# Heat-Assisted Batch Settling of Mineral Suspensions in Inclined Containers

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## Abstract

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## 1. Introduction

## 2. Governing Equations

## 3. Simulation Cases and Boundary Conditions

## 4. Results and Discussion

#### 4.1. Effect of Temperature

#### 4.2. Effect of Particle Diameter

#### 4.3. Effect of Particle Concentration

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Cell. The heated downward-facing, heated wall corresponds to the right side boundary of the domain. The horizontal arrows denote the heat flux on the downward facing wall, while the diagonal arrows show a scheme of the liquid (and partially fine solid) fraction flowing upwards near it.

**Figure 2.**Typical flow and particle progression for the case $\Delta {T}_{\mathrm{DFW}}=30\text{}{}^{\xb0}\mathrm{C}$, ${\varphi}_{0}=0.05$, and ${d}_{S}=10\text{}\mathsf{\mu}\mathrm{m}$ and (from left to right) $t=200\text{}\mathrm{s}$, 800 s, 1600 s, and 3200 s. (

**a**) Particle concentration ($\varphi $, with color scale in logarithmic scale) and (

**b**) local parallel mean velocity, defined as $\langle \mathbf{u}\rangle \phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\widehat{\mathit{p}}$, with $\widehat{\mathit{p}}$ a unit vector parallel to the upward facing wall.

**Figure 4.**Spatiotemporal diagram of concentration for (

**a**) $\Delta {T}_{\mathrm{DFW}}=0$, (

**b**) $\Delta {T}_{\mathrm{DFW}}=20\text{}{}^{\xb0}\mathrm{C}$, (

**c**) $\Delta {T}_{\mathrm{DFW}}=30\text{}{}^{\xb0}\mathrm{C}$, and (

**d**) $\Delta {T}_{\mathrm{DFW}}=50\text{}{}^{\xb0}\mathrm{C}$. The x and y axes represent time and vertical position ($y=0$ standing for the top of the cell). False color represents horizontally-averaged particle concentration.

**Figure 5.**Effect of temperature on the mean particle volume concentration within $\mathrm{A}1={Y}_{z}W$, for ${d}_{s}=10\text{}\mathsf{\mu}\mathrm{m}$, $\theta =45{}^{\xb0}$, ${\varphi}_{0}=5\%$, and $W=5\text{}\mathrm{c}\mathrm{m}$. The mass in A1 in these cases is equivalent to 60% of the total mass in the cell. (

**a**) ${Y}_{z}=0.6{Y}_{\infty}$ and (

**b**) ${Y}_{z}=0.9{Y}_{\infty}$.

**Figure 6.**Particle accumulation at zone A3 for ${d}_{s}=10\text{}\mathsf{\mu}\mathrm{m}$, $\theta =45{}^{\xb0}$, ${\varphi}_{0}=5\%$, and $W=5\text{}\mathrm{c}\mathrm{m}$. (

**a**) Particle mass and (

**b**) the vertical position (${Y}_{\mathrm{sed}}$) of the particle sediment above the upward facing wall.

**Figure 7.**Mass resuspended at zone A2 for ${d}_{s}=10\text{}\mathsf{\mu}\mathrm{m}$, $\theta =45{}^{\xb0}$, ${\varphi}_{0}=5\%$, and $W=5\text{}\mathrm{c}\mathrm{m}$.

**Figure 8.**Spatiotemporal diagram of concentration for (

**a**) $({d}_{s},\Delta {T}_{\mathrm{DFW}})=(10\text{}\mathsf{\mu}\mathrm{m},0\text{}{}^{\xb0}\mathrm{C})$, (

**b**) $({d}_{s},\Delta {T}_{\mathrm{DFW}})=(10\text{}\mathsf{\mu}\mathrm{m},30\text{}{}^{\xb0}\mathrm{C})$, (

**c**) $({d}_{s},\Delta {T}_{\mathrm{DFW}})=(50\text{}\mathsf{\mu}\mathrm{m},0\text{}{}^{\xb0}\mathrm{C})$, (

**d**) $({d}_{s},\Delta {T}_{\mathrm{DFW}})=(50\text{}\mathsf{\mu}\mathrm{m},30\text{}{}^{\xb0}\mathrm{C})$, (

**e**) $({d}_{s},\Delta {T}_{\mathrm{DFW}})=(100\text{}\mathsf{\mu}\mathrm{m},0\text{}{}^{\xb0}\mathrm{C})$, and (

**f**) $({d}_{s},\Delta {T}_{\mathrm{DFW}})=(100\text{}\mathsf{\mu}\mathrm{m},30\text{}{}^{\xb0}\mathrm{C})$, with $\theta =45{}^{\xb0}$, ${\varphi}_{0}=5\%$, and $W=5\text{}\mathrm{cm}$. The x and y axes represent time and vertical position ($y=0$ standing for the top of the cell). False color represents horizontally-averaged particle concentration.

**Figure 9.**Average concentration at zone A1 as a function of particle diameter for (

**a**) ${d}_{s}=50\text{}\mathsf{\mu}\mathrm{m}$ and (

**b**) ${d}_{s}=100\text{}\mathsf{\mu}\mathrm{m}$.

**Figure 10.**Average concentration at zone A2 for ${d}_{s}=10\text{}\mathsf{\mu}\mathrm{m}$, 50 $\mathsf{\mu}$m and 100 $\mathsf{\mu}$m, $\Delta {T}_{\mathrm{DFW}}=0$ and 50 ${}^{\xb0}\mathrm{C}$ as a function of the dimensionless time $\tau =t{w}_{0}/b$, with ${w}_{0}$ the Stokes velocity for each particle size.

**Figure 13.**Dependence of the initial particle concentration on the normalized accumulation in A1, defined as ${\langle \varphi \rangle}_{\mathrm{A}1}/{\varphi}_{0}$.

**Figure 14.**Effect of the initial particle concentration in the particle concentration in the clear water zone or A2.

**Table 1.**Range of variables studied in the numerical simulations. The acronym DFW stands for downward facing wall. In all the cases, the initial temperature prior to the start of the heating of the downward facing wall was ${T}_{0}=20\text{}{}^{\xb0}\mathrm{C}$. At the start of each experiment, the temperature at the downward facing wall was set as ${T}_{\mathrm{DFW}}={T}_{0}+\Delta {T}_{\mathrm{DFW}}$. Variables ${d}_{s}$, ${\varphi}_{0}$, $\theta $, and W denote particle diameter (monosized), initial concentration (constant), cell inclination, and horizontal projection of cell spacing, respectively. ${H}_{0}$ is equivalent to $\mathrm{cos}\left(45\right)$, and the length of the cell is 1 m.

Variable | Cases Considered |
---|---|

${H}_{0}$ | $0.707$$\mathrm{m}$ |

$\Delta {T}_{\mathrm{DFW}}$ | 20, 30, 40, and 50 ${}^{\xb0}\mathrm{C}$ |

${d}_{s}$ | 5, 10, and 50 $\mathsf{\mu}$$\mathrm{m}$ |

${\varphi}_{0}$ | 2%, 5%, 8%, and 15% |

$\theta $ | 45${}^{\xb0}$ |

W | 5 $\mathrm{c}$$\mathrm{m}$ |

**Table 2.**Dimensionless particle accumulation lapse in A1 for temperature difference $\Delta {T}_{\mathrm{DFW}}$ at the downward facing wall (${t}_{\Delta {T}_{\mathrm{DFW}}}^{*}$), normalized by the particle accumulation time without heating (${t}_{\Delta {T}_{\mathrm{DFW}}=0}^{*}$), ${\psi}_{\Delta {T}_{\mathrm{DFW}}}\equiv {t}_{\Delta {T}_{\mathrm{DFW}}}^{*}/{t}_{\Delta {T}_{\mathrm{DFW}}=0}^{*}$, where a and bare the case where ${Y}_{z}=0.6{Y}_{\infty}$ and ${Y}_{z}=0.9{Y}_{\infty}$, respectively. Here, ${d}_{s}=10\text{}\mathsf{\mu}\mathrm{m}$, $\theta =45{}^{\xb0}$, ${\varphi}_{0}=5\%$, and $W=5\text{}\mathrm{cm}$.

$\mathbf{\Delta}{\mathit{T}}_{\mathbf{DFW}}$ (K) | ${\mathit{\psi}}_{\mathbf{\Delta}{\mathit{T}}_{\mathbf{DFW}}\mathbf{,}\mathit{a}}$ | ${\mathit{\psi}}_{\mathbf{\Delta}{\mathit{T}}_{\mathbf{DFW}}\mathbf{,}\mathit{b}}$ |
---|---|---|

20 | 0.62 | 0.54 |

30 | 0.58 | 0.53 |

40 | 0.53 | 0.43 |

50 | 0.51 | 0.29 |

**Table 3.**Dimensionless particle accumulation lapse in A1 for temperature difference $\Delta {T}_{\mathrm{DFW}}=50\text{}{}^{\xb0}\mathrm{C}$ at the downward facing wall for ${d}_{s}=10\text{}\mathsf{\mu}\mathrm{m}$, ${d}_{s}=50\text{}\mathsf{\mu}\mathrm{m}$, and ${d}_{s}=100\text{}\mathsf{\mu}\mathrm{m}$ (${t}_{50,{d}_{s}}^{*}$), normalized by the particle accumulation time without heating (${t}_{0,{d}_{s}}^{*}$), ${\psi}_{50,{d}_{s}}\equiv {t}_{50,{d}_{s}}^{*}/{t}_{0,{d}_{s}}^{*}$. Here, $\theta =45{}^{\xb0}$, ${\varphi}_{0}=5\%$, and $W=5\text{}\mathrm{cm}$.

${\mathit{d}}_{\mathit{s}}$ ($\mathsf{\mu}$m) | ${\mathit{\psi}}_{\mathbf{50}\mathbf{,}{\mathit{d}}_{\mathit{s}}}$ |
---|---|

10 | 0.94 |

50 | 0.94 |

100 | 0.70 |

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**MDPI and ACS Style**

Reyes, C.; Ihle, C.F.; Apaz, F.; Cisternas, L.A. Heat-Assisted Batch Settling of Mineral Suspensions in Inclined Containers. *Minerals* **2019**, *9*, 228.
https://doi.org/10.3390/min9040228

**AMA Style**

Reyes C, Ihle CF, Apaz F, Cisternas LA. Heat-Assisted Batch Settling of Mineral Suspensions in Inclined Containers. *Minerals*. 2019; 9(4):228.
https://doi.org/10.3390/min9040228

**Chicago/Turabian Style**

Reyes, Cristian, Christian F. Ihle, Fernando Apaz, and Luis A. Cisternas. 2019. "Heat-Assisted Batch Settling of Mineral Suspensions in Inclined Containers" *Minerals* 9, no. 4: 228.
https://doi.org/10.3390/min9040228