## 1 Introduction  
In the last three decades, lithium-ion batteries (LIBs) have prevailed over the other types of secondary batteries due to their advantageous properties, such as high energy density or low self-discharge [1]. The growing use of LIBs encourages the development of reusing and recycling techniques aiming to minimize the environmental impact of the LIB technology and to favor the circular economy goal [2, 3]. Different technologies for LIBs recycling have been developed [2]. Among them, pyro- and hydrometallurgical methods are the most commonly used today. Biometallurgical and combined bio-electro-pyro-hydrometallurgical methods are also being developed [4, 5, 6]. Pyrometallurgical methods are based on the treatment of LIBs at high temperatures to produce pyrolysis, metal reduction, and subsequent gas incineration. Despite the easiness of the procedure and the optimal technology readiness, some important drawbacks associated with this method are high energy consumption, hazardous gaseous emissions, loss of metals (Li) in the slag, high amount of slags, and limited recovery rates of metals [7]. On the other hand, hydrometallurgy recycling processes consist of the recovery of metals via acidic leaching, extraction, and precipitation. Compared to pyrometallurgical methods, hydrometallurgy has higher recovery efficiency, lower energy consumption, and lower gaseous emissions. Hence, hydrometallurgical LIB recycling is gaining attention as a more environmentally friendly approach than pyrometallurgical [8]. In hydrometallurgical methods, it is common to use strong inorganic acids, such as HCl [9], HNO\({}_{3}\)[10], and H\({}_{2}\)SO\({}_{4}\)[11, 12], as leaching reagents. The use of organic acids is being explored as an alternative to reduce the footprint of recycling processes [13]. Moreover, the addition of reductant agents has been probed to significantly improve leaching efficiency. The understanding on the interaction between solid waste and extracting solution is essential to optimize the selectivity and extraction efficiency of the metals [14]. For that, the proposal of models to describe solid-liquid heterogeneous interactions is especially useful. Different versions of the shrinking core model (SCM) [15; 16; 17; 18; 19] have been proposed to evaluate the kinetics of heterogeneous reactions. The SCM considers that the reaction takes place initially at the outer surface of the solid particle and involves the particle size decreasing as the products are formed. That way, the deeper areas of the solid take part in the reaction as they are exposed to the extracting fluid, which involves the constant movement of the reaction zone. The core and the size of the solid particle decrease during the reaction time until they are entirely consumed. These kinds of heterogeneous models take into account the rate control mechanisms, from surface chemical reaction control, Equation (1), to mass transfer control, Equation (2)  
\[t/t^{*}=1-(1-X)^{1/3}\quad;\quad\text{for chemical reaction control} \tag{1}\]  
\[t/t^{*}=1-3(1-X)^{2/3}+2(1-X)\quad;\quad\text{for film diffusion control} \tag{2}\]  
where \(t\) is time and \(X\) is the extraction extent of the target metal. The unreacted shrinking core model (USCM) takes into account the conversion of the reactant into another solid phase, which accumulates in the outer part of the particles. In this model, the size of the unreacted core decreases but the overall particle size remains approximately constant [20]. Several studies have proposed the USCM to evaluate the leaching kinetics of the metals from [21; 22; 23]. From comparison of experimental and theoretical results, [PERSON] et al. in [15] concluded that these models were not suitable to describe the leaching of metals from the cathode scraps. They also evaluated the kinetics of the leaching processes using the Avrami Equation (3), which was initially proposed to describe crystallization kinetics and to give an indication of the crystal growth process.