BERNOULLI

TABLE OF CONTENT

1

CONTENT PAGE

ABSTRACT / SUMMARY 2

INTRODUCTION 2 – 3

AIMS / OBJECTIVE 4

THEORY 4 – 7

APPARATUS 7

PROCEDURE 8

RESULTS 9 – 11

CALCULATION 12 – 13

DISCUSSION 14

CONCLUSION 15

RECOMMENDATION 15

REFERENCES 16

APPENDICES 16

ABSTRACT / SUMMARY

From this experiment, we want to investigate the validity of the Bernoulli equation when

applied to the steady flow of water in a tapered duct. Secondly we want 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.

To run this experiment, firstly the Bernoulli equation apparatus on the hydraulic bench

was set up so that its base is horizontal for accurate height measurement from the

manometers. We used Δh manometer 50, 100 and 150 between Δh1 and Δh5 for both

converging and diverging tube. After that, the section diverging in the direction of flow

was set up. Then water inlet and outlet was connected. The time to collect 3 L water in

the tank was determined. Lastly calculate the flow rate, velocity, dynamic head, and total

head using the reading we get from the experiment and data given. The step was repeated

using converging in the direction of flow.

INTRODUCTION

In fluid dynamics, Bernoulli's principle states that for an inviscid 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 Dutch-Swiss mathematician

Daniel Bernoulli who published his principle in his book Hydrodynamica in 1738. [2]

Bernoulli's principle can be applied to various types of fluid flow, resulting in what is loosely

denoted as Bernoulli's equation. In fact, there are different forms of the Bernoulli equation

for different types of flow. The simple form of Bernoulli's principle is valid for

incompressible flows (e.g. most liquid flows) and also for compressible flows (e.g. gases)

moving at low Mach numbers. More advanced forms may in some cases be applied to

compressible flows at higher Mach numbers (see the derivations of the Bernoulli equation). [2]

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 that streamline. This requires that the sum of kinetic energy and

potential energy remain constant. If the fluid is flowing out of a reservoir the sum of all forms

of energy is the same on all streamlines because in a reservoir the energy per unit mass (the

sum of pressure and gravitational potential ρ g h) is the same everywhere. [2]

2

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 flowing

horizontally, the highest speed occurs where the pressure is lowest, and the lowest speed

occurs where the pressure is highest.[2]

Bernoulli's equation is sometimes valid for the flow of gases: provided that there is no transfer

of kinetic or potential energy from the gas flow to the compression or expansion of the gas. If

both the gas pressure and volume change simultaneously, then work will be done on or by the

gas. In this case, Bernoulli's equation -- in its incompressible flow form -- can not be assumed

to be valid. [1]

[4]

Suggested ShapesShape Title Comments

Narrow pipe widens

As cross-sectional area increases, velocity drops and pressure slightly increases

Rocket nozzle Exhaust is shot at high speed out of narrow opening

Drifting Rafters drift in lazy current between rapids

The relationship between the velocity and pressure exerted by a moving liquid is described by

the Bernoulli's principle: as the velocity of a fluid increases, the pressure exerted by that fluid

decreases. The Continuity Equation relates the speed of a fluid moving through a pipe to the

cross sectional area of the pipe. It says that as a radius of the pipe decreases the speed of fluid

flow must increase and visa-versa. [3]

3

AIMS / OBJECTIVE

1. To investigate the validity of the Bernoulli equation when applied to the steady

flow of water in a tapered duct.

2. 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.

THEORY [2]

In most flows of liquids, and of gases at low Mach number, the mass density of a fluid parcel

can be considered to be constant, regardless of pressure variations in the flow. For this reason

the fluid in such flows can be considered to be incompressible and these flows can be

described as incompressible flow. Bernoulli performed his experiments on liquids and his

equation in its original form is valid only for incompressible flow. A common form of

Bernoulli's equation, valid at any arbitrary point along a streamline where gravity is constant,

is:

(A)

where:

is the fluid flow speed at a point on a streamline,

is the acceleration due to gravity,

is the elevation of the point above a reference plane, with the positive z-direction

pointing upward — so in the direction opposite to the gravitational acceleration,

is the pressure at the point, and

is the density of the fluid at all points in the fluid.

For conservative force fields, Bernoulli's equation can be generalized as:

4

where Ψ is the force potential at the point considered on the streamline. E.g. for the Earth's

gravity Ψ = gz.

The following two assumptions must be met for this Bernoulli equation to apply:[5]

the fluid must be incompressible — even though pressure varies, the density must

remain constant along a streamline;

friction by viscous forces has to be negligible.

By multiplying with the fluid density ρ, equation (A) can be rewritten as:

or:

where:

is dynamic pressure,

is the piezometric head or hydraulic head (the sum of the elevation z

and the pressure head) and

is the total pressure (the sum of the static pressure p and dynamic

pressure q).

The constant in the Bernoulli equation can be normalised. A common approach is in terms of

total head or energy head H:

The above equations suggest there is a flow speed at which pressure is zero, and at even

higher speeds the pressure is negative. Most often, gases and liquids are not capable of

negative absolute pressure, or even zero pressure, so clearly Bernoulli's equation ceases to be

valid before zero pressure is reached. In liquids—when the pressure becomes too low --

cavitation occurs. The above equations use a linear relationship between flow speed squared

5

and pressure. At higher flow speeds in gases, or for sound waves in liquid, the changes in

mass density become significant so that the assumption of constant density is invalid.

In many applications of Bernoulli's equation, the change in the ρ g z term along the streamline

is so small compared with the other terms it can be ignored. For example, in the case of

aircraft in flight, the change in height z along a streamline is so small the ρ g z term can be

omitted. This allows the above equation to be presented in the following simplified form:

where q= v2

g

where p0 is called total pressure, and q is dynamic pressure. Many authors refer to the pressure

p as static pressure to distinguish it from total pressure p0 and dynamic pressure q. In

Aerodynamics, L.J. Clancy writes: "To distinguish it from the total and dynamic pressures, the

actual pressure of the fluid, which is associated not with its motion but with its state, is often

referred to as the static pressure, but where the term pressure alone is used it refers to this

static pressure."

The simplified form of Bernoulli's equation can be summarized in the following memorable

word equation:

static pressure + dynamic pressure = total pressure

Every point in a steadily flowing fluid, regardless of the fluid speed at that point, has its own

unique static pressure p and dynamic pressure q. Their sum p + q is defined to be the total

pressure p0. The significance of Bernoulli's principle can now be summarized as total pressure

is constant along a streamline.

If the fluid flow is irrotational, the total pressure on every streamline is the same and

Bernoulli's principle can be summarized as total pressure is constant everywhere in the fluid

flow. It is reasonable to assume that irrotational flow exists in any situation where a large

body of fluid is flowing past a solid body. Examples are aircraft in flight, and ships moving in

open bodies of water. However, it is important to remember that Bernoulli's principle does not

apply in the boundary layer or in fluid flow through long pipes.

6

If the fluid flow at some point along a stream line is brought to rest, this point is called a

stagnation point, and at this point the total pressure is equal to the stagnation pressure.

APPARATUS

The F1-10 Hydraulic Bench, which allows us to measure flow by, timed volume

collection.

The F1-15 Bernoulli’s Apparatus Test Equipment.

A stopwatch for timing the flow measurement.

7

PROCEDURE

1. The main switch and ump are switched on.

2. The venturi for the convergent flow position are setup.

3. The flow control valve are fully open to let the water flow into the venture and

manometer tubes.

4. The air bleed screw are adjusted.

5. The flow control valve and valve 1 are closed.

6. The air bleed screw are regulated until water level in manometer tubes reached 140

mmH2O.

7. The flow control valve are fully open.

8. Valve 1 are regulated slowly to get the different between water level in h1 and h5 that

is 50 mmH2O.

9. The reading from h1 until h5 are taken.

10. The ball in the water tank are dropped. The time are taken until the water reached 3 L.

11. Steps 8 until 10 are repeated for different value of ∆h that are 100 mmH2O and 150

mmH2O.

12. The experiment are repeated for divergent flow.

8

RESULTS

g = 9.81 m/s2

1. Divergence

∆h = 50 mm

Volume collected ( m3 ) Time ( s ) Flow rate ( m3/s )3 x 10-3 30.59 9.80 x 10-5

Distance into duct

( m )

Area of duct, A

x10-6 ( m2

)

Static head, h( mm )

Static head, h ( m )

Velocity, v x 10-3

( m/s )

Dynamic head, q ( v2/2g )

(m)

Total head, ho

( m )

h1 0.0000 490.9 150 0.150 0.200 0.0020 0.152

h2 0.0603 151.7 125 0.125 0.646 0.021 0.145

h3 0.0687 109.4 105 0.105 0.896 0.041 0.146

h4 0.0732 89.9 100 0.100 1.090 0.061 0.161

h5 0.0811 78.5 100 0.100 1.248 0.079 0.179

∆h = 100 mm

Volume collected ( m3 ) Time ( s ) Flow rate ( m3/s )3 x 10-3 23.62 1.27 x 10-4

Distance into duct

( m )

Area of duct, A

x10-6 ( m2

)

Static head, h( mm )

Static head, h ( m )

Velocity, v ( m/s )

Dynamic head, q ( v2/2g )

Total head, ho

( m )

h1 0.0000 490.9 165 0.165 0.259 0.0034 0.168

h2 0.0603 151.7 125 0.125 0.837 0.0357 0.161

h3 0.0687 109.4 70 0.070 1.161 0.069 0.139

h4 0.0732 89.9 65 0.065 1.413 0.102 0.167

h5 0.0811 78.5 65 0.065 1.618 0.133 0.198

9

∆h = 150 mm

Volume collected ( m3 ) Time ( s ) Flow rate ( m3/s )3 x 10-3 20.0 1.50 x 10-4

Distance into duct

( m )

Area of duct, A

x10-6 ( m2

)

Static head, h( mm )

Static head, h ( m )

Velocity, v ( m/s )

Dynamic head, q ( v2/2g )

Total head, ho

( m )

h1 0.0000 490.9 175 0.175 0.306 0.0047 0.180

h2 0.0603 151.7 115 0.115 0.989 0.050 0.165

h3 0.0687 109.4 45 0.045 1.371 0.096 0.141

h4 0.0732 89.9 25 0.025 1.669 0.142 0.167

h5 0.0811 78.5 25 0.025 1.910 0.186 0.211

2. Convergence

∆h = 50 mm

Volume collected ( m3 ) Time ( s ) Flow rate ( m3/s )3 x 10-3 48.5 6.19 x 10-5

Distance into duct

( m )

Area of duct, A

x10-6 ( m2

)

Static head, h( mm )

Static head, h ( m )

Velocity, v ( m/s )

Dynamic head, q ( v2/2g )

Total head, ho

( m )

h1 0.0000 490.9 165 0.165 0.126 0.00081 0.166

h2 0.0603 151.7 155 0.155 0.408 0.0085 0.164

h3 0.0687 109.4 145 0.145 0.566 0.016 0.161

h4 0.0732 89.9 130 0.130 0.689 0.024 0.154

h5 0.0811 78.5 115 0.115 0.789 0.032 0.147

10

∆h = 100 mm

Volume collected ( m3 ) Time ( s ) Flow rate ( m3/s )3 x 10-3 32.4 9.26 x 10-5

Distance into duct

( m )

Area of duct, A

x10-6 ( m2

)

Static head, h( mm )

Static head, h ( m )

Velocity, v ( m/s )

Dynamic head, q ( v2/2g )

Total head, ho

( m )

h1 0.0000 490.9 190 0.190 0.189 0.0018 0.192

h2 0.0603 151.7 165 0.165 0.610 0.019 0.184

h3 0.0687 109.4 145 0.145 0.846 0.036 0.181

h4 0.0732 89.9 120 0.120 1.030 0.054 0.174

h5 0.0811 78.5 90 0.090 1.180 0.071 0.161

∆h = 150 mm

Volume collected ( m3 ) Time ( s ) Flow rate ( m3/s )3 x 10-3 24.47 1.23 x 10-4

Distance into duct

( m )

Area of duct, A

x10-6 ( m2

)

Static head, h( mm )

Static head, h ( m )

Velocity, v ( m/s )

Dynamic head, q ( v2/2g )

Total head, ho

( m )

h1 0.0000 490.9 210 0.210 0.251 0.0032 0.213

h2 0.0603 151.7 180 0.180 0.811 0.034 0.214

h3 0.0687 109.4 145 0.145 1.124 0.064 0.209

h4 0.0732 89.9 105 0.105 1.368 0.095 0.200

h5 0.0811 78.5 60 0.060 1.567 0.125 0.185

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CALCULATION

1. Divergent

∆h = h1 – h5

= ( 150 – 100) mm

= 50 mm *

Flow rate = volume ( m3 ) time (s)

= 3 x 10-3 m3 30 .59 s

= 9.80 x 10-5 m3/s *

Velocity, v = flow rate m3 /sarea m2

= 9.80 x 10−5 m3 /s490.9 x 10−6 m2

= 0.200 *

Dynamic head , (m) = v2

2g

= 0.2002 m2/s2

2 x 9.81 m/ s2

= 0.00203 m. *

Total head = static head + dynamic head

= 0.150 m + 0.00203 m

= 0.152 m. *

12

2. Convergent

∆h = h1 – h5

= ( 165 – 115) mm

= 50 mm *

Flow rate = volume ( m3 ) time (s)

= 3 x 10-3 m3

48.5 s

= 6.19 x 10-5 m3/s *

Velocity, v = flow rate m3 /sarea m2

= 6.19 x 10−5 m3 /s490.9 x 10−6 m2

= 0.126 m. *

Dynamic head , (m) = v2

2g

= 0.1262 m2/s2

2 x 9.81 m/ s2

= 0.00081 m. *

Total head = static head + dynamic head

= 0.165 m + 0.00081 m

= 0.166 m. *

13

DISCUSSION

From the Bernoulli’s principle, he stated that that water is a fluid, and having the

characteristics of a fluid, it adjusts its shape to fit that of its container or other solid objects it

encounters on its path. Since the volume passing through a given length of pipe during a given

period of time will be the same, there must be a decrease in pressure. Hence Bernoulli's

conclusion: the slower the rate of flow, the higher the pressure, and the faster the rate of flow,

the lower the pressure.[5] For the divergence flow we can see that the total head at each tube h1

until h5 are not constantly increased but for the convergence flow, the total head are increased

due the increased of flow rate of water.

The flow rates of each flow also incresed that is :

∆h Divergence Convergence

50 mm 9.80 x10-5 m/s 6.19 x10-5 m/s

100 mm 1.270 x10-4 m/s 9.26 x10-5 m/s

150 mm 1.50 x 10-4 m/s 1.23 x10-4 m/s

From the experiment, we found that the total head pressure in both convergence and

divergence flow should be increased according to Bernoulli’s principle. So it shows that

Bernoilli’s equation is valid when applied to the steady flow of water in tapered duct and

velocity values increased along the same channel.

There must be an error during the divergence flow experiment as the total head are not

constantly increased. During the experiment, the pressure on h1 until h5 are not stable yet but

the reading are taken and time for 3 litres water are also recorded. These will affected the

calculation thus affected the total pressure at each tube. Other than that, the time should be

recorded when the water level reached at 0 litre but the time are started before the water level

reached 0 litre. Other than that, the position of eye during reading the manometer tube should

be staright to the meniscus. To get the constant desired pressure difference, the valve 1 and

bleed screw should be regulated smoothly and slowly.

14

CONCLUSION

From the experiment, we found that the total head pressure in both convergence and

divergence flow increased according to Bernoulli’s principle. So it shows that Bernoilli’s

equation is valid when applied to the steady flow of water in tapered duct and velocity

values increased along the same channel.

RECOMMENDATION

1. Make sure that there are no leakages on the connection between the pipes. The leakage

will cause the water bleed out from the pipes connection so its will make our

experiment have an errors.

2. Make sure that there are no bubbles in the manometer to get the accurate reading.

3. When take the reading of the volume of the water make sure that eyes parallel with the

volume meter.

4. Control the air bleed screw slowly and smoothly.

5. Waited until the pressure at each manometer tube are stable before the reading are

taken.

15

REFERENCES

1. http://www.absoluteastronomy.com/topics/Bernoulli%27s_principle

2. http://en.wikipedia.org/wiki/Bernoulli%27s_principle

3. http://library.thinkquest.org/27948/bernoulli.html

4. http://home.earthlink.net/~mmc1919/venturi.html

5. http://www.scienceclarified.com/everyday/Real-Life-Chemistry-Vol-3/Bernoulli-s- Principle.html

APPENDICES

16

h4

h3

V1

h1

h5

h2

venturi

Air bleed screw