Abstract :
It is of considerable practical importance to be able to calculate the interaction effect between two fissile lumps. In this paper, we compare reduced three-dimensional models of two interacting face-on cylinders, with exact numerical results from a finite element code. The parameter of interest is the overall multiplication factor of the two-body system as a function of separation distance when the surrounding medium is a void. There are two existing models for treating this problem; the first is based upon the integral transport equation in which the transverse spatial variation is approximated by a buckling, and the second is diffusion theory, supplemented by an effective boundary condition which approximates leakage from the central void between the cylinders. We have found severe limitations of these standard methods, especially when the cylinders are well-separated. In that case, instead of the interaction effect reducing according to the inverse square law, it is found to decrease exponentially as exp(-h/R) where h is the separation distance and R is the cylinder radius. We have introduced a new approach based on areal averaging in the transverse direction which removes this limitation and restores the physically correct inverse square law behaviour. This new model, and a number of related models, are compared. In addition we re-formulate the diffusion theory method in terms of a completely general transport theory boundary condition which enables the use of PN theory should this be considered appropriate. We also introduce a new and accurate effective boundary condition for use in diffusion theory. Extensive numerical results are given and compared with some three dimensional finite element results. Also some contact is made with experiment.
