# Erosion Failure of Horizontal Pipe Reducing Wall in Power-Law Fluid Containing Particles via CFD–DEM Coupling Method

### Abstract

A CFD–DEM-based two-phase flow model and a test-based erosion model are used to obtain the specific erosion on the reducing wall of sudden contraction section. The dimensionless filtered governing equations are adopted for incompressible power-law fluid flow, and the Hertz–Mindlin (no-slip) model for particle–particle and particle–wall contact. The annular reducing wall is divided into two erosion areas in radial direction based on erosion form and divided into four parts in the circumferential direction. The calculated result is verified with a full-scale experiment, and it shows a good agreement. The calculated results show that the erosion rate of the reducing wall is mainly determined by the flow velocity, and the erosion area is affected by liquid viscosity. The serious erosion region is located in the inner edge of the sample lower part, and this region expends to the outer circumference with the increasing flow velocity and the reducing liquid viscosity. The increase in flow velocity expands the flow region where the particle can impact the wall and thus increases the particle impact numbers.

### Graphical Abstract

## Keywords

Erosion analysis Sudden contraction section Power-law fluid flow CFD–DEM coupling approach## List of symbols

*A*_{p}Particle cross-sectional area (m

^{2})*C*_{d}Drag coefficient (–)

*D*Inside diameter of circular pipe (m)

*d*_{p}Diameter of particle (m)

*E*_{e}The measured erosion rate (mm/h)

*E*_{p}The calculated erosion rate (mm/h)

*F*_{i,n}Normal force of particle (N)

*F*_{t}Total force of particle (N)

*Fd*Fluid–particle drag force (N)

*F*_{d}Inter-particle contact force in tangential direction (N)

*F*_{n,ij}Inter-particle contact force in normal direction, N

*F*_{s}Particle shape coefficient (dimensionless)

*I*_{i}Moment of inertia of particle

*i*(N m)*K*Flow consistency coefficient (Pa s

^{n})*k*_{n}Normal spring stiffness (N/mm)

*L*Distance from the inner edge of the sample (m)

*m*_{p}Single particle mass (kg)

*nij*Normal unit vector (–)

*N*Number of particles in contact (–)

*r*_{p}Radius of particle (m)

*r*_{i}Radius of particle

*i*(m)*r*_{j}Radius of particle

*j*(m)*R*Relative error (%)

*R*_{l}Inside radius of circular pipe (m)

*R*_{p}Particle Reynolds number (–)

*Re*Reynolds number of fluid flow (–)

*T*_{t,i}Torque generated by tangential forces (N m)

*T*_{r,ij}Rolling friction torque (N m)

*u*_{x}Axial velocity of fluid (m/s)

*u*Fluid velocity (m/s)

*V*_{p}Particle volume (m

^{3})*v*Particle velocity (m/s)

- \( a,b,w,x,y,z \)
Empirical constants in angle functions (–)

## Greek symbols

*γ*Shear rate (s

^{−1})*τ*Shear stress (Pa)

*η*Apparent viscosity (mPa s)

*η*_{n}Normal damping coefficient (N/s/m)

*γ*_{d}Dimensionless shear rate (–)

*ρ*_{l}Liquid density (kg m

^{3})*φ*Scaling coefficient (–)

*α*Particle impact angle (°)

*θ*Critical angle of particle impact (°)

## Superscripts

*n*Flow behavior index

*m*Impact velocity power-law coefficient

## Subscripts

*i*First particle

*j*Second particle

*p*Particle

*l*Liquid

## Notes

### Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grant No. 51404198), and it was also performed by the group of profession and innovation for well testing integrity and safety of Xi’an Shiyou University.

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