Bernoulli WriteUp

Chemical Engineering Laboratory Experiment 2 Report

Dr J Tshuma

Experiment 2 : Bernoulli's Theorem

Adriano Q.PChikande-N01413534K

Department Of Chemical Engineering

Performed Wed 21 Oct 2015 Due Wed 4 Nov 2015

2 | A d r i a n o Q P C h i k a n d e N 0 1 4 1 3 5 3 4 K

Contents

1.Title…………………………………………………………………………………………..3

2.Aim…………………………………………………………………………………………...3

3.Apparatus……………………………………………………………………………………..3

4.Theory………………………………………………………………………………….……..6

5.Method………………………………………………………………………………….……..8

6.Results………………………………………………………………………………………...10

7.Analysis Of Results And Discussion………………………………………………………...15

8.Conclusion…………………………………………………………………………………...16

9.References…………………………………………………………………………………....16


Experiment 2 : Bernoulli’s Theorem

2. Aim : To investigate the validity of Bernoulli’s Theorem as applied to the flow of water in a

tapering circular duct.

3. Apparatus

Stop Watch

Bernoulli’s apparatus as shown on Figure 3.1

Hydraulic Bench


Installation of Bernoulli’s Apparatus

The test section is an accurately machined Perspex duct of varying circular cross section

provided with pressure tappings whereby the static pressure may be measured simultaneously at

each of six sections. The test section incorporates unions at either end to facilitate reversal for

convergent or divergent testing.

A hypodermic probe is provided which may be positioned to read the total head at any section of

the duct. The probe may be moved after slackening the gland nut. This nut should be retightened

by hand. To prevent damage, the probe should be fully inserted during transport/ storage. An

additional tapping is provided to facilitate setting up. All eight pressure tappings are connected to

a bank of pressurised manometer tubes. Pressurization of the manometers is facilitated by

removing the hand pump from the storagelocation at the rear of the manometer board and

connecting its flexible coupling to the inlet valve on the manometer manifold.

In use, the apparatus, mounted on base board, is stood on the work surface of the bench and the

adjustable feet are adjusted to level the apparatus.

The inlet pipe terminates in a female coupling which may be connected directly to the bench

supply.

A flexible hose attached to the outlet pipe is directed to the volumetric tank.

A flow control valve is incorporated downstream of the test section. Flow rate and pressure in

the apparatus, may be varied independently by adjustment of the flow control valve and the

bench control valve.

Figure 3.2 Bernoulli Apparatus Setup


Nomenclature

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

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

P = fluid pressure

Q = flow rate

Ai = cross sectional area at point i

H1 = head loss due to friction

di = diameter at point i


4. Theory In Brief

Bernoulli’s principle shows describes the conservation of energy within a fluid system.

The Theorem is based on 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.

Considering flow at two sections in a pipe, Bernoulli’s equation can be written as follows:

𝑧𝑖 +𝑣𝑖2

2𝑔+𝑝𝑖𝜌𝑔

= 𝑧𝑗 +𝑣𝑗2

2𝑔+𝑝𝑗

𝜌𝑔

In this equation, no account is taken for losses due to friction, that is, this is for inviscid flow.

Bernoulli proved his theorem for an inviscid fluid and for flow along a streamline. However, it

may be adopted for pipe flow, but since water has viscosity, there will be an energy loss

occurring. For practical purposes, the theorem is revised as follows:

𝑣𝑖2

2𝑔+𝑝𝑖𝜌𝑔

+ 𝑧𝑖 =𝑣𝑗2

2𝑔+𝑝𝑗

𝜌𝑔+ 𝑧𝑗 +𝐻1

H1 is the head loss due to friction between two points i and j, hence

𝐻1 =𝑝𝑖𝜌𝑔

+𝑣𝑖

2

2𝑔+ 𝑧𝑖 −

𝑝𝑗

𝜌𝑔−𝑣𝑗

2

2𝑔− 𝑧𝑗

The pipe is horizontal therefore 𝐻1 =𝑝𝑖−𝑝𝑗

𝜌𝑔+

𝑣𝑖2−𝑣𝑗

2

2𝑔


The manometers measure relative pressure. The velocity head is found using the continuity

equation: 𝑄 = 𝐴𝑖𝑣𝑖 = 𝐴𝑗𝑣𝑗

Fig 3 below shows the test section with measurements dimensioned. This test section is used as

both the convergent and divergent conduit, by just changing the direction of flow of water.

Figure 4.1 Test Section with measurements

Table 4.1 Apparatus measurements

Tapping position Manometer height Diameter (mm)

A 𝐡𝟏 25.0

B 𝐡𝟐 13.9

C 𝐡𝟑 11.8

D 𝐡𝟒 10.7

E 𝐡𝟓 10.1

F 𝐡𝟔 25.0


5. Method

Apparatus Preparation

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 the manometers.

The screw is re-tightened when the manometer reaches a convenient height.


Convergent Conduit

1. Close the discharge valve and the flow control valve.

2. Open the air bleed valve.

3. Position the flexible drain tubing connected to the air inlet to discharge to the

hydraulic bench volumetric well.

4. Open the hydraulic bench discharge valve to allow flow through the

manometers to purge all air.

5. Close the air bleed valve. Close the bench discharge valve and outlet control valve.

Stop the pump.

6. Open the air bleed valve to allow air to enter the top of the manometers. Close

the valve when the levels in the manometers reach approximately half-height.

7. For three different flow rates, determine the pressure head readings with manometers 1

through 6 and the total head readings at each station with the hypodermic manometer.

Note that if there is air along the system at one manometer, the measured head (h) is not

valid.

8. Measure each flow rate using the hydraulic bench and the stopwatch.

Divergent Conduit

1. To investigate the divergent conduit, stop the inlet feed, drain off the equipment, withdraw

the probe (full length), undo the couplings, reverse the test section and replace the couplings.

2. Repeat the above procedure for convergent flow.

Volume flow rate through the Bernoulli apparatus is controlled by adjusting the bench discharge

valve and the outlet flow control valve. Maximum flow rate is achieved when the minimum and

maximum manometer levels are just within the range of the manometer scale. Always start the

hydraulic bench pump with the discharge valve and the apparatus outlet control valve closed, and

then slowly open the valves.


6. Results

6.1. Convergent Section

Table 6.1 Convergent section Properties

Position Diameter

(m)

Cross

sectional

Area (×10-

4m)

Probe

distance (m)

Probe

Manometer

Level (m)

Velocity (ms-

1)

Velocity

Head (×10-4

m)

Total Head

(m)

A 0.0250 4.9087 0.00000 0.290 0.09633 4.7296 0.250

B 0.0139 1.5175 0.06028 0.277 0.31160 49.488 0.203

C 0.0118 1.0936 0.06868 0.267 0.43240 95.295 0.153

D 0.0107 0.8992 0.07258 0.256 0.52590 140.96 0.099

E 0.0100 0.7854 0.08108 0.235 0.60210 184.77 0.040

F 0.0250 4.9087 0.14154 0.249 0.09633 4.7296 0.110

The Calculations

Tapping position (A)

da= 25mm = 0.025m Q = Av

Aa = (ᴫ/4)×(da)2 va = Q/Aa

= (ᴫ/4)×0.0252 = 4.7287×10-5/4.9087×10-4

= 4.9087385 × 10-4 = 0.09633ms-1

= 4.9087×10-4m2

Velocity head = hi = vi2/2g

2g = 2×9.81 = 19.62

ha = 0.096332/19.62

= 4.7295967...×10-4

ha = 4.7296×10-4 m


Position (B)

db = 13.9 mm = 0.0139 m vb = Q/Ab

Ab = (ᴫ/4)×(0.01392) = 4.7287×10-5/1.5175×10-4

Ab = 1.5175×10-4 m2 vb = 0.3116 ms-1

Db = 76.08 – 15.8 Velocity head = hb

= 60.28 mm hb = 0.31162/19.62

Db = 0.06028 hb = 4.9488×10-3 m

Ub = 277 mm = 0.277 m

Position (C)

dc = 11.8 mm = 0.0118 m vb = Q/Ac

Ac = (π/4)×0.01182 = 4.7287×10-5/1.0936×10-4

Ac = 1.0936×10-4 m2 vc = 0.4324 ms-1

Dc = (76.08 – 7.4) mm Velocity head =hc

Dc = 0.06868 m hc = 0.43242/19.62

Uc = 267 mm = 0.267 m hc = 9.5295×10-3 m

Position (D)

dd = 10.7 mm = 0.0107 m vd = Q/Ad

Ad = (π/4)×0.01072 = 4.7287×10-5/8.9920×10-5

Ad = 8.9920×10-5 m2 vd = 0.5259 ms-1

Dd = (76.08 – 3.5) mm Velocity head = hd

Dd = 0.07258 m hd = 0.52592/19.62

Ud = 256 mm = 0.256 m hd = 1.4096×10-2 m


Position (E)

de = 10.0 mm = 0.01 m ve =Q/Ae

Ae = (π/4)×0.012 = 4.7287×10-5/7.8540×10-5

Ae = 7.8540×10-5 m2 ve = 0.6021 ms-1

De = (76.08 + 5) mm Velocity head = he

De = 0.08108 m he = 0.60212/19.62

Ue = 235 mm = 0.235 m he = 1.8477×10-2 m

Position (F)

df = 25 mm = 0.025 m vf = Q/Af

Af = (π/4)×0.0252 = 4.7287×10-5/4.9087×10-4

Af = 4.9087×10-4 m2 vf = 0.09633 ms-1

Df = (76.08 + 65.46) mm Velocity head = hf

Df = 0.14154 m hf = 0.096332/19.62

Uf = 249 mm = 0.249 m hf = 4.7296×10-4 m


6.2. Divergent Section

Table 6.2 Divergent Conduit Properties

Position Diameter

(m)

Crossectional

area (×10-4m)

Probe

distance

(m)

Probe

manometer

level (m)

Velocity

(ms-1)

Velocity

head

(×10-4m)

Total Head

(m)

A 0.025 4.9087 0.14154 0.25 1.2070 0.0743 0.269

B 0.0139 1.5175 0.08126 0.231 3.9043 0.7770 0.270

C 0.0118 1.0936 0.07286 0.213 5.4176 1.4959 0.271

D 0.0107 0.8992 0.06896 0.212 6.5889 2.2127 0.301

E 0.01 0.7854 0.06046 0.2 7.5435 2.9003 0.310

F 0.025 4.9087 0 0.287 1.2070 0.0743 0.304

The Calculations

Position (A)

va = Q/Aa Da = (76.08 + 65.46)mm

= 5.9247×10-4/4.9087×10-4 Da = 0.14154 m

va = 1.2070 ms-1

Ua = 250 mm = 0.25 m

Velocity head = ha

ha = 1.20702/19.62

ha = 0.07425 m

Position (B)

vb = Q/Ab Db = (65.46 + 15.8)mm

= 5.9247×10-4/1.5175×10-4 Db = 0.08126 m

vb = 3.9043 ms-1

Ub = 231 mm = 0.231m

Velocity head = hb

hb = 3.90432/19.62


hb = 0.7770 m

Position (C)

vc = Q/Ac Dc = (65.46 + 7.4) mm

= 5.9247×10-4/1.0936×10-4 Dc = 0.07286 m

vc = 5.4176 ms-1

Uc = 213 mm = 0.213 m

Velocity head

hc = 5.41762/19.62

hc = 1.4959 m

Position (D)

vd = Q/Ad Dd = (65.46 + 3.5) mm

= 5.9247×10-4/8.992×10-5 Dd = 0.06896 m

vd = 6.5889 ms-1

Ud = 212 mm = 0.212 m

Velocity head

hd = 6.58892/19.62

hd = 2.2127 m

Position (E)

ve = Q/Ae De = (65.46 – 5) mm

= 5.9247×10-4/7.854×10-3 De = 0.06046 m

ve = 7.5435 ms-1

Ue = 200 mm = 0.2 m

Velocity head

he = 7.54352/19.62

he = 2.9003 m


Position (F)

vf = Q/Af Df = 0 m

= 5.9247×10-4/4.9087×10-4

vf = 1.2070 ms-1 Uf = 287 mm = 0.287 m

Velocity head

hf = 1.20702/19.62

hf = 0.0743 m

7. Discussion And Analysis Of Results

The objective of this experiment was to investigate the validity of the Bernoulli equation when

applied to the steady flow of water in a tapered duct. Although the apparatus used was outdated,

the objective was successfully accomplished.

To achieve the objectives of this experiment, Bernoulli’s theorem demonstration apparatus along

with the hydraulic bench were used. This instrument was combined with a venturi meter and the

pad of manometer tubes which indicate the pressure of h1 to h8 but for this experiment only the

pressure in manometer h1 until h6 being measured. A venturi is basically a converging-diverging

section (like an hourglass), typically placed between tube or duct sections with fixed cross-

sectional area. The flow rates through the venturi meter can be related to pressure measurements

by using Bernoulli’s equation.

From the result obtained through this experiment, it is been observed that when the pressure

difference increase, the flow rates of the water increase and thus the velocities also increase for

both convergent and divergent flow. The result show a rise at each manometer tubes when the

pressure difference increases. As fluid flows from a wider pipe to a narrower one, the velocity of

the flowing fluid increases. This is shown in all the results tables, where the velocity of water that

flows in the tapered duct increases as the duct area decreases, regardless of the pressure difference

and type of flow of each result taken.


From the analysis of the results, it can be concluded that the velocity of water decrease as the

water flow rate decrease.

Sources Of Errors During Experiment

a) Errors due to inefficiency of equipment being used for example accumulation of bubbles

at vital tapping points and leakages at bleeding points

b) Parallax errors during manometer reading.

c) Pump vibrations may result in the development of turbulent conditions in the upstream.

d) Reaction time in measuring time elapsed for a certain height.

e) Reaction time in closing the water inlet valve when water reaches the set height.

8. Conclusion

The experimental heads were slightly different from the theoretical head probe in both the

converging and the divergent conduits. As a result, it can be said that Bernoulli’s equation is

valid for both convergent and divergent flow.

Recommendations

Regularly service the apparatus to deal with leakages

Make sure the trap bubbles must be removing first before start running the

experiment.

Repeat the experiment for several times to get the average values in order to ge t

more accurate results.

The valve must be control carefully to maintain the constant values of the pressure

difference as it is quite difficult to control.

The eye position of the observer must be parallel to the water meniscus when

taking the reading at the manometers to avoid parallax error.

The time keeper must be alert with the rising of water volume to avoid error and

must be only a person who taking the time.


9. References

1. National University of Science and Technology Zimbabwe. CHEMICAL

ENGINEERING LABOTORIES MANUAL, 2015

2. http://en.wikipedia.org/wiki/Bernoulli's_principle

3. http://www.scribd.com/doc/23106099/Bernoulli-Lab-Report

4. http://www.oneschool.net/Malaysia/UniversityandCollege/SPM/revisioncard/physics/forc

eandpressure/bernoulliprinciple.html#3

5. http://library.thinkquest.org/27948/bernoulli.html

6. Michael J. Moran, Engineering Fluid Mechanics, Wiley, 2008

http://en.wikipedia.org/wiki/Bernoulli's_principle

http://en.wikipedia.org/wiki/Bernoulli's_principle

http://www.scribd.com/doc/23106099/Bernoulli-Lab-Report

http://www.scribd.com/doc/23106099/Bernoulli-Lab-Report

http://www.oneschool.net/Malaysia/UniversityandCollege/SPM/revisioncard/physics/forceandpressure/bernoulliprinciple.html#3

http://www.oneschool.net/Malaysia/UniversityandCollege/SPM/revisioncard/physics/forceandpressure/bernoulliprinciple.html#3

http://library.thinkquest.org/27948/bernoulli.html

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