Analytical solution of modified point kinetics equations for linear reactivity variation in subcritical nuclear reactors adopting an incomplete gamma function approximation

Affiliation(s)

Nuclear Engineering Department, Federal University of Rio de Janeiro, Rio de Janeiro, Brazil;.

Nuclear Engineering Department, Federal University of Rio de Janeiro, Rio de Janeiro, Brazil;.

ABSTRACT

The present work aims to achieve a fast and accurate analytical solution of the point kinetics equations applied to subcritical reactors such as ADS (Accelerator-Driven System), assuming a linear reactivity and external source variation. It was used a new set of point kinetics equations for subcritical systems based on the model proposed by Gandini & Salvatores. In this work it was employed the integrating factor method. The analytical solution for the case of interest was obtained by using only an approximation which consists of disregarding the term of the second derivative for neutron density in relation to time when compared with the other terms of the equation. And also, it is proposed an approximation for the upper incomplete gamma function found in the solution in order to make the computational processing faster. In addition, for purposes of validation and comparison a numerical solution was obtained by the finite differences method. Finally, it can be concluded that the obtained solution is accurate and has fast numerical processing time, especially when compared with the results of numerical solution by finite difference. One can also observe that the gamma approximation used achieve a high accuracy for the usual parameters. Thus we got satisfactory results when the solution is applied to practical situations, such as a reactor startup.

The present work aims to achieve a fast and accurate analytical solution of the point kinetics equations applied to subcritical reactors such as ADS (Accelerator-Driven System), assuming a linear reactivity and external source variation. It was used a new set of point kinetics equations for subcritical systems based on the model proposed by Gandini & Salvatores. In this work it was employed the integrating factor method. The analytical solution for the case of interest was obtained by using only an approximation which consists of disregarding the term of the second derivative for neutron density in relation to time when compared with the other terms of the equation. And also, it is proposed an approximation for the upper incomplete gamma function found in the solution in order to make the computational processing faster. In addition, for purposes of validation and comparison a numerical solution was obtained by the finite differences method. Finally, it can be concluded that the obtained solution is accurate and has fast numerical processing time, especially when compared with the results of numerical solution by finite difference. One can also observe that the gamma approximation used achieve a high accuracy for the usual parameters. Thus we got satisfactory results when the solution is applied to practical situations, such as a reactor startup.

Cite this paper

Rebello Junior, A. , Martinez, A. and Goncalves, A. (2012) Analytical solution of modified point kinetics equations for linear reactivity variation in subcritical nuclear reactors adopting an incomplete gamma function approximation.*Natural Science*, **4**, 919-923. doi: 10.4236/ns.2012.431119.

Rebello Junior, A. , Martinez, A. and Goncalves, A. (2012) Analytical solution of modified point kinetics equations for linear reactivity variation in subcritical nuclear reactors adopting an incomplete gamma function approximation.

References

[1] Gandini, A. and Salvatores, M. (2002) The physics of subcritical multiplying systems. Journal of Nuclear Science and Technology, 6, 673-686. Hdoi:10.1080/18811248.2002.9715249

[2] Palma, D., Martinez, A.S. and Gon?alves, A.C. (2009) Analytical solution of point kinetics equations for linear reactivity variation during start-up of a nuclear reactor. Annals of Nuclear Energy, 36, 1469-1471. Hdoi:10.1016/j.anucene.2009.06.016

[3] Arfken, G. (1985) Mathematical Methods for Physicists. Academic Press, Waltham.

[4] Haglund, J. (2011) Some conjectures on the zeros of approximates to the Riemann ≡-function and incomplete gamma functions. Central European Journal of Mathematics, 9, 302-318. Hdoi:10.2478/s11533-010-0095-3

[5] Nemes, G. (2010) New asymptotic expansion for the Gamma function. Archiv der Mathematik, 95, 161-169.

[6] Gradshteyn, I.S. and Ryzhik, I.M., (2007) Table of integrals, series and products. Academic Press, California.

[1] Gandini, A. and Salvatores, M. (2002) The physics of subcritical multiplying systems. Journal of Nuclear Science and Technology, 6, 673-686. Hdoi:10.1080/18811248.2002.9715249

[2] Palma, D., Martinez, A.S. and Gon?alves, A.C. (2009) Analytical solution of point kinetics equations for linear reactivity variation during start-up of a nuclear reactor. Annals of Nuclear Energy, 36, 1469-1471. Hdoi:10.1016/j.anucene.2009.06.016

[3] Arfken, G. (1985) Mathematical Methods for Physicists. Academic Press, Waltham.

[4] Haglund, J. (2011) Some conjectures on the zeros of approximates to the Riemann ≡-function and incomplete gamma functions. Central European Journal of Mathematics, 9, 302-318. Hdoi:10.2478/s11533-010-0095-3

[5] Nemes, G. (2010) New asymptotic expansion for the Gamma function. Archiv der Mathematik, 95, 161-169.

[6] Gradshteyn, I.S. and Ryzhik, I.M., (2007) Table of integrals, series and products. Academic Press, California.