Abstract
This chapter aims to obtain analytical solutions for the neutron diffusion equation in three-dimensional Cartesian geometry by the separation of variables method, in homogeneous and heterogeneous domains, considering mono-energetic and two-energy groups, and a group of delayed neutron precursors. The present work is a continuation of the study of Oliveira et al. (Ann Nucl Energy 99: 253–257, 2017; Ann Nucl Energy 133:216–220, 2019) that uses the same methodology in the models but considering cylindrical geometry. Considering mono-energetic neutrons, we present simulations of the insertion of control rods at different values for the z variable. Considering two-energy groups, we assume the spatial functions of the fluxes and precursor concentration differ by a non-zero scale factor. The computational implementation of the algorithm associated with the obtained solution will be validated with the results of the literature.
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References
Aboanber, A.E., Nahla, A.A.: Solution of two-dimensional space-time multigroup reactor kinetics equations by generalized Padé and cut–product approximations. Ann. Nucl. Energy 33, 209–222 (2006)
Aboanber, A.E., Nahla, A.A.: Adaptive matrix formation AMF method of spacetime multigroup reactor kinetics equations in multidimensional model. Ann. Nucl. Energy 34, 103–119 (2007)
Aboanber, A.E., Hamada, Y.M.: Generalized Runge-Kutta method for two- and three-dimensional space-time diffusion equations with a variable time step. Ann. Nucl. Energy 35, 1024–1040 (2008)
Aboanber, A.E., Hamada, Y.M.: Computation accuracy and efficiency of a power series analytic method for two-and three- space-dependent transient problems. Prog. Nucl. Energy 51, 451–464 (2009)
Adomian, G.: Solving Frontier Problems of Physics: The Decomposition Method. Kluwer Academic Publishers, Dordrecht (1994)
Ceolin, C., Vilhena, M.T., Bodmann, B.E.J., Alvim, A.C.M.: On the analytical solution of the multi group neutron diffusion kinetic equation in a multilayered slab. In: Proceedings of the International Nuclear Atlantic Conference – INAC 2011, Belo Horizonte (2011)
Ceolin, C., Schramm, M., Vilhena, M.T., Bodmann, B.E.J.: On the neutron multi-group kinetic diffusion equation in a heterogeneous slab: an exact solution on a finite set of discrete points. Ann. Nucl. Energy 76, 271–282 (2015)
Corno, S.E., Dulla, S., Picca, P., Ravetto, R.: Analytical approach to the neutron kinetics of the non-homogeneous reactor. Ann. Nucl. Energy 50, 847–865 (2008)
Fernandes, J.C.L., Vilhena, M.T., Bodmann, B.E.J., Borges, V.: On the build-up factor from the multi-group neutron diffusion equation with cylindrical symmetry. World J. Nucl. Sci. Technol. 3, 1–5 (2013).
Fernandes, J.C.L., Vilhena, M.T., Bodmann, B.E.J.: On a comparative analysis of the solutions of the kinetic neutron diffusion equation by the Hankel transform formalism and the spectral method. Prog. Nucl. Energy 69, 71–76 (2013)
Nahla, A.A., Al-Malki, F.A., Rokaya, M.: Numerical techniques for the neutron diffusion equations in the nuclear reactors. Adv. Stud. Theor. Phys. 6, 640–664 (2012)
Oliveira, F.R., Bodmann, B.E.J., Vilhena, M.T., Carvalho, F.: On an analytical formulation for the mono-energetic neutron space-kinetic equation in full cylinder symmetry. Ann. Nucl. Energy 99, 253–257 (2017)
Oliveira, F.R., Fernandes, J.C.L., Bodmann, B.E.J., Vilhena, M.T.: On an analytical solution for the two energy group neutron space-kinetic equation in heterogeneous cylindrical geometry. Ann. Nucl. Energy 133, 216–220 (2019)
Petersen, C.Z., Bodmann, B.E.J., Vilhena, M.T., Barros, R.C.: Recursive solutions for multi-group neutron kinetics diffusion equations in homogeneous three dimensional rectangular domains with time dependent perturbations. Kerntechnik 79, 494–499 (2014)
Quintero-Leyva, B.: The multi-group integro-differential equations of the neutron diffusion kinetics. Solutions with the progressive polynomial approximation in multi-slab geometry. Ann. Nucl. Energy 37, 766–770 (2010)
Tumelero, F., Bodmann, B., Vilhena, M.T., Lapa, C.M.F.: On the solution of the neutron diffusion kinetic equation in planar geometry free of stiffness with convergence analysis. Ann. Nucl. Energy 125, 272–282 (2019)
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Tumelero, F., Vilhena, M.T., Bodmann, B.E.J. (2022). On the Mono-Energetic Neutron Space Kinetics Equation in Cartesian Geometry: An Analytic Solution by a Spectral Method. In: Constanda, C., Bodmann, B.E., Harris, P.J. (eds) Integral Methods in Science and Engineering. Birkhäuser, Cham. https://doi.org/10.1007/978-3-031-07171-3_23
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