Mench M.M. Fuel Cell Engines

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

210 Transport in Fuel Cell Systems

Figure 5.11 Through plane electrical resistance as function of compression pressure for common

carbon ﬁber paper used as gas diffusion layer in PEFCs. The gas diffusion layers typically have some

fraction of PTFE added to promote liquid water removal, which increases electrical bulk and contact

resistance. (Reproduced from Ref. [18].)

In general, the electrical contact resistance between the current collector and the diffusion

media (PEFC and AFC) or catalyst layer is the largest electrical conductivity loss. In PEFCs,

a common through-plane resistivity is around 0.07  ·cm, with a contact resistance of 0.002

 · cm

that depends on compression pressure from the lands and Teﬂon content in the

diffusion media, as shown in Figure 5.11.

Electrical conductivity is much more facile than ionic conductivity, with typical metal

bulk conductivity on the order of

≈ 10

− 10

(

S/cm

)

This compares to a typical ionic conductivity of 0.1–1 S/cm for a polymer electrolyte and

0.1–1 S/cm for a ceramic electrolyte and liquid electrolyte. For a 1-mm-thick metal, the

voltage loss from 1 A/cm

current would be very low:

V = iAR =

1 × 1



1000 mm

10,000 cm

100 cm



= 1 µV

Because of the low bulk conductivity losses, in practical low-temperature fuel cell applica-

tions, a passivation oxide layer on the metal can dominate the bulk electron transfer losses.

Metal fuel cell bipolar plates are highly robust and can be less than 0.5 mm in total thick-

ness. Because the current collectors and ﬂow ﬁelds are often used for mechanical support,

they must have higher electrical conductivity to assure low losses. Remember, each fuel

cell has only 1 V to work with, so even millivolts are important.

5.3 GAS-PHASE MASS TRANSPORT

5.3.1 General Diffusion

Similarly to mass diffusion in electrolytes, gas-phase mass diffusion is driven by concen-

tration gradients. If a spray of perfume is released at the front of a classroom, the students

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

5.3 Gas-Phase Mass Transport 211

at the front of the room will smell the perfume ﬁrst. Progressively, students at locations

farther back in the room will be able to smell the perfume. In time, the perfume will disperse

evenly throughout the room. The process by which random molecular motion acts to mix

and eliminate concentration gradients is called diffusion. The fundamental principle behind

bulk diffusion is that of intermolecular collisions, as illustrated in Figure 5.1. Imagine a

room full of moving basketballs representing molecules, initially with red basketballs at one

end of the room and white basketballs at the other end. As the molecules move, they will

occasionally collide with one another. The collisions change the trajectory of the basket-

balls, eventually resulting in a completely homogenous mix of white and red. The average

speed of red basketballs from one end to the other end of the court is representative of the

diffusion coefﬁcient for the red balls into white balls. The average diffusion coefﬁcient of

a given species is a function of the following:

1. The other species present. The other molecules will collide with the diffusing species

and affect the net rate of motion.

2. The number of molecules, which corresponds to pressure. The greater the number of

molecules, the greater the number of collisions, which reduces the average diffusion

rate.

3. The velocity of the molecules, which is proportional to temperature.

4. The size and mass of the molecules (e.g., molecular collision diameter and molecular

weight).

We shall see that the diffusion coefﬁcient for bulk diffusion is indeed a function of pressure,

temperature, molecular size, and weight. Diffusion is a spontaneous process that is a result

of the second law of thermodynamics. The second law of thermodynamics requires that

thermodynamic processes proceed in a way that maximizes entropy. This ultimately requires

uniform mixing of everything in the universe. When there is nonuniform mixing, diffusion

occurs to eliminate concentration gradients and can be written as

j,i

=−D

j,i

∂C

∂x

(5.27)

which is identical to the ion diffusion rate equation (5.2) except that it is expressed as a

total rate, by multiplying the ﬂux by the corss-sectional area, A. In general, this is known

as Fick’s law of diffusion, where D

j,i

is the diffusion coefﬁcient of species j with units

of cubic meters per second, A is the area through which diffusion occurs, C

is the molar

concentration of j, x

is the direction of transport (x, y,orz direction), and n

j,i

is the molar

rate of transport of j in the i direction. In principle, the diffusion coefﬁcient can be a function

of concentration, temperature, pressure, other species, and other molecular interactions. For

one-dimensional mass ﬂux, Fick’s law of diffusion can be written as

=−D

(5.28)

Diffusion coefﬁcients tend to be around 0.1 cm

/s for gases and 10

−5

/s for liquids.

Solid-state diffusion coefﬁcients are strong functions of temperature and are generally

less than 10

−10

/s but can vary by as much as 15 orders of magnitude. Diffusion

of gases into solid polymers is generally around 10

−8

/s. Some typical values for

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212 Transport in Fuel Cell Systems

Table 5.4 Some Experimental Gas-Phase Diffusion Coefﬁcients at 1 atm

Gasl Pair Temperature (K) D (cm

/s)

Air–O

273 0.176

Air–H

O 298 0.260

Air–He 282 0.658

–H

O 308 0.282

–O

293 0.220

–He 317 0.822

Air–H

282 0.710

–H

O 307 0.915

–He 317 1.706

–CO

298 0.646

CO–H

296 0.743

Source: From [19].

gas-phase, liquid-phase, and solid-phase diffusion coefﬁcients from [19] are given in Tables

5.4–5.6, respectively.

The diffusion coefﬁcient in a mixture is obviously affected by the other species present,

and to account for this a binary diffusion coefﬁcient D

is speciﬁcally related to the

diffusivity of species 1 into 2 can be used. Since diffusion of species 1 into 2 is identical to

species 2 into 1, D

= D

. In cases where the density is low and diffusion ﬂux is primarily

a result of self-interaction, the species self-diffusion coefﬁcient (D

= D) is used, and the

effect of interaction with other species is not included. Later in this chapter methods for

calculation of diffusion coefﬁcients are given.

Transient diffusion problems can be solved with Fick’s second law. If the diffusion

coefﬁcient is assumed to be independent of concentration:

∂C

∂t

= D

∂

∂x

(5.29)

where D

represents the proper diffusion coefﬁcient of j in its surrounding environment.

Table 5.5 Diffusion Coefﬁcients in Water at 298 K and Inﬁnite Dilution

Gas into Water Chemical Formula D (cm

/s)

Air (N

)

0.79

)

0.2l

2.00 × 10

−5

Carbon monoxide CO 2.03 × 10

−5

Carbon dioxide CO

1.92 × 10

−5

Hydrogen H

4.50 × 10

−5

Methane CH

1.49 × 10

−5

Oxygen O

2.10 × 10

−5

Methanol CH

OH 0.84 × 10

−5

Ethanol C

O 0.84 × 10

−5

Formic acid CH

1.50 × 10

−5

1-Propanol CH

OH 0.87 × 10

−5

Source: From [19].

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5.3 Gas-Phase Mass Transport 213

Table 5.6 Some Representative Solid-Phase Diffusion Coefﬁcients

System Temperature (K) D (cm

/s)

Hydrogen in SiO

293 4.0 × 10

−10

Cerium in tungsten 2000 95 × 10

−11

Hydrogen in nickel 358 1.16 × 10

−8

Sliver in aluminum 323 1.2 × 10

−9

Aluminum in copper 293 1.3 × 10

−30

Source: From [19].

A useful result can be found from this equation that allows us to determine the diffu-

sion penetration distance with time [19]. We can calculate the characteristic time scale of

diffusion τ

, which is the time for signiﬁcant diffusion to reach a given distance δ:

(5.30)

For example, in 1 h, oxygen initially released into water vapor at 308 K (Table 5.1) will

travel a penetration distance of



D = δ =



(3600 s)(0.282 cm

/s) = 31.86 cm

Example 5.6 Characteristic Times for Gas-, Liquid-, and Solid-State Diffusion Estimate

typical characteristic times for a gas into gas, a gas into a liquid, a gas into a solid, and a

gas into a polymer to diffuse a distance of 1 mm.

SOLUTION We assume a typical diffusion coefﬁcient of 0.1 cm

/s for gases, 10

−5

for liquids, 10

−8

/s for polymers, and 10

−10

/s for solids.

Solving Eq. (5.30)

(

0.1cm

)

Mode of Diffusion τ

for 1 mm Diffusion

Gas phase 0.1 s

Liquid phase 16.7 min

Gas in polymer 278 days

Solid state 76.1 years

COMMENTS: The second law of thermodynamics provides a driving force for everything

to become homogeneously mixed in time, so that any material differences (and in a larger

sense, every gradient) have some natural driving for mixing. The vast difference between

gas- and liquid-phase transport times is an important factor in the stability and control of

fuel cells.

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214 Transport in Fuel Cell Systems

Table 5.7 Critical Temperature and Pressure Data for Fuel Cell Gases

Gas Formula T

(K) P

(atm)

Water H

O 647.3 218.0

Hydrogen H

33.2 12.8

Oxygen O

154 49.8

Nitrogen N

126 33.5

Air Mix 133 37.2

Carbon dioxide CO

304 72.9

Carbon monoxide CO 133 34.5

Methanol CH

OH 513 78.5

Source: From [20].

Calculation of Binary Gas-Phase Diffusion Coefﬁcients There are many different meth-

ods to estimate gas-phase diffusion transport parameters. Based solely on molecular dy-

namic theory, we can derive

∝

3/2

P · MW

1/2

(5.31)

This is a simple expression of self-diffusion of species j into a mixture of j, independent

of other species, where σ is the collision diameter of the molecule.

In practice, these

qualitative dependencies are generally observed, with some deviation, especially for polar

molecules. Many other theories of different levels of complexity and accuracy have been

developed but are not discussed here for brevity.

For fuel cells with gas-phase reactants, an effective estimation of binary (two-species)

diffusion coefﬁcients can be taken from [21], developed from kinetic theory and the law of

corresponding states [22]:







√



(

)

1/3

(

)

5/12





1/2

(5.32)

where D

represents diffusivity of species 1 into species 2, temperature T is in Kelvin and

pressure P is in atmospheres, and T

and P

refer to the thermodynamic critical temperature

and pressure, respectively (some are listed in Table 5.7). For a nonpolar gas pair, a and b are

2.745 × 10

−4

and 1.823, respectively. For a nonpolar gas–H

O pair, a and b are 3.640 ×

−4

and 2.334, respectively. For commonly used gases, this simple equation can be used,

since everything but temperature and pressure are constants.

For example, for oxygen in water vapor, we can show that

O–O





4.19836 × 10

−7

(T )

2.334

(5.33)

where pressure P is in atmospheres and temperature T is in Kelvin. This theory matches

trends predicted by molecular collision theory as well as experimental data fairly well and

The reader is referred to advanced texts on transport theory, including Diffusion, by E. L. Cussler, Cambridge

University Press, 1997, and Transport Phenomena, 2nd ed., by R. B. Bird, W. E. Stewart, and E. N. Lightfoot,

Wiley, 2002.

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5.3 Gas-Phase Mass Transport 215

Table 5.8 Diffusion Volumes for Eq. (5.34)

Gas Formula



Water vapor H

O 12.7

Hydrogen H

7.07

Oxygen O

16.6

Nitrogen N

17.9

Air Mix 20.1

Carbon dioxide CO

26.9

Carbon monoxide CO 18.9

Source: From [19].

includes the capability to more accurately model polar interacting gases, such as H

O. It

should be noted that there is additional error for hydrogen and helium because these gases

do not exactly follow the law of corresponding states, as discussed in Chapter 3. Because

the hydrogen is usually not the source of diffusion limitations in a fuel cell, however, the

additional error is generally tolerable in light of the convenience of the model. If greater

accuracy for hydrogen is required, another method developed using molecular volumes can

be used [23]:

(T , P) = 10

−3

1.75

1/MW

+ 1/MW





)

1/3



)

1/3



(5.34)

where T and P are in Kelvin and atmospheres, respectively, and some of the molecular

volumes V are given in Table 5.8. This method can also be used for other gases besides

hydrogen if desired.

Another way to quickly obtain approximate data is to assume the functional dependence

of pressure and temperature in Eq. (5.31) and extrapolate from known experimental data

such as shown in Table 5.4. That is,

(T , P) = D

12,known

ref

, P

ref

)



ref



3/2



ref



(5.35)

This will provide fairly accurate results as well, especially for nonpolar molecules. In

summary, there are several approaches of varying complexity and precision available to

estimate the gas-phase diffusivity. The choice taken depends on the level of precision

required. Direct experimental data for all modes of transport are always preferred over a

generalized correlation.

Multicomponent Diffusion Approaches In most cases, more than two species will be

mixed, and calculation of the diffusion rate of a given species into a mixture will be

required. The approach for these calculations is as follows:

1. Mixture Property Approach If there are more than two gases involved, such as

in a cathode with nitrogen, oxygen, and water, a mole-fraction-based averaging

approach can be taken as an estimation. In fact, the critical properties of air are

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

216 Transport in Fuel Cell Systems

already tabulated this way (prove it to yourself by making the calculation), although

it is really a mixture property of nitrogen, oxygen, and other minor species. For

example, to calculate the diffusivity of oxygen into humidiﬁed air:

Ĺ Determine the mole fractions of the constituents using the saturation pressure and

psychrometric analysis tools of Chapter 3.

Ĺ Evaluate the diffusion coefﬁcient based on one of the many methods discussed.

Ĺ Evaluate the mixture diffusivity using molar averaging.

2. Ignore Inert Species Approach In this approach, we simply ignore the contribu-

tions of the inert species. In the cathode, that means we treat the gas as a binary

mixture of oxygen diffusing into water vapor. This is frequently employed in the

literature for simplicity.

3. Stefan–Maxwell Approach A more rigorous approach to multispecies diffusion

effects is known as the Stefan–Maxwell diffusion model and should be used for

higher order models seeking the greatest accuracy. The Stefan–Maxwell equation

for multicomponent diffusion ﬂux in the x direction is shown as



j=i

− y

(5.36)

where C is the total molar concentration of the gas mixture and the y

’s are the mole

fractions, which can be determined by the pressure and temperature based on the

ideal gas law,

C =

(5.37)

The n terms are the molar ﬂux of the components in the x direction. The effective

binary diffusion coefﬁcient of the various gas species combinations, D

, is cal-

culated under speciﬁed temperature and pressure according to one of the methods

already described. As an example, consider a mixture of oxygen, water vapor, and

nitrogen. From Eq. (5.36), we get following equations for the gradients for oxygen,

nitrogen, and water vapor mole fractions in the cathode:



− y

−H

− y

−N



(5.38)



− y

−H

− y

−O



(5.39)



− y

O−N

− y

O−O



(5.40)

Example 5.7 Estimation of Diffusivity of Hydrogen and Oxygen in Water Vapor Es-

timate the diffusivity of hydrogen in water vapor and oxygen in water vapor at 1 atm and

307 K and compare with experimental data in Table 5.4.

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5.3 Gas-Phase Mass Transport 217

SOLUTION From Eq. (5.32), for the water–nonpolar gas pair, a and b are 3.640 × 10

−4

and 2.334, respectively:







√



(

)

1/3

(

)

5/12





1/2

Plugging in the numbers yields

O−H





3.64 × 10

−4



307

√

647.4 × 33.3



2.334

(

217.5 × 12.8

)

1/3

(

647.4 × 33.3

)

5/12





1/2

= 1.36 cm

Due to hydrogen self-interaction, this value is substantially different (49% higher) from

the experimentally determined value of 0.915 cm

/s. We expect some error for hydrogen,

though, as discussed.

For the oxygen, we have

O−O





3.64 × 10

−4



307

√

647.4 × 154.4



2.334

(

217.5 × 49.7

)

1/3

(

647.4 × 154.4

)

5/12





1/2

= 0.268 cm

which is much closer to the 0.282 cm

/s found from experimental data in Table 5.4.

If we desire greater accuracy for hydrogen, we can use Eq. (5.34):

O−H

(T , P) = 10

−3

1.75







1/MW

+ 1/MW





(

)

1/3



(

)

1/3









= 10

−3

307

1.75



1/2 + 1/18



12.7

1/3

+ 7.07

1/3





= 0.69 cm

This estimation is closer but still has signiﬁcant (24%) error, in this case underestimating

the measured value.

COMMENTS: As discussed, the error in using hydrogen is usually tolerable in light of

the fact that it is not typically hydrogen that limits reaction. The molecular volume approach

also can be used for oxygen and other fuel cell species with high accuracy, although the

need for the tabulated molecular diffusion volumes is an inconvenience and therefore the

approaches shown in Eqs. (5.34) and (5.35) are generally preferred. Also note for later use

that the diffusivity of water vapor into hydrogen is 3–4 times greater than that of water

vapor into oxygen (and air). This fact is important in PEFCs, which will be discussed in

Chapter 6.

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

218 Transport in Fuel Cell Systems

Figure 5.12 Illustration of porous media with (a)lowand(b) high tortuosity.

5.3.2 Gas-Phase Flow in Porous Media

For gas-phase ﬂow in porous media, such as with the electrodes or gas diffusion media

of a PEFC or AFC, the porous nature inhibits the diffusion rate through the media. The

inhibition must be related to two factors:

1. Porosity At zero porosity (i.e., a complete solid), the effective gas-phase diffusivity

must be zero. At a porosity of 1 (i.e., an open volume with no solid), the effective

diffusivity must equal the bulk value.

2. Tortuosity Tortuosity is a measure of the effective average path length though the

porous media compared to the linear path length across the media in the direction of

transport. The more tortuous the path, the longer the effective path length through

the media, and the greater reduction in the effective diffusivity, as illustrated in

Figure 5.12.

The effective diffusivity for gas-phase ﬂow in porous media can be written as

eff

= D

(5.41)

where D

eff

is the effective bulk gas-phase diffusivity in the porous media φ is the porosity

(void volume fraction), D is the diffusivity in gas, and τ is the tortuosity. Since tortuosity is a

difﬁcult parameter to estimate except through direct experiment, a Bruggeman correlation

is often used for fuel cell studies. This relationship assumes τ is proportional to φ

−0.5

resulting in the simpler expression

eff

= Dφ

1.5

(5.42)

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5.3 Gas-Phase Mass Transport 219

Physically, the porosity correction is to adjust for the longer effective path length through

the porous media. Using another approach, Salem and Chilingarian estimated tortuosity to

be related to porosity for high-porosity material (φ = 0.62–0.88) [24], which is appropriate

for fuel cell media:

τ =−2.1472 +5.2438φ (5.43)

This relationship yields a value of τ ∼ 1.5φ for a typical PEFC diffusion layer (φ = 0.8),

eff

1.5

(5.44)

Comparing Eq. (5.44) to (5.42), there is not much difference between the result of using

either approximation at high porosities typical of fuel cell media, especially given the

inherent uncertainty of many other parameters involved.

Gas-Phase Limiting Current Density and Diffusion Resistance In Chapter 4, we derived

an expression for concentration polarization based on the concept of a mass-transport-

limited reaction rate at a given electrode. Now, we can derive an analytical expression for

this value, assuming one-dimensional transport to the catalyst surface. First, we equate the

rate of reactant consumption to the diffusive and adjective transport to the reaction surface:



Consumption

=−D



 

Diffusion

transport

+ C

  

Advective

transport

(5.45)

where i

is the mass transfer limiting current density. In some fuel cell designs, ﬂow is

intentionally forced to the surface by convection, which is the reason for the inclusion of

the advective transport in Eq. (5.45). Additionally, the suction of reactant to the surface

will naturally cause some bulk motion and convection near the electrode. However, this

blowing/suction effect is typically small relative to the diffusion.

The diffusion rate is limited by ﬂow through the porous media diffusion layer (PEFC

and AFC) and electrode. If we assume one-dimensional ﬂux to the electrode surface in the

x direction with no bulk ﬂow velocity (see Figure 5.13), we can write this as

= 0ati

=− nFD

eff

C − C

=− nFD

eff

=− nFD

eff

P/ R

(5.46)

where C

, the surface concentration, is reduced to zero at the limiting condition and C

∞

the concentration of the reactant at the boundary with the ﬂow channel and is calculated

with the ideal gas law. Here, δ is the distance to the electrode surface from the ﬂow channel

boundary. For high-temperature fuel cells, the diffusivity is usually high enough that the

diffusion limiting current is not a factor. For the PEFC and AFC, the diffusion coefﬁcient

through the diffusion media is typically modiﬁed by the Bruggeman relationship to account