Bubbles growth

The initial bubble is ideally a sphere that grows as a result of the interaction of the differential pressure, Ap, between the inside and outside of the cell and the interfacial surface tension, y. The radius, r, of the bubble at equilibrium is related to these factors as follows  

Nevertheless, in order to get some insight into the mechanism of bubble growth, and following the classic derivation of Scriven (34), we derive here the particular case for the rate of growth of a single bubble in a quiescent infinite liquid (Fig. 8.11), with the viscous forces acting as the rate-controlling step. 

The equation of continuity for an incompressible liquid, and with spherical symmetry, reduces to 

Since vrr2 is a function of time alone, it must hold everywhere in the liquid, including at the surface of the bubble. The surface itself moves at velocity R, while the liquid adjacent to the surface moves with velocity vr(R), which is different from that of the surface because some of the volatile material evaporates, and the mass flow rate is given by 4iiR2pL[R — vr(/ )], which is the rate of vaporization of volatile material into the bubble. We now can write a mass balance over the bubble surface  

Equation 8.9-2 suggests that the product of the radial velocity and the square of the radius is constant anywhere in the liquid phase, which gives the following continuity condition  

Next we turn to the Navier-Stokes equation, which for creeping flow of incompressible liquids, neglecting inertial and gravitational forces, reduces to 

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Fig. 11-19. The drop ejection process in an inkjet printer (a) bubble nucleation (b) bubble growth and drop ejection (c) refill. [From J. H. Bohoiquez, B. P. Canfield, K. J. Courian, F. Drogo, C. A. E. Hall, C. L. Holstun, A. R. Scandalis, and M. E. Shepard, Hewlett-Packard J. 45(1), 9-17 (Feb. 1994). Copyright 1994, Hewlett-Packard Company. Reproduced with permission.]...

Bubbles can grow to on the order of a meter in diameter in Group B powders in large beds. The maximum stable bubble size is limited by the size of the vessel or the stabiUty of the bubble itself. In large fluidized beds, the limit to bubble growth occurs when the roof of the bubble becomes unstable and the bubble spHts. EmpidcaHy, it has been found that the maximum stable bubble size may be calculated for Group A particles from... 

Based on rate of bubble growth away from fixed orifice. Approximately three times too high compared to experiments. 

Bubble growth will be hmited by the containing vessel and the bubble hydrodynamic stability. Bubbles in group-B systems can grow to several meters in diameter. Bubbles in group-A materials with high fines may reach a maximum stable bubble size of only several cm. 

For group B and D particles, nearly all the excess gas velocity (U — U,nj) flows as bubbles tnrough the bed. The flow of bubbles controls particle mixing, attrition, and elutriation. Therefore, ehitriation and attrition rates are proportional to excess gas velocity. Readers should refer to Sec. 17 for important information and correlations on Gel-dart s powder classification, minimum fluidization velocity, bubble growth and bed expansion, and elutriation. 

In Eq. (13), medium resistance to bubble compression-decompression depends on viscosity r, and is described by the second member in the right-hand part of the equation. It should be mentioned at this point that bubble growth in a Newtonian liquid was originally examined by the Soviet physicist Y. I. Frenkel [29], in a rarely cited work published in 1946. 

Yet, Eq. (14) does not describe the real situation. It must also be taken into account that gas concentration differs in the solution and inside the bubble and that, consequently, bubble growth is affected by the diffusion flow that changes the quantity of gas in the bubble. The value of a in Eq. (14) is not a constant, but a complex function of time, pressure and bubble surface area. To account for diffusion, it is necessary to translate Fick s diffusion law into spherical coordinates, assign, in an analytical way, the type of function — gradient of gas concentration near the bubble surface, and solve these equations together with Eq. (14). 

M. Amon and C. D. Denson [33-34] attempted a theoretical and experimental examination of molding a thin plate from foamed thermoplastic. In the first part of the series [33] the authors examined bubble growth, and in the second [34] — used the obtained data to describe how the thin plate could be molded with reference to the complex situation characterized in our third note. Here, we are primarily interested in the model of bubble growth per se, and, of course, the appropriate simplification proposals [33]. Besides the conditions usual for such situations ideal gets, adherence to Henry s law, negligible mass of gas as compared to mass of liquid, absence of inertia, small Reynolds numbers, incompressibility of liquid, the authors postulated [33] several things that require discussion ... 

These two considerations allowed M. Amon and C. D. Denson to avoid difficulties pertaining to the assignment of concentration gradients near the bubble wall. The authors called their model the cellular model . Setting the quantity of bubbles, they placed each bubble in correspondence with a spherical cell of surrounding liquid with a mass equal to the ratio of the entire liquid mass to the overall quantity of bubbles. This made it possible for them to solve the problem of bubble growth in this cell. 

Fig. 2.38a-h Bubble growth at low heat flux. G = 95kg/m s, q = 80kW/m. Reprinted from Hetsroni et al. (2003b) with permission... 

The results were obtained at heat flux = 10 kW/m. For both liquids at f = 1 ms the contact angle is approximately of 0 = 60°, which is very close to the equilibrium surface tension of water. Throughout bubble growth this value decreases approxi-... 

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... 

From the physical point of view it is possible to suggest that the rate of bubble growth in micro-channel is determined by the following parameters ... 

In accordance with (6.40) one can present the functional equation for rate of bubble growth as follows... 

According to Eq. (6.44) such a behavior may be analyzed using parameter Hi. In the range of Hi = 0.00791—0.0260 linear behavior of the bubble radius was observed, when Hi > 0.0260 exponential bubble growth took place (Fig. 6.24). 

Table 6.6 also demonstrates extraordinarily high bubble growth rates of 94.63, 72.8 and 95.3pm/ms. For these three cases, the growth rates are two orders of magnitude higher than the other cases. The authors noted that it is unclear why the bubble growth rate for such cases is much higher than the other cases. 

Visual observation of bubble growth in parallel triangular micro-channels showed that the majority of the first bubbles were observed on the channel bottom wall, though a few bubbles did appear on the side walls. After nucleation, bubbles first... 

Dimensional analysis shows that the behavior of the bubble radius with time depends on the parameter TI = qf (plUCpLATs). In the range of 17 = 0.0079-0.026, linear behavior was observed, and when 17 > 0.026 exponential bubble growth took place. 

Cole R, Shulman HL (1966) Bubble growth rates at high Jacob numbers Int J Heat Mass Transfer 9 1377-1390... 

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