applied sciences Article
Correlation between Material Properties and Breakage Rate Parameters Determined from Grinding Tests Evangelos Petrakis * and Konstantinos Komnitsas
ID
School of Mineral Resources Engineering, Technical University of Crete, University Campus, Kounoupidiana, 73100 Chania, Greece;
[email protected] * Correspondence:
[email protected]; Tel.: +30-28210-37608 Received: 22 December 2017; Accepted: 28 January 2018; Published: 31 January 2018
Abstract: The present study investigates four materials, namely quartz, marble, quartzite and metasandstone and aims to establish correlations, with the use of simple and multiple regression analysis, between their properties and breakage rate parameters. The material properties considered in this study derived from the application of destructive and non-destructive tests and include P-wave velocity (Vp ), Schmidt rebound value (RL ), uniaxial compressive strength (UCS) and tangent modulus of elasticity (Et ), while the breakage rate parameters determined from batch grinding tests, include breakage rate Si , maximum breakage rate Sm , αT and α, and optimum particle size xm . The results indicate that the properties of all materials examined show very good correlation and can be used to predict Si or αT . Furthermore, parameter α is well correlated with Vp , RL and Et using inverse exponential functions, while Sm is strongly correlated with RL and UCS. Overall, it is deduced that multiple regression analysis involving two independent variables is a reliable approach and can be used to identify correlations between properties and breakage rate parameters for quartz, quartzite and metasandstone, which are silica rich materials. The only exception shown is the determination of xm for marble. Keywords: comminution; grinding; material properties; population balance model; breakage rate parameters
1. Introduction Comminution and especially grinding is one of the most important operations of the materials processing industry. For many decades, it has been the subject of intensive research in several industrial sectors including mineral processing [1,2], metallurgy [3], cement, chemicals and ceramics production [4,5]. In addition, interest has been paid in recent years on the production of ultra-fine materials due to their improved physical and mechanical properties [6]. It is also known that comminution is an energy intensive process and has low efficiency, due to the fact that most energy consumed is absorbed by the device and only a small share is used for size reduction [7]. Considering these factors, it is deduced that improvement of process efficiency will definitely have a large impact on the operating cost of the mill, the environment and the conservation of resources. Scientific research, aiming to improve grinding efficiency, was initiated with the empirical relationships of Rittinger, Kick, Bond as well as Walker and Shaw, which described the energy-particle size relationship [8]. Later, theories which incorporated the concept of particle size distribution were developed [9,10], while recently research focused on the use of the mechanistic approach which is based on the recognition of physical events that take place during grinding [11–13]. The mechanistic approach includes the matrix model, which considers grinding as a series of breakage events and the population balance model, which considers grinding as a continuous process. The population Appl. Sci. 2018, 8, 220; doi:10.3390/app8020220
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balance model considers first order kinetics and uses two functions, namely the breakage rate Si and the breakage function bi,j [14,15]. These functions generate the fundamental size-mass balance equation which is applied for fully mixed batch grinding operations. It is mentioned that many researchers have underlined the advantages of these functions [16,17], while the scale-up from laboratory to industrial mills has also been discussed in a number of studies [18–21]. The variation of breakage rate Si (min−1 ) as a function of particle size xi (mm) has been considered by Austin [15] and is expressed with Equation (1), which is accepted by many researchers [22–24].
xi Si = α T · xo
α
· Qi
(1)
where, xi (mm) is the upper size of class i, x0 is the standard size (1 mm), αT (parameter that depends on milling conditions) is the breakage rate for size xi = 1 mm and α is a characteristic parameter depending on material properties. Qi is a correction factor, which is 1 for small particles (normal breakage) and smaller than one for large particles that need to be nipped and fractured by the grinding media. In the abnormal breakage region, the material behaves as it consists of a soft and a harder fraction [16]. Qi is calculated from Equation (2) [15], ! 1 Qi = (2) 1 + ( xi /µ)Λ where, µ is a parameter that depends on milling conditions and Λ is a positive number that depends on material type and shows how rapidly the breakage rate decreases as size increases (Λ ≥ 0). In addition, the particle size xm for which the breakage rate reaches its maximum value is related with parameter µ through the following equation [25,26], xm = µ·
α Λ−α
1/Λ , Λ>α
(3)
Equation (3) shows that xm is directly proportional to µ since for the same material α and Λ are constants. The physical and mechanical properties of a material play a crucial role in the design of a mineral processing circuit. The reliable determination of these properties enables the prediction of milling results and in this case no extensive milling tests that require large amounts of time and material are needed [27]. In this respect, several researchers [28–36] have attempted to identify the relationship between various material properties and P-wave velocity. These studies indicated that sound velocity is closely related to physico-mechanical properties such as uniaxial compressive strength, modulus of elasticity, hardness, effective porosity, water absorption and both saturated and dry density. P-wave velocity test is non-destructive, offers several advantages to researchers and can be easily applied both in laboratory and in situ. It is mentioned that P-wave velocity is related with the characteristics of the whole body of the material tested [29]. The Schmidt hammer test is equally acceptable and used for estimating indirectly material hardness and strength, through the determined RL value, which depends on the properties of the specimen. The operation of the device is based on the principle that the rebound of an elastic mass impacting on a surface depends on its hardness and thus the harder the surface the higher is the rebound distance [37]. Several societies, including the International Society for Rock Mechanics [38,39] and the American Society for Testing and Materials [40], have proposed test methods and standards, respectively, for the proper operation of the device [41]. Other options for determining such material properties with the use of the Schmidt hammer have also been proposed [42–44]. Breakage rate parameters depend on material properties, so it is very useful and practical from a laboratory and industrial point of view if correlations between them are established. In this regard, Deniz [45] used limestone as test material and identified the relationship between Bond’s grindability and breakage rate parameters, while Aras [46] established important correlations between the point
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load index and Bond’s grindability as well as breakage rate parameters. However, so far no correlations between other material properties such as uniaxial compressive strength or modulus of elasticity and breakage rate parameters have been established. This study has a certain degree of novelty since it attempts to establish potential correlations between material properties derived from the application of destructive and non-destructive tests, and breakage rate parameters. 2. Materials and Methods The materials used in the present experimental study were obtained from several Greek sites, namely quartz from Assiros, near Thessaloniki; marble from west Crete, Chania region; and quartzite and metasandstone from west Crete, Kissamos area. More than 100 kg of each material was sampled from several locations of the same site in order to obtain representative samples. Mineralogical analyses show that marble consists mainly of calcite, while quartzite and metasandstone of quartz and some mica. The quartz samples used in this study were of high purity white quartz. The main difference between quartzite and metasandstone is the degree of transformation. Quartzite, due to dynamic re-crystallization, consists of fine crystalline and elongated quartz grains, which are well oriented; due to the mechanical strain imposed, individual quartz crystals are present. However, metasandstone consists of a fine crystalline matrix with individual quartz crystals, which exhibit lack of orientation due to the lower degree of transformation compared to quartzite. In addition, large gaps are observed between its grains due to the high porosity. Table 1 presents the porosity and density of each raw material, as determined with the use of Archimedes method [47]; all measurements were carried out in triplicate and average values are shown. The standard deviation of porosity and density was 0.002 and 0.005 respectively, for all materials tested. Table 1. Porosity and density of the raw materials used. Material
Porosity %
Density g/cm3
quartz marble quartzite metasandstone
0.02 0.3 0.9 8.0
2.65 2.72 2.59 2.11
Cylindrical cores of representative samples with NX size (diameter 54.7 mm) were used to determine the physico-mechanical properties of each material. Both ends of the core specimens were cut flat and polished using a cut-off and a special polishing machine, respectively. The height to diameter ratio of all specimens was kept equal to 2 in order to minimize the size effect. From each material type 6–8 specimens were cored (Figure 1a,b). Specimens exhibiting cracks, discontinuities, or alteration phenomena as a result of coring were not used in tests and replaced with new specimens. In this study, the non-destructive P-wave velocity and Schmidt hammer rebound tests were carried out. The P-wave velocity test was performed using the PUNDIT 6 (Portable Ultrasonic Non-destructive Digital Indicating Tester) device (CNS Farnell, London, UK) at a frequency of 1 MHz. According to the procedure, two transducers were placed opposite the two parallel surfaces of the core samples (Figure 2a,b) and the travel time of the ultrasonic waves, from one transducer to the other, was recorded. By taking into account that the distance between the two surfaces of each specimen is known, the velocity of the ultrasonic waves can be calculated [48]. It is known that the ultrasonic velocity depends on density and the elastic properties of each material and is higher when the sample has adequate density, uniformity and homogeneity [49].
Cylindrical cores of representative samples with NX size (diameter 54.7 mm) were used to determine the physico-mechanical properties of each material. Both ends of the core specimens were cut flat and polished using a cut-off and a special polishing machine, respectively. The height to diameter ratio of all specimens was kept equal to 2 in order to minimize the size effect. From each material Appl. Sci. type 2018, 8,6–8 220 specimens were cored (Figure 1a,b). Specimens exhibiting cracks, discontinuities, 4 of 17 or alteration phenomena as a result of coring were not used in tests and replaced with new specimens.
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In this study, the non-destructive P-wave velocity and Schmidt hammer rebound tests were carried out. The P-wave velocity test was performed using the PUNDIT 6 (Portable Ultrasonic Nondestructive Digital Indicating Tester) device (CNS Farnell, London, UK) at a frequency of 1 MHz. According to the procedure, two transducers were placed opposite the two parallel surfaces of the core samples (Figure 2a,b) and the travel time of the ultrasonic waves, from one transducer to the other, was recorded. By taking into account that the distance between the two surfaces of each (a) (b) [48]. It is known that the specimen is known, the velocity of the ultrasonic waves can be calculated ultrasonic velocity depends on density and the (a) elastic properties of each material and is higher when Figure Figure1.1.Core Corespecimens specimensof of (a)marble marbleand and(b) (b)metasandstone. metasandstone. the sample has adequate density, uniformity and homogeneity [49].
(a)
(b)
Figure (b) metasandstone. metasandstone. Figure 2. 2. P-wave P-wave velocity velocity test test for for (a) (a) marble marble and and (b)
The Schmidt hammer rebound test was performed using an L-type hammer with impact energy The Schmidt hammer rebound test was performed using an L-type hammer with impact energy 0.735 Nm. Core specimens were tested in a V-shaped cradle block with an approximate weight of 23 0.735 Nm. Core specimens were tested in a V-shaped cradle block with an approximate weight of 23 kg. kg. Prior to test execution the circular ends of each specimen were divided into eight equal arcs, from Prior to test execution the circular ends of each specimen were divided into eight equal arcs, from the the ends of which eight straight parallel lines along their longitudinal axis were drawn (Figure 3). ends of which eight straight parallel lines along their longitudinal axis were drawn (Figure 3). The test The test was performed by placing the core specimens in the cradle and impacting the hammer was performed by placing the core specimens in the cradle and impacting the hammer vertically vertically along the drawn lines, starting from line No1 (Figure 4). In this manner, five individual along the drawn lines, starting from line No1 (Figure 4). In this manner, five individual impacts were impacts were conducted at equal spacing and consequently five rebound values RL were recorded conducted at equal spacing and consequently five rebound values RL were recorded along the first along the first line drawn. Then, the same procedure was repeated by rotating◦the specimen 45°. In line drawn. Then, the same procedure was repeated by rotating the specimen 45 . In total, depending total, depending on the number of the specimens used, 40 rebound values for each specimen and 240 on the number of the specimens used, 40 rebound values for each specimen and 240 or 280 for each or 280 for each material were recorded. The rebound values given in this study for each material are material were recorded. The rebound values given in this study for each material are the average of all the average of all recorded values. recorded values. After the execution of non-destructive tests, the cylindrical specimens were used for the determination of their uniaxial compressive strength (UCS) and tangent modulus of elasticity (Et). The uniaxial compression tests were performed using a 1600 KN hydraulic device, manufactured by MTS (Mechanical Testing System) corporation (Kalamazoo, MI, USA). According to the procedure, the cylindrical specimens were subjected to a load acting on both ends (Figure 5a), using a loading rate 0.01 mm/s under displacement control mode; the failure of specimens occurred in less than 5 min. The peak stress value and the tangent modulus of elasticity of each specimen were calculated from the complete stress-strain curve (Figure 5b).
Figure 3. Straight parallel lines drawn along the longitudinal axis of each specimen.
vertically along the drawn lines, starting from line No1 (Figure 4). In this manner, five individual impacts were conducted at equal spacing and consequently five rebound values RL were recorded along the first line drawn. Then, the same procedure was repeated by rotating the specimen 45°. In total, depending on the number of the specimens used, 40 rebound values for each specimen and 240 or 280 each material were recorded. The rebound values given in this study for each material Appl. Sci.for 2018, 8, 220 5 ofare 17 the average of all recorded values.
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Figure 3. Straight parallel lines drawn along the longitudinal axis of each specimen. Figure 3. Straight parallel lines drawn along the longitudinal axis of each specimen.
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Figure 4. The Schmidt hammer rebound test.
After the execution of non-destructive tests, the cylindrical specimens were used for the determination of their uniaxial compressive strength (UCS) and tangent modulus of elasticity (Et). The uniaxial compression tests were performed using a 1600 KN hydraulic device, manufactured by MTS (Mechanical Testing System) corporation (Kalamazoo, MI, USA). According to the procedure, the cylindrical specimens were subjected to a load acting on both ends (Figure 5a), using a loading rate 0.01 mm/s under displacement control mode; the failure of specimens occurred in less than 5 min. The peak stress value and the4.tangent modulus of elasticity of each specimen were calculated Figure The Schmidt hammer rebound test. Figure 4. The Schmidt from the complete stress-strain curve (Figure 5b).hammer rebound test. After the execution of non-destructive tests, the cylindrical specimens were used for the determination of their uniaxial compressive strength (UCS) and tangent modulus of elasticity (Et). The uniaxial compression tests were performed using a 1600 KN hydraulic device, manufactured by MTS (Mechanical Testing System) corporation (Kalamazoo, MI, USA). According to the procedure, the cylindrical specimens were subjected to a load acting on both ends (Figure 5a), using a loading rate 0.01 mm/s under displacement control mode; the failure of specimens occurred in less than 5 min. The peak stress value and the tangent modulus of elasticity of each specimen were calculated from the complete stress-strain curve (Figure 5b).
(a)
(b)
Figure 5. (a) Cylindrical specimen subjected to loading; (b) Stress-strain curve for the determination Figure 5. (a) Cylindrical specimen subjected to loading; (b) Stress-strain curve for the determination of of UCS and Et. UCS and Et .
For the determination of the breakage rate parameters, experiments were performed in a laboratory-scale ball mill with a volume of 5423 cm3 (Table 2) operating at a constant speed of Ν = 66 rpm (1.1 Hz), which is 70% of its critical speed. The core samples as well as other materials with the same properties (porosity, density, mineralogy etc.) were crushed and prepared for the grinding tests. The mill charge consisted of(a) 25.4 mm (1 inch) stainless steel (b) balls with density ρb = 7.85 g/cm3. The parameters J (ball filling volume) and fc (material filling volume), were calculated from Equations (4) Figure 5. (a) Cylindrical specimen subjected to loading; (b) Stress-strain curve for the determination andof(5), respectively. UCS and Et. 1
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For the determination of the breakage rate parameters, experiments were performed in a laboratory-scale ball mill with a volume of 5423 cm3 (Table 2) operating at a constant speed of N = 66 rpm (1.1 Hz), which is 70% of its critical speed. The core samples as well as other materials with the same properties (porosity, density, mineralogy etc.) were crushed and prepared for the grinding tests. The mill charge consisted of 25.4 mm (1 inch) stainless steel balls with density ρb = 7.85 g/cm3. The parameters J (ball filling volume) and fc (material filling volume), were calculated from Equations (4) and (5), respectively. 1 Volume o f solid balls · (4) J= mill volume 1−ε Volume o f solid material 1 fc = · (5) mill volume 1−ε where, ε is the bed porosity of balls and material (assumed to be 40%) [15]. The material filling volume fc was kept constant at 4%, corresponding to 345.0 g, 351.5 g, 342.5 g and 273.3 g of quartz, marble, quartzite and metasandstone, respectively. The fraction of the space between the balls at rest that is filled with material (interstitial filling, U) was calculated from Equation (6); in this study, U was kept constant at 0.5. fc (6) U= 0.4· J Four mono-size fractions of each material (−3.35 + 2.36 mm), (−1.7 + 1.18 mm), (−0.850 + 0.600 mm) and (−0.425 + 0.300 mm) were ground in a laboratory ball mill for various grinding times t (0.5, 1, 2, 4 min). The √ products obtained after each grinding time t were wet sieved using a series of screens with a ratio of 2 for the determination of particle size distribution. It should be noted that fresh feed was used in each test and the mill was loaded using successive layers of balls and feed [50]. Table 2. Grinding conditions. Mill
Balls
Material
diameter, D (cm)
20.4
diameter, d (mm)
25.4
length, L (cm)
16.6
number
77
volume, V
(cm3 )
5423
weight (g)
5149
quartz (2.65) marble (2.72) density
(g/cm3 )
quartzite (2.59) metasandstone (2.11)
operational speed, N (rpm)
66
density (g/cm3 )
7.85
critical speed, Nc (rpm)
93.7
porosity (%)
40
material filling volume, f c (%)
4
20
interstitial filling, U (%)
50
ball filling volume, J (%)
3. Results and Discussion 3.1. P-Wave Velocity of Materials The P-wave velocity for each material is shown in Figure 6 in Box and Whisker plots which were designed with the use of Statistica v.10 software (StatSoft Inc., Tulsa, OK, USA). In these plots the central rectangle spans the first to the third quartile, while the segment inside the rectangle represents the median (second quartile). Whiskers above and below the rectangle show the maximum and minimum values and identify the non-outlier range. Any value outside that range is plotted as independent point and is called outlier.
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9
8
P-wave velocity Vp, km/s
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P-wave velocity Vp, km/s
8
7
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6
5
4
3 Median
6
25%-75%
Non-Outlier Range
Outliers
2 Quartz
Marble
Quartzite
Metasandstone
Figure 6. P-wavevelocity velocity ofofallall materials used.used. Figure 6. P-wave materials
5
3.2. Schmidt Rebound Values 4
Figure 6 The shows thatrebound outliers are are present, just below minimum value, for quartz and Schmidt values shown in Figure 7. It the is seen that outliers, more only clear in this quartzite. case, It is also observed the descending order materials respect to theorder median are present only that for marble and quartzite. It isof also observedwith that the descending of Vp value 3 respect and to themetasandstone. median RL value is The quartz, quartzite, results marble and metasandstone. The also show is quartz, materials marble,with quartzite statistical of P-wave velocity Non-Outlier Range Outliers statistical results of RMedian for the25%-75% materials tested also the mean RL values follow the same that the mean Vp values of Lthe materials follow theshow samethat order as the median values. Furthermore, 2 as descending order observed for the median values. Finally, it is also indicated that the values Quartz Marble Quartzite Metasandstone low standard deviation is observed for Vpthefor all the materials tested and thusfor thequartz dataand are clustered obtained in this study are well within range of values reported in other studies around thequartzite mean [54], Vp values. It isand also mentioned[56]. that similar values were obtained in other studies for marble [55] metasandstone Figure 6. P-wave velocity of all materials used. marble and quartzite [51], sandstone [52] and quartz [53]. 65
3.2. 3.2. Schmidt Schmidt Rebound Rebound Values Values60 Schmidt rebound values, RL
The Schmidt rebound rebound55values valuesare areshown showninin Figure It seen is seen outliers, more in case, this The Schmidt Figure 7. 7. It is thatthat outliers, more clearclear in this case, are present only for marble and quartzite. is also observed that the descending order of are present only for marble and quartzite. It is alsoItobserved that the descending order of materials 50 materials with respect to the median R L value is quartz, quartzite, marble and metasandstone. The with respect to the median RL value is quartz, quartzite, marble and metasandstone. The statistical statistical results of RL for45the materials tested also show that the mean RL values follow the same results of R L for the materials tested also show that the mean RL values follow the same descending descending order as forvalues. the median values. Finally, it is that also the indicated that the values 40 order as observed forobserved the median Finally, it is also indicated values obtained in this obtained in this study are well within the range of values reported in other studies quartz [54], and study are well within the35range of values reported in other studies for quartz andfor quartzite quartzite [54], marble [55] and metasandstone [56]. marble [55] and metasandstone [56]. 30
65
60
25
20
Median
25%-75% Quartz
Schmidt rebound values, RL
55
Non-Outlier Range Marble
Outliers
Quartzite
Metasandstone
Figure 7. Schmidt rebound values (RL) of all materials used.
50 3.3. Uniaxial Compressive Strength and Modulus of Elasticity
The statistical results of UCS and Et for each material are shown in Tables 3 and 4, respectively. 45 Table 3 shows that the descending order of materials with respect to the mean UCS value is quartz, quartzite, marble and metasandstone. As far as the modulus of elasticity (Et) is concerned, the 40 descending order of materials with respect to the mean Et value is quartz, marble, quartzite and metasandstone (Table 4); this indicates, as expected, that quartz is the stiffest material and thus more 35
30
25
20
Median
25%-75% Quartz
Non-Outlier Range Marble
Outliers
Quartzite
Metasandstone
Schmidt rebound values (RLL)) of of all all materials materials used. used. Figure 7. Schmidt
3.3. Uniaxial Compressive Strength and Modulus of Elasticity The statistical results of UCS and Et for each material are shown in Tables 3 and 4, respectively. Table 3 shows that the descending order of materials with respect to the mean UCS value is quartz, quartzite, marble and metasandstone. As far as the modulus of elasticity (Et) is concerned, the descending order of materials with respect to the mean Et value is quartz, marble, quartzite and
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3.3. Uniaxial Compressive Strength and Modulus of Elasticity The statistical results of UCS and Et for each material are shown in Tables 3 and 4, respectively. Table 3 shows that the descending order of materials with respect to the mean UCS value is quartz, quartzite, marble and metasandstone. As far as the modulus of elasticity (Et) is concerned, the descending order of materials with respect to the mean Et value is quartz, marble, quartzite and metasandstone (Table 4); this indicates, as expected, that quartz is the stiffest material and thus more resistant to deformation. Similar UCS values have been reported in other studies for quartz [57], quartzite [58], marble and sandstone [59]. Table 3. The statistical results of uniaxial compressive strength (UCS) (MPa). Material Quartz Marble Quartzite Metasandstone
Observations 7 8 7 7
Mean
Minimum
Maximum
Median
Std. Dev.
135.1 68.0 80.9 57.6
122.0 58.6 68.0 42.6
151.5 100.4 110.0 74.6
132.0 63.5 76.7 64.0
12.4 13.7 15.4 14.0
Table 4. The statistical results of Et (GPa). Material Quartz Marble Quartzite Metasandstone
Observations Mean 7 8 7 7
59.5 48.2 41.7 27.1
Minimum
Maximum
Median
Std. Dev.
47.5 21.5 26.7 10.7
71.2 64.9 56.2 39.0
62.8 53.0 44.8 31.0
9.1 15.8 11.0 10.9
3.4. Breakage Rate Parameters Figure 8a–d shows in normal-log plots the relationship between the remaining mass fraction of each feed size vs. grinding time for all materials used. The feed size is the upper size of the tested size fractions. The results indicate that breakage follows a first order law i.e., the breakage rate Si is independent of time and the Si values can be determined from the slope of the straight lines. Each line refers to a different feed size. Figure 9 shows on log–log scale the variation of Si values, obtained from the first order plots, with the upper feed particle size. It is shown that for each material Si increases up to a specified size xm (optimum feed size), but above this size breakage rates decrease sharply, since particles are too large to be nipped and fractured by the grinding media used. The optimum size xm for each material is the size at which, under normal grinding conditions, the highest breakage rate Sm is reached. It is mentioned that the breakage rate parameters were back calculated using the Moly-Cop Tools™ v.1.0 software and the determined values are presented in Table 5. The back calculation method is described in detail in a previous recent publication of the authors [8]. Table 5. Breakage rate parameters for all materials used. Material Quartz Marble Quartzite Metasandstone
Si
αT
min−1
min−1
0.55 0.89 0.83 1.05
0.68 1.09 0.98 1.41
α
0.80 0.90 0.84 1.12
Sm
xm
min−1
mm
0.85 1.83 1.29 2.45
1.80 2.84 2.00 2.32
Mass % remaining, mi (t)
100
Quartz 10
1 0
2
4
6
8
100 Appl. Sci. 2018, 8, x FOR Grinding PEER REVIEW time, min
(a)Quartzite
1
0.1 0
100
Mass % remaining, mi (t)
10
Mass % remaining, mi (t)
Mass % remaining, mi (t)
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1
0.1
2
0
2
4
4
6
8
6
Grinding time, time, minmin Grinding
(c)
8
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100
Marble
10 1
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0.1 0.01 0
Mass % remaining, mi (t)
Mass % remaining, mi (t)
line refers to a different feed size. Figure 9 shows on log–log scale the variation of Si values, obtained from the first order plots, with the upper feed particle size. It is shown that for each material Si increases up to a specified size xm (optimum feed size), but above this size breakage rates decrease sharply, since particles are too large to be nipped and fractured by the grinding media used. The optimum size xm for each material is the size at which, under normal grinding conditions, the highest breakage rate Sm is reached. It is mentioned that the breakage rate parameters were back calculated using the Moly-Cop Tools™ v.1.0 software and the determined values are presented in Table 5. The Appl. Sci. 2018, 8, 220 back calculation method is described in detail in a previous recent publication of the authors [8].
2
100
4
6
8
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Grinding time, min
Metasandstone
(b)
10
100
Metasandstone
1
10 1
0.1
0.1
0.01
0.01
0.001
0.001 0
0
2
2
4
6
4
Grinding Grinding time, min
(c)
(d)
8
6
8
time, min
(d)
Figure 8. First order plots showing the mass % remaining for each size fraction of each material vs. First order plots showing mass%%remaining remaining for each of of each material vs. order showing thethe mass for eachsize sizefraction fraction each material grindingplots time; (a) quartz, (b) marble, (c) quartzite and (d) metasandstone.
8. FigureFigure 8. First grinding (a) quartz, (b) marble, quartziteand and(d) (d)metasandstone. metasandstone. grinding time; time; (a) quartz, (b) marble, (c) (c) quartzite Breakage rate Si, min-1
10
Quartz Marble
10
Breakage rate Si, min-1
vs.
Quartzite Quartz Metasandstone Marble
1
Quartzite Metasandstone
1
0.1 0.1
1
10
Upper feed particle size, mm
Figure 9. Variation of breakage rate Si vs. upper feed particle size. 0.1 0.1 1 10 Table 5. Breakage rate parameters for all materials used.
Upper feed particle size, mm αΤ α Sm xm Si min−1 rate min−1Si vs. upper min−1feed mm Figure 9. Variation of breakage size. Figure 9. Variation of breakage rate Si vs. upper feedparticle particle size. Quartz 0.55 0.68 0.80 0.85 1.80 Marble 0.89 1.09 0.90 1.83 2.84 Table 5.Quartzite Breakage rate parameters for all1.29 materials used. 0.83 0.98 Rate 0.84Parameters 2.00 between Material Properties and Breakage Metasandstone 1.05 1.41 1.12 2.45 2.32 Material
3.5. Correlation
Material
Si
αΤ
α
Sm
xm
−1 −1 3.5.1. Simple Regression min 3.5. CorrelationAnalysis between Material Propertiesmin and Breakage Rate Parametersmin−1 mm
Quartz
0.55
0.68
0.80
0.85
1.80
In the first3.5.1. part of Regression this study simple regression analysis involving one dependent and one Simple Analysis Marble 0.89 1.09 0.90 1.83 2.84 independent variable was carried out,study while Excel software was used to establish correlations In the first part of this simple regression dependent potential and one Quartzite 0.83 0.98analysis 0.84involving 1.29 one 2.00 independent variable wasbreakage carried out,rate whileparameters. Excel software was used to establish potential between material properties and During this part, linear or non-linear Metasandstone 1.41rate 1.12 2.45 2.32 correlations between material properties 1.05 and breakage parameters. During this part, linear or 2 was considered for functions were non-linear used and the equation with the highest correlation coefficient R 2 functions were used and the equation with the highest correlation coefficient R was considered for determining the breakageand rate Τ, α, Sm and xm). It is mentioned that no determining the breakage rate parameters (SiBreakage ,parameters αT , α, Rate S(Smi, αand xm ). It is mentioned that no similar 3.5. Correlation between Material Properties Parameters similar functions are given so far in other studies in literature. functions are givenThe socorrelations far in other studies in literature. between Si values and Vp, RL, UCS and Et are presented in Figure 10. It is seen 3.5.1. Simple Analysis thatRegression verybetween strong correlation between Si and (RL2 ,= UCS 0.98) and strong betweenin Si Figure and Vp 10. It is seen The correlations Si values and VpR,L R and Et correlations are presented (R2 = 0.85) and Et (R2 = 0.85) are obtained with the use of inverse linear functions. Furthermore, Si is 2 that very strong correlation Si simple and R regression (R = 0.98) and strong correlations between Si one and Vp In thevery first part of between this study analysis involving one dependent and well correlated with UCS with the useLof an inverse exponential function (R2 = 0.99). 2 = 0.85) independent variable was out, while was used to establish potentialSi is (R2 = 0.85) and Et (R arecarried obtained with the Excel use ofsoftware inverse linear functions. Furthermore, 2 = part, correlations between material properties and parameters. During(Rthis very well correlated with UCS with the use of anbreakage inverse rate exponential function 0.99).linear or non-linear functions were used and the equation with the highest correlation coefficient R2 was 11). Similar results are obtained for the correlations between αT and Vp , RL , UCS and Et (Figure considered for determining the breakage rate parameters (Si, αΤ, α, Sm and xm). It is mentioned that no Parameter αT , which defines the breakage rate for a feed size of 1 mm, is correlated very well with Vp , similar functions are given so far in other studies in literature. RL and Et using an inverse linear relationship. The very strong correlation (R2 = 1) between αT and The correlations between Si values and Vp, RL, UCS and Et are presented in Figure 10. It is seen RL is considered verycorrelation importantbetween for the Sdesign of grinding circuits. RL values are obtained from that very strong i and RL (R2 = 0.98) and strong correlations between Si and Vp the application of the convenient and low-cost Schmidt hammer test, which can be used both in laboratory (R2 = 0.85) and Et (R2 = 0.85) are obtained with the use of inverse linear functions. Furthermore, Si is and invery industrial mills, thus enabling theuse determination of breakage rate for well correlated with UCS with the of an inverse exponential function (R2a=specific 0.99). material in a grinding circuit.
Breakage rate Si
Breakage rate S
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0 10
10
(a)
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70
(b) 2 -1 Breakagerate rateSSi,i,min min-1 Breakage
y = -0.10x + 1.41 R² = 0.85
y = 21.00x-0.74 R² = 0.99
1
1
y = -0.02x + 1.74 R² = 0.98
y = -0.01x + 1.46 R² = 0.85
1
1
0.5
0.5
0
0
0
50
1.5
2
Breakage rate Si, min-1
Breakage rate Si, min-1
1.5
30
Schmidt rebound values RL
P-wave velocity Vp, km/s
20
0
70
5
120
0
10 30 50 70 90 10 30 of elasticity50 Tangent modulus Et, GPa
170
10
Uniaxial compressive strength UCS, MPa
70
Schmidt rebound values RL
P-wave velocity Vp, km/s
(c)
(d)
(a)
(b)
Breakage rate Si, min-1
Breakage rate Si, min-1
Figure 10. Correlations between breakage rate Si and (a) P-wave velocity Vp, (b) Schmidt rebound 2 2 values RL, (c) uniaxial compressive strength UCS and (d) tangent modulus of elasticity Et; horizontal -0.74 y = 21.00x y = -0.01x + 1.46 error bars show standard deviation of the measurements. R² = 0.99 R² = 0.85
Similar results are obtained for the correlations between αΤ and Vp, RL, UCS and Et (Figure 11). 1 1 Parameter αΤ, which defines the breakage rate for a feed size of 1 mm, is correlated very well with Vp, RL and Et using an inverse linear relationship. The very strong correlation (R2 = 1) between αT and RL is considered very important for the design of grinding circuits. RL values are obtained from the application of the convenient and low-cost Schmidt hammer 0 test, which can be used both in 0 laboratory industrial 120 mills, thus170 enabling the determination of 30 breakage 10 50 rate for 70 a specific 90 20 and in 70 Tangent modulus of elasticity Et, GPa material Uniaxial in a grinding circuit. compressive strength UCS, MPa Figure 12 shows(c) the correlations between parameter α and Vp, RL, UCS(d) and Et. It is seen that parameter α is very well correlated with Vp, RL and Et with the use of inverse exponential functions. Figure10. 10. Correlations Correlations between breakage rate SSand i and (a) P-wave , (b) Schmidt rebound Parameter α is constantbetween for the breakage same material therefore the velocity functionsVpobtained can be used Figure rate i and (a) P-wave velocity Vp , (b) Schmidt rebound 2 (c) uniaxial uniaxial compressive strength UCSand and (d)tangent tangent modulus(R ofelasticity elasticity E;t;horizontal horizontal valuesRRL,, (c) independently of grinding conditions. However, a moderate correlation = 0.67) between α and values compressive strength UCS (d) modulus of E t L error bars show standard deviation of the measurements. UCS is indicated. error bars show standard deviation of the measurements.
Parameter αΤ, min-1
1
y = 34.72x-0.81 y = -0.15x + 1.90 R² = 0.97 R² = 0.88 0 20
1
70
120
170
Uniaxial compressive strength UCS, MPa
(c)
1
2
Parameter αΤ, min-1
Parameter αΤ, min-1
parameter αΤ, min-1
2
Parameter αΤ, min-1
parameter αΤ, min-1
Similar results are obtained for the correlations between αΤ and Vp, RL, UCS and Et (Figure 11). 2 2 y = -0.03x + 2.37 y = -0.15x + 1.90 Parameter αΤ, which defines the breakage rate for a feed size of 1 mm, is correlated very well with Vp, R² = 1.00 R² = 0.88 RL and Et using an inverse linear relationship. The very strong correlation (R2 = 1) between αT and RL is considered 1very important for the design of grinding circuits. RL values are obtained from the 1 application of the convenient and low-cost Schmidt hammer test, which can be used both in laboratory and in industrial mills, thus enabling the determination of breakage rate for a specific material in a grinding circuit. 0 0 Figure 12 shows the correlations between parameter α10and Vp,30RL, UCS50and Et. 70It is seen that 0 5 10 Schmidt rebound values RL Appl. Sci. 2018, 8, x FOR PEERvelocity REVIEW 11 of 18 Vp, km/s parameter α is very P-wave well correlated with Vp, RL and Et with the use of inverse exponential functions. (a) the same material and therefore the functions (b) Parameter α is constant for obtained can be used 2 2 independently of grinding conditions. However, a moderate correlation (R2 = 0.67) between α and y = -0.02x + 1.96 UCS is indicated. R² = 0.88
y = -0.03x + 2.37 R² = 1.00
0
110
30 50 70 90 Tangent modulus of elasticity Et, GPa
(d)
Figure 11. Correlations between parameter αΤ and (a) P-wave velocity Vp, (b) Schmidt rebound values
Figure 011. Correlations between parameter αT and (a) P-wave velocity Vp , (b) Schmidt rebound values 0 RL, (c) uniaxial compressive strength UCS and (d) tangent modulus of elasticity Et; horizontal error 10 30 50 70 RL , (c) uniaxial compressive strength UCS of elasticity Et ; horizontal error 0 5 10 and (d) tangent modulus bars show standard deviation of the measurements. Schmidt rebound values RL P-wave velocity V , km/s bars show standard deviation ofp the measurements. 1.2
(a)
1.5
0.8 0.6 0.4
y= R² = 0.85
Parameter α
Parameter α
1
(b) y = 9.29x-0.62 R² = 0.93
1.74x-0.38 1
0.5
0
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Parameter αΤ, min-1
Parameter αΤ, min-1
Figure 12 shows the correlations between parameter α and Vp , RL , UCS and Et . It is seen that parameter α is very well correlated with Vp , RL and Et with the use of inverse exponential functions. Parameter α is Appl. Sci. 2018, 8, x FOR PEER REVIEW 11 of 18 constant for the same material and therefore the functions obtained can be used independently of grinding conditions. However, a moderate correlation (R2 = 0.67) between α and UCS is indicated. 2 2 The correlations between maximum breakage rate Sm and Vp , RL , UCS and Et are presented in y = -0.02x + 1.96 Figure 13. Very strong correlation (R2 = 0.96) is shown between Sm and RL with the R² = use 0.88 of an inverse linear function. Furthermore, Sm seems to follow an inverse exponential relationship with UCS, with correlation1 coefficient R2 = 0.95. However, the prediction of1 Sm from Vp and Et values is somehow risky due to the moderate correlation between these variables. y = 34.72x-0.81 Figure 14 shows the correlations between the optimum size xm and Vp , RL , UCS and Et . It is R² = 0.97 seen that weak correlations exist between xm and Vp (R2 = 0.08) as well as Et (R2 = 0.09) when inverse 0 0 functions are results between xm and are also with 10 RL or 30 UCS50 70 obtained, 90 20 used. Furthermore, 70 120 similar 170 Tangent modulus of elasticity Et, GPa optimum R2 0.35 and 0.57, respectively. It is mentioned that very good to excellent correlations between Uniaxial compressive strength UCS, MPa size xm and other variables analysis. Thus, it is (c) are obtained, if marble is excluded from regression (d) deduced that the optimum size xm increases with decreasing P-wave velocity, Schmidt rebound value, Figure 11. Correlations between Τ and (a) P-wave velocity Vp, (b) Schmidt rebound values uniaxial compressive strength andparameter tangent α modulus of elasticity, for all materials tested except for RL, (c) uniaxial compressive strength UCS and (d) tangent modulus of elasticity Et; horizontal error marble. It is therefore obvious that as strength or stiffness of a material increases the optimum size bars show standard deviation of the measurements. at which the breakage rate obtains its maximum value becomes finer. 1.5
y = 9.29x-0.62 R² = 0.93
y = 1.74x-0.38 R² = 0.85
1
Parameter α
Parameter α
1.2
0.8 0.6
1
0.5
0
0.4 0
5
10
10
30
(a)
70
(b) 1.2
1.2
y = 3.86x-0.33 R² = 0.67
0.8 0.6 0.4
y = 4.25x-0.41 R² = 0.85
1
parameter α
1
Parameter α
50
Schmidt rebound values RL
P-wave velocity Vp, km/s
0.8 0.6 0.4
0.2 20 70 120 170 Uniaxial compressive strength UCS, MPa
(c)
10
30
50
70
90
Tangent modulus of elasticity Et, GPa
(d)
Figure12. 12. Correlations Correlations between betweenparameter parameterααand and(a) (a)P-wave P-wavevelocity velocityVVp,, (b) (b) Schmidt Schmidt rebound rebound values values Figure p L, (c) uniaxial compressive strength UCS and (d) tangent modulus of elasticity Et; horizontal error R RL , (c) uniaxial compressive strength UCS and (d) tangent modulus of elasticity Et ; horizontal error barsshow showstandard standarddeviation deviationof ofthe themeasurements. measurements. bars
The correlations between maximum breakage rate Sm and Vp, RL, UCS and Et are presented in Figure 13. Very strong correlation (R2 = 0.96) is shown between Sm and RL with the use of an inverse linear function. Furthermore, Sm seems to follow an inverse exponential relationship with UCS, with correlation coefficient R2 = 0.95. However, the prediction of Sm from Vp and Et values is somehow risky due to the moderate correlation between these variables.
2
Sm, min-1
Sm, min-1
2
1
1
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3
3
y = -0.31x + 3.42 R² = 0.74
2 -1 Sm, min S , min-1
Sm, minS-1m, min-1
(b)
3
y = 297.79x-1.21 R² = 0.95
2
50
70
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Schmidt rebound values RL
P-wave velocity Vp, km/s 3
30
2
m
2 1 1
y = -0.07x + 4.59 R² = 0.96 y = -0.04x + 3.56 R² = 0.75
1 1 0
0 0 0 20
5
P-wave velocity Vp, km/s 70
10
10 120
170
0
(a) strength UCS, MPa Uniaxial compressive (c)
3
30
50
70
Schmidt rebound values RL
0
20
40
60
80
(b) of elasticity E , GPa Tangent modulus t (d)
3
y = -0.04x + 3.56 y = 297.79x-1.21 R² V =p0.75 Figure 13. Correlations between maximum breakage rate Sm and (a) P-wave velocity , (b) Schmidt R² = 0.95 Sm, min-1
Sm, min-1
2 2 rebound values RL, (c) uniaxial compressive strength UCS and (d) tangent modulus of elasticity Et; horizontal error bars show standard deviation of the measurements. 1
1
Figure 14 shows the correlations between the optimum size xm and Vp, RL, UCS and Et. It is seen that weak correlations exist between xm and Vp (R2 = 0.08) as well as Et (R2 = 0.09) when inverse functions0are used. Furthermore, similar results between xm0 and RL or UCS are also obtained, with R2 20 70 120 170 0 20 40 60 80 0.35 and 0.57, respectively. It is mentioned that very good to excellent correlations between optimum Tangent modulus of elasticity Et, GPa Uniaxial compressive strength UCS, MPa size xm and other variables analysis. Thus, it is (c) are obtained, if marble is excluded from regression (d) deduced that the optimum size xm increases with decreasing P-wave velocity, Schmidt rebound value, Figure 13. Correlations between maximum breakage Sm and (a) P-wave velocity Vp,tested (b) Schmidt for uniaxial tangentbreakage modulusrate of elasticity, all materials Figure 13.compressive Correlationsstrength betweenand maximum rate Sm and (a)for P-wave velocity Vp , (b)except Schmidt rebound values RL, obvious (c) uniaxial compressive strength UCS of and (d) tangent modulusthe of optimum elasticity Esize t; marble. It is therefore that as strength or stiffness a material increases rebound values RL , (c) uniaxial compressive strength UCS and (d) tangent modulus of elasticity Eatt ; horizontal error bars show standard deviation of the measurements. which the error breakage obtains its maximum value becomes finer. horizontal bars rate show standard deviation of the measurements.
xm, mm
xm, mm
Figure 14 shows the correlations between the optimum size xm and Vp, RL, UCS and Et. It is seen that weak3 correlations exist between xm and Vp (R2 = 0.08)3 as well as Et (R2 = 0.09) when inverse functions are used. Furthermore, similar results between xm and RL or UCS are also obtained, with R2 0.35 and 0.57, respectively. It is mentioned that very good to2excellent correlations between optimum 2 size xm and other variables are obtained, if marble is excluded from regression analysis. Thus, it is -0.50 y = 14.69x deduced that the optimum sizey x=m2.90x increases with decreasing P-wave velocity, Schmidt rebound value, -0.16 R² = 0.35 1 1 R² = 0.08 uniaxial compressive strength and tangent modulus of elasticity, for all materials tested except for marble. It is therefore obvious that as strength or stiffness of a material increases the optimum size at which the breakage rate obtains its maximum value becomes finer. 0
0
0
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xm, mm xm, mm
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Schmidt rebound values RL
3
y = 2.90x-0.16 R² = 0.08
y = 13.15x-0.41 R² = 0.57
1 0 0
5
0 20
70
(a)
120
10
30
50
70
Schmidt rebound values RL
0 170
Uniaxial compressive strength UCS, MPa
(c)
y = 4.28x-0.18 R² = 0.09
1 0
10
P-wave velocity Vp, km/s
y = 14.69x-0.50 R² = 0.35
2 1
10
30
(b)50
70
90
Tangent modulus of elasticity Et, GPa
(d)
Figure 14. Correlations between optimum size xm and (a) P-wave velocity Vp, (b) Schmidt rebound
Figure 14. Correlations between optimum size xm and (a) P-wave velocity Vp , (b) Schmidt rebound values RL, (c) uniaxial compressive strength UCS and (d) tangent modulus of elasticity Et; horizontal values RL , (c) uniaxial compressive strength UCS and (d) tangent modulus of elasticity Et ; horizontal error bars show standard deviation of the measurements. error bars show standard deviation of the measurements.
The results for marble are indeed peculiar. It is known that the behavior of a material during breakage depends on several factors including mainly rock size, grain size and microstructure, crystallinity, porosity, mixtures of grinding media, presence of secondary mineral phases and microcracks [26,60,61]. In addition, breakage of mineral grains may be affected by the presence of inter-granular cementations [62]. Furthermore, the energy required for plastic deformation is different for each material and this affects breakage rate [63]. In a recent study, a spheroid model was used to identify the role of shape in particle size analysis [64]. One major difference between marble and the other three materials examined in our study is that the formation of marble is the result of
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The results for marble are indeed peculiar. It is known that the behavior of a material during breakage depends on several factors including mainly rock size, grain size and microstructure, crystallinity, porosity, mixtures of grinding media, presence of secondary mineral phases and microcracks [26,60,61]. In addition, breakage of mineral grains may be affected by the presence of inter-granular cementations [62]. Furthermore, the energy required for plastic deformation is different for each material and this affects breakage rate [63]. In a recent study, a spheroid model was used to identify the role of shape in particle size analysis [64]. One major difference between marble and the other three materials examined in our study is that the formation of marble is the result of metamorphism of sedimentary carbonate rocks, mainly limestones or dolomites. Twinning planes (planes along which two crystals meet) are also present in marbles. Furthermore, during grinding of marble ultra-fine particles are generated and this may affect breakage rate and final particle size distribution. In a recent study, it has been shown that the initial breakage rate of coarse particles increases with increasing proportion of fines in the mixture; the presence of a big proportion of fines indicates that they exhibit a much smaller collision cross-section compared to coarser particles [65]. All these factors definitely affect the behavior of marble during breakage and thus additional experimental studies are required to elucidate this aspect. 3.5.2. Multiple Regression Analysis Multiple regression analysis is an extension of simple regression analysis and involves one dependent and two or more independent variables [66]. In our study, multiple regression analysis was performed, using two independent variables, to identify correlations between breakage rate parameters and Vp , RL , UCS and Et . The correlation matrix for breakage rate parameters (Si , αT , α and Sm ) and Vp , RL , UCS and Et is shown in Table 6. For each dependent variable the two independent variables with the highest correlation coefficients, as shown in the correlation matrix, were used to perform the analysis. Furthermore, because of the weak correlations between optimum size xm and other variables for all tested materials (Figure 14), this dependent variable was not taken into account in our study. Table 6. Correlation matrix for study variables. Property
Si
αT
α
Sm
Vp
Si αT α Sm Vp Et RL UCS
1 0.982 0.856 0.950 −0.922 −0.923 −0.990 −0.977
1 0.938 0.983 −0.938 −0.939 −0.991 −0.923
1 0.947 −0.886 −0.886 −0.896 −0.733
1 −0.865 −0.866 −0.983 −0.888
1 1.000 0.893 0.842
Et
RL
1 0.895 1 0.843 0.956
UCS
1
The equations obtained for each dependent variable are the following: Si = 1.440 − 0.007· R L − 0.002·UCS, R2 = 0.991
(7)
αT = 2.018 − 0.006·T − 0.011· R L , R2 = 0.995
(8)
α = 1.346 − 0.004· R L − 0.036·VP , R2 = 0.838
(9)
Sm = 3.977 − 0.054· R L − 0.012·UCS, R2 = 0.998
(10)
The results of multiple regression analysis indicate that when specific material properties are considered the breakage rate parameters of silica rich materials, namely quartz, quartzite and metasandstone, can be well determined. An exception is shown only for parameter xm for marble.
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4. Conclusions In this study, the correlation between the properties of four materials, namely quartz, marble, quartzite and metasandstone, and breakage rate parameters have been established. The properties of the materials that were taken into account were P-wave velocity, Schmidt rebound values, uniaxial compressive strength and tangent modulus of elasticity. The breakage rate parameters that were determined through batch grinding experiments were breakage rate Si , parameter αT , parameter α, maximum breakage rate Sm and optimum particle size xm . The results can be summarized as follows: Breakage rate Si or parameter αT increase with decreasing uniaxial compressive strength (UCS) and tangent modulus of elasticity (Et ). Therefore, it is deduced that as the strength or stiffness of a material is reduced this will be ground faster in a mill. Similar results can be obtained from the P-wave velocity (Vp ) and Schmidt rebound values (RL ), which are widely considered for the indirect estimation of material’s hardness and strength. The properties of the tested materials can be used for an accurate prediction of Si or αT of the ground materials since the obtained correlation coefficients are high. Parameter α which is constant for the same material and independent of grinding conditions is well correlated with Vp , RL and Et when inverse exponential functions are used. However, the results showed that α has a moderate correlation with UCS (R2 = 0.67) and therefore the results of the functions derived are not very reliable. Maximum breakage rate Sm is strongly correlated with RL and UCS using inverse functions and therefore either RL or UCS values can be used to predict the maximum breakage rate of each tested material during grinding. However, the prediction of Sm from Vp or Et is not fully reliable due to the moderate correlations between these parameters. As far as optimum size xm is concerned, the results showed that optimum size xm increases with decreasing P-wave velocity, Schmidt rebound value, uniaxial compressive strength and tangent modulus of elasticity. However, marble seems to deviate from this tendency and further research is required, by considering its properties and other aspects including the presence of twinning planes and the production of ultra-fine material during grinding, to elucidate its behavior. It is mentioned that the behavior of mixed materials and the effect of each material type on the overall breakage rate is now under investigation. Acknowledgments: The authors would like to thank the anonymous reviewers of this article for their constructive comments and recommendations. The authors also gratefully acknowledge S. Mavrigiannakis, Mining Engineer, Faculty of Rock Mechanics Laboratory, Technical University of Crete, for performing the UCS tests. Author Contributions: Evangelos Petrakis conceived, designed and performed the experiments, analyzed the data and wrote the paper; Konstantinos Komnitsas critically reviewed the experimental design, the analysis of the results and wrote the paper. Conflicts of Interest: The authors declare no conflict of interest.
Nomenclature D d Et fc J i N Qi RL R2 Si Sm t
internal diameter of the mill (m) ball diameter (mm) tangent modulus of elasticity (GPa) powder filling volume (%) ball filling volume (%) size class index rotational speed (rpm) correction factor (-) Schmidt hammer rebound value (-) correlation coefficient (-) breakage rate (min−1 ) maximum breakage rate (min−1 ) grinding time (min)
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U UCS Vp x0 xi xm α αT ε Λ µ ρb
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interstitial filling (-) uniaxial compressive strength (MPa) P-wave velocity (km/s) standard size 1 mm the upper size of size class i (mm) optimum feed size (mm) parameter depending on material properties (-) parameter depending on milling conditions (-) bed porosity (%) parameter depending on material properties (-) parameter depending on milling conditions (-) specific gravity of the balls (g/cm3 )
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