Diffusion in Ideal Gas Mixtures

Equations 8.3.52-8.3.57 were presented as an exact solution of the Maxwell-Stefan equations for diffusion in ideal gas mixtures by Burghardt (1984). Equations 8.3.52-8.3.57 are somewhat less useful than Eqs. 8.3.15-8.3.24 because we need to know the composition profiles in order to evaluate the matrizant. Even if the profiles are known, the computation of the fluxes from either of Eqs. 8.3.62 or 8.3.63 is not straightforward and not recommended. It is with the development in Section 8.4 in mind that we have included these results here. 

In 1964 Toor and Stewart and Prober independently put forward a general approach to the solution of multicomponent diffusion problems. Their method, which was discussed in detail in Chapter 5, relies on the assumption of constancy of the Fick matrix [D] along the diffusion path. The so-called Tinearized theory of Toor, Stewart, and Prober is not limited to describing steady-state, one-dimensional diffusion in ideal gas mixtures (as we have already demonstrated in Chapter 5) however, for this particular situation Eq. 5.3.5, with [P] given by Eq. 4.2.2, simplifies to... 

In Section 8.3 we presented a derivation of an exact matrix solution of the Maxwell-Stefan equations for diffusion in ideal gas mixtures. Although the final expression for the composition profiles (Eq. 8.3.12), is valid whatever relationship exists between the fluxes (i.e., bootstrap condition), the derivation given in Section... 

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. 

At a particular location in a distillation column, where the temperature is 350 K and the pressure 500 m Hg, the mol fraction of the more volatile component in the vapour is 0.7 at the interface with the liquid and 0.5 in the bulk of the vapour. The molar latent heat of the more volatile component is 1.5 times that of the less volatile. Calculate the mass transfer rates (kmol m 2s 1) of the two components. The resistance to mass transfer in the vapour may be considered to lie in a stagnant film of thickness 0.5 mm at the interface. The diffusivity in the vapour mixture is 2 x 10 5 m2s Calculate the mol fractions and concentration gradients of the two components at the mid-point of the film. Assume that the ideal gas law is applicable and that the Universal Gas Constant R = 8314 J/kmol K. 

In the case of transport in gases, Stokes law applies only if particle radius a is much larger than the mean free path (0.065 /im in air at 20°C and 1 atm). For smaller particles and molecules, a correction factor must be applied to Stokes law [30]. Alternatively, the friction coefficient may be approximated using Eq. 3.21 and the Stephan-Maxwell equation for diffusion in an ideal gas mixture composed of solute of molecular weight Mx in a carrier of molecular weight M2... 

Example 6.2 Diffusion in a ternary ideal gas mixture Methane is being absorbed from a mixture of argon and helium by a nonvolatile liquid in a wetted wall column operated at 25°C and 1 atm. The following boundary conditions and data may be used ... 

In a binary ideal gas mixture of species A and B, the diffusion coefficient of A in B is equal to the diffusion coefficient of S in A, and both increase with temperature... 

Properties The diffusion coeflicient of helium in air (or air in helium) at normal atmospheric conditions is Dgg = 7.2 x 10 m% (Table 14-2). The molar masses of air and helium are 29 and 4 kg/kmol, respectively (Table A-1). Analysis This is a typical equimolar counterdiffusion process since the problem involves two large reservoirs of ideal gas mixtures connected to each other by a channel, and the concentrations of species in each reservoir (the pipeline and the atmosphere) remain constant. 

In the case of an ideal gas mixture, the mole fraction of a gas is equal to its pressure fraction. Pick s law for the diffusion of a specie.s /I in a stationary binary mixture of species A and fl in a specified direction x is expressed as... 

Equation 2.1.8 is the Maxwell-Stefan equation for the diffusion of species 1 in a two-component ideal gas mixture. The symbol D 2 is the Maxwell-Stefan (MS) diffusivity. 

The results of the final integration are plotted in Figure 2.5 along with the data from Carty and Schrodt (1975). The agreement between theory and experiment is quite good and support the Maxwell-Stefan formulation of diffusion in multicomponent ideal gas mixtures. This conclusion was also reached by Bres and Hatzfeld (1977) and by Hesse and Hugo (1972). For further analysis of the Stefan diffusion tube see Whitaker (1991). ... 

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... 

Johns, L. E. and DeGance, A. E, Diffusion in Ternary Ideal Gas Mixtures. I. On the Solution of the Stefan-Maxwell Equation for Steady Diffusion in Thin Films, Ind. Eng. Chem. Fundam., 14, 237-245 (1975). 

Olivera-Fuentes, C. G. and Pasquel-Guerra, J., The Exact Penetration Model of Diffusion in Multicomponent Ideal Gas Mixtures. Analytical and Numerical Solutions, Chem. Eng. Commun., 51, 71-88 (1987). 

The ideal Thiele-Damkohler theory also assumes that mass transfer in the particle occurs exclusively by diffusion. In a gas reaction, however, the volume of the reacting mixture expands if the mole number increases, and contracts if the mole number decreases. If it expands, forced convection out of the particle counteracts reactant diffusion into it and thereby slows the reaction down. If the volume contracts, forced convection sucks reactant into the particle and speeds the reaction up [16,28]. 

Problems involving unsteady diffusion in more then one coordinete direction, such as a cylinder or finite length or a long slab of comparable width and depth dimensions, cun usually be solvnd by separation of variables. For example, for a cylinder of radius R and length L the governing equation and boundary conditions for an ideal gas mixture and constant surface conditions would be... 

The Maxwell-Stefan equations do not depend on choice of the reference velocity. For ideal gas mixtures, diffusivities Z) are independent of the composition, and equal to diffusivity D npf the hinary pair kl. In an w-component system, only n n-l)/2 different Maxwell-Stefan diffusivities are required as a result of the simple symmetry relations. Some advantages of the Maxwell-Stefan description of diffusion are ... 

The MaxweU-Stefan equations for describing the diffusion of gases in a multicomponent gas mixture have been developed from the kinetic theory of gases. A highly simplified illustration may be pursued as follows. Consider a system of a gas mixture of n species at constant T and P. Focus first on molecules of species i. The net force exerted on species i in the absence of any external forces is - Vpig / gmol of i. The net force exerted on species i per unit volume of the mixture is —(V/tjg) xig Ctg, where C,g is the total molar density of the gas mixture from equation (3.1.40) and an ideal gas mixture, the net force on species i per unit mixture volume is —xjg Ctg Rr V tn Pxtg), which is equivalent to... 

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