Skip to main content
SpringerLink
Log in

A multiphase solute diffusion model for dendritic alloy solidification

  • Published:
Metallurgical Transactions A Aims and scope Submit manuscript

Abstract

A solute diffusion model, aimed at predicting microstructure formation in metal castings, is proposed for dendritic solidification of alloys. The model accounts for the different length scales existing in a dendritic structure. This is accomplished by utilizing a multiphase approach, in which not only the various physical phases but also phases associated with different length scales are considered separately. The macroscopic conservation equations are derived for each phase using the volume averaging technique, with constitutive relations developed for the interfacial transfer terms. It is shown that the multiphase model can rigorously incorporate the growth of dendrite tips and coarsening of dendrite arms. In addition, the distinction of different length scales enables the inclusion of realistic descriptions of the dendrite topology and relations to key metallurgical parameters. Another novel aspect of the model is that a single set of conservation equations for solute diffusion is developed for both equiaxed and columnar dendritic solidification. Finally, illustrative calculations for equiaxed, columnar, and mixed columnar-equiaxed solidification are carried out to provide quantitative comparisons with previous studies, and a variety of fundamental phenomena such as recalescence, dendrite tip undercooling, and columnar-to-equiaxed transition (CET) are predicted.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Abbreviations

a :

constant, in the Appendix

A :

interfacial area, m2

As :

area of the solid/interdendritic liquid interface, m2

Ae :

area of the dendrite envelope, m2

b :

constant, in the Appendix

c :

constant, in the Appendix

C :

concentration of a chemical species, weight percent

cp :

volumetric specific heat, J m-3 K-1

d :

microscopic length scale, m

ds :

mean characteristic length or diameter of the solid phase, m

de :

mean characteristic diameter of the dendrite envelope, m

D :

diffusion coefficient, m2 s-1

Iv:

Ivantsov function

j:

species diffusion flux, kg m-2 s-1

J :

interfacial species transfer rate per unit of volume, kg m-3 s-1

k :

constants in the nucleation law, see Figure 6

l :

species diffusion length, m

L :

macroscopic length scale, m

ml :

liquidus line slope

n :

nuclei density, m3

n :

outwardly directed unit normal vector

Pe:

envelope Peclet number, -wneRf/Dl

Pet :

solutal Peclet number at the dendrite tip,V tRt/2Dl

r :

radial coordinate

R :

radius, m

S :

interfacial area concentration, Ak/Vo, m-1

Sv :

specific interfacial area, Ak/Vk, m-1 tet, time, s

T :

temperature, K

T :

cooling rate, Ks-1

v:

velocity, m s-1

Vk :

volume of phasek, m3

Vo :

averaging volume, m3

Vt :

dendrite tip velocity, m s-1

w:

interface velocity, m s-1

wn :

normal interface velocity, ms-1

x:

position vector

X :

phase function

α:

diffusion Fourier number, 4Dstf 2λ2

Β:

a factor

г:

interfacial mass transfer rate due to interface movement (kg m-3 s-1) or Gibbs-Thomson coefficient (mK)

δh:

volumetric latent heat of phase change, J m-3

ε:

volume fraction

κ:

partition coefficient

λ:

dendrite arm spacing, m

ρ:

density, kg m-3

Φ:

shape factor

Φ:

a field property

ψ:

a field property

Ω:

solutal supersaturation

d :

interdendritic liquid

e :

dendrite envelope

f :

final dimension of the dendrite envelope

j :

phasej

k :

phasek

]kj :

pertinent to phasek on thek- j interface

l :

extradendritic liquid

m :

melting point of pure metal

n :

normal direction

o :

initial state

r :

in the r-direction

s :

solid

t :

dendrite tip

j :

due to species gradients

г:

due to interface movement interfacial area—averaged

*:

effective or dimensionless

References

  1. C. Beckermann and R. Viskanta:Appl. Mech. Rev., 1993, vol. 46, pp. 1–27.

    Article  Google Scholar 

  2. M. Rappaz:Int. Mater. Rev., 1989, vol. 34, pp. 93–123.

    Article  CAS  Google Scholar 

  3. V.R. Voiler, M.S. Stachowicz, and B.G. Thomas:Materials Processing in the Computer Age, TMS, Warrendale, PA, 1991.

  4. J. Ni and C. Beckermann:Metall. Trans. B, 1991, vol. 22B, pp. 349–61.

    Article  CAS  Google Scholar 

  5. S. Ganesan and D.R. Poirier:Metall. Trans. B, 1990, vol. 21B, pp. 173–81.

    Article  CAS  Google Scholar 

  6. I. Dustin and W. Kurz:Z. Metallkd., 1986, vol. 77, pp. 265–73.

    CAS  Google Scholar 

  7. M. Rappaz and Ph. Thevoz:Acta Metall., 1987, vol. 35, pp. 1487–97.

    Article  CAS  Google Scholar 

  8. M. Rappaz and Ph. Thevoz:Acta Metall., 1987, vol. 35, pp. 2929–33.

    Article  CAS  Google Scholar 

  9. S.C. Flood and J.D. Hunt:Appl. Sci. Res., 1987, vol. 44, pp. 27–42.

    Article  CAS  Google Scholar 

  10. B. Giovanola and W. Kurz:Metall Trans. A, 1990, vol. 21A, pp. 260–63.

    Article  CAS  Google Scholar 

  11. R.C. Kerr, A.W. Woods, M.G. Worster, and H.E. Huppert:J. Fluid Mech., 1990, vol. 217, pp. 331–48.

    Article  CAS  Google Scholar 

  12. N. Wakao and J.M. Smith:Chem. Eng. Sci., 1962, vol. 17, pp. 825–34.

    Article  CAS  Google Scholar 

  13. O.A. Plumb and S. Whitaker: inDynamics of Fluids in Hierarchical Porous Media, J.H. Cushman, ed., Academic Press, London, 1990, pp. 97–148.

    Google Scholar 

  14. M. Kaviany:Principles of Heat Transfer in Porous Media, Springer-Verlag, New York, NY, 1991.

    Book  Google Scholar 

  15. C.Y. Wang and C. Beckermann:Int. J. Multiphase Flow, 1993, vol. 19, pp. 397–407.

    Article  Google Scholar 

  16. D.A. Drew:Annu. Rev. Fluid Mech., 1983, vol. 15, pp. 261–91.

    Article  Google Scholar 

  17. S.M. Hassanizadeh and W.G. Gray:Adv. Water Resour., 1979, vol. 2, pp. 131–44.

    Article  Google Scholar 

  18. S.M. Hassanizadeh and W.G. Gray:Adv. Water Resour., 1979, vol. 2, pp. 191–208.

    Article  Google Scholar 

  19. S.M. Hassanizadeh and W.G. Gray:Adv. Water Resour., 1980, vol. 3, pp. 25–41.

    Article  Google Scholar 

  20. M. Rappaz and V.R. Voller:Metall. Trans. A, 1990, vol. 21A, pp. 749–53.

    Article  CAS  Google Scholar 

  21. D.R. Poirier, P.J. Nandapurkar, and S. Ganesan:Metall. Trans. B, 1991, vol. 22B, pp. 889–900.

    Article  CAS  Google Scholar 

  22. S.P. Marsh and M.E. Glicksman: inModeling of Casting, Welding and Advanced Solidification Processes VI, T.S. Piwonka, V. Voller, and L. Katgerman, eds.,TMS, Warrendale, PA, 1993, pp. 55–62.

    Google Scholar 

  23. M.E. Glicksman, R.N. Smith, S.P. Marsh, and R. Kuklinski:Metall. Trans. A, 1992, vol. 23A, pp. 659–67.

    Article  CAS  Google Scholar 

  24. R.T. DeHoff and F.N. Rhines:Quantitative Microscopy, McGraw-Hill, New York, NY, 1968.

    Google Scholar 

  25. R.B. Bird, W.E. Stewart, and E.N. Lightfoot:Transport Phenomena, John Wiley, New York, NY, 1960, p. 167.

    Google Scholar 

  26. T.Z. Kattamis, J.C. Coughlin, and M.C. Flemings:Trans. TMS- A1ME, 1967, vol. 239, pp. 1504–11.

    CAS  Google Scholar 

  27. S.P. Marsh, M.E. Glicksman, L. Meloro, and K. Tsutsumi: inModeling and Control of Casting and Welding Processes IV, A.F. Giame and G.J. Abbashian, eds., TMS, Warrendale, PA, 1988, pp. 15–23.

    Google Scholar 

  28. M. Avrami:J. Chem. Phys., 1940, vol. 8, pp. 212–24.

    Article  CAS  Google Scholar 

  29. G.R. Speich and R.M. Fisher: inRecrystallization, Grain Growth and Textures, Margolin, H., ed., ASM, Metals Park, OH, 1966, pp. 563–98.

    Google Scholar 

  30. S. Ahuja, C. Beckermann, R. Zakhem, P.D. Weidman, and H.C. deGroh III: inMicro/Macro Scale Phenomena in Solidification, C. Beckermann, L.A. Bertram, S.J. Pien, and R.E. Smelser, eds., ASME, New York, NY, 1992, pp. 85–91.

    Google Scholar 

  31. W. Kurz and D.J. Fisher:Fundamentals of Solidification, Trans. Tech Publications, Aedermannsdorf, Switzerland, 1989.

  32. J. Lipton, M.E. Glicksman, and W. Kurz:Mater. Sci. Eng., 1984, vol. 65, pp. 57–63.

    Article  CAS  Google Scholar 

  33. C.Y. Wang and C. Beckermann:Mat. Sci. Eng. A, 1993, in press.

  34. I. Ohnaka:Trans. Iron Steel Inst. Jpn., 1986, vol. 26, pp. 1045–51.

    Article  CAS  Google Scholar 

  35. A.J. Campagna: Ph.D. Thesis, Massachusetts Institute of Technology, Cambridge, MA, 1970.

  36. M.C. Flemings:Metall. Trans. A, 1991, vol. 22A, pp. 957–81.

    Article  CAS  Google Scholar 

  37. L.A. Bendersky and W.J. Boettinger: inRapidly Quenched Metals, S.S. Warlimont, ed., 1985, vol. 1, pp. 887–90.

    Article  CAS  Google Scholar 

  38. C.Y. Wang and C. Beckermann:Metall. Trans. A, 1993, vol. 24A, in press.

  39. S.V. Patankar:Numerical Heat Transfer and Fluid Flow, Hemisphere Publishing Corp., New York, NY, 1980.

  40. J.D. Hunt:Mater. Sci. Eng., 1984, vol. 65, pp. 75–83.

    Article  CAS  Google Scholar 

  41. I. Ziv and F. Weinberg:Metall. Trans. B, 1989, vol. 20B, pp. 731–34.

    Article  CAS  Google Scholar 

  42. D.M. Stefanescu, G. Upadhya, and D. Bandyopahyoy:Metall. Trans. A, 1990, vol. 21A, pp. 997–1005.

    Article  CAS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mechanical Engineering, The University of Iowa, 52242, Iowa City, IA

    C. Y. Wang (Graduate Research Assistant) & C. Beckermann (Associate Professor)

Authors
  1. C. Y. Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. C. Beckermann
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Wang, C.Y., Beckermann, C. A multiphase solute diffusion model for dendritic alloy solidification. Metall Trans A 24, 2787–2802 (1993). https://doi.org/10.1007/BF02659502

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02659502

Keywords

Search

Navigation