# American Institute of Mathematical Sciences

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 and 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), 81-95. [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), 43-70. doi: 10.1016/S1359-6454(99)00287-6. [3] J. U. Brackbill, D. Kothe and C. Zemach, A Continuum method for modeling surface tension, J. Comput. Phys., 100 (1992), 335-354. doi: 10.1016/0021-9991(92)90240-Y. [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), 449-464. doi: 10.1006/jcph.1996.0072. [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), 4819-4836. doi: 10.1016/j.cma.2004.05.003. [6] J. H. Ferziger, Simulation of incompressible turbulent flows, Journal of Computational Physics, 69 (1999), 1-48. doi: 10.1016/0021-9991(87)90154-9. [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), 680-707. doi: 10.1002/num.20117. [8] F. H. Harlow and P. I. Nakayama, Turbulence transport equations, Phys Fluids, 10 (1967), 2323-2328. doi: 10.1063/1.1762039. [9] M. D. Gunzburger, "Finite Element Methods for Viscous Incompressible Flows. A Guide to Theory, Practice, and Algorithms," Computer Science and Scientific Computing, Academic Press, Boston, MA, 1989. [10] J. M. Hill and Y.-H. Wu, On a nonlinear Stefan problem arising in the continuous casting of steel, Acta Mechanica, 107 (1994), 183-198. doi: 10.1007/BF01201828. [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), 311-326. [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, John Wiley and Sons Ltd, (1996), 332-336. [13] A. Karma, Phase-field formulation for quantitative method of alloy solidification, Phys Rev. Lett., 8711 (2001), art. no. 115701. [14] H. Kim, J. Park, H. Jeong and J. Kim, Continuous casting of billet with high frequency electromagnetic field, ISIJ International, 42 (2002), 171-177. doi: 10.2355/isijinternational.42.171. [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), 844-850. doi: 10.2355/isijinternational.41.844. [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), 1521-1526. doi: 10.1016/j.fusengdes.2005.09.051. [17] B. Maury, Characteristics ALE method for the unsteady 3D Navier-Stokes Equations with a free surface, Comp. Fluid Dyn., 6 (1996), 175-188. doi: 10.1080/10618569608940780. [18] E. Olssen, G. Kreiss and S. Zahedi, A conservative level set method for two phase flow. II, J. Comput. Phys., 225 (2007), 785-807. [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), A380-A390. doi: 10.1149/1.1850339. [20] W. Rodi and D. B. Spalding, A two-parameter model of turbulence and its application to free jets, Warme-und Stofubertragung, 3 (1970), 85-95. doi: 10.1007/BF01108029. [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), 239-246. [22] B. G. Thomas, Metallurgical Transactions B, 21 (1990), 387-400. [23] B. G. Thomas, Continuous casting: Modelling, in "The Encyclopedia of Advanced Materials" (eds. J. Dantzig, A. Greenwell and J. Michalczyk), Pergamon Elsevier Science Ltd, UK, 2001. [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), 2615-2627. doi: 10.1016/S0017-9310(03)00038-3. [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), 307-319. doi: 10.1016/j.cam.2003.09.020. [26] B. Wiwatanapataphee, "Mathematical Modelling of Fluid Flow and Heat Transfer in Continuous Steel Casting Process," Ph.D Thesis, School of Mathematics, Curtin University of Technology, Australia, 1998. [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), 263-288. doi: 10.1017/S0334270000009292. [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), 577-584. doi: 10.1080/1061856031000137026. [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), 695-706. doi: 10.3934/dcdsb.2007.8.695. [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), 55-74. doi: 10.1016/j.jcp.2005.04.024.

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), 81-95. [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), 43-70. doi: 10.1016/S1359-6454(99)00287-6. [3] J. U. Brackbill, D. Kothe and C. Zemach, A Continuum method for modeling surface tension, J. Comput. Phys., 100 (1992), 335-354. doi: 10.1016/0021-9991(92)90240-Y. [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), 449-464. doi: 10.1006/jcph.1996.0072. [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), 4819-4836. doi: 10.1016/j.cma.2004.05.003. [6] J. H. Ferziger, Simulation of incompressible turbulent flows, Journal of Computational Physics, 69 (1999), 1-48. doi: 10.1016/0021-9991(87)90154-9. [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), 680-707. doi: 10.1002/num.20117. [8] F. H. Harlow and P. I. Nakayama, Turbulence transport equations, Phys Fluids, 10 (1967), 2323-2328. doi: 10.1063/1.1762039. [9] M. D. Gunzburger, "Finite Element Methods for Viscous Incompressible Flows. A Guide to Theory, Practice, and Algorithms," Computer Science and Scientific Computing, Academic Press, Boston, MA, 1989. [10] J. M. Hill and Y.-H. Wu, On a nonlinear Stefan problem arising in the continuous casting of steel, Acta Mechanica, 107 (1994), 183-198. doi: 10.1007/BF01201828. [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), 311-326. [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, John Wiley and Sons Ltd, (1996), 332-336. [13] A. Karma, Phase-field formulation for quantitative method of alloy solidification, Phys Rev. Lett., 8711 (2001), art. no. 115701. [14] H. Kim, J. Park, H. Jeong and J. Kim, Continuous casting of billet with high frequency electromagnetic field, ISIJ International, 42 (2002), 171-177. doi: 10.2355/isijinternational.42.171. [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), 844-850. doi: 10.2355/isijinternational.41.844. [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), 1521-1526. doi: 10.1016/j.fusengdes.2005.09.051. [17] B. Maury, Characteristics ALE method for the unsteady 3D Navier-Stokes Equations with a free surface, Comp. Fluid Dyn., 6 (1996), 175-188. doi: 10.1080/10618569608940780. [18] E. Olssen, G. Kreiss and S. Zahedi, A conservative level set method for two phase flow. II, J. Comput. Phys., 225 (2007), 785-807. [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), A380-A390. doi: 10.1149/1.1850339. [20] W. Rodi and D. B. Spalding, A two-parameter model of turbulence and its application to free jets, Warme-und Stofubertragung, 3 (1970), 85-95. doi: 10.1007/BF01108029. [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), 239-246. [22] B. G. Thomas, Metallurgical Transactions B, 21 (1990), 387-400. [23] B. G. Thomas, Continuous casting: Modelling, in "The Encyclopedia of Advanced Materials" (eds. J. Dantzig, A. Greenwell and J. Michalczyk), Pergamon Elsevier Science Ltd, UK, 2001. [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), 2615-2627. doi: 10.1016/S0017-9310(03)00038-3. [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), 307-319. doi: 10.1016/j.cam.2003.09.020. [26] B. Wiwatanapataphee, "Mathematical Modelling of Fluid Flow and Heat Transfer in Continuous Steel Casting Process," Ph.D Thesis, School of Mathematics, Curtin University of Technology, Australia, 1998. [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), 263-288. doi: 10.1017/S0334270000009292. [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), 577-584. doi: 10.1080/1061856031000137026. [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), 695-706. doi: 10.3934/dcdsb.2007.8.695. [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), 55-74. doi: 10.1016/j.jcp.2005.04.024.
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