Time of Bubble Collapse

Consider an empty bubble collapsing under the influence of a constant external pressure Pjj from an initial radius to a final radius R (Fig. A3,la). 

The work done by the (hydrostatic) pressure, Pjj, neglecting surface tension effects, is the product of the pressure and the change in volume and is given by 

This work will be equal to the kinetic energy (mv /2) of the liquid as it moves to fill the space vacated by the collapse of the bubble and is given by i p 47tr dr (dr/dt) (Fig. A3.lb) where p is the density of the liquid, 47t r d r = the volume of liquid moving and (dr/dt) is the velocity. Thus 

At first sight this integration appears somewhat difficult to perform since it contains both dr and (dr/dt). However, if it is assumed that the liquid is incompressible then the volume lost by the cavity (47t R dR) is equal to that filled by the liquid (47t r dr) - i. e. 47tR rdR = 47tr dr, or 

The time to collapse the bubble can now be obtained by integrating the right hand side of Eq. A.19 from R to zero to give 

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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 results obtained from the solution of the Raleigh-Plesset equation can be used to estimate the volume and intensity of the active zone downstream from the orifice. The volume of the zone can be calculated by multiplying the bubble lifetime (i.e., the total time used in solving the equation or the time of bubble collapse where the bubble wall velocity... 

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... 

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. 

Theoretical considerations by Noltingk and Neppiras [11], subsequently expanded by Flynn [12] and Neppiras [13], that assume adiabatic collapse of the bubbles, allow the temperature and pressure within the bubble at the time of total collapse to be calculated. 

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. 

To close this Section we comment on two papers that do not fit under any neat heading. The first of these is by Xiao et al,261 who study the final stages of the collapse of an unstable bubble or cavity using MD simulations of an equilibrated Lennard-Jones fluid from which a sphere of molecules has been removed. They find that the temperature inside this bubble can reach up to an equivalent of 6000 K for water. It is at these temperatures that sonolumines-cence is observed experimentally. The mechanism of bubble collapse is found to be oscillatory in time, in agreement with classical hydrodynamics predictions and experimental observation. The second paper, by Lue,262 studies the collision statistics of hard hypersphere fluids by MD in 3, 4 and 5 dimensions. Equations of state, self-diffusion coefficients, shear viscosities and thermal conductivities are determined as functions of density. Exact expressions for the mean-free path in terms of the average collision time and the compressibility factor in terms of collision rate are also derived. Work such as this, abstract as it may appear, may be valuable in the development of microscopic theories of fluid transport as well as provide insight into transport processes in general. 

Fig. 5.11 Final stage of bubble collapse, calculated for collapsing time of 0.1 ms when the bubble is filled with O2 and irradiated with ultrasound of frequency 1 MHz under 8 atm. pressure. The environment pressure is 1 atm. and initial bubble size is 0.5 mm. Panel (a) reports the temperature inside the bubble, panel (b) shows the number of molecules in it [246]...

A simplified description of the phenomenon of bubble collapse, first performed by Besant in 1859, begins with the case of a bubble maintaining perfect sphericity at all times and radially oscillating in an incompressible liquid (the acoustic approximation). Readers interested in advanced developments should refer to specialized papers and books. ... 

The time of complete collapse (the radius varies from Rm to R = 0) is inversely proportional to the bubble-wall speed (dR/dt) and proportional to R ... 

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. 

Fig. 31. Bubble wall velocity vs time during cavitational collapse for different values of the parameter X defined as X ss 0.4 c iTl] p./fri/2 (Ph — Pv)i/2). X permits us to account for the viscous and inertia effects of the polymer solution (redrawn according to Ref. [122]) ...

As the temperature and pressure dramatically increase inside a bubble at the end of the collapse, water vapor and oxygen, if present, are dissociated inside a bubble and oxidants such as OH, O, and H2O2 are created [12, 13]. They dissolve into the liquid and solutes are oxidized by them. This is called sonochemical reaction. For example, potassium iodide (KI) in aqueous solution is oxidized by the irradiation of ultrasound ((1.1)), and the solution is gradually colored by the product (I3 ) as the irradiation time increases. 

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. 

Transient cavitation is generally due to gaseous or vapor filled cavities, which are believed to be produced at ultrasonic intensity greater than 10 W/cm2. Transient cavitation involves larger variation in the bubble sizes (maximum size reached by the cavity is few hundred times the initial size) over a time scale of few acoustic cycles. The life time of transient bubble is too small for any mass to flow by diffusion of the gas into or out of the bubble however evaporation and condensation of liquid within the cavity can take place freely. Hence, as there is no gas to act as cushion, the collapse is violent. Bubble dynamics analysis can be easily used to understand whether transient cavitation can occur for a particular set of operating conditions. A typical bubble dynamics profile for the case of transient cavitation has been given in Fig. 2.2. By assuming adiabatic collapse of bubble, the maximum temperature and pressure reached after the collapse can be estimated as follows [2]. 

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