Two-Dimensional Flow of Liquid During the Late Stage of Solidification of Alloys

Abstract

This is a mathematical analysis of the two-dimensional interdendritic liquid flow occurring during late stages of solidification, i.e., when the liquidus front reaches the center of the slab. The liquid flow considered is that caused by density change accompanying freezing. The liquid-solid region is considered a porous medium where the liquid permeability is a function of the solid fraction. Analytical solution for the two-dimensional velocity and pressure fields in solidifying Al-4.5 percent Cu alloy is obtained from a mathematical model which includes the concept of conservation of mass and Darcy’s law to correlate the pressure and velocity. Assuming that there is no gas-bubble formation and that the solidified metal in the mushy zone remains rigid, the liquid moving with a creeping velocity in the interdendritic regions is under tension (when no external pressure is applied). The magnitude of this tension increases with increasing depth of the solidifying ingot and is a strong function of the cooling rate. For example, a negative pressure of the order of 100 atm in the interdendritic liquid of solidifying Al-4.5 percent Cu alloy is estimated at ∼90 percent solidification. On the basis of the present analysis, an estimate can be made of pressures required to suppress blow-hole formation during the later stages of freezing arising from the solidification shrinkage.