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Application of the Chebyshev polynomial (TN and UN) approximation to reflected slab geometry in the neutron transport equation and computation of critical half thicknesses

A. Bülbül and F. Anlı
From the journal Kerntechnik

Abstract

The transport of one-speed neutrons has been studied in isotropically scattering slab using ultraspherical polynomial approximation whose sub-cases are spherical harmonics approximation and Chebyshev polynomial approximation of first and second kind with reflective boundaries. An integral equation for the total flux has been derived and solved numerically for different values of c and reflection coefficients. In the solution, Marshak type vacuum boundary conditions were used. The critical thicknesses obtained by the Chebyshev polynomial method are in good agreement with literature values. Computations were made by using the TN, UN and PN approximations for comparison.

Kurzfassung

Anwendung der Tschebyscheff-Polynom-Approximation (TN und UN) auf reflektierende Stabgeometrie in der Neutronentransportgleichung und Berechnung kritischer Halbwertsdicken. Der Transport von Ein-Gruppen-Neutronen wurde bei isotroper Streuung in Stabgeometrie untersucht mit Hilfe ultraspärischer Polynom Approximation, die klassische polynomiellen Folgen, wie sphärische harmonische Approximation und Tschebyscheff-Polynom-Approximation erster und zweiter Ordnung mit reflektierenden Randbedingungen vereinheitlicht. Eine Integralgleichung für den gesamten Strahlungsfluss wurde hergeleitet und numerisch gelöst für verschiedene Werte von c und Reflektionskoeffizienten. Für den Lösungsweg wurden Vakuum-Randbedingungen vom Marshak-Typ verwendet. Die mit Hilfe der Tschebyscheff-Polynom-Approximation erhaltenen kritischer Halbwertsdicken stimmen gut mit Werten aus der Literatur überein. Zum Vergleich wurden Berechnungen mit TN, UN and PN Approximationen durchgeführt.


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References

1 Sahni, D. C.; Sjöstrand, N. G.: One-speed neutron transport in reflected spheres. Ann. Nucl. Energy24 (1997) 1157117010.1016/S0306-4549(97)00028-5 Search in Google Scholar

2 Garis, N. S.; Sjöstrand, N. G.: Eigenvalues for reflecting boundary conditions in one-speed neutron transport theory. Ann. Nucl. Energy21 (1994) 6780 Search in Google Scholar

3 Sahni, D. C.; Garis, N. S.; Sjöstrand, N. G.: Trans. Theory Statist. Phys.24 (1995) 62965610.1080/00411459508206019 Search in Google Scholar

4 Siewert, C. E.; Williams, M. M. R.: The effect of anisotropic scattering on the critical slab problem in neutron transport theory using a synthetic kernel.J. Phys. D: Appl. Phys.10 (1977) 20312040 Search in Google Scholar

5 Sharma, A.: Spherical Harmonics moments of neutron angular flux for spherically symmetric systems. Annals of Nuclear Energy28 (2001) 715721 Search in Google Scholar

6 Aspelund, O.: On a New Method for Solving the (Boltzmann) Equation in Neutron Transport Theory. PICG16 (1959) 530534 Search in Google Scholar

7 Conkie, W. R.: Polynomial Approximations in Neutron Transport Theory. Nucl. Scie. Eng.6 (1959) 260266 Search in Google Scholar

8 Yabushita, S.: Tschebyscheff Polynomial Approximation Method of the Neutron Transport Equation, Journ. of Math. Physics.2 (1961) 543549 Search in Google Scholar

9 Anlı, F.; Yaşa, F.; Güngör, S.; Öztürk, H.: TN approximation to neutron transport equation and application to critical slab problem. J. Quant. Spectroscopy Radiat. Transfer101 (2006) 129–134 Search in Google Scholar

10 Anlı, F.; Yaşa, F.; Güngör, S.; Öztürk, H.: TN approximation to reflected slab and computation of the critical half thickness. J. Quant. Spectroscopy Radiat. Transfer101 (2006) 135–140 Search in Google Scholar

11 Yilmazer, A.: Solution of one-speed neutron transport equation for strongly anisotropic scattering by TN approximation: Slab criticality problem. J. Quant. Spectroscopy Radiat. Transfer34 (2007) 743–751 Search in Google Scholar

12 Bülbül, A.; Ulutas, M.; Anlı, F.: UN approximation to the neutron transport equation in slab geometry. Kerntechnik73 (2008) 61–65 Search in Google Scholar

13 Öztürk, H.; Anlı, F.; Güngör, S.: Application of the UN method to the reflected critical slab problem for one-speed neutrons with forward and backward scattering. Kerntechnik72 (2007) 74–76 Search in Google Scholar

14 Öztürk, H.: The reflected critical slab problem for one-speed neutrons with strongly anisotropic scattering. Kerntechnik73 (2008) 6674 Search in Google Scholar

15 Bülbül, A.; Anlı, F.: Modified Chebyshev Polynomial approximation in the neutron transport theory and computation of the critical half thicknesses. Kerntechnik74 (2009) 6569 Search in Google Scholar

16 Case, K. M.; Zweifel, P. F.: Linear Transport Theory, Reading MA, Addison Wesley, 1967 Search in Google Scholar

17 Bell, G.; Glasstone, S.: Nuclear Reactor Theory. New York, VNR Company, 1972 Search in Google Scholar

Received: 2010-6-8
Published Online: 2013-04-25
Published in Print: 2010-09-01

© 2010, Carl Hanser Verlag, München