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United Arab Emirates University

College of Engineering Mechanical Engineering Department

Fluid Mechanics Lab

MECH-348

Experiment No. 6

Bernoullis Theorem

Student Name: Iman Abdulwaheed ID: 200820143 Wafa Mubarak 200302479 Sayeda Abboud Al Ameri 200812660

Submitted to: Prof. Abdallah Al Amiri Submission date: 10/4/2011

Spring 2011

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Contents

Abstract ...................................3

Introduction ........3

Materials & Methods:

Apparatus 4-5

Procedures ...5-6

Equations 6

Results 7-17

Discussion .......17-18

Conclusion ...18

References .................................................................................19

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Abstract:

This report presents analysis based on Bernoullis theorem to investigate the

validity of Bernoullis theorem as applied to the flow of water in a tapering circular duct.

A description of the apparatus used in the experiment, tables, calculations and

figures are provided through the report to reach the goals.

It was found that the total head is constant throughout the duct which is equal

the static head added to the velocity head. It was found also that Bernoullis theorem is

valid.

Introduction:

The Bernoulli equation is an approximate relation between pressure, velocity,

and elevation, and is valid in regions of steady, incompressible flow where net frictional

forces are negligible. It has proven to be a very powerful tool in fluid mechanics.

The Bernoulli equation states that the sum of the flow, kinetic, and potential

energies of a fluid particle along a streamline is constant. Therefore, the kinetic and

potential energies of the fluid can be converted to flow energy (and vice versa) during

flow, causing the pressure to change.

Some applications where the Bernoulli equation can be applied are spraying

water into the air, discharging water from a large tank, and the rise of the ocean due to

a hurricane.

The main objectives of this experiment are to invistigate:

o the total energy of a fluid flow through a passage of variable section.

o the validity of Bernoullis theorem as applied to the flow of water in a tapering

circular duct.

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Materials and Methods:

o Apparatus:

Bernoulli's Apparatus shown in Figures (1) to (3), consists essentially of a two

dimensional rectangular section convergent divergent. A seven tube static pressure

manometer bank is attached to the convergent divergent duct. The differential head

across the test section can be varied from zero up to a maximum height.

Figure (1): Bernoullis Apparatus

Figure (2): Bernoullis Apparatus

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Figure (3): Bernoullis Apparatus

Another apparatus used in the experiment is the stopwatch shown in Figure (4).

It was used to find the time during the tank is filled with water to a specified volume.

Figure (4): Stopwatch

o Procedures:

To achieve the aim of this experiment the following procedures were followed:

1. The apparatus was placed on a table or hydraulic bench.

2. The apparatus was leveled correctly using leveling feet (3H) and the spirit

level.

3. The control valve (1C) was closed and the pump was switched on.

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4. The valves (6K) and (1C) was opened. Then time was allowed to get the air

out of the system.

5. Valve (6K) was adjusted to give a steady flow rate. Time was allowed for the

water levels in the piezometer tubes to stabilize and the values of h were

recorded.

6. The tip of the total head probe was located at each of the selected sections

and the total values were recorded.

7. Steps 3 to 6 were repeated again for different flow rates.

8. Valves (6K) and (1C) were closed and the pump was shut.

9. The test section was reversed from convergent to divergent and vice-versa.

o Equations:

The following equations were used through the experiment to determine

the reqiured values:

(1)

Where V = fluid velocity, Z = elevation head

(2)

Where hv = velocity head, hp = static or pressure head and is equal to

, it is also

called hydraulic grade line HGL, H = total head and is also called energy grade line EGL,

htc = total calculated head

(3)

Where Q = energy flow rate, V = volume in m3

(4)

Where V = velocity in m/s

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Results:

Using the equations mentioned previosly, the following data in the tables were

obtained:

Table (1): Convergent section results

Volume=0.001m3 Tubes Velocity(m/s) P(pa) constant htotal a 0.812704192 998.63595 0.1356 0.1356 b 0.723308296 1067.2926 0.1356 0.1356 c 0.6364712 1126.3322 0.1356 0.1356 d 0.52855417 1189.1952 0.1356 0.1356 e 0.381871297 1255.9672 0.1356 0.1356 f 0.130033995 1320.4256 0.1356 0.1356

Volume=0.001m3 heights(m) heights(m)=Htotal h(total)m 0.01 0.072333333 0.1356 0.02 0.064333333 0.1356 0.03 0.053 0.1356 0.04 0.04 0.1356 0.05 0.023 0.1356 0.06 0.001333333 0.1356 Table (2): Calculations for the first readings in convergent section

Convergence

Vi(m3) Vf(m3) t(sec) ha hb hc hd he hf htm

0 0.001

13 61 73 86 98 117 134 137

16 60 71 80 99 111 130 136

18 69 70 82 90 110 139 134

Average 15.67 0.063 0.0713 0.0827 0.0957 0.1127 0.1343 0.1357 6.38298E-05

0 0.005 90 88 99 106 115 124 132 133

88 80 100 105 114 123 133 132

89 87 98 107 116 125 131 134

Average 89 0.085 0.099 0.106 0.115 0.124 0.132 0.133 5.61798E-05

0 0.005 33 50 84 130 173 218 264 268

36 49 83 129 172 219 263 267

38 51 85 131 174 217 265 269

Average 35.67 0.05 0.084 0.13 0.173 0.218 0.264 0.268 0.000140187

0 0.005 45 63 93 135 240 210 251 255

44 62 92 134 241 209 250 254

46 64 94 136 242 211 249 256

Average 45 0.063 0.093 0.135 0.241 0.21 0.25 0.255 0.000111111

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Figure5: Height Vs Heads and total head for the first readings

Volume=0.005m3

Tubes velocity(m/s) P(pa) constant htotal

a 0.715301442 1047.572 0.133 0.133

b 0.636619662 1100.758 0.133 0.133

c 0.560190007 1146.494 0.133 0.133

d 0.465206854 1195.191 0.133 0.133

e 0.33610395 1246.917 0.133 0.133

f 0.114449397 1296.851 0.133 0.133

Volume=0.005m3 heights(m) heights(m)=Htotal-h h(total)m 0.01 0.048 0.133 0.02 0.034 0.133 0.03 0.027 0.133 0.04 0.018 0.133 0.05 0.009 0.133 0.06 0.005 0.133 Table (3): Calculations for the second readings in convergent section

y = -16.25x2 - 0.2682x + 0.0764 R = 0.9996 0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0 0.02 0.04 0.06 0.08

He

igh

t(m

)

Height(m)

Height(m) Vs heads &total heads

heights(m)=Htotal

h(total)m

Poly. (heights(m)=Htotal)

Linear (h(total)m)

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Figure6: Height Vs Heads and total head for the second readings

Volume=0.005m3

Tubes velocity(m/s) P(pa) constant htotal

a 1.784911076 1033.446 0.268 0.268

b 1.588574296 1364.616 0.268 0.268

c 1.397857308 1649.397 0.268 0.268

d 1.160843271 1952.621 0.268 0.268

e 0.838689296 2274.7 0.268 0.268

f 0.285588681 2585.62 0.268 0.268

Volume=0.005m3 heights(m) heights(m)=Htotal-h h(total)m 0.01 0.218 0.268 0.02 0.184 0.268 0.03 0.138 0.268 0.04 0.095 0.268 0.05 0.05 0.268 0.06 0.004 0.268 Table (4): Calaculations for the third readings in convergent section

y = 7.5x2 - 1.3793x + 0.0604 R = 0.9944

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0 0.02 0.04 0.06 0.08

he

igh

t(m

)

height(m)

height(m) Vs heads &total heads

heights(m)=Htotal-h

h(total)m

Poly.(heights(m)=Htotal-h)

Linear (h(total)m)

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Figure7: Height Vs Heads and total head for the third readings

Volume=0.005m3

Tubes velocity(m/s) P(pa) constant htotal

a 1.414707297 1498.302 0.255 0.255

b 1.25909222 1706.343 0.255 0.255

c 1.107931348 1885.244 0.255 0.255

d 0.920075777 2075.73 0.255 0.255

e 0.664738924 2278.061 0.255 0.255

f 0.226355473 2473.382 0.255 0.255

Volume=0.005m3 heights(m) heights(m)=htotal-h h(total)m 0.01 0.192 0.255 0.02 0.162 0.255 0.03 0.12 0.255 0.04 0.088 0.255 0.05 0.045 0.255 0.06 0.005 0.255 Table (5): Calculations for the fourth readings in convergent section

y = -10x2 - 3.6286x + 0.257 R = 0.9994

0

0.05

0.1

0.15

0.2

0.25

0.3

0 0.02 0.04 0.06 0.08

He

igh

t(m

)

Height(m)

Height(m) Vs heads &total heads

heights(m)=Htotal-h

h(total)m

Poly. (heights(m)=Htotal-h)

Linear (h(total)m)

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Figure8: Height Vs Heads and total head for the fourth readings

Figure (9): Theoretical vs. Experimental head for convergent section

y = -9.6429x2 - 3.0907x + 0.2248 R = 0.9989

0

0.05

0.1

0.15

0.2

0.25

0.3

0 0.02 0.04 0.06 0.08

He

igh

t(m

)

Height(m)

Heights(m) Vs heads &total heads

heights(m)=htotal-h

h(total)m

Poly.(heights(m)=htotal-h)

Linear (h(total)m)

y = x R = 1

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0 0.05 0.1 0.15

The

ori

tica

l

Exp

Experimental total Vs Theoritical total

htotal

Linear (htotal)

volume=0.001m^3

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Figure (10): Theoretical vs. Experimental head for convergent section

Divergent section:

Using the equations mentioned previosly, the following data in the tables were obtained:

Divergence

Vi(m3) Vf(m3) t(sec) ha hb hc hd he hf htm

0 0.005

49 104 103 110 125 140 163 205

50 106 102 109 126 139 162 206

43 105 104 111 124 141 161 204

Average 47.33 0.105 0.103 0.11 0.125 0.14 0.162 0.205 0.0001056

0 0.005 38 94 80 99 121 142 181 249

49 93 81 100 120 140 180 248

40 95 79 101 119 141 179 250

Average 42.33 0.094 0.08 0.1 0.12 0.141 0.18 0.249 0.0001181

0 0.005 40 96 86 100 121 140 177 246

42 97 85 101 120 141 176 245

50 95 87 99 122 139 178 247

Average 44 0.096 0.086 0.1 0.121 0.14 0.177 0.246 0.0001136

0 0.005 51 125 120 129 143 155 176 221

48 126 121 128 142 156 177 220

59 124 119 130 144 154 175 219

Average 52.67 0.125 0.12 0.129 0.143 0.155 0.176 0.22 9.494E-05

Table (6): Divergent section results

y = x R = 1

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0 0.05 0.1 0.15

He

igh

t(m

)

Height(m)

Experimental total Vs theoritical total

htotal

Linear (htotal)

volume=0.005m^3

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Volume=0.005m3

Tubes velocity(m/s) P(pa) constant htotal

a 1.344968205 1104.5 0.205 0.205

b 1.197024293 1292.6 0.205 0.205

c 1.053315014 1454.3 0.205 0.205

d 0.874719929 1626.4 0.205 0.205

e 0.631970104 1809.3 0.205 0.205

f 0.215197105 1985.8 0.205 0.205

Volume=0.005m3 heights(m) heights(m)=Htotal-h h(total)m 0.01 0.1 0.205 0.02 0.102 0.205 0.03 0.095 0.205 0.04 0.08 0.205 0.05 0.065 0.205 0.06 0.043 0.205 Table (7): Calculations for the first readings in divergent section

Figure11: Height Vs Heads and total head for the first readings

y = -27.143x2 + 0.7257x + 0.0966

R = 0.9961 0

0.05

0.1

0.15

0.2

0.25

0 0.02 0.04 0.06 0.08

He

igh

t(m

)

height(m)

Height(m) Vs heads &total heads

heights(m)=Htotal-h

h(total)m

Poly.(heights(m)=Htotal-h)

Linear (h(total)m)

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Volume=0.005m3

Tubes velocity(m/s) P(pa) constant htotal

a 1.503822717 1309.459 0.249 0.249

b 1.338405115 1544.536 0.249 0.249

c 1.177722299 1746.685 0.249 0.249

d 0.978033307 1961.925 0.249 0.249

e 0.706612242 2190.55 0.249 0.249

f 0.240614086 2411.252 0.249 0.249

Volume=0.005m3 heights(m) heights(m)=Htotal-h h(total)m 0.01 0.177 0.249 0.02 0.169 0.249 0.03 0.149 0.249 0.04 0.129 0.249 0.05 0.108 0.249 0.06 0.069 0.249 Table (8): Calculations for the second readings in divergent section

Figure12: Height Vs Heads and total head for the second readings

y = -28.393x2 - 0.1354x + 0.1813 R = 0.9956

0

0.05

0.1

0.15

0.2

0.25

0.3

0 0.02 0.04 0.06 0.08

he

igh

t(m

)

height(m)

Height(m) Vs heads &total heads

heights(m)=Htotal-h

h(total)m

Poly. (heights(m)=Htotal-h)

Linear (h(total)m)

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Volume=0.005m3

Tubes velocity(m/s) P(pa) constant htotal

a 1.446859736 1364.098 0.246 0.246

b 1.287707952 1581.704 0.246 0.246

c 1.133111606 1768.829 0.246 0.246

d 0.940986591 1968.072 0.246 0.246

e 0.679846627 2179.704 0.246 0.246

f 0.231499916 2384.004 0.246 0.246

Volume=0.005m3 heights(m) heights(m)=Htotal-h h(total)m 0.01 0.165 0.246 0.02 0.151 0.246 0.03 0.146 0.246 0.04 0.125 0.246 0.05 0.106 0.246 0.06 0.069 0.246

Table (9): Calculations for the third readings in divergent section

Figure13: Height Vs Heads and total head for the third readings

y = -30.536x2 + 0.3204x + 0.1621 R = 0.9898

0

0.05

0.1

0.15

0.2

0.25

0.3

0 0.02 0.04 0.06 0.08

he

igh

t(m

)

Height(m)

Height(m) Vs heads &total heads

heights(m)=Htotal-h

h(total)m

Poly.(heights(m)=Htotal-h)

Linear (h(total)m)

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Volume=0.005m3 Tubes velocity(m/s) P(pa) constant htotal a 1.208768893 1425.439 0.22 0.22 b 1.075806643 1577.32 0.22 0.22 c 0.946650203 1707.927 0.22 0.22 d 0.786140696 1846.991 0.22 0.22 e 0.567973131 1994.703 0.22 0.22 f 0.193404993 2137.297 0.22 0.22

Volume=0.005m3 heights(m) heights(m)=Htotal-h h(total)m 0.01 0.11 0.22 0.02 0.1 0.22 0.03 0.091 0.22 0.04 0.077 0.22 0.05 0.065 0.22 0.06 0.044 0.22 Table (10): Calculations for the fourth readings in divergent section

Figure14: Height Vs Heads and total head for the fourth readings

y = -11.964x2 - 0.4454x + 0.1149 R = 0.9968

0

0.05

0.1

0.15

0.2

0.25

0 0.02 0.04 0.06 0.08

he

igh

t(m

)

height(m)

Height(m) Vs heads &total heads

heights(m)=Htotal-h

h(total)m

Poly.(heights(m)=Htotal-h)

Linear (h(total)m)

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Figure (15): Theoretical vs. Experimental head for divergent section

Discussion:

In this section, the previous tables and graphs will be discussed and compared.

The heights were recorded at a constant flow rate using the differents readings

rule. The readings were recorded three times for different volume flow rate at each

convergent section and divergent section which is shown in Table (1) and (5).

The average heights hp for the convergent section which are equals to pressure

heads are used in Tables (2) to (4), to find the total head for the three readings by first

calculating the velocity head hv. The same calculations were repeated in Tables (6) to

(8) for the divergent section.

Total theoretical head htc is compared with total experimental head htm in Table

(2,3,4) for convergent section and in Table (6,7,8) for divergent section. The total

theoretical and experimental heads are not very far from each other. That means the

total head is constant and is equal to the total head at each tap. This is true since the

pressure head increases while the velocity head decreases to remain the equality.

From the obtained results, the velocity in convergent section increases with the

diameter decreasing, while it decreases in divergent section due the increasing of the

y = x R = 1

0

0.05

0.1

0.15

0.2

0.25

0 0.05 0.1 0.15 0.2 0.25

The

ori

tica

l

Exp

Experimental total Vs Theoritical total

htotal

Linear (htotal)

volume=0.005m^3

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section diameter. As a result, the pressure in convergent section decreases and

increases in divergent section.

The total energy of the fluid flow increases since the velocity in the convergent

section increases, while it decreases in the divergent section with time.

Bernoullis theorem is applicable in this experiment, since the water flow during

the experiment was steady and continuous and the water is an incompressible and non

viscous fluid.

Conclusion:

The objectives of this experiment were finally achieved by investigating the total

energy of a fluid flow through a passage of variable section and by investigating the

validity of Bernoullis theorem as applied to the flow of water in a tapering circular duct.

It was found that the summation of the static head and the velocity head equals

the total head which is the energy grade line. The difference between the heights of the

energy grade line and the hydraulic grade line is equal to the dynamic head. It was also

found that Bernoullis equation is valid for this experiment.

Errors that appear in the experiment are almost personal such as being not

accurate during recoding the heights. It may occur also because of difficulty in making

the water flow steady. A systematic error was a leaking in water from the apparatus.

The Bernoulli equation is one of the most frequently used and misused equations

in fluid mechanics. Its simplicity and ease of use make it very valuable tool for use in

analysis, but the same attributes also make it very tempting to misuse.

This experiment is important since it is related to important topics in our life.

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References:

Cengel, Yunus, and John Cimbala. Fluid Mechanics. New York: McGraw-Hill,

2006.

http://www.ceet.niu.edu/faculty/kostic/bernoulli.html