S4 Solution of the Transport Equation for Eigenvalues: Isotropic, Forward and Backward Scattering in a Slab

Year 2017, Volume: 12 Issue: 2, 1 - 5, 30.11.2017

Hakan Öztürk

Abstract

Abstract: The neutron transport equation is solved
numerically for monoenergetic neutrons in a finite homogeneous slab with
backward and forward scattering for the eigenvalue spectrum. The
forward-backward-isotropic (FBI) scattering kernel is chosen for representing
the neutron scattering in transport equation. Then, the transport equation is
converted into a discrete ordinates form by using the integral transform
technique with the even-order Gauss-Legendre quadrature set. Finally, the
eigenvalues are calculated for a medium from weakly absorbing to highly
scattering condition using various values of the scattering, backward and
forward scattering parameters. Gauss-Legendre quadrature sets are used for all
calculations and the calculated eigenvalues are given in the tables.

Keywords

Transport Equation, Forward and Backward Scattering, Discrete Ordinates, Eigenvalues

Transport Denkleminin Özdeğerler için S4 Çözümü: Dilimde İzotropik, İleri ve Geri Saçılma

Abstract

Özet: İleri ve geri saçılmalı sonlu homojen bir dilimde tek enerjili nötronların özdeğer spektrumu için nötron transport denklemi nümerik olarak çözülmüştür. Transport denklemindeki nötron saçılmasını temsilen, ileri-geri-izotropic (FBI) saçılma fonksiyonu tercih edilmiştir. Daha sonra, çift-mertebeli Gauss-Legendre kuadratür seti ile integral dönüşüm tekniği kullanılarak transport denklemi diskret-ordinatlar haline dönüştürülmüştür. Son olarak, faklı saçılma, ileri ve geri saçılma parametreleri kullanılarak zayıf yutulmalı bir ortamdan kuvvetli saçılmalı bir ortama kadar özdeğerler hesaplanmıştır. Bütün hesaplamalarda Gauss-Legendre kuadratür setleri kullanılmış ve hesaplanan özdeğerler çizelgelerde verilmiştir.

Keywords

Transport Denklemi, İleri ve Geri Saçılma, Diskret Ordinatlar, Özdeğerler

References

• [1] N.S. Riyait, and R.T. Ackroyd, “The finite element method for multigroup neutron transport: Anisotropic scattering in 1‐D slab geometry”, Ann. Nucl. Energy, vol. 14, no. 3, pp. 113-133, 1987.
• [2] D.C. Sahni, N.G. Sjöstrand, and N.S. Garis, “Criticality and time eigenvalues for one-speed neutrons in a slab with forward and backward scattering”, J. Phys. D: Appl. Phys., vol. 25, no. 10, pp. 1381-1389, 1992.
• [3] E.W. Larsen, “Unconditionally Stable Diffusion-Synthetic Acceleration Methods for the Slab Geometry Discrete Ordinates Equations. Part I: Theory”. Nucl. Sci. Eng. Vol. 82, no. 1, pp. 47-63, 1982.
• [4] Los Alamos National Laboratory, Physics Division Progress Report, pp. 70-100, 1995.
• [5] K.M. Case, and P.F. Zweifel, Linear Transport Theory. Michigan: Addison-Wesley Company 1967, ch. 4.
• [6] F. Anlı, “Spectral Green’s function method for neutron transport: isotropic, forward, and backward scattering in 1-D slab geometry”, Ann. Nucl. Energy, vol. 28, no. 10, pp. 1033-1042, Jul. 2001.
• [7] H. Öztürk, F. Anlı, and S. Güngör, “Spectral Green's function method for neutron transport with backward and forward scattering in 1‐D slab and spherical geometry”, Trans. Theory Statisc. Phys., vol. 35, no. 5-7, pp. 229-250, 2006.