Critical bubble radius

The final result is very well known in experimental studies and is usually used to evaluate a32 from known and s values. Other studies in this field include the work by Shinnar and Church (S7) who used the Kolmogoroff theory of local isotropy to predict particle size in agitated dispersions, and an analysis by Levich (L3) on the breakup of bubbles. Levich derived an expression for the critical bubble radius (the radius at which breakup begins) ... 

An estimate may be made of this critical bubble radius, Rj, by differentiating the RHS of Eq. A.7, with respect to R, and equating to zero. 

Take the derivative of AGr with respect to r and set it to zero to find the critical bubble radius ... 

A striking feature is the way the so-called diseontinuity in the free rise velocity of a bubble varies with its size. Indeed, Astarita and Appuzzo [1965] noted a six to ten fold increase in the rise veloeity at a eritieal bubble size, as seen in Figure 5.6 for air bubbles in two polymer solutions. Subsequently, similar, though less dramatic results, summarised in a reeent review [DeKee et al., 1996], have been reported. The critical bubble radius appears to hover around 2.6 mm, irrespective of the type or the degree of non-Newtonian properties of the continuous phase, and this value is well predieted by the following... 

Obviously, some constraints will exist between the critical bubble radius (Rk, i.e., the radius R of a bubble at which a minimum reduction of the hydrostatic pressure causes a dramatic increase of R ) and the mean radius of the pores of the catalytic particle (rp). If the bubble radius is indicated as Re, then two classes of bubbles satisfy the following conditions ... 

It is known that the pores of a catalytic support can be divided into three classes according to their diameter dp in nm (or lOr m). Micropores dp have a diameter less than 2 nm in mesopores, 2 < dp < 50, and in macropores dp > 50. The values of the previously calculated critical bubble radius Rk change from 8.6 x 10 to... 

Table 2 Critical Bubble Radius Reported by Various Investigators...

Appendix 2 Critical Pressure (P, ) and Bubble Radius Relationship... 

Appendix 2 Critical Pressure fPJ and Bubble Radius Relationship I 63... 

Figure 1, Composition of the critical cluster (bubble of critical size), Xgas, thermodynamic driving force of critical bubble formation, CJ,radius of the critical bubble, Rc, and work of critical bubble formation, [J Gc, computed for the case of boiling in binary liquid-gas solutions in dependence on supersturation here expressed via the density of the liquid puq (for the details see Ref 21). By the number (1), the results are shown computed via the classical Gibbs approach employing the capillarity approximation, number (2) refers to computations via the generalized Gibbs approach and number (3) to computations via the van der Waals square gradient density functional method.

There is a critical drop radius, rcritkal, for equilibrium with the surrounding vapor pressure, because smaller drops have a higher vapor pressure and will spontaneously evaporate, and all drops larger than this size will grow at the expense of smaller (and unstable) droplets. For a water drop in air and also an air bubble in water, the effect of radius of curvature on equilibrium vapor pressures is given in Table 4.1. It can be seen that below a droplet size of lOnm, there is a considerable vapor pressure increase over the vapor pressure of flat surfaces due to the presence of curvature. 

Substituting this value into Eqs (10.6) and (10.7) yields an expression for the critical particle radius below which the inertia forces cannot cause the particle to approach the bubble. 

The negative effect of the centrifugal force can be summarised by the negative effect of SRHI, which is an essential deviation from Sutherland s formula. A common action of these factors appear if the limit trajectory ends not at the equator but at 0 = 0,. Results of such common action are shown in Fig. 10.15 for a fixed bubble radius a = 0.04cm and for a number of critical film thicknesses H,. = h,. / a,. 

Summarizing, we should note that the obtained results are only true in the limiting case when a change in the bubble size has a minor influence on its lift velocity. Later on, it will be shown that this case is possible when the bubble radius surpasses some critical value. 

FIG. 3 In this diagram, the Y axis represents AGtotai, and the X axis is the radius of the bubble, in units of r, the critical bubble size. The height of the curve at the point where x = 1, or r = r is the height of the kinetic barrier the system must overcome to undergo the phase change, AG. The curve crosses the X axis at the point r = 1.5r, and this is where the surface term exactly equals the volume term, and AG,otai = 0. 

FIG. 4 Log plot of the radius of the critical bubble as a function of the supersaturation, (7. The size increases very rapidly as the supersaturation decreases. 

In the computer simulation of the cell nucleation phenomena, the computation of 7 in Equation 1 requires an accurate determination of JV. By definition, fV is the amovmt of energy possessed by a cluster of gas molecules above which the cluster will grow spontaneously into a larger bubble. In other words, JV is the amount of energy required to create a bubble with its radius equal to the critical radius, (i.e., a critical bubble). Therefore, the values of Pbubbie,o in Equations (2) and (3) are equal to the gas pressure inside a critical bubble. 

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