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Discharge formula based on brink depth over sharp-crested weirs

Profile image of Ahmed Y . MohammedAhmed Y . MohammedProfile image of Nashwan K. AlomariNashwan K. Alomari

2024, water supply

https://doi.org/10.2166/WS.2024.021
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10 pages

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Abstract

Weirs are among the most essential hydraulic structures for measuring water discharge in open channels. The prediction of water discharge over weirs should be as precise and straightforward measured as feasible. The experimental investigation of flow prediction over varied heights of a conventional rectangular sharp-crested weir was conducted in the present work. The investigation evaluated five ratios of weir height to length, P/b, of 0.33, 0.4, 0.47, 0.53, and 0.6, different water discharges, Q, of up to 17.25 L/s, and different bed slopes, S, between 0.001 and 0.01. The experiment’s findings reveal that a change in the bed slope has no significant effect on the brink depth, hb, for a constant water discharge. However, it influences the head over the weir, h, which is usually measured upstream of the weir location and used to predict water discharge. A simple, accurate formula was developed for predicting water discharge over rectangular sharpcrested weirs depending on the brink depth with mean absolute percent error (MAPE) and root-mean-square error (RMSE) of 1.714% and 0.229, respectively. In addition to having a simple form, the developed formula performs well, is unaffected by the bed slope, and applies to a wide range of h/P values, from 0.158 to 0.945.

Figures (8)
Figure 1 | Schematic diagram of the laboratory flume. This study considered 160 experiments of various models of rectangular sharp-crested weirs with changing bed slopes, wate1 lows, and weir heights to develop an equation for discharge estimation using brink depth (,). The investigation also eval. 1ated several (h/P) values ranging from 0.158 to 0.945. PS bore snr eee EE. Nee See LORY | (peau, eee een En i Ry 3. enn. Ce ee pM) a er eme’ © RN kee y SSE oon premnees MMe geese? © PY COME MO Seo. (ore aE iy [ane ee, Nee CE. Smee Nes, ieee smear e ae eres 3. THEORETICAL METHODOLOGY
Figure 1 | Schematic diagram of the laboratory flume. This study considered 160 experiments of various models of rectangular sharp-crested weirs with changing bed slopes, wate1 lows, and weir heights to develop an equation for discharge estimation using brink depth (,). The investigation also eval. 1ated several (h/P) values ranging from 0.158 to 0.945. PS bore snr eee EE. Nee See LORY | (peau, eee een En i Ry 3. enn. Ce ee pM) a er eme’ © RN kee y SSE oon premnees MMe geese? © PY COME MO Seo. (ore aE iy [ane ee, Nee CE. Smee Nes, ieee smear e ae eres 3. THEORETICAL METHODOLOGY
Figure 2 | Schematic of sharp-crested weir.
Figure 2 | Schematic of sharp-crested weir.
Figure 3 | Relation between critical depth and brink depth over the weir crest. Substituting Equation (2) into Equation (1) results in a relationship between unit discharge (qg) and brink depth (,) as illus- rated in Equation (3), allowing direct estimation of unit discharge (qg) based on hy value:
Figure 3 | Relation between critical depth and brink depth over the weir crest. Substituting Equation (2) into Equation (1) results in a relationship between unit discharge (qg) and brink depth (,) as illus- rated in Equation (3), allowing direct estimation of unit discharge (qg) based on hy value:
Table 1 | Experimental data ranges Toc weir heights of 10 cm and 18 cm were investigated in this study. T ationships between the slope of the bed and the ratio of the water head to the height of the weir (,/P and h/P) are in Figure 5 for water discharge of 13.25 L/s and P of 10cm Figure 5(b)) as the sample. The other relations for different discharges almost The re shown 18 cm fea 5, the value; (energy conservation law) (Chow 1959). That affects the accuracy o bed slope has an inverse effect on the head over his is because increasing the flow velocity with dependent on the water head at a particular distance u ever, as shown in Figs. 4 and 5, changing the bed slope does not a from the weir. Therefore, using i, for the water discharge estimation of over the weir is more accurate t the weir upstream weir loca he increase in the bed slope pstream of f any equation for he weir (4), whic (Figure 5(a)) and water discharge of 1 follow the same trend eads to a decrease in estimating water disc early show the effect of bed slope, S, on the head over the weir, 4, 10 different bed slopes between 0.001 and 0.1 for he water head was measured at two locations (/ and hy). 3.025 L/s and P of s. From Figs. 4 and ion (/) for the constant water discharge the depth of water harge over the weir h does not consider the bed slope. How- ffect the brink depth (H,) as it is located at a zero distance han using h.
Table 1 | Experimental data ranges Toc weir heights of 10 cm and 18 cm were investigated in this study. T ationships between the slope of the bed and the ratio of the water head to the height of the weir (,/P and h/P) are in Figure 5 for water discharge of 13.25 L/s and P of 10cm Figure 5(b)) as the sample. The other relations for different discharges almost The re shown 18 cm fea 5, the value; (energy conservation law) (Chow 1959). That affects the accuracy o bed slope has an inverse effect on the head over his is because increasing the flow velocity with dependent on the water head at a particular distance u ever, as shown in Figs. 4 and 5, changing the bed slope does not a from the weir. Therefore, using i, for the water discharge estimation of over the weir is more accurate t the weir upstream weir loca he increase in the bed slope pstream of f any equation for he weir (4), whic (Figure 5(a)) and water discharge of 1 follow the same trend eads to a decrease in estimating water disc early show the effect of bed slope, S, on the head over the weir, 4, 10 different bed slopes between 0.001 and 0.1 for he water head was measured at two locations (/ and hy). 3.025 L/s and P of s. From Figs. 4 and ion (/) for the constant water discharge the depth of water harge over the weir h does not consider the bed slope. How- ffect the brink depth (H,) as it is located at a zero distance han using h.
Figure 4 | Water surface profile for bed slope, S = 0.001, 0.005, and 0.010 (Q = 17.25 L/s and P= 10 cm).
Figure 4 | Water surface profile for bed slope, S = 0.001, 0.005, and 0.010 (Q = 17.25 L/s and P= 10 cm).
where M; represents the measured values, P; represents the values, 1 is the dataset’s total number. Equation (4) has MAPE and RMSE values of 1.714% and 0.23, respectively. The MAPE and RMSE for Equation (5) are, respectively, 3.12% and 0.39. As depicted in Figure 6, a comparison was made between the proposed formulae (Equations (4)-(6)) with the measured ischarge values. Equations (7) and (8), which are often used to calculate the MAPE (Bowerman et al. 2005, Hanke & Vichern 2005, Kim & Kim 2016) and RMSE (Willmott et al. 2012, Alomari et al. 2018)), were utilized to evaluate the per- ormance of Equation (4). Two-thirds of the datasets were randomly selected to generate an empirical relationship equation etween the brink depth above the weir crest (4,) and the critical depth (y,). The other one-third of the datasets was used to xamine the following equations:
where M; represents the measured values, P; represents the values, 1 is the dataset’s total number. Equation (4) has MAPE and RMSE values of 1.714% and 0.23, respectively. The MAPE and RMSE for Equation (5) are, respectively, 3.12% and 0.39. As depicted in Figure 6, a comparison was made between the proposed formulae (Equations (4)-(6)) with the measured ischarge values. Equations (7) and (8), which are often used to calculate the MAPE (Bowerman et al. 2005, Hanke & Vichern 2005, Kim & Kim 2016) and RMSE (Willmott et al. 2012, Alomari et al. 2018)), were utilized to evaluate the per- ormance of Equation (4). Two-thirds of the datasets were randomly selected to generate an empirical relationship equation etween the brink depth above the weir crest (4,) and the critical depth (y,). The other one-third of the datasets was used to xamine the following equations:
Figure 5 | Effect of bed slope on water depths: (a) water discharge of 13.25 L/s and weir height of 10 cm; (b) water discharge of 13.025 L/s and 18 cm.
Figure 5 | Effect of bed slope on water depths: (a) water discharge of 13.25 L/s and weir height of 10 cm; (b) water discharge of 13.025 L/s and 18 cm.
Figure 6 | Comparison between measured and predicted discharges from new formula (Equation (4)), Equation (6), Aydin et a/. 2011, Nicosie et al. 2023, and Mahtabi & Arvanaghi (2018) equation (Equation (7)).
Figure 6 | Comparison between measured and predicted discharges from new formula (Equation (4)), Equation (6), Aydin et a/. 2011, Nicosie et al. 2023, and Mahtabi & Arvanaghi (2018) equation (Equation (7)).

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kiumars badr

KSCE Journal of Civil Engineering, 2014

One of the important problems which designers and engineers are faced with is proper measurement of water flow rates in open channels and rivers. To solve this problem, the water discharge to be precisely measured in water conveyance systems such as off takes and distribution ditches. The common procedure to meet this requirement is achieved by using water-measuring structures such as flumes, orifices, weirs and current meters. Among the mentioned structure, engineers prefers the weirs because of their simple structure and economical features. The broad-crested rectangular weirs are very common which are widely used in water ways and canals. In this study, the effect of steeping in the upstream side of rectangular broad-crested weirs over discharge coefficient and flow characteristics is investigated. Five weirs with 15, 30, 45, 60 and 90 weir angles were developed and flow discharge coefficient, negative velocity over the weir crest edge and water surface profiles along the weir crest evaluated in laboratory hydraulic flume. The obtained results showed by decreasing the slope of the upstream side, discharge efficiency and the discharge capacity of the weir are increased subsequently. Consideration of experimental data it is obvious that by using the proposed weir with slope of 15 degree in comparison with standard weir, the efficiency is increased up to 19.17%. It is noteworthy that by reducing the slope, return current on the crest is not observed.

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Mathematical modeling of flow discharge over compound sharp-crested weirs
reza zahiri

Journal of Hydro-environment Research, 2014

Water measurements play a pivotal role in hydraulic and environmental management of river systems, irrigation and drainage networks, and sewer conduits. Due to effective limitations of simple sharp crested weirs, recently compound sharp crested weirs have attracted great attention of civil engineers. With this type of weir, flow discharge is measured with a reasonable sensitivity over a wide range of flow rates. In this paper, the Shiono and Knight (SKM) quasi 2-D mathematical model has been numerically solved for discharge prediction of compound weirs, based on the assumption of flow similarity between compound sharp crested weirs and compound open channels. The modeled results are compared with the experimental data and the conventional method (linear combination of theoretical equations of simple weirs). The comparisons revealed that the proposed method shows improved discharge calculation within 3.8 percent absolute mean error while the traditional method has more than 12.5 percent error.

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Numerical investigation of the effect of geometric parameters on discharge coefficients for broad-crested weirs with sloped upstream and downstream faces
Farzin Salmasi

Applied Water Science

Weirs are structures that are important for measuring flow and controlling water levels. Research has shown that the discharge coefficient is not constant and depends on the crest length, the height of the weir, the upstream head, and the upstream and downstream slopes. In this study, the effect of these parameters on the discharge coefficient (Cd) is investigated by numerical simulation. The current study present numerical simulation using the ANSYS FLUENT software. The total number of simulations is 432 which includes: 4 upstream slopes, 4 downstream slopes, 3 weir heights, 3 upstream heads (h1) and 3 weir crest lengths. It was found that the downstream face slope has little effect on Cd. For 0.1

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Discharge coefficient of a rectangular sharp-edged broad-crested weir
Zbyněk Zachoval

Journal of Hydrology and Hydromechanics, 2014

This paper is concerned with the determination of the relationship for the calculation of the discharge coefficient at free overflow over a rectangular sharp-edged broad-crested weir without lateral contraction. The determination was made on the basis of new measurement in a range of the relative thickness of the weir from 0.12 to 0.30 and newly in a large range of relative height of the weir extremely from 0.24 to 6.8 which greatly expands the application possibilities of low weirs. In addition, the effects of friction and surface tension on the value of the discharge coefficient were evaluated as well as the effect of the relative thickness of the weir. The new equation for discharge coefficient, expressed using the relative height of the weir, was subjected to verification made by an independent laboratory which confirmed its accuracy.

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