Delayed Neutron Precursors Research Articles

On an analytical representation for the solution of the neutron point kinetics equation free of stiffness

In this work, we report on an analytical representation for the solution of the neutron point kinetics equation, free of stiffness and assuming that reactivity is a continuous or sectionally continuous function of time. To this end, we cast the point kinetics equation in a first order linear differential equation. Next, we split the corresponding matrix into a diagonal matrix plus a matrix that contains the remaining terms. Expanding the neutron density and the delayed neutron precursor concentrations in a truncated series allows one to construct a recursive system in form of a first order matrix differential equation with source. The initialization of the recursion procedure is of diagonal form and has no source but satisfies the initial conditions. The remaining equations are subject to null initial conditions and include the time-dependent diagonal elements together with the off-diagonal elements as a source term. The solution is obtained in an analytical representation which may be evaluated for any time value, because it is free of stiffness. We present numerical simulations and comparisons against results from the literature for a constant, a step, a ramp, a quadratic, and a sine shaped reactivity function.

A numerical solution to the point kinetic equations using Taylor–Lie series combined with a scaling and squaring technique

A system of differential equations is derived to model the dynamics of neutron density and the delayed neutron precursors within a point kinetics equation modeling framework for a nuclear reactor. The point kinetic equations are mathematically characterized as stiff, occasionally nonlinear, ordinary differential equations, posing significant challenges when numerical solutions are sought, and traditionally resulting in the need for smaller time step intervals within various computational schemes. In light of the above realization, the present paper proposes a new discretization/simulation method that is: (i) conceptually inspired by system-theoretic notions used in the analysis of sampled-date system dynamics (discrete-time system dynamics) under potentially irreducible large sampling periods/time steps, and (ii) technically based on Taylor–Lie series and the zero-order hold (ZOH). Within the proposed time discretization framework, the sampled-data representation of the original point kinetic system of equations is derived for an arbitrary size of the time-step used. Furthermore, within the context of the proposed approach, the integration of a scaling-and-squaring technique is pursued for the attainment of further performance enhancement of the proposed simulation technique. The performance of the proposed approach is evaluated in several case studies involving step and ramp-like reactivity profiles as well as multiple input examples that simultaneously account for variations in reactivity and neutron source function profiles. In particular, it is demonstrated, that by applying the proposed method, the inherent stiffness problem associated with the simulation challenges of the point kinetic equations can be adequately addressed within a wide range of reactor operating conditions, including large sampling periods dictated by physical and/or technical limitations associated with the current state of sensor and digital reactor control system technology.

ANALIZA MODELU UŁAMKOWEGO RZĘDU PROCESÓW SZYBKICH REAKTORA JĄDROWEGO

The paper presents the results concerning numerical solutions of the fractional point kinetics and heat exchange model for nuclear reactor. The fractional neutron point kinetics model with six groups of delayed neutron precursors was developed and numerical solutions were proposed. Mathematical model has been implemented in the Matlab environment and tested using typical step input change. The analysis of the impact of chosen parameters was conducted.

Development of an OpenFOAM model for the Molten Salt Fast Reactor transient analysis

In the paper, the development of a multiphysics model for the transient analysis of non-moderated Molten Salt Reactors is discussed. Particular attention is devoted to the description of the adopted time integration and physics coupling strategies. The proposed model features the adoption of an implicit Runge–Kutta scheme and the coupling among neutron diffusion, Reynolds-Averaged Navier–Stokes equations for mass and momentum conservation, and energy and delayed neutron precursor balance equations, in order to accurately catch thermal feedbacks on neutronics. The solver is aimed at performing fast-running simulations of the full-core three-dimensional Molten Salt Fast Reactor geometry. The neutronics modelling is assessed against Monte Carlo simulations and the results of a simplified case study are compared to those from multiphysics tools previously developed. As an example of the capability of the model, an unprotected MSFR single pump failure accidental scenario is simulated and discussed. The main purpose of the present model is to serve as fast-running computational tool in the phase of design optimization of fuel loop components. More in general, it is of valuable help in the study of reactor physics of circulating-fuel systems.

Fractional neutron point kinetics equation with Newtonian temperature feedback effects

In this paper we present the numerical analysis of the fractional neutron point kinetics (FNPK) equations with temperature feedback effects. The FNPK model considers a relaxation time associated with a rapid variation in the neutron flux density considered in the differential operator of fractional order known as anomalous diffusion exponent. Different values of the relaxation time with one-group delayed neutron precursor were used for this analysis. Results for neutron flux density dynamic behavior with temperature feedback for different values of anomalous diffusion exponent are shown and compared with the classical neutron point kinetics equations.

Three dimensional neutronic/thermal-hydraulic coupled simulation of MSR in steady state condition

MSR (molten salt reactor) uses liquid molten salt as the coolant and fuel solvent, making it the only liquid reactor among the six generation IV reactor types. As a liquid reactor the physical properties of the reactor are significantly influenced by the fuel salt flow therefore conventional analysis methods applied in solid fuel reactors are not applicable for this type of reactors. The present work developed a three dimensional neutronic/thermal-hydraulic coupled code and applied it to investigate the thermal-hydraulic characteristics of the core in steady state condition based on neutron diffusion theory and numerical heat transfer. The code consists of two group neutron diffusion equations for fast and thermal neutron fluxes and six group balance equations for delayed neutron precursors. The temperature distribution, neutron fluxes and delayed neutron precursors distribution of the core in steady state conditions was studied, and the result analyzed when inlet temperature and velocity were changed. From simulation it was found that the inlet temperature has little influence to neutron distribution however inlet velocity affects the delayed neutron distribution in steady state condition. The results provide some valuable information in design and research of this kind of reactor.

Calculating the effective delayed neutron fraction in the Molten Salt Fast Reactor: Analytical, deterministic and Monte Carlo approaches

This paper deals with the calculation of the effective delayed neutron fraction (βeff) in circulating-fuel nuclear reactors. The Molten Salt Fast Reactor is adopted as test case for the comparison of the analytical, deterministic and Monte Carlo methods presented. The Monte Carlo code SERPENT-2 has been extended to allow for delayed neutron precursors drift, according to the fuel velocity field. The forward and adjoint eigenvalue multi-group diffusion problems are implemented and solved adopting the multi-physics tool-kit OpenFOAM, by taking into account the convective and turbulent diffusive terms in the precursors balance. These two approaches show good agreement in the whole range of the MSFR operating conditions. An analytical formula for the circulating-to-static conditions βeff correction factor is also derived under simple hypotheses, which explicitly takes into account the spatial dependence of the neutron importance. Its accuracy is assessed against Monte Carlo and deterministic results. The effects of in-core recirculation vortex and turbulent diffusion are finally analysed and discussed.

Neutron Noise Calculations in Hexagonal Geometry and Comparison with Analytical Solutions

This paper presents the development of a neutronic and kinetic solver for neutron noise calculations in hexagonal geometries. The tool is developed based on diffusion theory with multienergy groups and several groups of delayed neutron precursors allowing the solutions of forward and adjoint problems of static and dynamic states. The tool is applicable to both thermal and fast systems with hexagonal geometries. In the dynamic problems, the small stationary fluctuations of macroscopic cross sections are considered as noise sources, then the induced first-order noise is solved fully in the frequency domain. Numerical algorithms for solving the static and noise equations are implemented using finite differences for spatial discretization and a power iterative solution. A coarse-mesh finite difference technique for accelerating the convergence has been adopted. Verification calculations have been performed and compared to analytical solutions based on a two-dimensional homogeneous system with two energy groups and one group of delayed neutron precursors, in which pointlike perturbations of thermal absorption cross section at central and noncentral positions are considered as noise sources.

Non-linear simulation and control of xenon induced oscillations in Advanced Heavy Water Reactor

The physical dimensions and the reactivity feedbacks of Advanced Heavy Water Reactor (AHWR) are such that, it is susceptible to xenon induced spatial oscillations. If these oscillations are not controlled, the power density and the rate of change of power at some locations in the reactor core may exceed their respective thermal limits, resulting into increased chances of fuel failure. Hence, it is essential to suppress xenon oscillations and achieve spatial stabilization of AHWR. Reactor core of AHWR is divided into 17 non-overlapping nodes. Non-linear model of AHWR is characterized by 90 first order differential equations. Total reactor power and 17 nodal powers are output variables. Four voltage signals to the Regulating Rods (RRs) and a feed flow rate are input variables. Applying a highly developed simulation is necessary for analysis and control of spatial oscillations developed in AHWR for safe operation. In this paper, after carrying out stability analysis, a control strategy based on feedback of total power and nodal powers in which RRs are placed is presented for spatial control of AHWR. For the same, a vectorized non-linear model of AHWR is developed and is implemented in the MatLab/Simulink environment which helps to understand the relationship between different variables of the system in a better way. With the proposed controller, non-linear model of AHWR is simulated and results are generated for different transient conditions. The behavior of delayed neutron precursor and xenon concentrations is also analyzed for each transient. From the simulation results, performance of the proposed controller is found to be satisfactory.

Qualitative and quantitative investigation of the propagation noise in various reactor systems

The space-dependent neutron noise, induced by propagating perturbations (propagation noise for short) is investigated in a one-dimensional homogeneous model of various reactor systems. By using two-group theory, the noise in both the fast and the thermal group is calculated. The purpose is to investigate the dependence of the properties of the space-dependent fast and thermal propagation noise on the static neutron spectrum as well as on the presence of the fluctuations of several cross sections. The motivation for this study arose in connection with recent work on neutron noise in molten salt reactors (MSR) with propagating fuel of various compositions. Some new features of the induced noise were observed, but it was not clear whether these were due to the propagating perturbation alone, or to the propagation of the fuel and hence that of the delayed neutron precursors. The present study serves to clarify the significance of the spectral properties of the different cores through calculating the propagation noise in four different reactor systems, as well as considering the influence of the perturbation of the various cross sections. By comparing the results with those obtained in MSR, the effect of the moving fuel on the propagation noise is clarified. It is shown that in fast systems the noise in the fast group is much larger than in the thermal group and hence can gain diagnostic importance. It is also shown that the co-existence of several cross section fluctuations leads to qualitatively and quantitatively new characteristics of the noise, hence it is important to model the effect of e.g. temperature fluctuations of the coolant in a proper way.

Modelling and analysis of the MSFR transient behaviour

Molten Salt Reactors (MSRs) were conceived at the early stages of nuclear energy in view of the favourable features fostered by a liquid fuel. They were developed as graphite-moderated thorium-fuelled breeder reactors till the seventies, when studies on this reactor concept were mostly abandoned in favour of the liquid–metal fast breeder concepts. A decade ago, the MSRs were included among the six GEN-IV systems and a core optimization process allowing for the GEN-IV main objectives led toward a fast-spectrum MSR concept (MSFR – Molten Salt Fast Reactor). Albeit advantageous in terms of U-233 breeding and/or radio-active waste burning, the new concept lacks the notable know-how available for the thermal-spectrum MSR technology. The present paper preliminarily investigates the MSR dynamics, based on the conceptual MSFR design currently available. A primary objective is to benchmark two different models of the MSFR primary circuit, both of them including a detailed and fully-coupled (node-wise) representation of turbulent fuel-salt flow, neutron diffusion, and delayed-neutron precursor diffusion and convection. A good agreement is generally observed between the adopted models, though some discrepancies exist in the temperature-field, with ensuing mild impacts on the reactor dynamics. The performed analyses are also used for a preliminary characterization of the MSFR steady-state and accidental transient response. Some points of enhancement needed in the MSFR conceptual design are identified, mainly related to in-core velocity and temperature fields. The reactor response following major accidental transient initiators suggests a generally benign behaviour of this reactor concept.

Random effects on reactivity in molten salt reactors

The problem of the effect of fissile lumps spatially appearing in a random fashion inside a fluid fuel reactor is addressed. The effect on static reactivity is evaluated by means of first-order perturbation theory. The analysis is carried out in diffusion theory with the presence of delayed neutron emissions, taking into account the fuel motion that introduces a distortion of the space distribution of the delayed neutron precursors. The method is applied to a one-dimensional configuration to investigate the general features of the random process. Afterwards, a more realistic two-dimensional cylindrical geometry is considered. The estimation of the mean value and standard deviation of the reactivity inserted is performed by Monte Carlo simulations and a deterministic quadrature approach, to compare the methods in terms of computational effort and accuracy of the results. The reconstruction of the probability density function of the reactivity is also performed by polynomial chaos expansion. The results presented show that random reactivity effects constitute an important issue in the assessment of these innovative molten salt systems.

Generalized and Stability Rational Functions for Dynamic Systems of Reactor Kinetics

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.

Time discretization of the point kinetic equations using matrix exponential method and First-Order Hold

A system of nonlinear differential equations is derived to model the dynamics of neutron density and the delayed neutron precursors within a point kinetics equation modeling framework for a nuclear reactor. The point kinetic equations are mathematically characterized as stiff, occasionally nonlinear, ordinary differential equations, posing significant challenges when numerical solutions are sought and traditionally resulting in the need for smaller time step intervals within various computational schemes. In light of the above realization, the present paper proposes a new discretization method inspired by system-theoretic notions and technically based on a combination of the matrix exponential method (MEM) and the First-Order Hold (FOH) assumption. Under the proposed time discretization structure, the sampled-data representation of the nonlinear point kinetic system of equations is derived. The performance of the proposed time discretization procedure is evaluated using several case studies with sinusoidal reactivity profiles and multiple input examples (reactivity and neutron source function). It is shown, that by applying the proposed method under a First-Order Hold for the neutron density and the precursor concentrations at each time step interval, the stiffness problem associated with the point kinetic equations can be adequately addressed and resolved. Finally, as evidenced by the aforementioned detailed simulation studies, the proposed method retains its validity and accuracy for a wide range of reactor operating conditions, including large sampling periods dictated by physical and/or technical limitations associated with the current state of sensor and digital reactor control system technology.

A multi-group neutron noise simulator for fast reactors

A neutron noise simulator has been developed for fast reactors based on diffusion theory with multi-energy groups and several groups of delayed neutron precursors. The tool is expected to be applicable for core monitoring of fast reactors and also for other reactor types with hexagonal fuel assemblies. The noise sources are modeled through small stationary fluctuations of macroscopic cross sections, and the induced first order noise is solved fully in the frequency domain. Numerical algorithms are implemented for solving both the static and noise equations using finite differences for spatial discretization, where a hexagonal assembly is radially divided into finer triangular meshes. A coarse mesh finite difference (CMFD) acceleration has been used for accelerating the convergence of both the static and noise calculations. Numerical calculations have been performed for the ESFR core with 33 energy groups and 8 groups of delayed neutron precursors using the cross section data generated by the ERANOS code. The results of the static state have been compared with those obtained using ERANOS. The results show an adequate agreement between the two calculations. Noise calculations for the ESFR core have also been performed and demonstrated with an assumption of the perturbation of the absorption cross section located at the central fuel ring.

A zero dimensional model for simulation of TRIGA Mark II dynamic response

Aim of this work is to reproduce the dynamic behavior of the TRIGA Mark II reactor of the University of Pavia on the entire operative power range (i.e. 0–250 kW) using a zero dimensional approach. In this work the coupling between neutronics and thermal-hydraulics in natural circulation has been considered. In specific, a point reactor kinetics model with one energy group and six delayed neutron precursors groups has been adopted while for thermal-hydraulics modeling two regions have been defined (i.e. the fuel and the coolant). The nonlinear system of coupled Ordinary Differential Equations has been solved by means of MATLAB Simulink®, which represents a reliable tool for dynamic and control analysis. The model has then been validated through the comparison with a set of experimental data collected in four different reactor power transients, obtaining a very satisfying agreement. Finally, the linear stability analysis of the TRIGA reactor has been performed by means of the root locus, finding out that the power level at which reactor is operating deeply influences the position of the poles of the transfer function between control rod height and neutron density. These results can then be employed as a reliable starting point in designing an automatic device for reactor power control.

Analysis of kinetic parameters of 3 MW TRIGA Mark-II research reactor using the SRAC2006 code system

The aim of this study is to evaluate the kinetic parameters of 3MW TRIGA Mark-II research reactor at AERE, Savar, Dhaka, Bangladesh from the viewpoint of reactor safety. The most important kinetic parameters of nuclear reactors are the effective delayed neutron fraction (βeff), the effective decay constant for ith family of delayed neutron precursor (λeff,i), the prompt neutron lifetime (lp) and the mean neutron generation time (Λ). These parameters are calculated using the 3-D diffusion code SRAC-CITATION of the comprehensive neutronics calculation code system SRAC2006 based on the evaluated nuclear data libraries JENDL-3.3 and ENDF/B-VII.0 in both cases. The calculated results of reactor kinetic parameters are compared to the available safety analysis report (SAR) values of 3MW TRIGA Mark-II reactor by General Atomic as well as the MCNP5 values (numerically benchmark) based on the evaluated nuclear data library ENDL/B-VII.0. It was found that in most cases, the calculated results of kinetic parameters demonstrate a good agreement between the JENDL-3.3 and the ENDF/B-VII libraries as well as the SAR and the MCNP5 values respectively. Therefore, this study will be essential to improve the basic nuclear data of reactor kinetic parameters for safe operation of 3MW TRIGA Mark-II research reactor.

An approach to the MSR dynamics and stability analysis

The first efforts in the development of the molten salt reactor technology were carried out in the sixties by the Oak Ridge National Laboratory and culminated with the design of the thermal-spectrum Molten Salt Breeder Reactor (MSBR). Only recently, the attention has been focused on fast-spectrum configurations, such as the Molten Salt Fast Reactor (MSFR) proposed in the framework of the Euratom EVOL (Evaluation and Viability of Liquid Fuel Fast Reactor System) Project, thanks to their favourable characteristics in terms of sustainability, waste minimization and improved safety. As a matter of fact, the MSFR has been recognized as a long term alternative to solid-fuelled fast neutron systems and has been identified as Gen-IV reference MSR configuration. From the dynamic behaviour point of view, the main feature that characterises this kind of systems is the presence of a liquid fuel that circulates in the primary circuit acting simultaneously as coolant. This feature leads to a complex and highly coupled behaviour, which requires a careful investigation, due to some peculiarities like the drift of delayed neutron precursors along the primary circuit. Although considerable studies have been carried out for the analysis of the graphite-moderated MSRs, the adoption of a fast spectrum configuration without graphite in the core is expected to notably modify the dynamic characteristics of the system, thus requiring further investigation. This work proposes an approach to the dynamics and stability analysis of molten salt reactors. In particular, the well-developed methods of the theory of linear systems are applied to the analysis of two case studies, namely: the MSBR and the recently proposed MSFR. This analysis is intended to provide a basic understanding of the inherent stability properties and of the dynamic characteristics of such kind of nuclear reactors, highlighting the main peculiarities of the new design compared with the more familiar graphite-moderated concept.

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

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.

Simulations of unprotected loss of heat sink and combination of events accidents for a molten salt reactor

The molten salt reactor (MSR) is one of the Generation IV reactors. The fuel is dissolved in the carrier salt and circulates in the loop. The technologies are different from that in the solid-fuel reactors. In this work, the attention is focused on development of safety analysis tool for the MSRs. A single channel model, a heat transfer model, a heat sink model and a liquid-fuel point kinetic model that takes into account the effect of circulation are employed. The validation of this code is done with the experimental data of pump coastdown and startup in MSRE. The unprotected loss of heat sink (ULOHS), combination of ULOHS and unprotected loss of flow (ULOF) are performed on the Molten Salt Actinide Recycler and Transmuter (MOSART). This work aims to study temperature fluctuation, corresponding power change, effect of flow delayed neutron precursors and temperature reactivity feedback in transient accident to examine the inherent safety design of MOSART. The transient results reveal that the large negative temperature feedback coefficients guarantee MOSART inherent safety and the range of temperature is within the safety margin in case of combination of accidental events.