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Applicationofresponsesurfacemethodologyformodelingofballmillsincoppersulphideoregrinding

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DOI:10.1016/j.powtec.2013.04.021

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Powder Technology 245 (2013) 292–296

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Application of response surface methodology for modeling of ball millsin copper sulphide ore grinding

A. Ebadnejad a,⁎, G.R. Karimi a, H. Dehghani b

a Department of Mining Engineering, School of Engineering, IKI University, Qazvin, Iranb Department of Mining Engineering, School of Engineering, University of Shahid Bahonar, Kerman, Iran

⁎ Corresponding author. Tel.: +98 9146119786; fax:E-mail address: a_ebadnejad@yahoo.com (A. Ebadne

0032-5910/$ – see front matter © 2013 Elsevier B.V. Allhttp://dx.doi.org/10.1016/j.powtec.2013.04.021

a b s t r a c t

a r t i c l e i n f o

Article history:Received 13 December 2012Received in revised form 12 April 2013Accepted 20 April 2013Available online 25 April 2013

Keywords:Copper sulphideBall millBox–Bhenken designRSMModeling

Modeling of some parameters of wet ball milling system of copper sulphide ore was performed in this study. Athree level Box–Bhenken design combining a response surface methodology (RSM) with quadratic model wasemployed for modeling of key operating parameters of ball mills. Grinding experiments were designed andexecuted by a laboratory ball mill, considering ball size, ball charge and solid content as variables. Grindingtests were performed changing these three variables (ball size, ball charge and solid content) in the rangeof 20–40 mm, 20–40% and 65–80% respectively. Product 80% passing size (d80) was defined as process re-sponse. A quadratic model was developed to demonstrate the effect of each parameter and its interactionwith d80 of product. Predicted values of response obtained using model equation were in good agreementwith the experimental values (R2 value of 0.994 for d80). Finer d80 was achieved using greater ball chargeswith smaller ball sizes. More favorable results were also obtained at the center of solid content level. Resultssuggest that RSM could be efficiently applied formodeling of ball milling system of some copper sulphide ores.

© 2013 Elsevier B.V. All rights reserved.

1. Introduction

In today's global markets suffering from the world crisis, mininggroups strive to optimize mill performances mainly by reducing pro-duction costs. Successful grinding with ball mills depends on selectingsuitable operating conditions. Therefore it is of great importance todetermine the operating parameters at which the response reachesits optimum.

Different variables have been studied to improve ball mill perfor-mance [1–3]. The slurry density and grinding media size are probablythe most frequently considered factors for the process optimization[3]. The efficiency of grinding depends on the surface area of thegrinding medium. Thus, balls should be as small as possible and thecharge should be graded in such manner that the largest balls arejust heavy enough to grind the largest and hardest particles in thefeed [4]. Harder ores and coarser feeds require high impact and largemedia while very fine ground sizes require substantial media surfacearea and small media [5].

The pulp density of the feed should be as high as possible, consis-tent with ease of flow through the mill. It is essential that the balls arecoated with a layer of ore; too dilute a pulp increases metal-to-metalcontact, giving increased steel consumption and reduced efficiency.Also slurry density certainly influences the distribution of the energy

+98 4112815477.jad).

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of impacts applied to the particles in a grinding mill [6]. The energyinput to a mill is also of great importance and valuable works are fo-cused on this area [7,8]. It increases with the ball charge, and reaches amaximum at a charge volume of approximately 50%, but for a numberof reasons, 40–50% is rarely exceeded [4].

As an essential approach it is very economical to adopt an experi-mental design methodology for extracting the maximum amount ofcomplex information while saving significant experimental time, ma-terial used for analysis and personnel costs [9]. Engineering experi-mentalists wish to find the conditions under which a certain processis optimized. The optimum could be either a maximum or a minimumof a function of the design parameters. One of the methodologies forobtaining the optimum is the response surface technique [10].

A large number of studies have been conducted trying to applydifferent types of modelling methods to a relatively wide range ofprocess optimization [11–14].

Experimental design methods and response surface methodolo-gies have been specifically applied formodeling of process parametersin mineral processing systems [15–21]. Response surface methodolo-gy (RSM) has been employed for modeling of some processes such asTurkish coal processing [16], flotation of celestite concentrate [17],chromite concentration [18], flotation of synthetic mixture of celestiteand calcite minerals [19] and sulphur grindability in a batch ball mill[20]. RSM also has been applied for optimization of yield at a desiredash level in coal flotation [21].

To optimize the milling process, influencing factors such as reduc-tion ratio or d80 of product were considered to be closely monitored.

Table 1Mineralogical composition of the feed ore sample using XRD.

Component Weight (%)

Cu2S 0.391CuS 0.284CuFeS2 0.431FeS2 6.628MoS2 0.037Fe2O3–Fe(OH)2 0.140Fe2O3 0.189Fe3O4 0.105SiO2 67.27Al2O3 14.38

293A. Ebadnejad et al. / Powder Technology 245 (2013) 292–296

In current study, grinding efficiency improvement is investigated bymaking key changes to milling system as follows:

- Using correct ball size, operating the mill in an appropriate mode(ball charge and solid content) and controlling milling circuit ina modified mode.

Reduction ratio is a determining factor in mill efficiency evalua-tion which can show how efficiently the energy is consumed. A higherreduction ratio can signal a more efficient milling in progress.

In this study RSM was used in conjunction with Box–Bhenken de-sign, which requires fewer tests than a full factorial design, to establishthe functional relationship between the three operating variables ofgrinding (ball size, ball charge and solid content) and d80 of product(response) for copper sulphide grinding. These relationships can bythat means be used to determine the optimal operating parameters.Subsequently, the application of RSM and Box–Bhenken design tothe modeling of the influence of three operating variables on the ballmill performance for grinding of copper sulphide is discussed.

2. Materials and methods

2.1. RSM

RSM is a collection of statistical and mathematical techniques use-ful for the modeling and analysis of problems in which a response ofinterest is influenced by several variables and the objective is to opti-mize this response [22]. In most RSM problems, the form of the rela-tionship between dependent and the set of independent variables isunknown. Thus, the first step in RSM is to find a suitable approxima-tion for a functional relationship between dependent and indepen-dent variables. Second-order models are widely used in RSM as theyhave several advantages. They are very flexible and can take on awide variety of functional forms so they will work well as an approx-imation to the true response surface [22–24].

The design procedure of RSM is as follows [14,16,25]:

I. Designing of a series of experiments for adequate and reliablemeasurement of the response of interest.

II. Developing a mathematical model of the second-order responsesurface with the best fittings.

III. Finding the optimal set of experimental parameters that pro-duce a maximum or minimum value of response.

IV. Representing the direct and interactive effects of process param-eters through two and three-dimensional (3D) plots.

If all variables are assumed to be measurable, the response surfacecan be expressed as follows:

y ¼ f x1; x2; x3; ::::; xkð Þ ð1Þ

Where y is the answer of the system, xi (i = 1−k) is the variableof action called factor and k is the number of variables. The goal is tooptimize the response variable (y).

2.2. Box–Bhenken design

It is assumed that the independent variables are continuous andcontrollable by experiments with negligible errors. It is required tofind a suitable approximation for the true functional relationshipbetween independent variables and the response surface. These vari-ables were changed during the tests with respect to the Box–Bhenkenexperimental design, whereas the other operational parameters ofgrinding were kept constant (feed amount, feed size, grinding timeand mill speed).

Usually a second-order model is utilized in RSM (Eq. (2)) [9,10,25].

y ¼ β0 þXki¼1

βixi þXki¼1

βiix2i þ

Xk−1

i¼1

Xkj¼2

βijxixj þ ε ð2Þ

Where x1, x2, …, xk are the input factors which influence the re-sponse y; β0, βii (i = 1,2,…,k), βij (i = 1,2, …, k; j = 1,2, …, k) areunknown parameters and ε is a random error. The β coefficients,which should be determined in the second-order model, are obtainedby the least square method. In general, Eq. (2) can be written in ma-trix form as Eq. (3):

Y ¼ bX þ ε ð3Þ

Where Y is defined to be a matrix of measured values, X to be a ma-trix of independent variables. The matrixes b and ε consist of coeffi-cients and errors, respectively. The solution of Eq. (3) can be obtainedby the matrix approach.

b ¼ X′X� �−1

X′Y ð4Þ

Where X′ is the transpose of the matrix X and (X′X)−1 is the in-verse of the matrix X′X.

2.3. Materials and experimental procedure

For this study, materials (copper sulphide ore) were sampled fromthe SAGmill feed in Sungun copper concentrator plant. The density ofore was 2.8 g/cm3. The ore was crushed in a laboratory scale jawcrusher and roll crusher successively to prepare materials for ballmill grinding tests. Mineralogical composition of the ore sample wasalso analyzed using XRD and the results were presented in Table 1.

The size distribution of ore prepared for ball mill feed was shownin Fig. 1. As shown in Fig. 1, the d80 of sample (as ball mill feed) was480 μm. The measured Bond work index of ore was 16 that obtainedfrom Bond grindability test.

Batch grinding tests were carried out using a 25.8*20.8 cm(length*diameter) ball mill equipped with 4 lifters. The L/D ratio inboth laboratory ball mill and the plant ball mill was similar.

The maximum ball size calculated from Bond formula for plantball mill (Eq. (5)) [11] was 30 mm.

dB mmð Þ ¼ 25:4F80k

� �0:5 s:g:Wi

100Cs

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3:281D

p !0:33" #

ð5Þ

Where F80: feed size 80% passing (μm), s.g.: specific gravity of orefeed, Wi: feed ball mill work index (kWh/t), D: the inside diameter ofthe mill (m), Cs: critical speed (Cs) is 13.5 rpm, k: a constant desig-nated as the mill factor (350).

To apply an average size of 30 mm, ball size range was consideredto be in a range of 20–40 mm.

0

20

40

60

80

100

10 100 1000

Cum

ulat

ive

Pass

ing

%

Particle size (µm)

Fig. 1. Size distribution of the laboratory ball mill feed.

Table 3Experimental plan of grinding tests and results.

No. A (ball size) B (ball charge) C (solid content) d80(μm)

Coded Actual (mm) Coded Actual (%) Coded Actual (%)

1 1 40 1 40 0 72.5 109.72 0 30 0 30 0 72.5 102.53 −1 20 0 30 −1 65 804 −1 20 −1 20 0 72.5 1085 1 40 −1 20 0 72.5 172.26 0 30 0 30 0 72.5 1027 0 30 −1 20 −1 65 152.88 0 30 0 30 0 72.5 1039 1 40 0 30 1 80 168.810 0 30 1 40 −1 65 86.911 0 30 1 40 1 80 123.512 −1 20 0 30 1 80 114.213 0 30 0 30 0 72.5 102.214 0 30 −1 20 1 80 175.715 −1 20 1 40 0 72.5 55.116 1 40 0 30 −1 65 142.417 0 30 0 30 0 72.5 102

294 A. Ebadnejad et al. / Powder Technology 245 (2013) 292–296

The amount and the residence time of feed was set at 1175 g and20 min. respectively which were both proportional to industrialscale. The ball mill speed was constantly set at 76.8% Cs. Experimentswere conducted at different ball sizes (between 20 and 40 mm), ballcharges (20–40%) and pulp densities (65–80%). Response surfacemethod and Box–Bhenken design were used for the experimental de-sign and optimization of these variables. The variables and their levelsare presented in Table 2.

Table 4Experimental and predicted values of d80.

Test no. Experimental d80 (μm) Predicted d80 (μm)

1 109.7 108.92 102.5 102.33 80 79.64 108 108.35 172.2 172.16 102 102.37 152.8 152.3

3. Results and discussion

Seventeen grinding experimentswere designed using Box–Bhenkendesign methodology to provide data to determine the relationship be-tween the response (i.e. d80 of product) and the 3 process parameters.The experimental conditions and their responses are shown in Table 3.The results inserted to “Design Expert (DX)” software and a quadraticmodel was chosen and fitted to the results. A model was fitted to d80of product.

From the experimental parameters in Table 2 and experimentalresults in Table 3, the second-order response function representingd80 of product can be expressed as a function of the three coded pro-cess parameters. The quadratic model found to adequately predict theresponse variables is given by the following equation:

d80 ¼ 102:3þ 29:5A−29:2Bþ 15C−2:4AB−1:9AC þ 3:4BCþ 8:7B2 þ 23:7C2 ð6Þ

In this model all variables are in coded values as A is ball size, B isball charge and C is solid content. Also AB, AC, BC are interaction ofmain parameters. The responses at any regime in the interval of ourexperiment design could be calculated from Eq. (6).

The observed and the predicted values obtained using modelequation (Eq. (6)) are given in Table 4 and Fig. 2. As can be seenfrom Fig. 2, predicted values of responses match the observed valuesreasonably well, with R2 of 0.994 for d80.

Table 2The level of variables in the Box–Bhenken design.

Variable Symbol Coded variable level

Low (−1) Center (0) High (+1)

Ball size (mm) A 20 30 40Ball charge (%) B 20 30 40Solid content (%) C 65 72.5 80

3.1. Effect of variables on d80

The main effects of variables on response are presented in Fig. 3.These results were gained in the mean point of other parameters. Asit was indicated, the d80 of product was decreased with decreasingball size, which reflects an increasing effect of surface area. The rela-tion between ball size and d80 was completely linear over the studiedrange. The effect of ball charge on grinding was positive since increas-ing ball charge increased the impact and abrasionmechanisms. The ef-fect of ball charge on d80 at lower values of ball charge was comparedto that of higher values of ball charge at studied levels. A very small ef-fect of solid content on d80 was observed at values of 65–72.5%, how-ever d80 was considerably increased with increasing solid content atvalues out of that range.

In order to gain a better understanding of the interaction effects ofthese grinding variables on d80, the predicted model is illustrated inFigs. 4–6 as 3D response surface plots. Also the relationship betweenthe dependent and independent variables can be further understoodby these plots. Since the model has more than two factors, one factorwas held constant at center level for each plot; therefore, a total of 3response surface plots were produced.

The effect of ball size and ball charge on d80 of product is shown inFig. 4. Data shows that finer d80 is obtained at lower values of ballsize and greater values of ball charge. The relation between the effectof ball size and solid content on d80 of product is plotted in Fig. 5.

8 103 102.39 168.8 168.610 86.9 87.111 123.5 123.912 114.2 113.413 102.2 102.314 175.7 175.515 55.1 54.716 142.4 142.417 102 102.3

R² = 0.994

0

20

40

60

80

100

120

140

160

180

200

0 20 40 60 80 100 120 140 160 180 200

Pred

icte

d d8

0 (µ

m)

Experimental d80 (µm)

Fig. 2. Relation between experimental and predicted values of d80.

Fig. 4. Effect of ball size and ball charge on d80 in solid content of 72.5%.

295A. Ebadnejad et al. / Powder Technology 245 (2013) 292–296

Fig. 6 also helps to clarify the effect of ball charge and solid content ond80. The best results are obtained at the center of the solid contentlevel. Plot data also indicate that decreasing ball size and increasingball charge have the same effect on d80 value. As can be seen fromthese figures, it seems to be a little interaction effect of variables ond80 of product.

This study demonstrates that the RSM can be successfully used forthe determination of ball mill parameters on grinding of a copper sul-phide ore. Also it is an economical way of obtaining the maximumamount of information in a short period of time and with the fewestnumber of experiments. The comparison of this research with otherpapers that used RSM method [15–21] also showed that this method

Fig. 3. Main effects of va

can be effectively beneficiated in improving process parameters ofmineral processing systems.

4. Conclusion

In this research, modeling of wet ball mill grinding process, usingRSM and Box–Bhenken design was investigated. Ball size, ball chargeand solid content were the process control variables while the d80 ofproduct was considered as response. Experiments were conducted oncopper sulphide ore samples collected from SAG mill feed in Sungun

riables on response.

Fig. 5. Effect of ball charge and solid content on d80 in residence time of 20 min.

Fig. 6. Effect of residence time and solid content on d80 in ball charge of 30%.

296 A. Ebadnejad et al. / Powder Technology 245 (2013) 292–296

copper concentrator plant. A quadratic model for d80 of product wasdeveloped. The 3D response surface plots for variables were drawn todetermine the interaction effect of variables on d80 of product.

Predicted values of response obtained using model equation werein good agreement with the experimental values (R2 value for d80was 0.994). The results showed that the effects of applying lowerball size and greater values of ball charge on d80 are almost equaland their effects are more pronounced than solid content effect. Finerd80 was achieved using higher values of ball charges with smaller ballsizes. Also better results are obtained at the center of the solid contentlevel. A comparative study on modeling and experimental data fromthis study proves that Box–Behnken design and RSM, in conjunction,could efficiently be applied for the modeling of ball mills in coppersulphide ore grinding. Promising results also suggest that RSM can betested and developed for grinding modeling of other ores.

Acknowledgments

The authors would like to thank the National Iranian Copper Indus-tries Company (N.I.C.I.Co.) for supporting this research and permission

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to publish this article. Special thanks is also extended to the Sungunmetallurgy and R&D personnel for their continued assistance.

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