Adaptive finite difference method for the simulation of batch crystallization

The modeling of batch crystallization involves a population balance to get the crystal size distribution which is one of the most important properties in this process. Thus, this model leads to a system of integral, partial differential and algebraic equations (IPDAE). This system can be easily solved by finite difference methods with uniform discretization. However to increase the calculation efficiency and the crystal size distribution accuracy, an adaptive finite difference method with a non-uniform discretization was developed. Results of both methods are compared in term of efficiency and are confronted to experimental data in term of accuracy.     

Simulation and analysis of industrial crystallization processes through multidimensional population balance equations. Part 2: a study of semi-batch crystallization

A bi-dimensional population balance model was presented in the previous part of this series of papers to simulate the time variations of two characteristic sizes of hydroquinone particles during crystallization. The multidimensional population balance equations combined with kinetic models and mass balance equations were shown to allow the simulation of the solution crystallization of hydroquinone characterized by a rod-like habit. Semi-continuous isothermal operations were performed at the lab-scale in the presence of various additive concentrations. Both the experimental solute concentration trajectory and the final bi-dimensional crystals size distribution were correctly predicted by the model. The simulated elongation shape factor characterizing the crystal shape was therefore in agreement with the experimental one. Due to the use of tailor-made additive, inhibition effects were observed to affect both primary nucleation and growth kinetics in the length direction. For secondary nucleation, indirect effects were assumed to occur which allowed satisfactory predictions of the final number of fine particles. The representation of the kinetics involved required the evaluation of a set of nine parameters. As a result it was observed that the elongation ratio characterizing the shape of the rod-like particles increases with the length in a nonlinear way. A major interest of the two-dimensional model lies in its ability to relate the time variations of the crystal habit: the particles lengthen in the first moments of their growth and then progressively get thicker until the end of the process.     

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.     

Investigation and simulation of crystallization of high aspect ratio crystals with fragmentation

Background

Crystallization is a widely used unit operation in the pharmaceutical industry where high aspect ratio crystals are often produced. To avoid undesired shape i.e. high aspect ratio crystals according to downstream operations, appropriate level of understanding process is necessary. Breakage often occurs during the process and gives rise to change the Crystal shape. A well-adjusted, detailed model can be usable tool for investigation and optimization of control.

Method

A two-dimensional population balance model of continuous cooling crystallization, involving nucleation, growth of two characteristic crystal facets and random binary breakage of high aspect ratio crystals is developed. The randomness of breakage is described by beta distribution of broken fractions. The population balance model is reduced into a closed moment equation model for the joint moments of the two size variables of crystals by means of which the influence of parameters of breakage on the dynamic and steady state behavior of crystallizer is analyzed according to understand the impact of complex kinetic on crystal shape.

Results

Analysis of the process provides information of breakage influence on aspect ratio of crystal. It is shown that the mean value of crystal width increases as the intensity of breakage along the crystal length resulted in increasing number of crystals.     

On the solution of population balance equations (PBE) with accurate front tracking methods in practical crystallization processes

The PBE (population balance equation) containing birth, growth, agglomeration and breakage kinetics is described by a conservation law with a moving source term. For the solution of the PBE, we compare two accurate front tracking methods such as a modified method of characteristics (MOC) and a finite difference method with the weighted essentially non-oscillatory (WENO) scheme. Both methods are applied to a potassium sulfate crystallization problem (K₂SO₄-H₂O system) with a discontinuous initial condition. Parameters of agglomeration and breakage kinetics are estimated on the basis of the experimental data of the K₂SO₄-H₂O system.Owing to moving axis along a crystal growth rate (i.e. elimination of the growth term), the modified MOC is able to provide a highly accurate solution even at discontinuous points without numerical diffusion error. However, in the case of stiff nucleation which can commonly appear in practical crystallization processes, it is necessary to adaptively determine time levels to add a mesh of the nuclei size. For solving PBEs involving agglomeration and breakage terms, the MOC can take more long computational time than the spatial discretization methods like the WENO scheme. It is pointed out that the MOC is not available to solve more than two coupled PBEs in general.WENO schemes for spatial discretization are firstly addressed in this study for the dynamic simulation of batch crystallization processes. The WENO schemes show improvements of accuracy and stability over conventional discretization methods (e.g., backward, central or common upwinding schemes). However the WENO schemes on fixed meshes show, to some extent, the numerical diffusion error near discontinuities or steep moving fronts like other finite difference methods. Hence, they require spatially-adaptive mesh techniques in order to track more accurately the moving fronts. Even though the WENO schemes are less accurate than the MOC, they are of practical use for solving complex PBEs owing to a short computational time and little limitation to use.     

聚集结晶动力学研究方法的改进

在Hounslow工作的基础上,推导了新的聚集机理,改进了聚集模型。通过粒数衡算和质量衡算,结合改进的聚集模型建立了聚集结晶动力学辨识的新方法。文中以头孢哌酮钠结晶过程为例,应用新方法确定了该物系的晶体生长、聚集以及成核动力学。结果表明新方法研究聚集结晶动力学具有一定的可靠性。     

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.     

苊冷却结晶动力学的间歇动态法研究

通过添加晶种的间歇冷却结晶实验研究了苊在乙醇中的结晶动力学。由苊晶体的粒数密度数据,通过矩量变换法按粒度无关生长模型求解粒数衡算方程,采用多元线性最小二乘法回归动力学数据,得到苊在乙醇中晶体生长及成核速率方程。对动力学方程的理论分析表明:搅拌速率对二次成核影响显著,随搅拌速率增加,晶体的成核速率明显增加。同时适宜的过饱和度及较低的悬浮密度,有利于苊晶体生长。该研究为苊冷却结晶特性的辨识、粒度分布的控制及工业放大提供了重要的理论指导。     

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.     

FINES DESTRUCTION DURING BATCH CRYSTALLIZATION

An analysis of the use of fines destruction during batch crystallization based on the population (number) balance is presented. The effects of operating policy and fines crystal cut size and destruction rate are examined in detail for two cases viz. natural and controlled cooling respectively. It is predicted that use of fines destruction has the potential as a control aid to significantly improve the product crystal size distribution from batch crystallizers and is supported by experimental data on the programmed cooling crystallization of potassium sulphate solutions.     

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.     

黑钛液中七水硫酸亚铁结晶动力学性质

采用平衡法研究了七水硫酸亚铁在黑钛液中的溶解度,通过拟合得到了其溶解度方程,并利用FBRM在线监测溶液结晶情况,测量了其结晶介稳区.通过间歇动态法研究了七水硫酸亚铁的结晶动力学,采用矩量变化法求解了粒数衡算方程,并利用最小二乘法对动力学实验数据进行多元线性回归后得到七水硫酸亚铁成核速率和生长速率方程,结果表明:搅拌速率和降温速率对七水硫酸亚铁结晶过程影响显著,因此,针对搅拌速率和降温速率对结晶工艺的影响进行了分析,并对优化前后七水硫酸亚铁产品进行了对比,发现优化后晶体呈规则的棱形,产品粒度明显增大,小颗粒晶体显著减小.     

A Laplace transformation based technique for reconstructing crystal size distributions regarding size independent growth

This article introduces a technique for reconstructing crystal size distributions (CSDs) described by well-established batch crystallization models. The method requires the knowledge of the initial CSD which can also be used to calculate the initial moments and initial liquid mass. The solution of the reduced four-moment system of ordinary differential equations (ODEs) coupled with an algebraic equation for the mass gives us moments and mass at the discrete points of the given computational time domain. This information can be used to get the discrete values of size independent growth and nucleation rates. The discrete values of growth and nucleation rates along with the initial distribution are sufficient to reconstruct the final CSD. In the derivation of current technique the Laplace transformation of the population balance equation (PBE) plays an important role. The proposed technique has dual purposes. Firstly, it can be used as a numerical technique to solve the given population balance model (PBM) for batch crystallization. Secondly, it can be used to reconstruct the final CSD from the initial one and also vice versa. The method is very efficient, accurate and easy to implement. Several numerical test problems of batch crystallization processes are considered here. For validation, the results of the proposed technique are compared with those from the high resolution finite volume scheme which solves the given PBM directly.     

环丙沙星冷却结晶动力学测定

依据粒数衡算方程,采用分离变量方法求解粒数密度函数表达式,依据该方法通过加晶种的间歇冷却结晶实验研究了环丙沙星在23％(体积)水/乙醇溶液中的结晶动力学数据,得到了结晶动力学方程,为工业结晶动力学的测定、结晶特性辨识、粒度分布预测和控制提供了新的方法.     

Analysis of ice crystallization in continuous crystallizers based on a particle size-dependent growth rate model

To predict the ice crystal size distribution in MSMPR type crystallizers easily, the population balance equation with respect to the crystal size was solved under steady state conditions with a size-dependent growth rate model, that is G = g₋₁/r + g₀. The kinetic parameters g₋₁ and g₀ obtained from the measurement of crystal size distribution in a dextran solution were consistent with those estimated by assuming that the growth process of ice crystals is governed by heat transfer process. We confirmed that the kinetic parameter β, determined from experiments in a batch crystallizer, can be adopted as the parameter for the nucleation rate per crystal in an MSMPR type crystallizer.     

Two-dimensional population balance modeling for shape dependent crystal attrition

Many suspension crystallization processes can be described by growth and secondary nucleation. The prevailing mechanism for secondary nucleation in industrial processes is attrition caused by crystal-impeller collisions. The understanding and prediction of attrition rates is of fundamental importance for product and process design.Attrition of a crystal is affected by the size of the crystals and, as experimental evidence reveals, by their shape. For crystals with the same mass, the attrition rate is significantly larger if crystals have distinct and sharp corners, than if they were spherical. Because of the associated modeling and computational effort, shape dependent behavior has mostly been disregarded in crystallization process modeling. In this work, a two-dimensional population balance model is formulated. The inner variables are the size and the shape of the crystals. Consequently, the model accounts for size and shape dependent process behavior. In order to close the model equation shape modification function are introduced. The reinforcement function specifies the increase in attrition resistance by rounding the sharp corners and increasing material strength. The face attrition ratio represents the differences of material removal from sharp corners and flat faces.The sensitivity of the results with respect to these shape modification functions is investigated. The results show that the model is capable of reflecting shape dependent attrition behavior in a physically meaningful way. To fit experimental data, mainly the parameters of the shape modification function need to be tuned.     

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     

Quantitative design of seed load for solution cooling crystallization based on kinetic analysis

Seed load of crystallization has a direct effect on the product qualities. To further reveal the effects of seed load on crystallization kinetics and improve the product size distribution, the aqueous solution of potassium nitrate (KNO₃–H₂O) is employed as a model system and the relevant kinetic experiments are conducted in a batch cooling crystallizer. The crystal nucleation and growth rate parameters are firstly estimated with the concentration and transmittance data using a mathematical model reported in our lab, and then the backward calculations with the help of the model parameters are successfully performed. It is found that the nucleation capacity decreases and growth capacity increases with increasing seed load, and the size distribution of crystal products tends to be more uniform. However, with the increasing of seed load, the linear growth rate of single crystal and the mean size of products both reduce accordingly. Based on the calculational and experimental results, a quantitative design scheme concerning seed load is proposed by further kinetic analyses, and the corresponding verification experiments are carried out. The results show that under the guidance of the proposed scheme, the size distribution of crystal products is more concentrated and the mean size of final particles can also escape from reducing obviously.     

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。