5.3.2 Radiative Transfer Equation

The radiative transfer equation (RTE) for an absorbing, emitting, and scattering medium at position ${\vec r}$ in the direction ${\vec s}$ is

where	${\vec r}$	=	position vector
	${\vec s}$	=	direction vector
	${\vec s} \; '$	=	scattering direction vector
	$s$	=	path length
	$a$	=	absorption coefficient
	$n$	=	refractive index
	$\sigma_s$	=	scattering coefficient
	$\sigma$	=	Stefan-Boltzmann constant (5.669 $\times$ 10 $^{-8}$ W/m $^2$-K $^4$)
	$I$	=	radiation intensity, which depends on position ( ${\vec r})$ and direction $({\vec s})$
	$T$	=	local temperature
	$\Phi$	=	phase function
	${\Omega}'$	=	solid angle

$(a+\sigma_s)s$ is the optical thickness or opacity of the medium. The refractive index $n$ is important when considering radiation in semi-transparent media. Figure  5.3.1 illustrates the process of radiative heat transfer.

Figure 5.3.1: Radiative Heat Transfer

The DTRM and the P-1, Rosseland, and DO radiation models require the absorption coefficient $a$ as input. $a$ and the scattering coefficient $\sigma_s$ can be constants, and $a$ can also be a function of local concentrations of H $_2$O and CO $_2$, path length, and total pressure. ANSYS FLUENT provides the weighted-sum-of-gray-gases model (WSGGM) for computation of a variable absorption coefficient. See Section  5.3.8 for details. The discrete ordinates implementation can model radiation in semi-transparent media. The refractive index $n$ of the medium must be provided as a part of the calculation for this type of problem. The Rosseland model also requires you to enter a refractive index, or use the default value of $1$.