Transfer in the Continuous Phase

Transfer from large bubbles and drops may be estimated by assuming that the front surface is a segment of a sphere with the surrounding fluid in potential flow. Although bubbles are oblate ellipsoidal for Re 40, less error should result from assumption of a spherical shape than from the assumption of potential flow. 

Transfer from a spherical segment in potential flow is described (Bl, B4, J2, L4) by 

High Reynolds Number For a spherical cap with a flat base  

At high Re the transfer through the base is not negligible. Weber (W4) showed that basal transfer may be estimated using the penetration theory, assuming complete renewal each time vortices are shed. He obtained 

Assuming that base and frontal transfer are independent, we obtain 

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Mass transfer in the continuous phase is less of a problem for liquid-liquid systems unless the drops are very small or the velocity difference between the phases is small. In gas-liquid systems, the resistance is always on the liquid side, unless the reaction is very fast and occurs at the interface. The Sherwood number for mass transfer in a system with dispersed bubbles tends to be almost constant and mass transfer is mainly a function of diffusivity, bubble size, and local gas holdup. 

There are no solutions for transfer with the generality of the Hadamard-Rybczynski solution for fluid motion. If resistance within the particle is important, solute accumulation makes mass transfer a transient process. Only approximate solutions are available for this situation with internal and external mass transfer resistances included. The following sections consider the resistance in each phase separately, beginning with steady-state transfer in the continuous phase. Section B contains a brief discussion of unsteady mass transfer in the continuous phase under conditions of steady fluid motion. The resistance within the particle is then considered and methods for approximating the overall resistance are presented. Finally, the effect of surface-active agents on external and internal resistance is discussed. 

Lochiel, a. C. Calderbank, P. H. 1964 Mass transfer in the continuous phase around axisymmetric bodies of revolution. Chemical Engineering Science 19,471 84. 

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... 

Ruckenstein, E. (1964), On mass transfer in the continuous phase from spherical bubbles or drops, Chemical Engineering Science, 19(2) 131-148. 

In 1988, Terry and coworkers attempted to homopolymerize ethylene, 1-octene, and 1-decene in supercritical C02 [87], The purpose of their work was to increase the viscosity of supercritical C02 for enhanced oil recovery applications. They utilized the free radical initiators benzoyl peroxide and fert-butyl-peroctoate and conducted polymerization for 24-48 h at 100-130 bar and 71 °C. In these experiments, the resulting polymers were not well studied, but solubility studies on the products confirmed that they were relatively insoluble in the continuous phase and thus were not effective as viscosity enhancing agents. In addition, a-olefins are known not to yield high polymer using free radical methods due to extensive chain transfer to monomer. 

The growth rate, characterized by the change of the radius with time, is proportional to the driving force for the phase separation, given by the differences between 2 > the chemical composition of the second phase in the continuous phase at any time, and, its equihbrium composition given by the binodal line. The proportionahty factor, given by the quotient of the diffusion constant, D, and the radius, r, is called mass transfer coefficient. Furthermore the difference between the initial amount of solvent, (])o, and c]) must be considered. The growth rate is mathematically expressed by [101]... 

The concentration is made dimensionless in one of several ways depending upon the situation considered. For example, for steady transfer to the continuous phase from a particle at constant concentration, the boundary conditions considered in this book are... 

Mass transfer rates are increased in the presence of eruptions because the interfacial fluid is transported away from the interface by the jets. For mass transfer from drops with the controlling resistance in the continuous phase, the maximum increase in the transfer rate is of the order of three to four times (S8), not greatly different from the estimate of Eq. (10-4) for cellular convection. This may indicate that equilibrium is attained in thin layers adjacent to the interface during the spreading and contraction. When the dispersed-phase resistance controls, on the other hand, interfacial turbulence may increase the mass transfer rate by more than an order of magnitude above the expected value. This is almost certainly due to vigorous mixing caused by eruptions within the drop. 

The J value denotes the absorption rate without the dispersed phase where the mass transfer rate can be accompanied by zero- or first-order chemical reactions in the continuous phase. These are well-known equations J = k° (O -i- - Ol)... 

Since bacteria are able to retain their dehalogenase activity after dehydration, this new process could allow direct continuous treatment of gaseous effluents. The two main points are that there is no need to transfer the pollutant in an aqueous phase and there is also no longer limitation by solubility, and secondly that microorganisms are no longer growing. If we consider that transfers in the gas phase are much more efficient than those in the liquid phase, this also means that the rate of degradation should be far less limited by transfer and diffusion rate of the... 

It will be assumed that the reaction A + B takes place only in the dispersed phase, the reactant A is assumed to be insoluble in the continuous phase, but the reactant B will diffuse into the drops. Further it is assumed that inside the drops there is no mixing but only pure molecular diffusion. The diffusivity of B in the dispersed phase is 33b while the mass transfer coefficient outside the drops is m. The partition coefficient of B is Hb and equals the ratio of the concentration of B in the dispersed phase at the interface over the concentration of B in the continuous phase at the interface. 

Fig. 8. Schematic course of the concentration profiles for a first-order reaction in the continuous phase with mass transfer limitation.

The molecular weights increase with conversion and the increase is much more pronounced for the VA runs. The strong increase of molecular weight with conversion results from the transfer of reaction loci from the solution to the polymer particles. At low conversion the polymers are mostly formed in the continuous phase in which the termination rate is high. In polymer particles the local concentration of monomer is much higher than that in the continuous phase and, therefore, the growth events are favored in the polymer particles. 

The ratio Mw/Mn (MWD) decreased with increasing PEO-MA fraction in the monomer feed and/or the number of EO units in the macromonomer. Generally, the Mw/Mn in bulk (homogeneous) systems is a function of the termination mode and the chain transfer events and varies between 1 and 2. In the present disperse systems, MWD is much broader (much above 2) as a result of further contributions, such as polymerization in the continuous phase, interface, and polymer particles. The chain transfer to PEO chains decreased the molecular weight, i,e., the Mw of copolymer decreased with increasing macromonomer concentration and PEO chain length. 

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