Multi-component diffusion model

Ammonia synthesis reaction is a complex multi-component and reversible reaction, and the kinetic equation can be expressed by power function. For simplification, the multi-component diffusion model is usually simply treated in engineering as a single-component diffusion model of key component, and the utilization ratio of internal surface is obtained by an approximate method from a simplified first-order reaction model. 

Zielinski, J.M. and Hanley, B.F. A.I.Ch.E.Jl. 45 (1999) 1. Practical friction-based approach lo modeling multi component diffusion. 

Cooper A.R. (1965) Model for multi-component diffusion. Phys. Chem. Glasses 6, 55-61. 

LPCVD reactor modeling involves many of the same issues of multi-component diffusion reactions that have been studied in the past decade in connection with heterogeneous catalysis. Complex fluid-flow phenomena strongly affect the performance of atmospheric-pressure CVD reactors. Two-dimensional and some three-dimensional flow structures in the classical horizontal and vertical CVD reactors have been explored through flow visual-... 

Scharfer et al. set up a multi-component transport model to describe the diffusion driven mass transport of water and methanol in PEM [170]. For a PEM in contact with liquid methanol and water on one side and conditioned air on the other, the corresponding differential equations and boundary conditions were derived taking into account the polymers three-dimensional swelling. Phase equilibrium parameters and unknown diffusion coefficients for Nafion 117 were obtained by comparing the simulation results to water and methanol concentration profiles measured with confocal Raman spectroscopy. The influence of methanol concentration, temperature and air flow rate was predicted by the model. Although there are indications for an influence of convective fluxes, the measured profiles are ascribed to a Fickean diffusion. Furthermore, the assumption to describe the thermodynamic phase equilibrium as liquid-type equilibrium also at the lower surface of the membrane, which is in contact with a gas phase, can be confirmed by their results. 

A model, frequently referred to as dusty-gas model [1-3], can be used to describe multi-component diffusion in porous media. This model is based on the Stefan-Maxwell approach for diluted gases which is an approximation of Boltzmann s equation. The pore walls are considered as consisting of giant molecules ( dust ) distributed in space. These dust molecules are treated as the n+l-th pseudo-species in a n-component gaseous mixture. The dust particles are kept fixed in space, and are treated like a gas component in the Stefan-Maxwell equations. This model analyzes the transport problem by distinguishing three separate components 1) diffusion, 2) viscous flow and 3) structure of the porous medium. 

There have been few studies reported in the literature in the area of multi-component adsorption and desorption rate modeling (1, 2,3., 4,5. These have generally employed simplified modeling approaches, and the model predictions have provided qualitative comparisons to the experimental data. The purpose of this study is to develop a comprehensive model for multi-component adsorption kinetics based on the following mechanistic process (1) film diffusion of each species from the fluid phase to the solid surface (2) adsorption on the surface from the solute mixture and (3) diffusion of the individual solute species into the interior of the particle. The model is general in that diffusion rates in both fluid and solid phases are considered, and no restrictions are made regarding adsorption equilibrium relationships. However, diffusional flows due to solute-solute interactions are assumed to be zero in both fluid and solid phases. 

Using these relations in equation (18), and applying the LRC model to predict the loadings and heats of adsorption, the model for "surface 1 diffusion of the i-th component in a multi-component mixture becomes ... 

Equation (7.145) results from the rigorous dusty gas model, but unfortunately, it is not easy to implement for a multi-component system. Therefore, we will use simplified equations for the flux relations (7.145). The ordinary diffusion term in formula (7.145) can be approximated by... 

Mass transport inside the catalyst has been usually described by applying the Fick equation, by means of an effective diffusivity Deff a Based on properties of the interface and neglecting the composition effect, composite diffusivity of the multi-component gas mixture is calculated through the simplified Wilke model [13], The effect of pore-radius distribution on Knudsen diffusivity is taken into account. The effective diffusivity DeffA is given by... 

The evaluation of D by fitting an error function, which is based upon an undisturbed diffusion model in a single component system, will not lead to a proper description of the fluorine uptake in a natural multi-component system. 

The dusty gas model (DGM) [21] is used most frequently to describe multi component transport in between the two limiting cases of Knudsen and molecular diffusion. This theory treats the porous media as one component in the gas mixture, consisting of giant molecules held fixed in space. The most important aspect of the theory is the statement that gas transport through porous media (or tubes) can be divided into three independent modes or mechanisms ... 

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] ... 

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 reactor modeling only the ordinary concentration diffusion term j° is generally considered. The rigorous kinetic theory model derivation for multi-component mixtures is outlined in chap 2. Meanwhile, the Tick s law for binary systems is used. 

Equation (5.177) resulting from the rigorous model (dusty gas model) for diffusion and flow is not easy to implement for a multi-component system. Some investigators (Soliman et al., 1988 Xu and Froment, 1989b Elnashaie and Abashar, 1992) have used simplified... 

So far in this chapter the diffusion problem has been treated in a relatively simple manner using the Fickian diffusion models, which are the most widely used, although they are not rigorous for multi-component systems. In the present section the diffusion... 

A first attempt to consider the role of the Debye counterion atmosphere on the transport of a surfactant ion through the DL was made by Mikhailovskij (1976, 1980) (cf. Kortilm 1966, Lyklema 1991). In contrast to a macro-kinetic model, Mikhailovskij derived kinetic equations for a multi-component system under the influence of an external electric field. The basis of this derivation was the set of Bogolubow equations for the partial distribution functions. As the result of the model derivation the following set of electro-diffusion equations is obtained. 

Koiwa, 1992 Bakker et al., 1992). Only recently a model, which is based on a combination of two mechanisms, has been proposed for describing the composition dependence of diffusion in B2 phases (Kao and Chang, 1993). It should be noted that the understanding of the basic diffusion processes for other intermetallics is still less than for NiAl and other B2 phases (Wever, 1992). Diffusion studies of multi-component systems are rare, and with respect to NiAl base alloys only data for the ternary phase (Ni,Fe)Al are available from a systematic study of the system Ni-Al-Fe (Cheng and Dayananda, 1979 Dayananda, 1992). Recently, the effect of Cr on diffusion in NiAl has been studied (Hopfe et al., 1993). 

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. 

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