Penetration model

The existence of a stagnant laminar fluid film adjacent to the interface is not difficult to visualize, in the case where the interface is stationary, as when the fluid flows along a solid surface. However, this situation seems rather unrealistic with the fluid-fluid interface, as when the surface of the liquid in an agitated vessel is in contact with a gas phase above, or if gas bubbles move upward through a liquid, or when one liquid phase is in contact with another liquid phase in an extractor. 

If this model is correct, should vary with the diffusivity D to the 0.5 power, but this does not agree with experimental data in general. Also, t is unknown except in the case of some specially designed equipment. 

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Note that both the penetration and the surface-renewal theories predict a square-root dependency on D. Also, it should be recognized that values of the surface-renewal rate s generally are not available, which presents the same problems as do 6 and t in the film and penetration models. 

Calculate ki from penetration model (use time for liquid to flow distance s). 

The set of equations was solved numerically for the carbonation of limewater with the conditions of batch operation. In an alternative approach, the penetration model has also been applied (Hostomsky and Jones, 1993). 

Hostomsky, J. and Jones, A.G., 1995. A penetration model of the gas-liquid reactive precipitation of calcium carbonate crystals. Transactions of the Institution of Chemical Engineers, 73A, 241-245. 

D. Film-Penetration Model with Simultaneous Chemical Reaction. 341... 

Fig. 7. Penetration model for diffusion with simultaneous irreversible first-order chemical reaction [after Lightfoot (L5)].

The penetration theory holds for the region where t is much less than L2jD, the film theory for the region where t is much greater than L2/D. This comparison is shown in Fig. 8, which clearly shows that the film and penetration theories are asymptotes of the film-penetration model. 

Fig. 8. Comparison of film-penetration model with film and penetration models [after Toor and Marchello (T5)].

In 1963 and in 1965, Huang and Kuo (H18, H19) applied the film penetration model to the mechanism of simultaneous mass transfer and chemical reaction. 

In the film-penetration model (H19), it is assumed that the reactant A penetrates through the surface element by one-dimensional unsteady-state molecular diffusion. Convective transport is assumed to be insignificant. The diffusing stream of the reactant A is depleted along the path of diffusion by its reversible reaction with the reactant B, which is an existing component of the liquid surface element. If such a reaction can be represented as... 

For mass transfer with irreversible and reversible reactions, the film-penetration model is a more general concept than the film or surface renewal models which are its limiting cases. 

As an example, it may be supposed that in phase 1 there is a constant finite resistance to mass transfer which can in effect be represented as a resistance in a laminar film, and in phase 2 the penetration model is applicable. Immediately after surface renewal has taken place, the mass transfer resistance in phase 2 will be negligible and therefore the whole of the concentration driving force will lie across the film in phase 1. The interface compositions will therefore correspond to the bulk value in phase 2 (the penetration phase). As the time of exposure increases, the resistance to mass transfer in phase 2 will progressively increase and an increasing proportion of the total driving force will lie across this phase. Thus the interface composition, initially determined by the bulk composition in phase 2 (the penetration phase) will progressively approach the bulk composition in phase 1 as the time of exposure increases. 

HARRIOTT 25 suggested that, as a result of the effects of interfaeial 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 thin 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 Him resistance in one phase and a penetration model resistance in the other. It is necessary in equation 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 C Ao — JPCAo, and the mass transfer rate at time t is given for a film thickness L by ... 

In a process where mass transfer takes place across a phase boundary, the same theoretical approach can be applied to each of the phases, though it does not follow that the same theory is best applied to both phases. For example, the film model might be applicable to one phase and the penetration model to the other. This problem is discussed in the previous section. 

It may be assumed that the penetration model may be used to represent the mass transfer process. The depth of penetration is small compared with the radius of the droplets and the effects of surface curvature may he neglected. From the penetration theory, the concentration C, at a depth y below the surface at time r is given by ... 

In a recent study involving 150 chemically diverse compounds [26] the following global BBB penetration model was obtained ... 

The authors noted, however, that the separation of compounds into chemically similar classes considerably improves the construction of predictive BBB penetration models. 

There is a thermally activated structural rearrangement of the protein such that channels appear and the quencher molecules are able to penetrate the protein—the penetration model.(67) This model distinguishes between external diffusion, kd(ext), and diffusion within the protein as follows ... 

The tape-stripping technique has been applied in vivo, as well as in vitro. The used in vitro incubation devices are the same as described for skin permeation studies. A specialized incubation device developed by Loth and coworkers, the Saarbriicken penetration model, allows investigation of skin penetration bypassing the normally occurring nonphysiological hydration of the dermis [64],... 

In 1951,Danckwerts [4] proposed the surface renewal model as an extension ofthe penetration model. Instead of assuming a fixed contact time for all fluid elements, Danckwerts assumed a wide distribution of contact time, from zero to infinity, and supposed that the chance of an element ofthe surface being replaced with fresh liquid was independent of the length of time for which it has been exposed. Then, it was shown, theoretically, that the averaged mass transfer coefficient at the interface is given as... 

Compared to the film model or the penetration model, the surface renewal approach seems closer to reality in such a case where the surface of liquid in an... 

It can be seen that a theoretical prediction of values is not possible by any of the three above-described models, because none of the three parameters - the laminar film thickness in the film model, the contact time in the penetration model, and the fractional surface renewal rate in the surface renewal model - is predictable in general. It is for this reason that the empirical correlations must normally be used for the predictions of individual coefficients of mass transfer. Experimentally obtained values of the exponent on diffusivity are usually between 0.5 and 1.0. 

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