Bubble simplified models

The general case of coupled heat-transfer and multicomponent mass-transfer in swarms of bubbles with residence-time and size distributions is treated in Section IV, L. In previous sections simplified cases of uncoupled mass transfer are considered starting from the most simplified models available for gas-liquid dispersions. 

The bubbles play the role of the gas phase. The role of the liquid is played by an emulsion phase that consists of solid particles and suspending gas in a configuration similar to that at incipient fluidization. The quasi-phases are in cocurrent flow, with mass transfer between the phases and with a solid-catalyzed reaction occurring only in the emulsion phase. The downward flow of solids that occurs near the walls is not explicitly considered in this simplified model. 

From the simplified model of Figure 5.17, the energy equation for the bubble layer is written in terms of average conditions across the liquid layer ... 

As an illustration, a simplified model will be considered here to show the importance of the nonisothermal effect in the dilute phase. We assume that

bubble phase is negligible, and temperature in the dense phase is uniform. Then the material balance equation for the dense phase is ... 

Furthermore, in many industrial systems the liquid phase is not operated in batch mode, a continuous flow of the liquid phase has to be allowed. However, due to numerical problems most reports on bubble column modeling introduce the simplifying assumption that the continuous phase is operated in batch mode. Further work is needed on the continuous mode boundary condition. 

To describe the contribution to the inter-electrode resistance a simplified model will be used. Let us assume that the bubble diffusion layer has a mean thickness d, and that the gas void fraction is constant and equal to e over the whole electrode height h. The efficient conductivity of the electrolyte is... 

The following assumptions are required to derive the simplified bubble point model ... 

Figure 4.6a-c plot experimental results for the pure reference fluid bubble point tests for the 325 x 2300, 450 x 2750, and 510 x 3600 screen samples, respectively. To compare with pore diameters based on SEM analysis and historical data, also plotted for reference is a best fit linear curve to the data. As shown, for all three screens, bubble point is directly proportional to the contact angle corrected surface tension of each pure reference fluid, thus vaiidating the simplified model in Chapter 3. From the bubble point versus surface tension plots, one can easily determine a best fit effective pore diameter (Method 1) for each screen sample. These pore diameters, along with those based on SEM analysis and historical data, are listed in Table 4.2 for comparison of the three methods proposed earlier. Uncertainties for pore diameters based on reference fluid tests here are based on uncertainty in the DPT reading. Uncertainties for historical values are estimated. 

Considering these circumstances, the potential method was chosen to predict the shape of a bubble in contact with a plate of poor wettability, and five simplified models were proposed for the shape of the bubble. 

In this article, PS/PMMA bilayer as a simplified model for PS/PMMA blend is used to study foaming dynamics and gain insight into phase interface effects on bubble nucleation and C02-polymer interaction effects on bubble nucleation and growth rates. 

Although the bilayer representation is a very over simplified model for describing polymer blends since the domain size of the dispersive phase is only in the range of micron, it gives us some invaluable information about bubble nucleation and growth in the early stage of blend foaming process. 

Gal-Or and Resnick (Gl) have developed a simplified theoretical model for the calculation of mass-transfer rates for a sparingly soluble gas in an agtitated gas-liquid contactor. The model is based on the average gas residencetime, and its use requires, among other things, knowledge of bubble diameter. In a related study (G2) a photographic technique for the determination of bubble flow patterns and of the relative velocity between bubbles and liquid is described. 

Analytical analyses for the growth of a single bubble have been performed for simple geometrical shapes, using a simplified heat transfer model. Plesset and Zwick (1954) solved the problem by considering the heat transfer through the bubble interface in a uniformly superheated fluid. The bubble growth equation was obtained... 

Glicksman and Farrell (1995) constructed a scale model of the Tidd 70 MWe pressurized fluidized bed combustor. The scale model was fluidized with air at atmospheric pressure and temperature. They used the simplified set of scaling relationships to construct a one-quarter length scale model of a section of the Tidd combustor shown in Fig. 34. Based on the results of Glicksman and McAndrews (1985), the bubble characteristics within a bank of horizontal tubes should be independent of wall effects at locations at least three to five bubble diameters away from the wall. Low density polyurethane beads were used to obtain a close fit with the solid-to-gas density ratio for the combustor as well as the particle sphericity and particle size distribution (Table 6). 

This approach is proven for design and prediction of the performance of multiple-bed down-flow reactors. The complication, and a critical difference between this and a bubble column, is that the gas bubbles are formed in situ. The gas flux, and thus gas hold-up, will vary over the bed height. For the down-flow beds, a simplified linear gas hold-up profile was inherent in the design models, but with no apparent penalty in design accuracy. 

The Kunii-Levenspiel model can be simplified by assuming that the derivative terms of eqs. (3.529) and (3.530) are unimportant compared to the rest of the terms. Furthermore, plug flow can be assumed for gas (bubble) phase. Under these assumptions, the set of equations reduces as follows (Carberry, 1976 Fogler, 1999) ... 

Emulsion phase in completely mixed state General solution—bubble phase free of solids (Orcutt model) In the following, the simplified Orcutt-Davidson-Pigford model (Orcutt et al., 1962) is presented. This model assumes in addition that the bubble phase is free of solids, and thus yh = 0. This means that the reaction takes place only in the emulsion (dense) phase. Then, the third term in eq. (3.516) disappears. The gas flow is in the inlet of the bed ,A, the particulate phase wfmA, and the bubble phase wswfmA, where A is the cross-sectional area of the bed and (eq. 3.476)... 

For simplifying the calculations, we consider that the gas flows only through the bubble phase (fh = 1) and that there are no solids in the bubble phase (yb = 0). Under these conditions, the model of the reactor is (eqs. (3.519) and (3.520))... 

P 11] Simulations were carried with a simplified chamber and air-bubble pocket geometry. Details on this geometry and the several assumptions taken for describing the fluid dynamics can be found in [23] and are not described further here. Generally, the experimental known fluid dynamic features were taken into account, e.g. the convective motion based on vortices was assumed also in the model. 

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