Surface Renewal Models

The surface renewal model, like the thin him model, yields a piston, or gas exchange, velocity that can be used to calculate chemical fluxes as previously described. In contrast with the thin him model, however, the surface renewal model predicts that the ratio of piston velocities for two different volatile chemicals depends on the square root of the ratio of their molecular diffusion coefficients (and thus approximately the fourth root of the inverse ratio of their molecular weights). Schwarzenbach et al. (1993) discuss molecular diffusion coefficients in more detail. 

Trichloroethylene has been spilled in a river so that the dissolved concentration is 1 ppb. Given a dimensionless Henry s law constant, H, of 0.4, and a 

Assuming Ca is essentially zero, diffusion through air is not a bottleneck due to the fairly high H, and Dw for TCE is approximately equal to D for propane multiplied by the square root of the inverse of the ratio of the square roots of the molecular weights  

Note that this flux is higher than that obtained in Example 2-6, but in practice the difference might be masked by experimental variability. 

In 1951, Danckwerts proposed the surface renewal model as an extension of the penetration model [4]. 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 of the surface being replaced 

Compared to the film model or the penetration model, the surface renewal approach seems closer to reality in such a case where the surface of liquid in an agitated tank is in contact with the gas phase above, or with the surface of a liquid flowing through an open channel. The values of s are usually unknown, although they could be estimated from the data acquired from carefully planned experiments. As with the penetration model, kL values should vary with diffusivity Do s. 

It can be seen that a theoretical prediction of kL 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 empirical correlations must normally be used for the prediction of individual coefficients of mass transfer. Experimentally obtained values of the exponent on diffusivity are usually between 0.5 and 1.0. 

Compared to the film model or the penetration model, the surface renewal approach seems closer to reality in such a case where the surface of liquid in an 

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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 ... 

Using the Danckwerts surface renewal model, estimate ... 

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. 

Danckwerts, P. V., Kennedy, A. M. and Roberts, D. Chem. Eng. Sci. 18 (1963) 63. Kinetics of CO2 absorption in alkaline solutions. II. Absorption in a packed column and tests of surface renewal models. 

In this surface-renewal model of gas exchange, the gas flux across the air-sea interfece is determined by the frequency at which the slab is replaced or renewed. Various parameterizations have been developed for this model. One example relates the net diffusive flux to the frequency of slab renewal (0) as follows... 

Surface Renewal Model Boundary Layer Model... 

An alternative approach, developed by chemical engineers as well, is the surface renewal model by Higbie (1935) and Danckwerts (1951). It applies to highly turbulent conditions in which new surfaces are continuously formed by breaking waves, by air bubbles entrapped in the water, and by water droplets ejected into the air. Here the interface is described as a diffusive boundary. 

Note With respect to transport time, the film and the surface renewal model are consistent except for the slightly different numerical factor in the denominator. 

It can be shown that if the flux is derived from the surface renewal model (Chapter 19.4), the result is identical with Eq. 20-49. Again, the corresponding transfer velocity v,w (this time given by Eq. 19-59 or 20-20) is enhanced by the factor FHAtot/HA. 

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 model based on the concept of pure limiting film resistance involves the steady-state concept of the heat transfer process and omits the essential unsteady nature of the heat transfer phenomena observed in many gas-solid suspension systems. To take into account the unsteady heat transfer behavior and particle convection in fluidized beds, a surface renewal model can be used. The model accounts for the film resistance adjacent to the heat transfer... 

G. W. Roberts, A Surface Renewal Model for Drying of Polymers during Screw Extrusion , AIChE J., 16, 878 (1970). 

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. 

The two-film model of gas exchange Chemical enhancement The Schmidt number Surface renewal model Micrometeorological models Bubble-mediated gas transfer Laboratory Studies of Air-Water Gas Exchange... 

A systematic study has been carried out in our laboratories over the last decade on the above three phenomena with the objective to design industrial reactors (See Figure 2). Three theoretical and semi-theoretical models have been developed. The heat transfer model, namely the "Surface Renewal Model" has been developed to predict the... 

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