Multicomponent Fick diffusion coefficients

The Fick diffusion coefficients may be termed practical in the sense that the binary coefficient P and the corresponding multicomponent diffusion coefficients can be obtained from composition profiles measured in a diffusion apparatus. The measurement of binary and multicomponent diffusion coefficients, a subject with an extensive literature, is beyond the scope of this book. The interested reader is referred to Dunlop et al. (1972), Cussler (1976) and Tyrrell and Harris (1984) for descriptions of techniques and summaries of experimental results. Most experimental data are reported for [P ]. This matrix must be 

TABLE 3.2 Fick Diffusion Coefficients in the System Acetone(l)-Benzene(2)-Methanol(3) at 25°C  

To give an indication of the magnitude of the cross-coefficients that may sometimes be encountered in practice we present in Table 3.2 some of the data of Alimadadian and Colver (1976) for [D ] for the system acetone(l)-benzene(2)-methanol(3) at 25°C and, in Table 3.3, some of the data of Cullinan and Toor (1965) for the system acetone(l)-benzene(2)-carbon tetrachloride(3). 

It is clear from this small selection of data that the matrix of multicomponent diffusion coefficients may be a complicated function of the composition of the mixture. The matrix [D] is generally nonsymmetric, except for two special cases identified below. The cross coefficients (/ = = k) can be of either sign indeed it is possible to alter the sign of these cross-coefficients by altering the numbering of the components. 

There are circumstances where the matrix [ ] is diagonal and the diffusion flux of species i is independent of the composition gradients of the other species. For an ideal mixture made up of chemically similar species the matrix of diffusion coefficients degenerates to a scalar times the identity matrix, that is. 

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Estimation of Multicomponent Fick Diffusion Coefficients for Liquid Mixtures... 

Relationship Between Effective, Maxwell-Stefan, and Multicomponent Fick Diffusion Coefficients... 

Some assumptions regarding the constancy of certain parameters are usually in order to facilitate the solution of the diffusion equations. For the binary diffusion problems discussed in Chapters 5 (as well as later in Chapters 8-10), we assume the binary Fick diffusion coefficient can be taken to be a constant. In the applications of the linearized theory presented in the same chapters, we assume the matrix of multicomponent Fick diffusion coefficients to be constant. If, on the other hand, we use Eq. 6.2.1 to model the diffusion process then we must usually assume constancy of the effective diffusion coefficient if... 

In order to calculate the composition at the interface we need to estimate the multicomponent Fick diffusion coefficients. This is usually done at the average composition for which the composition at the interface is required. We shall begin our illustration with the following values... 

The procedure for computing the multicomponent Fick diffusion coefficients in the mass average velocity reference frame was illustrated in Example 4.2.5 and the steps shown there have been repeated for this example with the result... 

The ratio of driving forces Axi/Ax2 plays an important role in enhancing diffusional interaction effects in multicomponent mass transfer. Thus, a small cross-coefficient k 2 may be linked to a large Ax2, resulting in large interaction effects. The criteria presented above are a little different from those discussed in Section 5.2, where Fick diffusion coefficients and the mole fraction gradients were used. The physical significance is, however, the same. 

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... 

However, the T>sr coefficients relate to the alternative set of consistent multicomponent Fick diffusivities, Dgr, by [18] ... 

In a binary mixture, diffusion coefficients are equal to each other for dissimilar molecules, and Fick s law can determine the molecular mass flows in an isotropic medium at isothermal and isobaric conditions. In a multicomponent diffusion, however, various interactions among the molecules may arise. Some of these interactions are (i) diffusion flows may vanish despite the nonvanishing driving force, which is known as the mass transfer barrier, (ii) diffusion of a component in a direction opposite to that indicated by its driving force leads to a phenomenon called the reverse mass flow, and (iii) diffusion of a component in the absence of its driving force, which is called the osmotic mass flow. 

The description of diffusion may be complex in mixtures with more than two components. Diffusion coefficients in multicomponent mixtures are usually unknown, although sufficient experimental and theoretical information on binary systems is available. The Maxwell-Stefan diffusivities can be estimated for dilute monatomic gases from D k Dkl when the Fick diffusivities are available. The Maxwell diflfusivity is independent of the concentration for ideal gases, and almost independent of the concentration for ideal liquid mixtures. The Maxwell-Stefan diffusivities can be calculated from... 

Fast and satisfactory mass transfer calculations are necessary since we may have to repeat such calculations many times for a rate-based distillation column model or two-phase flow with mass transfer between the phases in the design and simulation process. The generalized matrix method may be used for multicomponent mass transfer calculations. The generalized matrix method utilizes the Maxwell-Stefan model with the linearized film model for diffusion flux, assuming a constant diffusion coefficient matrix and total concentration in the diffusion region. In an isotropic medium, Fick s law may describe the multicomponent molecular mass transfer at a specified temperature and pressure, assuming independent diffusion of the species in a fluid mixture. Such independent diffusion, however, is only an approximation in the following cases (i) diffusion of a dilute component in a solvent, (ii) diffusion of various components with identical diffusion properties, and (iii) diffusion in a binary mixture. 

If we compare Eqs. 5.1.14 with the conservation equation (Eq. 5.1.2) for a binary system and the pseudo-Fick s law Eq. 5.1.15, with Eq. 3.1.1 then we can see that from the mathematical point of view these pseudomole fractions and pseudofluxes behave as though they were the corresponding variables of a real binary mixture with diffusion coefficient D-. The fact that the are real, positive, and invariant under changes of reference velocity strengthens the analogy. If the initial and boundary conditions can also be transformed to pseudocompositions and fluxes by the same similarity transformation, the uncoupled equations represent a set of independent binary-type problems, n - 1 in number. Solutions to binary diffusion problems are common in the literature (see, e.g.. Bird et al., 1960 Slattery, 1981 Crank, 1975). Thus, the solution to the corresponding multicomponent problem can be written down immediately in terms of the pseudomole fractions and fluxes. Specifically, if... 

In their original development of the linearized theory Toor (1964) and Stewart and Prober (1964) proposed that correlations of the type given by Eqs. 8.8.5 and 8.8.7 could be generalized by replacing the Fick diffusivity D by the charactersitic diffusion coefficients of the multicomponent system that is, by the eigenvalues of the Fick matrix [ >]. The mass transfer coefficient calculated from such a substitution would be a characteristic mass transfer coefficient an eigenvalue of [/c]. For example, the Gilliland-Sherwood correlation (Eq. 8.8.5) would be modified as follows ... 

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. 

The simplest approach is to calculate binary mass-transfer coefficients F.. from the corresponding empirical correlation, substituting the MS diffusivity D. for the Fick diffusivity in the Sc and Sh numbers. The Maxwell-Stefan equations are, then, written in terms of the binary mass-transfer coefficients. For ideal gas multicomponent mixtures and one-dimensional fluxes, they become... 

Wilke [161] proposed a simpler model for calculating the effective diffusion coefficients for diffusion of a species s into a multicomponent mixture of stagnant gases. For dilute gases the Maxwell-Stefan diffusion equation is reduced to a multicomponent diffusion flux model on the binary Fick s law form in which the binary diffusivity is substituted by an effective multicomponent diffusivity. The Wilke model derivation is examined in the sequel. 

The presence of nonzero cross-coefficients, D y = 0 (/ = j), in the Fick matrix [D] lends to multicomponent systems characteristics quite different from the corresponding binary system. These characteristics are best illustrated by considering a binary system for which the diffusion flux is given by Eq. 3.1.1... 

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