Modal decomposition method for BWR stability analysis using Alpha-modes

[EN] In-phase and out-of-phase oscillations have been observed in BWR reactors. To improve the safety of these reactors it is necessary to detect in a reliable way these oscillations from the neutronic signals. In this paper, a methodology to decompose the neutronic signals in its modal amplitudes is proposed. Usually, to compute this decomposition the Lambda eigenfunctions are used as expansion functions and their adjoint modes are used as weight factors. Different approaches using the Alpha modes are investigated to obtain the LPRM signals modal decomposition for a BWR unstability event. The calculation of Alpha eigenmodes is reviewed and the oscillation parameters for the modal decomposition of the neutronic signals from Ringhals NPP have been calculated.This work has been partially supported by the Spanish Ministerio de Educación y Ciencia under Project ENE2011-22823.Verdú Martín, GJ.; Ginestar Peiro, D. (2014). Modal decomposition method for BWR stability analysis using Alpha-modes. Annals of Nuclear Energy. 67:31-40. https://doi.org/10.1016/j.anucene.2013.07.035S31406

Preconditioning the solution of the time- dependent neutron diffusion equation by recycling Krylov subspaces

[EN] Spectral preconditioners are based on the fact that the convergence rate of Krylov subspace methods is improved if the eigenvalues of smallest magnitude of the system matrix are `removed'. In this paper, two preconditioning strategies are studied to solve a set of linear systems associated with the numerical integration of the time dependent neutron di usion equation. Both strategies can be implemented using the matrix-vector product as the main operation and succeed at reducing the total number of iterations needed to solve the set of systems.This work has been partially supported by the Spanish Ministerio de Educacion y Ciencia under projects MTM2010-18674 and ENE2011-22823.González Pintor, S.; Ginestar Peiro, D.; Verdú Martín, GJ. (2014). Preconditioning the solution of the time- dependent neutron diffusion equation by recycling Krylov subspaces. International Journal of Computer Mathematics. 91(1):42-52. https://doi.org/10.1080/00207160.2013.771181S425291

Using a simple model for water and nitrogentransport in soil in the practical works ofHorticulture

[EN] Simulation models for the transport of water and nitrogen in the soil are useful tools for evaluating the consequen-ces of di erent crop management practices without waiting until the end of the crop cycle. Particularly, simplecompartmental models for the transport of water and nitrogen are based on the solution of mass balance equationsfor the water and a system of ordinary di erential equations for the nitrogen cycle. In this work, these modelsare used to evaluate di erent possible scenarios of irrigation and nitrogen fertilization in a cauli ower crop, as apractical work for the students of Horticulture course. The results show that di erent irrigation and fertilizationstrategies lead to di erent results for the nitrate leaching, water drainage and nitrogen uptake[ES] Los modelos de simulación del transporte de agua y nitrógeno en el suelo son herramientas útiles para evaluar distintas prácticas de cultivo sin necesidad de tener que esperar a que se complete el ciclo de crecimiento. En concreto, los modelos compartimentales sencillos para el agua y el nitrógeno en el suelo se basan en ecuaciones de balance de masa y en la resolución de un sistema de ecuaciones diferenciales ordinarias. Se plantea la posibilidad de utilizar este tipo de modelos en las prácticas de la asignatura Horticultura, evaluando distintos escenarios posibles de riego y abonado en un cultivo de coliflor. Los resultados muestran que distintas prácticas de cultivo producen diferencias en la lixiviación de nitrato y en la extracción de nitrógeno por el cultivo.Lidón, A.; Ginestar, D.; Gómez De Barreda, D. (2017). Utilizacion de un modelo simple de transporte de agua y nitrógeno en el suelo en las practicas de Horticultura. Modelling in Science Education and Learning. 10(2):21-34. doi:10.4995/msel.2017.6600SWORD2134102Contreras, W. A., Lidón, A. L., Ginestar, D., & Bru, R. (2009). Compartmental model for nitrogen dynamics in citrus orchards. Mathematical and Computer Modelling, 50(5-6), 794-805. doi:10.1016/j.mcm.2009.05.008Lidón, A., Ramos, C., Ginestar, D., & Contreras, W. (2013). Assessment of LEACHN and a simple compartmental model to simulate nitrogen dynamics in citrus orchards. Agricultural Water Management, 121, 42-53. doi:10.1016/j.agwat.2013.01.008Cannavo, P., Recous, S., Parnaudeau, V., & Reau, R. (2008). Modeling N Dynamics to Assess Environmental Impacts of Cropped Soils. Advances in Agronomy Volume 97, 131-174. doi:10.1016/s0065-2113(07)00004-1Lidón, A., Ramos, C., & Rodrigo, A. (1999). Comparison of drainage estimation methods in irrigated citrus orchards. Irrigation Science, 19(1), 25-36. doi:10.1007/s002710050068Porporato, A., D’Odorico, P., Laio, F., & Rodriguez-Iturbe, I. (2003). Hydrologic controls on soil carbon and nitrogen cycles. I. Modeling scheme. Advances in Water Resources, 26(1), 45-58. doi:10.1016/s0309-1708(02)00094-5Wang, L., D’Odorico, P., Manzoni, S., Porporato, A., & Macko, S. (2009). Soil carbon and nitrogen dynamics in southern African savannas: the effect of vegetation-induced patch-scale heterogeneities and large scale rainfall gradients. Climatic Change, 94(1-2), 63-76. doi:10.1007/s10584-009-9548-

Solution of the Lambda modes problem of a nuclear power reactor using an h-p finite element method

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.This work has been partially supported by the Spanish Ministerio de Ciencia e Innovacion under project ENE2011-22823, the Generalitat Valenciana under projects PROMETEO/2010/039 and ACOMP/2013/237, and the Universitat Politecnica de Valencia under project UPPTE/2012/118.Vidal Ferràndiz, A.; Fayez Moustafa Moawad, R.; Ginestar Peiro, D.; Verdú Martín, GJ. (2014). Solution of the Lambda modes problem of a nuclear power reactor using an h-p finite element method. Annals of Nuclear Energy. 72:338-349. https://doi.org/10.1016/j.anucene.2014.05.026S3383497

Utilización de un modelo compartimental de transporte de pesticidas en las prácticas de Química Agrícola y Ambiental

[EN] Computational models are useful tools in most of the knowledge areas and can be easily used to visualize the behaviour of complex systems. In this work, we present a simple com- partmental model to simulate the dynamics of the concentration of a given pesticide in a rice crop. The model considers different compartments such as: air, water, rice plants and sediments, which are coupled with a soil column modelled by means of a convection-advection equation. The model has been implemented in MatlabQc and has a Graphical User Interface (Gui) that allows to change easily the value of different parameters of the model and to obtain graphical representations of the results. Different possible applications of the model for a practical class of the subject Environmental and Agricultural Chemistry are proposed.[ES] Los modelos computacionales son herramientas útiles en la mayoría de las áreas de conocimiento y pueden ser fácilmente utilizados para visualizar el comportamiento de sistemas complejos. En este trabajo se presenta un modelo sencillo de compartimentos para simular la dinámica de la concentración de un plaguicida en un cultivo de arroz. En el modelo se consideran diferentes compartimentos, tales como: aire, agua, plantas de arroz y sedimentos al que se acopla una columna de suelo mediante una ecuación de convección-advección. El modelo ha sido implementado en MatlabQc y tiene una interfaz gráfica de usuario (Gui) que permite cambiar fácilmente el valor de los diferentes parámetros del modelo y obtener representaciones gráficas de los resultados. Se proponen también diferentes posibilidades de aplicación del modelo para una clase práctica de Química Ambiental y Agrícola.Contreras, W.; Bautista-Carrascosa, I.; Lidón, A.; Ginestar, D. (2011). Utilización de un modelo compartimental de transporte de pesticidas en las prácticas de Química Agrícola y Ambiental. Modelling in Science Education and Learning. 4:147-157. doi:10.4995/msel.2011.3068SWORD147157

Updating the Lambda modes of a nuclear power reactor

[EN] 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. (C) 2010 Elsevier Ltd. All rights reserved.This work has been partially supported by the Spanish Ministerio de Educacion y Ciencia under projects ENE2008-02669 and MTM2007-64477-AR07, the Generalitat Valenciana under project ACOMP/2009/058, and the Universidad Politecnica de Valencia under project PAID-05-09-4285.González Pintor, S.; Ginestar Peiro, D.; Verdú Martín, GJ. (2011). Updating the Lambda modes of a nuclear power reactor. Mathematical and Computer Modelling. 54(7):1796-1801. https://doi.org/10.1016/j.mcm.2010.12.013S1796180154

Moving meshes to solve the time-dependent neutron diffusion equation in hexagonal geometry

To simulate the behaviour of a nuclear power reactor it is necessary to be able to integrate the time-dependent neutron diffusion equation inside the reactor core. Here the spatial discretization of this equation is done using a finite element method that permits h-p refinements for different geometries. This means that the accuracy of the solution can be improved refining the spatial mesh (h-refinement) and also increasing the degree of the polynomial expansions used in the finite element method (p-refinement). Transients involving the movement of the control rod banks have the problem known as the rod-cusping effect. Previous studies have usually approached the problem using a fixed mesh scheme defining averaged material properties. The present work proposes the use of a moving mesh scheme that uses spatial meshes that change with the movement of the control rods avoiding the necessity of using equivalent material cross sections for the partially inserted cells. The performance of the moving mesh scheme is tested studying one-dimensional and three-dimensional benchmark problems. (C) 2015 Elsevier B.V. All rights reserved.This work has been partially supported by the Spanish Ministerio de Ciencia e Innovacion under project ENE2011-22823, the Generalitat Valenciana under projects II/2014/08 and ACOMP/2013/237, and the Universitat Politecnica de Valencia under project UPPTE/2012/118.Vidal-Ferràndiz, A.; Fayez Moustafa Moawad, R.; Ginestar Peiro, D.; Verdú Martín, GJ. (2016). Moving meshes to solve the time-dependent neutron diffusion equation in hexagonal geometry. Journal of Computational and Applied Mathematics. 291:197-208. https://doi.org/10.1016/j.cam.2015.03.040S19720829

A time and frequency domain analysis of the effect of vibrating fuel assemblies on the neutron noise

[EN] The mechanical vibrations of fuel assemblies have been shown to give rise to high levels of neutron noise, triggering in some circumstances the necessity to operate nuclear reactors at a reduced power level. This work analyses the effect in the neutron field of the oscillation of one single fuel assembly. Results show two different effects in the neutron field caused by the fuel assembly vibration. First, a global slow variation of the total reactor power due to a change in the criticality of the system. Second, an oscillation in the neutron flux in-phase with the assembly vibration. This second effect has a strong spatial dependence that can be used to localize the oscillating assembly. This paper shows a comparison between a time-domain and a frequency-domain analysis of the phenomena to calculate the spatial response of the neutron noise. Numerical results show a really close agreement between these two approaches.This project has received funding from the Euratom research and training programme 2014-2018 under grant agreement No 754316. Also, this work has been partially supported by Spanish Ministerio de Economia y Competitividad under project BES-2015-072901 and financed with the help of a Primeros Proyectos de Investigation (PAID-06-18), Vicerrectorado de Investigacitin, Innovation y Transferencia of the Universitat Politecnica de Valencia (UPV).Vidal-Ferràndiz, A.; Carreño, A.; Ginestar Peiro, D.; Demazière, C.; Verdú Martín, GJ. (2020). A time and frequency domain analysis of the effect of vibrating fuel assemblies on the neutron noise. Annals of Nuclear Energy. 137:1-12. https://doi.org/10.1016/j.anucene.2019.107076S112137Akcasu, Z. (1958). General Solution of the Reactor Kinetic Equations without Feedback. Nuclear Science and Engineering, 3(4), 456-467. doi:10.13182/nse58-a25482Antonopoulos-Domis, M. (1976). Reactivity and neutron density noise excited by random rod vibration. Annals of Nuclear Energy, 3(9-10), 451-459. doi:10.1016/0306-4549(76)90030-xDemaziere, C. (2006). Analysis methods for the determination of possible unseated fuel assemblies in BWRs. International Journal of Nuclear Energy Science and Technology, 2(3), 167. doi:10.1504/ijnest.2006.010713Demazière, C. (2011). CORE SIM: A multi-purpose neutronic tool for research and education. Annals of Nuclear Energy, 38(12), 2698-2718. doi:10.1016/j.anucene.2011.06.010Demazière, C., & Andhill, G. (2005). Identification and localization of absorbers of variable strength in nuclear reactors. Annals of Nuclear Energy, 32(8), 812-842. doi:10.1016/j.anucene.2004.12.011Demazière, C., Dykin, V., & Jareteg, K. (2017). Development of a point-kinetic verification scheme for nuclear reactor applications. Journal of Computational Physics, 339, 396-411. doi:10.1016/j.jcp.2017.03.020Demazière, C., & Pázsit, I. (2009). Numerical tools applied to power reactor noise analysis. Progress in Nuclear Energy, 51(1), 67-81. doi:10.1016/j.pnucene.2008.01.010Ginestar, D., Verdú, G., Vidal, V., Bru, R., Marín, J., & Muñoz-Cobo, J. L. (1998). High order backward discretization of the neutron diffusion equation. Annals of Nuclear Energy, 25(1-3), 47-64. doi:10.1016/s0306-4549(97)00046-7Hébert, A. (1985). Application of the Hermite Method for Finite Element Reactor Calculations. Nuclear Science and Engineering, 91(1), 34-58. doi:10.13182/nse85-a17127Jonsson, A., Tran, H. N., Dykin, V., & Pázsit, I. (2012). Analytical investigation of the properties of the neutron noise induced by vibrating absorber and fuel rods. Kerntechnik, 77(5), 371-380. doi:10.3139/124.110258Kronbichler, M., & Kormann, K. (2012). A generic interface for parallel cell-based finite element operator application. Computers & Fluids, 63, 135-147. doi:10.1016/j.compfluid.2012.04.012Larsson, V., & Demazière, C. (2009). Comparative study of 2-group and diffusion theories for the calculation of the neutron noise in 1D 2-region systems. Annals of Nuclear Energy, 36(10), 1574-1587. doi:10.1016/j.anucene.2009.07.009Olmo-Juan, N., Demazière, C., Barrachina, T., Miró, R., & Verdú, G. (2019). PARCS vs CORE SIM neutron noise simulations. Progress in Nuclear Energy, 115, 169-180. doi:10.1016/j.pnucene.2019.03.041Park, J., Lee, J. H., Kim, T.-R., Park, J.-B., Lee, S. K., & Koo, I.-S. (2003). Identification of reactor internals’ vibration modes of a Korean standard PWR using structural modeling and neutron noise analysis. Progress in Nuclear Energy, 43(1-4), 177-186. doi:10.1016/s0149-1970(03)00021-0Pázsit, I. (1988). Control-rod models and vibration induced noise. Annals of Nuclear Energy, 15(7), 333-346. doi:10.1016/0306-4549(88)90081-3Pázsit, I., & Th.Analytis, G. (1980). Theoretical investigation of the neutron noise diagnostics of two-dimensional control rod vibrations in a PWR. Annals of Nuclear Energy, 7(3), 171-183. doi:10.1016/0306-4549(80)90082-1Pázsit, I., & Glöckler, O. (1983). On the Neutron Noise Diagnostics of Pressurized Water Reactor Control Rod Vibrations. I. Periodic Vibrations. Nuclear Science and Engineering, 85(2), 167-177. doi:10.13182/nse83-a27424Ravetto, P. (1997). Reactivity oscillations in a point reactor. Annals of Nuclear Energy, 24(4), 303-314. doi:10.1016/s0306-4549(96)00066-7Sunde, C., Demazière, C., & Pázsit, I. (2006). Calculation of the Neutron Noise Induced by Shell-Mode Core-Barrel Vibrations in a 1-D, Two-Group, Two-Region Slab Reactor Model. Nuclear Technology, 154(2), 129-141. doi:10.13182/nt06-1Tran, H.-N., Pázsit, I., & Nylén, H. (2015). Investigation of the ex-core noise induced by fuel assembly vibrations in the Ringhals-3 PWR. Annals of Nuclear Energy, 80, 434-446. doi:10.1016/j.anucene.2015.01.045Vidal-Ferràndiz, A., Carreño, A., Ginestar, D., & Verdú, G. (2019). A Block Arnoldi Method for the SPN Equations. International Journal of Computer Mathematics, 1-22. doi:10.1080/00207160.2019.1602768Vidal-Ferrandiz, A., Fayez, R., Ginestar, D., & Verdú, G. (2014). Solution of the Lambda modes problem of a nuclear power reactor using an h–p finite element method. Annals of Nuclear Energy, 72, 338-349. doi:10.1016/j.anucene.2014.05.026Vidal-Ferràndiz, A., Fayez, R., Ginestar, D., & Verdú, G. (2016). Moving meshes to solve the time-dependent neutron diffusion equation in hexagonal geometry. Journal of Computational and Applied Mathematics, 291, 197-208. doi:10.1016/j.cam.2015.03.040Viebach, M., Bernt, N., Lange, C., Hennig, D., & Hurtado, A. (2018). On the influence of dynamical fuel assembly deflections on the neutron noise level. Progress in Nuclear Energy, 104, 32-46. doi:10.1016/j.pnucene.2017.08.010Weinberg, A. M., & Schweinler, H. C. (1948). Theory of Oscillating Absorber in a Chain Reactor. Physical Review, 74(8), 851-863. doi:10.1103/physrev.74.85

Nodal collocation method for the multidimensional PL equations applied to neutron transport source problems

A PL spherical harmonics-nodal collocation method is applied to the solution of the multidimensional neutron source transport equation. Vacuum boundary conditions are approximated by setting Marshak's conditions. The method is applied to several 1D, 2D and 3D problems with isotropic fixed source and with isotropic and anisotropic scattering. These problems are chosen to test this method in limit conditions, showing that in some cases a high order $P_L$ approximation is required to obtain accurate results and convergence. Results are also compared with the ones provided by several reference codes showing good agreement. It is also shown that Marshak's approximation to vacuum boundary conditions gives the same results that simulating vacuum with a purely absorbing medium and setting zero flux boundary conditions.This work has been partially supported by the Spanish Ministerio de Economia y Competitividad under project ENE2011-22823, and the Generalitat Valenciana under project PROMETEO11/2014/008.Capilla Romá, MT.; Talavera Usano, CF.; Ginestar Peiro, D.; Verdú Martín, GJ. (2016). Nodal collocation method for the multidimensional PL equations applied to neutron transport source problems. Annals of Nuclear Energy. 87:89-100. https://doi.org/10.1016/j.anucene.2015.07.040S891008

Schwarz type preconditioners for the neutron diffusion equation

[EN] Domain decomposition is a mature methodology that has been used to accelerate the convergence of partial differential equations. Even if it was devised as a solver by itself, it is usually employed together with Krylov iterative methods improving its rate of convergence, and providing scalability with respect to the size of the problem. In this work, a high order finite element discretization of the neutron diffusion equation is considered. In this problem the preconditioning of large and sparse linear systems arising from a source driven formulation becomes necessary due to the complexity of the problem. On the other hand, preconditioners based on an incomplete factorization are very expensive from the point of view of memory requirements. The acceleration of the neutron diffusion equation is thus studied here by using alternative preconditioners based on domain decomposition techniques inside Schur complement methodology. The study considers substructuring preconditioners, which do not involve overlapping, and additive Schwarz preconditioners, where some overlapping between the subdomains is taken into account. The performance of the different approaches is studied numerically using two-dimensional and three-dimensional problems. It is shown that some of the proposed methodologies outperform incomplete LU factorization for preconditioning as long as the linear system to be solved is large enough, as it occurs for three-dimensional problems. They also outperform classical diagonal Jacobi preconditioners, as long as the number of systems to be solved is large enough in such a way that the overhead of building the pre-conditioner is less than the improvement in the convergence rate. (C) 2016 Elsevier B.V. All rights reserved.The work has been partially supported by the spanish Ministerio de Economía y Competitividad under projects ENE 2014-59442-P and MTM2014-58159-P, the Generalitat Valenciana under the project PROMETEO II/2014/008 and the Universitat Politècnica de València under the project FPI-2013. The work has also been supported partially by the Swedish Research Council (VR-Vetenskapsrådet) within a framework grant called DREAM4SAFER, research contract C0467701.Vidal-Ferràndiz, A.; González Pintor, S.; Ginestar Peiro, D.; Verdú Martín, GJ.; Demazière, C. (2017). Schwarz type preconditioners for the neutron diffusion equation. Journal of Computational and Applied Mathematics. 309:563-574. https://doi.org/10.1016/j.cam.2016.02.056S56357430