## 5 Constant Fuel Flow
### Constant Hydrogen-Oxygen Input Flow Ratio  
This subsection treats the operating condition of \(r_{HO}\) (Equation (17)) being held constant. The constancy of both \(q_{H_{2}}^{in}\) and \(r_{HO}\) means that the oxygen input molar flow rate, \(q_{O_{2}}^{in}\), is also constant. From Equations (10) and (14), we have  
\[p_{H_{2}}=\dfrac{1}{K_{H_{2}}}\left(q_{H_{2}}^{in}-\dfrac{I_{FC}}{2F}\right) \tag{32}\]  
The ratio of two different partial pressures, corresponding to two different current values, is obtained as  
\[\dfrac{p_{H_{2}}^{final}}{p_{H_{2}}^{initial}}=\dfrac{q_{H_{2}}^{in}-\dfrac{I_{ FC}^{final}}{2F}}{q_{H_{2}}^{in}-\dfrac{I_{FC}^{final}}{2F}} \tag{33}\]  
Figure 4: Validation results for the constant fuel utilization operation; mean absolute relative error = 3.74%. Next, substituting Equation (14) into Equation (19), we have  
\[p_{O_{2}} = \frac{1}{K_{O_{2}}}\left(\frac{1}{r_{HO}}q_{H_{2}}^{in}-\frac{1}{2} \frac{I_{FC}}{2F}\right) \tag{34}\] \[= \frac{1}{K_{O_{2}}r_{HO}}\left(q_{H_{2}}^{in}-r_{HO}\frac{I_{FC}}{ 4F}\right) \tag{35}\]  
Two different oxygen partial pressures, corresponding to \(I_{FC}^{initial}\) and \(I_{FC}^{final}\), are then related by:  
\[\frac{p_{O_{2}}^{final}}{p_{O_{2}}^{initial}}=\frac{q_{H_{2}}^{in}-r_{HO}\frac {I_{FC}^{final}}{4F}}{q_{H_{2}}^{in}-r_{HO}\frac{I_{FC}^{final}}{4F}} \tag{36}\]  
From Section 4, we know that the final and initial steam pressures have a ratio given by Equation (25). The change in voltage corresponding to the changes in the three pressures is given by  
\[\Delta V=\frac{RT}{2F}\ln\left(\frac{\frac{p_{H_{2}}^{final}\sqrt{p_{O_{2}}^{ final}}}{\frac{p_{H_{2}}^{final}}{p_{H_{2}O}^{final}}}}{\frac{p_{H_{2}}^{ final}\sqrt{p_{O_{2}}^{final}}}{p_{H_{2}O}^{final}}}\right) \tag{37}\]  
which, by Equations (25), (33) and (36), becomes  
\[\Delta V=\frac{RT}{2F}\ln\left(\frac{I_{FC}^{initial}}{I_{FC}^{final}}\times \frac{q_{H_{2}}^{in}-\frac{I_{FC}^{final}}{2F}}{q_{H_{2}}^{in}-\frac{I_{FC}^{ initial}}{2F}}\sqrt{\frac{q_{H_{2}}^{in}-r_{HO}\frac{I_{FC}^{final}}{4F}}{q_{H_{2}}^{in}-r_{HO} \frac{I_{FC}^{final}}{4F}}}\right) \tag{38}\]  
Figure 5: Variation in the input fuel flow rate with current in the two modes of operation: no variation (constant fuel flow mode) and linear variation (constant fuel utilization mode). The above expression shows that, for an increase in the cell current, \(\Delta V\) is negative, representing a loss. As in Section 4, expressing all currents in units of Amperes, and setting \(I_{\text{FC}}^{initial}=1\) A and \(I_{\text{FC}}^{final}=I_{\text{FC}}\) in the above equation, we have  
\[\Delta V=\frac{RT}{2F}\ln\left(\frac{1}{I_{FC}}\times\frac{q_{H_{2}}^{in}- \frac{I_{\text{FC}}}{2F}}{q_{H_{2}}^{in}-\frac{1}{2F}}\sqrt{\frac{q_{H_{2}}^{ in}-r_{HO}\frac{I_{\text{FC}}}{4F}}{q_{H_{2}}^{in}-r_{HO}\frac{1}{4F}}}\right) \tag{39}\]  
The output (load) voltage, \(V\), for the constant fuel flow mode is then given by  
\[\dfrac{p_{H_{2}}^{final}}{p_{H_{2}}^{initial}}=\dfrac{q_{H_{2}}^{in}-\dfrac{I_{ FC}^{final}}{2F}}{q_{H_{2}}^{in}-\dfrac{I_{FC}^{final}}{2F}} \tag{33}\]0  
where \(E_{\text{\emph{Nernst}}}\), the reversible (equilibrium) potential, is given by Equation (6), and the second term on the right side represents an irreversible loss (Nernstian loss). (Equation (40) holds for \(I_{\text{FC}}>1\) A.) This Nernstian loss is explained by the fact that, with \(q_{H_{2}}^{in}\) and \(r_{HO}\) held constant (note that the constancy of both \(q_{H_{2}}^{in}\) and \(r_{HO}\) implies that of \(q_{O_{2}}^{in}\), too), reactant concentrations decrease and product concentration increases with increasing current. Equation (32) shows that, for a fixed \(q_{H_{2}}^{in}\), hydrogen pressure \(p_{H_{2}}\) decreases linearly with increasing current. Similar is the behavior of \(p_{O_{2}}\) with respect to current, when the two input flow rates are kept fixed (see Equation (35)). The behavior of \(p_{H_{2}O}\) is opposite; as shown by Equation (24), \(p_{H_{2}O}\) grows linearly with current.