Mench M.M. Fuel Cell Engines
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c03 JWPR067-Mench December 18, 2007 1:59 Char Count=
110 Thermodynamics of Fuel Cell Systems
reaction. Inert species do have an effect on the Nernst voltage, though, through reduction
of the mole fraction of the active species.
Also note that the activity values use the pressure of the electrode where the reacting
species is located. That is, even though there is water at the anode, it is inert in terms of the
hydrogen oxidation reaction and only a participant in the electrochemical reaction at the
cathode.
While the Nernst voltage is easily calculated using a computer program, hand cal-
culations can be very tedious, especially considering the fuel cell temperature is usually
confined to a narrow range for a particular fuel cell type and the thermodynamic pressure
effect is small. Thus, constant specific heat assumptions are usually appropriate.
The Nernst equation can be used to determine the expected thermodynamic effect
of changes in species concentration. Looking at Example 3.10, however, the pressure-
dependent part of the equation is a relatively small component of the equilibrium voltage.
Essentially, this term corrects for the fact that the constituents are not all at unit activity.
As long as the ratio of reactant activity terms to product activity terms in Eq. (3.109) are
greater than unity, the net effect will be to increase the voltage. This is also consistent with
the results from application of Le Chatelier’s principle. That is, changes which increase
the ratio of reactant to products tend to shift the reaction to a higher voltage. So we expect
that any action to increase the partial pressure of hydrogen at the anode or oxygen at the
cathode in a hydrogen air fuel cell would increase the thermodynamic expected OCV. Keep
in mind that, under generation of current, reactions at each electrode take the system away
from a true thermodynamic equilibrium and kinetic effects will also impact the voltage, but
the qualitative effect of a gas concentration shift will remain the same.
Example 3.11 Thermodynamic Effect of Oxygen Enhancement Given an H
2
–air fuel
cell operating at 100
◦
C with vapor phase water as the product. Determine the approximate
expected change in voltage if air is replaced with oxygen, with all else remaining the same.
SOLUTION The only difference between air and oxygen is the mole fraction of oxygen
in the cathode gas:
E(T, P)
air
− E(T, P)
O
2
= E
◦
(T ) +
R
u
T
2F
ln
(y
H
2
P
anode
/P
◦
)(y
O
2
P
cathode
/P
◦
)
1/2
y
H
2
O
P
cathode
/P
sat
(T )
−
&
E
◦
(T ) +
R
u
T
2F
ln
(y
H
2
P
anode
/P
◦
)(y
O
2
P
cathode
/P
◦
)
1/2
y
H
2
O
P
cathode
/P
sat
(T )
'
E(T, P)
O
2
− E(T, P)
air
=
R
u
T
2F
ln
(y
H
2
P
anode
/P
◦
)(y
O
2
P
cathode
/P
◦
)
1/2
y
H
2
O
P
cathode
/P
sat
(T )
ln
(y
H
2
P
anode
/P
◦
)(y
O
2
P
cathode
/P
◦
)
1/2
y
H
2
O
P
cathode
/P
sat
(T )
=
R
u
T
2F
ln
(1)
1/2
(0.21)
1/2
=
8.314 × 353
2 × 96,485
ln
1
0.21
0.5
=+11.87 mV
COMMENTS: Based on thermodynamics alone, we expect about a 12-mV increase in
voltage going from an air to a pure oxygen oxidizer. Since air is essentially free, this seems
small. However, the actual increase in performance one gets from using pure oxygen is
much greater than this because of kinetic effects, which are discussed in Chapter 4. Despite
c03 JWPR067-Mench December 18, 2007 1:59 Char Count=
3.7 Maximum Expected Open-Circuit Voltage: Nernst Voltage 111
the increase in performance, however, use of pure oxygen is rarely justified in terrestrial fuel
cell applications due to increased cost, oxidizer storage requirements, and safety concerns.
Space applications, where pure oxygen is already available for liquid rocket propulsion, do
use pure oxygen, however.
Example 3.12 Thermodynamic Effect of a Pressure Increase Given an H
2
–air fuel
cell operating at 60
◦
C with vapor-phase water as the product. Determine the approximate
expected change in thermodynamic maximum OCV if the cathode pressure is doubled and
all else stays the same.
SOLUTION
E(T, P)
p
2
− E(T, P)
p
1
= E
◦
(T ) +
R
u
T
2F
ln
(y
H
2
P
anode
/P
◦
)(y
O
2
P
cathode
/P
◦
)
1/2
y
H
2
O
P
cathode
/P
sat
(T )
−
&
E
◦
(T ) +
R
u
T
2F
ln
(y
H
2
P
anode
/P
◦
)(y
O
2
P
cathode
/P
◦
)
1/2
y
H
2
O
P
cathode
/P
sat
(T )
'
=
R
u
T
2F
ln
(y
H
2
P
anode
/P
◦
)(y
O
2
P
cathode
/P
◦
)
1/2
y
H
2
O
P
cathode
/P
sat
(T )
ln
(y
H
2
P
anode
/P
◦
)(y
O
2
P
cathode
/P
◦
)
1/2
y
H
2
O
P
cathode
P
sat
(T )
=
R
u
T
2F
ln
(P
cathode,2
)
1/2
(P
cathode,1
)
1/2
=
8.314 × 353
2 × 96,485
ln(2
0.5
) =+5.27 mV
COMMENTS: The actual increase in performance expected from the thermodynamics
is smaller than what is typically observed in practice. There is also a strong kinetic effect,
which is discussed in Chapter 4. Also, we have assumed that at the catalyst surface, the
cathode is fully humidified, so that the activity of the cathode water vapor is unity for both
the high- and low-pressure cases at the cathode surface.
Example 3.13 Expected OCV Given a H
2
–O
2
PEFC with the following properties:
H
2
+
1
2
O
2
→ H
2
O
Anode: y
H
2
= 40%, y
H
2
O
= 20%, y
O
2
= 5%, y
N
2
= 35%, pressure = 4 atm, and λ
a
=
1.5.
Cathode: y
O
2
= 20%, y
H
2
O
= 10%, y
N
2
= 70%, pressure = 6 atm,and λ
c
= 2.5.
Find:
(a) Maximum expected OCV at 90
◦
C
(b) Change in voltage if we take our cathode down to 3 atm, all else the same
(c) Ratio of initial to final voltage expected if we boost oxygen mole fraction to 40%
and reduce nitrogen fraction to 50% in the cathode
SOLUTION (a) First we find the reversible voltage at 90
◦
C, E
◦
(363):
H = [−241,820 + 33.5(363 − 298)]
H
2
O
−
1
2
[0 + 29(363 − 298)]
O
2
− [0 + 26.5(363 − 298)]
H
2
=−249,907 J
c03 JWPR067-Mench December 18, 2007 1:59 Char Count=
112 Thermodynamics of Fuel Cell Systems
where the specific heats are assumed to be constant and evaluated at appropriate mean
temperatures.
Assuming constant specific heats and looking up the values for the reactants, we can
solve for the entropy change:
S =
&
188.72 + 33.5ln
363
298
'
H
2
O
−
1
2
&
205.03 + 29 ln
363
298
'
O
2
−
&
130.57 + 26.5ln
363
298
'
H
2
=−45.84 J/K
The Gibbs energy change is
G
◦
(T ) = H (T ) − T S(T ) =−249,907 J −(363 K)(−45.84 J/K) =−233,265 J
Now we can solve for the maximum reversible voltage at 90
◦
C, including only temperature
effects:
−G
◦
(T )
nF
= E(T )
◦
=−
−233,265 J
(2 e
−
eq/mol)(96,485 C/eq)
= 1.21 J/C = 1.21 V
The partial pressures of the reacting species are accounted for by the Nernst equation:
E(T, P) =
−G
◦
(T )
nF
+
R
u
T
nF
ln
$
a
ν
A
A
a
ν
B
B
a
ν
C
C
a
ν
D
D
%
Assuming ideal gas activities, we can plug in the numbers for the activity terms:
R
u
T
nF
ln
$
a
ν
A
A
a
ν
B
B
a
ν
C
C
a
ν
D
D
%
=
(8.314 J/mol · K)(363 K)
(2 e
−
eq/mol)(96,485 C/eq)
ln
&
(y
H
2
P
a
/P
◦
)(y
O
2
P
C
/P
◦
)
1/2
y
H
2
O
P
c
/P
sat
(T )
'
=
(8.314 J/mol · K)(363 K)
(2 e
−
eq/mol)(96,485 C/eq)
ln
&
(y
H
2
P
a
/P
◦
)(y
O
2
P
C
/P
◦
)
1/2
y
H
2
O
P
c
/P
sat
(T )
'
= (0.01564 J/C) ln
$
0.4 × 4
1
0.2 × 6
1
1/2
%
= 0.0088 V
So the various pressures and concentrations have the effect to boost the excepted OCV by
8.8 mV. Adding the temperature and activity contributions together, we have our Nernst
voltage:
E(T, P) =
−G
◦
(T )
nF
+
R
u
T
nF
ln
$
a
ν
A
A
a
ν
B
B
a
ν
C
C
a
ν
D
D
%
= 1.21 + 0.0088 = 1.22 V
c03 JWPR067-Mench December 18, 2007 1:59 Char Count=
3.7 Maximum Expected Open-Circuit Voltage: Nernst Voltage 113
(b) The change in voltage if we take our cathode down to 3 atm and all else stays the
same is calculated as follows:
E(T, P) =
−G
◦
(T )
nF
+
RT
nF
ln
&
a
ν
A
A
a
ν
B
B
a
νC
C
a
ν D
D
'
E
2
(T , P
2
) − E
1
(T , P
1
) =
−G
◦
(T )
nF
+
R
u
T
nF
ln
&
a
ν
A
A
a
ν
B
B
a
νC
C
a
ν D
D
'
2
−
−G
◦
(T )
nF
+
R
u
T
nF
ln
&
a
ν
A
A
a
ν
B
B
a
νC
C
a
ν D
D
'
1
=
R
u
T
nF
ln
&
a
ν
A
A
a
ν
B
B
a
νC
C
a
ν D
D
'
2
−
R
u
T
nF
ln
&
a
ν
A
A
a
ν
B
B
a
νC
C
a
ν D
D
'
1
Recall ln(a) − ln(b) = ln(a/b):
=
R
u
T
nF
ln
$
"
a
ν
A
A
a
ν
B
B
/a
νC
C
a
ν D
D
#
2
"
a
ν
A
A
a
ν
B
B
/a
νC
C
a
ν D
D
#
1
%
E
2
(T , P
2
) − E
1
(
T, P
1
)
=
R
u
T
nF
ln
(
y
H
2
P
a
/P
◦
)(
y
O
2
P
C
/P
◦
)
1/2
y
H
2
O
P
c
/P
sat
(T )
2
(
y
H
2
P
a
/P
◦
)(
y
O
2
P
C
/P
◦
)
1/2
y
H
2
O
P
c
/P
sat
(T )
1
=
8.314 × 363
2 × 96,485
ln
(
0.4×4/1
)(
0.2×3/1
)
1/2
1
2
(0.4×4/1)(0.2×6/1)
1/2
1
1
= 0.01564 ln
((1.5)
1/2
/1)
2
((3)
1/2
/1)
1
= 0.01564 ln
1.5
1/2
3
1/2
=−0.0827 V
(c) The ratio of initial to final equilibrium OCV expected if we boost oxygen mole
fraction to 40%, and reduce nitrogen fraction to 50% in the cathode is calculated as:
E
1
(T , P
2
)
E
2
(T , P
2
)
=
E
◦
(T ) +
R
u
T
nF
ln
(
y
H
2
P
a
/P
◦
)(
y
O
2
P
C
/P
◦
)
1/2
y
H
2
O
P
c
/P
sat
(T )
1
E
◦
(T ) +
R
u
T
nF
ln
(
y
H
2
P
a
/P
◦
)(
y
O
2
P
C
/P
◦
)
1/2
y
H
2
O
P
c
/P
sat
(T )
2
=
1.21 + 0.01564 ln
(0.4×4/1)(0.2×6/1)
1/2
0.1×6/0.75
1
1.21 + 0.01564 ln
(0.4×4/1)(0.4×6/1)
1/2
0.1×6/0.75
2
=
1.21 + 0.01564 ln(0.2
1/2
)
1
1.21 + 0.01564 ln(0.4
1/2
)
2
= 0.995 or 99.5%
c03 JWPR067-Mench December 18, 2007 1:59 Char Count=
114 Thermodynamics of Fuel Cell Systems
COMMENTS: Note the appropriate pressure and mole fractions are chosen based on the
global oxidation or reduction equation and from which electrode the products/reactants are
formed/consumed. If the species is not a reactant in the primary electrode reaction (e.g.,
the oxygen at the anode in the example), it does not appear directly in the Nernst equation
and only indirectly affects the voltage by changing the mole fractions for the active species.
Also, it is important to emphasize again that the Nernst equation accounts only for the
thermodynamic equilibrium effect of concentration change, and in practice, a change in
the mole fraction has a greater effect than predicted above due to kinetic effects not yet
accounted for. In Chapter 4, the voltage loss from nonequilibrium reaction kinetic losses
will be accounted for.
3.8 SUMMARY
The purpose of this chapter was to introduce the reader to the thermodynamics of electro-
chemical reactions and provide a linkage between macroscopic thermodynamic measurable
variables such as temperature and pressure and the expected voltage in electrochemical re-
actions. The ideal gas equation of state was examined, and several alternative methods with
improved accuracy were discussed. In some situations involving fuel cells, especially in
high-pressure gas storage, significant error can result if the ideal gas law is used without
some form of correction. The generalized compressibility chart can be used to determine the
compressibility factor Z for most gases, but unfortunately it is not well suited for hydrogen
without correction or use of a hydrogen-specific compressibility chart:
Pv
nR
u
T
= Z
The physical meanings of many thermodynamic parameters such as specific heat, enthalpy,
entropy, and the Gibbs function were discussed, and several methods for determining their
values for nonreacting and reacting liquid- and gas-phase mixtures were discussed. The
heat of formation, sensible enthalpy, and latent heat of phase change make up the total
enthalpy of a substance at a given temperature:
h
2
= h
◦
f
+ h
s
+ LH
The partial pressure of a species i is shown as
P
i
= y
i
P
where P is the total mixture pressure and y
i
is the mole fraction of species i, defined as
y
i
=
n
i
n
total
=
˙
n
i
˙
n
total
where n
i
is the number of moles of species i and n
total
is the total number of moles in the
mixture. The
˙
n
i
and
˙
n
total
terms represent the molar flow rates of species i and the mixture,
respectively. The sum of the partial pressures equals the total mixture pressure and the sum
of the mole fractions equals 1:
P
i
= P and
y
i
= 1
c03 JWPR067-Mench December 18, 2007 1:59 Char Count=
3.8 Summary 115
Nonreacting relationships used to calculate various molar specific mixture properties were
discussed, in particular
¯
x
mix
=
n
1=1
y
i
¯
x
i
The fundamental relationships of psychrometrics, the science of moist mixtures, were
presented, including discussion of the saturation pressure and relative humidity.
The relative humidity is defined as the ratio of actual water vapor pressure, P
v
, and
saturation water vapor pressure, P
sat
:
RH =
P
v
P
sat
(T )
where the saturation pressure is solely a function of temperature and has been curve fit to
the form for 15–100
◦
C
P
sat
(T )(Pa) =−2846.4 +411.24T (
◦
C) − 10.554T (
◦
C)
2
+ 0.16636T (
◦
C)
3
The maximum possible reversible voltage of an electrochemical cell is
−G
nF
= E
◦
The thermal voltage, derived assuming all the potential chemical energy for the reaction
went into electrical work, is
−H
nF
= E
th
= E
◦◦
The ratio of maximum expected voltage (E
◦
) and thermal voltage (E
◦◦
) represents the
maximum thermodynamic efficiency possible:
η
t,max
=
maximum electrical work
maximum available work
=
E
◦
E
◦◦
= 1 −
T S
H
For reactions involving water as a product, there is a choice in the calculation of thermo-
dynamic voltages between a HHV, where all water is assumed in the liquid phase, and a
LHV, where, from an energy perspective, all water is assumed to exist in the gas phase.
The Nernst equation was derived to show the reversible cell voltage as a function of
temperature and pressure. For an ideal gas mixture:
E(T, P) =
−G
◦
(T )
nF
+
R
u
T
nF
ln
&
(P
A
/P
◦
)
ν
A
(P
B
/P
◦
)
ν
B
(P
C
/P
◦
)
ν
C
(P
D
/P
◦
)
ν
D
'
It should be emphasized that the thermodynamic foundation established in this chapter is
valid only for true or quasi-equilibrium states. During fuel cell operation, true equilibrium
is only close to being achieved under zero-power conditions, and kinetic effects are also
prevalent in addition to thermodynamic effects. These are discussed in the following chapter.
c03 JWPR067-Mench December 18, 2007 1:59 Char Count=
116 Thermodynamics of Fuel Cell Systems
APPLICATION STUDY: FINDING APPROPRIATE
INTERNET RESOURCES
Obviously, not all Internet resources contain accurate information. From the first two
application studies in Chapters 1 and 2, you may have found a variety of information
that is sometimes conflicting. The Internet is an evolving source of technical information
available to engineers, and when reliable resources are used, the Internet greatly increases
the speed and portability of engineering analysis. Some questions should be asked when
consulting an Internet resource: (1) What is the motivation of the web page? (2) Is the
web page technically reviewed for accuracy? Recalling the application study in Chapter 1,
it is relatively easy to find resources that support the availability of platinum for fuel cell
use. Some of these resources, however, are based on studies supported by groups that
benefit from the sale of platinum and are not purely scientific in nature. Others, however,
are scientifically based and peer reviewed. It is often left to the engineer to discern the
difference when using Internet resources in analysis. In this assignment, you are asked to
find five reliable resources that contain technical information on fuel cells and five reliable
sources that contain thermodynamic data that can be used for analysis of fuel cells. A great
place to start for thermodynamic data is http://webbook.nist.gov/chemistry/fluid/. This site
contains accurate thermodynamic data for many species from the U.S. National Institute of
Standards and Technology. This site can supplant most thermodynamic tables in textbooks
and is therefore useful when a textbook is unavailable. There are many sites like this, of
varying utility. List each and summarize the content available on the fuel cell information
and thermodynamic data sites you choose and discuss how each site is motivated (why is
it even on the Web?) and how or if the veracity of the information presented is checked.
PROBLEMS
Calculation/Short Answer Problems
3.1 Consider a pure oxygen tank storage system for a fuel
cell used in a space application. What mass of oxygen can
be stored in a 2-m
3
tank at 34 MPa, 20
◦
C. Compare the re-
sults you get with the ideal gas law, the van der Waals EOS,
and the generalized compressibility chart. Using the gen-
eralized compressibility chart, determine at what storage
pressure the correction on the ideal gas law is 5%?
3.2 Using the generalized compressibility chart for hydro-
gen (with H
2
correction) and oxygen, in what temperature
and pressure range would you consider the ideal gas law to
be <95% accurate?
3.3 Complete a first-law analysis of a fuel cell system in
symbols only. Include flux of gas in and out as well as heat
and electricity generated. Consider the control volume to
be the fuel cell itself.
3.4 Plot the relationships for molar specific heat of hy-
drogen, air, and water vapor given in Eqs. (3.24), (3.27),
and (3.28), respectively. Over what temperature would you
estimate that you can consider the specific heats constant?
3.5 Do you expect gas-phase dimethyl ether (CH
3
OCH
3
)
to have a greater or lower constant-pressure specific heat
than hydrogen? Why?
3.6 Determine the mass fractions of a gas-phase mixture
with mole fractions of hydrogen, carbon dioxide, and water
vapor of 30, 30, and 40%, respectively.
3.7 For a mixture of 20% H
2
, 30% H
2
O
v
, and 50% CO
2
by volume, determine the mixture molecular weight and
enthalpy at 350 K.
3.8 Determine the change in enthalpy and entropy for a sto-
ichiometric (balanced) hydrogen–air redox reaction, H
2
+
1
2
O
2
→ H
2
O, with the hydrogen and oxygen reactants and
products at 1000
◦
C.
3.9 Given a flow inlet to a fuel cell cathode at 2 atm,
75% RH, 90
◦
C, and at a cathode stoichiometry of 2.0,
determine the maximum possible molar rate of water up-
take into the cathode flow if a 100-cm
2
active area fuel
cell is operating at 1.2 A/cm
2
. You can assume the flow
rate, pressure, and temperature are constant in the fuel
cell.
c03 JWPR067-Mench December 18, 2007 1:59 Char Count=
Problems 117
3.10 Given a PEFC fuel cell stack with each cell generating
100 A, if the air flow in the cathode is at a stoichiometry of
2.5, a back pressure of 2 atm, and an inlet RH of 0%, what
must the outlet RH be at 90
◦
C to remove exactly the amount
of water generated by reaction in vapor form? You can as-
sume the flow rate, pressure, and temperature are constant
in the fuel cell and the exit RH is 100%.
3.11 Given a fuel cell stack with each cell generating
1 A/cm
2
and a geometric area of 150 cm
2
per plate, if the
air flow in the cathode inlet RH is 25% at 80
◦
C and 1.5 atm,
what must the cathode stoichiometry be to remove exactly
the amount of water generated by reaction as water vapor at
80
◦
C? You can assume the flow rate, pressure, and temper-
ature are constant in the fuel cell and the exit RH is 100%.
3.12 Given a fuel cell stack with each cell generating
100 A, if the air flow in the cathode inlet RH is 20% at
80
◦
C and 1.5 atm at a stoichiometry of 2.0, what must the
exit temperature be to remove exactly the amount of water
generated by reaction as water vapor? You can assume the
flow rate and pressure are constant in the fuel cell and the
exit RH is 100%.
3.13 Determine the maximum thermodynamic efficiency
for a methane fuel cell at 298
◦
C based on LHV and HHV.
Does the efficiency increase or decrease with temperature?
With pressure?
3.14 Determine the maximum thermodynamic efficiency
for a methanol fuel cell at 100
◦
C based on LHV and HHV.
Does the efficiency increase or decrease with temperature?
With pressure?
3.15 Determine the maximum thermodynamic efficiency
for an ethanol fuel cell at 1000
◦
C based on LHV and HHV.
Does the efficiency increase or decrease with temperature?
With pressure?
3.16 Which has a higher maximum allowable thermody-
namic voltage, one calculated with a HHV or a LHV?
3.17 Consider a stationary hydrogen–air PEFC. Since hy-
drogen is not directly available, the fuel cell at 353 K is
to be fed by reformed natural gas. That is, natural gas will
be chemically converted into a mixture of hydrogen and
other nonreactive components. This gas mixture will be fed
into the anode side of the fuel cell, which will use the hy-
drogen in the mixture as fuel. The cathode is to be run on
humidified air of average composition (75% N
2
, 15% O
2
,
10% H
2
O
v
). Two types of reforming systems are offered,
steam-based reformation (SR) and autothermal reformation
(ATR). The mole fractions of all species to be input to the
anode are shown in the table below for the two different
reformation schemes. Consider the fuel cell to be operating
at 1.5 atm pressure on the anode, 3 atm pressure on the cath-
ode, a hydrogen stoichiometry (λ
a
) of 1.5, and a cathode
stoichiometry (λ
c
) of 3.0.
Anode Mole Fractions of Species for Two Reformation
Options
Species SR ATR
H
2
O 0.32 0.31
H
2
0.55 0.28
CO
2
0.13 0.12
N
2
0.00 0.29
¯
c
p,H
2
= 29 J/mol ·K
¯
c
p,O
2
= 30 J/mol ·K
¯
c
p,H
2
O
= 34 J/mol ·K
¯
h
◦
f,H
2
O,v
=−241,000 J/mol
¯
h
◦
f,H
2
O,l
=−285,000 J/mol
R
u
= 8.314 J/mol ·K F = 96,485 C/eq
¯
s
◦
H
2
O,v
= 188.83 J/mol ·K
¯
s
◦
H
2
O,l
= 70.05 J/mol ·K
(a) Determine the thermodynamically expected re-
versible Nernst OCV of the ATR case at 353 K?
Assume LHV.
(b) What is the change in expected voltage to switch
from ATR to SMR reformation for the anode gas
(assume all else stays the same besides anode feed
gas)?
(c) What is the maximum expected thermodynamic
efficiency at OCV for the ATR case (Assume
LHV)?
3.18 Consider the generic fuel cell complete redox reac-
tion:
Case (a): 3A + 2B → 2C + 1D
Case (b): 2A + 2B → 2C + 3D
Case (c): 3A + 2B → 2C + 3D
c03 JWPR067-Mench December 18, 2007 1:59 Char Count=
118 Thermodynamics of Fuel Cell Systems
Where A, B, C, and D are different gas-phase species. Draw
a qualitative plot showing the expected maximum thermo-
dynamic efficiency with respect to temperature.
3.19 One way to get hydrogen to operate a fuel cell is by
reforming gasoline into hydrogen. This is not totally effi-
cient, and CO
2
and N
2
endupintheflow.Youdesireto
compare this technique to using pure hydrogen. What is
the expected change in voltage for a hydrogen fuel cell if
the anode is switched from pure hydrogen to 30% hydro-
gen and the remainder is water vapor and CO
2
? Assume
T
cell
=100
◦
C.
3.20 In a SOFC, water vapor is created at the anode as
a product, just like it is created in a PEM fuel cell at the
cathode. What would the effect of the water content at the
anode be on fuel cell voltage (no numbers, just qualitative
response). Would voltage go up, down, or stay the same? Is
this similar to the effect of water vapor at the cathode of a
PEFC?
3.21 Le Chatelier’s principle is very powerful. It states that
any system initially at equilibrium, when subjected to a
change in temperature or pressure, will react in such a way
as to reduce stress. Discuss this principle for equilibrium
in relation to the expected shifts in voltage for changes in
reactant and product concentrations seen from the pressure-
dependent component of the Nernst equation. Can you tell
"just by looking" if the voltage will go up or down for a
given shift in concentration of reactants or products?
3.22 Given a hydrogen–air PEFC operating at 130
◦
C, a
cathode oxygen Faradic efficiency of 0.6, and an anode fuel
utilization fraction of 0.8. The average anode flow is 60%
hydrogen and 40% water vapor, and the average cathode
flow is 15% oxygen and 75% nitrogen and 10% water va-
por. Initially, the anode is at 5 atm pressure and the cathode
is at 3 atm total pressure.
Take the following specific heats to be constant:
¯
c
p,H
2
= 29 J/mol ·K
¯
c
p,O
2
= 30 J/mol ·K
¯
c
p,H
2
O
= 34 J/mol ·K
¯
h
◦
f,H
2
O,v
=−241,000 J/mol · K
¯
h
◦
f,H
2
O,l
=−285,000 J/mol · K
R
u
= 8.314 J/mol ·K F = 96,485 C/eq
(a) Using the Nernst equation, determine the expected
voltage at the given pressures and mole fractions.
Be careful how and where you put the water com-
ponents and what pressures you use. One of the
water vapors is "inert" and one is "active."
(b) How does the inert H
2
O affect the results? A qual-
itative answer is sufficient.
(c) From the Faradic efficiencies, you can see that
the reactants enter and deplete along the reaction
path. If you were to look at the calculated "Nernst
voltage” as you go along the channel from inlet
to outlet (as a function of x), what would happen
to the calculated value from inlet to exit (up or
down or stay the same)? A qualitative answer is
sufficient.
3.23 It is proposed to develop a fuel cell that runs directly
on propane (C
3
H
8,g
) at a propane stoichiometry (λC
3
H
8
)of
2.5 and a cathode oxygen stoichiometry (λ
c
) of 2 (note the
cathode is running on air, 79% N
2
, 21% O
2
by volume). In
the laboratory, the cell operates at 0.3 V at a current density
of 0.1 A/cm
2
. The superficial active area of the cell is 25
cm
2
.
The anode electrochemical reaction is
(
C
3
H
8
)
g
+ 6
(
H
2
O
)
g
→ 20H
+
+ 20e
−
+ 3CO
2
The basic cathode electrochemical reactions are
O
2
+ 4e
−
+ 4H
+
→ 2H
2
O
(
C
3
H
8
)
g
+ 5O
2
→ 4H
2
O +3CO
2
where the molecular weights are as follows:
C
3
H
8
= 44 g/mol
H
2
O 18 g/mol
O
2
32 g/mol
Air 28.85 g/mol
CO
2
28 g/mol
Other values needed can be found in the Appendix or the
text.
(a) Determine the balanced overall electrochemical
redox reaction.
(b) Find E
◦
, E
◦◦
,andη
th
at 298 K, 1 atm based on
HHV.
(c) Would the LHV result in a higher or lower η
th
?
(d) What is the voltaic efficiency of the fuel cell in
operation (ε
v
)?
(e) What is the overall fuel cell efficiency (ε
cell
)?
(f) Is the overall cell producing or consuming water?
At what rate in moles per second?
(g) What is the actual supply rate of air at the cathode
in grams per hour?
c03 JWPR067-Mench December 18, 2007 1:59 Char Count=
References 119
(h) Qualitatively sketch a plot of the expected max-
imum thermodynamic efficiency versus tempera-
ture for this type of fuel cell.
3.24 For a hydrogen PEFC based on the concepts in this
chapter, do you expect the open-circuit voltage to (circle
one):
(a) Increase/decrease/stay the same if the cathode
oxygen mole fraction increases
(b) Increase/decrease/stay the same if the cathode to-
tal pressure increases
(c) Increase/decrease/stay the same if the anode hu-
midity is increased
(d) Increase/decrease/stay the same if the operating
temperature increases
(e) Increase/decrease/stay the same if the cathode hu-
midity is increased
(f) Increase/decrease/stay the same if helium is mixed
with the air before entry into the cathode
3.25 Using constant specific heats, develop a simplified
expression for the variation of E
◦
with temperature for a
hydrogen–air PEFC in the range 300–400
◦
C. Would a sim-
ilar expression be valid for a hydrogen–air SOFC in the
range 900–100
◦
C?
3.26 Using constant specific heats, develop a simplified
expression for the variation of E
◦◦
with temperature for a
hydrogen–air PEFC in the range 300–400
◦
C. Would a sim-
ilar expression be valid for a hydrogen–air SOFC in the
range 900–100
◦
C.
Computer Program Problems
3.27 Develop a computer program to calculate the E
◦
,
E
◦◦
,andη
th
for a hydrogen–air fuel cell as a function of
temperature.
(a) Plot the LHV η
th
as a function of T between 300
and 1000
◦
C at 100
◦
C increments.
(b) On the same figure, plot the HHV η
th
as a function
of T between 300 and 1000
◦
C at 100
◦
Cincre-
ments.
(c) Check the validity of your model versus a hand
calculation at 500
◦
C. This is called model verifi-
cation and is a critical step to assure your computer
is giving good results.
(d) Compare the accuracy of your exact answers [done
by integrating the c
p
(T) equations] with those done
if you assume constant c
p
(easily done on a spread-
sheet by entering a constant value for c
p
). Is there
a lot of error associated with assuming c
p
=const?
3.28 Develop a computer program to calculate the E
◦
,
E
◦◦
,andη
th
for a methanol–air fuel cell as a function of
temperature.
(a) Plot the LHV η
th
as a function of T between 300
and 1000
◦
C at 100
◦
C increments.
(b) On the same figure, plot the HHV η
th
as a function
of T between 300 and 1000
◦
C at 100
◦
Cincre-
ments.
(c) Check the validity of your model versus a hand
calculation at 500
◦
C. This is called model verifi-
cation and is a critical step to assure your computer
is giving good results.
Open-Ended Problem
3.29 Go beyond the text and learn a little more about con-
densation and evaporation. Specifically, how are the pro-
cesses modeled? Do condensation and evaporation occur
at thermodynamic saturation points? Inside a PEFC at uni-
form temperature that is cooling, where would you expect
the water to condense first during steady operation?
REFERENCES
1. M. J. Moran and H. N. Shapiro, Fundamentals of Engineering Thermodynamics, 3rd ed., Wiley,
New York, 1995.
2. Y. A. C¸ engel and M. A. Boles, Thermodynamics, an Engineering Approach, 4th ed., McGraw-
Hill, New York, 2002.
3. F. D. Maslan and T. M. Littman, “Compressibility Chart for Hydrogen and Inert Gases,” Ind.
Eng. Chem., Vol. 45, No. 7, pp. 1566–1568, 1953.
4. ANR-1057B, New October 2000, Arlie A. Powell, Extension Horticulturist, Professor, and David
G. Himelrick, Extension Horticulturist, Professor, both in Horticulture at Auburn University.