The P-1 T3 radiation model is a simplified form of IMMERSOL that ignores the inter-wall distance term, so that the following equation is solved for the radiant temperature T3:
∇.[(16/3) ke-1σT33∇T3] + 4aσ(T4 - T34) = 0 Eqn 8.1
where a is the absorption coefficient per unit length; σ is the Stefan-Boltzmann constant,(=5.670367 -8 W m-2 K-4); ke is the extinction coefficient; and T is thermodynamic temperature, which like T3 is measured in degrees Kelvin. The extinction coefficient ke is given by:ke = a + s Eqn 8.2
where s is the scattering coefficient per unit length.
Equation (8.1) has the same form as the equation for the irradiance G in the P-1 radiation model (see Modest [1993]), which is the simplest case of the more general P-N model based on the first-order spherical-harmonic expansion of the radiation intensity. If anisotropic scattering is ignored in the P-1 model, then equation (8.1) is identical to the P-1 diffusion equation, and the irradiance G=4*σT34.
The radiative heat-flux vector is given by:
qr = -(16/3)ke-1σ T33 ∇T3 Eqn 8.3
This equation has the same form as Rosseland's diffusion model, which applies in the optically-thick limit when T3=T. If scattering is absent, i.e. s=0, the absorption coefficient a is exactly equal to the Rosseland mean absorption coefficient aR.
The P1-T3 model is essentially the same as the composite-radiosity model, but with the radiant temperature T3 as dependent variable, rather than the radiosity R.
The model can be used for conjugate-heat-transfer applications, whereas the composite radiosity model cannot be used for these cases.
The model is applicable to both optically-thin and optically-thick media, typically when the optical thickness τ>1, but especially when it is large, i.e. τ>>1. If τ is small or the media is transparent, IMMERSOL is recommended. Here, τ =a*L >>1, where L is the characteristic dimension of the solution domain, such as for example the width or height of a duct, or the diameter of a combustion chamber.
The boundary conditions for the P-1 T3 model are the same as those presented for the IMMERSOL model, except for wall boundaries, where temperature slip is accounted for by use of the Marshak boundary condition:
Sr,w = 2*[εw/(2-εw)]*σ*Ta3(Tw - T3) Eqn 8.4
where
Ta3 = (Tw + T3)*(Tw2 + T32) Eqn 8.5
and εw is the wall emissivity, and Tw is the wall temperature.
Modest, M.F.: Radiative Heat Transfer, 14.4 The P-1 Approximation., p511-519, McGraw Hill, (1993).
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