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.     

Optimization and nonlinear control of a batch crystallization process

Crystallization process has been widely used for separation in many chemical industries due to its capability to provide high purity product. To obtain the desired quality of crystal product, an optimal cooling control strategy is studied in the present work. Within the proposed control strategy, a dynamic optimization is first preformed with the objective to obtain the optimal cooling temperature policy of a batch crystallizer, maximizing the total volume of seeded crystals. Two different optimization problems are formulated and solved by using a sequential optimization approach. Owing to the complex and nonlinear behavior of the batch crystallizer, the nonlinear control strategy which is based on a generic model control (GMC) algorithm is implemented to track the resulting optimal temperature profile. The optimization integrated with nonlinear control strategy is demonstrated on a seeded batch crystallizer for the production of potassium sulfate.     

Population balance modelling and H∞—controller design for a crystallization process

In this paper we consider the robustly stabilizing control of a 1000l draft tube baffled crystallizer. When operated at high fines dissolution rates, the crystallizer exhibits sustained oscillations. A detailed population balance model for the process can be found in the literature. Based on this detailed model we develop a simpler population balance model. This, in turn, permits the derivation of an irrational transfer function from manipulated to measured variable. An H_∞ mixed sensitivity minimization problem is formulated and solved using an infinite-dimensional version of H_∞ theory. Two different controllers are designed and compared in simulation studies.     

A model-based control framework for industrial batch crystallization processes

Dynamic optimization is applied for throughput maximization of a semi-industrial batch crystallization process. The control strategy is based on a non-linear moment model. The dynamic model, consisting of a set of differential and algebraic equations, is optimized using the simultaneous optimization approach in which all the state and input trajectories are parameterized. The resulting problem is subsequently solved by a non-linear programming algorithm.     

An efficient numerical technique for solving one-dimensional batch crystallization models with size-dependent growth rates

In this work, an efficient numerical method is introduced for solving one-dimensional batch crystallization models with size-dependent growth rates. The proposed method consist of two parts. In the first part, a coupled system of ordinary differential equations (ODEs) for the moments and the solute concentration is numerically solved to obtain their discrete values in the time domain of interest. These discrete values are also used to get growth and nucleation rates in the same time domain. To overcome the issue of closure, a Gaussian quadrature method based on orthogonal polynomials is employed for approximating integrals appearing in the ODE system. In the second part, the discrete growth and nucleation rates along with the initial crystal size distribution (CSD) are used to construct the final CSD. The expression for CSD is obtained by applying the method of characteristics and Duhamel's principle on the given population balance model (PBM). The proposed method is efficient, accurate, and easy to implement in the computer. Several numerical test problems of batch crystallization processes are considered. For a validation, the results of the proposed technique are compared with those obtained using a high resolution finite volume scheme.     

Dynamic modeling of a batch crystallization process: A stochastic approach for agglomeration and attrition process

This paper deals with the dynamic modeling of a batch crystallizer. A complete model taking into account primary and secondary nucleations, crystal growth, agglomeration and attrition mechanisms is established. The proposed model is not restricted to binary agglomeration and breakage phenomena. From markovian considerations, continuous kernel functions are built and the basic balance equations are then presented. The complete model is solved using a finite difference method for the discretization of the size variable. As to distinguish agglomeration and breakage parameters from the others, on line measurement of the Crystal Size Distribution is necessary, a new on line measurement strategy is proposed. Finally, simulations of the crystal size distribution are compared with experimental results at different times. It appears that simulated curves are in good agreement with the experimental data.     

Prediction of the solubility of 6APA in aqueous phase and optimum control scheme for batch crystallization process through pH variation

Solubilities of 6APA in water at different pH and temperatures are measured. A three parameter model is proposed for predicting the aqueous solubility of this amino acid. Predicted values are in good agreement with the experimental data. The model may be used for quick and accurate evaluation of the aqueous solubility of other simple ampholytes. The proposed correlation is combined with the population balance and kinetic equations for predicting the optimum pH profiles required to maintain a constant supersaturation in a batch crystallizer. The suggested policy leads to a more uniform crystal size distribution for 6APA with C.V of 23%.     

Approximate ODE models for population balance systems

We propose an approximate polynomial method of moments for a class of first-order linear PDEs (partial differential equations) of hyperbolic type, involving a filtering term with applications to population balance systems with fines removal terms. The resulting closed system of ODEs (ordinary differential equations) represents an extension to a recently published method of moments which utilizes least-square approximations of factors of the PDE over orthogonal polynomial bases. An extensive numerical analysis has been carried out for proof-of-concept purposes. The proposed modeling scheme is generally of interest for control and optimization of processes with distributed parameters.     

Integrated batch‐to‐batch and nonlinear model predictive control for polymorphic transformation in pharmaceutical crystallization

Polymorphism, a phenomenon in which a substance can have more than one crystal form, is a frequently encountered phenomenon in pharmaceutical compounds. Different polymorphs can have very different physical properties such as crystal shape, solubility, hardness, color, melting point, and chemical reactivity, so that it is important to ensure consistent production of the desired polymorph. In this study, an integrated batch‐to‐batch and nonlinear model predictive control (B2B‐NMPC) strategy based on a hybrid model is developed for the polymorphic transformation of L ‐glutamic acid from the metastable α‐form to the stable β‐form crystals. The hybrid model comprising of a nominal first‐principles model and a correction factor based on an updated PLS model is used to predict the process variables and final product quality. At each sampling instance during a batch, extended predictive self‐adaptive control (EPSAC) is employed as a NMPC technique to calculate the control action by using the current hybrid model as a predictor. At the end of the batch, the PLS model is updated by utilizing the measurements from the batch and the above procedure is repeated to obtain new control actions for the next batch. In a simulation study using a previously reported model for a polymorphic crystallization with experimentally determined parameters, the proposed B2B‐NMPC control strategy produces better performance, where it satisfies all the state constraints and produces faster and smoother convergence, than the standard batch‐to‐batch strategy. © 2010 American Institute of Chemical Engineers AIChE J, 2011     

A dynamic programming based approach for explicit model predictive control of hybrid systems

This work presents an algorithm for explicit model predictive control of hybrid systems based on recent developments in constrained dynamic programming and multi-parametric programming. By using the proposed approach, suitable for problems with linear cost function, the original model predictive control formulation is disassembled into a set of smaller problems, which can be efficiently solved using multi-parametric mixed-integer programming algorithms. It is also shown how the methodology is applied in the context of explicit robust model predictive control of hybrid systems, where model uncertainty is taken into account. The proposed developments are demonstrated through a numerical example where the methodology is applied to the optimal control of a piece-wise affine system with linear cost function.     

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     

Improving scenario decomposition algorithms for robust nonlinear model predictive control

This paper deals with the efficient computation of solutions of robust nonlinear model predictive control problems that are formulated using multi-stage stochastic programming via the generation of a scenario tree. Such a formulation makes it possible to consider explicitly the concept of recourse, which is inherent to any receding horizon approach, but it results in large-scale optimization problems. One possibility to solve these problems in an efficient manner is to decompose the large-scale optimization problem into several subproblems that are iteratively modified and repeatedly solved until a solution to the original problem is achieved. In this paper we review the most common methods used for such decomposition and apply them to solve robust nonlinear model predictive control problems in a distributed fashion. We also propose a novel method to reduce the number of iterations of the coordination algorithm needed for the decomposition methods to converge. The performance of the different approaches is evaluated in extensive simulation studies of two nonlinear case studies.     

A control oriented study on the numerical solution of the population balance equation for crystallization processes

The population balance equation provides a well established mathematical framework for dynamic modeling of numerous particulate processes. Numerical solution of the population balance equation is often complicated due to the occurrence of steep moving fronts and/or sharp discontinuities. This study aims to give a comprehensive analysis of the most widely used population balance solution methods, namely the method of characteristics, the finite volume methods and the finite element methods, in terms of the performance requirements essential for on-line control applications. The numerical techniques are used to solve the dynamic population balance equation of various test problems as well as industrial crystallization processes undergoing simultaneous nucleation and growth. The time-varying supersaturation profiles in the latter real-life case studies provide more realistic scenarios to identify the advantages and pitfalls of a particular numerical technique.The simulation results demonstrate that the method of characteristics gives the most accurate numerical predictions, whereas high computational burden limits its use for complex real crystallization processes. It is shown that the high order finite volume methods in combination with flux limiting functions are well capable of capturing sharp discontinuities and steep moving fronts at a reasonable computational cost, which facilitates their use for on-line control applications. The finite element methods, namely the orthogonal collocation and the Galerkin's techniques, on the other hand may severely suffer from numerical problems. This shortcoming, in addition to their complex implementation and low computational efficiency, makes the finite element methods less appealing for the intended application.     

A unified framework for detection, isolation and compensation of actuator faults in uncertain particulate processes

This paper presents a methodology for the robust detection, isolation and compensation of control actuator faults in particulate processes described by population balance models with control constraints and time-varying uncertain variables. The main idea is to shape the fault-free closed-loop process response via robust feedback control in a way that enables the derivation of performance-based fault detection and isolation (FDI) rules that are less sensitive to the uncertainty. Initially, an approximate finite-dimensional system that captures the dominant process dynamics is derived and decomposed into interconnected subsystems with each subsystem directly influenced by a single manipulated input. The decomposition is facilitated by the specific structure of the process input operator. A robustly stabilizing bounded feedback controller is then designed for each subsystem to enforce an arbitrary degree of asymptotic attenuation of the effect of the uncertainty in the absence of faults. The synthesis leads to (1) an explicit characterization of the fault-free behavior of each subsystem in terms of a time-varying bound on an appropriate Lyapunov function and (2) an explicit characterization of the robust stability region in terms of the control constraints and the size of the uncertainty. Using the fault-free Lyapunov dissipation bounds as thresholds for FDI in each subsystem, the detection and isolation of faults in a given actuator is accomplished by monitoring the evolution of the system within the stability region and declaring a fault if the threshold is breached. The thresholds are linked to the achievable degree of asymptotic uncertainty attenuation and can therefore be properly tuned by proper tuning of the controllers, thus making the FDI criteria less sensitive to the uncertainty. The robust FDI scheme is integrated with a robust stability-based controller reconfiguration strategy that preserves closed-loop stability following FDI. Finally, the implementation of the fault-tolerant control architecture on the particulate process is discussed and the proposed methodology is applied to the problem of robust fault-tolerant control of a continuous crystallizer with a fines trap.     

Multi-scale product property model of polypropylene produced in a FBR: From chemical process engineering to product engineering

A multi-scale product model has been built to characterize the polypropylene (PP) formation dynamics in a catalytic FBR. For the first time, the gas–solid flow field, the morphological and molecular properties of particles, as well as their dynamics can be simultaneously obtained by solving the unique model that couples a CFD model, a population balance model (PBM) and moment equations. The quantitative relationships between the operating conditions and the multi-scale particle properties have been further established. The results demonstrate that the product model can be used to guide a multi-scale generalization of the polymer product from chemical process to product engineering.     

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.