Effect of Bubble Surface Properties on Bubble–Particle Collision Efficiency in Froth Flotation
Abstract
:1. Introduction
2. Theoretical Development
2.1. Schulze Model Theory
2.1.1. The Interceptional Effect
2.1.2. The Gravitational Effect
2.1.3. The Inertial Effect
2.2. GSE Model Theory
3. CFD-based Micro-Scale Modelling Method for Bubble-Particle Collision Efficiency
4. Results and Discussion
4.1. Effects of Bubble Surface Properties on Bubble Near-Wall Flow Field
4.2. Effects of Bubble Surface Properties on Particle-Bubble Relative Motion Behavior
4.3. Particle-Bubble Collision Efficiencies under Mobile and Immobile Bubble Surface Conditions
5. Conclusions
Author Contributions
Funding
Conflicts of Interest
Abbreviations
Roman | |
a | Fitting parameter |
b | Fitting parameter |
Db | Bubble diameter |
Dp | Particle diameter |
Ec | Collision efficiency |
Interceptional component of collision efficiency | |
Interceptional component of collision efficiency as given by Sutherland | |
Gravitational component of collision efficiency | |
Ecin | Inertial component of collision efficiency |
Etot | Total collision efficiency, as the sum of three components |
Collision efficiency as given by the GSE model | |
f | Parameter in the GSE model related to surface fluidity |
Fi | Force on the ith particle |
FD | Drag force on particle |
FG | Gravitational force on particle |
FP | Pressure force on particle |
FA | Added mass force on particle |
g | Acceleration due to gravity |
G | Non-dimensionalized terminal velocity of the particle, Vp/Ub |
K | Ratio of the particle inertial force to the drag force |
K3 | Modified Stokes number |
mp | Mass of particle |
p | Pressure |
r | Radial coordinate (i.e., distance from the center of the bubble) |
Rb | Bubble radius |
Rc | Distance of the critical flow line from vertical at a large distance |
Rp | Particle radius |
Re | Bubble Reynolds number |
xi | Position of the ith particle (vector) |
X | Non-dimensional radial co-ordinate, r/Rb |
u | Fluid velocity (vector) |
Ub | Velocity of bubble relative to surrounding liquid |
vi | Velocity of the ith particle (vector) |
Vp | Terminal velocity of particle |
Greek | |
β | Parameter in the GSE model |
ρ | Liquid density |
ρp | Particle density |
μ | Liquid dynamic viscosity |
ψ | Stokes (axisymmetric) stream-function |
ψc | Value of stream-function for critical (grazing) streamline |
Non-dimensionalized value of stream-function for critical streamline | |
θ | Angular coordinate (angle from the vertical) |
θc | Angle of the collision point of the critical grazing trajectory |
θt | Parameter in the GSE model |
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Model | Bubble Surface Mobility | Fluid Flow | Drawback(s) | |||
---|---|---|---|---|---|---|
Mobile and Immobile | ● | Stokes | It is valid only for particles with large Stokes Number | |||
Mobile | ● | Potential | It is not applicable for fine particles due to neglecting particle interceptional effect Particle inertial effect is entirely ignored | |||
Mobile and Immobile | ● | ● | Intermediate | It’s applicable to bubbles much larger than those used in flotation The hydrodynamic interaction between the particle and the fluid was presumed negligible | ||
Mobile | ● | ● | ● | Intermediate | The model does not take the negative inertial effect into account | |
Mobile | ● | ● | ● | Potential | The model poorly estimates the collision angle It disregards the microhydrodynamics and bubble wall effects It also neglects the higher Stokes numbers | |
Immobile | ● | Intermediate | It presumes a uniform distribution of collision over the entire upper half surface of bubbles It disregards the particle inertial effect | |||
Immobile | ● | ● | Stokes | It is only valid for Stokes flow conditions Drag force of fluid and the inertial forces of particles are ignored | ||
Immobile | ● | ● | Stokes and Potential | Inertial force is neglected No internal circulation and a non-slip condition at the bubble surface are assumed | ||
Immobile | ● | Stokes | It can be used just when the both particle (dp < 20 μm) and bubble sizes (db < 100 μm) are very fine | |||
Immobile | ● | ● | Stokes | The model can only used for small bubbles (db < 100 μm) | ||
Immobile | ● | ● | ● | Intermediate | It disregards the microhydyodynamics and bubble wall effects The impact of particle density along with cell turbulence are missing | |
Immobile | ● | Intermediate | It is only applicable when particle size (dp < 100 μm) and bubble size (db < 1 mm) | |||
Mobile and Immobile | ● | ● | ● | Intermediate | Complexity of the model makes it difficult to be used |
Bubble Diameter (mm) | Mesh Numbers | Particle Numbers | Inlet Velocity (m/s) | Re |
---|---|---|---|---|
0.6 | 5,102,342 | 4000 | 0.0656 | 39 |
0.9 | 6,590,614 | 6000 | 0.1 | 90 |
1.3 | 13,924,728 | 12,000 | 0.156 | 203 |
2.0 | 16,258,560 | 13,333 | 0.248 | 496 |
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Li, S.; Jue, K.; Sun, C. Effect of Bubble Surface Properties on Bubble–Particle Collision Efficiency in Froth Flotation. Minerals 2020, 10, 367. https://doi.org/10.3390/min10040367
Li S, Jue K, Sun C. Effect of Bubble Surface Properties on Bubble–Particle Collision Efficiency in Froth Flotation. Minerals. 2020; 10(4):367. https://doi.org/10.3390/min10040367
Chicago/Turabian StyleLi, Shuofu, Kou Jue, and Chunbao Sun. 2020. "Effect of Bubble Surface Properties on Bubble–Particle Collision Efficiency in Froth Flotation" Minerals 10, no. 4: 367. https://doi.org/10.3390/min10040367