The film-penetration theory

The mass transfer process is again governed by equation 10.66, but the third boundary condition is applied at y = L, the film thickness, and not at y = oo. As before, the Laplace 

Since there is no inverse of equation 10.136 in its present form, it is necessary to expand using the binomial theorem. Noting that, since 2y/(p/D)L is positive, e-2V p/D L i and from the binomial theorem  

The mass transfer rate across the interface per unit area is therefore given by  

Equation 10.139 converges rapidly for high values of L2/Dt. For low values of L2/Dt, it is convenient to employ an alternative form by using the identity 26, and  

It will be noted that equations 10.139 and 10.141 become identical in form and in convergence when L /Dt = n. 

A theory which incorporates some of the principles of both the two-film theory and the penetration theory has been proposed by TOOR and Marchello The whole of the resistance to transfer is regarded as lying within a laminar film at the interface, as in the two-film theory, but the mass transfer is regarded as an unsteady state process. It is assumed that fresh surface is formed at intervals from fluid which is brought from the bulk of the fluid to the interface by the action of the eddy currents. Mass transfer then takes place as in the penetration theory, except that the resistance is confined to the finite film, and material which traverses the film is immediately completely mixed with the bulk of the fluid. For short times of exposure, when none of the diffusing material has reached the far side of the layer, the process is identical to that postulated in the penetration theory. For prolonged periods of exposure when a steady concentration gradient has developed, conditions are similar to those considered in the two-film theory. 

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As noted previously, for equimolecular counterdiffusion, the film transfer coefficients, and hence the corresponding HTUs, may be expressed in terms of the physical properties of the system and the assumed film thickness or exposure time, using the two-film, the penetration, or the film-penetration theories. For conditions where bulk flow is important, however, the transfer rate of constituent A is increased by the factor Cr/Cgm and the diffusion equations can be solved only on the basis of the two-film theory. In the design of equipment it is usual to work in terms of transfer coefficients or HTUs and not to endeavour to evaluate them in terms of properties of the system. 

An important variation of the model of Mickley and Fairbanks is the film penetration model developed by Yoshida et al. [48] by treating packets as a continuum with a finite thickness (8em). The film penetration theory includes two extremes of emulsion behavior. On one extreme, the packet contacts the heating surface for a short time so that all the heat entering the packet is used to heat up the packet (penetration theory) while none passes through it. On the other extreme, the packet stays at the surface long enough to achieve steady state and simply provides a resistance for heat conduction. 

Z[ thickness of laminar surface layer in the film-penetration theory... 

Simplified Mass-Transfer Theories In certain simple situations, tne mass-transfer coefficients can be calculated from first principles. The film, penetration, and surface-renewal theories are attempts to extend tnese theoretical calculations to more complex sit-... 

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. 

The penetration and film-penetration theories have been developed for conditions of equimolecular counterdiffusion only the equations are too complex to solve explicitly for transfer through a stationary carrier gas. For gas absorption, therefore, they apply only when the concentration of the material under going mass transfer is low. On the other hand, in the two-fihn theory the additional contribution to the mass transfer which is caused by bulk flow is easily calculated and hp (Section 10.23) is equal to (D/L)(Cr/Cum) instead of D/L. 

These relations between the various coefficients are valid provided that the transfer rate is linearly related to the driving force and that the equilibrium relationship is a straight line. They are therefore applicable for the two-film theory, and for any instant of time for the penetration and film-penetration theories. In general, application to time-averaged coefficients obtained from the penetration and film-penetration theories is not permissible because the condition at the interface will be time-dependent unless all of the resistance lies in one of the phases. 

What are the general principles underlying the two-film, penetration and film-penetration theories for mass transfer across a phase boundary Give the basic differential equations which have to be solved for these theories with the appropriate boundary conditions. 

As can be seen from Figure 8, if Fo < 0.02, the concentration changes within the film are confined largely to the surface layer and the local mass transfer coefficient is given by the Higbie penetration theory (9) as... 

Huang and Kuo also solved two equations for a rapid first-order reversible reaction (i.e., equilibrium in the bulk liquid). The solutions are extremely lengthy and will not be given here. From a comparison of the film, surface renewal, and intermediate film-penetration theories it was found that for irreversible and reversible reactions with equal diffusivities of reactant and product, the enhancement factor was insensitive to the mass transfer model. For reversible reactions with product diffusivity smaller than that of the reactant, the enhancement factor can differ by a factor of two between the extremes of film and surface renewal theory. To conclude, it would seem that the choice of the model matters little for design calculations the predicted differences are negligible with respect to the uncertainties of prediction of some of the model or operation parameters. 

However, the absence of any parameter related to the channel size limits the application of the model to different two-phase systems. Van Baten and Krishna (2004) and Irandoust and Andersson (1989) included in their models the contributions of both bubble caps and film (Eq. 2.2.34). Van Baten and Krishna (2004) evaluated the contribution of the caps according to the Higbie penetration theory (Eq. 2.2.35), whilst the transfer through the film was obtained based on mass transfer in a falling film in laminar flow (Eqs. 2.2.36 and 2.2.37). 

Thie film-penetration theory, presented by Toor and Marchello, represents a combination of the three earlier theories reviewed above. The entire transfer resistance is conside to lie in a laminar surface layer of thickness Zt, where is uniform at ca. for all z greater than Zl- Surface renewal occurs by eddies that penetrate the surface from the bulk of the phase. Thus, transfer through young elements of surface obeys the penetration theory (k transfer through old elements follows the film theory (k D), and transfer through elertients of intermediate age is characterized by bmh rrrechanisms. 

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