Drop breakup capillary number

The previous discussion focused on the breakup of liquid thread suspended in a quiescent Newtonian fluid. In real mixing operations quiescent conditions will usually not occur, except perhaps for short periods of time. The more important issue, therefore, is how the breakup occurs when the system is subjected to flow. Good reviews on the breakup of liquid threads are available from Acrivos [304], Rallison [305], and Stone [306]. Probably the most extensive experimental study on drop breakup was performed by Grace [286] data was obtained over an enormous range of viscosity ratios 10- to 10 Grace determined the critical Weber (Capillary) number for breakup both in simple shear and in 2-D elongation the results are represented in Fig. 7.152. 

The tensor L defines the character of the flow. The capillary number for the drop deformation and breakup problem is... 

Many authors have worked on drop deformation and breakup, beginning with Taylor. In 1934, he published an experimental work [138] in which a unique drop was submitted to a quasi-static deformation. Taylor provided the first experimental evidence that a drop submitted to a quasi-static flow deforms and bursts under well-defined conditions. The drop bursts if the capillary number Ca, defined as the ratio of the shear stress a over the half Laplace pressure (excess of pressure in a drop of radius R. Pl = where yint is the interfacial tension) ... 

The breakup or bursting of liquid droplets suspended in liquids undergoing shear flow has been studied and observed by many researchers beginning with the classic work of G. I. Taylor in the 1930s. For low viscosity drops, two mechanisms of breakup were identified at critical capillary number values. In the first one, the pointed droplet ends release a stream of smaller droplets termed tip streaming whereas, in the second mechanism the drop breaks into two main fragments and one or more satellite droplets. Strictly inviscid droplets such as gas bubbles were found to be stable at all conditions. It must be recalled, however, that gas bubbles are compressible and soluble, and this may play a role in the relief of hydrodynamic instabilities. The relative stability of gas bubbles in shear flow was confirmed experimentally by Canedo et al. (36). They could stretch a bubble all around the cylinder in a Couette flow apparatus without any signs of breakup. Of course, in a real devolatilizer, the flow is not a steady simple shear flow and bubble breakup is more likely to take place. 

It is convenient to express the capillarity number in its reduced form K = K / K, where the critical capillary number, K., is defined as the minimum capillarity number sufficient to cause breakup of the deformed drop. Many experimental studies have been carried out to establish dependency of K on X. For simple shear and uniaxial extensional flow, De Bruijn [1989] found that droplets break most easily when 0.1 4 ... 

The mechanisms governing deformation and breakup of drops in Newtonian liquid systems are well understood. The viscosity ratio, X, critical capillary number, and the reduced time, t, are the controlling parameters. Within the entire range of X, it was found that elongational flow is more efficient than shear flow for breaking the drops. 

To make things more interesting, the experimental observations of De Bruijn [1989] seem to have contradicted the latter conclusion. The author found that the critical capillary number for viscoelastic droplets is always higher (sometimes much higher) than for Newtonian ones, whatever the -value De Bruijn concluded that drop elasticity always hinders drop breakup. 

Some authors report the next guide principles that may be applied for blend morphology after processing, (i) Drops with viscosity ratios higher than 3.5 cannot be dispersed in shear but can be in extension flow instead, (ii) The larger the interfacial tension coefficient, the less the droplets will deform, (iii) The time necessary to break up a droplet (Tj,) and the critical capillary number (Ca ) are two important parameters describing the breakup process, (iv) The effect of coalescence must be considered even for relatively low concentrations of the dispersed phase. 

Following the early work by Thorsen et al., focused on the formation of monodisperse aqueous droplets in an organic carrier fluid performed on a microfluidic chip, and then followed by others works, the breakup mechanism responsible of droplet formation was later analyzed by Garstecki et al. ° showing that when is order of 1 the dominant contribution to the dynamics of breakup at low capillary numbers is not dominated by shear stresses, but it is driven by the pressure drop across the emerging droplet. 

Figure 35.4 shows the variation of ellipticity with respect to the Weber, Reynolds, and capillary numbers at various axial locations. As observed fi-om these figures, the droplets are big close to the injector and the Ohnesorge numbers are small. The Weber number is much larger than the critical Weber number ( 6) and the drops undergo breakup. The deformation predicted by the above correlation... 

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... 

With regard to drop deformation and breakup, there are four regions of reduced capillary numbers, k, both in shear and elongation ... 

When values of the capillary number and the reduced time are within the region of drop breakup, the mechanism of breakup depends on the viscosity ratio, X. 

The microrheology discussed in Section 2.1.2.3 describes the breakup of isolated drops in a Newtonian system. The mechanisms leading to deformation and breakup take into account the three principal variables the viscosity ratio (X), critical capillary number (Kcni), and the reduced time (f ), defined in Eq. (2.19). For application of microrheology to polymer blends the theories developed for Newtonian emulsions need to be extended to viscoelastic systems in the fidl range of composition, that is, they must take into account coalescence. Since the microrheology evolution up to about the year 2000 has been summarized by Utradri and Kamal [3] the following text win focus on more recent developments. 

If the drops are such that the capillary number is smaller than its critical value, deformation will not lead to drop breakup and after the deformation has ended, the drop retracts (Fig. 5.4). 

In practice, in a mixture much larger drops can be found than predicted by the critical capillary number because Grace s observations were based on single drops. In actual systems, where many drops exist, coalescence will occur. Because material elements also undergo varying levels of shear forces in time, the mixing process in polymer systems can be considered as a complex interaction between deformation, drop breakup, coalescence, and retraction. 

Taylor [64] found that in simple shear flow, a dispersed drop with viscosity ratio p = 1 breaks up when the Ca > 0.5. Breakup seems to occur when the shear stress and the interfacial stress are of the same order of magnitude. The critical Capillary number depends on the type of flow and on the viscosity ratio. In the mixing process two regimes can typically be distinguished ... 

From the above equations it is possible to calculate the size of the largest drop that exists in a fluid undergoing distortion at any shear rate. In these equations, the governing parameters for droplet breakup are the viscosity ratio p (viscosity of the dispersed phase to that of the matrix) the type of flow (elongational, shear, combined, etc.) the capillary number Cfl, which is the ratio between the deforming stress (matrix viscosity x shear rate) imposed by the flow on the droplet and the interfacial forces a/R, where ais the interfacial... 

Grace [6] has constructed a plot of the critical capillary number as a function of the viscosity ratio, p, under two types of flow a simple shear flow and a hyperbolic (elongational) flow field (Figure 1.2). It is shown that droplets are stable when their Ca number is below a critical value the deformation and breakup are easier at P within a 0.25 to 1 range for shear flow, and the elongational flow field is more effective for breakup and dispersion than the shear flow. It can also be seen that at a viscosity ratio p > 4-5, it is not possible to break up the drop in simple shear flow. 

For shear flow in Newtonian liquids, oil drop breakup may be described by the capillary number Ca (viscosity of continuous phase tjc, shear rate y interfacial tension y, drop diameter x) ... 

The critical capillary number is the capillary number value below which no oil drop breakup occurs [10]. Grace [10] indicated that file viscosity ratio X (see (21.2)— viscosity of disperse //d and continuous phase is a crucial factor influencing the critical capillary number, especially for simple shear flow. Armhruster [11] and later Jansen et al. [12] showed that for high concentrated systems the viscosity of the emulsion rje, has to be used instead of the viscosity of cmitinuous phase. 

The equilibrium droplet size is proportional to the Capillary number, Ca, at a given viscosity ratio. The drop size reduces with an increase of the shear stress and a decrease in the droplet/matrix interfacial tension. 00 Droplet breakup cannot occur if the viscosity ratio exceeds 3.5 (Ca goes to infinity). 

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