Multigroup Neutron Diffusion Equation Research Articles

Effect of Radial Neutron Reflector on the Characteristics of Nuclear Fuel Burn-Up Wave in a Fast Neutron Energy Spectrum Multiplying Medium: A Consistent Parametric Approach

Abstract The influence of radial neutron reflector on the build-up and propagation of a nuclear fuel burn-up wave in a fast multiplying medium is investigated using a consistent parametric approach. Coupled multigroup neutron diffusion equations with a burn-up evolution model are simulated on the two-dimensional cylindrical reactor geometry with azimuthal symmetry. Uranium–plutonium transmutation model is considered, and the simulation is performed by using the finite element multiphysics software package comsol. Transient characteristics of the burn-up wave are represented by two new parameters, namely, transient time (TT) and transient length (TL). TT and TL are defined as the time and distance required for the burn-up wave to attain its steady-state nature. Steady-state phases are characterized in terms of wave velocity, full width half maximum (FWHM), and full width 10% of maximum (FW10M). A sensitivity study of steady-state and transient parameters is conducted for the different values of radial reflector thickness. The potential relevance of these characterization parameters on the development of optimal geometrical configuration of radial neutron reflector in breed and burn (B&B)-based reactor design is addressed based on the sensitivity study.

Reduced basis method for non-symmetric eigenvalue problems: application to the multigroup neutron diffusion equations

Abstract. In this article, we propose a reduced basis method for parametrized non-symmetric eigenvalue problems arising in the loading pattern optimization of a nuclear core in neutronics. To this end, we derive a posteriori error estimates for the smallest eigenvalue which is assumed to be simple and the associated left and right eigenvectors. The practical computation of these estimators requires the estimation of a constant called prefactor, which we can express as the spectral norm of some operator. We provide some elements of theoretical analysis which illustrate the link between the expression of the prefactor we obtain here and its well-known expression in the case of symmetric eigenvalue problems, either using the notion of numerical range of the operator, or via a perturbative analysis. Lastly, we propose a practical method in order to estimate this prefactor which yields interesting numerical results on actual test cases. We provide detailed numerical simulations on two-dimensional examples including a multigroup neutron diffusion equation. 1991 Mathematics Subject Classification. 65N15, 65N25, 65N30, 82D75.

An OpenFOAM solver for multiphysics modeling of fusion reactor design: The nemoFoam code

This paper introduces a multiphysics code, named nemoFoam, designed within the OpenFOAM environment to address the coupling of neutron and photon transport with thermal-hydraulics in nuclear reactor simulations. The code, conceived both for fusion and fission applications, employs a modular approach where the neutronic module is currently based on multi-group neutron diffusion equations, including a mono-kinetic treatment for photons. The thermal-hydraulic module is built on the standard OpenFOAM solver. The paper focuses on presenting the key features of nemoFoam and discussing the structure of the code, the nuclear and thermal-hydraulic models, and the coupling strategy between them.In order to assess the performance of the neutronic module, the code is applied to a fusion case study, indeed a benchmark against the Serpent Monte Carlo code for neutron and photon transport is performed, applying nemoFoam to the Affordable, Robust and Compact (ARC) fusion reactor design. Simulation results demonstrate good agreement with Monte Carlo benchmarks, emphasizing the potential of the code for multiphysics simulations. The modular design of nemoFoam allows users to extend the implemented model equations to study additional phenomena, focus only on selected aspects independently and, potentially, to add new solvers in each module.

Development and benchmarking of space-time kinetics solver in REDAC

A space–time kinetics solver based on finite volume approach has been developed for REactor Dynamics Analysis Code (REDAC). It solves 3D multi-group neutron diffusion equation, coupled with the heat transport equations. Code offers flexibility in terms of reactor geometry, boundary conditions and discretization. Efficient numerical methods have been employed for solving the governing equations. The performance of the code has been demonstrated by solving diverse benchmark problems. It is shown that REDAC is able to predict important static and dynamic core parameters with good accuracy. The framework of REDAC easily lends itself for sensitivity studies with respect to spatial discretization and temporal integration schemes. From these studies, it is observed that spatial resolution could have significant influence on accuracy and the semi-implicit method performs better than the implicit method. Improved quasi-static method requires less computational time compared to the direct method, however, its accuracy depends on the time step size.

Energy-Dependent, Self-Adaptive Mesh h(p)-Refinement of an Interior-Penalty Scheme for a Discontinuous Galerkin Isogeometric Analysis Spatial Discretization of the Multi-Group Neutron Diffusion Equation with Dual-Weighted Residual Error Measures

Energy-dependent self-adaptive mesh refinement algorithms are developed for a symmetric interior-penalty scheme for a discontinuous Galerkin spatial discretization of the multi-group neutron diffusion equation using NURBS-based isogeometric analysis (IGA). The spatially self-adaptive algorithms employ both mesh (h) and polynomial degree (p) refinement. The discretized system becomes increasingly ill-conditioned for increasingly large penalty parameters; and there is no gain in accuracy for over penalization. Therefore, optimized penalty parameters are rigorously calculated, for general element types, from a coercivity analysis of the bilinear form. Local mesh refinement allows for a better allocation of computational resources; and thus, more accuracy per degree of freedom. Two a posteriori interpolation-based error measures are proposed. The first heuristically minimizes local contributions to the discretization error, which becomes competitive for global quantities of interest (QoIs). However, for localized QoIs, over energy-dependent meshes, certain multi-group components may become under-resolved. The second employs duality arguments to minimize important error contributions, which consistently and reliably reduces the error in the QoI.

Energy-Dependent, Self-Adaptive Mesh h(p)-Refinement of a Constraint-Based Continuous Bubnov-Galerkin Isogeometric Analysis Spatial Discretization of the Multi-Group Neutron Diffusion Equation with Dual-Weighted Residual Error Measures

Energy-dependent self-adaptive mesh refinement algorithms are developed for a continuous Bubnov-Galerkin spatial discretization of the multi-group neutron diffusion equation using NURBS-based isogeometric analysis (IGA). The spatially self-adaptive algorithms employ both mesh (h) and polynomial degree (p) refinement. Constraint-based equations are established across irregular interfaces with hanging-nodes; they are based upon master-slave relationships and the conservative interpolation between surface meshes. A similar Galerkin projection is employed in the conservative interpolation between volume meshes to evaluate group-to-group source terms over energy-dependent meshes; and to evaluate interpolation-based error measures. Enforcing continuity over an irregular mesh does introduce discretization errors. However, local mesh refinement allows for a better allocation of computational resources; and thus, more accuracy per degree of freedom. Two a posteriori interpolation-based error measures are proposed. The first heuristically minimizes local contributions to the discretization error, which becomes competitive for global quantities of interest (QoIs). However, for localized QoIs, over energy-dependent meshes, certain multi-group components may become under-resolved. The second employs duality arguments to minimize important error contributions, which consistently and reliably reduces the error in the QoI.

Research on neutron diffusion equation and nuclear thermal coupling method based on gradient updating finite volume method

To solve the problem that the neutron diffusion equation has infinite solutions when the deterministic method is used to calculate the neutron diffusion equation, this paper uses the power calculation equation as a supplementary equation to form a differential and integral equation groups with the neutron diffusion equation, and designs a numerical analysis and machine learning coupling algorithm based on the gradient updating finite volume method that can be used to solve the above differential and integral equation groups. The algorithm decouples the multigroup neutron diffusion equations by source iteration. Then the finite volume method is used to solve the one group neutron diffusion equation and the reactor core power. Based on the power difference calculated by two adjacent iterations, the neutron flux density is updated by the gradient descent method until it converges. Finally, the algorithm can be used to calculate the neutron flux density and volumetric heat release rate of each energy group under the specified power. Taking 5 × 5 rod bundle channel as an example, the feasibility of the algorithm is verified in Fluent 18.0 by writing the UDF script. In addition, the thermal–hydraulic parameters such as temperature and velocity in the reactor can be further calculated according to the volumetric heat release rate, and a same set of grids can also be used to achieve the nuclear thermal coupling calculation. This algorithm can provide a new calculation method for the neutron diffusion equation, and also provide a theoretical basis for the development of application algorithms in the field of reactor physics using deep learning.

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.

Development and validation of multiphysics PWR core simulator KANT

KANT (KAIST Advanced Nuclear Tachygraphy) is a PWR core simulator recently developed at Korea Advance Institute of Science and Technology, which solves three-dimensional steady-state and transient multigroup neutron diffusion equations under Cartesian geometries alongside the incorporation of thermal-hydraulics feedback effect for multi-physics calculation. It utilizes the standard Nodal Expansion Method (NEM) accelerated with various Coarse Mesh Finite Difference (CMFD) methods for neutronics calculation. For thermal-hydraulics (TH) calculation, a single-phase flow model and a one-dimensional cylindrical fuel rod heat conduction model are employed. The time-dependent neutronics and TH calculations are numerically solved through an implicit Euler scheme, where a detailed coupling strategy is presented in this paper alongside a description of nodal equivalence, macroscopic depletion, and pin power reconstruction. For validation of the steady, transient, and depletion calculation with pin power reconstruction capacity of KANT, solutions for various benchmark problems are presented. The IAEA 3-D PWR and 4-group KOEBERG problems were considered for the steady-state reactor benchmark problem. For transient calculations, LMW (Lagenbuch, Maurer and Werner) LWR and NEACRP 3-D PWR benchmarks were solved, where the latter problem includes thermal-hydraulics feedback. For macroscopic depletion with pin power reconstruction, a small PWR problem modified with KAIST benchmark model was solved. For validation of the multi-physics analysis capability of KANT concerning large-sized PWRs, the BEAVRS Cycle1 benchmark has been considered. It was found that KANT solutions are accurate and consistent compared to other published works.

Analytical solution of the multigroup neutron diffusion equation coupled with an iterative method

Many numerical methods are being used to solve the multigroup neutron diffusion equation for different types of nuclear reactors. These methods solve this equation quite accurately and determine the neutron flux and power distribution in the reactor as well as the eigenvalue of the reactor core. In this paper, we are proposing the integration of an analytical solution with an iterative method to determine the neutron flux distribution in the reactor and the effective eigenvalue. To do this, we solve the one-dimensional neutron diffusion equation for two energy groups, where the nuclear parameters are uniform in both nuclear fuel and reflector regions. The eigenvalue will be determined from the analytical solution using the power method iteratively until reaching convergence in both flux and eigenvalue. The results obtained in this paper are compared with results obtained from numerical methods used to validate the proposed method.

NURBS Enhanced Virtual Element Methods for the Spatial Discretization of the Multigroup Neutron Diffusion Equation on Curvilinear Polygonal Meshes

The Continuous Galerkin Virtual Element Method (CG-VEM) is a recent innovation in spatial discretization methods that can solve partial differential equations (PDEs) using polygonal (2D) and polyhedral (3D) meshes. Recently, a new formulation of CG-VEM was introduced which can construct VEM spaces on polygons with curvilinear edges. This paper presents the application of the curved VEM to the multigroup neutron diffusion equation and demonstrates its benefits over the conventional straight-sided VEM for a number of benchmark verification test cases with curvilinear domains. These domains were constructed using a topological data-structure developed as part of this paper, based on the doubly-connected edge list, with curves and surfaces both represented using non-uniform rational B-splines (NURBS). This data-structure is used both to specify the geometry of the reactor and to represent the curvilinear polygonal mesh. We also present two separate methods of performing integrations on curvilinear polygons, one for homogeneous functions and one for non-homogeneous functions.

Asymptotic Diffusion Limits of the Multigroup Neutron Transport Equations

In this paper, the standard multigroup neutron diffusion equations are derived as an asymptotic approximation to the multigroup neutron transport equations. The asymptotic analysis employs a scaling that (1) is suggested by the multigroup neutron diffusion equations themselves and (2) generalizes the long-known asymptotic scaling for monoenergetic transport problems. Two other asymptotic scalings of the multigroup transport equations are also considered, both of which lead to a new “group-collapsed” (monoenergetic) “equilibrium” diffusion approximation. The standard multigroup and equilibrium diffusion approximations are shown to preserve certain nonasymptotic properties of the multigroup transport equations. Generalizations of the analyses in this paper, and possible practical applications, are discussed.

Higher order compact finite difference schemes for steady and transient solutions of space–time neutron diffusion model

One of the most difficult tasks in the numerical simulation of nuclear reactors is to find solution of a system arising from spatial discretization of multidimensional multigroup neutron diffusion equations by finite difference method. The big disadvantage is that it requires smaller mesh spacing. Using smaller mesh spacing produces reasonable results but it consumes huge computational effort. In this work, we overcome this flaw by constructing higher order compact finite difference schemes to approximate the neutron leakage terms. The technique is based on using concentrated points around a central point instead of minimizing the mesh spacing taking into account the location of the diffusion coefficients. Beside the accuracy of the proposed schemes, the dimension of the coefficient matrix and therefore the complexity of the system still the same. Five point stencils with equally distance h in each direction are used to produce more than one of standard central difference formulas to be combined producing the fourth order finite difference scheme with order of accuracy O(h4). Also, seven point stencils are used to get the sixth order scheme with accuracy O(h6). The proposed fourth and sixth order schemes are highlighted through two theories. The maximum error norms and the convergence rates of the proposed schemes are discussed. For the time discretization, multi-step differential transform method MDTM is employed to solve the neutron diffusion model. To illustrate the applicability and accuracy of the proposed schemes, results are tested on some multidimensional homogenous and heterogeneous benchmark problems. Our static normalized assembly power densities that are obtained by dividing the whole core into just few dozens or hundreds of fuel assemblies are compared with the Improved SIDK, FLUENT and BRBFCM methods that divide the core into many thousands of fuel assemblies. It is proved that the obtained results are very close to those methods.

Development and Processing of Thermal-Hydraulic Model for GFR 2400 Fast Reactor Design and NESTLE Coupled Transient Code

Abstract The paper investigates the influence of the used thermal-hydraulic approximations on the coupled calculations of gas-cooled fast reactor design (hereby GFR 2400). The NESTLE code is used as coupled simulation tool and solves the multigroup neutron diffusion equation by the finite difference method that is internally coupled with a thermal-hydraulic subchannel code. The in-house developed tempin code and the computational fluid dynamics (CFD) code fluent (from ANSYS code system, Canonsburg, PA) are used to prepare the thermal-hydraulic data for the GFR 2400 calculations. The tempin code solves the steady-state heat balance equation with flowing coolant in triangular lattice cell together with temperature dependent thermal-hydraulic properties of the fuel, cladding, and coolant. Based on the calculated fuel bundle temperature distributions by the tempin code, the thermal-hydraulic material properties (approximations) suitable for the NESTLE coupled code are processed for the GFR 2400 design. The influence of the constant and radial heat generation term within the fuel pin is studied within the paper. The performance of the NESTLE code with thermal-hydraulic approximations processed by both (tempin and fluent) methods is compared with the findings of the GoFastR project. Moreover, both the thermal-hydraulic approximations were compared for one steady-state and one transient state, related to the rapid withdrawal of one control rod assembly from the core. Changes in thermal-hydraulic distributions are described and visualized in the paper.

Opennode: A New Package to Simulate 3d Nuclear Reactor Cores by Solving the Multi-Group Neutron Diffusion Equations in Cartesian Geometry Using the Nodal Expansion Method

Opennode: A New Package to Simulate 3d Nuclear Reactor Cores by Solving the Multi-Group Neutron Diffusion Equations in Cartesian Geometry Using the Nodal Expansion Method

Simulation of the burnup in cell calculation using the wimsd-5b code considering different nuclear data libraries

This work proposes to implement the cell calculation considering the fuel burning using the WIMSD-5B code. The cell calculation procedure allows determining the nuclear parameters present in the multi-group neutron diffusion equation and for this purpose the neutron transport theory is used in a problem with dimensional reduction, but in contrast is considered a large number of groups associated with the neutron spectrum. There are a variety of reactor physics codes that determine the nuclear parameters by solving the neutron transport equation applied to an equivalent cell representing a fuel element. The WIMSD-5B code is a deterministic code that solves the transport equation using collision probability method. The simulation of fuel burning in the cell calculation took into account different nuclear data libraries. The WIMSD-5B code supports several nuclear data libraries and in the present work the following libraries were used: IAEA, ENDFB-VII.1, JENDL3.2, JEFF3.1 and JEF2.2, all formatted for 69 energy groups.

MATLAB PDE toolbox for 2-D and 3-D neutron diffusion simulations

This research extends MATLAB PDE Toolbox to model nuclear reactors by solving the 2-D and 3-D multigroup neutron diffusion equations using the finite element method. The developed scripts can simulate reactors with rectangular and hexagonal lattices and have the potential to handle complex core configurations, benefitting from the constructive solid geometry. The meshing subroutine decomposes the domain into unstructured triangular or tetrahedral elements that are either linear or quadratic. With many versatile MATLAB functions, data analysis and visualization can be accomplished within the same software environment. The proposed toolbox provides a graphical user interface for novices and simple models and a programmatic workflow for experienced users and complex problems. Multiple benchmark tests show that the developed framework produces accurate results with acceptable computation speed. The developed MATLAB reactor simulation engine is license friendly and readily accessible by researchers and students, making it an ideal research and educational tool.

Polynomial relaxation in the quasi-static approach and its implementation in nonlinear reactor transient analyses

The quasi-static (QS) method had been devised for efficient analyses of reactor transients, and two different approaches are widely implemented, which are namely Improved Quasi-Static Method (IQM) and Predictor-Corrector Quasi-Static Method (PCQM). Conventionally, both methods regard the shape to be static while deducing the variation of the amplitude. In this paper, the formulation of standard IQM and PCQM methods based on a multi-group neutron diffusion equation is presented along with a description for implementing such methods to feedback-encompassed nonlinear transient calculation. The idea of relaxation in the quasi-static treatment of the shape through polynomial interpolation is proposed, and comparison among the ramifications of having different conjectures, e.g., static, linear, and quadratic relaxation, has been made based on a thermal-hydraulics-coupled one-dimensional two-group reactor representing the pressurized water reactor. It is found that performance of both IQM and PCQM could be clearly improved with polynomial relaxations even for strongly non-linear reactor problems.

Variational nodal method for three-dimensional multigroup neutron diffusion equation based on arbitrary triangular prism

With the development of nuclear technology, new concepts of reactor core have been continuously proposed, and new types of geometric assembly and reactor core configuration would be adopted to achieve better characteristics. Traditional nodal method based on rectangle or hexagonal nodal is no longer applicable for complex geometry modeling. Given that triangles have the advantage of constructing arbitrarily complex geometry, in this study, variational nodal method of three-dimensional multigroup neutron diffusion theory based on arbitrary triangular prism was developed. ANSYS code is used to divide calculation region into arbitrary triangle mesh, which is then transformed into regular triangle mesh by coordinate transformation, and a set of orthogonal polynomial basis function is built. Variables such as nodal neutron scalar flux and neutron current on nodal surface are expanded based on those polynomial basis functions. The response relationship between neutron partial neutron current and neutron flux can be obtained by variational principle, and traditional power method is employed to solve those response matrix system. Based on above, TriVNM code was developed, and 4 types of typical benchmarks including rectangle, hexagonal assembly reactor and reactor with curved boundary were employed to verify TriVNM. Both power distributions and keff calculated by TriVNM code agree well with reference results, and the maximum relative error of all the benchmarks’ power doesn’t exceed 1% and error of keff doesn’t exceed 50 pcm. TriVNM code achieves the same or higher level of calculation accuracy compared with other diffusion codes. Mesh sensitivity analysis shows that for large mesh size TriVNM code still has good accuracy.