Abstract.
The spatial asymptotic approach is used to derive expressions for the diffusion coefficient from the neutron transport equation to improve the standard P1-based formula. A general explicit analytical expression is obtained in the frame of the PN model, which also allows an easy evaluation of the coefficient in the limit for \(N\rightarrow\infty\). The analytical formulae are presented at any order of the scattering anisotropy. An alternative formula is derived through the interpretation of the diffusive contribution of the exact transport solution obtained by means of Case’s method applied to a plane configuration in the presence of isotropic scattering. Some numerical results are presented to illustrate the performances of the different formulations once they are used in the diffusion model for the solution of source-injected and critical problems.
Similar content being viewed by others
References
R.V. Meghreblian, D.K. Holmes, Reactor Analysis (McGraw-Hill, New York, 1960)
J.J. Duderstadt, W.R. Martin, Transport Theory (John Wiley & Sons, New York, 1978)
E. Larsen, Infinite medium solutions to the transport equation, $S_{n}$ discretization schemes and the diffusion approximation, in Proceedings of the Joint International Topical Meeting on Mathematics and Computation and Supercomputing in Nuclear Applications (Mec, 2001), Salt Lake City, 2001 (American Nuclear Society, 2001)
M. Jarrett, B. Kochunas, A. Zhu, T. Downar, Nucl. Sci. Eng. 184, 208 (2016)
A. Yamamoto, T. Endo, A. Giho, Trans. Am. Nucl. Soc. 119, 1179 (2018)
Y. Ronen, Nucl. Sci. Eng. 146, 245 (2004)
D. Tomatis, A. Dall’Osso, Application of a numerical transport correction in diffusion calculations, in Proceedings of the International Conference on Mathematics and Computational Methods Applied to Nuclear Science and Engineering, Rio de Janeiro, 2011
V.G. Molinari, Diffusion coefficient in the $P_{3}$ approximation, in Seminario Nazionale di Fisica del Reattore e Teoria del Trasporto (Bologna, 1983) pp. 177--182
S. Dulla, S. Canepa, P. Ravetto, Math. Comput. Simul. 80, 2134 (2010)
K.M. Case, J.H. Ferziger, P.F. Zweifel, Nucl. Sci. Eng. 10, 352 (1964)
P. Ravetto, Atomkernenergie-Kerntechn. 44, 155 (1983)
G.I. Bell, S. Glasstone, Nuclear Reactor Theory (Van Nostrand Reinhold, New York, 1970)
E. Inönü, Nucl. Sci. Eng. 5, 248 (1959)
P.M. Morse, H. Feshbach, Methods of Theoretical Physics (MacGraw-Hill, New York, 1953)
S. Dulla, D.B. Ganapol, P. Ravetto, Ann. Nucl. Energy 33, 932 (2006)
R. Davison, Neutron Transport Theory (Oxford University Press, Oxford, 1957)
M. Abramowitz, I.A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965)
S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960)
K.M. Case, F. de Hoffmann, G. Placzek, Introduction to the Theory of Neutron Diffusion (Los Alamos Scientific Laboratory, Los Alamos, 1953)
K.M. Case, P.F. Zweifel, Linear Transport Theory (Addison-Wesley, New York, 1967)
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
The EPJ Publishers remain neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Chentre, N., Saracco, P., Dulla, S. et al. On Fick’s law in asymptotic transport theory. Eur. Phys. J. Plus 134, 516 (2019). https://doi.org/10.1140/epjp/i2019-13045-9
Received:
Accepted:
Published:
DOI: https://doi.org/10.1140/epjp/i2019-13045-9