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Interaction between Mixing, Chemical Reactions, and Precipitation
Abstract
The way in which reagents are mixed can have a large influence on the product distribution of chemical reactions and the size distribution of particles of the solid product. To model effects of mixing on various scales on the course of chemical reactions and precipitation processes, a nonequilibrium multiple-time-scale mixing model and a beta distribution of the mixture fraction are applied in combination with a simple conditional moment closure based on linear interpolation of local instantaneous reactant concentration values. The mixing model is linked to CFD (standard k−ε model) and the model predictions are compared with experimental data for fast parallel chemical reactions and barium sulfate precipitation both carried out in the single-feed semibatch stirred-tank reactor.
... The selectivity in the case of fast reactions, and the mean size in the case of precipitation, often shows a similar dependency on the mixing conditions. Still, it is easier to apply a complex mixing model to competing chemical reactions, since the reaction terminates quickly, making it possible to identify a chemically active region around the feed point in an otherwise chemically passive bulk (Baldyga et al., 2005). Crystal growth, on the other hand, operates on a longer time scale and is active in the whole bulk as long as the solution is supersaturated, and this makes the calculations more complicated. ...
... Nowadays, computational fluid dynamics (CFD) is often used to calculate the flow field. The population balance equations can either be solved together with the CFD equations (Baldyga et al., 2005;Piton et al., 2000;Vicum et al., 2004;Marchisio and Barresi, 2003;Baldyga and Orciuch, 2001), or the CFD simulation can be done separately to calculate energy dissipation rate values, which are then used in a mechanistic mixing model (Uehara-Nagamine, 2001;Akiti and Armenante, 2004;Zauner and Jones, 2000a). Another, more simple approach was used by Phillips et al. (1999), in which the reactor was divided into zones with different hydrodynamic properties and the flow map was used to calculate the corresponding variation in the mixing parameters as mixing proceeded. ...
... Multiscale Eulerian mixing models that account for all levels of mixing are usually combined with a CFD code to calculate the flow field. Baldyga et al. (2005) used the turbulent mixer model (Baldyga and Bourne, 1999) and a beta-distribution to describe micromixing for a single feed semi-batch process. The variation of selectivity with varying mean energy dissipation rate and feed concentration was well described, but the influence of feed time on the selectivity was less pronounced than in the experiments. ...
A population balance model is developed over single-feed semi-batch reaction crystallization of benzoic acid. The model is evaluated by comparison with experimental data, and simulations are carried out to advance the understanding of the process. The model accounts for chemical reaction, micro and mesomixing, primary nucleation, crystal growth and growth rate dispersion (GRD). Two mechanistic mixing models are evaluated: the segregated feed model and the engulfment model (E-model) with mesomixing. When the mixing is described by the E-model (engulfment model) and GRD is accounted for, the model quite well captures the influence of reactant concentrations, agitation rate, feed point location, feed pipe diameter, total feeding time and crystallizer volume, on the product weight mean size. When using the SF-model (segregated feed model) the results are less satisfactory. The kinetics of nucleation and crystal growth have a great impact on the results of the simulations, influencing the product weight mean size as well as the response to changes in the processing conditions. A new set of kinetic data for benzoic acid derived from semi-batch experimental results are presented.
... In RANS simulations such simplification cannot be used, since it leads to significant errors in predictions (Bałdyga et al., 2005). ...
... The aim of the work presented in this chapter is to understand better the influence of mixing on the course of precipitation process. A barium sulfate precipitation test process, commonly used in the literature (Bałdyga et al., 1995;Bałdyga and Orciuch, 1997;Wei and Garside, 1997;Barresi et al., 1999;Kim and Tarbell, 1999;Bałdyga and Orciuch, 2001;Jaworski and Nienow, 2003;Bałdyga et al., 2005;Makowski et al., 2012;Wojtas et al., 2014bWojtas et al., , 2015cWojtas and Makowski, 2015), can help in describing the influence of mixing on precipitation. ...
The thesis is focused on large eddy simulations (LES) of turbulent mixing and complex chemical processes in jet reactors. Several geometry types of jet reactors are used in this study and the effect of their construction differences on the results is discussed. Apart from large eddy simulations, the simulations were performed using the Reynolds-Averaged Navier-Stokes (k-ε) model, which was used as a reference and the first step of LES verification.
In the first part of the thesis, the thorough analysis of the mixing process carried out in the jet reactors is presented. The simulations’ results were verified using advanced laser techniques: particle image velocimetry and planar laser induced fluorescence. It was observed that both turbulence models predict experimental data correctly at higher Reynolds numbers. Clear differences between the RANS model and experiments were visible at lower Reynolds number values, while the LES results were in good agreement with measurement data in the whole tested range of flows. In the case of the inert tracer mixing modeling several literature models were tested. The differences between models’ predictions were most pronounced in the case of the tracer concentration variance. This was due to limitations of the considered models, and for this reason the new dynamic subgrid-scale variance model for LES was presented. The model takes into account a numerical cell size, as well as local values of the Reynolds and Schmidt numbers. The obtained results confirmed model correctness and usefulness, also for fluids characterized by high Schmidt numbers.
The second part includes the investigation of mixing effects on the course of parallel chemical reactions and precipitation. A closure based on a probability density function was used in simulations and a fast numerical procedure for computational fluid dynamics was presented. The results showed that in the case of the chemical reactions, the LES model predicts experimental reaction yield data very well in the whole tested range of flows. This verified the proposed numerical procedure and allowed modeling of the precipitation process. Barium sulfate precipitation was chosen as the second test process. The relevant particle formation mechanisms are analyzed and several kinetic relationships for nucleation and growth are presented and tested. The choice of a proper kinetics was justified basing on the experimental data. In the case of barium sulfate precipitation, both LES and RANS models quite accurately predict the effect of the mixing intensity on a mean particle size. Better agreement of the LES model with the experiments can be observed in the case of the effect of inlet reagents’ concentrations on the particles size.
The final part of this work is devoted to a practical application of large eddy simulations and jet reactors, i.e., molybdenum disulfide nanocrystals precipitation. Earlier modeling steps using the LES model enabled the selection of two most favorable jet reactors’ geometries in the case of the nanoparticles production. After a development of a manufacturing procedure, molybdenum disulfide crystals, obtained experimentally in these systems, were characterized by the desired properties. This proves that the LES method is a good tool to predict an effect of mixing conditions and process parameters on characteristics and properties of the manufactured product.
Large eddy simulations were successfully used in all stages of a numerical procedure conducted in this work, thereby allowing a detailed assessment of this method capabilities. The bottlenecks for accurate LES-based predictions of complex chemical processes are also discussed and identified. It has been confirmed that the LES model gave consistently the most accurate results in all considered processes across the work. This and growing interest in LES, as well as increasing available computing power, allow to consider large eddy simulations as a good tool for solving engineering problems.
... The SAS process involves mixing CO 2 and organic solution (solute dissolved in an organic solvent) from two outputs of a coaxial nozzle system, to obtain the desired conditions to produce thermodynamic supersaturation and particle formation, and the resulting PSD (particle size distribution) is affected by blending [4]. While blending effects are well known to those based on precipitation in liquids [5] [1][2][3][4][5][6][7][8], needed to explain the interactions between turbulence and particle formation are valid only for standard fluids, and its extension to supercritical fluids is not simple [11]. It is known that the fluid properties change drastically near and above the critical point, then the dynamic behavior of the fluid also changes and this directly influences the mixing at all scales. ...
... The SAS process involves mixing CO 2 and organic solution (solute dissolved in an organic solvent) from two outputs of a coaxial nozzle system, to obtain the desired conditions to produce thermodynamic supersaturation and particle formation, and the resulting PSD (particle size distribution) is affected by blending [4]. While blending effects are well known to those based on precipitation in liquids [5] [1][2][3][4][5][6][7][8], needed to explain the interactions between turbulence and particle formation are valid only for standard fluids, and its extension to supercritical fluids is not simple [11]. It is known that the fluid properties change drastically near and above the critical point, then the dynamic behavior of the fluid also changes and this directly influences the mixing at all scales. ...
To investigate the influence of pressure and temperature on the jet velocity of a three-dimensional flow was the main goal of this study. Using a precipitation chamber with approximate capacity of 600 mL, it was studied the thermodynamic behavior of supercritical carbon dioxide mixture, dichloromethane and grape seed extract via SAS (supercritical antisolvent process). For the numerical solution, the Navier-Stokes equations were used along with the model of turbulence k-ε and Peng-Robinson equation of state with quadratic mixing rules of Van der Waals. The method of Chung was employed to determine the viscosity, thermal conductivity and mass diffusivity of the flow numerically solved through commercial code based on CFD (computational fluid dynamics). Simulations for pressures between 80 bar and 160 bar and temperatures between 308.15 K and 318.15 K showed large variations in the jet velocity, an important property in the dynamic mixing process that involves the size, size distribution and particle morphology. Nomenclature k Turbulent kinetic energy (m 2 ·s -2) p Pressure (bar) R Universal gas constant (J·mol -1 ·K -1) R Radial direction (m) T Temperature (K) t Time (s) v Velocity (m·s -1) x Axial direction (m) ρ Density (kg·m -3) σ Prandtl number (-) ω Fugacity (-) λ Thermal conductivity (W·m -1 ·K -1) τ Tensor stress (pa) ε Rate of turbulent kinetic energy dissipation (m 2 ·s -3) ν Molar volume (m 3 ·mol -1)
... Most of the studies described thus far have been limited to mixing at the macroscale. However, mixing at the microscale is also proven to be very important in many cases, as a matter of fact it is well known that micromixing can influence particle size (Baldyga et al., 2005). And these effects are also observed in supercritical fluids (Henczka et al., 2005). ...
... The resulting nano-sized metal oxide particles are characterized by their PSD, that is in turn affected by the relative rates of nucleation, growth and aggregation (Dirksen and Ring, 1991). The final product quality of these processes is often determined by the interplay between mixing and these three phenomena (Marchisio et al., 2006; Baldyga et al., 2005). When particle formation occurs on a time-scale comparable with that of mixing, a model capable of describing that type of interaction is needed. ...
This paper presents a mathematical model able to quantify mixing efficiency in Supercritical Water Hydrothermal Reactors (SWHR) for the production of different types of nano particles. In fact, mixing plays a crucial role in determining the final particle size distribution and therefore the final product quality. In this work, mixing of super critical water streams is studied with Computational Fluid Dynamics (CFD) by using the Reynolds Averaged Navier Stokes (RANS) approach coupled with an equation of state and a micromixing model, to take into account the effect of molecular mixing. The performance of the model is investigated in three different scenarios, corresponding to very different values of the Richardson number and very different mixer configurations. The main results show how mixing can be quantified by means of a global mixing time and how turbulence enhances the process, leading to better final product characteristics, especially in terms of lower mean particle size and narrower particle size distributions. This confirms previous research on this topic, highlighting the fact that both the mean particle size and the particle size distribution are strongly dependent on the mixing features of the SWHR.
... Because the bridging strength of particle pairs are affected by the time the agglomerate is exposed to the supersaturated region and the degree of supersaturation in that region, its history should be considered. Baldyga and Makowski [23] used CFD to calculate the distribution of supersaturated regions in a reaction crystallizer and analyzed their effect on grain size growth in a model case of the reaction crystallization of barium sulfate. ...
NixMnyCoz(OH)2 is widely used as a precursor for the cathode active material LiNixMnyCozO2 in lithium ion batteries, and the precursor size, which determines the size of the active cathode material, affects the characteristics of lithium-ion batteries. This paper proposes a method for predicting the particle size of Ni1/3Mn1/3Co1/3(OH)2 precipitated by semi-batch reaction crystallization. The distribution of supersaturated components formed in the reactor varies with the reactor scale, feed conditions of the raw material solution, and agitation conditions. Therefore, the method presented in this paper considers the effects of these conditions on particle growth. First, to identify the turbulent dispersion and concentration distribution of the supersaturated components formed in the reaction crystallizer, a computational fluid dynamics (CFD) model was constructed consisting of the governing equations of hydrodynamics and the mass balance equations of the supersaturated components considering the production by neutralization and consumption by precipitation. Next, a model of agglomeration was constructed that focused on the balance of the binding and breaking up energies for the particle pairs. The binding energy was quantified based on the bridging between particle pairs by surface deposition. The breaking up-energy was quantified based on the hydrodynamic forces when the agglomerate passes through the impeller. These models were fitted using experimental results for the final average size of the Ni1/3Mn1/3Co1/3(OH)2 secondary aggregate, which was precipitated in a small-scale stirred-tank type semi-batch reaction crystallizer. The models predicted the experimental results of the final average size in a large-scale crystallizer, with the feeding conditions of the raw material solution and stirring conditions as experimental parameters, within ±20%. The models may be used to analyze semi-batch reaction crystallization systems of NixMnyCoz(OH)2 of any composition by adjusting the model parameters according to the procedure developed in this study.
... Experimental and numerical studies indicate the importance of CO2 and solute-solvent mixture flow rates for process efficiency [25,26]; in particular, from a numerical point of view, when high flow rates are considered in the boundary conditions it causes numerical instabilities, making convergence a challenge. Another numerical challenge that also characterizes a gap in the literature concerns the compressible and non-isothermal nature of the flow in supercritical state flow. ...
In this work with CFD simulations, the evaluation of the supercritical anti-solvent (SAS) process for producing nanoparticles from an expanded solution of ethanol/solute in carbon dioxide is reported. The influence of the solution and antisolvent flow rates on mean particle size, the flow dynamic, and the supercritical mixture's jet velocity must be well established in the literature and analyzed. The high flow rate of the anti-solvent resulted in increased mean particle sizes for all studied cases. At the lowest flow rate of CO<sub>2</sub> examined, an increase in the solvent flow rate [0.3-1.0] ml/min initially led to a decrease of 11.2% in the mean particle diameter (MPD); however, further increasing the solvent flow rate [1.0-2.0]ml/min was an increase of 33% in this parameter. At the highest CO<sub>2</sub> flow rate, the behavior of MPS was the opposite; it had a rise de 13.5% in MPD with an increase in solvent flow rate; further increasing the flow rate of the solvent, there was a drop of 8.6% in MPD. Significant variations in the temperature lead to large fluctuations in the particle diameters. At last, the contact zones between CO<sub>2</sub> and ethanol were delimited, favoring the understanding of the influence of the flow patterns generated by the variation of the flow rates in the mean particle diameters.
... These results showed that the rate of precipitation could be controlled by the mixing rate [26], as the rate of reaction was proportional to the mixing rate. In contrast, the size of precipitate particles was inversely proportional to the mixing rate [27]. The reason why the uranium concentration at 400 rpm was higher than at 200 rpm may be because at a higher mixing rate the particles could be broken thus disturbing the nucleation. ...
Applicability of uranium peroxide ((UO2)(O2)·4H2O; UO4) precipitation to remove uranium from secondary wastewaters, generated as part of a spent uranium catalyst treatment process, has been studied. Precipitation characteristics of UO4 were evaluated as a function of pH, stirring status, time, and initial uranium concentrations. Optimized conditions were applied to the real uranium wastewater achieving ≥ 99% removal efficiency within 24 hours. Dissolved impurities (Si, Sb, Fe) were co-precipitated with UO4 leading to increased particle sizes and faster precipitation kinetics. Uranium was separated effectively from the wastewater to below the 1 mg L-1 criteria necessary for secondary wastewater discharge.
... TMBOs are usually generated by the co-precipitation of metal ions with reducer (e.g., NaBH 4 ). It is generally suggested that a homogeneous supersaturation, which is generated through delicate mixing of two reactant solutions, is essential for the precipitation of nanomaterials with good product quality because of the highly nonlinear dependency of nucleation rate on supersaturation level (Bałdyga et al., 2005;Chen et al., 1996;Lafficher et al., 2018). Large-scale synthesis of nanomaterials is usually performed in stirred tank reactors (STRs). ...
Large-scale synthesis of graphene-based nanomaterials in stirred tank reactor (STR) often results in serious agglomeration because of the poor control during micromixing process. In this work, reactive impingement mixing is conducted in a two-stage impinging jet microreactor (TS-IJMR) for the controllable and scale-up synthesis of nickel-cobalt boride@borate core-shell nanostructures on RGO flakes (NCBO/RGO). Benefiting from the good process control and improved micromixing efficiency of TS-IJMR, NCBO/RGO nanosheet provides a large BET surface area, abundant of suitable mesopores (2–5 nm), fast ion diffusion, and facile electron transfer within the whole electrode. Therefore, NCBO/RGO electrode exhibits a high specific capacitance of 2383 F g⁻¹ at 1 A g⁻¹, and still retains 1650 F g⁻¹ when the current density is increased to 20 A g⁻¹, much higher than those of nickel boride@borate/RGO (NBO/RGO) and cobalt boride@borate/RGO (CBO/RGO) synthesized in TS-IJMR, as well as NCBO/RGO-S synthesized in STR. In addition, an asymmetric supercapacitor (NCBO/RGO//AC) is constructed with NCBO/RGO and activated carbon (AC), which displays a high energy density of 53.3 W h kg⁻¹ and long cyclic lifespan with 91.8% capacitance retention after 5000 charge-discharge cycles. Finally, NCBO/RGO is used as OER electrocatalyst to possess a low overpotential of 309 mV at a current density of 10 mA cm⁻² and delivers a good long-term durability for 10 h. This study opens up the potential of controllable and scale-up synthesis of NCBO/RGO nanosheets for high-performance supercapacitor electrode materials and OER catalysts.
... Mixing efficiency plays an important role in the chemical industry, as it can affect selectivity and yield [1], the size of crystals and precipitates [2], or the molecular weight of polymers [3]. More specifically micromixing, the mixing at the molecular scale, determines the product distribution when fast complex chemical reactions take place [4]. ...
In this paper, micromixing is investigated in two Spinning Disk Reactors for liquids with increased viscosity. Glycerol and CMC were used as viscosity modifiers and micromixing characterization was performed by means of the Villermaux-Dushman method. For the first time, the effects of increasing the pH of the bulk solution leading to the disproportionation reaction of iodine are successfully modeled. A kinetic model is proposed to account for the presence of glycerol as a co-solvent. Micromixing efficiency decreases with increasing viscosity, yet the results are in line with the proportionality to the Kolmogorov timescale. The viscosity can be significantly increased using CMC as a viscosity modifier; however, due to the viscoelastic behavior of CMC, the effective viscosity decreases in the high shear environment inside the reactor. This leads to a much higher micromixing efficiency. It is shown that micromixing times can be estimated provided that the rheology of the liquids is known.
... The analysis based on mixing timescale are only used in first approximation, for scale-up purpose, for example, precise spatial and timescales of turbulence are extracted from computational fluid dynamics (CFD) simulation results. Vicum et al. [46], Bałdyga et al. [47] found satisfactory agreement between the experimental results of a reactive mixing process in a semibatch stirredtank reactor and an engulfment model, as well as a CFDbased approach. More recently, Bal et al. [48] have developed a coupled CFD d Population Balance equation model that incorporate, mixing, reaction nucleation, growth, coagulation and Ostwald ripening. ...
The study of the mechanisms laying underneath reactive crystallization is a key subject in a myriad of natural and industrial processes. Here, control on particle nucleation and growth are active topics of research, in which important advances can be highlighted within the last years. While novel mechanisms for crystal nucleation have been recently unveiled, the effects of reagent mixing process and timescales on them, together with the effect of impurities and/or additives present during the reaction, still lack of fundamental comprehension. This work intends to provide a general view on the latest advances made on the critical parameters involved in reactive crystallization processes: nucleation, fast and efficient mixing and use of additives.
... Therefore, coupling the turbulent mixing with the reaction kinetics in an appropriate way to bridge the existing gap between CFD and phenomenological models is necessary. Many efforts have been devoted to investigate the interactions between mixing and chemical reactions (Baldyga and Makowski 2010;Bałdyga, Makowski, and Orciuch 2005;Han et al. 2012;Vicum and Mazzotti 2007). A detailed CFD-based process model was presented by Vicum and Mazzotti (2007) who accounted for all relevant phenomena of the microscale, the mesoscale, and the macroscale in the mixing-precipitation. Han et al. (2012) studied the effect of mixing on reaction rate in viscous systems. ...
Solid particle dispersion and chemical reactions in high-viscosity non-Newtonian fluid are commonly encountered in polymerization systems. In this study, an interphase mass transfer model and a finite-rate/eddy-dissipation formulation were integrated into a computational fluid dynamics model to simulate the dispersion behavior of particles and the mass transfer–reaction kinetics in a condensation polymerization-stirred tank reactor. Turbulence fields were obtained using the standard k – ε model and employed to calculate the mixing rate. Cross model was used to characterize the rheological property of the non-Newton fluid. The proposed model was first validated by experimental data in terms of input power. Then, several key operating variables (i.e. agitation speed, viscosity, and particle size) were investigated to evaluate the dispersive mixing performance of the stirred vessel. Simulation showed that a high agitation speed and a low fluid viscosity favored particle dispersions. This study provided useful guidelines for industrial-scale high-viscosity polymerization reactors.
... Mixing is particularly important when the supersaturation is generated rapidly, for example by a reaction between different compounds (i.e., so-called precipitation or reactive crystallization) or by the mixing of a feed stream with a miscible solvent that drastically reduces solubility (i.e., antisolvent crystallization). A lack of rapid mixing can lead to a high supersaturation locally, which affects crystal morphology, purity and size distribution often in an unpredictable way [64][65][66]. Mahajan and Kirwan [67] crystallized Lasparagine monohydrate from aqueous solution using 2-propanol as antisolvent in a device that was designed for rapid mixing. ...
Process intensification (PI) provides great opportunities to drastically improve the performance of chemical processes within many branches of chemical industry including the pharmaceutical industry. Crystallization is an important purification and separation technology within many pharmaceutical processes. This paper provides a systematic review on the recent developments of PI approaches that are applicable to pharmaceutical crystallization. The various approaches are categorized according to the four fundamental PI domains (space, time, function, energy) that have been proposed in literature. Each approach is illustrated with examples from literature with an emphasis on the opportunities to intensify pharmaceutical crystallization processes. Finally, some thoughts on the level of maturity for industrial implementation of the various approaches are provided.
... However, the establishment of the supersaturation is essential for a good control of the process, as it determines both nucleation and growth rates (Mersmann, 1999). As precipitation of sparingly soluble materials is generally associated with very fast reactions, the solid properties can be strongly influenced by mixing quality (Bałdyga et al., 2005). The importance of controlling the mixing during the precipitation is due to the difference between the mixing time scales and the various precipitation mechanisms time scale (Claassen and Sandenbergh, 2007). ...
The textural properties of γ-alumina catalyst supports largely depend on the solid properties of their precursors. Precipitation in fast contacting mixers of two precursors, boehmite and NH4-dawsonite, were investigated in order to obtain different and enhanced solid features. The mixing devices, a sliding surface mixing device (SSMD) and a rotor-stator mixer (RSM), were operated continuously. Micromixing was characterized by using a competitive parallel reaction system (iodide-iodate). It was possible to achieve micromixing time from 1 to 200 ms according to the type of mixer and the operating conditions. The micromixing times assessed experimentally are in correct agreement with the theoretical ones. For boehmite precursor, the micromixing time had not a decisive influence on the crystallite size. Better control of the particle surface area was obtained considering the shear rate level, maybe due to a disordered aggregation. Conversely, reduction of the crystallite size with a decrease of the micromixing time was observed with NH4-dawsonite. A possible explanation lies in a higher local supersaturation, leading to a more intense primary nucleation. Moreover, it was possible to adjust the pore volume of the NH4-dawsonite’s aggregates with the operating conditions to quite a large extent (0.2 up to 0.8 cm³ g–1).
... S4). Obviously, although a large convective flow is expected to be induced by the large temperature gradient, sufficient micromixing is only achieved in the presence of stirring. For more details on the influence of mixing on particle formation, the reader is referred to the works of Schwarzer, Gradl and Peukert, [58] Metzger and Kind [59] and in particular of Baldyga et al. [60,61]. However, below 100 rpm which corresponds to an impel- ler Reynolds number of 600, the onset of focusing is clearly less pronounced which is ascribed to the fact that mixing is becoming insufficient. ...
The hot injection technique for the synthesis of quantum dots (QDs) is a well-established and widely used method in the lab. However, scale-up rules do not exist. One reason is that in particular the role of process parameters like mixing on particle formation is largely unknown, as systematic examination of the latter is impossible for the laborious and complex manual synthesis. Herein we studied the mixing induced self-focusing of particle size distributions (PSDs) of CdSe QDs using automation in combination with a defined stirrer geometry. Basis for our study is a platform that allows parallelization with inline temperature monitoring, defined injection rate, accurate sampling times as well as controlled stirring. Reproducibility in terms of optical product properties was analyzed by absorption and emission whereas reproducibility in terms of the PSD was verified by deconvolution of UV/Vis absorbance spectra and especially by analytical ultracentrifugation (AUC) complemented by transmission electron microscopy. In line with previous results, AUC confirmed that even QDs made by hot injection in an automated setup are polydisperse with multimodal size distributions. Finally, reproducibility in combination with early stage sampling and controlled mixing allowed us for the first time to analyze the influence of stirring on focusing and defocusing of PSDs, that has been expressed in terms of the evolution of the relative standard deviation (RSD). Our work paves the way to gain in-depth understanding of often forgotten process-structure relationships of colloidal nanoparticles which eventually is a first step in the direction of the development of scalable synthesis and reliable application of high-quality QDs in technical applications.
... Ammonium dawsonite (NH 4 Al(OH) 2 CO 3 ) can be an interesting alternative choice for preparation of γ-Al 2 O 3 with high purity level and new textural properties [9]. Various methods can be used to synthesize this precursor [10], but the main one remains the precipitation in aqueous phase of aluminium salts (Al As precipitation of sparingly soluble materials, such as boehmite or NH 4-dawsonite, is generally associated with very fast reactions, the solid properties can be strongly influenced by mixing quality [11]. Therefore, the choice of the mixing technology is particularly important. ...
γ-Alumina is a widely used porous material for catalytic application. Possible routes for alumina improvement can be the use of alternative precursors as well as innovative precipitation technologies. In this study, we compare the influence of both precursor chemistry and mixing efficiency on γ-alumina properties. The conventionally used boehmite and the NH4-dawsonite precursors were precipitated using three mixing technologies: a conventional stirred-tank reactor, a rotor-stator mixer and a sliding surface mixing device. It was observed that, in the study conditions, γ-alumina mean pore diameter and porous volume were particularly sensitive to both precursor and mixing technology, while specific surface area was rather precursor dependent. A wide porosity range can thus be covered at isospecific surface area, using several precursor/mixing technology systems.
... Like all precipitation processes, this reaction is influenced by mixing. Mixing liquids to precipitate solid particles is a common multiphase chemical process that comprises several complex phenomena [47][48][49][50][51][52][53][54][55][56][57]. ...
Despite two decades of intensive laboratory investigations, several aspects of contaminant removal from aqueous solutions by elemental iron materials (e.g., in Fe0/H2O systems) are not really understood. One of the main reasons for this is the lack of a unified procedure for conducting batch removal experiments. This study gives a qualitative and semi-quantitative characterization of the effect of the mixing intensity on the oxidative dissolution of iron from two Fe0-materials (material A and B) in a diluted aqueous ethylenediaminetetraacetic solution (2 mM EDTA). Material A (fillings) was a scrap iron and material B (spherical) a commercial material. The Fe0/H2O/EDTA systems were shaken on a rotational shaker at shaking intensities between 0 and 250 min-1 and the time dependence evolution of the iron concentration was recorded. The systems were characterized by the initial iron dissolution rate (kEDTA). The results showed an increased rate of iron dissolution with increasing shaking intensity for both materials. The increased corrosion through shaking was also evidenced through the characterization of the effects of pre-shaking time on kEDTA from material A. Altogether, the results disprove the popular assumption that mixing batch experiments is a tool to limit or eliminate diffusion as dominant transport process of contaminant to the Fe0 surface.
... Like all precipitation processes, this reaction is influenced by mixing. Mixing liquids to precipitate solid particles is a common multiphase chemical process that comprises several complex phenomena4748495051525354555657. The reaction between Fe 3+ and OH − initially forms soluble Fe(OH) 3 , but in a supersaturated, metastable state relative to its equilibrium solubility product. ...
Despite two decades of intensive laboratory investigations, several aspects of contaminant removal from aqueous solutions by elemental iron materials (e.g., in Fe(0)/H2O systems) are not really understood. One of the main reasons for this is the lack of a unified procedure for conducting batch removal experiments. This study gives a qualitative and semi-quantitative characterization of the effect of the mixing intensity on the oxidative dissolution of iron from two Fe(0)-materials (materials A and B) in a diluted aqueous ethylenediaminetetraacetic solution (2 mM EDTA). Material A (fillings) was a scrap iron and material B (spherical) a commercial material. The Fe(0)/H2O/EDTA systems were shaken on a rotational shaker at shaking intensities between 0 and 250 min(-1) and the time dependence evolution of the iron concentration was recorded. The systems were characterized by the initial iron dissolution rate (k(EDTA)). The results showed an increased rate of iron dissolution with increasing shaking intensity for both materials. The increased corrosion through shaking was also evidenced through the characterization of the effects of pre-shaking time on k(EDTA) from material A. Altogether, the results disprove the popular assumption that mixing batch experiments is a tool to limit or eliminate diffusion as dominant transport process of contaminant to the Fe(0) surface.
Handbook of Industrial Crystallization - edited by Allan S. Myerson June 2019
The objective of this work is to investigate the interaction of turbulence with the nonlinear processes of particle nucleation and growth that occur in reaction crystallisation, also known as precipitation. A validated methodology for coupling the population balance equation with direct numerical simulation of turbulent flows is employed for simulating an experiment conducted by Schwarzer et al. ( Chem. Engng Sci. , vol. 61, no. 1, 2006, pp. 167–181), where barium sulphate nanoparticles are formed by mixing and reaction of barium chloride and sulphate acid in a T-mixer, with the spatial resolution resolved down to the Kolmogorov scale. A unity Schmidt number is assumed, since at present it is not possible to resolve the Batchelor scale for realistic Schmidt numbers (order of 1000 or more). The probability density function, filtered averages and spatial distribution of time and length scales are all examined in order to shed light on the interplay of turbulence and precipitation. Separate Damköhler numbers are defined for nucleation and growth and both are found to be close to unity, indicating that the process is neither mixing nor kinetics controlled. The nucleation length scales are also evaluated and compared with the Kolmogorov scale to show the importance of resolving nucleation bursts. In addition, zones of different rate-determining mechanisms are identified. The ultimate aim of precipitation is to obtain control over the product particle size distribution, and the present study elucidates the synergistic or competing roles of mixing, nucleation and growth on the process outcome and discusses the implications for modelling.
Sonochemical synthesis has flourished significantly in the last few decades for the preparation of photocatalysts. A large number of photocatalysts have been prepared through sonochemical techniques. This review highlights the scope of sonochemistry in the preparation of photocatalysts, and their applications in energy production and environmental remediation. Besides, the sonochemical degradation of pollutants is discussed in detail. The progress made in sonochemical synthesis and the future perspective for this technique are summarized here. This review may create more enthusiasm amongst researchers to pay extra attention to the sonochemical synthesis of materials and add their useful contribution to the investigation of new materials for photocatalytic and other applications. This will propel this technique toward commercial sonosynthesis of nanomaterials.
Melt crystallization has been considered as a green separation technique and widely applied in industry and manufacture due to several attractive features, including no need for solvent, achieving specific product...
The particle size distribution (PSD) of precipitated products is often influenced by mixing. This poses a serious challenge if stirred-tank reactors (STRs) are applied, because the complex fluid dynamics in STRs impedes process development and scale-up. Previous research has been limited to investigating the impact of dimensional physical or chemical parameters because their governing Π groups remained unclear. We aim within this work to fill this gap by transferring our recently developed theory of complete similitude to STRs. Hence, we propose that the dimensionless PSD depends solely on the STR’s Reynolds number, Re, the solids formation Damköhler number, Dasf and the feed velocity ratios, ζj if the same materials, educt concentration ratios and geometrically similar reactors are applied. Barium sulfate precipitation experiments in two semi-batch STRs with 0.4 and 10.8 L are conducted to validate this hypothesis. In addition to proving that complete similitude is possible for STRs, the results indicate that Re does not impact the dimensionless PSD for flows dominated by meso or macro mixing. This finding is further supported by a timescale analysis revealing that, contradictory to literature, meso and macro mixing are independent of Re. Furthermore, we demonstrate that complete similitude represents a fundamental framework from which other novel and established scale-up criteria can be derived.
The Villermaux–Dushman method, one of the most extensively used test reaction systems for micromixing characterization, has been widely criticized for years due to uncertainties regarding the incomplete dissociation of sulfuric acid and the proposed kinetic study by Guichardon et al. In this work, a renewed study of the kinetics of the iodide–iodate reaction is presented, using perchloric acid to avoid issues concerning incomplete acid dissociation. The experimental results are in good agreement with the fifth-order rate law for the iodide–iodate reaction. The reaction rate coefficient strongly depends on the ionic strength and can be modeled with a Davies-like equation. When implemented in the incorporation model, the kinetic model presented in this study can be used to estimate micromixing times that are in line with the theoretical engulfment time. This is observed in two different reactors with low and high intensities of mixing: an unbaffled stirred vessel and a rotor–stator spinning disk reactor. The results from the latter are also compared with the second Bourne reaction, giving very similar micromixing times. The kinetic model presented in this study in combination with strong monoprotic acids is suggested as an alternative to characterize micromixing behavior.
Reactive and antisolvent crystallization processes are efficient techniques for the production of fine chemicals and pharmaceuticals. These processes are usually characterized by high levels of supersaturation and fast primary crystallization kinetics. Understanding the mixing mechanism and its interaction with various crystallization phenomena is advantageous for achieving robust control of these crystallization processes and the desired crystal properties. Much modeling effort has been devoted to accounting for the combined effects of mixing on various scales and the complicated kinetic phenomena involved in these processes as complementary to experimental investigations. This article provides a review on the numerical simulation of reactive and antisolvent crystallization processes with an emphasis on modeling the mixing effect. Relevant Lagrangian mechanistic micromixing models and probability density function (PDF) approaches are introduced followed by simulation examples from the literature. Various examples are categorized into four groups according to the different coupling between mixing, crystallization kinetics and flow field. Also, some suggestions on further model development are given to enhance their predictive performances and broaden the range of application.
Numerous precipitation methods for creating nanoparticle dispersions that are based on mixing a solution with a miscible nonsolvent have been developed. Here, we show that for polymer particles, the formation is highly dependent on the rate of mixing. We also demonstrate the importance of the glass transition of the polymers on particle formation. A simple model of droplet formation during mixing provides a satisfactory description of the observed dependence of particle size on polymer molecular weight, concentration, solvent ratio, and mixing conditions.
Nickel‐cobalt borides (denoted as NCBs) have been considered as a promising candidate for aqueous supercapacitors due to their high capacitive performances. However, most reported NCBs are amorphous that results in slow electron transfer and even structure collapse during cycling. In this work, a nanocrystallized NCBs‐based supercapacitor is successfully designed via a facile and practical microimpinging stream reactor (MISR) technique, composed of a nanocrystallized NCB core to facilitate the charge transfer, and a tightly contacted Ni‐Co borates/metaborates (NCBi) shell which is helpful for OH⁻ adsorption. These merits endow NCB@NCBi a large specific capacity of 966 C g⁻¹ (capacitance of 2415 F g⁻¹) at 1 A g⁻¹ and good rate capability (633.2 C g⁻¹ at 30 A g⁻¹), as well as a very high energy density of 74.3 Wh kg⁻¹ in an asymmetric supercapacitor device. More interestingly, it is found that a gradual in situ conversion of core NCBs to nanocrystallized Ni‐Co (oxy)‐hydroxides inwardly takes place during the cycles, which continuously offers large specific capacity due to more electron transfer in the redox reaction processes. Meanwhile, the electron deficient state of boron in metal‐borates shells can make it easier to accept electrons and thus promote ionic conduction.
The paper presents an application of large eddy simulations (LES) to predict the course of barium sulfate precipitation carried out in jet reactors and basic guidelines for a reactor design. The reactors in question were of different geometries and made in different sizes in order to achieve high mixing intensities and low residence times in a mixing chamber, thus enabling to reflect and understand the effects of process conditions and influence of mixing on the course of precipitation better. The system’s behavior is explained using experimental and simulation results. Simulations were validated by comparing LES model predictions with experimental data, as well as the k-ε model supplemented with the multiple-time-scale mixing model. The BaSO4 precipitation results obtained with a CFD based precipitation model are in a very good agreement with experiments and are used to distinguish the jet reactors that allow obtaining a product with favorable characteristics, i.e., the smallest mean particle size. In general, the smallest studied reactors are found to best meet this criterion. A special attention is given to precipitation kinetics models, the choice of which can significantly affect predictions’ accuracy. Several literature precipitation kinetics models were tested and the ones working best in jet reactors were identified. Finally, a time-scale analysis of process controlling mechanisms is highlighted and the assumption of negligibility of the subgrid closure for precipitation in LES is validated.
A zone model for the semi-batch precipitation process of sparingly soluble salts in stirred-tank reactors is presented. The low product solubility of these substances leads to high levels of supersaturation during the process. Consequently, nucleation and growth rates are fast and solids formation only takes place in a part of the reactor close to the feed pipe. The presented local mixing environment (LME) model extends an existing model for semi-batch processes, which consists of two zones. In the mixing zone, a steady-state plug-flow reactor (PFR) is used to imitate the local flow environment of the feed. The tank is approximated as a well-mixed storage tank outside the mixing zone. Exchange streams between the two zones are estimated by dimensional analysis considering stirrer type, size and rotational speed. The correctness of the two-zone hypothesis and the accuracy of the LME model is validated by barium sulfate precipitation in an experimental comparison setup using different stirrer types, rotational speeds and feed rates. The PFR is represented in the experiments as a pipe-in-pipe reactor in jet-in-cross-flow (JICF) or coaxial-flow (COAX) arrangement. The experimental results show that the semi-batch precipitation of sparingly soluble salts in stirred-tanks can be successfully simplified by the assumption of a PFR mixing zone. The LME model is simple to implement, scalable and reaches acceptable results in the experimental validation. It is therefore a promising model for future application in process simulation of industrial precipitation processes.
Mixing of initially distinct substances plays an important role in our daily lives as well as in ecological and technological worlds. From the continuum point of view, which we adopt here, mixing is complete when the substances come together across smallest flow scales determined in part by molecular mechanisms, but important stages of the process occur via the advection of substances by an underlying flow. We know how smooth flows enable mixing but less well the manner in which a turbulent flow influences it; but the latter is the more common occurrence on Earth and in the universe. We focus here on turbulent mixing, with more attention paid to the postmixing state than to the transient process of initiation. In particular, we examine turbulent mixing when the substance is a scalar (i.e., characterized only by the scalar property of its concentration), and the mixing process does not influence the flow itself (i.e., the scalar is “passive”). This is the simplest paradigm of turbulent mixing. Within this paradigm, we discuss how a turbulently mixed state depends on the flow Reynolds number and the Schmidt number of the scalar (the ratio of fluid viscosity to the scalar diffusivity), point out some fundamental aspects of turbulent mixing that render it difficult to be addressed quantitatively, and summarize a set of ideas that help us appreciate its physics in diverse circumstances. We consider the so-called universal and anomalous features and summarize a few model studies that help us understand them both.
Crystallization has been applied to a broad range of industries such as bulk and fine chemicals, pharmaceutical and food industries. It is important to strategically control the in-situ purification process during crystallization to meet the regulatory and functional specifications of the crystals. While the control of the crystallization-purification process has been widely discussed for batch crystallizers, there has been little focus with the literature on controlling purification for continuous crystallizers. Continuous crystallization is a more intensified approach to crystallization, with lower capital footprint and potentially offering more consistent quality control. This review paper provides an in-depth discussion of the strategies and scientific understanding in controlling the crystallization-purification process in continuous crystallization. In particular, it describes how scientific understanding in the purification process generated so far for batch crystallization, can be translated to continuous crystallization.
This work aims to study the synthesis conditions effect on the photocatalytic properties of manganese tungstate (MnWO4) for H2 production by the water splitting reaction under visible light irradiation. This is achieved by relating the materials characterization and photocatalytic evaluation of MnWO4 at different synthesis conditions. MnWO4 was synthesized through a precipitation reaction between Mn²⁺ and (WO4)²⁻ ionic species, while adding oleic acid (OA) as surfactant at two concentrations (0.1% and 1% V) and using three different stirring methods: magnetic stirring (AM), ultrasound (US) and high-shear stirring (UT). Characterization was carried out by TGA, XRD, BET surface area, UV–Vis spectroscopy and FESEM. XRD patterns confirm the wolframite structure of MnWO4. BET surface area increased by using UT stirring. UV–Vis spectroscopy results revealed indirect transition Eg values of ≈2 eV, favorable for the MnWO4 photoactivation under visible light irradiation. During the photocatalytic evaluation, sample 1%-UT produced the highest H2 amount among all samples with a value of 72 μmolH2g⁻¹, which was far higher compared to WO3, which was taken as a reference photocatalyst.
La mécanique des fluides numérique (CFD) est utilisée pour modéliser la précipitation du carbonate de baryum dans un réacteur à lit fluidisé. L'étude est divisée en deux parties : la modélisation de l'hydrodynamique du réacteur et la modélisation de la précipitation du carbonate de baryum. Pour ces deux parties, les modèles sont validés par l'expérience. Dans la première partie de l'étude, des réactions instantanées de neutralisation en absence et en présence de particules solides inertes sont mises en oeuvre dans le réacteur à lit fluidisé. Pour représenter la réaction chimique dans la phase liquide, plusieurs modèles son testés : le modèle Eddy Dissipation (EDM), le modèle Eddy Dissipation Concept (EDC) et le modèle modifié Eddy Dissipation- Multiple Time Scale turbulent mixer (EDM-MTS). Le modèle qui donne la meilleure prédiction de la réaction chimique est choisi : il s'agit du modèle EDM-MTS, qui demande aussi le moins de temps de calcul. Dans la deuxième partie, l?équation de bilan de population est incorporée au code de calcul pour prédire la distribution de taille des particules (PSD). La méthode des classes est implémentée couplée avec le modèle des écoulements multiphasiques Eulérien-Eulérien, le modèle de turbulence k-e et le modèle EDM-MTS. Les cinétiques de précipitation de BaCO3 sont incluses dans le modèle. Des expériences de précipitation du BaCO3 en réacteur à lit fluidisé permettent de valider le modèle de CFD développé. La PSD donnée par le modèle de CFD est en bonne concordance avec les résultats expérimentaux
Superheated steam drying has been receiving research attention in recent years due to its increasing industry prevalence in the food and agricultural sector. There is however a gap of knowledge in superheated steam spray drying involving the drying of droplets with dissolved solids as most application to date are solely on solid materials. With this constraint, it is vital to understand and study the effect of superheated steam on the particle formation process as well as on the final particle. The aim of this work is to explore the potential of superheated steam in the spray drying of milk. Specifically, this report examines how superheated steam influences the migration of fats, protein, and lactose in milk during the particle formation process. Studies were conducted by drying fresh milk, using a single droplet drying technique in a superheated steam environment and a hot air environment at a fixed temperature of 110°C. The wettability of the dried single particle was examined using contact angle measurements. The surface of superheated steam-dried milk particles revealed a relatively higher wettability when compared to air-dried milk particles. This suggests that superheated steam promoted the presence of hydrophilic components such as lactose on the exterior surface of the particle. These results have therefore shown the possibility of using superheated steam to control component relocation in multicomponent solutions based on the component hydrophilicity. By recognizing the potential of application of superheated steam in spray drying, engineered multicomponent particles with specific features can be produced.
Mixing conditions were explored as a possible avenue for improvement of demulsifier performance in solvent diluted bitumen de-watering process. The effects of demulsifier bulk concentration, demulsifier injection concentration and mixing energy on water and solids removal from the oil phase were tested. All of the experiments were carried out in a confined impeller stirred tank (CIST), which provides well characterized mixing conditions and relatively uniform flow and turbulence. Results showed that lowering the injection concentration and increasing the mixing energy both improve demulsifier performance, allowing a 50% drop in the bulk concentration of demulsifier. This result agrees well with an earlier study by Laplante, et al. ¹ where a different demulsifier was investigated. In that study, it was shown that the product of mixing time and energy dissipation rate at the feed point (the mixing energy = J) provides an alternate mixing variable.
In the paper, a discussion on the basic problems of the population balance modelling is carried out and the application of the population balances to modelling of several chemical reactor processes is presented. This includes precipitation of particles from aqueous solutions, formulation of particulate products with supercritical fluids, liquid-liquid dispersions and effects of particulate populations on suspension rheology.
In this work the process of precipitation in the single and double feed semibatch stirred tank reactors is modeled by using CFD in combination with the closure scheme. Difficulties in modeling of semibatch process result from the fact that it is necessary to follow the process using CFD during long feed times, several orders of magnitude longer than the time period of revolution of the impeller. The problem is solved by discretizing the feed into portions and solving the differential balance equations for mass, momentum, species and populations of particles until the steady-state for each portion is obtained. Results of modeling are compared with experimental data for the semibatch precipitation of barium sulphate from aqueous solutions of BaCl2 and Na2SO4.
The influence of the addition of a drag reducing agent (100ppm PAA) to a pure liquid (water) in a stirred vessel has been studied. The vessel was stirred with two types of agitators, a Rushton turbine and an axial A310 impeller.This study was meant to implement previous findings with new data and various types of measurements were performed to determine a number of parameters. It is confirmed that the drag-reducing agent decreases power draw and increases mixing time. Mean flow and the fluctuating velocities were determined (with the PIV technique) and notable reduction in turbulence intensity was detected. The mechanism of tracer dispersion (followed by the PLIF technique) was also documented qualitatively. Finally, the influence of the drag-reducing agent on the particle size distribution of the solid product of a precipitation reaction (reaction of barium chloride and sodium sulphate) was shown.
The way in which reagents are mixed can affect significantly the size distribution and morphology of the solid product. The process of precipitation can be so fast that even the micromixing process that is usually the fastest of the turbulent mixing sequence can affect the course of precipitation and the quality of precipitation product. Effects of mixing are particularly high in the case of the double-feed precipitation process. The process of BaSO4 precipitation in a double-feed semibatch stirred tank reactor is modelled using a non-equilibrium multiple-time-scale mixing model and a beta distribution of the mixture fraction in combination with a simple conditional moment closure based on linear interpolation of local instantaneous reactant concentration values. Mixing, nucleation and crystal growth are modeled in the feed zone using CFD, whereas nucleation and growth of crystals in the bulk are simulated using a simple concept of the semibatch well-mixed tank. Modelling predicts all trends observed in experiments.
CFD techniques are used to study the precipitation of barium carbonate in a solid–liquid fluidized bed reactor. Experimental analysis of the hydrodynamic behaviour for a neutralization reaction in the fluidized bed column, followed by CFD simulations is carried out using different reaction models. The Eddy Dissipation model, the Eddy Dissipation model-MTS and the Eddy Dissipation Concept micro-mixing models are tested in order to simulate the acid–base instantaneous reaction.The modelling of the precipitation in a fluidized bed reactor is coupled with the Eddy Dissipation model-MTS micro-mixing model and the k-ɛ standard turbulence model. Barium carbonate is chosen as the model substance. The liquid phase reaction rate as well as the nucleation, growth and aggregation kinetics is included in the precipitation model. The discrete method is chosen to solve the population balance equation. The activity coefficient needed to calculate the supersaturation in the kinetic equations is determined using the Bromley's method. A good agreement is obtained between the experimental results and the CFD results.
Despite two decades of intensive laboratory investigations, the removal mechanism of several contaminants from aqueous solutions by elemental iron (e.g. in Fe0/H2O systems) are not really elucidated. Two of the major reasons for this are: (i) the failure to consider Fe0/H2O systems as consisted of the elemental iron material (Fe0) covered by a layer of corrosion products (oxide-film), and (ii) the failure to treat properly the combined problem of mass transport and chemical reaction in these complex systems. Well-mixed batch experiments that have been undertaken in order to circumvent the mass-transport problem associated with bulk solutions have not always adequately addressed these key issues. Mixing intensity may not only affect the hydrodynamic but also the chemical dynamics, in particular the formation of the oxide-film. The present work presents a critical review on the process of oxide-film formation and its impact on the process of mass-transport to the Fe0 surface. It is shown that well-mixed batch systems are not necessarily an effective tool for investigating the mechanism of contaminant removal by Fe0 since mixing may increase corrosion rate, avoid/delay the formation of oxide-films and/or provoke their abrasion. This discussion suggests that quantitative abiotic contaminant reduction in Fe0/H2O systems may mostly occur within the oxide-film as result of: (i) electron transfer from Fe0 surface, (ii) catalytic activity of secondary reductants (FeII, H2/H). Non-shaken batch experiments are proposed as a simple tool to investigate mass-transport limitation through oxide-films at laboratory scale. Working with stationary Fe0 samples and controlled stirring speeds may allow the investigation of oxide-film effect under more realistic conditions.
This review focuses on using precipitation (bottom-up) method to produce water-insoluble drug nanocrystals, and the stability issues of nanocrystals. The precipitation techniques for production of ultra-fine particles have been widely researched for last few decades. In this techniques, precipitation of solute is achieved by addition of a non-solvent for solute called anti-solvent to decrease the solvent power for the solute dissolved in a solution. The anti-solvent can be water, organic solvents or supercritical fluids. In this paper, efforts have been made to review the precipitation techniques involving the anti-solvent precipitation by simple mixing, impinging jet mixing, multi-inlet vortex mixing, the using of high-gravity, ultrasonic waves and supercritical fluids. The key to the success of yielding stable nanocrystals in these techniques is to control the nucleation kinetics and particle growth through mixing during precipitation based on crystallization theories. The stability issues of the nanocrystals, such as sedimentation, Ostwald ripening, agglomeration and cementing of crystals, change of crystalline state, and the approaches to stabilizing nanocrystals were also described in detail.
Solubilization of fluticasone propionate (FP) was effected using aqueous solutions of (i) different grades of poly(ethylene glycol) (PEG), (ii) methanol, and (iii) acetone to enable antisolvent crystallization by the addition of water. The solubility of FP in acetone was significantly higher than in PEG 400 or PEG 6000, and FP solubility was observed to be nonideal in either cosolvent. Crystallization of FP was instantaneous upon addition of water as antisolvent, with nucleation occurring during the mixing phase. The smallest crystals were produced in all cases from PEG solvents, which was attributed to a greater degree of nucleation and microcrystals of a size-range were produced. Crystals produced from PEG solvents also displayed a resistance to agglomeration and Ostwald ripening, which was observed to affect the morphology of FP crystallized from either methanol or acetone by the addition of water. In spite of the very rapid kinetics of solid formation, FP crystallized as the stable Form I polymorph from PEG 400 and PEG 6000. Conversely, mechanical milling of highly crystalline particles resulted in the generation of disorder in the crystals, which was apparent from surface dynamic vapor sorption analysis. The generation of FP microcrystals with a small size range from environmentally benign PEG solvents as the stable crystalline form represents an improvement over current crystallization and micronization techniques for the production of inhalable FP.
An experimental setup based on a static micromixer is used to determine nucleation and growth kinetics of L-asparagine monohydrate precipitated via antisolvent addition. Mixing in static micromixers was characterized previously using competitive-parallel reactions and computational fluid dynamics (Lindenberg et al., Chem Eng Sci. 2008;63:4135–4149). In this work, the mixer setup is used to determine nucleation and growth kinetics of L-asparagine at high supersaturations, i.e., true kinetics which are not affected by transport limitations. The method is based on measuring the particle size distribution obtained at different residence times. A population balance equation model of the process is used for the design of a continuous precipitation process. Finally, an analysis of the characteristic time scales of nucleation, growth, and mixing shows that, under the conditions in this study, mixing is much faster than precipitation and that the two processes can be decoupled. © 2010 American Institute of Chemical Engineers AIChE J, 2011
Crystallization processes in the pharmaceutical industry are usually designed to obtain crystals with controlled size, shape, purity, and polymorphic form. Knowledge of the process conditions required to fabricate crystals with controlled characteristics is critical during process development. In this work, continuous crystallization of ketoconazole, flufenamic acid, and l-glutamic acid in a nonconventional plug flow crystallizer was investigated. Kenics type static mixers were used to promote homogeneous mixing of active pharmaceutical ingredient solution and antisolvent. A strategy of multiple points of addition of antisolvent along the crystallizer was evaluated to control the size of the crystals. Interestingly, it was found that crystal size can be increased or decreased with an increased number of antisolvent addition points, depending on the kinetics of the system. It was also found that smaller crystals with a narrower size distribution can be obtained with the static mixers. A model to describe the continuous crystallization process was developed through the simultaneous solution of a population balance equation, kinetics expressions for crystal growth and nucleation, and a mass balance. The comparison of experimental and calculated values for crystal size distribution revealed that a growth rate dispersion model could describe accurately the continuous crystallization process. Collision of crystals with each other and with mixing elements inside the crystallizer may be the source of random fluctuation of the growth rate in the nonconventional plug flow crystallizer with static mixers.
Supercritical fluids have been extensively used for particle production of many natural and pharmaceutical substances providing useful alternatives for pharmaceutical and nutra-ceutical particulate system formulation. Among the different methods, the gas or supercritical antisolvent (GAS or SAS) process and its variants, have received a considerable interest due to the wide range of materials that can be micronized. Controlling particle formation in order to nucleate small particles is a key issue in GAS and SAS processes and this is directly related to mixing at all scales. In this work, we focus on numerical simulation of the process, emphasizing mixing modeling. Different mixing devices characterized by different nozzles are analyzed, to get an insight into mixing dynamics and its influence on the final particle size distribution. Results show that mixing is determinant in obtaining small particles, and that mixing at the microscale is a significant parameter to account for in the proper design of precipitators. (C) 2011 American Institute of Chemical Engineers AIChE J, 58: 385-398, 2012
In this study a model for the prediction of polyacrylic acid/protamine nanoparticle precipitation was developed, which constitutes the first numerical approach for modeling the precipitation of organic nanoparticles. Due to the complexity of the process, a spatial resolution (i.e., a simulation via computational fluid dynamics) was not the target. For the description of the precipitation process, population balance equations, accounting for nucleation, growth and aggregation were used and coupled with the engulfment model for mixing. Furthermore, experiments have been carried out by turbulent mixing of precursors in a vortex mixer. Measurements of the resulting particle size distributions and of the final concentrations in the liquid were performed. Finally, by selecting unknown parameters, it was possible to achieve a good agreement of experimental and numerical results. It was found that the formation of the electrostatic surface charge, caused by a layer of protamine on the particle surface, cannot be described instantaneously. Small deviations in the final liquid concentrations and particle size distributions are probably caused by the significant simplifications introduced by the engulfment model and by the description of the solid/liquid equilibrium. However, the presented model may serve as a basis for further development, i.e., by replacing the engulfment model by more sophisticated approaches, e.g., via computational fluid dynamics.
Mixing in static mixers is studied using a set of competitive-parallel chemical reactions and computational fluid dynamics (CFD) in a wide range of operating conditions. Two kinds of mixers, a wide angle Y-mixer and a two jet vortex mixer, referred to as Roughton mixer, are compared in terms of reaction yields and mixing times. It is found that the Roughton mixer achieves a better mixing performance compared to the Y-mixer. The effect of flow rate ratio on mixing in the Roughton mixer has been studied as well and it is shown that the mixing efficiency is not affected by the flow rate ratio. Moreover, experimental results and model predictions are in good agreement for all mixer geometries and operating conditions. CFD is used to calculate absolute mixing times based on the residence time in the segregated zone and it is shown that mixing times of less than 1 ms can be achieved in the Roughton mixer. In addition, CFD provides insight in local concentrations and reaction rates and serves as a valuable tool to improve or to scale-up mixers.
Assuming that complete information is available about (a) the chemical reactions (mechanism, stoichiometry, thermodynamics, and kinetics) and (b) the turbulent flow field in a reactor (mean velocities and turbulence parameters), we would like to calculate directly the time evolution of the temperature and the concentrations of all species present. No g e n e r a l method is yet available to satisfy this goal. Starting from the phenomena of reaction, molecular diffusion, and laminar deformation in laminated structures, which are formed by the action of vorticity, mixing in liquids at sub‐Kolmogorov scales can be quantitatively described. This approach requires the compositions of the fluids incorporated into the vortices to be known. Thus, for example, when a reagent solution (B) is slowly added to a huge volume of a second reagent solution (A) in a stirred tank reactor, fresh B mixes with the average A composition. (Slowly means that the addition time is much greater than the time needed for turbulent and molecular transport to homogenize the tank contents.) Progress of the reactions is then determined essentially by the interaction between the fine‐scale mixing (near the Batchelor concentration scale) and the chemical kinetics.
A convenient way to evaluate this modeling compares predicted and measured product distributions (or selectivities) for fast multiple reactions. (Up to now, this has been done for dilute solutions, which exhibited concentration, rather than also temperature segregation.) Advantages compared to working with single reactions can include enhanced sensitivity, inclusion of a wider range of flow fields than turbulent pipe flow, e.g., well‐stirred systems, and practical relevance of controlling product distributions in the manufacture of chemicals. The effects on the product distributions of fast reactions of operating parameters such as stoichiometric and volumetric feed ratios, reagent concentrations, feed location, impeller speed, size and shape, viscosity, and type of reaction can be well described. Present efforts concentrate on relaxing the assumption of s l o w feed addition and of describing reactors where not only fine‐scale, but also coarse‐scale inhomogeneities occur. A simplified model for the engulfment frequency between fluids having different compositions is being evaluated. Some progress has been made, but the way in which fluid elements mix with each other is still not fully known.
Effects of turbulent mixing on the course of two fast parallel chemical reactions (neutralisation of sodium hydroxide and hydrolysis of ethyl chloroacetate) carried out in a continuous flow and semibatch stirred tank reactors are experimentally investigated and theoretically interpreted. The flow field is computed using the k-ε model. Mixing effects are modelled using the non-equilibrium model for scalar dissipation and the conditional moment closure based on the Beta PDF. The models are implemented to the CFD code.
A CFD package was employed to simulate the steady state precipitation of barium sulphate in jet mixing devices. Both 2D and 3D descriptions were employed. Solutions of the Navier-Stokes equations along with the standard two equation k-ϵ turbulence model provide the velocity and energy fields while generation and development of the solid crystal phase were described by the moment equations of the population balance. Spatial distributions of species concentrations, supersaturation, total crystal number and magma density were calculated and the average size and coefficient of variation of the crystal product determined. Visualization of these distributions enables a rapid assessment of the effect of the actualflowpattern within a particular jet precipitator. The influence of the positions and orientation of the two inlet streams, as well as the length of the mixer on the mixing and the precipitation were investigated.
The effect of laminar micromixing on parallel reactions was studied experimentally and interpreted theoretically using a new model of laminar micromixing, developed by applying an integral transformation to material balance equations in the Lagrangian frame of reference. A solution of sodium hydroxide was contacted with a premixture of hydrochloric acid and ethyl chloroacetate solutions in two reactors: a semibatch tank reactor and a twin-screw extruder. The viscosities of the mixed solutions were increased to 0.3 Pa s with polyethylenepolypropylene glycol to obtain a laminar flow regime in both reactors. The selectivity of the parallel reactions was then dependent on the mode of operation, intensity of mixing, initial stoichiometric ratio and volume ratio of the mixed solutions. The model was used to determine an energetic efficiency of mixing based on the results of the parallel reaction experiments; the procedure proposed enables a direct and strict comparison to be made of systems with different geometries and/or mode of operation.
Precipitation of barium sulphate in the annular gap of a continuous Couette reactor with two unpremixed feeds has been experimentally investigated. The experiments have been carried out in the turbulent regime with an axial Rez varying from 100 to 240; the hydrodynamics can be described with the plug flow plus dispersion model, and in the condition tested the Peclet number varied from 2 to 6. The initial supersaturation has the strongest effect on both the particle size and the crystal morphology. The average size of the crystals is also influenced in some cases by the diameter of the feed tube and by the feed velocity, indicating the relevance of mesomixing. The rotation velocity has a relatively weak effect on crystal size and CSD, but a maximum has been observed increasing the rotation velocity.The performances of the system have been predicted in the limit cases of minimum and maximum age mixedness as a function of the Peclet number.
A general mixing-precipitation model describing interactions between mixing of various scales (macro-, meso- and micromixing) coupled with the population balance for crystallization is presented. Macro-, meso- and micromixing were interpreted as a circulation through the zones of different rate of energy dissipation, inertial-convective disintegration of large eddies and an engulfment of fluid from the environment, respectively. The model was verified experimentally for double feed semibatch BaSO4 precipitation. Experiments were carried out in a Rushton type stirred tank reactor. The effects of feed tube positions, addition feed rate, intensity of mixing, initial reagent concentration and volume ratio on particle size distribution and particle morphology were investigated. The model enables prediction of supersaturation profiles in the system and resulting CSD; the predicted history of supersaturation structure explains the influence of operating conditions on crystal morphology.
The literature gives little guidance on the factors (e.g. choice of reactor, sequence of adding reagents, imperfect mixing) determining the product distribution of parallel reactors. By considering two irreversible, second-order reactions between three substances, it is shown here how the sequence of adding reagents to a semi-batch reactor will influence the yield when the mixing is perfect as well as when segregation is complete. A mixing model is then applied to predict yields when segregation is partial. Such calculations were made for three mixing sequences when chemically equivalent quantities of the three reagents were employed and when one of the two reactions was instantaneous. The yield then depended upon the mixing sequence, the volume ratio of reagent solutions and the Damköhler number (this is a ratio of characteristic times for micromixing and chemical reaction). Sodium hydroxide, ferric sulphate and ethyl chloroacetate solutions were contacted in the sequences corresponding to cases 1 and 3 of the modelling. The alkali reacted competitively and the extents of the two parallel reactions were measured after it had been fully consumed. The independent variables were stirrer speed, volume ratio of the two reagent solutions and the concentration level at constant stoichiometric ratio. The effects of these operating variables on the product distribution were well predicted. The stoichiometry of the precipitation of ferric ions in alkaline solution is probably more complex than considered in the modelling and the parallel reactions used here need improvement or possibly replacement. A similar comparison was made between measured and predicted yields of nitrobenzene during the competitive nitration of benzene and toluene. Twelve out of 13 yields measured by Tolgyesi (1965) were satisfactorily predicted.
The similarity form of the scalar-variance spectrum at high Schmidt numbers is investigated for nonstationary turbulence. Theoretical arguments show that Batchelor scaling may apply only at high Reynolds numbers. At low Reynolds numbers, Batchlor scaling is not possible unless the turbulence is stationary or the enstrophy decays asymptotically as t−2. When this latter condition is satisfied, it is shown from an analysis using both the Batchelor and Kraichnan models for the scalar-variance transfer spectrum that the k−1 power law in the viscous-convective subrange is modified. Results of direct numerical simulations of high Schmidt number passive scalar transport in stationary and decaying two-dimensional turbulence are compared to the theoretical analysis. For stationary turbulence, Batchelor scaling is shown to collapse the spectra at different Schmidt numbers and a k−1 viscous-convective subrange is observed. The Kraichnan model is shown to accurately predict the simulation spectrum. For nonstationary turbulence decaying at constant Reynolds number for which the enstrophy decays as t−2, scalar fields for different Schmidt numbers are simulated in situations with and without a uniform mean scalar gradient. The Kraichnan model is again shown to predict the spectra in these cases with different anomalous exponents in the viscous-convective subrange.
A closure procedure based on the probability density functions of the concentrations of chemically inert species has been proposed and applied to derive the mathematical forms of the local average values of the nonlinear reaction, nucleation and crystal growth terms. Applied to experimental results for BaSO4 precipitation, satisfactory agreement between predicted and observed results was obtained for two tubular reactors. In experiments solutions of BaCl2 and Na2SO4 were contacted in tubular reactor equipped with an impingement mix-head.
Fast, competitive-consecutive reactions exhibit product distributions which are influenced by micro- and macroscale concentration gradients in a reactor. If a reaction is run many times under identical conditions in a semi-batch reactor, except that different feed rates are used, the product distribution is constant at low feed rates, but indicates increasing non-uniformity of composition at higher feed rates. Competition between micromixing and reaction determines product distribution at low feed rates. The additional inhomogeneity appearing at higher feed rates signals a mixing constraint at scales larger than microscopic. The relevant scale is not the macroscopic one of the whole vessel, but rather a mesoscale reflecting the interaction of the plume of fresh feed with its surroundings. Dispersion of this plume has been calculated for homogeneous turbulence, as well as when the mean plume velocity and the turbulence properties in its vicinity were spatially varying. Micromixing and chemical reaction could then be calculated in the concentration field resulting from turbulent dispersion. The relative importance of meso- and micromixing could be expressed by the ratio of their time constants. The description of simultaneous meso- and micromixing developed here successfully predicted product distributions from fast diazo-coupling reactions which had been measured at two scales, two feed points, three concentration levels and various stirrer speeds. The model requires knowledge of the reaction kinetics and the flow field, but does not, however, contain any arbitrary parameters.
Parallel reactions were used to investigate how product distributions can depend upon micromixing. A two-reaction test system consisted of the competition for sodium hydroxide between neutralization with hydrochloric acid and alkaline hydrolysis of ethyl monochloroacetate. A three-reaction system comprised these same reactions together with alkaline hydrolysis of methyl monochloroacetate. For stirred tank reactors, operated in the semibatch mode, the inhomogeneous turbulence was modeled by specifying the trajectory of the reacting fluid as well as the energy dissipation rates in the regions through which it was convected. The engulfment model of micromixing, combined with this experimental flow model, was then integrated to compute product distributions for the two- and three-reaction systems. This analysis compared well with experiments at three levels of initial concentrations, three volume ratios, various stirrer speeds, three feed positions, two sequences of reagent addition, tanks with dished and flat bottoms, and three tank sizes. The experimental flow model was extended to tanks with a turbine diameter larger than the standard one-third tank diameter, by proposing distributions of the velocity and energy dissipation fields. Measured product distributions using three sizes of turbine agreed reasonably well with the model predictions. Given reliable local hydrodynamic information, especially energy dissipation rates, it should now be possible to predict the product distribution of any parallel reaction that is influenced by micromixing.
The diazo coupling between 1-naphthol and diazotized sulfanilic acid at room temperature and pH 9.9 has been widely used to study the influence of mixing on the product distribution of fast chemical reactions. It is sufficiently fast for application in mixers whose rates of turbulent energy dissipation do not exceed 200-400 W.kg-1. Alternative reactions have been screened with a view to studying interactions between mixing and reaction in higher intensity mixers. Extension of the existing system to the simultaneous couplings between 1- and 2-naphthols and sulfanilic acid is recommended. Analysis by spectrophotometry was carried out, and extinction coefficients are tabulated. The kinetics of the five possible reactions have been studied, and second-order rate constants are given. This extended system should be suitable for energy dissipation rates in aqueous solutions up to about 10(5) W.kg-1. Both reaction systems were run under semibatch conditions in a stirred tank reactor, and the applicability of the extended reactions was confirmed.
Agglomeration effects, observed during precipitation of barium sulphate in the unpremixed feed two-dimensional tubular precipitator, are studied experimentally and interpreted theoretically. Effects of process parameters on precipitation–agglomeration phenomena are predicted using a CFD based model that describes micromixing (the multiple-time-scale turbulent mixer model is used) and precipitation (including nucleation, growth and agglomeration of crystals). Agglomeration rate is defined as a product of the collision frequency and the probability of agglomeration.
Partial segregation of reagents occurs when reaction rates exceed mixing rates and frequently causes product distributions to be mixing-dependent. A simultaneous temperature segregation, whereby the temperature in a reaction zone differs from that in its surroundings, has rarely been considered when mixing reagents in the liquid phase. The Prandtl number is usually sufficiently small that engulfment, not thermal conduction, determines the local temperature. The engulfment model of micromixing can then be extended by a heat balance to specify this temperature. To evaluate this model, a new pair of fast competitive reactions (neutralisation and acetal hydrolysis) has been characterised thermochemically and kinetically. A Mettder RC1 calorimeter was operated under isothermal and adiabatic conditions with various stirrer speeds and HCl was slowly added to a mixture of NaOH and 2,2-dimethoxypropane. Measured hydrolysis yields compared quite well with the extended engulfment model, although temperature segregation was of minor importance. It was also unimportant in other reaction systems (simultaneous neutralisation and ester hydrolysis; diazo coupling) employed earlier to study micromixing. Suggestions for further work are made.
In this work barium sulfate precipitation is studied in a tubular reactor in a wide range of operating conditions. The effect of reactant concentrations and barium or sulfate excess on crystal-size distribution and morphology is investigated. Experimental results show that ion excess has a strong influence and that at high concentration aggregation takes place. Computational fluid dynamics is coupled with the finite-mode probability density function approach for taking into account both macro- and micromixing, whereas the population balance is treated by using the standard moment method. Comparison with experimental data suggests that when micromixing and agglomeration are properly taken into account, the agreement is improved. However, this statement is partially affected by the lack of knowledge in barium sulfate kinetics.
Effects of turbulent mixing on the course of two fast parallel chemical reactions (neutralization of sodium hydroxide and hydrolysis of ethyl chloroacetate) carried out in a semibatch stirred tank reactor are experimentally investigated and numerically simulated. The flow pattern in the stirred tank is predicted using CFD and experimentally validated using Laser Doppler Anemometry. Mixing effects are modelled using three CFD based models. In the first and the second model the Beta probability distribution and the spiked distribution are used respectively; in the third model concentration fluctuations are neglected.
Thermodynamic models for aqueous Ba2+-SO42–-Na+-Cl–-solutions are compared in their accuracy to predict ion activities in saturated and supersaturated solutions. The Pitzer and the Bromley model are employed, taking into account ion pair formation of barium sulfate. Such models are then used to describe particle nucleation and growth, and finally they are imbedded in a mechanistic mixing-precipitation model for a single feed semibatch process. The effect of the key operating parameters on the mean particle size is analyzed through simulations. The results are compared with previous experimental data, thus highlighting the significance of a proper choice of the thermodynamic model.
A generalized analytic correlation is presented for activity coefficient. osmotic coefficient, enthalpy, and heat capacity of single and multicomponent strong aqueous solutions. A good correlation for each salt to an ionic strength of six is obtained by assigning a single parameter “B” value to each salt. These B values are well approximated by assigning two parameters for each ion. Values presented for the common ions at 25°C allow the estimation of activity and osmotic coefficients of many unmeasured salt solutions.
The solubilities of barite [BaSO4(c)] and celestite [SrSO4(c)] in Na2SO4 were studied and found to be significantly lower than the experimental values reported in the literature. Our new solubility data are in excellent agreement with the predictions of ion interaction models, which have previously been parameterized primarily from solubility data obtained in chloride media. Our solubility data were analyzed both in terms of aqueous thermodynamic models that included ion association species and in terms of ion interaction models that did not require the explicit recognition of such species. In the case of SrSO4, although both ion association and ion interaction models can accurately model our solubility data, the ion interaction approach is preferred because it is easier to extend to higher concentrations. In the case of BaSO4, the aqueous ion interactions appear to be stronger than those for SrSO4, and so the explicit recognition of a BaSO4(aq) ion association species is preferred. The logarithms of the thermodynamic solubility products (log K
sp
) for celestite and barite were –6.620.02 and –10.050.05, respectively. When the data were analyzed using models that include ion association species, the logarithms of the thermodynamic equilibrium constants for the SrSO4(aq) and BaSO4(aq) association reactions were 1.860.03 and 2.720.09, respectively.
Precipitation of barium sulphate in an unpremixed feed two-dimensional tubular precipitator is studied experimentally and interpreted theoretically using a closure procedure proposed previously by the authors. The closure employs the presumed beta PDF of the inert type composition variables formed with the local values of Ba2+ and SO42− concentrations and the turbulent mixer model. The flow field is computed using the k–ε model. The method enables the particle size and the particle size distribution to be predicted and compared with experimental data.
Precipitation of sparingly soluble materials from two aqueous ionic solutions is considered. It is well known that the flow conditions in the system can affect particle size distribution and particle morphology. In the paper, three problems related to interactions between hydrodynamics (turbulent flow) and the precipitation process are considered, namely, effects of mixing on crystal size using computational fluid dynamics (CFD), relation between turbulent diffusion of fluids and solid particles, and the problem of aggregation kernel for small particles.
The effect of micromixing limits on a process involving elementary chemical reaction and subsequent crystallization is considered. Two limiting cases are analysed; maximum species and age mixedness (Model I) and maximum species but minimum age mixedness (Model II). Although the conversion of the chemical reaction is similar in both situations, the crystal size distributions can be very different. Such differences arise from variations in the supersaturation profiles experienced by fluid elements. The sensitivity of the two models to the reaction rate constant and the nucleation kinetic parameters is explored.
A method to evaluate the decay of concentration variance in turbulent mixers is presented. The turbulent mixer model links together macromixing (large-scale convective motions and turbulent dispersion) with successive stages of micromixing process (inertial-convective disintegration of large eddies, formation of laminated structures within energy dissipating vortices and molecular diffusion within deforming laminated structures). A beta probability density function is used for an inert scalar type composition variable formed with reagents concentrations difference in order to estimate conversion of instantaneous reactions in the system. The results calculated from the model for the multijet mixers and reactors are compared with experimental results available in the literature.
In this work the influence of turbulent mixing on the course of two parallel chemical reactions (neutralization of sodium hydroxide and hydrolysis of ethyl chloroacetate) is investigated for a process carried out in a semibatch stirred tank reactor. The multi-scale nature of the process is highlighted by a time-scale analysis of controlling process mechanisms. Two reactive mixing models of different complexity, both accounting for all relevant phenomena from the micro- to the macroscale, are applied for the simulation of the process. Computational results of both models for various operating conditions are compared with experimental data, thus highlighting the strengths of the two models.
249. Figure 8. Predicted and measured crystal size distributions for c A0 ) c B0 ) 0
- Amsterdam
Amsterdam, 1979; p 249. Figure 8. Predicted and measured crystal size distributions for c A0 ) c B0 ) 0.0045 kmol m -3, N )
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