Nonlinear fractional diffusion model for space-time neutron dynamics

Abstract

In this work, a new fractional multidimensional neutron diffusion model is introduced using Cattaneo equation instead of the commonly used Fick's law for two neutron energy groups. The model represents a coupled system of nonlinear hyperbolic fractional partial differential equations considering finite fast and finite thermal neutron velocities. The model provides new quantities that relate some of the most important nuclear parameters together producing some newly physical concepts. These quantities include fast and thermal neutron relaxation times, Riemann-Liouville fractional integrals of the neutron densities and derivatives of macroscopic absorption cross sections. This model can be useful for mathematicians and physicists since the anomalous subdiffusion process describes the neutron movement closer to reality. The effect of the anomalous subdiffusion exponent (0<γ≤1) on the neutron flux is discussed. The model is successfully applied to simple homogenous and heterogonous reactor cores with step, ramp and sinusoidal perturbations for different values ofγ to estimate the average reactor powers in the fuel and moderator regions. Also, the new fractional model is applied to LRA BWR benchmark problem including adiabatic heating and Doppler feedback. On the other hand and based on the concept of prompt jump, the proposed fractional model is simplified and then analytically solved using Mittag-Liffler function. Solving fractional models numerically consumes considerable computational time costs due to the persistent memory of fractional operators, especially for smaller values of γ. For this regard, adaptively multi-step differential transform MDTM method is implemented to solve the proposed fractional model. The MDTM is developed using step size control technique for which the length of the step integration varies according to the solution behaviour.