Bubbles spherical

A foam can be considered as a type of emulsion in which the inner phase is a gas, and as with emulsions, it seems necessary to have some surfactant component present to give stability. The resemblance is particularly close in the case of foams consisting of nearly spherical bubbles separated by rather thick liquid films such foams have been given the name kugelschaum by Manegold [175]. 

I, have a fairly broad distribution of bubble sizes and can therefore maintain spherical bubbles with significantly less Hquid. Empirically, foams with greater than about 5% Hquid tend to have bubbles that are stiH approximately spherical, and are referred to as wet foams. Such is the case for the bubbles toward the bottom of the foam shown in Figure 1. Nevertheless, it is important to note that even in the case of these wet foams, some of the bubbles are deformed, if only by a small amount. 

Fig. 3. Two-dimensional schematic illustrating the distribution of Hquid between the Plateau borders and the films separating three adjacent gas bubbles. The radius of curvature r of the interface at the Plateau border depends on the Hquid content and the competition between surface tension and interfacial forces, (a) Flat films and highly curved borders occur for dry foams with strong interfacial forces, (b) Nearly spherical bubbles occur for wet foams where...

Cf, C y, and Cq are the concentrations of the substance in question (which may be a colligend or a surfactant) in the feed stream, bottoms stream, and foamate (collapsed foam) respectively. G, F, and Q are the volumetric flow rates of gas, feed, and foamate respectively, is the surface excess in equilibrium with C y. S is the surface-to-volume ratio for a bubble. For a spherical bubble, S = 6/d, where d is the bubble diameter. For variation in bubble sizes, d should be taken as YLnid fLnidj, where n is the number of bubbles with diameter dj in a representative region of foam. 

The bubbles shapes in gas purging vary from small spherical bubbles, of radius less than one centimen e, to larger spherical-cap bubbles. The mass transfer coefficient to these larger bubbles may be calculated according to the equation... 

Also the surface area. A, of the spherical bubble is given by... 

Now consider again the boundary condition referred to above. At the point when the softened sheet first enters the mould it forms part of a spherical bubble which does not touch the sides of the cone. The volume balance is therefore... 

This equation may also be used to calculate the wall thickness distribution in deep truncated cone shapes but note that its derivation is only valid up to the point when the spherical bubble touches the centre of the base. Thereafter the analysis involves a volume balance with freezing-off on the base and sides of the cone. 

Most theoretical studies of heat or mass transfer in dispersions have been limited to studies of a single spherical bubble moving steadily under the influence of gravity in a clean system. It is clear, however, that swarms of suspended bubbles, usually entrained by turbulent eddies, have local relative velocities with respect to the continuous phase different from that derived for the case of a steady rise of a single bubble. This is mainly due to the fact that in an ensemble of bubbles the distributions of velocities, temperatures, and concentrations in the vicinity of one bubble are influenced by its neighbors. It is therefore logical to assume that in the case of dispersions the relative velocities and transfer rates depend on quantities characterizing an ensemble of bubbles. For the case of uniformly distributed bubbles, the dispersed-phase volume fraction O, particle-size distribution, and residence-time distribution are such quantities. 

The estimation of the diffusional flux to a clean surface of a single spherical bubble moving with a constant velocity relative to a liquid medium requires the solution of the equation for convective diffusion for the component that dissolves in the continuous phase. For steady-state incompressible axisym-metric flow, the equation for convective diffusion in spherical coordinates is approximated by... 

In this section, a general formulation will be given for the effect of bubble residence-time and bubble-size distributions on simultaneous and thermodynamically coupled heat- and mass-transfer in a multicomponent gas-liquid dispersion consisting of a large number of spherical bubbles. Here one can... 

For swarms of spherical bubbles, the field may be expected to be approximately spherically symmetric when the origin of coordinates is fixed on the center of mass of a typical particle. Therefore, by using spherical coordinates and the initial condition ... 

For all flow conditions tested in that study, a bubbly flow pattern with bubbles much smaller than the channel diameter (100 pm) was never observed. While liquid-only flows (or liquid slugs) containing small spherical bubbles were not observed, small droplets were observed inside gas core flows. Furthermore, no stratified flow occurred in the micro-channel as reported in previous studies of two-phase flow patterns in channels with a diameter close to 1 mm (Damianides and Westwater 1988 Fukano and Kariyasaki 1993 Triplett et al. 1999a Zhao and Bi 2001a). 

The liquid alone pattern showed no entrained bubbles or gas-liquid interface in the field of view. The capillary bubbly flow, in the upper part of Fig. 5.14a, is characterized by the appearance of distinct non-spherical bubbles, generally smaller in the streamwise direction than at the base of the triangular channel. This flow pattern was also observed by Triplett et al. (1999a) in the 1.097 mm diameter circular tube, and by Zhao and Bi (2001a) in the triangular channel of hydraulic diameter of 0.866 mm. This flow, referred to by Zhao and Bi (2001a) as capillary bubbly... 

The first approach developed by Hsu (1962) is widely used to determine ONE in conventional size channels and in micro-channels (Sato and Matsumura 1964 Davis and Anderson 1966 Celata et al. 1997 Qu and Mudawar 2002 Ghiaasiaan and Chedester 2002 Li and Cheng 2004 Liu et al. 2005). These models consider the behavior of a single bubble by solving the one-dimensional heat conduction equation with constant wall temperature as a boundary condition. The temperature distribution inside the surrounding liquid is the same as in the undisturbed near-wall flow, and the temperature of the embryo tip corresponds to the saturation temperature in the bubble 7s,b- The vapor temperature in the bubble can be determined from the Young-Laplace equation and the Clausius-Clapeyron equation (assuming a spherical bubble) ... 

Figure 2.9 Representation of pressure-temperature relationship that exists during the growth period of a spherical bubble in a superheated liquid of infinite extent. (From Dwyer, 1976. Copyright 1976 by American Nuclear Society, LaGrange Park, IL. Reprinted with permission.)...

Figure 2.11 Comparison of calculated growth rates for a spherical bubble in a uniformly heated, large volume of superheated sodium. (From Board and Duffey, 1971. Copyright 1971 by Elsevier Science Ltd., Kidlington, UK. Reprinted with permission.)...

Fet us consider a spherical bubble of vapour inside its coexisting liquid. Again the gas phase is assumed to be ideal, and eq. (6.52) becomes... 

Sphere, flow across, 15 72 It Sphere-of-influence (SOI), 19 355-356, 358 Spherical bubbles, in foams, 12 7-8 Spherical fillers, phenolic resin,... 

Equation 9.6 determines the conditions of a mechanical equilibrium of the curved interface. This can be illustrated with an example of a spherical bubble of radius r. To compete the surface tension the pressure inside the bubble should exceed the external pressure with AP, which is determined from the work, W, for virtual change of r dll APdl adl. Under equilibrium, dll = 0 and APdU=erd4, thus... 

Equation (59), which is written around a spherical bubble, requires knowledge as to how Pg and R vary with time. The development of expressions for these variables proceeds along the following lines. First, a mass balance is made on each gas bubble ... 

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