Population balance models, drop size

Equation (A12) is widely used in RE, but it does not account for the specific interactions of the dispersed phase. In this respect current research is focused on drop population balance models, which account for the different rising velocities of the different-size droplets and their interactions, such as droplet breakup and coalescence (173-180). 

We shall briefly deliberate the choice of the particle state. Since the quantity of interest is the mass transfer rate from the droplets, particle state must be chosen to yield from it the instantaneous mass flux from the droplet. The mass flux by diffusion requires the concentration profile near the surface. Indeed, drop size (say radius) is clearly important if we choose the average solute concentration in the droplet as another variable, the two together cannot yield the surface mass flux. But since the concentration at birth is uniform, specification of drop age, the time elapsed since its birth, can be used as a third particle state variable. Shah and Ramkrishna (1973) provide the details of the calculation of how drop size x, the average solute concentration c, and drop age t together help to calculate the mass flux at the drop surface. The mass flux will directly provide the rate of change of average concentration C(x, c, t) so that the rate of change of particle state required for the population balance model is also completely identified. 

FIGURE 3.2.2 Concetration distributions, j (c), predicted by the population balance, model of Section 3.2.4 for different drop sizes (continuous lines) compared with predictions by model based on instantaneous breakage and exponential residence time distribution (dotted lines). Reprinted from Shah and Ramkrishna (1973) with permission from Elsevier Science. 

Spielman and Levenspiel (1965) appear to have been the earliest to propose a Monte Carlo technique, which comes under the purview of this section, for the simulation of a population balance model. They simulated the model due to Curl on the effect of drop mixing on chemical reaction conversion in a liquid-liquid dispersion that is discussed in Section 3.3.6. The drops, all of identical size and distributed with respect to reactant concentration, coalesce in pairs and instantly redisperse into the original pairs (after mixing of their contents) within the domain of a perfectly stirred continuous reactor. Feed droplets enter the reactor at a constant rate and concentration density, while the resident drops wash out at the same constant rate. Reaction occurs in individual droplets in accord with nth-order kinetics. 

The dimensionless spray flux parameter can be used both as a scale-up parameter and as a parameter to estimate nuclei starting sizes for population balance modelling (see Section 13.3.3). When combined with the drop penetration time, ha forms part of a nucleation regime map (see Figure 13.4) (Hapgood... 

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 Balance Approach. The use of mass and energy balances alone to model polymer reactors is inadequate to describe many cases of interest. Examples are suspension and emulsion polymerizations where drop size or particle distribution may be of interest. In such cases, an accounting for the change in number of droplets or particles of a given size range is often required. This is an example of a population balance. 

Delichatsios and Probstein (D4-7) have analyzed the processes of drop breakup and coagulation/coalescence in isotropic turbulent dispersions. Models were developed for breakup and coalescence rates based on turbulence theory as discussed in Section III and were formulated in terms of Eq. (107). They applied these results in an attempt to show that the increase of drop sizes with holdup fraction in agitated dispersions cannot be attributed entirely to turbulence dampening caused by the dispersed phase. These conclusions are determined after an approximate analysis of the population balance equation, assuming the size distribution is approximately Gaussian. 

A simplified homogeneous dispersed-phase mixing model was proposed by Curl (C16). Uniform drops are assumed, coalescence occurs at random and redispersion occurs immediately to yield equal-size drops of the same concentration, and the dispersion is assumed to be homogeneous. Irreversible reaction of general order s was assumed to occur in the drops. The population balance equations of total number over species concentration in the drop were derived for the discrete and continuous cases for a continuous-fiow well-mixed vessel. The population balance equation could be obtained from Eq. (102) by taking the internal coordinate to be drop concentration and writing the population balance equation in terms of number to yield... 

The attribute of the foregoing model is its remarkable simplicity and ability to assess the effect of drop mixing on conversion. Of course a drop population can have a broad spectrum of sizes. There have been attempts to improve this feature by incorporating a size distribution as measured experimentally and to view the coalescence of a pair of unequally sized droplets to result in the same pair of droplets except for the mixing of their contents It cannot be said that this viewpoint is an improvement, for the assumption of such memory in redispersion is less realistic than that of uniform size. However, it is of interest to see whether a uniformly distributed redispersion event can predict a size distribution that is anything like what is observed. We discuss this as another example in the formulation of population balances. 

In the second example, a stirred tank with two liquid phases was modeled. The tank was used to measure liquid drop size distributions for fitting of the population balance parameters. The tank was modeled with CFD in order to examine the flow patterns and turbulence levels in the tank. 

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