Diffusion of binary mixtures

For diffusion of a binary mixture in Silicalite equations 5.5 and 5.8 reduce to 

The interchange mechanism is often ignored and the following formulation used 

Recent work of Van de Graaf et al. [209] and Kapteijn et al. [210] has shown that for diffusion of binary mixtures in Silicalite, the complete Maxwell-Stefan formulation, equation 5.14, tciking interchange into account provides a much better description of binary permeation experimental results across a Silicalite membrane than with a model ignoring the interchange mechanism (portrayed by D12). 

The four elements of Fy can be obtained by analytic differentiation of equation 5.6. The result 

To demonstrate the consequences of the influence of Fy on diffusion, consider the permeation of hydrocarbon isomers across a Silicalite membrane (figure 5.8). To obtain the values of the permeation fluxes we need to solve the set of two coupled partial differenticd equations  

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In particular, for the species A in the case of Pick diffusion of binary mixtures [48] the flux is given ... 

Y.D. Chen and R.T. Yang, Preparation of carbon molecular sieve membrane and diffusion of binary mixtures in the membrane, Ind. Eng. Chem. Res. 55 3146 (1994). 

Paschek D and Krishna R. Diffusion of binary mixtures in zeolites Kinetic Monte Carlo versus molecular dynamics simulations. 

Chempath, S. Krishna, R. Snurr, R.Q. Nonequilibrium molecular dynamics simulations of diffusion of binary mixtures containing short n-alkanes in faujasite. J. Phys. Chem 2004, 108, 13481-13491. 

B. Srivastava, Mutual Diffusion of Binary Mixtures of Ammonia with He, Ne and Xe, Indian J. Phys, 36 193,1962. 

We have discussed the adsorption and diffusion of binary mixtures of hnear (n-hexane) and branched (2-methylpentane) alkanes in silicahte-1. It turned out that not only the size but also the siting of the molecules in the particular zeohte plays an important role in the behavior of the mixture components. A shght preference for the adsorption of n-hexane over 2-methyl-pentane was observed because of the higher packing efficiency of the hnear alkane. This is due to the preferential location of the branched alkane in the zeohte intersections. A consequence of this is that the diffusivity of n-hexane... 

Krishna R., Diffusion of Binary Mixtures in Zeolites Molecular Dynamics Simulations versus Maxwell-Stefan Theory. Chem. Phys. Lt. 326 (2000) pp ATI-AM... 

Chen YD, Yang RT (1994) Preparation of carbon molecular sieve membrane and diffusion of binary mixtures in the membrane. Ind Eng Chem Res 33 (12) 3146-3153 Moaddeb M, Koros WJ (1997) Gas transport properties of thin polymeric membranes in the presence of silicon dioxide particles. J Membr Sci 125 (1) 143-163 Ash R, Barrer RM, Lowson RT (1973) Transport of single gases and of binary gas mixtures in a microporous carbon membrane. J Chem Soc Faraday Trans I 69 (12) 2166-2178 Bird AJ, Trimm DL (1983) Carbon molecular sieves used in gas separation membranes. Carbon 21 (3) 177-163... 

Chen and Yang [9] prepared a large, crack-free carbon molecular sieve membrane (CMSM) supported on a macroporous substrate by coating a layer of PFA followed by controlled pyrolysis. Diffusion of binary mixtures was measured and the results were compared with the kinetic theory for predicting binary dififusivities from pure component dififusivities. Good agreement was obtained between theoretical predictions and experimental data for binary diffusion, as shown in Chap. 2. 

The type of treatment described here was originally introduced by Scott and Dullien [4], who confined attention to isothermal isobaric diffusion in binary mixtures. Similar equations were independently published shortly after by Rothfeld [5], and the method was later extended to multi-component mixtures by Silveston [6], Perhaps the most complete exposition is given by Mason and Evans [7],... 

For prediction of the gas diffusivity of binary air-hydrocarbon or nonhydrocarbon gas mixtures at low pressures, the method of Fuller et al. given by Eq. (2-152) is recommended. 

The methods outlined in Reid and Sherwood (87) have been used to estimate the diffusivities characteristic of binary mixtures of the various components of the fluid in the reactor. 

As indicated in my report, we now know the rates of lateral diffusion of phospholipids in lipid bilayers in the fluid state, and in a few cases the rates of lateral diffusion of proteins in fluid lipids are also known. At the present time nothing is known about the rates of lateral diffusion of phospholipids in the crystalline, solid phases of the substances. As mentioned in my report, there are reasons to suspect that the rates of lateral diffusion of phospholipids in the solid solution crystalline phases of binary mixtures of phospholipids may be appreciable on the experimental time scale. Professor Ubbelohde may well be correct in pointing out the possibility of diffusion caused by defects. However, such defects, if present, apparently do not lead to significant loss of the membrane permeability barrier, except at domain boundaries. 

Although there has not been much theoretical work other than a quantitative study by Hynes et al [58], there are some computer simulation studies of the mass dependence of diffusion which provide valuable insight to this problem (see Refs. 96-105). Alder et al. [96, 97] have studied the mass dependence of a solute diffusion at an infinite solute dilution in binary isotopic hard-sphere mixtures. The mass effect and its influence on the concentration dependence of the self-diffusion coefficient in a binary isotopic Lennard-Jones mixture up to solute-solvent mass ratio 5 was studied by Ebbsjo et al. [98]. Later on, Bearman and Jolly [99, 100] studied the mass dependence of diffusion in binary mixtures by varying the solute-solvent mass ratio from 1 to 16, and recently Kerl and Willeke [101] have reported a study for binary and ternary isotopic mixtures. Also, by varying the size of the tagged molecule the mass dependence of diffusion for a binary Lennard-Jones mixture has been studied by Ould-Kaddour and Barrat by performing MD simulations [102]. There have also been some experimental studies of mass diffusion [106-109]. 

Mode coupling theory of binary mixtures where the constituents are of rather different sizes is a challenging task, as we have already discussed while addressing the mass depenence of diffusion. In addition to the problem with proper formulation of mode coupling terms, there is an additional difficulty of the nonavailability of the equilibrium two-particle correlation functions The existing integral equation theories become unstable when the size ratio exceeds a certain (low) value, like 1.5 or so [195],... 

The Stokes-Einstein equation can also be used to estimate the diffusivity of binary liquid mixtures... 

Equimolar counter diffusion appears in the distillation of binary mixtures. In a distillation column the liquid falls downwards, and the vapour flows upwards, Fig. 1.43. As the liquid flowing down the column is colder than the vapour flowing upwards, chiefly the component with the higher boiling point, the so called least volatile component condenses, whilst the vapour from the boiling liquid mainly consists of the components with the lower boiling points, the more volatile components. The molar enthalpy of vaporization is, according to Trouton s rule, approximately constant for all components. If a certain amount of the least volatile component condenses out from the vapour, then the same number of moles of the more volatile substance will be evaporated out of the liquid. At the phase boundary between liquid and vapour we have cAwA = —cBwB. The reference velocity u is zero because cu = cAwA + cBwB. The molar flux transported to the phase boundary from (1.158) and (1.160) is... 

Figures 16.23a to d compare experimental profiles of mixtures of the enantiomers of 1-indanol on cellulose tribenzoate with those calculated with the GMS-GRM model of these authors [57]. For the numerical calculations, they assmned that surface diffusion plays the dominant role in mass transfer across the particles and neglected the contribution of pore diffusion to the fluxes. Unfortunately, it was impossible independently to measure or even estimate the surface diffusion parameters. So, the numerical values of the surface diffusion coefficients needed for the calculation were estimated by minimizing the discrepancies between the measured and the calculated band profiles i.e., by parameter adjustment). Yet, it is impressive that, using a unique set of diffusion coefficients, it was possible to calculate band profiles of single components of binary mixtures in the whole range of relative composition, for loading factors between 0 and 10%.

In the DGM model as presented by Mason and Malinauskas [11a] all the different contributions to the transport are taken into accoimt. The wall of the porous medium is considered as a very heavy component and so contributes to the momentum transfer. The model is schematically represented in Fig. 9.12 for a binary mixture (in analogy with an electriccd network). As can be seen from this figure, the flux contributions by Knudsen diffusion /k, and of molecular (continuum) diffusion of the mixture /m,i23re in series and so are coupled. The total flux of component i (i = 1,2) due to these contributions is /j km- Note that /k = /m,i2- The contribution of the viscous flow and of the surface diffusion are parallel with / km J d so are considered independent of each other (no coupling terms, e.g. no transport interaction between gas phase and surface diffusion). 

As discussed in Section 9.4.1, the contribution of Knudsen diffusion to the total flux decreases with decreasing pore radius of the membrane material. Initially the selectivity of binary mixtures of gases is constant and equal to the Knudsen value. 

Rathbun, R. E. and Babb, A. L., Empirical Method for Prediction of The Concentration Dependence of Mutual Diffusivities in Binary Mixtures of Associated and Nonpolar Liquids, Ind. Eng. Chem. Proc. Des. Dev., 5, 273-275 (1966). 

In equimolar diffusion, a binary mixture is assumed where the two components diffuse in opposite directions at equal rates. These conditions exist in binary distillation where a mole of component 1 is vaporized for each mole of component 2 that is condensed. In unimolar diffusion, one component diffuses through a second, stagnant one. This is typical of an absorption process where one component diffuses through the gas phase to the interface boundary, is absorbed by the liquid, and then diffuses to the bulk of the liquid. The other gas components are assumed to remain in the gas and the liquid components to remain in the liquid. 

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