Bubble pressure model

The virtual mass coefficient Cgp of an isolated spherical bubble is 0.5. The BP gradient is then added to the right-hand side of the gas momentum (Eq. (5.4)) and acts as a driving force for bubbles to move from areas of higher to areas of lower d and facilitates stabilization of the bubbly flow regime. However, Sankaranarayanan and Sundaresan (2002) indicate that as increases, the collisional and hydrodynamic contributions become important. 

Biesheuvel and Gorissen (1990) proposed a modified BP model of the form 

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Surface tension and density of liquid alloys have been studied by Moser et al. (2006). Measurements by maximum bubble pressure and dilatometric techniques were carried out in an extensive range of temperatures on liquid alloys close to the ternary eutectic Sn3 3Ag0 76Cu with different Sb additions, which decrease surface tension and density. The experimental data were discussed in comparison also with values calculated on the basis of different models. 

A similar equation to that of Eq. (43) was proposed by Bankoff (B6) on the basis of a bubble-flow model for vertical flow. His derivations are discussed in the following section (Section V, B). Finally, it should be mentioned that the momentum exchange model of Levy (L4), and the slip-ratio model of Lottes and Flinn (L7) are more readily applied for the determination of void fractions than for pressure drops. In general, these methods seem to give rather poorer accuracy than those already discussed. 

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 dissolved gas concentration at the bubble/melt interface, can be related to the bubble pressure P, through Henry s law. Gogos compared his model s predictions with the experimental data produced by Spence. The model predictions fit very well with experimental data when selecting a degree of saturation close to 100% (91.9-99.6%). An alternative approach has been proposed by Kontopoulou and Vlachopoulos, who modeled the dynamics of bubble dissolution into the melt using conservation of mass and momentum.f ... 

This program can be used in two ways, as shown in the tutorial that follows. If measured isothermal VLE data are available, the program can be used to calculate VLE at the measured liquid mole fractions with user-provided model parameters and to compare the calculated bubble pressures and vapor mole fractions with the experimental ones. Alternatively, the program can be used to optimize parameters of a selected model by fitting parameters to measured liquid mole fraction versus bubble pressure data. Pai ameter optimization is done using a simplex algorithm. In this mode the program reads previously stored data or accepts new data entered from the keyboard. The input data structure is identical to that described in... 

C Trondheim Bubble Column Model Pressure force ... 

The aim of this chapter is to present the fundamentals of adsorption at liquid interfaces and a selection of techniques, for their experimental investigation. The chapter will summarise the theoretical models that describe the dynamics of adsorption of surfactants, surfactant mixtures, polymers and polymer/surfactant mixtures. Besides analytical solutions, which are in part very complex and difficult to apply, approximate and asymptotic solutions are given and their range of application is demonstrated. For methods like the dynamic drop volume method, the maximum bubble pressure method, and harmonic or transient relaxation methods, specific initial and boundary conditions have to be considered in the theories. The chapter will end with the description of the background of several experimental technique and the discussion of data obtained with different methods. 

Adsorption Kinetics Model for the Maximum Bubble Pressure Method... 

The effective surface age has been determined on the basis of a diffusion controlled adsorption model. Similar to the maximum bubble pressure technique, where the bubble time is longer than teff, the drop formation time t is also significantly longer than the effective age teff and can be obtained as teff=3t/7 [185], This estimation assumes a radial flow inside and outside the growing drop and a homogeneous expansion of the drop surface. 

The graphs shown in Fig. 4.35 are the dynamic surface tensions of three mixtures of CioDMPO and CmDMPO measured with the maximum bubble pressure method MPT2 (O) and ring tensiometer TE2 (O). Although there is a general theoretical model to describe the adsorption kinetics of a surfactant mixture, model calculations are not trivial and a suitable software does not exists. 

The graph in Fig. 41 shows the dynamic surface tensions of a mixtured solution of CioDMPO and C14DMPO measured with the maximum bubble pressure method BPAl (O) and profile analysis tensiometer PATl ( ). The theoretical curves shown were calculated due to the adsorption kinetics model for surfactant mixtures discussed above (Miller et al. 2003). 

In this work, the binary interaction parameter (k,j) was fitted to experimental vapor-hquid equilibrium data. The objective fimction used to optimize the kij parameter for any binary mixture was the minimization of the deviation between the model s prediction of the bubble pressure and the experimental value. The optimized ky values are suimnarized in Table 1. For comparison, optimum ky values are also shown for the PR EoS. A semi-predictive approach was used to determine the sensitivity of the results to the interaction parameter using the CO2 coupled binary interaction parameters. 

A computer analysis was performed of the loss-of-load event with delayed reactor trip, similar to that used in safety valve capacity evaluation, except that a conservative 20% safety valve blowdown and initial conditions biased to maximize pressurizer liquid level were assumed. The purpose of this analysis was to determine the pressurizer liquid level response and the RCS subcooling under these conservative conditions. For additional conservatism, adjustments were made to the computer-calculated pressurizer level on the basis of a very conservative pressurizer model. This model assumed that the initial saturated pressurizer liquid did not mix with the cooler insurge liquid, that the initial liquid remained in equilibrium with the pressurizer steam space, and that the steam which flashed during blowdown remained dispersed in the liquid phase and caused the liquid level to swell. The adjusted pressurizer water level vs time curve showed a maximum level of 78%, Reference 2, (1874 ft" ), below the safety valve nozzle elevation which is greater than 100% level, so that dry saturated steam flow to the safety valves is assured throughout the blowdown. The computer analysis also showed that adequate subcooling was maintained in the RCS during the blowdown, so that steam bubble formation is precluded. 

Valtz, A. Laugier, S. Richon, D. Bubble pressures and saturated liquid molar volumes of trifluorotrichloroethane -fluorochlorohydrocarbon mixtures. Experimental data and modelization J. Chem. Eng. Data 1987,32, 397-400... 

Fig. 9 Dynamic surface tension of mixtures between a highly surface-active model suspension and solutions (1 g/L) of modified natural polymers at the same mixing ratio (volume suspension volume polymer) of 10 1 dynamic surface tension, was measured by bubble pressure tensiometer (t = 60 s)...

FIGURE 19.1 Dynamic surface-tension data for n-alkyldimethylphosphine oxides, as measured by maximum bubble-pressure technique (O), drop-volume tensiometry ( ), and de Nolly ring tensiometry (A), and model fit ignoring reorientation (dotted line) and incorporating reorientation (solid line). (From Fainerman, V. B., et al. 2000. Adv. Colloid Interface Sci. 86 (1-2) 83-101. With permission.)... 

The effect of the hydrostatic pressure is usually small, and the bubbles grow in size owing largely to coalescence. There are a number of bubble coalescence models in the literature. A few of the more well-known ones are discussed here. 

Chapter 10 presents the revised empirical static bubble point model for cryogenic liquids. Model dependencies are systematically presented to explain the trends in the data. Chapter 11 echoes Chapter 10 by presenting a refined reseal pressure drop model, using cryogenic data to build the model. Chapter 12 presents a new steady state quasi-3D... 

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