Collision Frequency Function Research Articles

Realization of Self-Preserving Size Distribution Theory for the Evolution of Tropospheric Atmospheric Aerosols Through an Inverse Gaussian Distribution

AbstractThe inverse Gaussian distributed method of moments (IGDMOM; J. Atmospheric Sci. 77 (9): 3011-3031, 2020) was developed to analytically solve the kinetic collection equation (KCE) for the first time. Using the IGDMOM, we obtained both new analytical and asymptotic solutions to the KCE. This is shown for both the free molecular and continuum regime collision frequency functions. The new analytical solutions are highly suitable for demonstrating the self-preserving size distribution (SPSD) theory. The SPSD theory is considered one of the most elegant research works in atmospheric science for aerosols or small cloud droplets. It was initially discovered by Friedlander (J. Meteorology 17 (5): 479-483, 1960) and then developed by Lee (J. Colloid Interface Sci. 92 (2): 315-325, 1983) with an assumption of the time-dependent lognormal size distribution function. In this study, we demonstrate that the SPSD theory of coagulating atmospheric aerosols can be presented in a simpler and more rigorous theoretical way, which is realized through the introduction of the IGDMOM for describing aerosol size distributions. Using the IGDMOM, the new formulas for the SPSD, as well as the time required for aerosols to reach the SPSD, are analytically provided and verified. Furthermore, we discover that the SPSD of atmospheric aerosols undergoing coagulation is only determined using a shape factor variable, 𝛺, which is composed of the first three moments at an initial stage. This study has critical implications for developing tropospheric atmospheric aerosol or small cloud droplet dynamics models and further verifies the SPSD theory from the viewpoint of theoretical analysis.

Self-preserving size distribution and collision frequency of flame-made nanoparticles in the transition regime

Coagulation of polydisperse primary particles is the dominant agglomerate growth mechanism in flame reactors and in the low-temperature or high-pressure regions of combustion engines. Here, the evolution of agglomerate size distribution (SD) and collision frequency function of coagulating soot and silica primary particles is investigated with simultaneous surface growth or coalescence using discrete element modeling (DEM). The growth of polydisperse soot and silica nanoparticles is captured from the free molecular up to the transition regime in excellent agreement with microscopy and mobility measurements in diffusion flames and flame spray pyrolysis (FSP) reactors. The broad agglomerate size distributions of soot and silica formed in the free molecular regime narrow by coagulation in the transition regime, attaining their mobility-based quasi-self-preserving or signature SD with the same geometric standard deviation of 1.48 ± 0.03, in excellent agreement with soot or silica SDs measured from diffusion flames and the CAST generator or from a FSP reactor, respectively. The coagulation rate of both soot and silica agglomerates is about twice as much higher as that of monodisperse agglomerates in the transition regime, regardless of material or combustion conditions. The DEM-derived collision kernel enhanced by a factor of 1.82 ± 0.35 can be interfaced with method of moments and monodisperse population balance models to improve the design of aerosol flame reactors and combustion engines.

Research on the coagulation law of dielectric nanoparticles in Stokes regime via Taylor expansion moment method

For the collision and coagulation of dielectric nanoparticles randomly oriented and moving in an electric field, the collision frequency function of the particles in Stokes regime is derived. The numerical solution obtained by the Taylor expansion moment method in the Stokes regime coagulation problem is very consistent with the numerical solution obtained by the previous study of the TEMOM model. In this paper, the first-order ordinary differential equations are constructed to obtain closed-form expressions. The fourth-order Runge-Kutta algorithm is used to derive the particle concentration of the dielectric nanoparticles under different electric field strengths and different polarities. The law of variation of the polydispersity of the system during coagulation and rupture. These expressions are used in simple overall balance expressions to illustrate the importance of their range of physical parameter values encountered in real systems. The results show that this method can be used to solve the accuracy of the general kinetic equation in the process of particles coagulation under electric field, and the calculation cost is low.

Modeling Collisional Transitions in Thermal Unimolecular Reactions: Successive Trajectories and Two-Dimensional Master Equation for Trifluoromethane Decomposition in an Argon Bath.

Collisional transition processes in thermal unimolecular reactions are modeled by collision frequency, Z, and probability distribution function, P(E, J; E', J'), which describes the probabilities of collisional transitions from the initial state specified by the total energy and angular momentum, (E', J'), to the final states, (E, J). The validity of the collisional transition model, consisting of Z and P(E, J; E', J'), is assessed here for the title reaction. The present model and its parameters are derived from the moments of transition probabilities calculated by classical trajectory simulations. The model explicitly accounts for coupling between the energy and angular momentum transfer and the dependence of transition probability on the initial state. The performance of the model is evaluated by comparing the rate constants calculated by solving the two-dimensional master equation with those obtained from the classical trajectory calculations of the sequence of successive collisions. The rate constants are also compared with available experimental data. The present collisional transition model is found to perform fairly well for predicting the pressure-dependent rate constants. The uncertainty in the prediction and sensitivities of the rate constants to the model parameters are discussed. A simplified version of the model is proposed, which performs as well as the full model. The simplifications and robust procedures for calculating the model parameters are described.

Effect of two temperature electrons in a collisional magnetized plasma sheath

A collisional magnetized plasma consisting of two temperature electrons has been investigated numerically to study the sheath structure and the ion flux to the wall. The low-temperature electrons are described by Maxwellian distribution, and the high-temperature electrons are described by truncated Maxwellian distribution. It has been observed that sheath width and ion flux to the wall are non monotonous functions of collision frequency as well as hot electron concentration. A sheath formation criterion has been analytically derived and is used to find the ion velocity at the sheath edge. The sheath edge velocity is found to decrease with the increase of hot electron concentration and hot electron temperature. The study of such a system can help in understanding of plasma surface interaction.

Numerical study on granule aggregation and breakage in fluidized bed granulation by a novel PBM with DEM-CFD coupling approach

A novel population balance model (PBM) in which particle collision frequency function was determined through the coupling of the discrete element method with computational fluid dynamics was proposed to investigate the mechanisms of granule growth and breakage in fluidized bed granulation. The collision analysis revealed that the collision frequency function of one particle C(t) can be derived using the Froude number. In addition, the results indicated that the two-interparticle collision frequency function Ci, j(t) is equal to the one-particle collision frequency function C(t). The particle size distribution calculated using the PBM agrees extremely well with the actual values obtained through experiments at an actual production scale. This PBM can also predict the particle size distribution under various operating conditions.

Population balance modeling: application in nanoparticle formation through rapid expansion of supercritical solution

The rapid expansion of supercritical solution (RESS) process is a novel method to produce free-solvent particles with narrow particle size distribution (PSD), and it is offered to replace conventional particle formation methods. The present study is aimed to investigate the numerical simulation of the RESS process comprising hydrodynamic and particle formation and solving population balance equation to compute PSD of solute-supercritical CO2 (SC-CO2) systems. Mass, momentum and energy balances in addition to an accurate equation of state are used to calculate hydrodynamic behavior of SC-CO2 at various conditions through a nozzle and supersonic free jet with considering heat exchange in the nozzle. The solubility of TBTPP (5, 10, 15, 20-tetrakis (3, 5-bis (trifluoromethyl) phenyl) porphyrin) in SC-CO2 is calculated using the Soave–Redlich–Kwong equation of state with van der Waals mixing rule. After that, supersaturation and nucleation of TBTPP and three other drugs, i.e., aspirin, salicylic acid and ibuprofen, are investigated. Eventually, the method of moments is used to solve the population balance equations to determine PSD of the solutes. The results are presented with and without coagulation. Furthermore, the effect of nozzle length and pre-expansion temperature on PSD is studied. Furthermore, to improve the results of coagulation model on PSD, collision frequency and lognormal function are modified. The modifications improve the coagulation results. The average absolute percent deviation of TBTPP solubility is 1.86. The CO2 hydrodynamic behavior shows the same trend as reported in the literature. Moreover, results of PSD calculation with coagulation indicated there is a good agreement with the experimental ones.

Numerical simulation of hydrate slurry flow behavior in oil-water systems based on hydrate agglomeration modelling

Hydrate agglomeration in oil-water systems could lead to pipeline hydrate plugging. Dynamic modeling of pipeline hydrate particle agglomeration is of great importance to hydrate management. For this purpose, a dynamic model of hydrate agglomeration was established based on population balance equation, which took both hydrate agglomeration and hydrate breakage into consideration. The calculation methods of collision frequency, agglomeration efficiency, breakage frequency and particle size distribution function of the sub particles resulting from particle breakage were given and fully discussed in the dynamic model. Then, combined with several traditional solid-liquid flow models, the dynamic model was solved by the CFD software FLUENT 14.5 to simulate the influences of flow rate and hydrate volume fraction on pipeline hydrate slurry flow behavior in oil-water systems. The simulation results show that hydrate particle size distribution in pipeline flowing systems can be well described by log-normal distribution. Hydrate particle diameters and hydrate volume concentrations in the near wall region are larger than that in the center region of the pipeline. The proportion of large hydrate particles and the mean hydrate particle diameter in pipeline increase with the increase of flow rate and hydrate volume fraction. Hydrate particle diameter and pressure gradient in oil-water systems are larger than that in water systems.

Semi-empirical model for indirect measurement of soot size distributions in compression ignition engines

This work proposes a semi-empirical model, which provides soot particle size distribution functions emitted by compression ignition engines. The model is composed of a phenomenological model based on the collision dynamics of particle agglomerates and an empirical model, which provides key input parameters such as primary particle size and a mathematical relationship between the size of the agglomerate and number of primary particles. The phenomenological model considers the relevant fluid-dynamics phenomena influencing the collision frequency function. It is observed that Brownian motion is the predominant phenomenon and in a much lesser degree inertial turbulent motion. The experimental model requires air/fuel ratio, engine speed, soot density and mean instantaneous in-cylinder pressure. A Dirac delta is used as a seed for the agglomerate size function whose magnitude depends on the soot volume concentration and the mean primary particle size at each engine operation condition. In a further step, the obtained modelled agglomerate size functions are fitted to lognormal size distributions defined by the modelled mean size and standard deviation. Modelled lognormal agglomerate size distribution functions are validated with respect to experimental distributions obtained using a Scanning Mobility Particle Sizer (SMPS).

Investigating the settling dynamics of cohesive silt particles with particle-resolving simulations

The settling of cohesive sediment is ubiquitous in aquatic environments, and the study of the settling process is important for both engineering and environmental reasons. In the settling process, the silt particles show behaviors that are different from non-cohesive particles due to the influence of inter-particle cohesive force. For instance, the flocs formed in the settling process of cohesive silt can loosen the packing, and thus the structural densities of cohesive silt beds are much smaller than that of non-cohesive sand beds. While there is a consensus that cohesive behaviors depend on the characteristics of sediment particles (e.g., Bond number, particle size distribution), little is known about the exact influence of these characteristics on the cohesive behaviors. In addition, since the cohesive behaviors of the silt are caused by the inter-particle cohesive forces, the motions of and the contacts among silt particles should be resolved to study these cohesive behaviors in the settling process. However, studies of the cohesive behaviors of silt particles in the settling process based on particle-resolving approach are still lacking. In the present work, three-dimensional settling process is investigated numerically by using CFD–DEM (Computational Fluid Dynamics–Discrete Element Method). The inter-particle collision force, the van der Waals force, and the fluid–particle interaction forces are considered. The numerical model is used to simulate the hindered settling process of silt based on the experimental setup in the literature. The results obtained in the simulations, including the structural densities of the beds, the characteristic lines, and the particle terminal velocity, are in good agreement with the experimental observations in the literature. To the authors’ knowledge, this is the first time that the influences of non-dimensional Bond number and particle polydispersity on the structural densities of silt beds have been investigated separately. The results demonstrate that the cohesive behavior of silt in the settling process is attributed to both the cohesion among silt particles themselves and the particle polydispersity. To guide to the macro-scale modeling of cohesive silt sedimentation, the collision frequency functions obtained in the numerical simulations are also presented based on the micromechanics of particles. The results obtained by using CFD–DEM indicate that the binary collision theory over-estimated the particle collision frequency in the flocculation process at high solid volume fraction.

Population Balance Equation Modeling of Crude Oil Demulsification Considering Demulsifier: Modification of Collision Frequency Function Based on Thermodynamic Model

A theoretical model is developed to consider the effect of demulsifier agent on demulsification of water-in-crude oil emulsion. A thermodynamic approach is considered to correlate the critical micelle concentration of a demulsifier to the collision frequency function in population balance equation (PBE). Based on the proposed correlation, the collision frequency function is modified to account the effect of demulsifier agent on water droplets coalescence. The water separation capability of a selected group of demulsifiers is investigated via the conventional bottle test method to obtain the adjustable parameter of the developed equation so that the absolute relative error of predicted demulsification efficiency and experimental data are considered as objective. Comparison between the model simulated results and the experimental data showed that the proposed model could appropriately reproduce the experimental data at different agent concentrations.

Temporal characteristics of electrostatic surface waves in a cold complex plasma containing collision-dominated ion flow

The influence of electron-ion collision frequency and dust charge on the growth rate of two-stream instability of the electrostatic surface wave propagating at the interface of semi-infinite complex plasma whose constituents are electrons, negatively charged dust, and streaming ions. It is found that the surface wave can be unstable if the multiplication of wave number and ion flow velocity is greater than the total plasma frequency of electrons and dusts. The analytical solution of the growth rate is derived as a function of collision frequency, dust charge, and ion-to-electron density ratio. It is found that the growth rate is inversely proportional to the collision rate, but it is enhanced as the number of electrons residing on the dust grain surface is increased. The growth rate of surface wave is compared to that of the bulk wave.

Collision frequency and radial distribution function in particle-laden turbulent channel flow

We performed Eulerian–Lagrangian direct numerical simulation of particle-laden channel flow at a frictional Reynolds number of 950. A fully parallelized deterministic particle collision algorithm is applied for elastic collisions between two particles and particles and the walls. A total number of 51 million mono-disperse particles is considered, resulting in a particle volume fraction close to 1×10−4. We studied the results of the simulation after a statistically steady turbulent state and particle concentration were reached. In this state the particles close to the walls are preferentially located in the low-speed streaks, whereas the particle distribution in the center of the channel is rather non-uniform as well, showing large void regions. The presence of the particles results in a decrease of the turbulence dissipation rate of 20% close to the walls. We studied in particular the particle collision frequency as a function of the wall-normal coordinate and compared the simulation results with two theoretical expressions involving the radial distribution function at contact and the mean relative velocity of two colliding particles. It appeared that the radial distribution function diverges if the distance between the particles approaches the particle diameter. This is not only caused by the drift mechanisms but also by repeated collision events, which occur relatively more often in the center of the channel. We proposed a collision criterion that distinguishes repeated collisions from single collisions, and which is easy to apply during a simulation and in the computation of the radial distribution function and the mean relative velocity of two colliding particles. Simulation results and the two theoretical expressions for the collision frequency are in good agreement if this criterion is applied.

Coagulation of Agglomerates Consisting of Polydisperse Primary Particles.

The ballistic agglomeration of polydisperse particles is investigated by an event-driven (ED) method and compared to the coagulation of spherical particles and agglomerates consisting of monodisperse primary particles (PPs). It is shown for the first time to our knowledge that increasing the width or polydispersity of the PP size distribution initially accelerates the coagulation rate of their agglomerates but delays the attainment of their asymptotic fractal-like structure and self-preserving size distribution (SPSD) without altering them, provided that sufficiently large numbers of PPs are employed. For example, the standard asymptotic mass fractal dimension, Df, of 1.91 is attained when clusters are formed containing, on average, about 15 monodisperse PPs, consistent with fractal theory and the literature. In contrast, when polydisperse PPs with a geometric standard deviation of 3 are employed, about 500 PPs are needed to attain that Df. Even though the same asymptotic Df and mass-mobility exponent, Dfm, are attained regardless of PP polydispersity, the asymptotic prefactors or lacunarities of Df and Dfm increase with PP polydispersity. For monodisperse PPs, the average agglomerate radius of gyration, rg, becomes larger than the mobility radius, rm, when agglomerates consist of more than 15 PPs. Increasing PP polydispersity increases that number of PPs similarly to the above for the attainment of the asymptotic Df or Dfm. The agglomeration kinetics are quantified by the overall collision frequency function. When the SPSD is attained, the collision frequency is independent of PP polydispersity. Accounting for the SPSD polydispersity in the overall agglomerate collision frequency is in good agreement with that frequency from detailed ED simulations once the SPSD is reached. Most importantly, the coagulation of agglomerates is described well by a monodisperse model for agglomerate and PP sizes, whereas the detailed agglomerate size distribution can be obtained by scaling the average agglomerate size to the SPSD.

Influence of electron-neutral collisions on the Compton scattering cross section and the Salpeter structure factor in warm collisional plasmas

The electron-neutral collision effects on the Compton scattering process are investigated in warm collisional plasmas. The Compton scattering cross section in warm collisional plasmas is obtained by the Salpeter structure factor with the fluctuation-dissipation theorem and the plasma dielectric function as a function of the electron-neutral collision frequency, Debye length, and wave number. It is shown that the influence of electron-neutral collision strongly suppresses the Compton scattering cross section in warm collisional plasmas. It is also found that the electron-neutral collision effect on the differential Compton scattering cross section is more significant in forward scattering directions. We show that the differential Compton scattering cross section has a maximum at the scattering angle φ=π/2. In addition, we find that the electron-neutral collision effect on the total Compton scattering cross section increases with increasing Debye length and wave number. The variation of the Compton scattering cross section due to the change of collision frequency and plasma parameters is also discussed.

Effect of the radio frequency discharge on the dust charging process in a weakly collisional and fully ionized plasma

A close analysis of dust charging process in the presence of radio frequency (RF) discharge on low pressure and fully ionized plasma for both weak and strong discharge's electric field is considered. When the electromagnetic waves pass throughout fully ionized plasma, the collision frequency of the plasma is derived. Moreover, the disturbed distribution function of plasma particles in the presence of the RF discharge is obtained. In this article, by using the Krook model, we separate the distribution function in two parts, the Maxwellian part and the perturbed part. The perturbed part of distribution can make an extra current, so-called the accretion rate of electron (or ion) current, towards a dust particle as a function of the average electron-ion collision frequency. It is proven that when the potential of dust grains increases, the accretion rate of electron current experiences an exponential reduction. Furthermore, the accretion rate of electron current for a strong electric field is relatively smaller than that for a weak electric field. The reasons are elaborated.

Modeling of silicon nanoparticle formation in inductively coupled plasma using a modified collision frequency function

A model is presented to describe particle growth in inductively coupled plasma. The model consists of plasma chemistry and a coagulation module that adopts a modified collision frequency function. The modified collision frequency function is modified by a collision correlation factor that reflects the repulsive force of the particle charge in plasma in order to describe the reduction of coagulation among medium size particles (around 100 nm). In this model, plasma state and concentration of nuclei are determined by a spatially averaged global model in the plasma chemistry module. Particle growth is calculated by a coagulation module. To verify the validity of the model, comparison analysis is performed between experimental data obtained with PBMS and models, some of which are modified by a collision correlation factor. The analysis is performed with respect to dependencies on synthesis time, plasma source power and chamber pressure. From the analysis, we confirm the validity of the model that adopts a modified collision frequency function for the plasma condition.

Multi-scale modeling of granulation processes: Bi-directional coupling of PBM with DEM via collision frequencies

Wet granulation is a complex particle design process often operated inefficiently in industrial applications. Enhanced process understanding is required to facilitate design, control, and optimization. In this study, a hybrid multi-scale model is presented using a bi-directional coupling approach between DEM and PBM. The hybrid model takes into account particle collision frequencies and liquid distribution, providing a framework suitable for the complex sub-processes in wet granulation. The effect of particle size distribution on the collision frequency function was demonstrated, indicating the need for a multi-scale model. Results of the hybrid model show an increase in particle size over time from an average diameter of 0.98mm to 2.5mm, which qualitatively agrees with experimental trends observed during the liquid addition and wet massing stages. Two-dimensional distributions in particle size and liquid fraction are also presented incorporating the key effect of liquid distribution on the evolution of granule PSD.

Calculation of collision frequency function for aerosol particles in free molecule regime in presence of force fields

The collision frequency function for aerosol particles has already been calculated for the free molecule regime and for the continuum range. The present work, taking into account the influence of internal force fields such as magnetic force, electric force and molecular forces, created by particles themselves, recalculated the collision frequency in the case of particles much smaller than the mean free path of the gas (free molecule regime). Attractive forces increase naturally the collision frequency, while repulsive forces decrease it. The calculation was performed for all types of central forces deriving from a potential, including Coulomb forces and Van der Waals forces.

Modeling particle collision processes in high Reynolds number flow

Shearing and inertial collision frequency functions are derived for spherical particles with motivation to high-speed flows where compressibility effects are present. Orthokinetic collision is considered under laminar flow where the radial component of relative velocity between two colliding particles is a source of collision. For a general laminar movement of flow, contraction and dilatation effects in compressible flow on shearing collision are assessed. The newly derived shearing collision frequency function consists of compressible and incompressible parts and is obtained from first principles using the local strain-rate tensor. It is shown that the contraction and dilatation of a fluid element significantly effects shearing collisions. The new shearing collision frequency function is compared with inertial collision frequency under laminar flow and the laminar total collision frequency is compared with the total collision frequency function for turbulent flow.