Multicomponent diffusion flux equations

Wilke [103] proposed a simpler model for calculating the effective diffusion coefficients for diffusion of a species s into a multicomponent mixture of stagnant gases. For dilute gases the Maxwell-Stefan diffusion equation is reduced to a multicomponent diffusion flux model on the binary Pick s law form in which the binary diffusivity is substituted by an effective multicomponent diffusivity. The Wilke model derivation is examined in the sequel. 

By solving the effective multicomponent diffusion flux as postulated by Wilke (2.319) for Vxs and equating the result to the simplified driving force (2.301) in the Maxwell-Stefan equations (2.320), we get ... 

Having shown the constitutive flux equation for binary and ternary systems, we can readily generalise the result to a multicomponent system containing n species by simply writing down the following diffusion flux equation for the component i ... 

The Chapman-Enskog kinetic theory cf gases (Hirschfelder et al., 1964) is used to describe the multicomponent diffusion flux of species i in a mixture of n gas species and expressed as the Stefan-Maxwell equation (Bird et al, 2002). The diffusion flux of species i is given as... 

Stefan-Maxwell Approach A more rigorous approach to multispecies diffusion effects is known as the Stefan-Maxwell diffusion model and should be used for higher order models seeking the greatest accuracy. The Stefan-Maxwell equation for multicomponent diffusion flux in the x direction is shown as... 

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). 

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-... 

Generally, a set of coupled diffusion equations arises for multiple-component diffusion when N > 3. The least complicated case is for ternary (N = 3) systems that have two independent concentrations (or fluxes) and a 2 x 2 matrix of interdiffusivities. A matrix and vector notation simplifies the general case. Below, the equations are developed for the ternary case along with a parallel development using compact notation for the more extended general case. Many characteristic features of general multicomponent diffusion can be illustrated through specific solutions of the ternary case. 

The balance over the ith species (equation IV. 5) consists of contributions from diffusion, convection, and loss or production of the species in ng gas-phase reactions. The diffusion flux combines ordinary (concentration) and thermal diffusions according to the multicomponent diffusion equation (IV. 6) for an isobaric, ideal gas. Variations in the pressure induced by fluid mechanical forces are negligible in most CVD reactors therefore, pressure diffusion effects need not be considered. Forced diffusion of ions in an electrical field is important in plasma-enhanced CVD, as discussed by Hess and Graves (Chapter 8). 

Multicomponent diffusion in the films is described by the Maxwell-Stefan equations, which can be derived from the kinetic theory of gases (89). The Maxwell-Stefan equations connect diffusion fluxes of the components with the gradients of their chemical potential. With some modification these equations take a generalized form in which they can be used for the description of real gases and liquids (57) ... 

The generic balance relations and the derived relations presented in the preceding section contain various diffusion flux tensors. Although the equation of continuity as presented does not contain a diffusion flux vector, were it to have been written for a multicomponent mixture, there would have been such a diffusion flux vector. Before any of these equations can be solved for the various field quantities, the diffusion fluxes must be related to gradients in the field potentials . 

Equations (10) are generally valid for both liquid and gas phases if reactions take place there. They represent nothing but a differential mass balance for the film region with the account of the source term due to the reaction. To link this balance to the process variables like component concentrations, some additional relationships - often called constitutive relations (see Ref. [16]) - are necessary. For the component fluxes Ni, these constitutive relations result from the multicomponent diffusion description (Eqs. (1), (2)) for the source terms, from the reaction kinetics description. The latter strongly depends on the specific reaction mechanism [27]. The reaction rate expressions lli usually represent nonlinear dependencies on the mixture composition and temperature of the corresponding phase. 

Recall this equation is similar to Equation (6.3.7) for the flux of A in one dimension. To solve the multicomponent diffusion/reaction problem of benzene hydrogenation in one dimension. Equation (6.3.66) must instead be used. First, let ... 

Thus, the transformed state variables Xi and transformed fluxes J for each i satisfy the flux equations and conservation equations of a binary system (Stewart and Prober 1964) with the same v function (laminar or turbulent) as the multicomponent system and with a material diffusivity Di = c/Xi, giving a Schmidt number fi/pDi = p/p) Xi/c). 

Multicomponent diffusion in pores is described by the dusty-gas model (DGM) [38,44,46 8]. This model combines molecular diffusion, Knudsen diffusion, viscous flux, and surface diffusion. The DGM is suitable for any model of porous structure. It was developed by Mason et al. [42] and is based on the Maxwell-Stefan approach for dilute gases, itself an approximation of Boltzmann s equation. The diffusion model obtained is called the generalized Maxwell-Stefan model (GMS). Thermal diffusion, pressmn diffusion, and forced diffusion are all easily included in the GMS model. This model is based on the principle that in order to cause relative motion between individual species in a mixture, a driving force has to be exerted on each of the individual species. The driving force exerted on any particular species i is balanced by the friction this species experiences with all other species present in the mixture. Each of these friction contributions is considered to be proportional to the corresponding differences in the diffusion velocities. 

Equations 9.3.15 and 9.3.16 represent an exact analytical solution of the multicomponent penetration model. For two component systems, these results reduce to Eqs. 92.1. Unfortunately, the above results are of little practical use for computing the diffusion fluxes because they require an a priori knowledge of the composition profiles (cf. Section 8.3.5). Thus, a degree of trial and error over and above that normally encountered in multicomponent mass transfer calculations enters into their use. Indeed, Olivera-Fuentes and Pasquel-Guerra did not perform any numerical computations with this method and resorted to a numerical integration technique. 

Cullinan, H. T., Analysis of the Flux Equations of Multicomponent Diffusion, Ind. Eng. Chem. 

It is possible to justify several alternative definitions of the multicomponent diffusivities. Even the multicomponent mass flux vectors themselves are expressed in either of two mathematical forms or frameworks referred to as the generalized Fick- and Maxwell-Stefan equations. 

In the literature the net momentum flux transferred from molecules of type s to molecules of type r has either been expressed in terms of the average diffusion velocity for the different species in the mixture [77] or the average species velocity is used [96]. Both approaches lead to the same relation for the diffusion force and thus the Maxwell-Stefan multicomponent diffusion equations. In this book we derive an approximate formula for the diffusion force in terms of the average velocities of the species in the mixture. The diffusive fluxes are introduced at a later stage by use of the combined flux definitions. 

The design of a complete set of governing equations for the description of reactive flows requires that the combined fluxes are treated in a convenient way. In principle, several combined flux definitions are available. However, since the mass fluxes with respect to the mass average velocity are preferred when the equation of motion is included in the problem formulation, we apply the species mass balance equations to a (/-component gas system with q — independent mass fractions Wg and an equal number of independent diffusion fluxes js. However, any of the formulations derived for the multicomponent mass diffusion flux can be substituted into the species mass balance (1.39), hence a closure selection optimization is required considering the specified restrictions for each constitutive model and the computational efforts needed to solve the resulting set of model equations for the particular problem in question. 

For multicomponent systems the diffusive flux term may be written in accordance with the Maxwell-Stefan equations (2.394) with the driving force... 

The applicability of the multicomponent mass diffusion models to chemical reactor engineering is assessed in the following section. Emphasis is placed on the first principles in the derivation of the governing flux equations, the physical interpretations of the terms in the resulting models, the consistency with Pick s first law for binary systems, the relationships between the molar and mass based fluxes, and the consistent use of these multicomponent models describing non-ideal gas and liquid systems. 

The rigorous Fickian multicomponent mass diffusion flux formulation is derived from kinetic theory of dilute gases adopting the Enskog solution of the Boltzmann equation (e.g., [17] [18] [19] [89] [5]). This mass flux is defined by the relation given in the last line of (2.281) ... 

A multicomponent Fickian diffusion flux on this form was first suggested in irreversible thermodynamics and has no origin in kinetic theory of dilute gases. Hence, basically, these multicomponent flux equations represent a purely empirical generalization of Pick s first law and define a set of empirical multi-component diffusion coefficients. 

The Stefan-Maxwell (Maxwell, 1860 Stefan, 1872) equation gives implicit relations for the fluxes when the system is isothermal and the wall effects are negligible, this means negligible viscous transport (i.e. constant pressure) and Knudsen diffusion. For multicomponent mixture the equation has the form ... 

For a multicomponent mixture of ideal gases such as this, the Maxwell-Stefan equations (1-35) must be solved simultaneously for the fluxes and the composition profiles. At constant temperature and pressure the total molar density c and the binary diffusion coefficients are constant. Furthermore, diffusion in the Stefan tube takes place in only one direction, up the tube. Therefore, the continuity equation simplifies to equation (1-69). Air (3) diffuses down the tube as the evaporating mixture diffuses up, but because air does not dissolve in the liquid, its flux N3 is zero (i.e., the diffusion flux J3 of the gas down the tube is exactly balanced by the diffusion-induced bulk flux x3N up the tube). 

Having presented the flux equations for a multicomponent system, we will apply the Stefan-Maxwell s approach to solve for fluxes in the Stefan tube at steady state. Consider a Stefan tube (Figure 8.2-3) containing a liquid of species 1. Its vapour above the liquid surface diffuses up the tube into an environment in which a species 2 is flowing across the top, which is assumed to be nonsoluble in liquid. 

The last two chapters have addressed the adsorption kinetics in homogeneous particle as well as zeolitic (bimodal diffusion) particle. The diffusion process is described by a Fickian type equation or a Maxwell-Stefan type equation. Analysis presented in those chapters have good utility in helping us to understand adsorption kinetics. To better understand the kinetics of a practical solid, we need to address the role of surface heterogeneity in mass transfer. The effect of heterogeneity in equilibria has been discussed in Chapter 6, and in this chapter we will briefly discuss its role in the mass transfer. More details can be found in a review by Do (1997). This is started with a development of constitutive flux equation in the presence of the distribution of energy of interaction, and then we apply it firstly to single component systems and next to multicomponent systems. 

For the reaction A — B at steady state, diffusion in the pore is equimolar counterdiffusion, Na = -Nb, and a = 0. The composition dependence in Equation 2.52 is passed on to particle diffu-sivities (Dg) and presents an important drawback in the integration of intraparticle diffusion-reaction equations. However, diffusion fluxes in multicomponent systems may be diverse, and it has been indicated that, on the whole, the dependence of Dcomb on composition is rather weak, and therefore, common practice has been to use the simpler composition-independent form of the equation [7, 9, 32] this additive resistance relationship is the Bosanquet formula [17, 33] ... 

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