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Fluid Mechanics LaboratoryDepartment of Civil Engineering and Construction Engineering Management

California State University, Long Beach

Lab # 3

Flow over Weirs(Prepared by Dr. Rebeka Sultana)

Objectives

The purpose of this experiment is to demonstrate the characteristics of flow over weirs. Weirs ofdifferent geometric designs are used and the theoretical flow over the weirs is calculated usingexperimental data in the theoretical equations. Then theoretically estimated values are comparedto the actual discharge determined by independent measurements. The ratio of actual totheoretical discharge is used to find the discharge coefficient for each type of weirs.

General Discussion

A weir is a partial obstruction on a channel bottom over which fluid must flow. Weirs provide aconvenient method of determining flowrate in open channel by a single measurement of flowdepth. Fluid accelerates over the obstruction with a free liquid surface. A definite relation exists

between the flow rate and the difference in elevation between the fluid surface ahead of the weirand the elevation of the fluid over the weir obstruction.

There are two types of weirs broad-crested and sharp-crested weirs. A broad-crested weir has a broad horizontal crest above which the fluid pressure is considered hydrostatic. A typical broad-crested weir is shown in Figure 1.

Sharp-crested weir, most common type of weir and used in this experiment, is essentially avertical sharp-edged flat plate across the channel. The weir is formed by a relatively thin vertical

plate which has a sharp edged top. This top portion of the restriction, over which the fluid flows,

Figure 1 Flow over a typical broad-crested weir (Munson et al., 2012)

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is called the crest. The stream of fluid flowing over the crest is called the nappe. Figure 2 showsa flow over a typical sharp-crested weir.

Figure 2 Flow over a typical sharp-crested weir (Munson et al., 2012)

The flow of fluid over the weir can be analyzed by applying continuity and Bernoullis equationsto a streamline approaching and then passing over the weir. For this analysis, Bernoullisequation is applied with the following assumptions (1) the velocity profile upstream of the weir

plate is uniform, (2) fluid streamlines at the nappe are parallel to one another so that the pressureat the nappe can be assumed atmospheric, and (3) velocity profile is non uniform at the nappe.

Figure 3 Flow over a sharp-crested weir with (a) velocity profile, and (b) front view of the weir.(Munson et al., 2012)

Bernoullis equation along an arbitrary streamline A B indicated in Figure 3 can be written as:

g V

z P

g V

z P B

B B A

A A

22

22

(1)

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g u

h P H g

V P H ww 2

)(02

22

21 (2)

where P A, z A, V A, and P B, z B, V B, are the pressure, elevation, and velocity at point A and B,respectively, P w is the height of the weir plate, H is the height of free surface above the weircrest, h is the distance that point B is below the free surface. The velocity at point A is same asvelocity at the section 1 which is assumed to be uniform and so V A = V 1. Velocity at point B is u 2 which is non uniform and pressure at B is assumed to be atmospheric (i.e., P B = 0). The locationof point A is arbitrary but total energy for any particular point along the vertical section (1) is

same, so g

V P H

g V

z P

w A A

22

21

21

. Therefore, from Equation (2) the following can be

obtained,

)2

(22

1

2 g

V h g u (3)

Applying continuity equation at the nappe section, theoretical flow rate Q t over the weir can becalculated as:

H h

h

H h

ht bdhudhudAuQ

0 20 22 (4)

where )(h is the cross-sectional width of a strip of the weir area, as shown in Figure 3(b).For rectangular weir is constant and is equal to width b. For other weirs, such as triangular or

circular weirs, is a function of h. From equation (4),

H t dh g V

hb g Q0

2/12

1 )2

(2 , or

2/32

12/32

1 )2

0()2

(232

g V

g V

H b g Q t (5)

The upstream velocity is negligible compared to the velocity at the nappe, thus g

V 2

21

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to calculate equation (6). So, actual flow rate Qa can be determined using a correction factor,known as discharge coefficient, C d .

t d a QC Q (7)

For rectangular weir, actual discharge coefficient can be derived from equation (6) as:

2/3232

bH g C QC Q dr tr dr ar (8)

where Qar , Q tr , C dr are the actual flow rate, theoretical flow rate, discharge coefficient from therectangular weir. For any type of weirs, a discharge coefficient ( C d ) must be experimentallydetermined for the respective weir for accurate flow rate estimates. Once C d is determined for a

particular weir, the weir is said to be calibrated and flow rates can be measured accurately. Thevalue of C dr can be approximately calculated as

wdr P

H C 075.0611.0 (9)

For small flow rates, the head over the nappe is small and does not spring clear over therectangular weir crest because of surface tension. Thus the pressure distribution in the nappe isnot completely ventilated and unknown. So, flow rate cannot be measured accurately. Thissituation is minimized by using a triangular weir which has a sharp edged triangular notchopening and for low flow rates, reasonable heads are developed and the nappe springs clear overthe crest. Analysis of the triangular weir yields the following flow rate equation

2/5

2tan2

158

H g C QC Q dt tt dt at

(10)

where Qat , Q tt , and are the actual flow rate, theoretical flow rate and the notch angle as shownin Figure 4.

Figure 4 A typical triangular weir

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Typical values of C dt for triangular weirs are in the range of 0.58 to 0.62.

Equipment

1. The weir experiment consists of a stilling baffle (Fig. 5), a rectangular/V-notch weir plate, a vernier hook and point gauge. The hydraulics bench incorporates a weir channelwhere the stilling baffle and the rectangular or V- notch weir plate are installed in thechannel carrier by thumb nuts as shown in the diagram.

2. The vernier hook and point gauge is mounted on the channel carrier which allows themeasurement of the depth of flow above the base of the notch.

Fig 5. Weir section instrument over the hydraulic bench

3. Hydraulic bench will be used to regulate flow over the weirs.4. A stopwatch is required to record time to collect water in the tank within the hydraulic

bench.

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Fig 6. Test section for flow over V-notch weir demonstration

5. Width of the rectangular weir b = 0.03 m and angle of V- notch weir = 90o

.

Procedure

1. After the weir equipment is set on the hydraulic bench, the height gauge will be used tomeasure the datum height and the height of the water level. Position the instrumentcarrier with the height gauge above the gauge and lower the gauge until the point is justabove the notch base. Then lock the coarse adjustment screw. Then, using the fineadjustment, adjust the gauge until the point just touches the notch bottom and take thereading of the datum height, h o.

2. Next move the instrument carrier with height gauge approximately halfway between thestilting baffle and the notch plate.

3. Gradually open the bench control valve to admit water in the channel. To get the firstexperimental data, adjust the valve to give approximately 10 mm depth of water abovethe notch base. It will be useful to pre-set the height gauge position to give a roughguide. Note: A very low flow is adequate to develop a depth of 10 mm of water above thenotch and a marker can be used to mark the 10 mm depth on the weir plate as a guide.

4. When the flow is steady, take the water level height, h using the scale in the instrumentcarrier.

5. Find the flow rate in the channel by recording the time to collect known volume water

(for example, 4 L) in the tank. Hydraulic bench has two tanks. For low flow, use the scale(i.e., the lower scale) for low flow tank and for high flow; use the scale (i.e., the upperscale) for high flow.

6. Gradually increase the flow rate in the channel by opening the control valve in thehydraulic bench. Record the water level height, h, and flow rate using the steps 5 and 6.

7. Repeat the steps 5, 6 and 7 to collect at least 4 more data sets.

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8. Next remove the rectangular notch from the hydraulic bench and mount the V-notch plate.

9. Get the first reading by opening the control valve in the hydraulic bench. Adjust thevalve to give approximately 10 mm depth of water in the channel.

10. Repeat the steps 5 through 8 to collect at least 6 more data sets.

Record experimental data in Table 1 and 2.

Calculations

For each flow rates, calculate the followings to complete the Table 1 and Table 2:

1. Experimental discharge in the channel (i.e., Q exp= volume /time t) column 5.2. Height above the notch, H ( H = h ho ) column 6.3. Calculate the discharge coefficient C d for the rectangular and V-notch weirs using the

relationships in equation 7, 8 and 10 column 8.

Discussions

Discuss your results by addressing the followings-1. Compare the experimental results of discharge coefficient to that of the theory. 2. What are the limitations of the theory?3. Why would you expect wider variations of C d values at lower flow rate?

References

Armfield, 2012, Flow over weirs, Instruction Manual.

Munson, B. R., T. H. Okiishi, W. W. Huebsch, A. P. Rothmayer, 2012, Fundamentals of FluidMechanics, 7 th edition, John Wiley, Chapter 10.

CE 336 Fluid Mechanics student manual, 1993, CSULB.

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Table 1: Data Table for Rectangular weir

Datumheight

WaterLevel

Volumecollected

Time tocollect

VolumeFlow rate

Heightabovenotch

H Dischargecoefficient

ho h t Q exp H T C dr (m) (m) (m ) (sec) (m /s) (m) (m )

Table 2: Data Table for V-notch weir

Datumheight

WaterLevel

Volumecollected

Time tocollect

VolumeFlow rate

Heightabove

notch

H Dischargecoefficient

ho h t Q exp H T C dt

(m) (m) (m ) (sec) (m /s) (m) (m )