Chapter 2, Problem 2.16P

Steady-state, one-dimensional conduction occurs in a rod of constant thermal conductivity k and variable cross-sectional area $A_x\left(x\right)=A_O{e}^{ax},$ where $A_O$ and a are constants. The lateral surface of the rod is well insulated.

(a) Write an expression for the conduction heat rate, $q_x\left(x\right).$ Use this expression to determine the temperature distribution $T\left(x\right)$ and qualitatively sketch the distribution for $T\left(0\right)>T\left(L\right).$
(b) Now consider conditions for which thermal energy is generated in the rod at a volumetric rate $\stackrel{.}{q}={\stackrel{.}{q}}_O\exp \left(-ax\right),$ where ${\stackrel{.}{q}}_O$ is a constant. Obtain an expression for $q_x\left(x\right)$ when the left face $\left(x=0\right)$ is well insulated.