Two coalescing bubbles

Figure 3.16 Geometry of two coalescing bubbles in the adherence region in the case of a (a) flat and (b) cylindrical electrode.

Figure 2 Bubbles bursting at the bed surface, which are ejecting solids into the freeboard (a) from the roof, (b) from the wake of a single bubble, and (c) from the wake of two coalescing bubbles.

There are many examples in science of reciprocal relationships of this kind. For example the focal length of a thin lens is related to the object and image distances, measured from the lens, by such a relationship. Also the total resistance due to two resistors in parallel is obtained by such a relationship, likewise for the sum of two series capacitors. The two coalescing bubbles could, in principle, be used as an analogue system for solving lens problems, resistor problems, etc. In the case of the lens problem it is necessary to produce two coalescing bubbles, one with its radius proportional to the object... 

In the case of two coalescing bubbles the radii of curvature of the three spherical surfaces, r, r, and rc, are related by the reciprocal relation... 

Using Fig. 4.17 we can obtain a geometric result for two coalescing bubbles that will enable us to use Eq. (4.31). The similar triangles AOC and BDC give... 

Rd Radius of the liquid disk between two coalescing bubbles (m)... 

Siemes and Weiss (SI4) investigated axial mixing of the liquid phase in a two-phase bubble-column with no net liquid flow. Column diameter was 42 mm and the height of the liquid layer 1400 mm at zero gas flow. Water and air were the fluid media. The experiments were carried out by the injection of a pulse of electrolyte solution at one position in the bed and measurement of the concentration as a function of time at another position. The mixing phenomenon was treated mathematically as a diffusion process. Diffusion coefficients increased markedly with increasing gas velocity, from about 2 cm2/sec at a superficial gas velocity of 1 cm/sec to from 30 to 70 cm2/sec at a velocity of 7 cm/sec. The diffusion coefficient also varied with bubble size, and thus, because of coalescence, with distance from the gas distributor. 

Figure 15.14 Coalescence of two air bubbles rising in a stirred tank reactor (From [29]).

A. i.e,. E — da/din A. If the film of liquid separating two neighboring bubbles in a foam develops a thin spot, the surface tension gradient in the vicinity of the thin spot will induce a Muraugoni flow of liquid toward Ihe direction of higher cr. This flow of liquid toward the thin spot helps heal the fluctuation and thus keeps the neighboring bubbles from coalescing. 

In addition Delnoij et al. used the VOF method to compute the coalescence of two coaxial gas bubbles of identical size generated at the same orifice. Figure 27a shows the temporal evolution of the positions of the two gas bubbles. From the sequence of bubble positions it can be seen that the trailing bubble moves faster than the leading bubble and eventually at t = 0.42 s coalescence of the two gas bubbles commences. From computer animations it could clearly be seen that just after completion of the coalescence process (at t = 0.45 s) a splashing liquid... 

Brereton, G., and Korotney, D., Co-axial and oblique coalescence of two rising bubbles. In Dynamics of Bubbles and Vortices Near a Free Surface, ASME, AMD 119, 1 (1991). 

Coalescence. This is caused by rupture of the film between two emulsion drops or two foam bubbles. The driving force is the decrease in free energy resulting when the total surface area is decreased, as occurs after film rupture. The Laplace equation (Section 10.5.1) plays a key role. 

Secondary Bjerknes forces arise when two oscillating bubbles are present in a pressure field. Attractive forces between bubbles with inphase pulsation cause coalescence. Bubbles oscillating out of phase are repelled. The bubble oscillation is in phase when both bubbles are smaller or larger than the resonance size and attractive forces dominate. If one bubble is smaller and one larger than the resonance size, they oscillate in and out of phase and repel one another. 

This dimensionless equation has recently appeared in the literature (Prince and Blanch 1990) to describe the physics of bubble coalescence. Here, y represents dimensionless film thickness (joining two touching bubbles) and x represents dimensionless time. The solution (credited to R.G. Rice) is a classic example on applications of the p-substitution method, so replace p = dy/dx, and for the second derivative, assume piy) since x is not explicit... 

FIGURE 12.2. When two gas bubbles are in contact, but do not rupture and coalesce, the radius of curvature of the septum between bubbles will be concave toward the smaller bubble and have a radius to equal to rir2l r - t2). 

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