Republic of the Philippines
UNIVERSITY OF SOUTHERN MINDANAO
College of Engineering and Information Technology (CEIT)
Main Campus
Kabacan, North Cotabato
HYDRAULICS
CE 313
LABORATORY NO. 2
BERNOULLI’S THEOREM DEMONSTRATION
Group 1 Members:
ABPI, NUF
BARAGUIR, DATU SAYYID HADI
BON, FRETCHIE M.
HALLARA, ROANNE JOY P.
JIMENEZ, RALPH C.
LAZARTE, KHENDALL BLYTE
3 Bachelor of Science in Civil Engineering – C
Instructor:
ENGR. CHRISLAM S. MANTAWIL
ENGR. CHRISTIAN MANGINSAY
LABORATORY NO. 2
BERNOULLI’S THEOREM DEMONSTRATION
I. Introduction
Energy presents in the form of pressure, velocity, and elevation in fluids with no
energy exchange due to viscous dissipation, heat transfer, or shaft work (pump or
some other device). The relationship among these three forms of energy was first
stated by Daniel Bernoulli (1700-1782), based upon the conservation of energy
principle. Bernoulli’s theorem pertaining to a flow streamline is based on three
assumptions: steady flow, incompressible fluid, and no losses from the fluid friction.
The validity of Bernoulli’s equation will be examined in this experiment.
II. Objectives
⚫ To investigate the validity of the Bernoulli equation when it is applied to a
steady flow of water through a tapered duct.
III. Materials
⚫ F1-10 hydraulics bench,
⚫ F1-15 Bernoulli’s apparatus test equipment
⚫ A stopwatch for timing the flow measurement
IV. Procedure
1.Place the apparatus on the hydraulics bench, and ensure that the outflow tube
is positioned above the volumetric tank to facilitate timed volume collections.
2.Level the apparatus base by adjusting its feet. (A sprit level is attached to the
base for this purpose.) For accurate height measurement from the manometers,
the apparatus must be horizontal.
3.Install the test section with the 14° tapered section converging in the flow
direction. If the test section needs to be reversed, the total head probe must be
retracted before releasing the mounting couplings.
4.Connect the apparatus inlet to the bench flow supply, close the bench valve and
the apparatus flow control valve, and start the pump. Gradually open the bench
valve to fill the test section with water.
5.The following steps should be taken to purge air from the pressure tapping
points and manometers:
⚫ Close both the bench valve and the apparatus flow control valve.
⚫ Remove the cap from the air valve, connect a small tube from the air
valve to the volumetric tank, and open the air bleed screw.
⚫ Open the bench valve and allow flow through the manometers to purge
all air from them, then tighten the air bleed screw and partly open the bench
valve and the apparatus flow control valve.
⚫ Open the air bleed screw slightly to allow air to enter the top of the
manometers (you may need to adjust both valves to achieve this), and
re- tighten the screw when the manometer levels reach a convenient
height. The maximum flow will be determined by having a maximum (h1)
and minimum (h5) manometer readings on the baseboard.
If needed, the manometer levels can be adjusted by using an air pump to pressurize
them. This can be accomplished by attaching the hand pump tube to the air bleed
valve, opening the screw, and pumping air into the manometers. Close the screw, after
pumping, to retain the pressure in the system.
6.Take readings of manometers h to h when the water level in the manometers
is steady. The total pressure probe should be retracted from the test section
during this reading.
7. Measure the total head by traversing the total pressure probe along the test
section from h1 to h6.
8. Measure the flow rate by a timed volume collection. To do that, close the ball
valve and use a stopwatch to measure the time it takes to accumulate a known
volume of fluid in the tank, which is read from the sight glass. You should
collect fluid for at least one minute to minimize timing errors. You may repeat
the flow measurement twice to check for repeatability. Be sure that the total
pressure probe is retracted from the test section during this measurement.
9. Reduce the flow rate to give the head difference of about 50 mm between
manometers 1 and 5 (h1 - h5). This is the minimum flow experiment. Measure
the pressure head, total head, and flow.
10. Repeat the process for one more flow rate, with the (h1 - h5) difference
approximately halfway between those obtained for the minimum and maximum
flows. This is the average flow experiment
V. Data Analysis
Table 1. Raw data of Position 1
Position 1: Tapering 14° to 21°
Test section Volume (Liter) Time (sec) Static Head (mm) Total Head (mm)
h1 225 228
h2 177 225
h3 115 225
20 133
h4 55 220
h5 25 220
h6 155 185
h1 210 195
h2 168 195
h3 144 195
17 162
h4 150 210
h5 145 235
h6 216 220
h1 195 195
h2 160 180
h3 125 173
15 132
h4 88 165
h5 70 155
h6 140 130
Table 2. Raw data of Position 2 (Reverse)
Position 2: Tapering 21° to 14°
Static Head Total Head
Test section Volume (Liter) Time (sec)
(mm) (mm)
h1 170 190
h2 130 216
h3 88 218
12 86
h4 70 220
h5 85 225
h6 235 235
h1 195 204
h2 170 223
h3 150 222
14 133
h4 135 224
h5 158 230
h6 235 240
h1 155 185
h2 125 211
h3 85 211
17 132
h4 80 210
h5 90 200
h6 210 210
The volume of water utilized in this experiment is 12L, 14L, and 17L. The static
head, which represents the pressure energy per unit weight of fluid, may be
measured using a variety of instruments, including: Piezometers; Piezometers are
simple tubes put into the fluid at certain places. The height of the fluid column in the
piezometer reveals the static pressure at that position, from which the static head
may be computed. Pressure Gauges; Pressure gauges, which are linked to the
system by taps or transducers, directly measure the pressure at a spot. The static
head is then determined using the pressure measurement and fluid density.
Once the raw data for static head and total head has been acquired, Bernoulli's
theorem may be used to compute the fluid's velocity and energy changes between
different points in the system. The study may also be used to detect possible areas of
energy loss and to evaluate the overall system efficiency.
Calculation Formula
Bernoulli’s theorem assumes that the flow is frictionless, steady, and
incompressible. These assumptions are also based on the laws of conservation
of mass and energy. Thus, the input mass and energy for a given control
volume are equal to the output mass and energy:
These two laws and the definition of work and pressure are the basis for
Bernoulli’s theorem and can be expressed as follows for any two points located
on the same streamline in the flow:
where:
P: pressure,
g: acceleration due to gravity,
v: fluid velocity, and
z: vertical elevation of the fluid.
In this experiment, since the duct is horizontal, the difference in height can be
disregarded, i.e., z1=z2
The hydrostatic pressure (P) along the flow is measured by manometers tapped
into the duct. The pressure head (h), thus, is calculated as:
Therefore, Bernoulli’s equation for the test section can be written as:
in which is called the velocity head (hd).
The total head (ht) may be measured by the traversing hypodermic probe. This
probe is inserted into the duct with its end-hole facing the flow so that the flow
becomes stagnant locally at this end; thus:
The conservation of energy or the Bernoulli’s equation can be expressed as:
The flow velocity is measured by collecting a volume of the fluid (V) over a time
period (t). The flow rate is calculated as:
The velocity of flow at any section of the duct with a cross-sectional area of is
determined as:
For an incompressible fluid, conservation of mass through the test section
should be also satisfied (Equation 1a), i.e.:
VI. Result and Discussion
Table 3. Calculated Result
Distance Pressure Velocity Calculated Measured
Test Flow Area Flow Rate Velocity Head Head Total Head Total Head
c into duct
Section (m²) (m³/s) (m/s)
(m) (m) (m) (m) (m)
h1 0 0.000490 0.307 0.225 0.0048 0.230 0.228
h2 0.06028 0.000150 1.003 0.177 0.0512 0.228 0.225
h3 0.06868 0.000110 1.367 0.115 0.0953 0.210 0.225
1 1.50E-04
h4 0.07318 0.000090 1.671 0.055 0.1423 0.197 0.220
h5 0.08108 0.000079 1.903 0.025 0.1847 0.210 0.220
h6 0.14154 0.000490 0.307 0.155 0.0048 0.160 0.185
h1 0 0.000490 0.214 0.210 0.0023 0.212 0.195
h2 0.06028 0.000150 0.700 0.168 0.0249 0.193 0.195
h3 0.06868 0.000110 0.954 0.144 0.0464 0.190 0.195
2 1.05E-04
h4 0.07318 0.000090 1.166 0.150 0.0693 0.219 0.210
h5 0.08108 0.000079 1.328 0.145 0.0899 0.235 0.235
h6 0.14154 0.000490 0.214 0.216 0.0023 0.218 0.220
h1 0 0.000490 0.232 0.195 0.0027 0.198 0.195
h2 0.06028 0.000150 0.758 0.160 0.0293 0.189 0.180
h3 0.06868 0.000110 1.033 0.125 0.0544 0.179 0.173
3 1.14E-04
h4 0.07318 0.000090 1.263 0.088 0.0813 0.169 0.165
h5 0.08108 0.000079 1.438 0.070 0.1055 0.175 0.155
h6 0.14154 0.000490 0.232 0.140 0.0027 0.143 0.130
h1 0 0.000490 0.285 0.170 0.0041 0.174 0.190
h2 0.06028 0.000150 0.930 0.130 0.0441 0.174 0.216
h3 0.06868 0.000110 1.268 0.088 0.0820 0.170 0.218
4 1.40E-04
h4 0.07318 0.000090 1.550 0.070 0.1225 0.193 0.220
h5 0.08108 0.000079 1.766 0.085 0.1590 0.244 0.225
h6 0.14154 0.000490 0.285 0.235 0.0041 0.239 0.235
h1 0 0.000490 0.215 0.195 0.0024 0.197 0.204
h2 0.06028 0.000150 0.702 0.170 0.0251 0.195 0.223
h3 0.06868 0.000110 0.957 0.150 0.0467 0.197 0.222
5 1.05E-04
h4 0.07318 0.000090 1.170 0.135 0.0697 0.205 0.224
h5 0.08108 0.000079 1.332 0.158 0.0905 0.248 0.230
h6 0.14154 0.000490 0.215 0.235 0.0024 0.237 0.240
h1 0 0.000490 0.263 0.155 0.0035 0.159 0.185
h2 0.06028 0.000150 0.859 0.125 0.0376 0.163 0.211
h3 0.06868 0.000110 1.171 0.085 0.0699 0.155 0.211
6 1.29E-04
h4 0.07318 0.000090 1.431 0.080 0.1044 0.184 0.210
h5 0.08108 0.000079 1.630 0.090 0.1355 0.225 0.200
h6 0.14154 0.000490 0.263 0.210 0.0035 0.214 0.210
Total head (calculated and measured), pressure head, and velocity head (y-
axis) vs. distance into duct (x-axis) from manometer 1 to 6 Graph
Test 1
0.250
0.200
0.150
Head (m)
0.100
0.050
0.000
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16
Distance into duct (m)
Pressure Head Velocity Head Calculated Total Head Measured Total Head
Test 2
0.250
0.200
0.150
Head (m)
0.100
0.050
0.000
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16
Distance into duct (m)
Pressure Head Velocity Head Calculated Total Head Measured Total Head
Test 3
0.250
0.200
0.150
Head (m)
0.100
0.050
0.000
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16
Distance into duct (m)
Pressure Head Velocity Head Calculated Head Measured Total Head
Test 4
0.300
0.250
0.200
Head (m)
0.150
0.100
0.050
0.000
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16
Distance into duct (m)
Pressure Head Velocity Head Calculated Total Head Measured Total Head
Test 5
0.300
0.250
0.200
Head (m)
0.150
0.100
0.050
0.000
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16
Distance into duct (m)
Pressure Head Velocity Head Calculated Total Head Measured Total Head
Chart Title
0.250
0.200
0.150
Head (m)
0.100
0.050
0.000
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16
Distance into duct (m)
Pressure Head Velocity Head Calculated Total Head Measured Total Head
Bernoulli's equation is a fundamental principle in fluid dynamics that establishes a
link between the pressure, velocity, and gravitational potential energy of a fluid particle.
While the equation is typically relevant to diverse fluid flow conditions, it does have
certain limits, notably in converging and diverging flows.
In cases when flow converges or diverges via a duct, steady flow along a
streamline, a critical assumption in deriving Bernoulli's equation, may not hold.
Furthermore, major variations in velocity, turbulence, or compressibility might have an
influence on the equation's validity.
According to Bernoulli's equation, in converging flows, as duct diameter reduces,
fluid velocity increases, resulting in a pressure reduction. However, ignoring elements
such as viscous effects and turbulence might result in departures from projected
performance. Similarly, in diverging flows with increasing duct width, a drop in velocity
should result in an increase in pressure, but actual behavior may differ owing to non-
ideal circumstances.
Despite its shortcomings, Bernoulli's equation is nonetheless a useful tool for
understanding fluid flow in real-world circumstances. Engineers frequently use it as a
starting point for study, acknowledging the necessity for more complicated models
when working with flows that deviate greatly from ideal circumstances.
In general, the difference between estimated and measured total heads should be
modest. Significant variances need an inquiry, which should include a review of
computation processes, measurement accuracy, and the discovery of unaccounted-
for losses. Inaccuracies in calculating methodologies, poor measurements, and
unaccounted-for system losses, such as those from leaks or fittings, can all contribute
to variances. Furthermore, if the estimated and observed total heads are nearly same,
it suggests that the experimental circumstances and application of Bernoulli's equation
are congruent with the theoretical framework. This consistency boosts confidence in
the equation's validity for the particular case and implies that the experiment's setup,
measurements, and computations were correct.
The fundamental motivating force behind Bernoulli's derivation was the rule of
conservation of energy. In steady-flowing fluids, the total of mechanical energies,
comprising kinetic energy, dynamic head, fluid pressure, and potential energy,
remains constant during the flow, according to Bernoulli's principle. According to the
concept, if one type of energy grows, others must decrease in order to preserve the
total sum.
Bernoulli's Theorem, which has been empirically verified, states that there are
three major components in a fluid flow: pressure head, kinetic energy, and potential
energy. Theorem demonstration findings indicated that as the diameter of the cross-
section reduced, velocities along the distribution tube increased, resulting in a
considerable pressure differential. Higher fluid flow speeds were also recorded in the
smaller tube, confirming Bernoulli's principle that relates high velocity with low
pressure. The data backed up the assumption that as velocity increases, so does total
head pressure, which is consistent with both Bernoulli's equation and the continuity
equation.
CONCLUSION
Finally, Bernoulli's equation is a fundamental principle in fluid dynamics,
providing important insights into the connection between pressure, velocity, and
potential energy in a fluid flow. However, it is critical to understand the equation's limits,
especially in cases involving converging or diverging flows, where the assumption of
continuous flow along a streamline may not be valid.
The equation's practical applicability is still significant, acting as a starting point
for engineers to examine fluid flow. Regardless of deviations from ideal circumstances,
engineers may apply Bernoulli's equation to acquire a basic knowledge of the system,
paving the way for more complicated models as necessary.
Disagreements between calculated and measured values should be thoroughly
investigated, taking into consideration aspects such as computation methodology,
measurement precision, and unaccounted-for losses. Bernoulli's derivation is based
on the law of conservation of energy, which emphasizes the interaction of kinetic
energy, dynamic head, fluid pressure, and potential energy in a steady-flowing fluid.
Bernoulli's theorem experimental demonstrations give actual evidence backing
its concepts. Variations in tube diameter resulted in commensurate changes in fluid
velocities and pressure in these tests, which aligned with both Bernoulli's equation and
the continuity equation. These data support the idea that higher velocity equals lower
pressure, emphasizing the dynamic link between pressure head, kinetic energy, and
potential energy in fluid flow.
Despite its limitations, Bernoulli's equation remains a useful tool in the toolkit of
fluid dynamicists and engineers, providing a realistic foundation for understanding and
studying fluid behavior in a variety of settings.
REFERENCES
Ahmari, H. (2019, August 14). Experiment #2: Bernoulli’s theorem demonstration.
Pressbooks.https://uta.pressbooks.pub/appliedfluidmechanics/chapter/experi
ment-2/
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