CACHE Modules on Energy in the Curriculum
Fuel Cells
Module Title: Fuel Cell Efficiency
Module Author: Michael D. Gross
Author Affiliation: Bucknell University
Course: Thermodynamics
Text Reference: Smith, Van Ness, and Abbott (2001), Sections 4.1 and 11.4; Sandler
(2006), Section 14.6; Larminie and Dicks (2003), Sections 2.1-2.4
Concepts: Gibbs Free Energy, Enthalpy, Efficiency
Problem Motivation:
Fuel cells are a promising alternative energy conversion technology. Most fuel cells use
hydrogen as a fuel which reacts with oxygen to produce electricity. Different types of
fuel cells are distinguished by the nature of the electrolyte. For example, a proton
exchange membrane fuel cell (PEMFC) contains a polymer electrolyte that transports
protons as shown in Figure 1.
The PEMFC reactions are:
H2
H2O
O2
H+
H2
H2O
H2
O2
O2
H+
H2
H2 → 2H+ + 2e½ O2 + 2H+ + 2e- → H2O
H2 + ½ O2 → H2O
Electron
Flow
(Current)
e-
e-
Anode:
Cathode:
Overall:
H2
H2
H2
H+
Cathode
Gas
Chamber
Fuel Cell
O2
Anode
Cathode
Electrolyte
Air
In
Anode
Gas
Chamber
H2O
H2O
H+
Cell Voltage
H2
In
H2
H2
Electric Load
H2
Out
Figure 1. Reactions with PEMFC
Air +
H 2O
Out
Figure 2. Flow Diagram for PEMFC
1st Draft
M.D. Gross
Page 1
April 19, 2008
For each mole of hydrogen consumed, two moles of electrons are passed through the
electric load. Faraday’s constant is commonly used to relate number of electrons and
charge ( F  96,485 coulombs/mole of electrons). The objective of a fuel cell is to deliver
power to the load:
P=E·I
Power = Voltage · Current
watt 
joule
coulomb
 volt 
s
s
Clearly the voltage of a fuel cell directly affects power output. The maximum voltage, E,
of a fuel cell is calculated with Gibbs Free Energy.
E
G
nF
Here n is the moles of electrons released for every mole of hydrogen reacted (2 mol e-/ 1
mol H2).
One of the major advantages of fuel cell technology is the potential for very high
efficiencies. In this module, we will calculate Gibbs Free Energy with heat capacity
integrals, show how Gibbs Free Energy can be used to calculate the maximum efficiency
of a fuel cell, and investigate how maximum efficiency is affected by fuel cell operating
temperature.
Problem Information
The Gibbs free energy of reaction, ΔGºrxn, at standard pressure and fuel cell operating
temperature can be calculated as follows.
G  rxn,T  H  rxn,T  T  S  rxn,T
The stoichiometry of the reaction must be taken into account when calculating enthalpy
and entropy of formation.
H2 + ½ O2 → H2O
1
H  rxn,T  H  f H 2O ,T  H  f H 2 ,T  H  f O2 ,T
2
1
S  rxn,T  S  f H 2O ,T  S  f H 2 ,T  S  f O2 ,T
2
1st Draft
M.D. Gross
Page 2
April 19, 2008
The enthalpy and entropy at standard pressure and temperature, 1 atm and 298K, can
easily be found in thermodynamic tables; however, heat capacities are often used to
determine enthalpy and entropy at other temperatures.
T
H  f T  H  f 298   C p dT
Smith, Van Ness, Abbott : Eqn 4.2
298
T
S  f T  S  f 298 

298
Cp
T
dT
Smith, Van Ness, Abbott : Eqn 5.14
Heat capacity is a function of temperature and is typically calculated with an empirical
expression. For ideal gases:
Cp
R
 A  BT  CT 2  DT -2
Appendix C Smith, Van Ness, Abbott
For liquids:
Cp
R
 A  BT  CT 2
Appendix C Smith, Van Ness, Abbott
The maximum efficiency limit of a fuel cell, often referred to as the thermodynamic
efficiency, is calculated as follows.
Maximum Efficiency  max 
G  rxn,T
100
H  rxn, 298
The Gibbs Free Energy of reaction is calculated at the operating temperature of a fuel cell
since this is the temperature the reaction takes place. However, the enthalpy of reaction is
typically calculated at standard temperature, 298 K, and for water in the liquid phase.
This allows us to take into account recoverable heat from cooling the water product to
298 K and condensing the water.
1st Draft
M.D. Gross
Page 3
April 19, 2008
Example Problem Statement:
The operating temperature of a PEMFC typically ranges from 30ºC to 100ºC. Assuming
an ideal gas:
a) Calculate the maximum efficiency limit of a PEMFC operating at standard
pressure and 30ºC.
b) Calculate the maximum voltage, E, of the fuel cell
Standard enthalpies and Gibbs Energies of formation are provided in Table C.4 of Smith,
Van Ness, and Abbott. Since the temperature of interest is below 100ºC, properties of
water in the liquid phase will be used.
Chemical Species
H2
O2
H2O
ΔHºf, 298 / J mol-1
0
0
-285830
State
Gas
Gas
Liquid
ΔGºf, 298 / J mol-1
0
0
-237129
From Table C.1 and C.3 of Smith, Van Ness, and Abbott
Chemical Species
H2
O2
H2O
State
Gas
Gas
Liquid
A
3.249
3.639
8.712
103 B
0.422
0.506
1.25
106 C
0
0
-0.18
10-5 D
0.083
-0.227
Example Problem Solution:
Part a)
Step 1) First, we should calculate the ΔSºf, 298 of liquid water. The ΔSºf, 298 of H2 and O2 is
referenced as zero since they are elemental.
G  f H2O, 298  H  f H2O, 298  T  S  f H 2O, 298
Rearranging,
1st Draft
M.D. Gross
Page 4
April 19, 2008
 G  f H 2O , 298  H  f H 2O , 298 

S  f H 2O , 298  


T



J  
J 
   237129
    285830

mol  
mol  





298 K




J
 163.43
mol  K
Step 2) The enthalpy of formation should calculated for PEMFC operation at 30ºC.
303
H  f H 2O ,303  H  f H 2O , 298   C p dT
298
303
 H  f H 2O , 298  R  ( A  BT  CT 2  DT  2 )dT
298
303
B
C
D

 H  f H 2O , 298  R  AT  T 2  T 3  
2
3
T  298


1.25 x10 3
 303 2  298 2
8.712  (303  298) 
2

J
J
 285830
 8.314

mol
mol  K 
 0.18 x10  6


 303 3  298 3  0

3


 285830



K



J
J
J
 377
 285453
mol
mol
mol
From similar calculations for H2 and O2 we get
1st Draft
H  f H 2 ,303  144
J
mol
H  f O 2 ,303  147
J
mol
M.D. Gross
Page 5
April 19, 2008
Step 3) The entropy of formation should calculated for PEMFC operation at 30ºC.
303
S  f H 2O ,303  S  f H 2O , 298 

Cp
298
T
dT
A

 S  f H 2O , 298  R    B  CT  DT 3 dT
T


298
303
303
C
D


 S f H 2O , 298  R  A ln( T )  BT  T 2  T  2 
2
2

 298



8.712  (ln( 303)  ln( 298))  1.25 x10 3  303  298  


J
J
 163.43
 8.314


mol
mol  K 

6

0
.
18
x
10


 303 2  298 2  0
2



 163.43

J
J
J
 1.25
 162.18
mol  K
mol  K
mol  K
From similar calculations for H2 and O2 we get
S  f H 2 ,303  0.48
J
mol  K
S  f O2 ,303  0.49
J
mol  K
The following table summarizes the calculated enthalpies and entropies at 303K.
Chemical Species
H2
O2
H2O
1st Draft
State
Gas
Gas
Liquid
ΔHºf, 303 / J mol-1
144
147
-285453
M.D. Gross
Page 6
ΔSºf, 303 / J mol-1
0.48
0.49
-162.18
April 19, 2008
Step 4) We can now calculate ΔHºrxn,303K, ΔSºrxn,303K, and ΔGºrxn,303K.
1 
H O2 ,303
2
J
J
1
J 
 285453
 144
 147

mol
mol 2 
mol 
J
 285671
mol
H  rxn,303  H  H 2O ,303  H  H 2 ,303 
1 
S O2 ,303
2
J
J
1
J 
 162.18
 0.48
  0.49

mol  K
mol  K 2 
mol  K 
J
 162.91
mol  K
S  rxn,303  S  H 2O ,303  S  H 2 ,303 
G  rxn,303  H  rxn,303  T  S  rxn,303
J
J
 303K  162.91
mol
mol  K
J
 236309
mol
 285671
Step 5) We can now calculate the maximum possible efficiency of a PEMFC operating at
30ºC.
G  rxn,303
 100
H  rxn, 298
J
 236309
mol  100  82.7%

J
 285830
mol
Maximum Efficiency   max 
1st Draft
M.D. Gross
Page 7
April 19, 2008
Part b)
The maximum voltage, E, of the fuel cell is calculated with Gibbs Free Energy.
E
G  rxn,303
nF
236309

2
J
mol H 2
mol e C
 96485
mol H 2
mol e -
 1.22
J
 1.22 V
C
Maximum fuel cell efficiencies typically range from 60% to 80%. Under most conditions,
the efficiency of fuel cells is significantly higher than the Carnot efficiency of heat
engines.
1st Draft
M.D. Gross
Page 8
April 19, 2008
Home Problem Statement:
A solid oxide fuel cell (SOFC) is a different type of fuel cell that contains a solid oxide
electrolyte and transports oxygen ions as shown in Figure 3.
The SOFC reactions are:
2 H2 + 2 O-2 2 H2O + 4 eO2 + 4 e-  2 O-2
2 H2 + O2  2 H2O
Anode:
Cathode:
Overall:
e-
eH2
O2
O2
O-2
H2O
O
H2
O2
O2
H2
O2
-2
H2O
H2O H2
O2
O-2
H2 H2O
O2
O-2
O2
Anode
Cathode
Electrolyte
Figure 3. Hydrogen reaction in the SOFC
The operating temperature of a SOFC typically ranges from 600ºC to 1000ºC. Calculate
the maximum efficiency limit of a SOFC operating at standard pressure and 700ºC.
Compare the maximum efficiency to the PEMFC results. Assuming an ideal gas:
a) Calculate the maximum efficiency limit of a SOFC operating at standard pressure
and 700ºC.
b) Calculate the maximum voltage, E, of the fuel cell
Standard enthalpies and Gibbs Energies of formation are provided in Table C.4 of Smith,
Van Ness, and Abbott. Since the temperatures of interest are above 100ºC, properties of
water in the gas phase will be used.
Chemical Species
H2
O2
H2O
1st Draft
State
Gas
Gas
Gas
ΔHºf, 298 / J mol-1
0
0
-241818
M.D. Gross
Page 9
ΔGºf, 298 / J mol-1
0
0
-228572
April 19, 2008
From Table C.1 and C.3 of Smith, Van Ness, and Abbott
Chemical Species
H2
O2
H2O
1st Draft
State
Gas
Gas
Gas
A
3.249
3.639
3.470
M.D. Gross
Page 10
103 B
0.422
0.506
1.45
106 C
0
0
0
10-5 D
0.083
-0.227
0.121
April 19, 2008