Ultraspherical-polynomials approximation to the radiative heat transfer in a slab with reflective boundaries

The ultraspherical-polynomials approximation or the method, which is obtained by incorporating all approximations that employ ultraspherical polynomials into a unified form, is applied to the radiative transfer problem in plane-parallel, absorbing, emitting, non-isothermal, gray medium with linearly anisotropic scattering. The unique formulation provides a simple means to make comparative assessments and to analyze some qualitative aspects of various ultraspherical-polynomials approximations. Effects of the order of approximation, optical thickness, specular reflection, anisotropic scattering, and change of the source term on results are investigated for different pre-selected values of λ, each leading to a different approximation. All results obtained by the method are consistent in themselves, equiconvergent, and in good agreement with the comparable data in literature and with the results obtained from the computational fluid dynamics code FLUENT.