# Trajectory Planning and Control of Preferential Crystallization Processes

## Contents

### Abstract

In this project, we investigate trajectory planning and feedback control of preferential crystallization processes for the separation of enantiomers. Our current research builds on prior work describing the use of orbital flatness properties for the control of certain classes of crystallization models.

### Description

This project aims at using preferential crystallization for the separation of enantiomers. Enantiomers are chiral substances, which share many physical and chemical properties, but can differ, e.g., with respect to their metabolic effects. Control and optimization of preferential crystallization has recently attracted increased attention. In previous joint work with the PCF group at MPI Magdeburg (e.g. [1]), two different operating modes were suggested. In cyclic operating mode, batch crystallization steps for the two substances to be separated alternate. In each step, seed crystals for one of the two enantiomer species are added to an oversaturated solution. For conglomerate forming enantiomers, these crystals will grow and stay pure even for approximately racemic solutions. If operated in the so called meta-stable area, secondary nucleation dominates primary nucleation. Hence, within each step, crystals of the seeded species will nucleate (and grow), but nucleation of the other species will be negligible. However, this will decrease the concentration of the seeded (preferred) enantiomer in the liquid phase, while the corresponding concentration of the other (counter) enantiomer will remain approximately constant. This, in turn, implies that eventually nucleation (and also growth) of the counter enantiomer will become significant, and, to maintain purity constraints, the process therefore has to be stopped.

Enantioseparation in two coupled vessels.

In coupled mode, crystal-free liquid is continuously exchanged between two crystallizer vessels, and sufficiently pure fractions of both enantiomer species can be obtained simultaneously by adding the respective seed crystals to the two vessels. The main advantage of the coupled mode is the approximately symmetric decrease in concentrations of both enantiomer species in the liquid phase. This allows for a control scheme where, by suitably manipulating the vessel temperatures, supersaturation of the preferred enantiomer in the respective vessel can be kept constant without causing the corresponding supersaturation of the counter enantiomer to steadily increase (as would be the case in a batch step in the cyclic operating mode).

Our research builds on prior work on the use of orbital flatness concepts for trajectory planning and control of single-substance crystallization. Although moment models describing preferential crystallization typically do not exhibit the orbital flatness property, we have developed an inversion procedure which, for a single vessel batch process, can be briefly summarized as follows:

(i) The model contains two population balance equations (PBEs), one for each enantiomer species. One species can be selected, and, as in the single substance case, scaling time with the growth rate of its crystal population,

Time scaling turns characteristic curves into straight lines.

$d\tau_{E_1}=G_{E_1}(t)\cdot dt,$

turns characteristic curves of the respective PBE into straight lines. Then, a specified final crystal size distribution (CSD) for that population can be easily mapped into a desired temporal evolution of the boundary condition, i.e., a desired temporal evolution of the (scaled) nucleation rate:

$\frac{B_{E_1}(t)}{G_{E_1}(t)}=f_{E_1}(0,t).$

(ii) Based on this information, a closed moment model for both species can be dynamically inverted in scaled time. In contrast to the single-substance case, this moment model exhibits nontrivial zero dynamics, even in transformed time. Therefore, differential equations have to be solved to finally obtain the desired temperature profile.

Depending on the application, a number of idealizing assumptions can be justified. For example, neglection of nucleation of the counter enantiomer may be appropriate if the focus lies on properties of the final CSD of the preferred enantiomer, and the assumption can be justified a posteriori. Dynamic inversion is then greatly simplified. Furthermore, optimal control solutions developed recently for single-substance crystallization [2] can be adopted.

The investigation of crystallization processes with size-dependent growth rate has profited from a cooperation with the PSE-group at MPI Magdeburg. This issue has been addressed in [3] [4] [5]. The latter suggests a finite ODE model which can approximate the underlying infinite-dimensional system with a required accuracy. This scheme is suited for use in control problems, such as dynamic inversion and optimal control. For instance, in [6] an efficient solution to optimal control of a single-substance batch crystallization process with size-dependent growth rate is found.

Feedback control becomes necessary in the face of uncertainties and disturbances:

• To reliably estimate purity from the available sensor data (temperature and concentration measurements), a nucleation observer was developed [7], which, for certain parametric model uncertainties and measurement uncertainty, provides a worst-case estimate of product purity.
• A controller for keeping concentrations of the two enantiomer species symmetric in the coupled mode was designed and experimentally validated [8] in cooperation with the PCF-group at MPI Magdeburg. Although, because of the poor quality of sensor data, symmetry could only be achieved in a very approximate manner, this turned out to be not a fundamental problem: investigations by the PCF-group seem to indicate that moderate asymmetries in the concentrations do not generally have a negative effect on purity [9].
• We also investigate the adaption of a number of robust optimization and control schemes developed for single-substance crystallization [10].

### Publications

1. Angelov, I., Raisch, J., Elsner, M. P., Seidel-Morgenstern, A.. Optimal operation of enantioseparation by batch-wise preferential crystallization. Chemical Engineering Science, 63 (5):1282–1292, 2008.
2. Hofmann, S., Raisch, J.. Application of Optimal Control Theory to a Batch Crystallizer using Orbital Flatness. In 16th Nordic Process Control Workshop, Lund, Sweden, 25-27th of August 2010; Nordic Working Group on Process Control, pages 61–67, 2010.
3. Bajcinca, N.. Forward and inverse integration of population balance equations with size-dependent growth rate. DYCOPS 2010, 9th International Symposium on Dynamics and Control of Process Systems, 2010, Leuven Belgium, pages 401–406, 2010.
4. Bajcinca, N., Qamar, S., Flockerzi, D.. Methods for integration and dynamic inversion of population balance equations with size-dependent growth rate. In 4th International Conference on Population Balance Modelling, Berlin, pages 227–243, 2010.
5. Bajcinca, N., Qamar, S., Flockerzi, D., Sundmacher, K.. Integration and dynamic inversion of population balance equations with size-dependent growth rate. Chemical Engineering Science, 66 (17):3711–3720, 2011.
6. Bajcinca, N., Hofmann, S.. Optimal control for batch crystallization with size-dependent growth kinetics. In American Control Conference 2011, San Francisco, USA, 2011.
7. Hofmann, S., Eicke, M., Elsner, M. P., Seidel-Morgenstern, A., Raisch, J.. A Worst-Case Observer for Impurities in Enantioseparation by Preferential Crystallization. In Proc. of ESCAPE-21, Chalkidiki, Greece, pages 860–864, 2011.
8. Hofmann, S., Eicke, M., Elsner, M. P., Raisch, J.. Innovative Control Strategies for Coupled Preferential Crystallization. In Proc. of WCPT6, April 2010.
9. Eicke, M., Hofmann, S., Raisch, J., Elsner, M. P., Seidel-Morgenstern, A.. Separation of Enantiomers by Coupled Preferential Crystallization - Impact of Initial Conditions on Process Performance. In Proc. of the 2010 AiChE Annual Meeting, 2010.
10. Bajcinca, N., Hofmann, S., Raisch, J., Sundmacher, K.. Robust and optimal control scenarios for batch crystallization processes. In Proc. of ESCAPE-20, Ischia, Naples, Italy, pages 1605–1610, 2010.