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TABLE OF CONTENTS

•  Abstract / summary CONTENT PAGE
•  Introduction 3 -
•  Aims / objectives
•  Theory 6 -
•  Apparatus 11 -
•  Experimental procedure 12 -
•  Result 15 -
•  Sample calculation 18 -
•  Discussion 21 -
•  Conclusion
•  Recommendation
•  Appendices 27 -

1.0 ABSTRACT

This experiment is aimed at investigating the validity of Bernoulli’s equation when applied to a steady flow of water in tapered duct and to measure the flow rate of steady flow rates. Based on (Bernoulli’s theorem, 2011) relates the pressure, velocity, and elevation in a moving fluid the compressibility and viscosity (internal friction) of which are negligible and the flow of which is steady, or laminar. For this experiment, by using the FM 24 Bernoulli’s Apparatus Test Equipment is to demonstrate the Bernoulli’s theorem. The experiment was conducted in order to find the time taken to collect 3L of water, the volumetric flow rates of the water, the pressure difference at all manometer tube at different cross section. The time to collect 0.003 m^3 water is recorded based on the different flow rate for each experiment. The combination of venture meter complete with manometer tube and hydraulic bench were used. During the experiment, water is fed through a hose connector and the flow rate can be adjusted at the flow regulator valve at the outlet of the test section. The venture can be demonstrated as a means of flow measurement and the discharge coefficient can be determined the results show the reading of each manometer tubes increase when the pressure difference increases. From the reading of height can be calculated the data by applied the Bernoulli equation to fin the velocity of the fluid moving. The pressure level and velocity reading for part A to E of the tube is recorded. From Bernoulli theory, the relation between the increase and decrease in the pressure value is inversely proportional to its velocity. Bernoulli's Principle tells that as the fluid flows more quickly through the narrow sections, the pressure actually decreases rather than increases. Thus, it proves the validity of Bernoulli’s theorem

2.0 Introduction

Bernoulli's Principle is a physical principle formulated that states that "as the speed of a moving fluid increases, the pressure within the fluid decreases. Bernoulli's principle is named after the Swiss scientist Daniel Bernoulli. 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's potential energy. Bernoulli's principle is named after the Swiss scientist Daniel Bernoulli who published his principle in his book Hydrodynamica in 1738.( Wikipedia, 2013) Bernoulli's principle can be derived from the principle of conservation of energy. Bernoulli’s Principle can be demonstrated by the Bernoulli equation. The Bernoulli equation is an approximate relation between pressure, velocity, and elevation. As a fluid passes through a pipe that narrows or widens, the velocity and pressure of the fluid vary. As the pipe narrows, the fluid flows more quickly. Surprisingly, Bernoulli's Principle tells that as the fluid flows more quickly through the narrow sections, the pressure actually decreases rather than increases. Bernoulli's principle can be explained in terms of the law of conservation of energy. As a fluid moves from a wider pipe into a narrower pipe or a constriction, a corresponding volume must move a greater distance forward in the narrower pipe and thus have a greater speed. Meanwhile, Continuity equation is about in physics is an equation that describes the transport of a conserved quantity. Continuity equations are a stronger, local form of conservation laws. (Wikipedia, 2013). However, Bernoulli’s Principle can only be applied under certain conditions. The conditions to which Bernoulli’s equation applies are the fluid must be frictionless (in viscid) and of constant density; the flow must be steady , continuous, incompressible, non-viscous fluid flow, the total energy or total head remains constant at all the section along the fluid flow provided there is no loss or addition of energy.

The Bernoulli equation:

. + + z = total head=constant (Equation 2.11)

Where, = Pressure Head (m)

= Velocity or kinetic head (m) (v=Q /A= m/s )

Z = Potential head (Height above some assumed level)

3.0 Objective /Aims

The objectives of the experiment are:-

Objectives

1. To investigate the validity of the Bernoulli equation when applied to the steady flow of water in a tapered duct.
2. To measure the flow rates with venture meter and to demonstrate Bernoulli.
3. To find the time taken to collect 3L of water, the volumetric flow rates of the water, the pressure difference at all manometer tube, velocity, dynamic head and also the total head
4. To determine a steady, continuous, incompressible, non-viscous fluid flow, the total energy or total head remains constant at all the section along the fluid flow provided there is no loss or addition of energy.

4.0 Theory

Figure 4.1 : Apparatus is used to investigate the validity of Bernoulli Equation

From Figure 4.1, show the apparatus is used to investigate the validity of the Bernoulli equation when applied to the steady flow of water in tapered duct. The apparatus consists of a clear acrylic duct of varying circular cross section, known as a Venturi. The duct has a series of wall tapings that allows measurement of the static pressure distribution along the converging duct, while a total head tube is provided to traverse along the center line of the test section. The venture meter is connected to the manometer with pipes. Wall pressure tapings are provided along the converging and diverging portions of the venturi to measure the static pressure distribution. The reading of the manometer from manometer 1 to manometer 5 shows the pressure head and manometer 6 shows the static head. Flow rate and pressure in the apparatus may be varied independently by adjustment of the flow control valve, and the bench supply control valve.

Figure 4.2: Venturi meter drawing. Tapping Point Distance (mm) Diameter (mm) A 60 26 B 83 21. C 105 16 D 148.6 20 E 166.4 22 F 215 216 Table 4.3: The dimension of cross section (Sources:- Lab manual)

From figure 4.2 and Table 4.3 respectively show the positions in mm of the pressure tapping’s and the dimensions of the cross-section. From this related figure, point A to F is the same along the venture meter. This data is use for to apply in Bernoulli equation and Continuity Equation. The test section is arranged so that the characteristics of flow through both a converging and diverging section can be studied. Water is fed through a hose connector and is controlled by a flow regulator valve at the outlet of the test section. The Venturi can be demonstrated as a means of flow measurement and the discharge coefficient can be determined. .From the table 4.2, the tapping point from A to F where the distance from point A to F is increase and the diameter is different for each of them.

.

Figure 4.4: Pipe of Varying cross section Source:-(Wikipedia,2005)

From the above figure, bernoulli’s principle relates much with incrompressible flow. Below is a common form or bernoulli’s equation, where is valid at point along a streamline when gravity is constant

Derivation Using Streamline Coordinates

Euler’s equation for steady flow along a streamline is

If a fluid particle moves a distance, ds, along a streamline,

Then, after multiplying Equation 3.1 by ds,

Integration of this equation gives:

∫

The relation between pressure and density must be applied in this equation. For the special case of incompressible flow, p = constant, and Equation 3.6 becomes the Bernoulli’s Equation.

Bernoulli’s law indicates that, if an inviscid fluid is flowing along a pipe of varying cross section (refer to figure 3.1), then the pressure is lower at constrictions where the velocity is higher, and higher where the pipe opens out and the fluid stagnates. The well-known Bernoulli equation is derived under the following assumptions:

 The liquid is incompressible. ( density ρ is constant );

 Along a stream line  The liquid is non-viscous.

 The flow is steady and the velocity of the liquid is less than the critical velocity for the liquid.  There is no loss of energy due to friction.

Bernoulli’s equation may be written as;

Where,

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. (Kostic, 2001)

In our experimental setup, the centre line if the cross sections lies on the same horizontal plane, taken as our datum, z = 0. As all the z value is equal to zero, the equation can be represented as:

This represents the total head at a cross section.

In this experiment, the pressure head is denoted as hi and the total head is denoted as h*I, where I represents the cross sections at different tapping points. A 1

Volume flow rate

= A 5 [ (^) ( ) ] Where, A 1 = and A 5 = (Equation 4.9)

Velocity measurement The velocity of the flow is measured by measuring the volume of the flow, V, over a time period, t. Thus gives the rate of volume flow: Qv = V/t m3/s, which in turn gives the velocity of the flow through a defined are :-

v = (Equation 4.10)

Continuity equation For an incompressible fluid, conservation of mass requires that volume is also conserved. A1V1 = A2V (Equation 4.11)

5.0 Apparatus

In order to complete the demonstration of Bernoulli’s theorem, several apparatus are needed. They are as follows:

1. Pad of Manometer tubes
2. Venturi tube
3. Air bleed screw
4. Water inlet
5. Pump and Pump switch
6. Stopwatch
7. Water hosts and tubes
8. Water tank equipped with valves, Control Valve and measuring tube
9. Hypodermic tube
10. Hydraulic Bench
11. Union
12. Air Bleed Screw
13. Discharge water
14. Sump tank
15. Volumetric Tank

Manometer tubes

Venturi tube

Venture Tube

Unions

Water Inlet Adjustable Feet

Air Bleed Screw

Discharge Valve

Control valve Pump

Hypodermic Tube

Sump Tank

Volutetric Tank

Measurement Tube

Pump Switch

Water

Hydraulic Bench

Figure 5.1:- Model: FM 24 Bernoulli’s Apparatus Test Equipment.

Figure 5.2:-Model FM-110 Hydraulic Bench which allows flow by timed volume collection to be measured.

6.0 Procedures General Start-up Procedures

The Bernoulli’s Theorem Demonstration (Model: FM 24) is supplied ready for use and only requires connection to the Hydraulic Bench (Model: FM110) as follows:

1. The clear acrylic test section was ensured installed with the converging section upstream. The unions tightened and checked.
2. The apparatus located on the flat top of the bench.
3. A spirit level was attached to the baseboard and the unit was level on the top of the bench by adjusting the feet.
4. The water was filled into the volumetric tank of the hydraulic bench until approximately 90% full.
5. The flexible inlet tube was connected using the quick released coupling in the bed channel.
6. The flexible hose was connected to the outlet and it directed into the channel.
7. The outlet flow control partially opened at the Bernoulli’s Theorem Demonstration Unit.
8. The bench flow control valve, V1 fully closed then the pump switched on.
9. V1 was gradually opened and water allowed filling until all air has been expelled from the system.
10. All the ‘trapped bubbles’ was checked in the glass tube or plastic transfer tube.
    • To remove air bubbles, the air were bleed out using a pen or screw driver to press the air bleed valve at the top right side of the manometer board.
11. Water flowing into the venturi and discharge into the collection tank of hydraulic bench.
12. The water flow rate proceeds to increased it. When all the water flow was steady and there were no trapped bubbles, the discharge valve closed to reduce the flow to the maximum measurable flow rate.
13. Water levels in the manometer were in different heights.
14. V1 and outlet control valve adjusted to obtain the flow through the test section and the static pressure profile observed along the converging and diverging sections is indicated on its respective manometers. The total head pressure along the venture tube being measured by traversing the hypodermic tube. Note: the manometer tube connected to the tapping adjacent to the outlet flow control valve is used as a datum when setting up equivalent conditions for flow through test section.
15. The actual flow rate measured by using the volumetric tank with a stop watch.

General Shut Down Procedures

1. Water supply valve and venturi discharge valve closed.
2. The water supply pump closed.
3. Water drain off from the unit when not in use.

Procedure

1. The general start up procedures was performed.
2. Before conducting the experiment, all the manometer tubing was checked to make sure air- bubbles free whether they are properly connected to the corresponding pressure taps. The discharge valve was adjusted to a high measurable flow rate
3. After the level stabilized, the water flow rate was measured using volumetric method and record the manometer reading.
4. The hypodermic tube that was connected to manometer #H was gently slide so that its end reaches the cross section of the venturi tube at #A. After some time, the readings from manometer #H and #A were noted. The reading shown by manometer #H is the sum of the static head and velocity heads, total head (h*), because the hypodermic tube is held against the flow of fluid forcing it to a stop (zero velocity). The reading in manometer #A measures just the pressure head (hi) because it is connected to the venturi tube pressure tap, which does not obstruct the flow thus measuring the flow static pressure.
5. Step 5 was repeated for other cross sections (#B, #C, #D, #E and #F).
6. Steps 3 to 6 were repeated at least with three other decreasing flow rates by regulating the venture discharge valve.
7. The actual flow Q from the volumetric flow measurement method was obtained.
8. The velocity was calculated using the Bernoulli’s equation where ViB =√
9. The velocity were also calculated using the continuity equation where, ViC =
10. The difference between the two velocities was calculated.

7.0 RESULT

Experiment 1

7.1 Experiment 1(flow rate: slow)

Volume (L) 3 L Average Time (s) to collect 19 s Q, Flow Rate ( /s) 1.58 x /s

Cross section

Using Bernoulli equation Using Continuity equation

difference

h* = hH

(mm)

hi (mm)

ViB = √ (m/s)

Ai =

(m^2 )

Vic =

(m/s)

ViB - Vic (m/s)

A 174 159 0.542 5.31 x 10-^4 0.298 0.

B 167 155 0.485 3.66 x 10-^4 0.432 0.

C 165 130 0.829 2.01 x 10-^4 0.786 0.

D 163 123 0.886 3.14 x 10-^4 0.503 0.

E 161 130 0.780 3.80 x 10-^4 0.416 0.

F 159 139 0.626 5.31 x 10-^4 0.298 0.

Experiment 2

7.2 Experiment 2 (flow rate: medium)

Volume (L) 3 L Average Time (s) to collect 14 s Q, Flow Rate ( /s) 2.143 x /s

Cross section

Using Bernoulli equation Using Continuity equation

difference

h* = hH

(mm)

hi (mm)

ViB = √ (m/s)

Ai =

(m^2 )

Vic =

(m/s)

ViB - Vic (m/s)

A 190 160 0.767 5.31 x 10-^4 0.404 0.

B 182 153 0.754 3.66 x 10-^4 0.586 0.

C 179 120 1.076 2.01 x 10-^4 1.066 0.

D 175 86 1.32 3.14 x 10-^4 0.683 0.

E 170 110 1.08 3.80 x 10-^4 0.564 0.

F 165 130 0.829 5.31 x 10-^4 0.404 0.

Experiment 3

7.3 Experiment 3 (flow rate: fast)

Volume (L) 3 L Average Time (s) to collect 19 s Q, Flow Rate ( /s) 3.75 x /s

Cross section

Using Bernoulli equation Using Continuity equation

difference

h* = hH

(mm)

hi (mm)

ViB = √ (m/s)

Ai =

(m^2 )

Vic =

(m/s)

ViB - Vic (m/s)

A 225 177 0.970 5.31 x 10-^4 0.706 0.

B 212 140 1.189 3.66 x 10-^4 1.025 0.

C 209 120 1.32 2.01 x 10-^4 1.866 - 0.

D 203 55 1.70 3.14 x 10-^4 1.194 0.

E 195 110 1.29 3.80 x 10-^4 0.987 0.

F 190 140 0.990 5.31 x 10-^4 0.706 0.

8.0 CALCULATIONS

Experiment 1: Flow rate of water = Sample Calculation (cross section A): Bernoulli equation:

ViB = (^) √

ViB = √ ViB = 0.542 m/s Continuity equation:

Ai =

Ai =

Ai = 5.31 x 10-4^ m^2

Vic =

Vic =

Vic = 0.298 m/s Therefore, the difference is = ViB - Vic = 0.542 m/s – 0.298 m/s = 0.244 m/s The step are repeated but use data for B, C, D ,E ,F for flow rate 1 only to get difference.

Experiment 2: Flow rate of water = Sample Calculation (cross section A): Bernoulli equation:

ViB = √

ViB = √ ViB = 0.767 m/s Continuity equation:

Ai =

Ai =

Ai = 5.31 x 10-4^ m^2

Vic =

Vic =

Vic = 0.404 m/s Therefore, the difference is = ViB - Vic = 0.767 m/s – 0.404 m/s = 0.363 m/s The step are repeated but use data for B, C, D ,E ,F for flow rate 2 only to get difference.

Experiment 3: Flow rate of water = Sample Calculation (cross section A): Bernoulli equation:

ViB = √

ViB = √ ViB = 0.970 m/s Continuity equation:

Ai =

Ai =

Ai = 5.31 x 10-4^ m^2

Vic =

Vic =

Vic = 0.706 m/s Therefore, the difference is = ViB - Vic = 0.970 m/s – 0.706 m/s = 0.264 m/s The step are repeated but use data for B, C, D ,E ,F for flow rate 3 only to get difference.

9.0 Discussion

From this experiment, the objective of this experiment is to investigate the validity of the Bernoulli equation when applied to the steady flow of water in a tapped duct and to measure the flow rates. This experiment is based on the Bernoulli’s principle which relates between velocities with the pressure for an invisid flow. The pressure different actually is determined from hA to hF meanwhile the flow rate is determined by recording the time taken to collect 3L of water in the tank.

To achieve the objectives of this experiment, Bernoulli’s theorem demonstration apparatus along with the hydraulic bench were used. This instrument was combined with a venturi meter and the pad of manometer tubes which indicate the pressure of hA until hF. A venturi is basically a converging-diverging section (like an hourglass), typically placed between tube or duct sections with fixed cross-sectional area. The flow rates through the venturi meter can be related to pressure measurements by using Bernoulli’s equation.

From the result , it is been observed that when the pressure difference increase, the flow rates of the water increase and thus the velocities also increase for both convergent and divergent flow.As fluid flows from a wider pipe to a narrower one, the velocity of the flowing fluid increases. This is shown in all the results tables, where the velocity of water that flows in the tapered duct increases as the duct area decreases, regardless of the pressure difference of each result taken.

Bernoulli’s equation used to calculate the velocity:

ViB = √

Where h* = total head

hi = pressure head at a point. From the experiment, to calculate the velocity using the continuity equation as it relates with area. The velocity was calculated with ViC = , where is the volumetric flow rate and is cross sectional area of pipe. From the data, the value of the volume flow rate is calculated based on the 3L volume collected per time in second .After calculated the value is to be

for the slow rate for flow rate 1, for the medium flow rate for flow rate 2 and

finally for fast flow rate for flow rate 3 by using the Bernoulli’s equation for velocity stated above.

Figure 9.1: The flow of velocity distribution in circular pipe with radius R. As we calculated the velocity by using the Bernoulli’s equation slightly higher compare than velocity calculated using the continuity equation. This is due to condition of the velocity taken, for Bernoulli’s equation, the velocity at stagnation point where the maximum velocity is taken at center of pipe meanwhile for the continuity equation where the velocity is taken by the average velocity. Thus there are different between the using Bernoulli equation and the Continuity equation.

From analysis of the results, it can be concluded that the velocity of the water decrease as the water flow rate decrease. The differences of velocity at cross section are from applying the different equation that is Bernoulli equation and Continuity equation. For slow flow rate, the velocity difference at cross section A for water flow rate is (0.244 m/s), B (0.053 m/s), C (0.043 m/s), D (0.383 m/s), E (0.364 m/s), and F (0.328 m/s).Also for medium flow rate the velocity difference at cross section A for water flow rate is (0.363 m/s), B (0.168 m/s), C (0.01 m/s), D (0.637 m/s), E (0.516 m/s), and F (0.425 m/s). At the same time, for fast flow rate, the velocity difference at cross section A for water flow rate is (0.264 m/s), B (0.164 m/s), C (0.546 m/s), D (0.506 m/s), E (0.303 m/s), and F (0.284 m/s).So from the data, it can be conclude that, the diameter of tube influence the differences in velocity of water flow. Based on the calculation made after the experiment, it can be concluded that the diameter of the tube will affect the differences in velocity as a bigger tube will cause the differences in velocity become bigger while the smaller tube cause the velocity differences between ViB and Vic to be smaller.From this experiment there are also happen the major and minor losses in the pipe. The major factor contributed to this loss is sudden expansion and enlargement across the point A to point F. Others, the friction factor also contributed in reducing the pressure inside the pipe.

Therefore, it can be concluded that the Bernoulli’s equation is valid when applied to steady flow of water in tapered duct and absolute velocity values increase along the same channel. Although the experiment proof that the Bernoulli’s equation is valid for both flow but the values obtain might be slightly differ from the actual value.

However, after the experiment, there are some errors on the results due to happen during the experiment is done. .This situation might due to some error or weaknesses when taking the measurement of each data. Some errors might occur due to the parallax error in taking the reading of monometer. One of them is the observer must have not read the level of manometer properly, where the eyes are not perpendicular to the water level on the manometer. Thus, possibility that the eye position of the readers is not parallel to the scale .Therefore, there are some minor effects on the calculations due to the errors. Besides that, the error also occurs due to manometer reading which not steady. This mean that water levels inside the manometer always keep moving but small changes and never stay at point, in fact its increase and decrease with time. This is due to water flow through the pipe not properly in stable state due to pump that not functions efficiently. Thus, the air bubble always comes out during conduct the experiment. That reasons some of data get negative result in differences of velocity. To overcome this problem the readings are taken on average value. In order to get the accurate value, the water level must be let to be really stable. Thus, a patient is needed in order to run this experiment successfully because sometimes the way the experiment is conduct may influence the result of the experiment.

10. Conclusion

In conclusion, the velocity of fluid will increase if the fluid is flowing from a wider to narrower tube and the velocity will decrease in the opposite case regardless of the type of flow and the pressure difference. As the velocity for all cases increases the dynamic head values are also seem to be increased. The velocity is dependent on the diameter of the tube also the pressure depends on the velocity of the flow. Increasing the flow rate leads to increasing the velocity at any point thus will cause the pressure drop. But there are some errors occurred in the results. There might be due to some errors occurred during the experiment. However, the results can be improved if some precautions are taken during the experiment for example the eyes level must be placed parallel to the scale when manometer readings are taken. Besides that, the valve is also need to be controlled slowly to stabilize the water level in the manometer. Furthermore, before conducting the experiment, make sure all the bubble inside the manometer tubes is remove completely .This is due to influence taking the reading of manometer and it also effect the pressure inside the manometer include the velocity flow. Bernoulli’s theorem has several applications in everyday lives. Carburetor is a device which apply Bernoulli Theorem where is in an internal combustion engine mixing air with a fine spray of liquid fuel. In conclusion, through this experiment all the objectives are successfully achieved. From all the data and results calculated it is proved the validity of Bernoulli’s equation. The second objective which is 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 is achieved. As the pressure difference increase, the time taken for 3L water collected increase and the flow rates of the water also increase.Thus, as the velocity of the same channel increase, the total head pressure also increase for both convergent and divergent flow.

11.0 RECOMMENDATION

1. Before start running the experiment, make sure the trap bubbles must be removing from manometer tubes to get accurate data by press air bleed valve lightly to allow fluid and trapped air to escape out.
2. The valve must be control carefully to maintain the constant values of the pressure difference as it is quite difficult to control.
3. When taking the reading at manometer, the eye position of the observer must be parallel to the water meniscus to avoid parallax error.
4. During taking the reading at manometer, the time keeper must be alert with the rising of water volume to avoid error.
5. Any leaking of water in the instrument must be avoided
6. The experiment should be repeat several times to get average values.
7. The valve should be controlled slowly to maintain the pressure difference.
8. The valve and bleed screw should regulate smoothly to reduce the errors.