(571b) Optimal Control of Batch Crystallization Processes: New Methods for New Challenges

Batch crystallization is a vital technique for the separation and purification of substances, especially in the production of pharmaceutical and fine/specialty chemicals. The operation of such processes can be optimized to improve crystal properties such as size and shape distribution. In general, to model a batch crystallization process it is necessary to solve a partial differential equation to determine how the crystal size distribution changes with time during the batch. If the process must be simulated repeatedly for optimization, this may be time consuming if a high resolution is desired. Furthermore, at the end of the batch the exact crystal size distribution is not known, only the moments or the values of the CSD at certain discrete points.

Raisch and coworkers [1-5] demonstrated that after a specific transformation of variables, optimal control theory can be applied to solve certain optimization problems in batch crystallization very efficiently and nearly-analytically, recovering the complete crystal size distribution. In the present work their method is modified, extended and applied to a variety of other problems of practical interest.

First, the method is applied to determine optimal operating trajectories for a large set of crystallization systems using a dimensionless framework [6]. The framework is used to calculate Pareto fronts for the tradeoff between the number of nuclei and the nucleated mass. The results show that for most systems operation with constant supersaturation gives a result similar to that of the knee point on the Pareto front suggesting that the constant supersatuation trajectory (which can be determined without a crystallization kinetic model) is a reasonable choice when the crystallization kinetics are unknown.

Second, the method is applied to systems with primary nucleation [7]. The original work of Raisch and coworkers considered only secondary nucleation where the nucleation rate is proportional to the growth rate and the magma density raised to certain powers. In the present work it is shown that the previous method can be extended to consider primary nucleation. The resulting two-point boundary value problem is difficult to solve using conventional shooting methods because of the equations are highly nonlinear. Instead the boundary value problem is solved using a gradient method. The algorithm is found to be fast, accurate and robust and permits the study of a tradeoff between number of nuclei and nucleated mass.

Third, the method is applied to batch cooling crystallization processes where the crystallization kinetics are temperature-dependent [8]. Three strategies for this case are considered and compared. The first strategy is to neglect the nucleated mass when calculating the nucleation rate as first suggested by Hofmann and Raisch [2]. The second strategy is to solve the two-point boundary value problem using a gradient method. The third strategy is to combine the two, using the approximation to determine an initial guess for the gradient method. The results show that the approximation introduces little error and the computational burden of the gradient method is dramatically reduced if the result from the approximation is used as an initial guess.

Fourth, the method is applied to problems with nucleation and crystal shape [9]. Pareto fronts for the tradeoff between number of nuclei and nucleated mass are plotted subject to constraints on the product crystal shape. The endpoints of the Pareto fronts map out a feasible region for the operation of the process, and the tradeoff between objective functions is most pronounced near the midpoint between the minimum and maximum achievable aspect ratio of the seed-grown crystals.

Finally, the method is applied to problems where cycles of growth and dissolution are used to tailor the product crystal size distribution [10]. In this case, the method can be used to efficiently determine the attainable region (in terms of seed size and seed variance). The results show that the attainable region expands as the number of cycles of growth and dissolution is increased, but the extent of the change of the crystal size depends on the sensitivity of the growth and dissolution rates to the crystal size.

References

1. Bajcinca, N.; Hofman, S. Optimal control for batch crystallization with size-dependent growth kinetics. In Proceedings of the 2011 American Control Conference, 2011; IEEE: pp 2558-2565.

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, 2010; pp 25-27.

3. Angelov, I.; Raisch, J.; Elsner, M. P.; Eidel-Morgenstern, A. S. Optimal operation of enantioseparation by batch-wise preferential crystallization. Chem. Eng. Sci. 2008, 63 (5), 1282-1292. DOI: 10.1016/j.ces.2007.07.023.

4. Vollmer, U.; Raisch, J. Control of batch crystallization - A system inversion approach. Chemical Engineering and Processing-Process Intensification 2006, 45 (10), 874-885. DOI: 10.1016/j.cep.2006.01.012.

5. Vollmer, U.; Raisch, J. Control of batch cooling crystallization processes based on orbital flatness. Int J Control 2003, 76 (16), 1635-1643. DOI: 10.1080/00207170310001626419.

6. Pan, H. J.; Ward, J. D. Dimensionless Framework for Seed Recipe Design and Optimal Control of Batch Crystallization. Ind Eng Chem Res 2021, 60 (7), 3013-3026. DOI: 10.1021/acs.iecr.0c06132.

7. Chien, W. T.; Pan, H. J.; Ward, J. D. Optimal Control of Batch Crystallization Processes with Primary Nucleation. Ind Eng Chem Res 2023, 62 (22), 8834-8846. DOI: 10.1021/acs.iecr.3c00286.

8. Pan, H. J.; Ward, J. D. Algorithms for solving boundary value problems in optimal control of seeded batch crystallization processes with temperature-dependent kinetics. Chem. Eng. Sci. 2023, 270. DOI: 1016/j.ces.2023.118517.

9. Pan, H. J.; Ward, J. D. Optimization of Simple Batch Crystallization Systems Considering Crystal Shape and Nucleation. Ind Eng Chem Res 2020, 59 (20), 9550-9561. DOI: 10.1021/acs.iecr.9b06842.

10. Pan, H.-J.; Ward, J. D. Computationally Efficient Algorithm for Solving Population Balances with Size-Dependent Growth, Nucleation, and Growth-Dissolution Cycles. Ind Eng Chem Res 2021, 60 (34), 12614-12628. DOI: 10.1021/acs.iecr.1c01947.