Mench M.M. Fuel Cell Engines

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

220 Transport in Fuel Cell Systems

Figure 5.13 Schematic of gas-phase limiting current density at electrode.

for the porosity:

=−nFD

eff

∞

=−nFDφ

1.5

P/R

(5.47)

This approach can easily be extended from just the diffusion media to include the catalyst

layer as well. If we assume that the average reaction location is about midway through the

catalyst layer of thickness γ , then we can write

=− nFD

eff,DM

C − C

=− nFD

eff,CL

− C

(5.48)

where C

is the concentration of the reacting species at the diffusion media and catalyst

layer interface. Solving for C

and plugging back in, we can show that

=−nFD

eff,DM

∞

− i

γ/(2nFD

eff,CL

)

nFD

eff,DM



−

2nFD

eff,CL



(5.49)

If the porosities in the catalyst layer and diffusion media are the same so that the effective

diffusivities in the diffusion media and catalyst layer are identical, we can show that

nFD

eff

(

P/R

)

δ + γ/2

(5.50)

which is a result we could have expected. If the porosities are not the same (as is often the

case), then Eq. (5.48) can be solved for the limiting current as

nFD

eff,DM

P/R

T )

δ + γ D

eff,DM

/(2D

eff,CL

)

nFDφ

1.5

P/R

T )

δ + (γ/2)(φ

1.5

/φ

1.5

)

(5.51)

Example 5.8 Calculation of Gas-Phase Transport Limited Current Density through

Diffusion Media Calculate the oxygen and hydrogen gas-phase transport limiting current

density in a hydrogen PEFC for the anode and cathode sides at 80

◦

C, 2 atm pressure

operation with fully humidiﬁed gas streams and high stoichiometry. Assume the anode

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

5.3 Gas-Phase Mass Transport 221

and cathode diffusion layers are 300 µm thick with 70% porosity. Ignore the catalyst layer

restriction in this problem.

SOLUTION We know the solution can be found with application of Eq. (5.47). However,

this problem also has several intermediate steps which integrate the previous chapters

nicely:

Step 1: Determine the mole fractions of reactants using mixture properties.From

Chapter 3, we know that

RH =

sat

(T )

sat

(T )

⇒ y

sat

(T )

where

sat

(T ) =−2846.4 +411.24T (

◦

C) − 10.554T (

◦

+ 0.16636T (

◦

and



= 1

At 80

◦

C, P

sat

∼

31,287 Pa, and

202,650

× 31,287 = 0.154

on both the anode and cathode. On the anode

= 1 − 0.154 = 0.846

On the cathode we have two equations and two unknowns:

0.21

0.79

and y

+ y

+ 0.154 = 1

Solving, we ﬁnd that y

= 0.178, y

= 0.668.

Step 2: Determine the diffusivities of the reactants. For the anode, this time we will use

the diffusion volume approach. For the cathode, from Eq. (5.34)

O−H

(T , P) = 10

−3

1.75







1/MW

+ 1/MW





(

)

1/3



(

)

1/3









= 10

−3

353

1.75



1/2 + 1/18



12.7

1/3

+ 7.07

1/3





= 0.44 cm

For the cathode, from Eq. (5.33)

O−O





4.19836 ×10

−7

(

atm

)

(T )

2.334

= 0.186 cm

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

222 Transport in Fuel Cell Systems

Step 3: Determine the effective diffusivities of the reactants through the porous media.

From the Bruggeman relationship [we could also use Eq. (5.44) with little change],

eff

= Dφ

1.5

For the anode we have

O−H

,eff

= 0.44 × 0.7

1.5

= 0.258 cm

which is lower than the open channel case, as expected. The porous media should

reduce the diffusivity. For the cathode:

O−O

,eff

= 0.186 × 0.7

1.5

= 0.109 cm

Step 4: Evaluate Eq. (5.47) for the anode and cathode:

=−nFD

eff

∞

=−nFD

eff

P/R

For the cathode

=−(4 eq/mol)(96,485 C/eq)(0.109 cm

/s)

(

0.178 mol O

/mol mix

)

(202,650 N/m

)/(8.314 N · m/mol · K)(353 K)

(300 × 10

−4

cm)

1 × 10





= 17.23 A/cm

For the anode

=−(2 eq/mol)(96,485 C/eq)(0.258 cm

/s)

(0.846 mol H

/mol mix)(202,650 N/m

)/(8.314 N · m/mol · K)(353 K)

300 × 10

−4

1 × 10





= 96.94 A/cm

The higher limiting current density on the anode compared to the cathode is typical

and a result of the relatively high diffusivity of hydrogen.

COMMENTS: In practical PEFCs, the limiting current density observed is much less,

around 2–4 A/cm

. The discrepancy is a result of factors not accounted for in this basic

gas-phase approach. Speciﬁcally, we have not included the diffusion resistances of liquid

water from ﬂooding and the effect of an ionomer ﬁlm on the catalyst. The assumption of

high stoichiometry is needed because at lower stoiciometry the consumption of reactant

means that the limiting current density near the inlet is different from the outlet because

of consumption of the reactant along the ﬂow path. If we would have used the expression

including the catalyst layer resistance in Eq. (5.51), we would expect a slightly lower

limiting current density.

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

5.3 Gas-Phase Mass Transport 223

Figure 5.14 Diffusion in large channel. The molecular interactions with the channel wall are

negligible compared to the collisions with other molecules.

5.3.3 Knudsen Diffusion

Compared to the molecules themselves, the channels and pores through which the molecules

diffuse are typically large, and the main interaction for a particular molecule is collision

with other molecules, the driving force for bulk diffusion. In some cases for ﬂow in

very small pores and channels, however, the molecular collisions with the solid walls

of the channel become a signiﬁcant component of the overall number of collisions the

molecule experiences, and the effective diffusion will deviate from that predicted with

Fickian diffusion theory. Consider “normal” diffusion in a cylindrical pore (Figure 5.14).

The molecular interactions with the channel wall are negligible compared to the collisions

with other molecules.

For diffusion through very small pores (Figure 5.15), however, Knudsen diffusion be-

comes dominant at small pore sizes where wall interactions become signiﬁcant. Physically,

the wall is acting as another interacting molecular species adding a “wall viscosity” to the

motion of the species, which acts to retard motion.

The key parameter in Knudsen diffusion is the average path length of the molecules

between collisions. If the path length becomes similar to the same length as the pore

diameter, then wall interaction (Knudsen diffusion) will be important. For liquids, the

density is so high that the path length is very small and Knudsen diffusion is unimportant.

For gases, however, the mean free path length, l, can be estimated based on molecular

dynamics [22]:

l =

√

2πσ

(5.52)

The relationship is linear in temperature and inversely proportional to pressure. Here, σ

is the collision diameter of the species, some of which are tabulated in Table 5.9, P is

the pressure in Pascals, T is the temperature in Kelvin, and k

is Boltzmann’s constant

Figure 5.15 Flow in a very small channel. The molecular interactions with the channel wall are no

longer negligible compared to the collisions with other molecules.

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

224 Transport in Fuel Cell Systems

Table 5.9 Collision Diameters σ

for Various Fuel Cell Species From

Species Formula σ

(

Carbon dioxide CO

3.941

Carbon monoxide CO 3.690

Hydrogen H

2.827

Oxygen O

3.467

Nitrogen N

3.798

Hydrogen peroxide H

4.196

Hydrogen sulﬁde H

S 3.623

Methanol CH

OH 3.626

Water H

O 2.641

Dimethyl ether CH

OCH

4.307

Air Mix 3.711

Source: From [19].

(1.3807 × 10

−23

J/K). For example, for oxygen at 1 atm pressure and 1000

◦

l =



1.3807 × 10

−23

N · m/K



(1273 K)

√

2π



3.467 × 10

−10



(101,325 N/m

)

= 0.325 µm

For hydrogen, we can solve for a path length of 0.489 µm under the same conditions.

In order to determine if Knudsen diffusion is signiﬁcant, the Knudsen number (Kn) must

be calculated according to the following criteria:

Kn =

√

2πσ

(5.53)

Ĺ When Kn > 10, Knudsen ﬂow dominates.

Ĺ When Kn < 0.01, bulk diffusion ﬂow dominates.

Ĺ For 0.01 < Kn < 10, both Knudsen and bulk diffusion are important and a combi-

nation ﬂow exists.

For Knudsen ﬂow, the effective diffusion coefﬁcient can be determined from the kinetic

theory of rigid spheres [19]:









1/2

(5.54)

where m

is the molecular mass of species i. Alternate theory yields a larger value [15]:





4850d

√

(5.55)

where MW

is the molecular weight of species i. The magnitude of Knudsen diffusion is a

direct function of pore size. It is around 3 × 10

−3

/s for a 50-nm pore with hydrogen

diffusion at 353 K. As the molecular weight increases, Knudsen diffusivity decreases, and

as pore diameter increases, diffusivity increases (less collisions to interfere with motion), to

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

5.3 Gas-Phase Mass Transport 225

the limit of transition to bulk diffusion. Also notice there is no pressure or multicomponent

factor in Knudsen diffusion. This is because in the Knudsen limit intermolecular collisions

are rare. Also notice bulk diffusion varies with T

1.5

, where Knudsen diffusivity varies with

0.5

. Knudsen diffusion ﬂux is modeled with the Fickian equation, with a modiﬁed Knudsen

diffusion coefﬁcient replacing the binary diffusion coefﬁcient:

=−D

(5.56)

Also note that the model assumes a straight pore path. As in bulk media, any tortuosity

will increase the real path length, and this must be added to the result as in Eq. (5.41). In

the context of the length scales involved with Knudsen diffusion, however, it is unlikely

signiﬁcant error will result without the use of a tortuosity correction.

Example 5.9 Knudsen Flow Estimate the pore diameters where Knudsen ﬂow will be

important as a function of temperature from 273 to 1000

◦

C for oxygen and hydrogen at

1atm.

SOLUTION We can easily solve for the mean molecular path length:

l =

√

2πσ

From this we can plot the regions where the ratio of l/d falls within the three ﬂow regimes.

COMMENTS: The range of pore diameters where Knudsen diffusion dominates is below

∼0.05 µm. Not many media in fuel cells have pore diameters less than 0.05 µm, so that

Knudsen diffusion does not likely dominate anywhere, besides possibly in very small pore

catalyst layers. However, for diameters less than around 10 µm, there should be some mixed

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

226 Transport in Fuel Cell Systems

bulk/Knudsen diffusion effect. For simple analysis, transition region ﬂow can be modeled

in several ways: (1) as parallel ﬂux of bulk and Knudsen ﬂow if the media has a bimodal

(bulk- and Knudsen-dominated) pore distribution, as occurs in some PEFC diffusion media;

(2) as an effective diffusion resistance with a decrease in the bulk diffusivity to account for

the Knudsen effect; or (3) by using more advanced mathematical model multicomponent

gas mixtures speciﬁcally designed to predict ﬂow behavior in the transitional ﬂow regime,

as in ref. [25].

5.3.4 Liquid-Phase Diffusion

As shown in Table 5.4, diffusion coefﬁcients of gases in liquids tend to fall in the 10

−5

-cm

range. In fuel cells, diffusion of gases in liquids occurs between the gas phase and a liquid

electrolyte (although there can be additional interactions with the electrolyte besides pure

diffusion) and in low-temperature PEFCs under slightly ﬂooded conditions or with a liquid

fuel solution feed such as a DMFC. Since diffusion in liquids is so slow relative to gases,

it can be limiting in these situations. The Stokes–Einstein equation is commonly used to

estimate liquid diffusion coefﬁcients:

D =

6πµR

(5.57)

where k

is the Boltzmann constant (1.3807×10

−23

J/K), µ is the solvent viscosity, and R

is the solute radius, some of which are tabulated in Table 5.10. The accuracy of diffusion

correlations for liquid is less than that for gases but is typically within 20% in most

applications. Other empirical correlations exist and can be used if higher accuracy is

desired. (See, e.g., [19].) To determine the viscosity of the solvent, we can use Eq. (5.13).

Example 5.10 Liquid Layer Penetration Time Estimate the time required for oxygen to

penetrate and diffuse across a 10-µm liquid ﬁlm on a catalyst in a partially ﬂooded PEFC

at 80

◦

Table 5.10 Estimated Solvent Radius

Gas Formula R

(

Hydrogen H

1.414

Oxygen O

1.734

Nitrogen N

1.899

Air Mix 1.856

Carbon dioxide CO

1.971

Carbon monoxide CO 1.845

Hydrogen peroxide H

2.098

Ethanol C

OH 2.265

Methanol CH

OH 1.813

Source: Adapted from data in [19].

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

5.3 Gas-Phase Mass Transport 227

SOLUTION The diffusivity of oxygen in water can be estimated from Eq. (5.57):





6πµR

(1.3807 × 10

−23

J/K)(353 K)

6πµ(1.734 × 10

−10

(10,000 cm

)

The viscosity of water can be estimated from a rearrangement of Eq. (5.13)

µ = exp



a + b





+ c







+ µ

= exp



−1.94 − 4.80



273.16

353



+ 6.74



273.16

353





(1.792 × 10

−3

kg/ms)

= (1.155 × 10

−3

kg/ms)

Now we can evaluate the diffusion coefﬁcient:





6πµR

(1.3807 × 10

−23

J/K)(353 K)

6π(1.155 × 10

−3

kg/ms)(1.734 × 10

−10

× (10,000 cm

) = 1.29 × 10

−5

and the characteristic time to penetrate this water layer is estimated from Eq. (5.30),



1 × 10

−4



1.29 × 10

−5

= 7.75 s

COMMENTS: Note there is no pressure dependence for diffusion through the liquid.

However, there is a pressure dependence on the concentration of oxygen in the liquid, as

discussed later in this chapter.

5.3.5 Diffusion through a Polymer Electrolyte

As we have discussed, the diffusion into the polymer electrolyte is an important consider-

ation for PEFCs for two conﬂicting reasons:

1. The catalyst is typically covered in a thin electrolyte layer, so that high diffusivity

of reactants in the electrolyte is desired.

2. Reactant crossover through the electrolyte reduces OCV and efﬁciency, so that low

diffusivity of reactants in the electrolyte is desired.

Based on the competing needs of these factors, an electrolyte with some diffusivity of

reactants is needed. For the PEFC, the diffusivity has been correlated from experimental

data for 1100-EW Naﬁon, the most well-studied polymer electrolyte. Other electrolyte

polymers should have similar trends, but actual values change with EW values. In general,

low-EW polymers that absorb more water will have higher gas-phase species diffusion

coefﬁcients than higher EW electrolytes. Transport properties in the polyﬂourosulfonic acid

(PFSA) based membranes such as Naﬁon are generally dependent on the water sorption

in the membrane. For dry membranes, ionic conductivity and water mobility are very low.

As water sorption is increased, the membrane swells, and particularly at over 60% RH,

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

228 Transport in Fuel Cell Systems

the water uptake increases dramatically, and transport properties approach that of a dilute

electrolyte solution.

The diffusivity of water vapor into PEFC electrolyte Naﬁon is a function of the ma-

terial’s water uptake, since the vapor must diffuse through the entire media. There is

signiﬁcant discrepancy between various authors on the measured values of the water diffu-

sivity coefﬁcient, since it is a difﬁcult parameter to accurately measure, and the membrane

itself swells with water uptake. The diffusion coefﬁcient of water in 1100-EW Naﬁon PFSA

polymer with λ>4 has been correlated as [12]:





= 10

−6

exp



2416



303

−



(2.563 − 0.33λ + 0.0264λ

− 0.000671λ

)

(5.58)

where λ is given in Eq. (5.18). The diffusion coefﬁcient of water in Naﬁon has been studied

by many authors, and a surprising degree of difference between results exists, Eq. (5.58)

is commonly used in modely however, and is considered a reasonable correlation. Oxygen

diffusivity into 1100-EW Naﬁon has been given empirically as [26]:

−Naﬁon





= 2.88 × 10

−6

exp



2933



313

−



(5.59)

For 1200-EW polymer [27]:

−Naﬁon





= 3.1 × 10

−3

exp



−

2768



(5.60)

Both approaches yield the same order of magnitude in results. Hydrogen diffusivity into

Naﬁon 1100-EW as a function of temperature (in Kelvin) for a fully moist Naﬁon 1100-EW

polymer has been correlated as [28]

−Naﬁon





= 4.1 × 10

−3

exp



−2602



(5.61)

It should be cautioned that the availability of precise transport coefﬁcients is incomplete

in the literature. There are signiﬁcant differences between research results for much of

these data, and complete details under a full range of temperature and humidity, and for all

materials are not yet available. Nevertheless, the values presented here serve as a reasonable

approximation to the diffusion coefﬁcients for calculation purposes.

5.3.6 Interfacial Flow between Phases and Film Resistance

From Example 5.7, we see that the pure gas-phase resistance is typically not the limiting

resistance to mass transport to the catalyst. The diffusion resistance model can be expanded

to make it more complete by adding in additional resistances, such as diffusion into the

electrolyte covering the catalyst or through a ﬁlm of liquid water ﬂooding the catalyst.

However, these ﬁlm effects are localized around only a fraction of the catalysts, and by

including an effective ﬁlm resistance, we are showing only qualitative aggregate effects.

However, in many instances in fuel cells, especially PEFCs, mass ﬂux across different

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

5.3 Gas-Phase Mass Transport 229

, gas side

, liquid side

Figure 5.16 Schematic mass transfer across a gas/liquid-phase boundary. There is a sharp drop in

molar concentration from the gas phase to the liquid phase that is a strong function of temperature.

phases occurs. For example, reactant ﬂux typically must penetrate a thin layer of electrolyte,

and possibly water, to reach the catalyst. In Example 5.7, we solved for the gas-phase

transport limited current density and found that, even when restricted by porous media, the

predicted mass transport limiting current density is much greater than actually observed for

PEFC systems. In this section, the ﬁlm resistance caused by mass transport across a phase

boundary is examined. Unlike a no-slip condition in ﬂuid mechanics or a no-temperature-

jump condition in heat transfer, there can be a signiﬁcant concentration discontinuity across

a phase interface.

Mass Flux across Phase Boundary: Henry’s Law When there is species transport across

a phase boundary (e.g., gas into a liquid), there is a discontinuous change in the molar

concentration of the species across the phase interface, as shown in Figure 5.16. The

concentration discontinuity across the phase boundary between a liquid and gas on across

a membrane to gas interface is typically modeled with Henry’s law:

i,liquid/membrane side

i,gas side

gas side

H(T )

(5.62)

where H is Henry’s constant, known to be a function of temperature. Assuming a di-

lute solution, the liquid-phase mole fraction can be converted into concentration by the

following:

i,liquid/membrane side



mol



= y

i,liquid/membrane side



mol

mol mix



liquid



kg/cm



liquid

(

kg mix/mol mix

)

(5.63)

Sometimes the conversion to solid- or liquid-side concentration is already made part of

Henry’s constant and termed the solubility. In this text, we will use this convention. That

is, we will deﬁne Henry’s law as

i,liquid/membrane side



mol



i,gas side

gas side

H(T )

(5.64)

Henry’s constant for a variety of common gas-phase species into liquid water is tabulated

in Table 5.11.