2.2.1 Thermodynamics
Fuel cell is more efficient than heat engines because it converts chemical energy to electrical energy directly. Carnot cycle shows the possible maximum efficiency of a heat engine, which can be expressed as:
(2-4) where TH is the maximum temperature and TL is the rejection temperature of the heat engine. For a typical heat engine that operates at 675K and rejects heat at 325k, the Carnot efficiency limit is 52%.
The ideal efficiency of a fuel cell is given by the ratio of the thermodynamically extractable energy of the reaction to the total heat energy that would have been released by the reaction in combustion:
(2-5) At standard conditions, ΔH0 for the hydrogen/oxygen reaction is -285.83KJ/mole and ΔG0 is -237.14KJ/mole, giving a theoretical efficiency limit of 83%.
Although ΔG0 represents the energy potential that can be extracted by a fuel cell at standard condition, this energy is expressed by the fuel cell as an electrical potential, or voltage (E ):
(2-6) At standard conditions, this equation gives 1.23V, which is referred as the fuel cell open circuit voltage (OCV).
ΔG is a function of temperature and pressure, therefore, the fuel cell open circuit voltage changes depending on the operating conditions.
These dependencies are described by the Nernst equation:
(2-7) The ideal fuel cell voltage decreases with increasing temperature, but usually the opposite trend is true for the actual fuel cell efficiency due to accelerated electrochemical reaction kinetics at higher temperatures.
2.2.2 Kinetics
An ideal fuel cell would supply an infinite current while maintaining the constant voltage determined by the Nernst equation. But in fact, the actual voltage output of a real fuel cell is less than the ideal thermodynamically predicted voltage. Furthermore, the more current is drawn from a real fuel cell, the more the voltage output of the cell declines, limiting the total power that the fuel cell can deliver. The performance of a real fuel cell device can be summarized with a graph of its current versus voltage characteristics. Such graph, called an I-V curve, shows the real voltage output of the fuel cell for a given current output. An example of a typical I-V curve for a PEMFC is shown in Fig.
2.1.
The voltage output of a real fuel cell is less than the
thermodynamically predicted voltage output due to irreversible kinetic losses. The more current is drawn from the cell, the greater these losses.
There are three major types of fuel cell losses, which give a fuel cell I-V curve to its characteristic shape as shown in Fig 2.2:
1. Activation losses 2. Ohmic losses
3. Concentration losses
An equation for the true fuel cell I-V behavior can thus be written by starting with the thermodynamically predicted voltage output of the fuel cell and then subtracting off the various loss terms:
(2-8) These three categories of irreversibility are considered one by one in the following sections.
2.2.2.1 Activation Losses
At low current density, the voltage of a fuel cell drops rapidly because of the sluggishness of the electrochemical half reactions occurring at the anode and the cathode electrodes. The oxygen reduction reaction at the cathode is especially sluggish and can account for most of the activation losses. Although the final state of product water is lower in energy than that in the initial reactants, an energy barrier impedes the conversion of reactants into products. A portion of the fuel cell voltage is sacrificed to lower this barrier and thus increases the rate at which reactants are converted into products, allowing the fuel cell to output more current. The relationship between the applied activation overvoltage and the current density output is exponential in nature, and can be described by the Butler-Volmer equation:
(2-9) When the overvoltage is greater than 50mV, the Butler-Volmer equation can be approximated by a much simpler form, called the Tafel equation:
(2-10) Activation losses are minimized by maximizing the exchange current density. The exchange current density is a function of the catalyst material and the total reaction surface area. As mentioned earlier, the electrodes are highly porous to maximize the total reaction surface area. Highly dispersing, nano-scale particles of platinum are mixed into the porous electrode so that they are in intimate contact with the gas phase pores, the electrically conductive electrode, and the ion conductive electrolyte. This maximizes the amount of triple phase boundary, thus maximizing the exchange current density. Platinum is currently the best known catalyst for PEMFC.
2.2.2.2 Ohmic Losses
Ohmic losses arise due to the internal resistance of the materials in the fuel cell to the flow of electrons and protons. These losses are called
“Ohmic losses” because they generally follow Ohms law, V=IR. Both the electrically conductive electrodes and the ion conductive electrolyte contribute to the resistance losses. Usually, ionic resistance is dominant in a well-designed fuel cell. The linear drop in the middle of the I-V curve in Fig. 2.3 distinctly manifests the Ohmic loss effects. In fuel cell systems, most of the Ohmic loss arises from the electrolyte.
2.2.2.3 Concentration Losses
At high current density, the voltage output of fuel cell once again drops rapidly and declines quickly to zero. The current density output at zero voltage is known as the short-circuit current, which represents the maximum current that can be produced by the fuel cell. However, at this current level, the voltage output of fuel cell is zero, so the total power delivered by the fuel cell is also zero. Therefore, power peak occurs somewhere in the middle of the I-V curve. The reason for the final steep decline in fuel cell voltage at high current density is due to mass transport limitations. At high current density, the fuel or the oxidant gases are consumed on the reaction surfaces faster than they can be replenished.
At a certain limiting current density, the partial pressures of the reactant gases at the reaction surfaces plummet towards zero. From the Nernst equation, it is clear that this dramatic decline in the partial pressures of reactants causes a dramatic decline in output voltage. The voltage drop from this mass transportation limit is:
(2-11) where A is a fitting parameter (V) and it is obvious that the concentration loss is dominant at high current density. Well-designed flow structures and thin, highly porous electrodes may reduce the concentration overvoltage.