DISCHARGE COEFFICIENT OF SIDE WEIRS.

EXPERIMENTAL STUDY AND COMPARATIVE ANALYSIS OF DIFFERENT FORMULAS

 

¹) ANTÓNIO N. PINHEIRO, ²) ISABEL N. SILVA

 

¹) Member I-5671, Ph.D., Assistant Professor, Department of Civil Engineering,

Instituto Superior Técnico (IST)

Address: IST, DECivil, Secção de Hidráulica

Avenida Rovisco Pais, 1049-001 Lisboa, Portugal

Tel.: 351 1 8418144 / 3; Fax: 351 1 8418144; e-mail: apinheir@civil.ist.utl.pt

²) Civil Engineer (IST) at MSF, Contractors, SA

MSc. Hydraulic and Water Resources (IST)

Tel.: 351 931 647259

 

 

ABSTRACT

The flow in channels with side weirs was studied by several authors since the beginning of this century. Most of these studies were motivated by the wide use of this type of hydraulic structure in irrigation and sewerage systems and by the need to improve the knowledge about its hydraulic performance.

The present paper presents six expressions, of different authors, to compute the discharge coefficient of side weirs based on experimental facilities with rather different sizes.

The main characteristics of an experimental facility built at the National Laboratory for Civil Engineering (LNEC) to study the discharge coefficient of side weirs is briefly described . A summary of the measurements carried out is included.

Finally, a new three parameter expression to compute the discharge coefficient of the side weir and its comparison with expressions proposed by other authors are presented. The discrepancies of the results are analysed considering the geometry differences between the several facilities.

 

Keywords: Side Weir, Discharge Coefficient

 

INTRODUCTION

There are many expressions to compute the discharge coefficient of side weirs. Nevertheless, it is not always present in the users' minds which conditions were considered to deduce them and the possible arising limitations that may restrain the corresponding field of application.

One of the major objectives of a more comprehensive study developed by the second author of the present paper under the supervision of the first author (SILVA 1997) was precisely to contribute to fill in that gap and to determine an expression based on experiments carried out on a fairly large facility, where the scale effects would have a minor influence on the results.

 

DISCHARGE COEFFICIENT EXPRESSIONS

The expressions proposed by six different authors are presented herein. The main characteristics of the experimental installations, as well as the values of Froude number and the range of tested discharges, used by these authors are mentioned in Table 1.

 

Table 1 - Side weirs used by different authors. Geometric characteristics and ranges of tested discharges and of F₁ values.

 

* not mentioned by the authors

 

SUBRAMANYA and AWASTHY (1972) present a study considering subcritical and supercritical flows. According to the authors, F₁ is the parameter having greater influence on the variation of C. Also according to these authors, the parameters representing the geometrical configuration, L/B, H₁/L and p/h₁, have low influence on the discharge coefficient. For subcritical regimes, SUBRAMANYA and AWASTHY (1972) propose

 

(1)

 

and for supercritical regimes

 

(2)

 

RAJU et al. (1979) carried out an experimental study in a channel with a rectangular cross-section, with subcritical regime and a side weir 0.20 to 0,50 m long, as is indicated in Table 1. The authors proposed to consider the effective weir length, L_e=L-0.05 m. For the smallest length used by the authors, the reach most affected by the boundary effect represents 20% of the weir length. In a longer weir such effect would have a much more reduced influence. The authors propose the following expression for broad crested weirs

 

(3)

 

where K is an empirical coefficient depending from (h₁-p)/e. For a sharp crested weir, K=1.

HAGER (1987) refers to an expression, where the discharge coefficient depends only on F₁, applicable whenever the channel slope and the convergence angle of the main channel walls are close to zero, and in case F₁ do not vary significantly along the side weir

 

(4)

 

SWAMEE (1988), based on experimental results obtained by other authors, presents an expression, with a formulation different from the previous ones, for the calculation of the discharge coefficient in sharp-crested weirs

 

(5)

 

CHEONG (1991) studied the discharge coefficient in a side weir of a channel with trapezoidal cross-section. However, he considers the results can be applied to rectangular channels and proposes the following expression

 

(6)

 

More recently, SINGH et al. (1994) proposed the following expression, deduced for rectangular channels and subcritical regimes

 

(7)

 

The expression (7) contradicts the conclusion presented by SUBRAMANYA and AWASTHY (1972) regarding the small influence of the geometric parameters.

 

EXPERIMENTAL FACILITY AND EXPERIMENTS

The experimental facility developed by SILVA (1997) includes a sharp-crested side weir, with confined flow in the downstream section, due to the different thickness of the side weir, with variable length with a maximum of 2.00 m, and with a height of 0.20 m. The main channel has a rectangular cross section with 0.50 m width. The lateral outflow is collected by a rectangular side-channel with 0.50 m width. The experimentation programme considered discharges in the main channel of 30, 40, 50, 75, 100, 125, 150 l/s. For each of these discharges, three lateral outflow situations were considered, corresponding to 25, 50 and 75 % of the discharge in the main channel.

 

DATA ANALYSIS AND RESULTS

Based on a dimensional analysis, SILVA (1997) concluded the discharge coefficient of side weirs was mainly influenced by three parameters: F₁ , which considers the influence of the approaching flow, h/p, the quotient between the flow height and the side weir height, and L_ndim= L/(V₁²/2g), the length of the side weir relatively to the upstream flow velocity

 

(8)

 

The experimental discharge coefficient, for each test, was calculated in accordance with

 

(9)

 

where Q_L is the lateral outflow, L the weir length, g the gravitational acceleration, p the weir height and is computed according to

(10)

where N is the total number of flow height measurements in the main channel along the side weir and Dx_i the distances between the measurements h_i and h_i+1.

The influence of each parameter on the discharge coefficient will be analysed separately, in order to show its relative importance.

The variation of the discharge coefficient, C, with the Froude number in the upstream section of the weir, F₁ is represented in Figure 1.

 

Figure 1 - Discharge coefficient as function of F₁.

 

The regression equation, obtained by the least squares method,

 

(11)

 

has a correlation coefficient of 0.74 and a standard deviation of 0.019.

The value of the correlation coefficient is comparatively low, which leads to the conclusion that this parameter is not sufficient to explain the variation of the discharge coefficient in a side weir.

Figure 2 shows the variation of the discharge coefficient, C, with the quotient between the mean flow height ()and the side weir height, h_m/p.

 

Figure 2 - Discharge coefficient as a function of h_m/p.

 

The discharge coefficient decreases with h_m/p. The regression equation determined by the least squares method is

(12)

with a correlation coefficient of 0.76 and a standard deviation of 0.026.

Figure 3 shows the variation of the discharge coefficient with the parameter L_ndim =L/(V₁²/2g). A non-linear behaviour and an asymptotic trend for a value C_max, which is considered to be close to 0.45, can be noticed. The meaning of this behaviour is that for relatively small side weirs, the boundary effects have an important contribute to reduce the discharge coefficient. For longer side weirs (L ³ 200 *V₁²/2g) the boundary effects do not influence C

 

Figure 3 - Discharge coefficient as a function of L/(V₁²/2g).

 

The behaviour presented by function C=C(L_ndim) is of the type

 

(13)

 

Linearizing and applying the least squares method, the next expression is obtained

 

(14)

 

The correlation coefficient and the standard deviation of expression (14) are, respectively, 0.84 and 0.023.

The conjugated influence of these three parameters should provide an expression with higher regression coefficient. A linear multiple regression analysis supplied the following expression

 

(15)

 

which presents a correlation coefficient of 0.93. This means the expression (15), which considers the influence of the three parameters, is rather more accurate than expressions (11), (12) and (14), which consider only one parameter each.

 

APPLICABILITY OF THE DISCHARGE COEFFICIENT EXPRESSIONS

The discharge coefficients computed according to SILVA (1997) and according to the expressions previously presented are plotted in Figure 4.

 

Figure 4 - Discharge coefficient of side weirs. Comparison between the values obtained by different authors and those obtained by SILVA (1997).

 

The analysis of these figures suggests the following comments:

the small length of the side weirs used by SUBRAMANYA and AWASTHY (1972) and by RAJU et al. (1979) makes it difficult to apply the results to larger side weirs, due to the importance of the boundary effects in such a short weir;

expressions such as the one of SWAMEE (1988), which only represents the influence of the geometric features in the discharge coefficient h/p, disregarding the influence of the approaching flow in the main channel, usually represented by the Froude number, F₁, do not present any satisfactory results;

expression (6), presented by CHEONG (1991), does not present satisfactory results;

expression (7), proposed by SINGH et al. (1994), which comprises two parameters that have influence on the discharge coefficient, h/p and F₁, is the one that presents more accurate results;

expression (15), proposed by SILVA (1997), considers the parameters proposed by SINGH et al. and an additional one, L/(V₁²/2g), which accounts for the influence of the velocity of the approaching flow relatively to the length of the weir; so, this parameter is considering the influence of the boundary effects, which will make this expression specially adequate to deal with side weirs of very different lengths.

 

CONCLUSIONS

The expressions analysed have been formulated mainly based on results obtained in small facilities, which may limit the application of these expressions to channels with larger side weirs, where the boundary effects have smaller importance.

It seems that, among the expressions analysed, the one that supplies more adequate discharge coefficients is (7), proposed by SINGH et al. (1994), since two parameters, h/p and F₁, which have influence on the discharge coefficient, are integrated.

It was thus justified the execution of a study that made possible proposing a three parameters expression, (15), for the calculation of the discharge coefficient in side weirs located in rectangular.

Further experiments should be made for supercritical flows and for smaller side weir lengths in order to test the adequacy of parameter L/(V₁²/2g) for flow conditions where the boundary effects are expected to be stronger.

 

AKNOWLEDGEMENTS

The authors wish to express their gratitude to the National Laboratory for Civil Engineering the support given to the construction of the experimental facility and to the second author during the experimentation period.

 

REFERENCES

CHEONG, H. (1991) - Discharge coefficient of lateral diversion from trapezoidal channel, J. of Irrigation and Drainage Engineering, ASCE, 117 (4), 461-475.

HAGER, W.H. (1987) - Lateral outflow of side weirs, Proceedings ASCE, J. of Hydraulic Enginnering, vol. 113, HY4, 491-504.

SILVA, N.- I. (1997) - Descarregadores laterais. Modelação física e matemática (Side weirs. Physical and mathematical modelling). MSc. Thesis, IST, Lisbon.

RAJU, R.K.G., PRASARD, B., and GUPTA, S.K. (1979) - Side rectangular channel, J. of Hydraulic Engineering, ASCE, 105(5), 547-554.

REHBOCK, T. (1929) - Discussion of "Precise measurements" by Turner, K.B., Translation, ASCE, 93, 1143-1162.

SINGH, R. MANIVANNAN, D. and SATYANARAYANA, T. (1994) - Discharge coefficient of rectangular side weirs. J. of Irrigation and Drainage Engineering, Vol 120, no 4, 814-819.

SUBRAMANYA, K. and AWASTHY, S.C. (1972) - Spatially varied flow over side weirs, J. of Hydraulic Engineering, ASCE, 98(1), 1-10.

SWAMEE, P.K. (1988) - Generalised rectangular weir equations, J. of Hydraulic Engineering, ASCE, 114(8), 945-949.

SWAMEE, P.K., PATHAK, S.K. e ALI, M.S. (1994) - Side weir analysis using elementary discharge coefficient, Proceeding. ASCE, J. of Irrigation and Drainage Division, vol. 120, no. 4, 743-755.