Yansıtıcılı reaktörlerin stokastik iki-nokta reaktör kinetik denklemlerinin sayısal simülasyonu

Nokta reaktör kinetik denklemlerinin sayısal çözümleri bize nötron nüfusu ve gecikmiş nötron öncü yoğunluklarının ortalama değerlerini vermektedir. Gerçek dinamik süreç stokastik bir süreç olduğu için, nötron nüfusu ve öncü yoğunlukları zamanla rastgele dalgalanmaktadır. Bu çalışmada, harici nötron kaynağı olmayan ve altı grup gecikmiş nötron öncüsü olan güçlü yansıyan reaktörlerin dinamik davranışını analiz etmek amacıyla iki-nokta reaktör kinetik denklemleri için yeni bir stokastik model geliştirilmiştir. Bu modele karşılık gelen Itô stokastik diferansiyel denklemler sistemini türetmek için iki-nokta reaktör kinetik denklemleri üç terime ayrılır: ani nötronlar, gecikmiş nötronlar ve yansıyan nötronlar. Geri besleme etkilerinin olup ve olmadığı farklı pertürbasyon durumlarında, stokastik diferansiyel denklemler sistemi Euler-Murayama sayısal yöntemini kullanarak çözülür. Sistemin ortalama yanıtının diğer deterministik sayısal yöntemlerin sonuçlarıyla karşılaştırılabilir halde olduğu görünmektedir.

Anahtar Kelimeler:

Reflected reactor, Euler-Murayama Method, Stochastic models, Feedback effect

Numerical simulation of stochastic two-point reactor kinetics equations for reflected reactors

Deterministic numerical solutions of point reactor kinetic equations give us the mean values of the neutron population and delayed neutron precursor concentrations, whereas the actual dynamical process is stochastic and the neutron population and precursor concentrations fluctuate randomly with time. In the present study, a novel stochastic model for two-point reactor kinetics equations is developed and used to analyze the dynamical behavior of the source-free strongly reflected reactors with six groups of delayed neutron precursors. To derive the Itô stochastic differential equations system corresponding to this model, the two-point reactor kinetics equations are separated into three terms: prompt neutrons, delayed neutrons and reflected neutrons. In the case of different perturbation scenarios, both with and without the Newtonian temperature reactivity feedback effects, this system of stochastic differential equations is solved using the Euler-Murayama numerical method. It is observed that the mean response of the system is comparable with the results of other deterministic numerical methods.

Keywords:

Reflected reactor, Euler-Murayama Method, Stochastic models,

PDF

___

[1]. Henry, A. F. 1975. Nuclear Reactor Analysis. MIT Press, Cambridge, MA.
[2]. Duderstadt, J. J. 1976. Nuclear reactor analysis. Wiley.
[3]. Hayes, J. G., & Allen, E. J. 2005. Stochastic point-kinetics equations in nuclear reactor dynamics. Annals of nuclear energy, 32(6), 572-587.
[4]. Da Silva, M. W., Vasques, R., Bodmann, B. E., & Vilhena, M. T. 2016. A nonstiff solution for the stochastic neutron point kinetics equations. Annals of Nuclear Energy, 97, 47-52.
[5]. Cohn, C. E. 1962. Reflected-reactor kinetics. Nuclear Science and Engineering, 13(1), 12-17.
[6]. van Dam, H. 1996. Inhour equation and kinetic distortion in a two-point reactor kinetic model. Annals of Nuclear Energy, 23(14), 1127-1142.
[7]. Spriggs, G. D., Busch, R. D., & Williams, J. G. 1997. Two-region kinetic model for reflected reactors. Annals of Nuclear Energy, 24(3), 205-250.
[8]. Aboanber, A. E., & Nahla, A. A. 2018. Mathematical treatment for two-point reactor kinetics model of reflected systems. Progress in Nuclear Energy, 105, 287-293.
[9]. Aboanber, A. E. 2009. Exact solution for the non-linear two-point kinetic model of reflected reactors. Progress in Nuclear Energy, 51(6-7), 719-726.
[10]. Aboanber, A. E., & Nahla, A. A. 2002. Solution of the point kinetics equations in the presence of Newtonian temperature feedback by Padé approximations via the analytical inversion method. Journal of Physics A: Mathematical and General, 35(45), 9609.
[11]. Aboanber, A. E., & El Mhlawy, A. M. 2008. A new version of the reflected core inhour equation and its solution. Nuclear engineering and design, 238(7), 1670-1680.
[12]. Aboanber, A. E., & El Mhlawy, A. M. 2009. Solution of two-point kinetics equations for reflected reactors using Analytical Inversion Method (AIM). Progress in Nuclear Energy, 51(1), 155-162.
[13]. Holschuh, T. V., Marcum, W. R., & Palmer, T. S. 2017. One-group analytical solution to two-region reactor kinetic model. Annals of Nuclear Energy, 99, 199-205.
[14]. Hayes, J. G., & Allen, E. J. 2005. Stochastic point-kinetics equations in nuclear reactor dynamics. Annals of nuclear energy, 32(6), 572-587.
[15]. Hayes, J. G., 2005. Stochastic point kinetics equations in nuclear reactor dynamics, thesis, Texas Tech University.
[16]. Ray, S. S. 2012. Numerical simulation of stochastic point kinetic equation in the dynamical system of nuclear reactor. Annals of Nuclear Energy, 49, 154-159.
[17]. Nahla, A. A., & Edress, A. M. 2016. Efficient stochastic model for the point kinetics equations. Stochastic Analysis and Applications, 34(4), 598-609.
[18]. Saha Ray, S., & Singh, S. 2019. Numerical solutions of stochastic nonlinear point reactor kinetics equations in presence of Newtonian temperature feedback effects. Journal of Computational and Theoretical Transport, 48(2), 47-57.
[19]. Nahla, A. A. 2017. Stochastic model for the nonlinear point reactor kinetics equations in the presence Newtonian temperature feedback effects. Journal of Difference Equations and Applications, 23(6), 1003-1016.
[20]. Cyganowski, S., Kloeden, P., & Ombach, J. 2001. From elementary probability to stochastic differential equations with MAPLE®. Springer Science & Business Media.
[21]. Buldygin, V. V., & Kozachenko, I. V. (2000). Metric characterization of random variables and random processes (Vol. 188). American Mathematical Soc.
[22]. Iacus, S. M. (2009). Simulation and inference for stochastic differential equations: with R examples. Springer Science & Business Media.