A New Formulation to the Point Kinetics Equations Considering the Time Variation of the Neutron Currents

Author(s)
Anderson Lupo Nunes^{1}^{*},
Aquilino Senra Martinez^{1},
Fernando Carvalho da Silva^{1},
Daniel Artur Pinheiro Palma^{2}

Affiliation(s)

^{1}
Department of Nuclear Engineering, COPPE/UFRJ, Rio de Janeiro, Brazil.

^{2}
Brazilian Commission for Nuclear Energy, Rio de Janeiro, Brazil.

ABSTRACT

The system of point kinetics equations describes the time behaviour of a nuclear reactor, assuming that, during the transient, the spatial form of the flux of neutrons varies very little. This system has been largely used in the analysis of transients, where the numerical solutions of the equations are limited by the stiffness problem that results from the different time scales of the instantaneous and delayed neutrons. Its derivation can be done directly from the neutron transport equation, from the neutron diffusion equation or through a heuristics procedure. All of them lead to the same functional form of the system of differential equations for point kinetics, but with different coefficients. However, the solution of the neutron transport equation is of little practical use as it requires the change of the existent core design systems, as used to calculate the design of the cores of nuclear reactors for different operating cycles. Several approximations can be made for the said derivation. One of them consists of disregarding the time derivative for neutron density in comparison with the remaining terms of the equation resulting from the P1 approximation of the transport equation. In this paper, we consider that the time derivative for neutron current density is not negligible in the P1 equation. Thus being, we obtained a new system of equations of point kinetics that we named as modified. The innovation of the method presented in the manuscript consists in adopting arising from the P1 equations, without neglecting the derivative of the current neutrons, to derive the modified point kinetics equations instead of adopting the Fick’s law which results in the classic point kinetics equations. The results of the comparison between the point kinetics equations, modified and classical, indicate that the time derivative for the neutron current density should not be disregarded in several of transient analysis situations.

The system of point kinetics equations describes the time behaviour of a nuclear reactor, assuming that, during the transient, the spatial form of the flux of neutrons varies very little. This system has been largely used in the analysis of transients, where the numerical solutions of the equations are limited by the stiffness problem that results from the different time scales of the instantaneous and delayed neutrons. Its derivation can be done directly from the neutron transport equation, from the neutron diffusion equation or through a heuristics procedure. All of them lead to the same functional form of the system of differential equations for point kinetics, but with different coefficients. However, the solution of the neutron transport equation is of little practical use as it requires the change of the existent core design systems, as used to calculate the design of the cores of nuclear reactors for different operating cycles. Several approximations can be made for the said derivation. One of them consists of disregarding the time derivative for neutron density in comparison with the remaining terms of the equation resulting from the P1 approximation of the transport equation. In this paper, we consider that the time derivative for neutron current density is not negligible in the P1 equation. Thus being, we obtained a new system of equations of point kinetics that we named as modified. The innovation of the method presented in the manuscript consists in adopting arising from the P1 equations, without neglecting the derivative of the current neutrons, to derive the modified point kinetics equations instead of adopting the Fick’s law which results in the classic point kinetics equations. The results of the comparison between the point kinetics equations, modified and classical, indicate that the time derivative for the neutron current density should not be disregarded in several of transient analysis situations.

Cite this paper

Nunes, A. , Martinez, A. , Silva, F. and Palma, D. (2015) A New Formulation to the Point Kinetics Equations Considering the Time Variation of the Neutron Currents.*World Journal of Nuclear Science and Technology*, **5**, 57-71. doi: 10.4236/wjnst.2015.51006.

Nunes, A. , Martinez, A. , Silva, F. and Palma, D. (2015) A New Formulation to the Point Kinetics Equations Considering the Time Variation of the Neutron Currents.

References

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http://dx.doi.org/10.1016/j.anucene.2010.10.012

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http://www.ans.org/pubs/journals/nse/a_17429

http://dx.doi.org/10.13182/NSE85-7

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http://www.sciencedirect.com/science/article/pii/S0306454907002368

http://dx.doi.org/10.1016/j.anucene.2007.08.015

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http://www.sciencedirect.com/science/article/pii/030645498590091X

http://dx.doi.org/10.1016/0306-4549(85)90091-X

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[12] Stacey, W.M. (2007) Nuclear Reactor Analysis. 2nd Edition, Wiley-VCH GmbH & CO KGaA, Weinheim.

[13] Kinard, M. and Allen, E.J. (2003) Efficient Numerical Solution of the Point Kinetics Equations in Nuclear Reactor Dynamics. Annals of Nuclear Energy, 31, 1039-1051.

http://www.sciencedirect.com/science/article/pii/S0306454904000027

http://dx.doi.org/10.1016/j.anucene.2003.12.008

[14] Palma, D.A.P., Martinez, A.S. and Goncalves, A.C. (2009) Analytical Solution of Point Kinetics Equations for Linear Reactivity Variation. Annals of Nuclear Energy, 36, 1469-1471.

http://www.sciencedirect.com/science/article/pii/S0306454909001947

http://dx.doi.org/10.1016/j.anucene.2009.06.016

[15] Jahanbin, A. and Malmir, H. (2012) Kinetic Parameters Evaluation of PWRs Using Static Cell and Core Calculation Codes. Annals of Nuclear Energy, 41, 110-114. http://www.sciencedirect.com/science/article/pii/S0306454911004464

[1] Duderstadt, J.J. and Hamilton, L.J. (1976) Nuclear Reactor Analysis. John Wiley & Sons Ltd., New York.

[2] Bell, G.I. and Glasstone (1970) Nuclear Reactor Theory. Van Nostrand Reinhold Ltd., New York.

[3] Henry, A.F. (1975) Nuclear Reactor Analysis. The MIT Press, Cambridge and London.

[4] Heizler, S.I. (2010) Asymptotic Telegrapher’s Equation (P1) Approximation for the Transport Equation. Nuclear Science and Engineering, 166, 17-35. http://physics.biu.ac.il/files/physics/shared/staff/u47/nse_166_17.pdf

[5] Espinosa-Paredes, G., Polo-Labarrios, M.A., Espinosa-Martinez, E.G. and del Valle-Gallegos, E. (2011) Fractional Neutron Point Kinetics Equations for Nuclear Reactor Dynamics. Annals of Nuclear Energy, 38, 307-330.

http://www.sciencedirect.com/science/article/pii/S0306454910003816

http://dx.doi.org/10.1016/j.anucene.2010.10.012

[6] Akcasu, Z., Lellouche, G. and Shotkin, L.M. (1971) Mathematical Methods in Nuclear Reactor Dynamics. Academic Press, New York and London.

[7] Chao, Y.A. and Attard, A. (1985) A Resolution of the Stiffness Problem of Reactor Kinetics. Nuclear Science and Engineering, 90, 40-46.

http://www.ans.org/pubs/journals/nse/a_17429

http://dx.doi.org/10.13182/NSE85-7

[8] Zhang, F., Chen, W.Z. and Gui, X.W. (2008) Analytic Method Study of Point-Reactor Kinetic Equation When Cold Start-Up. Annals of Nuclear Energy, 35, 746-749.

http://www.sciencedirect.com/science/article/pii/S0306454907002368

http://dx.doi.org/10.1016/j.anucene.2007.08.015

[9] Hoogenboom, J.E. (1985) The Laplace Transformation of Adjoint Transport Equations. Annals of Nuclear Energy, 12, 151-152.

http://www.sciencedirect.com/science/article/pii/030645498590091X

http://dx.doi.org/10.1016/0306-4549(85)90091-X

[10] Fuchs, D. and Tabachnikov, S. (2000) Mathematical Omnibus: Thirty Lectures on Classic Mathematics. American Mathematical Society, Rhode Island.

[11] Hetrick, D.L. (1971) Dynamics of Nuclear Reactor. The University of Chicago Press Ltd., Chicago and London.

[12] Stacey, W.M. (2007) Nuclear Reactor Analysis. 2nd Edition, Wiley-VCH GmbH & CO KGaA, Weinheim.

[13] Kinard, M. and Allen, E.J. (2003) Efficient Numerical Solution of the Point Kinetics Equations in Nuclear Reactor Dynamics. Annals of Nuclear Energy, 31, 1039-1051.

http://www.sciencedirect.com/science/article/pii/S0306454904000027

http://dx.doi.org/10.1016/j.anucene.2003.12.008

[14] Palma, D.A.P., Martinez, A.S. and Goncalves, A.C. (2009) Analytical Solution of Point Kinetics Equations for Linear Reactivity Variation. Annals of Nuclear Energy, 36, 1469-1471.

http://www.sciencedirect.com/science/article/pii/S0306454909001947

http://dx.doi.org/10.1016/j.anucene.2009.06.016

[15] Jahanbin, A. and Malmir, H. (2012) Kinetic Parameters Evaluation of PWRs Using Static Cell and Core Calculation Codes. Annals of Nuclear Energy, 41, 110-114. http://www.sciencedirect.com/science/article/pii/S0306454911004464