Parallel Transport Model

The parallel transport model considers the total flux as the contribution from the molecules travelling via surface diffusion and from the molecules travelling via Knudsen diffusion [27,36,49,50]. This model does not consider transition stages and is applicable to pores that remain roughly the same size throughout the entire membrane such as nanotube-based membranes. Gilron and Softer [27] presented the following expression. 

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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]. 

The parallel transport model assumes that surface diffusion and Knudsen diffusion are occurring simultaneously such that the total permeability is given by Equation (5.18). This model is explained in further detail earlier in this chapter and has been used by various groups [27,49,50]. Parallel transport is initially dominated by surface diffusion within the smaller pores where the surface concentration is high while the mode of Knudsen diffusion dominates within the larger pores. 

Q. To proceed further at this point one has to specify a pore model for the catalyst, and a model for the active site distribution. Froment and co-workers have examined a variety of cases such as single pore models (single-ended pores and pores open on both sides) with both deterministic and stochastic active site distributions, the bundle of parallel pores model and various tree-like models of the porous structure, which were earlier used by Pismen (40) to describe transport and reaction in porous systems. Such treelike models contain interconnected pores but lack any closed loops and are usually called Bethe networks or lattices. They are completely characterized by their coordination number Z, which is the number of pores connected to the same site of the network. 

To further illustrate the mechanisms of chemically and physically enhanced transdermal transport, a transdermal transport model will be discussed here. Generally, the permeability of the stratum corneum can be divided into parallel lipoidal and pore pathway... 

The transport properties of foods received much attention in the literature [184-188]. The main results presented by Saravacos and Maroulis [188] are summarized in this section. The results refer to moisture diffusivity and thermal condnc-tivity. Recently published values of moisture diffusivity and thermal conductivity in various foods were retrieved from the literature and were classified and analyzed statistically to reveal the influence of material moisture content and tempera-tnre. Empirical models relating moisture diffusivity and thermal conductivity to material moisture content and temperature were fitted to all examined data for each material. The data were screened carefully using residual analysis techniques. A promising model was proposed based on an Arrhenius-type effect of temperature, which uses a parallel structural model to take into account the effect of material moisture content. 

E. H. Cwirko and R. G. Carbonell, Interpretation of Transport-CoefTicients in Nafion Using a Parallel Pore Model, Journal of Membrane Science, 67,227 (1992). Y. M. Volfkovich, V. S. Bagotzky, V. E. Sosenkin, and I. A. Blinov, The Standard Contact Porosimetry, Colloids and Surfaces a-Physicochemical and Engineering Aspects, 187, 349 (2001). 

In a more recent study. Das and Chakraborty [9] presented analytical solutions for velocity, temperature, and concentration distribution in electroosmotic flows of non-Newtonian fluids in microchannels. A brief description of their transport model is summarized here, for the sake of completeness. A schematic diagram of the parallel plate microchannel configuration, as considered by the above authors, is depicted in Fig. 2. The bottom plate is denoted as y = H and top plate as y = +H. A potential gradient is applied along the axis of the channel, which provides the necessary driving force for electroosmotic flow. The governing equations appropriate to the physical problem are the equations for conservation... 

Damkohler (1935) is perhaps one of the pioneers to develop diffusion models to understand adsorption process within a particle. In his development he assumed a parallel transport of molecules occurring in both the void space as well as the adsorbed phase (Figure 9.2-1). 

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)...

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. 

This study proposes a simple approach to modelling the dynamics of solids transport within a flighted rotary dryer. The approach taken was to model the system in a series-parallel formulation of well-mixed tanks. The concept of active and passive solids is important, since it will lend itself well to the addition of mass and energy balance relations. This model formulation predicts the RTD of the system. Industrial RTD data was obtained from a 100 tonne per hour dryer and compared with the model predictions. gPROMS parameter estimation has delivered overall transport coefficients for this system. The transport coefficients are not independent, nor completely physically meaningful. However, they produce a very simple model formulation, which forms the basis for more detailed rotary dryer models incorporating mass and energy balances. Future work will see the development of a full dryer model based on the proposed solids transport model. Refinements will be made to the model to incorporate the effects of solids moisture and interaction with the counter current air stream. 

Cwirko, E. H. and CarboneU, R. G. 1992a. Interpretation of transport coefficients in Nafion using a parallel pore model. 67(2-3), 227-247. 

The same equation can be deriv on the basis of the parallel pore model which pictures transport as occurring through a number of parallel capillaries of the same size 14 The detailed derivation for this is describ elsewhere (Teramoto, M. et al., Ind, Eng, Chem. Res. in press.). 

Vidaver, G. A., 1966, Inhibition of parallel flux and augmentation of coimter flux shown by transport models not involving a mobile carrier, /. Theor. Biol. 10 301. 

Sampling from pneumatic conveyors parallels gas sampling. The exception is that soflds loadings can be as high as 50 kg of soHds per kg of gas. Commercially available samplers extract particles directly from a transport line. Fixed position samplers are mounted directly on the pneumatic conveyor pipe. Devices are available which extract samples from the product stream by the projection of a sample tube iato the flow. Particles impact on the tube and fill the open cavity. The tube is then withdrawn, and an internal screw discharges the collected material (20). In another model, the RX Sampler (manufactured by Gustafson) (29), samples are withdrawn usiag compressed air. 

Among the dynamical properties the ones most frequently studied are the lateral diffusion coefficient for water motion parallel to the interface, re-orientational motion near the interface, and the residence time of water molecules near the interface. Occasionally the single particle dynamics is further analyzed on the basis of the spectral densities of motion. Benjamin studied the dynamics of ion transfer across liquid/liquid interfaces and calculated the parameters of a kinetic model for these processes [10]. Reaction rate constants for electron transfer reactions were also derived for electron transfer reactions [11-19]. More recently, systematic studies were performed concerning water and ion transport through cylindrical pores [20-24] and water mobility in disordered polymers [25,26]. 

When a two- or higher-phase system is used with two or more phases permeable to the solute of interest and when interactions between the phases is possible, it would be necessary to apply the principle of local mass equilibrium [427] in order to derive a single effective diffusion coefficient that will be used in a one-equation model for the transport. Extensive justification of the principle of local thermdl equilibrium has been presented by Whitaker [425,432]. If the transport is in series rather than in parallel, assuming local equilibrium with equilibrium partition coefficients equal to unity, the effective diffusion coefficient is... 

In parallel with the identification of distinct transporters for GABA there has been continued interest in the development of selective blockers of these transporters and the therapeutic potential that could result from prolonging the action of synaptically released GABA. It has been known for a long time that certain pro-drugs of nipecotic add (e.g. nipecotic acid ethyl ester) are able to cross the blood-brain barrier and are effective anticonvulsants in experimental models of epilepsy. More recently, several different systemically active lipophillic compounds have been described that act selectively on GAT-1, GAT-2 or GAT-3 (Fig. 11.4). Of these, tiagabine (gabitiil), a derivative of nipecotic acid that acts preferentially on GAT -1, has proved clinically useful in cases of refractory epilepsy. 

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