Conditions for a Minimum Critical Mass in a Nuclear Reactor and Considerations on Goertzel’s Theorem in Transport Theory

Abstract

The classic minimum critical mass problem, posed and solved by Goertzel using multigroup diffusion theory, is revisited and reformulated in terms of the two-group transport equation with isotropic scattering. A new variational principle is constructed from which it is possible to derive the conditions necessary for a minimum critical mass. This condition is that the angular thermal flux ψ_t([bold]r, [bold]Ω) and a quantity ϕ_t([bold]r, [bold]Ω) related to the adjoint flux, must obey the constraint

[integral]dΩψ_t([bold]r, [bold]Ω)ϕ_t([bold]r, [bold]Ω) = constant.

Contrary to the behavior noted in diffusion theory, this condition does not correspond to a flat thermal flux in the core. This is a major conclusion of the present work.

To find the associated solutions, we develop a coupled set of integral equations for the components of the angular flux in the core. We then show that, for weakly absorbing moderators, the lowest order approximation to this set provides an accurate representation of the minimum mass conditions. It also emerges that the flat flux is a very good representation of the true flux. With the above assumptions, the problem reduces to that of solving a Fredholm equation of the first kind for the fuel mass distribution across the core. We solve this equation numerically for the case of an infinitely reflected, infinite slab and compare the results with those from diffusion theory. The transport theory results show one very interesting and important feature, namely, a steep rise in fuel concentration as the boundary is approached which goes to infinity at the boundary. This is in contrast to the diffusion theory result which requires an ad hoc addition of surface delta functions for a solution to exist. Thus we come to the conclusion that the increased surface concentration of fuel is a natural consequence of transport theory but not of diffusion theory. This is the second major conclusion of this work. Detailed numerical results are presented for ²³⁵U-graphite and ²³⁵U-water mixtures.