Mass transfer film theory for

Explain the basic concepts underlying the two-film theory for mass transfer across a phase boundary and obtain an expression for film thickness. 

This limiting case is sometimes referred to as the two-film theory for mass transfer. The resistances to mass transfer in the vapor and liquid phases are concentrated in the vapor and liquid films. Experimental evidence presented by Tung and Drickamer34 and Emmett and Pigford13 suggest that kt is very large except at very high rates of mass transfer. Thus, for all practical purposes,... 

Fig. 9.1-1. The film theory for mass transfer. In this model, the interfacial region is idealized as a hypothetical film or unstirred layer. Mass transfer involves diffusion across this thin film. Note that the constant value cio implies no resistance to mass transfer in the gas.

A relation between dy/dZ and (Ay)/ may be obtained on the basis of the two-film theory of mass transfer. For the vapour film, Fick s law, Volume 1, Chapter 10, gives ... 

As Sherwood and Pigford(3) point out, the use of spray towers, packed towers or mechanical columns enables continuous countercurrent extraction to be obtained in a similar manner to that in gas absorption or distillation. Applying the two-film theory of mass transfer, explained in detail in Volume 1, Chapter 10, the concentration gradients for transfer to a desired solute from a raffinate to an extract phase are as shown in Figure 13.19, which is similar to Figure 12.1 for gas absorption. 

The simplest problems are those in which the diffusion process is independent of the time. The solutions to these problems are important in the film theories of mass transfer and in steady state experiments for measuring diffusion and self-diffusion. 

The individual mass transfer and reaction steps occurring in a gas-liquid-solid reactor may be distinguished as shown in Fig. 4.15. As in the case of gas-liquid reactors, the description will be based on the film theory of mass transfer. For simplicity, the gas phase will be considered to consist of just the pure reactant A, with a second reactant B present in the liquid phase only. The case of hydro-desulphurisation by hydrogen (reactant A) reacting with an involatile sulphur compound (reactant B) can be taken as an illustration, applicable up to the stage where the product H2S starts to build up in the gas phase. (If the gas phase were not pure reactant, an additional gas-film resistance would need to be introduced, but for most three-phase reactors gas-film resistance, if not negligible, is likely to be small compared with the other resistances involved.) The reaction proceeds as follows ... 

As the film thickness 8 is not normally known, the mass transfer coefficient / , cannot be calculated from this equation. However the values for the cases used most often in practice can be found from the relevant literature (i.e. [1.23] to [1.26]) which then allows the film thickness to be approximated using (1.189). In film theory the mass transfer coefficient / for vanishing convection flux h( — 0 is proportional to the diffusion coefficient D. 

What are the general principles underlying the two-fihn penetration and film-penetration theories for mass transfer across a phase boundary Give the basic differential equations which have to be solved for these theories with the appropriate boundary conditions. 

The treatment that follows is based essentially on the analysis of gas-liquid reactions by Doraiswamy and Sharma (1984), Danckwerts (1970), Astarita (1967), Hikita and Asai (1964), Shah (1979), Shah et al. (1982), Ramachandran and Chaudhari (1983), Bisio and Kabel (1985), Deckwer (1985), and Joshi et al. (1988). A recent book by Kastanek et al. (1993) provides an extensive review of reactors for gas-liquid reactions. The film theory of mass transfer is the simplest and most extensively used, despite its many limitations. Our treatment too will be based on this theory (see Chapter 4), although reference will be made to the penetration theory in some cases. 

Calculating fiime scrubber efficiency requires an understanding of mass transfer principles. The reader is referred to Chapter 3 for a more detailed discussion of the two-film theory of mass transfer. 

To produce mass transfer of solute between the two solvents, a driving force is necessary. This driving force usually is expressed in terms of the concentrations of solute in the two phases and the departure from equilibrium values. The two-film theory of mass transfer developed in Chapter 3 for gas absorption also can be applied to liquid extraction. The rate of mass transfer of solute C from the feed (solvent A) to the extract (solvent B) is ... 

The presentation so far has been carried out in the context of a film theory of mass transfer (see Section 3.1.4) and steady state conditions. There is considerable literature on other models of mass transfer, e.g. surface renewal theory. Further, unsteady state analyses exist for a number of cases. Detailed treatments are available in Danckwerts (1970) and Sherwood et dL (1975). 

Fig. 9.2-1. The penetration theory for mass transfer. Here, the interfacial region is imagined to be a very thick film continuously generated by fiow. Mass transfer now involves diffusion into this film. In this and other theories, the interfacial concentration in the liquid is assumed to be in equilibrium with that in the gas.

Fig. 9.2-2. The surface-renewal theory for mass transfer. This approach tries to apply the mathematics of the penetration theory to a more plausible physical picture. The liquid is pictured as two regions, a large well-mixed bulk and an interfacial region that is renewed so fast that it behaves as a thick film. The surface renewal is caused by liquid flow.

Other Models for Mass Transfer. In contrast to the film theory, other approaches assume that transfer of material does not occur by steady-state diffusion. Rather there are large fluid motions which constantiy bring fresh masses of bulk material into direct contact with the interface. According to the penetration theory (33), diffusion proceeds from the interface into the particular element of fluid in contact with the interface. This is an unsteady state, transient process where the rate decreases with time. After a while, the element is replaced by a fresh one brought to the interface by the relative movements of gas and Uquid, and the process is repeated. In order to evaluate a constant average contact time T for the individual fluid elements is assumed (33). This leads to relations such as... 

Note that the transfer rate equation is based on an overall concentration driving force, (X-X ) and overall mass transfer coefficient, Kl. The two-film theory for interfacial mass transfer shows that the overall mass transfer coefficient, Kl, based on the L-phase is related to the individual film coefficients for the L and G-phase films, kL and ko, respectively by the relationship... 

The experiments were conducted at four different temperatures for each gas. At each temperature experiments were performed at different pressures. A total of 14 and 11 experiments were performed for methane and ethane respectively. Based on crystallization theory, and the two film theory for gas-liquid mass transfer Englezos et al. (1987) formulated five differential equations to describe the kinetics of hydrate formation in the vessel and the associate mass transfer rates. The governing ODEs are given next. 

By substituting the well-known Blasius relation for the friction factor, Eq. (45) in Table VII results. Van Shaw et al. (V2) tested this relation by limiting-current measurements on short pipe sections, and found that the Re and (L/d) dependences were in accord with theory. The mass-transfer rates obtained averaged 7% lower than predicted, but in a later publication this was traced to incorrect flow rate calibration. Iribame et al. (110) showed that the Leveque relation is also valid for turbulent mass transfer in falling films, as long as the developing mass-transfer condition is fulfilled (generally expressed as L+ < 103) while Re > 103. The fundamental importance of the Leveque equation for the interpretation of microelectrode measurements is discussed at an earlier point. 

The transport process, according to the two-film theory, of a volatile component across the air-water interface is depicted in Figure 4.3. The figure illustrates a concept that concentration gradients in both phases exist and that the total resistance for mass transfer is the sum of the resistance in each phase. 

Finally, a number of experimental studies have been conducted in a pressure range where the polymeric solution could boil. The vapor bubbles thus created would provide a much larger surface area for mass transfer than the surface area of the wiped film alone. And therefore, for fixed values of the diffusivity and the driving force, predicted values for mass transfer rates would be substantially lower than the measured values. Conversely, for a fixed mass transfer rate and driving force, use of the wiped film surface area alone would require unusually high values of the diffusivity in order to obtain agreement between theory and experiment. 

This chapter will first provide some basics on ozone mass transfer, including theoretical background on the (two-) film theory of gas absorption and the definition of over-all mass transfer coefficients KLa (Section B 3.1) as well as an overview of the main parameters of influence (Section B 3.2). Empirical correction factors for mass transfer coefficients will also be presented in Section B 3.2. These basics will be followed by a description of the common methods for the determination of ozone mass transfer coefficients (Section B 3.3) including practical advice for the performance of the appropriate experiments. Emphasis is laid on the design of the experiments so that true mass transfer coefficients are obtained. 

The resistance in each phase is made up of two parts the diffusional resistance in the laminar film and the resistance in the bulk fluid. All current theories on mass transfer, i. e. film, penetration, and surface renewal assume that the resistance in the bulk fluid is negligible and the major resistance occurs in the laminar films on either side of the interface (Figure 3-2). Fick s law of diffusion forms the basis for these theories proposed to describe mass transfer through this laminar film to the phase boundary. 

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