Marshak boundary condition

To solve Equation (38) boimdary conditions which describe the reflection and transmission of radiation at the boimdaries are required. In principle, boimdary conditions can only be established in a rigorous manner for the radiative intensity, not for G, because the optical properties of the interfaces depend on the direction of incidence of radiation. Because the PI approximation solves for an integrated quantity like G instead, approximate boundary conditions must be established (Modest, 2003). One possibility is the Marshak boundary condition (Marshak, 1947), which comes from considering the continuity of the radiative flux through the interface. If this continuity is considered together with the assumption (34) of the PI approximation and Equation (37), the following equation is obtained (Spott and Svaasand, 2000)... 

For the boundary conditions, Marshak s boundary relations are imposed. This means that the moments of the intensity at each of the boundaries are determined from ... 

The steady-state diffusion equation (Eq. (71)) is an ordinary differential equation of order 2, whose solution requires two boundary conditions. In radiative transfer, the value of the irradiance or the net flux at the boundary is rarely available. Therefore, a Hnear relation between G and its derivative, that is, between the irradiance and the flux (see Eq. (73)) is generally used for the boundary conditions this is what researchers in this field call the Marshak boundary conditions (Marshak, 1947). To our knowledge, in the existing Hterature, the expression for Marshak s boundary conditions is brought to the following functional form (Case and Zweifel, 1967 Durian, 1994 Ishimaru, 1999) ... 

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