Point Kinetics Equations Research Articles

Time-space fractional neutron point kinetics: Theory and simulations

In this work the mathematical derivation and numerical analysis of a fractional nuclear reactor point kinetics in time and space (TSFNPK) is presented. The TSFNPK model was derived considering a non-Fickian law for the neutron density current where the differential operators in space and time are of fractional order. The TSFNPK equations presented in this work constitutes a novel model for nuclear reactor kinetics, thus represent an extended model with respect to other fractional models and the classical neutron point kinetics equations. The model considers two diffusion exponents: one for the differential operator dependent in time and another for the dependent operator in space. Both exponents effect is numerically analyzed considering changes of reactivity step type, and temperature feedback reactivity. A first approach of the TSFNPK is presented, without temperature effects, and then a second approach considering temperature feedback effects is analyzed. In a following work, as a demonstration of application, a detailed analysis along with verification will be presented.

Simulation of IAEA Reactivity Initiated Transient Benchmarks Using Reactor Dynamics Code “REDAC”

Abstract This paper presents the analysis of reactivity initiated transients in an idealized, light water research reactor as a part of International Atomic Energy Agency (IAEA) safety related benchmark. The simulation model is based on point reactor kinetics coupled with one-dimensional (1D), two-channel model for thermal hydraulics. The point kinetics equations (PKEs) have been solved using an implicit Runge–Kutta (RK) method and the coolant transport equations have been solved using implicit finite difference formulation. Accuracy of the implemented models and methods has been demonstrated. Important safety parameters like peak power, peak fuel, and coolant temperatures have been predicted for a series of transients. Intercode comparison shows that the predictions of the present simulations are in good agreement with other codes. This approach provides a time efficient solution for safety analysis of reactors with tightly coupled core where point kinetics can be applied. To address the sensitivity of predictions with respect to important input parameters, simulations have been carried out with different sets of inputs reported in the literature. They indicate that predictions for fast transients are spread over a wider range compared to slow transients. For a given transient, predictions of peak power have a wider spread, while peak temperatures are relatively less sensitive to neutronic inputs. Also, for fast transients, prompt neutron generation time and delayed neutron fraction have dominant influence on the evolution of power. For slow transients, the reactivity feedback effects are equally important.

A new mathematical approach using Hermite approximations method for analysis of transient behavior in research reactors

In this paper, a new mathematical model based on Hermite approximations is presented to investigate the thermal behavior of a research reactor in transient conditions. For this purpose, the point kinetics equations are coupled to the thermal-hydraulic equations for a coolant and improved lumped heat conduction equation for the fuel and cladding. In contrast with the traditional method (such as finite difference method), the developed method can find exactly maximum temperatures of the fuel and cladding without considering numerical nodes in the fuel and cladding materials. This method obtains the peak fuel and cladding temperatures with only solving two ordinary differential equations. In complex thermal systems such as nuclear reactors, accurate prediction of the maximum fuel and cladding temperature is very important from a safety perspective. Hence, it is more desirable to provide a simple and efficient model for analyzing transient behavior of the system in a short time. The RELAP5 code has been used to validate the developed model. The comparison between the developed model and RELAP code results show that the physical models and the framework implemented in this model are reliable.

Generating Function Method for the solution of point reactor kinetic equations

Point kinetic equations (PKEs) represent a simplified approach to study the reactor transients at first hand, and help to develop an insight for reactor control, operation and safety. Several methods have been around for over last five decades for accurate, stable, and reliable solution of PKEs however, their realization has become possible with the advent of modern computers. Additionally, the stiffness of PKEs inspires researchers to seek further for more robust algorithms. In this paper, an explicit algorithm based upon Generating Function Method (GFM) is proposed for the solution of such a complex system of differential equations. The method was implemented to solve conventional reactivity problems, such as step, ramp, and sinusoidal reactivity insertions and compared with the other numerical algorithms. In these problems, GFM has been found to provide numerical estimates of neutron population with comparable accuracy to other algorithms, and it is computationally efficient and relatively stable. In order to explore the feasibility of the GFM for problems in reactor dynamics, a delayed-supercritical process with temperature feedback was also simulated, and results found were numerically equivalent to available techniques, however with lesser computational effort and time.

Comparatives Studies Of A Safety Analysis For Molten Salt Reactor (MSR)

The Generation IV nuclear power system has been developed by the Generation IV Forum (GIF) with its benefits such as highly economical, enhanced safety, low waste production, and proliferation resistant. MSR is one of Generation IV nuclear power system which uses molten salt as a fuel and a coolant, therefore its technology differs from the conventional solid-fuel reactors. In this study, the author focus on the safety characteristic analysis of the MSRs, in which a point kinetic equations and core heat transfer have been modeled by developing a mathematical model for MSRs. The founded model is applied to analyze the safety characteristics of the molten salt actinide recycler and transmuter system (MOSART) by simulating transient basic condition, and that is the unprotected loss of flow (ULOF) and unprotected overcooling accident (UOC). The result of reactor accident simulation is power distribution and temperature in the core reactor toward the time and showing that MOSART conceptual design is a reactor design which is stablee inherently. This reseach purposes to give basic understanding about safety characteristic of the MSR.

A coupled neutronic/thermal–hydraulic module for the transient analysis of VVER-1000 reactor during reactivity insertion accidents

For the safety analysis of nuclear power plants, it is essential to provide a simple and fast model to investigate the time-dependent behavior of reactor cores. Most research papers in this field have generally been limited to use coupling techniques between existing codes. In this paper, an integrated module called CMS (nuclear reactor Core Modular Simulator) is proposed to simulate the transient thermo-neutronic behavior of a VVER-1000 reactor core during reactivity insertion accidents. CMS contains three modules: a neutronic module based on the point kinetic equations, a thermal-hydraulic module based on the time-dependent single heated channel approximation and a reactivity calculation one. A two-stage fourth order implicit Runge-Kutta method was used for solving the point kinetics equations and the homogeneous two-phase flow model was considered for the thermal-hydraulic part. To verify the proposed module, two reactivity insertion accidents of the Bushehr VVER-1000 reactor, a design basis accident and an anticipated operational occurrence, have been simulated and the obtained results have been compared with its final safety analysis report showing a good agreement.

New semi-analytical algorithm for solving PKEs based on Euler-Maclaurin approximation

Point kinetics equations (PKEs) is an significant model used to describe dynamic behaviors of neutron in nuclear reactors. How to cope with its stiffness problems is very important to solve PKEs accurately and efficiently. In this paper, a new semi-analytical algorithm based on Euler-Maclaurin Approximation (SAEMA) is developed to solve PKEs. SAEMA algorithm is applied and tested in different reactors with three typical reactivity insertion cases, including step, ramp and sinusoidal insertions. Firstly, the investigations on the computational stability are performed under different time step sizes. Secondly, the performances on computational accuracy are evaluated by comparing the results by SAEMA algorithm with the ones by analytical method and excellent CATS algorithm. Finally, by comparing the CPU time consumed by SAEMA algorithm with the physical time, the studies on the computational efficiency are also carried out. Just as results shows, SAEMA algorithm is reasonably an attractive way to solve the PKEs.

Rod drop transient at VR-1 reactor – Experiment and Serpent transient calculation analysis

The rod drop experiment is an important transient in reactor operation. This study focuses on a comparison between experimental reactor physics data and respective calculation. The rod drop experiment was performed at the VR-1 reactor. A full core 3D model was used to calculate this transient using Serpent2. Additionally, a point kinetic solution is presented. The Serpent results were compared with the experiment and with the point kinetic calculation. Two different nuclear data libraries (ENDF/B-VIII.0, JEFF3.3) were used in the data sensitivity analysis. The Serpent 3D kinetics results were almost identical to the point kinetic equation solution. Experimental data differed in the rod worth. This study shows that the new Serpent dynamic toolkit provides an accurate description of the reactor behavior during this transient. The 3D calculation proved that the detector position during the transient has a direct impact on the measured control rod worth.

Neutron Density Calculation Using the Generalised Adams-Bashforth-Moulton Method

This paper presents a numerical solution to the equations of point kinetics for nuclear power reactors, a set of seven coupled differential equations that describe the temporal variation of neutron density and the concentration of delayed neutron precursors. Due to the nature of the system, we propose to numerically solve the point kinetics equations by implementing the Adams-Bashforth and Adams-Moulton methods, which are predictor-corrector schemes with their respective modifiers to increase precision. The proposed method was tested computationally for different forms of reactivity with up to six groups of delayed neutron precursors. This method was used in a recent publication to solve the inverse problem of finding the reactivity. In this work, it is shown that it can also be used for the calculation of nuclear power, that it is simple and easy to implement, and that it produces good results when compared with those in the literature for neutron population density and concentration of delayed neutron precursors.

Assessment of the fractional neutron point kinetic equation to simulate core transients with Newtonian temperature feedback

The model of the Fractional Neutron Point Kinetic equations (FNPK) considers a relaxation time for neutrons, as they travel in surrounding media. Relaxation time is associated with a rapid variation in neutron flux, as reflection of fast change of reactivity, and it can be implemented in the differential operator of fractional order related with non-Fick effects, also known as anomalous diffusion exponent and represents non-local time memory. In this work the model of FNPK extended, by adding the temporal term to including the neutron density derivative, is using with the objective of studying the effect of this added term on the behavior of neutron density during a transient, when there exists Newtonian temperature feedback reactivity, using different values of the anomalous diffusion exponent. Although the one-group delayed neutron precursor is use for this analysis, the results of the neutron density behavior were compared with the classical neutron point kinetics equations. The results show that the additional term implemented in this study led to show how the sub-diffusion phenomena have noticeable relevance in the neutron density behavior in those scenarios where the dynamic of a nuclear system is associated to slow transients.

Novel numerical solution to the fractional neutron point kinetic equation in nuclear reactor dynamics

In this work, a novel numerical solution to modified Fractional Neutron Point Kinetic (FNPK) equations is presented. The method is based on a numerical solution to linear multi-term fractional differential equations taking from scientific literature. Differential-integral operators of fractional order are numerically solved with the novel method. The impact of the order of the operators has been assessed during the process of order reduction of the fractional differential-integral equation. The numerical solution is applied to case with sinusoidal reactivity, and different values of the anomalous diffusion order are used to study the effect on the neutron density. The results of the neutron density behavior obtained with this proposed numerical novel solution were compared against the classical neutron point kinetics equations and with other results from scientific literature. The comparison showed a clear improvement of the numerical results when using a fractional differential-integral operator instead of an only fractional differential operator.

A new method for the solution of the point kinetics equations in the presence of Newtonian temperature feedback

The point reactor kinetics equation with temperature feedback simulates the time-dependent behavior of the nuclear reactors and resembles a stiff system of nonlinear coupled differential equations for the power density and delayed precursor concentrations. This paper develops a new method, based on Chebyshev Rational Approximation Method (CRAM), to solve the point kinetic equations in the presence of Newtonian temperature feedback. In addition, to gauge its accuracy, this investigation compares the result of the applied method with different ODE suites of MATLAB software as well as other solution methods of the literature. Also, numerical values printed with very high precision (in most of the cases up to 8 decimal places), and in some cases presented for the first time, thus these data can also be used for benchmark purposes. The proposed method can be considered as a new promising and very accurate tool for the analysis of nuclear reactor physics.

Core Power Control of a Nuclear Research Reactor During Power Maneuvering Transients Using Optimized PID-Controller Based on the Fractional Neutron Point Kinetics Model With Reactivity Feedback Effects

Core power control of a nuclear reactor during power maneuvering transients is an important issue in a nuclear power plant. This paper, for the first time, presents an optimized PID controller for a nuclear research reactor with a new model of the reactor core, which is based on the fractional neutron point kinetics (FNPK) equations. This model preserves the main dynamic properties of the neutron movement in which the relaxation time related to a quick change in the neutron flux includes a fractional order, acting as an exponent of the relaxation time, to show the dynamic characteristics of the nuclear reactor in the best way. The physical justification of the fractional exponent is associated with nonFickian impressions from the neutron diffusion equation point of view. In addition, in this paper, for the first time, feedback reactivity effects are considered for FNPK model. Genetic algorithm (GA) is used to optimize gains of the PID controller. The objective function is according to the minimization of the integral of time-weighted absolute error (ITAE). The simulation result demonstrates that this optimized control method has satisfactory performance and stability during output-tracking process for higher values of fractional order.

КОРРЕКЦИЯ УРАВНЕНИЙ ОБРАТНОЙ ТОЧЕЧНОЙ КИНЕТИКИ ДЛЯ ИЗМЕРЕНИЯ РЕАКТИВНОСТИ В ПРИБЛИЖЕНИИ МГНОВЕННОГО СКАЧКА

We consider methods for calculating the reactivity of a nuclear reactor from the measured dependence of the neutron flux in the reactor on time, based on the use of the inverse point kinetics equations, which relate the values of reactivity and neutron flux in the reactor. The main attention is paid to the correction of the equations of inverse point kinetics in the prompt-jump approximation (or in the theory of singular perturbations for equations with a small parameter with the highest derivative). The nonequivalence of the corresponding problems for the direct and inverse point kinetics equations in the prompt-jump approximation is noted, which consists in the fact that, contrary to expectations, the reactivity values appearing in these problems do not generally coincide with each other. The reasons for this non-equivalence are investigated and ways to eliminate it are considered. New inverse point kinetics equations are proposed in the prompt-jump approximation, devoid of the indicated disadvantage of the traditional equations. These equations are tested by substituting in them analytical solutions of the corresponding (direct) Cauchy problems for point kinetics equations using the apparatus of the theory of systems of ordinary differential equations, adapted to the problem in question. It is shown that the use of the equations proposed in the work leads to the correct determination of reactivity immediately after the instantaneous introduction of a disturbance into the reactor, and not in asymptotics, as is usually the case when applying traditional equations.

Design and stability analysis of an implicit non-standard finite difference scheme for fractional neutron point kinetic equation

This paper is concerned with the construction of a new numerical technique to obtain approximate solution for an extended fractional neutron point kinetic (EFNPK) model with six groups of delayed neutron precursors. The fractional derivative is approximated by means of Caputo's definition and the shifted Grünwald-Letnikov formula is used to obtain a finite difference approximation for the solution of this model. The discretization leads to an implicit finite difference scheme. The stability analysis of the method is discussed. It is proved that the method is unconditionally stable. Various numerical experiments are carried out for validity and efficiency of the suggested method. The neutron density is calculated for three types of reactivity: step, ramp and sinusoidal reactivity. The influences of anomalous diffusion exponent, reactivity functions, relaxation time and time steps on the neutron density are examined. Comparison is made between the results corresponding to the EFNPK model [4] and the results corresponding to the fractional neutron point kinetic model proposed in [17]. Comparison confirms that the difference between the values of neutron density increases as the anomalous diffusion exponent decreases or computational time increases. Comparison is also made between the approximate solution obtained by the suggested method and exact solution of the classical model for step and ramp reactivity. Moreover, our results are compared with those obtained by other methods.

Adjusted mean generation time parameter in the neutron point kinetics equations

This paper proposes an approximation for the modified point kinetics equations to obtain a mathematically simpler set of equations similar to those of the classic point kinetics. The new method, which is called the adjusted point kinetics method, discards the least important terms in the modified kinetics equations and adjusts the mean generation time parameter to tentatively incorporate the influence of the discarded terms. To validate the proposed model, it is compared to the modified kinetics model in numerical calculations of two very simple reactivity insertion cases.

On nuclear-coupled thermal-hydraulic instability analysis of Super-Critical-Light-Water-cooled-reactor (SCLWR)

The nonlinear stability analysis of a supercritical light water reactor (SCLWR) is presented using a nuclear-coupled thermal-hydraulic reduced-order model. The analytical model is developed by coupling 1. the point-kinetics equations with one group of delayed neutrons, 2. the fuel heat transfer and 3. a 1-D reduced order model which represents the heat absorption phenomenon during the coolant flow. Unlike the existing studies, which are limited to linear stability analysis, the primary objective of the work is to present the detailed nonlinear dynamics of the SCLWR system. The said goal is achieved at two levels. The first level is the linear stability analysis wherein the linear stability boundaries are shown in two sets of parameter space namely the two intrinsic reactivity feedbacks (Doppler reactivity feedback and density reactivity feedback) and the pseudo-phase-change number and pseudo subcooling number. The parametric effects show the sensitivity of the linear stability boundaries with the system parameters. In the second level, to discuss the nonlinear characteristics of the system, two types of Hopf bifurcations (subcritical and supercritical) are studied with the help of first Lyapunov coefficients of the system. Multiple numerical simulations are performed to verify the resultant limit cycle behavior associated with these bifurcations. Moreover, the occurrence of the generalized Hopf bifurcation is shown which represents the bifurcation between the subcritical Hopf and the supercritical Hopf regions. Further, inside the stable region of the linear stability boundary, the saddle-node bifurcation is found which represents the location of the turning point and the threshold of the globally stable region of the system.

Novel fluctuation reduction procedure for nuclear reactivity calculations based on the discrete fourier transform method

ABSTRACT A new method for calculating nuclear reactivity based on the Discrete Fourier Transform (DFT) – with two filters: a first-order delay low-pass filter and a Savitzky-Golay filter – is presented. The reactivity is calculated from an integrodifferential equation known as the inverse point kinetic equation, which contains the history of neutron population density. The new method can be understood as a convolution between the neutron population density signal and the response to the characteristic impulse of a linear system. The proposed method is based on the discrete Fourier transform (DFT) that performs a circular convolution. The fast Fourier transform algorithm (FFT) with the zero-padding technique is implemented to reduce the computational cost.

Novel consistent approach in controllability evaluations of point reactor kinetics models

The controllability of linearized equations of the equivalent normalized version of the reactor point kinetics equations with six delayed neutron groups and without neutron source at an equilibrium power is analyzed. There exists inconsistency between the results from two kinds of approaches. The equations analyzed are controllable according to the controllability evaluation by analytical derivation while they are uncontrollable by numerical calculation. The reason of this inconsistency is that the large difference between the time effect of prompt neutrons and that of delayed neutrons in reactors reduces the controllability matrix to become a rank-deficient matrix under the conditions of using finite-precision arithmetic. In order to improve the analysis approach and address the divergence in outcomes, a novel definition of degree of controllability is proposed by means of applying matrix norm and singular value decomposition (SVD). This definition resolves the inconsistent conclusions, i.e., the equations analyzed are controllable but their degree of controllability is very low. The calculation of the degree of controllability shows that in the multiple representations of point reactor kinetics, the representations of few-group models are more propitious to analyze and design the control system of nuclear reactor than the representation of six-group model.

Development of a displacement-reactivity feedback model for dynamic behavior simulation in fast burst reactor

In this study, a displacement-reactivity feedback model, which can directly represent the inherent “thermal expansion extinction effect” of fast burst reactors (FBRs), was developed with the aid of the static neutron transport component of the FBR-MPC code. Dynamic behaviors of bursts in the Godiva I reactor were simulated by coupling the simplified multiphysics models consisting of the point kinetic equations for neutronics, adiabatic equation for temperature, and thermoelastic equations for displacement/stress with the developed model. The results were compared with the corresponding experimental data and those obtained using the traditional fission yield (temperature rise)-reactivity feedback models. It was found that the developed model can provide good results for the bursts with no or a small inertia effect. For the bursts with a prominent inertia effect, the smaller burst width and asymmetric distribution of the fission rate curve, noticed in the experiments but not evident using the traditional models, can be reproduced. In addition, the realistic oscillations in reactivity and fission rate caused by the core vibration, as well as the deeper sub-prompt criticality in the plateau following the burst, can be observed. Therefore, the developed displacement-reactivity feedback model can be expected to be an effective tool for calculating the dynamic behaviors of bursts.