November  2011, 16(4): 1171-1183. doi: 10.3934/dcdsb.2011.16.1171

Numerical simulation of two-fluid flow and meniscus interface movement in the electromagnetic continuous steel casting process

1. 

Department of Mathematics, Faculty of Science, Mahidol University, 272 Rama 6 Road, Bangkok, ZIP 10400

2. 

Department of Mathematics, Faculty of Science, Mahidol University, Bangkok, 10400, Thailand

3. 

Department of Mathematics and Statistics, Curtin University of Technology, GOP Box U1987, Perth, WA 6845, Australia

Received  October 2010 Revised  May 2011 Published  August 2011

This paper presents a mathematical model and numerical technique for simulating the two-fluid flow and the meniscus interface movement in the electromagnetic continuous steel casting process. The governing equations include the continuity equation, the momentum equations, the energy equation, the level set equation and two transport equations for the electromagnetic field derived from the Maxwell's equations. The level set finite element method is applied to trace the movement of the interface between different fluids. In an attempt to optimize the casting process, the technique is then applied to study the influences of the imposed electromagnetic field and the mould oscillation pattern on the fluid flow, the meniscus shape and temperature distribution.
Citation: B. Wiwatanapataphee, Theeradech Mookum, Yong Hong Wu. Numerical simulation of two-fluid flow and meniscus interface movement in the electromagnetic continuous steel casting process. Discrete & Continuous Dynamical Systems - B, 2011, 16 (4) : 1171-1183. doi: 10.3934/dcdsb.2011.16.1171
References:
[1]

J. Archapitak, B. Wiwatanapataphee and Y. H. Wu, A finite element scheme for the determination of electromagnetic force in continuous steel casting,, Int. J. Computational and Numerical Analysis and Applications, 5 (2004), 81. Google Scholar

[2]

W. J. Boettinger, S. R. Coriell, A. L. Greer, A. Karma, W. Kurz, M. Rappaz and R. Trivedi, Solidification microstructure: Recent developments, future direction,, Acta Mater, 48 (2000), 43. doi: 10.1016/S1359-6454(99)00287-6. Google Scholar

[3]

J. U. Brackbill, D. Kothe and C. Zemach, A Continuum method for modeling surface tension,, J. Comput. Phys., 100 (1992), 335. doi: 10.1016/0021-9991(92)90240-Y. Google Scholar

[4]

Y. C. Chang, T. Y. Hou, B. Merriman and S. Osher, A level set formulation of Eulerian interface capturing methods for incompressible fluid flows,, J. Comput. Phys., 124 (1996), 449. doi: 10.1006/jcph.1996.0072. Google Scholar

[5]

F. Duarte, R. Gormaz and S. Natesan, Arbitrary Lagrangian-Eulerian method for Navier-Stokes equations with moving boundaries,, Comp. Methods Appl. Mech. Engrg., 193 (2004), 4819. doi: 10.1016/j.cma.2004.05.003. Google Scholar

[6]

J. H. Ferziger, Simulation of incompressible turbulent flows,, Journal of Computational Physics, 69 (1999), 1. doi: 10.1016/0021-9991(87)90154-9. Google Scholar

[7]

V. Girault, H. Lopez and B. Maury, One time-step finite element discretization of the equation of motion of two-fluid flows,, Numer Methods Partial Differential Eq., 22 (2006), 680. doi: 10.1002/num.20117. Google Scholar

[8]

F. H. Harlow and P. I. Nakayama, Turbulence transport equations,, Phys Fluids, 10 (1967), 2323. doi: 10.1063/1.1762039. Google Scholar

[9]

M. D. Gunzburger, "Finite Element Methods for Viscous Incompressible Flows. A Guide to Theory, Practice, and Algorithms,", Computer Science and Scientific Computing, (1989). Google Scholar

[10]

J. M. Hill and Y.-H. Wu, On a nonlinear Stefan problem arising in the continuous casting of steel,, Acta Mechanica, 107 (1994), 183. doi: 10.1007/BF01201828. Google Scholar

[11]

J. M. Hill, Y.-H. Wu and B. Wiwatanapataphee, Mathematical analysis of the formation of oscillation marks in the continuous steel casting,, Engineering Mathematics, 36 (1999), 311. Google Scholar

[12]

D. R. Jenkins and F. R. De Hoog, "Calculation of the Magnetic Field Due to the Electromagnetic Stirring of Molten Steel,", Numerical Methods in Engineering'96, (1996), 332. Google Scholar

[13]

A. Karma, Phase-field formulation for quantitative method of alloy solidification,, Phys Rev. Lett., 8711 (2001). Google Scholar

[14]

H. Kim, J. Park, H. Jeong and J. Kim, Continuous casting of billet with high frequency electromagnetic field,, ISIJ International, 42 (2002), 171. doi: 10.2355/isijinternational.42.171. Google Scholar

[15]

B. Li and F. Tsukihashi, Effect of static magnetic field application on the mass transfer in sequence slab continuous casting process,, ISIJ International, 41 (2001), 844. doi: 10.2355/isijinternational.41.844. Google Scholar

[16]

X-Y. Luo, M-J. Ni, A. Ying and M. Abdou, Application of the level set method for multi-phase flow computation in fusion engineering,, Fusion Engineering and Design, 81 (2006), 1521. doi: 10.1016/j.fusengdes.2005.09.051. Google Scholar

[17]

B. Maury, Characteristics ALE method for the unsteady 3D Navier-Stokes Equations with a free surface,, Comp. Fluid Dyn., 6 (1996), 175. doi: 10.1080/10618569608940780. Google Scholar

[18]

E. Olssen, G. Kreiss and S. Zahedi, A conservative level set method for two phase flow. II,, J. Comput. Phys., 225 (2007), 785. Google Scholar

[19]

U. Pasaogullari and C.-Y. Wang, Two-phase modeling and flooding prediction of polymer electrolyte fuel cells,, J. of the Electromhemical Society, 152 (2005). doi: 10.1149/1.1850339. Google Scholar

[20]

W. Rodi and D. B. Spalding, A two-parameter model of turbulence and its application to free jets,, Warme-und Stofubertragung, 3 (1970), 85. doi: 10.1007/BF01108029. Google Scholar

[21]

P. Sivesson, G. Hallen and B. Widell, Improvement of inner quality of continuously cast billets using electromagnetic stirring and thermal soft reduction,, Ironmaking & Steelmaking, 25 (1998), 239. Google Scholar

[22]

B. G. Thomas, Metallurgical Transactions B,, 21 (1990), 21 (1990), 387. Google Scholar

[23]

B. G. Thomas, Continuous casting: Modelling,, in, (2001). Google Scholar

[24]

H. S. Udaykumar, S. Marella and S. Krishman, Sharp-interface simulation of dendritic growth with convection: Benchmarks,, Int. J. Heat Mass Transfer, 46 (2003), 2615. doi: 10.1016/S0017-9310(03)00038-3. Google Scholar

[25]

B. Wiwatanapataphee, Y. H. Wu, J. Archapitak, P. F. Siew and B. Unyong, A numerical study of the turbulent flow of molten steel in a domain with a phase-change boundary,, Journal of Computational and Applied Mathematics, 166 (2004), 307. doi: 10.1016/j.cam.2003.09.020. Google Scholar

[26]

B. Wiwatanapataphee, "Mathematical Modelling of Fluid Flow and Heat Transfer in Continuous Steel Casting Process,", Ph.D Thesis, (1998). Google Scholar

[27]

Y.-H. Wu, J. M. Hill and P. Flint, A novel finite element method for heat transfer in the continuous caster,, J. Austral. Math. Soc. Ser. B, 35 (1994), 263. doi: 10.1017/S0334270000009292. Google Scholar

[28]

Y. H. Wu and B. Wiwatanapataphee, An Enthalpy control volume method for transient mass and heat transport with solidification,, Int. J. of Computational Fluid Dynamics, 18 (2004), 577. doi: 10.1080/1061856031000137026. Google Scholar

[29]

Y.-H. Wu and B. Wiwatanapataphee, Modelling of turbulent flow and multi-phase heat transfer under electromagnetic force,, Discrete and Continuous Dynamical System Series B, 8 (2007), 695. doi: 10.3934/dcdsb.2007.8.695. Google Scholar

[30]

Yi Yang and H. S. Udaykumar, Sharp interface Cartesian method III: Solidification of pure materials and binary solutions,, Journal of Computational Physics, 210 (2005), 55. doi: 10.1016/j.jcp.2005.04.024. Google Scholar

show all references

References:
[1]

J. Archapitak, B. Wiwatanapataphee and Y. H. Wu, A finite element scheme for the determination of electromagnetic force in continuous steel casting,, Int. J. Computational and Numerical Analysis and Applications, 5 (2004), 81. Google Scholar

[2]

W. J. Boettinger, S. R. Coriell, A. L. Greer, A. Karma, W. Kurz, M. Rappaz and R. Trivedi, Solidification microstructure: Recent developments, future direction,, Acta Mater, 48 (2000), 43. doi: 10.1016/S1359-6454(99)00287-6. Google Scholar

[3]

J. U. Brackbill, D. Kothe and C. Zemach, A Continuum method for modeling surface tension,, J. Comput. Phys., 100 (1992), 335. doi: 10.1016/0021-9991(92)90240-Y. Google Scholar

[4]

Y. C. Chang, T. Y. Hou, B. Merriman and S. Osher, A level set formulation of Eulerian interface capturing methods for incompressible fluid flows,, J. Comput. Phys., 124 (1996), 449. doi: 10.1006/jcph.1996.0072. Google Scholar

[5]

F. Duarte, R. Gormaz and S. Natesan, Arbitrary Lagrangian-Eulerian method for Navier-Stokes equations with moving boundaries,, Comp. Methods Appl. Mech. Engrg., 193 (2004), 4819. doi: 10.1016/j.cma.2004.05.003. Google Scholar

[6]

J. H. Ferziger, Simulation of incompressible turbulent flows,, Journal of Computational Physics, 69 (1999), 1. doi: 10.1016/0021-9991(87)90154-9. Google Scholar

[7]

V. Girault, H. Lopez and B. Maury, One time-step finite element discretization of the equation of motion of two-fluid flows,, Numer Methods Partial Differential Eq., 22 (2006), 680. doi: 10.1002/num.20117. Google Scholar

[8]

F. H. Harlow and P. I. Nakayama, Turbulence transport equations,, Phys Fluids, 10 (1967), 2323. doi: 10.1063/1.1762039. Google Scholar

[9]

M. D. Gunzburger, "Finite Element Methods for Viscous Incompressible Flows. A Guide to Theory, Practice, and Algorithms,", Computer Science and Scientific Computing, (1989). Google Scholar

[10]

J. M. Hill and Y.-H. Wu, On a nonlinear Stefan problem arising in the continuous casting of steel,, Acta Mechanica, 107 (1994), 183. doi: 10.1007/BF01201828. Google Scholar

[11]

J. M. Hill, Y.-H. Wu and B. Wiwatanapataphee, Mathematical analysis of the formation of oscillation marks in the continuous steel casting,, Engineering Mathematics, 36 (1999), 311. Google Scholar

[12]

D. R. Jenkins and F. R. De Hoog, "Calculation of the Magnetic Field Due to the Electromagnetic Stirring of Molten Steel,", Numerical Methods in Engineering'96, (1996), 332. Google Scholar

[13]

A. Karma, Phase-field formulation for quantitative method of alloy solidification,, Phys Rev. Lett., 8711 (2001). Google Scholar

[14]

H. Kim, J. Park, H. Jeong and J. Kim, Continuous casting of billet with high frequency electromagnetic field,, ISIJ International, 42 (2002), 171. doi: 10.2355/isijinternational.42.171. Google Scholar

[15]

B. Li and F. Tsukihashi, Effect of static magnetic field application on the mass transfer in sequence slab continuous casting process,, ISIJ International, 41 (2001), 844. doi: 10.2355/isijinternational.41.844. Google Scholar

[16]

X-Y. Luo, M-J. Ni, A. Ying and M. Abdou, Application of the level set method for multi-phase flow computation in fusion engineering,, Fusion Engineering and Design, 81 (2006), 1521. doi: 10.1016/j.fusengdes.2005.09.051. Google Scholar

[17]

B. Maury, Characteristics ALE method for the unsteady 3D Navier-Stokes Equations with a free surface,, Comp. Fluid Dyn., 6 (1996), 175. doi: 10.1080/10618569608940780. Google Scholar

[18]

E. Olssen, G. Kreiss and S. Zahedi, A conservative level set method for two phase flow. II,, J. Comput. Phys., 225 (2007), 785. Google Scholar

[19]

U. Pasaogullari and C.-Y. Wang, Two-phase modeling and flooding prediction of polymer electrolyte fuel cells,, J. of the Electromhemical Society, 152 (2005). doi: 10.1149/1.1850339. Google Scholar

[20]

W. Rodi and D. B. Spalding, A two-parameter model of turbulence and its application to free jets,, Warme-und Stofubertragung, 3 (1970), 85. doi: 10.1007/BF01108029. Google Scholar

[21]

P. Sivesson, G. Hallen and B. Widell, Improvement of inner quality of continuously cast billets using electromagnetic stirring and thermal soft reduction,, Ironmaking & Steelmaking, 25 (1998), 239. Google Scholar

[22]

B. G. Thomas, Metallurgical Transactions B,, 21 (1990), 21 (1990), 387. Google Scholar

[23]

B. G. Thomas, Continuous casting: Modelling,, in, (2001). Google Scholar

[24]

H. S. Udaykumar, S. Marella and S. Krishman, Sharp-interface simulation of dendritic growth with convection: Benchmarks,, Int. J. Heat Mass Transfer, 46 (2003), 2615. doi: 10.1016/S0017-9310(03)00038-3. Google Scholar

[25]

B. Wiwatanapataphee, Y. H. Wu, J. Archapitak, P. F. Siew and B. Unyong, A numerical study of the turbulent flow of molten steel in a domain with a phase-change boundary,, Journal of Computational and Applied Mathematics, 166 (2004), 307. doi: 10.1016/j.cam.2003.09.020. Google Scholar

[26]

B. Wiwatanapataphee, "Mathematical Modelling of Fluid Flow and Heat Transfer in Continuous Steel Casting Process,", Ph.D Thesis, (1998). Google Scholar

[27]

Y.-H. Wu, J. M. Hill and P. Flint, A novel finite element method for heat transfer in the continuous caster,, J. Austral. Math. Soc. Ser. B, 35 (1994), 263. doi: 10.1017/S0334270000009292. Google Scholar

[28]

Y. H. Wu and B. Wiwatanapataphee, An Enthalpy control volume method for transient mass and heat transport with solidification,, Int. J. of Computational Fluid Dynamics, 18 (2004), 577. doi: 10.1080/1061856031000137026. Google Scholar

[29]

Y.-H. Wu and B. Wiwatanapataphee, Modelling of turbulent flow and multi-phase heat transfer under electromagnetic force,, Discrete and Continuous Dynamical System Series B, 8 (2007), 695. doi: 10.3934/dcdsb.2007.8.695. Google Scholar

[30]

Yi Yang and H. S. Udaykumar, Sharp interface Cartesian method III: Solidification of pure materials and binary solutions,, Journal of Computational Physics, 210 (2005), 55. doi: 10.1016/j.jcp.2005.04.024. Google Scholar

[1]

Zhenlin Guo, Ping Lin, Guangrong Ji, Yangfan Wang. Retinal vessel segmentation using a finite element based binary level set method. Inverse Problems & Imaging, 2014, 8 (2) : 459-473. doi: 10.3934/ipi.2014.8.459

[2]

Sheng Xu. Derivation of principal jump conditions for the immersed interface method in two-fluid flow simulation. Conference Publications, 2009, 2009 (Special) : 838-845. doi: 10.3934/proc.2009.2009.838

[3]

Kun Wang, Yinnian He, Yueqiang Shang. Fully discrete finite element method for the viscoelastic fluid motion equations. Discrete & Continuous Dynamical Systems - B, 2010, 13 (3) : 665-684. doi: 10.3934/dcdsb.2010.13.665

[4]

Caterina Calgaro, Meriem Ezzoug, Ezzeddine Zahrouni. Stability and convergence of an hybrid finite volume-finite element method for a multiphasic incompressible fluid model. Communications on Pure & Applied Analysis, 2018, 17 (2) : 429-448. doi: 10.3934/cpaa.2018024

[5]

Daniele Boffi, Lucia Gastaldi. Discrete models for fluid-structure interactions: The finite element Immersed Boundary Method. Discrete & Continuous Dynamical Systems - S, 2016, 9 (1) : 89-107. doi: 10.3934/dcdss.2016.9.89

[6]

Cornel M. Murea, H. G. E. Hentschel. A finite element method for growth in biological development. Mathematical Biosciences & Engineering, 2007, 4 (2) : 339-353. doi: 10.3934/mbe.2007.4.339

[7]

Martin Burger, José A. Carrillo, Marie-Therese Wolfram. A mixed finite element method for nonlinear diffusion equations. Kinetic & Related Models, 2010, 3 (1) : 59-83. doi: 10.3934/krm.2010.3.59

[8]

Jiaping Yu, Haibiao Zheng, Feng Shi, Ren Zhao. Two-grid finite element method for the stabilization of mixed Stokes-Darcy model. Discrete & Continuous Dynamical Systems - B, 2019, 24 (1) : 387-402. doi: 10.3934/dcdsb.2018109

[9]

Barbara Lee Keyfitz, Richard Sanders, Michael Sever. Lack of hyperbolicity in the two-fluid model for two-phase incompressible flow. Discrete & Continuous Dynamical Systems - B, 2003, 3 (4) : 541-563. doi: 10.3934/dcdsb.2003.3.541

[10]

Jingzhi Li, Hongyu Liu, Qi Wang. Fast imaging of electromagnetic scatterers by a two-stage multilevel sampling method. Discrete & Continuous Dynamical Systems - S, 2015, 8 (3) : 547-561. doi: 10.3934/dcdss.2015.8.547

[11]

T. Tachim Medjo. On the Newton method in robust control of fluid flow. Discrete & Continuous Dynamical Systems - A, 2003, 9 (5) : 1201-1222. doi: 10.3934/dcds.2003.9.1201

[12]

Binjie Li, Xiaoping Xie, Shiquan Zhang. New convergence analysis for assumed stress hybrid quadrilateral finite element method. Discrete & Continuous Dynamical Systems - B, 2017, 22 (7) : 2831-2856. doi: 10.3934/dcdsb.2017153

[13]

Junjiang Lai, Jianguo Huang. A finite element method for vibration analysis of elastic plate-plate structures. Discrete & Continuous Dynamical Systems - B, 2009, 11 (2) : 387-419. doi: 10.3934/dcdsb.2009.11.387

[14]

So-Hsiang Chou. An immersed linear finite element method with interface flux capturing recovery. Discrete & Continuous Dynamical Systems - B, 2012, 17 (7) : 2343-2357. doi: 10.3934/dcdsb.2012.17.2343

[15]

Donald L. Brown, Vasilena Taralova. A multiscale finite element method for Neumann problems in porous microstructures. Discrete & Continuous Dynamical Systems - S, 2016, 9 (5) : 1299-1326. doi: 10.3934/dcdss.2016052

[16]

Qingping Deng. A nonoverlapping domain decomposition method for nonconforming finite element problems. Communications on Pure & Applied Analysis, 2003, 2 (3) : 297-310. doi: 10.3934/cpaa.2003.2.297

[17]

Runchang Lin. A robust finite element method for singularly perturbed convection-diffusion problems. Conference Publications, 2009, 2009 (Special) : 496-505. doi: 10.3934/proc.2009.2009.496

[18]

Stefan Berres, Ricardo Ruiz-Baier, Hartmut Schwandt, Elmer M. Tory. An adaptive finite-volume method for a model of two-phase pedestrian flow. Networks & Heterogeneous Media, 2011, 6 (3) : 401-423. doi: 10.3934/nhm.2011.6.401

[19]

Jing Xu, Xue-Cheng Tai, Li-Lian Wang. A two-level domain decomposition method for image restoration. Inverse Problems & Imaging, 2010, 4 (3) : 523-545. doi: 10.3934/ipi.2010.4.523

[20]

Ting Zhou. Reconstructing electromagnetic obstacles by the enclosure method. Inverse Problems & Imaging, 2010, 4 (3) : 547-569. doi: 10.3934/ipi.2010.4.547

2018 Impact Factor: 1.008

Metrics

  • PDF downloads (13)
  • HTML views (0)
  • Cited by (0)

[Back to Top]