Product quality improvement of batch crystallizers by a batch-to-batch optimization and nonlinear control approach

Batch crystallization is one of the widely used processes for separation and purification in many chemical industries. Dynamic optimization of such a process has recently shown the improvement of final product quality in term of a crystal size distribution (CSD) by determining an optimal operating policy. However, under the presence of unknown or uncertain model parameters, the desired product quality may not be achieved when the calculated optimal control profile is implemented. In this study, a batch-to-batch optimization strategy is proposed for the estimation of uncertain kinetic parameters in the batch crystallization process, choosing the seeded batch crystallizer of potassium sulfate as a case study. The information of the CSD obtained at the end of batch run is employed in such an optimization-based estimation. The updated kinetic parameters are used to modify an optimal operating temperature policy of a crystallizer for a subsequent operation. This optimal temperature policy is then employed as new reference for a temperature controller which is based on a generic model control algorithm to control the crystallizer in a new batch run.     

DYNAMIC MODELING AND OPERATION OF A SEEDED BATCH COOLING CRYSTALLIZER

In this study, a dynamic model of a batch cooling crystallizer is developed. The seeded crystallization of potash alum from aqueous solutions. Four different cooling policies namely natural cooling, linear cooling, optimal cooling, and controlled cooling (nonlinear geometric control (NGC) cooling) are presented. The simulation results indicate that both optimal and controlled cooling improve the weight mean size of the final product significantly. The influence of seed loading policy on the end product quality is also studied for the NGC cooling. It is found that the appropriate seed loading is important to achieve a good final CSD, especially for a fixed batch time.     

Multi-objective optimization of seeded batch crystallization processes

The optimization of a batch cooling crystallizer has been traditionally sought with respect to the cooling profile and seeding characteristics that keep supersaturation at an optimum level throughout the operation. Crystallization processes typically have multiple performance objectives and optimization using different objective functions leads to significantly different optimal operating conditions. Thus different temperature profiles and seeding characteristics impose a complex interplay on the crystallizer behavior and there is a trade-off between the performance objectives. Therefore, a multi-objective approach should be adopted for optimization of a batch crystallizer for best process operation. This study presents the solution of various optimal control problems for a seeded batch crystallizer within a multi-objective framework. A well known multi-objective evolutionary algorithm, the elitist Nondominated Sorting Genetic Algorithm, has been adapted here to illustrate the potential for the multi-objective optimization approach.     

Real‐time control of a semi‐industrial fed‐batch evaporative crystallizer using different direct optimization strategies

This article presents a model‐based control approach for optimal operation of a seeded fed‐batch evaporative crystallizer. Various direct optimization strategies, namely, single shooting, multiple shooting, and simultaneous strategies, are used to examine real‐time implementation of the control approach on a semi‐industrial crystallizer. The dynamic optimizer utilizes a nonlinear moment model for on‐line computation of the optimal operating policy. An extended Luenberger‐type observer is designed to enable closed‐loop implementation of the dynamic optimizer. In addition, the observer estimates the unmeasured process variable, namely, the solute concentration, which is essential for the intended control application. The model‐based control approach aims to maximize the batch productivity, as satisfying the product quality requirements. Optimal control of crystal growth rate is the key to fulfill this objective. This is due to the close relation of the crystal growth rate to product attributes and batch productivity. The experimental results suggest that real‐time application of the control approach leads to a substantial increase, i.e., up to 30%, in the batch productivity. The reproducibility of batch runs with respect to the product crystal size distribution is achieved by thorough seeding. The simulation and experimental results indicate that the direct optimization strategies perform similarly in terms of optimal process operation. However, the single shooting strategy is computationally more expensive. © 2010 American Institute of Chemical Engineers AIChE J, 57: 1557–1569, 2011     

On-line optimal control of a seeded batch cooling crystallizer

In this paper, an on-line optimal control methodology is developed for the optimal quality control of a seeded batch cooling crystallizer process. An extended Kalman filter is successfully implemented to predict seven unmeasured state variables based on three measurements in the batch process. A PI controller is used in a feedback control system to implement the optimal path. It is found that the PI controller can ensure tracking of the optimal path. The simulation results show that on-line optimal control strategy leads to a substantial improvement of the end product quality expressed in terms of the mean size and the width of the distribution. The effects of the plant/model mismatch and disturbances are also tested and discussed.     

DYNAMIC OPTIMAL CONTROL OF BATCH CRYSTALLIZATION PROCESSES

This work applies an on-line optimal control strategy developed by Zhang (2001) to two cooling batch crystallization processes. The algorithm initially finds the optimal crystallizer temperature and subsequently uses a feedback control system in order to achieve the desired final product quality of the crystals expressed in terms of the final crystal size distribution. In both batch processes, it is shown that the on-line optimal control approach provides better final product quality as compared with a simplified optimal cooling policy. The improvement is especially noticeable in the presence of plant/model mismatch or errors in the initial conditions.     

Extended kalman filter based nonlinear geometric control of a seeded batch cooling crystallizer

A nonlinear dynamic model of a seeded potash alum batch cooling crystallizer is presented. The model of the batch crystallizer is based on the conservation principles of mass, energy and population. In order to maintain constant supersaturation, a nonlinear geometric feedback controller is implemented. It is shown that compared to a natural and a simplified optimal cooling policies, the nonlinear geometric control (NCC) leads to a substantial improvement of the final crystal quality. An extended Kalman filter (EKF) is used as a closed loop observer for this nonlinear system to predict the non‐measurable state variables. It is found that the EKF is capable of effectively predicting the first four leading moments of the population density function. The effectiveness of the EKF based nonlinear geometric controller in the presence of plant/model mismatch is also studied. Simulation results show that the EKF based nonlinear geometric controller is reasonably robust in the presence of modeling error.     

Predictive control of particle size distribution in particulate processes

In this work, we focus on the development and application of predictive-based strategies for control of particle size distribution (PSD) in continuous and batch particulate processes described by population balance models (PBMs). The control algorithms are designed on the basis of reduced-order models, utilize measurements of principle moments of the PSD, and are tailored to address different control objectives for the continuous and batch processes. For continuous particulate processes, we develop a hybrid predictive control strategy to stabilize a continuous crystallizer at an open-loop unstable steady-state. The hybrid predictive control strategy employs logic-based switching between model predictive control (MPC) and a fall-back bounded controller with a well-defined stability region. The strategy is shown to provide a safety net for the implementation of MPC algorithms with guaranteed stability closed-loop region. For batch particulate processes, the control objective is to achieve a final PSD with desired characteristics subject to both manipulated input and product quality constraints. An optimization-based predictive control strategy that incorporates these constraints explicitly in the controller design is formulated and applied to a seeded batch crystallizer. The strategy is shown to be able to reduce the total volume of the fines by 13.4% compared to a linear cooling strategy, and is shown to be robust with respect to modeling errors.     

Modelling of crystallization process and optimization of the cooling strategy

To obtain a uniform and large crystal in seeded batch cooling crystallization, the cooling strategy is very important. In this study, an optimal cooling strategy is obtained through simulation and compared to linear and natural cooling strategies. A model for a crystallization process in a batch reactor is constructed by using population balance equation and material balance for solution concentration, and a prediction model for meta-stable limit is formulated by the dynamic meta-stable limit approach. Based on this model, an optimal cooling strategy is obtained using genetic algorithm with the objective function of minimizing the unwanted nucleation and maximizing the crystal growth rate. From the simulation results, the product from the optimal cooling strategy showed uniform and large crystal size distribution while products from the other two strategies contained significant amount of fine particles.     

Ammonium Sulfate Crystallization in a Cooling Batch Crystallizer

Abstract

Crystallization kinetics of ammonium sulfate crystals are determined in a 25-L draft tube baffled, agitated crystallizer from a series of batch cooling experiments performed in an integral mode. The method of s-plane analysis for relative nucleation kinetics and the method of initial derivatives for growth rate kinetics are used to establish conventional kinetic expressions. A simulation technique is developed to calculate the product population density functions for the same system in a seeded cooling batch crystallizer configuration.     

Optimal operation of batch enantiomer crystallization: From ternary diagrams to predictive control

In this work, the modeling and control of batch crystallization for racemic compound forming systems is addressed in a systematic fashion. Specifically, a batch crystallization process is considered for which the initial solution has been pre‐enriched in the desired enantiomer to enable crystallization of only the preferred enantiomer. A method for determining desired operating conditions (composition of the initial pre‐enriched solution and temperature to which the mixture must be cooled for maximum yield) for the batch crystallizer based on a ternary diagram for the enantiomer mixture in a solvent is described. Subsequently, it is shown that the information obtained from the ternary diagram, such as the maximum yield attainable from the process due to thermodynamics, can be used to formulate constraints for an optimization‐based control method to achieve desired product characteristics such as a desired yield. The proposed method is demonstrated for the batch crystallization of mandelic acid in a crystallizer with a fines trap that is seeded with crystals of the desired enantiomer. The process is controlled with an optimization‐based controller to minimize the ratio of the mass of crystals obtained from nuclei to the mass obtained from seeds while maintaining the desired enantioseparation. © 2017 American Institute of Chemical Engineers AIChE J, 64: 1618–1637, 2018     

A simplified approach to the operation of a batch crystallizer

A general model for a seeded cooling batch crystallizer based on first principles is developed, incorporating either size-dependent or size-independent growth rates. A simple approach is proposed to obtain temperature-time trajectories at constant supersaturation or nucleation rate, without resorting to optimization techniques. Cooling curves at constant supersaturation, which lead to a substantial improvement (a smaller coefficient of variation and a larger mean size) of the terminal crystal size distribution, can be determined even in the absence of precise nucleation and growth kinetics, whereas properties related to the crystal size distribution are sensitive to such kinetics. Experimental results for the potassium sulfate-water system, potash alum-water system, and hexamethylenetetramine in ethanol, methanol, and 2-propanol/water are predicted reasonably well by the model. Extension to any type of batch crystallization with super-saturation induced by means other than cooling, such as reactive precipitation and salting out, is briefly discussed.     

Handling sensor malfunctions in control of particulate processes

This work focuses on feedback control of particulate processes in the presence of sensor data losses. Two typical particulate process examples, a continuous crystallizer and a batch protein crystallizer, modeled by population balance models (PBMs), are considered. In the case of the continuous crystallizer, a Lyapunov-based nonlinear output feedback controller is first designed on the basis of an approximate moment model and is shown to stabilize an open-loop unstable steady-state of the PBM in the presence of input constraints. Then, the problem of modeling sensor data losses is investigated and the robustness of the nonlinear controller with respect to data losses is extensively investigated through simulations. In the case of the batch crystallizer, a predictive controller is first designed to obtain a desired crystal size distribution at the end of the batch while satisfying state and input constraints. Subsequently, we point out how the constraints in the predictive controller can be modified as a means of achieving constraint satisfaction in the closed-loop system in the presence of sensor data losses.     

Programmed cooling crystallization of potassium sulphate solutions

The crystal size distribution from a batch cooling crystallizer is predicted by the numerical solution of a mathematical model which uses empirical kinetics of nucleation and crystal growth. The predictions clearly point out the potential advantages of controlled cooling at a constant nucleation rate for improving the product crystal size over that obtained by either natural or linear cooling.Experimental runs following programmed cooling curves for seeded potassium sulphate solutions showed reasonable agreement with the theoretical predictions. A size dispersion of the crystals was observed which contributes to a slight deviation from theory. Nevertheless, controlled cooling significantly reduced the quantity of nuclei formed and improved the product crystal size distribution.     

Multivariable real‐time optimal control of a cooling and antisolvent semibatch crystallization process

This article presents an experimental study of simultaneous optimization with respect to two variables (cooling rate and flow‐rate of antisolvent) in an offline and online (real‐time) manner on a semibatch crystallizer. The nucleation and growth kinetic parameters of paracetamol in an isopropanol‐water cooling, antisolvent batch crystallizer were estimated by nonlinear regression in terms of the moments of the crystal population density. Moments of crystal population were estimated from the measured chord length distribution, generated by the FBRM®, and the supersaturation was calculated from the measured concentration by attenuated total reflectance‐fourier transform infrared spectroscopy. The results of real‐time optimization showed a substantial improvement of the end of batch properties (the volume‐weighted mean size and yield). For the same objective function, the real‐time optimization method resulted in an increase in the volume‐weighted mean size by ～100 μm and 15% of theoretical yield compared with the best result obtained in an offline optimization manner. © 2009 American Institute of Chemical Engineers AIChE J, 2009     

Optimal operation of enantioseparation by batch-wise preferential crystallization

This article describes batch-wise preferential crystallization separation of mixtures of L- and D-threonine. The use of online polarimetry combined with refractometry and microscopic investigation of the solid phase provides information on the crystallization kinetics. Results obtained for different crystallization conditions (supersaturation, temperature and enantiomeric excess) in a batch crystallizer are presented. Based on these results, a non-linear dynamic model has been developed. The control problem is to determine an optimal temperature profile which will result in a maximum amount of product with required quality. In this dynamic optimization problem B-splines have been used for interpolation of the temperature profile.     

Reactor modeling and recipe optimization of polyether polyol processes: Polypropylene glycol

The reactor modeling and recipe optimization of conventional semibatch polyether polyol processes, in particular for the polymerization of propylene oxide to make polypropylene glycol, is addressed. A rigorous mathematical reactor model is first developed to describe the dynamic behavior of the polymerization process based on first‐principles including the mass and population balances, reaction kinetics, and vapor‐liquid equilibria. Next, the obtained differential algebraic model is reformulated by applying a nullspace projection method that results in an equivalent dynamic system with better computational performance. The reactor model is validated against plant data by adjusting model parameters. A dynamic optimization problem is then formulated to optimize the process recipe, where the batch processing time is minimized, given a target product molecular weight as well as other requirements on product quality and process safety. The dynamic optimization problem is translated into a nonlinear program using the simultaneous collocation strategy and further solved with the interior point method to obtain the optimal control profiles. The case study result shows a good match between the model prediction and real plant data, and the optimization approach is able to significantly reduce the batch time by 47%, which indicates great potential for industrial applications. © 2013 American Institute of Chemical Engineers AIChE J, 59: 2515–2529, 2013     

Entropic analysis of a batch-crystallization process

The analysis of crystallization processes considering an entropic perspective is the primary purpose of this paper. Although the modeling of crystallization processes is well established in the literature, the entropic representation of these processes still needs to be studied. The modeling considering the second law of thermodynamics was investigated, which resulted in a model that represents the entropy production rate of a batch-crystallization process. The results indicate that the entropy production of the batch cooling crystallization is related to the variability of the crystal size distribution. The model of the entropy production rate could be used as a restriction criterion for multi-objective optimization of the cooling temperature profile of a batch crystallizer to improve the quality of the crystal size distribution of the final product.     

Nonlinear control of a batch crystallizer

A nonlinear geometric feedback controller with and without state estimation (the extended continuous-discrete Kalman filter) is developed and applied to a 0.027 m 3 potash alum batch crystallizer. The manipulated variable isthetemperature of the inlet cooling water supplied to the jacket of the crystallizer, and the controlled variable is the supersaturation. It is shown that the controller eliminates the large initial peak in the supersaturation (which results in excessive nucleation) and maintains the supersaturation at its set-point, provided that themanipulated variable does not reach its constraints. The controller performs well with only two measured states (the crystallizer temperature and the soluteconcentration) and results in larger terminal crystal mean size in com-parison with natural cooling and linear cooling policies with fines dissolution.     

NONLINEAR CONTROL OF A BATCH CRYSTALLIZER

A nonlinear geometric feedback controller with and without state estimation (the extended continuous-discrete Kalman filter) is developed and applied to a 0.027 m 3 potash alum batch crystallizer. The manipulated variable isthetemperature of the inlet cooling water supplied to the jacket of the crystallizer, and the controlled variable is the supersaturation. It is shown that the controller eliminates the large initial peak in the supersaturation (which results in excessive nucleation) and maintains the supersaturation at its set-point, provided that themanipulated variable does not reach its constraints. The controller performs well with only two measured states (the crystallizer temperature and the soluteconcentration) and results in larger terminal crystal mean size in com-parison with natural cooling and linear cooling policies with fines dissolution.