The aim of this research was to apply Box–Behnken experimental design and response surface methodology (RSM) for grinding modeling of some copper sulphide ores. In the present work, the effects of some key grinding parameters such as ball size (20–40mm), grinding time (10–30min), solids content (65–80%), and also ore work index (12–15.4kWh/t) on the grinding of some copper sulphide ore were investigated. Product 80% passing size (d80) was defined as process response.

Grinding experiments were designed and executed by a laboratory ball mill. These experiments were conducted on two feed sizes (480 and 1000μm). Predicted values of response obtained using model equation were in good agreement with the experimental values (R2 values were 0.997 and 0.996 for d80 of 480 and 1000μm feed sizes respectively).

The effect of parameters was explained as bellow:

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The d80 of product was decreased with increasing solids content from 65 to 71%.

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The relation between work index and also ball size with d80 were linear over the studied range.

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The effect of ball size and also grinding time on d80 was more at lower values of solids content compared to higher values of solids content at studied levels.

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The effect of grinding time on d80 was more at lower values of work index rather than higher values.

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There was no effect of feed size on the order of effective parameters but increasing feed size had effect on the interaction of variables.

Ball mills are usually the largest energy consumers within a mineral concentrator. Therefore, an efficient use has a great impact on performance and cost implications. Comminution is responsible for 50% of total mineral processing cost. In today's global markets suffering from the world crisis, mining groups are trying to optimize mill performances by mainly reducing production costs. Successful grinding with ball mills depends on the selection of suitable operating conditions. Therefore, it is important to determine the operating parameters at which the response reaches its optimum. The optimum could be either a maximum or a minimum of a function of the design parameters [1].

In the literature, there are many studies on the operating parameters affecting the grinding performance of ball mills [2–4]. These parameters are categorized as operating variables (such as: ball diameter, charge ratio, mill speed, pulp density, grinding time), which should be optimized to achieve the desired product size with minimum energy consumption. Among the available variables for improving ball mill efficiency, the pulp density and grinding media size are probably the most frequently considered factors for process optimization [3]. The efficiency of grinding depends on the surface area of the grinding medium. Thus, balls should be as small as possible and the charge should be graded in such manner that the largest balls are just heavy enough to grind the largest and hardest particles in the feed. Harder ores and coarser feeds require high impact and large media. Very fine grind sizes require substantial media surface area and small media [5].

The solids content of the pulp is a very important factor in wet grinding systems due to its direct influence on he ground product size. The fine particles present/produced during grinding cause high pulp viscosity and reduce the grinding performance.

The pulp density of the feed should be as high as possible, consistent with ease of flow through the mill. It is essential that the balls are coated with a layer of ore; too dilute a pulp increases metal-to-metal contact, leading to increased steel consumption and reduced efficiency. Ball mills should operate between 65 and 80% solids by weight, depending on the ore [6]. Additionally pulp density certainly influences the distribution of the energy of impacts applied to the particles in a grinding mill [7].

The general practice for determining the important process parameters for grinding is by varying one parameter and keeping the others at a constant level. This is the one-variable-at-a-time technique. The major disadvantage of this technique is that it does not include interactive effects among the variables and, eventually, it does not depict the complete effects of various parameters on the process [8]. A far more effective method is to apply a systematic approach to experimentation, one that considers all factors simultaneously. That approach is called design of experiments (DOE). DOE provides information about the interaction of factors and the way the total system works, something not obtainable through testing one factor at a time while holding other factors constant.

It is essential that experimental design methodology is a very economical way for extracting the maximum amount of complex information, a significant experimental time saving factor and moreover, it saves the material used for analyses and personal costs as well [9].

Recently different methods of DOE have been specifically applied for modeling of process parameters in mineral processing systems [10–16]. Factorial experimental design has been used to investigate the effects of some key hydrodynamic factors on the performance of the flotation of coarse coal particles [10]. Central composite design has been successfully used to modeling of a multi-gravity separator for chromite concentration [11]. Response surface methodology (RSM) has been employed for modeling of some processes such as Turkish coal processing [12], flotation of synthetic mixture of celestite and calcite minerals [14], flotation of celestite concentrate [15] and sulphur grindability in a batch ball mill [16].

The main purpose of grinding is treating the particles for extraction process by reducing their size. Reduction ratio is a determining factor in mill efficiency evaluation, which can show how efficiently the energy is consumed. A higher reduction ratio can signal a more efficient milling in progress. The 80% passing size (D80) of product has been considered for the response of different grinding processes [12,16,17]. However, d80 of product was considered to be closely monitored for evaluating the grinding process in this research.

Therefore, the main objective of this research was first to establish a functional relationship between three grinding variables (ball size, grinding time and solids content) and ore work index with the grinding response (d80 of product), using a statistical technique. Box–Behnken design was used to determine significant factors that affect the grinding of some copper sulphide ores and to develop quadratic mathematical model for the optimization of the process. The second purpose of this research was to investigate the effect of increasing feed size on the grinding modeling process of some copper sulphide ores.

2Materials and methods2.1RSM and Box–Behnken designRSM is a collection of statistical and mathematical methods that are useful for the modeling and analyzing engineering problems. In this technique, the main objective is to optimize the response surface that is influenced by various process parameters. RSM also quantifies the relationship between the controllable input parameters and the obtained response surfaces [1].

The design procedure for RSM is as follows [18,19]:

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Performing a series of experiments for adequate and reliable measurement of the response of interest.

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Developing a mathematical model of the second-order response surface with the best fit.

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Determining the optimal set of experimental parameters that produce a maximum or minimum value of response.

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

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

where y is the answer of the system, xi (i=1–k) is the variable of action called factor and k is the number of variables.The goal is to optimize the response variable (y). An important assumption is that the independent variables are continuous and controllable by experiments with negligible errors. The task then is to find a suitable approximation for the true functional relationship between independent variables and the response surface [18].

Box–Behnken factorial design was chosen to find out the relationship between the response function (d80) and four variables (ball size, grinding time, solids content and ore work index) on copper sulphide ore grinding. These variables were changed during the tests with respect to the Box–Behnken experimental design, whereas the other operational parameters of grinding were kept constant (feed amount, ball charge, mill speed).

Usually a second-order model (Eq. (2)) is utilized in response surface methodology [1,9].

where x1, x2, …, xk are the input factors which influence the response y; β0, βi, βii (i=1, 2, …, k), βij (i=1, 2, …, k; j=1, 2, …, k) are constant term, coefficients of the linear parameters, coefficients of the quadratic parameters and coefficients of the interaction parameters respectively. The ?? is a residual associated to the experiments. The β coefficients, which should be determined in the second-order model, are obtained by the least square method [20].2.2Materials and experimental procedureFor this study, materials (copper sulphide ore) were sampled from the feed in Sungun copper concentrator plant. Three different samples with different hardness were selected for the experiments. The ore was crushed in a laboratory scale jaw crusher and roll crusher successively. The d80 of ore reached to 480μm in all samples to prepare materials for ball mill grinding experiments, because d80 of plant ball mill feed in Sungun plant was 480μm.

As a base for this study, standard Bond grindability tests were initially done and Bond work index (Wi) values were calculated for samples from the Eq. (3)[21–23]:

where Wi is the work index of ore (kWh/t); Pi, test sieve size at which the test is performed; Gbg, Bond's standard ball mill grindability, net weight of ball mill product passing sieve size Pi produced per mill revolution (g/rev.); P80, sieve opening at which 80% of the product passes (μm); F80, sieve opening at which 80% of the feed passes (μm). The calculated Bond work indexes of samples were 12, 13.7 and 15.4 obtained from Bond grindability test.Mineralogical composition of the ore samples were also analyzed using XRD and the results were presented in Table 1.

Mineralogical composition of the feed ore samples using XRD.

Component | Weight % | ||
---|---|---|---|

Wi=12 | Wi=13.7 | Wi=15.4 | |

Cu2S | 0.387 | 0.391 | 0.404 |

CuS | 0.280 | 0.284 | 0.281 |

CuFeS2 | 0.427 | 0.431 | 0.445 |

FeS2 | 6.514 | 6.628 | 6.620 |

MoS2 | 0.039 | 0.037 | 0.037 |

Fe2O3–Fe(OH)2 | 0.145 | 0.140 | 0.140 |

Fe2O3 | 0.177 | 0.189 | 0.196 |

Fe3O4 | 0.102 | 0.105 | 0.114 |

SiO2 | 64.96 | 67.27 | 71.27 |

Al2O3 | 13.49 | 14.18 | 14.52 |

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

The maximum ball size for plant ball mill calculated from Bond formula (Eq. (4)) [24]:

where F80: feed size 80% passing (μm), s.g.: specific gravity of ore feed, Wi: feed ball mill work index (kWh/t), D: the inside diameter of the mill (m), Cs: critical speed (Cs) is 13.5rpm, k: a constant designated as the mill factor (350). The maximum ball size was calculated 30mm.To apply an average size of 30mm, ball size range was considered to be in a range of 20–40mm.

The amount of feed was set at 1175g, which was proportional to industrial scale. The ball mill speed was constantly set at 76.8%Cs. Experiments were conducted at different ball sizes (between 20 and 40mm), grinding time (10–30min), solids content (65–80%) and work index of copper sulphide ore (12–15.4). RSM and Box–Behnken design were used for the experimental design and modeling of these variables. The variables and their levels were presented in Table 2.

These experiments were conducted for the same 3 sample with providing d80 of 1000μm to show the effect of increasing feed size on the modeling process. The 1000μm feed size of same copper samples were obtained by grinding ores in laboratory jaw and roll crusher until the d80 of product reached 1000μm. Therefore, six ore samples were treated for grinding experiments that provided at Table 3. Also, the size distribution of 6 samples prepared for ball mill feed were shown in Fig. 1.

Using the Box–Behnken experimental design method, 29 sets of tests with appropriate combinations of ball diameter (A), grinding time (B), solids content (C) and ore work index (D) were conducted. Box–Behnken design with actual values and results was given in Table 4. Each run was performed in duplicate and thus the values of d80 given in Table 4 were the mean of two experiments, while the predicted values of response (d80) were obtained from quadratic model equations using the Design Expert 8 software.

Box–Behnken design with actual values of the grinding variables and results for two feed sizes.

Run no. | Variables | Experimental d80 (μm) | ||||
---|---|---|---|---|---|---|

A (ball size) | B (grinding time) | C (solids content) | D (work index) | F80=480 | F80=1000 | |

1 | 40 | 20 | 72.5 | 12 | 111 | 147 |

2 | 30 | 20 | 65 | 12 | 94 | 121 |

3 | 30 | 30 | 72.5 | 15.4 | 90 | 116 |

4 | 30 | 20 | 72.5 | 13.7 | 102 | 137 |

5 | 20 | 30 | 72.5 | 13.7 | 53 | 63 |

6 | 40 | 10 | 72.5 | 13.7 | 176 | 254 |

7 | 40 | 30 | 72.5 | 13.7 | 107 | 141 |

8 | 30 | 10 | 65 | 13.7 | 164 | 233 |

9 | 30 | 20 | 80 | 12 | 127 | 172 |

10 | 20 | 20 | 72.5 | 15.4 | 89 | 115 |

11 | 30 | 10 | 72.5 | 12 | 133 | 182 |

12 | 20 | 20 | 80 | 13.7 | 121 | 163 |

13 | 30 | 30 | 65 | 13.7 | 81 | 104 |

14 | 30 | 20 | 72.5 | 13.7 | 103 | 136 |

15 | 30 | 20 | 72.5 | 13.7 | 101 | 133 |

16 | 30 | 20 | 72.5 | 13.7 | 103 | 135 |

17 | 40 | 20 | 72.5 | 15.4 | 146 | 204 |

18 | 20 | 10 | 72.5 | 13.7 | 128 | 175 |

19 | 20 | 20 | 65 | 13.7 | 79 | 101 |

20 | 40 | 20 | 80 | 13.7 | 169 | 242 |

21 | 30 | 30 | 72.5 | 12 | 65 | 79 |

22 | 40 | 20 | 65 | 13.7 | 142 | 198 |

23 | 30 | 20 | 72.5 | 13.7 | 100 | 132 |

24 | 30 | 10 | 72.5 | 15.4 | 170 | 245 |

25 | 30 | 30 | 80 | 13.7 | 125 | 170 |

26 | 30 | 10 | 80 | 13.7 | 195 | 288 |

27 | 30 | 20 | 65 | 15.4 | 126 | 171 |

28 | 20 | 20 | 72.5 | 12 | 60 | 76 |

29 | 30 | 20 | 80 | 15.4 | 165 | 235 |

The results inserted to DX8 software and a quadratic model among several models were chosen and fitted to the results.

From the experimental parameters in Table 2 and experimental results in Table 4, the second-order response function representing d80 of product can be expressed as a function of the three coded process parameters. The quadratic model found to adequately predict the response variables was given by the following equations:

Model equation for f80 of 480μm: (model 1)

Model equation for f80 of 1000μm: (model 2)

In these models, all variables are in coded values and A is ball size, B is grinding time, C is solids content and D is work index of ore. Also, AC, AD, BC and BD are interaction of the main parameters. The response at any regime in the interval of this experiment design could be calculated from Eqs. (5) and (6) for f80 of 480 and 1000μm respectively.

The correlation between the observed and predicted results using above mentioned models were shown in Table 5 and Figs. 2 and 3, respectively. Values of R2 were 0.997 and 0.996 for the model 1 and model 2, respectively. The high value of R2 indicates that the quadratic equation is capable of representing the system under the given experimental domain. It can be seen that there was a good agreement between predicted and actual values.

Experimental and predicted values of d80 for two feed sizes.

Run no. | F80=480μm | F80=1000μm | ||
---|---|---|---|---|

Experimental d80 (μm) | Predicted d80 (μm) | Experimental d80 (μm) | Predicted d80 (μm) | |

1 | 111 | 112 | 147 | 145 |

2 | 94 | 93 | 121 | 122 |

3 | 90 | 92 | 116 | 118 |

4 | 102 | 102 | 137 | 134 |

5 | 53 | 51 | 63 | 58 |

6 | 176 | 179 | 254 | 257 |

7 | 107 | 105 | 141 | 139 |

8 | 164 | 163 | 233 | 228 |

9 | 127 | 129 | 172 | 178 |

10 | 89 | 91 | 115 | 114 |

11 | 133 | 133 | 182 | 184 |

12 | 121 | 123 | 163 | 168 |

13 | 81 | 83 | 104 | 111 |

14 | 103 | 102 | 136 | 135 |

15 | 101 | 102 | 133 | 135 |

16 | 103 | 102 | 135 | 135 |

17 | 146 | 144 | 204 | 206 |

18 | 128 | 125 | 175 | 175 |

19 | 79 | 80 | 101 | 102 |

20 | 169 | 168 | 242 | 240 |

21 | 65 | 65 | 79 | 80 |

22 | 142 | 140 | 198 | 193 |

23 | 100 | 102 | 132 | 135 |

24 | 170 | 172 | 245 | 248 |

25 | 125 | 126 | 170 | 168 |

26 | 195 | 193 | 288 | 286 |

27 | 126 | 126 | 171 | 173 |

28 | 60 | 59 | 76 | 72 |

29 | 165 | 162 | 235 | 230 |

The results of analysis of variance (ANOVA) for the model 1 were shown in Table 6. The model F-value of 847.97 implies the model is significant. There is only a 0.01% chance that a “model F-value” this large could occur due to noise. Values of “Prob>F” less than 0.0500 indicate model terms are significant. Then it illustrates that the fitted model is significant in 95% confidence level (p-value <0.05). In this case A, B, C, D, AC, BC, BD, B2, C2 are significant model terms for model 1.

The results of ANOVA analysis of the developed model 1.

Source d80 | Sum of squares | Degree of freedom | Mean square | F-value | p-value | |
---|---|---|---|---|---|---|

Model | 37,777.75 | 14 | 2698.41 | 847.97 | <0.0001 | Significant |

A-ball size | 8570.71 | 1 | 8570.71 | 2693.33 | <0.0001 | |

B-grinding time | 16,502.08 | 1 | 16,502.08 | 5185.74 | <0.0001 | |

C-solid content | 3909.63 | 1 | 3909.63 | 1228.59 | <0.0001 | |

D-work index | 3230.80 | 1 | 3230.80 | 1015.27 | <0.0001 | |

AB | 9.00 | 1 | 9.00 | 2.83 | 0.1148 | |

AC | 56.25 | 1 | 56.25 | 17.68 | 0.0009 | |

AD | 9.92 | 1 | 9.92 | 3.12 | 0.0992 | |

BC | 42.25 | 1 | 42.25 | 13.28 | 0.0027 | |

BD | 36.00 | 1 | 36.00 | 11.31 | 0.0046 | |

CD | 7.29 | 1 | 7.29 | 2.29 | 0.1524 | |

A2 | 0.33 | 1 | 0.33 | 0.10 | 0.7528 | |

B2 | 1184.35 | 1 | 1184.35 | 372.18 | <0.0001 | |

C2 | 4405.97 | 1 | 4405.97 | 1384.57 | <0.0001 | |

D2 | 1.46 | 1 | 1.46 | 0.46 | 0.5087 | |

Residual | 44.55 | 14 | 3.18 | |||

Lack of fit | 37.75 | 10 | 3.78 | 2.22 | 0.2297 | Not significant |

Pure error | 6.80 | 4 | 1.70 |

The results of ANOVA for the model 2 were shown in Table 7. It illustrates that the fitted model is significant in 95% confidence level (p-value <0.05). The model F-value of 457.10 implies the significance of the model 2. In this model, A, B, C, D, AC, AD, BD, B2, C2 are significant model terms.

The results of ANOVA analysis of the developed model 2.

Source – 37μm fraction | Sum of squares | Degree of freedom | Mean square | F-value | p-value | |
---|---|---|---|---|---|---|

Model | 93,301.75 | 14 | 6664.41 | 457.10 | <0.0001 | Significant |

A-ball size | 20,254.08 | 1 | 20,254.08 | 1389.19 | <0.0001 | |

B-grinding time | 41,301.33 | 1 | 41,301.33 | 2832.79 | <0.0001 | |

C-solid content | 9747.00 | 1 | 9747.00 | 668.53 | <0.0001 | |

D-work index | 7956.75 | 1 | 7956.75 | 545.74 | <0.0001 | |

AB | 0.25 | 1 | 0.25 | 0.017 | 0.8977 | |

AC | 81.00 | 1 | 81.00 | 5.56 | 0.0335 | |

AD | 81.00 | 1 | 81.00 | 5.56 | 0.0335 | |

BC | 30.25 | 1 | 30.25 | 2.07 | 0.1717 | |

BD | 169.00 | 1 | 169.00 | 11.59 | 0.0043 | |

CD | 42.25 | 1 | 42.25 | 2.90 | 0.1108 | |

A2 | 9.60 | 1 | 9.60 | 0.66 | 0.4306 | |

B2 | 3310.60 | 1 | 3310.60 | 227.07 | <0.0001 | |

C2 | 10,952.60 | 1 | 10,952.60 | 751.22 | <0.0001 | |

D2 | 3.98 | 1 | 3.98 | 0.27 | 0.6095 | |

Residual | 204.12 | 14 | 14.58 | |||

Lack of fit | 190.92 | 10 | 19.09 | 5.79 | 0.0527 | Not significant |

Pure error | 13.20 | 4 | 3.30 |

The main effects of variables on response for two feed sizes were presented in Figs. 4 and 5 respectively. These plots help to compare the effect of all the factors at a particular point in the design space. The response is plotted by changing only one factor over its range while holding all the other factors constant. These results were gained in the mean point of other variables. A steep slope or curvature in a factor shows that the response is sensitive to that factor. A relatively flat line shows insensitivity to change in that particular factor. If there are more than two factors, this plot could be used to find those factors that most affect the response. This plot does not show the effects of interactions.

As it can be seen from Figs. 4 and 5, in two sets of experiments, the most important factor for grinding was grinding time (B). The effects of other variables follow the order: ball size (A)>solids content (C)>work index of ore (D).

As it was indicated, the relation between ball size and d80 was completely linear over the studied range. The d80 of product was decreased with decreasing ball size, which reflects an increasing effect of surface area.

The effect of grinding time on d80 at lower values was more compared to that of higher values of grinding time at studied levels. At lower values of time the particles was more coarser compared to higher values of grinding time. It is indicated that the effect of time on d80 at lower values was high.

A very small effect of solids content on d80 was observed at values of 65–72.5%. The d80 of product was decreased with increasing solids content from 65 to 71%, but d80 was considerably increased with increasing solids content at values more than 71%.

Also, the relation between work index and d80 was linear over the studied range. It was obvious that the d80 of product was decreased with decreasing Wi of ore.

3.3Interaction effect of variablesIn order to gain a better understanding of the interaction effects of these grinding variables on d80, the predicted models were illustrated in Figs. 6–11 as contour plots for 480 and 1000μm feed sizes. Also, the relationship between the dependent and independent variables can be further understood by these plots. These results were gained in the mean point of other parameters. However, according to models interactions between variables have significant effects on responses therefore results were presented and discussed in terms of interactions.

The interaction effects of ball size and solids content on d80 of product were shown in Figs. 6 and 7 for f80 of 480 and 1000μm feed sizes, respectively. The effect of ball size on d80 was differed in different levels of solids content. The effect of solids content on d80 was almost constant at values of 68–71% of solids content. The effect of ball size on d80 was more at lower values of solids content compared to higher values of solids content at studied levels. These results were obtained from the slopes of curves at different points of ball size and solids content shown at Figs. 6 and 7. Data showed that finer d80 was obtained at lower values of ball size and middle values of solids content (almost solids content of 70%). Also, the interaction effect of ball size and solids content for both feed sizes were almost the same. It means that increasing feed size of ore had no considerably effect on the interaction effect of ball size and solids content.

The combined effect of grinding time and solids content was shown in Fig. 8 for f80 of 480μm feed size. The effect of grinding time on d80 at lower values was higher. Grinding time was more effective on d80 at lower values of solids content because abrasion of particles was increased at lower values of solids content. The maximum effects of ball size and also grinding time on d80 was obtained at middle values of solids content (almost 70%).

Therefore, the middle values of solids content resulted effective grinding of particles. The abrasion of balls was increased at lower values of solids content. Also, the detachment of particles among the balls was decreased at higher values of solids content so the grinding of particles was decreased.

There is no interaction between grinding time and solids content at 1000μm feed size.

Figs. 9 and 10 show the contour plots of the combined effects of grinding time and work index for the first and second series of experiments, respectively. It was obvious that d80 was decreased with increasing grinding time and decreasing work index, but the effect of grinding time on d80 was more at lower values of work index rather than higher values. Also, the feed size of ore had no effect on the interaction effect of grinding time and work index at desired levels.

3.4Effect of feed size on grinding modelThere is no effect of feed size on the order of effective parameters but increasing feed size had effect on the interaction of variables. For the feed size of 480μm the interaction between AC had more effect from other interaction effects of parameters but at 1000μm feed size the interaction effect of BD was higher.

The order of interaction parameters for 480μm feed size was: interaction between ball size and solids content (AC)>interaction between grinding time and solids content (BC)>interaction between grinding time and work index (BD).

The interaction parameters for f80 of 1000μm follow the order: interaction between grinding time and work index (BD)>interaction between ball size and solids content (AC)>interaction between ball size and work index (AD).

Also, the interaction between grinding time and solids content was not seen at 1000μm feed size but increasing feed size could make an interaction between ball size and work index (Fig. 11).

This study demonstrates that the RSM can be successfully used for the determination of ball mill parameters on grinding of some copper sulphide ores. Also, it is an economical way of obtaining the maximum amount of information in a short period of time and with the fewest number of experiments.

4ConclusionThe main objective of this research was first to establish a functional relationship between three grinding variables (ball size, grinding time and solids content) and ore work index with the grinding response (d80 of product), using a statistical technique. Box–Behnken design was used to determine significant factors that affect the grinding of some copper sulphide ores. The second purpose of this research was to investigate the effect of increasing feed size on the grinding modeling of some copper sulphide ores.

Grinding experiments were designed and executed by a laboratory ball mill and the DX8 software was used to analyze the results of the experiments. These experiments were conducted on 2 feed sizes (480 and 1000μm).

Predicted values of response obtained using model equation were in good agreement with the experimental values (R2 values were 0.997 and 0.996 for d80 of 480 and 1000μm feed sizes respectively).

According to DX8 software, it was concluded that the effects of the studied parameters follow the order: grinding time (B)>ball size (A)>solids content (C)>work index of ore (D).

The effect of parameters was explained as below:

- -
The effect of grinding time on d80 at lower values was higher compared to that of higher values of grinding time at studied levels.

- -
The d80 of product was decreased with increasing solids content from 65 to 71%, but d80 was considerably increased with increasing solids content at values more than 71%.

- -
The d80 of product was decreased with decreasing ball size, which reflects an increasing effect of surface area.

- -
The relation between work index and also ball size with d80 were linear over the studied range.

- -
The effect of ball size on d80 was higher at lower values of solids content compared to higher levels of solids content at studied levels.

- -
Grinding time is more effective on d80 at lower values of solids content. The maximum effects of ball size and also grinding time on d80 were obtained at middle values of solids content (almost 70%).

- -
It was obvious that d80 was decreased with increasing grinding time and decreasing work index, but the effect of grinding time on d80 was higher at lower values of work index rather than higher values.

- -
There was no effect of feed size on the order of effective parameters but increasing feed size had effect on the interaction of variables. For feed size of 480μm the interaction between AC had more effect from other interaction effect of parameters but at 1000μm feed size the interaction effect of BD was more.

Results suggested that Box–Behnken design and RSM could be efficiently applied for modeling of ball milling system of some copper sulphide ores.

Conflicts of interestThe author declares no conflicts of interest.

The author would like to thank National Iranian Copper Industries Company (N.I.C.I.Co.) for supporting this research. Special thanks is also extended to the Sungun metallurgy and R&D personnel for their continued assistance.