Discrete ordinate solutions associated with the finite Legendre transform for radiative transfer in a slab with sinusoidal refractive index

In this work, we develop an alternative discrete ordinate approximation for radiative transfer in a refractive slab. The present method treats the angular derivative term of the radiative transfer equation for a planar medium with varying refractive index (VRI) by using a finite Legendre transform which gives a simple expression of the angular derivative term. Thus, the solution procedure does not march along direction, and so is not restricted to a monotonic variation of refractive index. We apply this method to study radiative heat transfer in a cold slab with anisotropic scattering, diffuse boundaries and sinusoidal VRI. We also solve the problems by the discrete curved ray tracing (DCRT). The hemispherical reflectance and transmittance of slabs with irradiation from the upper surroundings obtained by the present method and those obtained by the DCRT are in excellent agreement. For a slab of a sinusoidal refractive index with the minimum at the center plane, the gradient of refractive index causes the internal reflection of a part of irradiation, which reduces the transmittance of the slab. Other effects of the VRI, the optical thickness, the scattering albedo, the anisotropically scattering coefficient and the boundary reflection are also investigated.