Population Balance Equation Model Research Articles

Aggregation and breakage dynamics of alumina particles under shear by coupled Computational Fluid Dynamics – Discrete Element Method

HypothesisShear affects simultaneous aggregation and fragmentation of fine particles. Understanding the effect of shear on the dynamics of particle aggregation and break-up is important to predict aggregate size and structure. It is hypothesized that there is a transition from pure breakage of large aggregates to regimes where restructuring and aggregation also play a role as aggregates become smaller. SimulationsHere, aggregation and fragmentation dynamics of alumina particles are investigated under laminar shear flow using Discrete Element Method (DEM) coupled with Computational Fluid Dynamics (CFD). The effect of the shear rate on the aggregation and breakage rates is quantified accounting for particle–particle and particle–fluid interactions. FindingsHigh shear rates promote the formation of small, compact aggregates. The collision efficiency decreases with increasing shear rate following a power law for shear rates higher than 1250 s−1. The transition from the pure breakage limit to the region dominated by breakage and restructuring has been observed for the first time. The breakage rate decreases significantly as the transition occurs upon decreasing aggregate size. CFD-DEM-derived collision efficiency and breakage rate equations are proposed that can be readily employed in detailed population balance equation models for industrial particle process design.

Hybrid modeling of hetero-agglomeration processes: a framework for model selection and arrangement

Modeling of hetero-agglomeration processes is invaluable for a variety of applications in particle technology. Traditionally, population balance equations (PBE) are employed; however, calculation of kinetic rates is challenging due to heterogeneous surface properties and insufficient material data. This study investigates how the integration of machine learning (ML) techniques—resulting in so-called hybrid models (HM)—can help to integrate experimental data and close this gap. A variety of ML algorithms can either be used to estimate kinetic rates for the PBE (serial HM) or to correct the PBE’s output (parallel HM). As the optimal choice of the HM architecture is highly problem-dependent, we propose a general and objective framework for model selection and arrangement. A repeated nested cross-validation with integrated hyper-parameter optimization ensures a fair and meaningful comparison between different HMs. This framework was subsequently applied to experimental data of magnetic seeded filtration, where prediction errors of the pure PBE were reduced by applying the hybrid modeling approach. The framework helped to identify that for the given data set, serial outperforms parallel arrangement and that more advanced ML algorithms provide better interpolation ability. Additionally, it enables to draw inferences to general properties of the underlying PBE model and a statistical investigation of hyper-parameter optimization that paves the way for further improvements.

Population balance equation for collisional breakage: A new numerical solution scheme and its convergence

A new numerical method for the solution of population balance equation modeling nonlinear collisional breakage is presented. The new scheme is obtained by modifying the birth term of discretized equation. Allocation of new born particles to their neighboring cells plays a vital role towards the efficiency of proposed model. A detailed investigation on mathematical properties of the scheme, e.g., consistency and stability, is presented for different mesh types. It shown that the new scheme is moment-consistent and has first order convergence rate. A new finite volume scheme for collisional breakage is also presented to validate the efficiency of new scheme over several physics embedded test problems.

Modeling of silica synthesis in a laminar flame by coupling an extended population balance model with computational fluid dynamics

In the present study, we propose a novel extended population balance equation (PBE) model for aggregation and sintering and couple it with computational fluid dynamics (CFD) to investigate synthesis of silica nanoparticles in a laminar diffusion flame. The extended PBE includes finite-rate sintering of primary particles by solving the PBE together with a transport equation for the number concentration of primary particles. In the process simulated, the particles are formed via the oxidation of a vapor precursor, hexamethyldisiloxane (HMDSO), and the aerosol processes include nucleation, condensation, aggregation and sintering. The model is validated with detailed experimental in-situ SAXS data found in the literature and is also compared with a monodisperse and a two-PBE approach. Good agreement is found between the extended one-PBE and two-PBE models, while both of them provide a substantial improvement over the monodisperse one. Furthermore, the coupled CFD-PBE simulation with the extended one-PBE model reduces substantially the computational time as compared with the two-PBE model and requires less than twice the time needed for the monodisperse model. Excellent agreement is found between numerical predictions and experimental data for temperature along the centerline and reasonably good agreement is found between numerical predictions and SAXS data for primary particle diameters. While results for the particle number concentration are in qualitative agreement with the experimental data, the particle formation rate is overpredicted, leading to an overestimation of the number concentration of the primary particles. This is attributed to uncertainties in the experimental data and precursor decomposition kinetics.

Modeling and Simulation of Photobioreactors with Computational Fluid Dynamics—A Comprehensive Review

Computational Fluid Dynamics (CFD) have been frequently applied to model the growth conditions in photobioreactors, which are affected in a complex way by multiple, interacting physical processes. We review common photobioreactor types and discuss the processes occurring therein as well as how these processes have been considered in previous CFD models. The analysis reveals that CFD models of photobioreactors do often not consider state-of-the-art modeling approaches. As a comprehensive photobioreactor model consists of several sub-models, we review the most relevant models for the simulation of fluid flows, light propagation, heat and mass transfer and growth kinetics as well as state-of-the-art models for turbulence and interphase forces, revealing their strength and deficiencies. In addition, we review the population balance equation, breakage and coalescence models and discretization methods since the predicted bubble size distribution critically depends on them. This comprehensive overview of the available models provides a unique toolbox for generating CFD models of photobioreactors. Directions future research should take are also discussed, mainly consisting of an extensive experimental validation of the single models for specific photobioreactor geometries, as well as more complete and sophisticated integrated models by virtue of the constant increase of the computational capacity.

Modeling the axial distribution of mass transfer coefficient in an agitated‐pulsed extraction column

AbstractThe dispersed phase holdup and drop size in solvent extraction columns vary along the column height and this affects the mass transfer coefficient and interfacial area. In this article, mass transfer study was performed experimentally using a 25 mm diameter agitated pulsed column. The axial distribution of mass transfer coefficient was determined by coupling population balance equation and axial dispersion model by taking the longitudinal variation in hydrodynamic performance into consideration. Feasibility of different mass transfer models in predicting concentration profiles was evaluated and a novel correlation based on effective diffusivity was developed. The results showed that both overall and volumetric mass transfer coefficients have significant change along the column height and greatly depends on the agitation speed and pulsation intensity. Increasing dispersed phase velocity also augments the overall mass transfer coefficient. The maximum number of transfer unit was measured to be 10 m−1 at agitation speed of 1000 rpm.

Solid-State Deracemization via Temperature Cycles in Continuous Operation: Model-Based Process Design.

Solid-state deracemization via temperature cycles converts a racemic crystal mixture into an enantiopure product by periodic cycling of the temperature in the presence of a racemization catalyst. A continuous counterpart of this conventional batch-operated process is proposed that can be performed in mixed suspension mixed product removal crystallizers (MSMPRCs). More specifically, three different configurations are described to perform periodic forcing via temperature cycles, which differ from each other in the type of the feed and in the withdrawal system. We have developed a model by extending our recent population balance equation model of batch solid-state deracemization via temperature cycles, and we exploit this tool to analyze the start-up and periodic steady-state behavior. Moreover, we compare the performance of the different configurations based on the selected key performance indicators, namely, average periodic steady-state enantiomeric excess and productivity. The process with solution feed yields pure enantiomers, while the solid and suspension-fed process alternatives result in highly enantiomerically enriched crystals. We further design an MSMPRC cascade to overcome this purity limitation. This work discusses guidelines on how to transform the batch process of temperature cycles into a continuous operation, which enables stable, unattended operation and chiral crystal production with consistent product quality.

Secondary Nucleation by Interparticle Energies. II. Kinetics.

This work presents a mathematical model that describes growth, homogeneous nucleation, and secondary nucleation that is caused by interparticle interactions between seed crystals and molecular clusters in suspension. The model is developed by incorporating the role of interparticle energies into a kinetic rate equation model, which yields the time evolution of nucleus and seed crystal populations, as in a population balance equation model, and additionally that of subcritical molecular clusters, thus revealing an important role of each population in crystallization. Seeded batch crystallization at a constant temperature has been simulated to demonstrate that the interparticle interactions increase the concentration of the critical clusters by several orders of magnitude, thus causing secondary nucleation. This explains how secondary nucleation can occur at a low supersaturation that is insufficient to trigger primary nucleation. Moreover, a sensitivity analysis has shown that the intensity of the interparticle energies has a major effect on secondary nucleation, while its effective distance has a minor effect. Finally, the simulation results are qualitatively compared with experimental observations in the literature, thus showing that the model can identify operating conditions at which primary or secondary nucleation is more prone to occur, which can be used as a useful tool for process design.

Model-based design of pressure-driven product removal from stirred suspensions

Achieving “mixed-product removal” in a mixed-suspension mixed-product removal crystallizer (MSMPRC), operating with a pressure-driven suspension transfer, is challenging. In this work, single and multiphase computational fluid dynamics (CFD) simulations were used to guide the choice of a suitable withdrawal position, which was found to be that of an immersed tube close to the vessel wall. The chosen withdrawal position was then tested and compared to other withdrawal positions experimentally. It performed as predicted, whereas for the other positions, the degree of dilution (size-independent classification) and of sieving (size-dependent classification) of the product removed increased. Furthermore, the influence of a non-representative withdrawal on an MSMPRC steady-state was investigated using a population balance equation (PBE) model.

Crystal Growth Kinetics of Benzoic Acid in Aqueous Ethanol Solution

Crystallization is one of the most important operations in the manufacturing of active pharmaceutical ingredients, where crystal growth is of vital importance for the final particle properties. The crystal growth kinetics of benzoic acid in antisolvent crystallization has been investigated by isothermal seeded desupersaturation experiments, applying the process analytical technology (PAT) tools and population balance modeling. The population balance equation (PBE) model selection incorporating different supersaturation expressions, growth rate expressions together with the solvent compositions was explored by fitting to corresponding concentration data sets. The results demonstrated that growth rate parameters depend heavily on solvent compositions. Therefore, a size-independent growth rate model concerning the effect of solvent compositions demonstrated the best fitness with respect to objective function values and confidence intervals. The estimated parameters with the corresponding model were consequently validated experimentally, demonstrating reasonable agreements with experimental concentrations, which illustrates the potential of the developed model in further experimental design and control.

Numerical simulation of ammonium perchlorate particles based on a population balance equation model in Taylor-Couette flow

Taylor-Couette flow, multiple vortex rings, is developed by rotating a co-axial rod in a cylindrical reactor. It is helpful in applying stable shear forces to suspended particles in the fluid domain. Recently, by using the Taylor-Couette flow, various experiments were performed to prepare micro-size particles at the laboratory level. However, unlike the simple concept of particle preparation by the Taylor-Couette flow, it is challenging to predict particle size distributions in the fluid domain, especially in the scale-up issue of a Taylor-Couette reactor. In this reason, computational methods are required to calculate particle growth in fluid flow. In this paper, ammonium perchlorate particles in the Taylor-Couette flow are simulated by using a population balance equation model, connected to a module of computational fluid dynamics. First of all, physical and empirical parameters for the population balance equation are determined by using experimental data in a Taylor-Couette flow reactor. Secondly, particle size distributions in a scale-up reactor are predicted by using the proposed method. Finally, validity and applicability of the suggested method are fully discussed on the basis of simulation results. As a major result, the simulation model for ammonium perchlorate particles was reliable in predicting their sizes along time. In addition, without changing the model parameters, the simulation results matched well with the experiments in the scale-up reactor. Based on the simulation results, it is expected that the suggested method can be utilized to estimate particle size distributions in various boundary and operating conditions.

Examining the physical and chemical contributions to size spectrum evolution during the development of hazes

China has experienced severe hazes with high concentrations of particulate matter in recent years. The understanding of the size spectrum evolution of submicron particulate matter is critical to making efficient remediation policies to minimize the regional and global environmental impacts from hazes. During a time period of about one month, we monitored five severe haze episodes in Xi’an and four severe haze episodes in Beijing, which were characterized by two distinct kinds of aerosol mass concentration growth processes: accumulative-rise and abrupt-rise. A new method was developed to quantitatively evaluate the physical and chemical contributions to growth processes by analysing the size spectrum evolution data. The results showed that the accumulative-rise processes are governed by primary emissions and the abrupt-rise processes are governed by secondary chemical reactions. The population balance equations (PBE) were used to describe the variation of size spectrum of fine particulate matter, and the respective contributions of the physical aggregation rate and the chemical growth rate. The PBE model is solved using the adjustable direct quadrature method of moments (ADQMOM) to simulate the abrupt-rise process of haze development and to calibrate the contribution of the physical and chemical effects on the size spectrum of aerosol particles.

Prediction of Droplet Size Distribution for High Pressure Homogenizers with Heterogeneous Turbulent Dissipation Rate

Droplet size distribution (DSD) of an emulsion is one of the important properties that affect product quality, stability, appearance, and rheology. Reliable prediction of DSDs in high-pressure homogenizers under various emulsion formulation and processing conditions is still challenging. A zero-dimensional population balance equation (PBE) model was developed to take account of the heterogeneous turbulent dissipation rate through its volume fraction distribution rather than the mean value as the coupling parameter between CFD and PBE. Furthermore, the effect of emulsifier on coalescence was also incorporated through the surface coverage of emulsifier. The model can give satisfactory DSDs under different oil fractions, homogenization pressures, and emulsifier concentrations for the oil-in-water emulsions. Moreover, the predictions were more sensitive to the breakage kernel parameters than to the coalescence ones or the number of daughter droplets. This approach is beneficial to the fast computational product design with less computational cost and reasonable prediction.

Key Factors Affecting Rheological Behavior of High-Concentration Graphene Oxide Dispersions and Population Balance Equation Model Analysis

Key Factors Affecting Rheological Behavior of High-Concentration Graphene Oxide Dispersions and Population Balance Equation Model Analysis

Modified kinetic rate equation model for cooling crystallization

The kinetic rate equation (KRE) model, unlike the population balance equation model, can describe growth, nucleation, and even Ostwald ripening simultaneously. However, the KRE model cannot be applied in cooling crystallization systems. In this work, we propose a modified KRE model to describe cooling crystallization. The modified KRE model can successfully describe crystal growth and nucleation in cooling crystallization systems. In addition, the metastable zone width was simulated using the modified KRE model and compared with the experimental data in references. The results revealed that the modified KRE model could express the effect of overheating prior to cooling on the metastable zone width. As the extent of overheating increases, the metastable zone width becomes wider, which phenomenon can be clearly simulated by the modified KRE model. This modeling capability is attributed to the behavior of particle clusters that are sized less than the size of sub-nuclei. Because the population balance equation model cannot describe the metastable zone width, the modified KRE model has certain competitive advantages in its application to various crystallization systems.

Modeling droplets swelling and escape in double emulsions using population balance equations

Monitoring the physical stability of double emulsions is considered in this work. A coupled population balance equation model for the inner and outer droplets is developed. The model involves a sub-model describing swelling of the inner, and so of the outer, droplets and a sub-model describing the escape of the inner droplets. The swelling phenomena is governed by the osmotic and Laplace pressures while the inner droplet escape rate is governed by the balance of the surface and viscous forces of the droplets. Experimental investigations demonstrate the usefulness of the model and its capability to describe the evolution of the inner and outer droplets size distributions and the release rate during storage of the double emulsions.

Population Balance Modelling of Zirconia Nanoparticles in Supercritical Water Hydrothermal Synthesis

Like any other precipitation process, in supercritical water hydrothermal synthesis (SWHS), the need to improve product quality and minimize production cost requires understanding and optimization of Particle Size Distribution (PSD). In this work, using Population Balance Equation (PBE) containing nucleation and growth terms, the reactive precipitation of zirconia nanoparticles prepared by SWHS in the batch reactor was modeled. An optimization method using genetic algorithm function in MATLAB environment was developed to find simultaneously the kinetic parameters of nucleation and crystal growth rates, used for predicting PSD in PBE. The methodology developed evaluated kinetic parameters at comparable order of magnitudes to those presented in the literature, indicating a reasonable validation of the modeling method adopted. PSD results, however, showed a weak convergence of experimental and those predicted, suggesting that here, aggregation most likely played a considerable role in the PBE modeling of SWHS preparation of the zirconia nanoparticles.

Conservative mathematical model and numerical simulation of batch gravity settling with coalescence of liquid-liquid dispersions

Population balance equations (PBE) provide a suitable a framework for dealing with drop breakage and coalescence. In addition, modeling of the solid particle sedimentation process based on Kynch’s theory has been successfully used and validated in mineral processing and wastewater treatment. In this work, we present a model that merges the coalescence process with hindered polydisperse sedimentation. The PBE model is projected onto a partial differential equation (PDE) system by discretizing the droplet volume. Because there is loss of mass in the system when the daughter droplets are greater than the larger species considered for the numerical solution, two terms that produce the conservation of the mass or total volume of the dispersed phase are incorporated into the PDE system. The resulting PDE system is split into two systems: homogeneous PDEs and ordinary differential equations (ODEs). The homogeneous PDEs and the ODEs are discretized using a first-order central differencing scheme and the second-order, two-stage Runge-Kutta method, respectively. The model predicts the vertical variation of the composition of the dispersed phase layer that forms at the top or bottom of the gravity settler. The proposed model was calibrated and validated through an experiment with an oil and water system. In particular, simulations illustrate the effects of: the continuity of the dispersion (oil-in-water and water-in-oil) and the standard deviation of the initial droplet volumes on phase separation quality, as well as the influence of the coalescence frequency on the average droplet volume.

Assessing the arrest of coalescence due to Marangoni effects in flowing emulsions using population balance

The conditions for coalescence arrest due to Marangoni effect of surfactant enriched emulsions flowing through a microfluidic device are analyzed. For that aim, we develop a Population Balance Equation model that allows the quantification of coalescence occurrence for emulsion systems at different surfactant concentration making use of isotherms. Besides, our model requires three parameters with physical significance to describe the behavior of the emulsions under shear to include Marangoni flow. The results were benchmarked by comparison to microfluidic experiments reported in the literature. Hydrodynamic coupling was observed at intermediate shear rates wherein coalescence was favored compared to the kinetics without surfactant. The coalescence kinetics was found to depend on the intertwined roles of both surfactant coverage and flow properties. These results represent a step towards the use of Population Balance Equation models not only for the study and prediction of conditions leading to coalescence arrest determined by either the external shear, repulsive barriers or Marangoni based forces but also for product design and control.

A Stochastic Population Balance Equation Model for Nucleation and Growth of Crystals with Multiple Polymorphs

Understanding polymorph crystallization is important due to the different properties, such as bioavailability or crystal shape, exhibited by the different crystal forms. In this work, we present a ...