Electrostatic collection of aerosol particles on a sphere at intermediate reynolds numbers

The experimental technique, equipment and measurements of charged and neutral drop collection efficiency by a single charged sphere (Re = 15–100) are described.     

Flow of power law liquids through particle assemblages at intermediate reynolds numbers

The complete equations of motion and continuity have been solved numerically using the finite element method for the flow of power law liquids through assemblages of rigid spherical particles. The inter-particle interactions have been simulated using the free surface cell model. Extensive results on drag coefficients have been obtained under a wide range of physical and operating conditions (0.9999 ≥ 0.3), 1 ≥ n ≥ 0.4 and 20 ≥ Re ≥ 1. The observed dependence of drag coefficient on voidage and non-Newtonian flow behaviour index have been explained qualitatively with the aid of order of magnitude considerations. Finally, the theoretical predictions have been validated using suitable experimental data available in the literature.     

Wall factor for acceleration and terminal velocity of falling spheres at high reynolds numbers

The wall factor for spheres in the acceleration and terminal velocity ranges was determined experimentally for very high Reynolds numbers (13 500 < Re < 70 000). Experiments were performed with 12, 15, 18, 21, 25 and 31.75 mm spheres, falling in water inside cylinders 3.4, 4.9, 7, 10, 14 and 19 cm in diameter. A published empirical equation was found to yield good results for the terminal velocity wall factor in the range of studied Reynolds numbers.     

Inertial impaction of fine particles at moderate reynolds numbers and in the transonic regime with a thin-plate orifice nozzle

Because the smallest size of very fine particles that can be separated in an impactor is proportional to the ratio Re/M², there is an interest in studying these instruments at moderate Reynolds numbers, Re, and at Mach numbers, M, up to sonic conditions. Here an inertial impactor of fixed geometry is tested at Reynolds numbers between 40 and 840, and in the transonic flow regime, at downstream to upstream pressure ratios between 0.42 and 0.79. Earlier theoretical and experimental work has shown that conventional impactors, with nozzles having relatively large length-to-diameter ratios, loose their resolution at Re smaller than several hundreds due to boundary layer growth at the nozzle walls. This effect is minimized here by means of nozzles having rapidly converging walls down to their exit, embodied by a thin-plate orifice with thickness to diameter ratio of 0.014. The nozzle to plate distance is equal to the orifice diameter. Test dioctyl sebacate (DOS) aerosols, with diameters ranging from 0.05 to 0.24 μm, were generated using a Differential Mobility Analyzer (DMA). Collection efficiencies were measured on-line at constant Reynolds and variable Mach numbers using two electrometers; one to monitor the current of charged particles impacting on the collecting plate, the other for the current of uncollected particles. A new method based on sweeping over the Mach number for a given test aerosol, rather than on varying the aerosol size at a fixed Mach number, provides an inversion technique which eliminates completely the effects of multiply charged particles from the DMA. Measurements show excellent resolution even at Re = 100, as well as through the transonic region, so that neither compressibility nor viscous phenomena limit the size of the smallest particle that can be inertially separated with high resolution. The limiting factor is Brownian diffusion. Its negative effects on particle collection efficiency curves are measured here for the first time using NaCl particles with diameters between 0.008 and 0.016 μm.     

Rising velocity of a swarm of spherical bubbles in power law fluids at high reynolds numbers

In this work, a theoretical scheme for estimating the rise velocity of a swarm of spherical bubbles through quiescent power law liquids at high Reynolds number is developed. The inter-bubble interactions have been accounted for by the use of a cell model. The effect of the power law index and the volume fraction of the gas on the rise velocity is elucidated. Depending upon the degree of shearthinning behaviour and the gas fraction, the swarm may rise slower or faster than a single bubble. This behaviour has been explained qualitatively in terms of two competing mechanisms.     

Modeling of unsteady flow around accelerating sphere at moderate reynolds numbers

The flow around an accelerating spherical particle of diameter ranging from 50 to 200 m?m is studied in the range of Reynolds number between 0.1 and 100. The flow around the sphere is assumed to be laminar and two-dimensional axisymmetric. The calculated drag coefficient is compared with the theoretical predictions of added mass term and Basset history term. Appropriate corrections for those two terms are proposed as function of the acceleration rate and the particle diameter.     

Coalescence of two bubbles rising in line at low reynolds numbers

Bubble coalescence phenomena have been examined at Reynolds numbers of 0·5–80 for five different classes of bubbles. The approach velocity of the following bubble (u₂) is experimentally obtained and its behaviour with respect to the leading bubble velocity (u₁) is examined. The coalescence of bubbles with Re < 7 follows the analysis of weightless solid spheres. Bubbles having toroidal wakes (Re > 7), coalesce with two additional velocities imparted due to the wake structure. Equations are developed to predict u₁/u₂ during coalescence.     

Surface remobilization of buoyancy-driven surfactant-laden drops at low reynolds and capillary numbers

It is well known that the terminal velocity of a drop settling in a viscous fluid is impacted by surface tension gradients. These gradients can develop because of nonuniform accumulation of surfactant on the surface as a result of a number of transport mechanisms. Here, a surfactant transport model based on a sorption-limited Frumkin framework is used to describe surfactant transport in the presence of both surface convection and diffusion at low Reynolds and capillary numbers. Constants characterizing surfactant transport in the Frumkin framework are experimentally determined and used to predict aqueous drop velocities with varying surfactant concentrations and volumes. Computation is carried out by satisfying equations governing mass, momentum, and interface species conservation. Experiments demonstrate qualitative and quantitative agreement between predicted and measured drop velocities. It is shown that surface remobilization explains some observed trends in measured velocity as the drop size decreases. © 2018 American Institute of Chemical Engineers AIChE J, 65: 294–304, 2019     

Viscous flow through particle assemblages at intermediate reynolds numbers — a cell model for transport in bubble swarms

The fluid mechanical behaviour of a bubble swarm was simulated using a cell model. The Navier-Stokes equations were solved numerically for the liquid flow in a uniform assemblage of circulating, spherical bubbles. Ranges of parameters studied included, Reynolds numbers, 0–1000 and porosities, 0.4–1. The numerical calculations show the effects of variations in Reynolds numbers and porosity on: surface vorticity and pressure distributions and form and friction drag coefficients. For all Reynolds numbers investigated a standing vortex ring was absent Predicted drag coefficients and Sherwood or Nusselt numbers agree with limiting analytical solutions for low and high Reynolds numbers. The theoretical results show good agreement with experimental data for porosity as a function of superficial gas velocity. Predicted and measured Sherwood and Nusselt Numbers were in substantial disagreement, making detailed comparison unwarranted The calculations should also be valid for dispersions of uniform, circulating, spherical droplets for the special case where the droplet viscosity is much less than the viscosity of the continuous fluid     

On the drag force of bubbles in bubble swarms at intermediate and high Reynolds numbers

An accurate and fast simulation of large-scale gas/liquid contact apparatusses, such as bubble columns, is essential for the optimization and further development of many (bio)chemical and metallurgical processes. Since it is not feasible to simulate an entire industrial-scale bubble column in full detail from first principles (direct numerical simulations), higher-level models rely on algebraic closure relations to account for the most important physical phenomena prevailing at the smallest length and time scales, while keeping computational demands low. The most important closure for describing rising bubbles in a liquid is the closure for the drag force, since it dominates the terminal rise velocity of the bubbles.Due to the very high gas loadings used in many industrial processes, bubble–bubble (or ‘swarm’) interactions need to be accounted for in the drag closure. An advanced front-tracking model was employed, which can simulate bubble swarms up to 50% gas hold-up without the problem of (numerical) coalescence. The influence of the gas hold-up for mono-disperse bubble swarms with different bubble diameters (i.e. Eötvös numbers) was quantified in a single drag correlation valid for the intermediate to high Reynolds numbers regime . Also the physical properties of the liquid phase were varied, but the simulation results revealed that the drag force coefficient was independent of the Morton number. The newly developed correlation has been implemented in a larger-scale model, and the effect of the new drag closure on the hydrodynamics in a bubble column is investigated in a separate paper (Lau et al., this issue).     

Steady incompressible fluid flow over a bundle of cylinders at moderate reynolds numbers

The complete Navier-Stokes equations describing the steady flow of incompressible Newtonian fluids normal to an array of long cylinders have been solved numerically using the finite difference method in terms of the stream function and vorticity variables. The inter-cylinder interactions have been mimicked using the well known free surface cell model. Extensive information on the detailed structure of the flow field in terms of the surface vorticity distribution, stagnation pressure, stream line and iso-vorticity line plots, as well as on the values of the integral quantities, such as pressure, friction and total drag coefficients, have been obtained under wide ranges of conditions as follows: 0.3 ≤ ? 0.99 and 0.01 ≤ Re ≤ 100. The numerical results presented herein have been validated using the appropriate theoretical and experimental results available in the literature; the match between the present predictions and the scant experimental results is good.     

Drag and mass transfer in non-newtonian flows through multi-particle systems at low reynolds numbers

An approximate solution for the motion of an assemblage of solid spheres moving through a power-law fluid at low Reynolds numbers is obtained using HapThe theoretical predictions based on Kuwabara's zero-vorticity cell model are very similar to those based on Happel's model. An approximation technique     

Numerical study on the hydrodynamics of agglomerates at intermediate Reynolds numbers

The flow pattern and hydrodynamics of a heterogeneous permeable agglomerate in a uniform upward flow at intermediate Reynolds numbers(1–40) are analyzed from three-dimensional(3 D) computational fluid dynamics simulations. Different from the homogeneous or stepwise-varying permeability models used in previous papers, a continuously radially varying permeability model is used in the present study. The effects of two dimensionless parameters, the Reynolds number and the permeability ratio, on the flow field and the hydrodynamics were investigated in detail. The results reveal that unlike the solid sphere, a small recirculating wake initially forms inside the agglomerate. The critical Reynolds number for the formation of the recirculating wake is lower than that of the solid sphere and it decreases with the increase of permeability ratio. A correlation of drag coefficient as a function of the Reynolds number and permeability ratio is proposed. Comparisons of drag coefficients obtained by different permeability models show that at intermediate Reynolds numbers(1–40),the effect of radially varying permeability on the drag coefficient must be considered.     

A numerical study of steady incompressible newtonian fluid flow over a disk at moderate reynolds numbers

The hydrodynamic behaviour of a single thin disk settling steadily, broad face-wise, in an incompressible Newtonian fluid has been investigated by solving the Navier-Stokes equations numerically using an iterative procedure. The finite difference method has been used to map out the complete flow domain in terms of the values of the stream function and vorticity. Finally, the drag coefficient was evaluated by considering the energy dissipation. Extensive results on drag for a disk of thickness to diameter ratio of 0.05 show excellent agreement with the experimental, as well as, previous scant numerical results in the range 1 ≤ Re ≤ 100.