Application of the spatial averaging theorem to radiative heat transfer in two-phase media

The spatial averaging theorem is applied to rigorously derive continuum-scale equations of radiative transfer in two-phase media consisting of either two semi-transparent or one semitransparent and one opaque homogeneous phases in the limit of geometric optics. The derivations are based on the equations of radiative transfer and the corresponding boundary conditions applied at the discrete-scale to each phase, and on the discrete-scale radiative properties of each phase and the interface between the phases. A Monte Carlo based methodology for the determination of average radiative properties is discussed in the light of previous pertinent studies. Nomenclature A Surface area fV Volume fraction I Average intensity L Discrete-scale intensity n Refractive index n̂ Normal unit vector r Global position vector s Thickness ŝ Direction unit vector T Temperature V Averaging volume x Averaging-volume position vector y Local position vector Greek symbols ε Emittance κ Absorption coefficient ρ Reflectance σ Stefan-Boltzman constant σs Scattering coefficient τ Trasmittance Φ Phase function ψ Scalar quantity Ω Solid angle Subscripts b blackbody c collision d discrete-scale i, j phase indices in incoming int interface k ray index r reflection w boundary of the twophase medium Superscripts “ bi-directional ‘ directional-hemispherical i, j averaging phase Proceedings of Eurotherm83 – Computational Thermal Radiation in Participating Media III 15-17 April 2009, Lisbon, Portugal