Concentration Of Delayed Neutron Precursors Research Articles

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.

A New Calculation Strategy for Molten Salt Reactor Neutronic–Thermal-Hydraulic Analysis Implemented with APOLLO3® and TRUST/TrioCFD

Molten salt nuclear reactors (MSRs) constitute a promising technology to produce safe, reliable, abundant low-carbon energy. To design MSR systems and perform safety analyses on them, numerical simulation is a powerful tool. Here, we implemented a coupling between several solvers of the deterministic neutronics code APOLLO3® (the MINARET SN transport and the MINOS diffusion and SPn-simplified transport solvers) and the computational fluid dynamics (CFD) code TRUST/TrioCFD, both developed at the French Alternative Energies and Atomic Energy Commission (CEA). The code coupling is orchestrated using the dedicated C3PO library of the open-source SALOME platform. A new code-coupling strategy is employed whereby the delayed neutron precursor concentrations are computed by the CFD code, which eases the use of traditional deterministic neutronics codes. We verified the correctness of our implementation by performing a numerical benchmark dedicated to fast spectrum MSRs originally devised by the French National Center for Scientific Research. The numerical results we obtained are in excellent agreement with those obtained by recent MSR-dedicated multiphysics simulation tools. This study provides a new convenient neutronic–thermal-hydraulic coupling strategy for MSR core simulation.

Continuous-Energy Time-Dependent Coarse Mesh Transport (COMET) Method for Kinetics Calculations

In this paper, the novel continuous-energy coarse mesh transport (COMET) method is extended to perform time-dependent neutronics calculations in highly heterogeneous reactor core problems. In this method, the time-dependent transport equation is converted into a series of steady-state transport equations by estimating the time derivative term using implicit finite differencing. The resulting steady-state transport equations, having additional terms that are imbedded in the total collision term and in the volumetric source terms, are then solved by the steady-state COMET method, in which all the phase-space variables, including energy, are treated continuously. Finally, the fission density distribution constructed by the steady-state COMET is used to solve a set of ordinary differential equations to obtain the delayed neutron precursor concentrations. The time-dependent COMET method is benchmarked against a direct continuous-energy Monte Carlo method (i.e., MCNP) in a set of infinite homogeneous problems and a set of single-assembly benchmark problems consisting of identical pin cells. It is found that the COMET results agree very well with the Monte Carlo reference solutions while maintaining its formidable computational speed (orders of magnitude faster than the Monte Carlo method).

A Simple Approach for Explicit Solution of The Neutron Diffusion Kinetic System

This paper introduces a new approach to directly solve a system of two coupled partial differential equations (PDEs) subjected to physical conditions describing the diffusion kinetic problem with one delayed neutron precursor concentration in Cartesian geometry. In literature, many difficulties arise when dealing with the current model using various numerical/analytical approaches. Normally, mathematicians search for simple but effective methods to solve their physical models. This work aims to introduce a new approach to directly solve the model under investigation. The present approach suggests to transform the given PDEs to a system of linear ordinary differential equations (ODEs). The solution of this system of ODEs is obtained by a simple analytical procedure. In addition, the solution of the original system of PDEs is determined in explicit form. The main advantage of the current approach is that it avoided the use of any natural transformations such as the Laplace transform in the literature. It also gives the solution in a direct manner; hence, the massive computational work of other numerical/analytical approaches is avoided. Hence, the proposed method is effective and simpler than those previously published in the literature. Moreover, the proposed approach can be further extended and applied to solve other kinds of diffusion kinetic problems.

A study on the solution of the spatial kinetics equations in the neutron diffusion theory

In this work, recent studies on the solution of the spatial kinetics equations in the neutron diffusion theory are presented. A nodal technique for the treatment of spatial variables is applied, resulting in the definition of average quantities with respect to neutron fluxes, delayed neutron precursor concentrations, and current densities. The current densities are expressed in terms of the average neutron fluxes to adjust the number of additional unknowns that originate in the nodal scheme. Thus, two systems of coupled differential equations, dependent on the temporal variable and involving the neutron fluxes and delayed neutron precursor concentrations must be solved. A proposal for an iterative scheme with independent solutions of these two systems is developed. Analytical solutions are derived for the delayed neutron precursor concentrations equations. For neutron fluxes equations, numerical and analytical solutions are discussed. The latter use Padé approximations and the Schur–Parlett algorithm to evaluate matrix exponentials. Numerical results obtained through the proposed techniques are compared with results available in the literature, showing satisfactory agreement and contributing to establishing benchmark results. The proposed approach of solving the two systems independently lead to a reduction of the stiffness of the system, a relevant issue for the global approach, and required lower computational costs.

Exact solution of the one-dimensional neutron diffusion kinetic equation with one delayed precursor concentration in Cartesian geometry

<abstract> <p>In this article we use one-dimensional, monoenergetic diffusion kinetic equation with one delayed neutron precursor concentration in Cartesian geometry to reconstruct the neutron flux of a reactor from the nuclear parameters, boundary, and initial conditions. The mathematical model is governed by a system of linear partial differential equations with prescribed boundary and initial conditions. As a matter of fact, the exact solution for any physical problem, if available, is of great importance which inevitably leads to a better understanding of the behavior of the involved physical phenomena. The present work represents an attempt for doing so, where the flux and precursor equations are solved by the help of Laplace transform in both spatial and time variables and consequently the exact expressions for the flux and concentration in space and time are established. We report numerical simulations as well study of numerical convergence of the obtained results.</p> </abstract>

Investigation of neutron noise in a micro-scale, natural circulation molten salt fission battery system

The Molten Salt Reactor (MSR) concept is a rapidly evolving Generation IV design that has recently attracted favorable attention due to the potential for reducing waste generation, realizing passive safety features, and seizing on the opportunity for cost effective economics. A specific novel micro-MSR, natural circulation, battery design concept involves placing all primary components within a single reactor vessel containment without the need of primary forced pumping. This small modular, integral design presents potential cost savings while producing safety, reliable, and transportable carbon-free power for decades. This investigation evaluates the neutron noise induced by density and flow fluctuations in a natural circulation MSR battery concept being developed at the University of Idaho. This study finds numerical solutions to the one-dimensional, one-group, coupled diffusion equations for a bare, homogeneous core, and evaluates neutron flux and delayed neutron precursor concentration noise due to core flow and fuel salt density perturbation sources. Noise analysis shows both point kinetic and space-dependent behavior is present despite the small size, low-flow, closely-coupled nature of the natural circulation MSR concept. Analysis shows that the low flow of the convective system results in noise behavior typically representative of a larger forced-circulation MSR system. Noise techniques may also be useful in MSR core diagnostics as a non-invasive, low-cost radiological option to traditional monitoring schemes.

Development and validation of PROTEUS-NODAL transient analyses capabilities for molten salt reactors

This paper presents the PROTEUS-NODAL capabilities developed for modeling and simulation of molten salt reactors (MSRs) with flowing fuel. Steady-state and transient solvers were developed to solve the coupled neutron flux and delayed neutron precursor concentration equations for a given velocity field. A variational nodal solver for cylindrical geometries was also developed. For thermal feedback calculations, a standalone thermal-hydraulics solver for flowing fuel was implemented. Verification and validation tests were performed using the Molten Salt Fast Reactor (MSFR) benchmark and the available experimental data of the Molten Salt Reactor Experiments (MSRE). The MSFR solution of PROTEUS-NODAL agreed well with the open literature solutions for all transients. The PROTEUS-NODAL results for static and transient experiments of the MSRE were in good agreement with measured values. These results indicate that the PROTEUS-NODAL code can be used reliably for steady-state and transient analyses of fast and thermal spectrum MSRs.

An efficient numerical method for fractional neutron diffusion equation in the presence of different types of reactivities

In this paper, we construct an efficient numerical technique for solving a fractional neutron diffusion equation with step, ramp and sinusoidal reactivity effects combined with one group of delayed neutron precursor concentration equation. This problem describes neutron transport in a nuclear reactor. We use L1 approximation technique for discretization of time derivative and a collocation method based on quintic B-spline (QBS) basis function for discretization of space derivative. Some numerical experiments are performed to show the accuracy and efficiency of the method. The effects of order of fractional derivative, relaxation time and radioactive decay constant on the neutron flux profile are examined. It is shown that the suggested method is of order O(k1+h4) convergence, where k and h represent the step size for time and space, respectively. The numerical result of fractional model is compared with that of classic one. The CPU time is provided to show the computational efficiency of the method.

Yansıtıcılı reaktörlerin stokastik iki-nokta reaktör kinetik denklemlerinin sayısal simülasyonu

Deterministic numerical solutions of point reactor kinetic equations give us the mean values of the neutron population and delayed neutron precursor concentrations, whereas the actual dynamical process is stochastic and the neutron population and precursor concentrations fluctuate randomly with time. In the present study, a novel stochastic model for two-point reactor kinetics equations is developed and used to analyze the dynamical behavior of the source-free strongly reflected reactors with six groups of delayed neutron precursors. To derive the Itô stochastic differential equations system corresponding to this model, the two-point reactor kinetics equations are separated into three terms: prompt neutrons, delayed neutrons and reflected neutrons. In the case of different perturbation scenarios, both with and without the Newtonian temperature reactivity feedback effects, this system of stochastic differential equations is solved using the Euler-Murayama numerical method. It is observed that the mean response of the system is comparable with the results of other deterministic numerical methods.

Coupled 3D neutron kinetics/thermal-hydraulics calculation of PWR core using nodal Green's function method on Neumann boundary condition

In this paper, we proposed a numerical method to perform the coupled three dimensional (3D) neutron kinetics/thermal-hydraulics calculations for the pressurized water reactor (PWR) core and verified its accuracy. In the spatial neutron kinetics, the nodal Green's function method (NGFM) on Neumann boundary condition is applied for the space discretization procedure, and the implicit difference scheme is used for the time differential. It is assumed that the fission source term in the delayed neutron precursor concentration equation is linearly changed within each time step. For the thermal-hydraulic analysis, the single channel model in which a fuel assembly corresponds to one average thermal-hydraulic channel is applied, and cross flow between channels is not taken into account. And we considered the coolant as a single phase incompressible fluid. The model proposed in here was tested by comparison with the two dimensional (2D) TWIGL problem and the NEACRP-L-335 transient benchmark for the 3D PWR core. The numerical results are in good agreement with reference solutions.

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.

Singular Eigenfunction Expansion Solution for Time-Dependent Neutron Transport with Delayed Neutrons

We develop an analytic solution for time-dependent neutron transport with delayed neutrons using the singular eigenfunction expansion method. Our approach is based on a technique for solving time-dependent neutron-transport problems without delayed neutrons (Case and Zweifel, Linear Transport Theory, Addison-Wesley, 1967), which we effectively generalize to include the presence of delayed-neutron precursors. In particular, we obtain eigenfunctions composed of two parts: one corresponding to the neutron angular flux and one corresponding to the delayed-neutron precursor concentration. We further demonstrate that these eigenfunctions are complete. We also provide numerical results for an example problem.

Hamming generalized corrector for reactivity calculation

Abstract This work presents the Hamming method generalized corrector for numerically resolving the differential equation of delayed neutron precursor concentration from the point kinetics equations for reactivity calculation, without using the nuclear power history or the Laplace transform. A study was carried out of several correctors with their respective modifiers with different time step calculations, to offer stability and greater precision. Better results are obtained for some correctors than with other existing methods. Reactivity can be calculated with precision of the order h5, where h is the time step.

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.

Sensitivity and uncertainty analysis of the fractional neutron point kinetics equations

The aim of the present work is to evaluate the sensitivity and uncertainty of the anomalous diffusion coefficient in the Fractional Neutron Point Kinetics (FNPK) equations. This analysis was carried out through Monte Carlo simulations of sizes up to 65,000; the size of 50,000 was considered as valid for routine applications. The sensitivity was evaluated in terms of 99% confidence intervals of the mean to understand the range of mean values that may represent the entire statistical population of performance variables. The regression analysis with anomalous diffusion coefficient as the predictor variable showed statistically valid quadratic relationship for neutronic density and the delayed neutron precursor concentration. The uncertainties were propagated as follows: in a 1% change in the anomalous diffusion exponent the responses for neutron density, and precursor density changed by 0.017% and 0.0000125% for short times, and for long times by 0.012% and 0.000267%, respectively.

Hamming method for solving the delayed neutron precursor concentration for reactivity calculation

We propose a new method for numerically solving the inverse point kinetic equation for a nuclear reactor using the Hamming method, without requiring the nuclear power history and without using the Laplace transform. This new method converges with accuracy of order h5, where h is the step in the computation time. The procedure is validated for different forms of the nuclear power and with different time steps. The results indicate that this method has a better accuracy and lower computational effort compared with other conventional methods that use the nuclear power history.

Spatial Reactor Dynamics and Thermo Hydraulic Behavior Simulation of a Large AGR Nuclear Power Reactor in Response to a Reactivity Step Change Disturbance

In this article, two-dimensional partial differential equations with time representation of nuclear power reactor kinetics are considered for spatial reactor dynamics and thermo hydraulic behavior analysis of a large thermal advanced gas cooled reactor (AGR) type used for nuclear power generation. The equations include the neutron flux equation and delayed neutron precursor concentration, together with taking into account the equations to represent the thermo hydraulic behavior of the fuel, coolant and moderator temperatures. These equations are solved numerically using the finite difference method. For time propagation, an implicit method is applied. The desired initial condition for the reactor to stay at stable critical condition is established by finding the correct value of reactivity. The reactivity disturbance effect in the reactor is studied for different cases and presented for high reactivity values. The model was developed for the analysis of a large AGR with 2000 MWe for future power generation. The results show that the model not only behaves stably but also predicts the results physically for all the various parameters.

ACCURACY AND EFFICIENCY OF A COUPLED NEUTRONICS AND THERMAL HYDRAULICS MODEL

This manuscript will discuss a numerical method where the six equations of two-phase flow, the solid heat conduction equations, and the two equations that describe neutron diffusion and precursor concentration are solved together in a tightly coupled, nonlinear fashion for a simplified model of a nuclear reactor core. This approach has two important advantages. The first advantage is a higher level of accuracy. Because the equations are solved together in a single nonlinear system, the solution is more accurate than the traditional "operator split" approach where the two-phase flow equations are solved first, the heat conduction is solved second and the neutron diffusion is solved third, limiting the temporal accuracy to <TEX>$1^{st}$</TEX> order because the nonlinear coupling between the physics is handled explicitly. The second advantage of the method described in this manuscript is that the time step control in the fully implicit system can be based on the timescale of the solution rather than a stability-based time step restriction like the material Courant limit required of operator-split methods. In this work, a pilot code was used which employs this tightly coupled, fully implicit method to simulate a reactor core. Results are presented from a simulated control rod movement which show <TEX>$2^{nd}$</TEX> order accuracy in time. Also described in this paper is a simulated rod ejection demonstrating how the fastest timescale of the problem can change between the state variables of neutronics, conduction and two-phase flow during the course of a transient.

Non-linear space-dependent kinetics for the criticality assessment of fissile solutions

A deterministic model for calculating the time dependent fission yield from solutions has been developed. The model is based on transient finite element methods and couples radiation transport modelling with computational fluid dynamics. Non-linear space dependent kinetic equations are derived, in which the non-linearities arise due to radiolytic gas generation, geometical changes in the liquid, the temperature dependent densities, cross sections and thermally/gas induced fluid motion. The latter advects the delayed neutron precursor concentrations together with the energy fields. Applications focus on the role of radiolytic gas evolution and buoyancy induced fluid motion on the criticality of fissile liquids with delayed and prompt supercritical step reactivity insertions. The analysis is performed with uranyl nitrate solutions. The theory behind the modelling is presented, together with numerical results, to validate the approach. The resulting computer code, which we call FETCH (finite element transient criticality), is validated against point kinetics based models for right cylinders and against experiment for both low (delayed supercritical) and high powered (usually prompt supercritical) transients. The term ‘low powered’ in this context will be used to describe transients in fissile liquids in which the rate of radiolytic gas generation is so small that it can be ignored without sacrificing the accuracy of the simulations. Modelling can then be conducted in a single fluid phase. The term “high powered’ refers to transients in which gas evolution and ensuing free surface motion play an important part in their dynamics and are thus solved using the multi-phase mode of FETCH. The simulations presented here provided extensive insights into the dynamics of these transients which can be difficult to study in detail with experiment.