Closed-form solutions for nodal formulations of two dimensional transport problems in heterogeneous media

We present a study of two dimensional, fixed-source neutron transport problems on a heterogeneous medium. It is an extension of a methodology used for solving neutron transport problems on an homogeneous medium, that combines nodal schemes with explicit solutions of the transversal integrated equations via the Analytical Discrete Ordinates method (ADO). We consider closed-form solutions of the integrated discrete ordinates transport equation on a two dimensional cartesian geometry, together with a level symmetric quadrature scheme, on each region of interest, in the domain, possibly characterised by different materials. In this work, each solution in a region is coupled with that of its neighbouring regions to provide the whole solution, without resorting to using iterative schemes. The terms involving unknown angular fluxes that arise using nodal schemes, assumed as constant function approximations, are added to the source term. The model proposed leads to a considerable reduction of the order of the associated eigenvalue problems written as perturbations of diagonal matrices, and the solutions obtained are explicit in terms of the spatial variables. Analytical expressions are also derived for the elementary solutions. The numerical results obtained are shown to be in good agreement with other results available in the literature.