Bubbles surface area

Oxygen transfer rate (OTR) The product of volumetric oxygen transfer rate kj a and the oxygen concentration driving force (C - Cl), (ML T ), where Tl is the mass transfer coefficient based on liquid phase resistance to mass transfer (LT ), a is the air bubble surface area per unit volume (L ), and C and Cl are oxygen solubility and dissolved oxygen concentration, respectively. All the terms of OTR refer to the time average values of a dynamic situation. 

Yet, Eq. (14) does not describe the real situation. It must also be taken into account that gas concentration differs in the solution and inside the bubble and that, consequently, bubble growth is affected by the diffusion flow that changes the quantity of gas in the bubble. The value of a in Eq. (14) is not a constant, but a complex function of time, pressure and bubble surface area. To account for diffusion, it is necessary to translate Fick s diffusion law into spherical coordinates, assign, in an analytical way, the type of function — gradient of gas concentration near the bubble surface, and solve these equations together with Eq. (14). 

Calderbank et al. (C6) studied the Fischer-Tropsch reaction in slurry reactors of 2- and 10-in. diameters, at pressures of 11 and 22 atm, and at a temperature of 265°C. It was assumed that the liquid-film diffusion of hydrogen from the gas-liquid interface is a rate-determining step, whereas the mass transfer of hydrogen from the bulk liquid to the catalyst was believed to be rapid because of the high ratio between catalyst exterior surface area and bubble surface area. The experimental data were not in complete agreement with a theoretical model based on these assumptions. 

For chemical reaction-rate constants greater than 10 sec-1, NT increases linearly with the total bubble surface area, i.e., linearly with the gas holdup. In other words, the agitation rate only affects the total bubble surface area and has almost no effect on the rate of absorption per unit area. This result is in accordance with the work of Calderbank and Moo-Young (C4), discussed in Section II. 

Characteristic length [Eq. (121)] L Impeller diameter also characteristic distance from the interface where the concentration remains constant at cL Li Impeller blade length N Impeller rotational speed also number of bubbles [Eq, (246)]. N Ratio of absorption rate in presence of chemical reaction to rate of physical absorption when tank contains no dissolved gas Na Instantaneous mass-transfer rate per unit bubble-surface area Na Local rate of mass-transfer per unit bubble-surface area Na..Average mass-transfer rate per unit bubble-surface area Nb Number of bubbles in the vessel at any instant at constant operating conditions N Number of bubbles per unit volume of dispersion [Eq. (24)] Nb Defined in Eq. (134)... 

In the narrow tubes used by Beek and van Heuven, the bubbles assumed the shape of Dumitrescu (or Taylor) bubbles. Using the hydrodynamics of bubble rise and the penetration theory of absorption, an expression was developed for the total absorption rate from one bubble. The liquid surface velocity was assumed to be that of free fall, and the bubble surface area was approximated by a spherical section and a hyperbola of revolution. Values calculated from this model were 30% above the measured absorption rates. Further experiments indicated that velocities are reduced at the rear of the bubble, and are certainly much less than free fall velocities. A reduction in surface tension was also indicated by extreme curvature at the rear of the 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) ... 

The mass transfer resistance at a liquid-vapor interface results from two resistances, the liquid boundary layer and the gas boundary layer. In conditions involving water and sparingly soluble gases, such as occurs here, the liquid-phase resistance is almost always predominant [71]. For this reason, equation (16) involves only k, the mass transfer coefficient across the liquid boundary, and a, which is the gas bubble surface area per unit volume of liquid. Often, as here, those factors cannot be estimated individually, so k is treated as a single parameter. 

The Davidson and Harrison approach concentrates solely on the resistance at the bubble/cloud boundary (or bubble/dense phase boundary for a < 1). The transfer coefficient, referred to bubble surface area, is... 

Increasing surfactant concentrations in the aeration cell has been found to decrease bubble diameter, bubble velocity, axial diffusion coefficient, but increase bubble s surface-to-volume ratio, and total bubble surface area in the system. The effect of a surface-active agent on the total surface area of the bubbles is also a function of its operating conditions. The surfactant s effect is pronounced in the case of a coarse gas diffuser where the chances of coalescence are great and the effectiveness of a surface-active solute in preventing coalescence increases with the length of its carbon chain. 

In gas-liquid mixing applications, solids are often present as a catalyst or a product. Inherently having small solids present on the bubble surface can diminish the effective intrinsic bubble surface area and significantly decrease the mass transfer coefficient. Though others report just the opposite effect, an illustrative example from Dow is shown in Table... 

Creation of bubble surface area for mass transfer... 

System 1 In a gas-liquid continuous-stirred tank reactor (Figure 13-If. the gaseous reactant was bubbled into the reactor while the liquid reactant was fed through an inlet tube in the reactor s side. The reaction took place at the gas-liquid interface of the bubbles, and the product was a liquid. The continuous liquid phase could be regarded as perfectly mi.xed. and the reaction rate was proportional to the total bubble surface area. The surface area of a particular bubble depended on the time it had spent in the reactor. Because of their different sizes, some gas bubbles escaped from the reactor almost immediately, while others spent so much time in the reactor that they were almost coin-... 

Frother Volume fraction of 0.2 mm bubbles Surface area fraction 0.2 mm bubbles... 

Another very important problem in maximum bubble pressure measurements at very short adsorption times is the initial amount of adsorbed material at the surface of a newly formed bubble. The larger this amount the higher is the initial surface pressure II = n = -y(t = 0). The time interval during which the bubble grows continuously from its hemispherical to the final size is Tj. The relative expansion 0 of the bubble surface area A is given by... 

Work done to expand bubble = Energy stored in creating new bubble surface area Fdr=ly(dA) w 2y 8n rdr)... 

FIGURE 6.8 The work required to expand a soap bubble is directly related to the creation of additional bubble surface area. 

C/ig- concentration of A in the bulk gas phase within the bubble CAg concentration of A in the gas at the gas-liquid inter phase Cal concentration of A in the liquid at the gas-liquid interphase Cal - concentration of A in the bulk liquid phase Cac - concentration of A on the surface of a catalyst pellet Uj, gas bubble surface area per unit reactor volume (m /m )... 

Most of the instruments allow only the measurement of surface and interfacial tensions, without a sufficient control of the drop/bubble size. Advanced models provide very accurate controlling procedures. The instrument described here in detail represents the state of the art of drop and bubble shape tensiometers. The possibility to study bubbles in addition to drops opens a number of features not available by other instruments less loss of molecules caused by adsorption from extremely diluted solutions (small reservoir in the small single drop), long time experiments with very small amounts of a sample, easy application of a pressure sensor for additional measurement of the capillary pressure inside the bubble. Moreover, high quality sinusoidal relaxation studies can be performed by inserting a piezo system which can be driven such that very smooth changes of the bubble surface area are obtained. 

Likewise, the drop/bubble surface area is estimated by summation of the partial zones ... 

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