# A New Method for Predicting Erosion Damage of Suddenly Contracted Pipe Impacted by Particle Cluster via CFD-DEM

^{1}

^{2}

^{*}

## Abstract

**:**

^{3}in density. The results included an erosion depth, particle-wall contact parameters, and a velocity decay rate of colliding particles along the radial direction at the target surface. Subsequently, the effect of interparticle collision mechanisms on particle cluster erosion was discussed. The calculated results demonstrate that collision interference between particles during one cluster impact was more likely to appear on the surface with large particle impact angles. This collision process between the rebounded particles and the following particles not only consumed the kinetic energy but also changed the impact angle of the following particles.

## 1. Introduction

## 2. Solution Methodology

#### 2.1. Particle-Particle Interaction

**F**

_{ij}), fluid interaction force (

**F**

_{l,i}), gravitational force (

**F**

_{g}) and buoyancy (

**F**

_{b}). The total contact force is split into two components: Normal (

**F**

_{n,ij}) and tangential (

**F**

_{t,ij}). The total torque

**T**

_{i}results from a vector summation of the torque at each particle-particle contact [20].

**F**

_{t}) is limited by Coulomb’s law of friction and meets the relationship ${\mathit{F}}_{\mathrm{t}}\le -{\mu}_{s}\left|{F}_{n,ij}\right|{\delta}_{t}/\left|{\delta}_{t}\right|$ in a system without sliding [18,22]. Therefore, the normal and tangential force based on the normal and tangential overlap can be represented as $-{S}_{n}{\delta}_{n}$ and $-{S}_{\mathrm{t}}{\delta}_{t}$, respectively. The normal and tangential damping forces are described in Tsuji’s research [24] and can be expressed as ${C}_{n}{v}_{n}^{rel}$ and ${C}_{t}{v}_{t}^{rel}$. Therefore, the total force is given by

_{n}and S

_{t}), damping coefficients (C

_{n}and C

_{t}), and other parameters are defined in Table 1. In addition to the contact force between the particles, there may be some non-contact forces between the particles, which can determine the particle movement state. Van der Waals forces, capillary forces and electrostatic forces are the major non-contact forces between the particles. For particles with a micrometer or nanometer diameter, Van der Waals forces may have a great effect on the movement of the particles but the effect on the particles (d

_{p}> 0.6 mm) in this study was ignored [25]. The reason for neglecting the other forces won’t be discussed here owing to the limited space.

**F**

_{l,i}often includes the drag force (

**F**

_{d}), virtual mass force (

**F**

_{v}), pressure gradient force (

**F**

_{p}), Saffman lift force (

**F**

_{s}), Basset force (

**F**

_{ba}), and Magnus lift force (

**F**

_{m}), respectively. The drag force plays a major role in the force acting on the particles by a fluid, which has been proven by literature [26]. The drag force model can be expressed by

_{p}is defined as

#### 2.2. Momentum Exchange and Energy Dissipation

^{rel}and K are shown in Table 1. There is an angle between

**v**and

**r**that can determine whether a collision is sliding or sticking. When this angle is greater than or equal to a critical angle θ*, a sliding collision occurs. Otherwise, a sticking collision takes place. The tangential coefficient of restitution can be expressed by [30]

_{n}), the tangential momentum decay coefficient (λ

_{t}) and the energy decay coefficient (λ

_{e}). The following definitions have been used in general.

#### 2.3. Particle-Wall Interaction

_{1}, for particle i and the moving time t

_{2}for particle j, if t

_{1}> t

_{2}, the particle j will collide with particle i before it leaves the wall region, on the contrary, it will not. The rebound process for particle i consists of two steps including a contact step and a release step. Therefore, the rebound time can be expressed as t

_{1}= t

_{e}+ t

_{r}. When the effect of elastic deformation is ignored, the depth of the extruding lips is given by

**v**

_{i,z}, closes to 0 m/s. Therefore, the contact time with the metal surface, based on the theorems of particle momentum, can be expressed by the following equation.

_{r}) should meet the following relationship in order to satisfy the time adaptive condition of particle collision.

#### 2.4. Governing Equations of Fluid Flow

**F**

_{p,i}) should be equal to the force of the liquid phase (

**F**

_{l,i}) acting on the solid phase in the opposite direction. In the above equations, the fluid volume fraction is obtained from the relation ${\alpha}_{l}1-\sum _{j=1}^{N}{V}_{p}/\Delta {V}_{l}$ and the viscous stress tensor

**τ**is defined as

#### 2.5. Erosion Model

_{n}·λ

_{t}·λ

_{e}; F

_{s}= 1.0 for sharp, 0.53 for semi-rounded, or 0.2 for fully rounded sand particles. The impact angle function f(α) is given by

## 3. Implementation and Verification

#### 3.1. Numerical Solving Step

_{p}/N

_{p})

^{1/3}, determined the number of CFD points within the mesh cell of a particle. It was usually necessary to ensure that N

_{p}≧ 1 and N

_{p}< N

_{l}. Meanwhile, a grid size that was too small would increase the amount of calculation and a grid size that was too large size may ignore collision details. Therefore, the grid size was set to 0.2 mm for the CFD mesh in this work.

_{D}= 0.2 × ∆t

_{Ra}and the shear modulus can be expressed as G = Y/2(2 + ν)(1 − ν). In order to obtain the contact parameters between the particles and the wall, the DEM time step needs less than the contact time (Equation (22)). Therefore, the time step was set to be 2 × 10

^{−6}s for DEM and 4 × 10

^{−5}s for CFD.

**Calculation of the critical interparticle distance:**The distance between adjacent particles was calculated when the front particle made contact with the wall. If L < L

_{p}, the subsequent impact on the wall was treated as a clustered particle impact; otherwise, the impact was considered as an independent particle impact.

**Acquisition of clustered particles**: The distance determination method was used to divide the form of particle impingement on the wall and impact parameters were calculated by classification.

**Determination of the decay coefficient:**For the clustered particles, the next step was to obtain the decay coefficient λ and the new impact angle for each colliding particle after a particle-particle collision.

#### 3.2. Boundary Conditions

_{r}|

_{r}= u

_{x}|

_{r}

_{= 0}= 0 and әu

_{r}/әx = 0 and a fully developed incompressible flow was considered at the velocity inlet and exit sections.

#### 3.3. Full-Scale Experiment

^{3}/h), an electric heating agitator, a temperature and pressure sensor, a magnetic flowmeter (8712HR, Emerson, Rosemount. Co., Boulder, CO， USA), four gate values, a control cabinet, a computer, and connecting pipes. After the test fluid containing the particles was mixed, stirring and electric heating units were turned on until the temperature reached the required value and then the pump was turned on. The gate valve in the test pipe was opened when the flow reached stability and the related data, including a particle motion image, flow pressure, and temperature were recorded.

^{3}and an average diameter of 0.65 mm, was used as an erosion abrasive (Figure 6c). The mass concentration of the particles in the suspension was 50 kg/m

^{3}. The flow velocities of the experimental liquid were set to 1.5, 2.5, and 3.5 m/s, respectively. The exposed surface was sealed with epoxy resin and ground using SiC emery paper grade 1200 prior to installation. A MEMRECAM SP-614 high-speed camera (NAC. Co., Ltd., Tokyo, Japan) was used to observe the particle clusters with a spatial resolution of 0.01 μm and the sample surface profiles were verified using an H1200WIDE confocal scanning laser microscope (Lasertec. Co., Ltd., Tokyo, Japan) with a scanning rate of 120 fps.

## 4. Results

#### 4.1. Particle-Wall Impingement

_{n}, which indicates the average pressure between a particle and the metal surface, needs to be obtained experimentally. The geometric dimensions of the 20 similar craters (Figure 11a) on the outer circumference of a sample surface were measured to determine the coefficient value. The obtained P

_{n}was equal to 2.1 × 10

^{6}Pa for super 13Cr stainless steel. Figure 11b shows the calculated erosion depth along the radial surface. The depth of an independent erosion crater gradually increased from the inner edge to the outer circumference, with maximum and minimum erosion depths of 17.8 and 2.9 μm, respectively. Hence, the calculated contact time using Equation (32) stabilized in the microsecond time scale and was notably lower than the particle release time (a difference of 1000 times) so that t

_{1}was approximately equal to t

_{r}. Figure 12 shows the release time for Particle I along the radial direction. The minimum release time for a particle that impacts the inner surface was 1.8 × 10

^{−4}s and the maximum release time was 3.3 × 10

^{−3}s. If a particle collision occurred after particle i rebounded from the wall, then a lower release time indicates that a shorter interparticle distance is required to allow two adjacent particles to collide.

#### 4.2. Particle Cluster Erosion

#### 4.3. Discussion

**Independent Impact Erosion (IIE)**: When particle j moves on the outer side of particle i (Figure 19a), the particle i initially impacts the wall and then rebounds back to the other side. Meanwhile, particle j follows particle i and does not collide with the latter. The impact between the two particles and the wall is independent of each other.

**Interferential Particle Erosion (IPE)**: When the trajectories of two particles are extremely close, as shown in Figure 19b, particle i impacts the wall and then collides with particle j. Particle j may impact the wall again after the collision, but the new impact energy of particle j should be attenuated during the particle-particle collision.

**Stacked Particle Erosion (SPE)**: When the motion path of particle j is inside particle i (Figure 19c), particle j may be subjected to a head-on collision after particle i rebounds off the reducing wall. Therefore, the second impact between a particle and the wall may be attributed to particle i instead of particle j. This result causes the actual velocity in the second wall impingement to decrease twice.

_{1}= α − φ (Figure 2a), the particle impact velocities in the second particle-wall impingement at different new impact angles are displayed in Figure 20. Given that the particle during the second impingement suffers twice the momentum loss in SPE and only once in IPE, the particle impact velocity in IPE is almost three times the velocity in SPE on the inner edge and almost 10 times on the outer circumference. Therefore, the effect of SPE on decreasing erosion extends far beyond the IPE and both erosion types exert more influence than IIE.

## 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

A_{p} | particle cross-sectional area, m^{2} | t_{e} | contact time, s |

C_{d} | drag coefficient, dimensionless | t_{r} | release time, s |

C_{n} | normal damping coefficient, kg·s^{−1} | t_{1} | rebound time, s |

C_{t} | tangential damping coefficient, kg·s^{−1} | t_{2} | moving time for particle j, s |

D | inside diameter of circular pipe, m | ∆t_{D} | DEM time step, s |

d_{p} | diameter of particle, m | ∆t_{Ra} | Rayleigh time step, s |

e_{n} | restitution coefficient in the normal direction, dimensionless | T_{r} | rolling friction torque, N·m |

e_{t} | restitution coefficient in the tangential direction, dimensionless | T_{t} | torque generated by tangential forces, N·m |

E | elastic modulus, Pa | u | fluid flow velocity, m |

E_{R} | total erosion rate, kg/kg | ∆v | the relative velocities of the centers of the spheres before and after a collision, m·s^{−1} |

E_{RA} | erosion rate caused by independent particle impact, kg/kg | v_{n} | particle normal velocity, m·s^{−1} |

E_{RI} | erosion rate caused by grouped particle impact, kg/kg | v_{t} | particle tangential velocity, m·s^{−1} |

E* | equivalent elastic modulus, Pa | v^{ref} | the relative velocity of the contact points, m·s^{−1} |

∆E_{n} | normal energy loss of a particle caused by particle-particle collision, J | V_{p} | volume of particle, m^{3} |

∆E_{t} | tangential energy loss of a particle caused by particle-particle collision, J | V_{l} | volume of liquid cell, m^{3} |

F_{b} | buoyancy, N | ∆x | space step, m |

F_{d} | drag force, N | Y | Young’s modulus, Pa |

F_{g} | gravitational force, N | a | empirical constants in angle functions, dimensionless |

F_{l,i} | the fluid force acting on a particle, N | b | |

F_{n} | inter-particle contact force in the normal direction, N | w | |

F_{p,i} | the particle force acting on the liquid, N | x | |

F_{ij} | contact force between particles, N | y | |

F_{t} | inter-particle contact force in the tangential direction, N | z | |

F_{s} | particle shape coefficient, dimensionless | Greek symbols | |

F_{n}^{d} | inter-particle damping force in the normal direction, N | ω_{i} | unit angular velocity vector of the object, rad·s^{−1} |

F_{t}^{d} | 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, m^{4} | β | The angle between the particle velocity and the centerline of the two particles, ° |

J | particle momentum, kg·m·s^{−1} | δ_{k} | identity tensor |

K | dimensionless coefficient, dimensionless | δ_{n} | normal overlap, m |

L | distance from the inner edge of the sample, m | δ_{t} | tangential overlap, m |

L_{p} | critical inter-particle distance of particle, m | ρ_{l} | liquid density, kg·m^{−3} |

L_{p,e} | the moving distance of the particle j along the centerline, m | μ_{s} | friction coefficient, dimensionless |

L_{p,r} | the critical inter-particle distance in the release process, m | μ_{r} | rolling friction coefficient, dimensionless |

L_{y} | particle spacing in the Y direction, m | φ | scattering angle for particle j, ° |

L_{z} | particle spacing in the Z direction, m | ψ | scattering angle for particle i, ° |

m_{p} | single particle mass, kg | α | particle impact angle, ° |

m* | equivalent mass, kg | α_{1} | new particle impact angle, ° |

N | total particle number, dimensionless | α_{l} | fluid volume fraction, % |

N_{p} | the number of sample points contained within the mesh cell of the particle, dimensionless | θ | critical angle of particle impact, ° |

N_{1} | the number of particles in the particle group, dimensionless | θ* | critical angle between particles, ° |

N_{t} | total number of sample points of the particle, dimensionless | ε | strain caused by the normal stress of indentation, m |

P_{n} | function of properties of the material, Pa | Superscripts | |

r | normal unit vector, dimensionless | n | flow behavior index |

r_{p} | radius of the erosion crater, m | m | impact velocity power law coefficient |

R | radius of particle, m | (1) | pre-collision |

R_{i} | maximum radius of deformation, m | (2) | post-collision |

R* | equivalent radius, m | Subscripts | |

R_{l} | inside radius of a circular pipe, m | i | particle i |

S_{n} | normal stiffness, N·m^{−1} | j | particle j |

S_{t} | tangential stiffness, N·m^{−1} | z | normal direction |

t | tangential unit vector, dimensionless | y | tangential direction |

## References

- Parsi, M.; Najmi, K.; Najafifard, F.; Hassani, S.; McLaury, B.S.; Shirazi, S.A. A comprehensive review of solid particle erosion modeling for oil and gas wells and pipelines applications. J. Nat. Gas. Sci. Eng.
**2014**, 21, 850–873. [Google Scholar] [CrossRef] - Wood, R.J.K.; Jones, T.F.; Ganeshalingam, J.; Miles, N.J. Comparison of predicted and experimental erosion estimates in slurry ducts. Wear
**2004**, 256, 937–947. [Google Scholar] [CrossRef] - Zhu, H.; Han, Q.; Wang, J.; He, S.; Wang, D. Numerical investigation of the process and flow erosion of flushing oil tank with nitrogen. Powder Technol.
**2015**, 275, 12–24. [Google Scholar] [CrossRef] - Njobuenwu, D.O.; Fairweather, M. Modelling of pipe bend erosion by dilute particle suspensions. Comput. Chem. Eng.
**2012**, 42, 235–247. [Google Scholar] [CrossRef] - Zhu, H.P.; Zhou, Z.Y.; Yang, R.Y.; Yu, A.B. Discrete particle simulation of particulate systems: Theoretical developments. Chem. Eng. Sci.
**2007**, 62, 3378–3396. [Google Scholar] [CrossRef] - Zhang, Y.; McLaury, B.S.; Shirazi, S.A. Improvements of particle near-wall velocity and erosion predictions using a commercial CFD code. J. Fluids Eng.
**2009**, 131, 031303. [Google Scholar] [CrossRef] - Bitter, J.G.A. Study of erosion phenomenon–1,2. Wear
**1963**, 5–21, 169–190. [Google Scholar] [CrossRef] - Finnie, I. Erosion of surfaces by solid particles. Wear
**1960**, 3, 87–103. [Google Scholar] [CrossRef] - Chen, J.K.; Wang, Y.S.; Li, X.F.; He, R.Y.; Han, S.; Chen, Y.L. Erosion prediction of liquid-particle two-phase flow in pipeline elbows via CFD–DEM coupling method. Powder Technol.
**2015**, 275, 182–187. [Google Scholar] [CrossRef] - Zhao, Y.; Xu, L.; Zheng, J. CFD-DEM simulation of tube erosion in a fluidized bed. AiChE J.
**2016**, 63, 418–437. [Google Scholar] [CrossRef] - Chen, X.; McLaury, B.S.; Shirazi, S.A. Application and experimental validation of a computational fluid dynamics (CFD)-based erosion prediction model in elbows and plugged tees. Comput. Fluids
**2004**, 33, 1251–1272. [Google Scholar] [CrossRef] - McLaury, B.S. Predicting Solid Particle Erosion Resulting from Turbulent Fluctuations in Oilfield Geometries. Ph.D. Thesis, The University of Tulsa, Tulsa, OK, USA, 1996. [Google Scholar]
- Ahlert, K. Effects of Particle Impingement Angle and Surface Wetting on Solid Particle Erosion of AISI 1018 Steel. Master’s Thesis, The University of Tulsa, Tulsa, OK USA, 1994. [Google Scholar]
- Lv, Z.; Huang, C.Z.; Zhu, H.T.; Wang, P.Y.; Liu, Z.W. FEM analysis on the abrasive erosion process in ultrasonic-assisted abrasive waterjet machining. Int. J. Adv. Manuf. Technol.
**2015**, 78, 1641–1649. [Google Scholar] [CrossRef] - Bedon, C.; Santarsiero, M. Laminated glass beams with thick embedded connections-Numerical analysis of full-scale specimens during cracking regime. Compos. Struct.
**2018**, 195, 308–324. [Google Scholar] [CrossRef] - Larcher, M.; Arrigoni, M.; Bedon, C.; Van Doormaal, J.C.A.M.; Haberacker, C.; Husken, G.; Millon, O.; Saarenheimo, A.; Solomos, G.; Thamie, L.; et al. Design of blast-loaded glazing windows and façades: A review of essential requirements towards standardization. Adv. Civ. Eng.
**2016**, 2016, 2604232. [Google Scholar] [CrossRef] - Malka, R.; Nešić, S.; Gulino, D.A. Erosion–corrosion and synergistic effects in disturbed liquid-particle flow. Wear
**2007**, 262, 791–799. [Google Scholar] [CrossRef] [Green Version] - Blais, B.; Lassaigne, M.; Goniva, C.; Fradette, L.; Bertrand, F. Development of an unresolved CFD-DEM model for the flow of viscous suspensions and its application to solid-liquid mixing. J. Comput. Phys.
**2016**, 318, 201–221. [Google Scholar] [CrossRef] - Iqbal, N.; Rauh, C. Coupling of discrete element model (DEM) with computational fluid mechanics (CFD): A validation study. Appl. Math. Comput.
**2016**, 277, 154–163. [Google Scholar] [CrossRef] - Gupta, P.; Sun, J.; Ooi, Y.J. DEM-CFD simulation of a dense fluidized bed: Wall boundary and particle size effects. Powder Technol.
**2016**, 293, 37–47. [Google Scholar] [CrossRef] - Mindlin, R.D.; Deresiewicz, H. Elastic spheres in contact under varying oblique forces. J. Appl. Mech.
**1953**, 20, 327–344. [Google Scholar] - Barrios, G.K.P.; Carvalho, R.M.D.; Kwade, A.; Tavares, L.M. Contact parameter estimation for DEM simulation of iron ore pellet handling. Powder Technol.
**2013**, 248, 84–93. [Google Scholar] [CrossRef] - Chung, Y.C.; Liao, H.H.; Hsiau, S.S. Convection behavior of non-spherical particles in a vibrating bed: Discrete element modeling and experimental validation. Powder Technol.
**2013**, 237, 53–66. [Google Scholar] [CrossRef] - Tsuji, Y.; Tanaka, T.; Ishida, T. Lagrangian numerical-simulation of plug flow of cohesionless particles in a horizontal pipe. Powder Technol.
**1992**, 71, 239–250. [Google Scholar] [CrossRef] - Chen, J.; Anandarajah, A. Van der Waals attraction between spherical particles. J. Colloid Interface Sci.
**1996**, 180, 519–523. [Google Scholar] [CrossRef] - Zhang, Y. Application and Improvement of Computational Fluid Dynamics (CFD) in Solid Particle Erosion Modeling; ProQuest, UMI Dissertations Publishing: Ann Arbor, MI, USA, 2006. [Google Scholar]
- Crowe, C.T.; Schwarzkopf, J.D.; Sommerfeld, M.; Tsuji, Y. Particle-particle interaction. In Multiphase Flows with Droplets and Particles, 2nd ed.; CRC Publishing Inc.: New York, NY, USA, 2011; pp. 119–153. [Google Scholar]
- Walton, O.R. Numerical simulation of inelastic, frictional particle-particle interactions. In Particulate Two-Phase Flow; Roco, M.C., Ed.; Butterworth-Heinemann: Boston, MA, USA, 1993; pp. 1249–1253. [Google Scholar]
- Jenkins, J.T.; Zhang, C. Kinetic theory for identical, frictional, nearly elastic spheres. Phys. Fluids
**2002**, 14, 1228–1235. [Google Scholar] [CrossRef] - Yang, L.; Padding, J.T.; Kuipers, J.A.M. Modification of kinetic theory for frictional spheres, Part I: Two-fluid model derivation and numerical implementation. Chem. Eng. Sci.
**2016**, 152, 767–782. [Google Scholar] [CrossRef] - Huang, C.K.; Chiovelli, S.; Minev, P.; Luo, J.; Nandakumar, K. A comprehensive phenomenological model for erosion of materials in jet flow. Powder Technol.
**2008**, 187, 237–279. [Google Scholar] [CrossRef] - Habib, M.A.; Ben-Mansour, R.; Badr, H.M.; Kabir, M.E. Erosion and penetration rates of a pipe protruded in a sudden contraction. Comput. Fluids
**2008**, 37, 146–160. [Google Scholar] [CrossRef] - Habib, M.A.; Badr, H.M.; Ben-Mansour, R.; Kabir, M.E. Erosion rate correlations of a pipe protruded in an abrupt pipe contraction. Int. J. Impact Eng.
**2007**, 34, 1350–1369. [Google Scholar] [CrossRef] - Grant, G.; Tabakoff, W. Erosion prediction in turbomachinery resulting from environmental solid particles. J. Aircraft
**1975**, 12, 471–478. [Google Scholar] [CrossRef] - Gidaspow, D. Transport Equations. In Multiphase Flow and Fluidization; Academic Press: San Diego, CA, USA, 1994; pp. 36–42. [Google Scholar]
- Meng, H.C.; Ludema, K.C. Wear models and predictive Equations: their form and content. Wear
**1995**, 181–183, 443–457. [Google Scholar] [CrossRef] - Zhao, Y.A.; Cai, W.B.; Cui, L.; Cheng, J.R.; Dou, Y.H. Erosion of premium connection cross-over joint in solid-liquid flow. In Proceedings of the International Conference on Engineering Technology and Application ICETA2015, Xia Men, China, November 2015. [Google Scholar]
- ANSYS FLUENT 15.0 User Guide; FLUENT Solutions; ANSYS Inc.: Canonsburg, PA, USA, 2014.
- EDEM 2.7 User Guide; DEM Solutions; DEM Solutions Ltd.: Edinburgh, UK, 2015.
- Ferziger, J.H.; Peric, M.; Leonard, A. Computational methods for fluid dynamics. Phys. Today
**1997**, 50, 80–84. [Google Scholar] [CrossRef] - Kremmer, M.; Favier, J.F. A method for representing boundaries in discrete element modelling—Part II: Kinematics. Int. J. Numer. Meth. Eng.
**2001**, 51, 1423–1436. [Google Scholar] [CrossRef] - Chen, X.; Zhong, W.; Zhou, X.; Jin, B.; Sun, B. CFD–DEM simulation of particle transport and deposition in pulmonary airway. Powder Technol.
**2012**, 228, 309–318. [Google Scholar] [CrossRef] - Hobbs, A. Simulation of an aggregate dryer using coupled CFD and DEM methods. Int. J. Comput. Fluid D
**2009**, 23, 199–207. [Google Scholar] [CrossRef] - Xu, Y.; Kafui, K.; Thornton, C.; Lian, G. Effects of material properties on granular flow in a silo using DEM simulation. Part. Sci. Technol.
**2002**, 20, 109–124. [Google Scholar] [CrossRef] - Cheng, J.R.; Zhang, N.S.; Li, Z.; Dou, Y.H.; Cao, Y.P. Erosion failure of horizontal pipe reducing wall in power-law fluid containing particles via CFD–DEM coupling method. J. Fail. Anal. Prev.
**2017**, 17, 1067–1080. [Google Scholar] [CrossRef] - Hutchings, I.M.; Macmillan, N.H.; Rickerby, D.G. Further studies of the oblique impact of a hard sphere against a ductile solid. Int. J. Mech. Sci.
**1981**, 23, 639–646. [Google Scholar] [CrossRef] - Hutchings, I.M.; Winter, R.E. Solid particle erosion studies using single angular particles. Wear
**1974**, 27, 121–128. [Google Scholar] [CrossRef]

**Figure 2.**The sketch of impingement between a particle and the wall, (

**a**) the particle-particle collision process; (

**b**) the particle-wall contact process.

**Figure 4.**Structure of the contraction section and the super 13Cr samples, (

**a**) the distribution and arrangement of the particles in two particle layers; (

**b**) the diagram of a sudden contraction section; (

**c**) the location of a particle cluster.

**Figure 5.**Schematic diagram of the experimental set-up: (1) Electrical control cabinet; (2) liquid storage tank containing the electric heater (2 m

^{3}); (3) screw pump; (4) (6) gate valve; (5) flow meter; (7) computer; (8) high-speed camera; (9) test section; (10) pressure transducer.

**Figure 6.**Physical map of the test section: (

**a**) Diagram of the contraction section; (

**b**) installation drawing of the samples; (

**c**) particle surface morphology scanned by SEM.

**Figure 7.**Distribution of particles in different fluids, (

**a**) dispersed particles in the base fluid; (

**b**) particle clusters in the crosslinking fluid.

**Figure 8.**The comparison of the erosion depth between the particles in the base fluid and the crosslinking fluid.

**Figure 9.**Schematic diagrams of the particle-wall impact and particle-particle collision in a cluster, (

**a**) distribution of the particles in pipe flow; (

**b**) initial contact between the front particle and the wall; (

**c**) the rebound of the front particle; (

**d**) mutual interference of the particles.

**Figure 10.**Velocity components of a particle in Particle Layer I along the radial surface. (

**a**) Normal impact velocities; (

**b**) Tangential impact velocities.

**Figure 11.**Depths for an erosion crater obtained by experimental measurement and numerical calculation, (

**a**) SEM micrograph of the single impact crater on the outer circumference surface; (

**b**) erosion depth caused by independent particle impacts and cluster impacts versus experimental results.

**Figure 12.**Release time of the particles after impacting on the wall versus the distance from the inner edge.

**Figure 16.**The critical inter-particle distance versus the distance from the inner edge at different flow velocities.

**Figure 17.**The velocity decay coefficient versus the distance from the inner edge at different flow velocities.

**Figure 19.**Schematic diagrams of particle-wall and particle-particle collisions between two particles.

**Figure 20.**The second particle impact velocity on the wall for interferential particle erosion (IPE) and stacked particle erosion (SPE). The second impact particle is particle j in IPE as well as is particle i in SPE.

Parameter | Expression |
---|---|

Normal stiffness constant (S_{n}) | ${\scriptscriptstyle \frac{4}{3}}{E}^{*}\sqrt{{R}^{*}{\delta}_{n}^{}}$ |

Tangential stiffness constant (S_{t}) | $8{G}^{*}\sqrt{{R}^{*}{\delta}_{n}}$ |

Normal damping coefficient (C_{n}) | $-2\sqrt{{\scriptscriptstyle \frac{5}{6}}}\gamma \sqrt{{S}_{n}{m}^{*}}$ |

Tangential damping coefficient (C_{t}) | $-2\sqrt{{\scriptscriptstyle \frac{5}{6}}}\gamma \sqrt{{S}_{t}{m}^{*}}$ |

Torque by tangential forces (T_{t,ij}) | ${\mathit{R}}_{i}\times {\mathit{F}}_{t,ij}$ |

Rolling friction torque (T_{r,i}_{j}) | $-{\mu}_{r}\left|{\mathit{F}}_{n,ij}\right|{R}^{*}\cdot {\omega}_{i}/\left|{\omega}_{i}\right|$ |

Equivalent elastic modulus (E*) | $\frac{1}{{E}^{*}}=\frac{1-{\nu}_{i}^{2}}{{E}_{i}}+\frac{1-{\nu}_{j}^{2}}{{E}_{j}}$ |

Equivalent shear modulus (G*) | $\frac{1}{{G}^{*}}=\frac{2(2+{\nu}_{i})(1-{\nu}_{i})}{{Y}_{i}}+\frac{2(2+{\nu}_{j})(1-{\nu}_{j})}{{Y}_{j}}$ |

Equivalent radius (R*) | $\frac{1}{{R}^{*}}=\frac{1}{{R}_{i}}+\frac{1}{{R}_{j}}$ |

Equivalent mass (m*) | $\frac{1}{{m}^{*}}=\frac{1}{{m}_{i}}+\frac{1}{{m}_{j}}$ |

Dimensionless coefficient (γ) | $\mathrm{ln}e/\sqrt{{\mathrm{ln}}^{2}e+{\pi}^{2}}$ |

The relative velocities of the centers of the spheres before and after a collision (∆v) | $\Delta \mathit{v}={\mathit{v}}_{i}-{\mathit{v}}_{j}$ |

The relative velocity of the contact points | ${\mathit{v}}^{rel}=\Delta \mathit{v}+\frac{{d}_{p}}{2}\left({\mathit{\omega}}_{i}+{\mathit{\omega}}_{j}\right)\times r$ |

The unit tangential vector (t) | $\mathit{t}=\frac{\left({\mathit{v}}^{rel}\times \mathit{r}\right)\times \mathit{r}}{\left|{\mathit{v}}^{rel}\times \mathit{r}\right|}$ |

Dimensionless coefficient (K) | $K=\frac{4I}{m{d}_{p}^{2}}$ |

K | F_{s} | 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/m^{3}) | 1020 |

Liquid viscosity (mPa·s) | 375 |

Solid phase | |

Diameter (mm) | 0.65 |

Mass (mg) | 0.26 |

Sphericity | 0.85 |

Particle density (kg/m^{3}) | 1850 |

Number of particles per calculation in each layer | 60 ± 5 |

Dimensionless coefficient K | 0.4 |

Coefficient of normal restitution e_{n} | 0.95 |

Coefficient of normal restitution e_{r} | 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 |

© 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**

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

**AMA Style**

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 Style**

Cheng, 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