Mench M.M. Fuel Cell Engines

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

170 Performance Characterization of Fuel Cell Systems

Porous diffusion layer

Catalyst layer electrode

Channel

Land Land

Suction of

into electrode

Figure 4.34 Schematic of channel ﬂow suction into catalyst layer of fuel cell.

Equations (4.81) and (4.83) can be used for any fuel and oxidizer reaction with different

reaction orders.

Considering ﬂow through a fuel cell channel, there is a ﬂux of products away from

the catalyst, a suction of reactants to the catalyst surface due to reaction, and inert species

not participating in the reaction, as shown in Figure 4.34. At higher current densities,

the mass transport limitations can reduce the concentration of the reactants at the catalyst

surface to well below the ﬂow ﬁeld channel concentration and cause a sharp decline in the

output voltage. This concentration loss can be precipitous and corresponds to (region III in

Figure 4.1). This region of the polarization curve is really a combined thermodynamic- and

kinetic-related phenomenon that occurs independently at both electrodes. To determine an

expression for η

m,a

and η

m,c,

the mass concentration polarizations at each electrode, consider

Eqs. (4.81) and (4.83). From these, a general form of the voltage change at an electrode for

concentration changes in reactant R from some state 2 to state 1 can be written as

V

−C

= η

= V

− V

(

n + γ

)



R,s,2

R,s,1



(4.84)

Example 4.10 Kinetic and Thermodynamic Effect of Change in Oxygen Concentration

Determine the change in voltage expected for an increase in oxygen mole fraction from a

fully humidiﬁed state at 90

◦

C, 1.5 atm, to a fully humidiﬁed state at 70

◦

C, 1.5 atm pressure.

Assume an ORR order γ of 0.75. For this example, ignore changes in exchange current

density with temperature.

SOLUTION In this example, we are considering increasing the operating temperature

of the fuel cell at constant relative humidity. This will have a positive effect in terms of

the exchange current density (ignored here) but will also decrease the available oxygen

mole fraction since the saturation pressure is strongly related to temperature, as we have

discussed in Chapter 3. First we need to calculate the mole fraction of oxygen in both cases.

From Chapter 3

sat

(T )(Pa) =−2846.4 + 411.24 T (

◦

C) − 10.554 T (

◦

+ 0.16636 T (

◦

Evaluation gives P

sat

(70) and P

sat

(90) = 0.31 and 0.69 atm, respectively.

From

RH =

sat

(

)

total

sat

(

)

we ﬁnd that

(

)

1 × 0.31

1.5

= 0.21 y

(

)

1 × 0.69

1.5

= 0.46

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4.4 Region III: Concentration Polarization 171

Next we need to covert the vapor mole fraction to the oxygen mole fraction. The ratio of

oxygen to nitrogen in dry air is constant in humidiﬁed air at y

,dry

= 0.21. For the moist

air at 70

◦

air,70

= 1 − y

v,70

= 0.79 y

= 0.21y

air

= 0.166

For the moist air at 90

◦

air,90

= 1 − y

v,90

= 0.54 y

= 0.21y

air

= 0.113

From simpliﬁcation of Eq. (4.84) at 70 and 90

◦

C, with γ = 0.75, we have

V

70−90



,70

,90



Finally:

− V

(

0.166

)

(

0.113

)

[

343 ln

(

0.166

)

− 363 ln

(

0.113

)

]

= 15 mV

So decreasing the temperature from 90 to 70

◦

C actually increases the expected voltage by

15 mV for a fully humidiﬁed system because the oxygen mole fraction will increase. There

will also be an expected decrease in the exchange current density which can partially or

completely offset this trend. However, this example illustrates another issue with operating

PEFCs at elevated temperatures where the oxygen mole fraction in a moist system is

decreased.

COMMENTS: High-temperature PEFC operation has several disadvantages, includ-

ing: (a) electrolyte material limitations (conventional electrolyte materials are limited to

<120

◦

C), (b) reduced oxygen mole fraction for humidiﬁed conditions, and (c) need for

larger humidiﬁcation system. The ideal operating temperature depends on the trade-offs

between enhanced kinetics and decreased reactant concentration, humidiﬁcation require-

ments, and longevity. For high-temperature fuel cell operation, an elevated pressure some-

times is preferred since it reduces the oxygen mole fraction decrease with temperature for

humidiﬁed ﬂow.

Mass-Limiting Current Density We previously examined ohmic limiting current density.

Now we consider the case of mass transfer limiting current density i

. At the mass transport

limiting current density, the rate of mass transport to the reactant surface is insufﬁcient to

promote the rate of consumption required for reaction. In this case, the local concentration

of reactant will be reduced to zero, which, from Eq. (4.84), must also reduce the cell

voltage to zero.

Assuming the surface concentration (C

) is zero at the limiting state (i

)

Small localized areas of fuel or oxidizer starvation in the fuel cell can exist, but if a signiﬁcant area is depleted

of fuel or oxidizer, the local voltage potential is reduced to zero, and since the catalyst layers are conductive, the

voltage of the entire cell will be reduced to zero. Thus, when we operate a fuel cell, the ﬂow stoichiometry of the

anode and cathode must always be greater than unity to enable operation at sufﬁcient voltage.

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

172 Performance Characterization of Fuel Cell Systems

and decreases linearly from state 1 to state 2, we can show that

R,s,2

= C

R,s,1

− C

R,s,1

= C

R,s,1



1 −



(4.85)

which we can plug into Eq. (4.84) and show that

V

−C

= η

=−

(

n + γ

)



1 −



(4.86)

If we assume the reaction occurs only at the catalyst layer interface, which is true for

high-current-density mass-transport-limited reactions, and we neglect kinetic effects, we

are left with the Nernst equation at each electrode:

=−



1 −



(4.87)

In practice, the concentration loss is more gradually observed than predicted by Eqs.

(4.87) or (4.86), but the qualitative result of the equation holds (see Figure 4.35). The

deviation from predicted values is a result of the dependence of the kinetics on reactant

concentration, the accumulation of liquid water or the accumulation of inert species such as

nitrogen, which cannot be easily predicted without inclusion of fuel cell geometry, material,

and other parameters. To account for this deviation, a semiempirical approach is often used

where Eq. (4.87) is changed slightly to include a constant (B) to better ﬁt with experimental

data:

=−B ln



1 −



(4.88)

Figure 4.35 Comparison of concentration polarization predicted with Nernst equation and that

predicted with semiempirical modiﬁcation with B factor of 0.05 V at 353 K and i

= 2.5 A/cm

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4.4 Region III: Concentration Polarization 173

Therefore, the total concentration polarization of the fuel cell can be written as

m,a

+ η

m,c

=−B



1 −

l,a



− B



1 −

l,c



(4.89)

Each electrode will have this loss (and a separate limiting current density), although the

anode loss is typically negligible for a hydrogen feed due to the high concentrations

and mass diffusivity, so that a single expression is often used to represent concentration

polarization. If the anode contribution to the concentration and activation polarization is

ignored, little error would result for air–H

systems. This is not the case for fuel cells

without neat hydrogen as fuel or with signiﬁcant dilution or impurities in the anode side,

such as when the hydrogen gas from a fuel reformer is used. It is important to understand

that each electrode has a different mass transport limiting current density, which can be

approximated with the concepts discussed in Chapter 5. However, the minimum value of

limiting current density between the electrodes will be the determining value on the fuel

cell polarization curve.

Example 4.11 Determine Concentration Polarization Given the anodic mass-transport-

limited current density is 15 A/cm

and the cathode mass-transport-limited current density

is 2.5 A/cm

, determine the anode and cathode concentration polarization at 0.1 and 1.0

A/cm

. Assume the B factor is 0.045 V on both electrodes and Eq. (4.88) is appropriate and

is determined from curve-ﬁt of several polarization curves.

SOLUTION On the anode

m,a

=−0.045 ln



1 −



which is 0.3 and 3.1 mV for 0.1 and 1.0 A/cm

, respectively. On the cathode

m,c

=−0.045 ln



1 −

2.5



which is 1.8 and 23.0 mV for 0.1 and 1.0 A/cm

, respectively.

COMMENTS: The fuel cell maximum current density would be limited by the cathode

in this case. It would also be possible that the maximum current density is limited by

kinetic or ohmic effects as well.

Alternate Empirical Approach Another completely empirical approach to describe the

overall fuel cell concentration polarization has been proposed [18–20]:

= m exp

(

)

(4.90)

If this equation is used, the constants m and n are typically ﬁt from several polarization

curves, and the total (anode +cathode) concentration polarization is included in this single

expression. According to [20], typical values of the m constant are around 3 × 10

−5

and the n constant is around 8 × 10

−3

/mA for a PEFC. Although this expression

completely loses physical meaning, it can be used to simply model the complex fuel cell

stack mass transport limitations if plentiful polarization curve data are available.

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

174 Performance Characterization of Fuel Cell Systems

Functionally, the mass-limiting current density at each electrode depends on:

1. Fuel cell design (discussed in other chapters for various fuel cells)

2. Operating conditions and stoichiometry

3. Mass transport to the catalyst layer (discussed in Chapter 5)

(a) Thickness and porosity of the diffusion media (PEFC and AFC) and catalyst

layer

(b) Transport mode to electrode (diffusion, forced convection)

4. Impurities/blockage/water accumulation/ﬂooding or liquid electrolyte can cover the

catalyst, hindering access to the catalyst

Flow Stoichiometry Until now, we have discussed the ﬂow into a fuel cell and not

explored the fact that the concentration of reactant is depleted inside the fuel cell from con-

sumption of reactant. Flow comes into a fuel cell with a molar ﬂow rate of reactant shown in

Chapter 2:

= λ

(4.91)

The reactant consumed can be determined from Faraday’s law:

consumed

(4.92)

Therefore the amount of reactant out of a fuel cell is

−

consumed

(

λ − 1

)

(4.93)

We can see now why the ﬂow stoichiometry must always be greater than unity in a fuel

cell, since in a unity condition, the ﬂow leaving the fuel cell would contain no reactant,

automatically assuring that the catalyst layer in this region would have no reactant and,

according to the Nernst equation, zero voltage. It is important to understand that the entire

fuel cell need not suffer reactant depletion. Since the fuel cell catalyst layer is electrically

conductive, any signiﬁcant region devoid of reactant can induce zero fuel cell voltage for

a single cell. Even smaller, locally starved regions can reduce performance and accelerate

degradation. If the cell is in a stack, then the individual fuel cell in series with severe reactant

depletion can suffer voltage reversal. Voltage reversal can result in rapid degradation of

the catalyst and support, and therefore local reactant depletion should be avoided.

To avoid depletion, the minimum requirement is an inlet anode and cathode stoichiom-

etry greater than unity. In a combustion engine, this would be like having a requirement

that the engine have unburned gasoline and air in the exhaust. For both an internal com-

bustion engine and a fuel cell this is obviously wasteful. If not recycled, the fuel cell loses

the potential for the energy released by the reaction of the unreacted fuel and oxidizer, in

addition to being a possible safety risk if released to the environment. In fuel cell stacks,

the hydrogen or other fuel used is typically recycled back into the anode, which carries a

parasitic energy penalty. In the cathode, since air is usually used, recycling is not neccesary.

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

4.5 Region IV: Other Polarization Losses 175

Some fuel cell designs have attempted to boost fuel utilization and reduce system

parasitic losses with a dead-ended hydrogen fuel compartment. That is, the hydrogen

ﬂow channels have no exit, and fuel is either continuously or sporadically supplied at the

consumption rate required for suitable performance. A major drawback of this approach

is that inerts and poisons in the ﬂow stream build up in the dead end over time, and at

least periodic purging is needed. In low-temperature systems, liquid accumulation is also

a common problem, and some ﬂow in the channels is beneﬁcial to remove liquid droplet

accumulations [20].

Returning to our ultimate goal of being able to analytically describe the fundamental

physics of the polarization curve, we now have an expression that includes the starting

equilibrium voltage, the departure from this voltage resulting from activation overpotential

at each electrode, the ohmic polarization, and concentration polarization:

cell

= E

◦

(T , P) − η

a,a

−



a,c



− η

m,a

−



m,c





 

Can now solve

−η

(4.94)

Only the crossover and shorting polarization losses need to be modeled, which are covered

in the next section.

4.5 REGION IV: OTHER POLARIZATION LOSSES

The ﬁnal piece of the polarization curve to be modeled is the departure from the expected

OCR given by the Nernst equation. For low-temperature PEFCs, the OCV is predicted to be

around 1.2 V, but in practice, only about 1.0 V is observed. For a high-temperature SOFC,

however, the actual OCV can be very close to the theoretical OCV. For the PEFC, the 0.2 V

represents an incredibly signiﬁcant efﬁciency loss before any useful current is even drawn.

The departure from the theoretical OCV is typically a result of two phenomena:

1. Electrical short circuits in the fuel cell

2. Crossover of reactants through the electrolyte and subsequent mixed-potential re-

action at the opposite electrode

Electrical Shorts Electrical short circuits can happen if the cell is poorly designed or

assembled or, more commonly, the electrolyte is not completely insulating for electrons. For

low-temperature fuel cells, this is usually not a major problem, but for higher temperature

fuel cells, especially SOFCs, the electrolyte phase can have a mixed conductivity for

electrons that is dependent on material properties, temperature, and oxygen partial pressure.

The transference number (t

) is the ratio of electrolyte ionic conductivity to the total

conductivity (ionic plus electronic), deﬁned as [22]

+ σ

(4.95)

Obviously, a transference number close to unity is desired, and a value greater than 0.9

is found in conventional SOFCs. When a signiﬁcant electrical conductivity exists in the

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

176 Performance Characterization of Fuel Cell Systems

electrolyte, current is short circuited through the electrolyte, causing a departure from the

Nernst potential at open-circuit conditions, as well as additional losses throughout the entire

polarization curve. At an open circuit, the current passed as a result of the electronic leakage

equals the ionic exchange, and the following equation can be developed:

OCV

= E

◦

(

T, P

)

Nernst

+ σ

= E

◦

(

T, P

)

× t

(4.96)

One goal of SOFC research is the development of electrolyte structures with high ionic

conductivity at low operating temperatures [22–26]. A problem with achieving this goal

has been the mixed conductivity of the electrolyte. Ironically, a long-term goal of PEFC

operation in to increase the operating temperature to provide tolerance to impurities and

better heat rejection, while a long-term goal of SOFCs is to reduce operating temperature

so that less expensive materials can be used. Phosphoric acid fuel cells do operate at an

intermediate temperature but have power density and cost limitations.

In fuel cells with coolant ﬂow, ionic impurities which accumulate in the coolant over

time can also cause short currents through the coolant to develop.

Species Crossover In a high-temperature SOFC, the Nernst potential is reduced compared

to a low-temperature PEFC, as discussed in Chapter 3. However, the normally observed

OCV of a SOFC and a PEFC are similar, due to crossover losses in the PEFC. Crossover of

reactants through the electrolyte can be a major issue degrading the performance of a PEFC.

The 0.2-V loss typically suffered at the open circuit in a PEFC represents ∼20% efﬁciency

loss without a useful electron being drawn. In a DMFC, the effect is even more extreme,

since the liquid solution used as a fuel has a higher molecular density, thus crossing over

more readily. For the DMFC, an actual OCV of 0.7 V (∼1.2 V is predicted) is normally

observed due to this loss. Intensive research and development has been invested to lessen

the crossover effect in DMFCs while not sacriﬁcing operating performance.

Physically, a large concentration gradient in the reactant between the anode and cath-

ode sides acts as a driving force for diffusion of reactants across the electrolyte that

compete with the opposing electrode reactions (Figure 4.36). Recall that with the Nernst

equation we establish the thermodynamic equilibrium potential between the two elec-

trodes. With crossover, the Nernst potential will be altered by slightly changing the surface

Anode Cathode

Thin-film

electrolyte

Figure 4.36 Reactant crossover in PEFC.

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4.5 Region IV: Other Polarization Losses 177

concentrations at the catalyst surface, but this effect is negligible. The major effect of

crossover on OCV is due to kinetics. At the hydrogen electrode, the exchange current

density is typically orders of magnitude higher for the hydrogen electro-oxidation than for

any reaction with crossover oxygen, despite the high reduction potential (see Table 4.1).

Therefore, there is little interference from oxygen crossover at the anode, and a nearly true

equilibrium can be established. However, at the cathode, the hydrogen crossover typically

has a much higher exchange current density for oxidation than the ORR, which is very slow.

At the cathode, a true thermodynamic equilibrium cannot be established due to hydrogen

crossover oxidation reaction and relatively slower oxygen reduction kinetics, resulting in a

mixed reaction and lowered OCV. It should be noted that, although the oxygen crossover

to the anode has a small impact on the OCV departure, it does have an important impact

on PEFC durability, discussed in Chapter 6. Because of these reasons, we seek to eliminate

crossover of reactants in PEFCs and other fuel cells in general.

From concepts discussed in the next chapter, the crossover can be related to the

diffusion coefﬁcient, thickness, and concentration gradient across the electrolyte through

the one-dimensional version of Fick’s law [27]:



− D

∂C

∂x

(4.97)

where D is a mass diffusivity coefﬁcient, C is the molar concentration, and x is the length

scale in the direction of transport. Even a very small amount of reactant crossover can have

a large impact on OCV.

Several approaches have been used to limit reactant crossover:

Ĺ Change in Material Properties of Electrolyte If the PEFC electrolyte is made less

permeable to the reactants [e.g., D in Eq. (4.97) is reduced], crossover losses will be

reduced. However, this approach has the severe limitation of reducing the reactant

transport to the catalyst, where it is needed, resulting in severe mass transport losses.

From the concept of the reaction surface of Chapter 1, some diffusion of reactants

through the electrolyte covering catalyst particles is needed for adequate perfor-

mance. Other alterations of the electrolyte material porosity or use of composite

structures have been tried with some success. Figure 4.37 shows the measured hydro-

gen crossover current density for several PEFC membranes. Notice that the thicker

membrane limits crossover, as expected. Experimentally, the hydrogen crossover is

oxidized until a mass transfer limiting condition is reached, which is proportional

to the rate of hydrogen crossover. The equivalent current density produced by the

crossover hydrogen can be solved from Faraday’s law, that is, i

2F/A

Ĺ Use of Thicker Electrolyte From Eq. (4.97), the longer the distance of diffusion,

the lower the ﬂux. This approach has been used extensively in the DMFC and

other alcohol-based liquid solution fuel cells. Although this is effective to reduce

crossover, the ohmic losses are directly proportional to electrolyte thickness, which

limits the use of this technique in high-power applications.

Ĺ Alteration of Morphology or Structure of Porous Media and/or Catalyst Lay-

ers to Limit Diffusion Through Electrolyte This approach is favored in DMFC

applications because this approach allows the use of high-methanol-concentration

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

178 Performance Characterization of Fuel Cell Systems

Figure 4.37 Measured hydrogen crossover current density with different electrolyte structures [28].

The crossover current density is measured directly with a hydrogen anode and an inert humidiﬁed

cathode compartment. The limiting current density of hydrogen oxidation at the cathode is determined

to be the crossover current density.

solution. A very simple concept, a diffusion barrier is put between the catalyst layer

and the diffusion media, causing a sharp diffusion gradient across this boundary

which does not add to overall ionic impedance since it is not in the electrolyte. Since

the methanol reaction is kinetically limited, the goal of this approach is to tailor the

diffusion barrier to permit only the amount of methanol to reach the catalyst layer

as is needed for the electrochemical oxidation reaction, since additional methanol is

assumed to cross to the cathode:

n CH

OHcatalyst −n CH

OHreacted = n CH

OHcrossover (4.98)

Ĺ Alteration of Electrolyte Composition to Consume Crossover before Reaching Elec-

trode A novel approach by Wantanabe [29] was used where platinum particles were

embedded within the PEFC electrolyte (Figure 4.38). The concept was designed to

react the crossover hydrogen with crossover oxygen within the electrolyte to gen-

erate water and maintain high membrane humidity levels while reducing crossover

losses. This approach is more costly since it involves additional platinum, and it may

also lead to reduced durability.

Ĺ Recirculation of Liquid Electrolyte In fuel cells with a liquid electrolyte, such as

an alkaline fuel cell, recirculation of the electrolyte can be used to ﬁlter impuri-

ties and prevent crossover. This adds ancillary equipment and other system-related

difﬁculties.

Obviously, each approach has design limitations that result in a balance depending

on the application. For example, for reduced ohmic losses, the ideal electrolyte thickness

is inﬁnitely thin. However, in this limit electrical shorts and crossover dominate. Some

thickness is required to provide a physical separation barrier. Ultimately, since a very small

amount (even milliamperes per cubic centimeter) of crossover will cause a large decrease

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

4.5 Region IV: Other Polarization Losses 179

Figure 4.38 Schematic of embedded catalyst for crossover elimination and internal water genera-

tion based on concept of Wantanabe [29].

in OCV, it is nearly impossible to completely eliminate this problem in polymer electrolyte

systems.

Modeling Crossover Losses There are several approaches used to empirically, semiem-

pirically, or analytically account for the noted OCV deviation caused by internal currents

or reactant crossover. For electrical conductivity effects, if the transference number (t

)is

known, it is a simple matter to add this effect to the performance deviation, since it can be

modeled as a short circuit in parallel with the external current, whose resistance is

short

electrolyte

Aσ

(



)

(4.99)

where δ

electrolyte

is the electrolyte thickness and A is the geometric area of the electrolyte.

For mass crossover losses, several approaches have been taken, with different levels of

accuracy and physicochemical relevance. The ﬁrst approach is purely empirical, based on

experimental data only. This is particularly useful for computational modeling that is not

overly concerned with conditions at OCV. The difﬁculty with this approach is it may not

be accurate and does not take into account the many related phenomena on OCV.

From the Nernst equation with unity reactant activity we can show the theoretical

equilibrium potential, which is close to accurate for the SOFC but assumes zero crossover

and only accounts for equilibrium effects. From experimental observation of a hydrogen

PEFC, Parthasarathy and co-workers [30] found that

OCV (cathode potential vs. RHE) = 0.0025T + 0.2329 (4.100)

where T is in Kelvin. From the Nernst equation, the OCV decreases with temperature, while

experimentally, the OCV is determined to increase slightly with the OCV for a PEFC. This

is due to crossover losses. As the temperature increases, the exchange current density on

the cathode increases dramatically, reducing the mixed potential effect of the crossover

hydrogen. The empirical trend is only valid over the low-temperature range of PEFCs (up

to ∼100

◦

C), however, and should also be a function of catalyst, electrolyte thickness and

material, age, and other factors not included in this simple ﬁt. However, Eq. (4.100) has

been used conveniently in modeling studies to simplify calculation. Other empirical forms

can be derived based on experimental or theoretically determined data as needed.