For a bubble

The preceding upper limit to particle size can be exceeded if more than one bubble is attached to the particle, t A matter relating to this and to the barrier that exists for a bubble to attach itself to a particle is discussed by Leja and Poling [63] see also Refs. 64 and 65. The attachment of a bubble to a surface may be divided into steps, as illustrated in Figs. XIII-8a-c, in which the bubble is first distorted, then allowed to adhere to the surface. Step 1, the distortion step, is not actually unrealistic, as a bubble impacting a surface does distort, and only after the liquid film between it and the surface has sufficiently thinned does... 

Calculate AGi, AG2, and H pract (Section XIII-4A) for a bubble in a flotation... 

The foregoing is an equilibrium analysis, yet some transient effects are probably important to film resilience. Rayleigh [182] noted that surface freshly formed by some insult to the film would have a greater than equilibrium surface tension (note Fig. 11-15). A recent analysis [222] of the effect of surface elasticity on foam stability relates the nonequilibrium surfactant surface coverage to the foam retention time or time for a bubble to pass through a wet foam. The adsorption process is important in a new means of obtaining a foam by supplying vapor phase surfactants [223]. 

The relative volatiHties Ot) are defined by Eq. (13-33), is the mini-mum-reflux ratio (L v + i/D)min,. nd q describes the thermal condition of the feed (e.g., 1.0 for a bubble-point feed and 0.0 for a saturated-vapor feed). The Xi p values are available from the given feed composition. The 0 is the common root for the top-section equations and the bottom-section equations developed by Underwood for a column at minimum reflux with separate zones of constant composition in each section. The common root value must fall between 06/, and Ot/, where hk and Ik stand for heavy key and light key respectively. The key components are the ones that the designer wants to separate. In the butane-pentane splitter problem used in Example 1, the light key is /1-C4 and the heavy key is i-C. ... 

Top-down calculations for tbe example problem are shown in Table 13-10 and bottom-up calculations in Table 13-11. Top-down and bottom-up calculations have provided values of and respectively. For a bubble-... 

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. 

For a bubble to be formed in a liquid, such as steam in water, for example, it is necessary for a surface of separation to be produced. Kelvin has shown that, as a result of the surface tension between the liquid and vapour, the vapour pressure on the inside of a concave surface will be less than that at a plane surface. As a result, the vapour pressure Pr inside the bubble is less than the saturation vapour pressure P, at a plane surface. The relation between Pr and P, is ... 

Similar analysis can be applied to side-by-side diffusion cell systems, where stirring is effected by bubbling an 02/C02 gas mixture. For a bubbling rate of 40 mL gas/min, each UWL was estimated to be 282 pm [515]. 

For a single-phase turbulent flow the ratio of the maximum to the average flow velocity is approximately 1.2, and the value of Co may also be close to 1.2 for a bubbly flow. Zuber and Findlay (1965) pointed out that, as the mixture velocity increases, the value of the exponent increases and flatter profiles result. 

If the supply of surfactant to and from the interface is very fast compared to surface convection, then adsorption equilibrium is attained along the entire bubble. In this case the bubble achieves a constant surface tension, and the formal results of Bretherton apply, only now for a bubble with an equilibrium surface excess concentration of surfactant. The net mass-transfer rate of surfactant to the interface is controlled by the slower of the adsorption-desorption kinetics and the diffusion of surfactant from the bulk solution. The characteris-... 

At low particle Reynolds numbers for a bubbling bed, the Ergun expression can be simplified using only the first term in Eq. (31). 

Table 6 presents the range of scaling parameters for tests undertaken by different investigators to verify the scaling relationships for a bubbling bed. 

Experimentally observed jet half-angle range from 8° to 12° for the experimental data mentioned above. These compare to 10° suggested by Anagbo (1980) for a bubbling jet in liquid. 

Figure 6. Split gas flow for a bubbling fluid bubbling bed.

Much of this work was carried out using a special distilling column called a bubble-plate column (Fig. 141). Each plate really does act like a distilling flask with a very efficient column, and one distillation is really carried out on one physical plate. To calculate the number of plates (separation steps, or distillations) for a bubble-plate column, you just count them ... 

A method similar to the falling-ball method is the bubble method. A vial is filled with liquid, leaving sufficient room for a bubble that can equal the diameter of the vial. The vial is inverted and the time required for the bubble to pass two predetermined marks is determined. The viscosity will be directly proportional to the time required for the bubble to rise. 

It has been argued (Appendix 3, Eq. A.21) that the collapse time for a bubble, initially of radius R, is considerably shorter than the time period of the compression cyde. Thus the external pressure Pj (= P + Pjj), in the presence of an acoustic field, maybe assumed to remain effectively constant (Pj ) during the collapse period. Neglecting surface tension, assuming adiabatic compression (i. e. very short compression time), and replacing R, by R, the size of the bubble at the start of collapse, the motion of the bubble wall becomes... 

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. 

The Young and Laplace equation, equation (10.14), reduces to equation (10.4) for the special case of a sphere with r 1 equal to r2. For a bubble, the right-hand side of equations (10.14) and (10.4) should be multiplied by 2 to allow for the fact that there are two surfaces being stretched, the interior and the exterior. 

The starting point for the Forster-Zuber theory (F4, F5, F6) is the Rayleigh equation (Rl) for a bubble growing in a liquid medium. In this... 

The critical radius f o for cavitation is found by considering the gas-law expression for a bubble surrounded by a liquid in tension. From 2a... 

For a bubble to grow, vapor must pass from the superheated liquid into the bubble. Thus latent heat of vaporization is removed from the surrounding liquid, and the liquid cools. The drop in liquid temperature near the bubble means a decrease in the driving force between liquid and bubble. This temperature drop strongly affects the bubble rate of growth. The rate can be shown to approach asymptotically a condition whereby the radius increases according to the square root of time. 

Estimate the liquid-phase volumetric coefficient of oxygen transfer for a bubble column fermentor, 0.8 m in diameter 9.0 m in height (clear liquid), containing the same liquid as in Problem 12.2. The superficial gas velocity is 150 m h . ... 

To calculate the critical gas velocity needed to suspend the solid catalyst in the reactor, we recommend the correlation published by Koide et al. [28], which refers either to flat-bottom or conical-bottom columns for a bubble column without liquid motion ... 

Although the most realistic model for a bubble column reactor is that of dispersed plug-flow in both phases, this is also the most complicated model in view of the uncertainty of some of the quantities involved, such a degree of complication may not be warranted. Because the residence time of the liquid phase in the column... 

Example 2.4. How large is the pressure in a spherical bubble with a diameter of 2 mm and a bubble of 20 nm diameter in pure water, compared with the pressure outside For a bubble the curvature is identical to that of a sphere R = R2 = R. Therefore... 

For a bubble in a liquid environment the two principal curvatures are negative C = C2 = —1/R. The pressure difference is negative and the pressure inside the liquid is lower than inside the bubble. 

Bubble in a liquid From Eq. (2.19) we see that a negative sign has to be used for a bubble because of the negative curvature of the liquid surface. As a result we get... 

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