Steady-state Particle Size Distribution Research Articles

Ostwald ripening under temperature gradient: A phase-field study

Ostwald ripening under a temperature gradient in binary model alloys is investigated using a quantitative phase-field model. The simulations show that a cube of average radius of a second-phase particle is proportional to time, and the particle size distribution shows self-similarity in a steady state, as with a uniform temperature field. It is found that the growth rate of particles under a temperature gradient is faster than that in the isothermal case, and the steady-state particle size distribution depends on the magnitude of the temperature gradient. Furthermore, the second-phase particles migrate from low temperature regions to high temperature regions when a non-uniform temperature field is applied. The migration velocity of particles, averaged over the whole system, increases with the magnitude of the temperature gradient. On the other hand, the velocity of each particle is not relevant to particle size. Hence, the particle migration is entirely ascribed to the diffusion flux driven by the concentration gradient originating from the temperature dependence of solute solubility.

Anderson acceleration method of finding steady-state particle size distribution for a wide class of aggregation–fragmentation models

A fast numerical method of finding steady-state distributions of particles sizes for a wide class of aggregation–fragmentation models, including the models with a source of monomers is developed. The method is based on a fast evaluation scheme for large sets of non-linear Smoluchowski-type ODE with an application of the Anderson acceleration method of the fixed point iterations. In the numerical tests of the suggested approach we demonstrate that huge sets of non-linear ODE may be solved with high precision in terms of Euclidian norm of the residual in modest times with use of a regular desktop computer. We compare our numerical solutions with the known analytical results for the steady-state distributions as well as with the other fast numerical schemes and prove the high accuracy of the novel method and its significant superiority with respect to the existing fast numerical method of solution of the addressed problems.

Comparison of Vehicle Exhaust Particle Size Distributions Measured by SMPS and EEPS During Steady-State Conditions

Fast-sizing spectrometers, such as the TSI Engine Exhaust Particle Sizer (EEPS), have been widely used to measure transient particle size distributions of vehicle exhaust. Recently, size distributions measured during different test cycles have begun to be used for calculating suspended particulate mass; however, several recent evaluations have shown some deficiencies in this approach and discrepancies relative to the gravimetric reference method. The EEPS converts electrical charge carried by particles into size distributions based on mobility classification and a specific calibration, and TSI recently released a matrix optimized for vehicle emissions as described by Wang et al. (Submitteda). This study evaluates the performance of the new matrix (soot matrix) relative to the original matrix (default matrix) and reference size distributions measured by a scanning mobility particle sizer (SMPS). Steady-state particle size distributions were generated from the following five sources to evaluate exhaust part...

Two phase steady-state particle size distribution in a gas-phase fluidized bed ethylene polymerization reactor

A population balance model is developed to investigate the steady state particle size distribution developments in a gas-phase fluidized-bed ethylene polymerization reactor. The model considers the combined effects of particle growth and elutriation for size distributed prepolymer feed. In the proposed model, the bed is divided into several sections consisting of bubble and emulsion phases. The population balance differential equations derived for each section were simultaneously solved to determine the density function of the size distribution of the polymer particles in each section. The model is able to estimate the axial profile of particle size distribution along the reactor height. It was shown that the mean size of the particles is larger at the bottom of the bed and becomes smaller when moving toward top of the reactor. The size distribution of the polymer particles in the product becomes sharper and their mean size decreases by increasing the superficial gas velocity. Also, mean size of the particles in the product increases by increasing the temperature of the reactor.

Particle-based modeling of aggregation and fragmentation processes: Fractal-like aggregates

The incorporation of particle inertia into the usual mean field theory for particle aggregation and fragmentation in fluid flows is still an unsolved problem. We therefore suggest an alternative approach that is based on the dynamics of individual inertial particles and apply this to study steady state particle size distributions in a 3D synthetic turbulent flow. We show how a fractal-like structure, typical of aggregates in natural systems, can be incorporated in an approximate way into the aggregation and fragmentation model by introducing effective densities and radii. We apply this model to the special case of marine aggregates in coastal areas and investigate numerically the impact of three different modes of fragmentation: large-scale splitting, where fragments have similar sizes, erosion, where one of the fragments is much smaller than the other and uniform fragmentation, where all sizes of fragments occur with the same probability. We find that the steady state particle size distribution depends strongly on the mode of fragmentation. The resulting size distribution for large-scale fragmentation is exponential. As some aggregate distributions found in published measurements share this latter characteristic, this may indicate that large-scale fragmentation is the primary mode of fragmentation in these cases.

Continuous crystallization of adipic acid with ultrasound

The application of a novel continuous cooling crystallization system coupled with ultrasound technology to crystallize adipic acid from an aqueous solution was investigated. Both continuous ultrasound irradiation and silent (control) regimes were used in order to elucidate the effects of unstable cavitation on the duration of the transient period, steady state operating conditions, crystal habit, and particle size distribution. The results reveal that under continuous ultrasonic irradiation the steady state particle size distribution is achieved after shorter times, with a consistent reduction of the steady state supersaturation resulting in increased product yields relative to silent continuous crystallization experiments. Continuous crystallization with ultrasonic irradiation results in significantly smaller crystal sizes, reduced agglomeration and an improved habit of crystals. Furthermore, the influence of mean residence time using continuous ultrasound produces a smaller change in particle size and size distribution. Finally, the experiments reveal a noticeable reduction of the fouling on non-cooling surfaces and underline the need for intermittent discharge to prevent classification on withdrawal product.

Quantification of gypsum crystal nucleation, growth, and breakage rates in a wet flue gas desulfurization pilot plant

AbstractThe aim of this work is to study the influence of nucleation, growth and breakage on the particle size distribution (PSD) of gypsum crystals produced by the wet flue gas desulfurization (FGD) process. The steady state PSD, obtained in a falling film wet FGD pilot plant during desulfurization of a 1000 ppm(v) SO2 gas stream, displayed a strong nonlinear behaviour (in a ln(n(l)) vs. l plot) at the lower end of the particle size range, compared to the well‐known linear mixed suspension mixed product removal model. A transient population balance breakage model, fitted to experimental data, was able to model an increase in the fraction of small particles, but not to the extent observed experimentally. A three‐parameter, size‐dependent growth model, previously used for sodium sulphate decahydrate and potash alum, was able to describe the experimental data, indicating either size‐dependent integration kinetics or growth rate dispersion. © 2009 American Institute of Chemical Engineers AIChE J, 2009

A basic population balance model for fluid bed spray granulation

A basic population balance approach is developed for a granulation process in a fluid bed spray granulator. The particle size distribution predicted by the model is confirmed by plant data. Hence this model is considered to be useful to describe and optimize an industrial process. The model depends on a limited number of parameters (most of these factors can be measured or are known): the spray volume flux, the nucleation fraction (the fraction of the spray volume flux which leads to new particles formed), the nucleation particle diameter, the product withdrawal threshold diameter, and the product withdrawal rate. Analysis of the model reveals a steady-state constraint; a steady state does not exist if the nucleation fraction is too large. For cases where the steady state does exist, the steady-state particle size distribution is solved analytically. A numerical implementation of the model is used to illustrate the transient evolution of the process. The steady-state solution appears to be stable for a constant nucleation fraction. However, if the nucleation fraction depends on the bed height the steady state can be unstable. Such a situation may occur if the spray inlet is near the height of the bed surface. Instead of convergence towards a steady state, the transient solution displays ongoing oscillatory behavior with an oscillation period of a number of hours. A linear stability analysis is performed to confirm the findings on the stability of the steady state.

On equilibrium solutions of aggregation–fragmentation problems

A characteristic feature of particulate systems that evolve due to competition between aggregation and breakage is that they sometimes produce non-trivial steady-state particle size distributions. If such solutions satisfy detailed balance conditions, then they are equilibrium solutions. The conditions that must be satisfied by aggregation and fragmentation rate kernels in order for equilibrium solutions to be produced are elaborated, and it is shown that the rate kernels are uniquely determined by the aggregation and breakage rate constants for the reactions involving monomers. Consequently, for equilibrium systems there is a significant reduction in the amount of information needed in order to infer the general form for aggregation or breakage kernels, and we explore implications for constructing rate kernels by using atomistic simulations such as molecular dynamics.

A discretized population balance for particle formation from gas antisolvent process: The combined Lax-Wendroff and Crank-Nicholson method

Population balance models describing particle formation kinetics in supercritical crystallization processes in general and of gas antisolvent crystallization (GAS) process in particular are rather complicated. Depending on the considered particle formation mechanism, population balance model of GAS process may include phenomena such as primary nucleation, secondary nucleation, crystal growth, as well as agglomeration and/or breakage (attrition) of crystals. The resulting equation is often of the integro-partial differential form. A powerful numerical algorithm for the treatment of the implemented population balance model structures is presented in this paper. The method/ algorithm is a combination of the Lax-Wendroff and Crank-Nicholson methods. Algorithm simulations were performed for changes in the main GAS process operating parameters, i.e., the antisolvent addition rate and saturation level. The simulations were performed at a process temperature of 25 °C, while the antisolvent addition rate, Q A , was varied between 1 and 100 ml/min, and the initial solute concentration was varied between 25% and 100% of the concentration ratio. The numerical algorithm was successfully able to treat the implemented population balance model structures, producing smooth dynamic and steady state particle size distributions. Moreover, simulation findings were reasonably consistent with the experimentally obtained data.

Arsenic distribution in the dissolved, colloidal and particulate size fraction of experimental solutions rich in dissolved organic matter and ferric iron

Due to the widespread contamination of groundwater resources with arsenic (As), controls on As mobility have to be identified. In this study we focused on the distribution of As in the dissolved, colloidal and particulate size fraction of experimental solutions rich in ferric iron, dissolved organic matter (DOM) and As(V). Size fractions between <5 kDa and >0.2 μm were separated by filtration and their elemental composition was analyzed. A steady-state particle size distribution with stable element concentration in the different size classes was attained within 24 h. The presence of DOM partly inhibited the formation of large Fe-(oxy)hydroxide aggregates, thus stabilized Fe in complexed and colloidal form, when initially adjusted molar Fe/C ratios in solution were <0.1. Dissolved As concentrations and the quantity of As bound to colloids (<0.2 μm) increased in the presence of DOM as well. At intermediate Fe/C ratios of 0.02–0.1, a strong correlation between As and Fe concentration occurred in all size fractions ( R 2 = 0.989). At Fe/C ratios <0.02, As was mainly present in the dissolved size fraction. These observations indicate that As mobility increased in the presence of DOM due to (I) competition between As and organic molecules for sorption sites on Fe particles; and (II) due to a higher amount of As bound to more abundant Fe colloids or complexes <0.2 μm in size. The amount of As contained in the colloidal size fractions also depended strongly on the initial size of the humic substance, which was larger for purified humic acids than for natural river or soil porewater samples. Arsenic in the particle size fraction >0.2 μm additionally decreased in the order of pH 4 ≫ 6 > 8. The presence of DOM likely increases the mobility of As in iron rich waters undergoing oxidation, a finding that has to be considered in the investigation of organic-rich terrestrial and aquatic environments.

Steady-state particle size distribution modeling of polypropylene produced in tubular loop reactors

A comprehensive steady-state population balance model is developed for the particle size distribution (PSD) of the polypropylene produced in tubular loop reactors. The model considers the flow type, the polymer particle dynamics, the particle growth and the attrition. Moreover, an empirical single particle model is used to describe the particle growth under internal and external heat and mass transfer limitations totally. The predicted PSD data obtained under steady-state polymerization conditions agree well with the actual data collected from industrial scale plant. The model are also used to predict the effects of some operational parameters on the polymer PSD produced under steady-state conditions. The results show that the polymer PSD are greatly affected by the feed catalyst size, the feed catalyst PSD, and the polymerization temperature.

Destructive aggregation: Aggregation with collision-induced breakage

Mean-field population balance equations are used to describe the evolution of particle size distributions in a wide variety of systems undergoing simultaneous aggregation and breakage. In this paper we develop a population balance that includes aggregation combined with collision-induced particle breakage for arbitrary fragment distribution functions, provided that this distribution function depends only on the total mass of the particles undergoing a collision. We then develop a specific distribution function for arbitrary two-body collisions by postulating that each collision produces a transition-state aggregate having the morphology of a linear polymer. The behavior of the resulting equation is then analyzed for the case in which the collision kernel is a constant, and partial analytical solutions are derived and compared to corresponding Monte-Carlo simulation results. The computer simulations are then used to validate a proposed scaling law for the steady-state particle size distribution. Lastly, the behavior of the aggregation with collision-induced-breakage population balance equation is compared and contrasted with the behavior of an analogous aggregation with linear-breakage population balance equation.

Part III: Dynamic evolution of the particle size distribution in batch and continuous particulate processes: A Galerkin on finite elements approach

The present study provides a comprehensive investigation on the numerical solution of the dynamic population balance equation (PBE) in batch and continuous flow particulate processes. The general PBE was numerically solved using the Galerkin on finite elements method (GFEM) for particulate processes undergoing simultaneous particle aggregation, growth and nucleation. The performance of the GFEM in terms of accuracy and stability was assessed by a direct comparison of the calculated particle size distributions (PSD) and/or their corresponding moments to available analytical solutions. Numerical simulations were carried out over a wide range of variation of particle aggregation and growth rates. Simulations of batch particulate processes revealed that the GFEM was generally more accurate than the standard fixed-grid sectional methods and more robust than the orthogonal collocation on finite elements method. In the case of continuous flow processes, numerical simulations were carried out till the dynamic solution reached its final steady-state. It was found that the calculated PSDs converged exactly to available analytical steady-state solutions of the PBE. The transient PSDs were often characterized by steep moving fronts that required careful discretization of the particle volume domain and the inclusion of an artificial diffusion term in order to eliminate numerical oscillations in the solution. Finally, for continuous particulate processes, the effect of various size-dependent particle growth models on the calculated steady-state PSDs was investigated.

Modelling particle size distribution dynamics in marine waters

Numerical simulations were carried out to determine the particle size distribution (PSD) in marine waters by accounting for particle influx, coagulation, sedimentation and breakage. Instead of the conventional rectilinear model and Euclidean geometry, a curvilinear collision model and fractal scaling mathematics were used in the models. A steady-state PSD can be achieved after a period of simulation regardless of the initial conditions. The cumulative PSD in the steady state follows a power-law function, which has three linear regions after log–log transformation, with different slopes corresponding to the three collision mechanisms, Brownian motion, fluid shear and differential sedimentation. The PSD slope varies from −3.5 to −1.2 as a function of the size range and the fractal dimension of the particles concerned. The environmental conditions do not significantly alter the PSD slope, although they may change the position of the PSD and related particle concentrations. The simulation demonstrates a generality in the shape of the steady-state PSD in the ocean, which is in agreement with many field observations. Breakage does not affect the size distribution of small particles, while a strong shear may cause a notable change in the PSD for larger and fractal particles only. The simplified approach of previous works using dimensional analysis still offers valuable approximations for the PSD slopes, although the previous solutions do not always agree with the simulation results. The variation in the PSD slope observed in field investigations can be reproduced numerically. It is argued that non-steady-state conditions in natural waters could be the main reason for the deviation of PSD slopes. A change in the nature of the particles, such as stickiness, and environmental variables, such as particle input and shear intensity, could force the PSD to shift from one steady state to another. During such a transition, the PSD slope may vary to some extent with the particle population dynamics.

Monte Carlo simulations of colloidal particle coagulation and breakup under turbulent shear

Both a Monte Carlo model and an algorithm were presented to simulate the particle coagulation and breakup phenomena taking place in a colloidal solution under turbulent fluid shear. The model is represented by the probability density functions that describe the stochastic coagulation and breakup phenomena taking place among numerous particles. From a dimensional analysis of the model two dimensionless groups, κc and κb, were derived that represent the relative intensity of the coagulation and breakup phenomena. In order to overcome the memory problem in saving the sizes of a large number of particles, the model was converted to a form suitable for carrying out a sectional mass balance. Detailed simulation steps were presented and applied to acrylonitrile-butadiene-styrene (ABS) latex coagulation. Numerical simulations revealed that the steady state particle size distribution does not depend on the initial distributions but on the κc/κb ratio. Setting the operation variables to increase the ratio was found to shift the particle size distribution toward larger particles.

Analytical solutions for steady and unsteady state particle size distributions in FBC and CFBC boilers for non-breaking char particles

Continuous analytical solutions for the particle size distributions of char in steady and unsteady states in fluidized beds, when the inlet fuel feed is presented by monosize, lognormal, Rosin-Rammler or gamma distributions, are derived from a population balance model. The stationary size distribution is directly related to the rate of reduction of the particle size. Combustion and attrition reduce the particle size. Thus, it is possible to extract the dependence of the rate of reduction of radius (affected by a fuel’s reactivity and attrition) on radius from a measured steady-state particle size distribution. Unsteady particle size distributions are derived for impulse, step and square pulse changes in the fuel feed, when the oxygen level in the reactor is maintained constant.

A two-scale PBM for modeling turbulent flocculation in water treatment processes

A population balance model (PBM) that incorporates two scales of turbulent motion in the breakup frequency function has been presented. The breakup frequency function is designed such that particles smaller than the impeller-region Kolmogoroff microscale will erode according to a critical velocity related to the local energy dissipation rate. Particles larger than the impeller-region Kolmogoroff microscale will fracture according to a critical velocity related to the impeller tip speed. The two-scale model was found to better predict the experimental steady-state particle size distribution in 5, 28, and 560 l tank sizes and with a Rushton turbine and A310 fluid foil impellers. The two-scale PBM was also used to investigate the most appropriate scale-up law for drinking water flocculation processes. In addition, the impact of higher tank average energy dissipation rate, primary particle concentration, and coagulant concentration on the volume mean particle size with increasing tank size and different impeller types was also presented.

Unsteady and steady-state particle size distributions in batch and continuous fluidized bed granulation systems

A two-dimensional balance model of populations distributed by property was developed using the ordinates of particle diameter and particle density. This model makes it possible to calculate the spectrums of granulation changing over time in the fluidized bed during batch operation and in addition to calculate the time-dependent size distribution of the product granulate for continuous operation. A physics-based model was devised for the submechanism of the growth of the granulates. On the one hand, the population balance model is divided into the particle balance around the granulator and, on the other hand, when granulation is continuous, particles are constantly removed from the fluidized bed. To this end, a separator model was used that subdivides the extractor device into two areas of balancing in accordance with the geometry of the apparatus and the properties of the particle. In the case of continuous operation, the results when the product is classified using a classifier tube inside the fluidized bed are discussed. The outcome is the influence of the various particle flows on the particle diameter and the product mass withdrawn. In particular, the problematic nature of the start-up is dealt with, i.e. reaching stable points of steady operation. The simulation results are validated using results from continuous tests on a semi-industrial plant.

Size Distributions of Motor Vehicle Exhaust PM: A Comparison Between ELPI and SMPS Measurements

Particle size measurements using the electrical low pressure impactor (ELPI) and scanning mobility particle sizer (SMPS) are compared from the perspective of characterizing the particulate matter in motor vehicle exhaust. Both steady state vehicle operation and transient drive cycles are considered, and both gasoline and diesel fueled vehicle emissions are compared. Although the ELPI and SMPS measure different physical properties, respectively, the aerodynamic diameter and mobility diameter, the steady state particle size distributions are in close agreement, except for the 37 nm impactor stage of the ELPI which may overestimate particle number by up to a factor of two relative to the SMPS. This has little effect on the volume, or mass, weighted distribution. These, too, are generally in good agreement, though discrepancies appear at large particle size due to multiple charging effects in the SMPS and to electrometer offsets and the small particle loss correction in the ELPI. Selecting particles based on their electrical mobility with the SMPS, and then measuring their aerodynamic diameter with the ELPI, reveals that diesel particulate matter with well-specified mobility diameter exhibits a wide range in aerodynamic diameter and, therefore, also in effective density. Over transient drive cycles, the ELPI provides second by second particle distributions, whereas the SMPS must be run in a fixed particle size mode and size distributions constructed from repeated tests. The ELPI registers higher instantaneous PM emission rates during transients than the SMPS due to the faster time responses of the ELPI. The time integrated ELPI and SMPS size distributions, however, remain in good agreement. The relative merits of the two instruments for steady state and transient tests are discussed.