Renewal models

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. 

This packet renewal model has been widely accepted and in the years since 1955 many researchers have proposed various modifications in attempts to improve the Mickley-Fairbanks representation. Several of these modifications dealt with the details of the thermal transport process between the heat transfer surface and the particle packet. The original Mickley-Fairbanks model treated the packet as a pseudo-homogeneous medium with a constant effective thermal conductivity, suggesting that... 

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... 

The problem of absorption accompanied by a chemical reaction in a liquid film flowing along a vertical wall was often treated by using the framework of the penetration theory [28 - 32]. This theory also constitutes the starting point of the renewal models of turbulence [31,33,34]. 

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. 

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. 

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