Mench M.M. Fuel Cell Engines

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310 Polymer Electrolyte Fuel Cells

COMMENTS: This calculation assumes all water generated goes directly to liquid and

none to vapor, which is unlikely to be true, so that the time scale required for liquid water

accumulation will be even greater than calculated. In a PEFC, this slow time scale, the slow

time scales of evaporation or drainage from the porous media and electrolyte, coupled with

the nearly instantaneous electrochemical and gas-phase response times lead to hysteresis

in a polarization curve and the fuel cell memory effect discussed.

6.2.2 Local Water Balance

We have just discussed the global water balance in a PEFC, but we have also mentioned that

actual ﬂooding loss is a localized phenomenon that can occur as a ﬁlm resistance and pore

ﬁlling in the catalyst layer, DM, and channels. To grasp the localized ﬂooding phenomenon,

it is also important to understand the macroscopic water transport processes which occur

within the fuel cell media.

Water Flux in Polymer Electrolyte Membranes Water ﬂux in the solid electrolyte mem-

brane of the PEFC must be understood to grasp the concept of a local water balance

in the fuel cell. From Chapter 5, we know that the ionic conductivity of perﬂuorosulfonic

acid–based solid polymer electrolytes is a strong function of water content. Within the elec-

trolyte, there are four basic modes of transport, as schematically illustrated in Figure 6.21:

1. Diffusion—concentration gradient driven ﬂow

2. Electro-osmotic drag—voltage gradient driven ﬂow

3. Hydraulic permeability—gas or capillary pressure gradient driven ﬂow

4. Thermo-osmosis—temperature driven ﬂow

vap

liq

Hydrophilic

Channel

Hydrophobic

Channel

liq

vapor

Diffusion

Medium

Micro-porous Layer Cathode

Catalyst Layer

Membrane

Condensed

Water Droplet

Hydraulic permeation

Diffusion

Electro -osmotic drag

vap

generated

Thermo-osmosis

Figure 6.21 Different modes of water transport inside fuel cell.

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6.2 Water Balance in PEFC 311

Diffusion in Naﬁon Diffusion into Naﬁon (and other perﬂuorosulfonic acid–based mem-

branes) can be modeled according to Fick’s law:

=−D

(6.20)

The water diffusion in the ionomer phase, D

, as a function of ionomer water content λ has

been measured by several groups, including Springer et al. [8]. Gong and co-workers [12]

also reported water self-diffusion coefﬁcients for Naﬁon with pulse ﬁeld gradient NMR.

Motupally and co-workers expanded the data from [8] to include changes in temperature

[13].

= 3.10 × 10

−3



−1 + exp

0.28λ



exp



−2436

T (K )



for (0 <λ≤ 3) (6.21)

= 4.17 × 10

−4



1 + 161 exp

−λ



exp



−2436

T (K )



for (3 ≤ λ<17) (6.22)

where λ is the water content of the membrane per sulfonic acid site:

λ =

(6.23)

and is deﬁned in Eq. (6.26). The water activity, a, needed to solve for λ is deﬁned as

a =

sat

(T )

= RH (6.24)

This relationship is shown in Figure 6.22 and is an alternative to the relationship given

in Chapter 5. At low water content, the diffusion coefﬁcient increases with water content

until a peak value around λ = 3, and then decreases with increasing water content until

Membrane water content,

,(H

O/SO

0 2 4 6 8 1012141618

Diffusivity x 10

-6

(cm

/s)

353 K

303 K

Figure 6.22 Diffusion coefﬁcient as a function of membrane water uptake from Zowadzinski et al.

[14].

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312 Polymer Electrolyte Fuel Cells

around λ = 6, and ﬁnally gradually increases. The diffusivity is a decreasing function of

temperature.

Water Uptake in Naﬁon In order to determine a relationship between the gas-phase

relative humidity and the ionomer water content, a relation between the ionomer water

concentration C

and the gas-phase water vapor mole fraction is needed. If we neglect

the volume change of the ionomer caused by water uptake, the water concentration in the

ionomer phase can be expressed by the ionomer water content λ as

dry

λ (6.25)

where ρ

dry

is the density of dry ionomer, EW is the equivalent molecular weight of the

ionomer material, and the water content λ is deﬁned as the ratio of the number of water

molecules to the number of sulfonic acid groups within the ionomer. As discussed in

Chapter 5, Zawodzinski et al. [14] showed

λ = 0.043 + 17.81a

− 39.85a

+ 36.0a

(6.26)

where a

= RH.

For gas-phase contact of vapor with the membrane, the water activity in the gas

phase is equivalent to the gas-phase relative humidity at the cell operating temperature, as

discussed in Chapter 5. For contact with liquid water, λ increases to 22, despite the same

thermodynamic activity as water vapor due to Schroeder’s paradox.

Hydraulic Permeability Hydraulic permeation of water through the membrane occurs as

a result of a pressure difference between the anode and cathode. The molar ﬂow rate of

water from the cathode to the anode can be written from Darcy’s law:

O,cathode

P

c−a

µl

(6.27)

where k is the effective permeability of the membrane µ is the liquid viscosity, k

the relative permeability of the membrane, l is the membrane thickness, and P

c−a

the gas-phase pressure difference between the cathode and anode. Water ﬂux through the

membrane can occur by gas- and liquid-phase transport, respectively. Transport of water

through the membrane can occur as a result of a gas pressure gradient across the membrane

or the capillary pressure gradient across the membrane. The gas-phase term is typically

small because the anode and cathode pressures are usually similar, so this effect can be

ignored. In the liquid phase, however, a capillary pressure difference between the anode and

cathode can result in a net ﬂux of water via this mode of transport [15]. The DM properties

are often tailored to achieve the desired capillary ﬂux effect, depending on the operating

conditions.

Electro-osmotic Drag Electro-osmotic drag of water is the mass ﬂux resulting from a

polar attraction of the water molecules to the positively charged protons moving from the

anode to the cathode through the electrolyte, as illustrated in Figure 6.23. As each proton

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6.2 Water Balance in PEFC 313

–

Figure 6.23 Schematic of electro-osmotic drag in electrolyte membrane under a current.

travels through the electrolyte from the anode to the cathode by Grotthuss and vehicular

mechanisms described in Chapter 5, the surrounding cloud of polar water molecules will

be dragged along.

Water transport by electro-osmotic drag is always from the anode to the cathode. Since

the drag is proportional to the current (protons), an expression for the ﬂux of water by

electro-osmotic drag is written as

= n

(6.28)

where n

is the electro-osmotic drag coefﬁcient in units of water molecules per proton. Many

studies have been conducted to determine the drag coefﬁcient. A drag coefﬁcient of 1–5

O/H

of Naﬁon membranes has been shown for a fully hydrated Naﬁon 117 membrane

[16]. Zawodzinski et al. [17] have investigated the electro-osmotic drag coefﬁcient of

Naﬁon 117 and other ionomeric polymer electrolyte membranes at 30

◦

C under conditions

relevant to PEFCs. For a Naﬁon 117 membrane fully hydrated in liquid water (λ = 22),

they obtained n

= 2.5. For a nearly fully hydrated membrane with a water content of

λ = 11, a much smaller value, n

= 0.9, was obtained. Fuller and Newman [18] obtained

= 1.4 for λ values of 5–14. From the same kind of measurement, however, Zawodzinski

et al. [17] obtained a convenient value of n

=1.0 for λ<14 at 30

◦

C. Ise and Kreuer et al.

[19] also measured the electro-osmotic drag coefﬁcients in Naﬁon and obtained n

= 2.6

for λ =20, which is very close to the value reported by Zawodzinski et al. [17]. In summary,

if equilibrated in moist vapor (0 <λ<14), the value of n

is around 1.0–1.5. For Naﬁon

equilibrated in liquid water, however, the membrane water uptake is much greater, and the

electro-osmotic drag coefﬁcient is around 2–5.

Example 6.4 Comparison of Generated and Drag Water Compare the total water

delivered to the cathode by electro-osmotic drag to that generated by reaction at a given

current density, assuming the membrane is in contact wit vapor-phase water only.

SOLUTION For water drag, the net molar ﬂux of water to the cathode is shown from

Eq. (6.28):

O,drag

= n

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314 Polymer Electrolyte Fuel Cells

where n

has been shown to be ∼1.0–1.5. For water generation, we have Faraday’s law:

O,gen

Comparing the two, we have

O,drag

O,gen

(iA/F)

iA/(2F )

(

1–1.5

)

× 2

= 2–3

Thus, water transport by electro-osmotic drag is always from the anode to cathode and is

two to three times greater than the water generation at the cathode for vapor-equilibrated

membranes! In terms of ﬂooding at the cathode, the electro-osmotic drag to the cathode

results in more water than the electrochemical generation.

COMMENTS: In hydrogen PEFCs, much of the water driven to the cathode by electro-

osmotic drag is removed by back diffusion to the anode, especially in thin membranes. In

DAFCs with a liquid fuel solution, however, back diffusion does not occur and the cathode

ﬂooding problem can be much more severe.

Temperature and Heat Flux Driven Flow A fourth mode of transport that has been shown

to drive water ﬂux in the membrane is heat ﬂux driven ﬂow. As a general rule, water will

move through the membrane toward a colder location. This occurs in a freezing process

due to capillary forces [20, 21], and nonfreezing processes [22, 23]. The nonfrozen mode

of transport is poorly understood but is likely a result of the combined effects of capillary

pressure change with temperature and thermo-osmosis in membranes.

There may also be a phase change phenomena associated with the motion of water in

the membrane under a temperature gradient, since Bradean et al. [22] showed an exponential

relationship between liquid ﬂux and the temperature gradient and heat ﬂux, but the exact na-

ture of this is not fully understood. This mode of transport has not commonly been included

in the analysis of normal operation, since this effect is obscured by the net diffusive and

electro-osmotic drag transfer. Under startup or shutdown conditions, however, where larger

gradients in temperature can exist, the net water ﬂux from this mode can be signiﬁcant, and

has been exploited to passively drain the DM on shutdown to a frozen state [22].

Net Transport Coefﬁcient The overall water transport through the membrane can be

written as the combination of the various modes of transport:

net,a−c

drag

diff

perm,gas

perm,cap

temp

(6.29)

The net drag coefﬁcient α

is a parameter used to express the net drag of water from the

anode to the cathode and accounts for the total effect of the modes in Eq. (6.29):

net,a−c

= α

(6.30)

The net drag coefﬁcient represents the total transport of water to the cathode. In an ideal

situation for water management, α

would be uniformly −0.5 along the electrode. In this

case, the exact amount of water generated by reaction is exactly balanced and moved

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6.2 Water Balance in PEFC 315

Figure 6.24 Measured local and net effective drag coefﬁcients along gas ﬂow channel of PEFC

operating at 0.7 V, with a dry anode inlet and cathode inlet at 50% RH at 9

◦

C. The net drag is negative,

indicating a net drag of water toward the dry anode. Toward the exit, however, the local effective drag

becomes positive. (Adapted from Ref. [24].)

through the electrolyte from the cathode to the anode. There would be the possibility of

anode ﬂooding, but the membrane would retain maximum moisture content. In practice, for

thethin(∼15–25-µm) membranes used in automotive applications with uniform anode and

cathode inlet humidity, the high back diffusion to the anode nearly completely compensates

for the electro-osmotic drag, and the net drag is nearly zero [25] or slightly negative [24].

For thicker membranes used in stationary applications, the net drag coefﬁcient can be

slightly positive because the back diffusion is limited.

Another point should be made that the assumption of uniform net drag within the fuel

cell is rarely justiﬁed. Figure 6.24 shows the measured net drag coefﬁcient distribution

along the ﬂow channel of a PEFC operating at either relatively dry anode or dry cathode

inlet conditions, with a thin 18-µm electrolyte membrane [11]. In this case, even though

the electrolyte is very thin, the net drag coefﬁcient is not near zero because of the initial

imbalance between anode and cathode humidity. For this dry anode inlet case, the overall net

drag coefﬁcient is −0.12, representing a net back-diffusion ﬂow toward the anode. However,

after the initial 40% of the fuel cell ﬂow channel, the net ﬂux is slightly positive, toward

the cathode, since the anode has become humidiﬁed, reducing the diffusion concentration

gradient.

As a result of Schroeder’s paradox, there is a very high electro-osmotic drag coefﬁcient

of 2–5 H

O/H

in applications where the anode is in contact with liquid-phase water such as

the DMFC. This and the lack of back diffusion result in a very positive net drag coefﬁcient

and more severe cathode ﬂooding without special engineering of the DM to provide a strong

capillary pressure gradient toward the anode.

Local Water Balance: Catalyst Layer Mass Balance In this section we discuss the local

water balance, which can be (and usually is) highly nonhomogeneous throughout the fuel

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316 Polymer Electrolyte Fuel Cells

Cathode

in,

out,

dn/dt

iA/2F

Figure 6.25 Control volume mass balance applied to cathode catalyst layer.

cell. From a mass balance on the cathode catalyst layer (Figure 6.25),



in,c

−

out,c

(6.31)

In steady state, the water generated will be exactly equivalent to the net water out of

the catalyst layer. The transport in and out of the catalyst layer involves many modes of

transport:

1. Transport into or from the electrolyte by diffusion, electro-osmotic drag, hydraulic

permeation, and temperature effects.

2. Transport of liquid and gas into the porous media between the catalyst layer and the

channel/land locations.

Example 6.5 Calculating Internal Water Balance Given a net drag coefﬁcient of 0.1,

determine the molar rate of water accumulation at the catalyst layer that must be removed

to prevent ﬂooding.

SOLUTION



net,a−c

gen

= α

(

+ 0.5

)

= 0.6

This value must be removed from the cathode exit of the fuel cell to obtain a condi-

tion of water balance. If this value is not removed, saturation of liquid in the catalyst

larger will increase, raising the local capillary pressure until it is pushed into the DM,

or electrolyte, or the active area is sufﬁcienty reduced to ﬂood the electrode and re-

duce the current. The balance can be achieved through a variety of methods already

discussed.

COMMENTS: For thin membranes in fully humidiﬁed conditions, we can typically

assume a net drag value close to zero. If there is an imbalance in the inlet RH values

between the anode and cathode, the diffusion ﬂux will change the net drag coefﬁcient. If

the anode ﬂow is relatively underhumidiﬁed compared to the cathode ﬂow, the net drag

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6.2 Water Balance in PEFC 317

Figure 6.26 Typical liquid water accumulation behavior under lands and channels in PEFC.

coefﬁcient generally becomes slightly negative. In the case of an underhumidiﬁed cathode,

the net drag value will become slightly positive.

Liquid Distribution and Transport in Catalyst, Microporous, and Diffusion Layers A

basic question of PEFCs is, where is the ﬂooding occurring? The answer is that it depends

on the porous media properties, ﬂow conditions, and fuel cell geometry. The exact nature

of the liquid water structure in the DM is not precisely known, but it is generally believed

from capillary theory that the liquid water ﬂows through large hydrophobic and hydrophilic

pores, and the gas phase ﬂows through small hydrophobic pores. However, evaporation and

condensation also play a role, so that a uniﬁed understanding based on capillary pressure

arguments alone is insufﬁcient.

Figure 6.26 is an illustration of the typical water distribution in the fuel cell porous

media under the lands and channels. Since the coldest location during operation is generally

under the lands, water condenses there ﬁrst. As the saturation increases, water pushes out

laterally from under the lands and forms connections between the lands under the channels,

resulting in DM ﬂooding and performance degradation. The removal of water from under

the lands into the channels is important to avoid ﬂooding, as lateral connectivity between the

water under adjacent lands in the DM can induce severe performance loss through reactant

blockage. Besides under the lands, there are other locations in a fuel cell where liquid

water tends to accumulate. These locations have been identiﬁed primarily using neutron

imaging, which enables a direct nonintrusive quantiﬁcation of the liquid water content

in the operating fuel cell and is used by several research institutions for this purpose

[26–28].

Liquid accumulation also commonly occurs around channel switchbacks, as shown in

Figure 6.27. This is a result of ﬂow recirculation, stagnation, and pressure drop at locations

of sudden momentum reversal. Additionally, the local ﬂow separation near the corner

accelerates the core ﬂow, promoting annular ﬂow of liquid water.

Interestingly, even in very dry operating conditions, there tends to be an accumulation

of liquid water under the lands, as shown in Figure 6.28. This is because the most efﬁ-

cient heat transfer is conduction through the lands, and access to the channel is blocked.

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318 Polymer Electrolyte Fuel Cells

Figure 6.27 Neutron radiograph showing liquid water accumulation along corners of 180

◦

switch-

back in operating PEFC. Water is often observed to preferentially accumulate at switchback locations

and along the channel walls. (Adapted from Ref. [9].)

Therefore, during normal operation, there is typically liquid water storage in the gas DM

under lands, since lands are the coolest location.

At low current density, channel-level ﬂooding is prevalent due to slug formation and the

lack of sufﬁcient gas-phase velocity to remove channel droplets via drag force. As the current

density (and channel ﬂow rates) increase, the channels are more efﬁciently cleared, and the

porous media begin to accumulate liquid from generation. At this point, ﬂooding may occur

in the catalyst layer, DM, or some portion of both depending on the materials and conditions.

From Chapter 5, we know that the capillary pressure liquid saturation relationship depends

on the net hydrophobicity of the DM. Since the carbon materials in the catalyst layer and

DM are hydrophilic and the PTFE additive is hydrophobic, there are regions of mixed

hydrophobic and hydrophilic behavior, and no single surface contact angle describes the

internal structure or multiphase ﬂow within the mixed wettability media. Instead, ﬂow

in these structures takes place along separate hydrophilic–hydrophobic pathways, each

with different behavior for liquids and gases (Figure 6.29). From porosimetry studies, a

typical range of hydrophobic to hydrophilic pores in PEFC diffusion media with PTFE

Land

Channel

Figure 6.28 Neutron image showing modest liquid water accumulation under the lands in a seven

channel parallel ﬂow conﬁguration fuel cell at an underhumidiﬁed condition. Lighter areas represent

the accumulation are under lands, while the generally black areas indicate a relatively liquid droplet-

free situation under the channels. The total water mass in the 14.5 cm

cell was 79 mg at 0.35 A/cm

(Adapted from Ref. [29].)

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6.2 Water Balance in PEFC 319

air

Water imbibed

automatically

Higher capillary

pressure

air

Hydrophilic

pore

Hydrophobic

pore

Figure 6.29 Schematic of different liquid water imbibitions and transport behavior in hydrophilic

and hydrophobic pores. The catalyst layer and diffusion media are typically mixed wettability media;

thus hydrophilic and hydrophobic pathways exist for transport of liquid and gas phases.

additive is 20–40% hydrophilic and 60–80% hydrophobic [30], so that the net condition of

a PTFE-treated DM is hydrophobic.

It is clear based on experimental studies that, while critical, ﬂooding is not solely a

result of capillary ﬂow, and phase change, morphology, and interfacial properties also play

key roles.

In order to develop a heuristic of the capillary ﬂow related ﬂooding in the catalyst

layer and DM, we should recall from Chapter 5 that the capillary pressure is inversely

proportional to the pore radius through the Laplace equation for capillary tubes:



2γ

∗



cos θ (6.32)

For hydrophobic media, the capillary pressure is elevated in the liquid phase and increases

with liquid saturation. In all cases, liquid tends to move along a path toward a lower capillary

pressure, which, for hydrophobic media, means liquid will move toward locations of

(a) lower liquid saturation,

(b) a more hydrophilic location, or

For a hydrophilic media, the liquid phase is nonwetting, a lower liquid-phase pressure exists

and provides a suction force to draw the connected liquid into smaller pores, where it will

remain until removed by convection or evaporation. Capillary ﬂow through a hydrophilic

porous media will ﬂow in the direction of

(a) higher liquid saturation (until fully saturated)

(b) a more hydrophilic surface, or