# A Study of the Effect of Medium Viscosity on Breakage Parameters for Wet Grinding

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

_{j}) for a ball grinding process in terms of the rheology of the system. In addition to this, a linear adjustment was established for the relationship between specific breakage rates with and without PAM addition, based on the reduced viscosity, μ

_{r}. Furthermore, within a certain interval of viscosity, it was proved that an increment of viscosity can increase the specific breakage rate, and consequently the grinding degree.

## 1. Introduction

_{i}and the breakage function (b

_{ij}), which provide the fundamental size-mass balance equation for fully mixed batch grinding operations.

_{j}) for a ball grinding process in terms of the rheology of the system. In addition to this, a linear adjustment could be established for the relationship between specific breakage rates with and without PAM addition, based on the reduced viscosity, μ

_{r}. Furthermore, within the validity interval of viscosity, it will be studied whether an increment of viscosity can increase the specific breakage rate, and consequently the grinding degree.

## 2. Methodology

#### 2.1. Materials

#### 2.2. Methods

^{−1}. This value was experimentally determined considering a fixed rotational speed of 100 rpm, and according to the Brookfield recommendations for the selected configuration.

_{j}, in terms of the variables that it is expected to have more influence on its behavior:

- viscosity of the suspending fluid, μ
_{l}; - viscosity of the suspension, μ
_{s}; - particle diameter, d
_{p}; - density of the grinding media, ρ
_{b}.

## 3. Results and Discussion

^{−1}.

_{j}, in the case of each monosize. Greater values of S

_{j}pose higher probability of particle breakage, so in general terms it seems that, within certain intervals, an increase in the viscosity of suspension can entail an increase in the breakage probability. This effect is more evident in the case of monosize 53 microns (Figure 5), while in the case of monosizes 45 and 38 microns over some value of suspension viscosity, an increase in suspension viscosity reduces the probability of particle breakage (Figure 6 and Figure 7). This behavior is different in the case of different ball sizes in the grinding charge, although in the case of 3 cm the variations in S

_{j}due to suspension viscosity changes is less noticeable. This probably can be caused by hydrodynamic reasons due to the combination of particle and ball dimensions, and rheological conditions, but this way of analysis evidences that the relationships among variables and their influence cannot be easily analyzed and understood, for further research should be performed to define a clear picture of the phenomenological model. This is the reason why an alternative way of analyzing these data was tried and introduced in this paper. To do this, the following notation is going to be used: We will refer as S

_{jw}in the case of specific breakage rate when the fluid is just water, and S

_{jd}in the rest of the cases in which the water properties have been modified with the dissolution of the abovementioned reagent.

_{r}is the reduced viscosity.

_{1}versus Π

_{2}/Π

_{3}(see Figure 8) we can analyze the behavior the specific grinding rate at different specific gravities as a function of μ

_{r}. Figure 8, Figure 9 and Figure 10 show the behavior of the abovementioned relationship for the systems studied. In all cases, an increase of μ

_{r}poses a decrease in Π

_{1}, that is, an increase in S

_{jd}. In most of the cases the linearity obtained can be satisfactory, although in the case of 2 cm diameter balls, the R

^{2}values are lower in the case of the monosizes 45 and 38 microns.

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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Al_{2}O_{3} | SiO_{2} | Fe_{2}O_{3} | TiO_{2} | CaO | MgO | Na_{2}O | K_{2}O | P_{2}O_{5} | LOI |
---|---|---|---|---|---|---|---|---|---|

6.49 | 89.42 | 0.73 | 0.67 | <0.1 | <0.1 | 0.03 | 1.10 | <0.1 | 1.54 |

As | Ba | Sr | Sb | Co | Cr | Cu | Cd | Hg | Pb | Zn | Zr | Ni | Mn | Sn |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

18 | 142 | 15 | <10 | 14 | 31 | 22 | <10 | <10 | <10 | <10 | 85 | <10 | 17 | <10 |

Sample | Quartz |
---|---|

Solids concentration, ϕ (%w/v) | 60 |

Mill lentgh, L (m) | 0.18 |

Mill diameter, D (m) | 0.16 |

L/D ratio | 1.16 |

Fraction of critical speed, ϕ_{C} | 0.75 |

Ball filling fraction, J | 0.3 |

Hole fraction, U | 1.0 |

Bed normal porosity | 0.4 |

Ball Diameter (cm) | |||||||
---|---|---|---|---|---|---|---|

2 | 3 | 4 | |||||

Monosize (μm) | Suspension Fluid Viscosity (cP) | S_{j} (min^{−1}) | R^{2} | S_{j} (min^{−1}) | R^{2} | S_{j} (min^{−1}) | R^{2} |

53 | 1 | 0.112 | 0.9793 | 0.152 | 0.9771 | 0.205 | 0.9826 |

53 | 4 | 0.215 | 0.9843 | 0.171 | 0.9455 | 0.155 | 0.9612 |

53 | 6 | 0.220 | 0.9739 | 0.175 | 0.9892 | 0.252 | 0.9890 |

53 | 8 | 0.198 | 0.9860 | 0.213 | 0.9796 | 0.271 | 0.9809 |

45 | 1 | 0.108 | 0.9532 | 0.153 | 0.9640 | 0.044 | 0.9820 |

45 | 4 | 0.141 | 0.9814 | 0.151 | 0.9834 | 0.092 | 0.9869 |

45 | 6 | 0.178 | 0.9619 | 0.158 | 0.9620 | 0.177 | 0.9871 |

45 | 8 | 0.087 | 0.9567 | 0.165 | 0.9829 | 0.199 | 0.9896 |

38 | 1 | 0.057 | 0.9638 | 0.107 | 0.9797 | 0.030 | 0.921 |

38 | 4 | 0.113 | 0.9824 | 0.130 | 0.9891 | 0.077 | 0.9706 |

38 | 6 | 0.161 | 0.9885 | 0.155 | 0.9870 | 0.100 | 0.9780 |

38 | 8 | 0.044 | 0.9820 | 0.163 | 0.9723 | 0.053 | 0.9891 |

Ball Size (cm) | Monosize (μm) | a | b | R^{2} | Range of Linearity |
---|---|---|---|---|---|

2 | 53 | −0.2511 | 15.9780 | 0.9946 | 1–6 cp |

45 | −0.1432 | 12.9380 | 0.8073 | 1–6 cp | |

38 | −0.2785 | 16.6670 | 0.8335 | 1–6 cp | |

3 | 53 | −0.0586 | 12.0930 | 0.9991 | 1–6 cp |

45 | −0.2529 | 16.0120 | 0.9953 | 4–8 cp | |

38 | −0.8789 | 24.3650 | 0.9819 | 4–8 cp | |

4 | 53 | −1.7286 | 30.8380 | 0.9983 | 4–8 cp |

45 | −2.2260 | 35.1930 | 0.9855 | 4–8 cp | |

38 | −0.2501 | 12.9700 | 0.9334 | 1–6 cp |

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**MDPI and ACS Style**

Osorio, A.M.; Bustamante, M.O.; Restrepo, G.M.; M. López, M.M.; Menéndez-Aguado, J.M.
A Study of the Effect of Medium Viscosity on Breakage Parameters for Wet Grinding. *Symmetry* **2019**, *11*, 1202.
https://doi.org/10.3390/sym11101202

**AMA Style**

Osorio AM, Bustamante MO, Restrepo GM, M. López MM, Menéndez-Aguado JM.
A Study of the Effect of Medium Viscosity on Breakage Parameters for Wet Grinding. *Symmetry*. 2019; 11(10):1202.
https://doi.org/10.3390/sym11101202

**Chicago/Turabian Style**

Osorio, Adriana M., Moisés O. Bustamante, Gloria M. Restrepo, Manuel M. M. López, and Juan M. Menéndez-Aguado.
2019. "A Study of the Effect of Medium Viscosity on Breakage Parameters for Wet Grinding" *Symmetry* 11, no. 10: 1202.
https://doi.org/10.3390/sym11101202