## 1 Introduction
These methodologies provide a structured framework for DOE, enabling the efficient exploration of the effects of multiple factors and their interactions on leaching outcomes. This is particularly advantageous in leaching processes, where variables such as pH, temperature, reagent concentration, and time significantly influence the extraction efficiency of metals like Co and Mn. RSM has proven effective in optimizing leaching conditions by evaluating the impact of multiple variables and their interactions. For instance, in the study of Co dissolution, RSM facilitated the identification of optimal conditions for Co recovery, achieving an impressive recovery rate of 95.79% under specific conditions of pH, temperature, and ferrous sulphate concentration [18]. Similarly, RSM has been utilized to optimize Mn leaching from low-grade ones by identifying the critical interaction between H\({}_{2}\)SO\({}_{4}\) concentration and the quantity of glucose [19]. Complementarily, DOE provides a systematic and efficient approach to exploring the experimental space, minimizing the number of experiments needed to comprehend the system. This is exemplified using central composite design within RSM, which effectively examines the influence of variables on leaching outcomes. Notably, both RSM and DOE excel in modeling complex interactions between variables, a critical feature in leaching processes where multiple factors may interact in a non-linear manner. For instance, the interaction between leaching time and the quantity of ferrous sulphate has been identified as significant in cobalt dissolution [18]. Recent studies have investigated optimization techniques for Cu and Co extraction from ores, emphasizing the use of RSM and DOE. Response surface methodology (RSM) and artificial neural networks (ANN) have been applied to model and predict leaching behavior [20; 21]. RSM has been used to optimize parameters such as particle size, temperature, and redox potential for Cu extraction from chalcopyrite [22; 23]. Similarly, RSM has been applied to optimize leaching parameters for Cu extraction from hematite-dominated ore, considering acid concentration, temperature, and contact time [24]. ANN models have shown high accuracy in predicting copper and cobalt recovery, with correlation coefficients exceeding 0.95 [20; 21]. DOE approaches, such as central composite design, have been employed to investigate and optimize operational conditions for Cu ion recovery using emulsion liquid membrane methods [25]. These modeling techniques provide valuable tools for enhancing extraction processes and underscore the effectiveness of employing and/or combining RSM, DOE, and ANN methods for optimizing and predicting outcomes in the hydrometallurgical extraction of copper and cobalt. This approach offers improved efficiency and higher recovery rates. Finally, understanding the impact of time and the \begin{table}
\begin{tabular}{c c c} \hline
**Reaction** & \(\mathbf{\Delta G^{\circ}\) **(kJ)** & **Reaction No.** \\ \hline Fe\({}_{3}\)O\({}_{4}\) (s) + 4H\({}_{2}\)SO\({}_{4}\) (l) \(\leftrightarrow\) FeSO\({}_{4}\) (aq) + Fe\({}_{2}\)(SO\({}_{4}\))\({}_{3}\) (s) + 4H\({}_{2}\)O (l) & \(-\)264.27 & (1) \\
2 FeSO\({}_{4}\) (aq) + 2H\({}_{2}\)SO\({}_{4}\) (aq) + MnO\({}_{2}\) (s) \(\leftrightarrow\) Fe\({}_{2}\)(SO\({}_{4}\))\({}_{3}\) (s) + 2H\({}_{2}\)O (l) + MnSO\({}_{4}\) (aq) & \(-\)221.44 & (2) \\ CoO + H\({}_{2}\)SO\({}_{4}\) \(\leftrightarrow\) CoSO\({}_{4}\) + H\({}_{2}\)O & \(-\)115.43 & (3) \\ Co\({}_{3}\)O\({}_{4}\) + 4H\({}_{2}\)SO\({}_{4}\) + 2 FeSO\({}_{4}\) \(\leftrightarrow\) 3 CoSO\({}_{4}\) + 4H\({}_{2}\)O + Fe\({}_{2}\)(SO\({}_{4}\))\({}_{3}\) & \(-\)354.18 & (4) \\ \hline \end{tabular}
\end{table} 
Table 1: Suggested reactions for the extraction of Co and Mn from deep-sea minerals [12]. crust/Fe ratio on the leaching process of Co and Mn from ferromanganese submarine crusts is crucial to optimizing efficiency, reducing costs, and improving process sustainability.