Particle Size Distribution Model Research Articles

Particle size distribution modeling and kinetic study for coagulation treatment of tannery industry wastewater at response surface optimized condition

The presence of considerable amounts of Total Suspended Solids (TSS) in wastewater streams can pose a severe threat to aquatic and human life. A high level of TSS will increase water temperature at an alarming rate and decrease dissolved oxygen levels. This study aims to remove TSS from raw tannery wastewater using an active coagulating protein extracted from Moringa oleifera (MO) seeds. The process variables such as, coagulant dosage and pH are varied to test the potential of the coagulant extracted using NaCl and KCl salt solutions, for TSS removal from tannery wastewater. The results show that the NaCl extracted coagulant could effectively remove 89.87% of TSS whereas the KCl extracted coagulant eliminated 76.23% of TSS, at the optimum condition of 40 mL coagulant dosage and 7 pH. The assessment of coagulation rate parameters using differential method of analysis reveals that at optimum condition, the TSS reduction process obeys second order kinetics with a rate constant 3.556 × 10– 5 (mg/L)– 1 min– 1. The volume distribution of suspended solids in raw wastewater is fitted with a log-normal distribution model and Gaussian distribution model. From the magnitude of the coefficient of determination (R2) and Akaike information criterion (AIC) of the Gaussian model (R2 = 0.6864 and AIC = 3.89), it is concluded that the Gaussian model is better than the log-normal distribution model (R2 = 0.4708 and AIC = 6.31) for explaining the volumetric distribution of suspended particles. The numeric distribution of suspended solids in raw wastewater is explained well by the variable β model (R2 = 0.668) rather than the power law model (R2 = 0.196). The 3D response surface plot indicates that maximum TSS removal can be achieved at an optimal condition of 40 mL coagulant dose and 7 pH. The quadratic model developed for the treatment process utilizing the NaCl extracted coagulant shows better response prediction ability (R2 = 0.8118) when compared to the model built for the KCl extracted coagulant (R2 = 0.7911). Both the model equations have a p-value less than 0.05 which indicates that the proposed models are statistically significant. The elliptical contours for TSS removal using NaCl extracted coagulant demonstrate the significant interactions between the process variables. This study exhibited the applicability of MO seed extract for real-time tannery wastewater treatment.

Influence of Shape and Particle Size Distribution on Discrete Element Simulation Flow Characteristics of Granular Compound Fertilizers

HighlightsThe relationship between the angle of repose and compound fertilizer particle size distribution and sphericity was revealed.The range of the rolling friction coefficient obtained by different modeling was calibrated by the golden section method.Scanning modeling required 80% of the rolling friction coefficient of spherical modeling to reflect actual motion.The relative error of scanning modeling (7.91%) was better than that of spherical modeling (30.84%).Abstract. The discrete element method (DEM) is widely used to simulate the behavior of granular materials. The accurate establishment of the particle model has a great influence on the accuracy of the simulation. Most of the existing discrete element simulations of granular compound fertilizers use spherical modeling without considering the shape or particle size distribution. To compare the simulation accuracy of spherical modeling (spherical particles with equivalent diameters) and scanning modeling (a three-dimensional model of fertilizers was obtained by a scanner and then automatically filled by EDEM2020 software) on compound fertilizer discharge, three kinds of granular compound fertilizers (Changqingshu, Xinshidai, and Munvhong) were selected as the research objects. The characterization parameters of fertilizer shape and fluidity were measured. Principal component analysis and Pearson correlation analysis were used to analyze the influence of the compound fertilizer characterization parameters on the angle of repose. The results showed that the particle shape and particle size distribution of compound fertilizer were the main factors affecting its angle of repose. Through the golden section method and a single-factor experiment, the rolling friction coefficients of the three granular fertilizers under the two granular modeling methods for spherical modeling and scanning modeling were determined. The results showed that the rolling friction coefficient when using scanning modeling to approximate the actual angle of repose was approximately 80% of that when using the spherical modeling method. The influence of different modeling methods on the total rotational kinetic energy and the formation of the angle of repose was discussed. For the same fertilizer, the total rotational kinetic energy was greater in spherical modeling than in scanning modeling, and the formation time of the angle of repose was longer in scanning modeling than in spherical modeling. The results of an external tank wheel fertilization simulation and bench test showed that considering the scanning modeling of fertilizer shape and particle size distribution, the fertilization variation coefficient obtained from the simulation test was closer to that of the bench test. This research helps to better understand the influence of particle shape and particle size distribution on DEM simulations and provides references for discrete element modeling of other granular fertilizers. Keywords: Compound fertilizer, Modeling method, Shape, Simulated flow characteristics, Size distribution.

Models for Counts and Particle Size Distributions of Subvisible Particle Data.

Traditional statistical analyses of subvisible particle data are usually based on either descriptive statistics, normal-based methods, or standard Poisson models. These methods often do not adequately describe the counts or particle size distribution. They usually ignore relevant information represented in the data, such as count correlation. Therefore, any meaningful analyses of subvisible particle data require a reasonable representation of counts and particle size distribution and the correlation in the data. Such comprehensive approaches are not widely available or used when analyzing subvisible particle data. In this article, we propose the use of generalized linear mixed models to analyze the counts and the particle size distribution of subvisible particle data. These models make optimal use of the information in the data and allow flexible approaches for the analyses of a wide range of data structures. They are readily accessible to practitioners through the use of modern statistical software. These models are demonstrated with two numerical examples using two different data structures.

Physically-Based Particle Size Distribution Models of Urban Water Particulate Matter

A particle size distribution (PSD) of particulate matter (PM) is a primary metric to examine PM transport and fate, as well as PM-bound chemicals and pathogens in urban waters. To facilitate physical interpretation and data sharing, a series of concise analytical models are examined to reproduce unit operation (UO) influent and effluent PSD data and indices. The models are a (1) single-parameter exponential and two-parameter (2) gamma, (3) lognormal, and (4) Rosin-Rammler distributions. Two-parameter models provide physical interpretations for the central tendency of PM diameters, and shape as an index of PSD hetero-dispersivity. Goodness-of-fit is used to test models and PSDs. For influent data from two disparate areas, a paved source area and a larger watershed delivering unique PSDs, lognormal and gamma models provide consistent representation of influent and effluent complexity. In these areas, contrasting UOs (a clarification basin and a volumetric filter), subject to type I settling, scour, and filter PM elution, are differentiated based on flow, surface area, volume, and residence time. Surface overflow rate (SOR) as a common heuristic design tool for only type I settling is used to further test PSD models by simulating effluent PSDs for a scaled basin design. Lognormal and gamma models of SOR-generated effluent PSDs were not statistically different. In conclusion, two-parameter PSD models have physical interpretations and lower errors compared to an exponential model. Gamma and lognormal distributions are physically-based models that reproduce actual complex influent or effluent or through SOR as a tool for PSD transformation. Results indicate that PSD models and parameters can be applied to evaluate behavior of common UOs.

Drop-weight impact fragmentation of gas-containing coal particles

Research on coal fragmentation can play an important role in understanding coal and gas outbursts. The study discussed in this paper explored the fragmentation of gas-containing coal particles using the drop-weight impact method. The effects of equilibrium gas pressures and type of adsorbate gas on particle size distributions and fragmentation energy were investigated in detail. We found that the Fractal particle size distribution model can most effectively describe the crushed coal particle sizes. The equilibrium pressure and type of gas can influence the Fractal distribution parameter. The crushing energy is composed of energy to create new surfaces and other forms of energy that are dissipated but the equilibrium gas pressure and type of adsorption gas can affect energy consumption and crushing efficiency. This research will be of guiding significance to the intensity evaluation and mechanism understanding of coal and gas outbursts.

Non-negative differential evolution for particle sizing from ultrasonic attenuation spectroscopy

Ultrasonic attenuation spectroscopy (UAS) is a potentially characterising technique for nano-meter size materials. It is the inversion of the ultrasonic attenuation spectrum to particle size distribution (PSD) by using an optimisation technique. Because the particle size is non-negative number, the inversion has to be optimised in the non-negative real number domain. However, many optimisation techniques have traditionally added the non-negative constrain into a cost function as a result the function becoming complex. In this article, the modification of the differential evolution (DE) technique has been proposed to narrow the searching solution domain bounded in the non-negative real number domain. An absolute function has been added to a mutant vector of the DE technique to guarantee the solution is non-negative real number. This technique is called DE-Nonneg technique. The technique has been demonstrated by characterising size of silica suspension. The optimisation results show that the PSD predictions are comparatively to simulated annealing (SA) technique. Moreover, the performance of DE-Nonneg could be improved via adjusting the parameters (expanding factor, cross over, number of parameters and mutation type) following the guideline addressed here. In addition, it is also possible to predict the PSD without using the structural PSD model. With this advantage, DE-Nonneg could possibly be applied for other non-negative optimisation problems.

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.

New Vipulanandan p-q Model for Particle Size Distribution and Groutability Limits for Sandy Soils

Abstract The particle size distribution (PSD) of the soil fines content and mean particle size (d50) are used in a number of soil property relationships and in the soil classification. To analyze and model PSD using a modified version of the Vipulanandan p-q model, some data collected from several research studies and the results of prediction were compared with the Fredlund and logistic growth models used in the literature. The Vipulanandan p-q model was modified and used to represent the PSD of soils. The Vipulanandan p-q model parameters were correlated very well to various soil properties, such as the diameter in the PSD curve corresponding to 10, 30, 60, and 90 % of finer (d10, d30, d60, and d90, respectively), mean particle size the diameter in the particle size distribution curve corresponding to 50 % finer (d50), and fines content (F%). There was no correlation between the Fredlund model and logistic growth model parameters with the soil properties. Based on the coefficient of determination and root mean square error, the two parameters of the Vipulanandan p-q model predicted the PSDs very well. A current study also had quantified the lower groutability limit based on the d50 and the Vipulanandan p-q model parameters. The relationship between fines content and d50 were also generalized using the Vipulanandan p-q model to quantify the upper and lower groutability limits for sandy soils.

Developing conceptual and empirical models for well- and gap-graded soil particle size distribution (PSD) curve

ABSTRACT Most of the previously introduced soil particle size distribution (PSD) models have at least one limitation. To overcome the limitations of the PSD models, two conceptual soil PSD models (unimodal- and bimodal-exponential), and one empirical model were suggested to represent PSD curve and evaluated based on the root mean square error (RMSE), adjusted coefficient of determination (R 2 adjusted), coefficient of determination (R 2), and corrected Akaike’s information criterion (AICc). The RMSE, R 2 adjusted and AICc were 0.068, 0.969, and −29.6; 0.073, 0.955 and −3.5; and 0.102, 0.905, and −13.9 for unimodal-exponential, bimodal-exponential and empirical models, respectively. Therefore, the unimodal and bimodal-exponential models were flexible over the entire range of soil PSD. The results were compared with those of 35 PSD models investigated in the previous studies. The results showed the superiority of the conceptual models and only eight percent of the models had an accuracy similar to the conceptual models. Therefore, fewer parameters (three, five and two parameters for unimodal-exponential, bimodal-exponential and empirical models, respectively), easy fitting procedure and high accuracy in terms of R2 adjusted, RMSE and AICc criteria are considered as the most important advantages of the proposed models to describe the soil PSD from 0 to 0.002 m.

Nucleation and crystal growth kinetic parameter optimization of a continuous Poiseuille flow struvite crystallizer using a discretized population balance and dynamic fluid model

Struvite crystallization is an important process in recovering phosphorus from waste streams, but kinetic parameters involve significant uncertainty. In this research a model is developed which includes population balance, fluid flow equations, and kinetic relations. Power law rate coefficients for struvite nucleation (knuc=7.509×107L-1.min-1) and growth (kg=16.72μm.min-1) are optimised by combining a Poiseuille flow reactor model (PFR) and experimental data for phosphorus concentration and particle size distribution (PSD) at the reactor outlet. Nucleation and crystal growth rates were optimized using both filtered and sonicated PSD samples. The model incorporated three key aspects: non-ideal thermodynamics; Poiseuille fluid flow (including advection, diffusion and reaction); and a discretized population balance model (PBM). Sensitivity analysis using deltas (min/max) showed diffusion effects to be negligible and indicated that power law rate constants for both nucleation and growth were more influential on PSD and phosphorus concentration than the associated rate orders. Optimized parameters agreed reasonably well with a previous study conducted by our group under different conditions, using a different model, providing confidence in results. The optimized nucleation rate, knuc, was only 12% lower than that found using a similar model for a seeded system, where secondary nucleation is dominant. Additionally, a primary nucleation model provided a poor fit to experimental data. These two factors indicate that secondary nucleation is an influential mechanism. A 2nd order growth model provided a better fit to data than a previously suggested 5th order growth model. The optimized growth rate coefficient, kg, was higher than in any previous studies, an effect attributed to sonication, which eliminated aggregates.

Modelling of particle size characteristics and specific energy demand for mechanical size reduction of wheat straw by knife mill

Mechanical size reduction, due to its significant effect on transport phenomena and biodegradability, is the crucial initial step in the production of advanced biofuels. As is generally known, particle size reduction is high energy demand operation. Energy requirements were experimentally studied and mathematically modelled for the processing of wheat straw with 4.6% moisture content wet basis that was reduced in size using a knife mill. The characteristic particle sizes for wheat straw before and after milling were analysed using the Rosin-Rammler-Sperling-Bennet model of particle size distribution. Energy demand 0.7–21.2 kWh t−1 was reported for size reduction of wheat straw with size reduction ratio D50 equal to 1.13–3.86. Applying the theory of energy demand modelled by Rittinger, it was confirmed that treated wheat straw displayed brittle behaviour. Energy demand for size reduction of wheat straw can be therefore be estimated using Rittinger theory with Rittinger constant value of 9.9 kWh mm t−1.

Particle size analysis of municipal solid waste for treatment process modeling.

Several unit operations used in municipal solid waste (MSW) processing facilities are based on physical properties of the waste materials, such as particle size, density and shape. Reliable expressions describing particle size distribution (PSD) of the different waste components present in MSW are not readily available in the context of process modeling. In this study, the characterization data for household wastes and construction and demolition (C&D) wastes were analysed with the purpose of selecting the most representative PSD expression for these waste streams. The Rosin-Rammler distribution was identified over the log-normal and the gamma distributions as the best-fitting PSD for the waste samples. This was demonstrated for both raw and processed waste samples. Parameters were derived and validated for every category of MSW materials considered in the characterization. A model for mixed household waste PSD was developed based on the summation of Rosin-Rammler expressions corresponding to each category of waste materials, as the composition was determined to be the main factor influencing particle size. A simplified model was also derived for mixed waste as a bimodal distribution since two main modes were observed in household waste - one for the "organic" fraction and one for the "inorganic" fraction.

Predicting spatial variability of selected soil properties using digital soil mapping in a rainfed vineyard of central Chile

Soil physical properties influence vineyard behavior, therefore the knowledge of their spatial variability is essential for making vineyard management decisions. This study aimed to model and map selected soil properties by means of knowledge-based digital soil mapping approach. We used a Random Forest (RF) algorithm to link environmental covariates derived from a LiDAR flight and satellite spectral information, describing soil forming factors and ten selected soil properties (particle size distribution, bulk density, dispersion ratio, Ksat, field capacity, permanent wilting point, fast drainage pores and slow drainage pores) at three depth intervals, namely 0–20, 20–40, and 40–60 cm at a systematic grid (60 × 60 m2). The descriptive statistics showed low to very high variability within the field. RF model of particle size distribution, and bulk density performed well, although the models could not reliably predict saturated hydraulic conductivity. There was a better prediction performance (based on 34% model validation) in the upper depth intervals than the lower depth intervals (e.g., R2 of 0.66; nRMSE of 27.5% for clay content at 0–20 cm and R2 of 0.51; nRMSE of 16% at 40–60 cm). There was a better prediction performance in the lower depth intervals than the upper depth intervals (e.g., R2 of 0.49; nRMSE of 23% for dispersion ratio at 0–20 cm and R2 of 0.81; nRMSE of 30% at 40–60 cm). RF model overestimated areas with low values and underestimated areas with high values. Further analysis suggested that Topographic position Index, Topographic Wetness Index, aspect, slope length factor, modified catchment area, catchment slope, and longitudinal curvature were the dominant environmental covariates influencing prediction of soil properties.

A forward approach for the validation of soot sizing models using laser-induced incandescence (LII)

While validating the numerical modeling of the primary particle size distribution (PPSD) in sooting flames, a common practice is to compare the numerical results to the corresponding experimental data obtained with the Time-Resolved Laser-Induced Incandescence (TiRe-LII) technique. Since the PPSD is not directly measured by TiRe-LII, but derived with a post-processing procedure, various uncertainties and errors can potentially affect the consistency of such comparison requiring the estimation of many input parameters. On the contrary, nowadays, detailed numerical simulations provide access to a more complete set of data, which can be used to reconstruct the incandescence signal. In this work, a forward approach for the generic validation of numerical models for the PPSD is performed. It is based on the numerical reconstruction of the temporal evolution of the incandescence from the numerical results and its comparison with the measured signal. First, two indexes are proposed to quantify the agreement between the numerically synthesized and the measured signals. Then, the effectiveness of the proposed approach is demonstrated a priori by quantifying the potential errors that can be avoided with this new strategy compared to the classical approach. Finally, the feasibility of the proposed procedure is proven by comparing synthesized signals to the experimental ones available in the literature for a laminar premixed flame. It is shown that the proposed approach can be used for strengthening the analysis on numerical model performances in addition to the classical approach.

Particle breakage of uniformly graded carbonate sands in dry/wet condition subjected to compression/shear tests

The behaviour of a granular material is primarily affected by its particle size distribution (PSD), which is not necessarily a soil constant as assumed in traditional soil mechanics. The PSD may change over time due to mechanical as well as environmental actions. In this study, a series of ring shear tests and one-dimensional compression tests were completed on carbonate sand, in both dry and saturated conditions. Samples were prepared with different initial uniform gradings, to investigate: (1) the influence of the saturation state and initial grading on mechanical and deformational behaviour of carbonate sands and (2) the evolution of the PSD as a result of breakage. The ring shear tests show that the residual friction angle remains almost constant, but dilatancy reduces with increasing saturation degree. In the one-dimensional compression test, the yield stress decreases with increasing saturation degree, but the compressibility (as defined by Cc) remains almost constant, irrespective of the saturation state. Moreover, saturated samples suffer more breakage than dry samples during ring shear tests, while there is no obvious effect of saturation state on particle breakage in one-dimensional compression. A recently proposed PSD model with only two parameters (λp and κp) is employed to model the evolution of PSD, as it can more broadly capture the whole PSD throughout the breakage process than existing breakage indices. Test results demonstrate that parameter λp is linearly related to Einav’s breakage index $$ B_{\text{r}}^{*} $$ and is dependent on initial grading, but independent of test mode. Parameter κp is in power relationship with $$ B_{\text{r}}^{*} $$ and is independent of initial grading or test mode. The evolution of parameters λp and κp is related to the input work for both ring shear and compression tests, with λp being hyperbolically related to input work and κp in power relationship with input work. Using such an evolution law provides an alternative approach to capture the effects of particle breakage in constitutive models.

Improved prediction of water retention curves for fine texture soils using an intergranular mixing particle size distribution model

Laboratory measurements to derive the soil water retention curve, θψ, are time consuming and expensive. We present a cost-effective alternative using particle size distribution (PSD) and saturated water content. We propose a novel physical conceptual intergranular mixing PSD model (IMP model) which derives θψ from PSD, exploiting the relation between particle size and pore size distributions and the intergranular arrangement of the soil particles. The IMP model successfully predicts θψ for fine texture soil, which is the most challenging soil texture to be modelled. With our novel model, reliable θψ can be obtained using only three general fitting parameters without needing to assume any particular type of soil particle packing, with mean Nash–Sutcliffe efficiency coefficient of 0.92 for 259 soils.The IMP model can accurately predict θψ for fine texture soils because: a) it implements an intergranular mixing function that accounts for soil pores not all being perfectly spherical and takes into consideration the intergranular rearrangement (mixing) of the particles, which allows neighbouring particles to have different sizes resulting in variations in pore radius and pore shape of the corresponding pore fraction; b) it overcomes the absence of PSD data for sizes smaller than the clay fraction by developing a normalised form of the Young–Laplace capillary equation; and c) the residual pore volume accounting for water strongly bound to solid particles or in very small pores is incorporated as a function of the clay fraction.

Modelling of particle size distribution in butyl acrylate emulsion polymerisation in a batch reactor

ABSTRACT A computer model for the development of particle size distribution in a batch reactor for the emulsion polymerisation of butyl acrylate was developed, solved and validated. Population balance model was represented by partial differential equations and these equations were solved by using the method of orthogonal collocation. The model was validated against two sets of experimental data taken from literature. The model’s reliability was demonstrated by successful simulation of the experimental data for the number of variables which included the number of particles formed at different emulsifier and initiator concentrations, the variation of conversion with time at one initiator concentration and one emulsifier concentration and number average diameter, volume average diameter at different emulsifier concentrations. The model is simpler and more accurate than the reported population balance models for this system in the literature. The simplicity is brought by studying the model in different levels of details in our earlier works.

Fitting soil particle-size distribution (PSD) models by PSD curve fitting software

This paper describes a particle-size distribution (PSD) curve fitting software for analyzing the soil PSD and soil physical properties. A better characterization of soil texture can be obtained by describing the soil PSD using mathematical models. The mathematical equations of soil PSD are mainly used as a basis to estimate the soil hydraulic properties. Until now, many attempts are made to represent PSD curves using mathematical models, but selecting the best PSD model requires fitting all models to the PSD data, which would be difficult and time-consuming. So far, no specific program has been developed to fit the PSD models to the experimental data. A practical user-friendly software called "PSD Curve Fitting Software" was developed and introduced to program a simultaneous fitting of all models on soil PSD data of all samples. Some of the capabilities of this software are calculating evaluation statistics for all models and soils and their statistical properties such as average, standard deviation, minimum and maximum for all models, the amount of models’ fitting parameters and their statistical properties for all soil samples, soil water retention curve by Arya and Paris (1981) and Meskini-Vishkaee et al. (2014) methods, soil hydraulic conductivity by Arya et al. (1999) method, different textural and hydraulic properties, specific surface area, and other descriptive statistics of PSD for all soil samples. All calculated parameters are presented in an output Excel file format by the software. The software runs under Windows XP/7/8/10.

Synthesis of Poly(N-vinylcaprolactam)-Based Microgels by Precipitation Polymerization: Pseudo-Bulk Model for Particle Growth and Size Distribution.

Particle size distribution and in particular the mean particle size are key properties of microgels, which are determined by synthesis conditions. To describe particle growth and particle size distribution over the progress of synthesis of poly(N-vinylcaprolactam)-based microgels, a pseudo-bulk model for precipitation copolymerization with cross-linking is formulated. The model is fitted and compared to experimental data from reaction calorimetry and dynamic light scattering, showing good agreement with polymerization progress, final particle size, and narrow particle size distribution. Predictions of particle growth and reaction progress for different experimental setups are compared to the corresponding experimental data, demonstrating the predictive capability and limitations of the model. The comparison to reaction calorimetry measurements shows the strength in the prediction of the overall polymerization progress. The results for the prediction of the particle radii reveal significant deviations and highlight the demand for further investigation, including additional data.

Use of the Swebrec Function to Model Particle Size Distribution in an Industrial-Scale Ni-Co Ore Grinding Circuit

Mathematical models of particle size distribution (PSD) are necessary in the modelling and simulation of comminution circuits. In order to evaluate the application of the Swebrec PSD model (SWEF) in the grinding circuit at the Punta Gorda Ni-Co plant, a sampling campaign was carried out with variations in the operating parameters. Subsequently, the fitting of the data to the Gates-Gaudin-Schumann (GGS), Rosin-Rammler (RRS) and SWEF PSD functions was evaluated under statistical criteria. The fitting of the evaluated distribution models showed that these functions are characterized as being sufficiently accurate, as the estimation error does not exceed 3.0% in any of the cases. In the particular case of the Swebrec function, reproducibility for all the products is high. Furthermore, its estimation error does not exceed 2.7% in any of the cases, with a correlation coefficient of the ratio between experimental and simulated data greater than 0.99.