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.     

Optimization of batch cooling crystallization

A novel framework has been proposed for optimizing batch cooling crystallization operation. The optimization strategy is effective for highly non-linear and dynamic mathematical systems such as batch cooling crystallization systems. The proposed framework can implement constrained multivariable optimization without the tendency of being trapped into local optima. Variables that can be optimized simultaneously include the cooling profile, seed size, mass of seeds, batch time, initial and final temperatures. Optimizations have been implemented on both seeded and unseeded processes of the citric acid-water system. The optimized operation leads to improvements that otherwise cannot be attained by using conventional approaches. Optimization results also suggest that seeding can increase the flexibility of the operation.     

Modeling induced nucleation processes during batch cooling crystallization: A sequential parameter determination procedure

An existing model is extended to simulate batch cooling crystallizations with induced nucleation processes like ultrasound or gassing. All important phenomena such as nucleation, growth, agglomeration and breakage are taken into account. A differentiation between ultrasound and gassing is necessary. Induced nucleation processes also require a modification of crystal growth mechanism.In general, the model parameters required for the kinetics are determined simultaneously by fitting them to experimental data. Mostly a correlation model results without physical basis. Here, the model parameters are sequentially determined by decoupling the mechanisms, saving effort and time, and make it possible to reduce the number of parameters also. The model and the model parameter determination procedure are validated using three different organic solute/solvent systems. The number of simulation runs for parameter fitting was reduced to less than 100 instead of simultaneous parameter determination, which requires several hundreds of thousands runs, resulting in physically reasonable solutions.     

Application of controlled cooling and seeding in batch crystallization

Some aspects of batch cooling crystallization in the industrial practice are analysed by computer simulations. The results indicate that without appropriate kinetics and very accurate process control, even the qualitative effects of applying controlled cooling and seeding are highly unpredictable. An increase in product size by applying controlled cooling is likely to be successful only rather randomly and the size distribution becomes broader. A linear or weakly non-linear cooling curve usually produces larger crystals than a natural cooling curve, and a better reproducibility than a controlled cooling curve. Seeding increases both size and reproducibility, but is also likely to increase significantly the coefficient of variation of the product distribution. The product weight mean size and coefficient of variation may increase or decrease at increasing amount of seeds, depending on governing kinetics.     

A study on crystallization kinetics of pentaerythritol in a batch cooling crystallizer

The crystallization kinetics of pentaerythritol (PeE) in aqueous solution in the presence of impurity or not in a batch cooling crystallizer was explored. Also, the solubility and the nucleation and crystal growth kinetics of PeE in aqueous solution were investigated. A second-order dependence of PeE growth rate on supersaturation is observed in pure PeE-water system. The crystal growth rate of PeE-water system in the presence of impurity is proportional to the supersaturation to the 3.5 power. The nucleation and crystal growth behaviors for PeE-water system in a batch cooling crystallizer were grasped according to Mersmann's criteria. The nucleation in this crystallizer was found to act with heterogeneous nucleation. In this system, it suggests that the crystal growth is controlled by a complex mechanism behavior of surface integrated and diffusion limited. Simplified relation was derived for calculation of mean crystal size of product crystals from the batch cooling crystallizer. The obtained relation was verified by a set of experiments.     

Dynamic modelling and scheduling of an industrial batch system

This paper addresses the optimal schedule of a resource constrained four-batch digester system of an industrial acid sulphite pulp mill. This involves the development of two different models, one to model the scheduling operational problem and the other the batch digester operation—process model. The first model uses a discrete-time Resource Task Network (RTN) based representation leading to a Mixed Integer Linear Program (MILP) formulation. In this, the main operational limitation, steam availability, is modelled through the definition of a superstructure including the most relevant heating alternatives. The duration of the generated heating tasks was estimated through the use of the process model—a distributed heterogeneous dynamic model in gPROMS, with a heat-transfer resistance at the solid–fluid-interface—validated with experimental plant data. The optimal schedules obtained showed that an increase in the total available steam from the boiler is vital to allow a higher level of productivity and that not much can be done regarding steam sharing improvements.     

On the nonisothermal melt crystallization kinetics of industrial batch crosslinked polyethylene

The chemical modification of commodity polymers such as polyethylene (PE) is a versatile synthetic approach for preparing materials that cannot be manufactured cost-effectively using conventional polymerization techniques. Aiming to improve PE character low contents of dicumyl peroxide (DCP), from 0% to 1.5% was added as crosslinker to an industrial batch (PEs mixture and additives). From tensile testing crosslinking provided higher elastic modulus most due to the restrained microstructure where XPEs macromolecular chains are interconnected also providing lower strain at break. Crosslinking effects on the nonisothermal melt crystallization rate (Cmax) and degree of crystallinity (Xc) were evaluated; Cmax increased with the cooling rates, whereas Xc increased upon DCP addition. The melt crystallization kinetics were thoroughly investigated applying Pseudo-Avrami, Ozawa, and Mo models. Ozawa failed to describe the crystallization most due to ignore the secondary crystallization and spherulites impingement at the end of crystallization while Pseudo-Avrami and Mo provided quite good fits. The activation energy was computed using Arrhenius' approach, crosslinked compounds presented higher energy consumption, whereas exception was verified for 0.5XPE which displayed the lowest energy and overall the best mechanical performance this is the most proper compound for industrial applications, such as packaging, and disposables as well as general goods.     

Novel strategies for optimization of batch, semi-batch and heating/cooling evaporative crystallization

A novel optimization framework based on a stochastic optimization algorithm is proposed for optimizing batch, semi-batch and non-isothermal evaporative crystallizations. The mathematical models for these dynamic processes are highly non-linear and hence involve prominent non-convexities in the solution space of optimization. The framework not only significantly reduces the likelihood of being trapped into local optima and convergence problems in optimization and simulations, but also employs a novel and efficient algorithm for optimization of control profiles. A new concept in combining heating/cooling and evaporation in a batch crystallization operation is also proposed in this work. Optimization results of the citric acid-water system using this concept indicate that the maximum average crystal size obtainable is 19% and 27% larger than those attainable in the optimized unseeded evaporative and cooling operations respectively. The crystal size distribution is significantly narrower than that under seeded operations. Yield increase of more than 42% is also observed in this operating methodology. Optimization results for the batch and semi-batch evaporative crystallizations also demonstrate improvements that otherwise cannot be attained by using conventional approaches and heuristic rules.     

Evaluation of optimisation techniques and control variable formulations for a batch cooling crystallization process

The mathematical optimisation of a batch cooling crystallization process is considered in this work. The objective is to minimize the standard deviation of the final crystal size distribution (CSD), which is an important feature in many industrial processes. The results with the problem written as a nonlinear programming and solved with the successive quadratic programming (SQP) coupled with the discretization of the control variable are compared with those obtained when SQP coupled with the parameterisation of the control variable is applied. Also it is proposed the implementation of the genetic algorithm (GA) coupled with parameterisation of the control variable. Extensive evaluations show that the SQP method is sensitive both to the parameterisation formulation and to the initial estimate. The solution with GA provided the control variable profile that leads to the minimum standard deviation of the final CSD. Nevertheless, it is a very time-consuming technique, which hampers its utilization in real time applications. However, its feature of global searching suggests its suitability in solving offline problems, in order to provide initial setup profiles. Bearing this in mind, it is proposed an algorithm which allows for the implementation of GA solution in a real time fashion, taking advantage of its robustness to find out the optimal solution.     

Determination of kinetics in batch cooling crystallization processes—A sequential parameter estimation approach

A comprehensive methodology to carry out a sequential parameter estimation approach has been developed and validated for the determination of the kinetic parameters of the crystallization of a generic organic compound. The strength of the approach lies in the thorough design of isothermal experiments which facilitate the isolation and/or decoupling of the different crystallization phenomena. This methodology has been applied for the parameter estimation of primary and secondary nucleation, growth and agglomeration kinetics. The resulting crystallization model has been able to reproduce the quantiles , and of the volume‐based particle size distribution of an independent seeded validation experiment with an error below 10 μm. The deviation in the prediction has been increased in the case of an independent unseeded experiment, although errors below the uncertainty of the measurement have been always obtained. The methodology here proposed is intended to be an efficient strategy for rapid modeling of batch crystallization processes. © 2016 American Institute of Chemical Engineers AIChE J, 62: 3992–4012, 2016     

Impeller geometry effect on crystallization kinetics of borax decahydrate in a batch cooling crystallizer

The effects of impeller type and diameter in a batch cooling crystallizer on the nucleation and crystal growth kinetics as well as on the shape and size distribution of borax decahydrate crystals were investigated. Two different types of impellers of various sizes were applied. Chosen impeller configurations generate completely different fluid flow patterns in the crystallizer what allows to investigate the influence of the axial and radial flow on the kinetic parameters as well. The nucleation in crystallizer was taking place by the heterogeneous nucleation mechanism at all mixing conditions. The number of crystals formed by this mechanism increases as ratio D/d_T decreases and it is higher when an axial flow pattern in crystallizer has been developed. The crystal growth rate increases with increasing the impeller size in observed supersaturation range. The radial impeller defined by ratio D₂/d_T = 0.58 could be considered as viable option for growth of borax crystal, since the further enlargement of this ratio does not increase growth rate and can only cause higher power consumption. The maxima in the coarser and finer fractions of CSD indicate a different influence of mixing conditions on the crystal grow and secondary nucleation. An axial flow pattern in crystallizer favors agglomeration of growing crystals increasing that way product mean crystal size, while radial flow results with more regular shape of borax crystals.     

Effect of cooling mode on product crystal size in seeded batch crystallization of potassium alum

Potassium alum was crystallized by seeding in a batch crystallizer under controlled and natural cooling modes. Regardless of the cooling mode, the product crystal size distribution (CSD) became bi-modal at low seed concentrations because of enormous secondary nucleation. The mean mass size of the product was smaller for the natural cooling mode compared to that for the controlled cooling mode with more intensive secondary nucleation. On the other hand, at high seed concentrations, the product CSD became uni-modal with the same mean mass size for both cooling modes, where the crystallization was dominated by seed growth. The low supersaturation caused by the growth of enough seeds plays a key role to produce uni-modal size distribution with suppressed nucleation. Adhering of small crystals (secondary nuclei) to growing seed crystals is also considered to be another mechanism for generating uni-modal CSD.     

氯化钾间歇冷却结晶生长规律及粒度控制实验研究

采用聚焦光束反射测量技术（FBRM）考察了氯化钾（KCl）间歇冷却结晶过程中晶体成核和生长规律,重点研究了降温速率对KCl水溶液冷却时产生过饱和度的影响,以及添加晶种的相关条件（如晶种粒径和添加量等因素）与KCl晶体产品粒度的关系。同时,采用直接冷却刺激起晶产生“晶种”,并控制其生长达到控制晶体产品粒度的目的。结果表明,在添加晶种条件下,程序降温过程产生的低过饱和度不易引起爆发成核,且晶种的添加量决定着晶体产品的平均粒度与理想生长模型的偏差。另外,降温速率是冷却刺激起晶产生“晶种”粒径的关键因素。     

The best objective function for seeded batch crystallization

Although many researchers have optimized the operation of seeded batch crystallization, there remains little understanding of the effect of the choice of objective function on the optimization and no consensus on which objective function is best. In this work, different objective functions for seeded batch crystallization are compared by calculating the supersaturation trajectory that minimizes each objective function and then comparing the results of each trajectory. We show that some objective functions lead to trajectories that cause excessive nucleated mass and should be avoided. We conclude that the best objective function in most cases is to minimize the nucleated mass. This also corresponds to maximizing the growth of seeds, and leads to a product with a large volume mean size. We also show that the width of the product crystal size distribution can be controlled much more effectively by manipulating seed properties than by adjusting the supersaturation trajectory. © 2012 American Institute of Chemical Engineers AIChE J, 59: 390–398, 2013     

Industrial batch crystallization of a plate-like organic product. In situ monitoring and 2D-CSD modelling. Part 2: Kinetic modelling and identification

The batch seeded cooling solution crystallization of a fine organic material, exhibiting a platelet-like habit, was investigated and a model of the time variations of the crystal size distribution (CSD) was designed using two-dimensional population balance equations. Activated surface secondary nucleation and attrition secondary nucleation mechanisms were considered, coupled with growth mechanisms of two main dimensions of the crystal, resulting in a set of eight kinetic parameters. The model relates the effects of the main batch operating conditions: seeding temperature, cooling rate and total area of the seed particles, on both the supersaturation profile and bi-dimensional CSD. Surface secondary nucleation occurs first since it is promoted by the introduction of seeds and remains active as long as the relative supersaturation exceeds a threshold value of about 16%. It vanishes below which could be expected as we deal with an activated mechanism. Contact secondary nucleation occurs later when the concentration of solid is sufficient. It is spread over time until supersaturation disappears at the end of the batch process. This contact secondary mechanism is assumed to be the dominant nucleation mechanism as it generates about two-thirds of the final crystal number. Sharp desupersaturation profile following the introduction of seeds, which was observed experimentally, is shown to be quantitatively described through the growth of seed particles. The termination of the batch process is more difficult to represent. Due to crystal attrition, distinct growth rates between initial and secondary crystals or growth rate dispersion might explain such difficulty.     

间歇结晶过程中几种可能的粒度分布

对间歇溶液结晶过程,依据粒数衡算方程,在忽略二次过程如聚集、破碎和老化等且混合良好情况的假设下,采用分离变量方法对粒度相关生长或生长分散的粒度分布进行理论分析,推导出间歇结晶过程中的几个粒数密度与结晶时间和晶体粒度的关系式.这些关系式可以用于间歇结晶过程中,选择合适的晶体生长机理与测定线性晶体生长速率,预测粒度分布与优化结晶过程,以及回归粒度分布.结果表明,本方法在溶液结晶过程中具有良好的应用价值.     

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.