Particle Size Distribution Model Research Articles

An Improved Comprehensive Atomization Model for Pressure Swirl Atomizers

This study presents an improved comprehensive atomization model for a pressure swirl atomizer. The model integrates internal flow predictions, linear instability analysis of a swirling annular liquid sheet, primary atomization sub-model, and droplet velocity sub-model. Measurement data combined with the inviscid theory model predict the internal flow, providing liquid sheet velocity and thickness at the atomizer outlet. The dispersion relation of surface disturbances is obtained through linear instability analysis. A primary breakup predictive model for particle size distribution is constructed based on the wavelength and growth rate within the full unstable wavenumber range of the dispersion relation. Assuming uniform circumferential distribution and a normal distribution of spray angles, the droplet velocity is assigned according to the liquid sheet velocity. The model is implemented into Eulerian–Lagrangian simulations as initial conditions for discrete phase droplets to simulate the spray field. Results show the model can accurately predict the Sauter mean diameter with an error of less than 6% and effectively predicts the spray structure and spray cone angle. The dependency of the model on its parameters is also studied, determining that the values of the ligament constant and dispersion angle have an obvious impact on the prediction of Sauter mean diameter and spray structure.

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.

3-D visual analysis and quantitative characterization on coal matrix based on Micro-CT

Coal matrix, the storage space for adsorbed gas, is highly related to gas recovery and gas outburst disasters. However, existing research predominantly relies on theoretical assumptions, lacking experimental validation. In this study, we employ X-ray computed tomography (CT) to obtain high-resolution (13.45–15.157 μm) internal structure images of various coal samples. The three-dimensional model of coal matrix elements is reconstructed using entropy threshold segmentation and watershed segmentation. A systematic quantification of the size, shape, and spatial arrangement of these matrix elements is then performed. The results reveal that coal matrix element equivalent diameters span 148.45–2800.35 μm, with volume-weighted average equivalent diameters ranging from 1320.82 to 1720.87 μm. Additionally, we explore particle size distribution (PSD) models to characterize size-quantity relationships and identify the Jaky model based on the logarithmic-exponential equation as the optimal PSD model. Observation of the matrix element models shows that large matrix elements typically exhibit flattened morphologies with rough surfaces, while smaller ones tend to be blocky with smooth surfaces. The dimensionless shape factor results suggest that, when scale effects are not considered, the shape of larger matrix elements is more conducive to matrix-fracture gas mass transfer. Based on this, a functional relationship between the matrix shape factor and size, considering changes in matrix morphology, is established. Moreover, we algorithmically determine that the average number of adjacent matrix elements per individual matrix element falls between 13 and 15. For matrix clusters with large volume fractions, the size ratio of the central matrix element to adjacent matrix elements primarily falls within the range of 0.4–0.6, approximately following a Weibull distribution. According to this feature, a method for building a matrix cluster model used for simulations is proposed.

Influence of particle size gradation on the preparation of highly concentrated coal-water slurry and the development of a gradation model

ABSTRACT Aiming at the problem that low-rank coal is unsuitable for making CWS (coal-water slurry) and lacks an accurate particle size gradation prediction model. This paper expounds on the effects of ball milling time and speed on the concentration and rheology of CWS under different grinding conditions. After size grading, the slurry’s bimodal parameters and particle size distribution are studied. Based on the BP neural network, a particle size distribution model more suitable for low-rank coal is established in this paper. The results show that the slurry concentration in the graded coal sample increases by 4% higher than that of the ungraded coal sample. The BP neural network model accurately predicts the relationship between bimodal parameters and slurry concentration based on bimodal parameters. In the context of grain size gradation, the optimized particle size cumulative distribution model, denoted as y = x n − x K n x 100 n − x K n + K { ( K + 10 % > x > K% } , demonstrates enhanced alignment with the attributes of low-rank coal. This underscores the paper’s contribution in providing a more fitting particle size distribution model for low-rank coal in the context of CWS production.

Modeling of Particle Size Distribution in the Presence of Flocculant

This study presents a mathematical description of the solid fraction aggregation process in the presence of a flocculant and its result. The basis is a population balance equation. The model is realized in Python language. Verification was carried out using red mud from the investigated enterprise; Flomin AL P 99 VHM was used as a flocculant. The mean square deviation for the parameter “mean aggregate diameter” is equal to 19.88 μm. The time required for the model calculation is about 3 min. The time spent on modeling depends on the number of calculation channels. In this study, 40 channels (20 with PSD source data, and 20 with empty values required for the calculation) were used for the calculation. The time spent on the model calculation is much shorter than the inertia via each of the communication channels for the studied symmetric radial type thickener. A user interface is developed, where the input parameters are the initial pulp particle size distribution, viscosity and density of pulp in the thickener, particle surface area, concentration and flow rate of flocculant, concentration of solid particles, inner diameter and height of the feed well, and simulation time. The result of the simulation is particle size distribution in the feed well of the washer and the mean flocculus diameter.

Comparative Study of the Rosin-Rammler and Gates-Gaudin-Schuhmann Particle Size Distribution Models for Microwave-Assisted Crushing Analysis

Microwaves selectively heat dielectric minerals within the ore particles during a relatively brief exposure period, inducing thermal stresses that generate cracks, thereby diminishing the structural integrity of the ore particles. The immediate effect of employing this technique is an alteration in the particle size distribution (PSD) of the comminution product. In this research, two types of diamond-bearing kimberlite ore underwent microwave irradiation before the crushing process. The microwave experimentation was carried out utilizing a multi-variable power system with a maximum capacity of 15 kW, within a single-mode cavity structure. The exposure time was adjustable to vary the input microwave energy, which ranged from 6 kJ to 270 kJ. Three distinct narrow feed size classes were examined: -31.5 +16 mm, -16 +6.7 mm, and -31.5 +6.7 mm. Each sample weighed five hundred grams. Both the microwave-treated samples and the untreated samples (as-received) were subjected to crushing using a single-roll crusher. The PSD of the crushed product was determined through a standard sieving test and subsequently analyzed using empirical PSD models. These models are advantageous for enhancing the accuracy of PSD analysis by linearizing datasets that often exhibit skewness towards the distribution's tail end. Two widely recognized empirical PSD models, the Rosin-Rammler (RR) and Gates-Gaudin-Schuhmann (GGS) models, were employed to assess how microwave pre-treatment influences the PSD of the crushed product. Various comparative parameters derived from these models, such as the P50, distribution uniformity (DP), and critical fine sizes, were evaluated for both untreated crushed samples and the samples treated with microwaves. The findings indicated that crushed products treated with microwaves generally displayed improved alignment with both PSD models in comparison to the untreated samples. Notably, the GGS model exhibited marginally higher overall r-squared values compared to the RR model. Moreover, the microwave-treated samples showcased a slightly coarser and more uniform PSD when contrasted with the untreated counterparts. Among the various microwave energy inputs tested, the lowest input at the highest power level yielded the most substantial increments in P50 and DP. Simultaneously, it led to the most significant reduction in fractions corresponding to critical fine sizes.

Inovadora brita graduada tratada com cimento de alto desempenho como concreto compactado com rolo

Abstract The main causes of poor performance of cement-treated crushed stones (CTCS) pavement layers are material-related heterogeneity, porosity, and brittle cementitious matrix, leading to fast fatigue degradation. Seeking to improve CTCS mechanical properties and structural response it was studied different grading curves such as DER-SP, EN 14227-1 CBGM 2 and RCC-ACI were studied. The aim of the laboratory study was to rank and select among the classical particle size distribution models like Talbot & Richart, and analytical models as compressible packing model – CPM, the best grading curve which was capable of producing mixtures with the best aggregate interlocking and improved mechanical resistances. The effect of cement content (4% and 5%) as well as the potential of using silica fume in suspension (percentages of 5, 7 and 10% of cement mass) to improve the interfacial transition zone were evaluated. It was found that CPM is a powerful tool for aggregates ranking aggregates and designing well-graded mixtures like CTCS; the CBGM 2 EN 14227-1 mean curve achieved the best packing density. The effect of increasing the cement content from 4 to 5% in terms of mechanical strength has the same result as the addition of 7% of silica fume in mixtures with 4% cement. Eventually, mixtures with 5% cement and 10% silica fume showed modulus of elasticity similar to traditional CTCS; nonetheless, the HP-CTCS (high performance CTCS) split tensile strength was significantly higher (about 28%), leading to a lower Et,sp/fct,sp relationship, which is a highly positive outcome to improve mechanical durability of this innovative dry compacted concrete.

Design method of high transport performance biochar-cement based materials based on particle size distribution and permeation probability calculation model

Cement-based materials are the most widely used and largest quantity of engineering materials. Enhancing carbon fixation efficiency and reducing carbon emissions in cement-based materials is an active method of coping with climate change, which has important environmental and social benefits. The amount of carbon dioxide transported into the cement-based material represents the maximum carbon fixation capacity. Enhancing the carbon transport performance of cement-based materials is the material guarantee and basic premise for improving their carbon fixation levels. Considering the impact of particle connections on the maximum cluster, a permeation probability calculation model of porous media system was established, providing the changing rule of the permeation probability with the mathematical expectation of the Poisson distribution, which was verified to be reasonable through experimental studies. The experimental results showed that under the same cement-based conditions, the water transport properties of biochar cement-based materials increased as the mathematical expectation increased. The carbon dioxide transport properties were influenced by the surface area of the biochar particles and exhibited a trend of first decreasing and then increasing with an increase in the mathematical expectation. When the water-cement ratio decreased, the differences in the microstructure of cement-based materials with different biochar particle size distributions increased, the change rate of permeation probability therefore was more sensitive, resulting that the differential effects of different particle size distributions on the transport properties were amplified, and the influence of the particle size distribution of biochar on the transport properties of cement-based materials was more significant.

Models for converting CLD to PSD for bimodal distributions of particles

Focused Beam Reflectance Measurement (FBRM) is a commonly used PAT tool that measures the Chord Length Distribution (CLD) of the particles in a suspension, providing a fingerprint of the size distribution of the suspended particles. Despite the extensive usage, there still exists a need for models for converting CLD to particle size distribution (PSD) particularly for bimodal distribution of particles with aspect ratio higher than one. In this work, we further developed and evaluated different models for converting the CLD to the PSD of particles with an aspect ratio greater than one and have discussed the applicability of the models to monomodal as well as bimodal distributions. The models have been verified at different aspect ratios using synthetically generated experimental data. It was observed that the geometric model proposed in this work performs better than the other models compared in this work. The presented models and results will be useful for selecting appropriate CLD to PSD models for a wide range of applications.

Relationship between the Intrinsic Properties of Sands and the Parameters of Mathematical Particle Size Distribution Models for Predicting Geotechnical Quantities

This work characterizes the relationships between the intrinsic properties of sands and the parameters of four mathematical models that best simulate the experimental curves and geotechnical properties of sands used in construction. Origin.Pro.2019" software was used to smooth the grading curves, define the parameters of the mathematical models and link them to the geotechnical data. To achieve this objective, the correlations between the intrinsic properties of the sands are developed using mathematical models with the highest coefficient of determination (R2) and the lowest statistical coefficient (χ²). The correlations used are those with a coefficient of determination greater than or equal to 0.9. The results obtained show that the models used provide a good description of the experimental curves. The model parameters are correlated with the granulometric fractions and the geotechnical parameters. The evolution of the points expressing the parameters of the Gaussian and exponential models (A1, Xc, A, W, Yo) and the parameter (t1) as a function of seven randomly chosen geotechnical quantities, are polylinear and linear fits, respectively. This study is important for predicting a geotechnical quantity from a modelled grading curve, by solving the mathematical expressions of the models used.

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.

Use of Particle Size Distribution Model to Produce High-Flowability Cementitious Composites

High packing density is desired for production of high-flowability cementitious composites. To achieve dense packing, various particle size distribution models have been proposed by researchers. In present study, with the use of two supplementary cementitious materials at different scale smaller than ordinary Portland cement (OPC), an ideal particle size distribution curve according to modified Andreasen’s formula has been produced. Analysis of particle size distribution showed that blending the cement with superfine cement (SFC) and/or condensed silica fume (CSF) would make the curve closer to the ideal curve, hinting that a better packing could be achieved. Packing densities of the cementitious materials were then directly obtained through a wet packing method, and results verified that cementitious materials mix blended with SFC and/or CSF own higher packing densities.

CHOICE OF STARTING VALUES OF PARTICLE-SIZE DISTRIBUTION PARAMETERS FOR THEIR CALCULATION FROM SMALL-ANGLE X-RAY SCATTERING DATA

Simple methods are proposed for determining the starting values of the parameters of particle-size distribution models (mean radius and its standard deviation), calculated from small-angle X-ray scattering curves. Estimates of these parameters from above based on the obtained analytical expression for the Guinier region of the scattering curve from a polydisperse system obeying the Schultz distribution are proposed for systems with narrow distributions. It is proposed to estimate the parameters and range of sizes from below based on the obtained expression of the Porod asymptotics for a polydisperse system. A method for calculating the generalized Guinier–Porod approximation in Kratky coordinates, from which independent estimates of the average size and variance can also be obtained, is proposed. The efficiency of the developed approach is demonstrated by an example of analyzing the scattering intensity from aqueous solutions of silicasol nanoparticles.

Blasting Fragmentation Study Using 3D Image Analysis of a Hard Rock Mine

Rock blasting with explosives is the first stage of rock fragmentation and plays a key role in the results of the mining chain. Fragmentation optimization is achieved by considering the energy efficiency of the explosive, the optimal distribution of explosive energy in the rocky mass, and the programmed and controlled release of explosive energy during blasting. The quality of fragmentation is usually performed using tools that estimate the sizes of fragments via the analysis of two- or three-dimensional images. This article presents a study of the rock fragmentation optimization of very compact itabirites, based on four blasting tests, performed in a different way from the conventional blasting design. The fragmentation analysis was performed using the PortaMetricsTM tool, and compared with the Kuznetsov and Rosin–Rammler particle size distribution model, to compare the reliability of this tool. Finally, the mine productivity after blasting was estimated from the particle size distribution obtained in the tests. The model presented idealistic results considering the technical parameters used in the equations. However, the PortaMetricsTM tool suggests good performance for the preliminary evaluation of blast design.

Research on Thermophoresis Deposition of Aerosol Particles

An aerosol is a system of liquid or solid particles in suspension in a gas, which includes both the particles and the suspended gas. The aim of this paper is to study the thermophoretic forces in aerosol systems. We introduce the concepts of particle size distribution, DLVO theory and multiphase flow models, followed by four case studies of thermophoretic deposition. This study has some limitations that need further investigation for a number of specific reasons. Firstly, the impact of thermophoresis is often overlooked in discussions about aerosols and their walls. Additionally, the focus has only been on solid walls, while the same problem applies to soft walls such as the human respiratory tract. The complexity of fluid models that involve multiple particles and forces also presents challenges. The task of examining the effects of temperature fields and temperature variations will be even more daunting. There is a pressing need for further research in this area.

Modeling of Particle Size Distributions in Industrial Poly(vinyl chloride) Suspension Polymerization Reactors

In the present paper, a mathematical model is built and implemented to describe the trajectories of mass inventories, pressures and polymer properties with emphasis on final particle size distributions of industrial scale poly(vinyl chloride) suspension polymerization reactors. The model comprises the mass balances, statistical moment balances, equilibrium relationships and population balance equations. A discretization scheme is employed to transform the integro-differential equations resulting from the population balance model into a system of differential equations. The obtained results show, for the first time, that classical breakage and coalescence kernels described in the literature can provide very good fittings of actual industrial scale data when coupled with proper parameter estimation procedures, so that the proposed model is able to represent the available operation data with good accuracy at distinct conditions. Particularly, it is also shown that the use of a top condenser for control of the reactor temperature can lead to changes of parameters that control the particle size distributions.

Study on Evaluation of Dust in Highway Construction on Qinghai-Tibet Plateau

With the rapid development of our country's economy and acceleration of urbanization construction, the original road construction to relieve the traffic pressure on the basis of should be improved. Tibetan plateau is located in southwest borderland in our country, the area of about 2.4 million square kilometers, occupying 1/4 of our country's land area , improving the traffic condition and road construction of this region have vital significance. Based on the mathematical model of particle size distribution and dust volume under different air humidity and wind speed during road construction, this paper evaluated the influence of dust volume brought by strong wind environment during road construction on the Qinghai-Tibet Plateau on the environment along the road, aiming at controlling the possible adverse impact of the project construction on the ecological environment to the lowest extent.

Development of soil Particle size distribution model and determination of all related coefficients

Development of soil Particle size distribution model and determination of all related coefficients

Constitutive relation and particle size distribution model of rock fragments in the goaf.

Permeability is considered a key parameter used in goaf flow simulation. Particle size of caved rock fragments is an important factor that affects permeability. Current literature treats the particle sizes of rock fragments as a constant, which is something that will inevitably lead to great errors. The sigmoid function is introduced, based on the existing main roof subsidence displacement model, to establish a new main roof subsidence displacement model, which will reflect the characteristics of the natural accumulation zone, load-affected zone, and compacted zone in the goaf. The constitutive relationship of the rock fragments meanwhile is established from a mechanical perspective. Particle sizes of the rock fragments in the goaf are controlled by the maximum median particle size, with the minimum particle size calculated as 69.25% of the maximum median particle size. In addition, with the same mining height, the stronger the overlying strata is, the larger the particle size, while under the same geological conditions, the larger the mining height, the smaller the particle size of the rock fragments. This paper provides a constitutive relation and particle size distribution model of rock fragments which is more consistent with the actual coal mine's characteristics.

Inversion Model of Nonspherical Aerosol Particle Size Distribution Based on Tikhonov’s Regularization With the PSO-GA Algorithm

Inversion Model of Nonspherical Aerosol Particle Size Distribution Based on Tikhonov’s Regularization With the PSO-GA Algorithm