Mass penetration model

In 1963 and in 1965, Huang and Kuo (H18, H19) applied the film penetration model to the mechanism of simultaneous mass transfer and chemical reaction. 

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. 

As an example, it may be supposed that in phase 1 there is a constant finite resistance to mass transfer which can in effect be represented as a resistance in a laminar film, and in phase 2 the penetration model is applicable. Immediately after surface renewal has taken place, the mass transfer resistance in phase 2 will be negligible and therefore the whole of the concentration driving force will lie across the film in phase 1. The interface compositions will therefore correspond to the bulk value in phase 2 (the penetration phase). As the time of exposure increases, the resistance to mass transfer in phase 2 will progressively increase and an increasing proportion of the total driving force will lie across this phase. Thus the interface composition, initially determined by the bulk composition in phase 2 (the penetration phase) will progressively approach the bulk composition in phase 1 as the time of exposure increases. 

HARRIOTT 25 suggested that, as a result of the effects of interfaeial tension, the layers of fluid in the immediate vicinity of the interface would frequently be unaffected by the mixing process postulated in the penetration theory. There would then be a thin laminar layer unaffected by the mixing process and offering a constant resistance to mass transfer. The overall resistance may be calculated in a manner similar to that used in the previous section where the total resistance to transfer was made up of two components—a Him resistance in one phase and a penetration model resistance in the other. It is necessary in equation 10.132 to put the Henry s law constant equal to unity and the diffusivity Df in the film equal to that in the remainder of the fluid D. The driving force is then CAi — CAo in place of C Ao — JPCAo, and the mass transfer rate at time t is given for a film thickness L by ... 

In a process where mass transfer takes place across a phase boundary, the same theoretical approach can be applied to each of the phases, though it does not follow that the same theory is best applied to both phases. For example, the film model might be applicable to one phase and the penetration model to the other. This problem is discussed in the previous section. 

It may be assumed that the penetration model may be used to represent the mass transfer process. The depth of penetration is small compared with the radius of the droplets and the effects of surface curvature may he neglected. From the penetration theory, the concentration C, at a depth y below the surface at time r is given by ... 

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. 

An interesting implication of the Toth-Freeze-Witherspoon model is the deep penetration of surface water into the basin over geologic time. This has been used to derive a geochemical mass balance model for the mixing of surface water and diagenetically modified sea water in the Western Canada sedimentary basin using deuterium as a tracer (204). 

Toor, R. L. and Marchello, J. M. (1958). Film-Penetration Model for Mass and Heat Transfer. AlChE J., 4,97. 

The removal of the acid components H2S and CO2 from gases by means of alkanolamine solutions is a well-established process. The description of the H2S and CO2 mass transfer fluxes in this process, however, is very complicated due to reversible and, moreover, interactive liquid-phase reactions hence the relevant penetration model based equations cannot be solved analytically [6], Recently we, therefore, developed a numerical technique in order to calculate H2S and CO2 mass transfer rates from the model equations [6]. 

In this investigation we carried out experiments with simultaneous absorption of H2S and CO2 into aqueous 2.0 M diisopropanol-amine (DIPA) solutions at 25 °C. The results are evaluated by means of our mathematical mass transfer model both in penetration and film theory form. The latter version has been derived from the penetration theory mass transfer model [5],... 

The mass transfer model. In our previous work [6] the mass transfer model equations and their mathematical treatment have been described extensively. The relevant differential equations, describing the process of liquid-phase diffusion and simultaneous reactions of the species according to the penetration theory, are summarized in table 1. Recently we derived from this penetration theory description a film model version, which is incorporated in the evaluation of the experimental results. Details on the film model version are given elsewhere [5]. 

Penetration theory equations for the mass transfer model (boundary conditions as usual in penetration theory [6 ]). ... 

If the amine depletion in the penetration zone in a simultaneous absorption situation is negligible, the mass transfer fluxes are independent of each other and the respective enhancement factors may be obtained easily from analytical solutions of single gas mass transfer models. 

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... 

The fundamental principles of the gas-to-liquid mass transfer were concisely presented. The basic mass transfer mechanisms described in the three major mass transfer models the film theory, the penetration theory, and the surface renewal theory are of help in explaining the mass transport process between the gas phase and the liquid phase. Using these theories, the controlling factors of the mass transfer process can be identified and manipulated to improve the performance of the unit operations utilizing the gas-to-liquid mass transfer process. The relevant unit operations, namely gas absorption column, three-phase fluidized bed reactor, airlift reactor, liquid-gas bubble reactor, and trickled bed reactor were reviewed in this entry. 

In Chapter 7 we define mass transfer coefficients for binary and multicomponent systems. In subsequent chapters we develop mass transfer models to determine these coefficients. Many different models have been proposed over the years. The oldest and simplest model is the film model this is the most useful model for describing multicomponent mass transfer (Chapter 8). Empirical methods are also considered. Following our discussions of film theory, we describe the so-called surface renewal or penetration models of mass transfer (Chapter 9) and go on to develop turbulent eddy diffusivity based models (Chapter 10). Simultaneous mass and energy transport is considered in Chapter 11. 

The calculation of k using Eqs. 9.2.11 and 9.2.12 requires a priori estimation of the exposure time or the surface renewal rate s. In some cases this is possible. For bubbles rising in a liquid the exposure time is the time the bubble takes to rise its own diameter. In other words, the jacket of the bubble is renewed every time it moves a diameter. If we consider the flow of a liquid over a packing, when the liquid film is mixed at the junction between the packing elements, then is the time for the liquid to flow over a packing element. For flow of liquid in laminar jets and in thin films, the exposure time is known but in these cases it may be important to take into account the distribution of velocities along the interface. In the penetration model, this velocity profile is assumed to be flat (i.e., plug flow). For gas-liquid mass transfer in stirred vessels, the renewal frequency in the Danckwerts model s may be related to the speed of rotation (see Sherwood et al. 1975). 

Equations 9.3.15 and 9.3.16 represent an exact analytical solution of the multicomponent penetration model. For two component systems, these results reduce to Eqs. 92.1. Unfortunately, the above results are of little practical use for computing the diffusion fluxes because they require an a priori knowledge of the composition profiles (cf. Section 8.3.5). Thus, a degree of trial and error over and above that normally encountered in multicomponent mass transfer calculations enters into their use. Indeed, Olivera-Fuentes and Pasquel-Guerra did not perform any numerical computations with this method and resorted to a numerical integration technique. 

Toor (1964) and Stewart and Prober (1964) did not use the method presented above they used the method described in Chapter 5. For the multicomponent penetration model, the following expression for the matrix of mass transfer coefficients is obtained (cf. Section 8.4.2) ... 

This is the value used by Krishna (1981a) in his simulations of Modine s experiments. This value is consistent with a penetration model of mass transfer in the liquid phase with a contact time of 0.065 s or a surface renewal frequency of 0.25 s The molar density of the liquid phase has been estimated as... 

The penetration model is used for the mass transfer coefficient in the liquid phase... 

The binary mass transfer coefficients for the liquid phase may be evaluated with a penetration model... 

The penetration model is used for the mass transfer coefficients in the liquid phase the contact time being the time required for the bubble to rise one diameter... 

The penetration model (Eq. 9.2.11) is used to predict the liquid-phase mass transfer coefficients with the exposure time assumed to be the time required for the liquid to flow between corrugations (a distance equal to the channel side)... 

Extend the film-penetration model of mass transfer developed by Toor and Marchello (1958) to multicomponent mixtures. See also, Krishna (1978a). 

A droplet containing a mixture of acetone(l)-benzene(2)-methanol(3) has a diameter of 8 mm and attains a velocity of 0.1 m/s in a sieve tray extraction column when it is dispersed in a continuous hydrocarbon phase. Use the penetration model to estimate the matrix of low flux mass transfer coefficients [A ] inside the droplet. 

---