P_N Blackness Theory in Plane Geometry

The derivation of blackness boundary conditions is reviewed and generalized into a standard matrix formalism that is valid for any order P_N approximation. It is then shown that for a finite slab effective diffusion and absorption matrices can be found which reproduce blackness boundary conditions at the interfaces. In the continuous or infinitely many mesh point description of the black region, the analysis leads to infinite series expressions for the equivalent matrices, which have been evaluated explicitly by means of the Caley-Hamilton theorem for the case of the P ₃ approximation. Equivalent matrices have also been derived for two- and three-mesh-point descriptions of the black region. Numerical calculations for three model problems indicate that P₃ blackness theory is a great improvement over conventional P₃ theory and is roughly equivalent to P₅ theory in the prediction of both the exterior scalar flux and the absorption rate in the black region.