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Simplified Spherical Harmonics (SPN ) Even-Parity Equation Set

Radiation transport equations:

$\displaystyle {\frac{{1}}{{c}}}$$\displaystyle {\frac{{\partial}}{{\partial t}}}$$\displaystyle \xi_{{m,g}}^{}$ + $\displaystyle \mbox{$\mbox{$\stackrel{^{\mathstrut}\smash{\longrightarrow}}{\nabla}$} \cdot$}$$\displaystyle \mbox{$\stackrel{^{\mathstrut}\smash{\longrightarrow}}{\Gamma}$}$m, g + $\displaystyle \sigma_{g}^{t}$$\displaystyle \xi_{{m,g}}^{}$ = $\displaystyle \sigma_{g}^{s}$$\displaystyle \phi_{g}^{}$ + $\displaystyle \sigma_{g}^{e}$Bg + $\displaystyle \mathcal {C}$sg  ,

$\displaystyle {\frac{{1}}{{c}}}$$\displaystyle {\frac{{\partial}}{{\partial t}}}$$\displaystyle \mbox{$\stackrel{^{\mathstrut}\smash{\longrightarrow}}{\Gamma}$}$m, g + $\displaystyle \mu^{2}_{m}$$\displaystyle \mbox{$\mbox{$\stackrel{^{\mathstrut}\smash{\longrightarrow}}{\nabla}$}$}$$\displaystyle \xi_{{m,g}}^{}$ + $\displaystyle \sigma_{g}^{t}$$\displaystyle \mbox{$\stackrel{^{\mathstrut}\smash{\longrightarrow}}{\Gamma}$}$m, g = $\displaystyle \mbox{$\stackrel{^{\mathstrut}\smash{\longrightarrow}}{\mathcal{C}}$}$vm, g

for m = 1, M , and g = 1, G .

Temperature equations:

Cvi$\displaystyle {\frac{{\partial T_i}}{{\partial t}}}$ = $\displaystyle \alpha$$\displaystyle \left(\vphantom{ T_e-T_i }\right.$Te - Ti$\displaystyle \left.\vphantom{ T_e-T_i }\right)$ + Qi  ,  
Cve$\displaystyle {\frac{{\partial T_e}}{{\partial t}}}$ = $\displaystyle \alpha$$\displaystyle \left(\vphantom{ T_i-T_e }\right.$Ti - Te$\displaystyle \left.\vphantom{ T_i-T_e }\right)$ + Qe + $\displaystyle \sum_{{g=1}}^{{G}}$$\displaystyle \left(\vphantom{ \sigma_g^a \phi^{(0)}_g- \sigma_g^e B_g }\right.$$\displaystyle \sigma_{g}^{a}$$\displaystyle \phi^{{(0)}}_{g}$ - $\displaystyle \sigma_{g}^{e}$Bg$\displaystyle \left.\vphantom{ \sigma_g^a \phi^{(0)}_g- \sigma_g^e B_g }\right)$  ,  

where
$\displaystyle \xi_{{m,g}}^{}$ = Even-parity pseudo-angular energy intensity,  
$\displaystyle \mbox{$\stackrel{^{\mathstrut}\smash{\longrightarrow}}{\Gamma}$}$m, g = Even-parity pseudo-angular energy current,  
$\displaystyle \mathcal {C}$sg = $\displaystyle \left(\vphantom{ \sigma_g^a - \sigma_g^s }\right.$$\displaystyle \sigma_{g}^{a}$ - $\displaystyle \sigma_{g}^{s}$$\displaystyle \left.\vphantom{ \sigma_g^a - \sigma_g^s }\right)$$\displaystyle \mbox{$\stackrel{^{\mathstrut}\smash{\longrightarrow}}{F}$}$(0)g . $\displaystyle {\frac{{\mbox{$\stackrel{^{\mathstrut}\smash{\longrightarrow}}{v}$}}}{{c}}}$  ,  
$\displaystyle \mbox{$\stackrel{^{\mathstrut}\smash{\longrightarrow}}{\mathcal{C}}$}$vm, g = 3$\displaystyle \mu^{2}_{m}$$\displaystyle \sigma_{g}^{t}$$\displaystyle \left(\vphantom{ P_g + \phi_g }\right.$Pg + $\displaystyle \phi_{g}^{}$$\displaystyle \left.\vphantom{ P_g + \phi_g }\right)$$\displaystyle {\frac{{\mbox{$\stackrel{^{\mathstrut}\smash{\longrightarrow}}{v}$}}}{{c}}}$  ,  
$\displaystyle \phi_{g}^{}$ = $\displaystyle \sum_{{m=1}}^{M}$$\displaystyle \xi_{{m,g}}^{}$ wm  ,  
Pg = $\displaystyle \sum_{{m=1}}^{M}$$\displaystyle \xi_{{m,g}}^{}$ $\displaystyle \mu^{2}_{m}$ wm  ,  
$\displaystyle \mbox{$\stackrel{^{\mathstrut}\smash{\longrightarrow}}{F}$}$g = $\displaystyle \sum_{{m=1}}^{M}$$\displaystyle \mbox{$\stackrel{^{\mathstrut}\smash{\longrightarrow}}{\Gamma}$}$m, g wm  ,  
$\displaystyle \phi^{{(0)}}_{g}$ = $\displaystyle \phi_{g}^{}$ -2$\displaystyle \mbox{$\stackrel{^{\mathstrut}\smash{\longrightarrow}}{F}$}$(0)g . $\displaystyle {\frac{{\mbox{$\stackrel{^{\mathstrut}\smash{\longrightarrow}}{v}$}}}{{c}}}$  ,  
$\displaystyle \mbox{$\stackrel{^{\mathstrut}\smash{\longrightarrow}}{F}$}$(0)g = $\displaystyle \mbox{$\stackrel{^{\mathstrut}\smash{\longrightarrow}}{F}$}$g - $\displaystyle \left(\vphantom{ P_g + \phi_g }\right.$Pg + $\displaystyle \phi_{g}^{}$$\displaystyle \left.\vphantom{ P_g + \phi_g }\right)$$\displaystyle {\frac{{\mbox{$\stackrel{^{\mathstrut}\smash{\longrightarrow}}{v}$}}}{{c}}}$  ,  
M = $\displaystyle \left(\vphantom{ N + 1 }\right.$N + 1$\displaystyle \left.\vphantom{ N + 1 }\right)$/2  .  


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Next: Diffusion (P1 ) Equation Up: Equation Sets Previous: Equation Sets
Michael L. Hall