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# Thermodynamic Voltage and Efficiency of Hydrogen Fuel Cells

This Demonstration carries out calculations of thermodynamic voltage and efficiency for hydrogen fuel cells. It shows how temperature, pressure and composition determine voltage and efficiency on both sides of the fuel cell. The thermodynamic voltage increases with increasing pressure and mole fraction of the reacting gases. Both thermodynamic voltage and efficiency decrease with increasing temperature. The thermodynamic efficiency changes only with temperature, since both the enthalpy and Gibbs free energy are functions of temperature alone. The calculations can be performed with lower heating value (LHV), where water is produced in the gaseous phase, and higher heating value (HHV), where water is produced in the liquid phase.

### DETAILS

The overall chemical reaction for hydrogen fuel cells is:
.
If the water is produced as steam, then the chemical reaction is of low heating value (LHV); if it is produced in liquid form, then the reaction is of high heating value (HHV).
The enthalpy or heat of reaction (J/mol) of the chemical reaction is the difference between the enthalpies of the product and reactants and :
.
The enthalpy of component (, or ) is the sum of standard enthalpy of formation and sensible enthalpy , namely:
.
can be found in thermodynamic tables and is obtained using the following formula:
,
where is the heat capacity at a constant pressure. (J/mol K) changes with temperature:
,
where , , , and are experimentally determined coefficients. Likewise, the entropy (J/mol K) of the chemical reaction is given by:
,
,
.
The Gibbs free energy , which represents the electric work of the fuel cell, is given as:
.
The reference thermodynamic voltage could be then obtained from:
,
where is the number of electrons transferred in the reaction for one mole of (here equal to 2) and is the Faraday constant (96485 ). The thermodynamic (Nernst) voltage, which incorporates the effects of pressure and composition of the reacting mixture, is obtained using the Nernst equation:
,
where is the universal gas constant (8.314 J/mol K) and is the activity of the component involved in the reaction. The activity of an ideal gas is the ratio between the partial pressure of the gas and the reference pressure. For example, the activity of is given as:
,
where is the mole fraction of hydrogen, is the pressure at the anode and is the reference pressure (1 atm for ideal gases). The activity of water () is normally approximated by 1. Based on the preceding, the Nernst equation becomes:
.
Finally, the thermodynamic efficiency is given using:
.
References
[1] M. M. Mench, Fuel Cell Engines, Hoboken, NJ: John Wiley and Sons, 2008 Chapter 3.
[2] F. Barbir, PEM Fuel Cells: Theory and Practice, 2nd ed., Boston: Elsevier/Academic Press, 2013 Chapter 2.
[3] R. H. Perry and D. W. Green (eds.), Perry's Chemical Engineers' Handbook, 7th ed., New York: McGraw-Hill, 1997 pp. 2–174.

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