## 4 Constant Fuel Utilization
The effect of the Nernstian gain can be seen rather conspicuously in the hump in the output voltage at low currents. The output voltage equals the Nernst voltage at the point where  
\[p_{H_{2}}=\frac{1}{K_{H_{2}}}\left(\frac{1}{u}-1\right)q_{H_{2}}^{react}. \tag{12}\]0  
an equation that is difficult to solve analytically. Figure 2 shows, for low currents, how the polarization plot's hump varies for different values of the cell resistance. (An \(r\) of 3.28125 \(\times\) 10\({}^{-3}\)\(\Omega\) is used in \[p_{H_{2}}=\frac{1}{K_{H_{2}}}\left(\frac{1}{u}-1\right)q_{H_{2}}^{react}. \tag{12}\]1) \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline \(T\) & 1273 K \\ \(u\) & 0.8 \\ \(K_{H_{2}}\) & 0.843 mol/(s-atm) \\ \(K_{H_{2}O}\) & 0.281 mol/(s-atm) \\ \(K_{O_{2}}\) & 2.52 mol/(s-atm) \\ \(r\) & 3.28125 \(\times\) 10\({}^{-4}\)\(\Omega\) \\ \(r_{HO}\) & 1.145 \\ \(n\) & 2 \\ Constants & \\ \(F\) & 96,485 Coulombs/mol \\ \(R\) & 8.31 J/(mol K) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Numerical values of parameters and constants. Figure 1: Polarization characteristics for the constant fuel utilization condition; the Nernstian gain and the ohmic loss are plotted separately. It can be argued that, in theory, in the constant hydrogen and oxygen utilization condition, the concentration loss (mass transport loss) is mitigated or almost eliminated. (Of course much depends on the physical characteristics of the reactant supply systems, reformer and control mechanisms, time-delays in the diffusion process between the channel and the three-phase-boundary, etc.) Under the assumption of the ideal conditions in our lumped model, the concentration loss is negligible or zero in Equation (30). For the constant fuel utilization operation, Equations (11), (12), (20) and (23) show that \(p_{H_{2}}\), \(p_{O_{2}}\), and \(p_{H_{2}O}\) all are directly proportional to (i.e., increase linearly with) the input fuel flow rate. This is shown in Figure 3, where the straight lines have been obtained using the parameter values in Table 1. All three lines have a zero intercept, and their slopes are given by  
\[p_{H_{2}} :\frac{1}{K_{H_{2}}}(1-u)\] \[p_{O_{2}} :\frac{1}{K_{O_{2}}}\left(\frac{1}{r_{HO}}-\frac{u}{2}\right)\] \[p_{H_{2}O} :\frac{u}{K_{H_{2}O}}\]  
The polarization characteristics produced by the constant fuel utilization model are similar in nature to those in the literature. Figure 2: Zoomed in version of Figure 1 at low current values, showing changes in the curvatures. The constant fuel utilization model is validated with experimental data, taken from Reference [87], on a Siemens-Westinghouse high-power-density solid oxide fuel cell with ten channels (APS HPD10). The data were obtained graphically (as in References [88; 89; 90; 91]) from the results of Test 1018 in Figures 9 and 10 of Reference [87]. Figure 4 presents a model-versus-experimental comparison. The average absolute relative error was found to be 3.74%, a figure that is generally considered in the literature to indicate good agreement between theory and practice. Since the experimental current reported in Figures 9 and 10 of Reference [87] starts at 87 A for Test 1018, the Nernst voltage was estimated, by an approximate extrapolation (as in Figure 7-17 of Reference [85]), to be 0.74 V. The temperature was 1273 K, and the cell area was 870 cm\({}^{2}\)[87]. The resistance was estimated (following Reference [61]), from the slope of the (approximately) linear part of the experimental polarization (current-voltage) curve, to be 0.0006 \(\Omega\). The slope was obtained from the equation of the straight line derived by a least-squares linear regression fit of all but the first five experimental data points.