Mass transfer surface renewal model

An apparent weakness of the film model is that it suggests that the mass transfer coefficient is directly proportional to the diffusion coefficient raised to the first power. This result is in conflict with most experimental data, as well as with more elaborate models of mass transfer [surface renewal theory considered in the next chapter, e.g., or boundary layer theory (Bird et al., I960)]. However, if we substitute the film theory expression for the mass transfer coefficient (Eq. 8.2.12) into Eq. 8.8.1 for the Sherwood number we find... 

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. 

On the assumption that the oxygen transfer can be represented by a surface renewal model, obtain the appropriate equation for mass transfer by starting with Tick s second law of diffusion and calculate ... 

When electrically insulated strip or spot electrodes are embedded in a large electrode, and turbulent flow is fully developed, the steady mass-transfer rate gives information about the eddy diffusivity in the viscous sublayer very close to the electrode (see Section VI,C below). The fluctuating rate does not give information about velocity variations, and is markedly affected by the size of the electrode. The longitudinal, circumferential, and time scales of the mass-transfer fluctuations led Hanratty (H2) to postulate a surface renewal model with fixed time intervals based on the median energy frequency. 

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. 

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 dependence of mass transfer coefficient on diffusivity is in agreement with surface renewal model prediction. 

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... 

A theoretical T.pproach to analyze the problem of heat transfer is to develop some form of surface renewal model. The rate of surface renewal is found from the knowledge of energy input per unit mass. Recently Deckwer ( ) has analyzed the problem of heat transfer in bubble columns on the basis of surface renewal model. The present study uses the earlier approach. 

Not only in the two-film model but also in the surface-renewal model, the separate mass-transfer parameters can be determined by chemical methods with a pseudo-first-order regime or an instantaneous regime. 

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 ... 

While the film and surface-renewal theories are based on a simplified physical model of the flow situation at the interface, the boundary layer methods couple the heat and mass transfer equation directly with the momentum balance. These theories thus result in anal3dical solutions that may be considered more accurate in comparison to the film or surface-renewal models. However, to be able to solve the governing equations analytically, only very idealized flow situations can be considered. Alternatively, more realistic functional forms of the local velocity, species concentration and temperature profiles can be postulated while the functions themselves are specified under certain constraints on integral conservation. Prom these integral relationships models for the shear stress (momentum transfer), the conductive heat flux (heat transfer) and the species diffusive flux (mass transfer) can be obtained. 

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. 

L. T. Fan, S. T. Lin, and N. Z. Azer, Surface Renewal Model of Condensation Heat Transfer in Tubes With In-Line Static Mixers, Int. J. Heat Mass Transfer (21) 849-854,1978. 

The enhancement of mass transfer due to chemical reaction depends on the order of the reaction as well as its rate. Order is defined as the sum of all the exponents to which the concentrations in the rate equation are raised. In elementary reactions, this number is equal to the number of molecules involved in the reaction however, this is only true if the correct reaction path has been assumed. Danckwerts presents a review of many cases of importance in gas absorption operations. He compares the results of using the film model and the Higbie and Danckweits surface-renewal models and concludes that, in general, the predictions based on the three models are quite similar. Mass transfer rate equations for a few of the cases encountered in a gas absorption operation are summarized in the following paragraphs, which are based primarily on discussions presented by Danckweits. ... 

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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