Population balance coalescence time

Solutions of the Population Balance. Solution of the population balance is not trivial. Analytical solutions are available for only a limited number of special cases. Table 8 lists analytical solutions for some special cases of practical importance. In general, analytical solutions are only available for specific initial or inlet size distributions. However, for batch coalescence, at long times the size distribution may become self preserving. The size distribution is self preserving if the normalized size distributions at long times are independent of mean size ... 

To predict the evolution of the droplet (floe) size distribution is the central problem in emulsion stability. It is possible, in principle, to predict the time dependence of the distribution of droplets (floes) if information concering the main subprocesses (flocculation, floe fragmentation, coalescence, creaming), constituting the whole phenomenon, is available. This prediction is based on consideration of the population balance equation (PBE). 

Despite the commercial importance of PVC particle morphology to its end-use applications, there has been little work done on the development of quantitative models relating the size evolution of primary particles in terms of process conditions. Kiparissides [57] developed a population balance model to describe the time evolution of the primary particle size distribution as a function of the process variables, such as temperature and ionic strength of the medium. However, for the solution of the population balance model, the coalescence rate constant between the primary particles needs to be known. This, in turn, requires the calculation of electrostatic and steric stabilization forces acting on these particles. 

The term /3(m, v) represents the coalescence rate constant of two colloidal particles of volume u and v. Note that the initial particle growth occurs mainly by particle aggregation and, to a smaller extent, by polymerization of the adsorbed monomer in the polymer-rich phase [58]. Thus, knowledge of analytical expressions for the coalescence rate constant is of profound importance to the solution of the population balance model (Equation 4.46), describing the time evolution of the primary particle size distribution. Such expressions have been derived by Kiparissides et al. [57, 59]. 

To follow the dynamic evolution of PSD in a particulate process, a population balance approach is commonly employed. The distribution of the droplets/particles is considered to be continuous in the volume domain and is usually described by a number density function, (v, t). Thus, n(v, f)dv represents the number of particles per unit volume in the differential volume size range (v, v + dv). For a dynamic particulate system, undergoing simultaneous particle breakage and coalescence, the rate of change of the number density function with respect to time and volume is given by the following non-linear integro-differential population balance equation (PBE) [36] ... 

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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