The Rosseland or diffusion approximation for radiation is valid
when the medium is optically thick (
), and is recommended for use
in problems where the optical thickness is greater than 3. It can
be derived from the P-1 model equations, with some approximations.
This section provides details about the equations used in the Rosseland
model. For information about setting up the model, see Using the Radiation Models in the User's Guide.
As with the P-1 model, the radiative heat flux vector in a gray medium can be approximated by Equation 5–25:
 | (5–49) |
where 
is given by Equation 5–24.
The Rosseland radiation model differs from the P-1 model in
that the Rosseland model assumes that the intensity is the black-body
intensity at the gas temperature. (The P-1 model actually calculates
a transport equation for 
.) Thus 
, where 
is the refractive index.
Substituting this value for 
into Equation 5–49 yields
 | (5–50) |
Since the radiative heat flux has the same form as the Fourier conduction law, it is possible to write
 | (5–51) |
 | (5–52) |
 | (5–53) |
where 
is the thermal conductivity and 
is the radiative
conductivity. Equation 5–51 is used in
the energy equation to compute the temperature field.
The Rosseland model allows for anisotropic scattering, using the same phase function (Equation 5–33) described for the P-1 model in Anisotropic Scattering.
Since the diffusion approximation is not valid near walls, it
is necessary to use a temperature slip boundary condition. The radiative
heat flux at the wall boundary, 
, is defined using the slip coefficient 
:
 | (5–54) |
where 
is the wall temperature, 
is the temperature
of the gas at the wall, and the slip coefficient 
is approximated by
a curve fit to the plot given in [596]:
 | (5–55) |
where 
is the conduction to radiation
parameter at the wall:
 | (5–56) |
and 
.
No special treatment is required at flow inlets and outlets for the Rosseland model. The radiative heat flux at these boundaries can be determined using Equation 5–51.