Surface-renewal theory

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  

By definition of a distribution function it follows that the fraction of the surface having been exposed to the gas for a time between t and t + dt is given by (r)dt = se dt. Also, since we are dealing with fractions, the distribution is normalized, so that 

Such an age-distribution function would arise when the rate of replacement of the surface elements is proportional to the number of elements or fraction of the surface having that age, c 

Integration of Eq. 6.4-3, taking Eq. 6.4-2 into account, leads to Eq. 6.4-1. s is a rate coeflBcient for surface element replacement and is the parameter of the model. Consequently, with this expression for (i), the average rate of absorption (Eq. 6.2-5) becomes  

as for the two-film theory, analytical integration of the equations is only possible for a few particular cases, especially since the equations are now partial differential equations with respect to both position and time, in contrast with what was done for the film theory, the instantaneous reaction will be discussed prior to the pseudo-first-order reaction — a more logical sequence to introduce further developments. 

Whereas Higbie assumed a constant renewal time, Dankwert s (1951) extension of the penetration theory employs a wide spectrum of eddy contact times. These eddy-like fluid packets are assumed to remain in contact with the interface for variable times from zero to infinity. He assumed a surface-age distribution function, which skews the contact times to small values it is 

Just as with the penetration theory The s is that one hard-to-find param- 

We will continue with the flux into an infinite slab, replacing with 

we have small surface elements remaining at the interface at a limited period of time. If we assume the interface behaving like a CSTR and recalling the residence time distribution of the CSTR from Chapter 3 as 

For a quickly renewing surface for small T, the surface is infinitely large such that 

Characteristic times for diffusion, reaction, and mass transfer 

From Equations 6.12 and 6.16, it is evident that the characteristic time for diffusion is = DIk i. On the other hand, for a first-order reaction, the characteristic time for reaction is = A , where is the reaction rate constant. Finally, the characteristic time for the mass transfer is 

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If, on the other hand, it is assumed that contact times for the individual fluid elements vary at random, an exponential surface age distribution characterized by a fractional rate of renewal s may be used (34). This approach is caUed surface renewal theory and results in... 

Neither the penetration nor the surface renewal theory can be used to predict mass transfer coefficients directiy because T and s are not normally known. Each suggests, however, that mass transfer coefficients should vary as the square root of the molecular diffusivity, as opposed to the first power suggested by the film theory. 

Simplified Mass-Transfer Theories In certain simple situations, tne mass-transfer coefficients can be calculated from first principles. The film, penetration, and surface-renewal theories are attempts to extend tnese theoretical calculations to more complex sit-... 

Note that both the penetration and the surface-renewal theories predict a square-root dependency on D. Also, it should be recognized that values of the surface-renewal rate s generally are not available, which presents the same problems as do 6 and t in the film and penetration models. 

Pasveer (PI) studied oxygen absorption from air bubbles by water and found that the experimental results could be adequately correlated in terms of a penetration or surface-renewal theory. 

Surface Renewal Theory. The film model for interphase mass transfer envisions a stagnant film of liquid adjacent to the interface. A similar film may also exist on the gas side. These h5q>othetical films act like membranes and cause diffu-sional resistances to mass transfer. The concentration on the gas side of the liquid film is a that on the bulk liquid side is af, and concentrations within the film are governed by one-dimensional, steady-state diffusion ... 

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. 

Surface renewal theory envisions a continuous exchange of material between the bulk fluid and the interface. Eddy diffusion brings material of uniform composition ai to the interface and exposes it to the gas phase for a period of time t. The exposed fluid is then replaced with fresh fluid. Diffusion during the exposure... 

Janssen and Hoogland (J3, J4a) made an extensive study of mass transfer during gas evolution at vertical and horizontal electrodes. Hydrogen, oxygen, and chlorine evolution were visually recorded and mass-transfer rates measured. The mass-transfer rate and its dependence on the current density, that is, the gas evolution rate, were found to depend strongly on the nature of the gas evolved and the pH of the electrolytic solution, and only slightly on the position of the electrode. It was concluded that the rate of flow of solution in a thin layer near the electrode, much smaller than the bubble diameter, determines the mass-transfer rate. This flow is affected in turn by the incidence and frequency of bubble formation and detachment. However, in this study the mass-transfer rates could not be correlated with the square root of the free-bubble diameter as in the surface renewal theory proposed by Ibl (18). 

Surface renewal theory (King, 1966) the theory describes the replacement of a surface liquid film by the action of eddies that move between the bulk water phase and the surface film. The surface renewal rate thereby determines the exchange between the surface and the bulk water. 

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... 

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. 

Figure 8.10. Conceptual sketches of surface renewal on the water side of the interface (top) and of the concentration boundary layer thickness at one location over time (bottom), as assumed by the surface renewal theory.

McCready et al., 1986). The surface renewal theory can be made to fit the transfer data at fluid-fluid interfaces. The exception to this is bubbles with a diameter less than approximately 0.5 mm. Even though there is a fluid on both sides, surface tension causes these small bubbles to behave as though they have a solid-fluid interface. There is also some debate about this 1 /2 power relationship at free surfaces exposed to low shear, such as wind-wave flumes at low wind velocity (Jahne et al., 1987) and tanks with surfactants and low turbulence generation (Asher et al., 1996). The difficulty is that these results are influenced by the small facilities used to measure Kl, where surfactants wiU be more able to restrict free-surface turbulence and the impact on field scale gas transfer has not been demonstrated. 

The penetration and surface renewal theories started out as conceptual, in that they were visualized to occur as such by individual theorists. These theories appeared to work successfully for a free interface, such as the air-water interface, but not for a fixed interface, such as solid-water. Now, the explanation is before us in equation (8.64). Surface renewal is a fairly accurate representation of Hanratty s jS at a free surface, and therefore can be seen to give representative results. It is Hanratty s p that we really should be measuring, and it happens that the mean surface renewal rate is a good representation of Hanratty s jS at a free surface. 

Of course, the concentration boundary layer does not grow indefinitely, but the penetration or the surface renewal theories give us a means of dealing with that fact. If we are going to use the penetration theory, t 4. If we are going to use the surface renewal theory, t 1/nr. With the penetration theory, the transfer enhancement ratio, Ke/Kl, is then given as... 

A similar substitution will give the transfer enhancement ratio for the surface renewal theory. 

What are the boundary conditions in the environment to which the stagnant film, penetration, surface renewal theories, and laminar boundary layer analogy should be applied Briefly explain why. 

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 mass transfer rates for the case when d > d can easily be obtained from Eqs. 9 or 12 (see [48]). Using the surface renewal theory this case is not relevant because the boundary layer thickness is here considered to be infinite. 

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). 

Several workers (Kolbel et al. [40, 41], Deckwer et al. [17], Michael and Reicheit [42]) have investigated the heat transfer in BSCR versus solid concentration and particle diameters. Deckwer et al. [17] applied Kolmogoroff s theory of isotropic turbulence in combination with the surface renewal theory of Higbie [43] and suggested the following expression for the heat transfer coefficient in the Fischer-Tropsch synthesis in BSCR ... 

Several different mechanisms have been proposed to provide a basis for a theory of interphase mass transfer. The three best known are the two-film theory, the penetration theory, and the surface renewal theory. 

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... 

Developing correlations to describe mass transfer in rotating packed beds has proven to be a challenge. Penetration theory (31), film-flow theory (32), and modified surface-renewal theory (12) are some examples of leveraging previous work... 

Liquid-solid mass transfer has also been studied, on a limited basis. Application to systems with catalytic surfaces or electrodes would benefit from such studies. The theoretical equations have been proposed based on film-flow theory (32) and surface-renewal theory (39). Using an electrochemical cell with rotating screen disks, liquid-solid mass transfer was shown to increase with rotor speed and increased spacing between disks but to decrease with the addition of more disks (39). Water flow over naphthalene pellets provided 4-6 times higher volumetric mass transfer coefficients compared to gravity flow and similar superficial liquid velocities (17). 

Chang and Rochelle [12-15] investigated some aspects of SO2 absorption in an agitated cell filled with aqueous solution or suspension, and compared their measurements with those of other authors with simulated absorbed mass flows, based on enhancement factors from the film and surface renewal theory. The results showed that mass transfer, when corrected by an enhancement factor from the surface renewal theory, agreed clearly with the measurements. 

Several theories have been developed to describe the rate of interphase mass transfer. These include film theory, boundary layer theory, penetration theory, and surface renewal theory. In this chapter we will review the first two, along with an overview of empirical correlations that are used to describe mass transfer. A more thorough overview of mass transfer theories can be found in Bird, Stewart and Lightfoot [48], Clark [49], Logan [50], and Weber and DiGiano [51]. 

Ravetkar and Kale (1981) have shown that Higbie s surface-renewal theory can be successfully employed for the calculation of the liquid-side mass-transfer coefficients for Newtonian and non-Newtonian liquids as... 

E. Systems for feL and aL Determinational Using Surface Renewal Theory ... 

Mass-Transfer Coefficient Denoted by kc, kx, Kx, and so on, the mass-transfer coefficient is the ratio of the flux to a concentration (or composition) difference. These coefficients generally represent rates of transfer that are much greater than those that occur by diffusion alone, as a result of convection or turbulence at the interface where mass transfer occurs. There exist several principles that relate that coefficient to the diffusivity and other fluid properties and to the intensity of motion and geometry. Examples that are outlined later are the film theory, the surface renewal theory and the penetration theory, all of which pertain to idealized cases. For many situations of practical interest like investigating the flow inside tubes and over flat surfaces as well as measuring external flow through banks of tubes, in fixed beds of particles, and the like, correlations have been developed that follow the same forms as the above theories. Examples of these are provided in the subsequent section on mass-transfer coefficient correlations. 

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... 

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