Determination of Surface Heating Condition for a Desired Thermal Growth Environment in an Industry-Size Hydrothermal Autoclave

Hydrothermal growth is the most common technique to grow piezoelectric single crystals such as quartz. Due to a high-temperature and high-pressure growth condition, hydrothermal autoclaves are designed to operate as a closed system. During operation, the only control mechanism that crystal growers have is adjusting the power input of the heaters, based on the temperature readings obtained by the thermocouples along the centerline inside the autoclaves. The power adjusting process, however, is purely experience dependent, and, normally, uniform heating conditions from electric heaters are employed along the autoclave wall. This study develops an inverse algorithm, with which the required heat flux distributions from the heaters can be obtained for a desired growth environment inside an autoclave. The algorithm involves solving three sub-models step by step. The first step is to solve a two-dimensional axisymmetric model of solution in the autoclave to obtain the temperature and heat flux on the solution/wall interface. Using these temperature and heat flux conditions as thermal boundary conditions, the second step solves an inverse heat conduction problem in the metal wall. The solution provides the heat flux and temperature on the outer surface of the metal wall. The final step is to solve a heat conduction problem in the insulation layer to obtain the heat flux on the inner surface of the insulation layer. The heat flux distributions for heaters are then determined by the heat flux on the outer surface of the metal wall and heat flux on the inner surface of the insulation layer. The paper describes the details of each model. As an example, the method is used to find the required heat flux distributions of heaters for the growth environment predicted by a 2-D isothermal wall model. The result is then used to develop a two-patch heater for industry autoclaves.