Maxwell-Stefan surface diffusion

The Maxwell-Stefan surface diffusivity is defined, by analogy to the definition of the Knudsen diffusivity [46], as... 

In this equation, Dj y is the Maxwell-Stefan surface diffusivity defined as... 

The GRM Formulated with the Maxwell-Stefan Surface Diffusion Model. 765... 

Surface diffusion of adsorbed molecular species along the pore wall surface. This mechanism of transport becomes dominant for micro-pores and for strongly adsorbed species. Extensions of the Maxwell-Stefan bulk diffusion model have been proposed in order to provide a realistic description of the combined bulk, Knudsen and surface diffusion mechanisms, apparently with limited success [70, 71]. 

More complicated and realistic models which allow the prediction of transport processes in porous media have been suggested, and have been validated in recent years. For example, it was realized that there might be significant contributions to the overall flux by components which are adsorbed at pore walls but possess a certain mobility [30]. To quantify such surface diffusion processes, a Generalized Stefan-Maxwell equation has been proposed [28] ... 

A similar model has been applied to the modeling of porous media with condensation in the pores. Capillary condensation in the pores of the catalyst in hydroprocessing reactors operated close to the dew point leads to a decrease of conversion at the particle center owing to the loss of surface area available for vapor-phase reaction, and to the liquid-filled pores that contribute less to the flux of reactants (Wood et al., 2002b). Significant changes in catalyst performance thus occur when reactions are accompanied by capillary condensation. A pore-network model incorporates reaction-diffusion processes and the pore filling by capillary condensation. The multicomponent bulk and Knudsen diffusion of vapors in each pore is represented by the Maxwell-Stefan model. 

The principle of the Maxwell-Stefen diffusion equations is that the force acting on a species is balanced by the ffiction that is exerted on that species. The driving force for diffusion is the chemical potential gradient. The Maxwell-Stefan equations were applied to surface diffusion in microporous media by Krishna [77]. During surface diffusion, a molecule experiences friction from other molecules and from the surface, which is included in de model as a pseudo-species, n+1 (Dusty-gas model). The balance between force and friction in a multi-component system can thus be written as [77] ... 

Differential mass balances across the membrane combine Stefan-Maxwell-type diffusive fluxes, a surface diffusion term, the Darcy expression for convective fluxes, and the reaction terms. 

The first term on the right-hand side of the equation denotes the friction between species the second term represents friction between a species and the surface. and )Ji+i are the Maxwell-Stefan diffusivities. The first term on the right side is often referred to as an exchange term that represents the probability of molecules exchanging places on the surface. Since this exchange is not likely to occur in narrow zeolite channels, it is commonly neglected. This is called single-file diffusion. 

The application of the Maxwell-Stefan theory for diffusion in microporous media to permeation through zeolitic membranes implies that transport is assumed to occur only via the adsorbed phase (surface diffusion). Upon combination of surface diffusion according to the Maxwell-Stefan model (Eq. 20) with activated-gas translational diffusion (Eq. 12) for a one-component system, the temperature dependence of the flux shows a maximum and a minimum for a given set of parameters (Fig. 15). At low temperatures, surface diffusion is the most important diffusion mechanism. This type of diffusion is highly dependent on the concentration of adsorbed species in the membrane, which is calculated from the adsorption isotherm. At high temperatures, activated-gas translational diffusion takes over, causing an increase in the flux until it levels off at still-higher temperatures. 

The temperature dependence of the methane permeation through a silicalite membrane, showing a maximum and a minimum as a function of temperature (Fig. 3 [14]), can not be predicted by using the Maxwell-Stefan description for surface diffusion only. Such a maximum and minimum in the permeation as a function of temperature can be predicted only when the total flux is described by a combination of surface diffusion and activated-gas translational diffusion (Fig. 15). 

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. 

Within micropores, surface forces are dominant and an adsorbed molecule never escapes completely from the force field of the surface. Diffusion within this regime has been called configurational diffusion, intra-crystalline diffusion, micropore diffusion, or simply surface diffusion. The Maxwell-Stefan formulation, which is generally accepted for diffusion in the bulk fluid phase, can be extended to describe surface diffusion by considering the vacant sites to be a (n + l)-th pseudospecies on the surface [38,47,49-52]. Using the Maxwell-Stefan diffusion formulation, the following relationship was obtained for surface diffusion. 

In order to accoimt correctly for surface diffusion in the GRM, we should consider the concentration-dependent generalized Maxwell-Stefan (GMS) matrix diffusiv-ities (see Chapter 5). Originally, the GMS approach was used to investigate gas adsorption on zeolites [54,55]. The same approach, however, can be used to describe surface diffusion in the packing materials used in HPLC, especially for the separation of macromolecules. 

In this case, the matrix of Pick diffusivities depends on the surface coverage and is given by the single file Maxwell-Stefan relationship. 

Kaczmarski et al. used a similar model for the calculation of the band profiles of the enantiomers of 1-indanol on a chiral phase in HPLC [29,57]. These authors ignored the external mass transfer and assumed that local equilibrium takes place for each component between the pore surface and the stagnant fluid phase in the macropores (infinite fast kinetics of adsorption-desorption). They also assumed that surface diffusion contribution is much faster than pore diffusion and neglected pore diffusion entirely. Instead of the single file Maxwell-Stefan diffusion, these authors used the generalized Maxwell-Stefan diffusion (see Chapter 5).The calculation (see below) requires first the selection of equations to calculate the surface molecular flux [29,57,58],... 

The competitive equilibrium isotherm model best fitting the FA experimental data for the R and S enantiomers of 1-phenyl-l-propanol on cellulose tiibenzoate was the Toth model. This model was used to calculate the elution profiles of samples of mixtures of the two enantiomers [29]. The General Rate model combined with the Generalized Maxwell-Stefan equation (GR-GMS) was used to model and describe surface diffusion (see Chapter 5). The mass transfer kinetics is slow and this model fits the experimental data well over a wide concentration range with one single set of numerical parameters to account for the band profiles in a wide range of concentrations, as shown in Figure 16.24. 

The Dusty Gas Model (DGM) is one of the most suitable models to describe transport through membranes [11]. It is derived for porous materials from the generalised Maxwell-Stefan equations for mass transport in multi-component mixtures [1,2,47]. The advantage of this model is that convective motion, momentum transfer as well as drag effects are directly incorporated in the equations (see also Section 9.2.4.2 and Fig. 9.12). Although this model is fundamentally more correct than a description in terms of the classical Pick model, DGM/Maxwell-Stefan models )deld implicit transport equations which are more difficult to solve and in many cases the explicit Pick t)q>e models give an adequate approximation. For binary mixtures the DGM model can be solved explicitly and the Fickian type of equations are obtained. Surface diffusion is... 

In an attempt to check the validity of the Maxwell-Stefan equations Carty and Schrodt (1975) evaporated a binary liquid mixture of acetone(l) and methanol(2) in a Stefan tube. Air(3) was used as the carrier gas. In one of their experiments the composition of the vapor at the liquid surface was x = 0.319, x = 0.528. The pressure and temperature in the vapor phase were 99.4 kPa and 328.5 K, respectively. The length of the diffusion path was... 

Equation 4.2.13 is shown in Figure 4.8 for a ternary system where it becomes clear that the limiting diffusivities are, in fact, the Maxwell-Stefan diffusion coefficients at the corners of the diffusivity-composition surface. Equation 4.2.13 should reduce to the binary Vignes... 

The binary K j may be calculated as a function of the appropriate Maxwell-Stefan diffusion coefficient from a suitable correlation or physical model (e.g., the surface renewal models of Chapter 10). These binary must also be used directly in the calculation of the rate factor matrix [ ] (cf. Eqs. 8.3.28 and 8.3.29). 

The K j may be estimated using an empirical correlation or alternative physical model (e.g., surface renewal theory) with the Maxwell-Stefan diffusivity of the appropriate i-j pair D-j replacing the binary Fick D. Since most published correlations were developed with data obtained with nearly ideal or dilute systems where F is approximately unity, we expect this separation of diffusive and thermodynamic contributions to k to work quite well. We may formally define the Maxwell-Stefan mass transfer coefficient k - as (Krishna, 1979a)... 

An important application of multicomponent mass transfer theory that we have not considered in any detail in this text is diffusion in porous media with or without heterogeneous reaction. Such applications can be handled with the dusty gas (Maxwell-Stefan) model in which the porous matrix is taken to be the n + 1th component in the mixture. Readers are referred to monographs by Jackson (1977), Cunningham and Williams (1980), and Mason and Malinauskas (1983) and a review by Burghardt (1986) for further study. Krishna (1993a) has shown the considerable gains that accrue from the use of the Maxwell-Stefan formulation for the description of surface diffusion within porous media. 

On the other hand, the more rigorous Maxwell-Stefan equations and the dusty gas model are seldom used in industrial reaction engineering applications. Nevertheless, the dusty gas model [64] represents a modern attempt to provide a more realistic description of the combined bulk and Knudsen diffusion mechanisms based on the multicomponent Maxwell-Stefan model formulation. Similar extensions of the Maxwell-Stefan model have also been suggested for the surface diffusion of adsorbed molecular pseudo-species, as well as the combined bulk, Knudsen and surface diffusion apparently with limited success [48] [49]. 

Typical examples where this extended analysis is necessary are conditions of possible back diffusion of components from the tube to the shell side. Additionally, more concentrated mixtures, Knudsen and/or surface diffusion may require the application of the Stefan-Maxwell approach in order to describe the mass transport in the porous medium (Krishna and Wesselingh, 1997). 

Krishna, R., Multicomponent surface diffusion of adsorbed species — a description based on the generalised Maxwell-Stefan equations, Chem. Eng. ScL, 45, 1779-1791, 1990. 

We have presented in this chapter an account on the development of diffusion theory. Various modes of flow are identified Knudsen, viscous, continuum diffusion and surface diffusion. Constitutive flux equations are presented for all these flow mechanisms, and they can be readily used in any mass balance equations for the solution of concentration distribution and fluxes. Treatment of systems containing more than two species will be considered in a more systematic approach of Stefan-Maxwell in the next chapter. 

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 considered the Stefan tube with pure liquid in the tube. Now we consider the case whereby the liquid contains two components. These two species will evaporate and diffuse along the tube into the flow of a third component across the top of the tube. The third component is assumed to be non-soluble in the liquid. What we will consider next is the Maxwell-Stefan analysis of this ternary system, and then apply it to the experimental data of Carty and Schrodt (1975) where they used a liquid mixture of acetone and methanol. The mole fractions of acetone and methanol just above the liquid surface of the tube are 0.319 and 0.528, respectively. 

To formulate the Stefan-Maxwell approach for surface diffusion, we will treat the adsorption site as the pseudo-species in the mixture, a concept put forwards by Krishna (1993). If we have n species in the system, the pseudo species is denoted as the (n+l)-th species, just like the way we dealt with Knudsen diffusion where the solid object is regarded as an assembly of giant molecules stationary in space. We balance the force of the species i by the friction between that species i with all other species to obtain ... 

It is noted that the total surface concentration is the same for all species. Using this definition, the Stefan-Maxwell equation to describe surface diffusion is ... 

You have learnt about the behaviour of adsorption kinetics of a single component in a single particle. Pore, surface diffusions and their combined diffusion have been studied in some details for linear as well as nonlinear isotherm and under isothermal as well as nonisothermal conditions. Here we will study a situation where there are more than one adsorbate present in the system and the interaction between different species will occur during diffusion as well as adsorption. Analysis of multicomponent system will require the application of the Maxwell-Stefan approach learnt in Chapter 8. To demonstrate the methodology as well as to show the essential features of how multiple species interact during the course of diffusion as well as adsorption onto adsorption sites, we will first consider a multicomponent adsorption system under isothermal conditions and dual diffusion mechanism is operating in the particle. 

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. 

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