Pore Model for Membrane Gas Transport

Once a geometric model for the pore and a model for the gas transport through the pores is chosen, this method (as the liquid displacement technique) allows determination of the absolute distribution (differential or integral) of the number of pores active to flux for the membrane. The model used should depend on the working con-... 

For gas transport in small pores (say, less than the 10 nm range) the sizes of which are no longer much larger than those of the gas molecules, the contribution of viscous flow can be neglected and other considerations need to be factored in the model. First, the gas molecules are considered to be hard sphere with a finite size and the gas diffusion process is assumed to proceed in the membrane pores by random walk. The membrane pores are assumed to consist of smooth-wall circular capillaries. In addition to gas molecules colliding with the membrane pore walls, adsorption on the pore wall and the associated surface flow or diffusion are considered. Adsorption also effectively reduces the pore size for diffusion. 

Figure 7 Five-step model for mass transfer through crystal membranes. Srep / adsorption from the gas phase to the external surface. Step 2 transport from the external surface into the pores. Step 3 intracrystalline diffusion. Step 4 transport out of the pores to the external surface. Step 5 desorption from the external surface into the gas phase. (From Ref. 35.)...

The modelling of gas permeation has been applied by several authors in the qualitative characterisation of porous structures of ceramic membranes [132-138]. Concerning the difficult case of gas transport analysis in microporous membranes, we have to notice the extensive works of A.B. Shelekhin et al. on glass membranes [139,14] as well as those more recent of R.S.A. de Lange et al. on sol-gel derived molecular sieve membranes [137,138]. The influence of errors in measured variables on the reliability of membrane structural parameters have been discussed in [136]. The accuracy of experimental data and the mutual relation between the resistance to gas flow of the separation layer and of the support are the limitations for the application of the permeation method. The interpretation of flux data must be further considered in heterogeneous media due to the effects of pore size distribution and pore connectivity. This can be conveniently done in terms of structure factors [5]. Furthermore the adsorption of gas is often considered as negligible in simple kinetic theories. Application of flow methods should always be critically examined with this in mind. 

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]. 

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... 

The accumulation and distribution of licpiid water in the polymer electrolyte membrane fuel cell (PEMFC) is highly dependent on the porous gas diffusion layer (GDL). The accmnulation of liquid water is often simply reduced to a relationship between liquid water saturation and capillary pressure however, recent experimental studies have provided valuable insights in how the microstmcture of the GDL as well as the dynamic behavior of the liquid play important roles in how water will be distributed in a PEMFC. Due to the importance of the GDL microstmcture, there have been recent efforts to provide predictive modeling of two-phase transport in PEMFCs including pore network modehng and lattice Boltzmann modeling, which are both discussed in detail in this chapter. Furthermore, a discussion is provided on how pore-scale infonnation is used to coimect microstmcture, transport and performance for macroscale upscaling. 

Intrinsic microporosity of polyphenylene oxides corresponds to the throat and cavity model, where a cavity may have several throats. Effective diameters of the pore throats are in the range of ca. 0.4 nm at 77 K, increasing up to ca. 0.5 nm as the temperature rises to ambient. At these temperatures the intrinsic micropores are accessible for the molecules of simple gases. Apparently it is through these micropores that the gas transport in the dense polyphenylene oxides membranes occurs, with pore throats playing a role of a size caliber for the molecular sieve effect that manifests itself in membrane separations of certain gaseous mixtures. 

Reverse osmosis, pervaporation and polymeric gas separation membranes have a dense polymer layer with no visible pores, in which the separation occurs. These membranes show different transport rates for molecules as small as 2-5 A in diameter. The fluxes of permeants through these membranes are also much lower than through the microporous membranes. Transport is best described by the solution-diffusion model. The spaces between the polymer chains in these membranes are less than 5 A in diameter and so are within the normal range of thermal motion of the polymer chains that make up the membrane matrix. Molecules permeate the membrane through free volume elements between the polymer chains that are transient on the timescale of the diffusion processes occurring. 

The above model has been refined based on the dusty gas model [Mason and Malinauskas, 1983] for transport through the gas phase in the pores and the surface diffusion model [Sloot, 1991] for transport due to surface flow. Instead of Equation (10-101), the following equation gives the total molar flux through the membrane pores which are assumed to be cylindrically shaped... 

Once the pore size and length I are given to the pore network, one can calculate the effective pressure field (by using iteration method), the temperature field through the network, and its effect on the vapor flux through the membrane. This model takes into account all molecular transport mechanisms based on the kinetic gas theory for a single cylindrical tube and could be applied to all forms of membrane distillation process [61]. 

For porous membranes the mass transport mechanisms that prevail depend mainly on the membrane s mean pore size [1.1, 1.3], and the size and type of the diffusing molecules. For mesoporous and macroporous membranes molecular and Knudsen diffusion, and convective flow are the prevailing means of transport [1.15, 1.16]. The description of transport in such membranes has either utilized a Fickian description of diffusion [1.16] or more elaborate Dusty Gas Model (DGM) approaches [1.17]. For microporous membranes the interaction between the diffusing molecules and the membrane pore surface is of great importance to determine the transport characteristics. The description of transport through such membranes has either utilized the Stefan-Maxwell formulation [1.18, 1.19, 1.20] or more involved molecular dynamics simulation techniques [1.21]. 

UF and RO models may all apply to some extent to NF. Charge, however, appears to play a more important role than for other pressure driven membrane processes. The Extended-Nemst Planck Equation (equation (3.28)) is a means of describing NF behaviour. The extended Nernst Planck equation, proposed by Deen et al. (1980), includes the Donnan expression, which describes the partitioning of solutes between solution and membrane. The model can be used to calculate an effective pore size (which does not necessarily mean that pores exist), and to determine thickness and effective charge of the membrane. This information can then be used to predict the separation of mixtures (Bowen and Mukhtar (1996)). No assumptions regarding membrane morphology ate required (Peeters (1997)). The terms represent transport due to diffusion, electric field gradient and convection respectively. Jsi is the flux of an ion i, Di,i> is the ion diffusivity in the membane, R the gas constant, F the Faraday constant, y the electrical potential and Ki,c the convective hindrance factor in the membrane. 

Water transport across the membrane is also described by two physical mechanisms electro-osmotic drag and diffusion. The balance between the electro-osmotic drag of water from anode to cathode and back diffusion from cathode to anode yields the net water content through the membrane. The present multi-phase model is capable of identifying important parameters for the wetting behavior of the gas diffusion layers and can be used to identify conditions that might lead to the onset of pore plugging, which has a detrimental effect of the fuel cell performance. 

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