Expanded channels

A.A. Vasiliev and V.V. Grigoriev, Critical conditions for gas detonations in sharply expanding channels, Fizika Gorenyia i Vzryva, 16,117-125,1980. 

Expanded channels (non-stoichiometric) Lattice planes Dehydrated hydrates Metal-ion coordinated water... 

Based on their structural characteristics, crystalline hydrates were broken into three main classes. These were (1) isolated lattice site water types, (2) channel hydrates, and (3) ion associated water types. Class 2 hydrates were further subdivided into expanded channel (nonstoichiometric) types, planar hydrates, and dehydrated hydrates. The classification of the forms together with a suitable phase diagram provides a rationale for anticipating the direction and likelihood of a transition, including transitions that may be solution mediated. 

When the membrane is neither fully liquid- nor vapor-equiUbrated, the transport mode is assumed to be a superposition between the two they ean be treated as separate transport modes occurring in parallel. For this ease, the transport-property values should be taken to be mode dependent and averaged depending on the fraction of channels that are expanded. By and their respective resistances to transport. By proposing this treatment, the model bridges the gap between the two transport modes and in essenee the two types of models in the literature. In addition, the physieal model goes further in that a structural parameter, namely the fraetion of expanded channels is used to do this averaging and thus a continuous function and transition between the two modes is obtained that is also physically based. 

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. 

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. 

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). 

Now that the contact angle and channel-size distribution are known, the fraction of expanded channels can be calculated as a function of the hydraulic pressure. In accordance with the physical model, the expanded channel-cluster network is treated as a bundle of capillaries. To calculate S, a critical... 

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). 

Figure 5.5. Fraction of expanded channels as a function of the liquid pressure using the parameters and equations given in the text the temperatures are 25, 45, 65, and 85°C.

The permeation coefficients, like the other transport properties, are expected to depend on water content, temperature, and the state of the membrane (i.e., collapsed or expanded channels). Fitting the experimental data [39] yields the following expressions for vapor- and liquid-equilibrated membranes respectively... 

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