The effect of macroscopic solute diffusion in the liquid upon surface macrosegregation
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I.
INTRODUCTION
MATHEMATICAL models have been successfully applied in the modeling of macrosegregation caused by solidthermosolutal ification shrinkage-induced flow,[1–7] convection,[6,8–15] and forced convection.[7,16–18] Also, macroscopic diffusion of solute can give rise to significant macrosegregation. Recently, Schneider and Beckermann[6] showed that macroscopic solute diffusion in the liquid leads to an extremely narrow negatively segregated band close to the chill. Also, Krane and Incropera[15] reported the formation of a solute-depleted boundary layer, but only at low heat-extraction rates. The formation of such a narrow negatively segregated layer is in contrast to the findings of Diao and Tsai,[3] who concluded that shrinkage-induced macrosegregation close to the chill remains positive, even when macroscopic solute diffusion in the liquid is taken into account. Although several authors have included solute diffusion in their macro-scale models, none of them reported any macrosegregation results similar to those of References 6 and 15. It should also be noted that in his analysis of growth rates and structures in alloy mushy zones, Worster[19] took into account the solute diffusion in the liquid. The purpose of the present article is to analyze the effect of macroscopic solute diffusion in the liquid on the macrosegregation formation close to a surface. A simple onedimensional mathematical model of the liquid diffusion is presented in Section II, and in Section III the influence of macroscale solute diffusion upon macrosegregation is discussed. II.
MATHEMATICAL FORMULATION
A one-dimensional model in which x denotes the position is considered (Figure 1). Initially, the solution domain con˚ VARD J. THEVIK, Research Scientist, and ASBJØRN MO, Senior HA Scientist, are with SINTEF Materials Technology, N-0314 Oslo, Norway. Manuscript submitted February 8, 1996. METALLURGICAL AND MATERIALS TRANSACTIONS B
tains liquid with a solute concentration ci. Due to the chill positioned at x 5 0, the system solidifies unidirectionally. By adding the volume-averaged conservation equations within the solid and liquid phase, the following macroscopic conservation equation for the solute concentration c is obtained:[20] ] ] ]j (rc) 1 (rl clUl) 5 2 c ]t ]x ]x
[1]
where t, r, r l, cl, Ul, and jc are the time, total density of the averaging volume, liquid density, liquid solute concentration, superficial liquid velocity, and solute diffusive flux, respectively. The solid phase is assumed to be stationary and the dispersion flux is neglected. The diffusive solute flux is modeled by the following:[20] jc 5 2gl rl Dl
]cl , ]x
x.0
[2]
where gl is the volume fraction of liquid and Dl is the effective solute diffusivity in the liquid. The macroscopic solute diffusion in the solid is not included because the diffusivity in the solid is usually very small. By assuming thermodynamic equilibrium at the solid-liquid interface and a uniform concentration locally in the liquid, the liquid concentration can be related to the temperatu
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