Velocity terminal: of drops

Many of the data on the gross terminal velocity of drops have been taken in vertical cylindrical glass tubes of limited size. To interpret such data in terms of a drop moving in an infinite medium, a wall correction factor is necessary. 

The generalized graphical correlation presented in Fig. 2.5 gives one method of estimating terminal velocities of drops and bubbles in infinite liquid media. For more accurate predictions, it is useful to have terminal velocities correlated explicitly in terms of system variables. To obtain such a correlation is especially difficult for the ellipsoidal regime where surface-active contaminants are important and where secondary motion can be marked. 

Fig. 9.8 Retarding effect of column walls on the terminal velocity of drops and bubbles of intermediate size.

Present knowledge of the terminal velocity of drops in liquids is very high. Small droplets often move a little bit faster than equivalent rigid spheres due to the mobility of the drop surface. Large drops, however, move significantly slower since they lose their spherical shape. Experimental data of the terminal velocity of some organic drops in water are shown in Fig. 6.4-1. The dimensionless terminal velocity is plotted vs. the dimensionless drop diameter. For comparison, the terminal velocity of equivalent rigid spheres is also shown. 

A generalized diagram of the terminal velocity of drops is shown in Fig. 3.6-3. From this diagram, the terminal velocity of drops can be predicted as a function of drop size and system properties. For large drops the velocity is nearly independent of drop size. Their velocity is (see Sect. 3.6.1)... 

In the case of a packed column, the terms on the right-hand side should each be divided by the voidage, ie, the volume fraction not occupied by the soHd packing (71). In unpacked columns at low values of the sHp velocity approximates the terminal velocity of an isolated drop, but the sHp velocity decreases with holdup and may also be affected by column internals such as agitators, baffle plates, etc. The sHp velocity can generally be represented by (73) ... 

FIG. 6-59 Terminal velocity of air bubbles in water at 20 C. (From Clift, Grace, and Weher, Bubbles, Drops and Particles, Academic, New York, 1978). 

As shown by Fig. 14-90, entrainment droplet sizes span a broad range. The reason for the much larger drop sizes of the upper curve is the short disengaging space. For this cui ve, over 99 percent of the entrainment has a terminal velocity greater than the vapor velocity. For contrast, in the lower cui ve the terminal velocity of the largest particle reported is the same as the vapor velocity. For the settling velocity to limit the maximum drop size entrained, at least 0.8 m (30 in) disengaging space is usually required. Note that even for the lower cui ve, less than 10 percent of the entrainment is in drops of less than... 

An evaluation of the retardation effects of surfactants on the steady velocity of a single drop (or bubble) under the influence of gravity has been made by Levich (L3) and extended recently by Newman (Nl). A further generalization to the domain of flow around an ensemble of many drops or bubbles in the presence of surfactants has been completed most recently by Waslo and Gal-Or (Wl). The terminal velocity of the ensemble is expressed in terms of the dispersed-phase holdup fraction and reduces to Levich s solution for a single particle when approaches zero. The basic theoretical principles governing these retardation effects will be demonstrated here for the case of a single drop or bubble. Thermodynamically, this is a case where coupling effects between the diffusion of surfactants (first-order tensorial transfer) and viscous flow (second-order tensorial transfer) takes place. Subject to the Curie principle, it demonstrates that this retardation effect occurs on a nonisotropic interface. Therefore, it is necessary to express the concentration of surfactants T, as it varies from point to point on the interface, in terms of the coordinates of the interface, i.e.,... 

Waslo and Gal-Or (Wl) recently generalized the Levich solution [Eq. (65)] by evaluating the effect of and y on the terminal velocity of an ensemble of spherical drops of bubbles. Their solution is... 

Eq. (207) can be derived also as a special case of a more general solution by Waslo and Gal-Or (Wl) which gives the terminal velocity of an ensemble of drops or bubbles in the presence of surfactants (see Sections IIF and 11IB for more details). 

At t = 0, V = 0 and the drag force is zero. As the particle accelerates, the drag force increases, which decreases the acceleration. This process continues until the acceleration drops to zero, at which time the particle falls at a constant velocity because of the balance of forces due to drag and gravity. This steady-state velocity is called the terminal velocity of the body and is given by the solution of Eq. (11-8) with the acceleration equal to zero ... 

The derivations of Hadamard and of Boussinesq are based on a model involving laminar flow of both drop and field fluids. Inertial forces are deemed negligible, and viscous forces dominant. The upper limit for the application of such equations is generally thought of as Re 1. We are here considering only the gross effect on the terminal velocity of a drop in a medium of infinite extent. The internal circulation will be discussed in a subsequent section. 

A plot of the terminal velocity of a drop moving in an infinite medium vs. drop size will show the features shown in Fig. 5. To exhibit all of these features, both drop and field liquids must be of very high purity. Using the pertinent fluid properties, a plot of Cd vs. Re will appear as in Fig. 6. In this plot the length term used is the very convenient De, in... 

Two papers (FI, M3) described the shapes of very large drops moving in non-Newtonian liquid fields of the pseudoplastic variety. The employment of a Reynolds number based on the power-function relationship permitted a good description of the variation of terminal velocity vs. drop size. 

Gross terminal velocity of carbon tetrachloride drops falling through... 

The Hadamard-Rybczynski theory predicts that the terminal velocity of a fluid sphere should be up to 50% higher than that of a rigid sphere of the same size and density. However, it is commonly observed that small bubbles and drops tend to obey Stokes s law, Eq. (3-18), rather than the corresponding Hadamard-Rybczynski result, Eq. (3-15). Moreover, internal circulation is essentially absent. Three different mechanisms have been proposed for this phenomenon, all implying that Eq. (3-5) is incomplete. 

Fig. 3.6 Effect of stagnant cap on terminal velocity of a bubble or inviscid drop.

Fig. 3.7 Effect of surfactant on the terminal velocity of small bubbles and drops.

Following a suggestion made by Davies (D2, D4), we define a degree of circulation Z such that the terminal velocity of a spherical bubble or drop in slow viscous flow is given by... 

Numerous determinations of the terminal velocities of water drops have been reported. The most careful measurements are those by Gunn and Kinzer (G13) and Beard and Pruppacher (B4). Figure 7.1, derived from these results," shows... 

Fig. 7.5 Terminal velocity of carbon tetrachloride drops falling through water, measured by different workers, with varying system purity.

There is a substantial body of data in the literature on the terminal velocities of bubbles and drops. In view of the influence of system purity discussed above, a separation of this data has been made. Cases where there is evidence that considerable care was taken to eliminate surfactants and where a sharp peak in the Uj vs. d curve at low M and k is apparent (as for the pure systems in Figs. 7.3 and 7.5) are discussed in Section 4. 

Many other correlations for calculating the terminal velocity of bubbles and drops are available [e.g. (EL12, J2, K3, Tl, VI, W2)]. None covers such a broad range of data as Eqs. (7-5) and (7-6). Moreover, a number of the earlier correlations require that values be read from graphs or that iterative procedures be used to determine Uj. 

It is an open question whether small quantities of surfactants, too small to influence the gross properties, affect the terminal velocity of liquid drops in air. This appears unlikely in view of the large values of /c, but Buzzard and Nedderman (B18) have claimed such an influence. Acceleration may have contributed to this observation. Quantities of surfactant large enough to lower a appreciably can lead to significantly increased deformation and hence to an increase in drag and a reduction in terminal velocity (R6). 

On disengaging from the wetting surface, the enlarged drops will now rise much taster in the water according to the well-known form of Stokes Law for the terminal velocity of spheres in a liquid medium. 

SGP T TWSEC vise VISCL Vt VTGOl VTWO wc wd specific gravity of production water at system temperature T system temperature, °F time for water drop in oil phase to fall, s viscosity of liquid, gas or oil at system temperature T, cP viscosity of liquid or oil at system temperature T, cP terminal velocity of particle, ft/s particle velocity in oil phase, ft/s terminal velocity of water particle in the oil phase, ft/s BS W of treated desalter outlet crude oil, percentage by volume Barrels of dilution water per day mixed with the crude oil fluid upstream of the desalter... 

Figure 4.10 displays a typical vertical vessel program run printout. The program calculation output gives a minimum required diameter. This diameter is based on the terminal velocity of liquid drop fall velocity, gas bubble in oil rise velocity, or the water drop fall velocity. The smaller this terminal velocity, the greater the vessel cross-section area required and thus the greater the vessel diameter required. In this example, the oil-phase gas bubble rise terminal velocity is controlling. If you reduce the oil flow to, say, 10,000 lb/h, then the gas-phase liquid... 

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