Journal of Materials Science

, Volume 53, Issue 13, pp 9771–9789 | Cite as

Interaction between primary dendrite arm spacing and velocity of fluid flow during solidification of Al–Si binary alloys

  • Hongda Wang
  • Mohamed S. Hamed
  • Sumanth Shankar


A new and more efficient numerical algorithm to simulate the solidification of binary metallic alloys, wherein for the first time, the undercooling of the liquidus temperature prior to solidification event and optimized thermo-physical properties was incorporated, has been recently developed and validated by various experiments. Subsequently, experiments were carried out to evaluate the validity of various theoretical models in the literature used to predict the dendrite arm spacing (DAS) and quantify the critical interaction between fluid flow and transient DAS during unsteady state solidification. Typically, models of solidification processes such as casting, welding and galvanizing assume a constant value of fluid flow to predict the DAS and in many cases unable to obtain validation. This practice is erroneous and the transient fluid flow developed during solidification has a significant effect on the transient DAS, thermal gradient (G), solidification velocity (R) and morphology of the mushy zone. The Bouchard–Kirkaldy model (DAS prediction) coupled with the Lehmann model to incorporate fluid flow velocity was the only valid theoretical model in binary alloy solidification.

List of symbols


Specific heat of solid as a function of temperature (J kg−1 K−1) [1]


Specific heat of liquid (J kg−1 K−1) [1]


Liquid concentration (wt%)


Average alloy composition (wt%)


Solid concentration (wt%)


Solute diffusivity coefficient of liquid [6.25 × 10−9 (m2 s−1)] [2]


Temperature gradient in liquid at the mushy zone/liquid interface (°C mm−1)


Average partition coefficient (0.116) [3]


Thermal conductivity of solid as a function of temperature (W m−1 K−1) [1]


Thermal conductivity of liquid (W m−1 K−1) [1]


Latent heat of fusion (J kg−1) [3]


The slope of liquidus line [− 6.675 (K wt%−1)] [3]


Pressure (Pa)


Velocity of mushy zone/liquid interface (mm s−1)


Time (s)


Temperature (°C)


Liquidus temperature (°C) [3]


Initial temperature of liquid (°C)


Melting temperature of pure aluminum (660 °C) [3]


Eutectic temperature (578.6 °C) [3]

\( \dot{T} \)

Instantaneous tip cooling rate = G × R (°C s−1)


Undercooling of Tliq (°C)


Velocity in r direction (mm s−1)


Velocity in y direction (mm s−1)


Flow velocity in the liquid of mushy zone/liquid interface (mm s−1)


Contraction ratio \( \left[ {\beta = \frac{{\rho_{s} - \rho_{l} }}{{\rho_{l} }}} \right] \) (volumetric shrinkage during solidification) [1]


Solute expansion coefficient [− 4.26 × 10−4 (K−1)] [1]


Thermal expansion coefficient [1.39 × 10−4 (K−1)] [1]


Gibbs–Thomson coefficient [1.97 × 10−7 (K m−1)] [4]


Liquid fraction


Liquid density (kg m−3) [1]


Solid density (kg m−3) [1]


Dynamic viscosity 1.3 × 10−3 (Pa s) [1]

\( \lambda_{1}^{0} \)

Primary arm spacing if no fluid flow effect is considered (µm)


Primary arm spacing (µm)



The authors wish to extend their gratitude to the Discovery Grant Program of the Natural Science and Engineering Research Council of Canada, for providing the funding for this research project.


  1. 1.
    JMatPro, version 4.1, Sente Software Ltd., Surrey Technology Centre, Surrey, UKGoogle Scholar
  2. 2.
    Sekulic DP, Galenko PK, Krivilyov MD, Walker L, Gao F (2005) Dendritic growth in Al–Si alloys during brazing. Part 2: computational modeling. Int J Heat Mass Transf 48:2385–2396CrossRefGoogle Scholar
  3. 3.
    Bale CW et al (2016) FactSage thermochemical software and databases—2010-2016. Calphad 54:35-53CrossRefGoogle Scholar
  4. 4.
    Gunduz M, Hunt JD (1985) The measurement of solid–liquid surface energies in the Al–Cu, Al–Si and Pb–Sn systems. Acta Metall 33:1651–1672CrossRefGoogle Scholar
  5. 5.
    Quaresma JMV, Santos CA, Garcia A (2000) Correlation between unsteady-state solidification conditions, dendrite spacings, and mechanical properties of Al–Cu alloys. Metall Mater Trans A 31:3167–3178CrossRefGoogle Scholar
  6. 6.
    Shabestari SG, Shahri F (2004) Influence of modification, solidification conditions and heat treatment on the microstructure and mechanical properties of A356 aluminum alloy. J Mater Sci Lett 39:2023–2032CrossRefGoogle Scholar
  7. 7.
    Bennon WD, Incropera FP (1987) A continuum model for momentum, heat and species transport in binary solid–liquid phase change systems—I. Model formulation. Int J Heat Mass Transf 30:2161–2170CrossRefGoogle Scholar
  8. 8.
    Bennon WD, Incropera FP (1987) A continuum model for momentum, heat and species transport in binary solid—liquid phase change systems—II. Application to solidification in a rectangular cavity. Int J Heat Mass Transf 30:2171–2178CrossRefGoogle Scholar
  9. 9.
    Bennon WD, Incropera FP (1988) Numerical analysis of binary solid—liquid phase change using a continuum model. Numer Heat Transf 13:277–296CrossRefGoogle Scholar
  10. 10.
    Heinrich JC, Poirier DR (2004) The effect of volume change during directional solidification of binary alloys. Model Simul Mater Sci Eng 12:881–899CrossRefGoogle Scholar
  11. 11.
    Ho C-J, Viskanta R (1984) Heat transfer during inward melting in a horizontal tube. Int J Heat Mass Transf 27:705–716CrossRefGoogle Scholar
  12. 12.
    Krane MJM, Incropera FP (1995) Analysis of the effect of shrinkage on macrosegregation in alloy solidification. Metall Mater Trans A 26A:2329–2339CrossRefGoogle Scholar
  13. 13.
    Magnusson T, Arnberg L (2001) Density and solidification shrinkage of hypoeutectic aluminum–silicon alloys. Metall Mater Trans A 32:2605–2613CrossRefGoogle Scholar
  14. 14.
    McBride E, Heinrich JC, Poirier DR (1999) Numerical simulation of incompressible flow driven by density variations during phase change. Int J Numer Meth Fluids 31:787–800CrossRefGoogle Scholar
  15. 15.
    Voller VR, Prakash C (1987) Fixed grid numerical modeling methodology for convection—diffusion mushy region phase—change problems. Int J Heat Mass Transf 30:1709–1719CrossRefGoogle Scholar
  16. 16.
    Wang H, Shankar S, Hamed MS (2007) Numerical model for binary alloy solidification. In: 5th international conference on computational heat and mass transfer, Canmore. pp 345–351Google Scholar
  17. 17.
    Xu D, Li Q (1991) Gravity- and solidification-shrinkage-induced liquid flow in a horizontally solidified alloy ingot. Numer Heat Transf Int J Comput Methodol A Appl 20:203–221CrossRefGoogle Scholar
  18. 18.
    Yao LS (1984) Natural convection effects in the continuous casting of a horizontal cylinder. Int J Heat Mass Transf 27:697–704CrossRefGoogle Scholar
  19. 19.
    Ramirez JC, Beckermann C, Karma A, Diepers H-J (2004) Phase-field modeling of binary alloy solidification with coupled heat and solute diffusion. Phys Rev E 69:051607-1–051607-16CrossRefGoogle Scholar
  20. 20.
    Warren JA, Boettinger WJ, Beckermann C, Karma A (2002) Phase-field simulation of solidification. Annu Rev Mater Sci 32:163–194CrossRefGoogle Scholar
  21. 21.
    Beckermann C, Li Q, Tong X (2001) Microstructure evolution in equiaxed dendritic growth. Sci Technol Adv Mater 2:117–126CrossRefGoogle Scholar
  22. 22.
    Elder KR, Grant M, Provatas N, Kosterlitz JM (2001) Sharp interface limits of phase-field models. Phys Rev E (Stat Nonlinear Soft Matter Phys) 64:021604/1–021604/18Google Scholar
  23. 23.
    Provatas N, Goldenfeld N, Dantzig J (1999) Modeling solidification using a phase-field model and adaptive mesh refinement. Solidification 1999. Proceedings. pp 151–160Google Scholar
  24. 24.
    Karma A, Rappel W-J (1996) Phase-field method for computationally efficient modeling of solidification with arbitrary interface kinetics. Phys Rev E (Stat Phys Plasmas Fluids Related Interdiscip Top) 53:R3017–R3020Google Scholar
  25. 25.
    Jeong JH, Goldenfield N, Dantzig JA (2001) Phase field model for three-dimensional growth with fluid flow. Phys Rev E 64:041602-1-14CrossRefGoogle Scholar
  26. 26.
    Lan CW, Shih CJ (2004) Efficient phase field simulation of a binary dendritic growth in a forced flow. Phys Rev E (Stat Nonlinear Soft Matter Phys) 69:31601-1–31601-10Google Scholar
  27. 27.
    Bouchard D, Kirkaldy JS (1997) Prediction of dendrite arm spacings in unsteady- and steady-state heat flow of unidirectionally solidified binary alloys. Metall Mater Trans B (Process Metall Mater Process Sci) 28B:651–663CrossRefGoogle Scholar
  28. 28.
    Lehmann P, Moreaub R, Camela D, Bolcatob R (1998) A simple analysis of the effect of convection on the structure of the mushy zone in the case of horizontal Bridgman solidification. Comparison with experimental results. J Cryst Growth 183:690–704CrossRefGoogle Scholar
  29. 29.
    Peres MD, Siqueira CA, Garcia A (2004) Macrostructural and microstructural development in Al–Si alloys directionally solidified under unsteady-state conditions. J Alloys Compd 381:168–181CrossRefGoogle Scholar
  30. 30.
    Spinelli JE, Peres MD, Garcia A (2005) Thermosolutal convective effects on dendritic array spacings in downward transient directional solidification of Al–Si alloys. J Alloys Compd 403:228–238CrossRefGoogle Scholar
  31. 31.
    Wang H (2009) Solidification simulation of binary Al–Si alloys: prediction of primary dendrite arm spacing with macro-scale simulations (~ 1 mm length scale). Ph.D. Thesis, Department of Mechanical Engineering, McMaster University, Hamilton, Ontario, Canada, Publication AGoogle Scholar
  32. 32.
    Wang H (2009) Solidification simulation of binary Al–Si alloys: prediction of primary dendrite arm spacing with macro-scale simulations (~ 1 mm length scale). Ph.D. Thesis, Department of Mechanical Engineering, McMaster University, Hamilton, Ontario, Canada, Publication CGoogle Scholar
  33. 33.
    Hunt JD (1979) Solidification and casting of metals. In: Proceedings of the international conference on solidification and casting of metals. The Metals Society, London. pp 3–9Google Scholar
  34. 34.
    Kurz W, Fisher DJ (1981) Dendritic growth and limit of stability tip radius and spacing. Acta Metall 29:11–20CrossRefGoogle Scholar
  35. 35.
    Trivedi R (1984) Interdendritic spacing: part II. A. Comparison of theory and experiment. Metall Trans A (Phys Metall Mater Sci) 15A:977–982CrossRefGoogle Scholar
  36. 36.
    Steinbach S, Ratke L (2005) The effect of rotating magnetic fields on the microstructure of directionally solidified Al–Si–Mg alloys. Mater Sci Eng A 413–414:200–204CrossRefGoogle Scholar
  37. 37.
    Flemings MC (1974) Solidification processing. McGraw-hill Book Co, New YorkGoogle Scholar
  38. 38.
    Felicelli SD, Heinrich JC, Poirier DR (1991) Simulation of freckles during vertical solidification of binary alloys. Metall Trans B (Process Metall) 22:847–859CrossRefGoogle Scholar
  39. 39.
    Burden MH, Hunt JD (1974) Cellular and dendritic growth. I. J Cryst Growth 22:99–108CrossRefGoogle Scholar
  40. 40.
    Burden MH, Hunt JD (1974) Cellular and dendritic growth. II. J Cryst Growth 22:109–116CrossRefGoogle Scholar
  41. 41.
    Carman PC (1937) Fluid flow through granular beds. Trans Inst Chem Eng 15:150–156Google Scholar
  42. 42.
    Carman PC (1938) The determination of the specific surface of powders. I. J Soc Chem Ind 57:225–234Google Scholar
  43. 43.
    Asai S, Muchi I (1978) Theoretical analysis and model experiments of the formation mechanism of channel—type segregation. Trans Iron Steel Inst Jpn 18:290–298Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  • Hongda Wang
    • 1
    • 2
  • Mohamed S. Hamed
    • 1
  • Sumanth Shankar
    • 2
  1. 1.Thermal Processing Laboratory (TPL), Department of Mechanical EngineeringMcMaster UniversityHamiltonCanada
  2. 2.Light Metal Casting Research Centre (LMCRC), Department of Mechanical EngineeringMcMaster UniversityHamiltonCanada

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