Multicomponent diffusion coefficient

Multicomponent Diffusion. In multicomponent systems, the binary diffusion coefficient has to be replaced by an effective or mean diffusivity Although its rigorous computation from the binary coefficients is difficult, it may be estimated by one of several methods (27—29). Any degree of counterdiffusion, including the two special cases "equimolar counterdiffusion" and "no counterdiffusion" treated above, may arise in multicomponent gas absorption. The influence of bulk flow of material through the films is corrected for by the film factor concept (28). It is based on a slightly different form of equation 13 ... 

Most distillation systems ia commercial columns have Murphree plate efficiencies of 70% or higher. Lower efficiencies are found under system conditions of a high slope of the equiHbrium curve (Fig. lb), of high Hquid viscosity, and of large molecules having characteristically low diffusion coefficients. FiaaHy, most experimental efficiencies have been for biaary systems where by definition the efficiency of one component is equal to that of the other component. For multicomponent systems it is possible for each component to have a different efficiency. Practice has been to use a pseudo-biaary approach involving the two key components. However, a theory for multicomponent efficiency prediction has been developed (66,67) and is amenable to computational analysis. 

For prediction of gas phase diffusion coefficients in multicomponent hydi ocarbon/nonKydi ocai bon gas systems, the method of Wilke shown in Eq. (2-154) is used. 

Multicomponent Mixtures No simple, practical estimation methods have been developed for predicting multicomponent hquid-diffusion coefficients. Several theories have been developed, but the necessity for extensive activity data, pure component and mixture volumes, mixture viscosity data, and tracer and binaiy diffusion coefficients have significantly limited the utihty of the theories (see Reid et al.). 

The generalized Stefan-Maxwell equations using binary diffusion coefficients are not easily applicable to hquids since the coefficients are so dependent on conditions. That is, in hquids, each Dy can be strongly composition dependent in binary mixtures and, moreover, the binaiy is strongly affected in a multicomponent mixture. Thus, the convenience of writing multicomponent flux equations in terms of binary coefficients is lost. Conversely, they apply to gas mixtures because each is practically independent of composition by itself and in a multicomponent mixture (see Taylor and Krishna for details). 

According to Maxwell s law, the partial pressure gradient in a gas which is diffusing in a two-component mixture is proportional to the product of the molar concentrations of the two components multiplied by its mass transfer velocity relative to that of the second component. Show how this relationship can be adapted to apply to the absorption of a soluble gas from a multicomponent mixture in which the other gases are insoluble and obtain an effective diffusivity for the multicomponent system in terms of the binary diffusion coefficients. 

The ordinary multicomponent diffusion coefficients D j and the viscosity and thermal conductivity are computed from appropriate kinetic theory expressions. First, pure species properties are computed from the standard kinetic theory expressions. For example, the binary diffusion coefficients are given in terms of pressure and temperature as... 

The task of the problem-independent chemistry software is to make evaluating the terms in Equations (6-10) as straightforward as possible. In this case subroutine calls to the Chemkin software are made to return values of p, Cp, and the and hk vectors. Also, subroutine calls are made to a Transport package to return the ordinary multicomponent diffusion matrices Dkj, the mixture viscosities p, the thermal conductivities A, and the thermal diffusion coefficients D. Once this is done, finite difference representations of the equations are evaluated, and the residuals returned to the boundary value solver. 

Diffusion of ions can be observed in multicomponent systems where concentration gradients can arise. In individnal melts, self-diffnsion of ions can be studied with the aid of radiotracers. Whereas the mobilities of ions are lower in melts, the diffusion coefficients are of the same order of magnitude as in aqueous solutions (i.e., about 10 cmVs). Thus, for melts the Nemst relation (4.6) is not applicable. This can be explained in terms of an appreciable contribntion of ion pairs to diffusional transport since these pairs are nncharged, they do not carry cnrrent, so that values of ionic mobility calculated from diffusion coefficients will be high. 

The diffusion coefficient (D) of ferf-butyloxycarbonyl-L-phenylalanine (Boc-Phe) was determined in a Merrifield network of polystyrene (PS) gels used as a solid phase reaction field.108 When probe molecules have multicomponents in diffusion on the measurement time scale, the total echo attenuation is given by a superposition of contributions from the individual components ... 

TRANFIT A Fortran Computer Code Package for the Evolution of Gas-Phase Multicomponent Transport Properties, Kee, R. J., Dixon-Lewis, G., Wamatz, J., Coltrin, M. E. and Miller, J. A. Sandia National Laboratories, Livermore, CA, Sandia Report SAND86-8246, 1986. TRANFIT is a Fortran computer code (tranlib.f, tranfit.f, and trandatf) that allows for the evaluation and polynomial fitting of gas-phase multicomponent viscosities, thermal conductivities, and thermal diffusion coefficients. 

The development of mixture sorption kinetics becomes increasingly Important since a number of purification and separation processes involves sorption at the condition of thermodynamic non-equilibrium. For their optimization, the kinetics of multicomponent sorption are to be modelled and the rate parameters have to be identified. Especially, in microporous sorbents, due to the high density of the sorption phase and, therefore, the mutual Influences of sorbing species, a knowledge of the matrix of diffusion coefficients is needed [6]. The complexity of the phenomena demands combined experimental and theoretical research. Actual directions of the development in this field are as follows ... 

Values of a diffusion coefficient matrix, in principle, can be determined from multicomponent diffusion experiments. For ternary systems, the diffusivity matrix is 2 by 2, and there are four values to be determined for a matrix at each composition. For quaternary systems, there are nine unknowns to be determined. For natural silicate melts with many components, there are many unknowns to be determined from experimental data by fitting experimental diffusion profiles. When there are so many unknowns, the fitting of experimental concentration... 

Loomis T.P., Ganguly J., and Elphick S.C. (1985) Experimental determination of cation diffusivities in aluminosilicate garnets, II multicomponent simulation and tracer diffusion coefficients. Contrib. Mineral. Petrol. 90, 45-51. 

It is not difficult to generalize the above formulation of the effective diffusion coefficient to the case in which there appear r different kinds of barriers or perturbation elements in the diffusion space. The result can be used to formulate the effective diffusion coefficient in a multicomponent solution. See [144] for a detailed explanation. 

In these equations is the partial molal free energy (chemical potential) and Vj the partial molal volume. The Mj are the molecular weights, c is the concentration in moles per liter, p is the mass density, and z, is the mole fraction of species i. The D are the multicomponent diffusion coefficients, and the are the multicomponent thermal diffusion coefficients. The first contribution to the mass flux—that due to the concentration gradients—is seen to depend in a complicated way on the chemical potentials of all the components present. It is shown in the next section how this expression reduces to the usual expressions for the mass flux in two-component systems. The pressure diffusion contribution to the mass flux is quite small and has thus far been studied only slightly it is considered in Sec. IV,A,6. The forced diffusion term is important in ionic systems (C3, Chapter 18 K4) if gravity is the only external force, then this term vanishes identically. The thermal diffusion term is impor-... 

Several remarks need to be made concerning the definition of the diffusion coefficients in Eqs. (32) to (36) above. The multicomponent diffusion coefficients Dij used here differ from those used in references (Hll) and (Bll). The latter must be multiplied by cRT/p to get the diffusion coefficients defined here. For perfect gases, of course, this difference in definition is unimportant since cRT/p is unity. For liquid, the definition used here is to be preferred, inasmuch as it is more in conformity with the customary definition used by experimentalists. The D ,- used here are defined in such a way that Da D, and Du = 0. 

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... 

Note that in Eq. (60) it is the multicomponent Dtj which appear, whereas in Eq. (61) the binary diffusion coefficients Sh, appear.13 For a system at constant temperature and pressure Eq. (61) may also be written... 

DiT Multicomponent thermal diffusion coefficients 3>ab Binary diffusion coefficients (37)... 

In thermal transport there was need of considering only temperature as the driving potential. However, in the case of multicomponent material transport it is necessary to evaluate the behavior of each component. Such relationships lead to rather complicated expressions when the diffusion of each of the components is considered individually. For this reason it is often appropriate to focus attention on one component and consider the characteristics of the remainder of the components in the phase as invariant. In the present discussion only the behavior of one component will be considered, but it should be realized that the effect of the properties of the phase upon the diffusion coefficient must be taken into account. 

In this text we are concerned exclusively with laminar flows that is, we do not discuss turbulent flow. However, we are concerned with the complexities of multicomponent molecular transport of mass, momentum, and energy by diffusive processes, especially in gas mixtures. Accordingly we introduce the kinetic-theory formalism required to determine mixture viscosity and thermal conductivity, as well as multicomponent ordinary and thermal diffusion coefficients. Perhaps it should be noted in passing that certain laminar, strained, flames are developed and studied specifically because of the insight they offer for understanding turbulent flame environments. 

Here D km represents a mixture-averaged diffusion coefficient for species k relative to the rest of the multicomponent mixture. The species mass-flux vector is given in terms of the mole-fraction gradient as... 

Here Dkj is the matrix of ordinary multicomponent diffusion coefficients, and Dj are the thermal diffusion coefficients. The vector dk represents the gradients in the concentration... 

Various forms of diffusion coefficients are used to establish the proportionality between the gradients and the mass flux. Details on determination of the diffusion coefficients and thermal diffusion coefficients is found in Chapter 12. Here, however, it is appropriate to summarize a few salient aspects. In the case of ordinary diffusion (proportional to concentration gradients), the ordinary multicomponent diffusion coefficients Dkj must be determined from the binary diffusion coefficients T>,kj. The binary diffusion coefficients for each species pair, which may be determined from kinetic theory or by measurement, are essentially independent of the species composition field. Calculation of the ordinary multicomponent diffusion coefficients requires the computation of the inverse or a matrix that depends on the binary diffusion coefficients and the species mole fractions (Chapter 12). Thus, while the binary diffusion coefficients are independent of the species field, it is important to note that ordinary multicomponent diffusion coefficients depend on the concentration field. Computing a flow field therefore requires that the Dkj be evaluated locally and temporally as the solution evolves. 

Chapter 12 provides a great deal more information about this and alternative formulations of multicomponent diffusion coefficients. 

In the foregoing discussion the diffusive mass fluxes are written in terms of the diffusion velocities, which in turn are determined from gradients of the concentration, temperature, and pressure fields. Such explicit evaluation of the diffusion velocities requires the evaluation of the multicomponent diffusion coefficients from the binary diffusion coefficients. 

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