## 3 Modeling
### Protontic Ceramic Fuel Cell  
In this study, IR-PCFC is used. The reactions in the fuel cell consist of steam reforming, water-gas shift reaction (WGS), and hydrogen fuel cell reaction. They can be written as follows:  
\[\mathrm{CH_{4}+H_{2}O\leftrightarrow CO+3H_{2}} \tag{1}\]  
\[\mathrm{C_{3}H_{8}+3H_{2}O\leftrightarrow 3 CO+7H_{2}} \tag{2}\]  
\[\mathrm{C_{4}H_{10}+4H_{2}O\leftrightarrow 4 CO+9H_{2}} \tag{3}\]  
\[\mathrm{CO+H_{2}O\leftrightarrow CO_{2}+H_{2}} \tag{4}\]  
\[\mathrm{H_{2}+0.5O_{2}\leftrightarrow H_{2}O} \tag{5}\]  
Equations (1) to (3) represent the major steam reforming reactions for methane, propane, and butane. These reactions are followed by a WGS reaction, and a hydrogen fuel cell reaction, as described in Equations (4) and (5). The steam reforming and WGS reactions are calculated by chemical reaction conditions, while a fuel cell reaction is calculated by considering the influence of fuel utilization factors.  
\begin{table}
\begin{tabular}{c c} \hline
**Parameter** & **Value** \\ \hline PCFC & \\ Material of anode/electrolyte/cathode & Ni-SrC\({}_{0.9}\)Y\({}_{0.1}\)O\({}_{3\text{-}\alpha}\) [34]/SrC\({}_{0.95}\)Y\({}_{0.05}\)O\({}_{3\text{-}\alpha}\) \\  & [35; 36; 37]/La\({}_{0.6}\)Sr\({}_{0.4}\)C\({}_{0.2}\)Fe\({}_{0.8}\)O\({}_{3\text{-}\alpha}\) [38] \\ Thickness of anode/electrolyte/cathode, \(\upmu\)m & 220 [39]/50 [32; 40]/50 [8; 32; 40] \\ Number of cells & 50,000 \\ Active membrane area per cell, m\({}^{2}\) & 0.01 [32] \\ Operating pressure, atm & 3 [32] \\ Operating temperature, K & 873-1073 [41] \\ Anode electrode conductivity, S/m & \(-8\times 10^{-7}\text{T}^{4}+0.002\text{T}^{3}-3.365\text{T}^{2}+1875.6\text{T}-366,965\)[34] \\ Anode electrode porosity & 0.33 [42] \\ Anode electrode tortuosity & 2 [42] \\ Anode electrode pore radius, \(\upmu\)m & 0.75 [42] \\ Cathode electrode conductivity, S/m & \(8\times 10^{-6}\text{T}^{2}-0.0022\text{T}-0.5053\)[38] \\ Cathode electrode porosity & 0.21 [43] \\ Cathode electrode tortuosity & 2.5 [43] \\ Cathode electrode pore radius, \(\upmu\)m & 0.07 [43] \\ Electrolyte conductivity, S/m & 225.92 exp(\(-6.3\times 10^{3}\text{T}\)) [35; 40] \\ Fuel utilization & 0.85 [22; 23; 24; 27] \\ Air flow rate, mol/s & 56 (air stoichiometry of 2) [44] \\ Fuel flow rate, mol/s & 3.2 (air stoichiometry of 1.1) \\ Steam to carbon ratio & 2 \\ Gas turbine & 3:1 [32] \\ Pressure ratio & 85 [27] \\ Isentropic efficiency, \% & 1:3 [32] \\ Air compressor & 85 [27] \\ Pressure ratio & 1:3 [32] \\ Isentropic efficiency, \% & 85 [27] \\ \hline \end{tabular}
\end{table}
Table 1: Balance of plant of PCFC/GT hybrid systems.  
In PCFC, the maximum electrical voltage can be written as Nernst or open circuit voltage (OCV) as follows:  
\[\text{E}^{\text{OCV}}=\text{E}^{0}_{\text{(T,P_{0})}}+\frac{\text{RT}}{\text{nF}} \ln\frac{\text{P}_{\text{H}_{2}}\text{P}_{\text{O}_{2}}^{0.5}}{\text{P}_{\text{ H}_{2}}\text{O}} \tag{6}\]  
where \(\text{P}_{\text{H}_{2}}\), \(\text{P}_{\text{O}_{2}}\), and \(\text{P}_{\text{H}_{2}\text{O}}\) are the partial pressure of hydrogen at the anode side, oxygen at the cathode side, and water at the cathode side, while n, R, and F are the number of electron transfer, gas constant, and the Faraday constant.  
The reversible cell voltage at a standard pressure can be expressed as follows:  
\[\text{E}^{0}_{\text{(T,P_{0})}}=-\frac{\Delta\text{G}}{\text{nF}} \tag{7}\]  
where delta \(\Delta\text{G}\) is the function of temperature.  
In the fuel cell, Nernst voltage cannot be achieved due to irreversibility or losses. The real voltage output can be written as follows:  
\[\text{V}=\text{E}^{\text{OCV}}-(\eta_{\text{act}}+\eta_{\text{ohm}}+\eta_{ \text{conc}}) \tag{8}\]  
where \(\eta_{\text{act}}\), \(\eta_{\text{ohm}}\), and \(\eta_{\text{conc}}\) are the activation polarization, ohmic polarization, and concentration polarization, respectively.