Exact Solution of Optimum Hydraulic Power-Law Section with General Exponent Parameter
This article has a reply.
VIEW THE REPLYThis article has a reply.
VIEW THE REPLYPublication: Journal of Irrigation and Drainage Engineering
Volume 144, Issue 12
Abstract
Power-law sections provide great flexibility in open channel design. However, in the literature the optimum hydraulic power-law section has been developed for only specific values of the exponent of the power-law formula. This paper presents a general exact solution of the optimum hydraulic section with as a parameter based on Gaussian hypergeometric mathematics and the Lagrange multiplier method. The relationships between and each of the optimum width-depth ratio and the side slope are derived. The explicit exact formulas of the shape factor, normal depth, critical depth, discharge, wetted perimeter, and flow area for different values are presented. The results show that the discharge of the optimum hydraulic section increases as and then decreases as for a given flow area or wetted perimeter. In addition, a super-best hydraulic power-law section with exists, where the discharge is largest. This super-best section represents a new discovery as it provides the global maximum discharge among all possible power-law section shapes. The characteristics of the super-best section are presented.
Get full access to this article
View all available purchase options and get full access to this article.
Acknowledgments
The authors are grateful to the editors and five anonymous reviewers for their thorough and most helpful comments. This research project is supported by the Natural Science Foundation of Shandong Province, China (ZR2017LEE028), the Key Research and Development Program of Shandong Province, China (2016GSF117038), the National Science and Technology Support Program of China (2015BAB07B02), the Development of Science and Technology Plan of Jinan City, China (201302052), and the Teaching and Research Projects of the University of Jinan (J1641).
References
Abdulrahman, A. 2007. “Best hydraulic section of a composite channel.” J. Hydraul. Eng. 133 (6): 695–697. https://doi.org/10.1061/(ASCE)0733-9429(2007)133:6(695).
Abramowitz, M., and I. A. Stegun. 1964. Handbook of mathematical functions. New York: Dover Publications.
Anwar, A. A., and D. Clarke. 2005. “Design of hydraulically efficient power-law channels with freeboard.” J. Irrig. Drain. Eng. 131 (6): 560–563. https://doi.org/10.1061/(ASCE)0733-9437(2005)131:6(560).
Anwar, A. A., and T. T. de Vries. 2003. “Hydraulically efficient power-law channels.” J. Irrig. Drain. Eng. 129 (1): 18–26. https://doi.org/10.1061/(ASCE)0733-9437(2003)129:1(18).
Babaeyan-Koopaei, K., E. M. Valentine, and D. C. Swailes. 2000. “Optimum design of parabolic-bottomed triangle canals.” J. Irrig. Drain. Eng. 126 (6): 408–411. https://doi.org/10.1061/(ASCE)0733-9437(2000)126:6(408).
Chanson, H. 1999. The hydraulics of open channel flow: An introduction. London: Edward Arnold.
Chow, V. T. 1959. Open-channel hydraulics. New York: McGraw-Hill.
Easa, S. M. 2009. “Improved channel cross section with two-segment parabolic sides and horizontal bottom.” J. Irrig. Drain. Eng. 135 (6): 357–365. https://doi.org/10.1061/(ASCE)IR.1943-4774.0000002.
Han, Y-C. 2015. “Horizontal bottomed semi-cubic parabolic channel and best hydraulic section.” Flow Meas. Instrum. 45 (5): 56–61. https://doi.org/10.1016/j.flowmeasinst.2015.04.001.
Han, Y-C., and S. M. Easa. 2016. “Superior cubic channel section and analytical solution of best hydraulic properties.” Flow Meas. Instrum. 50 (4): 169–177. https://doi.org/10.1016/j.flowmeasinst.2016.06.019.
Han, Y-C., and S. M. Easa. 2017. “New and improved three and one-third parabolic channel and most efficient hydraulic section.” Can. J. Civ. Eng. 44 (5): 387–391. https://doi.org/10.1139/cjce-2016-0535.
Han, Y-C., X.-P. Gao, and Z.-H. Xu. 2017. “The best hydraulic section of horizontal-bottomed parabolic channel section.” J. Hydrodyn. 29 (2): 305–313. https://doi.org/10.1016/S1001-6058(16)60740-0.
Jan, C. D., and C. L. Chen. 2013. “Gradually-varied open-channel flow profiles normalized by critical depth and analytically solved by using Gaussian hypergeometric functions.” Hydrol. Earth Syst. Sci. 17 (3): 973–987. https://doi.org/10.5194/hess-17-973-2013.
Liu, J., Z. Wang, and X. Fang. 2010. “Iterative formulas and estimation formulas for computing normal depth of horseshoe cross-section tunnel.” J. Irrig. Drain. Eng. 136 (11): 786–790. https://doi.org/10.1061/(ASCE)IR.1943-4774.0000254.
Loganathan, G. V. 1991. “Optimum design of parabolic canals.” J. Irrig. Drain. Eng. 17 (5): 716–735. https://doi.org/10.1061/(ASCE)0733-9437(1991)117:5(716).
Mironenko, A. P., L. S. Willardson, and A. S. Jenab. 1984. “Parabolic canal design and analysis.” J. Irrig. Drain. Eng. 110 (2): 241–246. https://doi.org/10.1061/(ASCE)0733-9437(1984)110:2(241).
Perez, M. A., C. G. Butler, and X. Fang. 2015. “Compute critical and normal depths of arch and elliptical pipes.” J. Irrig. Drain. Eng. 141 (9): 06015001. https://doi.org/10.1061/(ASCE)IR.1943-4774.0000884.
Strelkoff, T. S., and A. J. Clemmens. 2000. “Approximating wetted perimeter in power-law cross section.” J. Irrig. Drain. Eng. 126 (2): 98–109. https://doi.org/10.1061/(ASCE)0733-9437(2000)126:2(98).
Valiani, V., and V. Caleffi. 2009. “Analytical findings for power law cross-sections: Uniform flow depth.” Adv. Water Resour. 32 (9): 1404–1412. https://doi.org/10.1016/j.advwatres.2009.06.004.
Vatankhah, A. R. 2014. “Semi-regular polygon as the best hydraulic section in practice (generalized solutions).” Flow Meas. Instrum. 38 (4): 67–71. https://doi.org/10.1016/j.flowmeasinst.2014.05.016.
Vatankhah, A. R. 2015. “Normal depth in power-law channels.” J. Hydrol. Eng. 20 (7): 06014008. https://doi.org/10.1061/(ASCE)HE.1943-5584.0001083.
Wang, Z. 1998. “Formula for calculating critical depth of trapezoidal open channel.” J. Hydrol. Eng. 124 (1): 90–91. https://doi.org/10.1061/(ASCE)0733-9429(1998)124:1(90).
Zhang, X., and H. Lü. 2011. “Explicit solution for normal depth in parabolic-shape channel.” Trans. Chin. Soc. Agric. Eng. 28 (21): 121–125.
Information & Authors
Information
Published In
Journal of Irrigation and Drainage Engineering
Volume 144 • Issue 12 • December 2018
Copyright
©2018 American Society of Civil Engineers.
History
Received: May 19, 2017
Accepted: Jul 16, 2018
Published online: Oct 9, 2018
Published in print: Dec 1, 2018
Discussion open until: Mar 9, 2019
Authors
Metrics & Citations
Metrics
Citations
Download citation
If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.