# Breakage Characteristics of Heat-Treated Limestone Determined via Kinetic Modeling

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## Abstract

**:**

## 1. Introduction

_{3}), silica (SiO

_{2}) and clay minerals, so its CaO content is relatively low. Since the basic characteristics (low activity and inhomogeneity) of the raw materials are also less favorable, it is difficult to apply them to high value-added industries, e.g., in steelmaking and fine chemical processes.

## 2. Theory

#### Specific Rate of Breakage and Primary Breakage Distribution

_{j}(specific rate of breakage), and a distribution function, B

_{ij}(primary breakage distribution) [11]. These two functions use exponential forms with several parameters. Therefore, if the parameters of the functions are known, the particle size distribution of the ground particles can be determined at certain grinding times [12].

_{j}represents the probability that a particle will be broken by the application of force. Materials with higher strength have a smaller crushing probability [13], and a given material can show different probability depending on the particle size. In general, in a ball mill, the smaller the particle size, the smaller the S value. Below the maximum size, the relationship between the particle size and specific rate of breakage can be expressed as

_{j}is the particle size for size interval j; x

_{0}is the standard particle size (normally 1 mm); A is specific rate of breakage at x

_{0}; and α is the slope of the curve. By convention, x

_{1}and x

_{n}represent the size intervals for the largest particles and smallest particles, respectively. The variables in square brackets reflect the phenomenon that the S value decreases when the particle size becomes too large compared to the ball size [11].

_{ij}represents the distribution of fragments produced by each particle’s breakage. It is expressed as the sum of two exponential forms using Φ, γ, and β as follows:

_{i}(t) is the mass of the material in the size interval i after time t; S

_{i}and S

_{j}are specific rates of breakage for the material of size x

_{i}and x

_{j}, respectively; b

_{ij}is the breakage distribution, which represents the breakage of the material from size x

_{j}into size x

_{i}. The solution of Equation (3) using determined values of S

_{i}and b

_{ij}can provide the product size distribution for any grinding time [12].

## 3. Experimental

#### 3.1. Samples

#### 3.2. Conventional Ball Milling Tests

#### 3.3. Heat Treatment and Optical Sorting

## 4. Results and Discussion

#### 4.1. Effect of Heat Treatment on Limestone

_{3}) with whiteness > 86%. The brown sample group contains dolomite (CaMg(CO

_{3})

_{2}), with small amount of calcite and periclase (<2%) and a whiteness < 65%. Based on the optical separation, the brown and white groups were 45% and 55% of the total mass, respectively.

#### 4.2. Determination of Breakage Parameters by Experimental Methods

_{s}).

#### 4.3. Determination of Breakage Parameters by Back-Calculation Methods

_{i}(x) and n equations h

_{j}(x) [12].

_{i}is the weight fraction of original material of size i and p

_{i}is the weight fraction less than size i of the product.

## 5. Conclusions

- After heat treatment at 600 °C for 1 h, low-grade limestone can be divided into brown and white groups. XRD analysis showed that the brown group consists of mainly dolomite while the white limestone is composed mainly of calcite. These were separated by the optical sorting process based on color differences, and each sample group as well as the original ore showed various grinding characteristics in the ball mill.
- The breakage parameters of the kinetic model (specific rate of breakage and primary breakage distribution) were calculated from the size distribution of the product samples, and the predicted values of the size distribution agree well with the experimental data.
- The results from this study indicate that ball mill grinding of limestone can be described based on the population-balance model. This kinetic model approach can provide a basis for the design of milling processes for the utilization of low-grade limestone. In addition, it is expected to play a major role in optimizing the grinding process and improving the efficiency of limestone production.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 2.**Three types of sample material groups; original raw sample (O), brown sample (B) and white sample (W).

**Figure 6.**Experimental and simulation product size distributions for various batch grinding times (solid lines indicate the simulation results) for (O).

**Figure 7.**Experimental and simulation product size distributions for various batch grinding times (solid lines indicate the simulation results) for (B).

**Figure 8.**Experimental and simulation product size distributions for various batch grinding times (solid lines indicate the simulation results) for (W).

Component | SiO_{2} | Al_{2}O_{3} | Fe_{2}O_{3} | CaO | MgO | K_{2}O | Na_{2}O | TiO_{2} | MnO | P_{2}O_{5} | Ig loss |
---|---|---|---|---|---|---|---|---|---|---|---|

Content | 0.11 | 0.05 | 0.59 | 43.39 | 10.55 | 0.03 | 0.03 | <0.01 | 0.11 | 0.02 | 45.26 |

Sample Parameter | Value |
---|---|

Sample size (mesh) | 16 × 20 |

Mill diameter, D (cm) | 21 |

Mill length, L (cm) | 22 |

Ball type | Alumina |

Ball diameter, d (cm) | 2.5 |

Ball Charge, J | 0.3 |

Powder filling, U | 1.0 |

Rotation speed, φ_{c} | 69 rpm (70% of critical speed) |

Sample | Calcite | Dolomite | Periclase | Muscovite | Whiteness | wt % |
---|---|---|---|---|---|---|

(B) | 15 | 83.1 | 1.9 | - | 63.4 | 45 |

(W) | 100 | - | - | - | 85.7 | 55 |

Parameters | (O) | (B) | (W) |
---|---|---|---|

γ | 0.6 | 0.7 | 0.3 |

Φ | 0.4 | 0.6 | 0.5 |

β | 6.0 | 3.5 | 5.0 |

Breakage function | Parameters | Value | ||
---|---|---|---|---|

(O) | (B) | (W) | ||

Specific rate of breakage | A (1 mm) | 0.28 | 0.35 | 0.51 |

α | 0.37 | 0.85 | 1.39 | |

Primary distribution of breakage | γ | 0.58 | 0.67 | 0.15 |

Φ | 0.36 | 0.41 | 0.50 | |

β | 6.5 | 3.0 | 4.0 |

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**MDPI and ACS Style**

Lee, H.; Kim, K.; Kim, J.; You, K.; Lee, H.
Breakage Characteristics of Heat-Treated Limestone Determined via Kinetic Modeling. *Minerals* **2018**, *8*, 18.
https://doi.org/10.3390/min8010018

**AMA Style**

Lee H, Kim K, Kim J, You K, Lee H.
Breakage Characteristics of Heat-Treated Limestone Determined via Kinetic Modeling. *Minerals*. 2018; 8(1):18.
https://doi.org/10.3390/min8010018

**Chicago/Turabian Style**

Lee, Hoon, Kwanho Kim, Jeongyun Kim, Kwangsuk You, and Hansol Lee.
2018. "Breakage Characteristics of Heat-Treated Limestone Determined via Kinetic Modeling" *Minerals* 8, no. 1: 18.
https://doi.org/10.3390/min8010018