### Abstract:

An important improvement in the area of reactor core neutronic modeling is the development and use of the methods based on "quasi-diffusion" (QD) low-order equations. This family of methods takes into account the transport exactly using "functionals" computed by solving transport equations, and is amenable to solution with a variety of energy group structures and spatial discretizations. The methodology should provide transport-quality results, the only limitation being the quality of the transport-generated cross-sections and QD functionals.
The goal of this work is to develop a nodal differencing scheme for the solution of the 2-D, two-group, coarse-mesh, QD low-order equations and implement it into a code that will perform flux and k-eigenvalue calculations in Cartesian geometry and in hexagonal geometry applicable to high-temperature gas reactors. The development of the nodal differencing scheme is based on the transverse integration method. By transverse integration the 2-D problem is reduced to a 1-D problem which is then solved using a polynomial expansion of the neutron flux, fast and thermal. The QD functionals are computed using the moments of the angular neutron flux provided by the transport code Attila. Two-group macroscopic cross-sections are extracted from the Attila output files.
The results of this research show that the effective multiplication factors with and without QD (i.e., pure diffusion) are very close due to the fact that the diagonal Eddington functionals are very close to 1/3, and the off-diagonal functionals are a few orders of magnitude smaller. The calculations have shown that the 0-th moment of the angular flux (scalar flux) is much bigger than the second moments. The analysis of the results also points out that the errors between Mathematica QD and Attila decrease with number of hexagonal nodes in zero-flux boundary condition problems and the multiplication factor for infinite-medium (zero-current boundary conditions), homogeneous problems agree much better with Attila than those with zero flux boundary conditions. This finding may suggest that the methodology that was used does not model the leakage with sufficient accuracy. The effects of the QD are small and do not seriously affect the multiplication factor in either of the geometries; however the effects are somewhat more important in Cartesian geometry compared to hexagonal geometry, presumably due to the higher symmetry of the latter.
Future research may attempt to refine the computational grid (i.e., equilateral triangles instead of entire hexagons) and/or consider different sets of expansion functions. Another area of research may involve investigating the effect of isotopic composition and the nature of the fuel and the moderator on the magnitude of the quasi-diffusion effects. Various libraries of neutron cross-sections may be explored.