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id stringlengths 40 40	title stringlengths 15 120	text stringlengths 41 3.14k	source stringclasses 2 values
bd6784444af303e45ce98b38ce623542384a0e28	the above equation is nothing but emden–chandrasekhar equation,	the above equation is nothing but emden–chandrasekhar equation, which appears in astrophysics describing isothermal gas sphere. unlike planar and cylindrical case, the spherical vessel has infinitely many solutions for δ < δ c {\displaystyle \delta <\delta _{c}} oscillating about the point δ = 2 {\displaystyle \delta =...	wikipedia
be2a8b731f632af2c19dab1520e2de3715d68d6e	if the transformations Θ = θ m −	if the transformations Θ = θ m − θ {\displaystyle \theta =\theta _{m}-\theta } and ξ 2 = δ e θ m η 2 {\displaystyle \xi ^{2}=\delta e^{\theta _{m}}\eta ^{2}}, where θ m {\displaystyle \theta _{m}} is the maximum temperature which occurs at η = 0 {\displaystyle \eta =0} due to symmetry, are introduced	wikipedia
ca7fe5a9f640c9e447036348c70a086e4f1db468	for each value of δ {\displaystyle \delta },	for each value of δ {\displaystyle \delta }, we have two values of θ m {\displaystyle \theta _{m}} since b {\displaystyle b} is multi-valued. the maximum critical temperature is θ m, c = ln 4 {\displaystyle \theta _{m,c}=\ln 4}.	wikipedia
23e6de8a4d3bc0d8aaf03cd81f215234bebeb3d7	now if we use the other boundary condition	now if we use the other boundary condition θ (1) = 0 {\displaystyle \theta (1)=0}, we get an equation for b {\displaystyle b} as δ (b + 1) 2 − 8 b = 0 {\displaystyle \delta (b+1)^{2}-8b=0}. the maximum value of δ {\displaystyle \delta } for which solution is possible is when b = 1 {\displaystyle b=1}, so the critical f...	wikipedia
65f462f40966defed8dde6473b2ed019fad0409e	integrating once again, we have θ = a	integrating once again, we have θ = a − 2 ln (b η 2 + 1) {\displaystyle \theta =a-2\ln(b\eta ^{2}+1)} where b = 1 / b ′ {\displaystyle b=1/b'}. we have used already one boundary condition, there is one more boundary condition left, but with two constants a, b {\displaystyle a,\ b}. it turns out a {\displaystyle a} and ...	wikipedia
1897b84909c7fd18b5429f9e09fbf7781e2cce91	the general solution is ω 2 + 4	the general solution is ω 2 + 4 ω + c = − 2 δ χ {\displaystyle \omega ^{2}+4\omega +c=-2\delta \chi }. but c = 0 {\displaystyle c=0} from the symmetry condition at the centre. writing back in original variable, the equation reads,	wikipedia
3922b1d9786bd7ba359a63ac4b5dd70232518214	if the transformations ω = η d θ	if the transformations ω = η d θ / d η {\displaystyle \omega =\eta d\theta /d\eta } and χ = e θ η 2 {\displaystyle \chi =e^{\theta }\eta ^{2}} are introduced	wikipedia
fbb97ef4f2b95c3a69c339b32aae333c409ee95c	for cylindrical vessel, there is an exact solution.	for cylindrical vessel, there is an exact solution. though frank-kamentskii used numerical integration assuming there is no explicit solution, paul l. chambré provided an exact solution in 1952. h. lemke also solved provided a solution in a somewhat different form in 1913. here j = 1 {\displaystyle j=1}, then	wikipedia
4a7bd9042dc0e933514b41e6c430723ec6e80a92	so the critical frank-kamentskii parameter is δ c	so the critical frank-kamentskii parameter is δ c = 0.8785 {\displaystyle \delta _{c}=0.8785}. the system has no steady state (or explodes) for δ > δ c = 0.8785 {\displaystyle \delta >\delta _{c}=0.8785} and for δ < δ c = 0.8785 {\displaystyle \delta <\delta _{c}=0.8785}, the system goes to a steady state with very slo...	wikipedia
9e2e0a52944cc544bc155f78d64bf720c877e2ea	critical δ c {\displaystyle \delta _{c}} is obtained	critical δ c {\displaystyle \delta _{c}} is obtained by finding the maximum point of the equation (see figure), i.e., d δ / d θ m = 0 {\displaystyle d\delta /d\theta _{m}=0} at δ c {\displaystyle \delta _{c}}.	wikipedia
d79ba39aed8caad7e9271868ea2477498e668a9d	the above equation is the exact solution, but	the above equation is the exact solution, but θ m {\displaystyle \theta _{m}} maximum temperature is unknown, but we have not used the boundary condition of the wall yet. thus using the wall boundary condition θ = 0 {\displaystyle \theta =0} at η = 1 {\displaystyle \eta =1}, the maximum temperature is obtained from an ...	wikipedia
be2a8b731f632af2c19dab1520e2de3715d68d6e	if the transformations Θ = θ m −	if the transformations Θ = θ m − θ {\displaystyle \theta =\theta _{m}-\theta } and ξ 2 = δ e θ m η 2 {\displaystyle \xi ^{2}=\delta e^{\theta _{m}}\eta ^{2}}, where θ m {\displaystyle \theta _{m}} is the maximum temperature which occurs at η = 0 {\displaystyle \eta =0} due to symmetry, are introduced	wikipedia
bea90642383eb62ffa1a0eb27c471bbd6767e5e6	the only parameter which characterizes the explosion is	the only parameter which characterizes the explosion is the damköhler number δ {\displaystyle \delta }. when δ {\displaystyle \delta } is very high, conduction time is longer than the chemical reaction time and the system explodes with high temperature since there is not enough time for conduction to remove the heat. o...	wikipedia
76b95e22cdcc2b39910c37e00e974fc314ce829c	which implies that the ignition time τ →	which implies that the ignition time τ → ∞ {\displaystyle \tau \rightarrow \infty } as δ − δ c → 0 + {\displaystyle \delta -\delta _{c}\rightarrow 0^{+}} with a square-root singularity.	wikipedia
437f08a70b38d1979a41a738c3fe09dcd7b5d3a4	which immediately reveals that ψ → ∞ {\displaystyle	which immediately reveals that ψ → ∞ {\displaystyle \psi \rightarrow \infty } when ζ = 2 π δ c. {\displaystyle \zeta ={\sqrt {2}}\pi \delta _{c}.} writing this condition in terms of τ {\displaystyle \tau }, the explosion time in the near-critical condition is found to be	wikipedia
fc97961a21a79988de047b06fc51149be92eaa2d	in the near-critical condition, i.e., when δ −	in the near-critical condition, i.e., when δ − δ c → 0 + {\displaystyle \delta -\delta _{c}\rightarrow 0^{+}}, the system takes very long time to explode. the analysis for this limit was first carried out by frank-kamenetskii., although proper asymptotics were carried out only later by d. r. kassoy and amable liñán inc...	wikipedia
be1a495fbb6b1c72e12a41585d742c9b321b6754	at time τ = 1 {\displaystyle \tau =1},	at time τ = 1 {\displaystyle \tau =1}, the system explodes. this time is also referred to as the adiabatic induction period since the heat conduction term − θ / δ {\displaystyle -\theta /\delta } is neglected.	wikipedia
e97a48626d4b4e36c51bbef56c72e9afca0655ad	for δ c < δ < ∞ {\displaystyle	for δ c < δ < ∞ {\displaystyle \delta _{c}<\delta <\infty }, the system explodes since the exponential term dominates as time proceeds. we do not need to wait for a long time for θ {\displaystyle \theta } to blow up. because of the exponential forcing, θ → ∞ {\displaystyle \theta \rightarrow \infty } at a finite value ...	wikipedia
df69979b2266b25c4670dcaa91dcd22364449f3e	where w {\displaystyle w} represents the lambert w	where w {\displaystyle w} represents the lambert w function. from the properties of lambert w function, it is easy to see that the steady state temperature provided by the above equation exists only when δ ≤ δ c = 1 / e {\displaystyle \delta \leq \delta _{c}=1/e}, where δ c {\displaystyle \delta _{c}} is called as fran...	wikipedia
b87e0768fbb8f75f80011c5cba3fc8bedf94f221	when 0 < δ < e − 1	when 0 < δ < e − 1 {\displaystyle 0<\delta <e^{-1}}, the linear term eventually dominates and the system is able to reach a steady state as τ → ∞ {\displaystyle \tau \rightarrow \infty }. at steady state (d θ / d τ = 0 {\displaystyle d\theta /d\tau =0}), the balance is given by the equation	wikipedia
f5b1b8a17a17592f7ce305cd10aa95dfe77dc1a9	in which the exponential term e θ {\displaystyle	in which the exponential term e θ {\displaystyle e^{\theta }} will tend to increase θ {\displaystyle \theta } as time proceeds whereas the linear term − θ / δ {\displaystyle -\theta /\delta } will tend to decrease θ {\displaystyle \theta }. the relevant importance between the two terms are determined by the damköhler n...	wikipedia
55ee93552885912fe811f076b89ecef6b0c497c8	before frank-kamenetskii, his doctoral advisor nikolay semyonov (or	before frank-kamenetskii, his doctoral advisor nikolay semyonov (or semenov) proposed a thermal explosion theory with a simpler model with which he assumed a linear function for the heat conduction process instead of the laplacian operator. semenov's equation reads as	wikipedia
19472e428071d8c2c0620de81d404f58392d8d3a	at τ = 0 {\displaystyle \tau =0}, we	at τ = 0 {\displaystyle \tau =0}, we have θ (η, 0) = 0 {\displaystyle \theta (\eta,0)=0} and for τ > 0 {\displaystyle \tau >0}, θ {\displaystyle \theta } needs to satisfy θ (1, τ) = 0 {\displaystyle \theta (1,\tau)=0} and ∂ θ / ∂ η \| η = 0 = 0. {\displaystyle \partial \theta /\partial \eta \|_{\eta =0}=0.}	wikipedia
b185fd3a7c52bdc27a9217195cb6fc27b8e72609	since β ≫ 1 {\displaystyle \beta \gg 1},	since β ≫ 1 {\displaystyle \beta \gg 1}, the exponential term can be linearized e θ / (1 + θ / β) ≈ e θ {\displaystyle e^{\theta /(1+\theta /\beta)}\approx e^{\theta }}, hence	wikipedia
d616cb45a3b0562461882c91b9482765c3fd4ba4	an increment in temperature of order t −	an increment in temperature of order t − t o ∼ r t o 2 / e {\displaystyle t-t_{o}\sim rt_{o}^{2}/e}, where r t o 2 / e {\displaystyle rt_{o}^{2}/e} is the frank-kamenetskii temperature is enough to increase the chemical reaction by amount e {\displaystyle e}, as is evident from the ratio	wikipedia
94ab9d2d0e4b5cb7f17d42d3c19c8652ac6eb314	consider a vessel maintained at a constant temperature	consider a vessel maintained at a constant temperature t o {\displaystyle t_{o}}, containing a homogeneous reacting mixture. let the characteristic size of the vessel be a {\displaystyle a}. since the mixture is homogeneous, the density ρ {\displaystyle \rho } is constant. during the initial period of ignition, the con...	wikipedia
125d56c10f9ee787caa311db540e9cd82e04489d	in combustion, frank-kamenetskii theory explains the thermal explosion	in combustion, frank-kamenetskii theory explains the thermal explosion of a homogeneous mixture of reactants, kept inside a closed vessel with constant temperature walls. it is named after a russian scientist david a. frank-kamenetskii, who along with nikolay semenov developed the theory in the 1930s.	wikipedia
25fca8c6969cb61621fdeda39ffaeb5732f59901	lương duy cương is a vietnamese professional footballer	lương duy cương is a vietnamese professional footballer who plays as a centre back or a right back for v.league 1 club shb Đà nẵng and the vietnam national team. at the 2022 afc u-23 asian cup, he was also used as a defensive midfielder.	wikipedia
9169b0123871c8fdb09661aad854620daa807267	lương duy cương (born 7 november 2001) is	lương duy cương (born 7 november 2001) is a vietnamese professional footballer who plays as a centre back or a right back for v.league 1 club shb Đà nẵng and the vietnam national team.	wikipedia
7142ee88f97c1ca8f1c059452b20fb7dec1662e7	ernest a. calverley was an american professional basketball	ernest a. calverley was an american professional basketball player. he was an all-american while playing for the university of rhode island. he played professionally with the providence steamrollers of the basketball association of america for three seasons from 1946 to 1949. calverley led the league in assists and was...	wikipedia
cf5094541f4f4003cb30336acd11b21b30090e4e	he later returned to uri to coach the	he later returned to uri to coach the rams and led them to two ncaa division i men's basketball tournament appearances in 1961 and 1966, losing both times in the first round. calverely died of complications of an infection after insertion of a pacemaker on october 20, 2003.	wikipedia
03e585ea5c97097e7a244c1e4dce2d5630b55114	calverley led in assists per game in the	calverley led in assists per game in the league's first and second years, while playing for his home state team, the providence steamrollers. in his rookie season, calverley was selected to the all-baa second team. in his career, he missed only three games (one for each season). he had a 70.7% free throw percentage, wh...	wikipedia
b66df1a53ff19049794f9b9d1dc591cb31cbdaf9	calverley was retroactively declared the men's basketball season	calverley was retroactively declared the men's basketball season scoring leader for 1944. but calverley was most notable for one particular shot. dubbed the "shot heard round the world", it was a midcourt shot from 62 feet away, as time expired at madison square garden in the 1946 national invitation tournament against...	wikipedia
60a672c4768080c26a4e66059e1c44758734db79	ernest a. calverley (january 30, 1924 – october	ernest a. calverley (january 30, 1924 – october 20, 2003) was an american professional basketball player. he was an all-american while playing for the university of rhode island. he played professionally with the providence steamrollers of the basketball association of america for three seasons from 1946 to 1949. calve...	wikipedia
9591cafc343cf2f046851f700c78ba1ddb6f40b7	the 2023 north alabama lions football team represented	the 2023 north alabama lions football team represented the university of north alabama as a member of the united athletic conference during the 2023 ncaa division i fcs football season. led by first-year head coach brent dearmon, the lions played home games at the braly municipal stadium in florence, alabama.	wikipedia
9591cafc343cf2f046851f700c78ba1ddb6f40b7	the 2023 north alabama lions football team represented	the 2023 north alabama lions football team represented the university of north alabama as a member of the united athletic conference during the 2023 ncaa division i fcs football season. led by first-year head coach brent dearmon, the lions played home games at the braly municipal stadium in florence, alabama.	wikipedia
72c1036c01f85baf134b6d88e3ee3f447c87c15d	charles john priefer is an american former football	charles john priefer is an american former football coach who held a variety of defensive and special teams coaching positions at the college and professional levels of the sport, including 17 seasons in the national football league (nfl). he attended boromeo high school in wickliffe, ohio before matriculating to john ...	wikipedia
597a08603bb15c2f0a37a4bf1b43cc4f44368b95	priefer has helped mold pro bowl defensive tackle	priefer has helped mold pro bowl defensive tackle shaun rogers into an impressive kick-blocking force. since rogers arrived as a rookie in 2001, detroit has blocked a total of 13 kicks, eight by rogers. in 2005, rogers notched his seventh career blocked field goal when he re-directed a john kasay 52-yard attempt vs. ca...	wikipedia
2d50e5041ad8f7ff18f7f853f53f02fc0c54a29c	in 2004, the lions posted a single-season franchise	in 2004, the lions posted a single-season franchise record with five special teams return touchdowns. the previous record of four special teams return touchdowns was accomplished five times, with two of them coming under priefer's guidance (1951, 1992, 1994, 2002, and 2003).	wikipedia
c5b528805d5e060dbc48a0ddf12379f21962ce0c	priefer has formulated a superb combination of talent	priefer has formulated a superb combination of talent and scheme in developing the lions' return game into a force. five players have scored touchdowns on returns during priefer's tenure in detroit. in 1998, he guided terry fair to a league best 28.0 yards per kickoff average. in 2000, desmond howard's prolific punt an...	wikipedia