Maxwell-Stefan equation multicomponent systems

We may describe multicomponent diffusion by (1) the Maxwell-Stefan equation where flows and forces are mixed, (2) the Chapman-Cowling and Hirschfelder-Curtiss-Bird approaches where the diffusion of all the components are treated in a similar way, and (3) a reference to a particular component, for example, the solvent or mass average (baiycentric) definition. Frames of reference in multicomponent system must be clearly defined. Binary diffusion coefficients are often composition dependent in liquids, while they are assumed independent of composition for gases. 

In this textbook we have eoneentrated our attention on mass transfer in mixtures with three or more species. The rationale for doing this should be apparent to the reader by now multicomponent mixtures have characteristics fundamentally dijferent from those of two component mixtures. In fact, a binary system is peculiar in that it has none of the features of a general multicomponent mixture. We strongly believe that treatments of even binary mass transfer are best developed using the Maxwell-Stefan equations. We hope that this text will have the effect of persuading instructors to use the Maxwell-Stefan approach to mass transfer even at the undergraduate level. 

We also feel that portions of the material in this book ought to be taught at the undergraduate level. We are thinking, in particular, of the materials in Section 2.1 (the Maxwell-Stefan relations for ideal gases). Section 2.2 (the Maxwell-Stefan equations for nonideal systems). Section 3.2 (the generalized Fick s law). Section 4.2 (estimation of multicomponent diffusion coefficients). Section 5.2 (multicomponent interaction effects), and Section 7.1 (definition of mass transfer coefficients) in addition to the theory of mass transfer in binary mixtures that is normally included in undergraduate courses. 

A more rigorous derivation of these relations were given by Curtiss and Hirschfelder [16] extending the Enskog theory to multicomponent systems. FYom the Curtiss and Hirschfelder theory of dilute mono-atomic gas mixtures the Maxwell-Stefan diffusivities are in a first approximation equal to the binary diffusivities, Dgr Dsr- On the other hand, Curtiss and Bird [18] [19] did show that for dense gases and liquids the Maxwell-Stefan equations are still valid, but the strongly concentration dependent diffusivities appearing therein are not the binary diffusivities but merely empirical parameters. 

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

Write down the Maxwell-Stefan equations for a binary system and for multicomponent systems. 

These are the Maxwell-Stefan diffusion equations for multicomponent systems. They are named after the Scottish physicist James Clerk Maxwell and the Austrian scientist Josef Stefan, who were primarily reponsible for their development around 1870. It is important to point out that only n - 1 of the Maxwell-Stefan equations are independent because the d(- must sum to zero. Also, for a multicomponent ideal gas mixture a more elaborate analysis than that of Example 1.4 is needed to show that (Taylor and Krishna, 1993)... 

The Maxwell-Stefan equations for diffusion in multicomponent systems can become very complicated, and they are sometimes handled by using an effective dif -fusivity or pseudobinary approach. The effective diffusivity is defined by assuming that the rate of diffusion of component i depends only on its own composition gradient that is,... 

In practice, most of the mass transfer processes involve multicomponent, and the mass transfer rate should be calculated individually by each component. In some cases for simplifying the calculation, two influential components, called key components, are taken as if a two-component system. However, such simplification may lead to serious error, and the rigorous method is preferable. The calculation of multicomponent mass transfer is by the aid of Maxwell-Stefan equation which is introduced briefly in the section below. 

For the non-ideal multicomponent vapor-liquid system, the Maxwell-Stefan equation is usually employed to evaluate the mass transfer behaviors. The fundamentals of Maxwell-Stefan equation is briefly introduced in Sect. 3.4.2. 

Solute-solute Interactions may affect the diffusion rates In the fluid phase, the solid phase, or both. Toor (26) has used the Stefan-Maxwell equations for steady state mass transfer In multicomponent systems to show that, in the extreme, four different types of diffusion may occur (1) diffusion barrier, where the rate of diffusion of a component Is zero even though Its gradient Is not zero (2) osmotic diffusion, where the diffusion rate of a component Is not zero even though the gradient Is zero (3) reverse diffusion, where diffusion occurs against the concentration gradient and, (4) normal diffusion, where diffusion occurs In the direction of the gradient. While such extreme effects are not apparent in this system, it is evident that the adsorption rate of phenol is decreased by dodecyl benzene sulfonate, and that of dodecyl benzene sulfonate increased by phenol. 

Some of the molecular theories of multicomponent diffusion in mixtures led to expressions for mass flow of the Maxwell-Stefan form, and predicted mass flow dependent on the velocity gradients in the system. Such dependencies are not allowed in linear nonequilibrium thermodynamics. Mass flow contains concentration rather than activity as driving forces. In order to overcome this inconsistency, we must start with Jaumann s entropy balance equation... 

These are the Maxy ell-Stefan diffusion equations for multicomponent systems. These equations are named after the Scottish physicist James Clerk Maxwell and the Austrian scientist Josef Stefan who were primarily responsible for their development (Maxwell, 1866, 1952 Stefan, 1871). These equations appeared, in more or less the complete form of Eq. 2.1.15, in an early edition of the Encyclopedia Britannica (incomplete forms had been published earlier) in a general article on diffusion by Maxwell (see Maxwell, 1952). In addition to his major contributions to electrodynamics and kinetic theory. Maxwell wrote several articles for the encyclopedia. Stefan s 1871 paper is a particularly perceptive one and anticipated several of the multicomponent interaction effects to be discussed later in this book. 

There are few methods for predicting the Maxwell-Stefan diffusivities in multicomponent liquid mixtures. The methods that have been suggested are based on extensions of the techniques proposed for binary systems discussed in Section 4.1.5 (see, e.g., the works of Cullinan and co-workers, 1966-1975 Bandrowski and Kubaczka, 1982 Kosanovich, 1975). The Vignes equation, for example, may be generalized as follows (Wesselingh and Krishna, 1990 Kooijman and Taylor, 1991). 

In any event, we hope it is now well understood that mass transfer in multicomponent systems is described better by the full set of Maxwell-Stefan or generalized Fick s law equations than by a pseudobinary method. A pseudobinary method cannot be capable of superior predictions of efficiency. For a simpler method to provide consistently better predictions of efficiency than a more rigorous method could mean that an inappropriate model of point or tray efficiency is being employed. In addition, uncertainties in the estimation of the necessary transport and thermodynamic properties could easily mask more subtle diffusional interaction effects in the estimation of multicomponent tray efficiencies. It should also be borne in mind that a pseudobinary approach to the prediction of efficiency requires the a priori selection of the pair of components that are representative of the... 

When more than two components are present, the efficiencies of each are not necessarily the same. The rigorous approach to handling multicomponent mixtures, outlined by Taylor and Krishna, uses the Maxwell-Stefan diffusional equations. Chan and Fair used the rigorous approach to compare multicomponent system separations with those predicted by the use of the equivalent pseudobinary systems. They found that if the dominant pair of components present in the mixture is used to determine efficiency for all of the components, the separation determined is quite close to that resulting from rigorous multicomponent procedures. 

Chapter 5 is dedicated to the single particle problem, the main building block of the overall reactor model. Both porous and non-porous catalyst pellets are considered. The modelling of diffusion and chemical reaction in porous catalyst pellets is treated using two degrees of model sophistication, namely the approximate Fickian type description of the diffusion process and the more rigorous formulation based on the Stefan-Maxwell equations for diffusion in multicomponent systems. 

For non-porous catalyst pellets the reactants are chemisorbed on their external surface. However, for porous pellets the main surface area is distributed inside the pores of the catalyst pellets and the reactant molecules diffuse through these pores in order to reach the internal surface of these pellets. This process is usually called intraparticle diffusion of reactant molecules. The molecules are then chemisorbed on the internal surface of the catalyst pellets. The diffusion through the pores is usually described by Fickian diffusion models together with effective diffusivities that include porosity and tortuosity. Tortuosity accounts for the complex porous structure of the pellet. A more rigorous formulation for multicomponent systems is through the use of Stefan-Maxwell equations for multicomponent diffusion. Chemisorption is described through the net rate of adsorption (reaction with active sites) and desorption. Equilibrium adsorption isotherms are usually used to relate the gas phase concentrations to the solid surface concentrations. 

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. 

We have presented the necessary equation to relate flux and mole fraction gradient for a multicomponent system (eqs. 8.6-18) when both molecular diffusion and Knudsen diffusion are operating. Let us now treat a special case of binary systems. For such a case, the Stefan-Maxwell equations are ... 

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. 

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) ... 

The Ki, values for each species i and the enthalpies used in the energy balance equations for any stage ra are obtained from conventional approaches used in multistage distillation analysis. However, the species flux is expressed in terms of the sum of a convective component and a diffusive component. The diffusive component is modeled using the Maxwell-Stefan approach (Section 3.1.5.1) for this complex multicomponent system in a matrix framework. For an illustrative introduction, see Sender and Henley (1998). 

For multicomponent systems the diffusive flux terms may be written in accordance with the approximate Wilke bulk flux equation (2.450), the approximate Wilke-Bosanquet combined bulk and Knudsen flux for porous media (2.454), the rigorous Maxwell-Stefan bulk flux equations (2.421), and the consistent dusty gas combined bulk and Knudsen diffusion flux for porous media (2.504). The different mass based diffusion flux models are listed in Table 2.3. The corresponding molar based diffusion flux models are listed in Table 2.4. In most simulations, the catalyst pellet is approximated by a porous sphericai pellet with center point symmetry. For such spherical pellets a representative system of pellet model equations, constitutive laws and boundary conditions are listed in Tables 2.5,2.6 and 2.7, respectively. 

Maxwell-Stefan diffusion is a model for describing diffusion in multicomponent systems. The equations describing these processes were developed for dilute gases and fluids. The MaxweU-Stefan equation is ... 

In the formulation and solution of conservation equations, we tend to prefer the direct evaluation of the diffusion velocities as discussed in the previous section. However, it is worthwhile to note that the Stefan-Maxwell equations provide a viable alternative. At each point in a flow field one could solve the system of equations (Eq. 3.105) to determine the diffusion-velocity vector. Solution of this linear system is equivalent to determining the ordinary multicomponent diffusion coefficients, which, in this formulation, do not need to be evaluated. 

The Stefan-Maxwell equations (12.170 and 12.171) form a system of linear equations that are solved for the K diffusion velocities V. The diffusion velocities obtained from the Stefan-Maxwell approach and by evaluation of the multicomponent Eq. 12.166 are identical. 

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 ... 

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... 

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