A Scaling Analysis for Transport Phenomena during Alloy Solidification in a Rectangular Cavity

Abstract

A scale analysis of the equivalent single phase continuum conservation equations for mass, momentum, heat and species transport is performed for flow inside the mushy zone for the freezing of a binary mixture in a rectangular cavity, which isothermally cooled from one vertical side. Certain terms in the full momentum equations turn out Co be relatively insignificant, and an analysis of the remaining terms leads to the establishment of an advection velocity scale. The energy equation is then analysed, resulting in an estimation of the solid layer thickness. Different regimes corresponding to different dominant modes of transport are simultaneously identified. Stale analysis of the species conservation equation reveals how a non-equilibrium solidification model affects solute segregation and species distribution. It is shown that non-equilibrium effects result in a more severe macro-segregation than that dictated by the rules of equilibrium solidification. It is also observed that, in spite of low velocities, solute transport is driven primarily by advection and less significantly by species diffusion. To test the validity of the scale analysis, numerical calculations using CFD techniques are performed, which confirm. the trends suggested by the sealing analysis.