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Bernoulli Experiment for Fluid MEchanics
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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
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Contents
1.Titleβ¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦..3
2.Aimβ¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦...3
3.Apparatusβ¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦..3
4.Theoryβ¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦.β¦β¦..6
5.Methodβ¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦.β¦β¦..8
6.Resultsβ¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦...10
7.Analysis Of Results And Discussionβ¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦...15
8.Conclusionβ¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦...16
9.Referencesβ¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦β¦....16
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Experiment 2 : Bernoulliβs Theorem
2. Aim : To investigate the validity of Bernoulliβs Theorem as applied to the flow of water in a
tapering circular duct.
3. Apparatus
Stop Watch
Bernoulliβs apparatus as shown on Figure 3.1
Hydraulic Bench
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Installation of Bernoulliβs 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
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Nomenclature
p = fluid static pressure at the cross section in N/m2
Ο = 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
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4. Theory In Brief
Bernoulliβs 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, Bernoulliβs equation can be written as follows:
π§π +π£π2
2π+ππππ
= π§π +π£π2
2π+ππ
ππ
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:
π£π2
2π+ππππ
+ π§π =π£π2
2π+ππ
ππ+ π§π +π»1
H1 is the head loss due to friction between two points i and j, hence
π»1 =ππππ
+π£π
2
2π+ π§π β
ππ
ππβπ£π
2
2πβ π§π
The pipe is horizontal therefore π»1 =ππβππ
ππ+
π£π2βπ£π
2
2π
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The manometers measure relative pressure. The velocity head is found using the continuity
equation: π = π΄ππ£π = π΄ππ£π
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 π‘π 25.0
B π‘π 13.9
C π‘π 11.8
D π‘π 10.7
E π‘π 10.1
F π‘π 25.0
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5. Method
Apparatus Preparation
1. The Bernoulliβs 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.
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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.
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6. Results
6.1. Convergent Section
Table 6.1 Convergent section Properties
Position Diameter
(m)
Cross
sectional
Area (Γ10-
4m)
Probe
distance (m)
Probe
Manometer
Level (m)
Velocity (ms-
1)
Velocity
Head (Γ10-4
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 = (α΄«/4)Γ(da)2 va = Q/Aa
= (α΄«/4)Γ0.0252 = 4.7287Γ10-5/4.9087Γ10-4
= 4.9087385 Γ 10-4 = 0.09633ms-1
= 4.9087Γ10-4m2
Velocity head = hi = vi2/2g
2g = 2Γ9.81 = 19.62
ha = 0.096332/19.62
= 4.7295967...Γ10-4
ha = 4.7296Γ10-4 m
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Position (B)
db = 13.9 mm = 0.0139 m vb = Q/Ab
Ab = (α΄«/4)Γ(0.01392) = 4.7287Γ10-5/1.5175Γ10-4
Ab = 1.5175Γ10-4 m2 vb = 0.3116 ms-1
Db = 76.08 β 15.8 Velocity head = hb
= 60.28 mm hb = 0.31162/19.62
Db = 0.06028 hb = 4.9488Γ10-3 m
Ub = 277 mm = 0.277 m
Position (C)
dc = 11.8 mm = 0.0118 m vb = Q/Ac
Ac = (Ο/4)Γ0.01182 = 4.7287Γ10-5/1.0936Γ10-4
Ac = 1.0936Γ10-4 m2 vc = 0.4324 ms-1
Dc = (76.08 β 7.4) mm Velocity head =hc
Dc = 0.06868 m hc = 0.43242/19.62
Uc = 267 mm = 0.267 m hc = 9.5295Γ10-3 m
Position (D)
dd = 10.7 mm = 0.0107 m vd = Q/Ad
Ad = (Ο/4)Γ0.01072 = 4.7287Γ10-5/8.9920Γ10-5
Ad = 8.9920Γ10-5 m2 vd = 0.5259 ms-1
Dd = (76.08 β 3.5) mm Velocity head = hd
Dd = 0.07258 m hd = 0.52592/19.62
Ud = 256 mm = 0.256 m hd = 1.4096Γ10-2 m
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Position (E)
de = 10.0 mm = 0.01 m ve =Q/Ae
Ae = (Ο/4)Γ0.012 = 4.7287Γ10-5/7.8540Γ10-5
Ae = 7.8540Γ10-5 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 = 1.8477Γ10-2 m
Position (F)
df = 25 mm = 0.025 m vf = Q/Af
Af = (Ο/4)Γ0.0252 = 4.7287Γ10-5/4.9087Γ10-4
Af = 4.9087Γ10-4 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 = 4.7296Γ10-4 m
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6.2. Divergent Section
Table 6.2 Divergent Conduit Properties
Position Diameter
(m)
Crossectional
area (Γ10-4m)
Probe
distance
(m)
Probe
manometer
level (m)
Velocity
(ms-1)
Velocity
head
(Γ10-4m)
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
= 5.9247Γ10-4/4.9087Γ10-4 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
= 5.9247Γ10-4/1.5175Γ10-4 Db = 0.08126 m
vb = 3.9043 ms-1
Ub = 231 mm = 0.231m
Velocity head = hb
hb = 3.90432/19.62
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hb = 0.7770 m
Position (C)
vc = Q/Ac Dc = (65.46 + 7.4) mm
= 5.9247Γ10-4/1.0936Γ10-4 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
= 5.9247Γ10-4/8.992Γ10-5 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 β 5) mm
= 5.9247Γ10-4/7.854Γ10-3 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
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Position (F)
vf = Q/Af Df = 0 m
= 5.9247Γ10-4/4.9087Γ10-4
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, Bernoulliβs 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 Bernoulliβs 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.
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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 Bernoulliβs 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.
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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