Model-predictive control of the Czochralski crystallization process .2. Reduced-order convection model

A new reduced-order model (ROM) is proposed for the bulk-controller presented in Part I. This model considers convection in the melt based upon the assumption of an idealized flow geometry consisting of horizontal donuts and vertical tubes with radial and axial dispersion, respectively. Boundary and bulk layers are assumed to separate the core fluid from the crystal and crucible. Scale analysis and integral boundary-layer theory are utilized to estimate the boundary-layer thicknesses and the maximum stream function in the core fluid. A strategy is presented for adjusting the imperfect model online to reduce the process/model mismatch during MPC.