Stefan-Maxwell diffusion model

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. 

Fast and satisfactory mass transfer calculations are necessary since we may have to repeat such calculations many times for a rate-based distillation column model or two-phase flow with mass transfer between the phases in the design and simulation process. The generalized matrix method may be used for multicomponent mass transfer calculations. The generalized matrix method utilizes the Maxwell-Stefan model with the linearized film model for diffusion flux, assuming a constant diffusion coefficient matrix and total concentration in the diffusion region. In an isotropic medium, Fick s law may describe the multicomponent molecular mass transfer at a specified temperature and pressure, assuming independent diffusion of the species in a fluid mixture. Such independent diffusion, however, is only an approximation in the following cases (i) diffusion of a dilute component in a solvent, (ii) diffusion of various components with identical diffusion properties, and (iii) diffusion in a binary mixture. 

Figure 14 Temperature dependence of the flux according to the Maxwell-Stefan model for a one-component system. Activation energy for diffusion was varied, heat of adsorption was taken as 25 kJ mor . Other parameters arc A5 = -75 J mor -K", = 1 mmol g", DP = x 10 ...

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. 

Table 2 Single- and Multicomponent Diffusivities of Ethane and Ethene, Calculated Using the Maxwell-Stefan Model for Zeolitic Diffusion...

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. 

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... 

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. 

This relation is referred to as the Maxwell-Stefan model equations, since Maxwell [65] [67] was the first to derive diffusion equations in a form analogous to (2.302) for dilute binary gas mixtures using kinetic theory arguments (i.e., Maxwell s seminal idea was that concentration gradients result from the friction between the molecules of different species, hence the proportionality coefficients, Csk, were interpreted as inverse friction or drag coefficients), and Stefan [92] [93] extended the approach to ternary dilute gas systems. It is emphasized that the original model equations were valid for ordinary diffusion only and did not include thermal, pressure, and forced diffusion. 

To describe the combined bulk and Knudsen diffusion flrrxes the dusty gas model can be used [44] [64] [48] [49]. The dusty gas model basically represents an extension of the Maxwell-Stefan bulk diffusion model where a description of the Knudsen diffusion mechanisms is included. In order to include the Knudsen molecule - wall collision mechanism in the Maxwell-Stefan model originally derived considering bulk gas molecule-molecule collisions only, the wall (medium) molecules are treated as an additional pseudo component in the gas mixture. The pore wall medium is approximated as consisting of giant molecules, called dust, which are uniformly distributed in space and held stationary by an external clamping force. This implies that both the diffusive ffrrx and the concentration gradient with respect to the dust particles vanish. 

An alternative to the complete Maxwell-Stefan model is the Wilke approximate formulation [103]. In this model the diffusion of species s in a multicomponent mixture is written in the form of Tick s law with an effective diffusion coefficient instead of the conventional binary molecular diffusion coefficient. Following the ideas of Wilke [103] we postulate that an equation for the combined mass flux of species s in a multicomponent mixture can be written as ... 

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]. 

Fig. 14 Separation of C2H6/CH4 mixtures by permeation through a silicalite membrane, a Flux b selectivity. Continuous lines show the predictions of the Maxwell-Stefan model (Eq. 44) based on single-component diffusivities (Dqa> F>ob) with Dab from the Vignes correlation (Eq. 46). Dotted lines show predictions from the simplified Habgood model in which mutual diffusion effects are ignored (Eq. 45). From van de Graaf et al. [53] with permission...

Fundamentals of sorption and sorption kinetics by zeohtes are described and analyzed in the first Chapter which was written by D. M. Ruthven. It includes the treatment of the sorption equilibrium in microporous sohds as described by basic laws as well as the discussion of appropriate models such as the Ideal Langmuir Model for mono- and multi-component systems, the Dual-Site Langmuir Model, the Unilan and Toth Model, and the Simphfied Statistical Model. Similarly, the Gibbs Adsorption Isotherm, the Dubinin-Polanyi Theory, and the Ideal Adsorbed Solution Theory are discussed. With respect to sorption kinetics, the cases of self-diffusion and transport diffusion are discriminated, their relationship is analyzed and, in this context, the Maxwell-Stefan Model discussed. Finally, basic aspects of measurements of micropore diffusion both under equilibrium and non-equilibrium conditions are elucidated. The important role of micropore diffusion in separation and catalytic processes is illustrated. 

Section 15.6 describes the deficiencies in the Fickian model and points out why an alternative model (the fourth) is needed for some situations. The alternative Maxwell-Stefan model of mass transfer and diffusivity is explored in Section 15.7. The Maxwell-Stefan model has advantages for nonideal systems and multicomponent mass transfer but is more difficult to couple to the mass balances when designing separators. The fifth model of mass transfer, the irreversible thermodynamics model fde Groot and Mazur. 1984 Ghorayeb and Firoozabadi. 2QQQ Haase. 1990T is useful in regions where phases are unstable and can split into two phases, but it is beyond the scope of this introductory treatment. The... 

Maxwell-Stefan Model of Diffusion and Mass Transfer... 

Maxwell-Stefan model. The Maxwell-Stefan model is generally agreed to be a better model than the Fickian model for nonideal binary and all ternary systems. However, it is not as widely understood by chemical engineers, data collected in terms of Fickian diffusivities need to be converted to Maxwell-Stefan values, and the model can be more difficult to use. Use this model, coupled with a mass-transfer model, when the Fickian model fails or requires an excessive amount of data. 

Describe how the Maxwell-Stefan model differs from the Fickian model and use the Maxwell-Stefan model for ideal and nonideal binary and ideal ternary diffusion problems... 

Maxwell-Stefan model reduces to the Pick s first law (2.301). Moreover, it follows that for dilute gas systems D 2 Du- Similarly, it can be shown that the Pick s first law diffusivity is the same whether this law is formulated on a mass or molar basis. 

The multicomponent diffusivities are generally calculated from the Wilke equation, the Maxwell-Stefan model or the dusty gas model. The diffusion flux models are outlined in Chap. 2 and further investigated by Solsvik and Jakobsen [15] and Solsvik et al. [17]. T represents the tortuosity factor for the pellet. 

The set of two coupled partial differential equations (equation 5.18) subject to the initial and boundary conditions (equations 5.19 and 5.20) were solved using the method of lines [211] to determine the fluxes, as described in ref. [212]. In the calculations presented here we assume that the pure component Maxwell-Stefan diffusivities are identiccd for the isomers, i.e. D iz = D 2z this assumption is a conservative one from the viewpoint of sepeiration of the isomers as we expect the branched isomer to have a lower mobility within the Siliccdite structure. The simulations were carried out with the complete Maxwell-Stefan model for [D], i.e. equation 5.14. Since the interchange coefficient D12 has a value intermediate between Diz and D2Z [213] we must also have D iz = Dzz = D 12. A further point to note is that in the calculation of the fluxes we have made the assumption that the Maxwell-Stefan diffusivities are independent of the loading. Though this assumption is not always true (see refs. [214,215]), the values of the ratio of fluxes, i.e. selectivity for separation, is not expected to be influenced by this assumption. 

Kapteijn F, Moulijn J A and Krishna R, (2000), The generalized Maxwell-Stefan model for diffusion in zeolites sorbate molecules with different saturation loadings , Chem. Eng. Sci., 55,2923-2930. 

In order to design a zeoHte membrane-based process a good model description of the multicomponent mass transport properties is required. Moreover, this will reduce the amount of practical work required in the development of zeolite membranes and MRs. Concerning intracrystaUine mass transport, a decent continuum approach is available within a Maxwell-Stefan framework for mass transport [98-100]. The well-defined geometry of zeoHtes, however, gives rise to microscopic effects, like specific adsorption sites and nonisotropic diffusion, which become manifested at the macroscale. It remains challenging to incorporate these microscopic effects into a generalized model and to obtain an accurate multicomponent prediction of a real membrane. 

As the pore size decreases, molecules collide more often with the pore walls than with each other. This movement, intermediated by these molecule—pore-wall interactions, is known as Knudsen diffusion. Some models have begun to take this form of diffusion into account. In this type of diffusion, the diffusion coefficient is a direct function of the pore radius. In the models, Knudsen diffusion and Stefan—Maxwell diffusion are treated as mass-transport resistances in seriesand are combined to yield... 

The mass diffusive flux m, of Equation (3.2) generally depends on the operating conditions, such as reactant concentration, temperature and pressure and on the microstructure of material (porosity, tortuosity and pore size). Well established ways of describing the diffusion phenomenon in the SOFC electrodes are through either Fick s first law [21, 34. 48, 50, 51], or the Maxwell-Stefan equation [52-55], Some authors use more complex models, like for example the dusty-gas model [56] or other models derived from this [57, 58], A comparison between the three approaches is reported by Suwanwarangkul et al. [59], who concluded that the choice of the most appropriate model is very case-sensitive, and should be selected, according to the specific case under study. 

Thus the gas/vapor/liquid-liquid mass transfer is modeled as a combination of the two-film model and the Maxwell-Stefan diffusion description. In this stage model, the equilibrium state exists only at the interface. 

Single-stage simulations reveal that intermolecular friction forces do not lead to reverse diffusion effects, and thus the molar fluxes calculated with the effective diffusion approach differ only slightly from those obtained via the Maxwell-Stefan equations without the consideration of generalized driving forces. This result is as expected for dilute solutions and allows one to reduce model complexity for the process studied (143). 

Models lor PV from Solution-Diffusion to Maxwell-Stefan... 

---