Droplet breakup function

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. 

To understand how the dispersed phase is deformed and how morphology is developed in a two-phase system, it is necessary to refer to studies performed specifically on the behavior of a dispersed phase in a liquid medium (the size of the dispersed phase, deformation rate, the viscosities of the matrix and dispersed phase, and their ratio). Many studies have been performed on both Newtonian and non-Newtonian droplet/medium systems [17-20]. These studies have shown that deformation and breakup of the droplet are functions of the viscosity ratio between the dispersity phase and the liquid medium, and the capillary number, which is defined as the ratio of the viscous stress in the fluid, tending to deform the droplet, to the interfacial stress between the phases, tending to prevent deformation ... 

Figure 3.6 Droplet breakup as a function of viscosity ratio. The solid line represents the critical Weber number value above which droplet breakup will occur. Data from Isaacs and Chow [130].

Fig. 7.23 Critical capillary number for droplet breakup as a function of viscosity ratio p in simple shear and planar elongational flow. [Reprinted by permission from H. P. Grace, Chem. Eng. Commun., 14, 2225 (1971).]...

One variant of the GAS or SAS process (SCF as antisoivent) is solution enhanced dispersion by supercritical fluids (SEDS). Coaxial nozzles are used to introduce drug solution and carbon dioxide at the desired temperature and pressure. In this case, the SCF carries out both droplet breakup and antisoivent functions. SEDS has been tested for a number of pharmaceutical compounds. As noted above, this is a continuing effort. 

Unlike in NEMD models, the microstructures emerging due to competition between the breakup and coalescence processes can be studied by using DPD modeling. For example, in Figure 26.23, the four principal mechanisms, the same as those responsible for droplets breakup [ 118,119], can be observed in DPD simulation of the R-T instability. As shown in [116,119], moderately extended drops for capillary number close to a critical value, which is a function of dynamic viscosity ratio... 

Experimental work by Sundararaj and Macosko, nicely contrasted the competing effects of droplet breakup and droplet coalescence in both Newtonian and non-Newtonian mixtures. They concluded that the extent of interfacial tension reduction due to the presence of block copolymer was insufficient to be the primary reason for the reduction of the droplet size, and the primary effect of the copolymer was to prevent droplet coalescence through steric stabilization of the droplets. Sundararqj and Macosko also noted that the droplet size as a function of shear rate for a pure blend decreased to a minimum value, then increased at higher shear rates. No data was given for the compatibilized blend and they referred to this shear rate dependence of the dispersed phase size for the pure blend as "anomalous". Sundararaj and Macosko noted that this anomalous behavior has been observed previously by... 

Based on the critical droplet sizes for breakup and coalescence in Equations 19.12 and 19.20, the droplet size in polymer blends as a function of flow intensity (shear rate) can be mapped out [28], as shown in Figure 19.4. The critical droplet sizes for droplet breakup and coalescence become equal at a certain critical shear rate. For shear rates larger than this critical value, the critical droplet size for breakup is smaller than the critical droplet size for coalescence and the final droplet size is determined by a dynamic equilibrium between breakup and coalescence. However, below the critical shear rate, the critical droplet size for breakup is larger than the critical droplet size for coalescence, which results in a range of droplet sizes for which neither breakup nor coalescence will occur. This phenomenon is called morphological hysteresis and changing the flow conditions within this region... 

After breakup, droplets continue to interact with the surrounding environment before reaching thein final destination. In theory (24), each droplet group produced during primary breakup can be traced by using a Lagrangian calculation procedure. Droplet size and velocity can be deterrnined as a function of spatial locations. 

The time tb for a droplet to undergo deformation prior to secondary breakup is a function of Ohd and a characteristic time... 

Senda et al)335 415 also derived equations describing the thickness and diameter of the radial film formed on a hot surface as a function of the Weber number, and correlated the mean diameter of droplets resulted from the breakup of the radial film with the thickness of the radial film and the Weber number. 

It should be indicated that a probability density function has been derived on the basis of maximum entropy formalism for the prediction of droplet size distribution in a spray resulting from the breakup of a liquid sheet)432 The physics of the breakup process is described by simple conservation constraints for mass, momentum, surface energy, and kinetic energy. The predicted, most probable distribution, i.e., maximum entropy distribution, agrees very well with corresponding empirical distributions, particularly the Rosin-Rammler distribution. Although the maximum entropy distribution is considered as an ideal case, the approach used to derive it provides a framework for studying more complex distributions. 

Detailed modeling study of practical sprays has a fairly short history due to the complexity of the physical processes involved. As reviewed by O Rourke and Amsden, 3l() two primary approaches have been developed and applied to modeling of physical phenomena in sprays (a) spray equation approach and (b) stochastic particle approach. The first step toward modeling sprays was taken when a statistical formulation was proposed for spray analysis. 541 Even with this simplification, however, the mathematical problem was formidable and could be analyzed only when very restrictive assumptions were made. This is because the statistical formulation required the solution of the spray equation determining the evolution of the probability distribution function of droplet locations, sizes, velocities, and temperatures. The spray equation resembles the Boltzmann equation of gas dynamics[542] but has more independent variables and more complex terms on its right-hand side representing the effects of nucleations, collisions, and breakups of droplets. 

T.F. Wang, J.F. Wang, Y. Jin, A novel theoretical breakup kernel function for bubbles/droplets in a turbulent flow, Chem. Eng. Sci. 58 (2003) 4629-4637. 

Liquid-liquid mixing has been widely used in chemical industries. The state of dispersion is determined by the balance of the break-up and coalescence of droplets. In the case of liquid-liquid mixing, the breakup of the droplet is accelerated in the impeller region. Although the droplet size distribution in the operation has been expressed by various PSD functions, the PSD function that is utilized the most is the normal PSD function. However, there is no physical background to apply the normal PSD function to the droplet size distribution. Additionally, when the droplet size distribution is expressed by various PSD functions, it becomes difficult to discuss the relationship between the parameters in PSD and operation conditions. This is one of the obstacles for developing particle technology. 

Rumscheidt and Mason [14] described particle deformation in a shear field as a function of viscosity ratio (p). There is a minimum and a maximum viscosity ratio where it becomes impossible to reduce the droplet size. The limits described by Karam and Bellinger [15] are 0.005 and 4. Breakup of droplets readily occurs when the viscosity ratio is of the order of 0.2 1. Intuitively, a ratio of 1 would be best because in this case there is a maximum transfer of energy between the... 

Figure 9.9 Dimensionless critical droplet size for breakup (capillary number Ca<- = Crj a/r) as a function of viscosity ratio M of the dispersed to the continuous phase for two-dimensional flows in the four-roll mill. The data sets correspond from bottom to top to ff — 1.0(0), 0.8 (A), 0.6 (0), 0.4 (V), and 0.2 ( ), with a defined in Eq. (9-17). The fluids are those described in Fig. 9-7. The solid lines are the predictions of a small-deformation theory, while the dashed lines are for a large-deformation theory. The closed squares are from Rallison s (1981) numerical solutions (see also Rallison and Acrivos 1978). (From Bentley and Leal 1986, with permission from Cambridge University Press.)...

This formula is crude, and it does not account for differences in shear rates between the droplet and the medium (which are large when the viscosity ratio differs greatly from unity). Nevertheless, because of the shear-rate-dependence of, Eq. (9-22) can predict a.minimum in droplet size as a function of shear rate that is observed in some cases (Sundararaj and Macosko 1995 Plochocki et al. 1990 Favis and Chalifoux 1987). Viscoelastic forces have indeed been shown to suppress the breakup of thin liquid filaments that would otherwise rapidly occur via Rayleigh s instability (Goldin et al. 1969 Hoyt and Taylor 1977 Bousfield et al. 1986). Elongated filaments, for example, are observed in polymer blends (Sondergaard... 

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