A model of averaged radiation transfer in two-phase heterogeneous medium

Presently employed for describing the radiation transfer in heterogeneous media is the radiation transfer equation (RTE) which is rigorously validated only for homogeneous media, although the hypothesis of the validity of one and the same model for both homogeneous and heterogeneous media is questionable. The local intensities of radiation in different phases of heterogeneous medium may significantly differ; therefore, the only averaged radiation intensity employed in the RTE model is insufficient. A model of radiation transfer is obtained for two-phase heterogeneous medium in the geometrical optics limit, which consists of two transfer equations for partial radiation intensities averaged in each phase separately. These equations resemble ordinary RTEs but include the exchange of radiation between the phases; therefore, they are referred to as the vector model of RTE. This model reduces to ordinary RTE if one of two phases is nontransparent or one phase prevails in the volume. It is demonstrated that the vector model of RTE in these two extreme cases does not contradict the known results of calculations by the ray optics and Monte-Carlo methods, as well as the experimental data. The suggested vector model is necessary if both phases are transparent or semitransparent and their volume fractions are comparable, because no adequate mathematical models are available in this case. The vector model describes the known results of Monte-Carlo simulation of packed beds of semitransparent spheres. The use of the vector model of RTE for experimental identification of the radiative properties is illustrated with the example of normal-directional reflectance of packed bed of semitransparent SiC particles. The results of numerical calculations confirm the general experimentally observed tendency for increase in reflectance with increasing angle of reflection. The main contribution to the error is made by the boundary conditions for vector RTEs; therefore, detailed analysis of the suggested model in boundary regions is required.