A new SPn theory formulation with self-consistent physical assumptions on angular flux

The conventional SPn theory cannot provide the explicit angular flux solution. Chao and Yamamoto (2012) proposed the explicit angular flux representation for the SPn theory as cylindrically symmetric with respect to the net current direction at any point in space, such that the angular flux is an expansion in Legendre polynomials of the cosine of the polar angle with respect to the local net current. Such a model however cannot lead to the SPn equations without further ad hoc assumptions because the multiple directions of the spatial gradients of the flux moments cannot be all parallel to the net current. In this paper we relax the assumption of the total angular flux being locally one-dimensional and generalize it to each of the flux moments being locally one-dimensional along the direction of the spatial gradient of each individual flux moment. The angular distribution of the nth order flux moment is the nth order Legendre polynomial of the cosine of the polar angle with respect to the direction of the spatial gradient of the nth order flux moment, ∇ϕn(r). With this physical model one can rigorously derive the equations for the current and the boundary conditions. The SPn equations can also be derived with the additional assumption of the total cross-section being locally constant, which is practically always valid when the spatial variation is discretized in numerical calculations. However the boundary conditions turn out to be different from the conventional ones, containing some non-linear factors. The internal interface boundary conditions are not affected by the non-linear factors as they cancel out on the interface. But the external boundary condition does get affected by the non-linear factors. The effect of non-linear factors is of higher order and if neglected, the external boundary condition also reduces to the conventional one. The non-linear external boundary condition can be iteratively updated to estimate the correction effect. A numerical calculation problem is suggested to test the new SPn theory and to assess the non-linear effect.