Metallurgical and Materials Transactions B

, Volume 27, Issue 1, pp 101–113 | Cite as

Scaling of intragranuiar dendritic microstructure in ingot solidification

  • D. Bouchard
  • J. S. Kirkaldy


Analytic scaling formulas of complete constitutional generality for forced velocity cells and dendrites were in earlier research perfected forin situ steady-state solidification conditions involving binary organic alloys. As a further test, these were used, given the velocity and gradient control parameters, to predict the primary and secondary dendrite arm spacings of unidirectionally cooled Al-Cu alloys for which a large data set is available. Numerical methods were employed to determine the control parameters that exist under unsteady-state ingot solidification conditions according to the Scheil formulation. Primary and secondary arm spacings, corrected empirically for ripening, that by and large agree with the Al-Cu experimental data were obtained, demonstrating that the formulas are adequate for the prediction of dendrite scales in steady and unsteady-state conditions. The predictions have been incorporated into a computer program that displays the time-dependent columnar microstructure and mushy zone in an ingot cross section of an oriented single crystal together with the thermal and liquid-solid distributions.


Material Transaction Mushy Zone Dendrite Spacing Liquid Volume Fraction Primary Dendrite Spacing 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    F. Tyler:Intermediate Heat, Edward Arnold, London, 1949, p. 25.Google Scholar
  2. 2.
    M.C. Flemings:Solidification Processing, McGraw-Hill, New York, NY, 1974.Google Scholar
  3. 3.
    E. Scheil:Z Metallkd, 1942, vol. 34, pp. 70–72.Google Scholar
  4. 4.
    A. Hayes and J. Chipman:Trans. AIME, 1939, vol. 135, pp. 85–132.Google Scholar
  5. 5.
    J.S. Kirkaldy and W.V. Youdelis:Trans. AIME, 1958, vol. 212, pp. 833–40.Google Scholar
  6. 6.
    A. Kroupa and J.S. Kirkaldy: inModeling of Casting, Welding and Advanced Solidification Processes, T.S. Piwonka, V. Voller, and L. Katgerman, eds., 1993, TMS, Warrendale, PA, pp. 269–76.Google Scholar
  7. 7.
    P.E. Brown and C.M. Adams:Weld. Res. Suppl., 1960, Dec., pp. 520–24.Google Scholar
  8. 8.
    R.E. Spear and G.R. Gardner:Trans. Am. Foundrymen’s Soc., 1963, vol. 71, pp. 209–15.Google Scholar
  9. 9.
    B.P. Bardes and M.C. Flemings:Trans. Am. Foundrymen’s Soc., 1966, vol. 74, pp. 406–12.Google Scholar
  10. 10.
    S.F. Frederick and W.A. Bailey:Trans. TMS-AIME, 1968, vol. 242, pp. 2063–67.Google Scholar
  11. 11.
    J. Campbell:Castings, Butterworth-Heinemann, Oxford, United Kingdom, 1991, p. 264.Google Scholar
  12. 12.
    I.H. Nieswaag and A.J. Zuithof:Proc. Int. Foundry Cong., Belgrade, 1969.Google Scholar
  13. 13.
    W.V. Youdelis: Ph.D. Dissertation, McGill University, Montreal, 1958.Google Scholar
  14. 14.
    J.S. Langer:Science, 1989, vol. 243, 1150–56.CrossRefGoogle Scholar
  15. 15.
    S.C. Huang and M.E. Glicksman:Acta Metall., 1981, vol. 29, pp. 701–16.CrossRefGoogle Scholar
  16. 16.
    R. Kobayashi:Bull. Jpn. Soc. Ind. Appl. Math., 1991. vol. 1, pp. 22–27.Google Scholar
  17. 17.
    B. Chalmers:Principles of Solidification. John Wiley, New York, NY, 1964.Google Scholar
  18. 18.
    G.S. Cole: Ph.D. Dissertation, University of Toronto, Toronto, 1963.Google Scholar
  19. 19.
    L.X. Liu and J.S. Kirkaldy:Scripta Metall. Mater., 1993. vol. 28, pp. 1029–34.CrossRefGoogle Scholar
  20. 20.
    J.D. Hunt and K.A. Jackson:Trans. TMS-AIME, 1966. vol. 236, pp. 843–52.Google Scholar
  21. 21.
    J.S. Kirkaldy and D. Venugopalan:Scripta Metall., 1989, vol. 23, pp. 1603–08.CrossRefGoogle Scholar
  22. 22.
    J.S. Kirkaldy, L.X. Liu, and A. Kroupa:Acta Metall. Mater., 1995. vol. 43, pp. 2905–15.CrossRefGoogle Scholar
  23. 23.
    D. Venugopalan: Ph.D. Dissertation. McMastcr University, Hamilton, 1982.Google Scholar
  24. 24.
    L.X. Liu and J.S. Kirkaldy:Acta Metall. Mater., 1995, vol. 43, pp. 2891–2904.CrossRefGoogle Scholar
  25. 25.
    K. Somboonsuk, J.T. Mason, and R. Trivedi:Metall. Trans. A, 1984. vol. 15A, pp. 967–75.Google Scholar
  26. 26.
    T.Z. Kattamis, J.C. Coughlin, and M.C. Flemings:Trans. TMS-AIME, 1967, vol. 239, pp. 1504–11.Google Scholar
  27. 27.
    K.P. Young and D.H. Kirkwood:Metall. Trans., 1975, vol. 6A, pp. 197–205.Google Scholar
  28. 28.
    D.H. Kirkwood:Mater. Sci. Eng., 1985, vol. 73, pp. L1-L4.CrossRefGoogle Scholar
  29. 29.
    A. Mortensen:Metall. Trans. A, 1991, vol. 22A, pp. 569–74.Google Scholar
  30. 30.
    P.N. Hansen:Solidification and Casting of Metals, Proc. Conf., The Metals Society, London, 1979, pp. 350–56.Google Scholar
  31. 31.
    T.W. Clyne:Met. Sci., 1982, vol. 16, pp. 441–50.CrossRefGoogle Scholar
  32. 32.
    M. Samonds, K. Morgan, and R.W. Lewis:Appl. Math. Modelling, 1985, vol. 9, pp. 170–74.CrossRefGoogle Scholar
  33. 33.
    S.C. Flood and J.D. Hunt:J. Cryst. Growth, 1987, vol. 82, pp. 543–51.CrossRefGoogle Scholar
  34. 34.
    D.G. McCartney and V.A. Wills:Appl. Math. Modelling, 1988, vol. 12, pp. 354–60.CrossRefGoogle Scholar
  35. 35.
    V.A. Wills and D.G. McCartney:Mater. Sci. Technol., 1992, vol. 8, pp. 114–22.Google Scholar
  36. 36.
    Moving Boundary Problems in Heat Flow and Diffusion, J.R. Ockendon and W.R. Hodgkins, eds., Oxford University Press, Oxford, United Kingdom, 1975.Google Scholar
  37. 37.
    J. Crank: inNumerical Methods in Heat Transfer, R.W. Lewis, K. Morgan, and O.C. Zienkiewicz, eds., John Wiley, New York, NY, 1981, pp. 177–200.Google Scholar
  38. 38.
    J. Crank:Free and Moving Boundary Problems, Oxford University Press, New York, NY, 1984.Google Scholar
  39. 39.
    V. Alexiades and A.D. Solomon:Mathematical Modeling of Melting and Freezing Processes, Hemisphere Publishing Corp., New York, NY, 1993.Google Scholar
  40. 40.
    W. Kurz and D.J. Fisher:Fundamentals of Solidification. Trans Tech Publications, Aedermannsdorf, Switzerland, 1986, p. 129.Google Scholar
  41. 41.
    W.D. Murray and F. Landis:Trans. ASME, 1959, vol. 81, pp. 106–12.Google Scholar
  42. 42.
    C. Wagner:Trans. AIME, 1954, vol. 200, pp. 154–60.Google Scholar
  43. 43.
    Metals Handbook, 9th ed., ASM, Metals Park, OH, 1979, vol. 2, pp. 714–15.Google Scholar
  44. 44.
    M.P. Watson and J.D. Hunt:Metall. Trans. A, 1977, vol. 8A, pp. 1793–98.Google Scholar
  45. 45.
    Handbook of Materials Science, C.T. Lynch, ed., CRC Press, Cleveland, OH, 1974, vol. 1, p. 105.Google Scholar
  46. 46.
    T. Okamoto and K. Kishitake:J. Cryst. Growth, 1975, vol. 29, pp. 137–46.CrossRefGoogle Scholar
  47. 47.
    P.C. Dann, J.A. Eady, and L.M. Hogan:J. Austr. Inst. Met., 1974, vol. 19 (2), pp. 140–47.Google Scholar
  48. 48.
    D.G. McCartney and J.D. Hunt:Acta Metall., 1981, vol. 29, pp. 1851–63.CrossRefGoogle Scholar
  49. 49.
    M.A. Taha:Met. Sci., 1979, vol. 13, pp. 9–12.Google Scholar
  50. 50.
    T.F. Bower, H.D. Brody, and M.C. Flemings:Trans. TMS-AIME, 1966, vol. 236, pp. 624–34.Google Scholar
  51. 51.
    A.B. Michael and M.B. Bever:Trans. AIME, 1954, vol. 200, pp. 47–56.Google Scholar
  52. 52.
    J.A. Horwath and L.F. Mondolfo:Acta Metall., 1962, vol. 10, pp. 1037–42.CrossRefGoogle Scholar

Copyright information

© The Minerals, Metals & Materials Society - ASM International - The Materials Information Society 1996

Authors and Affiliations

  • D. Bouchard
    • 1
  • J. S. Kirkaldy
    • 1
  1. 1.McMaster UniversityHamiltonCanada

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