Abstract :
Solutions for a flat thermal flux in a two-group diffusion model are found for a range of assumptions, since this is a requisite for both a minimum and a maximum fuel loading in the model. It is shown that a maximum fuel loading then arises when such a region, at unity infinite multiplication factor, is complemented by an outer core to bring the finite reactor critical. Correspondingly, when a core with a flat flux is surrounded by a reflector, the core fuel may be so distributed as to flatten the flux using the ‘flux-trap’ phenomenon and provide a minimum loading. Critical conditions and necessary fuel density are derived in finite and infinitely thick reflectors. A curiosity is the possibility of observing ‘negative’ reflector savings. Fuel savings are estimated for a range of models for the minimum loading compared to a correspondingly critical uniform core loading. In some circumstances a saving of up to 70% is indicated with economic and safety implications. It is shown how the reduction of the minimum loading solution from a two- to a one-group model retains the flat flux in the core but fails to satisfy the thermal boundary condition unless a reflector, with the flux-trap, is used so that without it, the two-group minimum fuel loading solution cannot fully transform to a minimum loading in one group. The full solution for a two-group model flat flux core with finite thickness reflector is given. It is shown that as the reflector is reduced it becomes necessary to increase the fuel density at the centre to a point where this exceeds the capability of the chosen fuel, thus providing a secondary criticality equation. The increasing steepness (negative slope) of the flux distribution is required to make the flux trap phenomenon possible in the reducing reflector region. The range in which there may be two reflector thicknesses leading to the same size core, but different fuel distributions observed by Williams, is determined. Constrained solutions that limit either the size of core or the maximum fuel density are considered, generalising the original work of Goertzel to a practicable core design.
