Mass transfer spherical bubble

A film model for the transfer between the hquid slug and the catalytic surface kis can be given by the same expression. The mass transfer between bubbles and Uquid slugs through the approximately spherical caps (Figure 8.13) is predicted from the penetration theory by... 

The bubbles shapes in gas purging vary from small spherical bubbles, of radius less than one centimen e, to larger spherical-cap bubbles. The mass transfer coefficient to these larger bubbles may be calculated according to the equation... 

Yoshitome et al. (Y2) examined mass transfer from single samples of benzoic acid suspended in an air-water bubble-column. Spherical, cylindrical, and disk-shaped samples of diameters from 25 to 75 mm were used,... 

Most theoretical studies of heat or mass transfer in dispersions have been limited to studies of a single spherical bubble moving steadily under the influence of gravity in a clean system. It is clear, however, that swarms of suspended bubbles, usually entrained by turbulent eddies, have local relative velocities with respect to the continuous phase different from that derived for the case of a steady rise of a single bubble. This is mainly due to the fact that in an ensemble of bubbles the distributions of velocities, temperatures, and concentrations in the vicinity of one bubble are influenced by its neighbors. It is therefore logical to assume that in the case of dispersions the relative velocities and transfer rates depend on quantities characterizing an ensemble of bubbles. For the case of uniformly distributed bubbles, the dispersed-phase volume fraction O, particle-size distribution, and residence-time distribution are such quantities. 

In this section, a general formulation will be given for the effect of bubble residence-time and bubble-size distributions on simultaneous and thermodynamically coupled heat- and mass-transfer in a multicomponent gas-liquid dispersion consisting of a large number of spherical bubbles. Here one can... 

The rate of mass-transfer, unlike the terminal velocity, may reach its lower limit only when the whole surface of the drop or bubble is covered by the adsorbed film. In the absence of surface-active material, the freshly exposed interface at the front of the moving drop (due to circulation here) could well be responsible for as much mass transfer as occurs in the turbulent wake of the drop. The results of Baird and Davidson 67a) on mass transfer from spherical-cap bubbles are not inconsistent with this idea, and further experiments on smaller drops are in progress in the author s laboratory. In general, if these ideas are correct, while the rear half of the drop is noncirculating (and the terminal velocity has reached the limit of that for a solid sphere), the mass transfer at the front half of the drop may still be much higher, due to the circulation, than for a stagnant drop. Only when sufficient surface-active material is present to cover the whole of the surface and eliminate all circulation will the rate of mass-transfer approach its lower limit. 

Figure 12-15 Sketch of concentration profiles between a spherical bubble and a solid spherical catalyst particle in a continuous liquid phase (upper) in a gas-liquid sluny reactor or between a bubble and a planar solid wall (lower) in a catalytic w bubble reactor, It is assmned that a reactant A must migrate from the bubble, tirough the drop, md to tiie solid catdyst smface to react. Concentration variations may occur because of mass transfer limitations around both bubble and solid phases.

For the mass transfer coefficient between the bubble and its cloud due to diffusion, kbC, Davidson and Harrison (1963) also derived the following expression by assuming a spherical cap bubble with 0W = 50° (Problem 12.5)... 

Mass Transfer from "Spherical" Ends of the Cas Bubble... 

Another study carried out by these authors [93] modeled the collapsing motion of a single bubble near an electrode surface, and equations for the motion of a spherical gas bubble were obtained. The jet speed and water hammer pressure during jet flow (liquid jet) were calculated, and when the jet speed was 120 m/s, the water hammer pressure was approximately 200 MPa upon the electrode surface. This pressure played an important part in the fineness of the crystal deposits. Mass transfer during the electrode reaction was by turbulent diffusion. The diffusion layer thickness was reduced to approximately 1/10th its size in the presence of the ultrasonic field. The baths contained the ions Cl-, SO -, and Zn2+. The ultrasonic frequency employed in the experiments was 40 kHz and it was seen that ultrasound considerably increased the deposition rate and current efficiency, as well as the smoothness and hardness of the deposit. Microscopy studies showed that the... 

In this section we present expressions for the mass transfer coefficients for diffusion in spherical and cylindrical geometries. The results presented here are useful in the modeling of mass transfer in, for example, gas bubbles in a liquid, liquid droplets in a gas, or gas jets in a liquid as shown in Figure 9.6. 

Insofar as the mass-transfer coefficient for clean bubbles is concerned (see, for example, the review by Clift et al. (1978)), in the case of spherical bubbles moving under creeping (or Stokes) flow conditions, the following correlation has been proposed Sh = 1 + (1 + 0.564Pe ). For spherical particles with Rep > 70 the Sherwood number can be expressed by the following relationship (Lochiel Calderbank, 1964) ... 

Problem 9-10. Mass Transfer From a Spherical Gas Bubble for Re -C 1, Pc -C 1. Consider a spherical gas bubble and suppose that it contains a gaseous species A that is soluble in the liquid that surrounds the bubble and a second dominant species B that is insoluble in... 

Problem 9-20. Mass Transfer From a Spherical Bubble in an Extensional Flow. Consider a spherical gas bubble that is suspended in a liquid that undergoes an axisymmetric extensional flow. The bubble contains a component A that is soluble in the exterior liquid... 

Spherical bubble as Re —> 0, 0 < Pe < oo. The problem of mass transfer to a spherical bubble in a translational flow as Re 0 was studied numerically in [321], The results for the mean Sherwood number can be approximated well by the expression... 

Convective heat and mass transfer to spherical-cap bubbles was studied in... 

We consider a laminar steady-state flow past a solid spherical particle (drop or bubble) of radius a and study transient mass transfer to the particle surface. At the initial time t = 0, the concentration in the continuous phase is constant and equal to C, whereas for t > 0 a constant concentration Cs is maintained on the particle surface. 

Statement of the problem. In the preceding chapters we considered processes of mass transfer to surfaces of particles and drops for the case of an infinite rate of chemical reaction (adsorption or dissolution.) Along with the cases considered in the preceding chapters, finite-rate surface chemical reactions (see Section 3.1) are of importance in applications. Here the concentration on the surfaces is a priori unknown and must be determined in the course of the solution. Let us consider a laminar fluid flow with velocity U past a spherical particle (drop or bubble) of radius a. Let R be the radial coordinate relative to the center of the particle. We assume that the concentration is uniform remote from the particle and is equal to C. Next, the rate of chemical reaction on the surface is given by Ws = KSFS(C), where Ks is the surface reaction rate constant and the function F% is defined by the reaction kinetics and satisfies the condition Fs(0) = 0. 

Let us consider steady-state mass transfer on a spherical particle (drop or bubble) of radius o in a laminar fluid flow. We assume that a volume chemical reaction proceeds in the continuous phase with Wv = KVFV(C). The reactant transfer in the continuous phase is described in dimensionless variables by the equation... 

MASS TRANSFER TO DROPS AND BUBBLES. When small drops of liquid are falling through a gas, surface tension tends to make the drops nearly spherical, and the coefiBcients for mass transfer to the drop surface are often quite close to those for solid spheres. The shear caused by the fluid moving past the drop surface, however, sets up toroidal circulation currents in the drop that decrease the resistance to mass transfer both inside and outside the drop. The extent of the change depends on the ratio of the viscosities of the internal and external fluids and on the presence or absence of substances such as surfactants that concentrate at the interface. ... 

H. S. Lee and H. Merte, Spherical Vapour Bubble Growth in Uniformly Superheated Liquids, Int. J. Heat Mass Transfer (39) 2427-2447,1996. 

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