Wall effect for the fall of spheres in cylindrical tubes at high reynolds number

New experimental results on the wall effect for sphere motion in cylindrical tubes are presented and discussed for the conditions d/D ≤ 0.9 and Re_m ≤ 20000. Extensive comparisons with previous studies have been carried out to evaluate their predictability and to demonstrate the utility of the present results. The wall factor, defined as the ratio of settling velocity in an unbounded medium to that measured in a cylindrical tube, is found to depend on sphere-to-tube diameter ratio and on sphere Reynolds number. However, for small values of the Reynolds number (Re ≤ 0.5), as well for large values (Re ≥ 1000), the Reynolds number dependence of the wall factor disappears; in these regions, only the dependence on diameter ratio remains.     

An explicit equation for the terminal velocity of solid spheres falling in pseudoplastic liquids

An explicit equation is proposed which predicts directly the terminal velocity of solid spheres falling through stagnant pseudoplastic liquids from the knowledge of the physical properties of the spheres and of the surrounding liquid. The equation is a generalization of the equation proposed for Newtonian liquids. By properly defining the dimensionless diameter, d_*, a function of the Archimedes number, Ar, and the dimensionless velocity, U_*, a function of the generalized Reynolds number, Re, to account for the non-Newtonian characteristics of the liquid, the final equation relating these two variables has similar form to the Newtonian equation. The predictions are very good when they are compared to 55 pairs of Re—C_D for non-Newtonian data and 37 pairs for Newtonian data published previously. The root mean square error on the dimensionless velocity is 0.081 and much better than the only other equation previously proposed.     

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.     

Inertial deposition on front and rear sides of ribbons at intermediate and high reynolds numbers

Deposition of particles from gas flows on ribbons in normally expected on the front side. Collision efficiencies of ribbons are presented for this case, obtained by means of single particle trajectory computation, which is based on numerically determined flow fields in the Reynolds number range between 2 and 50 and on the potential flow model. Further, it is shown, that deposition of particles on the rear side can be caused by eddylike motion of the fluid in the wake of the ribbon. For this to occur, two conditions must be fulfilled: the free shear layer between the wake and the outer flow must be turbulent to cause lateral transfer of particles into the wake and the Stokes number may not exceed 0.5.     

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     

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.     

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.     

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     

Terminal settling velocity and drag coefficient of biofilm‐coated particles at high Reynolds numbers

The drag force (F_d) on bio‐coated particles taken from two laboratory‐scale liquid–solid circulating fluidized bed bioreactors (LSCFBBR) was studied. The terminal velocities (u_t) and Reynolds numbers (Re_t) of particles observed were higher than reported in the literature. Literature equations for determining u_t were found inadequate to predict drag coefficient (C_d) in Re_t > 130. A new equation for determining F_d as an explicit function of terminal settling velocity was generated based on Archimedes numbers (Ar) of the biofilm‐coated particle. The proposed equation adequately predicted the terminal settling velocity of other literature data at lower Re_t of less than 130, with an accuracy >85%. © 2010 American Institute of Chemical Engineers AIChE J, 2010     

Wall effects on terminal velocity of small drops in newtonian and non-newtonian fluids

This work investigated the extent of the wall effects on the free falling velocity of fluid spheres in quiescent Newtonian and pseudoplastic non-Newtonian media. The terminal velocity has been measured as a function of the physical properties of the both dispersed and continuous phases, and of falling tube diameter. It is shown that inthecreeDing flow region (Re « 1) and for the conditions when the viscosity of the dispersed phase is much smaller khan khat of khe continuous phase, the extent of wall effect is determined only by the ratio of the sizes of the settling fluid sphere and of the vessel. The same analytical relation correlates well the data for both Newtonian and non Newtonian continuous media in the range d/D < 0.45.     

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.