Penetration and Surface-Renewal Theories

Example 9.1-1 Finding the film thickness Carbon dioxide is being scrubbed out of a gas using water flowing through a packed bed of 1-cm Berl saddles. The carbon dioxide is absorbed at a rate of 2.3 10 mol/cm sec. The carbon dioxide is present at a partial pressure of 10 atm, the Henry s law coefficient //is 600 atms, and the diffusion coefficient of carbon dioxide in water is 1.9 10 cm /sec. Find the film thickness. 

Solution We first find the interfacial concentration of carbon dioxide  

Values around 10 cm are typical of many mass transfer processes in liquids. 

These theories provide a better physical picture of mass transfer than the film theory in return for a modest increase in mathematics. The net gain in understanding is often worth the price. Moreover, although the physical picture is still limited, similar equations can be derived from other, more realistic physical pictures. 

The model basic to this theory, suggested by Higbie in 1935, is shown schematically in Fig. 9.2-1. As before, we define the mass transfer coefficient into this film as 

The film and boundary layer theories presuppose steady transport, and can therefore not be used in situations where material collects in a volume element, thus leading to a change in the concentration with time. In many mass transfer apparatus fluids come into contact with each other or with a solid material for such a short period of time that a steady state cannot be reached. When air bubbles, for example, rise in water, the water will only evaporate into the bubbles where it is contact with them. The contact time with water which surrounds the bubble is roughly the same as that required for the bubble to move one diameter further. Therefore at a certain position mass is transferred momentarily. The penetration theory was developed by Higbie in 1935 [1.31] for the scenario described here of momentary mass transfer. He showed that the mass transfer coefficient is inversely proportional to the square root of the contact (residence) time and is given by 

Here (3m is the mean mass transfer coefficient from time /, 0 to time t. Experience 

Possible flow patterns for contact between two liquids or between a liquid and a gas 

---

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 and surface renewal theories started out as conceptual, in that they were visualized to occur as such by individual theorists. These theories appeared to work successfully for a free interface, such as the air-water interface, but not for a fixed interface, such as solid-water. Now, the explanation is before us in equation (8.64). Surface renewal is a fairly accurate representation of Hanratty s jS at a free surface, and therefore can be seen to give representative results. It is Hanratty s p that we really should be measuring, and it happens that the mean surface renewal rate is a good representation of Hanratty s jS at a free surface. 

The integral method thus leads to a mass transfer coefficient which vary with the 2/3 power of the diffusion coefficient. This parameter dependency is between the linear one of the film theory and the square-root variation of the penetration and surface-renewal theories. 

As can be appreciated by comparison the results for a solid surface are quite different from that for a fluid-fluid interface. The latter, in its functionality, is consistent with both the penetration and surface renewal theories. It appears that the approach of Hanratty et al. is based on a more realistic qualitative and quantitative characterization of the turbulence in a flowing fluid stream than the simpler ones. 

The sections of this chapter are different attempts to give a prediction of Equation 9.0-1. In Sections 9.1 and 9.2, we discuss the film theory, based on diffusion across a thin film and the penetration and surface-renewal theories, based on diffusion into a semi-infinite slab. In Section 9.3, we discuss why these theories do not predict Equation 9.0-1, and how this disagreement may be resolved. In Section 9.4, we talk about... 

Example 9.2-1 Finding the adjustable parameters of the penetration and surface-renewal theories What are the contact time /./v ax and the surface residence time x for the carbon dioxide scrubber described in Example 9.1-1 ... 

The mass transfer theories developed in the previous sections of this chapter are not especially successful. To be sure, the penetration and surface-renewal theories do predict that mass transfer does vary with the square root of the diffusion coefficient, consistent with many correlations. However, neither the film theory nor the surface-renewal theory predicts how mass transfer varies with flow. The penetration theory predicts variation with the square root of flow, less than that indicated by most correlations. This failure to predict the variation of mass transfer with flow is especially disquieting the film and penetration theories should bracket all behavior because a thin film and a semi-infinite slab bracket all possible geometries. 

The different theoretical efforts to achieve this goal are summarized in Table 9.6-1. These efforts predict mass transfer coefficients vary with diffusion coefficients to powers ranging from 0.0 to 1.0, and clustering around 0.5. This is close to the average of the various correlations. This implies that the physical picture of the penetration and surface-renewal theories is superior to that of the film theory. 

---