A macroscopic agglomeration kernel model for gibbsite precipitation in turbulent and laminar flows

A macroscopic agglomeration kernel model has been developed that is capable of describing gibbsite agglomeration over a broad range of process conditions, including both the laminar and turbulent flow regimes. The agglomeration kernel model was derived using chemical reaction engineering principles and data from an extensive experimental program covering a wide range of temperatures, supersaturations, seed sizes, shear rates and mixing regimes. The experimental precipitation data in the laminar flow regime were generated using a novel Taylor-Couette precipitator. Data in the turbulent regime were generated in a stirred reactor. The developed agglomeration kernel model incorporates terms for the collision, capture, rupture and cementation rates affecting the formation of agglomerates. The model is shown to be able to predict the data from independent experiments. The proposed model also captures the complex non-linear shear rate and size-dependency observed experimentally, e.g. (1) for small particles the agglomeration kernel exhibits a maximum with respect to the shear rate, increasing at low shear rates in the laminar flow, but decreasing in the unstable Taylor-Couette flows and turbulent regimes; (2) for large particles the agglomeration kernel decreases monotonically with increasing shear.     

Effects of Fine Scale Turbulent Flow and Mixing in Agglomerative Precipitation

Precipitation of a solid product from aqueous ionic solutions, including mixing, fast chemical reaction, nucleation, growth and agglomeration of crystals is considered. This work concentrates on the phenomenon of particle agglomeration which dominates the precipitation process at high supersaturation. Modeling of particle collision includes effects of bulk fluid motion, Brownian diffusion and colloidal forces for particles of equal and unequal size. The concept of probability of agglomeration based on multifractal formalism is introduced and applied. A simplified version of the model is linked to CFD and results of computations are compared with experimental data for barium sulfate precipitation.     

Modeling Entry of Micron-Sized and Submicron-Sized Particles into the Indoor Environment

A theoretical approach, based on particle dynamics, was used to examine the outdoor-to-indoor penetration coefficient ( P ) of fine particles inside thin rectangular cracks. Parallel-plate flow theory indicates that crack infiltration flow can be assumed laminar for long, thin rectangular cracks. Considering laminar crack flow, three particle penetration models were used to estimate P . They are the Licht model, the Fuchs model, and the Taulbee model. The first two models consider gravitational sedimentation as the particle deposition mechanism, while the third model considers particle deposition induced from both gravitational sedimentation and Brownian diffusion. Modeling results indicate that gravitational sedimentation governs particle deposition behavior for micron-sized particles, and that all three models can be used to model penetration for these particles. For submicron-sized particles, Brownian diffusion becomes the major deposition mechanism, and only the Taulbee model is suitable to model particle penetration. The Taulbee model was validated using published experimental results of other researchers. Model validation indicated that the Taulbee model satisfactorily estimates particle penetration for micron-sized and submicron-sized particles. Application of the three models to actual building penetration is discussed.     

An agglomeration efficiency model for gibbsite precipitation in a turbulently stirred vessel

Gibbsite crystal agglomeration is a critical size and morphology control operation in the commercially important Bayer process for producing alumina. There has long been a need for a mathematical gibbsite agglomeration model that is both capable of describing the process and is easy to implement. In this paper, a simple model is presented for the agglomeration efficiency of gibbsite crystals during precipitation from seeded sodium aluminate solutions in a turbulently stirred vessel. The model has been developed by viewing agglomeration as a combination of reversible and irreversible pseudo-reaction steps, involving the sub-processes of particle transport, collisions, capture, aggregate rupture, and aggregate cementation. A range of models for these sub-processes were trialled. The model was tuned using gibbsite precipitation data from an extensive experimental program covering a wide range of temperatures, supersaturations, particle sizes, shear rates and solids concentrations relevant to the Bayer process. The resultant agglomeration efficiency model is a function of liquor viscosity, gibbsite crystal growth rate, particles size, impeller tip speed and fluid shear rate. The effects of the key process variables of liquor composition, operating temperature and hydrodynamics are incorporated via these terms. The agglomeration efficiency model was validated by incorporating it into a population balance model of a laboratory precipitator and demonstrating good agreement between the simulated dynamic particle size distributions and the experimental values.     

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.     

Modelling of acoustic agglomeration processes using the direct simulation Monte Carlo method

The direct simulation Monte Carlo (DSMC) method is applied to simulate acoustic agglomeration in travelling sound waves; orthokinetic and hydrodynamic mechanisms are considered as well as Brownian coagulation. The method is based on the constant-volume Monte Carlo approach combined with a modified full-conditioning algorithm of Gillespie, and is proved accurate by simulating only Brownian coagulation. Using the DSMC method, the validities of the classic Mednikov orthokinetic kernel, the König hydrodynamic kernel and the Song's orthokinetic and hydrodynamic models are examined by simulating acoustic agglomeration with only individual kernels. The predictions obtained by applying simple additive combinations of Song's orthokinetic model with the König equation and Song's hydrodynamic model are validated against experimental data in the literature. It has been found that the individual Song's models and the two combinations yield fairly good agreement with the measurements in some experimental cases, but all have limitations in covering a wide range of experimental conditions. This suggests that the present modelling needs further improvement to correctly describe the coupled contributions of two or more mechanisms.     

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.     

Respiratory Deposition of Fibers in the Non-Inertial Regime—Development and Application of a Semi-Analytical Model

A semi-analytical model describing the motion of fibrous particles ranging from nano- to micro scale was developed, and some important differences in respiratory tract transport and deposition between fibrous particles of various sizes and shapes were elucidated. The aim of this work was to gain information regarding health risks associated with inhalation exposure to small fibers such as carbon nanotubes. The model, however, is general in the sense that it can be applied to arbitrary flows and geometries at small fiber Stokes and Reynolds numbers. Deposition due to gravitational settling, Brownian motion and interception was considered, and results were presented for steady, laminar, fully developed parabolic flow in straight airways. Regarding particle size, our model shows that decrease in particle size leads to reduced efficiency of sedimentation but increased intensity of Brownian diffusion, as expected. We studied the effects due to particle shape alone by varying the aspect ratios and diameters of the microfibers simultaneously, such that the effect of particle mass does not come into play. Our model suggests that deposition both due to gravitational settling and Brownian diffusion decreases with increased fiber aspect ratio. Regarding the combined effect of fiber size and shape, our results suggest that for particles with elongated shape the probability of reaching the vulnerable gas-exchange region in the deep lung is highest for particles with diameters in the size range 10–100 nm and lengths of several micrometers. Note that the popular multi-walled carbon nanotubes fall into this size-range.     

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。     

Agglomeration modeling of small and large particles by a diffusion theory approach

The interaction particle‐binder during the wet granulation process plays a major role in the agglomeration of particles. This interaction has been modeled by a force balance acting on the particle where the binder's viscous force increases the strength of liquid bridge and facilitates the particle agglomeration. In this work, agglomeration kernels based on Brownian movement approach of small particles in the binder layer, the size ratio between particles (monodispersed and polydispersed), and binder's viscous forces were considered to model the wet granulation process of pharmaceutical powders in a laboratory‐scale high shear mixer. The assumptions of no‐stationary and pseudostationary behavior were suitable to describe the growth kinetics of the two stages (fast and slow) observed. A volume ratio of 150 between large and small particles produces the most effective granulation growth. The developed kernels were tested simulating experimental data obtained from a high shear mixer. © 2009 American Institute of Chemical Engineers AIChE J, 2009     

Alumina trihydrate crystallization. Part 2. A model of agglomeration

In the second part of this paper on the kinetics of crystallization of aluminum trihydroxide from caustic aluminate solutions, interest is specially focused on the analysis of crystal size distribution as a function of time, in an isothermal batch crystallizer. Breakdown and agglomeration are shown to affect these curves simultaneously. While only qualitative information on breakdown was obtained, agglomeration could be isolated experimentally. Its rate was measured and fitted to a kinetic model which pictures agglomeration as the result of binary collisions between particles.     

Accelerated desupersaturation of reverse osmosis concentrate by chemically-enhanced seeded precipitation

A two-step chemically-enhanced seeded precipitation (CESP) process was demonstrated for accelerated desupersaturation of antiscalant-containing, gypsum-supersaturated model solutions, which mimicked reverse osmosis (RO) concentrate from RO desalting of agricultural drainage water of high mineral scaling propensity. In the CESP process, CaCO₃ precipitation is first induced via lime dosing for antiscalant scavenging, followed by subsequent CaSO₄ precipitation via gypsum seeding for concentrate desupersaturation. It was demonstrated that lime-precipitated CaCO₃ particles were able to scavenge generic and commercial polycarboxylic-acid antiscalants, thereby facilitating subsequent CaSO₄ precipitation to progress with minimal retardation. The study demonstrated via a series of batch CESP cycles that gypsum particle recycling can sustain CaSO₄ precipitation, suggesting that a continuous CESP process could be feasible. Process analysis suggests that CESP can be significantly less chemical-intensive than conventional precipitation softening and, with its integration as an intermediate RO concentrate demineralization process, can enable desalination water recovery enhancement via secondary RO desalting. For the present case of gypsum-saturated RO feed water, enhancement of overall water recovery from 63% up to 87% or higher appears to be feasible. The study suggests that there is merit for developing a continuous CESP process for high recovery RO desalting of brackish water of high gypsum scaling propensity.     

A method of characteristics for solving population balance equations (PBE) describing the adsorption of impurities during crystallization processes

Possible hindering effects of impurities on the crystal growth were shown to take place because of the adsorption of impurity species on the crystal surface. Transient features of this adsorption were observed, such that the growth of a given crystal does not depend on supersaturation only, but also on the time a given particle spent in contact with impurities present in the mother liquor. Meanwhile, few kinetic models describe the effect of impurities on the growth of crystals in solution, and published models are usually derived from data obtained, thanks to specific experiments based on the evaluation of the growth rate of single crystals. Such models are obviously questionable because, in the industrial practice, distributed properties of crystals are actually involved. Considering the “time of contamination” of particles as a new internal variable is thus made necessary. This is the reason why a specific PBE resolution algorithm is presented in this paper. The numerical scheme for the resolution of PBEs is based on the method of characteristics and shown to allow fast and accurate simulation of transient features of the crystal size distribution in the particular case when the growth or nucleation rates are assumed to exhibit unsteady-state dynamics. The algorithm is finally used to simulate the isothermal desupersaturation crystallization of citric acid in water.     

饲料级磷酸氢钙结晶过程中团聚的研究

在反应结晶过程中,一般过饱和度很大,很容易形成大量细小颗粒,这些小颗粒相互作用形成团聚体,会影响最后产品的粒度分布、纯度、晶体形貌、堆密度等参数。饲料级磷酸氢钙生产过程中极易形成团聚体,通过探索团聚体形成的pH范围,对团聚体形成的原因做了分析,并研究了中和条件对团聚形成的影响。结果表明,形成团聚体的pH范围为2.82~3.60。此外,提高反应温度、增加搅拌速度均对团聚体有减缓效果。实验最后建立了磷酸氢钙团聚尺寸模型。     

Particle Shape and Purity in Membrane Based Crystallization

Well‐controlled crystallization is the best method for preparing materials that are uniform in shape, size, structure and purity. The driving forces for crystallization are local gradients of supersaturation as the source and desupersaturation as the drain. Very high local supersaturation causes a high growth rate and represents a limiting factor for unstable modifications and product impurities. Hybrid membrane technology provides an interesting tool for controlling and limiting the maximum level of supersaturation due to defined mass transfer across the membrane. In this paper, the level of crystal growth rate in the system NaCl/KCl/water is varied by using different crystallization techniques. Vacuum evaporation crystallization (high growth rate) is compared to membrane based evaporation crystallization (low growth rate) and the results are interpreted in terms of product purity, particle shape and size. Membrane based crystallization in combination with effective solid/liquid separation as well as high performance analytics is suggested as a significant ultrapurification methodology.     

Particle deposition velocity onto a face-up flat surface in a laminar parallel flow considering Brownian diffusion and gravitational settling

The Gaussian Diffusion Sphere Model (GDSM) was developed and improved to reflect the effects of gravitational settling as well as Brownian diffusion of aerosol particles on deposition velocity onto a face-up flat surface in a laminar parallel flow. The model improvement also includes the applicability of the GDSM to a flat surface of any shape with finite dimensions. When deposition velocity for a face-up circular flat plate of 45 cm diameter, representing e.g. a semiconductor wafer in a laminar parallel flow, was calculated by the GDSM and compared with that by the theory of Liu and Ahn (1987). Particle deposition on semiconductor wafers. Aerosol Science and Technology, 6, 215–224, the agreement was good for the tested particle sizes ranging 0.003–1 μm and free stream velocities ranging 5–500 cm/s. Based on this result, deposition velocities onto the face-up square flat plates with different orientations in a laminar parallel flow, simulating e.g. photomasks, were predicted.     

Particle migration in a flow of nanoparticle suspensions

This paper is concerned with particle migration in pressure-driven laminar pipe flows of relatively dilute suspensions of nanoparticles (nanofluids), one of the most frequently used configuration in industries. The motivation behind the work is associated with the thermal behaviour of nanofluids, which can greatly exceed the values predicted by currently available macroscopic theories. A theoretical model is formulated to predict particle concentration, and velocity field of nanofluids in the transverse plane of the pipe. The model takes into account the effects of the shear-induced and viscosity gradient-induced particle migrations, as well as self-diffusion due to the Brownian motion. It is shown that particle concentration in the wall region can be much lower than that in the central core region. This indicates a highly non-uniform thermal conductivity profile across the transverse plane of the pipe, and thus has a significant implication to heat transfer intensification using nanofluids. The results also suggest the existence of an optimal particle size whereby the thermal conductivity is enhanced with little penalty due to the effect of pressure drop.     

采用粒数衡算模型研究沉淀过程中粒子的聚结和破裂

通过采用MSMPR反应沉淀器,研究了普鲁卡因青霉素反应沉淀过程中粒子的聚结和破裂对过程的影响机理,并对用生死函数表征聚结和破裂的粒数衡算模型采用新的分析方法。结果表明,将二元破裂模型应用于该粒数衡算模型是可行的。     

采用粒数衡算模型研究沉淀过程中粒子的聚结和破裂

通过采用MSMPR反应沉淀器,研究了普鲁卡因青霉素反应沉淀过程中粒子的聚结和破裂对过程的影响机理,并对用生死函数表征聚结和破裂的粒数衡算模型采用新的分析方法。结果表明,将二元破裂模型应用于该粒数衡算模型是可行的。