Breakage functions, modeling

Within Eq. (7), the selection function S(y) was approximated by Sty) = Kyc with adjustable parameters K and c and the breakage function b(x, y) was simply modeled by a triangular shaped function between a minimal breaking particle size of xq = 25 nm and x with a modal value at... 

It is encouraging that substantial progress has been made in analyzing the hydrodynamics of droplet interactions in dispersions from fundamental considerations. Effects of flow field, viscosity, holdup fraction, and interfacial surface tension are somewhat delineated. With appropriate models of coalescence and breakage functions coupled with the drop population balance equations, a priori prediction of dynamics and steady behavior of liquid-liquid dispersions should be possible. Presently, one universal model is not available. The droplet interaction processes (and... 

The preceding functions must be obtained by physical models. Collectively, the functions b x, r, Y, t), v(x, r, Y, t) and P(x, r x, r, Y, t) may be referred to as the breakage functions. We have been liberal with the choice of arguments for them in order to stress all their potential dependencies, but several ad hoc simplifications will guide applications. In particular the usefulness of phenomenological models of this kind lies in their being free of temporal dependence. However, the inclusion of time will serve as a remainder of the need for the assumption to be made consciously. 

The heart of the population balance model for breakage processes lies in the breakage functions described in Section 3.2.1. The breakage functions must be obtained either directly from experiments or by modeling considerations related to the processes causing the breakage. This is the subject of the next section. 

In either of the preceding categories, since the integrand contains the unknown number density, the mathematically rigorous choice for the pivot, which is consistent with the mean value theorem is of course not accessible. The finer the interval, the less crucial would be the location of the pivot in /f. The fineness required would depend on the extent to which the phenomenological functions of the population balance model such as the aggregation and breakage functions vary in the interval. 

Interaction term in Sg model equation for gas phase (kg/m s" ) Interaction term in kgp transport equation (kg/m s ) Weighted breakage function... 

The energy laws of Bond, Kick, and Rittinger relate to grinding from some average feed size to some product size but do not take into account the behavior of different sizes of particles in the mill. Computer simulation, based on population-balance models [Bass, Z. Angew. Math. Phys., 5(4), 283 (1954)], traces the breakage of each size of particle as a function of grinding time. Furthermore, the simu-... 

The function most often used in modelling breakage and disruption processes has the form... 

Their analysis of experimental data shows that tensile strength was the only parameter that varied as a function of particle size. Model simulation indicate that larger lumps were stronger than smaller lumps which is contradictory to Waters et al. [8], Teo and Waters [9], and Griffith [10] theory of fracture, which implies that larger particle are more likely to contain larger cracks and hence be more susceptible to breakage. 

The model assumes that drop coalescence is followed by immediate redispersion into two drops sized according to a uniform distribution. By assuming that the coalescence frequency is independent of drop size, the solution of the resulting form of Eq. (107) is exponential for the equilibrium drop volume distribution. Comparison of the distribution to experimental data is favorable. The analysis is useful in that a functional form for the distribution is obtained. The attendant simplifications necessary for solution, however, do not permit more rational forms of the interaction frequency of droplet pairs in order to account for the physical processes which lead to droplet coalescence and breakage as discussed in Section III. A similar work was presented by Inone et al. (II). 

The drops behave as segregated entities between flow and coalescence-redispersion simulation. The coalescence and breakage frequencies can be varied with vessel position. The computational time was related to coalescence frequency data available in the literature. Figure 15 shows the steady-state dimensionless droplet number size distribution as a function of rotational speed for continuous-flow operation. As expected the model predicts smaller droplet sizes and less variation of the size distribution with increase in rotational speed. Figure 16 is a comparison of the droplet number size distribution with drop size data of Schindler and Treybal (Sll). 

Mass-Transfer Models Because the mass-transfer coefficient and interfacial area for mass transfer of solute are complex functions of fluid properties and the operational and geometric variables of a stirred-tank extractor or mixer, the approach to design normally involves scale-up of miniplant data. The mass-transfer coefficient and interfacial area are influenced by numerous factors that are difficult to precisely quantify. These include drop coalescence and breakage rates as well as complex flow patterns that exist within the vessel (a function of impeller type, vessel geometry, and power input). Nevertheless, it is instructive to review available mass-transfer coefficient and interfacial area models for the insights they can offer. 

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