Solution of the Stefan Problem During Radiation-Conductive Heat Transfer in the Process of Growing Sapphire Single Crystals by a Modified Kyropoulos Method

The problem of creating an automated system to control growth of single crystal sapphires from melt as a problem of control and monitoring of a complex nonlinear and dynamic system

“…If we calculate the conducted-radiation parameter * = Thus, in solving the problem of temperature distribution along the axis of a growing crystal, the nonlinear parameter ( ) plays a significant role. In [5], we obtained expressions for the temperature distribution along the axis of a growing crystal depending on the geometric and nonlinear parameters 1 ( ) [9] and ( ). Based on this, the formulas for calculating the temperature gradient at the crystallization front and the maximum crystallization rate were obtained:…”

“…Taking into account the results obtained in [5], it is possible to conduct a qualitative analysis and compare the role of various factors in the formation of the thermally stressed state of a growing crystal.…”

“…Taking into account the formula for temperature distribution along the X axis [5], it is possible to calculate the axial component of thermal stresses using the formula [8]:…”

“…The complex structures in this model are unstable and can exist only with the correct specification of the initial data. In [5], we have investigated a similar model when calculating the temperature distribution along the crystal axis during growth and optimized some process parameters based on identification with experimental data.…”