Hydraulic permeation

What are the mechanisms and the transport coefficients of water fluxes (diffusion, convection, hydraulic permeation, electro-osmotic drag) ... 

The simple water charmel models can explain the ionomer peak and the small-angle upturn in the scattering data of fhe unoriented samples as well as of the oriented films. Interestingly, the helical structure of backbone segments is responsible for fhe sfabilify of fhe long cylindrical charmels. The self-diffusion behavior of wafer and protons in Nation is well described by the water channel model. The existence of parallel wide channels af high wafer uptake favors large hydrodynamic confributions to electro-osmotic water transport and hydraulic permeation. 

The experimental basis of sorption studies includes structural data (SANS, SAXS, USAXS), isopiestic vapor sorption isotherms,i and capillary isotherms, measured by the method of standard porosimetry. i 2-i44 Thermodynamic models for water uptake by vapor-equilibrated PEMs have been suggested by various groupThe models account for interfacial energies, elastic energies, and entropic contributions. They usually treat rate constants of interfacial water exchange and of bulk transport of water by diffusion and hydraulic permeation as empirical functions of temperature. 

Continuity of fhe wafer flux fhrough the membrane and across the external membrane interfaces determines gradients in water activity or concentration these depend on rates of water transport through the membrane by diffusion, hydraulic permeation, and electro-osmofic drag, as well as on the rates of interfacial kinetic processes (i.e., vaporization and condensafion). This applies to membrane operation in a working fuel cell as well as to ex situ membrane measuremenfs wifh controlled water fluxes fhat are conducted in order to study transport properties of membranes. 

An external gas pressure gradient applied between anode and cathode sides of the fuel cell may be superimposed on the internal gradient in liquid pressure. This provides a means to control the water distribution in PEMs under fuel cell operation. This picture forms the basis for the hydraulic permeation model of membrane operation that has been proposed by Eikerling et al. This basic structural approach can be rationalized on the basis of the cluster network model. It can also be adapted to include the pertinent structural pictures of Gebel et and Schmidt-Rohr et al. ... 

Notwithstanding any particular structural model, water transport in PEMs, in general, should be considered a superposition of diffusion in gradients of activity or concentration and hydraulic permeation in gradients of liquid or capillary pressure. Hydraulic permeation is the predominant mechanism xmder conditions for which water uptake is controlled by capillary condensation, whereas diffusion contributes significantly if water strongly interacts with the polymeric host. The molar flux of liquid water in the membrane, N, is thus given by... 

The hydraulic permeation model in Eikerling ef al. helped rationalize main dependence of fhe critical currenf densify on membrane parameters. A sharply peaked 5-function-like pore size disfribufion. 

The theoretical analysis of the hydraulic permeation model, moreover, provided an expression for the current density, at which membrane dehydra-... 

The diffusion model and the hydraulic permeation model differ decisively in their predictions of water content profiles and critical current densities. The origin of this discrepancy is the difference in the functions D (T) and /Cp (T). This point was illustrated in Eikerling et al., where both flux terms occurring in Equation (6.46) were converted into flux terms with gradients in water content (i.e., VA) as the driving force and effective transport coefficients for diffusion, A), and hydraulic permeation,... 

Direct comparison of D (A)and Dj( jj,(A) showed that hydraulic permeation dominates at high A, whereas diffusion prevails at low A. 

The hydraulic permeation model predicts highly nonlinear water content profiles, with strong dehydration arising only in the interfacial regions close to the anode. Severe dehydration occurs only at current densities closely approaching/p,. The hydraulic permeation model is consistent with experimental data on water content profiles and differential membrane resistance, i i as corroborated in Eikerling et al. The bare diffusion models exhibit marked discrepancies in comparison with these data. 

Recently, it was shown that the hydraulic permeation model could explain the response of the membrane performance to variations in external gas pressures in operating fuel cells. i Figure 6.15 shows data for the PEM resistance in an operational PEFC,... 

PEM resistance in operational PEFC as a function of the fuel cell current density, comparing experimental data (dots) and calculated results from a performance model based on the hydraulic permeation model for various applied gas pressure differences between anode and cathode compartments. (Reprinted from S. Renganathan et al. Journal of Power Sources 160 (2006) 386-397. Copyright 2006, with permission from Elsevier.)... 

The experimental data (dots) are reproduced very well within the framework of the hydraulic permeation model (solid lines). For the basic case with zero gas pressure gradient between cathode and anode sides, APe = 0, the model (solid line) predicts uniform water distribution and constant membrane resistance at )p < 1 A cm and a steep increase in R/R beyond this point. These trends are in excellent agreement with experimental data (open circles) for Nafion 112 in Figure 6.15. A finife positive gas pressure gradient, APs = P/ - P/ > 0, improves the internal humidification of fhe membrane, leading to more uniform water distribution and significantly reduced dependence of membrane resistance on X. The latter trends are consistent with the predictions of fhe hydraulic permeation model. 

Figure 3. Hydraulic permeation data jor PVA-water system (6)...

D.R. Paul, J.D. Paciotti and O.M. Ebra-Lima, Hydraulic Permeation of Liquids Through Swollen Polymeric Networks, J. Appl. Polym. Sci. 19, 1837 (1975). 

Structural models emerge from the notion of membrane as a heterogenous porous medium characterized by a radius distribution of water-filled pores. This structural concept of a water-filled network embedded in the polymer host has already formed the basis for the discussion of proton conductivity mechanisms in previous sections. Its foundations have been discussed in Sect. 8.2.2.1. Clearly, this concept promotes hydraulic permeation (D Arcy flow [80]) as a vital mechanism of water transport, in addition to diffusion. Since larger water contents result in an increased number of pores used for water transport and in larger mean radii of these pores, corresponding D Arcy coefficients are expected to exhibit strong dependencies on w. 

Note that diffusion models and hydraulic permeation models have their own caveats the membrane is neither a homogeneous acid solution, nor is it the well-structured porous rock. Critical comparison of the results of the two approaches with each other and with experiments, is of crucial importance for understanding the membrane functioning within the cell and developing the strategies on water management and optimized membrane properties. 

These concepts have been routinely employed to determine psds of genuine porous media [88]. A difficulty arises when they are applied to PEMs, since these membranes do not possess an intrinsic porosity. Instead, pores in them are created by the water of hydration, whereas in the dry state the pore network collapses. Gas permeability of PEM is very small. Thus, only with a certain degree of tolerance can one speak about three-phase capillary equilibria, implied in the Laplace equation. It is rather a semiempirical phenomenology, that allows one to relate the liquid pressure (the driving force of the hydraulic permeation)... 

Hydraulic Permeation versus Diffusion and Comparison with Experiments... 

In contrast to the diffusion approach, in the previous sections hydraulic permeation was considered as the effective mode of water transport. Transformed to the form of an effective diffusion coefficient the transport coefficient of the latter model becomes... 

Membrane performance characteristics in the hydraulic and diffusion limits are compared to each other in Fig. 9. Figure 9(a) illustrates that in the diffusion model considerable deviations from the purely ohmic performance of the saturated membrane arise already at small jv/Jj, well below the critical current density. This is in line with the comparison of the water-content profiles calculated in the diffusion model, Fig. 9(b), with those from the hydraulic permeation model, in Fig. 7. Indeed, membrane dehydration is much stronger in the diffusion model, affecting larger membrane domains at given values of jp/./j. Moreover, the profiles exhibit different curvature from those in Fig. 7. 

Experimental data by Biichi et al. [66, 92] and Mosdale et al. [93] speak in favor of hydraulic permeation as the basic mode of... 

Fig. 8 Parameterizations of effective dimensionless diffusion constants D(w>) in the model of hydraulic permeation (cf. Eq. 2.40), using parameterization (4) in Table 1) and in the diffusion model. In the latter, D is obtained from the dimensional diffusion constant V (in cm2s 1) via the identity fmV = FcvjWsV/L. Absolute values have been adjusted in such a way, that both parameterizations will give the same value of/ pC/J. ...

Fig. 10 Membrane resistance in H2/O2 fuel cell as a function of proton current density. Experimental data, normalized to the resistance 9ts of the saturated membrane at various temperatures have been extracted from Ref. 94. They are compared to the values calculated in the hydraulic permeation model (main figure) and to the results of the diffusion model, taken from Ref. 7 (inset).

A realistic approach should combine in some way elements of hydraulic permeation and diffusion, since generally the complex truth about water transport in PEM lies, presumably, somewhere between these two limits, with hydraulic permeation mechanism dominating at large and diffusion at small water contents. 

Furthermore, which mechanism prevails is also determined by the membrane microstructure and water/polymer interactions. A pronounced hydropho-bic/hydrophilic phase separation will result in a well-developed porous structure and, thereby promote hydraulic permeation as the relevant mechanism. In random polymer membranes, which exhibit a smaller extent of ion clustering, water fractions will be more dispersed in the... 

This notion is supported by a large number of independent experimental data, related to structure and mobility in these membranes. It implies furthermore a distinction of proton mobility in various water environments, strongly bound surface water and liquidlike bulk water, and the existence of water-filled pores as network forming elements. Appropriate theoretical treatment of such systems involves random network models of proton conductivity and concepts from percolation theory, and includes hydraulic permeation as a prevailing mechanism of water transport under operation conditions. On the basis of these concepts a consistent approach to membrane performance can be presented. 

Modeling approaches that explore membrane water management have been reviewed in [16]. Overall, the complex coupling between proton and water mobility at microscopic scale is replaced by a continuiun description involving electro-osmotic drag, proton conductivity and water transport by diffusion or hydraulic permeation. Essential components in every model are the two balance equations for proton flux (Ohm s law) and for the net water flux. Since local proton concentration is constant due to local electroneutrality of the membrane, only one variable remains that has to be solved for, the local water content. 

The hydraulic permeation model is appropriate for well-hydrated membranes. However, it cannot appropriately describe water transport in lowly hydrated membranes since it underestimates polymer-water correlations. Both model variants are mathematically similar and complementary in their range of applicability. Indeed, it is a straightforward task to merge them into a unified approach, as suggested in [11,16,144,147]. 

Fig. 9 Water content profiles in the membrane, calculated in the hydraulic permeation model, at various fuel-cell current densities. A typical value of the parameter / that determines the onset of membrane dehydration near the anode was estimated as / 5-10Acm for Nafion 117 [11,16]...

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