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Power law and composite power law friction factor correlations for laminar and turbulent non-Newtonian open channel flow

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Abstract

Extensive experimental results conducted in a 10-m flume for various types of non-Newtonian fluids spanning a range of cross-sectional open channel shapes are presented and analysed in depth in this work. Open channel flow of non-Newtonian slurries is relevant in mining and chemical engineering applications. This database coupled with the literature data is used to develop the generalised friction factor–Reynolds number correlations in a unified fashion. Much confusion still exists in the literature regarding the definition of non-Newtonian Reynolds numbers. This difficulty is circumvented by considering two widely accepted definitions of the Reynolds number, namely due to Haldenwang et al. (Hydrotransport 15: 15th international conference on the hydraulic transport of solids in pipes, Banff, pp 755–768, 2002) for open channel flow and the modified Metzner–Reed pipe flow Reynolds number adapted for open channel flow. Three different types of purely viscous non-Newtonian fluids in rectangular, trapezoidal, triangular and semi-circular channel shapes were tested. The modelling procedure of Garcia et al. (Int J Multiph Flow 29:1605–1624, 2003) used for pipe flow predictions was extended to the present work. The logistic dose curves based on the Reynolds number proposed by Haldenwang et al. [14] performed better than those based on the adapted Metzner–Reed Reynolds number. Correlations developed can be used for the design of open channels of various shapes to transport non-Newtonian fluids.

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Abbreviations

a :

Channel shape factor constant for laminar flow, Eq. (12) (–)

A :

Cross-sectional area of flow (m2)

b :

Power law exponent taken as −1, Eq. (12) (–)

c :

“Blasius” power law constant for turbulent flow, Eq. (13) (–)

d :

“Blasius” power law exponent for turbulent flow, Eq. (13) (–)

D :

Pipe diameter (m)

e :

Composite power law friction factor exponent, Eq. (11) (–)

j :

Composite power law friction factor exponent, Eq. (11) (–)

f :

Fanning friction factor (–)

F 1 :

Laminar flow power law friction factor (–)

F 2 :

Turbulent flow power law friction factor (–)

f mB :

Modified Blasius friction factor, Eq. (4) (–)

f pred :

Predicted Fanning friction factor based on Eq. (14) (–)

f exp :

Experimental Fanning friction factor Eq. (14) (–)

f exp(ave) :

Averaged experimental Fanning friction factor Eq. (14) (–)

g :

Acceleration due to gravity (m/s2)

K :

Laminar flow constant in the f vs. Re relationship (–)

k :

Consistency coefficient (Pa sn)

k′:

Apparent consistency coefficient, Eq. (7) (Pa s)

n :

Flow behaviour index (–)

N :

Number of points, Eq. (15) (–)

n′:

Apparent power law index, Eq. (6) (–)

P :

Wetted perimeter (m)

Q :

Volumetric flow rate (m3/s)

R 2 :

Correlation coefficient (–)

Re :

Reynolds number (–)

Re H :

Haldenwang et al. Reynolds number, Eq. (3) (–)

Re MR :

Metzner–Reed Reynolds number adapted, for channel flow Eq. (5) (–)

R h :

Hydraulic radius (m)

t :

Composite power law friction factor exponent, Eq. (11) (–)

V :

Average velocity (m/s)

η B :

Bingham plastic viscosity (Pa s)

Θ :

Channel angle from the horizontal (°)

ξ :

Ratio of wall shear stress to Bingham yield stress (–)

ρ :

Density (kg/m3)

τ 0 :

Wall shear stress Eqs. (9, 10) (Pa)

τ w :

Average wall shear stress (Pa)

τ y :

Yield stress (Pa)

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Acknowledgments

The authors would like to acknowledge the National Research Foundation of South Africa and the Cape Peninsula University of Technology for funding this research.

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Authors and Affiliations

  1. Flow Process and Rheology Centre, Cape Peninsula University of Technology, Cape Town, South Africa

    J. H. Burger & R. Haldenwang

  2. Department of Chemical Engineering, Indian Institute of Technology, Kanpur, India

    R. P. Chhabra

  3. BHR Group, The Fluid Engineering Centre, Cranfield, Bedfordshire, MK43 0AJ, UK

    N. J. Alderman

Authors
  1. J. H. Burger
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  2. R. Haldenwang
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  3. R. P. Chhabra
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  4. N. J. Alderman
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Correspondence to J. H. Burger.

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Technical Editor: Monica Feijo Naccache.

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Burger, J.H., Haldenwang, R., Chhabra, R.P. et al. Power law and composite power law friction factor correlations for laminar and turbulent non-Newtonian open channel flow. J Braz. Soc. Mech. Sci. Eng. 37, 601–612 (2015). https://doi.org/10.1007/s40430-014-0188-1

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