Triangular Short Crested Weir. Local Geometry ? Discharge Coefficients

International Journal of Engineering Sciences, 2(7) July 2013, Pages: 285-291

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International Journal of Engineering Sciences www.tijournals.com

ISSN 2306-6474

* Corresponding author. Email address: [email protected]

Triangular Short Crested Weir. Local Geometry – Discharge Coefficients

John Demetriou *1, Eugene Retsinis 2 1,2 Civil Engineer, National Technical University of Athens, Address: JD Research Hydrolab, 12 St. Polykarpou St., N. Smyrni, Athens, 17123, Greece.

A R T I C L E I N F O A B S T R A C T

Keywords: Crump Weir Sloped Channels Discharge Coefficients

In this experimental study the triangular short crested (Crump) weir is presented, analyzed and discussed from the point of views of some local geometrical flow elements,the discharge coefficient cd - for horizontal channels (slope, Jo=0) of rectangular cross sections, and the cd coefficient for adverse (Jo<0) and positive small channel slopes (Jo>0). A number of equations are given for those quantities, such as for cd as a function of dimensionless upstream head h1/L and downstream Froude number Fr2. Two graphs are presented, concerning iso-cd curves in horizontal or sloped channels, where all results appear to be systematic. The authors believe that the present work may help the hydraulic engineer dealing with the Crump weir as a water discharge measuring hydraulic structure in open channels.

© 2013 Int. j. eng. sci. All rights reserved for TI Journals.

1. Introduction

When dealing with free surface water flows in open channels, the hydraulic engineer has to select a structure convenient for discharge measurements. There are a number of such structures such as the broad crested weirs (of various control sections), the short crested weirs (e.g. cylindrical crested weirs), the critical depth flumes, the sharp crested weirs (of various control sections), etc. In this experimental paper some interesting characteristics of the so called “flat V triangular profile weir”, or “Crump weir” – from the name of E. S. Crump, who was the first to propose such a structure for rectangular water open channels – are examined. Fig. 1 schematically presents this simple and advantageous hydraulic structure and its details, of height w, length w7 , and slopes 1:2 (upstream) and 1:5 (downstream), which - because of its shape - does not allow any deposit of sediment material (e.g. silt), and thus no essential erosion or scour may damage the upstream part of it (upstream toe). The upstream uniform flow cross section (1) has a water depth y1, and the hydraulic head is h1=y1-w, while the uniform flow cross section 2 has a water depth y2, where three possible flow cases may appear: (1). Case A, with y2/w>1, subcritical flow and 75.0HH 12 , where, because of some submergence, a steady standing wave is present. (2). Case B, same as A but with 75.0HH 12 and, (3). Case (or type) C, with supercritical downstream flow, y2/w<1, and no standing wave. Hε1 and Hε2 are the energies’ heads above w, 75.0HH 12 is the modular limit (M.L.), while

,ygqFr 2/31

2/11

,ygqFr 2/3

22/1

2

and, with λ=y2/y1,

,FrFr 22/3

1 where q is the water discharge per unit channel width (b). The longitudinal channel slope is Jo (=sinφ, where φ is the inclination angle) which in the present work slightly varies from –0.009

( o51.0 ), to 0.00 (horizontal channel) and to Jo=0.0394 ( o26.2 ). The discharge q is measured through the well known formula

,Hg3/2cq 2/3

e2/12/3

d 1 (1)

where cd is the discharge coefficient. In situ, the hydraulic engineer cannot measure He1 and thus a correction coefficient is introduced

,h/Hc 2/311ev (2)

which gives the final practical formula,

.hg3/2ccq 2/31

2/12/3vd (3)

John Demetriou and Eugene Retsinis Internat ional Journal of Engineering Sci ences, 2(7) July 2013

286

Figure 1. Weir and flow details. The correction coefficient may determined from the combination of eq. (3), )yg2/q(hH 2

12

11e and eq. (2), leading to the equation

,01c)y/hc(c)3/2( 3/2v

211d

2v

32 (4)

giving cv as a function of )y/hc( 11d - rectangular channels. cv usually may found in tables for ,79.0)y/hc(0.1 11d in steps of 0.01. At the followings a very accurate equation for cv will be presented, having the advantage to hold for any step. The general onedimensional equations (per unit channel width) along the horizontal flow, are

the continuity equation ,yVyVq 2211

and the momentum equation

)],y/1()y/1[(qN)yy(g5.0 122

x22

21

where Nx is the total dynamic force from weir and boundary shear stresses opposite to x.

2. The Experiments

All measurements were performed in a small tilting rectangular water smooth channel, with slopes (adverse and positive), ,02197.0,00927.0,0025.0,001607.0,000.0,0147.0,009.0J o

,0394.0and0278.0 with corresponding angles

.26.2,59.1,26.1,53.0,143.0,092.0,0.0,84.0,51.0 ooooooooo The discharges were determined through a Venturi tube – with two accurate manometers (of water, for small discharges, and of mercury for larger discharges). The weir was made from transparent perspex with w=9.15cm and a total length of 64.0cm. The various depths were measured with a level gauge (with an accuracy of 0.1mm), while at the end of the channel a sluice gate was placed to regulate the various water depths. When the sluice gate was absent the flow beyond the weir was supercritical. The heads h1 were measured at a standard place max,1h2 upstream the weir toe, while in all measurements it was Fr1<0.5. Table 1 presents all basic experimental characteristics and some calculated quantities as well. For Jo=0.0 the flows within the M.L. are presented here, with Fr2>1 and no standing wave (C – type), while for 0J o all pertinent measurements are also elaborated in this paper.

Triangular Short Crested Weir. Local Geometry – Discharge Coefficients Internat ional Journal of Engineeri ng Sciences, 2(7) July 2013

287

3. Previous Results A lot of investigations have been dealt in the past with the Crump weir, such as the study by Hydraulic Research Station (HRS) – UK, 1967 [7], in which the discharge coefficient was measured on a similar but more specific weir with slopes 1:2, 1:2, the British Standards Institution (BSI) [3], and again the HRS, 1970 [8]. The same weir was also examined by the Water Resources Board, 1970 [11], by P. Ackers et al., 1971 [1] and 1978 [2], W. White, 1971 [12] and Delft Hydraulics Laboratory (DHL), 1976 [5]. A general report is also presented by Rao 1975 [10], and the International Standards Organisation, 1979 [9]. Most of the above authors give 15.1cd , independently of h1/L. Demetriou (2000), [6], has also determined a mean cd value,

,14.1cd for all h1/L flows within the M.L. and horizontal channels

( 0.00Jo ). He also presented a number of water free surface profiles over the weir.

Table 1. Experimental Results.

Run q

cm2/s

y1

cm

h1

cm

V1

cm/s H1

cm

H1/L

-

Fr1

-

y2

cm

Fr2

-

1 1,542 25.5 16.4 60.5 1.86 27.4 0.26 0.36 8.85 1.87

2 1,468 25.0 15.9 58.7 1.76 26.8 0.25 0.36 8.50 1.89

3 1,430 24.8 15.7 57.7 1.69 26.5 0.25 0.36 8.30 1.91

4 1,363 24.4 15.3 56.0 1.60 26.0 0.24 0.35 7.95 1.94

5 1,265 23.7 14.6 53.4 1.45 25.2 0.23 0.34 7.40 2.00

6 1,143 22.9 13.8 50.0 1.27 24.2 0.22 0.33 6.85 2.03

7 1,086 22.2 13.1 49.0 1.23 23.4 0.21 0.31 6.30 2.19

8 959.7 21.6 12.5 44.5 1.01 22.6 0.20 0.30 5.85 2.17

9 889.6 20.5 11.4 40.5 0.83 21.3 0.18 0.28 5.10 2.30

10 718.4 19.6 10.5 36.7 0.69 20.3 0.17 0.26 4.60 2.32

11 632.6 18.8 9.7 33.7 0.58 19.4 0.15 0.25 4.10 2.43

12 503.0 17.6 8.5 28.7 0.42 18.0 0.13 0.22 3.50 2.51

13 360.0 16.2 7.1 22.8 0.25 16.5 0.11 0.18 2.65 2.66

14 290.0 15.1 6.0 19.2 0.19 15.3 0.09 0.16 2.20 2.84

15 275.0 14.1 5.0 19.5 0.19 14.3 0.08 0.17 1.65 4.14

16 120.0 12.4 3.6 9.7 0.05 12.5 0.06 0.09 0.95 4.14

17 888.5 21.0 11.9 42.3 0.92 21.9 0.19 0.30 - -

18 414.7 16.5 7.4 25.1 0.32 16.8 0.12 0.20 3.20 2.31

19 1,190.3 23.0 13.9 51.8 1.37 24.4 0.22 0.33 7.10 2.00

20 284.6 15.0 6.9 19.0 0.13 15.2 0.09 0.16 2.20 2.78

4. Results. Analysis and Discussion Fig. 2 presents a curve between cv and 11d y/hcx (from eq. 4). This line is well represented by an empirical equation,

5.0x5.2v xfedxcxbac (5)

where the arithmetic coefficients may be found in Table 2.

Table 2. Arithmetic Coefficients

a b c d f

16.043702 19.6858 9.8807109 16.889682 1.2035574


288

Figure 2. cv vs 11d y/hcx .

Eq. (5) holds in the range 0.10<x<0.79, with a small error ( 0.03%) between x values from eq. (4) and predicted values from eq. (5). The advantage of eq. (5) is that it holds with 8 – digits coefficients and for any intermediate x ratio. cv are increasing with x to a max / min ratio 1.208/1.003 1.2. Table 3 presents the experimental measurements concerning h1/L ratios vs Fr2 and cd coefficients for Jo=0 (C-type flows).

Table 3. Experimental Measurements.

h1/L Fr2 cd

1 0.06 4.14 1.175

2 0.09 2.78 1.113

3 0.09 2.84 1.105

4 0.11 2.66 1.092

8 0.17 2.32 1.134

9 0.18 2.30 1.138

10 0.20 2.17 1.140

11 0.21 2.19 1.181

12 0.22 2.00 1.171

13 0.22 2.03 1.151

14 0.23 2.00 1.153

15 0.24 1.94 1.156

16 0.25 1.89 1.161

17 0.25 1.91 1.156

18 0.26 1.87 1.158

Table 4. Arithmetic Coefficients.

a b c d e f

0.51020668 0.049535473 0.0098436846 0.00055714191 0.55722081 0.13494174

A statistical elaboration of these experimental data gave the equation

,FrfFreLhdLhcLhbac 222

31

21

11d (6)

where the arithmetic coefficients are given in Table 4.


289

The above equation holds for 0.06<h1/L<0.26, 1.87<Fr2<4.14, Jo=0, and the results of the statistical elaboration (in relation to the experimental results) have an error of about ( 2.1%). For h1/L=const. cd are increasing with Fr2, while for Fr2=const. cd are increasing with increasing h1/L ratios. Fig. 3 presents a graph of eq. (6) and the values of Table 4 in the form of iso-cd curves. This graph is indicative only and eq. 6 should be rather used instead. However Fig. 3 qualitatively shows the strong dependence of cd on h1/L and Fr2 (all larger than 1) parameters, i.e. the dependence on both, the upstream and downstream flow conditions, since in older studies usually only the h1/L effect on cd was considered.

Figure 3. cd vs h1/L and Fr2, for Jo=0 and C-type of flows. Table 5 presents the experimental results connecting Fr2 and λ for Jo=0 and C-type of flows (y2/w<1, Fr2>1, no standing wave). These results are also shown in Fig.4, where a line is traced among the experimental points. After a suitable statistical elaboration a proper equation is given to this line

2

2Frba (7) where a=0.0048522188, b=1.1882386. This statistical elaboration presents an error of about 0.02% in relation to the experimental results of Table 5, and shows that when Fr2 are increasing λ are decreasing. Eq. (7) holds for 1.87< Fr2<4.14. Furthermore, for the same Crump weir a statistical elaboration among cd calculated from eq. (1), Jo (><0), and h1/w, gave the approximate – empirical equation

,whlwhkwhi

JgJfJdJcJbac3

12

11

5o

4o

3o

2o

1od

(8)

where the arithmetic coefficients are given in Table 6. This equation holds for-0.0147<Jo<+0.0394, 0.497< h1/w<1.934 and 1.10<cd<1.71.

Table 5. Experimental Results.

No Fr2 λ 1 1.87 0.34 2 1.89 0.33 3 1.91 0.33 4 1.94 0.32 5 2.00 0.30 6 2.00 0.31 7 2.03 0.29 8 2.17 0.27 9 2.19 0.27

10 2.30 0.25 11 2.31 0.19 12 2.32 0.23 13 2.43 0.22


290

Table 6. Arithmetic Coefficients.

a 2.8717829 g 7.615226.e-13 b 0.0026501913 i 3.873355 c 2.72895 e-5 k 3.3548676 d 1.25238 e-7 l 0.9165802 f 7.35979 e-10 -

Figure 4. λ vs Fr2 for Jo=0 and C-type of flow.

The max error between the experimental and the statistical results is of the order of 15%, but it is believed by the authors that it is the first time that the discharge coefficient cd, for a Crump weir, is correlated to adverse and positive channels’ slopes, and in the same time with h1/w.

Figure 5. iso-cd lines for h1/w ratios and Jo slopes. Finally, Fig. 5 presents a graph based on eq. 8 and Table 6, showing some iso-cd lines for adverse and positive channel slops and – in the same time – for h1/w ratios. This graph is rather complicate and indicative, and eq. (8) should be used instead. From a qualitative point of view for Jo<0, cd have rather small cd values (1.15 to 1.20 in the graph), while for Jo>0 cd coefficients reach higher values (1.55 to 1.60 in the graph).


291

5. Conclusions In this experimental study the triangular short crested (Crump) weir is presented, analyzed and discussed from the points of views of some local geometrical flow elements, the discharge coefficient cd for horizontal (Jo=0) rectangular channels, and cd for adverse (Jo<0) or positive channel slopes (Jo>0). The three possible flow cases, the pertinent parameters and the onedimensional equations are presented as well. The main conclusions are: (1). Eq. (5) gives a very accurate correlation for the well known correction coefficient, cv, vs 11d y/hc for the present weir. (2). Eq. (6) gives cd, as a function of both h1/L and Fr2, where cd are increasing with Fr2, in horizontal channels, Jo=0. (3). The dependence of this cd on h1/L and Fr2 is strong and it is believed that it is the first time to be presented. (4). Fig. 3 gives a graph of this cd dependence on h1/L and Fr2. (5). Eq. (7) gives the relationship between λ=y2/y1 and Fr2 in horizontal channels. (6). λ are decreasing when Fr2 are increasing. (7). Eq. (8) gives the discharge coefficient when Jo><0, as a function of Jo and h1/w. (8). Fig. 5 gives a graph concerning iso-cd lines for Jo><0, as a function of Jo and h1/w. The authors believe that the present results may help the hydraulic engineer dealing with the Crump weir as a discharge measuring hydraulic structure in open channels. References [1] Ackers P.: Flow measurement by Weirs and Flumes, Proceedings of Intern. Conference, England 1971, pp. 21-37. [2] Ackers P. - White W. - Perkins J. – Harrison A.: Weirs and flumes for flow measurement, J. Wiley, 1978, pp. 86-109. [3] British Standards Institution: Methods of measurements of liquid flow in open channels, Part 4, Weirs and Flumes (long-base weirs), BS 3680 (Part

4B), 1969, pp. 18-34. [4] Crump E.:A new method of gauging Stream Flow with little afflux by means of a submerged weir of Triangular Profile, Institution of Civil

Engineers, 1952, pp. 223-242. [5] Delft Hydraulics Laboratory: Discharge measurement structures, Publication No 161, 1976, pp. 191-197. [6] Demetriou J.: The Crump Weir in Horizontal and Sloping Channels, Journal, Tech. Chron. Scien. Tech. Chamber of Greece, I., No. 1, pp. 11-18. [7] Hydraulics Research Station: Triangular Profile weir with 1:2 upstream and downstream slopes, Report INT 64, Wallingford, 1967, pp. 41-47. [8] Hydraulics Research Station: The triangular profile crump weir (A re-examination of discharge characteristics), February 1970, Wallingford EX477,

pp. 1-11. [9] International Standard ISO 4360: Liquid flow measurements in open channels by weirs and flumes – Triangular profile weirs, 1979, pp.1-9. [10] Rao N.: Theory of weirs, Advances in Hydroscience, vol. 10, Academic Press, 1975, pp. 333-334. [11] Water Resources Board: Crump weir design, February 1970, pp. 3-17. [12] White W.: The performance of two-dimensional and flat-V triangular profile weirs, Proc. Inst. Civil Engineers, paper 7350, 1971, pp. 48.

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Triangular Short Crested Weir. Local Geometry ? Discharge Coefficients