Solution of integral equations of intensity moments for radiative transfer in an anisotropically scattering medium with a linear refractive index

In this work, we derive the integral equations of radiative transfer in terms of intensity moments for radiative transfer in an anisotropically scattering slab with a spatially varying refractive index (VRI). The integral equations are solved by the Nyström method. We apply this method to study radiative heat transfer in a cold slab with higher-degree anisotropic scattering and linearly VRI. The slab lays on an opaque substratum. The refractive index may have a jump at the interface between the surroundings and the slab, while the interface between the slab and the substratum is assumed to be non-reflecting. To exemplify the application of the integral formulation, we consider the case with irradiation from external source in the surroundings and the case with an emitting substratum. We also solve the problems by the Monte Carlo method (MCM). The hemispherical reflectance and transmittance of the slabs obtained by solving integral equations are in excellent agreement with those obtained by the MCM. A positive gradient of refractive index (n′) enhances forward radiative transfer, and so the dimensionless radiative heat flux increases with the increase of n′ for the cases with irradiation from the surroundings. Effects of the optical thickness, the scattering albedo and the scattering phase function are also investigated.