A fuel cell is being considered for a manned space flight. Because hydrogen and oxygen are needed for propulsion power, these are the preferred fuel and oxidant. The basic cell performance is shown in Figure 10.20. Our job is to recommend the current density and voltage at which the fuel cell operates. A high current density will result in higher power density and therefore a smaller cell stack. However, operation at high power means that the cell voltage is lower, and the efficiency lower, resulting in higher fuel consumption. We will seek a system where the combined mass of the fuel-cell stack, hydrogen, and oxygen is minimized.
Figure 10.20 Polarization curve for space fuel cell operating on hydrogen and oxygen. Utilization of hydrogen and oxygen are 0.95.
We’ll need to simplify this optimization, but we can nonetheless illustrate how to approach these challenges. First, let’s understand why minimizing the mass is so important. Any object that goes into orbit must be accelerated to a high velocity (about 7 km·s−1 for low earth orbit). This acceleration requires a large amount of energy and fuel, perhaps 2 kg of fuel for each kg of payload. For our purposes, the fuel-cell system as well as the oxygen and fuel needed for the mission duration can be considered payload. There is a tremendous incentive in keeping this mass low.
Next, let’s think about the key factors that go into fuel consumption besides the voltage efficiency. Two essential items are the average net conditioned electrical power and the length of the mission; these items are critical for a space mission because they determine the amount of fuel needed, and it is necessary to carry the fuel for the entire flight with you into space. Other items include those we have already discussed such as the mechanical efficiency, fuel efficiency, and power conversion efficiency. Baseline values for these parameters are given in Table 10.3. From these parameters and the cell potential, we can calculate the mass of hydrogen required.
where na is 2 for the hydrogen oxidation, and the energy (power × time) has been used rather than the power in order to calculate the total mass rather than the mass flow rate. Note that the amount of hydrogen used is inversely proportional to the voltage of the cell.
Table 10.3 Baseline Parameters for Case Study
Parameter Symbol Value
Power P 2000 W
Mission length t 106 s
Mechanical efficiency ηmech 0.90
Fuel efficiency, or hydrogen utilization ηfuel
uH2 0.98
Power conversion efficiency ηpc 0.95
Oxygen utilization uO2 0.95
Cell pitch γ 250 m−1
Active area ratio Ar 0.7
Apparent density of the fuel-cell stack ρs 2000 kgm−3
System/stack mass ratio mr 2
In contrast to terrestrial applications, the oxidant is pure oxygen not air, and its mass must be calculated and added to the total just like we did for hydrogen. Therefore, we need the utilization of oxygen to determine the amount of oxygen required, which is proportional to the amount of hydrogen.
Next, we need the mass of the cell stack. For a given potential of the cell, the current density is determined by the polarization curve. Because a fuel cell is a power conversion device rather than an energy storage device, the total cell area, A, is calculated from the average gross power of the cell stack:
This relationship works independent of whether the cells are in parallel or series, or a mixture of the two. Then, the volume of the cell stack is estimated as
The mass of the stack can now be calculated from the volume, the apparent stack density, and a ratio that accounts for the mass of system components required to support power conversion in the cell stack,
Thus, the combined mass for the cell stack, hydrogen (fuel) and oxygen for the required mission, is
Taking the parameters given in Table 10.3, the total mass can be determined as a function of cell potential or, equivalently, current density using the polarization curve. The results of these calculations are shown in Figure 10.21. A minimum mass is obtained for operation at a potential of about 0.88 V. As the cell potential increases, the fuel cell is more efficient, and therefore the mass of oxygen and hydrogen decrease. At the same time, the mass of the fuel-cell system increases sharply.
Figure 10.21 Optimized potential for planned space mission, where mass is the most important factor.
Closure
In this chapter, we have extended the analysis of fuel cells to include the overall system. The current–voltage relationship is still fundamental, but the amount of fuel and air provided relative to the stoichiometric value must be established. Material and energy balances are the main tools for understanding and designing full systems. Water and thermal balances have been shown to be critical to these designs. In addition, we have introduced some of the mechanical aspects that are critical to good cell stack and system designs.
Further Reading
Larminie, J. and Dicks, A. (2003) Fuel Cell Systems Explained, John Wiley & Sons, Inc., New York.
Mench, M. (2008) Fuel Cell Engines, John Wiley & Sons, Inc. New York.
Spiegel, C.S. (2007) Designing & Building Fuel Cells, McGraw Hill.
Problems
10.1. Using the definition of efficiency given by Equation 10.3, what is the maximum thermal efficiency of a hydrogen/oxygen fuel cell at 25 °C, standard conditions?
10.2. A fuel cell operating on methane produces 100 kW of gross electrical power. Calculate the voltage and thermal efficiency given the following:
The individual cells are operating at a potential of 0.65 V and gaseous water is produced at standard conditions.
75% of the fuel is converted to electricity (the balance is combusted external to the fuel cell to drive the reformation process).
5% of the electrical output is consumed by ancillary equipment and losses in power conversion.
10.3. How would the results of Problem 10.2 change if we assume that the water is produced as a liquid?
10.4. The quantity ΔG/ΔH was introduced in Equation 10.7. This quantity is needed because historically with fuel cells the efficiency is based on the heating value of the fuel rather than on the change in Gibbs energy. For a hydrogen/oxygen fuel cell, create a plot of ΔG/ΔH as a function of temperature from 25 to 200 °C. For temperatures below 100 °C, use the higher heating value of the fuel (product is liquid water); and above 100 °C, use the lower heating value (product is gaseous water). You’ll need to find data for heat capacity as a function of temperature to complete this problem.
10.5. Calculate mass of hydrogen needed to operate PEM FC at an average of 18 kW for a period of 3 hours. Assume the utilization of hydrogen is 0.98 and that each cell is operating at 0.7 V. Roughly18 kW power is needed to sustainably power a passenger vehicle on the highway. Compare the mass of hydrogen calculated with the mass of gasoline that would be needed. If helpful, you may assume that the average speed for the 3-hour period is 90 kmph.
10.6. A portable hydrogen fuel cell used by the military operates at an overall fuel-cell system thermal efficiency, ηth, of 55%. The stack has a specific power of 100 W·kg−1 and operates at 50 W, and the system delivers a net power of 35 W continuously. The ancillary systems (plumbing, controls, and electronics) weigh 0.210 kg. The theoretical energy content of a hydrogen storage system is 1200 W·h·kg−1.
Estimate the mass of the fuel required for providing 35 W of conditioned power for a 72-hour mission.
How does the mass of the fuel-cell system package (with fuel) compare to a lithium-ion battery with a specific energy of 150 W·h·kg−1?
Discuss two properties of the fuel-cell system you would improve to achieve 700 W·h·kg−1 for the fuel-cell system package for the same mission duration. (This problem was suggested by S.R. Narayan).
10.7. Create a plot of system thermal efficiency versus power level for PEM FC operating on pure hydrogen. Use the LHV of hydrogen, and assume that the utilization of hydrogen is 0.97. Power for ancillary equipment is 500 W + 5% of gross electrical output; that is,
The cell stack has 100 cells, each of 0.04 m2 area. The performance curve is represented by the model given in Problem 9.4 with the following parameters:
What do these results suggest about how the fuel cell might be best used in a fuel-cell hybrid electric vehicle?
10.8. A direct methanol fuel cell operates at 0.4 V and 2000 A·m−2. If the stack must produce 50 W at 12–24 V, what configuration would you propose? Assume a pitch of 4 mm/cell, an endplate thickness of 7.5 mm, and an active area ratio , where Ac is the active area of a single cell [m2]. What is the volume of the stack?
10.9. Estimate the mass-transfer limiting current density as a function of utilization of oxygen down the channel of a PEM FC. Treat the mass-transfer resistance as simple diffusion through a porous gas diffusion layer. The cell is operating at 70 °C, ambient pressure, and the air is saturated with water vapor. The thickness of the GDL is 170 μm, and the effective diffusivity of oxygen is 6 × 10−6 m2 s−1. What is the effect of increasing temperature on limiting current density for a PEM FC?
10.10. When analyzing the performance of a low-temperature fuel cell, it is often desirable to include the effect of oxygen utilization with a one-dimensional analysis. If the mole fraction of oxygen changes across the electrode, what value should be used? Assuming that the oxygen reduction reaction is first order in oxygen concentration, show that it is appropriate to use a log-mean mole fraction of oxygen as an approximation of the average mole fraction.
10.11. Express the log-mean term in Problem 10.10 in terms of oxygen utilization and the inlet mole fraction of oxygen, yin. Sketch the average current density as a function of utilization, keeping the overpotential for oxygen reduction fixed. How would this change if mass transfer is also included.
10.12. Discuss the advantages and disadvantages of using the alternative definition for fuel processing efficiency based on the heating value of the fuel as shown in Chapter 10 immediately following Equation 10.25.
10.13. The composition of industrial natural gas is as follows:
Methane 94.9% Ethane 2.5% Propane 0.2%
Butane 0.03 Nitrogen 1.67 CO2 0.7
A fuel-cell power plant is designed so that the fuel-processer efficiency is 78%, the mechanical efficiency is 92%, and the power conditioning efficiency is 96%. If the utilization of fuel (uf) is 0.85, what flow rate of natural gas is needed for a system that provides 400 kW of conditioned power? The LHV of the industrial natural gas is 53.08 MJ·kg−1.
10.14. A PEMFC operates at an average current density of 16 kA m−2, and humidified air (75 °C) is used with an oxygen utilization of 0.6. The dimensions of the channel are 2 mm wide, 2 mm deep, and the ribs are 3 mm wide. The planform size is 12 cm × 24 cm. What is the pressure drop on the air side for parallel channels across the shorter dimension of the planform? The longer dimension? What would be the pressure drop if the flow field consisted of three parallel serpentine channels?
10.15. The channel for a flow field has nominal dimensions of 2.0 mm × 1.5 mm. If the nominal utilization is 55%, the maximum permissible utilization is 65%, what are the minimum dimensions allowed for the channels?
10.16. Recreate Figure 10.17 for the case where the cell is operating at 200 and then 300 kPa absolute pressure. Explain the effect of operating pressure on the water balance for PEM fuel cells.
10.17. Dry air is fed to a PEMFC that is operating at 10 kA·m−2. The temperature is 55 °C and the pressure is 120 kPa. If the utilization of air is 40%, what fraction of the water produced by the cell is removed as a liquid? Assume that this exit stream is saturated at the cell temperature.
10.18. Whereas liquids are used to cool low-temperature fuel cells, cooling with gas is more appropriate at the high operating temperature of a solid oxide fuel cell. One strategy for cooling is to increase the air flow through the cathode (see the figure that follows). The additional power required to provide air for cooling in excess of what is required for the electrochemical reactions is a drain on the efficiency, but in contrast to PEM fuel cells, air stoichiometry has no effect on the water balance. Since temperature strongly affects the conductivity of the YSZ separator and because thermal gradients can cause failure of the seals, the temperature difference between the inlet air and the cell stack cannot be too large.
The fuel is reformed methane. Excess water is supplied to the reformer to prevent carbon deposition. Assume that the steam-to-carbon ratio is 2.5. There are two steps to the fuel processing. The first is reformation, which can be assumed to go to completion.
The second reaction is the water-gas shift reaction. Here, assume that 30% of the CO is converted to CO2.
ΔT = 200 K, what is the resulting air utilization? Perform an energy balance on the stack (enclosed by the dashed line). Assume that the fuel, fuel exhaust, and depleted air are at the temperature of the cell stack (Tc = 900 °C). How does this utilization change with a change in the inlet temperature?
Plot the air utilization as a function of the inlet temperature difference and explain the relationship between the air utilization and ΔT.
Both CO and H2 are oxidized in the fuel cell, assume that the utilization of each of these species is 0.9 and that the potential of each cell is 0.7 V.
Species Heat capacity [J·mol−1·K−1] ΔHf [kJ·mol−1]
H2 30.132 0
H2O 42.220 −241.8
Air 33.430 0
CO 29.511 −110.5
CO2 55.144 −393.5
10.19. One of the simplest models for a low-temperature hydrogen/oxygen fuel cell is
Using this model, which neglects mass transfer, how does the cell voltage change with oxygen utilization if the average current density is fixed? You may assume that the equilibrium potential, U, is constant. You will need to use an average oxygen partial pressure that accounts for the change in oxygen concentration along the length of the electrode.
10.20. In Section 10.5, 100% utilization of hydrogen is noted not to be practical even if the fuel is pure hydrogen. As shown in Figure 10.10, some of the fuel is recycled, and a small amount of fuel is vented through a pulse-width-modulated valve that can be scheduled to discharge gas in proportion to power. Holding the system utilization of fuel, ηf,s, constant at 0.9, plot the composition of the fuel entering the cell stack and the stack utilization, ηf, in the cell stack for recycle rates between 0 and 2.0. Assume the inlet composition to the system is 90% hydrogen.
10.21. By how much is the limiting current reduced when the gas-diffusion layer is compressed from 170 to 100 μm? Assume a Bruggeman relation (Equation 5.18) for tortuosity and an initial porosity of 0.7.
10.22. In the study of fuel consumption and utilization, several factors were neglected. Specifically, there can be leakage or permeation across the separator of a fuel cell. Additionally, there may be small shorts, for instance, due to the small electronic conductivity of a SOFC electrolyte or a small ionic conductivity in the interconnects (bipolar plates). Briefly discuss how these factors would affect the design of a fuel-cell system and propose a definition for the fuel utilization that accounts for these additional factors.
10.23. List advantages and disadvantages of the six flow fields shown in Figure 10.14.
10.24. A PEMFC stack in a fuel-cell vehicle is operating at 80 °C on hydrogen and air at a balanced pressure of 300 kPa and delivers 75 kW (gross power) at 42 V. The individual cells are operating at 0.7 V and 6 kA·m−2.
Determine the number of cells in the stack and the area of each electrode.
Find the minimum flow rates of reactants in g·s−1 to sustain this average power. State the reasons why this minimum rate is insufficient in practice?
Calculate the rate of accumulation of water at cathode from the reaction and electro-osmotic drag, assuming that back diffusion from the cathode to the anode is negligible. What are the consequences of not having back diffusion of water from the cathode to the anode and how do you propose to address these consequences to ensure stable operation? (This problem was suggested by S.R. Narayan).
10.25. Work out details for Illustration 10.7.
10.26. How would the results of the case study change if the mission were reduced to just 2 days?
10.27. Methanol can be oxidized in an aqueous fuel cell to carbon dioxide and water as per the following cell reaction:
Write the individual electrode reactions for the fuel cell.
Calculate the theoretical specific energy for the fuel and report it in W·h·kg−1.
Calculate the maximum thermal efficiency for this reaction at 298 K.
If such a methanol cell were operating at 298 K at 0.5 V and 0.1 A, calculate the voltage efficiency and the rate of heat output from the cell. (This problem was suggested by S.R. Narayan).
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