Diffusivity multicomponent mixtures

Though illustrated here by the Scott and Dullien flux relations, this is an example of a general principle which is often overlooked namely, an isobaric set of flux relations cannot, in general, be used to represent diffusion in the presence of chemical reactions. The reason for this is the existence of a relation between the species fluxes in isobaric systems (the Graham relation in the case of a binary mixture, or its extension (6.2) for multicomponent mixtures) which is inconsistent with the demands of stoichiometry. If the fluxes are to meet the constraints of stoichiometry, the pressure gradient must be left free to adjust itself accordingly. We shall return to this point in more detail in Chapter 11. 

In general, tests have tended to concentrate attention on the ability of a flux model to interpolate through the intermediate pressure range between Knudsen diffusion control and bulk diffusion control. What is also important, but seldom known at present, is whether a model predicts a composition dependence consistent with experiment for the matrix elements in equation (10.2). In multicomponent mixtures an enormous amount of experimental work would be needed to investigate this thoroughly, but it should be possible to supplement a systematic investigation of a flux model applied to binary systems with some limited experiments on particular multicomponent mixtures, as in the work of Hesse and Koder, and Remick and Geankoplia. Interpretation of such tests would be simplest and most direct if they were to be carried out with only small differences in composition between the two sides of the porous medium. Diffusion would then occur in a system of essentially uniform composition, so that flux measurements would provide values for the matrix elements in (10.2) at well-defined compositions. 

Multicomponent Mixtures No simple, practical estimation methods have been developed for predicting multicomponent hquid-diffusion coefficients. Several theories have been developed, but the necessity for extensive activity data, pure component and mixture volumes, mixture viscosity data, and tracer and binaiy diffusion coefficients have significantly limited the utihty of the theories (see Reid et al.). 

The generalized Stefan-Maxwell equations using binary diffusion coefficients are not easily applicable to hquids since the coefficients are so dependent on conditions. That is, in hquids, each Dy can be strongly composition dependent in binary mixtures and, moreover, the binaiy is strongly affected in a multicomponent mixture. Thus, the convenience of writing multicomponent flux equations in terms of binary coefficients is lost. Conversely, they apply to gas mixtures because each is practically independent of composition by itself and in a multicomponent mixture (see Taylor and Krishna for details). 

However, self-diffusion is not limited to one-component systems. As illustrated in Figure 4.4-1, the random walk of particles of each component in any composition of a multicomponent mixture can be observed. 

According to Maxwell s law, the partial pressure gradient in a gas which is diffusing in a two-component mixture is proportional to the product of the molar concentrations of the two components multiplied by its mass transfer velocity relative to that of the second component. Show how this relationship can be adapted to apply to the absorption of a soluble gas from a multicomponent mixture in which the other gases are insoluble and obtain an effective diffusivity for the multicomponent system in terms of the binary diffusion coefficients. 

Multicomponent diffusion, 1 43—46 Multicomponent mixtures, phase and chemical equilibrium criteria in, 24 675-678... 

First a derivative is given of the equations of change for a pure fluid. Then the equations of change for a multicomponent fluid mixture are given (without proof), and a discussion is given of the range of applicability of these equations. Next the basic equations for a multicomponent mixture are specialized for binary mixtures, which are then discussed in considerably more detail. Finally diffusion processes in multicomponent systems, turbulent systems, multiphase systems, and systems with convection are discussed briefly. 

Here D km represents a mixture-averaged diffusion coefficient for species k relative to the rest of the multicomponent mixture. The species mass-flux vector is given in terms of the mole-fraction gradient as... 

When considering the mass continuity of an individual species in a multicomponent mixture, there can be, and typically is, diffusive transport across the control surfaces and the production or destruction of an individual species by volumetric chemical reaction. Despite the fact that individual species may be transported diffusively across a surface, there can be no net mass that is transported across a surface by diffusion alone. Moreover homogeneous chemical reaction cannot alter the net mass in a control volume. For these reasons the overall mass continuity need not consider the individual species. At the conclusion of this section it is shown that that the overall mass continuity equation can be derived by a summation of all the individual species continuity equations. 

In practice, RSPs rarely operate at thermodynamic equilibrium. Therefore, some correlation parameters, such as tray efficiencies or HETS values, have been introduced to adjust the equilibrium-based theoretical description to reality. For multicomponent mixtures, however, this concept often fails, since diffusion interactions of several components result in unusual phenomena such as osmotic or reverse... 

The generic balance relations and the derived relations presented in the preceding section contain various diffusion flux tensors. Although the equation of continuity as presented does not contain a diffusion flux vector, were it to have been written for a multicomponent mixture, there would have been such a diffusion flux vector. Before any of these equations can be solved for the various field quantities, the diffusion fluxes must be related to gradients in the field potentials . 

In order to predict correctly the fluxes of multicomponent mixtures in porous membranes, simplified models based solely on Fields law should be used with care [28]. Often, combinations of several mechanisms control the fluxes, and more sophisticated models are required. A well-known example is the Dusty Gas Model which takes into account contributions of molecular diffusion, Knudsen diffusion, and permeation [29]. This model describes the coupled fluxes of N gaseous components, Ji, as a function of the pressure and total pressure gradients with the following equation ... 

The NMR technique for measuring self-diffusion in liquids was first reported in 1965390, and its variant for multicomponent mixtures became available with the introduction of the Fourier transform. Wider use had to wait until pulsed field gradients became generally... 

In a multicomponent mixture, the diffusion rate of a component depends not only on its own concentration in the mixture, but also on the concentration of other components. This may lead to coupling and interaction of the mass transfer among various components. Some examples are (192)... 

No strong differences were observed when multicomponent mixtures of phenols (phenol, benzyl alcohol, 1-phenyl-ethanol and m-cresol) were treated (Morao et al. 2004) the experimental results were interpreted well by assuming that all the components in the mixture were degraded in the same time and that the combustion of the compounds to C02 occurred at the anode surface by means of hydroxyl radicals electrogenerated. The instantaneous current efficiency was unity, until the reaction became diffusion controlled. 

Tables 2.12-2.14 show some values of diffusion coefficients in solids and polymers. In a flow of dilute solution of polymers, the diffusivity tensor is anisotropic and depends on the velocity gradient. The Maxwell-Stefan equation may predict the diffusion in multicomponent mixtures of polymers.

The Maxwell-Stefan diffusion coefficients represent binary diffusivities for ideal and many nonideal mixtures, they are independent of the concentration of the species in the multicomponent mixtures. 

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... 

Some of the molecular theories of multicomponent diffusion in mixtures led to expressions for mass flow of the Maxwell-Stefan form, and predicted mass flow dependent on the velocity gradients in the system. Such dependencies are not allowed in linear nonequilibrium thermodynamics. Mass flow contains concentration rather than activity as driving forces. In order to overcome this inconsistency, we must start with Jaumann s entropy balance equation... 

Low-Pressure/Multicomponent Mixtures These methods are outlined in Table 5-13. Stefan-Maxwell equations were discussed earlier. Smith and Taylor [23] compared various methods for predicting multicomponent diffusion rates and found that Eq. (5-214) was superior among the effective diffusivity approaches, though none is very good. They also found that linearized and exact solutions are roughly equivalent and accurate. 

Anderko and Lencka find. Eng. Chem. Res. 37, 2878 (1998)] These authors present an analysis of self-diffusion in multicomponent aqueous electrolyte systems. Their model includes contributions of long-range (Coulombic) and short-range (hard-sphere) interactions. Their mixing rule was based on equations of nonequilibrium thermodynamics. The model accurately predicts self-diffusivities of ions and gases in aqueous solutions from dilute to about 30 mol/kg water. It makes it possible to take single-solute data and extend them to multicomponent mixtures. 

For binary and multicomponent mixtures, the thermal conductivity depends on the concentrations as well as on temperature, and the formulas of the accurate kinetic theory are quite complicated [5]. Empirical expressions for X are therefore more useful for both binary [9] and ternary [6], [26] mixtures, although few data exist for ternary mixtures. Tabulations of available experimental and theoretical results for thermal conductivities may be found in [5], [6], [13], and [18]-[21], for example. The thermal diffusivity, defined as 2p/Cp, often arises in combustion problems its pressure and temperature dependences in gases are XjpCp T7p ( < a < 2), and its typical values in combustion lie between 10 cm /s and 1 cm s at atmospheric pressure. 

Aris (1991a), in addition to the case of M CSTRs in series, has also analyzed two other homotopies the plug flow reactor with recycle ratio R, and a PFR with axial diffusivity and Peclet number P, but only for first-order intrinsic kinetics. The values M = 1(< ), R = >(0), and P = 0( o) yield the CSTR (PFR). The M CSTRs in series were discussed earlier in Section IV,C,1. The solutions are expressed in terms of the Lerch function for the PFR with recycle, and in terms of the Niemand function for the PFR with dispersion. The latter case is the only one that has been attacked for the case of nonlinear intrinsic kinetics, as discussed below in Section IV,C,7,b. Guida et al. (1994a) have recently discussed a different homotopy, which is in some sense a basically different one no work has been done on multicomponent mixture systems in such a homotopy. 

Bailey, J. E., Diffusion of grouped multicomponent mixtures in uniform and nonuniforra media. AIChE J. 21, 192(1975). 

Guida, V., Ocone, R., and Astarita, G., Diffusion-convection-reaction in multicomponent mixtures. AIChEJ. 40, 1665 (1994b). 

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