Collision Efficiency Factor Research Articles

Describing the flocculation of PCC particles using population balance modelling approaches

Two different approaches for a two dimensional population balance modeling strategy are used and compared in this work to describe the flocculation of PCC particles induced by polyelectrolytes, the fixed pivot technique model and the model proposed by Hounslow (1988) and Spicer and Pratsinis (1996b). The median size diameter of the flocs, obtained by LDS, is compared with those obtained by the application of the two presented models. The models efficiency, when compared to the experimental data, is confirmed through a Goodness of Fit (GoF) above 98 % in all cases. Optimized fitting parameters are obtained for the collision efficiency factor, the fragmentation rate and the flocs restructuring parameter and related to the polyelectrolytes characteristics. The flocculation kinetics and scattering exponent data (related to the flocs structure) evolution with time, obtained from modelling, are discussed in relation with the flocculation process characteristics, including polyelectrolytes properties, and the different conditions considered.

The importance of chemical mechanisms in sonochemical modelling

A state-of-the-art chemical mechanism is introduced to properly describe chemical processes inside a harmonically excited spherical bubble placed in water and saturated with oxygen. The model uses up-to-date Arrhenius-constants, collision efficiency factors and takes into account the pressure-dependency of the reactions. Duplicated reactions are also applied, and the backward reactions rates are calculated via suitable thermodynamic equilibrium conditions. Our proposed reaction mechanism is compared to three other chemical models that are widely applied in sonochemistry and lack most of the aforementioned modelling issues. In the governing equations, only the reaction mechanisms are compared, all other parts of the models are identical. The chemical yields obtained by the different modelling techniques are taken at the maximum expansion of the bubble. A brief parameter study is made with different pressure amplitudes and driving frequencies at two equilibrium bubble sizes. The results show that due to the deficiencies of the former reaction mechanisms employed in the sonochemical literature, several orders of magnitude differences of the chemical yields can be observed. In addition, the trends along a control parameter can also have dissimilar characteristics that might lead to false optimal operating conditions. Consequently, an up-to-date and accurate chemical model is crucial to make qualitatively and quantitatively correct conclusions in sonochemistry.

Effective picture of bubble expansion

Recently the thermal friction on an expanding bubble from the cosmic first-order phase transition has been calculated to all orders of the interactions between the bubble wall and thermal plasma, leading to a γ2-scaling instead of the previously estimated γ1-scaling for the thermal friction exerted on a fast-moving bubble wall with a Lorentz factor γ. We propose for the first time the effective equation of motion (EOM) for an expanding bubble wall in the presence of an arbitrary γ-scaling friction to compute the efficiency factor from bubble collisions, which, in the case of γ2-scaling friction, is found to be larger than the recently updated estimation when the bubble walls collide after starting to approach a constant terminal velocity, leading to a slightly larger signal of the gravitational waves background from bubble collisions due to its quadratic dependence on the bubble collision efficiency factor, although the γ2-scaling friction itself has already suppressed the contribution from bubble collisions compared to that with γ1-scaling friction. We also suggest a phenomenological parameterization for the out-of-equilibrium term in the Boltzmann equation that could reproduce the recently found (γ2-1)-scaling of the friction term in the effective EOM of an expanding bubble wall, which merits further study in future numerical simulations of bubble expansion and collisions.

Demarcation of reaction effects on laminar burning velocities of diluted syngas–air mixtures at elevated temperatures

AbstractLaminar burning velocities of CO2/N2 (60%, 70%) diluted H2/CO/air mixtures were measured at elevated temperatures using the externally heated diverging channel method. The computed burning velocities using the two reaction mechanisms (Davis et al., Proc. Combust. Inst. 2005;30(1):1283–1292; FFCM‐1, http://nanoenergy.stanford.edu/ffcm1) are compared with the experimentally determined burning velocities. The accuracy of the chemical kinetic mechanisms at high dilution rates and elevated temperatures was investigated for various hydrogen fractions in the fuel composition. The burning velocity is observed to increase at high temperatures due to higher mixture enthalpy. The dilution effect on the variation of laminar burning velocity was stronger for the CO2 dilution case compared to N2 dilution. A comparison between the FFCM‐1 mechanism and experimental measurements shows an accurate depiction of the reaction chemistry regarding the prediction of laminar burning velocities. The role of third‐body reactions and direct inhibiting effect of N2 and CO2 molecules on burning velocity of diluted syngas–air mixtures is analyzed in detail. Detailed kinetic analysis revealed that the use of GRI 3.0 collision efficiency factors in the Davis mechanism helps in accurately predicting the burning velocities at elevated temperatures and high CO2 dilution rates. The thermal effect dominates the reduction in laminar burning velocity for N2 dilution case. The FFCM‐1 mechanism agrees well with the experiments for syngas flames diluted with N2 compared to the Davis mechanism. The addition of third‐body efficiency of N2 in the FFCM‐1 mechanism improved the predictions of laminar burning velocities for the N2 dilution case.

On the growth of pebble-accreting planetesimals

Pebble accretion is a new mechanism to quickly grow the cores of planets. In pebble accretion, gravity and gas drag conspire to yield large collisional cross sections for small particles in protoplanetary disks. However, before pebble accretion commences, aerodynamical deflection may act to prevent planetesimals from becoming large, because particles tend to follow gas streamlines. We derive the planetesimal radius where pebble accretion is initiated and determine the growth timescales of planetesimals by sweepup of small particles. We obtain the collision efficiency factor as the ratio of the numerically-obtained collisional cross section to the planetesimal surface area, from which we obtain the growth timescales. Integrations are conducted in the potential flow limit (steady, inviscid) and in the Stokes flow regime (steady, viscid). Only particles of stopping time $t_s \ll t_X$ where $t_X\approx10^3$ s experience aerodynamic deflection. Even in that case, the planetesimal's gravity always ensures positive collision factors. The maximum growth timescale occurs typically at around $R\approx100 \ \mathrm{km}$, but is less for colder disks, corresponding to interactions shifting to the Safronov focusing regime. For particles $t_s \gg t_X$ pebble accretion commences only after this phase and is characterized by a steep drop in growth timescales. Consequently, at distances beyond ~10 AU sweepup growth timescales are always longer than $10$ Myr, while in the inner disk (~<3 AU) the viability of the sweepup scenario is determined by the outcome of pebble-planetesimal collisions in the geometric regime. We present analytical fits for the collision efficiency factors and the minimum planetesimal size needed for pebble accretion. (Abridged)

Computational fluid dynamic modeling of the flame spray pyrolysis process for silica nanopowder synthesis

A computational fluid dynamic model that couples the fluid dynamics with various processes involving precursor droplets and product particles during the flame spray pyrolysis (FSP) synthesis of silica nanopowder from volatile precursors is presented. The synthesis of silica nanopowder from tetraethylorthosilicate and tetramethylorthosilicate in bench- and pilot-scale FSP reactors, with the ultimate purpose of industrial-scale production, was simulated. The transport and evaporation of liquid droplets are simulated from the Lagrangian viewpoint. The quadrature method of moments is used to solve the population balance equation for particles undergoing homogeneous nucleation and Brownian collision. The nucleation rate is computed based on the rates of thermal decomposition and oxidation of the precursor with no adjustable parameters. The computed results show that the model is capable of reproducing the magnitude as well as the variations of the average particle diameter with different experimental conditions using a single value of the collision efficiency factor α for a given reactor size.

Kinetics Model of Destabilization of Oil Droplets in Oily Wastewater Emulsions

In this article, a mathematical model that accounts for the kinetics of flocculation and coalescence of emulsion drops was developed to quantify the kinetics of emulsion stability. The model is similar to that developed by Borwankar et al. in 1992 with corrections in coalescence rate constant and collision efficiency factor. It includes a term to describe efficiency of collisions. Also, the coalescence rate constant was defined as a function of the total number of particles that were present in the system at the moment. The model has four adjustable parameters, and nonlinear least square optimization is used to obtain the adjustable parameters. The obtained adjustable parameters by the model served as parameters that can be used to quantitatively evaluate the effectiveness of a set of different demulsifiers by measuring total numbers of drops versus time and by determining the adjustable parameters and comparing them. The results of the presented model and the model of Borwankar et al. were compared with experimental data. The results of the new model were in good agreement with experimental data and showed the model's ability for dispalying the behavior of droplets during the destabilization processes of emulsions.

Analysis of the fate and transport of nC60 nanoparticles in the subsurface using response surface methodology

Predicting the distribution of engineered nanomaterials (ENMs) in the environment will provide critical information for risk assessment and policy development to regulate these emerging contaminants. The fate and transport of ENMs in natural subsurface environments are complicated by various factors, such as hydraulic gradient, initial release concentration, nanoparticle size, and collision efficiency factor. Based on advanced statistical methodologies (i.e., response surface methodology (RSM)), we explore simple relationships between key factors that control ENM transport (collision efficiency factor, particle size, hydraulic gradient, and initial release concentration) and key parameters that describe the ENM concentration distribution in porous media (maximum standardized concentration, the mass percentage of injected nanoparticle attached in the aquifer, the x-centroid of aqueous phase nC₆₀ plume, and the x-centroid of attached phase nC₆₀ distribution). Hypothetical scenarios for the release of nanoparticles into an aquifer were simulated numerically with randomly generated permeability fields that were based on mildly and highly heterogeneous sites. RSM was used to develop polynomial regression equations based on a statistical experimental design. High R-squared values (greater than 0.9) of the regression equations were obtained for all the models developed based on the mildly heterogeneous site. On the highly heterogeneous site, the R-squared value of the regression equation for the percentage of nanoparticles attached (by mass) was more than 0.9. The ability to accurately estimate aqueous phase ENM concentration distribution using simple regression equations is particularly critical for risk assessment. Even though the correlations developed in this study were site and scenario specific, this work represents a first effort of applying RSM for predicting the distribution of engineered nanomaterials in porous media.

Modeling the transport and retention of nC60 nanoparticles in the subsurface under different release scenarios

The escalating production and consumption of engineered nanomaterials may lead to their increased release into groundwater. A number of studies have revealed the potential human health effects and aquatic toxicity of nanomaterials. Understanding the fate and transport of engineered nanomaterials is very important for evaluating their potential risks to human and ecological health. While there has been a great deal of research effort focused on the potential risks of nanomaterials, a limited amount of work has evaluated the transport of engineered nanomaterials under different release scenarios in a typical layered geological field setting. In this work, we simulated the transport of fullerene aggregates (nC60), a widely used engineered nanomaterial, in a multi-dimensional environment. A Modular Three-Dimensional Multispecies Transport Model (MT3DMS) was modified to evaluate the transport and retention of nC60 nanoparticles. Hypothetical scenarios for the introduction of nanomaterials into the subsurface environment were investigated, including the release from an injection well and the release from a waste site. Under the conditions evaluated, the mobility of nC60 nanoparticles was found to be very sensitive to the release scenario, release concentration, aggregate size, collision efficiency factor, and dispersivity of the nanomaterial.

Modelling PCC flocculation by bridging mechanism using population balances: Effect of polymer characteristics on flocculation

A population balance model for flocculation of PCC particles with polyelectrolytes of very high molecular weight, medium charge density and different degree of branching is presented. The model considers simultaneously aggregation, breakage and flocs restructuring to describe the PCC flocculation by bridging mechanism. The maximum collision efficiency factor, a parameter related with the fragmentation rate and a time constant for flocs restructuring have been taken as fitting parameters. These fitting parameters are optimized to get the best fit between experimental data obtained by LDS in a previous study and the modelled results. The optimized parameters were correlated with flocculant concentration, flocs structure and polymer branching. The correlations obtained show well the effects of flocculant concentration, flocs structure and polymer structure on the flocculation kinetics and flocs restructuring which are translated in the model parameters. Moreover, the flocs break up due to polymer degradation was introduced in the model by decreasing, with time, the maximum collision efficiency factor. It was shown that this effect can be neglected since the improvement in the results is too small relatively to the high increase of the computational time required to perform the simulation.

Investigation of the Transport and Deposition of Fullerene (C60) Nanoparticles in Quartz Sands under Varying Flow Conditions

A coupled experimental and mathematical modeling investigation was undertaken to explore nanoscale fullerene aggregate (nC60) transport and deposition in water-saturated porous media. Column experiments were conducted with four different size fractions of Ottawa sand at two pore-water velocities. A mathematical model that incorporates nonequilibrium attachment kinetics and a maximum retention capacity was used to simulate experimental nC60 effluent breakthrough curves and deposition profiles. Fitted maximum retention capacities (S(max)), which ranged from 0.44 to 13.99 microg/g, are found to be correlated to normalized mass flux. The developed correlation provides a means to estimate S(max) as a function of flow velocity, nanoparticle size, and mean grain size of the porous medium. Collision efficiency factors, estimated from fitted attachment rate coefficients, are relatively constant (approximately 0.14) over the range of conditions considered. These fitted values, however, are more than 1 order of magnitude larger than the theoretical collision efficiency factor computed from Derjaguin-Landau-Verwey-Overbeek (DLVO) theory (0.009). Data analyses suggest that neither physical straining nor attraction to the secondary minimum is responsible for this discrepancy. Patch-wise surface charge heterogeneity on the sand grains is shown to be the likely contributor to the observed deviations from classical DLVO theory. These findings indicate that modifications to clean-bed filtration theory and consideration of surface heterogeneity are necessary to accurately predict nC60 transport behavior in saturated porous media.

Collision efficiency factor for heteroaggregation: Extension to soft interactions

An improved model for the collision efficiency factor of clusters of oppositely charged spheres has been developed, which accounts for repulsive and attractive interactions that occur at a finite distance from the colliding species, i.e., the so-called "soft" interactions. Trends in measured optimum dosages for rapid aggregation with increasing Debye length (a decrease at particle size ratios between 0.3 and 1 and an increase at particle size ratios less than 0.3) are explained qualitatively by employing the modified collision efficiency model. Several observations from the literature, specifically the formation of stringlike aggregates at low ionic strength and the uneven optimum dosage requirements of particles of equal size, are also explained in view of the model presented.

An improved collision efficiency model for particle aggregation

A generalized geometric model is presented which describes the collision efficiency factor of aggregation (the probability of a binary particle or aggregate collision resulting in adhesion) for systems comprised of two oppositely charged species. Application of the general model to specific systems requires calculation of the area of each species available for collision with a second species. This is in contrast to previous models developed for polymer-particle flocculation that are based on the fractional surface coverage of adsorbed polymer. The difference between these approaches is suggested as an explanation for previously observed discrepancies between theory and observation. In the current work the specific case of oppositely charged nondeformable spherical particles (heteroaggregation) is quantitatively addressed. The optimum concentration of oppositely charged particles for rapid aggregation (maximum collision efficiency) as a function of relative particle size is calculated and an excellent correlation is found with data taken from literature.

A population balance model for flocculation of colloidal suspensions by polymer bridging

A detailed population balance model for flocculation of colloidal suspensions by polymer bridging under quiescent flow conditions is presented. The collision efficiency factor is estimated as a function of interaction forces between polymer coated particles. The total interaction energy is computed as a sum of van der Waals attraction, electrical double layer repulsion and bridging attraction or steric repulsion due to adsorbed polymer. The scaling theory is used to compute the forces due to adsorbed polymer and the van der Waals attraction is modified to account for presence of polymer layer around a particle. The irregular structure of flocs is taken into account by incorporating the mass fractal dimension of flocs. When tested with experimental floc size distribution data published in the literature, the model predicts the experimental behavior adequately. This is the first attempt towards incorporating theories of polymer-induced surface forces into a flocculation model, and as such the model presented here is more general than those proposed previously.

Reaction‐limited aggregation in presence of short‐range structural forces

AbstractA geometrically discretized sectional population balance model for reaction‐limited aggregation of colloidal suspensions is presented. The two important model parameters are collision frequency factor and collision efficiency factor. The collision frequency factor is derived from physically realistic arguments proposed for collision of fractal aggregates. The collision efficiency factor is computed as a function of total interaction energy between particles, including short‐range structural repulsion forces. The irregular and open structure of aggregates is taken into account by incorporating their mass fractal dimension. The characteristic time constant of reaction‐limited aggregation, derived from dynamic scaling of mean aggregate size‐aggregation time data, is found to correlate with electrolyte concentration. The population balance model is tested with published experimental data for aggregation of γ‐alumina suspensions in the presence of different electrolytes. It is shown that the slow kinetics of aggregation under certain conditions of pH and electrolyte concentration require inclusion of short‐range structural repulsion forces along with van der Waals attraction and electrical double layer repulsion forces in an extended DLVO theory. The model predictions are in good agreement with experimental data for time evolution of mean aggregate diameter in the reaction‐limited aggregation regime. © 2005 American Institute of Chemical Engineers AIChE J, 2005

Modeling flocculation of colloidal mineral suspensions using population balances

Flocculation as well as dispersion of mineral suspensions has become increasingly important in mineral processing as we have to deal with fine and ultrafine particles. Population balance models for flocculation of such suspensions are discussed in this paper. Models ranging from those based on pure aggregation and fragmentation to those based on simultaneous aggregation–fragmentation are presented in detail. The two important parameters in the population balance model are the collision frequency factor and the collision efficiency factor. Differences among collision frequency models for impermeable and permeable aggregates are described. The main drawback of existing flocculation models is the disregard of forces between mineral particle surfaces. The collision efficiency is usually assumed to be unity or employed as a fitting parameter in a majority of the models. An approach for incorporating the influence of surface forces in the presence of salts and polymers is outlined. Another important lacuna of flocculation models is the assumption of homogeneous flow structure or uniform fluid velocity gradient in stirred flocculation tanks. Short-cut procedures adopted in the literature to address heterogeneous fluid flow structure are summarized. It is also stressed that an integrated approach be followed to tackle practically relevant issues such as selective flocculation of mineral mixtures, dual polymer flocculation, dynamics of polymer/surfactant adsorption, polymer–surfactant interactions, surface precipitation of dissolved species and presence of air bubbles in the suspension.

Collision efficiency factor in Brownian coagulation (αBr): calculation and experimental verification

The collision efficiency factor in Brownian motion (αBr) has been calculated by Han et al. (J. Water Sci. Technol. 36 (1997) 69) including both hydrodynamic and interparticle forces. The sensitivity to each contributing parameter was investigated and a coagulation zone diagram in a graphical plane of surface potential and ionic strength was developed for a particle diameter of 2 μm. The current paper describes a set of experiments performed in a well-controlled laboratory to compare experimental and predicted regions in the Brownian coagulation zone diagram. A polystyrene Latex suspension (nominal diameter 3.12 μm) was used and the change of particle size distribution was measured over a wide range of surface potential (pH) and ionic strength. The total number of particles and the shift of the centroids of the particle size distribution curves were compared to quantify the degree of coagulation with time. An abrupt change of coagulation status was observed at a critical ionic strength. The critical ionic strength increased as the surface potential increased. The experimental results compared well with the theory.

Modeling of DAF: the effect of particle and bubble characteristics

The characteristics of bubbles and particles in dissolved air flotation (DAF) will govern the collision efficiency between bubbles and particles and thereafter the removal efficiency of a suspension. In this paper, the collision efficiency factor for bubbles and particles ( α bp ) is calculated from a trajectory analysis by a method similar to that used for the calculation of the collision efficiency factor in differential sedimentation ( α ds ) (Han & Lawler 1991). The effects of the contributing parameters in the bubble-particle-solution system are investigated. The most important parameters in the DAF system are found to be the surface characteristics (zeta potential) of both bubbles and particles. The next most important parameters are the sizes of the bubbles and the particles. The density of the particles has a positive or negative effect on α bp depending on the sizes of particles and bubbles. The ionic strength of the solution affects the α bp slightly. The theoretical results can explain some of the current practices in the design and operation of the DAF process.

Modeling coagulation kinetics incorporating fractal theories: comparison with observed data

There are currently four possible approaches in modeling coagulation kinetics: the traditional Euclidean rectilinear; the Euclidean curvilinear; the fractal rectilinear; and the fractal curvilinear. The fractal model includes the Euclidean case as a subset. The primary purpose of this research is to investigate which of the rectilinear models among these best predicts the evolution of experimental observed particle size distribution (PSD). Using a fractal rectilinear model previously developed by the authors, model predictions were compared with a series of observed PSD data obtained from estuarine sediment particles in a 2 m settling column, where the average velocity gradient ( G) was 20 or 40 s −1. Nonlinear parameter estimation was performed to estimate two free parameters for the fractal model (the fractal dimension, D F, and the collision efficiency factor, α), and one free parameter (the collision efficiency factor, α) for the Euclidean model. Compared with the observed PSD, the simulation showed that the fractal rectilinear model was best, and that this model fit better for the larger size particles. The estimated D F was between 2.6 and 3.0. The research demonstrated that the α's have multiple values for the same observed data, depending on the coagulation model used. This finding is significant because α is currently used as a single value based on the conventional Euclidean rectilinear model.

Sorption irreversibility and coagulation behavior of 234Th with marine organic matter

Abstract The partitioning of 234 Th to natural organic matter (NOM) in the colloidal size range (1 kDa–0.1 μm) was evaluated in order to examine the sorption and coagulation behavior of marine colloidal organic matter. Colloids were isolated using large volume cross-flow ultrafiltration and the partitioning of 234 Th was quantified using stirred cell ultrafiltration and radioactive assay. The uptake of 234 Th by NOM is irreversible over a period of 5 days, implying that over the mean life of 234 Th, very little release of 234 Th would occur after binding to NOM. Furthermore, the Th–NOM complex is stronger than the Th–EDTA complex, as EDTA was unable to displace the 234 Th from its association with NOM. The extent of the initial partitioning of 234 Th to suspended matter and colloids is similar and independent of pH in the range from 3 to 9. Coagulation experiments show that 234 Th complexed with low molecular weight (1–10 kDa) colloids is transferred to larger (>0.1 μm) filter retained fractions. However, 234 Th is transferred to a greater extent than is organic matter and this results in greater partitioning coefficients for 234 Th onto particle phases with time. The final equilibrium between 0.1 μm filter-retained and filter-passing 234 Th activity is the same regardless of whether the Th was tagged initially to colloidal or suspended matter fractions. The coagulation of colloidal organic matter, COM, consists of both fast and slow steps, with kinetic rate constants on the order of 0.02–0.03 and 0.003–0.007 h −1 , respectively. The stickiness (or collision efficiency) factor, α , for COM was experimentally determined to be 0.7(−) for seawater conditions. Using the colloidal pumping model of Honeyman and Santschi [J. Mar. Res. 47 (1989): 951], the ‘predicted’ “fast-phase” coagulation rate coefficient is 0.03 h −1 in our coagulation experiments when the measured α value and the experimental conditions are used for model inputs. These experiments demonstrate that coagulation is the dominant step in the transport of 234 Th to the particulate phase.