The integral-transform method for solving neutron transport problems with general anisotropic scattering in a cylinder of finite height

Abstract

In this paper, we present a mathematical technique for solving the integral transport equation for the criticality of a homogeneous cylinder of finite height. The purpose of the present paper is two-fold : firstly, to show that our earlier formalism can be generalized to any order of anisotropy, and secondly to generate the numerical results, which could serve as benchmarks when scattering is linearly anisotropic. We expand the scattering function in spherical harmonics to retain the Lth order of anisotropy. Thereafter, we write the integral transport equations for the Fourier-transformed spherical harmonic moments of the angular flux. In conformity with the integral-transform method for multidimensional geometry, the kernels of these integral equations are represented in their respective factorized form, which consists of a series of products of suitable spherical Bessel functions. The Fourier-transformed spherical harmonic moments are also represented in their separable form by expanding them in a series of products of spherical Bessel functions, commensurate with the symmetry of finite cylindrical geometry. The criticality problem for the cylinder of finite height is then posed as a matrix eigen value problem whose eigen vector is composed of the expansion coefficients mentioned above. The general matrix element is expressed as a product of certain integrals of Bessel functions, which can be evaluated by recursion relations derived in this paper. Finally, a comparison between the present benchmark results and S N results ( twotran) in ( r–z) goemetry is presented when scattering is linearly anisotropic.