-triangle-and-binomial-coefficients-4901 | Expii
Pascal's Triangle and Binomial Coefficients - Expii
Pascal's triangle is a beautiful way to represent the "binomial coefficients" appearing in the binomial theorem.
Explanations (6)
Pascal's Triangle
The arithmetic motivation for the Pascal's Triangle (named after a french mathematician, Blaise Pascal) is so simple that even the youngest mathematicians can understand and replicate the pattern. However, as one begins to dig into the different patterns it becomes clear how this arithmetic motivation leads to some very deep and startling applications to counting theory. Let's begin by looking at the arithmetic.
The foundation of the triangle is 1s. The first row is a 1. The second row is two 1s. After that, the first and last entry in each row is a 1. Let's look at the first 3 rows:
Image source: by Trent Tormoehlen
The question mark is the first entry that is not a 1. It is easily found by looking at the two entries it is between that are in the row above it. In this case the "?" is 1+1=2. Here is a look at the first 4 rows. Note: the general convention is for the top row to be referred to as row 0. Thus the first 4 rows are rows 0 through 3.
Image source: by Trent Tormoehlen
As you can see, the next row is 1,3,3,1. The 3s come from 1+2=3 and 2+1=3.
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Expanding Polynomials: Factorial Notation
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Ruiran Xun
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Pascal's triangle is, as its name would suggest, a triangle of numbers arranged in rows of increasing size. It turns out that Pascal's triangle has many interesting properties, some of which we will explore later in this explanation. Its applications extend to number theory, combinatorics, and counting/probability.
Let's take a look at the first few rows of the triangle: 11112113311464115101051⋯
Let's think about where these numbers come from.
With the exception of the 1 at the top of the triangle, each number is the sum of the two numbers above it. Let's look at the above rows again, this time with some numbers colored for reference:
11112113311464115101051⋯
- 0+1=1. Note that 1s populate the far left and right sides of Pascal's triangle.
- 2+1=3.
- 4+6=10.
Note that the way Pascal's triangle is constructed leads to an inherent symmetry. Any given row of the triangle is the same forwards and backwards.
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Daniel Liu
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NOTE: read the other explanations to find out what a Binomial Coefficient is.
In this explanation we will be studying something called Pascal's Triangle, and how it relates to binomial coefficients. It is essentially a triangle of numbers that look like this:
1
1
1
1
1
2
3
3
1
1
6
4
4
1
1
⋮
⋮
⋮
⋮
⋮
Note that the numbers on the left and right edge of the triangle are always 1. In addition, each number is generated by adding the two numbers to the top-left and top-right. For example, the 2 is generated by adding the 1 and the 1 directly to its top-left and right. Similarly, the 3 is created by adding the 2 and the 1.
But what can this triangle possibly have to do with binomial coefficients?
You may have noticed in the third row that (40)=1, (41)=4, (42)=6, (43)=4, and (44)=1. Following this logic, the entire triangle can be rewritten like this:
(00)
(01)
(11)
(02)
(22)
(12)
(23)
(13)
(33)
(03)
(24)
(34)
(14)
(44)
(04)
⋮
⋮
⋮
⋮
⋮
You can confirm that each of the values of the binomials equals to the corresponding value of the number on the triangle. We can see that the top number of each binomial also corresponds to the row number of pascals triangle (WARNING: the very top row with one number is considered row 0) and the bottom number of each binomial is the element number in each row (element 0 being the first number in each row).
But how can this be so? Is this all a big coincidence, or is there an underlying reason?
There are several ways to see that this is indeed not a coincidence. The argument I will use is an algebraic argument.
Note that in Pascal Triangle, we have that the sum of two adjacent numbers forms the number that is directly below and adjacent to the two numbers.
So if the first number is (nk) and the second is (nk+1), then the number that they should sum to is (n+1k+1).
Recalls that the algebraic definition of a binomial is (nk)=n!k!(n−k)!.
Thus, we want to prove that n!k!(n−k)!+n!(k+1)!(n−k−1)!=(n+1)!(k+1)!(n−k)!
Setting a common denominator to the first two fractions, we see that n!k!(n−k)!+n!(k+1)!(n−k−1)!=(k+1)n!(k+1)!(n−k)!+(n−k)n!(k+1)!(n−k)!=(n+1)n!(k+1)!(n−k)!=(n+1)!(k+1)!(n−k)!
Thus, we have proved that (nk)+(nn+1)=(n+1k+1), and this proves that the relationship between Pascal's Triangle and binomial coefficients was not a coincidence.
This also brings to light a property of binomial coefficients, known as Pascal's Identity: (nk)+(nk+1)=(n+1k+1)
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Bladyda
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Pascal's Triangle, named after Blaise Pascal, is a triangle where two numbers added up, result in the next number:
Image source: by Bladyda
The top row of the triangle, containing only a single 1, is indexed as row 0. The next row of the triangle, containing two 1s, is therefore row 1.
Any of the numbers can be calculated via the expression (na) where n is the row, and a the place on that row. For example, (43)=6.
This also means that (00)=1.
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Stefan Cuevas
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(Video) Binomial Expansion Using Pascal's Triangle
by Mathispower4u
This video by Mathispower4u demonstrates how to expand using Pascal's Triangle and the binomial theorem.
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10993 | https://www.numberempire.com/45 | | |
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| Number 45 forty five |
| Arithmetic & Divisor Properties 3^2 5 1, 3, 5, 9, 15, 45 Count of divisors Sum of divisors Previous integer Next integer Is prime Previous prime 43 Next prime 47 45th prime 197 Prime gap 2 Twin prime No Sophie Germain prime No Safe prime No Euler's totient φ(45) 24 Sum of digits 9 Digital root 9 Number Properties Fibonacci number No Lucas number No Tribonacci number No Tetranacci number No Pell number No Highly composite number No Superior highly composite number No Bell number No Catalan number No Factorial No Regular number Yes Perfect number No Palindrome No Polygonal number (s < 11)? triangular(9), hexagonal(5) Tetrahedral number No Square pyramidal number No Cubic number No Pronic number No Number in Different Bases Binary 101101 Hamming weight 4 Ternary 1200 Quaternary 231 Quinary 140 Senary 113 Septenary 63 Octal 55 Nonary 50 Decimal 45 Duodecimal 39 Hexadecimal 2d Vigesimal 25 Base 36 19 Base 62 j Roman XLV Mathematical Functions Square 2025 Square root 6.7082039324994 Cube 91125 Cube root 3.5568933044901 Fourth power 4100625 Natural logarithm 3.8066624897703 Decimal logarithm 1.6532125137753 e^n 3.4934271057485E+19 2^n 35184372088832 10^n 1.0E+45 Factorial (n!) Double factorial (n!!) 1.1962222086548E+56 |
| Factorization Tree |
| Properties of the number 45 Number 45 is a composite number. Factors of 45 are 3^2 5. Number 45 has 6 divisors: 1, 3, 5, 9, 15, 45. Sum of the divisors is 78. Number 45 is a regular number (Hamming number) and a triangular number with n=9 and a hexagonal number with n=5. Number 45 is a deficient number. Mathematical properties: Euler's totient of 45 is 24, sum of digits = 9, digital root = 9. Number representations: binary: 101101 (hamming weight: 4), ternary: 1200, octal: 55, decimal: 45, duodecimal: 39, hexadecimal: 2d, base 36: 19, base 62: j, Roman: XLV. Mathematical functions: square: 2025, cube: 91125, square root: 6.7082039324994, cube root: 3.5568933044901, factorial: 1.1962222086548E+56, natural logarithm: 3.8066624897703, decimal logarithm: 1.6532125137753. Trigonometric functions: sine = 0.85090352453412, cosine = 0.52532198881773, tangent = 1.6197751905439. |
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10994 | https://aviation.stackexchange.com/questions/26265/in-raymer-diagram-take-off-parameter-take-off-distance-what-is-the-differenc | aircraft design - In Raymer diagram "Take off parameter- Take off distance", what is the difference among "balanced field length", "over 50 ft", "ground roll"? - Aviation Stack Exchange
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In Raymer diagram "Take off parameter- Take off distance", what is the difference among "balanced field length", "over 50 ft", "ground roll"?
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Raymer in his book suggests a method to find wing loading W/S (for example we consider jet with 2 engines) (starting from specific thrust T W T W, C L t o C L t o, density/reference density etc.).
In step 1: he suggests to use graphics "Fig. 5.4 Takeoff distance estimation": you enter takeoff distance (in ordinate), so .. in abscissa you can determine Takeoff Parameter ("TOP")
Step 2: From "TOP" you can find W S=T O P⋅σ⋅C L t o⋅t W W S=T O P⋅σ⋅C L t o⋅t W
The questions are: in figure 5.4 you can find many diagrams: takeoff distance for:
a) "Balanced field Length"
b) "over 50 ft"
c) "ground roll"
Which one I have to use? What is the difference among them? Which curve you have to use, in order to calculate "take off parameter"?
Source: Daniel P. Raymer, Aircraft design: a conceptual approach. Chapter 5 Thrust-to-weight ratio and wing loading, paragraph 5.3 Wing Loading, sub-paragraph Takeoff Distance, page 88-89 and "Fig. 5.4 Takeoff distance estimation".
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edited Mar 18, 2016 at 16:22
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1 I noticed that you often ask about Raymer, so I assume youtook it as a sort of "reference book" for Aircraft Design. I would suggest you to look at other books for completeness, for example: Roskam "Airplane Design" and Torenbeek "Synthesis of Subsonic Airplane Design", Nicolai "Fundamentals of Aircraft Design".GHB –GHB 2016-03-22 16:28:55 +00:00 Commented Mar 22, 2016 at 16:28
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Let's clarify:
Large commercial airplanes take off in a way which ensures there is sufficient runway length to stop in case of aborted takeoff. There is a speed threshold, V1 speed or decision speed, that determines whether the takeoff can still be aborted, or the airplane must continue and liftoff whatever happens. See balanced field takeoff on Wikipedia.
The balanced field length is the minimum runway length to execute a balanced field takeoff for a given aircraft. The V1 speed in this case is the lowest possible. Additional runway length allows greater V1 to be selected (greater V1 means a greater portion of the ground roll with a possibility to abort the takeoff).
V1 value and balanced field length (source)
50 ft distance: This is the distance from the start of the ground roll to the point where the aircraft is airborne and 50 ft over the runway, to ensure obstacle clearance. 35 ft is another typical obstacle clearance in the US, and indeed the distance to climb to 35 ft is shorter than the previous one (a 35 ft clearance is implicitly used to compute the balanced field distance)
Takeoff roll distance and Obstacle clearance distance (source)
Using the diagram
From this slideshow which reproduces the figure you refer to:
(Source)
This depicts the relationship between takeoff parameter and different distances, either the ground roll distance or the obstacle clearance distance (35 and 50 ft) in the three types of takeoff described above.
There are three groups from left to right:
Jet using balanced field T/O (the obstacle clearance is implicitly 35 ft).
Jet not using balanced field T/O.
Propeller (not using balanced field T/O).
In these groups, there are one or two data provided:
Jet using balanced field T/O. The only distance provided is the balanced field length (distance to accelerate, abort the T/O at V1 and decelerate). You must choose the right curve, depending on the number of engines (twin, tri or four-engine).
Jet / propeller. There are two distance provided and you may need both. Ground roll is the distance until the aircraft leaves the ground (liftoff); 50 ft is the ground roll + the distance to climb to 50 ft.
See also:
What would be the ground roll and total distance to clear a 50ft obstacle given these conditions?
How is minimum runway length related to V1?
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edited Apr 13, 2017 at 12:59
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answered Mar 18, 2016 at 20:16
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Thank you Mins for your wide explanation. I have to add a question in order to understand better: if jet is not using balanced field t/o how can be found the values (couples of takeoff distance and takeoff parameter) depicted for "Ground Roll" and "Over 50 ft" you can find in proper curve? Which different aircraft conditions you have in stead of balanced field t/o conditions (or hypothesis) ?d.pensopositivo –d.pensopositivo 2016-03-20 15:42:24 +00:00 Commented Mar 20, 2016 at 15:42
@d.pensopositivo: Not sure I understand fully the comment. If this is a jet not using a BF T/O, then you look at the group in the middle (5th and 4th curves from the left). You know your T/O parameter, report it on the horizontal axis, find the projection onto the two curves, report the intersections onto the vertical axis. This gives you the ground roll distance (5th curve) and the 50 ft obstacle clearance distance (4th curve). Feel free to ask more clarification if required.mins –mins 2016-03-20 15:54:17 +00:00 Commented Mar 20, 2016 at 15:54
Thank you; to understand better, do you know in which case the jet aircraft is not using balance field takeoff? In such case, historically which data the curve "ground roll" are taken from? How are they originated?d.pensopositivo –d.pensopositivo 2016-03-20 17:21:28 +00:00 Commented Mar 20, 2016 at 17:21
@d.pensopositivo: Balanced field T/O is required for the large commercial transport aircraft category in the US (CFR Title 14 → Chapter I → Subchapter C → Part 25 → Subpart B → §25.107 et seq.) Distances are taken from the certification test flights and are included in the AFM.mins –mins 2016-03-20 18:57:04 +00:00 Commented Mar 20, 2016 at 18:57
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10995 | https://www.chegg.com/homework-help/questions-and-answers/show-external-angle-bisectors-b-c-internal-angle-bisector-concurrent-point-concurrency-cal-q219420955 | Solved Show that the external angle bisectors of ∠B and ∠C | Chegg.com
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Show that the external angle bisectors of ∠B and ∠C and the internal angle bisectorof ∠A are concurrent. The point of concurrency is called an excenter for 4ABC. Denote it withEA (as is opposite to A in some sense).(d)(10 pts) Show that the external angle bisectors of (\angle B ) and (\angle C ) and the internal angle bisector of (\angle A )
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Question: Show that the external angle bisectors of ∠B and ∠C and the internal angle bisectorof ∠A are concurrent. The point of concurrency is called an excenter for 4ABC. Denote it withEA (as is opposite to A in some sense).(d)(10 pts) Show that the external angle bisectors of (\angle B ) and (\angle C ) and the internal angle bisector of (\angle A )
Show that the external angle bisectors of ∠B and ∠C and the internal angle bisector of ∠A are concurrent. The point of concurrency is called an excenter for 4 ABC. Denote it with EA (as is opposite to A in some sense).(d)(1 0 pts) Show that the external angle bisectors of (\angle B ) and (\angle C ) and the internal angle bisector of (\angle A ) are concurrent. The point of concurrency is called an excenter for (\triangle A B C ). Denote it with ( E {A})(as is opposite to ( A ) in some sense). Similarly there are two more excenters, which you denote ( E {B}) and ( E _{C}). Like the incenter, an excenter is equidistant to the (extended) sides of the triangle (why?), but it is situated in the exterior of the triangle. Corresponding to each excenter, we'll have three excircles associated to the triangle.
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10996 | https://www.statology.org/quartile-exc-vs-quartile-inc-excel/ | Published Time: 2021-06-23T13:39:24+00:00
QUARTILE.EXC vs. QUARTILE.INC in Excel: What's the Difference?
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QUARTILE.EXC vs. QUARTILE.INC in Excel: What’s the Difference?
by Zach BobbittPosted onLast updated on June 23, 2021
Quartiles are values that split up a dataset into four equal parts.
There are three different functions you can use to calculate quartiles in Excel:
1. QUARTILE.EXC: This function uses the following process to calculate the quartiles of a dataset:
Use the median to separate the dataset into two halves.
Calculate Q1 as the median value in the lower half and Q3 as the median value in the upper half. Be sure to exclude the median of the dataset when calculating Q1 and Q3.
2. QUARTILE.INC: This function uses the following process to calculate the quartiles of a dataset:
Use the median to separate the dataset into two halves.
Calculate Q1 as the median value in the lower half and Q3 as the median value in the upper half. Be sure to includethe median of the dataset when calculating Q1 and Q3.
3. QUARTILE: This function calculates the quartiles of a dataset as well. It will return the exact same value as the QUARTILE.INC function.
For example, suppose we have the following dataset:
Dataset: 4, 6, 6, 7, 8, 12, 15, 17, 20, 21, 21, 23, 24, 27, 28
The QUARTILE.EXC function will use the median to separate the dataset into two halves and calculate Q1 and Q3 as 7 and 23, respectively:
Q1: Median of 4, 6, 6, 7, 8, 12, 15 =7
Q3: Median of 20, 21, 21, 23, 24, 27, 28 = 23
The QUARTILE.INC function will use the median to separate the dataset into two halves and calculate Q1 and Q3 as 7.5 and 22, respectively:
Q1: Median of 4, 6, 6, 7, 8, 12, 15, 17 =7.5
Q3: Median of 17, 20, 21, 21, 23, 24, 27, 28 = 22
The following example shows how to use the various QUARTILE functions in Excel.
Example: QUARTILE.EXC vs. QUARTILE.INC in Excel
Suppose we have the following dataset in Excel:
The following screenshot shows how to calculate the quartiles for the dataset using the three different quartile formulas:
Using the QUARTILE or QUARTILE.INC functions, we calculate the lower and upper quartiles as:
Q1: 7.5
Q3: 22
Conversely, using the QUARTILE.EXC function we calculate the lower and upper quartiles as:
Q1: 7
Q3: 23
When to Use QUARTILE.EXC vs. QUARTILE.INC
There is no universally “correct” way to calculate the quartiles in a dataset.
In fact, different statistical softwares use different default formulas to calculate quartiles.
The R programming language uses a formula that matches the QUARTILE.INC function in Excel.
How to Calculate Interquartile Range in R
The Python programming language uses a formula that matches the QUARTILE.INC function in Excel.
How to Calculate Interquartile Range in Python
TI-84 calculators use a formula that matches the QUARTILE.EXC function in Excel.
How to Calculate Interquartile Range on a TI-84 Calculator
Fortunately, no matter which function you use to calculate quartiles the difference between the values calculated by QUARTILE.INC and QUARTILE.EXC will be very similar in most cases.
In some cases, it’s even possible that the two functions will return the same values depending on the sequence of numbers in the dataset.
Additional Resources
STDEV.P vs. STDEV.S in Excel: What’s the Difference?
VAR.P vs. VAR.S in Excel: What’s the Difference?
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Zach Bobbitt
Hey there. My name is Zach Bobbitt. I have a Masters of Science degree in Applied Statistics and I’ve worked on machine learning algorithms for professional businesses in both healthcare and retail. I’m passionate about statistics, machine learning, and data visualization and I created Statology to be a resource for both students and teachers alike. My goal with this site is to help you learn statistics through using simple terms, plenty of real-world examples, and helpful illustrations.
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4 Replies to “QUARTILE.EXC vs. QUARTILE.INC in Excel: What’s the Difference?”
Rodolfo Oviedosays: September 28, 2021 at 5:49 pm The explanation above is not right.
Consider the following 8 numbers: 1, 2, 4, 5, 7, 8, 10, 11.
According to the rules above, the first quartile should be the median of 1, 2, 4, and 5, which is 3. However, QUARTILE.EXC and QUARTILE.INC yield 2.5 and 3.5, respectively. Reply
2. Rachel Webbsays: July 19, 2022 at 5:19 pm The TI calculator does not use the same method as the =QUARTILE.EXC nor the .INC. The difference is that the TI calculator will find the median of the lower and upper halves, excluding the median. But Excel does a weighted median of the lower and upper halves. Try for a sample size of 14 and you will see that Q1 and Q3 will be 25% of the way between the 4th and 5th values in Excel, but the TI would use 50% of the way between the 4th and 5th values. Reply
3. zach pickeringsays: September 1, 2022 at 8:02 am You said that quartile.exc calculates the same as a ti calculator but it won’t for me. In fact, I can’t get either exc or inc to get a q3 that makes sense to anyone I know that teaches statistics for the following data set. All technologies and methods that I can find will give me q3 as 42.5 but excel gives me 41.5 and 53.5 for inc and exc respectively. Any help on why I am having this issue would be appreciated.
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4. Mark Dunnesays: August 21, 2023 at 8:45 am Orange Data Mining software appears to be calculating quartiles that represent the average of quartile.exc and quartile.inc values. Is this mathematically valid? Reply
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chuck bodeen
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Posted July 18, 2013 12 yr
Posted July 18, 2013 12 yr
The Kármán Vortex Street
By Chuck Bodeen
This article has taken a long time to write… mostly because it has turned into a research project.
I have found that I knew very little about the subject when I started.
Now, after a few weeks, I still know very little about it, but more than I used to.
It is probably the most interesting article I have written… for me at least.
I became so interested in the subject that the depth of it reminded me of the research I did in graduate school.
Take warning as you read this article. In myFlying in the Ground Effect piece I mixed up Kármán and Karmann in the discussion of the Volkswagon Ghia car.
Moral: Don’t believe everything you read on the internet.
How do you know you are at the end of a research project like this? It's when you can't find answers to your latest questions, e.g. “vortex shedding from airfoils at very high Reynolds numbers”.
On the X-Plane.org website, we are supposed to talk about airplanes and flight simulation. Kármán’s theory is so pervasive that it explains a wide variety of fluid dynamic phenomena. Generally, airplanes are associated with very high Reynolds numbers. The vast amount of the literature involves LOW Re. Why? Because large size objects in very fast wind tunnels are very expensive. Reports of such work are difficult to find. As it turns out, Kármán vortex streets occur only at relatively low Reynolds numbers…which was a surprise to me.
Luckily, there are common, everyday “marvels” that are explained by vortex streets at lower Re. These include such things as musical instruments and sirens – virtually anything that makes a noise or vibrates. There are also applications in the fields of meteorology and biology.
Several of the links in this article will take you to “abstracts” of research papers. For some you can freely download the complete work. For others you must “join” a distribution website and pay for the complete report. In such cases, I was able to learn enough from the abstracts so that I vetoed the full stories.
Theory
The Vortex Street
In a reply to my piece on “Flying in the Ground Effect” a friend commented, “Hey Chuck, forgive me if I'm ill-informed, but I thought the final analysis of the failure of the Tacoma Narrows bridge was due to aeroelastic flutter”. I wrote back and pointed out that aeroelastic flutter and the Kármán vortex street are closely related. But now I know that the bridge disaster was probably not caused by a vortex street. My article about “Aeroelastic Flutter” states that the Kármán vortex street is due to Theodore von Kármán. At least his name became attached to it after he published his analysis. However, now I know that vortex streets are not the only cause of flutter.
Physics Knowledge explains the relationship between Reynolds Number and the Kármán vortex street in these drawings, which I have modified for clarity. This means that the vortex street begins to form at the “critical” Reynolds Number (about 40), and that, for Re above 400, turbulence in the flow shatters the regular pattern of the “street”. Another source puts the critical number at 90 and some say 100. The Reynolds number beyond which turbulence destroys the vortex street may be even higher than 400 in certain cases. A videofrom the Earth System Sciences laboratory at the University of California, Irvine, shows that vortex streets can form behind cylinders at Reynolds numbers around 2,000, but that turbulence dominates at Re 15,000.
This does not mean that turbulent flow will not do damage to objects in such a wake, but the concomitant oscillations of the wake and the natural vibrational frequency of the body may still result in resonance and possible destruction. Until I found this information, I was not aware of the idea that turbulence at high Reynolds numbers would mitigate the formation of vortex streets. However, because we are all, presumably, interested in fluid dynamics I’ll keep writing.
Actually, at least two scientists studied vortex shedding before Kármán.Arnulph Mallock,(1851 – 1933) published a paper in 1907 on what would later be known by Kármán’s name.Kármánacknowledged that Henri Bénard (1874–1939), French physicist, published a paper on vortex in 1908. Though others had described and made studies of the vortex patterns before von Kármán, his 1911 quantitative theoretical analysis led to the naming of this arrangement ever since as the Kármán Vortex Street and sometimes as the Bénard-Kármán Vortex Street.1a.While he was at the University of Gottingen, he noticed Ludwig Prandt's graduate student, Karl Hiemenz, trying to achieve experimental symmetrical flow of water around a circular cylinder. Hiemenz’ daily comment was "It always oscillates." Von Kármán decided to compute the stability of the vortexes and showed that only the unsymmetrical arrangement was stable. Read about it here.Experiments that are more recent, digital simulations and analysis have shown that at very low Reynolds Numbers (less than 100) such a symmetrical flow can be achieved.
Vortex streets start on one side of a body or the other due to some slight random fluctuation from upstream symmetry in physical experiments and due to rounding error or purposeful small lack of symmetry in numerical integration research. There have been numerical simulations where the vortex street did not appear until the introduction of some small asymmetric feature. The text reads, “The famous von Karman vortex street can challenge the time-stepping properties of the numerical scheme.”
A very general report by L. Rosenhead, “Vortex Systems in Wakes”, points out that the brilliant papers by Kármán and others “have been interpreted
in such a way as to suggest that they explain completely the phenomena which they purport to describe.
Both were valuable in their time, but both are now open to criticism in light of new ideas…”
Here are a couple of videos showing how vortex streets get started. This one is a numerical simulation at Reynolds number 250…very low for us airplane fans. On the other hand real fluid flow and higher Reynolds numbers are used in the second one.
The Strouhal Number
A problem lies in the fact that if the frequency of vortex shedding is the same as one of the natural frequencies of the body from which they are being cast off, a resonant vibration may occur. That frequency is given by
f = St V / L
where
St is the dimensionless “Strouhal Number”
V is the free stream speed
L is a characteristic length
St is named afterVincenc Strouhal, a Czech physicist who experimented in 1878 with wires experiencingvortex sheddingand singing in the wind.
Notice that V / Lhas the dimensions of the reciprocal of time…that is “per second” or frequency. Let’s call that Hz - for Hertz. One Hertz is one cycle per second and it is named for Heinrich Hertz.
Therefore, every physical situation has a characteristic frequency determined by the fluid speed and an important length parameter. The frequency of any oscillations is determined by the product of that characteristic frequency and the Strouhal Number.
f = St V / L = St Hz
Note also that
V / f = L / St =λ
And λ has the dimensions of length. It is the spacing of the vortexes in the “street”. For a range of moderate Reynolds numbers the Strouhal number is about 0.2, so that means that the wave length, λ, is about five times as long as L. If we are talking about a vortex street shed by a cylinder one foot in diameter, the vortices in the street will be about five feet apart.
The Strouhal Number has been found to be a function of Re (Reynolds Number) which is the ratio of inertial forces to viscous forces.
Re = ρVL / µ
where
ρis the fluid density
µ is the dynamic viscosity
Vand Lare as above
Here is a very good free app that calculates Reynolds numbers using your choice of units and assumes you on Earth and flying in air. You can even download your own copy directly from NASA. Furthermore, here is a place to find calculators for almost anything, including Reynolds number and Strouhal number.
The relationship between the Strouhal and the Reynolds Number has been studied many times.
The chart on the left is from “ENSEEIHT”, École nationale supérieure d'électronique, d'électrotechnique, , d'hydraulique et des(a French engineering school). The plot on the right is from Wikipedia about the Strouhal number. The center graph represents the combined work of several scientists.
When plotted to the same scale, the left and right charts agree substantially with the “Rough Surface” line on the one in the center. The ranges of Reynolds Numbers on the right hand chart are for the following:
"D" "V" (mph) Hz(1/sec)
Auto Radio Antenna 25 inch 10 to 100 40 to 400
Biplane wires .20 inch 10 to 100 50 to 500
GA wing 5 feet 50 to 200 .83 to 3.33
ReStf(1/sec)Λ
Auto Radio Antenna 2,000 to 20,000 0.20 8 to 80 1.25 inch
Biplane wires 1,600 to 16,000 0.20 10 to 100 1.0 inch
GA wing 1.5E+6 to 1.0E+7 0.25 0.21 to .83 20 feet
IF a vortex street were to be shed from an airplane wing it would be shed with a lower frequency and longer wave length than those from wires or antennae.
Most of the reported work is for flow around a circular cylinder of diameter, D, or other shapes, which are NOT the shapes of airfoils.
.Many of the experiments in the literature are concerned with vortex streets created by vibrating bodies.
A study of Strouhal numbers forswimming animals produced the following St v Re plot. For comparison, the same data were added to the “center” Strouhal v Reynolds number plot which was shown above. This makes the fish business a little bit “fishy” compared with flow around a cylinder.
The same experimental paper presents this drawing showing thereverseBénard- Kármán Vortex Street that explains the thrust obtained from fish and other animals as they swim:
This next picture is from an article about plan form of flapping wings and the wakes they shed.
Airplanes
Wingtip Vortices
Let us get one thing straight before we go any further. There is a difference between a Kármán vortex street and trailing edge vortices.
The trailing edge vortex exists because it is a necessary result of producing lift.
Although drawings and pictures of these may resemble each other, you can search the Internet all day and never find a frequency related to wing tip vortices.
John Denker’s drawing (from his wonderful and complete See How it Flies) shows the vortex system of an airplane. It explains that ideally, both physically and mathematically, a vortex can have no end (except for viscosity). Think of the vortex structure of an airplane as if it were a smoke ring. The system consists of three parts: the bound vortex on the wing, the trailing vortices from of each wing, and the starting vortex back on the ground. Friction in the air will damp out the energy in the starting vortex and in the part of the trailing vortices that is far behind the aircraft, but don’t follow an airliner too closely! The Airbus 380, reportedly, has trailing vortices 15 miles long.
The drawings on the top show how the vortices necessary to produce lift are shed at the wing tips. The trailing vortices are visible in the photos at the bottom.
The Boeing M2-F1
R. Dale Reedwas the man behind the development of NACA’s “M2-F1” lifting body vehicle. He worked at the Dryden Flight Research Center for 52 years and was the author of “Wingless Flight”. The M2-F1 had no wings, but it turned out that it did produce a Kármán vortex street that was potentially damaging to the ship. Read about that in Chapter 3 of his book.
Reed explains the vortex street with the practical example of driving your car in the wake of a truck. The problem was eliminated by adding air scoops to adjust the pressure distribution on the body.
The Boeing X-53
In 2002, NASA experimented with aeroelastic controls of the wings on the Boeing X-53, a modified F-18. Making the control surfaces move in different ways produced maximum performance for the plane.
Turbine Blades
A jet engine is a part of many airplanes. Jet engines have turbines; turbines have blades, which are airfoils. Physical and numerical experiments have been made which include the study of vortex shedding from these blades.
The Strouhal number for cambered turbine blades is a function of the free-stream turbulence (Tu) and the roughness of the blade surface (ks/cx).
Other
Radar Antenna Much higher Reynolds numbers were encountered in this study for a Ground Tracking Radar Antenna. The graph on the left is for the circular cylinder data from ENSEEIHT. T he two red crosses (+) are for the antenna at low (100) and high (426,000) Reynolds numbers. At the center is the vertical velocity of the wake as a function of time. Finally, on the right is the last frame of a movie that shows the development of the wake vorticity. [There are really three short movies to watch.]
Hearing
You can check out your hearing frequency range with this free app. My cat was sitting beside me when I tried it and she perked up her ears at frequencies higher than I could perceive. Here are the ranges of frequencies that can be heard by different animals:
Low Hz High Hz
Human 20 20k
Cats 55 79k
Dogs 40 60k
Bats 15 90k
Mice 1 70k
To tell you the truth, I didn’t make the grade as a Human.
Swimming animals
Fish leave vortex streets in their wake. The details depend upon the species.
The top rendering is observed in eels, bream, trout, and mullet. In three dimensions these are vortex rings. The lower picture is seen in the zebra danio, water snakes, and Kuhli leaches. The diagram is from The Journal of Experimental Biology.
Medicine
You can buy "Karman" wheel chairs at your drug store, but that’s a different Karman.
Music
Unlike the Tacoma Narrows bridge disaster and aerodynamic flutter where bodies would be better off without the vortices they shed, there are other fluid flow situations where a purposely-vibrating body creates and sheds these vortices. Musical instruments are good examples. The beautiful sounds you here from pianos, harps, guitars, violins, and others are the result of Kármán vortex streets created by vibrating strings. The frequency of each tone is determined by the length and thickness of the string affected.An oscillating reed or the musician’s lips provide the source of energy for wind instruments. The different frequencies are the result of the player changing the effective length of the tube that makes up the body of the instrument. The slide trombone is the most obvious example.
Flow Meters
Vortex flow meters measure the vibrations of the downstream vortexes caused by a barrier in the moving stream. The vibrating frequency of the vortex shedding is related to the velocity of the flow. Seethis paper for more details.
Meteorology
Here are three Kármán vortex streets in clouds. Each is caused by turbulent air from a storm flowing over an island. The one on the left is off the coast of Chili. Heard Island, close to Antarctica, produced the one in the center. On the right is Jeju Do Island, off the coast of South Korea.
On a smaller scale, vortex streets were seen at the Sendai, Japan, airport during a tsunami. You can watch a video here.
Tacoma-Narrows Bridge
The exciting forces from such a vortex shedding are responsible for the vibration of electrical transmission lines, chimneys, and suspension bridges, as exemplified in the destruction of the Tacoma-Narrows Bridge (see videos here).
In the notes Von Kármán wrote at the time, he was convinced that his vortex street was the cause of the collapse. He gave the chore of analyzing the failure to Duncan Rannie, a PhD student at the time, who later became my graduate thesis professor.
I searched the Internet for Dr. Rannie’s results, but never found them.
An excellent paper by Billah and Scanlan clearly examines all of the theories, which try to explain the bridge failure, and shows that the Karman vortex frequency and the natural frequencies of the bridge structure were not close enough for a vortex street to have been the cause of failure. Another studyby various authors comes to the same conclusion.
Conclusion
This study has only touched on the interactions between moving fluids and stationary or vibrating bodies. If you want more information, the Internet is at your free disposal. I fully expect that those readers with more education than I on this subject will find errors I have made. I look forward to your comments.
In the end, everything is good. If it is not good, it is not the end." And remember, "It's not THEM". Even though it seems like it.
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nickster
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nicksterMember
August 14, 2013 12 yr
August 14, 2013 12 yr
Fascinating. Not that I understand much of it but I saw the Tacoma narrows film in science class about 50 years ago, and the explanation was less twisted back then. but it was a recreational science class.
Lest you thought nobody reads your articles I always learn something new.
asking silly questions since 2003
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chuck bodeen
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chuck bodeenCommunity Leader
August 16, 2013 12 yr
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August 16, 2013 12 yr
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Hey there, Nickster!
I have a grandson who goes by the same nickname.
I can see from the list that several people have read most of my articles - but not this one.
It was very interesting to me to find out that Karman Vortex Street was probably NOT the direct cause of the bridge failure.
I always thought my Professor, Dr. Duncan Rannie had analyzed it with that result.
I know that's what von Karman thought - at least at first.
Still, hard to tell with such a complex structure.
Chuck
In the end, everything is good. If it is not good, it is not the end." And remember, "It's not THEM". Even though it seems like it.
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Luke173
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August 18, 2013 12 yr
August 18, 2013 12 yr
Very interessting Post!
thank you
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chuck bodeen
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chuck bodeenCommunity Leader
August 26, 2013 12 yr
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August 26, 2013 12 yr
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Glad you liked it, Luke.
Thanks for your reply.
In the end, everything is good. If it is not good, it is not the end." And remember, "It's not THEM". Even though it seems like it.
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April 14, 2018 7 yr
Hello everybody.This page on the Strouhal is very interesting.I was looking for information about possible Karman vortices emission by the wings.But there is a problem with the calculations, for example of the auto-radio antenna : the frequency f is well (St V) / D, but in scientific unit.That makes 140 to 1400 hertz !! Where am I wrong? Friendly, Bernard
In fact, I find for f, at 8 mph, 140,8 hz, wich is 17,59 times more : (this is the division of 1609 m/mile by 3600s/h and by 0,0254 m/inches.
For 80 mph, this frequency is 1408 hz which is perfectly audible. (these calculations are based on an antenna diameter of 0.25 and not 25in, which is confirmed by the Reynolds.)
Anyway, I was happy to read the anecdote about the "turning vanes" of M2-F1. I thought that these "turning vanes" had been put just to reduce the negative pressure of base ((I mean to increase the pressure and decrease the drag) .What's funny is that the Saturn V rocket also had this kind of "turning vanes".What were they used for ?
By another chance, I came across this page because I write a text on the Strouhal. The first drawing that embellishes it is that of Zorro, because it is useful to remember that the sword of Zorro emits swirls of Karman when he signs a Z (which means Zorro).
Friendly, Bernard.
Edited April 16, 2018 7 yr by Bernard de Go mars
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10998 | https://zhuanlan.zhihu.com/p/627467549 | 第十三章 图像特征Vol.1：全局特征与区域特征 - 知乎
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第十三章 图像特征Vol.1：全局特征与区域特征
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第十三章 图像特征Vol.1：全局特征与区域特征
JoeyBG
既是一只Ph.D，也是一位音乐人，一位诗人，一个疯狂的人。
来自专栏 · 信号，机器学习与雷达 · 进阶篇
106 人赞同了该文章
目录
一、Introduction
从本章开始，我们总结的内容就越来越接近非计算机专业，尤其是雷达等信号处理领域高水平的智能算法工作瞄准的方向了，也就是所谓的提供更明确的特征表示、实现更接近系统本身成像原理的“特征提取”。在阅读了大量的论文后，小伙伴们应该对当前自己的研究领域有详尽的认识，通常更好的创新算法、更高水平的论文来自于对基本原理的阐释，而不完全只是修改分类、分割、参数拟合模型的结构（当然，我们应当承认这些工作的意义，他们构筑了整个全流程研究成果的框架，只是着眼的部分不一样，达到的效果往往有差异罢了）。究其根本，我们研究的是一种“新的特征表示”、“特征提取的方法”、“特征存在的意义”，研究预处理的方法，研究更新颖的可以表示特征的数据形式，不将Raw或者原始成像的图片、视频、时序数据等直接丢给网络暴力训练。
了解集成学习的朋友们应该直到，模型推理的三大宏观阶段可分为数据层级（Data Level）、特征层级（Feature/Signature Level）和决策层级（Decision Level），在各个不同层级做创新，需要不一样的思维方式。从数据、系统层面创新，这是改变行业或者改变领域级别的创新，只有Born Genius和少数被苹果砸中的幸运儿才能做到。遗憾的是，如笔者一样，大多数的我们只是普通人，只能实现“微调的进步与试错的过程”，这之中，从特征层面的创新是“更有可能存在其意义”的工作。
本文的部分内容和图片参考自：
Krig S . Computer Vision Metrics[J]. Apress, 2014.
M. Farbod, G. Akbarizadeh, A. Kosarian, and K. Rangzan, “Optimized fuzzy cellular automata for synthetic aperture radar image edge detection,”Journal of Electronic Imaging, vol. 27, no. 01, p. 1, Feb. 2018.
Carreño Conde, Francisco, and María De Mata Muñoz. 2019. "Flood Monitoring Based on the Study of Sentinel-1 SAR Images: The Ebro River Case Study"_Water_11, no. 12: 2454.
Ma, Xiaole, Shaohai Hu, and Shuaiqi Liu. 2017. "SAR Image De-Noising Based on Shift Invariant K-SVD and Guided Filter"_Remote Sensing_9, no. 12: 1311.
M. N. Sumaiya and R. Shantha Selva Kumari, "Gabor filter based change detection in SAR images by KI thresholding," Optik, vol. 130, pp. 114–122, Feb. 2017.
在此表示感谢！下面先从背景简介开始讲起吧。
二、背景简介
1、本系列的计划
王小谟、左群声院士领衔的国防工业出版社雷达与探测前沿技术丛书中，《雷达图像解译》以660+页的厚度独树一帜，且书中大部分只介绍了针对二维成像的处理算法，足可见我们对图像特征研究的意义。基础篇第七章讲解了雷达领域的经典成像算法：【第七章 成像算法（雷达、射频信号成像） - 知乎 (zhihu.com)】。本系列会从该文的基础上往后探讨图像特征度量与描述，计划核心包括下面四章的内容：
图像特征Vol.1：全局特征与区域特征；
图像特征Vol.2：局部特征；
图像特征Vol.3：兴趣点检测与特征描述子；
图像特征Vol.4：特征学习。
内容希望不止是局限于抄书吧，一边学习一边思考，再把已经嚼碎的东西重新总结一遍，会是一个很大的收获。
2、图像特征度量分类
图像，计算机视觉特征度量是计算机视觉领域中的一个重要问题，它涉及到如何定义和计算用于描述图像或视频的特征的相似性或差异性。视觉特征度量可以用于各种任务，如图像检索、目标跟踪、人脸识别、物体识别等。其中，所谓的视觉特征通常是由计算机从图像或视频中提取出来的数值表示，例如颜色直方图、梯度直方图、形状描述符等。这些特征可以用来表示图像或视频的不同方面，如颜色、纹理、形状等。
视觉特征度量的目标是计算两个特征之间的相似度或距离。相似度度量通常用于比较两个特征的相似程度，而距离度量用于比较两个特征之间的差异程度。常见的视觉特征度量方法包括欧氏距离、余弦相似度、汉明距离、马氏距离等。下表给出了常见的计算机视觉特征度量方法：
图2-1. 计算机视觉特征度量方法分类，摘自Scott Krig著《Computer Vision Metrics - Survey, Taxonomy, and Analysis》
视觉特征度量的选择取决于任务的具体要求和特征的性质。在实际应用中，需要根据具体情况选择最适合的度量方法，以提高检测、识别、分割、定位跟踪、参数拟合等任务的准确性和效率。在本章中提及的区域特征中，包含了纹理区域度量、统计区域度量两类；在本章中提及的全局特征中，主要针对统计区域度量中设计全局统计量计算的部分和基空间度量展开。
三、纹理区域度量
纹理是图像中重要的视觉特征之一，它通常指由像素之间的亮度、颜色、形状等局部特征构成的、重复出现的图案或结构。在计算机视觉中，纹理区域度量是指对图像中的纹理区域进行特征提取和相似性度量的过程。纹理区域度量的目的是衡量图像中纹理区域之间的相似性或差异性，以在图像检索、图像分类、目标跟踪等应用中进行区分和匹配。常见的纹理区域度量方法包括以下 7 种：
边缘特征；
互相关特征；
Fourier谱、小波谱特征；
共生矩阵、Haralick特征与扩展SDM特征；
Laws纹理特征；
局部二值模式（Local Binary Pattern，LBP）；
动态纹理。
接下来，我们逐个展开总结其度量特征的定义、数学计算理论的细节和一些简单的应用举例。个别比较经典的特征会给出图示。
1、边缘特征
在每个像素处计算梯度 G(d) ，选择适当的梯度算子 G() 并选择适当的卷积核大小或距离 d 以表征图像上的区域或全局的边缘特征。改变 d 或卷积核的大小可以获得不同的度量特征，另外，将每个边缘的梯度方向分组为一个直方图，可以计算边缘的走向分布特性。图像边缘检测的常用方法有：
（1）Sobel算子：
Sobel算子是一种基于梯度的边缘检测算法，它通过计算图像中每个像素的梯度值，得到图像中的边缘信息。Sobel算子通常通过卷积操作来实现，其计算公式如下：
G_x = \begin{bmatrix} -1 & 0 & 1\ -2 & 0 & 2\ -1 & 0 & 1 \end{bmatrix} I,\quad G_y = \begin{bmatrix} -1 & -2 & -1\ 0 & 0 & 0\ 1 & 2 & 1 \end{bmatrix} I,\quad G = \sqrt{G_x^2 + G_y^2}\ 其中， G_x 和 G_y 分别代表图像在 x 和 y 方向上的梯度值， G 是两者的平方和的平方根， I 是图像矩阵。
（2）Canny算子：
Canny算子是一种基于多阶段的边缘检测算法，它通过滤波、非极大值抑制、双阈值分割等步骤来检测图像中的边缘。为了利用Canny算子实现边缘检测，我们首先对图像进行高斯滤波：
G(x,y) = \frac{1}{2\pi\sigma^2} e^{-\frac{x^2+y^2}{2\sigma^2}} I(x,y)\其中， G(x,y) 是高斯滤波器的输出图像， \sigma 是高斯滤波器的标准差， I(x,y) 是输入图像矩阵。而后，我们计算梯度幅值和方向：
G_x = \begin{bmatrix} -1 & 0 & 1\ -2 & 0 & 2\ -1 & 0 & 1 \end{bmatrix} G,\quad G_y = \begin{bmatrix} -1 & -2 & -1\ 0 & 0 & 0\ 1 & 2 & 1 \end{bmatrix} G,\quad M(x,y) = \sqrt{G_x^2 + G_y^2} \quad \theta(x,y) = \arctan\left(\frac{G_y}{G_x}\right)\这一步和Sobel算子计算是很类似的， G_x 和 G_y 分别代表图像在 x 和 y 方向上的梯度值， M(x,y) 是梯度幅值， \theta(x,y) 是梯度方向。然后，对梯度幅值做非极大值抑制：
M_\mathrm{nms}(x,y) = \begin{cases} M(x,y) & \text{if } M(x,y) \geq M(x+\Delta x, y+\Delta y) \text{ and } M(x,y) \geq M(x-\Delta x, y-\Delta y)\ 0 & \text{otherwise} \end{cases}\其中， M_\mathrm{nms}(x,y) 是经过非极大值抑制后的梯度幅值， \Delta x 和 \Delta y 是沿着梯度方向的单位步长。最后，利用双阈值分割实现边缘像素的筛选：
M_\mathrm{th}(x,y) = \begin{cases} M_\mathrm{nms}(x,y) & \text{if } M_{nms}(x,y) \geq T_1\ 0 & \text{if } M_{nms}(x,y) < T_2\ M_\mathrm{conn}(x,y) & \text{otherwise} \end{cases}\其中， M_\mathrm{th}(x,y) 是经过双阈值分割后的梯度幅值， T_1 和 T_2 分别是高低阈值超参数， M_\mathrm{conn}(x,y) 是在高阈值检测后与边缘相连的像素。以上公式中， 代表卷积操作， G 是经过高斯滤波后的图像， \arctan 是反正切函数。虽然Canny算子方法相对最复杂，但确是研究者最喜欢用的高精度边缘检测算法之一。
（3）LoG算子：
LoG算子是一种基于拉普拉斯算子的边缘检测算法，它通过对图像进行高斯滤波和拉普拉斯变换，得到图像中的边缘信息。LoG算子的计算公式如下：
\mathrm{LoG}(x,y) = -\frac{1}{\pi\sigma^4}\left[1 - \frac{x^2+y^2}{2\sigma^2}\right]e^{-\frac{x^2+y^2}{2\sigma^2}}I(x,y)\其中， x 和 y 分别代表图像中的像素位置， \sigma 是高斯滤波器的标准差。阈值检测方法的计算公式为：
M(x,y) = \begin{cases} \mathrm{LoG}(x,y) & \text{if } |\mathrm{LoG}(x,y)| \geq T\ 0 & \text{otherwise} \end{cases}\其中， M(x,y) 是经过阈值检测后的图像，所得就是边缘检测的结果，公式里的 T 是阈值超参数。
（4）Robert算子：
Robert算子是一种基于差分的边缘检测算法，它通过计算图像中每个像素邻域内的差分值，得到图像中的边缘信息。其计算公式如下：
G_x = \begin{bmatrix} 1 & 0\ 0 & -1 \end{bmatrix} I, \quad G_y = \begin{bmatrix} 0 & 1\ -1 & 0 \end{bmatrix} I ,\quad G = \sqrt{G_x^2 + G_y^2}\其中， G_x 和 G_y 分别代表图像在 x 和 y 方向上的梯度值， G 是两者的平方和的平方根， I 是图像矩阵。从公式显而易见Robert算子方法是速度最占优的边缘检测思路。
（5）Prewitt算子：
Prewitt算子是一种基于差分的边缘检测算法，它通过计算图像中每个像素邻域内的差分值，得到图像中的边缘信息。其计算公式如下：
G_x = \begin{bmatrix} -1 & 0 & 1\ -1 & 0 & 1\ -1 & 0 & 1 \end{bmatrix} I, \quad G_y = \begin{bmatrix} -1 & -1 & -1\ 0 & 0 & 0\ 1 & 1 & 1 \end{bmatrix} I,\quad G = \sqrt{G_x^2 + G_y^2}\其中， G_x 和 G_y 分别代表图像在 x 和 y 方向上的梯度值， G 是两者的平方和的平方根， I 是图像矩阵。该方法和Sobel算子方法唯一的区别在于卷积核不一样。相对于Robert算子而言，Prewitt增加了更多的邻域像素对梯度的贡献，因此对噪声的响应较弱，边缘检测效果也相对更好。但对于某些特定方向的边缘，Prewitt算子可能无法很好地检测到。
（6）Classic CA方法：
Classic CA（Cellular Automata）方法是一种基于元胞自动机的边缘检测算法，它通过对图像中的每个像素进行局部规则的迭代，得到图像中的边缘信息。算法首先将图像中的每个像素视为一个元胞，用 x_i 表示第 i 个元胞的灰度值。其次，算法定义一个局部规则，用于根据元胞周围的灰度值计算该元胞的边缘概率 p_i 。常用的“局部规则”包括Von Neumann邻域： p_i = \begin{cases} 1 & \text{if } x_i - x_j > T\ 0 & \text{otherwise} \end{cases} 或者Moore邻域： p_i = \begin{cases} 1 & \text{if } x_i - x_j > T\ 0 & \text{otherwise} \end{cases} ，其中， T 是阈值， x_j 是元胞 i 的 8 个邻居元胞的灰度值。然后，根据局部规则，对每个元胞的边缘概率进行计算，并将其保存在一个概率图像 P 中。最后，将 P 进行二值化处理，得到边缘检测结果 E 。通常可以使用阈值法将 P 二值化：
E(x,y) = \begin{cases} 1 & \text{if } P(x,y) > T\ 0 & \text{otherwise} \end{cases}\其中， T 是阈值。Classic CA方法是一种简单而有效的边缘检测算法，但需要选择合适的局部规则和阈值才能得到较好的检测效果。
（7）OFCA方法：
OFCA（Optimized Fuzzy Cellular Automata）方法是一种基于模糊元胞自动机的边缘检测算法，它通过对图像中的每个像素进行模糊化和迭代，得到图像中的边缘信息。类似于Canny算子检测方法，OFCA首先利用高斯滤波或者均值滤波实现图像模糊，其次，将模糊图像中的每个像素视为一个元胞，用 x_i 表示第 i 个元胞的灰度值。然后，定义一个模糊规则，用于根据元胞周围的灰度值计算该元胞的边缘概率 p_i 。常见的模糊规则包括Crisp模型： p_i = \frac{1}{1 + e^{-(x_i - t)/s}} 与Discrete模型： p_i = \begin{cases} 0 & \text{if } x_i < t - w\ \frac{x_i - t + w}{2w} & \text{if } t - w \leq x_i \leq t + w\ 1 & \text{if } x_i > t + w \end{cases} 等，其中， t 是阈值， s、w 是模糊因子， e 是自然对数的底数。当 x_i 接近阈值 t 时， p_i 的值接近 0.5 ，表明该元胞可能是边缘。最后，根据模糊规则，对每个元胞的边缘概率进行计算，并将其保存在一个概率图像 P 中。同传统CA方法，将 P 进行二值化处理，得到边缘检测结果 E 。通常可以使用阈值法将 P 二值化：
E(x,y) = \begin{cases} 1 & \text{if } P(x,y) > T\ 0 & \text{otherwise} \end{cases}\其中， T 是阈值。OFCA方法相对于Classic CA方法而言，引入了模糊化的概念，使得边缘检测结果更加稳定和准确。但需要选择合适的模糊规则和阈值才能得到较好的检测效果。
下图给出了在SAR二维成像上应用上述各类边缘检测方法的效果对比：
图3-1. 仿真SAR图像及边缘检测效果对比：（a）无噪声SAR图像，（b）有噪声SAR图像，（c）Sobel边缘检测，（d）Roberts边缘检测，（e）Canny边缘检测，（f）Prewitt边缘检测，（g）Classic CA边缘检测，以及（h）OFCA边缘检测
2、互相关及自相关特征
图像的互相关性和自相关性都是一种衡量图像相似度的方法，常用于匹配、跟踪和识别等应用中。图像的互相关性指的是两幅图像之间的相似度。设两幅图像为 I 和 J ，它们的互相关性 R_{IJ} 可以通过以下公式计算：
R_{IJ}(u, v) = \sum_{x=-\infty}^{\infty} \sum_{y=-\infty}^{\infty} I(x, y) J(x-u, y-v)\其中， u、v 是位移量， I(x,y)、J(x-u,y-v) 分别是两幅图像在坐标 (x,y) 和 (x-u,y-v) 处的像素值。公式中的无限计算范围的双重求和表示对给定两幅图像的所有像素进行遍历。
图像的自相关性指的是一幅图像内部不同位置之间的相似度。设一幅图像为 I ，它的自相关性 R_{II} 可以通过以下公式计算：
R_{II}(u, v) = \sum_{x=-\infty}^{\infty} \sum_{y=-\infty}^{\infty} I(x, y) I(x-u, y-v)\其中，u、v 是位移量， I(x,y)、I(x-u,y-v) 分别是图像在坐标 (x,y) 和 (x-u,y-v) 处的像素值。公式中的双重求和表示对所有像素进行遍历。写程序的时候，以上公式中的求和的“ ±∞ ”范围可根据具体应用进行限定，以加快计算速度和防止堆栈溢出。
3、Fourier谱与小波谱
图像的傅里叶谱和小波谱都是一种表示图像频率分布的方法，常用于图像处理和分析中。图像的傅里叶谱表示的是图像在频域内的分布情况。设一幅图像为 I(x,y) ，它的傅里叶变换 F(u,v) 可以通过以下公式计算：
F(u,v) = \mathcal{F}[I(x,y)] = \sum_{x=0}^{M-1} \sum_{y=0}^{N-1} I(x,y) e^{-j2\pi(\frac{ux}{M} + \frac{vy}{N})}\其中， u、v 是频率变量， M、N 分别是图像的宽度和高度， j 是虚数单位。傅里叶谱的可视化图像形式可以通过计算傅里叶变换的幅度谱 |F(u,v)| 得到，表示图像在不同频率下的像素强度分布情况。熟悉雷达动目标识别的小伙伴们看出来，如果将慢时间维度拼接而成的时域的雷达回波矩阵视作一个图像，其的Fourier谱就是距离多普勒强度图RDM（Range-Doppler Intensity Map）。
类似地，图像的小波谱表示的是图像在小波域内的分布情况。设一幅图像为 I(x,y) ，它的小波变换 W(a,b) 可以通过以下公式计算：
W(a,b) = \sum_{x=0}^{M-1} \sum_{y=0}^{N-1} I(x,y) \psi_{a,b}(x,y)\其中， a、b 是尺度和位移变量， \psi_{a,b}(x,y) 是小波基函数，可以通过不同的小波基函数选择得到。小波变换的详细原理和意义讲解可以参考基础篇信号变换一章：【第四章 信号变换 - 知乎 (zhihu.com)】。
图3-2. 一些点目标图像的FFT谱示例
P. S. FFT谱作为纹理度量或描述子时会具备旋转不变性，也就是说，原图像空间旋转多少，频率空间也会相应旋转多少。
4、共生矩阵、Haralick与扩展SDM特征
图像的共生矩阵和Haralick特征是一种用于描述图像纹理特征的方法，常用于图像分类和分割等应用中。图像的共生矩阵描述的是图像中不同灰度级别之间的出现次数和位置关系。设一幅图像为 I ，其灰度级别为 G ，共生矩阵 P 可以通过以下公式计算：
P_{\delta,\theta}(i,j) = \sum_{x=1}^{N} \sum_{y=1}^{M} \begin{cases} 1 & \text{if } I(x,y) = i \text{ and } I(x+\delta,y+\theta) = j\ 0 & \text{otherwise} \end{cases}\其中， i、j 是灰度级别， \delta、\theta 是位移量， N、M 分别是图像的宽度和高度。共生矩阵描述了图像中不同灰度级别之间的空间关系，可以通过计算共生矩阵的统计特征来描述图像的纹理特征。
图像的Haralick特征是一组描述图像纹理特征的统计特征，可以通过共生矩阵计算得到。常用的Haralick特征包括：
（1）对比度：
\text{Contrast} = \sum_{i,j=1}^{N_g} P_{\delta,\theta}(i,j)(i-j)^2\（2）能量：
\text{Contrast} = \sum_{i,j=1}^{N_g} P_{\delta,\theta}(i,j)(i-j)^2\（3）相关度：
\text{Correlation} = \frac{\sum_{i,j=1}^{N_g} P_{\delta,\theta}(i,j)(i-\mu)(j-\mu)}{\sigma^2}\其中， N_g 是灰度级别数， \mu 和 \sigma 分别是共生矩阵的均值和标准差。位移量 \delta 和 \theta 可以通过不同的取值得到不同方向和距离的纹理特征。
图像的扩展SDM（Spatial Distribution of Multi-scale features）特征是一种用于描述图像纹理特征的方法，常用于图像分类和识别等应用中。图像的扩展SDM特征的获取基于图像的灰度共生矩阵和小波变换共同实现，所以说本质上接近上面几类特征的糅合改进，其可以同时描述图像的纹理特征和空间分布特征。给定一幅图像 I ，其扩展SDM特征可以通过以下步骤计算：对于一幅图像 I ，假设其小波变换得到的小波系数矩阵为 W_{a,b} ，其中 a 和 b 分别表示尺度和位置。对于每个小波系数矩阵 W_{a,b} ，可以计算其灰度共生矩阵 P_{a,b} 和特征向量 l_{a,b} 。具体地，灰度共生矩阵可以通过以下公式计算：
P_{a,b}(i,j) = \sum_{x=1}^{N} \sum_{y=1}^{M} \begin{cases} 1 & \text{if } W_{a,b}(x,y) = i \text{ and } W_{a,b}(x+\delta,y+\theta) = j\ 0 & \text{otherwise} \end{cases}\其中， i、j 是灰度级别， \delta、\theta 是位移量， N、M 分别是图像的宽度和高度。这一步和求共生矩阵是完全一样的，不同的是，特征向量 l_{a,b} 可以通过计算灰度共生矩阵的统计特征得到，例如均值、方差、能量、熵等。常用的特征向量计算公式包括：
（1）均值：
\mu_{a,b} = \frac{1}{N_g} \sum_{i=1}^{N_g} i P_{a,b}(i)\（2）方差：
\sigma_{a,b}^2 = \frac{1}{N_g} \sum_{i=1}^{N_g} (i-\mu_{a,b})^2 P_{a,b}(i)\（3）能量：
\sigma_{a,b}^2 = \frac{1}{N_g} \sum_{i=1}^{N_g} (i-\mu_{a,b})^2 P_{a,b}(i)\（4）熵：
H_{a,b} = -\sum_{i=1}^{N_g} P_{a,b}(i) \log_2 P_{a,b}(i)\扩展SDM特征向量 F 可以通过将所有小波系数矩阵的特征向量 l_{a,b} 按照空间位置聚合得到。常用的聚合方法包括加权平均、取最大值或中位数等。具体地，假设图像 I 被分割为 K 个空间区域，第 k 个区域的特征向量为 l_k ，则扩展SDM特征向量 F 可以通过以下公式计算：
F(i) = \frac{1}{K} \sum_{k=1}^{K} w_k l_k(i)\其中， w_k 是第 k 个区域的权重，这些权重用于表示我们之前选择的聚合方法的策略。
5、Laws纹理特征
图像的Laws特征是一种基于图像滤波和能量统计的方法，用于描述图像的纹理特征。Laws纹理特征可以用于图像分类、识别、检索等任务中，具有较好的性能。Laws纹理特征的基本思想是将图像分解为不同的小块，然后对每个小块进行一组滤波器的卷积操作，得到一组滤波响应（滤波器可以是多种不同方案组合得到的）。在得到滤波响应后，可以通过计算其能量特征来描述图像的纹理特征。具体地，假设对于一幅图像 I ，通过一组滤波器得到的滤波响应为 F_{i,j} ，其中 i、j 分别表示滤波器的编号和图像块的编号（这些块可以是 5\times5、7\times7 等等这一类的方形或者各种由研究人员指定的形状大小的图像子集），则Laws纹理特征可以通过以下公式计算：
T_{i,j} = \frac{1}{N} \sum_{x=1}^{N} (F_{i,j}(x) - \mu_{i,j})^2\其中， \mu_{i,j} 是滤波响应的均值， N 是滤波响应的长度。Laws纹理特征 T 是一个向量，包含了所有图像块和滤波器组合的能量特征。
6、局部二值模式（LBP）
图像的局部二值模式（Local Binary Pattern，LBP）是一种基于图像灰度值的局部纹理特征描述子，常用于图像分类、识别和检索等应用中，具有良好的性能和鲁棒性。对于一幅图像 I 中的每个像素点 x ，可以计算其对应的局部二值模式 \mathrm{LBP}(x) ，表示其周围像素点与中心像素点的灰度值大小关系。具体地，对于一个半径为 r 的圆形邻域，以中心点的灰度值为阈值，将周围的 8 个像素点分别与中心点进行比较，得到一个 8 位二进制数。将这个二进制数转换为十进制数，即得到 x 点的局部二值模式 \mathrm{LBP}(x) 。在得到所有像素点的局部二值模式后，可以通过计算其直方图或统计特征来描述图像的纹理特征。常用的统计特征包括LBP值的均值、方差、能量、熵等。
图像的旋转不变局部二值模式（Rotation Invariant Local Binary Pattern，RILBP）与局部二值模式（LBP）类似，对于一幅图像 I 中的每个像素点 x ，可以计算其对应的RILBP值 \mathrm{RILBP}(x) ，表示其周围像素点与中心像素点的灰度值大小关系。与LBP不同的是，RILBP在计算时考虑了图像中的旋转不变性。具体地，对于一个半径为 r 的圆形邻域，以中心点的灰度值为阈值，将周围的 8 个像素点分别与中心点进行比较，得到一个 8 位二进制数。将这个二进制数按顺时针或逆时针方向旋转，使得其最小值在最前面，即得到一个旋转不变的二进制数。将这个旋转不变的二进制数转换为十进制数，即得到 x 点的RILBP值 \mathrm{RILBP}(x) 。在得到所有像素点的RILBP值后，可以通过计算其直方图或统计特征来描述图像的纹理特征。常用的统计特征包括RILBP值的均值、方差、能量、熵等，后面的操作就和LBP一样了。
7、动态纹理特征
在动态图像中，纹理通常会随着时间的推移而发生变化。因此，可以通过分析图像序列中的纹理变化来描述图像的动态纹理特征。常用的动态纹理特征包括光流、时空纹理、动态纹理模型等。光流是指图像中像素点随着时间的推移而发生的运动变化。可以通过计算相邻帧之间的像素点位移来估计光流。光流可以用于描述图像中的运动轨迹和速度等信息，进而用于动态纹理特征描述。时空纹理是指图像序列中的纹理随着时间的推移而发生的变化。可以通过计算图像序列中每个像素点的时空灰度变化来描述时空纹理特征。常用的时空纹理特征包括时空LBP、时空SIFT等。动态纹理模型是指用于描述动态纹理变化的模型，例如基于隐马尔可夫模型（HMM）的动态纹理模型、基于高斯混合模型（GMM）的动态纹理模型等。这些模型可以用于对动态纹理进行建模和分类。这些内容相对于静态纹理特征复杂很多，其各种改进类型也很多，后面再单独列成章节来详细介绍。
四、统计区域度量
图像像素的统计度量是一种用于描述图像像素值分布或变化情况的方法。它通常用于图像处理和计算机视觉领域中，例如图像增强、图像分割和图像分类等任务。常见的图像像素统计度量包括以下 8 种：
图像矩特征；
点度量特征；
全局直方图；
局部区域直方图；
散点图与3D直方图；
多尺度直方图；
径向直方图；
轮廓或边缘直方图。
这些统计度量可以通过对图像像素值进行数学运算和分析来计算得到。它们可以为图像处理和计算机视觉算法提供重要的特征和信息，从而提高算法的性能和鲁棒性。接下来，我们逐个展开总结其度量特征的定义、数学计算理论的细节和一些简单的应用举例。同样地，个别比较经典的特征会给出图示。
1、图像矩特征
图像矩特征是一种用于描述图像像素统计特性的方法，可以描述成一个函数在其基空间上的投影。常见的图像矩特征包括几何矩和中心矩等。几何矩是指通过对图像像素值进行加权求和来计算图像的形状特征，其中权重是像素的坐标值的幂次方。例如， p 阶几何矩可以表示为：
M_{pq}=\sum_{x}\sum_{y}x^py^qI(x,y)\其中， I(x,y) 表示图像在位置 (x,y) 处的像素值， p 和 q 分别表示几何矩的幂次。当 p+q=0 时， M_{pq} 称为零阶几何矩，它等于图像中所有像素值的和；当 p+q=1 时， M_{pq} 称为一阶几何矩，它可以用来计算图像的重心；当 p+q=2 时， M_{pq} 称为二阶几何矩，它可以用来计算图像的方差和协方差等。
中心矩是指通过对图像像素值进行加权求和来计算图像的纹理特征，其中权重是像素坐标值与图像重心的差的幂次方。例如， p 阶中心矩可以表示为：
\mu_{pq}=\sum_{x}\sum_{y}(x-\bar{x})^p(y-\bar{y})^qI(x,y)\其中， \bar{x} 和 \bar{y} 分别表示图像的重心坐标。中心矩可以用来计算图像的方向和形状等特征，不管是一维的序列分布还是二维的图像，不同的阶次均具备下述视觉上的规律性性质：
零阶矩：表示一维均值或二维质心；
中心矩：描述均值或二维质心周围的变化情况；
一阶中心矩：包含二维面积、质心和物体/目标大小等相关信息；
二阶中心矩：与方差和2D椭圆度量相关；
三阶中心矩：提供了二维形状（或偏度）的对称信息；
四阶中心矩：用来度量二维分布，如高、矮、细、短、胖等形态；
更高阶的矩：可由多个矩的比值组成，比如协方差。
图像矩特征可以应用于图像处理、计算机视觉和模式识别等领域。例如，在目标检测和识别中，可以使用图像矩来提取目标的形状和纹理特征，以实现目标的自动识别和跟踪。除了几何矩和中心矩外，还有许多其他类型的图像矩特征，例如旋转不变矩和尺度不变矩等。这些图像矩特征可以根据不同的应用需求进行选择和组合，以实现更加准确和鲁棒的图像处理和分析。
2、点度量特征
图像的点度量特征是一种用于描述图像中单个像素的特征的方法。它通常用于图像分割、图像增强和图像处理等任务中。图像分割是点度量特征的核心下游任务。常见的图像点度量特征包括灰度值、梯度、方向、角点和纹理特征等。此处，纹理特征是多个可以组合的点的度量特征的接续，已经在上面一节中详细介绍。角点属于兴趣点特征，这个内容比较重要且复杂，会在后面的章节深入分析，这里主要阐释灰度、梯度、方向三者。
其实灰度、梯度、方向三个点度量特征已经在上面反复使用过了，这些是绝大部分图像特征的计算基础，数学定义分别是 I(x,y) 、 \nabla I(x,y)=\begin{pmatrix} G_x\ G_y \end{pmatrix}=\begin{pmatrix} \frac{\partial I}{\partial x}\ \frac{\partial I}{\partial y} \end{pmatrix} 、 \theta(x,y)=\arctan\frac{G_y}{G_x} ，其中， I(x,y) 表示图像在位置 (x,y) 处的像素值， G_x 和 G_y 分别表示图像在 x 和 y 方向上的梯度值， \frac{\partial I}{\partial x}、\frac{\partial I}{\partial y} 分别表示图像在 x 和 y 方向上的灰度值变化率， \theta(x,y) 表示图像在位置 (x,y) 处的梯度方向， \arctan 表示反正切函数。
3、全局直方图
图像的全局直方图是一种描述图像像素值分布的方法。全局直方图是由图像中所有像素的灰度级别组成的直方图，它可以用来表示图像中每个灰度级别的像素数量。通常，全局直方图是一个一维的向量，其长度等于图像的灰度级别数。该特征可以用下式计算：
H(i)=\sum_{x}\sum_{y}[I(x,y)=i]\其中， H(i) 表示图像中灰度级别为 i 的像素数量， [I(x,y)=i] 表示判断图像在位置 (x,y) 处的像素值是否等于 i 。全局直方图可以用来表示图像的灰度分布情况，从而提供图像的全局信息。它可以用于图像分类、图像检索、图像分割和图像增强等任务中。下面给出了一个SAR图像上计算全局直方图的实例。
图4-1. SAR图像的全局直方图示例，其中右边列为对应Image的全局直方图，直方图向量的分布变化可以作为诸如SAR图像变化检测的依据
4、局部区域直方图
顾名思义，局部区域直方图就是加窗图像之后的窗内小图像的像素变化规律。这个窗可以是某个由研究人员指定大小形状的规则区域，也可以是通过某种策略从图像上筛出来的点的集合。如果是常规的矩形窗口，那么局部区域直方图可以用来表征图像中某个像素周围局部区域的像素分布情况。类似于全局直方图，局部区域直方图也是一个向量，其大小等于窗口内灰度级别数，可以用下式计算：
H_{i,j}(x,y)=\sum_{u,v\in N_{x,y}}[I(u,v)=i][I(x,y)=j]\其中， H_{i,j}(x,y) 表示图像在位置 (x,y) 处，灰度级别为 i 和 j 的像素在邻域 N_{x,y} 中的数量， [I(u,v)=i] 和 [I(x,y)=j] 分别表示图像在位置 (u,v) 和 (x,y) 处的像素值是否等于 i 和 j 。该特征可以用于图像分割、目标检测和图像识别等任务中。
5、散点图与3D直方图
把这两者放在一起是因为，散点图和3D直方图都是描述图像特征之间关系的方法。它们可以用来表示图像中不同特征之间的相关性和分布情况等，从而提供图像的多维特征信息。下面将对图像的散点图和3D直方图进行详细介绍。
（1）散点图：
图像的散点图是一种用于表示图像中两个特征之间关系的方法，它可以用来描述图像中不同特征之间的相关性和分布情况等。通常，散点图是一个二维坐标系，其中每个点表示图像中两个特征之间的一个数据点。例如，在图像的RGB颜色空间中，可以使用散点图来表示图像中不同颜色通道之间的关系：
S(i,j)={(f_i(x,y),f_j(x,y))|I(x,y)\in\Omega}\其中， S(i,j) 表示图像中特征 f_i 和 f_j 之间的散点图，例如我们要表征RGB三通道间的相互关系，这里的图像特征就可以定义为不同通道下的像素值。 I(x,y) 表示图像在位置 (x,y) 处的点， \Omega 表示图像的像素坐标集合。这只是散点图的一种定义方式，常用的定义方式还有很多。比如最简单的散点图定义方式就是将图像下采样，但坐标轴不作缩放，那么所得的图像在空间上就是离散的多个点各自有数值组成，形如“点云状”。
（2）3D直方图：
图像的3D直方图是一种用于表示图像中三个特征之间关系的方法，它可以用来描述图像中不同特征之间的相关性和分布情况等。通常，画3D直方图需要一个三维坐标系，其中每个立方体表示图像中三个特征之间的一个数据点。例如，在图像的HSI颜色空间中，可以使用3D直方图来表示图像中颜色和饱和度之间的关系。计算公式：
H(i,j,k)=\sum_{x}\sum_{y}[f_1(x,y)=i][f_2(x,y)=j][f_3(x,y)=k]\其中， H(i,j,k) 表示图像中特征 f_1 、 f_2 和 f_3 之间的3D直方图， [f_1(x,y)=i] 、 [f_2(x,y)=j] 和 [f_3(x,y)=k] 分别表示图像在位置 (x,y) 处的特征值是否等于 i 、 j 和 k 。
6、多尺度直方图
图像的多分辨率、多尺度直方图是一种用于描述图像不同分辨率和尺度下的特征分布情况的方法。它可以用来表示图像中的多尺度特征和纹理信息等。图像的多分辨率直方图是由多个不同尺度的全局直方图组成的，每个全局直方图对应着图像在不同分辨率下的特征分布情况，可以用下式计算：
H_i^s(k)=\sum_{x,y\in R_i^s}[I(x,y)=k]\其中， H_i^s(k) 表示图像在第 s 个尺度下，灰度级别为 k 的像素在区域 R_i^s 中的数量， I(x,y) 表示图像在位置 (x,y) 处的像素值。
类似于上面提到的局部区域直方图，如果由多个不同尺度的局部区域直方图组成，这样的特征也属于多尺度直方图，也可以用来表示图像中的多尺度特征和纹理信息等。计算公式如下：
H_{i,j}^{s}(x,y)=\sum_{u,v\in N_{x,y}^s}[I(u,v)=i][I(x,y)=j]\其中， H_{i,j}^{s}(x,y) 表示图像在第 s 个尺度下，灰度级别为 i 和 j 的像素在邻域 N_{x,y}^s 中的数量， I(u,v) 和 I(x,y) 分别表示图像在位置 (u,v) 和 (x,y) 处的像素值是否等于 i 和 j 。该方法可以用来进行物体检测和定位，也可以用来进行特征匹配和配准，应用范围很广，且因为金字塔形（也即图像缩放倍率 s 可调）的图像处理思路，可以自适应各种不同“大小”的目标，是“多尺度”系列共性的优势。
7、径向直方图
图像的径向直方图是一种用于描述图像中像素到中心点的距离和方向之间关系的方法，它可以用来表示图像中的径向纹理和特征信息等。通常径向直方图是由一系列的环形区域直方图组成的，每个环形区域直方图对应着图像中不同距离和方向下的像素分布情况。计算公式如下：
H_r(\theta,k)=\sum_{(x,y)\in R_{r,\theta}}[I(x,y)=k]\其中， H_r(\theta,k) 表示图像中到中心点距离为 r 、方向为 \theta 的环形区域内，灰度级别为 k 的像素的数量， I(x,y) 表示图像在位置 (x,y) 处的像素值， R_{r,\theta} 表示半径为 r 、方向为 \theta 的环形区域。径向直方图一般用于具备多个环形特征或者极坐标下图像的目标特征描述、检测与定位。
8、轮廓或边缘直方图
图像的边缘直方图是一种用于描述图像中边缘信息分布情况的方法。它可以用来表示图像中的目标轮廓像素点的分布特性，是基于上一节中“边缘检测”之后的二级特征。可以用下式计算：
H_{\theta}(k)=\sum_{(x,y)\in E_{\theta}}[I(x,y)=k]\其中， H_{\theta}(k) 表示图像在方向为 \theta 的边缘直方图中，灰度级别为 k 的像素的数量， I(x,y) 表示图像在位置 (x,y) 处的像素值， E_{\theta} 表示在方向为 \theta 的边缘区域内的所有像素点。边缘直方图可以用来进行图像检索和相似度匹配，也可以用来进行物体检测和定位。
五、基空间度量
图像的基空间度量是一种用于描述图像中像素之间距离和相似性的方法。它基于图像的灰度值或者其他参数，将图像表示为高维的向量空间 z ，然后在该向量空间中计算像素之间的距离和相似性，从而实现对图像的度量和比较。基空间度量方法在图像检索、图像分类、图像匹配等领域具有广泛的应用。简单来说，基空间度量就是“图像特征投影”。其中，基变换方法的选取是基空间度量的核心讨论问题，不止是在图像处理领域，在经典数字信号处理领域基变换也是老生常谈的话题。常用的用于图像的基变换分矩形基、统计基、方向基、正弦基四大类数十小类如下思维导图：
图5-1. 常用基变换方法列表，摘自Scott Krig著《Computer Vision Metrics - Survey, Taxonomy, and Analysis》
下面主要介绍其中最经典好用的几类基变换方法，包括下面 9 种：
Fourier变换；
Walsh-Hadamard与Haar变换
斜变换；
Zernike多项式；
导向滤波器；
Karhunen-Loeve与Hotelling变换；
小波变换（Wavelet）与Gabor滤波器；
Hough变换与Radon变换。
基空间度量方法在图像处理和计算机视觉中具有广泛的应用。例如，在基于内容的图像检索中，可以使用基空间度量方法来计算图像之间的相似性，从而实现对图像的检索和分类。在基于特征的图像匹配中，可以使用基空间度量方法来计算不同图像之间的相似性和距离，从而实现图像的配准和匹配。更通俗的说法是，让图像换个域换个形态，可能更利于某些在当前时域、空间域下的不清晰的目标特征变得清晰。接下来，我们逐个展开总结诸变换方法定义、数学计算理论的细节和一些简单的应用举例。同样地，个别比较经典的会给出图示。
1、Fourier变换
这和第三大节的Fourier谱分析内容是完全一致的，这里简单重述一遍。设一幅图像为 I(x,y) ，它的傅里叶变换 F(u,v) 可以通过以下公式计算：
F(u,v) = \mathcal{F}[I(x,y)] = \sum_{x=0}^{M-1} \sum_{y=0}^{N-1} I(x,y) e^{-j2\pi(\frac{ux}{M} + \frac{vy}{N})}\其中， u、v 是频率变量， M、N 分别是图像的宽度和高度， j 是虚数单位。傅里叶谱的可视化图像形式可以通过计算傅里叶变换的幅度谱 |F(u,v)| 得到，表示图像在不同频率下的像素强度分布情况。
2、Walsh-Hadamard与Haar变换
图像的Walsh-Hadamard变换（WHT）是一种用于图像变换和特征提取的方法。它可以将图像表示为一组Walsh-Hadamard系数，从而实现对图像的特征投影。WHT可以用于图像压缩、图像特征提取和图像识别等领域。具体来说，WHT是一种正交变换，它可以将一个长度为 N 的向量 x 变换为一个长度为 N 的向量 y ，变换公式如下：
y=Hx\其中， H 是一个 N\times N 的Walsh-Hadamard变换矩阵， x 和 y 分别表示输入和输出的向量（习惯上都把 N 定义为 2 的幂次）。对于一个 N\times N 的图像 I ，可以将其展开从而转换为一个 N^2 维的列向量 x ，然后进行WHT变换，得到一个 N^2 维的列向量 y ，再将其转换为一个 N\times N 的图像 J 。WHT可以用下式计算：
J=H_I^{-1}IH_I\其中， H_I 是图像 I 的Walsh-Hadamard变换矩阵， H_I^{-1} 表示其逆矩阵。一些经常被用到的Walsh-Hadamard变换矩阵有：\begin{equation} \begin{aligned} & H_0=+(1) \ & H_1=\frac{1}{\sqrt{2}}\left(\begin{array}{rr} 1 & 1 \ 1 & -1 \end{array}\right) \ & H_2=\frac{1}{2}\left(\begin{array}{rrrr} 1 & 1 & 1 & 1 \ 1 & -1 & 1 & -1 \ 1 & 1 & -1 & -1 \ 1 & -1 & -1 & 1 \end{array}\right) \ & H_3=\frac{1}{2^{3 / 2}}\left(\begin{array}{rrrrrrrr} 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \ 1 & -1 & 1 & -1 & 1 & -1 & 1 & -1 \ 1 & 1 & -1 & -1 & 1 & 1 & -1 & -1 \ 1 & -1 & -1 & 1 & 1 & -1 & -1 & 1 \ 1 & 1 & 1 & 1 & -1 & -1 & -1 & -1 \ 1 & -1 & 1 & -1 & -1 & 1 & -1 & 1 \ 1 & 1 & -1 & -1 & -1 & -1 & 1 & 1 \ 1 & -1 & -1 & 1 & -1 & 1 & 1 & -1 \end{array}\right) \ & \left(H_n\right)_{i, j}=\frac{1}{2^{n / 2}}(-1)^{i \cdot j} \ & \end{aligned} \end{equation}\其中， N=2^n ， i，j 代表Hadamard变换矩阵的行列角标。利用Walsh-Hadamard变换实现图像压缩的原理是，原始图像 I 中有大量数据信息冗余，从而可以投影成一个尺度大小相同但稀疏度更高的 J 表示，图像占用空间自然就变小了。Walsh-Hadamard变换与压缩感知的区别是，变换前后的特征向量尺度大小一致，且投影矩阵固定为Hadamard矩阵。
Walsh-Hadamard变换与Haar变换的区别就在矩阵 H 上。Haar矩阵是一种正交矩阵，用于将信号从时域转换到频域，严格意义上来说，Haar变换就是Haar小波基函数条件下的小波变换。具体讲，Haar矩阵是一个 N\times N 的矩阵，其中 N 是 2 的整数次幂。Haar矩阵可以通过递归的方式定义。当 N=1 时，Haar矩阵为：
H_1=\begin{bmatrix}1\end{bmatrix}\当 N=2 时，Haar矩阵为：
H_2=\begin{bmatrix}1 & 1\1 & -1\end{bmatrix}\当 N=4 时，Haar矩阵的递归定义：
H_4=\begin{bmatrix}H_2 & H_2\H_2 & -H_2\end{bmatrix}\当 N=8 时，Haar矩阵的递归定义参考上面 N=4 时候的方式给出：
H_8=\begin{bmatrix}H_4 & H_4\H_4 & -H_4\end{bmatrix}\以此类推，当 N=2^k 时，Haar矩阵可以通过递归地将 H_N 分成两个子矩阵，然后将它们分别填充到四个象限中，其中左上和右上象限填充子矩阵 H_{N-1} ，左下象限填充零矩阵，右下象限填充子矩阵 -H_{N-1} 。最终得到的矩阵就是 N\times N 的Haar矩阵。得到Haar矩阵后，替换上述Walsh-Hadamard变换中的 H ，所得的图像变换方法就是Haar变换。
3、Skew（斜）变换
图像的斜变换（Skew Transformation）是指将图像在水平方向或者竖直方向上进行平移，从而使图像呈现出斜的形态。斜变换可以通过对图像进行仿射变换实现，本质上还是一种卷积操作，其卷积核的矩阵表示如下：
M = \begin{bmatrix} 1 & \tan\theta & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{bmatrix}\其中， \theta 表示斜变换的角度。对于水平方向上的斜变换，可以将 \theta 设置为斜率的反正切值；对于竖直方向上的斜变换，则将 \theta 设置为负斜率的反正切值。斜变换主要应用于文本识别和OCR等场景中，用于将文本图片中的倾斜文本进行纠正，提高识别准确率。此外，在计算机视觉中也常常使用斜变换对图像进行预处理，以便于后续的目标检测、图像分类等任务的处理。
4、Zernike多项式
图像的Zernike多项式是一种用于描述圆形区域上函数的正交基函数。不同于其它的图像特征描述算子，该数值理论由荷兰天文学家Frits Zernike在20世纪30年代提出，最早用于光学拍摄数据中的相位重建和目标表面形貌分析。对于一个圆形区域上的函数 f(r,\theta) ，其Zernike多项式可以表示为：
Z_{n,m}(r,\theta) = \sqrt{n+1}\sum_{k=0}^{\lfloor(n-m)/2\rfloor}\frac{(-1)^k(n-k)!}{k!(\frac{1}{2}(n+m)-k)!(\frac{1}{2}(n-m)-k)!}r^{n-2k}\其中， n 和 m 是非负整数， r 是极坐标系下的径向距离， \theta 是极角， \lfloor\cdot\rfloor 表示向下取整操作。Zernike多项式在图像处理中的应用非常广泛，主要应用于图像的形状分析、配准、纹理分析和特征提取等方面。例如，可以通过计算图像的Zernike矩来描述图像的形态、纹理和边缘信息，进而实现图像分类、目标检测等任务。此外，Zernike多项式也可以用于图像的去噪、模糊和失真校正等方面。
5、导向滤波器
图像的导向滤波器（Guided Filter）是一种基于图像辅助信息进行滤波的方法，可以用于图像去噪、图像增强、图像分割等任务。该方法是由老何（Kaiming He，何恺明）等人在2013年提出的，是一种快速、简单而有效的滤波方法。导向滤波器的基本思想是利用一张高质量的引导图像（或称为导向图像）来指导滤波过程，从而实现对目标图像的滤波。具体来说，导向滤波器通过对目标图像和引导图像之间的相关性进行建模，来计算出每个像素的加权平均值。其中，像素之间的权值是根据引导图像和目标图像之间的相似程度来计算的。用数学公式描述上面的理论就是：
\begin{aligned} &I_{mean}(p)=\frac{1}{|\Omega|}\sum_{q\in\Omega}I(q) \ &I_{corr}(p)=\frac{1}{|\Omega|}\sum_{q\in\Omega}(I(q)\cdot G_p(q)) \ &\alpha(p)=\frac{1}{|\Omega|}\sum_{q\in\Omega}G_p(q) \ &\hat{I}(p)=\frac{I_{corr}(p)}{\alpha(p)}+\frac{b}{\alpha(p)}(I(p)-\frac{1}{|\Omega|}\sum_{q\in\Omega}I(q)) \end{aligned}\其中， I 为目标图像， G 为Guided Image也就是高质量的引导图像， G_p 为引导图像在 p 处的局部均值， \alpha 为归一化因子， b 为平滑参数， \hat{I} 为滤波后的图像。导向滤波器在图像处理中的应用非常广泛，主要应用于图像的去噪、图像增强、图像分割等方面。例如，在图像去噪中，可以利用导向滤波器对图像进行平滑处理，从而去除噪声。在图像增强中，可以利用导向滤波器增强图像的细节和结构信息，使图像更加清晰和鲜明。在图像分割中，可以利用导向滤波器对图像进行预处理，提取出图像的边缘和纹理信息，从而实现图像分割的目的。
下面这组图片给出了在含有强烈噪声的SAR成像上运用Guided Filter实现去噪的效果范例：
图5-2. 对SAR图像运用Guided Filter的对比效果，（a）为经过Shift Invariant K-SVD后具备稀疏表征的预处理图像，（b）为经过Fast Guided Filter处理后的去噪图像结果
6、Karhunen-Loeve与Hotelling变换
图像的Karhunen-Loeve变换（KLT）和Hotelling变换（HT）都是一种基于统计学方法的图像变换技术，可以用于图像的特征提取和降维处理。它们都是在协方差矩阵的特征值和特征向量的基础上进行的，所以总结到一块，但是其应用场景和计算公式有所不同。
（1）Karhunen-Loeve变换：
KLT是一种基于正交变换的线性变换技术，它可以将输入图像转换为一组正交的基函数，从而实现图像的特征提取和降维处理。其计算公式如下：
X_\mathrm{KLT} = E^{-1/2}U^T(X-E)\其中， X 为输入图像， E 为输入图像的协方差矩阵， U 为协方差矩阵的特征向量， E^{-1/2} 为协方差矩阵的特征值的平方根的逆矩阵。 X_\mathrm{KLT} 表示KLT变换后的图像。Karhunen-Loeve变换在图像压缩和图像识别等方面有广泛的应用。例如，在图像压缩中，可以利用KLT变换将图像的冗余信息去除，从而实现图像的压缩；在图像识别中，可以利用KLT变换提取图像的特征信息，从而实现对图像的识别。
（2）Hotelling变换：
Hotelling变换是一种基于投影的线性变换技术，它可以将输入图像投影到一个新的特征空间中，从而实现图像的降维处理。其计算公式如下：
X_\mathrm{HT} = V^TX\该公式的格式和上面的Walsh-Hadamard、Haar变换类似，只是其中 X 为输入图像， V 为输入图像的协方差矩阵的特征向量矩阵。 X_\mathrm{HT} 表示HT变换后的图像。Hotelling变换主要应用于图像的分类和识别等方面，例如，在人脸识别中，可以利用HT变换将人脸图像投影到一个新的特征空间中，从而实现对人脸的分类和识别。
7、小波变换（WT）与Gabor滤波器
图像的小波变换和Gabor滤波器都是一种基于频域分析的图像处理技术，可以用于图像的特征提取和分析。虽然它们都是基于时频分析的技术，本质上和Fourier分析也是同一类的，但是其应用场景和计算公式有所不同，下面我们分别讲解其原理。
（1）小波变换：
前面第三节纹理区域特征部分已经提到了小波谱，这里的小波变换内核是一致的，正是因为Fourier和Wavelet方法同时隶属于纹理区域度量和基空间度量，所以这里才简单重述一下。该变换是一种基于多尺度分析的图像变换技术，它可以将图像分解成不同尺度的小波函数，从而实现对图像的特征提取和分析。其计算公式如下：
W(a,b) = \int_{-\infty}^{\infty}x(t)\psi_{a,b}(t)dt\严格意义上来说，小波变换和短时傅里叶变换一样是时频分析手段，不是纯粹的频域分析手段，但大家习惯将它们笼统地归为频域分析手段之一。上式中， x(t) 为输入图像， \psi_{a,b}(t) 为小波基函数， a 和 b 分别表示尺度和平移参数。 W(a,b) 表示经过小波变换后的系数。小波变换主要应用于图像压缩和图像增强等方面。例如，在图像压缩中，可以利用小波变换将图像分解成多个尺度的小波系数，并对系数进行量化和编码，从而实现图像的压缩；在图像增强中，可以利用小波变换对图像进行分解和重构，从而实现图像的增强。
（2）Gabor滤波器：
Gabor滤波器是一种基于滤波的图像处理技术，它可以将图像通过一组Gabor滤波器进行滤波，从而提取图像的纹理和结构信息。该滤波器的数学表达式如下：
G(x,y;\lambda,\theta,\psi,\sigma,\gamma) = \frac{1}{2\pi\sigma^2}\exp(-\frac{x'^2+\gamma^2y'^2}{2\sigma^2})\cos(2\pi\frac{x'}{\lambda}+\psi)\其中， x 和 y 为空间坐标， \lambda 为波长， \theta 为方向参数， \psi 为相位， \sigma 为标准差， \gamma 为椭圆度。 x' 和 y' 为 x 和 y 在滤波器方向上的投影。Gabor滤波器主要应用于图像的纹理分析和特征提取等方面。例如，在图像分类中，可以利用Gabor滤波器提取图像的纹理信息，从而实现对图像的分类和识别；在图像分割中，可以利用Gabor滤波器对图像进行预处理，提取出图像的纹理和结构信息，从而实现图像分割的目的。
下面给出了一个简单的利用Gabor滤波器实现SAR图像变化检测的实例：
图5-3. 参数条件下的基于Gabor滤波器的SAR图像特征提取以及变化检测结果：（a）σ = 0.1，（b）σ = 0.25，（c）σ = 0.5，（d）σ = 0.7，（e）σ = 1
8、Hough变换与Radon变换
图像的Hough变换和Radon变换都是一种基于投影的图像处理技术，可以用于图像的特征提取和分析。这两种方法的一个比较明显的共性是能够改变图像上“目标的形态”，但不影响整个图像上的目标的分布。
（1）Hough变换：
Hough变换是一种基于投影的图像处理技术，它可以针对图像中的直线和圆等几何形状实现良好地检测和提取。其计算公式如下：
\rho = x\cos\theta + y\sin\theta\其中， \rho 和 \theta 分别表示极径和极角， x 和 y 为图像中的像素坐标。简单来说，就是将图像投影到极坐标上，或者也叫大家高中数学选修课熟悉的“参数空间”上。由于其变换后图像的极坐标特性，Hough变换对直线和圆特征是比较敏感的，例如，在自动驾驶领域，可以利用Hough变换提取图像中的直线，从而实现对车道线的快速准确检测和识别。
（2）Radon变换：
Radon变换也是一种基于投影的图像处理技术，它可以将图像通过一系列的投影操作的组合，将图像转换为另一种特征形式。其计算公式如下：
p(t,\theta) = \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}f(x,y)\delta(x\cos\theta+y\sin\theta-t)dxdy\其中， p(t,\theta) 为投影图像， \delta 为狄拉克函数， f(x,y) 为输入图像， t 和 \theta 分别表示投影距离和投影角度。Radon变换和Hough变换的工程实践意义一样强。例如，在医学影像领域中，可以利用Radon变换将图像转换为一组投影图像，从而实现对图像中的骨骼和血管等细长弯曲、多枝的线性结构的分析和识别。
下面给出了一个针对SAR目标检测任务使用Radon变换的处理实例：
图5-4. SAR图像的Radon变换示例。上半段图像：输入SAR成像，图像上只能显示出微弱的明暗水平变化迹线特征。左下方：Radon变换的结果。右下方：RDRT的结果。
其中，旋转区域二分法Radon变换（Rotation Division and Region-based Truncation Radon Transform，RDRT）是将图像划分为多个子区域，并对每个子区域进行Radon变换。在进行变换时，可以根据每个子区域的特点和重要性，对Radon变换结果进行截断和抑制，从而实现对图像的特征提取和分析。该方法类似于上面提到过的局部区域直方图一样，是传统Radon变换的局部区域分析模式改进。
六、下期预告
下一章会总结图像的局部特征。图像的局部特征是指在图像中具有独特性、可区分性和重复性的小区域或点或者某种存在于图像上肉眼可分辨的模式。这些内容在本章其实已经有了比较概览性的总结。为了不让进阶篇显得重复累赘，下面一章我们主要会瞄准一些非常利于图像匹配、物体识别、3D重建等任务的局部特征展开，筛选那些在计算机视觉中被广泛应用的例子。与此同时，尽量使得所提取的特征满足不受图像缩放、旋转、平移等变换影响、具有鲁棒性好的优势。
另外，下面一章的一个核心的总结内容是特征描述子，也就是对图像中的局部特征进行数学表征的向量或矩阵。这些描述子是我们帮忙高效能地完成下游任务的规范化基础。
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1: Functions
Precalculus 1e (OpenStax)
{ }
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Mon, 02 May 2022 02:11:47 GMT
1.R: Functions (Review)
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1.R: Functions (Review)
Last updated May 2, 2022
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1.E: Functions (Exercises)
2: Linear Functions
Page ID 19619
This page is a draft and is under active development.
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OpenStax
( \newcommand{\kernel}{\mathrm{null}\,})
Table of contents
1. 1.1: Functions and Function Notation
2. 1.2: Domain and Range
3. 1.3: Rates of Change and Behavior of Graphs
4. 1.4: Composition of Functions
5. 1.5: Transformation of Functions
6. 1.6: Absolute Value Functions
7. 1.7: Inverse Functions
8. Practice Test
9. Contributors and Attributions
1.1: Functions and Function Notation
For the exercises 1-4, determine whether the relation is a function.
1) {(a,b),(c,d),(e,d)}
Answer
function
2) {(5,2),(6,1),(6,2),(4,8)}
3) y 2+4=x,for x the independent variable and y the dependent variable
Answer
not a function
4) Is the graph in the Figure below a function?
For the exercises 5-6, evaluate the function at the indicated values: f(−3);f(2);f(−a);−f(a);f(a+h)
5) f(x)=−2x 2+3x
Answer
f(−3)=−27;f(2)=−2;f(−a)=−2a 2−3a;−f(a)=2a 2−3a;f(a+h)=−2a 2+3a−4ah+3h−2h 2
6) f(x)=2|3x−1|
For the exercises 7-8, determine whether the functions are one-to-one.
7) f(x)=−3x+5
Answer
one-to-one
8) f(x)=|x−3|
For the exercises 9-11, use the vertical line test to determine if the relation whose graph is provided is a function.
9)
Answer
function
10)
11)
Answer
function
For the exercises 12-13, graph the functions.
12) f(x)=|x+1|
13) f(x)=x 2−2
Answer
For the exercises 14-17, use the Figure below to approximate the values.
14) f(2)
15) f(−2)
Answer
2
16) If f(x)=−2, then solve for x
17) If f(x)=1, then solve for x
Answer
x=−1.8 or x=1.8
For the exercises 18-19, use the function h(t)=−16t 2+80t to find the values.
18) h(2)−h(1)2−1
19) h(a)−h(1)a−1
Answer
−64+80a−16a 2−1+a=−16a+64
1.2: Domain and Range
For the exercises 1-4, find the domain of each function, expressing answers using interval notation.
1) f(x)=2 3x+2
2) f(x)=x−3 x 2−4x−12
Answer
(−∞,−2)∪(−2,6)∪(6,∞)
3)
4) Graph this piecewise function: f(x)={x+1 x<−2−2x−3 x≥−2
Answer
1.3: Rates of Change and Behavior of Graphs
For the exercises 1-3, find the average rate of change of the functions from x=1 to x=2
1) f(x)=4x−3
2) f(x)=10x 2+x
Answer
31
3) f(x)=−2 x 2
For the exercises 4-6, use the graphs to determine the intervals on which the functions are increasing, decreasing, or constant.
4)
Answer
increasing (2,∞); decreasing (−∞,2)
5)
6)
Answer
increasing (−3,1); constant (−∞,−3)∪(1,∞)
7) Find the local minimum of the function graphed in Exercise 4.
8) Find the local extrema for the function graphed in Exercise 5.
Answer
local minimum (−2,−3); local maximum (1,3)
9) For the graph in the Figure in Exercise 10, the domain of the function is [−3,3]. The range is [−10,10]. Find the absolute minimum of the function on this interval.
10) Find the absolute maximum of the function graphed in the Figure below.
Answer
(−1.8,10)
1.4: Composition of Functions
For the exercises 1-5, find (f∘g)(x) and (g∘f)(x) for each pair of functions.
1) f(x)=4−x,g(x)=−4x
2) f(x)=3x+2,g(x)=5−6x
Answer
(f∘g)(x)=17−18x;(g∘f)(x)=−7−18x
3) f(x)=x 2+2x,g(x)=5x+1
4) f(x)=x+2,g(x)=1 x
Answer
(f∘g)(x)=1 x+2;(g∘f)(x)=1 x+2
5) f(x)=x+3 2,g(x)=1−x
For the exercises 6-9, find (f∘g) and the domain for (f∘g)(x) for each pair of functions.
6) f(x)=x+1 x+4,g(x)=1 x
Answer
(f∘g)(x)=1+x 1+4x,x≠0,x≠−1 4
7) f(x)=1 x+3,g(x)=1 x−9
8) f(x)=1 x,g(x)=x
Answer
(f∘g)(x)=1 x,x>0
9) f(x)=1 x 2−1,g(x)=x+1
For the exercises 10-11, express each function H as a composition of two functions f and g where H(x)=(f∘g)(x)
10) H(x)=2x−1 3x+4
Answer
sample: g(x)=2x−1 3x+4;f(x)=x
11) H(x)=1(3x 2−4)−3
1.5: Transformation of Functions
For the exercises 1-8, sketch a graph of the given function.
1) f(x)=(x−3)2
Answer
2) f(x)=(x+4)3
3) f(x)=x+5
Answer
4) f(x)=−x 3
5) f(x)=−x 3
Answer
6) f(x)=5−x−4
7) f(x)=4[|x−2|−6]
Answer
8) f(x)=−(x+2)2−1
For the exercises 9-10, sketch the graph of the function g if the graph of the function f is shown in the Figure below.
9) g(x)=f(x−1)
Answer
10) g(x)=3f(x)
For the exercises 11-12, write the equation for the standard function represented by each of the graphs below.
11)
Answer
f(x)=|x−3|
12)
For the exercises 13-15, determine whether each function below is even, odd, or neither.
13) f(x)=3x 4
Answer
even
14) g(x)=x
15) h(x)=1 x+3x
Answer
odd
For the exercises 16-18, analyze the graph and determine whether the graphed function is even, odd, or neither.
16)
17)
Answer
even
18)
1.6: Absolute Value Functions
For the exercises 1-3, write an equation for the transformation of f(x)=|x|.
1)
Answer
f(x)=1 2|x+2|+1
2)
3)
Answer
f(x)=−3|x−3|+3
For the exercises 4-6, graph the absolute value function.
4) f(x)=|x−5|
5) f(x)=−|x−3|
Answer
6) f(x)=|2x−4|
For the exercises 7-8, solve the absolute value equation.
7) |x+4|=18
Answer
x=−22,x=14
8) |1 3x+5|=|3 4x−2|
For the exercises 9-10, solve the inequality and express the solution using interval notation.
9) |3x−2|<7
Answer
(−5 3,3)
10) |1 3x−2|≤7
1.7: Inverse Functions
For the exercises 1-2, find f−1(x) for each function.
1) f(x)=9+10x
2) f(x)=x x+2
Answer
f−1(x)=−2x x−1
3) For the following exercise, find a domain on which the function f is one-to-one and non-decreasing. Write the domain in interval notation. Then find the inverse of f restricted to that domain. (1.R.1)f(x)=x 2+1
4) Given f(x)=x 3−5 and g(x)=x+5 3 :
Find f(g(x)) and g(f(x)).
What does the answer tell us about the relationship between f(x) and g(x)?
Answer
1. f(g(x))=x and g(f(x))=x
2. This tells us that f and g are inverse functions
For the exercises 5-8, use a graphing utility to determine whether each function is one-to-one.
5) f(x)=1 x
Answer
The function is one-to-one.
6) f(x)=−3x 2+x
Answer
The function is not one-to-one.
7) If f(5)=2, find f−1(2)
Answer
5
8) If f(1)=4, find f−1(4)
Practice Test
For the exercises 1-2, determine whether each of the following relations is a function.
1) y=2x+8
Answer
The relation is a function.
2) {(2,1),(3,2),(−1,1),(0,−2)}
For the exercises 3-4, evaluate the function f(x)=−3x 2+2x at the given input.
3) f(−2)
Answer
−16
4) f(a)
5) Show that the function f(x)=−2(x−1)2+3 is not one-to-one.
Answer
The graph is a parabola and the graph fails the horizontal line test.
6) Write the domain of the function f(x)=3−x in interval notation.
7) Given f(x)=2x 2−5x, find f(a+1)−f(1)
Answer
2a 2−a
8) Graph the function f(x)={x+1 if−2<x<3−x if x≥3
9) Find the average rate of change of the function f(x)=3−2x 2+x by finding f(b)−f(a)b−a
Answer
−2(a+b)+1
For the exercises 10-11, use the functions f(x)=3−2x 2+x and g(x)=x to find the composite functions.
10) (g∘f)(x)
11) (g∘f)(1)
Answer
2
12) Express H(x)=5x 2−3x 3 a composition of two functions, f and g, where (f∘g)(x)=H(x)
For the exercises 13-14, graph the functions by translating, stretching, and/or compressing a toolkit function.
13) f(x)=x+6−1
Answer
14) f(x)=1 x+2−1
For the exercises 15-17, determine whether the functions are even, odd, or neither.
15) f(x)=−5 x 2+9x 6
Answer
even
16) f(x)=−5 x 3+9x 5
17) f(x)=1 x
Answer
odd
18) Graph the absolute value function f(x)=−2|x−1|+3.
19) Solve |2x−3|=17.
Answer
x=−7 and x=10
20) Solve −|1 3x−3|≥17. Express the solution in interval notation.
For the exercises 21-22, find the inverse of the function.
21) f(x)=3x−5
Answer
f−1(x)=x+5 3
22) f(x)=4 x+7
For the exercises 23-26, use the graph of g shown in the Figure below.
23) On what intervals is the function increasing?
Answer
(−∞,−1.1) and (1.1,∞)
24) On what intervals is the function decreasing?
25) Approximate the local minimum of the function. Express the answer as an ordered pair.
Answer
(1.1,−0.9)
26) Approximate the local maximum of the function. Express the answer as an ordered pair.
For the exercises 27-29, use the graph of the piecewise function shown in the Figure below.
27) Find f(2).
Answer
f(2)=2
28) Find f(−2).
29) Write an equation for the piecewise function.
Answer
f(x)={|x|if x≤2 3 if x>2
For the exercises 30-35, use the values listed in the Table below.
x F(x)
0 1
1 3
2 5
3 7
4 9
5 11
6 13
7 15
8 17
30) Find F(6).
31) Solve the equation F(x)=5
Answer
x=2
32) Is the graph increasing or decreasing on its domain?
33) Is the function represented by the graph one-to-one?
Answer
yes
34) Find F−1(15).
35) Given f(x)=−2x+11, find f−1(x).
Answer
f−1(x)=−x−11 2
Contributors and Attributions
Jay Abramson (Arizona State University) with contributing authors. Textbook content produced by OpenStax College is licensed under aCreative Commons Attribution License 4.0license.Download for free at
This page titled 1.R: Functions (Review) is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by OpenStax via source content that was edited to the style and standards of the LibreTexts platform.
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