Delayed Neutron Precursors Research Articles

Analytical representation of the solution of the space kinetic diffusion equation in a one-dimensional and homogeneous domain

In this work we solve the space kinetic diffusion equation in a one-dimensional geometry considering a homogeneous domain, for two energy groups and six groups of delayed neutron precursors. The proposed methodology makes use of a Taylor expansion in the space variable of the scalar neutron flux (fast and thermal) and the concentration of delayed neutron precursors, allocating the time dependence to the coefficients. Upon truncating the Taylor series at quadratic order, one obtains a set of recursive systems of ordinary differential equations, where a modified decomposition method is applied. The coefficient matrix is split into two, one constant diagonal matrix and the second one with the remaining time dependent and off-diagonal terms. Moreover, the equation system is reorganized such that the terms containing the latter matrix are treated as source terms. Note, that the homogeneous equation system has a well known solution, since the matrix is diagonal and constant. This solution plays the role of the recursion initialization of the decomposition method. The recursion scheme is set up in a fashion where the solutions of the previous recursion steps determine the source terms of the subsequent steps. A second feature of the method is the choice of the initial and boundary conditions, which are satisfied by the recursion initialization, while from the first recursion step onward the initial and boundary conditions are homogeneous. The recursion depth is then governed by a prescribed accuracy for the solution.

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.

Investigation of the chance structure of the fission cross section of 238U by neutrons in energy range from 14 to 18 Mev using the delayed neutrons

In the present work, delayed neutrons are used for the investigation of the chance structure of the fission cross section. For this, time parameters of delayed neutron precursors emitted from the fission of 238U by neutrons with the energies from 14 to 18 MeV have been measured. The T(d,n) reaction was used as a neutron source. The experimental data have been processed to account the effect of degradation of detector counting rate in high intensity neutron flux and the effect of concomitant neutron source D(d,n) caused by implanted deuterium ions in neutron target. The data obtained based on the time parameters of delayed neutron precursors have been used in the analysis of the chance structure of the fission cross section of the 238U.

On an analytical solution for the two energy group neutron space-kinetic equation in heterogeneous cylindrical geometry

The present work is a continuation for the multigroup scalar neutron flux and multigroup precursor concentration model, here two energy groups and one delayed neutron precursor concentration. We determine a general solution for this problem considering cylinder geometry. The solution is determined using variable separation. The complete spectrum is analysed with respect to each variable separation and truncated such as to allow a parametrisation of the solution for the considered homogeneous problem. For a specific choice of nuclear parameter we present the three dimensional solution for the fast, the thermal angular neutron fluxes and the delayed neutron precursor concentration. Possible extensions of the model and solutions are indicated.

An implicit finite difference method for solving the corrected fractional neutron point kinetics equations

It is well known that Espinosa-parades et al. (Espinosa-Paredes et al., 2011) proposed a fractional neutron point kinetic (FNPK) model with multi-group of delayed neutrons to describe dynamic behavior in a nuclear reactor. In (Aboanber and Nahla, 2016b), Aboanber and Nahla presented an extension of Espinosa-parades et al. FNPK model. This new model is called as the corrected fractional neutron point kinetic (CFNPK) model. The present study is concerned with the numerical solution of the CFNPK model (Aboanber and Nahla, 2016b) with six groups of delayed neutron precursors. The fractional derivative is described in the sense of Grünwald-Letnikov. An implicit finite difference method (FDM) is constructed for the solution of CFNPK model. The stability analysis of the method is carried out. We analyze the results of neutron density for different values of anomalous diffusion order, reactivity function, relaxation time and time step size. In addition, we compare the results corresponding to CFNPK model (Aboanber and Nahla, 2016b) with the results corresponding to the FNPK model proposed by Espinosa-parades et al. (Espinosa-Paredes et al., 2011). It is shown that as anomalous diffusion order decreases or simulation time increases the difference between the values of neutron density increases. We have investigated the effects of each term of the CFNPK.

ВРЕМЕННАЯ ЗАВИСИМОСТЬ АКТИВНОСТИ ЗАПАЗДЫВАЮЩИХ НЕЙТРОНОВ И ГРУППОВЫЕ ПАРАМЕТРЫ В 6- И 8-ГРУППОВОМ ПРЕДСТАВЛЕНИИ ПРИ ДЕЛЕНИИ 235U, 238U И 239Pu ТЕПЛОВЫМИ И БЫСТРЫМИ НЕЙТРОНАМИ

In the present work the digital data on decay curves of delayed neutron intensity, measured for the fission of 235U, 238U and 239Pu by thermal and fast neutrons, is presented. The estimation of time parameters of delayed neutrons in 6 and 8 groups representation have been made on the base of individual experimental series using iterative least squares technique. It have been shown, that uncertainties of the parameters estimation in the 8 group representation, obtained on the basis of the experimental data are comparable with the uncertainties of the 6-group representation. The experimental data measured in the wide range of the time after the end of irradiation - (0,1-700 s) are of significant interest for intercomparison and validation of the different mathematical methods and programs, used for the processing of the decay curves of delayed neutron intensity, and also for analysis of the experimental data in the process of formulating of the evaluated data on relative abundances and half-lives of the delayed neutron precursors for the wide range of the fissioning nuclides.

Mid-Weighed Kinetic Parameters for Use in the Two-Group Diffusion Model of Reactor Dynamics with Fuel Based on a Mixture of Fission Isotopes

In the model of reactor kinetics based on the description of neutron transport in the two-group diffusion approximation, the number of equations describing the change in the concentration of delayed neutron precursors depends not only on the number of groups of delayed neutrons, but also on the number of fissile isotopes present in nuclear fuel. Since each isotope is characterized by six groups of delayed neutrons, the total number of differential equations describing concentrations of delayed neutron precursors is equal to the product of the number of fissile isotopes (M) and the number of groups of delayed neutrons for each isotope (i = 6). This is true provided that the decay constant of the concentrations of delayed neutron precursors that were formed from the division by fast or thermal neutrons can be taken in the same way. In fact, there is a difference, though small, in these values for the two energy groups. Therefore, the number of the corresponding equations is twice as high. In this paper, a mathematical expression is obtained for the weighted average decay constant of delayed neutron predecessors from fission by fast and thermal neutrons in a multiplying medium with several fissile isotopes. This, together with the conventional procedure of weighing the fraction of delayed neutrons from fission by fast or thermal neutrons in a similar multiplying medium, allows the two-group diffusion model of the reactor kinetics to be limited to only six equations for the concentrations of delayed neutron precursors and thus the kinetic model of the reactor to be simplified.

Reactivity and Delayed Neutron Precursors’ Concentration Estimation Based on Recursive Nonlinear Dynamic Data Reconciliation Technique

Reactivity is widely used as the paramount means for defining nuclear reactor status, but there is no physical sensor available to measure this parameter directly. Similarly delayed neutron precursors’ concentrations, the source of the delayed neutrons which play important role in reactor control cannot be measured directly. Hence, these quantities must be calculated or estimated from observed neutron flux transients. Recently, Kalman filter (KF) and its variants have been used for the reactivity estimation. A key limitation of KF and all of its variants is that they cannot account for the constraints on state variables systematically, which may cause the filter’s instability. In this paper, a constrained estimator based on the recursive nonlinear dynamic data reconciliation technique is proposed for the estimation of reactivity and delayed neutron precursors’ concentrations in the nuclear reactor. The effectiveness of the proposed technique has been evaluated through the step test signal in the presence of measurement noise and experimental power variation data sets collected from one of the Indian reactors.

Application of Monte Carlo method to calculate the effective delayed neutron fraction in molten salt reactor

Delayed neutron loss is an important parameter in the safety analysis of molten salt reactors. In this study, to obtain the effective delayed neutron fraction under flow condition, a delayed neutron precursor transport was implemented in the Monte Carlo code MCNP. The molten-salt reactor experiment (MSRE) model was used to analyze the reliability of this method. The obtained flow losses of reactivity for 235U and 233U fuels in the MSRE are 223 pcm and 100.8 pcm, respectively, which are in good agreement with the experimental values (212 pcm and 100.5 pcm, respectively). Then, six groups of effective delayed neutron fractions in a small molten salt reactor were calculated under different mass flow rates. The flow loss of reactivity at full power operation is approximately 105.6 pcm, which is significantly lower than that of the MSRE due to the longer residence time inside the active core. The sensitivity of the reactivity loss to other factors, such as the residence time inside or outside the core and flow distribution, was evaluated as well. As a conclusion, the sensitivity of the reactivity loss to the residence time inside the core is greater than to other parameters.

Adaptive state-observer for monitoring flexible nuclear reactors

Since nuclear power still fulfills nearly 11% of world electricity demand nowadays, the flexible operation of nuclear reactors can be positive to improve the penetration level of intermittent renewable energy (IRE) sources. Meanwhile, the operational flexibility of nuclear reactors results in the frequent variation of reactor process variables such as the neutron flux, fuel temperature and primary coolant pressure and temperature, which may fasten the degradation of equipment, and further leads to the necessity of developing performance monitoring methods for nuclear reactors in the context of fast and deep IRE penetration. Motivated by the importance of performance monitoring, an adaptive state observer (ASO) for nuclear reactors is newly proposed, which only needs the measurements of neutron flux and coolant temperature at reactor inlet and can provide globally asymptotic estimation for the normalized concentrations of delayed neutron precursors, 135Xe and 135I, the average temperatures of fuel elements and primary coolant as well as the total reactivity disturbance. The ASO is then applied for reconstructing the unmeasurable state variables and total reactivity disturbance of a nuclear heating reactor (NHR). Numerical simulation results verify the theoretical analysis, and show the satisfactory observation performance with its influence given by the ASO parameters. Furthermore, a hardware-in-loop (HIL) verification experiment is performed, and the corresponding results show that this newly-built ASO is not sensitive to the measurement noises, which leads to the feasibility of deploying the ASO on those digital control system platforms.

Development of a dynamics model for graphite-moderated channel-type molten salt reactor

A molten salt reactor (MSR) is one of the six advanced reactor concepts selected by the generation IV international forum because of its advantages of inherent safety, and the promising capabilities of Th-U breeding and transuranics transmutation. A dynamics model for the channel-type MSR is developed in this work based on a three-dimensional thermal–hydraulic model (3DTH) and a point reactor model. The 3DTH couples a three-dimensional heat conduction model and a one-dimensional single-phase flow model that can accurately consider the heat conduction between different assemblies. The 3DTH is validated by the RELAP5 code in terms of the temperature and mass flow distribution calculation. A point reactor model considering the drift of delayed neutron precursors is adopted in the dynamics model. To verify the dynamics model, three experiments from the molten salt reactor experiment are simulated. The agreement of the experimental data and simulation results was excellent. With the aid of this model, the unprotected step reactivity addition and unprotected loss of flow of the 2 MWt experimental MSR are modeled, and the reactor power and temperature evolution are analyzed.

Development of a control-oriented power plant simulator for the molten salt fast reactor

In this paper, modelling and simulation of a control-oriented plant-dynamics tool for the molten salt fast reactor (MSFR) is presented. The objective was to develop a simulation tool aimed at investigating the plant response to standard control transients, in order to support the system design finalization and the definition of control strategies. The simulator was developed employing the well tested, flexible and open-source object-oriented Modelica language. A one-dimensional modelling approach was used for thermal-hydraulics and heat transfer. Standard and validated thermal-hydraulic Modelica libraries were employed for various plant components (tubes, pumps, turbines, etc.). An effort was spent in developing a new MSR library modelling the 1D flow of a liquid nuclear fuel, including an ad-hoc neutron-kinetics model which properly takes into consideration the motion of the Delayed Neutron Precursors along the fuel circuit and the consequent reactivity insertion due to the variation of the effective delayed fractions. An analytical steady-state 2-D model of the core and the fuel circuit was developed using MATLAB in order to validate the Decay Neutron Precursors model implemented in the plant simulator. The plant simulator was then employed to investigate the plant dynamics in response to three transients (variation of fuel flow rate, intermediate flow rate and turbine gas flow rate) that are relevant to control purposes. Simulation outcomes highlight the typical reactor-follows-turbine behavior of the MSFR, and they show the small influence of fuel and intermediate flow rate on the reactor power and their strong effects on the temperatures in their respective circuits. Starting from the insights on the reactor behavior gained from the analysis of its free dynamics, the plant simulator here developed will provide a valuable tool in support to the finalization of the design phase, the definition of control strategies and the identification of controlled operational procedures for reactor startup and shutdown.

Reactor Kinetics Using Partial Equations

This paper proposes partial neutron transport equations for stationary and transient calculations. The partial equations of neutron transport are based on separately following neutrons born from external source, prompt fission neutrons, and delayed neutrons. The delayed neutrons are described by a system of equations containing one equation for each group. The paper defines the parameters of these equations and presents the results of fast neutron reactor benchmark calculations.Determination of the field of the external source neutrons in the system of partial equations provides a natural transfer of the source power (in units of neutrons per second) to the core power of energy release from the interaction of the external source neutrons in the reactor core (in units of watt). Thus, an external source neutron is used for the initial normalization of the neutron field based on the required reactor power. Operating with the field of delayed neutrons, in contrast to the field of concentrations of delayed neutron precursors, provides a quantitative assessment of the interaction of these neutrons with the reactor environment, and thus, assesses their contribution to the reactivity effects in fast reactors.Partial neutron transport equations allow us to extract additional information about the time behavior of the fast neutron reactor.

A quasi-static Monte Carlo algorithm for the simulation of sub-prompt critical transients

A pseudo-time-dependent Monte Carlo algorithm suitable for simulating thermal reactor transients with timescales on the order of seconds is presented. An initial experimental implementation has been written into the Monte Carlo Application ToolKit (MCATK) developed at Los Alamos National Laboratory. The algorithm is essentially a combination of the fully time-dependent Monte Carlo algorithm, including the explicit simulation of delayed neutron precursors, and a quasi-static algorithm that does not use the point-kinetics equations. The fully time-dependent calculation is performed for some number of inactive and active micro time steps. The dynamic reactor time constant and delayed neutron precursor production rates are calculated directly from the active cycles, rather than from point-kinetics parameters. These quantities are then used to advance the neutron population and delayed neutron precursors forward in time across a macro time step. The method is applied to the C5G7-TD benchmark operating in steady state and in transient mode.

On the solution of the neutron diffusion kinetic equation in planar geometry free of stiffness with convergence analysis

In this work we solve the space kinetic diffusion equation in a one-dimensional geometry considering a homogeneous domain, for two energy groups and six groups of delayed neutron precursors. The proposed methodology makes use of a Taylor expansion in the space variable of the scalar neutron flux (fast and thermal) and the concentration of delayed neutron precursors, allocating the time dependence to the coefficients. Upon truncating the Taylor series at quadratic order, one obtains a set of recursive systems of ordinary differential equations, where a modified decomposition method is applied. The coefficient matrix is split into two, one constant diagonal matrix and the second one with the remaining time dependent and off-diagonal terms. Moreover, the equation system is reorganized such that the terms containing the latter matrix are treated as source terms. Note, that the homogeneous equation system has a well known solution, since the matrix is diagonal and constant. This solution plays the role of the recursion initialization of the decomposition method. The recursion scheme is set up in a fashion where the solutions of the previous recursion steps determine the source terms of the subsequent steps. A second feature of the method is the choice of the initial and boundary conditions, which are satisfied by the recursion initialization, while from the first recursion step onward the initial and boundary conditions are homogeneous. The recursion depth is then governed by a prescribed accuracy for the solution. To this end we present error estimates and a convergence and stability analysis.

An improved zero-dimensional model for simulation of TRIGA Mark II dynamic response

The primary aim of this work is to improve the analysis of the dynamic behaviour of the TRIGA Mark II reactor at the University of Pavia through a zero-dimensional approach. Besides the coupling between neutronics (point-reactor kinetics with six delayed neutron precursors group) and thermal-hydraulics (two-region model, with fuel and coolant) implemented in earlier works, the new model considers also the time behaviour of the mass flow rate due to natural circulation, of the neutron poisons and of the primary and secondary pool temperature. The system of coupled first-order differential equations is non-linear, as some state variables, such as the mass flow rate and the coolant temperature, multiply each other. The Simulink™ programming environment for dynamic analysis and control purposes is used to solve the system. A comparison with experimental data collected on-site for different reactor power transients and with measurements of the poison anti-reactivity during reactor shut-down and of the pool temperature allows the validation of the model. The model results and the experimental data reach a remarkable agreement. In addition, a linear stability analysis of the reactor is performed through the root locus and the stability map in terms of the thermal feedback coefficients. This analysis shows how the power level influences the dynamic of the system, and that, for certain values (always negative) of the fuel thermal feedback coefficient, positive values of the one for the moderator still ensures the system stability.

Mathematical foundation of the neutron diffusion problem for a reflected nuclear reactor

The analytical approach to the neutron kinetics of a reflected multiplying system constitutes a challenging problem in mathematical physics and provides useful results for a full physical understanding of the multiplication process in non-homogeneous media, opening the possibility to establish reliable benchmarks. In this work, the basic mathematical features of the neutron diffusion operator in a reflected configuration are investigated. The completeness of the eigenstates is discussed and proven. The case of the infinite reflector is taken into consideration, highlighting its peculiarities that lead to the appearance of a continuum spectrum. The limitation of the critical eigenfunctions is also discussed. The analysis is carried out at first disregarding the presence of delayed emissions. The work is then extended to include also in the model the presence of delayed neutron precursors.

A study of the impact of mixed mode of fuel salt in plenum on steady-state performance of channel-type molten salt reactor

In this study, a coupled neutronics and thermal-hydraulics (THs) code named MOREL2.0, based on multi-group diffusion theory and multi-channel THs model, was employed to analyze a channel-type molten salt reactor TMSR-LF and the impact of different fuel mix mode in plenum on its steady-state performance was also discussed. MOREL2.0 adopts “two-step” calculation scheme, in which few-group homogenized parameters are generated by lattice code PIJ, and diffusion equation considering the drift of delayed neutron precursors is coupled with multi-channel THs model, and the least square method is implemented for cross-section feedback in coupling scheme. The numerical results indicate that TMSR-LF has negative reactivity coefficients in term of inlet fuel temperature and core power level, and the response of keff to mass flow is different for cases of nominal power and zero power, and keff decreases with the rise of time of fuel salt spent in external loop and gradually tends to steady. Different mix mode in plenum have a greater impact on the mass flow distribution and delayed neutron precursors distribution, but less impact on core lumped parameter such as keff and effective delayed neutron fraction.

Ex-Vessel Delayed Neutron Detection Systems for the ASTRID Sodium-Cooled Fast Reactor

Monitoring fuel integrity during sodium-cooled fast reactor operation becomes a crucial issue when sodium uranoplutonate formed consecutively to a clad aperture will foster cooling defaults. Delayed neutron detection (DND) systems are the key tools allowing to prevent and to mitigate such an accident by monitoring the concentration of delayed neutron precursors contained into the coolant. DND systems operate in hard gamma-rays fluxes and low-neutron activity levels. They have to be carefully optimized in order to detect, as early as possible, potentially harmful failures. A discussion is conducted on privileged ways to design DND systems, and some conclusions are stated.

Sensitivity analysis of prompt neutron decay constant using perturbation theory

ABSTRACT Experimental results of prompt neutron decay constant is useful information to validate numerical results of -eigenvalue for spatial and energetic fundamental mode. In order to accomplish the data assimilation technique using , it is desirable to establish an efficient numerical calculation method for sensitivity coefficient analysis of . For this purpose, the numerical calculation method using the first-order perturbation theory is investigated. A specific theoretical formula is derived to evaluate the sensitivity coefficient of to nuclear data. The derived rigorous formula utilizes forward and adjoint eigenfunctions which consist of neutron flux and delayed neutron precursor densities. Using the prompt approximation, the derived formula can be simplified without the term involving the delayed neutron precursor densities. By calculating using the multi-energy-group neutron transport code for an ICSBEP benchmark problem, the derived formula for sensitivity analysis using the perturbation theory is verified by comparing with the reference results using the direct method. Consequently, the efficient numerical procedures for uncertainty quantification of can be established by the aid of the sensitivity coefficients based on the perturbation theory.