Diffusion in binary gas mixtures

Several experimental techniques have been developed for the investigation of the mass transport in porous catalysts. Most of them have been employed to determine the effective diffusivities in binary gas mixtures and at isothermal conditions. In some investigations, the experimental data are treated with the more refined dusty gas model (DGM) and its modifications. The diffusion cell and gas chromatographic methods are the most widely used when investigating mass transport in porous catalysts and for the measurement of the effective diffusivities. These methods, with examples of their application in simple situations, are briefly outlined in the following discussion. A review on the methods for experimental evaluation of the effective diffusivity by Haynes [1] and a comprehensive description of the diffusion cell method by Park and Do [2] contain many useful details and additional information. 

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 fundamental binary diffusivity is evaluated using the Chapman-Enskog model. This model describes the diffusion in binary gas mixtures at low to moderate pressures and it results from solving the Boltzmarm equation. The derived working equation is ... 

Bulk diffusion coefficients in binary gas mixture are almost independent of the ratio of components of the mixture. Therefore, it was supposed that if diffusion in the measurements described above is of the bulk type, i.e., the free path of molecules is much lesser than the diameter of pores, then the first gas diffuses into the second gas at the same rate as the second gas diffuses into the first. 

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... 

The two bulb diffusion cell is a simple device that can be used to measure diffusion coefficients in binary gas mixtures. Figure 5.3 is a schematic of the apparatus. Two vessels containing gas mixtures with different compositions are connected by a capillary tube. At the start of the experiment (at t = 0), the valve is opened and the gases in the two bulbs allowed to diffuse along the capillary tube. Samples from each bulb are taken after some time and this information is used to calculate the binary diffusion coefficient. 

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. 

An alternate method for gas diffusivity of binary gas mixtures at low pressures is the method of Hirschfelder et al. ° The method requires several molecular parameters and, when evaluated, gives an average absolute error of about 10 percent. The method is discussed in detail in the Data Prediction Manual. 

An injection port is added at the closed end of the diffusion column when diffusion coefficients in binary gas mixtures are measured. 

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.

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). 

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. 

Mass Transport. An expression for the diffusive transport of the light component of a binary gas mixture in the radial direction in the gas centrifuge can be obtained directly from the general diffusion equation and an expression for the radial pressure gradient in the centrifuge. For diffusion in a binary system in the absence of temperature gradients and external forces, the general diffusion equation retains only the pressure diffusion and ordinary diffusion effects and takes the form... 

In a binary gas mixture, the diffusion coefficient of the species i at a mole fraction jc, widr respect to tlrat of the species j is given after evaluating the constants by tire equation... 

At moderate pressures the diffusion coefficient of a binary gas mixture of molecules i and j is well described by the Chapman-Enskog theory, discussed in Section 12.4 ... 

The sorption and diffusion behaviour of gas mixtures is of particular interest from the point of view of membrane gas separation, which is steadily gaining in importance by virtue of its low energy requirements. On the basis of the dual mode sorption model, one may reasonably expect that sorption of a binary gas mixture A, B in the polymer matrix will exhibit little gas-gas interaction and hence will tend to occur essentially additively. In the Langmuir-like mode of sorption, on the other hand, there will be competition between A and B for the limited number of available sites. These considerations led 67) to the following reformulation of Eqs. (8) and (9)... 

Fick s law is derived only for a binary mixture and then accounts for the interaction only between two species (the solvent and the solute). When the concentration of one species is much higher than the others (dilute mixture), Fick s law can still describe the molecular diffusion if the binary diffusion coefficient is replaced with an appropriate diffusion coefficient describing the diffusion of species i in the gas mixture (ordinary and, eventually, Knudsen, see below). However, the concentration of the different species may be such that all the species in the solution interact each other. When the Maxwell-Stefan expression is used, the diffusion of... 

P 21] The mixing of gaseous methanol and oxygen was simulated. The equations applied for the calculation were based on the Navier-Stokes (pressure and velocity) and the species convection-diffusion equation [57]. As the diffusivity value for the binary gas mixture 2.8 x 10 m2 s 1 was taken. The flow was laminar in all cases adiabatic conditions were applied at the domain boundaries. Compressibility and slip effects were taken into account The inlet temperature was set to 400 K. The total number of cells was —17 000 in all cases. 

The effective binary coefficient of ordinary diffusion for the reactive species in a gas mixture may be expressed as. °... 

But the concentration of a species in a gas mixture or liquid or solid solution can be defined in. several ways such a.s density, mass fraction, molar concentration, and mole fraction, as already discussed, and thus Pick s law can be expressed mathematically in many ways. It turns out that it is best to exprc.ss the concentration gradient in terms of the mass or mole fraction, and the most appropriate formulation of Pick s law for the diffusion of a species A in a stationary binary mixture of species A and fi in a specified direction x is given by (Fig. 14-10)... 

The binary diffusion coefticients for several binary gas mixtures and solid and liquid solutions are given in Tables 14-2 and 14-3, We make two obser-valious fiom these tables ... 

An interesting technique for the measurement of intraparticle diffusivity as well as longitudinal diffusion in the particle bed has been described by Deisler and Wilhelm (21). It deviates from all other techniques mentioned in that it is based on a dynamic flow study, analyzing the effect of the particles on the propagation of a sinusoidal variation of composition of a binary gas mixture passed through the catalyst bed. The authors have demonstrated the versatility of their general technique for determination of diffusion properties, as well as adsorption equilibria between the solids and the gas composition employed. If this general technique were modified to measure specifically the particle diffusivity, a very convenient and accurate method may result. 

The treatment in Section 9.2.4 will first start with some simple limiting cases (Knudsen diffusion and viscous flow in mixtures), followed by a comparison of an extended Pick model with the DGM model derived equations for binary gas mixtures. Subsequently a treatment will be given of a direct application to membrane separation of a set of equations derived from the model of Present and Bethune by Wu et al. [18] and by Eichmann and Werner [19]. 

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. 

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. 

The MSC membranes are produced by carbonization of polyacrylonitrile, polymide, and phenolic resins [30]. They contain nanopores (typically <5 A in diameter) that allow some of the molecules of a feed gas mixture to enter the pores at the high-pressure side, adsorb, and then diffuse to the low-pressure side where they desorb into the gas phase. The other molecules of the feed gas are excluded from entering the pores and they are enriched in the high-pressure side. Thus the separation is based on the differences in the molecular sizes and shapes of the feed gas components. The smaller molecules preferentially diffuse through the membrane as schematically depicted by Fig. 22.7(a). Table 22.7 gives the permeance and the permselectivity of the smaller species (component 1) of several binary gas mixtures by the MSC membrane [25, 26, 30]. 

In the absence of condensation, the concentration di.stribution in the gas is determined by the ec uatioti of convective diffusion for a binary gas mixture ... 

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] ... 

Gas mixtures. Nomenclature to be used in describing the flow of a binary gas mixture through a diffusion barrier is shown in Fig. 14.4. The problem is to determine how the molar velocities of li t and heavy components, Gi and Gj, respectively, depend on upstream and downstream... 

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. 

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