Full Particle Size Distribution Research Articles

Population balance modelling and reconstruction by quadrature method of moments for wet granulation

Population balance methods utilised in multiphase flow simulations mark a significant advancement in computational fluid dynamics. However, existing approaches exhibit shortcomings, such as being prone to inaccuracies or being computationally prohibitive. Addressing these challenges, a recent innovation in closure for the method of moments is the introduction of quadrature based moments methods (QBMM). Discretising a distribution by a number of discrete elements, QBMM facilitate efficient and accurate tracking of density distributions, particularly for particle size distributions (PSD). However, obtaining the full particle size distribution information using these methods requires reconstructing the distribution from a finite set of moments, which is not a trivial step.This study introduces a novel combination of the maximum entropy reconstruction (MER) and QBMM, establishing a robust and rapid framework for the time evolution and reconstruction of PSDs. As proof of concept for this framework, we focus on the direct quadrature method of moments (DQMOM) with spatially homogeneous and monovariate distributions. We show that coupling of MER with DQMOM has numerous advantages. To verify the framework, special cases of constant growth, aggregation, and breakage are considered for which analytical solutions can be found. Furthermore, we show the advantage of using DQMOM with volume-based over length-based distributions, and address numerical as well as theoretical issues.Application of the framework is successfully conducted on the evolution of the PSD from a twin-screw wet granulation dataset, considering all active primary physical mechanisms in a wet granulation process, namely growth, aggregation, and breakage. This showcases the consistency of the proposed framework and underscores its applicability to real-world scenarios.

Synthesis of gold nanoparticles in emulsion–emulsion precipitation route: Experiment, mechanism and simulation

Soft templates like water-in-oil emulsion drops (of 1-100μm diameter), have been used for synthesis of nanoparticles; with specific mean particle diameter and narrow particle size distribution. In this context, two separate emulsions have been reacted by an emulsion–emulsion precipitation route. Consequently, two different kinds of interacting populations coexist — emulsion drops evolving through drop coalescence and breakage, with multiple nanoparticle populations growing inside each emulsion drop. However, no model framework currently exists to simulate these two highly interacting populations, for predicting evolution of nanoparticle size. To this end, presently gold nanoparticles (GNPs) have been used as a model system, to explore the nanoparticle formation mechanism. The metal salt precursor and reducing agent (NaBH4 or N2H4) are prepared in two separate water-in-oil emulsions and are reacted upon mixing to form GNPs. A kinetic Monte Carlo (kMC) scheme accounting for drop coalescence, drop breakage, nucleation, particle growth and particle–particle coagulation was developed to handle simultaneous size and number evolution of aqueous emulsion drops and GNP populations. We find that, the mean GNP diameter decreases (from 6.9 to 3.5 nm) with increasing molar ratio (M) of reducing agent to precursor concentration. Further, increasing the precursor concentration at a fixed M, does not affect the final particle size. Our kMC simulation results not only agree well with experimental mean particle diameter and coefficient of variation (COV) of particle diameter, but was also validated with full particle size distribution of GNPs. So, M can be used as a tunable parameter for controlling GNP diameter experimentally.

Size-resolved environmentally persistent free radicals in cold region atmosphere: Implications for inhalation exposure risk.

Environmental persistent free radicals (EPFRs) have attracted more attentions recently due to their potential adverse effects to human. EPFRs in full-size range particles were comprehensively investigated in this study. The average EPFRs concentration during heating season was 3.01×1014 spins/m3, which was much higher than that in non-heating season (4.30×1013 spins/m3). The highest concentration of EPFRs presented in 0.56-1.0µm particles during heating season, while it shifted to 5.6-10µm particles during non-heating season. Besides, the contributions of EPFRs on PM>10 to the total concentration of EPFRs cannot be neglected, especially in the non-heating season. The International Commission on Radiological Protection model and the specific factors of the Chinese population were applied to evaluate the inhalation exposure risk of EPFRs. The results indicated that the exposure levels of EPFRs to the upper respiratory tract were much higher. The daily exposure dose of EPFRs suggested the inhalation exposure risk of 3-4 years old was higher than other age groups. In summary, these finding provided new insights for the full range particle size distribution and the inhalation exposure risk of EPFRs, which improved our understanding on the environmental fate and the health risk of EPFRs in atmosphere.

Impact of solvent properties on the precipitation of active pharmaceutical ingredients

In antisolvent precipitation, water and an organic drug-containing solvent induces particle formation. For each water-solvent mixture, the fluid properties such as viscosity, density and diffusion coefficient but also the drug solubility depend nonlinearily on the fluid mixture compositions. Each property in itself has a strong impact on the solid formation, and thus on the outcome of the precipitation. The simulation framework recently developed by Schikarski et al. [1] allows investigating separately the impact of each fluid property on the precipitation. For a T-mixer and a novel 3-inlet-mixer, we first show that the viscosity and density variations largely determine the macroscopic mixing behavior and, thus, the build-up of supersaturation. The composition-dependent diffusion properties of the drug molecules largely govern the transport-controlled nucleation and particle growth at small scales. In a second step, we numerically predict the experimentally obtained full particle size distribution of the precipitated Ibuprofen nanoparticles using different water-solvent mixtures and different operating conditions. Our numerical results are in excellent agreement with our experimental measurements using ethanol, methanol and 2-propanol as solvents.

Highly accurate and numerical tractable coupling of nanoparticle nucleation, growth and fluid flow

Supersaturation as the thermodynamic driving force of particulate precipitation couples the mixing of solute compounds with the solid formation consisting of nucleation and molecular growth. Because the process spans many orders of magnitude in space and time, gathering spatio-temporal information by experiments is hardly achievable. Numerical simulation of the nonlinear, non-local balance equations for mass, momentum and dispersed phase provide detailed solution information in space and time. We here introduce a novel, highly accurate and very efficient numerical formulation of the population balance equations for the dispersed phase coupled to the concentration and velocity field. We reformulate the problem in terms of a fixed-point equation solving the population balance equation along Lagrangian paths with the recently developed exact Method of Moments (eMoM, Pflug et al., 2020) coupled to the concentration field (Eulerian frame). The analytical reformulation provides the information about the full particle size distribution while the computational effort remains in the order of classical moment methods and is much less than numerical schemes approximating the full particle size distribution such as finite volume methods. Furthermore, we explore the difference between an Euler-Euler and Euler-Lagrange coupling considering moderate to large P é clet numbers for the solutes and particles. It turns out that the mean particle size as a solution of the population balance is much more sensitive to the numerical discretization of the advection term than the approximation of the number density function. Based on this finding, we eventually explore the impact of particle diffusion on the solid formation process and give guidelines, when particle diffusion should be taken into account.

Surface area and size distribution of cement particles in hydrating paste as indicators for the conceptualization of a cement paste representative volume element

The conceptualization of a representative volume element (RVE) of hardened cement paste for numerical homogenization of mechanical problems rests on identifying the largest discernible microstructural feature, i.e. unreacted cement grains. While the particle size distribution (PSD) of anhydrous cement is a well-controlled production parameter, the size evolution of a representative cement grain throughout hydration remained unresolved. This study analyzes digitized 3D cement paste microstructures obtained from X-ray micro-computed tomography, coupled with CEMHYD3D hydration model, and segmented by image-processing tools, to obtain the full PSD and specific surface area evolutions of unreacted grains throughout hydration. Results provided indicate a representative grain size in the range of 30−40μm regardless of hydration elapsed, implying a cement paste RVE should amount to 150−200μm to realistically represent cement grains. The PSD shape remained self-similar and two distinctive hydration regimes were identified, differing in dissolution rate and specific surface area decrease, correlating with calcium sulfate reactivity peak. Both measures provide easily accessible microstructural features that may be used for constructing artificial RVEs of hardened cement paste in micromechanical models and related simulations, resting on experimental data.

Quantitative modeling of precipitation processes

Precipitation from the liquid phase is a powerful and common unit operation for the continuous, highly reproducible production of nanoparticles. However, a general, predictive and quantitative modeling framework is still missing due to the inherent multiscale nature of the precipitation process and the complex interplay between the relevant sub-processes. We apply direct numerical simulation of the fluid flow coupled with a population balance framework to investigate the precipitation of stabilized ibuprofen nanoparticles in a T-mixer. Our findings suggest that the Damköhler number (the ratio between the mixing time and solid formation time) determines the precipitation outcome. We demonstrate how the primarily unknown solid formation kinetics can be estimated in the simulations with the guidance of experimental input at a single process condition. We subsequently vary the Damköhler number by changing the inflow rates (global mixing time) and the initial ibuprofen concentration. In doing so, excellent agreement between numerical simulations and experiments in the full particle size distribution at different process conditions (from laminar to turbulent flow and different initial ibuprofen concentrations) is obtained using the beforehand estimated solid formation kinetics. Our model opens avenues for the predictive simulation of particle-formation dynamics and is a stepping stone for the tailored, scalable production of nanoparticles.

Full particle size distribution characteristics of land surface sediment and their effect on wind erosion resistance in arid and semiarid regions of Northwest China

Particle size distribution (PSD) is one of the fundamental physical properties of clastic sediment. Although many mathematical models have been proposed to describe the PSD of sediment, the physical meaning behind model parameter characterization remains poorly understood, which means it is seldom used in studies of the environment, geomorphologic evolution, and related surface wind–sand processes in arid areas. In this study, based on the Li model of the PSD of sediment, the parameters of sediment obtained from different landscapes in Northwest China were investigated and their relationship with the resistance of sediment to wind erosion was analyzed. Results showed that the full PSD parameters (μ and Dc) of the Li model not only revealed the full PSD of the surface sediment of the various landscapes, but also reflected the differences in sediment structure. The full PSD parameters of fine silty, sandy, and sandy gravel surface sediment were −0.2 < μ < 0 and Dc < 0.08, −0.2 < μ < 0 and 0.08 ≤ Dc ≤ 0.2, and μ < 0 and Dc > 0.2, respectively. Parameter Dc not only represents the mean particle size of sediment, but it also indicates the degree of sorting, that is, it has a positive linear correlation with the component content of coarser particles in the sediment structure. Results also showed that vegetation cover markedly enhances surface resistance to wind erosion. Under different vegetation cover scenarios, the resistance of sandy and sandy gravel surfaces to wind erosion increased as the Dc value increased.

Particle Size and Particle Size Distribution of Emulsion Polymers: A Review of Modeling Studies

The aim of this work is to provide a literature review in the area of modeling of particle size and particle size distribution of emulsion polymers. Modeling studies of emulsion polymerization are of great interest due to the industrial importance of latexes produced. There are two broad strategies adopted to model particle size distribution: 1) Lumped model in which average particle size assuming the monodispersed distribution is modelled. 2) Distributed model or population balance model in which full particle size distribution is modelled. Lumped model serves as a learning model and a stepping stone to develop a population balance model.

Population balance approach to model Ostwald ripening of silica using Gram – Charlier series expansion based closure

Ostwald ripening is a main phenomenon responsible for growth of particles manufactured using colloidal methods. This paper considers mathematical description of the process using population balance approach solved by method of moments. However, in the case of particles’ dissolution which is a part of the process a closure problem is generated. A novel closure method for the moment – transformed population balance equations based on Gram – Charlier expansions is introduced. Due to its simplicity it allows to reconstruct the full particle size distribution in a stepwise manner giving an access to the pointwise values of the density function required to close the model. The introduced numerical scheme is further used to simulate several carefully chosen test cases including dissolution of solid particles and ageing of particles’ population. The mathematical model presented in the paper is then applied to interpret experimental results for batch precipitation of silica.

Zonal modelling of a counter-current spray drying tower

A generic multi-zonal modelling methodology for spray drying towers is proposed to provide relatively quick (seconds) simulations on desktop computers for decision making on changing process parameters for automatic process control. A multi-zonal model comprises combinations of all or some of six differing zones: plug-flow, semi-plug-flow and well-mixed zones in either co- or counter-current flow. This approach is demonstrated by predicting the dried powder characteristics of a detergent powder from a pilot-scale counter-current spray drying tower. The types, sizes and locations of the different zones are obtained by a detailed analysis of predictions from a previous 3-dimensional Computational Fluid Dynamics (CFD) simulation of gas and particle flow dynamics and drying kinetics within the tower. The multi-zonal model consists of seven zones – five in counter-current and two in co-current flow. The trends of the predicted gas temperature profiles are close to those from the CFD results. The particle exit temperature, moisture content and residence times over the full particle size distribution (PSD) range are very close to the CFD values unlike the previous simulated results from our plug-flow model. The outcome clearly demonstrates that the prerequisite to have a sound conceptual model of spray drying towers, such as the multi-zonal model developed here, is a detailed knowledge of the gas and particle flow fields within the tower.

Semibatch emulsion polymerisation modelling: polybutyl acrylate case study

ABSTRACTA comprehensive mathematical model is developed for emulsion polymerisation of butyl acrylate to polybutyl acrylate in a semi batch reactor. The mathematical framework is based on the population balance approach. The model includes such mechanistic details as: (i) particle generation from radicals entering micelles, (ii) coupling of the radical concentration in the aqueous phase and particle phase, (iii) determination of the particle phase radical concentration by radical entry into, exit from and termination inside the particle, (iv) thermodynamic equilibrium between the monomer concentration in the aqueous phase and the particle phase. The population balance equations are solved efficiently using orthogonal collocation. Dynamic simulations are compared to the experimental data taken from the literature. The predictions for the various average diameters of the particle size distribution, polydispersity index, standard deviation and full particle size distribution are also given.

Experimental Determination of Co2 Mineralization Depth of Slag Minerals at Co2 Concentrations Available in Iron and Steelmaking Flue Gases

We report the diffusion coefficient relevant to CO2 mineralization using minerals commonly encountered in ironmaking and steelmaking slag. Diffusion coefficients were gained from CO2 mineralization achieved under passive conditions consistent with processes that are likely to maintain net CO2 emission reduction. In other words, no chemical, thermal, or mechanical enhancement was applied. Tests were performed at 30 ˚C with molar CO2 concentrations of 5%, and 20%. We applied the diffusion coefficients using the shrinking core model with a modification to account for the full particle size distribution of a sample and the fractal surface roughness. Results indicate that CO2 mineralization efficacy is highly variable among minerals common to slag; thus, the slow kinetics and low conversion common to slag-based CO2 mineralization is likely due in part to blockage of reactive species by unreactive species. Further, the modified shrinking core model matches early phase CO2 mineralization rates where the traditional models fail. Analysis of slag-based CO2 mineralization methods should thus account for the specific mineralogy, particle size distribution, and micromorphology of tested slag.

Accuracy control in Monte Carlo simulations of particle breakage

Monte Carlo (MC) methods are an important tool for the numerical solution of the population balance equation, allowing the optimisation and control of particulate processes on laboratory or plant scales. We investigate in this work a family of MC methods for particle breakage proposed by Kotalczyk et al. (2017, pp.417-429). The authors reported that specific breakage schemes (defined by a combination factor R) allow rendering the full particle size distribution. They also showed that specific ranges of the combination factor R might lead to severe systematic errors, but did not investigate measures of control or prevention. In this paper, a strategy which allows estimating the magnitude of the systematic error from the simulation data is presented. It is also shown how the simulation parameters can be set in order to keep the systematic error at an acceptable level.

Accuracy control in Monte Carlo simulations of particle breakage

Monte Carlo (MC) methods are an important tool for the numerical solution of the population balance equation, allowing the optimisation and control of particulate processes on laboratory or plant scales. We investigate in this work a family of MC methods for particle breakage proposed by Kotalczyk et al. (2017, pp.417-429). The authors reported that specific breakage schemes (defined by a combination factor R) allow rendering the full particle size distribution. They also showed that specific ranges of the combination factor R might lead to severe systematic errors, but did not investigate measures of control or prevention. In this paper, a strategy which allows estimating the magnitude of the systematic error from the simulation data is presented. It is also shown how the simulation parameters can be set in order to keep the systematic error at an acceptable level.

Effect of powder size distribution on densification and microstructural evolution of binder-jet 3D-printed alloy 625

Binder-jet 3D-printing is a powder bed additive manufacturing process that selectively deposits binder on a powder bed layer-by-layer to fabricate a green part followed by a sintering step for densification. Gas-atomized alloy 625 powders of three different powder size distributions including 16–63 μm (full), 16–25 μm (fine) and 53–63 μm (coarse) powders were 3D-printed with green relative bulk densities of about 52%, 45% and 48%, respectively, followed by vacuum-sintering at temperatures between 1225 and 1300 °C for 4 h. For the fine and coarse powders with narrow size distribution, printing defects with high pore coordination numbers may form during the binder jetting process which cannot be removed during the final sintering stage even during supersolidus liquid phase sintering. However, the full particle size distribution gave higher green density with fewer large, highly coordinated pores so supersolidus liquid phase sintering was able to reach near-full density. Additionally, the fine powders gave non-uniform, anisotropic linear shrinkage during sintering which is unfavorable for designing complex structures. The results suggest that particle size distribution is a determining factor for supersolidus liquid phase sintering, pore removal and final microstructure, if printing parameters such as layer thickness, binder saturation, printhead binder droplet size and drying time are similar.

Experimental evaluation of the effect of particle properties on the segregation of ternary powder mixtures

Segregation of components, especially the low content level of a highly active substance, has serious deleterious effects on powder formulation. This study investigates the effect of particle properties, particularly size, shape, density and cohesion, on the segregation of laundry detergent powders. Tetraacetylethylenediamine (TAED) particles, spray-dried synthetic detergent powder referred to as Blown Powder (BP) and Enzyme Placebo Granules (EP granules) are used as a model formulation. The segregation of components is evaluated using image processing of the photographic records taken from the front face of a two-dimensional heap of powders. Coefficient of variation concept is considered as segregation index. It is shown that EP granules, the component used as low-level ingredient (< 2 wt%) in the ternary mixture, are prone to extensive segregation due to their higher density as compared to BP and TAED particles. Desired properties for segregation minimisation of EP granules have been further investigated. It is found that the segregation of EP granules can be reduced noticeably by applying a thin layer of a sticky liquid on them before mixing with the rest of powders with full particle size distribution of the components. Polyethylene glycol has been used for this purpose. Addition of 2.5 wt% gives an optimum level to reduce the segregation of EP granules without compromising the flowability of the mixture as well as EP granules themselves.

Evaluation of Different Predictor Models for Detailed Soil Particle-Size Distribution

An accurate mathematical representation of soil particle-size distribution (PSD) is required to estimate soil hydraulic properties or to compare texture measurements using different classification systems. However, many databases do not contain full PSD data, but instead contain only the clay, silt, and sand mass fractions. The objective of this study was to evaluate the abilities of four PSD models (the Skaggs model, the Fooladmand model, the modified Gray model GM (1,1), and the Fredlund model) to predict detailed PSD using limited soil textural data and to determine the effects of soil texture on the performance of the individual PSD model. The mean absolute error (MAE) and root mean square error (RMSE) were used to measure the goodness-of-fit of the models, and the Akaike's information criterion (AIC) was used to compare the quality of model fits. The performance of all PSD models except the GM (1,1) improved with increasing clay content in soils. This result showed that the GM (1,1) was less dependent on soil texture. The Fredlund model was the best for describing the PSDs of all soil textures except in the sand textural class. However, the GM (1,1) showed better performance as the sand content increased. These results indicated that the Fredlund model showed the best performance and the least values of all evaluation criteria, and can be used using limited soil textural data for detailed PSD.

Large eddy simulation of particle aggregation in turbulent jets

Aggregation is an inter-particle process that involves a multitude of different physical and chemical mechanisms. Aggregation processes often occur within turbulent flows; for example in spray drying, soot formation, or nanoparticle formation. When the concentration of particles is very large, a direct simulation of individual particles is not possible and alternative approaches are needed. The present work follows the stochastic aggregation modelling based on a Lagrangian framework by Pesmazoglou, Kempf, and Navarro-Martinez (2016) and implements it in the Large Eddy Simulation context. The new coupled model is used to investigate particle aggregation in turbulent jets. Two cases are considered: an existent Direct Numerical Simulation of nanoparticle agglomeration in a planar jet and an experimental configuration of SiO2 nanoparticles in a round jet. The results show a good agreement in both cases, demonstrating the advantages of the Lagrangian framework to model agglomeration and it capacity to describe the full particle size distribution.

Using semidefinite programming to calculate bounds on particle size distributions

Many chemical engineering processes involve a population of particles with a distribution of sizes that changes over time. Because calculating the time evolution of the full particle size distribution (PSD) is computationally expensive, it is common to instead calculate the time evolution of only finitely many moments of the distribution. The problem with moments is that they provide only a summary description of the PSD. In particular, they do not contain enough information to answer industrially relevant questions such as: How many particles are there in the size range [a,b]? What is the shape of the distribution? What is its D10? While these questions cannot be answered exactly, in this paper, we demonstrate that one can efficiently calculate rigorous bounds on the answers by solving semidefinite programs. To the best of the authors’ knowledge this natural application of semidefinite programming to PSDs has, until now, gone unnoticed.