## 2 Model Development
Equation (4) can be rewritten as \[E=E_{OCV}-IR_{ohmic}-\eta_{c}-\eta_{a}\] (5)  
with the resistance of the separator (\(R_{ohmic}\)), the overpotential in the cathode space (\(\eta_{c}\)) and the overpotential in the anode space (\(\eta_{a}\)). The overpotential in the separator can be calculated based on [PERSON]'s law as well as the corresponding material parameters. However, the dependencies of the overpotential on electrode losses are more complex, in particular regarding the losses in the porous cathode area. That means that Na ions enter through the solid separator into the cathode space. In the case of discharging, the ions are transported by the liquid electrolyte towards the side of the reaction. At the interface of the active material and the electrolyte, the electrochemical charge transfer occurs dependent on the current density (\(j\)). In cases where all active material has reacted at the separator interface, the ions need to migrate deeper into the cathode area. Restricted by the porous structure of the cathode, the ion path is not direct (described by the tortuosity (\(\tau\))) and a certain resistance needs to be overcome. Likewise, the electrons flow through the complex metal matrix towards the side of the reaction. To account for this complex interplay of electrochemical reactions and charge transport in the model, the cathodic overpotential (\(\eta_{c}\)) is calculated according to Orchard and Weaving [15] based on an equivalent circuit shown in Figure 2.  
The cathode space is separated into \(N\) segments with equal volume (\(v_{n}\)) along the cathode radius (\(r_{n}\)). Each segment is represented by a combination of three electrical components. All three components contribute to the entire cathodic overpotential, \(\eta_{c}\). * \(j_{n}\) is the current density due to the charge transfer reaction and the corresponding overpotential, \(\eta_{n}\), in the cathode segment \(n\) (described by the Butler-Volmer equation) [11; 13; 15; 24]. * \(R_{\kappa,n}\) represents the ionic resistance of the secondary electrolyte in the cathode segment \(n\), leading to an Ohmic potential loss. * \(R_{\sigma,n}\) represents the electrical resistance of the nickel and iron matrix in the cathode segment \(n\), leading to an Ohmic potential loss. Either the discharge or charge processes are divided into \(j\) timesteps. For every timestep \(j\), the properties of each segment \(n\) are calculated with respect to the segment \(n-1\) and the time step \(j-1\). Initially, each segment is assumed to have a uniform quantity of active species (\(MCI_{2}\) with \(M=Ni\) or \(Fe\)) and \(NaCl\). At the separator-electrode interface, the current flowing into the network is purely ionic, \(i_{\kappa,1}\), and hence must be equal to the electrical current flowing out of the network, \(i_{\sigma,N}\), as summarized in Equation (6). \[i_{\sigma,N,j}=i_{\kappa,1,j}=-\sum_{n=1}^{N}j_{n,j}v_{n,j} \tag{6}\]  
Any oxidation (charge) or reduction (discharge) of the active compound includes a charge transfer process, which can be described by the Butler-Volmer equation (Equation (7)), where the contribution of a single segment is given by the current density, \(j_{n,j}\)[11; 15; 25]. This process represents the phase change of ionic current in the electrolyte to electronic current in the metal matrix. \[j_{n,j,M}=j_{0,n,j,M}[(e^{(\frac{(1-a)\pi_{n,j}F}{RT})})-(e^{(\frac{-a\pi_{n, j}F}{RT})})] \tag{7}\]  
with  
\[\alpha:\text{Transfer coefficient}\]  
\[E_{OCV}=\frac{\Delta G}{zF} \tag{3}\]0  
\[E_{OCV}=\frac{\Delta G}{zF} \tag{3}\]1  
In Equation (7), \(j_{0,n,j,M}\) represents the standard exchange current density. Orchard and Weaving showed that a dependency on the local depth of discharge (\(d_{n,j,M}\)) must be considered:  
\[E_{OCV}=\frac{\Delta G}{zF} \tag{3}\]2  
\[E_{OCV}=\frac{\Delta G}{zF} \tag{3}\]3 Equivalent circuit representing the segmented cathode space based on Orchard and Weaving [15].