An accurate solution of point kinetics equations of one-group delayed neutrons and an extraneous neutron source for step reactivity insertion

Abstract

The continuous indication of the neutron density and its rate of change are important for the safe startup and operation of reactors. The best way to achieve this is to obtain analytical solutions of the neutron kinetics equations because none of the developed numerical methods can well satisfy the need for real-time or even super-time computation for the safe startup and operation of reactors in practice. In this paper, an accurate analytical solution of point kinetics equations with one-group delayed neutrons and an extraneous neutron source is proposed to calculate the change in neutron density, where the whole process from the subcritical stage to critical and supercritical stages is considered for step reactivity insertions. The accurate analytical solution can also be used as a benchmark of all numerical methods employed to solve stiff neutron kinetics equations. neutron kinetics, accurate solution, extraneous neutron source, step reactivity Citation:Li H F, Shang X L, Chen W Z. An accurate solution of point kinetics equations of one-group delayed neutrons and an extraneous neutron source for step reactivity insertion.Equations of point reactor neutron kinetics describe variations in the densities of neutrons and delayed neutron precursors, and are coupled but stiff equations. Huge disparities between the life-spans of the prompt neutron and delayed neutron lead to time constants with different orders of magnitude for the solutions of fast and slow components. It is well accepted that analytical solutions can be obtained only under simplified conditions, such as there being no extraneous neutron source, there being a prompt jump [1-3], and there being a constant neutron source [4]. These analytical solutions can only be used for qualitative rather than quantitative computation. For the latter, researchers have to turn to a numerical method. However, a very small step size has to be adopted for stability when applying a general numerical method such as the Euler, Runge-Kutta, or Adams method. A small step results in a long computing time, and more importantly, there is large accumulated error due to *These authors contributed equally to this work †Corresponding author (email: Cwz2@21cn.com) the many computation steps. Many researchers have attempted to solve this problem and some relatively effective numerical methods have been proposed, such as the finite-difference method [5], finite-element method [6], Runge-Kutta procedure [7], quasistatic method [8,9], piecewise polynomial approach [10], singular perturbation method [11], stiffness confinement method [12], power series solution [13-15], and Padé approximation [16-18]. Other methods [19] are being proposed and discussed.However, none of these methods can satisfy the need for real-time or even super-time computation in the safe operation of a reactor in practice. With so many numerical methods available, which should we choose? Can we in fact obtain an accurate analytical solution to the equations of point reactor neutron kinetics for step reactivity inse...

Explicit appropriate basis function method for numerical solution of stiff systems

“…And there are several methods especially adapted for solving the initial value problems for stiff systems of ordinary differential equations (Aboanber and Hamada, 2003;Aboanber, 2004;Tashakor et al, 2010). Among the methods are numerical integration using Simpson's rule, finite element method, Runge-Kutta procedures, quasi-static method, piecewise polynomial approach and other methods (Li et al, 2010;Abdallah and Nahla, 2011;Hamada, 2013). Most of these methods are successful in some specific problems, but still suffer, more or less, from disadvantages as mentioned by Chen et al, 2013. In this paper, an explicit numerical method for stiff systems is developed and tested.…”