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Analysis of flow through lateral rectangular orifices in open channels

https://doi.org/10.1016/j.flowmeasinst.2014.02.002Get rights and content

Highlights

  • We have revised the concept used by Ramamurthy et al. [14] in deriving the equations for flow through side orifice.

  • Relationships for flow through side rectangular orifice were developed.

  • Computed discharges using the proposed relationships are well within ±5% of observed discharge.

  • The equation for coefficient of discharge given by Carballada [3] is modified.

Abstract

A side lateral orifice in open channel is hydraulic control structure widely used in hydraulic, irrigation and environmental engineering for diverting the flow from main channel to a secondary channel. In this paper, analytical relationships for the discharge through side orifice are developed accounting for the pressure distribution over the area of the orifice. The computed discharges using the proposed relationship are within ±5% of the observed values; however percentage error is more in case the discharge is computed using earlier equations.

Introduction

Side orifices are used to divert flow from main channel to secondary channels and widely used in hydraulic, environmental and irrigation engineering. The flow is diverted from a channel to different fields through prescribed side orifices for the irrigation purpose. In water and wastewater treatment plants, side orifices are often used to distribute incoming flow to parallel process units such as flocculation basins, sedimentation tanks, and aeration basins [14]. The mechanics of flow through a single orifice located in the side of open channels have been studied by many researchers. Detailed study on side rectangular orifice or slot has been carried out by Gill [4], Ramamurthy et al. [14], Ojha and Subbaiah [12] and Hussain et al. [8]. Hussain et al. [7] carried out an experimental study on side circular orifice and highlighted a discharge equation. Other than side slot or side orifice, side sluice gate and side weir are used to divert the flow from main channel to a secondary channel. Studies on such devices are extensively reported in literature [13], [1], [9], [5], [15].

Ramamurthy et al. [14] derived an expression for the flow through a rectangular side orifice discharging from a rectangular channel. Fig. 1 shows a typical sketch of the side orifice in an open channel. They considered the velocity of jet Vj issuing out from the orifice as resultant of the velocity in the channel V1 and velocity normal to the channel due to static head H and expressed discharge through orifice asQ=H2H1CdVjLdHwhere Vj=V12+2gH; H1=YmW; and H2=YmWb

Here H1 is the head above the lower crest of the orifice; H2 is the head above the upper crest of the orifice; Cd is the coefficient of discharge; L is the width of the rectangular orifice; b is the height of the rectangular orifice; Ym is the depth of flow in the main channel; g is the acceleration due to gravity and W is sill height. They used the following equation for the coefficient of discharge [3]:Cd=0.611+C1η2+C2η4+C3η6where velocity ratioη=V1Vj=V1V12+2gHandC1=0.254(LB)0.538;C2=0.234(LB)+0.058;C3=0.489(LB)0.129

where B is the width of the channel.

Using Eqs. 1–4, they derived the following relation for the discharge through the orifice:Q=V13Lg[f(η1,LB)f(η2,LB)]wheref(η1,LB)=(1η13)(C33+0.6113η13)+(1η1)(C1η1+C2)andf(η2,LB)=(1η23)(C33+0.6113η23)+(1η2)(C1η2+C2)Considering that the velocity of the upper layers will be higher than mean velocity V1, Ramamurthy et al. [14] introduced a flow reduction factor 0.95 in discharge computation which partly accounts for the effect of non-uniform velocity distribution,Q=0.95V13Lg[f(η1,LB)f(η2,LB)]

The assumptions made in deriving the above discharge equation were that the channel is horizontal and the flow is subcritical. The validity of Eq. (8) was checked by [8] using experimental data collected on square side orifices of size L=0.044 m, 0.089 m and 0.133 m in an open channel of 9.15 m length, 0.50 m width and 0.60 m depth. The crest heights of the orifices were at 0.5, 0.10, 0.15 and 0.20 m. They have used ultrasonic flow meter for the calibration of weir used for measuring discharge from the side orifice and main channel. They reported that the Ramamurthy et al. [14] equation underestimates the discharge as shown in Fig. 2. This could be due to erroneous formulation of Eq. (1) where angle between the area and velocity vectors are not taken into consideration. The present paper deals with analytical consideration of discharge equation for the side orifices with due consideration of directions of velocity and area vector while computing the discharge i.e., Q=VA.

Section snippets

Analytical consideration

Considering varying pressure head over the flow area of a lateral rectangular orifice of size L fitted in the side of an open channel at sill height W, discharge through the lateral orifice may be written asQ=H2H1CdVLdHQ=H2H1Cd2gHLdHSubstitution of Eqs. (2), (3) into Eq. (10) yieldsQ=0.611L2gH2H1H1/2dH+C1L2gH2H1V12(V12+2gH)H1/2dH+C2L2gH2H1V14(V12+2gH)2H1/2dH+C3L2gH2H1V16(V12+2gH)3H1/2dHAfter the integration of the terms on the right hand side of Eq. (11), it may be written asQ=I1+I2+I3+I4

Results and discussions

Generally, 70–80% of the total data are used for the calibration of an equation, and 20–30% for its validation [10], [8], [7], [11], [2], [6]. Hence Eq. (19) is validated using the remaining unused 20% data sets of Hussain et al. [8] for rectangular side orifices. Such validation is shown in Fig. 5, which depicts that the computed discharge is well within ±5% of the observed values. While the computed discharge from Ramamurthy et al. [14] equation for 20% unused data have more than 5% error as

Conclusions

A discharge equation for lateral orifices in open channel has been derived with due consideration of direction of velocity and area vectors and also polynomial variation of coefficient of discharge with velocity ratio. The derived discharge equation is checked for its accuracy using data of Hussain et al. [8] for rectangular orifices. The computed discharges using the proposed equation were within ±5% of the observed values while computed discharge from Ramamurthy et al. [14] equation has

References (15)

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