Particle Size Distribution Function Research Articles

Non-destructive measurement of the dispersion of High-density polyethylene/Polystyrene blends using near-infrared diffuse reflectance spectroscopy based on deep learning

The dispersion of polymer blends significantly influences their overall performance. However, current methods for characterizing dispersion are destructive and time-consuming. In this study, a near-infrared (NIR) regression model was first developed for the measurement of cumulative particle size distribution function (CDF) of dispersed particles in polymer blends, thus indirectly and quantitively characterizing dispersion. High-density Polyethylene/Polystyrene (HDPE/PS) with varying dispersed particle sizes were prepared by adjusting the content of PS, flow field intensity during processing, and viscosity ratio between HDPE and PS. Scanning electron microscopy (SEM) was used to provide reference particle size values for calibrating the NIR data. Deep-learning methods were used for the model building. Compared to traditional partial least-square regression, deep learning methods demonstrated superior fitting ability, The best-performing model, an MLP with three hidden layers, achieved an coefficient of determination (R2) of approximately 0.93 on the independent test set. The predicted CDFs closely matched the actual values, with no significant deviations observed. This study provides a non-destructive and time-saving new method for characterizing the dispersion of polymer blends.

Determination of the particle size distribution of cube-shaped colloidal perovskite quantum dots from photoluminescence spectra: A combined theoretical-experimental approach.

A theoretical-experimental approach is proposed to convert the photoluminescence spectra of colloidal perovskite quantum dot ensembles into accurate estimates for their intrinsic particle size distribution functions. Two main problems were addressed and properly correlated: the size dependence of the first excitonic transition in a single cube-shaped quantum dot and the inhomogeneous broadening of the fluorescence line shape due to the size nonuniformity of the chemically prepared quantum dot suspension in addition to the single-dot homogeneous broadening. By applying the reported methodology to CsPbBr3 quantum dot samples belonging to the strong and intermediate confinement regimes, the calculated size distributions exhibited close agreement with those obtained from transmission electron microscopy, with precise estimates for the average particle size and standard deviation. Specifically for strongly confined ultrasmall CsPbBr3 quantum dots, the presented spectroscopic model for size distribution computation is based on a new analytical expression for the size-dependent bandgap, which was developed within the framework of the finite-depth square-well effective mass approximation accounting for band nonparabolicity effects. Such a quantum mechanical approach correctly predicts the expected transition to the intermediate confinement regime in sufficiently large quantum dots, which are traditionally described by the well-known bandgap equation in the infinite potential barrier limit with a spatially correlated electron-hole wavefunction and nonparabolic carrier effective masses. The proposed calculation scheme originates from general theoretical considerations so that it can be readily adapted to semiconductor quantum dots of many other systems, from all inorganic metal halides to hybrid perovskite materials, regardless of the adopted chemical synthesis route.

Insights into phthalocyanine blue pigment dispersion: Modeling particle size distribution and grinding rate constants

The study presents an advanced mathematical model aimed at enhancing the understanding of pigment dispersion processes, with a particular focus on analyzing the dynamics of particle size distribution and determining grinding rate constants. Pigmented dispersions are crucial in various industries, including paints, coatings, and inks, where achieving optimal particle size distribution is vital for product quality. This investigates the relationship between grinding duration and pigment dispersion, aiming to establish empirical equations for the grinding rate constant as a function of particle size distribution and feed rate. Through experimental validation, the developed kinetic model provides insights into the dynamics of pigment dispersion and offers predictive capabilities for optimizing dispersion processes. Understanding the underlying mechanisms of pigment dispersion through mathematical modeling contributes to enhancing the performance and quality of final products in diverse industrial applications. The development of a new kinetic model, effective mathematical model is proposed to accurately simulate the evolution was applicable for the particle size from 32.67 to 122.4 nm distribution of pigment during comminution processes. In addition, it was observed that the kinetic model was not applicable for the particle size from 166.2 to 190.1 nm and also confirmed that the grinding was completed within 40 min, which was possible for uniform particle size distribution.

Two-step crystallization in a supersaturated solution with application to protein crystal growth: Theory and analytical solutions

A theory of two-step nucleation and evolution of crystals with allowance for their diffusion in the space of particle radii is developed. The theory is based on a two-step growth law of crystals appearing in dense liquid precursors, initially evolving within them with a diffusion-limited growth law, and then leaving them and evolving with another growth law. This two-step crystal growth law influences the integrodifferential system of kinetic and balance equations for the crystal-size distribution function and liquid supersaturation. The system of integrodifferential equations for the evolution of polydisperse ensemble of crystals is solved using the integral Laplace transform and saddle-point methods for the Weber–Volmer–Frenkel–Zel’dovich and Meirs kinetic mechanisms. A complete analytical solution to this non-linear system has been constructed in a parametric form. Namely, the particle-size distribution function, liquid supersaturation and time have been found as the functions of decision variable — the maximum size of crystals. The theory is in agreement with the experimental data on the growth of beta-lactoglobulin and insulin crystals.

The effects of processing parameters on the shape of particle size distribution and grinding rate in a stirred mill

The study of particle size distribution (PSD) of materials is a crucial aspect of mineral processing. It not only determines the appropriate beneficiation process and equipment but also has a significant impact on the grade and recovery of concentrate. This study utilizes a pin-stirred mill for the wet fine grinding of limonite to examine four different widths of PSD functions and compares their quantitative descriptions of the PSD. Skewness is a parameter introduced to quantitatively describe the asymmetry of the PSD. This study aims to investigate the effect of the stirrer speed, the density and the size of the grinding media on the shape of the PSD and on the grinding rate. The results show that a lower stirrer speed and a lower density of the grinding media result in a narrower and a more symmetrical PSD of the grinding products. Moreover, the time-dependent approach succeeds in simulating and proving the evolution behavior of the median particle size. It was found that the optimum mixing ratio of the grinding media can effectively adjust the shape of the PSD and improve the grinding efficiency.

Unification of particle size analysis results, part 1 − comparison of particle size distribution functions obtained by various measurement methods

This article concerns comparing the grain size distribution functions of mineral powder samples and analyzes errors obtained by various measurement methods. The research used three standard and currently most frequently used methods: sieve analysis, laser diffraction analysis and vision analysis. The most important research, from the point of view of unification of results, consisted in performing precise analyses of the grain size of mineral powders (fly ash, gneiss, quartz sand and Cu ore) in various grain classes. The measurement techniques used in the research and issues related to measurement analytics were characterized. The distribution functions of the grain size distributions of the tested material samples were obtained by different measurement methods and the measurement errors were compared. Using the coefficient of variation CV calculated for characteristic particle diameters, the accuracy of measurements made using various methods was analyzed. It has been shown that different particle size measurement techniques give significantly different particle size distributions for the same material, with the grain size distribution of the measured samples having a greater impact on the results than the material itself. The lowest values of the coefficients of variation were obtained for wet sieve analysis and the highest for laser diffraction. Reliable data informing about grain size distributions were generated for further model calculations necessary to standardize the results between measurement methods.

Depth from Defocus Technique For High Number Density Particle Images

The Depth from Defocus (DFD) imaging technique is used to measure the size and number concentration of particles in dispersed two-phase flows, but until now it has primarily been applied to low concentration particle images. This study explores how the technique can be extended to handle overlapping images caused by neighboring particles, significantly broadening the application scope of the DFD technique and enabling measurements at higher particle number/volume concentrations. The processing algorithms are experimentally validated using a dedicated apparatus that can systematically vary particle size, shape, and degree of image overlap. Additionally, this study explores the use of Convolutional Neural Networks (CNN) for this task, comparing these results with those obtained using conventional analyses in terms of accuracy, tolerable concentration limits, and computational speed. This approach requires a large teaching dataset of images, which is only practical and feasible if the dataset can be synthetically generated. An image generation procedure for out-of-focus neighboring spherical particles, resulting in a known blurred image overlap, is therefore first developed. This procedure is validated using laboratory images with known particle size distribution, position, and image overlap before creating the teaching dataset. The trained processing scheme is then applied to both synthetic datasets and experimental data, allowing the evaluation of the technique’s limits in terms of image overlap and tolerable volume concentration, as a function of particle size distribution.

A Single Camera Calibration-Free Approach For Dispersion Size Measurement Using Depth From Defocus

The Depth from Defocus (DFD) imaging technique is used to measure the size and number concentration of particles in dispersed two-phase flows, but until now it has primarily been applied to low concentration particle images. This study explores how the technique can be extended to handle overlapping images caused by neighboring particles, significantly broadening the application scope of the DFD technique and enabling measurements at higher particle number/volume concentrations. The processing algorithms are experimentally validated using a dedicated apparatus that can systematically vary particle size, shape, and degree of image overlap. Additionally, this study explores the use of Convolutional Neural Networks (CNN) for this task, comparing these results with those obtained using conventional analyses in terms of accuracy, tolerable concentration limits, and computational speed. This approach requires a large teaching dataset of images, which is only practical and feasible if the dataset can be synthetically generated. An image generation procedure for out-of-focus neighboring spherical particles, resulting in a known blurred image overlap, is therefore first developed. This procedure is validated using laboratory images with known particle size distribution, position, and image overlap before creating the teaching dataset. The trained processing scheme is then applied to both synthetic datasets and experimental data, allowing the evaluation of the technique’s limits in terms of image overlap and tolerable volume concentration, as a function of particle size distribution.

Establishment of fog droplet distribution model and study on canopy deposition uniformity

In plant protection operations, the distribution of droplets affects the atomization effect. To make the distribution of fog droplets more uniform in the air field, a fog droplet distribution model was established based on a three-dimensional motion model of droplets and the particle size distribution function, combined with a two-dimensional normal distribution function. The effects of the initial incidence angle and the additional wind speed on the distribution of fog droplets were analyzed. The fog droplet distribution was simulated to analyze the droplet distribution in the spatial layer, which was compared with the experimental results. To investigate the impacts of different factors on the atomization distribution, the Lagrangian interpolation method was employed, and the optimal initial incidence angle and external wind speed were found. When the initial angle of incidence was 17°, the slope of the fitted curve was the smallest, with a coefficient of determination of 0.9622 and a relative error of 3.12%. With an additional wind speed of 0.1 m/s, the coefficient of determination was 0.9782, with an average error of 4.61%. The simulation results were consistent with the experimental findings, and the accuracy of the fog droplet distribution model was verified. In summary, this research provides a novel method to improve the uniformity of the droplet distribution, which can provide a theoretical basis for determining the operating parameters.

Optimal analysis of the proportion between modified lime mud and active mineral additive in mortar: Insights from particle size distribution and packing density modeling

The use of lime mud in building materials is becoming more popular. However, its application has been limited by optimizing the proportion of lime mud and other active admixtures. In this study, the lime mud was firstly modified by a mechanochemical method to refine the grain size. Secondly, various particle size distribution functions were used to described the particle size distribution of cementitious materials. Further, based on the modifier Anderson and Andreasen (MAA) function, the optimal mixing proportion of cementitious materials was designed, and its feasibility was confirmed by the orthogonal test and range analysis. According to the hydration analysis through X-Ray Diffraction analysis (XRD), Thermogravimetry-Derivative Thermogravimetry (TG-DTG), Scanning Electron Microscope (SEM), and Mercury Intrusion Porosimetry (MIP), a significant increase in the harmless pore content was observed in the mortar with the optimal blend of MLM, MK, and SF.

Combining performances of E(m)-corrected LII and absorption for in situ measurements of the volume fraction of 2–4 nm soot particles.

Determining the soot volume fraction (fv) in combustion environments requires detailed knowledge of the optical properties of the soot particles, and in particular of their absorption function E(m). This study addresses a fundamental lack of information on the optical properties of 2–4 nm soot particles. Recent works based on the modeling of the photoelectron emission yields and UV-vis-NIR-absorption measurements found a sharp decrease of E(m) with the particle size in the vis-NIR spectral region, which is inconsistent with the in situ detection of 2–4 nm particles in the near-infrared region by laser-induced incandescence (LII) or sensitive absorption methods like cavity ring-down extinction (CRDE). The objective of this study is twofold: first, an original method for the determination of E(m) of soot particles, including 2–4 nm particles is proposed. Then, the dynamic of two widespread in situ diagnostics, LII and CRDE, are compared over three orders of magnitude of fv in atmospheric premixed ethylene/air flames with different flow rates and C/O. The determination of the absolute value of E(m) and of its variation in the flames is derived from an original analysis, which does not require complex LII modeling. This analysis is based on the comparison between the experimental and calculated LII/LIImax signals in the low fluence regime, LIImax being the plateau value of the fluence curve, which is reached at fluence larger than 1 J/cm2 for the smallest C/O. E(m) is found to vary between 0.15 at low C/O up to 0.36 for the richest flames. Concerning the comparison of the dynamics of LII and CRDE, an excellent agreement is found above a threshold (C/O)limit, while LII exhibits a stronger decrease with C/O below (C/O)limit. This discrepancy is attributed to the spectral dependence of E(m) which is negligible above (C/O)limit, but increases when C/O decreases below (C/O)limit. The particle size distribution function (PSD), measured by scanning mobility particle sizing, reveals monomodal or bimodal PSDs with soot having mobility diameter in the range 2.3–7.5 nm depending on the flame conditions. It is suggested that the particles contained in the first PSD mode, which is dominant in the low C/O range, could be affected by a significant spectral dependence of E(m) in comparison with the second PSD mode.

Response of mineral particles in inland lakes to water optical properties and its influence on chlorophyll-a estimation.

Many chlorophyll-a (Chl-a) remote sensing estimation algorithms have been developed for inland water, and they are proposed always based on some ideal assumptions, which are difficult to meet in complex inland waters. Based on MIE scattering theory, this study calculated the optical properties of mineral particles under different size distribution and refractive index conditions, and the Hydrolight software was employed to simulate remote sensing reflectance in the presence of different mineral particles. The findings indicated that the reflectance is significantly influenced by the slope (j) of particle size distribution function and the imaginary part (n') of the refractive index, with the real part (n) having a comparatively minor impact. Through both a simulated dataset containing 18,000 entries and an in situ measured dataset encompassing 2183 data from hundreds of lakes worldwide, the sensitivities of band ratio (BR), fluorescence baseline height (FLH), and three-band algorithms (TBA) to mineral particles were explored. It can be found that BR showed the best tolerance to mineral particles, followed by TBA. However, when the ISM concentration is less than 30 g m-3, the influence of CDOM cannot be ignored. Additionally, a dataset of over 400 entries is necessary for developing the BR algorithm to mitigate the incidental errors arising from differences in data magnitude. And if the amount of developing datasets is less than 400 but greater than 200, the TBA algorithm is more likely to obtain more stable accuracy.

Discrete pore and particle size distributions during pore blocking and cake filtration

Modeling membrane filtration is important for plenty industrial applications. The understanding of the phenomena and related mechanisms aims at the improvement of techniques in different areas. Therefore, different authors have proposed models to describe the mechanisms related to the filtration process involving either cake build-up or blocking processes, a few of them however have proposed both concepts simultaneously, although none considered pore and particle size distribution during the process, which is the proposal brought up in this paper. Transient solutions for filtrate volume and filter cake thickness as a function of initial pore and particle size distributions are obtained. Good matching between the studied experimental data and the proposed model was observed. In general, the results showed that permeability reduction and filtrate volume are strongly influenced by the initial pore and particle size distributions. In addition, the cake build-up rate depends on both the particle size distribution and the residual flow through blocked pores. It was observed that, after all the largest particles have blocked the smaller pores, effective permeability and flow rate vary only due to cake build-up. Finally, because the proposed empirical parameters do not depend on the pore size distribution, this model can be applied for predicting permeability decline for any feed concentration.

On evaluating the hypothesis of shape similarity between soil particle-size distribution and water retention function

Two pedotransfer functions (PTFs) are available in the literature enabling the soil water retention function (WRF) to be estimated from knowledge of the soil particle-size distribution (PSD), oven-dry soil bulk density (b), and saturated soil water content (s): i) the Arya and Heitman model (PTF-AH) and ii) the Mohammadi and Vanclooster model (PTF-MV). These physico-empirical PTFs rely on the hypothesis of shape similarity between PSD and WRF, and do not require the calibration of the input parameters. In the first stage, twenty-seven PSD models were evaluated using 4,128 soil samples collected in Campania (southern Italy). These models were ranked according to the root mean square residuals (RMSR), corrected Akaike Information Criterion (AICc), and adjusted coefficient of determination (R2adj). In the second stage, three subsets of PSD and WRF data (DS-1, DS-2, and DS-3), comprising 282 soil samples, were used to evaluate the two PTFs using the best three PSD models selected in the first stage. The hypothesis of shape similarity was assumed as acceptable only when the RMSR value was lower than the field standard deviation of the WRFs (*), which is viewed as a tolerance threshold and computed from the physically-based scaling approach proposed by Kosugi and Hopmans (1998). In the first study area (DS-1), characterized by a fairly uniform, loamy textured volcanic soil, the PTF-AH outperformed the PTF-MV and both PTFs provided reasonable performance within the acceptance threshold (i.e., RMSR < *). In the other two heterogeneous field sites (DS-2 and DS-3, characterized by soil textural classes that span from clay and clay-loam to loam and even sandy-loam soils), the PTF-MV (with 3% to 6% RMSR surpassing *) outperformed the PTF-AH (with 8% to 30% RMSR surpassing *) and the majority of RMSR values were larger than those obtained in the original studies. The mean relative error (MRE) revealed that the PTF-MV systematically underestimates the measured WRFs, whereas the PTF-AH provided negative MRE values indicating an overall overestimation. The outcomes of our study provide a critical evaluation when using calibration-free PTFs to predict WRFs over large areas.

Particle Size Distributions and Extinction Coefficients of Aerosol Particles in Land Battlefield Environments

In land battlefield environments, aerosol particles can cause laser beams to undergo attenuation, thus deteriorating the operational performance of military laser devices. The particle size distribution (PSD) and extinction coefficient are key optical properties for assessing the attenuation characteristics of laser beams caused by aerosol particles. In this study, we employed the laser diffraction method to measure the PSDs of graphite smoke screen, copper powder smoke screen, iron powder smoke screen, ground dust, and soil explosion dust. We evaluated the goodness of fit of six common unimodal PSD functions and a bimodal lognormal PSD function employed for fitting these aerosol particles using the root mean square error (RMSE) and adjusted R2, and selected the optimal PSD function to evaluate their extinction coefficients in the laser wavelength range of 0.249~12 μm. The results showed that smoke screens, ground dust, and soil explosion dust exhibited particle size ranges of 0.7~50 µm, 1~400 µm, and 1.7~800 μm, respectively. The lognormal distribution had the best goodness of fit for fitting the PSDs of these aerosol particles in the six unimodal PSD functions, followed by the gamma and Rosin–Rammler distributions. For the bimodal aerosol particles with a lower span, the bimodal lognormal PSD functions exhibited the best goodness of fit. The graphite smoke screen exhibited the highest extinction coefficient, followed by the copper and iron powder smoke screens. In contrast, the ground dust and soil explosion dust exhibited the lowest extinction coefficients, reaching their minimum values at a wavelength of approximately 8.2 μm. This study provides a basis for analyzing and improving the detection and recognition performance of lasers in land battlefield environments.

Fluidized Bed Reactor Modeling and Simulation with Aspen Plus

Fluidized bed reactor is useful for heterogeneous (multiphase) reactions, separations, size enlargement, coating, and blending in chemical and pharmaceuticals industries. To develop a model and simulation of this reactor with advanced systems process engineering software (Aspen Plus). Aspen plus has the capability to simulate real factory performance, design improved plants, and intensify profitability in present plants. This modeling and simulation are applied Geldart B particle classification, GGC is used for particle size distribution function, Newton parameters for mass convergence, and Ergun equation is used for minimum fluidization velocity calculation and pressure drop through the fluidized bed reactor. Generally, most of the predictions model from Aspen plus is a reasonable agreement through the experimental results. Advantage of this investigation is used small quantity of bed particles and identified their melting point temperature is above 340°C. This fluidized bed reactor model and simulation is functional for aluminum hydroxide decomposition reaction. The feed 275 kg/hr of Al(OH)3is converted into 169.6 kg/hr of aluminum oxide (Al2O3) and 89.9 kg/hr of water. A total mass conversion percent is 94.4% of aluminum hydroxide.

A computationally efficient approach for soot modeling with discrete sectional method and FGM chemistry

A novel approach for the prediction of soot formation in combustion simulations within the framework of discrete sectional method (DSM) based univariate soot model and Flamelet Generated Manifold (FGM) chemistry, referred to as FGM-CDSM, is proposed in this study. The FGM-CDSM considers the clustering of soot sections derived from the original soot particle size distribution function (PSDF) to minimize the computational cost. Unlike conventional DSM, in FGM-CDSM, governing equations for soot mass fractions are solved for the clusters, by using a pre-computed lookup table with tabulated soot source terms from the flamelet manifold, while the original soot PSDF is re-constructed in a post-processing stage. The flamelets employed for the manifold are computed with detailed chemistry and the complete sectional soot model. A comparative assessment of FGM-CDSM is conducted in laminar diffusion flames for its accuracy and computational performance against the detailed kinetics-based classical sectional model. Numerical results reveal that the FGM-CDSM can favorably reproduce the global soot quantities and capture their dynamic response predicted by detailed kinetics with a good qualitative agreement. Furthermore, compared to detailed kinetics, FGM-CDSM is shown to substantially reduce the computational cost of the complete reacting flow simulation with soot particle transport. Primarily, the use of FGM reduces the overall calculation by about two orders of magnitude compared to detailed kinetics, which is advanced further with the clustering of sections at a low memory footprint. Therefore, the present work demonstrates the promising capabilities of FGM-CDSM in the context of computationally efficient soot calculations and provides an excellent framework for extending its application to the simulations of turbulent sooting flames.

Anomalous Absorption of Smoke Aerosol in the Visible and Near-Infrared Regions of the Spectrum

Anomalous absorption of finely dispersed smoke aerosol was recorded in large-scale smoke haze during mass fires in the boreal forests of Alaska in July 2019 in the visible and near-infrared regions of the spectrum (440‒1020 nm) according to the data of monitoring of the spectral dependences in the refractive index imaginary part on the network of AERONET stations. The variations in the spectral dependences of the aerosol optical depths (extinction and absorption), as well as the size distribution function of aerosol particles, are analyzed. During the anomalous absorption, the imaginary part of the refractive index increased by a factor of 1.8‒7.2 with an increasing optical wavelength from 440 to 1020 nm, reaching a value of 0.315 for the wavelength of 1020 nm. A power approximation is proposed for the spectral dependence of the imaginary part of the refractive index with indices of power approximately from 0.7 to 2.3.

ANOMALOUS ABSORPTION OF SMOKE AEROSOL IN THE VISIBLE AND NEAR-INFRARED OF SPECTRUM INTERVALS

In large-scale smoke haze during mass fires in the boreal forests of Alaska in July 2019 in the visible and near infrared of spectrum intervals 440–1020 nm, according to monitoring data on the network of AERONET stations of the spectral dependences of the refractive index imaginary part, anomalous absorption of the fine smoke aerosol was found. Variations in the spectral dependences of the aerosol optical extinction and absorption depths, as well as the size distribution function of aerosol particles, are analyzed. With anomalous absorption, the imaginary part of the refractive index increased by a factor of 1.8–7.2 with increasing wavelength from 440 to 1020 nm, reaching a value of 0.315 for the wavelength of 1020 nm. A power-law approximation of the spectral dependence of the refractive index imaginary part with exponents approximately from 0.7 to 2.3 is proposed.

ON TEMPORAL VARIATIONS IN THE GRANULOMETRIC CHARACTERISTICS OF BINARY MIXED MICROPOWDERS TREATED IN A PLANETARY-TYPE BALL MILL

Integral and differential particle size distribution functions have been experimentally determined for two-component mixed micropowders, which contain stoichiometric amounts of polycrystalline aluminum, nickel, and titanium, as well as amorphous boron, and have been mechanically treated in a planetary-type ball mill under different time conditions of the process. The influence of the duration of treatment of the above mixtures on the mathematical parameters of the found distribution functions has been analyzed. It has been shown that, in all considered cases, these functions can be represented in a lognormal form. The most informative statistical characteristics (moments) of these functions have been determined. The dependences of these characteristics on the duration of the mechanical treatment of the mixtures have been revealed. Within the framework of an approximation based on the use of generalized dynamic-stochastic Langevin-type equations, a kinetic model has been proposed for the process of mechanical treatment of metallic and non-metallic mixed micropowders in planetary-type mills. The model makes it possible to describe, as a first approximation, the temporal evolution of the fractional composition of the mixtures during their machining, as well as to record temporal variations of the main statistical characteristics of the corresponding distribution functions in this process.