Single-bubble-size model

Fryer and Potter (1972), using the model of Davidson and Harrison, reported that a bubble size found at about 0.4h could be used as the single bubble size in that model. Earlier in this paper, the bubble size found at 0.5h was used arbitrarily in calculating the conversion in an ammonia oxida tion system using the K-L model. 

Instead of arbitrarily considering two bubble classes, it may be useful to incorporate a coalescence break-up model based on the population balance framework in the CFD model (see for example, Carrica et al., 1999). Such a model will simulate the evolution of bubble size distribution within the column and will be a logical extension of previously discussed models to simulate flow in bubble columns with wide bubble size distribution. Incorporation of coalescence break-up models, however, increases computational requirements by an order of magnitude. For example, a two-fluid model with a single bubble size generally requires solution of ten equations (six momentum, pressure, dispersed phase continuity and two turbulence characteristics). A ten-bubble class model requires solution of 46 (33 momentum, pressure. 

The additional work required to model this variation in the bubble size is most often unwarranted. In any case, variations in bubble sizes with time and radial position at a given height may be as important, and no effort has been made to Incorporate these bubble size variations in reactor models. When a single bubble size is adopted to represent the entire bed, calculations (35) indicate that this should be the bubble size at x=0.4H. 

Figure 4 also compares the single-bubble-size (SBS) model and DBS model. In the former case, only one bubble class is introduced in the resolution of structure and energy dissipation, and hence there are only three structure parameters and two conservation equations. The three energy dissipation terms also only hold for one bubble class. The SBS model calculation shows only a monotonous increase of gas holdup and there is no jump change. This can be understood since the mesoscale mechanism, i.e., the compromise between the two dominant mechanisms pertinent to the TBCs, cannot be reflected in the SBS model, and therefore, the capabUity of reflecting the structure heterogeneity and evolution at macroscale is lost. 

The first approach developed by Hsu (1962) is widely used to determine ONE in conventional size channels and in micro-channels (Sato and Matsumura 1964 Davis and Anderson 1966 Celata et al. 1997 Qu and Mudawar 2002 Ghiaasiaan and Chedester 2002 Li and Cheng 2004 Liu et al. 2005). These models consider the behavior of a single bubble by solving the one-dimensional heat conduction equation with constant wall temperature as a boundary condition. The temperature distribution inside the surrounding liquid is the same as in the undisturbed near-wall flow, and the temperature of the embryo tip corresponds to the saturation temperature in the bubble 7s,b- The vapor temperature in the bubble can be determined from the Young-Laplace equation and the Clausius-Clapeyron equation (assuming a spherical bubble) ... 

The rise velocity of bubbles is another important parameter in fluidized-bed models, but it can be related to bubble size (and bed diameter, D). For a single bubble, the rise... 

Another study carried out by these authors [93] modeled the collapsing motion of a single bubble near an electrode surface, and equations for the motion of a spherical gas bubble were obtained. The jet speed and water hammer pressure during jet flow (liquid jet) were calculated, and when the jet speed was 120 m/s, the water hammer pressure was approximately 200 MPa upon the electrode surface. This pressure played an important part in the fineness of the crystal deposits. Mass transfer during the electrode reaction was by turbulent diffusion. The diffusion layer thickness was reduced to approximately 1/10th its size in the presence of the ultrasonic field. The baths contained the ions Cl-, SO -, and Zn2+. The ultrasonic frequency employed in the experiments was 40 kHz and it was seen that ultrasound considerably increased the deposition rate and current efficiency, as well as the smoothness and hardness of the deposit. Microscopy studies showed that the... 

The Cb parameter takes values between 0 and 1, and generally depends on bubble size and shape, and on the turbulent length scale. The empirical coefficients in the turbulence model were kept equal to the standard values for the original single phase model. 

BSA was also used as a model protein by Brown et al. [75] for protein recovery by foam flotation. They investigated the influence of feed concentration, superficial gas velocity, feed flow rate, bubble size, pH and ionic strength on the enrichment and recovery of BSA in a single-stage continuous flotation column. 

Bubble size is required to calculate, for example, the drag force imparted on a bubble. Most Eulerian-Eulerian CFD codes assume a single (average) bubble size, which is justified if one is modeling systems in which the bubble number density is small (e.g., bubbly flow in bubble columns). In this case, the bubble-bubble interactions are weak and bubble size tends to be narrowly distributed. However, most industrially relevant flows have a very large bubble number density where bubble-bubble interactions are significant and result in a wide bubble size distribution that may be substantially different from the average bubble size assumption. In these cases, a bubble population balance equation (BPBE) model may be implemented to describe the bubble size distribution (Chen et al., 2fX)5). 

Various models are available to calculate liquid side mass transfer coefficients kj. The value of this hydrodynamic parameter and the equations that apply to its calculation largely depend on bubble size and the constitution of the bubble surface. Fig. 6 presents some recent measurements on mass transfer from single bubbles (19) which demonstrate the above influences. The evaluated kL values are plotted as Sherwood numbers vs. Peclet numbers. Large circulating bubbles with mobile surface yield kj, values which approach the... 

Gas-liquid mixed tanks are used for various operations in industrial practise. The design of gas-liquid mixing units and reactors is still done by empirical correlations, which are usually valid for specific components, mixing conditions and geometries. Computational Fluid Dynamic (CFD) techniques have been used successfully for single-phase flow, but gas-liquid flow calculations are still tedious for computers. Therefore, simpler and more accurate multiphase models are needed. In order to verify multiphase CFD calculations and to fit unknown parameters in the multiphase models, experimental local bubble size distributions and flow patterns are needed. 

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. 

One of the major contributions from this group has been the development of a unified model, which could explain both bubble and drop formation through a single set of equations.This model has been tested for a variety of systems. It has been modified to take into account the complexities, drainage between bubbles, etc. New models have been developed for the prediction of drop and bubble sizes in complex situtations like sieve plates, sintered disks, etc. The model has been also extended to predict pneumatic atomisation, nucleate boiling, etc. 

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