A scaling analysis of the unidirectional solidification of a binary alloy

A scaling analysis of the conservation equations for momentum, heat and species transport inside the solidifying mushy zone of a binary alloy is performed to examine systematically common assumptions and predict general behavior of the mixture during freezing. Several terms in the momentum equation are found to be negligible throughout the solidifying domain, and the use of D'Arcy's law to approximate the momentum equations in the mushy zone is found to be valid except in the region near the liquidus isotherm. A criterion is developed to define this region, and the dependence of the streamfunction and buoyancy driven velocity on material properties and fraction solid is determined. The energy equation is examined to provide scaling laws for the mushy zone and solid region thicknesses, as well as the transient chill wall temperature. Advection is shown to dominate solute transport throughout the mush, although, in the denser regions of the solid-liquid region, liquid velocities are so small as to have a negligible effect on macrosegregation. Numerical calculations performed for Pb-Sn and Al-Mg alloys at different cooling rates confirm trends suggested by the scaling analysis. Copyright (C) 1996 Elsevier Science Ltd.