Numerical calculation of the permeability in a dendritic mushy zone
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I. INTRODUCTION
IN most macroscopic models, the two-phase dendritic columnar mushy zone that commonly occurs during directional solidification of binary alloys is generally modeled as a porous medium, and the interdendritic flow is described using Darcy’s law or its extensions.[4–10] In such a macroscopic representation, the permeability is clearly a determining parameter for an accurate description of thermosolutal natural convection giving rise to macrosegregation. However, due to the very complex microstructure of the dendrites, its determination still remains a challenge. Indeed, the dendritic columnar region is characterized, first, by a strong anisotropy, which requires a knowledge of the different components of the permeability tensor,[3,11,12] and, second, by the nonuniform macroscopic properties such as the volume liquid fraction, which continuously varies from unity in the melt to zero in the solid region. Although these continuous variations, or evolving heterogeneities, can obviously lead to important differences in the convective flows,[10] rarely have they been explicitly taken into account in the theoretical models. In these macroscopic descriptions, liquid volume fraction and permeability are usually related by the classical Kozeny– Carman relationship, which has been derived for an ideal porous medium constituted of a bundle of regular capillary tubes of equal length. Because of the strong irregularity of the dendrites and because of the nonhomogeneity of the mushy layer, this relation may be inappropriate, particularly in the vicinity of the liquidus interface, where important variations of the liquid fraction take place. Due to the complexity of the problem, relatively few studies have been B. GOYEAU, Assistant Professor D. GOBIN, and T. BENIHADDADENE, Postdoctoral Student, are with FAST, University of Paris, 91405 Orsay, Cedex, France. D. GOBIN and M. QUINTARD, Research Directors, are with IMFT, 31400 Toulouse, France. Manuscript submitted July 7, 1998. METALLURGICAL AND MATERIALS TRANSACTIONS B
carried out in order to quantitatively estimate the permeability tensor. Some experimental studies have been performed for columnar[3,11,13,14] or equiaxed[13,15] dendritic mushy zones, and these proposed relationships for the permeability tensor. In the columnar configuration, data in different dendritic structures have been summarized by Poirier,[3] and quantitative relationships for parallel and normal permeability have been derived using a regression analysis. In the range of liquid fraction considered (0.19 # « # 0.66), the regressions provided were in good agreement with the classical physical models, but the extrapolations beyond the upper limit failed. Numerical experiments for parallel and normal flows to the primary dendritic arms in columnar structures with a high liquid volume fraction (« . 0.6) have been found to be more successful.[16,17] In Reference 17, Bhat et al. calculated, from digitized images, the permeability for flow normal to the primary dendrite arms using a Navier– Stokes finit
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