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1.0 INTRODUCTION
Bernoulli's Principle is a physical principle formulated that states that "as the speed of a
moving fluid (liquid or gas) increases, the pressure within the fluid decreases. Bernoulli's
principle is named after the Swiss scientist Daniel Bernoulli who published his principle in
his book Hydrodynamica in 1738.Bernoullis Principle can be demonstrated by the
Bernoulli equation. The Bernoulli equation is an approximate relation between pressure,
velocity, and elevation. While the Continuity equation relates the speed of a fluid that
moving through a pipe to the cross sectional area of the pipe. It says that as a radius of the
pipe decreases the speed of fluid flow must increase and vice-versa. However, Bernoullis
Principle can only be applied under certain conditions. The conditions to which Bernoullis
equation applies are the fluid must be frictionless (inviscid) and of constant density; the
flow must be steady, and the relation holds in general for single streamline. In general,
frictional effects are always important very close to solid wall (boundary layers) and
directly downstream of bodies (wakes). Thus, the Bernoulli approximation is typically
useful in flow regions outside of boundary layers and wakes, where the fluid motion is
governed by the combined effects of pressure and gravity forces. Bernoulli's principle can
be explained in terms of the law of conservation of energy. As a fluid moves from a wider
pipe into a narrower pipe or a constriction, a corresponding volume must move a greater
distance forward in the narrower pipe and thus have a greater speed. At the same time, the
work done by corresponding volumes in the wider and narrower pipes will be expressed by
the product of the pressure and the volume. Since the speed is greater in the narrower pipe,
the kinetic energy of that volume is greater. Then, by the law of conservation of energy,
this increase in kinetic energy must be balanced by a decrease in the pressurevolume product, or, since the volumes are equal, by a decrease in pressure.

2.0 THEORY
The well-known Bernoulli equation is derived under the following assumptions:
The liquid is incompressible.
The liquid is non-viscous.
The flow is steady and the velocity of the liquid is less than the critical velocity for
The liquid.
There is no loss of energy due to friction.
Then, it is expressed with the following equation: Where (in SI units):
p= fluid static pressure at the cross section in N/m2

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## = density of the flowing fluid in kg/m3

g = acceleration due to gravity in m/s2 (its value is9.81 m/s=9810 mm/s2)
v= mean velocity of fluid flow at the cross section in m/s
z = elevation head of the center of the cross section with respect to a datum z=0
h*= total (stagnation) head in m
The terms on the left-hand-side of the above equation represent the pressure head
(h), velocity head (hv), and elevation head (z), respectively. The sum of these terms is
known as the total head (h*).According to the Bernoullis theorem of fluid flow through a
Pipe, the total head h*at any cross section is constant (based on the assumptions given).
Pressure head is a term used in fluid mechanics to represent the internal energy of a fluid
due to the pressure exerted on its container. It may also be called static pressure head
or simply static head (but not static head pressure). It is mathematically expressed as:
Where:
is pressure head (Length, typically in units of m);

## p is fluid pressure (Force per unit area );

is the specific weight(Weight per unit volume ,typically Nm3units)
is the density of the fluid (Mass per unit volume ,typically kgm3)
g is acceleration due to gravity
In this experimental, the center line of the entire cross sections we are considering lie on
the same horizontal plane (which we may choose as the datum, z=0), and thus, all the
z values are zeros so that the above equation reduces to:
The Bernoulli equation:
Kinetic energy + potential energy + flow energy = constant

p/ + v^2/2+gz=constant ;z=0
p/ + v^2/2= constant

3.0 Experimental
3.1 Apparatus
a. the Bernoullis apparatus test equipment that consists 7 vertical tubes
b. The Bernoullis apparatus that consist of venturi meter that function as manipulated variables

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c. The stopwatch that used for the timing to the flow measurement of the water
d. The hydraulic bench which allows water flow by time volume collection to be measured.

## 3.2 Experimental set up

A schematic diagram of the experimental setup is given in Figure 1.

## Figure 3.2.1: Bernoullis theorem demonstration apparatus.

3.3 Procedure
Equipment Set Up
1. The Bernoullis equation apparatus is first set up on the hydraulic bench so that the
Base is in the horizontal position.
2. The test section is ensured to have the 14- tapered section converging in the direction of the
flow.

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## 3. The rig outflow tube is positioned above the volumetric tank.

4. The rig inlet is connected to the bench flow supply, the bench valve and the apparatus flow
control are closed and then the pump is started.
5. Gradually, the bench valve is opened to fill the test rig with the water.
6. In order to bleed air pressure tapping point and the manometers, both the bench valves and the
rig flow control valves are closed. Then, the air bleed screw is opened and the cap from the
7. A length of small-bore tuning from the air valve is connected to the volumetric tank.
8. The bench valve is opened and allowed to flow through the manometer to purge all air from
them.
9. After that, the air bleed screw is tightened and both the bench valve and rig flow control valve
are partly opened.
10. Next, the air bleed is opened slightly to allow the air to enter the top of themanometers. The
screw is re-tightened when the manometer reach a convenientheight.6.2

## Taking a Set of Results

1. We take five observations each time we change the flow rate using regulator.
2. Then when we get steady flow we take down pitot tube reading and piezometer tube
4 Then from the flow rate and area of cross-sections of the tubes we achieve the theoretical
5 Then we compare the theoretical and experimental values.
6 Steps 1 to 5 are repeated for both converging and diverging sections.

## 4.0 Observed data

Table 1: Observed Data for Manometer and Pitot tube Readings at Different Tappings, Weight of
Water and Bucket and Time

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## Observati Manometer & Pitot tube readings at different tapping

on
in millimeter (mm)
number

Tapping
position

Weig
ht of
water
(Kg)

Time
in
secon
d
(s)

154

156

152

156

149

156

145

156

142

154

146

147

1.1

20.38

174

175

164

175

150

175

136

174

118

172

139

147

1.6

20.48

201

203

175

203

148

201

121

196

87

194

130

142

3.3

20.56

208

212

180

214

146

213

109

213

64

210

125

140

2.6

20.1

227

231

190

239

145

239

90

236

33

232

120

130

3.1

20.60

Diameter
D in
millimete
r
(mm)

25

13.9

11.8

10.7

10

25

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## 5.0 Treatment of data

5.1 Calculated data
Table 02:- Table of calculated Data for volumetric flow rate, velocity head, pressure head and

-tion
flow c flow
number rate
rate
(kg/s m^3/s
)

Tube
numbe
r

Cross
Velocit
sectiona y (V)
l area of m/s
tapping
position
A
10
m

01

02

03

.055

.08

.115

5.510-5

810-5

1,1510-4

Theoretic
al velocity
tapping
position,
V2
2g

Compute
d total

Measure
(m)

p V
+
g 2 g
(m)

(m)

4.909

0.11

0.155

.156

0.36

0.000617
3
0.0061

1.5175

.158

.156

1.0935

0.502

.013

.162

.156

0.8992

0.612

.019

.164

.156

0.7853

0.700

0.025

.167

.154

4.909

0.112

.00064

.147

.147

4.909

0.163

.00136

.175

.175

1.5175

0.53

0.014

.178

.175

1.0935

0.732

0.027

.177

.175

0.8992

0.8897

0.04

.176

.174

.7853

1.019

0.053

.171

.172

4.909

0.163

0.001356

.140

.147

4.909

0.234

.00279

.203

.203

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04

05

.13

.155

1,310-4

1.5510-4

1.5175

0.758

.0293

.204

.203

1.0935

1.052

0.056

.204

.201

0.8992

1.279

0.083

.204

.196

.7853

1.46

0.109

.19.6

.194

4.909

0.234

0.00279

.133

.142

4.909

0.265

0.00358

.212

.212

1.5175

0.857

0.0375

.218

.214

1.0935

1.189

0.072

.218

.213

0.8992

1.146

0.067

.213

.213

.7853

1.655

0.139

.203

.210

4.909

0.265

0.00358

.129

.140

4.909

0.316

0.00509

.232

.231

1.5175

1.021

0.053

.243

.239

1.0935

1.417

0.102

.247

.239

0.8992

1.724

0.152

.242

.236

.7853

1.974

0.199

.232

.232

4.909

0.316

0.00509

.125

.130

## 5.2 Graphical representation

Graph 1-Tapping positions VS Theoretical total head

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0.3

0.25

0.2

0.15

0.1

0.05

tapping position

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10 | P a g e

(Observation-1)

0.17

0.17

0.16

0.16

## Theoritical and experimental total head

0.15

0.15

0.14

0.14

01234567

Tapping position

Figure-5.2.3: Tapping positions vs. experimental and theoretical total head for first observation

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## Graph 4-Tapping Positions VS Experimental And Theoretical Total Head

(Observation-2)

Figure-5.2.4: Tapping Positions VS Experimental and Theoretical Total Head for second
observation

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## Graph 5-Tapping Positions VS Experimental And Theoretical Total Head

(Observation-3)

Figure-5.2.5: Tapping Positions VS Experimental and Theoretical Total Head for third
observation

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## Graph 6-Tapping Positions VS Experimental And Theoretical Total Head

(Observation-4)

Figure-5.2.6: Tapping Positions VS Experimental and Theoretical Total Head for fourth
observation

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## Graph 7-Tapping Positions VS Experimental And Theoretical Total Head

(Observation-5)

Figure-5.2.7: Tapping Positions VS Experimental and Theoretical Total Head for fifth
observation

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## 5.3 Sample calculation:

Sample calculation for third observation:
Calculation of volumetric flow rate:
Mass of the bucket with water =3.3 kg
Mass of the empty bucket =0 kg
Mass of water, m= 3.3 Kg
Time of flow, t = 20.56s
m
3.3
=
= 0.115 kg / s
=
Mass flow rate, m
t
20.56
Volumetric flow rate , v

m
0.115
-5
3
= = 1000 =1.15 10 m /s

Velocity calculation:
For diameter, a:
3

Diameter, D =25.0 10
D
Area, A= 4

-4
= 4.91 10

m2

-5
v 1.15 10
=
=0.234 ms-1
A 4.9110 -4

Velocity, V =

For diameter, b:
-3
Diameter, D =13.9 10 m

D
Area, A= 4

-4
= 1.52 10

m2

v 1.15 10-5
-1
Velocity, V = A = 1.52 10-4 =0..758 ms
For diameter, c:

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Diameter, D =11.8 10

-3

D2
-4
Area, A= 4
= 1.09 10

m2

v 1.155 10-5
-1
Velocity, V = A = 1.09 10-4 =1.052 ms
For diameter, d:
Diameter, D =10.7 10
D
Area, A= 4

-3

-4
= 0.899 10

m2

-5
v 1.15 10
=
Velocity, V = A 0.899 10-4 =1.279 m/s

For diameter, e:
-3
Diameter, D =10.0 10 m

D
Area, A= 4

-4
= 0.785 10

m2

v 1.155 10-5
-1
Velocity, V = A = 0.785 10-4 =1.46 ms
For diameter, f:
-3
Diameter, D =25.0 10 m

D2
-4
Area, A= 4
= 4.91 10

m2

v 1.15 10-5
-1
Velocity, V = A = 4.9110-4 =0.234 ms
For cross sectional diameter, a

17 | P a g e

v2
2g

( 0.234)
29.81

P

v2 P
+
2g g

10 -3 m
m = 2.79
= 20110-3 m

) = 0.203 m

## For cross sectional diameter, b

v2 (0.758)2
Velocity head, 2g = 29.81 =0.0293m
P

H=(

v2 P
+
2g g

= 17510-3 m

) m = .204 m

## For cross sectional diameter, c

2

v
(1.052)
Velocity head, 2g = 29.81 =.056m
P

H= (

v2 P
+
2g g

= 14810-3 m

) m = 0.204m

## For cross sectional diameter, d

v2 (1.279)2
Velocity head, 2g = 29.81 =0.083m
P

= 12110-3 m

18 | P a g e

H=(

v2 P
+
2g g

) m = 0.204m

## For cross sectional diameter, e

2

v
(1.46)
Velocity head, 2g = 29.81 =0.109m
P

H=(

v2 P
+
2g g

=8710-3 m

)m = 0.196 m

## For cross sectional diameter, f

2

v
(0.234)
=
=0.00279m
2g 29.81

P

H=(

v2 P
+
2g g

= 13010-3 m

) m = 0.133 m

6.0 Result :
The total pressure found for the different flow rates has given below in tabular for a quick glance
Tub
e

Observation

Observation

Observation

Observation

Observation

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num number 01

number 02

number 03

number 04

number 05

ber/
Tap
ping
posi
tion
a
b
c
d
e
f

theor
etical

experi
mental

theor
etical

experi
mental

theor
etical

experi
mental

theor
etical

experi
mental

theor
etical

experi
mental

.155
.158
.162
.164
.167
.147

.156
.156
.156
.156
.154
.147

.175
.178
.177
.176
.171
.140

.175
.175
.175
.174
.172
.147

.203
.204
.204
.204
.196
.133

.203
.203
.201
.196
.194
.142

.212
.218
.218
.213
.203
.129

.212
.214
.213
.213
.210
.140

.232
.243
.247
.242
.232
.125

.231
.239
.239
.236
.232
.130

7.0 Discussion
The

possible

reasons

for

discrepancies

are:

For the sake of Bernoullis theorem it was assumed that the fluid that used (water) was an
ideal fluid that means there should be no loss between the tubes due to friction. The fluid is
considered as in viscid and incompressible but, practically it is not possible for the real fluids
like water. Friction appears in boundary layers because the work done by shear forces. So the
total head dropped gradually as the fluid went down stream. The friction increases as velocity
increases. This energy was converted into heat energy and changed the viscosity and density of
the fluid. The experimental data showed discrepancies mainly due to this fact.

Expansion loss: In sudden expansion there is a state of excessive turbulence. The loss due to

sudden expansion is greater than the loss due to a corresponding contraction. This is so because
of the inherent instability of flow in expansion where the diverging path of the flow tend to
encourage the formation of eddies within the flow. In converging flow there is a dampening

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effect on eddy formation and hence loss is less than diverging flow. It is reflected by the drastic
decrease of total head from point e (diameter 0.01m) to f (diameter 0.025m) in figure
7.1.experimentally the head losses varies a lot and which indicates that Bernoullis equation is
not valid.

## Figure 7.1: Flow at sudden enlargement of cross

Contraction loss: There is a drop in pressure due to the increase in velocity and to the loss of
energy in turbulence at the entrance of the pitot tube due to sudden contraction. But loss at
contraction is negligible.

## Figure 7.2: Flow at sudden contraction of cross section

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While taking the readings, the manometer fluid levels were oscillating vigorously. We were
compelled to take the average readings. These readings gave a mean result. But to determine the
pressure head at that very moment the instantaneous pressure head was needed. Since
instantaneous pressure head cannot be measured using a Bernoullis apparatus, there will always
remain a level of error in the experiment.

There might be some water bubble in the manometer tube and pitot tube. So the reading was
fluctuating and we did not get the accurate one.

Since the venturi tube cannot be thermally isolated from the surrounding completely, there are
some possible heat transfer between tube and surrounding which is not account in the theorem. It
introduces a permanent frictional resistance in the pipeline.
Bernoullis theorem was generated for steady flow. But in lab water was supplied by pump so
The use of mean velocity without kinetic energy correction factor () introduces some error in
the results. Here we assume that = 1. But it varies with Reynolds number
The piezometer readings were fluctuating continuously during the experiment due to unsteady
supply. Since capillarity makes water rise in piezometer tube, it introduce some error in
The tubes of manometer should be completely air-free, but it was not completely air-free due
to leakage or experimental limitation.
The mass of water collected in a definite time interval had been measured manually.So it was
not possible to get the accurate mass flow rate for each steady flow
The weight-measuring machine was manually operated machine. With that machine it is not
possible to get the exact mass of the water accurately. This is why a little error may come due to
the mass flow rate.

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The scaling of the manometer bank did not indicate smaller unit. The smallest unit it showed
was 5 mm only. So we took the possible nearest value but not the accurate one.
The starting of the stop watch and the falling of water into the bucket should be coincident.
But we were disabled to do that at the same time.
These are the possible causes for not getting the total head same at all points. Some modification
and introducing of some additional energy term will make the equation more accurate and
general.

8.0 Reference:
Robert l. Daughtery, Joseph B. Franzini, E. JohnFinnermore, (fluid mechanics with engineering
applications)
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