Mass transfer coefficients from surface renewal 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 theories vary in the assumptions and boundary conditions used to integrate Fick s law, but all predict the film mass transfer coefficient is proportional to some power of the molecular diffusion coefficient D", with n varying from 0.5 to 1. In the film theory, the concentration gradient is assumed to be at steady state and linear, (Figure 3-2) (Nernst, 1904 Lewis and Whitman, 1924). However, the time of exposure of a fluid to mass transfer may be so short that the steady state gradient of the film theory does not have time to develop. The penetration theory was proposed to account for a limited, but constant time that fluid elements are exposed to mass transfer at the surface (Higbie, 1935). The surface renewal theory brings in a modification to allow the time of exposure to vary (Danckwerts, 1951). 

Liquid-solid mass transfer has also been studied, on a limited basis. Application to systems with catalytic surfaces or electrodes would benefit from such studies. The theoretical equations have been proposed based on film-flow theory (32) and surface-renewal theory (39). Using an electrochemical cell with rotating screen disks, liquid-solid mass transfer was shown to increase with rotor speed and increased spacing between disks but to decrease with the addition of more disks (39). Water flow over naphthalene pellets provided 4-6 times higher volumetric mass transfer coefficients compared to gravity flow and similar superficial liquid velocities (17). 

The value of a varies with the system under consideration. For example, in equimolar counter diffusion, Na and Nb are of the same magnitude, but in opposite direction. As a result, a is equal to 1 and hence, Eq. (2) reduces to Eq. (1), where is equal to Convective mass transfer coefficients are used in the design of mass transfer equipment. However, in most cases, these coefficients are extracted from empirical correlations that are determined from experimental data. The theories, which are often used to describe the mechanism of convective mass transfer, are the film theory, the penetration theory, and the surface renewal theory. 

The mass transfer model is based on a physical picture of surface renewal that was developed for describing mass transfer across mobile interfaces. The mass transfer coefficient is then based on the theory for non-steady state diffusion. For relatively short periods of time, the time dependent mass transfer coefficient, according to the penetration theory follows from (see also section 4.62,1)... 

To this end, there are many theories which attempt to interpret or explain the behavior of mass-transfer coefficients, such as the film, penetration, surface-renewal, and other theories. They are all speculations, and are continually being revised. It is helpful to keep in mind that transfer coefficients, for both heat and mass transfer, are expedients used to deal with situations which are not fully understood. They include in one quantity effects which are the result of both molecular and turbulent diffusion. The relative contribution of these effects, and indeed the detailed character of the turbulent diffusion itself, differs from one situation to another. The ultimate interpretation or explanation of the transfer coefficients will come only when the problems of the fluid mechanics are solved, at which time it will be possible to abandon the concept of the transfer coefficient. 

The penetration theory can be viewed as the original surface-renewal model. This model was formulated by Higbie [51]. This model is based on the assumption that the liquid surface contains small fluid elements that contact the gas phase for a time that is equal for all elements. After this contact time they penetrate into the bulk liquid and each element is then replaced by another element from the bulk liquid phase. The basic mechanism captured in this concept is that at short contact times, the diffusion process will be unsteady. Considering that the fluid elements may diffuse to an infinite extend into the liquid phase, the model formulation developed earlier for diffusion into a semi-infinite slab can be applied describing this system. After some time the diffusion process will reach a steady state, thus the penetration theory predictions will then correspond to the limiting case described by the basic film theory. However, when the transient flux development is determining a notable amount of the total flux accumulated, the two models will give rise to different mass transfer coefficients. 

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. 

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