SLAG QUANTITY MINIMIZATION IN THE PYROMETALLURGICAL ... · least 99,95 % Cu [9 - 11]. In the...

Dimitar Borisov

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SLAG QUANTITY MINIMIZATION IN THE PYROMETALLURGICAL PRODUCTION OF ANODE COPPER

Dimitar Borisov

ABSTRACT

Approximate mathematical models, numerical data and graphical results of conducted virtual planned experiments on a verified analytical mathematical model are presented in this article. Two adequate approximating polynomial mathematical models have been developed that can be used to optimize the beneficial product (mate) and to minimize the quantities of industrial waste (slag) in the pyrometallurgical processes of flash smelting of sulphide copper raw materials. The proposed mathematical models allow finding a number of Pareto - optimal solutions for components in charge in the metallurgical processes of sulfide copper concentrates flash smelting, which provide requirements for a maximum amount of mate and a minimum amount of slag. The approximation models can be used for operational planning, product optimization, process control and minimization of waste material flows.

Keywords: Pareto-optimal solution, compromise sollutions, regressional model, pyrometallurgy of copper, ap-proximate mathematical models.

Received 02 April 2019Accepted 21 June 2019

Journal of Chemical Technology and Metallurgy, 54, 5, 2019, 1047-1060

University of Chemical Technology and Metallurgy8 Kliment Ohridski, 1756 Sofia, BulgariaE-mail: [email protected]

INTRODUCTION

The functioning of the production systems is related to the use of natural resources, whose main components determine the composition and the consumer qualities of the obtained market product. The main criterion for the environmental level of an industrial system is the degree of its adverse impact on the surrounding ecosystems [1 - 3]. Defining meaning for this impact are quantitative indicators of the waste materials flows and the environmentally hazardous components within.

In the metallurgical and chemical industry, the quantitative and qualitative characteristics of generated industrial waste are identifiers for the efficiency of raw materials utilization and the degree of extraction of useful components in a technological process. The most preferred method for industrial solid waste management from technically and ecologically point of view is to reduce the quantities generated at the source [5 - 8] by using all the technical and technological opportunities.

Technological development of the human society

leads to the creation of opportunities for utilization of natural raw materials (ores, minerals, energy carriers, etc.), containing ever lower concentrations of useful components [1, 2]. This trend is deepening and definitely leads to increasing the mass of industrial wastes both after each technological operation and as a result of the overall production process. In many industries in the industrial solid wastes remain significant amounts of useful components due to physico-chemical limitations for complete extraction. As a typical example, confirm-ing the described ascertainment can be considered the pyrometallurgical processing of copper ores containing about 1 mass % Cu to electrolytic copper containing at least 99,95 % Cu [9 - 11]. In the industrial system, more than 100 t of solid wastes are generated in the production of 1 t of the final product, which is a serious ecological problem.

The characteristic features of the processes in the metallurgical and chemical industry, complicating the creation of a complete and common mathematical model for optimization and process control of the technological

Journal of Chemical Technology and Metallurgy, 54, 5, 2019

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process, are:• The processes taking place in the production instal-

lations are complex and with interconnected input and output parameters;

• Some of the thermo-physical and physicochemical processes in extractive metallurgy are not theoretically clarified and therefore an indirect evaluation of the actual process condition is necessary;

• Most of the controling and controles parameters of the technologicall processes are not subject to continuous and direct control;

• Dynamic process modeling is complicated by the lack of a unique impact on the input parameters;

• Non-stationarity of processes and consequent variation within a wide range of measurable values.

These features may have serious difficulties in con-cordance of composition and quantity of the individual technological material and energy flows.

In pyrometallurgical processing of mineral raw materials, large quantities of waste material flows containing toxic components are generated. For flash smelting of sulphide copper raw materials, flotation concentrates are used, the main components of which are sulphides of Cu, Fe and other metals, such as SiO2, Al2O3, CaO, etc. Concentrations of the major constituents, which make up more than 95 % of the composition, vary within the following limits [9]: copper - from 20 to 35 mass %; iron - from 20 to 30 mass %; sulphur - from 20 to 35 mass %; SiO2 - from 5 to 15 mass %.

The technological objective of the oxidation smelt-ing process is to obtain a product (mate) containing 60 - 65 % Cu, which is a melt containing Cu2S, FeS and less than 1 % dissolved magnetite (Fe3O4). The main reactions in the technological torch are the following:

dissociation of the higher sulphides of copper and iron:

2 2CuFeS Cu S FeS= + H∆+ (1)

22 21 SFeSFeS += H∆+ (2)

oxidation of elemental sulfur:

2 2 212

S O SO+ = H∆− (3)

oxidation of FeS and slag formation:

2 2 2 22FeS O SiO FeO.SiO SO+ + = + H∆− (4)

2 3 4 2FeS O Fe O SO+ = + H∆− (5)

The producing slag phase, in addition to oxide compounds (2FeO.SiO2, Fe3O4) contains up to 1.5 % Cu, in the form of dissolved Cu2S, as well as other heavy metals.

Due to the content of toxic components, slag is a hazardous waste which is treated to extract the useful component (Cu) into it by flotation enrichment. As a result of the flotation, a secondary copper concentrate and non-hazardous solid waste are produced – “fayalite sand”. The secondary copper concentrate is processed at high temperature in the flash smelting furnace. The fayalite is finally disposed by landfilling in a tailing pond.

The main components forming the slag phase are iron oxides and SiO2. This also determines the possibilities of reducing the amount of these technological wastes.

The studied physicochemical characteristics of the smelting processes of sulphide raw materials make it possible to determine the possibilities for reducing their quantity, partially utilizing the useful components and reducing the content of hazardous substances in them.

The concentration of iron in the charge materials for the smelting process determines the mass of waste slag. The use of sulphide concentrates with the lowest possible Fe content is the most effective way to reduce the slag phase [10].

On the other hand, the process of autogenous smelting of sulphide concentrates is carried out at tem-peratures higher than 1500°C, the main heat-generating process being the oxidation of the iron sulfides by reac-tion (4) and (5). The use of low-Fe feedstocks (copper concentrates) must be consistent to ensure the heat bal-ance of the process.

Silicon dioxide is the second most important slag-forming component. Its concentration in the charge is determined by the Fe content. Conducting the smelting process operation at optimal temperature regime and close to the stoichiometrically required concentration of SiO2 creates possibilities to reduce the relative amounts of slag phase [10, 11].

The high temperatures in which the process is car-ried out (above 1500ºС in the technological torch) allow maximum (over 90 %) main toxic components of charge S, As, Pb, Zn, etc. to pass into the gas phase.

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EXPERIMENTALModel description

The main processes occurring in the process of pyrometallurgical production of anode copper are physical, chemical, physico-chemical, hydro- and aerodynamic and thermodynamic. These processes are interrelated thus determining the overall behavior of the flash smelting furnace as a multi-linked control object [12].

In many cases, from engineering practice, analytical and techno-economic mathematical models are complex, require many calculations and take a long time to solve them. The great advantage of such models is that they are „accurate“ models because they are based on physico-chemical dependencies that are universal. The use of such real-time process control models is greatly complicated by the heavy computational procedure and the time to find a solution that is longer than the time required to make a quick decision about a particular process situation. Especially for automated control systems, this is very important, as it is necessary to quickly change the assignments of the control parameters that can be found in solving the mathematical model for the new situation. In such situations, so-called approximation model (“short cut” models, “tailored” models) are recommended. Such models can be obtained as experimental - statistical (regression) models based on numerical data obtained from the analytical model and approximated to the desired precision by a regression model. The area of change of input (control) parameters for “numerical experiments” is chosen in practice the significant area of the particular process.

Approximate mathematical models, numerical data and graphical results of conducted virtual planned experi-ments on a verified analytical mathematical model are presented in this article. The approximation models are of the regression type and their adequacy is evaluated according to the known statistical criteria [12 - 14]. Two experimental designs proposed for exploring a limited practical area are the plans of McLean and Anderson [12].

On the basis of expert experience and reference data, two possible limitations of the main components in the charge, Cu, Fe and S are identified. The main indicators studied are: quantity of mate (Y1, t/h) and amount of slag (Y2, t/h). In the virtual experiment some simplifications have been introduced that do not affect the results of the research and the correctness of the experiments. They are limited to the following:

The percentage of SiO2 in the experiments was kept

constant (SiO2 ch% = 6.00 mass %).

In the phase composition (Xi) - Cu, Fe and S form 90 mass %, all other components of the charge except the above mentioned are grouped in the form of “Other” = 4 %.

Experimental DesignTwo possible experimental designs for the virtual

experiments are described below.Experimental Design No 1:10 ≤ Cu ≤ 35 mass %; 15 ≤ Fe ≤ 45 mass %; 10 ≤ S ≤ 40 mass %.Experimental Design No 2:5 ≤ Cu ≤ 40 mass %; 10 ≤ Fe ≤ 50 mass %; 5 ≤ S ≤ 50 mass %.

The experimental designs of numerical experiments were performed using the method of McLean and Anderson [12] for the study of multi-component systems that have been subject to constraints in component changes through inequalities of the following type,

iii bXa ≤≤ (i = 1, 2, …, n). The necessary conditions for the layout of the McLean and Anderson plan are as follows:

1001

=∑=

n

iiX ; 100

1<∑

=

n

iia ; 100

1>∑

=

n

iib (6)

The above mentioned four inequalities in the barientricular factor space form a multipote whose shape depends on the values ai of and bi.

The components of the system are:Percentage of Cu: X1

Percentage of Fe: X2 Percentage of S: X3 Percentage of SiO2: X4 = 6Percentage of others: X5 = 4

Two experimental designs of McLean and Anderson were prepared for the three combinations of different limit values of the components described above. After calculating the numerical experiments on the plans, the data were processed by the regression analysis method.

To obtain approximation models, one of the most used polynomials for describing the studied dependencies is as follows:

2 2 20 1 1 2 2 3 3 12 1 2 13 1 3 23 2 3 11 1 22 2 33 3Y b b X b X b X b X X b X X b X X b X b X b X= + + + + + + + + +

2 2 20 1 1 2 2 3 3 12 1 2 13 1 3 23 2 3 11 1 22 2 33 3Y b b X b X b X b X X b X X b X X b X b X b X= + + + + + + + + +

(7)


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This article explores various possible descriptions of “experimental” data. From the different types of regres-sion models studied it was found that the best description is obtained with a regression equation of the type:

1 1 2 2 3 3 12 1 2 13 1 3 23 2 3Y b X b X b X b X X b X X b X X= + + + + +

1 1 2 2 3 3 12 1 2 13 1 3 23 2 3Y b X b X b X b X X b X X b X X= + + + + + (8)

In (7) and (8) Y is the predicted value of the function, xi are the components of the charge and bij are the coefficients of the model.

Numerical experiment for Experimental design No 1The following system of components have been

considered:10 ≤ X1 ≤ 35 mas. %15 ≤ X2 ≤ 45 mas. % (9)10 ≤ X3 ≤ 40 mas. %

According to the requirements (6), the existing

sub-area is eligible for McLean and Anderson‘s plan

of experiments: 1

35n

ii

a=

=∑ and 1201

=∑=

n

iib mass %.

The experimental design consists of 9 experiments located at the peaks of the limited area, centroids of the border points and one point in the center of the polygon.

The created experimental design of McLean and Anderson is given on the Table 1 and the graphical layout of the points - in Fig. 1.Numerical experiment for experimental design No 2

The following system of components has been considered: 5 ≤ X1 ≤ 40 mas. %10 ≤ X2 ≤ 50 mas. % (10) 5 ≤ X3 ≤ 50 mas. %

According to the conditions (6) there is a limited sub-area, as the conditions (6) are fulfilled:

120

n

ii

a=

=∑ and 1401

=∑=

n

iib mass %.

Table 1. McLean and Anderson design of experiments No 1.

No X1, % X2, % X3, % X4, % X5, % Y1, t/h Y2, t/h

Peak

s of t

he li

mite

d ar

ea

1. 10,00 45,00 35,00 6,00 4,00 13.937766 81.280508

2. 35,00 15,00 40,00 6,00 4,00 55.121794 20.227236

3. 35,00 45,00 10,00 6,00 4,00 42.163232 88.767355

4. 10,00 40,00 40,00 6,00 4,00 14.150004 72.366505

Cen

ter

poin

t

5. 22,50 36,25 31,25 6,00 4,00 34.264482 60.891754

Cen

troid

of t

he b

orde

r po

ints

6. 10,00 42,50 37,50 6,00 4,00 14.043885 76.823506

7. 35,00 30,00 25,00 6,00 4,00 54.485079 44.960001

8. 22,50 45,00 22,50 6,00 4,00 33.893065 76.491259

9. 22,50 27,50 40,00 6,00 4,00 34.635899 45.292248

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No X1, % X2, % X3, % X4, % X5, % Y1, t/h Y2, t/h

Peak

s of t

he li

mite

d ar

ea

1. 5,00 50,00 35,00 6 4 5.743408 92.110211

2. 40,00 10,00 40,00 6 4 63.316152 12.051780

3. 5,00 35,00 50,00 6 4 6.380122 65.368201

4. 40,00 45,00 5,00 6 4 43.673606 88.426292

5. 30,00 10,00 50,00 6 4 47.351912 14.944144

6. 35,00 50,00 5,00 6 4 38.534689 97.687420

Cen

ter

poin

t

7. 25,8333 33,3333 30,8333 6 4 39.709700 54.414786

Cen

troid

of t

he b

orde

r poi

nts

8. 5,0000 42,5000 42,5000 6 4 6.061765 78.739206

9. 40,0000 27,5000 22,5000 6 4 62.573318 38.587299

10. 20,0000 50,0000 20,0000 6 4 29.689767 86.363112

11. 35,0000 10,0000 45,0000 6 4 55.334032 13.497962

12. 17,5000 22,5000 50,0000 6 4 26.866017 38.293944

13. 37,5000 47,5000 5,0000 6 4 41.104147 93.056856

Fig. 1. Graphical layout of the points for experimental design No 1.

Table 2. McLean and Anderson design of experiments No 2.

Fig. 2. Graphical layout of the points for experimental design No 2.


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The experimental design is created as a McLean and Anderson design. The experiments are located at the boundary points of the polygon defined by (9). Six points are on “peaks” of the limited area, six points on centroids of the border points and one point in the center of polygon. The number of possible combinations is 13.

The design of McLean and Anderson is given in the Table 2 and the graphical layout of the points - in Fig. 2.

Creation of approximate regression modelsQStatLab v.5 program product [13] is used to obtain

and statistically analyze the regression models that ap-proximated the data from the virtual experiments.

Regression models for Experimental design No 1 Model 1The regression equation for the obtained mate (У1) is:

1 1 2

3 1 2

2 3 1 3

1.2083156 0.278361

0.907446 0.00102

0.0232562 0.0277845

Y X X

X X X

X X X X

= − −

− + +

+ +

The calculated and predicted the model values of

mate, t/h (Y1) and of slag, t/h (Y2) for experimental design No 1 are presented on Table 3.

The statistical characteristics of the approximate model for mate (Y1) at α=0.05 are: Ftabl(0.05, 5, 3) = 9.01346 < Fcalc = 335.95635.

The calculated value of the Fisher criterion Fcalc = 335.95635 is greater than the table value, indicating a good approximation. The multiple correlation coefficient is high and significant R2 = 0.99822, which also shows very good approximation. The average absolute error of Model 1 for matte (Y1) is 0.5247 t/h.

The compliance between the calculated values of mate (Y1) in the analytical model (Y1exper) and by fitting (approximation) model (Y1calc) is given in Fig. 3. The figure also shows the statistical characteristics that con-firm the good approximation between “experimental” and “calculated” for mate (Y1).

The regression equation for the slag (У2) is:

2 1 2

3 1 2

2 3 1 3

0.3202852 2.310249

1.432696 0.0062668

0.037963 0.0407528

Y X X

X X X

X X X X

= + +

+ − −

− −

No Y1 exper Y1 calc Y1 exper – Y1 calc Y2 exper Y2 calc Y2 exper – Y2 calc

1. 13.9378 14.6083 -0.6706 81.2805 80.4332 0.8473

2. 55.1218 55.2053 -0.0835 20.2272 20.0498 0.1775

3. 42.1632 42.4864 -0.3231 88.7674 88.2811 0.4863

4. 14.1500 13.3825 0.7675 72.3665 73.3720 -1.0055

5. 34.2645 35.4515 -1.1870 60.8918 58.9541 1.9377

6. 14.0439 14.1408 -0.0969 76.8235 76.6653 0.1582

7. 54.4851 54.0785 0.4066 44.9600 45.6237 -0.6637

8. 33.8931 32.8887 1.0044 76.4913 77.9895 -1.4982

9. 34.6359 34.4532 0.1827 45.2922 45.7317 -0.4395

Mean absolute error for matte: 0.5247 t/h Mean absolute error for slag: 0.8015 t/h

Table 3. Calculated and predicted by the model values of mate, t/h (Y1) and of slag, t/h (Y2) for experimental design No 1.

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The statistical characteristics of the approximate model for slag (Y2) at 05.0=α=0.05 are: Ftabl(0.05, 5, 3) = 9.01346 < Fcalc = 271.97395.

The calculated value of the Fisher criterion Fcalc = 271.974 is greater than the table value, indicating a good approximation. The multiple correlation coefficient is

high and significant R2 = 0.99780, which also shows very good approximation. The average absolute error of Model 1 for slag (Y2) is 0.8015 t/h. The compliance between the calculated values of slag (Y2) in the analyti-cal model (Y2exper) and by fitting (approximation) model (Y2calc) are given in Fig. 4.

Fig. 3. Compliance between experimental and calculated values of mate, t/h (Y1) for ex-perimental design No 1. (У1exper mean = 32.9661; У1calc mean = 32.9661; R = 0.9991; Y1calc mean = 0.0567 + 0.9982У1exper)

Fig. 4. Compliance between experimental and calculated values of slag, t/h (Y2) for experimental design No 1. (У2exper mean = 63.0112; У2calc mean = 63.0112; R = 0.9989; Y2calc = 0.1387 + 0.9978У2exper)

Xa = 32,9661; Ya = 32.9661; r = 0,9991; Y = 0,06 + 1,00X

Xa = 63,0112 ; Ya = 63,0112 ; r = 0,9989 ; Y = 0,14 + 1,00X


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Regression models for Experimental design No 2 Model 2The regression equation for the obtained mate (У1) is:

1 1 2

3 1 2

2 3 1 3

1.02899 0.34697915

0.5456387 0.00754044

0.01828791 0.02349546

Y X X

X X X

X X X X

= − −

− + +

+ +

The calculated and predicted the model values of mate, t/h (Y1) and of slag, t/h (Y2) for experimental design No 2 are presented on Table 4.

The statistical characteristics of the approximate model for mate (Y1) at 05.0=α 0.05 are: Ftabl(0.05, 5, 4) = 3.97152 < Fcalc = 276.9195.

The calculated value of the Fisher criterion Fcalc = 276.9195 is greater than the table value, indicating a good approximation. The multiple correlation coeffi-cient is high and significant R2 = 0.99497. The average absolute error of Model 2 for matte (Y1) is 1.1349 t/h. The compliance between the calculated values of mate (Y1) in the analytical model (Y1exper) and by fitting (ap-proximation) model (Y1calc) are given in Fig. 5.

The regression equation for the slag (У2) is:

2 1 2

3 1 2

2 3 1 3

0.42038079 2.3380658

0.71694356 0.01519034236

0.025236 0.0259377

Y X X

X X X

X X X X

= + +

+ − −

− −

Table 4. Calculated and predicted by the model values of mate, t/h (Y1) and of slag, t/h (Y2) for experimental design No 2.

No Y1 exper Y1 calc Y1 exper – Y1 calc Y2 exper Y2 calc Y2 exper – Y2 calc

1. 5.7434 6.7019 -0.9585 92.1102 91.5985 0.5117

2. 63.3162 63.7913 -0.4752 12.0518 11.2028 0.8490

3. 6.3801 4.9198 1.4603 65.3682 66.4756 -1.1074

4. 43.6736 45.2044 -1.5308 88.4263 87.4047 1.0216

5. 47.3519 46.7710 0.5809 14.9441 15.7576 -0.8135

6. 38.5347 37.8173 0.7174 97.6874 97.7701 -0.0827

7. 6.0618 6.8395 -0.7778 78.7392 77.6176 1.1217

8. 62.5733 60.0985 2.4748 38.5873 41.5752 -2.9879

9. 29.6898 27.5461 2.1437 86.3631 88.8484 -2.4852

10. 55.3340 55.8686 -0.5345 13.4980 12.8317 0.6662

11. 26.8660 27.0236 -0.1576 38.2939 38.7431 -0.4492

12. 41.1041 41.5580 -0.4538 93.0569 92.4925 0.5644

13. 39.7097 42.1986 -2.4889 54.4148 51.2234 3.1913

Mean absolute error for matte: 1.1349 t/h Mean absolute error for slag: 1.2194 t/h

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The statistical characteristics of the approximate model for slag (Y2) at 05.0=α 0.05 are: Ftabl(0.05,5,4) = 6.25606 < Fizch = 572.11787.

The calculated value of the Fisher criterion Fcalc = 572.11787 is greater than the table value, indicating a good approximation. The multiple correlation coeffi-cient is high and significant R2 = 0.99756. The average absolute error of Model 2 for the slag (Y2) is 1.2194 t/h. The compliance between the calculated values of slag (Y2) in the analytical model (Y2exper) and by fitting (ap-

proximation) model (Y2calc) is given in Fig. 6.

RESUTLS AND DISCUSSIONComparative analysis of the results of the proposed two experimental plans

The average absolute errors (Δav) of the proposed models are presented on Table 5.

If we assume the average absolute error ( av∆ ) be-tween the values of the analytical model and the values obtained by the approximate model for mate (Y1) and for

Fig. 5. Compliance between experimental and calculated values of mate, t/h (Y1) for experimental design No 2. (У1exper mean = 52.8722; У1calc mean = 52.87221; R = 0.9975; Y1calc = 0.180 + 0.995У1exper)

Fig. 6. Compliance between experimental and calculated values of slag, t/h (Y2) for experimental design No 2.(У2exper mean = 59.5032; У2calc mean = 59.5032; R = 0.999; Y2calc = 0.1452 + 0.99756 У2exper)

Xa = 35,8722 ; Ya = 35.8722 ; r = 0,9975 ; Y = 0,18 + 0,99X

Xa = 59,5032 ; Ya = 59.5032 ; r = 0,9988 ; Y = 0,15 + 1,00X


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slag (Y2) as an indicator of the acceptable accuracy of the approximation model, from Table 5 it can be seen that Model 1, obtained from Experimental design No 1 by McLean and Anderson, shows smaller error in the range of 10 ≤ Cu ≤ 35, 15 ≤ Fe ≤ 45 and 10 ≤ S ≤ 40 mass %.

For Model 2 (Experimental design No 2), the aver-age absolute errors ( av∆ ) are about 2 times greater than those for Model 1, but the area of study is larger.

For Model 1 (Experimental design No 1) for mate (Y1), the normal plot of the standardized residues, r, is given in Fig. 7. For Model 1 (Experimental design 1) for slag mass (Y2), the normal plot of the standardized residues is presented in Fig. 8.

Determination of the optimum phase composition for the mate

Using the QStatLab software product, by the ap-proximate mathematical Model 1, the following optimal phase composition is obtained:

Х1* (Cu*) = 35.00 %; X2

* (Fe*) = 22.50 %; X3

* (S*) = 32.50 %.With this optimal composition, the maximum value

for a mate is obtained: У1max = 55.95 t/h.In Fig. 9, the lines of constant values for the mate

(Y1) calculated by the approximate Model 1 are repre-sented in the allowable area defined by the McLean and Anderson plan.

Table 5. Average absolute error ( av∆ ) of the proposed models.

Model 1

(Experimental design No 1)

Model 2

(Experimental design No 2)

У1,

mate, t/h

У2,

slag, t/h

У1,

mate, t/h

У2,

slag, t/h

0.5247 0.8015 1.1349 1.2194

Fig. 7. Normal plot of the standardized residues for mate (Y1).AD = 0,1544; p = 0,9308

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Determination of the optimum phase composition for the slag

By the approximate mathematical Model 1, with software product QStatLab the optimal phase composi-tion for slag is obtained:

X1* (Cu*) = 35.00 %; X2

* (Fe*) = 15.00 %; X3

* (S*) = 40.00 %.

Table 5. Average absolute error ( av∆ ) of the proposed models.

With this optimal composition, a minimum value for slag has been obtained: Y2min = 20.05 t/h.

In Fig. 10, the lines of constant values for the slag (Y2) calculated by the approximate Model 1 are repre-sented in the allowable area defined by the McLean and Anderson plan. From the obtained optimal phase com-positions for mate and slag as well as Fig. 9 and Fig. 10

Fig. 8. Normal plot of the standardized residues for slag (Y2).

Fig. 9. Lines of constant values for mate (Y1) calculated by the approximate Model 1 in the eligible area.

AD = 0,2028; p = 0,8204


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show that for optimum process control is necessary to find a compromise solutions (Pareto - optimal solutions).

Determining the Pareto - optimal region of the phase composition

From the optimization procedure and from Fig. 9 and Fig. 10 it can be seen that in this two-goal task of process control of the phase compositions, the desired/optimal values for mate,for which we seek maximum values and

for the slag, for which we seek a minimum values, the optimum phase compositions do not coincide. To obtain a variety of possible compromise solutions an optimi-zation program with a genetic algorithm of QstatLab is used. A number of 10 Pareto-optimal (compromise) solutions for the Cu, Fe and S (%) phase compositions are obtained for the target indicators: mate (Y1) and slag (Y2). The results are given in Table 6.

The Pareto-front of the compromise solutions for the

Fig. 10. Lines of constant values for slag (Y2) calculated by the approximate Model 1 in the eligible area.

Solution No X1, Cu X2, Fe X3, S Y1, mate Y2, slag - % % % t/h t/h 1. 35.00 15.50 39.50 55.3277 20.5868 2. 35.00 17.50 37.50 55.7494 23.1695 3. 35.00 17.20 37.80 55.6904 22.6977 4. 35.00 18.80 36.20 55.9100 24.9189 5. 35.00 19.50 35.50 55.9655 25.9159 6. 35.00 15.30 39.70 55.2795 20.3679 7. 35.00 15.80 39.20 55.3994 20.9322 8. 35.00 16.30 38.70 55.5221 21.5918 9. 35.00 16.70 38.30 55.5970 22.0486

10. 35.00 18.20 36.80 55.8466 24.1122

Table 6. Pareto - optimal decisions for phase compositions phase compositions and corresponding targets: mate (У1) and slag (У2).

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targets indicators mate and slag are obtained and pre-sented in Fig. 11. For illustration, three compromise solutions (1), (2) and (3) of the Pareto-front are

selected. Their corresponding compositions for Cu, Fe and S (%) are given on Table 7 and Figs. 12, 13 and 14.

Fig. 11. Pareto - optimal front for mate (У1) and slag (У2) and three selected compromise solutions (1), (2) and (3).

Fig. 12. Compositions of Х1 (Cu) and X2 (Fe) for the selected solutions (1), (2) и (3).

Fig. 13. Compositions of Х1 (Cu) and X3 (S) for the selected solutions (1), (2) и (3).

Fig. 14. Compositions of Х2 (Fe) and X3 (S) for the selected solutions (1), (2) и (3).

Solution Х1, Cu Х2, Fe Х3, S У1, mate У2, slag % % % t/h t/h Solution (1) 35.00 15.13 40.00 55.24 20,20 Solution (2) 35.00 20.21 34.80 56.00 26,98 Solution (3) 35.00 19.00 36.83 55,84 24,03

Table 7. Exemplary Pareto-optimal solutions of Fig. 17.


1060

CONCLUSIONSTwo experimental designs for virtual experiments

are proposed for different areas of the three main com-ponents in the charge Cu - Fe - S:

Experimental deign No 1: 10 ≤ Cu ≤ 35; 15 ≤ Fe ≤ 45; 10 ≤ S ≤ 40, (mass %);

Experimental deign №2: 5 ≤ Cu ≤ 40; 10 ≤ Fe ≤ 50; 5 ≤ S ≤ 50, (mass %).

In the virtual experiments of the mathematical model, the quantity of slag (Y2), t/h and of the mate (Y1), t/h for each of the experimental plan is numeri-cally determined.

Two adequate approximation mathematical models (Model 1 and Model 2) have been developed that can be used to optimize the beneficial product (mate) and to minimize the amount of industrial waste (slag) in the metallurgical flash smelting processes of sulphide copper raw materials. The approximation models are verified by the results of the numerical “experiments” in the limited factor space.

The smallest average absolute errors for the product and the waste are obtained on Model 1 in the following range of concentrations of major components: 10 ≤ Cu ≤ 35; 15 ≤ Fe ≤ 45; 10 ≤ S ≤ 40 (mass %).

The approximated Model 2, obtained by Experi-mental deign No 2, can also be used as it covers a sig-nificantly wider range of phase compositions, although the model error is greater than Model 1.

The approximation models can be used for operational planning, optimization and process control of other objective products and wastes.

The proposed mathematical models allow to find a number of Pareto - optimal solutions for components in charge in the metallurgical processes of sulfide copper concentrates flash smelting, which provide requirements for a maximum amount of mate and a minimum amount of slag. А number of Pareto-optimal solutions for charge composition of mathematical model 1 in the limited area for the phase compositions: 10 ≤ Cu ≤ 35; 15 ≤ Fe ≤ 45; 10 ≤ S ≤ 40 (mass %) are presented.

AcknowledgementsThe author would like to thank for the financial

support from the National programme “Young Scientists and Post-doctoral Students”.

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