Drag coefficient—reynolds number transition for gas bubbles rising steadily in viscous fluids

Recently, the author (Rodrigue, 2001) developed a generalized correlation for the motion of gas bubbles rising steadily in uncontaminated viscous Newtonian fluids of infinite extent. It is the purpose of this note to show that this model can be written in an explicit form for the drag coefficient as a function of the Reynolds and Morton numbers. This new correlation is then used to predict the position of the minimum in a graphical representation of C_D versus Re. It is shown that the model can predict quite nicely this hydrodynamic transition for viscous fluids.     

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.     

Modelling of transport of two proppant‐laden immiscible power‐law fluids through an expanding fracture

Hydraulic fracturing technology is one of the most popular and efficient reservoir stimulation techniques in oil and gas industry since it allows significant increase of the hydrocarbon production rate. A possible advancement of this technology involves consequent pumping of several immiscible fluids laden with solid particles (proppant) into a fracture. This technique has a potential for better control of the proppant placement and, thus, a better control of the fracture shape. The major result of the present work is a mathematical model and a computational code for investigation of a flow of two immiscible slurries inside an expanding fracture. © 2012 Canadian Society for Chemical Engineering     

Rising velocity of a swarm of spherical bubbles in a power law non-newtonian liquid

Previous work on the rise of a swarm of bubbles in non-Newtonian media has been reviewed. Variational principles have been combined with the Happel free surface cell model to obtain upper and lower bounds on the swarm velocities of bubbles rising slowly through power law liquids. The predictions presented herein encompass wide ranges of gas holdup and shear-thinning behaviour.     

Three‐dimensional numerical simulation of coalescence and interactions of multiple horizontal bubbles rising in shear‐thinning fluids

The dynamics of multiple horizontal bubbles rising from different orifice arrangements in shear‐thinning fluids was simulated numerically by three‐dimensional Volume of Fluid method. The effects of bubble size, rheological properties of shear‐thinning fluids, and orifice structure arrangements on multiple bubbles interaction and coalescence were analyzed, and the mechanisms of bubble coalescence and breakup were fully discussed and elucidated. The variation of bubble rising velocity during coalescence process and freely rising processes for different orifice arrangements was also deeply investigated. The critical initial horizontal intervals for coalescence of multiple horizontal bubbles with various orifice arrangements were attained by simulation, which could serve as the critical criterion of bubble coalescence or noncoalescence. Furthermore, the critical bubble interval was predicted based on the film drainage model, the prediction accords well with the simulation result and is quite conducive for the design and optimization of perforated gas–liquid contact equipment. © 2015 American Institute of Chemical Engineers AIChE J, 61: 3528–3546, 2015     

Unified entry length for newtonian and power–law fluids in laminar pipe flow

A numerical method based on finite differencing is used for investigating the steady–state entrance region laminar flow of incompressible Newtonian and power–law fluids in a circular pipe. The Solution method is validated by comparing the results for Newtonian fluids with those reported in the literature. For power–law fluids, the entry length results are compared with other approximate solutions in the literature. On the basis of the calculated results, a generalized entry length ξ99 = 0.056 is shown to be valid for the laminar flow at Re > 200 of both Newtonian and power–law fluids with 0.75 < n < 1.5.     

Semianalytical solution for power‐law polymer solution flow in a converging annular spinneret

A semianalytical solution for a power‐law fluid flowing through a conical annulus was derived to estimate the velocity profile in the axial direction, the shear rate and the elongation rate within a spinneret during the spinning of hollow fiber membranes. The angle coefficient was introduced as a new parameter to account for the effect of radial flow and to modify the governing equation, which initially neglected the effect of radial flow. The results estimated from this semianalytical solution agreed more closely with computational fluid dynamics simulation results than those obtained from the approximate analytical solution in our previous study. By accurately predicting the velocity profile in the axial direction and the shear and elongation rates in a conical annulus, the solution derived in this study is expected to provide a reliable criterion for spinneret design to achieve a specified membrane morphology with a desired performance. © 2015 American Institute of Chemical Engineers AIChE J, 61: 3489–3499, 2015     

Breakup dynamics of slender bubbles in non‐newtonian fluids in microfluidic flow‐focusing devices

This study aims to investigate the breakup of slender bubbles in non‐Newtonian fluids in microfluidic flow‐focusing devices using a high‐speed camera and a microparticle image velocimetry (micro‐PIV) system. Experiments were conducted in 400‐ and 600‐μm square microchannels. The variation of the minimum width of gaseous thread with the remaining time before pinch‐off could be scaled as a power‐law relationship with an exponent less than 1/3, obtained for the pinch‐off of bubbles in Newtonian fluids. The velocity field and spatial viscosity distribution in the liquid phase around the gaseous thread were determined by micro‐PIV to understand the bubble breakup mechanism. A scaling law was proposed to describe the size of bubbles generated in these non‐Newtonian fluids at microscale. The results revealed that the rheological properties of the continuous phase affect significantly the bubble breakup in such microdevices. © 2012 American Institute of Chemical Engineers AIChE J,, 2012     

Pressure drop for single and two‐phase flow of non‐newtonian liquids in helical coils

New experimental results on pressure loss for the single and two‐phase gas‐liquid flow with non‐Newtonian liquids in helical coils are reported. For a constant value of the curvature ratio, the value of the helix angle of the coils is varied from 2.56° to 9.37°. For single phase flow, the effect of helix angle on pressure loss is found to be negligible in laminar flow regime but pressure loss increases with the increasing value of helix angle in turbulent flow conditions. On the other hand, for the two‐phase flow, the well‐known Lockhart‐Martinelli method correlates the present results for all values of helix angle (2.56‐9.37°) satisfactorily under turbulent/laminar and turbulent/turbulent conditions over the following ranges of variables as: 0.57 ≤ n′ ≤ 1; Re′ < 4000; Re_l < 4000; Re_g < 8000; 8 ≤ x ≤ 1000 and 0.2 ≤ De′ ≤ 1000.     

Time‐resolved ultrasonic spectroscopy for bubbles

We show that ultrasound can provide time‐resolved measurements of the size distribution and the concentration of bubbles in a liquid. The potential of the technique is demonstrated by following disappearance of bubbles having an average radius of 20 μm with a 10 ms time resolution. We show that our technique can detect small concentrations of bubbles, with a large spectrum of accessible bubble radii (from 80 nm to 40 μm for a gas volume fraction of ), and with a sub‐millisecond time resolution. This new technique could be a valuable tool for investigating rapid processes such as nucleation or dissolution of bubbles. © 2017 American Institute of Chemical Engineers AIChE J, 63: 4666–4672, 2017     

Effect of gas expansion on the velocity of individual Taylor bubbles rising in vertical columns with water: Experimental studies at atmospheric pressure and under vacuum

This study was designed to determine the effect of gas expansion on the velocity of Taylor bubbles rising individually in a vertical column of water. This experimental study was conducted at atmospheric pressure or under vacuum (33.3 and ) using three different acrylic columns with internal diameters of 0.022, 0.032, and 0.052 m, and more than 4.0 m high. A non-intrusive optical method was used to measure velocity and length of Taylor bubbles at five different locations along the columns. The operating conditions used correspond to inertial controlled regime.In experiments performed under vacuum, there is considerable gas expansion during the rise of Taylor bubbles, particularly when they approach the liquid free surface where the pressure drop (due to the hydrostatic pressure) is of the order of magnitude of the absolute pressure. The liquid ahead of the bubble is displaced upward by an amount proportional to the gas expansion resulting in increased bubble velocity. The calculated Reynolds number suggests a laminar regime in the liquid ahead of the bubble. However, the experimentally determined velocity coefficient C for each column was much smaller than 2, which would be expected for laminar flow. The value of C obtained ranges from 1.13±0.09, for the narrowest column, to 1.40±0.24, for the widest column. This suggests that a fully developed laminar flow in the liquid ahead of the bubble is never achieved due to continuous bubble expansion at a variable rate, regardless of column height.The velocity coefficient C can be used to calculate the contribution of liquid motion to bubble velocity. Subtracting this contribution from the measured bubble velocity defines a constant value which is nearly identical to the bubble rise velocity measured in the same column operated as a constant volume system (two ends closed) where gas expansion is absent.     

宽Re范围内气泡和液滴曳力系数的关联模型

流粒(气泡或液滴)的曳力系数CD和上升/终端速度因有助于准确预测反应器内相含率分布、液相速度分布、流粒停留时间和传质速率而具有重要意义.但现有用于估算流粒CD的关联式大多分段且只在低雷诺数Re区间内有效,并难以同时准确预测不同实验体系和操作条件下的实验结果.针对这些不足,基于实验测量和理论分析,本工作提出了一个能够在整...     

Flow around individual Taylor bubbles rising in stagnant CMC solutions: PIV measurements

The flow around single Taylor bubbles rising in non-Newtonian solutions of Carboxymethylcellulose (CMC) polymer was studied using simultaneously particle image velocimetry (PIV) and shadowgraphy. This technique made it possible to determine the correct position of the bubble interface. Solutions of polymer with weight percentage varying from 0.1 to 1.0 wt% were used to cover a wide range of flow regimes. The rheological fluid properties and pipe dimension yielded Reynolds numbers between 4 and 714 and Deborah numbers for the higher concentration solutions between 0.001 and 0.236. The shape of the bubbles rising in the different solutions was compared. The flow around the nose of the bubbles was found to be similar in all the studied conditions. Velocity profiles in the liquid film around the bubble were measured and different wake structures were found. With increasing solution viscosity, the wake flow pattern varied from turbulent to laminar, and a negative wake was observed for the higher polymer concentration solutions.     

An experimental investigation of the cross‐flow of power law liquids past a bundle of cylinders and in a bed of stacked screens

The steady flow of power‐law polymer solutions normal to arrays of cylinders and in a bed of screens has been investigated experimentally. Extensive pressure drop measurements have been made for three different test cells (two tube bundles and one bed made of screens) of different voidages for a series of inelastic carboxymethyl cellulose and sodium alginate aqueous solutions. The resulting values of friction factor correlate well with a modified definition of the Reynolds number based on the capillary bundle representation of the bed. The new data extend considerably the range of voidage values from ～0.6 to ～0.87. Extensive comparisons have also been made between the present experimental data and our previous calculations based on the use of simple cell models. The close correspondence between the two demonstrates the utility of such idealized analyses. All in all, the present results embrace the following ranges of physical and kinematic conditions: Reynolds number: 0.01 to ～1200; power‐law flow behaviour index: 0.38 to 1; and three values of voidage, namely, 0.74, 0.78 and 0.87.     

Mathematical predictions of gas permeability for automotive air‐bags

The mathematical relationship combining an applied pressure drop and the resultant gas velocity through a woven fabric is important to the air‐bag industry to predict the performance of new materials before they are woven. The main difficulty in formulating a mathematical solution is the complexity of the woven fabric structure. In available publications, fabric pores had normally been represented as a series of cylindrical pipes. This article considers the same approach and analyzes some of the equations to review their industrial applicability. Because none of the equations have been found adaptable in predicting the permeability behavior of air‐bag fabrics, experimental data have been used to generate empirical equations. The data were generated using a dynamic air permeability tester that was used to project air at high pressure through a variety of air‐bag fabrics. A static permeability tester was also used to generate results through the same fabrics at lower pressures. The final equations combine the fabric cover factor and the pressure differential to give the resultant gas velocity. © 2000 John Wiley & Sons, Inc. J Appl Polym Sci 77: 2104–2112, 2000     

Natural convection in shear‐thinning yield stress fluids in a square enclosure

The influence of viscoplastic rheological features on the Rayleigh‐Bénard convection is investigated by numerical means in order to compare with first experimental results given by Darbouli et al. The fluid is modeled by a regularized Herschel‐Bulkley law which is often used to fit numerous pasty fluids. Natural convection in a two‐dimensional square cavity heated from below is considered. Critical values of Oldroyd number Od and yield number Y are provided. Numerical results highlight a stabilizing effect of the yield stress as well as a destabilizing effect of increasing shear‐thinning coefficient n as the increase in n enhances the heat transfer in the range of our calculations. Unyielded regions are located in the square corners of the cavity and in the cavity where convection occurs. The unyielded zones size increases with the increase in Od and can invade all the cavity for sufficiently large values of Od. © 2015 American Institute of Chemical Engineers AIChE J, 62: 1347–1355, 2016     

Hindered settling in non-newtonian power law liquids

New experimental results on the hindered settling of model glass bead suspensions in non-Newtonian suspending media are reported. The data presented encompass the following ranges of variables: 7.38 × 10^?4 ≤ Re_1∞ ≤ 2; 0.0083 ≤ d/D ≤ 0.0703; 0.13 ≤ C ≤ 0.43 and 1 ≥ n ≥ 0.8. In these ranges of conditions, the dependence of the hindered settling velocity on concentration is adequately represented by the corresponding Newtonian expressions available in the literature. The influence of the power law flow behaviour index is completely embodied in the modified definition of the Reynolds number used for power law liquids.