Renewal models Danckwerts

Using the Danckwerts surface renewal model, estimate ... 

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. 

A more realistic description that avoids the countercurrent flow in the region between two successive paths is represented in Fig. 6b, but we will assume that the regions 2 have a much smaller extent than the regions 1. It is appropriate to cite here references [59,60] in which the Danckwerts renewal idea was used to describe the turbulent boundary layer near a wall, as well as the 1969 paper of Black [61] in which a model similar to that of Ruckenstein [58] is considered. 

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... 

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. 

The so-called rate-based stage model presents a different way to the modeling of separation processes, by directly considering actual mass and heat transfer rates (Seader, 1989 Taylor and Krishna, 1993). A number of models fall into the general framework of the rate-based stage. In most cases, the film (Lewis and Whitman, 1924) or penetration and surface renewal (Higbie, 1935 Danckwerts, 1951) models find application, whereas the necessary model parameters are estimated by means of correlations. In this respect, the film model appears advantageous due to numerous correlation data available in the literature (see, e.g., Billet and Schultes, 1999). 

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... 

Therefore, from a comparison with Eq. (29), the three models give nearly comparable results in this case. Also, when = D, the instantaneous enhancement factor , has the same form for the film and the surface-renewal models (D2). Recently De Coursey (DIO) derived an approximate solution for the Danckwerts model, given in the next section, which can help considerably when this model is used for design. Therefore, even though analytical expressions for the average rate of absorption based on the three models look very different, nevertheless the three will give the same value of the enhancement factor to within a few percent for all values of Ha between 0.1 and >. 

The basis for the Danckwerts (1951) surface renewal model is the idea that the chance of... 

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 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. 

Traditional turbulence-diffusion models (based on the boundary layer adjoining a solid wall) imply that n = 2/3, but a value n = 1/2 is appropriate for a boundary layer adjoining a free surface (Jahne and Haussecker 1998). The appropriate value of n depends on the wind stress and the surfactant loading of the surface. Soloviev and Schluessel (1994) have described a procedure for estimating gas transfer velocities from measurements of heat transport, assuming that transport is adequately described by the classical (Danckwerts) surface renewal model. The key relationship can be written in the form ... 

In Figure 8 (right) the transfer coefficients are plotted as a function of the waterside friction velocity. The theoretical curves for a smooth surface based on a diffusion model (Deacon 1977) and for a rough surface based on a surface renewal model (Danckwerts 1970) are also included. The theoretical curve for a smooth surface (no waves were visually observed) is in good agreement with the measurements when a surfactant was present. For a clean interface, the transfer coefficient follows the theoretical curve for a surface renewal model with a wavy surface. 

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 ... 

The surface renewal models only consider the liquid phase. In Sec. 6.3 on the film model the resistances of both gas and liquid phase were combined into one single expression like Eq. 6.3.b-5. The same can be done here Danckwerts [24] has shown that in most cases the surface renewal models combined with a gas side resistance lead to the same rules for the addition of resistances as the two-film theory. 

Another possibility is to use a pseudo-first-order reaction, rather slow so that little A reacts in the film, yet sufficiently fast to make Cm zero. This approach has been used by Danckwerts et al. [36] who interpreted their results in terms of the surface renewal theory. The system they investigated was CO2 absorption in C03 /HC03"-buffers of different compositions. This is a pseudo-first-order reaction for which, the surface renewal model leads to the following rate of absorption, Eq. 6.4.b-5 N A = A,y/D (k -1- s)Cm- Danckwerts et al. plotted (Nji A versus the different values of k corresponding to the different com-... 

It is now justifiable to solve this equation using the boundary conditions of the simplified physical absorption models (1,2). Solution of Eqn (2) then enables us to calculate the absorption rate at a particular "point" in the absorber. These results are usually expressed in terms of an enhancement factor, i.e. a factor by which the rate of absorption is increased by the chemical reaction. It is well known that this enhancement factor differs little in value whether film or the Higbie or Danckwerts surface renewal models are used as the basis of calculation. Figure (2) shows the typical representation of the effect of chemical reaction for a second order (r=k2A B) irreversible reaction. With the exception of Region IV, all regions are amenable to analytical solutions. In fact, the enhancement factor predic-... 

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. ... 

A graphical representation of these equations is given in Fig. 6.4-14. van Kievelen and Hoftijzer originally developed their correlation only for irreversible second-order reactions (first-order in each reactant) and for equal difiusivities of the two reactants. Danckwetts pointed out that the results also are applicable to the case where is not equal to Dg. Decoursey developed an approximate solution for absorption with irreversible second-order reaction based on the Danckwerts surface-renewal model. The resulting expression, which is somewhat easier to use than the van Krevelen-HofUJzer approach, is... 

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... 

Average rates of mass transfer can be obtained, as previously, by using either the Higbie or the Danckwerts model for surface renewal. 

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. 

In an attempt to test the surface renewal theory of gas absorption, Danckwerts and Kennedy measured the transient rate of absorption of carbon dioxide into various solutions by means of a rotating drum which carried a film of liquid through the gas. Results so obtained were compared with those for absorption in a packed column and it was shown that exposure times of at least one second were required to give a strict comparison this was longer than could be obtained with the rotating drum. Roberts and Danckwerts therefore used a wetted-wall column to extend the times of contact up to 1.3 s. The column was carefully designed to eliminate entry and exit effects and the formation of ripples. The experimental results and conclusions are reported by Danckwerts, Kennedy, and Roberts110 who showed that they could be used, on the basis of the penetration theory model, to predict the performance of a packed column to within about 10 per cent. 

Alternatives to the film theory are also in use. These models [Higbie (1935) Danckwerts (1950, 1955)] view that the liquid at the interface is continually washed away and replaced by fresh fluid from the main body of the liquid, and that this is the means of mass transport. These unsteady-state surface renewal theories all predict... 

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