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The equation was found to match the Colebrook–White equation within 0.0497%. |
Praks and Brkić show one approximation of the Colebrook equation based on the Wright formula_40-function, a cognate of the Lambert W-function |
The equation was found to match the Colebrook–White equation within 0.0012%. |
Since Serghides's solution was found to be one of the most accurate approximation of the implicit Colebrook–White equation, Niazkar modified the Serghides's solution to solve directly for the Darcy–Weisbach friction factor "f" for a full-flowing circular pipe. |
Niazkar's solution is shown in the following: |
Niazkar's solution was found to be the most accurate correlation based on a comparative analysis conducted in the literature among 42 different explicit equations for estimating Colebrook friction factor. |
Early approximations for smooth pipes by Paul Richard Heinrich Blasius in terms of the Moody friction factor are given in one article of 1913: |
Johann Nikuradse in 1932 proposed that this corresponds to a power law correlation for the fluid velocity profile. |
Mishra and Gupta in 1979 proposed a correction for curved or helically coiled tubes, taking into account the equivalent curve radius, Rc: |
The following table lists historical approximations to the Colebrook–White relation for pressure-driven flow. Churchill equation (1977) is the only equation that can be evaluated for very slow flow (Reynolds number < 1), but the Cheng (2008), and Bellos et al. (2018) equations also return an approximately correct value for friction factor in the laminar flow region (Reynolds number < 2300). All of the others are for transitional and turbulent flow only. |
A continuity equation or transport equation is an equation that describes the transport of some quantity. It is particularly simple and powerful when applied to a conserved quantity, but it can be generalized to apply to any extensive quantity. Since mass, energy, momentum, electric charge and other natural quantities are conserved under their respective appropriate conditions, a variety of physical phenomena may be described using continuity equations. |
Continuity equations more generally can include "source" and "sink" terms, which allow them to describe quantities that are often but not always conserved, such as the density of a molecular species which can be created or destroyed by chemical reactions. In an everyday example, there is a continuity equation for the number of people alive; it has a "source term" to account for people being born, and a "sink term" to account for people dying. |
Any continuity equation can be expressed in an "integral form" (in terms of a flux integral), which applies to any finite region, or in a "differential form" (in terms of the divergence operator) which applies at a point. |
Continuity equations underlie more specific transport equations such as the convection–diffusion equation, Boltzmann transport equation, and Navier–Stokes equations. |
Flows governed by continuity equations can be visualized using a Sankey diagram. |
A continuity equation is useful when a "flux" can be defined. To define flux, first there must be a quantity which can flow or move, such as mass, energy, electric charge, momentum, number of molecules, etc. Let be the volume density of this quantity, that is, the amount of per unit volume. |
The way that this quantity is flowing is described by its flux. The flux of is a vector field, which we denote as j. Here are some examples and properties of flux: |
The integral form of the continuity equation states that: |
Mathematically, the integral form of the continuity equation expressing the rate of increase of within a volume is: |
In a simple example, could be a building, and could be the number of people in the building. The surface would consist of the walls, doors, roof, and foundation of the building. Then the continuity equation states that the number of people in the building increases when people enter the building (an inward flux through the surface), decreases when people exit the building (an outward flux through the surface), increases when someone in the building gives birth (a source, ), and decreases when someone in the building dies (a sink, ). |
By the divergence theorem, a general continuity equation can also be written in a "differential form": |
This general equation may be used to derive any continuity equation, ranging from as simple as the volume continuity equation to as complicated as the Navier–Stokes equations. This equation also generalizes the advection equation. Other equations in physics, such as Gauss's law of the electric field and Gauss's law for gravity, have a similar mathematical form to the continuity equation, but are not usually referred to by the term "continuity equation", because in those cases does not represent the flow of a real physical quantity. |
In the case that is a conserved quantity that cannot be created or destroyed (such as energy), and the equations become: |
In electromagnetic theory, the continuity equation is an empirical law expressing (local) charge conservation. Mathematically it is an automatic consequence of Maxwell's equations, although charge conservation is more fundamental than Maxwell's equations. It states that the divergence of the current density (in amperes per square metre) is equal to the negative rate of change of the charge density (in coulombs per cubic metre), |
Current is the movement of charge. The continuity equation says that if charge is moving out of a differential volume (i.e. divergence of current density is positive) then the amount of charge within that volume is going to decrease, so the rate of change of charge density is negative. Therefore, the continuity equation amounts to a conservation of charge. |
If magnetic monopoles exist, there would be a continuity equation for monopole currents as well, see the monopole article for background and the duality between electric and magnetic currents. |
In fluid dynamics, the continuity equation states that the rate at which mass enters a system is equal to the rate at which mass leaves the system plus the accumulation of mass within the system. |
The differential form of the continuity equation is: |
The time derivative can be understood as the accumulation (or loss) of mass in the system, while the divergence term represents the difference in flow in versus flow out. In this context, this equation is also one of the Euler equations (fluid dynamics). The Navier–Stokes equations form a vector continuity equation describing the conservation of linear momentum. |
If the fluid is incompressible (volumetric strain rate is zero), the mass continuity equation simplifies to a volume continuity equation: |
which means that the divergence of the velocity field is zero everywhere. Physically, this is equivalent to saying that the local volume dilation rate is zero, hence a flow of water through a converging pipe will adjust solely by increasing its velocity as water is largely incompressible. |
In computer vision, optical flow is the pattern of apparent motion of objects in a visual scene. Under the assumption that brightness of the moving object did not change between two image frames, one can derive the optical flow equation as: |
Conservation of energy says that energy cannot be created or destroyed. (See below for the nuances associated with general relativity.) Therefore, there is a continuity equation for energy flow: |
An important practical example is the flow of heat. When heat flows inside a solid, the continuity equation can be combined with Fourier's law (heat flux is proportional to temperature gradient) to arrive at the heat equation. The equation of heat flow may also have source terms: Although "energy" cannot be created or destroyed, "heat" can be created from other types of energy, for example via friction or joule heating. |
If there is a quantity that moves continuously according to a stochastic (random) process, like the location of a single dissolved molecule with Brownian motion, then there is a continuity equation for its probability distribution. The flux in this case is the probability per unit area per unit time that the particle passes through a surface. According to the continuity equation, the negative divergence of this flux equals the rate of change of the probability density. The continuity equation reflects the fact that the molecule is always somewhere—the integral of its probability distribution is always equal to 1—and that it moves by a continuous motion (no teleporting). |
Quantum mechanics is another domain where there is a continuity equation related to "conservation of probability". The terms in the equation require the following definitions, and are slightly less obvious than the other examples above, so they are outlined here: |
With these definitions the continuity equation reads: |
Either form may be quoted. Intuitively, the above quantities indicate this represents the flow of probability. The "chance" of finding the particle at some position and time flows like a fluid; hence the term "probability current", a vector field. The particle itself does "not" flow deterministically in this vector field. |
The total current flow in the semiconductor consists of drift current and diffusion current of both the electrons in the conduction band and holes in the valence band. |
This section presents a derivation the equation above for electrons. A similar derivation can be found for the equation for holes. |
Consider the fact that the number of electrons is conserved across a volume of semiconductor material with cross-sectional area, "A", and length, "dx", along the "x"-axis. More precisely, one can say: |
Total electron current is the sum of drift current and diffusion current: |
Aplying the product rule reults in the final expression: |
The key to solving these equations in real devices is whenever possible to select regions in which most of the mechanisms are negligible so that the equations reduce to a much simpler form. |
The notation and tools of special relativity, especially 4-vectors and 4-gradients, offer a convenient way to write any continuity equation. |
The density of a quantity and its current can be combined into a 4-vector called a 4-current: |
where is the speed of light. The 4-divergence of this current is: |
where is the 4-gradient and is an index labelling the spacetime dimension. Then the continuity equation is: |
in the usual case where there are no sources or sinks, that is, for perfectly conserved quantities like energy or charge. This continuity equation is manifestly ("obviously") Lorentz invariant. |
Examples of continuity equations often written in this form include electric charge conservation |
where is the electric 4-current; and energy–momentum conservation |
In general relativity, where spacetime is curved, the continuity equation (in differential form) for energy, charge, or other conserved quantities involves the "covariant" divergence instead of the ordinary divergence. |
For example, the stress–energy tensor is a second-order tensor field containing energy–momentum densities, energy–momentum fluxes, and shear stresses, of a mass-energy distribution. The differential form of energy–momentum conservation in general relativity states that the "covariant" divergence of the stress-energy tensor is zero: |
This is an important constraint on the form the Einstein field equations take in general relativity. |
However, the "ordinary" divergence of the stress–energy tensor does "not" necessarily vanish: |
The right-hand side strictly vanishes for a flat geometry only. |
As a consequence, the "integral" form of the continuity equation is difficult to define and not necessarily valid for a region within which spacetime is significantly curved (e.g. around a black hole, or across the whole universe). |
Quarks and gluons have "color charge", which is always conserved like electric charge, and there is a continuity equation for such color charge currents (explicit expressions for currents are given at gluon field strength tensor). |
There are many other quantities in particle physics which are often or always conserved: baryon number (proportional to the number of quarks minus the number of antiquarks), electron number, mu number, tau number, isospin, and others. Each of these has a corresponding continuity equation, possibly including source / sink terms. |
One reason that conservation equations frequently occur in physics is Noether's theorem. This states that whenever the laws of physics have a continuous symmetry, there is a continuity equation for some conserved physical quantity. The three most famous examples are: |
See Noether's theorem for proofs and details. |
In differential calculus, the Reynolds transport theorem (also known as the Leibniz–Reynolds transport theorem), or in short Reynolds theorem, is a three-dimensional generalization of the Leibniz integral rule which is also known as differentiation under the integral sign. |
The theorem is named after Osborne Reynolds (1842–1912). It is used to recast derivatives of integrated quantities and is useful in formulating the basic equations of continuum mechanics. |
Consider integrating over the time-dependent region that has boundary , then taking the derivative with respect to time: |
If we wish to move the derivative within the integral, there are two issues: the time dependence of , and the introduction of and removal of space from due to its dynamic boundary. Reynolds transport theorem provides the necessary framework. |
Reynolds transport theorem can be expressed as follows: |
in which is the outward-pointing unit normal vector, is a point in the region and is the variable of integration, and are volume and surface elements at , and is the velocity of the area element ("not" the flow velocity). The function may be tensor-, vector- or scalar-valued. Note that the integral on the left hand side is a function solely of time, and so the total derivative has been used. |
In continuum mechanics, this theorem is often used for material elements. These are parcels of fluids or solids which no material enters or leaves. If is a material element then there is a velocity function , and the boundary elements obey |
This condition may be substituted to obtain: |
If we take to be constant with respect to time, then and the identity reduces to |
as expected. (This simplification is not possible if the flow velocity is incorrectly used in place of the velocity of an area element.) |
The theorem is the higher-dimensional extension of differentiation under the integral sign and reduces to that expression in some cases. Suppose is independent of and , and that is a unit square in the -plane and has limits and . Then Reynolds transport theorem reduces to |
which, up to swapping and , is the standard expression for differentiation under the integral sign. |
The Fanning friction factor, named after John Thomas Fanning, is a dimensionless number used as a local parameter in continuum mechanics calculations. It is defined as the ratio between the local shear stress and the local flow kinetic energy density: |
In particular the shear stress at the wall can, in turn, be related to the pressure loss by multiplying the wall shear stress by the wall area ( formula_12 for a pipe with circular cross section) and dividing by the cross-sectional flow area ( formula_13 for a pipe with circular cross section). Thus formula_14 |
This friction factor is one-fourth of the Darcy friction factor, so attention must be paid to note which one of these is meant in the "friction factor" chart or equation consulted. Of the two, the Fanning friction factor is the more commonly used by chemical engineers and those following the British convention. |
The formulas below may be used to obtain the Fanning friction factor for common applications. |
The Darcy friction factor can also be expressed as |
For laminar flow in a round tube. |
From the chart, it is evident that the friction factor is never zero, even for smooth pipes because of some roughness at the microscopic level. |
The friction factor for laminar flow of Newtonian fluids in round tubes is often taken to be: |
where Re is the Reynolds number of the flow. |
For a square channel the value used is: |
For turbulent flow in a round tube. |
Blasius developed an expression of friction factor in 1913 for the flow in the regime formula_21. |
Koo introduced another explicit formula in 1933 for a turbulent flow in region of formula_23 |
When the pipes have certain roughness formula_25, this factor must be taken in account when the Fanning friction factor is calculated. The relationship between pipe roughness and Fanning friction factor was developed by Haaland (1983) under flow conditions of formula_26 |
As the roughness extends into turbulent core, the Fanning friction factor becomes independent of fluid viscosity at large Reynolds numbers, as illustrated by Nikuradse and Reichert (1943) for the flow in region of formula_30. The equation below has been modified from the original format which was developed for Darcy friction factor by a factor of formula_31 |
For the turbulent flow regime, the relationship between the Fanning friction factor and the Reynolds number is more complex and is governed by the Colebrook equation which is implicit in formula_2: |
Various explicit approximations of the related Darcy friction factor have been developed for turbulent flow. |
Stuart W. Churchill developed a formula that covers the friction factor for both laminar and turbulent flow. This was originally produced to describe the Moody chart, which plots the Darcy-Weisbach Friction factor against Reynolds number. The Darcy Weisbach Formula formula_35, also called Moody friction factor, is 4 times the Fanning friction factor formula_36 and so a factor of formula_37 has been applied to produce the formula given below. |
Due to geometry of non-circular conduits, the Fanning friction factor can be estimated from algebraic expressions above by using hydraulic radius formula_41 when calculating for Reynolds number formula_42 |
The friction head can be related to the pressure loss due to friction by dividing the pressure loss by the product of the acceleration due to gravity and the density of the fluid. Accordingly, the relationship between the friction head and the Fanning friction factor is: |
In fluid mechanics and astrophysics, the relativistic Euler equations are a generalization of the Euler equations that account for the effects of general relativity.They have applications in high-energy astrophysics and numerical relativity, where they are commonly used for describing phenomena such as gamma-ray bursts, accretion phenomena, and neutron stars, often with the addition of a magnetic field. "Note: for consistency with the literature, this article makes use of natural units, namely the speed of light" formula_1 "and the Einstein summation convention." |
For most fluids observable on Earth, traditional fluid mechanics based on Newtonian mechanics is sufficient. However, as the fluid velocity approaches the speed of light or moves through strong gravitational fields, or the pressure approaches the energy density (formula_2), these equations are no longer valid. Such situations occur frequently in astrophysical applications. For example, gamma-ray bursts often feature speeds only formula_3 less than the speed of light, and neutron stars feature gravitational fields that are more than formula_4 times stronger than the Earth's. Under these extreme circumstances, only a relativistic treatment of fluids will suffice. |
The equations of motion are contained in the continuity equation of the stress–energy tensor formula_5: |
where formula_7 is the covariant derivative. For a perfect fluid, |
Here formula_9 is the total mass-energy density (including both rest mass and internal energy density) of the fluid, formula_10 is the fluid pressure, formula_11 is the four-velocity of the fluid, and formula_12 is the metric tensor. To the above equations, a statement of conservation is usually added, usually conservation of baryon number. If formula_13 is the number density of baryons this may be stated |
These equations reduce to the classical Euler equations if the fluid three-velocity is much less than the speed of light, the pressure is much less than the energy density, and the latter is dominated by the rest mass density.To close this system, an equation of state, such as an ideal gas or a Fermi gas, is also added. |
In the case of flat space, that is formula_15 and using a metric signature of formula_16, the equations of motion are, |
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