Mass transfer spherical interface

Concerning the structure of dispersed CLAs, the model originally proposed by Sebba [57] of a spherical oil-core droplet surrounded by a thin aqueous film stabilized by the presence of three surfactant layers is, in our opinion, essentially correct. However, there is still little direct evidence for the microstructure of the surfactant interfaces. From an engineering point of view, however, there is now quantitative data on the stability of CLAs which, together with solute mass transfer kinetics, should enable the successful design and operation of a CLA extraction process. 

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. 

Study of the eflSciency of packed columns in liquid-liquid extraction has shown that spontaneous interfacial turbulence or emulsification can increase mass-transfer rates by as much as three times when, for example, acetone is extracted from water to an organic solvent (84, 85). Another factor which may be important for flow over packing has been studied by Ratcliff and Reid (86). In the transfer of benzene into water, studied with a laminar spherical film of water flowing over a single sphere immersed in benzene, they found that in experiments where the interface was clean... 

In the previous section, we demonstrated the micrometer droplet size dependence of the ET rate across a microdroplet/water interface. Beside ET reactions, interfacial mass transfer (MT) processes are also expected to depend on the droplet size. MT of ions across a polarized liquid/liquid interface have been studied by various electrochemical techniques [9-15,87], However, the techniques are disadvantageous to obtain an inside look at MT across a microspherical liquid/liquid interface, since the shape of the spherical interface varies by the change in an interfacial tension during electrochemical measurements. Direct measurements of single droplets possessing a nonpolarized liquid/liquid interface are necessary to elucidate the interfacial MT processes. On the basis of the laser trapping-electrochemistry technique, we discuss MT processes of ferrocene derivatives (FeCp-X) across a micro-oil-droplet/water interface in detail and demonstrate a droplet size dependence of the MT rate. 

The contributions due to the mass transfer at the particle boundary will be decreased 49] with monodisperse spherical particles of small diameter (1-2 //m). However, with very small particles a regular packing of the columns is difficult to achieve. Moreover, the corresponding increase of the column capacity will lead to a decrease of the contribution for the binding rate process Eq. (14)]. A compromise is thus necessary between these two contradictory requirements. Experiments by varying particle size are needed to ascertain the validity of adsorption rate measurements at the gas-liquid interface. 

Assuming that the reaction rate, represented by Iq. (m g/(m ss)) is limited by the external gas-to-solid interface mass transfer to spherical particles, k,=kg a=kg 6/dp yielding ... 

These boundary conditions are in spherical coordinate for the spherical particles. The first condition corresponds to symmetry at the center of die pore. The second condition corresponds to the mass transfer at the interface between mobile and stationary phase. 

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. ... 

Consider creeping viscous flow of an incompressible Newtonian fluid past a stationary gas bubble that is located at the origin of a spherical coordinate system. Do not derive, but write an expression for the tangential velocity component (i.e., vg) and then linearize this function with respect to the normal coordinate r within a Ihin mass transfer boundary layer in the liquid phase adjacent to the gas-liquid interface. Hint Consider the r-9 component of the rate-of-strain tensor ... 

As an incompressible fluid of infinite extent approaches and flows past either a spherical solid pellet or a gas bubble, a mobile component undergoes inteiphase mass transfer via convection and diffusion from the sphere to the fluid phase. The overall objective is to calculate the mass transfer coefficient and the Sherwood number at any point along the interface (i.e., the local transfer coefficients), as well as surface-averaged transfer coefficients. The results are applicable in the laminar flow regime (1) when the sphere is stationary and the fluid moves,... 

The second condition, given by (11-12 ), is referred to as the boundary layer boundary condition (BLBC). The mass transfer boundary layer is infinitely thick at the separation point where 0 = tt, so, in principle, one could travel an infinite distance away from the spherical interface without measuring the bulk concentration of A that is characteristic of the approach stream. Hence, the separation point is not considered in this discussion. A better statement of the boundary layer boundary condition is... 

In other words, if the mass transfer boundary layer is very thin, then there is a tremendous change in the radial concentration gradient as one moves a very short distance radially outward from the spherical interface. Hence, at large Schmidt numbers where Sc(9) R, the following locally flat approximation is valid for spherical interfaces because curvature within the boundary layer is negligible ... 

Now the simplified mass transfer equation that accounts for convection normal and parallel to the spherical interface, and radial diffusion into a thin boundary layer, is... 

Locally Flat Description of Boundary Layer Mass Transfer in Spherical Coordinates. Since the radial coordinate r does not change much as one moves from the fluid-solid interface to the outer edge of the mass transfer boundary layer, it is acceptable to replace r by in the two-dimensional mass transfer equation and the equation of continuity ... 

MASS TRANSFER ACROSS A SPHERICAL GAS-LIQUID INTERFACE... 

Answer For boundary layer mass transfer in an incompressible liquid that contacts a zero-shear interface, a previous example problem on pages 311 and 312 reveals that the relative importance of the second term on the right side of the spherical coordinate expression for radial diffusion,... 

The primary focus of this chapter is to analyze the dimensionless equation of motion in the laminar flow regime and predict the Reynolds number dependence of the tangential velocity gradient at a spherical fluid-solid interface. This information is required to obtain the complete dependence of the dimensionless mass transfer coefficient (i.e., Sherwood number) on the Reynolds and Schmidt numbers. For easy reference, the appropriate correlation for mass transfer around a solid sphere in the laminar flow regime, given by equation (11-120), is included here ... 

The radial variable r is dimensionalized to isolate the Damkohler number in the mass balance. It is important to emphasize that dimensional analysis on the radial coordinate must be performed after implementing the canonical transformation from Ca to iJia- If the surface area factors of and 1/r are written in terms of as defined by equation (13-9), prior to introducing the canonical transformation given by equation (13-4), then the mass transfer problem external to the spherical interface retains variable coefficients. If diffusion and chemical reaction are considered inside the gas bubble, then the order in which the canonical transformation and dimensional analysis are performed is unimportant. Hence,... 

The appropriate mass transfer coefficient in the boundary layer on the liquid side of spherical interfaces with first-order or pseudo-first-order irreversible chemical reaction predominantly in the liquid phase is... 

Although the microparticles circulate in the fluid bulk as a result of external agitation, it is assumed that the particles contained in the fluid elements in transient contact with the interface remain stationary. Thus it is possible to assume a Sherwood number of 2, a generally accepted value for mass transfer to and from a spherical particle in an infinite medium. 

Brauer, H. Unsteady state mass transfer through the interface of spherical particles, InL J. Heat Mass Transfer 21 (1978) p. 445/453,455/465... 

Finally, the cases of spherical, hemispherical and cylindrical electrodes will be tackled, which cover the use of wire electrodes, mercury drops and microhemispheres, and liquid-liquid interfaces. These geometries enable us to introduce the effects due to convergent diffusion on the mass transport and voltammetric response. Moreover, as in the case of planar electrodes, because of the symmetry of the mass transfer field the problems can each be reduced to only one dimension the distance to the electrode surface in the normal direction. 

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