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Effect of Shrinkage on Primary Dendrite Arm Spacing during Binary Al-Si Alloy Solidification

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Abstract

Upward and downward directional solidification of hypoeutectic Al-Si alloys were numerically simulated inside a cylindrical container. Undercooling of the liquidus temperature prior to the solidification event was introduced in the numerical model. The finite-volume method was used to solve the energy, concentration, momentum, and continuity equations. Temperature and liquid concentrations inside the mushy zone were coupled with local equilibrium assumptions. An energy equation was applied to determine the liquid fraction inside the mushy zone while considering the temperature undercooling at the solidifying dendrite/liquid interface. Momentum and continuity equations were coupled by the SIMPLE algorithm. Flow velocity distribution at various times, G, R, λ 1, and solidification time at mushy zone/liquid interface during solidification were predicted. The effect of shrinkage during solidification on these solidification parameters was quantified. Transient temperature distribution, solidification time for the mushy zone/liquid interface, and λ 1 were validated by laboratory experiments. It was found that better agreement could be achieved when the fluid flow due to solidification shrinkage was considered. Considering shrinkage in upward solidification was found to only have a marginal effect on solidification parameters, such as G, R, and λ 1; whereas, in the downward solidification, fluid flow due to shrinkage had a significant effect on these solidification parameters. Considering shrinkage during downward solidification resulted in a smaller R, stronger fluid flow, and increased solidification time at the mushy zone/liquid interface. Further, the flow pattern was significantly altered when solidification shrinkage was considered in the simulation. The effect of shrinkage on G and λ 1 strongly depended on the instantaneous location of the mushy zone/liquid interface in the computational domain. The numerical results could be validated by experimental data only when both the undercooling of the liquidus temperature prior to solidification and fluid flow in the liquid caused by the effect of shrinkage during solidification were included in the model.

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Abbreviations

c ps :

specific heat of solid as a function of temperature (J Kg−1 C−1)[1]

c pl :

specific heat of liquid (J Kg−1 C−1)[1]

C L :

liquid concentration (wt pct)

C o :

average alloy composition (wt pct)

C S :

solid concentration (wt pct)

D :

coefficient of solute diffusion in the liquid (6.25 × 10−9 (m2 s−1))[2]

G :

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

k :

average partition coefficient of the Al-Si binary hypoeutectic alloy (0.116)[3]

K s :

thermal conductivity of solid as a function of temperature (W m−1 C−1)[1]

K l :

thermal conductivity of liquid (W m−1 C−1)[1]

L :

latent heat of fusion (J Kg−1)[3]

m :

the slope of the liquidus line in the Al-Si binary hypoeutectic alloy phase diagram (−6.675 (K wt pct−1))[3]

p :

pressure (Pa)

R :

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

t :

time (s)

T :

temperature (°C)

T liq :

liquidus temperature (°C)[3]

T ini :

initial temperature of liquid (°C)

T m :

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

T eut :

eutectic temperature (578.6 °C)[3]

\( \dot{T} \) :

instantaneous tip cooling rate = G·R (°C s−1)

ΔT :

undercooling of T liq (°C)

u r :

velocity in r direction (mm s−1)

u y :

velocity in y direction (mm s−1)

U :

magnitude of the flow velocity in the liquid at the mushy zone/liquid interface (mm s−1); \( U = \sqrt {\left( {u_{r}^{2} + u_{y}^{2} } \right)} \)

β :

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

β C :

solute expansion coefficient (4.26 × 10−4 (wt pct−1))[1]

β T :

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

Γ:

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

ϕ :

liquid fraction

ρ l :

liquid density (Kg m−3)[1]

ρ s :

solid density (Kg m−3)[1]

μ :

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

λ 01 :

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

λ 1 :

primary arm spacing (μm)

References

  1. JMatPro, Sente Software Ltd., Ver. 4.1, Guildford, United Kingdom, 2007.

  2. D.P. Sekulic, P.K. Galenko, M.D. Krivilyov, L. Walker, and F. Gao: Int. J. Heat Mass Transfer, 2005, vol. 48, pp. 2385–96.

    Article  CAS  Google Scholar 

  3. Factsage, Computherm LLC., Madison, Ver. 5.5, WI, 2007.

  4. M. Gunduz and J.D. Hunt: Acta Metall., 1985, vol. 33, pp. 1651–72.

    Article  CAS  Google Scholar 

  5. W.D. Bennon and F.P. Incropera: Int. J. Heat Mass Transfer., 1987, vol. 30, pp. 2161–70.

    Article  CAS  Google Scholar 

  6. S.D. Felicelli, J.C. Heinrich, and D.R. Poirier: Metall. Trans. B, 1991, vol. 22B, pp. 847–59.

    Article  CAS  Google Scholar 

  7. V.R. Voller and C. Prakash: Int. J. Heat Mass Transfer, 1987, vol. 30, pp. 1709–19.

    Article  CAS  Google Scholar 

  8. H. Wang, S. Shankar, and M.S. Hamed: 5th Int. Conf. on Computational Heat and Mass Transfer, Canmore, AB, Canada, 2007, pp. 345–51.

    Google Scholar 

  9. W.D. Bennon and F.P. Incropera: Numer. Heat Transfer, 1988, vol. 13, pp. 277–96.

    Article  Google Scholar 

  10. S.D. Felicelli, J.C. Heinrich, and D.R. Poirier: Numer. Heat Transfer, Part B: Fundamentals, 1993, vol. 23, pp. 461–81.

    Article  CAS  Google Scholar 

  11. J.C. Heinrich and D.R. Poirier: Modell. Simul. Mater. Sci. Eng., 2004, vol. 12, pp. 881–99.

    Article  CAS  Google Scholar 

  12. E. McBride, J.C. Heinrich, and D.R. Poirier: Int. J. Numer. Meth. Fluids, 1999, vol. 31, pp. 787–800.

    Article  Google Scholar 

  13. D. XU and Q. LI: Numer. Heat Transfer, Part A, 1991, vol. 20, pp. 203–21.

    Article  Google Scholar 

  14. K.C. Chiang and H.L. Tsai: Int. J. Heat Mass Transfer, 1992, vol. 35, pp. 1763–70.

    Article  CAS  Google Scholar 

  15. K.C. Chiang and H.L. Tsai: Int. J. Heat Mass Transfer, 1992, vol. 35, pp. 1771–78.

    Article  CAS  Google Scholar 

  16. M.J.M. Krane and F.P. Incropera: Metall. Trans. A, 1995, vol. 26A, pp. 2329–39.

    Article  CAS  Google Scholar 

  17. S. Steinbach and L. Ratke: Mater. Sci. Eng. A, 2005, vols. 413–414, pp. 200–04.

  18. T. Magnusson and L. Arnberg: Metall. Mater. Trans. A, 2001, vol. 32A, pp. 2605–13.

    Article  CAS  Google Scholar 

  19. M.C. Schneider and C. Beckermann: Metall. Trans. A, 1995, vol. 26A, pp. 2373–88.

    Article  CAS  Google Scholar 

  20. M.C. Schneider and C. Beckermann: Summary of a Method for Numerically Simulating the Columnar Dendritic Solidification of Binary Alloys, University of Iowa, Iowa City, IA, 1993.

    Google Scholar 

  21. M.D. Peres, C.A. Siqueira, and A. Garcia: J. Alloys Compd., 2004, vol. 381, pp. 168–81.

    Article  CAS  Google Scholar 

  22. J.E. Spinelli, M.D. Peres, and A. Garcia: J. Alloys Compd., 2005, vol. 403, pp. 228–38.

    Article  CAS  Google Scholar 

  23. M.C. Flemings: Solidification Processing, McGraw-Hill Book Co., New York, NY, 1974.

    Google Scholar 

  24. M.H. Burden and J.D. Hunt: J. Cryst. Growth, 1974, vol. 22, pp. 99–108.

    Article  CAS  Google Scholar 

  25. M.H. Burden and J.D. Hunt: J. Cryst. Growth, 1974, vol. 22, pp. 109–16.

    Article  CAS  Google Scholar 

  26. P.C. Carman: Trans. Inst. Chem. Eng., 1937, vol. 15, pp. 150–56.

    CAS  Google Scholar 

  27. P.C. Carman: I. J. Soc. Chem. Indus., 1938, vol. 57, pp. 225–34.

    CAS  Google Scholar 

  28. S. Asai and I. Muchi: Trans. Iron Steel Inst. Jpn., 1978, vol. 18, pp. 290–98.

    Google Scholar 

  29. D. Bouchard and J.S. Kirkaldy: Metall. Mater. Trans. B, 1997, vol. 28B, pp. 651–63.

    Article  CAS  Google Scholar 

  30. P. Lehmann, R. Moreaub, D. Camela, and R. Bolcatob: J. Cryst. Growth, 1998, vol. 183, pp. 690–704.

    Article  CAS  Google Scholar 

  31. H. Wang: Ph.D. Thesis, McMaster University, Hamilton, ON, Canada, 2009.

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Author information

Authors and Affiliations

  1. Light Metal Casting Research Centre (LMCRC), McMaster University, Hamilton, ON, L8S 4L7, Canada

    Hongda Wang (Postdoctoral Candidate) & Sumanth Shankar (Associate Professor)

  2. Thermal Processing Laboratory (TPL), McMaster University, Hamilton, ON, L8S 4L7, Canada

    Hongda Wang (Postdoctoral Candidate) & Mohamed S. Hamed (Associate Professor)

Authors
  1. Hongda Wang
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  2. Mohamed S. Hamed
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  3. Sumanth Shankar
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Corresponding author

Correspondence to Mohamed S. Hamed.

Additional information

Manuscript submitted July 8, 2010.

Appendix

Appendix

Thermophysical Material Properties

 

Properties

Al-3 Wt pct Si

Al-5 Wt pct Si

Al-7 Wt pct Si

K s

228.08 − 0.061055 × T

226.01 – 0.077488 × T

223.93 – 0.093920 × T

K l

85.476

84.568

83.661

C ps

887.23 + 0.50227 × T

883.54 + 0.50227 × T

879.85 + 0.50227 × T

C pl

1168.9

1163.7

1158.6

ρ s

2627.9

2621.6

2614.5

ρ l

2415.0

2422.8

2430.6

L

4.05 × 105

4.25 × 105

4.45 × 105

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Wang, H., Hamed, M.S. & Shankar, S. Effect of Shrinkage on Primary Dendrite Arm Spacing during Binary Al-Si Alloy Solidification. Metall Mater Trans A 42, 2331–2345 (2011). https://doi.org/10.1007/s11661-011-0611-5

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