A stochastic, four energy group model of a nuclear assembly

The formulation of a stochastic, four energy group, space-independent model of a nuclear assembly was studied. The model included one group of delayed neutron precursors and an extraneous source. The model treated the evolutionary birth and death processes of a neutron chain in a nuclear assembly as a multi-dimensional Markovian process. The probability balance equation was derived by accounting for all the birth and death processes occurring in the assembly and then was changed into its differential-difference form. Using the probability generating function technique, the differential-difference equation was transformed into a first order partial differential equation in the probability generating function. Equations for the neutron and delayed neutron precursor population probability distribution functions, the deterministic functions (means, variances, and covariances), and the extinction probabilities of the neutron chains were derived from the characteristic equations for the various moments of the first order P.D.E. for the probability generating function.Numerical results for the population probability distribution functions, deterministic functions, and extinction probabilities of neutron chains were calculated for three different cases: (1) An infinite fissile assembly with delayed neutrons ignored and no extraneous sources present; (2) A finite fissile assembly with delayed neutrons ignored and no extraneous sources present; (3) A finite fissile assembly with delayed neutrons considered and no extraneous sources present. The four energy group cross sections used in the calculations were a collapsed version of the Hansen-Roach sixteen energy group cross sections (42). The finite sized assembly was approximated in the space-independent model by estimated the neutron leakage probabilities from output from the program TWODANT (3) and treating them as loss reactions.The results were compared to show the effects of neutron leakage, delayed neutrons, and different energy neutrons starting the chain reactions in the fissile assembly. The four energy group model results for the absolute extinction probability of a neutron chain in an infinite U-235 fissile assembly compared favorably with the one energy group model found in the literature (14). For all cases, the asymptotic limit that the time dependent neutron population extinction probability approaches underpredicts the absolute extinction probability of the neutron chain by 30-50%.U of I OnlyETDs are only available to UIUC Users without author permissio