A hydrodynamic thermal-capillary model (HTCM) for heat transfer in Czochralski crystal growth systems is used to calculate steady-state, axisymmetric solutions for heat transfer and fluid mechanics while incorporating a self-consistent description of the free boundaries of the melt/crystal interface, the melt meniscus, and the crystal diameter. The model employs a Galerkin finite-element method to discretize the model equations, and solutions are obtained using a Newton-Raphson iterative scheme. Sample results are presented for the growth of a large-dimension oxide crystal with thermophysical properties similar to those of gadolinium gallium garnet (GGG). Calculations with the HTCM show the effects of crystal rotation on heat transfer, flow in the melt, and melt/crystal interface shape. Severe deflections of the melt/crystal interface are calculated for moderate rotation rates, and limit points in the steady-state solutions are found with respect to crystal rotation.
Bibliographical noteFunding Information:
Computational resources were supplied by Lawrence Livermore National Laboratory under the auspices of the US Department of Energy, Contract W-7405-Eng-48, by the Army High performance Computing Research Center (AHPCRC) under a grant from the US Army, and by the Minnesota Supercomputer Institute. J.J.D. acknowledges support from the Shell Oil Company Foundation (as a Shell Faculty Fellow) and the National Science Foundation PYI award program. Q.X. was supported in part by AHPCRC. Parts of this paper were presented at the 27th National Heat Transfer Conference, Minneapolis, July 1991.