Coalesced-sphere model

The latter assumption has been verified within experimental error from an analysis of the total scattering invariant which has been calculated from the absolute intensity of scattering. The results for n listed in Table II show an apparent increase at low water contents and then a slight decrease at large water contents. It is noted that this decrease in H implying particle coalescence is in apparent contradiction to the hard sphere model used above. 

Chen et al. (179) used optieal microscopy to study the same crude-oil emulsions in an electric field. They modeled the system with a computer simulation based on a hard-sphere model describing the droplets as stabilized by a rigid asphaltene film. They identified two different types of coalescence. The mobile interfaeial films led to low emulsion conductivities owing to immediate droplet/droplet coalescence, and the incompressible interfaeial film led to low emulsion conductivities due to droplet-chain formation and... 

Ostwald first modelled catastrophic inversions as being caused by the complete coalescence of the dispersed phase at the close packed condition (corresponding to a dispersed phase fraction of 0.74 in Ostwald s uniform hard sphere model). Other studies, e.g. Marzall, have shown that catastrophic inversions (though these inversions were not called catastrophic inversions by that author) can occur over a wide range of WOR. It has been suggested that this may be due to the formation of double emulsion drops (0/W /0), boosting the actual volume of the dispersed phase. 

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. 

To describe the formation of a nanosheU, we do not need to change the model equations - we change only the initial conditions. Namely, let us consider a sphere of pure B of radius tba surrounded by a shell of pure A. To avoid solving the nucleation problem, let us assume that the initial pure sphere A already contains a small void in the center. Of course, for a very big initial core, this assumption seems unreasonable since the first voids should nucleate in the vicinity of the initial contact between A and B. Yet, for nanoparticles, it is natural that the initial nanovoids coalesce very fast into a single central void. Thus, in our model, the initial B-profile is... 

Structure within a sphere of mesophase is modeled in Figure 2.36 where the lamellar polycyclic aromatic mesogens are stacked parallel to each other to create the anisotropy. On coalescence, these structures are maintained when the two adjoining structures merge into each other to create bulk anisotropy. 

Although difficult to apply in practice, models for coalescence rate provide an appreciation for the physical phenomena that govern coalescence. They also provide an appreciation for why it is difficult to interpret stirred tank data or even to define the appropriate experiment. For instance, it can be clearly seen from eq. (12-49) to (12-51) that the collision frequency increases with e, whereas the coalescence efficiency decreases with e. For constant phase fraction, the number of drops also increases with e. The models for coalescence of equal-sized drops are quite useful to guide the interpretation of data that elucidate the time evolution of both mean diameter and drop size distribution during coalescence. To this end, Calabrese et al. (1993) extended the work of Coulaloglou and Tavlarides (1977) to include turbulent stirred tank models for rigid spheres and deformable drops with immobile and partially mobile interfaces. The later model accounts for the role of drop viscosity. In practice, models for unequal-sized drops are even more difficult to apply, but they do suggest that rates are size dependent. They are useful in the application of the population balance models discussed in Section 12-4. 

Table 12.4. Examples of coalescences in agreement with the model of tangential spheres...

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