Cavitation bubble dynamics equations

Prediction of cavitational activity distribution based on theoretical analysis of the bubble dynamics equations can be used to identify the regions with maximum pressure fields in a large scale reactor and then may be small reactors can... 

The bubble-dynamics equations are very similar to acoustic cavitation the only difference being the fact that the surrounding fluctuating pressure field is driven by hydrodynamic conditions existing downstream of the constriction, whereas in the case of acoustic cavitation, it is dependent on the frequency and intensity of the ultrasonic irradiation (sinusoidal variation). There are two approaches used for the estimation of the local pressure at any location downstream of the constriction (the typical pressure recovery profiles are shown in Fig. 8.2.6). 

In this chapter we will deal with those parts of acoustic wave theory which are relevant to chemists in the understanding of how they may best apply ultrasound to their reaction system. Such discussions tvill of necessity involve the use of mathematical concepts to support the qualitative arguments. Wherever possible the rigour necessary for the derivation of the basic mathematical equations has been kept to a minimum within the text. An expanded treatment of some of the derivations of key equations is provided in the appendices. For those readers who would like to delve more deeply into the physics and mathematics of acoustic cavitation numerous texts are available dealing with bubble dynamics [1-3]. Others have combined an extensive treatment of theory with the chemical and physical effects of cavitation [4-6]. 

The first and second derivatives of R with respect to time are represented by R and R, respectively. Solving this equation for different values of R0 can be quite illustrative on the complex nonlinear dynamics of cavitation bubbles. Fig. 2 shows two different cases when a frequency of 20 kHz and an intensity equivalent to Py — 2.7 bar are used. In the first case (Fig. 2a), a relatively large (R0 = 2 mm) bubble couples with the sonic field through small-amplitude growth and compression cycles (stable cavitation). In contrast, a smaller bubble (P0 = 20 pm) experiences resonant coupling, which results... 

Another flaw of the earlier investigations on the bubble dynamics was that al though there has been ample discussion on whether the compressibility of the cavitating medium plays an important role on the cavity motion, there have been no illustrations, where the difference between the predictions of equations considering the compressibility of the medium (e.g. Gilmore model [Gilmore, 1954]... 

By combining the Rayleigh-Plesset equation with a mass and energy balance over the bubble, the temperature and pressure in the bubble can be calculated (16,17). The model also describes the dynamic movement of the bubble wall, which results in a calculated radius of the cavitation bubble as a fimction of time (see Fig. 2). The explosive growth phase and the collapse phase of the bubble can clearly be distinguished. Moreover, in case dynamic effects are more important than the surface tension, the cavitation threshold can be calculated with the dynamic model, while the Blake threshold pressure cannot be used at these conditions. 

Question by R. J. Good, General Dynamics/Astronautics Do you have an explanation (1) for your observation that the critical negative pressure for cavitation-or. noncavitation is so much larger for liquid nitrogen than for water and (2) for the independence of air saturation, for water The expectation based on the Kelvin equation (i.e., that the vapor pressure in a bubble is less than that over a flat surface) would be the reverse of your observation. 

Therefore, sonochemistry can be described as the result of acoustic cavitation consisting of three events creation, expansion, and implosive collapse of bubbles in ultrasonically irradiated liquids (Apfel, 1981 also see Suslick, 1986). The microbubble, or cavitational, field is characterized by spherical bubbles partially filled with noncondensable gases. The radius of the bubble rf, at any time deviates from its initial value /-, o in a periodic fashion. This dynamic behavior of the bubble is well described by the Raleigh-Plesset equation given by... 

---