Interfacial transfer penetration theory

The penetration theory has been used to calculate the rate of mass transfer across an interface for conditions where the concentration CAi of solute A in the interfacial layers (y = 0) remained constant throughout the process. When there is no resistance to mass transfer in the other phase, for instance when this consists of pure solute A, there will be no concentration gradient in that phase and the composition at the interface will therefore at all Limes lie the same as the bulk composition. Since the composition of the interfacial layers of the penetration phase is determined by the phase equilibrium relationship, it, too. will remain constant anil the conditions necessary for the penetration theory to apply will hold. If, however, the other phase offers a significant resistance to transfer this condition will not, in general, be fulfilled. 

In many of these experiments, interfacial turbulence was the obvious visible cause of the unusual features of the rate of mass transfer. There are, however, experimental results in which no interfacial activity was observed. Brian et al. [108] have drawn attention to the severe disagreement existing between the penetration theory and data for the absorption of carbon dioxide in monoethanolamine. They have performed experiments on the absorption of C02 with simultaneous desorption of propylene in a short, wetted wall column. The desorption of propylene without absorption of C02 agrees closely with the predictions of the penetration theory. If, however, both processes take place simultaneously, the rate of desorption is greatly increased. This enhancement must be linked to a hydrodynamic effect induced by the absorption of C02 and the only one which can occur appears to be the interfacial turbulence caused by the Marangoni effect. No interfacial activity was observed because of the small scale and small intensity of the induced turbulence. 

Equations (I3-II5) to (13-117) contain terms, e, for rates of heat transTer from the vapor phase to the liquid phase. These rates are estimated from convective and bulk-flow contributions, where the former are based on interfacial area, average-temperature driving forces, and convective heat-transfer coefficients, which are determined from the Chilton-Colburn analogy for the vapor phase and from the penetration theory for the liquid phase. 

The penetration theory was proposed by Higbie19 who observed that the mass transfer of a solute A from a liquid to a gas in industrial processes could be regarded as a succession of intermittent processes wherein each bubble of vapor is exposed to a succession of liquid surfaces. Each liquid surface is visualized as forming at the top of the bubble and remaining in contact with it for amount of time t, equal to that required for the bubble to move through a distance equal to one bubble diameter. The mass transfer process over the time period t was described by the partial differential equation known as Fick s second law. The solution obtained for a suitable set of boundary conditions led to the following expression for NA, the rate of transfer of A from the liquid to the vapor per unit of interfacial area... 

Harriott suggested that, as a result of the effects of interfacial tension, the layers of fluid in the immediate vicinity of the interface would frequently be unaffected by the mixing process postulated in the penetration theory. There would then be a tiiin laminar layer unaffected by the mixing process and offering a constant resistance to mass transfer. The overall resistance may be calculated in a manner similar to that used in the previous section where the total resistance to transfer was made up of two components—a film resistance in one phase and a penetration model resistance in the other. It is necessary in equatjon 10.132 to put the Henry s law constant equal to unity and the diffusivity Df in the film equal to that in the remainder of the fluid D. The driving force is then Cai — Cao in place of — 3 Cao, and the mass transfer rate at time t is given for a film thickness L by ... 

Using a simple scaling analysis, involving (1) viscous pressure drop, (2) hydrostatic pressure drop, (3) interfacial pressure drop and (4) penetration theory for mass transfer, it has been demonstrated that two-phase laminar bubble-train flow in small channels can exhibit better mass transfer for a given power input than turbulent contactors. 

However, as with the penetration theory analysis, the difference in magnitude of the mass and thermal diffusivities with cx 100 D, means that the heat transfer film is an order of magnitude thicker than the mass transfer film. This is depicted schematically in Fig. 8, The fall in temperature from T over the distance x is (if a = 100 D) about 0% of the overall interface excess temperature above the datum temperature T.. Furthermore, in considering the location of heat release oue to reaction in the mass transfer film, this is bound to be greatest closest to the interface, and this is especially the case when the reaction becomes fast. Therefore, two simplifications can be introduced as a result of this (i) the release of heat of reaction can be treated as am interfacial heat flux and (ii) the reaction can be assumed to take place at the interfacial temperature T. The differential equation for diffusion and reaction can therefore be written... 

E. Mass-Transfer Considerations. The earliest application of the theories of mass transfer to gas absorption did not recognize the influence of chemical reaction in the liquid phase on mass-transfer coefflcients. This influence is explained in the film and penetration theories of absorption by showing that the concentration of the absorbate is depressed everywhere near the gas-liquid interface by its reaction. The interfacial concentration gradient is sharper as a result, and this increases the liquid-phase mass-transfer coefficient. 

The main transport parameters to be estimated are the mass transfer coefficients (gas-liquid (liquid side) fc , gas-liquid (gas side) kg, and liquid-solid fc )). Coupled to that is the estimation of the interfacial area per unit volume a, and often it is the combination (i.e., kia or kgO) that is estimated in a certain experimental procedure. Thermodynamic parameters, such as Henry s law constant (fZ) can be estimated in a simpler manner since their estimation on the flow or on any time-dependent phenomenon. Mass transfer coefticients may be evaluated in well-defined geometries with known flow fields using classical theories like film theory, penetration theory, surface renewal... 

Under the condition of no Rayleigh convection, the unsteady interfacial mass transferred can be calculated from penetration theory by the following equation ... 

Fig. 9.2-1. The penetration theory for mass transfer. Here, the interfacial region is imagined to be a very thick film continuously generated by fiow. Mass transfer now involves diffusion into this film. In this and other theories, the interfacial concentration in the liquid is assumed to be in equilibrium with that in the gas.

Fig. 9.2-2. The surface-renewal theory for mass transfer. This approach tries to apply the mathematics of the penetration theory to a more plausible physical picture. The liquid is pictured as two regions, a large well-mixed bulk and an interfacial region that is renewed so fast that it behaves as a thick film. The surface renewal is caused by liquid flow.

The Gouy-Chapman theory for metal-solution interfaces predicts interfacial capacities which are too high for more concentrated electrolyte solutions. It has therefore been amended by introducing an ion-free layer, the so-called Helmholtz layer, in contract with the metal surface. Although the resulting model has been somewhat discredited [30], it has been transferred to liquid-liquid interfaces [31] by postulating a double layer of solvent molecules into which the ions cannot penetrate (see Fig. 17) this is known as the modified Verwey-Niessen model. Since the interfacial capacity of liquid-liquid interfaces is... 

In 2005, an interesting critical summary of the current knowledge on capacitance measurements and potential distribution, together with a new model, was published by Monroe et al. [64]. In this work, the good old Verwey-Niessen theory was extended to allow ionic penetration at the interface. With this adaptation, several features could be accounted for, such as asymmetry and shifts of the capacitance minimnm, that could not be described by the classical Gouy-Chapman or Verwey-Niessen theories. Gibbs energies of ion transfer were used as input parameters to describe ionic penetration into the mixed-solvent interfacial layer, and experimental data were successfully reproduced. 

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