## 43 Fuel Cell Thermodynamics an Introduction

4.3.1 Fuel cell efficiency and efficiency limits

One of the attractions of fuel cells is that they are not heat engines. Their thermodynamics are different, and in particular their efficiency is potentially greater as they are not limited by the well-known Carnot limit that impinges on IC and other types of fuel burning engines. However, as we shall see, they do have their own limitations, and while fuel cells are often more efficient than IC engines, the difference is sometimes exaggerated.

Hydrogen Energy = ?

Oxygen Energy = ?

Hydrogen Energy = ?

Electricity Energy = VI t

Figure 4.8 Fuel cell inputs and outputs

Electricity Energy = VI t

Figure 4.8 Fuel cell inputs and outputs

At first we must acknowledge that the efficiency of a fuel cell is not straightforward to define. In some electrical power generating devices it is very clear what form of energy is being converted into electricity. With a fuel cell such energy considerations are much more difficult to visualise. The basic operation has already been explained, and the input and outputs are shown in Figure 4.8. The electrical power and energy output are easily calculated from the well known formulas:

Power = VI and Energy = VIt

However, the energy of the chemical inputs and output is not so easily defined. At a simple level we could say that it is the chemical energy of the H2, O2 and H2O that is in question. The problem is that chemical energy is not simply defined, and terms such as enthalpy, Helmholtz function and Gibbs free energy are used. In recent years the useful term 'exergy' has become quite widely used, and the concept is particularly useful in high temperature fuel cells, though we are not concerned with these here. There are also older (but still useful) terms such as calorific value.

In the case of fuel cells it is the Gibbs free energy that is important. This can be defined as the energy available to do external work, neglecting any work done by changes in pressure and/or volume. In a fuel cell the external work involves moving electrons round an external circuit; any work done by a change in volume between the input and output is not harnessed by the fuel cell.3 Exergy is all the external work that can be extracted, including that due to volume and pressure changes. Enthalpy, simply put, is the Gibbs free energy plus the energy connected with the entropy. The enthalpy H, Gibbs free energy G and entropy S are connected by the well-known equation:

The energy that is released by a fuel cell is the change in Gibbs energy before and after a reaction, so the energy released can be represented by the equation:

AG = G outputs G inputs

However, the Gibbs free energy change is not constant, but changes with temperature and state (liquid or gas). Table 4.2 below shows AG for the basic hydrogen fuel cell reaction

3 It may be harnessed by some kind of turbine in a combined cycle system, as discussed in Chapter 6.

 Form of water product Temperature (°C) AG (kJ/mole) Liquid 25 -237.2 Liquid 80 -228.2 Gas 80 -226.1 Gas 100 -225.2 Gas 200 -220.4 Gas 400 -210.3 Gas 600 -199.6 Gas 800 -188.6 Gas 1000 -177.4

for a number of different conditions. Note that the values are negative, which means that energy is released.

If there are no losses in the fuel cell, or as we should more properly say, if the process is reversible, then all this Gibbs free energy is converted into electrical energy. We could thus define the efficiency of a fuel cell as:

electrical energy produced Gibbs free energy change

However, this is not very useful, and is rarely done, not least because the Gibbs free energy change is not constant.

Since a fuel cell uses materials that are usually burnt to release their energy, it would make sense to compare the electrical energy produced with the heat that would be produced by burning the fuel. This is sometimes called the calorific value, though a more precise description is the change in enthalpy of formation. Its symbol is AH. As with the Gibbs free energy, the convention is that AH is negative when energy is released. So to get a good comparison with other fuel using technologies, the efficiency of the fuel cell is usually defined as:

electrical energy produced per mole of fuel

However, even this is not without its ambiguities, as there are two different values that we can use for AH. For the burning of hydrogen:

Whereas if the product water is condensed back to liquid, the reaction is:

The difference between these two values for AH (44.01 kJ/mole) is the molar enthalpy of vaporisation4 of water. The higher figure is called the higher heating value (HHV), and the lower, quite logically, the lower heating value (LHV). Any statement of efficiency should say whether it relates to the higher or lower heating value. If this information is not given, the LHV has probably been used, since this will give a higher efficiency figure.

We can now see that there is a limit to the efficiency, if we define it as in equation (4.4). The maximum electrical energy available is equal to the change in Gibbs free energy, so:

Maximum efficiency possible =- x 100% (4.5)

This maximum efficiency limit is sometimes known as the thermodynamic efficiency. Table 4.3 gives the values of the efficiency limit, relative to the higher heating value, for a hydrogen fuel cell. The maximum voltage obtainable from a single cell is also given.

The graphs in Figure 4.9 show how these values vary with temperature, and how they compare with the Carnot limit, which is given by the equation:

Carnot limit =-

Ti where T1 is the higher temperature, and T2 the lower, of the heat engine. The graph makes clear that the efficiency limit of the fuel cell is certainly not 100%, as some supporters of fuel cells occasionally claim. Indeed, above the 750°C the efficiency limit of the hydrogen fuel cell is actually less than for a heat engine. Nevertheless, the PEM fuel cells used in vehicles operate at about 80°C, and so their theoretical maximum efficiency is actually much better than for an IC engine.

### 4.3.2 Efficiency and the fuel cell voltage

A very useful feature of fuel cells is that their efficiency can be very easily found from their operating voltage. The reasoning behind this is as follows. If one mole of fuel is reacted in the cell, then two moles of electrons are pushed round the external circuit; this

 Form of water product Temp °C AG id/mole-1 Max. EMF Efficiency limit Liquid 25 -237.2 1.23 V 83% Liquid 80 -228.2 1.18 V 80% Gas 100 -225.3 1.17 V 79% Gas 200 -220.4 1.14V 77% Gas 400 -210.3 1.09 V 74% Gas 600 -199.6 1.04 V 70% Gas 800 -188.6 0.98 V 66% Gas 1000 -177.4 0.92 V 62%

4 This used to be known as the molar 'latent heat'

4 This used to be known as the molar 'latent heat'

Fuel cell, liquid product s0

Fuel cell, liquid product s0

200 400 600 S00 1000

Operating temperature/Celsius

Figure 4.9 Maximum hydrogen fuel cell efficiency at standard pressure, with reference to the higher heating value. The Carnot limit is shown for comparison, with a 50°C exhaust temperature

200 400 600 S00 1000

Operating temperature/Celsius

Figure 4.9 Maximum hydrogen fuel cell efficiency at standard pressure, with reference to the higher heating value. The Carnot limit is shown for comparison, with a 50°C exhaust temperature can be deduced from Figure 4.3. We also know that the electrical energy is given by the fundamental energy equation:

The Faraday constant F gives the charge on one mole of electrons. So, when one mole of hydrogen fuel is used in a fuel cell, if it were 100% efficient, as defined by equation (4.4), then we would be able to say that:

Using standard values for the Faraday constant (96 485 Coulombs), and the two values for AH given above, we can easily calculate that the '100% efficient' voltage for a single cell is 1.48 V if using the HHV or 1.25 V if using the LHV.

Now of course a fuel cell never is, and we have shown in the last section never can be, 100% efficient. The actual fuel cell voltage will be a lower value, which we can call Vc. Since voltage and electrical energy are directly proportional, it is clear that

Energy = Charge x Voltage

Clearly it is very easy to measure the voltage of a fuel cell. In the case of a stack of many cells, remember that the voltage of concern is the average voltage one cell, so the system voltage should be divided by the number of cells. The efficiency can thus be found remarkably easily.

It is worth noting in passing that the maximum voltage of a fuel cell occurs when 100% of the Gibbs free energy is converted into electrical energy. Thus we have a 'sister' equation to equation (4.4), giving the maximum possible fuel cell voltage:

This is also a very important fuel cell equation, and it was used to find the figures shown in the fourth column of Table 4.3.

### 4.3.3 Practical fuel cell voltages

Equation (4.7) above gives the maximum possible voltage obtainable from a single fuel cell. In practice the actual cell voltage is less than this. Now of course this applies to ordinary batteries too, as when current is drawn out of any electric cell the voltage falls, due to internal resistances. However, with a fuel cell this effect is more marked than with almost all types of conventional cell. Figure 4.10 shows a typical voltage/current density curve for a good PEM fuel cell. It can be seen that the voltage is always less, and is often much less, than the 1.18 V that would be obtained if all of the Gibbs energy were converted into electrical energy.

There are three main reasons for this loss of voltage, as detailed below.

"No loss" voltage of 1.2 Volts

"No loss" voltage of 1.2 Volts

Even the open circuit voltage is less than the theoretcial no loss value

° 0.4 - Voltage begins to fall faster at higher currents

Even the open circuit voltage is less than the theoretcial no loss value

° 0.4 - Voltage begins to fall faster at higher currents

200 400 600 800 1000

Current density/mA.cm-2

Figure 4.10 Graph showing the voltage from a typical good quality PEM fuel cell operating on air at about 80° C

• The energy required to drive the reactions at the electrodes, usually called the activation energy, causes a voltage drop. This is especially a problem at the air cathode, and shows itself as a fairly constant voltage drop. This explains the initial fall in voltage even at quite low currents.

• The resistance of the electrolyte and the electrodes causes a voltage drop that more-or-less follows Ohm's law, and causes the steady fall in voltage over the range of currents. This is usually called the Ohmic voltage loss.

• At very high currents, the air gets depleted of oxygen, and the remnant nitrogen gets in the way of supplying fresh oxygen. This results is a fall in voltage, as the electrodes are short of reactant. This problem causes the more rapid fall in voltage at higher currents, and is called mass transfer or concentration voltage loss.

A result of the huge effort in fuel cell development over the last ten years or so has resulted in great improvement in the performance of fuel cells, and a reduction in all these voltage losses. A fuel cell will typically operate at an average cell voltage of about 0.65 to 0.70 V, even at currents approaching 1A per cm2. This represents an efficiency of about 50% (with respect to the HHV), which is considerably better than any IC engine, though some of the electrical energy is used up driving the fuel cell ancillary equipment to be discussed in the sections that follow.

We should point out that a consequence of the higher cell voltage at lower currents is that the efficiency is higher at lower currents. This is a marked contrast to the IC engine, where the efficiency is particularly poor at low powers.

In the opening section of this chapter we pointed out that a fuel cell could be compared to an IC engine running on hydrogen fuel, which would also give out very limited pollution. This section has shown that fuel cells do have the potential to have a considerably higher efficiency than IC engines, and so they would, all other things being equal, be preferred. The problem is that all other things are not equal. At the moment fuel cells are vastly more expensive than IC engines, and this may remain so for some time. It is by no means clear-cut that fuel cells are the better option. A hydrogen powered IC engine in a hybrid drive train would be not far behind a fuel cell in efficiency, and the advantages of proven and available technology might tip the balance against higher efficiency and even less pollution. Time will tell.

### 4.3.4 The effect of pressure and gas concentration

The values for the changes in the Gibbs free energy given in Tables 4.2 and 4.3 all concern pure hydrogen and oxygen, at standard pressure, 100 kPa. However, as well as changing with temperature, as shown in these tables, the Gibbs energy changes with pressure and concentration.

A full treatment of these issues is beyond a book such as this, and it can easily be found elsewhere (e.g. Chapter 2 of Larminie and Dicks 2003). Suffice to say that the relationship is given by a very important fuel cell equation derived from the work of Nernst. It can be expressed in many different forms, depending on what issue is to be analysed. For example, if the change of system pressure is the issue, then the Nernst equation takes the form:

Where AV is the voltage increase if the pressure changes from P1 to P2. Other causes of voltage change are a reduction in voltage caused by using air instead of pure oxygen. The use of hydrogen fuel that is mixed with carbon dioxide, as is obtained from the 'reforming' of fuels such as petrol, methanol or methane (as described in Chapter 5), also causes a small reduction in voltage.

For high temperature fuel cells the Nernst equation predicts very well the voltage changes. However, with lower temperature cells, such as are used in electric vehicles, the changes are nearly always considerably greater than the Nernst equation predicts. This is because the 'activation voltage drop' mentioned in the last section is also quite strongly affected by issues such as gas concentration and pressure. This is especially the case at the air cathode.

For example, equation (4.8) would predict that for a PEM fuel cell working at 80°C, the voltage increase resulting from a doubling of the system pressure would be:

4 x 96485 F

However, in practice the voltage increase would typically be about 0.04 V, nearly ten times as much. Even so, we should note that the increase is still not large, and that there is considerable energy cost in running the system at higher pressure. Indeed, it is shown elsewhere (e.g. Larminie and Dicks 2003, Chapter 4) that the energy gained from a higher voltage is very unlikely to be greater than the energy loss in pumping the air to higher pressure.

Nevertheless, it is the case that most PEM fuel cells in vehicle applications are run at a pressure distinctly above air pressure, typically between 1.5 and 2.0 bar. The reasons for this are not primarily because of increasing the cell voltage. Rather, they are because it makes the water balance in the PEM fuel cell much easier to maintain. This complicated issue is explained in Section 4.5 below.

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