Abstract
Fractional point kinetics has been discussed recently as one of the novel approaches that describes the short-term evolution of neutron densities as well as precursor concentrations in nuclear reactor theory. Kinetics may be derived from an original transport problem introducing simplifications that allow to decouple the time from spatial degrees of freedom. One of the motivations to extend the traditional point kinetics by additional terms that contain a fractional derivative is to improve the solution in the sense to compensate effects due to the simplifications mentioned above. Works on the fractional derivative point kinetics equation found in the literature (Espinosa-Paredes et al., Ann. Nucl. Eng. 35:1963–1967, 2008; Espinosa-Paredes et al., Energy 38:207–330, 2011; Nec and Nepomnyashchy, J. Phys. A Math. Theor. 40(49):14687–14702, 2007; Saha Ray and Patra, Ann. Nucl. Eng. 41:61–66, 2012) treat the extended kinetics problem, whereas in the present work we consider additionally temperature feedback on the reactivity (El Tokhy and Mahmoud, Ann. Nucl. Eng. 68:228–233, 2014; Nahla, Nucl. Eng. Des. 241:1502–1505, 2011; Silva et al., Int. J. Nucl. Eng. Sci. Technol. 8(2):131–140, 2014), which mimics influences of thermohydraulics on neutronics and thus may be considered beside being a novelty also a more realistic model in comparison with the previous works.
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References
Adomian, G.: Solving frontier problems of physics: the decomposition method. Kluwer Academic Publishers, Aarhus C., Denmark (1994)
Debnath, L.: Recent applications of fractional calculus to science and engineering International Journal of Mathematics and Mathematical Sciences 2003, 3413–3442 (2003)
El Tokhy, M., Mahmoud, I.I.: Parameter analysis of the neutron point kinetics equations with feedback temperature effects. Annals of Nuclear Energy 68, 228–233 (2014)
Espinosa-Paredes, G., Morales-Sandoval, J., Vásquez-Rodríguez, R. and Espinosa-Martínez, E.-G.: Constitutive laws for the neutron density current. Annals of Nuclear Energy 35, 1963–1967 (2008)
Espinosa-Paredes, G., Polo-Labarrios, M., Espinosa-Martínez, E.-G., and Valle-Galegos, E.: Fractional neutron point kinetics equations for nuclear reactor dynamics, Annals of Nuclear Energy 38, 207–330 (2011)
Ganapol, B.D.: A highly accurate algorithm for the solution of the point kinetics equations. Annals of Nuclear Energy 62, 564–571 (2013)
Nahla, A.A.: Taylor’s series method for solving the nonlinear point kinetics equations. Nuclear Engineering and Design 241, 1502–1505 (2011)
Nec, Y., Nepomnyashchy, A.A.: Turing instability in sub-diffusive reaction-diffusion system. Journal of Physics A: Mathematical and Theoretical 40(49), 14687–14702 (2007)
Oldham, K., Spanier, J.: The Fractional Calculus. Academic Press, New York, USA (1974)
Palma, D.A.P., Martinez, A.S., Gonalves, A.: Analytical solution of point kinetics equations for linear reactivity variation during the start-up of a nuclear reactor. Annals of Nuclear Energy 36, 1469–1471 (2009)
Saha Ray, S., Patra, A.: An explicit finite difference scheme for numerical solution of fractional neutron point kinetic equation. Annals of Nuclear Energy 41, 61–66 (2012)
Silva, J.J.A., Alvim, A.C.M., Vilhena, M.T.M.B., Bodmann, B.E.J., Petersen, C.Z.: On a closed-form solution of the point kinetics equations with reactivity feedback of temperature. International Journal of Nuclear Energy Science and Technology 8(2), 131–140 (2014)
da Silva, M.W., Leite, S.B., Vilhena, M.T.M.B., Bodmann, B.E.J.: On an analytical representation for the solution of the neutron point kinetics equation free of stiffness. Annals of Nuclear Energy 71, 97–102 (2014)
Trenčevski, K.: New Approach to The Fractional Derivatives. International Journal of Mathematics and Mathematical Sciences 2003(5), 315–325 (2003)
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Schramm, M., Alvim, A.C.M., Bodmann, B.E.J., Vilhena, M.T.B., Petersen, C.Z. (2015). The Neutron Point Kinetics Equation: Suppression of Fractional Derivative Effects by Temperature Feedback. In: Constanda, C., Kirsch, A. (eds) Integral Methods in Science and Engineering. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-16727-5_46
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DOI: https://doi.org/10.1007/978-3-319-16727-5_46
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