A reduced dynamic simulation model is presented to accurately predict in real time the global heat transfer and the shape of the solidification front, for single crystals grown by the Czochralski process. This model, which is robust, reliable and fast, is devoted to be incorporated in a general control system, where it will behave as an observer/predictor in order to provide estimations for nonmeasurable process variables from which crystal quality can be inferred.
Olivari, E., Jacmin, P., Vandenbogaert, N., Wertz, V., & Dupret, F. (1997). The use of a reduced model for on-line simulations of the Czochralski growth of single crystals. Journal of Crystal Growth, 180(3-4), 627-637. https://doi.org/10.1016/S0022-0248(97)00258-3 (Original work published 1997)