Experimental and theoretical study of sharp-crested weir located at the end of horizontal pipes

Weir is one of the most applicable instruments for flow control and measurement in open channels. Accurate measurement of discharge in irrigation channels is vital for irrigation management. Weirs are designed as control sections for providing a unique relationship between the discharge and water head. In current study, theoretical and experimental characteristics of flow through a weir located in a horizontal circular channel were investigated. Choosing of this structure is due to lack of adequate studies on governing hydraulic equations of this kind of channels and its practical characteristics. In this research, discharge through the weir located at the end of a horizontal circular channel was developed from a theoretical viewpoint, and then calibration of this theoretical discharge equation was done via the discharge coefficient. The experimental data obtained in the study were also used to develop an empirical discharge coefficient relationship.
Theoretical discharge equation:The discharge equation of the circular weir ends up incomplete elliptic integrals of the first and second kinds, which are not very suitable for practical purposes due to their complexity. The theoretical discharge over the weir, Qt, can be expressed as:in which F(a,b) is approximated using the method of undetermined coefficients as:where g is the gravitational acceleration, h is the flow depth above the weir crest, D is the channel diameter, , , , and P is the weir height.
Introducing the discharge coefficient, Cd, yields the following equation for actual discharge, Q, of the weir.
Experimental setup and discharge coefficient equation: Determination of discharge coefficient, Cd, needs measuring the actual discharge (experimental or field data). The experiments were performed at the hydraulic laboratory of the Irrigation and Reclamation Engineering Department, University of Tehran. In order to collect experimental data, the weir was installed at the end of two horizontal circular channels with nominal diameters of 200 mm and 300 mm, and numerous measurements of different parameters that may have effect on discharge coefficient were carried out. An opening was provided on top of the pipes for flow head measurements by the point gauge. Water head over the weir crest was measured at a distance of 0.45 m upstream of the weir section. The weir plate was made of 10-mm-thick PVC sheet with a crest thickness of about 1 mm, and the downstream edge bevelled to a 45° angle (sharp-crested weir). For water heads over the weir crest smaller than 1.5 cm, in some cases clinging flow did occur and thus there was no atmospheric pressure under the lower nappe. Though, for heads greater than 1.5 cm a clear lower nappe profile with atmospheric pressure underneath was formed. Effects of surface tension and viscosity forces on the discharge coefficient, Cd, became pronounced at low water heads. Viscous and surface tension effects on discharge coefficient are neglected in this research. This is owing to operating range of the weir (minimum water head of 2 cm), which forbids surface tension effects and eliminates viscous effects. The experimental coefficient of discharge is computed for each data set collected in this study for known values of D, h, P and Q. A total of 350 tests were conducted to collect the data in the discharge ranges of 0.5 and 45 l/s. Suitable empirical equations for discharge coefficient were proposed using the 70% of the experimental data, and remaining data (30%) were used for validation procedure.
Proposed empirical equation for discharge coefficient: Using the curve-fitting technique, the following equation is obtained for CdThis empirical equation is valid for 0.095≤h/P≤1.8, 0.27≤P/D≤0.73 and 0.05≤h/D≤0.56. The assumptions made in deriving the above empirical equation of discharge coefficient are that the channel slope is horizontal, the flow is free and surface and viscous tension effects on Cd are negligible. A comparison of the experimental discharge coefficient with computed ones using the proposed empirical equation for all of the experimental data, the calibration data set and the validation data set showed that the computed discharge coefficient values are well within ±5% of the observed data for all three data sets. The average relative error of computed discharge coefficient using the proposed empirical equation for discharge coefficient Cd compared with the observed values is less than 5% for 95% of all experimental data. The results of this study reveals that a weir with low flow height is more sensitive to water flow depth, and weirs with more height are less sensitive to upstream flow head variation and thus are more reliable for flow measurement.