Search

The time-dependent neutron diffusion equation approximates the neutronic power evolution inside a nuclear reactor core. Applying a Galerkin finite element method for the spatial discretization of these equations leads to a stiff semi-discrete system of ordinary differential equations. For time discretization, an implicit scheme is used, which implies solving a large and sparse linear system of equations for each time step. The GMRES method is used to solve these systems because of its fast convergence when a suitable preconditioner is provided. This work explores several matrix-free strategies based on different updated preconditioners, which are constructed by low-rank updates of a given initial preconditioner. They are two tuned preconditioners based on the bad and good Broyden's methods, initially developed for nonlinear equations and optimization problems, and spectral preconditioners. The efficiency of the resulting preconditioners under study is closely related to the selection of the subspace used to construct the update. Our numerical results show the effectiveness of these methodologies in terms of CPU time and storage for different nuclear benchmark transients, even if the initial preconditioner is not good enough. [ABSTRACT FROM AUTHOR]

• This work presents four methodologies to simulate the neutron noise induced by mechanical vibrations. • These methodologies employ the diffusion approximation, with and without a previous homogenization model, to simulate the neutron noise in the time and the frequency domain. • All methodologies provide consistent results and can reproduce typical features of the neutron noise induced by mechanical vibrations of core components. • Numerical results for mechanical vibrations of fuel assemblies and core barrel vibrations are shown. Mechanical vibrations of core internals are among the main perturbations that induce oscillations in the neutron flux field, also known as neutron noise. In this work, different simulation models for the study of the influence of the mechanical vibrations of fuel assemblies on the neutron flux in the reactor core have been discussed. These methodologies employ the diffusion approximation, with or without a previous homogenization model, to simulate the neutron noise in the time or the frequency domain. The diffusion-based approach is expected to be less accurate in the vicinity of the vibrating fuel assemblies, but correct when considering distances larger than a few diffusion lengths away from the perturbation. All methodologies provide consistent results and can reproduce typical features of the neutron noise induced by mechanical vibrations of core components. First, FEMFFUSION can perform simulations in both the time and frequency domains. Second, CORE SIM + can be used to study various neutron noise scenarios in realistic three-dimensional reactor configurations. The third methodology is centred on using commercial codes as CASMO-5, SIMULATE-3 and SIMULATE-3K. This methodology allows time domain simulations of the neutron noise induced by different neutron noise sources in a nuclear reactor. Finally, a model for time-dependent geometry is implemented for the code system ATHLET/QUABOX-CUBBOX employing a cross-section-based approach for encoding water gap width variations at the reflector. [ABSTRACT FROM AUTHOR]

Abstract: Proper generalized decomposition is a recently developed technique that solves some multidimensional problems using auxiliar models in a lower dimensional space. A method based on this technique together with the Newton method for the computation of eigenvalues is proposed to obtain the dominant eigenvalue of a nuclear power reactor core and its corresponding eigenvector. The method is tested using the bidimensional reactor Biblis. [Copyright &y& Elsevier]

Abstract: Starting from a steady state configuration of a nuclear power reactor some situations arise in which the reactor configuration is perturbed. The Lambda modes are eigenfunctions associated with a given configuration of the reactor, which have successfully been used to describe unstable events in BWRs. To compute several eigenvalues and its corresponding eigenfunctions for a nuclear reactor is quite expensive from the computational point of view. Krylov subspace methods are efficient methods to compute the dominant Lambda modes associated with a given configuration of the reactor, but if the Lambda modes have to be computed for different perturbed configurations of the reactor more efficient methods can be used. In this paper, different methods for the updating Lambda modes problem will be proposed and compared by computing the dominant Lambda modes of different configurations associated with a Boron injection transient in a typical BWR reactor. [Copyright &y& Elsevier]

Abstract: To study the behavior of nuclear reactors like the Russian VVER reactors it is necessary to solve the time dependent neutron diffusion equation using a hexagonal mesh. Two methods are proposed to solve this equation. In both methods the spatial part of the equations is discretized using a high order spectral element method, based on assuming that the neutron flux can be expanded in terms of the modified Dubiner’s polynomials. For the time discretization of the equations two different strategies have been considered. For the first a finite differences one-step implicit method has been used and the other method is based on a modal expansion of the flux in terms of the dominant Lambda modes of the reactor core. The performance of both methods has been tested for an hypothetical transient in a 2-dimensional VVER 440 reactor. [ABSTRACT FROM AUTHOR]

Abstract: The equations are classical approximations to the neutron transport equation that admit a diffusive form. The diffusive form of the approximation is known as the neutron diffusion equation. Different methods based on the expansion of the neutron flux in terms of a continuous basis of polynomials have been developed for the neutron diffusion equation and tested using two 1D benchmark problems. [Copyright &y& Elsevier]

Abstract: A High Order Finite Element Method to approximate the Lambda modes problem for reactors with hexagonal geometry has been developed. This method is based on the expansion of the neutron flux in terms of the modified Dubiner’s polynomials on a triangular mesh. This mesh is fixed and the accuracy of the method is improved increasing the degree of the polynomial expansions without the necessity of remeshing. The performance of method has been tested obtaining the dominant Lambda modes of different 2D reactor benchmark problems. [Copyright &y& Elsevier]

Abstract: The Jacobi–Davidson method is a modification of Davidson method, which has shown to be very effective to compute the dominant eigenvalues and their corresponding eigenvectors of a large and sparse matrix. This method has been used to compute the dominant Lambda modes of two configurations of Cofrentes nuclear power reactor, showing itself a quite effective method, especially for perturbed configurations. [Copyright &y& Elsevier]

Fast codes, capable of dealing with three-dimensional geometries, are needed to be able to simulate spatially complicated transients in a nuclear power reactor. In this paper, we propose a modal method to integrate the neutron diffusion equation in which the spatial part has been previously dicretized using a nodal collocation method. For the time integration of the resulting system of differential equations it is supposed that the solution can be expanded as a linear combination of the dominant Lambda modes associated with a static configuration of the reactor core and, using the eigenfunctions of the adjoint problem, a system of differential equations of lower dimension is obtained. This system is integrated using a variable time step implicit method. Furthermore, for realistic transients, it would be necessary to calculate a large amount of modes. To avoid this, the modal method has been implemented making use of an updating process for the modes at each certain time step. Five transients have been studied: a homogeneous reactor, a non-homogeneous reactor, the 3D Langenbuch reactor and two transients related with in-phase and out-of-phase oscillations of Leibstadt NPP. The obtained results have been compared with the ones provided by a method based on a one-step backward discretization formula. [Copyright &y& Elsevier]

The mechanical vibrations of core internals such as fuel assemblies (FAs) cause oscillations in the neutron flux that require in some circumstances nuclear power plants to operate at a reduced power level. This work simulates and analyzes the changes of the neutron flux throughout a nuclear core due to the oscillation of a single FA without considering thermal-hydraulic feedback. The amplitude of the FA vibration is bounded to a few millimeters, and this implies the use of fine meshes and accurate numerical solvers due to the different scales of the problem. The results of the simulations show a main oscillation of the neutron flux with the same frequency as the FA vibration along with other harmonics at multiples of the vibration frequency much smaller in amplitude. Also, this work compares time domain analysis and frequency domain analysis of the mechanical vibrations. Numerical results show a close match between these two approaches for the fundamental frequency. [ABSTRACT FROM AUTHOR]

Different spatial modes can be defined for the neutron diffusion equation such as the λ , α and γ -modes. These modes have been successfully used for the analysis of nuclear reactor characteristics. In this work, these modes are studied using a high order finite element method to discretize the equations and also different methods to solve the resulting algebraic eigenproblems, are compared. Particularly, Krylov subspace methods and block-Newton methods have been studied. The performance of these methods has been tested in several 3D benchmark problems: a homogeneous reactor and several configurations of NEACRP reactor. [ABSTRACT FROM AUTHOR]

• This work presents the implementation of a frequency-domain finite element method in the FEMFFUSION code to simulate mechanical vibrations in hexagonal reactors. • The perturbations are introduced on the edges of the cells without the necessity of using locally refined meshes. • The proposed edge-wise method is tested against the usual cell-wise model in the frequency-domain (CORE SIM+) and time-domain methodologies (FEMFFUSION-TD and PARCS). • The proposed code have been verified against a one-dimensional homogeneous reactor, simplified two-dimensional rectangular and hexagonal reactors and a VVER-1000 reactor. The mechanical vibrations of fuel assemblies have shown to give high levels of neutron noise, triggering in some circumstances the necessity to operate nuclear reactors at a reduced power level. This behaviour can be modelled using the neutron noise diffusion approximation in the frequency-domain. This work presents an extension of the finite element method code FEMFFUSION, to simulate mechanical vibrations in hexagonal reactors in the frequency domain. This novel strategy in neutron noise simulation is based on introducing perturbations on the edges of the cells associated with the vibrating fuel assemblies, allowing to model the movement of these fuel assemblies accurately and efficiently, without the necessity of using locally refined meshes. Numerical results verify the edge-wise methodology in the frequency-domain against the usual cell-wise frequency-domain model and the time-domain model. The edge-wise frequency-domain methodology has also been compared to other neutronic codes, as CORE SIM+ and PARCS. [ABSTRACT FROM AUTHOR]

Lambda modes of a nuclear power reactor have interest in reactor physics since they have been used to develop modal methods and to study BWR reactor instabilities. An h-p-Adaptation finite element method has been implemented to compute the dominant modes the fundamental mode and the next subcritical modes of a nuclear reactor. The performance of this method has been studied in three benchmark problems, a homogeneous 2D reactor, the 2D BIBLIS reactor and the 3D IAEA reactor. [ABSTRACT FROM AUTHOR]

Abstract: P L equations are classical approximations to the neutron transport equations, which are obtained expanding the angular neutron flux in terms of spherical harmonics. These approximations are useful to study the behavior of reactor cores with complex fuel assemblies, for the homogenization of nuclear cross-sections, etc., and most of these applications are in three-dimensional (3D) geometries. In this work, we review the multi-dimensional P L equations and describe a nodal collocation method for the spatial discretization of these equations for arbitrary odd order L, which is based on the expansion of the spatial dependence of the fields in terms of orthonormal Legendre polynomials. The performance of the nodal collocation method is studied by means of obtaining the k eff and the stationary power distribution of several 3D benchmark problems. The solutions are obtained are compared with a finite element method and a Monte Carlo method. [Copyright &y& Elsevier]

CORTEX is a EU H2020 project (2017-2021) devoted to the analysis of 'reactor neutron noise' in nuclear reactors, i.e. the small fluctuations occurring around the stationary state due to external or internal disturbances in the core. One important aspect of CORTEX is the development of neutron noise simulation codes capable of modeling the spatial variations of the noise distribution in a reactor. In this paper we illustrate the validation activities concerning the comparison of the simulation results obtained by several noise simulation codes with respect to experimental data produced at the zero-power reactors AKR-2 (operated at TUD, Germany) and CROCUS (operated at EPFL, Switzerland). Both research reactors are modeled in the time and frequency domains, using transport or diffusion theory. Overall, the noise simulators managed to capture the main features of the neutron noise behavior observed in the experimental campaigns carried out in CROCUS and AKR-2, even though computational biases exist close to the region where the noise-inducing mechanical vibration was located (the so-called "noise source"). In some of the experiments, it was possible to observe the spatial variation of the relative neutron noise, even relatively far from the noise source. This was achieved through reduced uncertainties using long measurements, the installation of numerous, robust and efficient detectors at a variety of positions in the near vicinity or inside the core, as well as new post-processing methods. For the numerical simulation tools, modeling the spatial variations of the neutron noise behavior in zero-power research reactors is an extremely challenging problem, because of the small magnitude of the noise field; and because deviations from a point-kinetics behavior are most visible in portions of the core that are especially difficult to be precisely represented by simulation codes, such as experimental channels. Nonetheless the limitations of the simulation tools reported in the paper were not an issue for the CORTEX project, as most of the computational biases are found close to the noise source. • Validation activities with dedicated experimental data from zero-power reactors. • Modeling in the time and frequency domain using transport or diffusion theory. • The noise simulators capture the main features of the neutron noise behavior. • Computational biases exist close to the noise source. [ABSTRACT FROM AUTHOR]

• Neutron noise problems with vibrating pins in a nuclear fuel assembly are analyzed. • MC and deterministic transport methods are studied to generate reference solutions. • Diffusion-based solutions are assessed via higher-order transport solutions. Vibrations of fuel assemblies are an important issue in the safe operation of nuclear reactors, because they can challenge the integrity of the fuel with potential for radioactive releases. Reactor neutron noise-based techniques for monitoring vibrations are valuable for core diagnostic since they are not intrusive and make use of ordinary neutron flux measurements from ex-core and in-core detectors. The application of these techniques involves the solution of inverse problems that require numerical simulations capable of estimating the reactor neutron noise, given a model of the vibrations. For this purpose, several novel reactor neutron noise solvers have been developed in the CORTEX project using either Monte Carlo or deterministic methods, such as the discrete ordinates method, the method of characteristics, and the diffusion approximation. In the current work, these solvers have been scrutinized by computing the neutron noise induced by vibrations of one or multiple fuel pins in a simplified UOX fuel assembly benchmark, via proper variations of macroscopic neutron cross sections. The comparison of these neutron noise solutions obtained from the different methods shows novel insights into the simulation of neutron noise induced by mechanical vibrations, such as the challenges posed by the Monte Carlo method, the impact of the angular discretization on the application of the discrete ordinates method, and the accuracy of the diffusion approximation assessed via the higher-order neutron transport methods. [ABSTRACT FROM AUTHOR]