Averaged solute transport during solidification of a binary mixture: Active dispersion in dendritic structures
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I. INTRODUCTION
MACROSCOPIC solidification modeling has been the subject of intense research in the last few decades,[1,2,3] and it is now recognized that one of the most challenging improvements lies in the microscopic-macroscopic description of transport phenomena within the dendritic mushy zone. Generally, this dendritic region is described as a porous medium, where the macroscopic conservation equations have been derived by using the mixture theory[4,5,6] or the volume-averaging method.[1,7–11] This latter up-scaling method has often been preferred due to its ability to incorporate microscopic information (microstructure of the geometry of the dendrites, local inertial and dispersion phenomena, etc.) in the conservation equations and in the definition of the effective transport properties (permeability, diffusiondispersion, etc.). In this sense, intense experimental and numerical research has been carried out in order to provide a more realistic description of the spatial evolution of the permeability tensor in columnar and equiaxed dendritic structures than that given by using the traditional Kozeny– Carman relationship, which has been derived for an ideal, regular, and periodic porous medium.[12–20] On the other hand, the effective diffusion coefficients in the heat and species macroscopic transport equations in solidification models remain very schematic, and most of the time, the mass-diffusion rate in the solid phase is assumed to be extremely fast (lever rule) or very slow (Scheil’s behavior). In this latter case, the effective mass-diffusion coefficient is generally represented by D* ⫽ Ᏸ, where  P. BOUSQUET-MELOU, formerly Ph.D. Student at Fluides, Automatique et Syste`mes Thermiques, FAST University of Paris 6, 91405 Orsay, France, and Institut de Protection et de Suˆrete´ Nucle´aire, CEA Cadarache, 13108 Saint Paul lez Durance, France, is with ONERA, 92332 Chatillon, France. A. NECULAE, Ph.D. Student, is with Fluides, Automatique et Syste`mes Thermiques, FAST, University of Paris 6, and the Department of Physics, University of the West Timisoara, Romania. B. GOYEAU, Associate Professor, is with FAST, UMR CNRS 7608, France. Contact e-mail: [email protected] M. QUINTARD, Research Director, is with the Institut de Me´canique des Fluides de Toulouse, IMFT, 31400 Toulouse, France. Manuscript submitted June 27, 2001. METALLURGICAL AND MATERIALS TRANSACTIONS B
and Ᏸ are the liquid volume fraction (porosity) and the molecular diffusion coefficient in the liquid phase, respectively. However, a recent numerical study regarding passive dispersion in the dendritic structures (i.e., dispersion without interfacial mass transfer) has shown that the effective solutediffusion coefficient can be much more complicated, including at least tortuosity and local dispersion phenomena.[21] Nevertheless, this passive representation still remains an approximation, since it is based on the hypothesis that there is no mass exchange at the solid/liquid interface. This limitation has been overcome by Ni and Beckermann,[9]
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