Volume scattering of radiation in packed beds of large, opaque spheres

A simple model is proposed for radiative properties of close-packed large, opaque spheres that accounts for nonvanishing volume of the particles, i.e., volume scattering as opposed to point scattering. It is based on the mean-beam-length concept applied to an assembly of particles, as illustrated by Mills. The resulting particle-scattering properties differ from those of classical pseudocontinuum theory based on point scattering by the simple factor of void fraction, and reduce to the point-scattering expressions in the limit of small particle volume fraction. The volume-scattering model matches detailed Monte Carlo results for extinction obtained by Kaviany and Singh and by Coquard and Baillis, which explicitly accounted for particle volume. The present model also confirms the Monte Carlo finding that the effects of nonvanishing particle volume are felt primarily in the extinction coefficient; albedo and phase function are relatively unaffected. These findings pertain only to the geometric optics regime where dependent scattering (wave coherence effects) are negligible.