## 2 Methodology
### Chemical Equilibrium Modeling  
The isomerization reactions of C\({}_{4}\), C\({}_{5}\), and C\({}_{6}\) are reversible, necessitating the consideration of chemical equilibrium data, which is obtained from experimental results [19; 20]. These data provide the equilibrium mole fractions at different temperatures, from which the experimental equilibrium constants (\(k_{e,j,exp}\)) can be calculated using the following expression [17]:  
\[k_{e,j,exp}=\left(\frac{P}{P_{0}}\right)^{v_{j}}\prod_{i}y_{i,exp}{}^{v_{i,j}} \tag{2}\]  
where \(v_{i,j}\) is the stoichiometric coefficient of component \(i\) in reaction \(j\), \(P\), and \(P_{0}\) denote the system and reference pressures, respectively. \(y_{i}\) is the mole fraction of component \(i\) in the mixture. In all cases, \(\ln\bigl{(}k_{e,j,exp}\bigr{)}\) varies linearly with the reciprocal of temperature (\(T\)):  
\[\ln\bigl{(}k_{e,j,exp}\bigr{)}=Ae_{j}+\frac{Be_{j}}{T} \tag{3}\]  
Equation (3) is of the van't Hoff type, where \(Ae_{j}\) and \(Be_{j}\) represent the change in entropy (\(\Delta S^{\circ}/R\)) and enthalpy \((-\Delta H^{\circ}\,/R)\) of the reaction, respectively.  
Once \(Ae_{j}\) and \(Be_{j}\) are determined, a temperature range is established in which, for each value, \(k_{e,j,cal}\) is calculated. This allows for the validation of the methodology and verification of the fit with experimental data through the solution of a reactor at equilibrium using Equation (4) [17].  
\[k_{e,j,cal}=\left(\frac{P}{P_{0}}\right)^{v_{j}}\prod_{i}y_{i,cal}{}^{v_{i,j}} \tag{4}\]  
The mole fractions, \(y_{i,cal}\), are determined using the molar flow rates of component \(i\) (\(F_{i}\)) and the total molar flow rate (\(F\)) at the reactor exit. In a system with multiple reactions, \(y_{i,cal}\) can be expressed as a function of the extent of reaction [21]:  
\begin{table}
\begin{tabular}{c c} \hline
**No.** & **Reaction** \\ \hline
1 & nC\({}_{4}\leftrightarrow\) iC\({}_{4}\) \\
2 & nC\({}_{5}\leftrightarrow\) iC\({}_{5}\) \\
3 & C\({}_{6}\) isomerization Rxn \\
4 & C\({}_{6}\) isomerization Rxn \\
5 & C\({}_{6}\) isomerization Rxn \\
6 & C\({}_{6}\) isomerization Rxn \\
7 & C\({}_{6}\) isomerization Rxn \\
8 & B + 3H\({}_{2}\rightarrow\) CH \\
9 & CH \(\rightarrow\) MCP \\
10 & MCP \(\rightarrow\) CH \\
11 & MCP + H\({}_{2}\rightarrow\) nC\({}_{6}\) \\
12 & CP + H\({}_{2}\rightarrow\) nC\({}_{5}\) \\
13 & 2 22 \(-\) DMB + H\({}_{2}\rightarrow\) 3 nC\({}_{5}\) \\
14 & 22 \(-\) DMB + 2 H\({}_{2}\rightarrow\) 3 C\({}_{2}\)H\({}_{6}\) \\
15 & 22 \(-\) DMB + H\({}_{2}\rightarrow\) 2 C\({}_{3}\)H\({}_{8}\) \\
16 & CH + H\({}_{2}\rightarrow\) nC\({}_{6}\) \\
17 & 22 \(-\) DMB + 5 H\({}_{2}\rightarrow\) 6 CH\({}_{4}\) \\ \hline \end{tabular}
\end{table}
Table 2: Set of reactions considered for kinetic modeling. ‘C\({}_{6}\) isomerization Rxn’ represents the respective reaction in the scheme under study.  
\[y_{i,cal}=\frac{F_{i}}{F}=\frac{F_{i,o}+\sum_{j}\
u_{i,j}\varepsilon_{j}}{F_{o}+ \sum_{j}\
u_{i}\varepsilon_{j}} \tag{5}\]  
\(F_{i,o}\) and \(F_{o}\) represent the initial molar flow rates of component \(i\) and the total feed, respectively, \(\
u_{j}\) is the overall stoichiometric coefficient for the reaction, and \(\varepsilon_{j}\) denotes the extent of reaction \(j\).  
Equation (4) yields a system of \(j\) algebraic equations. By solving this system, the \(j\) reaction extents are obtained, which are then used to calculate \(y_{i,jcal}\). The comparison of these compositions with experimental data allows for the evaluation of the model's accuracy.