Critical capillary number

The degree of deformation and whether or not a drop breaks is completely determined by Ca, p, the flow type, and the initial drop shape and orientation. If Ca is less than a critical value, Cacri the initially spherical drop is deformed into a stable ellipsoid. If Ca is greater than Cacrit, a stable drop shape does not exist, so the drop will be continually stretched until it breaks. For linear, steady flows, the critical capillary number, Cacrit, is a function of the flow type and p. Figure 14 shows the dependence of CaCTi, on p for flows between elongational flow and simple shear flow. Bentley and Leal (1986) have shown that for flows with vorticity between simple shear flow and planar elongational flow, Caen, lies between the two curves in Fig. 14. The important points to be noted from Fig. 14 are these ... 

Fig. 14. Critical capillary number (Cac ) as a function of the viscosity ratio (p) for two-dimensional linear flows with varying vorticity (Bentley and Leal, 1986).

Figure 1.21. Comparison of the critical capillary number for simple shear, when the shear is applied in quasi-static conditions or suddenly.

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).]...

Fig. 7.24 Breakup of a droplet of 1 mm diameter in simple shear flow of Newtonian fluids with viscosity ratio of 0.14, just above the critical capillary number. [Reprinted by permission from H. E.H. Meijer and J. M. H. Janssen, Mixing of Immiscible Fluids, in Mixing and Compounding of Polymers, I. Manas-Zloczower and Z. Tadmor, Eds., Hanser, Munich (1994).]...

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. 

In quasi-static conditions, it has been established both theoretically [7, 12] and experimentally [6, 12] that a drop breaks when the applied stress a overcomes the product of the critical capillary number Cacr and the Laplace pressure... 

A third, related limit on the capillary pressure is created by the existence of an upper critical capillary pressure above which the life times of thin films become exceedingly short. Values of this critical capillary number were measured by Khistov and co-workers for single films and bulk foams (72). The importance of this phenomenon for dispersions in porous media was confirmed by Khatib and colleagues (41). Figure 5 shows the latter authors plot of the capillary pressures required for capillary entry by the nonwetting fluid and for lamella stability versus permeability of the porous medium. 

Figure 9.7 Photographs of droplet shapes in planar extensional flow for various viscosity ratios M of the dispersed to the continuous phase. The droplets are viewed in the plane normal to the velocity gradient direction. The critical capillary numbers Cac and droplet deformation parameters Dc at breakup are also given. The droplet fluids are silicon oils with viscosities ranging from 5 to 60,000 centistokes, while the continuous fluids are oxidized castor oils both phases are Newtonian. (From Bentley and Leal 1986, with permission from Cambridge University Press.)...

In the case of discontinuous oil, L may be equal to Db, which is the diameter of a single oil blob. The capillary number required to mobilize the single oil blob is calculated using the preceding equation with L = Db. In the case of continuous oil whose size could be several times of Db, and L would be several times of Db, then the capillary number required to mobihze the continuous oil would be several times lower than that required to mobilize a single oil blob. In other words, the critical capillary number required to mobilize discontinuous oil is higher than that to mobilize continuous oil. This is another justification that chemical flood should be conducted early in the secondary recovery mode instead of in the tertiary recovery mode. 

To further reduce waterflood residual oil saturation, the capillary number must be higher than the preceding calculated value. In general, the capillary number must be higher than a critical capillary number, (Nc)c, for a residual phase to start to mobilize. Practically, this (Nc)c is much higher than the capillary number at normal waterflooding conditions. Another parameter is maximum desaturation capillary number, (Nc)max, above which the residual saturation would not be further reduced in practical conditions even if the capillary number is increased. Lake (1989) used the term total desaturation capillary number for (Nc)max. In practical conditions, total desaturation (i.e., zero residual saturation) may not occur due to some films or blobs trapped in pores. 

Morrow and coworkers (Morrow and Songkran, 1981 Morrow et al., 1988) used the terms capillary number for mobilization and capillary number for prevention of entrapment for (Nc)c and (Nc)max, respectively. In UTCHEM, lower and higher critical capillary numbers are used for (Nc)c and (Nc)max, respectively. Table 7.8 summarizes some of the published experimental data for these critical capillary numbers. In principle, the critical capillary numbers should be system specific. Experiments should always be conducted to determine the capillary desaturation curves (CDC) for the particular application whenever possible. The summarized data could be useful only when no experimental data are available. From Table 7.8, the following observations can be made regarding capillary number ... 

The critical capillary number for bead packs (unconsolidated) is higher than that for sandstones (Morrow et al., 1988). 

The critical capillary number required to mobilize discontinuous oil is higher than that to mobilize continuous oil. 

Now we have discussed the two important capillary numbers critical and maximum. The general relationship between residual saturation of a nonaque-ous or aqueous phase and a local capillary number is called capillary desaturation curve (CDC). The residual saturations start to decrease at the critical capillary number as the capillary number increases, and cannot be decreased further at the maximum capillary number. As discussed earlier, the range of capillary numbers for residual phases to be mobilized is, for example, 10 to... 

Nc)max mean at critical capillary number and maximum desaturation capillary number (Nc) is capillary number and Tp is the parameter used to fit the laboratory measurements. The definition of capillary number used in the preceding equation must be the same as that used in the simulation model. One example of CDC using Eq. 7.121 is shown by the curves in Figure 7.35, and some of the CDC parameters are presented in Table 7.9. The data points in Figures 7.35 and 7.36 are calculated using Eq. 7.124, to be discussed later. 

It is assumed that the end-point relative permeabilities depend on the residual saturation of the other conjugate phase. If we assume that kjp at any capillary number can be interpolated between those at the critical capillary number and at the maximum capillary number, we have... 

Finally, we consider the case of a sohd particle attached to a hquid-fluid interface. This configuration is depicted in Figure 5.17e note that the position of the particle along the normal to the interface is determined by the value of the three-phase contact angle. Stoos and Leal investigated the case when such an attached particle is subjected to a flow directed normally to the interface. These authors determined the critical capillary number, beyond which the captured particle is removed from the interface by the flow. 

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 ... 

Note that in shear for A, = 1, the critical capillary number = 1, whereas for A, > 1, increases with X and becomes infinite for X > 3.8. This means that the breakup of the dispersed phase in pure shear flow becomes impossible for X > 3.8. This limitation does not exist in extensional flows. 

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. 

The shear deformation of viscoelastic drops in a Newtonian medium has been the subject of several studies. Gauthier et al. [1971] found higher values of the critical capillary number than... 

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. 

UCST - upper critical solubility K crit - critical capillary number... 

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. 

Using Eqs. 24b and 24c in Eq. 24a and denoting the initial extension as 2o = a critical capillary number of droplet breakage at the T-junction can be estimated as... 

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