Numerical simulation of molten silicon flow; comparison with experiment

Numerical simulation containing fluid flow, heat conduction and heat exchange by radiation has been performed using the geometry of a real Czochralski furnace for silicon single crystal growth. The flow velocity fields of molten silicon are obtained from extrapolation of the stream function, which has been newly developed using the velocity boundary layer theory. The calculated flow velocity and particle path are semi-quantitatively identical to the results obtained from X-ray radiography experiment. The calculated value of the characteristic velocity is about 10^-2 m/s. The same order of flow velocity which is obtained from the experiment has been already reported. It has also become clear from a comparison of flow velocities between experimental and calculated results that the order of the volume expansion coefficient of the molten silicon (β) is 10^-4 K^-1. The flow was almost axisymmetric and steady for a specific case with low crystal and crucible rotation rates and with a shallow melt. We also found that a flow with larger azimuthal velocity component exists just beneath a crystal, while that with opposite flow direction exists near the crucible wall.