## 2 Modeling
### Fuel Processor Description
Since the entire system must operate at a steady state, the total reactive energy required (\(\mathrm{Q_{RE}}\)) is equal to the amount of energy supplied by the burner through combustion (\(\mathrm{Q_{CE}}\)). \[Q_{CE}=\sum_{ \begin{subarray}{c}Thermal\ energy\\ supply\ part\end{subarray} }\dot{m}_{i}h_{i}=\dot{m}_{fuel}\Delta h_{fuel}+\dot{m}_{Air} \Delta h_{Air}-\dot{m}_{Exhauset\ gas}\Delta h_{Exhauset\ gas} \tag{6}\]  
\[Q_{RE}=\sum_{ \begin{subarray}{c}Reaction\ part\end{subarray} }\dot{m}_{i}h_{i}= \dot{m}_{PNG}\Delta h_{PNG}+\dot{m}_{Water}\Delta h_{water}-\dot{m}_{Reformate} \Delta h_{Reformate} \tag{7}\]  
The required energy of the reaction section can be calculated by summing the heat of the reaction and the heat energy for heating the reactants, or for cooling the products as shown in Equation (8). In Equation (8), each energy is calculated by the enthalpy difference between the inlet stream and outlet stream. The enthalpy of each stream is a function of the temperature, flow rate, and degree of reaction. \[Q_{RE}=-Q_{H1}-Q_{H2}-Q_{SR}+Q_{C1}+Q_{C2}+Q_{C3}+Q_{HTS}+Q_{LTS} \tag{8}\]  
Figure 5 shows the above explanation intuitively. \begin{table}
\begin{tabular}{c c c} \hline \hline \(\mathbf{T_{SR}}\) & \multicolumn{2}{c}{**Equilibrium Constant**} \\ (\(\mathbf{{}^{\circ}C}\)) & **Steam Reforming (\(K_{SR}\))** & **Water-Gas Shift (\(K_{WGS}\))** \\ \hline
100 & 0 & 2289 \\
200 & 0 & 210.8 \\
300 & 0 & 38.83 \\
400 & 0.0001 & 11.72 \\
500 & 0.0098 & 4.904 \\
600 & 0.5212 & 2.551 \\
700 & 12.54 & 1.541 \\
800 & 168.87 & 1.036 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The equilibrium constants for reactor temperatures. Figure 4: Schematic diagram of the energy balance of the fuel processor system. The thermal efficiency of the fuel processor system (\(\eta_{FP}\)) is defined as Equation (9) [32; 33; 34],  
\[\eta_{FP}=\frac{n_{H_{2}}LHV_{H_{2}}}{(n_{BNG}+n_{PNG})*LHV_{NG}+n_{AOG~{}H_{2}} LHV_{AOG~{}H_{2}}+n_{AOG~{}CH_{4}}LHV_{CH_{4}}} \tag{9}\]  
The thermal efficiency of the fuel processor is calculated by dividing the energy of the produced hydrogen by the total energy supplied to the process (PNG, natural gas to the burner, AOG). Energy is expressed as the product of the moles and lower heating value (\(LHV\)). The \(n_{BNG}\) is the number of moles of natural gas introduced to the burner, and the \(n_{PNG}\) is the number of moles of natural gas introduced to the reaction part. The hydrogen and methane moles in AOG are denoted by \(n_{AOG~{}H_{2}}\) and \(n_{AOG~{}CH_{4}}\). The product gas containing the produced hydrogen is cooled in the cooler Q\({}_{\rm C3}\) system to remove the moisture in the product gas and is then supplied to the PEMFC. Hydrogen is supplied to the anode side and oxygen from the air is supplied to the cathode side of the PEMFC. Using an electrochemical reaction, electrical energy is produced. To calculate the amount of electrical energy produced by the PEMFC, fuel utilization efficiency and the irreversibility of the PEMFC should be considered. The detailed simulation condition for the fuel processing system is summarized in Table 2.  
The S/C ratio is defined as in Equation (10) and is a value obtained by dividing the moles of steam introduced by the moles of carbon. \[S/C~{}\text{ratio}=\frac{Steam~{}\text{moles}}{Carbon~{}\text{moles}} \tag{10}\] \begin{table}
\begin{tabular}{c c} \hline \hline
**Variables** & **Condition** \\ \hline Temperature of steam reformer (T\({}_{\rm SR}\)) & 500–900 \({}^{\circ}\)C \\ Temperature of HTS (T\({}_{\rm HTS}\)) & 300–400 \({}^{\circ}\)C \\ Temperature of LTS (T\({}_{\rm LTS}\)) & 200–300 \({}^{\circ}\)C \\ Steam-to-carbon ratio (S/C) & 1–5 \\ The molar flow rate of natural gas supplied to steam reformer (\(\dot{n}_{PNG}\)) & 0.7\(\sim\)1.2 mol/min \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Simulation conditions for the fuel processing system. Figure 5: Detailed thermal energy flow of the reaction part.