ASIAN INSTITUTE OF TECHNOLOGY SCHOOL OF ENGINEERING AND TECHNOLOGY WATER ENGINEERING AND MANAGEMENT
CE74.9001A (RESEARCH DESIGN AND EXPERIMENTAL METHODS)
LABORATORY EXERCISE: 1 CALIBRATION OF WEIR (STEADY AND UNSTEADY CASES)
Submitted by Mr. Sajal Kumar Adhikary Student ID: 105599 Group No: 6
March 10, 2008
Lab Exercise: 1
Calibration of Weir
1 Introduction The weir is one of the oldest structures used to measure the flow rates of water in open channels and rivers in connection to water supply and sewage plants. The discharge over a weir is a function of the weir geometry and of weir head. There are different types of weirs, classified according to the geometric shape of the portion of the weir over which the flowing water is in contact. Rectangular and V-notch (triangular) are the most common shapes for weirs. If properly calibrated, weirs can measure the flow rate to within plus or minus 2 percent. By simply knowing the dimensions of the notch of the weir, as well as the height of water above the crest (head of water, H), it is possible calculate the amount of flow over the weir, by using the theoretical equations for discharge over diff erent type weirs. 2 Objectives The overall objective of this lab exercise is to familiar with a hands-on experience about the different techniques of weir calibration in the laboratory and to obtain a through knowledge to compare the results us ing corresponding analytical solutions to find limitations (if any) and its application in practical problems of measurement techniques. The specific objectives of this exercise is: •
To determine the head-discharge relationship (calibration equation) of a 90 0 V-notch weir using the falling head technique (unsteady flow conditions).
•
To calibrate the same weir again under steady state flow conditions (constant head technique) to compare the results obtained for both cases.
3 Experimental Procedures 3.1 Falling Head Method (Unsteady Flow Condition) •
The reference head of the weir is m easured using the manometer, which is attached, on the weir tank.
•
The weir is blocked with a wooden board so that it is free from leakage of water.
•
The tank is filled with water up to the level appropriate for measurement.
•
The clock is kept close to the gauge attached to the side of the weir tank.
•
The video camera is focused on both the gauge and the clock.
•
The video is started for recording and at the same time the clock is turned on while swiftly removing the wooden covers from its position to allow water to flow over weir.
•
Recording is stopped when there is no more appreciable fall of water level in the gauge.
•
The video recording is watched and the water level for each time interval is recorded in the laboratory notebook.
Prepared by: Sajal Kumar Adhikary (105599) / Group No: 6
2
Lab Exercise: 1
Calibration of Weir
3.2 Constant Head Method (Steady Flow Condition) •
The weir reference head is measured using the manometer attached on the weir tank.
•
The desired head is determined for each trial that would cover the range from minimum to maximum flow (10 different head has been taken to have a reasonable range). The appropriate discharge is set in the weir corresponding to the chosen head by
•
adjusting the valve on the by-pass pipe and the flow is allowed to become steady. Now, the measurement of the actual weir head is taken and the corresponding
•
discharge reading on the Electromagnetic Flow Meter attached on the pipeline is recorded. The discharge is changed after each measurement has been finished and is repeated
•
steps 3 and 4 until the whole range of weir head (minimum to maximum) is covered. 4 Experimental Data 4.1 Falling Head Method Table 1: Observed Data (From Video) for Falling Head Method Time
Head, H (cm) for Test No.
Time
Head, H (cm) for Test No.
(sec)
1
2
3
4
(sec)
1
2
3
4
0
43.70
43.20
43.80
43.80
20
15.90
16.00
15.80
15.90
1
39.00
39.50
39.40
39.20
22
15.00
15.10
14.90
14.80
2
35.00
36.00
35.40
35.00
24
14.40
14.50
14.10
14.20
3
32.50
32.30
33.20
32.20
26
13.80
14.00
13.80
13.40
4
30.50
30.00
31.00
31.80
28
13.20
13.30
13.00
12.90
5
28.00
28.00
28.80
28.90
30
13.00
13.00
12.90
12.30
6
26.50
26.50
26.50
26.90
35
12.00
12.10
12.20
11.90
7
25.00
25.10
25.00
25.10
40
11.00
11.20
11.10
10.30
8
24.00
24.00
24.10
24.00
45
10.30
10.00
10.20
9.90
9
23.00
23.10
22.80
22.90
50
9.80
9.80
9.80
9.20
10
22.10
22.50
21.90
21.90
55
9.20
9.20
9.1
8.80
11
21.00
21.00
20.90
20.80
60
8.50
8.80
8.5
8.00
12
20.50
20.70
20.00
19.90
65
8.20
8.30
8.1
7.90
13
19.60
19.50
19.20
19.10
70
7.80
8.00
7.8
7.40
14
19.20
19.30
18.60
19.00
75
7.50
7.60
7.4
7.20
15
18.30
18.20
18.00
17.90
80
7.20
7.20
7.2
7.10
16
17.80
17.40
17.90
17.50
85
7.00
7.10
7.1
6.90
17
17.20
17.10
17.00
16.90
90
6.80
6.60
6.7
6.70
18
16.80
16.70
16.80
16.50
95
6.50
6.40
6.4
6.20
19
16.40
16.30
16.10
16.00
100
6.10
6.10
6.3
6.10
Prepared by: Sajal Kumar Adhikary (105599) / Group No: 6
3
Lab Exercise: 1
Calibration of Weir
4.2 Constant Head Method Table 2: Observed Data for Constant Head Method Test
Head
Discharge
Discharge
No. 1
H (cm) 18.50
Q (m3/h) 97.78
Q (m3/s) 0.02716
2
18.00
77.86
0.02163
3
17.50
70.96
0.01971
4
16.50
64.00
0.01778
5
16.00
59.98
0.01666
6
14.50
48.05
0.01335
7
15.00
49.70
0.01381
8
13.00
34.59
0.00961
9
11.50
19.95
0.00554
10
7.00
6.73
0.00187
11
8.00
13.53
0.00376
Prepared by: Sajal Kumar Adhikary (105599) / Group No: 6
4
Lab Exercise: 1
Calibration of Weir
5 Data Analysis and Results 5.1 Falling Head Method Table 3: Observed and calculated data for 1 st experiment Time (sec) 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 22 24 26 28 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100
∆t
(sec) 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 5 5 5 5 5 5 5 5 5 5 5 5 5 5
H (cm) 43.70 39.00 35.00 32.50 30.50 28.00 26.50 25.00 24.00 23.00 22.10 21.00 20.50 19.60 19.20 18.30 17.80 17.20 16.80 16.40 15.90 15.00 14.40 13.80 13.20 13.00 12.00 11.00 10.30 9.80 9.20 8.50 8.20 7.80 7.50 7.20 7.00 6.80 6.50 6.10
H (m) 0.437 0.390 0.350 0.325 0.305 0.280 0.265 0.250 0.240 0.230 0.221 0.210 0.205 0.196 0.192 0.183 0.178 0.172 0.168 0.164 0.159 0.150 0.144 0.138 0.132 0.130 0.120 0.110 0.103 0.098 0.092 0.085 0.082 0.078 0.075 0.072 0.070 0.068 0.065 0.061
V (m3)
Prepared by: Sajal Kumar Adhikary (105599) / Group No: 6
1.603 1.430 1.284 1.192 1.119 1.027 0.972 0.917 0.880 0.844 0.811 0.770 0.752 0.719 0.704 0.671 0.653 0.631 0.616 0.601 0.583 0.550 0.528 0.506 0.484 0.477 0.440 0.403 0.378 0.359 0.337 0.312 0.301 0.286 0.275 0.264 0.257 0.249 0.238 0.224
Q (m 3
-0.1180 -0.0703 -0.0856 -0.0764 -0.0520 -0.0458 -0.0342 -0.0336 -0.0397 -0.0281 -0.0251 -0.0232 -0.0232 -0.0281 -0.0189 -0.0189 -0.0128
/s)
3 Q (m /s) 0.1180 0.0703 0.0856 0.0764 0.0520 0.0458 0.0342 0.0336 0.0397 0.0281 0.0251 0.0232 0.0232 0.0281 0.0189 0.0189 0.0128
0.0101 -0.0101
0.0098 -0.0098 -0.0065 -0.0039 -0.0037
0.0065 0.0039 0.0037 0.0037
-0.0037
-0.0022 -0.0018 -0.0012
0.0022 0.0018 0.0012
0.1180 0.0703
5
Lab Exercise: 1
Calibration of Weir
Table 4: Observed and calculated data for 2 nd experiment Time (sec) 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 22 24 26 28 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100
∆t
(sec) 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 5 5 5 5 5 5 5 5 5 5 5 5 5 5
H (cm) 43.20 39.50 36.00 32.30 30.00 28.00 26.50 25.10 24.00 23.10 22.50 21.00 20.70 19.50 19.30 18.20 17.40 17.10 16.70 16.30 16.00 15.10 14.50 14.00 13.30 13.00 12.10 11.20 10.00 9.80 9.20 8.80 8.30 8.00 7.60 7.20 7.10 6.60 6.40 6.10
H (m) 0.432 0.395 0.360 0.323 0.300 0.280 0.265 0.251 0.240 0.231 0.225 0.210 0.207 0.195 0.193 0.182 0.174 0.171 0.167 0.163 0.160 0.151 0.145 0.140 0.133 0.130 0.121 0.112 0.100 0.098 0.092 0.088 0.083 0.080 0.076 0.072 0.071 0.066 0.064 0.061
Prepared by: Sajal Kumar Adhikary (105599) / Group No: 6
V (m3) 1.584 1.449 1.320 1.185 1.100 1.027 0.972 0.921 0.880 0.847 0.825 0.770 0.759 0.715 0.708 0.667 0.638 0.627 0.612 0.598 0.587 0.554 0.532 0.513 0.488 0.477 0.444 0.411 0.367 0.359 0.337 0.323 0.304 0.293 0.279 0.264 0.260 0.242 0.235 0.224
Q (m 3 /s)
3 Q (m /s)
-0.1394 -0.1131 -0.0733 -0.0630 -0.0520 -0.0465 -0.0367
0.1394 0.1131 0.0733 0.0630 0.0520 0.0465 0.0367
-0.0330 -0.0263 -0.0257 -0.0196
0.0330 0.0263 0.0257 0.0196
-0.0177
0.0177
-0.0159
0.0159
-0.0119
0.0119
-0.0098
0.0098
-0.0089 -0.0050
0.0089 0.0050
-0.0040 -0.0033 -0.0029 -0.0023
0.0040 0.0033 0.0029 0.0023
-0.0013
0.0013
6
Lab Exercise: 1
Calibration of Weir
Table 5: Observed and calculated data for 3 rd experiment Time (sec) 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 22 24 26 28 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100
∆t
(sec) 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 5 5 5 5 5 5 5 5 5 5 5 5 5 5
H (cm) 43.80 39.40 35.40 33.20 31.00 28.80 26.50 25.00 24.10 22.80 21.90 20.90 20.00 19.20 18.60 18.00 17.90 17.00 16.80 16.10 15.80 14.90 14.10 13.80 13.00 12.90 12.20 11.10 10.20 9.80 9.1 8.5 8.1 7.8 7.4 7.2 7.1 6.7 6.4 6.3
H (m) 0.438 0.394 0.354 0.332 0.310 0.288 0.265 0.250 0.241 0.228 0.219 0.209 0.200 0.192 0.186 0.180 0.179 0.170 0.168 0.161 0.158 0.149 0.141 0.138 0.130 0.129 0.122 0.111 0.102 0.098 0.091 0.085 0.081 0.078 0.074 0.072 0.071 0.067 0.064 0.063
Prepared by: Sajal Kumar Adhikary (105599) / Group No: 6
V (m3) 1.606 1.445 1.298 1.218 1.137 1.056 0.972 0.917 0.884 0.836 0.803 0.767 0.734 0.704 0.682 0.660 0.656 0.623 0.616 0.590 0.579 0.546 0.517 0.506 0.477 0.473 0.447 0.407 0.374 0.359 0.334 0.312 0.297 0.286 0.271 0.264 0.260 0.246 0.235 0.231
Q (m 3 /s)
3 Q (m /s)
-0.1112
0.1112
-0.0801
0.0801
-0.0740
0.0740
-0.0422
0.0422
-0.0361 -0.0318 -0.0251 -0.0238
0.0361 0.0318 0.0251 0.0238
-0.0196
0.0196
-0.0147
0.0147
-0.0107
0.0107
-0.0084 -0.0042 -0.0035
0.0084 0.0042 0.0035
-0.0027 -0.0024
0.0027 0.0024
-0.0018
0.0018
7
Lab Exercise: 1
Calibration of Weir
Table 6: Observed and calculated data for 4 th experiment Time (sec) 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 22 24 26 28 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100
∆t
(sec) 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 5 5 5 5 5 5 5 5 5 5 5 5 5 5
H (cm) 43.80 39.20 35.00 32.20 31.80 28.90 26.90 25.10 24.00 22.90 21.90 20.80 19.90 19.10 19.00 17.90 17.50 16.90 16.50 16.00 15.90 14.80 14.20 13.40 12.90 12.30 11.90 10.30 9.90 9.20 8.80 8.00 7.90 7.40 7.20 7.10 6.90 6.70 6.20 6.10
H (m) 0.438 0.392 0.350 0.322 0.318 0.289 0.269 0.251 0.240 0.229 0.219 0.208 0.199 0.191 0.190 0.179 0.175 0.169 0.165 0.160 0.159 0.148 0.142 0.134 0.129 0.123 0.119 0.103 0.099 0.092 0.088 0.080 0.079 0.074 0.072 0.071 0.069 0.067 0.062 0.061
V (m3) 1.606 1.438 1.284 1.181 1.166 1.060 0.987 0.921 0.880 0.840 0.803 0.763 0.730 0.700 0.697 0.656 0.642 0.620 0.605 0.587 0.583 0.543 0.521 0.491 0.473 0.451 0.436 0.378 0.363 0.337 0.323 0.293 0.290 0.271 0.264 0.260 0.253 0.246 0.227 0.224
Q (m3 /s)
3 Q (m /s)
-0.1406
0.1406
-0.1064 -0.0685 -0.0520
0.1064 0.0685 0.0520
-0.0391 -0.0379 -0.0342
0.0391 0.0379 0.0342
-0.0220
0.0220
-0.0189 -0.0177
0.0189 0.0177
-0.0245 -0.0122
0.0245 0.0122
-0.0084
0.0084
-0.0049 -0.0033
0.0049 0.0033
-0.0032
0.0032
-0.0031
0.0031
Sample of calculation (Table 6 only): Size of the tank = 2.445 m x 1.50 m At time 2 sec: Volume, V (m 3) = 0.35 x 2.445 x 1.50 = 1.284 m 3 1 Discharge, Qi = [5(∀ t + ∆t − ∀t − ∆t ) − (∀ t +2 ∆t − ∀ t −2 ∆t )] 6 ∆t 1 Discharge, Q2 = [5(1.181 − 1.438) − (1.166 − 1.606 )] = −0.1406 m 3 / s = 0.1406 m 3 / s 6 x1 (Negative sign indicates that the discharge is reducing with the decrease of head) Prepared by: Sajal Kumar Adhikary (105599) / Group No: 6
8
Lab Exercise: 1
Calibration of Weir
st
Discharge v s Head Plot (1 experiment) for Variable Head Method 0.100
2.6023
y = 1.6986x 2
) s / 3 m ( Q , 0.010 e g r a h c s i D
R = 0.9834
0.001 0.01
0.10
1.00
Head, H (m)
Figure 1: Relationship between discharge and head on arithmetic scale (First test)
Discharge v s Head Plot (2
nd
experiment) for Variable Head Method
1.000
2.5893
) s / 3 0.100 m ( Q , e g r a h c 0.010 s i D
0.001 0.01
y = 1.7456x 2
R = 0.986
0.10
1.00
Head, H (m)
Figure 2: Relationship between discharge and head on arithmetic scale (Second tes t)
Prepared by: Sajal Kumar Adhikary (105599) / Group No: 6
9
Lab Exercise: 1
Calibration of Weir
rd
Discharge v s Head Plot (3 experiment) for Variable Head Method 1.000
2.5886
y = 1.8346x ) s / 3 0.100 m ( Q , e g r a h c 0.010 s i D
2
R = 0.9865
0.001 0.01
0.10
1.00
Head, H (m)
Figure 3: Relationship between discharge and head on arithmetic scale (Third tes t)
th
Discharge v s Head Plot (4 experiment) for Variable Head Method 1.000
2.3531
y = 1.5043x ) s / 3 0.100 m ( Q , e g r a h c 0.010 s i D
0.001 0.01
2
R = 0.9744
0.10
1.00
Head, H (m)
Figure 4: Relationship between discharge and head on arithmetic scale (Fourth test) Prepared by: Sajal Kumar Adhikary (105599) / Group No: 6
10
Lab Exercise: 1
Calibration of Weir
Sample Calculation (4th experiment): The head discharge relationship is given by Q Equation of the line: y
= 1.5043 x
2.3531
=
KH n (by theory)
(from graph)
So, by comparing two equations, K = 1.5043 and n = 2.3531 5.2 Constant Head Method Table 7: Data analysis for constant head technique Discharge, Q (m3/h)
18.50
Head, H (m) 0.185
97.78
Discharge, Q (m3/s) 0.02716
2
18.00
0.180
77.855
0.02163
3
17.50
0.175
70.96
0.01971
4
16.50
0.165
64.00
0.01778
5
16.00
0.160
59.98
0.01666
6
14.50
0.145
48.05
0.01335
7
15.00
0.150
49.7
0.01381
8
13.00
0.130
34.59
0.00961
9
11.50
0.115
19.95
0.00554
10
7.00
0.070
6.73
0.00187
11
8.00
0.080
13.53
0.00376
No.
Head, H (cm)
1
Sample Calculation: The head discharge relationship is given by Q Equation of the line: y
= 1.632 x
2.5089
=
KH n (by theory)
(from graph)
So, by comparing two equations, K = 1.632 and n = 2.5089
Prepared by: Sajal Kumar Adhikary (105599) / Group No: 6
11
Lab Exercise: 1
Calibration of Weir
Discharge vs Head Plot (Constant Head Method) 0.100
y = 1.632x 2.5089 R2 = 0.9753
1 ) s / 3 m ( Q0.010 , e g r a h c s i D
0.001
0.01
0.10
1.00
Head, H (m)
Figure 5: Calibration Graph for Constant Head Method 5.3 Results Table 8: The value of weir constant (K and n) from plotting data Test no.
Falling Head Method K
n
1st
1.6986
2.6023
2nd
1.7456
2.5893
3rd
1.8346
2.5886
4th
1.5043
2.3531
Average
1.6958
2.5333
Constant Head Method K
n
1.632
2.5089
6 Discussions and Conclusions 6.1 Discussions A weir is an obstruction in an open channel over which flow occurs. Although a dam spillway is an example of a weir, most weirs are intended specifically for flow measurement. The weir is placed in a channel so that fluid backs up behind it and then falls through a notch cut into the weir, normally causing a waterfall effect over the crest of the weir. There are many kinds of weirs. In this experiment, the triangular weir has been used. The bottom of the notch is called the crest of the weir, and the crest height is measured from the bottom of the Prepared by: Sajal Kumar Adhikary (105599) / Group No: 6
12
Lab Exercise: 1
Calibration of Weir
channel to the crest. By simply knowing the dimensions of the notch of the weir, as well as the height of water above the crest (head of water, H), one can calculate the amount of flow over the weir, by using the theoretical equations for discharge over different type weirs. Theoretical equation: From the general equation of head discharge relationship of weir, Q
=
KH n
This test is V-notch (θ = 900) weir. So, theoretical discharge equation considered loss is:
Q
=
C d
8
5
θ
2 g tan( ) H 2 2
15
,
n = 5/2 = 2.50
From V-notch (θ = 900) weir, Lenz (Lecture note, Hydraulic) show that :
C d ν
=
0.56 +
0.70
R
0.165
0.170
W
Where R
=
H gH
ν
and W =
ρ gH 2 σ
= Viscosity, σ = Surface tension (C d is vary by H in above equation)
We know that C d
K = C d
≈
8 15
0.585 (Henderson, F.M. , 1966) 2 g tan(45°) = 0.585 x
8 15
2 x9.81 x tan(45°) = 1.382 ; So, K = 1.382
Table 9: Comparison of K and n values Constant
Theoretical
n K
Experimental Falling head
Constant head
2.50
2.5333
2.5089
1.382
1.6958
1.632
From the Henderson concept so the errors are: •
Falling head error n value : K value:
•
2.50 − 2.5333 2.50
x100 = 1.332%
1.382 − 1.6958 1.382
x100 = 22.706%
Constant head error n value : K value:
2.50 − 2.5089 2.50 1.382 − 1.632 1.382
x100 = 0.356%
x100 = 18.089%
From the result, it is observed that the experimental values are different from the theoretical value. There is a little variation of n for both cases but more variation is seen for K in both cases.
Prepared by: Sajal Kumar Adhikary (105599) / Group No: 6
13
Lab Exercise: 1
Calibration of Weir
The uncertainties associated with the resulting weir calibration equation may be the fluctuation of water level in the weir and the accuracy of reading while testing, as there are many possibilities of error during experiments. Some of them are given below: Possible sources of errors for falling head method due to •
Leakage of weir tank is observed and valve system is not prefect.
•
Reading the analog manometer from video and pulling out the wooden cover.
Possible sources of errors for constant head method •
The operation system: weir tank, EFM (Electromagnetic flow meter) and main Lab valve are not installed in the same place. So, it is difficult to measure all the data at the same time.
•
It is observed that the flows always fluctuate in EFM. So, error may arise during taking reading. Moreover, it takes so much time to be steady flow. During taking the reading, flow may not be steady which is needed in this method.
•
Equipment error: When the head is about 6 cm, the EFM shows zero flow (we see flow from water supply pipe but EFM can not read flow rate).
Figure 6: Error sources in the experiment
Prepared by: Sajal Kumar Adhikary (105599) / Group No: 6
14
Lab Exercise: 1
Calibration of Weir
Figure 7: Weir flow (actual) In data analysis, some value of discharge has been ignored (Table 4 to 6) because those values are less than the next time step discharge values calculated using Taylor series equation. But practically, it doesn’t happen. W hen the head decreases, the discharge should be decreased continuously in sequential manner as discharge is directly related to head of water. By using this concept, those discharge values are neglected.
6.2 Conclusions In the field, the calibration of weir is essential to measure the flow rate in the open channel, which is different from the theoretical value as theoretical formulation uses some assumption. For the falling head technique the value of K is 1.6958 and n = 2.5333 and for constant head technique the value of K = 1.632 and n = 2.5089. for both cases, the variation for n is very less and K is little bit higher than the theoretical value. Based on the result, it can be finally concluded that the calibration equation developed for both cases in this experiment is good enough.
References 1. Laboratory Manual (AIT) and Handout, Supplied by Laboratory supervisor. 2. Likhi, S.K.1987. Hydraulics Laboratory Manual, Tata Mc Graw-Hill Publishing Company Ltd., New Delhi. 3. F.M. Henderson. (1966), Open Channel Flow, Macmillan Publishing Co.,Inc., New York. 4. R. H. French (1986), Open Channel Hydraulics, McGraw Hill, New York, U.S.A 5. Chow, V.T. (1992), Open Channel Hydraulics, McGraw Hill, New York, U.S.A. 6. Robert L. Street, Gary Z. Watters, John K. Vennard, Elementary Fluid Mechanics (1996, 7th edition), P.668 Prepared by: Sajal Kumar Adhikary (105599) / Group No: 6
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