Abstract
Some features of the thermally concentrated convective motion of a binary melt, hardening in a closed rectangular region with movable boundaries, and the impurity distribution are investigated numerically.
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Abbreviations
- x0 :
-
characteristic dimension
- xi (i=1, 2):
-
a dimensional coordinate
- li (i=1, 2):
-
height and width of the crystallizer cavity
- ri, ei (i=1, 2):
-
dimensional coordinates of the phase transition in the Ox1x2 coordinate system
- T, T0, and TK :
-
current temperature, initial temperature, and melt crystallization temperature
- ρ :
-
density of the melt
- P, Pmax, and Pmin :
-
current pressure, maximum pressure, and minimum pressure in the system
- c, c0 :
-
current and initial impurity concentration
- e2 :
-
unit vector having the same direction as the direction as the force of gravity
- ¯g:
-
acceleration due to gravity
- β :
-
coefficient of thermal expansion
- γ :
-
diffusion broadening coefficient
- ¯u:
-
velocity of convective motion
- ν :
-
kinematic viscosity
- k:
-
equilibrium impurity distribution coefficient
- t:
-
current time
- D:
-
diffusion coefficient
- a :
-
thermal diffusivity
- ΔT=t0-tK :
-
initial overheating of the melt
- η i=Xi/x0 (i=1, 2):
-
dimensionless coordinate
- ι i=L1/x0 (i=1, 2):
-
relative height and width of the crystallizer cavity in the coordinate system Oη 1 η 2
- Ri=ri/x0, εi=εi/x0 :
-
dimensionless coordinates of the phase-transition boundary in the Oν 1 η 2 coordinate system
- Ū=ū/u0 :
-
dimensionless velocity of convective motion
- Gr=¦¯g¦βΔTx 30 /ν 2 :
-
Grashof hydrodynamic number
- GrD=¦¯g¦γc0X 30 /ν2 :
-
Grashof diffusion number
- Fo=Dt x 20 ):
-
dimensionless time, Sm=ν/D, Schmidt number
- Lu=D/a:
-
Lewis number
Literature cited
I. O. Kulik and G. E. Zil'berman, “The impurity distribution when a crystal is grown from the melt,” in: Crystal Growth [in Russian], Moscow, Vol. 3 (1961), pp. 85–89.
P. F. Zavgorodnii, F. V. Nedopekin, and I. L. Povkh, “Hydrodynamics and heat and mass transfer in a hardening melt,” Inzh.-Fiz. Zh.,33, No. 5, 922–930 (1977).
B. Ya. Lyubov, Theory of Crystallization in Large Volumes [in Russian], Nauka (1975).
B. I. Vaiman and E. L. Tarunin, “The effect of crystallization on the process of free convection in melted metals,” in: Hydrodynamics [in Russian], Perm, No. 4, 107–118 (1972).
É. A. Iodko et al., “Investigation of convective flows in hardening ingots,” Izv. Akad. Nauk SSSR, Metally, No. 2, 102–108 (1971).
P. F. Zavgorodnii, I. L. Povkh, and G. M. Sevast'yanov, “Intensity of thermal convection as a function of the Grashof members and the hardening kinetics of a melt,” Teplofiz. Vys. Temp.,14, No. 4, 823–828 (1976).
A. A. Samarskii, Introduction to the Theory of Difference Schemes [in Russian], Nauka, Moscow (1973).
N. N. Yanenko, The Method of Fractional Steps in Multivariate Problems of Mathematical Physics [in Russian], Nauka, Novosibirsk (1957).
P. F. Zavgorodnii, “Numerical investigation of thermally concentrated convection in the liquid nucleus of a crystallizing binary melt,” Inzh.-Fiz. Zh.,35, No. 1, 155–162 (1978).
P. F. Zavgorodnii et al., “Calculation of the impurity distribution in a crystallizing ingot,” Izv. Vyssh, Uchebn. Zaved., Metall., No. 3, 47–50 (1977).
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Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 39, No. 1, pp. 118–125, July, 1980.
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Zavgorodnii, P.F. Some features of the thermally concentrated convective motion of a hardening binary melt and the impurity distribution. Journal of Engineering Physics 39, 797–803 (1980). https://doi.org/10.1007/BF00821839
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DOI: https://doi.org/10.1007/BF00821839