1.0 ABSTRACT/ SUMMARY Bernoulli’s Theorem Demonstration unit consists of a classical venturi made of clear acrylic. A series of wall tappings allow measurement of the static pressure distribution along the converging duct, while a total head tube is provided to traverse along the centre line of the test section. Basic hydraulic bench or any water supply also attached to this apparatus to operate together. The apparatus can show the flow comparison by using venturi tube. The objective of this experiment is to demonstrate Bernoulli’s Theorem. Generally all the apparatus are set up and the experiment only running on the Bernoulli’s Theorem Demonstration unit and hydraulic bench. The readings of the manometer tube or the height of the water levels that is concerned and recorded in these experiments. The readings from probe A to H will be taken to calculate the V iB by using Bernoulli equation, V iC using continuity equation and the difference between two. While doing the experiments, the presence of bubbles in the tube will affect the height of the water level in the manometer tube and will cause error in the calculation for pressure. This problem can be solved by ‘bleeding processes’. The results show the pressure at the throat of the venturi is the lowest while its velocity is the highest. The results obtain are according to the theory. There are two equations used to obtain the different velocity which are Bernoulli equation and Continuity equation. The results calculated from this two equation then are subtract to find the difference. As the conclusion, the experiments are success where the supposed values have been obtained. All the objectives are achieved and the theories of these experiments are well- understood. 1
Transcript
1.0 ABSTRACT/ SUMMARY
Bernoulli’s Theorem Demonstration unit consists of a classical venturi made of clear acrylic. A series of wall tappings allow measurement of the static pressure distribution along the converging duct, while a total head tube is provided to traverse along the centre line of the test section. Basic hydraulic bench or any water supply also attached to this apparatus to operate together. The apparatus can show the flow comparison by using venturi tube. The objective of this experiment is to demonstrate Bernoulli’s Theorem.
Generally all the apparatus are set up and the experiment only running on the Bernoulli’s Theorem Demonstration unit and hydraulic bench. The readings of the manometer tube or the height of the water levels that is concerned and recorded in these experiments. The readings from probe A to H will be taken to calculate the V iB by using Bernoulli equation, ViC using continuity equation and the difference between two. While doing the experiments, the presence of bubbles in the tube will affect the height of the water level in the manometer tube and will cause error in the calculation for pressure. This problem can be solved by ‘bleeding processes’.
The results show the pressure at the throat of the venturi is the lowest while its velocity is the highest. The results obtain are according to the theory. There are two equations used to obtain the different velocity which are Bernoulli equation and Continuity equation. The results calculated from this two equation then are subtract to find the difference.
As the conclusion, the experiments are success where the supposed values have been obtained. All the objectives are achieved and the theories of these experiments are well-understood.
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2.0 INTRODUCTION
Bernoulli’s Theorem Demonstration unit consists of a classical venturi made of clear acrylic. A series of wall tappings allow measurement of the static pressure distribution along the converging duct, while a total head tube is provided to traverse along the centre line of the test section. These tappings are connected to a manometer bank incorporating a manifold with air bleed valve. Pressurization of the manometers is facilitated by hand pump.
The venturi can be demonstrated as a means of flow measurement and the discharge coefficient can be determined. This test section can be used to demonstrate those circumstances to which Bernoulli’s Theorem may be applied as well as in other circumstances where the theorem is not sufficient to describe the fluid behaviour.
Figure 2.1: Bernoulli’s Theorem Demonstration unit
Note: Refer the apparatus section for labeled part of the Bernoulli’s Theorem Demonstration unit.
The main test section is an accurately machined acrylic venturi of different circular cross section. It is provided with a number of side hole pressure tappings, which are connected to the manometer tubes on the rig. These tappings allow the measurement of static pressure head simultaneously at each 6 sections. The test section incorporates two unions, one at either end, to facilitate reversal for convergent or divergent.
A hypodermic tube, the total pressure head probe, is provided which may be positioned to read the total pressure head at any section of the duct. This total preesure probe may be moved after slacking the gland nut; this nut should be retightened by hand after adjustment. An additional tapping is provided to facilitate setting up. All eight pressure tappings are connected to a bank of pressurized manometer tubes. Pressurization of manometer is facilitated by connecting any hand pump to the inlet valve on the manometer manifold.
The unit is connected to the hydraulic bench using flexible hoses. The hoses and the connections are equipped with rapid action couplings. The flexible hose attached to the outlet pipe which should be directed to the volumetric measuring tank on the hydraulic bench. 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 supply control valve.
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3.0 AIM/OBJECTIVE
To demonstrate Bernoulli’s Theorem
4.0 THEORY
4.1 Derivation using Streamline Coordinates
The Euler's equation for steady flow of an ideal fluid along a streamline is a relation between the velocity, pressure and density of a moving fluid. It is based on the Newton's Second Law of Motion. The integration of the equation gives Bernoulli's equation in the form of energy per unit weight of the following fluid (www.codecogs.com, 2013).
It is based on the following assumptions:
The fluid is non-viscous (i,e., the frictional losses are zero).
The fluid is homogeneous and incompressible (i.e., mass density of the fluid is constant).
The flow is continuous, steady and along the streamline.
The velocity of the flow is uniform over the section.
No energy or force (except gravity and pressure forces) is involved in the flow.
Derivation Of Equation
Let us consider a steady flow of an ideal fluid along a streamline and small element AB of the flowing fluid as shown in figure (www.codecogs.com, 2013).
Figure 4.1
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Let,
dA = Cross-sectional area of the fluid element
ds = Length of the fluid element
dW = Weight of the fluid element
P = Pressure on the element at A
P+dP = Pressure on the element at B
v = velocity of the fluid element
We know that the external forces tending to accelerate the fluid element in the direction of the streamline
(1)
We also know that the weight of the fluid element,
From the geometry of the figure, we find that the component of the weight of the fluid element in the direction of flow,
(2)
Mass of the fluid element =
We see that the acceleration of the fluid element
(3)
Now, as per Newton's second law of motion, we know that Force = Mass *Acceleration
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Dividing both sides by
or,
(4)
This is the required Euler's equation for motion as in the form of a differential equation. Integrating the above equation,
or in other words,
(5)
which proves the Bernoulli's equation (http://www.codecogs.com, 2013).
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4.2 Bernoulli’s Equation
The Bernoulli Equation can be considered to be a statement of the conservation of energy principle appropriate for flowing fluids. The qualitative behavior that is usually labeled with the term "Bernoulli effect" is the lowering of fluid pressure in regions where the flow velocity is increased. This lowering of pressure in a constriction of a flow path may seem counterintuitive, but seems less so when you consider pressure to be energy density. In the high velocity flow through the constriction, kinetic energy must increase at the expense of pressure energy (http://hyperphysics.phy-astr.gsu.edu,
Bernoulli’s law states that if a non-vicious fluid is flowing along a pipe of varying cross section, then the pressure is lower at the constrictions where the velocity is higher and the pressure is higher where the pipes opens out and the fluid is stagnate. Many people find this situation paradoxical when they first encounter it (higher velocity, lower pressure). This can be seen in the equation below:
Pρg
+ V2
2g+z=h¿ (6)
Where,
P = Fluid pressure static at the cross sectionρ = Density of the flowing fluidg = Acceleration due to gravityz = Elevation head of the center at the cross section with respect to a datumh* = Total (stagnation) head
The terms on the left-hand-side of the above equation represent the pressure head (h), velocity head (hv) and the elevation head (z), respectively. The sum of these terms is known as the total head (h*). According to the Bernoulli’s theorem of fluid flow through a pipe, the total head at any cross section is constant. In a real flow due to friction and other imperfections, as well as measurement uncertainties, the results will deviate from the theoretical ones.In the experimental setup, the centerline of all the cross sections that is considered lie on the same horizontal plane where z = 0, the equation reduces to:
Pρg
+ V2
2g=h¿ (7)
This represents the total head at a cross section. For this experiments, the pressure head is denoted as hi and the total head as h*
i, where i represents the cross sections at different tapping points.
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4.3 Static, Stagnation and Dynamic Pressures
The pressure, P which have been used in deriving the Bernoulli’s equation,
Pρ+V
2
2+gz=constant (8)
is the thermodynamic pressure which commonly called as static pressure. The static pressure is that pressure which would be measured by an instrument moving with the flow. However, such a measurement is rather difficult to make in a practical situation.
There was no pressure variation normal to straight streamlines. This fact makes it possible to measure the static pressure in a flow fluid using a wall pressure tapping, placed in a region where the flow streamlines are straight, as shown in figure 4.2(a). The pressure tap is a small hole, drilled carefully in the wall, with its axis perpendicular to the surface. If the hole is perpendicular to the duct wall and free from burrs, accurate measurements of static pressure can be made by connecting the tap to a suitable pressure measuring instrument.
Figure 4.2: Measurement of static pressure
In a fluid stream far from a wall, or where streamlines are curved, accurate static pressure measurements can be made by careful use of a static pressure probe, shown in figure 4.2(b). Such probe must be designed so that the measuring holes are place correctly with respect to the probe tip and stem to avoid erroneous results. In use, the measuring section must be aligned with the local flow direction.
Static pressure probes or any variety of forms are available commercially in sizes as small as 1.5mm in diameter. The stagnation pressure is obtained when a flowing fluid is deceleratedto zero speed by a frictionless process. In incompressible flow, the Bernoulli equation can be used to relate changes in speed and pressure along a streamline for such a process. Neglecting the elevation differences, equation (8) becomes
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Pρ+V
2
2=constant (9)
4.4 Pressures Varies Along the Pipe
A number of factors can cause for pressure to vary along the pipe such as:
Friction the pipe’s inner surface The diameter of the pipe (small, the pressure is lower) Density of the fluid in the pipe The height of the pipe at which the pipe stands or height at which the flow through
(gravity) Turbulence of the fluid
4.5 Venturi Meter
The venturi meter consists of a venturi tube and a suitable differential pressure gauge. The venturi tube has a converging portion, a throat and a diverging portion as shown in the figure below. The function of the converging portion is to increase the velocity of the fluid and lower its static pressure. A pressure difference between inlet and throat is thus developed, which pressure difference is correlated with the rate of discharge. The diverging cone serves to change the area of the stream back to the entrance area and convert velocity head into pressure head. A differential pressure measurement between two sections of s pipe of different diameters and joined by a smooth change in diameter can be interpreted by using Bernoulli’s equation, to provide a measure of momentum change and thus velocity. In order to calculate the flow rate from the differential pressure measurement of the venturi meter, it is necessary to know the density of the fluid which in this case, the density of water (1000kg/m3) (Hunt, 1989).
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Figure 4.3: Venturi Meter
Assume incompressible flow and no frictional losses, from Bernoulli’s equation
P1
γ+V 1
2
2g+Z1=
P2
γ+V 2
2
2 g+Z2
(10)
Use of the continuity Equation Q = A1V1 = A2V2, equation (10) becomes
P1−P2
γ+Z1−Z2=
V 22
2 g [1−( A2
A1)
2] (11)
Ideally,
Q = A2V2 = A2 [1−( A2
A1)
2]−12
[2g( P1−P2
γ+Z1−Z2)]
12 (12)
However, in the case of real fluid flow, the flow rate will be expected to be less than that given by equation (12) because of frictional effects and consequent head loss between inlet and throat.
Qa = (Cd) (A2)[1−( A2
A1)
2]−12
[2g( P1−P2
γ+Z1−Z2)]
12 (13)
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In metering practice, this non-ideality is accounted by insertion of an experimentally determined coefficient, Cd that is termed as the coefficient of discharge. With Z1=Z2 in this apparatus, the discharge coefficient is determined as follow:
Cd = QaQi
(14)
Discharge coefficient, Cd usually lies in the range between 0.9 and 0.99.
5.0 APPARATUS
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Figure 5.1: Bernoulli’s Theorem Demonstration unit
6.0 EXPERIMENTAL PROCEDURE
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1) General Start-up Process
1. The clear acrylic section is installed with the converging section upstream. The union is also checked to make sure it is tighten.
2. The apparatus is located on the flat top of the bench.3. A spirit level is attached to baseboard and the level unit on top of the bench by
adjusting the feet.4. The water is filled to the volumetric tank of the hydraulic bench until approximately
90% full.5. The flexible inlet tube is connected using the release coupling in the bed of channel.6. A flexible hose is connected to the outlet and make sure that it is directed into the
channel.7. The outlet flow control valve is partially opened.8. The bench flow control valve, V1 is fully closed the pump is switched on. 9. The V1 is gradually opened and the piping is allowed to fill with water until all air has
been expelled from the system.10. The trapped bubbles are also checked in the glass tube or plastic transfer tube. If the
bubble exists, the bleeding process is carrying out.11. The water is flowed into the venturi and discharged into the collection tank.12. The water flow rate is increased. When the flow in the pipe is steadied and no trapped
bubble, the discharge valve is closed to reduce the flow to the maximum measurable flow rate.
13. The V1 and the outlet control valve are adjusted to obtain the flow through the test section and observed that the static profile along the converging and the diverging section is indicated on its respective manometers.
14. The total head pressure along the venture tube is measured by traversing the hypodermic tube.
15. The actual flow of water is measured using the volumetric tank with a stop watch.
2) General Shut-down Process
1. The water supply valve and the venturi discharge valve are closed.2. The water supply pump is turned off.3. The water is drained off from the unit when not in use.
3) Bernoulli’s Theorem Demonstration
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1. The general start-up procedures above are performed.2. All manometer tubings are checked whether it is properly connected to the
corresponding pressure taps and air-bubble free.3. The discharge valve is adjusted to a high measurable flow rate.4. After the level stabilizes, the water flow rate is measured using volumetric method.5. The hypodermic tube is slide gently (total head measuring) connected to manometer
#H, so that its end reaches the cross section of the venturi tube at #A. after sometimes, the reading is recorded from manometer #H and #A. The manometer reading for manometer #H is the sum of the static head and velocity heads (h *). The reading in manometer #A measured the pressure head (hi) because it is connected to venturi tube pressure tap, which does not obstruct the flow, thus measuring the flow static pressure.
6. The step 5 is repeated for cross sections #B, #C, #D, #E and #F.7. Steps 3 to 6 are repeated with three other decreasing flow rates by regulating the
venturi discharge valve.8. The velocity, ViB are calculated using the Bernoulli’s equation where V iB =
√2g (hB−hi ) .9. The velocity, ViC is calculated using the equity equation where ViC = Qav / Al.10. The differences between two calculated velocities are determined.
7.0 RESULTS
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1) Data Analysis
1st Flowrate 2nd Flowrate 3rd FlowrateVolume (m3) 0.003 0.003 0.003Average Time (s) 33.0 13.8 7.8Flowrate (m3/s) 9.09x10-5 2.17x10-4 3.85x10-4
1) The flowrates are calculated from the volume and average time recorded.
Q = Volume / Time (m3/s)
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Q = 0.003 / 33.0
Q = 9.09x10-5 m3/s
* All the flowrates are calculated using the equation above and can be seen in the table above.
2) Using the Bernoulli equation, the values ViB are calculated using the formula below:
V iB=√2¿ g¿ (h¿−hi ) (m/s)
Where, h* = hH
V iB=√2( 9.81ms
2
)(0.177−0.165 )
= 0.485 m/s
* All the V iB values are calculated and tabulated in the table above according to its flowrates.
3) To calculate the ViC values, the area first determined using diameter of the venturi (refer to figure 4.3)
Al=π Dl
2
4 (m2)
Al=π 0.0262
4
= 5.309x10-4 m2
* The area for venturi from A to F is the same throughout this experiment because it is the constant values.
V iC=QAV
Al (m/s)
V iC=9.09 x10−5
5.309x 10−4
= 0.171 m/s
4) The difference are then calculated after the values of V are determined
Difference = ViB – ViC (m/s)
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= (0.485 – 0.171) m/s
= 0.314 m/s
9.0 DISCUSSION
This experiments also highlighted about the pressure, same as the previous one (flowmeter measurement). In this experiment, the equipment used consists of a classical venturi made of clear acrylic. The concept of the venturi meter is when the fluid flow through the inlet or diverging part, the pressure is increased while the velocity is decreased. Opposite to the latter facts, when the fluid flows through the converging part, which is at the throat where the diameter is the smallest, the pressure is decreased while the velocity is at its maximum. This can be seen in the table below,
After the first readings are taken, where the actual readings of the manometer of the first flow rates successfully recorded, the hypodermic tube for measuring the total head measuring is connected to manometer #H, so that its end reaches the cross section of the venturi tube at #A, where it will become resistance in the water, and effected the reading in manometer where the level of water becomes lower than before. The readings are also the same for other tubes where the water levels decreased. The results are recorded and calculated for the velocity for total head and the velocity for pressure head.
The experiments are repeated twice with different flowrates but still the reading showing the same patterns. The difference is when the flowrates are increased, the results calculated also increase. For the calculations using the continuity equations, the area of the venturi is calculated. In this case, the area for the three flowrates is the same as the same equipment and venturi are used. What differentiate them is only the flowarates because the flowrates on this experiment are increased. This is because, if the flowrates is let to decrease, there will be error in taking the measurement as the level of water will lower. Then the 2nd and 3rd experiment cannot be carrying out respectively to the first experiment.
The flowrates of the water are determined by dividing the volume with time. All the results calculated are in SI units. The last thing to do after the two velocities are obtained is to subtract them to find the value of the difference between the two head.
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10.0 CONCLUSION
All in all, this experiment obeys the theory and the objectives also achieved. The objective of this experiment only one, which is to demonstrate Bernoulli’s Theorem. Most of all,the Bernoulli theorem explain about the pressure of liquid. The venturi tube also involve in this experiment because basically venturi is used to determine the pressure and head of the water. It is suitable the most because it have different diameter across the tube of venturi where it have throat which we can see the different in pressure and velocity of water.
11.0 RECOMMENDATIONS
There are some recommendations in order to improve this experiment. Firstly, make sure the experiment is started with the highest level of water or the height of water is at its maximum so that when the flow rate is decreased, the water will not too low from the manometer tube. The observer also has to differentiate and familiarized themselves with the valves that they supposed to handle while running the experiment. The observer tend to get confuse with the valve, the right way to open or close. The small mistakes will affect the results.
Secondly, avoid any error as possible when doing the experiment. The manometer reading should be taken perpendicular to the eyes of the observer. The observer also has to use ruler in order to take the reading because the scale at the apparatus is small. The manometer itself should have calibration mark to get accurate value of reading.
Lastly, the students should ensure the pipes are cleared from the bubbles when running the experiment because the bubbles inside the pipe will affect the pressure and height of the water in the manometer tube resulting the inaccurate reading. To avoid this, bleeding process can be done. Also to make the experiment more interesting, other types of fluid should be used to see the pressure difference between using water and other liquid.
12.0 REFERENCE
1. Codecogs (2013). Eulers Equation. Retrieved from http://www.codecogs.com/reference/engineering/fluid_mechanics/fundamentals/eulers_equation.php
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2. Hunt, A. (1989). U. S. Patent No. 4856344. Houston, Tex: U.S. Patent and Trademark Office.
3. Hyperphysics. Bernoulli Equation. Retrieved from http://hyperphysics.phy-astr.gsu.edu/hbase/pber.html#bcal
4. Utexas (2013). Fox McDonal Excerpt Bernoulli equation. Retrieved from http://www.me.utexas.edu/~longoria/me383Q/notes/3_fluid_hydraulic/Fox_McDonalM_Excerpt_Bernoulli_equation.pdf
