Journal of Nuclear Science, Engineering and Technology (JONSAT)
In cooperation with the Iranian Nuclear Society

Simulation of two-phase flow drift flux model in a vertical channel using Newton–Krylov method

Document Type : Research Paper

Authors

H. Esmaili 1
H. Kazeminejad 2
H. Khalafi 1
S.M. Mirvakili 1

1 Nuclear Science and Technology Research Institute (NSTRI), AEOI, P.O. Box 11365-3486, Tehran, Iran

2 Radiation Application Research School, Nuclear Science and Technology Research Institute , ,AEOI, P.O.Box:11365-3486, Tehran Iran

https://doi.org/10.24200/nst.2021.1179
Abstract
In the present work, a numerical method is proposed in order to model the subcooled boiling flow in a vertical channel using the Drift Flux Model. The system of nonlinear equations is solved with the fully-implicit scheme using the Jacobian-free Newton–Krylov (JFNK) method. In order to improve the efficiency of the JFNK method and its numerical stability, a semi-implicit physics based preconditioning (PBP) is implemented. Recently, the JFNK method has been widely used to solve large and sparse system of equations. To validate the proposed method, the results were compared with the experimental data, the results of modeling by using the RELAP5 code, and the available numerical results in the literature. It was found that the results corresponding to the present work have a good agreement with those of the other mentioned methods. Also, it was found that the convergence rate of the JFNK method with the PBP is at least 50% higher than the JFNK method, and the void fraction mean absolute percentage error (MAPE) is less than 7.34% over a wide range of flow and pressure

Highlights

1.     J. C. Collier, Convective Boiling and Condensation, oxford engineering science series (1994).

 

2.    M. Ishii, T. Hibiki, Thermo-Fluid Dynamics of Two-Phase Flow, Springer (2006).

 

3.      D. Liles, W. H. Reed, A Semi-implicit Method for Two phase Fluid Dynamics, J. Comput. Phys., 26, 390-407 (1978).

 

4.     Y. G. Lee, G. C. Park, TAPINS: a thermal-hydraulic system code for transient analysis of a fully-passive integral PWR, Nucl. Eng. Technol. 45,172-185 (2013).

 

5.       S. Talebi, H. Kazeminejad, H. Davilu, A numerical technique for analysis of transient two-phase flow in a vertical tube using the drift flux model, Nucl. Eng. Des. 242, 316–322 (2012).

 

6.   RELAP5 Code Development Team, RELAP/MOD3 Code manual, Idaho national engineering and environmental laboratory, vol. 1-6. Idaho 83415 (2001).

 

7.       RETRAN-3D – A Program for Transient Thermal-Hydraulic Analysis of Complex Fluid Flow Systems: Volume 1: Theory and Numerics (Revision 3), NP-7450-V1R3, Electric Power Research Institute, October, (1998).

 

8.     C. H. Frepoli, J. Mahaffy, K. Ohkawa, Notes on the implementation of a fully implicit numerical scheme for a two-phase three-field flow model, Nucl. Eng. Des. 225 (2003) 191–217.

 

9.   D.A. Knoll, D.E. Keyes, Jacobian-free Newton-Krylov methods: a survey of approaches and applications, J. Comput. Phys. 193 (2004) 357–397.

 

10.  A. Segal, M. Rehman, C. Vuik, Review article: preconditioners for incompressible navier-stokes solvers, Numer. Math. Theory, Methods Appl. 3 (2010) 245-275.

 

11.   H. Elman, et al., Block preconditioners based on approximate commutators, SIAM J. Sci. Comput. 27 (2006) 1651-1668.

 

12.   A. Hajizadeh, H. Kazeminejad, S.  Talebi, A new numerical method for solution of boiling flow using combination of SIMPLE and Jacobian-free Newton-Krylov algorithms, Progr. Nucl. Energy, 95 (2017) 48-60.

 

13. L. Zou, H. Zhao, H. Zhang, Numerical implementation, verification and validation of two-phase flow four-equation drift flux model with

Jacobian free Newton-Krylov method, Ann. Nucl. Energy 87 (2015) 707-719.

 

14.   V. A. Mousseau, Implicitly balanced solution of the two-phase flow equations coupled to nonlinear heat conduction, J. Comp. Phys. 200 (2004) 104–132.

 

15.   H. Elman, et al., A taxonomy and comparison of parallel block multi-level preconditioners for the incompressible Naviere Stokes equations, J. Comput. Phys. 227 (2008) 1790-1808.

 

16. J. Reisner, et al., An efficient physics-based preconditioner for the fully implicit solution of small-scale thermally driven atmospheric flows, J. Comput. Phys. 189 (2003) 30–44.

 

17.   L. Hu, et al., JFNK method with a physics-based preconditioner for the fully implicit solution of one-dimensional drift-flux model in boiling two-phase flow, App. Therm. Eng.  116 (2017) 610–622.

 

18. Y. Saad, Iterative Methods for Sparse Linear Systems, second ed., (2003).

 

19. T. Hibiki, M. Ishii, One-dimensional drift-flux model and constitutive equations for relative motion between phases in various two-phase flow regimes, Int. J. Heat Mass Transf. 46 (2003) 4935–4948.

 

20.  W. Wagner, A.  Pruss, The IAPWS formulation 1995 for thermodynamic properties of ordinary water for general and science use, J. Phys. Chem. Ref. Data 31 (2002) 387–535.

 

21.  B. Chexal, G. Lellouche, A Full-range Drift-flux Correlation for Vertical Flows, EPRI-NP-3989-SR (Revision 1), EPRI (1986).

 

22.  R. T. Lahey, A Mechanistic Subcooled Boiling Model, Proc. 6th International Heat Transfer Conference, 1 (1978) 293-297.

 

23.  Y. Saad, M. H.  Schultz, GMRES: a generalized minimal residual algorithm for solving non-symetric linear systems, SIAM J. Sci. Stat. Comp. 7 (1986) 856-569.

 

24.    G. G. Bartolomei, V. M.  Chanturiya, Experimental study of true void fraction when boiling subcooled water in vertical tubes, Therm. Eng. 14 (1967) 123–128.

 

25. G. G. Bartolomej, et al., An experimental investigation of the true volumetric vapor content with subcooled boiling tubes, Therm. Eng. 29 (1982) 132–1

Keywords

Two-phase flow
Drift Flux Model
JFNK method
1.     J. C. Collier, Convective Boiling and Condensation, oxford engineering science series (1994).
 
2.    M. Ishii, T. Hibiki, Thermo-Fluid Dynamics of Two-Phase Flow, Springer (2006).
 
3.      D. Liles, W. H. Reed, A Semi-implicit Method for Two phase Fluid Dynamics, J. Comput. Phys., 26, 390-407 (1978).
 
4.     Y. G. Lee, G. C. Park, TAPINS: a thermal-hydraulic system code for transient analysis of a fully-passive integral PWR, Nucl. Eng. Technol. 45,172-185 (2013).
 
5.       S. Talebi, H. Kazeminejad, H. Davilu, A numerical technique for analysis of transient two-phase flow in a vertical tube using the drift flux model, Nucl. Eng. Des. 242, 316–322 (2012).
 
6.   RELAP5 Code Development Team, RELAP/MOD3 Code manual, Idaho national engineering and environmental laboratory, vol. 1-6. Idaho 83415 (2001).
 
7.       RETRAN-3D – A Program for Transient Thermal-Hydraulic Analysis of Complex Fluid Flow Systems: Volume 1: Theory and Numerics (Revision 3), NP-7450-V1R3, Electric Power Research Institute, October, (1998).
 
8.     C. H. Frepoli, J. Mahaffy, K. Ohkawa, Notes on the implementation of a fully implicit numerical scheme for a two-phase three-field flow model, Nucl. Eng. Des. 225 (2003) 191–217.
 
9.   D.A. Knoll, D.E. Keyes, Jacobian-free Newton-Krylov methods: a survey of approaches and applications, J. Comput. Phys. 193 (2004) 357–397.
 
10.  A. Segal, M. Rehman, C. Vuik, Review article: preconditioners for incompressible navier-stokes solvers, Numer. Math. Theory, Methods Appl. 3 (2010) 245-275.
 
11.   H. Elman, et al., Block preconditioners based on approximate commutators, SIAM J. Sci. Comput. 27 (2006) 1651-1668.
 
12.   A. Hajizadeh, H. Kazeminejad, S.  Talebi, A new numerical method for solution of boiling flow using combination of SIMPLE and Jacobian-free Newton-Krylov algorithms, Progr. Nucl. Energy, 95 (2017) 48-60.
 
13. L. Zou, H. Zhao, H. Zhang, Numerical implementation, verification and validation of two-phase flow four-equation drift flux model with

Jacobian free Newton-Krylov method, Ann. Nucl. Energy 87 (2015) 707-719.
 
14.   V. A. Mousseau, Implicitly balanced solution of the two-phase flow equations coupled to nonlinear heat conduction, J. Comp. Phys. 200 (2004) 104–132.
 
15.   H. Elman, et al., A taxonomy and comparison of parallel block multi-level preconditioners for the incompressible Naviere Stokes equations, J. Comput. Phys. 227 (2008) 1790-1808.
 
16. J. Reisner, et al., An efficient physics-based preconditioner for the fully implicit solution of small-scale thermally driven atmospheric flows, J. Comput. Phys. 189 (2003) 30–44.
 
17.   L. Hu, et al., JFNK method with a physics-based preconditioner for the fully implicit solution of one-dimensional drift-flux model in boiling two-phase flow, App. Therm. Eng.  116 (2017) 610–622.
 
18. Y. Saad, Iterative Methods for Sparse Linear Systems, second ed., (2003).
 
19. T. Hibiki, M. Ishii, One-dimensional drift-flux model and constitutive equations for relative motion between phases in various two-phase flow regimes, Int. J. Heat Mass Transf. 46 (2003) 4935–4948.
 
20.  W. Wagner, A.  Pruss, The IAPWS formulation 1995 for thermodynamic properties of ordinary water for general and science use, J. Phys. Chem. Ref. Data 31 (2002) 387–535.
 
21.  B. Chexal, G. Lellouche, A Full-range Drift-flux Correlation for Vertical Flows, EPRI-NP-3989-SR (Revision 1), EPRI (1986).
 
22.  R. T. Lahey, A Mechanistic Subcooled Boiling Model, Proc. 6th International Heat Transfer Conference, 1 (1978) 293-297.
 
23.  Y. Saad, M. H.  Schultz, GMRES: a generalized minimal residual algorithm for solving non-symetric linear systems, SIAM J. Sci. Stat. Comp. 7 (1986) 856-569.
 
24.    G. G. Bartolomei, V. M.  Chanturiya, Experimental study of true void fraction when boiling subcooled water in vertical tubes, Therm. Eng. 14 (1967) 123–128.
 
25. G. G. Bartolomej, et al., An experimental investigation of the true volumetric vapor content with subcooled boiling tubes, Therm. Eng. 29 (1982) 132–1

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