On the Build-Up Factor from the Multi-Group Neutron Diffusion Equation with Cylindrical Symmetry

Julio Cesar Lombaldo Fernandes, Marco Túllio Vilhena, Bardo Ernst Bodmann, Volnei Borges

DepartmentofAppliedMath, Universidade Federal do Rio Grande do Sul, Porto Alegre, Brazil.

DOI: 10.4236/wjnst.2013.31001                

Abstract

We consider the time dependent neutron diffusion equation for one energy group in cylinder coordinates, assuming translational symmetry along the cylinder axis. This problem for a specific energy group is solved analytically applying the Hankel transform in the radial coordinate r. Our special interest rests in the build-up factor for a time dependent linear neutron source aligned with the cylinder axis, which in the limit of zero decay constant reproduces also the static case. The new approach to solve the diffusion equation by integral transform technique is presented and results for several parameter sets and truncation in the solution for the flux and build-up factor are shown and found to be compatible to those of literature [1,2].

Keywords

Build-Up Factor, Cylindrical Geometry; Hankel Transform

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	J. Wood, “Computational Methods in Reactor Shielding,” Pergamon Press, Oxford, 1982.
[2]	B. D. A. Rodriguez, M. T. Vilhena and V. Borges, “The Determination of the Exposure Build-Up Factor Fomultion in a Slab Using the LTSN Method,” Kerntechnik, Vol. 71, No. 4, 2006, pp. 182-184.
[3]	H. Hirayama and K. Shin, “Application of the EGS4 Monte Carlo Code to a Study of Multilayer Gamma-Ray Exposure Build-Up Factors,” Journal of Nuclear Science and Technology, Vol. 35, No. 11, 1998, pp. 816-829. doi:10.1080/18811248.1998.9733949
[4]	B. D. A. Rodriguez, M. T. Vilhena and V. Borges, “A Solution for the Two-Dimensional Transport Equation for Photons and Electrons in a Rectangular Domain by the Laplace Transform Technique,” International Journal of Nuclear Energy Science and Technology, Vol. 5, No. 1, 2010, pp. 25-40.
[5]	G. C. Pomraning, “Flux-Limited Diffusion and FokkerPlanck Equations,” Nuclear Science and Engineering, Vol. 85, No. 2, 1983, pp. 116-126.
[6]	M. T. Vilhena, C. F. Segatto and L. B. Barichello, “A Particular Solution for the SN Radiative Transfer Problems,” Journal of Quantitative Spectroscopy and Radiative Transfer, Vol. 53, No. 4, 1995, pp. 467-469. doi:10.1016/0022-4073(95)90020-9
[7]	V. Borges, J. C. L. Fernandes, M. T. Vilhena, B. Bodmann and B. D. A. Rodriguez, “A Closed-Form Formulation for the Build-Up Factor and Absorbed for Photons and Electrons in the Compton Energy Range in Cartesian Geometry,” World Journal of Nuclear Science and Technology, Vol. 1, No. 2, 2012, pp. 23-28. doi:10.4236/wjnst.2012.21004
[8]	C. Borges and W. Larsen, “The Transversed Integrated Scalar Flux of a Narrowly Focused Particle Beam,” SIAM Journal on Applied Mathematics, Vol. 55, No. 1, 1995, pp. 1-22.
[9]	J. Lamarsh, “Introduction to Nulcear Reactor Theory,” McGrawn-Hill Company, New York, 1972.