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Non-Newtonian open channel flow: the effect of shape
Open channels, flumes or launders are used in the mining industry to transport slurries during processing and to disposal sites. Water plays a major part in the makeup of these slurries, its usage and availability is critical in countries where there are strict water usage management programs. The optimisation of flume design involves the maximisation of solids transport efficiency whilst, at the same time reduces water usage. The design of open channels is complex as it is dependent on both the slurry rheology and the channel shape. Very little has been reported in the literature for predicting non-Newtonian laminar flow in open channels of arbitrary cross-section. The only method available was that proposed by Kozicki and Tiu (1967, 1986). The shape factors they used were those evaluated from analytical solutions for flow of Newtonian fluids in open channels of the same cross-section. However, they carried out no experimental work to validate their model. Few experimental studies have been made on the effect of shape on non-Newtonian flow in open channels. Naik (1983) tested kaolin in water suspensions in a rectangular channel. Coussot (1994) provided some data for the flow of a Herschel-Bulkley fluid in rectangular and trapezoidal channels. Fitton (2007; 2008) obtained data for flow of three different non-Newtonian fluids (carboxymethylcellulose, carbopol and thickened tailings) in a semi-circular channel. A large experimental database for non-Newtonian flow in rectangular open channels was published by Haldenwang (2003) at the Flow Process Research Centre, Cape Peninsula University of Technology. Guang et al. (2011) performed Direct Numerical Simulations of turbulent flow of a yield- pseudoplastic fluid in a semi-circular channel. They compared their simulations with actual field measurements and found them to over-predict the flow velocity by approximately 40%. The source for this discrepancy was difficult to ascertain. A comprehensive database was compiled during this research of the flow of three non–Newtonian fluids in rectangular, trapezoidal, semi-circular and triangular channels. The flow of carboxymethylcellulose solutions and aqueous kaolin and bentonite suspensions was investigated in a 10 meter long flume at angles ranging from 1° to 5° from the horizontal plane. The effect of channel shape on the friction factor-Reynolds number relationship for laminar and turbulent open channel flow of these three fluids was investigated. New models for the prediction of laminar and turbulent flow of non-Newtonian fluids in open channels of different cross-sectional shapes are proposed. The new laminar and turbulent velocity models are compared with three previously-published velocity models for laminar flow and five previously-published velocity models for turbulent flow using average velocity as comparison criteria. For each channel shape, the laminar flow data can be described by a general relationship, f = K/Re where f is the Fanning friction factor and Re is the appropriate Haldenwang et al. (2002) Reynolds number. The K values were found to be 14.6 for triangular channels with a vertex angle of 90°, 16.2 for semi-circular channels, 16.4 for rectangular channels and 17.6 for trapezoidal channels with 60 degree sides. These K values were found to be in line with those reported by Straub et al. (1958) and Chow (1969) for open channel laminar flow of Newtonian fluids as opposed to the assumption made by Haldenwang et al. (2002; 2004) of using a constant value of 16 based on the pipe flow paradigm for all channel shapes. This new laminar model gave a closer fit to the laminar flow data than those from the three previously-published models. However, the presence of the yield stress still presents a problem, which makes the flow prediction in laminar flow for such fluids not very accurate. The investigation on non-Newtonian turbulent flow of the three fluids in the four different shaped open channels revealed that the data was described by the modified Blasius equation f = a Re b where a and b are constant values determined for each channel shape and Re is the Haldenwang et al. (2002) Reynolds number. Values of a and b for a rectangular channel were found to be 0.12 and -0.330, for a semi- circular channel 0.048 and -0.205, for a trapezoidal channel with 60° sides, 0.085 and -0.266 and for a triangular channel with vertex angle of 90°, 0.042 and -0.202. New laminar and turbulent velocity models were derived from using the new laminar f = K/Re and turbulent f = a Re b, friction factor-Reynolds number relationship. The laminar velocity model did not always give the best result, but the majority of the time it did, compared to the three previously published models. The new turbulent velocity model yielded the best results when compared to the five previously published models using average velocity as comparison criteria. The composite power law modelling procedure of Garcia et al. (2003) used for pipe flow predictions was extended to the present work on non-Newtonian flow in open channels of various cross-sections. The results show that the modelling technique used by Garcia et al. (2003) for pipe flow can be used to adequately predict flow in an open channel of a given cross-sectional shape provided that an appropriate Reynolds number is used to take into account the non-Newtonian behaviour of the test fluid. It was found that the results using the Haldenwang et al. (2002) Reynolds number yielded better results than those based on the adapted Metzner-Reed Reynolds number. The correlations and models developed and experimentally validated during this research can be used to further improve the design of rectangular, semi-circular, trapezoidal and triangular open channels to transport non-Newtonian fluids.
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