Particle Size Distribution Model Research Articles

Influence of mixing on particle formation of fast precipitation reactions—A new coarse graining method using CFD calculations as a “measuring” instrument

Methods based on Computational Fluid Dynamics (CFD) are a promising tool to gain insight into poorly accessible and swift solid formation processes such as granulation or precipitation. Unfortunately, CFD modeling of transient particle size distributions in such reactors requires enormous computational effort because complex and multi-scale interactions have to be considered. Hence, a new coarse graining approach is presented with which these complex interactions can be handled with reasonable computational effort and which enables to extract short cut methods from complex CFD simulations.The method is exemplified for precipitation crystallization of nanoscaled solid particles from the liquid phase. This process is governed by fast primary processes, such as supersaturation build-up, nucleation and growth. Experimental access to internal parameters is very difficult or even impossible.The new methodology which we call “Spatially and Temporally Averaged Reduced Numeric Measurement” (STAR NM) is based on a reasonable averaging and correlation of process dominating state variables or rates which are consigned to a fast 1D population balance solver. Precipitation of the sparingly soluble barium sulfate in water is used as a model process. A confined impinging jet mixer is used as a benchmark apparatus which allows for the adjustment of highly reproducible mixing conditions. Experimental results are compared to those gained with the new STAR NM approach.

A discussion of the importance of particle size distribution data for characterizing Athabasca Oil Sands

Abstract Predicting bitumen recovery from the McMurray Formation oil sands is required to optimize resource management and development planning. While there are many parameters that affect the recovery predictions, bitumen grade and fines in the oil sands mining, and permeability in the in situ projects, are the most important geological parameters. In this paper, we discuss how particle size distribution data can aid in the estimation of these and some other parameters and what value it brings in improving our understanding of oil sands quality and recovery. We conclude with a short review of the methods currently available for particle size distribution modeling, while paying special attention to a practical particle size distribution modeling approach based on look-up tables. The proposal is illustrated using Telephone Lake data.

Particle size distribution modeling of milled coals by dynamic image analysis and mechanical sieving

ABSTRACT Particle size distribution (PSD) of lignite and hard coals ground by ball and Gy-Ro mills were determined and compared using mechanical sieving (MS; direct, width-classification) and dynamic image analysis (DIA; indirect, length-classification), expressed in terms of sieves opening diameter and equivalent circular area diameter. Ground coal PSD data (eight combinations) were visualized and studied using length-transformation on MS and DIA using log-normal distribution plots. Data were also analyzed and compared by developing PSD models, namely Gaudin–Schuhmann (GS) and Rosin–Rammler (RR). In general, in the fine particle range below 100 μm, all ground coal samples exhibited similar PSD, for both MS and DIA methods. On PSD modeling, RR model ( R 2 ≥ 0.81; AIC ≤ − 8.6) fitted well the observation over the entire range of particles sizes produced by both ball and Gy-Ro mills compared to GS model ( R 2 ≥ 0.69; AIC ≤ − 7.3). DIA can easily determine the particle size ranges below 100 μm, while MS have only limited sieves in this range. Besides, DIA has produced more accurate PSD results than MS especially 38 μm and below. Thus, DIA and RR model can be recommended for PSD analysis of fine particulate coals, minerals, and similar products.

Effects of various input levels and different soil water retention curve models on water content estimation using different statistical methods

This study investigated the impact of different input variables on the predictability of the water content using soil water retention curve (SWRC) models. The particle and aggregate size distribution model parameters were calculated by fitting the Perrier model to the related distributions for 75 soil samples. Nine SWRC models were fitted to the experimental data and their coefficients were obtained. The regression method was used to estimate the coefficients for nine SWRC models at three input levels. Cluster analysis classified the SWRC models into more homogeneous groups according to the accuracy of their predictions. The SWRC estimated using the Gardner model had the highest accuracy, but it was not an appropriate model for the soils because of its low fitting accuracy. Boltzman, Campbell, and Fermi models obtained the highest accuracy after the Gardner model. The Durner model yielded the lowest prediction accuracy due to the lack of correlation between the input variables and coefficients in this model. Thus, the water content predictions obtained using different SWRC models varied because different input variables were employed.

Optimal Design of a Gas Antisolvent Recrystallization Process of Cyclotetramethylenetetranitramine (HMX) with Particle Size Distribution Model

Physical properties and characteristics of a solid particle highly depend on the particle shape and size. The Gas AntiSolvent (GAS) process is a recently developed process that can control the size and morphology of solid particles using supercritical fluids as an antisolvent to resrystallize desired products in solution. Though many lab-scale experiments with different choices of antisolvents exist, no study for the large-scale operation of such a system has been reported yet. This study proposes different design options of a GAS process for recrystallization of HMX (1,3,5,7-tetranitro-1,3,5,7-tetrazocane or cyclotetramethylenetetranitramine) using CO2 as an antisolvent. A detailed mathematical model for particle size distribution is integrated into the process flow sheet model. An optimal process which includes the particle size data of product and process specifications are proposed with economic evaluation and comparison of various alternative processes.

Receptor modelling of both particle composition and size distribution from a background site in London, UK

Abstract. Positive matrix factorisation (PMF) analysis was applied to PM10 chemical composition and particle number size distribution (NSD) data measured at an urban background site (North Kensington) in London, UK, for the whole of 2011 and 2012. The PMF analyses for these 2 years revealed six and four factors respectively which described seven sources or aerosol types. These included nucleation, traffic, urban background, secondary, fuel oil, marine and non-exhaust/crustal sources. Urban background, secondary and traffic sources were identified by both the chemical composition and particle NSD analysis, but a nucleation source was identified only from the particle NSD data set. Analysis of the PM10 chemical composition data set revealed fuel oil, marine, non-exhaust traffic/crustal sources which were not identified from the NSD data. The two methods appear to be complementary, as the analysis of the PM10 chemical composition data is able to distinguish components contributing largely to particle mass, whereas the number particle size distribution data set – although limited to detecting sources of particles below the diameter upper limit of the SMPS (604 nm) – is more effective for identifying components making an appreciable contribution to particle number. Analysis was also conducted on the combined chemical composition and NSD data set, revealing five factors representing urban background, nucleation, secondary, aged marine and traffic sources. However, the combined analysis appears not to offer any additional power to discriminate sources above that of the aggregate of the two separate PMF analyses. Day-of-the-week and month-of-the-year associations of the factors proved consistent with their assignment to source categories, and bivariate polar plots which examined the wind directional and wind speed association of the different factors also proved highly consistent with their inferred sources. Source attribution according to the air mass back trajectory showed, as expected, higher concentrations from a number of source types in air with continental origins. However, when these were weighted according to their frequency of occurrence, air with maritime origins made a greater contribution to annual mean concentrations.

Kinetic modeling of particle size distribution of soot in a premixed burner-stabilized stagnation ethylene flame

A detailed model of soot formation is proposed, which consists of a gas-phase kinetic model for the pyrolysis and oxidation of selected hydrocarbon fuels and a kinetic mechanism of soot nucleation and mass/size growth through coagulation and surface reactions. The gas-phase model (Ranzi et al., 2012) was expanded to include the chemistry of Polycyclic Aromatic Hydrocarbons (PAHs) up to four-to-five ring PAHs, with a modular and hierarchical approach. The discrete sectional method was employed to solve the size evolution of the particle size distribution function (PSDF). Analogy and similarity rules were employed to describe heterogeneous reaction kinetics of soot surface reactions. A variable collision efficiency was assumed for the coalescence of small soot particles. Larger particles were assumed to undergo aggregation. The predicted PSDFs are found to be in reasonably good agreement with the experimental data for nascent soot measured in an atmospheric-pressure premixed ethylene–oxygen–argon flame in the burner-stabilized stagnation flame configuration. Sensitivity analyses of the PSDF, number density, and volume fraction were carried out with respect to the rate parameters of addition reactions of acetylene, PAHs, resonantly stabilized radical reactions, and coalescence and aggregation. The results show that the reaction of PAHs and acetylene with soot surfaces and the kinetics of coalescence and aggregation exhibit dominant effects on the detailed and global soot properties for the flame studied, in agreement with conclusions of a large range of previous modeling studies.

Critical evaluation of particle size distribution models using soil data obtained with a laser diffraction method.

Mathematical descriptions of classical particle size distribution (PSD) data are often used to estimate soil hydraulic properties. Laser diffraction methods (LDM) now provide more detailed PSD measurements, but deriving a function to characterize the entire range of particle sizes is a major challenge. The aim of this study was to compare the performance of eighteen PSD functions for fitting LDM data sets from a wide range of soil textures. These models include five lognormal models, five logistic models, four van Genuchten models, two Fredlund models, a logarithmic model, and an Andersson model. The fits were evaluated using Akaike’s information criterion (AIC), adjusted R2, and root-mean-square error (RMSE). The results indicated that the Fredlund models (FRED3 and FRED4) had the best performance for most of the soils studied, followed by one logistic growth function extension model (MLOG3) and three lognormal models (ONLG3, ORLG3, and SHCA3). The performance of most PSD models was better for soils with higher silt content and poorer for soils with higher clay and sand content. The FRED4 model best described the PSD of clay, silty clay, clay loam, silty clay loam, silty loam, loam, and sandy loam, whereas FRED3, MLOG3, ONLG3, ORLG3, and SHCA3 showed better performance for most soils studied.

Conversion between Soil Texture Classification Systems using the Random Forest Algorithm

This study focuses on the reclassification of a soil texture system following a hybrid approach in which the conventional particle-size distribution (PSD) models are coupled with a random forest (RF) algorithm for achieving more generally applicable and precise outputs. The existing parametric PSD models that could be used for this purpose have various limitations; different models frequently show unequal degrees of precision in different soils or under different environments. The authors present in this article a novel ensemble modeling approach in which the existing PSD models are used as ensemble members. An improvement in precision was proved by better statistical indicators for the results obtained, and the article documents that the ensemble model worked better than any of its constituents (different existing parametric PSD models). This study is verified by using a soil dataset from Slovakia, which was originally labeled by a national texture classification system, which was then transformed to the USDA soil classification system. However, the methodology proposed could be used more generally, and the information provided is also applicable when dealing with the soil texture classification systems used in other countries.

Predicting the soil moisture retention curve, from soil particle size distribution and bulk density data using a packing density scaling factor

Abstract. A substantial number of models predicting the soil moisture characteristic curve (SMC) from particle size distribution (PSD) data underestimate the dry range of the SMC especially in soils with high clay and organic matter contents. In this study, we applied a continuous form of the PSD model to predict the SMC, and subsequently we developed a physically based scaling approach to reduce the model's bias at the dry range of the SMC. The soil particle packing density was considered as a metric of soil structure and used to define a soil particle packing scaling factor. This factor was subsequently integrated in the conceptual SMC prediction model. The model was tested on 82 soils, selected from the UNSODA database. The results show that the scaling approach properly estimates the SMC for all soil samples. In comparison to the original conceptual SMC model without scaling, the scaling approach improves the model estimations on average by 30%. Improvements were particularly significant for the fine- and medium-textured soils. Since the scaling approach is parsimonious and does not rely on additional empirical parameters, we conclude that this approach may be used for estimating SMC at the larger field scale from basic soil data.

Performance of four-parameter analytical models of atmospheric aerosol particle size distribution

We compared the ability of five analytical models to approximate size distributions of particles on the basis of a large three-year dataset of ground level measurements of atmospheric aerosols in Hyytiälä, Finland. The collection of models includes: modified gamma distribution; inverse modified gamma distribution; regularized power law; two-power law (called the KL-distribution in previous publications); and the two-power-square law, which is introduced in this study. Comprehensibility, approximation accuracy, and information transfer efficiency are considered to be the main attributes of a model. A model is comprehensible if its parameters have meaningful interpretation. Accuracy is measured by the divergence between the measured distribution and its closest approximation. Information transfer efficiency is characterized by the loss of measurement information at interpretation of model parameters when the part of information contained in the correlations between the parameters is abolished. The analysis allows us to recommend the two-power law to fit the distributions of atmospheric aerosol particles measured in Northern Europe and regions with a similar climate.

Quantitative Analysis of Uncoated Eszopiclone Tablets by Near-Infrared Spectroscopy

This study employed near-infrared (NIR) spectroscopy to analyze content uniformity, moisture content, compression force, tablet hardness, average particle size, and particle-size distribution. The content uniformity, moisture content, compression force, tablet hardness, and average particle size models yielded high correlation coefficients (R2) of 0.99582, 0.99725, 0.99620, 0.96294, and 0.98421, respectively, whereas the particle size distribution models showed good predictive ability. Conventional criteria such as R2, root-mean-square error of calibration, and the root-mean-square error of prediction were used to evaluate the accuracy and precision of the model. To ensure the accuracy and predictability of the content model for low-dose tablets, additional validation and reliability evaluations were performed using 70%, 80%, 100%, 120%, and 130% drug concentrations as well as 90% and 110% active content formulations. Near-infrared spectroscopy with multivariate modeling is a rapid, nondestructive technique for the characterization of the manufacturing process.

Particle size distribution and optimal capture of aqueous macrobial eDNA

Summary Using environmental DNA (eDNA) to detect aquatic macroorganisms is a new survey method with broad applicability. However, the origin, state and fate of aqueous macrobial eDNA – which collectively determine how well eDNA can serve as a proxy for directly observing organisms and how eDNA should be captured, purified and assayed – are poorly understood. The size of aquatic particles provides clues about their origin, state and fate. We used sequential filtration size fractionation to measure the particle size distribution (PSD) of macrobial eDNA, specifically Common Carp (hereafter referred to as Carp) eDNA. We compared it to the PSDs of total eDNA (from all organisms) and suspended particle matter (SPM). We quantified Carp mitochondrial eDNA using a custom qPCR assay, total eDNA with fluorometry and SPM with gravimetric analysis. In a lake and a pond, we found Carp eDNA in particles from >180 to <0·2 μm, but it was most abundant from 1 to 10 μm. Total eDNA was most abundant below 0·2 μm, and SPM was most abundant above 100 μm. SPM consisted of ≤0·1% total eDNA, and total eDNA consisted of ≤0·0004% Carp eDNA. 0·2 μm filtration maximized Carp eDNA capture (85% ± 6%) while minimizing total (i.e. non‐target) eDNA capture (48% ± 3%), but filter clogging limited this pore size to a sample volume <250 mL. To mitigate this limitation, we estimated a continuous PSD model for Carp eDNA and derived an equation for calculating isoclines of pore size and water volume that yield equivalent amounts of Carp eDNA. Our results suggest that aqueous macrobial eDNA predominantly exists inside mitochondria or cells, and that settling may therefore play an important role in its fate. For optimal eDNA capture, we recommend 0·2 μm filtration or a combination of larger pore size and water volume that exceeds the 0·2 μm isocline. In situ filtration of large volumes could maximize detection probability when surveying large habitats for rare organisms. Our method for eDNA particle size analysis enables future research to compare the PSDs of eDNA from other organisms and environments, and to easily apply them for ecological monitoring.

Agglomeration of FePO<sub>4</sub> Prepared by Continuous Reaction Precipitation

In order to control the FePO4 particles size, the reaction precipitation between Fe (NO3)3 and H3PO4 was performed by the controlled crystallization technique. The influences of the feed concentration, stirring speed and reaction temperature on the agglomeration size of FePO4 were investigated. The particles produced were amorphous and can be crystallied to pure phase FePO4 after calcining at 550 °C for 5 hours. The agglomerates size and morphology were characterized by the laser granulometer and scanning electron microscope, respectively. The results showed that the products were spherical and the agglomerate size was in the range of 2~9 μm at the solution PH 2.30 and the other experiment conditions were as follows, the reactant concentration was ranged from 0.5 to 1.0mol/L, stirring speed from 500 to 3000 rpm and reaction temperature from 15 to 80°C. It was found that the agglomerates size increased with the increase of the reaction temperature, decreased with the increase of the feed concentration and stirring speed. Based on the discretized population balance, the particle size distribution model was developed. There was a good agreement between the calculated values and the experimental data, which indicated that this model can be used for predicting the agglomerates size distribution at any desired levels of the variables studied.

Grinding of Coriander Seeds: Modeling of Particle Size Distribution and Energy Studies

Coriander seeds were ground in an impact type hammer mill and a pin mill to study the pattern of particle size distribution and their relationship with energy consumption. The particle size distribution for all samples was best described by the Rosin-Rammler-Bennett (RRB) and model with high degree of correlation coefficient compared to the Gaudin-Shumann (GS) model and Log-normal function. Particle size distribution was also characterized by various size parameters such as: uniformity index, size range variation coefficient, mass relative span, skewness, kurtosis, size guide number, coefficient of uniformity, and the coefficient of gradation and geometric standard deviation. Energy consumption for grinding was studied based on classical grinding laws, namely, Bond's, Rittenger's and Kick's law. Bond's Work index varied from 0.5 to 4.3 kJ kg−1 depending on the size reduction ratio. Energy consumption followed a linear relationship with size reduction ratio.

Numerical simulation of precipitation in multicomponent Ni-base alloys

A comprehensive particle size distribution model has been developed for the simulation of γ′ precipitation in multicomponent Ni alloys. Nucleation, growth and coarsening of the precipitates are described by a particle size distribution. The growth rate of each precipitate class is calculated with a multi-component diffusion model formulated for non-diagonal matrices of diffusion coefficients. The model is fully coupled with CALPHAD calculations of the thermodynamic equilibrium at the interface, including a direct treatment of the effect of curvature through modification of the Gibbs free energy. An optimization strategy was developed to minimize the computational cost. The model was used to simulate ageing heat treatment at 600°C of Ni–7.56at.% Al–8.56at.% Cr, which was studied experimentally by Booth-Morrison and others (Booth-Morrison C, Weninger J, Sudbrack CK, Mao Z, Noebe RD, Seidman DN. Acta Mater 2008;56:3422; Mao Z, Booth-Morrison C, Sudbrack CK, Martin G, Seidman DN. Acta Mater 2012;60:1871). The comparisons showed that the precipitation stages of γ′ precipitates are correctly captured by the numerical model. It was shown that non-diagonal diffusion coefficients substantially influence the selection of the operating tie-line and the overall transformation kinetics. With non-diagonal diffusion matrices, complex phenomena such as uphill diffusion of Cr due to the Al gradients were evidenced and explained.

On the Generative Equations of Fractal Self‐Similarity in Granular Media and the Related PSD Models

The physical appearance of a great number of granular media shows, together with the heterogeneity of grain sizes, suggests the existence of geometrical scale invariance. This note focuses its attention on discussing how such physicoempirical property, visually perceived, can be mathematically formulated and used in the modeling of particle size distribution (PSD). Two ways of encoding that property by means of respective equations of different natures are analyzed. They lead to respective fractal particle size distribution (PSD) models coming from sources from the literature. The nature and features of the two different models are analyzed and the simulation of the PSD by means of both models are compared. In addition, the parameters that both models provide and their role in characterizing PSD are discussed. This note aims mainly to contribute to the clarification of some conceptual, methodological, and historical aspects within the frame of the fractal modeling of PSD of granular media.

Grinding Studies of Mango Ginger: Mathematical Modelling of Particle Size Distribution and Energy Consumption

Mango ginger was ground in hammer mill with three different classifying screens and pin mill to study the particle size distribution and energy consumption. The Rosin-Rammler Bennet (RRB) model fitted well the particle size distribution data over the entire range of the size distribution for grinding in both hammer mill and pin mill with high coefficient of determination (R2) and low values of residual sum square, root mean square error and Chi-square. Relationship between RRB model parameters with hammer mill screen size was obtained with high R2. All the three classical models such as Rittinger’s, Kick’s and Bond’s law were found suitable to explain the energy consumption for grinding. Energy consumption increased exponentially with decrease in classifying screen size of hammer mill. The Work index for grinding increased with increase in size reduction ratios and were in the range of 0.075-0.58 kW/kg.

Particle size distribution models for soils of the humid tropics

Standardisation of particle size distribution (PSD) is a prerequisite to achieve compatibility of soil data among various countries with different texture classification systems. Therefore, several mathematical models have been proposed to accurately represent PSD. Previous studies evaluated the performance of such models to describe PSD of soils from temperate regions. This study aims at evaluating the performance of models for describing PSD of soils from the humid tropics based on a large dataset. A dataset of 1,412 soils from Central Africa representing 11 different FAO Soil Groups was used. Ten PSD models with two to four fitting parameters were selected: simple log-normal (LN_2p), van Genuchten-type1 (VG_2p), van Genuchten-type2 (vG_3p), Fredlund-type1 (F_3p), Fredlund-type2 (F_4p), Weibull (W_3p), Skaggs (Sk_3p), Gompertz-type1 (G_2p), Gompertz-type2 (G_4p) and Andersson (A_4p). The fitting performance of the PSD models was evaluated by three statistical indices: the adjusted coefficient of determination, the Akaike information criterion and the relative error. Clustered columns and box plots were also used to get more insights. The predictive ability of the best PSD models was tested using a leave-one-out method and 1:1 plots. A table of initial values for the fitting parameters of each PSD model was provided for future applications. Some models like VG_2p, VG_3p, Sk_3p and G_4p were not suitable to describe PSD of soils in the humid tropics. On the other hand, F_3p, F_4p, W_3p and A_4p models showed outstanding fitting performance. The fitting performance of PSD models was also dependent of the textural class, the broad textural group and the bimodal character of the soil. For the most frequent textural classes in the dataset, the F_3p and A_4p models were the best closely followed by the W_3p model. While the F_3p model performed better than the A_4p model for coarse-textured soils, the opposite was observed for fine-textured soils. The W_3p model showed acceptable fitting performance for fine, medium and coarse-textured soils. The performance of the PSD models was found to be better for bimodal soils, which are common in the humid tropics, than for unimodal soils. Great differences in fitting and prediction performance were found between the PSD models. Soil texture as well as the bimodal character of the soil significantly affect their respective performance. Some models like VG_2p, VG_3p, Sk_3p and G_4p are not suitable to describe PSD of soils of the humid tropics. On the other hand, F_3p, F_4p, W_3p and A_4p models showed outstanding fitting performance. Therefore, they are highly recommended in order to get a better description of the PSD of soils of the humid tropics.

Regression Modeling of Particle Size Distributions in Urban Storm Water: Advancements through Improved Sample Collection Methods

A new sample collection system was developed to improve the representation of sediment entrained in urban storm water by integrating water quality samples from the entire water column. The depth-integrated sampler arm (DISA) was able to mitigate sediment stratification bias in storm water, thereby improving the characterization of suspended-sediment concentration and particle size distribution at three independent study locations. Use of the DISA decreased variability, which improved statistical regression to predict particle size distribution using surrogate environmental parameters, such as precipitation depth and intensity. The performance of this statistical modeling technique was compared to results using traditional fixed-point sampling methods and was found to perform better. When environmental parameters can be used to predict particle size distributions, environmental managers have more options when characterizing concentrations, loads, and particle size distributions in urban runoff.