Penetration/surface renewal model

Considering homogeneous RSPs, mass transfer at the gas/vapor/liquid-liquid interface can be described using different theoretical concepts (57,59). Most often the two-film model (87) or the penetration/surface renewal model (27,88) is used, in which the model parameters are estimated via experimental correlations. In this respect the two-film model is advantageous since there is a broad spectrum of correlations available in the literature, for all types of internals and systems. For the penetration/surface renewal model, such a choice is limited. 

The predictions of correlations based on the film model often are nearly identical to predictions based on the penetration and surface-renewal models. Thus, in view of its relative simphcity, the film model normally is preferred for purposes of discussion or calculation. It should be noted that none of these theoretical models has proved adequate for maldug a priori predictions of mass-transfer rates in packed towers, and therefore empirical correlations such as those outlined later in Table 5-28. must be employed. 

For mass transfer with irreversible and reversible reactions, the film-penetration model is a more general concept than the film or surface renewal models which are its limiting cases. 

In 1951,Danckwerts [4] proposed the surface renewal model as an extension ofthe penetration model. Instead of assuming a fixed contact time for all fluid elements, Danckwerts assumed a wide distribution of contact time, from zero to infinity, and supposed that the chance of an element ofthe surface being replaced with fresh liquid was independent of the length of time for which it has been exposed. Then, it was shown, theoretically, that the averaged mass transfer coefficient at the interface is given as... 

It can be seen that a theoretical prediction of values is not possible by any of the three above-described models, because none of the three parameters - the laminar film thickness in the film model, the contact time in the penetration model, and the fractional surface renewal rate in the surface renewal model - is predictable in general. It is for this reason that the empirical correlations must normally be used for the predictions of individual coefficients of mass transfer. Experimentally obtained values of the exponent on diffusivity are usually between 0.5 and 1.0. 

II is a function of hydrodynamic parameters of the model. Unfortunately, these parameters which describe the effect of hydrodynamics do not correspond to any physical quantity nor can they be Independently evaluated. For some models, the value of w is a constant. For example, the penetration and surface renewal models (Danckwerts, 31) predict w 0.5, while for the boundary layer model w 2/3. The film-penetration model, on the other hand, predicts that w varies between 0.5 and 1 (Toor and Marchello, 32). Knowledge of the effect of dlffuslvlty on k Is needed in evaluating the various mass transfer models. Calderbank (13) reported a value of 0.5 Linek et al. (22) used oxygen, Helium and argon. The reported diffusion coefficients for helium and similar gases vary widely. Since in the present work three different temperatures have been used, the value of w can be determined much more accurately. Figure 4... 

The gas-liquid and gas-solid reaction processes can be analyzed by several different physical models, namely film, penetration, surface renewal, Danckwerts, film-penetration, etc. These models are described by Danckwerts.39 Although each of these models gives a somewhat different physical picture of the reaction process, in many instances the final desired answer for the rate of absorption of gas in the presence of a liquid- or a solid-phase reaction is similar. Since film and penetration theories are most widely used, we review their applications here. 

One very important restriction in the development of the surface renewal models is the assumption that the penetrating, or diffusing, component does not see the bulk fluid, which, to all intents and purposes, is located at an infinite distance from the interface. This assumption is implicit in the boundary condition (Eq. 9.1.6) and is strictly true only for short Fourier times... 

The results showed that mass transfer through the gas-side boundary layer could be described by the penetration theory (Hygbie 1935) or by the surface renewal model (Danckwerts 1951). It was found that ... 

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. 

To make Pick s law tractable for engineering calculations, several theories, or models, have been discussed. Their application usually introduces some empiricism, since distance of diffusion and extent of interfacial area are usually indeterminate. For most calculations, the time-tested film model is appropriate, not because it represents with validity the physical situation, but rather because it is supported by a wealth of experience as well as the ready availability of data, particularly molecular diffusion coefficients. The penetration and surface-renewal models have specialized applications and also depend on empirically derived parameters. 

Four of the simplest and best known of the theories of mass transfer from flowing streams are (1) the stagnant-film model, (2) the penetration model, (3) the surface-renewal model, and (4) the turbulent boundary-layer model... 

The various forms of the penetration theory can be classified as surface-renewal models, implying either formation of new surfaee at frequent intervals or replacement of fluid elements at the surface with fresh fluid from the bulk. The time or its reciprocal, the average rate of renewal, are functions of the fluid velocity, the fluid properties, the the geometry of the system and can be accurately predicted in only a few special cases. However, even if tj must be determined empirically, the surface-renewal models give a sound basis for correlation of mass-transfer data in many situations, particularly for transfer to drops and bubbles. The similarity between Eqs. (21.44) and (15,20) is an example of the close analogy between heat and mass transfer. It is often reasonable to assume that tj-is the same for both processes and thus to estimate rates of heat transfer from measured mass-transfer rates or vice versa. 

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 ... 

These considerations have led to other models, called "penetration" or "surface renewal" models. In these models the surface at any point is considered... 

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 net result is a physical representation of turbulence in conceptual terms that is more realistic than those of the film, penetration, or surface renewal models. As with these models, this one also depends upon measurements in this case, it is P(t). The end result of the analysis yields a correlation for the MTC at a solid interface, which is a function of f)(t) ... 

The definitions for the mass transfer coefficients can be used to theoretically predict them using the diffiisivity, concentrations, length scales, and fluid flow characteristics, thus rendering the two mass transfer approaches equivalent. This can easily be done in the cases of equimolar counterdiffusion (Maz + A bz = 0) and diffusion of A through a stagnant film (Ab = 0) (Hines and Maddox, 1985, p. 140). Also, the theoretical models of film, penetration, surface renewal, and film penetration have been proposed for the estimation of the mass transfer coefficients at a fluid-fluid interface (Hines and Maddox, 1985, pp. 146-151). 

Danckwerts [Jnd. Eng. Chem., 42, 1460(1951)] proposed an extension of the penetration theoiy, called the surface renewal theoiy, which allows for the eddy motion in the liquid to bring masses of fresh liquid continually from the interior to the surface, where they are exposed to the gas for finite lengths of time before being replaced. In his development, Danckwerts assumed that every element of fluid has an equal chance of being replaced regardless of its age. The Danck-werts model gives... 

Note that both the penetration and the surface-renewal theories predict a square-root dependency on D. Also, it should be recognized that values of the surface-renewal rate s generally are not available, which presents the same problems as do 6 and t in the film and penetration models. 

Marchello and Toor (M2) proposed a mixing model for transfer near a boundary which assumes that localized mixing occurs rather than gross displacement of the fluid elements. This model can be said to be a modified penetration-type model. Kishinevsky (K6-K8) assumed a surface-renewal mechanism with eddy diffusion rather than molecular diffusion controlling the transfer at the interface. 

As an example, it may be supposed that in phase 1 there is a constant finite resistance to mass transfer which can in effect be represented as a resistance in a laminar film, and in phase 2 the penetration model is applicable. Immediately after surface renewal has taken place, the mass transfer resistance in phase 2 will be negligible and therefore the whole of the concentration driving force will lie across the film in phase 1. The interface compositions will therefore correspond to the bulk value in phase 2 (the penetration phase). As the time of exposure increases, the resistance to mass transfer in phase 2 will progressively increase and an increasing proportion of the total driving force will lie across this phase. Thus the interface composition, initially determined by the bulk composition in phase 2 (the penetration phase) will progressively approach the bulk composition in phase 1 as the time of exposure increases. 

Kishinev ski/23 has developed a model for mass transfer across an interface in which molecular diffusion is assumed to play no part. In this, fresh material is continuously brought to the interface as a result of turbulence within the fluid and, after exposure to the second phase, the fluid element attains equilibrium with it and then becomes mixed again with the bulk of the phase. The model thus presupposes surface renewal without penetration by diffusion and therefore the effect of diffusivity should not be important. No reliable experimental results are available to test the theory adequately. 

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. 

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. 

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