Bubble collapse time

One solution that was considered by Rayleigh (Lamb, 1945) for the determination of bubble collapse time, tm, used the model of a bubble with initial size Rm, suddenly subjected to a constant excess liquid pressure pL. Neglecting the surface tension and the gas pressure in the bubble, Eq. (2-29) may be rearranged to... 

Cavitation has three negative side effects in valves—noise and vibration, material removal, and reduced flow. The bubble-collapse process is a violent asymmetrical implosion that forms a high-speed microjet and induces pressure waves in the fluid. This hydrodynamic noise and the mechanical vibration that it can produce are far stronger than other noise-generation sources in liquid flows. If implosions occur adjacent to a solid component, minute pieces of material can be removed, which, over time, will leave a rough, cinderlike surface. 

X = 10 4 r0 = initial bubble radius tcou = collapse time scale, for an inviscid fluid, collapse occurs at t = 0.915 xco ... 

Finally, the bubble collapse stops when the pressure inside a bubble (pg) in the right hand side of (1.13) dramatically increases as the density inside a bubble nearly reaches that of a condensed phase (A bubble is almost completely occupied by the van der Waals hard-cores of gas and vapor molecules at that moment). At the same time, the temperature and pressure inside a bubble dramatically increase. 

Characterization of the cavitational phenomena and its effects in sonochemical reactors are generally described through mapping. Mapping of sonochemical reactor is a stepwise procedure where cavitational activity can be quantified by means of primary effect (temperature or pressure measurement at the time of bubble collapse) and/or secondary effect (quantification of chemical or physical effects in terms of measurable quantities after the bubble collapse) to identify the active and passive zones. 

The dynamic process of bubble collapse has been observed by Lauter-born and others by ultrahigh speed photography (105 frames/second) of laser generated cavitation (41). As seen in Fig. 4, the comparison between theory and experiment is remarkably good. These results were obtained in silicone oil, whose high viscosity is responsible for the spherical rebound of the collapsed cavities. The agreement between theoretical predictions and the experimental observations of bubble radius as a function of time are particularly striking. 

Eq. 2.25 is useful in that it allows an estimate of the point in the compression cycle where total collapse is likely to occur. For example, the collapse time of a bubble of radius 10 cm (R ), in water at an ambient pressure (Pq) of 1 atm, is approximately 1 ps. Since an applied acoustic field of 20 kHz has a compression cycle of 25 ps, it is expected that total collapse will occur in the first 4 % of the cycle. 

In general an increase in intensity (I) will provide for an increase in the sonochemical effects. Cavitation bubbles, initially difficult to create at the higher frequencies (due to the shorter time periods involved in the rarefaction cycles) will now be possible, and since both the collapse time (Eq. 2.27), the temperature (Eq. 2.35) and the pressure (Eq. 2.36) on collapse are dependent on P i(=Ph + PA)> bubble collapse will be more violent. However it must be realised that intensity cannot be increased indefinitely, since (the maximum bubble size) is also dependent upon the pressure amplitude (Eq. 2.38). With increase in the pressure amplitude (P ) the bubble may grow so large on rarefaction (R g, ) that the time available for collapse is insufficient. 

It has been argued (Appendix 3, Eq. A.21) that the collapse time for a bubble, initially of radius R, is considerably shorter than the time period of the compression cyde. Thus the external pressure Pj (= P + Pjj), in the presence of an acoustic field, maybe assumed to remain effectively constant (Pj ) during the collapse period. Neglecting surface tension, assuming adiabatic compression (i. e. very short compression time), and replacing R, by R, the size of the bubble at the start of collapse, the motion of the bubble wall becomes... 

Flotation. Flotation (qv) is used alone or in combination with washing and cleaning to deink office paper and mixtures of old newsprint and old magazines (26). An effective flotation process must fulfill four functions. (/) The process must efficiently entrain air. Air bubble diameter is about 1000 Jim. Typically air bubbles occupy 25—60% of the flotation cell volume. Increasing the air liquid ratio in the flotation cell is said to improve ink removal efficiency (27). (2) Ink must attach to air bubbles. This is primarily a function of surfactant chemistry. Air bubbles must have sufficient residence time in the cell for ink attachment to occur. (3) There must be minimal trapping of cellulose fibers in the froth layer. This depends on both cell design and surfactant chemistry. (4) The froth layer must be separated from the pulp slurry before too many air bubbles collapse and return ink particles to the pulp slurry. 

Once formed, a bubble nucleus is exceedingly stable.. ..There appears to be no time limit beyond which it is not possible to regenerate a bubble from a hydrophobic collapse site. In these experiments as much as 48 hours were permitted to elapse between bubble collapse and regeneration (ref. 84 cf. synthetic microbubble disappearance and formation in Section 10.4). 

Fig. 42 Time evolution of a bubble collapsing near an electrode surface.

The sonication intensity is directly proportional to the square of the vibration amplitude of the ultrasonic source. As a rule, increasing the intensity increases the sonoohemioal effeots however, the ultrasonic energy a system can take is limited. Thus, cavitation bubbles, which are initially difficult to create at the higher frequencies as a result of the shorter duration of rarefaction cycles, are now possible by virtue of the collapse time, temperature and pressure on collapse being mutually dependent. However, the sonication intensity cannot be increased indefinitely as the maximum possible bubble size is also dependent on the pressure amplitude. As the pressure amplitude is increased, bubbles may grow so large on rarefaction that the time available for collapse will be inadequate. In fact, it has been unequivocally established that ... 

Chemat et al. [14] found the ]oint use of US and microwaves for the treatment of edible oils for the determination of copper to shorten the time taken by this step to about a half that was required in the classical procedure (heating in a Buchi digester) or with microwave assistance, nitric acid and hydrogen peroxide. However, they did not state the specific medium where the microwave-US-assisted method was implemented and assumed US to have mechanical effects only, even though they mentioned a cavitational effect. This is a very common mistake in working with US that is clarified in an extensive discussion by Chanon and Luche [15] of the division of sonochemistry applications into reactions which were the result of true and false effects. Essentially, these terms refer to real chemical effects induced by cavitation and those effects that can be ascribed to the mechanical impact of bubble collapse. The presence of one of these phenomena only has not been demonstrated in the work of Chemat et al. [14] — despite the illustrative figure in their article — so their ascribing the results to purely mechanical effects of US was unwarranted. 

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