Three-dimensional finite element simulation of transport phenomena in alloy solidification

The development of a numerical algorithm based on a three-dimensional finite element mesh for simulating the transport phenomena in alloy solidification is completed. The model equations are implemented in a single-phase finite element computational fluid dynamics code, FEAST (Finite Element Analysis & Solution Tool), which is being developed at the Institut fur Angewandte Mathematik of the Universitat Heidelberg, Germany. Weighted least squares operators are used in the discrete operator module to obtain the high-order divergences and gradients of the velocities and pressure when either solidification or melting is present. The species and enthalpy conservation equations are coupled through the phase change rate, which is estimated using an expression derived from the binary phase diagram and the species conservation equations when convection is present. The algorithm is verified with several analytical, similarity, and numerical benchmark solutions as well as previous numerical solutions.; Using the developed algorithm, an investigation of freckle formation during vertical directional solidification is performed. The three dimensional model enables a detailed study of the mechanisms that produce convection in the melt and the mushy zone and the effect of convection on macrosegregation. Analysis of the liquid concentration and vertical velocity distributions at different 2-D cuts and locations indicates that the discretization of the computational domain can significantly affect the computed freckles and insufficient grid solution leads to significant errors in the predicted results. Based on a parametric study, a critical Rayleigh number for freckle formation of approximational 0.25 is found. This critical Rayleigh number is in agreement with one found from previously conducted experiments. With increasing temperature gradient, while keeping the solidification speed at a low value, the extent of the compositional boundary layer ahead of the mush-liquid interface increases. The numerical results show that a convective instability can occur within the compositional boundary. This instability occurs at a lower Rayleigh number than the one for freckle formation. Additional numerical simulations are needed to investigate the resulting convection.