Modeling and Predictive Control of Cooling Crystallization of Potassium Sulfate by Dynamic Image Analysis: Exploring Phenomenological and Machine Learning Approaches

Representative mathematical modeling is essential for understanding the batch cooling crystallization processes. Efficient process design and operation are relevant to achieving high-quality criteria and minimizing variation between batches. This work first presents the modeling of batch cooling crystallization based on online dynamic image analysis. A flow-through microscope was used to track the temporal evolution of the crystal population. A population balance modeling (PBM) approach, parameter estimation, and validation were obtained for the batch cooling crystallization of potassium sulfate in water. The performed experiments provided new experimental data, giving dynamic information about the crystal size throughout each run. The kinetic model parameters for crystal nucleation and growth were estimated using a hybrid optimization algorithm, followed by the confidence region construction using a more exploratory particle swarm algorithm. In the parameter estimation framework, in addition to solute concentration, the first fourth-order moments computed throughout all experiments were included in the objective function. A linear size-dependent growth rate was found to capture well the dynamics of the potassium sulfate crystal size distribution. The experimental results evidenced that the crystal shape of potassium sulfate is predominantly constant, allowing the adequacy of the developed model. The validated PBM was also employed as a digital twin of the crystallization process to develop a machine-learning-based control for the process. Then, a surrogate model based on a recurrent neural network, called an echo state network (ESN), was applied in a nonlinear model predictive controller approach (ESN-NMPC). The ESN model could predict the moments of the population balance model up to five steps (5 min) forward. The ESN-NMPC achieved the desired control scenarios for the crystal size and its coefficient of variation. Its performance was comparable to the controller that uses the PBM as the internal model (PB-NMPC).