1 Introduction
2 Theory
2.1 The Wave Equation
2.2 Bubble Dynamics
2.3 Acoustic Cavitation Modeling
2.4 Particle Modeling
2.5 Particle–Particle Forces
2.6 Adhesion Theories
Hertz |
\( 0 \)
|
---|---|
JKR |
\( 3\pi \gamma R_{p} \)
|
DMT |
\( 4\pi \gamma R_{p} \)
|
Hertz |
\( 3PR_{p} /4E \)
|
---|---|
JKR |
\( 3R_{p} /4E\left[ {P + 6\pi \gamma R_{p} + \sqrt {12P\pi \gamma R_{p} + 36\pi^{2} \gamma^{2} R_{p}^{2} } } \right] \)
|
DMT |
\( 3R_{p} /4E\left[ {P + 4\pi \gamma R_{p} } \right] \)
|
2.7 Oblique Contact
2.8 Oblique Contact with JKR Adhesion
2.9 Oblique Contact with DMT Adhesion
2.10 Particle–Fluid Forces
Force | Model | Comments |
---|---|---|
Drag force (Di Felice model) |
\( \begin{array}{*{20}c} {F_{d} = \frac{1}{2}\rho_{f} (v - u)^{2} {\text{C}}_{d} \pi R_{p}^{2} \varepsilon^{ - \beta } } \\ {Re_{p} = \frac{{\rho_{f} }}{{\mu_{f} }}\alpha_{f} R_{p} \left| {v - u} \right| } \\ {{\text{C}}_{\text{d}} = \left( {0.63 + \frac{4.8}{{\sqrt {{\text{Re}}_{\text{p}} } }}} \right)^{2} } \\ {\beta = 3.7 - 0.65e^{{ - 0.5\left( {1.5 - \log_{10} Re_{p} } \right)^{2} }} } \\ \end{array} \)
|
R
p
—particle radius
Re
p
—particle Reynolds number
C
d
—drag coefficient
β—empirical coefficient
ε—void volume fraction
u, v velocities of particle and fluid
ρ
f
—fluid density |
Gravity/buoyancy |
\( F_{g} = \left( {1 - \rho_{f} /\rho_{p} } \right)mg \)
|
ρ
p
—particle material density |
Pressure gradient force |
\( F_{p} = \frac{{\rho_{f} }}{{\rho_{p} }}m\left( {\frac{du}{dt} - \left[ {\left( {v - u} \right)\cdot\nabla } \right]u} \right) \)
| — |
Saffman lift force |
\( F_{s} = - 2.18\frac{{\rho_{f} }}{{\rho_{p} }}m \frac{{({\text{v}} - {\text{u}}) \times \omega }}{{\sqrt {Re_{p} \left| \omega \right|R_{p} |v - u|} }} \)
|
ω—angular velocity of the particle |
Magnus lift force |
\( F_{m} = - \frac{3}{4}\frac{{\rho_{f} }}{{\rho_{p} }}m\left( {\frac{1}{2}\omega - \varOmega } \right) \times (v - u) \)
|
Ω—vorticity of the fluid flow |
Magnus torque |
\( M_{m} = 8\pi \mu R_{p}^{3} \left( {\frac{1}{2}\omega - \varOmega } \right) \)
| — |
2.11 Coupling of Acoustic Solver with the DEM Model
3 Problem Description
3.1 Material Properties
Material Property | Value |
---|---|
Sound speed (\( c \) m s−1) | 4600 |
Density (\( \rho \) kg m−3) | 2375 |
Dynamic viscosity (\( \mu \) mPa s) | 1.0 |
Surface tension (\( \sigma \) N m−1) | 0.860 |
Vapor pressure (\( p_{v} \) Pa) | 0 |
Bulk modulus (\( K \) GPa) | 41.2 |
Ratio of specific heats (\( \gamma \)) | 1.4 |
Particle Property | Value | Units |
---|---|---|
Diameter | 10 |
µm |
Young’s modulus | 450 | GPa |
Poisson’s ratio | 0.185 | — |
Density | 2600 | kg/m3
|
Friction coefficient | 0.3 | — |
Interfacial energy | 0.2, 2.0 | J/m2
|
3.2 Ultrasonic Treatment Setup
3.3 Particles Positioning
3.4 De-agglomeration
4 Results
4.1 Acoustic Cavitation
4.2 De-agglomeration of Particles
Valley | Time (ms) | dR/dt (m/s) | Peak | Time (ms) | dR/dt (m/s) | p-v (m/s) |
---|---|---|---|---|---|---|
v
1
| 0.123 | 0.57 |
p
1
| 0.125 | 2.2 | 1.53 |
v
2
| 0.127 | −2.32 |
p
2
| 0.128 | 0.53 | 2.85 |
v
3
| 0.142 | −2.29 |
p
3
| 0.144 | 1.42 | 3.71 |
v
4
| 0.149 | −0.5 |
p
4
| 0.152 | 3.68 | 4.18 |
v
5
| 0.181 | −2.13 |
p
5
| 0.183 | 4.52 | 6.65 |
v
6
| 0.188 | 1.71 |
p
6
| 0.190 | 3.60 | 1.89 |
v
7
| 0.232 | −5.57 |
p
7
| 0.234 | 10.96 | 16.53 |
v
8
| 0.235 | −17.4 |
p
8
| 0.237 | 1.33 | 18.73 |