Rosseland Approximation Theory
The Rosseland approximation is available as a radiation discretization method as part of the Optically Thick Participating Medium subnode.
Rosseland approximation relies on the hypotheses that the participating medium is optically thick — that is, τ >>1 — where τ is the optical thickness defined by the integral of absorption coefficient, κ, along a typical optical path:
From a computational cost point of view this approximation has a limited impact because it does not introduce any extra degree of freedom to the heat equation. Instead it adds nonlinear contribution to the thermal conductivity. This is why this method is popular for some applications where the optical thickness is large. Nevertheless, because it gives a simple approximation of heat transfer by radiation in a participating media, it should be carefully validated.
In this case, the radiative heat flux can be evaluated by (Ref. 22):
For a gray media it leads to
Assuming a constant refractive index, this can be rewritten as qr = − kR ∇T with
and
Rosseland mean extinction coefficient, βR, can be defined from the absorption and scattering coefficients of the media from βR=κ+σs.