Cluster coalescence model

Five bubble size distributions were selected from the literature to compare with the binary and cluster coalescence models. Four were measured in small-scale bubble column test sections and one in a sieve tray. Newtonian systems of air-water, oxygen-water, and oxygen-glycol, and a non-Newtonian oxygen-PAA solution were used. The data sets are described in Table 1. The mean bubble diameter is the equivalent diameter of a sphere of equal volume. Bubble shapes are generally ellipsoidal, though larger cap bubbles of -2 cm in diameter appear in several tests. 

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. 

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... 

The Pareto distribution fits this relation just as it did the cluster coalescence model. Recall that the tail of the Pareto distribution is so fat in the latter case that the mean is not even defined. The situation is not quite so severe with the 3/2 exponent—the mean is defined, but the variance is not. We conclude that the Pareto distribution need not require that all coalescences occur simultaneously in clusters, but can also apply to a binary process. 

The bubble size distribution resulting from purely random binary coalescence is well-represented by the geometric and exponential distributions. But these distributions completely miss the behavior of measured bubble size distributions that show relatively fewer of the smallest bubbles and more of the very largest ones. But a simple cluster coalescence model follows the Pareto distrihution, which matches these characteristic trends quite well. We conclude that multiple bubble interaction... 

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 mechanisms of the crystal-building process of Cu on Fe and A1 substrates were studied employing transmission and scanning electron microscopy (1). These studies showed that a nucleation-coalescence growth mechanism (Section 7.10) holds for the Cu/Fe system and that a displacement deposition of Cu on Fe results in a continuous deposit. A different nucleation-growth model was observed for the Cu/Al system. Displacement deposition of Cu on A1 substrate starts with formation of isolated nuclei and clusters of Cu. This mechanism results in the development of dendritic structures. 

Another way to prepare model catalysts (under UHV conditions) is to grow the clusters in gas phase and depose them on the substrate. However, to avoid implantation, fragmentation, and dynamic coalescence, it is necessary to soft-land the clusters. A first possibility is to decrease the kinetic energy of the clusters to less... 

Figure 3.11 (a) Cluster (hatched) -tissue (dotted) texture of glass, (b) coalesced potential wells and energy separations and (c) variation of in model calculations as a function of RT/AE for various sets of n and a values. Doted portions show steep fall of to zero. (After Rao, 1984). 

A new model for the clustering of charges in dry ionomers is presented. The basic idea is that, under the influence of electrostatic interactions, the multiplets of charges coalesce in clusters that have an internal structure compatible with the steric hindrances due to the polymeric material. The size of the cluster is shown to be independent of the concentration of charges. The tension of the chains within the matrix is discussed, and it is suggested that the clusters are arranged in small hypercrystallites with a local order of the diamond type. 

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