This thesis presents a range of mathematical and computational models for use in transient nuclear criticality safety assessment. A mathematical model for quantifying the uncertainty in the wait-time probability distributions of criticality excursions initiated in the presence of weak intrinsic neutron sources is presented. This model is used to demonstrate the potential influence of parametric uncertainty on the wait-time probability distributions of the 1958 Y-12 criticality accident and experiments on the Caliban reactor.

Also presented in this thesis is a new mathematical and computational model of radiolytic gas production and evolution in fissile liquids. This model has been validated against nuclear criticality safety benchmark experiments on fissile solution reactors and has been shown to accurately predict features of the fission power profiles related to the appearance and advection of radiolytic gas voids in the solution. The model has also demonstrated efficacy in predicting the timing and magnitude of secondary peaks in the fission power output. The purpose of this new mathematical and computational radiolytic gas model was to improve the simulation of fissile liquid criticality transients while removing the need for the adjustable heuristic parameters used by existing fissile liquid simulation codes. These parameters, which must be appropriately adjusted to criticality safety benchmark experiments, are dependent on the geometry and composition of the system being analysed. The need for these heuristic parameters therefore precludes the use of these codes as predictive modelling and simulation tools. The new mathematical and computational model, presented in this thesis, offers valuable insights into the behaviour of radiolytic gas in fissile liquid systems.