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.     

The growth and nucleation of ice in a batch Couette flow crystallizer

An experimental investigation into the crystallization characteristics of ice crystals from brine solution was carried out using a batch Couette flow crystallizer. Particle size distributions were determined by a photographic technique. The nucleation rate was correlated by the power law model: B⁰ = 26.2(P/V)^0.52 (δC^*)^1.8 μ^0.2₂ Qualitative growth characteristics were also inferred and found to be consistent with a diffusion controlling mechanism. The results obtained here suggest that early time data be emphasized in future batch crystallization experiments to more accurately determine the nucleation rate model.     

铁离子对磷酸二氢钾结晶过程的影响

采用激光散射法测定了含有不同铁离子含量的磷酸二氢钾水溶液介稳区宽度,在介稳区宽度内测得不同铁离子含量对磷酸二氢钾结晶生长速率的影响。并采用间歇冷却结晶法在未加晶种条件下测定了铁离子对磷酸二氢钾结晶形态、晶体堆积密度及晶体铁含量的影响。结果表明：随着掺入铁离子含量的增大,结晶介稳区宽度变宽,生长速率下降; 晶体由规则棱柱状变为针状,晶体出现开裂,光泽不均匀;晶体堆积密度逐渐减小,包裹铁含量逐渐增大。     

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.     

Kinetics of nucleation and growth of L-sorbose crystals in a continuous MSMPR crystallizer with draft tube: Size-independent growth model approach

The experimental data concerning kinetics of a continuous mass crystallization in L-sorbose - water system are presented and discussed. Influences of L-sorbose concentration in a feeding solution and mean residence time of suspension in a working volume of laboratory DT MSMPR crystallizer on the resulting crystal size distributions, thus on the nucleation and growth kinetics, were determined. The kinetic parameter values were evaluated on the basis of size-independent growth (SIG) kinetic model (McCabe’s ΔL law). It was observed that within the investigated range of crystallizer productivity (220–2,200 kg of L-sorbose crystals m⁻³ h⁻¹), a crystal product of mean size Lm from 0.22 to 0.28 mm and CV from 68.8 to 44.0% was withdrawn. The values of linear growth rate show increasing trend (from 6.6·10⁻⁸ to 7.6·10⁻⁸ m s⁻¹) with the productivity enlargement (assuming constant residence time τ=900 s). Occurrence of secondary nucleation phenomena within the circulated suspension, resulting from the crystals attrition and breakage was observed. The parameter values in a design equation, matching linear growth rate and suspension density with nucleation rate were determined.     

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 impeller type and position in a batch cooling crystallizer on the growth of borax decahydrate crystals

The effects of impeller type and position in a batch cooling crystallizer on the crystal growth kinetic as well as the shape, agglomeration and crystal size distribution of disodium tetraborate decahydrate (borax) were investigated in detail. Three types of impellers applied in this work generated completely different fluid flow patterns in the crystallizer allowing investigation of the influence of the liquid flow on the kinetic parameters as well. The ratio of impeller off-bottom clearance, C, to the liquid height, H, was varied from 0.1 to 0.5 for all impeller types used. Results indicate that the impeller type and off-bottom clearance influence the overall crystal growth rate and the crystal size distribution of the final product. It was also found that agglomeration of crystals produced was less pronounced as the radial fluid flow in the crystallizer became more dominant. In that system borax crystals tend to be smaller, but more regularly shaped.     

Modeling, simulation and stabilizing H∞-control of an oscillating continuous crystallizer with fines dissolution

In this contribution, a detailed model for a continuous crystallizer with fines dissolution is derived. The main focus of this article is the identification of physical reasons responsible for oscillations occurring in these crystallization plants. In contrast to many other crystallization models used in literature for the investigation of such limit cycles, detailed kinetic expressions for crystal growth and attrition, as well as for the separation of fines in the annular zone, are incorporated. By dynamic simulations of the model and by comparison with measured data, an undesired dissolution of larger crystals can be identified as a possible reason for the appearance of sustained oscillations. Finally, a stabilizing feedback controller is designed using H_∞-theory. It is demonstrated in simulations that this controller enables stable operation of the crystallizer even at a high fines dissolution rate.     

Modeling of growth rate dispersion of citric acid monohydrate in continuous crystallizers

A mathematical model for prediction of the crystal size distribution from a continuous crystallizer is presented. The kinetic data used for the model were obtained from batch contact nucleation experiments with citric acid monohydrate. In these experiments, the distribution of growth rates as well as the initial size distribution were estimated. Results from the model indicate that the excess number of crystals usually present at small sizes in continuous crystallizers is due to growth rate dispersion (where crystals of the same size may have different growth rates) and not size dependent growth.     

Membrane assisted cooling crystallization: Process model,nucleation, metastable zone,and crystal size distribution

The membrane assisted cooling crystallization was proposed and investigated by the simulation and experiments. The developed process model concerned the supersaturation evolution on the membrane interface, the combined nucleation rate in the crystallizer. The impact of different membrane on reducing the nucleation barrier was investigated by introducing the metastable zone width theory. The influence of membrane distillation conditions on the crystal nucleation and growth kinetic was uncovered based on the simulation and experiments results. The experimental results indicated that membrane assisted cooling mode with optimized profiles did improve the crystal size distribution and crystal habit comparing with conventional cooling mode. Terminal coefficient of variation decreased from 55.4 to 33.9 under similar mean crystal growth rate, 2.27 × 10^?7 m s^?1 (conventional cooling) and 1.98 × 10^?7 m s^?1 (membrane assisted cooling). Finally, the brief summary on the advantages and key issues of this propose membrane assisted crystallization operation were concluded. © 2015 American Institute of Chemical Engineers AIChE J, 62: 829–841, 2016     

Kinetics of Primary Nanoparticle Agglomeration in Precipitation of Silver

The precipitation kinetics of silver was studied using a lab‐scale batch crystallizer with special attention to characterization of agglomeration of primary nanoparticles. The vessel was operated at different feed concentrations, molar ratios, and stirrer speed. Nucleation, volume average crystal growth rates, and agglomeration kernels were determined. Empirical relations were obtained between the rates of the different crystallization steps with supersaturation, magma density, and energy dissipation rate. The results show that larger crystals will be obtained at high supersaturation due to dominance of agglomeration, because the rate of agglomeration also increases with supersaturation, thereby suppressing the primary nucleation rate and enhancing the observed average volume growth rate.     

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.     

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.     

Kinetic study on the anisotropic grain growth of elongated iron-containing mullite

Elongated iron-containing mullite exhibiting anisotropic grain growth was prepared using fused silica, α-alumina, and Fe₂O₃ powders as raw materials at elevated temperatures. The effects of different Fe₂O₃ contents on the anisotropic growth of mullite columnar crystals were investigated, and phase compositional analysis and structural characterisation were carried out by X-ray diffraction and field emission scanning electron microscopy. The kinetic index and activation energy for the anisotropic grain growth of elongated mullite crystals were calculated, and the surface energy of each crystal plane of the elongated crystal and the solid solution behavior of the iron ions in the mullite were calculated using the density functional theory (DFT) approach. It was found that the addition of Fe₂O₃ promotes the anisotropic grain growth of mullite, and the aspect ratio of the elongated crystals increased upon increasing the Fe₂O₃ content. The solid solubility upper limit of the iron-containing mullite in this system is 9 wt% of iron addition. After sintering at 1973 K, for Fe₂O₃ contents of 3, 6, and 9 wt%, the growth kinetics indices obtained by calculations in the longitudinal direction were 2.22, 2.50, and 3.85, while the reaction activation energies were 457.3, 442.1, and 411.9 kJ mol⁻¹. The calculation results on the surface energy of the elongated mullite crystal showed that the smallest surface energy is on the (001) crystal plane, which is the preferred growth direction. Upon comparison of the surface energies for different Fe₂O₃ contents on the (001) surface, a decrease was observed upon increasing the quantity of Fe₂O₃, and both the growth tendency of the mullite crystals and the aspect ratio of the elongated crystals also increased. The refractoriness under load of the prepared elongated iron-containing mullite was 1664 °C, demonstrating its excellent high-temperature mechanical properties.     

Simulation of batch crystallization

The closed mathematical model of a well-mixed batch crystallizer has been presented. This model takes into consideration crystals growth rate fluctuations producing an increased spread of crystal sizes. The numerical methods for solving partial integro-differential system of this model equations are proposed for the cases of when growth rate is expressible as product of supersaturation and size functions.Simultaneous population and mass balance equations have been calculated together with power law kinetics of nucleation and growth to determine crystal-size distribution functions (CSD), variation coefficients (CV), produced crystal sizes and numbers. The case of size-dependent crystal growth determined by sequential processes of diffusion and surface reaction is also investigated.The computer results are presented in the dimensionless form as functions on both diffusion and fluctuation Peclet numbers and dimensionless cooling rate. In particular, it has been established that the CV of weight CSD increases approximately twice due to fluctuations and becomes close to 40%; the CV also increases owing to the decrease of cooling rate and approaches 28%. Some formulae are suggested for predicting the kinetics of crystallization (or precipitation) and determining the kinetic parameters of crystals growth and nucleation.     

The Investigation of the Influence of Impeller Blade Inclination on Borax Nucleation and Crystal Growth Kinetics

An influence of impeller blade angle on the crystallization kinetics of borax has been investigated in detail. Its importance was studied in a batch crystallizer equipped with a four-blade turbine, whose angle was varied from 30° to 90°. Heterogeneous nucleation mechanism was dominant at all examined conditions, but the rate of nucleation significantly differed. Kinetic parameters of crystal growth were determined as well. Numerical values of order of crystal growth remain almost unchanged regardless of the blade angle. Different values of crystal growth rate constants are the result of the fluid dynamics in the crystallizer, caused by the different impeller blade inclinations. In all examined systems, crystal growth is integration limited. The transition from axial to radial flow, due to different blade angles, reflects on the crystal size distribution, agglomeration ratio, and crystallization yield.     

Crystal habit modification studies with ammonium and potassium dihydrogen phosphate

Face rates of growth of ammonium and potassium dihydrogen phosphate crystals have been measured for crystallisation from pure solutions and from solutions that contained known amounts of foreign ions. The level of supersaturation and the solution pH both effect the crystal habit. Low supersaturation at normal pH (3.8) tends to give tapered crystals, and at a slightly higher pH (?5), the linear growth rates of both principal axes are considerably increased. The presence of foreign ions such as Cr³⁺, Fe³⁺ and Al³⁺ in low concentrations (? 1 × 10^?3 g ion/1 or ? 50 ppm) also causes a tapering habit. A mechanism of habit modification based on the presence of aquo ions at the crystal face is suggested.     

Growth rates of ibuprofen crystals grown from ethanol and aqueous ethanol

To quantify the crystallization of racemic ibuprofen [2-(4-isobutyl-phenyl)-propionic acid] from aqueous ethanol it is necessary to know the growth rate kinetics. Growth rates were measured by adding SPG (size proportional growth) seed crystals to an isothermal non-nucleating batch crystallization and sampling during the batch. The supersaturation was measured by refractive index and the crystal size by laser light scattering (Malvern). To ensure the batch was non-nucleating the supersaturation was kept within the narrow secondary MZ (metastable zone). Measurements were made at three temperatures, 10, 25 and 40 °C. The growth rate was proportional (first order kinetics) to the solution supersaturation, expressed in mass ratio of ibuprofen to ethanol units. Despite the narrow MZ, the initial growth rates were substantial (up to 1 μm/min). Temperature has a large effect on growth rate with an activation energy of 13.7 kJ/mol. The water content affects the growth rate coefficient differently depending on the temperature.     

Simulation of crystal size and shape by means of a reduced two-dimensional population balance model

Commonly, population balance modeling for crystallization processes only considers one inner variable. Usually, a variable characterizing particle size, like a sphere-equivalent diameter is employed. However, crystal structures are obviously not spherical but exhibit a complex habit. Often, the habit is even crucial for the quality of the product or for the operability of downstream filter units. To describe the transient behavior of crystals in a batch process considering two-dimensional growth and nucleation, a multidimensional population balance needs to be employed. Considering two characteristic lengths of a crystal, the standard discretization of such a system (by, e.g. finite differences) leads to quite a large model size, which may be unsuited for model-based control and parameter identification purposes. In this contribution a model reduction which is based on two steps is proposed. First, a coordinate transformation is performed to model the system not in terms of two characteristic lengths but by means of the crystal volume and a shape factor. In a second step the actual model reduction is performed by generating cross-moments for the two-dimensional representation. An ansatz for the two-dimensional crystal size distribution, which gives full flexibility in the volume coordinate but restricts the dependence in the shape factor κ, allows the closure of the system of moments. The reduced system consists of three coupled population balance equations in which all three are structurally similar to a single one-dimensional population balance equation, where growth and nucleation only are considered. Solving this reduced system allows the detailed simulation of the easily measurable volume-based number density distribution and preserves average and dispersity information on the crystal shape. The resulting model size, however, scales only linearly with the number of discretization grid points instead of the quadratic scaling for standard discretizations. Numerical results for the crystallization of potassium dihydrogen phosphate (KH₂PO₄) in a batch process are presented for illustration.