Leaching state prediction for battery black mass using online sensor data†

Wei Song*^a, Andrey Yasinskiy^a, Fabian Diaz^a, Tobias Kleinert^b and Bernd Friedrich^a
aIME Process Metallurgy and Metal Recycling, Institute of RWTH Aachen University, Intzestr. 3, 52056 Aachen, Germany. E-mail: wsong@ime-aachen.de
^bChair of Information and Automation Systems for Process and Material Technology, RWTH Aachen University, 52064 Aachen, Germany

First published on 21st July 2025

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Green foundation 1. This study proposes a new approach for predicting the leaching state using easily measurable online sensor data. Compared to leaching kinetic models, this approach enhances model generalizability by considering the dynamic conditions of a new process and providing timely feedback for potential process optimization. 2. The leaching state of Al, Fe, Ni, Co, Li, and Mn can be accurately predicted using a multiple linear regression model with pH, conductivity, and temperature as input features. The model demonstrated an average adjusted R-squared score of 0.995 and an average prediction error of 1.1%. 3. The proposed method avoids the need for complex material characteristics and structural limitations and provides valuable insights for online process monitoring by predicting leaching state using sensor data. Advanced methods such as machine learning can be applied to simply the process and address more complex scenarios in future research.

1 Introduction

Hydrometallurgy is an extractive metallurgical technique and has been widely used to recycle valuable components from spent lithium-ion batteries (LIBs) with various aqueous solutions.¹ Leaching, frequently the initial step of hydrometallurgical processes, selectively dissolves target metals from solid matrix into a liquid phase using a leaching agent. This process is crucial for extracting valuable metals from both primary and secondary materials. Inorganic acids, such as sulfuric acid and hydrochloric acid, have been used to leach battery cathode materials, as hydrogen ions facilitate lithium extraction from spent LIBs.² Organic acids, such as citric acid and ascorbic acid, have less environmental impact and serve as alternative leaching agents for battery recycling. Hydrogen peroxide (H₂O₂) is commonly used as a reductant in black mass leaching, and its influence has been investigated in many studies.^3–5

Thermodynamics and kinetics are fundamental for understanding the leaching mechanisms. Thermodynamics analyzes the directions, energies, products, and equilibrium conditions of chemical reactions, but it does not consider the time factor. Conversely, kinetics considers time changes and thus provides a more realistic explanation of leaching phenomena.⁶ The chemical reaction rate, which describes the progress of a reaction, can be expressed as a function of conversion. Kinetic studies interpret reaction mechanisms by examining the influence of experimental conditions on the reaction rate. This understanding is crucial for process control, optimization, and reactor design. As leaching is often the initial step in hydrometallurgical processes, comprehending its mechanisms is essential.⁷ Specifically, in battery recycling, the leaching kinetics of critical metals (e.g. Ni, Mn, Co, and Li) from black mass is a key consideration for industrial-scale recycling.⁸

Various mathematical models simplify kinetic investigations of heterogeneous reactions.^9,10 Reuter et al. employed a generalized neural-net kinetic rate equation to simulate a metallurgical and mineral processing system and identify optimal process conditions.¹¹ In leaching kinetic modeling, the progressive conversion model (PCM) is a simple approach used to predict leaching completion time. It assumes a constant particle size and is applicable when the diffusion rate through the particle's pores is high. The shrinking core model (SCM), which accounts for changes in solid material structure and surface area, offers a more realistic representation of solid–liquid reactions and is the most commonly used model for leaching kinetics.⁶ For homogeneous nucleation, the Avrami model can also be applied.^12,13 These models provide simplified analytical solutions for complex reactions.⁶

Correlation analysis is a valuable tool in experimental research. It helps identify and quantify the relationship between variables, providing insight into the strength and direction of the data relationship.¹⁴ Correlation analysis efficiently determines linear and monotonic relationships, serving as a fundamental step in developing forecasting models. By examining the potential to predict one variable with another, research can gain valuable insights.^15–17 Correlation analysis can also be used for validation purposes by assessing the agreement between two measurement methods.¹⁸ The coefficient of determination (R²) quantifies the difference between predicted and experimental data, and is widely used to evaluate the performance of regression models.^19–21 However, relying solely on correlation coefficient is insufficient and can cause misinterpretation. It is essential to employ additional analysis methods that highlight differences between pairs of observations.²² Correlation analysis is frequently combined with kinetics analysis to optimize experimental conditions. By evaluating the correlation between process parameters and outcomes, researchers can identify optimal settings. Furthermore, correlation and statistical analyses are crucial for selecting and assessing suitable kinetic models.

Table 1 summarizes related studies that have employed kinetic modeling and correlation analysis in leaching processes. For rotating disk electrode experiments, mechanistic kinetic models were developed to study the rate-controlling steps and to estimate the rate of gold dissolution. Correlation analysis revealed that the gold dissolution rate is proportional to temperature and the concentration of either cupric or ferric ions, depending on whether cupric chloride or ferric chloride was used as the leaching agent.^23,24 The leaching kinetics of copper extraction from waste printed circuit boards (WPCBs) can be effectively described using the Avrami model. Multiple linear regression (MLR) analysis was employed to investigate the correlation between experimental conditions and copper leaching efficiency. The results indicated that temperature (T), solid–liquid ratio, and H₂O₂ concentration exert significant effects on the copper leaching efficiency.²⁵ In the context of scrap magnet recycling, the shrinking core model was utilized to model the kinetics of neodymium dissolution. The overall leaching process was determined to be governed by surface chemical control, with the dissolution rate of neodymium primarily influenced by temperature and acetic acid concentration.²⁶

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The SCM is also frequently applied to describe the leaching kinetics of spent lithium-ion batteries. Kinetic modeling of cobalt recovery from cathodic powder using nitric acid revealed that the process is controlled by product layer diffusion, whereas cobalt leaching with citric acid is chemically controlled. Notably, temperature and solvent concentration are key parameters significantly influencing cobalt leaching efficiency.^27,28 When leaching battery cathode materials using hydrochloride acid, the dissolution of nickel, manganese, and cobalt (NMC) is controlled by chemical reaction. The dissolution rate of lithium is faster than that of the transition metals. Cathode materials with higher nickel content exhibit faster dissolution rates compared to those with lower nickel content. Both the SCM and a first-order rate law were employed to calculate the time required for complete digestion.²⁹ A rapid leaching method was developed by using a mixture of hydrochloric acid and ascorbic acid. The SCM, Avrami model, and parabolic product layer diffusion control models were used to interpret the reaction mechanisms. The Pearson's correlation coefficient was employed to quantify the correlations of both reagents to leaching rates.³⁰ In the sulfuric acid leaching system, ferrous ions serve as an essential reducing agent for cobalt in the recycling of spent LIBs. Iron scrap can be utilized as a cost-effective source of ferrous ions during leaching to enhance cobalt recovery. Kinetic analysis using the SCM indicated that the addition of ferrous ions accelerates metal transfer by mitigating the formation of the ash layer. The results of MLR demonstrated that the pH of the reagent has the most significant impact on the leaching efficiency of lithium and cobalt, followed by ferrous concentration, temperature, and agitation.³¹ In addition to iron scrap, industrial black liquor, a by-product of the Kraft pulping process in papermaking, can also be employed as a renewable reducing agent for the leaching of battery black mass with sulfuric acid. A linear correlation was observed between black liquor concentration and the leaching efficiency of manganese. Based on the better fit of diffusion control models and high activation energies, the reaction mechanism for cobalt, nickel, and manganese appears to be a mixed diffusion-reaction process.³²

Kinetic investigations of leaching can be challenging due to the complexities of data collection and kinetic modeling. Experimental data, particularly leachate concentrations, must be collected over time and analyzed using precise analytical techniques. To facilitate the selection of an appropriate kinetic model, it is essential to characterize the material's structure both before and after leaching operation. Techniques such as transmission electron microscopy (TEM), scanning electron microscopy (SEM), and Brunauer–Emmett–Teller (BET) analysis are commonly employed for this purpose.^33–35 Once a kinetic model is developed, it can be used to predict the leaching behavior in subsequent experiments. This prediction is achieved by applying equations that incorporate previously determined kinetic parameters and the specific conditions of the new experiment. However, existing theoretical models and correlation analyses often rely on fixed values for experimental conditions. Consequently, due to the inherent complexity of real-world systems, which frequently involve heterogeneous materials and phase changes, predictions made without considering the dynamic nature of the process may exhibit deviations from actual observed values.³⁶

To address the need to consider process dynamics and improve prediction accuracy, in-line sensors can be utilized to capture process parameters during leaching. pH is mathematically defined as the negative logarithm of the hydrogen ion activity. In dilute solutions, the hydrogen ion activity can be simplified and approximated by its concentration, which is influenced by factors such as temperature, dissolved CO₂ levels, and the completion of reactions within the solution.³⁷ Redox potential indicates a solution's capability for electron transfer. Similar to pH, it represents the ratio of the activities of oxidizing and reducing species within the solution.³⁸ During acid leaching, alkaline cathode metal oxides react with hydrogen ions, forming soluble metal salts. The change in hydrogen ion concentration within the solution is reflected by pH measurements, while redox potential quantifies the oxidation and reduction activities. Conductivity is mathematically defined as the product of conductance and the cell constant of the measurement cell. A solution's conductivity is dependent on the concentration and electrochemical properties of all dissolved ions. It is also strongly influenced by temperature. Increasing temperature enhances ion mobility, leading to increased conductivity. In binary solutions containing only one electrolyte and water, conductivity can be used as a surrogate for concentration.³⁹ However, due to the presence of multiple electrolytes in the pregnant leach solution (PLS) derived from battery black mass, conductivity cannot be directly used as a substitution for concentration. Nevertheless, monitoring conductivity can provide valuable real-time insights into the progression of the leaching process and the ion concentrations within the solution. These parameters can be utilized as features of chemical reactions.

In biotechnology, the correlation between redox potential and state variables was investigated during ornithine production. The ornithine concentration can be estimated using online data of redox potential.⁴⁰ To study the leaching kinetics of pyrite, May et al. developed an experimental technique for on-line redox potential measurement. The relationship between the ferric leach rate of pyrite and measured redox potential was studied qualitatively.⁴¹ In electronic waste recycling, an Arduino program was developed to monitor and control pH and potential within a constant range. This control was implemented to ensure successful leaching of copper, lead, and silver using nitric acid. Furthermore, response surface methodology (RSM) was employed to determine the optimal stirring speed and solid–liquid ratio for maximizing copper leaching efficiency.⁴² In battery recycling, researchers monitored pH, potential, and dissolved oxygen concentration during leaching processes to elucidate the fundamental dissolution mechanisms of cathode materials from spent LIBs. The results indicated that the progressive delithiation of the materials is associated with an increase in operating potentials.^30,43

However, previous studies investigating process behaviors predominantly relied on visual inspection to infer relationships between measured parameters, rather than applying rigorous quantitative correlation analysis.⁴⁴ Furthermore, the potential of sensor-derived parameters as reliable indicators of process progression has not been systematically explored. While techniques such as RSM are widely used for optimizing experimental conditions and guiding design, they are not inherently suited for exploring underlaying data relationships. In contrast, methods like correlation analysis and linear regression offer robust tools for quantifying associations between continuous variables.^14,22 The absence of such quantitative analytical approaches in the context of process monitoring and modeling represents a relevant research gap.

This study aims to effectively track and predict metal leaching states during the sulfuric acid leaching of pyrolyzed battery black mass using online sensor data. Unlike previous work that relied on visual inspection without quantitative correlation analysis, this research employs statistical methods and kinetic models to investigate the relationships between sensor data and metal dissolution.

In this study, pyrolyzed battery black mass was leached with sulfuric acid at varying temperatures. Multiple samples were collected during the leaching process to track the dissolution progress of metal elements. pH, temperature, electrode potential, and conductivity of the leaching solution were monitored continuously. Both sensor data and leaching state are non-fixed variables. The strength and direction of the relationships between these variables were initially investigated through correlation analysis. Subsequently, linear regression models were developed to estimate leaching states based on the online sensor data. The performance of the models was statistically evaluated. The observed linear relationships between online sensor data and metal leaching states were explained by combining kinetic analysis with SCM theory. The activation energies for nickel and cobalt were calculated using the Arrhenius equation. These values were subsequently used, in conjunction with the SCM conversion equations, to determine their respective leaching states. Finally, the morphology and crystal structure of the materials before and after leaching were characterized to support the kinetic analysis.

2 Experimental

2.1 Materials

A commercially vacuum thermally treated black mass (BM) derived from lithium-ion batteries was utilized in this study. The holding temperature during the thermal treatment ranged from 500 to 550 °C. Further details regarding the thermal treatment are proprietary and not disclosed by the company. However, according to the inductively coupled plasma optical emission spectroscopy (ICP-OES) elemental analysis and combustion analysis, the black mass consists of 4.6% aluminum, 1.2% iron, 14.2% nickel, 10.8% cobalt, 4.2% lithium, 9.5% manganese, 2.2% copper, and 27% carbon, as presented in Table 2. The chemical composition suggests that the black mass originated from a blend of various NMC battery cells.

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2.2 Leaching experiments

Leaching experiments were conducted using 1 molar sulfuric acid. A charge of 150 grams of pyrolyzed black mass was introduced into a 2-liter double-wall batch reactor and leached for 60 minutes. A solid–liquid ratio of 100 g L⁻¹ was maintained throughout experiments. Following the leaching period, the mixture was separated into the undissolved solid residue and the pregnant leach solution by filtration using a paper filter and a vacuum pump.

To ensure the stability of the experimental data and minimize operational errors, temperature was maintained as the sole variable during the leaching process. Furthermore, to reduce operator-induced variability across experiments, no reducing agents, such as hydrogen peroxide, were employed. This methodology ensured consistency in the results and eliminated potential confounding factors. Eight leaching experiments were conducted, varying the temperature from 25 °C to 75 °C. The temperature was manually set and maintained constant throughout the experiment by a double-wall reactor connected to an external thermostat. During each leaching experiment, 7 mL liquid samples were collected at regular intervals of five minutes using a pipette. For subsequent analysis, solid–liquid separation of these samples was achieved through centrifugation followed by filtration using a syringe filter. Samples collected at 5, 10, 15, 30, and 60 minutes were selected for analysis. Prior to analysis, all collected samples were acidified with HNO₃ and diluted with distilled water at a ratio of 1:10. The elemental composition of the diluted samples was then determined using ICP-OES.

In all experiments, a single pH sensor (InPro4260i) and one inductive conductivity sensor (InPro7250), both from Mettler-Toledo GmbH, were used to continuously monitor pH, temperature, electrode potential, and conductivity. The pH sensor provided measurements of pH, temperature, and electrode potential (mV), while the conductivity sensor monitored conductivity. Calibration of the pH sensor was performed using a two-point method with buffer solutions at pH 4 and pH 7. The conductivity sensor was calibrated using a zero-point calibration procedure. The specified measurement accuracies for these parameters were ±0.02 pH, ±0.25 °C, ±1 mV, and ±1%, respectively. Sensor signals were processed by dedicated transmitters before being collected by a datalogger. Data were subsequently transmitted to a measurement laptop via serial communication. A custom data acquisition program was developed to visualize and record the sensor data. The complete experimental setup and the measurement devices are illustrated in Fig. 1.

3 Methodology

3.1 Correlation analysis

Current hydrometallurgical battery recycling processes often rely on studies conducted under static conditions. This is largely due to the inherent complexity of the chemical reactions involved and the typical delay in obtaining comprehensive information about the process state through offline analysis. Online process monitoring, however, offers significant advantages by capturing transient process states, thereby enhancing process transparency and providing deeper insights into underlaying leaching mechanisms. Furthermore, online monitoring facilitates more dynamic and adaptive process control strategies, which can lead to notable advancements in process optimization and innovation.⁴⁵

Given that correlations may exist between parameters such as pH, temperature, electrode potential, and conductivity and the actual leaching states, these online sensor data have potential for estimating leaching progression in real-time. To investigate this potential, a systematic correlation analysis was performed.

As a preliminary step in correlation analysis, the relationships between the metal leaching state and the continuously monitored parameters were visually inspected. An initial focus was placed on the relationship between pH and the leaching state, which was examined using scatter plots. Subsequently, the relationships between all monitored sensor parameters and the leaching state were quantified using correlation coefficients. Since both the sensor data and the metal leaching state are continuous variables, Pearson's correlation coefficient was initially employed to assess the strength and direction of their linear associations. Additionally, Spearman's rank correlation coefficient was calculated to evaluate the strength of monotonic relationships. Compared to Pearson's correlation coefficient, Spearman's is less sensitive to outliers, making it a more robust alternative for analyzing data that may not fully satisfy the assumptions of a linear relationship.

3.2 Multiple linear regression

Linear regression models utilize independent variables to predict dependent variables. In this study, sensor data serve as the independent variables, and used as inputs to the models. While these were treated as independent features for model input, potential relationships among the sensor readings were examined. To mitigate issues arising from multicollinearity, a threshold of 5 was established for the variance inflation factor (VIF). Only combinations of features yielding a VIF below this threshold were utilized in the models. Given that the leaching states of multiple metal elements are described, the linear regression models employed in this study are multivariant linear regression or multiple linear regression.

The primary objective of the regression model was to predict the leaching states of metal elements based on the provided sensor data. However, the inherent complexity of the chemical reactions and the inhomogeneity of the solution posed significant challenges in obtaining consistent sampling results through manual procedures. Variations in sampling time and the dynamic nature of the reaction process could substantially affect the reliability and reproducibility of manually collected data. Consequently, the leaching states inferred from discrete sampling analysis primarily reflected the overall trend of the chemical reactions rather than providing precise, real-time insights. Therefore, linear regression models were utilized to describe this overall trend of the leaching process.

Unlike a purely correlation analysis, the linear regression modeling included data collected from the commencement of the leaching experiments. A total of 46 data points were utilized for modeling the leaching process. Based on the statistical guideline known as the “one in ten rule”,⁴⁶ which suggests that the number of observations should be at least ten times the number of independent variables, a sample size of 46 is considered adequate for multiple linear regression models incorporating up to four independent variables. To ensure the robustness and generalizability of the models to unseen data, 10-fold cross-validation was implemented during the model fitting process using the scikit-learn library. The entire dataset was portioned into 10 subsets. In each iteration, 9 subsets were used for training the model, while the remaining subset was reserved for validation. After each subset had served as the validation set, the average of the validation results was calculated to assess the overall model performance.

3.3 Kinetic analysis

To further elucidate the linear relationships and distinct leaching behaviors observed among the metal elements, the shrinking core model was employed. The SCM provides a theoretical framework for determining the rate-controlling step in heterogeneous leaching reactions. Pyrolyzed black mass typically exhibits a relatively porous structure, which facilitates internal mass transfer while maintaining particle size integrity. Upon the addition of black mass to the acidic solution, a thin gas film initially forms around the solid particles. However, this gas film is rapidly replaced by a liquid film due to electrical agitation. Within this liquid film, insoluble metal oxides (e.g. MgO, Al₂O₃) and contaminants can accumulate on the surface of the transition metal oxides particles, forming an ash layer.

The leaching mechanism starts with the diffusion of sulfuric acid ions through the liquid film to the surface of the ash layer. Subsequently, sulfuric acid penetrates and diffuses through the ash layer to reach the surface of the unreacted core. At the core surface, sulfuric acid reacts with the transition metal oxides, forming soluble products. These products then diffuse through the ash layer and the liquid film, ultimately returning to the bulk solution.

Based on shrinking core model, three potential rate-controlling steps were investigated: surface chemical reaction control, film diffusion control, and ash diffusion control. The corresponding conversion equations for these control mechanisms are presented in eqn (1)–(3).^47,48 These SCM equations indicate that the conversion on the left side is proportional to time. Conversion values were calculated based on the leaching rates.

1 − (1 − X)⅓ = kct	(1)

X = kd,lt	(2)

1 – 3 × (1 − X)⅔ + 2 × (1 − X) = kd,st	(3)

where X denotes the leaching rate; k_c is the apparent rate constant for chemical reaction control; k_d,l is the apparent rate constant for film diffusion control; k_d,s is the apparent rate constant for ash diffusion control.

The Arrhenius equation, shown in eqn (4), describes the temperature dependence of reaction rates in physical chemistry. It has been utilized to determine the chemical reaction rate and calculate the activation energy.⁴⁹ When the apparent rate constant of a chemical reaction obeys the Arrhenius equation, its Arrhenius plot is expected to exhibit a linear relationship.⁵⁰

	(4)

where k is the rate constant; E_a is the activation energy; R is the universal gas constant 8.314 J mol⁻¹ K⁻¹; T is the absolute temperature in Kelvin; A is the frequency factor.

4 Results and discussion

4.1 Data preparation

The leaching state was calculated based on eqn (5). In this equation, V and C represent the volume and concentration of the leaching solution at the sampling point, respectively. ω is the mass fraction of the element in the input material, and M represents the total mass of the input material. Leaching efficiency is defined as the leaching state determined for the final liquid sample collected from an experiment.

Leaching state LS = (V × C)/(ω × M)	(5)

Table 3 summarizes the experimental conditions and the corresponding leaching efficiencies. Overall, the results indicate that leaching efficiency increased with increasing temperature. The highest average leaching efficiency was obtained at 65 °C, as shown in Fig. 2(a).

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Fig. 2(b) illustrates the influence of leaching temperature on the leaching efficiencies for individual elements (Al, Fe, Ni, Co, Li, and Mn). Specially, the leaching efficiencies of Al, Fe, Ni, Co increased with increasing leaching temperature up to 65 °C, while the leaching efficiencies of Li and Mn exhibited less sensitivity to temperature variations. The comparatively lower leaching efficiencies observed for Ni and Co may be attributed to the low acid concentration employed and the absence of H₂O₂. The concentration of copper in all samples remained below 1 mg L⁻¹ and was therefore not considered further in this study.

Fig. 3(a) illustrates the changes in leaching states of Al, Fe, Ni, Co, Li, and Mn over time during the leaching experiment conducted at 65 °C. The leaching efficiencies of Li and Mn exceeded 95% within a short duration and subsequently remained relatively stable. Similarly, the rate of increase in leaching efficiencies for Al, Fe, Ni, and Co also decreased following an initial period of rapid dissolution. Fig. 3(b) depicts the measured profiles of pH, temperature, electrode potential, and conductivity throughout the leaching process. The leaching process was initiated upon the addition of black mass to the solution. Strong exothermic reactions occurred within the initial three minutes, resulting in a temperature increase exceeding 10 °C. Subsequently, the rate of chemical reaction decreased and the leach solution gradually cooled due to the temperature differential between the leach solution and the surrounding water bath. To eliminate the cooling effect arising from the temperature difference between the water bath and the reaction solution, the target leaching temperature was increased from 45 °C to 55 °C after the temperature reached the peak in one specific experiment. However, comparable leaching efficiencies were achieved in this instance compared to experiments conducted at a constant temperature of 55 °C. Apart from periodic sampling, no other external interruptions were introduced during the leaching experiments.

4.2 Correlation analysis

Fig. 4 presents a scatter plot illustrating the relationship between nickel leaching state and pH across different leaching temperatures. A linear trend was observed, with the data points from a single experiment exhibiting a strong linear relationship.

Fig. 5 illustrates the calculated Pearson's and Spearman's correlation matrices based on 38 sampling points. For this analysis, the initial sampling points of each experiment were excluded because they uniformly showed a leaching state of 0%, which would artificially inflate the perceived linear relationship between variables. Correlation coefficients range from −1 to 1. A positive coefficient indicates that two variables tend to increase or decrease together, while a negative coefficient suggests an inverse relationship. The absolute value signifies the strength of the correlation, with values approaching 1 indicating a stronger linear or monotonic relationship.

The leaching states of Al, Fe, Ni, and Co exhibited strong positive correlations with pH values (correlation coefficients > 0.7). Furthermore, the leaching states of Al, Ni, and Co showed strong negative correlations with conductivity. In contrast, the leaching state of Fe displayed a strong negative correlation with electrode potential (mV). The leaching states of Li and Mn exhibited moderate correlations with the electrode potential (correlation coefficients between 0.5 and 0.7).

In addition to the relationships between leaching states and sensor data, the correlation matrix revealed strong correlations among certain metal ions: Al, Fe, Ni, and Co exhibited high correlations among themselves, as did Li and Mn. These results suggest similar leaching behaviors for Al, Fe, Ni, and Co as a group, distinct from the behaviors of Li and Mn. Additionally, the correlation matrices indicated strong correlations between pH and conductivity, and between both of these variables and temperature. The correlation matrix calculated based on Spearman's rank correlation coefficient yielded results consistent with those obtained using Pearson's correlation.

Both visual inspection and the quantitative correlation analysis indicate that the relationship between online acquired sensor data and the leaching state is not only monotonic, but also potentially linear. However, as correlation analysis intrinsically focuses on the relationship between only two variables, the leaching state may be more accurately described by considering a combination of multiple features.

4.3 Multiple linear regression

4.4 Kinetic analysis

4.5 Leaching state prediction

4.6 Material characterization and its influence on leaching behaviors

Regression modeling demonstrated a strong correlation between sensor measurements and the leaching states of the metal elements. Based on kinetic analysis, this linear relationship can be attributed to the diffusion of metal-ion products through the ash or liquid film layers. However, due to limited linearity in the SCM equations, the statistical data were insufficient to definitively differentiate between the rate-controlling steps for Li and Mn. Furthermore, the applicability of the shrinking core model relies on the assumption that the spherical structure of the metal particles remains intact before and after the leaching process.

To further investigate the distinct leaching behaviors of the metal elements, the thermal pretreatment process of the employed battery black mass was analyzed. Subsequently, scanning electron microscope and energy-dispersive X-Ray spectroscopy (EDS) were employed to characterize the morphology and crystal structure of both the employed black mass and the solid residue obtained after leaching.

4.7 Leaching predictive models

Predictive models, including statistical models, chemical kinetic models, and machine learning models, have been applied to the leaching process of primary and secondary materials, as detailed in Table 7. However, research specially focused on predicting leaching states in battery recycling is limited. Multiple linear regression has been commonly employed in previous studies to examine data relationships and predict leaching efficiency, particularly when the available dataset is small. Nevertheless, the majority of these studies have used static experimental conditions for prediction, thus neglecting the dynamic nature of the leaching process.

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Chemical kinetic models have been utilized in previous leaching kinetic studies to determine the rate-controlling step and describe the dissolution rate of metallic elements. However, the predictive capabilities of these models have not been sufficiently evaluated.^20,24,25,48 The coefficient of determination, R², does not directly quantify prediction error and should therefore not serve as the sole evaluation metric for regression models. Conversely, the root mean squared error, RMSE, measures the average magnitude of the difference between predicted and actual values, thus providing a more robust evaluation of regression model performance.

The proposed approach utilizes sensor measurements as model inputs and the leaching states of metallic elements as model outputs, thereby accounting for the dynamic conditions of the leaching process. In comparison to predictions derived from traditional kinetic models, this approach avoids the need for intricate material characterization and circumvents structural limitations, while simultaneously providing valuable insights for process monitoring. This study represents a pioneering effort in predicting leaching states using online sensor data during the leaching process of battery black mass.

Data relationships are quantitatively investigated through correlation analysis and multiple linear regression, and are also elucidated by principles of leaching kinetics theory. The performance of the models developed in this study is evaluated using both R² and RMSE. The MLR model, using pH, conductivity, and temperature as input variables, demonstrates the capability to predict the leaching state of six metallic elements with the highest adjusted R² value and the lowest RMSE. Furthermore, the observed variations in the leaching behaviors of the metallic elements are investigated based on thermal pretreatment mechanisms and material morphology, aspects not typically addressed in other leaching kinetic studies of battery black mass.

The proposed approach exhibits adaptability and can be readily applied to other leaching processes where sensor measurements are accessible. More advanced methodologies, such as machine learning algorithms, can be implemented to streamline the process and address more complex scenarios in future research, contingent on the availability of larger datasets.

5 Conclusion

This study combines correlation analysis and regression modeling to propose a novel approach for predicting leaching states using readily measurable sensor data. Pearson's and Spearman's correlation coefficients were employed to explore the linear relationships between sensor data and the metal leaching state. Regression modeling extended this correlation study by incorporating the effect of feature combinations. Predictions of the leaching sate derived from the multiple linear regression model outperformed the traditional calculation methods based on the Arrhenius equation and shrinking core model theory, achieving an average prediction error of less than 1.1%.

The leaching kinetics of nickel and cobalt were effectively described by the shrinking core model. A strong, chemically controlled reaction phase was observed during the initial minutes following the addition of black mass. Subsequently, the leaching of nickel and cobalt proceeded via a diffusion-controlled mechanism. During this diffusion-controlled phase, a strong linear relationship between sensor data and leaching state was observed. The calculated activation energies for nickel and cobalt were 29.8 kJ mol⁻¹ and 22.6 kJ mol⁻¹, respectively. Given that aluminum and iron do not possess a spherical structure, their leaching kinetics could not be modeled using the shrinking core model. The rapid reaction kinetics observed for lithium and manganese were attributed to the unique physiochemical properties of their respective metal compounds, which were influenced by the thermal pretreatment. To further investigate the kinetics of lithium and manganese, particularly during the initial five minutes of the reaction, faster sampling methods are required.

Due to the complexity of the leaching process and the inhomogeneity of the materials, the findings of this study are specific to the black mass and the experimental conditions employed. To enhance the generalizability of these findings, future research should systematically vary other parameters, such as acid concentration, solid–liquid ratio, and input material. Nevertheless, this methodology, which integrates statistical analysis of online sensor data with kinetic modeling and material characterization techniques, offers a valuable framework for further research and practical applications in this field. The proposed predictive strategy is applicable to leaching processes involving other types of input materials. The monitoring of leaching operations can be optimized through the systematic analysis and modeling of sensor and experimental data. Such optimization is crucial not only for advancing fundamental research but also for enhancing efficiency in industrial applications.

Conflicts of interest

There are no conflicts to declare.

Data availability

The data supporting this article have been included as part of the ESI.†

Acknowledgements

This work was supported by the German Bundesministerium für Bildung und Forschung (BMBF) in the frame of the project “DiRectION”, grant number 03XP0358C.

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Footnote

† Electronic supplementary information (ESI) available. See DOI: https://doi.org/10.1039/d5gc00581g

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