Starting with an energy balance on a rectangular volume element, derive the one-dimensional transient heat conduction equation for a plane wall with constant thermal conductivity and no heat generation.

Question

What will be an ideal response?

Answer 1

We consider a thin element of thickness ?x in a large plane wall. The density of the wall is p1, the specific heat is c, and the area of the wall normal to the direction of heat transfer is A. In the absence of any heat generation, an energy balance on this thin element of thickness ?x during a small time interval ?t can be expressed as



where



Substituting,



Dividing by A?x gives



Taking the limit as ?x - 0 and ?t - 0 yields



since from the definition of the derivative and Fourier’s law of heat conduction,



Noting that the area A of a plane wall is constant, the one-dimensional transient heat conduction equation in a plane wall with constant thermal conductivity k becomes



where the property a=k/pc is the thermal diffusivity of the material.