Diffusion Coefficient for Binary Gas Mixture

The kinetic theory of dilute gases accounts for collisions between spherical molecules in the presence of an intermolecular potential. Ordinary molecular diffusion coefficients depend linearly on the average kinetic speed of the molecules and the mean free path of the gas. The mean free path is a measure of the average distance traveled by gas molecules between collisions. When the pore diameter is much larger than the mean free path, collisions with other gas molecules are most probable and ordinary molecular diffusion provides the dominant resistance to mass transfer. Within this context, ordinary molecular diffusion coefficients for binary gas mixtures are predicted, with units of cm /s, via the Chapman-Enskog equation (see Bird et al., 2002, p. 526) ... 

Diffusion coefficients for binary gas mixtures at low pressures Empirical... 

The explanation of Graham s law given by Hoogschagen is not complete, as subsequent authors (17,18) stated. However, the attempts of these authors to give a more complete explanation for the law are not convincing. It is known that at conventional measurements of diffusion coefficients in binary gas mixtures using wide capillaries, equal velocities of counterdiffusion of the components are observed. From the considerations developed... 

Fig. 1 Schematic representation of columns and gas connections for studying (a) diffusion coefficients in binary gas mixtures, (b) interaction between gases and liquids, and (c) interaction between gases and solids.

Note that for extremely low pressures the mean free path becomes the order of magnitude of the vessel diameter, which is then limiting and has to be used instead of A in Eqs. (3.1.68)-(3.1.70). For air (at 20 °C), a pressure of less than 10 mbar is needed to obtain a mean free path of the order of magnitude of a cm [Eq. (3.1.72)] Xg is then proportional to p, and thus this effect is used for superinsulations by highly evacuated casings. In addition, note that the diffusivities given in Table 3.1.7 are only valid for pure gases (self-diffusion coefficients). In binary gas mixtures, the binary coefficient D g g has to be used (Table 3.1.8). Note that in a binary gas mixture the diffusion coefficient is independent of the content of both components and that the diffusion coefficient of A in B is equal to the diffusion coefficient of B in A. 

Fuller s equation, applied for the estimation of the coefficient of diffusion of a binary gas mixture, at a pressure greater than 10 bar, predicts values that are too high. As a first approximation, the value of the coefficient of diffusion can be corrected by multiplying it by the compressibility of the gas /... 

To use this formula, the assumption has been made that the fuel consists of a binary mixture of hydrogen and water, while the cathodic gas is a binary mixture of oxygen and nitrogen. The diffusion coefficient for binary mixtures D y eff is estimated by the equation proposed by Hirschfelder, Bird and Spotz [12], and the Knudsen diffusion coefficient for species i is given by free molecule flow theory [11], Finally, combining Equations (6.15-6.18) the anodic and the cathodic concentration overvoltages are given by (see also Equations (A3.20) and (A3.21)) ... 

Similar equations can be written for components J2 and J3. The coefficients Dx x and D22 are the main coefficients they are not self-diffusion coefficients. Du and D2l are the cross-coefficients and assumed to be equal to each other for binary gas mixtures. 

The Stefan tube, depicted schematically in Figure 2.4, is a simple device sometimes used for measuring diffusion coefficients in binary vapor mixtures. In the bottom of the tube is a pool of quiescent liquid. The vapor that evaporates from this pool diffuses to the top of the tube. A stream of gas across the top of the tube keeps the mole fraction of diffusing vapor there to essentially nothing. The mole fraction of the vapor at the vapor-liquid interface is its equilibrium value. 

Diffusion coefficients in binary liquid mixtures are of the order 10 m /s. Unlike the diffusion coefficients in ideal gas mixtures, those for liquid mixtures can be strong functions of concentration. We defer illustration of this fact until Chapter 4 where we also consider models for the correlation and prediction of binary diffusion coefficients in gases and liquids. 

Note that unlike the case for binary gas mixtures the diffusion coefficient for a dilute solution of. 4 in 5 is not the same as for a dilute solution of B in A, since fi, Mb, and will be different when the solute and solvent are exchanged. For intermediate concentrations, an approximate value of is sometimes obtained by interpolation between the dilute solution values, but this method can lead to large errors for nonideal solutions. 

The Stefan tube, depicted schematically in Figure 1.8, is a simple device sometimes used for measuring diffusion coefficients in binary vapor mixtures. In the bottom of the tube is a pool of quiescent liquid. The vapor that evaporates from this pool diffuses to the top of the tube. A stream of gas across the top of the tube keeps the mole fraction of the diffusing vapors there to essentially zero. The compositon of the vapor at the vapor-liquid interface is its equilibrium value. Carty and Schrodt (1975) evaporated a binary liquid mixture of acetone (1) and methanol (2) in a Stefan tube. Air (3) was used as the carrier gas. In one of their experiments the composition of the vapor at the liquid interface was yx - 0.319, y2 - 0.528, and y3 = 0.153. The pressure and temperature in the gas phase were 99.4 kPa and 328.5 K, respectively. The length of the diffusion path was 0.24 m. The MS diffusion coefficients of the three binary pairs are ... 

Experimental determination of diffusion coefficients. A number of different experimental methods have been used to determine the molecular diffusivity for binary gas mixtures. Several of the important methods are as follows. One method is to evaporate a pure liquid in a narrow tube with a gas passed over the top as shown in Fig. 6.2-2a. The fall in liquid level is measured with time and the diffusivity calculated from Eq. (6.2-26). 

At ambient temperature and pressure, gas-phase diffusion coefficients are of the order of tO -tO ft /s (10 -10 mVs). Table 2.3-1 presents some rqiresentative values for binary gas mixtures at I atm. Marrero and Mason" have provided an extensive review of experimental values of gas phase binary diffiisivities. 

For binary gas mixtures at low pressures, e.g., below 10 atm, the diffusion coefficient is inversely proportional to the pressure, increases with increasing temperature, and is almost independent of composition for a given gas pair. Slattery and Bird [17] have proposed the following expression for low pressures ... 

For a two-component mixture the multicomponent diffusion coefficients D, become the ordinary binary diffusion coefficients Sh,. For these quantities 2D,-, = 2D,- and 2D = 0. For a three-component system the multicomponent diffusion coefficients are not equal to the ordinary binary diffusion coefficients. For example, it has been shown by Curtiss and Hirschfelder (C12) in their development of the kinetic theory of multicomponent gas mixtures that... 

To estimate the Maxwell-Stefan and effective diffusion coefficients, diffusion data for binary mixtures is necessary. For gas systems under low pressure, the model of Fuller et al. is used most frequently [51]. The method of Wilke and Lee [40] is also valid for low pressures. Both of these methods generally agree with experimental data with an accuracy of up to 10 %, although discrepancies of about 20 % cannot be excluded [40],... 

For a multicomponent gas mixture, the effective binary diffusion coefficient for species j diffusing through the mixture may be found by equating the driving forces Ay, in Eqs. (9.4) and (9.5)... 

Let us now turn our attention to n-component mixtures. Exact analytical solutions of the Maxwell-Stefan equations for a film model can be obtained for a mixture of ideal gases for which the binary diffusion coefficients are independent of composition and identical to the diffusivity of the binary gas i-k pair. Solutions of the Maxwell-Stefan equations for certain special cases involving diffusion in ternary systems have been known for a long time (Gilliland (1937) = 0) Pratt (1950) (TV, = 0) Cichelli et al. (1951) Toor (1957),... 

This relation is referred to as the Maxwell-Stefan model equations, since Maxwell [65] [67] was the first to derive diffusion equations in a form analogous to (2.302) for dilute binary gas mixtures using kinetic theory arguments (i.e., Maxwell s seminal idea was that concentration gradients result from the friction between the molecules of different species, hence the proportionality coefficients, Csk, were interpreted as inverse friction or drag coefficients), and Stefan [92] [93] extended the approach to ternary dilute gas systems. It is emphasized that the original model equations were valid for ordinary diffusion only and did not include thermal, pressure, and forced diffusion. 

Diffusion coefficients for some binary gas mixtures have been measured and are reported in various compendia, such as Perry s Handbook [2]. Of concern here are the models available for estimating the coefficients, or for extrapolating the values of measured coefficients. A number of predictive models have been presented for the case of binary gas mixtures. The models are based on experimental data, where the movement of one component is measured under carefully controlled laminar conditions. A model combining both accuracy and ease of use is due to Fuller et al. [3] ... 

The D j ate the binary diffusion coefficients (or an / — j mixture thus, no additional information is required for computations of multicomponent diffusion in dilute gas mixtures although one might prefer a form in which the fluxes appeared explicitly. Generalization of the Stefan-Maxweli form to dense gassa and to liquids has been suggested. 1 but in these cases there is no rigorous relationship to the binety diffusivilies. Furthermore, Ihe form of Eq- (2.3-10) in which the fluxes du not appear explicitly has little to recommend it. 

The theory describing diffusion in binary gas mixtures at low to moderate pressures has been well developed. Modem versions of the kinetic theory of gases have attempted to account for the forces of attraction and repulsion between molecules. Hirschfelder et al. (1949), using the Lennard-Jones potential to evaluate the influence of intermolecular forces, presented an equation for the diffusion coefficient for gas pairs of nonpolar, nonreacting molecules ... 

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... 

The diffusion coefficient of a gas A into another gas B, Dab, is a function of temperature, T, pressure, p, and composition, x, even for binary mixtures at low pressure Dab is almost independent of the gas composition. Several empirical equations describing the dependence of Dab on T and p are available, among which the most important are the following ... 

Diffusion coefficients, for various binary gas mixtures, at various temperatures and pressures with their accuracy and precision, measured by gas chromatographic broadening techniques (GC-BTs continuous, as well as arrested elution methods), are given in Table 3 of Ref. and in Table 1 of Ref. Representative data are collected here in Table 1. 

Diffusion coefficients from the stopped-flow technique, which are very sensitive to the precision with which L is measured, since Dab is proportional to L, are given in literature for some binary gas mixtures. 

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