Fig. 4 displays the particle axial velocity at three specularity coefficients of 1, 0.0005 and 0, when the EMMS-based drag
model was used, showing that the specularity coefficient has a pronounced effect on particles velocity. For the free-slip wall boundary
condition (ϕ = 0), the particle velocities, either downward near the
wall or upward in the core, are higher than those for the no-slip wall
boundary condition (ϕ = 1). When ϕ = 0, there is no friction between
particles and wall, and the free slip of particles on the wall leads to more particles flowing downward, thus resulting in higher particle velocity and consequently a high particles concentration in the
bottom of the riser. For the no-slip wall boundary condition (ϕ = 1),
there is strong friction between particles and wall, resisting the
downward flow of particles, thus resulting in low particle velocity
near the wall. Specularity coefficient therefore strongly influences
the movement of particles near the wall, thus affecting the internal
circulation rate of particles in the riser: strong internal circulation
of particles is predicted for small specularity coefficients