Experiment to Study Flow Through a Venturimeter

EXPERIMENT TO STUDY FLOW THROUGH A VENTURI METER

HYDRAULICS PRACTICALS FLUIDS MECHANICS LABORATORY

MODULE: FLUIDS MECHANICS1

REPORT OF EXPERIMEN EXPERIMENT: T: EXPERIMENT TO STUDY FLOW THROUGH A VENTURI METER

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Table of Content Introduction Objectives Apparatus Description of apparatus & Experimental Procedures Tabulation of results v

Calculation of area of the different cross sections of the Venturimeter Result and Analysis v

Calculating the ideal pressure at each section of the Venturimeter

v

A graph of ‘Ideal Pressure,’ against ‘Section of Venturimeter, a n’

v

Calculating the actual pressure at each section of the Venturimeter

v

A graph of ‘Actual pressure’ against ‘Sections of Venturimeter’ for Flow rate No 1.

v

A graph of ‘Actual pressure’ against ‘Sections of Venturimeter’ for Flow rate No 4.

v

Calculation of the coefficient of discharge, C d, of the Venturimeter, for each flow rate.

v

A graph of √h1-h2 against Q.

v

Calculation of the value of coefficient of discharge , Cd for each value of volumetric flow rate, Q.

v

A graph of Cd against Q

Conclusions References

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Introduction In this experiment, we will investigate how the pressure varies at different positions along a tube of varying cross-sectional area (the convergent-divergent Venturimeter), for different volumetric flow rates.

A Venturimeter (fig 1.1) consists of a short converging conical tube leading to a cylindrical portion, called the throat, of smaller diameter of that of the pipeline, which is followed by a diverging section in which the diameter increases again to that of the main pipeline.

Convergent-Divergent Venturi meter

Piezometers

Fig 1.1 Venturi Meter Diagram

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EXPERIMENT TO STUDY FLOW THROUGH A VENTURI METER The Venturimeter is used to measure the flow rate of compressible and incompressible fluid, through a pipeline. The change in pressure in the fluid as it passes through the meter can be used to determine the flow rate. The meter, in its simplest form, consists of a short length of pipe, tapering to a narrow throat in the middle. At a number of points along the length of the convergent-divergent passage of the Venturi, piezometer tubes are drilled into the wall and connections are made from each of these to vertical manometer tubes, which are mounted in front of a scale marked in millimeters. The manometer tubes are connected at their top ends to a common manifold in which the amount of air may be controlled by a small air valve at one end.

The procedure, of theoretically quantifying the discharge for a Venturimeter, lies basically in the combined aid of the Bernoulli and Continuity equations, assessing both the coefficient of discharge, (C d ) and the flow coefficients. The coefficient of discharge is the ratio of the actual flow rate to the theoretical flow rate. It is the fraction of the theoretical flow rate which gives the actual flow rate.

The discharge, Q is given by

Q=

Cd a2

√[2g (h1 - h2)]

√1 – (a2/a1)

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EXPERIMENT TO STUDY FLOW THROUGH A VENTURI METER Where Cd is the coefficient of discharge of the meter and that The ideal pressure is equal to 2

hn – h1 = (a2) 2 2 u2 /2g (a1)

2

- (a2) 2 (an)

Where, hn: the actual piezometer reading at a particular position h1: the actual piezometer reading at position A u2: the velocity of flow in Venturimeter at piezometer position D g: the acceleration due to gravity = 9.810 ms

-2

an: the cross sectional area of the particular piezometer a1: the cross sectional area of Venturimeter at piezometer position A = 0.053093 m

2

a2: the cross sectional area of Venturimeter at piezometer position D = 0.020106 m

2

an: the cross sectional area of the Venturimeter at piezometer position where ideal pressure is to be

determined

Actual pressure =

hn - h1 2

u2 /2g

Where, hn: the actual piezometer reading at a particular position h1: the actual piezometer reading at position A u2: the velocity of flow in Venturimeter at piezometer position D g: the acceleration due to gravity = 9.810 ms

-2

2

u2 /2g : head loss at piezometer position D

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EXPERIMENT TO STUDY FLOW THROUGH A VENTURI METER Bernoulli’s equation relates the pressure, velocity, and elevation between any two points in the

flow field of a flow that is steady, not rotational, non-viscous, and incompressible. This implies that, the equation links pressure energy, kinetic energy and potential energy per unit weight from point to point in the fluid. It also states that they obey the law of conservation of energy. a constant Pressure energy Kinetic energy Potential energy Total energy per + + = = per unit weight per unit weight per unit weight unit weight

In this experiment, the axis of the Venturimeter is kept horizontal, with the help of the adjusting screws, to avoid any change in elevation of the fluid flow. Thus, there is no change in the potential energy per unit weight, as the fluid does not lose elevation during flow. The horizontal plane is taken as the datum which reduces the potential energy variable in the equation to zero. Thus the equation alters as follows:

Pressure energy + Kinetic energy = Total energy = A constant per unit weight

per unit weight

Bernoulli’s equation:

per unit weight

P1/ρg + V1 /2g = P2/ρg + V2 /2g

Continuity equation involves flow streams having a uniform velocity across the flow section of

a steady, incompressible, and one-dimensional flow. Continuity Equation:

=

A1V1 A2V2

Where, P/ρg: Pressure Energy 2

V /2g: Kinetic Energy A*V = flow rate of a fluid passing through a passage of cross sectional area A with velocity V (Q) A1: Cross Sectional Area of Venturimeter at point 1 A2: Cross Sectional Area of Venturimeter at point 2 V1: Velocity of fluid at point 1 V2: Velocity of fluid at point 2

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Objectives The aims of this experiment:

1. To determine the coefficient of discharge of meter and its variation with discharge. 2. To determine the variation of energy loss through the meter as a function of discharge and main pipe velocity.

Apparatus Apparatus needed: Venturi meter apparatus (Fig 1.2), stop watch, mass (2.5 Kg) Fig1.2 Piezometer position

Diameter (mm)

A B C D E F G H J K L

26.00 23.20 18.40 16.00 16.80 18.47 20.16 21.84 23.53 25.24 26.00

Table I: Diameter of crosssection of venture metre at different piezometer stations

Convergent-Divergent Venturi meter

A

B C

D

E F

G

H

J

K

L

26.00 mm 16.00 mm

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Description of apparatus & Experimental Procedures The apparatus consist of a flow bench that allows water to flow to the Venturimeter. Inside the flow bench there is a weighing tank connected to one end of a lever arm. The end of the lever arm protrudes from the side of the flow bench so that the amount of weight on this end of the lever arm may be adjusted (as shown in the photo below). The purpose of the lever arm is to measure the actual mass flow rate of water flowing through the measuring devices. When using the hydraulic bench, placing weight on the lever arm closes the trip valve of the inner tank. When water entering the tank is sufficiently heavy enough to counterbalance the weight on the arm, the arm will rise and the trip valve will open. Dividing, the mass of water contained in the tank, by the amount of time it takes for the internal tank to fill, will yield the actual mass flow rate. Since the adjustable weight end of the lever arm has a three-to-one advantage over the water tank end, the mass of the water in the tank will equal three times the mass added to the lever arm. The weight of the hanger is accounted for in the design of the equipment; therefore, do not add the weight of the hanger to weights placed on the hanger. The manometer scale is levelled by first opening both the control valve downstream of the meter and bench supply valve so as to flow for a few seconds to clean air pockets from the supply system. The control valve is adjusted to obtain a proper value on the lowest piezometer reading. The control valve is kept in positions so that the flow is constant and therefore the readings on the piezometer are noted. Having levelled the scales, measurements of a series of values of (h1-h2) and Q may be made, where (h1-h2) is the difference in the levels in manometer limbs A and D, and Q is the flow rate. The first reading may be taken at the maximum available value of (h2-h1), i.e. with h1 close to the top of the scale and h2 close to bottom, where h 2 is noted as the lowest value and the other piezometer readings as h1.

To measure the flow rate, water is collected in the weighing tank; the ratio of the weight of water in the tank to the load placed on the other edge of the pivoted beam is 3:1. The beam initially points downwards but when it becomes in equilibrium state (horizontal), the time (measured using a stop watch) is started and a 2.5 Kg load is placed on the hanger found on the edge of the beam.

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EXPERIMENT TO STUDY FLOW THROUGH A VENTURI METER The time, t, for which the beam becomes back again to equilibrium is recorded. For the 2.5 Kg load, 7.5 Kg of water collected on the other side of the beam in a laps of time, t seconds. The control valve is adjusted to obtain different flows and piezometer readings. The ideal pressure and the actual pressure at each piezometer position, for two values of Q, are calculated.

Tables are tabulated as follows: 1. Table 1: The values of diameter and area of cross sections of venturimeter 2. Table 2: Flow rate in each manometer in millimeters(convert in meters) 3. Table 3: The values of a n and ideal pressure for each section of the Venturimeter 2

4. Table 4: The values of Time, Volume Flow rate and u 2 /2g 5. Table 5: For Flow rate No 1, the values of h n, h1 and hhnn--hh11 , for each section of the Venturimeter.

u22/2g

6. Table 6: For Flow rate No 4, the values of h n, h1 and hhnn--hh11 , for each section of the Venturimeter.

u22/2g

7. Table 6: The values of h 1, h2, √h1-h2 and Q. 8. Table 8: The values of Volumetric Flow rate, Q, h 1, h2, and Coefficient of discharge, Cd.

Plot graph of y-axis against x-axis and draw the line/curve of best fit: 1. A graph of ‘Ideal Pressure’ is plotted against ‘Sections of Venturimeter’ 2. A graph of ‘Actual pressure’ is plotted against ‘Sections of Venturimeter’ for Flow rate No 1. 3. A graph of ‘Actual pressure’ is plotted against ‘Sections of Venturimeter’ for Flow rate No 4. 4. A graph of √h1-h2 is plotted against Q. 5. A graph of Cd is plotted against Q.

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Tabulation of results v

Calculation of area of the different cross sections of the Venturimeter 2

Area = ╥ x (diameter of the cross section of the venturimeter/2)

Where ╥ = 22/7 The values of diameter and area of cross sections of venturimeter are tabulated below. Table 1

v

2

Piezometer position

Diameter (m)

Area (m )x 10

A

0.02600

5.3093

B

0.02320

4.2273

C

0.01840

2.6590

D

0.01600

2.0106

E

0.01680

2.2167

F

0.01847

2.6793

G

0.02016

3.1921

H

0.02184

3.7462

J

0.02353

4.3484

K

0.02524

5.0034

L

0.02600

5.3093

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Theory of Venturimeter The discharge Q is given by Q = Cd a2 √ [2g (h1 - h2)] √1 – (a2/a1)2 Where Cd is the coefficient of discharge of the meter and that The ideal pressure is equal to 2

2

hn – h1 = (a2) - (a2) u2 2/2g (a1)2 (an)2

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Result and Analysis Table 2: Flow rate in each manometer in millimetres (convert in meters)

Manometer scale readings in millimetres Flow rate No Piezometer Position A(h1) B C D(h2) E F G H J K L v

1

2

3

4

5

6

7

8

9

10

219 218 210 192 196 198 208 205 210 214 212

224 220 205 180 195 204 204 205 204 212 219

225 220 200 168 174 189 199 202 207 211 203

230 225 195 148 154 179 193 200 201 202 214

235 226 188 132 172 188 188 199 205 210 204

237 228 185 120 134 168 187 196 205 214 216

241 232 180 105 121 159 184 196 206 203 217

243 234 176 94 112 154 179 195 204 213 218

246 237 173 81 103 150 176 194 205 213 218

249 238 169 67 93 142 173 192 204 214 219

Calculating the ideal pressure at each section of the Venturimeter

Using the equation,

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EXPERIMENT TO STUDY FLOW THROUGH A VENTURI METER Table 3: The values of a n and ideal pressure for each section of the Venturimeter are tabulated below. Ideal Pressure at each Piezometer Position

Cross sectional area of -4

Venturimeter, anx10 2

(m )

section of Venturimeter -1

-1

(Kg m s ) 2

2

(a2) - (a2) 2 2 (a1) (an)

A(a1)

5.3093

0

B

4.2273

-0.0828

C

2.6590

-0.4284

D(a2)

2.0106

-0.8566

E

2.2167

-0.6793

F

2.6793

-0.4197

G

3.1921

-0.2533

H

3.7462

-0.1446

J

4.3484

-0.0704

K

5.0034

-0.0181

L

5.3093

0

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EXPERIMENT TO STUDY FLOW THROUGH A VENTURI METER A graph of ‘Ideal Pressure,’ is plotted against ‘Section of Venturimeter, a n’

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EXPERIMENT TO STUDY FLOW THROUGH A VENTURI METER v

Calculating the actual pressure at each section of the Venturimeter

For the sake of accuracy near the maximum flow, we will be using the set of values for flow rate No 1 and flow rate No 4, to calculate the actual pressure at each section of the Venturimeter. Using the equation,

Actual pressure =

hn - h1 u2 /2g

2

Calculation of u2 /2g: U2 = Volume Flow rate /Area of cross sections of the Venturimeter at piezometer position D Volume Flow rate = Volume of water collected/Time taken to collect that volume Note: The volume of water collected is equal for all flow rate and is equal to -3

7.5 kg/1000 kgm = 0.0075 m

3

Density of water = 1000kg/m

Q=AV

Volume Flow rate = 0.0075 /Time taken to collect 0.0075m 3 of water

Therefore, 3

u2 = (0.0075/Time taken to collect 0.0075m of water) / 2.0106 x 10

-4

Thus, 2

3

-4 2

u2 /2g = [ (0.0075/Time taken to collect 0.0075m of water) / 2.0106 x 10 ] ÷ (2 x 9.810) Unit: m

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EXPERIMENT TO STUDY FLOW THROUGH A VENTURI METER 2

Table 4: The values of Time, Volume Flowrate and u 2 /2g are tabulated below .

Flow rate No

Time, t(s)

Volume Flow rate, -4 3 Q*10 (m /s)

(u2 /2g) m

1

47.13

1.5913431

0.031928

2

37.41

2.0048115

0.050675

3

33.25

2.2556391

0.064149

4

27.38

2.7392257

0.094603

5

26.17

2.8658769

0.103553

6

23.70

3.1645569

0.126263

7

21.79

3.4419458

0.149368

8

20.66

3.6302033

0.166154

9

19.35

3.8759689

0.189413

10

19.55

3.8363171

0.185558

2

hn - h1 Table 5: For Flow rate No 1, the values of h n, h1 and hn - h , for each section of the Venturimeter 21 u2 /2g

Note: For flow rate No 1, the value of

Piezometer Position

u 22/2g

= 0.031928

m

Actual Pressure h1 (m)

hn (m)

hn – h1 2

U2 /2g

A

0.219

0.219

0

B

0.219

0.218

-0.0313

C

0.219

0.210

-0.2819

D

0.219

0.192

-0.8457

E

0.219

0.196

-0.7204

F

0.219

0.198

-0.6577

G

0.219

0.208

-0.3445

H

0.219

0.205

-0.4385

J

0.219

0.210

-0.2819

K

0.219

0.214

-0.1566

L

0.219

0.212

-0.2192

Table 5

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EXPERIMENT TO STUDY FLOW THROUGH A VENTURI METER A graph of ‘Actual pressure’ is plotted against ‘Sections of Venturimeter’ for Flow rate No 1.

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EXPERIMENT TO STUDY FLOW THROUGH A VENTURI METER hn - h1

Table 6: For Flow rate No 4, the values of h n, h1 and hn - h1 , for each section of the Venturimeter.

u22/2g

2

Note: For flow rate No 4, the value of u2 /2g = 0.094603 m Piezometer Position

Actual Pressure h1 (m)

hn (m)

hn – h1 2

U2 /2g

A

0.230

0.230

0

B

0.230

0.225

-0.0529

C

0.230

0.195

-0.3700

D

0.230

0.148

-0.8668

E

0.230

0.154

-0.8034

F

0.230

0.179

-0.5391

G

0.230

0.193

-0.3911

H

0.230

0.200

-0.3171

J

0.230

0.201

-0.3065

K

0.230

0.202

-0.2960

L

0.230

0.214

-0.1691

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EXPERIMENT TO STUDY FLOW THROUGH A VENTURI METER A graph of ‘Actual pressure’ is plotted against ‘Sections of Venturimeter’ for Flow rate No 4.

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Calculation of the coefficient of discharge, Cd, of the Venturimeter, for each flowrate.

The above equation is put in the form of Y = mX

Where, Y = √h1-h2 X=Q

m= 1

√1-( a2 / a1)2

Cd

= gradient

a2 √ 2g

The values of h1, h2, √h1-h2 and Q are tabulated below. Note: h1 is the actual piezometer reading at position A

h2 is the actual piezometer reading at position D Table 7

Flow rate No

h1 (m)

h2 (m)

h1-h2

1

0.219

0.192

0.1643

1.591

2

0.224

0.180

0.2098

2.005

3

0.225

0.168

0.2387

2.256

4

0.230

0.148

0.2864

2.739

5

0.235

0.132

0.3209

2.866

6

0.237

0.120

0.3421

3.164

7

0.241

0.105

0.3688

3.442

8

0.243

0.094

0.3860

3.630

9

0.246

0.081

0.4062

3.876

10

0.249

0.067

0.4266

3.836

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Volume flow rate, -4 3 Q*10 (m /s)

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EXPERIMENT TO STUDY FLOW THROUGH A VENTURI METER A graph of √h1-h2 is plotted against Q

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EXPERIMENT TO STUDY FLOW THROUGH A VENTURI METER Using the graph, the gradient of the plotted line is calculated.

CALCULATION From the equation,

(X2, Y2) - (4.00 x 10 -4, 0.424) -4

(X1, Y1) - (1.5x10 , 0.152) M = (0.424 - 0.152) / (4.00 x 10

-4

-4

- 1.5x10 )

= 1088 1

√1-( a

2/

a1)2

= gradient of plotted line

a2 √ 2g

Cd Then,

1 Cd

√1-( a2 / a1)2 a2 √ 2g

= 1088

Where, Cd is the coefficient of discharge a1 is the cross sectional area of Venturimeter at piezometer position A = 5.3093 x 10 -4 m2 a2 is the cross sectional area of Venturimeter at piezometer position D = 2.0106 x 10 -4 m2 g is the acceleration due to gravity = 9.81ms -2 1

x 1039.235 = 1088

Cd Cd = 0.95518 Cd ≈ 0.96 (2 S.F)

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Calculation of the value of coefficient of discharge, Cd for each value of volumetric flow rate, Q

Q=Cda2√2g(h1-h2)  √1-(a2/a1)

2

Therefore, Cd=Qx√1-(a 2/a1)

2

a 2√2g(h 1-h2)

Where, a1 is the cross sectional area of Venturimeter at piezometer position A = 5.3093 x 10

-4

m

2

a2 is the cross sectional area of Venturimeter at piezometer position D = 2.0106 x 10

-4

m

2

h1 is the actual piezometer reading at position A h2 is the actual piezometer reading at position D Table 8: The values of Volumetric Flow rate, Q; h 1; h2; and Coefficient of discharge, Cd are

tabulated below. Flow rate No

Volumetric Flow -4 3 rate, Q x 10 (m /s)

h1 (m)

h2 (m)

Coefficient of discharge, Cd

1

1.591

0.219

0.192

1.006

2

2.005

0.224

0.180

0.993

3

2.256

0.225

0.168

0.982

4

2.739

0.230

0.148

0.994

5

2.866

0.235

0.132

0.928

6

3.164

0.237

0.120

0.961

7

3.442

0.241

0.105

0.969

8

3.630

0.243

0.094

0.981

9

3.876

0.246

0.081

0.992

10

3.836

0.249

0.067

0.935

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EXPERIMENT TO STUDY FLOW THROUGH A VENTURI METER A graph of Cd is plotted against Q

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Conclusions The shapes of the graphs of ideal and actual pressures are not similar. There is a large difference between the values of ideal pressure and actual pressure. One possible reason for that difference is that in practice there are head losses which are partly due to friction and partly due to shock caused by a change of section. Another reason is that the venturimeter is not accurate for low velocities. The value of the coefficient of friction Cd ranges from 0.6-0.9. The graph of C d against Q is not a straight line and this shows that the coefficient of discharge is not constant. The coefficient Cd will have a different value for the converging and diverging cones of the meter. The flow rate is independent of the inclination of the meter. The venturimeter has a coefficient of discharge of 0.91, which means that the actual flow rate if 91% of the theoretical flow rate. Therefore, this value of discharge is a very good estimation for flow rates and this is why venturimeters are commonly used to measure flow in pipes.

The graph of ideal and actual pressure against for Q1 and Q2 shows that the actual fluid pressure is always smaller that the ideal flow rate. The difference in pressure between ideal and actual becomes greater as we move along the length of the piezometer tubes. This is so because of greater energy losses along the length of the tube, occurring due to turbulence at the diverging section and due to greater frictional losses with the inner walls of the tube. Energy losses accompanying the change of flow pattern within the converging section is lower than the diverging section. Moreover, at the lowest pressures (at the trough) as the graph goes down to its lowest values, the pressure is lower since the cross-sectional areas of the tubes are lower and the velocities are high. The graph of √ (h1-h2) against the flow rate Q is a straight line, meaning that these two variables are proportional to each other. This confirms the validity of the flow rate equation used before. The graph of C d against flow rate Q is a curve showing the variation of C d with an increasing flow rate.

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EXPERIMENT TO STUDY FLOW THROUGH A VENTURI METER In this experiment, a horizontal venturimeter was used and the reference points were all at the same elevation with respect to any horizontal datum. Therefore, the elevation head is similar and cancel out in Bernoulli’s equation (as used before) and is not included for calculations. If an inclined venturimeter was used, the elevation would have been different and hence the elevation head would have to be included in the calculations. The actual and theoretical discharge will both decrease since energy is used (or lost) to move the fluid to a higher elevation along the venturimeter and is converted to potential energy. The readings on the piezometer tubes will be lower from A to L than obtained here.

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References •

Hydraulics and Fluid Mechanics by E.H. Lewitt

•

http://www.cwru.edu

•

fie.engrng.pitt.edu/fie2003/papers/1086.pdf

•

www.ambalalab.com/engg_1024_6.htm

•

www.ngu.ac.in/academic/engineering/ME402.htm

•

Fluid Mechanics Lab Manual

•

http://www.loai-naamani.com/Academics/Venturi_Meter.htm

•

http://webpages.eng.wayne.edu/~aa6287/ce3250/

•

The University of Edinburgh School of Engineering and Electronics, Fluid Mechanics 3 Flow Measurement Methods

•

Fluids mechanics third edition, J.F Douglas, J.M Gasiorek, J.A Swaffield

•

www.ngu.ac.in/academic/engineering/ME402.htm

•

http://www.loai-naamani.com/Academics/Venturi_Meter.htm

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Experiment to Study Flow Through a Venturimeter

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