Point Reactor Kinetics Model Research Articles

Hybrid mechanistic and neural network modeling of nuclear reactors

An accurate and efficient model is the foundation for dynamic characteristics analysis and control design of nuclear reactors. Nevertheless, the widely-used mechanistic models of nuclear reactors inevitably have some deviations from the actual reactors due to inaccurate model parameters and model assumptions. This paper proposes a hybrid mechanistic and neural network modeling method for nuclear reactors to eliminate or minimizing these deviations. The high-efficiency point reactor kinetics model was adopted as the mechanistic reactor core model, and the simulation results obtained using the commercial program RELAP5 were taken as the reference operational data, based on which the reactivity feedback in the mechanistic model that is difficult to be accurately determined during reactor operations was calibrated using a genetic algorithm optimized BP neural network. Dynamic simulation results of a Generation III large pressured water reactor under three typical reactivity change transients show that the core power and outlet coolant temperature obtained with the hybrid model match much better with the reference results than those of the mechanistic model, demonstrating that the hybrid modeling method can improve the accuracy of the mechanistic reactor model effectively. This study can provide a valuable reference for accurate and efficient modeling of complex systems.

State-space model construction and dynamic analysis for a space thermionic nuclear reactor

The space nuclear reactor (SNR) is an important energy pattern for spacecrafts due to its high power density, long life and stabilization. A linear dynamic model is necessary to analyze the dynamics and design the appropriate control system analytically. A simplified nonlinear model based on conservation equations including point reactor kinetics model, thermal-hydraulic model, coolant model and heat pipe and radiator model was constructed. The state-space linear model was derived using perturbation theory. To verify the linear model, comprehensive comparisons were made in both time domain and frequency domain. The dynamic characteristics with respect to reactor power and reactivity were obtained in frequency and time domains. It is concluded that the reactor is an unstable system with a large time delay and strong nonlinearity at different power levels. This study provides solid foundation for future control system design.

Design and Development of Neutronics and Thermal Hydraulics Modeling Code for ACP1000 Nuclear Reactor Dynamics in LabVIEW

An advanced neutronics and thermal hydraulics nuclear code, called GNTHACP code, is designed and developed in LabVIEW as Graphical Neutronics and Thermal Hydraulics toolkit for 1100 MWe Advanced Chinese PWR (ACP-1000) nuclear power plant. The reactor neutronics model is developed using a nonlinear point reactor kinetics model, while, the reactor thermal hydraulics model is developed based on nonlinear fuel and coolant temperature dynamics. The heart of the GNTHACP code is the control rod reactivity model. Control rod reactivity banks are comprised of four power compensation banks G1, G2, N1, N2 and one temperature compensation bank R. The reactivity control configuration of these banks is highly nonlinear, complex and challenging in nature. The control rod reactivity model as a function of G1, G2, N1, N2 and R banks is optimized using two distinct techniques. The control rod reactivity model is optimized using Simplex Linear Programming (SLP) technique under constraints of reactor power as safety limit and control rod speed as maximum speed limit in LabVIEW. The control rod reactivity model is also optimized and investigated using nonlinear Sequential Quadratic Programming (SQP) technique under same constraints in LabVIEW. All the models are integrated and the state-of-the-art virtual instruments (VIs) are designed for cost function optimization, configuring models and calibration of model parameters in LabVIEW. The integrated model as graphical coupled neutronics and thermal hydraulics modeling code is optimized and validated against the Final Safety Analysis Report (FSAR) and different parameters of interest are investigated and proved within design limits as reported with CORCA and CORTH benchmark nuclear codes. The proposed code is stable, highly efficient and accurate as compared to other nuclear codes.

Modeling nonlinear fractional-order subdiffusive dynamics in nuclear reactor with artificial neural networks.

This paper presents the development and analysis of artificial neural network (ANN) models for the nonlinear fractional-order (FO) point reactor kinetics model, FO Nordheim-Fuchs model, inverse FO point reactor kinetics model and FO constant delayed neutron production rate approximation model. These models represent the dynamics of a nuclear reactor with neutron transport modeled as subdiffusion. These FO models are nonlinear in nature, are comprised of a system of coupled fractional differential equations and integral equations, and are considered to be difficult for solving both analytically and numerically. The ANN models were developed using the data generated from these models. The work involves the iterative process of ANN learning with different combinations of layers and neurons. It is shown through extensive simulation studies that the developed ANN models faithfully capture the transient and steady-state dynamics of these FO models, thereby providing a satisfactory representation for the nonlinear subdiffusive process of neutron transport in a nuclear reactor.

HEART, a specific code for thermal-electrical analysis of heat pipe cooled nuclear reactor

Heat pipe cooled reactor (HPR) features high power density, compact structure, long operation life and high reliability, which is an excellent candidate for small nuclear power source for deep space or underwater missions. In this paper, a conceptual Nuclear Silent Thermal-Electrical Reactor (NUSTER) is preliminarily designed, using 109 sodium heat pipes for passive cooling. The thermoelectric generators (TEGs) are employed in the reactor to convert fission heat to electric power. Based on the design, a HEat pipe cooled Advanced Reactor Transient analysis code (HEART) is developed, including point reactor kinetics model, multi-channel model, core heat transfer model, heat pipe model, thermoelectric generator model and coolant model. The models proposed in HEART are preliminarily verified against the design value and experiment data, and the maximum deviation is less than 12.3%. Steady-state performance of the NUSTER are successfully predicted by HEART, and results indicate that the solid-state core has good temperature flattening ability. The surface temperature of the heat pipe is less than 1300 K, and the average temperature drop of the TEG module in central channel is 724.1 K, which can produce an electrical power of 301.94 W. The models developed in HEART can also be applied to other heat pipe cooled reactors, providing valuable experience in design and thermal-hydraulic evaluation of HPRs.

Power control of CiADS core with the intensity of the proton beam

This paper reports the control method for the core power of the China initiative Accelerator Driven System (CiADS) facility. In the CiADS facility, an intense external neutron source provided by a proton accelerator coupled to a spallation target is used to drive a sub-critical reactor. Without any control rod inside the sub-critical reactor, the core power is controlled by adjusting the proton beam intensity. In order to continuously change the beam intensity, an adjustable aperture is considered to be used at the Low Energy Beam Transport (LEBT) line of the accelerator. The aperture size is adjusted based on the Proportional Integral Derivative (PID) controllers, by comparing either the setting beam intensity or the setting core power with the measured value. To evaluate the proposed control method, a CiADS core model is built based on the point reactor kinetics model with six delayed neutron groups. The simulations based on the CiADS core model have indicated that the core power can be controlled stably by adjusting the aperture size. The response time in the adjustment of the core power depends mainly on the adjustment time of the beam intensity.

Impact of neutron generation time on released energy in falling fuel debris at supercritical condition

ABSTRACT The risk of criticality accidents by fuel debris in the decommissioning of Fukushima Dai-ichi Nuclear Power Station is unlikely to be negligible. Therefore, an estimation of the consequences of such accidents is important. According to the latest study, estimating the energy that would be released if fuel debris became supercritical while falling into water is thought to be important. In falling fuel debris systems, the geometry changes substantially with time, which could cause large changes in neutron generation time. Treating the change in neutron generation time in a kinetics analysis might have a large impact on results such as released energy. Clarifying such phenomena could contribute to conducting a reasonable analysis and avoiding misleading results. Thus, in this study we seek to clarify whether it is necessary to consider changes of neutron generation time when evaluating released energy in a falling fuel debris system at supercritical conditions in water with a kinetics analysis based on a point reactor kinetics model. We found that change of neutron generation time should be considered when evaluating released energy in falling fuel debris systems at supercritical conditions; otherwise, obtained results could be unrealistic.

Transient heat pipe failure accident analysis of a megawatt heat pipe cooled reactor

Heat pipe cooled reactors, which use heat pipes rather than fluid flow for passive cooling, are an excellent candidate for micro nuclear power sources. Heat pipes provide excellent built-in redundancy and safety margin. However, a heat pipe failure is quite likely over the reactor lifetime, which is one of the essential design basis accidents in the reactor design. This work analyzes heat pipe failure accidents to investigate the system transient performance and the response of a heat pipe cooled reactor. Single heat pipe failure and cascading heat pipe failure are simulated and analyzed by establishing a transient analysis code, including a point reactor kinetics model, a core thermal-mechanical model, and a heat pipe model. The simulation shows that in the single heat pipe failure accident, the peak monolith temperature at the failed channel increases and is only 8 K lower than the fuel center temperature. In the cascading heat pipe failure accident, the temperatures in each channel suddenly increase over a short period and rise to the maximum at 22,900 s as the heat losses increase and the core power decreases. The sensitivity of the external convection for residual heat removal is also analyzed. For small external convection coefficients less than 100 W/m2K, increasing the external convection coefficient significantly reduces the maximum temperatures. By contrast, for external convection coefficient greater than 500 W/m2K, increasing external convection coefficient still reduces the core peak temperature, but the effect is weak.

Estimation of the fuel temperature reactivity coefficient of the UMass-Lowell research reactor based on empirical and theoretical point reactor kinetics models

Modeling nuclear reactors for simulating transients can be very challenging for a lot of reasons, e.g. space–time representation, heterogeneity, etc. Reactor-specific design and data may also be required for the facility being analyzed. Although reactor point kinetics theory is usually a method of choice in analyzing such transients, a reactor’s specifications are required to achieve meaningful results. This work describes a unique method for estimating the fuel temperature reactivity coefficient for use with the point reactor kinetics model of the UMass-Lowell Research Reactor. The computed coefficient is later validated with experimental data.

Supercritical Transient Analysis for Ramp Reactivity Insertion Using Multiregion Integral Kinetics Code

To proceed with the decommissioning of the Fukushima Daiichi Nuclear Power Station, analyses of unexpected fuel debris criticality accidents are needed. Supercritical transient analyses have been conducted for fuel debris using the Multiregion Integral Kinetic (MIK) code, which can take the space dependence of fuel debris into account. In those analyses, reactivity is assumed as stepwise insertion because the MIK code does not include delayed neutron effects, which might be negligible. However, reactivity insertion may not always be stepwise. Therefore, it is important to clarify an applicable range of the MIK code for nonstepwise insertion, such as ramp reactivity insertion. To show that kinetics codes without delayed neutron effects could be applied for a supercritical transient induced by ramp reactivity insertion, we established a method to clarify its applicable range. An analysis using the point reactor kinetics model was introduced as a pre-analysis to clarify this range in the case of ramp reactivity insertion in terms of the contribution of delayed neutrons. We applied the methodology to a simple cylindrical fuel debris system and successfully demonstrated a supercritical transient analysis for ramp reactivity insertion using the MIK code.

Inverse method to obtain reactivity in nuclear reactors with P1 point reactor kinetics model using matrix formulation

The aim of this work considers a second order point reactor kinetics model based on the P1 approximation of transport theory, called in this work as P1 point reactor model. The P1 point reactor model implicitly considers the time derivative of the neutron source which has not been thus considered previously. The inverse method to calculate the reactivity in nuclear reactors -chosen because its high accuracy- Matrix Formulation. The numerical results shown that the Matrix Formulation for the reactivity estimation constitutes a method with insignificant calculation errors.

Stabilization of the stochastic point reactor kinetic equation through power feedback

Most basic models for the power (or equivalently, the neutron population) in a nuclear core consider the power as a function of time (with an energetic and spatial distribution) and lead to deterministic description of the reactor kinetics. While these models are of common use and are undoubtedly the main analytic tool in understanding the reactor kinetics, the true nature of the power in a reactor core is stochastic and should be considered as a stochastic process in time. The stochastic fluctuations of the power around the mean field (which is given by the deterministic models) are referred to as “reactor noise”, and understanding them is a basic topic in nuclear science and engineering. Traditionally, most models for reactor noise consider a sub-critical core, reaching steady state after exposure to an external source. The focus on a sub-critical setting is driven by two main factors. First, from a practical point of view, measuring the power fluctuations in a sub-critical core (known as “noise experiments”) has proven to be a very efficient tool for estimating the static and kinetic parameters of the core. Second, once we assume a critical setting, the current models become statistically unstable, while the mean field solution has a stationary solution, the variance tends to $$\infty $$ linearly in time. The instability of the stochastic models is a known problem, and it has been conjectured in the past that this (some what strange) increase in the variance—that is not observed in physical systems—can be restrained by power feedback. However, this conjecture was never proven. The outline of the present study is to present a stochastic analysis to the point reactor kinetics model, proving that once the reactivity has a negative feedback, it not only forces a specific steady-state solution (in terms of the mean field equation), but also prevents the variance to “explode”, and the variance is bounded in time.

Transient thermal-hydraulic analysis for fuel assembly blockage accident by coupling CFD code with the neutron kinetics model

Lead-based reactor coolants have many excellent characteristics but the coolants may seriously erode the fuel cladding and the structural materials and the flow area will be reduced by the corrosion products. This phenomenon is called blockage accident.In this paper, the blockage accident of single subassembly for a lead-based reactor has been analyzed by applying the commercial computational fluid dynamics (CFD) software coupled with the point reactor kinetics model. Firstly, the neutronics/thermal-hydraulics coupling model was established by user defined function (UDF) of FLUENT software based on the point reactor kinetics function with one group delayed neutron. Then the neutronics/thermal-hydraulics coupling model was validated with a single fuel pin model by comparing the fuel power, the cladding temperature and the coolant temperature with the calculation results of the RELAP5 code. Secondly, after considering mesh independence, turbulence models and other factors, the steady state of a single subassembly with 61 fuel pins at full power for a lead-based reactor designed by INEST/FDS Team was simulated in the normal operating conditions. Finally, a single subassembly for the center blockages of the flow channel was simulated by the neutronics/thermal-hydraulics coupling model. The temperature distributions and the maximum temperature were compared with that of the normal operating conditions and the effect of axial position for blockage was analyzed. It shows that the core outlet temperature and the highest temperature of cladding are increasing after blockage accident happening. And the highest temperature of cladding is lower calculated by neutronics/thermal-hydraulics coupling model than the condition without considering reactivity feedback. And the maximum cladding temperature for the 500 mm axially location blockage accident is higher.

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.

Probability distribution for neutron density of two energy group point kinetics system

The probability distribution of the number of neutrons and delayed neutron precursors in a multiplying assembly of various types of reactivities is developed for two energy groups. The space independent point reactor kinetics model for six precursors group of delayed neutrons is considered. The problem is formulated in terms of the probability distribution, generating function which satisfies a partial differential equation, derived in this paper. The probability distribution of two energy group delayed neutrons and the density of the precursors are obtained by solving this system of reactor kinetics model by adopting the mathematical methods. At the point, when the system is formulated in an operator or matrix form, the Magnus expansion furnishes an elegant setting to build up approximate exponential representations of the solution of the kinetics system. It provides a power series expansion for the corresponding exponent and is sometimes referred to as time-dependent exponential perturbation theory. Every Magnus approximate corresponds in perturbation theory to a partial re-summation of infinite terms with the important additional property of preserving, in any order, certain symmetries of the exact solution. The first, second and third Magnus expansions are described and used to predict the first moment of fast, thermal and multi-group of delayed neutrons precursor for the two-energy point kinetics reactor system. The validity of the presented method is tested with the aid of the eigenvectors and eigenvalues of the kinetics system in the matrix form by comparing with the conventional methods.

Spectrum behavior for the nonlinear fractional point reactor kinetics model

The nonlinear fractional point reactor kinetics equation in the presence of Newtonian temperature reactivity feedback with a multi-group of delayed neutrons, which describes the spectrum behavior of neutron density into the homogenous nuclear reactors, is developed. This system is one of the most important stiff coupled nonlinear fractional differentials for nuclear reactor dynamics. The generalization of Taylor’s formula that involves Caputo fractional derivatives is developed in an attempt to overcome the difficulty of the stiffness of the nonlinear fractional differential model. Moreover, the general fractional derivatives are calculated analytically throughout this work. Furthermore, the local and global estimated errors were analyzed, which suggest that the error quantification should take into account the possible grow in time of the error. This observation provides a motivation for going beyond more classical local-in-time concepts of error (local truncation error). The neutron density response with time is analyzed for the anomalous diffusion, sub-diffusion, and super-diffusion processes.

Analytical solution of the Telegraph Point Reactor Kinetics model during the cold start-up of a nuclear reactor

We solve the new model of the Point Reactor Kinetics (PRK) equations developed based on the Telegraph approximation of the neutron transport equation analytically for a linear insertion of reactivity typically introduced by lifting the control rods discontinuously and manually during the cold start-up of a subcritical nuclear reactor. The Telegraph model introduces a new parameter called the Relaxation Time (τ) and we study its impact on the analytical solution for this case of reactivity insertion and for several speeds of lifting the control rods. We find that as the Relaxation time increases, the solution response is relaxed behind that of the diffusion model which was solved earlier in the literature for the same case. On the other hand, as the Relaxation time tends to zero, we find that the response of the neutron density tends to that of the diffusion case yielding a verification for the new proposed solution. Moreover, when reducing the Control Rod lifting speed, the effect of the relaxation time and hence the Telegraph approximation is reduced and it approaches that of the diffusion. We discuss Both mathematical and physical reasons for the cause of these behaviors.

Response of the point-reactor telegraph kinetics to time varying reactivities

The new model of the Point Reactor Kinetics (PRK) equations developed based on the Telegraph approximation of the neutron transport equation, is solved for several cases of time varying Reactivities insertions and Temperature feedback while comparing it to that of the diffusion PRK model in an infinite Thermal Homogenous Nuclear Reactor. Diffusion PRK is based on the Neutron Diffusion Equation which is a parabolic differential equation and hence it assumes an infinite velocity of propagation, while neutrons propagate with a finite velocity. By the introduction of the hyperbolic type Telegraph equation which is a more accurate representation of the neutron transport than the diffusion equation and in which neutrons propagate with a finite velocity, one could overcome this paradox that contradicts causality. The new model introduces a new parameter called the relaxation time (τ), which is not present in the diffusion approximation, and affects the neutron density calculations. Both Ramp insertions of reactivity and Sinusoidal insertions of reactivity were studied, as well as the effect of The Adiabatic Temperature feedback. The general phenomena in the solution of the new model is a Relaxation in the time response of the solution. It is found that the Telegraph model with its extra second order time derivative, will give observable different values than that of the diffusion even when we used small (τ) especially for the cases at which the neutron density changes rapidly.

Nuclear reactor with subdiffusive neutron transport: development of linear fractional-order models

This paper deals with fractional-order (FO) modeling of a nuclear reactor. Modeling neutron transport in a nuclear reactor as subdiffusion results in the development of FO neutron telegraph equation. This model predicts subdiffusive behaviour for long-time. This fact is used to develop various linear control-oriented FO models for nuclear reactor. Development and analysis of three FO linear models is reported: FO point reactor kinetics model, Zero power FO transfer function and FO transfer function with temperature feedback of reactivity. In addition to the detailed stability analysis of these FO models, a comparative study of time-domain indices is carried out with respective integer-order models. The proposed FO models give a better representation of the neutron transport in the heterogeneous reactor core and therefore can be used to develop better model-based control strategies for a nuclear reactor.

Accident analysis of heat pipe cooled and AMTEC conversion space reactor system

A space power with high power density, light weight, low cost and high reliability is of crucial importance to future exploration of deep space. Space reactor is an excellent candidate because of its unique characteristics of high specific power, low cost, strong environment adaptability and so on. Among all types of space reactors, heat pipe cooled space reactor, which adopts the passive heat pipe (HP) as core cooling component, is considered as one of the most promising choices and is widely studied all over the world.This paper develops a transient analysis code (TAPIRS) for heat pipe cooled space reactor power system (HPS) based on point reactor kinetics model, lumped parameter core heat transfer model, combined HP model (self-diffusion model, flat-front startup model and network model), energy conversion model of Alkali Metal Thermal-to-Electric Conversion units (AMTEC), and HP radiator model. Three typical accidents, i.e., control drum failure, AMTEC failure and partial loss of the heat transfer area of radiator are then analyzed using TAPIRS. By comparing the simulation results of the models and steady state with those in the references, the rationality of the models and the solution method is validated. The results show the following. (1) After the failure of one set of control drums, the reactor power finally reaches a stable value after two local peaks under the temperature feedback. The fuel temperature rises rapidly, however it is still under safe limit. (2) The fuel temperature is below a safe limit under the AMTEC failure and partial loss of the heat transfer area of radiator. This demonstrates the rationality of the system design and the potential applicability of the TAPIRS code for the future engineering application of heat pipe cooled space reactor.