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

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1 Water Siene and Engineering, 2012, 5(1): doi: /j.issn Modified theoretial stage-disharge relation for irular sharp-rested weirs Rasool GHOBADIAN*, Ensiyeh MERATIFASHI Department of Water Engineering, Razi University, Kermanshah , Iran Abstrat: A irular sharp-rested weir is a irular ontrol setion used for measuring flow in open hannels, reservoirs, and tanks. As flow measuring devies in open hannels, these weirs are plaed perpendiular to the sides and bottoms of straight-approah hannels. Considering the omplex patterns of flow passing over irular sharp-rested weirs, an equation having experimental orrelation oeffiients was used to extrat a stage-disharge relation for weirs. Assuming the ourrene of ritial flow over the weir rest, a theoretial stage-disharge relation was obtained in this study by solving two extrated non-linear equations. To study the preision of the theoretial stage-disharge relation, 58 experiments were performed on six irular weirs with different diameters and rest heights in a 30 m-wide flume. The results show that, for eah stage above the weirs, the theoretially alulated disharge is less than the measured disharge, and this differene inreases with the stage. Finally, the theoretial stage-disharge relation was modified by exerting a orretion oeffiient whih is a funtion of the ratio of the upstream flow depth to the weir rest height. The results show that the modified stage-disharge relation is in good agreement with the measured results. Key words: irular weir; stage-disharge relation; analytial method 1 Introdution Regardless of their performane, properties, ages, or onditions, it should be noted that weirs are engineering strutures that have to funtion in diffiult onditions (Rikard et al. 2003). As one of the main omponents of dam-buildings and water projets, weirs are important strutures built for various purposes. Two of the most important funtions of weirs are measurement of water disharge and adjustment of the water level in primary and seondary hannels. Considering the omplex work they do, weirs should be strong, reliable, and highly effiient so that they an readily be put to use. Broad-rested, sharp-rested, ylindrial-rested, and ogee weirs are the most ommon types of weirs. The advantages of irular sharp-rested weirs are that the rest an be turned and beveled with preision in a lathe, and more partiularly that they do not have to be leveled (Bos 1989). Aording to different standards, weirs an be lassified into different ategories. For example, weirs are of the following types: primary, anillary, or emergent based on their *Corresponding author ( [email protected]) Reeived De. 11, 2010; aepted Feb. 24, 2012

2 performane, and overflow, hute, or tunnel based on strutural omponents. With onsideration of the type of entrane weirs, they are lassified as siphon, lateral, orifie, and morning-glory weirs. Although muh researh has been done on sharp-rested weirs, there are few studies that have foused on irular sharp-rested weirs. A irular ontrol setion loated in a vertial thin plate, whih is plaed at a right angle to the sides and bottom of a straight-approah hannel, is defined as a irular thin plate weir. Cirular sharp-rested weirs, in pratie, are fully ontrated so that the bed and sides of the approah hannel an be suffiiently remote from the ontrol setion to have no influene on the development of the nappe (Bos 1989). Also, a irular orifie installed at the end of the disharge pipe would be running partly for most of time and beame a irular weir (Steven 1957). Greve (1924) analyzed sharp-edged irular weirs and showed that if the ross-setion upstream of the weir is large, the depth of water nearly reahes the energy head. He developed an empirial equation between disharge and energy head. Greve (1932) investigated the harateristis of flow through irular, paraboli, and triangular weirs with diameters ranging from m to 0.76 m. Panuzio and Ramponi (1936) (reported in Lenastre (1961)) investigated irular sharp-rested weirs and developed a different equation for the overflow with a disharge oeffiient μ being a funtion of the relative depth. Staus (1931) determined experimental values for a disharge oeffiient, whih is a funtion of the filling ratio, of irular sharp-rested weirs with different weir diameters. Stevens (1957) derived a funtion relationship between the theoretial disharge and water head in terms of the omplete ellipti integrals of the first and seond kinds. This omplex equation is not very suitable for pratial purposes. Stevens also tabulated his solution. Rajarathnam and Muralidhar (1964) investigated the end depth in a ylindrial hannel. They proposed a funtion between disharge and the water depth at the end of the hannel. Vatankhah (2010), using experimental data, presented a theoretial disharge equation and a suitable disharge oeffiient equation for a irular sharp-rest weir. Thus, atual disharge an be omputed via his proposed equations. With a theoretial formula, the relationship between disharge and the wetted area for free overflow in a semi-irular hannel was developed by Qu et al. (2010). Their results provide a basis for irular weir development. Although a handful of simple and aurate equations in the tehnial literature an be used to analytially predit the stage-disharge relation for irular weirs, due to the omplex patterns of the flow passing over irular sharp-rested weirs, the stage-disharge relation for these weirs annot be estimated merely analytially. In order to extrat a stage-disharge relation for weirs, it is neessary to apply an equation having experimental orretion oeffiients. Assuming the ourrene of ritial flow over a weir rest, in this study, a theoretial stage-disharge relation was obtained by solving two non-linear equations. To modify the relation, an experimental orretion oeffiient, whih was a funtion of the ratio of the flow depth of the upstream anal to the height of the weir rest and was obtained from experimental results, was applied. Rasool GHOBADIAN et al. Water Siene and Engineering, Mar. 2012, Vol. 5, No. 1,

3 2 Materials and methods 2.1 Governing equations For a irular sharp-rested weir, the disharge is given by Panuzio and Ramponi (1936) (reported in Lenastre (1961)) as follows: 52 Q = μϕd (1) where D is the diameter (dm), Q is the disharge (dm 3 /s), ϕ is a funtion of the water level, and μ is the disharge oeffiient, alulated from Eq. (2), in whih h is the water head: μ = D h h + D (2) Panuzio and Ramponi (1936) obtained another equation for irular weirs with the distane between the lowest points of weirs and the bottom of the anal ranging from 0.4 m to 0.8 m: Q = μ S 2gh (3) where S is the flow area between the rest and the free surfae related to the water head h, and g is the gravitational aeleration. μ was obtained from the following formula: 2 D S μ = h (4) S where S is the anal flow area. In this study, assuming that the flow depth reahed the ritial depth while flowing downward over the weir, for irular hannels, the values of flow disharge and total head above the weir rest were alulated from Eqs. (5) and (6), respetively (Chow 1959): D 12 g ( θ sinθ) 3 ga 8 Q = = (5) T D sin ( θ 2) A D osθ 1 sin 1 D θ θ H = y + = + (6) 2T sin( θ 2) where H is the total head upstream of the weir, A is the flow area between the weir rest and the free surfae speified to a ritial depth y, T is the width of the water surfae over the weir rest speified to the ritial depth, and θ is the entral angle of the irular weir orresponding to the ritial depth. A theoretial stage-disharge relation is obtained by substituting hypothetial values of θ in Eqs. (5) and (6). 2.2 Experimental setup To examine the preision of the theoretial stage-disharge relation, this study made six irular weirs with different diameters (D = 15 m, 20 m, and 25 m) and different rest heights (P = 20 m and 25 m). The weirs were sharp-rested and made of plexiglas materials. 28 Rasool GHOBADIAN et al. Water Siene and Engineering, Mar. 2012, Vol. 5, No. 1, 26-33

4 In the hydrauli laboratory of the Department of Water Engineering in Razi University, 58 experimental tests were performed on these weirs at different disharge values in a 9 m-long, 0.30 m-wide, and 0.55 m-high flume. Weir harateristis and flow onditions in the experimental tests are provided in Table 1. No. Upstream flow depth Table 1 Weir harateristis and flow onditions in experimental tests Disharge (L/s) Weir diameter Crest height No. Upstream flow depth Disharge (L/s) Weir diameter Crest height The height of the water surfae above weirs was measured with a point gauge devie with a preision of 0.1 mm. The flume disharge was measured after drainage of water inside a ubi metal tank equipped with a triangular weir with a noth angle of 53. The pumping system supplied a maximum disharge of 15 L/s. Fig. 1 shows the experimental setup. Rasool GHOBADIAN et al. Water Siene and Engineering, Mar. 2012, Vol. 5, No. 1,

5 3 Results and disussion Fig. 1 Plan view of experimental setup (Unit: m) The stage-disharge relations alulated by Eqs. (5) and (6), along with those measured using weirs with different diameters (D) and different rest heights (P), are illustrated in Fig. 2. As seen in the figure, for eah upstream stage of the weir, the theoretially alulated disharge is less than the measured value, and this differene inreases with the stage. Fig. 2 Calulated (before modifiation) and measured stage-disharge relations for weirs with different diameters (D) and rest heights (P) 30 Rasool GHOBADIAN et al. Water Siene and Engineering, Mar. 2012, Vol. 5, No. 1, 26-33

6 To modify the alulated stage-disharge relation (Eqs. (5) and (6)), a orretion oeffiient was defined as C = Qm Q, where Q and Q m were alulated and measured disharge for the same upstream stage of the weir, respetively. Using geneti programming, Eq. (7) an be obtained to alulate the orretion oeffiient. The oeffiient of determination (R 2 ) of Eq. (7) is The appliation limits for Eq. (7) were y1 P between 1 and 1.7 and the maximum flow disharge was equal to 15 L/s. y p y 1 C = os tan (7) y y1 p 1 1 p p For eah of the 58 tests performed, the values of C are plotted against the ratios of the upstream flow depth to the weir rest height ( y 1 P) in Fig. 3. Fig. 3 Changes of orretion oeffiient (C) against ratio of upstream flow depth to weir rest height Following alulation of C, a alulated disharge value was obtained from the following equation, whih is a modified form of Eq. (5): 3 2 D g ( θ sinθ) 8 Q= C Dsin ( θ 2) Measured disharge values are plotted against modified alulated values in Fig. 4, indiating a high preision of Eq. (7) in determining the orretion oeffiient C. Additional evidene of the preision of Eq. (7) in determining the orretion oeffiient is the omparison of the measured stage-disharge relation with the alulated one presented in Fig. 5. In order to ompare the results from the present study with those of earlier researh, the disharge values measured and alulated using Eq. (8) and the equation presented by Panuzio and Ramponi (Eq. (3)), respetively, are shown in Fig. 6, for weirs with D = 0.25 m and P = 0.25 m, and D = 0.15 m and P = 0.15 m. 12 (8) Rasool GHOBADIAN et al. Water Siene and Engineering, Mar. 2012, Vol. 5, No. 1,

7 Fig. 4 Measured disharge values vs. values alulated with Eq. (8) Fig. 5 Stage-disharge relations measured and alulated with Eqs. (6) and (8) for weir with P = 0.2 m and D = 0.2 m Fig. 6 Comparison of measured and alulated disharge values using Eq. (8) and Eq. (3) As observed, the disharge values alulated from Eq. (3) are always slightly lower than measured values, while Eq. (8) presented in this study estimates the disharge values with high preision. 4 Conlusions A new method for determination of the stage-disharge relation for irular sharp-rested weirs is outlined in this study. Assuming the ourrene of ritial flow over the weir rest, a theoretial stage-disharge relation was obtained in this study through solutions of two extrated non-linear equations. The alulated disharge, using the proposed relationship, is less than the measured disharge, and this differene inreases with the stage. Using the data from 58 experiments performed on six irular weirs with different diameters and rest heights in a 30 m-wide flume, a orretion oeffiient was extrated, whih is a funtion of the ratio of the upstream flow depth to the weir rest height. The modified stage-disharge relation, after appliation of the orretion oeffiient, shows good agreement with the data sets derived from experiments. Referenes Bos, M. G Disharge Measurement Strutures. Wageningen: International Institute for Land 32 Rasool GHOBADIAN et al. Water Siene and Engineering, Mar. 2012, Vol. 5, No. 1, 26-33

8 Relamation and Improvement (ILRI). Chow, V. T Open-hannel Hydraulis. New York: MGraw-Hill. Greve, F. W Semi-irular Weirs Calibrated at Purdue University. Engineering News-Reord, 93(5). Greve, F. W Flow of Water Through Cirular, Paraboli, and Triangular Vertial Noth-weirs. Lafayette: Purdue University. Lenastre, A Manuel D'hydraulique Générale. Paris: Eyrolles. Panuzio, F. L., and Ramponi, F Cirular Measuring Weirs. Bureau of Relamation. Qu, L. Q., Yu, X. X., Xiao, J., and Lei, T. W Development and experimental verifiation of a mathematial expression for the disharge rate of a semi-irular open hannel. International Journal of Agriulture and Biology Engineering, 3(3), [doi: /j.issn ] Rajaratnam, N., and Muralidhar, D End depth for irular hannels. Journal of the Hydraulis Division, 90(2), Rikard, C., Day, R., and Purseglove, J River Weirs:Good Pratie Guide. Swindon: R&D Publiation. Staus, A Der Beiwert kreisrunder Uberfalle. Wasserkraft und Wasserwirtshaft, 25(11), Stevens, J. C Flow through irular weirs. Journal of Hydrauli Engineering, 83(6), Vatankhah, A. R Flow measurement using irular sharp-rested weirs. Flow Measurement and Instrumentation, 21(2), [doi: /j.flowmeasinst ] Rasool GHOBADIAN et al. Water Siene and Engineering, Mar. 2012, Vol. 5, No. 1,