Bubble cloud

For a better understanding of the interactions between parameters, it is often helpful to calculate the effective bubble rise velocity from measurea valves of for example, the data of Mersmann (loc. cit.) indicated = 0.6 for = 0.05 iti/s, giving U, = 0.083 m/s, which agrees with the data reported in Fig. 14-43 for the rise velocity of bubble clouds. The rise velocity of single bubbles, for d - 2 mm, is about 0.3 m/s, for liquids with viscosities not too different from water. Using this value in Eq. (14-220) and comparing with Fig. 14-104, one finds that at low values of the rise velocity of the bubbles... 

The effectiveness of a fluidized bed as a ehemical reactor depends to a large extent on the amount of convective and diffusive transfer between bubble gas and emulsion phase, since reaction usually occurs only when gas and solids are in contact. Often gas in the bubble cloud complex passes through the reactor in plug flow with little back mixing, while the solids are assumed to be well mixed. Actual reactor models depend greatly on kinetics and fluidization characteristics and become too complex to treat here. 

Fig. 1.16 Calculated radius of a bubble in a bubble cloud as a function of time for one acoustic cycle at 29 kHz and 2.36 bar in frequency and pressure amplitude of ultrasound, respectively. The ambient radius is 5 pm. The dotted curve is the calculated result for an isolated bubble. The dashed one is the calculated result with the interaction only with neighboring bubbles. The solid one is the calculated result taking into account all the interactions with surrounding bubbles. Reprinted figure with permission from Yasui K, Iida Y, Tuziuti T, Kozuka T, Towata A (2008) Strongly interacting bubbles under an ultrasonic horn. Phys Rev E 77 016609 [http /Aink.aps.org/abstract/PRE/v77/ e016609]. Copyright (2008) by the American Physical Society...

Parlitz U, Mettin R, Luther S, Akhatov I, Voss M, Lauterbom W (1999) Spatio-temporal dynamics of acoustic cavitation bubble cloud. Philos Trans R Soc London A 357 313-334... 

Servant G, Laborde JL, Hita A, Caltagirone JP, Gerard A (2003) On the interaction between ultrasound waves and bubble clouds in mono and dual-frequency sonoreactors. Ultrason Sonochem 10 347-355... 

The performance equation for the model is obtained from the continuity (material-balanoe) equations for A over the three main regions (bubble, cloud + wake, and emulsion), as illustrated schematically in Figure 23.7. Since the bed is isothermal, we need use only the continuity equation, which is then uncoupled from the energy equation. The latter is required only to establish the heat transfer aspects (internally and externally) to achieve the desired value of T. 

In Figure 23.7, the bubble, cloud, and emulsion regions are represented by b,c + iv, and e, respectively. The control volume is a thin horizontal strip of height dx through tiie vessel. The overall depth of the bed is Lfl, which is related to the holdup of catalyst, Wcat. The performance equation may be used to determine Wcat for a given conversion /A (and production rate), or the converse. 

Further investigations of chemical kinetics and transformation products will be carried out during the final phase of the project. In order to truly understand sonochemical effects, the behavior of the individual bubbles and the bubble clouds must be more finely resolved. Physical characterization of cavitation bubble clouds will also be performed. Thus, a more fundamental link will be established between bulk, observable parameters and sonochemistry, via the physics and hydrodynamics of the cavitating cloud. 

Using Davidson s theoretical expression for bubble-cloud circulation and the Higbie theory for cloud-emulsion diffusion the interchange of gas between bubble and cloud is then found to be... 

Changing bubble size with height in the bed Negligible bubble-cloud resistance Negligible cloud-emulsion resistance Nonspherical bubbles... 

Material balances can be written over a differential section of the bed (dz) for a reactant, in each of the three-phases (bubble, cloud, and emulsion). Then, the equations of the model are as follows ... 

Most bubbles in gas-solid fluidized beds are of spherical cap or ellipsoidal cap shape. Configurations of two basic types of bubbles, fast bubble (clouded bubble) and slow bubble (cloudless bubble), are schematically depicted in Fig. 9.7. The cloud is the region established... 

Figure 9.7. Bubble configurations and gas flow patterns around a bubble in gas-solid fluidized beds (a) Fast bubble (clouded bubble) Ub > /mf/ mf (b) Slow bubble (cloudless bubble)...

Phases. Both two-phase and three-phase representations are widely used as shown schematically in Figure 1. In two-phase representations the dilate phase may represent bubbles alone, jets (in the grid region), or bubbles plus clouds. Three-phase representations generally use the scheme followed by Kunii and Leven-spiel (19) whereby bubbles, clouds, and "emulsion" (i.e. that part of the non-bubble bed not included in the clouds) are each treated as separate regions. As shown in Table II, all of these possibilities are represented in the models adopted by the authors in this symposium. There appears, however, to be an increasing tendency to adopt three phase models, probably as a result of experimental results (20) which showed that the Kunii and Levenspiel model gave a better representation of measured concen-... 

Phases 2 Bubble (or jet) dense or 1 (CSTR) 2 Bub/Cloud emulsion or 3 Bubble, cloud emulsion 2 Bubble dense 3 Bubble, cloud and emulsion... 

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... 

Kunii and Levelspiel (19) again use Equation (2) to describe bubble/cloud transfer. Based on the penetration theory, they propose the following expression for cloud/emulsion transfer ... 

For most practical conditions, a comparison of k and k from Equations (4) and (5) would suggest that the principal resistance to transfer resides at the outer cloud boundary. However, when (a), (b) and (c) are taken into account, this is no longer the case. In fact, experimental evidence (e.g. 30,31,32) indicates strongly that the principal resistance is at the bubble/ cloud interface. With this in mind, it is probably more sensible to include the cloud with the dense phase (as in the Orcutt (23, 27) models) rather than with the bubbles (as in the Partridge and Rowe (37) model) if a two-phase representation is to be adopted (see Figure 1). If three-phase models are used, then Equations (2) and (5) appear to be a poor basis for prediction. Fortunately the errors go in opposite directions. Equation (2) overpredicting the bubble/cloud transfer coefficient, while Equation (5) underestimates the cloud/emulsion transfer coefficient. This probably accounts for the fact that the Kunii and Levenspiel model (19) can give reasonable predictions in specific instances (e.g.20),... 

The concentration profiles in the bubbles, cloud and emulsion phases are plotted in Figure 1 for a set of parameter values. For the sake of comparison, the profiles for the same values of parameters obtained using the Fryer-Potter model are shown in Figure 2. Figures 3-6 show the influence of parameters such as... 

Calculating The Reaction Parameters. The overall dimensionless rate constant is expressed in terms of exchange coefficients between the bubble, cloud, and emulsion, and in terms of the volumes of catalyst per volume of bubble in the bubble, cloud, and emulsion ... 

Figure 4. Ratio of the average bubble-cloud transport coefficient to the transport coefficient evaluated at h/2 as a function of the aspect ratio for various ratios the maximum to minimum bubble diameter, r...

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