(545g) Determining Recipes for Seeded Batch Crystallization for Many Chemical Systems with Optimal Control Theory and a Dimensionless Framework

An adapted version of a dimensionless framework for batch crystallization processes (Ward et al., 2011) is developed to solve optimization problems for 32 chemical systems (Garside and Shah, 1980; Tseng and Ward, 2014) with different crystal growth and nucleation kinetic parameters. The optimization variable is the supersaturation trajectory during the batch while nucleated volume and nucleated number are selected as objective functions. Constraints are placed on the seed-grown volume, maximum admissible growth rate, and batch time. Optimization problems formulated with differential equations are solved nearly-analytically by application of a coordinate transformation and optimal control theory (Vollmer and Raisch, 2006; Hofmann and Raisch, 2010; Tseng and Ward, 2019).

The results suggest that seed properties have a greater effect on the nucleated mass and number of nuclei than the supersaturation trajectory. However, since product mean size depends strongly on seed mean size, there is a significant trade-off between objectives concerning nucleation and product mean size. The trade-off between nucleated number and nucleated volume is strong especially when nucleation rate is much more sensitive to supersaturation than growth rate as shown by plotting Pareto-optimal fronts. Objective values under constant growth trajectory are also evaluated since this control strategy can be implemented without prior knowledge of the kinetic parameters (Nagy, 2017). Results show that the difference in objective values is small when the nucleation rate is proportional to magma density or sensitivity of nucletion to crystal growth is small.

If the seed mean size is 50 microns and batch time is 1 hour, 22 of the 32 systems require 10% or lower seed loading to effectively inhibit nucleation (so that the nucleated mass is 1% of the seed-grown mass) after application of a constant growth trajectory. However, when the seed mean size is 200 microns, nucleation be suppressed with a seed loading less than 10 % in only four cases. Furthermore, increasing the batch time does not provide a significant benefit for most systems.

Though the kinetic models considered are relatively simple (size- and temperature-dependence of kinetic parameters as well as breakage and aggregation are neglected), this work provides understanding for optimal seeding and growth rate control policy for many chemical systems which can serve as a basis for understanding general trends and outliers in the synthesis of batch crystallization recipes.

References

Garside, J.; Shah, M. B., Crystallization kinetics from MSMPR crystallizers. Industrial & Engineering Chemistry Process Design and Development 1980, 19 (4), 509-514.

Ward, J. D.; Yu, C. C.; Doherty, M. F., A new framework and a simpler method for the development of batch crystallization recipes. AIChE journal 2011, 57 (3), 606-617.

Randolph, A. D., Theory of particulate processes: analysis and techniques of continuous crystallization. Academic Press: New York, 1971.

Vollmer, U.; Raisch, J., Control of batch crystallization—A system inversion approach. Chemical Engineering and Processing: Process Intensification 2006, 45 (10), 874-885.

Hofmann, S.; Raisch, J. In Application of optimal control theory to a batch crystallizer using orbital flatness, 16th Nordic Process Control Workshop, Lund, Sweden, 2010; pp 25-27.

Tseng, Y.-T.; Pan, H.-J.; Ward, J. D., Pareto-optimal fronts for simple crystallization systems using Pontryagin’s minimum principle. Industrial & Engineering Chemistry Research 2019, 58 (31), 14239-14251.

Nagy, Z. K., Crystallization Control Approaches and Models. In Engineering Crystallography: From Molecule to Crystal to Functional Form, Springer: 2017; pp 289-300.

Tseng, Y. T.; Ward, J. D., Critical seed loading from nucleation kinetics. AIChE Journal 2014, 60 (5), 1645-1653.