Approximate Methods for Inverting Generating Functions from the Pál-Bell Equations for Low Source Problems

A number of approximate probability distribution functions (pdf’s) for the neutron density are examined with reference to low source startup. The most accurate method for determining the safe source strength, to reduce the likelihood of a rogue transient during startup, is that arising from the Pál-Bell equations. When these equations are extended to include space and energy dependence the numerical work becomes extensive. A pdf is developed which gives results that compare favorably with those from the exact solution but requires very much less numerical work. The method is applicable to space- and energy-dependent problems. Extensive numerical examples are given of the new method and of others which have been proposed over the years. In addition, we explore other approximations, unrelated to the generating function, that can lead to substantial computational savings. We have additionally described the principles behind, and provided a simple review of, the low source algorithm from which anyone unfamiliar with low source concepts can benefit.