Delayed Neutron Precursors Research Articles

Improvements of the dynamics code TMSR3D for molten salt reactor leveraging triangular prism nodal method

Dynamic simulation with coupled model of neutronics/thermal-hydraulics (TH) is one of the crucial aspects in safety analysis for molten salt reactor (MSR), which requires a deeper insight into the interactions among hydrodynamics, heat transfer and neutron kinetics because of its special characteristics induced by fuel drift, in comparison to the traditional reactors using solid fuels. To improve the geometric adaptability and calculation precision of the MSR specific dynamics code TMSR3D based on quadrilateral nodal method and TH module including the multi-channel flow and single-channel heat transfer models, the analytic basis function expansion nodal (ABFEN) method using triangular prism nodes is implemented to solve the static neutron diffusion equations, an implicit backward difference method along with exponential transformation is adopted to deal with the neutron kinetics problems, and the concentration equations of delayed neutron precursors (DNP) considering fuel flow are solved by the method of characteristics. The VVER-440 transient benchmark is first selected to validate capacity of the updated code for simulating core with hexagonal assemblies. Subsequently, the Molten Salt Reactor Experiment (MSRE) is simulated for further validation, in which the steady-state characteristics including delayed neutron loss induced by circulating fuel, profiles of reactor temperature and DNP, impact of fuel residence time in external loop on the effective yields of delayed neutrons, two fuel pump driven transients, and the natural convection are analyzed and discussed in detail. The results agree well with that from similar codes and the experimental values, which indicate that the improvements conducted for TMSR3D are effective and correct.

Accurate measurements of delayed neutron data for reactor applications: methodology and application to 235U(nth,f)

Large inconsistencies still exist in nuclear data libraries regarding the kinetic parameters of delayed neutron (DN) precursors. As an example, there is a 17% gap between ENDF-B/VIII.0 and JEFF-3.3 on the average lifetime T1/2 of DN precursors from thermal fission of 235U. This parameter is of major importance for reactivity predictions of nuclear reactors in nominal or accidental configurations. In this context, CEA is actively participating to the ALDEN project (Average number and Lifetime of DElayed Neutrons) which aims at providing the nuclear data community with new data sets of DN from thermal and fast neutron induced fission of various actinides. A dedicated experimental setup was designed and optimized for that purpose and is presented in this paper. It consists of a “long counter” detector containing 16 proportional counters filled with 3He, embedded in a high density polyethylene matrix. The detector surrounds a fissile target prepared in the form of a miniature fission chamber, containing a few hundreds of micro-grams of fissile material. This set-up is connected to fast and efficient neutron shutters that can produce step-irradiations of variable durations. The equations driving the DN counting following step-irradiations of the fissile target are established and discussed in the perspective of DN yield or group parameter measurement. A comprehensive analysis of the different steps of data reduction is detailed: dead time characterization, Region of Interest (ROI) determination, absolute and relative efficiency calibration, fission rate estimation, irradiation time and background determination, DN decay curve production and physical parameter fitting. Following a prototype experiment performed in 2018 at the PF1b cold neutron beam line of Institute Laue Langevin (ILL, Grenoble, France), we discuss here the analysis of two campaigns occurring in 2019 and 2021 in which significant improvements were achieved in terms of background minimization, counting statistics and fission rate determination. The achievements of this work are the measurement of the delayed neutron emission per fission for the thermal neutron induced fission of 235U, estimated at (1.625 ± 0.010) % and the group parameters leading to an estimated lifetime of T1/2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${T}_{1/2}$$\\end{document} = (8.87 ± 0.10) s. Those results are consistent with the values recommended by the IAEA/CRP work and they come with reduced uncertainties compared with previously published results.

Development of a New Thermal-Hydraulic Module for FRENETIC, a Code for the Multiphysics Analysis of Liquid Metal–Cooled Reactors

This paper describes the development of a new thermal-hydraulic (TH) module for the Fast REactor NEutronics/Thermal-hydraulICs (FRENETIC) multiphysics code for the full-core simulation of liquid metal–cooled fast reactors, developed at Politecnico di Torino. The code performs steady-state and transient neutronic (NE) and TH coupled calculations while maintaining a relatively low computational cost thanks to the adoption of simplified physical models. The NE module implements the nodal formulation of the multigroup neutron diffusion equations with delayed neutron precursors whereas the TH module treats the reactor hexagonal assemblies as separate channels, which are individually modeled as one-dimensional in the axial direction, accounting for the thermal coupling in the horizontal direction. The new TH module is more robust and portable while providing improved performance with respect to the previous implementation—also thanks to the adopted OpenMP parallelization. Some physical aspects that were previously neglected, such as the thermal inertia of nonfuel rods, have also been included in the model. The development was carried out in accordance with current best practices for code design, implementation, and testing, thus rendering the code easier to be maintained and possibly to be extended in the future. The code usability has also been improved by means of a set of Python classes purposely developed to simplify the input generation and postprocessing phases. This can potentially widen the usage of FRENETIC within the fast reactor community for the simulation of full-core coupled NE-TH transients and/or as a platform to test new solution methods. The paper also includes the application of this new FRENETIC version to a representative configuration of the Advanced Lead-cooled Fast Reactor European Demonstrator (ALFRED) core design. First, the nominal operating conditions of the ALFRED reactor were simulated, proving that the new upgrade of the code is faster and more robust with respect to the previous one. Then, two accidental transients where NE-TH feedback effects are relevant—an unprotected loss of flow and a total instantaneous blockage of a single fuel assembly—were considered.

Stochastic differential equations for non-analog Monte Carlo model of point kinetics equations

The stochastic form of one-point and two-point reactor kinetics models have been extensively investigated in the literature. We have noticed that, in some studies, the probabilistic nature of neutron interaction mechanisms was not incorporated truly. For instance, in the case of negative reactivity insertions, one of the presented probability functions becomes negative, making the Monte Carlo (MC) simulation impossible. In this manuscript, the non-analog stochastic point reactor kinetics models are developed to simulate the stochastic behavior of nuclear systems. To reduce the imposed variance from the leakage and capture of neurons, both capture and leakage events are sampled implicitly. Precursors are also forced to decay in each simulation step. The true variances of the neutron’s and delayed neutron precursor’s populations are determined from the solution of the developed non-analog stochastic models. Also, they are compared with those of both analog and non-analog Monte Carlo methods. The results show that the analog Monte Carlo (MC) has a larger variance than the non-analog MC.

Application of Lie symmetry to find an analytical solution for reactor point kinetics equation with prompt jump approximation and power feedback

The reactor Point Kinetics Equations (PKE) are simpler zero-dimensional approximation to space dependent dynamical models of nuclear reactor core, that are accurate enough to describe transients in small to medium size fast reactor cores. Even these simplified equations can be solved only by numerical methods, except in a very few restrictive cases, where they are amenable to analytical solution. Symmetry methods using Lie's point symmetry have been shown to be a systematic and powerful tool to solve any given ordinary or partial differential equation. An approximation of the PKE, known as Prompt Jump Approximation (PJA) converts the coupled system of first order ODEs with one delayed-neutron precursor group and power feedback, into a single first order nonlinear ordinary differential equation. In this study, we demonstrate an application of Lie symmetry method for solving the point kinetics equation under PJA. The analytical solution obtained is compared with benchmark numerical solution of PKE with PJA.

Extended development of the Monte Carlo code MCNP for effective delayed neutron fraction calculation in molten salt reactors

In this study the Monte Carlo code MCNP is extended to account for the fuel circulation effect in molten salt reactors. An algorithm is created to track delayed neutron precursors and the effective delayed neutron fraction is calculated using predicted fuel velocity fields from the computational fluid dynamics solver. Delayed neutron precursors are monitored throughout various sections of a closed fuel loop in a molten salt reactor. The Moten Salt Fast Reactor is chosen as the benchmark for the extended code, with the calculated multiplication factor and reactivity feedback coefficients agreeing well with referenced results. Dependence of the effective delayed neutron fraction on the fuel flow rate is studied. Validation of the extended code is performed using the Molten Salt Reactor Experiment, showing a good agreement between the calculated delayed neutron fraction loss and the experimental data. These results demonstrate that the extended MCNP code can be reliably used for delayed neutron fraction calculations in both fast and thermal spectrum molten salt reactors.

RBF-based event-triggered fixed-time stable and chattering-free controller for load following of the pressurized water reactor

Preventing chattering phenomenon in load following control of the Pressurized Water Reactor (PWR) is still a main challenge. This article focuses on the chattering-free and rapid load following control strategy for PWR. Firstly, by proposing a newly adaptive updated law, the Radial Basis Function Neural Networks (RBFNNs) is employed to compensate the unmeasured states of relative delayed neutron precursor density, average fuel temperature and uncertain disturbances simultaneously online. On this basis, a fixed-time stable controller including saturation function is constructed to guarantee that the actual output power (OP) of nuclear reactor can follow the desired power (DP) within a fixed-time and the chattering phenomenon is alleviated. Additionally, to save the limited network resource and reduce the cost of computation, an event-triggered mechanism is constructed. Based on it, a chattering-free fixed-time stable controller is structured to ensure that the actual reactivity of control rod (ARCR) converges to the desired control rod reactivity (DCRR) within a fixed-time, meanwhile, it can eliminate Zeno phenomenon during the control process. Finally, the effectiveness and feasibility of the theoretical results are demonstrated through simulation examples.

Symmetry Determining Equations and Lie Groups of the Neutron Transport Equation

The Grigoriev-Meleshko Method, an indirect Lie group theory method, is used to derive the symmetry determining equations (SDEs) of the neutron transport equation (NTE) and the coupled delayed neutron precursor equations (DNPEs). A solution to the SDEs is a Lie group of transformations that can be used to reduce the order of the NTE and DNPEs or outright solve the equations. We found several solutions of the SDEs and worked through the mathematical algorithm to demonstrate relationships of instantiations of the NTE and its known solutions with the Lie groups. Examples of solutions include the Lie group that allows for the transformation of the differential form of the NTE to the integral form of the NTE; the Lie groups that permit Case’s solution; and the Lie group used to transform from the NTE to the α -eigenvalue form of the NTE.

Verification of the SAM Code by Means of the Method of Manufactured Solutions: Application to Molten Salt Reactor Multiphysics Simulation

Software verification and validation constitute crucial phases in the development of simulation computer codes, particularly in the context of nuclear reactor safety analysis codes, where stringent safety requirements govern the development and deployment of nuclear technologies. This paper focuses on numerical verification study of the System Analysis Module (SAM) computer code, currently under development at Argonne National Laboratory. Specifically, we employed the Method of Manufactured Solutions (MMS) and proposed a verification technique tailored to the multiphysics simulation of molten salt reactors (MSRs). This research accomplished three main objectives. First, we have addressed key challenges associated with applying the MMS to MSR systems, arising from (1) the complex multiphysics coupling inherent in this problem and (2) the necessity to model the entire coolant loop for describing the drift of delayed neutron precursors outside the reactor core. The paper provides recommendations and guidelines to overcome these challenges, enabling the successful application of the MMS for simulating MSRs. Second, we have presented a comprehensive set of verification examples, serving as an exhaustive benchmark for code verification within the nuclear community. Third, we have established a robust verification of the SAM code’s capability to model MSR systems.

MSR Transient Simulation and MSRE Transient Benchmark with SAM and SPECTRA

In recent years, there has been renewed interest in molten salt reactors (MSRs) for their potential advantages compared to reactors that rely on solid fuel. In response to such interest, many methods and codes have been developed to capture the unique features of MSRs. Among them, SPECTRA and SAM are two system analysis codes that have been enhanced to include MSR-specific modeling capabilities, including delayed neutron precursor drift and modified point kinetics equations. This paper discusses the efforts taken to verify and validate these features. A standard MSR system test problem was developed to verify and demonstrate the capability of SPECTRA and SAM on the MSR transient simulation. Sixteen transients were simulated. The results obtained from SPECTRA and SAM show good agreement. The Molten Salt Reactor Experiment transient experiments were reviewed and selected to validate the SPECTRA and SAM codes. The experiments included pump startup and coastdown tests at zero power, reactivity insertion tests at different power levels, frequency tests, and a natural convection test. The simulation results from SPECTRA and SAM show good agreement with the experimental data.

Discrete-time sliding mode prescribed performance controller via Kalman filter and disturbance observer for load following of a pressurized water reactor

In practical implementations, most nuclear reactors employ digital controllers to accomplish the load following operation. However, most of the pre-existing works only focus on continuous time controller design, which may yield degraded control performance when the control strategy is employed by a digital controller owing to the finite sampling frequency and the inevitable discrete-time approximations. To this end, this paper proposes a discrete-time sliding mode prescribed performance controller (DTSMPPC) via Kalman filter (KF) and disturbance observer (DO) for a pressurized water reactor (PWR), aiming to achieve an easier implementation on the practical nuclear reactor systems. Specifically, the continuous mathematical model of the PWR system with consideration of model errors and disturbances is first transformed to a discrete-time form by virtue of Euler’s discretization technique. Then, based on this transformed discrete-time model, the KF and DO, which provide the estimates of unmeasured system states (relative density of delayed neutron precursor, average fuel temperature, xenon concentration, and iodine concentration) and uncertainties, respectively, are constructed using the input/output measurement information acquired from the nuclear reactor system merely. By incorporating the observed information provided by the KF and DO into the control framework, the proposed DTSMPPC enables the load following error to fulfill the accurate and robust performance requirement even in the presence of unmeasured system states and uncertainties, while at the same time ensuring the transient and steady-state load following error within a prescribed zone described by performance functions. The system stability and the aforementioned theoretical findings are proved through rigorous analysis and simulations. Simulation results also reveal that the proposed DTSMPPC via the KF and DO yields a superior load following performance compared with some previous control strategies.

Verification of Griffin-Pronghorn-Coupled Multiphysics Code System Against CNRS Molten Salt Reactor Benchmark

The molten salt reactor (MSR) flowing-fuel simulation capability of the Griffin-Pronghorn-coupled multiphysics code system developed by Idaho National Laboratory (INL) was verified against the Center National de la Recherche Scientifique (CNRS) MSR benchmark problem. Griffin and Pronghorn, which are INL’s neutronics and thermal-hydraulics codes built upon the Multiphysics Object-Oriented Simulation Environment (MOOSE) framework, have been recently extended to handle the flowing fuel of MSRs causing the drift of delayed neutron precursors (DNP). In the Griffin-Pronghorn code system, Griffin provides the fission rate density to Pronghorn, which simulates the generation, decay, and transport of DNPs along with the fluid, and the redistributed DNP densities are fed back to Griffin. The coupling and transfers are largely automatically managed at the framework level by the powerful MultiApp system of MOOSE. The verification results against the CNRS benchmark problem demonstrate that the Griffin-Pronghorn code system can accurately simulate the unique physics phenomena of MSRs in both steady-state and transient conditions.

Cross section modeling using Monte Carlo tally derivatives for transient multiphysics prediction

Monte Carlo neutron transport is often seen as the standard for accurate neutronics simulation of nuclear reactors in steady-state. However, the time dependent equation includes time derivatives of flux and delayed neutron precursors which are difficult to model and tally. In this work, a high-order/low-order approach is adopted that uses the omega method to approximate the time derivatives as frequencies. While this scheme has been previously applied to prescribed transients, thermal feedback is now incorporated to provide a fully self-propagating Monte Carlo transient multiphysics solver which can be applied to transients of several seconds long. Such multiphysics problems are less straight-forward than prescribed transients that can be forced to be axially symmetric or otherwise artificially constrained. This work leverages tally derivatives, which are a Monte Carlo perturbation technique that can identify how a tally will change with respect to a small change in the system, to successfully predict thermal feedback and greatly improve accuracy in an axially non-symmetric, flow-initiated transient. Results show good agreement with a reference solution where very fine outer time steps are used for Monte Carlo calculations.

Coupled neutronics and thermal-hydraulics using TRUST-NK for high performance computing in molten salt reactor simulation

Molten Salt Reactors display a strong coupling between neutronics and thermal-hydraulics for which various simulation tools have been developed. In this paper we describe TRUST-NK, a neutronics code based on TRUST, CEA’s High Performance Computing (HPC) platform. TRUST-NK provides a multi-group diffusion solver with transport of delayed neutron precursors, for both steady-state and transient simulations. Being based on TRUST allows TRUST-NK to take advantage of the platform’s HPC capabilities and to easily couple to CFD solvers provided by other derivatives of TRUST. After a brief description of the code, we present an example of application to a simplified MSR model.

Fast simulation of neutron noise using the Transient Fission Matrix approach and validation on the CROCUS reactor

This article presents the development of a neutron noise calculation technique with a reduced calculation time based on the TFM (Transient Fission Matrix) approach. The latter contains different information such as the system transfer function (fission and absorption probability density functions and corresponding neutron multiplicity) of prompt and delayed neutrons, and the corresponding propagation time. This information is used to reconstruct neutron showers based on precursor decays up to the shower disappearance by absorption, leakage or delayed neutron precursor creation. Then pseudo-detectors such as fission chamber count rate histograms are reconstructed from these neutron showers and the αprompt is calculated from the cross-correlation function. A very good agreement has been obtained on the uranium-fueled light water reactor CROCUS between the model developed and the reference value.

On an extended variable separation method approach to the neutron space kinetics equations with two energy groups and six groups of delayed neutron precursors

On an extended variable separation method approach to the neutron space kinetics equations with two energy groups and six groups of delayed neutron precursors

A finite volume method based multi-group neutron diffusion model for space-time nuclear reactor kinetics problems in RZ geometry

The transient analysis of Fast Breeder Reactor (FBR) requires the development of high fidelity space-time nuclear reactor kinetics codes for the realistic prediction of spatial and temporal behavior of neutron flux and fission power along with the feedback effects. So, a space-time nuclear reactor kinetics model has been developed for solving the coupled time dependent multi-group neutron diffusion equations and the delayed neutron precursor equations in axi-symmetric cylindrical (r-z) geometry. The numerical model employs an orthogonal mesh finite volume method for spatial discretization and implicit method for time discretization. A computer code named MANDIR-KRZ (Multi-group Analysis of Neutron DIffusion in Reactor for Kinetics problems in RZ geometry) has been developed and employed to reconstruct the keff, transient neutron flux and power results of different benchmark problems. Mathematical formulation of the numerical model and results of the code for Dodds and FBR transient benchmark problems are discussed in this paper.

Development and benchmarking of SIMULATE5-K-VVER

SIMULATE5-K-VVER (S5K-VVER) is the newest addition to the CMS5-VVER modeling suite developed by Studsvik Scandpower, Inc. (SSP). In S5K-VVER, the coupled time-dependent multigroup neutron diffusion equation, delayed neutron precursor concentrations, and thermal hydraulic models are solved in triangular nodal geometry to model transients in Water-Water Energetic Reactors (VVERs).The neutron kinetics equations in S5K-VVER are solved in a two-step process: temporal integration and spatial integration. In S5K-VVER there are two options for temporal integration: the Frequency Transform Method (FTM) and a Backward-Euler Method (BEM).S5K-VVER is then used to solve a series of benchmark problems including the AER-DYN-001 and AER-DYN-002 numerical transient benchmarks. Finally, the Kalinin-3 benchmark is modeled which describes the shutting-off of a single Main Coolant Pump (MCP) in the Kalinin-3 VVER-1000/320 nuclear power plant. Compared to plant data, the S5K-VVER solution agrees within 1.5% of core total power during this transient.

Reactivity calculation in molten salt reactors with inverse kinetics model

Most molten salt reactor designs present a liquid fuel that, mixed with the coolant composed of molten salts, circulates through the reactor. This circulation introduces important phenomena in the study of reactor physics, thus requiring changes in the existing physical models applied in conventional reactors with solid fuel. One of the main influences of the fuel flow in the reactor is in the delayed neutron precursors’ concentration, which partially decay outside the core, not contributing to the fission chain reaction. In this paper, we investigate the physical models used for this type of reactor, especially the neutron point kinetics, in order to develop an inverse kinetics model, which has practical applications such as core reactivity monitoring. We develop a semi-analytical solution for the proposed model, which is used to solve two transients: a linear variation between power levels, and the determination of criticality regions during a decrease in flow velocity.

Analysis of dynamic responses of circulating fuel reactors using in-house 3D space-time kinetics code ARCH

Molten Salt Reactors (MSRs) are characterized by power generation in circulating liquid fuel salt rather than in static fuel pins of traditional nuclear reactors like in water cooled reactors. The fuel flow in MSRs drift the delayed neutron precursors (DNP) and results in lower delayed neutron fraction. Thus, MSRs show added complexity in dynamic analysis due to inherently coupled neutronics and thermal hydraulics. With regard to the reactor safety analysis, accurate predictions of DNPs distribution and its effect on reactor transients turn out to be an interesting subject of research. Due to these unusual features of MSRs, the traditional safety analysis codes are not applicable in dynamic analysis of reactors with circulating fuel. Challenges in accurate modelling and analysis of dynamic responses of MSRs have been described in present work using new capabilities in 3D space-time kinetics code ARCH. The numerical results of the implemented methodology presented in this study are compared with data available from Molten Salt Reactor Experiment (MSRE), a reactor operated at ORNL.