Steady-state heat conduction in slabs, cylindrical and spherical shells with non-uniform heat generation

Abstract

A comparison is made between temperature drops in cylindrical and spherical shells and in slabs in the cases of uniform and non-uniform internal heat generation with the same total amount of heat. One boundary is assumed to. be insulated; the rate of internal heat generation may be an arbitrary function of the radius. The (variational) ratio between the relative difference in temperature drops (in the non-uniform and uniform cases) and a (dimensionless) measure of the non-uniformity in heat generation is found to vary between narrow bounds which depend only upon the geometry. When the power density is better known, the bounds on the variational ratio get closer together. Similar derivations are made for the average temperature when one boundary is insulated or when both boundaries are at the same temperature. Kernels allowing the exact computation of temperatures of interest are listed in tables for various geometries and boundary conditions.