# THERMODYNAMICS OF THE PEM FUEL CELL

Thermodynamics mainly deals with the change in energy of the system during a process. A fuel cell is a control volume system because mass is entering and leaving the system boundary. As we know, the fuel cell converts chemical energy into direct electric current. This current flows out of the control volume and does electrical work. To have an idea about the energy conversion efficiency of fuel, its thermodynamic analysis is important [7].

## Reversible Voltage

All practical processes contain a certain amount of irreversibility, which degrades the quality available with energy. If we consider that all the chemical energy is converted into electrical work, then the work obtained is known as reversible, or no-loss work. The energy released during the oxidation of hydrogen is defined by Gibbs free energy.

The Gibbs free energy of formation can be calculated as Consider the simple reaction for the hydrogen/oxygen fuel cell: which is equivalent to:

One mole of hydrogen reacts with a half mole of oxygen to form one mole of water, and energy is released during the process; hence, Gibbs free energy of formation per unit mole can be calculated as:

Hence,

If the fuel cell is working on pure hydrogen and oxygen, then one molecule of hydrogen gives two electrons. According Avogadro’s Principle, one mole contains N (6.022 x 10^{23}) molecules; hence, the total charge produced by one mole of hydrogen is -2Ne, where -e is a charge on one electron (1.602 x 10^{_I9}°C).

If E is a voltage of the fuel cell, then electrical work by E to move 2Ne charge is given by

If the system involves no irreversibility, then all work should be equal to the energy released during the reaction.

Thus,

If we take z as the number of electrons, then Eq. (9.1) can be written as

## Fuel Cell Efficiency

Hydrogen is oxidized in a fuel cell, and energy is released, which is finally converted into electricity. Hence, the efficiency of a fuel cell can be defined as the ratio of electrical work produced to the energy released during the oxidation of fuel (hydrogen).

Oxidation processes in which fuel combines with oxygen and produces heat. This heat is termed as the calorific value of a fuel. This is defined by the enthalpy of formation Ah, .

Hence, mathematically, the efficiency of the fuel cell is

The maximum electrical work equals to Gibbs free energy of formation, that is, Ag_{f }when there is no loss of energy due to irreversibility in the system.

Theoretically, fuel can have a maximum possible efficiency as described by Eq. (9.4). This is also known as the thermodynamic limit (without violating the law of thermodynamics) of energy conversion that is possible by a fuel cell.

## Efficiency in Terms of Cell Voltage

Assuming we have a fuel cell for which q_{MAX} = 100%, then Gibbs free energy is equal to its calorific value, and we can write the cell voltage as

Ah_{f} For hydrogen = -241.83 kJ mol^{-1} (for steam as output)

Ah_{f} For hydrogen = -285.84 kJ mol^{-1} (for water as output)

Or

From the above discussion, we can define the fuel efficiency as the ratio of actual voltage of fuel cell to the maximum possible voltage. Mathematically, it is written as

If some amount of fuel gets unused during reaction, then we can define the fuel utilization factor as the ratio of the mass of fuel used in reaction to the total mass of fuel supplied to the system.

Now we can rewrite Eq. (9.6) by considering fuel utilization factor u_{f}

## Effect of Gas Concentration and Pressure

### Nernst Equations

The performance of the fuel cell is greatly affected by pressure and concentration of gas. As fuel is used in a reaction, the partial pressure of gas decreases, which affects the cell voltage. This variation can be greatly explained by the Nernst equation.

Let a general chemical reaction be

where p moles of P reacts with к moles of К to produce m moles of M.

Activity is associated with each reactant and product involved in the chemical reaction. Let the activity associated with P, K, and M denoted by a_{p}, a_{k}, and a_{m}. One should note for an ideal gas activity, a function of the pressure of the gas can be written as

where P is the pressure or partial pressure of the gas, and P° is standard atmospheric pressure, O.l MPa.

According to Gibbs free energy equation,

where Ag? is the change in molar Gibbs free energy of formation at standard atmospheric pressure.

For the fuel cell supplied with hydrogen and oxygen, Eq. (9.10) can be modified as

From the above equation, it is clear that as the activity of product increases (i.e., a_{H},0) Ag_{f} becomes less negative; hence, the energy released during a reaction is decreased. Now cell voltage can be expressed by putting the value of Ag, in Eq. (9.11).

where E° is the electromotive force (EMF) at standard atmospheric pressure.

Equation (9.12) is the governing equation of cell voltage. It gives the relation between cell voltage and activity of reactant and product. This equation is known as the Nernst equation, assuming that the water coming out from cell is in vapor form and acts as ideal gas. Then activity for an ideal gas is given as

Then Eq. (9.12) will become
If P^{(>} = I atm then

If the operating system pressure is P, then we can say that

where a, (3, and 8 are constants depending on the molar masses and concentrations of H_{2}. 0_{2}, and H_{2}0.

Equation (9.13) can be written as

Equation (9.14) is the modified form of the Nernst equation when the system is operating under total pressure p.

### Effect of Partial Pressure of Hydrogen

Hydrogen can be used in pure form or as a mixture of gases. Its pressure changes as hydrogen is utilized in a reaction, so the effect of change in pressure can be calculated by the Nernst equation. Equation (9.13) can be written as

Let the hydrogen pressure change from p, to p_{2}, then the difference in voltage can be calculated as

### Fuel and Oxidant Utilization

The fuel cell utilizes fuel and oxidants, and their respective partial pressures decrease, which affect the values of a, [), and 8. In a fuel cell reaction, hydrogen is oxidized; hence, the value of *a* decreases and 8 increases, which affects the cell voltage.

The value of a is varied from inlet to an outlet, but due to the high conductivity of the bipolar plate, we cannot have different voltages at different points, which binds current density to decrease [8].

To get a higher system efficiency, fuel utilization should be maximum. But according to the preceding discussion, fuel utilization affects the cell voltage, so it becomes an important parameter to control for optimum performance spatially when fuel is supplied by the reformer.