static simulation program of copper solvent extraction ...

STATIC SIMULATION PROGRAM OF COPPER SOLVENT EXTRACTION CONFIGURATIONS USING MICROSOFT EXCEL SOLVER Joseph Kafumbila This textbook provides the procedure for design static simulation

programs

using

Excel

Solver

for

optimization on an Excel worksheet. This procedure will allow the design of a static simulation program for all copper solvent extraction configurations. The static simulation program will be designed to size a new plant or track performance of an old plant.

Kafumbila Kasonta Joseph Process designer

Design of simulation program of copper solvent extraction configuration using Microsoft Excel Solver © 2016 Joseph Kafumbila [email protected]

1

Contents

1.

INTRODUCTION..................................................................................................................... 3

2.

EQUILIBRIUM LINE SIMULATION MODELS ............................................................................ 6 2.1.

EXTRACTION STEP ....................................................................................................................... 6

2.2.

STRIPPING STEP ........................................................................................................................ 15

3.

EQUILIBRIUM LINE AND MACCABE THIELE DIAGRAM ......................................................... 19 3.1.

EQUILIBRIUM LINE..................................................................................................................... 19

3.2.

MACCABE THIELE DIAGRAM ....................................................................................................... 21

4.

CONSTRAINTS OF COPPER SX-EW PLANT ............................................................................ 28 4.1.

EQUILIBRIUM CONSTRAINTS BETWEEN EXTRACTION AND STRIPPING STEPS ............................................ 28

4.2.

MAXIMUM VALUE OF EXTRACTANT VOLUME PERCENTAGE................................................................. 28

4.3.

FREE ACID CONCENTRATION IN PLS .............................................................................................. 28

4.4.

MAXIMUM FREE ACID CONCENTRATION IN SPENT ELECTROLYTE .......................................................... 28

4.5.

MINIMUM COPPER CONCENTRATION IN SPENT ELECTROLYTE ............................................................. 28

4.6.

MAXIMUM COPPER CONCENTRATION IN ADVANCE ELECTROLYTE ........................................................ 29

4.7.

OPTIMUM VALUE OF RATIO OF ORGANIC TO AQUEOUS OF EXTRACTION STEP ......................................... 30

4.8.

SATURATION RATIO SR .............................................................................................................. 30

5.

6.

SIMULATION PROGRAM USING EXCEL SOLVER PROGRAM ................................................. 31 5.1.

DESCRIPTION ........................................................................................................................... 31

5.2.

STATIC SIMULATION PROGRAM DESIGN ......................................................................................... 31 BIBLIOGRAPHY .................................................................................................................... 74

2

1.

Introduction Copper production technology changes drastically in the last 25 years with introduction of solvent

extraction-electrowinning circuit as a copper production method. The technology of copper solvent extraction produces the most economical copper from low-grade copper ore. Copper solvent extraction technology consists of two circuits connected by a common organic phase. In the first step, called extraction step, metal is extracted from aqueous phase by organic phase. In the second step, called stripping step, metal is recovered from organic phase. The second aqueous phase is more pure and concentrated.

At the beginning, copper solvent extraction configuration operating on dilute aqueous phases were constituted with two stages respectively to extraction and stripping steps. Design of this 2Ex2S configuration was simple and based on the value of copper transfer per extractant volume percentage of 0.22 (g/l/1% v/v). This value gives copper extraction efficiency greater than 98% and copper stripping efficiency of 60%. Afterwards, understanding that copper extraction efficiency was not the most important parameter than the cost of copper production plant, one stage of stripping step was removed and the value of copper transfer was increased to 0.26 (g/l/1%v/v). Design of this 2Ex1S configuration was based on the expected value of copper extraction efficiency of 90%.

MacCabe Thiele method was introduced in design of copper solvent extraction configuration when copper solvent extraction technology started to be used for high-grade copper ore. A large number of laboratory tests were required before obtaining the optimal configuration by using MacCabe Thiele method. It was at this level that a simulation model of equilibrium line of extraction and stripping steps was introduced. Several simulation models of equilibrium line of copper solvent extraction using chelating reagents were made. These simulation models were based either on equilibrium constant of copper solvent extraction chemical reaction [1] or on extrapolation curves of equilibrium lines of extraction and stripping steps [2]. Simulation model based on chemical reaction equilibrium constant was required knowledge of phenomenology of chemical reaction. Currently, most of static simulation programs of copper solvent extraction configuration use equilibrium line simulation models based on chemical reaction equilibrium constants because it is easy to simulate impact of iron co-extraction to extraction step and temperature to stripping step. Design of this static simulation program requires a certain level of knowledge in computer programming. This blocks innovation in design of new reagent or complex copper solvent extraction configurations because static simulation programs are developed by extractant suppliers far from plant designers.

Equilibrium line simulation model based on chemical reaction equilibrium constant requires knowledge of all intermediate chemical reactions. Example of intermediate chemical reactions occurring during copper solvent extraction with MOC- 45 is the following [1]: Cu2+ + 2HR ↔ CuR 2 + 2H +

(a)

3

Cu2+ + (HR)2 ↔ CuR 2 + 2H +

(b)

2HR ↔ (HR)2

(c)

HR and (HR)2 are the molecular forms of MOC- 45 extractant in organic phase. Equilibrium conditions of each chemical reaction are given by mathematical expressions (1), (2), and (3). [CuR ]x[H+ ]2

K a =[Cu+22]x[HR]2

(1)

[CuR ]x[H+ ]2

2 K b =[Cu2+ ]x[(HR)

Kc =

(2)

2 2]

[(HR)2 ]

(3)

[HR]2

where K a , K b , and K c are respectively chemical reaction equilibrium constants of chemical reactions (a), (b), and (c).

Static simulation model will be defined by finding the equilibrium constants values using experimental data of equilibrium lines. It must involve the following mathematical expressions giving mass balances.

-

Copper mass Balance: [Cu2+ ]i = [Cu2+ ]f +

-

Vo Va

x [CuR 2 ]

(5)

Oxime radical R mass balance: [HR]t = [HR] + 2 x [(HR)2 ] + 2 x [CuR 2 ]

-

(6)

Hydrogen ion mass balance: [H + ]f = [H + ]i + 2 x

Vo Va

x [CuR 2 ]

(7)

Where the indices i, f and t respectively mean initial, final and total. Vo and Va respectively mean volume of organic and aqueous phases.

4

Once the values of chemical reaction equilibrium constants are obtained, the equilibrium line simulation model will use these mathematical expressions from (1) to (7) to predict the value of copper concentration in organic phase from the value of copper and free acid concentrations in aqueous and the values of Va and Vo . This design method of equilibrium line simulation model is used by reagent suppliers and its design is complex.

For dynamic modeling used in industrial copper solvent extraction plant, plant metallurgists use extrapolation curves of equilibrium lines [2].

In most cases, the conditions for using these extrapolation

equations are limited to this copper solvent extraction plant.

The goal of this textbook is to give a simple procedure for designing a static simulation program of copper solvent extraction configuration. This simple procedure will allow designers to quickly have a static simulation program for complex configuration or new product that have not yet simulation program developed by supplier. It will also allow plant metallurgists to have a static simulation program for checking plant performances. This static simulation program uses Excel solver program for optimization. Excel Solver is the Microsoft add-in program used for what-if analysis. Excel solver program allows finding the optimum value for a formula in one cell. Knowledge of the use of Excel Solver is not sufficient. The textbook also gives advice that prevents static simulation programs to crash when initial data change.

In the second chapter, the textbook offers equilibrium line simulation models of extraction and stripping steps not requiring knowledge of intermediate chemical reactions. Equilibrium condition of global chemical reaction is resulted on the assumption that ionic strength varies very little only in aqueous phase. On the other hand, the ionic strength varies in organic phase. The textbook gives results of experimental tests. Extraction and stripping equilibrium lines were generated by contacting a series of different volumes of aqueous phase and stripped organic phase by mechanical stirring. Following attainment of equilibrium after 15 min, phases were allowed to separate. After phase disengagement, copper was analyzed in raffinate by AAS, whereas free acid was determined by titration. After more than 10 years of research, the new theory of equilibrium condition of copper solvent extraction chemical reaction begins to be clear.

In the third and fourth chapters, the textbook also gives theoretical concepts of copper solvent extraction and industrial constraints which allow, together with equilibrium line simulation models, to design a static simulation program of simple or complex copper solvent extraction configuration.

In the fifth chapter, the textbook gives the design procedure of the static simulation program of copper solvent extraction configuration from an example. The procedure involves two options. The first concerns the designers and the second concerns the plant metallurgists. The simulation program called “SimSXCu Full version 2.0” gives static simulation programs of eighteen copper solvent extraction configurations.

5

2.

Equilibrium line simulation models

2.1.

Extraction step

2.1.1.

Equilibrium condition Extraction step is guided by thermodynamic disequilibrium between aqueous and organic phases.

Copper mass transfer is stopped when thermodynamic property reaches equilibrium condition in both phases. Global copper solvent extraction chemical reaction with Lix984N extractant is followed chemical reaction (d) [3]. LIX 984N reagent, a 1:1 volume blend of LIX 860N-I and LIX 84N-I, is a mixture of 5nonylsalicylaldoxime and 2-hydroxy-5-nonylacetophenone oxime.

Cu2+ + 2HR ↔ CuR 2 + 2H +

(d)

where Cu2+ and H + are copper and hydrogen ionic species in aqueous phase, HR is acid form of Lix984N extractant, and CuR 2 is copper complex form in organic phase Thermodynamic equilibrium condition of global chemical reaction (a) is given by the mathematical expression (8) [4].

μCuR2 + 2μH+ - μCu+2 - 2μHR = 0

(8)

where μρ is chemical potential of species ‘ρ’ Chemical potential of species ‘ρ’ is given by the mathematical expression (9).

μρ = RTln(βρ x γρ x

Cρ C0ρ

(9)

)

where βρ is standard-state activity of species ‘ρ’ and is a function of solvent nature and temperature, γρ is chemical activity coefficient of specie ‘ρ’, Cρ is molar concentration of species ‘ρ’, Cρ0 is reference molar concentration which by convention is 1 molar concentration, R is the perfect gas constant, and T is the temperature

The substitution of the mathematical expression (9) for all species in the mathematical expression (8) gives the mathematical expression (10) which is the thermodynamic equilibrium condition.

6

[Cueor ] x [Heaq ]2 [Cueaq ] x [HReor ]2

=

βCu x βHR 2 βCuR2 x βH

x γHR 2 2= CuR2 x γH

γ

2

x γ Cu

Ke

2.1.2.

Value of 𝐊 𝐞 from molar concentration

2.1.2.1.

Definition

(10)

The value of thermodynamic equilibrium condition K e from molar concentrations called K e1 is given by the mathematical expression (11).

K e1 =

2.1.2.2.

[Cueor ]x[Heaq ]2

(11)

[Cueaq ]x[HReor ]2

Copper molar concentrations in organic and aqueous phases The values of copper molar concentrations at the steady-state in organic and aqueous phases are

respectively given by the mathematical expressions (12) and (13). Cue

or [Cueor] = 63.55

(12)

Cueaq

[Cueaq] = 63.55

(13)

where Cueaq and Cueor are respectively copper concentrations (g/l) in aqueous and organic phases

The mathematical expression (14) gives copper mass balance at the steady-state.

Cueor = Cuior + (Cuiaq - Cueaq ) x

Vaq

(14)

Vor

where Cuiaq and Cuior are respectively initial copper concentrations (g/l) in aqueous and organic phases. Vaq and Vor are respectively volumes of aqueous and organic phases.

2.1.2.3.

Free extractant molar concentration in organic phase The value of free extractant molar concentration in organic phase is given by the mathematical

expression (15).

[HReor ] =

V% x 0.91 x 1000 100 x 270

Cue

or - 2 x 63.55

(15)

7

where v% is extractant volume percentage (v/v) in organic phase. 0.91 is the density of Lix984N extractant. 270 is the mass molar of Lix984N extractant.

2.1.2.4.

Hydrogen ion molar concentration in aqueous phase The value of hydrogen ion molar concentration in aqueous phase comes from sulfuric acid

dissociation reaction. Sulfuric acid dissociation reaction has followed chemical reactions (e) and (f). + H2 SO4 ↔ HSO− 4 +H

K1ac = 104

(e)

−2 + HSO− 4 ↔ SO4 + H

−2 K ac 2 = 1.25 10

(f)

If C1 is free sulfuric acid molar concentration in aqueous phase and C2 is molar concentration of anion SO4−2 associated with copper and buffers in aqueous phase. Sulfuric acid dissociation reaction (e) is complete because the value of chemical reaction equilibrium constant K1ac is big. Following mass balance comes from sulfuric dissociation reaction (e). H2 SO4

HSO4−

H+

Initial state

C1

0

0

Final state

-C1

C1

C1

Mass balance

0

C1

C1

Sulfuric acid dissociation reaction (f) is not complete because the value of chemical reaction equilibrium constant K ac 2 is not big enough. Following mass balance comes from sulfuric acid dissociation reaction (f). Y is molar concentration of anion HSO4− which goes into dissociation. HSO4−

SO−2 4

H+

Initial state

C1

C2

C1

Final state

-Y

Y

Y

Mass balance

C1-Y

C2+Y

C1+Y

Thermodynamic equilibrium condition of sulfuric acid dissociation reaction (f) is given by the mathematical expression (16). Resolution of the mathematical expression (16) gives second-degree equation. Hydrogen ion molar concentration in aqueous phase is given by the mathematical expression (17). The value of molar concentration Y is given by the mathematical expression (18) and the values of constants A and B are given by the mathematical expressions (19) and (20).

K ac 2 =

(C2 +Y)(C1 +Y) (C1 −Y)

(16)

8

e [Haq ] = C1 + Y

Y=

(17)

(−A+ √A2 −4 x B) 2

(18)

A = C1 + C2 + K ac 2

(19)

B = C1 x (C2 - K ac 2 )

(20)

From the mathematical expression (18), the value of molar concentration Y becomes zero when the value of molar concentration C2 is equal to the value of chemical reaction equilibrium constant K ac 2 . When the value of molar concentration C2 is greater than the value of chemical reaction equilibrium constant K ac 2 , the value of molar concentration Y is lower than zero. In this condition, the value of molar concentration Y in the mathematical expression (17) is zero. When the value of molar concentration C2 is lower than the value of chemical reaction equilibrium constant k ac 2 , the value of molar concentration Y is greater than zero.

2.1.3.

Value of 𝐊 𝐞 from chemical activity coefficients

2.1.3.1.

Definition The value of thermodynamic equilibrium condition K e from chemical activity coefficients of species

called K e2 is given by the mathematical expression (21).

K e2 =

βCu x βHR 2

βCuR2 x βH 2

x γHR 2 2 CuR2 x γH

γ

x γ Cu

(21)

The value of K e2 is multiplication of two ratios. First ratio is the ratio of standard – state activities of species which is a constant and second ratio is the ratio of chemical activity coefficients of species. The second ratio of chemical activity coefficients also is multiplication of ratios of chemical activity coefficients in organic and aqueous phases respectively.

2.1.3.2.

Ratio of chemical activity coefficients in aqueous phase Depending on molar concentrations of species in aqueous phase, three cases are possible [5]:

-

Aqueous phase very diluted. In this condition, the values of chemical activity coefficients tend to 1.

-

Ionic strength in aqueous phase varies very little. In the condition, the value of ratio of chemical activity coefficients is a constant.

-

Ionic strength in aqueous phase varies. In this condition, the value of ratio of chemical activity coefficients is a variable.

9

The assumption is that ionic strength varies very little. So, ratio of chemical activity coefficients is a constant in aqueous phase.

2.1.3.3.

Ratio of chemical activity coefficients in organic phase In organic phase, the study done on the extractant Kelex 100, which is chelating copper solvent

extraction reagent, shows that ratio of chemical activity coefficients of species in organic phase is a function of copper concentration in organic phase [4].

The assumption is that this thermodynamic property is also true for chelating copper extractant.

2.1.4.

Observations Conclusion from preceding observations is that the value of thermodynamic equilibrium condition K e2

is a function of copper concentration in organic phase.

2.1.5.

Mathematical relationship between the values of thermodynamic equilibrium condition and copper concentration in organic phase

2.1.5.1.

Pure copper sulfate aqueous phase Test conditions of lab test 1 done with pure copper sulfate aqueous phase are the following:

-

Initial copper concentration in aqueous phase: 8 g/l.

-

Initial free acid concentration in aqueous phase: 5 g/l.

-

Extractant: Lix984N.

-

Extractant volume percentage in organic: 20%.

Table 1 gives equilibrium line data of lab test 1 from which the values of thermodynamic equilibrium condition K e1 of each steady-state position are calculated. Data of steady-state position having the value of copper concentration in aqueous of 0.79 g/l are derived from condition that the value of molar concentration C2 is equal to the value of chemical reaction equilibrium constant K ac 2 . Corresponding copper concentration in organic phase is obtained by extrapolation on equilibrium line. Figure 1 gives the values of thermodynamic equilibrium condition K e1 versus the values of copper concentration in organic phase. Results show that: -

The values of thermodynamic equilibrium condition K e1 are not the same for steady-state positions of equilibrium line.

-

The value of thermodynamic equilibrium condition K e1 is a linear function of the value of copper concentration in organic phase.

10

-

The values of line slope are not the same in the ranges (0.8 to 5.75 g/l Cu) and (5.75 to 10.8 g/l Cu) where molar concentration Y is greater or lower than zero respectively. Table 1: Value of K e1 from equilibrium line data of lab test 1 Lab test 1 Aq Org

Organic

Cu

Cu

Cu

g/l 0.04 0.09 0.35 0.79 1.19 3.38 5.92 6.92

g/l 0.80 1.58 3.83 5.75 6.76 9.26 10.45 10.80

moles/l 0.013 0.025 0.060 0.090 0.106 0.146 0.164 0.170

Free Lix984N moles/l 0.649 0.624 0.554 0.493 0.461 0.383 0.345 0.334

Aqueous K e1

Cu

Y

C1

C2

H

moles/l 0.001 0.001 0.006 0.012 0.019 0.053 0.093 0.109

moles/l 0.010 0.010 0.006 0.000 -0.005 -0.032 -0.048 -0.046

moles/l 0.176 0.175 0.171 0.164 0.158 0.124 0.084 0.068

moles/l 0.001 0.001 0.006 0.012 0.019 0.053 0.093 0.109

moles/l 0.187 0.185 0.178 0.164 0.158 0.124 0.084 0.068

1.666 1.550 1.125 0.805 0.668 0.287 0.104 0.065

Figure 1: Values of K e1 versus values of copper concentrations in organic phase of lab test 1

2.1.5.2.

Industrial copper sulfate aqueous phase Test conditions of lab test 2 done with industrial copper sulfate aqueous phase are the following:

-

Elements grade of industrial copper sulfate aqueous phase is given in Table 2.

-


-

Extractant volume percentage in the organic: 18%.

Table 3 gives equilibrium line data from which the values of thermodynamic equilibrium condition K e1 of each steady-state position are calculated. Figure 2 gives the values of thermodynamic equilibrium condition K e1 versus the value of copper concentration in organic phase.

11

Table 2: Element grades of industrial copper sulfate aqueous phase of lab test 2

Cu g/l 5.23

Co g/l 4.68

Al g/l 0.53

Fe g/l 0.97

Mg g/l 1.58

Mn g/l 1.64

Zn g/l 0.46

Ni g/l .0.02

Acid g/l 4.99

Table 3: Value of K e1 from equilibrium line data of lab test 2 Lab test 2 Aq Org

Organic

Cu

Cu

Cu

g/l 0.07 0.12 0.19 0.40 1.55 3.22 4.34

g/l 1.72 2.55 3.36 4.83 7.36 8.60 8.88

moles/l 0.027 0.040 0.053 0.076 0.116 0.135 0.140

Free Lix984N moles/l 0.553 0.526 0.501 0.455 0.375 0.336 0.327

Aqueous K e1

Cu

Y

C1

C2

H

moles/l 0.001 0.002 0.003 0.006 0.024 0.051 0.068

moles/l -0.107 -0.107 -0.106 -0.105 -0.093 -0.073 -0.058

moles/l 0.132 0.131 0.130 0.127 0.109 0.083 0.065

moles/l 0.229 0.230 0.231 0.234 0.252 0.279 0.297

moles/l 0.132 0.131 0.130 0.127 0.109 0.083 0.065

1.467 1.323 1.186 0.945 0.400 0.161 0.080

Figure 2: Values of K e1 versus values of copper concentration in organic phase of lab test 2 Results show that: -

In the industrial aqueous phase containing various sulfate buffers, molar concentration C2 is greater than the value of chemical reaction equilibrium constant K ac 2 . So, molar concentration Y is lower than zero.

12

-

The value of thermodynamic equilibrium condition K e1 also is a linear function of the value of copper concentration in organic phase.

2.1.5.3.

Observations In order to get closer to industrial conditions, only the range where molar concentrations Y are lower

than zero will be taken in simulation model. Hydrogen ion concentration in aqueous phase is given now by the mathematical expression (22). The mathematical expression (23) gives sulfuric acid concentrations in aqueous phase after copper solvent extraction.

e [Hor ]=

ACeaq

(22)

98

e i Acaq = Acaq + (Cuiaq - Cueaq) x 1,54

(23)

e i where Acaq is sulfuric acid concentration (g/l) in aqueous phase at the steady-state and Acaq is initial

sulfuric acid concentration (g/l) in aqueous phase.

2.1.6.

Equilibrium line simulation model of extraction step The substitution of the mathematical expressions (12), (13), (15) and (22) in the mathematical

expression (11) gives the mathematical expression (24) which gives the value of thermodynamic equilibrium condition K1e1 as a function of extractant volume percentage in organic phase, free acid concentration in aqueous phase and copper concentrations in organic and aqueous phases.

K1e1 =

Cueorg Cueaq

x

[Aceaq ]2

(24)

[3.3030 x V%−3.0842 x Cueorg ]2

The value of thermodynamic equilibrium condition K1e1 obtained from the mathematical expression (24) is plotted versus the values of copper concentration in organic phase in the range where the value of molar concentration Y is lower than zero using pure copper sulfate aqueous phase. Table 4 gives equilibrium line data of lab test 1, 3, 4 and 5. Figure 3 gives the value of thermodynamic equilibrium condition K1e1 versus the values of copper concentration in organic phase. Results show that: -

Initial free acid concentration does not affect the value of line slope.

-

The value of line slope is a function only of extractant volume percentage in organic phase.

Line mathematical expression of thermodynamic equilibrium condition obtained from equilibrium line data has a shape given by the mathematical expression (25).

13

K1e1 =D x Cueor + E

(25)

Table 4: Equilibrium line data of lab tests 1, 3, 4 and 5 of extraction step

Cuiaq i Acaq V%

Lab test 1 8 5 20 Cueaq 0.04 0.09 0.35 1.19 3.38 5.92 6.92

g/l g/l % Cueor 0.80 1.58 3.83 6.76 9.26 10.45 10.80

Cuiaq i Acaq V%

Lab test 3 8 15 20 Cueaq 0.10 0.23 0.76 2.00 4.20 6.30 7.10

g/l g/l % Cueor 0.80 1.56 3.63 6.01 7.80 9.00 9.30

Cuiaq i Acaq V%

Lab test 4 8 10 15 Cueaq 0.10 0.24 0.94 2.58 4.70 6.58

g/l g/l % Cueor 0.80 1.60 3.60 5.40 6.58 7.06

Cuiaq i Acaq V%

Lab test 5 8 20 15 Cueaq 0.27 0.48 1.56 3.30 5.16 6.80

g/l g/l % Cueor 0.78 1.50 3.24 4.68 5.62 6.20

Figure 3: Values of K1 versus the values of copper concentrations in organic phase from equilibrium line data of lab tests 1, 3, 4, and 5

The values of constants D and E are functions of extractant volume percentage. Mathematical expressions giving the values of constants D and C are extrapolation curves of the value of constants D and C obtained at different values of extractant volume percentage in the range of 8-32%. Equilibrium line simulation model of extraction step is given by the mathematical expression (26). Thermodynamic equilibrium conditions of K 2e1 and K1e1 given by the mathematical expressions (26) and (24) are equals for each steady-state position of equilibrium line.

K12 =(-28.511 x v%(−1.746) ) x Cueor+ 11.711 x v%(−0.646)

14

(26)

2.2.

Stripping step

2.2.1.

Equilibrium condition Stripping step is guided by thermodynamic disequilibrium between aqueous and organic phases.

Copper mass transfer is stopped when thermodynamic property reaches equilibrium condition in both phases. Copper stripping chemical reaction with Lix984N extractant is followed chemical reaction (g) which is the reverse reaction of copper extraction chemical reaction.

CuR 2 + 2H + ↔ Cu2+ + 2HR

(g)

The mathematical expression (27) gives the thermodynamic equilibrium condition. [Cueor ] x [Heaq ]2 [Cueaq ] x [HReor ]2

2.2.2.

=

βCu x βHR 2 βCuR2 x βH

2

x γHR 2 2= CuR2 x γH

γ

x γ Cu

Ks

(27)

Simulation with equilibrium line model of extraction step Thermodynamic equilibrium condition of extraction and stripping steps are similar. Equilibrium line

simulation model of extraction step is used to simulate equilibrium line of stripping step. Test conditions of lab test 6 of stripping step are the following: -

Initial copper concentration in aqueous phase: 35 g/l

-

Initial free acid concentration in aqueous phase: 180 g/l.

-

Copper concentration in loaded organic: 15.98 g/l.

-


-

Extractant volume percentage in organic: 32%.

Table 5 gives equilibrium line data and the values of copper concentration in organic phase from equilibrium line simulation model of extraction step. Figure 4 gives the values of copper concentration in aqueous phase from equilibrium line data and Extraction step simulation model versus the values of copper concentration in organic phase

Results show that there is a gap between the values of copper concentration from experimental data and equilibrium line simulation model of extraction step. The gap is due probably to the difference of ionic strength between extraction and stripping aqueous phases.

15

Table 5: Copper concentration in organic simulated with extraction step simulation model

Equilibrium line data Aqueous Organic Cu (g/l) Cu (g/l) 37.24 4.45 39.61 4.74 41.58 4.90 47.63 5.71 49.74 5.84 53.95 6.42

Model Aqueous Cu (g/l) 22.14 23.89 24.89 29.99 30.83 34.68

Figure 4: Copper concentration from equilibrium line data and Extraction step simulation model in aqueous phase versus copper concentration in organic phase of lab test 6

2.2.3.

Equilibrium line simulation model of stripping step Table 6 gives equilibrium line data and the values of thermodynamic equilibrium condition K s1 of lab

test 6. Figure 5 gives the value of thermodynamic equilibrium condition K s1 versus the values of copper concentration in organic phase.

Results show that: -

The value of thermodynamic equilibrium condition K s1 is a linear function of the values of copper concentration in organic phase.

16

Accuracy of the value of thermodynamic equilibrium condition K s1 depends on accuracy in chemical

-

analysis of copper concentration in aqueous phase. -

The value of molar concentration Y is lower than zero. So, line slope is a function of extractant volume percentage only. The values of K1s1 is given by the mathematical expression (28).

K1s1 =

Cueorg Cueaq

x

[Aceaq ]2

(28)

[3.3030 x V%−3.0842 x Cueorg ]2

Table 6: Value of K s1 from equilibrium line data of lab test 6 Lab test 1 Aq Org

Organic

Cu

Cu

Cu

g/l 37.2 39.6 41.6 47.6 49.7 53.9

g/l 4.45 4.74 4.90 5.71 5.84 6.42

moles/l 0.070 0.075 0.077 0.090 0.092 0.101

Free Lix984N moles/l 0.938 0.929 0.924 0.899 0.895 0.876

Aqueous K s1

Cu

Y

C1

C2

H

moles/l 0.586 0.623 0.634 0.750 0.783 0.849

moles/l -0.562 -0.598 -0.628 -0.718 -0.748 -0.809

moles/l 1.802 1.764 1.733 1.638 1.605 1.539

moles/l 0.586 0.623 0.654 0.750 0.783 0.849

moles/l 1.802 1.764 1.733 1.638 1.605 1.539

0.441 0.432 0.415 0.398 0.378 0.367

Figure 5: Values of K s1 versus the values of copper concentration in organic phase of lab test 6 Line mathematical expression of thermodynamic equilibrium condition obtained from equilibrium line data has a shape given by the mathematical expression (29).

K1s1 =F x Cueor + G

(29)

17

The values of constants D and E are functions of extractant volume percentage. Mathematical expressions giving the values of constants D and C are extrapolation curves of the value of constants D and C obtained at different values of extractant volume percentage in the range 8-32%. Equilibrium line simulation model of stripping step is given by the mathematical expression (30). Thermodynamic equilibrium conditions of K 2s1 and K1s1 given by the mathematical expressions (30) and (28) are equals for each steady-state position of equilibrium line.

K 2s1 =(4.8579 x 10−3x v% - 0.19183) x Cueor + 11.365 x v%(−0.85)

18

(30)

3.

Equilibrium line and MacCabe Thiele diagram

3.1.

Equilibrium line

3.1.1.

Description Equilibrium line gives the distribution of dissolved component, between two phases. In the case of

copper solvent extraction, it shows how copper distributes between aqueous and organic phases under the different conditions experienced in the process. Equilibrium line data are obtained from lab test or by computer modeling.

3.1.2.

Computer modeling of equilibrium line Thermodynamic equilibrium conditions given by the mathematical expressions (24) and (26) for

extraction step and by the mathematical expression (28) and (30) for stripping step are used for computer modeling of extraction and stripping equilibrium lines respectively. Copper and free acid concentrations in initial aqueous phase and also extractant volume percentage in organic phase are data which must be known for computer modeling of equilibrium lines. Thermodynamic equilibrium conditions K1e1 and K 2e1 for extraction step and K1s1 and K 2s1 for stripping step are rearranged to have copper concentration in aqueous phase on one side and copper concentration in organic phase on the other side. It appears two news equilibrium corrections α1 and α2 for extraction step and π1 and π2 for stripping step. The mathematical expressions of news equilibrium corrections are given by the mathematical expressions (31) and (32) for extraction step and (33) and (34) for stripping step. At the steadystate, the equilibrium corrections α1 and α2 are equals for extraction step and π1 and π2 are equals for stripping step.

α1 = α2 = π1 =

π2 =

[Aciaq +(Cuiaq −Cueaq ) x 1.54]2

(31)

Cueaq [−28.511 x v%(−1.746) x Cueor +11.711 x v%(−0.646) ] x (3.303 x V%−3.0842 x Cueor )2

(32)

[Aciaq +(Cuiaq −Cueaq ) x 1.54]2

(33)

Cueor

Cueaq

[(4.8579 x 10−3 x v% − 0.19183) x Cueor +11.365 x v%(−0.85) ]x(3.303 x V% − 3.0842 x Cueor )2 Cueor

19

(34)

There are two ways for computer modeling of equilibrium line. The first way is started from the known value of copper concentration in aqueous phase from which copper concentration in organic phase is calculated. Grouping of equilibrium corrections α1 and α2 of extraction step or π1 and π2 of stripping step gives a cubic equation as a function of copper concentration in organic phase.

The second way is started from the known value of copper concentration in organic phase from which copper concentration in aqueous phase is calculated. Grouping of equilibrium corrections α1 and α2 of extraction step or π1 and π2 of stripping step gives a second-degree equation as a function of copper concentration in aqueous phase.

For extraction step, the value of copper concentration in aqueous phase is given by the mathematical expression (35) where the values of constants H and J are given by the mathematical expression (36) and (37).

Cueaq =

−H−√H2 −4 x J 2

(35)

i H = -1.299 x Acaq – 2 x Cuiaq – 0.422 x α2

(36)

i J = (0.644 x Acaq + Cuiaq )2

(37)

For stripping step, the value of copper concentration in aqueous phase is given by the mathematical expression (38) where the values of constants L and M are given by the mathematical expressions (39) and (40).

Cueaq =

3.1.3.

−L−√L2 −4 x M 2

(38)

i L = -1.299 x Acaq – 2 x Cuiaq – 0.422 x π2

(39)

i M = (0.644 x Acaq + Cuiaq )2

(40)

Maximum loading Maximum loading (ML) is the value of copper concentration in organic phase corresponding to the

value of initial copper concentration in aqueous phase on equilibrium line at the steady-state.

Maximum loading is a function of the values of copper and free acid concentrations in aqueous phase and also extractant volume percentage in organic phase. The value of maximum loading is given by the first away method for computer modeling of equilibrium line in which the value of copper concentration in aqueous phase at steady-state is the value of initial copper concentration Cuiaq . The value of equilibrium correlation α1 is changed to α1ML . The value of equilibrium correlation α1ML is given by the mathematical expression (41).

20

α1ML =

3.1.4.

[Aciaq ]2

(41)

Cuiaq

Absolute maximum loading Absolute maximum loading is the value of maximum loading when the value of initial free acid

concentration in aqueous phase is zero. Therefore, the values of equilibrium correlations α1 and α2 are also zeros. The value of absolute maximum loading (AML) comes from equilibrium correlation α2 and is given by the mathematical expression (42). The value of absolute maximum loading is not a linear function of extractant volume percentage.

AML = 0.4108 x (V%)1.1

(42)

3.2.

MacCabe Thiele diagram

3.2.1.

Stage scheme of extraction and stripping steps Extraction and stripping steps of copper solvent extraction are frequently carried out at industrial scale

with mixer-settlers. This equipment includes a mixer in one stage or two stages in series to disperse one phase into other to provide interfacial contact for mass transfer, followed by a settler to allow phases to coalesce and separate.

3.2.1.1.

Extraction step Figure 6 gives scheme of stage of rank ‘n’ of extraction step in cascade configuration. Stage of rank

n−1 n+1 ‘n’ of extraction step receives aqueous phase Eaq from stage of rank ‘n-1’ and organic phase Eor from stage of n n rank ‘n+1’. Stage of rank ‘n’ produces aqueous phase Eaq and organic phase Eor .

Figure 6: Scheme of stage of rank ‘n’ of extraction step

21

o For cascade configuration containing ‘m’ stages, stage of rank ‘1’ receives PLS (Eaq ) and produces 1 m+1 Loaded organic LOe (Eor ). Stage of rank ‘m’ receives stripped organic SOe (Eor ) and produces raffinate Raf m (Eaq ).

3.2.1.2.

Stripping step Figure 7 gives scheme of stage of rank ‘n’ of stripping steps in cascade configuration. Stage of rank

n+1 n−1 ‘n’ of stripping step receives aqueous phase Saq from stage of rank ‘n+1’ and organic phase Sor from stage of n n rank ‘n-1’. Stage of rank ‘n’ produces aqueous phase Saq and organic phase Sor .

0 For cascade configuration containing ‘p’ stages, stage of rank ‘1’ receives loaded organic LOs (Sor ) p+1

1 and produces advance electrolyte AD (Saq ). Stage of rank ‘p’ receives spent electrolyte SP (Saq ) and produces p

stripped organic SOs (Sor ).

Figure 7: Scheme of stage of rank ‘n’ of stripping step

3.2.2.

Parameters and designation

3.2.2.1.

Stage of rank ‘n’ Each aqueous phase leaving stage of rank ‘n’ of extraction or stripping step is characterized by the

following independent parameters: -

n Vaq

: Flowrate (m3/h).

-

Cunaq

: Copper concentration (g/l).

-

n Acaq

: Free acid concentration (g/l).

Each organic phase leaving stage of rank ‘n’ of extraction or stripping step is characterized by the following independent parameters:

22

-

n Vor

: Flowrate (m3/h).

-

Cunor

: Copper concentration (g/l).

-

v%n

: Extractant volume percentage in organic phase (%).

The important parameters of stage of rank ‘n’ of extraction and stripping are: -

Ratios Rne and Rns are ratios of organic to aqueous flowrates of extraction and stripping stage of rank ‘n’ respectively.

-

Parameters Mefen and Mefsn are respectively mixer efficiency of stage of rank ‘n’ of extraction and stripping steps.

3.2.2.2.

Cascade The important parameters of cascade configuration are:

-

Ratios Rce and Rcs are respectively ratios of organic to aqueous flowrates of extraction and stripping cascade.

-

Parameter v%c is extractant volume percentage in organic phase (%) of cascade.

-

Parameters Effec and Effsc are respectively extraction and stripping efficiency of cascade.

-

Parameter Cut is net copper transfer from organic phase to copper electrolyte per 1% of extractant volume percentage of stripping cascade.

3.2.3.

Mass balance

3.2.3.1.

Extraction step Stage of rank ‘n’ Conservation of copper mass flowrate at stage of rank ‘n’ is given by the mathematical expression

(43). The value of ratio Rne of stage of rank ‘n’ is given by the mathematical expression (44). The value of free acid concentration in outlet aqueous phase of stage of rank ‘n’ is given the mathematical expression (45). n n n+1 n n n n Vaq x Cun−1 aq + Vor x Cuor = Vaq x Cuaq + Vor x Cuor

Vn

n−1 Cun aq −Cuaq

(43)

Rne = Vnor = Cun+1 −Cun

(44)

n n−1 n Acaq = Acaq + (Cun−1 aq - Cuaq ) x 1.54

(45)

aq

or

or

23

Cascade

Conservation of copper mass flowrate of cascade is given by the mathematical expression (46). The value of ratio of Rce

of cascade is given by the mathematical expression (47). The value of free acid

concentration in outlet aqueous phase of cascade is given by the mathematical expression (48). Copper extraction efficiency of cascade is given by the mathematical expression (49). c c c c Vaq x PLS + Vor x SOe = Vaq x Raf + Vor x LOe

Vc

PLS−Raf e −SOe

Rce = Vcor = LO aq

(47)

m 0 Acaq = Acaq + (PLS-Raf) x 1.54

Effec =

3.2.3.2.

(PLS−Raf) PLS

(46)

(48)

(49)

x 100

Stripping step Stage of rank ‘n’ Conservation of copper mass flowrate of stage of rank ‘n’ is given by the mathematical expression

(50). The value of ratio Rns is given by the mathematical expression (51). The value of free acid concentration in outlet aqueous phase of stage of rank ‘n’ is given by the mathematical expression (52). n n n+1 n n n n Vor x Cun−1 or + Vaq x Cuaq = Vor x Cuor + Vaq x Cuaq

Rns =

Vn or Vn aq

=

n+1 Cun aq −Cuaq

(50)

(51)

n Cun−1 or −Cuor

n n+1 n Acaq = Acaq + (Cun+1 aq - Cuaq ) x 1.54

(52)

Cascade

Conservation of copper mass flowrate of cascade is given by the mathematical expression (53). The value of ratio Rcs is given by the mathematical expression (54). The value of free acid concentration in outlet aqueous phase of cascade is given by the mathematical expression (55). Copper stripping efficiency of cascade is given by the mathematical expression (56). Net copper transfer from organic phase to aqueous phase is given by the mathematical expression (57).

24

c c c c Vor x LOs + Vaq x SP = Vor x SOs + Vaq x AD

Vc

(53)

AD−SP s −SOs

Rcs = Vcor = LO aq

p+1

(54)

p+1

1 Acaq = Acaq + (Cuaq - Cu1aq) x 1.54

Effsc =

Cut =

(LOs −SOs ) LOs

(LOs −SOs ) V%c

(55)

x 100

(56)

AD−SP

(57)

= V%c x Rc (g/l/v%) s

3.2.4.

MacCabe Thiele diagram

3.2.4.1.

Extraction step MacCabe Thiele diagram of stage of rank ‘n’ of extraction step is shown in Figure 8. The Point ‘An ’

gives feed coordinates, the Point ‘B n ’ gives outlet coordinates and the Point ‘C n ’ gives outlet equilibrium coordinates. Triangle ‘E n B n – B n Dn – Dn E n ’ gives MacCabe Thiele diagram of stage of rank ‘n’ of extraction step. Slope of line ‘E n Dn ’ is given by the mathematical expression (58). Slope of line ‘An B n ’ is given by the mathematical expression (59).

Slope ‘E n Dn ’=

n+1 Cun or −Cuor n−1 Cuaq −Cun aq

=

Cun −Cun+1

1 Rn e

(58)

1

Slope ‘An B n ’= Cunor −Cuor n−1 = - Rn aq

aq

(59)

e

Copper concentrations in aqueous and organic phases, leaving stage of rank ‘n’ are given by the mathematical expressions (60) and (61). Mefn e 100

+ Cun−1 aq x (1-

Mefn e 100

+ Cun+1 or x (1-

Cunaq= Cun(aq/e) x Cunor= Cun(or/e) x

Mefn e ) 100

(60)

Mefn e ) 100

(61)

25

Figure 8: MacCabe Thiele diagram of stage of rank ‘n’ of extraction step

3.2.4.2.

Stripping step MacCabe Thiele diagram of stage of rank ‘n’ of stripping step is shown in Figure 9. The Point ‘An ’

gives feed coordinates, the Point ‘B n ’ gives outlet coordinates and the Point ‘C n ’ gives outlet equilibrium coordinates. Triangle ‘Dn B n – B n E n – E n Dn ’ gives MacCabe Thiele diagram of stage of rank ‘n’ of stripping step. Slope of line ‘E n Dn ’ is given by the mathematical expression (62). Slope of line ‘An B n ’ is given by the mathematical expression (63). Cun−1 −Cun

1

or Slope E n Dn = Cunor −Cun+1 = Rn aq

aq

Cun−1 −Cun

(62)

s

1

or or Slope An B n = Cun+1 = - Rn −Cun aq

aq

(63)

s

Copper concentrations in aqueous and organic phases, leaving stage of rank ‘n’ are given by the mathematical expressions (64) and (65).

Cunaq= Cun(aq/e) x

Mefn s 100

+ Cun+1 aq x (1-

Mefn s ) 100

(64)

Cunor= Cun(or/e) x

Mefn s 100

+ Cun−1 or x (1-

Mefn s ) 100

(65)

26

Figure 9: MacCabe diagram of stage of rank ‘n’ of stripping step

27

4.

Constraints of copper SX-EW plant

4.1.

Equilibrium constraints between extraction and stripping steps In copper solvent extraction plant having extraction step containing one or more cascades and

stripping step containing one or more cascades, equilibrium constraints between extraction and stripping steps are given by the mathematical expressions (66) and (67).

4.2.

LOe = LOs

(66)

SOe = SOs

(67)

Maximum value of extractant volume percentage Organic phase is constituted with extractant and diluent. Extractant is organic compound which

extracts metal from aqueous phase. Diluent is organic liquid in which extractant is dissolved. In the industrial practices, the maximum value of extractant volume percentage in organic phase in copper solvent extraction is between 30 – 33% [6]. Organic phase viscosity increases with increasing of extractant volume percentage in organic phase.

4.3.

Free acid concentration in PLS Free acid concentration in PLS depends on leaching technique. Free acid concentration in PLS can go

from 0.5 to 80 g/l.

4.4.

Maximum free acid concentration in spent electrolyte The value of maximum free acid concentration in spent electrolyte is 180 g/l. This value of maximum

free acid concentration is fixed by anode corrosion rate in EW circuit. It has been observed that increasing of free acid concentration in copper electrolyte increases disconnection of PbO2 to Pb metallic. Therefore, it increases anode corrosion rate [7].

4.5.

Minimum copper concentration in spent electrolyte In copper EW circuit, limiting current density is current density beyond which copper powder is

produced. The value of limiting current density increases with increasing copper concentration in copper

28

electrolyte. Critical current density is current density beyond which copper cathode structure starts to be granular. It has been observed that the value of critical current density is 35% of limiting current density [8].

The maximum value of current density in copper EW circuit is fixed by anode corrosion rate due to oxygen evolution on anode. Evolution of oxygen gas on anode increases with increasing current density. Increasing of oxygen evolution on anode also increases unhooking of PbO2 on anode. The maximum value of current density, which allows to anodes to have a life of 5 years is 320 A/m2. In the presence of cobalt in copper electrolyte at the level of 100 mg/l (cobalt stabilizes PbO2 on anode), the maximum value of current density can reach 370 A/m2. The value of maximum current density must be lower than critical current density of copper concentration in spent electrolyte. It has been observed that the minimum value of copper concentration in spent electrolyte which respects this constraint is 30 g/l. In the industrial practice, the value of copper concentration in spent electrolyte is varied between 30 and 35 g/l.

4.6.

Maximum copper concentration in advance electrolyte In the past time, ratio Rcs /Rce was fixed. This ratio is the enrichment factor of copper concentration

from leach solution to copper electrolyte. Under this condition, copper concentration in advance electrolyte increases with increasing amount of copper transferred. The value of maximum copper concentration in advance electrolyte is reached when copper concentration in loaded organic reaches the value of maximum loading (ML). Copper concentration in advance electrolyte is given by the mathematical expression (68)

AD = SP +

Rcs Rce

x (PLS – Raf)

(68)

Currently, copper tankhouse consists of two parts; the first, called scavenger cells, is treating advance electrolyte to oxidize soluble organic phase coming with advance electrolyte. The second, called commercial cells, is treating mixture of outlet electrolyte from scavenger cells and recycled electrolyte from commercial cells. Design of copper tankhouse is done such as flowrate of copper electrolyte to commercial cells is 4 times greater than advance electrolyte flowrate to scavenger cells. This means that 20% of cells in copper tankhouse work as scavenger cells. Maximum copper concentration drop across cell is fixed at 3 g/l because copper electrolyte flowrate in conventional cell is fixed by cathode face velocity for a better quality of copper cathode surface. In the industrial practice, cathode face velocity is 0.10 m3/hrs. per m2 of total cathode area in the cell [9].

Copper concentration in spent electrolyte is fixed at 35 g/l. In this condition, copper concentration drop between spent and advance electrolytes is 15 g/l and maximum copper concentration in advance electrolyte is 50 g/l. These conditions give smallest size of copper tankhouse. The value of ratio Rcs is given by the mathematical expression (69). (AD−SP)

Rcs = Rce x (PLS−Raf)

(69)

29

4.7.

Optimum value of ratio of organic to aqueous of extraction step Ratios Rce and Rcs are external ratios. Internal ratio is ratio of volume flowrates inside of mixer. When

the value of internal ratio (O/A) is greater than 1.1, aqueous phase is dispersed in organic phase. Mixer works in organic continuity regime. On the other side, when ratio (A/O) is greater than 1.1, organic phase is dispersed in aqueous phase. Mixer works in aqueous continuity regime. It has been observed that crud formation decreases when mixer works in organic continuity regime [10]. Therefore, all mixers of extraction and stripping steps work in organic continuity regime and the optimum value of ratio Rce is now 1.1 which gives smallest size of mixer and settler without any organic or aqueous stage recycle.

4.8.

Saturation ratio SR Currently, saturation of organic phase with copper (high value of (LO/ML) is important parameter in

design of copper solvent extraction plant. This parameter takes into account good use of organic phase (high value of Net copper transfer) and also rejection of iron from organic phase.

In copper solvent extraction, iron is transferred to copper electrowinning circuit by physical entrainment. In addition, Ferric is transferred by chemical entrainment. Studies on iron transfer show that the average of iron chemical entrainment is between 35 to 60% of total entrainment [11]. Iron presence in copper electrolyte causes reduction of current efficiency. Maximum iron concentration in copper electrolyte is 2.5g/l which will give current efficiency greater than 90%. Copper electrolyte bleed is used to maintain this iron level in electrolyte. Increasing of flowrate of copper electrolyte bleed increases cost of cobalt reagent which is added to copper EW circuit.

In all copper solvent extraction plants having cascade configuration with two stages to extraction step, it has been observed that iron concentration in loaded organic out of stage of rank 1 is lower than iron concentration in partially loaded organic out of stage of rank 2. This effect is called ‘crowding’ [11].

The value of ratio (

LO ML

) is a new constraint called saturation ratio ‘SR’. This constraint is applicable to

all copper solvent extraction configurations (2Ex1S, 2Ex2S, 3Ex1S, series-parallel, interlaced, double seriesparallel and others). The mathematical expression (70) gives the value of saturation ratio.

SR =

LO ML

(70)

x 100 (%)

30

5.

Simulation program using Excel solver program

5.1.

Description Simulation program is done on Excel spreadsheet in format of Excel solver program. Excel solver is

the Microsoft add-in program used for what-if analysis. Excel solver program allows finding the optimum value for a formula in cell called the objective cell.

First step of static simulation program design is creation of simulation program table on Excel spreadsheet. Simulation program table is made with three small tables which are table of extraction step, table of stripping step, and table of simulation program constraint. In simulation program table, data have red color, solver constraints have blue color and solver constraints have green color.

There are two static simulation program options for each copper solvent extraction configuration. For each option of each copper solvent extraction configuration, there is a static simulation program table. Two options are: -

Option 1: Unknown parameter in copper solvent extraction simulation program is extractant volume percentage in organic phase. This static simulation program is for designer of copper solvent extraction configuration.

-

Option 2: Unknown parameters in copper solvent extraction simulation program are extractant volume percentage, saturation ratio, and mixer efficiencies of extraction and stripping steps for existing plant. This simulation program is for plant metallurgist.

5.2.

Static simulation program design

5.2.1.

General In this book for good understanding, static simulation program design will be done step by step using

a conventional configuration containing two stages in cascade to extraction step and two stages in cascade to stripping step as an example.

31

5.2.2.

Option 1 . In this option, unknown parameter is extractant volume percentage. Table 7 gives data of chosen

example for explanation of simulation program conception.

Table7: Data of chosen example of copper solvent extraction configuration

Description PLS Flow Copper concentration in PLS Acid concentration in PLS Ratio O/A Saturation ratio Number of extraction stage Stage of rank 1 mixer efficiency Stage of rank 2 mixer efficiency

Symbol Extraction step PLS Flow PLS Cu PLS Ac Rce SR

Stripping step Copper concentration in spent electrolyte Acid concentration in spent electrolyte Copper concentration in advance electrolyte Number of stripping stage Stage of rank 1 mixer efficiency Stage of rank 2 mixer efficiency

Unity

Value

m3/h g/l g/l

400 7 1.96 1.1 80 2 92 95

%

Mefe1 Mefe2

% %

SP Cu SP Ac AD Cu

g/l g/l g/l

Mefs1 Mefs2

% %

35 180 50 2 98 85

Static simulation program table of option 1 of 2Ex2S configuration is given in Table 8 as it appears on Excel Microsoft spreadsheet. Data from Table 7 have red color. In extraction table, designation αn2 , H n and J n are respectively the values of equilibrium correlation 𝛼2 and constants H and J of stage of rank ‘n’ for calculating the value of copper concentration in aqueous phase from the value of copper concentration in organic phase. Designation An , C n , Dn and E n are the Points A, B, C, and D from extraction step MacCabe Thiele diagram of stage of rank ‘n’. In stripping table, designation πn2 , Ln and M n are respectively the values of equilibrium correlation 𝜋2 and constants L and M of stage of rank ‘n’ for calculating the value of copper concentration in aqueous phase from the value of copper concentration in organic phase. Designation An , C n , Dn and E n are the Points A, B, C, and D from stripping step MacCabe Thiele diagram of stage of rank ‘n’.

5.2.2.1.

Extraction step Data

Simulation program design is started with solver variables of data. These solver variables will be data of all mathematical expressions of extraction step.

32

-

In the cell ‘F6’ of solver variable of PLS flowrate (PLS flow), enter the number giving the value of PLS flowrate data that is in the cell ‘C6’.

-

In the cell ‘F7’ of solver variable of PLS copper concentration (PLS Cu), enter the number giving the value of PLS copper concentration data that is in the cell ‘C7’.

-

In the cell ‘F8’ of solver variable of PLS free acid concentration (PLS Ac), enter the number giving the value of PLS acid concentration data that is in the cell ‘C8’.

-

In the cell ‘F9’ of solver variable of saturation ratio (SR), enter the number giving the value of saturation ratio data that is in the cell ‘C9’.

-

In the cell ‘F10’ of solver variable of ratio (Rce ), enter the number giving the value of ratio (Rce ) data that is in the cell ‘C10’.

-

In the cell ‘F11’ of solver variable of mixer efficiency ( Mefe1 ), enter the number giving the value of mixer efficiency ( Mefe1 ) that is in the cell ‘C11’.

-

In the cell ‘F12’ of solver variable of mixer efficiency (Mefe2 ), enter the number giving the value of mixer efficiency (Mefe2 ) that is in the cell ‘C12’. Solver constraints of solver variables of extraction step data are the following:

-

In the cell ‘I6’, enter formula: ‘=F6-C6’.

-


-


-


-


-


-


Starting value

The starting value of extractant volume percentage of simulation program is random number between 1 and 32%. The starting value of extractant volume percentage is used in all mathematical expression of extraction step as extraction volume percentage data. The optimum value of extractant volume percentage is known at the end of simulation program by running Excel Solver program. The starting value of extractant volume percentage is the expected value of extractant volume percentage for those who have experience. For this case, chosen starting value is 25%. -

In the cell ‘F15’, enter the number ‘25’.

General The value of organic flowrate is in the cell ‘C18’. This value is given by the mathematical expression (47) where the values of ratio (Rce ) is in the cell ‘F10’ and PLS flowrate is in the cell ‘F6’.

33

-

In the cell ‘C18’, enter formula: ‘=F10*F6’. The value of absolute maximum loading is in the cell ‘C19’. This value is given by the mathematical

expression (42) where the value of extractant volume percentage is in the cell ‘F15’. -

In the cell ‘C19’, enter formula: ‘=0.4108*F15^(1.1)’.

The value of maximum loading is solver variable because resolution of cubic equation is complicated. The value of maximum loading is in the cell ‘F21’. The starting value of maximum loading is the value of absolute maximum loading. -

In the cell ‘F21’, enter the number ’14.170’. Solver constraint of the value of maximum loading is that the values of equilibrium correlations α1ML

and α2 must be equals. Solver constraint of the value of maximum loading is in the cell ‘I21’. The value of equilibrium correlation α1ML is in the cell ‘B21’ and is given by the mathematical expression (41). The value of equilibrium correlation α2 is in the cell ‘C21’ and is given by the mathematical expression (32) where the value of copper concentration in organic phase at the steady-state is the value of maximum loading. -

In the cell ‘B21’, enter formula: ‘=F8^2/F7’.

-

In the cell ‘C21’, enter formula: ‘=((-28.511*F15^(-1.746)*F21+11.711*F15^(-0.646))*(3.303*F153.0842*F21)^2)/F21’.

-

In the cell ‘I21’, enter formula: ‘=B21-C21’.

Extraction stage 1 The value of copper concentration in organic phase of the Point E1 is in the cell ‘C28’ and is the value of loaded organic. The value of loaded organic is given by the mathematical expression (70) where the values of saturation ratio (SR) is in the cell ‘F9’ and the value of maximum loading is in the cell ‘F21’. -

In cell ‘C28’, enter formula: ‘=F9*F21/100’. The value of copper concentration in aqueous phase of the Point E1 is in the cell ‘D28’ and is the

value of copper concentration in PLS which is in the cell ‘F7’. -

In the cell ‘D28’, enter formula: ‘=F7’ The value of copper concentration in organic phase of the Point C1 is solver variable and is in the cell

‘F29’. In the cell ‘F29’, enter a random number between the values of copper concentration in organic phase of

34

the Point E1 and the value of maximum loading. The starting value of copper concentration in organic phase of the Point C1 is the first whole number above the value of copper concentration in organic phase of the Point E1 . In this case, a random number is 12. -

In the Cell ‘F29’, enter the number ‘12’.

-

In the cell ‘C29’, enter formula: ‘=F29’ The value of copper concentration in aqueous phase of the Point C1 is in the cell ‘D29’. This value is

given by the mathematical expression (35) where the value of constant H1 is in the cell ‘C26’ and the value of constant J1 is in the cell ‘D26’. The value of constant H1 is given by the mathematical expression (36) where the value of initial free acid concentration is in the cell ‘F8’, the value of initial copper concentration is in the cell ‘F7’, and the value of equilibrium correlation α12 is in the cell ‘B26’. The value of equilibrium correlation α12 is given by the mathematical expression (32) where the value of extractant volume percentage is in the cell ‘F15’ and the value of copper concentration in organic phase is in the cell ‘C29’. -

In the cell ‘B26’, enter formula: ‘=((-28.511*F15^(-1.746)*C29+11.711*F15^(-0.646))*(3.303*F153.0842*C29)^(2))/C29’.

-

In the cell ‘C26’, enter formula: ‘=-1.299*F8-2*F7-0.422*B26’. The value of constant J1

is given by the mathematical expression (37) where initial free acid

concentration is in the cell ‘F8’ and initial copper concentration is in the cell ‘F7’. -

In the cell ‘D26’, enter formula: ‘=(0.644*F8+F7)^(2)’.

-

In the cell ‘D29’, enter formula: ‘=[-C26-(C26^(2)-4*D26)^(0.5)]/2’. . The values of copper concentrations in aqueous phase of the Points E1 and A1 are equals. The value

of copper concentration in aqueous phase of the Point A1 is in the cell ‘D32’. -

In the cell ‘D32’, enter formula: ‘=D28’. The value of copper concentration in aqueous phase of the Point B1 is in the cell ‘D30’. This value is

given by the mathematical expression (60) where the value of Cu1(aq/e) is the value of copper concentration in aqueous phase of the Point C1 , the value of Cu0aq is the value of copper concentration in aqueous phase of the Point A1 and the value of mixer efficiency of stage of rank 1 is in the cell ‘F11’. -

In the ‘D30’, enter formula: ‘=D29*F11/100 + D32*(1-F11/100)’. The value of copper concentration in organic phase of the Point B1 is in the cell ‘C30’. This value is

given by the mathematical expression (59) where the value of ratio (Rce ) is in the cell ‘F10’, the value of Cu1or is the value of copper concentration in organic phase of the Point B1 , the value of Cu2or is changed by the value of

35

Cu1or/e which is the value of copper concentration in organic phase of the Point C1 , the value of Cu1aq is the value of copper concentration in aqueous phase of the Point B1 and the value of Cu0aq is changed by the value of Cu1aq/e which is the value of copper concentration in aqueous phase of the Point C1 . -

In the cell ‘C30’, enter formula: ‘=C29+(1/F10)*(D29-D30)’. The value of copper concentration in organic phase of the Point A1 is in the cell ‘C32’. This value is

given by the mathematical (59) where the value of ratio (Rce ) is in the cell ‘F10’, the value of Cu1or is changed by the value of Cu1or/e which is the value of copper concentration in organic phase of the Point C1 , the value of Cu2or is the value of copper concentration in organic phase of the Point A1 , the value of Cu1aq is changed by the value of Cu1aq/e which is the value of copper concentration in aqueous phase of the Point C1 and the value of Cu0aq is the value of copper concentration in aqueous phase of the Point A1 . -

In the cell ‘C32’, enter formula: ‘=C29+(1/F10)*(D29-D32)’. The values of copper concentrations in organic phase of the Points D1 and A1 are equals. The value of

copper concentration in organic phase of the Point D1 is in the cell ‘C31’. -

In the cell ‘C31’, enter formula: ‘=C32’. The values of copper concentration in aqueous phase of the Points B1 and D1 are equals. The value of

copper concentration in aqueous phase of the Point D1 is in the cell ‘D31’. -

In the cell ‘D31’, enter formula: ‘=D30’.

Solver constraint of solver variable which is the value of copper concentration in organic phase of the 1

Point C is that the value of copper concentrations in organic phase of the Points B1 and E1 must be equals. The value of solver constraint of the value of copper concentration in organic phase is in the cell ‘I29’. -

In the cell ‘I29’, enter formula: ‘=C30-C28’.

Extraction stage 2 The value of copper concentration in organic phase of the Point E 2 is in the cell ‘C38’. This value is the value of copper concentration in organic phase of the Point D1 . -

In the cell ‘C38’, enter formula: ‘=C31’.

36

The value of copper concentration in aqueous phase of the Point E 2 is in the cell ‘D38’. This value is the value of copper concentration in aqueous phase of the Point D1 . -

In the cell ‘D38’, enter formula: ‘=D31’ The value of copper concentration in organic phase of the Point C 2 is solver variable. This value is in

the cell ‘F39’. In the cell ‘F39’, enter a random number between the values of copper concentration in organic phase of the Point E 2 and the value of copper concentration in organic phase of the Point C1 . The starting value of copper concentration in organic phase of the Point C 2 is the first whole number above the value of copper concentration in organic phase of the Point E 2 . In this case, the random number is 8. -

In the Cell ‘F39’, enter number ‘8’.

-

In the cell ‘C39’, enter formula: ‘=F39’ The value of copper concentration in aqueous phase of the Point C 2 is in the cell ‘D39’. This value is

given by the mathematical expression (35) where the value of constant H 2 is in the cell ‘C36’ and the value of constant J 2 is in the cell ‘D36’. The value of constant H 2 is given by the mathematical expression (36) where the value of initial free acid concentration is in the cell ‘F8’, the value of initial copper concentration is in the cell ‘F7’, and the value of equilibrium correlation α22 is in the cell ‘B36’. The value of equilibrium correlation α22 is given by the mathematical expression (32) where the value of extractant volume percentage is in the cell ‘F15’ and the value of copper concentration in organic phase is in the cell ‘C39’. -


-

In the cell ‘C36’, enter formula: ‘=-1.299*F8-2*F7-0.422*B36’. The value of constant J 2




-

In the cell ‘D39’, enter formula: ‘=[-C36-(C36^(2)-4*D36)^(0.5)]/2’. . The values of copper concentrations in aqueous phase of the Points E 2 and A2 are equals. The value


In the cell ‘D42’, enter formula: ‘=D38’. The value of copper concentration in aqueous phase of the Point B 2 is in the cell ‘D40’. This value is

given by the mathematical expression (60) where the value of Cu2(aq/e) is the value of copper concentration in

37

aqueous phase of the Point C 2 , the value of Cu1aq is the value of copper concentration in aqueous phase of the Point A2 and the value of mixer efficiency of stage of rank 2 is in the cell ‘F12’. -

In the ‘D40’, enter formula: ‘=D39*F12/100 + D42*(1-F12/100)’. The value of copper concentration in organic phase of the Point B 2 is in the cell ‘C40’. This value is

given by the mathematical expression (59) where the value of ratio (Rce ) is in the cell ‘F10’, the value of Cu2or is the value of copper concentration in organic phase of the Point B 2 , the value of Cu3or is changed by the value of Cu2or/e which is the value of copper concentration in organic phase of the Point C 2 , the value of Cu2aq is the value of copper concentration in aqueous phase of the Point B 2 and the value of Cu1aq is changed by the value of Cu2aq/e which is the value of copper concentration in aqueous phase of the Point C 2 . -


given by the mathematical (59) where the value of ratio (Rce ) is in the cell ‘F10’, the value of Cu2or is changed by the value of Cu2or/e which is the value of copper concentration in organic phase of the Point C 2 , the value of Cu3or is the value of copper concentration in organic phase of the Point A2 , the value of Cu2aq is changed by the value of Cu2aq/e which is the value of copper concentration in aqueous phase of the Point C 2 and the value of Cu1aq is the value of copper concentration in aqueous phase of the Point A2 . -



In the cell ‘C41’, enter formula: ‘=C42’. The values of copper concentration in aqueous phase of the Points B 2 and D2 are equals. The value of



Solver constraint of solver variable which is the value of copper concentration in organic phase of the Point C 2 is that the value of copper concentrations in organic phase of the Points B 2 and E 2 must be equals. The value of solver constraint of the value of copper concentration in organic phase is in the cell ‘I39’. -


38

Performance The value of free acid concentration in raffinate is in the cell ‘C45’. This value is given by the 0 mathematical expression (48) where the value of Acaq is in the cell ‘F8’, the value of PLS is in the cell ‘F7’ and

the value of Raf is in the cell ‘D41’. -

In the cell ‘C45’, enter formula: ‘=F8 + (F7-D41)*1.54’. The value of copper extraction efficiency is in the cell ‘C46’. This value is given by the mathematical

expression (49) where the value of PLS is in the cell ‘F7’ and the value of Raf is in the cell ‘D41’. -

In the cell ‘C46’, enter formula: ‘=(F7-D41)/F7*100’

5.2.2.2.

Stripping step Data

Design of simulation program is started with solver variables of stripping step data. These solver variables will be data of all mathematical expressions of stripping step. -

In the cell ‘F52’ of solver variable of spent electrolyte copper concentration (SP Cu), enter the number giving the value of spent electrolyte copper concentration data that is in the cell ‘C52’.

-

In the cell ‘F53’ of solver variable of spent electrolyte free acid concentration (SP Ac), enter the number giving the value of spent electrolyte free acid concentration data that is in the cell ‘C53’.

-

In the cell ‘F54’ of solver variable of advance electrolyte copper concentration (AD Cu), enter the number giving the value of advance electrolyte copper concentration data that is in the cell ‘C54’.

-

In the cell ‘F55’ of solver variable of mixer efficiency of stage of rank 1 (Mefs1 ), enter the number giving the value of stage of rank 1 mixer efficiency data that is in the cell ‘C55’.

-

In the cell ‘F56’ of solver variable of mixer efficiency of stage of rank 2 (Mefs2 ), enter the number giving the value of stage of rank 2 the mixer efficiency data that is in the cell ‘C56’.

Solver constraints of solver variables of data of stripping step are the following: -

In the cell ‘I52’, enter formula ‘=F52-C52’.

-


-


-


-


39

General The value of ratio (Rcs ) is in the cell ‘C59’. This value is given by the mathematical expression (54) where value of AD is in the cell ‘F54’, the value of SP is in the cell ‘F52’, the value of LOs is changed by the value of LOe which is in the cell ‘C28’ and the value of SOs is changed by the value of SOe which is in the cell ‘C41’. These changes follow the mathematical expressions (66) and (67). -

In the cell ‘C59’, enter formula: ‘=(F54-F52)/(C28-C41)’. The value of spent electrolyte flowrate is in the cell ‘C60’. This value is given by the mathematical

expression (54) where the value of ratio (Rcs ) is in the cell ‘C59’ and the value of organic flowrate is in the cell ‘C18’. -

In the cell ‘C60’, enter formula: ‘=C18/C59’.

Stripping stage 1 The value of copper concentration in organic phase of the Point D1 is in the cell ‘C66’. This value is the value of copper concentration in organic phase of the Point E1 of extraction step (loaded organic). -

In the cell ‘C66’, enter formula: ‘=C28’ The value of copper concentration in aqueous phase of the Point D1 is in the cell ‘D66’. This value is

the advance electrolyte copper concentration which is in the cell ‘F54’. -


‘F67’. The starting value of copper concentration in organic phase of the Point C1 is half the value of copper concentration in organic phase of the Point D1 . In this case, half the value is 5.668. -

In the Cell ‘F67’, enter number ‘5.668’.

-


given by the mathematical expression (38) where the value of constant L1 is in the cell ‘C64’ and the value of constant M1 is in the cell ‘D64’. The value of constant L1 is given by the mathematical expression (39) where the value of initial free acid concentration is in the cell ‘F53’, the value of initial copper concentration is in the cell ‘F52’, and the value of equilibrium correlation π12 is in the cell ‘B64’. The value of equilibrium correlation π12 is

40

given by the mathematical expression (34) where the value of extractant volume percentage is in the cell ‘F15’ and the value of copper concentration in organic phase is in the cell ‘C67’. -

In

the

cell

‘B64’,

enter

formula:

‘=(((4.8579*10^(-3)*F15-0.19183)*C67+11.365*F15^(-

0.85))*(3.303*F15-3.0842*C67)^(2))/C67’. -

In the cell ‘C64’, enter formula: ‘=-1.299*F53-2*F52-0.422*B64’. The value of constant M1 is given by the mathematical expression (40) where initial free acid



-

In the cell ‘D67’, enter formula: ‘=[-C64-(C64^(2)-4*D64)^(0.5)]/2’. . The values of copper concentrations in organic phase of the Points D1 and A1 are equals. The value

of copper concentration in organic phase of the Point A1 is in the cell ‘C70’. -

In the cell ‘C70’, enter formula: ‘=C66’. The value of copper concentration in organic phase of the Point B1 is in the cell ‘C68’. This value is

given by the mathematical expression (65) where the value of Cu1(or/e) is the value of copper concentration in organic phase of the Point C1 , the value of Cu0or is the value of copper concentration in organic phase of the Point A1 and the value of mixer efficiency of stage of rank 1 is in the cell ‘F55’. -

In the ‘C68’, enter formula: ‘=C67*F55/100 + C70*(1-F55/100)’. The value of copper concentration in aqueous phase of the Point B1 is in the cell ‘D68’. This value is

given by the mathematical expression (63) where the value of ratio (Rc𝑠 ) is in the cell ‘C59’, the value of Cu1or is the value of copper concentration in organic phase of the Point B1 , the value of Cu0or is changed by the value of Cu1or/e which is the value of copper concentration in organic phase of the Point C1 , the value of Cu1aq is the value of copper concentration in aqueous phase of the Point B1 and the value of Cu2aq is changed by the value of Cu1aq/e which is the value of copper concentration in aqueous phase of the Point C1 . -

In the cell ‘D68’, enter formula: ‘=D67-C59*(C68-C67)’. The value of copper concentration in aqueous phase of the Point A1 is in the cell ‘D70’. This value is

given by the mathematical (63) where the value of ratio (Rce ) is in the cell ‘F59’, the value of Cu1or is changed by the value of Cu1or/e which is the value of copper concentration in organic phase of the Point C1 , the value of Cu0or is the value of copper concentration in organic phase of the Point A1 , the value of Cu1aq is changed by the value

41

of Cu1aq/e which is the value of copper concentration in aqueous phase of the Point C1 and the value of Cu2aq is the value of copper concentration in aqueous phase of the Point A1 . -

In the cell ‘D70’, enter formula: ‘=D67-C59*(C70-C67)’. The values of copper concentrations in aqueous phase of the Points E1 and A1 are equals. The value of

copper concentration in organic phase of the Point E1 is in the cell ‘D69’. -

In the cell ‘D69’, enter formula: ‘=D70’. The values of copper concentration in organic phase of the Points B1 and E1 are equals. The value of

copper concentration in aqueous phase of the Point E1 is in the cell ‘C69’. -


Solver constraint of solver variable which is the value of copper concentration in organic phase of the Point C1 is that the value of copper concentrations in aqueous phase of the Points B1 and D1 must be equals. The value of solver constraint of the value of copper concentration in organic phase is in the cell ‘I67’. -

In the cell ‘I67’, enter formula: ‘=D68-D66’.

Stripping stage 2 The value of copper concentration in organic phase of the Points D2 and E1 are equals. The value of copper concentration in organic phase of the Point D2 is in the cell ‘C76’ -

In the cell ‘C76’, enter formula: ‘=C69’. The value of copper concentration in aqueous phase of the Points D2 and E1 are equals. The value of

copper concentration in aqueous phase of the Point D2 is in the cell ‘D76’ -

In the cell ‘D76’, enter formula: ‘=D69’ The value of copper concentration in organic phase of the Point C 2 is solver variable and is in the cell

‘F77’. The starting value of copper concentration in organic phase of the Point C 2 is two-thirds the value of copper concentration in organic phase of the Point D2 . In this case, two-thirds the value is 3.854. -

In the Cell ‘F77’, enter the number ‘3.854’.

-

In the cell ‘C77’, enter formula: ‘=F77’

42

The value of copper concentration in aqueous phase of the Point C 2 is in the cell ‘D77’. This value is given by the mathematical expression (38) where the value of constant L2 is in the cell ‘C74’ and the value of constant M 2 is in the cell ‘D74’. The value of constant L2 is given by the mathematical expression (39) where the value of initial free acid concentration is in the cell ‘F53’, the value of initial copper concentration is in the cell ‘F52’, and the value of equilibrium correlation π22 is in the cell ‘B74’. The value of equilibrium correlation π22 is given by the mathematical expression (34) where the value of extractant volume percentage is in the cell ‘F15’ and the value of copper concentration in organic phase is in the cell ‘C77’. -

In

the

cell

‘B74’,

enter

formula:

‘=(((4.8579*10^(-3)*F15-0.19183)*C77+11.365*F15^(-

0.85))*(3.303*F15-3.0842*C77)^(2))/C77’. -

In the cell ‘C74’, enter formula: ‘=-1.299*F53-2*F52-0.422*B74’. The value of constant M 2 is given by the mathematical expression (40) where initial free acid



-



In the cell ‘C80’, enter formula: ‘=C76’. The value of copper concentration in organic phase of the Point B 2 is in the cell ‘C78’. It is given by

the mathematical expression (65) where the value of Cu2(or/e) is the value of copper concentration in organic phase of the Point C 2 , the value of Cu1or is the value of copper concentration in organic phase of the Point A2 and the value of mixer efficiency of stage of rank 2 is in the cell ‘F56’. -

In the ‘C78’, enter formula: ‘=C77*F56/100 + C80*(1-F56/100)’. The value of copper concentration in aqueous phase of the Point B 2 is in the cell ‘D78’. This value is

given by the mathematical expression (63) where the value of ratio (Rc𝑠 ) is in the cell ‘C59’, the value of Cu2or is the value of copper concentration in organic phase of the Point B 2 , the value of Cu1or is changed by the value of Cu2or/e which is the value of copper concentration in organic phase of the Point C 2 , the value of Cu2aq is the value of copper concentration in aqueous phase of the Point B 2 and the value of Cu3aq is changed by the value of Cu2aq/e which is value of copper concentration in aqueous phase of the Point C 2 . -

In the cell ‘D78’, enter formula: ‘=D77-C59*(C78-C77)’.

43

The value of copper concentration in aqueous phase of the Point A2 is in the cell ‘D80’. This value is given by the mathematical (63) where the value of ratio (Rce ) is in the cell ‘F59’, the value of Cu2or is changed by the value of Cu2or/e which is the value of copper concentration in organic phase of the Point C 2 , the value of Cu1or is the value of copper concentration in organic phase of the Point A2 , the value of Cu2aq is changed by the value of Cu2aq/e which is the value of copper concentration in aqueous phase of the Point C 2 and the value of Cu3aq is the value of copper concentration in aqueous phase of the Point A2 . -

In the cell ‘D80’, enter formula: ‘=D77-C59*(C80-C77)’. The values of copper concentrations in aqueous phase of the Points E 2 and A2 are equals. The value of

copper concentration in organic phase of the Point E 2 is in the cell ‘D79’. -

In the cell ‘D79’, enter formula: ‘=D80’. The values of copper concentration in organic phase of the Points B 2 and E 2 are equals. The value of

copper concentration in aqueous phase of the Point E 2 is in the cell ‘C79’. -



Point C is that the value of copper concentrations in aqueous phase of the Points B 2 and D2 must be equals. The value of solver constraint of the value of copper concentration in organic phase is in the cell ‘I77’. -


Performance The value of stripping copper efficiency is in the cell ‘C83’. This value is given by the mathematical expression (56) where the value of LOs is in the cell ‘C66’ and the value of SOs is in the cell ‘C79’. -

In the cell ‘C83’, enter formula: ‘=(C66-C79)/C66*100’ The value of net copper transfer is in the cell ‘C84’. This value is given by the mathematical

expression (57) where the value of LOs is in the cell ‘C66’, the value of SOs is in the cell ‘C79’ and the value of the extractant volume percentage is in the cell ‘F15’. -

In the cell ‘C84’, enter formula: ‘=(C66-C79)/F15’

44

5.2.2.3.

Simulation program constraint The value of simulation program constraint is in the cell ‘I88’. Simulation program constraint is that

copper concentrations in stripped organic of extraction and stripping steps must be equals. This simulation program constraint is the set objective of Excel solver program. -

In the cell ‘I88’, entre formula: ‘=C79-C41’

At this level, it appears Table 9 as it appears on Excel Microsoft spreadsheet. Table 9 gives results of static simulation program with the starting values of extractant volume percentage, maximum loading, copper concentration in organic phase of the Points C1 and C 2 of extraction step, and copper concentration in the organic phase of the Points C1 and C 2 of stripping step. Solver variables have blue color and solver constraints have green color.

5.2.2.4.

Excel solver program Excel solver program execution is as follows:

1) On the ‘Data’, in the ‘Analysis group’ click solver (if the solver command is not available, you must activate the solver add-in). 2) In the ‘Set objective’ box, enter the cell reference ‘I88’ of simulation program constraint. 3) Click ‘Value of’ and then type the number ‘0’ in the box. 4) In the ‘By Changing Variable Cells’ box, enter reference for each solver variable (in blue color in Table 9). Separate references with commas (English version). 5) In the ‘Subject to the constraints’ box, enter solver constraints by doing the following: a.

In the ‘Solver Parameters’ dialog box, click ‘Add’.

b.

In the ‘Cell Reference’ box, enter cell reference of PLS flowrate solver constraint (in green color in Table 9).

c.

Click the ‘relationship’ ‘=’, in the ‘Constraint’ box, type the number ‘0’.

d.

Click ‘Add’ for the second solver constraint. When the last solver constraint is added (cell ‘I77’), click ‘OK’ to return to ‘Solver Parameters’ dialog box.

6)

Click ‘Solve’. To keep the solution values on the worksheet, in the ‘Solver Results’ dialog box, click ‘Keep solver solution’.

At this level, it appears Table 10 as it appears on Excel Microsoft spreadsheet. Table 10 gives results of static simulation program with the optimum values of extractant volume percentage, maximum loading, copper concentration in organic phase of the Points C1 and C 2 of extraction step, and copper concentration in the organic phase of the Points C1 and C 2 of stripping step.

Simulation program of option 1 of copper solvent extraction configuration (2Ex2S) is done. For this option 1, data (having red color) will be changed only for others simulations.

45

A 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

B

C

Table

8

PLSflow PLS Cu PLS Ac SR Rce Mefe1 Mefe2

Data 400 7.00 1.96 80 1.1 92 95

D

E

F

G

H

I

Simulation program table for option 1 (part 1)

m3/h g/l g/l % % %

Extraction step Solver variables PLSflow m3/h PLS Cu g/l PLS Ac g/l SR % Rce % Mefe1 % Mefe2

Solver constraints PLSflow PLS Cu PLS Ac SR Rce Mefe1 Mefe2

Starting value V% General Orgflow AML α1ML

% Solver variables

α2 ML

α12 E1 C1 B1 D1 A1 α22 E2 C2 B2 D2 A2

Stage 1 H1

J1

Cuor

Cuaq

g/l

Solver variables

C1 Cuor

Stage 2 H2

J2

Cuor

Cuaq

Solver variables

C 2 Cuor

46

C1 Cuor

Solver constraints

g/l

g/l %

ML

Solver constraints

g/l

Performance Raf Ac Eff1c

Solver constraints

m3/h g/l

C 2 Cuor

J

A 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89

B

C

Table

8

SP Cu SP Ac AD Cu Mefs1 Mefs2

Data 35 180 50 98 85

R𝑐s SPflow π12 D1 C1 B1 E1 A1 π22 2

D

E

G

H

I


g/l g/l g/l % %

Stripping step Solver variables SP Cu SP Ac AD Cu Mefs1 Mefs2

g/l g/l g/l % %

Solver constraints SP Cu SP Ac AD Cu Mefs1 Mefs2

General m3/h Stage 1 L1

M1

Cuor

Cuaq

Solver variables

C1 Cuor

Stage 2 L2

M2

Cuor

Cuaq

D C2 B2 E2 A2 Effs𝑐 Cut

F

Solver constraints

g/l

Solver variables

C 2 Cuor

C1 Cuor

Solver constraints

g/l

C 2 Cuor

Performance % g/l/v% Simulation constraint Set objective

47

J

A 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

B

C

Table

9


Data 400 7.00 1.96 80 1.1 92 95

D

E

α12 38.76 E1 C1 B1 D1 A1

General 440.00 14.170 α2 -0.020

Stage 1 H1 -32.90 Cuor 11.336 12.000 11.653 7.659 7.659

E2 C2 B2 D2 A2

Stage 2 H2 129.26 Cuor 7.659 8.000 7.906 6.112 6.112

Raf Ac Eff1c

Performance 11.763 90.94

α22 267.09

G

H

I


m3/h g/l g/l % % %

Extraction step Solver variables PLSflow 400 m3/h PLS Cu 7.00 g/l PLS Ac 1.96 g/l SR 80 % 1.1 Rce 92 % Mefe1 95 % Mefe2

V%

Orgflow AML α1ML 0.549

F

Starting value 25

Solver constraints PLSflow 0.000 PLS Cu 0.000 PLS Ac 0.000 SR 0.000 0.000 Rce 0.000 Mefe1 0.000 Mefe2

%

Solver variables

Solver constraints

m3/h g/l ML

J1 68.26 Cuaq 7.000 2.225 2.607 2.607 7.000 J2 68.26 Cuaq 2.607 0.530 0.634 0.634 2.607

14.170

g/l

Solver variables

C1 Cuor

12.000

8.000

g/l %

48

0.568

Solver constraints

g/l

Solver variables

C 2 Cuor

ML

C1 Cuor

0.317

Solver constraints

g/l

C 2 Cuor

0.246

J

A 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89

B

C

Table

9


Data 35 180 50 98 85

g/l g/l g/l % %

R𝑐s SPflow

General 2.871 153.24

m3/h

Stage 1 L1 -410.39 Cuor 11.336 5.668 5.781 5.781 11.336

M1 22776.9 Cuaq 50.00 66.17 65.84 49.89 49.89

Stage 2 L2 -558.51 Cuor 5.781 3.854 4.143 4.143 5.781

M2 22776.9 Cuaq 49.89 44.29 43.46 38.76 38.76

π12 225.54 D1 C1 B1 E1 A1 π22 603.53 D2 C2 B2 E2 A2 Effs𝑐 Cut

D

E

F

G

H

I

Simulation program table for option 1 (part 2) Stripping step Solver variables SP Cu 35 SP Ac 180 AD Cu 50 98 Mefs1 85 Mefs2

g/l g/l g/l % %

Solver variables

C1 Cuor

5.668

Solver constraints

g/l

Solver variables

C 2 Cuor

3.854

Solver constraints SP Cu 0.000 SP Ac 0.000 AD Cu 0.000 0.000 Mefs1 0.000 Mefs2

C1 Cuor

15.843

Solver constraints

g/l

C 2 Cuor

-6.430

Performance 63.45 % 0.29 g/l/v% Simulation constraint Set objective

49

-1.969

J

A 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

B

C

Table

10


Data 400 7.00 1.96 80 1.1 92 95

D

E

α12 44.44 E1 C1 B1 D1 A1

General 440.00 11.041 α2 0.549

Stage 1 H1 -35.30 Cuor 8.801 9.161 8.801 4.663 4.663

E2 C2 B2 D2 A2

Stage 2 H2 239.86 Cuor 4.663 4.762 4.663 2.795 2.795

Raf Ac Eff1c


α22 529.17

G

H

I


m3/h g/l g/l % % %

Extraction step Solver variables PLSflow 400 m3/h PLS Cu 7.00 g/l PLS Ac 1.96 g/l SR 80 % 1.1 Rce 92 % Mefe1 95 % Mefe2

V%


F

Starting value 19.93

Solver constraints PLSflow 0.000 PLS Cu 0.000 PLS Ac 0.000 SR 0.000 0.000 Rce 0.000 Mefe1 0.000 Mefe2

%

Solver variables

Solver constraints

m3/h g/l ML

J1 68.26 Cuaq 7.000 2.053 2.449 2.449 7.000 J2 68.26 Cuaq 2.449 0.285 0.393 0.393 2.449

11.001

g/l

Solver variables

C1 Cuor

9.161

4.762

g/l %

50

0.000

Solver constraints

g/l

Solver variables

C 2 Cuor

ML

C1 Cuor

0.000

Solver constraints

g/l

C 2 Cuor

0.000

J

A 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89

B

C

Table

10


Data 35 180 50 98 85

g/l g/l g/l % %

R𝑐s SPflow

General 2.497 176.18

m3/h

Stage 1 L1 -503.42 Cuor 8.801 3.545 3.650 3.650 8.801

M1 22776.9 Cuaq 50.00 50.26 50.00 37.14 37.14

Stage 2 L2 -644.69 Cuor 3.650 2.644 2.795 2.795 3.650

M2 22776.9 Cuaq 37.14 37.51 37.14 35.00 35.00

π12 472.99 D1 C1 B1 E1 A1 π22 807.74 D2 C2 B2 E2 A2 Effs𝑐 Cut

D

E

F

G

H

I

Simulation program table for option 1 (part 2) Stripping step Solver variables SP Cu 35 SP Ac 180 AD Cu 50 98 Mefs1 85 Mefs2

g/l g/l g/l % %

Solver variables

C1 Cuor

3.545

Solver constraints

g/l

Solver variables

C 2 Cuor

2.644

Solver constraints SP Cu 0.000 SP Ac 0.000 AD Cu 0.000 0.000 Mefs1 0.000 Mefs2

C1 Cuor

0.000

Solver constraints

g/l

C 2 Cuor

0.000


51

0.000

J

5.2.3.

Option 2 . In this option, unknown parameters are extractant volume percentage, saturation ratio, mixer

efficiencies of stages of rank 1 and 2 of extraction and stripping steps. Table 11 gives data of chosen example for explanation of simulation program design.

Table 11: Data of copper solvent extraction configuration of option 2

Description PLS flowrate Copper concentration in PLS Acid concentration in PLS Ratio Rce Maximum loading Raffinate from stage of rank 1 Raffinate from stage of rank 2 Number of extraction stage

Symbol Extraction step PLS flow PLS Cu PLS Ac Rce ML Raf E1 Raf E2

Stripping step Copper concentration in spent electrolyte Acid concentration in spent electrolyte Copper concentration in advance electrolyte Number of stripping stage Stripped organic of stage of rank 1 Stripped organic of stage of rank 2

Unity

Value

m3/h g/l g/l

700 6.0 1.96 1.07 10.5 2.00 0.43 2

g/l g/l g/l

SP Cu SP Ac AD Cu

g/l g/l g/l

SO S1 SO S2

g/l g/l

35 190 50 2 3.60 2.59

Simulation program table for option 2 of 2Ex2S configuration is given in Table 12 as it appears on Excel Microsoft spreadsheet. Data from Table 11 have red color. In extraction table, designation αn2 , H n and J n are respectively the values of equilibrium correlation 𝛼2 and constants H and J of stage of rank ‘n’ for calculating the value of copper concentration in aqueous phase from the value of copper concentration in organic phase. Designation An , C n , Dn and E n are the Points A, B, C, and D from extraction step MacCabe Thiele diagram of stage of rank ‘n’. In stripping table, designation πn2 , Ln and M n are respectively the values of equilibrium correlation 𝜋2 and constants L and M of stage of rank ‘n’ for calculating the value of copper concentration in aqueous phase from the value of copper concentration in organic phase. Designation An , C n , Dn and E n are the Points A, B, C, and D from stripping step MacCabe Thiele diagram of stage of rank ‘n’.

Plant constraints of extraction and stripping steps are simulation program results to match with plant data.

52

5.2.3.1.

Extraction step Data

Simulation program design is started with solver variables of data. These solver variables will be data of all mathematical expressions of extraction step. -

In the cell ‘F6’ of solver variable of PLS flowrate (PLS flow), enter the number giving the value of PLS flow data that is in the cell ‘C6’.

-

In the cell ‘F7’ of solver variable of PLS copper concentration (PLS Cu), enter the number giving the value of PLS copper concentration data that is in the cell ‘C7’.

-

In the cell ‘F8’ of solver variable of PLS free acid concentration (PLS Ac), enter the number giving the value of PLS free acid concentration data that is in the cell ‘C8’.

-

In the cell ‘F9’ of solver variable of ratio (Rce ), enter the number giving the value of ratio ( Rce ) data that is in the cell ‘C9’.

Solver constraints of solver variables of extraction step data are the following: -


-


-


-


Starting values The starting value of extractant volume percentage of simulation program is in the cell ‘F12’. This value is the expected value of extractant volume percentage. For this case, the expected value of extractant volume percentage is 20%. -

In the cell ‘F12’, enter the number ‘20’. The starting value of saturation ratio of simulation program is in the cell ‘F13’. This value is the

expected value of saturation ratio. For this case, the expected value of saturation ratio is 70%. -

In the cell ‘F13’, enter the number ‘70’. The starting value of mixer efficiency of stage of rank 1 of simulation program is in the cell ‘F14’.

This value is the expected value of mixer efficiency of stage of rank 1. For this case, the expected value is 90%.

53

-


This value is the expected value of mixer efficiency of stage of rank 2. For this case, the expected value is 90%. -


General The value of organic flowrate is in the cell ‘C18’. This value is given by the mathematical expression (47) where the values of ratio (Rce ) is in the cell ‘F9’ and PLS flowrate is in the cell ‘F6’. -

In the cell ‘C18’, enter formula: ‘=F9*F6’. The value of absolute maximum loading is in the cell ‘C19’. This value is given by the mathematical

expression (42) where the value of extractant volume percentage is in the cell ‘F12’. -

In the cell ‘C19’, enter formula: ‘=0.4108*F12^(1.1)’.

The value of maximum loading is solver variable because resolution of cubic equation is complicated and is in the cell ‘F21’. The starting value of maximum loading is the value of absolute maximum loading. -

In the cell ‘F21’, enter the number ’11.086’. Solver constraint of the value of maximum loading is that the value of equilibrium correlations α1ML

and α2 must be equals. Solver constraint of the value of maximum loading is in the cell ‘I21’. The value of equilibrium correlation α1ML is in the cell ‘B21’ and is given by the mathematical expression (41). The value of equilibrium correlation α2 is in the cell ‘C21’ and is given by the mathematical expression (32) where the value of copper concentration in organic phase at the steady-state is the value of maximum loading. -

In the cell ‘B21’, enter formula: ‘=F8^2/F7’.

-

In the cell ‘C21’, enter formula: ‘=((-28.511*F12^(-1.746)*F21+11.711*F12^(-0.646))*(3.303*F123.0842*F21)^2)/F21’.

-

In the cell ‘I21’, enter formula: ‘=B21-C21’.

Extraction stage 1 The value of copper concentration in organic phase of the Point E1 is in the cell ‘C28’ and is the value of loaded organic. The value of loaded organic is given by the mathematical expression (70) where the values of saturation ratio (SR) is in the cell ‘F13’ and the value of maximum loading is in the cell ‘F21’.

54

-

In cell ‘C28’, enter formula: ‘=F13*F21/100’. The value of copper concentration in aqueous phase of the Point E1 is in the cell ‘D28’ and is the

value of copper concentration in PLS which is in the cell ‘F7’. -

In the cell ‘D28’, enter formula: ‘=F7’ The value of copper concentration in organic phase of the Point C1 is solver variable. This value is in

the cell ‘F29’. In the cell ‘F29’, enter a random number between the values of copper concentration in organic phase of the Point E1 and the value of maximum loading. The starting value of copper concentration in organic phase of the Point C1 is the first whole number above the value of copper concentration in organic phase of the Point E1 . In this case, the random number is 8. -

In the Cell ‘F29’, enter number ‘8’.

-


given by the mathematical expression (35) where the value of constant H1 is in the cell ‘C26’ and the value of constant J1 is in the cell ‘D26’. The value of constant H1 is given by the mathematical expression (36) where the value of initial free acid concentration is in the cell ‘F8’, the value of initial copper concentration is in the cell ‘F7’, and the value of equilibrium correlation α12 is in the cell ‘B26’. The value of equilibrium correlation α12 is given by the mathematical expression (32) where the value of extractant volume percentage is in the cell ‘F12’ and the value of copper concentration in organic phase is in the cell ‘C29’. -


-

In the cell ‘C26’, enter formula: ‘=-1.299*F8-2*F7-0.422*B26’. The value of constant J1




-

In the cell ‘D29’, enter formula: ‘=[-C26-(C26^(2)-4*D26)^(0.5)]/2’. . The values of copper concentrations in aqueous phase of the Points E1 and A1 are equals. The value



55

The value of copper concentration in aqueous phase of the Point B1 is in the cell ‘D30’. This value is given by the mathematical expression (60) where the value of Cu1(aq/e) is the value of copper concentration in aqueous phase of the Point C1 , the value of Cu0aq is the value of copper concentration in aqueous phase of the Point A1 and the value of mixer efficiency of stage of rank 1 is in the cell ‘F14’. -

In the ‘D30’, enter formula: ‘=D29*F14/100 + D32*(1-F14/100)’. The value of copper concentration in organic phase of the Point B1 is in the cell ‘C30’. This value is

given by the mathematical expression (59) where the value of ratio (Rce ) is in the cell ‘F9’, the value of Cu1or is the value of copper concentration in organic phase of the Point B1 , the value of Cu2or is changed by the value of Cu1or/e which is the value of copper concentration in organic phase of the Point C1 , the value of Cu1aq is the value of copper concentration in aqueous phase of the Point B1 and the value of Cu0aq is changed by the value of Cu1aq/e which is the value of copper concentration in aqueous phase of the Point C1 . -





In the cell ‘C31’, enter formula: ‘=C32’. The values of copper concentration in aqueous phase of the Points B1 and D1 are equals. The value of




Point C is that the value of copper concentrations in organic phase of the Points B1 and E1 must be equals. The value of solver constraint of the value of copper concentration in organic phase is in the cell ‘I29’.

56

-


Extraction stage 2 The value of copper concentration in organic phase of the Point E 2 is in the cell ‘C38’. This value is the value of copper concentration in organic phase of the Point D1 . -

In the cell ‘C38’, enter formula: ‘=C31’. The value of copper concentration in aqueous phase of the Point E 2 is in the cell ‘D38’. This value is

the value of copper concentration in aqueous phase of the Point D1 . -

In the cell ‘D38’, enter formula: ‘=D31’ The value of copper concentration in organic phase of the Point C 2 is solver variable. This value is in

the cell ‘F39’. In the cell ‘F39’, enter a random number between the values of copper concentration in the organic phase of the Point E 2 and the value of copper concentration in organic phase of the Point C1 . The starting value of copper concentration in organic phase of the Point C 2 is the first whole number above the value of copper concentration in organic phase of the Point E 2 . In this case, the random number is 4. -

In the Cell ‘F39’, enter the number ‘4’.

-


given by the mathematical expression (35) where the value of constant H 2 is in the cell ‘C36’ and the value of constant J 2 is in the cell ‘D36’. The value of constant H 2 is given by the mathematical expression (36) where the value of initial free acid concentration is in the cell ‘F8’, the value of initial copper concentration is in the cell ‘F7’, and the value of equilibrium correlation α22 is in the cell ‘B36’. The value of equilibrium correlation α22 is given by the mathematical expression (32) where the value of extractant volume percentage is in the cell ‘F12’ and the value of copper concentration in organic phase is in the cell ‘C39’. -


-

In the cell ‘C36’, enter formula: ‘=-1.299*F8-2*F7-0.422*B36’. The value of constant J 2




-

In the cell ‘D39’, enter formula: ‘=[-C36-(C36^(2)-4*D36)^(0.5)]/2’.

57

. The values of copper concentrations in aqueous phase of the Points E 2 and A2 are equals. The value of copper concentration in aqueous phase of the Point A2 is in the cell ‘D42’. -

In the cell ‘D42’, enter formula: ‘=D38’. The value of copper concentration in aqueous phase of the Point B 2 is in the cell ‘D40’. This value is

given by the mathematical expression (60) where the value of Cu2(aq/e) is the value of copper concentration in aqueous phase of the Point C 2 , the value of Cu1aq is the value of copper concentration in aqueous phase of the Point A2 and the value of mixer efficiency of stage of rank 2 is in the cell ‘F15’. -

In the ‘D40’, enter formula: ‘=D39*F15/100 + D42*(1-F15/100)’. The value of copper concentration in organic phase of the Point B 2 is in the cell ‘C40’. This value is

given by the mathematical expression (59) where the value of ratio (Rce ) is in the cell ‘F9’, the value of Cu2or is the value of copper concentration in organic phase of the Point B 2 , the value of Cu3or is changed by the value of Cu2or/e which is the value of copper concentration in organic phase of the Point C 2 , the value of Cu2aq is the value of copper concentration in aqueous phase of the Point B 2 and the value of Cu1aq is changed by the value of Cu2aq/e which is the value of copper concentration in aqueous phase of the Point C 2 . -


given by the mathematical (59) where the value of ratio (Rce ) is in the cell ‘F9’, the value of Cu2or is changed by the value of Cu2or/e which is the value of copper concentration in organic phase of the Point C 2 , the value of Cu3or is the value of copper concentration in organic phase of the Point A2 , the value of Cu2aq is changed by the value of Cu2aq/e which is the value of copper concentration in aqueous phase of the Point C 2 and the value of Cu1aq is the value of copper concentration in aqueous phase of the Point A2 . -



In the cell ‘C41’, enter formula: ‘=C42’. The values of copper concentration in aqueous phase of the Points B 2 and D2 are equals. The value of

copper concentration in aqueous phase of the Point D2 is in the cell ‘D41’.

58

-


Solver constraint of solver variable which is the value of copper concentration in organic phase of the Point C 2 is that the value of copper concentrations in organic phase of the Points B 2 and E 2 must be equals. The value of solver constraint of the value of copper concentration in organic phase is in the cell ‘I39’. -


Plant constraints

Plant constraints are simulation program results to match with plant data. The plant data of maximum loading come from lab test by contacting stripped organic with PLS aqueous phase (follow procedure). Plants data of extraction step are the value of maximum loading that is in the cell ‘C45’, the value of copper concentration in raffinate of stage of rank 1 that is in the cell ‘C46’ and the value of copper concentration in raffinate of stage of rank 2 that is in the cell ‘C47’. -

In the cell ‘F45’ of maximum loading plant constraint, enter formula: ‘=F21’

-

In the cell ‘F46’ of raffinate E1 plant constraint, enter formula: ‘=D31’

-

In the cell ‘F47’ of raffinate E2 plant constraint, enter formula: ‘=D41’ Solver constraints of plants constraints of extraction step are the following:

-


-


-


Performance The value of free acid concentration in raffinate is in the cell ‘C50’. This value is given by the 0 mathematical expression (48) where the value of Acaq is in the cell ‘F8’, the value of PLS is in the cell ‘F7’ and

the value of Raf is in the cell ‘D41’. -

In the cell ‘C50’, enter formula: ‘=F8 + (F7-D41)*1.54’. The value of copper extraction efficiency is in the cell ‘C51’. This value is given by the mathematical

expression (49) where the value of PLS is in the cell ‘F7’ and the value of Raf is in the cell ‘D41’. -

In the cell ‘C51’, enter formula: ‘=(F7-D41)/F7*100’

59

5.2.3.2.

Stripping step Data

Design of simulation program is started with solver variables of data of stripping step. These solver variables are data of all mathematical expressions of stripping step. -

In the cell ‘F57’ of solver variable of spent electrolyte copper concentration (SP Cu), enter the number giving the value of spent electrolyte copper concentration data that is in the cell ‘C57’.

-

In the cell ‘F58’ of solver variable of spent electrolyte free acid concentration (SP Ac), enter the number giving the value of spent electrolyte free acid concentration data that is in the cell ‘C58’.

-

In the cell ‘F59’ of solver variable of advance electrolyte copper concentration (AD Cu), enter the number giving the value of advance electrolyte copper concentration data that is in the cell ‘C59’.

Solver constraints of solver variables of data of stripping step are the following: -


-


-


Starting values The starting value of mixer efficiency of stage of rank 1 of simulation program is in the cell ‘F62’. This value is the expected value of mixer efficiency of stage of rank 1. For this case, the expected starting value is 90%. -


This value is the expected value of mixer efficiency of stage of rank 2. For this case, the expected starting value is 90%. -


General The value of ratio (Rcs ) is in the cell ‘C66’. This value is given by the mathematical expression (54) where value of AD is in the cell ‘F59’, the value of SP is in the cell ‘F57’, the value of LOs is changed by the

60

value of LOe which is in the cell ‘C28’ and the value of SOs is changed by the value of SOe which is in the cell ‘C41’. These changes follow the mathematical expressions (66) and (67). -

In the cell ‘C66’, enter formula: ‘=(F59-F57)/(C28-C41) The value of spent electrolyte flowrate is in the cell ‘C67’. This value is given by the mathematical

expression (54) where the value of ratio (Rcs ) is in the cell ‘C66’ and the value of organic flowrate is in the cell ‘C18’. -

In the cell ‘C67’, enter formula: ‘=C18/C66’

Stripping stage 1 The value of copper concentration in organic phase of the Point D1 is in the cell ‘C73’. This value is the value of copper concentration in organic phase of the Point E1 of extraction step. -

In the cell ‘C73’, enter formula: ‘=C28’ The value of copper concentration in aqueous phase of the Point D1 is in the cell ‘D73’. This value is

the copper concentration of advance electrolyte which is in the cell ‘F59’. -


‘F74’. The starting value of copper concentration in organic phase of the Point C1 is half the value of copper concentration in organic phase of the Point D1 . In this case, half the value is 3.88. -


-


given by the mathematical expression (38) where the value of constant L1 is in the cell ‘C71’ and the value of constant M1 is in the cell ‘D71’. The value of constant L1 is given by the mathematical expression (39) where the value of initial free acid concentration is in the cell ‘F58’, the value of initial copper concentration is in the cell ‘F57’, and the value of equilibrium correlation π12 is in the cell ‘B71’. The value of equilibrium correlation π12 is given by the mathematical expression (34) where the value of extractant volume percentage is in the cell ‘F12’ and the value of copper concentration in organic phase is in the cell ‘C74’. -

In

the

cell

‘B71’,

enter

formula:

‘=(((4.8579*10^(-3)*F12-0.19183)*C74+11.365*F12^(-

0.85))*(3.303*F12-3.0842*C74)^(2))/C74’.

61

-

In the cell ‘C71’, enter formula: ‘=-1.299*F58-2*F57-0.422*B71’. The value of constant M1 is given by the mathematical expression (40) where initial free acid



-



In the cell ‘C77’, enter formula: ‘=C73’. The value of copper concentration in organic phase of the Point B1 is in the cell ‘C75’. This value is

given by the mathematical expression (65) where the value of Cu1(or/e) is the value of copper concentration in organic phase of the Point C1 , the value of Cu0or is the value of copper concentration in organic phase of the Point A1 and the value of mixer efficiency of stage of rank 1 is in the cell ‘F62’. -

In the ‘C75’, enter formula: ‘=C74*F62/100 + C77*(1-F62/100)’. The value of copper concentration in aqueous phase of the Point B1 is in the cell ‘D75’. This value is

given by the mathematical expression (63) where the value of ratio (Rc𝑠 ) is in the cell ‘C66’, the value of Cu1or is the value of copper concentration in organic phase of the Point B1 , the value of Cu0or is changed by the value of Cu1or/e which is the value of copper concentration in organic phase of the Point C1 , the value of Cu1aq is the value of copper concentration in aqueous phase of the Point B1 and the value of Cu2aq is changed by the value of Cu1aq/e which is the value of copper concentration in aqueous phase of the Point C1 . -



In the cell ‘D77’, enter formula: ‘=D74-C66*(C77-C74)’.

62

The values of copper concentrations in aqueous phase of the Points E1 and A1 are equals. The value of copper concentration in organic phase of the Point E1 is in the cell ‘D76’. -

In the cell ‘D76’, enter formula: ‘=D77’. The values of copper concentration in organic phase of the Points B1 and E1 are equals. The value of

copper concentration in aqueous phase of the Point E1 is in the cell ‘C76’. -



Point C is that the value of copper concentrations in aqueous phase of the Points B1 and D1 must be equals. The value of solver constraint of the value of copper concentration in organic phase is in the cell ‘I74’. -


Stripping stage 2 The value of copper concentration in organic phase of the Point D2 and E1 are equals. The value of copper concentration in organic phase of the Point D2 is in the cell ‘C83’ -

In the cell ‘C83’, enter formula: ‘=C76’. The value of copper concentration in aqueous phase of the Point D2 and E1 are equals. The value of

copper concentration in aqueous phase of the Point D2 is in the cell ‘D83’ -

In the cell ‘D83’, enter formula: ‘=D76’ The value of copper concentration in organic phase of the Point C 2 is solver variable and is in the cell

‘F84’. The starting value of copper concentration in organic phase of the Point C 2 is two-thirds the value of copper concentration in organic phase of point D2 . In this case, two-thirds the value is 2.845. -


-


given by the mathematical expression (38) where the value of constant L2 is in the cell ‘C81’ and the value of constant M 2 is in the cell ‘D81’. The value of constant L2 is given by the mathematical expression (39) where the value of initial free acid concentration is in the cell ‘F58’, the value of initial copper concentration is in the cell ‘F57’, and the value of equilibrium correlation π22 is in the cell ‘B81’. The value of equilibrium correlation π22 is

63

given by the mathematical expression (34) where the value of extractant volume percentage is in the cell ‘F12’ and the value of copper concentration in organic phase is in the cell ‘C84’. -

In

the

cell

‘B81’,

enter

formula:

‘=(((4.8579*10^(-3)*F12-0.19183)*C84+11.365*F12^(-

0.85))*(3.303*F12-3.0842*C84)^(2))/C84’. -

In the cell ‘C81’, enter formula: ‘=-1.299*F58-2*F57-0.422*B81’. The value of constant M 2 is given by the mathematical expression (40) where initial free acid



-



In the cell ‘C87’, enter formula: ‘=C83’. The value of copper concentration in organic phase of the Point B 2 is in the cell ‘C85’. This value is

given by the mathematical expression (65) where the value of Cu2(or/e) is the value of copper concentration in organic phase of the Point C 2 , the value of Cu1or is the value of copper concentration in organic phase of the Point A2 and the value of mixer efficiency of stage of rank 2 is in the cell ‘F63’. -

In the ‘C85’, enter formula: ‘=C84*F63/100 + C87*(1-F63/100)’. The value of copper concentration in aqueous phase of the Point B 2 is in the cell ‘D85’. This value is

given by the mathematical expression (63) where the value of ratio (Rc𝑠 ) is in the cell ‘C66’, the value of Cu2or is the value of copper concentration in organic phase of the Point B 2 , the value of Cu1or is changed by the value of Cu2or/e which is the value of copper concentration in organic phase of the Point C 2 , the value of Cu2aq is the value of copper concentration in aqueous phase of the Point B 2 and the value of Cu3aq is changed by the value of Cu2aq/e which is the value of copper concentration in aqueous phase of the Point C 2 . -


given by the mathematical (63) where the value of ratio (Rce ) is in the cell ‘F66’, the value of Cu2or is changed by the value of Cu2or/e which is the value of copper concentration in organic phase of the Point C 2 , the value of Cu1or is the value of copper concentration in organic phase of the Point A2 , the value of Cu2aq is changed by the value

64

of Cu2aq/e which is the value of copper concentration in aqueous phase of the Point C 2 and the value of Cu3aq is the value of copper concentration in aqueous phase of the Point A2 . -

In the cell ‘D87’, enter formula: ‘=D84-C66*(C87-C84)’. The values of copper concentrations in aqueous phase of the Points E 2 and A2 are equals. The value of

copper concentration in organic phase of the Point E 2 is in the cell ‘D86’. -

In the cell ‘D86’, enter formula: ‘=D87’. The values of copper concentration in organic phase of the Points B 2 and E 2 are equals. The value of

copper concentration in aqueous phase of the Point E 2 is in the cell ‘C86’. -


Solver constraint of solver variable which is the value of copper concentration in organic phase of the Point C 2 is that the value of copper concentrations in aqueous phase of the Points B 2 and D2 must be equals. The value of solver constraint of the value of copper concentration in organic phase is in the cell ‘I84’. -


Plant constraints

Plant constraints are simulation program results to match with plant data. Plant data of stripping step are the value of copper concentration in stripped organic of stage of rank 1 that is in the cell ‘C90’ and the value of copper concentration in stripped organic of stage of rank 2 that is in the cell ‘C91’. -

In the cell ‘F90’ of plant constraint of stripped organic 𝑆1 , enter formula: ‘=C76’.

-

In the cell ‘F91’ of plant constraint of stripped organic 𝑆2 , enter formula: ‘=C86’. Solver constraints of plant constraints of stripping step are the following:

-


-


Performances The value of copper stripping efficiency is in the cell ‘C94’. This value is given by the mathematical expression (55) where the value of LOs is in the cell ‘C73’ and the value of SOs is in the cell ‘C86’.

65

-

In the cell ‘C84’, enter formula: ‘=(C73-C86)/C73*100’ The value of net copper transfer is in the cell ‘C95’. This value is given by the mathematical

expression (56) where the value of LOs is in the cell ‘C73’, the value of SOs is in the cell ‘C86’ and the value of extractant volume percentage is in the cell ‘F12’. -

In the cell ‘C95’, enter formula: ‘=(C73-C86)/F12’

5.2.3.3.

Simulation constraint The value of simulation program constraint is in the cell ‘I99’. Simulation program constraint is that

the copper concentrations in stripped organic of extraction and stripping steps must be equals. This simulation program constraint is the set objective of Excel solver program. -

In the cell ‘I99’, entre formula: ‘=C86-C41’

At this level, it appears Table 13 as it appears on Excel Microsoft spreadsheet. Table 13 gives results of simulation program with the starting values of extractant volume percentage, saturation ratio, maximum loading, copper concentration in organic phase of the Point C1 and C 2 of extraction step and copper concentration in organic phase of the Point C1 and C 2 of stripping step and mixer efficiencies of stage of rank 1 and 2 of extraction and stripping steps. Simulation results are not matched with plant data. Solver variables have blue color and solver constraints have green color.

5.2.3.4.

Excel solver program Excel solver program execution is as follows:

1)

On the ‘Data’, in the ‘Analysis group’ click solver (if the solver command is not available, you must activate the solver add-in).

2)

In the ‘Set objective’ box, enter the cell reference ‘I99’ of simulation program constraint.

3)

Click ‘Value of’ and then type the number ‘0’ in the box.

4)

In the ‘By Changing Variable Cells’ box, enter the reference for each solver variable (blue color in Table 13). Separate the references with commas (English version).

5)

In the ‘Subject to the constraints’ box, enter solver constraints by doing the following: a.

In the ‘Solver Parameters’ dialog box, click ‘Add’.

b.

In the ‘Cell Reference’ box, enter the cell reference of PLS flowrate solver constraint (green color in Table 13).

c.

Click the ‘relationship’ ‘=‘, in the ‘Constraint’ box, type the number ‘0’.

d.

Click ‘Add’ for the second solver constraint. When the last solver constraint is added (cell ‘I91’), click ‘OK’ to return to ‘Solver Parameters’ dialog box.

66

6)

Click ‘Solve’. To keep the solution values on the worksheet, in the ‘Solver Results’ dialog box, click ‘Keep solver solution’.

At this level, it appears Table 14 as it appears on Excel Microsoft spreadsheet. Table 14 gives results of simulation program with the optimum value of the starting values. Plant data are matched with simulation results.

Simulation program of copper solvent extraction configuration (2Ex2S) of option 2 is done. For this option 2, data (in red color) will be changed only for others simulations.

67

A 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52

B

C

Table

12

PLSflow PLS Cu PLS Ac Rce

Data 700.0 6.00 1.96 1.07

D

E

F

G

H

I

Simulation program table option 2 (part1)

m3/h g/l g/l

Extraction step Solver variables PLSflow m3/h PLS Cu g/l PLS Ac g/l Rce

Solver constraints PLSflow PLS Cu PLS Ac Rce

Starting values V% SR Mefe1 Mefe2

% % % %

ML

g/l

General Orgflow AML α1ML

α12 1

m3/h g/l α2

Stage 1 H1

J1

Cuor

Cuaq

E C1 B1 D1 A1 α22 2

C1 Cuor

Stage 2 H2

J2

Cuor

Cuaq

E C2 B2 D2 A2

ML Raf E1 Raf E2

Solver variables

Plant data 10.50 2.00 0.43

g/l g/l g/l

Solver variables

Plant constraints ML g/l g/l Raf E1 g/l Raf E2

g/l %

68

C1 Cuor

Solver constraints

g/l

Performance Raf Ac Effec

Solver constraints

g/l

C 2 Cuor

ML

C 2 Cuor

Solver constraints ML Raf E1 Raf E2

J

A 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100

B

C

Table

12

SP Cu SP Ac AD Cu

Data 35 190 50

D

E

F

π12 1

g/l g/l g/l

D2 C2 B2 E2 A2 SO S1 SO S2 Effs𝑐 Cut

I

Stripping step Solver variables SP Cu g/l SP Ac g/l AD Cu g/l

Solver constraints SP Cu SP Ac AD Cu

Starting values % %

General m3/h Stage 1 L1

M1

Cuor

Cuaq

D C1 B1 E1 A1 π22

H


Mefs1 Mefs2 Rcs SPflow

G

Solver variables

C1 Cuor

Stage 2 L2

M2

Cuor

Cuaq

g/l

Solver variables

C 2 Cuor

Plant data 3.60 2.59

g/l g/l

Solver constraints

C1 Cuor

Solver constraints

g/l

Plant constraints g/l SO S1 g/l SO S2

C 2 Cuor

Solver constraints SO S1 SO S2

Performance % g/l/v% Simulation constraint Set objective

69

J

A 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52

B

C

Table

13


Data 700.0 6.00 1.96 1.07


α12 100.74 E1 C1 B1 D1 A1

General 746.67 11.086 α2 -0.022

Stage 1 H1 -57.06 Cuor 7.760 8.000 7.526 3.256 3.256

D

F

G

H

I


m3/h g/l g/l

Extraction step Solver variables PLSflow 700.0 m3/h PLS Cu 6.00 g/l PLS Ac 1.96 g/l 1.07 Rce

V% SR Mefe1 Mefe2

Starting values 20 70 90 90

% % % %

ML

11.086

g/l

Solver constraints PLSflow 0.000 PLS Cu 0.000 PLS Ac 0.000 0.000 Rce

m3/h g/l

J1 52.74 Cuaq 6.000 0.940 1.446 1.446 6.000

E2 C2 B2 D2 A2

Stage 2 H2 343.62 Cuor 3.256 4.000 3.879 2.789 2.789

ML Raf E1 Raf E2

Plant data 10.50 2.00 0.43

g/l g/l g/l

Raf Ac Effec


g/l %

α22 779.80

E

J2 52.74 Cuaq 1.446 0.154 0.283 0.283 1.446

Solver variables

C1 Cuor

8.000

4.000

g/l

C1 Cuor

-0.235

Solver constraints

g/l

Plant constraints ML 11.086 g/l 1.446 g/l Raf E1 0.283 g/l Raf E2

70

0.662

Solver constraints

Solver variables

C 2 Cuor

ML

C 2 Cuor

0.623

Solver constraints ML 0.586 -0.554 Raf E1 -0.147 Raf E2

J

A 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100

B

C

Table

13

SP Cu SP Ac AD Cu

Data 35 190 50

D

E

F

g/l g/l g/l

General 3.017 247.48

m3/h

Stage 1 L1 -483.35 Cuor 7.760 3.880 4.268 4.268 7.760

M1 24762.2 Cuaq 50.00 58.25 57.08 46.54 46.54

D2 C2 B2 E2 A2

Stage 2 L2 -619.23 Cuor 4.268 2.845 2.987 2.987 4.268

M2 24762.2 Cuaq 46.54 42.97 42.54 38.68 38.68

SO S1 SO S2


g/l g/l

π12 394.64 D1 C1 B1 E1 A1 π22 716.63

Effs𝑐 Cut

H

I

Simulation program table option 2 (part2) Stripping step Solver variables SP Cu 35 g/l SP Ac 190 g/l AD Cu 50 g/l Mefs1 Mefs2

Rcs SPflow

G

Solver constraints SP Cu 0.000 SP Ac 0.000 AD Cu 0.000

Starting values 90.00 % 90.00 %

Solver variables

C1 Cuor

3.880

Solver constraints

g/l

Solver variables

C 2 Cuor

2.845

C1 Cuor

7.080

Solver constraints

g/l

Plant constraints 4.268 g/l SO S1 2.987 g/l SO S2

C 2 Cuor

-4.002

Solver constraints 0.668 SO S1 0.397 SO S2


71

0.199

J

A 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52

B

C

Table

14


Data 700.0 6.00 1.96 1.07


α12 58.94 E1 C1 B1 D1 A1

General 746.67 10.544 α2 0.640

Stage 1 H1 -39.42 Cuor 7.812 8.387 7.812 4.062 4.062

D

F

G

H

I


m3/h g/l g/l

Extraction step Solver variables PLSflow 700.0 m3/h PLS Cu 6.00 g/l PLS Ac 1.96 g/l 1.07 Rce

V% SR Mefe1 Mefe2

Starting values 19.11 74.40 86.71 87.05

% % % %

ML

10.500

g/l

Solver constraints PLSflow 0.000 PLS Cu 0.000 PLS Ac 0.000 0.000 Rce

m3/h g/l

J1 52.74 Cuaq 6.000 1.387 2.000 2.000 6.000

E2 C2 B2 D2 A2

Stage 2 H2 268.62 Cuor 4.062 4.281 4.062 2.590 2.590

ML Raf E1 Raf E2

Plant data 10.50 2.00 0.43

g/l g/l g/l

Raf Ac Effec


g/l %

α22 602.07

E

J2 52.74 Cuaq 2.000 0.196 0.430 0.430 2.000

Solver variables

C1 Cuor

8.387

4.281

g/l

C1 Cuor

0.000

Solver constraints

g/l

Plant constraints ML 10.50 g/l 2.00 g/l Raf E1 0.43 g/l Raf E2

72

0.000

Solver constraints

Solver variables

C 2 Cuor

ML

C 2 Cuor

0.000

Solver constraints ML 0.000 0.000 Raf E1 0.000 Raf E2

J

A 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100

B

C

Table

14

SP Cu SP Ac AD Cu

Data 35 190 50

D

E

F

g/l g/l g/l

General 2.873 259.93

m3/h

Stage 1 L1 -536.72 Cuor 7.812 3.259 3.600 3.600 7.812

M1 24762.2 Cuaq 50.00 50.98 50.00 37.90 37.90

D2 C2 B2 E2 A2

Stage 2 L2 -684.04 Cuor 3.600 2.434 2.590 2.590 3.600

M2 24762.2 Cuaq 37.90 38.35 37.90 35.00 35.00

SO S1 SO S2


g/l g/l

π12 521.10 D1 C1 B1 E1 A1 π22 870.22

Effs𝑐 Cut

H

I

Simulation program table option 2 (part2) Stripping step Solver variables SP Cu 35 g/l SP Ac 190 g/l AD Cu 50 g/l Mefs1 Mefs2

Rcs SPflow

G

Solver constraints SP Cu 0.000 SP Ac 0.000 AD Cu 0.000

Starting values 92.52 % 86.61 %

Solver variables

C1 Cuor

3.259

Solver constraints

g/l

Solver variables

C 2 Cuor

2.434

C1 Cuor

0.000

Solver constraints

g/l

Plant constraints 3.60 g/l SO S1 2.59 g/l SO S2

C 2 Cuor

0.000

Solver constraints 0.000 SO S1 0.000 SO S2


73

0.000

J

6.

Bibliography 1.

Alguacil F.J., Modelling copper solvent extraction from acidic sulphate using MOC 45, Rev.Metal Madrid, 1998, p381-384.

2.

Komulainen T.,Pekkla P., Ramtala A. and Jämsa-Jounela S., Dynamic modeling of an industrial copper solvent extraction process, Hydrometallurgy 81, 2006, p52-61.

3.

Hossen Aminian, Modélisation et simulation des opérations d’extraction par solvant et d’électrolyse du Cuivre, thèse, université de Laval, Canada, 1999.

4.

Gerald L. Bauer and Thomas W. Chapman, Measurement and correlation of solvent extraction equilibria. The extraction of Copper by kelex 100, Metallurgical transaction B, 1976.

5.

AMEL A., Etude thermodynamique de l’extraction des Métaux de transition par la Salicylidèneaniline, thèse, Université Mohamed Khider, 2013.

6.

Ritcey G.M., Ashbrook A.W., Solvent extraction: principles and application of process metallurgy, Part I, Elsevier, 1984.

7.

Jeffers T.H, Groves R.D, Minimizing lead contamination in Copper produced by solvent extraction – electrowinning, Salt Lake City Research Center.

8.

Winand R., Electrocristallisation. Théorie et application, Journal de physique IV, 1994.

9.

Beukes N.T. and Badenhorst J., Copper electrowinning; theoretical and practical design, Hydrometallurgy conference, SAIMM, 2009.

.

10. Liu Jiarshe, Lan Zhuo Yue, Qiu Guarr Zhou, Wang Diarr Zuo, Mechanism of crud formation in Copper solvent extraction, journal CSUT, Vol 9, 2002. 11. Hans Hein, Importance of a wash stage in Copper solvent extraction, HydroCopper, 2005.

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