Operation of a methanol fuel cell is illustrated in Figure 9.1. This particular fuel cell uses a proton-exchange membrane separator in which the ionic current is carried by protons. In contrast to batteries, where we were careful to use the terms negative and positive for electrodes rather than anode and cathode, the terms anode and cathode are appropriate and unambiguous when applied to fuel cells. In fact, while either set of terms is acceptable, anode and cathode are more commonly used. At the anode (negative electrode), methanol is oxidized to produce carbon dioxide and protons. The protons move toward the cathode (positive electrode) through the electrolyte. At the cathode, oxygen is reduced to form water. Electrons move through an external circuit from the anode to the cathode:

The overall reaction is the reaction of methanol and oxygen to form water and carbon dioxide, which is the same reaction that would have resulted from burning the methanol. The increased efficiency of the fuel cell comes from using the fuel directly to produce electrical work rather than combusting the fuel in a thermal process, such as is done in an internal combustion engine.

Methanol is just one of a number of possible fuels, the most common of which is hydrogen gas. The fuel cell in Figure 9.1 is known as a direct methanol fuel cell because the fuel is oxidized at the anode without first being converted to hydrogen by a chemical process called reformation. Figure 9.1 also identifies a number of basic components used in fuel cells. These components and their functions will be discussed in this and the next chapter. In contrast to a combustion process that only releases heat, the conversion of methanol to carbon dioxide and water occurs in the fuel cell through two separate electrochemical reactions. As a consequence, fuel cells are not thermal devices and are not limited by the Carnot efficiency of heat engines.

Thermodynamics (Chapter 2) informs us about the equilibrium potential of the half-cell reactions and overall cell, as well as the enthalpy of the overall reaction. The equilibrium potential, U, is related to the change in Gibbs energy according to Equation 2.3. For the methanol fuel cell at standard conditions,

A second important thermodynamic quantity is the change in enthalpy associated with the reaction. For the methanol reaction, this change in enthalpy at standard conditions is calculated with the data from Appendix C:

In calculating both ΔG° and ΔH°, liquid water and liquid methanol were assumed. Since the overall reaction is the same as the combustion of methanol, we expect the reaction to be highly exothermic with a change in enthalpy that is a large negative value. Let’s use the first law of thermodynamics to examine the fuel cell, as depicted in Figure 9.2. For an open system, the change in enthalpy is equal to the heat transfer to the system, Q, minus the electrical work done by the system:

(9.1)

Note that the signs for Q and W depend on an arbitrary convention; here, positive indicates heat transfer to the system and work done by the system. ΔG, and therefore U, represent the maximum electrical work that can be obtained from the cell corresponding to a reversible process occurring at an infinitesimally slow rate. The electrochemical engineer is interested in designing a practical device where rate, size, and cost are important considerations. Much of this chapter will focus on how the rate (think of current density) affects the useful electrical work that can be extracted. Chapter 10 will expand this discussion and apply the principles to more complete systems. Mass and energy balances will illustrate many key ideas that will be addressed in more detail in the subsequent sections.

The efficiency of the fuel cell can be expressed as the electrical work produced divided by the energy available in the reactants. In contrast to the typical battery, fuel cells are not cycled but operate indefinitely with the continuous flow of fuel and oxidant. Thus, our analysis will involve rates of energy (power). Assuming that we provide  [kg·s⁻¹] of reactants, the voltage efficiency of a fuel cell is

(9.2)

We have used both Equation 2.3 and Faraday’s law to eliminate the current. The net electrical work done by the cell is simply the cell current multiplied by the operating potential of the cell. The efficiency we have defined, , is a type of voltage efficiency. Here, we have presumed that the reactants are supplied in exactly stoichiometric quantities. Clearly, we’ll want the cell voltage to be as high as possible in order to maximize efficiency. Thus, minimization of ohmic, kinetic, and mass-transfer losses in the cell is desired, just as was with batteries. Compare this definition with the voltage efficiency used for batteries. Why are they different?

ILLUSTRATION 9.1

A hydrogen oxygen fuel cell is operating at 0.7 V. What is the fuel-cell voltage efficiency? You may assume standard conditions.

SOLUTION:

Assuming that the hydrogen and oxygen are supplied in exactly stoichiometric amounts and at standard conditions at 25 °C, the change in Gibbs energy is given by Equation 2.3. For the H₂/O₂ fuel cell this value is 1.229 V. The efficiency of this specific fuel cell is therefore

or 57%. In general, efficiency is a function of the specific operating conditions, which differ from the standard state.

We’ll see in Chapter 10 that the analysis of efficiency done here, while appropriate from a fundamental perspective, is not adequate for the characterization of fuel-cell systems. For example, it does not account for the fraction of the current or power from the fuel cell that goes to operating ancillary equipment, such as a blower to supply air; this power is subtracted from the gross power, IV, to get the net electrical power produced. Also, by convention, the change in enthalpy (ΔH) is often used as the benchmark rather than the change in Gibbs energy. This convention is rooted in history more than science. The analysis of efficiency for fuel cells is explored further in Chapter 10. In this chapter we next describe several types of fuel cells. We will then look in detail at the polarization curve from a fuel cell.