TABLE OF CONTENT

TITLE PAGE

Table of Content

1.0 Abstract 3

2.0 Introduction 4-5

3.0 Objectives 5

4.0 Theory 6-7

5.0 Apparatus and material 7-8

6.0 Experimental procedure 8

7.0 Results and calculation 9-11

8.0 Discussion 11-12

9.0 Conclusion 13

10.0 Recommendations 13

References 13

Appendices 14

1

ABSTRACT

First of all, the objectives that wanted to success in this experiment are to determine

the discharge coefficient of the venturi meter, to calculate flow rate with venturi meter and to

demonstrate Bernoulli’s Theorem. The combination of venturi meter complete with the

manometer tubes and hydraulic bench were used. The experiment was proceed in order to

find the time taken to collect 3L of water, the volumetric flow rates of the water, difference of

the pressure at all manometer tubes which known as static head, dynamic head and last but

not least the total head. The data was recorded at three adjusted head differences which were

78mm, 116mm and 160mm. All the related procedure in this experiment is referred to

Bernoulli’s principle. 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’s potential energy. The values for the total head when 78mm pressure

difference is 0.162m (Ha), 0.152m (Hb), 0.080m (Hc), 0.121m (Hd), 0.129m (He) and 0.136m

(Hf). Besides that, at the pressure difference is 116mm H20, the value of the total head is

0.146m (Ha), 0.139m (Hb), 0.079m (Hc), 0.114m (Hd), 0.122m (He) and 0.130m (Hf). Lastly,

the value of total head that measured at 160mm H20 is 0.175m (Ha), 0.164m (Hb), 0.076m

(Hc), 0.132m (Hd), 0.144m (He) and 0.154m (Hf).

2

INTRODUCTION

In the Bernoulli’s Theorem, states that an increase in the speed of moving air or a

flowing fluid is accompanied by a decrease in the air fluid’s pressure. This theorem also

known as Bernoulli’s principle. Daniel Bernoulli which is Swiss scientist (1700-1782),

demonstrated that, in most cases the pressure in a liquid or gas decreases as the liquid or gas

move faster. This is an important principle involving the movement of a fluid through the

pressure difference. Normally, a fluid is moving in a horizontal direction and encounters a

pressure difference. This pressure difference will result in a net force, which is by Newton’s

Second Law will cause an acceleration of the fluid.

Bernoulli’s Theorem also states that the total energy involves the pressure energy,

potential energy and kinetic energy of an incompressible and non-viscous fluid in steady flow

through a pipe remains constant throughout the flow, provided there is no source or sink of

the fluid along the length of the pipe. This statement is depend to the assumption that there is

no loss energy due to friction.

P + gh + ½ V 2 = constant

3

The converging-diverging nozzle apparatus also can be used to identify the validity of

Bernoulli’s equation. It is also used to show the validity of the continuity equation where the

fluid flows is relatively incompressible. In addition, the results that have been recorded will

show the presence of fluid energy losses, often attributed to friction and the turbulence with

eddy currents associated with a separation of the flow from the conduit walls.

OBJECTIVES

1. To determine the discharge coefficient of the venturi meter

2. To measure flow rate with venturi meter

3. To demonstrate Bernoulli’s Theorem

4

THEORY

Clearly state that the assumption made in driving Bernoulli’s principle equation is:

1. The model calculation here assumes laminar flow(no tubulance)

2. The distance from the larger diameter to the smaller is short enough that viscous

losses can be neglected

3. The velocity profile follows that of theoretical laminar flow

4. The flow is steady and the velocity of the liquid is less than the critical velocity for

the liquid.

5. There is no loss energy due to friction.

Then, it is expressed with the following equation:

            

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

hT = total (stagnation) head in m

The terms on the left-hand-side of the above equation represent the pressure head (h),

velocity head (hi ), and elevation head (z), respectively. The sum of these terms is known as

the total head (hT). According to the Bernoulli’s theorem of fluid flow through a pipe, the

total head hT 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.

5

In our experimental setup, the center line 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

ρ' g + v2

2. g = hT = constant (This is the total head at a cross section).

Total head, hT = hs + hv

For our experiment, we denote the pressure head as h and the total head as h*i, where ‘i’

represents the cross section we are referring to.

APPARATUS

1. Venture meter

2. Pad of manometer tube

3. Hydraulic bench

4. Stop watch

5. Water

6. Water tank equipped with water controller

7. Water host and tubes

6

PROCEDURE

1. The main switch on the pump is switched on.

2. The flow control valve is fully opened to let the water flow into the venture meter and

manometer tubes.

3. The control valve and valve are closed.

4. The air bleed screw is regulated until water level in manometer tubes reach 150 mm.

5. The flow control valve is fully opened and waited for some time for the level in

manometer tube h is in steady state.

6. After the steady state is achieved, the water tank is closed with water controller and

the time to for volume of water to reach the 3 litre is recorded.

7. The Pitot (total head measuring) tube that connected to manometer h is pushed gently

and its end reaches the cross section of the venture tube at a. After waited some time,

the reading of manometer h and a are taken.

8. The step 5 to 7 is repeated with difference flow rate.

7

RESULT

Experiment 1

Volume (m3) 0.003

Average Time (s) 20

Flow rate (m3/s) 1.50 x 10-4

Cross

Section

Using Bernoulli Equation Using Continuity Equation Difference

h*= hH

(m)

hi

(m)

ViB = √ 2g (hH-hi)

(m/s)

Ai = πDi2 / 4

(m2)

Vic = QAv /Ai

(m/s)

ViB – Vic

(m/s)

A 168 159 0.420 5.309 x 10-4 0.283 0.137

B 163 151 0.485 3.664 x 10-4 0.409 0.076

C 160 130 0.767 2.011 x 10-4 0.746 0.021

D 158 141 0.678 3.142 x 10-4 0.477 0.101

E 156 150 0.343 3.801 x 10-4 0.395 -0.052

F 154 153 0.140 5.309 x 10-4 0.283 -0.143

Experiment 2

Volume (m3) 0.003

Average Time (s) 17

Flow rate (m3/s) 1.764 x 10-4

Cross

Section

Using Bernoulli Equation Using Continuity Equation Difference

h*= hH

(m)

hi

(m)

ViB = √ 2g (hH-hi)

(m/s)

Ai = πDi2 / 4

(m2)

Vic = QAv /Ai

(m/s)

ViB – Vic

(m/s)

A 169 159 0.442 5.309 x 10-4 0.372 0.110

B 166 155 0.465 3.660 x 10-4 0.482 -0.070

C 171 126 0.852 2.011 x 10-4 0.878 -0.026

D 169 144 0.560 3.142 x 10-4 0.562 -0.002

E 168 147 0.485 3.801 x 10-4 0.464 0.021

F 167 152 0.313 5.309 x 10-4 0.332 -0.019

8

Experiment 3

Volume (m3) 0.003

Average Time (s) 16

Flow rate (m3/s) 1.875 x 10-4

Cross

Section

Using Bernoulli Equation Using Continuity Equation Difference

h*= hH

(m)

hi

(m)

ViB = √ 2g (hH-hi)

(m/s)

Ai = πDi2 / 4

(m2)

Vic = QAv /Ai

(m/s)

ViB – Vic

(m/s)

A 180 165 0.543 5.309 x 10-4 0.353 0.190

B 175 158 0.578 3.664 x 10-4 0.512 0.066

C 171 116 1.039 2.011 x 10-4 0.932 0.107

D 169 142 0.728 3.142 x 10-4 0.597 0.131

E 168 148 0.626 3.801 x 10-4 0.493 0.133

F 167 154 0.505 5.314 x 10-4 0.353 0.152

9

CALCULATION

Experiment 1

Volume (m3) 0.003

Average Time (s) 20

Flow rate (m3/s) 1.50 x 10-4

Cross

Section

Using Bernoulli Equation Using Continuity Equation Difference

h*= hH

(m)

hi

(m)

ViB = √ 2g (hH-hi)

(m/s)

Ai = πDi2 / 4

(m2)

Vic = QAv /Ai

(m/s)

ViB – Vic

(m/s)

A 168 159 0.420 5.309 x 10-4 0.283 0.137

B 163 151 0.485 3.664 x 10-4 0.409 0.076

C 160 130 0.767 2.011 x 10-4 0.746 0.021

D 158 141 0.678 3.142 x 10-4 0.477 0.101

E 156 150 0.343 3.801 x 10-4 0.395 -0.052

F 154 153 0.140 5.309 x 10-4 0.283 -0.143

Experiment 2

Volume (m3) 0.003

Average Time (s) 17

Flow rate (m3/s) 1.764 x 10-4

Cross

Section

Using Bernoulli Equation Using Continuity Equation Difference

h*= hH

(m)

hi

(m)

ViB = √ 2g (hH-hi)

(m/s)

Ai = πDi2 / 4

(m2)

Vic = QAv /Ai

(m/s)

ViB – Vic

(m/s)

A 169 159 0.442 5.309 x 10-4 0.372 0.110

B 166 155 0.465 3.660 x 10-4 0.482 -0.070

C 171 126 0.852 2.011 x 10-4 0.878 -0.026

D 169 144 0.560 3.142 x 10-4 0.562 -0.002

E 168 147 0.485 3.801 x 10-4 0.464 0.021

F 167 152 0.313 5.309 x 10-4 0.332 -0.019

10

Experiment 3

Volume (m3) 0.003

Average Time (s) 16

Flow rate (m3/s) 1.875 x 10-4

Cross

Section

Using Bernoulli Equation Using Continuity Equation Difference

h*= hH

(m)

hi

(m)

ViB = √ 2g (hH-hi)

(m/s)

Ai = πDi2 / 4

(m2)

Vic = QAv /Ai

(m/s)

ViB – Vic

(m/s)

A 180 165 0.543 5.309 x 10-4 0.353 0.190

B 175 158 0.578 3.664 x 10-4 0.512 0.066

C 171 116 1.039 2.011 x 10-4 0.932 0.107

D 169 142 0.728 3.142 x 10-4 0.597 0.131

E 168 148 0.626 3.801 x 10-4 0.493 0.133

F 167 154 0.505 5.314 x 10-4 0.353 0.152

11

DISCUSSION

Based on the objective, this experiment is being conduct to investigate the validity of

the Bernoulli’s equation when applied to the steady flow of water in a tapered duct. As we

already know water is fluid and all fluid have properties to take the shape of the container or

wherever they flow or stored. Since the volume passing through at given length of pipe

during a given period of time will be the same, there must be a decrease in pressure. From the

Bernoulli’s principle, it states that the slower the rate of flow, the higher the pressure, and the

fastest the rate flow the lower the pressure.

The Bernoulli theorem is an approximate relation between pressure, velocity, and

elevation, and is valid in regions of steady, incompressible flow where net frictional forces

are negligible. The equation is obtained when the Euler’s equation is integrated along the

streamline for a constant density for incompressible fluid. The constant of integration (called

the Bernoulli’s constant) varies from one streamline to another but remains constant along a

streamline in steady, frictionless, incompressible flow. Despite its simplicity, it has been

proven to be a very powerful tool for fluid mechanics. Bernoulli’s equation states that the

“sum of the kinetic energy (velocity head), the pressure energy (static head) and Potential

energy (elevation head) per unit weight of the fluid at anypoint remains constant” provided

the flow is steady, irrotational, and frictionless and the fluid used is incompressible. This is

however, on the assumption that energy is neither added to nor taken away by some external

agency. The key approximation in the derivation of Bernoulli’s equation is that viscous

effects are negligibly small compared to inertial, gravitational, and pressure effects.

The Bernoulli’s equation forms the basis for solving a wide variety of fluid flow

problems such as jets issuing from an orifice, jet trajectory, flow under a gate and over a weir,

flow metering by obstruction meters, flow around submerged objects, flows associated with

pumps and turbines etc. The equipment is designed as a self-sufficient unit it has a sump tank,

measuring tank and a pump for water circulation as shown in figure1. The apparatus consists

of a supply tank, which is connected to flow channel. The channel gradually contracts for a

length and then gradually enlarges for the remaining length.

From the experiment, we knew that as the fluid flow from wider to narrower one, the

velocity of flowing fluid increases. This shown in all the results table, where the velocity of

water that flows in the tapered duct increases as the duct area decreases. From the analysis

12

we can conclude that for this flow, the difference velocity increases as the pressure difference

increases.

In addition to find the velocity difference we use Bernoulli Equation and Continuity

Equation. As for the Bernoulli Equation : ViB = √ 2g (hH-hi)

Where g – gravitational force = 9.81 m/s2

hH – Reading at Tube A

hi – Reading at Tube H

As for the continuity Equation : , Vic = QA i

Where Q – Flow Rate

A – Area of the tube

Thus, from these values the velocity difference can be obtained.

Pressure Difference (mm H20) Velocity Difference, ViB – Vic (m/s)

210 0.137

116 0.110

160 0.119

As we can see from table above, as the pressure difference increase, the velocity difference

increase. From the Bernoulli’s Principle Theorem, as the pressure increase, the velocity must

be decrease. Thus, we can conclude that there must be error during the experiment as the

value of difference velocity are not constantly decreased. One of them is, the observer must

have not read the level of static head properly, where the eyes are not perpendicular to the

water level on the manometer. Other than that, the pressure on A until F are not stable yet but

the reading are already taken. Thus, there are some minor effects on the circulations due to

the errors. From the experiment has been found that the Bernoulli’s principle is valid for

steady flow of fluid in tapered duct .

13

CONCLUSION

As a conclusion, the objective of this experiment was achieved that is to demonstrate

the Bernoulli’s Theorem experiment. The result collected from the experiment is according to

the Bernoulli’s Theorem which is the highest speed is the one at the lowest pressure, whereas

the lowest speed is present at the most highest pressure. Due to the highest pressure of the

water, it causes the reading of manometer become the highest. This principle complies with

the principle of conservation of energy which it is the sum if all forms of mechanical energy

along the streamline.

RECOMMENDATION

Repeat the experiment several times to get an accurate value.

The eye of the observer must be parallel to the water level to avoid parallax error.

Make sure the bubbles in the manometer are completely removed by adjusting the

bleed screw.

Make sure there is no leakage along the tube to prevent the water flowing out.

Make sure the pressure difference be maintained by control the valve slowly.

REFERENCES

John F.Douglas, (2001), Fluid Mechanics (4th ed.), Pearson Education Limited.

B.R. Munson, D.F. Young, and T.H. Okiishi, (1998), Fundamentals of Fluid

Mechanics (3rd ed.), Wiley.

Lab Manual: Bernoulli’s Theorem demonstration Unit.

http://www.scribd.com/doc/39165346/Bernoulli-s-Theorem-Distribution-Experiment

http://www.solution.com.my/pdf/FM24(A4).pdf

14

APPENDICES

15