General Formula for Water Surface Profile over Side Weir in the Combined, Trapezoidal and Exponential, Channels

Authors: Abdulrahman Abdulrahman

Abstract:

A side weir is a hydraulic structure set into the side of a channel. This structure is used for water level control in channels, to divert flow from a main channel into a side channel when the water level in the main channel exceeds a specific limit and as storm overflows from urban sewerage system. Computation of water surface over the side weirs is essential to determine the flow rate of the side weir. Analytical solutions for water surface profile along rectangular side weir are available only for the special cases of rectangular and trapezoidal channels considering constant specific energy. In this paper, a rectangular side weir located in a combined (trapezoidal with exponential) channel was considered. Expanding binominal series of integer and fraction powers and the using of reduction formula of cosine function integrals, a general analytical formula was obtained for water surface profile along a side weir in a combined (trapezoidal with exponential) channel. Since triangular, rectangular, trapezoidal and parabolic cross-sections are special cases of the combined cross section, the derived formula, is applicable to triangular, rectangular, trapezoidal cross-sections as analytical solution and semi-analytical solution to parabolic cross-section with maximum relative error smaller than 0.76%. The proposed solution should be a useful engineering tool for the evaluation and design of side weirs in open channel.

Keywords: Analytical solution, combined channel, exponential channel, side weirs, trapezoidal channel, water surface profile.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1340262

References:

[1] Allen J. W. The discharge of water over side weirs in circular pipes. In: Proceedings of the institution of civil engineers, vol. 6. London, England; 1957: p. 270-87.
[2] Collinge VK. The discharge capacity of side weirs. Proceedings of the institution of civil engineers, vol. 6. London, England; 1957: p. 288-304
[3] Ranga Raju, K. G., Prasad, B., and Kupta, S. K., Side weir in rectangular channel. J. Hydraul. Div., 1977; 105(5), 547-554.
[4] Delo E. A., Saul A. J. Charts for the hydraulic design of high side-weirs in storm sewage of overflows. Proceeding of the Institution of Civil Engineers 1988;87 (2):175-193.
[5] Uymaz A., Smith R. H. Design procedure for flow over side weirs. Journal of Irrigation and Drainage Engineering 1991;117(1):79-90.
[6] Uymaz A, Muslu Y. Flow over side weirs in circular channels. Journal of Hydraulic Engineering 1985;111(1):144-160.
[7] Vatankhah A. R. New solution method for water surface profile along a side weir in circular channel. Journal of Irrigation and Drainage Eng. (ASCE) 2012; 138(9-c) 948-954.
[8] Uymaz A. Side weir in triangular channel. Journal of Irrigation and Drainage Eng. (ASCE) 1992; 118(6) 965-970.
[9] Vatankhah A. R. Analytical solution method for water surface profile along a side weir in triangular channel. Flow Measurement and Instrumentation 2012; 23(1) 76-79.
[10] Vatankhah A. R. Water surface profile over side weir in trapezoidal channel. Proceeding of the Institution of Civil Engineers (ICE) Water Management 2012; 165(5) 247-252.
[11] Vatankhah AR. Water surface profile along a side weir in parabolic channel. Flow Measurement and Instrumentation 2013; 32(1) 90-95.
[12] Vatankhah AR. Water Surface profile along a Side weir in a U- Shaped Channel (Analytical Findings). Journal of Hydrologic Eng. (ASCE) 2013; 18(5) 595-602.