Control of Crystallization Processes Based on Population Balances

Industrial crystallization plants can be operated in basically two different ways: continuous or batch. These two modes of operation result in two different control problems, which are both addressed in this dissertation. The thesis is composed of two major parts corresponding to the two fundamentally different control problems arising in crystallization. It is the aim of this work to demonstrate that sophisticated up-to-date control synthesis methods can be applied to crystallization processes on the basis of population balance models. Stabilizing Control of Continuous Crystallization The first part of the thesis comprises Chapters 2 to 4. It deals with continuously operated crystallization plants. Continuous processes run for very long periods of time. They are desired to operate at a steady state. Product quality is determined by the steady state crystal size distribution (CSD). This quantity can be influenced by fines dissolution. Unfortunately, apart from the desired effect on the CSD this also affects the dynamics of the system in an undesirable way. Extensive fines dissolution may lead to instability of the steady state. In Chapter 2, a detailed population balance model introduced by [Mit02] is presented, which describes a continuous crystallizer. This model considers nucleation, growth and attrition of crystals. It comprises two population balances, one for the crystallizer vessel itself and one for the settling zone used for fines separation. It captures the destabilizing effect of fines dissolution and predicts the resulting sustained oscillations of CSD and solute concentration. As a basis for controller design, however, this model is too complex. Therefore, in Chapter 3, the detailed model is simplified by replacing the rather complex detailed growth and attrition laws by simpler expressions. Furthermore, the settling zone is regarded quasi-stationary. Thus, a population balance model is obtained from which a transcendental transfer function can be derived relating the control input (fines dissolution rate) and the measured variable (third moment of CSD). In Chapter 4, this transfer function serves as a basis for the design of stabilizing feedback controllers. Since the controllers are designed on basis of a simplified model it is essential that they are robust with respect to plant model mismatch. H-infinity-theory provides a framework for the systematic consideration of such robustness issues. In this work, a specific version of H-infinity-theory for infinite-dimensional systems is used [FOT96]. The resulting controllers are tested in simulations with the simplified and the detailed model. What distinguishes this approach from most other contributions in the area of continuous crystallizer control is that controllers are designed on basis of an infinite-dimensional model. The population balance model is not discretized prior to controller design. Control of Batch Crystallization Using System Inversion The second part of the thesis, dealing with batch crystallization, comprises Chapters 5 and 6. In batch cooling crystallization the product quality is determined by the CSD at the end of the batch, which can be influenced by the cooling profile, i.e. the temperature trajectory during the batch run. In an experiment or by numeric simulation of a population balance model, it is obviously possible to determine the product CSD created by a given temperature trajectory. The solution of the reverse problem, i.e. the design of a temperature trajectory which produces a specific CSD defined as a function of crystal length, requires inversion of the system model. In Chapter 5, a relatively simple standard population balance model [MR94] is presented which allows the derivation of a finite-dimensional moment model. The notion of differential flatness [FLMR92] is introduced and it is shown that the moment model is not flat immediately but can be rendered flat by an appropriate state dependent time scaling. Such systems are called orbitally flat. In Chapter 6, it is shown that the system model can be inverted, exploiting the flatness property of the time scaled moment model and the structure of the time scaled population balance. Thus, for a given desired final CSD it is possible to check whether it can be produced under the given conditions and, if so, the corresponding temperature trajectory can be computed analytically. In the literature, the most common approach to batch crystallizer control is the dynamic optimization of CSD properties. It is shown that this problem can be simplified considerably taking advantage of the system’s flatness property. Finally, the design of nonlinear feedback tracking controllers is presented, which control the system along desired trajectories of the moments. The distinctive feature of this system inversion approach is the possibility to design a temperature trajectory achieving one specific CSD defined as a function of crystal length rather than merely optimizing a scalar property of the CSD. Moreover, temperature trajectories are determined analytically whereas most other approaches rely on numerical algorithms. References [FLMR92] M. Fliess, J. Levine, P. Martin, and P. Rouchon. On differentially flat nonlinear systems. In Nonlinear Control Systems Design, pages 408-412. Pergamon Press, 1992. [FOT96] C. Foias, H. Ozbay, and A. Tannenbaum. Robust Control of Infinite Dimensional Systems. Springer, 1996. [Mit02] A. Mitrovic. Population Balance Based Modelling, Simulation, Analysis and Control of Crystallization Processes. PhD thesis, ISR, Universität Stuttgart, 2002. [MR94] S.M. Miller and J.B. Rawlings. Model identification and control strategies for batch cooling crystallizers. AIChE Journal, 40(8):1312-1327, 1994