Natural convection in the hale-shaw cell of horizontal bridgman solidification
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I.
INTRODUCTION
CONVECTIVE flow has a great impact on the solidification process and solidified microstructures; therefore, this has been a very active research field in recent years. El4] Understanding the nature and behavior of convection is crucial to the successful control of many important crystal growth and solidification processes. The horizontal Bridgman directional solidification process is widely used in research and crystal growth. In transparent model alloy study, the thin Hale-Shaw cells are used to contain the growth material. The cells are very thin (a couple of hundred micrometers); therefore, many believe that either there is no convection in the cell or that the convection is so weak that its effect can be neglected. However, there was no conclusive evidence to prove whether these assumptions were valid or under what conditions they would be valid. Nevertheless, without proper justification, the results or analysis deduced from such experiments with the preassumption that there is no convection and only diffusion processes are active could be misleading or incorrect. In this article, we will first present our findings from numerical simulations of convective flow in the Hale-Shaw cell in the horizontal Bridgrnan process, and then proceed with experimental verification. II.
MATHEMATICAL MODEL
When dealing with natural convection problems, because the melt density change is very small, we usually only consider the density change in the buoyancy terms in the momentum equation. The density will be treated as constant in other places. We adopt the Boussinesq assumption in this article. For Hale-Shaw cells, as shown in Figure 1, the governing equations for natural convection are as follows. Continuity equation V. V = 0
DV Po Dt
[2]
Energy equation DT Dt
=/O72T
[3]
where {p = po[1 - a (T - To)]} = melt density at given temperature T; Po = melt density at a reference temperature; P = pressure; T = temperature; K = thermal diffusivity; a = the volume expansion coefficient of melt; t = time; V = the velocity vector; and /z = hydrodynamic viscosity. The solute transport equation has been omitted, for only pure SCN is considered. Numeric simulation could be carried out when the appropriate boundary conditions are applied. A constant temperature gradient is assumed between the hot and cold stages of the Bridgman apparatus. The Rayleigh number, depending on the vertical temperature gradient, of the cell is about 1, which is much smaller than the critical Rayleigh number (1708). Consequently, natural convection induced by vertical temperature difference could be omitted. Furthermore, the width- to-thickness ratio of the cell is very large (30 to 200), apart from the area very close to the sidewall, the convection problems could be simplified to a two-dimensional problem. Equations [1] through [3] were first nondimensionalized:
[1] ell
Momentum equation
YILI LU, Lecturer, JIAN LIU, Professor, and YAOHE ZHOU, Professor, are with the National Solidification Processing Laboratory, Northwestern Polytechn
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