Freely rising bubbles

Because of its large velocity, a freely rising bubble has a diffusion layer much thinner that in the described experiments. This effect can manifest itself only if the particles are small enough so that their thermal motion becomes significant. Thus, electro- and diffusiophoresis should be taken into account in describing the Brownian diffusion of sub-micron particles towards the bubble s mobile surface under the conditions of a sufficiently low electrolyte concentration. The influence of diffusiophoretic transport to the surface of a rising bubble through its diffusion layer is theoretically proved by Zholkovsky et al. (1983). 

Figure 4.6. Scheme of a two-compartment model for oxygen transfer in a stirred tank. The liquid phase is structured into a mixed zone (LJ and into a zone with freely rising bubbles in analogy to a bubble column (L2), with a connecting exchange (Fp) due to impeller pumping. (From Oosterhuis, 1984.)... 

The mechanisms by which freely rising bubbles interact with each other in relatively low-viscosity liquids and, specifically, how they approach, contact, and coalesce or break up are important aspects of multi-phase flow. Coalescence and breakup can control the interfacial area and mass transfer rate in bubble columns and gas-sparged chemical and biological reactors. Bubble interaction is fundamental in two-phase flow instability that plagues boilers and oil and gas wells. But bubble interaction remains a relatively mysterious area. 

When combined with boundary layer analysis, potential flow theory provides estimates for the velocity of a rising bubble. The lowest-order result for the terminal velocity of a freely rising bubble is... 

To disperse gas, the gas is usually injected into the liquid from the bottom of the tank or near the impeller to enhance dispersion. Disk style turbines are found to be most convenient for gas dispersion because the disk disturbs the freely rising gas bubbles. The turbines with flat blades give radial flow and are very useful for gas dispersion where the gas is introduced just below the impeller at its axis and drawn up to the blades and chopped into fine bubbles. 

In reactor modeling we need to deal with the behavior of the bubbling bed as a whole rather than single rising bubbles. In extending the simple two-phase theory, Davidson and Harrison [29] proposed that the average velocity of bubbles in a freely bubbling bed can be approximated by... 

Small bubbles in polar liquids can be treated as rigid spheres because of the effects of surfactants. Provided that the bubble Reynolds number is 0(1) or smaller, one can use the Maxey-Riley equation, which is given in Eq. (14). However, this may be a very restrictive assumption. For example, based on its rise velocity, a 120-/im bubble in water has a Reynolds number roughly equal to unity. It reasonable to assume that bubbles as large as 1 mm are spherical in water. Using Ryskin and Leal s [94,95,96] finite difference methods, McLaughlin [68] found that the axis ratio of a freely rising 1-mm bubble in pure water (i.e., a mobile interface) was 1.12. However, the axis ratio of a fully contaminated 1 -mm bubble was 1.01. In the latter case, the Reynolds number based on the rise velocity and the equivalent spherical diameter was 110. This... 

For bubbles and drops rising or falling freely in infinite media it is possible to prepare a generalized graphical correlation in terms of the Eotvds number, Eo Morton number, M and Reynolds number. Re (Gl, G2) ... 

As noted in Chapter 2, bubbles and drops remain nearly spherical at moderate Reynolds numbers (e.g., at Re = 500) if surface tension forces are sufficiently strong. For drops and bubbles rising or falling freely in systems of practical importance, significant deformations from the spherical occur for all Re > 600 (see Fig. 2.5). Hence the range of Re covered in this section, roughly 1 < Re < 600, is more restricted than that considered in Section II for solid spheres. Steady motion of deformed drops and bubbles at all Re is treated in Chapters 7 and 8. 

With a rising voltage across the electrolyzer the back electromotive force will rise in proportion, because gases will be evolved and dissolved at a higher pressure in the platinum electrodes. When this pressure rises to 1 atm., the gases will be able to overcome the ambient atmospheric pressure and begin to escape freely into the atmosphere in the form of bubbles. Should this process proceed under reversible conditions and should the concentration of hydrochloric acid during electrolysis remain constant, the back electromotive force... 

The drag coefficient for freely falling spherical droplets (or rising gas bubbles) of Newtonian fluids in power-law liquids at low Reynolds munber has been approximately evaluated and, in the absence of smface tension effects, it is given by equation (5.4), i.e. 

Haberman, W.L., Morton, R.K. (1953). An experimental investigation of the drag and shape of air bubbles rising freely in various liquids. TMB Report 802. Washington DC. 

MODELING COALESCENCE OF BUBBLE CLUSTERS RISING FREELY IN LOW-VISCOSITY LIQUIDS... 

Stewart, C. W., Coalescence of Ellipsoidal Bubbles Rising Freely in Low-Viscosity Liquids, Ph.D. Dissertation, Washington State University, Pullman, Washington (1993). 

On studying the tables, it will be seen that there is a number of links between them. For example, knowledge of how small bubbles rise through a liquid will be of use in the design of immiscible liquid-liquid and solid-liquid gravity settlers as well as gas-liquid gravity separators, since all three depend upon the same basic principle namely, the application of Newton s laws of motion to a freely moving body. 

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