A New Method for Predicting Erosion Damage of Suddenly Contracted Pipe Impacted by Particle Cluster via CFD-DEM
Abstract
:1. Introduction
2. Solution Methodology
2.1. Particle-Particle Interaction
2.2. Momentum Exchange and Energy Dissipation
2.3. Particle-Wall Interaction
2.4. Governing Equations of Fluid Flow
2.5. Erosion Model
3. Implementation and Verification
3.1. Numerical Solving Step
3.2. Boundary Conditions
3.3. Full-Scale Experiment
4. Results
4.1. Particle-Wall Impingement
4.2. Particle Cluster Erosion
4.3. Discussion
5. Conclusions
- (1)
- Formation of particle clusters results in interparticle collisions when certain particles bounce back from the wall and the succeeding particle erosion rate decreases with the change in particle impact velocity.
- (2)
- During particle impact and rebound, the contact time of the target surface is less than that of the particle rebound time before the particle collides with another particle. The decay rate of a normal impact velocity changes from 60 to 90% along the radial surface and the decay rate of the tangential velocity changes in the range of 30 to 40%.
- (3)
- A smaller critical interparticle distance, which indicates that particles must be closer to each other for a collision to exist, results in a lower clustered particle percentage of the total impact particles. Therefore, the probability and frequency of a particle cluster impact on the outer circumference are larger than those on the inner edge. Meanwhile, by updating the particle impact velocity and angle, the relative error between the calculation results and the experimental results was reduced by almost 11% compared with that of the complete independent impact setting.
- (4)
- The second impact velocity of the particle in IPE is almost three times the velocity in SPE on the inner edge and almost 10 times on the outer circumference because the particle velocity decays once in IPE and twice in SPE. The growth of flow velocity not only decreases the critical interparticle distance but also the decay coefficient in particle collision, indicating that the effect of a cluster impact on erosion can be weakened by an increasing flow velocity.
Author Contributions
Acknowledgments
Conflicts of Interest
Nomenclature
Ap | particle cross-sectional area, m2 | te | contact time, s |
Cd | drag coefficient, dimensionless | tr | release time, s |
Cn | normal damping coefficient, kg·s−1 | t1 | rebound time, s |
Ct | tangential damping coefficient, kg·s−1 | t2 | moving time for particle j, s |
D | inside diameter of circular pipe, m | ∆tD | DEM time step, s |
dp | diameter of particle, m | ∆tRa | Rayleigh time step, s |
en | restitution coefficient in the normal direction, dimensionless | Tr | rolling friction torque, N·m |
et | restitution coefficient in the tangential direction, dimensionless | Tt | torque generated by tangential forces, N·m |
E | elastic modulus, Pa | u | fluid flow velocity, m |
ER | total erosion rate, kg/kg | ∆v | the relative velocities of the centers of the spheres before and after a collision, m·s−1 |
ERA | erosion rate caused by independent particle impact, kg/kg | vn | particle normal velocity, m·s−1 |
ERI | erosion rate caused by grouped particle impact, kg/kg | vt | particle tangential velocity, m·s−1 |
E* | equivalent elastic modulus, Pa | vref | the relative velocity of the contact points, m·s−1 |
∆En | normal energy loss of a particle caused by particle-particle collision, J | Vp | volume of particle, m3 |
∆Et | tangential energy loss of a particle caused by particle-particle collision, J | Vl | volume of liquid cell, m3 |
Fb | buoyancy, N | ∆x | space step, m |
Fd | drag force, N | Y | Young’s modulus, Pa |
Fg | gravitational force, N | a | empirical constants in angle functions, dimensionless |
Fl,i | the fluid force acting on a particle, N | b | |
Fn | inter-particle contact force in the normal direction, N | w | |
Fp,i | the particle force acting on the liquid, N | x | |
Fij | contact force between particles, N | y | |
Ft | inter-particle contact force in the tangential direction, N | z | |
Fs | particle shape coefficient, dimensionless | Greek symbols | |
Fnd | inter-particle damping force in the normal direction, N | ωi | unit angular velocity vector of the object, rad·s−1 |
Ftd | inter-particle damping force in the tangential direction, N | λ | decay coefficient in the collision, dimensionless |
G | shear modulus, Pa | κ | particle velocity loss rate in process of particle-wall impact, % |
G* | equivalent shear modulus, Pa | ν | poisson ratio, dimensionless |
H | maximum depth of indentation, m | δ | tangential displacement, m |
Δh | depth of the extruding lips, m | μ | dynamic viscosity, Pa·s |
I | moment of inertia, m4 | β | The angle between the particle velocity and the centerline of the two particles, ° |
J | particle momentum, kg·m·s−1 | δk | identity tensor |
K | dimensionless coefficient, dimensionless | δn | normal overlap, m |
L | distance from the inner edge of the sample, m | δt | tangential overlap, m |
Lp | critical inter-particle distance of particle, m | ρl | liquid density, kg·m−3 |
Lp,e | the moving distance of the particle j along the centerline, m | μs | friction coefficient, dimensionless |
Lp,r | the critical inter-particle distance in the release process, m | μr | rolling friction coefficient, dimensionless |
Ly | particle spacing in the Y direction, m | φ | scattering angle for particle j, ° |
Lz | particle spacing in the Z direction, m | ψ | scattering angle for particle i, ° |
mp | single particle mass, kg | α | particle impact angle, ° |
m* | equivalent mass, kg | α1 | new particle impact angle, ° |
N | total particle number, dimensionless | αl | fluid volume fraction, % |
Np | the number of sample points contained within the mesh cell of the particle, dimensionless | θ | critical angle of particle impact, ° |
N1 | the number of particles in the particle group, dimensionless | θ* | critical angle between particles, ° |
Nt | total number of sample points of the particle, dimensionless | ε | strain caused by the normal stress of indentation, m |
Pn | function of properties of the material, Pa | Superscripts | |
r | normal unit vector, dimensionless | n | flow behavior index |
rp | radius of the erosion crater, m | m | impact velocity power law coefficient |
R | radius of particle, m | (1) | pre-collision |
Ri | maximum radius of deformation, m | (2) | post-collision |
R* | equivalent radius, m | Subscripts | |
Rl | inside radius of a circular pipe, m | i | particle i |
Sn | normal stiffness, N·m−1 | j | particle j |
St | tangential stiffness, N·m−1 | z | normal direction |
t | tangential unit vector, dimensionless | y | tangential direction |
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Parameter | Expression |
---|---|
Normal stiffness constant (Sn) | |
Tangential stiffness constant (St) | |
Normal damping coefficient (Cn) | |
Tangential damping coefficient (Ct) | |
Torque by tangential forces (Tt,ij) | |
Rolling friction torque (Tr,ij) | |
Equivalent elastic modulus (E*) | |
Equivalent shear modulus (G*) | |
Equivalent radius (R*) | |
Equivalent mass (m*) | |
Dimensionless coefficient (γ) | |
The relative velocities of the centers of the spheres before and after a collision (∆v) | |
The relative velocity of the contact points | |
The unit tangential vector (t) | |
Dimensionless coefficient (K) |
K | Fs | m | a | b | x | y | z | w | θ |
---|---|---|---|---|---|---|---|---|---|
7.8 × 10−8 | 0.35 | 1.57 | 5.9 × 10−5 | −7.2 × 10−5 | 0.75 | −0.21 | 0.83 | −1.2 | 70 |
DEM Parameter | Value |
---|---|
Liquid phase | |
Liquid density (kg/m3) | 1020 |
Liquid viscosity (mPa·s) | 375 |
Solid phase | |
Diameter (mm) | 0.65 |
Mass (mg) | 0.26 |
Sphericity | 0.85 |
Particle density (kg/m3) | 1850 |
Number of particles per calculation in each layer | 60 ± 5 |
Dimensionless coefficient K | 0.4 |
Coefficient of normal restitution en | 0.95 |
Coefficient of normal restitution er | 0.36 |
Coefficient of friction µ | 0.1 |
Geometry | |
Upstream pipe length in the axial direction (mm) | 400 |
Downstream pipe length in the axial direction (mm) | 200 |
Upstream pipe diameter (mm) | 50 |
Downstream pipe diameter (mm) | 25 |
Total grid cell number | 845, 732 |
Length of virtual plane (mm) | 10 |
Width of virtual plane (mm) | 3 |
The distance from virtual planes to target wall (mm) | 300 |
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Cheng, J.; Dou, Y.; Zhang, N.; Li, Z.; Wang, Z. A New Method for Predicting Erosion Damage of Suddenly Contracted Pipe Impacted by Particle Cluster via CFD-DEM. Materials 2018, 11, 1858. https://doi.org/10.3390/ma11101858
Cheng J, Dou Y, Zhang N, Li Z, Wang Z. A New Method for Predicting Erosion Damage of Suddenly Contracted Pipe Impacted by Particle Cluster via CFD-DEM. Materials. 2018; 11(10):1858. https://doi.org/10.3390/ma11101858
Chicago/Turabian StyleCheng, Jiarui, Yihua Dou, Ningsheng Zhang, Zhen Li, and Zhiguo Wang. 2018. "A New Method for Predicting Erosion Damage of Suddenly Contracted Pipe Impacted by Particle Cluster via CFD-DEM" Materials 11, no. 10: 1858. https://doi.org/10.3390/ma11101858