Diffusive bubble growth

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). 

At a later stage of bubble growth, heat diffusion effects are controlling (as point c in Fig. 2.9), and the solution to the coupled momentum and heat transfer equations leads to the asymptotic solutions and is closely approximated by the leading term of the Plesset-Zwick (1954) solution,... 

Diffusion-Controlled Bubble Growth S. G. Bankoff Evaporative Convection... 

Westwater (W4, W5) has written a detailed review of boiling in liquids with emphasis on nucleation at surfaces. Although written in 1956, this is still very useful and it provides a detailed description of the factors affecting nucleation. In a more recent review, Leppert and Pitts (L2) have described the important factors in nucleate boiling and bubble growth, and Bankoff (B2) has reviewed the field of diffusion-controlled bubble growth in nonflowing batch systems. 

Relative importance of coalescence and rectified diffusion in the bubble growth is still under debate. After acoustic cavitation is fully started, coalescence of bubbles may be the main mechanism of the bubble growth [16, 34], On the other hand, at the initial development of acoustic cavitation, rectified diffusion may be the main mechanism as the rate of coalescence is proportional to the square of the number density of bubbles which should be small at the initial stage of acoustic cavitation. Further studies are required on this subject. 

Fig. 3. Thresholds of cavitation. Region A Bubble growth through rectified diffusion only. Region B Bubble growth through transient cavitation. RD, Threshold for rectified diffusion Rlt threshold for predomination of inertial effects RB, Blake threshold for transient cavitation. [After R. E. Apfel (S).]...

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.

On the other hand, if bubble growth is diffusion controlled, then the mass transfer coefficient may be meaningful. However, in this case, the surface area for mass transfer is the surface area of the bubbles entrained in the solution and this depends on the volume of liquid in the extraction zone and not on the surface area of the extraction zone. Clearly, attempts to correlate experimental data for the extraction of a volatile component from a polymeric solution containing entrained bubbles using mass transfer coefficients can be misleading or totally erroneous. 

Diffusion-Controlled Bubble Growth S. G. Bankoff Evaporative Convection John C. Berg, Andreas Acrivos, and Michel Boudart... 

The scope of kinetics includes (i) the rates and mechanisms of homogeneous chemical reactions (reactions that occur in one single phase, such as ionic and molecular reactions in aqueous solutions, radioactive decay, many reactions in silicate melts, and cation distribution reactions in minerals), (ii) diffusion (owing to random motion of particles) and convection (both are parts of mass transport diffusion is often referred to as kinetics and convection and other motions are often referred to as dynamics), and (iii) the kinetics of phase transformations and heterogeneous reactions (including nucleation, crystal growth, crystal dissolution, and bubble growth). 

Geochemical kinetics is stiU in its infancy, and much research is necessary. One task is the accumulation of kinetic data, such as experimental determination of reaction rate laws and rate coefficients for homogeneous reactions, diffusion coefficients of various components in various phases under various conditions (temperature, pressure, fluid compositions, and phase compositions), interface reaction rates as a function of supersaturation, crystal growth and dissolution rates, and bubble growth and dissolution rates. These data are critical to geological applications of kinetics. Data collection requires increasingly more sophisticated experimental apparatus and analytical instruments, and often new progresses arise from new instrumentation or methods. 

The physical transport of mass is essential to many kinetic and d3mamic processes. For example, bubble growth in magma or beer requires mass transfer to bring the gas components to the bubbles radiogenic Ar in a mineral can be lost due to diffusion pollutants in rivers are transported by river flow and diluted by eddy diffusion. Although fluid flow is also important or more important in mass transfer, in this book, we will not deal with fluid flow much because it is the realm of fluid dynamics, not of kinetics. We will focus on diffusive mass transfer, and discuss fluid flow only in relation to diffusion. 

For the dissolution of many crystals when their diffusion profiles overlap, the bulk melt can no longer be treated as an infinite reservoir. An approximate treatment is to assume that the crystals are regularly distributed in the melt, and every crystal is enclosed by a spherical melt shell. The problem may then be solved using the method developed for bubble growth by Proussevitch et al. (1993) and Proussevitch and Sahagian (1998) (Section 4.2.S.2). 

Proussevitch A.A. and Sahagian D.F. (1996) Dynamics of coupled diffusive and decompressive bubble growth in magmatic systems. /. Geophys. Res. 101, 17447-17455. 

---