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.
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:
· The theoretical maximum voltage of this fuel cell is 1.2 V. This is called the “open circuit voltage” .
· 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.
· 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 DV 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 DV = 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.
Example Problem Statement: The following data has been obtained for the “overvoltage” for the hydrogen electrode in a proton exchange membrane fuel cell.
40 / 72.1
43 / 75.1
47 / 76.2
50 / 78.8
54 / 81.8
58 / 83.2
61 / 84.1
64 / 86.7
69 / 88.8
At low current densities, the kinetics are commonly modeled by the Tafel equation, given by:
(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:
(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.
Figure 4. Tafel Plot for Overvoltage.
Part b)
Step 1) A best-fit line of the plot in Figure 4 gives the equation:
(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 , so that
or mA/cm2.
Step 4) Put it together to give:
(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:
(5)
Step 2) Determine the cell voltage given as:
(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.
Polymath solution for example problem.
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).
20 / 253 / 46.3
33 / 298 / 56.6
41 / 332 / 60.9
50 / 350 / 62.2
59 / 359 / 65.6
63 / 366 / 69.1
78 / 391 / 74.4
At low current densities, the kinetics are commonly modeled by the Tafel equation, given by:
(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?
1st Draft J.M. Keith July 30, 2008
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