Binary friction model

In the case of supported membranes also, the support can play an important role in the separation performance of the membrane in the gas as well as in the Hquid phase [101-103]. Transport in these support pores can be accurately described by the Dusty Gas Model [100, 104] although it is put forward by Kerkhof and Geboers that their Binary Friction Model is physically more correct [105]. 

Here,yjm is obtained from the binary friction model. 

As an alternative to the DGM, the Binary Friction Model (BFM) is of interest. This principally new model has been developed recently [22]. The BFM flux equation, contending with Equation 3.20, is ... 

Reprinted from Chemical Engineering Journal, 64, P.J.A.M. Kerkhof, A modified Maxwell-Stefan model for transport through inert membranes the binary friction model, 319-344,1996, with kind permission from Elsevier Science S.A., P.O. Box 564,1001 Lausanne, Switzerland. 

Chapter 3 described a new model for transport through porous media, developed recently by Kerkhof [5] and called the binary friction model (BFM). It is of interest to see how this model can be applied to the description of available experiments and to compare the results with those of the dusty gas model (DGM). Kerkhof [5] took the experimental data of Evans et al. [6,7] for the permeation of He and Ar through a low-permeability porous graphite septum. The experimental set-up, similar to the Wicke-Kallenbach diffusion cell, is sketched in Figure 9.7. Of interest are the steady... 

Figure 9.11 Net pressure difference in the counterdiffusion of N2 and CjH4. Symbols show the experimental data of Waldmann and Schmitt [11] drawn line show simulation with the binary friction model. Highest pressure on the nitrogen side. The DGM predicts no pressure difference (from Kerkhof [5]). Reprinted from Chemical Engineering Journal, 64, PJ.A.M. Kerkhof, A modified Maxwell-Stefan model for transport through inert membranes the binary friction model, 319-344,1996, with kind permission from Elsevier Science S.A., RO. Box 564,1001 Lausanne, Switzerland.

The model of TMT is one of the few models solely targeted at predieting conductivity behaviour of a membrane, and in contrast to the model of SZG, is based on physical rather than purely empirical considerations [22]. It is in this vein that they invoke the dusty fluid model (DFM) to model transport in the membrane. Before considering the model of TMT we examine the baek-ground of the DFM and the binary friction model (BFM). 

We now proceed to reduce the general binary friction model, Eq. (4.4), to the binary friction membrane model (BFM2) by means of scaling arguments. 

We have presented a review of experimental and macroscopic modelling aspects of transport phenomena in polymer electrolyte membranes. This included examination of the connection between the hydration scheme and the behaviour of the membrane, a discussion of the so-called Schroeder s paradox, and the influence of the membrane phase on transport mechanisms. We also provided a critical examination of various approaches to modelling transport phenomena in membranes, and established that binary friction model provide a correct and rational framework for modelling membrane transport. 

P.J.A. M. Kerkhof A Modified Maxwell-Stefan Model for Transport through Inert Membranes The Binary Friction Model, Chem. Eng. J. 64 319-343 (1996). 

For the present purpose it is most suitable to use the formulation of the binary friction model, expressing the total force by ... 

The results of the last three sections can now be combined to yield the Binary Friction Membrane Transport Model (BFM2) which, after reinserting the dimensions, can be written as... 

Using Eqs. (4.25) and (4.26), we obtain the Binary Friction Conductivity Model (BFCM)... 

Based on this framework, a Binary Friction Membrane Model (BFM2) was developed to account for coupled transport of water and hydronium ions in polymer electrolyte membranes. The BFM2 was cast in a general form to allow for broad applicability to the PFSA family of membranes. As a tool to determine the model parameters, a simplified Binary Friction Conductivity Model (BFCM) was derived to represent conditions found in AC impedance conductivity measurements. 

While the focus of the model performance assessment was on Nafion 1100 equivalent weight (EW), the binary friction membrane transport model is quite general and should be applicable to other PFSA membranes. This is supported by preliminary results in applying the model to Dow membranes and membrane C, whereby rational changes in a single model parameter based on physical considerations of structural differences from Nafion yield conductivity predictions that are in good agreement with experimental measurements [61]. 

B. Carnes and N. Djilali, Modelling and Simulation of Conduction of Protons and Liquid Water in a Polymer Electrolyte Membrane Using the Binary Friction Membrane Model, lESVic Report, University of Victoria (2005). 

We have implicitly allowed the friction coefficients to be independent of the magnitude and the nature of applied forces, that is to say these coefficients are completely defined by the equilibrium properties of the solution as shown for example by Bearman for self-diffusion processes in binary liquid solutions [14]. Nevertheless, for ionic solutions polarization effects resulting from the application of an external field of forces may give rise to distorted ionic atmospheres and the identification of a unique interaction parameter in electrical and self-diffusion processes becomes questionable. However, it has been proved that as far as polyelectrolytes are concerned, the perturbation of the counter-ion distribution with respect to the equilibrium situation is fairly small despite the high polarizability of polyelectrolyte solutions [18]. Moreover, linear forces - fluxes relations have usually been reported from experimental investigations and for both polyelectrolyte and pure salt solutions electrical and self-diffusion determinations have led to nearly identical frictional parameters [19-20]. The friction model might therefore be used with confidence as long as systems not too far from equilibrium are concerned. 

To simulate the particle-particle collision, the hard-sphere model, which is based on the conservation law for linear momentum and angular momentum, is used. Two empirical parameters, a restitution coefficient of 0.9 and a friction coefficient of 0.3, are utilized in the simulation. In this study, collisions between spherical particles are assumed to be binary and quasi-instantaneous. The equations, which follow those of molecular dynamic simulation, are used to locate the minimum flight time of particles before any collision. Compared with the soft-sphere particle-particle collision model, the hard-sphere model accounts for the rotational particle motion in the collision dynamics calculation thus, only the translational motion equation is required to describe the fluid induced particle motion. In addition, the hard-sphere model also permits larger time steps in the calculation therefore, the simulation of a sequence of collisions can be more computationally effective. The details of this approach can be found in the literature (Hoomans et al., 1996 Crowe et al., 1998). 

For a large particle in a fluid at liquid densities, there are collective hydro-dynamic contributions to the solvent viscosity r, such that the Stokes-Einstein friction at zero frequency is In Section III.E the model is extended to yield the frequency-dependent friction. At high bath densities the model gives the results in terms of the force power spectrum of two and three center interactions and the frequency-dependent flux across the transition state, and at low bath densities the binary collisional friction discussed in Section III C and D is recovered. However, at sufficiently high frequencies, the binary collisional friction term is recovered. In Section III G the mass dependence of diffusion is studied, and the encounter theory at high density exhibits the weak mass dependence. 

In a hard sphere approach, particles are assumed to interact through instantaneous binary collisions. This means particle interaction times are much smaller than the free flight time and therefore, hard particle simulations are event (collision) driven. For a comprehensive introduction to this type of simulation, the reader is referred to Allen and Tildesley (1990). Hoomans (2000) used this approach to simulate gas-solid flows in dense as well as fast-fluidized beds. There are three key parameters in such hard sphere models, namely coefficient of restitution, coefficient of dynamic friction and coefficient of tangential restitution. Coefficient of restitution is discussed later in this chapter. Detailed discussion of these three model parameters can be found in Hoomans (2000). 

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 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 ... 

A first approach to take into account the solvent s effect on the absolute mobility of an ion was made by Walden. It is based on the Stokes law of frictional resistance. Walden s rule states that the product of absolute mobility and solvent viscosity is constant. It is clear that the serious limitation of this model is that it does not consider specific solvation effects, because it is based on the sphere-in-continuum model. However, it delivers an appropriate explanation for the fact that, within a given solvent, the mobility depends on temperature to the same extent as the viscosity (in water, for example, the mobility increases by about 2.5% per degree Kelvin).The mobilities do not deviate too much from Walden s rule in some binary mixtures of water with organic solvents. This model is, on the other hand, not appropriate for forecasting or explaining the effect of the solvent on the mobility in a more general manner (see Table 1). 

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