scholarly journals Solution of Point Reactor Neutron Kinetics Equations with Temperature Feedback by Singularly Perturbed Method

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Wenzhen Chen ◽  
Jianli Hao ◽  
Ling Chen ◽  
Haofeng Li
Keyword(s):  
Nuclear Reactor ◽  
Large Range ◽  
Accurate Solution ◽  
Average Density ◽  
Singularly Perturbed ◽  
Neutron Density ◽  
Delayed Neutron ◽  
Small Step ◽  
Neutron Kinetics ◽  
Temperature Feedback

The singularly perturbed method (SPM) is proposed to obtain the analytical solution for the delayed supercritical process of nuclear reactor with temperature feedback and small step reactivity inserted. The relation between the reactivity and time is derived. Also, the neutron density (or power) and the average density of delayed neutron precursors as the function of reactivity are presented. The variations of neutron density (or power) and temperature with time are calculated and plotted and compared with those by accurate solution and other analytical methods. It is shown that the results by the SPM are valid and accurate in the large range and the SPM is simpler than those in the previous literature.

An Integral Method for Solving the Point Reactor Neutron Kinetics Equations with Newtonian Temperature Feedback

2013 ◽  
Vol 732-733 ◽  
pp. 83-89
Author(s):  
Gen Fan ◽  
Wen Bin Liu
Keyword(s):  
Basis Function ◽  
Integral Method ◽  
Numerical Evaluation ◽  
Power Reactor ◽  
Neutron Density ◽  
Delayed Neutron ◽  
Time Step ◽  
Neutron Kinetics ◽  
Temperature Feedback ◽  
Point Kinetics

A numerical integral method to efficiently solve the point kinetics equations with Newtonian temperature feedback is described and investigated, which employs the better basis function (BBF) for the approximation of the neutron density in integral of one time step. The numerical evaluation is performed by the developed BBF code. The code can solve the general non-linear kinetics problems with six groups of delayed neutron. For the application purposes, the developed code and the method are tested by using a variety of problems, including ramp reactivity input with or without temperature feedback. The results are shown that the BBF method is clearly an effective and accurate numerical method for solving the point kinetics equations with Newtonian temperature feedback, and it can be used in real time power reactor forecasting in order to prevent the reactivity accidents.

scholarly journals A New Accurate Numerical Method Based on Shifted Chebyshev Series for Nuclear Reactor Dynamical Systems

2018 ◽  
Vol 2018 ◽  
pp. 1-15 ◽  
Author(s):  
Yasser Mohamed Hamada
Keyword(s):  
Nuclear Reactor ◽  
Approximate Solutions ◽  
Time Interval ◽  
Neutron Density ◽  
Chebyshev Series ◽  
System A ◽  
Temperature Feedback ◽  
Exponential Rate Of Convergence ◽  
Point Kinetics ◽  
The Given

A new method based on shifted Chebyshev series of the first kind is introduced to solve stiff linear/nonlinear systems of the point kinetics equations. The total time interval is divided into equal step sizes to provide approximate solutions. The approximate solutions require determination of the series coefficients at each step. These coefficients can be determined by equating the high derivatives of the Chebyshev series with those obtained by the given system. A new recurrence relation is introduced to determine the series coefficients. A special transformation is applied on the independent variable to map the classical range of the Chebyshev series from [-1,1] to [0,h]. The method deals with the Chebyshev series as a finite difference method not as a spectral method. Stability of the method is discussed and it has proved that the method has an exponential rate of convergence. The method is applied to solve different problems of the point kinetics equations including step, ramp, and sinusoidal reactivities. Also, when the reactivity is dependent on the neutron density and step insertion with Newtonian temperature feedback reactivity and thermal hydraulics feedback are tested. Comparisons with the analytical and numerical methods confirm the validity and accuracy of the method.

A Comparative Study of Ten Different Methods on Numerical Solving of Point Reactor Neutron Kinetics Equations

2018 ◽  
Vol 4 (2) ◽  
Author(s):  
Yining Zhang ◽  
Haochun Zhang ◽  
Kexin Wang
Keyword(s):  
Nuclear Power ◽  
Nuclear Reactor ◽  
Reactor Core ◽  
Density Variation ◽  
Euler Method ◽  
Reactor Neutron ◽  
Computational Time ◽  
Neutron Density ◽  
Power Series Method ◽  
Neutron Kinetics

Point reactor neutron kinetics equations describe the time-dependent neutron density variation in a nuclear reactor core. These equations are widely applied to nuclear system numerical simulation and nuclear power plant operational control. This paper analyzes the characteristics of ten different basic or normal methods to solve the point reactor neutron kinetics equations. The accuracy after introducing different kinds of reactivity, stiffness of methods, and computational efficiency are analyzed. The calculation results show that: considering both the accuracy and stiffness, implicit Runge–Kutta method and Hermite method are more suitable for solution on these given conditions. The explicit Euler method is the fastest, while the power series method spends the most computational time.

The Controllability of the Point Reactor Neutron Kinetics Equations

2014 ◽  
Author(s):  
Dong Liu ◽  
Xiuchun Luan ◽  
Tao Yu ◽  
Weining Zhao ◽  
Lei Liu
Keyword(s):  
Nuclear Reactor ◽  
Radioactive Decay ◽  
Pole Placement ◽  
Reactor Neutron ◽  
Final Conclusion ◽  
Delayed Neutron ◽  
Neutron Kinetics ◽  
Controllability Matrix ◽  
The Matrix ◽  
Point Kinetics

In this paper, the conditions to ensure the controllability of the point reactor neutron kinetics equations are studied. In a nuclear reactor, due to the delayed neutron precursor concentration and the internal reactivity, the kinetics equations of the nuclear reactor are nonlinear. To solve the problem of the pole placement, the controllability of the point kinetics equations must be guaranteed. Then, a new method to analysis of the controllability conditions of the point kinetics equations of a reactor is carried out here. The method is based on the controllability matrix directly denoted by relevant symbols, and a formula used for controllability analysis is showed with symbols by calculating the determinant of the matrix. First, with using the linearization technique, the equations are linearized with respect to any possible equilibrium point. Subsequently, an analysis of the controllability of the general linear model that includes only one group delayed neutron precursor is performed, obtaining the interesting result that the controllability of the equations are controllable except when the effective precursor radioactive decay constant and the reciprocal of the fuel-to-coolant heat transfer mean time have the same value, which does not occur in practice. Thus, with the same method, the other analysis obtained the conditions to guarantee the controllability of the point kinetics equations with different groups delayed neutron precursor, which includes two-group, three-group and six-group models. Then, the results are compared with that of the numerical controllability matrix, obtaining the final conclusion that the results of the new analysis method give the closer results to the actual situation and list the restrictions that guarantee the controllability of the point reactor neutron kinetics equations.

A parameter-uniform finite difference scheme for singularly perturbed parabolic problem with two small parameters

2021 ◽  
Author(s):  
Tesfaye Aga Bullo ◽  
Guy Aymard Degla ◽  
Gemechis File Duressa
Keyword(s):  
Finite Difference ◽  
Difference Scheme ◽  
Finite Difference Scheme ◽  
Variable Parameter ◽  
Accurate Solution ◽  
Singularly Perturbed ◽  
Finite Difference Approximation ◽  
Parabolic Problems ◽  
Difference Approximation ◽  
Uniform Mesh

A parameter-uniform finite difference scheme is constructed and analyzed for solving singularly perturbed parabolic problems with two parameters. The solution involves boundary layers at both the left and right ends of the solution domain. A numerical algorithm is formulated based on uniform mesh finite difference approximation for time variable and appropriate piecewise uniform mesh for the spatial variable. Parameter-uniform error bounds are established for both theoretical and experimental results and observed that the scheme is second-order convergent. Furthermore, the present method produces a more accurate solution than some methods existing in the literature.   

Coupled Analysis of Thermal Hydraulics and Neutronics for a Molten Salt Reactor

2017 ◽  
Author(s):  
Jian Ge ◽  
Dalin Zhang ◽  
Wenxi Tian ◽  
Suizheng Qiu ◽  
G. H. Su
Keyword(s):  
Molten Salt ◽  
Nuclear Reactor ◽  
Coupled Analysis ◽  
Molten Salt Reactor ◽  
Thermal Hydraulics ◽  
Delayed Neutron ◽  
Analysis Model ◽  
Liquid Salt ◽  
Energy Equations ◽  
Volume Method

As one of the six selected optional innovative nuclear reactor in the generation IV International Forum (GIF), the Molten Salt Reactor (MSR) adopts liquid salt as nuclear fuel and coolant, which makes the characteristics of thermal hydraulics and neutronics strongly intertwined. Coupling analysis of neutronics and thermal hydraulics has received considerable attention in recent years. In this paper, a new coupling method is introduced based on the Finite Volume Method (FVM), which is widely used in the Computational Fluid Dynamics (CFD) methodology. Neutron diffusion equations and delayed neutron precursors balance equations are discretized and solved by the commercial CFD package FLUENT, along with continuity, momentum and energy equations simultaneously. A Temporal And Spatial Neutronics Analysis Model (TASNAM) is developed using the User Defined Functions (UDF) and User Defined Scalar (UDS) in FLUENT. A neutronics benchmark is adopted to demonstrate the solution capability for neutronics problems using the method above. Furthermore, a steady state coupled analysis of neutronics and thermal hydraulics for the Molten Salt Advanced Reactor Transmuter (MOSART) is performed. Two groups of neutrons and six groups of delayed neutron precursors are adopted. Distributions of the liquid salt velocity, temperature, neutron flux and delayed neutron precursors in the core are obtained and analyzed. This work can provide some valuable information for the design and research of MSRs.

A Space-Time Parallel Method to Solve Space-Dependent Neutron Kinetics Equations in Hexagonal-Z Geometry

2018 ◽  
Author(s):  
Zhizhu Zhang ◽  
Yun Cai ◽  
Xingjie Peng ◽  
Qing Li
Keyword(s):  
Nuclear Reactor ◽  
Time Integration ◽  
Euler Method ◽  
Space Time ◽  
Parallel Method ◽  
Sequential Method ◽  
Neutron Kinetics ◽  
Nodal Method ◽  
Backward Euler ◽  
Parareal Method

Neutron kinetics plays an important role in reactor safety and analysis. The backward Euler method is the most widely used time integration method in the calculation of space-dependent nuclear reactor kinetics. Diagonally Implicit Runge-Kutta (DIRK) method owns high accuracy and excellent stability and it could be applied to the neutron kinetics for hexagonal-z geometry application. As solving the neutron kinetics equations is very time-consuming and the number of available cores continues to increase with parallel architectures evolving, parallel algorithms need to be designed to utilize the available resources effectively. However, it is difficult to parallel in time axis since the later moment is strongly dependent on the previous moment. In this paper, the Parareal method which is a time parallel method and implemented by MPI in the processor level is studied in the hexagonal-z geometry with the help of DIRK method. In order to make good use of the parallelism, a parallel strategy in the space direction is also used. In the coarse nodal method, many same operations are finished in the nodes and these operations could be parallel by OpenMP in the thread level since they are independent. Several transient cases are used to validate this method. The results show that the Parareal method gets a fast-convergent speed such as only 2∼3 iterations are needed to convergent. This space-time parallel method could reduce the cost time compared to the sequential method.

scholarly journals Generalized and Stability Rational Functions for Dynamic Systems of Reactor Kinetics

2013 ◽  
Vol 2013 ◽  
pp. 1-12 ◽  
Author(s):  
Ahmed E. Aboanber
Keyword(s):  
Dynamic Systems ◽  
Nuclear Reactor ◽  
High Efficiency ◽  
Rational Functions ◽  
Neutron Density ◽  
Time Saving ◽  
General Validity ◽  
Reactor Kinetics ◽  
Stiff Ordinary Differential Equations ◽  
Different Types

The base of reactor kinetics dynamic systems is a set of coupled stiff ordinary differential equations known as the point reactor kinetics equations. These equations which express the time dependence of the neutron density and the decay of the delayed neutron precursors within a reactor are first order nonlinear and essentially describe the change in neutron density within the reactor due to a change in reactivity. Outstanding the particular structure of the point kinetic matrix, a semianalytical inversion is performed and generalized for each elementary step resulting eventually in substantial time saving. Also, the factorization techniques based on using temporarily the complex plane with the analytical inversion is applied. The theory is of general validity and involves no approximations. In addition, the stability of rational function approximations is discussed and applied to the solution of the point kinetics equations of nuclear reactor with different types of reactivity. From the results of various benchmark tests with different types of reactivity insertions, the developed generalized Padé approximation (GPA) method shows high accuracy, high efficiency, and stable character of the solution.

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