# 4.4: The Heat Conduction Equation

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- 7231

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.

In this case the temperature gradient is written as a partial derivative, \( \frac{ \delta T}{ \delta x} and is not uniform down the length of the rod. We'll suppose it is \frac{ \deta T}{ \delta x} at *x* and \( \frac{ \delta T}{ \delta x} + \frac{ \delta^2 T}{ \delta x^2} \delta x\) 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 \( -KA \frac{ \delta T}{ \delta x}\), and the rate of flow out at x + δx is \( -KA \left( \frac{ \delta T}{ \delta x} + \frac{ \delta^2 T}{ \delta x^2} \delta x \right)\), so that the net flow of heat into that portion is \( KA \frac{ \delta^2 T}{ \delta x^2} \delta x\). This must be equal to \( C \rho A \delta x \frac{ \delta T}{ \delta t}\), where ρ is the density (and hence ρ*A*δ*x* is the mass of the portion), and *C* is the specific heat capacity.

Therefore

\[ C \rho \frac{ \delta T}{ \delta t} = K \frac{ \delta^2 T}{ \delta x^2}.\]

This can be written

\[ \frac{ \delta T}{ \delta t} = D \frac{ \delta^2 T}{ \delta x^2},\]

where

\[ D = \frac{K}{C \rho}\]

is the *thermal diffusivity* (m^{2} s^{−1}).

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

\[ T = D \nabla^2 T.\]