Theories of Mass Transfer

Lewis and Whitman (1924) proposed that this resistance to mass transfer across an interface is the sum of the resistances in each phase. They called this concept the two-film theory. As Treybal (1968) pointed out, their two-film theory does not depend on which model is used to describe the mass transfer in each phase, therefore, the two-resistance theory would be a more appropriate name. It would also cause less confusion, since the names film theory (mass transfer in one phase) and two-film theory (mass transfer between 

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Other correlations based partially on theoretical considerations but made to fit existing data also exist (71—75). A number of researchers have also attempted to separate from a by measuring the latter, sometimes in terms of the wetted area (76—78). Finally, a number of correlations for the mass transfer coefficient itself exist. These ate based on a mote fundamental theory of mass transfer in packed columns (79—82). Although certain predictions were verified by experimental evidence, these models often cannot serve as design basis because the equations contain the interfacial area as an independent variable. 

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

On the. assumptions involved in the penetration theory of mass transfer across a phase boundary, the concentration Ca of a solute A at a depth v below the interface at a time l after the formation of the interlace is given by ... 

When the half-life time of reaction and the half-life time of micromixing in the absence of chemical reaction are of the same order or the former is less than the latter, the role of micromixing may become crucial. For instance, nitration or bromination of resorcinol, even when the ratio of moles of resorcinol to moles of bromine is high, can lead to predominantly disubstituted product contrary to the general belief. In such cases, in many respects, the theory of coupling between reaction and micromixing has parallels with the formalism of theory of mass transfer with chemical reaction (Bourne, 1983). 

The basic theory of mass transfer to a RHSE is similar to that of a RDE. In laminar flow, the limiting current densities on both electrodes are proportional to the square-root of rotational speed they differ only in the numerical values of a proportional constant in the mass transfer equations. Thus, the methods of application of a RHSE for electrochemical studies are identical to those of the RDE. The basic procedure involves a potential sweep measurement to determine a series of current density vs. electrode potential curves at various rotational speeds. The portion of the curves in the limiting current regime where the current is independent of the potential, may be used to determine the diffusivity or concentration of a diffusing ion in the electrolyte. The current-potential curves below the limiting current potentials are used for evaluating kinetic information of the electrode reaction. 

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. 

It is generally agreed that mass transfer coefficients are only correlated for negligibly small convectional motion of the transitional component, which is vertical to the interface. However, when the mass transfer is mutual and equimolar, no such convections normal to the interface result otherwise the transfer coefficient and the driving force must be corrected with the aid of theories of mass transfer [18]. The transitional rates and, accordingly, convectional flow rates normal to the interface are only low for the extraction process, so that the uncorrected Eq. (9.31) may be used. 

The eddy diffusion coefficients that we introduced in Chapter 5 were steady quantities, using mean turbulence quantities (e.g., the temporal mean of u C). This temporal mean character of eddy diffusion coefficients can be misleading in determining the thickness of a diffusive boundary layer because of the importance of unsteady characteristics. We will review some conceptual theories of mass transfer that have been put forward to describe the interaction of the diffusive boundary layer and turbulence. 

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. 

Continuous changes in compositions of phases flowing in contact with each other are characteristic of packed towers, spray or wetted wall columns, and some novel equipment such as the FHGEE contactor (Fig. 13.14). The theory of mass transfer between phases and separation of mixtures under such conditions is based on a two-film theory. The concept is illustrated in Figure 13.15(a). 

The absorption of a gas by a liquid with simultaneous reaction in the liquid phase is the most important case. There are several theories of mass transfer between two fluid phases (see Volume 1, Chapter 10 Volume 2, Chapter 12), but for the purpose of illustration the film theory will be used here. Results from the possibly more realistic penetration theory are similar numerically, although more complicated in their mathematical form0,4. ... 

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 ... 

The kinetic equations for the volume phase of the solid body are equations of the diffusion type (63). Much attention has been given to them in the literature [154,155], therefore here will be reminded only those aspects of the theory of mass transfer for which the lattice-gas model has been used. These are problems involved in the construction of expressions for the diffusion the coefficients and boundary conditions of the diffusion equations. 

The quantity (RA) can be further elaborated if interface mass and heat transfer coefficients are known, and with the theory of mass transfer with reaction inside porous catalysts as treated in Chapters 6 and 7 ... 

The concept of radical capture efficiency was further elaborated on by Hansen et al. [15-17]. By applying the theory of mass transfer with simultaneous chemical reactions, they proposed the following expression to represent the net rate of radical absorption by a particle, introducing an ""absorption efficiency factor F into Eq. 8... 

Another significant effect is connected to the bulk concentration of the external solution (Cq). It is shown (Sec. IV) that the bulk concentration effects not only the concentration (Cg) of the invading counterion in the bead but, more importantly, the selectivity of exchanger for the exchanging counterions. According to the theory of mass transfer in sorption systems [63] the motion of the sorption concentration front of substances depend on the curvature of their sorption isotherm. For IE selective systems this effect is connected to the selectivity factor since this factor controls the shape of the ion-exchange isotherm. 

Correlations of numbers of transfer units developed for binary systems may be used to compute numbers of transfer units for multicomponent systems as described in Section 12.1.5. An alternative method that follows the ideas put forward by Toor in his development of the linearized theory of mass transfer is to generalize binary correlations by replacing the binary diffusivity with the matrix of Fick diffusion coefficients (in much the same way that we generalized correlations of binary mass transfer coefficients in Section 8.8.2). Let the number of transfer units in a binary system be expressed as... 

We also feel that portions of the material in this book ought to be taught at the undergraduate level. We are thinking, in particular, of the materials in Section 2.1 (the Maxwell-Stefan relations for ideal gases). Section 2.2 (the Maxwell-Stefan equations for nonideal systems). Section 3.2 (the generalized Fick s law). Section 4.2 (estimation of multicomponent diffusion coefficients). Section 5.2 (multicomponent interaction effects), and Section 7.1 (definition of mass transfer coefficients) in addition to the theory of mass transfer in binary mixtures that is normally included in undergraduate courses. 

Representative lists for gas-liquid and liquid-liquid reactions are given in Tables 11.15 and 11.16, respectively. Doraiswamy and Sharma (1984) have covered in detail the theory of mass transfer with simple and complex reactions. A summary of simple irreversible reactions is given below. 

Theory of Mass Transfer Accompanied by Irreversible Chemical Reaction... 

Many commercial absorption processes involve a chemical reaction between the solute and the solvent. The occurrence of a reaction affects not only gas-liquid equilibrium relationships but also the rate of mass tmasfer. Sioce the reaction occurs in the solvent, only the liquid mass transfer rate is affected. Normally, the effect is an iocrease in the liquid mass tmaster coefficianl kL. The development of correlations for predicting the degree of enhancement for various types of chemical reaction and system configuration has been the subject of numerous studies. Comprehensive discussions of the theory of mass transfer with ckemical reaction are presented in recent books by Aslarita,27 Danckwarts,2 and Aslarita et al.J... 

Four of the simplest and best known of the theories of mass transfer from flowing streams are (1) the stagnant-film model, (2) the penetration model, (3) the surface-renewal model, and (4) the turbulent boundary-layer model... 

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