4.4: The Heat Conduction Equation

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{\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 $\frac{\partial T}{\partial x} + \frac{\partial ^2 T}{\partial x^2} \delta x$ 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 $-KA \frac{\partial T}{\partial x}$, and the rate of flow out at x + δx is $-KA \left( \frac{\partial T}{\partial x} + \frac{\partial ^2 T}{\partial x^2} \delta x \right)$, so that the net flow of heat into that portion is $KA \frac{\partial ^2 T}{\partial x^2} \delta x$. This must be equal to $C \rho A \delta x \frac{\partial T}{\partial t}$, where ρ is the density (and hence ρAδx is the mass of the portion), and C is the specific heat capacity.

Therefore

$$C \rho \frac{\partial T}{\partial t} = K \frac{\partial ^2 T}{\partial x^2}.$$

This can be written

$$\frac{\partial T}{\partial t} = D \frac{\partial ^2 T}{\partial x^2},$$

where

$$D = \frac{K}{C \rho}$$

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

$$T = D \nabla^2 T.$$