A medium is said to be optically thick if the reciprocal of the absorption/extinction coefficient is very small compared with a characteristic dimension of the medium.
When the medium is optically thick, the radiation can be approximated as an isotropic "diffusion" process by use of the well-known Rosseland approximation for radiative heat transfer ( see for example Deissler [1964], Ozisik [1973] and Siegel and Howell [1992]).
This was shown in Section 3 and 4 above.
In PHOENICS, the radiosity model or P-1 T3 model can conveniently be used as the vehicle for invoking Rosseland's model.
All that is needed is to set the scattering coefficient s to zero and the absorption coefficient a to a relatively large value so as to force radiative equilibrium.
The diffusion model implies a radiative conductivity in the energy equation defined by:
λr =(16/3)σ(a+s)-1T3 Eqn 7.1
and so the medium behaves like a conductor with a temperature-dependent conductivity.
It should be noted that, as conventionally formulated, the diffusion approximation is not valid near a boundary solid-fluid boundary ( see for example Deissler [1964] and Viskanta [1966] ).
However, this is difficulty is overcome in the PHOENICS implementation, because the wall boundary condition (6.7) for the radiosity allows for a temperature jump at the wall.
R.G.Deissler, 'Diffusion approximation for thermal radiation in gases with jump boundary condition', ASME J.Heat Transfer, p240, (1964).
H.C.Hottel and A.F.Sarofim, 'Radiative transfer', McGraw Hill, New York, (1967).
M.N.Ozisik, 'Radiative heat transfer', John Wiley, New York, (1973).
S.Rosseland, 'Theoretical astrophysics', Oxford Univ. Press, Clarendon, London and New York, (1936).
R.Siegel and J.R.Howell, 'Thermal Radiation Heat Transfer', 3rd Edition, Hemisphere Publishing Corporation, (1992).
R.Viskanta, 'Radiation transfer and interaction of convection with radiation heat transfer', In Advances in Heat Transfer, Vol.3, Ed. T.F.Irvine and J.P.Hartnett, Academic Press, p175, (1966).