Mench M.M. Fuel Cell Engines

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c02 JWPR067-Mench December 19, 2007 17:26 Char Count=

60 Basic Electrochemical Principles

where methanol density is 700 kg/m

and methanol molec-

ular weight is 32 g/mol.

(a) Calculate the minimum volume (in cubic centime-

ters) of a pure methanol fuel tank required to run

a soldier’s uniform equipment for three days. The

nominal power is 20 W. There are 10 cells in the

stack in series and the total stack voltage is 5 V.

(b) What is the molar rate of water consumption at the

anode?

cathode?

(d) Is there a water ﬂow rate required at the anode?

(e) What is the net molar rate of water production per

mole of methanol for the cell?

(f) What is the minimum molar rate of air required

for reaction?

2.14 Consider a dimethylether [DME, (CH

)

O] fuel cell

stack running at a DME stoichiometry (λ

DME

)of2.4anda

cathode oxygen Faradic efﬁciency (ε

) of 0.3. There are 15

cells (connected in series) in the stack, all operating at 0.4 V.

The active area of all cells in the stack is 25 cm

, and the

current density of each cell is 0.1 A/cm

The anode electrochemical reaction is

(CH

)

O + 3H

O → 12H

+ 12e

−

+ 2CO

The cathode electrochemical reaction is

+ 4e

−

+ 4H

→ 2H

with the following molecular weights:

DME 44 g/mol

O 18 g/mol

32 g/mol

Air 28.85 g/mol

(a) How many moles of water are created at the cath-

ode per mole of DME?

(b) What is the theoretical consumption rate of DME

at the anode in grams per second? What is the

actual supply rate in grams per second?

at the anode in grams per second?

(d) What is the actual supply rate of air at the cathode

in grams per hour?

(e) Would a DME fuel cell theoretically need a water

storage tank? Explain your answer.

2.15 Consider the following reactions typical of many fuel

cells:

→ 2H

+ 2e

−

+ 4e

−

+ O

→ 2H

(a) Which reaction occurs at the anode of the fuel cell

and which reaction occurs at the cathode?

(b) Is this a galvanic or electrolytic cell?

2.16 Consider a 25-cell hydrogen/air PEM fuel cell stack

producing a total of 2.0 kW at 10 V.

(a) What is the total stack mass ﬂow rate of hydrogen

if the anode stoichiometry is 1.3?

(b) What is the total stack rate of generation of water

at the cathodes in grams per hour?

is 1.23 V, what is the voltaic efﬁciency of a single

cell. Assume all cells have the same voltage.

2.17 Consider an ideal fuel cell to be run on pure oxygen

and hydrogen for a given length of time. Determine the ra-

tio of the minimum size of fuel to oxidizer storage tanks,

fuel

oxidizer

, assuming they are stored as gases at the same

pressure and the anode and cathode stoichiometries (λ

and

) are 1.5 and 2.0, respectively.

2.18 Considering the concepts discussed for the generic

fuel cell, list three reasons why the original fuel cells in-

vented by Grove worked so poorly relative to modern fuel

cells? The Grove fuel cell operated with the use of ﬂat-

plate platinum electrodes in an aqueous dilute sulfuric acid

electrolyte solution, as shown below:

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References 61

Open-Ended Problems

2.19 Consider the generic fuel cell of Figure 2.9.

(a) List three to ﬁve qualities that each of the com-

ponents shown should have. Note: For repeating

units (e.g., cathode and anode catalyst layers) you

can just say “same as the other.”

(b) Using your intuition about the function of these

components, describe a loss or limitation that can

occur with (a) the electrodes and (b) the bipo-

lar plates and how you would solve or improve

it. Then discuss what limitations or other off-

shoot advantages you might face with your so-

lution/improvement.

As an example the bipolar plate needs to be noncorrosive.

So we could use some kind of plastic composite or coated

metal. The limitations encountered could be that the plat-

ing is expensive or will not last or the plastic composite has

high electrical resistance compared to the metal it replaced.

Other advantages would be that it is potentially cheaper to

produce, faster to machine, and much lighter.

2.20 Estimate how much weight savings in terms of fuel

and oxidizer would be realized by replacing a 100 W, 20 A

fuel cell stack designed for 4000 h service with a reversible

fuel cell recharged by a solar panel for a space application.

Because fuel and oxidizer are recycled, you can assume an

effective stoichiometry of 1.0 for the anode and cathode in

both cases.

REFERENCES

1. G. J. Binczewski, “The Point of a Monument: A History of the Aluminum Cap of the Washington

Monument,” J. Met.,Vol.47, No. 11, pp. 20–25, 1995.

2. Battery and EV Industry Review, Business Communications Co, Formington, CT Distributed by

Global Information, 2005.

3. J. R. Hofmann, Andr

e-Marie Amp

ere: Enlightenment and Electrodynamics, Cambridge Univer-

sity Press, New York, 1996.

4. G. Pancaldi, Volta: Science and Culture in the Age of Enlightenment, Princeton University Press,

Princeton, NJ, 2005.

5. T. L. Brown and H. E. LeMay, Chemistry, 4th ed., Prentice-Hall, Englewood Cliffs, NJ, 1988.

6. A. J. Bard and L. R. Falkner, Electrochemical Methods, Fundamentals and Applications, 2nd

ed., Wiley, New York, 2001.

7. W. Feldenkirchen, Werner Von Siemens: Inventor and International Entrepreneur,OhioState

University Press, Columbus, OH, 1994.

8. J. S. Newman, Electrochemical Systems, 2nd ed., Prentice-Hall, Englewood Cliffs, NJ, 1991.

9. G. Prentice, Electrochemical Engineering Principles , Prentice-Hall, Englewood Cliffs, NJ, 1991.

10. D. Thompsett, “Pt Alloys as Oxygen Reduction Catalysts,” in Handbook of Fuel Cells—

Fundamentals, Technology and Applications,Vol.3, W. Vielstich, A. Lamm, and H. A. Gasteiger,

Eds., Wiley, New York, 2003, pp. 467–480.

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Thermodynamics of Fuel

Cell Systems

I believe fuel cell vehicles will ﬁnally end the hundred-year reign of

the internal combustion engine as the dominant source of power for

personal transportation. It’s going to be a winning situation all the

way around—consumers will get an efﬁcient power source, com-

munities will get zero emissions, and automakers will get another

major business opportunity—a growth opportunity.

—William C. Ford, Jr., Ford Chairman, International Auto

Show, January 2000

3.1 PHYSICAL NATURE OF THERMODYNAMIC

VARIABLES

Thermodynamics is the study of equilibrium at a macroscopic level. When a system is

in mechanical equilibrium, there is no net force imbalance that causes motion. Complete

thermodynamic equilibrium is more extensive and requires not only mechanical equilibrium

but also thermal, phase, and chemical equilibrium. We can use classical thermodynamics

to analyze chemically reacting and nonequilibrium ﬂows, such as those in fuel cells, but

are restricted to only the quasi-equilibrium beginning and end states of the process, with

no details of the reaction itself. Thermodynamics can tell us the potential for reaction

and direction of spontaneous reaction, but not how fast the reaction will occur. Classical

thermodynamics also assumes a continuous ﬂuid, meaning that there are enough molecules

of a substance to yield accurate values of thermodynamic variables like pressure and

temperature. As such, classical thermodynamics is generally inappropriate for use with

microscopic-level molecular charge transfer processes and electrochemical reactions.

In this chapter, the fundamentals of classical thermodynamics as it applies to the study

of fuel cells is introduced. Although the reader is assumed to have a background in basic

thermodynamics, this chapter includes a review of the physical meaning of several param-

eters used frequently in electrochemistry and how calculations of their values can be made.

This chapter concludes by applying the thermodynamic concepts presented to determine

the maximum expected thermodynamic efﬁciency and open-circuit voltage expected for a

fuel cell at a given condition.

Fuel Cell Engines

Matthew M. Mench

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3.1 Physical Nature of Thermodynamic Variables 63

Figure 3.1 Concept of temperature and a continuum: (a) continuum assumption not valid;

(b) continuum assumption valid. CV = control volume.

3.1.1 Physical Meaning of Parameters

Temperature (T) Temperature is a thermodynamic parameter that everyone has common

experience with, yet many people have no concept of what temperature physically repre-

sents. Temperature is a measure of the mean kinetic energy of the continuum of molecules

being measured. For an individual molecule, temperature has little physical meaning in a

macroscopic sense (imagine trying to measure the temperature of an individual molecule).

Consider a very small box representing the space of interest, as in Figure 3.1. When the

box is very small (Figure 3.1a), only a few molecules travel in and out of it, and any mea-

sure of temperature would be erratic and unsteady, as the number of molecules in the box

changes with time. Now consider that the box is large enough so that the average number

of molecules in the box remains constant over time, which is the continuum assumption,

illustrated in Figure 3.1b. In this case, a measurement of the average kinetic energy of the

molecules is a meaningful quantity represented by the temperature.

Pressure (P) Pressure is similar to temperature in that, from a macroscopic perspective,

there is no physical meaning for pressure of an individual molecule. For a continuous

mixture, pressure is a measure of the molecular momentum transfer from collision on the

plane of measurement. Consider a balloon ﬁlled with helium, as in Figure 3.2. To expand

the balloon against the restraining force of the elastic balloon material, there must be an

internal pressure greater than the atmospheric pressure. At the balloon’s internal surface,

the molecules are colliding and reﬂecting off of the wall. Since the internal pressure must

c03 JWPR067-Mench December 18, 2007 1:59 Char Count=

64 Thermodynamics of Fuel Cell Systems

Figure 3.2 Molecular collision and concept of pressure at a balloon wall.

be greater (or the balloon would not inﬂate), either there is a greater density of molecules

colliding on the inside of the balloon than the outside, or the average momentum of the

molecules inside is larger, which corresponds to a higher temperature. This also explains

why pressure is linearly related to temperature and the number of moles in the ideal gas

equation of state (EOS):

PV = nR

T (3.1)

where n is the number of moles of the mixture, P is the absolute pressure, R

is the universal

gas constant (8.314 kJ/kmol·K), V is the system volume, and T is the absolute temperature.

Nonideal behavior Although much of the thermodynamic state behavior for the gas-

phase species in fuel cells can be well approximated with the ideal gas law, it is important

to realize that Eq. (3.1) is not a perfect representation of the true physics, and there are

some applications pertinent to fuel cells where the assumption of ideal gas behavior is not

accurate. The ideal gas law assumes that (1) there are no net intermolecular interaction

forces and (2) the volume of the molecules is very small relative to the volume of the

containment. While these assumptions are accurate at low pressure and high temperature

(where density is low), they are not accurate at low temperature and high pressure, or

near the critical point of a substance. Obviously, ideal gas behavior is not appropriate in a

two-phase regime or to describe condensing or vaporizing water in a low-temperature fuel

cell. Thermodynamic parameters for the water vapor in mixtures should generally be taken

from thermodynamic steam tables, except at very low vapor pressure with no phase change

where ideal gas behavior can sometimes be used. In fact, tabulated thermodynamic data are

always preferred to empirical or semiempirical correlations or generalized charts. Another

application where the assumption of the ideal gas law is not perfectly accurate is at the high

pressures used for fuel or oxidizer storage, where the intermolecular interactions and ﬁnite

molecular volume can become signiﬁcant.

There are many methods to correct for nonideal gas behavior, including use of empiri-

cally or semiempirically modiﬁed EOSs. Actually, hundreds of EOSs have been developed

to describe the pressure–density–temperature relation for a wide variety of gas-, liquid-, and

solid-phase substances. For additional background, the reader is referred to a fundamental

thermodynamics textbook [e.g., 1]. An early attempt to improve the ideal gas EOS was

c03 JWPR067-Mench December 18, 2007 1:59 Char Count=

3.1 Physical Nature of Thermodynamic Variables 65

proposed by Johannes Diderik van der Waals in 1873. Van der Waals was awarded the 1910

Nobel Prize in Physics for his work, which can also be applied to compressible ﬂuids with

modiﬁcation. For gases, the van der Waals EOS is shown as

P =

v − b

−

(3.2)

where v is the molar speciﬁc volume, V/n, and n is the number of moles. The van der Waals

EOS achieves a higher accuracy than the ideal gas law because the two constants a and

b correct for the main assumptions implicit in the ideal gas law. The positive constant a

represents the net intermolecular attractive forces and therefore acts to reduce the effective

pressure in Eq. (3.2). Theoretically, this constant can be shown to be [2]

a =

(3.3)

where T

and P

are the critical temperature and pressure, respectively. The constant b

accounts for the ﬁnite volume of the molecules; thus it is subtracted from the molar speciﬁc

volume of the system. Theoretically, this can be shown to be [2]

b =

(3.4)

Table 3.1 provides some theoretical correction factors for the van der Waals EOS, calculated

based on the critical-point data. Although inconvenient to use, improved accuracy can be

achieved by using empirically derived correction factors, rather than the theoretical values

determined from Eqs. (3.3) and (3.4). Such data are available for many species but are

rarely, if ever, needed in the study of fuel cells.

While the van der Waals EOS has improved accuracy compared to the ideal gas law and

is historically quite important, it is not frequently utilized because more accurate approxi-

mations are now available, especially for behavior near the critical point. Other approaches

include two or more parameters that are empirically deﬁned by ﬁtting experimental data,

or the so-called virial EOSs, which have a series expansion form with coefﬁcients based on

molecular theory, statistical mechanics, or experimental data.

Table 3.1 Van der Waals EOS Coefﬁcients for Various Species Calculated from Critical-Point Data

Force Parameter, a Volume Parameter, b

Species Formula [kPa · (m

/kmol)

](m

/kmol)

Hydrogen H

24.73 0.02654

Oxygen O

136.95 0.03169

Water vapor H

553.12 0.03045

Carbon dioxide CO

364.68 0.04275

Nitrogen N

136.57 0.03863

Air Mixture property 136.83 0.03666

Methanol CH

OH 965.32 0.06706

Methane CH

229.27 0.04278

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66 Thermodynamics of Fuel Cell Systems

Perhaps the simplest approach to EOS modiﬁcation is that of a compressibility factor

Z. Here, the ideal gas EOS is modiﬁed with a compressibility factor:

= Z (3.5)

Obviously, for a true ideal gas, Z = 1. Analytical expressions for Z can be derived as a func-

tion of various parameters, based on the van der Waals correction factors or those in various

other EOS formulations. Compressibility charts for many species have been experimentally

generated and can be used to estimate the compressibility factor. For a wide variety of gas-

phase species such as air and water vapor, the compressibility factor follows a consistent

behavior when correlated by the reduced pressure (P

) and reduced temperature (T

(3.6)

where P

and T

are the critical pressure and temperature, respectively. This behavior is

known as the law of corresponding states, although it is not really a law of nature. Many

species follow this relationship, and the ideal gas correction factor can be represented on a

generalized compressibility chart. Figure 3.3 is a generalized compressibility chart that can

be used in lieu of species-speciﬁc data and provides a good estimate of the compressibility

factor. For hydrogen and some noble gases like helium, which do not follow the generalized

compressibility chart trends well, a speciﬁc chart based on measured data should be used.

Figure 3.4 is a hydrogen-speciﬁc compressibility chart. If a speciﬁc chart is not available,

hydrogen and other noble gases can be approximated with a generalized compressibility

Figure 3.3 Generalized compressibility chart. (Reproduced from E. F. Obert, Concepts of Ther-

modynamics, McGraw-Hill, New York, 1960.)

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3.1 Physical Nature of Thermodynamic Variables 67

Reduced Pressure, P

4.0

3.5

3.02.5

2.01.5

1.0

0.50

0.1

0.3

0.4

0.5

0.6

0.7

0.50

0.90

0.95

=100

105

110

115

120

130

140

150

160

180

200

250

300

400

600

500

300

700

1000

500

350

400

500

0.8

0.9

1.0

1.00

Compressibility Factor, Z

1.05

1.10

4.5 5.0 5.5 6.0

6.5 7.0

7.5

8.58.0 9.5 10.09.0

Figure 3.4 Experimentally measured hydrogen compressibility chart. (Reproduced with permis-

sion from [3]. Copyright American Chemical Society.)

chart if 8 K and 8 atm are added to the critical temperature and pressure used in Eq. (3.6)

respectively. Although more accurate, this is a rather arbitrary empirical correction and

only appropriate over a limited range [3].

Example 3.1 Hydrogen Storage Volume Consider a hydrogen tank storage system for

a fuel cell automobile. Seven kilograms of hydrogen gas compressed to 68 MPa (approxi-

mately 10,000 psig) and stored at 20

◦

C is required to provide a driving range of about 480 km

(approximately 300 miles). Using the ideal gas law, the van der Waals EOS, and the gen-

eralized compressibility chart, determine the interior volume required for the hydrogen

storage tanks.

SOLUTION (a) Ideal gas law EOS:

V =

⇒ V =

kg · N · m/kmol · K · K

kg/kmol · N/m

= m

7 × 8314 × 293

2 × 68,000,000

= 0.1254 m

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68 Thermodynamics of Fuel Cell Systems

(b) Van der Waals EOS: We ﬁrst need to rearrange Eq (3.2) to suit the problem, which

becomes a cubic expression in V:







−







V − ab = 0

where n is the number of moles, which in this case is 3.5 kmol.

For hydrogen, we look up T

= 33.2 K from the Appendix and P

= 1.3 MPa. From

Eq. (3.3)

a =



(N · m/kmol · K) · K

N/m



8,314

33.2

1,300,000

= 24,725 N · m

/kmol

The constant b is deﬁned in Eq. (3.4):

b =

8314 × 33.2

8 × 1,300,000

= 0.02654 m

/kmol

Solving the cubic expression we ﬁnd V = 0.2064 m

This result is substantially higher than using the ideal gas assumption due to the

intermolecular attractive forces and ﬁnite molecular volume effects.

chart with the 8 K and 8 atm correction for hydrogen along with some unit conversion

(1 MPa = 9.869 atm):

671.1atm

12.83 atm + 8atm

= 32.21 T

293 K

33.2K+ 8K

= 7.111

From the generalized compressibility chart, we see that Z ∼ 1.42. Then

V = Z

⇒ V = Z

kg · N · m/kmol · K · K

kg/kmol · N/m

= 1.42 ×

7 × 8314 × 293

2 × 68,000,000

= 0.178 m

If we had not used the correction, we would have found that

671.1atm

12.83 atm

= 52.31 T

293 K

33.2K

= 8.83

and we could extrapolate Z to be ∼1.58 from the generalized compressibility chart:

V = 1.58 ×

(7 kg)(8314 N · m/kmol · K)(293 K)

(2 kg/kmol)(68,000,000 N/m

)

= 0.198 m

The most accurate result would be from the hydrogen-speciﬁc compressibility chart in

Figure 3.4. We would have found that

671.1atm

12.83 atm

= 52.31 T

293 K

33.2K

= 8.83

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3.1 Physical Nature of Thermodynamic Variables 69

Z ∼ 1.4

V = 1.4 ×

(7 kg)(8314 N · m/kmol · K)(293 K)

(2 kg/kmol)(68,000,000 N/m

)

= 0.176 m

COMMENTS: Note that there is signiﬁcant deviation for each approach. Of them, the

hydrogen-speciﬁc compressibility chart should be the most accurate since it is based on

directly measured experimental data. The corrected generalized compressibility chart values

were also quite close. Considering the vehicle range is directly related to the mass of fuel

that can be stored, use of the ideal gas law alone would greatly underestimate the storage

volume required, by almost 30% in this case.

If the hydrogen were stored in a liquid form, the tank volume could be decreased, but

insulation would be needed to prevent excessive boil-off. The ideal gas law would deﬁnitely

not be appropriate for liquid H

storage, and direct saturated and liquid thermodynamic

data tables for hydrogen would be appropriate.

Internal Energy (U) We should ﬁrst distinguish the nomenclature used in this text for the

intensive (mass-related) variables, including internal energy. For these variables, a capital

letter refers to the absolute value of the parameter of interest. For example, U, the internal

energy, is taken to be in units of energy, or joules. A lowercase parameter represents a

mass-intensive quantity. For example, u represents the internal energy per unit mass in

kilojoules per kilogram. An overscored lowercase letter is representative of a molar speciﬁc

quantity. Therefore

u represents the internal energy in kilojoules per kilomole.

Internal energy U is a macroscopic measure of the total thermal energy stored in a

thermodynamic system. Considering a closed container with gas as our system, we need to

understand the way thermal energy can be stored in the gas-phase constituents to understand

the concept of total internal energy. From the ﬁrst law of thermodynamics for the system,

the internal energy can be converted into potential energy or kinetic energy or transferred

out of the system as heat or as work:

dE = dKE +dPE + dU = δ Q − δW (3.7)

Where dE is the change in energy of the system and dPE and dKE represent the change in

potential and kinetic energy of the system, respectively. The internal energy U is a measure

of the system total thermal energy stored in the individual, nonreacting species at the given

temperature and pressure condition.

In individual molecules, the internal energy can be stored in the following ways:

Ĺ Translational Kinetic Energy This is different from the kinetic energy of the system

in Eq. (3.7), which is a measure of the change in kinetic energy of the system as a

whole. As an example, consider a stationary container of gas. There is no change in

the kinetic energy of the system (it is zero); however, the individual molecules can

store energy in the form of their translational motion. The thermodynamic measure

of this form of energy is the temperature.

Ĺ Vibrational Motion A molecule can store energy via the oscillation of the bond

distance between atoms. The more bonds in a molecule, the greater the vibrational

motion contribution to the total stored energy. A monatomic species such as argon

has no bonds and therefore cannot store energy in this form.