Mench M.M. Fuel Cell Engines

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

180 Performance Characterization of Fuel Cell Systems

On a semiempirical basis, the crossover current density i

can be modeled as a base

current density that occurs even under open-circuit conditions. In this approach the current

density at the cathode is modeled as i + i

. At OCV, the external current i = 0, but i

is still

active. For instance, if a Tafel kinetics approach is used on the cathode, we would model

the cathode kinetic overpotential in the absence of concentration affects as

a,c

=−



i + i

o,c



(4.101)

This has the desired effect of decreasing the OCV to observed levels but is really not an

accurate representation of the true mixed potential reaction, in part because the exchange

current density on the cathode for the ORR is different than for the HOR on the cathode.

If a full analytical model of the surface is desired, the mixed potential must be properly

accounted for and include separate terms to account for the kinetic losses associated with the

individual reactions present on the particular electrode. Except in the cases where detailed

analysis of the OCV effects is desired, this is often not necessary, since the OCV departure

affects the starting potential at zero current but has little effect beyond the OCV loss.

Example 4.12 Calculating Crossover Losses In ref. [9], the authors noted a hydrogen

crossover loss of 3.3 mA/cm

for their automotive H

PEFC applications. Calculate the mass

crossover rate of hydrogen through the membrane. Also, calculate and plot the cathode

activation overpotential loss at open circuit and 1 A/cm

as a function of cathodic exchange

current density. Assume the cathodic charge transfer coefﬁcient at the cathode is 1.5 at a

temperature of 353 K, and the fuel cell has a 50 cm

geometric area.

SOLUTION The mass ﬂux of hydrogen can be calculated from Faraday’s law and a unit

conversion using the molecular weight of hydrogen. Note the need to keep consistent units

of current density in the expression:

3.3/1000 × 50

2 × 96,485

= 8.55 × 10

−7

mol/s

To convert to mass crossover rate, we use the MW:

= n

· MW

= (8.55 × 10

−7

mol/s)(2 g/mol = 1.71 × 10

−6

g/s

We can assume Tafel kinetics at the cathode; then

a,c

=−



i + i

o,c



=−0.0203 ln



i + 0.0033

o,c



At OCV this becomes

a,c

=−



o,c



=−0.0203 ln



0.0033

o,c



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4.6 Polarization Curve Model Summary 181

Plotting the activation losses as a function of cathodic exchange current density, we ﬁnd:

COMMENTS: Even a very small crossover results in an extremely signiﬁcant polarization

loss near an open circuit. In the PEFC at low temperatures, 0.1–0.2 V is typically sacriﬁced

to crossover. At high current density, the added effect of the crossover is minimal, though,

as i >> i

4.6 POLARIZATION CURVE MODEL SUMMARY

Returning again to our overall model, we now have a complete representation of the

polarization curve. If we know several key paramaters relating to the kinetic, ohmic, and

mass transfer processes, we can predict the overall polarization curve of the fuel cell.

Much more complex models exist in the literature to cover multidimensional, multiphase,

and transient aspects as well as approach the problem from various length scales from

molecular to full-size stack simulation. However, the approach taken here does include the

most important physicochemical phenomena that affect fuel cell performance:

cell

= E

◦

(T , P) − η

a,a

−



a,c



− η

m,a

−



m,c



− η

(4.102)

Extension of these relations into multi-dimensional is a matter of additional mathematics,

not fundamental understanding.

From Chapter 3

◦

(T , P) =−

G





fuel

∗

fuel



fuel



∗



oxidizer



(4.103)

The anode and cathode activation polarization losses (η

a,a

and η

a,c

) for most fuel cell

reactions can be determined from Eq. (4.35), the BV equation or a simpliﬁed form [i.e.,

Eqs. (4.52)–(4.55)].

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182 Performance Characterization of Fuel Cell Systems

The ohmic polarization, η

can be determined from Eq. (4.71), including contact, ionic,

and electronic resistances.

The concentration polarization values, η

m,a

and η

m,c

from Eq. (4.86), and the crossover

losses can be included by adding either the fuel crossover current density to the cathode

current (mass transfer) or an internal short resistor for the case of a mixed conductivity in

the electrolyte.

We still cannot fully investigate the concentration polarization without the tools of

Chapter 5, which will allow us to predict the mass transfer limiting current density.

Alternative Simpliﬁed Empirical Model The complexity of the model shown requires

calculation or estimation of many parameters that are themselves functions of operating

conditions. Additionally, due to the many complexities and material, geometry, ﬂuid mani-

folding, and so on, not included in our bulk model, the calculated results often have a large

deviation from experimental results. As stated, much more complex computational models

exist, but even these have signiﬁcant deviations from measured results because not every

phenomenon can be accurately modeled and many of the transport parameters needed still

have signiﬁcant uncertainty in their values. It must be emphasized that the basic model

presented in this chapter is a mere starting point and is not intended as a quantitative or

exact solution for all operating fuel cells. Instead, it serves to promote understanding of the

underlying physicochemical phenomena that control performance, so that an understanding

of the engineering trade-offs in design optimization can be achieved based on the qualitative

trends predicted.

Given the error and experimental effort associated with estimation of all the parameters

in the analytical model and the complexity and time required for computational simulation,

which, ultimately, still relies on the accuracy of the input parameters, there is a need for

an entirely empirical approach to quickly characterize performance of fuel cells and stack

designs. The following semiempirical model can be used to quickly glean some comparative

information:

cell

= E

OCV

− A ln

(

)

− iR− B ln



1 −



(4.104)

or, using Eq. (4.90) [20],

cell

= E

OCV

− A ln

(

)

− iR− m exp

(

)

(4.105)

where A, R, B, i

, m, and n, are parameters taken from curve-ﬁts of experimental data.

Each form has ﬁt parameters for activation, ohmic, and concentration polarizations. For

the OCV, an empirical equation or constant can be used based on experimental data. This

approach is useful to quickly model real fuel cell performance, and by comparison with

other similar curve its, the relative impact of the ohmic, concentration, and activation

polarizations between different designs can be compared.

Example 4.13 Predicting Change in Polarization Curve Based on our understanding of

fuel cell polarizations, we can now diagnose and predict basic polarization curve behavior

as a function of the relevant parameters. We should be careful not to generalize too much,

as many other minor effects and differently combined variables can lead to similar bulk

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

4.7 Summary 183

polarization curve results. Nevertheless, at this stage basic conclusions and predictions can

be made based on the understanding of this chapter:

(a) Sketch a typical hydrogen PEFC polarization curve with electrolyte A. Then,

sketch a polarization curve with the same operating conditions, but with a thicker

electrolyte B. Be sure to think about all of the effects of this change.

(b) Sketch a typical high-temperature fuel cell (SOFC, MCFC) polarization curve

operating at temperature A. Then, sketch a polarization curve B with elevated

temperature conditions. Be sure to think about all of the effects of this change.

(Ignore the effects on mass transport region until Chapter 5.)

for the hydrogen PEFC.

SOLUTION

COMMENTS: When thinking about changes in operating conditions, the effects on all the

regions of the polarization must be considered. Some of these we cannot yet fully understand

(without Chapter 5, that is). Note that in many cases what seems like a completely positive

effect has other limitations that require an engineering optimization.

4.7 SUMMARY

This chapter presents a fundamental background in the various polarization losses on a fuel

cell system. The main ﬁgure of merit for a fuel cell system is the polarization curve, shown in

Figure 4.1. Starting at the maximum (unobtainable) thermal voltage E

◦◦

, the potential is re-

duced to the predicted Nernst open circuit potential due to entropy and concentration effects.

From this initial maximum possible starting point, four main modes of polarization occur:

1. Activation polarization dominates at low current density and occurs at each elec-

trode.

2. Ohmic polarization occurs throughout the polarization curve as a result of ionic

and electronic bulk and contact resistance through all components of the fuel cell,

but typically is dominated by the electrolyte ionic transfer resistance.

3. Concentration polarization dominates at high current density and occurs at each

electrode.

4. Crossover or mixed-conductivity polarization is mainly responsible for the depar-

ture from the expected Nernst potential at open-circuit conditions.

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184 Performance Characterization of Fuel Cell Systems

The activation polarization is commonly modeled with the BV expression, which

assumes the electrochemical reaction is rate limited by a charge transfer process:

cell

= i



exp





− exp



−α



Several simpliﬁcations that allow explicit calculation of the activation overpotential are

possible from this expression, including:

1. Tafel kinetics for high polarization (branch ratio 10 : 1):

η =±





2. Linerarized kinetics for low polarization (α

Fη/R

T < 0.15):

η =

(

+ α

)

3. A sinh simpliﬁcation for the case where the anodic and cathodic charge transfer

coefﬁcients are equal:

η =

αF

sinh

−1



cell



In electrochemical reactions, the dominating kinetic parameter is the exchange current

density i

, which is a strong function of temperature and reactant concentration. The

exchange current density is not an intrinsic parameter of a catalyst and is also a strong

function of the morphology of the electrode structure:

(T , C

) = i

o,ref

exp



−



R,ref



The ohmic polarization loss can be represented symbolically as

= iA





k=1



where the summation is performed over the contacts and bulk ionic and electronic resis-

tances of all components. The losses are typically dominated by ionic resistance in the

electrolyte, which is discussed in Chapter 5.

The concentration polarization is a result of decreasing surface concentration of reac-

tant. The restriction can be a result of gas-phase transport limits, impurity absorption at the

catalyst surface, liquid blockage in low-temperature fuel cells, or other reasons. The ther-

modynamic (Nernst) and kinetic concentration polarization at an electrode can be written

=−B ln



1 −



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

Application Study: Polarization Curve Losses 185

where B is an empirical constant and i

is the mass transfer limiting current density at that

electrode. Another approach is to use an empirically derived expression:

= m exp

(

)

The departure from Nernst OCV is a result of reactant crossover through the electrode

and mixed reaction potential, or electrical shorts from mixed electrolyte conductivity. For

most fuel cell systems, the major concern is reactant crossover, although high-temperature

SOFC systems suffer from some electrical conductivity in the electrolyte. For the mass

crossover case, the effect can be modeled by inclusion of the crossover current density

with the fuel cell current density activation overpotential at the electrode with the mixed

potential reaction.

The various polarization losses are summarized in the following equation for fuel cell

voltage as a function of current density:

cell

= E

◦

(T , P) − η

a,a

−



a,c



− η

m,a

−



m,c



− η

where E

◦

(T, P) is solved from the Nernst equation.

APPLICATION STUDY: POLARIZATION CURVE LOSSES

Now that this chapter is completed, we have some real analytical tools to work with to

describe fuel cells in operation. The methods shown in the chapter can be applied to

any fuel cell system, given appropriate transport and electrochemical parameters. First,

the empirical method can be used to curve-ﬁt any experimental polarization curve and to

understand a little about the general performance of a fuel cell. The basic analytical model

we described can be applied to more fundamentally understand the roles of particular types

of materials and geometry on the performance, provided all of the unknown quantities can

be properly ﬁt or estimated from existing resources. Fortunately, there are a lot of existing

data available that are continuously updated in technical journals, which can be used as a

starting point for evaluation of polarization behavior. For this assignment:

(a) Find a recent fuel cell technical journal article or publication with some experi-

mental polarization curves in it at different experimental conditions for either a

fuel cell or stack. Perform an empirical curve ﬁt of the form of Eq. (4.105). Find a

best-ﬁt kinetic, ohmic, and mass transfer parameter ﬁt.

(b) Try to estimate (or use vales provided) for a more complete analytical model from

this chapter. Use given parameters where available and reasonable assumptions or

approximations where there is not enough information.

operating conditions. Discuss how reasonable the results are and what would be

needed to have a more accurate predictive tool.

Some good journals to refere to are the Journal of Power Sources, Journal of the Electro-

chemical Society, Electrochimica Acta, and others your professor can provide. Your school

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

186 Performance Characterization of Fuel Cell Systems

library likely has online access to some of these resources or can get the relevant texts

for you. The websites of these journals provide free topical searches and abstracts to get

you started. In lieu of these data, you can also use the data provided in the table below

for a PEFC operating at 65

◦

C, 1 atm back pressure, 100% RH inlets, anode and cathode

stoichiometry of 1.5 and 2.0, respectively.

Current Density (A/cm

) Voltage (V)

0.0000 0.9317

0.0539 0.8493

0.1817 0.7995

0.4129 0.7490

0.6944 0.6992

0.9132 0.6493

1.0850 0.5989

1.2894 0.4993

1.4452 0.3988

1.5612 0.2990

1.6509 0.1987

1.7300 0.0000

PROBLEMS

Calculation/Short Answer Problems

4.1 Draw a typical polarization curve and label the regions

dominated by activation polarization (A), ohmic polariza-

tion (B), and concentration polarization (C). Also label the

speciﬁc locations of the OCV (D), Mass-limiting current

density i

(E), heat-generating region (F), and useful elec-

tric region (G). Discuss the factors that result in the various

polarizations (e.g., the catalyst choice has a strong effect on

the activation region, as do many other factors).

4.2

(a) Consider a fuel cell stack with poor overall com-

pression. What region of the polarization curve

would be most affected?

(b) Consider a poor catalyst layer. What region of the

polarization curve would be most affected?

PEFC and the ionic conductivity decreases. What

region of the polarization curve would be most

affected?

4.3 Describe what is happening when a low-temperature

–O

PEM fuel cell is “ﬂooded” and what region of the

polarization curve in Problem 4.1 (A, B,orC) would be

most affected?

4.4 Explain why the operating stoichiometry of a fuel cell

reactant can never be equal to or less than 1 (without recy-

cling efﬂuent gas).

4.5 Consider a hydrogen fuel cell with 80% H

and 20%

water vapor in the anode at 2 atm total pressure. If the hy-

drogen mole fraction is reduced to 70% because the water

content has increased to 30%, what must your anode total

pressure be to offset the effect of the increased water con-

tent on expected OCV? Consider Nernst and kinetic effects

with a hydrogen reaction order of 1.

4.6 Fill out the plot for a typical PEFC. The plot is of fuel

cell voltage at a given steady-state current versus compres-

sion pressure:

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

Problems 187

4.7 Consider the following typical polarization curve for

a fuel cell. Be careful to consider all the effects of what the

question poses!

(a) Consider the case where we have half the thickness

of the electrolyte. Draw the expected polarization

curve. Label the curve A.

(b) Consider the case where we increase the exchange

current density of both electrodes. Draw the ex-

pected polarization curve. Label the curve B.

ture of operation in reference to the baseline case

already drawn. Label the curve C.

4.8 Consider the polarization curve of an electrode below.

If we decrease the activity of this electrode by using a worse

catalyst, what will the same curve look like? Redraw the

curve below with a worse catalyst.

log i

activation

4.9 On the single-electrode polarization plot below indi-

cate the regions where (a) Butler–Volmer kinetics, (b) Tafel

kinetics, (c) the sinh solution, and (d) facile kinetics (lin-

earized Butler–Volmer) apply.

log i

activation

4.10 Consider a hydrogen fuel cell with 75% N

, 19%

oxygen, and 6% H

O in the cathode at 1 atm total pressure.

If the oxygen mole fraction is reduced because water vapor

has been added to the cathode (new balance 15% O

, 25%

O, and 60% N

), the expected OCV will fall. To counter

this loss, you suggest changing the cathode pressure. To

what value must your cathode total pressure be changed to

in order to maintain the OCV? Conditions on the anode are

not to be changed.

4.11 Given the table below of parameters for a 50-cm

active-area fuel cell, solve the following:

(a) Using the appropriate kinetics, calculate the ca-

thodic activation overpotential η

a,c

at 0.8 A/cm

(b) Calculate the total ohmic overpotential η

at 0.5

A/cm

, including contact resistance.

Parameter Value Unit

cell

340 K

Ionomer fraction, anode 0.4 —

Ionomer fraction, cathode 0.3 —

Anode thickness 30 µm

Cathode thickness 20 µm

Average electrolyte

conductivity

7.0 S/m

Electrolyte thickness 50 µm

Anode GDL thickness 250 µm

Cathode GDL thickness 300 µm

(for anode) 0.6 —

(for anode) Solve —

(for cathode) Solve —

(for cathode) 0.4 —

E(T,P) 1.15 V

Anode GDL porosity 0.3 —

Cathode GDL porosity 0.4 —

Anode i

1.5 A/cm

Cathode i

5.0 × 10

−4

A/cm

Anode pressure 101,325 Pa

Cathode pressure 300,000 Pa

B (replaced RT/nF in

concentration loss)

0.05 V

n anode elementary reaction 1 —

n cathode elementary reaction 2 —

R contact, total for cell 35 m·cm

Landing to channel width ratio 1:1 Unitless

(crossover) 3.0 mA/cm

average mole fraction 0.15 Unitless

average mole fraction 0.75 Unitless

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

188 Performance Characterization of Fuel Cell Systems

4.12 Do you expect the voltage of a hydrogen–air fuel

cell to increase, decrease, or stay the same if the anode and

cathode pressures are both doubled.

4.13 Given the table of parameters included in problem 11

for a 100-cm

-active-area fuel cell, determine the follow-

ing:

(a) Using the appropriate kinetics, calculate the an-

odic activation overpotential η

a,a

at 0.5 A/cm

(b) Using the appropriate kinetics, calculate the ca-

thodic activation overpotential η

a,c

at 0.5 A/cm

at 0.5

A/cm

4.14

(a) Discuss under what conditions the Butler–Volmer

kinetics model is valid.

(b) Discuss under what conditions the Tafel kinetics

model is valid.

Butler–Volmer model is valid.

(d) Discuss under what conditions the sinh solution to

the Butler–Volmer model is valid.

(e) Deﬁne and discuss the meaning of the charge

transfer coefﬁcient.

(f) Deﬁne the exchange current density and list what

factors affect it.

(g) Deﬁne and discuss the basic assumptions of the

Temkin and Langmuir kinetics model in contrast

to the Butler–Volmer model.

4.15 Consider you are designing a 50-cm

-active-area

hydrogen–oxygen PEM fuel cell system to operate at a

nominal current density of 1.2 A/cm

at 100

◦

C. The current

collector landing–channel width ratio is 1 : 2 on the anode

and cathode sides.

(a) In order to enhance power output, you would like

to increase operating pressure from 1 to 3 atm

on the cathode with 1 atm constant on the an-

ode. However, in doing so, the contact resistance

between the cathode and gas diffusion layer in-

creases by 30 m·cm

. What is the net effect of

changing the anode from 1 to 3 atm pressure on

voltage at 1.2 A/cm

(b) Determine the operating current density at which

the positive effect of the pressure increase is ex-

actly offset by the negative impact from the in-

creased contact resistance.

4.16 Consider a fuel cell with 100 cm

superﬁcial active

area and a measured contact resistance of 20 m · cm

the anode and 10 m · cm

at the cathode. Consider the

Naﬁon 115 (0.005-in.thick) electrolyte as fully saturated

with moisture everywhere. Consider the anode and cathode

electrodes to be 20 and 30 µm thick, respectively, and 25

and 30% equivalent ionomer fraction. Find the fraction of

the total potential power lost at 1.2 A/cm

current density

from ohmic losses if E

◦

=1.25 V. Since the ionic resistance

is much greater than the electronic resistance, you may

neglect the electronic resistance. Assume the area ratio is

1 : 1.3 between the landing and channel.

4.17 In Equation (4.96), it was shown that

OCV

= E

(

T, P

)

Nernst

+ σ

= E

◦

(

T, P

)

× t

Derive this relationship. Recall it is for open-circuit con-

ditions where the ionic and electronic current exchange is

equivalent. That is, i

= i

4.18 Sketch a typical hydrogen PEFC polarization curve

with electrolyte A. Then, sketch a polarization curve with

the same operating conditions, but with a thicker electrolyte

B. Be sure to think about all of the effects of this change.

4.19 Sketch a typical high-temperature fuel cell (SOFC,

MCFC) polarization curve operating at temperature A.

Then, sketch a polarization curve B with reduced temper-

ature conditions. Be sure to think about all of the effects

of this change. (Ignore the effects on mass transport region

until Chapter 5.)

4.20 Sketch the shape of the anode and cathode electrode

polarizations versus current for the hydrogen PEFC on the

same plot as a DMFC.

4.21 Using total kinetics, plot the ORR over potential ver-

sus current density for a PEFC with a PT-C catalyst at 300 K,

1 atm for roughness factors of 1 (smooth electrode), 500,

and 2000.

Computer Problems

4.22 Using a spreadsheet or other computer program, put

together a basic fuel cell model program for a chosen fuel

cell system that includes the various terms and parameters

discussed in this chapter for the activation, ohmic, con-

centration, and crossover polarization losses. For now, use

reasonable parameters for the unknown quantities based

on the examples in this chapter. After Chapters 5–7, more

speciﬁc parameters for a given fuel cell can be added.

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

References 189

(a) Select appropriate anodic and cathodic kinetics

based on the fuel cell chosen and plot the acti-

vation polarization for each electrode as a func-

tion of current density for each electrode. Check

your code by a hand calculation at a given cur-

rent density. The act of checking the computed

values with hand calculations or exact analyti-

cal solutions is called model veriﬁcation and is

a necessary step every time a computer model is

used.

(b) Plot the ohmic polarization as a function of current

density. Check your code by a hand calculation at

a given current density.

of current density for both electrodes. Check your

code by a hand calculation at a given current den-

sity.

(d) Now add crossover to the calculation and show a

plot of the OCV as a function of crossover ﬂux.

Check your code by a hand calculation at a given

current density.

(e) Plot the polarization curve for this fuel cell.

4.23 Using a spreadsheet or other computer program, ﬁnd

a polarization curve for a given fuel cell from the litera-

ture and curve ﬁt the results using the empirical fuel cell

model. Discuss the meaning of the mass transfer, kinetic,

and ohmic parameters. How could this model be expanded

to use for other test conditions? Hint: Your professor can

help you ﬁnd some typical polarization curves to use.

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