Bubble surface, deformation

Fig. 9.2. A sketch of the three consecutive stages of the binary coalescence process. Two bubbles are approaching each other. The bubble surfaces deform and a thin liquid film is created between them. The liquid drains thinning the film, and hydrodynamic instabilities cause film rupture. The final result of binary bubble coalescence is a new larger bubble.

In devolatilizing systems, however, Ca 1 and the bubbles deform into slender S-shaped bodies, as shown in Fig. 8.12. Hinch and Acrivos (35) solved the problem of large droplet deformation in Newtonian fluids. They assumed that the cross section of the drop is circular, of radius a, and showed that the dimensionless bubble surface area, A, defined as the ratio of the surface area of the deformed bubble A to the surface area of a spherical bubble of the same volume, is approximated by (36) ... 

Bubble deformation in shear flow increases mass transfer because of the increase in surface area and because of convection. The latter brings volatile-rich liquid to the bubble surface. Favelukis et al. (39) studied the (identical but experimentally easier) reverse problem of dissolution of a gas bubble in a sheared liquid, both theoretically and experimentally, and they confirmed the increase of mass transfer with increasing shear rate. They also showed that the rate of dissolution, da/dt, where a is the equivalent radius of the bubble, is given by... 

The theory of effective viscosity has been developed by Betherton [172], Hirasaki and Lawson [173], Falls et al. [171] and Kovscek and Radke [153]. It was shown that the effective viscosity is a sum of three terms the first accounts for the contribution of the slugs of liquid between bubbles, the second is the resistance against surface deformation in the advancement of bubbles through the capillaries (pores) and the third is the gradient of surface tension caused by the withdrawal of the surfactant (from the bubble front to the bubble back). The experimental data of Falls et al. [171], Hirasaki and Lawson [173], Ettinger and Radke [166] for bead packs and Berea core agree with the calculations from Hirasaki and Lawson s models [173],... 

One complication is that the boundary conditions (4-264)-(4-266) must be applied at the bubble surface, which is both unknown [that is, specified in terms of functions R(t) and fn(9,tangent unit vectors n and t, that appear in the boundary conditions are also functions of the bubble shape. In this analysis, we use the small-deformation limit s 1 to simplify the problem by using the method of domain perturbations that was introduced earlier in this chapter. First, we note that the unit normal and tangent vectors can be approximated for small e in the forms... 

It will be noted that this result is independent of the azimuthal index /, and it is thus convenient to drop this subscript so that au is denoted simply as a. Further, the mode k = 1 is a special case as we can see by writing the expression for the bubble surface for a pure k = 1 deformation, namely,... 

On deformation of the system, the bubbles are deformed, which increases their Laplace pressure p. Moreover, some films between particles are stretched and others are compressed, causing surface tension gradients to form, which also needs energy. Above a certain stress, yielding may occur, which means that bubbles (or drops) start to slip past each other, which generally occurs in planes about parallel to the direction of flow. Calculation of the shear modulus and the yield stress from first principles is virtually impossible because of the intricacy of the problem for a three-dimensional and polydisperse system, but trends can be predicted. One relation is that these parameters are proportional to the average apparent Laplace pressure... 

Beside the capillary wave techniques, the oscillating bubble method belongs to the first experiments for measuring the surface dilational elasticity (Lunkenheimer Kretzschmar 1975, Wantke et al. 1980, 1993). For soluble adsorption layers it allows of the exchange of matter at a harmonically deformed bubble surface to be determined. 

Comparing (9.51) with (8.107) it is easy to verify that under the condition (7.29) the bubble surface velocity decreases. This can be explained in the following way. For a completely retarded surface, AT is required to be of the same order of magnitude as in the absence of electrostatic retardation. However, a substantially smaller deformation of the adsorption layer at the bubble and surface velocity v are required to create such adsorption gradient because the supply of surfactant anions from the bulk solution is retarded. 

Unlike the original flotation whose elementary act is complicated by an inertia impact and the accompanying deformation of bubble surface, microflotation is completely a colloid chemical process and it can be described in terms of modem colloid chemistry as orthokinetic heterocoagulation (Deijaguin Dukhin, 1960). 

With very large particles the liquid interlayer thinning process is complicated by the deformation of the bubble surface by an inertia impact of the particle. It was shown by Derjaguin et al. (1977) and Dukhin Rulyov (1977) that in the inertia-free deposition of small particles on a bubble surface its deformation under the influence of the hydrodynamic pressing force is insignificant. This third important feature facilitates the development of a quantitative kinetic theory of flotation of small particles. 

Inertia forces manifest themselves in at least in three effects 1 - shift of particles trajectories away from the liquid stream-lines from the bubble (Section 10.1) 2 - deformation of the bubble surface by a particle at the instant of impact (cf. Section 11.1) 3 - prevention of particle deposition on some parts of the bubble surface due to centrifugal forces (cf Section 11.3). In the present section these effects are considered simultaneously (Dukhin et al. 1986), although some restrictions are imposed on the properties of the bubbles and the particles. 

At St > St the inertial impact of a particle deforms the bubble surface, creates a thin water layer between the particle and bubble and makes the particle jump back from the bubble surface. This decreases the collision efficiency which otherwise rises with the particles size. For particles of subcritical diameter the collision efficiency increases with particle diameter. The proposed theory of inelastic collision unlike other theories describes the coupling of inertial bubble-particle interaction and water drainage from the liquid interlayer. 

The important factor influencing on specific surface area of phase interface is deformation of drops (bubbles) surface that in general case is caused by dynamic head under the effect of turbulent pulsations of disperse medium rate and (or) phases movement rate because of the difference in their densities. In this case the minimal size of dispersion phase particles dcr undergoing to deformation may be calculated from the ratio characterizing stability of phase interface (1.23) and (1.24). 

With their lower viscosity, bubbles will deform more readily than emulsion droplets and, therefore, be relatively more prone to depart from Stokes law behaviour. Hadamard and Rybczynski developed a terminal velocity equation for the creaming of bubbles with a mobile surface ... 

Surface tension is a liquid property that tends to counter bubble deformation and encourages bubble breakup (Akita and Yoshida, 1974 Mehmia et al 2005 Walter and Blanch, 1986). The result is a more stable bubble interface that leads to smaller bubble diameters, a more stable flow regime (Lau et al., 2004 Schafer et al., 2002), and higher gas holdups and interfacial areas (Kluytmans et al., 2001). It is also thought that a lower surface tension leads to a higher contact time because the liquid flow over the bubble surface is slowed (Lau et al., 2004). 

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