Summary:

The main objective of this experiment is -

To examine in depth on the validity of Bernoulli’s theorem when applied to the steady

flow of water in tapered circular duct.

To measure the flow rates and both static and total pressure heads in a rigid convergent

or divergent tube of known geometry for a range of steady flow rate.

To investigate the effect of flow rate on head loss in a flowing system.

To learn the application of Bernoulli’s theorem in practical life.

To calculate the fluctuation between theoretical velocity head & practical velocity head

and determine its reasons.

Bernoulli’s theorem states that, for a frictionless ideal fluid (for which density and viscosity is

constant) flowing through a system the total energy entering a system is equal to the total energy

coming out of the system. It is true only if there is no energy loss due to friction or any other

reasons and no energy is added as shaft work or any other form. Here the three types of energies

that are taken under consideration are flow energy, kinetic energy and potential energy. The

energies are expressed in the form of height of water column (4o Celsius) at ground level and are

called pressure head, velocity head and potential head respectively. So, under ideal conditions

(no frictional loss), for ideal fluid and no shaft work, the summation of the changes in heads is

zero. Bernoulli's apparatus is designed actually to demonstrate visually the interchange between

pressure head and velocity head as water flows through a tube of variable cross sectional area in

a horizontal plane. In this experiment, fluid was flowing through a tapering circular duct and

there was six open end manometer (piezometer) to observe the steady head at different points. It

was found that at the converging point where pressure head was larger, velocity head was

smaller and opposite was the case for the diverging point. By taking the mass of water flown the

volumetric flow rate was measured and velocity head was calculated. Adding velocity head with

the steady head, the theoretical total head was obtained and compared with the practical one

obtained from the pitot tube of experimental apparatus. There was a little fluctuation of value

between the practical and theoretical one as all the condition to satisfy Bernoulli’s theorem

wasn’t fulfilled.

Experimental Setup:

Bernoulli’s Theorem demonstration unit consists of:

Hydraulic Bench

Discharge pipe

Piezometer

Pitot tube

Stop watch

Venture tube with 6 measurement points

Probe for measuring overall pressure (can be moved axially)

Hose connection, water supply

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

Observed Data:

Room temperature 29◦C

Water density at 29◦C= 996 kg/m3 [Franzini & Finnemore, SI mertric edn , Page-571]

Mass of the empty bucket = 650 g or .65 kg

Table 01: Observed data in Piezometer and Pitot tube for flow rate-1,2&3

Observation

No

Tube

No

Diameter

of cross

section

(mm)

Manometer levels (mm) Velocity

Head (mm)

Mass of

(Water+

Bucket)

(kg)

Time

(sec)

Static Head

(mm)

Total Head

(mm)

1

1 25 320 325 5

3.6

30

2 13.9 305 325 20

3 11.8 290 320 30

4 10.7 280 315 35

5 10.0 270 310 40

6 25 275 285 10

2

1 25 345 350 5

3.7

30

2 13.9 325 350 25

3 11.8 300 350 50

4 10.7 290 345 55

5 10.0 245 345 100

6 25 280 295 15

3

1 25 365 370 5

4.6

30

2 13.9 330 370 40

3 11.8 300 370 70

4 10.7 280 365 85

5 10 320 365 140

6 25 305 300 25

290

Calculated Data:

Table 2: Table for calculated data for different head

No of

observations

Volumetric

Flow rate

V

(mm3)

Cross

Sectional

Area

A

(mm2)

Velocity

V (mm/s)

Theoretical

Velocity

Head

V2/2g

(mm)

H2O

Practical

Velocity

Head

V2/2g

(mm)

H2O

Theoretical

Total

Head

V2/2g+

P2/ρg +Z

(mm)

H2O

Practical

Total

Head

(from

pitot tube

Reading)

V2/2g+

P2/ρg +Z

(mm)

H2O

1 98300 490.87 200.32 2.045

5 322.045 325

151.75 647.98 21.40

20 326.40 325

109.36 899.15 41.21

30 331.21 320

89.92 1093.53 60.94

35 340.94 315

78.53 1251.97 79.89

40 349.89 310

490.87 200.32 2.045

10 277.05 285

2 101670 490.87 207.12 2.186 5 347.186 350

151.75 669.99 22.88 25 347.88 350

109.36 929.69 44.05 50 344.05 350

89.92 1130.67 65.16 55 355.16 345

78.53 1294.50 85.40 100 330.41 345

490.87 207.12 2.186 15 282.186 295

3 131670 490.87 268.23 3.67 5 368.67 370

151.75 867.69 38.37 40 368.37 370

109.36 1204.02 73.88 70 373.88 370

89.92 1464.29 109.28 85 389.285 365

78.53 1676.47 143.25

140 368.25

365

490.87 268.23 3.67

25 278.67

300

Sample Calculation: Sample calculation for third observation:

Calculation of volumetric flow rate:

Mass of the bucket with water =4.6 kg

Mass of the empty bucket =0.65 kg

Mass of water, m= 3.95 Kg

Time of flow, t = 30.0 s

Mass flow rate, m= m

t=

3.95

30.0 = 0.132 kg s⁄

Density at 29˚c, ρ =996 Kg m3⁄

Volumetric flow rate , v = m

ρ=

0.132

996= 1.316×10

-4m3 s⁄ = 131670

Velocity calculation:

For diameter, a:

Diameter, D =25.0× 10−3 m

Area, A= πD2

4= 4.91× 10

-4 m2

Velocity, V = v

A=

1.316×10-4

4.91×10-4=0.268 ms-1 = 268.02 𝑚𝑚/𝑠

For diameter, b:

Diameter, D =13.9× 10-3

m

Area, A= πD2

4= 1.52×10

-4 m2

Velocity, V = v

A=

1.316×10-4

1.52×10-4=0.865 ms-1 = 865.789 𝑚𝑚/𝑠

For diameter, c:

Diameter, D =11.8×10-3

m

Area, A= πD2

4= 1.09×10

-4 m2

Velocity, V =v

A=

1.316×10-4

1.09×10-4=1.204 ms-1=1204.02mm/s

For diameter, d:

Diameter, D =10.7×10-3

m

Area, A= πD2

4= 0.899×10

-4 m2

Velocity, V =v

A=

1.316×10-4

0.899×10-4= 1.464 ms-1 = 1464.29

For diameter, e:

Diameter, D =10.0×10-3

m

Area, A= πD2

4= 0.785×10

-4 m2

Velocity, V =v

A=

1.316×10-4

0.785×10-4=1.676ms-1 = 1676.47𝑚𝑚/𝑠

For diameter, f:

Diameter, D =25.0×10-3

m

Area, A= πD2

4= 4.91×10

-4 m2

Velocity, V =v

A=

1.316×10-4

4.91×10-4=0.268 ms-1 = 268.02 𝑚𝑚/𝑠

Total Head Calculation for third observation:

For cross sectional diameter, a

Velocity head =g

v

2

2

= (268.02)

2

2×9.81×1000 m = 3.67 mm

Static head (observed), P

ρg +z= 365 mm

Total head, H= (gρ

P+

g2

v2

+z) = (3.67 +365 ) mm = 368.67 mm

For cross sectional diameter, b

Velocity head, v2

2g=

(865.789)2

2×9.81×1000=38.21 mm

Static head (observed), P

ρg +z= 330 mm

H= (330+38.21) mm= 368.21 mm

For cross sectional diameter, c

Velocity head, v2

2g=

(1204.02)2

2×9.81×1000= 73.88 mm

Static head (observed), P

ρg +z= 300 mm

H= (73.88 +300) mm = 373.88 mm

For cross sectional diameter, d

Velocity head, v2

2g=

(1464.29)2

2×9.81×1000= 109.28 mm

Static head (observed), P

ρg+ 𝑧 = 280 mm

H= (109.28 +280) mm = 389.28 mm

For cross sectional diameter, e

Velocity head, v2

2g=

(1676.47)2

2×9.81×1000= 143.25 mm

Static head (observed), P

ρg +z=225 mm

H = (143.25+ 225) m = 368.25 mm

For cross sectional diameter, f

Velocity head, v2

2g=

(268.02)2

2×9.81×1000= 3.66 mm

Static head (observed), P

ρg+ 𝑧 = 275 mm

H = (3.66 +275) m = 278.66 mm

Result: The theoretical and experimental values are very much close to each other. So we can say that

the Bernoulli’s theorem is valid for steady flow of ideal fluids.

Graphical Representation:

Here head loss is actually the difference between theoretical total head that we have got from the

volumetric flow rate and calculated total head that we have got from the manometer levels. It is

evident from the graph that as the flow rate increases the head loss between two different point

also increases. With increase in velocity frictional losses like contraction and expansion at the

converging & diverging path increases. That’s why head loss also increases.

90000

100000

110000

120000

130000

140000

1 6 11 16

Flow Rate

Head Loss

Flow Rate vs Head Loss

Discussions:

Bernoulli’s theorem states that, for a frictionless ideal fluid flowing through a system the total energy

entering a system is equal to the total energy coming out of the system. So the assumptions in this

theorem are,

The fluid have to be an ideal one.

The surface have to be frictionless.

The fluid needed to be incompressible.

There should be no exchange of energy between the points.

Though the experiment was done very carefully and properly, there is some discrepancy in the result.

The total head was not same in different points. The possible reasons are-

In Bernoulli’s theorem the fluid is considered as inviscid and incompressible. But

practically the water, used in this experiment as the working fluid does not satisfy

this assumption.

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.

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 variation of α with Reynolds number has to be accounted if we want to get

correct result.

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.

Fig: Flow at sudden contraction of cross section

Directional velocity fluctuation due to turbulence increase pitot tube readings and

hence we got large value of total head.

A large head loss occurs at the entrance of the pitot tube due to sudden

contraction.

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 in following figure .

Fig: Flow at sudden enlargement of cross section

There was a leak in the venturi tube which induced clogging in the flow and some error

occurred as a result. The piezometer readings were fluctuating continuously during the experiment due to

unsteady supply of the flowing system by the pump. Since capillarity makes water rise in

piezometer tube, it introduce some error in the calculation of the static head.

However ,it is also have to be noted that there might have been some human (parallax

error while taking reading of manometer levels) and unintentional errors done in the

experiment like misreading stop watch and volume meter which might have given us

some deviated results from the actual results.

Now ,let us talk about some of remedies to eradicate all those discrepancies and

application of Bernoulli’s theorem:

Recommendations:

However, the results can be improved if some precautions are taken during the

experiment for example the eyes level must be placed parallel to the scale when

manometer readings are taken. Besides that, the valve is also needed to be

controlled slowly to stabilize the water level in the manometer. The human

reaction error while noting the time using a stop watch can be avoided by using

light gates to give out highly accurate results for the time measured.

Applications of Bernoulli’s theorem:

Bernoulli’s theorem has several applications in everyday lives.In certain problems

in fluid flows when given the velocities at two points of the streamline and

pressure at one point, the unknown is the pressure of the fluid at the other point.

In practical life, if at a point in a pipeline sensors or any other pressure measuring

device can’t be used then we can use Bernoull’s theorem (if they satisfy the

required condition for Bernoulli's Equation) to find the unknown pressure. One

such example is the flow through a converging nozzle.