Isotropic scaling limits for one-dimensional radiative heat transfer with collimated incidence

As high-anisotropic scattering of materials induces difficulty in the resolution of radiative transfer equation, the isotropic assumption or approximate anisotropic phase functions are very useful. Isotropic scattering approximations need to be solved with a scaled optical depth and albedo. Isotropic scaling models involve the transformation of an anisotropic problem to an isotropic one. In this paper, the scaled optical depth and albedo are derived from the zero and first moment calculation of the scaled albedo and phase function product. Comparisons between the isotropic scaling, the anisotropic Henyey-Greenstein phase function approximations, and the exact solution are studied for one-dimensional steady state radiative heat transfer. The results show that both the isotropic scaling and the Henyey-Greenstein approximation are accurate for highly or weakly scattering media. For absorbing/scattering medium, the accuracy of the two approximations depends on the optical thickness and on the scattering function of the medium. The maximum errors between the two approximations and the benchmark solution are for albedo values around 0.5.