Series transport model, resistance

Membranes with ordered structures such as zeolites or nanotubes have considerable potential as gas separation membranes [46-48], In addition to having thermal and chemical stability, the porosity of these structures is ordered, and therefore there is usually more control over the separation properties. The pores within these structures are such that gas transport can not be completely explained by the transition state theory. This is because, in nanotubes for example, there is only one transition, from outside of the tube to inside of the tube. Two alternative models are outlined here, the parallel transport model and the resistance in series transport model, which are illustrated in Figure 5.5, and they are explained in detail by the work of Gilron and Softer [27]. 

Modelling Gas Separation in Porous Membranes 95 5.7 J, Resistance in Series Transport Model... 

CH4 because of its lighter mass resulting in a higher molecular velocity and does not change with pore size. Parallel transport follows the same trend as surface diffusion in small pores and tends toward Knudsen behaviour as the pore sizes increase. Finally, the resistance in series transport model predicts a decrease in selectivity as the permeability of CH4 increases more rapidly than for CO2 with increasing pore size. 

Figure 5.5 Schematic models for (a) parallel transport and (b) resistance in series transport [27], Reprinted from Journal of Membrane Science, 209, j. Gilron and /4. Soffer, Knudsen diffusion in microporous carbon membranes with molecular sieving character, 339-352, Copyright (2002), with permission from Elsevier...

Figure 5.12 Model prediction of permeability as a function of temperature. Modes of transport are indicated for the following pore sizes activated diffusion (d = 6.8 A), surface diffusion (d = 10 A), Knudsen diffusion (d = 10 A), parallel transport (d = 10A), and resistance In series transport (d rraii = 6.8A, d/a,ge = loA, Xk = 0.8)...

Figure 5.13 Model predictions of CO2/CH4 selectivity versus CO2 permeability for varying pore size d. Arrows indicate the direction of increasing pore size. The pore size range, 7.22 > d > 30A, was chosen for all the modes of transport, apart from the resistance in series transport where the constriction size varied while the large pore size...

Figure 10. Kleitz s reaction pathway model for solid-state gas-diffusion electrodes. Traditionally, losses in reversible work at an electrochemical interface can be described as a series of contiguous drops in electrical state along a current pathway, for example. A—E—B. However, if charge transfer at point E is limited by the availability of a neutral electroactive intermediate (in this case ad (b) sorbed oxygen at the interface), a thermodynamic (Nernstian) step in electrical state [d/j) develops, related to the displacement in concentration of that intermediate from equilibrium. In this way it is possible for irreversibilities along a current-independent pathway (in this case formation and transport of electroactive oxygen) to manifest themselves as electrical resistance. This type of chemical valve , as Kleitz calls it, may also involve a significant reservoir of intermediates that appears as a capacitance in transient measurements such as impedance. Portions of this image are adapted from ref 46. (Adapted with permission from ref 46. Copyright 1993 Rise National Laboratory, Denmark.)...

Figure 1.4 gives an example of the adsorption of a compound to suspended sediment, modeled as two resistances in series. At first, the compound is dissolved in water. For successful adsorption, the compound must be transported to the sorption sites on the surface of the sediment. The inverse of this transport rate can also be considered as a resistance to transport, Ri. Then, the compound, upon reaching the surface of the suspended sediment, must find a sorption site. This second rate parameter is more related to surface chemistry than to diffusive transport and is considered a second resistance, R2, that acts in series to the first resistance. The second resistance cannot... 

For relatively porous nanofiltration membranes, simple pore flow models based on convective flow will be adapted to incorporate the influence of the parameters mentioned above. The Hagen-Poiseuille model and the Jonsson and Boesen model, which are commonly used for aqueous systems permeating through porous media, such as microfiltration and ultrafiltration membranes, take no interaction parameters into account, and the viscosity as the only solvent parameter. It is expected that these equations will be insufficient to describe the performance of solvent resistant nanofiltration membranes. Machado et al. [62] developed a resistance-in-series model based on convective transport of the solvent for the permeation of pure solvents and solvent mixtures ... 

Modeling of H F contactors is in most papers based on a simple diffusion resistance in series approach. In many systems with reactive extractants (carriers) it could be of importance to take into account the kinetics of extraction and stripping reactions that can influence the overall transport rate, as discussed in refs. [30,46], A simple shortcut method for the design and simulation of two-phase HF contactors in MBSE and MBSS with the concentration dependent overall mass-transfer and distribution coefficients taking into account also reaction kinetics in L/L interfaces has been suggested [47]. 

As the first insulating/semiconducting higher boride series, the electrical transport of these compounds has been carefully investigated (e.g. Slack et al., 1977 Golikova, 1987 Werheit et al., 1991). The temperature dependence of the resistivity p follows the dependency of Mott s variable range hopping (VRH) model for 3 dimensional systems (Mott, 1968 Efros and Shklovskii, 1985), where... 

In this model, the steps can be classified into two categories, mass transport and surface reaction steps. The slowest of these steps determines if the process is mass transport or surface reaction limited. At lower temperatures the deposition rate is generally surface reaction limited. As the temperature increases, the surface reaction rate rises exponentially, resulting in a mass transport limited because transport becomes the slowest step in the series of deposition steps. Reaction resistances are often used to predict rate-limiting steps in CVD process. 

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