The Multicomponent Diffusion Equations

Both formulations of the constitutive equations for multicomponent diffusion, the Maxwell-Stefan equations and the generalized Fick s law, are most compactly written in matrix form. It might, therefore, be as well to begin by writing the continuity equations (Eq. 1.3.9) in n - 1 dimensional matrix form as well 

Inserting the generalized Fick s law, (Eq. 3.2.5) into this result gives 

Equations 5.1.6 represent a set of n - 1 coupled partial differential equations. Since the Fick matrix [ )] is a strong function of composition it is not always possible to obtain exact solutions to Eqs. 5.1.6 without recourse to numerical techniques. The basis of the method put forward by Toor and by Stewart and Prober is the assumption that c and [D] can be considered constant. (Actually, Toor worked with the generalized Fick s law formulation, whereas Stewart and Prober worked with the Maxwell-Stefan formulation. Toor et al. (1965) subsequently showed the two approaches to be equivalent.) With this assumption Eqs. 5.1.6 reduce to 

The theory of Toor and of Stewart and Prober is referred to as the linearized theory of multicomponent mass transfer because the set of nonlinear Eqs. 5.1.6 is linearized to give 

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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). 

Despite these differences both solutions of the multicomponent diffusion equations will give identical results if... 

It may be shown that this equation is equivalent to the phenomenological equations derived from irreversible thermodynamics, as weU as the multicomponent diffusion equations derived from the Stefan-Maxwell equations, which were first used to describe diffusion in multicomponent gases. 

Solution The chemical reaction produces a ternary mixture of ethane, ethylene, and hydrogen. Such a mixture may require consideration of the multicomponent diffusion equations in Chapter 7. However, if conversion is low, the diffusion coefficient... 

Self-diffusion and tracer diffusion are described by Equation 3-10 in one dimension, and Equation 3-8 in three dimensions. For interdiffusion, because D may vary along a diffusion profile, the applicable diffusion equation is Equation 3-9 in one dimension, or Equation 3-7 in three dimensions. The descriptions of multispecies diffusion, multicomponent diffusion, and diffusion in anisotropic systems are briefly outlined below and are discussed in more detail later. 

When one refers to the diffusion equation, it is usually the binary diffusion equation. Although theories for multicomponent diffusion have been extensively developed, experimental studies of multicomponent diffusion are limited because of instrumental analytical error and theoretical complexity, and there are yet no reliable diffusivity matrix data for practical applications in geology. Multicomponent diffusion is hence often treated as effective binary diffusion by treating the component under consideration as one component and combining all the other components as the second component. 

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-... 

In equation IV. 6, Dik represents the ordinary diffusion coefficient for binary interactions, and a, is the thermal diffusion ratio. The reactants are often present in small amounts (<1%) relative to the carrier gas thus, the multicomponent diffusion expression (equation IV. 6) may be replaced by a simple Fickian diffusion expression that includes thermodiffusion... 

For this study, mass transfer and surface diffusions coefficients were estimated for each species from single solute batch reactor data by utilizing the multicomponent rate equations for each solute. A numerical procedure was employed to solve the single solute rate equations, and this was coupled with a parameter estimation procedure to estimate the mass transfer and surface diffusion coefficients (20). The program uses the principal axis method of Brent (21) for finding the minimum of a function, and searches for parameter values of mass transfer and surface diffusion coefficients that will minimize the sum of the square of the difference between experimental and computed values of adsorption rates. The mass transfer and surface coefficients estimated for each solute are shown in Table 2. These estimated coefficients were tested with other single solute rate experiments with different initial concentrations and different amounts of adsorbent and were found to predict... 

The methods of determination of the reaction matrix [AT] are considered in Refs. 167, 181, 183, 184 and 186. Another important matrix parameter entering into the linearized film mass transport equation is the multicomponent diffusion matrix /). The latter results from the transformation of the Maxwell-Stefan Eqs. (1) to the form of the generalized Fick s law (83). Matrix [D] is generally a function of... 

The convective diffusion equation is analogous to equations commonly used in dealing with heat and mass transfer. Similarly, if migration can be neglected in a multicomponent solution, then the convective diffusion equation can be applied to each species,... 

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. 

If Eqs. (5-200) and (5-201) are combined, the multicomponent diffusion coefficient may be assessed in terms of binary diffusion coefficients [see Eq. (5-214)]. For gases, the values Dy of this equation are approximately equal to the binary diffusivities for the ij pairs. The Stefan-Maxwell diffusion coefficients may be negative, and the method may be applied to liquids, even for electrolyte diffusion [Kraaijeveld, Wesselingh, and Kuiken, Ind. Eng. Chem. Res., 33, 750 (1994)]. Approximate solutions have been developed by linearization [Toor, H.L., AlChE J., 10,448 and 460 (1964) Stewart and Prober, Ind. Eng. Chem. Fundam., 3,224 (1964)]. Those differ in details but yield about the same accuracy. More recently, efficient algorithms for solving the equations exactly have been developed (see Taylor and Krishna, Krishnamurthy and Taylor [Chem. Eng. J., 25, 47 (1982)], and Taylor and Webb [Comput Chem. Eng., 5, 61 (1981)]. 

Diffusion equations often are written differently from those given in Section E.2.1 [6]. In particular, multicomponent diffusion coefficients differing from D j often are introduced so that diffusion velocities may be expressed directly as linear combinations of gradients. The multicomponent diffusion coefficients are defined so that they reduce to D for binary mixtures [6]. Use of diffusion equations involving multicomponent diffusion coefficients is being made increasingly frequently. 

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 ... 

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 multicomponent diffusivities are generally calculated from the Wilke equation, t represents the tortuosity factor for the pellet. 

Here, Dy is an empirical, radial dispersion coefficient and e is the void fraction. The units of diffusivity Dy are square meters per second. The major differences between this model and the convective diffusion equation used in Chapter 8 is that the velocity profile is now assumed to be flat and Dy is an empirically determined parameter instead of a molecular diffusivity. The value of Dy depends on factors such as the ratio of tube to packing diameters, the Reynolds number, and (at least at low Reynolds numbers) the physical properties of the fluid. Ordinarily, the same value for Dy is used for all reactants, finessing the problems of multicomponent diffusion and allowing the use of stoichiometry to eliminate Equation 9.1 for some of the components. Note that Us in Equation 9.1 is the superficial velocity, this being the average velocity that would exist if the tube had no packing. 

At about the same time that Maxwell and Stefan were developing their ideas of diffusion in multicomponent mixtures, Adolf Fick and others were attempting to uncover the basic diffusion equations through experimental studies involving binary mixtures (Fick, 1855). The result of Fick s work was the law that bears his name. The Fick equation for a binary mixture in an isothermal, isobaric system is... 

Since D plays the same role as the kinematic viscosity v, we may expect for large Schmidt numbers (v>D) that the viscous boundary layer thickness should be considerably larger than the diffusion boundary layer thickness. A consequence of this is that the velocity seen by the concentration layer at its edge is not the free stream velocity U but something much less, which is more characteristic of the velocity close to the wall (Fig. 4.2.1). We note also that since c is understood to be c, then in a multicomponent solution there may be as many distinct boundary layers as there are species, with the thickness of each defined by the appropriate diffusion coefficient. With this caveat in mind, we may write the convective diffusion equation for a two-dimensional diffusion boundary layer and estimate the diffusion layer thickness. 

For the simulation of RD columns in which the chemical reactions take place at heterogeneous catalysts, it is important to keep in mind that a macrokinetic expression (5.55) has to be applied. Therefore, the microkinetic rate has to be combined with the mass transport processes inside the catalyst particles. For this purpose a model for the multicomponent diffusive transport has to be formulated and combined with the microkinetics based on the component mass balances. This has been done by several authors [50-53] by use of the generalized Maxwell-Stefan equations. 

The effects of capillary condensation were included in the network model, by calculating the critical radius below which capillary condensation occurs based on the vapor composition in each pore using the multicomponent Kelvin Equation (23.2). Then the pore radius was compared with the calculated critical radius to determine whether the pore is liquid- or vapor-filled. As a significant fraction of pores become filled with capillary condensate, regions of vapor-filled pores may become locked off from the vapor at the network surface by condensate clusters. A Hoshen and Kopelman [30] algorithm is used to identify vapor-filled pores connected to the network surface, in which diffusion and reaction continue to take place after other parts of the network filled with liquid. It was assumed that, due to the low hydrogen solubility in the liquid, most of the reaction takes place in the gas-filled pores. The diffusion/reaction simulation is repeated, including only vapor-filled pores connected to the network surface by a pathway of other vapor-filled pores. 

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