Mench M.M. Fuel Cell Engines

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130 Performance Characterization of Fuel Cell Systems

Total exothermic release

available for work

(electrical or chemical)

Extent of reaction

Initial “Push”

Metastable State I

Metastable State II

“Activated complex”

Figure 4.8 Schematic of reaction coordinate pathway for given reaction.

the initial state which corresponds to nonreacted fuel and oxidizer, the reactants are at a

metastable equilibrium with a higher energy than the product state. Under the driving force

of a voltage over potential at the electrode (v/a), the reactant is moved along the reaction

coordinate toward an activated complex state, where the reactant is partially converted to

product. This state corresponds to the elementary charge transfer reaction step discussed

in Chapter 2. The intermediate species at this state is highly reactive and cannot remain

in a stable condition. This transition-state-activated complex separates the reactant from

the product states along the reaction coordinate. As the reaction moves to completion, the

activated complex moves to a product state, and a net chemical energy conversion occurs

as the products reach a ﬁnal metastable equilibrium. The difference between the initial

reactant and ﬁnal product chemical energy is the enthalpy of reaction, which is converted

to electrical work and heat in an electrochemical reaction.

Now, consider an individual electrode, initially at equilibrium. For example, consider

the global HOR:

Reduced

↔2H

+2e

−

←−−−−−−−−→ Oxidized (4.7)

At equilibrium (open circuit), this reaction actually proceeds in both directions across

the anode double layer, with no net reaction in either direction. Some hydrogen is being

oxidized, and an equivalent amount is being reduced. This is analogous to a frictionless

pendulum, shown in Figure 4.9. At equilibrium, the pendulum swings back and forth with

Figure 4.9 Mechanical pendulum analogy to describe electrode reaction and equilibrium.

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4.2 Region I: Activation Polarization 131

Figure 4.10 Schematic of pendulum analogy of small net anodic current.

no net motion in either direction. The energy in the system is conserved and oscillates

between potential energy and kinetic energy, just as the species at the electrode will

oscillate between oxidized and reduced hydrogen, with no net change in the reactants.

This equilibrium current exchange is termed the exchange current density i

, and is an

important parameter discussed in the following sections.

Next, consider moving the electrode out of equilibrium and into a condition of net

hydrogen oxidation and current generation. This reaction produces a net ﬂow of electrons

which are transported to the external circuit. For a very low current, a small overpotential is

required to drive the net reaction in the anodic direction. As the level of current is increased

beyond some equilibrium exchange value, the overpotential required to sustain the reaction

rate is greatly increased. Consider again the frictionless pendulum analogy; in order to

leave equilibrium and move to a net anodic reaction, if the pendulum were reﬂected at

the midpoint in the oscillation by a perfectly elastic barrier, as shown in Figure 4.10, the

pendulum would rise back to the initial height in the anodic direction to conserve the system

energy. The net result is a small anodic current.

In order to push the pendulum beyond the level allowed by the initial system oscillation

energy, additional external energy must be input to the pendulum, as illustrated in Figure

4.11. In this analogy, the additional work input is converted to a higher height of the

pendulum in the anodic direction and a greater net reaction (i.e., more current).

Figure 4.11 Schematic of pendulum analogy of larger net anodic current and energy input.

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132 Performance Characterization of Fuel Cell Systems

Figure 4.12 Schematic of activation overpotential with respect to current.

With the pendulum analogy in mind, we can now examine the typical electrode activa-

tion polarization behavior, as shown in Figure 4.12. At low current density, the activation

overpotential η

activation

required to maintain a net reaction rate in a given direction is small.

Beyond a threshold value in current density related to the equilibrium reaction exchange

rate of the electrode, the additional polarization requiredfor increasing current is greatly

increased. The exchange current density i

is of preeminent importance in the activation

over potential for a given electrochemical reaction rate, as it is a measure of the effec-

tiveness of the electrode in promoting the electrochemical reaction and is the electrode

reaction exchange at equilibrium. At zero net cell current density, the electrode current is

the exchange current density. The higher the exchange current density at a given electrode,

the lower the overall activation polarization losses for a given current.

4.2.1 Butler–Volmer Model of Kinetics

In this section, we derive a general expression to describe activation polarization losses at a

given electrode, known as the Butler–Volmer (BV) kinetic model. The BV model is not the

only (or necessarily the most appropriate) model to describe a particular electrochemical

reaction process. Nevertheless, it is a classical treatment of electrode kinetics that is widely

applied to study and model a majority of the electrode kinetics of fuel cells. The BV

model describes an electrochemical process limited by the charge transfer of electrons,

which is appropriate for the ORR, and in most cases the HOR with pure hydrogen. The

fundamental assumption of the BV kinetic model is that the reaction is rate limited by a

single electron transfer step, which may not actually be true. Some reactions may have

two or more intermediate charge transfer reactions that compete in parallel or another

intermediate step such as reactant adsorption (Tafel reaction from Chapter 2) may limit

the overall reaction rate. Nevertheless, the BV model of an electrochemical reaction is

standard fare for a student of electrochemistry and can be used to reasonably ﬁt most fuel

cell reaction behavior.

Instructors may wish to skip to Eq. (4.35) and assign the derivation of the Butler–Volmer kinetics to advanced

undergraduate or graduate students.

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4.2 Region I: Activation Polarization 133

In a fuel cell, at each electrode, there is an equilibrium reaction that can be written as

+ ne

−



→= Reduction (cathodic)

←= Oxidation (anodic)

(4.8)

where

Ĺ The stoichiometric coefﬁcients of the rate-limiting elementary charge transfer reac-

tion at a given electrode are A and B. The elementary reaction should be distinguished

from the global reaction occurring at a given electrode, as discussed in Chapter 2.

The elementary charge transfer reaction is the intermediate reaction responsible for

the charge transfer. Although the BV model assumes a single charge transfer reaction

step, there can be several charge transfer steps occurring in parallel. The BV model

can still accommodate this, as will be discussed.

Ĺ At each electrode in the fuel cell at open circuit condition, an equilibrium as in

Eq. (4.8) is occurring with no net current through the circuit (recall the equilibrium

pendulum of Figure 4.9). That is, both the anode and the cathode have completely

separate equilibrium reactions occurring, linked only by the net charge transfer

through the circuit. At open circuit, there is no net current ﬂowing through the

electrodes, and the anode and cathode reactions are independent.

For a given purely chemical reaction, we can change the temperature and pressure to affect

the reaction rate. For an electrochemical reaction, there is an additional factor: the electrode

overpotential across the double layer. The overpotential at the electrode surface controls the

direction and rate of the net reaction. When net current is drawn, an overpotential at each

electrode forces the electrode reactions out of the equilibrium condition and toward the

desired direction. At the anode, the electrode potential becomes higher than its equilibrium

potential (see Figure 4.5), resulting in a net oxidation reaction. At the cathode, the electrode

potential becomes lower than its equilibrium potential, resulting in a net reduction reaction.

Figure 4.13 shows this on a reaction coordinate. At the initial surface electrode potential

, the forward reaction is not favored. At φ

, the reaction is now favored because the

ﬁnal energy state is below the initial energy state. For electrochemical reaction circuits,

spontaneous galvanic (exothermic) reactions can be reversed simply by applying an external

potential to change the polarity of the electrodes. This is the principle of the reversible fuel

cell discussed in Chapter 1.

Consider an electrode at state φ

in Figure 4.14. To induce this electrode to have a net

spontaneous reduction reaction, we must go from φ

to φ

. Although the potential energy

of the electrode is increased to promote this reaction and support charge transfer across

the double layer of the electrode, the actual surface overpotential relative to the SHE will

decrease (see Figure 4.5), since we are moving toward a cathodic reduction reaction product

in this example.

Symmetry Factor We have added nF(φ

– φ

) in electrical potential to the system. Only

a fraction of this energy will go toward reducing the activation energy of the cathodic

(reduction) reaction at the electrode. In a general case, the fraction of the additional energy

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134 Performance Characterization of Fuel Cell Systems

Energy

Reactants Products

Extent of Reaction

is endothermic for the forward reaction

is exothermic for the forward reaction

is the surface potential (V) of the electrode

Figure 4.13 Schematic of activation overpotential with current at given electrode.

imparted to the electrode that promotes the reduction reaction is called β, the symmetry

factor:

0 <β<1 (4.9)

If β = 1 (see Figure 4.15), the additional overpotential at the electrode goes completely

toward promoting the reduction reaction. If β = 0 (see Figure 4.16), all of the additional

potential is applied toward promotion of the anodic oxidation reaction. In the particular

case shown in Figure 4.15, all of the additional voltage potential increases the oxidized

state energy, and none is applied to lower the reduced state energy. In practice, β varies for

a given electrode and reaction and lies between 0 and 1 but is not necessarily 0.5, since

certain catalysts promote oxidation of a given species more readily than others.

The activation energy (polarization) required to promote charge transfer across the

double layer for the reduction reaction at an electrode can be shown as

a2,c

= E

a1,c

+ βnF(φ

− φ

) (4.10)

For a reduction reaction, the sign of φ

is more negative than φ

(Figure 4.5), so φ

− φ

is negative.

For the oxidation reaction on the electrode, 1 − β is the anodic symmetry factor, the

fraction of the polarization energy which promotes the oxidation reaction at the electrode,

and the activation energy (polarization) required to promote charge transfer across the

Energy

Oxidized state Reduced state

Extent of reaction

Transition state

Figure 4.14 Activation overpotential increase to initiate reaction at given electrode.

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4.2 Region I: Activation Polarization 135

Energy

Extent of Reaction

Oxidized State Reduced State

nF φ

−

()

1, c

nF φ

−

()

2, c

= cathodic reaction activation energy

Figure 4.15 Symmetry factor β = 1.

double layer for the oxidation reaction at an anode can be shown as

a2,a

= E

a1,a

− (1 − β)nF(φ

− φ

) (4.11)

So when we decrease the electrode potential, the activation energy for reduction goes down

by βn

F(φ

−φ

), and the oxidation activation energy is increased by

(1 − β)nF(φ

− φ

) (4.12)

Returning to the elementary charge transfer reaction,

+ ne

−



BR (4.13)

For each reaction, k

and k

, a commonly used model for the rate equation is an Arrhenius

expression:

k = k

ref

exp



−E



(4.14)

where the reference reaction rate (k

ref

) and the reaction activation energy (E

) are different

for k

and k

, the forward and reverse reactions at the electrode, respectively.

Energy

Extent of Reaction

Oxidized State Reduced State

nF φ

−

()

No help in promoting

reduction reaction

1, c

2,c

Figure 4.16 Symmetry factor β = 0.

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136 Performance Characterization of Fuel Cell Systems

A rate equation for the electrochemical reaction can be written considering Faraday’s

law and the reaction rate constant:

|r|=

mol

s · cm

= k



[C]

exp



−E



(4.15)

where γ is the reaction order of the elementary electron transfer step, r is the rate of

consumption/formation of species, and C is the concentration of species in reaction of

interest in units of moles per cubic centimeter. For an ideal gas,

C =

(4.16)

Taking the anodic (oxidation) reaction branch of Eq. (4.13),

−−−−−−−→ AO

+ ne

−

(4.17)

Then the oxidation rate of the elementary charge transfer reaction can be written as

mol

s · cm

= k



]

exp



−E

a,a

−

(

1 − β

)

(

φ − φ

ref

)



(4.18)

Let the reference potential for the anode and cathode electrodes (ø

ref

) be 0 V (SHE potential)

= k

]

exp



(

1 − β

)

nF(φ)



(4.19)

where we have absorbed E

a,a

, the oxidation reaction reference activation energy, into the



term to become k

, the anode reaction rate constant.

For the reduction reaction rate at the cathode

= k

]

exp



−βnF(φ)



(4.20)

Some derivations also absorb the electron transfer number n into β as

nβ = α

(4.21)

where α

is known as the cathodic charge transfer coefﬁcient, and

n(1 − β) = α

(4.22)

where α

is known as the anodic charge transfer coefﬁcient.

It is important to note that n is the number of electrons transferred in the elementary

charge transfer step. This is very different from the global reaction n we deﬁned for

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4.2 Region I: Activation Polarization 137

Faraday’s law and in the Nernst equation. Typically, n here is less than 2. Based on the

single charge transfer reaction step used in Eq. (4.8), the value of n should be an integer.

However, experimentally this value is often determined to be a noninteger, which can be

true if more than one reaction is acting as a limiting charge transfer step in parallel. In this

case, the BV model can still be applied to model an electrode’s reaction kinetics, but the

value of n determined experimentally will be a noninteger value, a result of the cumulative

effect of the multiple charge transfer reaction steps.

Charge Transfer Coefﬁcient From the deﬁnition of the cathode charge transfer coef-

ﬁcient α

, it physically represents the fraction of additional energy that goes toward the

cathodic reduction reaction at an electrode. It can also be thought of as a symmetry coefﬁ-

cient of the electrode reaction. In terms of the pendulum analogy shown in Figure 4.9, the

pendulum does not have to be balanced. If the pendulum is tilted sideways with respect to

the ground reference, additional work input to the system will asymmetrically add potential

energy to the maximum height on each swing. Many reactions tend toward symmetry, so

with no information it is usually reasonable to assume a value of 0.5 = α

. The effect of

the charge transfer coefﬁcient on the electrode polarization symmetry is shown in Figure

4.17. The negative values of overpotential correspond to the cathode polarization, and the

positive values of polarization correspond to the anode reaction. The vertical axis corre-

sponds to the current density. For a charge transfer coefﬁcient of n × 0.5, polarization

affects the anodic and the cathodic reaction at a given electrode equally, and the curve is

symmetric around the axis. For a cathodic charge transfer coefﬁcient that is n × 0.75, the

polarization required for the cathodic reduction reaction for a given current is much less

than the polarization required for the anodic oxidation. Obviously, a cathode with α

> n ×

0.5 would be preferred for the cathode. However, as an engineering design parameter, there

is little we can do to alter this value in practice for a given electrode, and other parameters

we can affect through engineering, such as the exchange current density i

, have a much

greater impact.

= 0.75n

= 0.50n

= 0.25n

= 0.75n

= 0.50n

= 0.25n

Positive

Negative

Figure 4.17 Effect of charge transfer coefﬁcient on symmetry of current–overpotential curves.

(Reproduced from [1].)

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138 Performance Characterization of Fuel Cell Systems

Note that we are still not considering the entire fuel cell yet! We are discussing

a single electrode with simultaneously occurring oxidation and reduction reactions. For

nonequilibrium operating conditions, we desire a net current density. For a net oxidation

current density at the anode we consider the net oxidation reaction rate:

net,anodic

= r

−r

−

= k

]

exp



F(φ)



− k

]

exp



−α

F(φ)



(4.23)

At open circuit, the fuel cell net current density is zero, but the rates of exchange r

and r

are not zero but equal. We can solve for this resting exchange current density i

= r

= k

]

exp



Fφ

◦



= k

]

exp



−

Fφ

◦



(4.24)

where ø

◦

is the equilibrium potential (OCV) at i = 0. If we rearrange Eq. (4.24) and take

the natural log of both sides,

◦

−





or E(OCV) = E

(

)

−



νr

νo



(4.25)

which is the Nernst equation! This makes sense, since any kinetic theory must reduce to

the thermodynamic theory at equilibrium. So at i

net

= 0, we are at equilibrium, and the

expected maximum voltage is determined from the Nernst equation, as we have already

shown in Chapter 3.

Now, consider the overpotential at an electrode, η, which represents a departure from

this equilibrium potential:

η = φ − φ

(4.26)

Plug this into Eq. (4.23), and we can show that

net

= k

]

exp



F(φ)



− k

]

exp



−α

F(φ)



(4.27)

and the net current density in the oxidation (anodic) direction, i

net

, at a single electrode can

be shown as

net

= nFk

exp





η +



−nFk

exp



−



η +



(4.28)

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4.2 Region I: Activation Polarization 139

Simplifying yields

net

= nFk

(1−β)

(−β)

(1−β) P

β P



exp

η − exp

−α



(4.29)

At open-circuit conditions, i

net

= 0, but each component of anode/cathode current must

equal the exchange current density i

. So the ﬁrst term in brackets at OCV (η = 0) is the

oxidation branch at the electrode, and the second term is the reduction branch. At η = 0,

each branch must be at the exchange current density, so that i

– i

= i

net

= 0. So we see

that the term outside the brackets in Eq. (4.29) is really the exchange current density.

We can now rewrite our standard BV model of kinetics for an individual electrode:

net

= i



exp

η − exp

−α



(4.30)

= i

o,ref



∗



= i



∗



(4.31)

where α

and α

refer to the anodic and cathodic charge transfer coefﬁcients at the electrode,

respectively. This is to be applied at each electrode. For example, for the anode

cell

= i

net,anode

= i

o,a



exp

a,a

− exp

−α

c,a



(4.32)

For the cathode

cell

= i

net,cathode

= i

o,c



exp

a,c

− exp

−α

c,c



(4.33)

By conservation of charge, i

= i

cell

, so that

o,a



,surface

∗





exp

a,a

− exp

−α

c,a



= i

o,c



,surface

∗





exp

a,c

− exp

−α

c,c



(4.34)

In words, this means that the net current at the cathode and anode is conserved, providing

the linkage between the two electrodes. The overpotential at each electrode will adjust to