Solutions of the diffusion equation in cones and wedges

Abstract

The neutron diffusion equation has been solved for cones and finite wedges. Criticality conditions are derived linking the dimensions of the cones and wedges to the nuclear properties of the system. The fundamental mode flux is evaluated numerically for a number of cases. We have also discovered an interesting minimum in the leakage rate as the cone or wedge angle passes through a certain value for a fixed volume. Some practical implications of this result are pointed out.The exact solutions derived have been used as benchmarks to assess the accuracy of the finite element code TRIPAC in R-Z and X-Y-Z geometries for a four-group citicality problem in cones and wedges. Excellent agreement is found between the analytical results and the code for both eigenvalues and fluxes.