ABSTRACT
This experiment is carried out to examine in depth on the validity of Bernoulli’s theorem when applied to the steady flow of water in tapered duct and to measure the flow rates and both static and total pressure heads in a rigid convergent or divergent tube of known geometry for a range of steady flow rates. The relation among the pressure, velocity and elevation in a moving fluid (liquid or gas), the compressibility and viscosity (internal friction) of which are negligible and the flow of which is steady or laminar is indicated in Bernoulli’s theorem. The F1-15 Bernoulli’s Apparatus Test Equipment is used in this, in order to demonstrate the Bernoulli’s theorem. The reading shown by manometer h* is the sum of the pressure and velocity heads and the reading in manometer hi measured the pressure head only. The time to collect 3L water in the tank was measured. Lastly, the flow rate and total velocity was calculated by using both Bernoulli and Continuity equation and the difference in velocity for both equations was also calculated from the data of the results. Based on the results taken, it has been analyzed that the velocity of the fluid is increased when it is flowing from a wider to a narrower tube as the pressure is lower at constrictions and the pressure increased as the cross-sectional area increases.
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INTRODUCTION
In fluid dynamics, Bernoulli’s principle states that for an in viscid flow, an increase in the speed of the fluid occurs simultaneously with a decrease in pressure or a decrease in the fluid potential energy. Bernoulli’s principle is named after the Dutch-Swiss mathematician, Daniel Bernoulli who published his principle in his book, Hydrodynamics, in 1738. Bernoulli’s principle can be applied to various types of fluid flow, resulting in what is loosely denoted as Bernoulli equation. 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 the streamline. This requires that the sum of kinetic energy and potential energy remain constant. Thus, an increase in the speed of fluid occurs proportionately with an increase in both its dynamic pressure and kinetic energy, and a decrease in its static pressure and potential energy. 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 moving horizontally, the highest speed occurs when the pressure is lowest and the lowest speed occur when the pressure is highest. Bernoulli’s equation holds that for fluids in an ideal state, pressure and density are inversely related, in other words, a slow-moving fluid exerts more pressure than a fast-moving fluid. Since fluid in this context applies equally to liquid and gases. The principle has many as many applications with regard to airflow as to the flow of liquids. One of the everyday examples of Bernoulli’s principle can be found on the airplane, which stays aloft due to pressure differences on the surface of its wing.
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OBJECTIVES
The objectives of this Bernoulli’s Theorem experiment is verify experimentally the validity of the Bernoulli equation for fluid flow in a tapered duct and to measure flow rates, static and total pressure heads in a rigid convergent or divergent tube of known geometry for a range of steady flow rates.
THEORY
Bernoulli’s principle involves these laws which are the conservation of mass, energy and momentum and, it each application, these laws can be simplified in an attempt to describe quantitatively the behavior of fluid. For a horizontal device, the continuity equation shows that for an incompressible fluid, the reduction in diameter will cause an increase in the fluid flow speed which shows that there must be a decrease in the pressure in the reduced diameter region. Thus, this phenomenon is known as venturi effect. Bernoulli equation is derived under the following assumptions:
1.
fluid is incompressible ( density r is constant );
2.
flow is steady:
3.
flow is frictionless (t = 0);
4.
along a streamline; Then, it is expressed with the following equation:
3
Where (in SI units): p= fluid static pressure at the cross section in N/m2. r= density of the flowing fluid in kg/m3 g= acceleration due to gravity in m/s2 (its value is 9.81 m/s2 = 9810 mm/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 The terms on the left-hand-side of the above equation represent the pressure head (h), velocity head (hv), and 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 h* at any cross section is constant (based on the assumptions given above). In a real flow due to friction and other imperfections, as well as measurement uncertainties, the results will deviate from the theoretical ones. In this experimental setup, the centerline of all the cross sections on the same horizontal plane (which we may choose as the datum, z=0), and thus, all the ‘z’ values are zeros so that the above equation reduces to:
(This is the total head at a cross section). For our experiment, we denote the pressure head as hi and the total head as h*i, where ‘i’ represents the cross section we are referring to.
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APPARATUS Bernoulli’s Theorem demonstration unit:
Venturi Manometer Water tank Stopwatch Discharge valve
5
PROCEDURE 1. The pump switch was switched on 2. The discharge valve was adjusted to high measurable flow rate to filled the manometer tube to make sure the air bubble inside the manometer is free. 3. The water level was reduced from 200mm to 150mm by using air bleed screw. 4. The first flow rate was started by contolling the control valve. 5. After the level stabilizes, gently slide the hypodemic tube connected to manometer H. So that its end reaches the cross section of the venturi tube at A.The readings from manometer H and A was taken. 6. The step 5 was repeated for cross section ( B, C, D, E and F ) 7. After that, the water flow rate is filled into the measuring tube to 3L. The time was taken by using the stopwatch from 0L to 3L. 8. The step 4 to 7 was repeated with two other flow. 9. The velocity, ViB was calculated using the Bernoulli’s equation where; ViB =
√ 2 x g x (h∗−hi)
10. The velocity, ViC was calculated using the continuity equation where ViC =
Qav Ai
11. The difference between two calculated velocities was determined.
RESULTS 6
(i)
Reading 1 (Slow flow rate)
Volume (L) Average Time (min) Flow rate (LPM)
3.000 0.283 10.588
Using Bernuolli Equation
Continuity Equation
Cross sectio n
Difference Vib - Vic
(m/s)
A = πDi
h* = hH (m)
hi (m)
ViB =
A
0.187
0.165
(m/s) 0.657
5.31× 10
−4
0.331
0.326
B
0.172
0.154
0.594
3.66 ×10−4
0.481
0.113
C
0.169
0.146
0.672
2.01× 10−4
0.876
-0.204
D
0.165
0.115
0.990
3.14 ×10−4
0.561
0.429
E
0.163
0.124
0.875
3.80 ×10
−4
0.463
0.412
F
0.160
0.135
0.700
5.31× 10
−4
0.331
0.369
√ 2 g ( h−hi )
2
/
4 (m²)
Table 1.1: Data recorded at 10.588 L/min
(ii)
Reading 2 (Medium flow rate)
Volume (L) Average Time (min)
3.000 0.150 7
Vic = QAV / Ai (m/s)
Flow rate (LPM)
20.000
Using Bernuolli Equation
Continuity Equation
Cross sectio n
Difference Vib - Vic
(m/s)
A = πDi
h* = hH (m)
hi (m)
ViB =
A
0.219
0.184
(m/s) 0.829
5.31× 10−4
0.628
0.201
B
0.210
0.173
0.852
3.66 ×10−4
0.911
-0.509
C
0.207
0.158
0.980
2.01× 10
−4
1.658
-0.678
D
0.200
0.091
1.462
3.14 ×10−4
1.062
0.400
E
0.196
0.125
1.393
3.80 ×10−4
0.877
0.516
F
0.186
0.145
0.804
5.31× 10−4
0.628
0.176
√ 2 g ( h−hi )
2
/
4 (m²)
Table 1.2: Data recorded at 20.000 L/min (iii)
Reading 3 (Higher flow rate)
Volume (L) Average Time (min) Flow rate (LPM)
3.000 0.117 25.714
8
Vic = QAV / Ai (m/s)
Using Bernuolli Equation
Continuity Equation
Cross sectio n
Difference Vib - Vic
(m/s)
A = πDi
h* = hH (m)
hi (m)
ViB =
A
0.278
0.215
(m/s) 1.112
5.31× 10−4
0.808
0.304
B
0.264
0.193
1.180
3.66 ×10−4
1.172
0.008
C
0.260
0.164
1.372
2.01× 10−4
2.134
-0.762
D
0.250
0.035
2.054
3.14 ×10−4
1.366
0.688
E
0.243
0.130
1.489
3.80 ×10−4
1.129
0.360
F
0.236
0.165
1.180
5.31× 10−4
0.808
0.372
√ 2 g ( h−hi )
2
/
4 (m²)
Table 1.3: Data recorded at 25.714 L/min
CALCULATIONS Example: Reading 1 (Q = 10.588 L/min at Manometer A) Volume collected = 3 L 1000 L = 1m³ 3.00 L = 3.00 L x 1m³ 1000 L= 3.0 x 10ˉ3 m³ Flow Rate, Q = Volume collected / times collection 9
Vic = QAV / Ai (m/s)
Q=
10.588 L 1 min 1 m3 × × min 60 s 1000 L
−4 3 = 1.76 ×10 m
For cross section A,
√ 2 g ( h−h i)
Velocity, ViB =
√ 2 ( 9.81 ) ( 0.187−0.165 )
=
m/s
= 0.657 m/s Area = πDi
2
/ 4 m²
= π (0.026) −4
= 5.31× 10
2
/4 m2
Velocity, Vic = Q/A m/s −4 −4 = 1.76 ×10 / 5.31× 10
= 0.331 m/s Vib – Vic = 0.657 – 0.331 = 0.326 m/s
DISCUSSION
The purpose of this experiment is to verify the validity of Bernoulli’s equation for fluid flow upon the converging and diverging flow passage at the tube. Bernoulli's law indicates that, if an in viscid fluid is flowing along a pipe of varying cross section, then the pressure is lower at
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constrictions with respect to its velocity which is higher, and pressure will become higher at large cross-sectional area and the fluid stagnates. Overall, from the tables provided previously, it can be seen that by using Bernoulli’s equation, the highest velocity, Vib is achieved at cross section D. This is because the tube has small diameter second only to C, which is equivalent to 20mm, providing its large cross – sectional area, thus resulting low pressure and high velocity. However, by using continuity equation, the highest velocity, Vic is achieved at cross section C which has the smallest diameter of 16mm. Therefore, we had calculated the difference between Vib and Vic for each section to further investigate the significance of these two equations (i.e. Bernoulli equation and Continuity equation) and how they are related to each other.
According to Bernoulli’s, as the speed of the liquid increased, the pressure is lower. Taking example from Table 1.3, this statement can be proved. Constant flow rate of 25.714 L/min equivalent to
4.29 ×10−4
m3/s has the lowest speed which was calculated at the cross
section A. The velocity, Vib at A is recorded 1.112 m/s and hi was recorded at 0.215 m. This is because the difference between height (level) of liquid, (h – hi) with respect to tube A is high resulting the highest pressure in the system. Hence, the velocity at A is lower. Next, is where the fastest velocity in the tube, at cross section D with velocity of 2.054 m/s and hi at 0.035 m, this time with the lowest pressure in the system. Thus, the Bernoulli’s statement was proven.
A
B
C 11
D
E
F
For continuity, it is based on a condition that is, what goes in equal to what goes out (Q = Q
in
). Plus that the fluid is considered as incompressible. For this case, water can be
out
considered as incompressible. Looking at the calculation using continuity equation, in all flow rates, the highest velocity is also achieved at cross section C, due to its largest cross-sectional area. It obeys the theory where the cross sectional area is larger, the pressure is lower. Hence, when the pressure is lower, the velocity is increased. In this experiment, this situation happened accordingly to cross-section C at all readings. The condition is vice versa at low velocity. If it was gas, continuity cannot be applied as it can be compressed, changing its density, thus resulting in Q in ≠ Q out. However, while demonstrating the significance of both Bernoulli equation and Continuity equation, there is though one abnormally results which can be seen at each table, at cross section C. The difference of velocity (Bernoulli’s minus Continuity) is negative. Basically, this condition cannot happen as explained earlier, continuity equation was derived on one basic condition, what goes in equal to what goes out. But in Bernoulli’s, the kinetic energy of the fluid was also calculated. So actually, the Bernoulli’s should have a bigger velocity. The reason for this may happen is because; there is a bubble formation in the venturi tube. This may be cause by low flow rate in of the water, but high flow out of the tube. When this happen (the air inside tube), the reading of hi will not be accurate.
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CONCLUSION
From the experiment conducted, there are different cross-sections for each tube A, B, C, D, E and F. These differences resulted in varieties of value obtained for stagnation head h* and pressure head hi. By using Bernoulli equation to calculate the velocity, it can be said that the velocity of fluid increase as the fluid is flowing from a wider to narrower tube and the velocity decrease in the opposite direction. This also indicates that the pressure of fluid decreases as the velocity increases. The Bernoulli’s principle is proven where the highest velocity Vib, 2.054 m/s is achieved at cross section D because of the small tube diameter. As for the larger diameter tube at A, the velocity is the lowest which is 1.112 m/s. The first inclination might be to say that, where the velocity is the greatest, the pressure is higher. A big force could be feel on the hand in the flow where it’s going the fastest. However, the force does not come from the pressure there from the hand taking away the momentum from the liquid. The continuity equation is also used in this experiment to relate the pressure in pipes to their changes in diameter. The equation of continuity shows that liquid flows at constant mass rate and can also relates speed to pressure. There are few readings which go against the continuity equation. These circumstances occurred due to errors when the experiment was conducted. In order to prevent error, proper precautions must be taken before the experiment starts. The Bernoulli equation forms the basis for solving a wide variety of fluid flow problems such as jets issuing from an orifice, flows associated with pumps and also turbines. Bernoulli’s equation is also useful in demonstration of aerodynamic properties such as drag and lift. From the result obtained, we can conclude that the Bernoulli equation is valid for flow as it obeys the equation. As the area decreases at a section, velocity increase and the pressure decrease.
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RECOMMENDATION
1
The eyes of the observer may not be parallel to the scale and will cause parallax error. To prevent this from happen during the experiment, the eyes of the observer must be perpendicular to the reading scale.
2
The factors such as temperature, pressure and other things especially for the air bubble inside the tubes should be stabilized first before conducting the experiment for the accurate results.
3
The reading of the venture meter level should be taken more than three times in order to get an accurate value.
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REFERENCES
1. Holzner, S. (n.d.). Use Bernoulli’s Equation to Calculate Pressure Difference between Two Points. Retrieved from http://www.dummies.com/how-to/content/use-bernoullis-equation-to-calculate-pressurediff.html 2. Fitzpatrick, R. 2012. Bernoulli’s Theorem. Retrieved from http://farside.ph.utexas.edu/teaching/336L/Fluid.pdf 3. Anonymous. (n.d.). Bernoulli’s Theorem Lab. Retrieved from http://www.markedbyteachers.com/university-degree/engineering/bernoulli-s-theoremdistribution-experiment.html
APPENDICES
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