CACHE Modules on Energy in the Curriculum
Fuel Cells
Module Title: Tafel Equation and Fuel Cell Kinetic Losses
Module Author: Jason Keith
Author Affiliation: Michigan Technological University
Course: Kinetics and Reaction Engineering
Text Reference: Fogler (4th edition), Section 3.3
Concepts: Given the hydrogen electrode voltage drop as a function of current, determine
the parameters in the Tafel Equation and use them to perform some basic fuel cell
calculations.
Problem Motivation:
Fuel cells are a promising alternative energy conversion technology. One type of fuel
cell, a proton exchange membrane fuel cell (PEMFC) reacts hydrogen with oxygen to
produce electricity (Figure 1). Fundamental to the design of fuel cells is an understanding
of the effect of kinetics on the fuel cell performance.
Consider the schematic of a compressed hydrogen tank (2000 psi, regulated to 10 psi)
feeding a proton exchange membrane fuel cell, as seen in Figure 2 below. We will now
focus on the voltage / current relationship of the fuel cell.
-
-
e
e
H2
H2O
O2
H+
H2
H2O
H2
O2
O2
+
H
H2
Computer
(Electric
Load)
Pressure
regulator
H2 feed line
Air in
H2
H2
H2
H2
H2O
H2O
+
H
H2
H+
O2
Anode
Cathode
Electrolyte
Figure 1. Reactions in the PEMFC
H2 out
H2 tank
Fuel Cell
Air / H2O out
Figure 2. Diagram for fueling a laptop.
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J.M. Keith
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July 30, 2008
Background
Figure 3 shows the relationship between current density i (fuel cell current per unit area
of the electrode, in units of milliamps per square centimeter) and cell voltage Vc (in units
of volts). There are several things to note here:
Open circuit voltage
(@ i = 0) is less than
theoretical value
“No loss” voltage
1.2 V
Overvoltage
Rapid drop
(kinetic losses)
Linear drop
(ohmic losses)
Rapid drop at
higher currents
(mass transfer
losses)
Figure 3. Polarization Plot





The theoretical maximum voltage of this fuel cell is 1.2 V. This is called the
“open circuit voltage” VOCV .
The hydrogen reaction rate is directly proportional to the current, since for each
hydrogen molecule that reacts, two electrons are formed.
Any drop from this maximum value is termed “overvoltage.” It is desired to
minimize the overvoltage so that the fuel cell can operate as efficiently as
possible.
There is a critical current density called the “exchange current density” with
symbol io. For current densities i < io, the cell voltage is equal to the theoretical
value. For current densities i < io, there is a rapid fall in cell voltage, due to a slow
reaction rate constant (kinetics). It is desired to have as high a value of io as
possible, and as rapid kinetics as possible.
At current densities between 100 mA/cm2 and 800 mA/cm2, there is a linear fall
in voltage as the current density increases. This effect is due to the fact that there
is a resistance to current flow within the fuel cell. As the current increases, the
voltage drop will increase. In physics and electrical engineering, this effect is
referred to as Ohm’s law. It is desired to have as small a resistance as possible.
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July 30, 2008

At very high current densities (greater than 800 mA/cm2), the hydrogen reaction
rate is high. However, the hydrogen cannot diffuse to the electrode fast enough to
react. Thus, mass transfer is limiting and the voltage rapidly drops to zero.
In chemical engineering analysis reaction rates are often described by the Arrhenius
relationship k = A exp (-E/RT) where k is the reaction rate, A is a pre-exponential factor,
E is the activation energy, R is the gas constant, and T is the absolute temperature.
Section 3.3 of Fogler describes methods of obtaining A and E.
In fuel cells, the overvoltage V due to kinetics is a function of the current density (which
is directly proportional to the reaction rate) and is often described by the Tafel equation
given by V = A ln (i/io) where A is called the Tafel constant (with typical units of Volts)
and io is called the exchange current density (with typical units of mA/cm2).
In this module we will use the Tafel equation to model and predict overvoltages in proton
exchange fuel cell systems (which operate at temperatures of about 80 oC). Note that in a
previous module you may have studied ohmic losses in solid oxide fuel cells. These fuel
cells operate at higher temperature (between 500 and 1000 oC) and have negligible
kinetic losses.
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Example Problem Statement: The following data has been obtained for the
“overvoltage” for the hydrogen electrode in a proton exchange membrane fuel cell.
Current Density (mA/cm2)
40
43
47
50
54
58
61
64
69
Overvoltage (mV)
72.1
75.1
76.2
78.8
81.8
83.2
84.1
86.7
88.8
At low current densities, the kinetics are commonly modeled by the Tafel equation, given
by:
i 
V  A ln  
 io 
(1)
where the Tafel constant A is higher for an electrochemical reaction that is slow (a slow
reaction leads to a higher overvoltage) and io is the exchange current density.
a) Construct an appropriate plot to prove the Tafel equation can be used
b) Use the given data to determine the values of A and io
c) Use your values of A and io to estimate the cell voltage at a current density of
125 mA/cm2.
d) You have run an experiment at a current density of 125 mA/cm2 and found a
cell voltage of 1.00 V. Why do you think you should run this experiment?
Also, explain the difference between these numbers and your model
prediction.
Example Problem Solution:
Part a)
Step 1) Transform equation 1 to the form:
V  A ln i  A ln io
(2)
Step 2) Plot the overvoltage as a function of the natural log of the current density (a
semilog plot) to show a linear trend, as indicated in Figure 4 below.
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Overvoltage (mV)
100
90
80
70
60
50
40
30
20
10
0
3.6
3.7
3.8
3.9
4
4.1
4.2
4.3
ln (Current Density, mA/cm^2)
Figure 4. Tafel Plot for Overvoltage.
Part b)
Step 1) A best-fit line of the plot in Figure 4 gives the equation:
V  30.0 ln( i )  38.6
(3)
where the current density is in mA/cm2 and the numbers and overvoltage have units of
mV.
Step 2) Solve for A. Setting Equations 2 and 3 equal yields an A value equal to the slope
of Equation 3 and Figure 4, A = 30.0 mV (A ~ 0.03 V).
Step 3) Solve for io. The intercept of Equation 3 is equal to  A ln io , so that
ln io  
 38.6
 1.29 or io  3.6 mA/cm2.
 30.0
Step 4) Put it together to give:
 i 
V  30.0 ln 

 3.6 
(4)
where the overvoltage is in mV and the current density is in mA/cm2.
Part c)
Step 1) Use the Tafel equation to determine the overvoltage:
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 125 
V  30.0 ln 
  106mV
 3.6 
(5)
Step 2) Determine the cell voltage given as:
V  VOCV  V  1.2  0.106  1.09 V
(6)
Part d)
Step 1) Analysis. It is noted that we are extrapolating the current density beyond the
limits measured in the laboratory. This is not a good idea, and it is recommended to
verify the voltage with an additional experimental data point.
Step 2) The cell voltage of 1.00 V is less than that predicted by the Tafel equation. The
difference of ~90 mV may be due to a poor fit of the experimental data or more likely is
due to Ohmic losses within the fuel cell.
Part e) Polymath Solution
Step 1) Launch Polymath on your computer.
Step 2) Click on REG (regress and analyze data)
Step 3) Enter the natural log of the current density in the first column (C01) and the
overvoltage in the second column (C02).
Step 4) Click on the regression tab.
Step 5) Select linear & polynomial, and choose a polynomial of degree 1. Make sure the
graph and report check boxes are checked. Click the right arrow. View the report to
determine the fit and make sure the graph shows an adequate agreement between the data
and the model.
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J.M. Keith
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Polymath solution for example problem.
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Home Problem Statement: You have developed a new catalyst for use in a proton
exchange membrane fuel cell. You have tested the material using two different hydrogen
supplies (99% purity with 0.1% carbon monoxide) and 99.9% purity with 0.001% carbon
monoxide).
Current Density (mA/cm2)
20
33
41
50
59
63
78
Overvoltage with Overvoltage with
99% H2 gas (mV) 99.9% H2 gas
(mV)
253
46.3
298
56.6
332
60.9
350
62.2
359
65.6
366
69.1
391
74.4
At low current densities, the kinetics are commonly modeled by the Tafel equation, given
by:
i 
V  A ln  
 io 
(1)
where the Tafel constant A is higher for an electrochemical reaction that is slow (a slow
reaction leads to a higher overvoltage) and io is the exchange current density.
a) Construct an appropriate plot to prove the Tafel equation can be used for each
data set
b) Use the given data to determine the values of A and io for each data set
c) What do you think is happening to the catalyst?
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