bernoulis theorem

UNIVERSITI TEKNOLOGI MARA

FAKULTI KEJURUTERAAN KIMIACHEMICAL ENGINEERING LABORATORY II

(CHE523)

NAME : WAN ZULKARIM BIN WAN NADZRI

STUDENT NO. : 2004624899

EXPERIMENT : BERNOULLI’S THEOREM DEMONTRATION

DATE PERFORMED

: 8 August 2005

SEMESTER : JULY-OCTOBER 2005

PROGRAMME / CODE

: Bachelor of Engineering (Hons.) in Chemical Engineering/ EH220

Remarks:

Checked by: En.Alawi bin Sulaiman Rechecked

TABLE OF CONTENT

PK.FKK.PPM.MANUAL MAKMAL CHE523

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No. Title Allocated marks % Marks %1 Abstract/Summary 5 2 Introduction 5 3 Aims/Objectives 5 4 Theory 5 5 Procedures 3 6 Apparatus 5 7 Results 20 8 Calculations 10 9 Discussions 20

10 Conclusions 10 11 Recommendations 5 12 References 5 13 Appendices 2

TOTAL 100

Summary 3

Introduction 3

Objectives 4

Theory 5

Procedures 6

Apparatus 8

Results 8

Sample of calculations 10

Discussions 11

Conclusions 12

Recommendations 12

Reference 13

Appendices 13

SUMMARY

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Roughly, in 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/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. 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.

Based on tabulated result, we can see that both flow rate for diverging and

converging are increase. Based on plotted graft we find out the shape of graft for

diverging like ‘u’ and for converging the graft is decrease.

For the conclusion the experiment we have done obey the theory of the

Bernoulli’s theorem equation. So the Bernoulli equation is valid.

INTRODUCTION

To demonstrate a particular aspect of hydraulic theory the hydraulics bench service module, F1-10, was designed to provide the necessary facilities to support a comprehensive range of hydraulic model.

The specific hydraulic model that we are concerned with for this experiment is the Bernoulli’s Theorem Demonstration Apparatus, F1-15. This consists of a classical static pressure. A probe can be traversed a long the center of the section to obtain total head readings.

The test section is an accurately machined clear acrylic duct of varying circular cross section. It is provided with a number of side hole pressure tapping, which are connected to the manometers housed on the rig, these tapping allow the measurement of static pressure head simultaneously at each of six sections. To allow the calculation of the dimensions of the test section, the tapping positions and the test section diameters are shown on the following diagram:

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The dimensions of the tube are detailed below: -

Tapping Position Manometer Legend Diameter(mm) A h1 25.0

B h2 13.9C h3 11.8D h4 10.7E h5 10.0F h6 25.0

Note: The assumed datum position is at tapping A associated with h1

The test section incorporates two unions, one at either end, to facilitate reversal for convergent or divergent testing.

A hypodermic, the total pressure head probe, is provided which may be positioned to read the total pressure head at any sections of the duct. This total pressure head probe may be moved after slackening the gland nut; this nut should be re-tightened by hand. To prevent damaged, the total pressure head probe should be fully inserted during transport/storage. An additional tapping is provided to facilitated by removing the hand pump from its storage location at the rear of the manometer board and connecting its flexible coupling to the inlet valve on the manometer manifold.

In use, the apparatus is mounted on a baseboard, which is stood on the work surface of the bench. This baseboard has feet, which may be connected directly to the bench supply. A flexible hose is attached to the outlet pipe, which should be directed to the volumetric measuring tank on the hydraulics bench.

A flow control valves is incorporated downstream of the test section. Flow rate and pressure in the apparatus may be varied independently by adjustment of the flow control valve, and the bench supply control valve.

OBJECTIVES

Our objectives done this experiment are to investigate the validity of the Bernoulli

equation when applied to the steady flow of water in a tapered duct. We also want to

measured flow rates and both static and total pressure heads in a rigid convergent or

divergent tube of unknown geometry for a range of steady flow rates.

THEORY

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The Bernoulli equation represents the conservation of mechanical energy for a steady, incompressible, frictionless flow: -

P1 + V12 + Z = P 2 + V 2 + Z2

ρg 2g ρg 2g

Where:

P = static pressure detected at a side hole.V = fluid velocity.Z = vertical elevation on the fluid,Z1 = Z2 horizontal tube.

The equation may be derived from the Eular Equations by integration.It also may be derived from energy conservation principles.Derivation of the Bernoulli Equation is beyond the scope of this theory.With the Armfield F1 – 15 apparatus, the static pressure head P is measured using a manometer directly from a side hole pressure tapping.The manometer actually measures the static pressure head, h, in meter, which is related to P using the relationship:

h1 = P ρg

This allows the Bernoulli equation to be written in a revised from, ie:

h1 + V12 = h2 + V2

2

2g 2g

The velocity related portion of the total pressure head is called the dynamic pressure head.

The total pressure head, ho, can be measured from a probe with an end hole facing into the flow such that it brings the flow to rest locally at the probe end.Thus,

ho = h + V2 (meters) and, from the Bernoulli equation, it follow that 2g

h1o = h2

o

The velocity of the flow is measured by measuring the volume of the flow, V, over a time period, t. This gives the rate of volume flow as: Qv = V , which in turn gives the velocity of flow through a defined area, A, t

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V = Qv A

For an incompressible fluid, conservation of mass requires that volume is also conserved,

A1V1 = A2V2

Other forms of the Bernoulli equation,If the tube is horizontal, the difference in height can be disregarded,

Z1 = Z2

Hence:

P1 + V12 = P2 + V2

2 ρg 2g ρg 2g

PROCEDURES

A) Equipment set up

Level the apparatus.

Bernoulli equation apparatus on the hydraulic bench was set up, so that its base is horizontal.

Set the direction of the test section.

The test section was ensuring to have the 14o –tapered section converging in the direction of flow. The total pressure head probe was withdrawn before releasing the mounting couplings when reversed the test section.

Connect the water inlet and outlet.

The rig outflow was ensuring is positioned above the volumetric tank, in order to facilitate timed volumes collections. The rig inlet was connected to the bench flow supply. The bench valve and apparatus flow control valve was closed and the pump started. The bench valve was gradually opened to fill the test rig with water.

Bleeding the manometers.

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Both the bench valve, the rig flow control valve, the air bleed screw was opened, and the cap from the adjacent air valve was removed in order to bleed air from pressure tapping points and manometer. A length of small-bore tubing from the air valve to the volumetric tank was connected. The bench valve was opened and flow was allowed through the manometers to purge all air from them then the air bleed screw was tightened. The bench valve and test rig flow control valve was partly opened. Next, the air bleed screw was slightly opened to allow air to enter the top of the manometers. The screw was re-tighten when the manometer levels reach a convenient height.

B) Taking a set of results

3-flow rate reading was taken then the test section was reversed in order to see the effects of a more rapid converging section.

Setting the flow rate.

First set of reading at the maximum flow rate was taken then the volume flow rate was reduced to give the h1 – h5 head difference of about 50mm. The whole process for one further flow rate was repeated, using Δ(h1 – h5) 100 and 150.

Timed volume collection.

The ball valve was closing and the time taken to accumulate a known volume of fluid in the tank was measured (with stop watch).

Reading the total head distribution.

The total pressure head distribution was measured by traversing the total pressure probe along the length of the test section. The datum line is the side hole pressure tapping associated with the manometer h1. a suitable starting point is 1 cm upstream of the beginning of the 14o tapered section and measurements should be made at 1 cm intervals along the test-section length until the end of the divergent (21o) section.

Reversing the test section.

The total pressure probe was ensured fully withdrawn from the test-section. The two couplings were unscrewed, the test-section was removed and it then re-assembles reversed by tightening the coupling.

APPARATUS

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The apparatus that we used to demonstrate the Bernoulli apparatus, we needed a number

of pieces of equipment: -

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.

RESULTS

Divergence

Flow rate = 7.81 x 10-5 m3/s Time = 38.41 s Distance into duct(m)

Area of duct, A(m2)X 10-6

Static head, h (m)X 10-

1

Velocity, v (m/s)

Dynamic head (m)X 10-2

Total headHo (m)

1 h1 0.00 490.9 1.55 0.159 0.129 0.1562 h2 0.0603 151.7 1.33 0.515 1.4 0.1473 h3 0.0687 109.4 1.14 0.714 2.6 0.144 h4 0.0732 89.9 1.13 0.869 3.8 0.1515 h5 0.0811 78.5 1.05 0.995 5.0 0.1556 h6 0.1415 490.9 1.90 0.159 0.129 0.191

Flow rate = 1.26 x 10-4 m3/s Time = 23.82 sDistance into duct(m)



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Velocity,v(m/s)


Total headHo (m)

1 h1 0.00 490.9 1.65 0.257 0.337 0.1682 h2 0.0603 151.7 1.25 0.832 3.5 0.163 h3 0.0687 109.4 0.87 1.152 6.8 0.1554 h4 0.0732 89.9 0.75 1.40 9.9 0.1745 h5 0.0811 78.5 0.65 1.605 13.1 0.1966 h6 0.1415 490.9 2.375 0.257 0.337 0.24

Flow rate = 1.56 x 10-4 m3/s Time = 19.25 s

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Distance into duct(m)



1

Velocity,v(m/s)


Total headHo (m)

1 h1 0.00 490.9 1.75 0.317 0.5122 0.182 h2 0.0603 151.7 1.15 1.030 5.40 0.1693 h3 0.0687 109.4 0.67 1.426 10.40 0.1714 h4 0.0732 89.9 0.40 1.735 15.30 0.1935 h5 0.0811 78.5 0.25 1.987 20.10 0.2266 h6 0.1415 490.9 2.70 0.317 0.5122 0.275

Convergence




1

Velocity,v(m/s)


Total headHo (m)

1 h1 0.00 490.9 1.75 0.128 0.835 0.1752 h2 0.0603 151.7 1.65 0.413 8.69 0.1743 h3 0.0687 109.4 1.55 0.573 16.7 0.1714 h4 0.0732 89.9 1.42 0.697 24.7 0.1675 h5 0.0811 78.5 1.25 0.799 32.5 0.1586 h6 0.1415 490.9 1.35 0.128 0.835 0.136




1

Velocity,v(m/s)

Dynamic head (m)

Total headHo (m)

1 h1 0.00 490.9 2.05 0.199 0.2018 0.2102 h2 0.0603 151.7 1.80 0.645 2.12 0.2013 h3 0.0687 109.4 1.57 0.894 4.1 0.1984 h4 0.0732 89.9 1.35 1.088 6.0 0.1955 h5 0.0811 78.5 0.95 1.277 7.9 0.1746 h6 0.1415 490.9 1.20 0.199 0.2018 0.122

Flow rate = 1.235 x 10-4 m3/s Time = 24.29 s

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Distance into duct(m)



1

Velocity,v(m/s)


Total headHo (m)

1 h1 0.00 490.9 2.20 0.252 0.3237 0.2232 h2 0.0603 151.7 1.88 0.814 3.4 0.2223 h3 0.0687 109.4 1.59 1.129 6.5 0.2244 h4 0.0732 89.9 1.23 1.374 9.6 0.2195 h5 0.0811 78.5 0.70 1.573 12.6 0.1966 h6 0.1415 490.9 1.15 0.252 0.3237 0.118

SAMPLE OF CALCULATIONS

Volume collected = 3 L

1000L = 1m3

3.00L = 3.00L x 1m 3

1000L

= 3.0 x 10-3 m3

Flow rate, Qv = Volume collected Time

= 3.0 x 10 -3 m 3 38.41s

= 7.81 x 10-5 m3/s

Velocity, v = Flow rate Area into duct

= 7.81 x 10 -5 m 3 /s 490.9 x 10-6 m2

= 0.159 m/s

Dynamic head = v 2 2g

= (0.159 m/s) 2 2 x 9.81m/s2

= 1.29 x 10-3 m

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Total head = Static head + Dynamic head

= (1.55 x 10-1 + 1.29 x 10-3) m

= 0.156 m

DISCUSSIONS

There are a few discussions we can make about our experiment. Analysis form the

tabulated result that we get are: Divergence: - The flow rate are increased from 7.81 x 10-

5 m3/s, 1.26 x 10-4 m3/s, and 1.56 x 10-4 m3/s. The times to collect 3 L water are decrease

from 38.41s, 23.82s, and 19.25s. The value of total head from h1 – h6 for Δh = 50 are

0.156 m, 0.147 m, 0.140 m, 0.151 m, 0.155 m, and 0.191 m. For Δh = 100 are 0.168 m,

0.160 m, 0.155 m, 0.174 m, 0.196 m, and 0.240 m. For Δh = 150 are 0.18 m, 0.169 m,

0.171 m, 0.193 m, 0.226 m, and 0.275 m. We can see the shape of the plotted graph like “

U “ as in the appendices. Overall the velocities are increases from h1 – h5 and h6 is equal

with the h1 for every Δh 50 – 150. Overall dynamic head are increases from h1 – h5 and h6

is equal with the h1 for every Δh 50 – 150.

Convergence: - The flow rate for convergence also increased from 6.27 x 10-5

m3/s, 9.79 x 10-5 m3/s, and 1.235 x 10-4 m3/s. Timed to collect 3 L water also decrease

from 47.84 s, 30.62 s, and 24.29 s. The value of total head from h1 – h6 for Δh = 50 are

0.175 m, 0.174 m, 0.171 m, 0.167 m, 0.158 m, and 0.135 m. For Δh = 100 are 0.210 m,

0.201 m, 0.198 m, 0.195 m, 0.174 m, and 0.122 m. For Δh = 150 are 0.223m, 0.222 m,

0.224 m, 0.219 m, 0.196 m, and 0.118 m. We can see that the values of total head are

decreases for every Δh. We can see the shape of the plotted graft in the appendices.

Overall the velocities are increases from h1 – h5 and h6 is equal with the h1 for every Δh

50 – 150. Overall dynamic head are increases from h1 – h5 and h6 is equal with the h1 for

every Δh 50 – 150.

CONCLUSIONS

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Based on experiment we have done, we can conclude that both convergence flow

and divergence flow obey the Bernoulli equation. It is because we can see by the shape of

the plotted graft. For the diverging flow graft the shape is like ’U’, initially it decreases

until one point and then its increase.

For the converging flow graft all the state minimum, middle, and maximum flow

rate its will decrease as in the graph.

We can conclude that both of flow rate for diverging and converging are decrease.

The total head for diverging flow is increase initially and then at one point it will increase

weather the total head for converging flow are decrease. The static head for both

diverging flow and converging flow are decrease initially and then its will increases.

The area of the all manometers is difference except for h1 and h6 the area is same.

It’s that why the velocity of all manometers is different because the velocity is depending

on the flow rate divide by the area.

RECOMMENDATIONS

There are a few recommendations to make sure that our result more accurate and

to void from mistake. Firstly 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. Secondly make sure that there are no

bubbles in the manometer to get the accurate reading. Thirdly when take the reading of

the volume of the water make sure that eyes parallel with the volume meter.

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REFERENCES

1. Fluid Mechanics, 3rd edition, Longmans Singapore Publisher, (1995),

Douglas. J.F., Gasiorek. J.M. and Swaffield.

2. Giles, R.V., Evett, J.B. and Cheng Lui, Schaumm’s Outline Series Theory

and Problems of Fluid Mechanics and Hydraulic, McGraw-Hill intl., (1994).

3. Chemical Engineering Laboratory II, by Ms Dayang Aisyah Chi, (2005).

APPENDICES

h1 h2 h3 h4 h5 h60

0.050.1

0.150.2

0.250.3

divergence graft

Series1

Series2

Series3

h1 h2 h3 h4 h5 h60

0.05

0.1

0.15

0.2

0.25

convergence graft

Series1

Series2

Series3

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bernoulis theorem

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