Equiaxed dendritic solidification with convection: Part I. Multiscale/multiphase modeling
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I.
INTRODUCTION
ThE solidification microstructure of many metal alloys can generally be categorized into two groups: (1) columnar structures consisting of long, aligned dendrite arms that are attached to the mold wall; and (2) equiaxed structures consisting of crystals that grow radially inside an undercooled melt. In the recent past, columnar solidification with thermosolutal convection in the liquid melt and the resulting macrosegregation have been extensively studied, t2,31In contrast, modeling of equiaxed dendritic solidification with convection has not been widely attempted because of the complications associated with the transport of free equiaxed crystals in the melt. The gravity-induced settling or flotation of free crystals is fundamental to the development and extent of the equiaxed zone and greatly affects the columnar to equiaxed transition. Moreover, grain transport may cause severe macrosegregation and structural inhomogeneities. Toward predicting composition and structure in alloy castings, this article presents a general framework for modeling microstructure evolution during equiaxed alloy dendritic solidification. The modeling task is to incorporate descriptions of fundamental microscopic phenomena, such as nucleation, undercooling, and grain growth mechanisms, into macroscopic heat and fluid flow calculations. An extensive review of the micro-macroscopic modeling approach has been provided by Rappazt41 and Thevoz et aLl 51 and recent developments have been reported in conference proceedings.t6,71 The two most recent approaches include the probabilistic modeling proposed by Brown and Spittle, I81 Zhu and Smith,tgl and Rappaz and Gandinu~ and the multiscale/multiphase model developed by Wang and BeckermannJ 1,~r'121 However, none of these previous models has considered melt convection and solid transport occurring C.Y. WANG, Assistant Professor, is with the Department of Mechanical Engineering, University of Hawaii at Manoa, Honolulu, HI 96822. C. BECKERMANN, Associate Professor, is with the Department of Mechanical Engineering, The University of Iowa, Iowa City, IA 52242. Manuscript submitted February 9, 1995. 2754--VOLUME 27A, SEPTEMBER 1996
during solidification, thereby greatly limiting their utility. The present work is intended to extend the multiphase model of Wang and Beckermannt~l by accounting for both melt convection and solid-phase transport. The present work is a continuation of Ni and Beckermann's two-phase modeling study,t131which deals with globulitic solidification only but does account for melt convection and solid-phase transport. The general multiscale/multiphase modeling framework is first introduced in Section II. A specific model obtained using the general approach is subsequently presented in Section III, where all the necessary supplementary relations are also supplied to complete the mathematical system. In the second article of this series on equiaxed dendritic solidification,E~4] calculations for two-dimensional (2-D) equiaxed dendritic solidification of an A1-4 wt pct
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