(a) A spherical shell has inner and outer radii a and b, respectively, and the temperatures at the inner and outer surfaces are T2 and T1. The thermal conductivity of the material of which the shell is made is k. Derive an equation for the total heat current through the shell.
(b) Derive an equation for the temperature variation within the shell in part (a); that is, calculate T as a function of r, the distance from the center of the shell.
(c) A hollow cylinder has length L, inner radius a, and outer radius b, and the temperatures at the inner and outer surfaces are T2 and T1. (The cylinder could represent an insulated hot-water pipe, for example.) The thermal conductivity of the material of which the cylinder is made is k. Derive an equation for the total heat current through the walls of the cylinder.
(d) For the cylinder of part (c), derive an equation for the temperature variation inside the cylinder walls.
(e) For the spherical shell of part (a) and the hollow cylinder of part (c), show that the equation for the total heat current in each case reduces to Eq. (17.21) for linear beat flow when the shell or cylinder is very thin.