Integral equation solutions based on exact ray paths for radiative transfer in a participating medium with formulated refractive index

The exact analytical path length of radiation traveling in a slab with formulated variable refractive index is derived. Based on the analytical path lengths, the integral equations in terms of intensity moments for radiative transfer in a participating slab with one of the family of spatially varying refractive indices are developed. We solve the integral equations for radiative transfer in a slab at radiative equilibrium or for radiative transfer in an isothermal slab. The boundaries are assumed to be black for the slab at radiative equilibrium and the index jumps at both boundaries for the isothermal slab are considered. For comparison purpose, we also solve the radiative equilibrium problems by the discrete ordinates method (DOM). The nondimensional emissive power and nondimensional radiative heat flux obtained by solving integral equations show an excellent agreement with those obtained by the DOM. For the slab at radiative equilibrium and with positive gradient of refractive index, the jump of the emissive power at bottom boundary decreases with the increase of optical thickness for the cases with slightly varying refractive index, but the trend may not hold for the cases with significantly varying refractive index. For the non-scattering slab with positive gradient of refractive index and fixed refractive indices at the boundaries, the directional emittances at both boundaries for the case with linear refractive index are smaller than those for the case with a refractive index of slope-increasing profile. Effects of the scattering albedo and the scattering phase function coefficient are investigated too.