Bubble population

Here, J (a, t) is the instantaneous interfacial flux expressed as a state vector whose components are the diffusional and heat fluxes. Therefore, the expected value < J(a)> takes into account the variations in residence time among the entire bubble population. 

The moment equations of the size distribution should be used to characterize bubble populations by evaluating such quantities as cumulative number density, cumulative interfacial area, cumulative volume, interrelationships among the various mean sizes of the population, and the effects of size distribution on the various transfer fluxes involved. If one now assumes that the particle-size distribution depends on only one internal coordinate a, the typical size of a population of spherical particles, the analytical solution is considerably simplified. One can define the th moment // of the particle-size distribution by... 

Because the bubble population increases with heat flux, a point of peak flux may be reached in nucleate boiling where the outgoing bubbles jam the path of the incoming liquid. This phenomenon can be analyzed by the criterion of a Hemholtz instability (Zuber, 1958) and thus serves to predict the incipience of the boiling crisis (to be discussed in Sec. 2.4.4). Another hydrodynamic aspect of the boiling crisis, the incipience of stable film boiling, may be analyzed from the criterion for a Taylor instability (Zuber, 1961). 

These different approaches are complementary to each other in basic concept. However, these analyses have not provided clear insight information of the bubble layer at the CHF about the bubble shape (spherical or flat elliptical), bubble population and its effect on turbulent mixing, and bubble behavior. The bubble behavior in a bubble layer could involve bubble rotation caused by flow shear, normal bubble velocity fluctuation, and bubble condensation in the bubble layer caused by the subcooled water coming from the core. Further visual study and measurements in this area may be desired. 

Iida Y, Ashokkumar M, Tuziuti T, Kozuka T, Yasui K, Towata A, Lee J (2010) Bubble population phenomena in sonochemical reactor II Estimation of bubble size distribution and its number density by simple coalescence model calculation. Ultrason Sonochem 17 480-486... 

Segebarth N, Eulaerts O, Reisse J, Crum LA, Matula TJ (2002) Correlation between acoustic cavitation noise, bubble population, and sonochemistry. J Phys Chem B 106 9181-9190... 

Xu H, Eddinsaas NC, Suslick KS (2009) Spatial separation of cavitatin bubble populations The nanodroplet injection model. J Am Chem Soc 131 6060-6061... 

Fig. 21. Variation of the extraction efficiency with dimensionless bubble radius for diffusion-controlled and hydrodynamically controlled bubble growth when the bubble population is constant = 0.10, Xo = 0.10, = 5.87.

Jakob and Linke (J5) observed nucleate boiling at low bubble populations for water and carbon tetrachloride from flat vertical and horizontal chromium surfaces. 

Wu, J., Bubble populations and spectra in near-surface ocean Summary and review of field measurements , J. Geophys. Res., 86,457-463 (1981). 

We applied this technique to make an initial estimate of the timing and extent of Cenozoic uplift of the Colorado Plateau (Sahagian et al. 2002a). Because the technique measures paleoatmospheric pressure, it is not subject to uncertainties stemming from the use of proxies that depend on environmental factors other than elevation alone. Vesicular lavas preserve a record of paleopressure at the time and place of lava emplacement because the difference in internal pressure in bubbles at the base and top of a lava flow depends on atmospheric pressure and lava flow thickness. The modal size of the vesicle (bubble) population is larger at the top than at the bottom. This leads directly to paleoatmospheric pressure and thus elevation because the thickness of the flow can easily be measured in the field, and the vesicle sizes can be measured in the lab. All proxies have their limitations and hence are not applicable in all places and at all times. Vesicular basalts are no exception. For a lava flow to record atmospheric pressure, the flow thickness must remain constant between the time the upper and lower crusts (10-20 cm) form, and the time of complete solidification of the flow. 

H. Medwin, In situ acoustic measurements of bubble populations in coastal ocean waters, J. Geophys. Res. 75 (1970) 599-611. 

The population balance simulator has been developed for three-dimensional porous media. It is based on the integrated experimental and theoretical studies of the Shell group (38,39,41,74,75). As described above, experiments have shown that dispersion mobility is dominated by droplet size and that droplet sizes in turn are sensitive to flow through porous media. Hence, the Shell model seeks to incorporate all mechanisms of formation, division, destruction, and transport of lamellae to obtain the steady-state distribution of droplet sizes for the dispersed phase when the various "forward and backward mechanisms become balanced. For incorporation in a reservoir simulator, the resulting equations are coupled to the flow equations found in a conventional simulator by means of the mobility in Darcy s Law. A simplified one-dimensional transient solution to the bubble population balance equations for capillary snap-off was presented and experimentally verified earlier. Patzek s chapter (Chapter 16) generalizes and extends this method to obtain the population balance averaged over the volume of mobile and stationary dispersions. The resulting equations are reduced by a series expansion to a simplified form for direct incorporation into reservoir simulators. 

A simplified one-dimensional transient solution of the bubble population balance equations, verified by experiments, has been presented elsewhere (5) for a special case of bubble generation by capillary snap-off. 

Interaction parameters, a(x,t), calculated from the bubble population balance itself, e.g., the total bubble density in flowing and stationary foam, the higher moments of the bubble number density distribution, etc. 

As mentioned before. Equations (5) and (6) are the differential transport equations of average bubbles and could be written from scratch without the convoluted derivations invoked here. Unfortunately, modeling of foam flow in porous media is a lot more complicated than Equations (3) and (6) lead us to believe. Having started from a general bubble population balance, we discovered that flow of foams in porous media is governed by Equations (2) and (3), and that Equations (5) and (6) are but the first terms in an infinite series that approximates solutions of (2) and (3). 

The continuum form of the bubble population balance, applicable to flow of foams in porous media, can be obtained by volume averaging. Bubble generation, coalescence, mobilization, trapping, condensation, and evaporation are accounted for in the volume averaged transport equations of the flowing and stationary foam texture. 

A meaningful simplification of the bubble population balance can be achieved by expanding the bubble number density into the moments of bubble mass. 

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