Prediction Of Particle Size Distribution Research Articles

DEM-PBM approach to predicting particle size distribution in tumbling mills

Tumbling mills using balls as grinding media are used extensively in the mining and cement industries to produce fine powders; however, it is well known that the process of size reduction is highly energy-intensive. For this reason, much of today’s comminution research is aimed at understanding grinding mechanisms, by estimating power draft and modeling milling circuits using the discrete element method (DEM). DEM predicts individual particle trajectories, the distribution of energy and contact forces at each collision, lifter wear and power draw. This paper presents a prediction of the discharge particle size distribution using three components, mass balance model (sometimes referred to as a population balance model, PBM), impact energy distribution of the mill obtained from the simulation of the charge motion using DEM and breakage characteristics of particles determined from drop-ball tests DEM simulation of the charge of a tumbling mill was done using the code MinProSim 1.0 to obtain impact spectra at different operative conditions. Then, thepopidation balance model (PBM), based on impact energy distribution, was used to predict the final particle size distribution in a batch mill and compared with experimental data. It is demonstrated that the DEM-PBM approach allows a reliable solution to predict particle size distribution for tumbling mills. Ideally, the model has to predict size distribution for any operational condition and feed characteristic; however, it has been shown that predictions include some degree of error.

A priori prediction of aggregation efficiency and rate constant for fluidized bed melt granulation

This paper presents a predictive aggregation rate model for spray fluidized bed melt granulation. The aggregation rate constant was derived from probability analysis of particle–droplet contact combined with time scale analysis of droplet solidification and granule–granule collision rates. The latter was obtained using the principles of kinetic theory of granular flow (KTGF). The predicted aggregation rate constants were validated by comparison with reported experimental data for a range of binder spray rate, binder droplet size and operating granulator temperature. The developed model is particularly useful for predicting particle size distributions and growth using population balance equations (PBEs).

Nonhomogeneous Mixing Population Balance Model for the Prediction of Particle Size Distribution in Large Scale Emulsion Polymerization Reactors

The present study describes the development of a nonhomogeneous two-compartment model for the prediction of particle size distribution in a semibatch emulsion ter-polymerization reactor. The multicompartment model accounts for spatial variations of particle size distribution (PSD) in the reactor due to nonideal mixing conditions. A comprehensive emulsion polymerization model is applied to each compartment, which allows the calculation of the various species concentrations in the aqueous and particle phases in each compartment. Moreover, a particle population balance equation is solved for each compartment to determine the individual PSDs as well as the overall PSD in the reactor. The effects of the two-compartment nonhomogeneous model parameters, that is, the volume ratio of the two compartments, the compartment exchange flow rates, and the partitioning of the monomer and initiator feed streams into the two compartments, on the overall polymerization rate and PSD are analyzed in detail. It is shown that d...

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.

Nucleation and Growth Kinetics of MgO in Molten Steel

The size, number, morphology and type of inclusion particles are the key factors to estimate the quality of steel product. Although considerable efforts have been made in the mathematical modeling of inclusion growth, few papers were involved in inclusion's nucleation and collision-growth, and all the existing researches about the behaviors of magnesia inclusion were based on the experiments. Thus, a mathematical model was developed to investigate the nucleation, Ostwald ripening and collision-growth of magnesia inclusion in the molten steel. Numerical results showed that the predicted particle size distributions are consistent with the previous experimental data. For the magnesia inclusions smaller than 100 nm, Brownian collision is the main collision modes. For the inclusions ranging from 0.1 to 10 μm, Brownian collision and turbulent collision are the main collision modes. For the inclusions ranging from 10 to 100 μm, turbulent collision and Stokes collision are the main collision modes. Thus, the strong turbulent flow can decrease the peak-value diameter of the magnesia inclusion effectively.

On the solution of bivariate population balance equations for aggregation: X–discretization of space for expansion and contraction of computational domain

A new structured discretization of 2D space, named X-discretization, is proposed to solve bivariate population balance equations using the framework of minimal internal consistency of discretization of Chakraborty and Kumar [2007, A new framework for solution of multidimensional population balance equations. Chem. Eng. Sci. 62, 4112–4125] for breakup and aggregation of particles. The 2D space of particle constituents (internal attributes) is discretized into bins by using arbitrarily spaced constant composition radial lines and constant mass lines of slope −1. The quadrilaterals are triangulated by using straight lines pointing towards the mean composition line. The monotonicity of the new discretization makes is quite easy to implement, like a rectangular grid but with significantly reduced numerical dispersion. We use the new discretization of space to automate the expansion and contraction of the computational domain for the aggregation process, corresponding to the formation of larger particles and the disappearance of smaller particles by adding and removing the constant mass lines at the boundaries. The results show that the predictions of particle size distribution on fixed X-grid are in better agreement with the analytical solution than those obtained with the earlier techniques. The simulations carried out with expansion and/or contraction of the computational domain as population evolves show that the proposed strategy of evolving the computational domain with the aggregation process brings down the computational effort quite substantially; larger the extent of evolution, greater is the reduction in computational effort.

Modelling and Simulation of Particle Breakage in Jaw Crushers

Jaw crushers are commonly-used machines in aggregate production. The aim of the research is to present a model which is able to predict particle size distribution in jaw crushers and to provide a new method for effectively controlling the quality of crushed product. The object of study was a domestic PEF 400×600 jaw crusher. A performance model was developed based on the standard mass balance formulation and chamber dividing results. Some important parameters that influence the particle output of the crushers were taken into account in the model. The variations of the particle size distribution resulting from changes in the eccentric shaft rotational speed and the parallel strip length were investigated.

A more efficient simulator of particle size distribution in slurry phase polyolefin systems

The Segregation Approach is used as an alternative to Population Balance Modelling (PBM) to predict particle size distributions in olefin polymerisation reactors. The Segregation Approach affords simpler formulation, more explicit integration of mixing characteristics, particle growth and initial size distribution, and easier extension to unsteady-state conditions and complex kinetics. The evolution of a non-uniformly sized feed of catalyst particles in a polymer reactor system is simulated. While the two approaches yield identical predictions of the particle size distribution, in terms of computation time the Segregation model (0.09–1.7 s) far outperforms the PBM (1.8–6.2 s). This is because the sets of differential equations required by the PBM are avoided by the Segregation Approach. Based on this result, the Segregation model is described in the context of developing a model which can be used for the real-time model-based control and grade transition trajectory optimisation of olefin polymerisation processes.

Stochastic Approach for the Prediction of PSD in Crystallization Processes: Formulation and Comparative Assessment of Different Stochastic Models

A stochastic formulation for the description of antisolvent mediated crystal growth processes is discussed. In the proposed approach, the crystal size growth dynamics is driven by a deterministic growth factor coupled to a stochastic component. The evolution in time of the particle size distribution (PSD) is then described in terms of a Fokker−Planck equation. In this work, we investigate and assess comparatively the performance of the FPE approach to model the crystal size distribution based on different expressions for the stochastic component. In particular, we investigate the one-dimensional Fokker−Planck equation with a nonlinear diffusion coefficient to represent the crystal growth process. Validations against experimental data are presented for the NaCl−water−ethanol antisolvent crystallization system. It is shown that the stochastic model better suited to describe the experiments is given by the Geometric Brownian Motion (GBM), which gives an excellent agreement, with the experiments for a wide range of process conditions (i.e., antisolvent feed rate).

Comparison of different chemometric and analytical methods for the prediction of particle size distribution in pharmaceutical powders

This work compares the estimation of the particle size distribution of a pharmaceutical powder using near-infrared spectroscopy (NIRS), powder flowability properties, and components concentration. The estimations were made by considering the former data blocks separately and together using a multi-block approach. The powders were based on a formulation of paracetamol as the pharmaceutical active ingredient. The reference method used to determine particle size distribution was sieving. Partial least squares methods were used to estimate the multivariate regression models, and the results were compared in terms of figures of merit. It was shown that the partial least squares methods gave similar prediction errors. Regarding the data blocks used, the NIRS block was proven the most advantageous to estimate the particle size distribution. The prediction error of the NIRS block was similar to the other data blocks with additional advantages such as less generalization problems and the possibility of its use to predict additional physical and chemical properties with an improvement to analysis time. The multi-block approach produced the worst results but nevertheless allowed a deeper understanding of the individual contributions of the data blocks in the prediction of the particle size distribution.

A Combined Precipitation, Yield Strength, and Work Hardening Model for Al-Mg-Si Alloys

In the present article, a new two-internal-variable model for the work hardening behavior of commercial Al-Mg-Si alloys at room temperature is presented, which is linked to the previously developed precipitation and yield strength models for the same class of alloys. As a starting point, the total dislocation density is taken equal to the sum of the statistically stored and the geometrically necessary dislocations, using the latter parameters as the independent internal variables of the system. Classic dislocation theory is then used to capture the overall stress-strain response. In a calibrated form, the work hardening model relies solely on outputs from the precipitation model and thus exhibits a high degree of predictive power. In addition to the solute content, which determines the rate of dynamic recovery, the two other microstructure parameters that control the work hardening behavior are the geometric slip distance and the corresponding volume fraction of nonshearable Orowan particles in the base material. Both parameters are extracted from the predicted particle size distribution. The applicability of the combined model is illustrated by means of novel process diagrams, which show the interplay between the different variables that contribute to work hardening in commercial Al-Mg-Si alloys.

Evaluation of the Gray Model GM(1,1) Applied to Soil Particle Distribution

Particle size distribution (PSD) is a fundamental soil physical property. The conventional approaches for representing PSD use empirical models with two to four parameters. We developed an alternative way to predict PSD that differs from conventional approaches by using the gray model GM(1,1), which does not depend on the model shape as empirical approaches do. The performance of GM(1,1) was compared with Skaggs model by using four statistical criteria. From nine textures of soil samples in our study, the results reveal that the GM(1,1) is superior for making PSD predictions. The results show that for the overall textures, the GM(1,1) model makes better predictions than the Skaggs model except for sand. Therefore, the performance of the GM(1,1) is fairly reliable and efficient and is not affected by soil textures in general.

Grinding kinetics of amorphous powder obtained by sol–gel process

Pre-mullite powder has been synthesized by sol–gel process followed by calcination. Aluminium nitrate and tetraethyl ortosilicate (TEOS) were used as the starting materials. X-ray diffraction (XRD) analysis showed that calcined pre-mullite powder is completely amorphous. The powders were wet milled in planetary ball-mill within various time intervals. The particle size distribution of ground samples has been determined using particle size analyzer (PSA). The powder samples morphology was investigated with Scanning Electron Microscopy (SEM). Comminution kinetics of calcined pre-mullite gel is modeled using a classical batch grinding equation based on selection and breakage functions. The analytical solution of the batch grinding equation, as well as parameters of grinding kinetics was obtained through Nakajima and Tanaka approach. Analysis of experimental data resulted with model suitable for prediction of particle size distribution of calcined amorphous gel comminuted in planetary ball mill. Model is suitable if the duration of grinding is no longer than 128 min. Obtained product size distribution, for shorter grinding duration (≤ 128 min) is consistent with massive fracture mechanism. For longer grinding periods (≥ 256 min) changes of breakage mechanism indicated parallel fracture and attrition mechanism. The parameters of grinding kinetics obtained by the linear regression analysis were used to reproduce the product size distributions. The good agreement between experimental and calculated size distributions confirms the applicability of the applied rate equation to the description of grinding of calcined amorphous gel.

Population balance modelling of protein precipitates exhibiting turbulent flow through a circular pipe

Abstract The breakage of liquid–liquid, solid–liquid and solid–gas dispersions occurs in many industrial processes during the transport of particulate materials. In this work, breakage of whey protein precipitates passing through a capillary pipe is examined and an experimentally derived breakage frequency is applied to construct a suitable population balance model to characterize the breakage process. It has been shown that the breakage frequency of precipitate particles is highly dependent on their shear history and on the turbulent energy dissipation rate in the pipe. The population balance equation (PBE) uses a volume density based discrete method which is adapted from mass density based discretization. In addition to comparing the model with experimental data, predicted results at different velocities are presented. It was found that the population balance breakage model provides satisfactory results in terms of predicting particle size distributions for such processes.

Coagulation of Organic Carbon Nanoparticles in Exhaust Conditions

The concentration of combustion-generated Nanoparticles of Organic Carbon (NOC) in vehicle emissions and urban atmospheres was evaluated using particle measurement techniques sensitive to NOC (Atomic Force Microscopy, Differential Mobility Analysis and Far UV Absorption Spectroscopy) and a theoretical model was used to predict particle size distributions throughout the exhaust plume. A significant fraction, and in some cases, the majority of particulate formed in combustion conditions of modern engines are NOC, which are an order of magnitude smaller than graphitic soot particles. NOC are mostly smaller than the detection limit of commercial particle sizing instrument used for atmospheric measurements (d < 5–10 nm). The smallest NOC have a higher affinity for water, higher mobility, higher number concentration, and longer lifetime in flames than larger soot particles. Therefore, NOC may contribute to observed human health effects associated with pollution. In previous works, these observations were explained by a theoretical model that predicts the particle coagulation rate as steeply increasing with particle size. The same theoretical model was used in this study to simulate aerosol dynamics in vehicle exhausts and dilution plumes as they mix with urban air and estimate their size distributions. The model predicts bimodal particle size distributions near roadways with a modal diameter of 2–3 nm, similar to the size of combustion-generated NOC measured in flames and engine exhausts. Predictions indicate the presence of NOC in urban atmospheres, similar to measurements by cryogenic sampling and AFM, presented in this work, and characterization studies of organic species in collected rain and fog samples.

Modelling soot formation in a premixed flame using an aromatic-site soot model and an improved oxidation rate

An updated rate of O2 oxidation of one to four ring polyaromatic hydrocarbons in premixed flames is presented based on density function theory simulations of oxygen attack at different radical sites on various PAHs. The rate is in agreement with other rates found in the literature; however, it is several orders of magnitude lower than the currently accepted oxidation rate of multi-ring aromatic species, including soot. Simulations are presented of a premixed flame using this improved rate and a new advanced soot particle model, which is developed in this paper. This model includes unprecedented detail of the particles in the ensemble, including the aromatic content, C/H composition and primary-particle aggregate structure. The O2 oxidation rate calculated in this paper is shown to give a better prediction of particle number density and soot volume fraction for a premixed flame. The predicted particle size distributions are shown also to describe better the experimental data. Predicted C/H ratio and PAH size distributions are shown for the flame. Computed TEM-style images are compared to experimental TEM images, which show that the aggregate structure of the particles is well predicted.

Effects of mixing in pharmaceutical product design

SUMMARY The product design procedure is considered in this work in relation to process design and choice of equipment with a goal to match the desired product specification. The procedure is based on inverse analysis and scale up methods; it includes mathematical modeling and experimental investigations. Two problems of pharmaceutical product design are presented in detail, namely manufacturing of pure products at high reaction selectivity by applying clever mixing procedures, and designing mixing strategy to control particle size distribution (PSD). The problem of reactive mixing is solved in this work using hierarchy of methods, starting from a simple time scale analysis, applying then directly mechanistic models, and finally combining mechanistic models with CFD. Examples of CFD simulations and related experimental data for test reactions are presented to illustrate proposed methodology. The problem of manufacturing particles characterized by a well defined size distribution (PSD) is discussed and methods of process design are presented. In particular methods for prediction of PSD in the case when precipitation is carried out using supercritical fluids are considered; mathematical modeling is used afterwards to interpret related experimental data, illustrating influence of process parameters on PSD. More traditional processes (reactive precipitation) are briefly considered for comparison as well.

Experimental identification of entropy model of comminution process

The entropy model based on population balance enables theoretical prediction of particle size distribution of comminution product. Selection and breakage functions occur in this model. The form of the selection function was determined experimentally. Informational entropy was used for the breakage function determination. Parameters of both functions can be estimated from experiments. The parameter identification of the entropy model was carried out on the basis of research on limestone comminution. Grinding tests were carried out in a laboratory fluidised bed jet mill. The results of the experimental identification confirm the accuracy of the entropy model.

Estimation of the size distribution of particles moving on a conveyor belt

In a previous study, a model to predict particle size distribution from data extracted from particle profiles by measuring two profile parameters, namely mean Feret’s diameter ( d F) and equivalent area diameter ( d eq.) were developed. The application of this model which has been called joint analysis (Nottsizer) on the estimation of static particle heaps of known size distribution showed some improvement in the estimation of size distribution parameters ( D 20, D 50 and D 80). In this paper, further validation of Nottsizer was carried out by estimating sieve size distribution of particles moving on a conveyor belt taking into account some well known problems associated with imaging moving particles such as camera location, particle overlap, image blurring, conveyor speed, dust generation and treatment.

Energy and population balances in comminution process modelling based on the informational entropy

The results of theoretical and experimental studies of a comminution process are presented. There are two random functions: the selection function and the breakage function in the stochastic model based on a population balance. This model enables prediction of particle size distributions of comminution products after determination of both random functions. Maximum entropy method is used in the entropy model for determining the breakage function. Two cases are analysed, based on continuous and discrete particle size distribution functions of the fed material. Apart from mass balance, the energy balance of comminution process is also used. Searched form of breakage function is determined with the application of methodology of calculus of variations. The results of experimental identification of both models are presented. The parameters that occur in the discrete form of the selection and breakage functions were the identification objects. The results of experimental investigations of quartz sand single comminution in a laboratory jet mill provided an identification base. The experimentally identified results of the entropy model confirmed the adequacy of the theoretical analysis and demonstrated the possibility of adequate prediction of particle size distributions resulting from single comminution.