Homogenization of radiation transfer in two-phase media with irregular phase boundaries

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Abstract
The model currently applied to radiation transfer in heterogeneous media is the radiation transfer equation (RTE) rigorously derived for homogeneous media only. However, the hypothesis that heterogeneous media can be described by the same model as homogeneous media is questionable. Indeed, local radiation intensities in different phases of a heterogeneous medium can be of considerably different values, so that the unique homogenized radiation intensity employed in the RTE model is not sufficient. A mathematical model of radiation transfer is obtained for a two-phase heterogeneous medium in the limit of geometrical optics. It consists of two transport equations for the partial homogenized radiation intensities defined by averaging in each phase. The equations are similar to the conventional RTE but contain terms taking into account exchange of radiation between phases. This model is referred to as the vector RTE model. It reduces to the conventional RTE in the cases when one of the two phases is opaque or one phase prevails in volume. In these two limiting cases, the vector RTE model is shown not to contradict the known calculations by ray optics and Monte Carlo simulation and to agree with the known experimental data. The proposed vector RTE model is crucial when the two phases are transparent or semitransparent and their volume fractions are comparable because of the absence of satisfactory mathematical models for this case. It describes the known results of Monte Carlo simulation for packed beds of semitransparent spheres. The application of the vector RTE model for an experimental identification of radiative properties is illustrated by normal-directional reflectance of the packed bed of semitransparent SiC particles. The numerical calculations confirm the general experimental tendency of increasing the reflectance with the reflection angle. The principal uncertainty arises from the boundary conditions applied to the vector RTE equations, so that a detailed analysis of the model near the boundaries would be worthwhile.