Nuclear Engineering and Technology

Korean Nuclear Society (한국원자력학회)
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A comprehensive examination of the linear and numerical stability aspects of the bubble collision model in the TRACE-1D two-fluid model applied to vertical disperse flow in a PWR core channel under loss of coolant accident conditions

  • Satya Prakash Saraswat (Department of Mechanical and Nuclear Engineering, Khalifa University) ;
  • Yacine Addad (Department of Mechanical and Nuclear Engineering, Khalifa University)
  • Received : 2024.01.08
  • Accepted : 2024.02.29
  • Published : 2024.08.25
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Abstract

The one-dimensional Two-Fluid concept uses an area-average approach to simplify the time and phase-averaged Two-Fluid conservation equations, making it more suitable for addressing difficulties at an industrial scale. Nevertheless, the mathematical framework has inherent weaknesses due to the loss of details throughout the averaging procedures. This limitation makes the conventional model inappropriate for some flow regimes, where short-wavelength perturbations experience uncontrolled amplification, leading to solutions that need to be physically accurate. The critical factor in resolving this problem is the integration of closure relations. These relationships play a crucial function in reintroducing essential physical characteristics, thus correcting the loss that occurs during averaging and guaranteeing the stability of the model. To improve the accuracy of predictions, it is essential to assess the stability and grid dependence of one-dimensional formulations, which are particularly affected by closure relations and numerical schemes. The current research presented in the text focuses on improving the well-posedness of the TFM, specifically within the TRACE code, which is widely utilized for nuclear reactor safety assessments. Incorporating a bubble collision model in the momentum equations is demonstrated to enhance the TFM's resilience, especially in scenarios with high void fractions where conventional TFMs may face challenges. The analysis presents a linear stability analysis performed for the transient one-dimensional Two-Fluid Model of system code TRACE within the framework of vertically dispersed flows. The main emphasis is on evaluating the stability characteristics of the model while also acknowledging its susceptibility to closure relations and numerical techniques.

Keywords

Acknowledgement

This investigation was conducted with financial support from the project, entitled, Validation of Safety Analysis Code and Assessment of Thermal-Hydraulic Behaviours of APR1400 via OECD-ATLAS Phase 3, Grant No. 8434000468, funded by the Federal Authority for Nuclear Regulation (FANR) in the United Arab Emirates. The opinions presented herein are exclusively those of the authors and do not necessarily reflect the perspectives of the OECD-ATLAS collaboration or its team members. The responsibility for these views lies solely with the authors, and OECD-ATLAS and its team members are not accountable for them.

References

  1. M. Ishii, Thermo-fluid dynamic theory of two-phase flow, France: Eyrolles (1975). 
  2. J. Liao, R. Mei, J.F. Klausner, A study on the numerical stability of the two-fluid model near ill-posedness, Int. J. Multiphas. Flow 34 (2008) 1067-1087, https://doi.org/10.1016/j.ijmultiphaseflow.2008.02.010. 
  3. S.P. Saraswat, P. Munshi, C. Allison, Linear stability analysis of RELAP5 two-fluid model in nuclear reactor safety results, Ann. Nucl. Energy 149 (2020) 107720. 
  4. S.P. Saraswat, P. Munshi, C. Allison, Characteristics and linear stability analysis of RELAP5 two-fluid model for two-component, two-phase flow, Ann. Nucl. Energy 151 (2021) 107948. 
  5. M. Leporini, A. Bonzanini, M. Ferrari, P. Poesio, The extension of the one-dimensional two-fluid slug capturing method to simulate slug flow in vertical pipes, Int. J. Numer. Methods Fluid. 93 (2021) 816-833, https://doi.org/10.1002/fld.4909. 
  6. S.P. Saraswat, P. Munshi, C. Allison, Non-hyperbolicity of conservation equations of RELAP5 two-fluid model in nuclear reactor safety results (investigation and eigenvalue analysis), J. Nucl. Eng. Radiat. Sci. 7 (1) (2021) 011402. 
  7. C. Pauchon, S. Banerjee, Interphase momentum interaction effects in the averaged multifield model: Part I: void propagation in bubbly flows, Int. J. Multiphas. Flow 12 (4) (1986) 559-573. 
  8. S. Evje, W. Wang, H. Wen, Global well-posedness and decay rates of strong solutions to a non-conservative compressible two-fluid model, Arch. Ration. Mech. Anal. 221 (2016) 1285-1316. 
  9. Y. Chen, Q. He, B. Huang, X. Shi, Global strong solution to a thermodynamic compressible diffuse interface model with temperature-dependent heat conductivity in 1D, Math. Methods Appl. Sci. 44 (17) (2021) 12945-12962. 
  10. F. Chao, W. Yang, L. Li, J. Qiu, J. Shan, Eigenvalue analysis of well-posedness of two-fluid single pressure model with virtual mass force and interfacial pressure, in: International Conference on Nuclear Engineering, 85253, American Society of Mechanical Engineers, 2021 V002T07A016. 
  11. A. Vaidheeswaran, M.L. de Bertodano, Stability and convergence of computational Eulerian two-fluid model for a bubble plume, Chem. Eng. Sci. 160 (2017) 210-226. 
  12. B. Sanderse, J.F.H. Buist, R.A.W.M. Henkes, A novel pressure-free two-fluid model for one-dimensional incompressible multiphase flow, J. Comput. Phys. 426 (2021) 109919. 
  13. M.R. Ansari, V. Shokri, Numerical modeling of slug flow initiation in a horizontal channel using a two-fluid model, Int. J. Heat Fluid Flow 32 (1) (2011) 145-155. 
  14. E.M.G. Fontalvo, R.L.C. Branco, J.N.E. Carneiro, A.O. Nieckele, Assessment of closure relations on the numerical predictions of vertical annular flows with the Two-Fluid Model, Int. J. Multiphas. Flow 126 (2020) 103243, https://doi.org/10.1016/j.ijmultiphaseflow.2020.103243. 
  15. Y. Lou, T. Kawaue, I. Yow, Y. Toyama, J. Prost, T. Hiraiwa, Interfacial friction and substrate deformation mediate long-range signal propagation in tissues, Biomech. Model. Mechanobiol. 21 (5) (2022) 1511-1530. 
  16. K. Sekoguchi, K. Hori, M. Nakazatomi, K. Nishikawa, On ripple of annular two-phase flow: 2. Characteristics of wave and interfacial friction factor, Bulletin of JSME 21 (152) (1978) 279-286. 
  17. A.M. Aliyu, Y.D. Baba, L. Liao, H. Yeung, K.C. Kim, Interfacial friction in upward annular gas-liquid two-phase flow in pipes, Exp. Therm. Fluid Sci. 84 (2017) 90-109, https://doi.org/10.1016/j.expthermflusci.2017.02.006. 
  18. A. Vaidheeswaran, W.D. Fullmer, K. Chetty, R.G. Marino, M. Lopez de Bertodano, Stability analysis of chaotic wavy stratified fluid-fluid flow with the 1D fixed-flux two-fluid model, in: Fluids Engineering Division Summer Meeting, 50299, American Society of Mechanical Engineers, 2016 V01BT33A013. 
  19. R.J. Belt, J.M.C. Van't Westende, H.M. Prasser, L.M. Portela, Prediction of the interfacial shear-stress in vertical annular flow, Int. J. Multiphas. Flow 35 (2009) 689-697, https://doi.org/10.1016/j.ijmultiphaseflow.2008.12.003. 
  20. W. Bian, X. Chen, J. Wang, Assessment of the interphase drag coefficients considering the effect of granular temperature or solid concentration fluctuation via comparison of DNS, DPM, TFM and experimental data, Chem. Eng. Sci. 223 (2020) 115722. 
  21. Y. Zhang, G. Vinay, A. Poubeau, Q. Van Hoang, Development of a hybrid Lagrange-Euler transition model for the film formation and dynamics of an impinging liquid spray, Comput. Fluid 251 (2023) 105756. 
  22. F. Song, W. Wang, K. Hong, J. Li, Unification of EMMS and TFM: structure-dependent analysis of mass, momentum and energy conservation, Chem. Eng. Sci. 120 (2014) 112-116. 
  23. P.G. Saffman, On the rise of small air bubbles in water, J. Fluid Mech. 1 (1956) 249-275. 
  24. J. Mercier, A. Lyrio, R. Forslund, Three-dimensional study of the nonrectilinear trajectory of air bubbles rising in water, J. Appl. Mech. 40 (1973) 650-654. 
  25. C. Brucker, Structure and dynamics of the wake of bubbles and its relevance for bubble interaction, Phys. Fluids 11 (1999) 1781-1796. 
  26. C. Pauchon, S. Banerjee, Interphase momentum interaction effects in the averaged multifield model, part II: kinematic waves and interfacial drag in bubbly flows, Int. J. Multiphas. Flow 14 (3) (1988) 253-264. 
  27. S.P. Antal, R.T. Lahey, J.E. Flaherty, Analysis of phase distribution in fully developed laminar bubbly two-phase flow, Int. J. Multiphas. Flow 17 (1991) 635-652. 
  28. T. Haley, D. Drew, R. Lahey, An analysis of the Eigenvalues of bubbly two-phase flows, Chem. Eng. Commun. 106 (1991) 93-117. 
  29. M. Lopez de Bertodano, R.T. Lahey Jr., O.C. Jones, Development of a k-epsilon model for bubbly two-phase flow, J. Fluid Eng. 116 (1994) 128-134. 
  30. "RELAP5/MOD3.3 Code Manual, Vol. 1: Code Structure, System Models, and Solution Methods," NUREG/CR-5535/RevP3-Vol I, Information Systems Laboratories, 2003. 
  31. TRACE V5.0: Theory Manual, U.S. Nuclear Regulatory Commission, 2008. 
  32. S. Ogawa, A. Umemura, N. Oshima, On the equations of fully fluidized granular materials, Zeitschrift fur angewandte Mathematik und Physik ZAMP 31 (1980) 483-493. 
  33. S.B. Savage, D.J. Jeffrey, The stress tensor in a granular flow at high shear rates, J. Fluid Mech. 110 (1981) 255-272. 
  34. N.F. Carnahan, K.E. Starling, Equation of state for nonattracting rigid spheres, J. Chem. Phys. 51 (2) (1969) 635-636. 
  35. C.K. Lun, S.B. Savage, D.J. Jeffrey, N. Chepurniy, Kinetic theories for granular flow: inelastic particles in Couette flow and slightly inelastic particles in a general flowfield, J. Fluid Mech. 140 (1984) 223-256. 
  36. C.K. Lun, S.B. Savage, The effects of an impact velocity dependent coefficient of restitution on stresses developed by sheared granular materials, Acta Mech. 63 (1-4) (1986) 15-44. 
  37. A. Boelle, G. Balzer, O. Simonin, Second-order prediction of the particle-phase stress tensor of inelastic spheres in simple shear dense suspensions, ASME-Publications-Fed 228 (1995) 9-18. 
  38. A. Alajbegovic, D.A. Drew, R.T. Lahey Jr., An analysis of phase distribution and turbulence in dispersed particle/liquid flows, Chem. Eng. Commun. 174 (1) (1999) 85-133. 
  39. A. Ayodeji, M.A. Amidu, S.A. Olatubosun, Y. Addad, H. Ahmed, Deep learning for safety assessment of nuclear power reactors: reliability, explainability, and research opportunities, Prog. Nucl. Energy 151 (2022) 104339, https://doi.org/10.1016/j.pnucene.2022.104339. ISSN 0149-1970. 
  40. X. Wu, W. Li, Y. Wang, Y. Zhang, W. Tian, G. Su, H. Huang, Preliminary safety analysis of the PWR with accident-tolerant fuels during severe accident conditions, Ann. Nucl. Energy 80 (2015) 1-13. 
  41. Y.P. Zhang, S.P. Niu, L.T. Zhang, S.Z. Qiu, G.H. Su, W.X. Tian, A review on analysis of LWR severe accident, J. Nucl. Eng. Radiat. Sci. 1 (4) (2015) 041018. 
  42. A. Vaidheeswaran, M.A. Lopez de Bertodano, Stability and convergence of computational eulerian two-fluid model for a bubble plume, Chem. Eng. Sci. (2016), https://doi.org/10.1016/j.ces.2016.11.032. 
  43. M.A. Lopez de Bertodano, W. Fullmer, A. Vaidheeswaran, One-dimensional two-equation two-fluid model stability, Multiphas. Sci. Technol. 25 (2) (2013) 133-167. 
  44. A. Zaruba, E. Krepper, H.-M. Prasser, E. Schliecher, Measurement of bubble velocity profiles and turbulent diffusion coefficients of the gaseous phase in rectangular bubble column using image processing, Exp. Therm. Fluid Sci. 29 (2005) 851-860. 
  45. S. Chapman, T.G. Cowling, The Mathematical Theory of Non-uniform Gases, Cambridge University Press, London, 1970. 
  46. S.P. Saraswat, F. Galleni, M. Eboli, N. Forgione, A. Del Nevo, Numerical Investigation of LIFUS5/Mod3 Series E Experiment Test 5.1 towards Thermal-Hydraulic System Code" SIMMER-III" Validation (No. 9711), EasyChair, 2023. https://easychair.org/publications/preprint/ZlmB.
  47. S.P. Saraswat, V. Cossu, F. Galleni, M. Eboli, A. Del Nevo, N. Forgione, Progress towards validating the SIMMER-III code model for lead-lithium water chemical interaction, Fusion Eng. Des. 193 (2023) 113819, https://doi.org/10.1016/j.fusengdes.2023.113819. 
  48. S.P. Saraswat, N. Forgione, M.E. Angiolini, Advancement towards the experimental measurement of the Lbe thermal properties using DSC technique, J. Nucl. Eng. Radiat. Sci. (2023), https://doi.org/10.1115/1.4063572. 
  49. S.P. Saraswat, P. Munshi, A. Khanna, C. Allison, Thermal hydraulic and safety assessment of first wall helium cooling system of a generalized test blanket system in ITER using RELAP5 code, ASME J. Nucl. Rad. Sci. 3 (1) (2016) 014503, https://doi.org/10.1115/1.4034680. 
  50. S.P. Saraswat, P. Munshi, A. Khanna, C. Allison, Thermal hydraulic and safety assessment of LLCB test blanket system in ITER using modified RELAP/SCDAPSIM/MOD4.0 code, ASME J. Nucl. Eng. Radiat. Sci. 4 (2) (2018) 21001-21010, https://doi.org/10.1115/1.4038823. 
  51. D. Ray, S.P. Saraswat, M. Kumar, O.P. Singh, P. Munshi, Build up and characterization of ultraslow nuclear burn-up wave in epithermal neutron multiplying medium, J. Nucl. Eng. Radiat. Sci. 10 (2022) 1-4049727. 
  52. S.P. Saraswat, P. Munshi, A. Khanna, C. Allison, Ex-vessel loss of coolant accident analysis of ITER divertor cooling system using modified RELAP/SCADAPSIM/mod 4.0, J. Nucl. Eng. Radiat. Sci. 3 (4) (2017) 041009. 
  53. S.P. Saraswat, D. Ray, P. Munshi, C. Allison, Analysis of loss of heat sink for ITER divertor cooling system using modified RELAP/SCDAPSIM/MOD 4.0, J. Nucl. Eng. Radiat. Sci. 5 (4) (2019) 042202.