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# Paul Slatter, ATC Williams - Laminar flows in open channels and launders

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Professor Paul Slatter, Principal Engineer, Rheology and Slurry Engineering, ATC Williams presented this at the 3rd Annual Slurry Pipeline Conference. The Conference focuses on the design, construction, operation and maintenance of mineral slurry pipelines.

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### Paul Slatter, ATC Williams - Laminar flows in open channels and launders

1. 1. LAMINAR FLOWS IN OPEN CHANNELS AND LAUNDERS Paul Slatter, ATC Williams, Australia.
2. 2. INTRODUCTION  The free surface flow behaviour is of critical importance in many industrial contexts Previous studies have shown • • • A sheet flow diagram can be constructed Similar to tube flow Unresolved issues remain • • • Role that the yield stress may play Location of the laminar/turbulent transition region
3. 3. INTRODUCTION       Objectives:develop and evaluate the free surface sheet flow approach for viscoplastic sheet flows The previously developed approach for the laminar-turbulent transition should now been extended to the characteristic length independent approach approach should be validated against experimental viscoplastic data
4. 4. THEORY AND LITERATURE y x τ τ0 h H Flow direc tion α
5. 5. NEW MODEL Shear stress distribution for sheet flow y x τ τ0 h H Flow direc tion α τ 0  ρgH sin α
6. 6. NEW MODEL   For a Newtonian fluid 3V 0  H For a general time independent nonNewtonian fluid 0 3V 3  2   ( )d H 0 0
7. 7. NEW MODEL Table 1: Comparison of key elements of the rheometric analysis of tube and sheet flow Bulk Shear Rate Tube Flow Sheet Flow 8V D 3V H Wall Shear Stress D p 4L gH sin  R-M Factor 3n   1 4n   2n  1  3n
8. 8. NEW MODEL  Taking the approach further  Follow the approach of Metzner-Reed for pipe flow:-  3V  τ 0  K*'   H  n*' .
9. 9. NEW MODEL  In order to accommodate the reality  Actual channels may approach sheet flow  But will always have side edges at some point  Replace H with the Hydraulic radius Rh so that: n'  3V   τ 0  K*'  R   h * .
10. 10. TRANSITIONAL FLOW – RE4 MODEL  We can define a Reynolds number after the Newtonian paradigm:- 8 V Re 4  . τ0 2  Or better still as Re4  8 V 2  3V  K* '  R    h n* ' .
11. 11. NEW MODEL - EXTENSION TO VISCOPLASTIC MATERIAL • Herschel-Bulkley constitutive relationship  τ  τ y  K • n The bulk shear rate 3V/H for Herschel-Bulkley sheet flow can be expressed as 3V 3nK   0     H  0 2n  1  K   n 1    n    y 1    0       n 1    n     n  y 1     n  1   0   .  
12. 12. TRANSITIONAL FLOW – RE4 MODEL • Transitional Flow Criterion • Transition occurs at Re4 = 700 • From previous work
13. 13. TRANSITIONAL FLOW – CHARACTERISTIC LENGTH INDEPENDENT Vc 26 • y  Wasp, E. and Slatter, P. 2006, 'Transition velocity estimation for viscoplastic fluids', in 13th International Conference Transport and Sedimentation of Solid Particles, Akademia Rolnicza, Wroclaw, Poland, pp. 291-299
14. 14. TRANSITIONAL FLOW – CHARACTERISTIC LENGTH INDEPENDENT Vc 26 • • y  Transition occurs at V = Vc From previous work in large pipes
15. 15. EXPERIMENTAL WORK  Test Rig 10m Tilting Flume In-Line Tube viscometer Conventional rheometer Haldenwang, R. and Slatter, P.T. (2006) Experimental procedure and database for non-Newtonian open channel flow, Journal of Hydraulic Research, Vol. 44, Issue 2, pp. 283–287.
16. 16. Rheology and Materials Processing Centre Flow meter calibration tank 10 M F L U M E Hydraulic Ram to tilt flume Positive displacement pump 23 l/s Mixing tank 2000 litres Stirrer
17. 17. ON-LINE PIPE VISCOMETER DP cells Heat Exchanger Magnetic Flow meters Mass Flow meter 13 mm pipe Pressure tappings 28 mm pipe 80 mm pipe
18. 18. MATERIAL  6% Kaolin suspension  Viscoplastic  Model rheology Paste Material
19. 19. 25 Wall shear stress (H) (Pa) 3 degrees 4 degrees 20 5 degrees New Model 15 10 5 0 0 100 200 300 3V/H (s-1) 400 500
20. 20. DISCUSSION Using the criterion Re4 = 700  For the 3 and 4 degree empirical data  More deviation at 5 degrees.  Using the characteristic length independent approach Vc,  The reverse is true  Best prediction at 5 degrees  Significant deviation for the 3 and 4 degree empirical data. 
21. 21. Discussion It should be noted that – as for critical pipe flow – there may well be a zone or range of values within which transition can occur.  The main benefits of the new approach  Laminar flow data can be scaled up directly for engineering design purposes  Or  From the rheology as measured by standard bench-top methods. 
22. 22. DISCUSSION The principal unresolved issues revolve around  Edge effects with a yield stress  Froude number or free surface effects  These relate to critical, tranquil or shooting flow. 
23. 23. DISCUSSION Extension to visco-plastic fluids has been achieved.  The apparent flow behaviour index is not constant.  Using the tangent method at the relevant wall shear stress as proposed by Metzner-Reed is viable 
24. 24. DISCUSSION Has been validated against experimental data.  This remains to be validated against more data.  Negative slope in the turbulent region ---
25. 25. TURBULENT FLOW 90 80 70 60 To 50 5 deg 4 deg 40 3 deg 30 20 10 0 0 50 100 150 3V/H 200 250 300 350
26. 26. CONCLUSIONS        A new analytical approach for the sheet flow of a power law fluid has been extended to viscoplastic Yield Stress material. Exploits the fact that the bulk shear rate is a unique function of the rheogram and the wall shear stress Can be used for scale-up and design at any required slope and depth, in laminar flow. Onset of turbulent flow can be predicted from Re4 = 700 or V = Vc approach This approach is analogous to that of Metzner and Reed (1955) for laminar pipe flow. Unresolved issues have been highlighted.
27. 27. FUTURE WORK Further work on the transition using a wider data base  Entrance, end and edge effects for sheet flow  Identification and analysis of critical flow conditions    Related to surface disturbances and Froude number effects. Influence, identification and analysis of slip flow effects
28. 28. ACKNOWLEDGEMENT    The support and encouragement of the ATC Williams team is gratefully acknowledged – without which this work would not have been possible.