Hydrodynamic Studies of the Non-Spherical Particles Settling in Annular and Non-Annular Channels Filled with Newtonian and non-Newtonian Fluids

Abstract

The wall factor (f) and drag coefficient (CD) of hollow cylinders, hollow frustum, clusters, solid and hollow disks, and solid cylinder, frustum, hemi-cylinder, and hemi- frustum settling in cylindrical annular and non-annular channels were investigated. Fluid flows through the space between two concentric cylinders in the annular channel. The particles' terminal velocity decreased linearly with the blockage ratio. Terminal velocity was higher for non-annular channels filled with Newtonian fluid, and a reverse trend was observed for non-Newtonian fluids. The wall factor varied with the Reynolds number (Re), blockage ratio (deq/D for non-annular and deq/L, where deq is the equivalent diameter and L = (Do - Di)/2 for the annular channel), and di/do (hollow particle’s inner to outer diameter ratios). It increased initially to reach a constant value at a higher Reynolds number. The wall factor was higher for the non-annular channel. The drag coefficient declined with Re and increased with the blockage ratio. The correlations for the experimental drag coefficient were developed. Moreover, the current drag coefficient data were successfully predicted by the laminar flow model available in Ansys (Fluent 18.1). In the case of the hollow cylindrical particles, f and CD were estimated for 0.2 ≤ di/do ≤ 0.8, 0.08 ≤ deq/D ≤ 0.47, and 0.14 ≤ deq/L ≤ 0.46. The estimated drag coefficient using the Newtonian fluids was 0.782 ≤ CD ≤ 3249.75 and 39.62 ≥ CD ≥ 0.8 for 0.064 ≤ Re ≤ 101.34 and 1.22 ≤ Re ≤ 100.21 for non-annular and annular channels, respectively in Newtonian fluids. The same was estimated in the range of 0.73721 ≤ CD ≤ 125507 and 3104 ≥ CD ≥ 1.27 for 0.0017 ≤ Re ≤ 64.07 and 0.054 ≤ Re ≤ 47.67 for the settling the particle in the non-annular and annular channel in the non-Newtonian fluid. For hollow frustum particles, the experimental f and CD are limited to 0.22 ≤ di/do ≤ 0.77, 0.22 ≤ deq/L ≤ 0.43, and 0.12 ≤ deq/D ≤ 0.44. CD, varied in the range of 0.80 ≤ CD ≤ 15.49 and 0.82 ≤ CD ≤ 7.12 for 2.798 ≤ Re ≤ 90 and 5.29 ≤ Re ≤ 89.97 while settling in Newtonian fluids in the non- nnular and annular channel, respectively. The same was varied over 0.983 ≤ CD ≤ 1292.47 and 1.48 ≤ CD ≤ 191.55 for 0.168 ≤ Re ≤ 50 and 0.1506 ≤ Re ≤ 20 for the non-annular and annular channels, respectively using non-Newtonian fluid as the process fluid. For the cluster particles, f and CD were estimated for 2 ≤ N (number of particles used) ≤ 7, 0.04 ≤ deq/D ≤ 0.24, and 0.10 ≤ deq/L ≤ 0.23. The estimated drag coefficient, CD, was varied in the range of 1.66 ≤ CD ≤ 43.30 and 1.92 ≤ CD ≤ 36.67 for 0.76 ≤ Re ≤ 22.09 and 0.74 ≤ Re ≤ 20.51 while settling in the non-annular and annular channels, respectively filled with Newtonian fluids. The same was varied for non-Newtonian fluids over 2.84 ≤ CD ≤ 4002 and 2.21 ≤ CD ≤ 1638 for 0.028 ≤ Re ≤ 10.14 and 0.052 ≤ Re ≤ 11.6 for the settling of the particles in the non-annular and annular channel, respectively. In the case of the disk/cylinder particles settling in non-annular and annular channels, f and CD were estimated for 0.12 ≤ H/d ≤ 1.09, 0.10 ≤ deq/D ≤ 0.57, and 0.17 ≤ deq/L ≤ 0.55. The wall factor increased with the sphericity. The estimated drag coefficient, CD, appeared in the range of 1.56 ≤ CD ≤ 503.88 and 0.851 ≤ CD ≤ 133.65 for 0.20 ≤ Re ≤ 46.75 and 0.40 ≤ Re ≤ 63.44 for the settling of the disk/cylinder in the non-annular and annular channel, respectively in Newtonian fluids. The same was varied in the range of 1.35 ≤ CD ≤ 31934 and 0.69 ≤ CD ≤ 1203.91 for 0.005 ≤ Re ≤ 30 and 0.05 ≤ Re ≤ 43.02 for the settling the particle in the non-annular and annular channel, respectively in the non- Newtonian fluid. For hollow disk particles, the f and CD were estimated for 0.12 ≤ H/do ≤ 0.27, 0.16 ≤ di/do ≤ 0.58, 0.06 ≤ deq/D ≤ 0.35, and 0.15 ≤ deq/L ≤ 0.34. f increased with H/do (Height/Outer diameter) ratio. The experimental CD varied over 3.061 ≤ CD ≤ 106.29 and 3.11 ≤ CD ≤ 34.41 for 0.463 ≤ Re ≤ 16.30 and 0.81 ≤ Re ≤ 16.16 while settling in the non- annular and annular channels, respectively in Newtonian fluids. The same varied in the range of 6.81 ≤ CD ≤ 57293 and 6.93 ≤ CD ≤ 12952 for 0.0042 ≤ Re ≤ 6.23 and 0.011 ≤ Re ≤ 6.17 while settling in the non-annular and annular channel, respectively in the non- Newtonian fluids. The estimated CD varied over 1.96 ≤ CD ≤ 390.05 and 1.12 ≤ CD ≤ 25.53 for the settling of the cylindrical particles in the Newtonian fluids in non-annular and annular channels for 0.33 ≤ Re ≤ 31.71 and 1.31 ≤ Re ≤ 41.91, respectively and 1.87 ≤ CD ≤ 25961.28 and 1.84 ≤ CD ≤ 1719.75 for 0.0086 ≤ Re ≤ 19.15 and 0.05 ≤ Re ≤ 19.32, respectively while settling in non- Newtonian fluids. For frustum particles the estimated drag coefficient was 1.83 ≤ CD ≤ 350.65 and 1.07 ≤ CD ≤ 24.83 for 0.35 ≤ Re ≤ 32.73 and 1.33 ≤ Re ≤ 42.92 in Newtonian fluids and 1.78 ≤ CD ≤ 24143.72 and 1.75 ≤ CD ≤ 1433.97 for 0.0091 ≤ Re ≤ 19.70 and 0.06 ≤ Re ≤ 19.87 in non-Newtonian fluids in the non-annular and annular channels, respectively. For hemi-cylinder particles, the range of the drag coefficients was 1.65 ≤ CD ≤ 397.24 and 0.93 ≤ CD ≤ 22.52 for 0.37 ≤ Re ≤ 32.80 and 1.35 ≤ Re ≤ 43.65 in Newtonian fluids and 1.56 ≤ CD ≤ 20539.46 and 1.59 ≤ CD ≤ 1097.72 for 0.0096 ≤ Re ≤ 19.95 and 0.07 ≤ Re ≤ 19.78 in non- Newtonian fluids in the non-annular and annular channels, respectively. For hemi-frustum particles, the range was 1.55 ≤ CD ≤ 269.91 and 0.89 ≤ CD ≤ 17.45 for 0.39 ≤ Re ≤ 33.82 and 1.53 ≤ Re ≤ 44.68 in Newtonian fluids and 1.48 ≤ CD ≤ 19209.97 and 1.51 ≤ CD ≤ 943.45 for 0.0101≤ Re ≤ 20.50 and 0.08 ≤ Re ≤ 20.33 while settling in non- Newtonian fluids in the non-annular and annular channels, respectively. The cluster particles experienced a higher wall effect for the Newtonian fluids. The hollow particles showed a higher drag coefficient.