Effect of fluid flow on macrosegregation in axi-symmetric ingots
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I. I N T R O D U C T I O N M E T A L L U R G I S T S are continually attempting to overcome the macrosegregation problems faced in the different casting processes, e.g., vacuum arc refining (VAR), electroslag remelting (ESR), and continuous casting. It is common to observe, within an ingot or a casting, one or several types of macrosegregation (taking into account shape, location or concentration). The various types of segregation are commonly referred to as "center-line segregation", "A-segregation", "V-segregation", "inverse segregation", "negative-cone segregation", and so forth. The most relevant fact that needs to be kept in mind is that the mechanism which causes a vast majority of these heterogeneities has been shown to be interdendritic fluid flow, ~-m driven by 1) solidification contraction, 2) the force of gravity acting on a liquid of variable density, and 3) penetration of bulkliquid flow from the fully liquid region in front of the liquidus isotherm into the "mushy" zone. Previous work on quantitative predictions of macrosegregation have considered the first two effects listed above, while limiting the calculation of interdendritic fluid velocity to a constant permeability coefficient, 7, to be discussed below. However, the third effect--bulk liquid convection--has been entirely ignored. In a recent paper, 8 a steady-state axi-symmetric mathematical model of fluid flow and solidification-segregation was presented. The research reported herein expands and extends this theory to include the effects of convective flow in the liquid metal pool ahead of the liquidus isotherm in an axi-symmetric ingot cast under steady state conditions and incorporates the use of a variable, ~,, to account for permeability changes induced by variations in the dendrite arm spacing across the ingot. The theoretical calculations were verified on a low-
S. D. RIDDER and R. MEHRABIAN, formerly Graduate Student and Professor at the University of Illinois at Urbana-Champaign, IL, are now Metallurgist and Chief of the Metallurgy Division, respectively, at the National Bureau of Standards, Washington, DC 20234. S. KOU, formerly Research Associate at the University of Illinois in Urbana-Champaign, IL, is now Assistant Professor in the Department of Metallurgy and Materials Science at Carnegie-Mellon University in Pittsburgh, PA 15213. Manuscript submitted July 18, 1980. METALLURGICAL TRANSACTIONS B
temperature laboratory scale apparatus capable of producing small (90 mm diam), cylindrical ingots under closely controlled conditions that simulate many of the characteristics found in VAR, ESR and continuous casting. II. T H E O R Y 1. Fluid Flow a . I n t e r d e n d r i t i c F l u i d F l o w . It is assumed that the "mushy" region of the casting can be treated as a porous medium where flow is described by the following equation: 8 v =
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K /xgz --
(Vp
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PLg)
[1]
where gL = volume fraction liquid, K = permeability of porous medium, P = pressure, g = acceleration due to gravity, v = interdendritic fluid flow velocity,/~ = viscos
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