# Correlation between Material Properties and Breakage Rate Parameters Determined from Grinding Tests

^{*}

## Abstract

**:**

_{p}), Schmidt rebound value (R

_{L}), uniaxial compressive strength (UCS) and tangent modulus of elasticity (E

_{t}), while the breakage rate parameters determined from batch grinding tests, include breakage rate S

_{i}, maximum breakage rate S

_{m}, α

_{T}and α, and optimum particle size x

_{m}. The results indicate that the properties of all materials examined show very good correlation and can be used to predict S

_{i}or α

_{T}. Furthermore, parameter α is well correlated with V

_{p}, R

_{L}and E

_{t}using inverse exponential functions, while S

_{m}is strongly correlated with R

_{L}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 x

_{m}for marble.

## 1. Introduction

_{i}and the breakage function b

_{i,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 [18,19,20,21].

_{i}(min

^{−1}) as a function of particle size x

_{i}(mm) has been considered by Austin [15] and is expressed with Equation (1), which is accepted by many researchers [22,23,24].

_{i}(mm) is the upper size of class i, x

_{0}is the standard size (1 mm), α

_{T}(parameter that depends on milling conditions) is the breakage rate for size x

_{i}= 1 mm and α is a characteristic parameter depending on material properties. Q

_{i}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]. Q

_{i}is calculated from Equation (2) [15],

_{m}for which the breakage rate reaches its maximum value is related with parameter µ through the following equation [25,26],

_{m}is directly proportional to μ since for the same material α and Λ are constants.

_{L}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 [42,43,44].

## 2. Materials and Methods

_{L}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 for each material were recorded. The rebound values given in this study for each material are the average of all recorded values.

_{t}). 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).

^{3}(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 25.4 mm (1 inch) stainless steel balls with density ρ

_{b}= 7.85 g/cm

^{3}. The parameters J (ball filling volume) and f

_{c}(material filling volume), were calculated from Equations (4) and (5), respectively.

_{c}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.

## 3. Results and Discussion

#### 3.1. P-Wave Velocity of Materials

_{p}value is quartz, marble, quartzite and metasandstone. The statistical results of P-wave velocity also show that the mean V

_{p}values of the materials follow the same order as the median values. Furthermore, low standard deviation is observed for V

_{p}for all the materials tested and thus the data are clustered around the mean V

_{p}values. It is also mentioned that similar values were obtained in other studies for marble and quartzite [51], sandstone [52] and quartz [53].

#### 3.2. Schmidt Rebound Values

_{L}value is quartz, quartzite, marble and metasandstone. The statistical results of R

_{L}for the materials tested also show that the mean R

_{L}values follow the same descending order as observed for the median values. Finally, it is also indicated that the values obtained in this study are well within the range of values reported in other studies for quartz and quartzite [54], marble [55] and metasandstone [56].

#### 3.3. Uniaxial Compressive Strength and Modulus of Elasticity

_{t}for each material are shown in Table 3 and Table 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 (E

_{t}) is concerned, the descending order of materials with respect to the mean E

_{t}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].

#### 3.4. Breakage Rate Parameters

_{i}is independent of time and the S

_{i}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 S

_{i}values, obtained from the first order plots, with the upper feed particle size. It is shown that for each material S

_{i}increases up to a specified size x

_{m}(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 x

_{m}for each material is the size at which, under normal grinding conditions, the highest breakage rate S

_{m}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].

#### 3.5. Correlation between Material Properties and Breakage Rate Parameters

#### 3.5.1. Simple Regression Analysis

^{2}was considered for determining the breakage rate parameters (S

_{i}, α

_{Τ}, α, S

_{m}and x

_{m}). It is mentioned that no similar functions are given so far in other studies in literature.

_{i}values and V

_{p}, R

_{L}, UCS and E

_{t}are presented in Figure 10. It is seen that very strong correlation between S

_{i}and R

_{L}(R

^{2}= 0.98) and strong correlations between S

_{i}and V

_{p}(R

^{2}= 0.85) and E

_{t}(R

^{2}= 0.85) are obtained with the use of inverse linear functions. Furthermore, S

_{i}is very well correlated with UCS with the use of an inverse exponential function (R

^{2}= 0.99).

_{Τ}and V

_{p}, R

_{L}, UCS and E

_{t}(Figure 11). Parameter α

_{Τ}, which defines the breakage rate for a feed size of 1 mm, is correlated very well with V

_{p}, R

_{L}and E

_{t}using an inverse linear relationship. The very strong correlation (R

^{2}= 1) between α

_{T}and R

_{L}is considered very important for the design of grinding circuits. R

_{L}values are obtained from the 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.

_{p}, R

_{L}, UCS and E

_{t}. It is seen that parameter α is very well correlated with V

_{p}, R

_{L}and E

_{t}with the use of inverse exponential functions. Parameter α is constant for the same material and therefore the functions obtained can be used independently of grinding conditions. However, a moderate correlation (R

^{2}= 0.67) between α and UCS is indicated.

_{m}and V

_{p}, R

_{L}, UCS and E

_{t}are presented in Figure 13. Very strong correlation (R

^{2}= 0.96) is shown between S

_{m}and R

_{L}with the use of an inverse linear function. Furthermore, S

_{m}seems to follow an inverse exponential relationship with UCS, with correlation coefficient R

^{2}= 0.95. However, the prediction of S

_{m}from V

_{p}and E

_{t}values is somehow risky due to the moderate correlation between these variables.

_{m}and V

_{p}, R

_{L}, UCS and E

_{t}. It is seen that weak correlations exist between x

_{m}and V

_{p}(R

^{2}= 0.08) as well as E

_{t}(R

^{2}= 0.09) when inverse functions are used. Furthermore, similar results between x

_{m}and R

_{L}or UCS are also obtained, with R

^{2}0.35 and 0.57, respectively. It is mentioned that very good to excellent correlations between optimum size x

_{m}and other variables are obtained, if marble is excluded from regression analysis. Thus, it is deduced that the optimum size x

_{m}increases with decreasing P-wave velocity, Schmidt rebound value, 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.

#### 3.5.2. Multiple Regression Analysis

_{p}, R

_{L}, UCS and E

_{t}. The correlation matrix for breakage rate parameters (S

_{i}, α

_{Τ}, α and S

_{m}) and V

_{p}, R

_{L}, UCS and E

_{t}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 x

_{m}and other variables for all tested materials (Figure 14), this dependent variable was not taken into account in our study.

_{m}for marble.

## 4. Conclusions

_{i}, parameter α

_{T}, parameter α, maximum breakage rate S

_{m}and optimum particle size x

_{m}. The results can be summarized as follows:

_{i}or parameter α

_{Τ}increase with decreasing uniaxial compressive strength (UCS) and tangent modulus of elasticity (E

_{t}). 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 (V

_{p}) and Schmidt rebound values (R

_{L}), 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 S

_{i}or α

_{Τ}of the ground materials since the obtained correlation coefficients are high.

_{p}, R

_{L}and E

_{t}when inverse exponential functions are used. However, the results showed that α has a moderate correlation with UCS (R

^{2}= 0.67) and therefore the results of the functions derived are not very reliable.

_{m}is strongly correlated with R

_{L}and UCS using inverse functions and therefore either R

_{L}or UCS values can be used to predict the maximum breakage rate of each tested material during grinding. However, the prediction of S

_{m}from V

_{p}or E

_{t}is not fully reliable due to the moderate correlations between these parameters.

_{m}is concerned, the results showed that optimum size x

_{m}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

## Author Contributions

## Conflicts of Interest

## Nomenclature

D | internal diameter of the mill (m) |

d | ball diameter (mm) |

E_{t} | tangent modulus of elasticity (GPa) |

f_{c} | powder filling volume (%) |

J | ball filling volume (%) |

i | size class index |

N | rotational speed (rpm) |

Q_{i} | correction factor (-) |

R_{L} | Schmidt hammer rebound value (-) |

R^{2} | correlation coefficient (-) |

S_{i} | breakage rate (min^{−1}) |

S_{m} | maximum breakage rate (min^{−1}) |

t | grinding time (min) |

U | interstitial filling (-) |

UCS | uniaxial compressive strength (MPa) |

V_{p} | P-wave velocity (km/s) |

x_{0} | standard size 1 mm |

x_{i} | the upper size of size class i (mm) |

x_{m} | optimum feed size (mm) |

α | parameter depending on material properties (-) |

α_{T} | parameter depending on milling conditions (-) |

ε | bed porosity (%) |

Λ | parameter depending on material properties (-) |

μ | parameter depending on milling conditions (-) |

ρ_{b} | specific gravity of the balls (g/cm^{3}) |

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**Figure 5.**(

**a**) Cylindrical specimen subjected to loading; (

**b**) Stress-strain curve for the determination of UCS and E

_{t}.

**Figure 8.**First order plots showing the mass % remaining for each size fraction of each material vs. grinding time; (

**a**) quartz, (

**b**) marble, (

**c**) quartzite and (

**d**) metasandstone.

**Figure 10.**Correlations between breakage rate S

_{i}and (

**a**) P-wave velocity V

_{p}, (

**b**) Schmidt rebound values R

_{L}, (

**c**) uniaxial compressive strength UCS and (

**d**) tangent modulus of elasticity E

_{t}; horizontal error bars show standard deviation of the measurements.

**Figure 11.**Correlations between parameter α

_{Τ}and (

**a**) P-wave velocity V

_{p}, (

**b**) Schmidt rebound values R

_{L}, (

**c**) uniaxial compressive strength UCS and (

**d**) tangent modulus of elasticity E

_{t}; horizontal error bars show standard deviation of the measurements.

**Figure 12.**Correlations between parameter α and (

**a**) P-wave velocity V

_{p}, (

**b**) Schmidt rebound values R

_{L}, (

**c**) uniaxial compressive strength UCS and (

**d**) tangent modulus of elasticity E

_{t}; horizontal error bars show standard deviation of the measurements.

**Figure 13.**Correlations between maximum breakage rate S

_{m}and (

**a**) P-wave velocity V

_{p}, (

**b**) Schmidt rebound values R

_{L}, (

**c**) uniaxial compressive strength UCS and (

**d**) tangent modulus of elasticity E

_{t}; horizontal error bars show standard deviation of the measurements.

**Figure 14.**Correlations between optimum size x

_{m}and (

**a**) P-wave velocity V

_{p}, (

**b**) Schmidt rebound values R

_{L}, (

**c**) uniaxial compressive strength UCS and (

**d**) tangent modulus of elasticity E

_{t}; horizontal error bars show standard deviation of the measurements.

Material | Porosity % | Density g/cm^{3} |
---|---|---|

quartz | 0.02 | 2.65 |

marble | 0.3 | 2.72 |

quartzite | 0.9 | 2.59 |

metasandstone | 8.0 | 2.11 |

Mill | Balls | Material | |||
---|---|---|---|---|---|

diameter, D (cm) | 20.4 | diameter, d (mm) | 25.4 | density (g/cm^{3}) | quartz (2.65) |

length, L (cm) | 16.6 | number | 77 | marble (2.72) | |

volume, V (cm^{3}) | 5423 | weight (g) | 5149 | quartzite (2.59) | |

operational speed, Ν (rpm) | 66 | density (g/cm^{3}) | 7.85 | metasandstone (2.11) | |

critical speed, Ν_{c} (rpm) | 93.7 | porosity (%) | 40 | material filling volume, f_{c} (%) | 4 |

ball filling volume, J (%) | 20 | interstitial filling, U (%) | 50 |

Material | Observations | Mean | Minimum | Maximum | Median | Std. Dev. |
---|---|---|---|---|---|---|

Quartz | 7 | 135.1 | 122.0 | 151.5 | 132.0 | 12.4 |

Marble | 8 | 68.0 | 58.6 | 100.4 | 63.5 | 13.7 |

Quartzite | 7 | 80.9 | 68.0 | 110.0 | 76.7 | 15.4 |

Metasandstone | 7 | 57.6 | 42.6 | 74.6 | 64.0 | 14.0 |

Material | Observations | Mean | Minimum | Maximum | Median | Std. Dev. |
---|---|---|---|---|---|---|

Quartz | 7 | 59.5 | 47.5 | 71.2 | 62.8 | 9.1 |

Marble | 8 | 48.2 | 21.5 | 64.9 | 53.0 | 15.8 |

Quartzite | 7 | 41.7 | 26.7 | 56.2 | 44.8 | 11.0 |

Metasandstone | 7 | 27.1 | 10.7 | 39.0 | 31.0 | 10.9 |

Material | S_{i} | α_{Τ} | α | S_{m} | x_{m} |
---|---|---|---|---|---|

min^{−1} | min^{−1} | min^{−1} | mm | ||

Quartz | 0.55 | 0.68 | 0.80 | 0.85 | 1.80 |

Marble | 0.89 | 1.09 | 0.90 | 1.83 | 2.84 |

Quartzite | 0.83 | 0.98 | 0.84 | 1.29 | 2.00 |

Metasandstone | 1.05 | 1.41 | 1.12 | 2.45 | 2.32 |

Property | S_{i} | α_{Τ} | α | S_{m} | V_{p} | E_{t} | R_{L} | UCS |
---|---|---|---|---|---|---|---|---|

S_{i} | 1 | |||||||

α_{Τ} | 0.982 | 1 | ||||||

α | 0.856 | 0.938 | 1 | |||||

S_{m} | 0.950 | 0.983 | 0.947 | 1 | ||||

V_{p} | −0.922 | −0.938 | −0.886 | −0.865 | 1 | |||

E_{t} | −0.923 | −0.939 | −0.886 | −0.866 | 1.000 | 1 | ||

R_{L} | −0.990 | −0.991 | −0.896 | −0.983 | 0.893 | 0.895 | 1 | |

UCS | −0.977 | −0.923 | −0.733 | −0.888 | 0.842 | 0.843 | 0.956 | 1 |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Petrakis, E.; Komnitsas, K. Correlation between Material Properties and Breakage Rate Parameters Determined from Grinding Tests. *Appl. Sci.* **2018**, *8*, 220.
https://doi.org/10.3390/app8020220

**AMA Style**

Petrakis E, Komnitsas K. Correlation between Material Properties and Breakage Rate Parameters Determined from Grinding Tests. *Applied Sciences*. 2018; 8(2):220.
https://doi.org/10.3390/app8020220

**Chicago/Turabian Style**

Petrakis, Evangelos, and Konstantinos Komnitsas. 2018. "Correlation between Material Properties and Breakage Rate Parameters Determined from Grinding Tests" *Applied Sciences* 8, no. 2: 220.
https://doi.org/10.3390/app8020220