Modified theoretical stage-discharge relation for circular sharp-crested weirs

Rasool GHOBADIAN*, Ensiyeh MERATIFASHI

Department of Water Engineering, Razi University, Kermanshah 6715685438, Iran

Modified theoretical stage-discharge relation for circular sharp-crested weirs

Rasool GHOBADIAN*, Ensiyeh MERATIFASHI

Department of Water Engineering, Razi University, Kermanshah 6715685438, Iran

A circular sharp-crested weir is a circular control section used for measuring flow in open channels, reservoirs, and tanks. As flow measuring devices in open channels, these weirs are placed perpendicular to the sides and bottoms of straight-approach channels. Considering the complex patterns of flow passing over circular sharp-crested weirs, an equation having experimental correlation coefficients was used to extract a stage-discharge relation for weirs. Assuming the occurrence of critical flow over the weir crest, a theoretical stage-discharge relation was obtained in this study by solving two extracted non-linear equations. To study the precision of the theoretical stage-discharge relation, 58 experiments were performed on six circular weirs with different diameters and crest heights in a 30 cm-wide flume. The results show that, for each stage above the weirs, the theoretically calculated discharge is less than the measured discharge, and this difference increases with the stage. Finally, the theoretical stage-discharge relation was modified by exerting a correction coefficient which is a function of the ratio of the upstream flow depth to the weir crest height. The results show that the modified stage-discharge relation is in good agreement with the measured results.

circular weir; stage-discharge relation; analytical method

1 Introduction

Regardless of their performance, properties, ages, or conditions, it should be noted that weirs are engineering structures that have to function in difficult conditions (Rickard et al. 2003). As one of the main components of dam-buildings and water projects, weirs are important structures built for various purposes. Two of the most important functions of weirs are measurement of water discharge and adjustment of the water level in primary and secondary channels. Considering the complex work they do, weirs should be strong, reliable, and highly efficient so that they can readily be put to use. Broad-crested, sharp-crested, cylindrical-crested, and ogee weirs are the most common types of weirs. The advantages of circular sharp-crested weirs are that the crest can be turned and beveled with precision in a lathe, and more particularly that they do not have to be leveled (Bos 1989).

According to different standards, weirs can be classified into different categories. For example, weirs are of the following types: primary, ancillary, or emergent based on theirperformance, and overflow, chute, or tunnel based on structural components. With consideration of the type of entrance weirs, they are classified as siphon, lateral, orifice, and morning-glory weirs.

Although much research has been done on sharp-crested weirs, there are few studies that have focused on circular sharp-crested weirs. A circular control section located in a vertical thin plate, which is placed at a right angle to the sides and bottom of a straight-approach channel, is defined as a circular thin plate weir. Circular sharp-crested weirs, in practice, are fully contracted so that the bed and sides of the approach channel can be sufficiently remote from the control section to have no influence on the development of the nappe (Bos 1989). Also, a circular orifice installed at the end of the discharge pipe would be running partly for most of time and became a circular weir (Steven 1957).

Greve (1924) analyzed sharp-edged circular weirs and showed that if the cross-section upstream of the weir is large, the depth of water nearly reaches the energy head. He developed an empirical equation between discharge and energy head. Greve (1932) investigated the characteristics of flow through circular, parabolic, and triangular weirs with diameters ranging from 0.076 m to 0.76 m. Panuzio and Ramponi (1936) (reported in Lencastre (1961)) investigated circular sharp-crested weirs and developed a different equation for the overflow with a discharge coefficientμbeing a function of the relative depth. Staus (1931) determined experimental values for a discharge coefficient, which is a function of the filling ratio, of circular sharp-crested weirs with different weir diameters. Stevens (1957) derived a function relationship between the theoretical discharge and water head in terms of the complete elliptic integrals of the first and second kinds. This complex equation is not very suitable for practical purposes. Stevens also tabulated his solution. Rajarathnam and Muralidhar (1964) investigated the end depth in a cylindrical channel. They proposed a function between discharge and the water depth at the end of the channel. Vatankhah (2010), using experimental data, presented a theoretical discharge equation and a suitable discharge coefficient equation for a circular sharp-crest weir. Thus, actual discharge can be computed via his proposed equations. With a theoretical formula, the relationship between discharge and the wetted area for free overflow in a semi-circular channel was developed by Qu et al. (2010). Their results provide a basis for circular weir development.

Although a handful of simple and accurate equations in the technical literature can be used to analytically predict the stage-discharge relation for circular weirs, due to the complex patterns of the flow passing over circular sharp-crested weirs, the stage-discharge relation for these weirs cannot be estimated merely analytically. In order to extract a stage-discharge relation for weirs, it is necessary to apply an equation having experimental correction coefficients. Assuming the occurrence of critical flow over a weir crest, in this study, a theoretical stage-discharge relation was obtained by solving two non-linear equations. To modify the relation, an experimental correction coefficient, which was a function of the ratio of the flow depth of the upstream canal to the height of the weir crest and was obtained from experimental results, was applied.

2 Materials and methods

2.1 Governing equations

For a circular sharp-crested weir, the discharge is given by Panuzio and Ramponi (1936) (reported in Lencastre (1961)) as follows:

whereDis the diameter (dm),Qis the discharge (dm3/s),?is a function of the water level, andμis the discharge coefficient, calculated from Eq. (2), in whichhis the water head:

Panuzio and Ramponi (1936) obtained another equation for circular weirs with the distance between the lowest points of weirs and the bottom of the canal ranging from 0.4 m to 0.8 m:

whereSis the flow area between the crest and the free surface related to the water headh, andgis the gravitational acceleration.μwas obtained from the following formula:

whereS′ is the canal flow area.

In this study, assuming that the flow depth reached the critical depth while flowing downward over the weir, for circular channels, the values of flow discharge and total head above the weir crest were calculated from Eqs. (5) and (6), respectively (Chow 1959):

whereHis the total head upstream of the weir,Acis the flow area between the weir crest and the free surface specified to a critical depthyc,Tcis the width of the water surface over the weir crest specified to the critical depth, andθcis the central angle of the circular weir corresponding to the critical depth.

A theoretical stage-discharge relation is obtained by substituting hypothetical values ofθcin Eqs. (5) and (6).

2.2 Experimental setup

To examine the precision of the theoretical stage-discharge relation, this study made six circular weirs with different diameters (D= 15 cm, 20 cm, and 25 cm) and different crest heights (P= 20 cm and 25 cm). The weirs were sharp-crested and made of plexiglas materials.

In the hydraulic laboratory of the Department of Water Engineering in Razi University, 58 experimental tests were performed on these weirs at different discharge values in a 9 m-long, 0.30 m-wide, and 0.55 m-high flume. Weir characteristics and flow conditions in the experimental tests are provided in Table 1.

Table 1 Weir characteristics and flow conditions in experimental tests

The height of the water surface above weirs was measured with a point gauge device with a precision of 0.1 mm. The flume discharge was measured after drainage of water inside a cubic metal tank equipped with a triangular weir with a notch angle of 53°. The pumping system supplied a maximum discharge of 15 L/s. Fig. 1 shows the experimental setup.

Fig. 1 Plan view of experimental setup (Unit: m)

3 Results and discussion

The stage-discharge relations calculated by Eqs. (5) and (6), along with those measured using weirs with different diameters (D) and different crest heights (P), are illustrated in Fig. 2. As seen in the figure, for each upstream stage of the weir, the theoretically calculated discharge is less than the measured value, and this difference increases with the stage.

Fig. 2 Calculated (before modification) and measured stage-discharge relations for weirs with different diameters (D) and crest heights (P)

To modify the calculated stage-discharge relation (Eqs. (5) and (6)), a correction coefficient was defined as, whereQcandQmwere calculated and measured discharge for the same upstream stage of the weir, respectively.

Using genetic programming, Eq. (7) can be obtained to calculate the correction coefficient. The coefficient of determination (R2) of Eq. (7) is 0.889 3. The application limits for Eq. (7) werey1Pbetween 1 and 1.7 and the maximum flow discharge was equal to 15 L/s.

For each of the 58 tests performed, the values ofCare plotted against the ratios of the upstream flow depth to the weir crest height (y1P) in Fig. 3.

Fig. 3 Changes of correction coefficient (C) against ratio of upstream flow depth to weir crest heightFollowing calculation ofC, a calculated discharge value was obtained from the following equation, which is a modified form of Eq. (5):

Measured discharge values are plotted against modified calculated values in Fig. 4, indicating a high precision of Eq. (7) in determining the correction coefficientC.

Additional evidence of the precision of Eq. (7) in determining the correction coefficient is the comparison of the measured stage-discharge relation with the calculated one presented in Fig. 5.

In order to compare the results from the present study with those of earlier research, the discharge values measured and calculated using Eq. (8) and the equation presented by Panuzio and Ramponi (Eq. (3)), respectively, are shown in Fig. 6, for weirs withD= 0.25 m andP= 0.25 m, andD= 0.15 m andP= 0.15 m.

Fig. 4 Measured discharge values vs. values calculated with Eq. (8)

Fig. 5 Stage-discharge relations measured and calculated with Eqs. (6) and (8) for weir withP= 0.2 m andD= 0.2 m

Fig. 6 Comparison of measured and calculated discharge values using Eq. (8) and Eq. (3)

As observed, the discharge values calculated from Eq. (3) are always slightly lower than measured values, while Eq. (8) presented in this study estimates the discharge values with high precision.

4 Conclusions

A new method for determination of the stage-discharge relation for circular sharp-crested weirs is outlined in this study. Assuming the occurrence of critical flow over the weir crest, a theoretical stage-discharge relation was obtained in this study through solutions of two extracted non-linear equations. The calculated discharge, using the proposed relationship, is less than the measured discharge, and this difference increases with the stage. Using the data from 58 experiments performed on six circular weirs with different diameters and crest heights in a 30 cm-wide flume, a correction coefficient was extracted, which is a function of the ratio of the upstream flow depth to the weir crest height. The modified stage-discharge relation, after application of the correction coefficient, shows good agreement with the data sets derived from experiments.

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*Corresponding author (e-mail:r_ghobadian@razi.ac.ir)

Received Dec. 11, 2010; accepted Feb. 24, 2012