Drops internal circulation

Although small drops behave like rigid spheres, this similarity of behaviour does not extend beyond a Reynolds number of around 10. For larger drops internal circulation of the Hadamard-Rybczynski type sets in which reduces the drag coefficient to a value below that of a corresponding solid sphere. However, larger drops are also subject to deformation, the extent of which depends on the Weber number (8)... 

Flow Past Deformable Bodies. The flow of fluids past deformable surfaces is often important, eg, contact of Hquids with gas bubbles or with drops of another Hquid. Proper description of the flow must allow for both the deformation of these bodies from their shapes in the absence of flow and for the internal circulations that may be set up within the drops or bubbles in response to the external flow. DeformabiUty is related to the interfacial tension and density difference between the phases internal circulation is related to the drop viscosity. A proper description of the flow involves not only the Reynolds number, dFp/p., but also other dimensionless groups, eg, the viscosity ratio, 1 /p En tvos number (En ), Api5 /o and the Morton number (Mo),giJ.iAp/plG (6). 

Mass transfer in a gas-liquid or a liquid-liquid reactor is mainly determined by the size of the fluid particles and the interfacial area. The diffusivity in gas phase is high, and usually no concentration gradients are observed in a bubble, whereas large concentration gradients are observed in drops. An internal circulation enhances the mass transfer in a drop, but it is still the molecular diffusion in the drop that limits the mass transfer. An estimation, from the time constant, of the time it wiU take to empty a 5-mm drop is given by Td = d /4D = (10 ) /4 x 10 = 6000s. The diffusion timescale varies with the square of the diameter of the drop, so... 

The surface characteristics of these species are determined by the particulates and stress transfer across the membrane will tend to be low, reducing internal circulation within the drop. The structure of the interface surrounding the drop plays a significant role in determining the characteristics of the droplet behaviour. We can begin our consideration of emulsion systems by looking at the role of this layer in determining linear viscoelastic properties. This was undertaken by... 

GARNER, F. H. and Skelland, A. H. P. Trans. Inst. Chem. Eng. 29 (1951) 315. Liquid-liquid mixing as affected by the internal circulation within drops. 

Fig. 9.11 Stationary relative velocity, Vp, of pentachloroethane drops in motionless water is dependent on drop diameter, dp. Very small drops behave bice rigid spheres, as shown. Larger drops have an internal circulation and are bnaUy deformed ebipticaUy. When they have reached a certain diameter, the drops in the end oschlate along and across their major axis. Their axial velocity is nearly independent of the diameter. Once the drop size is higher than a maximum value dp raax., the drop will break, owing to the drag forces. (From Ref. 2.)...

Interfacial tension is the parameter in equations influencing the drop size, as discussed in preceding sections. The smaller the value of a, the smaller are the resulting drops, if all the other conditions are the same, and the larger is the transfer area per unit volume. On the other hand, small drops may show little or no internal circulation, which implies equivalent consequences for the mass transfer coefficient and a lower rising velocity and, accordingly, a lower flow rate at the flooding point. 

The value of Rl within a falling drop of liquid is of interest in view of the applications of spray absorbers. A wind-tunnel (59) for the study of individual liquid drops, balanced in a stream of gas, has shown (60) that Rl for a drop depends on its shape, velocity, oscillations, and internal circulation. The drop will remain roughly spherical only if... 

This natural circulation occurs by a direct transfer of momentum across the interface, and the presence of a monolayer at the interface will affect it in two ways. Firstly, the surface viscosity of the monolayer may cause a dissipation of energy and momentum at the surface, so that the drop behaves rather more as a solid than as a liquid, i.e., the internal circulation is reduced. Secondly, momentum transfer across the surface is reduced by the incompressibility of the film, which the moving stream of gas will tend to sweep to the rear of the drop (Fig. 14b) whence, by its back-spreading pressure n, it resists further compression and so damps the movement of the surface and hence the transfer of momentum into the drop. This is discussed quantitatively below, where Eq. (32) should apply equally well to drops of liquid in a gas. 

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. 

Horton (H9, K2), using a tapered tube, was able to match the velocity of fall of the drop with the velocity of the rising field and thereby observe the behavior of the system for an hour or more. Using a dark field trace photography technique, the semi-vectorial velocities were recorded. In every case the internal circulation was slowly damped out as the interface changed its character and became more contaminated. 

Large drops (De =1 cm) of chlorobenzene will fall through water with a somewhat erratic oscillatory motion (L3). The drop pitches and rolls. The flight is not vertical but is erratically helical in nature. A series of oscillations, accompanied by waves moving over the interface, can cause the drop to drift several inches in a horizontal direction in a range of a foot or two of fall. Such drops can not oscillate violently as described above, due to the damping action of such movement by the sliding side-wise motion of the wobble. Motion pictures indicate that internal circulation is also considerably damped out by this type of oscillation. Rate of... 

The surface viscosity effect on terminal velocity results in a calculated drag curve that is closer to the one for rigid spheres (K5). The deep dip exhibited by the drag curve for drops in pure liquid fields is replaced by a smooth transition without a deep valley. The damping of internal circulation reduces the rate of mass transfer. Even a few parts per million of the surfactant are sometimes sufficient to cause a very radical change. 

Not until the above effects can be mathematically related can we expect to progress beyond the experimental stage. To predict such items as size of drop formed at a nozzle, terminal velocity, drag curves, changes of oscillations, and speed of internal circulation, one must possess experimental data on the specific agent in the specific system under consideration. Davies (Dl, D2) proposes the use of the equation... 

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. 

Boussinesq (B4) proposed that the lack of internal circulation in bubbles and drops is due to an interfacial monolayer which acts as a viscous membrane. A constitutive equation involving two parameters, surface shear viscosity and surface dilational viscosity, in addition to surface tension, was proposed for the interface. This model, commonly called the Newtonian surface fluid model (W2), has been extended by Scriven (S3). Boussinesq obtained an exact solution to the creeping flow equations, analogous to the Hadamard-Rybczinski result but with surface viscosity included. The resulting terminal velocity is... 

Internal circulation patterns have been observed experimentally for drops by observing striae caused by the shearing of viscous solutions (S7) or by photographing non-surface-active aluminum particles or dyes dispersed in the drop fluid [e.g. (G2, G3, J2, L5, Ml, SI)]. A photograph of a fully circulating falling drop is shown in Fig. 3.5a. Since the internal flow pattern for the Hadamard-Rybczynski analysis satisfies the complete Navier-Stokes equation... 

Fig. 3.5 Internal circulation in a water drop falling through castor oil [from Savic (SI), reproduced by permission of the National Research Council of Canada] (a) d = 1.77 cm, Uj = 1.16 cm/s, exposure 1/2 s, fully circulating (b) d= 1.21 cm, Uj = 0.62 cm/s, exposure 1 s, stagnant cap at top of drop.

Accounting for the influence of surface-active contaminants is complicated by the fact that both the amount and the nature of the impurity are important in determining its effect (G7, L5, Rl). Contaminants with the greatest retarding effect are those which are insoluble in either phase (L5) and those with high surface pressures (G7). A further complication is that bubbles and drops may be relatively free of surface-active contaminants when they are first injected into a system, but internal circulation and the velocity of rise or fall decrease with time as contaminant molecules accumulate at the interface (G3, L5, R3). Further effects of surface impurities are discussed in Chapters 7 and 10. For a useful synopsis of theoretical work on the effect of contaminants on bubbles and drops, see the critical review by Harper (H3). Attention here is confined to the practically important case of a surface-active material which is insoluble in the dispersed phase. The effects of ions in solution or in double layers adjacent to the interface are not considered. 

When a fluid sphere exhibits little internal circulation, either because of high K = Pp/p or because of surface contaminants, the external flow is indistinguishable from that around a solid sphere at the same Re. For example, for water drops in air, a plot of versus Re follows closely the curve for rigid spheres up to a Reynolds number of 200, corresponding to a particle diameter of approximately 0.85 mm (B5). In fact, many of the experimental points used in Section II to determine the standard drag curve refer to spherical drops in gas streams, where high values of k ensure negligible internal circulation. 

All the work discussed in the preceding sections is subject to the assumptions that the fluid particles remain perfectly spherical and that surfactants play a negligible role. Deformation from a spherical shape tends to increase the drag on a bubble or drop (see Chapter 7). Likewise, any retardation at the interface leads to an increase in drag as discussed in Chapter 3. Hence the theories presented above provide lower limits for the drag and upper limits for the internal circulation of fluid particles at intermediate and high Re, just as the Hadamard-Rybzcynski solution does at low Re. 

As for other types of fluid particle, the internal circulation of water drops in air depends on the accumulation of surface-active impurities at the interface (H9). Observed internal velocities are of order 1% of the terminal velocity (G4, P5), too small to affect drag detectably. Ryan (R6) examined the effect of surface tension reduction by surface-active agents on falling water drops. 

The formation of an attached wake and the subsequent onset of wake shedding tend to be promoted by increasing oblateness (see Chapter 6) and by the tendency of surface-active contaminants to damp out internal circulation (see Chapter 5). Experiments have been conducted with dyes added to enable attached wakes and shedding phenomena to be visualized (H8, Ml, M2, S2) and wake volumes to be measured (H8, Y4) for drops and bubbles. Since dyes tend to be surface active, the results of these experiments are probably relevant... 

Surface-active contaminants play an important role in damping out internal circulation in deformed bubbles and drops, as in spherical fluid particles (see Chapters 3 and 5). No systematic visualization of internal motion in ellipsoidal bubbles and drops has been reported. However, there are indications that deformations tend to decrease internal circulation velocities significantly (MI2), while shape oscillations tend to disrupt the internal circulation pattern of droplets and promote rapid mixing (R3). No secondary vortex of opposite sense to the prime internal vortex has been observed, even when the external boundary layer was found to separate (Sll). 

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