Mechanism of dispersed‐phase mass transfer in viscous, single‐drop extraction systems

Abstract

The theory of solute extraction in viscous single-drop systems is extended to show (1) the dependence of the asymptotic Nusselt number on the Peclet number from N p , = 0, the molecular diffusion limit, to N p , = 00, the Kronig and Brink limit, and (2) the dependence of the diffusion entry region Nusselt number on the Peclet number and the initial concentration profile.A numerical solution of the diffusion equation, limited to dilute solute concentrations and salute transport by viscous convection and molecular diffusion, is presented from which the nature of the Nusselt number is deduced. The observed oscillatory behavior of the Nusselt number in the diffusion entry region, as N p , +cs, is given a simple physical interpretation in terms of the circulation period of the drop liquid.The model is based upon the Hadamard stream function which theoretically is limited to creeping flow; however some experimental evidence indicates that flow fields similar to the Had0ma.d stream function exist at continuous phase Reynolds numbers of the order of ten. It is customary to analyze and correlate the results of single-drop extraction experiments in terms of mathematical models. For example, experiments with viscous drops normally are related to either the stagnant-drop model, at the extreme of vanishing circulation or to the Kronig and Brink (10) model at the opposite extreme; whereas ex-L. J ! $ Johns, Jr., is with Dow ChFmical Company, Midland, Michigan. &ann is with the University of Maryland, College Park, Mary-R. B. land. periments with turbulent drops frequently are related to the Handlos and Baron model ( 1 5 ) .This paper presents the solution to a viscous %ow model which reduces to the stagnant-drop and the Kronig and Brink models in the respective limits, that is, N p s = 0 and Np. + co, and complements these models on the interval 0 < N p e < co. A mathematical formulation of the model will be given after a brief summary of the problem and a presentation of the major assumptions.

Influence of sieve trays on the mass transfer of single drops

“…1 is very wide, for example, the circulation inside a drop that is induced hydrodynamically by the sedimentation can be taken into account with Es 2.5, if the convection is fast as compared to the diffusion. Then a separate consideration of con-Ž vection terms is not necessary Kronig and Brink, 1950;Johns and Beckmann, 1966;Brauer, 1979;Brander and Brauer, . 1993 .…”

Multiscale models of nuclear waste reprocessing : from the mesoscale to the plant-scale.

“…Reliable production of monodisperse oil in water and water in oil droplets has been well documented for a variety of channel geometries [Cristini and Tan, 2004;Gunther and Jensen, 2006;Teh et al, 2008]. Unlike falling droplet experiments where motion is driven by gravity [Kronig and Brink, 1951;Johns and Beckmann, 1966], residence and reaction times for droplets in microfluidic channels are on the order of those observed for centrifugal contactors.…”

Decompositional burning of a droplet in a small Peclet number flow.

“…(15)(16)(17)(18)(19) into Eqs. (3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14) yields to lowest order the radially symmetric problem for the adiabatic vaporization of an inert pure spherical droplet in the absence of free and forced convection.…”

“…(15)(16)(17)(18)(19) into Eqs. (11)(12)(13)(14) and by matching expansions (15)(16)(17)(18)(19) to (20)(21)(22)(23)(24). The generalization of Eqs.…”