## 3 Results and Discussion  
In this section, the experimental and simulation results are presented and discussed for the two experimental cases. The parameters used for the simulations are listed in Table 2. The software COMSOL Multiphysics 5.5 was used for the numerical integration of the partial differential equations describing the dissolution of LiCoO\({}_{2}\) particles and for the parameter optimization. Figure 2 shows the percentage of extracted Co and Li (mol/mol%) based on the initial total amount of Co and Li in the solid for Experiment A (the extraction experiment with S/L = 5 g/L in 0.1 M HCl). The extracted Co and Li correspond to the Co\({}^{2+}\) and Li\({}^{+}\) in the solution. According to the results, the extraction is very fast in the first hours, with a nearly lineal tendency. Then, the extraction rate decreases, and it becomes asymptotically stabilized. This behavior is consistent with the described switch in the rate controlling mechanisms. As observed before, the extraction of Li is approximately twice as much as that of Co, and the maximum extracted Li is around 65%. The pH evolution has a similar tendency: Protons are consumed during the first hours of extraction. After approximately 50 h, the pH stabilizes to a value of pH \(\approx\) 1.5. This also indicates that the reaction was not stopped due to depletion of protons but for an increase on the resistance to the mass transport. The simulation results are also presented in Figure 2 in order to be compared with the experimental data. It can be observed that the simulation results match fairly well with the experimental observations. The presented model predicts the switch of process rate control as the resistance of the diffusion of protons increases with the thickness of the crust. To better understand the change of the controlling mechanisms, in Figure 3a, the calculated Damkohler number for the simulation related to Experiment A is presented. It can be observed that the reaction rate changes from kinetic to diffusion control \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Parameter** & **Value (Case A)** & **Value (Case B)** & **Unit** & **Description** \\ \hline \(k_{r}\) & \(8\times 10^{-10}\) & \(6\times 10^{-10}\) & mol\({}^{-0.5}\)m s\({}^{-1}\) & Kinetic constant \\ \(k_{D}\) & \(-5.2\times 10^{7}\) & \(-4.1\times 10^{7}\) & m\({}^{-1}\) & Diffusion resistance factor \\ \(\alpha\) & & 3/2 & (–) & Order of reaction for H\({}^{+}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Parameters used in the simulation results. Figure 2: Experimental (marks) and simulation (lines) results for the case of \(S/L=5\) g/L in 0.1 M HCl. (\(Da_{II}=1\)) at approximately 50 h. Figure 3b shows the simulation results on the radius decrease of the unreacted core. This value is used to calculate the increase of the resistance modelled in Equation (14), as a consequence of the increase of the thickness of the outer crust. Figure 4 shows the experimental and simulation results for the experiment with \(S/L=50\) in 2.5 M HCl. In this case, proton concentration is presented instead of pH due to the high concentration of protons, producing negative pH values. It can be observed that in experiment B, with 10 times higher solid to liquid ratio and 25 times higher acid concentration than in experiment A, asymptotic stabilization occurs much faster. Experiment B reaches asymptotic stabilization in approximately 25 min. However, despite the faster behavior, the same features are observed in these conditions: The maximum Li recovery is near 70%, and the amount of Li extracted is twice as much as that of Co. The Li%, Co%, and proton concentration profiles obtained in the simulations presented in Figure 4 were obtained with the parameters listed in Table 2, case B. These values are very similar to those obtained for experiment A. There is a slight variation on the most suitable value for the kinetic constant, which can be attributed to small temperature differences between the experiments.