Model coalescence frequency

Coalescence frequencies can have a pronounced effect on the rate of mass transfer or chemical reaction in a liquid-liquid dispersion. Various investigators have attempted to model and measure coalescence frequencies in agitated vessels. A review of the experimental techniques is given by Rietema (R12) and Shah et al. (S16). 

These methods for obtaining coalescence frequencies in batch systems have been extended to continuous-flow systems (C13, K13). Komasawa (K13) derived the appropriate model for flow systems assuming uniform drops present and two dispersed-phase feed streams entering the vessel, one with dye and the other without. The result is... 

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). 

As for the collision density in the macroscopic model formulation, the average collision frequency of fluid particles is usually described assuming that the mechanisms of collision is analogous to collisions between molecules as in the kinetic theory of gases. The volume average coalescence frequency, ac d d, Y), can thus be defined as the product of an effective swept volume rate hc d d, Y) and the coalescence probability, pc d d, Y) (e.g., [16, 92, 114, 39, 46, 118]) ... 

One of the earliest attempts to quantify coalescence frequencies was the work of Howarth (1967). A procedure was used that is similar to the one described in Section 12-3.1.5. A steady dispersion was established at a high agitation rate. The stirrer speed was then lowered so that only coalescence occurred, at least initially. Howarth defined a global or macroscopic coalescence frequency as the initial slope of a plot of interfacial area (related to d32> versus time and demonstrated that systematic experiments could be conducted to determine the effect of various system variables on coalescence rate. Since the coalescence frequency depends strongly on drop diameter, most models are based on the approach discussed below. 

Numerous authors have developed models for coalescence frequency. These include the models of Muralidhar and Ramkrishna (1986), Das et al. (1987), Muralidhar et al. (1988), Tsouris and Tavlarides (1994), and Wright and Ramkrishna (1994), for turbulent stirred tanks, as well as those of Davis et al. (1989),... 

Models for coalescence frequency show the importance of agitation rate, physicochemical phenomena, and interfacial properties on coalescence. This information is broadly useful for explaining the behavior of stirred vessels, decanters, extractors, and centrifuges, as well as how to prevent coalescence. It is also useful in the determination of which phase will tend to dominate as the continuous phase and in the interpretation of phase inversion phenomena. 

Steam-liquid flow. Two-phase flow maps and heat transfer prediction methods which exist for vaporization in macro-channels and are inapplicable in micro-channels. Due to the predominance of surface tension over the gravity forces, the orientation of micro-channel has a negligible influence on the flow pattern. The models of convection boiling should correlate the frequencies, length and velocities of the bubbles and the coalescence processes, which control the flow pattern transitions, with the heat flux and the mass flux. The vapor bubble size distribution must be taken into account. 

Kapur and Fuerstenau (K6) have presented a discrete size model for the growth of the agglomerates by the random coalescence mechanism, which invariably predominates in the nuclei and transition growth regions. The basic postulates of their model are that the granules are well mixed and the collision frequency and the probability of coalescence are independent of size. The concentration of the pellets is more or less fixed by the packing... 

Bubble Dynamics. To adequately describe the jet, the bubble size generated by the jet needs to be studied. A substantial amount of gas leaks from the bubble, to the emulsion phase during bubble formation stage, particularly when the bed is less than minimally fluidized. A model developed on the basis of this mechanism predicted the experimental bubble diameter well when the experimental bubble frequency was used as an input. The experimentally observed bubble frequency is smaller by a factor of 3 to 5 than that calculated from the Davidson and Harrison model (1963), which assumed no net gas interchange between the bubble and the emulsion phase. This discrepancy is due primarily to the extensive bubble coalescence above the jet nozzle and the assumption that no gas leaks from the bubble phase. 

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. 

Aerosols are unstable with respect to coagulation. The reduction in surface area that accompanie.s coalescence corresponds to a reduction in the Gibbs free energy under conditions of constant temperature and pressure. The prediction of aerosol coagulation rates is a two-step process. The first is the derivation of a mathematical expression that keeps count of particle collisions as a function of particle size it incorporates a general expression for tlie collision frequency function. An expression for the collision frequency based on a physical model is then introduced into the equation Chat keep.s count of collisions. The collision mechanisms include Brownian motion, laminar shear, and turbulence. There may be interacting force fields between the particles. The processes are basically nonlinear, and this lead.s to formidable difficulties in the mathematical theory. 

Lei Nij be the number of collisions occurring per unit lime per unit volume between the two classes of particles of volumes u,- and Uj. All particles are assumed to be spherical, which means that i and j are uniquely related to particle diameters. When two particles collide, according to this simplified model, they coalesce instantaneously to form a third whose volume is equal to the sum of the original two. In terms of the concentrations of particles and with volumes u,- and Vj, the collision frequency is... 

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