Vapor-equilibrated transport mode

One comment should be made regarding the form of the transport equations. In the literature, two-phase flow has often been modeled using Schlogl s equation [50, 51]. This equation is similar in form to Eq. (5.9), but it is empirical and ignores the Onsager cross coefficients. Equations (5.8) and (5.9) stem from concentrated-solution theory and take into account all the relevant interactions. Furthermore, the equations for the liquid-equilibrated transport mode are almost identical to those for the vapor-equilibrated transport mode making it easier to compare the two with a single set of properties (i.e., it is not necessary to introduce another parameter, the elec-trokinetic permeability). 

As in the case for the vapor-equilibrated transport mode, the properties of the liquid-equilibrated transport mode depend on the water content and temperature of the membrane. For a fully liquid-equilibrated membrane, the properties are uniform at the given temperature. This is because the water content remains constant for the liquid-equilibrated mode unlike in the vapor-equilibrate one. From experimental data, the value of A, for liquid-equilibrated Nafion is around 22, assuming the membrane has been pretreated correctly [6, 7, 52]. In agreement with the physical model, the water content is only a very weak function of temperature for extended (E)-form membranes (as assumed in our analysis) and can be ignored [6]. For other membrane forms, this dependence is much stronger and cannot be ignored, as discussed in the Section 5.10.1. 

When the membrane is in contact with liquid water on one side and vapor on the other (i.e., it is neither fully liquid nor vapor equilibrated), as can often occur during fuel-cell operation, both the liquid- and vapor-equilibrated transport modes will occur. This results in a transition between modes that exists in the membrane. As discussed in the physical model, a continuous transition between the two transport modes is assumed. Thus, transport in the transition region is a superposition between the two transport modes they are treated as separate transport mechanisms occurring in parallel (i.e., the middle region in Figure 5.3). In this section, an approach to modeling the transition region is introduced followed by a discussion of its limitations, other approaches, and points to consider. 

In summary, when both the liquid- and vapor-equilibrated transport modes occur in the membrane they are assumed to occur in parallel. In other words, there are two separate contiguous pathways through the membrane, one with liquid-filled channels and another that is a one-phase-type region with collapsed channels. To determine how much of the overall water flux is distributed between the two transport modes, the fraction of expanded channels is used. As a final note, at the limits of S = 1 and S = 0, Eqs. (5.17) and (5.18) or their effective property analogs collapse to the respective equations for the single transport mode, as expected. 

Using the above equations, isotherms of the fraction of expanded channels versus liquid pressure can be generated as shown in Figure 5.5. From the curves, the temperature dependence of the saturation is not strong since the transition still occurs over a small liquid-pressure range. All of the curves show that, at a liquid pressure of 1 bar, the channels are completely expanded and filled with liquid in agreement with experimental observations. If the liquid pressure falls below about 0.15 bar, then the liquid water phase ceases to exist at all temperatures and the transport of water is solely by the vapor-equilibrated transport mode, which also agrees with the physical model. If the liquid pressure is above around 0.6 bar, then X is 22 (only the liquid-equilibrated transport mode). 

Weber and Newman do the averaging by using a capillary framework. They assume that the two transport modes (diffusive for a vapor-equilibrated membrane and hydraulic for a liquid-equilibrated one) are assumed to occur in parallel and are switched between in a continuous fashion using the fraction of channels that are expanded by the liquid water. Their model is macroscopic but takes into account microscopic effects such as the channel-size distribution and the surface energy of the pores. Furthermore, they showed excellent agreement with experimental data from various sources and different operating conditions for values of the net water flux per proton flux through the membrane. 

In summary, the transport mode of a vapor-equilibrated membrane is that of a single membrane phase in which protons and water are dissolved. The chemical-potential gradient is used directly since it precludes the necessity of separating it into pressure and activity terms. Thus, Eqs. (5.8-5.11) are used directly without any modifications. Although it makes sense to use the chemical-potential driving force, most of the experimental data are a function of water content or X. Thus, a way is needed to relate X to the chemical potential. 

When both transport modes occur, the governing equations above do not form a closed set of equations because both /Xw and pl appear in Eqs. (5.17) and (5.18) as separate driving forces another relationship is needed. If local equilibrium between the vapor- and liquid-equilibrated parts of the membrane is assumed, then the necessary additional equation becomes... 

To examine the transport-mode-transition region in more detail, simulations were run at different current densities [71]. The resultant membrane water profiles are shown in Figure 5.11 where a vapor-equilibrated membrane at unit activity has a water content of L = 8.8 as calculated by the modified chemical model (see Section 5.5.1). The profiles in the figure demonstrate that the higher the current density the sharper the transition from the liquid-equilibrated to the vapor-equilibrated mode as well as the lower the value of the water content at the anode GDL/membrane interface. The reason why the transition occurs at the same point in the membrane is that the electro-osmotic flow and the water-gradient flow are both proportional to the current... 

If the membrane itself is partially in contact with liquid and vapor, as can commonly be the case in an operating fuel cell, the water content and uptake in the membrane can vary with location, although in equilibrium the water content in the membrane will become homogeneous with uptake depending on the overall water avadabihty. Transport in this case can be modeled as occurring in parallel between gas and liquid equilibrated modes, with a suitable fraction denoting the liquid and gas phase fractions of contact with the membrane. 

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