Motion of bubbles and drops

Drops and bubbles, in addition, imdergo deformation because of the differences in the pressmes acting on various parts of the surface. Thus, when a drop is falling in a quiescent medium, both the hydrostatic and impact pressures will be greater on the forward than on the rear face this will tend to flatten the drop. Conversely, the viscous drag will tend to elongate it. 

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. 

This equation is valid for only mildly shear-thinning behaviom (n 0.6 or so). For a Newtonian fluid (n = 1), equation (5.25) gives X = 1/Qi = 2/3. This simple expression does not, however, account for either smface tension 

The usefulness of these studies is severely limited by the various simplifying assumptions which have been made concerning shape and siuface tension effects. Indeed, a variety of shapes of drops and bubbles has been observed imder free fall conditions in non-Newtonian fluids and these differ significantly from those in Newtonian fluids [Clift et al., 1978 Chhabra, 1993a]. 

In addition, many workers have reported experimental data on various aspects (such as shapes, coalescence, terminal falling velocity) of bubble and drop motion in a non-Newtonian continuous phase. 

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There has been relatively little work on the motion of bubbles and drops in well-characterized turbulent flow fields. There is some evidence (B3, K7) that mean drag coefficients are not greatly altered by turbulence, although marked fluctuations in velocity (B3) and shape (K7) can occur relative to flows which are free of turbulence. The effect of turbulence on splitting of bubbles and drops is discussed in Chapter 12. 

As for steady motion, shape changes and oscillations may complicate the accelerated motion of bubbles and drops. Here we consider only acceleration of drops and bubbles which have already been formed formation processes are considered in Chapter 12. As for solid spheres, initial motion of fluid spheres is controlled by added mass, and the initial acceleration under gravity is g y - l)/ y + ) (El, H15, W2). Quantitative measurements beyond the initial stages are scant, and limited to falling drops with intermediate Re, and rising... 

An excellent review of work through 1972 is J. F. Harper, The motion of bubbles and drops through liquids, Adv. Appl. Mech. 12, 59-129 (1972). 

V G. Levich, Motion of gas bubbles with high Reynolds numbers, Zh. Eksp. Teor. Fiz. 19, J8-24 (1949). An excellent summary of theoretical and experimental work (up to 1972) on the motion of bubbles and drops is, J. F. Elarper, The motion of bubbles and drops through liquids, Adv. Appl. Mech. 12, 59-129 (1972). 

Liquid-liquid reactors are similar to gas-liquid reactors. In the former case, the dispersed phase is in the form of droplets as against bubbles in the latter. The motion of bubbles and drops can be described using a unified approach. A spray column (or a drop column) is the equivalent of a bubble column but with one difference. The dispersed gas phase is always lighter than the continuous liquid phase (p < Pl)- However, the dispersed liquid phase in spray columns may be lighter or heavier than the continuous immiscible liquid phase. Nevertheless, spray columns can be easily described similar to bubble columns. Furthermore, packed bubble columns and sectionalized bubble columns can be considered equivalent to packed extraction columns and plate extraction columns. External-loop and internal-loop reactors are also possible (for equivalent gas-liquid reactors, refer to Section 11.4.2.1.4). 

Subramanian, R. S., The motion of bubbles and drop in reduced gravity. In Transport Processes in Bubbles, Drop and Particle, pp. 1-42, Hemisphere, New York, 1992. 

Little is known about the effect of visco-elasticity on the motion of bubbles and drops in non-Newtonian fluids, though a preliminary study suggests that spherical bubbles are subject to a larger drag in a visco-elastic than in an inelastic liquid. Recent surveys clearly reveal the paucity of reliable experimental data on the behaviour of fluid particles in non-Newtonian liquids [Chhabra, 1993a DeKee et al., 1996]. 

Subramanian, R.S. 1992 The motion of bubbles and drops in reduced gravity, in Transport Processes in Bubbles, Drops and Particles, Subramanian, Chabra and DeKee, Eds., Hemisphere Publishing, New York, NY. 

Subramanian, R.S. Balasubramaniam, R. The Motion of Bubbles and Drops in Reduced Gravity, Cambridge University Press Cambridge, England, 2000. 

A considerable amount of experimental results on the free motion of bubbles and drops in quiescent non-Newtonian media is available in the literature. In most cases, the two-parameter power-law model has been used to model the ambient liquid rheology. It is worthwhile to mention here that in the bulk of the work dealing with the free fall of liquid drops in polymer solutions, the ratio of the viscosity of the dispersed phase to that of the continuous phase is in the range lO -lO" and, hence, these drops may effectively be treated as gas bubbles. At this juncture, it is also important to recall that bubbles in the pretransition region (prior to the abrupt change) behave more like solid spheres and, hence, the drag under these conditions is approximated... 

Electrophoresis of bubbles and drops is a story on its own. As long ago as 1861 Quincke ) observed the electrophoresis of small air bubbles in water. Such a motion is possible only when there is a double layer at the Interface, containing free ions. It is extremely difficult to keep oil-water or air-water Interfaces rigorously free from adsorbed ionic species. When these are present, especially for surfactants, Marangoni effects make the surface virtually inexten-slble then the drops or bubbles behave as solid spheres. Electrophoretic studies... 

A second problem, closely related to Stokes problem, is the steady, buoyancy-driven motion of a bubble or drop through a quiescent fluid. There are many circumstances in which the buoyancy-driven motions of bubbles or drops are of special concern to chemical engineers. Of course, bubble and drop motions may occur over a broad spectrum of Reynolds numbers, not only the creeping-flow limit that is the focus of this chapter. Nevertheless, many problems involving small bubbles or drops in viscous fluids do fall into this class.23... 

The coupling of the transport of momentum with the mass transport practically excludes any analytical solution in the field of physico-chemical hydrodynamics of bubbles and drops. However, a large number of effective approximate analytical methods have been developed which make solutions possible. Most important is the fact, that the calculus of these methods allows to characterise different states of dynamic adsorption layers quantitatively weak retardation of the motion of bubble surfaces, almost complete retardation of bubble surface motion, transient state at a bubble surface between an almost completely retarded and an almost completely free bubble area. 

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) ... 

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. 

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. 

Bubbles and drops of intermediate size show two types of secondary motion ... 

Fig. 7.12 Simple model to show nature of shape oscillations for bubbles and drops in free motion.

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