The Maxwell-Stefan Model

Krishna and Paschek [91] employed the Maxwell-Stefan description for mass transport of alkanes through silicalite membranes, but did not consider more complex (e.g., unsaturated or branched) hydrocarbons. Kapteijn et al. [92] and Bakker et al. [93] applied the Maxwell-Stefan model for hydrocarbon permeation through silicalite membranes. Flanders et al. [94] studied separation of C6 isomers by pervaporation through ZSM-5 membranes and found that separation was due to shape selectivity. 

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 26. Teinpeiatuie dependence of the flux according to the Maxwell-Stefan model for one-component Activation energy for diffiision Ed was varied, heat of adsorption Q was taken to be 25 kJ-mof Other parameters AS=-75 J-mof -K, csat=l mmol-g-1, D =M0, membrane thickness f=50 pm,...

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...

Two other dependences were suggested by Kaczmarski et al. [123]. First, they used a model borrowed from gas-solid adsorption, the Maxwell-Stefan model, that is valid if the mobile phase is much less strongly adsorbed than the compoimd studied, which is generally true in RPLC. This model gives... 

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. 

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. 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... 

Application of the Maxwell-Stefan model to dense gases and liquids... 

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. 

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