Interplay of axis ratio on neutron flux in a spheroid nuclear reactor core using jacobi elliptic theta functions

Abstract

The instantaneous neutron’s density in a reactor core is influenced by several factors. Some of them include the reactor material’s characteristics and the reactor configuration geometry properties. The role of the former has been well explored and understood while the latter continues to arouse interest in research and applications despite being poorly understood for some configuration types. In particular, spheroid configuration exhibits relatively higher robustness compared to other. However, the behavior of time dependent neutron flux at varying axis ratios and how the latter affects neutron leakage rates has not been well explored for this type of configuration. Therefore, this study is aimed at establishing how the axis ratio determines the behavior of neutron flux and neutron leakage rates. Specifically; modeling and determining the behavior of time dependent neutron diffusion flux in a spheroid reactor core at varying axis ratios, formulating the relationship between the axis ratio and neutron leakage rates and elaborating the behavior of neutron leakage rates for both spheroids at axis ratios equal, smaller and larger than unity. In order to carry out this, Fick’s law of diffusion was modified into a Jacobi elliptic theta function to describe the desired time dependent neutron diffusion problem in spheroid coordinates system. The quasi- radial component was adapted to represent the axis ratio and thereafter appropriate boundary conditions were imposed. Secondly, a relationship between neutron leakage rate and the axis ratio of spheroids was formulated using geometric buckling and neutrons thermal life time equations, and the results were evaluated for axis ratios equal, smaller and larger than unity with software used to solve all the formulated equations. It was found that neutrons diffuse outwards from the core towards the boundaries of the spheroid exhibiting the characteristics of Jacobi elliptic theta curves of the third kind. Various configurations of diffusion configurations were obtained that included ternary surfaces, continuous and discontinuous surfaces of various characteristics as the value of ‘n’ was varied. In addition, neutrons diffusion behavior along the quasi-angular component and the time component was found to be largely similar. In the investigation of neutron leakage rate versus the axis ratio, both configurations (with the same volume and same neutron leakage constant (k)) exhibited similar profile, although the neutron leakage rate for prolate was lower compared to that of oblate at axis ratios smaller than unity. In contrast, at axis ratios larger than unity, it was found that the neutrons leakage rate for prolate became greater than that of an oblate of the same volume. The results further showed that, at axis ratio larger than unity, the neutron leakage rate was mildly affected by the axis ratio of the spheroid. Finally, the values for neutron leakage rates for both prolate and oblate spheroids converged when the axis ratio was unity, for instance, the neutron leakage rates for both types of spheroids was 2.5 neutrons/square unit for neutron leakage constant of k = 200. The findings of this study could be utilized in the design of superior reactors with enhanced safety that can mitigate against nuclear accidents by varying core axis ratios in order to alter reactor criticality conditions. Further research needs to be conducted on multigroup neutron diffusion for a similar problem and determining the flux behavior for each type of spheroid separately