Collision Frequency Function Research Articles

Collision Kinetics and Electrostatic Dispersion of Airborne Submicrometer Fractal Agglomerates

Collision and electrostatic dispersion rates of airborne submicrometer TiO2 agglomerates were measured and compared with the classical collision theory for spheres as well as with models accounting for the agglomerate structure in terms of the fractal dimension and electrostatic effects such as Coulomb and van der Waals interactions. According to the authors' knowledge, this is the first time that the agglomerate fractal dimension and electrostatic effects have been considered simultaneously in determining the collision frequency function of agglomerates. The observed enhancement in the collision frequency of agglomerates was found mainly to be a result of electrostatic particle interactions. Nonspherical particle shape has only a comparatively small influence on the collision probability, on the order of 10–20%. Electrostatic dispersion coefficients of agglomerates were found to be similar to those of spheres.

Analysis of Electrical Coagulation of Unipolar Charged Particles in an Alternating Electric Using Moment Method

A numerical study has been carried out on the evolution of the particle size distribution for unipolar charged particles that experience coagulation in an alternating electric field. The collision frequency function of charged particles was analytically derived. The log-normal size distribution function is utilized for representing a poly-disperse size distribution and the moments of the particle size distribution are used to solve the general dynamic equation considering only AC electric force effect. The results are compared with the effects of brownian coagulation.

Excitation of current-driven electrostatic ion-cyclotron waves in presence of a transverse direct current electric fields in a magnetized plasma

The temporal evolution of the current-driven electrostatic ion–cyclotron (CDEIC) instability is investigated in the presence of a transverse dc electric field in a collisional magnetized plasma. It was found that the inclusion of a transverse dc electric field in addition to the magnetic field changes the dispersion characteristics of the ion cyclotron waves. The growth rate of the instability increases with the mode frequency and has the largest value at the mode frequencies 43 kHz (for ω≫kzvte) and 7.53 kHz (for kzvti≪ω≪kzvte) when the electron drift velocity is less than the critical value for the CDEIC instability. The growth rate also increases with the guide magnetic field and has the largest value at the magnetic fields 1.0 kG (kyρi∼0.35) for ω≫kzvte and 0.1 kG (kyρi∼0.2) for kzvti≪ω≪kzvte. The growth rate is a sensitive function of electron collision frequency. The results of the theory are applied to explain some of the experimental observations of Koepke et al. [IEEE Trans. Plasma Sci. 20, 631 (1992)].

A simple model for turbulence induced flocculation of cohesive sediment

The transport and fate of fine grained cohesive sediments in estuarine and coastal waters is influenced by the settling velocity of the sediment, which in turn is affected by flocculation effects. The flocculation processes depend on the physico-chemical properties of the sediment and the water, and on several physical mechanisms, of which turbulence is a major one. In the present paper the effects of turbulence are further analyzed. The floes are treated as self-similar fractal entities. As a result, the settling velocity is shown not to scale with the diameter D squared, as predicted by Stokes' law, but with Dnf-1 , where nf is the fractal dimension. Based on a collision frequency function and a floe breakup formulation, a simple flocculation model is proposed that can be solved analytically for uniform conditions. The flocculation and floe breakup coefficients were obtained from experiments described in literature. It is shown that the maximal attainable floe size and settling velocity are determined to a large extent, especially at low turbulence levels, by the available residence time, i.e. water depth.

Collision Frequencies between Fractal Aggregates and Small Particles in a Turbulently Sheared Fluid

Three groups of aggregates with fractal dimensions of 1.89 ± 0.06, 2.21 ± 0.06, and 2.47 ± 0.10 were generated by coagulation of latex microspheres (2.85 μm) in a Jar-test (paddle-mixing) device. The collision rates between these fractal aggregates (200−1000 μm) and small (1.48 μm) particles were measured in the turbulent shear environ ment of the paddle mixer at mean shear rates of 2.1, 7.3, and 14.7 s-1. Collision frequencies were 5 orders of magnitude higher than predicted by a curvilinear model but 2 orders of magnitude lower than predicted by a rectilinear model. Collision frequencies much higher than predicted by the curvilinear collision kernel were attributed to significant flow through the interior of the fractal aggregates. The fluid shear rate (G) and the aggregate fractal dimension (D) affected the collision frequency function (β) between fractal aggregates and small particles, resulting in β ∼ G1-0.33D. According to this relationship, as D → 0, the ag gregates become infinitely porous and β becomes proportional to G1 as described by a rectilinear collision model based on aggregates sweeping out all fluid within their pathway. As aggregates become less fractal and D → 3, β becomes relatively insensitive to the magnitude of G as predicted by a curvilinear model.

An Approximate Method to Calculate the Collision Rates of a Discrete—Sectional Model

An approximate method is proposed to improve the computational efficiency of discrete-sectional representations of aerosol dynamics. The expense that is associated with the numerical calculation of intersectional conversion rates is reduced by using a binomial expansion of the nondimensional collision frequency function in the summand so that an analytical integral form for the inner collision summation is obtained. A considerable saving in computational expense is achieved in comparison with a double summation with a very small sacrifice in accuracy.

Quantum transport of strongly correlated electrons on a liquid helium surface

We report a new approach to the description of quantum transport phenomena of strongly correlated electrons on the liquid helium surface in a magnetic field. The self-consistent Born approximation (SCBA) theory was re-established for a generalized collision frequency function ν instead of conductivity tensor components, to be valid for any ratio of the cyclotron frequency ωc to ν. It was found that for electron-vapour atom scattering the collision frequency increases faster with the magnetic field than ωc, which explains the discrepancy between experimental data and the previous SCBA theory.

Effect of Cumulus Maturity on Sperm Penetration in the Golden Hamster

The influence of the cumulus on gamete interaction was investigated in naturally and precociously ovulated eggs from superstimulated (eCG) hamsters. Inseminations were performed in which the absolute gamete ratio (sperm/egg) was held constant, but the relative gamete ratio (sperm x egg-1 x ml-1) was varied 10-fold. Sperm penetration to the perivitelline space was significantly higher at the higher relative gamete ratio, indicating that there is no chemoattractant in the cumulus and that gamete interaction in cumulus-intact eggs is a function of collision frequency, which varies according to the relative gamete ratio. The level of successful gamete interaction as a function of the relative gamete ratio was determined for naturally and precociously ovulated eggs. The level of sperm penetration reached saturation at a relative gamete ratio of 10(3.3) for naturally ovulated eggs and 10(4.0) sperm x egg-1 x ml-1 for precociously ovulated eggs. At suboptimal relative gamete ratios, there was no difference in the level of penetration between partially and fully capacitated sperm. These results indicate that the larger, more mature cumulus of naturally ovulated eggs is successfully penetrated at lower collision frequencies than the denser, less mature cumulus of precociously ovulated eggs. To determine whether the maturity of the cumulus affects the timing of sperm penetration, cumulus-free and cumulus-intact eggs were inseminated at their optimal relative gamete ratios. The rate of sperm penetration of partially capacitated sperm was not different between cumulus-free and cumulus-intact eggs, regardless of ovulation method. However, the period of time between insemination and when half the eggs were penetrated (T50) was longer in cumulus-intact eggs.(ABSTRACT TRUNCATED AT 250 WORDS)

The (Relative) Insignificance of G in Flocculation

The kinetics of flocculation of heterodisperse suspensions like those in water treatment plants are usually described by the Smoluchowski equation, which incorporates collision frequency functions for particle collisions by Brownian motion, fluid shear, and differential sedimentation. These collision‐frequency functions are based on a rectilinear view of collisions, i.e., one that ignores short‐range forces and changes in fluid motion as particles approach one another. With this rectilinear approach, the velocity gradient, G, is the principal design parameter for flocculation units. A curvilinear approach, i.e., one that accounts for short‐range effects in particle collisions, is presented as a set of corrections to the rectilinear collision frequency functions for each mechanism. The primary ramification of this curvilinear, heterodisperse approach is that G is found to be not nearly so important. Previous experimental work in which the role of G has been examined is reviewed in light of this finding.

Interactions of Two Settling Spheres: Settling Rates and Collision Efficiency

The behavior of two spheres settling in a low Reynolds' number region is described from the hydrodynamic equations developed by others. Even when they are not close enough to form a floe, neighboring particles increase their settling velocity above that described by Stokes. The collision efficiency factor in differential sedimentation (αDS) is calculated from trajectory analysis that includes the hydrodynamic effects and particle attraction. αDS is a function of many parameters and the sensitivity of αDS to each governing parameter is determined. αDS increases as particle sizes decrease (at constant size ratio), as particle size ratio (small to large) approaches unity, as the gravity acceleration decreases, and as the density of the particle decreases. Viscosity does not affect αDS. Comparisons to previously published values of αDS are made and the differences are discussed. Ramifications of αDS to the collision frequency function and the modeling of flocculation of heterodisperse suspensions are noted.

Change of Weight K-alum Crystals with Breakage in a Crystallizer.

The morphology of suspended crystals of K-alum in a crystallizer was investigated by microscopic observation. These crystals were broken, and their form and surface conditions were changing continuously. When a crystal suspended in the crystallizer was constantly colliding with an impeller, the crystal weight W decreased with elapsed time θ. This change in crystal weight was expressed as W = W0 exp (-K. θ), where W0 was the initial crystal weight and K was a rate parameter which was determined experimentally and was considered to be a function of collision energy and collision frequency, etc. In the case of suspended crystals, where the crystal weight decreased with elapsed time θ, its change was also expressed by the above equation.

Plasma transport coefficients for nonsymmetric toroidal confinement systems

A variational principle is developed for computing accurate values of local plasma transport coefficients in nonsymmetric toroidal confinement configurations. Numerical solutions of the linearized drift Fokker–Planck equation are used to obtain the thermodynamic fluxes as functions of collision frequency and the radial electric field. Effects resulting from the variation of the longitudinal adiabatic invariant J along an orbit (resulting from particle transitions from helically trapped to toroidally trapped orbits) are treated. The velocity-space distribution resulting from trapped, circulating, and transition particle orbits is well represented by a Legendre polynomial expansion in the pitch angle coordinate. The computational effort is significantly reduced from that required with Monte Carlo methods through use of an efficient treatment of the disparity between the time scales of collisionless and collisional particle dynamics. Numerical computations for a stellarator configuration are presented.

Finite hard rod systems and their thermodynamic limit. III. Collision frequency, no-collision probability, and velocity autocorrelation function

We use the methods developed for the one-dimensional hard rod gas in the preceding papers to derive exact expressions for the average number of collisions per unit time, the probability that a tagged particle does not collide with other ones within a given time interval, and the velocity autocorrelation function. It is discussed how these expectation values, obtained for finite systems and three kinds of ensembles (smallest stationary, microcanonical, and canonical), behave as the size of the system increases (thermodynamic limit).

Copolymerization Behavior of N-Vinylimidazole with Various Diaikyl Maleates and Fumarates

Abstract N-Vinylimidazole was copolymerized with diaikyl maleates and fumarates in order to investigative quantitatively the deviation from the Alfrey-Price Q,e scheme as a function of steric hindrance or collision frequency. There was no particular correlation observed between deviations from the Alfrey-Price Q,e-scheme of alkyl maleates (cis isomers) and fumarates (trans isomers) and the bulkiness of those monomers. The copolymerization reactivities of these comonomers can be understood by the so-called “collision theory.” Maleates were more affected than fumarates in this regard.

Effect of potential shape on isomerization rate constants for the BGK model

As a model for isomerization reactions in liquids, we consider the motion, in a one-dimensional bistable potential, of a particle which suffers BGK collisions. We investigate how the relaxation rate constant as a function of collision frequency depends upon the shape of the potential. We discuss our results in relation to recent high-pressure NMR experiments involving cyclohexane isomerization. For a potential with narrow wells and a wide barrier, we find qualitative agreement between theory and experiment.

Solution of the integral equation relating collisions with power reflection in the ionosphere

The integral equation for the quasi-transverse approximation of the ionospheric propagation, relating the collision frequency function of height with the measured power reflection coefficients for different signal frequencies, is formulated and solved in a closed form, using the Wentzel-Kramer-Brillouin (WKB) approximation.

乱流場における液中造粒の速度論的考察

Agglomerates growth in turbulent flow is theoretically investigated. Coalescence probability in the process of agglomerates growth is formulated on the basis of Kolmogoroffs theory under the assumption that agitation field is in an isotropic turbulent flow, and also on the basis of physical properties of agglomerates. And characteristic diameter (CD) of agglomerate is defined as a diameter at which coalescence probability of two agglomerates in the same diameter is zero, and CD is generally expressed by Eqs.(34) and (35). A population balance equation is constructed using the collision frequency function based on the Kolmogoroff s theory and the coalescence probability.Simulation results are as follows:(1) Regardless of variation of cw/w1, cumulative size distributions normalized by each weight mean diameter approach a specified distribution curve with lapse of agglomeration time, within a limitted time in the R>λν.(2) CS is proportional to CD.(3) Accordance with the results of (1) and (2), it was proved that it is capable to predict the cumulative size distribution at steady state.(4) When CD suddenly increases in agglomeration process, growth pattern of agglomerates changes from slow growth to rapid growth.(5) lf σst is equal to Qt arid the saturation degree is relatively high, CD is expressed by Eqs.(50) and (51). And also, CD is given by Eqs.(52) and (53), when each of B1 and B2 in Eqs (50) and (51) is kept constant and ε is proportional to N3. The following characteristics in the agglomeration are theoretically proved from Eqs.(52) and (53).i) CD increases, as γ, cosθ and ψB, increase, and as γ andε decrease. The effect of these factors on the CD is larger in R>λν than in R>λν.ii) On the fixed condition of N, agglomerates growth rate becomes faster, as γ, cosθ and ψB and N (0) V increase, and as ε decreases. ln the cases resulting the same CD at different N, the higher N, the faster in agglomerates growth rate.

A highly accurate and simple expression of electron drift velocity in gases and semiconductors

The drift velocity for electrons (or holes) in a scattering medium is obtained as the sum of the usual first-order expression plus a correction term. Both terms are expressed as integrals over a single variable and the integrands are known functions of the electron differential collision frequency. Since the correction term is small compared with the principal, usual term, the expression obtained is in practice equivalent to an explicit rigorous solution.

On ferromagnetic resonance damping in metals

The Landau-Lifshitz damping coefficient is defined in terms of spin-orbital torque correlations in the one-electron model and estimated as a function of the orbital collision frequency. The formal results are compared with previous studies and the results of a numerical calculation for nickel are compared with known experimental data.

Attenuation of longitudinal electro-acoustic waves in a plasma

There are several practical situations in partially ionized plasmas when both collisionless (Landau) damping and electron-neutral collisions contribute to the attenuation of longitudinal waves. The longitudinal-wave dispersion relation is derived from Maxwell's equations and the linearized Boltzmann equation, in which electron-neutral collisions are represented by a Bhatnagar–Gross–Krook model that conserves particles locally. (The dispersion relation predicts that, for a given signal frequency ώ), an infinite number of complex wavenumbers kn can exist. Using Fourier–Laplace transform techniques, an integral representation for the electric field of the longitudinal waves is readily derived. Then, using theorems from complex variable theory, a modal expansion of the electric field can be made in terms of an infinite sum of confluent hypergeometric functions, whose arguments are proportional to the complex wavenumbers kn. It is demonstrated numerically that the spatial integral of the square of the electric field amplitude decreases as the electron-neutral collision frequency increases. Also, the amount of energy contained in the first few (lowest) modes, and the coupling between the modes, is examined as a function of plasma frequency, signal frequency and collision frequency.