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# 4.4: The Heat Conduction Equation

• • Contributed by Jeremy Tatum
• Emeritus Professor (Physics & Astronomy) at University of Victoria

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

$T = D \nabla^2 T.$