Optimal fissile distribution in multiplying systems: Illustrative examples with Monte Carlo simulation and Pontryagin's maximum principle

In multiplying systems, such as nuclear reactors and criticality experiments, it is desirable to place the fissile material in the optimal or ‘best’ way to reduce the critical mass of the system as well as to achieve uniform fuel burnup. This paper considers two methods, namely Pontryagin’s maximum principle (PMP) and Monte Carlo (MC) perturbation for estimating a minimum critical mass configuration. These methods are applied to an elementary multizone model of a pressurized water reactor (PWR) and a criticality experiment to estimate the minimum critical mass. It is found that while two-group diffusion theory with PMP predicts a minimum critical mass, more detailed MC simulations with MCNP5 show a consistent reduction in critical mass when fissile fuel is placed in inner zones. Such a distribution reduces the fissile material requirement but is undesirable due to the higher power peaking. MC simulations show that for a three-zone model of the KORI 1 PWR, a uniform fissile distribution gives criticality for 1.09 atomic percent (at.%) enrichment, whereas non-uniform fissile distribution (0.6, 1.6, 0.6 at.%) reduces the critical mass by 14%. The changes found from MC simulations were subsequently predicted from first- and second-order derivative sampling. It was found that substantial computational savings can be achieved for large-scale optimization problems. In the case of a criticality experiment, MC derivative sampling was also used to estimate optimal fissile distribution for minimizing the critical mass.