Abstract
This paper describes the simulation of thermosolutal convection in directionally solidified (DS) alloys. A linear stability analysis is used to predict marginal stability curves for a system that comprises a mushy zone underlying an all-liquid zone. In the unperturbed and nonconvecting state .e.}, the basic state), isotherms and isoconcentrates are planar and horizontal. The mushy zone is realistically treated as a medium with a variable volume fraction of liquid that is con-sistent with the energy and solute conservation equations. The perturbed variables include tem-perature, concentration of solute, and both components of velocity in a two-dimensional system. As a model system, an alloy of Pb-20 wt pct Sn, solidifying at a velocity of 2 X 10-3 cm s-1 was selected. Dimensional numerical calculations were done to define the marginal stability curves in terms of the thermal gradient at the dendrite tips,G L ,vs the horizontal wave number of the perturbed quantities. For a gravitational constant of 1g,0.5 g, 0.1g, and 0.01g, the marginal stability curves show no minima; thus, the system is never unconditionally stable. Nevertheless, such calculations quantify the effect of reducing the gravitational constant on reducing convection and suggest lateral dimensions of the mold for the purpose of suppressing convection. Finally, for a gravitational constant of 10-4 g, calculations show that the system is stable for the thermal gradients investigated (2.5 ≤G L ≤ 100 K-cm-1).
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J.S. Kirkaldy and W.V. Youdelis:Trans. AIME, 1958, vol. 212, pp. 833–40.
M.C. Flemings and G.E. Nereo:Trans. TMS-AIME, 1967, vol. 239, pp. 1449–61.
M.C. Flemings, R. Mehrabian, and G.E. Nereo:Trans. TMS- AIME, 1968, vol. 242, pp. 41–49.
M.C. Flemings and G.E. Nereo:Trans. TMS-AIME, 1968, vol. 242, pp. 50–55.
R. Mehrabian, M. Keane, and M.C. Flemings:Metall. Trans., 1970, vol. 1, pp. 1209–20.
R. Mehrabian, M. Keane, and M.C. Flemings:Metall. Trans., 1970, vol. 1, pp. 3238–41.
S. Kou, D.R. Poirier, and M.C. Flemings:Electric Furnace Proceedings, ISS-AIME, New York, NY, 1977, vol. 35, pp. 221–28.
S. Kou, D.R. Poirier, and M.C. Flemings:Metall. Trans. B, 1978, vol. 9B, pp. 711–19.
S.D. Ridder, F.C. Reyes, S. Chakravorty, R. Mehrabian, J.D. Nauman, J.H. Chen, and H.J. Klein:Metall. Trans. B, 1978, vol. 9B, pp. 415–25.
T. Fujii, D.R. Poirier, and M.C. Flemings:Metall. Trans. B, 1979, vol. 10B, pp. 331–39.
D. Petrakis, M.C. Flemings, and D.R. Poirier: inModeling of Casting and Welding Processes, H.D. Brody and D. Apelian, eds., TMS-AIME, Warrendale, PA, 1981, pp. 285–312.
C. Jeanfills, J.H. Chen, and H.J. Klein: inModeling of Casting and Welding Processes, H.D. Brody and D. Apelian, eds., TMS- AIME, Warrendale, PA, 1981, pp. 313–32.
S.D. Ridder, S. Kou, and R. Mehrabian:Metall. Trans. B, 1981, vol. 12B, pp. 435–47.
A.L. Maples and D.R. Poirier:Metall. Trans. B, 1984, vol. 15B, pp. 163–72.
D.R. Poirier, S. Kou, T. Fujii, and M.C. Flemings:Electroslag Remelting, Report No. AMMRC TR 78-28, Army Materials and Mechanics Research Center, Watertown, MA, June 1978.
D.R. Poirier, M.C. Flemings, R. Mehrabian, and H.J. Klein:Advances in Metal Processing, J.J. Burke, R. Mehrabian, and V. Weiss, eds., Plenum Publishing, New York, NY, 1981, pp. 277–318.
S.M. Copley, A.F. Giamei, S.M. Johnson, and M.F. Hornbecker:Metall. Trans., 1970, vol. 1, pp. 2193–204.
S.R. Coriell, M.R. Cordes, W.J. Boettinger, and R.F. Sekerka:J. Cryst. Growth, 1980, vol. 49, pp. 13–28.
G.B. McFadden, R.G. Rehm, S.R. Coriell, W. Chuck, and K.A. Morrish:Metall. Trans. A, 1984, vol. 15A, pp. 2125–37.
G.B. McFadden and S.R. Coriell:Phys. Fluids, 1987, vol. 30, pp. 659–71.
J.C. Heinrich:Comput. Meth. Appl. Mech. Eng., 1988, vol. 69, pp. 65–88.
A.F. Giamei and B.H. Kear:Metall. Trans., 1970, vol. 1, pp. 2185–92.
A.K. Sample and A. Hellawell:Metall. Trans. A, 1984, vol. 15A, pp. 2163–73.
J.R. Sarazin and A. Hellawell:Metall. Trans. A, 1988, vol. 19A, pp. 1861–71.
R.N. Hills, D.E. Loper, and P.H. Roberts:Q.J. Mech. Appl. Math., 1983, vol. 36, pp. 505–39.
H.E. Huppert and M.G. Worster:Nature, 1985, vol. 314, pp. 703–07.
M.G. Worster:J. Fluid Mech., 1986, vol. 167, pp. 481–501.
K.S. Yeum and D.R. Poirier:Cast Metals, 1988, vol. 1, pp. 161–70.
W.D. Bennon and F.P. Incropera:Int. J. Heat Mass Transfer, 1987, vol. 30, pp. 2161–70.
W.D. Bennon and F.P. Incropera:Int. J. Heat Mass Transfer, 1987, vol. 30, pp. 2171–87.
W.D. Bennon and F.P. Incropera:Metall. Trans. B, 1987, vol. 18B, pp. 611–16.
C. Beckermann and R. Viskanta:Int. J. Heat Mass Transfer, 1988, vol. 31, pp. 35–46.
J.C. Heinrich, S. Felicelli, P. Nandapurkar, and D.R. Poirier:Metall. Trans. B, in press.
J. Szekely and A.S. Jassal:Metall. Trans. B, 1978, vol. 9B, pp. 389–98.
V.R. Voller and G. Prakash:Int. J. Heat Mass Transfer, 1987, vol. 30, pp. 1709–19.
D.R. Poirier:Metall. Trans. B, 1987, vol. 18B, pp. 245–55.
M.C. Flemings:Solidification Processing, McGraw-Hill, New York, NY, 1974, pp. 77–83.
D.R. Poirier:Metall. Trans. A, 1988, vol. 19A, pp. 2349–54.
D.R. Poirier and P. Nandapurkar:Metall. Trans. A, 1988, vol. 19A, pp. 3057–61.
H.R. Thresh and A.F. Crawley:Metall. Trans., 1970, vol. 1, pp. 1531–35.Metals Handbook, 8th ed., ASM, Metals Park, OH, 1973, vol. 8, p. 330.
Y.S. Touloukian, R.W. Powell, C.Y. Ho, and P.G. Klemens:Thermophysical Properties of Matter, IFI/Plenum, New York, NY, 1970, vol. 1, pp. 653–54 and 840–41.
Metals Handbook, 9th ed., ASM, Metals Park, OH, 1979, vol. 2, pp. 505-06 and 620–21.
J.T. Mason, J.D. Verhoeven, and R. Trivedi:J. Cryst. Growth, 1982, vol. 59, pp. 516–24.
G.B. McFadden, S.R. Coriell, and R.F. Boisvert:Phys. Fluids, 1985, vol. 28, pp. 2716–22.
R.Z. Guerin, B. Billia, P. Maldenwang, and B. Roux:Phys. Fluids, 1988, vol. 31, pp. 2086–92.
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Nandapurkar, P., Poirier, D.R., Heinrich, J.C. et al. Thermosolutal convection during dendritic solidification of alloys: Part i. Linear stability analysis. Metall Trans B 20, 711–721 (1989). https://doi.org/10.1007/BF02655929
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DOI: https://doi.org/10.1007/BF02655929