## 4 Constant Fuel Utilization  
Following Reference [7], the following two parameters are kept constant in this section: \(u\) (Equation (11)) and \(r_{HO}\) (Equation (17)). This corresponds to the operating condition known as constant fuel utilization ratio/factor condition or, equivalently, constant stoichiometry condition. Hydrogen partial pressure is given by [28; 38]  
\[p_{H_{2}}=\frac{1}{K_{H_{2}}}\left(q_{H_{2}}^{in}-q_{H_{2}}^{react}\right) \tag{10}\]  
where \(K_{H_{2}}\) is a valve molar constant [28], \(q_{H_{2}}^{in}\) represents the input molar flow rate (mol/s) of hydrogen fuel, and \(q_{H_{2}}^{react}\) is the hydrogen flow rate (mol/s) that takes part in the reaction. Using the definition of fuel utilization ratio \(u\),  
\[u=\frac{q_{H_{2}}^{react}}{q_{H_{2}}^{in}} \tag{11}\]  
we have  
\[p_{H_{2}}=\frac{1}{K_{H_{2}}}\left(\frac{1}{u}-1\right)q_{H_{2}}^{react}. \tag{12}\]  
The ratio of two different hydrogen partial pressures is then given by  
\[\frac{p_{H_{2}}^{final}}{p_{H_{2}}^{initial}}=\frac{q_{H_{2}}^{react,final}}{ q_{H_{2}}^{react,initial}}. \tag{13}\]  
Since the reaction rate is given (from electrochemistry) by  
\[q_{H_{2}}^{react}=\frac{I_{\mathrm{FC}}}{2F}, \tag{14}\]  
the pressure ratio reduces to  
\[\frac{p_{H_{2}}^{final}}{p_{H_{2}}^{initial}}=\frac{I_{\mathrm{FC}}^{final}}{ I_{\mathrm{FC}}^{initial}}. \tag{15}\]Next, the partial pressure for oxygen is expressed as  
\[p_{O_{2}}=\frac{1}{K_{O_{2}}}\left(q_{O_{2}}^{in}-q_{O_{2}}^{react}\right) \tag{16}\]  
where \(K_{O_{2}}\) is a valve molar constant [28], \(q_{O_{2}}^{in}\) denotes the oxygen input molar flow rate, and \(q_{O_{2}}^{react}\) is the oxygen flow rate that reacts. Noting that \(r_{HO}\) is the ratio of hydrogen-to-oxygen input flow rates,  
\[r_{HO}=\frac{q_{H_{2}}^{in}}{q_{O_{2}}^{in}} \tag{17}\]  
and that  
\[q_{H_{2}}^{react}=2q_{O_{2}}^{react} \tag{18}\]  
we have  
\[p_{O_{2}}=\frac{1}{K_{O_{2}}}\left(\frac{1}{r_{HO}}q_{H_{2}}^{in}-\frac{1}{2} q_{H_{2}}^{react}\right). \tag{19}\]  
Now, by using the definition of \(u\), we obtain  
\[u=\frac{q_{H_{2}}^{react}}{q_{H_{2}}^{in}} \tag{11}\]0  
Two different oxygen partial pressures, then, have the following ratio:  
\[u=\frac{q_{H_{2}}^{react}}{q_{H_{2}}^{in}} \tag{11}\]1  
Water (steam) partial pressure is expressed as  
\[u=\frac{q_{H_{2}}^{react}}{q_{H_{2}}^{in}} \tag{11}\]2 \[u=\frac{q_{H_{2}}^{react}}{q_{H_{2}}^{in}} \tag{11}\]3 (23) \[u=\frac{q_{H_{2}}^{react}}{q_{H_{2}}^{in}} \tag{11}\]4  
where \(K_{H_{2}O}\) is a valve molar constant [28], and \(q_{H_{2}O}^{react}\) is the flow rate of the water vapor (steam) that is produced in the reaction. The final and initial pressures are then related by  
\[u=\frac{q_{H_{2}}^{react}}{q_{H_{2}}^{in}} \tag{11}\]5  
The change in the voltage corresponding to the changes in the pressures is given by  
\[u=\frac{q_{H_{2}}^{react}}{q_{H_{2}}^{in}} \tag{11}\]6which, by Equations (15), (21) and (25), yields  
\[u=\frac{q_{H_{2}}^{react}}{q_{H_{2}}^{in}} \tag{11}\]7 \[u=\frac{q_{H_{2}}^{react}}{q_{H_{2}}^{in}} \tag{11}\]8  
The above expression for voltage change shows that, for an increase in the cell current, \(\Delta V\) is positive, representing a gain. (Following the present approach, an expression for \(\Delta V\) is derived in the next section for the constant fuel flow condition. For an increase in current, \(\Delta V\) for the constant fuel flow case is negative, representing a loss.) Now, setting \(I_{\text{FC}}^{final}=I_{\text{FC}}\), and calling  
\[u=\frac{q_{H_{2}}^{react}}{q_{H_{2}}^{in}} \tag{11}\]9  
where \(I_{\text{FC}}^{ratio}  
found from Equation (6) by plugging in the initial partial pressures and the \(E_{0}\), is taken to be 1 V in Figures 1 and 2. Figure 1 shows the two irreversibilities--Nernstian gain and ohmic loss--separately. The magnitude of the ohmic loss is plotted, ignoring the sign. The Nernstian gain, being a logarithmic function, is monotone increasing. The shape of the output voltage curve is determined by how much of the Nernstian gain is offset by the ohmic loss for a given current.