A modified spherical harmonic approximation for radiative transfer in a refractive slab

A modified spherical harmonics (P₃) approximation which applies the ordinary P₃ approximation to solve the fairly diffuse part of radiation due to in-scattering and treats the attenuated incident intensity rigorously is developed. The modified third-order P_n approximation (MP₃) is applied to analyze radiative heat transfer in a refractive slab exposed to diffuse irradiation. The refractive index of the medium is varying linearly. The results obtained by the MP₃ approximation are in good agreement with the solutions of the integral equations and those obtained by the Monte Carlo method for the cases with various combinations of optical thicknesses, scattering albedos and variable refractive indices. The results show that the modified P₃ approximation offers the possibility of more accurately treating the incident radiation at the boundaries which dominates the optically moderate and thin cases and the weak scattering cases.