### 1. Introduction

_{2}SO

_{4}acid with pH 2 contained 10 g/L Cu

^{2+}ions. The electrolysis was performed at a potential value of −0.4 V vs. SCE (corresponding to the cell voltage of 2.4 V). Copper was recovered after 2 h with a current efficiency of about 95% and a specific energy consumption of 2.13 kWh/kg [16]. Similarly, copper with the high purity of 98.93% was recovered at the constant cell voltage of 2.2 V and the highest current efficiency of 99.45% [19]. However, the electrolysis manner at a constant potential or at a constant cell voltage is normally suitable on lab-scale rather industrial scale operations because production and operation costs are relatively expensive. Besides, another electrolysis technique, viz. cyclone electrowinning with an optimum circulation flow which enabled minimizing polarization effects, was also used to increase the current efficiency up to 97% [20] for the copper recovery from a leached solution containing 45.7 g/L Cu

^{2+}ions in the presence of Zn

^{2+}, Cd

^{2+}and Fe

^{2+}impurities. Sudipta Roy’s group [21] investigated the copper recovery in a parallel-plate cell using a multi-step current electrolysis technique. Based on the calculation on the diminution in the concentration of copper ions, the research group established a list of current densities after each 1 h of electrolysis, which are approximately limiting current densities. However, this technique was only effective in initial 10 h, and about 67% of copper in the sludge was recovered. The achieved current efficiency of the electrolysis process was 82.1% corresponding to the initial concentration of 0.3 M Cu

^{2+}. Also, in the research of Sudipta Roy and co-worker, the cathodic mass transfer coefficient was determined from the initial electrolyte and regarded as a constant value during the electrolysis process. Accordingly, the practical limiting current density and the theoretical limiting current density were quite different, resulting in the decrease in the current efficiency of the electrolysis process. Hence, in the present work, we consider the change in the mass transfer coefficient against the change in the Cu

^{2+}concentration. Accordingly, the electrolysis parameters of the step current electrolysis process would be determined more exactly. This is beneficial for increasing the current efficiency and decreasing the specific consumption energy.

### 2. Experimental

### 2.1. Electrolyte preparation for electrolytic recovery of copper

_{2}SO

_{4}solution under magnetically stirring at 600 rpm for 75 min. After filtration, the obtained pregnant leach solution was added into a 20 vol.% solution of LIX 984N in purified kerosene as an organic extractant. Their volumetric ratio was controlled to 1:1. After subjected to stirring for 30 min, the mixture was statically stored in 30 min to separate into two liquor layers by gravity. The obtained raffinate after separation was re-mixed with the fresh organic extractant. The extraction process was repeated three times. Meanwhile, the remaining copper-loaded organic extractants were collected and mixed with a 0.4 M H

_{2}SO

_{4}solution for stripping copper. After stirring for 30 min and aging for 30 min, the mixture was gravitationally separated to attain a copper-stripped organic extractant and a copper-enriched aqueous electrolyte. The stripped organic extractant was returned to renewed contact with the acidic solution. This copper stripping process was repeated three times. Remarkably, for the copper stripping, the volumetric ratio of the copper-loaded organic extractant to acidic solution was fixed to 3:1. Finally, the copper-enriched aqueous solution was collected and analyzed the concentration using the ICP-MS method. From the ICP-MS result, the leaching solution was determined to contain 0.3 M CuSO

_{4}, 0.4 M H

_{2}SO

_{4}and 0.575 ppm iron.

### 2.2. Electrolytic recovery of copper

*η*

*,*

_{a}*η*

*), solution resistance (R*

_{c}_{s}), cell voltage (V

_{cell}), recovery efficiency (H

_{r}), current efficiency (H

_{c}) and special energy consumption (W

_{s}) all were calculated based on the simulation model. Their calculation equations are described in detail in the below section.

### 2.3. Model development of copper electrowinning

*E*

_{0}values given above are the standard equilibrium potentials for the two half reactions. Normally, if copper electrolysis is conducted at the current density smaller than or equal a limiting current density, the 100% of the cathodic current is used for copper production. In the present work, the copper electrolysis was designed to operate at limiting current densities by using the step current mode. It was supposed that only the reduction reaction of Cu

^{2+}to produce metal copper occurred at the cathode. Therefore, the reduction reaction (2) was also assumed not to occur.

*(i)*thermodynamic and kinetic relations,

*(ii)*electrolyte resistance and concentration correlation, and

*(iii)*electrochemical equations. Each section will be examined respectively as follows.

#### a) Thermodynamic and kinetic relations

^{2+}/Cu,

*a*

_{Cu}_{2+}is the activity of Cu

^{2+},

_{2}/H

_{2}O,

*a*

_{H}_{+}is the activity of H

^{+}, R is the gas constant (R = 8.314 Jmol

^{−1}K

^{−1}), F is the Faraday constant (F = 96 500 C mol

^{−1}). The activities of O

_{2}, H

_{2}O and Cu are assigned a unit activity (

*a*

*= 1,*

_{Cu}*a*

_{H}_{2}

*= 1,*

_{O}*a*

_{O}_{2}= 1).

*j*solute species, for example, Cu

^{2+}and H

^{+}ions, the activity,

*a*

*(mol/L), is a product of a concentration (*

_{j}*C*

*) and an activity coefficient (*

_{j}*γ*

*):*

_{j}*I*

*is the ionic strength of the aqueous solution (mol/kg) and*

_{γ}*I*

*= 0.5*

_{γ}_{j}is the integer charge of the ion, Aγ and Bγ are constant with A = 0.5365 kg

^{0.5}mol

^{−0.5}and B = 0.3329 kg

^{0.5}mol

^{−0.5}Å

^{−1}; B is constant (Bγ = 0.0430 kg mol

^{−1}) and r

_{j}is the ion radius of j species (Å). Regarding Cu

^{2+}and H

^{+}ions, their radii are 0.073 Å and 0.01 Å, respectively.

*η*) and is defined as the difference between the working electrode potential (

*E*) and the equilibrium potential (

*E*

*). For the cathode and anode, their overpotentials will be written:*

^{e}_{c}and E

_{a}are the cathodic and anodic working potentials, respectively.

_{c}), which is proposed to be controlled by both charge transfer rate and mass-transfer rate, will be calculated by [24]:

*i*

_{0, }

*is the cathodic exchange current density (A/m*

_{c}^{2}),

*i*

*is the limiting cathode current density (A/m*

_{L,c}^{2}),

*α*

_{c}is the cathodic charge transfer coefficient, η

_{c}is the cathode overpotential (V), T is the absolute temperature (K), F is the Faraday constant, R is the universal gas constant.

^{2+}in electrolyte (mol/m

^{3}), k

_{m}is the cathodic mass transfer coefficient (m/s),

*n*is the number of exchanged electron (n = 2).

*i*

_{0,}

*is the anodic exchange current density (A/m*

_{a}^{2}),

*α*

*is the anodic charge transfer coefficient,*

_{a}*η*

*is the anode overpotential (V).*

_{a}#### b) Electrolyte resistance and concentration correlation

*R*

*) is calculated by,*

_{s}*σ*is the specific conductance (Ω

^{−1}m

^{−1}),

*d*is the cathode-anode distance (m), A is the cross section area (m

^{2}).

*σ*parameter could be calculated:

*C*

_{Cu}_{2+}and

*C*

_{H}_{2}

_{SO}_{4}are the concentration of Cu

^{2+}and H

_{2}SO

_{4}present in the electrolyte (kg/m

^{3}), T is the electrolyte temperature (°C).

#### c) Electrochemical equations

_{c}) must equal the total anodic current (I

_{a}) and equal the applied current (

*I*

*):*

_{app}*i*

*and*

_{Cu}*i*

_{O}_{2}are the current densities related to Reactions (1) and (3). A

_{c}and A

_{a}are the cathode and anode areas, respectively.

*H*

*, of the copper electrolysis is defined by,*

_{c}*m*

*is the mass of copper collected from the cathode after electrolysis (g),*

_{t}*I*

*is the applied current (A),*

_{app}*t*is electrolysis time (h),

*ɛ*

*is the electrochemical equivalent of copper (g/Ah),*

_{Cu}*ɛ*

*= 1.186 g/Ah.*

_{Cu}_{r}, at a given time (t) is calculated according to the below equation:

*m*

*is the mass of copper in the electrolyte before recovery (g),*

_{0}*m*

*is the mass of remaining copper in the electrolyte after the electrolysis time*

_{t}*t (*g).

*H*

*, and the cell voltage*

_{r}*V*

*was to solve mathematical equations that used loops for calculation of consecutive electrolysis times.*

_{cell}### 3. Results and Discussion

### 3.1. Estimation of the input parameters in the model

#### a) Determination of exchange current density and transfer coefficient

_{4}and 0.4 M H

_{2}SO

_{4}, using a three-electrode cell configuration. In fact, the three-electrode cell was composed of a 11 L-volume bath, which is the same to the pilot cell, a stainless steel as a working electrode, and a titanium electrode as a counter electrode and vice versa. The area of the stainless steel and titanium electrodes were fixed to be 0.44 cm

^{2}. Meanwhile, the Ag/AgCl electrode served as a reference electrode. The solution was agitated using a circulation pump at a flow rate of 3 mL/min. From the obtained results in Fig. 3, by extrapolation of the two Tafel lines of the two anodic and cathode branches, the exchange current density of the cathode and anode,

*i*

_{0,}

*and*

_{c}*i*

_{0,}

*, were determined to be 12.6 and 0.1 A/m*

_{a}^{2}, respectively. Similarly, the transfer coefficients,

*α*

*and*

_{c}*α*

*were 0.55 and 0.198.*

_{a}#### b) Determination of mass transfer coefficient

^{2+}concentration in the electrolyte after electrolysis reaches approximately 0 g/L. Noticeably, during the electrolysis the Cu

^{2+}concentration decreases gradually. Accordingly, the limiting current density of the copper electrolysis decreases as well. Therefore, for the copper recovery, use of the electrolysis technique at a constant current mode during a long period of time firmly leads to high energy consumption, followed with high copper recovery cost. Such an electrolysis technique is likely ineffective. If so, applying the step electrolysis current mode with gradually decreasing values during the electrolysis process will enable high current efficiency and copper recovery efficiency.

^{2+}. To find the mass transfer coefficient in Eq. (11), the polarization curves of the copper electrodeposition in the electrolytes of 0.4 M H

_{2}SO

_{4}with the different concentrations of CuSO

_{4}, where the concentration accounted for Cu

^{2+}ranged from 5 to 25 g/L, were recorded by the linear voltammetry polarization technique. The measurements were conducted on Autolab PGSTAT302N using the three-electrode cell configuration as described above. The obtained results are displayed in Fig. 4.

*i*

_{c}_{,}

*) for the copper electrodeposition in the potential range of 0.2–0.5 V*

_{L}*vs*. Ag/AgCl corresponding to the given concentrations of

*Cu*

^{2+}were determined. After that, based on Eq. (11), the mass transfer coefficient of the

*Cu*

^{2+}containing-solutions (

*k*

*) was calculated and presented in Fig. 5 and Table S2.*

_{m}*k*

*are not an unchanged number, but*

_{m}*k*

*is regarded as a function of the concentration of Cu*

_{m}^{2+}. After fitting with a regression model,

*k*

*can be expressed as,*

_{m}*V*is the electrolysis cell volume (dm

^{3}),

*y*

*is the concentration of species*

_{j}*j*in the electrolyte (M),

*Q*

*is the volumetric flowrate of the feed (dm*

_{f}^{3}/s),

*x*

*is the concentration of species*

_{j}*j*in the feed (M);

*r*

*is the generation rate of species*

_{gen,j}*j*(mol/s),

*Q*

*is the volumetric flow leaving the bath (dm*

_{d}^{3}/s),

*r*

*is the evaporation rate of species*

_{evap,j}*j*(mol/s), and

*r*

*is the consumption rate of species*

_{con,j}*j*(mol/L).

^{2+}against electrolysis time is given:

^{2+}as Eq. (30), the concentration of Cu

^{2+}present in the electrolyte after each 30 min as well as the limiting electrolysis current was determined. After that, the practical electrolysis current could be designed properly. The total of the electrolysis time for the copper recovery was 20 h. About 5 mL of the electrolyte was drained after each 30 min of electrolysis for determination of the copper concentration via the UV-Vis method. The calculation results from Eq. (30) are listed in Table S3.

#### c) Determination of activity coefficients

*CuSO*

_{4}and 0.4 M

*H*

_{2}

*SO*

_{4}, the dissociation of the compounds could be written as:

*K*

*is the dissociation constant, [*

_{f}*H*

^{+}], [

*H*

^{+},

### 3.2. Validation of the real solution model

^{2+}after each 30 min during 20 h of electrolysis observed from model predictions and experimental data. From Table S3, it is observable that the difference between the concentrations calculated via the simulation model and the experimental concentration is negligible, smaller than 3% for the initial 14.5 h of electrolysis. After that, this deviation increases, but still maintains at values smaller than 8% at the electrolysis time of 20 h. This verified the high accuracy of the simulation equation showing the electrolysis timedependence of the copper concentration.

^{+}ions) which were generated at the anode (Reaction (3)) were accumulated sufficiently in the electrolyte. This promoted the discharge competition of H

^{+}ions against Cu

^{+2}ions at the cathode surface. As a result, the initial hypothesis that the current efficiency equals 100% and only Cu

^{+2}ions participate the reduction reaction (Reaction (1)) at the cathode is no longer suitable.

*y*= 1.00053

*x*with a high correlation coefficient R

^{2}= 0.99989. Meanwhile, concerning the copper recovery efficiency and specific energy consumption parameters these functions are

*y*= 1.01635

*x*with R

^{2}= 0.99996 and

*y*= 0.9845

*x*with R

^{2}= 0.99942. So, from the linear regression equations obtained in Fig. 7, the correlation efficient between the simulated and experimental data for the output parameters of the copper recovery efficiency, cell voltage, and specific energy consumption are 98.39%, 99.95% and 98.45%, respectively. These values are extremely high, over 98.3%, indicating the high precision of the designed simulation program. The below 1.7% deviation between the simulated and real data likely originated from ignoring the resistances of electrical connection and the conducting wires as well as excluding the parasitic reaction of H

^{+}ions discharging at the cathode. In addition, another reason, the errors induced in the measurements of the input parameters of the pilot cell system, should be accounted as well.

*α*

*) for the oxygen evolution reaction has a large influence on the model predictions. In detail, the predicted values of both simulated cell voltage and specific energy consumption fluctuated within a deviation range of around ±5% when the*

_{a}*α*

*variable was determined with an error range of ±10%. Meanwhile, the deviation in determination of other variables involving the cathodic exchange coefficient and cathodic/anodic exchange current densities, only has negligible effect on either the cell voltage or the specific energy consumption. The ±10% changes in the cathodic exchange coefficient and cathodic/anodic exchange current densities merely caused a narrow error range of below ±1% for the cell voltage or the specific energy consumption.*

_{a}