As was noted in Section 10.4, the fuel and oxidant must be directed over the surface of anode and cathode in the fuel-cell stack. Most often this is done with rectangular channels. The set of channels make up the flow field. Drawing on what we have learned in the previous sections, we now have a good idea of the reactant flow required. The flow to individual cells is in parallel and manifolds are used to distribute flow evenly to individual cells. In order to get the same flow to each cell, we would like the resistance to flow through individual cells to be high relative to the resistance in the manifold. On the other hand, high resistance requires additional pumping power to move the reactants. Remember, efficiency is the key driver in fuel-cell technology. As you might then expect, knowing the resistance to flow or, equivalently, the pressure drop through the flow field is vital.
Let’s begin by considering the pressure drop associated with flow through a closed conduit, typically a rectangular channel. The most important parameter is the Reynolds number:
(10.26)
where vz is the average velocity in the channel and the equivalent diameter is
(10.27)
where l and w are the height and width of the channel, respectively. The critical Reynolds number is 2300. As illustrated in the example below, the Re is typically small (about 100) and the flow is laminar. We can calculate the pressure drop through a channel of length L along the direction of flow as
(10.28)
The fanning friction factor, f, is 16/Re for fully developed laminar flow. This equation does not account for entrance effects in the channel as the flow field develops (see Figure 10.12). For laminar flow, the entrance length, Le, is estimated as
(10.29)
As shown in Figure 10.12, the pressure drop will be a bit higher than that predicted by Equation 10.28 due to entrance effects. We will typically ignore entrance effects for initial calculations, but you should be aware of the implications of this assumption.
Figure 10.12 Pressure drop for laminar flow including the entrance effect.
Finally, work is required to overcome friction losses in the channel and move the reactants through the fuel cell. The greater the pressure drop, the greater the power needed:
(10.30)
Changes in the density with pressure have been ignored, a relatively good assumption for the magnitude of pressure drops typically observed. The power required to move the reactants is part of the ancillary power that enters into the mechanical efficiency and directly impacts the overall efficiency of the system.
In addition to the pumping power and its impact on system efficiency, there are some other considerations. As was noted previously, providing equal flow to each cell is important. One of the most important challenges in flow-field design is the distribution of reactants to multiple channels and multiple cells. Reactants are supplied to cells in parallel, and within a single cell there are often many flow channels in parallel. It is important that each receives nearly the same flow of reactants. What are the consequences of supplying too much or too little reactant? The main problem is not providing enough fuel to a cell or to an isolated region of a cell. This phenomenon is called fuel starvation. Since all the cells of a bipolar stack are connected electrically in series, the current through each cell is identical. With many cells in series, the voltage differences across the stack can be tens or hundreds of volts. So, even if one cell has too little fuel, the large potential will drive current through the cell anyway. If there is no fuel available to be oxidized, then part of the cell components are oxidized to provide the electrons to allow current to flow. Oxidation of the catalyst carbon support is typical in low-temperature fuel cells,
Clearly, fuel starvation is a serious problem that must be avoided.
ILLUSTRATION 10.6
The cathode of a fuel cell has straight parallel channels. If the current density is 10 kA m−2, and air is used with a utilization of 0.6, what is the pressure drop through the channels? The dimensions of the channel are 2 mm deep, 2 mm wide (l and w), and the ribs (also called lands) are 1.5 mm wide. The planform size is 30 cm by 30 (L × W) cm.
SOLUTION:
Since the planform is square and the channels are straight, the length of the channels = L = W. We will assume that the channels are in along the L dimension. The velocity is determined from the volumetric flow rate of air, . The volumetric flow rate of the air can be determined from the cell current density and the utilization value given. The density of air at the operating conditions of the fuel cell is 0.9865 kg m−3. The viscosity of air at the same conditions is .
The average velocity in the channel is
where a correction has been made for the fact that the air only flows through the channels (rib area is not available for flow). The hydraulic diameter is
And, therefore,
The flow is laminar, and the fanning friction factor is
The allowable variability (tolerances) in channel dimensions must be specified and carefully controlled when manufacturing these flow fields. Imagine that there are multiple cells connected electrically in series and flow of reactants through these cells is in parallel. What is the impact of having the dimensions of one cell low compared to the others? The overall pressure drop will only be slightly altered by having one of many flow fields out of tolerance, and we will neglect this effect. For a given overall pressure drop, we can use Equation 10.28 to calculate the average velocity:
(10.31)
Of course, the flow rate is proportional to the cross-sectional area, which also changes with the channel dimensions. Accordingly, the volumetric flow through each channel is
(10.32)
As shown in Equation 10.32, the volumetric flow rate depends on the channel dimensions, and can be significantly different in channels that are out of specification. As a result, the utilization may vary over the planform. Figure 10.13 shows the utilization in a channel if its dimensions are slightly different from the nominal values. Here, the designed utilization of the reactant is 0.6. A 5% change in the dimensions of a channel can cause the utilization to increase from 0.6 to more than 0.7. Severe damage can occur if at any point there is insufficient fuel in the channel. In contrast, channels whose dimensions cause the utilization to decrease relative to the design value are much less of an issue. The flow rate is higher than nominal, but otherwise the performance is good.
Figure 10.13 Effect of poor tolerance control of channel dimensions in the flow field.
There are a large number of flow-field designs. The style chosen depends on the type of fuel cell, the application, and specific design preferences of the manufacturer. These flow-field designs vary greatly in complexity. Several of the approaches are shown in Figure 10.14. You can easily verify that the flow fields with multiple channels will have lower pressure drops; however, the chance of maldistribution is increased. The interdigitated design forces flow through the gas diffusion and electrode layers. The main objective is always to distribute the reactants in a controlled manner without excessive pressure drop.
Figure 10.14 Examples of flow-field designs.
The simple analysis above can be used to estimate the pressure drop for the straight-through, spiral, and serpentine designs. More complex flow fields require numerical simulation, typically using computational fluid dynamics (CFD), or experiments to characterize their performance.
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