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Department of Civil and Environmental Engineering

CIVL 2510 Fluid Mechanics

Laboratory Exercise 2: Flow Rate Measurements

Group: 3B2

Members:

20036043 Lam Kin On

20017011 Lee Kai Fung

10100092 Lee Chun Yat

20020628 Chow Jun Kang

Objectives:

The objectives of this experiment laboratory exercise are:

1. To understand how two different weirs can be used to determine the flow rate in an openchannel.

2. To determine the theoretical relationships for the flow rate over a weir.3. To carry out measurement to determine an experimental relationship for the flow rate over a

weir.

4. To obtain estimates for the discharge coefficient, relating the actual flow rate to the ideal flowrate.

Apparatus:

1. Hydraulics bench2. Rectangular-notch weir3. V-notch weir4. Vernier Height Gauge5. StopwatchProcedures:

1. The rectangular notch weir was inserted into the hydraulic bench. The weir was fitted tightly byusing screws to prevent leakage.

2. The Vernier Height Gauge was adjusted so that its pointer just touched the bottom of the notch,then it was set to zero. The bottom was taken as datum, attention was given to make that

Vernier Height Gauge did not slide along the mast during the experiment.

3. The water supply was opened and water was allowed to flow over the weir. (H was not equal tozero).

4. The water supply was adjusted. About a minute was waited for until the water flow over theweir had reached steady state (no longer changed with time). The head, H, was measured using

the Vernier Height Gauge. Then a second measurement for H was obtained by different

member of the group for better accuracy.

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5. The stopwatch was used to record the time it took for 10 litres of water to flow over the weir.This step was repeated for better accuracy. For small H, volume of water was reduced (to 5

litres).

6. Steps 4 & 5 were repeated for at 10 different settings of the water supply. For each setting, theH and the corresponding times were measured and recorded.u

7. Then, the rectangular notch weir was changed to V-notch weir. The weir was also fitted tightlyby using screws to prevent leakage.

8. Steps 2 to 6 were repeated for the V-notch weir.9. For the calculation of the discharge coefficient, record the breadth of the rectangular notch and

the vertex angle for the V-notch weir.

Task 1 Derivation of equations

The Bernoulli equation is:

To find the velocity of the flow through the elementary area, assume point 1 is at the free surface of

the reservoir, while point 2 is at the centre of elementary area, a vertical distance h below point 1. In

addition assume that the velocity of the water behind the weir is negligible and the pressure at both

points is atmospheric.

The equation becomes

( )

For flow rate over a rectangular notch weir, From the result obtained above, Where

is the apparent ideal flow rate through elementary area

is the elementary area in the plane of crest with a horizontal slot of length B and heightdh.

To obtain the total ideal flow rate, integrate over the whole area, from to .

*

+

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While for the actual flow rate, a coefficient of discharge, will be introduced, the flow rateequation becomes

While for the flow rate over a V-notch weir, an additional parameter, the vertex angle isintroduced. The length of the crest B is no longer a constant, but depends on distance h below the

level of crest.

Write length of crest at different level (in terms ofh),

The ideal flow rate equation becomes .To obtain the total ideal flow rate, integrate over the whole area, from to .

*

+

[ ]

While for the actual flow rate, a coefficient of discharge, will be introduced, the flow rateequation becomes

Task 2 Estimation of experimental accuracy

Few errors arise throughout the experiment.

Sources of error:

1. Parallax error occurred when measuring the meniscus of the water level. To avoid it, the eye levelshould be placed at the bottom of the meniscus of water level.

2. Readings were taken when the water flow had not reached steady state. To avoid it, water supplyshould be adjusted and wait for about a minute to ensure the water reach the steady state.

3. Water vibration occur which affect the reading ofH. To avoid it, care needs to be taken to ensurethe no source could vibrate the water level.

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4. Different people have different reaction time which caused the time taken was not accurate.5. Vernier Height Gauge may not be exactly adjusted on the surface on water and this affected the

reading ofH.

6. Rectangular and V-notch weir may not be tightly fitted and this caused the leakage of water thusaffected the total volume of water flow.

7. The water surface may not a horizontal plane. Reading was affected while reading was taken atdifferent points of the water surface.

Random errors versus Systematic errors:

1. Random error, also known as precision error, is the difference between the value of an individualmeasurement and the average of a number or repeated measurements. Smaller random error

indicates better repeatability of the measurement. Random errors could be decreased by taking

the average of a large number of individual measurements.

2. While systematic error, also known as accuracy error, is the difference between the value of anindividual measurement and true value. It is related to experimental setup and procedure hence,

it would not decrease by obtaining an average reading. When systematic errors are discovered,

setup and procedure should be modified to obtain improvements before new measurements are

obtained.

Assumption on systematic errors:

This assumption ignores the errors from measurement tools, which is Vernier Height Gauge for this

laboratory exercise. Consequently, this would increase the different between actual value and the

measured value since tools are assumed with high accuracy and precision with no error. However in

all circumstances, measurement tools are not as accurate as we measured.

Appropriate Number of Significant digits:

The least significant numerical in a number implies the precision of the measurement (or the

calculation based on the measurement). Below are the accuracy of the measurement tools used:

Volume measurement - Stop watch - Vernier Height Gauge -

After performing calculations of several parameters, final result is expressed with 3 significantfigures.

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Task 3 Obtain experimental measurements

Rectangular-notch Weir

H (mm) H (mm) Volume (L) Time (s) Time (s)

Trial 1 64.4 64.4 10 12.05 12.35

Trial 2 62.8 62.9 10 12.66 12.62Trial 3 59.2 59.2 10 14.31 14.27

Trial 4 56.5 56.5 10 15.64 15.41

Trial 5 55.5 55.5 10 17.58 17.74

Trial 6 53.0 53.0 10 18.55 18.35

Trial 7 50.0 50.0 10 20.65 20.02

Trial 8 46.5 46.5 10 25.50 26.11

Trial 9 41.5 41.5 10 31.31 30.03

Trial 10 35.4 35.4 10 48.04 49.13

V-notch WeirH (mm) H (mm) Volume (L) Time (s) Time (s)

Trial 1 39.0 39.0 5 11.24 10.91

Trial 2 37.3 37.3 5 12.63 12.62

Trial 3 35.5 35.6 5 12.81 12.97

Trial 4 34.4 34.4 5 15.01 15.25

Trial 5 33.0 33.0 5 17.15 16.83

Trial 6 31.0 31.0 5 20.11 20.15

Trial 7 30.0 30.0 5 21.41 21.40

Trial 8 28.7 28.7 5 23.74 24.07

Trial 9 27.0 27.0 5 27.35 27.20

Trial 10 25.9 25.9 5 30.92 30.88

Task 4 Calculations and Discussion

Consider rectangular notch weir. For analysis purpose, average H and time were calculated.

H

(mm)

H

(mm)

Mean of H

(mm)

Volume

(L)

Time

(s)

Time

(s)

Mean of Time

(s)

Trial 1 64.4 64.4 64.4 10 12.05 12.35 12.2

Trial 2 62.8 62.9 62.9 10 12.66 12.62 12.6

Trial 3 59.2 59.2 59.2 10 14.31 14.27 14.3

Trial 4 56.5 56.5 56.5 10 15.64 15.41 15.5

Trial 5 55.5 55.5 55.5 10 17.58 17.74 17.7

Trial 6 53.0 53.0 53.0 10 18.55 18.35 18.5

Trial 7 50.0 50.0 50.0 10 20.65 20.02 20.3

Trial 8 46.5 46.5 46.5 10 25.50 26.11 25.8

Trial 9 41.5 41.5 41.5 10 31.31 30.03 30.7

Trial 10 35.4 35.4 35.4 10 48.04 49.13 48.6

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Data for graph of flow rate Q as a function ofH.

Mean of H (m)

Volume

(m3)

Mean of Time

(s)

Flow rate, Q

(m3/s)

Trial 1 0.064 0.0100 12.2 0.000820

Trial 2 0.063 0.0100 12.6 0.000791Trial 3 0.059 0.0100 14.3 0.000700

Trial 4 0.057 0.0100 15.5 0.000644

Trial 5 0.056 0.0100 17.7 0.000566

Trial 6 0.053 0.0100 18.5 0.000542

Trial 7 0.050 0.0100 20.3 0.000492

Trial 8 0.047 0.0100 25.8 0.000388

Trial 9 0.042 0.0100 30.7 0.000326

Trial 10 0.035 0.0100 48.6 0.000206

However, it is difficult to determine the accuracy of the theoretical relationship without an

independent way to determine the coefficient of discharge. Thus, in order to confirm thedependence of Q on H, take the logarithm of both sides of the equation for the rectangular notch

weir, the equation becomes:

( )

0.000100

0.000200

0.000300

0.000400

0.000500

0.000600

0.000700

0.000800

0.000900

0.030 0.035 0.040 0.045 0.050 0.055 0.060 0.065 0.070

Q against HQ (m3/s)

H (m)

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Data for graph of as a function of.Mean of H

(m)

Volume

(m3)

Mean of Time

(s) Flow rate, Q (m3/s) log H log Qd

Trial 1 0.0644 0.0100 12.2 0.000820 -1.19 -3.09

Trial 2 0.0629 0.0100 12.6 0.000791 -1.20 -3.10Trial 3 0.0592 0.0100 14.3 0.000700 -1.23 -3.16

Trial 4 0.0565 0.0100 15.5 0.000644 -1.25 -3.19

Trial 5 0.0555 0.0100 17.7 0.000566 -1.26 -3.25

Trial 6 0.0530 0.0100 18.5 0.000542 -1.28 -3.27

Trial 7 0.0500 0.0100 20.3 0.000492 -1.30 -3.31

Trial 8 0.0465 0.0100 25.8 0.000388 -1.33 -3.41

Trial 9 0.0415 0.0100 30.7 0.000326 -1.38 -3.49

Trial 10 0.0354 0.0100 48.6 0.000206 -1.45 -3.69

Theoretically, if we plot the log (Qd) against log (H) graph, the slope of the curve should be . Fromthe graph, the slope obtained is 2.279358

Absolute error = | | . The percentage of absolute error is relatively high. It ismainly due to the random errors occurred during the experiment. These random errors included the

different reaction time in taking the time for 10 L of water to flow,

Using ideal flow rate equation,

-3.90

-3.70

-3.50

-3.30

-3.10

-2.90

-2.70

-2.50

-1.50 -1.40 -1.30 -1.20 -1.10 -1.00

log (Qd) against log (H)

log (H)

log (Qd

)

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Substitute the discharge rate, Q found into the above equation we obtain the following data:

Mean of H (mm) H (m) H3/2

(m3/2

) (Discharge rate) Q (m3/s)

Trial 1 64.4 0.0644 0.0163 0.000820

Trial 2 62.9 0.0629 0.0158 0.000791

Trial 3 59.2 0.0592 0.0144 0.000700

Trial 4 56.5 0.0565 0.0134 0.000644

Trial 5 55.5 0.0555 0.0131 0.000566

Trial 6 53.0 0.0530 0.0122 0.000542

Trial 7 50.0 0.0500 0.0112 0.000492

Trial 8 46.5 0.0465 0.0100 0.000388

Trial 9 41.5 0.0415 0.0085 0.000326

Trial 10 35.4 0.0354 0.0067 0.000206

From the graph, we found that the slope is 0.0640, which means:

; where

While directly substitute the different values ofH found into the ideal flow rate equation we obtain

the following data:

0.000300

0.000400

0.000500

0.000600

0.000700

0.000800

0.000900

0.0200 0.0300 0.0400 0.0500 0.0600 0.0700

Q against H3/2Q (m3/s)

H3/2 (m3/2)

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Mean of H

(mm) H (m)

H3/2

(m3/2

)

Ideal Flow rate,

Qi

Discharge rate,

Q (m3/s) Q/Qi

64.4 0.0644 0.0163 0.00145 0.000820 0.566

62.9 0.0629 0.0158 0.00140 0.000791 0.567

59.2 0.0592 0.0144 0.00128 0.000700 0.548

56.5 0.0565 0.0134 0.00119 0.000644 0.541

55.5 0.0555 0.0131 0.00116 0.000566 0.489

53.0 0.0530 0.0122 0.00108 0.000542 0.501

50.0 0.0500 0.0112 0.00099 0.000492 0.497

46.5 0.0465 0.0100 0.00089 0.000388 0.436

41.5 0.0415 0.0085 0.00075 0.000326 0.435

35.4 0.0354 0.0067 0.00059 0.000206 0.349

Where mean of Q/Qi = 0.493, which indicates

which is deviates largely from the value

found above. This could be due to the error occurred throughout the experiment.

While consider V-notch weir. For analysis purpose, average H and time were calculated.

H (mm) H (mm)

Mean of H

(mm)

Volume

(L)

Time

(s)

Time

(s)

Mean of time

(s)

Trial 1 39.0 39.0 39.0 5 11.24 10.91 11.1

Trial 2 37.3 37.3 37.3 5 12.63 12.62 12.6

Trial 3 35.5 35.6 35.6 5 12.81 12.97 12.9

Trial 4 34.4 34.4 34.4 5 15.01 15.25 15.1

Trial 5 33.0 33.0 33.0 5 17.15 16.38 16.8Trial 6 31.0 31.0 31.0 5 20.11 20.15 20.1

Trial 7 30.0 30.0 30.0 5 21.41 21.4 21.4

Trial 8 28.7 28.7 28.7 5 23.74 24.07 23.9

Trial 9 27.0 27.0 27.0 5 27.35 27.2 27.3

Trial 10 25.9 25.9 25.9 5 30.92 30.88 30.9

Data for graph of flow rate Q as a function ofH.

Mean of H (m)

Volume

(m3)

Mean of time

(s)

Flow rate, Q

(m3/s)

Trial 1 0.0390 0.00500 11.1 0.000451

Trial 2 0.0373 0.00500 12.6 0.000396

Trial 3 0.0356 0.00500 12.9 0.000388

Trial 4 0.0344 0.00500 15.1 0.000330

Trial 5 0.0330 0.00500 16.8 0.000298

Trial 6 0.0310 0.00500 20.1 0.000248

Trial 7 0.0300 0.00500 21.4 0.000234

Trial 8 0.0287 0.00500 23.9 0.000209

Trial 9 0.0270 0.00500 27.3 0.000183Trial 10 0.0259 0.00500 30.9 0.000162

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However, it is difficult to determine the accuracy of the theoretical relationship without an

independent way to determine the coefficient of discharge. Thus, in order to confirm the

dependence of Q on H, take the logarithm of both sides of the equation for the rectangular notch

weir, the equation becomes:

( ) Data for graph of as a function of.

Mean of H

(m)

Volume

(m3)

Mean of Time

(s)

Flow rate, Q

(m3/s) log H log Qd

Trial 1 0.0644 0.0100 12.2 0.000820 -1.19 -3.09

Trial 2 0.0629 0.0100 12.6 0.000791 -1.20 -3.10

Trial 3 0.0592 0.0100 14.3 0.000700 -1.23 -3.16

Trial 4 0.0565 0.0100 15.5 0.000644 -1.25 -3.19Trial 5 0.0555 0.0100 17.7 0.000566 -1.26 -3.25

Trial 6 0.0530 0.0100 18.5 0.000542 -1.28 -3.27

Trial 7 0.0500 0.0100 20.3 0.000492 -1.30 -3.31

Trial 8 0.0465 0.0100 25.8 0.000388 -1.33 -3.41

Trial 9 0.0415 0.0100 30.7 0.000326 -1.38 -3.49

Trial 10 0.0354 0.0100 48.6 0.000206 -1.45 -3.69

0.000100

0.000150

0.000200

0.000250

0.000300

0.000350

0.000400

0.000450

0.000500

0.0200 0.0250 0.0300 0.0350 0.0400

Q against HQ (m3/s)

H (m)

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Theoretically, if we plot the log (Qd) against log (H) graph, the slope of the curve should be. From

the graph, the slope obtained is 2.52652.

Absolute error = | | , which is acceptable.Substitute the discharge rate, Q found into the above equation we obtain the following data:

Mean of H

(mm)

Mean of H

(m) H5/2

(m5/2

)

Ideal flow

rate, Qi

Discharge

rate, Q (m3/s) Q/Qi

39.0 0.0390 0.000300 0.000710 0.000451 0.636

37.3 0.0373 0.000269 0.000635 0.000396 0.624

35.6 0.0356 0.000238 0.000563 0.000388 0.689

34.4 0.0344 0.000219 0.000518 0.000330 0.637

33.0 0.0330 0.000198 0.000467 0.000298 0.638

31.0 0.0310 0.000169 0.000400 0.000248 0.621

30.0 0.0300 0.000156 0.000368 0.000234 0.634

28.7 0.0287 0.000140 0.000330 0.000209 0.634

27.0 0.0270 0.000120 0.000283 0.000183 0.648

25.9 0.0259 0.000108 0.000255 0.000162 0.634

-3.90

-3.70

-3.50

-3.30

-3.10

-2.90

-2.70

-2.50

-1.50 -1.40 -1.30 -1.20 -1.10 -1.00

log (Qd) against log (H)

log (H)

log (Qd)

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From the graph, we found that the slope 1.52, which means:

, where

While directly substitute the different values ofH found into the ideal flow rate equation we obtain

the following data:

Mean of H (m) H5/2

(m5/2

) Discharge rate, Q (m3/s)

Trial 1 0.0390 0.000300 0.000710

Trial 2 0.0373 0.000269 0.000635

Trial 3 0.0356 0.000238 0.000563

Trial 4 0.0344 0.000219 0.000518

Trial 5 0.0330 0.000198 0.000467

Trial 6 0.0310 0.000169 0.000400Trial 7 0.0300 0.000156 0.000368

Trial 8 0.0287 0.000140 0.000330

Trial 9 0.0270 0.000120 0.000283

Trial 10 0.0259 0.000108 0.000255

Where mean of Q/Qi = 0.640, which indicates , which is almost identical with resultfound above.

0.000200

0.000250

0.000300

0.000350

0.000400

0.000450

0.000500

0.000050 0.000100 0.000150 0.000200 0.000250 0.000300 0.000350

Q against H5/2Q (m3/s)

H5/2 (m5/2)

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Discussions

From the graph of log (Qd) against log (H), we could observe that the coefficient of discharge should

be 1.50 and 2.50 for rectangular-notch weir and V-notch weir respectively. However, value obtained

through the experiment is 2.28 and 2.52 for rectangular-notch weir and V-notch weir respectively.

Many errors were arisen for the set of experiment using rectangular-notch weir. This could be

probably due to random errors occurred (eg: parallax error occurred when measuring the meniscus

of the water level, different reaction time of each members when measuring time taken for 5L/10L

of water to flow).

Furthermore, assumptions made could also contribute to the errors. For example, fluid is assumed

to be incompressible, which indicates its density remains unchanged throughout the experiment.

Besides, fluid is assumed to be non-viscous, which indicates no resistance to the motion.

There are other conditions which caused the error to arise. Difficult in reading the desired water

level while water is rising is one of the situations. The surface tension of the water in the tube couldalso reduce the accuracy of recording the time taken for water to flow.

Conclusions

For rectangular-notch weir, its discharge coefficient is 72.2% (49.3% for 2nd

method). While for the V-

notch weir, its corresponding discharge coefficient is 64.3% (64.0% for 2nd

method).

Since both values are less than 1, they are accepted solution.