Danckwerts theory

The reaction/mass-transfer technique is based on Danckwerts theory of mass transfer accompanied by a fast pseudo first-order reaction (10) ... 

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. 

All three of these proposals give the mass transfer rate N A directly proportional to the concentration difference (CAi — CAL) so that they do not directly enable a decision to be made between the theories. However, in the Higbie-Danckwerts theory N A a s/Dj whereas NA film theory. Danckwerts applied this theory to the problem of absorption coupled with chemical reaction but, although in this case the three proposals give somewhat different results, it has not been possible to distinguish between them. 

P. V. Danckwerts, Theory of Mixtures and Mixing, Research (London), 6, 355-361 (1953) see also P. V. Danckwerts, General Aspects of Mixtures and Mixing, paper presented at the Symposium on Mixing of Thick Liquids, Pastes, and Slurries, British Society of Rheology, London, September 1953 cf. N. Wookey, Mixing of Thick Liquids, Pastes and Slurries , Nature, 172, 846-847 (1953). 

A number of successful devices have been in use for finding mass-transfer coefficients, some of which are sketched in Fig. 23-29, and all of which have known or adjustable interfacial areas. Such laboratoiy testing is reviewed, for example, by Danckwerts (Ga.s-Liquid Reac-tion.s, McGraw-Hih, 1970) and Charpentier (in Ginetto and Silveston, eds., Multiphase Chemical Reactor Theory, De.sign, Scaleup, Hemisphere, 1986). 

In calculating Ihe mass transfer rate from the penetration theory, two models for the age distribution of the surface elements are commonly used — those due to Higbie and to Danckwerts, Explain the difference between the two models and give examples of situations in which each of them would be appropriate. 

In the Danckwerts model, it is assumed that elements of the surface have an age distribution ranging from zero to infinity. Obtain the age distribution function for this model and apply it to obtain the average, mass Iransfer coefficient at the surface, given that from the penetration theory the mass transfer coefficient for surface of age t is VlD/(7rt, where D is the diffusivity. 

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. 

Penetration theories (Higbie, 1935 Danckwerts, 1951 Dobbins, 1956) according to the penetration theory, diffusion of gases takes place into elements of water transported by turbulence to the surface. 

One of the early successes of the CRE approach was to show that RTD theory suffices to treat the special case of non-interacting fluid elements (Danckwerts 1958). For this case, each fluid element behaves as a batch reactor. 

An alternative theory described in detail in Volume 1, Chapter 10, has been put forward by Higbie 2 , and later extended by Danckwerts and Danckwerts and Kennedy14 in which the liquid surface is considered to be composed of a large number of small elements each of which is exposed to the gas phase for an interval of time, after which they are replaced by fresh elements arising from the bulk of the liquid. 

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. 

There are several theories concerned with mass transfer across a phase boundary. One of the most widely used is Whitman s two-film theory in which the resistance to transfer in each phase is regarded as being located in two thin films, one on each side of the interface. The concentration gradients are assumed to be linear in each of these layers and zero elsewhere while at the interface itself, equilibrium conditions exist (Fig. 5). Other important theories are Higbie s penetration theory and the theory of surface renewal due to Danckwerts. All lead to the conclusion that, in... 

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... 

For plane, unstirred cells, the theoretical equations are (21) and (25) and for plane stirred cells there are the theories of Lewis and Whitman, Kishinevskii, Danckwerts, and Levich, and Eqs. (14) and (16) of the present writer. In particular, if Eqs. (7) and (11) represent processes... 

Developed by Danckwerts (1951), the surface renewal theory states that 4 is not constant, but the renewal of the interface by turbulent eddies will have a period that is represented by a Gaussian distribution with respect to time. Then, KlOt Kg are given by the equation... 

Instead of determining 4, in equation (8.33), we must determine r in equation (8.34). Although the difference between Higbie s penetration theory and Danckwerts surface renewal theory is not great, the fact that a statistical renewal period would have a similar result to a fixed renewal period brought much credibility to Higbie s penetration theory. Equation (8.34) is probably the most used to date, where r is a quantity that must be determined from the analysis of experimental data. 

One other measurement technique that has been used to measure Kl over a shorter time period, and is thus more responsive to changes in wind velocity, is the controlled flux technique (Haupecker et al., 1995). This technique uses radiated energy that is turned into heat within a few microns under the water surface as a proxy tracer. The rate at which this heat diffuses into the water column is related to the liquid film coefficient for heat, and, through the Prandtl-Schmidt number analogy, for mass as well. One problem is that a theory for heat/mass transfer is required, and Danckwert s surface renewal theory may not apply to the low Prandtl numbers of heat transfer (Atmane et al., 2004). The controlled flux technique is close to being viable for short-period field measurements of the liquid film coefficient. 

The theories vary in the assumptions and boundary conditions used to integrate Fick s law, but all predict the film mass transfer coefficient is proportional to some power of the molecular diffusion coefficient D", with n varying from 0.5 to 1. In the film theory, the concentration gradient is assumed to be at steady state and linear, (Figure 3-2) (Nernst, 1904 Lewis and Whitman, 1924). However, the time of exposure of a fluid to mass transfer may be so short that the steady state gradient of the film theory does not have time to develop. The penetration theory was proposed to account for a limited, but constant time that fluid elements are exposed to mass transfer at the surface (Higbie, 1935). The surface renewal theory brings in a modification to allow the time of exposure to vary (Danckwerts, 1951). 

Surface renewal theory (Danckwerts, 1951) proposes that there is an infinite range of ages for elements of the surface and the surface age distribution function (t) can be expressed as... 

Thus, we recover the Danckwerts model only if no distinction is made between the cup-mixing and spatial average concentrations (with this assumption, the effective axial dispersion coefficient is given by the Taylor-Aris theory). This derivation also shows that the concept of an effective axial dispersion coefficient and lumping the macro- and micromixing effects into one parameter is valid only at steady-state, constant inlet conditions and when the deviation from plug flow is small. [Remark Even with all these constraints, the error in the model because of the assumption (cj) — cym is of the same order of magnitude as the dispersion effect ]... 

Danckwerts [Ind. Eng. Chem., 42,1460(1951)] proposed an extension of the penetration theory, called the surface renewal theory, 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 Danckwerts model gives... 

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. 

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