Development of a Dynamic Semi-empirical Model for Simulation of Copper Electrowinning Processes

Abstract

Dynamic process models present an opportunity to investigate optimization and control strategies for the energy-intensive copper electrowinning process. In this paper, a dynamic semi-empirical model of the copper electrowinning process was developed to predict the copper yield, current efficiency, and specific energy consumption. The model uses input variables readily measured in industrial tankhouses and incorporates the ability to induce step or pulse disturbances in the electrolyte composition or flow rate. Dynamic bench-scale electrowinning data were used to show how the model may be calibrated and validated for use in predicting electrowinning performance. Overall, the performance of the developed dynamic model lends credence to the application thereof for operator training, process monitoring, and early fault detection. The model also represents a further step towards investigating advanced control strategies for the electrowinning process.

Introduction

The hydrometallurgical method for extraction of copper is expected to see an increase in popularity due to its perceived reduced environmental impact and ability to extract copper from low-grade ores and scrap.¹ The hydrometallurgical method consists of leaching, solvent extraction, and electrowinning. Electrowinning is an energy-intensive process with approximately 2 MWh used per ton of copper produced.² All the energy is, however, not used in the deposition of copper. Up to 30% of the energy is wasted because of electrolyte resistance, subsidiary reactions, and other process inefficiencies.³ The reactive-based control strategies, which are typically employed by copper electrowinning tankhouses, often contribute to the sub-optimal process performance.⁴

The availability of a dynamic model for the copper electrowinning process would support the investigation of improved optimization and control strategies, as previously done for the preceding leaching and solvent extraction steps.^4,5,6,7 Likewise, the value of using dynamic models for process optimization and control has previously been outlined for zinc electrowinning operations.^8,9,10,11,12 Both steady-state and dynamic zinc electrowinning models are reported in literature, with the dynamic models being able to accurately predict the performance of full-size zinc electrowinning cells.^9,10,11,12 Although several steady-state models have been developed for the copper electrowinning process, limited research has been conducted on the development of dynamic models.^13,14,15,16 Furthermore, the validity of the dynamic models that do exist is restricted to specific industrial copper production plants, or their predictive performance is inadequate.^17,18,19,20 The review conducted by Zhang and Free⁸ emphasized that achieving a high-fidelity dynamic model for zinc electrowinning involved a stepwise increase in model complexity. Similarly, there is scope for the development of a general dynamic copper electrowinning model that can be incrementally enhanced to improve its fidelity.

In this paper, a general dynamic semi-empirical model of the copper electrowinning process is developed for the prediction of key performance indicators (KPIs). Three KPIs are used to define the process efficiency: copper quality, copper yield, and energy consumption. The energy consumption is quantified by the current efficiency and specific energy consumption (SEC). The KPI relating to the quality of copper is excluded from the model scope. The backbone of the dynamic model is the steady-state model developed by Tucker et al.,^15,16 which in turn used the research conducted by Aminian and Bazin¹³ and Blackett and Nicol²¹ as a foundation. Subsequently, dynamic bench-scale electrowinning data are used as an example to show how the model may be calibrated and validated for use in predicting electrowinning performance. Based on the validation results, the developed model is expected to be a useful tool for operator training, process monitoring, and early fault detection.

Dynamic Model

Electrowinning Model

This section details the adaptation of the steady-state model developed by Tucker et al.^15,16 for use in dynamic simulation. The original steady-state model used a combination of operational input variables (electrolyte composition, voltage, temperature, and electrolyte flow rate) and fixed input variables (electrode surface area, interelectrode spacing, hardware resistance, and number of cathodes per cell) to predict the KPIs (copper yield, current efficiency, SEC). For the dynamic model, the same input and output variables were selected, except for replacing the voltage with the applied current as an operational input variable. This enabled the simulation of a current-controlled electrowinning process, which is representative of the current state-of-the-art in most industrial electrowinning tankhouses.^22,23 The dynamic model, furthermore, also includes the ability to induce step or pulse disturbances in any one of the operational input variables at a time.

The dynamic model includes the reduction of copper at the cathode (Eq. 1) and the oxidation of water at the anode (Eq. 2). Iron is a major impurity in copper-containing ore and is present in the electrolyte because the preceding solvent extraction does not reject all of it. Consequently, the cyclic reduction and oxidation of iron at the cathode and anode, respectively, are also included (Eq. 3).

$${{\text{Cu}}}_{\left({\text{aq}}\right)}^{2+}+2{{\text{e}}}^{-}\to \mathrm{ C}{{\text{u}}}_{\left({\text{s}}\right)}$$

(1)

$$2{{\text{H}}}_{\left({\text{aq}}\right)}^{+}+\frac{1}{2}{{{\text{O}}}_{2}}_{({\text{g}})}+2{{\text{e}}}^{-}\to {{\text{H}}}_{2}{{\text{O}}}_{\left({\text{l}}\right)}$$

(2)

$${{\text{Fe}}}_{\left({\text{aq}}\right)}^{3+}+{{\text{e}}}^{-} \leftrightarrow {{\text{Fe}}}_{\left({\text{aq}}\right)}^{2+}$$

(3)

A resistance network approach, adapted from Aminian and Bazin,¹³ was used to develop the original steady-state model.^15,16 Tucker et al.^15,16 originally used a capacitor-resistor pair to represent each electrochemical reaction. Blackett and Nicol,²¹ however, showed that two diodes in parallel had a voltage-current characteristic similar to the Butler–Volmer equation when simulating the current distribution in an electrowinning cell. Moreover, Blackett and Nicol²¹ report that the inclusion of mass transport is also possible by suitable modification of the diode characteristics. Accordingly, the resistance network representation of an electrowinning cell used in the dynamic model, showing the use of parallel diodes in combination with a voltage source to represent a single electrode pair (plated on both sides), is given in Fig. 1.

Fig. 1

Schematic representation of the simplified resistance network diagram for a single electrode pair (plated on both sides) (adapted from Ref. 13).

The relationships between the electrical components in Fig. 1 were used to inform the relationships between the electrochemical equations used to model the single electrode pair. The electrochemical reactions that occur at an electrode-side are connected in parallel. Consequently, the current that flows through each electrode could be calculated by summing the current that is used in each reaction. The current required by each reaction was calculated by multiplying the respective current density by the surface area of the electrode. Moreover, the series connection between each respective cathode side and anode side means that the current flowing through the cathode must equal that flowing through the corresponding anode (assuming both sides of the respective electrodes can be taken as identical).

As the branches are connected in parallel, the total potential (${U}_{{\text{T}}}$) is equal to the potential over each main branch. The potential over each main branch is, in turn, equal to the sum of the anodic potential (${U}_{{\text{a}}}$), cathodic potential (${U}_{{\text{c}}}$), and a voltage loss term (${U}_{{\text{L}}}$) (Eq. 4). The electrolyte resistance has been reported to account for up to 24% of the SEC.^24,25,26,27 Consequently, the voltage loss term was approximated as a parabolic function of the normalized electrolyte resistance (${R}_{{\text{s}},{\text{norm}}}$) and regressed parameters ($A, B, C$) (Eq. 5). The electrolyte resistance (${R}_{{\text{s}}}$) was determined as a function of the electrolyte conductivity ($\kappa$), interelectrode spacing ($d$), and the electrode area ($A$) (Eq. 6). An empirical correlation developed by Price and Davenport²⁶ was, in turn, used to determine the conductivity as a function of the electrolyte component concentrations ($x$) and the temperature ($T$) (Eq. 7).

$${U}_{{\text{T}}}= {U}_{{\text{a}}}+ {U}_{{\text{c}}}+ {U}_{{\text{L}}}$$

(4)

$${U}_{{\text{L}}}= A\cdot {\left({R}_{{\text{s}},\mathrm{ norm}}\right)}^{2}+B\cdot {R}_{{\text{s}},\mathrm{ norm}}+C$$

(5)

$${R}_{{\text{s}}}=\frac{d}{\kappa \cdot A}$$

(6)

$$\kappa ={\left(3200+7.3{\cdot x}_{{\text{Cu}}}+4.5{\cdot x}_{{\text{Fe}}}-5.6{\cdot x}_{{{\text{H}}}_{2}{{\text{SO}}}_{4}}-14.6\cdot T\right)}^{-1}\cdot {10}^{5}$$

(7)

The kinetics of the included electrochemical reactions (copper reduction (Eq. 1), water oxidation (Eq. 2), iron reduction, and iron oxidation (Eq. 3) form an integral part of the dynamic model. In the original steady-state model, the kinetics of each reaction were iteratively determined using an initial guess of the component concentration at the electrode surface.^15,16 This was possible because the steady-state assumption resulted in the reaction rate being equal to the mass-transfer rate for each reaction. This approach is, therefore, not feasible for the dynamic state. Instead, it was assumed that the cyclic reduction and oxidation of iron was mass-transfer limited and that the copper reduction and water oxidation reactions were reaction-rate limited. It was, therefore, assumed that mass transfer effects could be neglected for the copper reduction and water oxidation reactions. This assumption is supported^16,28,29 for the typical concentration ranges of copper and sulfuric acid reported by Robinson et al.³⁰ and Sole et al.³¹

Accordingly, the current densities ($i$) associated with the iron reduction and oxidation reactions were determined using the mass-transfer equation (Eq. 8). The mass-transfer equation is a function of the number of electrons involved in the reaction ($n$), Faraday’s constant ($F$), the mass-transfer coefficient (${m}_{i}$), the molar concentration of the component in the bulk electrolyte (${C}_{i,{\text{bulk}}}$), and the molar concentration of the component at the surface (${C}_{i,{\text{surface}}}$, assumed to be zero).

$$i =n\cdot F\cdot {m}_{i}\cdot ({C}_{i,{\text{bulk}}}-{C}_{i,{\text{surface}}})$$

(8)

The Butler–Volmer equation was used to model the reaction-rate limited copper reduction and water oxidation kinetics (Eq. 9). The Butler–Volmer equation requires the overpotential ($\eta$) associated with the respective copper reduction and water oxidation reactions. Figure 1 shows that the electrochemical reactions that occur at each anode side are connected in series with those that occur at the respective cathode side. Consequently, the overpotentials were determined iteratively using the constraint that the current flowing through each cathode side must equal that flowing through the connected anode side. Additional parameters and constants required in the Butler–Volmer equation are the exchange current density (${i}_{0}$), charge-transfer coefficient ($\alpha$), and universal gas constant ($R$).

$$i={i}_{0}\cdot \left[{\text{exp}}\left(\frac{-\alpha \cdot n \cdot F}{R \cdot T} \cdot \eta \right)-{\text{exp}}\left(\frac{\left(1-\alpha \right) \cdot n \cdot F}{R \cdot T}\cdot \eta \right)\right]$$

(9)

Mass Conservation Model

This section details the dynamic mass conservation equations required to model the electrowinning process. The overall mass conservation equation is given in Eq. 10, where $m$ is the mass, $Q$ is the flowrate, and $\rho$ is the density. Assuming a constant volume of electrolyte ($V$) and excluding the losses ($L$) due to evaporation and oxygen formation at the anode (Eq. 2) resulted in Eq. 11. The rate of generation or consumption (${P}_{i}$) was determined as a function of the current ($I$) associated with the electrochemical reactions (Eqs. 1, 2, and 3), using Faraday’s law (Eq. 12). The current associated with each reaction was previously determined in the electrowinning model section. Additional constants required in Faraday’s law are the stoichiometric coefficient of the component (${s}_{i}$) and the molar mass of the component (${M}_{i}$). The mass conservation equation for a general component is given in Eq. 13. The mass conservation equations were solved as a system of differential-algebraic equations (DAEs) and a set of ordinary differential equations (ODEs) at every time step (of 1 s). The MATLAB solver ode15s was used to solve both the system of DAEs and set of ODEs.

$$\frac{{\text{d}}m}{{\text{d}}t}={m}_{{\text{in}}}-{m}_{{\text{out}}}-{P}_{{\text{Cu}}}={Q}_{{\text{in}}}\cdot {\rho }_{{\text{in}}}-{Q}_{{\text{out}}}{\cdot \rho }_{{\text{out}}}-{P}_{{\text{Cu}}}-L$$

(10)

$$\frac{{\text{d}}m}{{\text{d}}t}=\frac{{\text{d}} \cdot \left({\rho }_{{\text{out}}} \cdot V\right)}{{\text{d}}t}={Q}_{{\text{in}}} \cdot {\rho }_{{\text{in}}}-{Q}_{{\text{out}}} \cdot {\rho }_{{\text{out}}}-{P}_{{\text{Cu}}}$$

(11)

$${P}_{i}=-\frac{{s}_{i} \cdot {M}_{i} \cdot I}{n \cdot F}$$

(12)

$$\frac{{\text{d}}{m}_{i}}{{\text{d}}t}=\frac{{\text{d}}\left({x}_{i,{\text{out}}} \cdot V\right)}{{\text{d}}t}={Q}_{{\text{in}}} \cdot {x}_{i,{\text{in}}}-{Q}_{{\text{out}}} \cdot {x}_{i,{\text{out}}}-{P}_{i}$$

(13)

Modeling Overview

In this section, an overview of the modeling algorithm, shown in Fig. 2, is discussed. The algorithm consists of three levels of nested while loops, coded in MATLAB. The two innermost “current loops” use the electrowinning model to determine the current density associated with each reaction. The “concentration loop” uses the current densities to determine the rate of generation or consumption of each component and, correspondingly, the spent electrolyte concentration, using the mass conservation model. The outermost “time loop” repeats all the calculations completed in the current loops and concentration loop at every time step (of 1 s) for the duration of the electrowinning process. The KPIs are calculated in two parts to optimize the execution of the code. The first part consists of the calculations for the cumulative mass copper plated. After the time loop converges, the voltage is determined and the second part of the KPI calculations (current efficiency and SEC) is executed.

Fig. 2

Modeling algorithm for the dynamic electrowinning model.

Modeling Calibration

Approach

The original steady-state model was accompanied by a rigorous parameter-fitting approach to calibrate the model for use in a specific application.^15,16 In a similar vein, a parameter-fitting approach was developed for calibrating the dynamic model. The parameters required in the dynamic model were divided into the kinetic parameters and the voltage loss parameters. The kinetic parameters consisted of the Butler–Volmer equation parameters for copper reduction and water oxidation (Eq. 9) and the mass-transfer equation coefficients for iron reduction and oxidation (Eq. 8). The voltage loss parameters consisted of the parabolic coefficients of the voltage loss equation (Eq. 5). Nine parameters in total were fit to the dynamic model.

Dynamic electrowinning data are required to calibrate and validate the semi-empirical model. In practice, industrial electrowinning data (or pilot-plant data if historical tankhouse data are not available) would ideally be used to calibrate the model for use in a specific tankhouse. In this paper, a single case study, consisting of dynamic bench-scale electrowinning experiments, is used as an example of how the model may be calibrated. A detailed discussion of the parameter-fitting approach will not form part of this paper.

Experimental Method

In total, 24 dynamic bench-scale electrowinning experiments were conducted to generate case study data for use in calibrating (and later validating) the developed dynamic model. As done by Tucker et al.,¹⁶ the input and output variables for the dynamic model were either fixed (independent variables) or measured (dependent variables) in the electrowinning experiments. The advance electrolyte copper and iron concentrations and applied current density were selected as manipulated variables because of their effect on the electrowinning kinetics. The range of each manipulated variable was based on typical industrial values (Table I). A classical design of experiments approach was adapted to include guidelines for generating model validation data.^32,33 A small face-centered central composite design, comprising the three manipulated variables at three levels (Table II), was repeated three times. Each replicate incorporated a step disturbance (of one coded level) in a different manipulated variable. All experiments in which the step disturbance led to a final coded level of more than one in the respective variable were excluded from the design, resulting in a total of 24 experiments. The magnitudes of the induced disturbances represent significant fluctuations in electrolyte composition and applied current for a typical industrial tankhouse (operating under standard conditions). The generated case study data could, therefore, be used to comment on the model’s performance when simulating realistic dynamic operating conditions.

Table I Manipulated variables for bench-scale electrowinning experiments, with ranges

Table II Manipulated variables for bench-scale electrowinning experiments as coded variables

The bench-scale electrowinning setup used consisted of a 5-L electrowinning cell, a power supply (Manson, SIM-9106 1-15V 60A switching mode power supply), a pump (Blue-White Industries, Flexflo A1N31F-2T peristaltic metering pump), and a heating bath (Julabo, CORIO C heating immersion circulator) (Fig. 3). The electrowinning cell and all pipes and connections were manufactured using polyvinyl chloride because of the material’s resistance to acid corrosion at the required operating temperature. The electrolyte entered the cell from the bottom through a perforated horizontal plate and exited by overflowing over the weirs situated on two opposite sidewalls near the top of the cell. The electrolyte leaving the cell was recycled back to the stock solution bottle. A recycled stainless-steel cathode and two cold-rolled lead-calcium-aluminum alloy anodes were inserted into slots cut into the top of the cell. Both electrodes had dimensions of 12 cm by 15 cm, with a working surface area of 300 cm² and thickness of 0.3 cm. The interelectrode spacing was 2.5 cm, with both sides of the cathode and anodes exposed to the electrolyte. The electrodes were riveted to copper hanger bars with dimensions of 2.5 cm by 19 cm and thickness of 0.3 cm.

Fig. 3

Isometric projection of the bench-scale electrowinning setup.

The experimental procedure consisted of preparing the synthetic electrolyte, setting up the equipment, running the experiment, and analyzing the samples. The procedure closely resembles that used by Tucker¹⁵ for conducting steady-state electrowinning experiments, with key differences highlighted; 10 L of a synthetic electrolyte solution containing 170 g/L sulfuric acid and the desired concentrations of copper and iron was prepared in a stock solution container, which was placed in a heating bath to ensure that a steady-state operating temperature of 45°C was achieved in the electrowinning cell. The synthetic electrolyte was circulated through the solution container and cell at 3.2 L/h to ensure a typical industrial interfacial velocity of between 0.05 and 0.1 m³/h/(m² of available cathode surface area).²⁸ The resulting electrowinning cell residence time, accounting for the volume of the three electrodes, was 91 min. Once thermal equilibrium was reached the power supply to the electrodes was switched on in current-controlled mode, signifying the start of the experiment.

Operating in current-controlled mode is the first significant difference from the experiments conducted by Tucker¹⁵ in which the power supply was used in potential-controlled mode and manually adjusted to ensure that the desired current was applied. Tucker¹⁵ furthermore terminated the steady-state experiments after 4 h. The dynamic experiments were, however, only terminated after 8 h, with a step disturbance introduced 4 h after electrowinning commenced. It was assumed that the system was at steady state at the 4-h mark (i.e., before the disturbance was introduced), as supported by the experiments conducted by Tucker.¹⁵ One-step disturbance was introduced per experiment by either adding additional solid copper sulfate or ferric sulfate (increasing the advance electrolyte copper or iron concentration) or adjusting the value of the current setpoint (increasing the current density). It was elected to induce the disturbance by adding solid salts, instead of a volume of concentrated solution, to ensure that the disturbance was only in the intended variable. The transparent feed line was monitored to ensure that solids did not deport to the electrowinning cell and become incorporated into the cathode. After termination of the experiment, the cathode was rinsed with demineralized water and allowed to air dry before being weighed.

Sampling took place at set intervals during the experiments. At each sampling interval the current and voltage displayed on the power supply were noted, and the voltage was measured over each anode-cathode pair using a multimeter. The advance and spent electrolyte streams were sampled for concentration analysis. Inductively coupled plasma optical emission spectroscopy (ICP-OES) was used to determine the concentrations of copper and iron in the collected electrolyte samples. The mass copper plated at each sampling interval was determined via a mass conservation balance over the electrolyte cell, using the advance and spent electrolyte copper concentrations.

Calibration Method

The parameter-fitting approach for fitting the kinetic parameters consisted of sequential non-linear regressions for each reduction and oxidation reaction, as done in the original steady-state approach developed by Tucker et al.¹⁶ The separate voltage loss parameter-fitting approach, in contrast, consisted of only one non-linear regression. The same resampling method, k-fold cross validation, was incorporated into both approaches. This allowed for a set of data, previously unseen by the model to be used for model validation.

Kinetic Parameters

The model calibration for the kinetic parameters was initiated by defining the required variables at every sampling interval for the duration of the dynamic experiments. Thereafter, the total overpotential was calculated using the constituents of the total cell voltage from the resistance network diagram. To determine what fraction of the total overpotential should be allocated for copper reduction, a fixed ratio was used. It was assumed that 30% of the total overpotential was associated with copper reduction, based on a fictional cell developed by Beukes and Badenhorst²⁸ This ratio is also within the approximate range of 0.09 to 0.67 specified by Schlesinger et al.³ for a typical industrial electrowinning tankhouse. The intermediary variables required for fitting the copper reduction and iron reduction parameters were determined using the current efficiency and overpotential. The intermediary variables associated with the iron oxidation were, thereafter, calculated using a system of ODEs. The system consisted of the mass conservation equations for ferric and ferrous iron. Lastly, the overpotential and current density associated with the oxidation of water were determined as intermediary variables. A series of non-linear regressions were performed using all the intermediary variables, together with the integrated resampling method, to determine the kinetic parameters given in Table III.

Table III Summary of kinetic parameters fitted to the dynamic bench-scale electrowinning experiments alongside parameters reported in literature

The kinetic parameters are dependent on the individual electrowinning cell being modeled. The charge-transfer coefficient is the fraction of the electrostatic potential energy affecting the reduction rate in an electrode reaction, with the remaining fraction affecting the corresponding oxidation rate.³⁴ Typical values of the charge-transfer coefficient for copper reduction lie between 0.2 and 2.³⁵ The charge-transfer coefficient for copper reduction fitted for the case study dynamic experiments is within this range. Although the fitted parameter is lower than the value reported by Aminian and Bazin,¹³ it corresponds well with the value reported by Tucker et al.¹⁶ The charge-transfer coefficient for water oxidation fitted for the case study system is higher than the values reported by Aminian and Bazin¹³ and Tucker et al.¹⁶ but below the value of 1.46 reported by Werner et al.³⁶

The exchange current density is dependent on the concentrations of reactants and products, temperature, the electrolyte-electrode interface, and impurities on the electrode surface.³⁷ Additionally, Cifuentes and Simpson³⁸ reported that the kinetic parameters obtained in previous studies on copper electrodeposition also varied with cell geometry and cathode material. Each of these factors can change the value of the exchange current density by several orders of magnitude.³⁵ The exchange current densities for real-life systems can vary from < 1 × 10⁻⁸ A/m² to > 100,000 A/m².³⁹ The exchange current densities obtained for the dynamic experiments are within this broad range of expected values.

The exchange current density for copper reduction corresponds well with the value obtained by Tucker et al.¹⁶ during calibration of the developed steady-state model. It is, however, lower than the value reported by Aminian and Bazin.¹³ The exchange current density for water oxidation obtained in this paper is significantly higher than the values reported by both Aminian and Bazin¹³ and Tucker et al.¹⁶ Werner et al.³⁶ developed a correlation for the exchange current density for water oxidation at different temperatures, based on the work done by Laitinen and Pohl.⁴⁰ Using the developed correlation, a value of 2.3 × 10⁻⁷ A/m² was obtained for this parameter based on the temperature used for the case study experiments. Again, the value reported for the case study system in this paper is significantly higher. Nevertheless, as the parameters are dependent on the individual system it is expected that variation will occur when calibrating the model for a specific case study.

The fitted parameters for copper reduction and water oxidation were used to demonstrate that the resulting total potential corresponds with the anticipated potential range for a typical industrial electrowinning tankhouse. The total cell potential in an industrial copper electrowinning tankhouse encompasses various components, including the thermodynamic potential requirement (0.89 V), cathodic overpotential (ranging from 0.05 V to 1 V), anodic overpotential (approximately 0.5 V), electrolyte potential drop (approximately 0.275 V), and a potential drop due to hardware resistance (approximately 0.3 V).³ By summing these components, an expected total potential of between 2 V and 3 V is obtained. At a current density of 287.5 A/m² (mean value from the investigated range, as shown in Table I), the fitted parameters for copper reduction resulted in a cathodic overpotential of − 0.39 V. The negative overpotential arises from the positive current density associated with the reduction reaction.³⁹ Similarly, the fitted parameters for water oxidation yielded an anodic overpotential of 0.85 V at the same current density, considering the negative current density associated with oxidation reactions.³⁹ Notably, both the cathodic and anodic overpotentials align well with the respective values provided by Schlesinger et al.³ Additionally, by incorporating the remaining components reported by Schlesinger et al.,³ a total potential of 2.7 V is obtained. This value falls within the expected potential range for a typical copper electrowinning tankhouse.

The fitted mass transfer coefficients for iron reduction and oxidation exhibit notable disparities compared to the corresponding coefficients reported by Aminian and Bazin.¹³ Nonetheless, it is important to acknowledge that mass transfer coefficients are influenced by system-specific factors. Factors that could possibly have contributed to the observed variations in coefficient values include the fluid flow, cell geometry, temperature, electrolyte composition, and formation of oxygen gas.^14,29,41,42

Voltage Loss Parameters

The model calibration for the voltage loss parameters was initiated by defining the normalized electrolyte resistance and difference in model-predicted and experimental potential (as read from the power supply) for the duration of each dynamic experiment. In other words, the inaccuracy of the model-predicted potential (with the voltage loss parameters set to 0) was defined as a function of the normalized electrolyte resistance. The model predictions for the potential were, therefore, improved by assuming that the bulk of the voltage loss could be attributed to electrolyte resistance. This indirect approach for quantifying the voltage loss associated with electrolyte resistance was necessary because of the lack of measurements available for the case study experimental system. Dynamic measurements of the electrolyte cell temperature and sulfuric acid concentration, two factors that have a significant effect on the electrolyte conductivity,^25,26 were not taken during the experiments. Consequently, the effect of changes in these two factors on electrolyte resistance could not be directly accounted for using the empirical correlation given in Eq. 7.

The relationship between the normalized electrolyte resistance and the difference in model-predicted and the experimental potential is shown in Fig. 4, alongside the parabolic approximation. The graph shows the relationship between the variables averaged at each time-step, for all 24 dynamic electrowinning experiments. Non-linear regression was performed, again with the incorporated resampling method to determine the voltage loss parameters given in Table IV. The non-linear trend observed for the relationship between the difference in model-predicted and experimental potential, and the normalized electrolyte resistance (which did not accurately account for the effect of temperature and sulfuric acid concentration), motivated the inclusion of the described approach.

Fig. 4

Relationship between the normalized electrolyte resistance (${R}_{\rm s, norm}$) and the difference in the model-predicted and experimental potentials, together with the parabolic approximation thereof. Error bars denote the standard error of the mean for the 24 experiments.

Table IV Summary of voltage loss parameters fitted to the dynamic bench-scale electrowinning experiments

Model Validation

The performance of the dynamic model was evaluated by comparing the model-predicted KPIs with the corresponding experimental values. This enabled validation of the developed model to ensure that the model had a satisfactory range of accuracy for the proposed intended purpose. The proposed purpose of the predictive dynamic model is to enable the effective training of operators and the prediction of plant performance for monitoring and early fault detection. The predictive performance of the model was evaluated using three experiments, previously unseen by the model during calibration, termed the “validation experiments.” A disturbance was induced in a different manipulated variable for each of the three validation experiments. For the first validation experiment (Test a), the copper concentration was increased from 25 g/L to 40 g/L 4 h into the 8-h run. Likewise, for the second validation experiment (Test b), the iron concentration was increased from 1 g/L to 3 g/L. For the third validation experiment (Test c), the current density was increased from 200 A/m² to 285 A/m².

Experimental Versus Predicted Cumulative Mass Copper Plated

The accuracy of the model-predicted cumulative mass copper plated was evaluated by comparison with the experimentally determined version (Fig. 5). The model predicted the cumulative mass copper plated for all three validation experiments with good accuracy. The normalized residual mean square error (nRMSE) values were 2.2%, 6.0%, and 4.8% for Test a, Test b, and Test c, respectively. A baseline of 30% was selected for the nRMSE values, meaning that values < 30% indicate a good correlation. For the experiment in which an increased disturbance in copper concentration occurred (Test a), both the model-predicted cumulative mass copper plated and the experimental version thereof showed a linear trend, even initially during the start-up phase. This resulted from an assumption made for all experiments in which an increase disturbance in copper concentration occurred when processing the generated experimental data. It was assumed that the copper reduction reaction was reaction-rate limited. Accordingly, a linear line was used to represent the experimental cumulative mass copper plated in both the calibration and validation experiments.

Fig. 5

Experimental versus predicted cumulative mass copper plated for the validation experiments with a disturbance in xCu (Test a), xFe (Test b), and the current density (Test c).

The experimental cumulative mass copper plated for the validation experiment in which an increase disturbance in the advance electrolyte iron concentration occurred (Test b) showed a decrease (Point 1) followed by an increase (Point 2) in response to the disturbance. This behavior is not mirrored by the model-predicted cumulative mass copper plated, but the instantaneous plating rate did decrease by approximately 5.5%, before slightly increasing. The observed trend in the experimental data is suggested to be the result of the method used to induce the disturbance during the experiment. The additional ferric sulfate required to reach the final iron concentration was added to the stock solution bottle which fed the cell. The ferric iron concentration was, therefore, very high in the advance electrolyte just after the disturbance was induced. Gradually, through mixing with the recycled spent electrolyte stream, and to a lesser extent reduction of iron, the ferric iron concentration in the advance electrolyte decreased.

The experimental and model-predicted cumulative mass copper plated both showed an increase (Point 3) in response to the increase disturbance in current density (Test c). The observed behavior supports the assumption that the copper reduction reaction was reaction-rate limited.

Experimental Versus Predicted Current Efficiency

The model-predicted current efficiency was compared with the experimental version (Fig. 6). The nRMSE values were 18.5%, 27.2%, and 36.6% for Test a, Test b, and Test c, respectively. Although the nRMSE value for Test c (36.6%) exceeded the baseline of 30%, the prediction accuracy was overall reasonable.

Fig. 6

Experimental versus predicted current efficiency for the validation experiments with a disturbance in xCu (Test a), xFe (Test b), and the current density (Test c).

The reaction-rate limited assumption made for the experiments in which the copper concentration was increased (Test a) resulted in an experimental current efficiency that was independent of copper concentration and time. Literature supports the independence of the mass copper plated and, to an extent, the current efficiency from the advance electrolyte copper concentration for typical industrial operating conditions.^16,28,29 It is, however, unlikely that the current efficiency will remain constant over time, especially during start-up. The model-predicted trend, showing a slight initial increase in current efficiency, seems to better fit with what would occur in the bench-scale system. Moats and Khouraibchia²⁹ determined that for a synthetic electrolyte similar to the electrolyte used in the bench-scale experiments, between 90% and 95% of the iron was in the ferric form at the start of electrowinning. This means that, initially, a greater portion of the applied current is required for iron reduction because the iron reduction reaction is mass-transfer limited. Gradually, as more ferric iron is converted to ferrous iron and the concentration of ferric iron decreases, more current becomes available for copper reduction.

The experimental current efficiency for the validation experiment in which an increase disturbance in advance electrolyte iron concentration occurred (Test b) showed the same decrease (Point 1) followed by an increase (Point 2) observed for the corresponding experimental cumulative mass copper plated. The similarities between the behavior observed for the cumulative mass copper plated and current efficiency is expected as the current efficiency was calculated as a function of the mass copper plated. As before, it is suggested that the observed increase after the initial decrease in response to the induced disturbance is the result of the method used to increase the iron concentration. The model-predicted current efficiency decreased slightly because of the disturbance before showing a subtle increase.

Notably, the experimental current efficiency for Test b exceeds 100% between t = 3 h and t = 5.5 h. The cumulative mass copper plated used to calculate the current efficiency was determined using the instantaneous copper plated rate. The instantaneous copper plated rate was, in turn, determined via mass conservation balance using the measured advance and spent electrolyte copper concentrations, as previously mentioned. Only limited data reconciliation was possible as the system was underspecified. Consequently, it is acknowledged that the current efficiencies > 100% indicate possible shortcomings in the data validation approach.

Similarly, although the initial trend of increasing current efficiencies observed for Test b and Test c could be a result of the system not yet being at steady state, it is possible that this could also be a consequence of potential shortcomings in the data processing and validation approach. Initially, the difference between the advance and spent electrolyte copper concentrations, used to determine the mass copper plated (and, therefore, current efficiency), was less pronounced. This less pronounced difference meant that variations in the measurements had a significant effect on the calculated copper plated rate. For the purposes of model validation, the experimental data were, therefore, only analyzed for the trends observed because of the induced disturbances. For future work, it is suggested that additional variables (such as the sulfuric acid concentration) be measured to ensure more rigorous data reconciliation can be performed. Nonetheless, the observed trends are expected to remain useful for model validation purposes.

The experimental and model-predicted current efficiency showed an overall increase in response to the increase in current density (Test c), similar to the behavior described for the corresponding cumulative mass copper plated. The model-predicted current efficiency showed a slight increase compared to the constant current efficiency predicted for Test a. The observed gradual increase in experimental current efficiency resulting from the step increase in current density is suggested to indicate that the system is reaching a new steady-state operating point after adapting to the induced disturbance. It is recommended that the duration of future experiments following a similar format be extended to better comment on the observed behavior.

Experimental Versus Predicted Specific Energy Consumption

The accuracy of the model-predicted SEC was evaluated by comparison with the experimentally determined version (Fig. 7). The model accuracy is quantified using the nRMSE and mean absolute error percentage (MAPE) values presented in Table V. Although the nRMSE values for Test b and Test c show reasonable model accuracy, the value for Test a alludes to a bad model correlation (254.1%). The sharp initial decrease predicted by the model for the SEC contributed to the high nRMSE. The decrease results from the initial increase in copper plated rate, as shown by the model-predicted current efficiency. Consequently, a possible limitation of the nRMSE method is highlighted by the high value; the nRMSE is sensitive to model overpredictions, as is the case for the SEC. As all the actual SEC values are significantly > 0, the MAPE values could be used as an alternative metric to infer model accuracy. The MAPE values for all three tests indicate an acceptable model fit.

Fig. 7

Experimental versus predicted SEC for the validation experiments with a disturbance in xCu (Test a), xFe (Test b), and current density (Test c).

Table V MAPE and nRMSE values for the prediction of SEC for the validation experiments with a disturbance in xCu (Test a), xFe (Test b), and the current density (Test c)

The SEC was calculated as a function of the current, voltage, and mass of copper plated. The bench-scale experiments were conducted under current-controlled conditions, and it was previously confirmed that the mass of copper plated is independent of the copper concentration. It is, therefore, expected that the behavior of the SEC would mirror the behavior of the predicted voltage.

The experimental SEC for the experiment in which an increase disturbance in the advance electrolyte copper concentration occurred (Test a) showed a decrease in response to the disturbance (Point 1). In contrast, the corresponding model-predicted SEC showed an increase (Point 2). It is suggested that the inclusion of the voltage loss factor when predicting the voltage, and therefore indirectly the effect of electrolyte resistance, led to the model-predicted increase. Although some literature sources support the model-predicted behavior, it is only valid for copper concentrations above approximately 40 g/L, and the findings are not definitive.^29,43 If the electrolyte resistance, however, accounts for a significant portion of the energy consumption, as is expected for the case of an industrial tankhouse, it is likely that an increase in copper concentration would increase the energy consumption.

The experimental and model-predicted SEC both showed a slight increase for the increase disturbance in advance electrolyte iron concentration (Test b). The experimental SEC showed a sharp increase (Point 3) because of the induced increase disturbance in current density (Test c). The predicted SEC, however, showed a significantly smaller increase (Point 4) than the observed experimental increase. It is suggested that the increase in experimental SEC was amplified by the approximately 30-min deadtime before the mass of copper plated showed an increase (Fig. 5).

Conclusion

In the present work a dynamic semi-empirical model of the copper electrowinning process was developed. The model uses input variables that are readily measured in industrial tankhouses (or have the potential to be) to predict the KPIs of copper yield, current efficiency, and SEC. The response of the KPIs to step or pulse disturbances in the electrolyte composition, flow rate, and current density can be investigated using the model, fostering an understanding of how process variables interact.

An example has been provided of how the model may be calibrated for a specific application using dynamic bench-scale electrowinning data. The experimental work involved testing the effect of an induced step disturbance in the advance electrolyte copper or iron concentrations, or the current density, on the electrowinning performance. The calibration method was used to fit kinetic and voltage loss parameters to the bench-scale dynamic experimental data.

The accuracy of the model predictions for the electrowinning performance was evaluated by comparison with the bench-scale dynamic experimental data as an example. The model was able to predict similar responses to industry-specific disturbances as those shown in the dynamic experimental data. The only exception was the model-predicted increase in SEC for an increase in advance electrolyte copper concentration. It is expected that the electrolyte resistance will have a more significant effect on the SEC for industrial electrowinning operations. If this is the case, the model-predicted increases will be correct. The model can, however, be adapted to exclude the effect of the electrolyte resistance if required. Overall, the performance of the developed dynamic model lends credence to the application thereof. It is acknowledged that a single experimental case study is not sufficient to validate the model for use in industry. Comprehensive validation will require the availability of suitable industrial or pilot-plant electrowinning data and is left for future work. Although it is not possible to comment on the validity of the model for industrial-scale application, the limited validation conducted has indicated the model has potential for predicting the performance of an industrial tankhouse consisting of multiple electrode pairs. The developed model could be used directly in industry for operator training, process monitoring, and early fault detection. The model also represents a step towards investigating alternative control strategies for the electrowinning process.