Exact Solution of the Radiation Heat Transport Equation in a Gas-Filled Spherical Cavity
This memorandum treats the case 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 problem is formulated in terms of an integral equation with a difference kernel over a finite interval. A complex function is introduced for the source function. This in turn leads to a singular integral equation of the principal-value type that can be solved by standard techniques. An exact solution is found that yields the entire distribution of the gas emissive power. When this exact transport theory solution is compared with the Rosseland diffusion theory, the latter is shown to be capable of describing only the overall spatial dependence of the emissive power distribution. The solution has applications for high-temperature systems such as military rockets, reentry vehicles, gaseous-fueled cavity reactors, and modern power plants.