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

Abstract

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).

Statistical Analyses of a Population Balance Model of a Batch Crystallization Process

“…where i indicates the moment order. 5,6,17 Each moment of a specific order has a well-defined physical meaning: μ 0 is the total number of particles per volume of suspension; r 1 = μ 1 /μ 0 , r 2 = μ 2 /μ 0 , and r 3 = μ 3 /μ 0 are the average characteristic length, surface area, and volume of the particles, respectively. The population balance equations used in the present study consider a perfect mixture and linear size-dependent growth of the potassium sulfate crystals.…”

“…Moraes et al 17 proposed the following differential equations for the population balance model of the potassium sulfate batch crystallization using the method of moments…”

“…13−15 Considering, in particular, the crystallization of salts, population balance models were also developed to describe the crystallization of potassium dihydrogen phosphate 16 and potassium sulfate. 17 From experimental data, it becomes possible to estimate appropriate values for the kinetic parameters, enabling the evaluation of mechanisms such as nucleation, growth, agglomeration, and crystal breakage. This approach provides a comprehensive understanding of the complex phenomena involved in the crystallization processes.…”

“…Based on the previous remarks, the present work investigates an experimental data set on batch-cooling crystallization of potassium sulfate previously obtained by Moraes et al 17 In general, measurements of transient systems display high correlation between consecutive samples (as the dynamic trajectory fundamentally indicates that the next sample depends on the previous one), and measurements of moments can also be subject to a high correlation between different moments measured at each time because of the measurement procedures. For this reason, a statistical characterization was performed, revealing, as expected, a high correlation among the measured variables.…”

“…Section 2 introduces the population balance model used to describe potassium sulfate crystallization. Section 3 presents the experimental data obtained by Moraes et al 17 and points out possible sources of random errors in the measured variables. Section 4 presents a discussion of the probability distribution of these errors.…”