Moving Drops

It is discovered that in the cooling tower the water moving downward from the jets changes its direction to upward after drop formation. There is an effective heat transfer process when the drops move upward heat transfers from the outlet air to the drops through convection and condensation. 

Drops collide with the drop separator and drain down to the lower part of the tower. These drops are large, so their total surface area is small and insignificant. The effective heat transfer process takes place when the drops move with the air flow, so this arrangement has to be treated as a parallel flow heat transfer. 

This response time should be compared to the turbulent eddy lifetime to estimate whether the drops will follow the turbulent flow. The timescale for the large turbulent eddies can be estimated from the turbulent kinetic energy k and the rate of dissipation e, Xc = 30-50 ms, for most chemical reactors. The Stokes number is an estimation of the effect of external flow on the particle movement, St = r /tc. If the Stokes number is above 1, the particles will have some random movement that increases the probability for coalescence. If St 1, the drops move with the turbulent eddies, and the rates of collisions and coalescence are very small. Coalescence will mainly be seen in shear layers at a high volume fraction of the dispersed phase. 

A dropping electrolyte electrode (fig. 9.3) was also developed for the study of electrolysis at ITIES. When the density of the organic phase is greater than that of the aqueous phase (for example, as with nitrobenzene), the aqueous drops move upwards. 

However, no matter how rough the surface, the forces will be the same as those that exist between a solid and a liquid. The surface roughness may show contact angle hysteresis if one makes the drop move, but this will arise from other parameters (e.g., wetting and dewetting). Further, in practice, the surface roughness is not easy to define. A fractal approach has been used to achieve a better understanding (Feder, 1988 Birdi, 1993). 

Any hydrodynamic consideration of a drop moving in a liquid field starts with the Navier-Stokes equations of motion, as given in representative books on fluid mechanics (L2, Sll). Using vector notation to conserve space, these equations may be written (B3, B4)... 

For a liquid drop falling in air or for a very viscous drop in a low viscosity field liquid, the correction term reduces to unity, and Eq. (23) becomes equivalent to Stokes law. For a gas bubble rising through a liquid or a low viscosity drop moving in a very viscous liquid field, the limiting correction factor of 1.5 may be realized for a fully circulating drop. These two limiting values have been confirmed by many experi-... 

Most drop situations in extraction are far above the upper limit of application of the preceding equations. A drop moving through a liquid at a velocity such that the viscous forces could be termed negligible can not exist. It will break up into two or more smaller droplets (HIO, K5). Most real situations involve both viscous and inertial terms, and the Navier-Stokes equations can not then be solved. 

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. 

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 above equations are limited to the creeping-flow range. For large drops moving under conditions such that inertial terms are not negligible, an empirical equation equation based on experimental data (S12) is... 

The shape of a liquid drop moving in a liquid field is dependent upon the balance between the hydrodynamic pressure exerted because of the relative velocities of drop and field and the surface forces which tend to make the drop a sphere (H7). 

As has been previously shown in this chapter, the velocity field around a submerged shape can be mathematically predicted in some cases and experimentally demonstrated in all. Such a field is shown in Fig. 9. But as a large drop moves through a stationary liquid field, it will carry... 

The shape of a drop moving under the influence of gravity may be affected by interfacial motions the drop may also wobble and move sideways (S27, W3). In one system (S22) the terminal velocity was reduced yielding a drag coefficient nearly equal to that of a solid particle. Interfacial convection tends to increase the rate of mass transfer above that which would occur in the absence of interfacial motion. The interaction between mass transfer and interfacial convection has been treated by Sawistowski (S7) and Davies (D4, D5). 

Fig 3.7. Components on the optical bench of a generalized four-parameter flow cytometer. (The drop charging, the deflection plates, and the drops moving into separate test tubes apply only to sorting cytometers [see Chapter 9] and not to benchtop instruments.) Adapted from Becton Dickinson Immunocytometry Systems. 

Put a drop of oil and a drop of water on a piece of wax paper. Tilt the wax paper so the drops run down the paper. Does the oil drop spread out in a thin layer Does the water stay together in a drop Does the oil drop move more slowly than a water drop The nonpolar oil is attracted to the nonpolar wax paper so it spreads out. The attraction between the oil drop and the wax paper makes the oil drop move more slowly than the water drop. The polar water is not attracted to the nonpolar wax paper so it stays in a drop. 

The DME, invented by Heyrovsky is a historically important technique which forms the basis for many of the electrochemical techniques discussed here. This electrode is composed of a capillary through which mercury flows forming a spherical drop of approximately 1 mm, at which point its weight snrpasses the surface tension and the drop falls into solntion. There are a few drawbacks to this techniqne. The mercnry drop has a finite lifetime, typically 2 to 6 s, and with the drop moving through solution, the area can continnonsly change which makes accounting for mass transport difficnlt. 

The negative sign means that the oil drops move upward instead of downward. 

Chisnell RE (1987) The unsteady motion of a drop moving vertically under gravity. J Fluid Mech 176 443-464... 

Magnaudet JJM, Takagi S, Legendre D (2003) Drag, deformation and lateral migration of a buoyant drop moving near a wall. J Fluid Mech 476 115-157 Maxey MR, Riley JJ (1983) Equation of motion for a a mall rigid sphere in a non-uniform flow. Phys Fluids 26 (4) 883-889. 

Wellek, R. M., Arawal, A. K. Skelland, A. H. P. 1966 Shapes of liquid drops moving in liquid media. AIChE Journal 12, 854-862. 

M. Kojima, E. J. Hindi, and A. Acrivos, The formation and expansion of a toroidal drop moving in a viscous fluid, Phys. Fluids 27, 19-32 (1984). 

A more efficient approach is to base the boundary-integral formulation on a fundamental solution (or more accurately a Green s function) that incorporates the relevant boundary conditions at one or more of the surfaces. In the case of a particle or drop moving near an infinite plane wall, this means finding a solution for a point force that exactly satisfies the no-slip and kinematic boundary conditions at the wall. If we were to consider the motion of a particle or drop in a tube, it would be useful to have the solution for a point force satisfying the same conditions on the tube walls. 

Drops in a typical hydraulic nozzle are formed in a liquid sheet that travels at 15-25 m/s. Following sheet disintegration, drops move in an air-jet caused by the interaction of the spray plume and the surrounding air. Close to the nozzle,... 

The dependence of drop deformation on the Weber number and the vorticity inside the drop was studied in [336]. It was shown that the drop is close in shape to an oblate ellipsoid of revolution with semiaxis ratio > 1 If there is no vortex inside the drop, then this dependence complies with the function We(x) given in (2.8.3). The ratio x decreases as the intensity of the internal vortex increases. Therefore, the deformation of drops moving in gas is significantly smaller than that of bubbles at the same Weber number We. The vorticity inside an ellipsoidal drop, just as that of the Hill vortex, is proportional to the distance TZ from the symmetry axis,... 

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