Diffusion in a Multicomponent System

The high quality of the fits and similarity in Do values (1.89 imi ls for H2O profile and 1.95 fim ls for profile) confirm the assumption. The dashed curve in (h) is an error function 

Uphill diffusion of some components is reported in silicate melts (e.g., Sato, 1975 Watson, 1982a Zhang et al., 1989 Lesher, 1994 Van Der Laan et al., 1994). Recall that uphill diffusion in binary systems is rare and occurs only when the two-component phase undergoes spinodal decomposition. In multicomponent systems, uphiU diffusion often occurs even when the phase is stable, and may be explained by cross-effects of diffusion by other components. 

Several different ways have been developed to deal with diffusion in a multi-component system. In order of increasing complexity and increasing accuracy, they are 

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The entropy production during diffusion in a multicomponent system is (Appendix 1)... 

Diffusion in multicomponent system is difficult to analyze. Transport of one component is affected by the presence of the other component due to their mumal interaction. This results in the coupling of fluxes. Thus, single-component diffusion equation cannot be used to predict diffusion in a multicomponent system. Greenlaw et al. [38] proposed a simple relationship in which the diffusion coefficients for components i and j are interdependent on both component concentrations ... 

When we combine Eqs. 4.2.15 and 4.2.17 we obtain the following elegant expression for the binary Maxwell-Stefan diffusivity in a multicomponent system. 

The parameter D is known as the axial dispersion coefficient, and the dimensionless number, Pe = uL/D, is the axial Peclet number. It is different than the Peclet number used in Section 9.1. Also, recall that the tube diameter is denoted by df. At high Reynolds numbers, D depends solely on fluctuating velocities in the axial direction. These fluctuating axial velocities cause mixing by a random process that is conceptually similar to molecular diffusion, except that the fluid elements being mixed are much larger than molecules. The same value for D is used for each component in a multicomponent system. 

If the difference in concentration is in only two exchangeable components, such as FeO and MgO, the interdiffusion in a multicomponent system may be treated as effective binary. The diffusion of other components in the system may or may not be treated as effective binary diffusion. 

In multicomponent systems, the single diffusivity is replaced by a multicomponent diffusion matrix. By going through similar steps, it can be shown that the [D] matrix must have positive eigenvalues if the phase is stable. In a multicomponent system, the diffusive flux of a component can be up against its chemical potential gradient except for eigencomponents. 

After this formal discussion of chemical diffusion, let us now turn to some more practical aspects. In order to compare the formal theory with experiment, we have to carefully define the reference frame for the diffusion process, which is not trivial in the case of binary or multicomponent diffusion. To become acquainted with the philosophy of this problem, we deal briefly with defining a suitable reference frame in a binary system. Since only one (independent) transport coefficient is needed to describe chemical diffusion in a binary system, then according to Eqn. (4.57) we have in a one-dimensional system... 

Thermal diffusion, also known as the Ludwig-Soret effect [1, 2], is the occurrence of mass transport driven by a temperature gradient in a multicomponent system. While the effect has been known since the last century, the investigation of the Ludwig-Soret effect in polymeric systems dates back to only the middle of this century, where Debye and Bueche employed a Clusius-Dickel thermogravi-tational column for polymer fractionation [3]. Langhammer [4] and recently Ecenarro [5, 6] utilized the same experimental technique, in which separation results from the interplay between thermal diffusion and convection. This results in a rather complicated experimental situation, which has been analyzed in detail by Tyrrell [7]. 

When considering ordinary diffusion only, the momentum conservation equation for the species i in a multicomponent system can be written as [17]... 

In the theoretical treatment of diffusive reactions, one usually works with diffusion coefficients, which are evaluated from experimental measurements. In a multicomponent system, a large number of diffusion coefficients must be evaluated, and are generally interrelated functions of alloy composition. A database would, thus, be very complex. A superior alternative is to store atomic mobilities in the database, rather than diffusion coefficients. The number of parameters which need to be stored in a multicomponent system will then be substantially reduced, as the parameters are independent. The diffusion coefficients, which are used in the simulations, can then be obtained as a product of a thermodynamic and a kinetic factor. The thermodynamic factor is essentially the second derivative of the molar Gibbs energy with respect to the concentrations, and is known if the system has been assessed thermodynamically. The kinetic factor contains the atomic mobilities, which are stored in the kinetic database. 

The requirement [Sc] = [/] for a multicomponent system is a much more special case than for a corresponding binary system for it requires that all binary pair diffusivities in the multicomponent system be equal to one another and, furthermore, that r/D = 1, a situation realizable only for ideal mixtures made up of species of similar size and nature. 

For diffusion in isothermal multicomponent systems the generalized driving force was written as a linear function of the relative velocities (m/ — My). In the general case, we must allow for coupling between the processes of heat and mass transfer and write constitutive relations for and q in terms of the (m — My) and V(l/r). With this allowance, the complete expression for the conductive heat flux is... 

The presence of temperature gradients in a multicomponent system introduces an additional complication in the analysis of the mass transfer process such gradients influence the values of physical, thermodynamic, and transport properties, such as the diffusion coefficients. These property variations may be taken care of by introducing temperature dependent property functions or by using average values of the properties (as is done here). The consequence of this simplification is that the basic mass transfer analysis remains essentially unchanged from those in Chapters 8-10 and we need only consider the effect of mass transfer on the heat transfer process. 

Small particles in a temperature gradient are driven from the high- to low-temperature regions. This effect was first observed in the nineteenth century when it was discovered that a dust-free or dark space surrounded a hot body, suitably illuminated. Particle transport in a temperature gradient has been given the name thermophoresis, which means being carried by heat." Thermophoresis is closely related to the molecular phenomenon thermal diffusion, transport produced by a temperature gradient in a multicomponent system. 

Now, in a multicomponent system, the variation of the chemical potential with space can be expressed in terms of the molar fractions, or concentrations as function of space. Further the velocity of the particles can be expressed in terms of a material flux across an imaginary perpendicular surface to the respective axis. In this way, the equation of diffusion can be derived from thermodynamic arguments. We emphasize that we have now silently crossed over from equilibrium thermodynamics to irreversible thermodynamics. 

In a multicomponent system, the diffusivity of A is estimated by assuming that it diffuses through a stagnant film of the other gases. Then the overall diffusivity, say, of component j can be calculated from the various constituent binary diffusivities using one of the following two equations (the second is slightly more accurate) ... 

In this section we shall once again be concerned with binary metallic systems. In the previous section, some general remarks were made regarding diffusion in multiphase multicomponent systems. The complexity of the problem was pointed out, and the difficulties entailed in a quan-... 

Diffusion Coefficients in Multicomponent Systems. The value of the diffusion coefficient of a species in a binary system is often not the same as the value in a multicomponent system. The diffusion coefficients can be modified in multicomponent systems as a result of added frictional forces at the atomistic scale. The multiple diffusing species interact in various complex ways that can be described using equation 9, which is derived from the so-called Stefan-Maxwell relations (4) ... 

Show that in a multicomponent system the summation of the diffusive mass flux of all the components is zero ... 

Eq. (1.12) is the generalization of eq. (1.1) i.e. of the Fick s empirical relationship. It is clear that in a multicomponent system, (r >2), the diffusion flux of component i is driven not only by its concentrations gradients but also by the concentration gradients of the other components. 

For diffusion in a mulhcomponent system, the gas mixture involves three or more species. For example, the gas supply in the anode side may involve multicomponent diffusion of both hydrogen and water species. If oxygen is supplied as air in the cathode side of the fuel cell, then it involves multicomponent diffusion of three components oxygen, nitrogen, and water vapor. The diffusion coefficient of species a depends not only on the concentration gradient... 

Thus, the diffusion process with the highest AE should become the main rate determinant in the nucleation process. Diffusion within the crystals is not well known, but it is sometimes important, for example, nucleation in a multicomponent system [11], and the chain sliding diffusion of polymer chains, which mainly controls the formation of folded-chain crystals (FCCs) and extended-chain crystals (ECCs) [14,15]. 

A major disadvantage of the Maxwell-Stefan theory is that the diffusion coefficients, with the exception of the diffusion of dilute gases, do not correspond to the Pick s diffusion coefficients and are therefore not tabulated. Only the diffusion coefficients for the binary and ternary case can be determined with reasonable effort. In a multicomponent system, a set of approximate formulae exist to predict the Maxwell-Stefan diffusion coefficient. 

When concentration gradients exist in a multicomponent system, there is a natural tendency for the concentration differences to be reduced and, ultimately, eliminated by mass transfer. This is the process of diffusion, and mass transfer occurs by molecular means. Thus, water evaporates from an open dish and increases the humidity of the air. However, the rate of mass transfer can be increased by blowing air past the dish. This is called convective mass tranter or mass transfer due to flow. 

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. 

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