Improvements to the Theory of Resonance Escape in Heterogeneous Fuel: II. The Dancoff Factor and the Equivalence Theorem in Cluster-Type Fuel

In a previous paper Leslie, Hill and Jonsson put forward a method for the rapid evaluation of the Dancoff factor in regular arrays of fuel rods. They also showed how extended rational approximations to the fuel nonescape probability could be used to improve the form of the equivalence theorem based on Wigner's rational approximation. This form of equivalence asserts that the resonance integral is a function of the geometry through the excess potential scattering 1/N only, where N is the number density of the absorber and is the mean chord. In the modification proposed by Leslie, Hill and Jonsson, this function is generalized to a/N; the Bell factor a is found to vary with coolant density. By making use of an approximate analytic method for the calculation of collision probabilities in geometries more general than regular arrays, the present authors extend this work to cluster-type fuel elements. The basic procedure is the same as in the work referred to above. An analytic expression for the fuel-to-fuel collision probability is derived using arguments about its behavior in the black and white limits (i.e. in the limits of high and low cross sections). The Dancoff factor is derived from the behavior in the black limit. It is shown, by comparison with exact calculations, that for two types of cluster geometry of current interest in fuel element design, the proposed Dancoff factor is in error by at most 2%. Improved equivalence relations for cluster geometry are also considered. It has been customary to assume that the cluster is equivalent to an isolated rod of diameter d_p/Γ, where d_p is the diameter of a single pin in the cluster and Γ is the Dancoff factor. Such a procedure implies that the Bell factor of the cluster is constant and equal to its value for an isolated rod. It is shown in this paper that the Bell factor is a function of coolant density and that, in a particular case, the cluster is almost equivalent to an isolated rod at low density. As the density increases, the Bell factor drops rapidly by 6% and then increases slowly.