Lab1 Bernoulli_s Equation

TABLE OF CONTENT

1

No . Title Pages

1 Abstract/summary 2

2 Introduction 3-5

3 Aims/objectives 5

4 Theory 5-7

5 Procedures 8-9

6 Apparatus 10

7 Results 10-16

8 Sample of Calculations 17-18

9 Discussions 18-19

10 Conclusion 19

11 Recommendation 20

12 References 20

13 Appendices 21

ABSTRACT

Generally, the objectives for this experiment are to determine the validity of the Bernoulli’s

equation when applied to the steady flow of water in a tapered duct. Besides that, the objective is

to measure flow rates and both static and total pressure heads in a rigid convergent or divergent

tube of known geometry for a range of steady flow rates.

To conduct this experiment, first we must set up the apparatus which is the hydraulic

bench, so that its base is horizontal for accurate height measurement form the manometers. The

readings should be taken at 3 flow rates for convergent and divergent section. Then, we must

adjust the head difference of about 150mm, 100mm, and 50mm. First, we set up the diverging in

the direction of flow, and then we set up the converging section. We must determine a time to

collect the 3L of water and determine the volumetric flow rate. Then, take the readings of h1-h6.

After get all the data, we must calculate the velocity, dynamic head, static head and the total

head. We can see that the flow rate for both convergent and divergent duct are decrease from

difference of about 150mm to 50mm. The times collected for both convergent and divergent are

increase from difference of about 150mm to 50mm. Actually, this is a incompressible flow, so,

form the theory, the converging duct it follows that as area reduced, the velocity must increase

while for diverging duct, it follows that as area increase the velocity must decrease. The overall

velocities for both convergent and divergent are decrease. So, what we can say that this

experiment is success.

2

Tube Cross Section

0.00 50.00 100.00Position Along Tube (mm)

INTRODUCTION

The Hydraulics bench service module, F1-10, provides the necessary facilities to support a

comprehensive range of hydraulic models each of which is designed to demonstrate a particular

aspect of hydraulic theory.

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 along 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 13.9

C h3 11.8

D h4 10.7

E h5 10.0

F 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 section 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 facilitate setting up. All eight-pressure tapping are connected to a bank of

pressurized manometer tubes. Pressurization of the manometer is 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.

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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.

AIMS/OBJECTIVES

The objectives of this experiment are to determine the validity of the Bernoulli’s equation when

applied to the steady flow of water in a tapered duct. Besides that, the objective is to measure

flow rates and both static and total pressure heads in a rigid convergent or divergent tube of

known geometry for a range of steady flow rates.

THEORY

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.

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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.

Total 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

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Velocity Measurement

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

V = Qv A

Continuity equation

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

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PROCEDURES

A) Equipment set up

1. Level the apparatus

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

horizontal.

2. 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.

3. 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.

4. Bleeding the manometers.

Close both the bench valve, the rig flow control valve, open the air bleed screw,

andremove the cap from the adjacent air valve in order to bleed air from pressure

tapping points and manometers. A length of small bore tubing from the air valve was

connected to the volumetric tank. To purge all air from them, open the bench valve

and allow flow through the manometers. Then, tighten the air bleed screw and partly

open the bench valve and test rig flow control valve. To allow air to enter the top of

the manometers, open the air bleed screw slightly. Re-tighten the screw when the

manometer levels reach a convenient height. The maximum (h1) and minimum (h5)

manometer readings both on scale. If required, adjust the manometer levels by using

the air bleed screw and the hand pump supplied. The air bleeds screw control the

airflow through the air valve, so when using the hand pump, the bleed screw must be

open. The screw must be closed after pumping to retain the hand pump pressure in the

system.

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B) Taking a set of results

Take the readings at 3 flow rates. Finally, you may reverse the test section in order to

see the effect of a more rapid converging section.

1. Setting the flow rate

At the maximum flow rate, take the first set of readings, then the volume flow rate

was reduced to give the h1-h5 head difference of about 50mm. Finally, the whole

process for one further flow rate was repeated set to give the h1-h5 difference

approximately half way between that obtained in the above two test.

2. Time volume collection

In order to determine the volume flow rate, we should carry out at a timed volume

collection, using the volumetric tank. This is achieved by closing the ball valve and

measuring (with a stop watch) the time taken to accumulate a known volume of fluid

in the tank, which is read from the sight glass. To minimize timing errors, we should

collect fluid for at least one minute. Again, the total pressure probe should be

retracted from the test-section during these measurements.

3. Reading the total head distribution.

Measure the total pressure head distribution 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.

4. 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 Hydraulics Bench service module, F1-10.

The F1-15 Bernoulli’s Apparatus Test Equipment.

A stopwatch for timing the flow measurement.

RESULTS

For divergent (difference of about 150mm)

No Volume collected, V (m3)

Time to collect, t (sec)

Flow rate, Qv (m3/sec)

Distance into duct (m)

Area of duct, A (m2)

Static head, h (m)

Velocity, v (m/s)

Dynamic head (m)

Total head, ho (m)

1. 0.003 20.72 1.448e-4

h1 0.00 490.9 x 10-6

0.180 0.312 4.961e-3 0.185

2. 0.003 18.53 1.619e-4

h2 0.0603 151.7 x 10-6

0.125 1.009 51.90e-3 0.177

3. 0.003 19.65 1.527e-4

h3 0.0687 109.4 x 10-6

0.050 1.400 99.90e-3 0.150

4. Average flow rate= 153.13e-6

Average time= 19.63

h4 0.0732 89.9 x 10-6

0.035 1.703 147.8e-3 0.183

5. h5 0.0811 78.5 x 10-6

0.030 1.951 194.0e-3 0.224

6. h6 0.1415 490.9 x 10-6

0.250 0.312 4.961e-3 0.255


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Static head, h (m)

Velocity, v (m/s)

Dynamic head (m)

Total head, ho (m)

1. 0.003 23.10 1.299e-4

h1 0.00 490.9 x 10-6

0.160 0.274 3.827e-3 0.164

2. 0.003 24.62 1.219e-4

h2 0.0603 151.7 x 10-6

0.120 0.888 40.19e-3 0.160

3. 0.003 26.03 1.523e-4

h3 0.0687 109.4 x 10-6

0.070 1.231 77.24e-3 0.147


Average time= 24.58

h4 0.0732 89.9 x 10-6

0.075 1.498 114.4e-3 0.189

5. h5 0.0811 78.5 x 10-6

0.060 1.716 150.1e-3 0.210

6. h6 0.1415 490.9 x 10-6

0.210 0.274 3.827e-3 0.214


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Static head, h (m)

Velocity, v (m/s)

Dynamic head (m)

Total head, ho (m)

1. 0.003 31.25 9.60e-5 h1 0.00 490.9 x 10-6

0.145 0.192 1.879e-3 0.147

2. 0.003 31.22 9.609e-5

h2 0.0603 151.7 x 10-6

0.120 0.602 18.47e-3 0.138

3. 0.003 33.25 9.023e-5

h3 0.0687 109.4 x 10-6

0.090 0.860 37.70e-3 0.128


Average time= 31.91

h4 0.0732 89.9 x 10-6

0.095 1.047 55.87e-3 0.151

5. h5 0.0811 78.5 x 10-6

0.095 1.199 73.27e-3 0.168

6. h6 0.1415 490.9 x 10-6

0.180 0.192 1.879e-3 0.182

For convergent (difference of about 150mm)

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Static head, h (m)

Velocity, v (m/s)

Dynamic head (m)

Total head, ho (m)

1. 0.003 22.10 1.325e-4

h1 0.00 490.9 x 10-6

0.215 0.259 3.419e-3 0.218

2. 0.003 24.37 1.231e-4

h2 0.0603 151.7 x 10-6

0.180 0.838 35.79e-3 0.216

3. 0.003 23.88 1.256e-4

h3 0.0687 109.4 x 10-6

0.160 1.162 68.82e-3 0.229


Average time= 23.45

h4 0.0732 89.9 x 10-6

0.110 1.413 101.8e-3 0.212

5. h5 0.0811 78.5 x 10-6

0.065 1.619 133.6e-3 0.199

6. h6 0.1415 490.9 x 10-6

0.120 0.259 3.419e-3 0.123

For convergent (difference of about 100)

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Static head, h (m)

Velocity, v (m/s)

Dynamic head (m)

Total head, ho (m)

1. 0.003 27.85 1.077e-4

h1 0.00 490.9 x 10-6

0.190 0.208 2.205e-3 0.192

2. 0.003 30.04 9.987e-5

h2 0.0603 151.7 x 10-6

0.165 0.675 23.22e-3 0.188

3. 0.003 30.16 9.947e-5

h3 0.0687 109.4 x 10-6

0.145 0.936 44.65e-3 0.190


Average time= 29.35

h4 0.0732 89.9 x 10-6

0.120 1.138 66.01e-3 0.186

5. h5 0.0811 78.5 x 10-6

0.090 1.304 86.67e-3 0.177

6. h6 0.1415 490.9 x 10-6

0.125 0.208 2.205e-3 0.127

For convergent (difference of about 50mm)

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Static head, h (m)

Velocity, v (m/s)

Dynamic head (m)

Total head, ho (m)

1. 0.003 42.91 6.991e-5 h1 0.00 490.9 x 10-6

0.160 0.142 1.028e-3 0.1610

2. 0.003 41.56 7.218e-5 h2 0.0603 151.7 x 10-6

0.150 0.459 10.74e-3 0.1607

3. 0.003 45.00 6.667e-5 h3 0.0687 109.4 x 10-6

0.145 0.636 20.62e-3 0.166


Average time= 43.16

h4 0.0732 89.9 x 10-6

0.125 0.774 30.53e-3 0.156

5. h5 0.0811 78.5 x 10-6

0.110 0.886 40.01e-3 0.150

6. h6 0.1415 490.9 x 10-6

0.130 0.142 1.028e-3 0.131

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SAMPLE OF CALCULATIONS

Example: For divergent (difference of about 150mm)

Volume collected = 3 L

1000L = 1m3

3.00L = 3.00L x 1m 3

1000L

= 3.0 x 10-3 m3

Flow rate = volume collected

time

= 0.003

20.72

= 1.448 x 10-4 m3/sec

Velocity, v = Flow rate

Area duct

= 1.448 x 10- 4

490.9 x 10-6

= 0.312 m/s

Dynamic head = v 2 g

= (0.312) 2

9.81

= 4.961 x 10-3

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Total head = static head + dynamic head

= 0.180 + 4.961 x 10-3

= 0.185 m

DISCUSSIONS

For this experiment, we are able to determine the flow rate of the steady flow of water in

rigid convergent and divergent duct. Divergence- For the divergent duct, the flow rate is decrease

from the difference of about 150mm to 50mm. The flow rates are decreased from 153.13 x 10-6,

134.7 x 10-6 and 94.11 x 10-6. The time collected for divergent is increase from the difference of

about 150mm to 50mm. The values of total head from h1 –h6 for difference of about 150mm are

0.185, 0.177, 0.150, 0.183, 0.224, and0.255. The values of total head from h1 –h6 for difference

of about 100mm are 0.164, 0.160, 0.147, 0.189, 0.210 and 0.214. The values of total head from

h1 –h6 for difference of about 50mm are 0.147, 0.138, 0.128, 0.151, 0.168 and 0.182. From the

plotted graph, we can see the shape of the graph is like ‘ U’. For the overall velocities are

increases.

Convergent- For the convergent duct, the flow rate is decrease from the difference of

about 150mm to 50mm. The flow rates are decreased from 127.07 x 10-6, 102.35 x 10-6 and

69.59 x 10-6. The time collected for convergent is increase from the difference of about 150mm

to 50mm. The values of total head from h1 –h6 for difference of about 150mm are 0.218, 0.216,

0.229, 0.212, 0.199 and 0.123. The values of total head from h1 –h6 for difference of about

100mm are 0.192, 0.188, 0.190, 0.186, 0.177 and 0.127. The values of total head from h1 –h6 for

difference of about 50mm are 0.1610, 0.1607, 0.166, 0.156, 0.150 and 0.131. From the plotted

graph, we can see the shape of graph at the results part. For the overall velocities are increases.

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For this experiment, it is the incompressible fluid which is for the converging flow, it

follows that as the area reduces then the velocity must increase, while in a diverging duct as the

area increases then the velocity must decrease. The eddies and shear stress are created as the

flow through a convergent duct then, it creating head loss. So, the result is quite successful.

CONCLUSIONS

As a conclusion, we can say that this experiment is success because the convergent and divergent

flows are obeying the Bernoulli’s equation. We can see the shape of the graph at the results part.

We can say that both of flow rate for converging and diverging duct are decrease. The times

collected for both convergent and divergent are increase from difference of about 150mm to

50mm. Total head loss is created when the water flow through a varies area then it creating

eddies and shear stress. We get the different value of velocity because it depending on the flow

rate divided by area.

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RECOMMENDATIONS

Before conduct this experiment, make sure there is no bubble in the manometer because the

readings will be affected. Then, measure the height of the water level in each manometer tube by

marking the paper positioned behind the tubes and record on the test sheet. To get the consistent

results, the time collected should be taken more than once and get the average. Besides that, be

careful when control the valve, because it quite difficult. Then, make sure the position of eyes is

parallel to the level of reading.

REFERENCES

Laboratory manual, Chemical Engineering Laboratory II, (CHE 523), faculty Of

Chemical Engineering, Uitm.

Yahoo and google search engine. (keyword:

www3.interscience.wiley.com

www.physicsforums.com-convergent

www.patentstorm.us

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APPENDICES

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Lab1 Bernoulli_s Equation

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