Stefan-Maxwell equations molecular diffusion

All these different mechanisms of mass transport through a porous medium can be studied experimentally and theoretically through classical models (Darcy s law, Knudsen diffusion, molecular dynamics, Stefan-Maxwell equations, dusty-gas model etc.) which can be coupled or not with the interactions or even reactions between the solid structure and the fluid elements. Another method for the analysis of the species motion inside a porous structure can be based on the observation that the motion occurs as a result of two or more elementary evolutions that are randomly connected. This is the stochastic way for the analysis of species motion inside a porous body. Some examples that will be analysed here by the stochastic method are the result of the particularisations of the cases presented with the development of stochastic models in Sections 4.4 and 4.5. 

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 ... 

To rigorously model the effects of multicomponent molecular diffusion, an alternative diffusion equation should be used, such as the Stefan-Maxwell equation [9, 10] ... 

Molecular diffusion of a component is driven by a composition gradient. In a mixture of n components, the partial pressure gradient of component i is given by the Stefan-Maxwell equation ... 

Equations (2.15) or (2.16) are the so-called Stefan-Maxwell relations for multicomponent diffusion, and we have seen that they are an almost obvious generalization of the corresponding result (2.13) for two components, once the right hand side of this has been identified physically as an inter-molecular momentum transfer rate. In the case of two components equation (2.16) degenerates to... 

The principles and basic equations of continuous models have already been introduced in Section 6.2.2. These are based on the well known conservation laws for mass and energy. The diffusion inside the pores is usually described in these models by the Fickian laws or by the theory of multicomponent diffusion (Stefan-Maxwell). However, these approaches basically apply to the mass transport inside the macropores, where the necessary assumption of a continuous fluid phase essentially holds. In contrast, in the microporous case, where the pore size is close to the range of molecular dimensions, only a few molecules will be present within the cross-section of a pore, a fact which poses some doubt on whether the assumption of a continuous phase will be valid. 

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. 

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 ... 

Molecular diffusion is the mechanism of transfer of a substance either through a fluid which is motionless or, if the fluid is in laminar flow, in a direction perpendicular to the velocity of the fluid. The phenomenon has been studied from many points of view7ffequently conflicting, the most important of which are those of Fick and of Maxwell-Stefan. Fick (7) applied the well-known Fourier equation for rate of heat flow to the problem of diffusion. Unfortunately the mechanism of the two processes is not identical, since in the penetration of a liquid by a diffusing solute there will necessarily be displacement of the liquid and consequent volume changes arising for which the Fourier equation does not account. As an approxi-... 

Wilke and Lee [7] have developed an approximate equation for the molecular diffusion coefficient starting from the Stefan-Maxwell theory ... 

The introductory Section 3.1.2.5 in Chapter 3 identifies the negative chemical potential gradient as the driver of targeted separation, and the relevant species flux expression is developed in Section 3.1.3.2 (see Example 3.1.9 also). Section 3.1.4 introduces molecular diffusion and convection and basic mass-transfer coefficient based flux expressions essential to studies of distillation and other phase equilibrium based separation processes. Section 3.1-5.1 introduces the Maxwell-Stefan equations forming the basis of the rate based approach of analyzing distillation column operation. After these fundamental transport considerations (which are also valid for other phase equilibrium based separation processes), we encounter Section 3.3.1, where the equality of chemical potential of a species in all phases at equilibrium is illustrated as the thermodynamic basis for phase equilibrium (Le. = /z ). Direct treatment of distillation then begins in Section 3.3.7.1, where Raouit s law is introduced. It is followed by Section 3.4.1.1, where individual phase based mass-transfer coefficients are reiated to an overall mass-transfer coefficient based on either the vapor or liquid phase. 

Transport of components within the porous membrane can be expressed using the dusty gas model, which is based on the Maxwell-Stefan equations for multi-component molecular diffusion [8] ... 

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... 

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],... 

A similar model often used by reaction engineers is derived for the limiting case in which all the convective fluxes can be neglected. Consider a dilute component s that diffuses into a homogeneous mixture, then J 0 for r 7 s. To describe this molecular transport the Maxwell-Stefan equations given by the last line in (2.298) are adopted. With the given restrictions, the model reduces to ... 

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]. 

To establish the relationship between self- and transport diffusion it is necessary first to consider diffusion in a binary adsorbed phase within a micropore. This can be conveniently modeled using the generalized Maxwell-Stefan approach [45,46], in which the driving force is assumed to be the gradient of chemical potential with transport resistance arising from the combined effects of molecular friction with the pore walls and collisions between the diffusing molecules. Starting from the basic form of the Maxwell-Stefan equation ... 

For a dilute solution of polymer Aina low molecular weight solvent B, the polymer molecules are modeled as bead-spring chains. Resistance in the motion of beads is characterized by a friction coefficient As the number of beads is proportional to the polymer molecular weight M, we have Dab 1 / Vm. Table 2.8 shows some values of diffusion coefficients in polymers. In a flow of dilute solution of polymers, the diffiisivity tensor is anisotropic and depends on the velocity gradient. The Maxwell-Stefan equation may predict the diffusion in multicomponent mixtures of polymers. 

A molecular dynamics approach can also be used to predict mixed gas diffusivities in microporous materials, at the expense of computation cost (e.g., Qureshi and Wei, 1990 Chitra and Yashonath, 1995 Trout et al., 1997 Snurr and Karger, 1997). The empirical correlation of Vignes (1966) for binary diffusivities in liquid solutions and also metallic alloys has been used extensively for calculating binary diffusivities, using the Maxwell-Stefan formalism for flux equations (e.g., Krishna, 1990). 

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 [109] or the average species velocity is used [148]. 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. Nevertheless, the given momentum flux formula (2.537) is not useful before the unknown average velocity after the collisions v has been determined. For elastic molecular collisions this velocity can be calculated, in... 

For the transport of gas mixtures, the generalised Maxwell-Stefan equation (Krishna and WesseUngh, 1997) has been widely adopted to describe multi-component diffusion. Although quantitative descriptions of gas diffusion in various microporous or mesoporous ceramic membranes based on statistical mechanics theory (Oyama et al., 2004) or molecular dynamic simulation (Krishna, 2009) have been reported, the prediction of mixed gas permeation in porous ceramic membranes remains a challenging task, due to the difficulty in generating an accurate description of the porous network of the membrane. 

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