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Numerical studies on columnar-to-equiaxed transition in directional solidification of binary alloys

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Abstract

A numerical study on columnar-to-equiaxed transition (CET) during directional solidification of binary alloys is presented using a macroscopic solidification model. The position of CET is predicted numerically using a critical cooling rate criterion reported in literature. The macroscopic solidification model takes into account movement of solid phase due to buoyancy, and drag effect on the moving solid phase because of fluid motion. The model is applied to simulate the solidification process for binary alloys (Sn–Pb) and to estimate solidification parameters such as position of the liquidus, velocity of the liquidus isotherm, temperature gradient ahead of the liquidus, and cooling rate at the liquidus. Solidification phenomena under two cooling configurations are studied: one without melt convection and the other involving thermosolutal convection. The numerically predicted positions of CET compare well with those of experiments reported in literature. Melt convection results in higher cooling rate, higher liquidus isotherm velocities, and stimulation of occurrence of CET in comparison to the nonconvecting case. The movement of solid phase aids further the process of CET. With a fixed solid phase, the occurrence of CET based on the same critical cooling rate is delayed and it occurs at a greater distance from the chill.

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Abbreviations

c p :

Specific heat (J/kg-K)

C :

Species concentration (kg/kg)

f :

Mass fraction

D :

Mass diffusion coefficient of the species (m2/s)

g :

Acceleration due to gravity (m/s2)

g l :

Volume fraction of liquid

g s :

Volume fraction of solid

G L :

Temperature gradient (K/m)

ΔH :

Latent enthalpy content of a control volume (J/kg)

k p :

Partition coefficient

L :

Latent heat of fusion (J/kg)

p :

Pressure (N/m2)

r :

Cooling rate (K/s)

S :

Source term

t :

Time (s)

T :

Temperature (K)

u :

Continuum velocity vector (m/s)

V L :

Tip velocity (m/s)

x, y :

Coordinates

β T :

Thermal expansion coefficient (1/K)

β C :

Solutal expansion coefficient

μ :

Dynamic viscosity (kg/s-m)

ρ :

Mixture density (kg/m3)

cr:

Critical

eff:

Effective

i:

Initial

l:

Liquid phase

mix:

Mixture

ref:

Reference

s:

Solid phase

References

  1. Quested TE, Greer AL (2005) Acta Mater 53:4643

    Article  CAS  Google Scholar 

  2. Gandin CA (2000) Acta Mater 48:2483

    Article  CAS  Google Scholar 

  3. Hunt JD (1984) Mater Sci Eng 65:75

    Article  ADS  CAS  Google Scholar 

  4. Martorano MA, Beckermann C, Gandin CA (2003) Metall Mater Trans A 34:1657

    Article  Google Scholar 

  5. Mathiesen R, Arnberg L, Ramsoskar K, Weitkamp T, Rau Ch, Snigirev A (2002) Metall Mater Trans B 33:613

    Article  Google Scholar 

  6. Kumar A, Dutta P, Rayleigh A (2008) J Phys D 41:155501, 9 pp

    Article  ADS  Google Scholar 

  7. Mathiesen RH, Arnberg L, Bleuet P, Somogyi A (2006) Metall Mater Trans A 37:2515

    Article  Google Scholar 

  8. Griffiths WD, Xiao L, McCartney DG (1996) Mater Sci Eng A 205:31

    Article  Google Scholar 

  9. Kumar A, Dutta P (2006) J Phys D 39:3058

    Article  ADS  CAS  Google Scholar 

  10. Flood SC, Hunt JD (1988) Metals handbook, vol 15. ASM International, Materials Park, p 130

    Google Scholar 

  11. Herlach DM, Eckler K, Karma A, Schwarz M (2001) Mater Sci Eng A 304–306:20

    Google Scholar 

  12. Siqueira CA, Cheung N, Garcia A (2002) Metall Mater Trans A 33:2107

    Article  Google Scholar 

  13. Siquiera CA, Cheung N, Garcia A (2003) J Alloys Compd 351:126

    Article  Google Scholar 

  14. Suri VK, El-Kaddah N, Berry JT (1991) AFS Trans 99:187

    CAS  Google Scholar 

  15. Mahapatra RB, Weinberg F (1987) Metall Trans B 18:425

    Article  Google Scholar 

  16. Ziv I, Weinberg F (1989) Metall Trans B 20:731

    Article  Google Scholar 

  17. Ares AE, Schvezov CE (2000) Metall Mater Trans A 31:1611

    Article  Google Scholar 

  18. Hunt JD, Lu SZ (1996) Metall Mater Trans A 27:611

    Article  Google Scholar 

  19. Wang CY, Beckermann C (1994) Metall Mater Trans A 25:1081

    Article  Google Scholar 

  20. Spinelli JE, Ferreira IL, Garcia A (2004) J Alloys Compd 384:217

    Article  CAS  Google Scholar 

  21. Li L, Zhang Y, Esling C, Zhao Z, Zuo Y, Zhang H, Cui J (2009) J Mater Sci 44(4):1063. doi:10.1007/s10853-008-3158-0

    Article  ADS  CAS  Google Scholar 

  22. Zuo M, Liu X, Sun Q (2009) J Mater Sci 44(8):1952. doi:10.1007/s10853-009-3287-0

    Article  ADS  CAS  Google Scholar 

  23. Tang J, Xue X (2009) J Mater Sci 44(3):745. doi:10.1007/s10853-008-3157-1

    Article  ADS  CAS  Google Scholar 

  24. Oryshchyn DB, Dogan ON (2008) J Mater Sci 43(4):1471. doi:10.1007/s10853-007-2325-z

    Article  ADS  CAS  Google Scholar 

  25. Dong HB, Yang XL, Lee PD, Wang W (2004) J Mater Sci 39(24):7207. doi:10.1023/B:JMSC.0000048733.96958.c3

    Article  ADS  CAS  Google Scholar 

  26. Dong HB, Lee PD (2005) Acta Mater 53(3):659

    Article  CAS  Google Scholar 

  27. Ludwig A, Wu M (2005) Mater Sci Eng A 413–414:109

    Google Scholar 

  28. Appolaire B, Combeau H, Lesoult G (2008) Mater Sci Eng A 487(1–2):33

    Google Scholar 

  29. Bennon WD, Incropera FP (1987) Int J Heat Mass Transf 30(16):2161

    Article  MATH  CAS  Google Scholar 

  30. Voller VR, Brent AD, Prakash C (1989) Int J Heat Mass Transf 32(9):1719

    Article  ADS  CAS  Google Scholar 

  31. Prescott PJ, Incropera FP (1996) In: Poulikakos D (ed) Advances in heat transfer, vol 28. Academic Press, New York, pp 231–338

    Google Scholar 

  32. Voller VR (2006) In: Minkowycz WJ, Sparrow EM, Murthy JY (eds) Handbook of numerical heat transfer, 2nd edn. Wiley, New York, chap 19

    Google Scholar 

  33. Kumar A, Dutta P (2005) Int J Heat Mass Transf 48:3674

    Article  CAS  Google Scholar 

  34. Vreeman CJ, Krane MJM, Incropera FP (2000) Int J Heat Mass Transf 43:677

    Article  MATH  CAS  Google Scholar 

  35. Chowdhury J, Ganguly S, Chakraborty S (2005) J Phys D 38:2869

    Article  ADS  CAS  Google Scholar 

  36. Chowdhury J, Ganguly S, Chakraborty S (2007) Int J Heat Mass Transf 50:2692

    Article  MATH  Google Scholar 

  37. Manninen M, Taivassalo V, Kallio S (1996) On the mixture model for multiphase flow. VTT Publications 288, Technical Research Centre of Finland, Espoo

  38. Yang BJ, Leon-Torres J, Stefanescu DM (1999) Int J Cast Met Res 11:527

    CAS  Google Scholar 

  39. Kumar A, Dutta P, Walker MJ, Sundarraj S (2007) Numer Heat Transf A 51:59

    Article  ADS  CAS  Google Scholar 

  40. Patankar SV (1980) Numerical heat transfer and fluid flow. Hemisphere, Washington, DC

    MATH  Google Scholar 

Download references

Acknowledgement

The authors gratefully acknowledge the financial support of General Motors Corporation, USA.

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  1. Arvind Kumar

    Present address: Ecole des Mines de Nancy, Parc de Saurupt, 54042, Nancy Cedex, France

Authors and Affiliations

  1. Department of Mechanical Engineering, Indian Institute of Science, Bangalore, 560012, India

    Arvind Kumar & Pradip Dutta

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  1. Arvind Kumar
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  2. Pradip Dutta
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Correspondence to Pradip Dutta.

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Kumar, A., Dutta, P. Numerical studies on columnar-to-equiaxed transition in directional solidification of binary alloys. J Mater Sci 44, 3952–3961 (2009). https://doi.org/10.1007/s10853-009-3539-z

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  • DOI: https://doi.org/10.1007/s10853-009-3539-z

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