Hydraulic permeation model modeling

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. ... 

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,... 

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. 

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. 

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. 

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).

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]...

It should be emphasized again that hydraulic permeation models do not rule out water transport by diffusion. Both mechanisms contribute concurrently. The water content in the PEM determines relative contributions of diffusion and hydraulic permeation to the total backflux of water. Hydraulic permeation prevails at high water contents, that is, under conditions for which water uptake is controlled by capillary condensation. Diffusion prevails at low water contents, that is, under conditions for which water strongly interacts with the polymeric host (chemisorption). The critical water content that marks the transition from diffusion-dominated to hydraulic permeation-dominated transport depends on water-polymer interactions and porous network morphology. Sorption experiments and water flux experiments suggest that this transition occurs at A. 3 for Nafion with equivalent weight 1100. 

The hydraulic permeation model in Eikerling et al. (1998, 2007a) assumed a negligible contribution of diffiisional water flux. It is valid at sufficiently high water content. The model helped rationalizing main dependences of the critical current density on membrane parameters. A 5-function-like pore size distribution dk r)/dr = itmaxS (r - ri), which is completely determined by the maximum water uptake A- ax. and by the first moment of the pore size distribution (i.e., the average pore size), r, provided an explicit expression for jpc. ... 

Rpem/Rs beyond this point. These trends are in excellent agreement with experimental data for Nafion 112. A finite positive gas pressure gradient, = Pf — P > 0, improves the internal humidification of the membrane, leading to a more uniform water distribution and a significantly reduced dependence of membrane resistance on k. The latter trends are consistent with predictions of the hydraulic permeation model. 

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. 

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... 

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