Photon path length analysis of radiative heat transfer in planar layers with arbitrary temperature distributions

The internal source analytical technique is extended to predict the radiative heat transfer for a layer having an arbitrary temperature distribution. By combining a number of internal sources distributed at various optical depths in the layer and weighting them appropriately, a nonisothermal layer is modeled. Heat flux and intensity distributions within layers having a single internal source are presented. The distributions are found to present trends unique to the internal source problem. Isothermal layers are modeled and compare very well with published results. Increased accuracy is attained for all cases and particularly for larger optical depths and smaller albedos by increasing the number of internal sources. The technique is applied to a nonisothermal layer having a temperature distribution similar to that for a hot medium with a cold boundary region. The effect of the boundary region on the normalized heat flux leaving the layer is seen to collapse to a single line for small layer optical thicknesses and large albedos, the slope of which is governed by the temperature ratio Tmax/Tmin.