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13.3.4 Rosseland Radiation Model Theory

The Rosseland or diffusion approximation for radiation is valid when the medium is optically thick ( $(a + \sigma_s)L \gg 1$), 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.

The Rosseland Model Equations

As with the P-1 model, the radiative heat flux vector in a gray medium can be approximated by Equation  13.3-4:

 q_r = - {\Gamma} \nabla G (13.3-22)

where ${\Gamma}$ is given by Equation  13.3-3.

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 $G$.) Thus $G = 4 \sigma n^2 T^4$, where $n$ is the refractive index. Substituting this value for $G$ into Equation  13.3-22 yields

 q_r = - 16 \sigma {\Gamma} n^2 T^3 \nabla T (13.3-23)

Since the radiative heat flux has the same form as the Fourier conduction law, it is possible to write

$\displaystyle q$ $\textstyle =$ $\displaystyle q_c + q_r$ (13.3-24)
  $\textstyle =$ $\displaystyle - (k + k_r) \nabla T$ (13.3-25)
$\displaystyle k_r$ $\textstyle =$ $\displaystyle 16 \sigma {\Gamma} n^2 T^3$ (13.3-26)

where $k$ is the thermal conductivity and $k_r$ is the radiative conductivity. Equation  13.3-24 is used in the energy equation to compute the temperature field.

Anisotropic Scattering

The Rosseland model allows for anisotropic scattering, using the same phase function (Equation  13.3-7) described for the P-1 model in Section  13.3.3.

Boundary Condition Treatment for the Rosseland Model at Walls

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, $q_{r,w}$, is defined using the slip coefficient $\psi$:

 q_{r,w} = - \; \frac{\sigma\left(T_{w}^4 - T_{g}^4 \right)}{\psi} (13.3-27)

where $T_w$ is the wall temperature, $T_g$ is the temperature of the gas at the wall, and the slip coefficient $\psi$ is approximated by a curve fit to the plot given in [ 332]:

 \psi = \left\{ \begin{array}{ll} 1/2 & N_w < 0.01 \\ \frac{... ...4} & 0.01 \leq N_w \leq 10 \\ 0 & N_w > 10 \end{array}\right. (13.3-28)

where $N_w$ is the conduction to radiation parameter at the wall:

 N_w = \frac{k(a + \sigma_s)}{4\sigma T_w^3} (13.3-29)

and $x = \log_{10} N_w$.

Boundary Condition Treatment for the Rosseland Model at Flow Inlets and Exits

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  13.3-24.

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