Experimental Study of Operational Parameters on Product Size Distribution of Tumbling Mill

: To assess the effects of the mill operating parameters such as mill speed, ball filling, slurry concentration and slurry filling on grinding process and size distribution of mill product, it was endeavored to build a pilot model with smaller size than the mill. For this aim, a pilot mill with 1m × 0.5m was implemented. There are 15 lifters with 50mm height and face angle of 30˚. In the present work, the combination of the balls (40% of the balls with 60mm diameter, 40% of the balls with 40mm diameter and 20% of the balls with 25mm diameter) was used as grinding media with 10%, 15%, 20% and 25% of the total volume of the mill. The experiments were carried out at 60%, 70%, 80% and 90% of the critical speed. The feed of the mill is copper ore with the size smaller than 25.4 mm, which d 80 and d 50 of them are 12.7 and 8 mm, respectively and slurries with 40%, 50%, 60%, 70% and 80% of solid and the slurry filling between 0.5 and 2.5. The results showed that the best grinding and grading occurs at 70-80% of the critical speed and ball filling of 20-25%. Optimized grinding was observed when the slurry volume is 1-1.5 times of the ball bed voidage volume and the slurry concentration is between 60% and 70%. The mill grinding mechanism in this work is a combination of both impact and abrasion mechanisms.

fine domain is usually the costliest part of the production of mineral processing plants and mining industries due to high energy consumption [1][2].
A number of investigators have studied the effect and the role of slurry density on the grinding behavior of ball mills [3][4]. To control slurry rheology and therefore the grinding rate, Klimpel and Austin [5][6][7] have employed a viscosity modifying agents. Tangsathitkulchai [8] analyzed the influence of slurry density on the breakage parameters of quartz in a laboratory ball mill. He also investigated the role of slurry density on the dynamics and behavior of wet grinding in a tumbling ball mill [9].
It is generally observed that dry and wet grinding of materials in tumbling ball mills to very fine sizes can lead to the slowing down of the overall grinding process [10][11][12]. Austin and Bagga [13] postulated that the slowing down of grinding rates observed in the dry systems could result from the inefficiency of particle capture by grinding media, caused by the ability of the cohesive fine particles to flow away from the ball collision zone. In this article, the mills grinding mechanism is a combination of impact and abrasion breakage mechanisms. But in previous works abrasion breakage is major mechanism in grinding.

2-Grinding Model of the Mill
For the product size distributions in a ball mill, the well-known batch grinding kinetic model proposed by Austin [10,14] is used. The following equation represents the discrete form of the model: where t is grinding time, n is the number of size intervals, w i is weight fraction of particles of size i, S i is a constant termed the specific rate of breakage of size i or selection function of size i and b ij is breakage function (weight fraction of particles of size i due to breakage from size j). The size distribution caused by a stone breakage from an impact is expressed via breakage function [10]. The selection function is one of the main parameters in the modeling of the ball mills. The selection function is an index of material milling process that depends on different elements such as ores properties, mill diameter, mill rotational speed, size and material of the balls [14]. In a continuum system like the mill, all materials are not subjected to the grinding at the same period, Mean Residence Time (MRT) of the particles is obtained by dividing of materials volume inside the mill (V) into the rate of flow charge (Q) as follows [1]: In the continuum system, some particles have a residence time less than τ, and some others have more.
To determine the residence time, the value of PH is measured at the inlet and outlet of the mill at various time [1]. Fig. 1 exhibits Residence Time Distribution (RTD) in a ball mill. There are two main breakage mechanisms in a SAG mill: the impact breakage (high energy) and the abrasion (low energy). The JKMRC institution uses t 10 is the percentage of breakage product that passes 1/10th of the initial particle size. Eq. 3 correlates the breakage index, t 10 , to the specific input energy. E cs is the specific breakage energy (kWh/t) as calculated from the input energy of the falling weight and the average weight of the impacted particles [15].
where A is the maximum value of t 10 , the highest level of size reduction from a single impact event, typically varying from 35% to 70% and b is the slope of the curve. It is noted that A and b are dependent parameters which they are responsible for the ore stiffness in the self-breakage mechanism. In fact, the multiplication of the aforementioned parameters is the slope of curve E cs -t 10 which is a value of ore breakage at low energy level of the SAG mill. A high value of A×b means that the rock has a low resistance to impact breakage and vice versa. The parameters of impact breakage are determined using a high energy impact breakage apparatus which is called drop weight tester. The specific breakage energy is calculated from the following relation [15]: is the medium mass of per particle size class and h(cm) is the height of fall of drop weight. The JKMRC has defined the following equation for the abrasion breakage mechanism (t a ) [15]: The parameter of abrasion mechanism is obtained by the test of ore abrasion exploiting a rotating experimental mill of dimensions 300mm×300mm with four lifting bars of 10mm length. It is noted that this experiment differs from the Bond work index experiment. Napier Munn et al. [15] have obtained the following equations between the impact parameters (A×b) and abrasion breakage parameters (t a ) with work index of Bond (W i ): It is important to note that a few studies have been carried out for the assessment of grinding in tumbling mills and available data in this area are rather limited. However, some articles report results on the kinetic analysis and a mechanism underlying the slowing down of breakage rates in fine wet grinding in a batch laboratory ball mill [16][17][18] and abrasion breakage is major mechanism in that works. The numerical simulation of pulp flow investigated in the SAG mills [19], but the influences of operational parameters on the grinding of SAG mill have not been observed. In this study, the optimization of operational parameters of the SAG mill to decrease the size of final product or to increase the mill capacity regarding to downstream floatation has been carried out by an experimental pilot mill. The mill grinding mechanism in this work is a combination of both impact and abrasion mechanisms.

3-Experimental
Experimental model of the mill is illustrated in Fig. 2. In the rig, there are 15 lifters with 50mm height and face angle of 30˚. In the present work, the combination of the balls (40% of the balls with 60mm diameter, 40% of the balls with 40mm diameter and 20% of the balls with 25mm diameter) was used as grinding media with 10%, 15%, 20% and 25% of the total volume of the mill. The rotational speed of the mill motor can be adjusted continually to 100% of critical speed. The applied speeds of this study were 60%, 70%, 80% and 90% of the critical speed. In Table 1, the conditions of experiments are listed.  The feed of the mill is copper ore with the size smaller than 25.4 mm, which 80 and 50 of them are 12.7 and 8 mm, respectively. The slurry concentration used in the tests was 40%, 50%, 60%, 70% and 80%. Moreover, to investigate the influence of slurry filling (U) on the grading, the amount of slurry is increased till the slurry volume becomes 2.5 times of the balls bed voidage (the balls volume disregarding their voids). The concept of slurry filling in the mills was first introduced by Austin et al. [10]. The procedure of the experiments is such that for the investigation of four parameters; the mill speed, the ball filling, slurry concentration and slurry filling, three parameters are kept constant and the fourth parameters is varied. For each experimental condition, the mill is allowed to rotate for a set period of time, and then a specimen is taken from the product. After filter pressing, the specimens are put in the electric oven. After the cakes are dried, they are weighted. The weighted materials were then wet screened on a 325 mesh (44μm) sieve to reduce screen blinding, after which they are dried and weighted again to calculate the mass of particles smaller than 44μm. Next, the specimens are put in the shaker and screened with different sieve fractions 25400-44μm and the product particle size distribution is calculated. The sieve analysis of the product was computed based on the recorded weights of material retained on each screen and in the bottom pan.
In a pilot mill with 1m diameter, in cascading motion, the ball velocity rapidly approaches to 4m/s and it impacts at high speed in the toe region [20]. The kinetic energy of the balls with 60mm diameter and 0.88kg mass is approximately 7J. The energy of balls to grinding the copper ore feed with dimensions less than 1 inch and the average hardness is enough [21].
On experiment has been carried out to investigate whether the mill has been well designed in terms of operational parameters such as size and volume of the balls, number, height and face angle of the lifters, speed, etc or not. Therefore, 20% of the mill volume was filled with the balls. Besides, the slurry concentration is 60% and the slurry having the same volume of the balls was poured into the mill (U=1).
Then, the mill worked for 10min with 75% of the critical speed.
In Fig. 3, the sizing analysis of product from the mill grinding mechanism is depicted. As seen from the figure, the mill grinding mechanism is a combination of impact and abrade breakage mechanisms. If there is just abrade mechanism, the materials remain in their initial size or a bit smaller than one, and a huge amount of small particles are produced. However, in impact mechanism, the amounts of very large and very small particles are small. Obviously from Fig. 3, there is no particle with large or medium size in the mill product while most of the particles are tiny, consequently, it can be stated that the mill grinding mechanism is a combination of both impact and abrasion mechanisms.
The results of 27 different experiments for the assessment of the effects of the mill operational parameters such as mill speed (Φ c ), ball filling (J b ), slurry concentration (C) and slurry filling (U) on the size distributions of product are presented in the next sections. All experiments are repeated (totally 54 tests) and 2700kg of the feeding charges and 1800lit of water (totally 4.5 tons) were used in all of the tests.

4-1 The mill speed effects
In Fig. 4, the size distribution of the mill product has been illustrated at different speeds. To investigate the influence of speed on the grinding, the balls with 20% occupation of the mill volume (approximately 352kg) and the slurry with 60% of solid (approximately 71kg) were used. Also, the slurry volume and volume of the balls bed voidage were the same (U=1). Table 2 shows 80% passing size, P 80 , of the product at different speeds. Regarding to Fig. 4 and Table 2, it can be stated that the best grinding occurs at the speeds range between 70-80% of the critical speed.

Fig. 4. Product size distributions at different mill speeds
To have better understanding of the product grinding in terms of the mill speed, the percentage of produced particles smaller than 44μm has been depicted in Fig. 5. As seen, initially with the increase of speed, the amount of particles smaller than 44μm increases, then after approaching to the peak point, the amount of such particles decreases sharply. The pick point relies at the speeds range from 70% to 80% of the critical speed depending on the slurry filling. Moreover, before the maximum point of the curve, due to the speed increase and its influence on the variation of cascade path and also due to the impact intensity, the grinding caused by the impact increases. When the mill speed increases, the shoulder of load move up, and the falling height of particles are increased in the cascading motion and as a result, the materials impact to the toe region with more speed and more energy that leads to the increase of impact grinding. At the speeds larger than 80% of the critical speed, the cascade path is changed due to the lifting of shoulder angle and moves toward the end of the toe. Therefore, the falling height, impact speed and impact force are lowered. Consequently, the particles and the balls have direct impact to the lifters, but due to the reduction of the speed and pressure, the grinding caused by the abrasion is lowered. Moreover, the centrifugal forces are augmented due to the speed increase. As a result, the relative velocity between the materials and the balls increase which causes the reduction of the abraded grinding rate. When the speed increases, the shoulder angle is transfered to the top and the toe angle remains almost constant. As a result, the sliding region extends and takes thin and the abraded product decreases.

4-2 The ball filling effects
The size distribution of the mill products under different ball filling is depicted in Fig. 6. To study the influence of the balls percentage on the grinding, the slurry with 60% solid was used while the mill speed was kept constant at 75% of the critical speed. Besides, for each experimental condition, the slurry volume and the balls bed voidage volume were the same (U=1). As observed from Fig. 6, the grinding rate increases initially, then it decreases. Also, under 20% of the charge, the grinding rate is maximum.
The rise of grinding rate is due to the increase of impacts before the peak point. It is however important that when the charge volume is small and the mill speed is high, the balls land upper than load toe and strike the lifters directly which causes the rapid failure and wear of them. Moreover, with the rise of charge and transmission of the load toe to the top, direct impact between the balls and the mill shell reduces and the balls land on the toe region, consequently, the impact grinding increases. Furthermore, after the peak point, the grinding rate reduces with the increase of ball filling. With the increase of ball charge, the shoulder angle remains almost constant while the toe height increases, thus, the balls have short falling height and in turn the grinding due to the impact is reduced. In addition, with the increase of the load weight and the increase of pressure on the sliding region, the displacement and slide of the materials are lowered and thus the grinding rate due to abrasion is decreased. Addition of the balls charge from 10% to 25% requires that the powder charge should rise to keep the value of U constant. Thus, the slurry weight should be increased from 35kg to 89kg. As the slurry volume increases, by assuming that the rate of flow charge of the mill remains constant, the residence time of the materials in the mill rises ( = ⁄ ). To examine the influence of residence time in this test, the experiment were repeated for different balls charges, however, in each test, the residence time was different with respect to slurry volume. At 10% of the balls charge and the slurry weight of 35kg, the residence time became 7.5min. Also, this time for the other charges were 11.25, 15 and 18.75min, respectively. In Table 3, 80% passing size, P 80 , and in Fig. 7 the percentage of produced particles smaller than 44μm for two cases of similar and variable residence time are presented which demonstrate that the grinding is optimum at charges between 20% and 25%.  Fig. 7. Weight percent of product less than 44μm at different ball filling

4-3 The slurry filling effects
To examine, the influence of U on the grinding, the mill speed was set to 75% of the critical speed and the slurry with 60% of solid was used. Besides, the balls filled 20% of the mill volume (almost 352kg). In Fig. 8, the variation of grinding in terms of slurry filling was depicted. Apparently from Fig. 8, the best grinding occurs at U=0.5, in which with the increase of slurry volume and the pool formation, the grinding decreases. It is noted that with the increase of slurry volume, the pool is formed in the mill for U≥1, consequently, the impact loads is absorbed and damped by the pool. Also, with the decrease of impact loads, the contribution of grinding due to the impact is lowered and with the constancy of grinding due to the abrasion, the total grinding is reduced. It is important however to state that on the one hand, with the increase of slurry volume and the increase of materials weight in the mill, the normal force on the materials should be increases, but on the other hand, with the formation of pool, the buoyancy (Archimedes) force on the materials is increased in which the resultant normal force is lowered. Therefore, the normal force effects can be ignored.  Fig. 9 for different values of the residence time. The residence time was chosen to be 5min and 10min for U=0.5 and 1, respectively. In Fig. 10, the percentage of the produced particles smaller than 44μm is shown in terms of the slurry volume under various residence times and under the same residence time. According to Fig. 10 and Table   4, the best grinding occurs at the slurry filling between 1 and 1.5. Also, if the residence time is similar, with the increase of U, the size distribution of product becomes larger. Instead, if the residence time is not similar, initially with the increase of U, the grinding is improved in which its maximum occurs at U=1.5, after that, the grinding is lowered again.   Fig. 10. Weight percent of product less than 44μm at different slurry filling Furthermore, the formed slurry pool in the mill has profound effect on the absorption of the balls energy [22,23]. In other words, with the formation of slurry pool, the energy due to the impact loads is lowered and the contribution of impact to the grinding is reduced. Fig. 11 exhibits that with the increase of U from 0.5 to 1.5 and then to 2.5, the values of impact forces reduce. These impact forces were measured using a load cell mounted under one of the lifters for one rotation of the mill. It is noted that the coordinate was chosen to be at 12:00 o'clock and all angles were measured based on that point. According to Fig. 11, with the increase of formed pool level and also with the increase of slurry density, the pool acts like a damper at which the impact loads decrease more and consequently the grinding due to the impact is lowered.

4-4 The slurry concentration effects
The particle size distribution of the products at different density of slurry (or slurry concentration) was depicted in Fig. 12. In order to assess the influence of density on the grinding, the mill is allowed to rotate with 75% of the critical speed and the balls with the weight of 352kg have occupied 20% of the mill volume. Also, the slurry with 40-80% of the solid as a feed charge with approximate weight of 59-88kg was used. In each tests, the slurry filling was chosen to be unity. In Table 5, 80% passing size, P 80 , has been shown under various concentrations. Comparison between the results obtained from different densities reveals that the grinding rate is more at low concentration due to the low viscosity and density of the slurry and consequently the increase of relative speed between the balls and the materials.
According to Fig.12 and Table 5, it is clear that the grinding is maximum for the slurry with 40% of solid compared to the other concentrations. It is because of the fact that whatever the slurry density and viscosity are low, the particles move more simply among the balls and cross the liners with higher speed, therefore, the abrasion grinding improves. At the concentration of 40%, 352 kg of the balls should grind 59kg of the slurry while this amount of the balls at the concentration of 50% should grind 64 kg of the slurry. Therefore, the possibility of the balls impact to the mill materials is higher at the concentration of 40%, so better grading is obtained. It is noted that the same amount of the balls should grind 71, 79 and 88 kg of the slurry at the concentration of 60%, 70% and 80%, respectively.
In addition, if the percentage of slurry solid is low, it causes formation of a pool in the mill, the contact between metal to metal, the increase of the balls consumption and the escape of the material among the balls. Also, a large volume of the mill is filled with water which practically reduces the mill efficiency.
Besides, in the next stages, the drying process is costly. On the other hand, at very high density and viscosity, the slurry acts as a damper and creates resistance of the balls movements against impact.
Consequently, the impact grinding speed improves with the reduction of slurry viscosity. So, the best viscosity for the minimum abrasion time is for slurry with 60-70% of the solid.
From Fig. 13, the percentage of produced materials smaller than 44μm for concentration of 40%, 50%, 60% and 70% are 55.68%, 52.68%, 51.05% and 47.85%, respectively. However, the best grinding is obtained at 40% of concentration, but it is noted that for a condition that the mill speed, the balls volume, the slurry filling and the residence time are similar, at concentration of 70%, 34% more solid feed can be ground compared to concentration of 40% (79/59=1.34). This ratio is 8% and 20% more than the concentration of 40% for the concentration of 50% and 60%, respectively. In the other word, P 80 at the concentration of 40% is 80μm and at the concentration of 70% is 130μm. But, the rate of feed flow in the slurry of 70% is almost 34% more than the slurry of 40%.   With the formation of slurry pool and the increase of its density, the energy of impact forces reduces and thus, the contribution of impact in grinding is lowered. With the increase of slurry concentration from 40% to 60% and then to 80%, the values of impact forces are reduced according to Fig. 14. The materials packing between lifters and the walls at the concentration of 80% are plotted in Fig. 15. The packing is due to the fact that the grading of slurry of 80% is quite different than the other results.

5-Conclusions
The present paper aimed to study the effect of mill speed, ball filling, slurry filling and slurry concentration on the particle size distribution of product in the SAG mills. The pilot mill (1000×500 mm), initially loaded with balls at four different charges from 10% to 25% volumetric filling, was run at four different speeds varied from 60% to 90% of the critical speed. The tests covered a range of slurry filling (U) from 0.5 to 2.5 using of a feed of -1 inch copper ore with 40% to 80% slurry concentration. The following conclusions can be drawn from the present work: The best grinding and grading occurs at 70-80% of the critical speed. The grinding decreased before and after of this range.
Under 20-25% of the ball filling, the grinding rate is maximum.
Optimized grinding is observed when the slurry volume is 1-1.5 times of the ball bed voidage volume.
The best grinding and grading occurs at the slurry concentration between 60% and 70%. The materials' packing between lifters and the walls is at the concentration of 80%.
With the increase of slurry volume, the pool is formed in the mill, with the increase of formed pool level and also with the increase of slurry concentration and density, the pool acts like a damper at which the impact loads decrease more and consequently the grinding is lowered.

Availability of data and materials
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