# Modeling Neutron Dynamics in Nuclear Reactor using Fractional-order Point Reactor Kinetics Model with Adiabatic Temperature Feedback

### Vishwesh Vyawahare, P. S. V. Nataraj

#### Abstract

This paper deals with the development and analysis of fractional-order (FO) point reactor kinetics (FPRK) model with reactivity feedback for a nuclear reactor. Incorporation of adiabatic temperature feedback of reactivity makes this model nonlinear. It basically forms a system of coupled, nonlinear ordinary differential equations. The nonlinear subprompt critical FPRK model is developed and analyzed in detail. Fundamental motivation for this model is the fact that neutron transport inside the core of a reactor is truly a subdiffusion. The work presented here analyzes the effect of temperature feedback on the neutron concentration dynamics inside reactor core which is modeled using fractional differential equations. The system of nonlinear differential equations is solved numerically. The analysis clearly establishes the fact that the proposed model is ‘stable’ in the sense that it predicts self-limitting power excursions. The model presented in this paper constitutes an important step in the development of fractional-order model for a nuclear reactor, which can be used to achieve better control and operation.

#### References

- Aboanber, A. E. and Nahla, A. A. (2004). On pade' approximations to the exponential function and application to the point kinetics equations. Progress in Nuclear Energy, 44(4):347-368.
- Beckurts, K. H. and Wirtz, K. (1964). Neutron Physics. Springer-Verlag, Germany.
- Compte, A. and Metzler, R. (1997). The generalized Cattaneo equation for the description of anomalous transport processes. Journal of Physics A: Mathematical and General, 30:7277-7289.
- Connolly, J. A. (2004). The numerical solution of fractional and distributed order differential equations. PhD thesis, University of Liverpool, UK.
- Daftardar-Gejji, V. and Jafari, H. (2005). Adomian decomposition: a tool for solving a system of fractional differential equations. Journal of Mathematical Analysis and Applications, 301:508-518.
- Das, S. (2011). Functional Fractional Calculus for System Identification and Controls. Springer, Germany.
- Das, S. and Biswas, B. B. (2007). Fractional divergence for neutron flux profile in nuclear reactor. International Journal of Nuclear Energy Science and Technology, 3(2):139-159.
- Das, S., Das, S., and Gupta, A. (2011). Fractional order modeling of a PHWR under step-back condition and control of its global power with a robust PI?Dµ controller. IEEE Transactions on Nuclear Science, 58(5):2431-2441.
- Diethelm, K. (2010). The Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type. Springer, Germany.
- Diethelm, K., Ford, N. J., and Freed, A. D. (2002). A Predictor-Corrector approach for the numerical solution of fractional differential equations. Nonlinear Dynamics, 29:3-22.
- Diethelm, K., Ford, N. J., Freed, A. D., and Luchko, Y. (2005). Algorithms for the fractional calculus: A selection of numerical methods. Computer Methods in Applied Mechanics and Engineering, 194:743-773.
- Duderstadt, J. J. and Hamilton, L. J. (1976). Nuclear Reactor Analysis. John Wiley & Sons, USA.
- Espinosa-Paredes, G., del Valle-Gallegos, E., Nún¯ezCarrera, A., Polo-Labarrios, M.-A., EspinosaMartínez, E.-G., and Vázquez-Rodríguez, R. (2014). Fractional neutron point kinetics equation with newtonian temperature feedback effects. Progress in Nuclear Energy, 73:96-101.
- Espinosa-Paredes, G., Morales-Sandoval, J. B., VázquezRodríguez, R., and Espinosa-Martínez, E.-G. (2008). Constitutive laws for the neutron transport current. Annals of Nuclear Energy, 35:1963-1967.
- Espinosa-Paredes, G., Polo-Labarrios, M.-A., EspinosaMartínez, E.-G., and del Valle-Gallegos, E. (2011). Fractional neutron point kinetics equations for nuclear reactor dynamics. Annals of Nuclear Energy, 38:307- 330.
- Glasstone, S. and Sesonske, A. (2002). Nuclear Reactor Engineering: Vol. 1. CBS Publishers & Distributors, India.
- Hetrick, D. L. (1993). Dynamics of Nuclear Reactors. American Nuclear Society, USA.
- Kadem, A. (2009). The fractional transport equation: an analytical solution and a spectral approximation by Chebyshev polynomials. Applied Sciences, 11:78-90.
- Kadem, A. and Baleanu, D. (2010). Analytical method based on Walsh function combined with orthogonal polynomial for fractional transport equation. Communications in Nonlinear Science and Numerical Simulation, 15(3):491-501.
- Klages, R., Radons, G., and Sokolov, I. M., editors (2008). Anomalous Transport. WILEY-VCH Verlag GmbH & Co.
- Li, C. and Peng, G. (2004). Chaos in Chen's system with a fractional order. Chaos, Solitons & Fractals, 22(2):443-450.
- Machado, J. T., Kiryakova, V., and Mainardi, F. (2011). Recent history of fractional calculus. Communications in Nonlinear Science and Numerical Simulation, 16(3):1140-1153.
- Magin, R. L. (2006). Fractional Calculus in Bioengineering. Begell House Publishers, USA.
- Mathworks (2005). MATLAB Manual. The Mathworks Inc., MATLAB version 7.1 (R14), USA.
- Meghreblian, R. V. and Holmes, D. K. (1960). Reactor Analysis. McGraw-Hill Book Company, USA.
- Metzler, R. and Klafter, J. (2000). The random walk's guide to anomalous diffusion: A fractional dynamics approach. Physics Reports, 339:1-77.
- Monje, C. A., Chen, Y. Q., Vinagre, B. M., Xue, D., and Feliu, V. (2010). Fractional-order Systems and Control: Fundamentals and Applications. SpringerVerlag London Limited, UK.
- Nahla, A. A. (2009). An analytical solution for the point reactor kinetics equations with one group of delayed neutrons and the adiabatic feedback model. Progress in Nuclear Energy, 51:124-128.
- Odibat, Z. and Momani, S. (2006). Application of Variational Iteration Method to nonlinear differential equations of fractional order. International Journal of Nonlinear Sciences and Numerical Simulation, 1(7):15- 27.
- Ruskeepaa, H. (2009). Mathematica Navigator: Mathematics, Statistics and Graphics. Academic Press, USA.
- Samko, S. G., Kilbas, A. A., and Marichev, O. I. (1997). Fractional Integrals and Derivatives. Gordon and Breach Science Publishers, Netherlands.
- Sardar, T., Ray, S. S., Bera, R., Biswas, B., and Das, S. (2010). The solution of coupled fractional neutron diffusion equations with delayed neutrons. International Journal of Nuclear Energy Science and Technology, 5(2):105-113.
- Tavazoei, M. S. and Haeri, M. (2007). A necessary condition for double scroll attractor existence in fractionalorder systems. Physics Letters A, 367:102-113.
- Vyawahare, V. A. and Nataraj, P. S. V. (2010). Modeling neutron transport in a nuclear reactor as subdiffusion: The neutron fractional-order telegraph equation. In The 4th IFAC Workshop on Fractional Differentiation and its Applications, Badajoz, Spain.
- Vyawahare, V. A. and Nataraj, P. S. V. (2012). Development and analysis of the fractional point reactor kinetics model for a nuclear reactor with slab geometry. In The 5th IFAC Workshop on Fractional Differentiation and its Applications, Nanjing, China.
- Vyawahare, V. A. and Nataraj, P. S. V. (2013a). Development and analysis of some versions of the fractionalorder point reactor kinetics model for a nuclear reactor with slab geometry. Communications in Nonlinear Science and Numerical Simulation, 18:1840-1856.
- Vyawahare, V. A. and Nataraj, P. S. V. (2013b). Fractionalorder Modeling of Neutron Transport in a Nuclear Reactor. Applied Mathematical Modelling, 37:9747- 9767.
- Vyawahare, V. A. and Nataraj, P. S. V. (2014). Development and analysis of fractional-order NordheimFuchs model for nuclear reactor. In Daftardar-Gejji, V., editor, Fractional Calculus: Theory and Applications. Narosa Publishing House, India.

#### Paper Citation

#### in Harvard Style

Vyawahare V. and Nataraj P. (2014). **Modeling Neutron Dynamics in Nuclear Reactor using Fractional-order Point Reactor Kinetics Model with Adiabatic Temperature Feedback** . In *Proceedings of the 4th International Conference on Simulation and Modeling Methodologies, Technologies and Applications - Volume 1: SIMULTECH,* ISBN 978-989-758-038-3, pages 352-360. DOI: 10.5220/0005038103520360

#### in Bibtex Style

@conference{simultech14,

author={Vishwesh Vyawahare and P. S. V. Nataraj},

title={Modeling Neutron Dynamics in Nuclear Reactor using Fractional-order Point Reactor Kinetics Model with Adiabatic Temperature Feedback},

booktitle={Proceedings of the 4th International Conference on Simulation and Modeling Methodologies, Technologies and Applications - Volume 1: SIMULTECH,},

year={2014},

pages={352-360},

publisher={SciTePress},

organization={INSTICC},

doi={10.5220/0005038103520360},

isbn={978-989-758-038-3},

}

#### in EndNote Style

TY - CONF

JO - Proceedings of the 4th International Conference on Simulation and Modeling Methodologies, Technologies and Applications - Volume 1: SIMULTECH,

TI - Modeling Neutron Dynamics in Nuclear Reactor using Fractional-order Point Reactor Kinetics Model with Adiabatic Temperature Feedback

SN - 978-989-758-038-3

AU - Vyawahare V.

AU - Nataraj P.

PY - 2014

SP - 352

EP - 360

DO - 10.5220/0005038103520360