## 2 System Establishment and Modeling
### Modeling
#### 2.2.3 CPOx Model  
The calculations for the reformates are all supposed to be at equilibrium since the reaction is fast. The chemical equilibrium is established using the minimization of the Gibbs free energy method (MGFE). The method is based on minimizing the Gibbs free energy of the system reacting [37]. This results in finding the molar values of the species in the reaction when that total energy reaches a minimum value at a particular temperature and pressure. We set constraints on the mass balance of the species. The product species of the CPOx reaction between air and C\({}_{12}\)H\({}_{26}\) will be C, CO, CO\({}_{2}\), H\({}_{2}\), CH\({}_{4}\), and H\({}_{2}\)O, essentially following the following reaction:  
\[\text{C}_{12}\text{H}_{26}\left(\text{g}\right)+a(\text{O}_{2}+3.76\text{N}_{ 2})\left(\text{g}\right)\to b\text{CO}(\text{g})+c\text{H}_{2\left(\text{g} \right)}+3.76a\text{N}_{2\left(\text{g}\right)}+d\text{CO}_{2\left(\text{g} \right)}+e\text{CH}_{4}+f\text{H}_{2}\text{O}_{\left(\text{g}\right)}+g \text{C}_{\left(\text{g}\right)}\]As air is being added to the reaction, only the oxygen (O\({}_{2}\)) portion (approximately 21% of all inlet air) is considered for this reaction. The O\({}_{2}\) is added to the fuel at the oxygen-to-carbon (O/C) ratio defined as the ratio of twice the molar oxygen flow rate versus the molar flow rate of the fuel carbon within a hydrocarbon reaction scheme [38]. A value of O/C of 1 denotes the stoichiometric partial oxidation point. \[\frac{O}{C}ratio=\frac{2\dot{n}_{O_{2}}}{\dot{x}\dot{n}_{C_{2}}H_{y}} \tag{43}\]  
The air source is the HPC outlet air flow, which has its own enthalpy, and the fuel added to the reformer is assumed to be vaporized on entry with a temperature of approximately 630 K. If the fuel inlet temperature exceeds this temperature, heterogeneous reactions would initiate before the catalyst, which is undesirable [39]. The exit flow enthalpy is therefore dependent on the variable HPC outlet air flow enthalpy, which changes with the operational points. Therefore, the O/C ratio for reforming is variable to satisfy the outlet temperature requirements and, in adiabatic reforming, the heat equilibrium must be satisfied. \[ \begin{split}\sum\limits_{j_{product}}^{J_{product}}\dot{n}_{ j_{product}}\left\{\Delta h^{\circ}_{f_{j_{product}}}(298.15)+\left(h^{\circ}_{j_{product}} \left(T_{final}\right)-h^{\circ}_{j_{product}}(298.15)\right)\right\}\\ =\\ \sum\limits_{j_{reactant}}^{T_{reactant}}\dot{n}_{i_{reactant}} \left\{\Delta h^{\circ}_{f_{reactant}}(298.15)+\left(h^{\circ}_{i_{reactant}}(T_{i_{reactant}})-h^{\circ}_{i_{reactant}}(298.15)\right)\right\}\end{split}  \tag{44}\]  
The energy balance equation is used to determine \(T_{final}\) through iteration with the equations below:  
\[\dot{m}_{in}h_{\tau,in}-\dot{m}_{out}h_{\tau,out}-\dot{Q}_{loss}=0 \tag{45}\]  
\[T_{final}=\frac{\sum\dot{m}_{in}cp_{in}T_{in}-Q_{loss}}{\sum\dot{m}_{out}Cp_{ out}} \tag{46}\]  
The iteration between the reformer outlet flow rate \(\dot{m}_{out}\) and the \(T_{final}\) is performed with the MGFE method, which utilizes the \(T_{final}\) to perform chemical equilibrium reactions at a set reactor pressure, which is preferably set at 300 kPa to maximize the outlet flow composition of H\({}_{2}\) and CO.  
\[g^{0}(T)=h^{0}(T)-T\mathbb{S}^{0}(T) \tag{47}\]  
\[\overline{g}_{i}(T)=g^{0}(T)+RT\ln\frac{y_{i}P}{P_{0}} \tag{48}\]  
\[Z=\sum\dot{n}_{i}\overline{g}_{i}=\sum\dot{n}_{i}g^{0}_{i}+RT\sum\dot{n}_{i} \ln\frac{y_{i}P}{P_{0}} \tag{49}\]  
where \(g^{0}_{i}(T)\) is the Gibbs free energy of the reactant and product species calculated at \(T=T_{final}\). Efficiencies are obtained based on the medium of fuel utilization, and in this case, the SOFC. The efficiency of the reformer is expressed by:  
\[\eta_{refo}=\frac{n_{CH_{4}}LHV_{CH_{4}}+n_{CO}\cdot LHV_{CO}+n_{H_{2}}\cdot LHV _{H_{2}}}{n_{C_{12}}H_{26}\cdot LHV_{C_{12}H_{26}}} \tag{50}\]  
where \(LHV\) is the lower heating value of each of the species' components in the equilibrium reaction.