## 2 Model Development  
The following section describes the major steps of the model development and the implementation itself. Table 2 shows a full list of the used symbols. The reactions in Equations (1) and (2) are balanced for a given equilibrium voltage, also called the open circuit voltage (\(E_{OCV}\)). The respective \(E_{OCV}\) for the nickel and iron \begin{table}
\begin{tabular}{c c l} \hline \hline
**Symbol** & **Unit** & **Description** \\ \hline \(\Delta G\) & J & Gibbs free energy change \\ \(F\) & A s mol\({}^{-1}\) & Faraday constant \\ \(R\) & J K\({}^{-1}\) mol\({}^{-1}\) & Universal gas constant \\ \(T\) & K & Temperature \\ \(z\) & & Number of electrons transferred \\ \(E_{OCV}\) & V & Open circuit v t V voltage \\ \(E\) & V & Cell voltage \\ \(\eta\) & V & Overpotential \\ \(\eta_{C}\) & V & Cathode overpotential \\ \(\eta_{A}\) & V & Anode overpotential \\ \(\eta_{m}\) & V & Overpotential in segment \(n\) \\ \(I\) & A & Current \\ \(i_{e,n}\) & A & Ionic current in segment \(n\) \\ \(i_{e,n}\) & A & Electronic current in segment \(n\) \\ \(j_{a,0}\) & A cm\({}^{-3}\) & Anodic standard exchange current density \\ \(j_{a}\) & A cm\({}^{-3}\) & Anodic exchange current density \\ \(j_{a,0M}\) & A cm\({}^{-3}\) & Charge exchange current density of active material M \\ \(j_{b,i}\) & A cm\({}^{-3}\) & Sum of exchange current densities \\ \(j_{0}\) & A cm\({}^{-3}\) & Standard exchange current density \\ \(j_{0,n,jM}\) & A cm\({}^{-3}\) & Standard exchange current density of active material M \\ \(d_{n,jM}\) & & Local depth of discharge \\ \(R_{0\mathrm{min}}\) & \(\Omega\) & Separator resistance \\ \(R_{e,n}\) & \(\Omega\) & Ionic resistance \\ \(R_{e,n}\) & \(\Omega\) & Electronic resistance \\ \(\kappa\) & \(\Omega^{-1}\) cm\({}^{-1}\) & Standard conductivity \\ \(\kappa_{n,j}\) & \(\Omega^{-1}\) cm\({}^{-1}\) & Ionic conductivity in segment \(n\) at timestep \(j\) \\ \(\sigma_{e,n}\) & \(\Omega^{-1}\) cm\({}^{-1}\) & Standard conductivity \\ \(\sigma_{e,n}\) & \(\Omega^{-1}\) cm\({}^{-1}\) & Electronic conductivity in segment \(n\) \\ \(\epsilon_{n,j}\) & & Cathode porosity \\ \(\epsilon_{M,n}\) & & Metal matrix porosity \\ \(\tau\) & & Totosity \\ \(a\) & & Charge transfer coefficient \\ \(l_{n}\) & cm & Length of each segment \\ \(A_{n}\) & cm\({}^{2}\) & Exchange area \\ \(q\) & W cm\({}^{-3}\) & Volumetric heat generation rate \\ \(\Phi_{1}\) & V & Potential in solid phase \\ \(\Phi_{2}\) & V & Potential in the electrolyte \\ \(\sigma_{MCl,nj}\) & mol cm\({}^{-3}\) & Molar volume of active material M in segment \(n\) at timestep \(j\) \\ \hline Subscripts & & \\ \hline \(j\) & Time step index \\ \(n\) & Segment number \\ \(N\) & Total number of segments \\ \(M\) & Metal indexing (\(N\),\(i\),\(Fe\)) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: List of symbols. reactions can be calculated according to Equation (3). Here, the Gibb's free energy (\(\Delta G\)) represents the convertible amount of chemical energy into electrical energy. \[E_{OCV}=\frac{\Delta G}{zF} \tag{3}\]  
with  
\[F:\text{Faraday constant}\]  
\[z:\text{numbers of electrons transferred in electrode reaction}\]  
While discharging the battery cell, electrons flow across the load, corresponding to the discharge current (\(I\)). The amount of electrons is directly proportional to the reaction rate. To overcome reaction and transport resistances, an additional amount of electrical energy is required, which is expressed with the so-called overpotential, \(\eta\). More precisely, the overpotential is the amount by which the cell voltage (\(E\)) deviates from the equilibrium voltage (\(E_{OCV}\)). Therefore, the overall cell voltage can be written as  
\[E=E_{OCV}-\eta \tag{4}\]  
The overpotential (\(\eta\)) can be distinguished into two main parts:  
1. Ohmic drop due to electronic and ionic conducting structures, e.g., separator and current collector. 2. Electrode losses/electrode overpotential: Combines electrode processes such as transport processes and charge transfer kinetics.