Nonlocal fractal neutrons transport equation and its implications in nuclear engineering

In this study, the nonlocal fractal neutrons transport equation in fractal dimensions (alpha, beta) is constructed based on the concepts of fractal derivative and a spatially symmetric kernel function which measures the influence of neighbours. The resulting equation is comparable to the Swift-Hohenberg equation which plays a central role in studies of patterns formation. The new transport equation generalizes the neutron telegraph equation based on Cattaneo's constitutive law. Both the dominant and subdominant higher-order moments are discussed for dependent and time-independent fractal neutrons transport equations. Estimates of the numerical values of fractal dimensions have been obtained for both cases. For the time-independent transport it was observed that for the dominant case plausible solutions limit the fractal dimensions to 0.636364 < beta < 0.716542 or 0.919821 < beta < 0.951938 whereas the subdominant case settles 0.5 < beta < 1. Whereas for a time-dependent transport stationary solutions are obtained for 0 < alpha < 0.5, and the system is subdiffusive. This result indicates that for lower fractal dimensions, burly absorbing regions exist next to the fuel bundles and control rods of the nuclear reactor. We have also discussed the time-dependent subdominant higher-order moments for the case of parallelepiped nuclear reactor. Our analysis indicates that the ratio between the maximum and the average flux is much lesser than the ratio obtained in the Euclidean dimensions. This has important implications in nuclear engineering and designs since in that case the heat production and the fuel burn-up are uniform since the nuclear materials such Uranium burns-up non-uniformly in the reactor core. This study represents a crucial step in the process of building models for a nuclear reactor based on fractal derivatives and dimensions.