Non-Linear Radiative-Conductive Heat Transfer in a Heterogeneous Gray Plane-Parallel Participating Medium

Radiative transfer considers problems that involve the physical phenomenon of energy transfer by radiation in media. These phenomena occur in a variety of realms (Ahmad & Deering, 1992; Tsai & Ozisik, 1989; Wilson & Sen, 1986; Yi et al., 1996) including optics (Liu et al., 2006), astrophysics (Pinte et al., 2009), atmospheric science (Thomas & Stamnes, 2002), remote sensing (Shabanov et al., 2007) and engineering applications like heat transport by radiation (Brewster, 1992) for instance or radiative transfer laser applications (Kim & Guo, 2004). Furthermore, applications to other media such as biological tissue, powders, paints among others may be found in the literature (see ref. (Yang & Kruse, 2004) and references therein). Although radiation in its basic form is understood as a photon flux that requires a stochastic approach taking into account local microscopic interactions of a photon ensemble with some target particles like atoms, molecules, or effective micro-particles such as impurities, this scenario may be conveniently modelled by a radiation field, i.e. a radiation intensity, in a continuous medium where a microscopic structure is hidden in effective model parameters, to be specified later. The propagation of radiation through a homogeneous or heterogeneous medium suffers changes by several isotropic or non-isotropic processes like absorption, emission and scattering, respectively, that enter the mathematical approach in form of a non-linear radiative transfer equation. The non-linearity of the equation originates from a local thermal description using the Stefan-Boltzmann law that is related to heat transport by radiation which in turn is related to the radiation intensity and renders the radiative transfer problem a radiative-conductive one (Ozisik, 1973; Pomraning, 2005). Here, local thermal description means, that the domain where a temperature is attributed to, is sufficiently large in order to allow for the definition of a temperature, i.e. a local radiative equilibrium. The principal quantity of interest is the intensity I, that describes the radiation energy flow through an infinitesimal oriented area dΣ = ndΣ with outward normal vector n into the solid angle dΩ = ΩdΩ, where Ω represents the direction of the flow considered, with angle θ of the normal vector and the flow direction n · Ω = cos θ = μ. In the present case we focus on the non-linearity of the radiative-conductive transfer problem and therefore introduce the simplification of an integrated spectral intensity over all wavelengths or equivalently all frequencies that contribute to the radiation flow and further ignore possible effects due to polarization. Also possible effects that need in the formalism properties such as coherence 8