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4.4: The Heat Conduction Equation

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The situation described in Section 4.2 and in figure IV.1 was a steady-state situation, in which the temperature was a function of x but not of time. We are now going to consider a more general situation in which the temperature may vary in time as well as in space.

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In this case the temperature gradient is written as a partial derivative, \( \frac{\partial T}{\partial x} and is not uniform down the length of the rod. We'll suppose it is \frac{ \deta T}{\partial x} at x and Tx+2Tx2δx at x + δx.

Consider the heat flow into and out of the portion between x and x + δx. The rate of flow into this portion at x is KATx, and the rate of flow out at x + δx is KA(Tx+2Tx2δx), so that the net flow of heat into that portion is KA2Tx2δx. This must be equal to CρAδxTt, where ρ is the density (and hence ρAδx is the mass of the portion), and C is the specific heat capacity.

Therefore

CρTt=K2Tx2.

This can be written

Tt=D2Tx2,

where

D=KCρ

is the thermal diffusivity (m2 s−1).

Equation 4.3.2 is the heat conduction equation. In three dimensions it is easy to show that it becomes

T=D2T.


This page titled 4.4: The Heat Conduction Equation is shared under a CC BY-NC license and was authored, remixed, and/or curated by Jeremy Tatum.

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