# A Ternary Model for Particle Packing Optimization

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

## 3. Theory and Results

_{i}be the volume fraction of the coarser material of any two consecutive powder samples (1).

_{1}as follows (2):

_{i}, can be defined in terms of the specific true gravity of the sample. For the coarsest powder sample packed initially into our hypothetical vessel, we define ${\mathrm{w}}_{1}={\mathrm{G}}_{1}{\mathrm{v}}_{\mathrm{s}1}=\left(1-{\mathrm{e}}_{1}\right){\mathrm{G}}_{1}$ where G

_{i}is the specific gravity.

_{i}, is not to be confused with the percent composition, ${\mathsf{\omega}}_{\mathrm{i}}$. The apparent specific gravity, ${\mathrm{G}}_{\mathrm{a}}=\left(1-\mathrm{e}\right)\mathrm{G}$, where G is the true specific gravity. Thus (4),

_{B}, which is the sum of the above three expressions (5)–(7), becomes (8):

_{d}is a reduction factor ranging from 0 and 1 and can only be determined experimentally. Furnas experimented with different diameter ratios of different mixtures and, in each case, determined the corresponding contraction in volume that resulted [47]. With this data, he generated a quadratic fitting and an equation for the parameter y. It follows that thisparameter is exactly our parameter k

_{d}. Below is a table of measured and calculated values for y and k

_{d}. The measured values for y for corresponding K

^{1/n}values are from the paper by Furnas (see Table 1):

_{d}and y are related (Figure 5):

_{d}and y, we define k

_{d}similarly as (11):

_{d}) is (15):

## 4. Discussion

_{i}), specific gravities (G

_{i}), and average diameters of the components (d

_{i}) were given as follows (see Table 2).

_{2}) as in the table above.

_{f}), compared with 0.8260 for the Furnas model [47]. The proportion by volume of each component required to produce the maximum density of the system compare as follows (Table 3):

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Wall and Loosening Effect [44].

**Figure 2.**(

**a**) Portion of a random packing of spherical particles (porosity = 30%); (

**b**) Scheme of the sphere arrangement used to represent an agglomerate of spheres [45].

**Figure 3.**Relationship between packing density and particle size [46].

**Figure 5.**A graph of the volume reduction factor (vertical axis) vs. the ratio of consecutive sizes, k (horizontal axis).

K^{1/n} | y_{obs} | y_{computed} | k_{d} |
---|---|---|---|

0 | 1 | 1 | 1 |

0.025 | 0.945 | 0.9355125 | 0.9355125 |

0.05 | 0.83 | 0.87305 | 0.87305 |

0.1 | 0.71 | 0.7542 | 0.7542 |

0.2 | 0.51 | 0.5408 | 0.5408 |

0.3 | 0.375 | 0.3598 | 0.3598 |

0.4 | 0.245 | 0.2112 | 0.2112 |

0.5 | 0.12 | 0.095 | 0.095 |

1 | 0 | 0 | 0 |

Limestone | Fine Sand | Cement | |
---|---|---|---|

d_{i} (in) | 2 | $\sqrt{2\times 0.001}=0.045$ | 0.001 |

e_{i} | 0.48 | 0.42 | 0.52 |

G_{i} | 2.5 | 2.65 | 3.10 |

Proportions for Max. Density: | Proportions --> Furnas Model | % Error | ||
---|---|---|---|---|

limestone | v_{1}/V_{f} = 59.5% | limestone | v_{1}/V_{f} = 59.7% | 0.37% |

fine sand | v_{2}/V_{f} = 28.5% | fine sand | v_{2}/V_{f} = 28.3% | 0.75% |

cement | v_{1}/V_{f} = 12.0% | cement | v_{1}/V_{f} = 12% | 0.43% |

Name of Powder | Boron Powder (Coarse) | Boron Powder (Middle) | Boron Powder (Fine) |
---|---|---|---|

Diameter (${\mathrm{d}}_{\mathrm{n}}$), μm | 106 | 90 | 75 |

Bulk Volume (${\mathrm{V}}_{\mathrm{b}}$), cm^{3} | 10.68 ± 0.17 | 9.58 ± 0.17 | 9.14 ± 0.12 |

Tap Volume (${\mathrm{V}}_{\mathrm{s}}$), cm^{3} | 8.50 ± 0.14 | 7.80 ± 0.14 | 7.60 ± 0.14 |

Volume of void (${\mathrm{V}}_{\mathrm{v}}$), cm^{3} | 2.18 ± 0.18 | 1.78 ± 0.18 | 1.54 ± 0.18 |

Specific Gravity, S | 2.34 | 2.34 | 2.34 |

Void ratio/void percentage (e) | 0.20 ± 0.08 | 0.19 ± 0.10 | 0.17 ± 0.12 |

Method | Mix Ratios | Density (g/cm^{3}) | |
---|---|---|---|

1 | Proposed model | 27.67: 5.67: 1 | 1.69 ± 0.11 |

2 | Experimental 1 | 27.67: 5.67: 1 | 1.63 ± 0.04 |

3 | Experimental 2 | 27.67: 1: 5.67 | 1.44 |

4 | Experimental 3 | 1: 5.67: 27.67 | 1.26 |

5 | Experimental 4 | 1: 27.67: 5.67 | 1.29 |

6 | Experimental 5 | 5.67: 27.67: 1 | 1.26 |

7 | Experimental 6 | 5.67: 1: 27.67 | 1.24 |

8 | Experimental 7 | 11.44: 11.44: 11.44 | 1.27 |

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**MDPI and ACS Style**

Abu-Lebdeh, T.M.; Damptey, R.; Ungureanu, L.M.; Petrescu, F.I.T.
A Ternary Model for Particle Packing Optimization. *J. Compos. Sci.* **2022**, *6*, 113.
https://doi.org/10.3390/jcs6040113

**AMA Style**

Abu-Lebdeh TM, Damptey R, Ungureanu LM, Petrescu FIT.
A Ternary Model for Particle Packing Optimization. *Journal of Composites Science*. 2022; 6(4):113.
https://doi.org/10.3390/jcs6040113

**Chicago/Turabian Style**

Abu-Lebdeh, Taher M., Ransford Damptey, Liviu Marian Ungureanu, and Florian Ion Tiberiu Petrescu.
2022. "A Ternary Model for Particle Packing Optimization" *Journal of Composites Science* 6, no. 4: 113.
https://doi.org/10.3390/jcs6040113