Higher-order approximate solutions of fractional stochastic point kinetics equations in nuclear reactor dynamics

Stochastic point kinetics equations(SPKEs) are a system of Ito? stochastic differential equations whose solution has been obtained by higher-order approximation.In this study, a fractional model of SPKEs has been analyzed. The efficiency of the proposed higher-order approximation scheme has been discussed in the results section. The solutions of SPKEs in the presence of Newtonian temperature feedback have also been provided to further discuss the physical behavior of the fractional model.     

A new integral method for solving the point reactor neutron kinetics equations

A numerical integral method that efficiently provides the solution of the point kinetics equations by using the better basis function (BBF) for the approximation of the neutron density in one time step integrations is described and investigated. The approach is based on an exact analytic integration of the neutron density equation, where the stiffness of the equations is overcome by the fully implicit formulation. The procedure is tested by using a variety of reactivity functions, including step reactivity insertion, ramp input and oscillatory reactivity changes. The solution of the better basis function method is compared to other analytical and numerical solutions of the point reactor kinetics equations. The results show that selecting a better basis function can improve the efficiency and accuracy of this integral method. The better basis function method can be used in real time forecasting for power reactors in order to prevent reactivity accidents.     

Developed mathematical technique for fractional stochastic point kinetics model in nuclear reactor dynamics

Fractional stochastic kinetics equations have proven to be valuable tools for the point reactor kinetics model, where the nuclear reactions are not fully described by deterministic relations. A fractional stochastic model for the point kinetics system with multi-group of precursors,including the effect of temperature feedback, has been developed and analyzed. A major mathematical and inflexible scheme to the point kinetics model is obtained by merging the fractional and stochastic technique. A novel split-step method including mathematical tools of the Laplace transforms, Mittage–Leffler function, eigenvalues of the coefficient matrix, and its corresponding eigenvectors have been used for the fractional stochastic matrix differential equation. The validity of the proposed technique has been demonstrated via calculations of the mean and standard deviation of neutrons and precursor populations for various reactivities: step, ramp, sinusoidal, and temperature reactivity feedback. The results of the proposed method agree well with the conventional one of the deterministic point kinetics equations.     

Stochastic point-kinetics equations in nuclear reactor dynamics

A system of Itô stochastic differential equations is derived that model the dynamics of the neutron density and the delayed neutron precursors in a point nuclear reactor. The stochastic model is tested against Monte Carlo calculations and experimental data. The results demonstrate that the stochastic differential equation model accurately describes the random behavior of the neutron density and the precursor concentrations in a point reactor.     

Analytical solution of point kinetics equations for linear reactivity variation during the start-up of a nuclear reactor

The analytical solution of point kinetics equations with a group of delayed neutrons is useful in predicting the variation of neutron density during the start-up of a nuclear reactor. In the practical case of an increase of nuclear reactor power resulting from the linear insertion of reactivity, the exact analytical solution cannot be obtained. Approximate solutions have been obtained in previous articles, based on considerations that need to be verifiable in practice. In the present article, an alternative analytic solution is presented for point kinetics equations in which the only approximation consists of disregarding the term of the second derivative for neutron density in relation to time. The results proved satisfactory when applied to practical situations in the start-up of a nuclear reactor through the control rods withdraw.     

Analytical solution to solve the point reactor kinetics equations

New analytical solution for solving the point reactor kinetics equations with multi-group of delayed neutrons is presented. This solution is based on the roots of inhour equation, eigenvalues of the coefficient matrix. The inhour equation presents in new sample formula. The analytical solution represents the exact analytical solution for the point kinetics equations of multi-group of delayed neutrons with constant reactivity. Also, it represents the accurate solution for solving the point kinetics equations of multi-group of delayed neutrons with ramp and temperature feedback reactivities. This method are applied to different types of reactivity and compared to the traditional methods.     

On the stability of the point reactor kinetics equations

As the basic neutronic problem is unstable by nature, maintaining a reactor critical is a task that requires a lot of effort. This work presents and discusses some aspects related to the stability of the basic physics behind a nuclear reactor core based on the well-known point reactor kinetics equations. First, the linear non-feedback case is studied where differences between Lyapunov and BIBO stability are found. These differences are shown both numerically and analytically, and explained using a reactor physics based reasoning. Finally, a simple model is used to analyse the intrinsic stability of the point reactor equations when reactivity feedbacks are taken into account. A method for constructing conceptual stability design maps is proposed and a basic interpretation of the simple results obtained is given.     

代中子时间法求解点堆中子动力学方程

由于点堆中子动力学方程是个刚性方程,因此准确、快速、稳定地求解方程是困难的。得益于现代计算机技术的进步,本文直接采用代中子时间计算法求解点堆中子动力学方程,并用C++语言编制了计算程序。经过基准例题和动态-逆动态对比计算,验证了模型、程序计算的准确性和稳定性,而计算时间也是可接受的。     

一个求解点堆中子动力学方程组的数值积分方法

在求解点堆中子动力学方程组中,对中子密度N(t)使用分段全隐式一阶泰劳多项式近似技术,给出一个求解的数值积分方法。用FORTRAN语言在COMPAQ机上计算实例的数值结果表明:对给定的反应性输入,此方法能够取得较高精确度的数值结果,计算过程简洁且计算速度快,可适宜于反应堆中子动力学控制的设计分析和仿真计算。     

求解点堆动态方程的IGear方法

求解点堆中子动力学方程的改进的Gear方法——IGear,以其高阶和大稳定域的性质保持了Gear方法适应性强,计算精度较高的优点,并改进了它的缺点。因此能够灵活应用于各种反应性的输入。计算结果表明:从计算时间和计算精度上看,该方法都可以满足实际工程的需要,是求解点堆动态方程的一种较好的方法。     

Simulation of nuclear reactor dynamics equations using reconfigurable computing

This paper describes the application of a multilayer discrete-time cellular neural network (DT-CNN¹) and its hardware implementation on a field programmable gate array (FPGA²) to model and simulate the nuclear reactor dynamics equations. A new computing architecture model based on FPGA and its detailed hardware implementation are proposed for accelerating the solution of nuclear reactor dynamics equations. The proposed FPGA-based reconfigurable computing platform is implemented on a Xilinx FPGA device and is utilized to simulate step and ramp perturbation transients in typical 2D nuclear reactor cores. Properties of the implemented specialized architecture are examined in terms of speed and accuracy against the numerical solution of the nuclear reactor dynamics equations using MATLAB and C programs. Steady state as well as transient simulations, show a very good comparison between the two methods. Also, the validity of the synthesized architecture as a hardware accelerator is demonstrated.     

An efficient technique for the point reactor kinetics equations with Newtonian temperature feedback effects

The point reactor kinetics equations of multi-group of delayed neutrons in the presence Newtonian temperature feedback effects are a system of stiff nonlinear ordinary differential equations which have not any exact analytical solution. The efficient technique for this nonlinear system is based on changing this nonlinear system to a linear system by the predicted value of reactivity and solving this linear system using the fundamental matrix of the homogenous linear differential equations. The nonlinear point reactor kinetics equations are rewritten in the matrix form. The solution of this matrix form is introduced. This solution contains the exponential function of a variable coefficient matrix. This coefficient matrix contains the unknown variable, reactivity. The predicted values of reactivity in the explicit form are determined replacing the exponential function of the coefficient matrix by two kinds, Backward Euler and Crank Nicholson, of the rational approximations. The nonlinear point kinetics equations changed to a linear system of the homogenous differential equations. The fundamental matrix of this linear system is calculated using the eigenvalues and the corresponding eigenvectors of the coefficient matrix. Stability of the efficient technique is defined and discussed. The efficient technique is applied to the point kinetics equations of six-groups of delayed neutrons with step, ramp, sinusoidal and the temperature feedback reactivities. The results of these efficient techniques are compared with the traditional methods.     

Numerical treatment for the point reactor kinetics equations using theta method,eigenvalues and eigenvectors

The point reactor kinetics model is a stiff system of linear/nonlinear ordinary differential equations. In fact, the numerical solutions of this stiff model need a smaller time step intervals within various computational schemes. The aim of this work is an accurate numerical solution without need to the smaller time step intervals. Theta method is the most popular, simplest and widely used method for solving the first order ordinary differential equations. In light of this fact, theta method is treated for solving the matrix form of this model via the eigenvalues and corresponding eigenvectors of the coefficient matrix. In this work, the matrix form of the stiff point kinetics equations with multi-group of delayed neutrons is introduced. The treatment theta method is applied to solve the stiff point kinetics equations with six groups of delayed neutrons. The performance of the treatment theta method is evaluated in several case studies involving step, ramp, sinusoidal and pulse reactivities. The results of the treatment theta method are more accurate than the theta method comparing with the conventional methods.     

Numerical solution of the point reactor kinetics equations with fuel burn-up and temperature feedback

Point reactor kinetics equations are solved numerically using one group of delayed neutrons and with fuel burn-up and temperature feedback included. To calculate the fraction of one-group delayed neutrons, a group of differential equations are solved by an implicit time method. Using point reactor kinetics equations, changes in mean neutrons density, temperature, and reactivity are calculated in different times during the reactor operation. The variation of reactivity, temperature, and maximum power with time are compared with the predictions by other methods.     

Analysis on the neutron kinetics for a molten salt reactor

The neutron kinetics of the molten salt reactor is significantly influenced by the fuel salt flow, which leads to the analysis methods for the conventional reactors using solid fuels not being applicable for the molten salt reactors. In this study, a neutron kinetic model considering the fuel salt flow is established based on the neutron diffusion theory, which consists of two-group neutron diffusion equations for the fast and thermal neutron fluxes and six-group balance equations for delayed neutron precursors. The temperature feedback on the neutron kinetics is considered by introducing a heat transfer model in the core, in which the group constants which are dependent on the temperature are calculated by the code DRAGON. The mathematical equations are discretized and numerically calculated by developing a code, in which the fully implicit scheme is adopted for the time-dependent terms, and the power law scheme is for the convection–diffusion terms. The neutron kinetics is conducted during three transient conditions including the rods drop transient, the pump coastdown transient and the inlet temperature drop transient. The relative power changes and the distributions of the temperature, neutron fluxes and delayed neutron precursors under these three different transient conditions are obtained in the study. The results provide some valuable information for the research and design of this new generation reactor.     

Methods for solving the stochastic point reactor kinetic equations

In this paper we present two new methods to analyze the statistical properties of nonlinear outputs of a point reactor to stochastic non-white reactivity inputs. They are Bourret's approximation and logarithmic linearization. The results have been compared with the exact results, previously obtained in the case of Gaussian white reactivity input. It was found that when the reactivity noise has short correlation time, Bourret's approximation should be recommended because it yields results superior to those yielded by logarithmic linearization. When the correlation time is long, Bourret's approximation is not valid, but in that case, if one can assume the reactivity noise to be Gaussian, one may use the logarithmic linearization.