Study of reactor constitutive model and analysis of nuclear reactor kinetics by fractional calculus approach

Abstract

The diffusion theory model of neutron transport plays a crucial role in reactor theory since it is simple enough to allow scientific insight, and it is sufficiently realistic to study many important design problems. The neutrons are here characterized by a single energy or speed, and the model allows preliminary design estimates. The mathematical methods used to analyze such a model are the same as those applied in more sophisticated methods such as multi-group diffusion theory, and transport theory. The neutron diffusion and point kinetic equations are most vital models of nuclear engineering which are included to countless studies and applications under neutron dynamics. By the help of neutron diffusion concept, we understand the complex behavior of average neutron motion. The simplest group diffusion problems involve only, one group of neutrons, which for simplicity, are assumed to be all thermal neutrons. A more accurate procedure, particularly for thermal reactors, is to split the neutrons into two groups; in which case thermal neutrons are included in one group called the thermal or slow group and all the other are included in fast group. The neutrons within each group are lumped together and their diffusion, scattering, absorption and other interactions are described in terms of suitably average diffusion coefficients and cross-sections, which are collectively known as group constants. We have applied Variational Iteration Method and Modified Decomposition Method to obtain the analytical approximate solution of the Neutron Diffusion Equation with fixed source. The analytical methods like Homotopy Analysis Method and Adomian Decomposition Method have been used to obtain the analytical approximate solutions of neutron diffusion equation for both finite cylinders and bare hemisphere. In addition to these, the boundary conditions like zero flux as well as extrapolated boundary conditions are investigated. The explicit solution for critical radius and flux distributions are also calculated. The solution obtained in explicit form which is suitable for computer programming and other purposes such as analysis of flux distribution in a square critical reactor. The Homotopy Analysis Method is a very powerful and efficient technique which yields analytical solutions. With the help of this method we can solve many functional equations such as ordinary, partial differential equations, integral equations and so many other equations. It does not require enough memory space in computer, free from rounding off errors and discretization of space variables. By using the excellence of these methods, we obtained the solutions which have been shown graphically.