טכניון מכון טכנולוגי לישראל
הטכניון מכון טכנולוגי לישראל - בית הספר ללימודי מוסמכים  
Ph.D Thesis
Ph.D StudentRosenstein Yaron
SubjectHydrodynamic Stabilty in Czochralski Process of Crystal
Growth
DepartmentDepartment of Applied Mathematics
Supervisor Professor Emeritus Pinhas Bar-Yoseph


Abstract

Hydrodynamic stability analysis was performed on Czochralski's process of crystal growth. The governing equations are the Navier-Stokes equations coupled with the equation of energy through the Boussinesq appoximation.

The hydrodynamics and heat transport in this process are fairly complex, thus most published works perform full 3D time dependent analysis which is very CPU consuming.

In this work the basic solution is steady axisymmetric and is solved first using the spectral element method. Penalty method is applied to eliminate pressure terms and finally preconditioned GMRES is employed to solve the system.

In the linear stability analysis  3D perturbations decomposed into Fourier series in the azimuthal direction are considered.


Stability diagrams are presented for parameter ranges of Prandtl number (Pr)  between 0.005 and 0.02 and for aspect ratios between 0.4 and 1.0. The dependence of dominant wave numbers and critical Grashof numbers on geometry was studied.

Results show that dominant modes are the first 5 Fourier wave numbers (0 - 4).

Dependence of modes on aspect ratios was studied for Pr=0.01 (Silicone melt). It was found through analysis of dispersion relation of the critical frequencies as functions of modes and geometry that for aspect ratios lower than 0.85 convective heat transfer dominates the instability mechanism while for aspect ratios greater or equal to 0.85 rotational effect take over.

Many mode transitions were observed at different aspect ratios. Some modes were observed approaching each other closely in some ranges of the aspect ratio parameter.

Further analysis was carried out for the case where heat convection is absent (Pr=0). This analysis strengthens the conclusions about the role of convective heat transfer in the instability mechanism.

Analysis performed varying the ratios of rotational velocities of crucible to seed has shown greater instability thresholds in co-rotation in agreement with previous published works.