The macroscopic conservation equations governing solute transport during solidification of binary alloys are derived using an averaging procedure in the context of macroscale nonequilibrium. Special attention is focused on the derivation of the associated closure problems, leading to the determination of the effective dispersion tensor and macroscopic interphase coefficients that characterize active dispersion phenomena. These closure problems are solved numerically using schematic structures and digitized images of real columnar dendritic structures observed experimentally during solidification of succinonitrile-4 wt pct acetone. The influence of the geometry and dispersion on the effective solute-properties transport is analyzed, and comparison with passive dispersion is provided. The theoretical and numerical results indicate, first, that tortuosity effects are small, and this is related to the impact of the boundary condition at the interface between the two phases, and, second, that dispersion becomes very important only at very large Péclet numbers.
This paper deals with the derivation of a macroscopic model for columnar dendritic solidification of binary mixtures using the volume averaging method with closure. The main originalities of the model are first related to the explicit description of evolving heterogeneities of the dendritic structures and their consequences on the derivation of averaged conservation equations, where additional terms involving porosity gradients are present, and on the determination of effective transport properties. These average properties are defined by the associated closure problems taking into account the geometry of the dendrites and the local intensity of the flow. The macroscopic solute transport is obtained by considering macroscale non-equilibrium giving rise to macroscopic dispersion and interfacial exchange phenomena. Mass exchange coefficients are accurately explicited as a function of the local geometry.
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