Christian Borchert

Optimal control solutions for crystal shape manipulation

Abstract Optimal control algorithms and strategies for crystal shape manipulation are explored in this paper. We focus on minimum time trajectories with the temperature as the control input. A challenge results from the inherent constraints in the particle growth vector, which is confined to lie within a cone in the model state-space. As a consequence, switching trajectories with a number of subsequent growth and dissolution phases may be indispensable. Such a strategy employs the unequal growth and dissolution rates in order to achieve crystal morphologies which do not result directly from a pure growth process only. For instantaneous switching between subsequent phases we need to extend the temperature solution profiles to piecewise-continuous. By a suitable specification of the switching manifolds, we are then able to avoid unnecessary long paths in the state-space. Diverse solution approaches to the optimal control problem are suggested using the minimum principle and efficient numerical techniques which utilize the invertibility property of the particle growth dynamics. In particular, we prove that minimum-time scenarios are composed of constant supersaturation trajectory sections.

Model reduction techniques for the simulation of particle populations in fluid flow

Crystallization processes are characterized by a close interaction between particle formation and fluid flow. A detailed physical description of these processes leads to complicated high-order models whose numerical solution is challenging and expensive. For advanced process control and other model-based online applications, reduced-order models are required. In this work, a reduced model for a urea crystallizer is developed using the method of moments for the internal coordinate and proper orthogonal decomposition for the external coordinate. Simulations are carried out to compare the reduced model with the detailed reference model.

Model Based Prediction of Crystal Shape Distributions

Abstract In classical crystallization as well as in nanoparticle engineering crystal shape is an important property of crystalline products. The emergent properties of bulk material are closely related to the crystal shape. We present a single crystal model describing shape evolution. The model has a hybrid character since the appearance and disappearance of crystal facets is a discrete event in the continuous process of crystal growth. Tight control of not only the single crystals shape and size is desirable but with regards to mass production shape and size distribution control is required. A first step in this direction is a model representing the dynamical evolution of the shape distribution of a crystal population. We show that a population balance approach can be used. In the proposed framework the hybrid nature of the single crystal evolution has to be taken care of. We also sketch a numerical scheme with which the proposed model equations can be solved.