18 Pags., 1 Tabl., 3 Figs.  The definitive version is available at: http://ascelibrary.org/toc/jidedh/140/3Triangular weirs are commonly used to measure discharge in open channel flow, representing an inexpensive, reliable methodology to monitor water allocation. In this work, a low-speed photographic technique was used to characterize the upper and lower nappe profiles of flow over fully aerated triangular weirs. A total of 112 experiments were performed covering a range of weir vertex angles (from 30° to 90°), crest elevations (8 or 10 cm), and discharges (0.01–7.82  l s−1). The experimental nappe profiles were mathematically modeled and combined with elements of free-vortex theory to derive a predictive equation for the weir discharge coefficient. Comparisons were established between measured Cd, the proposed discharge coefficient equation, and discharge coefficient equations identified in the literature. The proposed equation predicts Cd with a mean estimation error (MEE) of 0.001, a root-mean square error (RMSE) of 0.004, and an index of agreement (IA) of 0.984. In the experimental conditions of this study, this performance slightly improves that of the equation proposed by Greve in 1932, and showed the same absolute value of MEE but lower values of RMSE and IA.This research was funded by the Secretaría de Agricultura, Ganadería, Desarrollo Rural, Pesca y Alimentación of the Mexican Government (SAGARPA, Mexico) and the Secretaría del Campo of Zacatecas State Government (SECAMPO, Zacatecas). Thanks are also expressed to the Universidad Autónoma of Zacatecas, Mexico. Cruz Octavio Robles Rovelo received a scholarship from the Mexican Consejo Nacional de Ciencia y Tecnología (CONACYT).Peer reviewe

Bautista-Capetillo, Carlos F.

Júnez-Ferreira, H.

Playán Jubillar, Enrique

Robles Rovelo, Octavio

Journal of Irrigation and Drainage Engineering

Digital.CSIC

	 1Discharge coefficient analysis for triangular sharp-crested  1	weirs using a low-speed photographic technique 2	by 3	 4	Bautista–Capetillo C. 1 * M. ASCE, Robles O.2, Júnez-Ferreira H.3, Playán E.4 5	Abstract 6	Triangular weirs are commonly used for the measurement of discharge in open channel 7	flow, representing an inexpensive, reliable methodology for the monitoring of water 8	allocation. In this work, a low-speed photographic technique was used to characterize 9	the upper and lower nappe profiles of flow over fully aerated triangular weirs. A total of 10	112 experiments were performed covering a range of weir vertex angles (from 30º to 11	90º), crest elevations (8 or 10 cm) and discharges (0.01 – 7.82 l s-1). The experimental 12	nappe profiles were mathematically modeled and combined with elements of free-13	vortex theory to derive a predictive equation for the weir discharge coefficient. 14	Comparisons were established between measured Cd, the proposed discharge 15	coefficient equation and discharge coefficient equations identified in the literature. The 16	proposed equation can predict Cd with a Mean Estimation Error (MEE) of 0.001, a Root 17																																																									1 Professor, Recursos Hidráulicos, Universidad Autónoma de Zacatecas, Av. Ramón López Velarde 801, 98000 Zacatecas, Mexico. 2  Master Degree Student, Maestría en Ingeniería Aplicada Orientación en Recursos Hidráulicos, Universidad Autónoma de Zacatecas, Av. Ramón López Velarde 801, 98000 Zacatecas, Mexico. 3 Professor, Recursos Hidráulicos, Universidad Autónoma de Zacatecas, Av. Ramón López Velarde 801, 98000 Zacatecas, Mexico. 4 Research Professor, Departamento Suelo y Agua, Estación Experimental de Aula Dei, CSIC. P. O. Box 13034. 50080 Zaragoza, Spain. * Corresponding author: baucap@uaz.edu.mx 	 2Mean Square Error (RMSE) of 0.004, and an Index of Agreement (IA) of 0.984. In the 18	experimental conditions of this study, this performance slightly improves that of the 19	equation proposed by Greve in 1932, showing the same absolute value of MEE, but 20	lower values of RMSE and IA. 21	Keywords: weir vertex angle, flow measurement, hydrometry, free-vortex theory 22		 3Introduction 23	Weirs are elevated barriers located perpendicular to the main direction of water 24	movement to cause the fluid to rise above the obstruction in order to flow through an 25	opening of regular shape. For a properly designed and operated weir of a given 26	geometry there is a unique discharge corresponding to each measurement of flow depth 27	(El-Hady 2011). The geometrical parameters involved in the hydraulic operation of 28	weirs are the length of the weir crest and the shape of the flow control section (Emiroglu 29	et al. 2010; USBR 2001). In sharp-crested or thin-plate weirs the upstream head (h) to 30	length of crest in the direction of flow (L) ratio is greater than 15 (Fig. 1). Specific 31	assumptions are adopted to estimate the relation between discharge and upstream head 32	(Bagheri and Heidarpour 2010; Sotelo 2009; El-Alfy 2005; Bos 1989). These structures 33	have been extensively studied using classical physics and experimental analyses to 34	understand the characteristics of flow as well as to determine the coefficient of discharge 35	(Cd). This coefficient represents the effects not taken into consideration in the derivation 36	of the equations used to estimate discharge from flow depth. Such effects include 37	viscosity, capillarity, surface tension, velocity distribution in the approach section and 38	streamline curvature due to weir contraction (Aydin et al. 2011; El-Hady 2011). 39	In the particular case of triangular sharp-crested weirs, Shen (1981) described 40	experimental procedures used by different authors to determine Cd. El-Alfy (2005) 41	experimentally evaluated the effect of vertical flow curvature on the discharge 42	coefficient, and reported that Cd is inversely proportional to the V-notch angle (θ) and 43	directly proportional to the relative head (h/P). Recently, Bagheri and Heidarpour 44		 4(2010) obtained a discharge coefficient equation for rectangular sharp-crested weirs 45	based on the upper and lower nappe profiles and free-vortex theory. 46	Photography has been used for the characterization of flow over hydraulic structures, 47	particularly weirs. For instance, Del Giudice et al., (1999) used photographs to illustrate 48	complex flow patters near a sewer sideweir. Novak et al. (2013) photographed the 49	planes displayed by a laser on the flow near a side weir, and used these images to 50	determine flow depth profiles and flow velocity (from the movement of hydrogen 51	bubles). Photography was recently applied to a different hydraulic problem: the 52	characterization of sprinkler irrigation drops moving in the air. Salvador et al. (2009) 53	and Bautista et al. (2009) performed out-door and in-door experiments to evaluate drop 54	geometrical and kinematic characteristics using a low-speed photographic technique. 55	The objective of this study was to determine a discharge coefficient equation for 56	triangular sharp-crested weirs based on: 1) the free vortex theory as described by 57	Bagheri and Heidarpour (2010); and 2) measurements of the upper and lower nappe 58	profiles using an adaptation of the low-speed photographic technique proposed by 59	Salvador et al. (2009).  60		 5Materials and methods 61	Governing equations 62	For a sharp-crested weir of any geometrical section with the crest elevation (P) being 63	high enough to neglect the velocity head (Figure 1), discharge equations are usually 64	obtained from the mathematical integration of an elemental flow strip over the nappe 65	(Singh et al. 2010). The total discharge flowing between elevations 0 and h can be 66	obtained solving the following expression: 67	   h021d dyyhxC2g2Q  [1] where Q is the discharge over the weir (m3 s-1); g is the gravitational acceleration (m s-2); 68	Cd is the discharge coefficient (dimensionless); h is the water head (m); x is the flow 69	width, with x= f(y) depending of the weir geometry; and dy is the vertical thickness of 70	elemental flow strip. A sharp-crested weir with symmetrical triangular section and 71	vertex angle (θ) entails that 2θtanyx , as shown in Figure 1b. The resulting discharge 72	equation is: 73	25d h2θtan2gC158Q   [2] Considering free-vortex motion theory, Bagheri and Heidarpour (2010) proposed an 74	expression to derive the discharge coefficient of flow passing over a rectangular sharp-75	crested weir. Following the reasoning of these authors, a similar expression could be 76	obtained for a triangular sharp-crested weir (Equation 3): 77		 6bbbb RRRRkYlnkY2θtanV2Q b  [3] where Vb is the lower nappe velocity, obtained at the section of maximum elevation of 78	the lower nappe (m s-1); Rb is the radius of streamline curvature at lower nappe of the 79	profile in segment OB (m); k is the non-concentricity coefficient; and Y is the flow depth 80	at the section of maximum elevation of lower nappe (m) (Figure 1). 81	Experimental setup and measuring techniques 82	Experiments were performed in a horizontal rectangular recirculating plexiglass 83	laboratory channel 7.2 m long, 0.3 m wide, and 0.3 m high. Canal cross section was 84	designed for a maximum discharge of 10 l s-1, having in mind a common application of 85	this type of weirs: the analysis of furrow irrigation inflow and outflow. Mild steel plates 86	(galvanized sheet metal) with a thickness of 1.5 mm were used to manufacture weirs. 87	Vertex angles were 30º, 45º, 60º and 90º, each of them with 8 and 10 cm of crest height. 88	Water was supplied to the channel through an overhead tank provided with an 89	overflow arrangement to maintain constant head. A grid wall was installed into the 90	channel to dissipate flow velocity. To avoid the area of water surface draw-down, head 91	over the weir was measured 1.0 m upstream of the vertical weir plane using a point 92	gage with accuracy of ± 0.1 mm. Discharge over the weirs was volumetrically measured, 93	using a prismatic steel measuring tank with base dimensions of 0.75 m x 0.75 m. Weirs 94	were installed at the end of the channel to provide an unrestricted supply of air under 95	the nappe. Consequently, all data for this study correspond to the conditions of fully 96	aerated flow. Equations of flow nappe profiles and discharge coefficients for triangular 97		 7sharp-crested weirs were obtained for four different models. Table 1 presents a 98	summary of the weir characteristics and test conditions. Weir models were tested using 99	14 flow rates. A total of 112 experiments were conducted (4 vertex angles x 2 values of P 100	x 14 flow rates). Additionally, each discharge was measured five times. The average of 101	these replications was used to obtain the discharge coefficient. 102	An adaptation of the low-speed photographic technique proposed by Salvador et al. 103	(2009) was implemented in order to identify a set of points (z, y) along the upper and 104	lower nappes to characterize the profiles. Coordinate z corresponds to the horizontal 105	distance downstream from the weir. All coordinate values were initially registered in 106	pixels and then transformed to millimeters using the pixel per millimeter ratio obtained 107	from image analysis (all images included a reference ruler). In order to assess the 108	differences between measured-estimated values and different estimation equations 109	proposed by other authors, the following statistic parameters were used: mean 110	estimation error (MEE), root mean square error (RMSE), and index of agreement (IA) 111	(Willmott, 1982).  112		 8Results 113	The points obtained from the photographs were plotted as shown in Figure 2, where y is 114	the vertical depth of flow at z distance downstream from the weir. Plotted information 115	corresponds to all measured upper and lower flow nappe profiles for the different 116	values of vertex angle. Figure 2 shows pairs (z, y) relative to head (h) as well as the 117	polynomials that best fit each case. The upper and lower nappe profiles could be 118	successfully adjusted to quadratic equations. Polynomials were used to determine 119	distances OA, OB, AC, and AE (Figure 1) for each weir model using the general 120	regression equations in Figure 2. The same procedure was used to determine the mean 121	radius of curvature of the streamline along the distance of OB at lower nappe profiles 122	(Rb), the flow depth at the section of maximum elevation of the lower nappe (Y), and the 123	correction coefficient of non-concentricity streamline (k) (Bagheri and Heidarpour, 124	2010). The analysis of ratios Rb/h and Y/h against weir vertex angle expressed as 125	tan (θ/2) shows potential relations in both cases. Regarding the non-concentricity 126	coefficient, the best relation between k and h tan (θ/2) is represented by a potential 127	equation. Substituting Rb/h, Y/h, and k expressions into Equation 3 results in Equation 128	4: 129	10110125ZZkZlnZZkZ2θtanh8.859Q  [4] where 130	0.0440 2θtan0.682Z    [5] 	 9and 131	0.0981 2θtan0.445Z   [6] Combining Equations 2 and 4, the discharge coefficient can be expressed as Equation 7: 132	 101101d ZkZZlnZkZZ3.750C  [7] Estimated discharge coefficients (for head over the weir ranging from 1.5 cm to 15 cm) 133	ranged between 0.669-0.607, 0.674-0.614, 0.677-0.618, and 0.680-0.624 for weir angles of 134	30º, 45º, 60º, and 90º respectively. Measured discharge coefficients (for heads over the 135	weir of 1.5-15 cm for weir angles of 30º, 45º, and 60º; and for heads over the weir of 1.5-136	12 cm for a weir angle of 90º) ranged between 0.665-0.614, 0.668-0.616, 0.672-0.620, and 137	0.677-0.624 for the same weir vertex angles. Figure 3 presents a comparison of the 138	experimental data, the proposed discharge coefficient (Equation 7) and the estimates 139	obtained using some references discussed by Shen (1981). The proposed equation can 140	predict Cd for the range or 30º-90º weir vertex angles with MME = 0.001, RMSE = 0.004, 141	and IA = 0.984. In the experimental conditions of this study, this performance can only 142	be compared to that of the equation proposed by Greve (1932), which showed the same 143	absolute value of MEE but lower values of RMSE and IA.   144		 10Conclusions 145	An experimental analysis was performed to estimate the discharge coefficient for four 146	triangular sharp-crested weir models. Regression equations of the upper and lower 147	nappe profiles developed from experimental data and free-vortex theory were used to 148	derive a discharge coefficient equation as a function of head over the weir (h) and weir 149	vertex angle expressed as tan (θ/2). Experimental data showed that both nappe profiles 150	can be successfully represented by second-degree polynomials. Results also indicated 151	that the non-dimensional mean radius of curvature of the streamline along the distance 152	OB at lower nappe profiles (Rb/h) and the non-dimensional flow depth at the section of 153	maximum elevation of the lower nappe (Y/h) show potential relations with the weir 154	vertex angle expressed as tan (θ/2). To take into account the non-concentricity of the 155	streamlines, a correction coefficient was proposed as a function of h and θ. Comparisons 156	between measured Cd, the proposed discharge coefficient equation and discharge 157	coefficient equations proposed by a number of authors were established. In the 158	experimental conditions, the proposed equation represents an improvement in the 159	estimation of discharge from triangular weirs, and confirms the validity of a predictive 160	equation proposed by Greve in 1932.   161		 11Acknowledgements 162	This research was funded by the Secretaría de Agricultura, Ganadería, Desarrollo Rural, 163	Pesca y Alimentación of the Mexican Government (SAGARPA, Mexico) and the 164	Secretaría del Campo of Zacatecas State Government (SECAMPO, Zacatecas). Thanks 165	are also due to the Universidad Autónoma of Zacatecas, Mexico. Cruz Octavio Robles 166	Rovelo received a scholarship from the Mexican Consejo Nacional de Ciencia y 167	Tecnología (CONACYT).  168		 12References 169	Aydin, I., Altan-Sakarya, A. B., and Sisman, C. (2011). “Discharge formula for 170	rectangular sharp-crested weirs.” Flow Measurement and Instrumentation, 22: 144-151. 171	Bagheri, S., and Heidarpour, M. (2010). “Flow over rectangular sharp-crested weirs” 172	Irrigation Science, 28:173-179. 173	Bautista-Capetillo, C. F., Salvador, R., Burguete, J., Montero, J., Tarjuelo, J. M., Zapata, 174	N., González, J., and Playán, E. (2009). “Comparing methodologies for the 175	characterization of water drops emitted by an irrigation sprinkler.” Transactions of the 176	ASABE 52(5): 1493-1504. 177	Bos, M. G. (1989). Discharge measurement structures. International Institute for Land 178	Reclamation and Improvement, ILRI, Wageningen, Netherlands. 179	Del Giudice, G., and Hager, W. H. (1999). “Sewer sideweir with throttling pipe.” Journal 180	of Irrigation and Drainage Engineering-ASCE 125(5):298-306. 181	El-Alfy, K. M. (2005). “Effect of vertical curvature of flow at weir crest on discharge 182	coefficient”. Ninth International Water Technology Conference, Sharm El-Sheikh, Egypt, 183	249-262. 184	El-Hady, R. M. A. (2011). “2D-3D modeling of flow over sharp-crested weirs.” Journal of 185	Applied Sciences Research, 7(12): 2495-2505. 186		 13Emiroglu, M. E., Kaya, N., and Agaccioglu, H. (2010). “Discharge capacity of labyrinth 187	side weir located on a straight channel.” Journal of Irrigation and Drainage Engineering 188	ASCE, 136(1): 37-46. 189	Novak, G., Kozelj, D., Steinman, F. and Bajcar, T. (2013). “Study of flow at side weir in 190	narrow flume using visualization techniques.” Flow Measurement and Instrumentation 191	29:45-51. 192	Salvador R., Bautista-Capetillo C., Burguete J., Zapata N., and Playán, E. (2009). “A 193	photographic methodology for drop characterization in agricultural sprinklers.” 194	Irrigation Science, 27(4): 307-317. 195	Shen, J. (1981). Discharge characteristics of triangular-notch thin-plate weirs. Studies of flow 196	of water over weirs and dams. Geological Survey Water-Supply Paper 1617-B, 197	Washington D. C. 198	Singh, N. P., Yada, S. M., and Singh, R. (2010). “Behavior of sharp crested weirs of 199	various curvilinear shapes.”Proceedings of the 13th Asian Congress of Fluid Mechanics, 200	17-21, Dhaka, Bangladesh. 201	Sotelo, G. (2009). Hidráulica general I: Fundamentos. Ed. LIMUSA, Mexico. 202	United States Bureau of Reclamation (2001). Water measurement manual. Revised reprint. 203	Denver, Colorado. 204	Willmott C. J. (1982). “Some comments on the evaluation of model performance.” 205	Bulletin American Meteorological Society, 63(11): 1309-1313.   206		 14List of Tables 207	Table 1. Triangular sharp-crested weir characteristics and test conditions. 208	 209	List of Figures 210	Figure 1. Experimental parameters: a) direction of flow view, b) frontal view. 211	Figure 2. Upper and lower nappe profiles. Weir vertex angles: a) 30°, b) 45°, c) 60°, and d) 90°. 212	Figure 3. Discharge coefficient (Cd) vs. head over triangular sharp-crested weir. Weir vertex 213	angles: a) 30°, b) 45°, c) 60°, and d) 90°. 214	Table 1. Triangular sharp-crested weir characteristics and test conditions. Weir model Vertex angle (θ, °) P (cm) Q (l s-1) h (cm) 1 30 8, 10 0.01-3.56 1.5-15.0 2 45 8, 10 0.02-5.52 1.5-15.0 3 60 8, 10 0.03-7.74 1.5-15.0 4 90 8, 10 0.04-7.82 1.5-12.0 Figure 1. Experimental parameters: a) direction of flow view, b) frontal view. YV²/2g D EEnergy linehdyxXyhO ACYB FVbR PPO'bFigure 2. Upper and lower nappe profiles. Weir vertex angles: a) 30°, b) 45°, c) 60°, and d) 90°.y/h = -0.316 (x/h)2 - 0.578 (x/h) + 0.958R² = 0.9710.60.81.0y/h = -0.375 (x/h)2 - 0.455 (x/h) + 0.918R² = 0.968a by/h = -1.179 (x/h)2 + 0.651 (x/h) - 0.003R² = 0.8250.20.4y/h = -1.077 (x/h)2 + 0.630 (x/h) - 0.001R² = 0.958y/h0.01.0y/h = -0.430 (x/h)2 - 0.406 (x/h) + 0.881R² = 0.9700.60.8y/h = -0.442 (x/h)2 - 0.295 (x/h) + 0.823R² = 0.962y/h dcy/h = -1.166 (x/h)2 + 0.640 (x/h) - 0.001R² = 0.9230 00.20.4y/h = -1.213 (x/h)2 + 0.528 (x/h) + 0.002R² = 0.822y.0 0.2 0.4 0.6 0.8 1 1.2 0 0.2 0.4 0.6 0.8 1 1.2x/h x/hFigure 3. Discharge coefficient (Cd) vs. head over triangular sharp-crested weir. Weir vertexangles: a) 30°, b) 45°, c) 60°, and d) 90°.0.800.700.75Cda b0 600.65500.750.80.c d500.650.70Cd00.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.160.600.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16Head (m) Head (m)  Barr and Strickland (1910)  Lenz (1943) Cone (1916)  King (1954) Greve (1932)  Equation suggested Experimental data 

Discharge coefficient analysis for triangular sharp-crested weirs using a low-speed photographic technique

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