Coalescence efficiency model

An attempt has been made by Tsouris and Tavlarides[5611 to improve previous models for breakup and coalescence of droplets in turbulent dispersions based on existing frameworks and recent advances. In both the breakup and coalescence models, two-step mecha-nisms were considered. A droplet breakup function was introduced as a product of droplet-eddy collision frequency and breakup efficiency that reflect the energetics of turbulent liquid-liquid dispersions. Similarly, a coalescencefunction was defined as a product of droplet-droplet collision frequency and coalescence efficiency. The existing coalescence efficiency model was modified to account for the effects of film drainage on droplets with partially mobile interfaces. A probability density function for secondary droplets was also proposed on the basis of the energy requirements for the formation of secondary droplets. These models eliminated several inconsistencies in previous studies, and are applicable to dense dispersions. 

Valentas et al. (VI) also used a coalescence efficiency of exponential form, but gave no rigorous arguments to support their model. 

Intuitively, bubble coalescence is related to bubble collisions. The collisions are caused by the existence of spatial velocity difference among the particles themselves. However, not all collisions necessarily lead to coalescence. Thus modeling bubble coalescence on these scales means modeling of bubble collision and coalescence probability (efficiency) mechanisms. The pioneering work on coalescence of particles to form successively larger particles was carried out by Smoluchowski [109, 110]. 

The task of modeling binary droplet collisions in Euler-Lagrangian simulations of spray flows was first taken up by O Rourke and coworkers. Their model in [83] first estimates the coalescence efficiency, which is the probability that coalescence occurs after the collision, once it has taken place ... 

The influence of a block copolymer on the droplet breakup and coalescence in model immiscible PEP/PPO polymer blends was investigated by Ramie et fd. [18], who found that the addition of 0.1 wt% or 1.0 wt% of PEO-b-PPO-b-PEO [poly (ethylene oxide)-poly(propylene oxide) copolymer] triblock copolymers facilitated breakup and inhibited coalescence. The steady-state droplet size resulting from breakup was reduced only slightly by the addition of 0.1 wt% copolymer, but more substantially by addition of lwt%. However, the kinetics of coalescence were suppressed effectively even when 0.1 wt% of copolymer was added. In these systems, the copolymer seems to reduce the efficiency of both droplet collision and film drainage and/or rupture. 

More complex computational models using Monte Carlo methods have attempted to predict bubble size distributions for a combination of breakup and coalescence. These models typically treat bubble coalescence by analogy with the kinetic theory where bubbles are assumed to act as solid particles [18,19]. They use a binary collision rate (probability) and a collision efficiency factor to account for collisions that do not lead to coalescence. Since collision is assumed to be a random process in these models, turbulence of the same scale as the bubbles or smaller would increase collisions and, therefore, also increase the coalescence rate. 

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... 

If drops adhere for sufficient time to allow them to deform, and to permit drainage of the continuous phase that is trapped between them, then coalescence may occur [26], By taking account of these events, expressions can be obtained for the coalescence efficiency [50], Expressions, for coalescence and for collision rates are not easy to use because they often contain parameters that are difficult to quantify. Alvarez et al. [50] constructed a model for drop breakage and coalescence, in the suspension polymerization of styrene, which takes account of viscosity effects. 

They show further that the approach force is given by F (d 8 ) and the contact time is given by tc (d /e). These quantities, derived from turbulence theory, are required inputs to models for the drainage rate of the laminar film and the coalescence efficiency, as described below. 

The coalescence efficiency is determined by comparing the time to reach critical thickness with the available contact time determined by the external flow model. The approach of Figure 12-20 allows development of a variety of models. Whereas the form of (d, dO depends on the process flow field, that for X(d, dO... 

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. 

Mechanistic models developed for coalescence efficiency that include physical and turbulence information regarding drop motion, rest time, etc. 

If the solvent is nonvolatile, it can cause accumulation of heavy impurities in an extraction loop that are surface-active. Even in trace concentrations, these culprits can have a devastating effect on extractor performance. They can reduce the coalescing rates of drops—and thus reduce column capacity. Since most flooding models are based on pure-component tests, these models tend to be overly optimistic. Relative to a clean system, the presence of impurities can lower column capacity by 20% or more and efficiency by as much as 60%. 

Prince and Blanch [92] modeled bubble coalescence in bubble columns considering bubble collisions due to turbulence, buoyancy, and laminar shear, and by analysis of the coalescence probability (efficiency) of collisions. It was assumed that the collisions from the various mechanisms are cumulative. The collision density resulting from turbulent motion was expressed as a function of bubble size, concentration and velocity in accordance with the work of Smoluchowski [111] ... 

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