Coalescence of bubble clusters

MODELING COALESCENCE OF BUBBLE CLUSTERS RISING FREELY IN LOW-VISCOSITY LIQUIDS... 

Modeling Coalescence of Bubble Clusters 423 Exponential Model... 

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]. 

A duster is defined as a group of neighbouring occupied sites. Bubble coalescence is allowed. Recall that bubbles coalesce if they have mechanical contact. Therefore, all sites of a cluster are considered to have coalesced and formed a new larger bubble. 

Clusters of a number of bubbles have been observed in a great number of experiments. Coalescence depends on the number of bubbles interacting in the clusters and on the number of clusters. Let us investigate further by comparing models that assume simple binary and clustering coalescence mechanisms with bubble size distribution data. 

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. 

Cluster coalescence changes the dependence on number from linear to quadratic. That is, we should expect to find the number of bubbles decreasing much more rapidly with size because more of the smaller bubbles combine into a few much larger ones. To find the distribution function corresponding to (16), we replace with the exceedance by dividing by N as before, and let the exponent on the right side be a parameter, t) > 0, rather than an exact quadratic, Tj = 2. 

That the Pareto model fits the data much better than the binary exponential model is shown very dramatically in Figures 10 and 11, which compare predicted to measured probabilities for all five data sets. The Pareto clustering coalescence model shows a surprisingly good match over the entire range of bubble sizes. On the other hand, the exponential binary coalescence model hardly shows any correlation to the data. 

Prince and Blanch s [19] bubble size distribution is the only one that appears to match the exponential model in Figure 10. They designed an experiment to achieve an equilibrium with equal breakup and coalescence rates. If coalescence is cluster-wise and breakup is binary, it would take many binary breakup events to balance one coalescence of a large cluster, and the binary process, which is characterized by an exponential distribution would dominate. 

Finally, recall the assertion that the coalescence rate and bubble size distribution resulting from coalescence can be modeled by the Pareto distribution for which the coalescence rate is proportional to a power of the number of bubbles. A simple cluster coalescence model makes this power equal to two. This means that the number of coalescences observed in a release ought to be roughly proportional to the square of the number of bubbles released. Figure 14 shows that this is true. The... 

It is not necessary to assume simultaneous coalescence of clusters to derive a model with a coalescence rate proportional to some power of the number of bubbles. Recall that we quantified the coalescence probability as the product of a collision probability and coalescence efficiency, = P 5 T1 5. The coalescence density, or coalescence rate per unit volume, N, , is directly proportional to Then, with approximately constant (ignoring effects of bubble size) and P, proportional to a , we have... 

Bubble interaction in swarms is a complex process. Bubble clusters commonly form that coalesce more or less simultaneously into very large bubbles. Recent experiments have revealed some of the details behind this behavior. A bubble contacts another only by following its wake to an overtaking collision. Coalescence or breakup occurs only after the collision, when one bubble is pulled into the near wake of the other. Interaction of three or more bubbles in clusters leads to increased coalescence rates. We have also shown analytically that bubbles do not collide like solid particles, but rather are drawn together by the dynamics of the surrounding fluid. Gravity and fluid acceleration drive bubble motion small-scale turbulence tends to prevent rather than enhance coalescence. 

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. 

Lee J, Tuziuti T, Yasui K, Kentish S, Grieser F, Ashokkumar M, Iida Y (2007) Influence of surface-active solutes on the coalescence, clustering, and fragmentation of acoustic bubbles confined in a microspace. J Phys Chem C 111 19015-19023... 

Figure 7.12 shows a cluster of partially coalesced fat droplets on the surface of an air bubble. Typically, 30% of the fat is partially coalesced, but this can vary substantially. 

In ice cream the fat content is lower, so there is not enough fat to cover the whole surface of the air bubbles (Figure 7.15b). The discrete and partially coalesced fat droplets are somewhat hydrophobic because they are partly coated with emulsifier. As a result they adsorb at the air bubble surface. They stabilize the air bubbles by forming a barrier between them. They also increase the matrix viscosity (since they are suspended solid particles), which strengthens the films of matrix between the bubbles and hinders coalescence. The extent to which the partially coalesced fat in ice cream exists as discrete clusters or an extended network (as in whipped cream) is an area of current research. 

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