Hybrid Finite-element/finite-volume Method for Integrated Hydrodynamic Thermal Capillary Analysis of Turbulence and Heat Transfer of Czochralski Crystal Growth of Silicon

Modeling turbulent convection in the melt is the most di cult part of the simulation of the transport processes in the growth of large-diameter silicon crystals by the Czochralski (CZ) process. We report the application of hybrid nite-element/ nitevolume methods for the integrated modeling of turbulence in the melt and heat transfer by conduction, convection and radiation throughout the CZ system. Turbulence calculations are based on a low Reynolds number model proposed by Jones and Launder. The integrated hybrid method is based on coupled solution of energy transport in every phase in the CZ system by an integrated nite element analysis with the velocity eld in the melt and turbulence variables interpolated from a nite volume solution of the coupled convection and energy transport problem in the melt alone. Sample calculations make clear the advantages of the hybrid method. NOMENCLATURE Cp Heat capacity K Thermal conductivity T temperature Tm melting temperature V crystal pulling rate g acceleration due to gravity k turbulent kinetic energy p pressure r radial cylindrical coordinate u melt velocity z vertical cylindrical coordinate coe cient of thermal expansion of the melt dissipation rate of turbulent energy srf surface emissivity dimensionless temperature, T=Tm molecular dynamic viscosity t turbulent dynamic viscosity molecular kinimatic viscosity t turbulent kinimatic viscosity density stress tensor rotation rate