At the inlet and outlet, all properties should be defined based on the specific physics and assumption of the problem. For the gas phase, no-slip and non-penetrating wall conditions may be considered. For the solid phase, the slip boundary condition is the recommended boundary condition (Johnson and Jackson 1987):
where vs,para is the particle slip velocity parallel to the wall. ф is the specularity coefficient between the particle and the wall, which is defined as the average fraction of relative tangential momentum transferred between the particle and the wall during a collision. The specularity coefficient varies from zero (smooth walls) to one (rough walls). A proper value based on the particles and wall properties should be assumed. For a specularity coefficient tending toward zero, a free slip boundary condition for the solids tangential velocity is imposed at a smooth wall boundary as explained by Benyahia et al. (2005).
Johnson and Jackson (1987) proposed the following wall boundary condition for the total granular heat flux as
The dissipation of solids turbulent kinetic energy by collisions with the wall is specified by the particle-wall restitution coefficient, esw. A high value of specularity coefficient implies high production at the wall, and a value of esw close to unity implies low dissipation of granular energy at the wall. It is expected that the specularity coefficient and the particle-wall restitution coefficient need to be calibrated for a given gas/particle flow system because the specularity coefficient cannot be measured and esw can be measured only with some difficulty (Benyahia et al. 2005). Equations (2.63) and (2.64) could be written as