A Nodal-iterative Technique for Criticality Calculations in Multigroup Neutron Diffusion Models

R. Zanette, L. B. Barichello, C. Z. Petersen

Abstract


In this work, a nodal and iterative technique to evaluate the effective multiplication factor as well as the neutron flux, in multigroup diffusion problems, is presented. An iterative scheme, similar to the source iteration method, is implemented to decouple the system of differential equations which is the fundamental mathematical model. Then, analytical solutions are derived for the one-dimensional transverse integrated equations, of each energy group, resulting from a nodal approach. Constant approximations are assumed for the unknown transverse leakage terms in the contours of the nodes. In addition, constant and linear representations are investigated to express the fluxes in the source term to be updated in the iterative process.  Numerical results for the effective multiplication factor were obtained for a series of two-dimensional multigroup problems with up-scattering and down-scattering. The procedure is simple, fast, the analysis of the results indicated a satisfactory agreement with results available in the literature and the use of different approximations to the source term seems to be a good alternative, instead of using higher-order approximations on the contour of the nodes, to improve accuracy.

Keywords


multigroup neutron diffusion equation; source iteration method; nodal technique; effective multiplication factor

Full Text:

PDF

References


J. J. Duderstadt and L. J. Hamilton, Nuclear Reactor Analysis. New York: John Wiley, 1976.

W. M. Stacey, Nuclear Reactor Physics. New York: John Wiley & Sons, 2001.

M. Vagheian, D. R. Ochbelagh, and M. Gharib, “A new moving-mesh finite volume method for the efficient solution of two-dimensional neutron diffusion equation using gradient variations of reactor power,” Nuclear Engineering and Technology, vol. 51, no. 5, pp. 1181 – 1194, 2019.

Z. Chunyu and C. Gong, “Fast solution of neutron diffusion problem by reduced basis finite element method,” Annals of Nuclear Energy, vol. 111, pp. 702 – 708,

A. Bernal, J. E. Roman, R. Miró, and G. Verdú, “Calculation of multiple eigenvalues of the neutron diffusion equation discretized with a parallelized finite volume method,” Progress in Nuclear Energy, vol. 105, pp. 271–278, 2018.

S. A. Hosseini, “Development of galerkin finite element method threedimensional computational code for the multigroup neutron diffusion equation with unstructured tetrahedron elements,” Nuclear Engineering and Technology, vol. 48, no. 1, pp. 43–54, 2016.

M. Vagheian, N. Vosoughi, and M. Gharib, “Enhanced finite difference scheme for the neutron diffusion equation using the importance function,” Annals of Nuclear Energy, vol. 96, pp. 412 – 421, 2016.

W. Wu, Y. Yu, Q. Luo, D. Yao, Q. Li, and X. Chai, “Calculation of higher eigen-modes of the forward and adjoint neutron diffusion equations using iram algorithm based on domain decomposition,” Annals of Nuclear Energy, vol. 143, p. 107463, 2020.

A. C. Silva, A. S. Martinez, and A. C. Gonçalves, “Reconstruction of the neutron flux in a slab reactor,” World Journal of Nuclear Science and Technology, vol. 2, pp. 181–186, 2012.

X. Zhou and F. Li, “General nodal expansion method for multi-dimensional neutronics/thermal-hydraulic coupled problems in pebble-bed core systems,” Annals of Nuclear Energy, vol. 116, pp. 10 – 19, 2018.

N. Gupta, “Nodal methods for three-dimensional simulators,” Progress in Nuclear Energy, vol. 7, no. 3, pp. 127 – 149, 1981.

R. Lawrence, “Progress in nodal methods for the solution of the neutron diffusion and transport equations,” Progress in Nuclear Energy, vol. 17, no. 3, pp. 271 – 301, 1986.

R. Shober and A. Henry, Nonlinear methods for solving the diffusion equation. MIT Report NE-196, 1976.

K. S. Smith, An Analytic Nodal Method for Solving the Two-group, Multidimensional, Static and Transient Neutron Diffusion Equations. PhD thesis, Massachusetts Institute of Technology, Department of Nuclear Engineering, Cambridge, MA, 1979.

A. Hébert, “A simplified presentation of the multigroup analytic nodal method in 2-d cartesian geometry,” Annals of Nuclear Energy, vol. 35, no. 11, pp. 2142 – 2149, 2008.

A. Carreño, A. Vidal-Ferràndiz, D. Ginestar, and G. Verdú, “Block hybrid multilevel method to compute the dominant λ-modes of the neutron diffusion equation,” Annals of Nuclear Energy, vol. 121, pp. 513–524, 2018.

R. Zanette, C. Z. Petersen, M. Schramm, and J. R. Zabadal, “A modified power method for the multilayer multigroup two-dimensional neutron diffusion equation,” Annals of Nuclear Energy, vol. 111, pp. 136 – 140, 2018.

H. Sekimoto, Nuclear Reactor Theory. Tokyo: Tokyo Institute of Technology Press, 2007.

E. Muller and Z. Weiss, “Benchmarking with the multigroup diffusion highorder response matrix method,” Annals of Nuclear Energy, vol. 18, no. 9, pp. 535 – 544, 1991.




DOI: https://doi.org/10.5540/tcam.2022.023.02.00315

Article Metrics

Metrics Loading ...

Metrics powered by PLOS ALM

Refbacks

  • There are currently no refbacks.



Trends in Computational and Applied Mathematics

A publication of the Brazilian Society of Applied and Computational Mathematics (SBMAC)

 

Indexed in:

                       

 

Desenvolvido por:

Logomarca da Lepidus Tecnologia