Analysis of the effect of shrinkage on macrosegregation in alloy solidification
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INTRODUCTION
RECENTLY, numerous studies have addressed means by which the solidification of binary metal alloys may be modeled, with particular attention given to continuum formulations based on classic mixing theory or volume averaging methods.t1-7] In these models, it is possible to track the freezing fronts without specifying interface conditions or explicitly computing the size of the different regions (solid, liquid, and mush). Important features of such models include prediction of the nonuniform redistribution of solute (macrosegregation) during freezing, as well as determination of growth rates for the solid and two-phase (mushy) regions and convection in the mush and bulk liquid. Recent successes include the prediction of channel development in the mushy zone for unidirectional solidification of an off-eutectic solution cooled from below and its relationship to the formation of freckle segregates in the final casting, t8,91The freckles are regions of high concentration gradients in the frozen ingot and, therefore, of severe nonuniformity in mechanical properties. Although the foregoing studies assume that the density of the alloy does not change during solidification, it is well known that most alloys shrink as they are cast. The attendant volume change can cause the casting to pull away from the mold wall and ~caninduce defects such as hot tears and porosity. The objective of this study is to determine conditions under which shrinkage-induced flow significantly alters macrosegregation in a casting through its influence on concentration fields during freezing from a sidewall. Many early attempts to model macrosegregation in binary alloys examined shrinkage effects as the only source of interdendritic liquid motion.[~O.l~.~2]In these studies, all of which used globally stable AI-Cu mixtures chilled from below, experiments and analysis were in good agreement, showing the sometimes significant effect of solidification contraction on macrosegregation. Mehrabian e t al. ti31 ex-
MATTHEW JOHN M. KRANE, Research Assistant, and FRANK P. INCROPERA, Professor, are with the Heat Transfer Laboratory, School of Mechanical Engineering, Purdue University, West Lafayette, 1N 47907. Manuscript submitted August 19, 1994. METALLURGICAL AND MATERIALS TRANSACTIONS A
tended the model to study side-cooled A1-Cu ingots in which buoyancy effects had a major role. They predicted that at higher solidification rates, shrinkage-induced flows would have a significant effect on the velocity field but that such effects would be negligible at lower rates. Lead-tin solutions were solidified from below at moderate freezing rates by Streat and Weinberg,t~4J whose results indicated that for the solutally unstable cases (20 pct Sn), shrinkage does not play a significant role in macrosegregation. The effect of shrinkage on the temperature field was considered by Chang and Tsai,t15] who numerically solved the governing equations for one-dimensional (l-D) freezing of a pure substance. An approximate solution to this problem is described by
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