Ellipsoidal bubble

Growing gas content makes bubbles merge until ultimately they occupy the central part of the pipe section, reaching 1 metre and more in length these long bubbles are separated from the pipe wall by a thin layer of water. They have the shape of an ellipsoid at the front, and are cut off at the back, not unlike artillery shells. Hence the term, shell mode. 

Moreover, ellipsoidal bubbles and drops commonly undergo periodic dilations or random wobbling motions which make characterization of shape particularly difficult. Chapter 7 is devoted to this regime. 

The conditions under which fluid particles adopt an ellipsoidal shape are outlined in Chapter 2 (see Fig. 2.5). In most systems, bubbles and drops in the intermediate size range d typically between 1 and 15 mm) lie in this regime. However, bubbles and drops in systems of high Morton number are never ellipsoidal. Ellipsoidal fluid particles can often be approximated as oblate spheroids with vertical axes of symmetry, but this approximation is not always reliable. Bubbles and drops in this regime often lack fore-and-aft symmetry, and show shape oscillations. 

Because of their practical importance, water drops in air and air bubbles in water have received more attention than other systems. The properties of water drops and air bubbles illustrate many of the important features of the ellipsoidal regime. 

The generalized graphical correlation presented in Fig. 2.5 gives one method of estimating terminal velocities of drops and bubbles in infinite liquid media. For more accurate predictions, it is useful to have terminal velocities correlated explicitly in terms of system variables. To obtain such a correlation is especially difficult for the ellipsoidal regime where surface-active contaminants are important and where secondary motion can be marked. 

General criteria for determining the shape regimes of bubbles and drops are presented in Chapter 2, where it is noted that the boundaries between the different regions are not sharp and that the term ellipsoidal covers a variety of shapes, many of which are far from true ellipsoids. Many bubbles and drops in this regime undergo marked shape oscillations, considered in Section F. Where oscillations do occur, we consider a shape averaged over a small number of cycles. 

Few observations have been reported on wakes of ellipsoidal bubbles and drops at Re > 1000. Yeheskel and Kehat (Y4) characterized shedding in this case as random. However, Lindt (L7, L8) studied air bubbles in water and distinguished a regular periodic component of drag associated with an open helical vortex wake structure. Strouhal numbers (defined as 2af/Uj, where / is the frequency and 2a is the maximum horizontal dimension) increase with Re, to level off at about 0.3 as bubbles approach the transition between the ellipsoidal and spherical-cap regimes. 

Surface-active contaminants play an important role in damping out internal circulation in deformed bubbles and drops, as in spherical fluid particles (see Chapters 3 and 5). No systematic visualization of internal motion in ellipsoidal bubbles and drops has been reported. However, there are indications that deformations tend to decrease internal circulation velocities significantly (MI2), while shape oscillations tend to disrupt the internal circulation pattern of droplets and promote rapid mixing (R3). No secondary vortex of opposite sense to the prime internal vortex has been observed, even when the external boundary layer was found to separate (Sll). 

The flow and shape transitions for small and intermediate size bubbles and drops are summarized in Fig. 7.13. In pure systems, bubbles and drops circulate freely, with internal velocity decreasing with increasing k. With increasing size they deform to ellipsoids, finally oscillating in shape when Re exceeds a value of order 10. In contaminated systems spherical and nonoscillating ellipsoidal... 

Fig. 8.6 Dimensionless wake volumes for ellipsoidal-cap and spherical-cap bubbles and drops, compared with solid spherical-caps.

Fig. 8.7 Streamlines in the outer fluid relative to an ellipsoidal-cap and a spherical-cap bubble, after Bhaga (B3). Points on streamlines show positions at intervals of 0.03 sec.

Transfer from large bubbles and drops may be estimated by assuming that the front surface is a segment of a sphere with the surrounding fluid in potential flow. Although bubbles are oblate ellipsoidal for Re < 40, less error should result from assumption of a spherical shape than from the assumption of potential flow. 

There is considerable evidence (D3, G7, PI, P4, SI) that bubbles in liquid metals show the behavior expected from studies in more conventional liquids. Because of the large surface tension forces for liquid metals, Morton numbers tend to be low (typically of order 10 ) and these systems are prone to contamination by surface-active impurities. Figure 8.10a shows a two-dimensional nitrogen bubble in liquid mercury. For experimental convenience, the bubbles studied have generally been rather large, so that there are few data available for spherical or slightly deformed ellipsoidal bubbles in liquid metals. Data... 

For large bubbles where inertia effects are dominant, enclosed vertical tubes lead to bubble elongation and increased terminal velocities (G7). The bubble shape tends towards that of a prolate spheroid and the terminal velocity may be predicted using the Davies and Taylor assumptions discussed in Chapter 8, but with the shape at the nose ellipsoidal rather than spherical. The maximum increase in terminal velocity is about 16% for the case where 2 is small (G6) and 25% for a bubble confined between parallel plates (G6, G7) and occurs for the enclosed tube relatively close to the bubble axis. 

Most bubbles in gas-solid fluidized beds are of spherical cap or ellipsoidal cap shape. Configurations of two basic types of bubbles, fast bubble (clouded bubble) and slow bubble (cloudless bubble), are schematically depicted in Fig. 9.7. The cloud is the region established... 

When the hydrodynamic conditions correspond to Re = 1500 (R = 2 - 10 mm) bubbles become strongly deformed when rising. They acquire the shape of a flat ellipsoid and begin to vibrate and move on a spiral trajectory. In fact their size does not influence the velocity of rise [14]. The following relation was derived from the results on the velocity of rise of large bubbles reported in [16]... 

Fig. 2.25 illustrates a device for the study of the deformation of a spherical foam film in an electric field, proposed in [130]. The surfactant solution is fed into the glass cell through the tube 2, so that its level reaches the porous plate 4. When air is blown through the tube 3, a foam bubble forms at the capillary orifice. An electric field is created between electrode 1 and 6, which deforms the foam film (bubble). The bubble transforms from spherical to ellipsoidal shape. The value of the deformation A/ depends on the surface tension of the film... 

For ellipsoidal gas bubbles, Mendelsen (1967) has given the following correlation for terminal bubble rise velocity ... 

Up — Uc represents the resultant slip velocity between the particulate and continuous phase. Some other commonly used drag coefficient correlations are listed in Appendix 4.2. For fluid particles such as gas bubbles or liquid drops, the drag coefficient may be different than that predicted by the standard drag curve, due to internal circulation and deformation. For example, Johansen and Boysen (1988) proposed the following equation to calculate Cd, which is valid for ellipsoidal bubbles in the range 500 < Re < 5000 ... 

---