Mench M.M. Fuel Cell Engines

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c04 JWPR067-Mench January 28, 2008 17:28 Char Count=

190 Performance Characterization of Fuel Cell Systems

14. M. L. Perry, J. Newman, and E. J. Cairns, “Mass Transport Gas-Diffusion Electrodes: A Diag-

nostic Tool for Fuel-Cell Cathodes”, J. Electrochem. Soc., Vol. 145, p. 5, 1998.

15. C. Boyer, S. Gamburzev, O. Velev, S. Srinivasan, and A.J. Appleby, “Measurements of Proton

Conductivity in the Active Layer of PEM Fuel Cell Gas Diffusion Electrodes,” Electrochim.

Acta, Vol. 43, p. 3703, 1998.

16. J. Scott, The Development and Operation of a 466 cm

Direct Methonol Fuel Cell, M. S. Thesis,

The Pennsylvansa State University, University Park, PA, 2002.

17. M. F. Mathias, J. Roth, J. Fleming, and W. Lehnert, “Diffusion Media Materials and Charac-

terization,” 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. 517–537.

18. J. Kim, S-M. Lee., S. Srinivasan, and C. E. Chamberlin, “Modelling of Proton Exchange Mem-

brane Fuel Cell Performance Using An Empirical Equation,” J. Electrochem. Soc., Vol. 142, No.

6, pp. 1895–1901, 1995.

19. F. Laurencelle, R. Chahine, J. Hamelin, T. K. Bose, and A. Laperriere, “Characterization of a

Ballard MK5-E Proton Exchange Membrane Stack,” Fuel Cells, Vol. 1, No. 1, pp. 66–71, 2001.

20. J. Larmine and A. Dicks, Fuel Cell Systems Explained, 2nd ed., Wiley, New York, 2003.

21. E. C. Kumbur, K. V. Sharp, and M. M. Mench, “Liquid Droplet Behavior and Instability in a

Polymer Electrolyte Fuel Cell Flow Channel,” J. Power Sources, Vol. 161, pp. 335–345, 2006.

22. N. Q. Minh and T. Takahashi, Science and Technology of Ceramic Fuel Cells, 2nd ed., Elsevier,

New York, 2005.

23. R. Doshi, V. L. Richards, J. D. Carter, X. Wang, and M. Krumpelt, “Development of Solid-Oxide

Fuels That Operate at 500

◦

C,” J. Electrochem. Soc., Vol. 146, No. 4, pp. 1273–1278, 1999.

24. J. M. Ralph, C. Rossignol, and R. Kumar, “Cathode Materials for Reduced-Temperature SOFCs,”

J. Electrochem. Soc., Vol. 150, No. 11, pp. A1518–A1522, 2003.

25. T. Hibino, A. Hashimoto, T. Inoue, J. Tokuno, S. Yoshida, and M. Sano, “Single-Chamber

Solid Oxide Fuel Cells at Intermediate Temperatures with Various Hydrocarbon-Air Mixtures,”

J. Electrochem. Soc., Vol. 147, No. 8, pp. 2888–2892, 2003.

26. O. Yamamoto, “Low Temperature Electrolytes and Catalysts,” In Handbook of Fuel

Cells—Fundamentals, Technology and Applications, Vol. 4, W. Vielstich, A. Lamm, and H.

A. Gasteiger, Eds.,Wiley, New York, 2003, pp. 1002–1014.

27. R. B. Bird, W. E. Stewart, and E. N. Lightfoot, Transport Phenomena, 2nd ed., Wiley, New York,

2002.

28. M.-K. Song, Y.-T. Kim, J. M. Fenton, H. R. Kunz, and H.-W. Rhee, “Chemically-Modiﬁed

Naﬁon

R

/Poly(vinylidene ﬂuoride) Blend Ionomers for Proton Exchange Membrane Fuel Cells,”

J. Power Sources, Vol. 117, pp. 14–21, 2003.

29. M. Watanabe, H. Uchida, Y. Seki, M. Emori and P. Stonehart, “Self-Humidifying Polymer

Electrolyte Membrane for Fuel Cells,” J. Electrochem. Soc., Vol. 143, pp. 3847–3852, 1996.

30. A. Parthasarathy, S. Srinivasan, A. J. Appleby, and C. Martin, “Temperature Dependence of the

Electrode Kinetics of Oxygen Production of the Platinum/Nation

R

Interface-A Microelectrode

Investigation,” J. Electrochem. Soc., Vol. 139, pp. 2530–2537, 1992.

c05 JWPR067-Mench January 23, 2008 18:58 Char Count=

Transport in Fuel Cell Systems

General Motors absolutely sees the long-term future of the world

being based on a hydrogen economy. Forty-ﬁve percent of For-

tune 50 Companies will be affected, impacting almost two trillion

dollars in revenue.

—Larry Burns, Vice President of R&D and Planning,

General Motors Corporation, February 2003

In the previous chapter we discussed the polarization curve and all of the losses associated

with the generation of current that result in decreased operating efﬁciency and generation of

heat. At a fundamental level, all of these polarizations are a result of transport limitations.

The ohmic polarization is a result of ion and electron transport losses, and the concentra-

tion and activation polarization is a result of mass transport limitations of the reactant to

the catalyst surface and charged particles across the double layer, respectively. Even the

crossover and internal short current loss from the expected Nernst potential is a result of

transport. Optimization of the fuel cell design therefore must include an optimization of

the (desired) modes of transport and minimization of the undesired modes of transport. In

this chapter, the modes of transport relevant to fuel cells are described in greater detail.

5.1 ION TRANSPORT IN AN ELECTROLYTE

Ion transport in an electrolyte is necessary for current ﬂow. Because the ions carry charge,

the net rate of ion transport through the electrolyte is directly proportional to the current

ﬂow:

i =

F (5.1)

where

is the molar rate of transfer of the ion through the electrolyte, F is Faraday’s

constant, and z

is the charge number of the ion (e.g., for H

, z = 1, and for O

2−

, z =−2).

Transfer of ions through the electrolyte occurs through several mechanisms of electrical,

chemical, and thermodynamic origin.

1. Mass Diffusion Mass transfer of ions (and any other species) occurs as a result of

a concentration gradient in the material. Figure 5.1 shows diffusion of oxygen and

191

Fuel Cell Engines

Matthew M. Mench

c05 JWPR067-Mench January 23, 2008 18:58 Char Count=

192 Transport in Fuel Cell Systems

Net O

diffusion direction

Net H

diffusion direction

Figure 5.1 Diffusion mass transfer results from concentration gradient.

hydrogen into each other:



j,i



mol

s · cm



=−D

j,i

∂C

∂x

(5.2)

where



j,i

is the mass ﬂux rate of the j ioninthex, y, and z directions, and

represents the molar concentration of species j. The net hydrogen motion is a

result of collisions with other hydrogen molecules (self-diffusion) and collisions

with oxygen molecules. Equation (5.2) therefore represents a total of three scalar

equations. That is,



j,x

=−D

j,x

∂C

∂x



j,y

=−D

j,y

∂C

∂y



j,z

=−D

j,z

∂C

∂z

The negative sign represents the fact that mass transport of j will occur in this mode

in the direction of decreasing concentration of j. Here, D

j,i

is the mass diffusivity

coefﬁcient of j in the i direction and ∂C

/∂x

is the concentration gradient of j in

the i direction.

2. Convection Mass transfer of ions (and any other species) occurs as a result of the

net motion of the electrolyte (e.g., stirred liquid solutions). Forced convection is

a result of externally controlled ﬂuid motion, and natural convection is a result of

buoyancy forces resulting from a density gradient. Figure 5.2 illustrates convective

Net O

diffusion direction

Net H

diffusion direction

Flow Velocity

Figure 5.2 Convection mass transfer results from net motion of ﬂuid.

c05 JWPR067-Mench January 23, 2008 18:58 Char Count=

5.1 Ion Transport in an Electrolyte 193

Figure 5.3 Migration mass transfer results from charged particles subjected to potential ﬁeld

gradient.

mixing of hydrogen into oxygen by a velocity ﬁeld:



j,i



mol

s · cm



= C

(5.3)

where v

is the solution velocity ﬁeld vector. Equation (5.3) represents a total of

three scalar equations, one in each direction.

3. Migration Mass transport of charged species is driven by an electrical potential

difference (Figure 5.3). In a fashion that is similar to diffusion mass transfer driven

by a concentration gradient and heat transport driven by a temperature gradient,

charged species will be driven by an electrical potential gradient:



j,i



mol

s · cm



=−

j,i

∂φ

∂x

(5.4)

Here ∂φ/∂x

is the electrical ﬁeld gradient in all directions. Simply stated, a negative

ion will migrate toward the positively charged electrode under a potential ﬁeld

gradient. The negative sign represents the fact that mass transport of positive ion j

will occur in the direction of decreasing electrical potential.

By putting all the relevant modes together, we have the Nernst–Planck equation gov-

erning ion transport:



j,i

=−D

j,i

∂C

∂x

+ C

−

j,i

∂φ

∂x

(5.5)

which reduces to simple diffusion and convection in the absence of a potential gradient, as

it must. In an electrolyte with no convection or concentration gradient, Eq. (5.5) reduces

to Ohm’s law. Thus, as discussed in Chapter 4, Ohm’s law is not strictly valid in elec-

trolytes unless there is no concentration gradient or motion, which is not always the case.

If there is a concentration gradient in the electrolyte, then an additional ohmic concentration

polarization will exist. Convective ﬂow enhances mixing and reduces any concentration

ohmic polarization. In most fuel cell systems (or as part of an introductory analysis), this

additional effect is neglected or is minor. Equation (5.5) is in compact form and expands to

c05 JWPR067-Mench January 23, 2008 18:58 Char Count=

194 Transport in Fuel Cell Systems

the following three equations in the x, y, and z directions:



j,x

=−D

j,x

∂C

∂x

+ C

−

j,x

∂φ

∂x

(5.6)



j,y

=−D

j,y

∂C

∂y

+ C

−

j,y

∂φ

∂y

(5.7)



j,z

=−D

j,z

∂C

∂z

+ C

−

j,z

∂φ

∂z

(5.8)

where the electrolyte velocity vector of the ﬂuid is

v = v

i + v

j + v

k (5.9)

From Eqs. (5.1) and (5.5), we can show in general that

i =



−D

j,i

∂C

∂x

+ C

−

j,i

∂φ

∂x



F (5.10)

The mobility u

of ion j in the electrolyte is an important parameter related to the diffusion

coefﬁcient D

through the Nernst–Einstein relation [1]:



(5.11)

The ion mobility is a function of the ionic charge and the operating temperature, pres-

sure, and ionic concentration and ion size. Despite the form of Eq. (5.11), the mobility

typically increases with temperature due to a decrease in the viscosity of the electrolyte,

which increases diffusivity. The ionic mobilities of various ions in solution are highly varied

and available in various literature, but generally on the order of 10

−3

–10

−4

(cm

/Vs) at

◦

C. The absolute value is necessary in Eq. (5.11) due to the charge on the ionic species j

since mobility is always a positive quantity. Using ionic mobility, Eq. (5.10) can be written

i =−z

j,i

∂C

∂x

+ z

− u

∂φ

∂x

(5.12)

This is a general form appropriate for all electrolyte systems. In static electrolyte systems,

the electrolyte velocity vector reduces to zero, and only diffusion and migration terms re-

main. At open-circuit conditions in a static electrolyte, the diffusion and migration transport

will balance each other, so there is no net current.

Liquid viscosity (µ) is a function of temperature [2]:

= a + b





+ c





(5.13)

Values for the empirically derived constants can be found in various resources, and mixture

relations can be used to estimate solution viscosities. For water, a =−1.94, b =−4.80,

and c = 6.75 with T

= 273.16 K and µ

= 0.001792 kg/(m · s).

c05 JWPR067-Mench January 23, 2008 18:58 Char Count=

5.1 Ion Transport in an Electrolyte 195

Conductivity A convenient relationship exists between the ionic conductivity (σ

mobility (u

), and ion concentration (C

) in a single ion carrier electrolyte [1]:

= F|z

(5.14)

This expression provides physical insight into the ion conduction process in an electrolyte:

1. As the charge number z

is increased, the total current carried per ion increases

proportionally, increasing the effective conductivity.

2. As the mobility of the charge carriers increases, the conductivity increases.

3. As the concentration of charge carriers (participants in the ion exchange) increases,

the ionic conductivity increases, although this trend does not hold for highly con-

centrated solutions.

Plugging in our expression for mobility, Eq. (5.11), we can glean a little more physical

insight into the ion transport process.

(5.15)

From this expression, we can see that the conductivity of an electrolyte is strongly re-

lated to the diffusivity, ion concentration, and charge number. This relationship predicts a

monotonically increasing ionic conductivity for electrolyte solutions as ion concentration

is increased. This trend is not actually seen in strong solutions, however, due to ion–ion

interactions. The model in Eq. (5.14) is derived for dilute solutions with low ion–ion in-

teraction. Concentrated solution theory is treated in other electrochemistry texts, such as

[1]. Ion conductivity is a different phenomenon than electrical conductivity. In electrical

conductivity, the valence electrons have a very high relative mobility along the surface of

an electrical conductor. In contrast, ion conductivity relies on the mobility of all ions in the

electrolyte with a much lower relative charge carrier concentration compared to electron

conductors. Not all molecules in an electrolyte are charge carriers. Almost all molecules in

a metal are participants and are very closely packed.

In most types of fuel cells, ohmic losses are dominated by ionic conductivity losses

through the electrolyte. Contacts and electrical conductivity typically play a small, although

signiﬁcant, overall role in the ohmic polarization. The electrolyte is also responsible for

many of the durability limitations and overall system cost.

5.1.1 Solid Polymer Electrolytes

In a solid polymer electrolyte, such as used in the PEFC, ion mobility is a result of an

electrolyte solution integrated into an inert polymer matrix. Early electrolyte membranes

developed for the United States space program consisted of treated hydrocarbons, which

resulted in poor longevity due to the relatively weaker hydrocarbon bonds [3]. Most modern

solid electrolytes are perﬂourinated ionomers with a ﬁxed side chain of sulphonic acid

bonded covalently to the inert, but chemically stable, polymer polytetraﬂuoroethylene

(PTFE) structure.

As a result, the membrane consists of two very different sub-structures:

1) a hydrophilic and ionically conductive phase related to the bonded sulphonic acid groups

Additional details on membrane manufacturing can be found in [3].

c05 JWPR067-Mench January 23, 2008 18:58 Char Count=

196 Transport in Fuel Cell Systems

FFFFFFFF

CCCCCCCC

PTFE

FFFFFFFF

FFFFFFF

CCCCCCCC

CFF

SOO

Sulfonated

side chain

FFFFFFFF

CCCCCCCC

FFFFFFF

FFFFFFFF

FFFFFFF

CCCCCCCC

CFF

SOO

CFF

SOO

Repeating

Figure 5.4 Schematic of sulfonated polytetraﬂuoroethylene (PTFE) structure used as ion-

conducting electrolyte in PEFCs.

that absorbs water, and 2) a hydrophobic and relatively inert polymer backbone that is not

ionically conductive but provides chemical stability and durability.

Perhaps the most ubiquitously studied perﬂourinated ionomers for fuel cells is Naﬁon

R

developed by E. I. DuPont de Nemours and Company. Many other companies such as

W. L. Gore and Associates, Asahi Glass Co, Ltd., Dow Chemical, and 3M have developed

ionically conductive solid polymers as well. Due to the desire for higher temperature

operation and low humidity high durability capabilities, many other alternative electrolyte

technologies have emerged [4]. In this text, for the purposes of understanding and brevity,

we will focus on understanding the fundamental transport modes and known transport

parameters for Naﬁon that can be used as a starting point for advanced study. Transport

in other polymer membranes used in fuel cells is generally similar to that of Naﬁon,

but material-speciﬁc relationships for most of the transport parameters discussed here

are available in the literature. Naﬁon, like other perﬂourinated ionomers is created by

sulphonation (SO

−

) of the basic PTFE structure, as shown in Figure 5.4.

When the structure is hydrated, H

-SO

−

groups enable motion of H

ions. Dry

perﬂourinated ionomers are almost completely non-conductive, so PEFCs typically operate

with humidiﬁed reactant ﬂow to boost conductivity and reduce ohmic losses. Studies of the

fundamental nature of proton transport in the ionomer have suggested that two modes of

transport exist, as illustrated in (Figure 5.5):

1. Under low water content, the ionically conductive hydrated portion of the mem-

branes behave as nearly isolated clusters, and proton transport is dominated by a

vehicular mechanism or diffusion, where proton transport is direct, and by purely

physical means.

c05 JWPR067-Mench January 23, 2008 18:58 Char Count=

5.1 Ion Transport in an Electrolyte 197

Figure 5.5 Schematic of connected sulfonated side chains which enable proton conduction through

(a) Grotthuss and (b) vehicular mechanisms in wet and dry PEFC electrolyte membranes, respectively.

(Adapted from Weber and Newman [5].)

c05 JWPR067-Mench January 23, 2008 18:58 Char Count=

198 Transport in Fuel Cell Systems

2. With high hydration in the electrolyte, a proton hopping, or Grotthuss mechanism

[5] is observed, with concomitantly higher effective proton conductivity. In this

mode of transport, protons “hop” from one H

to another along a connected

pathway in the ionomer structure.

The ionic conductivity of the electrolyte is related to the clustering of the sulfonic acid

side groups and hydration level. The structural relationship between the non-conductive

polymer backbone and the conductive side chains is the critical factor in the electrolyte

water uptake, conductivity, and swelling behavior. A measure of this structural relationship

is the equivalent weight (EW) of the ionomeric membrane [3]:





= 100 × k + 446 (5.16)

Where k is the number of tetraﬂouroethylene groups per polymer chain. The higher the

EW, the higher the amount of inert backbone relative to conducting side chains. A higher

EW should then theoretically lead to:

1. higher durability through a greater fraction of inert structure,

2. lower ionic conductivity due to reduced conducting acid content,

3. reduced membrane water uptake and swelling due to reduced hydrophilic sulfonic

acid side chain clusters, and

4. higher reactant solubility due to reduced water content [6].

The EW for fuel cell electrolytes is typically from 800–1200. The most commonly used

electrolyte is Naﬁon, 11×, representing an EW of 11 × 100, or 1100. The last number

in the Naﬁon designation represents dry electrolyte thickness in thousandths of an inch.

That is, Naﬁon 112 represents a 0.002



(51 µm) thick dry membrane with 1100 EW.In

general, lower EW electrolytes tend to have better low-humidity performance due to higher

moisture uptake, but suffer increased swelling in moist environments. At very low EW

(EW < 800), these membranes suffer reduced ionic conductivity from excessive dilution

of the acid. Swelling of the electrolyte in the PEFC can cause mechanical stresses in the

membrane, and lead to more rapid degradation.

Water Uptake Water uptake, λ, is commonly referenced in terms of water molecules per

sulfonic acid site:

λ =

(5.17)

Water-soaked values range from ∼30 for an EW of 900, to ∼20 for an EW of 1100 (see

Table 5.1). Additional data for commonly used Naﬁon electrolytes is given in Table 5.2.

Note that these values are for a membrane soaked in liquid water solution and represents a

maximum value. For membranes in a low humidity environment, the water uptake is much

lower.

For contact with a gas-phase ﬂow stream, the water uptake of Naﬁon 1100 EW at 30

◦

has been correlated as [7]:

λ = 0.043 + 17.18a − 39.85a

+ 36.0a

for 0 < a ≤ 1 (5.18)

c05 JWPR067-Mench January 23, 2008 18:58 Char Count=

5.1 Ion Transport in an Electrolyte 199

Table 5.1 Water Uptake, Swelling, and Ionic Conductivity versus EW in a PEFC

Effective Ionic Ionic Conductivity

EW (g/eq) Water Uptake (wt %) Concentration at 23

◦

C(S/cm)

1500 13.3 1.245 0.0123

1350 19.4 1.338 0.0253

1200 21.0 1.492 0.0636

1100 25.0 1.591 0.0902

980 27.1 1.764 0.1193

834 53.1 1.761 0.1152

785 79.1 1.539 0.0791

Source: Adapted from [8].

Table 5.2 Water Uptake, Weight Percent, and Swelling for Naﬁon Membranes

Naﬁon Water Uptake, Water Weight Thickness Strain from

Designation λ (H

O/SO

H) Percentage (%) Water Uptake, t

wet

dry

112 21–22 24–26 14–21

115 21–22 24–26 14–18

117 21–22 24–26 13–15

105 27–28 32–33 26–30

Source: Adapted from [9].

where a is water vapor activity, which is the relative humidity RH:

a = RH =

sat

(T )

(5.19)

The relationship in Eq. (5.18) has a maximum value at fully humidiﬁed conditions of

λ = 14. Although the relationship in Eq. (5.18) was derived for 30

◦

C, it has been commonly

applied with little reservation for modeling fuel cells at much higher temperatures. However,

although the qualitative shape of the uptake curve is similar, the maximum λ achievable

actually decreases with increasing temperature, to a value around λ = 10 at 80

◦

C. A more

rigorous treatment of the water uptake including temperature and EW effects is given in

the literature [10].

When equilibrated with liquid water, the water uptake for expanded form Naﬁon

much higher, and nearly invariant over the range of normal PEFC operating temperatures

at a value of around λ = 22. Other forms of Naﬁon also show an abrupt increase in water

content when equilibrated with liquid water, although the maximum value decreases with

increasing temperature.

The sharp difference in water uptake between a membrane equilibrated with vapor

and liquid water is known as Schroeder’s paradox. This important phenomenon is of

Perﬂourosulfonic acid membranes such as Naﬁon can be oriented in three basic forms of S (Shrunken) N

(Normal) and E (Expanded) types, depending on the pretreatment and environment [11]. The E form of the

membrane uptakes the most water for its EW, and is the most conductive and is normally what is in operation in

the fuel cell.