## 2 Materials and Methods
### Model  
The loss of energy efficiency in a chlor-alkali process is mostly related to the activation overpotentials and ohmic overpotentials associated with the membrane, hardware and electrolyte. At each electrode, the activation overpotential, \(\eta_{\text{act}}\), is related to the electrochemical kinetics and affects the total current density, \(j\), which according to the Butler Volmer equation, relates to the activation overpotential as follows [18]:  
\[j=j_{0}\bigg{[}e^{(\frac{\sigma+F\cdot\eta}{kT})}-e^{(\frac{-(1-\alpha)+F\cdot \eta}{kT})}\bigg{]} \tag{7}\]  
where \(\eta\) represents a potential loss due to activation, \(j_{0}\) is the exchange current density of the reaction, \(\alpha\) is the transfer coefficient, \(n\) is the number of the electrons exchanged in the reaction, \(F\) is the Faraday  
Figure 1: Scheme of a bipolar electrolyser where the stack consists of four cells connected in series: the potential between the terminals, \(E\), is the sum of the individual potential of the cells and the same current density, \(j\), flows through the cells. constant, \(T\) is the absolute temperature and \(R\) is the gas constant. At high current densities, the second term in Equation (7) can be neglected and the overpotential is given by the Tafel equation [18]:  
\[\eta=-\frac{R\cdot T}{\alpha\cdot n\cdot F}\ \ln j_{0}+\frac{R\cdot T}{\alpha \cdot n\cdot F}\ \ln j \tag{8}\]  
Parameters \(\alpha\) and \(j_{0}\) depend on the composition and pH of the electrolyte and on nature and surface characteristics of the coatings of the electrodes. The overpotential can, thus, be obtained by a linear regression:  
\[\eta=b\cdot(-\ln j_{0}+\ \ln j) \tag{9}\]  
where the parameter \(b\) is called Tafel slope. Under Tafel conditions, normally for \(\eta>50\) mV [6], Equation (9) can be introduced into Equation (6):  
\[E=[E_{\text{therm}}-\ (b_{\text{a}}\cdot\ln j_{0,\text{a}})+(b_{\text{c}} \cdot\ln j_{0,\text{c}})]+[(b_{\text{a}}-\ b_{\text{c}})\cdot\frac{\ln j}{j}+( \ R_{\text{mem}}+R_{\text{hw}}+R_{\text{sol}})]\cdot j \tag{10}\]  
\(E\) as a function of \(j\), as described by Equation (10), is not linear but experimentally it follows a mostly linear relationship over the current density range of 2 kA\(\cdot\)m\({}^{-2}\) to 5 kA\(\cdot\)m\({}^{-2}\), corresponding to the Tafel region. In this way, \(E\) vs. \(j\) values can be fitted using Equation (11), where \(k\)-factor corresponds to the slope of the \(E\) vs. \(j\) curve and \(E_{0}\) is the interception or the open circuit potential [6]:  
\[E=E_{0}+k\cdot j \tag{11}\]  
where:  
\[E_{0}=E_{\text{therm}}-(b_{\text{a}}\cdot\ln j_{0,\text{a}})+(b_{\text{c}} \cdot\ln j_{0,\text{c}}) \tag{12}\]  
\[k=(b_{\text{a}}-b_{\text{c}})\cdot\frac{\ln j}{j}+(R_{\text{mem}}+R_{\text{hw }}+R_{\text{sol}}) \tag{13}\]  
This linear approach is usually called \(k\)-factor analysis. Since the overall ohmic resistance is normally dominant, it is usually considered that the \(k\)-factor depends mostly on the ohmic overpotential, where the membrane has the major contribution [6]. [PERSON] [19] proposed to record the potential of small groups of cells connected in series and to compare it with a reference potential. An alarm would be activated, and the supply of energy to the cell disconnected, whenever the measured potential varies from the reference potential by more than a preset potential [19]. In a day to day operation, the imposed current density fluctuations result in a potential change. Consequently, it is important to remove these fluctuations from the overpotential analysis. [PERSON] [20] proposed an improvement to the previous approach by [PERSON] correcting the real potential for a reference constant current density, according to the Equation (14). \[E_{j,\text{ref}}=E_{0}+(E-E_{0})\cdot\frac{j_{\text{ref}}}{j} \tag{14}\]  
where \(E_{j,\text{ref}}\) is the corrected potential for a reference current density (\(j_{\text{ref}}\)). Here, it is assumed again that the ohmic overpotentials dominate within the Tafel region.