# Effect of Particle Size and Shape on Separation in a Hydrocyclone

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Experimental Modes and Methods

#### 2.1. Hydrocyclone Model

#### 2.2. Numerical Simulation

#### 2.2.1. Particle Characteristics

#### 2.2.2. Mathematical Model and Simulation Method

^{−1}, The continuous medium flow is calculated from the continuity and the Navier–Stokes equations based on the local mean variables over a computational cell, which are given by:

_{p−f}is Interaction forces between fluid and solid phases, equal to $\sum _{\mathrm{i}=1}^{{\mathrm{k}}_{\mathrm{c}}}{\mathrm{f}}_{\mathrm{P}-\mathrm{f},\mathrm{i}}/\Delta {\mathrm{V}}_{\mathrm{c}},\mathrm{N}/{\mathrm{m}}^{3}$; the flow solved in Equations (1) and (2) represents the mixture flow of medium and sand, and was obtained by use of the models in a commercial ANSYS software package, i.e., Fluent. The details of the medium flow calculation and its validation can be found elsewhere [13,14]. In this work, we only give an overall description of the ANSYS model for the hydrocyclone.

_{p}

_{−f},

_{i}):

_{t})

_{0}and C

_{1}were determined from the correlation of the particle Reynolds numbers and measured drag coefficients using the ratio of the particle diameter D

_{2}to thickness T as the particle shape factor

_{D})

#### 2.3. Separation Experiments

^{−3}and 40 kg·m

^{−3}, respectively.

**Experiment**

**1.**

**Experiment**

**2.**

## 3. Analysis of Experimental Results

#### 3.1. Flow Field, Flow Rate and Particle Distribution

#### 3.2. Relationship between Maximum Projected Area and Separation Results

#### 3.2.1. Radial Concentration Distribution of Particles

#### 3.2.2. Separation Results in Numerical Simulation

#### 3.3. Separation Mechanism of Single and Mixed Particles

#### 3.3.1. Distribution of Axial Velocity and Radial Concentration

#### 3.3.2. Particle Separation Results

## 4. Experimental Tests

#### 4.1. Test on the Effect of Cross-Section Change on Separation

^{2}, respectively. Accordingly, the size of spherical particles was 100.75, 103.11, 104.65, 69.81, and 70.93 µm, whose volume was less than that of spherical particles with the same projected area. Smaller particles were more than larger particles. Moreover, most were lamelliform particles with a similar size.

#### 4.2. Separation Test of Sand Particles 60 and 120 μm

^{−3}and 40 kg·m

^{−3}, respectively, to ensure an equal number of sand particles put into the hydrocyclone. The capacity of the container was 1 m

^{3}. The separation experiment lasted for 16 min.

## 5. Discussion

#### 5.1. Influence of Maximum Projected Area and Volume Changes on Separation Results

#### 5.2. Analysis of the Following Phenomenon of Fine Particles

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Nomenclature

A | measure of area |

A_{p} | maximum projected area of sand particles in the direction of fluid flow, m^{2} |

C_{LS} | Saffman lit coefficient |

C_{LM} | magnus lift coefficient |

D_{1} | drag |

D_{2} | particle diameter |

d_{p} | particle diameter, m |

d | damping |

d_{i} | diameter of sand particles, m |

f | fluid phase |

F_{A} | added mass force per unit mass, m·s^{−2} |

F_{B} | basset force per unit mass, m·s^{−2} |

F_{D} | total drag force; |

F_{M} | Magus force per unit mass, m·s^{−2} |

F_{P} | pressure gradient force per unit mass, m·s^{−2} |

F_{p-f} | interaction forces between fluid and solids phases |

F_{S} | Saffman lift force per unit mass, m·s^{−2} |

g | gravity acceleration vector, 9.81 m·s^{−2} |

i(j) | corresponding to i(j)th particle |

k_{c} | number of particles in a computational cell, dimensionless |

N_{downstream} | number of sand particles separated from the downstream |

N_{upstream} | number of sand particles separated from the upstream |

P_{downstream} | fractional flow of the downstream |

P_{upstream} | fractional flow of the upstream |

P | pressure, Pa |

pg | pressure gradient |

p | particle phase |

Re | Reynolds number, dimensionless |

T | particle shape factor (thickness) |

t | time, s |

u | fluid velocity vector, m·s^{−1} |

u_{t} | settling velocity, m·s^{−1} |

u_{r} | relative velocity, m·s^{−}^{1} |

V | volume, m^{3} |

v | fluid kinematic viscosity, kg·m^{−1} s^{−1} |

ΔV_{c} | volume of a computational cell, m^{3} |

ΔP | pressure drop, Pa |

∇p | pressure gradient, kg·m^{−2}s^{−2} |

Greek letters | |

ε | porosity, dimensionless |

μ | fluid viscosity, kg·m^{−}^{1}·s^{−}^{1} |

τ | viscous stress tensor, N·m^{−3} |

ρ | density, kg·m^{−3} |

β | Empirical coefficient defined in Equation (3), dimensionless |

ρ_{p} | particle density, kg·m^{−3} |

γ | fluid strain rate, s^{−1} |

ω_{r} | relative angular velocity, rad·s^{−}^{1} |

ω | particle rotation angular velocity, rad·s^{−}^{1} |

ξ | drag force coefficient |

Ω | fluid rotation angular velocity, rad·s^{−}^{1} |

f | force |

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**Figure 4.**Structure of the cyclone, water flow velocity (m·s

^{−1}) and pressure (Pa) distribution: (

**a**) structure of hydrocyclone; (

**b**) flow velocity in X-axis; (

**c**) flow velocity in Y-axis; (

**d**) flow velocity in Z-axis; (

**e**) water flow velocity distribution; (

**f**) water flow pressure distribution.

**Figure 5.**Sand particle distribution in simulation: (

**a**) the separation simulation of 60 and 120 μm particles; (

**b**) the simulation of the effect of the cross-section change of sand particles in the separation; (

**c**) the distribution of sand particles along the Z-axis; (

**d**) the particle distribution at the D-D section in (

**b**).

**Figure 6.**Average velocity distribution in a positive direction of Z axis: (

**a**) radial velocity of 60 and 120 µm particles; (

**b**) partially enlarged image of (

**a**).

Title 1 | Type A | Type B | Type C | 60 µm | 70 µm | 90 µm | 106 µm | 120 µm |
---|---|---|---|---|---|---|---|---|

Mass (μg) | 0.28 | 0.28 | 0.28 | 0.28 | 0.44 | 0.95 | 1.54 | 2.23 |

Thickness (mm) | 0.015 | 0.030 | 0.045 | 0.060 | 0.070 | 0.090 | 0.106 | 0.120 |

Maximum projected area (S) | 3.11 | 2.24 | 1.36 | 1.00 | 1.36 | 2.24 | 3.11 | 4.00 |

Phase | Parameter | Symbol | Units | Value |
---|---|---|---|---|

Solid | Density distribution | ρ | kg·m^{−3} | 2500 |

Rolling friction coefficient | μ_{r} | - | 0.01 | |

Sliding friction coefficient | μ_{s} | - | 0.30 | |

Poisson’s ratio | V | - | 0.40 | |

Young’s modulus | E | N·m^{−2} | 2 × 10^{−7} | |

Coefficient of Restitution | c_{r} | - | 0.55 | |

fluid | Density | ρ | kg·m^{−}^{3} | 998.20 |

Viscosity | μ | kg·m^{−1}·s^{−1} | 0.001 |

Symbol | Units | Value | |
---|---|---|---|

Particle velocity at inlet | v | m·s^{−1} | 2.00 |

Viscosity of Water Phase | v | m·s^{−1} | 2.00 |

Turbulent intensity | I | - | 5% |

Hydraulic radius | D | mm | 2.00 |

Pressure at upstream | - | Pa | 0.00 |

Pressure at downstream | - | Pa | 0.00 |

Back-flow turbulence intensity | I_{h} | - | 5% |

Number of particle | - | N·s^{−1} | 1000 |

Particle diameter | d_{i} | μm | - |

S/N | Name | Formula | Description |
---|---|---|---|

1 | The normal force (F_{n}) | ${\mathrm{F}}_{\mathrm{n}}=\frac{4}{3}{\mathrm{E}}^{\ast}{\left({\mathrm{R}}^{\ast}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}{\mathsf{\alpha}}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$ | R* is the equivalent radius α is the normal overlap |

2 | The equivalent elastic modulus (E*) | $\frac{1}{{\mathrm{E}}^{\ast}}=\frac{1-{\mathrm{V}}_{1}^{2}}{{\mathrm{E}}_{1}}+\frac{1-{\mathrm{V}}_{2}^{2}}{{\mathrm{E}}_{2}}$ | E_{1}, v_{1} and E_{1}, v_{1} are elastic modulus and Poisson’s ratio of sand 1 and sand 2 |

3 | The damping force (F_{n}^{d}) | ${\mathrm{F}}_{\mathrm{n}}^{\mathrm{d}}=-2\sqrt{\frac{5}{6}}\mathsf{\beta}\sqrt{{\mathrm{S}}_{\mathrm{n}}{\mathrm{m}}^{\ast}}{\mathrm{V}}_{\mathrm{n}}^{\mathrm{rel}}$ | ${\mathrm{V}}_{\mathrm{n}}^{\mathrm{rel}}$ is the normal relative velocity; S_{n} is the normal stiffnessβ is coefficient |

4 | The equivalent mass (m*) | ${\mathrm{m}}^{\ast}=\frac{{\mathrm{m}}_{1}{\mathrm{m}}_{2}}{{\mathrm{m}}_{1}+{\mathrm{m}}_{2}}$ | m_{1} and m_{2} are the mass of sand 1 and sand 2 |

5 | The tangential force among the sands (F_{t}) | ${\mathrm{F}}_{\mathrm{t}}=-{\mathrm{S}}_{\mathrm{t}}\mathsf{\delta}$ | δ is the tangential overlap |

6 | The tangential stiffness (S_{t}) | ${\mathrm{S}}_{\mathrm{t}}={8\mathrm{G}}^{\ast}\sqrt{{\mathrm{R}}^{\ast}\mathsf{\alpha}}$ | |

7 | The equivalent shear modulus (G*) | ${\mathrm{G}}^{\ast}=\frac{2-{\mathrm{V}}_{1}^{2}}{{\mathrm{G}}_{1}}+\frac{2-{\mathrm{V}}_{2}^{2}}{{\mathrm{G}}_{2}}$ | G_{1} and G_{2} are shear modulus of sand 1 and sand 2 |

- | V_{1} and V_{2} are velocity of sand 1 and sand 2 | ||

8 | The tangential damping force among sand particles (F_{t}) | ${\mathrm{F}}_{\mathrm{t}}=-2\sqrt{\frac{5}{6}}\mathsf{\beta}\sqrt{{\mathrm{S}}_{\mathrm{t}}{\mathrm{m}}^{\ast}}{\mathrm{V}}_{\mathrm{t}}^{\mathrm{rel}}$ | ${\mathrm{V}}_{\mathrm{t}}^{\mathrm{rel}}$ is the tangential relative velocity |

9 | The rolling friction (T_{i}) | ${\mathrm{T}}_{\mathrm{i}}=-{\mathsf{\mu}}_{\mathrm{r}}{\mathrm{F}}_{\mathrm{n}}{\mathrm{R}}_{\mathrm{i}}{\mathsf{\omega}}_{\mathrm{i}}$ | μ_{r} is coefficient of rolling frictionR _{i} is the distance between the center of mass to the point of contact; ω_{i} is unit angular velocity vector of object at the contact point |

Particles | Particle Size (µm) | Concentration (kg·m^{−3}) | Separation Time (s) |
---|---|---|---|

Singe | 60 | 5 | 16 |

120 | 40 | 16 | |

Mixed | 60 + 120 | 5 + 40 | 16 |

Radial Distance | Type A | Type B | Type C | 60 µm | 70 µm | 90 µm | 106 µm |
---|---|---|---|---|---|---|---|

Percentage of Content (%) | |||||||

0–1 | 8.10 | 6.82 | 3.96 | 2.65 | 2.11 | 1.81 | 1.01 |

1–2 | 9.01 | 7.12 | 5.39 | 5.14 | 3.23 | 2.34 | 1.96 |

2–3 | 12.87 | 11.12 | 9.01 | 8.63 | 7.05 | 5.91 | 3.21 |

3–4 | 18.59 | 17.36 | 16.20 | 13.85 | 11.61 | 7.32 | 6.89 |

4–5 | 51.43 | 57.58 | 65.44 | 69.73 | 76.00 | 82.62 | 86.93 |

Radial Distance | Maximum Projected Area (S) | t_{upstream}(s) | t_{downstream}(s) | V_{upstream}(m s ^{−1}) | V_{downstream}(m s ^{−1}) | L_{upstream}(m) | L_{downstream}(m) | p (%) |
---|---|---|---|---|---|---|---|---|

Type A | 3.11 | 0.63 | 2.21 | 0.41 | 0.59 | 0.26 | 1.30 | 24.0 |

Type B | 2.24 | 0.66 | 2.15 | 0.41 | 0.59 | 0.27 | 1.27 | 17.8 |

Type C | 1.36 | 0.69 | 2.11 | 0.41 | 0.59 | 0.28 | 1.25 | 14.9 |

60 µm | 1.00 | 0.72 | 2.08 | 0.40 | 0.58 | 0.29 | 1.21 | 9.4 |

70 µm | 1.36 | 0.75 | 2.02 | 0.39 | 0.59 | 0.29 | 1.19 | 7.4 |

90 µm | 2.24 | 0.79 | 1.98 | 0.39 | 0.60 | 0.31 | 1.19 | 3.2 |

106 µm | 3.11 | 0.81 | 1.92 | 0.39 | 0.60 | 0.32 | 1.15 | 0.9 |

_{upstream}, the average residence time of particles crossing the upstream; t

_{downstream}is the average residence time of particles crossing the downstream; V

_{upstream}is the average absolute velocity of particles crossing the upstream; V

_{downstream}is the average absolute velocity of particles crossing the downstream; L

_{upstream}is the average path length of particles crossing the upstream; L

_{downstream}is the average path length of particles crossing the downstream; and p is the percentage of the particles crossing the upstream in the separated particles, which was obtained using Equation (19).

0–1 mm | 1–2 mm | 2–3 mm | 3–4 mm | 4–5 mm | ||
---|---|---|---|---|---|---|

60 µm particles | Single | 2.65 | 5.14 | 8.63 | 13.85 | 69.73 |

Mixed | 1.41 | 1.16 | 1.19 | 5.78 | 90.46 | |

Difference | −1.24 | −3.98 | −7.44 | −8.07 | 20.73 | |

120 µm particles | Single | 1.33 | 1.50 | 1.64 | 2.30 | 93.23 |

Mixed | 1.21 | 1.32 | 1.79 | 2.95 | 92.73 | |

Difference | −0.12 | −0.18 | 0.15 | 0.65 | −0.5 |

t_{avearge} | N_{downstream} | N_{upstream} | p (%) | ||
---|---|---|---|---|---|

Single | X | 2.00 | 906 | 94 | 90.6 |

D | 1.70 | 1000 | 0 | 100 | |

Mixed | X | 1.80 | 947 | 53 | 94.7 |

D | 1.70 | 1000 | 0 | 100 |

_{avearge}is the average time duration in which particles were separated from the downstream staying in the hydrocyclone in 3–4 s; N

_{downstream}is the number of particles passing the downstream; N

_{upstream}is the number of particles passing the upstream in 3–4 s; and p is the percentage of particles in the downstream, which was calculated using Formula (19).

Single | Mixed | |||
---|---|---|---|---|

60 µm | 120 µm | 60 µm | 120 µm | |

Upstream | 0.312 | 0 | 0.255 | 0 |

Downstream | 28.85 | 29.25 |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Tang, Z.; Yu, L.; Wang, F.; Li, N.; Chang, L.; Cui, N.
Effect of Particle Size and Shape on Separation in a Hydrocyclone. *Water* **2019**, *11*, 16.
https://doi.org/10.3390/w11010016

**AMA Style**

Tang Z, Yu L, Wang F, Li N, Chang L, Cui N.
Effect of Particle Size and Shape on Separation in a Hydrocyclone. *Water*. 2019; 11(1):16.
https://doi.org/10.3390/w11010016

**Chicago/Turabian Style**

Tang, Zhaojia, Liming Yu, Fenghua Wang, Na Li, Liuhong Chang, and Ningbo Cui.
2019. "Effect of Particle Size and Shape on Separation in a Hydrocyclone" *Water* 11, no. 1: 16.
https://doi.org/10.3390/w11010016