# Coupling Flotation Rate Constant and Viscosity Models

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

^{−1}.

^{−1}were 0.17, 0.03, and 0.02 Pa s, respectively.

^{−1}were 0.038 and 0.012 Pa s, respectively. Forbes et al. (2014) [28] studied the effect of rheology and slime coating on the natural floatability of chalcopyrite in a clay-rich flotation pulp and reported viscosity values between 0.001 and 0.15 Pa s. They reported copper recovery of 92% at quartz/kaolinite content of 100/0; however, with the change in quartz/kaolinite content (i.e., 70/30, and 30/70), the copper recovery was reduced to 87% and 82%, respectively. Table 1 summarizes the literature discussing the influence of the suspension viscosity on the flotation performance.

**Table 1.**Summary of the literature discussing the influence of change in viscosity on flotation performance.

Researcher | Reported Suspension Viscosity | Flotation Results (Rate Constant or Recovery/Grade) | References |
---|---|---|---|

Farrokhpay et al., 2011 | The viscosity of 50 vol.% glycerol–water mixture, used in their study, was 0.0076 Pa s. | The recovery of coarse composite copper-bearing particles (+210 μm) of porphyry copper ore, recovered in the tailings of rougher, at a grind size of d_{80} = 250 μm, increased from 83 to 90% with the increase in viscosity from 0.001 to 0.0076 Pa s. | [24] |

Shabalala et al., 2011 | The viscosity of kaolin ore slurry increased exponentially to the maximum values of between 0.03 and 0.08 Pa s with the increase in solid concentration from 15 to 40 wt.% of kaolin. | Bubble size generated within kaolin ore suspension decreased from 1 to 0.65 mm with the increase in solid concentration from 30 to 40 wt.% at an impeller speed of 650 rpm. | [26] |

Forbes et al., 2014 | The viscosity of pulp containing chalcopyrite and clay minerals (kaolinite and quartz) was found between 0.001 and 0.15 Pa s. | The recovery of chalcopyrite (copper) was 92% at quartz/kaolinite content of 100/0; however, with the change in quartz/kaolinite ratio (i.e., 70/30 and 30/70), the chalcopyrite recovery reduced to 87% and 82%, respectively. | [28] |

Cruz et al., 2015 | The viscosity of copper–gold ore slurry increased from 0.0035 to 0.014 Pa s with the increase of solid concentration of bentonite from 0 to 15 wt.% at a shear rate of 100 s^{−1}. | The baseline flotation of their ore (i.e., 100 wt.% ore without bentonite/kaolinite) resulted in a copper recovery of 92% at a grade of 10% copper, and 81% gold recovery at a grade of 7 ppm gold. The addition of 15 wt.% bentonite to the ore (100 wt.%) decreased the recovery (i.e., copper recovery from 92 to 83% and gold recovery from 81 to 64%) and slightly decreased in copper and gold grades from 10 to 8% and 7 ppm to 5 ppm, respectively. The addition of 30 wt.% kaolinite to the ore (100 wt.%) did not decrease copper and gold recoveries but did decrease copper and gold grades from 10 to 2% and 7 ppm to 1 ppm, respectively. | [20] |

Wang et al., 2015 | The apparent viscosity of Telfer clean ore increased from 0.001 to 0.008 Pa s with the increase in solid concentration of bentonite from 5 to 25 wt.% at a shear rate of 100 s^{−1}. | The copper recovery decreased from 76 to 25% with the increase in solid concentration of bentonite from 0 to 20 wt.%; however, it slightly decreased the copper grade from 5.1 to 5%. The copper recovery decreased from 80 to 67% with the increase in solid concentration of kaolin from 0 to 20 wt.%; however, it decreased the copper grade from 5 to 4%. | [16] |

Zhang & Peng, 2015 | The apparent viscosity of a copper–gold ore increased from 0.0018 to 0.0076 Pa s with the increase in solid concentration of bentonite from 0 to 15 wt.% at a shear rate of 100 s^{−1}. | The copper recovery decreased from 82 to 60% with the increase in solid concentration of bentonite from 0 to 15 wt.%. The gold recovery decreased from 78 to 65% with the increase in solid concentration of bentonite from 0 to 15%. | [14] |

Farrokhpay et al., 2016 | The apparent viscosity of their copper ore slurry in the presence of 15 wt.% montmorillonite, 30 wt.% of kaolinite, and 30 wt.% illite at a shear rate of 100 s^{−1} was 0.17, 0.03, and 0.02 Pa s, respectively. | The copper recovery decreased (90 to 80%) in the presence of 15 wt.% swelling clay (montmorillonite); however, in the presence of 15 wt.% non-swelling clays (illite and kaolinite), the copper recovery decreased slightly to 87% and 88%, respectively. The copper grade decreased from 18% to about 1% in the presence of both 30 wt.% of kaolinite and 15 wt.% of montmorillonite, respectively; however, it decreased to about 5% in the presence of 30% illite. The copper ore flotation rate constants were 0.51 s ^{−1} and 0.49 s^{−1} in the presence of 15 wt.% of kaolinite and 15 wt.% of illite, respectively, and 0.33 s^{−1} in the presence 15 wt.% of montmorillonite, as compared with the copper ore flotation rate constant of 0.70 s^{−1} in the absence of clay minerals. | [25] |

Basnayaka et al., 2017 | The viscosity of gold ore increased from 0.0018 to 0.0035 Pa s by the addition of 10 wt.% kaolin at pH 7, a shear rate of 100 s^{−1}, and polyacrylate depressant concentration of 0 and 200 g/t, respectively; however, by the addition of 5 wt.% bentonite, the viscosity increased to 0.0060 Pa s, under the same conditions. | The flotation rate constant of their gold-bearing pyrite ore was decreased from 13.71 to 3.37 s^{−1} (822.6 to 202.2 min^{−1}) without and with the presence of 10 wt.% kaolin at pH 7, air rate of 5 L/min, and polyacrylate depressant concentration of 0 and 200 g/t, respectively. The presence of bentonite under same conditions reduced the flotation rate constant to 4.14 s^{−1} (248.4 min^{−1}). | [33] |

Chen et al., 2017 | The apparent viscosity of amorphous silica and quartz suspension increased from 0.109 to 0.147 Pa s with the increase in the concentration of amorphous silica (in amorphous silica and quartz suspension) from 30 to 50 vol.%, at a shear rate of 100 s^{−1}. | The copper recovery dropped sharply from 95 to 63.6% after the percentage of amorphous silica increased from 30 to 50 vol.%. The copper grade decreased slightly from 3.9 to 3.8% with the increase in the concentration of amorphous silica (in amorphous silica and quartz suspension) from 30 to 50 vol.%. | [18] |

Farrokhpay et al., 2018 | The viscosity of their copper ore slurry increased from 0.010 to 0.038 Pa s with the increase in solid concentration of muscovite from 0 to 30 wt.% at a shear rate of 100 s^{−1}; however, with the increase in solid concentration of talc from 0 to 7.5 wt.%, there was a slight increase in the slurry viscosity from 0.010 to 0.012 Pa s, at a shear rate of 100 s^{−1}. | The flotation grade decreased (19 to 2%) with the increase in solid concentration of muscovite from 0 to 30 wt.%; however, the change in the recovery was reported negligible. The flotation recovery (90 to 83%) and grade (19 to 2%) decreased with the increase in solid concentration of talc from 0 to 7.5 wt.%. | [17] |

## 2. Theory

#### 2.1. Collection Efficiency (${E}_{\mathrm{coll}}$)

#### 2.1.1. Collision Efficiency (${E}_{\mathrm{c}}$)

#### 2.1.2. Attachment Efficiency $\left({E}_{a}\right)$

_{ind}) is defined as the time for the liquid thin film to get thin and rupture, and for the three-phase line of contact to expand until an equilibrium value is obtained [44].

#### 2.1.3. Stability Efficiency $\left({E}_{\mathrm{s}}\right)$

_{s}, is exponentially distributed [8], then E

_{s}can be determined from Equation (17).

#### 2.2. Flotation Rate Constant under Turbulent Flow Condition

#### 2.3. Viscosity Modeling and Factors Affecting Viscosity

^{−3}μm < d < 10

^{1}μm), the flow behavior is determined by a combination of hydrodynamic forces, Brownian motion, and interparticle forces [6]. The idea of this study is to propose a viscosity equation incorporating those three kinds of forces based on previously suggested equations.

#### 2.3.1. Hard Sphere Suspensions

#### 2.3.2. Effect of Shear Rate

#### 2.3.3. Colloidal Suspensions

## 3. Results and Discussion

^{−2}M KNO

_{3}, respectively. pH 10 is a typical pH for quartz flotation [79]. The zeta potential value of quartz particle was extracted from our previous report [80]. After our preliminary calculations (Appendix A Figure A1, presented in Section 3.1), the flotation efficiencies and rate constant were calculated using the calculated suspension viscosity and for the following conditions: turbulence dissipation energy $\epsilon =10$ m

^{2}s

^{−3}, gas flow rate as 0.0035 m

^{3}min

^{−1}, bubble diameter as 0.0014 m, volume of cell as 0.00225 m

^{3}, quartz particle density as 2650 kg/m

^{3}, bubble velocity as 0.18 m s

^{−1}, liquid–vapor surface tension as 72.8 m Nm

^{−1}, particle contact angle as 80°, and flotation particle diameter as 80 μm [5].

#### 3.1. Calculated Flotation Collection Efficiencies ${E}_{\mathrm{c}}$, ${E}_{\mathrm{a}}$, ${E}_{\mathrm{s}}$, and Rate Constant $k$

^{−3}m in diameter introduced at a gas flow rate of 3.5 × 10

^{−3}m

^{3}min

^{−1}and at an agitation speed of 650 rpm in a Rushton flotation cell. According to them, the best fit of the experimental and calculated flotation rate constant was obtained with a fluid velocity of 0.18 ± 0.01 ms

^{−1}and a dissipation energy of 38 ± 7 m

^{2}s

^{−3}(95% confidence limits).

#### 3.2. Effect of Suspension Viscosity on ${E}_{\mathrm{c}}$, ${E}_{\mathrm{a}}$, ${E}_{\mathrm{s}}$, and Rate Constant $k$

^{−1}at 80 µm and 0.0009 Pa s. The $k$ decreased above the viscosity value 0.0009 Pa s (Figure 3d). However, the value of $k$ increases below 0.00091 Pa s, that is, the viscosity of water at 25 °C [78]. Therefore, considering water is used in flotation operations, 0.00091 Pa s was selected to calculate the maximum value of $k$. This also agrees with the study performed by Pyke et al. (2003) [5] and Duan et al. (2003) [11] who reported the optimum flotation rate constant of 7 min

^{−1}with the quartz particles with an advancing water contact angle of 80° interacting in a Rushton flotation cell with gas bubbles 1.4 × 10

^{−3}m in diameter introduced at a gas flow rate of 3.5 × 10

^{−3}m

^{3}min

^{−1}and at an agitation speed of 650 rpm. Chen et al. (2017) [18] reported the decrease in copper flotation recovery from 95 to 63.6% with an increase in pulp viscosity from 0.109 to 0.147 Pa s in the presence of 30 vol.% and 50 vol.% of amorphous silica in a suspension of the mixture of amorphous silica and quartz, respectively. This agrees with the $k$ calculated and shown in Figure 3d, and $k$ decreased from 1.347 to 1.227 min

^{−1}with increase in the suspension viscosity from 0.109 to 0.147 Pa s. As seen in Figure 3d, k is more affected by ${E}_{\mathrm{a}}$ since the magnitude of ${E}_{\mathrm{a}}$ is higher than ${E}_{\mathrm{c}}$, while ${E}_{\mathrm{s}}$ remain unchanged.

#### 3.3. Suspension Viscosity Calculation

#### 3.3.1. Modified Krieger and Dougherty Model—Hard Sphere Suspensions

#### 3.3.2. Our Predictive Model—Hard Sphere and Interacting Colloidal Particle Suspensions

#### Homogeneous Case (i.e., Same Particle with Same Sizes)

^{−1}. Schubert (1999) [81] reported that the suspensions containing higher percent of solids produce higher slurry viscosity, particularly in the case of interacting particles. Ndlovu et al. (2011) [83] reported an increase in the suspension viscosity from 0.010 to 0.074 Pa s with an increase in the solid concentration from 25 to 35 vol.% (62.5 to 87.5 wt.%) of pure vermiculite, respectively. They also reported an increase in the suspension viscosity from 0.011 to 0.023 Pa s with an increase in the solid concentration from 30 to 35 vol.% (79.5 to 92.75 wt.%) of pure quartz, respectively.

^{3}, Table 2) due to the decrease in interparticle distance (Equation (26)), which results in abnormal values of suspension viscosity. On the other hand, for the lower solid fractions (i.e., $\varphi $ ≤ 0.3) and the case of same quartz particles with same particle sizes, the DLVO calculations result in acceptable total potential energies and were thus applied in the calculations of suspension viscosities as well as flotation efficiencies and rate constants.

#### Homogeneous Case (i.e., Same Particles with Different Particle Sizes)

^{−1}) at both solid fractions ($\varphi =0.1$ and $0.5$). The increase in the suspension viscosity was more at higher solid fraction ($\varphi =0.5$, Figure 6c) as compared with the lower solid fraction ($\varphi =0.1$, Figure 6a). This was obvious because particles crowding enhances thermal diffusion and colloidal interactions, and the suspension shows a high viscosity at low shear rate (≤0.001 s

^{−1}), while it decreased and becomes constant due to the domination of hydrodynamic interactions at higher shear rate (>0.001 s

^{−1}). Genovese [21] discussed the similar behavior of interacting particles and reported that the suspension viscosity decreased significantly from 0.2548 to 0.00254 Pa s between varied shear rates from 0.01 to 1000 s

^{−1}when the solid fraction $\left(\varphi \right)$ was 0.345. Schubert (1999) [81] also reported that suspensions containing a higher percent of solids produce higher slurry viscosity, particularly in the case of interacting particles.

^{−1}, when the solid fraction $\left(\varphi \right)$ was as low as 0.176. With the higher solid fraction ($\varphi =0.5$), there is a slight increase (2%) in the suspension viscosity at the lowest shear rate (0.0001 s

^{−1}), possibly due to thermal diffusion and colloidal interactions enhanced by many particle systems (Figure 6d).

^{−1}to 9.27 × 10

^{−1}with an increase in solid fraction from 0.1 to 0.5. The reason behind this decrease is that with the increase in solid fraction, interparticle distance (Equation (26)) decreases, resulting in stronger van der Walls attraction. Similarly, behavior can be seen with the increase in fine/ultra-fine particle size (8 nm to 100 µm); for example, with the increase in fine/ultra-fine particle size (8 nm to 100 µm) at solid fraction 0.1, the total potential energy decreased from 9.98 × 10

^{−1}to 8.92 × 10

^{−1}. The DLVO calculations shown in Table 3 were thus applied in the calculations of suspension viscosities as well as flotation efficiencies and rate constants.

#### 3.4. Flotation Efficiencies and Rate Constant Calculations

#### 3.4.1. Incorporation of the Modified Krieger and Dougherty Model—Hard Sphere Suspensions

^{−1}) changed significantly; however, ${E}_{\mathrm{s}}$ remained the same. The collision efficiency $\left({E}_{\mathrm{c}}\right)$ increases with the decrease in ultra-fine/fine particle size (i.e., 100,000 to 8 nm) at both solid fractions (i.e., $\varphi =0.1$, Figure 7a and $0.5$, Figure 8a). This can be explained due to the fact that with the decrease in fine/ultra-fine particles (i.e., 100,000 to 8 nm), the suspension viscosity increases (from 0.00123 to 0.00128 Pa s for $\varphi =0.1$, and from 0.01047 to 0.02136 Pa s for $\varphi =0.5$) (Figure 7e and Figure 8e), and as explained earlier, the increase in suspension viscosity decreases the Stokes number (Equation (6)), which in turn increases the maximum collision angle (${\theta}_{\mathrm{t}}$) (Equation (4)). As shown in Figure 2, a higher ${\theta}_{\mathrm{t}}$ means a higher collision probability between particle and bubble.

^{−1}(Figure 7c). Similarly, ${E}_{\mathrm{a}}$ increased from 0.475 to 0.488 for 1000 nm ultra-fine particle, with an increase in shear rate from 0.0001 to 0.01 s

^{−1}(Figure 7c). On the other hand, at the higher solid fraction ($\varphi =0.5$ Figure 8c), the effect of shear rate on ${E}_{\mathrm{a}}$ was also dominant in the intermediate fine/ultra-fine particle sizes but in a different range (i.e., 120 nm–10 µm) from $\varphi =0.1$ (Figure 7c), as ${E}_{\mathrm{a}}$ increased from 0.3422 to 0.3429 for 120 nm particles, with an increase in shear rate from 10 to 1000 s

^{−1}. ${E}_{\mathrm{a}}$ also increased from 0.3422 to 0.3438 (Figure 8c) for 10 µm particles, with an increase in shear rate from 0.0001 to 0.01 s

^{−1}. The dominancy of shear rate affecting ${E}_{\mathrm{a}}$ was observed because the hydrodynamic forces were more dominant as compared with the thermal diffusion with increasing shear rate.

^{−1}with the decrease in the presented particle size from 100 µm to 8 nm at a shear rate of 100 s

^{−1}(Figure 7d). Similarly, at the higher solid fraction $\varphi =0.5$ (Figure 8d), the flotation rate constant decreased from 2.42 to 2.11 min

^{−1}with the decrease in particle size from 100 µm to 8 nm at a shear rate of ≤100 s

^{−1}. As stated earlier and discussed in this article and the literature, the presence of fine/ultra-fine particles has deleterious effects on the flotation rate constant/flotation performance [17,18]. Farrokhpay et al. (2016) [25] reported the copper ore flotation rate constant of 0.68 s

^{−1}(40.2 min

^{−1}) in the presence of 10 wt.% kaolinite or illite, and 0.60 s

^{−1}(36 min

^{−1}) in the presence of 10 wt.% montmorillonite, as compared with the copper ore flotation rate constant of 0.70 s

^{−1}(42 min

^{−1}) in the absence of clay minerals. They also increased the content of fine/ultra-fine clay mineral particles as 30 wt.% kaolinite, 30 wt.% illite, or 15 wt.% montmorillonite and found further decrease in flotation rate constant to 0.51, 0.49, and 0.33 s

^{−1}, respectively.

^{−1}. At $\varphi =0.5$ (Figure 8e), $\eta =0.0104$ Pa s for particle size of 100 µm, while $\eta =0.0212$ Pa s for particle size of 8 nm at a shear rate of 100 s

^{−1}. This increase in the suspension viscosity causes decrease in the flotation rate constant. For example, at $\varphi =0.1$, $k$ decreased very slightly from 3.37 to 3.34 s

^{−1}with the decrease in particle size from 100 µm to 8 nm, at a shear rate of 100 s

^{−1}, while at $\varphi =0.5$, $k$ noticeably decreased from 2.42 to 2.12 s

^{−1}with the decrease in particle size from 100 µm to 8 nm, at a shear rate of 100 s

^{−1}. Basnayaka et al., 2017 [33] reported that the suspension viscosity increased from 0.0018 to 0.0035 Pa s by the addition of 10 wt.% kaolin at pH 7, a shear rate of 100 s

^{−1}, and a polyacrylate depressant concentration of 0 and 200 g/t, respectively. This increase in viscosity caused decrease in the flotation rate constant of the gold-bearing pyrite ore from 13.71 to 3.37 s

^{−1}(822.6 to 202.2 min

^{−1}).

#### 3.4.2. Incorporation of Our Predictive Model—Hard Sphere and Interacting Colloidal Particle Suspensions

#### Homogeneous Case—Same Particles with the Same Sizes

^{−2}M KNO

_{3}, as well as the other conditions mentioned in Section 3. The flotation efficiencies and rate constant (Figure 9 and Figure 10) almost follow the same general trends as of the modified Krieger and Dougherty model coupling (Figure 7 and Figure 8), with a slight change in their absolute values (increase in ${E}_{\mathrm{c}}$ and decrease in ${E}_{\mathrm{a}}$ and $k$, but ${E}_{\mathrm{s}}$ remained same). For instance, at solid fraction $\varphi =0.1$, with the decrease in fine/ultra-fine particle size (100 μm to 8 nm) ${E}_{\mathrm{c}}$ increased from 0.06729 to 0.06830 (1.5% increase), ${E}_{\mathrm{a}}$ decreased from 0.4788 to 0.4702 (1.8% decrease), $k$ decreased from 3.3775 to 3.3196 min

^{−1}(1.71% decrease), and ${E}_{\mathrm{s}}$ remained unchanged. As our predictive model incorporates the total potential energies calculated by using the DLVO theory (Table 2), it leads to this additional increase in the magnitude of flotation efficiencies and rate constant. The DLVO calculations were made using Equations (25)–(30) for the conditions stated in the beginning of this section. At a lower solid fraction ($\varphi =0.1$), with the increase in particle size (8 nm to 100 μm), the interparticle distance (calculated from Equation (26)) increased; therefore, the total potential energies (van der Walls and electric double-layer) decreased, resulting in a decrease in the suspension viscosity from 0.001285 to 0001232 Pa s (4.12% decrease). Otsuki and Hayagan (2020) [73] studied the surface properties of hematite and gangue minerals and their mixtures with the bentonite binder to understand and properly control their surface properties for pelletization of fine hematite ores. They calculated the total potential between quartz–quartz particles and reported the similar trend of decrease in total potential with the increase in interparticle distance, as we reported here. As shown in Table 2, at higher solid fractions ($\varphi >0.3)$, the DLVO calculations become inapplicable because the calculated viscosity values from Equation (33) are too high; thus, the following flotation efficiencies and rate constant calculations were made at $\varphi $ ≤ 0.3, and their results are shown in Figure 9 ($\varphi =$ 0.1) and Figure 10 ($\varphi =$ 0.3) in this section. At the higher solid fraction ($\varphi =$ 0.3), the interparticle distance decreases (e.g., for 8 nm and 80 μm particles, interparticle distance decreased from 5.86 to 1.61 nm, with the increase in solid fraction from $\varphi =0.1$ to $\varphi =0.3$), and thus the van der Waals potential energy ${V}_{A}$ decreased from −0.472 × 10

^{−21}to −1.71 × 10

^{−21}K

_{B}T, while electrical double-layer potential ${V}_{R}$ increased from 5.46 × 10

^{−21}to 22.1 × 10

^{−21}K

_{B}T, and the total potential energy ${V}_{T}$ increased from 4.99 × 10

^{−21}to 20.4 × 10

^{−21}K

_{B}T with the increase in solid fraction from $\varphi =0.1$ to $\varphi =0.3$). This resulted in higher values of the suspension viscosity (i.e., 0.002729–0.003314 Pa s for $\varphi =$ 0.3, as compared to 0.00123–0.00128 Pa s for $\varphi =$ 0.1) (Figure 9e and Figure 10e). This indicates that electrostatic repulsive interaction could be responsible for the increase in suspension viscosity, as Sakairi et al. (2005) [84] reported that electrostatic repulsion was the cause of increase in their montmorillonite suspension yield stress. The higher suspension viscosity leads to the change in flotation efficiencies (Figure 10a,c) and rate constant (Figure 10d). For example, ${E}_{\mathrm{c}}$ increased from 0.06729–006830 for $\varphi =$ 0.1 to 0.09849–0.09146 for $\varphi =$ 0.3, ${E}_{\mathrm{a}}$ decreased from 0.4788–0.4702 for $\varphi =$ 0.1 to 0.3768–0.3658 for $\varphi =$ 0.3, $k$ decreased from 3.377–3.319 min

^{−1}for $\varphi =$ 0.1 to 2.720–2.7163 min

^{−1}for $\varphi =$ 0.3, and ${E}_{\mathrm{s}}$ remained constant.

^{−1}, and a polyacrylate depressant concentration of 0 and 200 g/t, respectively. This increase in viscosity caused a decrease in the flotation rate constant of the gold-bearing pyrite ore from 13.71 to 3.37 s

^{−1}(822.6 to 202.2 min

^{−1}, 75.4% decrease). The presence of bentonite under the same conditions reduced the flotation rate constant to 4.14 s

^{−1}(248.4 min

^{−1}, 69.8% decrease). Using their suspension viscosity values for the case of kaolin, in our predictive model, the flotation rate constant clearly indicates its decrease from 2.982 to 2.703 min

^{−1}(9.36% decrease), although the literature system and our system are different. Our predictive model calculations were made on the quartz particles.

^{−1}(15.69% decrease). The possible reasons for such decreases in flotation rate constant were reported as the occurrence of slime coating or physical adsorption of clay mineral particles onto the copper-containing mineral surface [28].

#### Homogeneous Case—Same Particle with Different Particle Sizes

^{−1}). However, it can be seen that the decrease in collision efficiency took place up to the shear rate of 0.01 s

^{−1}, and above 0.01 s

^{−1}shear rate, the collision efficiency became almost constant. This can be explained by the effect of suspension viscosity (Figure 11e), as below the shear rate of 0.01 s

^{−1}, the suspension viscosity very slightly increases (from 0.00123289 to 0.00123295 Pa s with the decrease in fine/ultra-fine particle size of 100 μm to 8 nm). The increase in suspension viscosity decreases the Stokes number (Equation (6)), which in turn increases the maximum collision angle (${\theta}_{\mathrm{t}}$) (Equation (4)). As shown in Figure 2, a higher ${\theta}_{t}$ means higher collision probability between a particle and a bubble.

^{−1}(0.003% decrease) with the decrease in the fine/ultra-fine particle size (100 µm to 8 nm). This can be explained by the effect of suspension viscosity and the flotation efficiencies (${E}_{\mathrm{c}}$ and ${E}_{\mathrm{a}}$). The decrease in fine/ultra-fine particles very slightly increased the suspension viscosity from 0.00123289 to 0.00123295 Pa s (0.005% increase), especially at low shear rates (below 0.01 s

^{−1}). This degree of suspension viscosity increase caused negligible change in the flotation rate constant, from 3.3775 to 3.3774 min

^{−1}(0.003% decrease).

^{−1}at $\varphi =0.1$ to its range between 2.137 and 2.421 min

^{−1}at $\varphi =0.5$.

^{−1}) causes decrease in the suspension viscosity from 0.02035 to 0.01040 Pa s (48.9% decrease, Figure 13e). This shows that the hydrodynamic forces are dominant on the interparticle forces above 0.01 s

^{−1}shear rate, and the viscosity does not change significantly, showing that the hydrodynamic forces become fully dominant on the interparticle forces. Genovese (2012) [21] calculated a relative viscosity (${\eta}_{\mathrm{r}}$) and reported that ${\eta}_{\mathrm{r}}$ increased with solid fraction and decreased with shear rate, but he mentioned that the shear thinning effect was only noticeable at approximately $\varphi >0.25$ and at high shear rates (>100 s

^{−1}) where hydrodynamic forces become more significant than interparticle forces. He further explained that the viscosity decrease in this shear-thinning region was due to the ordering of particles along the flow direction.

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

$A$ | Hamaker constant |

$a$ | Radius of hard spheres |

${B}_{0}^{*}$ | Bond number |

$\mathrm{C}$ | Concentration of ions |

${d}_{\mathrm{b}}$ | Bubble diameter |

${d}_{\mathrm{p}}$ | Particle diameter |

$e$ | Elementary charge |

${E}_{\mathrm{coll}}$ | Collection efficiency |

${E}_{\mathrm{c}}$ | Collision efficiency |

${E}_{\mathrm{a}}$ | Attachment efficiency |

${E}_{\mathrm{s}}$ | Stability efficiency |

${F}_{\mathrm{att}}$ | Attractive forces |

${F}_{\mathrm{b}}$ | Buoyancy force |

${F}_{\mathrm{ca}}$ | Capillary force |

${F}_{\mathrm{d}}$ | Machine acceleration force |

${F}_{\mathrm{det}}$ | Detachment forces |

${F}_{\mathrm{g}}$ | Gravitational force |

${F}_{\mathrm{hyd}}$ | Hydrostatic force |

${F}_{\u2134}$ | Capillary pressure force |

${G}_{f}$ | Gas flow rate |

$H$ | Interparticle separation distance |

$k$ | Flotation rate constant |

${k}_{B}$ | Boltzman’s constant |

${N}_{A}$ | Avogadro’s number |

$n$ | Number concentration of ions |

$Pe$ | Peclet number |

$P{e}_{\mathrm{c}}$ | Characteristic Peclet number |

${r}_{\mathrm{b}}$ | Bubble radius |

${R}_{\mathrm{eb}}$ | Reynolds number |

${r}_{\mathrm{p}}$ | Particle radius |

$T$ | Absolute temperature |

${t}_{\mathrm{ind}}$ | Induction time |

${V}_{cell}$ | Volume of flotation cell |

$\upsilon $ | Kinematic viscosity |

${v}_{\mathrm{b}}$ | Bubble velocity |

${v}_{\mathrm{p}}$ | Particle velocity |

$z$ | Ionic valence |

${\theta}_{\mathrm{a}}$ | Collision angle |

${\theta}_{\mathrm{t}}$ | Maximum collision angle |

$\phi $ | Contact angle |

${\rho}_{\mathrm{p}}$ | Particle density |

${\rho}_{\mathrm{f}}$ | Fluid density |

$\Delta {\rho}_{\mathrm{p}}$ | ${\rho}_{p}-{\rho}_{f}$ |

$\mu $ | Dynamic viscosity |

$\eta $ | Suspension viscosity |

${\eta}_{\mathrm{L}}$ | Liquid’s viscosity |

${\eta}_{\mathrm{r}}$ | Relative viscosity |

$\left[\eta \right]$ | Intrinsic viscosity of the particles |

$\varphi $ | Solid volume fraction |

${\varphi}_{\mathrm{m}}$ | Maximum packing fraction |

$\stackrel{.}{\mathsf{\gamma}}$ | Shear rate |

$\gamma $ | Reduced surface potential |

$\kappa $ | Debye–Huckel reciprocal length |

$\epsilon $ | Dielectric constant of the medium |

${\epsilon}_{0}$ | Permittivity of free space |

$\zeta $ | Zeta potential |

## Appendix A

**Figure A1.**Calculated (

**a**) collision $\left({E}_{\mathrm{c}}\right)$, (

**b**) attachment $\left({E}_{\mathrm{a}}\right)$, and (

**c**) stability $\left({E}_{\mathrm{s}}\right)$ efficiencies as well as (

**d**) flotation rate constant (k) as a function of particle diameter $\left({d}_{\mathrm{p}}\right)$ and dissipation energy $\left(\epsilon \right)$ (10, 20, and 38 m

^{2}s

^{−3}), (${d}_{\mathrm{b}}=0.0014\mathrm{m},{\rho}_{\mathrm{p}}=2650\mathrm{k}\mathrm{g}{\mathrm{m}}^{-3},{v}_{\mathrm{b}}=0.18\mathrm{m}{\mathrm{s}}^{-1},\phi =80\xb0,\mu =0.001\mathrm{N}\mathrm{s}{\mathrm{m}}^{-2}$).

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**Figure 1.**Sub-processes of particle–bubble interactions along with their predominant regions, modified from [6].

**Figure 2.**Schematic of different angles involved in particle–bubble attachment, modified from [47].

**Figure 3.**Effect of quartz particle suspension viscosity on calculated (

**a**) ${E}_{\mathrm{c}}$, (

**b**) ${E}_{\mathrm{a}}$, (

**c**) ${E}_{\mathrm{s}}$, and (

**d**) $k$, as a function of particle diameter, at ε = 10 m

^{2}s

^{−3}.

**Figure 4.**Suspension viscosity as a function of shear rate calculated by the modified Krieger and Dougherty model at (

**a**) solid fraction, ϕ = 0.1, and (

**b**) solid fraction, ϕ = 0.5. Note: the major y-axis of subfigure (

**a**) is made for the purpose of comparison.

**Figure 5.**Viscosity of quartz particle suspension as a function of shear rate calculated by using our predictive model, at solid fraction (

**a**) ϕ = 0.1 and (

**b**) ϕ = 0.3. Note: the major y-axis of subfigure (

**a**) is made for the purpose of comparison.

**Figure 6.**Viscosity of quartz particle suspension as a function of shear rate by a predictive model, (

**a**) $\varphi =0.1,{a}_{1}:{a}_{2}=1:9$, (

**b**) $\varphi =0.1,{a}_{1}:{a}_{2}=9:1$, (

**c**) $\varphi =0.5,{a}_{1}:{a}_{2}=1:9$, and (

**d**) $\varphi =0.5,{a}_{1}:{a}_{2}=9:1$. Note: the major y-axis of subfigure (

**d**) is made for the purpose of comparison.

**Figure 7.**Flotation efficiencies: (

**a**) collision, (

**b**) stability, (

**c**) attachment, and (

**d**) rate constant by using the (

**e**) respective suspension viscosity calculated by modified Krieger and Dougherty model at $\varphi =0.1$. Note: the major y-axes of subfigure (

**a**,

**e**) are made for the purpose of comparison.

**Figure 8.**Flotation efficiencies: (

**a**) collision, (

**b**) stability, (

**c**) attachment, and (

**d**) rate constant by using the (

**e**) respective suspension viscosity calculated by modified Krieger and Dougherty model at $\varphi =0.5$. Note: the major y-axes of subfigures (

**c**,

**d**) are made for the purpose of comparison.

**Figure 9.**Flotation efficiencies: (

**a**) collision, (

**b**) stability, (

**c**) attachment, and (

**d**) rate constant by using (

**e**) the suspension viscosity calculated by the predictive model (same quartz particles with same sizes) at $\varphi =0.1$. Note: the major y-axes of subfigures (

**a**,

**e**) are made for the purpose of comparison.

**Figure 10.**Flotation efficiencies: (

**a**) collision, (

**b**) stability, (

**c**) attachment, and (

**d**) rate constant by using (

**e**) the suspension viscosity calculated by the predictive model (same quartz particles with same sizes) at $\varphi =0.3$. Note: the major y-axes of subfigures (

**c**,

**d**) are made for the purpose of comparison.

**Figure 11.**Flotation efficiencies: (

**a**) collision, (

**b**) stability, (

**c**) attachment, and (

**d**) rate constant by using (

**e**) the suspension viscosity calculated by the predictive model (same particles with different sizes (${a}_{1}:{a}_{2}=1:9$), at $\varphi =0.1$).

**Figure 12.**Flotation efficiencies: (

**a**) collision, (

**b**) stability, (

**c**) attachment, and (

**d**) rate constant by using (

**e**) the suspension viscosity calculated by the predictive model (same particles with different sizes (${a}_{1}:{a}_{2}=9:1$), at $\varphi =0.1$).

**Figure 13.**Flotation efficiencies: (

**a**) collision, (

**b**) stability, (

**c**) attachment, and (

**d**) rate constant by using (

**e**) the suspension viscosity calculated by the predictive model (same particles with different sizes (${a}_{1}:{a}_{2}=1:9$), at $\varphi =0.5$).

**Figure 14.**Flotation efficiencies: (

**a**) collision, (

**b**) stability, (

**c**) attachment, and (

**d**) rate constant by using (

**e**) the suspension viscosity calculated by the predictive model (same particles with different sizes (${a}_{1}:{a}_{2}=9:1$), at $\varphi =0.5$).

**Table 2.**DLVO total potential energies between two quartz particles at different solid fractions and particle sizes for the same quartz particles with the same particle sizes.

ϕ/a | 8 (nm) | 12 (nm) | 16 (nm) | 20 (nm) | 24 (nm) | 28 (nm) | 32 (nm) | 36 (nm) | 40 (nm) | 60 (nm) | 120 (nm) | 500 (nm) | 1000 (nm) | 10,000 (nm) | 50,000 (nm) | 100,000 (nm) |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

0.1 | 3.36 | 1.91 | 1.31 | 1.07 | 9.70 × 10^{−1} | 9.26 × 10^{−1} | 9.06 × 10^{−1} | 8.98 × 10^{−1} | 8.94 × 10^{−1} | 8.92 × 10^{−1} | 8.92 × 10^{−1} | 8.92 × 10^{−1} | 8.92 × 10^{−1} | 8.92 × 10^{−1} | 8.92 × 10^{−1} | 8.92 × 10^{−1} |

0.176 | 1.51 × 10^{1} | 9.63 | 5.24 | 3.04 | 1.99 | 1.47 | 1.20 | 1.05 | 9.57 × 10^{−1} | 8.35 × 10^{−1} | 8.24 × 10^{−1} | 8.24 × 10^{−1} | 8.24 × 10^{−1} | 8.24 × 10^{−1} | 8.24 × 10^{−1} | 8.24 × 10^{−1} |

0.2 | 2.40 × 10^{1} | 1.80 × 10^{1} | 1.01 × 10^{1} | 5.53 | 3.30 | 2.19 | 1.62 | 1.30 | 1.11 | 8.33 × 10^{−1} | 7.99 × 10^{−1} | 7.99 × 10^{−1} | 7.99 × 10^{−1} | 7.99 × 10^{−1} | 7.99 × 10^{−1} | 7.99 × 10^{−1} |

0.3 | 1.42 × 10^{2} | 3.19 × 10^{2} | 3.69 × 10^{2} | 2.85 × 10^{2} | 1.76 × 10^{2} | 9.81 × 10^{1} | 5.30 × 10^{1} | 2.91 × 10^{1} | 1.67 × 10^{1} | 2.39 | 6.91 × 10^{−1} | 6.59 × 10^{−1} | 6.59 × 10^{−1} | 6.59 × 10^{−1} | 6.59 × 10^{−1} | 6.59 × 10^{−1} |

0.4 | 5.20 × 10^{2} | 5.55 × 10^{3} | 3.16 × 10^{4} | 1.07 × 10^{5} | 2.39 × 10^{5} | 3.78 × 10^{5} | 4.54 × 10^{5} | 4.38 × 10^{5} | 3.57 × 10^{5} | 3.12 × 10^{4} | 1.63 × 10^{1} | 3.99 × 10^{−1} | 3.99 × 10^{−1} | 3.99 × 10^{−1} | 3.99 × 10^{−1} | 3.99 × 10^{−1} |

0.5 | 1.13 × 10^{1} | 7.41 × 10^{2} | 4.18 × 10^{4} | 2.05 × 10^{6} | 8.72 × 10^{7} | 3.25 × 10^{9} | 1.06 × 10^{11} | 3.05 × 10^{12} | 7.73 × 10^{13} | 1.40 × 10^{20} | 3.80 × 10^{32} | 4.60 × 10^{25} | 5.73 × 10^{3} | 1.69 × 10^{−3} | 1.69 × 10^{−3} | 1.69 × 10^{−3} |

**Table 3.**DLVO total potential energies between coarse (80 µm) and fine/ultra-fine particles at different solid fractions and particle sizes for the same quartz particles with different particle sizes.

ϕ/a | 8 (nm) | 12 (nm) | 16 (nm) | 20 (nm) | 24 (nm) | 28 (nm) | 32 (nm) | 36 (nm) | 40 (nm) | 60 (nm) | 120 (nm) | 500 (nm) | 1000 (nm) | 10,000 (nm) | 50,000 (nm) | 100,000 (nm) |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

0.1 | 9.98 × 10^{−}^{1} | 9.97 × 10^{−}^{1} | 9.97 × 10^{−}^{1} | 9.96 × 10^{−}^{1} | 9.96 × 10^{−}^{1} | 9.96 × 10^{−}^{1} | 9.95 × 10^{−}^{1} | 9.95 × 10^{−}^{1} | 9.95 × 10^{−}^{1} | 9.94 × 10^{−}^{1} | 9.91 × 10^{−}^{1} | 9.82 × 10^{−}^{1} | 9.75 × 10^{−}^{1} | 9.30 × 10^{−}^{1} | 8.94 × 10^{−}^{1} | 8.92 × 10^{−}^{1} |

0.176 | 9.96 × 10^{−}^{1} | 9.95 × 10^{−}^{1} | 9.95 × 10^{−}^{1} | 9.94 × 10^{−}^{1} | 9.93 × 10^{−}^{1} | 9.93 × 10^{−}^{1} | 9.92 × 10^{−}^{1} | 9.92 × 10^{−}^{1} | 9.91 × 10^{−}^{1} | 9.89 × 10^{−}^{1} | 9.85 × 10^{−}^{1} | 9.70 × 10^{−}^{1} | 9.58 × 10^{−}^{1} | 8.86 × 10^{−}^{1} | 8.29 × 10^{−}^{1} | 8.25 × 10^{−}^{1} |

0.2 | 9.96 × 10^{−}^{1} | 9.95 × 10^{−}^{1} | 9.94 × 10^{−}^{1} | 9.93 × 10^{−}^{1} | 9.92 × 10^{−}^{1} | 9.92 × 10^{−}^{1} | 9.91 × 10^{−}^{1} | 9.91 × 10^{−}^{1} | 9.90 × 10^{−}^{1} | 9.88 × 10^{−}^{1} | 9.83 × 10^{−}^{1} | 9.65 × 10^{−}^{1} | 9.52 × 10^{−}^{1} | 8.69 × 10^{−}^{1} | 8.04 × 10^{−}^{1} | 8.00 × 10^{−}^{1} |

0.3 | 9.92 × 10^{−}^{1} | 9.90 × 10^{−}^{1} | 9.88 × 10^{−}^{1} | 9.87 × 10^{−}^{1} | 9.86 × 10^{−}^{1} | 9.85 × 10^{−}^{1} | 9.83 × 10^{−}^{1} | 9.82 × 10^{−}^{1} | 9.82 × 10^{−}^{1} | 9.77 × 10^{−}^{1} | 9.68 × 10^{−}^{1} | 9.37 × 10^{−}^{1} | 9.12 × 10^{−}^{1} | 7.69 × 10^{−}^{1} | 6.66 × 10^{−}^{1} | 6.60 × 10^{−}^{1} |

0.4 | 9.82 × 10^{−}^{1} | 9.78 × 10^{−}^{1} | 9.74 × 10^{−}^{1} | 9.71 × 10^{−}^{1} | 9.69 × 10^{−}^{1} | 9.66 × 10^{−}^{1} | 9.64 × 10^{−}^{1} | 9.62 × 10^{−}^{1} | 9.60 × 10^{−}^{1} | 9.51 × 10^{−}^{1} | 9.31 × 10^{−}^{1} | 8.66 × 10^{−}^{1} | 8.16 × 10^{−}^{1} | 5.61 × 10^{−}^{1} | 4.09 × 10^{−}^{1} | 4.01 × 10^{−}^{1} |

0.5 | 9.27 × 10^{−}^{1} | 8.87 × 10^{−}^{1} | 8.56 × 10^{−}^{1} | 8.32 × 10^{−}^{1} | 8.12 × 10^{−}^{1} | 7.95 × 10^{−}^{1} | 7.80 × 10^{−}^{1} | 7.67 × 10^{−}^{1} | 7.54 × 10^{−}^{1} | 7.06 × 10^{−}^{1} | 6.10 × 10^{−}^{1} | 3.67 × 10^{−}^{1} | 2.44 × 10^{−}^{1} | 1.81 × 10^{−}^{2} | 2.01 × 10^{−}^{3} | 1.76 × 10^{−}^{3} |

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**MDPI and ACS Style**

Sajjad, M.; Otsuki, A.
Coupling Flotation Rate Constant and Viscosity Models. *Metals* **2022**, *12*, 409.
https://doi.org/10.3390/met12030409

**AMA Style**

Sajjad M, Otsuki A.
Coupling Flotation Rate Constant and Viscosity Models. *Metals*. 2022; 12(3):409.
https://doi.org/10.3390/met12030409

**Chicago/Turabian Style**

Sajjad, Mohsin, and Akira Otsuki.
2022. "Coupling Flotation Rate Constant and Viscosity Models" *Metals* 12, no. 3: 409.
https://doi.org/10.3390/met12030409