A new SPn theory formulation with self-consistent physical assumptions on angular flux. (January 2016) - Chao, Yung-An
Highlights: A new SP n theory formulation is proposed, which can provide the explicit angular flux for the SP n solution. The n -th order Legendre polynomial contains the cosine of the angle with respect to the gradient of the n -th flux moment. The SP n equation and the interface boundary condition remain the same. The external boundary condition is different except for the reflective case. The external boundary condition contains non-linear factors depending on the angle between the boundary normal vector and the flux gradient vector. The external boundary conditions can be iteratively updated. A numerical test problem is proposed. Abstract: The conventional SP n theory cannot provide the explicit angular flux solution. Chao and Yamamoto (2012) proposed the explicit angular flux representation for the SP n 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 SP n 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 n th order flux moment is the n th order Legendre polynomial of the cosine of the polar angle with respect to the direction of the spatial gradient of the n th order flux moment, ∇ ϕ n ( r ) . With this physical model one can rigorously derive the equations for the current and the boundary conditions. The SP n 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 SP n theory and to assess the non-linear effect.