BERNOULLI
Transcript of BERNOULLI
TABLE OF CONTENT
1
CONTENT PAGE
ABSTRACT / SUMMARY 2
INTRODUCTION 2 – 3
AIMS / OBJECTIVE 4
THEORY 4 – 7
APPARATUS 7
PROCEDURE 8
RESULTS 9 – 11
CALCULATION 12 – 13
DISCUSSION 14
CONCLUSION 15
RECOMMENDATION 15
REFERENCES 16
APPENDICES 16
ABSTRACT / SUMMARY
From this experiment, we want to investigate the validity of the Bernoulli equation when
applied to the steady flow of water in a tapered duct. Secondly we want to measure flow
rates and both static and total pressure heads in a rigid convergent and divergent tube of
known geometry for a range of steady flow rates.
To run this experiment, firstly the Bernoulli equation apparatus on the hydraulic bench
was set up so that its base is horizontal for accurate height measurement from the
manometers. We used Δh manometer 50, 100 and 150 between Δh1 and Δh5 for both
converging and diverging tube. After that, the section diverging in the direction of flow
was set up. Then water inlet and outlet was connected. The time to collect 3 L water in
the tank was determined. Lastly calculate the flow rate, velocity, dynamic head, and total
head using the reading we get from the experiment and data given. The step was repeated
using converging in the direction of flow.
INTRODUCTION
In fluid dynamics, Bernoulli's principle states that for an inviscid flow, an increase in the
speed of the fluid occurs simultaneously with a decrease in pressure or a decrease in the
fluid's potential energy. Bernoulli's principle is named after the Dutch-Swiss mathematician
Daniel Bernoulli who published his principle in his book Hydrodynamica in 1738. [2]
Bernoulli's principle can be applied to various types of fluid flow, resulting in what is loosely
denoted as Bernoulli's equation. In fact, there are different forms of the Bernoulli equation
for different types of flow. The simple form of Bernoulli's principle is valid for
incompressible flows (e.g. most liquid flows) and also for compressible flows (e.g. gases)
moving at low Mach numbers. More advanced forms may in some cases be applied to
compressible flows at higher Mach numbers (see the derivations of the Bernoulli equation). [2]
Bernoulli's principle can be derived from the principle of conservation of energy. This states
that in a steady flow the sum of all forms of mechanical energy in a fluid along a streamline is
the same at all points on that streamline. This requires that the sum of kinetic energy and
potential energy remain constant. If the fluid is flowing out of a reservoir the sum of all forms
of energy is the same on all streamlines because in a reservoir the energy per unit mass (the
sum of pressure and gravitational potential ρ g h) is the same everywhere. [2]
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Fluid particles are subject only to pressure and their own weight. If a fluid is flowing
horizontally and along a section of a streamline, where the speed increases it can only be
because the fluid on that section has moved from a region of higher pressure to a region of
lower pressure; and if its speed decreases, it can only be because it has moved from a region
of lower pressure to a region of higher pressure. Consequently, within a fluid flowing
horizontally, the highest speed occurs where the pressure is lowest, and the lowest speed
occurs where the pressure is highest.[2]
Bernoulli's equation is sometimes valid for the flow of gases: provided that there is no transfer
of kinetic or potential energy from the gas flow to the compression or expansion of the gas. If
both the gas pressure and volume change simultaneously, then work will be done on or by the
gas. In this case, Bernoulli's equation -- in its incompressible flow form -- can not be assumed
to be valid. [1]
[4]
Suggested ShapesShape Title Comments
Narrow pipe widens
As cross-sectional area increases, velocity drops and pressure slightly increases
Rocket nozzle Exhaust is shot at high speed out of narrow opening
Drifting Rafters drift in lazy current between rapids
The relationship between the velocity and pressure exerted by a moving liquid is described by
the Bernoulli's principle: as the velocity of a fluid increases, the pressure exerted by that fluid
decreases. The Continuity Equation relates the speed of a fluid 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 visa-versa. [3]
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AIMS / OBJECTIVE
1. To investigate the validity of the Bernoulli equation when applied to the steady
flow of water in a tapered duct.
2. To measure flow rates and both static and total pressure heads in a rigid
convergent and divergent tube of known geometry for a range of steady flow rates.
THEORY [2]
In most flows of liquids, and of gases at low Mach number, the mass density of a fluid parcel
can be considered to be constant, regardless of pressure variations in the flow. For this reason
the fluid in such flows can be considered to be incompressible and these flows can be
described as incompressible flow. Bernoulli performed his experiments on liquids and his
equation in its original form is valid only for incompressible flow. A common form of
Bernoulli's equation, valid at any arbitrary point along a streamline where gravity is constant,
is:
(A)
where:
is the fluid flow speed at a point on a streamline,
is the acceleration due to gravity,
is the elevation of the point above a reference plane, with the positive z-direction
pointing upward — so in the direction opposite to the gravitational acceleration,
is the pressure at the point, and
is the density of the fluid at all points in the fluid.
For conservative force fields, Bernoulli's equation can be generalized as:
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where Ψ is the force potential at the point considered on the streamline. E.g. for the Earth's
gravity Ψ = gz.
The following two assumptions must be met for this Bernoulli equation to apply:[5]
the fluid must be incompressible — even though pressure varies, the density must
remain constant along a streamline;
friction by viscous forces has to be negligible.
By multiplying with the fluid density ρ, equation (A) can be rewritten as:
or:
where:
is dynamic pressure,
is the piezometric head or hydraulic head (the sum of the elevation z
and the pressure head) and
is the total pressure (the sum of the static pressure p and dynamic
pressure q).
The constant in the Bernoulli equation can be normalised. A common approach is in terms of
total head or energy head H:
The above equations suggest there is a flow speed at which pressure is zero, and at even
higher speeds the pressure is negative. Most often, gases and liquids are not capable of
negative absolute pressure, or even zero pressure, so clearly Bernoulli's equation ceases to be
valid before zero pressure is reached. In liquids—when the pressure becomes too low --
cavitation occurs. The above equations use a linear relationship between flow speed squared
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and pressure. At higher flow speeds in gases, or for sound waves in liquid, the changes in
mass density become significant so that the assumption of constant density is invalid.
In many applications of Bernoulli's equation, the change in the ρ g z term along the streamline
is so small compared with the other terms it can be ignored. For example, in the case of
aircraft in flight, the change in height z along a streamline is so small the ρ g z term can be
omitted. This allows the above equation to be presented in the following simplified form:
where q= v2
g
where p0 is called total pressure, and q is dynamic pressure. Many authors refer to the pressure
p as static pressure to distinguish it from total pressure p0 and dynamic pressure q. In
Aerodynamics, L.J. Clancy writes: "To distinguish it from the total and dynamic pressures, the
actual pressure of the fluid, which is associated not with its motion but with its state, is often
referred to as the static pressure, but where the term pressure alone is used it refers to this
static pressure."
The simplified form of Bernoulli's equation can be summarized in the following memorable
word equation:
static pressure + dynamic pressure = total pressure
Every point in a steadily flowing fluid, regardless of the fluid speed at that point, has its own
unique static pressure p and dynamic pressure q. Their sum p + q is defined to be the total
pressure p0. The significance of Bernoulli's principle can now be summarized as total pressure
is constant along a streamline.
If the fluid flow is irrotational, the total pressure on every streamline is the same and
Bernoulli's principle can be summarized as total pressure is constant everywhere in the fluid
flow. It is reasonable to assume that irrotational flow exists in any situation where a large
body of fluid is flowing past a solid body. Examples are aircraft in flight, and ships moving in
open bodies of water. However, it is important to remember that Bernoulli's principle does not
apply in the boundary layer or in fluid flow through long pipes.
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If the fluid flow at some point along a stream line is brought to rest, this point is called a
stagnation point, and at this point the total pressure is equal to the stagnation pressure.
APPARATUS
The F1-10 Hydraulic Bench, which allows us to measure flow by, timed volume
collection.
The F1-15 Bernoulli’s Apparatus Test Equipment.
A stopwatch for timing the flow measurement.
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PROCEDURE
1. The main switch and ump are switched on.
2. The venturi for the convergent flow position are setup.
3. The flow control valve are fully open to let the water flow into the venture and
manometer tubes.
4. The air bleed screw are adjusted.
5. The flow control valve and valve 1 are closed.
6. The air bleed screw are regulated until water level in manometer tubes reached 140
mmH2O.
7. The flow control valve are fully open.
8. Valve 1 are regulated slowly to get the different between water level in h1 and h5 that
is 50 mmH2O.
9. The reading from h1 until h5 are taken.
10. The ball in the water tank are dropped. The time are taken until the water reached 3 L.
11. Steps 8 until 10 are repeated for different value of ∆h that are 100 mmH2O and 150
mmH2O.
12. The experiment are repeated for divergent flow.
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RESULTS
g = 9.81 m/s2
1. Divergence
∆h = 50 mm
Volume collected ( m3 ) Time ( s ) Flow rate ( m3/s )3 x 10-3 30.59 9.80 x 10-5
Distance into duct
( m )
Area of duct, A
x10-6 ( m2
)
Static head, h( mm )
Static head, h ( m )
Velocity, v x 10-3
( m/s )
Dynamic head, q ( v2/2g )
(m)
Total head, ho
( m )
h1 0.0000 490.9 150 0.150 0.200 0.0020 0.152
h2 0.0603 151.7 125 0.125 0.646 0.021 0.145
h3 0.0687 109.4 105 0.105 0.896 0.041 0.146
h4 0.0732 89.9 100 0.100 1.090 0.061 0.161
h5 0.0811 78.5 100 0.100 1.248 0.079 0.179
∆h = 100 mm
Volume collected ( m3 ) Time ( s ) Flow rate ( m3/s )3 x 10-3 23.62 1.27 x 10-4
Distance into duct
( m )
Area of duct, A
x10-6 ( m2
)
Static head, h( mm )
Static head, h ( m )
Velocity, v ( m/s )
Dynamic head, q ( v2/2g )
Total head, ho
( m )
h1 0.0000 490.9 165 0.165 0.259 0.0034 0.168
h2 0.0603 151.7 125 0.125 0.837 0.0357 0.161
h3 0.0687 109.4 70 0.070 1.161 0.069 0.139
h4 0.0732 89.9 65 0.065 1.413 0.102 0.167
h5 0.0811 78.5 65 0.065 1.618 0.133 0.198
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∆h = 150 mm
Volume collected ( m3 ) Time ( s ) Flow rate ( m3/s )3 x 10-3 20.0 1.50 x 10-4
Distance into duct
( m )
Area of duct, A
x10-6 ( m2
)
Static head, h( mm )
Static head, h ( m )
Velocity, v ( m/s )
Dynamic head, q ( v2/2g )
Total head, ho
( m )
h1 0.0000 490.9 175 0.175 0.306 0.0047 0.180
h2 0.0603 151.7 115 0.115 0.989 0.050 0.165
h3 0.0687 109.4 45 0.045 1.371 0.096 0.141
h4 0.0732 89.9 25 0.025 1.669 0.142 0.167
h5 0.0811 78.5 25 0.025 1.910 0.186 0.211
2. Convergence
∆h = 50 mm
Volume collected ( m3 ) Time ( s ) Flow rate ( m3/s )3 x 10-3 48.5 6.19 x 10-5
Distance into duct
( m )
Area of duct, A
x10-6 ( m2
)
Static head, h( mm )
Static head, h ( m )
Velocity, v ( m/s )
Dynamic head, q ( v2/2g )
Total head, ho
( m )
h1 0.0000 490.9 165 0.165 0.126 0.00081 0.166
h2 0.0603 151.7 155 0.155 0.408 0.0085 0.164
h3 0.0687 109.4 145 0.145 0.566 0.016 0.161
h4 0.0732 89.9 130 0.130 0.689 0.024 0.154
h5 0.0811 78.5 115 0.115 0.789 0.032 0.147
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∆h = 100 mm
Volume collected ( m3 ) Time ( s ) Flow rate ( m3/s )3 x 10-3 32.4 9.26 x 10-5
Distance into duct
( m )
Area of duct, A
x10-6 ( m2
)
Static head, h( mm )
Static head, h ( m )
Velocity, v ( m/s )
Dynamic head, q ( v2/2g )
Total head, ho
( m )
h1 0.0000 490.9 190 0.190 0.189 0.0018 0.192
h2 0.0603 151.7 165 0.165 0.610 0.019 0.184
h3 0.0687 109.4 145 0.145 0.846 0.036 0.181
h4 0.0732 89.9 120 0.120 1.030 0.054 0.174
h5 0.0811 78.5 90 0.090 1.180 0.071 0.161
∆h = 150 mm
Volume collected ( m3 ) Time ( s ) Flow rate ( m3/s )3 x 10-3 24.47 1.23 x 10-4
Distance into duct
( m )
Area of duct, A
x10-6 ( m2
)
Static head, h( mm )
Static head, h ( m )
Velocity, v ( m/s )
Dynamic head, q ( v2/2g )
Total head, ho
( m )
h1 0.0000 490.9 210 0.210 0.251 0.0032 0.213
h2 0.0603 151.7 180 0.180 0.811 0.034 0.214
h3 0.0687 109.4 145 0.145 1.124 0.064 0.209
h4 0.0732 89.9 105 0.105 1.368 0.095 0.200
h5 0.0811 78.5 60 0.060 1.567 0.125 0.185
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CALCULATION
1. Divergent
∆h = h1 – h5
= ( 150 – 100) mm
= 50 mm *
Flow rate = volume ( m3 ) time (s)
= 3 x 10-3 m3 30 .59 s
= 9.80 x 10-5 m3/s *
Velocity, v = flow rate m3 /sarea m2
= 9.80 x 10−5 m3 /s490.9 x 10−6 m2
= 0.200 *
Dynamic head , (m) = v2
2g
= 0.2002 m2/s2
2 x 9.81 m/ s2
= 0.00203 m. *
Total head = static head + dynamic head
= 0.150 m + 0.00203 m
= 0.152 m. *
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2. Convergent
∆h = h1 – h5
= ( 165 – 115) mm
= 50 mm *
Flow rate = volume ( m3 ) time (s)
= 3 x 10-3 m3
48.5 s
= 6.19 x 10-5 m3/s *
Velocity, v = flow rate m3 /sarea m2
= 6.19 x 10−5 m3 /s490.9 x 10−6 m2
= 0.126 m. *
Dynamic head , (m) = v2
2g
= 0.1262 m2/s2
2 x 9.81 m/ s2
= 0.00081 m. *
Total head = static head + dynamic head
= 0.165 m + 0.00081 m
= 0.166 m. *
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DISCUSSION
From the Bernoulli’s principle, he stated that that water is a fluid, and having the
characteristics of a fluid, it adjusts its shape to fit that of its container or other solid objects it
encounters on its path. Since the volume passing through a given length of pipe during a given
period of time will be the same, there must be a decrease in pressure. Hence Bernoulli's
conclusion: the slower the rate of flow, the higher the pressure, and the faster the rate of flow,
the lower the pressure.[5] For the divergence flow we can see that the total head at each tube h1
until h5 are not constantly increased but for the convergence flow, the total head are increased
due the increased of flow rate of water.
The flow rates of each flow also incresed that is :
∆h Divergence Convergence
50 mm 9.80 x10-5 m/s 6.19 x10-5 m/s
100 mm 1.270 x10-4 m/s 9.26 x10-5 m/s
150 mm 1.50 x 10-4 m/s 1.23 x10-4 m/s
From the experiment, we found that the total head pressure in both convergence and
divergence flow should be increased according to Bernoulli’s principle. So it shows that
Bernoilli’s equation is valid when applied to the steady flow of water in tapered duct and
velocity values increased along the same channel.
There must be an error during the divergence flow experiment as the total head are not
constantly increased. During the experiment, the pressure on h1 until h5 are not stable yet but
the reading are taken and time for 3 litres water are also recorded. These will affected the
calculation thus affected the total pressure at each tube. Other than that, the time should be
recorded when the water level reached at 0 litre but the time are started before the water level
reached 0 litre. Other than that, the position of eye during reading the manometer tube should
be staright to the meniscus. To get the constant desired pressure difference, the valve 1 and
bleed screw should be regulated smoothly and slowly.
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CONCLUSION
From the experiment, we found that the total head pressure in both convergence and
divergence flow increased according to Bernoulli’s principle. So it shows that Bernoilli’s
equation is valid when applied to the steady flow of water in tapered duct and velocity
values increased along the same channel.
RECOMMENDATION
1. Make sure that there are no leakages on the connection between the pipes. The leakage
will cause the water bleed out from the pipes connection so its will make our
experiment have an errors.
2. Make sure that there are no bubbles in the manometer to get the accurate reading.
3. When take the reading of the volume of the water make sure that eyes parallel with the
volume meter.
4. Control the air bleed screw slowly and smoothly.
5. Waited until the pressure at each manometer tube are stable before the reading are
taken.
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REFERENCES
1. http://www.absoluteastronomy.com/topics/Bernoulli%27s_principle
2. http://en.wikipedia.org/wiki/Bernoulli%27s_principle
3. http://library.thinkquest.org/27948/bernoulli.html
4. http://home.earthlink.net/~mmc1919/venturi.html
5. http://www.scienceclarified.com/everyday/Real-Life-Chemistry-Vol-3/Bernoulli-s- Principle.html
APPENDICES
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h4
h3
V1
h1
h5
h2
venturi
Air bleed screw