Particle shape manipulation and optimization in cooling crystallization involving multiple crystal morphological forms

A population balance model for predicting the dynamic evolution of crystal shape distribution is further developed to simulate crystallization processes in which multiple crystal morphological forms co‐exist and transitions between them can take place. The new model is applied to derive the optimal temperature and supersaturation profiles leading to the desired crystal shape distribution in cooling crystallization. Since tracking an optimum temperature or supersaturation trajectory can be easily implemented by manipulating the coolant flowrate in the reactor jacket, the proposed methodology provides a feasible closed‐loop mechanism for crystal shape tailoring and control. The methodology is demonstrated by applying it to a case study of seeded cooling crystallization of potash alum. © 2009 American Institute of Chemical Engineers AIChE J, 2009     

基于PBM的L-谷氨酸粒度分布控制优化

针对β型L-谷氨酸冷却结晶过程,为获取期望粒度分布,采用特征曲线法（MOCH）来建立关于粒度相关生长率的种群平衡方程（PBE）,然后通过对种群平衡模型（PBM）参数辨识后确定最优过饱和度及控温曲线。由于辨识模型参数的目标函数具有非线性和非凸型性,因而采用少量经济性的批量冷却结晶实验,结合图像分析晶种和产品粒度分布得到的统计数据,拟合模型参数。根据实际要求的结晶过程时间,为达到目标粒度分布,通过优化结晶过程的过饱和度获得最优调温曲线,实现基于恒定过饱和度的晶体生长过程优化控制。实验结果表明通过优化的控温曲线,实现了基于最优过饱和度控制的期望目标粒度分布。     

Optimization and control of crystal shape and size in protein crystallization process

Large molecule protein crystals have shown significant benefits in the delivery of biopharmaceuticals to achieve high stability, high concentration of active pharmaceutical ingredients (API), and controlled release of API. However, among the about 150 biopharmaceuticals on the market by 2004, only insulin has been marketed in crystalline form. A major technological challenge is that protein crystallization has a very complicated environment and is affected by many factors. There is currently a lack of knowledge on large scale production of protein crystals. In contrast to the majority of previous work on protein crystallization that was centered on single crystal scale, the current research is focused on computational study of protein crystallization at process scale, investigating the growth behavior of a population of crystals in a crystallizer. Using a newly developed morphological population balance model that can simulate the multidimensional size distributions of a population of crystals, known as shape distribution, an optimization technique is applied to optimize the growth of individual faces with the aim of obtaining desired crystal shape and size distributions. Using a target shape as the objective function, optimal temperature and supersaturation profiles leading to the desired crystal shape were derived. Genetic algorithm was investigated and found to be an effective optimization technique for the current application. Since tracking an optimum temperature or supersaturation trajectory can be easily implemented by manipulating the coolant flowrate in the reactor jacket, the methodology provides a feasible closed-loop mechanism for protein crystal shape tailoring and control.     

Iterative learning control for the systematic design of supersaturation controlled batch cooling crystallisation processes

The paper presents an approach to improve the product quality from batch-to-batch by exploiting the repetitive nature of batch processes to update the operating trajectories using process knowledge obtained from previous runs. The data based methodology is focused on using the linear time varying (LTV) perturbation model in an iterative learning control (ILC) framework to provide a convergent batch-to-batch improvement of the process performance indicator. The major contribution of this work is the development of a novel hierarchical ILC (HILC) scheme for systematic design of the supersaturation controller (SSC) of seeded batch cooling crystallizers. The HILC is used to determine the required supersaturation setpoint for the SSC and the corresponding temperature trajectory required to produce crystals with desired end-point property. The performance and robustness of these approaches are evaluated through simulation case studies. These results demonstrate the potential of the ILC approaches for controlling batch processes without rigorous process models.     

Robust model predictive control and fault handling of batch processes

This work considers the control of batch processes subject to input constraints and model uncertainty with the objective of achieving a desired product quality. First, a computationally efficient nonlinear robust Model Predictive Control (MPC) is designed. The robust MPC scheme uses robust reverse‐time reachability regions (RTRRs), which we define as the set of process states that can be driven to a desired neighborhood of the target end‐point subject to input constraints and model uncertainty. A multilevel optimization‐based algorithm to generate robust RTRRs for specified uncertainty bounds is presented. We then consider the problem of uncertain batch processes subject to finite duration faults in the control actuators. Using the robust RTRR‐based MPC as the main tool, a robust safe‐steering framework is developed to address the problem of how to operate the functioning inputs during the fault repair period to ensure that the desired end‐point neighborhood can be reached upon recovery of the full control effort. The applicability of the proposed robust RTRR‐based controller and safe‐steering framework subject to limited availability of measurements and sensor noise are illustrated using a fed‐batch reactor system. © 2010 American Institute of Chemical Engineers AIChE J, 2011     

A systematic framework for design of process monitoring and control (PAT) systems for crystallization processes

A generic computer-aided framework for systematic design of a process monitoring and control system for crystallization processes has been developed to study various aspects of crystallization operations. The systematic design framework contains a generic crystallizer modelling toolbox, a tool for generation of the supersaturation set-point for supersaturation control, as well as a tool for design of a process monitoring and control system (also called Process Analytical Technology (PAT) system). This systematic design allows one to generate the necessary problem-chemical system specific model, the necessary supersaturation set-point as well as a PAT system design including implementation of monitoring tools and control strategies in order to produce the desired target product properties notably crystal size distribution (CSD) and shape for a wide range of crystallization processes. Application of the framework is highlighted through a case study involving the design of a monitoring and control system for a potassium dihydrogen phosphate (KDP) crystallization process, where also the one-dimensional CSD and two-dimensional CSD modelling features are highlighted.     

On the prediction of crystal shape distributions in a steady-state continuous crystallizer

Zhang and Doherty [2004. Simultaneous prediction of crystal shape and size for solution crystallization. A.I.Ch.E. Journal 50, 2101-2112] have provided a one-dimensional analysis of crystallization based on the assumption that the relative face-specific growth rates of a (2-D) crystal are independent of supersaturation and hence invariant with time. Subsequent work by these authors [Zhang, Y., Sizemore, J.P., Doherty, M.F., 2006. Shape evolution of 3-dimensional faceted crystals. A.I.Ch.E. Journal 52, 1906-1915) consider shape evolution of single three-dimensional crystals with morphological changes. In this work, we present a multidimensional population balance approach accounting for dependence of the relative face-specific growth rates on supersaturation, a situation more commonly encountered. For example, Joshi and Paul [1974. Effect of supersaturation and fluid shear on habit and homogeneity of potassium dihydrogen phosphate crystals. Journal of Crystal Growth 22, 321-327] and Mullin and Whiting [1980. Succinic acid crystal-growth rates in aqueous solution. Industrial & Engineering Chemistry Fundamentals 19, 117-121] report face-specific growth rates with different dependence on the supersaturation. Thus it has been observed that there exists significantly different crystal shapes in a crystallizer [Yang, G., Kubota, N., Sha, Z., Louhi-Kultanen, M. Wang, J., 2006. Crystal shape control by manipulating supersaturation in batch cooling crystallization. Crystal Growth and Design 6, 2799-2803]. Consequently, the population of crystals at any instant will have widely varying crystal shapes and sizes depending upon the initial crystal shape and size distribution. Computations are presented for the shape distributions of the crystal population emerging from a steady-state continuous crystallizer for two cases: (1) feed without crystals including nucleation for the formation of new crystals, and (2) feed with seed crystals of known shape, with suppressed nucleation. In the range of mean residence times investigated, the calculated crystal volume distributions for the first case show geometrically dissimilar shapes without morphological variations. However, in the second case, because the feed crystals of the chosen shape were susceptible to morphological changes, the volume distributions display this feature with shape and size distributions for each of a number of different morphologies. By varying operating conditions such as the flow rate, the inlet supersaturation, and the shapes of feed crystals, the proposed model can clearly be used to manipulate the crystal shape and size distributions and their morphologies.     

Comparison of different solubility equations for modeling in cooling crystallization

In this paper we evaluate the sensitivity of using different solubility models on cooling crystallization. Specifically, the cooling crystallization of acetaminophen in ethanol is investigated. Empirical, correlative thermodynamic (namely van Laar, Wilson, and NRTL) and predictive thermodynamic (namely MOSCED, NRTL-SAC, and UNIFAC) models are considered. Equilibrium solubility model prediction determines the predicted supersaturation profile. The different solubility equations are used within a population balance model for prediction of crystal size properties. Incorrect prediction of the supersaturation profile results in incorrect prediction of crystal size distribution. The NRTL model was found to be more accurate at predicting equilibrium solubility and consequently crystal size. After the solubility sensitivity is evaluated, two methods are proposed to make the crystallization model more robust against solubility model errors.     

Measurement based modeling and control of bimodal particle size distribution in batch emulsion polymerization

In this article, a novel modeling approach is proposed for bimodal Particle Size Distribution (PSD) control in batch emulsion polymerization. The modeling approach is based on a behavioral model structure that captures the dynamics of PSD. The parameters of the resulting model can be easily identified using a limited number of experiments. The resulting model can then be incorporated in a simple learning scheme to produce a desired bimodal PSD while compensating for model mismatch and/or physical parameters variations using very simple updating rules. © 2010 American Institute of Chemical Engineers AIChE J, 2010     

单分散纳米硫酸锶粉体的制备

利用过饱和溶液存在介稳区且其宽度可调节的原理,使反应物在沉淀形成前充分微混,制备出纳米硫酸锶粉体.分析了反应条件的改变对产品粒径和粒度分布的影响,利用正交试验确定适宜的工艺条件.产品用TEM及X射线粉末衍射进行表征,并测定其粒度分布.结果表明：通过改变溶液介稳区宽度可以制备粒度分布窄、晶型一致的纳米硫酸锶粉体.并从唯象层面上推测了向反应体系中加入EDTA和乙醇的作用机理.     

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.     

Modeling and control of protein crystal shape and size in batch crystallization

In this work, the modeling and control of a batch crystallization process used to produce tetragonal hen egg white lysozyme crystals are studied. Two processes are considered, crystal nucleation and growth. Crystal nucleation rates are obtained from previous experiments. The growth of each crystal progresses via kinetic Monte Carlo simulations comprising of adsorption, desorption, and migration on the (110) and (101) faces. The expressions of the rate equations are similar to Durbin and Feher. To control the nucleation and growth of the protein crystals and produce a crystal population with desired shape and size, a model predictive control (MPC) strategy is implemented. Specifically, the steady‐state growth rates for the (110) and (101) faces are computed and their ratio is expressed in terms of the temperature and protein concentration via a nonlinear algebraic equation. The MPC method is shown to successfully regulate both the crystal size and shape distributions to different set‐point values. © 2013 American Institute of Chemical Engineers AIChE J, 59: 2317–2327, 2013     

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.     

Crystallization of high purity ammonium meta-tungstate for production of ultrapure tungsten metal

The growth mechanism of Ammonium Meta-Tungstate (AMT) crystal was interpreted as two-step model. Growth rates of AMT crystals were measured in a fluidized bed crystallizer. The effects of temperature, supersaturation and crystal size on the crystal growth were investigated. The contribution of the diffusion step increased with the increase of temperature, crystal size and supersaturation. The nucleation kinetics from measurements on the width of the metastable region of Ammonium Meta-Tungstate (AMT) was also evaluated. The crystal size distribution from a programmed cooling crystallization system was predicted by the numerical simulation of a mathematical model using the kinetics of nucleation and crystal growth. It was also observed that the shape of AMT crystals was changed during the growth period. This paper was presented at The 5th International Symposium on Separation Technology-Korea and Japan held at Seoul between August 19 and 21, 1999.     

On-Line control of supersaturation in a continuous cooling KCL crystallizer

Control of supersaturation in a 1-L continuous cooling KC1 crystallizer was investigated. The supersaturation was determined from on-line measurements of the density and temperature of clear liquor samples. A cascade control scheme was used to control the supersaturation through the manipulation of the co-saturated feed temperature set-point while maintaining the crystallizer temperature at 303.2 K. Experimental results showed that due to the suppression of spontaneous nuclearion, a decrease in the measured supersaturation resulted in a 23% increase in the mean crystal size and a 12% decrease in the amount of NaCl impurity in the KCI crystals.     

On-line determination of supersaturation of a kcl-nacl aqueous solution based on density measurement

A method is developed for the calculation of the saturation temperature of a KCI-NaCI aqueous solution, based on the measurement of the density and temperature of a sample solution, and prior knowledge of the NaCl concentration. Experimental density and solubility data for solutions saturated with KCI in the temperature range of 299 to 321 K with concentrations of NaCl greater than 0.200 kg/kg H₂O were used to develop the empirical correlation allowing the calculation of the solution saturation level. The method is applicable in the on-line determination of the level of supersaturation in a KCI crystallizer in which the NaCl concentration is known. Knowledge of the prevailing supersaturation is necessary for the control of crystal purity and crystal size distribution in industrial potash crystallizers.     

NONLINEAR MODEL PREDICTIVE CONTROL

Nonlinear Model Predictive Control (NMPC), a strategy for constrained, feedback control of nonlinear processes, has been developed. The algorithm uses a simultaneous solution and optimization approach to determine the open-loop optimal manipulated variable trajectory at each sampling instant. Feedback is incorporated via an estimator, which uses process measurements to infer unmeasured state and disturbance values. These are used by the controller to determine the future optimal control policy. This scheme can be used to control processes described by different kinds of models, such as nonlinear ordinary differential/algebraic equations, partial differential/algebraic equations, integra-differential equations and delay equations. The advantages of the proposed NMPC scheme are demonstrated with the start-up of a non-isothermal, non-adiabatic CSTR with an irreversible, first-order reaction. The set-point corresponds to an open-loop unstable steady state. Comparisons have been made with controllers designed using (1) nonlinear variable transformations, (2) a linear controller tuned using the internal model control approach, and (3) open-loop optimal control. NMPC was able to bring the controlled variable to its set-point quickly and smoothly from a wide variety of initial conditions. Unlike the other controllers, NMPC dealt with constraints in an explicit manner without any degradation in the quality of control. NMPC also demonstrated superior performance in the presence of a moderate amount of error in the model parameters, and the process was brought to its set-point without steady-state offset.     

Optimal seed recipe design for crystal size distribution control for batch cooling crystallisation processes

The paper presents a novel quality-by-design framework for the design of optimal seed recipes for batch cooling crystallisation systems with the aim to produce a desired target crystal size distribution (CSD) of the product. The approach is based on a population balance model-based optimal control framework, which optimises the compositions of seed blends, based on seed fractions that result from standard sieve analysis. The population balance model is solved using a combined quadrature method of moments and method of characteristics (QMOM-MOCH) approach for the generic case of apparent size-dependent growth. Seed mixtures are represented as a sum of Gaussian distributions, where each Gaussian corresponds to the seed distribution in a particular sieve size range. The proposed methods are exemplified for the model system of potassium dichromate in water, for which the apparent size-dependent growth kinetic parameters have been identified from laboratory experiments. The paper also illustrates the simultaneous application of in situ process analytical tools, such as focused beam reflectance measurement (FBRM) for nucleation detection, attenuated total reflection (ATR) UV/Vis spectroscopy for concentration monitoring, as well as the in-line use of laser diffraction particle sizing for real-time CSD measurement.     

Numerical investigation of the influence of kinetics and shape factor on barium sulfate precipitation in a continuous stirred tank

The effect of kinetics and shape factor on barium sulfate precipitation in a continuous stirred tank has been investigated numerically through solving the standard momentum and mass transport equations in combination with the moment equations for crystal population balance. The numerical method was validated with the literature data. The simulated results include the distribution of the local supersaturation ratio in the reactor, the mean crystal size, and the coefficient of variation. The simulation results show that the value of shape factor used in the model affected greatly the mean crystal size and the moments of the crystal size distribution. The influence of the kinetic expressions on the simulation is also analyzed. It is important to investigate the relationship of the shape factor with the precipitator type and other operation conditions to obtain reliable simulation results and suitable kinetic equations of crystal nucleation and growth rates.     

On‐line control of crystal properties in nonisothermal antisolvent crystallization

The issues regarding the design and implementation of on‐line optimal control strategies of crystal properties in nonisothermal antisolvent crystallization processes to control particles’ mean size and standard deviation are dealt. The one‐dimensional Fokker–Planck equation is used to represent the dynamic characteristics of the crystal growth and generate iso‐mean and iso‐standard deviation curves. Using controllability tools it is demonstrated that the system is ill conditioned in the whole operational range, posing limitations on the achievable control performance. To circumvent the problem, a control strategy is formulated by pairing crystals’ mean size with antisolvent feed rate and manipulating temperature to control the standard deviation. A novel digital image‐texturing analysis approach is discussed and implemented to track crystals’ size distribution along the experiment and providing the on‐line information for further feedback control action. Subsequently, alternative control strategies are implemented and tested to achieve a desired crystal size distribution. © 2015 American Institute of Chemical Engineers AIChE J, 61: 2188–2201, 2015