Steady-state, one-dimensional conduction occurs in a rod of constant thermal conductivity k and variable cross-sectional area As(x) = Aoe ax, where Ao and a are constants. The lateral surface of the rod is well insulated.

(a) Write an expression for the conduction heat rate, qx(x). Use this expression to determine the temperature distribution T(x) and qualitatively sketch the distribution for T (0) > T (L).

(b) Now consider conditions for which thermal energy is generated in the rod at a volumetric rate q = qo exp (-ax), where qo is a constant. Obtain an expression for qx(x) when the left face (x = 0) is well insulated.

A,(x) = A,eax Ao Lox