1. The flow field will change by the oblique angle of impeller blades. Small oblique angle will bring axial flow and large oblique angle will bring radial flow. When the oblique angle increase to a certain angle, the flow field will change from axial flow to radial flow. The significant angle is defined as critical angle.
2. The definition of the multiple reference frames does really affect the computational prediction result. In Shen’s woke, the rotational frame is defined only at the range of impeller blade in radius direction and in axial direction; stationary frame is used on the other places. In present work, the rotational frame is defined encompass all the impeller and the flow surrounding it, and the other place is defined as stationary frame. For the second condition the interface between the rotational frame and stationary frame may be a nearly-stationary flow, and it would be better for MRF technique to apply to. The result shows that the Power number of present work has better agreement on experimental data.
3. When the flow field is axial flow, the flow below or above of the disc is baffled by the disc. The disc makes the fluid flow toward radial direction but the pitched impeller blade force them flow down to form the axial flow. When the flow field is radial flow, the fluid flow out the disk and the impeller blades continues force them toward the same direction to from radial flow. The pressure act on impeller blades in first condition would be bigger then that in second condition. This may be the reason why the power number decreases rapidly when the flow field changed from axial flow to radial flow. As the disc is bigger, this circumstance will be more obvious. Bigger impeller disc brings bigger decrease of power number when the flow field changes from axial flow to radial flow.
4. In the condition of D=T/3, d=2D/3, C=T/2, the power number increase rapidly when the oblique angle reaches critical angle. This is because the effect of disk is smaller then the effect that when the flow field change from radial flow to axial flow the pressure act on blades increase rapidly reported by Chou[18].
5. The less the clearance from impeller to vessel bottom is, the bigger the critical angle is. The less the clearance is, the stronger the axial flow will be. This might because when the impeller is closer to the vessel bottom, the fluid volume below the impeller is smaller. At the same impeller rotational speed, the fluid velocity of smaller volume will be faster and small oblique angle can produce large axial velocity component, this may be the reason why the flow field is easy to become axial flow when the clearance is small.
6. In cases of dual impeller turbines, MRF technique can predict the parallel flow well, but not the merging flow or diverging flow. That might because there is no sufficient space between the lower impeller and the vessel bottom in diverging flow or between the two impellers in merging flow. When the impellers and walls are sufficiently separated, the region between them may contain nearly-stationary flow, we have sufficient physical meaning to define the interface between rotational frame and stationary frame on it, but not on merging flow and diverging flow.
Though the MRF technique can predict the velocity and turbulence field well on merging and diverging flow, but there are still inaccuracies on power number.
Reference
[1] J.COSTES and J.P.COUDERC, “Study by laser Doppler anemometry of the turbulent flow induced by a Rushton turbine in a stirred tank: influence of the size of the units- Part 1: mean flow and turbulence”, Chemical Engineering Science, Vol 43, No10, pp.2751-2764, 1988.
[2] H.WU and G.K.PATTERSON, “Laser-Doppler measurements of turbulent flow parameters in a stirred mixer”, Chemical engineering science, Vol.44, No.10, pp.2207-2221, 1989.
[3] V.V.RANADE and J.B.JOSHI, “Flow generated by a disc turbine: part 1 experimental”, Transactions of the Institution of Chemical Engineers, Vol.68, part A, pp.19-33, 1990.
[4] M. YIANNESKIS AND J. H. WHITELAW, “On the structure of the trailing vortices around Rushton turbine blades”, Transactions of the Institution of Chemical Engineering, Vol. 71, Part A, pp. 543-550,1993.
[5] V. P. MISHRA and J. B. JOSHI, “Flow generated by a disc turbine: part : Effect Ⅲ of impeller diameter, Impeller Location and Comparison with other radial flow turbines”, Transactions of the Institution of Chemical Engineering, Vol. 71, Part A, pp. 563-573, 1993.
[6] V. P. MISHRA and J. B. JOSHI, “Flow generated by a disc turbine: part : Ⅳ Multiple impellers”, Transactions of the Institution of Chemical Engineering, Vol.72, Part A, pp. 657-668, 1994.
[7] K. C. LEE and M. YIANNESKIS, “Turbulence properties of the impeller stream of a Rushton turbine, American Institute of Chemical Engineering Journal, Vol. 44, No.1, pp.13-24, 1998.
[8] V. V. Ranade, M. Perard, N. Lesauze, C. Xuereb and J. Bertrand, “Trailing
vortices of Rushton turbine: PIV measurements and CFD simulations with snapshot approach” Transactions of Institution of Chemical Engineering, Vol. 79 part A, pp. 3-12, 2001.
[9] K. RUTHERFORD, K. C. LEE, S. M. S. MAHMOUDI and M. YIANNESKI,
“Hydrodynamic characteristics of dual Rushton impeller stirred vessels”, American Institute of Chemical Engineering Journal, Vol. 42, No.2, pp. 332-346, 1996.
[10] J. MARKOPOULOS, E. BABALONA, and E. TSILIOPOULOU, “Power consumption in agitated vessels with dual Rushton turbines: Baffle length and impeller spacing effects”, Chemical Engineering Technology, Vol.27, No.11, pp.
1212-1215, 2004.
[11] V. V. RANADE and J. B. JOSHI, “Flow generated by a disc turbine: part : Ⅱ Mathematical modelling and comparison with experimental data”, Transactions of the Institution of Chemical Engineering, Vol. 68, Part A, pp.34-50, 1990.
[12] J. Y. LUO, A. D. Gosman, R. I. Issa, J. C. Middleton, and M. K. Fitzgerald. “Full flow field computation of mixing in baffled stirred vessels”, Institution of Chemical Engineers, vol.71, part A, 1993.
[13] J. Y. LUO, R, I, ISSA and A. D. GOSMAN, “Prediction of impeller induced flow in mixing vessels using multiple frame of reference”, ICHEME Symposium Series, No. 136, pp. 549-556, 1994.
[14] Albert D. Harvey III, Stewart P. Wood and Douglas E. Leng. “Experimental and computational study of multiple impeller flows”, Chemical Engineering Science, vol. 52, No. 9, pp. 1479-1491, 1997
[15] G. TABOR, A. D. GOSMAN and R. I. ISSA, “Numerical simulation of the flow in a mixing vessel stirred by a Rushton turbine”, I. Chem. E. Fluid Mixing : UK Ⅴ
conference on Mixing, Bradford, 1996.
[16] R. M. JONES, A. D. HARVEY , Ⅲ and S. ACHARYA, “Two-equation turbulence modeling for impeller stirred tanks”, Transactions of the ASME, Journal of Fluids Engineering, Vol.123, pp. 640-648, 2001
[17] Yu-Chang Hu, “Calculation of the Flow in Impeller Stirred Tanks”, National Chiao Tung University, Thesis for Master Degree, 2003.
[18] Jian-Ren Chou, “Flow analysis in a Tank Agitated by Pitched-Blade Impellers”, National Chiao Tung University, Thesis for Master Degree, 2004.
[19] Shih-Jhen Shen, “Analysis of the Flow Agitated by Disc Turbines with Pitched Blades”, National Chiao Tung University, Thesis for Master Degree, 2005.
[20] Z. Jaworski, K. N. Dyster and A. W. Nienow, “The effect of size, location and Pumping direction of pitched blade turbine impellers on flow patterns: LDA measurements and CFD predictions”, Transactions of Institution of Chemical Engineers, vol. 79, part A, pp. 887-894, 2001.
[21] Albert D. Harvey III and Cassian K. Lee, “Steady-state modeling and experimental measurement of a baffled impeller stirred tank”, American Institute of Chemical Engineering Journal, vol. 41, No. 10, pp. 2177-2186, 1995.
[22] G. MICALE, A. BRUCATO, and F. GRISAFI, “Prediction of flow fields in a dual-impeller stirred vessel”, American Institute of Chemical Engineering Journal, Vol. 45, No 3, pp. 445-464, 1999.
[23] D. B. Spalding and B. E. Launder, “The Numerical computation of turbulence flows”, Computer Method in Applied Mechanics and Engineering, Issue 3, pp.
269-289,1974
(a) Pitched blade impeller (b) Propeller
(c) Rushton turbine (d) Straight blade impeller
Figure 1.1 Various impellers
0 50 100 150 0
50 100 150 200 250 300
0 50 100 150
0 50 100 150 200 250 300
(a) axial flow (b) radial flow
Figure 1.2 Axial flow and radial flow
(a)C1=T/4, C2=T/2, C3=T/4
(b)C1=T/3, C2=T/3, C3=T/3
(c)C1=0.15T, C2=0.5T, C3=0.35T
Figure 1.3 Flow patterns for three geometrical configuration studied: (a) parallel flow;
(b) merging flow; (c)diverging flow
Figure 1.4 Dual Rushton turbine vessel with variable baffle length
Φ
Ω
(a) Stationary Frame
Stationary Frame
Rotational Frame Rotational Frame
(b)
Figure 1.5 Rotational and stationary frame of Shen’s model (a) Horizontal plane (b) Vertical plane
Ω
Φ
(a) Stationary Frame
Stationary Frame
Rotational Frame
Rotational Frame
(b)
Figure 1.6 Rotational and stationary frame of present work (a) Horizontal plane (b) Vertical plane
Ω
Periodic Boundary
Figure 2.1 Periodic boundary conditions
δrK
d
f
JG
S JJG
P
Cf
Figure 3.1 Over-relaxed approach
f1
f2
f3 1
P
Sf
JJJG
2
Sf
JJJG
3
Sf
JJJG
Sa
JJG δJJGr
Figure 3.2 Calculation of boundary pressure
P
δUJG δUJG⊥
δUJG&
δn
SW
JG τGW
Figure 3.3 Calculation of wall shear stress
P
C r1
C r2
C r3
C s Ω
Stationary Frame Rotational Frame
Figure 3.4 Calculated grids in rotational frame
P
C s3
C r C s2
C s1
Ω Rotational Frame
Interface
Stationary Frame
Interface
Figure 3.5 Calculated grids in stationary frame
Figure 4.1 Geometric of single impeller stirred tank
α
α
Figure 4.2 Geometrical parameters of stirred tank
(a)
(b)
Figure 4.3 (a) Calculation domain of straight-blade turbine (b) Grids of straight-blade turbine
(a)
(b)
Figure 4.4 (a) Calculation domain of pitched-blade turbine (b) Grids of pitched-blade turbine Vr/Vtip
2r/T
Figure 4.5 Radial profiles of (a) radial, (b) tangential, and (c) axial velocity compare prediction of Shen [19] and experimental data of Ranade[3]. (D=T/3, C=T/2, α=90)
Z/(H/2)=0.2
Z/(H/2)=0.5 Z/(H/2)=0.667 Z/(H/2)=0.833 Z/(H/2)=0.933
Figure 4.6 The position of various Z/R
2r/T
Figure 4.7 Radial profiles of (a) radial, (b) tangential, and (c) axial velocity of four kinds of grids. (D=T/3, C=T/2, α=90)
0 50 100 150
Figure 4.8 Stream lines of various blade angles at the vertical plane of φ= -30° (D=T/3 d=2D/3 C=T/2)
Angle(deg.)
Figure 4.9 The Power number, Pumping number, and Pumping efficiency of various blade angles. (D=T/3 d=2D/3 C=T/2)
Angle(deg.)
Powernumber
30 40 50 60 70 80 90
0 1 2 3 4 5 6
Present Shen
Angle(deg.)
Pumpingnumber
30 40 50 60 70 80 90
-0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2
Present Shen
Figure 4.10 The angular profiles of Power number and Pumping number compare with Shen’s work. (D=T/3 d=2D/3 C=T/2)
α=47° α=48°
Figure 4.11 The pressure contour of α= 47° and α= 48° at the vertical plane of Φ=-30°.
(D=T/3 d=2D/3 C=T/2)
0 50 100 150
Figure 4.12 Stream lines of various blade angles at the vertical plane of φ= -30°.
(D=T/3 d=2D/3 C=T/3)
Angle(deg.)
Figure 4.13 The Power number, Pumping number, and pumping efficiency of various blade angles. (D=T/3 d=2D/3 C=T/3)
Angle(deg.)
Powernumber
30 40 50 60 70 80 90
0 1 2 3 4 5 6
Present Shen
Angle(deg.)
Pumpingnumber
30 40 50 60 70 80 90
-0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2
Present Shen
Figure 4.14 The angular profil of Power number and Pumping number compare with Shen’s work. (D=T/3 d=2D/3 C=T/3)
α=72° α=73°
Figure 4.15 The pressure contour of α= 72° and α= 73° at the vertical plane of Φ=-30°.
(D=T/3 d=2D/3 C=T/3)
0 50 100 150
Figure 4.16 Stream lines of various blade angles at the vertical plane of φ= -30°.
(D=T/3 d=2D/3 C=T/4)
Angle(deg.)
Figure 4.17 The Power number, Pumping number, and Pumping efficiency of various blade angles. (D=T/3 d=2D/3 C=T/4)
Angle(deg.)
Powernumber
30 40 50 60 70 80 90
0 1 2 3 4 5 6
Present Shen
Angle(deg.)
Pumpingnumber
30 40 50 60 70 80 90
-0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2
Present Shen
Figure 4.18 The angular profile of Power number and Pumping number compare with Shen’s work. (D=T/3 d=2D/3 C=T/4)
α=88° α=89°
Figure 4.19 The pressure contour of α= 88° and α= 89° at the vertical plane of Φ=-30°.
(D=T/3 d=2D/3 C=T/4)
0 50 100 150
Figure 4.20 The flow field changes from axial flow to radial flow of different impeller location.
C=T/2
0 50 100 150
0 50 100 150 200 250 300
C=T/3
0 50 100 150
0 50 100 150 200 250 300
C=T/4
0 50 100 150
0 50 100 150 200 250 300
Figure 4.21 The flow field of blade angle α=60° when the impeller is posited at three different location
0 50 100 150
Figure 4.22 Stream lines of various blade angles at the vertical plane of Φ= -30.
(D=T/3 d=3D/4 C=T/2)
Angle(deg.)
Figure 4.23 The Power number, Pumping number, and Pumping efficiency of various blade angles. (D=T/3 d=3D/4 C=T/2)
Angle(deg.)
Powernumber
20 30 40 50 60 70 80 90
0 1 2 3 4 5 6
Present Shen
Angle(deg.)
Pumpingnumber
20 30 40 50 60 70 80 90
-0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2
Present Shen
Figure 4.24 The angular profile of Power number and Pumping number compare with Shen’s work. (D=T/3 d=3D/4 C=T/2)
α=37° α=38°
Figure 4.25 The pressure contour of α= 37° and α= 38°at the vertical plane of Φ=-30°.
(D=T/3 d=3D/4 C=T/2)
0 50 100 150
Figure 4.26 Stream lines of various blade angles at the vertical plane of Φ= -30.
(D=T/3 d=3D/4 C=T/3)
Angle(deg.)
Figure 4.27 The Power number, Pumping number, and Pumping efficiency of various blade angles. (D=T/3 d=3D/4 C=T/3)
Angle(deg.)
Powernumber
30 40 50 60 70 80 90
0 1 2 3 4 5 6
Present Shen
Angle(deg.)
Pumpingnumber
30 40 50 60 70 80 90
-0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2
Present Shen
Figure 4.28 The angular profile of Power number and Pumping number compare with Shen’s work. (D=T/3 d=3D/4 C=T/3)
α= 65° α= 66°
Figure 4.29 The pressure contour of α= 65° and α= 66° at the vertical plane of Φ=-30°.
(D=T/3 d=3D/4 C=T/3)
0 50 100 150
Figure 4.30 Stream lines of various blade angles at the vertical plane of φ= -30°.
(D=T/3 d=3D/4 C=T/4)
Angle(deg.)
Figure 4.31 The Power number, Pumping number, and Pumping efficiency of various blade angles. (D=T/3 d=3D/4 C=T/4)
Angle(deg.)
Powernumber
30 40 50 60 70 80 90
0 1 2 3 4 5 6 7
Present Shen
Angle(deg.)
Pumpingnumber
30 40 50 60 70 80 90
-0.5 0 0.5 1
Present Shen
Figure 4.32 The angular profile of Power number and Pumping number compare with Shen’s work. (D=T/3 d=3D/4 C=T/4)
α= 84° α= 85°
Figure 4.33 The pressure contour of α= 84° and α= 85° at the vertical plane of Φ=-30°.
(D=T/3 d=3D/4 C=T/4)
0 50 100 150
Figure 4.34 Stream lines of various blade angles at φ= -30°. (D=T/2 d=3D/4 C=T/2)
Angle(deg.)
Figure 4.35 The Power number, Pumping number, and Pumping efficiency of various blade angles. (D=T/2 d=3D/4 C=T/2)
Angle(deg.)
Figure 4.36 The angular profile of Power number and Pumping number compare with Shen’s work. (D=T/2 d=3D/4 C=T/2)
0 50 100 150
Figure 4.37 Stream lines of various blade angles at the vertical plane of φ= -30°.
(D=T/2 d=3D/4 C=T/3)
Angle(deg.)
Figure 4.38 The Power number, Pumping number, and Pumping efficiency of various blade angles. (D=T/2 d=3D/4 C=T/3)
Angle(deg.)
Figure 4.39 The angular profile of Power number and Pumping number compare with Shen’s work. (D=T/2 d=3D/4 C=T/3)
0 50 100 150
Figure 4.40 Stream lines of various blade angles at the vertical plane of φ= -30°.
(D=T/2 d=3D/4 C=T/4)
Angle(deg.)
Figure 4.41 The Power number, Pumping number, and Pumping efficiency of various blade angles. (D=T/2 d=3D/4 C=T/4)
Angle(deg.)
Figure 4.42 The angular profile of Power number and Pumping number compare with Shen’s work. (D=T/2 d=3D/4 C=T/4)
r
Figure 4.43 Flow field behind the blade (D=T/3 d=3D/4 C=T/3)
α=50° α=65°
α=66° α=90°
Figure 4.44 Pressure contours of different blade angles (D=T/3 d=3D/4 C=T/3)
(a)α=50°
(b)α=65°
(c)α=66°
(d) α=90°
Figure 4.45 Pressure contours on blade surface for different blade angles (D=T/3 d=3D/4 C=T/3)
r
Figure 4.46 Stream lines on blade surface for different blade angles (D=T/3 d=3D/4 C=T/3)
0 100 200 0
50 100 150 200 250 300
Figure 4.47 The definition of flow out angle
Angle(deg.)
Figure 4.48 Angular profiles of flow out angle for different off-bottom clearance
Angle(deg.)
Figure 4.49 Radial profiles of Power number for different off-bottom clearance
Angle(deg.)
Pumpingnumber
30 40 50 60 70 80 90
0.4 0.6 0.8
1 C=T/2
C=T/3 C=T/4
(a)D=T/3 d=2D/3
Angle(deg.)
Pumpingnumber
30 40 50 60 70 80 90
0.4 0.6 0.8
1 C=T/2
C=T/3 C=T/4
(b) D=T/3 d=3D/4
Angle(deg.)
Pumpingnumber
30 40 50 60 70 80 90
0.4 0.6 0.8
1 C=T/2
C=T/3 C=T/4
(c) D=T/2 d=3D/4
Figure 4.50 Radial profiles of Pumping number for different off-bottom clearance
Angle(deg.)
Figure 4.51 Radial profiles of k* for different off-bottom clearance
Angle(deg.)
Figure 4.52 Radial profiles of ε* for different off-bottom clearance
Angle(deg.)
Figure 4.53 Radial profiles of pumping efficiency for different off-bottom clearance
Φ=0° Φ=-10° Φ=-20° Φ=-30°
Figure 4.54 The stream lines and contours of stirred tank at different vertical plane when the blade angle α=70° (D=T/3 d=3D/4 C=T/3)
Figure 5.1 Dual impeller stirred tank configuration
Stationary Frame Rotational Frame
Figure 5.2 Multiple reference of parallel flow
Rotational Frame Stationary Frame
Figure 5.3 Multiple reference of merging flow
Stationary Frame Rotational Frame
Figure 5.4 Multiple reference of diverging flow
.
(a) (b)
(c) (d)
Figure 5.5 Parallel flow: comparison of velocity vector plots in a plane midway between baffles. (a) Experimental results [9] (b) Predictions, IO technique [22] (c) Predictions, SG technique [22] (d) Predictions, MRF technique (present)
(a) (b)
(c) (d)
Figure 5.6 Parallel flow: Comparison of turbulence energy distribution in a plane midway between baffles (a) Experimental results [9] (b) Predictions, IO technique [22] (c) Predictions, SG technique [22] (d) Predictions, MRF technique (present)
(a) (b)
(c) (d)
Figure 5.7 Merging flow: Comparison of velocity vector plots in a plane midway between baffles. (a) Experimental results [9] (b) Predictions, IO technique [22] (c) Predictions, SG technique [22] (d) Predictions, MRF technique (present)
(a) (b)
(c) (d)
Figure 5.8 Merging flow: Comparison of turbulence energy distributions in a plane midway between baffles. (a) Experimental results [9] (b) Predictions, IO technique [22] (c) Predictions, SG technique [22] (d) Predictions, MRF technique (present)
(a) (b)
(c) (d)
Figure 5.9 Diverging flow: comparison of velocity vector plots in a plane midway between baffles. (a) Experimental results [9] (b) Predictions, IO technique [22] (c) Predictions, SG technique [22] (d) Predictions, MRF technique (present)
(a) (b)
(c) (c)
Figure 5.10 Diverging flow: Comparison of turbulence energy distributions in a plane midway between baffles. (a) Experimental results [9] (b) Predictions, IO technique [22] (c) Predictions, SG technique [22] (d) Predictions, MRF technique (present)