Mass transfer spherical drop

Equations which predict the volume or equivalent spherical diameter of a formed drop are not sufficient for extraction calculations, in the light of the very high rate of mass transfer during drop formation. It is desirable that the equation also lend itself to mathematical manipulation for the calculation of instantaneous interfacial area. To do this, the shape of the drop throughout the formation period must be defined. 

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. ... 

The relationship between adsorption capacity and surface area under conditions of optimum pore sizes is concentration dependent. It is very important that any evaluation of adsorption capacity be performed under actual concentration conditions. The dimensions and shape of particles affect both the pressure drop through the adsorbent bed and the rate of diffusion into the particles. Pressure drop is lowest when the adsorbent particles are spherical and uniform in size. External mass transfer increases inversely with d (where, d is particle diameter), and the internal adsorption rate varies inversely with d Pressure drop varies with the Reynolds number, and is roughly proportional to the gas velocity through the bed, and inversely proportional to the particle diameter. Assuming all other parameters being constant, adsorbent beds comprised of small particles tend to provide higher adsorption efficiencies, but at the sacrifice of higher pressure drop. This means that sharper and smaller mass-transfer zones will be achieved. 

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. 

In a drop extractor, liquid droplets of approximate uniform size and spherical shape are formed at a series of nozzles and rise eountercurrently through the continuous phase which is flowing downwards at a velocity equal to one half of the terminal rising velocity of the droplets. The flowrates of both phases are then increased by 25 per cent. Because of the greater shear rate at the nozzles, the mean diameter of the droplets is however only 90 per cent of the original value. By what factor will the overall mass transfer rate change ... 

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. 

Figure 12-15 Sketch of concentration profiles between a spherical bubble and a solid spherical catalyst particle in a continuous liquid phase (upper) in a gas-liquid sluny reactor or between a bubble and a planar solid wall (lower) in a catalytic w bubble reactor, It is assmned that a reactant A must migrate from the bubble, tirough the drop, md to tiie solid catdyst smface to react. Concentration variations may occur because of mass transfer limitations around both bubble and solid phases.

J. Mass Transfer to a Continuous Phase from a Single Spherical Drop... 

Moreover, it is difficult to find one s way in the overwhelming amount of literature on this subject because the major part of it is focussed on d.c. polarography and thus to the mass transfer problem at the dropping mercury electrode (DME). Neglecting the sphericity, the expansion of the drop has still to be accounted for in the diffusion equation for a species i. Equation (19b), which we have adopted thus far, should therefore be replaced by [11, 147]... 

A spherical benzoic acid particle with a diameter of 2 mm is dropped into a quiescent air duct at ambient conditions (25°C). During the fall, sublimation of the particle occurs. Taking the molecular diffusivity of benzoic acid in air as 4 x 10-6 m2/s, calculate the convective mass transfer coefficient when the particle is falling at its terminal velocity condition. 

Table 6.8 presents the details of calculations for spherical particles with an equivalent diameter of 2.4mm. It may be observed that the pore diffusion considerably affects the process rate, particularly at higher temperatures. The external mass transfer plays a minor role. Their combination leads to a global effectiveness that drops from 75% to 35% when the temperature varies from 160 to 220°C. Based on the above elements the apparent reaction constant may be expressed by the following Arrhenius law ... 

On the very small scale of individual lines, vias, pole pieces, etc., current-distribution behavior is dominated by three characteristics. Firstly, ohmic potential drop becomes nearly insignificant because the distances are so short. Secondly, since small features can be smaller than or at least comparable in size to diffusion boundary layers, geometric concentration-field effects, such as radially and spherically enhanced diffusion, can play an important role. Thirdly, since the lateral dimensions of the feature are often comparable to the final thickness of the deposit, the geometry of the problem (which is responsible for mass-transfer nonuniformity) constantly evolves during deposition. Hence it may be necessary to include cumulative shape-change effects in studies on the feature scale. 

Maximum stable drop diameter, m Impeller diameter, m Diffusivity of dissolved component or reactant in liquid, m /s Gravitational acceleration, m/s Height of liquid in vessel, m Mass transfer coefficient, m/s Mass transfer coefficient for a single spherical droplet immersed in a liquid flowing at constant velocity past the droplet, m/s Mass of liquid, kg Rate of mass transfer of solute or reactant, kg/s Impeller speed, rotations/s Minimum speed to just suspend solid particles in vessel, rotations/s Minimum impeller speed to completely incorporate dispersed phase into continuous phase in liquid-liquid systems, rotations/s Power dissipation, W Time, s... 

For mass transfer in a rigid spherical drop the matrix [F] is given by the n - 1 dimensional matrix generalization of Eq. 9.4.6... 

Particle diameter, dp, and pressure drop. For a given column performance, the pressure drop is lowest when the adsorbent particles are spherical and of closely uniform size. The external mass-transfer rate increases. inversely as dpM, and the internal rate increases inversely as dfl. The pressure drop variation will depend upon the Reynolds number, but is... 

Spherical drop at high Peclet numbers for Re > 35. For high Re, the fluid velocity distribution in the boundary layer near the drop surface was obtained in [180]. These results were used in [504], where mass transfer to a spherical drop in a translational flow was investigated. The results for the mean Sherwood number are well approximated by the formula [94]... 

We consider a laminar steady-state flow past a solid spherical particle (drop or bubble) of radius a and study transient mass transfer to the particle surface. At the initial time t = 0, the concentration in the continuous phase is constant and equal to C, whereas for t > 0 a constant concentration Cs is maintained on the particle surface. 

In the case of nonstationary mass transfer in a steady-state translational Stokes flow past a spherical drop with limiting resistance of the continuous phase, the steady-state value Shst is presented in the first row of Table 4.7. By substituting this value into (4.12.3), we obtain... 

Statement of the problem. Preliminary remarks. Let us consider the transient convective mass and heat transfer between a spherical drop of radius a and a translational Stokes flow where the resistance to the transfer exists only in the disperse phase. We assume that at the initial time t = 0 the concentration inside the drop is constant and equal to Co, whereas for t > 0 the concentration on the interface is maintained constant and equal to Cs. 

We note that in [421], the cell flow model was used for the investigation of mass and heat transfer in monodisperse systems of spherical drops, bubbles, or solid particles for Re < 250 and 0 < < 0.5. 

Statement of the problem. In the preceding chapters we considered processes of mass transfer to surfaces of particles and drops for the case of an infinite rate of chemical reaction (adsorption or dissolution.) Along with the cases considered in the preceding chapters, finite-rate surface chemical reactions (see Section 3.1) are of importance in applications. Here the concentration on the surfaces is a priori unknown and must be determined in the course of the solution. Let us consider a laminar fluid flow with velocity U past a spherical particle (drop or bubble) of radius a. Let R be the radial coordinate relative to the center of the particle. We assume that the concentration is uniform remote from the particle and is equal to C. Next, the rate of chemical reaction on the surface is given by Ws = KSFS(C), where Ks is the surface reaction rate constant and the function F% is defined by the reaction kinetics and satisfies the condition Fs(0) = 0. 

Let us consider steady-state mass transfer on a spherical particle (drop or bubble) of radius o in a laminar fluid flow. We assume that a volume chemical reaction proceeds in the continuous phase with Wv = KVFV(C). The reactant transfer in the continuous phase is described in dimensionless variables by the equation... 

Let us now study the inner mass transfer problems involving a volume chemical reaction. We assume that the diffusion process is quasi-stationary and takes place inside a solid spherical inclusion or a drop of radius a filled with a stagnant or moving medium. 

Brignell, A. S., Solute extraction from an internally circulating spherical liquid drop, Int. J. Heat Mass Transfer, Vol. 18, No. 1, pp. 61-68,1975. 

Weber, M. E., Mass transfer from spherical drops at high Reynolds numbers, Ind. Eng. Chem. Fundam., Vol. 14, No. 4, pp. 365-366,1975. 

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