General Explicit Solutions of Most Economic Sections and Applications for Trapezoidal and Parabolic Channels

  • Yan-Cheng Han (韩延成)Email author
  • Said M. Easa
  • Ping-ping Chu (初萍萍)
  • Wei Tang (唐伟)
  • Meng-yuan Liang (梁梦媛)
  • Xue-Ping Gao (高学平)


A channel section that has minimum construction cost is known as the most economic section. Such a section has important implications for economic efficiency. However, the most economic section is a complex optimization model with nonlinear objective function and constraints that is difficult to use by ordinary engineers. A general simple formula for the most economic section has not been attempted. In this paper, the general differential equation for the most economic section is derived using Lagrange multiplier optimization method. A simple method to solve the most economic section is proposed that converted the optimization model into a general equation for the most economic section of any shape. By solving this equation, the dimensions of the most economic section are directly obtained. To illustrate, the direct formula for trapezoidal section is derived. To aid application in practice, a simple explicit iterative formula for trapezoidal sections is presented. The direct and explicit iterative formulas were validated. The proposed method is superior to the classical optimization method and as such represent a valuable tool for open channel design. To illustrate the versatility of the presented method, a direct formula for the parabolic section was also derived.

Key words

Most economic section channel parabolic explicit hydraulic section 


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  1. [1]
    Shao Y., Zuo L., Zhang Q.–L. Optimization design of channel concrete lining plate in South–to–North Water Diversion Middle Route Project [J], Yellow River, 2011, 33(10): 81–85(in Chinese).Google Scholar
  2. [2]
    Li W. L., Yan F., Li Y. T. Optimal design method of trapezoidal channel based on slope stability and minimum cost [J]. China Rural Water and Hydropower. 2011, 3: 140–142(in Chinese).Google Scholar
  3. [3]
    Bhattacharjya, R. K., SatishMysore, G. Optimal design of a stable trapezoidal channel section using hybrid optimization techniques [J]. Journal of Irrigation and Drainage Engineering. 2007, 133(4): 323–329.CrossRefGoogle Scholar
  4. [4]
    Jain, A., Bhattacharjya, R. K., Srinivasulu, S. Optimal design of composite channels using genetic algorithm [J]. Journal of Irrigation and Drainage Engineering. 2004, 30(4): 286–295.CrossRefGoogle Scholar
  5. [5]
    Chahar, B. R. Optimal Design of Parabolic Canal Section [J]. Journal of Irrigation and Drainage Engineering. 2005, 131(6): 546–554.CrossRefGoogle Scholar
  6. [6]
    Huai W.–X., Chen Zh.–B. Mathematical Model for The Flow With Submerged and Emerged Rigid Vegetation [J], Journal of Hydrodynamics, 2009, 21(5): 722–729.CrossRefGoogle Scholar
  7. [7]
    Huai W.–X., Gao M., Zeng Y.–H., Li D. Two–Dimensional Analytical Solution for Compound Channel Flows with Vegetated Floodplains [J], Applied Mathematics and Mechanics, 2009, 30(9), 1121–1130Google Scholar
  8. [8]
    Das J. A. Optimal design of channel having horizontal bottom and parabolic sides [J]. Journal of Irrigation and Drainage Engineering. 2007, 133(2): 192–197.CrossRefGoogle Scholar
  9. [9]
    Easa S. M. Improved channel cross section with two–segment parabolic sides and horizontal bottom [J]. Journal of Irrigation and Drainage Engineering. 2009, 135(3), 357–365.Google Scholar
  10. [10]
    Easa S. M., Vatankhah A. R. New Open Channel with Elliptic Sides and Horizontal Bottom [J], KSCE Journal of Civil Engineering, 2014, 18(4): 1197–1204.Google Scholar
  11. [11]
    Easa S.M. Versatile general elliptic open channel cross section [J], KSCE Journal of Civil Engineering, 2016, 20(4): 1572–1581.CrossRefGoogle Scholar
  12. [12]
    Bhattacharjya R. K. Optimal design of open channel section incorporating critical flow condition [J], Journal of Irrigation and Drainage Engineering, 2006, 132(5): 513–518.CrossRefGoogle Scholar
  13. [13]
    Vatankhah A.R. Semi–regular polygon as the best hydraulic section in practice (generalized solutions) [J], Flow Measurement and Instrumentation, 2014, 38: 67–71CrossRefGoogle Scholar
  14. [14]
    Han Y–C. Horizontal bottomed semi–cubic parabolic channel and best hydraulic section [J], Flow Measurement and Instrumentation, 2015, 45(2015): 56–61.CrossRefGoogle Scholar
  15. [15]
    Han Y–C., Easa S. M. Superior cubic channel section and analytical solution of best hydraulic properties [J], Flow Measurement and Instrumentation, 2016, 50: 169–177.Google Scholar
  16. [16]
    Han Y–C., Xu Z–H. Easa S.M., Wang S., Fu L. Optimal hydraulic section of ice–covered open trapezoidal channel [J], Journal of Cold Regions Engineering, 2017, 31 (3): 06017001.Google Scholar
  17. [17]
    Han Y–C., Easa S. M. New and improved three and one–third parabolic channel and most efficient hydraulic section [J], Canadian Journal of Civil Engineering, 2017, 44(5): 387–391.CrossRefGoogle Scholar
  18. [18]
    Han Y–C., Xu Z–H, Gao X–P., Easa S. M. Design of two and a half parabola–shaped canal and its effect in improving hydraulic property [J], Transactions of the Chinese Society of Agricultural Engineering, 2017, 33(4): 131–136(in Chinese).Google Scholar
  19. [19]
    Das A. Optimal channel cross section with composite roughness [J], Journal of Irrigation and Drainage Engineering, 2000, 126(1): 68–72.CrossRefGoogle Scholar
  20. [20]
    Han Y, C. Gao X. P., Xu Z. H. The best hydraulic section of horizontal–bottomed parabolic channel section [J], Journal of Hydrodynamics, 2017, 29(2): 305–313.CrossRefGoogle Scholar
  21. [21]
    Maleki S. F., Khan A. A. Effect of channel shape on selection of time marching scheme in the discontinuous Galerkin method for 1–D open channel flow [J], Journal of Hydrodynamics, 2015, 27(3): 413–426.CrossRefGoogle Scholar
  22. [22]
    Mohanty P. K., Khatua K. K. Estimation of discharge and its distribution in compound channels [J], Journal of Hydrodynamics, 2014, 26(1): 144–154.CrossRefGoogle Scholar
  23. [23]
    Han Y–C., Chu P. Liang M., Tang W., Gao X. Explicit iterative algorithm of normal water depth for trapezoid and parabolic open channels under ice cover [J], Transactions of the Chinese Society of Agricultural Engineering, 2018, 34(14): 101–106 (in Chinese).Google Scholar
  24. [24]
    Han Y. C., Easa S. M. Exact Solution of Optimum Hydraulic Power–Law Section with General Exponent Parameter [J], Journal of Irrigation and Drainage Engineerin(ASCE), 2018, 144(12): 04018035CrossRefGoogle Scholar
  25. [25]
    Bertsekas D. P. Nonlinear programming [M]. 2nd Edition, Athena Scientific, Belmont, 1999.zbMATHGoogle Scholar
  26. [26]
    Gerald C. F., Wheatley P. O. Applied numerical analysis [M]. 7th Edition Addison–Wesley, New Jersey, 2003.Google Scholar
  27. [27]
    Vatankhah A. R. Direct solutions for normal depth in parabolic and rectangular open channels using asymptotic matching technique [J], Flow Measurement and Instrumentation, 2015, 46: 66–71.CrossRefGoogle Scholar
  28. [28]
    Hu H., Huang J., Qian Z., Huai W., Yu G. Hydraulic analysis of parabolic flume for flow measurement [J], Flow Measurement and Instrumentation, 2014, 37: 54–64.CrossRefGoogle Scholar
  29. [29]
    Zhou Ch., Numerical algorithm for science and Engineering (Visual Basic) [M]. Tsinghua University Press, Beijing, 2002(in Chinese).Google Scholar

Copyright information

© China Ship Scientific Research Center 2018

Authors and Affiliations

  • Yan-Cheng Han (韩延成)
    • 1
    Email author
  • Said M. Easa
    • 2
  • Ping-ping Chu (初萍萍)
    • 1
  • Wei Tang (唐伟)
    • 1
  • Meng-yuan Liang (梁梦媛)
    • 1
  • Xue-Ping Gao (高学平)
    • 3
  1. 1.Department of Hydraulic Engineering, School of Water Conservancy and EnvironmentUniversity of JinanJinanChina
  2. 2.Department of Civil EngineeringRyerson UniversityTorontoCanada
  3. 3.State Key Laboratory of Hydraulic Engineering Simulation and SafetyTianjin UniversityTianjinChina

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