A Numerical Investigation of Directional Binary Alloy Solidification Processes Using a Volume-averaging Technique

A numerical investigation of directional binary alloy solidification processes is presented. In particular, a mathematical model is developed to study macrosegregation patterns as a consequence of thermal-solutal convection in the melt and mushy zone. A good understanding of the basic mechanisms of macrosegregation is helpful in designing and controlling solidification processes in order to achieve better quality cast products. A volume-averaging technique is used to derive the macroscopic conservation equations for momentum, energy and species transport. This set of governing equations is applicable to the solid, mushy and liquid regions of the solidification system. Hence, the analysis is performed in a single-domain with a fixed numerical grid and a single set of boundary conditions. Three finite element schemes for the solution of the momentum equation have been implemented. These schemes include a stabilized Galerkin equal-order-interpolation algorithm, a penalty function formulation and the fractional step method. Whereas the penalty method appears to yield the best efficiency, the fractional step method was shown to be the most reliable and hence is used for all the simulations in the present study. The numerical model is developed in two stages. First, the set of macroscopic conservation equations is applied to natural convection and double diffusive convection problems in porous media. The simulation of fluid flow through porous media could help validate the application of the volume-averaging equations to solidification process modelling where the mushy region is typically considered as a porous medium. The numerical results are compared with various porous media flow predictions reported in the literature. Good agreement with the results of earlier studies shows that the macroscopic equations derived through the volume-averaging technique are indeed valid in both singleand two-phase regions. Secondly, the above model is applied to the simulation of directional binary-alloy solidification processes. The developed single-domain model is shown to predict well the extended mushy zone and the formation of channels. It is found that the variation of the alloy composition in the cast product reaches 40% of the initial composition. This finding supports the theory that thermal-solutal convection can induce severe macrosegration. The effects of anisotropic permeability of the mushy zone on the obtained segregation patterns are also studied. A simulation of pure aluminum solidification is also carried out as a limiting case to test the versatility of the present model with the results compared with those achieved using a front tracking method. Finally, a discussion on further work needed to extend the current model to model more complex solidification processes is provided.