Breakup Rate Research Articles

About bubble breakup models to predict bubble size distributions in homogeneous flows

We present results of a study of the equilibrium between coalescence and breakup of bubbles in homogeneous media with isotropic turbulence. The Boltzmann equation for the particle distribution function (pdf) was evaluated in steady state, using a multigroup approach. Binary bubble breakup was assumed. We used uniform function, delta function, and the model proposed by Luo and Svendsen (1996) for the bubble size distributions resulting from a breakup. The bubble breakup rate was calculated with Luo and Svendsen (1996) and Prince and Blanch (1990) models. Significant differences in bubble breakup rate, and therefore in bubble size distribution, are predicted by both models. The models were compared to the bubble size distributions measured by Boyd and Varley (1998) in air-water flow. The transient response of the bubble size distribution and interfacial area density was also analyzed. This work is of significance in the prediction of reaction rates when they are dependent on bubble size distribution.

Experimental study on the number density distribution function in turbulent bubbly flows with coalescence and break-up

The population balance equation considers the change of the bubble size and the bubble number density due to coalescence and break-up in bubbly flows. Although a number of theoretical models exist for coalescence and break-up rates, nearly no experimental data for the validation of these models exist for turbulent flow with high void fraction. To overcome this lack, vertical air–water pipe flows were examined. A comprehensive validation of the population balance equation is based on the knowledge of the development of the number density distribution function and of the liquid flow turbulence along the pipe flow. The experimental and analytical procedure to obtain these parameters is presented. The void fraction, interfacial area density and mean bubble volume were calculated as statistical moments of the number density distribution function. For its measurement a combination of single and double fibre optical probes was used that allowed measurement in high void fraction flows. X-hotfilm probes were used to measure the liquid flow turbulence level of the two-phase flow. The flow inlet condition was influenced with the aid of turbulence grids. The results show that the number of coalescence and break-up events depend strongly on the number of bubbles per unit volume and the on bubble volume that differ even for the same gas and liquid superficial velocities. From the axial development of the number density distribution function coalescence and break-up events could be identified within certain bubble volume ranges and the source terms of the population balance equation could be quantified. In a first step the data were used to check two existing theoretical models with respect to coalescence and break-up rates.

Bubble‐Size distributions and flow fields in bubble columns

AbstractBubble‐size distributions and flow fields in bubble columns are calculated numerically. The population balance is simplified and reduced to a balance equation for the average bubble volume. Models developed predict the rate of bubble breakup and coalescence based on physical principles. The flow fields are numerically calculated for bubble columns with cylindrical cross sections using the Euler‐Euler method. The newly derived balance equations for the average bubble volumes are implemented into a commercial CFD code. The solutions of the balance equation for high superficial gas velocities result mainly in two fractions: one for the fraction with small and the other for the fraction with large bubble diameters. Both are considered pseudocontinuous phases, in addition to the liquid phase. The calculated flow fields are characterized by several large‐scale vortices. The local volume fractions of gas and liquid are locally inhomogeneous and highly time‐dependent. The time‐averaged flow field is axisymmetric and stationary. The calculated volume fractions, velocities, and bubble‐size distributions agree well with existing and previously published experimental results for bubble columns up to 0.3 m in diameter.

A model for turbulent binary breakup of dispersed fluid particles

Accurate predictions of particle size distributions, and therefore of the underlying processes of fluid particle breakup and coalescence are of vital importance in process design, but reliable procedures are still lacking. The present paper aims at developing a modular formulation for the turbulent particle breakup process. The model is to be included in a population balance model which is formulated such as to facilitate the direct future implementation into a full multifluid CFD model. The breakup process is described without introducing adjustable parameters. The current model is a further development of an existing model by Luo and Svendsen (AIChE J. 42 (5) (1996) 1225), which has been expanded and refined, and where an inherent weakness regarding the breakup rate for small particles and small daughter particle fragments are removed. A new criterion regarding the kinetic energy density of the colliding turbulent eddy causing breakup has been introduced. This new criterion is a novel concept describing the breakup process. The details are thoroughly discussed together with possible further modifications. The results from the new model are encouraging because the breakup rate is greatly reduced when the dispersed fluid particles are reduced in size. Further, the response to changes in system variables is reasonable and the distribution of daughter sizes vary in a reasonable way for the different collision possibilities.

STOCHASTIC FRAGMENTATION AND SOME SUFFICIENT CONDITIONS FOR SHATTERING TRANSITION

We investigate the fragmentation process developed by Kolmogorov and Filippov, which has been studied extensively by many physicists (independently for some time). One of the most interesting phenomena is the shattering (or disintegration of mass) transition which is considered a counterpart of the well known gelation phenomenon in the coagulation process. Though no masses are subtracted from the system during the break-up process, the total mass decreases in finite time. The occurrence of shattering transition is explained as due to the decomposition of the mass into an infinite number of particles of zero mass. It is known only that shattering phenomena occur for some special types of break-up rates. In this paper, by considering the n-particle system of stochastic fragmentation processes, we find general conditions of the rates which guarantee the occurrence of the shattering transition.

Gravitational waves from rotating neutron stars and evaluation of fast chirp transform techniques

X-ray observations suggest that neutron stars in low mass x-ray binaries (LMXB) are rotating with frequencies in the range 300–600 Hz. These spin rates are significantly less than the break-up rates for essentially all realistic neutron star equations of state, suggesting that some process may limit the spin frequencies of accreting neutron stars to this range. If the accretion-induced spin up torque is in equilibrium with gravitational radiation losses, these objects could be interesting sources of gravitational waves. I present a brief summary of current measurements of neutron star spins in LMXBs based on the observations of high-Q oscillations during thermonuclear bursts (so-called ‘burst oscillations’). Further measurements of neutron star spins will be important in exploring the gravitational radiation hypothesis in more detail. To this end, I also present a study of fast chirp transform (FCT) techniques as described by Jenet and Prince (Prince T A and Jenet F A 2000 Phys. Rev. D 62 122001) in the context of searching for the chirping signals observed during x-ray bursts.

Liquid breakup at the surface of turbulent round liquid jets in still gases

An experimental study of liquid column breakup lengths and turbulent primary breakup properties at the surface of turbulent round liquid jets in still air at standard temperature and pressure is described. Jet exit conditions were limited to non-cavitating water and ethanol flows, long length/diameter ratio (greater than 40:1) constant-diameter round injector passages, jet exit Reynolds numbers of 5000–200,000, jet exit Weber numbers of 235–270,000 and liquid/gas density ratios of 690 and 860, at conditions where direct effects of viscosity were small (e.g., liquid jet Ohnesorge numbers were smaller than 0.0053). Three liquid column breakup modes were observed, as follows: a weakly turbulent Rayleigh-like breakup mode observed at small jet exit Weber and Reynolds numbers, a turbulent breakup mode observed at moderate jet exit Weber numbers, and an aerodynamic bag/shear breakup mode observed at large jet Weber numbers. The turbulent primary liquid column breakup mode was associated with conditions where drop diameters resulting from turbulent primary breakup along the liquid surface were comparable to the diameter of the liquid column itself. The bag/shear liquid column breakup mode was observed when liquid turbulence caused large deformations of the liquid column so that portions of it were in a gaseous cross flow; this resulted in bag or shear liquid column breakup, very similar to the breakup of non-turbulent liquid jets in gaseous cross flow. Mean streamwise drop velocities after breakup were comparable to mean streamwise velocities within the jet whereas mean cross stream drop velocities after breakup were comparable to cross stream velocity fluctuations within the liquid. Rates of primary breakup at the liquid surface are reported as surface efficiency factors (the fraction of the maximum possible cross stream drop liquid flux at the surface based on the mean relative cross stream velocity of the drops at the surface and the liquid density). The resulting surface efficiency factors varied from small values near the onset of liquid surface breakup to values of the order of unity as conditions near breakup of the liquid column as a whole were approached.

Rb-Xe spin relaxation in dilute Xe mixtures

We present measurements of spin-relaxation of Rb atoms in He(98%)-N2(1%)-Xe(1%) mixtures similar to those used in magnetic resonance imaging polarizers. The pressure dependence allows us to separate out contributions from Rb-Xe van der Waals molecules and binary collisions. For the first time, we observe the predicted increase in the molecular contribution at high pressure. Our data suggest that the deduced molecular breakup rate has a strong temperature dependence. The realization of magnetic resonance imaging of human subjects using hyperpolarized noble gases @1# has stimulated the intense development of Xe-Rb spin-exchange optical pumping @2# in high pressure buffer gases @3‐5#. The need to minimize the mismatch between the Rb absorption linewidth and the linewidth of inexpensive, high power but broadband diode laser sources requires pressure broadening the Rb resonance lines with many atmospheres of He gas, chosen for its weak spin-relaxation properties @6#. Despite the widespread use of high pressure He for Xe-Rb spin exchange, there exist no systematic studies of spin relaxation and spin exchange rates under such conditions, as most of the pioneering work in Xe spin-relaxation and spin-exchange either had no buffer gas @7# or used N2 buffer gas @8#. Two recent experiments report relaxation rates measured at a single pressure and temperature @4,5#. In this paper we report measurements of spin-relaxation rates at 80 and 150 °C, in

Kinetics of the aggregation of polyethylene oxide at temperatures above the polyethylene oxide–water cloud point temperature

The kinetics of the aggregation of high molecular weight polyethylene oxide (PEO) at temperatures above the PEO–water cloud point temperature (CPT) was studied by monitoring the particle size profile with time using photometric dispersion analysis and light diffraction. Upon PEO–water phase separation, hydrophobic inter/intra molecular PEO–PEO interactions cause the random PEO coils to collapse to a compact configuration and aggregate with each other. The rate of aggregation was found to increase with increasing temperature deviation from the CPT, PEO concentration and molecular weight. Higher temperature and lower shear gave rise to larger equilibrium PEO aggregates. These aggregates can reach a size of 600 μm in diameter. Aggregation was also found to proceed until the aggregation rate is balanced out by the breakup rate. It was observed that the aggregation process could be reversed repeatedly by shifting across the phase boundary. It is suggested that the main mechanism for the breakup of the aggregates is surface erosion, which is caused by the shearing action of the water surrounding the aggregates.

A transport equation for the interfacial area density applied to bubble columns

A population balance equation is used to describe the evolution of bubble sizes in two-phase flow. New kernel functions for the rate of bubble break-up and coalescence are presented. The model is able to predict the bubble size distribution in bubble columns including the formation of large bubbles at high superficial gas velocities. Applying the concept of self-similarity an approximate analytical solution for the bubble size distribution can be found. With this solution the population balance is reduced to a transport equation for the average bubble volume. We implemented the transport equation in a commercial CFD-code to demonstrate the possibility of combining population balances with CFD. A dynamic two-phase flow model of the Eulerian–Eulerian type is used and calculations for three space coordinates are performed for cylindrical bubble columns.

Confocal microscopy studies of shear-induced coalescence in blends of poly(butyl methacrylate) and poly(2-ethylhexyl methacrylate)

This article reports the results of confocal fluorescence microscopy studies of shear-induced coalescence in binary blends of poly(2-ethylhexyl methacrylate) (PEHMA; 90 wt %) and poly(butyl methacrylate) (PBMA; 10 wt %). We prepared the blends by casting a mixture of latex dispersions of the components onto a substrate and allowing the film to dry under ambient conditions. The initial morphology of the film was a dispersion of 120-nm PBMA spheres in a continuous PEHMA matrix. One-fifth of the PBMA particles were labeled with anthracene, the emission of which we observed with confocal microscopy. The blends were sheared in a parallel-plate rheometer at 80 and 100 °C for 1 and 10 h. Careful image analysis allowed us to estimate the mean size of the dispersed phase and the width of the size distribution. The results were compared with the theoretical limits of Wu and Taylor. After 10 h of shearing, the mean particle size decreased and the particle distribution became narrower in comparison with the results obtained after 1 h of shearing. We explain this result by inferring that before the sample reached steady-state morphology, its rate of coalescence was greater than the rate of particle breakup. © 2001 John Wiley & Sons, Inc. J Polym Sci Part B: Polym Phys 39: 2317–2332, 2001

Analytical Solutions for Bikinetic Coagulation: Incorporation of a Maximum Size Class

Analytical solutions are derived for both orthokinetic and perikinetic coagulation rate mechanisms by considering bikinetic collisions of the agglomerating particles. The bikinetic behavior assumes that only equal-sized particles collide to form larger aggregates. In addition to the bikinetic assumption used to simplify the rate equations, the coagulating particles are assumed to approach a maximum size class with increasing time. Mechanisms of perikinetic, orthokinetic, differential sedimentation, and flow-induced breakup were examined in the analyses. A solution incorporating flow-induced breakup forced the equilibrium population to a maximum size class that was less than an initially assumed maximum size class, consistent with coagulation modeling. This breakup equation is shown to be beneficial to researchers seeking to develop predictive models for orthokinetic coagulation by establishing estimates for breakup constants. The new solutions are contrasted with the existing simplified models and found to be more characteristic of actual coagulation behavior. The new solutions approach a maximum particle size class and conserve mass at increasing time, rather than an asymptotic approach to zero particles or zero mass as predicted by the existing models. Applications of the new coagulation rate solutions are useful in estimating particle collision efficiency and breakup rate constants and in determining the accuracy of numerical simulations of coagulation processes.

Oscillations of Accretion Disks and Boundary Layers in Cataclysmic Variables. II. A Local, Linear Stability Analysis of Accretion Disk Boundary Layers

We present the results of a linearized perturbation analysis for the models of cataclysmic variable (CV) disks and their boundary layers (BLs), discussed in a previous companion paper. For the case of large-scale, azimuthal oscillations in the BL, we find a triplet of unstable modes consisting primarily of perturbations of the azimuthal velocity, plus a singlet mode consisting mainly of a pressure perturbation. In the disk, the torsional modes are modified gravity waves, with rise times short enough for significant amplification before entering the BL; in the BL, the frequencies of these modes are close enough in magnitude to produce beating. In the BL region where the effective temperature has its maximum, the beat frequencies, for a star rotating at 5% of its breakup rate, span a range from the stellar rotation frequency Ω* up to an order of magnitude larger than that, i.e., from periods of P ~ 20 s to (2-3) × 100 s, with the underlying fundamental oscillations having much smaller periods, ~1 s. These beat-frequency oscillations of the torsional modes have attributes similar to the quasi-periodic oscillations (QPOs) observed in many CVs. Furthermore, if a particular region of the BL is excited, fairly high Q (where Q ≡ |dP/dt|-1) beat-frequency oscillations are produced, which are similar to some observed dwarf nova oscillations (DNOs). We propose two observational tests to determine whether some QPOs can be identified with the modes studied here. The tests require an extension of the search for CV oscillations to the 0.2-2 Hz frequency range.

Time and Force Dependence of the Rupture of Glycoprotein IIb-IIIa-Fibrinogen Bonds between Latex Spheres

We studied the shear-induced breakup of doublets of aldehyde/sulfate (A/S) latex spheres covalently linked with purified platelet GPIIb-IIIa receptor, and cross-linked by fibrinogen. Flow cytometry with fluorescein isothiocyanate-fibrinogen showed than an average of 22,500 molecules of active GPIIb-IIIa were captured per sphere, with a mean K d = 56 nM for fibrinogen binding. The spheres, suspended in buffered 19% Ficoll 400 containing 120 or 240 pM fibrinogen, were subjected to Couette flow in a counter-rotating cone-plate rheoscope. Doublets, formed by two-body collisions at low shear rate (G = 8 s −1) for ≤15 min, were subjected to shear stress from 0.6 to 2.9 Nm −2, their rotations recorded until they broke up or were lost to view. Although breakup was time dependent, occurring mostly in the first 2 rotations after the onset of shear, the percentage of doublets broken up after 10 rotations were almost independent of normal hydrodynamic force, F n: at 240 pN, 15.6, 16.0, and 17.0% broke up in the force range 70–150 pN, 150–230 pN, and 230–310 pN. Unexpectedly, at both [fibrinogen], the initial rate of breakup was highest in the lowest force range, and computer simulation using a stochastic model of breakup was unable to simulate the time course of breakup. When pre-sheared at low G for >15 min, no doublets broke up within 10 rotations at 70 < F n < 310 pN; it required >3 min shear (>1110 rotations) at F n = 210 pN for significant breakup to occur. Other published work has shown that binding of fibrinogen to GPIIb-IIIa immobilized on plane surfaces exhibits an initial fast reversible process with relative low affinity succeeded by transformation of GPIIb-IIIa to a stable high-affinity complex. We postulate that most doublet breakups observed within 10 rotations were from a population of young doublets having low numbers of bonds, by dissociation of the initial receptor complex relatively unresponsive to force. The remaining, older doublets with GPIIb-IIIa in the high-affinity complex were not broken up in the time or range of forces studied.

Fragmentation of Kaolinite Aggregates Induced by Ion-Exchange Reactions within Adsorbed Humic Acid Layers

Mixing of humic acids and kaolinite clay suspended in aqueous solution containing aluminum ions promotes aggregation of the clay instantaneously. To this fast clay destabilization succeed reversible fragmentation processes which display varying kinetic ranges for which the changes in the average masses of fragments are determined. We show the rate of formation of fragments to decay with the mass i as i−2 and the maximal rate of aggregate breakup to increase with the fragment mass as iλ, where λ lies between 0.35 and 0.40. At short terms the rate of variation of the average masses of the fragments is lower than expected from the theory. Aggregate fragmentation is attributed to slow progressive modifications in the positive and negative charge distributions within the amphoteric humic acid adsorbed layer, which are induced by the high density of positive charge existing on the clay surface after aluminum ion adsorption. Kaolinite–aluminum ion–humic acid complexes aggregate by forming initially linear small portions of stacked platelets. These portions are connected together to form larger agglomerates of fractal dimension close to 2.5.

Incorporation of Aggregate Breakup in the Simulation of Orthokinetic Coagulation

The agglomeration and breakup of floc aggregates formed in orthokinetic coagulation is examined. By considering local flow strain-rate, a breakup rate kernel is derived based on flow-induced normal stresses. The new breakup kernel is included in a population size class balance for floc aggregates. The resulting population balance was solved numerically over a wide range of parameters to obtain a variety of floc size distributions. Results indicate that the inclusion of a breakup kernel in orthokinetic coagulation modeling eliminates the computational growth to a maximum size class, producing more realistic distributions. The breakup kernel was rigorously compared to prior research and found to be consistent with the earlier theories of coagulation agglomeration and breakup.

Hydrodynamic behavior of slurry bubble column at high solids concentrations

Hydrodynamics of a slurry bubble column have been investigated up to slurry concentrations of 40 vol.% and gas velocities of 0.26 m/s. High slurry concentrations represent high catalyst loading to increase reactor productivity, while high gas velocities would be required to increase reactor throughputs. The solid particles used are 35 μm glass beads representing typical particle size for catalytic slurry reactors. The two important hydrodynamic parameters investigated are gas holdup and solids concentration profiles. The average gas holdups decreased with increasing slurry concentration but the rate of decrease slowed down for higher slurry concentrations. The analysis of axial gas holdup profiles indicated that decrease in gas holdup due to solids addition could be attributed to a decrease in bubble breakup rates. The experimental gas holdups are compared with literature correlations and models. The axial distribution of slurry concentration followed the classical sedimentation–dispersion model. Effects of gas velocity on axial solids distribution are found to be minimal over the range of gas velocities investigated.

Particle reduction study of flocculation mixing by means of grids

A flocculation study was completed by means of a vertically oscillating grid mixing device. Five types of single grids with different solidity ratios were vertically oscillated inside a 2 L jar to promote floc aggregation. Kaolin was used as the simulated turbid particles and alum was applied as the chemical coagulant. The method of measurement was completed based on the particle reduction represented by the settled water turbidity. It was found that low turbidity readings could be achieved at a wide range of average volume velocity gradient [Formula: see text], especially in the case of high solidity ratio types of grids. This indicates that the grid mixing had a stable performance and was not greatly affected by mixing variations in the vessel. The floc aggregation and erosion rate coefficient analysis showed that the grid mixing produced particle contacts with low break-up rate. A general relationship among [Formula: see text], flocculation performance parameters, and grid physical characteristics was found, indicating that the flocculation performance was easily controlled. This study has shown the potential of grids as a mixing device to produce an excellent mixing environment for floc aggregation.Key words: flocculation, grid, solidity ratio, kaolin, alum, [Formula: see text], turbidity.

The kinetics and mechanism of Corynebacterium glutamicum aggregate breakup in bioreactors

Cultivation of bacteria requires high levels of agitation and aeration to satisfy the mass transfer requirements of the cells. Associated with these conditions are turbulent forces which may act on the surfaces of cells and be detrimental to their growth, metabolism and morphology. The kinetics and mechanism of hydrodynamic trauma has been investigated for the breakup of aggregates of Corynebacterium glutamicum ( ATCC 13032) in a stirred-tank reactor and a capillary flow loop system. The effect of forces associated with turbulent eddies in the impeller discharge zone of the stirred tank reactor has been compared with that of collapsing air bubbles at the air medium interface. A model is presented to describe the initial rate of aggregate breakup caused by fluid–aggregate interactions. It assumes that aggregate disruption is caused by the interaction of aggregates with similarly sized turbulent eddies. The extent of aggregate breakup is a function of the magnitude of the turbulent force as well as the total duration of the force event. The applicability of the model to animal cell systems has been investigated. Results showed that both the interaction of microbial cells with turbulent eddies in the viscous dissipation subrange in the impeller discharge zone, as well as with collapsing air bubbles at the air medium interface contributed to the total force acting on the cells.

Shear-Induced Flocculation of Colloidal Particles in Stirred Tanks

Colloidal polystyrene and paramagnetic particles consisting of mixtures of polystyrene and magnetite are used to experimentally investigate flocculation kinetics in a stirred tank under turbulent shear flow. The effects of various parameters—agitation speed, solution pH, ionic strength, particle size, and particle concentration—on the flocculation rate are investigated. A trajectory model applicable for shear-flow systems is formulated to describe particle flocculation in stirred tanks. The collision efficiency of particles is obtained from the limiting trajectory of one particle moving toward another and is a function of interparticle forces and flow properties. The collision frequency is determined as a function of particle size and energy dissipation. The flocculation frequency is then determined by multiplying the collision frequency by the collision efficiency and is incorporated into a population balance model to predict the particle size evolution. Results suggest that the flocculation rate is enhanced by increasing the agitation speed, even though the collision efficiency is decreased at a higher agitation speed. It is also found that the collision rate increases and the collision efficiency decreases as the particle size ratio is increased. Results also suggest that the breakup rate of aggregates in a turbulent shear flow could be significant and may need to be included in the population balance modeling to correctly predict the evolution of particle size distribution.