This study proposed a fully coupled computational fluid dynamics-discrete element method (CFD-DEM) model based on a diffusion averaging algorithm for the hydraulic transport of dense particles, integrally considering particle--liquid interphase force and complex particle-turbulence interaction. The proposed model overcame the limitation that fluid mesh needs to be several times the size of the particles in the traditional CFD-DEM model. Moreover, experimental and numerical studies were conducted mainly on horizontal and vertical pipes, and few were conducted on inclined pipes.

Calculation of particle volume fraction was divided into two steps. First, each particle was randomly and uniformly divided into several feature points, and the initial value of the particle volume fraction was calculated based on the number of feature points occupied in each mesh. Subsequently, a diffusion-based averaging method was employed to solve the particle volume fraction with the initial field and no-flux condition on all physical boundaries in the computational domain. Furthermore, the source terms were added to the k-ε turbulence model to account for the modulation of the turbulence from particles, and the discrete random walk model was used to calculate the stochastic effect of turbulence on particle motion. A drag force considering porosity modification was applied to the two-phase flow through densely packed particle beds. Other particle-liquid forces and particle torques caused by the fluid were also included in the model. The fully coupled CFD-DEM model predicted the hydraulic conveying of dense particles in the pipeline system well. Moreover, this model was used to investigate the effects of pipe inclination on the hydraulic transport of coarse particles (2 mm), including the effects on the spatial distribution of particles, axial velocity of each phase, fluid turbulent kinetic energy, and pressure drop.

The results are summarized as follows: 1) The spatial distribution of particles gradually transformed from a relatively densely packed distribution at the bottom of the horizontal pipe to a nearly uniform distribution in the vertical pipe with increasing inclination angle. The distributions of axial liquid velocity and turbulent kinetic energy along the vertical direction were gradually asymmetric and then returned to symmetry, reaching the maximum degree of asymmetry at 60°. 2) In the inclined pipes, the axial velocity of particles was lower and higher at the bottom and top of the pipe, respectively. Meanwhile, the axial velocity of the particles in the vertical pipe was parabolically distributed, with higher velocity at the center of the pipe and lower velocity near the wall. 3) The number of collisions between particles and between particles and walls increased slightly and then decreased rapidly with increasing inclination angle. 4) Moreover, pressure drop in the two-phase flow initially increased and then decreased with increasing inclination angle, reaching the maximum at 60°.

This study demonstrates that the inclination angle significantly affects the distributions of particles, the number of collisions between particles and between particles and walls, liquid turbulent kinetic energy, and pressure drop. A small or large inclined angle is suggested for the hydraulic transport of particles, and a 60° inclined pipe should be avoided to reduce energy consumption.