Predictive simulation of nanoparticle precipitation based on the population balance equation

Nanoparticle precipitation is an interesting process to generate particles with tailored properties. In this study we investigate the impact of various process steps such as solid formation, mixing and agglomeration on the resulting particle size distribution (PSD) as representative property using barium sulfate as exemplary material. Besides the experimental investigation, process simulations were carried out by solving the full 1D population balance equation coupled to a model describing the micromixing kinetics based on a finite-element Galerkin h-p-method. This combination of population balance and micromixing model was applied successfully to predict the influence of mixing on mean sizes (good quantitative agreement between experimental data and simulation results are obtained) and gain insights into nanoparticle precipitation: The interfacial energy was identified to be a critical parameter in predicting the particle size, poor mixing results in larger particles and the impact of agglomeration was found to increase with supersaturation due to larger particle numbers. Shear-induced agglomeration was found to be controllable through the residence time in turbulent regions and the intensity of turbulence, necessary for intense mixing but undesired due to agglomeration. By this approach, however, the distribution width is underestimated which is attributed to the large spectrum of mixing histories of fluid elements on their way through the mixer. Therefore, an improved computational fluid dynamics-based approach using direct numerical simulation with a Lagrangian particle tracking strategy is applied in combination with the coupled population balance-micromixing approach. We found that the full DNS-approach, coupled to the population balance and micromixing model is capable of predicting not only the mean sizes but the full PSD in nanoparticle precipitation.     

TAILORING PARTICLE SIZE THROUGH NANOPARTICLE PRECIPITATION

Precipitation of nanoscaled particles and the size-determining precipitation parameters are investigated experimentally as well as numerically using barium sulfate as a reference substance. The objective of this work is to successfully understand and predict precipitation kinetics. Optimization and tailoring of product properties to specific needs would then be possible without the need of extensive experimentation and its costs. Special attention is paid to the influences of mixing as well as stabilization on the formed PSD. To simulate particle formation the population balance equation, including the terms for nucleation, growth, and agglomeration, is coupled with an specially developed extended version for equi-volumetric mixing of the Engulfment-Deformation-Diffusion-model of micromixing of Baldyga and Bourne (1999). The proposed predictive model for nanoparticle precipitation is explained in detail and simulation results are presented, discussed, and compared to experimental results.     

TAILORING PARTICLE SIZE THROUGH NANOPARTICLE PRECIPITATION

Precipitation of nanoscaled particles and the size-determining precipitation parameters are investigated experimentally as well as numerically using barium sulfate as a reference substance. The objective of this work is to successfully understand and predict precipitation kinetics. Optimization and tailoring of product properties to specific needs would then be possible without the need of extensive experimentation and its costs. Special attention is paid to the influences of mixing as well as stabilization on the formed PSD. To simulate particle formation the population balance equation, including the terms for nucleation, growth, and agglomeration, is coupled with an specially developed extended version for equi-volumetric mixing of the Engulfment-Deformation-Diffusion-model of micromixing of Baldyga and Bourne (1999) Baldyga, J. and Bourne, J. R. 1999. Turbulent Mixing and Chemical Reactions, Chichester: John Wiley.  [Google Scholar]. The proposed predictive model for nanoparticle precipitation is explained in detail and simulation results are presented, discussed, and compared to experimental results.     

Simultaneous determination of nucleation and crystal growth kinetics of gypsum

The chemical treatment of industrial effluent, containing sulphuric acid, is typically completed by neutralising it with lime through solid/liquid separation. The mineral precipitation processes developed by Veolia Environnement use sludge recirculation. The production of a high sludge density characterises these processes, at a minimum of at least 25% weight in solids. The first laboratory tests studied the precipitation reaction, confirming the production of gypsum (CaSO₄·2H₂O). Then the precipitation mechanisms, nucleation and crystal growth kinetics, are determined in order to establish a model which predicts population density in reactor in accordance with fluid dynamics. The determined nucleation and growth kinetics are then used to feed a reaction model, coupling computational fluid dynamics (CFD) and population balance modelling to simulate the precipitation process.     

Towards predictive simulation of single feed semibatch reaction crystallization

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.     

Actinides oxalate precipitation in emulsion modeling: From the drop scale to the industrial process

An original process of actinides coprecipitation based on pulsed flow column is studied. The novelty of this process lies in the confinement of the aqueous reagents in separated droplets, dispersed in an inert organic phase (W/O emulsion). Precipitation occurs inside drops when they coalesce. Besides the implementation of well-known technologies of the nuclear industry, this precipitation in emulsion process is particularly convenient for the control of supersaturation, and ensures the sticky precipitates’ confinement within drops, thereby limiting the fouling risk and its adverse consequences on productivity and safety.     

Kinetic modelling of nickel powder precipitation by high-pressure hydrogen reduction

The active mechanisms in the precipitation of nickel powder by hydrogen reduction were investigated by means of mathematical models based on the moment form of the population balance equation (PBE). The objective of the work was to establish the mechanisms involved in powder formation and how these are affected by the presence of impurities. The effects of two major impurities were considered namely; iron, inherently present as ferrous sulphate, and a morphology modifier, a polyacrylic acid derivative used as an additive. Experiments were conducted on a laboratory and pilot-plant scale using a 0.5 and 75 L stainless steel autoclave, respectively. Nickel powder samples were collected from the autoclaves after each successive batch reduction (densification) and the particle size distribution (PSD) analysed using a laser diffraction technique. The PSD data was then transformed into moments and the experimental values were compared with those simulated using different models based on the moment form of the PBE. Five models were tested namely: (a) aggregation-only; (b) aggregation and growth; (c) nucleation and aggregation; (d) nucleation and growth and (e) aggregation and breakage. Under standard operating conditions, the process was best simulated by a size-independent aggregation and growth model in the early stages of the cycle, with breakage and growth becoming significant in later stages of the cycle when large particles have been formed. Crystallisation in the presence of Fe was characterised by a size-independent aggregation and growth model with varying degrees of nucleation depending on the Fe concentration and the available surface area. At modifier dosages of 0.25 and 5 vol% the process was best modelled by a size-independent aggregation and growth model coupled with a constant breakage frequency model. Based on the mathematical modelling results and evidence from scanning electron micrographs, the spherically shaped nickel powder particles were proposed to be formed through the formation of a pre-cursor by secondary aggregation followed by spherulitic growth. The degree of compactness of the spherulites was proposed to be determined by the number of active growth sites on the nickel particle surface. The morphology modifier was found to act as a growth inhibitor, decreasing the number of growth sites leading to more open spherulites. Iron was found to induce surface nucleation, thus, creating more growth sites on the particle surface and leading to more compact spherulites.     

Application of in situ adaptive tabulation to CFD simulation of nano-particle formation by reactive precipitation

Reactive precipitation involves four fundamental processes: mixing-limited reaction, nucleation, growth, and aggregation. A novel algorithm, in situ adaptive tabulation (ISAT), has been implemented in a code for micromixing simulations, which is often applied together with computational fluid dynamics (CFD), using full probability density function (PDF) methods to incorporate these fundamental processes in the formation of nano-particles by reactive precipitation in a plug-flow reactor. The quadrature method of moments is applied to solve population balance equations for turbulent aggregation of the growing particles. The various performance issues (error control, accuracy, number of records, speed-up) for ISAT are discussed. Based on a large number of simulations, an error tolerance of 10⁻⁴-10⁻⁵ is found to be satisfactory for carrying out time-evolving full PDF simulations of nano-particle formation by reactive precipitation. Our results show that CFD simulation of reactive precipitation requires a much smaller computational effort when the ISAT algorithm is implemented than when direct integration is used. Finally, the effects of initial species concentrations, micromixing time, and turbulent shear rate on the reactive precipitation of barium sulfate are studied.     

Derivation of supersaturation during precipitation from the mixing pattern of an inert tracer in the same device: case of partially premixed feed streams

The average supersaturation field is predicted for precipitations in the case of partially premixed feed streams by a simple mixing model from a couple of successive experiments with the same inert tracer by Planar Laser Induced Fluorescence and using a single camera. Then, the mathematical derivation is different from that one used for the unmixed feed case. The corresponding experiments have been achieved in a 90° impinging jets premixer and the supersaturation field was predicted. Due to the non-simultaneity of tracer experiments, a calculation of the averaged nucleation flux is not possible in the premixed feed case. Nevertheless, a comparison between different premixers can be done on the basis of the generated supersaturation levels and fouling risks.     

On the mixing properties of spherically symmetric helical coherent structures in turbulence

A paradigmatic family of flow fields for localized, spherically symmetrical flow with maximal helicity—a model for helical coherent structures that are localized—is introduced. The Lagrangian mixing of the lowest order member of the family that is truly 3-D due to spiral arms is analyzed with linear theory, demonstrating that trajectory growth rates for the short, convective time scale are exponential and bounded by the extremal eigenvalues of the Jacobian. However, these rates show strong inhomogeneity and anisotropy associated with anomalous mixing. It was found for nonlinear Lagrangian mixing times that for this paradigm helical coherent structure, 22% of the trajectory pairs were bounded by the initial separation (non-mixing) and 78% mixed in various classifications of convective dispersal. All non-local studies of 10,000 Lagrangian trajectories could be categorized into five classes of growth (decay) patterns which exhibit the effects of localized, finite helicity/momentum associated with this class of velocity field. A scalar dispersion simulation confirms that the “patch” of fluid near the origin is slowly mixing—on the diffusive time scale—and is convected “unmixed” when the influence of molecular diffusion is still not pronounced (short times relative to Pe=100).     

Derivation of supersaturation and nucleation flux during precipitation from the mixing pattern of an inert tracer in the same device: case of unmixed feed streams

Inert liquid tracer concentrations in a continuous mixer are analyzed by planar laser induced fluorescence (PLIF). Special attention is paid to the two separated entering feed streams containing the tracer solution or pure solvent. From the instantaneous tracer concentration fields, the method proposed allows one to easily calculate the instantaneous supersaturation fields, which would be obtained in the same mixing device with reagents instead of an inert tracer solution and pure solvent. A typical mixing situation in a stirred tank with separated feed streams is investigated. Maps of averaged supersaturation and averaged nucleation flux are yielded with high spatial resolution of a few tens of micrometers for each pixel. The method gives interesting indications about the ability of a given precipitator geometry to generate coarse or fine particles. However, it cannot be extended straightforward to partially premixed feed streams without the use of a mixing model.     

Estimation of the micromixing time in the torus reactor by the application of the incorporation model

On the basis of previous experimental results in a torus reactor, micromixing time is determined using the incorporation model. Obtained results allowed the characterisation of the performances of this new configuration of reactor in comparison to other reactors, such as the stirred tank reactor. In addition, a correlation is proposed for each incorporation law, in order to determine the micromixing time from the experimental micromixedness ratio (α). Finally, in terms of Kolmogorov's turbulence theory, a relationship between micromixing time and the local energy dissipation rate is obtained and compared to those previously published.     

Development of a networks-of-zones fluid mixing model for an unbaffled stirred vessel used for precipitation

Reactive acid-alkali tracers have been deployed to capture the macromixing and partial segregation behaviour in an unbaffled stirred vessel. This configuration is often used in precipitators to avoid inadvertent solid accretions on vessel internals. The macromixing behaviour for semi-batch addition with visualisation of reactive (acid-alkali) tracers has been acquired via video images which are rendered visible using phenolphthalein as indicator. By means of visual reality modelling, in which computer graphics are used to reconstruct and closely mimic the experimentally visualised fluid mixing “scenes”, the parameters for a networks-of-zones mixing model for the unbaffled semi-batch case have been established. The model can then be used for predicting precipitation behaviour for single-jet and other modes of operation. Some illustrative examples for barium sulphate, showing the underlying supersaturation fields in 3-D and the consequent time evolving particle size distributions, are presented and discussed for a single jet case.     

Two state estimators for the barium sulfate precipitation in a semi-batch reactor

The on-line determination of particle property distributions by direct measurements is often difficult, because the measurement equations are not invertible or because the inverse problem is ill-posed. If the process is observable, one can use state estimation techniques in order to reconstruct unmeasurable internal states of the process. This is discussed here for a semi-batch precipitation reactor. A square root unscented Kalman filter and state estimation by online minimisation are studied for the case of a measurable average particle size. Both estimators use a one-dimensional population balance model. The two approaches are compared in simulations.     

The influence of iron on the precipitation behaviour of nickel powder

The effect of iron on the precipitation behaviour of nickel powder was investigated. Reduction experiments were conducted on a 0.5 L laboratory autoclave fitted with a Teflon reaction beaker and a double impeller configuration consisting of an upper axial impeller and lower Rushton turbine. Reduction was conducted in the presence of a morphology modifier at a temperature between 180 and and 2800 kPa pressure using a synthetic nickel ammine sulphate solution (, free NH₃:Ni²⁺ and (NH₄)₂SO₄:Ni²⁺ molar ratios of 2.0-2.1 and 2.2, respectively). Nickel seed was used to initiate reduction and iron was added to the reduction solution as ferrous sulphate solution (acidified to pH 2.5 to prevent oxidation) to give a reduction solution with Fe²⁺ concentration of 6, 20 and 200 mg/L. The effect of iron was investigated by studying the evolution of the moments, volume or mass moment mean D(4.3), number based mean size , nickel depletion rate and population balance in the absence of sampling between batch reductions. Iron was found to act as a growth promoter and nucleation agent through reversible adsorption and hydrolysis on the surface of the seed particles. Growth was preferentially favoured over nucleation up to a Fe²⁺ concentration of 6 mg/L, thereafter the extent of nucleation increased with increasing Fe²⁺ concentration up to 200 mg/L. Nucleation and growth promotion in the presence of high shear rates gave rise to rapid aggregation, which ceased at a critical size of approximately and in the presence of iron and without. However, the sharp increase in the D(4.3) towards the end of the cycle and the general decrease in surface area shows that aggregation of larger particles plays a major role in size enlargement. Comparison of the scanning electron microscopy (SEM) micrographs of the powder with undesirable morphology produced in industrial practice and that produced in the laboratory in the presence of iron showed that iron was one of the factors responsible for the production of powder with undesirable morphology. Based on these laboratory scale experiments, iron levels in reduction solutions should not exceed 6 mg/L for effective control of particle morphology.     

On the use of bi-variate population balance equations for modelling barium titanate nanoparticle precipitation

In this work barium titanate hydrothermal synthesis is studied from the modelling point of view. In fact, a mathematical model can be used to investigate the origin of the typical multi-crystalline particles obtained during this precipitation process. Their origin is not completely clear, since it is not understood which is the controlling mechanism (i.e., secondary nucleation and/or aggregation). Previous works on this subject tried to retrieve the kinetics underlying this particulate process but being based on mono-varied population balance equations failed to give a definitive answer. In this work a bi-variate model is presented, discussed and solved. Results clearly show that such an approach can overcome the limitations of previous modelling works and provides an useful tool for more detailed kinetic parameters estimation. Moreover the model shows that secondary nucleation is indeed very important but aggregation cannot be neglected.     

Simulation of reactive precipitation processes using the network-of-zones model

A 2-D network-of-zones model is extended and applied to a reactive precipitation process in batch mode. The simulations are performed for a network of size 2 × (10 × 10) for an elementary reaction through the solution of 1400 ODEs. The complicated interactions between mixing efficacy and the system kinetics are systematically investigated. When the stirrer speed is very slow, the crystal size distribution (CSD) of the product in the precipitator is determined by the intensity of mixing. Conversely, at higher stirrer speed, the CSD is controlled by the system kinetics. More effective mixing leads to an increase in the number of crystals, a reduction of the average size and a narrower crystal size distribution. The extended network-of-zones model presented in this work can be used conveniently for integrating computational fluid dynamics and reactive precipitation processes.     

Understanding the role of restructuring in flocculation: The application of a population balance model

The effect of shear on floc properties was observed through population balance to comprehend the mechanisms of flocculation, in particular the role of restructuring. Little fundamental attention has been given before to the shear influence responsible for creating compact aggregates, while the floc characteristics might differ in other conditions. It is crucial to understand how aggregates evolve to steady state, if their properties are to be ‘tailored’ to suit subsequent solid-liquid separation. From a previous experimental study (Langmuir 18(6) (2002) 1974), restructuring was observed to occur extensively in the flocculation of latex particles in couette-flow, and was proposed to be responsible for the decrease in floc size on their transition to equilibrium. On the other hand, flocs of larger primary particles were more susceptible to breakage, with densification occurring as a result of fragmentation and re-aggregation. Denser flocs were found when structural deformation dominated, particularly in the initial stage of the process, while comparatively tenuous ones were observed when formation and breakage kinetics were the governing mechanisms. The distinct manners in which aggregates of different primary particle sizes evolved with time, were replicated with a population balance that incorporated the floc structural variation; verifying that restructuring indeed played a crucial role under certain flocculation conditions.     

Modeling and simulation of mixing in water-in-oil emulsion flow through a valve-like element using a population balance model

Emulsion flows are very common in natural processes as well as in several engineering areas, such as in the process of desalting crude oil that occurs in refineries. This kind of flow is described as a polydispersed multiphase flow. In this work, we evaluated the behavior of water-in-oil emulsion flowing through a duct with an element used to mimic the effect of a globe valve. An Eulerian multi-fluid approach was employed by solving the population balance equation coupled with computational fluid dynamics. Coalescence and breakage models recently developed were extended to this inhomogeneous model. A bivariate population balance problem was also solved to demonstrate the mixing caused by the valve-like element. The simulated results showed good agreement with the available experimental data for the Sauter and DeBroukere mean diameters.