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13.3.2 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


 \frac{dI({\vec r}, {\vec s})}{ds} + (a + \sigma_s) I({\vec r... ...ec s} \; ') \; \Phi({\vec s}\cdot {\vec s} \; ') \; d{\Omega}' (13.3-1)


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 (5.672 $\times$ 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  13.3.1 illustrates the process of radiative heat transfer.

Figure 13.3.1: Radiative Heat Transfer
figure

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. FLUENT provides the weighted-sum-of-gray-gases model (WSGGM) for computation of a variable absorption coefficient. See Section  13.3.8 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 $1$.


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© Fluent Inc. 2006-09-20