Mass Transfer to Falling Drops

For a stationary drop of radius Rp for continuum regime transport, we have seen that steady-state transport is rapidly achieved and the flux to the drop is given by (12.12). The flux per unit droplet surface, JA = Jc/(4nR2), is then 

When the drop is in motion, calculation of the flux of gas molecules to the droplet surface is considerably more involved. The flux is usually defined in terms of a mass transfer coefficient kc as 

For a stationary drop comparing (12.124) and (12.125) kc = Dg/Rp. In an effort to estimate kc for a moving drop one defines the dimensionless Sherwood number in terms of 

For diffusion to a stationary drop Sh = 2. When the drop is falling, one usually resorts to empirical correlations for Sh as a function of the other dimensionless groups of the problem, for example (Bird et al. 1960) 

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MASS TRANSFER TO DROPS AND BUBBLES. When small drops of liquid are falling through a gas, surface tension tends to make the drops nearly spherical, and the coefiBcients for mass transfer to the drop surface are often quite close to those for solid spheres. The shear caused by the fluid moving past the drop surface, however, sets up toroidal circulation currents in the drop that decrease the resistance to mass transfer both inside and outside the drop. The extent of the change depends on the ratio of the viscosities of the internal and external fluids and on the presence or absence of substances such as surfactants that concentrate at the interface. ... 

Mass transfer rates from drops are obtained by measuring the concentration change in either or both of the phases after passage of one or more drops through a reservoir of the continuous phase. This method yields the average transfer rate over the time of drop rise or fall, but not instantaneous values. For measurements of the resistance external to the drop this is no drawback, because this resistance is nearly constant, but the resistance within the drop frequently varies with time. The fractional approach to equilibrium, F, is calculated from the compositions and is then related to the product of the overall mass transfer coefficient and the surface area ... 

Mass transfer from a single spherical drop to still air is controlled by molecular diffusion and. at low concentrations when bulk flow is negligible, the problem is analogous to that of heat transfer by conduction from a sphere, which is considered in Chapter 9, Section 9.3.4. Thus, for steady-state radial diffusion into a large expanse of stationary fluid in which the partial pressure falls off to zero over an infinite distance, the equation for mass transfer will take the same form as that for heat transfer (equation 9.26) ... 

In a liquid-liquid extraction unit, spherical drops of solvent of uniform size are continuously fed to a continuous phase of lower density which is flowing vertically upwards, and hence countercurrently with respect to the droplets. The resistance to mass transfer may be regarded as lying wholly within the drops and the penetration theory may be applied. The upward velocity of the liquid, which may be taken as uniform over the cross-section of the vessel, is one-half of the terminal falling velocity of the droplets in the still liquid. 

The moving-drop method [2] employs a column of one liquid phase through which drops of a second liquid either rise or fall. The drops are produced at a nozzle situated at one end of the column and collected at the other end. The contact time and size of the drop are measurable. Three regimes of mass transport need to be considered drop formation, free rise (or fall) and drop coalescence. The solution in the liquid column phase or drop phase (after contact) may be analyzed to determine the total mass transferred, which may be related to the interfacial reaction only after mass transfer rates have been determined. 

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. 

When an interfacial film has reduced the circulation within a drop, the wake vortex becomes more marked, while the extraction rate falls to that for a stagnant sphere (74) More detailed studies of the hydrodynamics of naturally moving drops have recently been carried out (75). The mass-transfer rate in 2-component systems should correlate 76) with... 

The rates at which drops and bubbles rise and fall are rather more sensitive to traces of surface-active materials than are the mass-transfer coeflScients 77a, 77b). Whereas, for example, the rate of fall of CCh drops... 

Mass transfer during formation of drops or bubbles at an orifice can be a very significant fraction of the total mass transfer in industrial extraction or absorption operations. Transfer tends to be particularly favorable because of the exposure of fresh surface and because of vigorous internal circulation during the formation period. In discussing mass transfer in extraction, it has become conventional (H12) to distinguish four steps (1) formation, (2) release, (3) free rise or fall, (4) coalescence. Free rise or fall has been treated in previous chapters. Steps 1 and 2 are considered here. 

In flow situations, empirical analogies between mass and heat transfer are usually employed. For single-particle mass transfer, the boundary layer analysis for mass transfer is similar to that for heat transfer and thus is used for typical applications such as sublimation of a solid (e.g., naphthalene ball) or evaporation of a liquid drop falling in air. For a single sphere of diameter dp moving in a fluid, in terms of a boundary layer analysis analogous... 

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