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Application of an Equiaxed Grain Growth and Transport Model to Study Macrosegregation in a DC Casting Experiment

  • Akash Pakanati
  • Knut Omdal Tveito
  • Mohammed M’HamdiEmail author
  • Hervé Combeau
  • Miha Založnik
Article
First Online:

Abstract

A simplified three-phase, multiscale macrosegregation model which describes the growth kinetics of equiaxed grains and the coupling between microstructure morphology and the macroscopic transport has been proposed previously. In this paper, the model is validated by comparing the numerical model predictions to the experimental data from DC casting of an AA7050 alloy billet. The morphology of the equiaxed grains has an important influence on the macrosegregation, and we show that the model predictions are accurate when the grain morphology is described correctly.

Nomenclature

〈Ci,〉

Average mass concentration of solute i (wt pct)

〈Ci*,〉

Average equilibrium mass concentration of solute i (wt pct)

Co, i

Mean concentration of solute i (wt pct)

cp

Specific heat (J kg−1 K)

CD

Drag co-efficient (–)

d

Diameter of inoculant particle (m)

D, i

Diffusion coefficient of solute i (m2 s−1)

g

Volume fraction (–)

gpack

Packing fraction (–)

gintern

Internal solid fraction (–)

\( \vec{\varvec{g}} \)

Acceleration due to gravity (m s−2)

〈hl〉l

Averaged liquid enthalpy (J kg−1)

〈hs〉s

Averaged solid enthalpy (J kg−1)

hm

Mixture enthalpy (J kg−1)

hprimary

Primary cooling heat-transfer coefficient (W m−2 K−1)

hsecondary

Secondary cooling heat-transfer coefficient (Wm−2 K−1)

kp, i

Partition coefficient of solute i (–)

K

Permeability (m2)

Lf

Latent heat of fusion (J kg−1)

lkc

Characteristic length for permeability (m)

ml, i

Liquidus slope of solute i, K (wt pct−1)

Nnuci

Volumetric number density (m−3)

Ng

Grain density (m−3)

pl

Liquid pressure (N m−2)

P

Perimeter of the ingot (m)

Qwater

Water flow rate (m3 s−1)

Renv

Radius of the envelope (m)

Rs,eq

Radius of the solid grain (m)

Re

Reynolds number

Sv

Interfacial area density (m−1)

Sc

Schmidts number

t

Time (s)

T

Temperature (K)

Twater

Temperature of cooling water (K)

Tsat

Temperature of boiling water (K)

Tcast

Casting temperature (K)

Tliq

Temperature of liquidus (K)

Tm

Melting temperature of pure Al (K)

Teut

Eutectic temperature (K)

ΔT

Undercooling (K)

ΔTc

Critical undercooling for nucleation (K)

\(\langle \vec{v}_{\text{l}}\rangle^{\text{l}} \)

Intrinsic average velocity of liquid phase (ms−1)

\( \langle\vec{v}_{\text{s}}\rangle^{\text{s}} \)

Intrinsic average velocity of solid phase (ms−1)

\( \vec{V}_{\text{cast}} \)

Casting velocity (ms−1)

Vtip

Velocity of dendrite tip (ms−1)

βT

Thermal expansion coefficient (K)

βC, i

Solutal expansion coefficient of solute i, (pct w−1)

δi

Diffusion length of solute i (− m)

δ(t)

Dirac function

ΓGT

GIBBS–Thomson co-efficient (Km)

Γ

Growth rate (kg m−3 s−1)

κ

Thermal conductivity (W m−1 K)

ρl

Liquid density (kg m−3)

ρs

Solid density used to account for shrinkage (kg m−3)

ρl,b

Liquid buoyancy density used to account for Bousinessq approximation (kg m−3)

ρs,b

Solid buoyancy density used to account for grain motion (kg m−3)

ρm

Mixture density (kg m−3)

μl

Liquid dynamic viscosity (Pa s)

Subscripts and Superscripts

l

Liquid

s

Solid

env

Envelope

e

Extragranular liquid

d

Intragranular liquid

s–d

Solid–liquid interface

e–d

Intra-extra granular liquid interface

*

equilibrium

l,b

Liquid buoyancy

s,b

Solid buoyancy

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Notes

Acknowledgments

This study was conducted within the framework of PRIMAL project, of which Hydro Aluminium ASA, Alcoa Norway ANS, Aleris Rolled Products Germany GmbH, Institute of Energy Technology (IFE), NTNU, and SINTEF are the partners. This project is supported by the Research Council of Norway. A.P and M.M acknowledge the support of NOTUR High Performance Computing program. H.C and M.Z. acknowledge the support by the French State through the program “Investment in the future” run by the National Research Agency (ANR) and referenced by ANR-11 LABX-0008-01 (LabEx DAMAS).

References

  1. 1.
    A. V. Reddy and N.C. Beckermann: Metall. Mater. Trans. B, 1997, vol. 28, pp. 479–89.Google Scholar
  2. 2.
    G. Lesoult, V. Albert, B. Appolaire, H. Combeau, D. Daloz, A. Joly, C. Stomp, G.U. Grün, and P. Jarry: Sci. Technol. Adv. Mater., 2001, vol. 2, pp. 285–91.Google Scholar
  3. 3.
    K.O. Tveito, A. Pakanati, M. M’Hamdi, H. Combeau, and M. Založnik: Metall. Mater. Trans. A, 2018, vol. 49, pp. 2778–94.Google Scholar
  4. 4.
    L. Heyvaert, M. Bedel, M. Založnik, and H. Combeau: Metall. Mater. Trans. A, 2017, vol. 48, pp. 4713–34.Google Scholar
  5. 5.
    A. Olmedilla, M. Založnik, B. Rouat, and H. Combeau: Phys. Rev. E., 2018, 1:1–12. https://doi.org/10.1103/physreve.97.012910.Google Scholar
  6. 6.
    M. Bedel, K.O. Tveito, M. Založnik, H. Combeau, and M. M’Hamdi: Comput. Mater. Sci., 2015, vol. 102, pp. 95–109.Google Scholar
  7. 7.
    M. Rappaz: Int. Mater. Rev., 1989, vol. 34, pp. 93–124.Google Scholar
  8. 8.
    J. Ni and C. Beckermann: Metall. Trans. B 1991, vol. 22, pp. 349–61.Google Scholar
  9. 9.
    C.Y. Wang and C. Beckermann: Metall. Mater. Trans. A, 1996, vol. 27A, pp. 2754–64.Google Scholar
  10. 10.
    M. Wu, A. Ludwig, A. Buhrig-Polaczek, M. Fehlbier, and P.R. Sahm: Int. J. Heat Mass Transf., 2003, vol. 46, pp. 2819–32.Google Scholar
  11. 11.
    M. Wu and A. Ludwig: Acta Mater., 2009, vol. 57, pp. 5621–5631.Google Scholar
  12. 12.
    M. Wu, A. Fjeld, and A. Ludwig: Comput. Mater. Sci., 2010, vol. 50, pp. 43–58.Google Scholar
  13. 13.
    M. Založnik and H. Combeau: Comput. Mater. Sci., 2010, vol. 48, pp. 1–10.Google Scholar
  14. 14.
    H. Combeau, M. Založnik, S. Hans, and P.E. Richy: Metall. Mater. Trans. B, 2009, vol. 40, pp. 289–304.Google Scholar
  15. 15.
    A. Kumar, M. Založnik, and H. Combeau: Int. J. Adv. Eng. Sci. Appl. Math., 2010, vol. 2, pp. 140–8.Google Scholar
  16. 16.
    J. Li, M. Wu, A. Ludwig, and A. Kharicha: Int. J. Heat Mass Transf., 2014, vol. 72, pp. 668–79.Google Scholar
  17. 17.
    M. Wu, A. Ludwig, and A. Kharicha: Steel Res. Int., 2018, vol. 89, pp. 1–14.Google Scholar
  18. 18.
    T. Jalanti: PhD thesis, Ecole Polytechnique Fédérale de Lausanne, Laussanne, Switzerland, 2000.Google Scholar
  19. 19.
    C.J. Vreeman and F.P. Incropera: Int. J. Heat Mass Transf., 2000, vol. 43, pp. 687–704.Google Scholar
  20. 20.
    M. Založnik and B. Šarler: Model. Cast. Weld. Adv. Solidif. Process. XI, 2006, pp. 243–50.Google Scholar
  21. 21.
    L. Zhang, D.G. Eskin, A. Miroux, T. Subroto, and L. Katgerman: IOP Conf. Ser. Mater. Sci. Eng., 2012, vol. 33, pp. 1–8.Google Scholar
  22. 22.
    Q. Du, D.G. Eskin, and L. Katgerman: Metall. Mater. Trans. A Phys. Metall. Mater. Sci., 2007, vol. 38, pp. 180–89.Google Scholar
  23. 23.
    A. V. Reddy and C. Beckermann: Mater. Process. Comput. Age II, 1995, pp. 89–102.Google Scholar
  24. 24.
    M. Založnik, A. Kumar, H. Combeau, M. Bedel, P. Jarry, and E. Waz: Adv. Eng. Mater., 2011, vol. 13, pp. 570–80.Google Scholar
  25. 25.
    A. Hakonsen, D. Mortensen, S. Benum, and H.E. Vatne: in Light Metals, TMS, Warrendale, PA, 1999, pp. 821–7.Google Scholar
  26. 26.
    M. Bedel, L. Heyvaert, M. Založnik, H. Combeau, D. Daloz, and G. Lesoult: IOP Conf. Ser. Mater. Sci. Eng., 2015, vol. 84 (1), pp. 1–8.Google Scholar
  27. 27.
    H. Combeau, M. Založnik, and M. Bedel: Jom, 2016, vol. 68, pp. 2198–206.Google Scholar
  28. 28.
    L. Zhang, D.G. Eskin, A. Miroux, T. Subroto, and L. Katgerman: Metall. Mater. Trans. B, 2012, vol. 43, pp. 1–9.Google Scholar
  29. 29.
    A.L. Greer, A.M. Bunn, A. Tronche, P. V. Evans, and D.J. Bristow: Acta Mater., 2000, vol. 48, pp. 2823–35.Google Scholar
  30. 30.
    M. Rappaz and P.H. Thévoz: Acta Metall., 1987, vol. 35, pp. 2929–33.Google Scholar
  31. 31.
    M. Rappaz and W.J. Boettinger: Acta Mater., 1999, vol. 47, pp. 3205–19.Google Scholar
  32. 32.
    D. Weckman and P. Niessen: Metall. Trans. B, 1982, vol. 13, pp. 593–602.Google Scholar
  33. 33.
    A.L. Dons, E.K. Jensen, Y. Langsrud, E. Trømborg, and S. Brusethaug: Metall. Mater. Trans. A, 1999, vol. 30, pp. 2135–2146.Google Scholar
  34. 34.
    Y. Du, Y.A. Chang, B. Huang, W. Gong, Z. Jin, H. Xu, Z. Yuan, Y. Liu, Y. He, and F.Y. Xie: Mater. Sci. Eng. A, 2003, vol. 363, pp. 140–51.Google Scholar
  35. 35.
    C.J. Vreeman, M.J.M. Krane, and F.P. Incropera: Int. J. Heat Mass Transf., 2000, vol. 43, pp. 677–86.Google Scholar
  36. 36.
    I. Vušanović and M.J.M. Krane: Mater. Sci. Eng. 1:1-12 (2011). doi:10.1088/1757-899x/27/1/012069.Google Scholar
  37. 37.
    A. Pakanati, K.O. Tveito, M. M’Hamdi, H. Combeau, and M. Založnik: in Light Metals 2018. TMS 2018. The Minerals, Metals & Materials Series, Martin O. ed., Springer, Cham, 2018, pp. 1089–96.Google Scholar
  38. 38.
    A. Tronche: PhD thesis, University of Cambridge, Cambridge, England, 2000.Google Scholar
  39. 39.
    M. Založnik, S. Xin, and B. Šarler: Int. J. Numer., 2008, vol. 18, pp. 308–324.Google Scholar
  40. 40.
    C.Y. Wang and C. Beckermann: Metall. Trans. A, 1993, vol. 24, pp. 2787–802.Google Scholar
  41. 41.
    Ø. Nielsen, B. Appolaire, H. Combeau, and A. Mo: Metall. Mater. Trans. A, 2001, vol. 32, pp. 2049–60.Google Scholar
  42. 42.
    M.A. Martorano, C. Beckermann, and C.A. Gandin: Metall. Mater. Trans., 2003, vol. 32, pp. 1657–74.Google Scholar
  43. 43.
    B. Appolaire, H. Combeau, and G. Lesoult: Mater. Sci. Eng. A, 2008, vol. 487, pp. 33–45.Google Scholar
  44. 44.
    M. Wu and A. Ludwig: Acta Mater., 2009, vol. 57, pp. 5632–44.Google Scholar
  45. 45.
    M. Bedel: PhD theses, Université de Lorraine, Nancy, France, 2014.Google Scholar
  46. 46.
    M. Torabi Rad, M. Založnik, H. Combeau, and C. Beckermann: Materialia, 2019, article in press.Google Scholar

Copyright information

© The Minerals, Metals & Materials Society and ASM International 2019

Authors and Affiliations

  • Akash Pakanati
    • 1
  • Knut Omdal Tveito
    • 1
  • Mohammed M’Hamdi
    • 1
    • 2
    Email author
  • Hervé Combeau
    • 3
  • Miha Založnik
    • 3
  1. 1.Department of Materials TechnologyNTNUTrondheimNorway
  2. 2.SINTEF Materials and ChemistryOsloNorway
  3. 3.Université de Lorraine, CNRS, Institut Jean Lamour – IJLNancyFrance

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