Surface renewal theory Higbie penetration model

Two rather similar models have been devised to remedy the problems of simple film theory. Both the penetration theory of Higbie and the surface renewal theory of Danckwerts replace the idea of steady-state diffusion across a film with transient diffusion into a semi-inhnite medium. We give here a brief account of surface renewal theory. 

In Chapter 7 we discussed the basics of the theory concerned with the influence of diffusion on gas-liquid reactions via the Hatta theory for flrst-order irreversible reactions, the case for rapid second-order reactions, and the generalization of the second-order theory by Van Krevelen and Hofitjzer. Those results were presented in terms of classical two-film theory, employing an enhancement factor to account for reaction effects on diffusion via a simple multiple of the mass-transfer coefficient in the absence of reaction. By and large this approach will be continued here however, alternative and more descriptive mass transfer theories such as the penetration model of Higbie and the surface-renewal theory of Danckwerts merit some attention as was done in Chapter 7. 

In surface renewal models the liquid surface is assumed to consist of a mosaic of elements with different age at the surface. The rate of absorption at the surface is then an average of the rates of absorption in each element, weighted with respect to a distribution function (t)—see Eq. 6.2-5. Under this heading of surface renewal theory we will also occasionally mention results of Higbie s [23] so-called penetration-theory, which can be considered as a special case in which every element is exposed to the gas for the same length of time before being replaced. The main emphasis of this section is on the Danckwerts [24] approach using the distribution function for completely random replacement of surface elements ... 

Other models such as the penetration model (surface renewal theory) developed by Higbie and Danckwerts (Westerterp, van Swaaij, and Beenackers, 1998) consider the mass transfer process to be essentially non-stationary, and the surface is assumed to consist of elements of different age at the surface, returning into the bulk phase while new elements originating from the bulk phase take their place. Results of calculations by the two-film theory and the surface renewal theory are similar. Thus, only the two-film theory, which is easier to understand and which is therefore used most, is considered here. 

Given that, from the penetration theory for mass transfer across an interface, the instantaneous rale ol mass transfer is inversely proportional to the square root of the time of exposure, obtain a relationship between exposure lime in the Higbie mode and surface renewal rate in the Danckwerts model which will give the same average mass transfer rate. The age distribution function and average mass transfer rate from the Danckwerts theory must be deri ved from first principles. 

The mass-transfer coefficient in each film is expected to depend upon molecular diffusivity, and this behavior often is represented by a power-law function k . For two-film theory, n = 1 as discussed above [(Eq. (15-62)]. Subsequent theories introduced by Higbie [Trans. AIChE, 31, p. 365 (1935)] and by Dankwerts [Ind. Eng. Chem., 43, pp. 1460-1467 (1951)] allow for surface renewal or penetration of the stagnant film. These theories indicate a 0.5 power-law relationship. Numerous models have been developed since then where 0.5 < n < 1.0 the results depend upon such things as whether the dispersed drop is treated as a rigid sphere, as a sphere with internal circulation, or as oscillating drops. These theories are discussed by Skelland [ Tnterphase Mass Transfer, Chap. 2 in Science and Practice of Liquid-Liquid Extraction, vol. 1, Thornton, ed. (Oxford, 1992)]. 

The penetration theory can be viewed as the original surface-renewal model. This model was formulated by Higbie [57]. This model is based on the assump>-tion that the liquid surface contains small fluid elements that contact the gas... 

The classical Danckwerts surface-renewal model is analogous to the penetration theory. The improvement is in the view of the eddy replacement process. Instead of Higbies assumption that all elements have the same recidence time at the interface, Danckwerts [29] proposed to use an averaged exposure time determined from a postulated time distribution. The recidence time distribution of the surface elements is described by a statistical distribution function E(t), defined so that E(t)d,t is the fraction of the interface elements with age between t and t + dt. The rest of the formulation procedure is similar to that of the penetration model. 

This model, proposed originally by Danckwerts [18], is an extension of the penetration model. Whereas Higbie assumed that all exposure times were the same, Danckwerts provided for a range of times, based on probability theory. After a given exposure, the surface (interface) was renewed, leading to the name of the theory. The elements of the model are... 

Penetration theory can also be applied to turbulent conditions by assuming the turbulence spectrum to consist of large eddies, capable of surface renewal, and small eddies responsible for the presence of eddy diffusivity The small eddies are damped when an element of liquid reaches the interface so that, during its residence time there, mass transfer occurs in accordance with the assumptions of the penetration theory If all the eddies stay at the interface for the same interval of time we talk about penetration theory with regular surface renewal or the Higbie model If there is random distribution of residence times with an age-independent fractional rate of surface renewal, s, the term penetration theory with random surface renewal, or the Danckwerts nK)del, is employed In the case of the Higbie model, the mass transfer coefficient is the same as that given by eqn (18). For the Danckwerts model it takes the form... 

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. 

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