1. Laminar non-Newtonian flow in open channels of different cross-sectional shapes: An alternative approach
Dr Neil J Alderman

2. Background Laminar flow of a Newtonian fluid in an open channel
Primary variables – linear co-ordinates
Dimensionless analysis – ln ƒ vs ln ReN plot (Moody chart)

3. Background Laminar flow of a non-Newtonian fluid in an open channel
Dimensionless analysis – ln ƒ vs ln ReH plot
Primary variables – linear co-ordinates?

4. Professor Peter N Rowe 25/12/1919 – 27/4/2014
A cautionary tale!
He showed how incorrect dimensional analysis enabled fictitious heat transfer data for gas flow through a packed bed of particles based on random numbers to be correlated successfully!
Rowe, P. N. (1963)  The correlation of engineering data, The Chemical Engineer, p CE69, March.
Professor Peter N Rowe 25/12/1919 – 27/4/2014

5. Laminar flow of a Herschel-Bulkley fluid in an open channel
5.4% kaolin suspension flowing in a 300 mm semi‑circular flume at slope angles of 1 to 5o

6. Laminar flow of a Herschel-Bulkley fluid in an open channel
C = numerical constant dependent on channel shape
ƒ = Fanning friction factor
ReHHB = Haldenwang Reynolds Number for a Herschel- Bulkley fluid

7. Laminar flow of a Herschel-Bulkley fluid in an open channel
Q=VA

8. Laminar flow of a Herschel-Bulkley fluid in an open channel

9. Laminar flow of a Bingham plastic fluid in an open channel
Set tyHB = tyB , K = p and n = 1
Obtain

10. Laminar flow of a Power law fluid in an open channel
Set tyHB = 0
Obtain

11. Laminar flow of a Newtonian fluid in an open channel
Set y = 0, K = N and n = 1
Obtain

12. Y vs X plots
The Y vs X plot for each flow curve model will give a slope of 1/C with a zero intercept

13. FPRC 10m FLUME Flow depth measuring position
Hydraulic Ram to tilt flume
Flow meter calibration tank
Positive displacement pump 23 l/s
Mixing tank
2000 litres
The tests were carried out in 10 m long tilting flume designed and built by the Flow Process Research Centre at the Cape Peninsula University of Technology. This flume can be hydraulically tilted at various angles up to 5 degrees from the horizontal. The width of this rectangular flume can be changed from 300 to 150 mm by placing a partition mid-section lengthways down the flume. By inserting an appropriate cross sectional insert, the rectangular flume can be changed into a flume with a triangular, semi-circular or trapezoidal cross-section.Flow depth measurement was done with two digital depth gauges.
Stirrer

14. IN-LINE PIPE VISCOMETER
DP cells
Heat Exchanger
Magnetic Flow meters
Mass Flow meter
13 mm pipe
. Flow curve measurements of the test material were also made in-situ during the flume test using and in-line tube viscometer fitted with three tubes of different diameters 13,28 and 80mm.Flow from supply pumps was measured by electromagnetic flow meter.
Pressure tappings
28 mm pipe
80 mm pipe

15. Flume shapes and sizes used in test rig

16. Laminar flow of a Newtonian fluid in an open channel
Mineral oil flowing in a 75mm rectangular flume at slope angles of 1 to 4o

17. Laminar flow of a Newtonian fluid in an open channel
Critical ReH > 700

18. Laminar flow of a Newtonian fluid in an open channel
Mineral oil flowing in a 75mm rectangular open channel at slope angles of 1 to 4o
C = 24.2

19. Laminar flow of a power law fluid in an open channel
4.9% CMC solution in a 300mm triangular flume at slope angles of 1 to 5o

20. Laminar flow of a power law fluid in an open channel
Critical ReH ~ 500

21. Laminar flow of a power law fluid in an open channel
Mineral oil flowing in a 75mm rectangular open channel at slope angles of 1 to 4o

22. Laminar flow of a Bingham plastic fluid in an open channel
Mineral oil flowing in a 75mm rectangular open channel at slope angles of 1 to 4o
4.5% bentonite slurry flowing in a 150mm trapezoidal flume at slope angles of 1 to 5o

23. Laminar flow of a Bingham plastic fluid in an open channel
Critical ReH ~ 500

24. Laminar flow of a Bingham plastic fluid in an open channel

25. Laminar flow of a Herschel-Bulkley fluid in an open channel
Mineral oil flowing in a 75mm rectangular open channel at slope angles of 1 to 4o
5.4% kaolin suspension flowing in a 300 mm semi‑circular flume at slope angles of 1 to 5o

26. Laminar flow of a Herschel-Bulkley fluid in an open channel
Critical ReH ~ 500

27. Laminar flow of a Herschel-Bulkley fluid in an open channel

28. Results Flow curve model Flume size and shape No of data points
f vs Re
Y vs X
Critical ReH
Newtonian
75mm rectangular 44
24.3
24.2
> 700
Power law
300mm triangular 51
15.0
15.5
~500
Bingham Plastic
150mm trapezoidal 40
15.1
14.3
Herschel-Bulkley
300mm semi-circular
16.7
16.4

29. Conclusions
General expression for laminar open channel flow of a Herschel-Bulkley fluid derived in key primary variables.
This enabled channel flow data, traditionally presented as a f vs ReH plot on logarithmic coordinates, to be presented as a Y vs X plot on linear coordinates where values of Y and X are given according to flow curve model used.

30. Conclusions
Good agreement found between C values from f vs ReH plots on logarithmic coordinates and Y vs X plots on linear coordinates for laminar flow of Newtonian, power law, Bingham plastic and Herschel-Bulkley fluids.
Y vs X plot is the better plot for detecting demarcation between laminar and transitional flow.

31. Acknowledgements
Professor Rainer Haldenwang and Dr Johan Burger of Cape Peninsula University of Technology, Cape Town, South Africa for the provision of their channel flow data.

32. Laminar non-Newtonian flow in open channels of different cross-sectional shapes: An alternative approach Dr Neil J Alderman BHR Group, Cranfield, Bedfordshire, UK