18.2.2.1 Solution stanexh-a

Question:

This is a continuation of a corresponding question in the subsection on the Laplace equation. See there for a definition of terms.

Derive the heat equation for unsteady heat conduction in a two-di­men­sion­al plate of thickness $\delta$, Do so by considering a little Cartesian rectangle of dimensions ${\Delta}x\times{\Delta}y$.

In particular, derive the heat conduction coefficient $\kappa$ in terms of the material heat coefficient $k$, the plate thickness $t$, and the specific heat of the solid $C_p$.

Answer:

The amount of thermal energy residing in the little rectangle will equal its volume times its density times its specific heat times its temperature (plus a constant that is not important):

$$E = \rho C_p u \Delta x \Delta y t$$

The time derivative of this energy is of course the net heat energy flowing in per unit time. Which is minus the net heat energy flowing out. That heat flow was derived in the corresponding question for the Laplace equation.

Put the two together and divide by ${\Delta}x{\Delta}y\delta$ to get the heat equation.

You should be able to find arguments like the above in many books on engineering mathematics.