Hydrogen Fuel Cells
The maximum efficiency for a hydrogen fuel cell at 25ºC is 83%. This decreases with temperature, and is 79% at 100ºC. This maximum efficiency occurs when an infinitesimal amount of work is being done by the cell, or when no load is imposed. The efficiency under load can be calculated from V, the voltage of an individual cell:
(1) EHHV = 100*V/1.481
This equation has been derived from the definition of efficiency, and is based on the High Heating Value (HHV) for hydrogen. Many researchers report efficiencies based on the more favorable Low Heating Value (LHV):
(2) ELHV = V/1.253
Since there is an exact relationship between the current (I) and the chemical consumption of hydrogen (n gmol/sec), the formulae are a simple ones. Computed values of EHHV and ELHV are chemical/thermodynamic efficiencies and not overall efficiencies. They do not include leaks, parasitic power or purge. Overall efficiency must be obtained from actual hydrogen usage and power output.
The voltage of a multi-cell stack is simply the value for a single cell multiplied by the number of cells per stack, so that single cell voltages are easy to obtain.
Efficiencies from single cells with H2 and O2 from bottles usually fit these equations well. Multi-cell stacks with hydrogen and air under pressure will have more gas leaks, and purge, and must provide power for compression and cooling.
Power output from a cell is:
(3) Power out = V*I
Where I is current in amperes, and power is in watts. Power output will determine the number of cell stacks to do a job. Output voltage is tied to cell efficiency, and will determine the cell count per stack to achieve a given voltage. Most importantly, hydrogen usage, and power loss, will be determined by efficiency. The table below gives efficiencies based on single cell voltage from equations (1) and (2):
SINGLE CELL VOLTAGES
Voltage ELHV EHHV
1.184 95(max) 80
1.00 80 68
0.9 72 61
0.8 64 54
0.7 56 47
0.6 48 41
0.5 40 34
The above efficiencies are for any Fuel Cell with hydrogen as fuel. As discussed, they do not include efficiency loss due to gas leakage, parasitic power, and process purge, which can be quite significant. In the following analysis it will be assumed that air (21% O2) is used, along with 100% pure hydrogen.
Efficiency and Cell Output
The following table was taken from the literature, for an actual cell stack operating on air and hydrogen. Both were at an inlet pressure of 50 psig. Hydrogen and air were fed at 1.15x and 2.0x stoichiometric, respectively. The cell membrane was Nafion 117.
EFFICIENCY AND CELL OUTPUT
Single cell Single Cell Single cell 10 Cells in
Loading Cell Efficiency Output Series Output
Amps/sq.ft Voltage EHHV W/sq.ft W/sq.ft
100 0.88 59 88 880
300 0.8 54 240 2400
600 0.68 46 408 4080
900 0.54 36 486 4860
The area specified in square feet (sq.ft) is the area of one side of the membrane of one cell. Each cell in a stack will pass the same current. Single cell loading and voltage are measured quantities. Cell efficiency was calculated from equation (1). Single cell output is single cell loading multiplied by single cell voltage.
Output for ten cells in series is 10 times the single cell output. While the loading in Amps/(sq ft) will still be the same for each cell, the overall voltage will increase by a factor of 10.
The table was taken from some good and consistent numbers obtained in 1989, and no doubt there have been improvements since. The table does illustrate that there is a trade- off between efficiency and output. The fuel cell idles well, but suffers low efficiency at high loading. The efficiencies above do not include parasitic power, purge, or process leaks. Parasitic power, for higher pressure air in particular, will be high, and purge of hydrogen will be significant for impure hydrogen.
High and Low Heating Values
Heating values are based on precise chemical reactions.
For High Heating Value, H2 (gas) + 0.5O2 (gas) >> H2O (liquid)
For Low Heating Value, H2 (gas) + 0.5O2 (gas) >> H2O (gas).
The difference is H2O liquid >> H2O gas, or the equation for the the heat of vaporization of water. Heating values can be obtained from common thermodynamic tables.
The generally accepted Fuel Cell maximum efficiency is 83% at 25° C, obtained from thermodynamic properties utilizing HHV. No one has seriously suggested that the maximum is 94.5%, obtained from ΔHLHV and ΔGLHV. Recall that:
(4) Maximum efficiency = Emax = 100*ΔG/ΔH.
Thermodynamic properties for hydrogen are given below:
HYDROGEN THERMODYNAMIC PROPERTIES AT 25°C
Property to LHV to HHV Units
----------------- --------------- -----------
Water Product Gas Liquid
ΔH -241,820 -285,830 J/mol
ΔG -228,570 -237,130 J/mol
Vmax -1.184 -1.228 volts
Emax 94.5 83.1 %
An Emax of -1.184V would be generally accepted if it were true, supporting LHV. But workers have repeatedly found a maximum voltage of near 1.228V in experiments at 25º, and their sensitive equipment can measure thousandths of a volt.
The reactions at the anode and cathode are ionic reactions, again indicating that water must be present, and the product formed as liquid water. It follows that the product is confirmed as liquid water, so that ΔHHHV should be used, leading to equation (1) above.
Cell design, stack design, membrane material, operating pressure, and other factors will contribute to process leaks. The effect can be estimated by measuring hydrogen input and power output and computing:
(5) Efficiency=100*VI / (n*ΔHHHV)
where n is in moles of hydrogen per second. This can be compared to the number from equation (1), which incorporates an exact relationship between n and I for chemical conversion only:
(6) I/n = 2*F = 2*96,520 coulombs per mole.
In equation 6, F is the Faraday, I is in coulombs/s = amperes, and n in mol/s. Two coulombs are needed for each hydrogen molecule, H2 , thus the 2 in equation (6).
After all of this theory, there is not much else to say about leaks. There will be leaks, as that is the nature of pressure systems, and particularly of hydrogen.
Because of the relatively low efficiency of the fuel cell stack, provisions have to be made for cooling, necessitating a pump. Furthermore, as air is used at 40 to 100 psig, there must be an onboard compressor.
One manufacturer gives parasitic power losses for a 150 kW plant. In this plant, air is at 40 psig, and circulation is 2 times stoichiometric. The gross efficiency has been measured at 48%, but the parasitic power is 39 kW, which gives a net efficiency of only 36%. This is attributed to compression and pumping power only, as the manufacturer uses an ingenious device to recirculate hydrogen, and not a blower.
The computation for parasitic power is not a simple one, as all parasitic power must be generated by the fuel cell. Furthermore, parasitic power is computed from the total energy input which includes parasitic power.
The compressor power can be computed using equations which give the theoretical adiabatic power required. One such equation is given in Perry’s Chemical Engineers Handbook, edition 50, equation 6-23e. Using this equation, the table below was prepared.
POWER FOR COMPRESSION
For air, with k=1.4.
Theoretical Actual Power
Outlet Pressure Power Required Required
psi kW/100scfm kW/100scfm
10 2.7 4.5
40 7.6 12.7
75 11.3 18.8
100 13.3 22.2
The power requirement is directly proportional to the inlet air flow rate, which is measured at 14.7 psi and 32°F. The inlet pressure is 14.7 psi, or one atmosphere. The actual power is based on 60% efficiency for DC to AC conversion, driver, and compressor. Large compressor sets with large efficient electric motors can have efficiencies in the 70 to 80% range. Smaller compressors and drivers, and the requirement for variable load, will move efficiency down to about 65%. A DC to AC conversion efficiency of 92% was used. Liquid sealed compressor sets will go up to 70% efficiency, but the liquid will end up in the air stream.
The pumping requirement is about 120 gallons/minute at 30 feet of head for a fuel cell operating at a chemical efficiency, EHHV = 48%, and input power 412 kW(chemical). This assumes a 20°F difference between the cooling fluid and the Fuel Cell,. The theoretical power to pump this fluid is 0.65 kW, and for a pump efficiency of 50%, the parasitic power would be 0.65/0.5 = 1.3 kW. This is quite small when compared to the compression power, but high enough to be included.
Hydrogen recirculation is necessary to keep a minimum flow through the unit and across the membranes, otherwise impurities can build up. Hydrogen can be supplied individually to each cell in a stack, but this is impractical. If hydrogen feeds a series of cells, the first cell will get the highest flow, and the last cell will be dead- ended. Impurities will build up, and cell output will decline, starting with the last cell. Furthermore, cell output is influenced by the flow perpendicular to the cell, which affects the transfer and dissolution of hydrogen. This mass transfer problem related to hydrogen flow is solved in experimental cells by feeding more hydrogen than is needed, and venting the excess, usually 15%. This is obviously not a good scheme for a working unit. In a working cell, recirculation mixes the gas from the last cell with the feed, using an ejector. This ejector is run with hydrogen, which is available at high pressure, so there will be no parasitic power debit. Impurities are removed by a purge.
Hydrogen purge simply removes gaseous impurities which come in with the feed hydrogen. The less pure the feed hydrogen, the more purge. With 100% hydrogen, there will be no purge requirement. With <100% hydrogen, the purge will depend on the desired hydrogen content of the recirculating gas. In this analysis it is assumed that H2 recycle is substantial, and all cells in the stack “see” this concentration. This is a conservative assumption.
The table below is based on a balance around impurities.
PURGE REQUIRED IN % OF FEED
Feed Gas Hydrogen Content of
%Hydrogen Recycle (Circulating) Hydrogen
95 90 80 70
99.9 2.0 1.0 0.5 0.3
99.8 4.0 2.0 1.0 0.7
99.5 10.0 5.0 2.5 1.7
99.0 20 10 5 3.3
98.0 40 20 10 6.7
95 100 50 25 16.7
Very simply, Feed*(impurities in feed) = Purge*(impurities in purge), so that:
Purge/Feed = (100-Feed H2%) / (100-Recycle H2%)
Suppose that the feed is 95% hydrogen, and a minimum of 90% hydrogen is needed at the cells, meaning 90% or less for the recirculating gas. The purge must be 50% of feed! If 70% hydrogen is circulated, the purge has to be 16.7% of feed. But the hydrogen content of the recirculating gas is what the cells ‘see’. Lowering the hydrogen content will affect cell productivity and loading. The solution is to use pure hydrogen! If not, report the purge losses in quoted efficiency.
Overall Fuel Cell Efficiency
As discussed above, the fuel cell voltage will give cell efficiency. Overall efficiency must include parasitic power and purge. The analysis below illustrates these effects.
All numbers are for 100% hydrogen.
(Single cell LHV efficiency = 57%)
Single cell HHV efficiency= 48%
Total energy input to Fuel Cell System = 418 kW chemical energy
Hydrogen in purge = 16 kW chemical energy (3% purge)
Hydrogen to Fuel Cells = 402 kW chemical energy
Air to Fuel Cells at 2x stoichiometric = 318 cubic feet per minute at 32°F and 1 Atm.
Power to compress air to 50 psig = 43.2 kW
Power to circulate cooling water @120 gallons per minute, and 30 feet of head = 1.3 kW
Fuel Cell outlet power = 402*0.48 = 193.0 kW
Fuel Cell System Net Power out = 193.0-43.2-1.3 = 148.5 kW
Net Fuel Cell System Efficiency = 148.5/(402+16)*100 = 35.5%
This is the true efficiency of the fuel cell system, when all parasitic power and purge are included. There is a huge difference between an advertised LHV efficiency of 57%, and the net fuel cell system efficiency of 35.5% based on HHV.
Hydrogen from reforming is at 65% efficiency (HHV) for a medium size plant, giving hydrogen at 350 psig.
For transportation, compression from 350 to 6000 psig will require about 15 kW, or 15/148.5*100 = 10% of outlet power from the fuel cell.
The overall efficiency of Fuel Cell plus transportation is 133.5/418*100 = 32%.
The overall efficiency of Fuel Cell Systems, starting with natural gas at 350 psig, is only 32*65/100 = 20.7%.
Inexpensive internal combustion engines using natural gas can be purchased off the shelf for $~500/kW. A primary (continuous use) unit @200 kW for $520/kW with efficiency 29% at maximum output, and a cheaper stand-by unit @100 kW with efficiency 23% are offered for sale. Natural gas turbines (about 400 MW) have efficiencies of about 33%, and combined cycle can go to 55%, for a capital cost of about $450 /kW. Smaller turbines (80 MW) are at 29% with costs ~$400/kW. All efficiencies are on the same basis, but capital costs include different things for each manufacturer, and are probably plus or minus 20%. All efficiencies quoted are natural gas to electric power, and no credit is taken for waste heat utilization.
Diesel generators cost about $800/kW for primary machines, and have an efficiency of 40%. Standby machines are about $200 /kW, with efficiency of ~37%.
Gasoline powered generators are available in the 1 to 10 kW range, and have efficiencies of ~18% for multi cylinder, four-stroke configuration. Single cylinder two stroke engines with low compression- ratio for < 1 kW can be lower than 10% efficient Gasoline powered generators are all standby machines, not intended for more than 10kW or a few days continuous service, and cost from $350 to $650/kW. Less on sale!
The above were all found on the internet, except for the large (multi MW) machines, which were from plants in construction, or just started up, in 2001.
The major development problem will be to increase efficiencies while reducing costs by at least an order of magnitude, as fuel cells cost over $10,000/kW.