Analytical layered solution of radiation and non–Fourier conduction problems in optically complex media

Abstract

The analytical layered solution for radiative transfer through participating optically complex media is developed. The angular processing of radiation is based on double spherical harmonics method (DPN) which split up the radiative intensity into two stream components before expanding in Legendre polynomial basis. The proposed layered radiative solution assumes that the optically complex media is a set of thin layers dealing with homogeneous properties. Therefore, the analytical solution for radiative transfer is performed and then coupled to the finite volume method (FVM), to solve the non – linear hyperbolic thermal conduction problems, formulated thanks to Cattaneo–Vernotte flux. In developing the FVM, the Roe′s corrections of interface fluxes is adopted in order to enhance the performances of the method. The accuracy of the proposed model for dealing with inhomogeneous radiation/conduction problems is investigated by considering participating media such as slab, solid or hollow spheres, with temperature – dependent thermal conductivity. The effects of different parameters, known as scattering albedo, graded index function, thermal conductivity, boundary emissivity and the conduction–radiation parameter on both temperature and heat flux distributions for purely radiation, steady and transient states are studied. Results of the present work compare well with those available in the literature with maximum relative error less than 1%. These results show that the listed parameters have a significant effect on both temperature profiles and hyperbolic sharp wave front. It also comes from this study that the proposed layered approach is an efficient, robust, and accurate method for radiative flow analysis in inhomogeneous media while the Roe′s correction of interface fluxes in FVM is suitable to accommodate thermal wave front in non – Fourier analysis.