Derivation and solution of multifrequency radiation diffusion equations for homogeneous refractive lossy media

Starting from the radiation transport equation for homogeneous, refractive lossy media, we derive the corresponding time-dependent multifrequency diffusion equations. Zeroth and first moments of the transport equation couple the energy density, flux and pressure tensor. The system is closed by neglecting the temporal derivative of the flux and replacing the pressure tensor by its diagonal analogue. The radiation equations are coupled to a diffusion equation for the matter temperature. We are interested in modeling heating and cooling of silica (SiO₂), at possibly rapid rates. Hence, in contrast to related work, we retain the temporal derivative of the radiation field. We derive boundary conditions at a planar air–silica interface taking account of reflectivities obtained from the Fresnel relations that include absorption. The spectral dimension is discretized into a finite number of intervals leading to a system of multigroup diffusion equations. Three simulations are presented. One models cooling of a silica slab, initially at 2500 K, for 10 s. The other two are 1D and 2D simulations of irradiating silica with a CO₂ laser, λ = 10.59 μm. In 2D, a laser beam (Gaussian profile, r₀ = 0.5 mm for 1/e decay) shines on a disk (radius = 0.4, thickness = 0.4 cm).