Exact solutions to the radiation heat transport equation in gaseous media using singular integral equation theory

Abstract

This paper treats the problem of steady-state, radiative heat transport through an absorbing, emitting, and heat-generating gray gas contained inside a black-wall spherical cavity. An exact analytical solution to the radiative transport equation is obtained for the case of uniform heat generation throughout the gaseous medium. The integral equation governing heat transfer by radiation is solved by introducing a complex function for the source distribution, which in turn leads to a singular integral equation of the principal-value type. This equation is solved by standard techniques. It is found that the exact solution involves the iteration of two Fredholm equations which have rapidly convergent solutions for all optical radii. A closed-form asymptotic solution is developed for the case of an optically thick medium. Numerical studies show that this closed-form solution is valid over a wide range of optical radii. Comparisons of the exact transport theory solution with diffusion theory are made and the accuracy and range of validity of diffusion theory is found. The heat flux radiated from the surface of a gaseous medium is determined exactly for all optical thicknesses. Applications of the theory to problems in plane geometry are also briefly discussed.