Resistance in Series Transport Model

Resistance model for transport in composite hollow fibre membranes based on polysul-fone with siloxane coating has been described in a classical work by Henis and Tripody [51], The resistance in series model assumes that the gas molecules encounter constrictions at certain positions throughout the pore which control the rate of diffusion [20,27,33]. For this scenario the total permeability is inversely related to the total resistance, thus 

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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]. 

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 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]. 

Various parameters in Equation 31.17 have been defined earlier. Danesi et al. [92] described a simple correlation between permeability coefficient in FSSLM and HFSLM configuration. At very large values of ( ) (as compared to 1), Equation 31.17 is transformed into the one used for FSSLM by Danesi et al. [92]. Hence, the smaller the value of ( ), the higher will be the negative value of the left-hand side of Equation 31.17, which suggests the higher rate of mass transfer. Later on, D Elia et al. [93] considered the resistance in series model where they have studied the mass transport across hollow-fiber contactors in NDSX mode. They showed that the overall mass transfer resistance is equal to the sum of individual mass transfer resistances across the aqueous boundary layer and membrane phase. Mathematically, it can be written as follows ... 

Finally, Machado et al. [21] developed a resistances-in-series model and proposed that solvent transport through the MPF membrane consists of three main steps (1) transfer of the solvent into the top active layer, which is characterized by surface resistance (2) viscous flow through NF pores and (3) viscous flow through support layer pores, all expressed by viscous resistances, i.e. 

The data in Table II was used along with the theoretical transport equations given for the individual process coefficients in Table I to make predictions for the Fox River. These calculated results appear in Table Illb. Equation 7 was used to combine coefficients for a resistance-in-series computation. In addition to these calculated coefficients Table Ilia contains those obtained from field modeling studies on the Fox and the two additional... 

Extensive theoretical and experimental work has previously been reported for supported liquid membrane systems (SLMS) as effective mimics of active transport of ions (Cussler et al., 1989 Kalachev et al., 1992 Thoresen and Fisher, i995 Stockton and Fisher, 1998). This was successfully demonstrated using di-(2-ethyl hexyl)-phosphoric acid as the mobile carrier dissolved in n-dodecane, supported in various inert hydrophobic microporous matrices (e.g., polypropylene), with copper and nickel ions as the transported species. The results showed that a pH differential between the aqueous feed and strip streams, separated by the SLMS, mimics the PMF required for the emulated active transport process that occurred. The model for transport in an SLMS is represented by a five-step resistance-in-series approach, as follows (1) diffusion of the ion through a hydrodynamic boundary layer (2) desolvation of the ion, where it expels the water molecules in its coordination sphere and enters the organic phase via ion exchange with the mobile carrier at the feed/membrane interface (3) diffusion of the ion-carrier complex across the SLMS to the strip/membrane interface (4) solvation of the ion as it enters... 

The reaction mechanism corresponding to the equivalent circuit shown in Figure 5.26 does not take mass transport or adsorption phenomena into account and therefore provides an incomplete picture of electrode reactions. Figure 5.28 shows a more realistic equivalent circuit. It still includes the ohmic resistance R and the double layer capacity C. However, the transfer resistance / , is replaced by he, faradaic impedance Zp, which may contain one or more circuit elements, in series or in parallel. In order to evaluate the faradaic impedance, one needs a physical reaction model. 

Either the anodic or cathodic half-cell reaction can become mass transport limited and restrict the rate of corrosion at co,r- The presence of diffusion controlled corrosion processes does not invalid the EIS method, but does require extra precaution and a modification to the circuit model of Fig. 4. In this case, the finite diffusional impedance is added in series with the usual charge transfer parallel resistance shown in Fig. 4. The transfer function for the frequency dependent finite diffusional impedance, Z fco), has been described [43] ... 

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