Fick Diffusion Coefficients

Fick (F2) proposed one of the more useful relationships for describing molecular material transport. The Fick diffusion coefficient is identified... 

In the case of transport in the gas phase it is often convenient to use the Maxwell diffusion coefficient (M4), which is related to the Fick diffusion coefficient in the following way (04) for a phase that may be treated as an ideal solution (L4) ... 

The conservation principle applied to unsteady, nonuniform, laminar, two-dimensional flow results in the following expression if the Fick diffusion coefficient is considered to be isotropic ... 

Schmidt numbers vary widely depending on the value of the Fick diffusion coefficient, which in the case of gases is strongly dependent on the concentration of the diffusing component. It should be noted that in a mixture the Schmidt number is not a property solely of either the component or... 

If the above treatment is formulated in terms of Fick diffusion coefficients Dls D2 53) (cf. the introductory section for the difference between a thermodynamic and a Fick diffusion coefficient), the expressions equivalent to Eqs. (13) and (14) are respectively... 

The behavior of the Fick diffusion coefficient in nonideal systems may be complicated, while the Maxwell-Stefan diffusion coefficients behave quite well, and are always positive for binary systems. In nonideal binary systems, the Fick diffusivity varies with concentration. As seen in Figure 6.1, water-acetone and water-ethanol systems exhibit a minimum diffusivity at intermediate concentrations. Table 6.1 displays the dependency of binary diffusivity coefficients on concentration for selected alkenes in chloroform at 30°C and 1 atm. As the nonideality increases, mixture may split into two liquid phases at certain composition and temperature. 

A few typical values of the Fick diffusion coefficients are listed in Table 3.1. Although it may not be discerned from this small sample of values, the diffusion coefficient in an ideal gas mixture is independent of the mixture composition, inversely proportional to pressure, and varies with the absolute temperature to around the 1.5 power. More extensive listings are provided by Reid et al. (1987) and by Cussler (1984). The most comprehensive collection of 50... 

The matrix [D] of Fick diffusion coefficients is a square matrix of dimension n — 1 X n — 1... 

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"... 

Example 3.2.1 Fick Diffusion Coefficients for the System Acetone-Benzene-Methanol... 

The Fick diffusion coefficients for the system acetone(l)-benzene(2)-methanol(3) in the volume average reference velocity frame are given in Table 3.2. Calculate the elements of [D] in the molar average reference velocity frame. 

It follows that the Fick diffusion coefficient must tend towards zero as the spinodal curve is approached. This has been experimentally confirmed for a few systems, the data of Haase and Siry (1968) for the systems water-triethylamine and n-hexane-nitrobenzene are shown in Figs. 3.2 and 3.3 (see, also, Claesson and Sundeldf, 1957 Myerson and Senol, 1984). Vitagliano et al. (1980) and Clark and Rowley (1986) determined spinodal compositions by extrapolating diffusivity data to zero. 

Figure 3.2. Fick diffusion coefficient D as a function of temperature for the system water-triethyl-amine. Measured data for Fick diffusivity D at constant composition = critical composition X2 = 0.0874. Critical temperature = 18.3°C. Data from Haase and Siry (1968).

Examples illustrating the use of Eq. 4.1.1 and 4.1.2 are given by Reid et al. (1977, 1987) and by Danner and Daubert (1983). The same authors describe methods for estimating Fick diffusion coefficients for gases at high pressure. 

Figure 4.1. a Concentration dependence of the Fick diffusivity D and the Maxwell-Stefan D for the system ethanol(l)-water(2). Data from Tyn and Calus (1975b). (6) Composition dependence of Fick D and Maxwell-Stefan D for the system acetone(l)-benzene(2). Data from Anderson et al. (1958) and Cullinan and Toor (1965). (c) Composition dependence of Fick D and Maxwell-Stefan D for diffusion in triethylamine(l)-water(2). Data from Dudley and Tyrell (1973). (J) Fick diffusion coefficient for the system methanol-n-hexane at 40°C measured by Clark and Rowley (1986). 

Estimate the Maxwell-Stefan and Fick diffusion coefficients for an acetone(l)-benzene(2) mixture of composition Xj = 0.7808 at a temperature of 25°C. The NRTL equation may be... 

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... 

Figure 4.5. Four Fick diffusion coefficients plotted as a function of composition for the system H2-N2-CCI2F2 for the three possible choices of component 3. ...

Estimate the Fick diffusion coefficients in the mass average velocity reference frame for a mixture containing carbon monoxide (1), hydrogen (2), methane (3), and water vapor (4) flowing inside a tubular reactor at a point where the mole fractions are... 

The Fick diffusion coefficients in the mass average reference velocity frame [D°] may now be calculated directly from the expression above as... 

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

Figure 4.10. The Fick diffusion coefficients as a function of composition and component numbering for a nonideal liquid mixture.

In most cases the general Eq. 5.1.2 is simplified somewhat before being solved. It is common, for example, to assume constant molar density and Fick diffusion coefficient D. Unless these (or other) assumptions are made it may be difficult or impossible to solve... 

The matrices of Fick diffusion coefficients differ slightly in Example 5.3.1 from those used here. In this example the equilibrium composition was used to calculate [D], whereas in Example 5.3.1, [D] was calculated at the arithmetic average composition. 

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... 

Equation 7.1.6 should be compared to Eq. 3.1.1. It appears that may be related directly to the binary Fick diffusion coefficient D. Indeed, this will be shown to be the case when we examine various specific hydrodynamic models for mass transfer later in this book. 

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