Vertical solidification of dendritic binary alloys

This paper addresses the effect of thermosolutal convection in the formation of defects in directionally solidified (DS) alloys. Three different models have been used as the basis for numerical simulations of the solidification process, and all use the Boussinesq approximation. In increasing order of complexity, the three are the plane-front model, an extension of the plane-front model that includes a dendritic region with a time-independent volume fraction of liquid, and a model where the dendritic region is free to develop according to local equilibrium conditions and that is capable of predicting channel segregates or freckles. Finite element models of thermosolutal convection have been developed for all three cases and have been used to investigate nonlinear convection when the systems are unstable. The dendritic or mushy region is treated as a porous medium with variable porosity; the algorithm is based on a Petrov-Galerkin and penalty function formulation using the four-noded bilinear Lagrangian element with reduced integration of the penalty term. Calculations are presented for lead-tin alloys at different concentrations of tin and for various temperature gradients. Also shown are calculations in which freckles develop in DS alloys - the first of their kind. The results of the numerical calculations are discussed and compared to experimental observations and other theoretical analyses.