A key objective in this chapter is to gain a detailed understanding of what is known as the polarization curve. This curve represents the steady-state relationship between the potential of the cell, V_cell, and its current density, i. It is typically measured experimentally. During this measurement, the temperature and pressure are held constant, and the flow of reactants is either fixed or proportional to the current density. Figure 9.3 shows example curves for three types of fuel cells: AFC, PEMFC, and SOFC. As we expect for a galvanic cell, the potential decreases as the current density increases. The current–voltage characteristics of a fuel cell can be understood with use of the principles that we have already learned, namely, thermodynamics, kinetics, ohmic losses, and mass transfer. It turns out that, to a good approximation, each of these determines the behavior of a part or section of the polarization curve as follows: (i) open-circuit potential (thermodynamics), (ii) low-current behavior (kinetics), (iii) behavior at moderate currents (ohmic losses), and (iv) the behavior at high currents (mass transfer).

Thermodynamics

Looking at Figure 9.3, we see that the potentials at open circuit (zero current density) vary. Also, note that none of the curves reaches 1.229 V (the equilibrium value found in Appendix A) at zero current. There are several factors that can explain these data. First, let’s consider the impact of temperature, which is different than 25 °C for all three cases. The influence of the temperature on the equilibrium potential can be estimated by integrating Equation 2.16b. Assuming that the change in enthalpy does not vary with temperature, the resulting expression is

(9.3)

where the temperature ratios must be in absolute units. Figure 9.4 shows the equilibrium potential, U⁰, as a function of temperature as given by Equation 9.3. Also included are the results of a more accurate calculation that does not assume a constant enthalpy change and is needed for increased fidelity at high temperatures. This graph was created assuming that the product water is formed as a vapor rather than a liquid, so the standard potential at 25 °C is 1.184 instead of 1.229 V. The equilibrium potential is 1.171 V at 80 °C, and drops to 0.9794 V at 800 °C.

Let’s compare the results of these calculations with the open-circuit potentials observed in Figure 9.3 for each type of fuel cell. Starting with the SOFC data, we see that the potential at zero current is actually higher than the value calculated at 800 °C. How can these data be explained? In addition to the influence of temperature on the equilibrium potential, the effect of gas composition must be considered. Using the methodology of Chapter 2, we arrive at the following equation for a hydrogen–oxygen fuel cell, where the activity has been retained:

(9.4)

Note that the activity correction term is also a function of temperature. The equilibrium potential, U⁰, is at the operating temperature. The nature of the electrolyte does not affect this equation, which applies to all three types of fuel cells. At 100 kPa, the activity of each gas species can be taken simply as its mole fraction, and the equilibrium potential can be calculated. For the SOFC data, the anode gas stream is 94% hydrogen and 6% water. The cathode is pure oxygen. Thus, at 800 °C

which is close to the value found in Figure 9.3. It is typical that for many high-temperature fuel cells, the open-circuit potential is close to the thermodynamic value.

For the alkaline system operating with pure oxygen, the mole fractions of water, hydrogen, and oxygen are all about 0.5. However, the pressure is above standard conditions (p = 100 kPa), and the activity for an ideal gas is . Thus, at 80 °C and 414 kPa,

In this case, the OCV is a bit less than the thermodynamic value. The reduced value can be explained by permeation of oxygen and hydrogen across the separator (see Problem 9.16). Finally, we examine the PEM fuel cell. Here, the cell is operating on air. Both the air and fuel are humidified, containing 47 mol% water vapor. Thus, the mole fractions are roughly y_H2O = 0.47, y_H2 = 0.53, and y_O2 = 0.21 × 0.53 = 0.11.

The data in Figure 9.3 for the PEMFC are well below this value. There are two important reasons for the observed difference in potential. As in the AFC example, there is some permeation of reactants across the electrolyte, which results in a mixed-potential and depression of the OCV. A second, more important, reason is the sluggishness of the oxygen reduction reaction in acid at low temperatures. The reaction is so slow that even minute impurities and contaminants can compete with the oxygen reduction reaction. Therefore, in low-temperature acid systems, the thermodynamic potential is difficult to achieve experimentally. To summarize, the open-circuit potential (zero current) is predicted theoretically by thermodynamics. However, quite often these thermodynamic values are not observed because of finite permeation of reactants through the separator, impurities, and side reactions. In particular, the reaction for oxygen in acid media at low temperatures is highly irreversible, and the thermodynamic potential is hard to achieve in practice. Finally, we note that the equilibrium potential is about 1 V for a fuel cell, independent of the chemistry. This uniformity is in contrast to batteries, where large variations in the equilibrium potential are observed for different chemistries.

Kinetics

As we know from Chapter 3, the kinetics of electrochemical reactions depends strongly on overpotential, catalysts, and temperature. We will focus initially on the low-temperature fuel cells: PEM and AFC. Referring to Figure 9.3, drawing a small current from these cells causes the cell potential to decrease rapidly, followed by a more gradual decline. An alternative way to examine these same data is with a Tafel plot as shown in Figure 9.5. There are two main differences between the polarization curve and the Tafel plot. First, the cell potential is plotted as a function of the logarithm of the current density. Second, the ohmic resistance is removed as discussed in Chapter 6. Figure 9.5 shows the original data for the PEMFC along with the iR-corrected data. At high current densities, mass-transfer effects are present; but at lower current densities, the iR-corrected curve clearly shows the kinetics and the corresponding surface overpotential. Generally, the kinetics of low-temperature fuel cells is described well by the Butler–Volmer equation. Recall that the oxygen reduction reaction (ORR) is highly irreversible, whereas the hydrogen oxidation is fast. Consistent with this, the kinetic polarization at the hydrogen anode is negligible, and we only need to consider the polarization at the oxygen electrode. With the slow ORR kinetics, a large overpotential is needed to achieve any appreciable current; therefore, the reaction operates in the Tafel regime and polarization losses vary linearly with the logarithm of current density. As a result, we can readily obtain basic kinetic information for the oxygen reduction reaction from the data as plotted in Figure 9.5. The slope of the curve can be measured and is described as the Tafel slope in V per decade (see Section 3.5 and Illustration 9.2). Often it is assumed that for ORR α_c ≈ 1, and the value of the Tafel slope is estimated as follows:

(9.5)

By comparison, the data in Figure 9.5 indicate a value of 0.06 V/decade. Referring back to Figure 9.3, when plotted on a linear scale, the polarization curve is not straight. At low current densities, ohmic and mass-transfer polarizations are small and the cell potential is dominated by the reaction kinetics. Ohmic losses, which scale linearly with the current density, increase in relative importance as the current increases. Thus, strong curvature at low current densities is an indication of sluggish reaction kinetics.

The Tafel slope is a key descriptor of the kinetics of oxygen reduction in low-temperature fuel cells.

The units are V per decade.

Next, we can examine the data for AFC (Figure 9.3). Here the story is much the same, but note that the bowed portion at low current density (kinetic control) is smaller. Tafel kinetics still applies; but, compared to the PEMFC, the kinetics for oxygen reduction are much faster in alkaline media.

In the case of SOFC, the temperature is so high that the overpotential for the reaction is much less important than for low-temperature fuel cells. There is only a slightly noticeable curvature in the polarization curve at low current densities. Use of the Butler–Volmer equation for solid oxide fuel cells is explored in Problem 9.20.

ILLUSTRATION 9.2

If oxygen reduction kinetics are described by a Tafel equation that is first-order in oxygen pressure:

estimate the change in potential for a one-decade change in current density assuming that the fuel cell is operating in the kinetic region at 25°C. Write the Tafel equation for i₁ and i₂, which are rearranged to give

If α_c = 1, then for a change of 10 in the current density, the potential change is

This is equivalent to the Tafel slope calculation from Chapter 3. The slope is negative since it is a cathodic reaction and a lower voltage gives a higher current. In practice, only the magnitude of the Tafel slope is typically reported since the sign is understood from the reaction.

Ohmic Region

At moderate current densities, the importance of ohmic polarization increases compared to the activation polarization. Ohmic losses increase linearly with current, whereas kinetic losses are proportional to the logarithm of current density. The absolute magnitude of the kinetic polarization is still large for the PEMFC, but it increases much more slowly than the ohmic polarization at moderate current densities, resulting in a linear decrease in cell voltage with increasing current density. This linear region of the polarization curve is the ohmic region. Referring to Figure 9.3 again, we see that above a few thousand A·m⁻², the polarization curves for all three cells are roughly linear. However, the slopes of the curves are not the same. This slope, which we will refer to as the ohmic resistance, is largely dependent on the conductivity and thickness of the electrolyte.

(9.6)

Thus, either decreasing the thickness of the separator or increasing its conductivity will reduce the slope of the polarization curve and lower ohmic losses. Using the example polarization curves from Figure 9.3, the resistance of this SOFC (0.04 Ω·m²) is a little more than three times higher than the resistance of the PEMFC.

Mass Transfer

Finally, as the current density increases further, mass-transfer effects become important. Reactants and products are transported to and from the catalyst sites due to molecular diffusion and bulk fluid motion. For instance, consider oxygen at the cathode. Assuming that the source of oxygen is atmospheric air, the initial concentration of oxygen is only 21%, with the balance mostly N₂. Depending on the type of fuel cell, water vapor or other gases can be present. For instance, gases are saturated with water vapor in a typical PEM fuel cell, which at 80 °C corresponds to almost 50% water vapor. As oxygen is consumed at the electrode surface, a concentration gradient develops between the bulk stream and the surface. In the extreme, the concentration of oxygen approaches zero at the surface and the limiting current is reached. Of the three examples shown in Figure 9.3, the PEMFC shows the most obvious effect of mass transfer near 25 kA·m⁻², where the potential of the cell drops rapidly with current density. The SOFC appears to have small additional polarization due to mass transfer near 18 kA·m⁻², but this is largely masked by the large ohmic polarization. Unfortunately, the data for the AFC do not extend much beyond 10 kA·m⁻². However, the use of pure oxygen in the AFC will reduce the mass-transfer resistance relative to that of the PEMFC, which uses air.

Recall that the efficiency of a fuel cell is proportional to the potential of the cell as shown in Equation 9.1. Thus, the fuel cell is more efficient at low power or low current densities. This behavior is a key feature of fuel cells. In a well-designed system, operation in or near the mass-transfer region is avoided.

The polarization of SOFC and PEMFC are summarized pictorially in Figure 9.6. For the data in Figure 9.3, the operating potentials of the two cells are nearly the same, ∼0.7 V, at a current density of 8 kA·m⁻². The thermodynamic potentials are close too. However, the individual contributions to the overall polarization are quite different. It is important that the electrochemical engineer know the dominant source of polarization in order to determine the types of changes in cell design that are likely to improve performance. For instance, looking at Figure 9.6 and holding the operating potential at 0.7 V, we would conclude that the current density of the SOFC could be improved dramatically with a thinner electrolyte. In contrast, most of the polarization for the PEMFC is at the cathode, and a thinner electrolyte would cause minimal improvement in current density at 0.7 V.