Liquid-equilibrated

What happens upon equilibration with liquid water instead of water vapor According to Equation (6.13), the capillary radius would go to infinity for PVP —> 1. Thus, in terms of external conditions, swelling would be thermodynamically unlimited, corresponding to the formation of an infinitely dilute aqueous solution of ionomer. However, the self-organized polymer is an effectively cross-linked elastic medium. Under liquid-equilibrated conditions, swelling is not controlled by external vapor... 

Equation (6.20) determines the maximum degree of swelling and the maximum pore radius of a liquid-equilibrated membrane. This relation suggests that the external gas pressure over the bulk water phase, which is in direct contact with the membrane, controls membrane swelling. The observa-hon of different water uptake by vapor-equilibrated and by liquid water-equilibrated PEMs, denoted as Schroeder s paradox, is thus not paradoxical because an obvious disparity in the external conditions that control water uptake and swelling lies at its root cause. 

In the physical model, there are two separate structures for the membrane depending on whether the water at the boundary is vapor or liquid these are termed the vapor- or liquid-equilibrated membrane, respectively. The main difference between the two is that, in the vapor-equilibrated membrane, panel c, the channels are collapsed, while, in the liquid-equilibrated case, panel d, they are expanded and filled with water. These two structures form the basis for the two types of macroscopic models of the membrane. 

In opposition to the single-phase treatment of the membrane system mentioned above are the models that assume the membrane system is two phases. This type of model corresponds to the liquid-equilibrated membrane shown in panel d of Figure 6. In this structure, the membrane is treated as having pores that are filled with liquid water. Thus, the two phases are water and membrane. 

Unlike the cases of the single-phase models above, the transport properties are constant because the water content does not vary, and thus, one can expect a linear gradient in pressure. However, due to Schroeder s paradox, different functional forms might be expected for the vapor- and liquid-equilibrated membranes. 

Instead of the dilute solution approach above, concentrated solution theory can also be used to model liquid-equilibrated membranes. As done by Weber and Newman, the equations for concentrated solution theory are the same for both the one-phase and two-phase cases (eqs 32 and 33) except that chemical potential is replaced by hydraulic pressure and the transport coefficient is related to the permeability through comparison to Darcy s law. Thus, eq 33 becomes... 

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. 

L.C. Clark Jr., F. Gollan, Survival of mammals breathing organic liquids equilibrated with oxygen at atmospheric pressure. Science 152 (1966) 1755-1756. 

Figure 2. Sketch of the falling-film gas-liquid equilibrator.

Oxidation Kinetics. The results of kinetic experiments carried out in the laboratory in the gas-liquid equilibrator are illustrated in Figure 6. For these experiments, unfiltered epilimnetic water from Lake RBR was adjusted to pH 6.5 and amended with Mn(C104)2 to bring the concentration... 

Mobile Phase—The solvent mixture pumped through the column carrying the injected sample the liquid phase of the solid-liquid equilibration. 

When a condensed phase (the solvent), solid or liquid, equilibrates with a gas phase (the solute), some concentration of the gaseous species will be dispersed in the solid or liquid (i.e., some gas will be dissolved). Solution is the most general way in which a noble gas will interact with other materials. Note, however, that the term solution implies a more or less uniform microscopic-scale admixture of solvent and solute molecules or complexes of molecules this assumption is presumably reasonable for liquid solvents but perhaps not for solids and is difficult to test experimentally. 

C, porous membrane gas-liquid equilibration system, Dong Dasgupta 1986)... 

An electro-osmotic drag due to proton migration is defined as the number of water molecules moved with each proton in the absence of a concentration gradient For comparison, the electro-osmotic drag coefficient for vapor or liquid-equilibrated Nafion membranes ranges from 0.9 to 3.2 at room temperature [146]. For phosphoric acid-doped PBI membranes, however, the water drag coefficient is dose to zero [149,... 

Table 4. Isotope salt effects determined by vapor-liquid equilibration.

Liquids are necessarily more complicated than gases. To start with, they have much greater cohesiveness than gases for example, a liquid equilibrated with its gaseous vapor develops a meniscus. This boundary has a measurable surface tension caused by the asymmetry of particle interactions in the liquid and minimal interaction of particles in the gas phase above it. Hence a liquid must have significant interaction... 

Kilinc A., Carmichael I.S.E., Rivers M.L. and Sack R.O., 1983, The ferric-ferrous rario of natural silicate liquids equilibrated in air. Contrib. Mineral. Petrol., 83, 136-140. Kramers J.D., Smith C.B., Lock N.P., Harmon R.S, and Boyd F.R., 1981, Can Kimberlites be generated firom ordinary mande Nature, 291, 53-56. 

Figure 4.1. Schematics of (a) vapour-equilibrated membrane showing the collapsed interconnecting channel, (b) liquid-equilibrated membrane showing interconnecting channel swollen (after Ref. [27]).

Both membranes in Figure 4.4 exhibit the so-called Schroeder s paradox, an observed difference in the amount of water sorbed by a liquid-equilibrated membrane and a saturated vapour-equilibrated membrane, with both reservoirs at the same temperature and pressure [27, 35, 36]. This difference leads to the jump in lambda when the membrane is water equilibrated (activity = 1), as shown in Figure 4.4. The underlying mechanisms for this behaviour are not completely resolved, but Choi and Datta [29] proposed a good explanation, arguing that an additional capillary pressure causes the vapour-equilibrated membrane to sorb less water than the liquid-equilibrated membrane from an external solvent with the same activity. 

To simplify our modelling efforts we will assume that one water is carried through the membrane per proton over a wide range of water vapour activities, which is commonly done [40, 22], and approximately 2.5 water are carried through per proton when liquid-equilibrated. 

An approach that is conceptually simpler and does not require the prescription of transport to hydraulic or diffusion mechanisms was proposed by Janssen [47], and Thampan et al. [22] (hereafter TMT) based on the use of chemical potential gradients in the membrane. More recently, Weber and Newman [27] developed a novel model where the driving force for vapour-equilibrated membranes is the chemical potential gradient, and for liquid-equilibrated membranes it is the hydraulic pressure gradient. A continuous transition is assumed between vapour- and liquid-equilibrated regimes with corresponding transition from 1 to 2.5 for the electro-osmotic drag coefficient. 

In developing the transport equations, TMT make several assumptions that need to be critically re-examined. Though it is probably reasonable to assume that hydronium ions are the charge carriers for vapour-equilibrated membranes, this is not valid for liquid-equilibrated membranes, where the transport number is found to be around 2.5 [52]. For more realistic predictions in the liquid-equilibrated regime considered by TMT, this assumption needs to be modified. 

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. 

In terms of a capillary framework, the fraction of expanded channels is similar to a saturation. Although averaging the two equations by this fraction is not necessarily rigorous, it has a physical basis. Furthermore, it has the correct limiting behavior (i.e., all vapor-equilibrated when there are no expanded channels (i.e., no bulk-like water), 5 = 0, and all liquid-equilibrated when all the channels are expanded (i.e., bulk-like water throughout), 5=1) and a relatively sharp transition, as expected for a phase transition. 

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 determine the fraction of expanded channels, T and the channel-size distribution must be known. The channel-size distribution gives the fully expanded channel radii and is taken to be the same for different operating conditions and the same as the distribution measured for a liquid-equilibrated membrane. The reasons that this distribution is assumed to be constant are that it should not vary significantly with pressure or temperature xmder typical fuel-cell operating conditions and is used only when there is a separate liquid-water phase. This assumption has been used and proved valid within error tolerances [13, 18, 57]. The pore-size distribution for Nafion has been measured by the method of standard contact porosimetry [29, 58, 59]. In those studies, the distribution included both the channels and the clusters. Since only the channel-size distribution is of interest, only that regime of data is fit using the log-normal distribution [39]. The average channel radius is around 1.5 nm as expected from the physical model and other studies [23, 60, 61). 

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). 

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