Clusters of bubbles

Additionally, macroscopic flow structure of 3-D bubble columns were studied [10]. The results reported can be resumed as follows (a) In disperse regime, the bubbles rise linearly and the liquid flow falls downward between the bubble stream, (b) If gas velocity increases, the gas-liquid flow presents a vortical-spiral flow regime. Then, cluster of bubbles (coalesced bubbles) forms the central bubble stream moving in a spiral manner and 4-flow region can be identified (descending, vortical-spiral, fast bubble and central flow region). Figure 10 shows an illustrative schemes of the results found in [10]. 

The bubble size distribution from a completely random, binary coalescence process is modeled well by the geometric and exponential distributions. We now develop a simple model for non-binary, clusterwise coalescence. In the random binary model (13), that leads to the exponential distribution, the effect of coalescence is linear. But now assume that coalescence occurs not between pairs of bubbles, but simultaneously among clusters of bubbles. Then the change in the number of bubbles with volume, m, is the product of the number in the cluster (dN dN, ) and the change in the number of clusters with volume. That is. 

This basic model seems contrived, in that no direct experimental justification is given for it by Pelton and Goddard [47], Indeed these workers argue [47] that groups in this treatment are a mathematical convenience to facilitate the derivation and not a physically observable cluster of bubbles. However, in a later paper, which uses a similar model, Pelton [48] claims to observe formation of secondary bubbles in the foam column. It is claimed that they expand in size by coalescence until the buoyancy force exceeds the yield stress in the foam whereupon they rise rapidly to the top of the foam column and rupture. 

In the Steiner problem, Chapter 3, and the minimum surface area problems, discussed earlier in this chapter, the Laplace-Young equation had a simple form. This resulted from the zero excess pressure across any point on the surface of the soap film. In the case of a bubble, or clusters of bubbles, the excess pressure across any surface is not in general zero. However the Laplace-Young equation can be applied under these more general conditions. Plateau s rules concerning the angles at which surfaces and lines of soap films intersect apply also to the surfaces and lines of soap film produced by clusters of bubbles. 

Clusters of bubbles illustrate Plateau s three rules. In the two cases studied so far the points, such as O and S in Fig. 4.16 and P, Q, R and O in Fig. 4.18, are formed from the intersection of three surfaces. The angles of intersection of the tangent planes to these surfaces are 120°. 

Bubbles can produce hemispherical clusters when formed on a wet surface or on the surface of a bath of soap solution. Figure 4.21 shows symmetrical hemispherical clusters of bubbles that are symmetrical about a plane. A symmetrical vertical section has been taken through the bubbles and the 120 ° angles of intersection of the surfaces have been indicated by dots. 

An interesting and practically valuable result was obtained in [21] for PE + N2 melts, and in [43] for PS + N2 melts. The authors classified upper critical volumetric flow rate and pressure with reference to channel dimensions x Pfrerim y Qf"im-Depending on volume gas content

channel entrance (pressure of 1 stm., experimental temperature), x and y fall, in accordance with Eq. (24), to tp 0.85. At cp 0.80, in a very narrow interval of gas concentrations, x and y fall by several orders. The area of bubble flow is removed entirely. It appears that at this concentration of free gas, a phase reversal takes place as the polymer melt ceases to be a continuous phase (fails to form a continuous cluster , in flow theory terminology). The theoretical value of the critical concentration at which the continuous cluster is formed equals 16 vol. % (cf., for instance, Table 9.1 in [79] and [80]). An important practical conclusion ensues it is impossible to obtain extrudate with over 80 % of cells without special techniques. In other words, technology should be based on a volume con-... 

After the bubble venting the liquid remains attached to the wall as droplets or clusters of droplets. It evaporates during the period of the cycle. 

Figure 6.18d shows the appearance of liquid droplets or clusters of liquid droplets on the wall after the bubble venting. The pressure in the micro-channel decreases and water starts to move into it from the inlet manifold (Fig. 6.18e). Figure 6.18f shows the start of a new cycle. 

Figure 29 (Qin and Liu, 1982) shows the behavior of individual particles above the distributor recorded by video camera of small clusters of particles, coated with a fluorescent material and spot-illuminated by a pulse of ultra violet light from an optical fiber. The sequential images, of which Fig. 29 just represents exposures after stated time intervals, were reconstructed to form the track of motion of the particle cluster shown in Fig. 30. Neither this track nor visual observation of the shallow bed while fluidized, reveal any vestige of bubbles. Instead, the particles are thrown up by the high velocity jets issuing from the distributor orifices to several times their static bed height.

Free Energy of a Cluster For clarity of discussion, crystal nucleation from a melt is used to derive the following relations. For nucleation of liquid droplets, the derivation is similar. For nucleation of bubbles, the formulation is slightly different and is summarized separately below. Let the Gibbs free energy difference between the crystalline and the melt state per mole of the crystalline composition be AGc m = where Hc and /im are the chemical potential (partial... 

Equation (293) cannot be applied to gas fluidized beds because in the latter case, the fluidized bed contains a large number of bubbles. The rate of heat transfer between the bed and wall is determined in the latter case by the heat transfer in the packets (clusters) of solid particles (through which the gas flows at the minimum fluidization velocity) which are exchanged, because of bubbling, between the wall and the bulk of the fluidized bed [74], The heat transfer coefficient is given in the latter case by an expression similar to Eq. (282) ... 

Particle irradiation effects in halides and especially in alkali halides have been intensively studied. One reason is that salt mines can be used to store radioactive waste. Alkali halides in thermal equilibrium are Schottky-type disordered materials. Defects in NaCl which form under electron bombardment at low temperature are neutral anion vacancies (Vx) and a corresponding number of anion interstitials (Xf). Even at liquid nitrogen temperature, these primary radiation defects are still somewhat mobile. Thus, they can either recombine (Xf+Vx = Xx) or form clusters. First, clusters will form according to /i-Xf = X j. Also, Xf and Xf j may be trapped at impurities. Later, vacancies will cluster as well. If X is trapped by a vacancy pair [VA Vx] (which is, in other words, an empty site of a lattice molecule, i.e., the smallest possible pore ) we have the smallest possible halogen molecule bubble . Further clustering of these defects may lead to dislocation loops. In contrast, aggregates of only anion vacancies are equivalent to small metal colloid particles. 

Of special interest in the recent years was the kinetics of defect radiation-induced aggregation in a form of colloids-, in alkali halides MeX irradiated at high temperatures and high doses bubbles filled with X2 gas and metal particles with several nanometers in size were observed [58] more than once. Several theoretical formalisms were developed for describing this phenomenon, which could be classified as three general categories (i) macroscopic theory [59-62], which is based on the rate equations for macroscopic defect concentrations (ii) mesoscopic theory [63-65] operating with space-dependent local concentrations of point defects, and lastly (iii) discussed in Section 7.1 microscopic theory based on the hierarchy of equations for many-particle densities (in principle, it is infinite and contains complete information about all kinds of spatial correlation within different clusters of defects). 

As a phase-reversed counterpart of bubbling fluidization, bubbles are replaced by clusters, which are subject to rapid dissolution and reformation, thus leading to improved G/S contacting. 

An appropriate understanding of particle-fluid two-phase flow rests on an adequate analysis of the local hydrodynamics corresponding to the scale of bubbles or clusters, as well as the overall hydrodynamics corresponding to the scale of the retaining vessel. Local hydrodynamics, designated as phases, is rooted in the intrinsic characteristics of particle-fluid systems,... 

Contrary to bubbling fluidization, gas flow in the form of bubbles is transformed to a continuous dilute phase, while solids in the emulsion are transformed into a discontinuous phase as clusters. 

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