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

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)

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?

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

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

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

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

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

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

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

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

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

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

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

Flume shapes and sizes used in test rig

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

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

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

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

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

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

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

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

Laminar flow of a Bingham plastic fluid in an open channel

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

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

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

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

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.

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.

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. www.cput.ac.za/flowpro/rectangularflume_april2005.pdf www.cput.ac.za/flowpro/allshapesflumedata_june2009.pdf

Laminar non-Newtonian flow in open channels of different cross-sectional shapes: An alternative approach Dr Neil J Alderman BHR Group, Cranfield, Bedfordshire, UK nalderman@bhrgroup.co.uk www.bhrgroup.com