A finite volume method for solute segregation in directional solidification and comparison with a finite element method Original Research Article

A finite volume method is presented for the calculation of solute transport in directional solidification. Gallium and silicon doped germanium growth in the vertical Bridgman process are used as examples. The method is based on the stream function/vorticity formulation of the Navier-Stokes equation, energy and mass equations, and their associated boundary conditions in generalized curvilinear coordinates. Fluid flow, heat and mass transfer and the growth interface are solved simultaneously by Newtonʹs method with a nearly quadratic convergence. A consistent implementation of solute boundary conditions, which is crucial to the global conservation of solute, is used. Calculated results are compared with those obtained from the Galerkin finite element method for various process conditions, and they are in excellent agreement. With the present approach, the global conservation of solute is preserved, even with a coarse mesh, and is not affected by meshes and convection. However, fine meshes are required for the finite element method to achieve an acceptable global accuracy for the cases with stronger convection. Multiple steady states of double-diffusive convection and constitutional supercooling in the non-dilute solution are also illustrated.