## 2 Methodology
### Mathematical Model of the Isomerization Process  
[PERSON] et al. [15] describe an industrial isomerization process conducted in a series of three plug-flow reactors (PFRs). The PFR design equation, expressed in terms of reaction extent, was employed in this work to simulate the reactors [22]:  
\[\frac{d\varepsilon_{j}}{dV_{r}}=f(\varepsilon_{j},T)=r_{j,\varepsilon} \tag{6}\]  
\(V_{r}\) is the reactor volume. Equation (6) shows that \(f\) is a function of temperature along the reactor (\(T\)) and the reaction extent (\(\varepsilon_{j}\)). Therefore, the reaction rate (\(r_{j,\varepsilon}\)) must also exhibit this dependence, as shown in the following equation [17]:  
\[r_{j,\varepsilon}=\frac{1}{v_{i,j}}r_{j} \tag{7}\]  
Given that reaction rates are assumed to be elementary and expressed in terms of mole fractions, that are dependent on \(\varepsilon\) as discussed earlier, the mass balance represented by Equation (5) must be included. For irreversible reactions, the generalized elementary reaction rate is formulated as follows:  
\[r_{j,\varepsilon}=k_{j}(\prod y_{i}v_{i,j})_{reac} \tag{8}\]  
and for reversible reactions, the generalized expression is as follows:  
\[r_{j,\varepsilon}=k_{j}\bigg{[}(\prod y_{i}v_{i,j})_{reac}-\frac{(\prod y_{i}v _{i,j})_{prod}}{k_{e,j}}\bigg{]} \tag{9}\]  
The subscripts _reac_ and _prod_ represent reactants and products, respectively. \(k_{j}\) are Arrhenius rate constants, given by [23]:  
\[k_{j}=A_{j}\text{exp}\bigg{(}\frac{-E_{j}}{RT}\bigg{)} \tag{10}\]  
where \(A_{j}\) represents the pre-exponential factor and \(E_{j}\) denotes the activation energy. Equations (8) and (9) illustrate the dependence of \(r_{j,\varepsilon}\) with respect to \(T\). Given that the isomerization process is adiabatic, the energy balance must be included to determine \(T\) and solve the model. This balance is calculated using residual enthalpies, considering a hypothetical ideal gas state with the same composition as the respective stream, as shown in the following equation [17]:  
\[\Delta H_{PFR}=-F_{o}\Big{[}H_{in}^{R}+\Delta H_{T_{ref},T_{0}}^{ig}\Big{]}+ \sum_{j}\varepsilon_{j}\Delta H_{j}^{25^{\prime}C}+F\Big{[}\Delta H_{T_{ref},T }^{ig}+H_{out}^{R}\Big{]}=Q \tag{11}\]where \(Q=0\) for the adiabatic process and \(\Delta H_{PFR}\) is the enthalpy change in the reactor. \(H_{in}^{R}\) and \(H_{out}^{R}\) are the residual enthalpies at the reactor inlet and outlet [24], respectively. \(\Delta H_{T_{ref},~{}T_{0}}^{ig}\) and \(\Delta H_{T_{ref},T}^{ig}\) correspond to the ideal gas enthalpy changes evaluated from the reference state to at the inlet (\(T_{0}\)) and outlet (\(T\)) temperatures of the reactor, respectively. Finally, \(H_{i}^{i}\)[25] is the enthalpy of formation of component \(i\) at 25 \({}^{\circ}\)C.  
The design equation of the PFR, Equation (6), generates a system of differential equations, which is solved using the explicit fourth-order Runge-Kutta (RK4) method, based on initial conditions. The model solution requires the simultaneous solution of the system of differential equations and the energy balance at each integration step (\(n\)). At each \(n\), a value for \(T_{n}\) is determined, and the flows are calculated based on values of \(\epsilon_{j,n}\). The energy balance for the integration step \(n\) results in the following:  
\[\Delta H_{PFR,n}=-F_{n-1}\Big{[}H_{n-1}^{R}+\Delta H_{T_{ref},T_{n-1}}^{ig} \Big{]}+\sum_{j}\big{(}\epsilon_{j,n}-\epsilon_{j,n-1}\big{)}\Delta H_{j}^{25 \,\mathrm{C}}+F_{n}\Big{[}H_{n}^{R}+\Delta H_{T_{ref},T_{n}}^{ig}\Big{]}=Q_{n} \tag{12}\]  
\(n-1\) corresponds to the outlet conditions of the previous integration step. The term \((\epsilon_{j,n}-\epsilon_{j,n-1})\) represents the flows that react in reaction \(j\). For the first integration step, \(\epsilon_{j,n-1}=\epsilon_{j,0}=0\). As a first approximation for \(T_{n}\), and considering the exothermic nature of the process, it is assumed that \(T_{n,s}=T_{n-1}+2\)\({}^{\circ}\)C, where the subscript \(s\) denotes an assumed temperature.