Basic Model Equations

As radiation travels through a medium, its radiant intensity in the direction is absorbed and incremented by the intervening material. This process is governed by the radiative transfer equation (RTE) which, written in terms of radiant intensity for a specific wavelength , is given by:

(1)

where

(2)

and , .

Note that the in-scattering component is assumed to be isotropic.

When the absorption and scattering coefficients of the media are independent of wavelength, the media is called gray. In that case, the RTE can be integrated over wavelength (or, equivalently, wave number) to produce a wavelength-independent equation.

The boundary condition applied to the RTE for diffusely emitting (with emissivity ) and reflecting (with reflectivity ) boundaries is, for each wavelength :

(3)

The radiant heat flux in a particular direction is given by the integration of the radiant intensity over all solid angles and over the wavelength spectrum:

(4)

The radiation solution is coupled to the fluid dynamic solution through the divergence of the radiative heat flux. This term exchanges energy between the fluid and the radiant energy field. Given the intensity field, the divergence of the heat flux is computed as:

(5)