Microscopic-Macroscopic Modelling of Transport Phenomena during Solidification in Heterogeneous Systems

  • Piotr Furmański
Part of the International Centre for Mechanical Sciences book series (CISM, volume 449)


Microscopic-macroscopic approach to modelling of solidification in heterogeneous systems (the mushy zone of binary alloys, in manufacture of metal-matrix composites and in porous media) is addressed in the paper. Microscopic phenomena accompanying solidification and microscopic equations are presented. Averaging procedures (volume and ensemble averaging techniques) and their limitations are discussed. Macroscopic equations are derived and macroscopic phenomena described. Conditions for existence of local thermal and chemical equilibrium during solidification on the macroscopic scale are shown. Some examples of equilibrium and non-equilibrium solidification based on macroscopic approach are presented.


Solidification Process Mushy Zone Effective Property Solidification Front Macroscopic Scale 
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  1. Amberg, G. (2002). Solidification, microstructure and convection. Lecture Notes from CISM Course “Phase Change with Convection: Modelling and Validation, Coordinated by T. Kowalewski and D. Gobin, September 2–6, CISM, Udine.Google Scholar
  2. Amberg, L., Chai, G., and Backerud, L. (1993). Determination of dendritic coherency in solidifying melts by rheological measurements. Materials Science and Engineering, A173: 101–103.CrossRefGoogle Scholar
  3. Aziz, M. J. (1996). Interface attachment kinetics in alloy solidification. Metallurgical and Materials Trans., 27A: 671–686.CrossRefGoogle Scholar
  4. Banaszek, J., Jaluria, Y., Kowalewski, T.A., and Rebow, M. (1999). Semi-implicit FEM analysis of natural convection in freezing water. Num. Heat Transfer, A36: 449–472.CrossRefGoogle Scholar
  5. Banaszek, J., and Furmanski, P. (2000). FEM analysis of binary dilute system solidification using the anisotropic porous medium model of a mushy zone. Computer Assisted Mechanics and Engineering Sciences, 7: 343–362.MATHGoogle Scholar
  6. Banaszek, J., Browne, D.J., and Furmanski, P. (2002). Some aspects of modelling of binary system solidification on a fixed grid. Proceedings of International Conference on Phase Change Processes, Kielce, June, Poland.Google Scholar
  7. Batchelor, G.K. (1974). Transport properties of two-phase materials with random structure. Ann. Rev. Fluid Mech., 6: 227–255.CrossRefGoogle Scholar
  8. Beckermann, C., and Viskanta, R. (1993). Mathematical modelling of transport phenomena during alloy solidification. Appl. Mech. Rev., 46: 1–27.MathSciNetCrossRefGoogle Scholar
  9. Beckermann, C., Diepers, H.-J., Steinbach, I., Karma, A., and Tong, X. (1999). Modelling melt convection in phase-field simulations of solidification. J. of Computational Physics, 154: 468–496.CrossRefMATHGoogle Scholar
  10. Bennon, W.D., and Incropera, F.P. (1987). A continuum model for momentum, heat and species transport in binary solid-liquid phase change systems–I and–II. Int. JHeat & Mass Transfer, 30: 2161–2187.CrossRefMATHGoogle Scholar
  11. Bouissou, Ph., and Pelcé, P. (1989). Effect of a forced flow on dendritic growth. Physical Review A, 40: 6673–6680.CrossRefGoogle Scholar
  12. Bousquet-Melou, P., Goyeau, B., Quitard, M., Fichot, F., and Gobin, D. (2002). Average momentum equation for interdendritic flow in a solidifying columnar mushy zone. Inter. J of Heat and Mass Transfer, 45: 3651–3665.CrossRefMATHGoogle Scholar
  13. Browne, D. J., Hunt, J. D. (2000). A model of columnar growth using a front tracking technique. Modelling of Casting, Type=“Italic”>Welding and Advanced Solidification Processes IX, Springer Verlag, Aachen, Germany, 437–444.Google Scholar
  14. Browne, D. J.,Banaszek, J., and Hunt, J. D. (2002). Front tracking method on a fixed grid versus enthalpy approach in modelling of binary alloy solidification. Proceedings of IMECE 2002: International Mechanical Engineering Congress and Exhibition,November 17–22, New Orleans, USA.Google Scholar
  15. Browne, D.J., and O’Mahoney, D. (2001). Interface heat transfer in investment casting of aluminium alloys. Metallurgical and Materials Trans., 32A: 3055–3063.CrossRefGoogle Scholar
  16. Burden, M.H., Hunt, J. D. (1974). Cellular and dendritic growth. Journal of Crystal Growth, 22: 99–116.CrossRefGoogle Scholar
  17. Buyevich, Yu. A. (1992). Heat and mass transfer in disperse media. I. Averaged Field Equations. Int. J. Heat & Mass Transfer, 35: 2445–2452.CrossRefMATHGoogle Scholar
  18. Chung, J.D., Lee, J.S., and Yoo, H. (1998). An extended similarity solution for the alloy solidification system. Proceedings of 11th IHTC, 7: 187–192.Google Scholar
  19. Furmanski P. (1994). A mixture theory for heat conduction in heterogeneous media. Int. J. Heat & Mass Transfer, 37: 2993–3002.CrossRefMATHGoogle Scholar
  20. Furmanski, P. (1995). Influence of laminar convection of fluid on effective thermal conductivity of some porous media. Advances in Engineering Heat Transfer. Computational Mechanics Publications, 513–524.Google Scholar
  21. Furmanski, P. (1997). Heat conduction in composites. Homogenization and macroscopic behavior. Appl. Mech. Rev., 50: 327–356.CrossRefGoogle Scholar
  22. Furmanski, P. (1999). Thermal properties and local heat sources in composite materials. Thermal Conductivity, 24: 581–594.Google Scholar
  23. Furmanski, P. (2000). Modelling of transport phenomena during solidification of binary systems. Computer Assisted Mechanics and Engineering Sciences, 7: 391–402.MATHGoogle Scholar
  24. Ganesan, S., Chan, C.L., and Poirier, D.R. (1992). Permeability for flow parallel to primary dendrite arms. Mater. Sci.and Eng., A151: 97–105.CrossRefGoogle Scholar
  25. Garda, B., Hodaj, F., Durand, F. (1993). Semi-analytical calculation of equiaxed grain-size in recoalescence conditions. Mater. and Eng., A173: 105–108.CrossRefGoogle Scholar
  26. Geindreau, C., and Auriault, J-L. (2001). Transport phenomena in saturated porous media undergoing liquid-solid phase change. Computer Assisted Mechanics and Engineering Sciences, 8: 391–402.Google Scholar
  27. Hills, R.N., Loper, D.E., and Roberts, P.H. (1992). On continuum models for momentum, heat and species transport in solid-liquid phase change systems. Int. Comm. Heat & Mass Transfer, 19: 585–594.CrossRefGoogle Scholar
  28. Hunt, J..D., and Lu, S. Z. (1996). Numerical modelling of cellular/dendritic array growth: spacing and structure predictions. Metallurgical and Materials Transactions, 27A: 611–623.CrossRefGoogle Scholar
  29. Junk, D. and Tryggvason, G. (1996). A front-tracking method for dendritic solidification. Journal of Computational Physics, 123: 127–148.MathSciNetCrossRefGoogle Scholar
  30. Kaviany, M. (1995). Principles of Heat Transfer in Porous Media. Springer Verlag, 2nd edition, New York.Google Scholar
  31. Khan, J.A., and Tong, X. (1998). Unidirectional infiltration and solidification/remelting of Al-Cu alloy. J. of Thermophysics and Heat Transfer. 12: 100–106.CrossRefGoogle Scholar
  32. Koch, D.L., and Brady, J.F. (1987). A non-local description of advection-diffusion with application to dispersion in porous media. J. Fluid Mech., 387–403.Google Scholar
  33. Kunin, I.A. (1984). On foundations of the theory of elastic media with microstructure. Int. J.Engng Sci., 22: 969–978.CrossRefMATHGoogle Scholar
  34. Krieger, I.M. (1972). Rheology of monodispersed lattices. Advances in Colloid Interface Science, 3: 111–136.CrossRefGoogle Scholar
  35. Langer, J.S. (1989). Dendrites, viscous fingers, and the theory of pattern formation. Science, 243: 1150–1156.CrossRefGoogle Scholar
  36. Loulou, T., Artyukhin, E.A., and Bardon, J.P. (1999). Estimation of thermal contact resistance during the first stages of metal solidification process: I–experiment principle and modelisation. International J. of Heat & Mass Transfer, 42: 2119–2127.CrossRefGoogle Scholar
  37. Loulou, T., Artyukhin, E.A., and Bardon, J.P. (1999). Estimation of thermal contact resistance during the first stages of metal solidification process: II–experimental set-up and results. International J. of Heat & Mass Transfer, 42: 2129–2142.CrossRefGoogle Scholar
  38. Mat, M.D., and Ilegbusi, O.J. (2002). Application of a hybrid model of mushy zone to macrosegregation in alloy solidification. Int. J. of Heat and Mass Transfer, 45: 279–289.CrossRefMATHGoogle Scholar
  39. Matsumoto, K., Okada, M., Murakami, M., and Yabushita, Y. (1193). Solidification of porous medium saturated with aqueous solution in a rectangular cell. Int. J. Heat and Mass Trans., 36: 2869–2880.CrossRefGoogle Scholar
  40. Mortensen, A., and Flemings, M.C. (1996). Solidification of binary hypoeutectic alloy matrix composite castings. Metallurgical and Materials Trans., 27A: 595–609.CrossRefGoogle Scholar
  41. Murakami, K., and Okamoto, T. (1984). Fluid flow in the mushy zone composed of granular grains. Acta Metall., 32: 1741–1744.CrossRefGoogle Scholar
  42. Naterer, G.F., and Schneider, G.E. (1995). Phases model for binary-constituent solid-liquid phase transition. Part 1: Numerical Methods. Numerical Heat Transfer, Part B, 28: 111–126.Google Scholar
  43. Naumann, R. (1995). Marangoni convection around voids in Bridgman growth. J. of Crystal Growth, 154: 156–162.CrossRefGoogle Scholar
  44. Okada, M, Ochi, M., and Tateno, M. (1998). Solidification of a supercooled aqueous solution in a porous medium. Proceedings of 11 th IHTC, Kongjiu, Korea, 7: 169–174.Google Scholar
  45. O’Mahoney, D., and Browne, D.J. (2000). Use of experiment and an inverse method to study interface heat transfer during solidification in the investment casting process. Experimental Thermal and Fluid Science, 22: 111–122.CrossRefGoogle Scholar
  46. Poirier, D.R. (1987). Permeability for flow of interdendritic liquid in columnar-dendritic alloys. Metallurgical Trans., 18B: 245–255.CrossRefGoogle Scholar
  47. Poirier, D.R., Nandapurkar, P.J., and Ganesan, S. (1991). The energy and solute conservation equations for dendritic solidification. Metallurgical. Trans., 22B: 889–900.CrossRefGoogle Scholar
  48. Prakash, C., and Voller, V. (1989). On the numerical solution of continuum mixture model equations describing binary solid-liquid phase change. Numerical Heat Transfer B, 15: 171–189.CrossRefMATHGoogle Scholar
  49. Prescott, P.J., Incropera, F.P., and Bennon, W.D. (1991). Modelling of dendritic solidification systems: reassessment of the continuum momentum equation. Int. J. of Heat and Mass Transfer, 34: 2351–2359.CrossRefGoogle Scholar
  50. Quintard, M., and Whitaker, S. (1993). One and two equations models for transient diffusion processes in two-phase systems. Advances in Heat Transfer, 23: 369–464.CrossRefGoogle Scholar
  51. Rappaz, M. (1989). Modelling of microstructure formation in solidification processes. International Materials Reviews, 34: 93–123.Google Scholar
  52. Rappaz, M., and Voller, V.R. (1990). Modelling of micro-macrosegregation in solidification processes. Metall. Trans., 21A: 749–753.CrossRefGoogle Scholar
  53. Rappaz, M., Gandin, Ch. A., Desbiolles, J. L., and Thevoz, Ph.. (1996). Prediction of grain structures in various solidification processes. Metallurgical and Materials Trans., 27A: 695–705.CrossRefGoogle Scholar
  54. Sältzer, W.D., and Schultz, B. (1983). Theory and measurement of the viscosity of suspensions. High Temperatures–High Pressures, 15: 289–298.Google Scholar
  55. Santoli, L., Cumo, F., and Menno, I. (1998). Freezing of liquid-saturated porous media of building materials. Proceedings of 11th IHTC, Kyongiu, Korea, 4: 387–391.Google Scholar
  56. Schrage, D.S. (1999). A simplified model of dendritic growth in the presence of natural convection. J. of Crystal Growth, 205: 410–426.CrossRefGoogle Scholar
  57. Shyy, W., Udaykumar, H. S., Rao, M..M., and Smith, R. W. (1996), Computational Fluid Dynamics with Moving Boundaries, Taylor and Francis, Washington DC, USA.Google Scholar
  58. Simpson, J.E., Garimella, S. V., and Guslick, M.M. (1998). Interface propagation in the presence of a fibrous phase in alloy solidification. Proceedings of II th IHTC, Kyongiu, Korea, 7: 235–240.Google Scholar
  59. Sinha, S.K., Sundararajan, T., and Garg, V.K. (1992). A variable property analysis of alloy solidification using the anisotropic porous medium approach. Int. J. of Heat and Mass Transfer: 2865–2877.Google Scholar
  60. Swaminathan, C.R., and Voller, V.R. (1992). General enthalpy method for modelling solidification processes. Metall. Trans., 23B: 651–65.CrossRefGoogle Scholar
  61. Swaminathan, C.R., and Voller, V.R. (1997). Towards a general numerical scheme for solidification systems. Int. J. of Heat and Mass Transfer, 40: 2959–2868.Google Scholar
  62. Timchenko, V.,. Chen, P.Y.P., de Vahl Davies, G, and Leonardi, E. (1998). Directional solidification in microgravity. Proceedings of 11th IHTC, Kyongju, Korea, 7: 241–246.Google Scholar
  63. Tönhardt, R., and Amberg, G. (1998). Phase-field simulation of dendritic growth in a shear flow. J. of Crystal Growth, 194: 406–425.CrossRefGoogle Scholar
  64. Wang, W., and Qiu, H.H. (2002). Interfacial thermal conductance in rapid contact solidification process. International J. of Heat & Mass Transfer, 45: 2043–2053.MathSciNetCrossRefGoogle Scholar
  65. Warren, J. A., and Boettinger, W. J. (1995). Prediction of dendritic microsegregation patterns using a diffuse interface phase field model. Modelling of Casting, Type=“Italic”>Welding and Advanced Solidification Processes VII, M. Cross and J. Campbell eds, TMS, Warrendale, PA, USA, 601–607.Google Scholar
  66. Voller, V.R., Brent, A.D., Prakash, C. (1989). The modelling of heat, mass and solute transport in solidification systems. Int. J. Heat & Mass Transfer, 32: 1719–1731.CrossRefGoogle Scholar
  67. Voller, V.R., and Swaminathan, C.R. (1991). General source-based method for solidification phase change. Numerical Heat Transfer, 19: 175–189.CrossRefGoogle Scholar
  68. Viskanta, R. (1988). Heat transfer during melting and solidification of metals. Trans. of ASME. J. of Heat Transfer, 110: 1205–1219.CrossRefGoogle Scholar

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© Springer-Verlag Wien 2004

Authors and Affiliations

  • Piotr Furmański
    • 1
  1. 1.Institute of Heat EngineeringWarsaw University of TechnologyWarsawPoland

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