Radiative equilibrium in a rectangular enclosure bounded by gray walls

Two-dimensional temperature and heat-flux distributions are calculated for an absorbing-emitting gray medium at radiative equilibrium in a rectangular enclosure. The bounding walls are gray and diffuse with arbitrary surface-temperature distributions, and heat generation may take place inside the medium. As a first approximation, the problem is solved for optically thick systems (differential approximation). These results are subsequently improved by the introduction of a number of geometrical parameters to yield good accuracy for all optical thicknesses. As examples, two cases are discussed in detail: (1) uniform heat generation in a black enclosure, and (2) an enclosure with one gray surface at constant temperature. Comparison with some numerical solutions generated by Hottel's /Hottel and Cohen (1958) and Einstein (1963)/ zonal method shows excellent agreement.