Terminal drop velocity

The r.m.s. deviation between measured and predicted terminal velocities is about 15% for the 774 points with H < 59.3 and 11% for the 709 points with H > 59.3. This correlation is recommended for calculations of bubble and drop terminal velocities when the criteria outlined above are satisfied and where some surface-active contamination is inevitable. The predictions from this correlation for carbon tetrachloride drops in water are shown on Fig. 7.5. 

Coverage has been limited to horizontal three-phase separators up to this point. Considering Fig. 4.9, oil and water must flow vertically downward and gas vertically upward. The same laws of buoyancy and drag force apply. Equation (4.3) may therefore be used in the oil phase for water separation. Equations (4.12), (4.13), and (4.7) (see Fig. 4.8) are applied to the gas phase and oil phase for oil-gas particle separations, as was equally done for horizontal separators. The equations for the horizontal separator from Fig. 4.8 may also be used for the water drop terminal velocity in the vertical separator. 

Once the drop terminal velocity is found, the time taken for the dispersed phase to reach the interface is given by Equation 6.15.8 in Table 6.15, and the decanter length required for the droplets to settle is given by Equation 6.15.9. The maximum distance that the disperse phase droplets have to travel to reach the interface, which is located at the center of the separator, is D/2. The distance varies from zero to D/2. Also, the path of the droplets is not straight down or up but will curve because of the motion of the phases. 

The flow capacity of the sieve-tray tower depends upon drop-formation characteristics of the system, drop terminal velocities, dispersed-phase holdup, and pressure drop. IMth these estabUshed, the design can be developed and mass-transfer rates estimated. 

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) ... 

Droplet trajectories for limiting cases can be calculated by combining the equations of motion with the droplet evaporation rate equation to assess the likelihood that drops exit or hit the wall before evaporating. It is best to consider upper bound droplet sizes in addition to the mean size in these calculations. If desired, an instantaneous value for the evaporation rate constant may also be used based on an instantaneous Reynolds number calculated not from the terminal velocity but at a resultant velocity. In this case, equation 37 is substituted for equation 32 ... 

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). 

Terminal velocities for water drops in air have been correlated by Berry and Pruager (J. Appl. Meteoml., 13, 108-113 [1974]) as... 

Commonly, the most important feature of a nozzle is the size of droplet it produces. Since the heat or mass transfer that a given dispersion can produce is often proportional to (1/D ) , fine drops are usually favored. On the other extreme, drops that are too fine will not settle, and a concern is the amount of liquid that will be entrained from a given spray operation. For example, if sprays are used to contact atmospheric air flowing at 1.5 m/s, drops smaller than 350 [Lm [terminal velocity = 1.5 m/s (4.92 ft/s)] will be entrained. Even for the relative coarse spray of the hoUow-cone nozzle shown in Fig. 14-88, 7.5 percent of the total hquid mass will be entrained. 

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 ... 

This section is a continuation of Section 21.3.2 dealing with pressure drop (-AP) for flow through a fixed bed of solid particles. Here, we make further use of the Ergun equation for estimating the minimum superficial fluidization velocity, ump In addition, by analogous treatment for free fall of a single particle, we develop a means for estimating terminal velocity, ur as a quantity related to elutriation and entrainment. 

There are two main periods of evaporation. When a drop is ejected from an atomiser its initial velocity relative to the surrounding gas is generally high and very high rates of transfer are achieved. The drop is rapidly decelerated to its terminal velocity, however, and the larger proportion of mass transfer takes place during the free-fall period. Little error is therefore incurred in basing the total evaporation time on this period. 

A detailed analysis by GriflSth (77/) of the velocity of fall of a clean fluid drop gives the following equation for the terminal velocity v ... 

The rate of mass-transfer, unlike the terminal velocity, may reach its lower limit only when the whole surface of the drop or bubble is covered by the adsorbed film. In the absence of surface-active material, the freshly exposed interface at the front of the moving drop (due to circulation here) could well be responsible for as much mass transfer as occurs in the turbulent wake of the drop. The results of Baird and Davidson 67a) on mass transfer from spherical-cap bubbles are not inconsistent with this idea, and further experiments on smaller drops are in progress in the author s laboratory. In general, if these ideas are correct, while the rear half of the drop is noncirculating (and the terminal velocity has reached the limit of that for a solid sphere), the mass transfer at the front half of the drop may still be much higher, due to the circulation, than for a stagnant drop. Only when sufficient surface-active material is present to cover the whole of the surface and eliminate all circulation will the rate of mass-transfer approach its lower limit. 

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. 

Correlations have been developed to relate terminal velocity, drop size, peak size, peak velocity, and maximum drop size to the physical properties of the system (El, HIO, K5). Some of the accumulated information in these areas is given below. 

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... 

---