## 2 Materials and Methods
### Electrical Equivalent Circuit Model  
To reproduce the changes of impedance spectrum due to stratification a spatially-resolved equivalent electrical circuit (srEEC) was used. It consists of two layers to simulate inhomogeneous current distribution over the height of the electrodes, which is intensified by the stratification. Every layer contains a resistance connected in series to two RC-elements with the charge-transfer resistances and the double-layer capacitances for negative and positive electrode, respectively. The current dependency of the charge-transfer process in the lead-acid battery is significant in a wide range of charging and discharging currents and is considered here [23]. With the corresponding double-layer capacitance the characteristic form of the impedance spectrum is generated. A voltage source indicates the open-circuit voltage (OCV), which depends on the local acid density. The resulting srEEC is shown in Figure 3.  
\[V_{0,x}=\rho_{x}+0.84V\] \begin{table}
\begin{tabular}{c c c c} \hline \hline  & **Density** & **Volume** & **Amount of Substance** \\ \hline homo. acid _Ref. 1_ & 1.284 g cm\({}^{-3}\) & 1160 cm\({}^{3}\) & 5.72 mol \\ low conc. acid _Strat. 1_ & 1.246 g cm\({}^{-3}\) & 600 cm\({}^{3}\) & 2.56 mol \\ high conc. acid _Strat. 1_ & 1.308 g cm\({}^{-3}\) & 580 cm\({}^{3}\) & 3.19 mol \\ \hline homo. acid _Ref. 2_ & 1.284 g cm\({}^{-3}\) & 1150 cm\({}^{3}\) & 5.67 mol \\ low conc. acid _Strat. 2_ & 1.20 g cm\({}^{-3}\) & 605 cm\({}^{3}\) & 1.48 mol \\ high conc. acid _Strat. 2_ & 1.36 g cm\({}^{-3}\) & 630 cm\({}^{3}\) & 4.08 mol \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Amount of acid with different densities to generate _Strat. 1_ and _Strat. 2_. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Position** & _Ref. 1_ & **Before Strat. 1** & **After Strat. 1** & _Ref. 2_ & **Before Strat. 2** & _After Strat. 2_ & _Ref. 3_ \\ \hline top & 1.283 g cm\({}^{-3}\) & 1.256 g cm\({}^{-3}\) & 1.264 g cm\({}^{-3}\) & 1.283 g cm\({}^{-3}\) & 1.239 g cm\({}^{-3}\) & 1.246 g cm\({}^{-3}\) & 1.276 g cm\({}^{-3}\) \\ bottom & 1.284 g cm\({}^{-3}\) & 1.299 g cm\({}^{-3}\) & 1.289 g cm\({}^{-3}\) & 1.284 g cm\({}^{-3}\) & 1.346 g cm\({}^{-3}\) & 1.320 g cm\({}^{-3}\) & 1.276 g cm\({}^{-3}\) \\ difference & 0.001 g cm\({}^{-3}\) & 0.043 g cm\({}^{-3}\) & 0.025 g cm\({}^{-3}\) & 0.001 g cm\({}^{-3}\) & 0.107 g cm\({}^{-3}\) & 0.074 g cm\({}^{-3}\) & 0.000 g cm\({}^{-3}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Measured acid densities in the top and bottom of the test cell at different states of measurement procedure. The relationship between OCV and the density \(\rho_{x}\) is assumed to be linear (see Equation (2)), which is a good approximation for a wide range of the acid density [8]. The ohmic resistance of a lead-acid battery comprises of the lead grid resistance, the active mass and the electrolyte resistance as well as the resistances of the intersections from grid to active mass and to the electrolyte. Here the grid resistance is considered separately and is named \(R_{g,t}\) for the top part and \(R_{g,b}\) for the bottom part. The other components of the ohmic resistance are summarized in the resistances \(R_{i,AM,t}\) and \(R_{i,AM,b}\). The charge-transfer resistances of the negative \(R_{ct,neg}\) and positive \(R_{ct,pos}\) electrodes were connected in parallel to the corresponding double-layer capacitances (\(C_{dl,neg}\) and \(C_{dl,pos}\)). On the negative electrode a second electro-chemical process is present, which can be modeled with an additional RC-element [23]. However, this process could not be described yet [16] and the share to the total battery impedance is small, so that this second RC-element is not considered here. The Butler-Volmer equation describes the dependency of the charge-transfer rate \(I\) on the potential difference to equilibrium potential of the electrode \(\eta\) and is presented in Equation (3).