Hydraulic permeation model diffusion

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. 

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).

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 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... 

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. 

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... 

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. ...

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 most complete description of variations in local distributions and fluxes of water is provided by combination models, which allow for concurrent contributions of diffusion and hydraulic permeation to the water backflux that competes with the electro-osmotic drag. The main conclusions from these models for membrane water management under operation are... 

In this Section, it is implicitly assumed that the mass transport resistance at the fluid-membrane interface on either side of the membrane is negligible. Also the following is information that is made available publicly by the membrane manufacturers, when not otherwise noted. As in technical processes, mass transport across semipermeable medical membranes is conveniently related to the concentration and pressme driving forces according to irreversible thermodynamics. Hence, for a two-component mixture the solute and solvent capacity to permeate a semipermeable membrane under an applied pressure and concentration gradient across the membrane can be expressed in terms of the following three parameters Lp, hydraulic permeability Pm, diffusive permeability and a, Staverman reflection coefficient (Kedem and Katchalski, 1958). All of them are more accurately measured experimentally because a limited knowledge of membrane stmcture means that theoretical models provide rather inaccurate predictions. 

An effect not considered in the above models is the added resistance, caused by fouling, to solute back-diffusion from the boundary layer. Fouling thus increases concentration polarization effects and raises the osmotic pressure of the feed adjacent to the membrane surface, so reducing the driving force for permeation. This factor was explored experimentally by Sheppard and Thomas (31) by covering reverse osmosis membranes with uniform, permeable plastic films. These authors also developed a predictive model to correlate their results. Carter et al. (32) have studied the concentration polarization caused by the build-up of rust fouling layers on reverse osmosis membranes but assumed (and confirmed by experiment) that the rust layer had negligible hydraulic resistance. 

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