Population balance collision time

Again the assumption on the aggregation rate are that the most frequent collisions are between the larger particle and the small particles. This partial differential equation can be approximated by an ordinary one by creating a new characteristic time variable, t (= t -RIG = R G], which is constant. With this variable change the population balance becomes... 

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. 

Population balances are a set of mathematical tools that enable one either to predict the time evolution of the DSD or to determine specific information, such as breakage frequency and daughter size distribution, or collision frequency and coalescence efficiency, from an analysis of time-variant drop size data. They were first developed by Valentas et al. (1966) and Valentas and Amundson (1966), as applied to liquid-liquid dispersions. These techniques have been used for both batch and continuous systems and for steady state as well as unsteady conditions. 

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