Estimation of Multicomponent Diffusion Coefficients

The NRTL parameters for ethanol-water at 40°C are taken from the collection of Gmehling and Onken (1977ff Vol. I/l p. 172). 

SOLUTION The Maxwell-Stefan diffusivity is computed using the Vignes equation (Eq. 4.1.10) as 

The parameter F is computed from the NRTL equation as illustrated in Example 4.1.2 with the result 

Strictly speaking, the rules of matrix algebra do not allow us, on the basis of Eqs. 3.2.5 and 2.2.10, to assert that [D] and [B] [r] are equal. The equality of these two matrices is an assumption, albeit the only reasonable way to relate the Fick diffusion coefficients to the Maxwell-Stefan diffusion coefficients ,y. The equality 

Equation 4.2.1 is an important result for it allows us to predict the Fick matrix [D] from information on the binary Maxwell-Stefan diffusivities and activity coefficients. 

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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. 

It is possible to use alternative formulations considering mole fractions rather than mass fractions. For most cases, mass fraction formulations will be adequate. An estimation of the diffusion coefficient (of component k) in a multicomponent mixture Dkm) however, is not straightforward. For mixtures of ideal gases, the diffusion coefficient in a mixture can be estimated as (Hines and Maddox, 1985)... 

The method of Blanc [16] permits calculation of the gas-phase effective multicomponent diffusion coefficients based on binary diffusion coefficients. A conversion of binary diffusivities into effective diffusion coefficients can be also performed with the equation of Wilke [54]. The latter equation is frequently used in spite of the fact that it has been deduced only for the special case of an inert component. Furthermore, it is possible to estimate the effective diffusion coefficient of a multicomponent solution using a method of Burghardt and Krupiczka [55]. The Vignes approach [56] can be used in order to recalculate the binary diffusion coefficients at infinite dilution into the Maxwell-Stefan diffusion coefficients. An alternative method is suggested by Koijman and Taylor [57]. 

Even the binary system diffusivities in liquid mixtures are composition dependent. Therefore, in multicomponent liquid mixtures with n components, predictions of the diffusion coefficients relating flows to concentration gradients are empirical. The diffusion coefficient of dilute species i in a multicomponent liquid mixture, Dim, may be estimated by Perkins and Geankoplis equation... 

SOLUTION The diffusivities >23 and D 2 nearly equal and we will make this assumption in the estimation of the multicomponent diffusion coefficients. With >23 = >12 = ) we calculate, with reference to Eqs. 4.2.7, the elements of [D] as... 

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

The result obtained from the film theory is that the mass transfer coefficient is directly proportional to the diffusion coefficient. However, the experimental mass transfer data available in the literature [6], for gas-liquid interfaces, indicate that the mass transfer coefficient should rather be proportional with the square root of the diffusion coefficient. Therefore, in many situations the film theory doesn t give a sufficient picture of the mass transfer processes at the interfaces. Furthermore, the mass transfer coefficient dependencies upon variables like fluid viscosity and velocity are not well understood. These dependencies are thus often lumped into the correlations for the film thickness, 1. The film theory is inaccurate for most physical systems, but it is still a simple and useful method that is widely used calculating the interfacial mass transfer fluxes. It is also very useful for analysis of mass transfer with chemical reaction, as the physical mechanisms involved are very complex and the more sophisticated theories do not provide significantly better estimates of the fluxes. Even for the description of many multicomponent systems, the simplicity of the model can be an important advantage. 

When multicomponent diffusion is significant, it is best described with a generalized form of Pick s law containing (n - 1) diffusion coefficients in an -component system. This form of diffusion equation can be rationalized using irreversible thermodynamics. Concentration profiles in these multicomponent cases can be directly inferred from the binary results. However, multicomponent diffusion coefficients are difficult to estimate, and experimental values are fragmentary. As a result, you should make very sure that you need the more complicated theory before you attempt to use it. 

Multicomponent Diffusion. In multicomponent systems, the binary diffusion coefficient has to be replaced by an effective or mean diffusivity Although its rigorous computation from the binary coefficients is difficult, it may be estimated by one of several methods (27—29). Any degree of counterdiffusion, including the two special cases "equimolar counterdiffusion" and "no counterdiffusion" treated above, may arise in multicomponent gas absorption. The influence of bulk flow of material through the films is corrected for by the film factor concept (28). It is based on a slightly different form of equation 13 ... 

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 system of equations (1) to (10) provide the basis for predicting multicomponent rate profiles. The input parameters required are the mass transfer and diffusion coefficients for each solute, the single solute isotherm constants, and the mixture equilibria correlation coefficients. Estimation of these equilibrium and rate parameters are discussed in the following sections. 

The rate parameters of importance in the multicomponent rate model are the mass transfer coefficients and surface diffusion coefficients for each solute species. For accurate description of the multicomponent rate kinetics, it is necessary that accurate values are used for these parameters. It was shown by Mathews and Weber (14), that a deviation of 20% in mass transfer coefficients can have significant effects on the predicted adsorption rate profiles. Several mass transfer correlation studies were examined for estimating the mass transfer coefficients (15, jL6,17,18,19). The correlation of Calderbank and Moo-Young (16) based on Kolmogaroff s theory of local isotropic turbulence has a standard deviation of 66%. The slip velocity method of Harriott (17) provides correlation with an average deviation of 39%. Brian and Hales (15) could not obtain super-imposable curves from heat and mass transfer studies, and the mass transfer data was not in agreement with that of Harriott for high Schmidt number values. 

For this study, mass transfer and surface diffusions coefficients were estimated for each species from single solute batch reactor data by utilizing the multicomponent rate equations for each solute. A numerical procedure was employed to solve the single solute rate equations, and this was coupled with a parameter estimation procedure to estimate the mass transfer and surface diffusion coefficients (20). The program uses the principal axis method of Brent (21) for finding the minimum of a function, and searches for parameter values of mass transfer and surface diffusion coefficients that will minimize the sum of the square of the difference between experimental and computed values of adsorption rates. The mass transfer and surface coefficients estimated for each solute are shown in Table 2. These estimated coefficients were tested with other single solute rate experiments with different initial concentrations and different amounts of adsorbent and were found to predict... 

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... 

Zielinski and Hanley [AlChE J. 45,1 (1999)] developed a model to predict multicomponent diffusivities from self-diffusion coefficients and thermodynamic information. Their model was tested by estimated experimental diffusivity values for ternary systems, predicting drying behavior of ternary systems, and reconciling ternary selfdiffusion data measured by pulsed-field gradient NMR. 

In the five chapters that make up Part II (Chapters 7-11) we consider the estimation of rates of mass and energy transport in multicomponent systems. Multicomponent mass transfer coefficients are defined in Chapter 1, Chapter 8 develops the multicomponent film model, Chapter 9 describes unsteady-state diffusion models, and Chapter 10 considers models based on turbulent eddy diffusion. Chapter 11 shows how the additional complication of simultaneous mass and energy transfer may be handled. 

Estimate the effective diffusivity of a component in a multicomponent mixture of gases from its binary diffusion coefficients with each of the other constituents of the mixture. 

From the Maxwell-Stefan theory for multicomponent diffusion, Taylor and Krishna (1993) developed the following scheme to estimate the matrices of zero-flux multicomponent mass-transfer coefficients from binary-pair mass-transfer coefficients, Fy. These are obtained from correlations of experimental data, with the Chan and Fair (1984) correlations being the most widely used. For an ideal gas solution ... 

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