Liquid velocity distribution in two-phase bubble flow

An improved analytical treatment is developed which makes possible the satisfactory prediction of the liquid velocity distribution in two-phase bubble flow.In the analysis, the shear stress in the liquid phase is regarded as important. When the fluctuation of turbulent velocity can be subdivided into two components: one due to the inherent liquid turbulence independent of the existence of the bubble, (u′, υ′), and the other due to the additional liquid turbulence by the bubble agitation, (u″, υ″), it is possible to split the shear stress into two components, - ?$\text{u′υ′}$ and - ?$\text{u″υ″}$ corresponding to (u′, υ′) and (u″, υ″), respectively.A basic equation for the liquid velocity distribution is derived from further development of this treatment. The agreement between the measured velocity profiles and those calculated is quite close especially in the core region of a duct.     

Dynamics of a Taylor bubble in steady and pulsatile co-current flow of Newtonian and shear-thinning liquids in a vertical tube

A computational analysis is carried out to ascertain the effects of steady and pulsatile co-current flow, on the dynamics of an air bubble rising in a vertical tube containing water or a solution of Carboxymethylcellulose (CMC) in water. The mass fraction (m_f) of CMC in the solution is varied in the range 0.1% ⩽ m_f ⩽ 1% to accommodate zero-shear dynamic viscosities in the range 0.009–2.99 Pa-s. It was found that the transient and time-averaged velocities of Taylor bubbles are independent of the bubble size under both steady as well as pulsatile co-current flows. The lengths of the Taylor bubbles under the Newtonian conditions are found to be consistently greater than the corresponding shear-thinning non-Newtonian conditions for any given zero-shear dynamic viscosity of the liquid. In contrast to observations in stagnant liquid columns, an increase in the dynamic viscosity of the liquid (under Newtonian conditions) results in a concomitant increase in the bubble velocity, for any given co-current liquid velocity. In shear-thinning liquids, the change in the bubble velocity with an increase in m_f is found to be relatively greater at higher co-current liquid velocities. During pulsatile shear-thinning flows, distinct ripples are observed to occur on the bubble surface at higher values of m_f, the locations of which remain stationary with reference to the tube for any given pulsatile flow frequency, while the bubble propagated upwards. In such a pulsatile shear-thinning flow, a localised increase in dynamic viscosity is accompanied near each ripple, which results in a localised re-circulation region inside the bubble, unlike a single re-circulation region that occurs in Newtonian liquids, or shear-thinning liquids with low values of m_f. It is also seen that as compared to frequency, the amplitude of pulsatile flow has a greater influence on the oscillating characteristics of the rising Taylor bubble. The amplitude of oscillation in the bubble velocity increases with an increase in the CMC mass fraction, for any given value of pulsatile flow amplitude.     

Local properties of vertical mercury-argon two-phase flow in a circular tube under transverse magnetic field

Experimental data are presented in this paper on the profiles of local void fraction, bubble impaction rate, bubble velocity and its spectrum, and also bubble length and its spectrum, of mercury-argon two-phase slug flow flowing upwards in a vertical circular tube in the presence of a transverse magnetic field. Decrease in void fraction and increase in bubble velocity are significant when the magnetic flux density is larger than $\text{0.3～0.4}$$\text{T}\text{(Ha ? 100)}$. This effect is discussed by analyzing the bubble size distribution. Recovery of local void fraction profile in the downstream of an obstacle and diffusion of void injected from only one nozzle in the presence of magnetic field are also discussed.     

Void fraction,bubble velocity and bubble size in two-phase flow

Experiments were performed in atmosphereic vertical air-water flows, for void fractions between 0.25 and 0.75 (cross-sectional averages) and superficial liquid velocities of 1.3, 1.7 and 2.1 m/s. Local values of void fraction and bubble velocity as well as the bubble diameter were measured by means of a resistivity probe technique. Reliable values were obtained for the local void fraction over the entire range $\text{0 ≤ α ≤ 1}$. The void fraction profiles appeared to have a local maximum at the pipe center, local maxima close to the wall were obviously absent. The resistivity probes are shown to measure the velocity of the interface between the conducting and nonconducting phases, which equals the gas velocity only for low void fractions. The measured data for void fraction and bubble velocity were correlated by means of power law distribution functions, with exponents given by a function of the cross-sectionally averaged void fraction. The Sauter mean diameters for the bubble size spectra found, agree reasonably well with diameters predicted by a theoretical model based on the energy dissipation in the flow.     

Slip velocity measurements in an upward bubbly flow by combined LDA and electrodiffusional techniques

 An experimental technique for the measurement of the local slip velocity of spherical bubbles is reported. It is based on the measurement of the local liquid velocity by an electrodiffusional method, and the bubble velocity by a specially adapted LDA (Laser Doppler anemometer) with a short measuring volume. The bubble velocity is measured taking into account the shift between the bubble centre and the centre of the LDA measuring volume. The slip velocity is obtained by subtracting the liquid velocity from the bubble velocity at the point corresponding to the bubble centre. The technique is applicable for flows with high velocity gradients. Results of the slip velocity measurements in an upward bubbly flow at laminar pipe Reynolds numbers are presented. Received: 25 July 1996/Accepted: 13 April 1998     

Hydrodynamic and mass transfer characteristics of slug flow in a vertical pipe with and without dispersed small bubbles

In this work, we present a numerical study to investigate the hydrodynamic characteristics of slug flow and the mechanism of slug flow induced CO₂ corrosion with and without dispersed small bubbles. The simulations are performed using the coupled model put forward by the authors in previous paper, which can deal with the multiphase flow with the gas–liquid interfaces of different length scales. A quasi slug flow, where two hypotheses are imposed, is built to approximate real slug flow. In the region ahead of the Taylor bubble and the liquid film region, the presence of dispersed small bubbles has less impacts on velocity field, because there are no non-regular intensive disturbance forces or centrifugal forces breaking the balance of the liquid and the dispersed small bubbles. In the liquid slug region, the strong centrifugal forces generated by the recirculation below the Taylor bubble lead to the effect of heterogeneity, which makes the profile of the radial liquid velocity component sharper with higher volume fraction of dispersed small bubbles. The volume fraction has a maximum value in the range of r/R = 0.5–0.6. Meanwhile, it is usually higher than 0.35, which means that larger dispersed bubbles can be formed by coalescences in this region. These calculated results are in good agreement with experimental results. The wall shear stress and the mass transfer coefficient with dispersed small bubbles are higher than those without dispersed small bubbles due to enhanced fluctuations. For short Taylor bubble length, the average mass transfer coefficient is increased when the gas or liquid superficial velocity is increased. However, there may be an inflection point at low mixture superficial velocities. For the slug with dispersed small bubbles, the product scales still cannot be damaged directly despite higher wall shear stress. In fact, the alternate wall shear stress and the pressure fluctuations perpendicular to the pipe wall with high frequency are the main cause for breaking the product scales.     

Creeping flow through cubic arrays of spherical bubbles

We consider creeping flow through a cubic array of identical spherical bubbles and compute the drag force exerted on a representative bubble in the array using a method originally employed by Hasimoto (1959) and recently modified by Sangani & Acrivos (1982). In addition to deriving analytic expressions for the drag to $\text{0(c}{}^2\text{)}$, we present numerical results for the complete range of bubble volume fractions $\text{c}$ for the three cubic arrays.     

Liquid flow patterns about barbotage bubbles

This paper summarizes the results of a flow visualization study on the liquid motion around barbotage bubbles during growth and departure. Flow patterns, as well as for the first time, instantaneous velocities, are reported as a function of time and location about the bubbles. The experiments, employing the hydrogen-bubble technique and high-speed cine photography, were with: water as the liquid, air as the bubbled gas, orifice diameters of 0.116 and 0.252 cm, and different air flow rates; the two limiting cases of constant supply pressure and constant volumetric flow rate were covered. It was found that the liquid around a barbotage bubble assumes two velocity maxima, the first an outward maximum during bubble growth and the second in the opposite direction approximately at the time of bubble departure; further, liquid velocities were found to be higher close to the bubbling site. Certain differences in liquid velocities between the constant pressure and constant flow cases are explained in terms of available theoretical solutions to the bubble growth rate. Qualitative comparisons of the barbotage liquid flow patterns and those recently reported for boiling flow patterns are also presented.     

Local liquid velocity measurements in air-water bubbly flow

Local measurements of axial liquid velocity were performed for vertical upward air-water bubbly flow in a 101.6-mm inner-diameter round pipe by using a laser Doppler anemometer (LDA) and a hot-film anemometer (HFA). The data reduction approaches for both the LDA and HFA are discussed in detail. A threshold scheme with the information of local void fraction and velocity distribution in single-phase flow was applied to the LDA to approximately discriminate liquid velocity signals from those of the bubble interface velocity. Furthermore, a formulation was given to account for the effect of the bubble relative velocity on the liquid in the front and wake regions of the bubbles. For the HFA, an amplitude threshold scheme and a slope criterion were used to extract liquid velocity information. To reduce the measurement uncertainty, the experiments were performed in flow conditions where the area-averaged void fraction was less than 20%. The experimental results showed satisfactory agreement between the liquid volumetric flow rates calculated by area integration of the local liquid velocity and void fraction measurements, and the measured value by a magnetic flow meter. Also, the area-averaged relative velocity between the gas and liquid phases obtained from the current measurements agreed well with previous research.     

Dynamics of a liquid with gas bubbles

The hydrodynamics and diffusion of an admixture near an isolated bubble, which simulates the rise of either a chain of identical bubbles or a system of regularly arranged bubbles of the same volume, are analyzed by solving the Navier-Stokes equations numerically. Data are presented for a specific liquid. It is shown that in both cases the maximum flow velocity on the surface of identical bubbles is practically the same, although in the former case the ascent velocity is considerably higher. The stationary admixture diffusion from a bubble also proves to be nearly the same.In relation to the bubbling of a gas through a liquid layer, it is shown that the total admixture diffusion is maximum for regularly arranged bubbles whose diameter is comparable with the liquids capillary constant. Although the flow past the bubble remains continuous, the values of the hydrodynamic parameters are no longer small.Novosibirsk. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 3, pp. 75–88, May–June, 1996.     

Interaction between Taylor bubbles rising in stagnant non-Newtonian fluids

The interaction between Taylor bubbles rising in stagnant non-Newtonian solutions was studied. Aqueous solutions of carboxymethylcellulose (CMC) and polyacrylamide (PAA) polymers were used to study the effect of different rheological properties: shear viscosity and viscoelasticity. The solutions studied covered a range of Reynolds numbers between 10 and 714, and Deborah numbers up to 14. The study was performed with pairs of Taylor bubbles rising in a vertical column (0.032 m internal diameter) filled with stagnant liquid. The velocities of the leading and trailing bubbles were measured by sets of laser diodes/photocells placed along the column. The velocity of the trailing bubble was analysed together with the liquid velocity profile in the wake of a single rising bubble (Particle Image Velocimetry data obtained from the literature). For the less concentrated CMC solutions, with moderate shear viscosity and low viscoelasticity, the interaction between Taylor bubbles was similar to that found in Newtonian fluids. For the most concentrated CMC solution, which has high shear viscosity and moderate viscoelasticity, a negative wake forms behind the Taylor bubbles, inhibiting coalescence since the bubbles maintain a minimum distance of about 1D between them. For the PAA solutions, with moderate shear viscosity but higher viscoelasticity than the CMC solutions, longer wake lengths are seen, which are responsible for trailing bubble acceleration at greater distances from the leading bubble. Also in the PAA solutions, the long time needed for the fluid to recover its initial shear viscosity after the passage of the first bubble makes the fluid less resistant to the trailing bubble flow. Hence, the trailing bubble can travel at a higher velocity than the leading bubble, even at distances above 90D.     

Calculating liquid droplets settling on a cylinder in gas-liquid spray flow

Adding atomized liquid to air flowing around a cylinder gives an appreciable increase in heat transfer by forming a liquid film on the cylinder surface. The heat transfer coefficient depends upon the amount of liquid forming the film, which is limited by two phenomena: droplet deflection from the liquid film on the surface and droplets not striking the cylinder. This paper presents a method of calculating the quantity of liquid droplets settling on a cylinder surface in a gas-liquid spray flow. A coefficient $\text{k}$, the volume ratio of the liquid entering the film to the amount of liquid directed at the cylinder, is introduced. $\text{k}$ values were calculated by means of numerical computation and the theory verified experimentally. The calculation method permits estimation of the dependence of the amount of liquid settling on a cylinder on the droplet diameter distribution parameters and on the linear gas velocity     

A novel numerical scheme for the investigation of surface tension effects on growth and collapse stages of cavitation bubbles

In the present study the effects of surface tension on the growth and collapse stages of cavitation bubbles are studied individually for both spherical and nonspherical bubbles. The Gilmore equation is used to simulate the spherical bubble dynamics by considering mass diffusion and heat transfer. For the collapse stage near a rigid boundary, the Navier–Stokes and energy equations are used to simulate the flow domain, and the VOF method is adopted to track the interface between the gas and the liquid phases. Simulations are divided into two cases. In the first case, the collapse stage alone is considered in both spherical and nonspherical situations with different conditions of bubble radius and surface tension. According to the results, surface tension has no significant effects on the flow pattern and collapse rate. In the second case, both the growth and collapse stages of bubbles with different initial radii and surface tensions are considered. In this case surface tension affects the growth stage considerably and, as a result, the jet velocity and collapse time decrease with increasing surface tension coefficient. This effect is more significant for bubbles with smaller radii.     

Local gas and liquid phase velocity measurement in a miniature stirred vessel using PIV combined with a new image processing algorithm

A method which combines standard two-dimensional particle image velocimetry (PIV) with a new image processing algorithm has been developed to measure the average local gas bubble velocities, as well as the local velocities of the liquid phase, within small stirred vessel reactors. The technique was applied to measurements in a gas–liquid high throughput experimentation (HTE) vessel of 45 mm diameter, but it is equally suited to measurements in larger scale reactors. For the measurement of liquid velocities, 3 μm latex seeding particles were used. For gas velocity measurements, a separate experiment was conducted which involved doping the liquid phase with fluorescent Rhodamine dye to allow the gas–liquid interfaces to be identified. The analysis of raw PIV images enabled the detection of bubbles within the laser plane, their differentiation from obscuring bubbles in front of the laser plane, and their use in lieu of tracer particles for gas velocity analysis using cross-correlation methods. The accuracy of the technique was verified by measuring the velocity of a bubble rising in a vertical glass column. The new method enabled detailed velocity fields of both phases to be obtained in an air–water system. The overall flow patterns obtained showed a good qualitative agreement with previous work in large scale vessels. The downward liquid velocities above the impeller were greatly reduced by the addition of the gas, and significant differences between the flow patterns of the two-phases were observed.     

Robust numerical analysis of the dynamic bubble formation process in a viscous liquid

The dynamics of bubble formation from a submerged nozzle in a highly viscous liquid with relatively fast inflow gas velocity is studied numerically. The numerical simulations are carried out using a sharp interface coupled level set/volume-of-fluid (CLSVOF) method and the governing equations are solved through a hydrodynamic scheme with formal second-order accuracy. Numerical results agree well with experimental results and it is shown that the sharp interface CLSVOF method enables one to reproduce the bubble formation process for a wide range of inflow gas velocities. From numerical results, one can improve their understanding of the mechanisms regarding the dynamics of bubble formation. For example, it is found that for some sets of parameters that the bubble formation process reaches steady state after several bubbles are released from the nozzle. At steady state, bubbles uniformly rise freely in the viscous liquid. It is observed that the fluid flow around a formed bubble has a significant role in determining the overall dynamic process of bubble formation; e.g. the effect of the fluid flow from the preceding bubble can be seen on newly formed bubbles.     

Eulerian–Lagrangian 3-D simulations of unsteady two-phase gas–liquid flow in a rectangular column by considering bubble interactions

This work discusses the development of a three-dimensional Eulerian–Lagrangian CFD model for a gas–liquid flow in a rectangular column. The model resolves the time-dependent, three-dimensional motion of small gas bubbles in a liquid to simulate the dynamic characteristics of the oscillating bubble plume. Our model incorporates drag, gravity, buoyancy, lift, pressure gradient and virtual mass forces acting on a bubble rising in a liquid, and accounts for two-way momentum coupling between the phases. We use MUSIG model that provides a framework in which the population balance method together with the break up and coalescence models can be incorporated into three-dimensional CFD calculations. We use turbulent flow to describe liquid flow field. The standard κ–ε of turbulence is selected for calculating the properties of turbulent flow. The effect of aspect ratio of the column on the flow pattern, liquid velocity and gas hold-up profiles is discussed.     

Small amplitude pressure wave in subsonic and supersonic bubble flows in convergent-divergent nozzles

This paper makes a theoretical analysis of the propagation phenomena of the small amplitude pressure wave in the subsonic and supersonic bubble flow with a velocity slip between bubble and liquid in the convergent-divergent nozzle. From an analysis of the time-mean flow, the nondimensional parameter $\text{m = {u}\text{2}\text{G}\text{·α(1 ? α)ρ}\text{l}\text{β(2 ? 1/S)/P·[αβS + (1 ? α)βS}{}^2\text{+ α(1 ? α)]}}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$ corresponds to Mach number is gasdynamics where u_G is the gas velocity, α: the void fraction, ρ_L: the liquid density, P: the pressure, S: the velocity ratio of the gas and liquid flows and β: the proportional constant for the virtual mass. From a theoretical analysis of the small disturbance field, it is clarified that the parameter m also plays an essential and important role as Mach number, although the propagation performance of the disturbance is very complicated compared with that in gasdynamics. It is also shown that the pressure waves are divided into four groups depending on the velocity ratio S. Two of them are rather realistic, but the other two are required of a further investigation in future.     

Boundary layer in compressible flow of wet water vapour

In this paper the influence of small droplets, with radius $\text{10}{}^{\mathrm{?8}}\text{m}\text{< r < 10}{}^{\mathrm{?6}}\text{m}$, on laminar and turbulent boundary layer behavior is considered. It is found that the laminar boundary layer in a two-phase flow with strongly dispersed liquid retains dissipation energy and that the recovery factor of enthalpy is greater than unity. In turbulent boundary layers small droplets are transported by turbulent diffusion and this leads to the recovery factor being less than unity. Its value in both cases depends mainly on the nondimensional number $\text{Ds}\text{= C}{}_{\mathrm{Le}}\text{L/(U}{}_{\mathrm{e}}{}^2\text{/2)}$. The laminar boundary layer solution for non-equilibrium two-phase flow is obtained. Profiles of the droplet mass fraction, vapour and droplets temperatures and droplet radius are computed for the case of a steady two-dimensional flow. The turbulent boundary layer is treated using a semi-empirical theory assuming thermodynamic equilibrium.     

Numerical study on the effect of initial flow velocity on liquid film thickness of accelerated slug flow in a micro tube

Numerical simulation of air–water slug flows accelerated from steady states with different initial velocities in a micro tube is conducted. It is shown that the liquid film formed between the gas bubble and the wall in an accelerated flow is significantly thinner than that in a steady flow at the same instantaneous capillary number. Specifically, the liquid film thickness is kept almost unchanged just after the onset of acceleration, and then gradually increases and eventually converges to that of an accelerated flow from zero initial velocity. Due to the flow acceleration, the Stokes layer is generated from the wall, and the instant velocity profile can be given by superposition of the Stokes layer and the initial parabolic velocity profile of a steady flow. It is found that the velocity profile inside a liquid slug away from the bubble can be well predicted by the analytical solution of a single-phase flow with acceleration. The change of the velocity profile in an accelerated flow changes the balance between the inertia, surface tension and viscous forces around the meniscus region, and thus the resultant liquid film thickness. By introducing the displacement thickness, the existing correlation for liquid film thickness in a steady flow (Han and Shikazono, 2009) is extended so that it can be applied to a flow with acceleration from an arbitrary initial velocity. It is demonstrated that the proposed correlation can predict liquid film thickness at Re < 4600 within the range of ±10% accuracy.