ABSTRACT This experiment is about Bernoulli‟s theorem. The objective of this experiment is to demonstrate the Bernoulli‟s theorem. This experiment use the Bernoulli„s Theorem Demonstration Apparatus. The apparatus contains of many part which are venture meter, pad of manometer tube, pump, and water tank equipped with pump water controller, water host and tubes. This experiment uses the venturi meter to measure the fluid flow through a pipe. The venturi tube has converging and diverging portion that function of increasing and decreasing the velocity of fluid and lower and higher the static pressure respectively. The experiment is conducted for three times with different flow rates starting the first after the level pressure of each manometer A, B, C, D, E and F is equal and stable. The flow rate was taken as the water indicator reached at 3 liters of water level. Level pressure of each manometer is measured as head loss. From the experiment conducted, we observed than when fluids moves from a region of higher pressure to lower pressure, the velocity of the fluid increases. There are a little bit differences on the speed of the flow and the pressure when we use Bernoulli‟s equation and Continuity equation to calculate velocity. Thus, we concluded that the Bernoulli‟s theorem is accepted. The experiment was a successfully and the objectives are obtained.
INTRODUCTION Bernoulli‟s principle tells us that windows tend to explode rather than implode when hurricanes. During the hurricanes, a very high speed of air outside the window leading to low air pressure compare to inside, which the air is still. The differences of forces causing the windows push outward and explode. That it is why to better open all windows during the hurricane. Another example of Bernoulli‟s principle application is to determine the flow speed of a fluid. The devices such as venturi meter or an orifice plate is used and can be placed into a pipeline to reduce the diameter of the flow. The reduction in diameter will cause an increase in the fluid flow speed, thus according to Bernoulli‟s there must be decrease in pressure. In his book, Daniel Bernoulli states that “as the speed of a moving fluid (liquid or gas) increases, the pressure within the fluid decreases. He comes up this principle and published in his book Hydrodynamica in 1738. The book describes the theory of water flowing through a tube and water flowing from a hole in a container which explained the hydronomic pressure. In order to demonstrate the Bernoulli‟s principle, model: FM 24 – Bernoulli‟s Theorem Demonstration apparatus is used. The apparatus consist of a classical Venturi made of clear acrylic. A series of wall tapping allow measurement of the static pressure distribution along the converging duct, while a total head tube is provided to transverse along the centre line of the test section. These tapings are connected to a manometer bank incorporating a manifold with air bleed valve. Pressurization of the manometers is facilitated by a hand pump. This unit has been designed to be used with Hydraulics Bench for students to study the characteristics of flow through both converging and diverging sections. During the experiment, water is fed through a hose connector and students may control the flow rate of the water by adjusting a flow regulator valve at the outlet of the test section.
AIM This aim of the experiment is to determine Bernoulli‟s theorem which can be applied to a venturi meter consist of converging portion and diverging cone.
THEORY Bernoulli‟s principle is essentially a work energy conservation principle which states that an ideal fluid or for situations where effects of viscosity are neglected, with no work being performed on the fluid, the total energy is constant. This principle also states that the sum of all forms of energy in a fluid flowing along an enclosed path (a streamline) is the same at any two points in that path. The pressure, speed and height (h) at two points in a steady-flowing, non-viscous, incompressible fluid are related by the equation:
This equation is the sum of the kinetic energy, potential energy and gravitational energy at height. If the equation was multiplied through by the volume, the density can be replaced by mass and the pressure replaced by force times distance. As the replacement, we can see that the difference in pressure does work which can be used to change the kinetic energy and the potential energy of the fluid for this equation. The continuity equation of continuity states that for an incompressible fluid flowing in a tube of varying cross-section, the mass flow rate is the same everywhere in the tube. The equation of continuity can be written as:
ρ = density , A = cross sectional area , v = velocity The density stay at the constant and then it is simply the flow rate (Av) that is constant. The velocity of fluid is greater in the narrow section as we applied the continuity equation. Bernoulli's principle also states that if a non-viscous flow along a pipe of varying cross section. Then, an increment in the speed of the fluid simultaneously with a drop in pressure or a decrease in the fluid‟s potential energy and the pressure increases when the pipe opens out and the fluid stagnate. Thus, pressure would decreases when the velocity increases. This is expressed with the following equation: p v2 z h * Constant g 2g
Where, p
= Fluid static pressure at the cross section
ρ
= Density of the flowing fluid
g
= Acceleration due to gravity
v
= Mean velocity of fluid flow at the cross section
z
= Elevation head of the center at the cross section with respect to a datum
h* = Total (stagnation) head
The sum of the pressure head (h), velocity head (hv ) and the elevation head (z) is known as the total head (h*). The total head h* at any cross section is constant according to the Bernoulli‟s theorem of fluid flow through a pipe. The results will deviate from the theoretical ones in real flow due to friction, other imperfections and the measurement uncertainties.
The centerline of all the cross sections we are considering lie 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: p v2 h * Constant g 2g
This represents the total head at cross section.
Venturi Meter The venturi meter is a tool used to measure the flow through a pipeline. This apparatus consists of a venturi tube and differential pressure gauge. The venturi tube has a converging portion, a throat and a diverging portion as shown in the figure below. The converging portion‟s function 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.
The picture above shows a venture tube.
APPARATUS Bernoulli‟s Theorem Demostration Unit (FM 24)
Figure 1: Parts Identification Diagram
1 Manometer 2 Test Section 3 Water Inlet 4 Unions 5 Air Bleed Screw 6 Discharge Valve 7 Gland Nut 8 Hypodermic Probe 9 Adjustable Feet
PROCEDURE General Start-Up Procedures 1. The clear acrylic is ensured that the test section is installed with the converging section upstream. The unions are tighten 2. The apparatus is located on the flat top of the bench. 3. A spirit level is attached to baseboard and on top of the bench is levelled by adjusting the feet. 4. The flexible inlet tube is connected using the quick release coupling in the bed of the channel. 5. A flexible hose is connected to the outlet and directed into the channel. 6. The outlet flow control valve at the Bernoulli‟s Theorem Demonstration Unit is gradually opened to allow the volumetric tank of the hydraulic bench is filled. 7. The water is allowed to fill in the manometer tubes until all trapped air is removed. 8. Air bleed screw is opened and controlled to get a certain volumetric flow. The screw is re-tightened when the manometer has reached a convenient height.
Experimental Procedures 1. The general Start-up Procedures is performed 2. All manometer tubes are checked properly connected to the corresponding pressure taps are air-bubble free. 3. The discharged valve is adjusted to a high measureable flow rate. 4. After the level is stabilized, the water flow rate is measured using volumetric method. 5. The hyperdomic tube connected to manometer #H is gently slide, so that the cross section of venturi tube at #A reached its end. 6. Step 5 is repeated for other cross sections (#B, #C, #D, #E and #F) 7. Step 3 to 6 is repeated with three other decreasing flow rates by regulating the venturi discharged. 8. The velocity,
is calculated using the Bernoulli‟s equation where;
√ 9. The velocity,
is calculated using the continuity equation where;
10. The difference between two calculated velocities is determined.
=
General Shut-down Procedures 1. Water supply valve and venturi discharge valve are closed. 2. The water supply pump is turned off. 3. Water from the unit is drained off when not in use
RESULTS 1st trial Volume of water: 3L Time taken: 43.2 s Flowrate: 6.944 x 10-5 m3/s Cross Section
i
Using Bernoulli‟s equation
(mm)
(mm)
Using Continuity
√ (m/s)
⁄ (
Difference
⁄
(m/s)
(m/s)
A
161
156
0.3132
0.000531
0.1308
0.1824
B
159
154
0.3132
0.000366
0.1897
0.1235
C
158
145
0.505
0.000201
0.3455
0.1595
D
157
150
0.3706
0.000314
0.2211
0.1495
E
156
151
0.3132
0.000380
0.1827
0.1305
F
156
153
0.2426
0.000531
0.1308
0.1118
2nd Trial Volume of water: 3L Time taken: 31.55 s Flowrate: 9.509 x 10-5 m3/s Cross Section
i
Using Bernoulli‟s equation
(mm)
(mm)
Using Continuity
√ (m/s)
⁄ (
Difference
⁄
(m/s)
(m/s)
A
162
155
0.3706
0.000531
0.1791
0.1915
B
160
154
0.3431
0.000366
0.2598
0.0833
C
158
117
0.8969
0.000201
0.4731
0.4238
D
156
146
0.4429
0.000314
0.3028
0.1401
E
156
140
0.5609
0.000380
0.2502
0.3107
F
154
144
0.4429
0.000531
0.1791
0.2638
3rd Trial Volume of water: 3L Time taken: 9.30 s Flowrate: 3.226 x 10-4 m3/s Cross Section
i
Using Bernoulli‟s equation
(mm)
(mm)
Using Continuity
√ (m/s)
⁄ (
Difference
⁄
(m/s)
(m/s)
A
209
184
0.7003
0.000531
0.6075
0.0928
B
205
164
0.8968
0.000366
0.8814
0.0154
C
200
68
1.6093
0.000201
1.6049
0.0044
D
188
135
1.0197
0.000314
1.0273
-0.0076
E
185
148
0.8520
0.000380
0.8489
0.0031
F
184
160
0.6862
0.000531
0.6075
0,0787
CALCULATIONS During the experiment, reading of h* , hi ,and time taken of water to be filled until 3L are recorded. There are a few things need to be calculated in order to determine the difference value between ViB and ViC : 1. 2. 3. 4.
ViB by using the Bernoulli‟s equation Area of each of the cross section (A, B, C, D, E and F) Volumetric flow rate of each trial ( 1st, 2nd and 3rd) ViC by using continuity equation
1. To calculate ViB , using the Bernoulli‟s equation
p v2 z h * Constant g 2g We eliminate the pressure and z , thus we have this equation : √ As for example, we calculated the ViB of the cross section A of the 1st trial experiment; g = 9.81 m/s2 √ 0.3132 m/s
2. The area of each cross section of the tube are calculated using this equation ⁄
⁄
As for example; DA = 26 m therefore
= 0.000531 m2 DB = 21.6 m DC = 16.0 m DD = 20.0 m DE = 22.0 m
DF = 26.0 m
3. Volumetric flow rate of each trial ( 1st, 2nd and 3rd) are calculated using equation; = As for example we calculated the Q of the cross section A of the 1st trial experiment; = = 6.944 × 10-5 m3/s 4. Lastly, we substituted the value of Ai and Q into the continuity equation
𝑉𝑖𝑐
𝑄𝑎𝑣 ⁄𝐴 𝑖
Thus ViC = 0.3108 m/s We calculated the difference value between ViB and ViC . For cross section A; ViB - ViC = 0.1824 m/s
All of the calculations are repeated for every trial.
DISCUSSION We had demonstrated the Bernoulli‟s Theorem and from the result we had calculated that when the diameter of the cross section smaller, there were increasing in velocities along the distribution tube. The smallest cross section (which is cross section C) gave huge pressure difference and had the highest velocities. The results had shown that larger pressure difference gave out the most speed of fluid flow. Furthermore through the experiment, we had observed as fluid flow through the venturi meter, the head is higher at the narrower tube and vice versa. The narrower tube leads to high velocities and low pressure. The volumetric flow rates are calculated for each trial. The speed of water was controlled by using valve and times taken are recorded as the volumetric tank reached 3L. Every trial had different volumetric flow rate as to manipulate the result of the velocities and the calculated volumetric flow rates were substituted into the continuity equation to determine the velocities. From the calculation, the highest volumetric flow rates gave the highest velocities. In order to determine the velocities of each cross section, two different equation were used; by using Bernoulli‟s Equation (ViB) and by using continuity equation (ViC). From the calculations we observed that all of calculated ViB had highest value compared to ViC . The continuity equation is the conservation of mass on a steady flow process through a control volume. With uniform density of water, the velocity is equal to volumetric flow rate over area. ViC took area of cross section into consideration, which when area increase velocity decrease. ViC is directly proportional to area. Whereas the Bernoulli‟s equation is when an increment in the speed of the fluid simultaneously with a drop in pressure and the pressure increases when the pipe opens out and the fluid stagnate. Thus, pressure would decreases when the velocity increases. This equation took pressure difference into consideration. As we discussed about both equations, both equations were suitable to be used to calculate the velocities. However, there some parameters needed to be considered like pressure difference and area. We also had calculated the difference between ViB and ViC (ViB - ViC ), to show that the value of ViB are greater than ViC. As we calculated, there was one negative value which was should not happen. This may be because of parallax error during taking the reading and air bubble trapped inside the tube.
CONCLUSION As Bernoulli state high velocity of fluid flow result in low pressure and based on continuity equation, smaller area result in high velocity. Thus, the result proved both equation could be used to determine the velocity of fluid flow in manometer. Even the value is not exactly the same, but the pattern of increasing and decreasing at the converging and diverging portion is same. So that, as the velocity increases, the total head pressure also increases for both convergent and divergent flow.
RECOMMENDATIONS In order to obtain accurate results, there are a few recommendations and precautions that should be taken during undergo the experiment. Firstly, the equipment must be drained before starting and after the experiment, to prevent the equipment damaged. Next, make sure there is no air bubbles trapped before and during running the experiment. The air bubbles will affect the result of the experiment. Other than that, the experiment should be run after we have fully understands the unit and the procedures to avoid some simple mistakes. In addition, to get the accurate results, the eye level must be perpendicular to the reading when recording the data to avoid parallax error, by the reading at the right meniscus. Besides that, the control valve should be maintained at a constant flow so that each at every readings of each manometer has the same value of pressure. Furthermore, when taking the time taken using the stopwatch, the suction pipe has to be started simultaneously to get the accurate volumetric flow rate. Next, the experiment should be repeated for a few times so that an average value could be obtained. Lastly, the leakage of water in the instrument must be avoided to reduce the error during conducting the experiment.
REFERENCE 1. Coulson & Richardson‟s Chemical Engineering, Volume 1, Sixth Edition, J.M. Coulson, J.F. Richardson, J.R. Backhurst, J.H Harker, Elsevier ButterworthHeinemann (1954) 2. “Bernoulli‟s Principle” from http://en.wikipedia.org/wiki/Bernoulli's_principle. Retrieved on 13th Oct 2014 3. Elementary Fluid Mechanics 7th Edition, Robert L. Street, Gary Z. Watters, John K. Vennard, John Wiley & Sons Inc. 4. “Bernoulli‟s Equation” from http://www.princeton.edu/~asmits/Bicycle_web/Bernoulli.html. Retrieved on 18th May 2014 5. Venturi Meters from www.thermopedia.com/content/1241. Retrieved on 13h Oct 2014
APPENDIX
