Spanning the Core Design Space with Binary Perturbations - A Mathematical Investigation
A reactor core design engineer, even one with many years of experience or one using advanced fuel pattern optimization software, will likely fine tune the pattern before finalizing the design. Fine tuning can further minimize local enthalpy peaks, optimize reactivity to extend burnup, and add additional margin to limits on reactivity coefficients or heat flux. For reactors that have reached an equilibrium fuel cycle, the number of feeds each cycle is typically the similar to previous cycles as is their placement in the core. It is then the engineer's responsibly to efficiently select the placement of burned assemblies to ensure optimal fuel utilization, satisfy safety limits, and meet cycle energy requirements. One of the engineer's most commonly used tools in this fine tuning investigation is to swap the positions of two assemblies, most likely two previously irradiated ('burned') assemblies since their reactivity is lower than fresh fuel. Such an operation is referred to in this work as a binary perturbation or a 'swap'. If quarter core symmetry is maintained, the perturbation translates to the other three quadrants. Likewise if eighth core symmetry is maintained, the perturbation translates to other seven octants. The question is then, how much of the core design space can binary perturbations possibly span? This work seeks to quantitatively define the space spanned by m number of binary perturbations as a function of variable size through combinatorial principles and computer simulation. The results of this study are useful in understanding the in-core fuel management problem faced by both professionals who employ hand manipulations and those who attempt stochastic solutions of the problem.