Aggregation Kernel Research Articles

Modelling of Polymer Particle Formation Using Population Balance Model

The paper presents a study of formation of the primary particle size distribution in suspension “powder” polymerization of vinyl chloride. The process is modelled by means of a population balance model, and the primary particle size distribution inside the polymerizing monomer droplets is determined by analysing the population balance equation, governing the nucleation, growth, and aggregation of the primary particles, using the moment method. The infinite set of moment equations obtained by moment transformation was closed using a sum aggregation kernel, and for numerical experimentation a second order moment equation model was used. The results show how important are to choose the correct parameters in production of poly(vinyl chloride) by suspension polymerization. Changing the parameters a bit the quality of product may change significantly. The results presented in the paper illustrate well that the population balance model can be used for describing the process and a number its properties with sufficient accuracy.

Mathematical modeling of DNA-mediated selective aggregation of CdS quantum dots

We report a mathematical modeling of the DNA-mediated selective aggregation of CdS quantum dots. Addition of hybridized double-stranded DNAs into the suspensions of CdS quantum dots at the optimal salt concentration causes a selective aggregation and a fluorescence-quenching phenomenon depending on the target DNA sequence. We monitor the aggregation process with quasi-elastic light scattering (QELS), zeta potential, and conductivity measurements at different salt concentrations. To model the aggregation process, we use the constant-number Monte Carlo method with the aggregation kernel that accounts for the interparticle interaction from the classical DLVO model. We find that the calculated results are in good agreement with the experiments, and that the fractal dimension of the quantum dot aggregates is 2.3. Modeling also allows us to estimate that the total number of initial quantum dots in aggregates at the beginning of the fluorescence-quenching phenomenon is approximately 200. The insights gained in this study should be useful in the design of biosensors based on the fluorescence-quenching phenomenon caused by the quantum dot aggregation.

Light Scattering Study of Aggregation Kinetics in Dense, Gelling Aerosols

We report the results of a study of the kinetics of a dense aggregating aerosol system that show that the predictions of the Diffusion Limited Cluster Aggregation (DLCA) model no longer hold when the system is dense. We studied a soot aerosol using the small angle light scattering technique created by exploding a mixture of a hydrocarbon gas and oxygen in a closed chamber. The soot particles started as individual monomers, ca. 38 nm radius, grew to bigger fractal clusters with time and finally stopped evolving after spanning a network across the whole system volume. This spanning is aerosol gelation. The study of the kinetics of the aggregating system showed that as the system evolved from a cluster dilute to cluster dense system, the aggregation kernel homogeneity λ evolved from the dilute limit, DLCA value of zero to a value 0.42 +/− 0.05 at the gel point. This evolution is consistent with previous simulation and theory. The experimental value at the gel point is nearly equal to the value of 0.5 predicted by simulation and theory. In addition, the magnitude of the aggregation kernel showed an increase with increasing volume fraction.

CFD Modeling of Nucleation, Growth, Aggregation, and Breakage in Continuous Precipitation of Barium Sulfate in a Stirred Tank

In this work, the precipitation of barium sulfate (BaSO4) in a continuous stirred tank reactor (CSTR) is modeled. The flow field is obtained through solving the single-phase Reynolds averaged Navier−Stokes equations with a standard single-phase k−ε turbulence model. The population balance equation is solved through the standard method of moments (SMM) and the quadrature method of moments (QMOM) both with and without aggregation and breakage terms. In the cases of precipitation simulation without aggregation and breakage, the results predicted from 2-node QMOM, 3-node QMOM, and SMM are very close. Thus, 2-node QMOM could replace SMM and be well-incorporated into an in-house CFD code to simulate the precipitation in CSTR with acceptable accuracy. The predicted area-averaged crystal size d32 decreases almost linearly with increasing feed concentration, and the deviation from experimental data becomes significant at high feed concentration. Numerical simulation using 2-node QMOM with the Brownian motion and shear-induced aggregation kernels as well as a power-law breakage kernel indicates that the predicted d32 shows good qualitative agreement with experimental results, and the quantitative agreement is achieved when the appropriate breakage rate equation is adopted.

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.

Modelling breakage and reagglomeration during fine dry grinding in ball milling devices

Modelling of grinding of fine powders in ball milling devices is addressed. The model quantitatively describes breakage and agglomeration phenomena by considering two populations, i.e. primary particles and porous aggregates. The population balance approach takes into account breakage and aggregation kernels which are considered functions of the size of the two populations. The proposed model is able to properly simulate the inversion from the breakage to the agglomerative regime typical of a fragile material powder system undergoing ball milling. A suitable fitting procedure is performed for separately determining the adjustable parameters of the model. Model reliability is tested against experimental data, while the proposed breakage/agglomeration kernels are related to the quantitative description of ball milling apparatus dynamics.

Experimental validation studies on a multi-dimensional and multi-scale population balance model of batch granulation

In this study, a dynamic model is presented for the granulation process, employing a three-dimensional population balance framework. As a first attempt to account for the multi-scale character of the process, the nucleation and aggregation kernels used in the population balance model are derived using mechanistic representations of the underlying particle physics such as wetting kinetics and energy dissipation effects. Thus, the fundamental properties of the powder and the liquid were used as parameters in the model to predict the granulator dynamics and granule properties. The population balance model is validated against experimental data from a calcite / PVOH – H 2 O recipe obtained using a lab-scale drum granulator for granule size, fractional binder content and porosity. A reasonably good agreement between experimental and simulation results were obtained for the granule size distribution under different experimental conditions. In addition, accurate model predictions were made for the evolution of the average properties (i.e., size, fractional binder content and porosity) for various operating conditions.

Modeling flocculation processes of fine-grained particles using a size-resolved method: Comparison with published laboratory experiments

The transport of fine-grained particles in estuarine and coastal waters is influenced by flocculation processes (aggregation and floc breakup). As a consequence, the particle size varies with time in the water column, and can be orders of magnitude larger than those of primary particles. In this study the variations in floc size is simulated using a size-resolved method, which approximates the real size distribution of particles by a range of size bins and solves a mass balance equation for each bin. To predict the size distribution both aggregation and breakup processes are included. The conventional rectilinear aggregation kernel is used which considers both turbulent shear and differential settling. The breakup kernel accounts for the fractal dimension of the flocs. A flocculation simulation is compared to the settling column lab experiments of Winterwerp [1998. A simple model for turbulence induced flocculation of cohesive sediment, Journal of Hydraulic Research, 36, 309–326], and a one-dimensional sediment transport model is verified with the observed variations in floc size and concentration over tidal cycles in a laboratory flume experiment of Bale et al. [2002. Direct observation of the formation and break-up of aggregates in an annular flume using laser reflectance particle sizing. In: Winterwerp, J.C., Kranenburg, C. (Eds.), Fine Sediment Dynamics in the Marine Environment. Elsevier, pp. 189–201]. The numerical simulations compare qualitatively and quantitatively well with the laboratory measurements, and the analysis of the two simulation results indicates that the median floc size can be correlated to the sediment concentration and Kolmogorov microscale. Sensitivity studies are conducted to explore the role of settling velocity and erosion rate. The results are not sensitive towards the formulation of settling velocity, but the parameterization of erosion flux is important. The studies show that for predicting the sediment deposition flux it is crucial to include flocculation processes.

Kinetic study of coupled field-induced aggregation and sedimentation processes arising in magnetic fluids

In this work, the kinetics of coupled aggregation and sedimentation processes arising in magnetic fluids has been studied. Aggregation was induced applying a constant uniaxial magnetic field. The time evolution of the cluster-size distribution and the weight-average chain length was monitored using optical microscopy and digital image analysis. The experimental results are compared with the corresponding solutions of Smoluchowski's equation. For this purpose, a recently proposed aggregation kernel was employed. When sedimentation effects are taken into account, the fits improve especially at long aggregation times.

The kinetics of the granulation process: Right from the early stages

In this study, experimentation and modelling were carried out to understand the granulation process. This work assesses whether there is a significant difference in the aggregation rate of the wet granulation process between the very early stages and later stages of high shear granulation. Measurements of the size distribution and binder content from the beginning of the process, just after liquid binder addition, were carried out. A population balance model based on two different kernels, the Equi Kinetic Energy (EKE) kernel and two-dimensional population balance equations with a Size Independent (SI) kernel, was applied to the high shear granulation process. It was concluded that the population balance equations with SI kernel best described the aggregation in the high shear granulation process. The value of aggregation rate constant in the early stages is smaller than aggregation kernel in the later stages.

On Brownian Dynamics Simulation of Nanoparticle Aggregation

Accurate simulation and control of nanoparticle aggregation in chemical reactors requires that population balance equations be solved by using realistic expressions for aggregation and breakage rate kernels. Obtaining such expressions requires that atomistic simulation approaches that can account for microscopic details of particle collisions be used. In principle, molecular dynamics simulations can provide the needed microscopic information, but because of the separation in length scales between the aggregates and solvent molecules, such simulations are too costly. Brownian dynamics simulations provide an alternative to the molecular dynamics approach for simulation of particle aggregation, but there has been no systematic attempt to validate the Brownian dynamics method for this class of problems. In this work we attempt to develop a better understanding of Brownian dynamics simulations of aggregation by (1) developing convergence criteria, (2) determining criteria for aggregation to occur in BD simulations using dimensionless variables, and (3) directly comparing BD and MD simulation predictions for a model aggregation problem.

A new discretization of space for the solution of multi-dimensional population balance equations

In this work, a novel radial grid is combined with the framework of minimal internal consistency of discretized equations of Chakraborty and Kumar [2007. A new framework for solution of multidimensional population balance equations. Chemical Engineering Science 62, 4112–4125] to solve n-dimensional population balance equations (PBEs) with preservation of ( n + 1 ) instead of 2 n properties required in direct extension of the 1-d fixed pivot technique of Kumar and Ramkrishna [1996a. On the solutions of population balance equation by discretization-I. A fixed pivot technique. Chemical Engineering Science 51, 1311–1332]. The radial grids for the solution of 2-d PBEs are obtained by intersecting arbitrarily spaced radial lines with arcs of arbitrarily increasing radii. The quadrilaterals obtained thus are divided into triangles to represent a non-pivot particle in 2-d space through three surrounding pivots by preserving three properties, the number and the two masses of the species that constitute the newly formed particle. Such a grid combines the ease of generating and handling a structured grid with the effectiveness of the framework of minimal internal consistency. A new quantitative measure to supplement visual comparison of two solutions is also introduced. The comparison of numerical and analytical solutions of 2-d PBEs for a number of uniform and selectively refined radial grids shows that the quality of solution obtained with radial grids is substantially better than that obtained with the direct extension of the 1-d fixed pivot technique to higher dimensions for both size independent and size dependent aggregation kernels. The framework of Chakraborty and Kumar combined with the proposed 2-d radial grid, which offers flexibility and achieves both reduced numerical dispersion and the ease of implementation, appears as an effective extension of the widely used 1-d fixed pivot technique to solve 2-d PBEs.

Influence of Cementitious Materials and Aggregates Content on Compressive Strength of Palm Kernel Shell Concrete

This study reports the effect of cementitious materials, fine and coarse aggregates content on workability and compressive strength of palm kernel shell concrete. Palm kernel shells a by product of the production of palm oil, were used as lightweight aggregates. The following cementitious materials were added: 10 silica fume as additional cementitious material and 5 fly ash as cement replacement on weight of cement. The influence of varying fine aggregate and palm kernel shell contents on workability and compressive strength has been studied. The specimens have been cured under three different curing environments to study the effect on compressive strength. The effect of cementitious materials and curing conditions on compressive strength for a period of 90 days was analyzed. The fresh density of concrete was found to be in the range of 1810 to 1940 kg m-3. The strength of Palm Kernel Shells (PKS) was found to be the primary factor controlling the strength. However, the addition of silica fume was found to have influence on compressive strength. An increase in fine aggregate content and subsequent decrease in PKS content had positive effect on both workability and compressive strength. The 28 day compressive strengths of the mixes containing cementitious materials were found in the range of 26 to 36 MPa. The difference in strength between water cured and specimens cured under controlled environment was found to vary between 3 and 5.

Large time behavior of nonlocal aggregation models with nonlinear diffusion

The aim of this paper is to establish rigorous results on thelarge time behavior of nonlocal models for aggregation, includingthe possible presence of nonlinear diffusion terms modeling localrepulsions. We show that, as expected from the practicalmotivation as well as from numerical simulations, one obtainsconcentrated densities (Dirac $\delta$ distributions) asstationary solutions and large time limits in the absence ofdiffusion. In addition, we provide a comparison for aggregationkernels with infinite respectively finite support. In the firstcase, there is a unique stationary solution corresponding toconcentration at the center of mass, and all solutions of theevolution problem converge to the stationary solution for largetime. The speed of convergence in this case is just determined bythe behavior of the aggregation kernels at zero, yielding eitheralgebraic or exponential decay or even finite time extinction. Forkernels with finite support, we show that an infinite number ofstationary solutions exist, and solutions of the evolution problemconverge only in a measure-valued sense to the set of stationarysolutions, which we characterize in detail.Moreover, we also consider the behavior in the presence of nonlinear diffusion terms,the most interesting case being the one of small diffusion coefficients. Via the implicit function theorem we give aquite general proof of a rather natural assertion for such models, namely that there exist stationary solutionsthat have the form of a local peak around the center of mass. Our approach even yields the order of the size ofthe support in terms of the diffusion coefficients.All these results are obtained via a reformulation of theequations considered using the Wasserstein metric for probabilitymeasures, and are carried out in the case of a single spatialdimension.

Integrated parallel performance views

The influences of the operating system and system-specific effects on application performance are increasingly important considerations in high performance computing. OS kernel measurement is key to understanding the performance influences and the interrelationship of system and user-level performance factors. The KTAU (Kernel TAU) methodology and Linux-based framework provides parallel kernel performance measurement from both a kernel-wide and process-centric perspective. The first characterizes overall aggregate kernel performance for the entire system. The second characterizes kernel performance when it runs in the context of a particular process. KTAU extends the TAU performance system with kernel-level monitoring, while leveraging TAU's measurement and analysis capabilities. We explain the rational and motivations behind our approach, describe the KTAU design and implementation, and show working examples on multiple platforms demonstrating the versatility of KTAU in integrated system/application monitoring.

Compositional distributions in multicomponent aggregation

We consider the granulation of two components, a “solute” (the component of interest) and an excipient. We specifically focus on cases such that the aggregation kernel is independent of the composition of the aggregating granules. In this case, theory predicts that the distribution of components is a Gaussian function such that the mean concentration of solute in granules of a given size is equal to the overall mass fraction of solute in the system, and the variance is inversely proportional to the granule size. To study these effects, we perform numerical simulations of the bicomponent population balance equation using a constant aggregation kernel as well as a kernel based on the kinetic theory of granular flow (KTGF). If the solute and excipient are initially present in the same size (monodisperse initial conditions), both kernels produce identical distributions of components. With different initial conditions, the KTGF kernel leads to better mixing of components, manifested in the form of narrower compositional distributions. These behaviors are in agreement with the predictions of the theory of aggregative mixing. We further demonstrate that the overall mixedness of the system is controlled by the initial degree of segregation in the feed and show that the size distribution in the feed can be optimized to produce the narrowest possible distribution of components during granulation.

Kinetics and dynamics of gelation reactions

Gelation, which can occur when molecules react or colloids aggregate, is defined as the divergence of second and higher moments of molecular-weight and cluster-size distributions. Distribution kinetics and dynamics, described by population balance equations for aggregation and fragmentation, provide a framework for investigating quantitatively when and how gelation occurs and the conditions for existence of a nongelling steady state. We solve these integrodifferential equations for the size distribution by a numerical method and confirm the results by comparison with moment solutions. The results show how pregelation behavior is influenced by parameters in the aggregation and breakage kernels.

Modelling of flocculation using a population balance equation

In this paper, a model based on a population balance equation (PBE) is developed. It aims at reproducing experimental floc size distributions obtained at steady state in a jar-test. The objective is to develop a simple model, based on physical phenomena, and that does not contain any adjustable parameters. Floc size distributions obtained using a part of a particle image velocimetry (PIV) device and image analysis are used to develop mathematical expressions for the aggregation and breakage kernels. A critical volume beyond which breakage is of significant importance is identified and related to the hydrodynamics. Hydrodynamic sequencing allows the distribution of the daughter particles resulting from a breakage event to be established. The model is finally successfully validated against experimental results.

Computer simulation of diffusion-limited cluster-cluster aggregation with an Epstein drag force

The motion of particles, dispersed in a medium, between collisions with each other can, in limiting situations, be either ballistic (straight line) or diffusive (random walker). The diffusive regime can be divided into two distinct subregimes. The "continuum regime" exhibits Stokes-Einstein-type diffusion (no-slip surface boundary condition) with a frictional coefficient proportional to the particle size (linear dimension). The "Epstein regime," as we shall refer to it, is characterized by a frictional coefficient proportional to the particle cross-sectional area, hence an Epstein-type diffusion (slip surface). The purpose of the current study is to illuminate the dynamics of dilute-limit aggregation in the Epstein regime. We present results from low volume fraction Monte Carlo simulations of cluster-cluster aggregation in the Epstein regime with the particle motion based on each particle's cross-sectional area. Our findings indicate that aggregates grown under Epstein conditions have a fractal dimension of approximately 1.8, similar to that of diffusion-limited cluster-cluster aggregates (DLCA) in the continuum regime. The kinetic exponent z in the Epstein regime is found to be z approximately 0.8, lower than its value for both the continuum regime DLCA (z = 1) and for the ballistic cluster aggregation regime (z approximately 2). Cluster size distribution data for Epstein systems are found to scale at large cluster sizes with exponents consistent with the kinetic data. A scaling argument for predicting the kinetic exponent and kernel homogeneity based on the mass or size dependence of the particle velocity and collision cross section is presented and is seen to give accurate results for dilute and intermediate values of particle volume fractions not only for the current study, but also for work done by other researchers with various choices for the aggregation kernel.

Regression of Multicomponent Sticking Probabilities Using a Genetic Algorithm

A genetic algorithm (GA) was developed for the purpose of regressing composition-dependent aggregation kernels from time series of experimentally measured component or size distributions. The GA evolves initially random populations of kernel models in accordance with the principles of microevolution. To test the robustness of the GA, functionally diverse kernelssincluding one describing the shear-mediated aggregation of blood cellsswere constructed. The stochastic time evolution of their corresponding aggregation processes were then simulated under physiological conditions via Monte Carlo. The GA successfully regressed the kernels underlying these “gold standard” datasets swhere we employ the term in the sense of a “trusted reference” s from these simulation results, reproducing the multicomponent kernels to a maximum relative deviation of less than 9% over their entire composition ranges. Finally, ramifications of these cases pertinent to experimental design are considered, including the effects of extreme initial population ratios for multicomponent aggregation experiments with extreme population ratios.