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1.0 INTRODUCTIONBernoulli'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.Bernoulli’s 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, Bernoulli’s Principle can only be applied under certain conditions. The conditions to which Bernoulli’s 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 pressure-volume product, or, since the volumes are equal, by a decrease in pressure.

2.0 THEORYThe 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):

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 p= fluid static pressure at the cross section in N/m2

 ρ= 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 Bernoulli’s 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:

ψ is pressure head ( Length, typically in units of m);

 p is fluid pressure (Force per unit area );

γ is the specific weight (Weight  per unit volume ,typically N·m−3units)

ρ is the density of the fluid (Mass per unit volume ,typically kg·m−3)

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/ ρ+ v^2/2+gz=constant ;z=0

p/ ρ+ v^2/2= constant

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3.0 Experimental

3.1 Apparatus

a. the Bernoulli’s apparatus test equipment that consists 7 vertical tubes

b. The Bernoulli’s apparatus that consist of venturi meter that function as manipulated variables

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: Bernoulli’s theorem demonstration apparatus.

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3.3 Procedure

Equipment Set Up

1. The Bernoulli’s 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.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 adjacent air valve is removed.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 reading by adjusting pitot tubes in the test section.

3 The pitot tube reading gives the total head reading and piezometer tube reading gives us the reading of potential head reading. Subtracting the values we get the experimental velocity head reading.

4 Then from the flow rate and area of cross-sections of the tubes we achieve the theoretical velocity head

5 Then we compare the theoretical and experimental values. 6 Steps 1 to 5 are repeated for both converging and diverging sections.

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4.0 Observed data

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

Observationnumber

Manometer & Pitot tube readings at different tappingin millimeter (mm)

Weight of water(Kg)

Timein second(s)

Tapping position

a b c d e f

m p m p m p m p m p m p

1 154 156 152 156 149 156 145 156 142 154 146 147 1.1 20.38

2 174 175 164 175 150 175 136 174 118 172 139 147 1.6 20.48

3 201 203 175 203 148 201 121 196 87 194 130 142 3.3 20.56

4 208 212 180 214 146 213 109 213 64 210 125 140 2.6 20.1

5 227 231 190 239 145 239 90 236 33 232 120 130 3.1 20.60

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Diameter D inmillimeter(mm)

25 13.9 11.8 10.7 10 25

5.0 Treatment of data

5.1 Calculated data

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

Observa-tionnumber

Mass flow rate(kg/s)

Volumetric flow ratem^3/s

Tube number

Cross sectional area of tapping positionA×104

m2

Velocity (V) m/s

Theoretical velocity head at tapping position,

V 2

2 g(m)

Computed total head

pρg

+ V 2

2 g(m)

Measured head(m)

01 .055 5.5×10-5 A 4.909 0.11 0.0006173

0.155 .156

B 1.5175 0.36 0.0061 .158 .156

C 1.0935 0.502 .013 .162 .156

D 0.8992 0.612 .019 .164 .156

E 0.7853 0.700 0.025 .167 .154

F 4.909 0.112 .00064 .147 .147

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02 .08 8×10-5 A 4.909 0.163 .00136 .175 .175

B 1.5175 0.53 0.014 .178 .175

C 1.0935 0.732 0.027 .177 .175

D 0.8992 0.8897 0.04 .176 .174

E .7853 1.019 0.053 .171 .172

F 4.909 0.163 0.001356 .140 .147

03 .115 1,15×10-4 A 4.909 0.234 .00279 .203 .203

B 1.5175 0.758 .0293 .204 .203

C 1.0935 1.052 0.056 .204 .201

D 0.8992 1.279 0.083 .204 .196

E .7853 1.46 0.109 .19.6 .194

F 4.909 0.234 0.00279 .133 .142

04 .13 1,3×10-4 A 4.909 0.265 0.00358 .212 .212

B 1.5175 0.857 0.0375 .218 .214

C 1.0935 1.189 0.072 .218 .213

D 0.8992 1.146 0.067 .213 .213

E .7853 1.655 0.139 .203 .210

F 4.909 0.265 0.00358 .129 .140

05 .155 1.55×10-4 A 4.909 0.316 0.00509 .232 .231

B 1.5175 1.021 0.053 .243 .239

C 1.0935 1.417 0.102 .247 .239

D 0.8992 1.724 0.152 .242 .236

E .7853 1.974 0.199 .232 .232

F 4.909 0.316 0.00509 .125 .130

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5.2 Graphical representation

Graph 1-Tapping positions VS Theoretical total head

0 1 2 3 4 5 6 70

0.05

0.1

0.15

0.2

0.25

0.3

tapping position vs theoritical head

theoritical total head 1theoritical total head 2theoritical total head 3theoritical total head 4theoritical total head 5

tapping position

theo

ritica

l hea

d

Figure-5.2.1: Tapping positions VS Theoretical total head

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Graph 2-Tapping positions VS Experimental total head

Figure-5.2.2: Tapping position vs. experimental total head

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Graph 3-Tapping Positions VS. Experimental And Theoretical Total Head

(Observation-1)

0 1 2 3 4 5 6 70.135

0.14

0.145

0.15

0.155

0.16

0.165

0.17

Tapping position versus theoritical and experimental total head

theoritical total head Experimental total head

Tapping position

Theo

ritica

l and

exp

erim

enta

l tot

al h

ead

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

Mass flow rate, m= mt

=3.3

20.56 = 0.115 kg /s

Volumetric flow rate, ˙ v = mρ

=0.1151000

=1.15× 10-5 m3/s

Velocity calculation:

For diameter, a:

Diameter, D =25.0×10−3m

Area, A= πD2

4= 4.91×10 -4 m2

Velocity, V = vA

=1.15×10 -5

4.91× 10-4 =0.234 ms-1

For diameter, b:

Diameter, D =13.9×10 -3 m

Area, A= πD2

4= 1.52×10-4 m2

Velocity, V =vA

=1.15× 10-5

1.52× 10-4 =0..758 ms-1

For diameter, c:

Diameter, D =11.8×10-3 m

Area, A= πD2

4= 1.09×10-4 m2

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Velocity, V =vA

=1.155× 10-5

1.09× 10-4 =1.052 ms-1

For diameter, d:

Diameter, D =10.7×10-3 m

Area, A= πD2

4= 0.899×10-4 m2

Velocity, V =vA

=1.15× 10-5

0.899×10-4 =1.279 m/s

For diameter, e:

Diameter, D =10.0×10-3 m

Area, A= πD2

4= 0.785×10-4 m2

Velocity, V =vA

=1.155× 10-5

0.785×10-4 =1.46 ms-1

For diameter, f:

Diameter, D =25.0×10-3 m

Area, A= πD2

4= 4.91×10-4 m2

Velocity, V =vA

=1.15× 10-5

4.91×10 -4 =0.234 ms-1

Total Head Calculation

For cross sectional diameter, a

Velocity head =

v2

2 g =

(0.234)2

2×9.81 m = 2.79 ×10 -3 m

Pressure head (observed), Pρg

= 201×10-3 m

Total head, H= (

v2

2 g+ P

ρg ) = 0.203 m

For cross sectional diameter, b

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Velocity head, v2

2g=

(0.758)2

2×9.81=0.0293m

Pressure head (observed), Pρg

= 175×10-3 m

H=(

v2

2 g+ P

ρg ) m = .204 m

For cross sectional diameter, c

Velocity head, v2

2g=

(1.052)2

2×9.81=.056m

Pressure head (observed), Pρg

= 148×10-3 m

H= (

v2

2g+ P

ρg ) m = 0.204m

For cross sectional diameter, d

Velocity head, v2

2g=

(1.279)2

2×9.81=0.083m

Pressure head (observed), Pρg

= 121×10-3 m

H=(

v2

2 g+ P

ρg ) m = 0.204m

For cross sectional diameter, e

Velocity head, v2

2g=

(1.46)2

2×9.81=0.109m

Pressure head (observed), Pρg

=87×10-3 m

H = (

v2

2g+ P

ρg )m = 0.196 m

For cross sectional diameter, f

Velocity head, v2

2g=

(0.234)2

2×9.81=0.00279m

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Pressure head (observed), Pρg

= 130×10-3 m

H = (

v2

2 g+ P

ρg ) 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 about the experiment.

Table 03: A Table of Computed Total Head and Experimental Head.

Tub

e

num

ber/

Tappingposition

Computed and experimental Total head

Observation number 01

Observation number 02

Observation number 03

Observation number 04

Observation number 05

theoretical

experimental

theoretical

experimental

theoretical

experimental

theoretical

experimental

theoretical

experimental

a .155 .156 .175 .175 .203 .203 .212 .212 .232 .231b .158 .156 .178 .175 .204 .203 .218 .214 .243 .239c .162 .156 .177 .175 .204 .201 .218 .213 .247 .239d .164 .156 .176 .174 .204 .196 .213 .213 .242 .236e .167 .154 .171 .172 .196 .194 .203 .210 .232 .232f .147 .147 .140 .147 .133 .142 .129 .140 .125 .130

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7.0 Discussion

The possible reasons for discrepancies are:

For the sake of Bernoulli’s 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

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 Bernoulli’s equation is

not valid.

Figure 7.1: Flow at sudden enlargement of cross

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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

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 Bernoulli’s 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.

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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.

Bernoulli’s theorem was generated for steady flow. But in lab water was supplied by pump so

it was not actually steady.

The use of mean velocity without kinetic energy correction factor (α) introduces some error in

the results. Here we assume that α = 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

calculation of the static head.

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.

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.

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