Abstract
In this paper, the Ronen method is developed, implemented, and applied to resolve the neutron flux and the criticality eigenvalue in simple one-dimensional homogeneous and heterogeneous problems. The Ronen method is based on iterative calculations of correction factors to use in a multigroup diffusion model, where the factors are actually given by the integral transport equation. In particular, spatially dependent diffusion constants are modified locally in order to reproduce new estimates of the surface currents obtained by the integral transport operator. The diffusion solver employed in this study uses finite differences, and the transport-corrected currents are introduced into the numerical scheme as drift terms. The corrected solutions are compared against reference results from a discrete ordinate code. The results match well with the reference solutions, especially in the limit of fine meshes, but slow convergence of the scalar flux is reported.
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References
J. Kim, Y. Kim, Development of 3-D HCMFD algorithm for efficient pin-by-pin reactor analysis. Ann. Nuclear Energy 127, 87–98 (2019)
W.R. Martin, Challenges and prospects for whole-core Monte Carlo analysis. Nuclear Eng. Technol. 44(2), 151–160 (2012)
K. Smith, Monte Carlo for practical LWR analysis: what’s needed to get to the goal? in American Nuclear Society Summer Meeting, M&C Division, Computational Roundtable, Hollywood, FL, June 27, 2011 (2011)
R.D. Lawrence, Progress in nodal methods for the solution of the neutron diffusion and transport equations. Prog. Nuclear Energy 17(3), 271–301 (1986)
K.S. Smith, Assembly homogenization techniques for light water reactor analysis. Prog. Nuclear Energy 17(3), 303–335 (1986)
F. Bertrand, N. Marie, G. Prulhière, J. Lecerf, J.M. Seiler, Comparison of the behaviour of two core designs for ASTRID in case of severe accidents. Nuclear Eng. Des. 297, 327–342 (2016)
IAEA, Best Estimate Safety Analysis for Nuclear Power Plants: Uncertainty Evaluation. Number 52 in Safety Reports Series (International Atomic Energy Agency, Vienna, 2008)
G.I. Bell, S. Glasstone, Nuclear Reactor Theory (Van Nostrand Reinhold, New York, 1970)
J.M. Pounders, F. Rahnema, On the diffusion coefficients for reactor physics applications. Nuclear Sci. Eng. 163, 243–262 (2009)
P.S. Brantley, E.W. Larsen, The simplified \(P_3\) approximation. Nuclear Sci. Eng. 134, 1–21 (2000)
D. Tomatis, A. Dall’Osso, Application of a numerical transport correction in diffusion calculations, in International Conference on Mathematics and Computational Methods Applied to Nuclear Science and Engineering (M&C 2011), Rio de Janeiro, RJ, Brazil, May 8–12, 2011, 05 (2011)
Y. Ronen, Accurate relations between the neutron current densities and the neutron fluxes. Nuclear Sci. Eng. 146(2), 245–247 (2004)
K.S. Smith, Nodal method storage reduction by nonlinear iteration. Trans. Am. Nuclear Soc. 44, 265–266 (1983)
B. Davison, J.B. Sykes, Neutron Transport Theory (Oxford University Press, Oxford, 1957)
K.M. Case, P.F. Zweifel, Linear Transport Theory (Addison-Wesley, Reading, 1967)
G.C. Pomraning, The Equations of Radiation Hydrodynamics (Pergamon Press, Oxford, 1973)
J.J. Duderstadt, W.R. Martin, Transport Theory (Wiley, New York, 1979)
E.E. Lewis, W.F. Miller, Computational Methods of Neutron Transport (Wiley, New York, 1984)
M. Abramowitz, I.A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover, New York, 1964)
Y.A. Cengel, M.N. Özişik, Integrals involving Legendre polynomials that arise in the solution of radiation transfer. J. Quant. Spectrosc. Radiat. Transf. 31(3), 215–219 (1984)
J.J. Settle, Some properties of an integral that occurs in problems of radiative transfer. J. Quant. Spectrosc. Radiat. Transf. 52(2), 195–206 (1994)
I.S. Gradshteyn, I.M. Ryzhik, Table of Integrals, Series, and Products, 7th edn. (Academic Press, London, 2007)
J.R. Lamarsh, Introduction to Nuclear Reactor Theory (Addison-Wesley, Reading, 1966)
F. Rahnema, E.M. Nichita, Leakage corrected spatial (assembly) homogenization technique. Ann. Nuclear Energy 24(6), 477–488 (1997)
Acknowledgements
The authors express their gratitude to Prof. Yigal Ronen who initiated the idea for this study. R.G. is partially supported by the Israel Ministry of Energy, Contract No. 216-11-008.
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Gross, R., Tomatis, D. & Gilad, E. High-accuracy neutron diffusion calculations based on integral transport theory. Eur. Phys. J. Plus 135, 235 (2020). https://doi.org/10.1140/epjp/s13360-020-00216-y
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DOI: https://doi.org/10.1140/epjp/s13360-020-00216-y