## 3 The Problem  
The Nernst equation, which is the cornerstone of fuel cell thermodynamics, provides an expression for the reversible thermodynamic potential, also known as the equilibrium voltage or the open-circuit electromotive force (EMF), of the fuel cell [1]:  
\[E_{Nernst}=E_{0}+\frac{RT}{nF}\ln\left(\frac{\prod_{i}a_{\mbox{reactant}_{i}}^{ c_{i}}}{\prod_{j}a_{\mbox{product}_{j}}^{c_{j}}}\right) \tag{1}\]  
where \(E_{0}\) is the reference (standard) EMF at unit activity and atmospheric pressure, \(i\) and \(j\) are the numbers of reactant and product species, \(a\) represents the _activity_, \(c_{i}\) is the stoichiometric coefficient of species \(i\), \(R\) is the universal gas constant, \(F\) is [PERSON]'s constant, \(n\) is the number of electrons transferred for each molecule of the fuel participating in the reaction, and \(T\) is the temperature. For a hydrogen-oxygen fuel cell (e.g., solid oxide fuel cell (SOFC) or proton exchange membrane fuel cell (PEMFC)), hydrogen and oxygen are the reactants, and the product is water (or steam). The reference EMF, \(E_{0}\), depends on temperature \(T\):  
\[E_{0}=E_{0}^{0}+(T-T_{0})\frac{\Delta s}{nF} \tag{2}\]  
where \(E_{0}^{0}\) is the standard EMF at temperature \(T_{0}\), and \(\Delta s\) is the change in entropy. The activity \(a\) of an ideal gas is expressed in terms of its pressure (or partial pressure) \(p\):  
\[a_{H_{2}} = \frac{p_{H_{2}}}{p^{0}}, \tag{3}\] \[a_{O_{2}} = \frac{p_{O_{2}}}{p^{0}}, \tag{4}\]where \(p^{0}\) is the standard-state pressure (1 atm). At high temperatures, such as 1000 \({}^{\circ}\)C (as in solid oxide fuel cells), steam can be assumed to behave as an ideal gas, and therefore  
\[a_{H_{2}O}=\frac{p_{H_{2}O}}{p^{0}}. \tag{5}\]  
Using \(p^{0}=1\) atm, and noting that \(n=2\) for a hydrogen fuel cell, we have the following version of the Nernst equation for solid oxide fuel cells:  
\[E_{Nernst}=E_{0}+\frac{RT}{2F}\ln\left(\frac{p_{H_{2}}\sqrt{p_{O_{2}}}}{p_{H_{ 2}O}}\right). \tag{6}\]  
If the fuel cell is operated below 100 \({}^{\circ}\)C, so that liquid water is produced (as in proton exchange membrane fuel cells), the activity of water can be taken to be unity (\(a_{H_{2}O}=1\)). In that case, the Nernst equation takes the form  
\[E_{Nernst}=E_{0}+\frac{RT}{2F}\ln\left(p_{H_{2}}\sqrt{p_{O_{2}}}\right). \tag{7}\]  
The terminal (load) voltage is generally obtained by subtracting from \(E_{Nernst}\) the following types of losses (or \"irreversibilities\"):  
* activation loss
* concentration loss
* ohmic loss
* losses due to fuel crossover and internal current. The practice of expressing--either directly or indirectly via the partial pressures--the Nernst EMF as a function of cell current is rather common in the literature (e.g., [33; 34]). Building upon References [28; 29], a major piece of work [7] on control strategies for SOFC operation developed a model for the constant fuel utilization condition using the following form of the Nernst equation:  
\[E_{Nernst}=E_{0}+\frac{RT}{2F}\left\{\ln\left(\frac{K_{H_{2}O}}{K_{H_{2}}} \sqrt{\frac{K_{r}}{r_{HO}K_{O_{2}}}}\right)+\frac{1}{2}\ln\left(I_{FC}\left( \frac{1}{u}-1\right)^{2}\left(\frac{2}{u}-r_{HO}\right)\right)\right\} \tag{8}\]  
where \(K_{H_{2}}\), \(K_{O_{2}}\), and \(K_{H_{2}O}\) are valve molar constants [28], \(K_{r}=1/(4F)\)[28; 38; 39], \(r_{HO}\) is the ratio of hydrogen to oxygen input molar flow rates [7], \(u\) is the fuel utilization ratio [7; 28], and \(I_{FC}\) is the load current. This model, or one of its variants, has been used in many papers in the fuel cell literature (e.g., [6; 38]). (A detailed re-derivation of Equation (8) can be found in, for example, Reference [8].) Of the four types of losses mentioned above, the models in References [7; 28] consider only the ohmic loss and obtain the output (load) voltage, \(V\), as  
\[V=E_{Nernst}-rI_{FC} \tag{9}\]  
where \(E_{Nernst}\) is given by Equation (8), and \(r\) is the ohmic resistance of the cell.