Differential Equations for Heat Conduction in Different Coordinates
Explore the general differential equations for heat conduction in Cartesian and cylindrical coordinates, along with steady state conditions and Laplace equation. Fourier's law of heat conduction and the thermal diffusivity are discussed, providing a comprehensive overview of heat conduction principles.
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CHAPTER TWO- GENERAL DIFFERENTIAL EQUATION FOR HEAT CONDUCTION
GENERAL DIFFERENTIAL EQUATION FOR HEAT CONDUCTION 1.1 General differential equation for heat conduction in Cartesian coordinates We can write it mathematically as ??? ????+ ????= ??? .1 where, ???= energy entering the control volume per unit time. ????= energy leaving the control volume per unit time. ????= energy generated within the control volume per unit time ???= energy storage within the control volume per unit time
???= ?? + ?? + ??.2 ????= ??+??+ ??+??+ ??+?? .3 Now, from calculus, we know that ??+??etc. can be expressed by a Taylor series expansion, where, neglecting the higher order terms, we can write, ??+??= ??+??? ?? ??+??= ??+??? ?? ??+??= ??+??? ?? ????= ????.??.?? .5 .?? .4.1 .?? .4.2 .?? .4.3 ?? ?? .6 ???= ?.?? ?? ??.?? Now, substituting for all terms in Eq. 1, we get, ??? ????+ ????= ??? ?? ?? ??+ ??+ ?? ??+??+ ??+??+ ??+?? + ????.??.?? = ?.?? ?? ??.?? ??? ?? ?? ?? .?? +??? .?? +??? ?? ???? ?? ?? ..7 .?? + ????.??.?? = ? ??
Now let us bring in Fourier's law of heat conduction. ?? ??= ??????? ?? ??= ??????? ?? ??= ??????? ??= ??? ?? 8.1 ??= ??? ?? 8.2 ??= ??? ?? 8.3 Substituting Eq.(8), in Eq.(7), and dividing by ??.??.??, we obtain, ? ?? ??? ?? ? ?? ??? ?? ??2+?2? +? ??? ?? ?? ?? + + ??= ? ?? ?? ?2? ??2+?2? ?? ?? ? + ??= ? ?? ??2 ?2? ??2+?2? ?2? ??2+?2? ??2+?2? ??2+?2? +?? ?=? ?? +?? ?? ??=1 ?? ?? 9 ?? ?? ??2 ? ? ?=1 ??2 ? Where, ? = ? ???is thermal diffusivity.
Steady state: This means that the temperature at any position does not change with time, i.e. ?? ??= 0 Eq.9, becomes: ?2? ??2+?2? ??2+?2? +?? ?= 0 ??2 With no Internal heat generation: This means that qgterm is zero. So, Eq. 9 becomes, ?2? ??2+?2? ??2+?2? =1 ?? ?? ??2 ? Steady state, with no Internal heat generation: This means that qgand ?? ??are zero. So, Eq. 9 becomes. ?2? ??2+?2? ??2+?2? = 0 ??2 This is known as Laplace equation, and it represents steady state, three-dimensional heat conduction with no internal heat generation, with constant thermal conductivity, in Cartesian coordinates. One-dimensional, steady state, with no internal heat generation: This means that. ?2? ??2=?2? ?? ??= 0 ??2= 0 , ??= 0 ??? So Eq. 9 becomes, ?2? ??2
1.2 General differential equation for heat conduction in cylindrical coordinates ? = ? ??? ? = ? ??? ? = ? = ??? 1 ? ? The resulting general differential equation in cylindrical coordinates is, ?2? ? 2+?2? ??2+?? 1 ? ? ?? ??? ?? +1 ?=1 ?? ?? 10 ?2 ?
General differential equation for heat conduction in spherical coordinates? = ? ???? ??? ? = ? ???? ??? ? = ? ???? The resulting general differential equation in spherical coordinates is, ?2? ? 2+?? 1 ? ? ?? ?2?? 1 ? ?????? 1 ?=1 ?? ?? ..13 + + ?2???? ?2???? ?? ?? ?? ?
