Convection modeling in directional solidification

Juan C. Heinrich; David R. Poirier

doi:10.1016/j.crme.2004.02.001

Microgravity and Transfer/Solidification, crystal growth from the melt

Convection modeling in directional solidification
[Modélisation de la convection au cours de la solidification directionnelle]

Juan C. Heinrich ¹ ; David R. Poirier ²

¹ Department of Aerospace and Mechanical Engineering, The University of Arizona, Tucson, AZ 85721, USA
² Department of Materials Science and Engineering, The University of Arizona, Tucson, AZ 85721, USA

Comptes Rendus. Mécanique, Volume 332 (2004) no. 5-6, pp. 429-445.

Résumé
Abstract

Nous exposons des modèles mathématiques et numériques de la solidification d'alliages dendritiques, binaires et multiconstituants, qui peuvent décrire la dynamique de la zone pâteuse ainsi que la région liquide. La discussion est centrée sur des modèles développés par les auteurs du présent article et basés sur la discrétisation en éléments finis des équations qui régissent les phénomènes. Nous discutons la capacité des programmes de simulation existants à modéliser les effets de la convection et de la macroségrégation qui en résulte dans les pièces coulées, et plus particulièrement la formation de « freckles » dans les monocristaux dendritiques solidifiés verticalement. Nous faisons ressortir les limites actuelles des modèles ainsi que les domaines dans lesquels des améliorations sont nécessaires. Des exemples numériques illustrent les différent aspects des simulations.

Mathematical and numerical models of solidification of binary and multicomponent dendritic alloys that can model the dynamics of the mushy zone as well as the all liquid region are examined. The discussion is centered around models based on finite element discretization of the governing equations that have been developed by the authors during the last fifteen years. The capabilities of existing simulation codes to model the effects of convection and the resulting macrosegregation in castings, and in particular, the development of ‘freckles’ in vertically solidified dendritic monocrystals are discussed. The current capabilities of the models as well as the areas in which more improvement is needed are noted. Numerical examples are presented to illustrate the different aspects of the simulations.

Publié le : 2004-04-12

DOI : 10.1016/j.crme.2004.02.001

Keywords: Dendritic solidification, Multicomponent alloys, Binary alloys, Directional solidification, Macrosegregation
Mot clés : Solidification dendritiques, Alliages multiconstituants, Alliages binaires, Solidification directionnelle, Macroségrégation

Affiliations des auteurs :

Juan C. Heinrich ¹ ; David R. Poirier ²

¹ Department of Aerospace and Mechanical Engineering, The University of Arizona, Tucson, AZ 85721, USA
² Department of Materials Science and Engineering, The University of Arizona, Tucson, AZ 85721, USA

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Juan C. Heinrich; David R. Poirier. Convection modeling in directional solidification. Comptes Rendus. Mécanique, Volume 332 (2004) no. 5-6, pp. 429-445. doi : 10.1016/j.crme.2004.02.001. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/j.crme.2004.02.001/

Bibliographie
Cité par

[1] G.J.S. Higginbotham From research to cost-effective directional solidification and single-crystal production – an integrated approach, Mater. Sci. Tech., Volume 2 (1986), pp. 442-460

[2] C. Frueh; D.R. Poirier; S.D. Felicelli Predicting freckle-defects in directionally solidified Pb–Sn alloys, Mat. Sci. Engrg. A, Volume 328 (2002), pp. 245-255

[3] M. McLean Directionally Solidified Materials for High Temperature Service, The Metals Society, London, 1983

[4] T.C. Midea; D. Schmidt Casting simulation software survey, Modern Casting, Volume 89 (1999), pp. 47-51

[5] M. Cruchaga; D. Celentano Numerical analysis of thermally coupled flow problems with interfaces and phase-change effects, Int. J. Comp. Fluid Dyn., Volume 16 (2002), pp. 247-262

[6] D. Juric; G. Tryggvason A front-tracking method for dendritic solidification, J. Comput. Phys., Volume 123 (1996), pp. 127-148

[7] H.S. Udaykumar; R. Mittal; W. Shyy Computation of solid–liquid phase fronts in the sharp interface limit on fixed grids, J. Comput. Phys., Volume 153 (1999), pp. 535-574

[8] P. Zhao; J.C. Heinrich Front – tracking finite element method for dendritic solidification, J. Comput. Phys., Volume 173 (2001), pp. 765-796

[9] A. Karma; W.J. Rappel Quantitative phase-field modeling of dendritic growth in two and three dimensions, Phys. Rev. E, Volume 57 (1998), pp. 4323-4349

[10] L.H. Ungar; R.A. Brown Cellular interface morphologies in directional solidification IV. The formation of deep cells, Phys. Rev. B, Volume 31 (1985), pp. 5931-5940

[11] J.A. Warren; W.J. Boettinger Prediction of dendritic growth and microsegregation patterns in a binary alloy using the phase-field method, Acta Metall. Mater., Volume 43 (1995), pp. 689-703

[12] P. Zhao; M. Vénere; J.C. Heinrich; D.R. Poirier Modeling dendritic growth of a binary alloy, J. Comput. Phys., Volume 188 (2003), pp. 434-461

[13] J.S. Kirkaldy; W.V. Youdelis Contribution to the theory of inverse segregation, Trans. TMS-AIME, Volume 212 (1958), pp. 833-840

[14] M.C. Flemings; G.E. Nereo Macrosegregation: Part I, Trans. Metall Soc. AIME, Volume 239 (1967), pp. 1449-1461

[15] M.C. Flemings; G.E. Nereo Macrosegregation: Part II, Trans. Metall Soc. AIME, Volume 242 (1968), pp. 41-49

[16] M.C. Flemings; G.E. Nereo Macrosegregation: Part III, Trans. Metall Soc. AIME, Volume 242 (1968), pp. 50-55

[17] R. Mehrabian; M. Keane; M.C. Flemings Interdendritic fluid flow and macrosegregation: Influence of gravity, Metall Trans., Volume 1 (1970), pp. 1209-1220

[18] R. Mehrabian; M. Keane; M.C. Flemings Experiments on macrosegregation and freckle formation, Metall. Trans., Volume 1 (1970), pp. 3238-3241

[19] S. Kou; D.R. Poirier; M.C. Flemings Macrosegregation in ESR ingots, Electric Furnace Proc. ISS-IAME, Volume 35 (1977), pp. 221-228

[20] S. Kou; D.R. Poirier; M.C. Flemings Macrosegregation in rotated remelted ingots, Metall. Trans. B, Volume 9 (1978), pp. 711-719

[21] S.D. Ridder; F.C. Reyes; S. Chakravorty; R. Mehrabian; J.D. Nauman; J.H. Chen; H.J. Klein Steady-state segregation and heat flow in ESR, Metall. Trans. B, Volume 9 (1978), pp. 415-425

[22] T. Fujii; D.R. Poirier; M.C. Flemings Macrosegregation in a multicomponent low alloy steel, Metall. Trans. B, Volume 10 (1979), pp. 331-339

[23] D. Petrakis; M.C. Flemings; D.R. Poirier Some effects of forced convection on macrosegregation (H.D. Brody; D. Apelian, eds.), Modeling of Castings and Welding Processes, TMS-AIME, Warrendale, PA, 1981

[24] C.D.R. Poirier Convection of interdendritic liquid in unidirectionally solidified alloys, Proceedings of the U.S.–Japan Cooperative Science Program Seminar on Solidification Processing, Dedham, MA, 1983

[25] S.D. Ridder; S. Kou; R. Mehrabian Effect of fluid flow on macrosegregation in axi-symmetric ingots, Metall. Trans. B, Volume 12 (1981), pp. 435-447

[26] A.L. Maples; D.R. Poirier Convection in the two-phase zone of solidifying alloys, Metall. Trans. B, Volume 15 (1984), pp. 163-172

[27] D.R. Poirier; M.C. Flemings; R. Mehrabian; V. Weiss Modeling macrosegregation in electroslag remelted ingots (J.J. Burke; R. Mehrabian; V. Weiss, eds.), Advances in Metal Processing, Plenum, New York, 1981

[28] W.D. Bennon; F.P. Incropera A continuum model for momentum, heat and species transport in binary solid–liquid phase change systems-I. Model formulation, Int. J. Heat Mass Transfer, Volume 30 (1987), pp. 2161-2170

[29] S. Ganesan; D.R. Poirier Conservation of mass and momentum for the flow of interdendritic liquid during solidification, Metall. Trans. B, Volume 21 (1990), pp. 173-181

[30] D.R. Poirier; P.J. Nandapurkar; S. Ganesan The energy and solute conservation equations for dendritic solidification, Metall. Trans. B, Volume 22 (1991), pp. 889-900

[31] C. Beckermann; R. Viskanta Mathematical modeling of transport phenomena during solidification, Appl. Mech. Rev., Volume 46 (1993), pp. 1-27

[32] W.D. Bennon; F.P. Incropera A continuum model for momentum, heat and species transport in binary solid–liquid phase change systems-II; applications to solidification in a rectangular cavity, Int. J. Heat Mass Transfer, Volume 30 (1987), pp. 2171-2187

[33] S.D. Felicelli; J.C. Heinrich; D.R. Poirier Simulation of freckles during vertical solidification of a binary alloy, Metall. Trans. B, Volume 22 (1991), pp. 847-859

[34] G. Amberg Computation of macrosegregation in an iron-carbon cast, Int. J. Heat Mass Transfer, Volume 34 (1991), pp. 217-227

[35] D. Xu; Q. Li Numerical method for solution of strongly coupled binary alloy solidification problems, Numer. Heat Transfer A, Volume 20 (1991), pp. 181-201

[36] K.C. Chiang; H.L. Tsai Shrinkage induced fluid flow and domain change in two-dimensional alloy solidification, Int. J. Heat Mass Transfer, Volume 35 (1992), pp. 1763-1770

[37] M.C. Schneider; C. Beckermann A numerical study of the combined effects of microsegregation, mushy zone permeability and flow, caused by volume contraction and thermosolutal convection, on macrosegregation and eutectic formation in binary alloy solidification, Int. J. Heat Mass Transfer, Volume 38 (1995), pp. 3455-3473

[38] C.R. Swaminathan; V.R. Voller Towards a general numerical scheme for solidification systems, Int. J. Heat Mass Transfer, Volume 40 (1997), pp. 2859-2868

[39] G.F. Naterer Simultaneous pressure-velocity coupling in the two-phase zone for solidification shrinkage in an open casting, Model. Simul. Mater. Sci. Engrg., Volume 5 (1997), pp. 595-613

[40] N. Ahmad; H. Combeau; J.-L. Desbiolles; T. Jalanti; G. Lesoult; J. Rappaz; M. Rappaz; C. Stomp Numerical simulation of macrosegregation: a comparison between finite volume method and finite element method predictions and a confrontation with experiments, Metall. Mater. Trans. A, Volume 29 (1998), pp. 617-630

[41] V. Voller; C.M. Swaminathan; B.G. Thomas Fixed grid techniques for phase change problems: a review, Int. J. Numer. Methods Engrg., Volume 30 (1990), pp. 875-898

[42] C. Beckermann Modeling of macrosegregation: applications and future needs, Int. Mater. Rev., Volume 47 (2002), pp. 243-261

[43] J.C. Heinrich, D.R. Poirier, Effect of volume change during directional solidification of binary alloys, Model. Simul. Mater. Sci. Engrg. (2004), in press

[44] D.A. Nield The boundary correction for the Rayleigh–Darcy problem: limitations of the Brinkman equation, J. Fluid Mech., Volume 128 (1983), pp. 37-46

[45] D.A. Nield; D.D. Joseph Effect of quadratic drag on convection in a saturated porous medium, Phys. Fluids, Volume 28 (1985), pp. 995-997

[46] C. Beckerman; R. Viskanta Double-diffusive convection during dendritic solidification of a binary mixture, Phys.-Chem. Hydrodyn., Volume 10 (1988), pp. 195-213

[47] S.D. Felicelli; J.C. Heinrich; D.R. Poirier Finite element analysis of directional solidification of multicomponent alloys, Int. J. Numer. Methods Fluids, Volume 27 (1998), pp. 207-227

[48] J.C. Heinrich; E. McBride Calculation of pressure in a mushy zone, Int. J. Numer. Methods Engrg., Volume 47 (2000), pp. 735-747

[49] G. Amberg Parameter ranges in binary solidification from vertical boundaries, Int. J. Heat Mass Transfer, Volume 40 (1997), pp. 2565-2578

[50] J.C. Heinrich; D.W. Pepper Intermediate Finite Element Method; Fluid Flow and Heat Transfer Applications, Taylor & Francis, Philadelphia, 1999

[51] S.D. Felicelli; J.C. Heinrich; D.R. Poirier Numerical model for dendritic solidification of binary alloys, Numer. Heat Transfer B, Volume 23 (1993), pp. 461-481

[52] S.D. Felicelli; J.C. Heinrich; D.R. Poirier Finite element analysis of directional solidification of multicomponent alloys, Int. J. Numer. Methods Fluids, Volume 27 (1998), pp. 207-227

[53] E. McBride; J.C. Heinrich; D.R. Poirier Numerical simulation of incompressible flow driven by density variations during phase change, Int. J. Numer. Methods Fluids, Volume 31 (1999), pp. 787-800

[54] D.G. Nielson; F.P. Incropera Three-dimensional considerations of unidirectional solidification in a binary liquid, Numer. Heat Transfer, Volume 23 (1993), pp. 1-20

[55] S.D. Felicelli; J.C. Heinrich; D.R. Poirier Three-dimensional simulations of freckles in binary alloys, J. Crystal Growth, Volume 191 (1997), pp. 879-888

[56] S.D. Felicelli; D.R. Poirier; J.C. Heinrich Modeling freckle formation in three dimensions during solidification of multicomponent alloys, Metall. Mater. Trans. B, Volume 29 (1998), pp. 847-855

[57] D. Xu; Q. Li Gravity- and solidification-shrinkage-induced liquid flow in a horizontally solidified alloy ingot, Numer. Heat Transfer A, Volume 20 (1991), pp. 203-221

[58] H. Shahani; G. Amberg; H. Fredriksson On the formation of macrosegregations in unidirectionally solidified Sn–Pb and Pb–Sn alloys, Metall. Trans. A, Volume 23 (1992), pp. 2301-2311

[59] D.G. Westra, Simulation of directional solidification in a binary alloy using the fractional step method, Ph.D. Dissertation, The University of Arizona, Tucson, 2003

[60] A. Bejan Convective heat transfer in porous media (S. Kakacs; R.K. Shah; W. Aung, eds.), Handbook of Single Phase Convective Heat Transfer, Wiley, New York, 1987

[61] D.R. Poirier Permeability for flow of interdendritic liquid in columnar-dendritic alloys, Metall. Trans. B, Volume 18 (1987), pp. 245-255

[62] S. Ganesan; C.L. Chan; D.R. Poirier Permeability of flow parallel to dendrite arms, Mater. Sci. Engrg. A, Volume 151 (1992), pp. 97-105

[63] M.S. Bhat; D.R. Poirier; J.C. Heinrich Permeability for cross flow through columnar-dendritic alloys, Metall. Mater. Trans. B, Volume 26 (1995), pp. 1049-1056

[64] J.C. Heinrich; D.R. Poirier; D.F. Nagelhout Mesh generation and flow calculations in highly contorted geometries, Comp. Methods Appl. Mech. Engrg., Volume 133 (1996), pp. 79-92

[65] D.R. Poirier; J.C. Heinrich Continuum model for predicting macrosegregation in dendritic alloys, Mater. Character., Volume 32 (1994), pp. 287-298

[66] S.K. Sinha; T. Sundararajan; V.K. Garg A variable property analysis of alloy solidification using the anisotropic porous medium approach, Int. J. Heat Mass Transfer, Volume 35 (1992), pp. 2865-2877

[67] B. Goyeau; T. Bennihaddadene; D. Gobin; M. Quintard Averaged momentum equation for flow through a nonhomogeneous porous structure, Transp. Porous Media, Volume 28 (1997), pp. 19-50

[68] T. Motegi; A. Ohno Inverse segregation in unidirectionally solidified Al–Cu alloy ingots, Trans. Japan Inst. Metals, Volume 25 (1984), pp. 122-132

[69] M.G. Worster Instabilities of the liquid and mushy regions during solidification of alloys, J. Fluid Mech., Volume 237 (1992), pp. 649-669

[70] C.F. Chen; F. Chen Experimental study of directional solidification of aqueous ammonium chloride solutions, J. Fluid Mech., Volume 227 (1991), pp. 567-586

[71] T.P. Schulze; M.G. Worster A numerical investigation of steady convection in mushy layers during the directional solidification of binary alloys, J. Fluid Mech., Volume 356 (1998), pp. 199-220

[72] H.W. Huang; J.C. Heinrich; D.R. Poirier Simulation of directional solidification with steep thermal gradients, Model. Simul. Mater. Sci. Engrg., Volume 4 (1996), pp. 245-259

[73] S.N. Tewari; R. Shah Macrosegregation during steady-state arrayed growth of dendrites in directionally solidified Pb–Sn alloys, Metall. Trans., Volume 23A (1992), pp. 3383-3392

[74] S.D. Felicelli; D.R. Poirier; J.C. Heinrich Macrosegregation patterns in multicomponent Ni-base superalloys, J. Crystal Growth, Volume 177 (1996), pp. 145-161

[75] S.D. Felicelli; D.R. Poirier; A.F. Giamei; J.C. Heinrich Simulation of the solidification of DS and SC superalloys, JOM, Volume 49 (1997), pp. 21-25

[76] M.C. Schneider; C. Beckermann Formation of macrosegregation by multicomponent thermosolutal convection during solidification of steel, Metall. Mater. Trans. A, Volume 26 (1995), pp. 2373-2388

[77] S.D. Felicelli; P.K. Sung; D.R. Poirier; J.C. Heinrich Transport properties and transport phenomena in casting nickel superalloys, Int. J. Thermophys., Volume 19 (1998), pp. 1657-1669

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