## 3 Results
### Kinetic Analysis  
In order to perform a comparative analysis of the kinetics in both treatments (AS and EK), the experimental leaching data for the different metals have been fitted to a simple mathematical model proposed in the literature [8,16]. Such kinetic modeling analysis is important to determine important kinetic features of bioleaching, such as the reaction time, reaction rate, or reaction favorability. During leaching or bioleaching, oxidants such as H\({}^{+}\) and Fe\({}^{3+}\) react to extracted metal. This oxidation process can be controlled by one of the three following steps, with the slowest step typically being the rate-controlling process: (i) formation of the product layer (around the material), which creates resistance to the diffusion of oxidants to the material's surface (diffusion-controlled process); (ii) reaction of the oxidants at the particle surface, which will be slower than their diffusion when no product layer is formed or it does not affect the free movement of oxidants to the particles' surface (chemically controlled process); and (iii) liquid film diffusion, in which the leaching solution may resist the transport of oxidants to the surface, controlling the overall observed rate. [PERSON] et al. [8] suggest that considering film diffusion as the controlling step may not be prudent; instead, the chemical and product layer diffusion processes should be evaluated to determine the rate-controlling step. If the reaction rate is controlled by solid product layer diffusion, it can be described  
Figure 4: Metal removal yields (%) at the end of experiments (AS: agitated slurry; EKP: EK-percolation tests; RT: reference test; BL: biologically produced leaching medium). using the shrinking core model theory (Equation (1)). Conversely, if the leaching rate is controlled by a chemical reaction at the particle's surface, Equation (2) is used to represent leaching kinetics [16]. Both equations describe a linear plot with gradient k, where Ft represents the fraction of metal mobilized at time t, calculated from the solubilization data and the initial total metal concentration in tailings. It was assumed that the concentration of the leaching agent (Fe\({}^{3+}\)) was high enough to not be a limiting reagent, which is a necessary condition for using these two conventional models [16]. \[kt=1-\frac{2}{3}F_{t}-\left(1-F_{t}\right)^{2/3} \tag{1}\]  
\[kt=1-\left(1-F_{t}\right)^{1/3} \tag{2}\]  
Figure 5 shows the fitting of experimental data to Equations (1) and (2) in the case of Mn leaching (Mn is used as an example, and the following discussion applies to other metals as well). [PERSON] [28] proposed that plotting these equations in a bioleaching process reveals two stages: the first stage shows a progressive increase in the slope (k), while the second stage exhibits linear behavior with a constant and maximum k value. The first stage would correspond to an initial microbial colonization or activation phase, during which an acidophilic microbial biofilm is forming. In the second stage, the biofilm would be fully developed, colonization is complete, and the leaching rate reaches its maximum and remains constant, depending on the availability of the mineral surface area. In the present work, the first stage was not considered to occur for obvious reasons: leaching was induced by the addition of chemical oxidants in the BL medium and, therefore, the microbiological contribution may be considered negligible. Figure 5: Kinetic modeling of Mn leaching data: (**Left Column**): agitated slurry system; (**Right Column**): EK-percolation. Reference test (white points) and using BL medium (black points). Two models are plotted: (**a**) Top figures: considering the leaching rate controlled by a solid product layer diffusion (shrinking core model); (**b**) Bottom figures: considering the leaching rate controlled by a chemical reaction at the particles’ surface (shrinking particle model).