Calculation of dendrite settling velocities using a porous envelope
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INTRODUCTION
IT has been shown that the settling (or floating) of dendrites and equiaxed grains can cause severe macrosegregation in metal castings ~1-41 and that the formation of free, unattached grains and their movement is critical to the formation and structure of the equiaxed zone in a variety of solidification arrangements, t5-8~ An understanding of the settling characteristics of dendritic grains is needed to understand and predict this type of segregation and the development of the equiaxed zone. Attempts have been made to use Stokes' law to assist in the analysis of dendrite settling processes, t8,9,~~ However, since dendrites are highly nonspherical, the use of Stokes' law has yielded only rough estimates of dendrite velocities. In a recent experimental study by Zakhem et a l . , t~l~ the effects of shape on the settling rate of model dendrites were examined. In this study, the low Reynolds number drag and settling speed ratio of equiaxed dendritic grains and dendrite fragments were determined. Plastic dendrite models patterned after the shapes of real dendrites observed in metallic alloys and metal analogs were constructed and tested in a large Stokes' flow facility. The size, shape, and density of these were varied so that the drag and terminal velocities of 34 different models undergoing free fall along their axis of symmetry were measured. It was concluded in this work that the interdendritic liquid of settling grains is effectively immobilized, and thus, the added surface area of secondary H.C. de GROH Ill, Materials Research Engineer, is with NASA Lewis, Cleveland, OH. P.D. WEIDMAN, Associate Professor, and R. ZAKHEM, Graduate Student, are with the Department of Mechanical Engineering, University of Colorado, Boulder, CO. C. BECKERMANN, Associate Professor, is with the Department of Mechanical Engineering, University of Iowa, Iowa City, IA. S. AHUJA, formerly Graduate Student at the University of Iowa, is now Software Development Engineer, Engineering Mechanics Research Corporation, Troy, MI. Manuscript submitted December 4, 1992. METALLURGICAL TRANSACTIONS B
dendrite arms does not significantly add to the drag coefficient of the particle. Ahuja e t al. t121 advanced this work through use of a conceptual envelope which (1) surrounds the dendrite, (2) is defined as a porous particle, and (3) is used to determine an effective density and sphericity for the particle. Figure 1 shows a typical shape of an equiaxed dendrite and two possible envelopes, one idealized. The nonspherical envelope contains both the solid dendrite and the interdendritic liquid. The fluid flow around the dendrite is mainly controlled by the nonspherical shape of the dendrite envelope, while the fluid flow through the dendrite is determined by the porosity and permeability of the dendrite. These two flows affect the drag force experienced by the dendrite and hence its settling velocity. In Ahuja e t a l . , t~2] relevant dimensionless parameters useful in quantifying the effects of the shape and porosity of dendrites on sett
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