Bubble motion

If the bed is slugging, bubble motion is retarded by the bed wall, and the bed or tube diameter, Z9, rather than the actual bubble diameter, determines the bubble rise velocity, ie... 

For NRt < 1, the problem of bubble motion is closely related to that of the motion of a liquid drop in a liquid medium, and can consequently be derived from the Rybczynski-Hadamard formula (H2, R13) ... 

Figure 9. Bubble radius and pressure transients of the water vapor inside the bubbles. The first maximum in pressure at 650 ps marks the collapse of the bubbles. The following modulations are only expected for oscillatory bubble motion.

The boiling Reynolds number or bubble Reynolds number (Re,) is defined as the ratio of the bubble inertial force to the liquid viscous force, which indicates the intensity of liquid agitation induced by the bubble motion ... 

Fig. 4. Dynamics of bubble motion. Laser-induced cavitation in silicone oil upper portion is the experimental observations at 75,000 frames/second lower curve compares the experimentally observed radius versus theory. [W. Lauterborn (47).]...

Prediction of Critical Sizes. In order to use the above model for actual predictions, it is necessary to assign values to the relative velocity U0 this is, at the present level of knowledge, an extremely difficult task since, due to bubble motion (and perhaps the presence of fixed and moving internals in a fluid bed such as, for example, draft tubes) the particle movement in a fluidized bed is extremely complex. Some crude estimates of the relative velocity between particles have been made (Ennis etal., 1991) and these were expressed as... 

Vaux (1978), Ulerich et al. (1980) and Vaux and Schruben (1983) proposed a mechanical model of bubble-induced attrition based on the kinetic energy of particles agitated by the bubble motion. Since the bubble velocity increases with bed height due to bubble coalescence, the collision force between particles increases with bed height as well. The authors conclude that the rate of bubble-induced attrition, Rbub, is then proportional to the product of excess gas velocity and bed mass or bed height, respectively,... 

Zhang Y. and Finch J.A. (2001) A note on single bubble motion in surfactant solutions. /. Fluid Mech. 429, 63-66. 

The fundamental approach used was that of hydrodynamics to obtain solutions of equations for the conservation of mass, momentum arid energy. It is convenient to express these equations in vector notation and to consider small amplitude waves separately from waves of finite amplitude. In what follows, we will first discuss the shock effects of underwater expins and then proceed to a quantitative description of gas bubble motion... 

So far we have been considering theoretical treatments of underwater shock effects. Now we turn our attention to a theoretical description of bubble motion... 

These observations about the bubble motion are the basis of all the bubble theories which lead to numerical predictions of bubble radius, migration and period. It is a common characteristic of such theories that changes in density of the water surrounding the bubble are neglected (the noncompressive approximation), and it is further assumed that the bubble retains a spherical form thruout its motion. From what has been said, it is evident that both these assumptions are plausible as far as the expanded phase of the motion is concerned. They must, however, be increasingly poor as the bubble approaches its minimum radius for which very much larger pressures and acceleration are involved... 

Most of the features of the theoretical treatment of bubble motion are present in the treatment that considers the water incompressible and neglects gravity effects. We quote from Cole (Ref 1, Chapt 8) The simplest approximation to the true motion of the bas bubble is the one in which it is assumed that the motion of the surrounding water is entirely radial and there is no vertical migration. In this approximation, which has been discussed by a number of writers, the hydrostatic buoyance resulting from differences in hydrostatic pressure at different depths is neglected. It is thus assumed that at an infinite distance from the bubble in any direction the pressure has the same value as the initial hydrostatic pressure P0 at the depth of the charge... 

For pressure changes of the order 15 lb/in2, such as prevail over most of the bubble motion, the corresponding changes in density are of the order 10-4po, where p0 is the equilibrium density. Under these conditions, the derivatives of density p are easily seen to be negligible in the first of Eqs (8.1), which then becomes... 

Figure 6-60 gives the drag coefficient as a function of bubble or drop Reynolds number for air bubbles in water and water drops in air, compared with the standard drag curve for rigid spheres. Information on bubble motion in non-Newtonian liquids may be found in Astarita and Apuzzo (AIChE J., 11, 815-820 [1965]) Calderbank, Johnson, and Loudon (Chem. Eng. Sci., 25, 235-256 [1970]) and Acharya, Mashelkar, and Ulbrecht (Chem. Enz. Sci., 32, 863-872 [1977]). 

In bubbling fluidization, bubble motion becomes increasingly vigorous as the gas velocity increases. This behavior can be reflected in the increase of the amplitude of the pressure fluctuations in the bed. With further increase in the gas velocity, the fluctuation will reach a maximum, decrease, and then gradually level off, as shown in Fig. 9.16. This fluctuation variation marks the transition from the bubbling to the turbulent regime. 

Bubble motion appears to be more random with enhanced interphase exchange and hence intimate gas-solid contact and high heat and mass transfer. 

It should be mentioned that this deviation is more model-dependent than mechanistic because the real gas-solid contact is much poorer than that portrayed by plug flow, on which Eq. (12.36) is based [Kunii and Levenspiel, 1991]. The deviation can also be related to the effects of the particle boundary layer reduction due to particle collision and the generation of turbulence by bubble motion and particle collision [Brodkey et ah, 1991]. 

For the bed-to-surface heat transfer in a dense-phase fluidized bed, the particle circulation induced by bubble motion plays an important role. This can be seen in a study of heat transfer properties around a single bubble rising in a gas-solid suspension conducted... 

The bubble motion will also determine the solid circulation in the bed, as a consequence of the entrainment of solids in the bubble wake. For an assumed ratio of wake to bubble volume, the volumetric circulation rate of solids per unit cross-sectional area of bed is equal to (uQ-umf)a. The solid circulation time is then equal to h/(uo-um )a. For the values uq = 2m/s, = 0.4m/s,... 

A slight variant on the plume model of Park et al (42) is shown in Fig. 9. Coal particles injected at the bottom of the bed are convected upward and diffuse radially as a consequence of bubble motion. Volatiles will be released at a decaying rate for several seconds, depending upon particle size and bed temperature. The radius r of the cross-section over which the volatiles are released is determined by the solid diffusivity VQ. To a first approximation... 

The problem of bubble motion in a liquid is fundamental in chemical reaction engineering because about 25% of all chemical reactions occur in bi-phase systems. As we have shown, the gas-liquid two-phase flow prevailing in a bubble column is extremely complex. It is dominated by a rich variety of logical configurations and exhibits inherent unsteadiness. As a consequence, the modelling of this flow is an attractive subject and constitutes an excellent subject for stochastic modelling. 

Fig. 12. Typical results reported by Tomiyama ei al. (1993) on the effect of the Morton number M (atEotvSs number Eo = 10) on the shape and dynamics of a single bubble rising in (a) a Newtonian liquid, and (b) graphical correlation due to Grace (1973) and Grace et al. (1976). [Part (a) reprinted from Nuclear Engineering and Design, Volume 141, Tomiyama, A., Zun, I., Sou, A., and Sakaguchi, T., Numerical analysis of bubble motion with the VOF method, pp. 69-82, Copyright 1993, with permission from Elsevier Science. Part (b) reprinted from Grace, R., Clift, R., and Weber, M.E., Bubbles, Drops, and Particles. Academic Press, Orlando, 1976. Reprinted by permission of Academic Press.)...

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