Effect of pipe orientation on dense-phase transport .1. Critical angle in inclined upflow

The dense-phase region of the Zenz-type phase diagram in the pneumatic transport of medium-sized glass beads with a diameter of 0.45 mm along a 26.6 mm i.d. pipeline oriented at different inclination angles from 5 degrees to 80 degrees has been predicted by modifying the earlier model of Hong and Tomita (Int. J. Multiphase Flow, 21 (1995) 649-665) for inclined gas-particle stratified upflow. It is confirmed that the pressure gradient of inclined upward transport is consistently greater than that of horizontal transport and a maximum pressure gradient exists corresponding to the critical inclination angle around 65 degrees as the pipeline orientation changes from the horizontal to nearly vertical. Lowering the solids mass flow rate and/or loading ratio leads to a higher critical inclination angle. These findings are in good agreement with the existing experimental results reported in the literature. For the first time, an expression for estimating the critical inclination angle is derived. In comparison with the phase diagram for horizontal transport, a similar contour in the dense-phase region of the Zenz-type phase diagram has been predicted for inclined upward transport under given operating parameters. Furthermore, regardless of the parameter (i.e. solids mass flow rate or loading ratio) employed in the phase diagram, the minimum pressure-gradient point is found to shift up to a higher gas velocity and also to be different from the theoretical saltation point.