Nuclear Engineering and Technology

  • 제56권4호
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  • Pages.1296-1309
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  • 2024
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  • 1738-5733(pISSN)
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  • 2234-358X(eISSN)
한국원자력학회 (Korean Nuclear Society)
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Stability analysis in BWRs with double subdiffusion effects: Reduced order fractional model (DS-F-ROM)

  • Gilberto Espinosa-Paredes (Area de Ingenieria en Recursos Energeticos, Universidad Autonoma Metropolitana-Iztapalapa) ;
  • Ricardo I. Cazares-Ramirez (Universidad Nacional Autonoma de Mexico, Facultad de Ingenieria) ;
  • Vishwesh A. Vyawahare (Department of Electronics Engineering, Ramrao Adik Institute of Technology, DY Patil deemed to be University) ;
  • Erick-G. Espinosa-Martinez (Department of Engineering, CIIDETEC-Coyoacan, Universidad del Valle de Mexico)
  • 투고 : 2023.05.22
  • 심사 : 2023.11.18
  • 발행 : 2024.04.25
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초록

The aim of this work is to explore the effect of the double subdiffusion on the stability in BWRs. A BWR novel reduced order model with double subdiffusion effects: reduced order fractional model (DS-F-ROM) to describe the neutron and heat transfer processes was proposed for this study. The double subdiffusion was developed with a fractional-order two-equation model, and with different fractional-orders and relaxation times. The stability analysis was carried out using the root-locus method and change from the s to the W domain and were confirmed using the time-domain evolution of neutron flux for a unit step change in reactivity. The results obtained using the reduced fractional-order model are presented for different anomalous diffusion coefficient values. Results are compared with normal diffusion and P1 equations, which are obtained straightforwardly with DS-ROM when relaxation time tends to zero, and when the anomalous diffusion coefficient tends to one, respectively.

키워드

과제정보

We would like to express our sincere gratitude to the anonymous reviewers for their invaluable contributions to the improvement of this manuscript. Their thoughtful comments, insightful suggestions, and constructive feedback have played a pivotal role in enhancing the quality and rigor of our research. We greatly appreciate the time and effort dedicated to the review process, which has undoubtedly strengthened the final version of this paper. R. Cazares-Ramirez (CVU 387444) gratefully acknowledges for the financial support of CONAHCyT provided through the postdoctoral fellowship EPM 2023(1).

참고문헌

  1. J. March-Leuba, A reduced-order model of boiling water reactor linear dynamics, Nucl. Technol. 75 (1986) 15-22, https://doi.org/10.13182/NT86-A15973.
  2. J. March-Leuba, D.G. Cacuci, R.B. Perez, Nonlinear dynamics and stability of boiling water reactors: Part 2 - quantitative analysis, Nucl. Sci. Eng. 93 (1986) 124-136, https://doi.org/10.13182/NSE86-A17664.
  3. J. March-Leuba, D.G. Cacuci, R.B. Perez, Nonlinear dynamics and stability of boiling water reactors: Part 1 - qualitative analysis, Nucl. Sci. Eng. 93 (1986) 111-123, https://doi.org/10.13182/NSE86-A17663.
  4. G. Espinosa-Paredes, M.I. Sanchez-Romero, E.C. Herrera-Hernandez, Nonlinear analysis of a reduced-order model with relaxation effects for BWRs, Prog. Nucl. Energy 117 (2019), 103089, https://doi.org/10.1016/j.pnucene.2019.103089.
  5. S. Glasstone, A. Sesonske, Nuclear Reactor Engineering: Reactor Systems Engineering, Springer, 2012.
  6. J. Dorning, Nuclear reactor kinetics: 1934-1999 and beyond, in: Nuclear Computational Science, Springer Netherlands, Dordrecht, 2010, pp. 375-457, https://doi.org/10.1007/978-90-481-3411-3_8.
  7. S. Saha Ray, Numerical simulation of stochastic point kinetic equation in the dynamical system of nuclear reactor, Ann. Nucl. Energy 49 (2012) 154-159, https://doi.org/10.1016/j.anucene.2012.05.022.
  8. R. Ponciroli, A. Bigoni, A. Cammi, S. Lorenzi, L. Luzzi, Object-oriented modelling and simulation for the ALFRED dynamics, Prog. Nucl. Energy 71 (2014) 15-29, https://doi.org/10.1016/j.pnucene.2013.10.013.
  9. V. Singh, M.R. Lish, O. Chvala, ' B.R. Upadhyaya, Dynamics and control of molten-salt breeder reactor, Nucl. Eng. Technol. 49 (2017) 887-895, https://doi.org/10.1016/j.net.2017.06.003.
  10. G. Espinosa-Paredes, C.G. Aguilar-Madera, Scaled neutron point kinetics (SUNPK) equations for nuclear reactor dynamics: 2D approximation, Ann. Nucl. Energy 115 (2018) 377-386, https://doi.org/10.1016/j.anucene.2018.01.020.
  11. Y.M. Hamada, Solution of a new model of fractional telegraph point reactor kinetics using differential transformation method, Appl. Math. Model. 78 (2020) 297-321, https://doi.org/10.1016/j.apm.2019.10.001.
  12. M.A. Polo-Labarrios, S. Quezada-Garcia, G. Espinosa-Paredes, L. Franco-Perez, J. Ortiz-Villafuerte, Novel numerical solution to the fractional neutron point kinetic equation in nuclear reactor dynamics, Ann. Nucl. Energy 137 (2020), 107173, https://doi.org/10.1016/j.anucene.2019.107173.
  13. E. Schiassi, M. De Florio, B.D. Ganapol, P. Picca, R. Furfaro, Physics-informed neural networks for the point kinetics equations for nuclear reactor dynamics, Ann. Nucl. Energy 167 (2022), 108833, https://doi.org/10.1016/j.anucene.2021.108833.
  14. G. Espinosa-Paredes, J.B. Morales-Sandoval, R. Vazquez-Rodriguez, E.-G. Espinosa-Martinez, Constitutive laws for the neutron density current, Ann. Nucl. Energy 35 (2008) 1963-1967, https://doi.org/10.1016/j.anucene.2008.05.002.
  15. M.R. Altahhan, M.S. Nagy, H.H. Abou-Gabal, A.E. Aboanber, Formulation of a point reactor kinetics model based on the neutron telegraph equation, Ann. Nucl. Energy 91 (2016) 176-188, https://doi.org/10.1016/j.anucene.2016.01.011.
  16. M.R. Altahhan, A.E. Aboanber, H.H. Abou-Gabal, Analytical solution of the Telegraph Point Reactor Kinetics model during the cold start-up of a nuclear reactor, Ann. Nucl. Energy 109 (2017) 574-582, https://doi.org/10.1016/j.anucene.2017.06.001.
  17. M.R. Altahhan, A.E. Aboanber, H.H. Abou-Gabal, M.S. Nagy, Response of the point-reactor telegraph kinetics to time varying reactivities, Prog. Nucl. Energy 98 (2017) 109-122, https://doi.org/10.1016/j.pnucene.2017.03.008.
  18. D. Suescun-Diaz, G. Espinosa-Paredes, F.H. Escobar, Inverse method to obtain reactivity in nuclear reactors with P1 point reactor kinetics model using matrix formulation, Nucl. Eng. Technol. 53 (2021) 414-422, https://doi.org/10.1016/j.net.2020.07.003.
  19. A.E. Aboanber, A.A. Nahla, A.M. El-Mhlawy, O. Maher, An efficient exponential representation for solving the two-energy group point telegraph kinetics model, Ann. Nucl. Energy 166 (2022), 108698, https://doi.org/10.1016/j.anucene.2021.108698.
  20. M. Zarei, An adjoint sensitivity analysis of the Telegrapher's neutron kinetic equations, Ann. Nucl. Energy 167 (2022), 108839, https://doi.org/10.1016/j.anucene.2021.108839.
  21. G. Espinosa-Paredes, D. Suescun-Diaz, Point reactor kinetics equations from P1 approximation of the transport equations, Ann. Nucl. Energy 144 (2020), 107592, https://doi.org/10.1016/j.anucene.2020.107592.
  22. S.I. Heizler, Asymptotic telegrapher's equation (P1) approximation for the transport equation, Nucl. Sci. Eng. 166 (2010) 17-35, https://doi.org/10.13182/NSE09-77.
  23. G. Espinosa-Paredes, M.A. Polo-Labarrios, Time-fractional telegrapher's equation (P1) approximation for the transport equation, Nucl. Sci. Eng. 171 (2012) 258-264, https://doi.org/10.13182/NSE11-58.
  24. P. Van Uffelen, J. Hales, W. Li, G. Rossiter, R. Williamson, A review of fuel performance modelling, J. Nucl. Mater. 516 (2019) 373-412, https://doi.org/10.1016/j.jnucmat.2018.12.037.
  25. F. Cappia, D. Pizzocri, M. Marchetti, A. Schubert, P. Van Uffelen, L. Luzzi, D. Papaioannou, R. Macian-Juan, V.V. Rondinella, Microhardness and Young's modulus of high burn-up UO2 fuel, J. Nucl. Mater. 479 (2016) 447-454, https://doi.org/10.1016/j.jnucmat.2016.07.015.
  26. K.A. Terrani, M. Balooch, J.R. Burns, Q.B. Smith, Young's modulus evaluation of high burnup structure in UO2 with nanometer resolution, J. Nucl. Mater. 508 (2018) 33-39, https://doi.org/10.1016/j.jnucmat.2018.04.004.
  27. D.D. Joseph, L. Preziosi, Heat waves, Rev. Mod. Phys. 61 (1989) 41-73, https://doi.org/10.1103/RevModPhys.61.41.
  28. F. Xu, K.A. Seffen, T.J. Lu, Non-Fourier analysis of skin biothermomechanics, Int. J. Heat Mass Tran. 51 (2008) 2237-2259, https://doi.org/10.1016/j.ijheatmasstransfer.2007.10.024.
  29. J. Shiomi, S. Maruyama, Non-Fourier heat conduction in a single-walled carbon nanotube: classical molecular dynamics simulations, Phys. Rev. B 73 (2006), 205420, https://doi.org/10.1103/PhysRevB.73.205420.
  30. A. Singh, E.B. Tadmor, Thermal parameter identification for non-Fourier heat transfer from molecular dynamics, J. Comput. Phys. 299 (2015) 667-686, https://doi.org/10.1016/j.jcp.2015.07.008.
  31. J.H. Choi, S.-H. Yoon, S.G. Park, S.-H. Choi, Analytical solution of the Cattaneo - vernotte equation (non-Fourier heat conduction), Journal of the Korean Society of Marine Engineering 40 (2016) 389-396, https://doi.org/10.5916/jkosme.2016.40.5.389.
  32. C. Cattaneo, Sur une Forme de I'equation de la Chaleur Eliminant le Paradoxe d'une Propagation Instantane'e, C. R. Acad. Sci. 247 (1958) 431-433.
  33. P. Vernotte, Les paradoxes de la theorie continue de l'equation de la chaleur, Comptes Rendus de 1'Academie Des Sciences de Paris, France. 246 (1958) 3154-3155.
  34. G. Espinosa-Paredes, E.-G. Espinosa-Martinez, Fuel rod model based on Non-Fourier heat conduction equation, Ann. Nucl. Energy 36 (2009) 680-693, https://doi.org/10.1016/j.anucene.2009.01.006.
  35. I.M. Sokolov, J. Klafter, A. Blumen, Fractional kinetics, Phys. Today 55 (2002) 48-54, https://doi.org/10.1063/1.1535007.
  36. M.A. Ezzat, A.A. El-Bary, M.A. Fayik, Fractional fourier law with three-phase lag of thermoelasticity, Mech. Adv. Mater. Struct. 20 (2013) 593-602, https://doi.org/10.1080/15376494.2011.643280.
  37. P. Pal, A. Sur, M. Kanoria, Thermo-viscoelastic interaction subjected to fractional Fourier law with three-phase-lag effects, Journal of Solid Mechanics 7 (2015) 400-415.
  38. C. Li, L. Zheng, X. Zhang, G. Chen, Flow and heat transfer of a generalized Maxwell fluid with modified fractional Fourier's law and Darcy's law, Comput. Fluids 125 (2016) 25-38, https://doi.org/10.1016/j.compfluid.2015.10.021.
  39. Y. Povstenko, T. Kyrylych, Time-fractional heat conduction in a plane with two external half-infinite line slits under heat flux loading, Symmetry (Basel) 11 (2019) 689, https://doi.org/10.3390/sym11050689.
  40. A.E. Abouelregal, H. Ahmad, A modified thermoelastic fractional heat conduction model with a single-lag and two different fractional-orders, Journal of Applied and Computational Mechanics 7 (2021) 1676-1686.
  41. G. Espinosa-Paredes, Fractional-Order Models for Nuclear Reactor Analysis, 2020.
  42. E.-G. Espinosa-Martinez, J.-L. Francois, C. Martin-del-Campo, N. Maleki Moghaddam, Time-space fractional neutron point kinetics: theory and simulations, Ann. Nucl. Energy 143 (2020), 107448, https://doi.org/10.1016/j.anucene.2020.107448.
  43. A. Zameer, M. Muneeb, S.M. Mirza, M.A.Z. Raja, Fractional-order particle swarm based multi-objective PWR core loading pattern optimization, Ann. Nucl. Energy 135 (2020), 106982, https://doi.org/10.1016/j.anucene.2019.106982.
  44. N. Zare, G. Jahanfarnia, A. Khorshidi, J. Soltani, Robustness of optimized FPID controller against uncertainty and disturbance by fractional nonlinear model for research nuclear reactor, Nucl. Eng. Technol. 52 (2020) 2017-2024, https://doi.org/10.1016/j.net.2020.03.002.
  45. A.E. Aboanber, A.A. Nahla, S.M. Aljawazneh, Fractional two energy groups matrix representation for nuclear reactor dynamics with an external source, Ann. Nucl. Energy 153 (2021), 108062, https://doi.org/10.1016/j.anucene.2020.108062.
  46. G. Fernandez-Anaya, S. Quezada-Garcia, M.A. Polo-Labarrios, L.A. Quezada-Tellez, Novel solution to the fractional neutron point kinetic equation using conformable derivatives, Ann. Nucl. Energy 160 (2021), 108407, https://doi.org/10.1016/j.anucene.2021.108407.
  47. O. Nikan, Z. Avazzadeh, J.A.T. Machado, Numerical approximation of the nonlinear time-fractional telegraph equation arising in neutron transport, Commun. Nonlinear Sci. Numer. Simul. 99 (2021), 105755, https://doi.org/10.1016/j.cnsns.2021.105755.
  48. M.A. Polo-Labarrios, F.A. Godinez, S. Quezada-Garcia, Numerical-analytical solutions of the fractional point kinetic model with Caputo derivatives, Ann. Nucl. Energy 166 (2022), 108745, https://doi.org/10.1016/j.anucene.2021.108745.
  49. R.M. Refeat, R.A. Fahmy, Optimized fractional-order PID controller based on nonlinear point kinetic model for VVER-1000 reactor, Kerntechnik 87 (2022) 104-114, https://doi.org/10.1515/kern-2021-0038.
  50. G. Espinosa-Paredes, R.-I. Cazares-Ramirez, J.-L. Francois, C. Martin-del-Campo, On the stability of fractional neutron point kinetics (FNPK), Appl. Math. Model. 45 (2017) 505-515, https://doi.org/10.1016/j.apm.2016.12.015.
  51. A.G. Radwan, A.M. Soliman, A.S. Elwakil, A. Sedeek, On the stability of linear systems with fractional-order elements, Chaos, Solit. Fractals 40 (2009) 2317-2328, https://doi.org/10.1016/j.chaos.2007.10.033.
  52. R.-I. Cazares-Ramirez, Vishwesh.A. Vyawahare, G. Espinosa-Paredes, P.S. V. Nataraj, On the feedback stability of linear FNPK equations, Prog. Nucl. Energy 98 (2017) 45-58, https://doi.org/10.1016/j.pnucene.2017.02.007.
  53. Vishwesh.A. Vyawahare, G. Espinosa-Paredes, BWR stability analysis with sub-diffusive and feedback effects, Ann. Nucl. Energy 110 (2017) 349-361, https://doi.org/10.1016/j.anucene.2017.06.059.
  54. I. Podlubny, Fractional Differential Equations an Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, 1998.
  55. E.C. de Oliveira, J.A. Tenreiro Machado, A review of definitions for fractional derivatives and integral, Math. Probl Eng. 2014 (2014) 1-6, https://doi.org/10.1155/2014/238459.
  56. M.D. Ortigueira, J.A. Tenreiro Machado, What is a fractional derivative? J. Comput. Phys. 293 (2015) 4-13, https://doi.org/10.1016/j.jcp.2014.07.019.
  57. G. Espinosa-Paredes, R.-I. Cazares-Ramirez, Source term in the linear analysis of FNPK equations, Ann. Nucl. Energy 96 (2016) 432-440, https://doi.org/10.1016/j.anucene.2016.06.038.
  58. J.J. Duderstadt, L.J. Hamilton, Nuclear Reactor Analysis, 1976.
  59. G. Espinosa-Paredes, M.A. Polo-Labarrios, E.G. Espinosa-Martinez, E. Del Valle-Gallegos, Fractional neutron point kinetics equations for nuclear reactor dynamics, Ann. Nucl. Energy 38 (2011) 307-330, https://doi.org/10.1016/j.anucene.2010.10.012.
  60. D.S. Chandrasekharaiah, Thermoelasticity with second sound: a review, Appl. Mech. Rev. 39 (1986) 355-376, https://doi.org/10.1115/1.3143705.
  61. T.Q. Qiu, C.L. Tien, Heat transfer mechanisms during short-pulse laser heating of metals, J. Heat Tran. 115 (1993) 835-841, https://doi.org/10.1115/1.2911377.
  62. H. Kim, M.H. Kim, M. Kaviany, Lattice thermal conductivity of UO2 using ab-initio and classical molecular dynamics, J. Appl. Phys. 115 (2014), 123510, https://doi.org/10.1063/1.4869669.
  63. C.A. Monje, Y. Chen, B.M. Vinagre, D. Xue, V. Feliu, Fractional-order Systems and Controls, Springer London, London, 2010, https://doi.org/10.1007/978-1-84996-335-0.
  64. Mukesh D. Patil, Vishwesh.A. Vyawahare, Manisha K. Bhole, A new and simple method to construct root locus of general fractional-order systems, ISA (Instrum. Soc. Am.) Trans. 53 (2) (2014) 380-390, https://doi.org/10.1016/j.isatra.2013.09.002.