## 2 Materials and Methods
### Two-Stage Counter-Current Acid Leaching Method with the Addition of Hydrogen Peroxide  
The two-stage counter-current leaching tests were performed for all three materials--I, II, III--as shown in Figure 2. Thus, the experiments were carried out as follows: In the first cycle, a fresh portion of the battery mass was leached with the use of 5 dm\({}^{3}\) of 20% sulfuric acid solution--primary leaching solution. The material in the amount of 1 kg was added to the solution in portions over a period of about 5 min (the leaching solution was heated to a temperature around 10 \({}^{\circ}\)C lower than the set temperature before leaching). The tests were carried out at a temperature of 60 \({}^{\circ}\)C, for 1 h, with a solid-to-liquid-phase ratio of 1:5. After leaching, the suspension was filtered using the Buchner funnel, which allowed for obtaining a filtrate after the first stage (directed to the recovery of valuable components) and unwashed sludge, which was sent to the second stage of leaching. In the second stage of leaching, 20% aqueous solution of sulfuric acid was also used as the leaching agent--the primary leaching solution--but in this cycle, 30% H\({}_{2}\)O\({}_{2}\) solution was added in an amount of 1 dm\({}^{3}\). The sludge (after the first stage of leaching) was placed in a reactor with a capacity of 10 dm\({}^{3}\) by quantitatively transferring the entire mass to a beaker, using 5 dm\({}^{3}\) of the original leaching solution (20% sulfuric acid solution). The suspension was stirred and heated to 60 \({}^{\circ}\)C. After reaching the set temperature, 1 dm\({}^{3}\) of hydrogen peroxide was added within 30-60 min. The tests were carried out at a temperature of 60 \({}^{\circ}\)C for 1 h. After that, the obtained suspension was filtered using the Buchner funnel and the remaining material was washed with 0.5 dm\({}^{3}\) of water. The combined filtrate and washings after the second stage of leaching were used as the leaching solution for the next portion of the battery mass. In the first leaching stage of the next cycle, the tests were also carried out at 60 \({}^{\circ}\)C, for 1 h, with a solid-to-liquid-phase ratio of 1:5. After leaching, the suspension was filtered, obtaining a filtrate and an unwashed sludge, which was sent to the second stage of leaching, creating a cyclic method. For all tests, the volumes of filtrates were measured, in which the concentration of Co, Ni, Li, Cu, Mn, Fe, P and F ions was analyzed. For the solutions formed after leaching, pH was also measured and content of free sulfuric acid was determined. The leaching efficiency of the individual components was calculated as described below. Leaching efficiency for the first stage of the first cycle:  
\[W_{Me}=\frac{V_{x}\cdot C_{x}}{Z_{Me\%}\cdot m_{y}}\cdot 100\% \tag{2}\]  
where \(W_{Me}\) is the leaching efficiency of individual element \(Me\) (%), \(V_{x}\) is the volume of the solution after the \(x\)-th leaching (\(\mathrm{dm}^{3}\)), \(C_{x}\) is the concentration of element \(Me\) in the solution after the \(x\)-th leaching (\(\mathrm{g}/\mathrm{dm}^{3}\)), \(Z_{Me\%}\) is the weight percentage of the element in the battery mass (%) and \(m_{y}\) is the mass of the material used in cycle \(x\) (g). Leaching efficiency for the second stage of the first cycle:  
\[W_{Me}=\frac{V_{x}\cdot C_{x}+V_{x-1}\cdot C_{x-1}}{Z_{Me\%}\cdot m_{y}}\cdot 100\% \tag{3}\]  
where \(W_{Me}\) is the leaching efficiency of individual element \(Me\) (%), \(V_{x}\) is the volume of the solution after the \(x\)-th leaching (\(\mathrm{dm}^{3}\)), \(C_{x}\) is the concentration of element \(Me\) in the solution after the \(x\)-th leaching (\(\mathrm{g}/\mathrm{dm}^{3}\)), \(Z_{Me\%}\) is the weight percentage of the element in the battery mass (%), \(m_{y}\) is the mass of the material used in cycle \(x\) (g), \(V_{x-1}\) is the volume of the solution after the \(x-1\)-th leaching (\(\mathrm{dm}^{3}\)) and \(C_{x-1}\) is the concentration of element \(Me\) in the solution after the \(x-1\)-th leaching (\(\mathrm{g}/\mathrm{dm}^{3}\)).