## 5 Constant Fuel Flow
### Constant Hydrogen-Oxygen Input Flow Ratio
Interestingly, given current, \(p_{H_{2}O}\) is (conditionally) independent of \(q_{H_{2}}^{in}\). Figure 6 shows an example of how the three pressures change with current when the flow rates are held constant (the plots are produced with the parameter values in Table 1 and a fixed \(q_{H_{2}}^{in}=0.1\) mol/s). The polarization curve for the constant flow rate condition (see Equation (40)) is shown in Figure 7 where the Nernst loss and ohmic loss are shown separately. The positive magnitudes of the two losses are shown; their signs are accounted for in the output voltage calcul  
\[\dfrac{p_{H_{2}}^{final}}{p_{H_{2}}^{initial}}=\dfrac{q_{H_{2}}^{in}-\dfrac{I_{ FC}^{final}}{2F}}{q_{H_{2}}^{in}-\dfrac{I_{FC}^{final}}{2F}} \tag{33}\]1 Linear dependence of the three partial pressures upon current (hydrogen and oxygen input flow rates are held constant). the parameter values in Table 1 along with \(E_{Nernst}=1\) V, \(q_{H_{2}}^{in}=0.004\) mol/s and \(r=3.28125\times 10^{-3}\)\(\Omega\). The polarization plot shows the same trend that is generally seen in constant flow polarization in the literature. \[\dfrac{p_{H_{2}}^{final}}{p_{H_{2}}^{initial}}=\dfrac{q_{H_{2}}^{in}-\dfrac{I_{ FC}^{final}}{2F}}{q_{H_{2}}^{in}-\dfrac{I_{FC}^{final}}{2F}} \tag{33}\]2 Effect of the input hydrogen flow rate on polarization (both input flow rates are held constant, with the oxygen flow rate proportional to the hydrogen flow rate). \[\dfrac{p_{H_{2}}^{final}}{p_{H_{2}}^{initial}}=\dfrac{q_{H_{2}}^{in}-\dfrac{I_{ FC}^{final}}{2F}}{q_{H_{2}}^{in}-\dfrac{I_{FC}^{final}}{2F}} \tag{33}\]3 Polarization curve when hydrogen and oxygen input flow rates are held constant; the Nernst loss and the ohmic loss are plotted separately. The effect of the (constant) input fuel flow rate on the polarization behavior is presented in Figure 8, where three different \(q_{H_{2}}^{in}\) values are used: 0.02 mol/s, 0.004 mol/s, and 0.001 mol/s. The figure shows that a higher input flow rate leads to a higher voltage for the same load current, an observation that fits in with the concept of a higher reactant concentration leading to a higher potential. Figure 9 shows the variation in power characteristics corresponding to different (constant) fuel flow rates (the fuel flow rates in this figure are the same as the ones in Figure 8); a higher fuel flow rate results in higher power, as expected. Figures 8 and 9 use \(E_{Nernst}\) = 1 V and \(r\) = 3.28125 \(\times\) 10\({}^{-3}\)\(\Omega\). Results of model validation are shown in Figure 10, where the experimental polarization data (on a planar single SOFC at ENEA laboratories) were graphically obtained from Reference [61]. The results show reasonably good agreement, with an average absolute relative error of 3.96%. The validation was done under the following operational condition [61] for both the model and the experiment: temperature: 923 K; pressure: atmospheric; \(q_{H2}^{in}\): 0.00112 mol/s; \(q_{Q2}^{in}\): 0.00038 mol/s; anode: 96% hydrogen, 4% water; cathode: 21% oxygen, 79% nitrogen; cell area: 121 cm\({}^{2}\). The calculated value of the model \(E_{Nernst}\) was 1.122 V, with the value of \(E_{0}\) in Equation (2) estimated from the following relationship [61; 95; 96]:  
\[\dfrac{p_{H_{2}}^{final}}{p_{H_{2}}^{initial}}=\dfrac{q_{H_{2}}^{in}-\dfrac{I_{ FC}^{final}}{2F}}{q_{H_{2}}^{in}-\dfrac{I_{FC}^{final}}{2F}} \tag{33}\]4  
Given the \"systematic error of about 0.018 V\" in Reference [61] between the theoretical Nernst voltage and its experimental counterpart, the experimental \(E_{Nernst}\) in the present validation was taken as 1.122 V \(-\) 0.018 V = 1.104 V. A resistance of 0.0043 \(\Omega\) was used for validation; this value was obtained  
\[\dfrac{p_{H_{2}}^{final}}{p_{H_{2}}^{initial}}=\dfrac{q_{H_{2}}^{in}-\dfrac{I_{ FC}^{final}}{2F}}{q_{H_{2}}^{in}-\dfrac{I_{FC}^{final}}{2F}} \tag{33}\]5 Power-versus-current behavior (both input flow rates are held constant, with the oxygen flow rate proportional to the hydrogen flow rate).