Nuclear Engineering and Technology

Korean Nuclear Society (한국원자력학회)
권호 표지
DOI QR코드

DOI QR Code

Time-dependent simplified spherical harmonics formulations for a nuclear reactor system

  • Carreno, A. (Instituto Universitario de Seguridad Industrial, Radiofisica y Medioambiental, Universitat Politecnica de Valencia) ;
  • Vidal-Ferrandiz, A. (Instituto Universitario de Matematica Multidisciplinar, Universitat Politecnica de Valencia) ;
  • Ginestar, D. (Instituto Universitario de Matematica Multidisciplinar, Universitat Politecnica de Valencia) ;
  • Verdu, G. (Instituto Universitario de Seguridad Industrial, Radiofisica y Medioambiental, Universitat Politecnica de Valencia)
  • Received : 2021.02.05
  • Accepted : 2021.06.07
  • Published : 2021.12.25
Download PDF
⟨ Previous Next ⟩

Abstract

The steady-state simplified spherical harmonics equations (SPN equations) are a higher order approximation to the neutron transport equations than the neutron diffusion equation that also have reasonable computational demands. This work extends these results for the analysis of transients by comparing of two formulations of time-dependent SPN equations considering different treatments for the time derivatives of the field moments. The first is the full system of equations and the second is a diffusive approximation of these equations that neglects the time derivatives of the odd moments. The spatial discretization of these methodologies is made by using a high order finite element method. For the time discretization, a semi-implicit Euler method is used. Numerical results show that the diffusive formulation for the time-dependent simplified spherical harmonics equations does not present a relevant loss of accuracy while being more computationally efficient than the full system.

Keywords

Acknowledgement

This work has been partially supported by Spanish Ministerio de Economia y Competitividad under projects ENE2017-89029-P and MTM2017-85669-P. Furthermore, this work has been financed by the Generalitat Valenciana under the project PROMETEO/2018/035.

References

  1. W.M. Stacey, Nuclear Reactor Physics, Wiley, Weinheim, Germany, 2007, https://doi.org/10.1002/9783527611041.
  2. B. Sjenitzer, J. Hoogenboom, Dynamic Monte Carlo method for nuclear reactor kinetics calculations, Nucl. Sci. Eng. 175 (2013) 94-107. https://doi.org/10.13182/NSE12-44
  3. N. Shaukat, M. Ryu, H. Shim, Dynamic Monte Carlo transient analysis for the organization for economic co-operation and development nuclear energy agency (OECD/NEA) C5G7-TD benchmark, Nuclear Engineering and Technology 49 (2017) 920-927. https://doi.org/10.1016/j.net.2017.04.008
  4. M. Mazaher, A. Salehi, N. Vosoughi, A time dependent Monte Carlo approach for nuclear reactor analysis in a 3D arbitrary geometry, Prog. Nucl. Energy 115 (2019) 80-90. https://doi.org/10.1016/j.pnucene.2019.03.024
  5. A. Vidal-Ferrandiz, R. Fayez, D. Ginestar, G. Verdu, Solution of the lambda modes problem of a nuclear power reactor using an h-p finite element method, Ann. Nucl. Energy 72 (2014) 338-349. https://doi.org/10.1016/j.anucene.2014.05.026
  6. A. Avvakumov, V. Strizhov, P. Vabishchevich, A. Vasilev, Numerical Modeling of Neutron Transport in SP3 Approximation by Finite Element Method, 2019 arXiv preprint arXiv:1903.11502.
  7. K. Ivanov, M. Manolova, T. Apostolov, An effective solution scheme of a three-dimensional reactor core model in hexagonal geometry, Comput. Phys. Commun. 82 (1994) 1-16. https://doi.org/10.1016/0010-4655(94)90126-0
  8. M. Capilla, C. Talavera, D. Ginestar, G. Verdu, Numerical analysis of the 2D C5G7 MOX benchmark using PL equations and a nodal collocation method, Ann. Nucl. Energy 114 (2018) 32-41. https://doi.org/10.1016/j.anucene.2017.12.002
  9. A. Carreno, A. Vidal-Ferrandiz, D. Ginestar, G. Verdu, Spatial modes for the neutron diffusion equation and their computation, Ann. Nucl. Energy 110 (2017) 1010-1022. https://doi.org/10.1016/j.anucene.2017.08.018
  10. W. Stacey, Space-time Nuclear Reactor Kinetics, vol. 5, Academic Press, 1969.
  11. D. Ginestar, G. Verdu, V. Vidal, R. Bru, J. Marin, J. Munoz-Cobo, High order backward discretization of the neutron diffusion equation, Ann. Nucl. Energy 25 (1998) 47-64. https://doi.org/10.1016/S0306-4549(97)00046-7
  12. K. Ott, Quasistatic treatment of spatial phenomena in reactor dynamics, Nucl. Sci. Eng. 26 (1966) 563-565. https://doi.org/10.13182/NSE66-A18428
  13. S. Dulla, E. Mund, P. Ravetto, The quasi-static method revisited, Prog. Nucl. Energy 50 (2008) 908-920. https://doi.org/10.1016/j.pnucene.2008.04.009
  14. R. Miro, D. Ginestar, G. Verdu, D. Hennig, A nodal modal method for the neutron diffusion equation. Application to BWR instabilities analysis, Ann. Nucl. Energy 29 (2002) 1171-1194. https://doi.org/10.1016/S0306-4549(01)00103-7
  15. A. Carreno, A. Vidal-Ferrandiz, D. Ginestar, G. Verdu, Adaptive Time-step Control for Modal Methods to Integrate the Neutron Diffusion Equation, Nuclear Engineering and Technology, 2020.
  16. B. Ganapol, A more efficient implementation of the discrete-ordinates method for an approximate model of particle transport in a duct, Ann. Nucl. Energy 86 (2015) 13-22. https://doi.org/10.1016/j.anucene.2015.01.015
  17. Y. Chen, B. Zhang, L. Zhang, J. Zheng, Y. Zheng, C. Liu, ARES: a Parallel Discrete Ordinates Transport Code for Radiation Shielding Applications and Reactor Physics Analysis, Science and Technology of Nuclear Installations, 2017. 2017.
  18. Y. Ma, Y. Wang, J. Yang, ntkFoam: an OpenFOAM based neutron transport kinetics solver for nuclear reactor simulation, Comput. Math. Appl. 81 (2021) 512-531, https://doi.org/10.1016/j.camwa.2019.09.015.
  19. J. Fletcher, A solution of the neutron transport equation using spherical harmonics, J. Phys. Math. Gen. 16 (1983) 2827. https://doi.org/10.1088/0305-4470/16/12/028
  20. M. Capilla, C. Talavera, D. Ginestar, G. Verdu, Validation of the SHNC time-dependent transport code based on the spherical harmonics method for complex nuclear fuel assemblies, J. Comput. Appl. Math. (2020) 112814.
  21. R. Harel, S. Burov, S. Heizler, The Time-dependent Asymptotic PN Approximation for the Transport Equation, 2020, p. 1, arXiv preprint arXiv: 2006.11784.
  22. M. Halsall, CACTUS, a Characteristics Solution to the Neutron Transport Equations in Complicated Geometries, Technical Report UKAEA Atomic Energy Establishment, 1980.
  23. J. Yan, B. Kochunas, M. Hursin, T. Downar, Z. Karoutas, E. Baglietto, Coupled computational fluid dynamics and MOC neutronic simulations of Westinghouse PWR fuel assemblies with grid spacers, in: The 14th International Topical Meeting on Nuclear Reactor Thermalhydraulics (NURETH-14), 2011.
  24. F. Alexander, A. Almgren, J. Bell, A. Bhattacharjee, J. Chen, P. Colella, D. Daniel, J. DeSlippe, L. Diachin, E. Draeger, et al., Exascale applications: skin in the game, Philosophical Transactions of the Royal Society A 378 (2020) 20190056. https://doi.org/10.1098/rsta.2019.0056
  25. E. Gelbard, Application of Spherical Harmonics Method to Reactor Problems, Bettis Atomic Power Laboratory, West Mifflin, PA, 1960. Technical Report No. WAPD-BT-20.
  26. G. Pomraning, Asymptotic and variational derivations of the simplified Pn equations, Ann. Nucl. Energy 20 (1993) 623-637. https://doi.org/10.1016/0306-4549(93)90030-S
  27. E. Larsen, J. Morel, J. McGhee, Asymptotic derivation of the simplified PN equations, in: Proceedings of the Joint International Conference on Mathematical Methods and Supercomputing in Nuclear Applications, vol. 1, 1993, p. 718.
  28. R. McClarren, Theoretical aspects of the simplified PN equations, Transport Theor. Stat. Phys. 39 (2010) 73-109. https://doi.org/10.1080/00411450.2010.535088
  29. E. Larsen, J. Morel, J. McGhee, Asymptotic derivation of the multigroup p1 and simplified Pn equations with anisotropic scattering, Nucl. Sci. Eng. 123 (1996) 328-342. https://doi.org/10.13182/NSE123-328
  30. A. Klose, E. Larsen, Light transport in biological tissue based on the simplified spherical harmonics equations, J. Comput. Phys. 220 (2006) 441-470. https://doi.org/10.1016/j.jcp.2006.07.007
  31. A. Vidal-Ferr andiz, A. Carreno, D. Ginestar, G. Verdu, A block arnoldi method for the sPN equations, Int. J. Comput. Math. 97 (2020b) 341-357.
  32. M. Altahhan, M. Nagy, H. Abou-Gabal, A. Aboanber, Formulation of a point reactor kinetics model based on the neutron telegraph equation, Ann. Nucl. Energy 91 (2016) 176-188. https://doi.org/10.1016/j.anucene.2016.01.011
  33. M. Altahhan, A. Aboanber, H. Abou-Gabal, M.S. Nagy, Response of the point-reactor telegraph kinetics to time varying reactivities, Prog. Nucl. Energy 98 (2017) 109-122. https://doi.org/10.1016/j.pnucene.2017.03.008
  34. A. Baudron, J. Lautard, Simplified PN transport core calculations in the Apollo3 system, in: International Conference on Mathematics and Computational Methods Applied to Nuclear Science and Engineering (M&C 2011). Latin American Section, American Nuclear Society, 2011.
  35. D. Lee, T. Kozlowski, T. Downar, Multi-group SP3 approximation for simulation of a three-dimensional PWR rod ejection accident, Ann. Nucl. Energy 77 (2015) 94-100. https://doi.org/10.1016/j.anucene.2014.10.019
  36. C. Hauck, R. McClarren, Positive PN closures, SIAM J. Sci. Comput. 32 (2010) 2603-2626. https://doi.org/10.1137/090764918
  37. S. Hamilton, T. Evans, Efficient solution of the simplified PN equations, J. Comput. Phys. 284 (2015) 155-170. https://doi.org/10.1016/j.jcp.2014.12.014
  38. O. Zienkiewicz, R. Taylor, P. Nithiarasu, J. Zhu, The Finite Element Method, vol. 3, McGraw-hill, London, 1977.
  39. V. Hernandez, J. Roman, V. V, SLEPc: a scalable and flexible toolkit for the solution of eigenvalue problems, ACM Trans. Math Software 31 (2005) 351-362. https://doi.org/10.1145/1089014.1089019
  40. R. Morgan, D. Scott, Generalizations of Davidson's method for computing eigenvalues of sparse symmetric matrices, SIAM J. Sci. Stat. Comput. 7 (1986) 817-825. https://doi.org/10.1137/0907054
  41. S. Balay, S. Abhyankar, M. Adams, J. Brown, P. Brune, K. Buschelman, L. Dalcin, A. Dener, V. Eijkhout, W. Gropp, et al., PETSc Users Manual, Argonne National Laboratory, 2019.
  42. W. Bangerth, R. Hartmann, G. Kanschat, deal.II - a general purpose object oriented finite element library, ACM Trans. Math Software 33 (2007), 24/1-24/27.
  43. D. Ropp, J. Shadid, Stability of operator splitting methods for systems with indefinite operators: Advection-diffusion-reaction systems, J. Comput. Phys. 228 (2009) 3508-3516. https://doi.org/10.1016/j.jcp.2009.02.001
  44. M. Capilla, C. Talavera, D. Ginestar, G. Verdu, A nodal collocation method for the calculation of the lambda modes of the pl equations, Ann. Nucl. Energy 32 (2005) 1825-1853. https://doi.org/10.1016/j.anucene.2005.07.004
  45. G. Niederauer, Neutron Kinetics Based on the Equation of Telegraphy, Ph.D. thesis, Iowa State University, 1967.
  46. A. Vidal-Ferr andiz, A. Carreno, D. Ginestar, G. Verdu, FEMFFUSION: A Finite Element Method Code for the Neutron Diffusion Equation, 2020. https://www.femffusion.imm.upv.es.
  47. L. Hageman, J. Yasinsky, Comparison of alternating-direction time-differencing methods with other implicit methods for the solution of the neutron group-diffusion equations, Nucl. Sci. Eng. 38 (1969) 8-32. https://doi.org/10.13182/NSE38-8
  48. K. Smith, An Analytic Nodal Method for Solving the Two-Group, Multidimensional, Static and Transient Neutron Diffusion Equations, Ph.D. thesis, Massachusetts Institute of Technology, 1979.
  49. M. Capilla, D. Ginestar, G. Verdu, Applications of the multidimensional PL equations to complex fuel assembly problems, Ann. Nucl. Energy 36 (2009) 1624-1634. https://doi.org/10.1016/j.anucene.2009.08.008
  50. C. Cavarec, J. Perron, D. Verwaerde, J. West, Benchmark Calculations of Power Distribution within Assemblies, Technical Report Nuclear Energy Agency, 1994.
  51. V. Boyarinov, P. Fomichenko, J. Hou, K. Ivanov, A. Aures, W. Zwermann, K. Velkov, Deterministic Time-dependent Neutron Transport Benchmark without Spatial Homogenization (C5G7-TD), Nuclear Energy Agency Organisation for Economic Co-operation and Development (NEA-OECD), Paris, France, 2016.
  52. T. Downar, Y. Xu, T. Kozlowski, D. Carlson, PARCS V2. 7 US NRC Core Neutronics Simulator User Manual, Purdue University, 2006.
  53. A. Vidal-Ferr andiz, A. Carreno, D. Ginestar, C. Demaziere, G. Verdu, A time and frequency domain analysis of the effect of vibrating fuel assemblies on the neutron noise, Ann. Nucl. Energy 137 (2020a) 107076. https://doi.org/10.1016/j.anucene.2019.107076