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13.3.3 P-1 Radiation Model Theory

The P-1 radiation model is the simplest case of the more general P-N model, which is based on the expansion of the radiation intensity $I$ into an orthogonal series of spherical harmonics [ 55, 332]. This section provides details about the equations used in the P-1 model.



The P-1 Model Equations


As mentioned above, the P-1 radiation model is the simplest case of the P-N model. If only four terms in the series are used, the following equation is obtained for the radiation flux $q_r$:


 q_r = -\frac{1}{3(a+\sigma_s) - C \sigma_s} \nabla G (13.3-2)

where $a$ is the absorption coefficient, $\sigma_s$ is the scattering coefficient, $G$ is the incident radiation, and $C$ is the linear-anisotropic phase function coefficient, described below. After introducing the parameter


 {\Gamma} = \frac{1}{(3(a+\sigma_s) - C\sigma_s)} (13.3-3)

Equation  13.3-2 simplifies to


 q_r = - {\Gamma} \nabla G (13.3-4)

The transport equation for $G$ is


 \nabla \cdot \left( {\Gamma} \nabla G \right) -aG + 4a\sigma T^4 =S_G (13.3-5)

where $\sigma$ is the Stefan-Boltzmann constant and $S_G$ is a user-defined radiation source. FLUENT solves this equation to determine the local radiation intensity when the P-1 model is active.

Combining Equations  13.3-4 and  13.3-5, the following equation is obtained:


 -\nabla \cdot q_r = aG - 4a\sigma T^4 (13.3-6)

The expression for $-\nabla \cdot q_r$ can be directly substituted into the energy equation to account for heat sources (or sinks) due to radiation.



Anisotropic Scattering


Included in the P-1 radiation model is the capability for modeling anisotropic scattering. FLUENT models anisotropic scattering by means of a linear-anisotropic scattering phase function:


 \Phi({\vec s} \; ' \cdot {\vec s}) = 1 + C {\vec s} \; ' \cdot {\vec s} (13.3-7)

Here, ${\vec s}$ is the unit vector in the direction of scattering, and ${\vec s} \; '$ is the unit vector in the direction of the incident radiation. $C$ is the linear-anisotropic phase function coefficient, which is a property of the fluid. $C$ ranges from $-1$ to 1. A positive value indicates that more radiant energy is scattered forward than backward, and a negative value means that more radiant energy is scattered backward than forward. A zero value defines isotropic scattering (i.e., scattering that is equally likely in all directions), which is the default in FLUENT. You should modify the default value only if you are certain of the anisotropic scattering behavior of the material in your problem.



Particulate Effects in the P-1 Model


When your FLUENT model includes a dispersed second phase of particles, you can include the effect of particles in the P-1 radiation model. Note that when particles are present, FLUENT ignores scattering in the gas phase. (That is, Equation  13.3-8 assumes that all scattering is due to particles.)

For a gray, absorbing, emitting, and scattering medium containing absorbing, emitting, and scattering particles, the transport equation for the incident radiation can be written as


 \nabla \cdot ({\Gamma} \nabla G) + 4\pi \left(a \frac{\sigma T^4}{\pi} + E_p\right) - (a + a_p) G = 0 (13.3-8)

where $E_p$ is the equivalent emission of the particles and $a_p$ is the equivalent absorption coefficient. These are defined as follows:


 E_p = \lim_{V \rightarrow 0} \sum_{n=1}^{N} \epsilon_{pn} A_{pn} \frac{\sigma T_{pn}^4}{\pi V} (13.3-9)

and


 a_p = \lim_{V \rightarrow 0} \sum_{n=1}^{N} \epsilon_{pn} \frac{A_{pn}}{V} (13.3-10)

In Equations  13.3-9 and 13.3-10, $\epsilon_{pn}$, $A_{pn}$, and $T_{pn}$ are the emissivity, projected area, and temperature of particle $n$. The summation is over $N$ particles in volume $V$. These quantities are computed during particle tracking in FLUENT.

The projected area $A_{pn}$ of particle $n$ is defined as


 A_{pn} = \frac{\pi d_{pn}^2}{4} (13.3-11)

where $d_{pn}$ is the diameter of the $n$th particle.

The quantity ${\Gamma}$ in Equation  13.3-8 is defined as


 {\Gamma} = \frac{1}{3 (a + a_p + \sigma_p)} (13.3-12)

where the equivalent particle scattering factor is defined as


 \sigma_p = \lim_{V \rightarrow 0} \sum_{n=1}^{N} (1 - f_{pn})(1 - \epsilon_{pn}) \frac{A_{pn}}{V} (13.3-13)

and is computed during particle tracking. In Equation  13.3-13, $f_{pn}$ is the scattering factor associated with the $n$th particle.

Heat sources (sinks) due to particle radiation are included in the energy equation as follows:


 -\nabla \cdot q_r = - 4\pi \left(a \frac{\sigma T^4}{\pi} + E_p\right) + (a + a_p) G (13.3-14)



Boundary Condition Treatment for the P-1 Model at Walls


To get the boundary condition for the incident radiation equation, the dot product of the outward normal vector ${\vec n}$ and Equation  13.3-4 is computed:


$\displaystyle q_r \cdot {\vec n}$ $\textstyle =$ $\displaystyle - {\Gamma} \nabla G \cdot {\vec n}$ (13.3-15)
$\displaystyle q_{r,w}$ $\textstyle =$ $\displaystyle - {\Gamma} \frac{\partial G}{\partial n}$ (13.3-16)

Thus the flux of the incident radiation, $G$, at a wall is $- q_{r,w}$. The wall radiative heat flux is computed using the following boundary condition:


$\displaystyle I_w({\vec r}, {\vec s})$ $\textstyle =$ $\displaystyle f_w({\vec r}, {\vec s})$ (13.3-17)
$\displaystyle f_w({\vec r}, {\vec s})$ $\textstyle =$ $\displaystyle \epsilon_w \frac{\sigma T_w^4}{\pi} + \rho_w I({\vec r}, -{\vec s})$ (13.3-18)

where $\rho_w$ is the wall reflectivity. The Marshak boundary condition is then used to eliminate the angular dependence [ 275]:


 \int^{2\pi}_0 I_w({\vec r}, {\vec s}) \; {\vec n} \cdot {\ve... ... f_w({\vec r}, {\vec s}) \; {\vec n} \cdot {\vec s} \; d\Omega (13.3-19)

Substituting Equations  13.3-17 and  13.3-18 into Equation  13.3-19 and performing the integrations yields


 q_{r,w}= - \; \frac{4 \pi \epsilon_w \frac{\sigma T_w^4}{\pi} - (1 - \rho_w) G_w}{2 (1 + \rho_w)} (13.3-20)

If it is assumed that the walls are diffuse gray surfaces, then $\rho_w = 1 - \epsilon_w$, and Equation  13.3-20 becomes


 q_{r,w} = - \frac{\epsilon_w}{2 \left(2 - \epsilon_w \right)} \left( 4 \sigma T_w^4 - G_w \right) (13.3-21)

Equation  13.3-21 is used to compute $q_{r,w}$ for the energy equation and for the incident radiation equation boundary conditions.



Boundary Condition Treatment for the P-1 Model at Flow Inlets and Exits


The net radiative heat flux at flow inlets and outlets is computed in the same manner as at walls, as described above. FLUENT assumes that the emissivity of all flow inlets and outlets is 1.0 (black body absorption) unless you choose to redefine this boundary treatment.

FLUENT includes an option that allows you to use different temperatures for radiation and convection at inlets and outlets. This can be useful when the temperature outside the inlet or outlet differs considerably from the temperature in the enclosure. See Section  13.3.15 for details.


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