Metallurgical and Materials Transactions A

, Volume 34, Issue 12, pp 3009–3019

Modeling of marangoni-induced droplet motion and melt convection during solidification of hypermonotectic alloys

  • Menghuai Wu
  • Andreas Ludwig
  • Lorenz Ratke
Article

Abstract

A two-phase volume averaging approach to model Marangoni-induced droplet motion of the minority liquid phase and the convection in the parent melt during solidification of the hypermonotectic alloys is presented. The minority liquid phase decomposed from the parent melt as droplets in the miscibility gap was treated as the second-phase L2. The parent melt including the solidified monotectic matrix was treated as the first phase L1. Both phases were considered as different and spatially interpenetrating continua. The conservation equations of mass, momentum, solute, and enthalpy for both phases, and an additional transport equation for the droplet density, were solved. Nucleation of the L2 droplets, diffusion-controlled growth, interphase interactions such as Marangoni force at the L1-L2 interface, Stokes force, solute partitioning, and heat release of decomposition were taken into account by corresponding source and exchange terms in the conservation equations. The monotectic reaction was modeled by adding the latent heat on the L1 phase during monotectic reaction, and applying an enlarged viscosity to the solidified monotectic matrix. A two-dimensional (2-D) square casting with hypermonotectic composition (Al-10 wt pct Bi) was simulated. This paper focused on Marangoni motion, hence gravity was not included. Results with nucleation, droplet evolution, Marangoni-induced droplet motion, solute transport, and macrosegregation formation were obtained and discussed.

Nomenclature

c0

alloy concentration

cc

critical concentration

c1, c2

volume-averaged species concentration

c*1, c*2

interface concentration under thermal equilibrium

Δcd

c*2c*1

cL2

L2 monotectic concentration

cm

monotectic concentration

c*

interface species

Δc

c1c*1

C12(= −C21)

species exchange rate

C12d(= −C21d)

species transfer at L1-L2 interface

C12p(= −C21p)

solute partitioning due to phase change

cmix

mix concentration

cp(1), cp(2)

specific heat

D1, D2

diffusion coefficient

d2

droplet diameter

f1, f2

volume fraction

fM

Marangoni force on single droplet

fst

Stokes force on a single droplet

FM

volume-averaged Marangoni force

g

gravity

H

heat-transfer coefficient at casting-mold interface

H*

volume heat-transfer coefficient between two liquid phases

h1, h2

enthalpy

h1ref, h2ref

enthalpy at Tref

h*

interface enthalpy

Δhd

heat of decomposition

ΔhM

latent heat of monotectic reaction

K21(=K12)

momentum exchange coefficient

k

solute partitioning coefficient

k1, k2

thermal conductivity

L1, L2

two liquid phases

M12 (= −M21)

mass-transfer rate per volume

m

slope of liquidus in phase diagram at c0

m12

mass-transfer rate for a single droplet

N

droplet nucleation rate

n

droplet density

nmax

maximum droplet density

p

pressure

Q12(= −Q21)

energy exchange rate

Q12d(= −Q21d)

energy exchange by heat transfer

Q12p(= −Q21p)

energy exchange due to phase change

R

droplet radio

SA

solid-phase Al

SB

solid-phase Bi

Tc

critical temperature

T, T1, T2

temperature

TfA

melting point of pure metal (Al)

TfB

melting point of pure metal (Bi)

Tmf

monotectic temperature

Tref

reference temperature for enthalpy definition

T

temperature gradient

ΔT

undercooling

ΔTN

Gaussian distribution width of droplet nucleation law

ΔTσ

undercooling for maximum droplet nucleation rate

t

time

U12 (= −U21)

momentum exchange rate

U12d (= −U21d)

momentum exchange due to Stokes force

U12p (= −U21p)

momentum exchange due to phase change

u1, u2

velocity component in x direction

u1, u2

velocity vector

u12, u21

interphase velocity

u*

interface velocity

v1, v2

velocity component in y direction

ρ1, ρ2

density

σ

surface tension at liquid-liquid interface

σ0

experimental parameter in Eq. [19]

μ1, μ2

viscosity

τ1, τ2

stress-strain tensors

Subscripts 1, and 2

indicate first and second liquid phases.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    L. Ratke and S. Diefenbach: Mater. Sci. Eng., 1995, vol. 15 (R), pp. 263–347.CrossRefGoogle Scholar
  2. 2.
    B. Predel, L. Ratke, and H. Fredriksson: in Decomposition of Alloys: The Early Stages, H.U. Walter and M.F. Ashhy, eds., Springer-Verlag, Berlin, 1987, pp. 517–65.Google Scholar
  3. 3.
    B. Prinz, A. Romero, and L. Ratke: J. Mater. Sci., 1995, vol. 30, pp. 4715–19.CrossRefGoogle Scholar
  4. 4.
    L. Ratke, G. Korekt, and S. Drees: Adv. Space Res., 1998, vol. 22, pp. 1227–36.CrossRefGoogle Scholar
  5. 5.
    L. Ratke and G. Korekt: Z. Metallkd., 2000, vol. 91, pp. 919–27.Google Scholar
  6. 6.
    N.O. Young, J.S. Goldstein, and M.J. Block: J. Fluid Mech., 1959, vol. 6, pp. 350–56.CrossRefGoogle Scholar
  7. 7.
    W. Günter: Über die Dynamik von Fluidpartikeln Aufgrund des Marangoni-Effektes, VDI-Verlag, Düsseldorf, Germany, 1993, pp. 1–71.Google Scholar
  8. 8.
    M.G. Velarde: in Materials and Fluids under Low Gravity, L. Ratke, H. Walter, and B. Feuerbacher eds., Springer-Verlag, Berlin, 1995, pp. 283–98.Google Scholar
  9. 9.
    C. Beckermann and R. Viskanta: Appl. Mech. Rev., 1993, vol. 46, pp. 1–27.CrossRefGoogle Scholar
  10. 10.
    J. Ni and C. Beckermann: Metall. Trans. B, 1991, vol. 22B, pp. 349–61.Google Scholar
  11. 11.
    C.Y. Wang and C. Beckermann: Metall. Mater. Trans. A, 1996, vol. 27A, pp. 2754–64.Google Scholar
  12. 12.
    C.Y. Wang and C. Beckermann: Metall. Mater. Trans. A, 1996, vol. 27A, pp. 2765–83.Google Scholar
  13. 13.
    C.Y. Wang and C. Beckermann: Metall. Mater. Trans. A, 1996, vol. 27A, pp. 2784–95.Google Scholar
  14. 14.
    C. Beckermann: JOM, 1997, vol. 49, pp. 13–17.Google Scholar
  15. 15.
    A.V. Reddy and C. Beckermann: Metall. Mater. Trans. B, 1997, vol. 28B, pp. 479–89.CrossRefGoogle Scholar
  16. 16.
    J. Ni and F.P. Incropera: Int. J. Heat Mass Transfer, 1995, vol. 38, pp. 1271–84.CrossRefGoogle Scholar
  17. 17.
    J. Ni and F.P. Incropera: Int. J. Heat Mass Transfer, 1995, vol. 38, pp. 1285–96.CrossRefGoogle Scholar
  18. 18.
    A. Ludwig and M. Wu: Metall. Mater. Trans. A, 2002, vol. 33A, pp. 3673–83.CrossRefGoogle Scholar
  19. 19.
    M. Wu and A. Ludwig: Adv. Eng. Mater., 2003, vol. 5, pp. 62–66.CrossRefGoogle Scholar
  20. 20.
    M. Wu, A. Ludwig, A. Bührig-Polaczek, M. Fehlbier, and P.R. Sahm: Int. J. Heat Mass Transfer, 2003, vol. 46, pp. 2819–32.CrossRefGoogle Scholar
  21. 21.
    S. Diefenbach: Ph.D. Thesis, Ruhr-University Bochum, Bochum, Germany, 1993.Google Scholar
  22. 22.
    W. Oldfield: Trans. ASM, 1966, vol. 59, pp. 945–61.Google Scholar
  23. 23.
    M. Rappaz: Int. Mater. Rev., 1989, vol. 34, pp. 93–123.Google Scholar
  24. 24.
    F. Falk: in Immiscible Liquid Metals and Organics, L. Ratke, ed., DGM Informationsgesellschatt mbH, Oberursel, Germany, 1993, pp. 93–100.Google Scholar
  25. 25.
    S.V. Patankar: Numerical Heat Transfer and Fluid Flow, Hemisphere Publishing Corp., Washington, DC, 1980.Google Scholar
  26. 26.
    L. Ratke, S. Dree, S. Diefenbach, B. Pronz, and A. Ahlborn: in Materials and Fluids under Low Gravity, L. Ratke, H. Walter, and B. Feuerbacher, eds., Springer-Verlag, Berlin, 1995, pp. 115–33.Google Scholar
  27. 27.
    P. Thevoz, J.L. Desbiolles, and M. Rappaz: Metall. Trans. A, 1989, vol. 20A, pp. 311–22.Google Scholar
  28. 28.
    N. Uebber and L. Ratke: Scripta Metall. Mater., 1991, vol. 25, pp. 1133–37.CrossRefGoogle Scholar
  29. 29.
    L. Granasy and L. Ratke: Scripta Metall. Mater., 1993, vol. 28, pp. 1329–34.CrossRefGoogle Scholar
  30. 30.
    H.U. Walter: Proc. RIT/ESA/SSC Workshop-ESA SP 219, Stockholm, Sweden, 1984, Järva Krog, ed., pp. 47–64.Google Scholar

Copyright information

© ASM International & TMS-The Minerals, Metals and Materials Society 2003

Authors and Affiliations

  • Menghuai Wu
    • 1
  • Andreas Ludwig
    • 1
  • Lorenz Ratke
    • 2
  1. 1.Simulation and Modeling of Metallurgical ProcessesUniversity of LeobenLeobenAustria
  2. 2.the Institute for Space SimulationGerman Aerospace Research Establishment DLRCologneGermany

Personalised recommendations