# Effect of Agitation on the Dissolution of APAM with Different Molecular Weights and the Equivalent Diameter of Coal Slime Settling Floc with Different Particle Sizes

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Test

#### 2.1. Test Materials and Instruments

#### 2.2. Test Methods and Significance

#### 2.3. Data Calculation

^{2}/s; μ is the dynamic viscosity of the solution after dissolution, Pa.s; $\rho $ is the density of solution after dissolution, kg/m

^{3}; Re is the Reynolds number.

_{v}is the stirring power per unit volume, W/m

^{3}; V is the solution volume, m

^{3}; E

_{v}is the input energy per unit volume of solution, J/m

^{3}; T is the stirring time, s.

^{2}; C is the floc perimeter, μm; n is the floc aspect ratio; S

_{0}is the floc mesopore area, μm

^{2}; k

_{1}, k

_{2}and k

_{3}are the discount factors for perimeter, aspect ratio and mesopore area respectively, which are decimals from 0 to 1, and are taken as 0.5; k is the constant of proportionality; D

_{f}is the mean fractal dimension.

## 3. Results and Analysis

- (1)
- Table 1 and Table 2 indicate that the optimum stirring speed N = 800 r/min for 3–6 million molecular weight APAM corresponds to an impeller linear velocity v=0.84 m/s; The optimum stirring speed N = 600 r/min for 8–16 million molecular weight APAM corresponds to an impeller linear velocity v = 0.63 m/s; The optimum stirring speed N = 800 r/min for 18–25 million molecular weight APAM corresponds to an impeller linear velocity v = 0.84 m/s.
- (2)
- Figure 2, Figure 3 and Figure 4 indicate that when the stirring speed ranges from 200 to 1600 r/min, the corresponding impeller linear velocity ranges from 0.21 to 1.68 m/s, the dissolution parameters are positively correlated: kinematic viscosity and molecular weight, Reynolds number and linear velocity, power number and molecular weight, stirring power and linear velocity, stirring power and molecular weight, stirring time and molecular weight; There are negative correlations between Reynolds number and molecular weight, power number and linear velocity, stirring time and linear velocity.
- (3)
- Based on the original stirring power formula, the single factor d of impeller diameter is replaced by the two factors D of stirring bucket diameter and the ratio of the impeller to stirring bucket diameter d/D as the influencing parameters of stirring power. Through data fitting, the improved stirring power formula applicable to the dissolution of APAM is $P=K\rho {N}^{3}{D}^{4}{\left(\frac{d}{D}\right)}^{5.45}$, which is obtained by fitting all the data from the stirring dissolution test of 3–25 million APAM.

#### 3.1. Kinematic Viscosity versus Stirring Speed

#### 3.2. Reynolds Number & Stirring Power per Unit Volume versus Stirring Speed

**Figure 3.**(

**a**) Reynolds number versus stirring speed; (

**b**) Stirring power per unit volume versus stirring speed.

#### 3.3. Power Number & Stirring Time versus Stirring Speed

#### 3.4. APAM Flocculation Performance Determination at Different Stirring Speeds

_{2}) and a certain amount of kaolinite (AlSi

_{2}O

_{5}(OH)

_{4}). From Table 3 and Figure 5b, when the stirring speed was 800 r/min, the flocculation settlement was the fastest, and the clarification and compressible layer were the highest. The optimal speed of 18 million molecular weight APAM for treating 100 g/L coal slime water is 800 r/min, the optimal volume dosage is 3.2–3.6 mL/L, and the optimal solid dosage is 16–18 g/t (Formula (14)). It verifies that the flocculant solution configured at the optimal stirring speed selected in APAM stirring dissolution test has the optimal flocculation performance. The flocculant solution at the optimal stirring speed selected by APAM with other molecular weights is verified by analogy to have the optimal flocculation performance theoretically.

## 4. Equivalent Diameter of Settling Floc with Different Particle Sizes of Slime

#### 4.1. Mixing Speed versus Equivalent Diameter of Settling Floc (−0.5 mm Slime)

#### 4.2. Mixing Speed versus Equivalent Diameter of Settling Floc (0.5–0.25 mm Slime)

#### 4.3. Mixing Speed versus Equivalent Diameter of Settling Floc (0.25–0.125 mm Slime)

#### 4.4. Mixing Speed versus Equivalent Diameter of Settling Floc (0.125–0.075 mm Slime)

#### 4.5. Mixing Speed versus Equivalent Diameter of Settling Floc (0.075–0.045 mm Slime)

#### 4.6. Mixing Speed versus Equivalent Diameter of Settling Floc (−0.045 mm Slime)

#### 4.7. Results and Discussion

- (1)
- When the mixing speed N* = 600 r/min (impeller linear velocity v* = 0.63 m/s), the equivalent diameter of settling floc of −0.5 mm slime is the largest; When N* = 200 r/min or 400 r/min (v* = 0.21 m/s or 0.42 m/s), the equivalent diameters of 0.5–0.25 mm, 0.25–0.125 mm, 0.125–0.075 mm, 0.75–0.045 mm and −0.045 mm slime settling floc are the largest.
- (2)
- The equivalent diameter is positively correlated with the size of slime, and the equivalent diameter of −0.045 mm slime settling floc is the largest; The size of slime is the dominant factor of fractal dimension change, and the fractal dimension of −0.5 mm slime settling floc is the largest.
- (3)
- When the mixing speed is 0–1000 r/min, the characteristic parameters are positively correlated with: statistical diameter and floc area; The negative correlations are as follows: particle size and floc area, floc perimeter, and statistical diameter.

## 5. Flocculant Dissolution—Stirring Time Derivation Method

#### 5.1. Derivation Method and Purpose

#### 5.2. Application Formula of Stirring Time Method

^{2}is the largest and the fitting error is the smallest. Therefore, APAM of 18 million molecular weight is recommended as the flocculating agent for dissolution, and the general formula of Re-K relationship applicable to each molecular weight is proposed: K = 10Re

^{−}

^{0.5}. Similarly, through the numerical fitting of stirring power per unit volume and stirring time, it is found that they are linearly correlated, and the fitting slope and intercept are in a fluctuating trend. A general formula for the relationship between P

_{v}and T applicable to each molecular weight is put forward: T = −1.4P

_{v}+ 69.4.

#### 5.3. Contents of Derivation

^{3}. By multiplying P and (V/0.01), the stirring power per unit volume P

_{v}is obtained. Finally, the stirring time T is obtained according to the fitting relation between the corresponding molecular weight P

_{v}and T.

#### 5.4. Industrial Application of Stirring Time Derivation Method

^{−}

^{3}was taken and Re = 841.7 was obtained according to the Reynolds number formula. According to the fitting formula of K and Re in Table 10, K = 10Re

^{−0.5}, K = 0.34; Using the improved stirring formula and solution volume ratio of 20:1, the unit volume of stirring power P

_{v}was calculated to be 7.56 kW/m

^{3}. Finally, according to the fitting formula T = −1.4P

_{v}+ 69.4, the stirring time T = 58.8 min was calculated.

## 6. Conclusions

- (1)
- The optimum stirring speed is 800 r/min for 3–6 million molecular weight APAM corresponds to an impeller linear velocity is 0.84 m/s; The optimum stirring speed is 600 r/min for 8–16 million molecular weight APAM corresponds to an impeller linear velocity is 0.63 m/s; The optimum stirring speed is 800 r/min for 18–25 million molecular weight APAM corresponds to an impeller linear velocity is 0.84 m/s. Through data fitting, the improved stirring power formula for APAM dissolution is $P=K\rho {N}^{3}{D}^{4}{\left(\frac{d}{D}\right)}^{5.45}$.
- (2)
- When the stirring speed is 200 to 1600 r/min, the corresponding impeller linear velocity is 0.21 to 1.68 m/s, the positively correlated dissolution parameters are kinematic viscosity and molecular weight, Reynolds number and linear velocity, power number and molecular weight, stirring power and linear velocity, stirring power and molecular weight, stirring time and molecular weight; There are negative correlations between Reynolds number and molecular weight, power number and linear velocity, stirring time and linear velocity.
- (3)
- When the mixing speed is 600 r/min, the equivalent diameter of −0.5 mm slime settling floc is the largest. When the mixing speed is 200 r/min or 400 r/min, the equivalent diameters of 0.5–0.25 mm, 0.25–0.125 mm, 0.125–0.075 mm, 0.075–0.045 mm and −0.045 mm slime settling floc are the largest. The equivalent diameter is positively correlated with the size of slime, and the equivalent diameter of −0.045 mm slime settling floc is the largest. The size of slime is the dominant factor in the change of fractal dimension. The fractal dimension of −0.5 mm slime settling floc is the largest, and the fractal dimension is little affected by mixing speed.
- (4)
- The general formula of the relation between Reynolds number Re and power number K is: K = 10Re
^{−0.5}. The general formula of the relation between stirring power per unit volume P_{v}and stirring time T is: T = −1.4P_{v}+ 69.4, and a flocculant dissolution—stirring time derivation method for different geometric parameters of agitator drum model or industrial agitator drum is proposed.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 5.**(

**a**) Analysis of mineral composition of coal sample; (

**b**) Flocculation determination of main parameters versus stirring speed (18 million molecular weight APAM).

**Figure 6.**(

**a**) Area diagram of −0.5 mm coal slime floc at different mixing speeds. (

**I**) 0, (

**II**) 200 r/min, (

**III**) 400 r/min, (

**IV**) 600 r/min, (

**V**) 800 r/min, (

**VI**) 1000 r/min; (

**b**) Fractal dimension fitting diagram of settling floc of −0.5 mm coal slime.

**Figure 7.**(

**a**) Area diagram of 0.5–0.25 mm coal slime floc at different mixing speeds. (

**I**) 0, (

**II**) 200 r/min, (

**III**) 400 r/min, (

**IV**) 600 r/min, (

**V**) 800 r/min, (

**VI**) 1000 r/min; (

**b**) Fractal dimension fitting diagram of settling floc of 0.5–0.25 mm coal slime.

**Figure 8.**(

**a**) Area diagram of 0.25–0.125 mm coal slime floc at different mixing speeds. (

**I**) 0, (

**II**) 200 r/min, (

**III**) 400 r/min, (

**IV**) 600 r/min, (

**V**) 800 r/min, (

**VI**) 1000 r/min; (

**b**) Fractal dimension fitting diagram of settling floc of 0.25–0.125 mm coal slime.

**Figure 9.**(

**a**) Area diagram of 0.125–0.075 mm coal slime floc at different mixing speeds. (

**I**) 0, (

**II**) 200 r/min, (

**III**) 400 r/min, (

**IV**) 600 r/min, (

**V**) 800 r/min, (

**VI**) 1000 r/min; (

**b**) Fractal dimension fitting diagram of settling floc of 0.125–0.075 mm coal slime.

**Figure 10.**(

**a**) Area diagram of 0.075–0.045 mm coal slime floc at different mixing speeds. (

**I**) 0, (

**II**) 200 r/min, (

**III**) 400 r/min, (

**IV**) 600 r/min, (

**V**) 800 r/min, (

**VI**) 1000 r/min; (

**b**) Fractal dimension fitting diagram of settling floc of 0.075–0.045 mm coal slime.

**Figure 11.**(

**a**) Area diagram of −0.045 mm coal slime floc at different mixing speeds. (

**I**) 0, (

**II**) 200 r/min, (

**III**) 400 r/min, (

**IV**) 600 r/min, (

**V**) 800 r/min, (

**VI**) 1000 r/min; (

**b**) Fractal dimension fitting diagram of settling floc of −0.045 mm coal slime.

Molecular Weight /(Million) | Stirring Speed N/(r/min) | Linear Velocity v/(m/s) | Stirring Time T/(min) | Kinematic Viscosity γ/(×10 ^{−3}m^{2}/s) | Reynolds Number Re | Power Number K | Stirring Power per Unit Volume P _{v}/(kW/m^{3}) | Energy Input per Unit Volume E _{v}/(×10^{3}kJ/m^{3}) |
---|---|---|---|---|---|---|---|---|

3 | 800 | 0.84 | 65.50 | 1.32 | 12.73 | 2.57 | 1.94 | 7.62 |

6 | 800 | 0.84 | 64.27 | 1.53 | 10.98 | 2.74 | 2.06 | 7.94 |

8 | 600 | 0.63 | 65.28 | 1.69 | 7.46 | 3.35 | 1.06 | 4.15 |

10 | 600 | 0.63 | 68.63 | 1.63 | 7.73 | 3.29 | 1.04 | 4.28 |

12 | 600 | 0.63 | 68.27 | 1.63 | 7.73 | 3.29 | 1.05 | 4.30 |

16 | 600 | 0.63 | 69.48 | 2.15 | 5.86 | 3.87 | 1.22 | 5.09 |

18 | 800 | 0.84 | 65.23 | 2.34 | 7.18 | 3.43 | 2.58 | 10.10 |

20 | 800 | 0.84 | 65.82 | 2.04 | 8.24 | 3.17 | 2.39 | 9.44 |

22 | 800 | 0.84 | 66.63 | 1.93 | 8.70 | 3.08 | 2.31 | 9.23 |

25 | 800 | 0.84 | 67.25 | 2.30 | 7.30 | 3.39 | 2.54 | 10.25 |

APAM Molecular Weight /(Million) | Optimal Stirring Speed /(r/min) | Optimal Linear Velocity /(m/s) | Optimal Stirring Time /(min) | Optimal Kinematic Viscosity /(×10 ^{−3}m^{2}/s) | Stirring Power per Unit Volume /(kW/m ^{3}) | Energy Input per Unit Volume /(×10 ^{3}kJ/m^{3}) |
---|---|---|---|---|---|---|

3–6 | 800 | 0.84 | 64–66 | 1.3–1.6 | 1.9–2.1 | 7.6–8.0 |

8–16 | 600 | 0.63 | 65–70 | 1.6–2.2 | 1.0–1.3 | 4.1–5.1 |

18–25 | 800 | 0.84 | 65–68 | 1.9–2.4 | 2.3–2.6 | 9.2–10.3 |

APAM Stirring Speed /(r/min) | APAM Stirring Time /(min) | Time of Interface Settlement /s | Velocity of Free Settlement /(m/h) | Degree of Clarification after 30 min /% | Height of Compression Layers /mm | APAM Volume Dosage /(mL/L) | APAM Solid Dosage /(g/t) |
---|---|---|---|---|---|---|---|

200 | 72.40 | 32.77 | 13.77 | 23.2 | 42.7 | 3.2–3.6 | 16–18 |

400 | 70.12 | 30.02 | 14.92 | 30.8 | 45.2 | 3.2–3.6 | 16–18 |

600 | 68.03 | 29.98 | 14.93 | 36.5 | 45.1 | 3.2–4.0 | 16–20 |

800 | 65.23 | 19.82 | 22.66 | 43.0 | 48.1 | 3.2–3.6 | 16–18 |

1000 | 62.42 | 25.88 | 16.92 | 38.6 | 46.6 | 3.2–4.0 | 16–20 |

1200 | 60.15 | 27.49 | 16.53 | 35.6 | 44.8 | 2.8–3.2 | 14–16 |

1400 | 56.30 | 28.10 | 16.45 | 33.1 | 44.2 | 2.4–2.8 | 12–14 |

1600 | 52.42 | 30.67 | 14.60 | 31.2 | 41.3 | 3.2–3.6 | 16–18 |

Slime Size/mm | Percentage of This Level/% | Ash/% |
---|---|---|

−0.5 | 100 | 44.30 |

0.5–0.25 | 52.17 | 39.20 |

0.25–0.125 | 20.03 | 45.26 |

0.125–0.075 | 21.45 | 46.93 |

0.075–0.045 | 6.29 | 49.60 |

−0.045 | 0.06 | 40.50 |

Mixing Speed N*/(r/min) | Floc Area S/μm ^{2} | Floc Perimeter C/μm | Floc Aspect Ratio | Mesopore Area S _{0}/μm^{2} | Equivalent Diameter Φ/μm | Statistical Diameter D*/μm | Floc Fractal Dimension |
---|---|---|---|---|---|---|---|

0 | 13.844 | 13.517 | 1.764 | 0.020 | 3.247 | 2.952 | 2.106 |

200 | 332.100 | 100.191 | 1.418 | 0.614 | 14.405 | 19.691 | 2.121 |

400 | 487.159 | 117.184 | 1.465 | 0.900 | 17.455 | 23.625 | 2.113 |

600 | 621.536 | 147.912 | 1.453 | 1.919 | 18.938 | 27.140 | 2.137 |

800 | 250.463 | 97.007 | 1.573 | 1.709 | 11.487 | 16.916 | 2.202 |

1000 | 217.873 | 82.418 | 1.523 | 1.219 | 11.246 | 15.504 | 2.164 |

Mixing Speed N*/(r/min) | Floc Area S/μm ^{2} | Floc Perimeter C/μm | Floc Aspect Ratio | Mesopore Area S _{0}/μm^{2} | Equivalent Diameter Φ/μm | Statistical Diameter D*/μm | Floc Fractal Dimension |
---|---|---|---|---|---|---|---|

0 | 8.796 | 15.154 | 1.911 | 0.155 | 2.140 | 2.515 | 1.893 |

200 | 210.463 | 71.005 | 1.472 | 0.643 | 11.832 | 15.733 | 1.868 |

400 | 232.714 | 90.271 | 1.454 | 1.049 | 11.588 | 16.437 | 1.925 |

600 | 161.343 | 71.092 | 1.418 | 0.924 | 9.952 | 13.533 | 1.893 |

800 | 147.104 | 70.947 | 1.528 | 1.287 | 9.051 | 12.741 | 1.933 |

1000 | 133.455 | 57.871 | 1.596 | 0.744 | 9.026 | 11.729 | 1.891 |

Mixing Speed N*/(r/min) | Floc Area S/μm ^{2} | Floc Perimeter C/μm | Floc Aspect Ratio | Mesopore Area S _{0}/μm^{2} | Equivalent Diameter Φ/μm | Statistical Diameter D*/μm | Floc Fractal Dimension |
---|---|---|---|---|---|---|---|

0 | 8.809 | 15.046 | 1.855 | 0.124 | 2.174 | 2.603 | 2.011 |

200 | 318.213 | 117.278 | 1.731 | 1.450 | 12.192 | 18.481 | 2.059 |

400 | 526.273 | 134.425 | 1.765 | 1.172 | 16.253 | 24.452 | 2.027 |

600 | 347.649 | 104.718 | 1.638 | 0.835 | 13.801 | 19.316 | 1.995 |

800 | 319.651 | 97.070 | 1.630 | 0.480 | 13.441 | 18.969 | 1.983 |

1000 | 213.312 | 87.115 | 1.663 | 0.698 | 10.501 | 15.223 | 2.019 |

Mixing Speed N*/(r/min) | Floc Area S/μm ^{2} | Floc Perimeter C/μm | Floc Aspect Ratio | Mesopore Area S _{0}/μm^{2} | Equivalent Diameter Φ/μm | Statistical Diameter D*/μm | Floc Fractal Dimension |
---|---|---|---|---|---|---|---|

0 | 10.155 | 13.978 | 1.784 | 0.019 | 2.534 | 3.151 | 2.068 |

200 | 1234.052 | 209.218 | 1.559 | 3.533 | 25.911 | 38.391 | 2.066 |

400 | 1488.903 | 294.515 | 1.756 | 9.195 | 24.941 | 41.258 | 2.149 |

600 | 516.473 | 129.028 | 1.410 | 3.016 | 17.747 | 24.637 | 2.035 |

800 | 412.032 | 127.182 | 1.811 | 2.416 | 13.876 | 21.151 | 2.139 |

1000 | 126.904 | 64.890 | 1.710 | 0.788 | 8.110 | 11.986 | 2.125 |

Mixing Speed N*/(r/min) | Floc Area S/μm ^{2} | Floc Perimeter C/μm | Floc Aspect Ratio | Mesopore Area S _{0}/μm^{2} | Equivalent Diameter Φ/μm | Statistical Diameter D*/μm | Floc Fractal Dimension |
---|---|---|---|---|---|---|---|

0 | 3.379 | 8.300 | 1.708 | 0.021 | 1.463 | 1.930 | 2.096 |

200 | 1602.686 | 261.753 | 1.660 | 1.841 | 27.892 | 43.195 | 2.091 |

400 | 1054.716 | 194.764 | 1.782 | 2.550 | 22.729 | 33.940 | 2.094 |

600 | 1008.134 | 187.500 | 1.474 | 2.127 | 24.032 | 34.498 | 2.043 |

800 | 481.434 | 133.881 | 1.542 | 2.811 | 16.085 | 23.397 | 2.072 |

1000 | 293.505 | 93.803 | 1.612 | 1.128 | 12.876 | 18.047 | 2.059 |

Mixing Speed N*/(r/min) | Floc Area S/μm ^{2} | Floc Perimeter C/μm | Floc Aspect Ratio | Mesopore Area S _{0}/μm^{2} | Equivalent Diameter Φ/μm | Statistical Diameter D*/μm | Floc Fractal Dimension |
---|---|---|---|---|---|---|---|

0 | 2.354 | 8.692 | 1.909 | 0.027 | 1.066 | 1.637 | 1.958 |

200 | 1358.856 | 197.174 | 1.670 | 4.624 | 27.597 | 39.726 | 1.954 |

400 | 2143.444 | 285.286 | 1.436 | 4.370 | 34.865 | 51.044 | 1.954 |

600 | 702.405 | 162.639 | 1.740 | 1.173 | 18.558 | 28.231 | 1.994 |

800 | 521.996 | 100.775 | 1.317 | 1.514 | 20.422 | 24.848 | 1.832 |

1000 | 450.453 | 132.859 | 1.550 | 2.575 | 15.384 | 23.013 | 1.968 |

Molecular Weight /(Million) | Re versus K Power Fitting Relationship | P_{v} versus T LinearFitting Relationship |
---|---|---|

3 | K = 9.3797Re^{−0.487} | T = −1.5359P_{v} + 68.71 |

6 | K = 9.8157Re^{−0.512} | T = −1.0376P_{v} + 66.148 |

8 | K = 9.6223Re^{−0.5} | T = −1.1622P_{v} + 66.272 |

10 | K = 10.018Re^{−0.521} | T = −1.5059P_{v} + 69.669 |

12 | K = 10.3Re^{−0.535} | T = −1.6264P_{v} + 68.53 |

16 | K = 10.397Re^{−0.544} | T = −1.4681P_{v} + 70.688 |

18 | K = 10.486Re^{−0.55} | T = −1.3668P_{v} + 69.917 |

20 | K = 10.122Re^{−0.534} | T = −1.5372P_{v} + 70.586 |

22 | K = 9.952Re^{−0.526} | T = −1.6101P_{v} + 71.961 |

25 | K = 10.048Re^{−}^{0.532} | T = −1.1669P_{v} + 71.84 |

Average Fitting General Formula | K = 10Re^{−0.5} | T = −1.4P_{v} + 69.4 |

Core Components | Diameter/cm | Length, Width, Height/cm | Note |
---|---|---|---|

Impeller | 16 | 10 × 8 × 6 | Number 3, Angle 36° |

Mixing drum | 68 | 68 × 68 × 68 | Treatment 0.2 m^{3} |

Stirring rod | 10 | 10 × 6 × 64 | Connecting impeller |

Motor | 26 | High 30 | Fixed speed 1440 r/min |

Reduction gear | 26 | High 23 | Increase output torque |

Coupling | Upper 26, Lower 30 | High 25 | Cushioning and damping |

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## Share and Cite

**MDPI and ACS Style**

Li, J.; Zhou, W.; Cai, C.; Wang, S.; Zhu, J.
Effect of Agitation on the Dissolution of APAM with Different Molecular Weights and the Equivalent Diameter of Coal Slime Settling Floc with Different Particle Sizes. *Minerals* **2023**, *13*, 204.
https://doi.org/10.3390/min13020204

**AMA Style**

Li J, Zhou W, Cai C, Wang S, Zhu J.
Effect of Agitation on the Dissolution of APAM with Different Molecular Weights and the Equivalent Diameter of Coal Slime Settling Floc with Different Particle Sizes. *Minerals*. 2023; 13(2):204.
https://doi.org/10.3390/min13020204

**Chicago/Turabian Style**

Li, Jianbo, Wei Zhou, Chuanchuan Cai, Shujie Wang, and Jinbo Zhu.
2023. "Effect of Agitation on the Dissolution of APAM with Different Molecular Weights and the Equivalent Diameter of Coal Slime Settling Floc with Different Particle Sizes" *Minerals* 13, no. 2: 204.
https://doi.org/10.3390/min13020204