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The energy and solute conservation equations for dendritic solidification

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Abstract

The energy equation for solidifying dendritic alloys that includes the effects of heat of mixing in both the dendritic solid and the interdendritic liquid is derived. Calculations for Pb-Sn alloys show that this form of the energy equation should be used when the solidification rate is relatively high and/or the thermal gradients in the solidifying alloy are relatively low. Accurate predictions of transport phenomena in solidifying dendritic alloys also depend on the form of the solute conservation equation. Therefore, this conservation equation is derived with particular consideration to an accounting of the diffusion of solute in the dendritic solid. Calculations for Pb-Sn alloy show that the distribution of the volume fraction of interdendritic liquid (g L) in the mushy zone is sensitive to the extent of the diffusion in the solid. Good predictions ofg L are necessary, especially when convection in the mushy zone is calculated.

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References

  1. R.N. Hills, D.E. Loper, and P.H. Roberts:Q. J. Mech. Appl. Math., 1983, vol. 36, pp. 505–39.

    Article  Google Scholar 

  2. W.D. Bennon and F.P. Incropera:Int. J. Heat Mass Transfer, 1987, vol. 30, pp. 2161–70.

    Article  Google Scholar 

  3. W.D. Bennon and F.P. Incropera:Int. J. Heat Mass Transfer, 1987, vol. 30, pp. 2171–87.

    Article  Google Scholar 

  4. W.D. Bennon and F.P. Incropera:Metall. Trans. B, 1987, vol. 18B, pp. 611–16.

    Article  Google Scholar 

  5. C. Beckermann and R. Viskanta:Int. J. Heat Mass Transfer, 1988, vol. 31, pp. 35–46.

    Article  Google Scholar 

  6. V.R. Voller and G. Prakash:Int. J. Heat Mass Transfer, 1987, vol. 30, pp. 1709–19.

    Article  Google Scholar 

  7. P. Nandapurkar, D.R. Poirier, J.C. Heinrich, and S. Felicelli:Metall. Trans. B, 1989, vol. 20B, pp. 711–21.

    Article  Google Scholar 

  8. J.C. Heinrich, S. Felicelli, P. Nandapurkar, and D.R. Poirier:Metall. Trans. B, 1989, vol. 20B, pp. 883–91.

    Article  Google Scholar 

  9. J. Szekely and A.S. Jassal:Metall. Trans. B, 1978, vol. 9B, pp. 389–98.

    Article  Google Scholar 

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

    Article  Google Scholar 

  11. P.J. Nandapurkar, D.R. Poirier, and J.C. Heinrich:Numerical Heat Transfer, Part A: Applications, 1991, vol. 19, pp. 297–311.

    Article  Google Scholar 

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

    Article  Google Scholar 

  13. B. Schulz: inThermal Conductivity 18, T. Ashworth and D.R. Smith, eds., Plenum Press, New York, NY, 1985, pp. 33–43.

    Chapter  Google Scholar 

  14. W.D. Kingery, H.K. Bowen, and D.R. Uhlmann:Introduction to Ceramics, 2nd ed., John Wiley and Sons, New York, NY, 1976, pp. 634–37.

    Google Scholar 

  15. D.R. Poirier and P. Nandapurkar:Metall. Trans. A, 1988, vol. 19A, pp. 3057–61.

    Article  Google Scholar 

  16. K.S. Yeum and D.R. Poirier:Cast Metals, 1988, vol. 1, pp. 161–70.

    Article  Google Scholar 

  17. R.B. Bird, W.E. Stewart, and E.N. Lightfoot:Transport Phenomena, John Wiley, New York, NY, 1960, pp. 567–68.

    Google Scholar 

  18. M.C. Flemings and G.E. Nereo:Trans. TMS-A1ME, 1967, vol. 239, pp. 343–47.

    Google Scholar 

  19. M.C. Flemings:Solidification Processing, McGraw-Hill, New York, NY, 1974, pp. 34–35.

    Google Scholar 

  20. S. Ganesan and D.R. Poirier:J. Cryst. Growth, 1989, vol. 97, pp. 851–59.

    Article  Google Scholar 

  21. H.D. Brody and M.C. Flemings:Trans. TMS-AIME, 1966, vol. 236, pp. 615–24.

    Google Scholar 

  22. M.C. Flemings, D.R. Poirier, R.V. Barone, and H.D. Brody:J. Iron Steel Inst., 1970, vol. 208, pp. 371–81.

    Google Scholar 

  23. J.I. Nurminen and H.D. Brody: inTitanium Science and Technology, R.I. Jaffee and H.M. Burte, eds., Plenum Press, New York, NY, 1973, vol. 3, pp. 1893–1914.

    Google Scholar 

  24. M. Basaran:Metall. Trans. A, 1981, vol. 12A, pp. 1235–43.

    Article  Google Scholar 

  25. D.H. Kirkwood:Mater. Sci. Eng., 1984, vol. 65, pp. 101–09.

    Article  Google Scholar 

  26. A.J.W. Ogilvy and D.H. Kirkwood:Appl. Sei. Res., 1987, vol. 44, pp. 43–49.

    Article  Google Scholar 

  27. T.P. Battle and R.D. Pehlke:Metall. Trans. B, 1990, vol. 21B, pp. 357–75.

    Article  Google Scholar 

  28. K.S. Yeum, V. Laxmanan, and D.R. Poirier:Metall. Trans. A, 1989, vol. 20A, pp. 2847–56.

    Article  Google Scholar 

  29. R. Mehrabian, M. Keane, and M.C. Flemings:Metall. Trans., 1970, vol. 1, pp. 1209–20.

    Article  Google Scholar 

  30. D.R. Poirier:Metall. Trans. A, 1988, vol. 19A, pp. 2349–54.

    Article  Google Scholar 

  31. H.R. Thresh and A.F. Crawley:Metall. Trans., 1970, vol. 1, pp. 1531–35.

    Article  Google Scholar 

  32. Metals Handbook, 8th ed., ASM, Metals Park, OH, 1961, p. 1064.

  33. E.T. Turkdogan:Physical Chemistry of High Temperature Technology, Academic Press, New York, NY, 1980, p. 124.

    Google Scholar 

  34. Y.S. Touloukian, R.W. Powell, C.Y. Ho, and P.G. Klemens:Thermophysical Properties of Matter, Vol. 1, Thermal Conduc- tivity, Metallic Elements and Alloys, IFI/Plenum, New York, NY, 1970, pp. 652–54 and 839-41.

    Google Scholar 

  35. Metals Handbook, 9th ed., ASM, Metals Park, OH, 1979, vol. 2, pp. 505–06.

  36. T. Iida and R.I.L. Guthrie:The Physical Properties of Liquid Metals, Clarendon Press, Oxford, United Kingdom, 1988, p. 234.

    Google Scholar 

  37. D.R. Poirier:Metall. Trans. B, 1987, vol. 18B, pp. 245–56.

    Article  Google Scholar 

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Authors and Affiliations

  1. Department of Materials Science and Engineering, The University of Arizona, 85721, Tucson, AZ

    D. R. Poirier (Professor)

  2. Dow Chemical, 77541, Freeport, TX

    P. J. Nandapurkar (Senior Research Engineer)

  3. Motorola, Inc., 85008, Phoenix, AZ

    S. Ganesan (Senior Staff Engineer)

Authors
  1. D. R. Poirier
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  2. P. J. Nandapurkar
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  3. S. Ganesan
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Additional information

Formerly Research Associate, The University of Arizona.

Formerly Graduate Student, The University of Arizona.

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Poirier, D.R., Nandapurkar, P.J. & Ganesan, S. The energy and solute conservation equations for dendritic solidification. Metall Trans B 22, 889–900 (1991). https://doi.org/10.1007/BF02651165

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