Mass transfer coefficients from 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-... 

In the Danckwerts model, it is assumed that elements of the surface have an age distribution ranging from zero to infinity. Obtain the age distribution function for this model and apply it to obtain the average, mass Iransfer coefficient at the surface, given that from the penetration theory the mass transfer coefficient for surface of age t is VlD/(7rt, where D is the diffusivity. 

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). 

Penetration theory (Higbie, 1935)assumes that turbulent eddies travel from the bulk of the phase to the interface where they remain for a constant exposure time te. The solute is assumed to penetrate into a given eddy during its stay at the interface by a process of unsteady-state molecular diffusion. This model predicts that the mass-transfer coefficient is directly proportional to the square root of molecular diffusivity... 

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... 

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 thickness of the fictitious film in the film theory can never be measured. The film theory predicts that the convective mass transfer coefficient k is directly proportional to the diffusivity whereas experimental data from various studies show that k is proportional to the two-third exponent of the diffusivity. In addition, the concept of a stagnant film is unrealistic for a fluid-fluid interface that tends to be unstable. Therefore, the penetration theory was proposed by Higbie to better describe the mass transfer in the liquid phase... 

In the previous sections, stagnant films were assumed to exist on each side of the interface, and the normal mass transfer coefficients were assumed proportional to the first power of the molecular diffusivity. In many mass transfer operations, the rate of transfer varies with only a fractional power of the diffusivity because of flow in the boundary layer or because of the short lifetime of surface elements. The penetration theory is a model for short contact times that has often been applied to mass transfer from bubbles, drops, or moving liquid films. The equations for unsteady-state diffusion show that the concentration profile near a newly created interface becomes less steep with time, and the average coefficient varies with the square root of (D/t) [4] ... 

Pictures of bubbles and clouds have inspired some workers to develop reactor models based on the predicted behavior of individual bubbles [3,10]. In these models, the equations for gas interchange include a term for flow out of the bubble and a second term for mass transfer by molecular diffusion to the dense phase. In some models, the cloud is included as part of the bubble in others, diffusion from bubble to cloud and cloud to dense phase are treated as mass transfer steps in series. In these models, the mass transfer coefficient is assumed to vary with following the penetration theory, and the diffusion contribution is the major part of the predicted gas interchange rate. 

As far as the gas-liquid systems are concerned, a number of models for the prediction of the kLtt have been developed based on both empirical correlations, and on film and penetration theory. These models provide estimates of the mass transfer coefficient in the continuous liquid phase, while the mass transfer resistance in the gas phase is considered negligible. In these models the individual contributions of the caps of the plugs, and of the fully developed film separating the plugs from the channel wall are estimated. Ber6i6 and Pintar (1997) proposed a model for the calculation of the mass transfer coefficient in small channels (Eq. 2.2.33), that includes only the contribution of the caps because of the rapid saturation of the film, which is given by... 

An analysis of chemical desorption has recently been published (Chem.Eng.Sci., 21 0980)), which is based on a number of simplifying assumptions the film theory model is assumed, the diffusivities of all species are taken to be equal to each other, and in the solution of the differential equations an approximation which is second order with respect to distance from the gas-liquid interface is used this approximation was introduced as early as 1948 by Van Krevelen and Hoftizer. However, the assumptions listed above are not at all drastic, and two crucial elements are kept in the analysis reversibility of the chemical reactions and arbitrary chemical mechanisms and stoichiometry.The result is a methodology for developing, for any given chemical mechanism, a highly nonlinear, implicit, but algebraic equation for the calculation of the rate enhancement factor as a function of temperature, bulk-liquid composition, interface gas partial pressure and physical mass transfer coefficient The method of solution is easily gene ralized to the case of unequal diffusivities and corrections for differences between the film theory and the penetration theory models can be calculated. 

From figure 8 it can be concluded that forced desorption of CO2 can easily be realized under practical conditions and can be also predicted by the models. Measured H2S mole fluxes fall between penetration and film theory calculations. The forced desorption of CO2 agrees better with the film theory than with the penetration theory. It should be kept in mind however that the calculations are extremely sensitive to mass-transfer coefficients, diffusion and equilibrium constants which were obtained from separate experiments and open literature. 

One of the most significant contributions of the penetration theory is the prediction that the mass transfer coefficient varies as D. As will be seen in Section 2.4-3, experimental mass transfer coefficients generally are correlated with an exponent on Pab ranging from to. The penetration theory model has also been successfully used to predict the effect of simultaneous chemical reaction on mass transfer in gas absorption and in carrier-facilitated membranes. Stewart solved the penetration theoiy model by taking into account bulk flow at the interface. The results, as in the film theory case, are conveniently expressed as the dependence of the ratio kj/k on the dimensionless total flux, (Nf, + Na)k°. This curve is also shown in Fig. 2.4-2 and generally predicts greater effects of convection than does the film theoiy. 

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. 

In the study of mass transfer, the fluid element (microcell) can be used to describe the behaviors of the process. Under the condition of no Marangoni effect, according to the penetration theory, the fluid element flows randomly from fluid phase to the interface and stays there within residence time T for unsteady mass transfer and then go back to the bulk fluid. The liquid-phase mass transfer coefficient is given by the following ... 

This equation, due to Higbie, was originally derived to describe mass transfer between rising gas bubbles and a surrounding liquid Tran. AIChE, 31,368 [1935]). It applies quite generally to situations where the contact time between the phases is short and the penetration (or depletion) depth is so small that transfer may be viewed as taking place from a plant to a semiinfinite domain. In Section 4.1.2.3 we will provide a quantitative criterion for this approach, which is also referred to as the Penetration Theory. It also describes both the short- and long-term behavior in diffusion between a plane and a semi-infinite space, and we used this property in Chapter 1, Table 1.4, to help us set upper and lower bounds to mass transfer coefficients and "film" thickness Zp j. 

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