Free bubbles

The mixing impeller is used primarily to subdivide the incoming gas into bubbles and to disperse these bubbles throughout the agitated liquid phase. The shear produced by the impeller blades on both liquid and gas causes the incoming gas to be subdivided into numerous bubbles which have relatively small diameters compared to the diameter obtained by free bubbling. In general the results are (G2) ... 

Increase in interfacial area. The total surface area for diffusion is increased because the bubble diameter is smaller than for the free-bubbling case at the same gas flow rate hence there is a resultant increase in the overall absorption rate. The overall absorption rate will also increase when the diffusion is accompanied by simultaneous chemical reaction in the liquid phase, but the increase in surface area only has an appreciable effect when the chemical reaction rate is high the absorption rate for this case is then controlled by physical diffusion rather than by the chemical reaction rate (G6). 

Janssen and Hoogland (J3, J4a) made an extensive study of mass transfer during gas evolution at vertical and horizontal electrodes. Hydrogen, oxygen, and chlorine evolution were visually recorded and mass-transfer rates measured. The mass-transfer rate and its dependence on the current density, that is, the gas evolution rate, were found to depend strongly on the nature of the gas evolved and the pH of the electrolytic solution, and only slightly on the position of the electrode. It was concluded that the rate of flow of solution in a thin layer near the electrode, much smaller than the bubble diameter, determines the mass-transfer rate. This flow is affected in turn by the incidence and frequency of bubble formation and detachment. However, in this study the mass-transfer rates could not be correlated with the square root of the free-bubble diameter as in the surface renewal theory proposed by Ibl (18). 

Flow of trains of surfactant-laden gas bubbles through capillaries is an important ingredient of foam transport in porous media. To understand the role of surfactants in bubble flow, we present a regular perturbation expansion in large adsorption rates within the low capillary-number, singular perturbation hydrodynamic theory of Bretherton. Upon addition of soluble surfactant to the continuous liquid phase, the pressure drop across the bubble increases with the elasticity number while the deposited thin film thickness decreases slightly with the elasticity number. Both pressure drop and thin film thickness retain their 2/3 power dependence on the capillary number found by Bretherton for surfactant-free bubbles. Comparison of the proposed theory to available and new experimental... 

Consider the vacuum forming of a polymer sheet into a conical mold as shown in Figure 7.84. We want to derive an expression for the thickness distribution of the final, conical-shaped product. The sheet has an initial uniform thickness of ho and is isothermal. It is assumed that the polymer is incompressible, and it deforms as an elastic solid (rather than a viscous liquid as in previous analyses) the free bubble is uniform in thickness and has a spherical shape the free bubble remains isothermal, but the sheet solidifies upon confacf wifh fhe mold wall fhere is no slip on fhe walls, and fhe bubble fhickness is very small compared fo ifs size. The presenf analysis holds for fhermoforming processes when fhe free bubble is less than hemispherical, since beyond this point the thickness cannot be assumed as constant. 

In Figure 7.84, we note that after a certain time the free bubble contacts the mold at height z and has a spherical shape of radius R. The radius is determined by the mold geometry and bubble position and is given by... 

The bubble size may also be estimated on the basis of the correlation of Darton et al. (1977) under conditions of free bubbling beds without slugging and maximum stable bubble size as... 

In a free bubbling bed, the average bubble rise velocity Ubb can be described by [Davidson and Harrison, 1963]... 

Ostergaard and Michelsen104 measured the gas holdup in beds of 0.25-, 1-, and 6-mm glass particles using a radioactive tracer technique. They found that hG °c Uqg, where U0G is the superficial gas velocity, and n took values of 0.88, 0.78, and 0.93, respectively, for three particle sizes. The solid-free bubble-column gave n = 1.05. They also found that, in the solid-free system and in beds of 6-mm particles, the gas holdup decreased with increasing liquid flow rate whereas in beds of 0.25- and 1-mm particles, the gas holdup increased with increasing liquid flow rate. 

Ostergaard and Fosbol103 also showed that the volumetric mass-transfer coefficient was a strong function of bed position. In solid-free bubble-columns and in beds of 1-mm particles, as shown in Figs. 9-18 and 9-19, the volumetric mass-transfer coefficient decreased with the distance away from the gas distributor. In beds of 6-mm particles, as shown in Fig. 9-20, the volumetric mass-transfer coefficient first increased and then decreased with the distance away from the gas distributor. 

This result is obtained in the same way as Equation 8 (see Bretherton (28)), but because of the bubble contact and flattening, the curvature in the Plateau border regions is 2R/(R2-Rc2) instead of 2/R in the free bubble case. It is interesting to note that the results in both cases are the same. 

Figure 9.8. BF image (g = lOTl) showing prismatic dislocation loops (b = <1120 and strain-free bubbles in wet synthetic quartz after heating at 550°C for 2 hours. (From McLaren et al. 1989.)...

On heating, these clusters evolve into strain-free bubbles and associated dislocation loops (Figure 9.8). The following mechanism of growth was... 

Figure 12.2. Idealized view of the free bubbling regime on a distillation tray.

In the free bubbling zone a distribution of bubble sizes is usually obtained. The model parameters for the bubbling zone are... 

The splash zone above the free bubbling zone consists of entrained droplets. We may model this zone as being made up of droplets of uniform size rising in plug flow through the splash zone. The model parameters are... 

The splash zone at the top of the dispersion has the same characteristics as the one described above for the free bubbling zone its contribution to mass transfer can usually be neglected. 

An obvious question that may occur to the reader is why the very simple method of integrating the viscous dissipation function has not been used earlier for calculation of the force on a solid body. The answer is that the method provides no real advantage except for the motion of a shear-stress-free bubble because the easily attained inviscid or potential-flow solution does not generally yield a correct first approximation to the dissipation. For the bubble, Vu T=0(l) everywhere to leading order, including the viscous boundary layer where the deviation from the inviscid solution yields only a correction of 0(Re x 2). For bodies with no-slip boundaries, on the other hand, Vu T is still 0(1) outside the boundary layer, but inside the boundary layer Vu T = O(Re). When integrated over the boundary layer, which is G(Re k2) in radial thickness, this produces an ()( / Re) contribution to the total dissipation,... 

Since the conductivity of electrolytes and the cross section and thickness of the membrane are known, a can be determined from the voltage drops across the three pairs of probe electrodes 1-2, 3-4 and 5-6. The sodium current efficiency (CE) can also be determined by titrating the amount of caustic soda generated over a given period of time. The confinement chambers around the working electrodes are used to eliminate free bubbles near the membrane. Our normalized transport data for sulfonate, carboxylate and sulfonamide ionomers are plotted In Figure 5 the universal percolative nature of perfluorinated ionomers can be clearly eeij. The prefactor sulfonate ionomers. The exponent t is 1.5 0.1 in reasonable agreement with theory and the thresholds are between 8 to 10 vol. %, which are consistent with the bimodal distribution in cluster size postulated by the cluster-network model (5.18). This theory has also been applied recently to delineate sodium selectivity of perfluorinated ionomers (20). 

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