experiment to study flow through a venturimeter

EXPERIMENT TO STUDY FLOW THROUGH A VENTURI METER

FLUIDS MECHANICS1 1

HYDRAULICS PRACTICALS

FLUIDS MECHANICS LABORATORY

MODULE: FLUIDS MECHANICS1

REPORT OF EXPERIMENT:



FLUIDS MECHANICS1 2

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, an’

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, Cd, 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


FLUIDS MECHANICS1 3

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


FLUIDS MECHANICS1 4

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, (Cd) 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)2


FLUIDS MECHANICS1 5

Where Cd is the coefficient of discharge of the meter and that

The ideal pressure is equal to

hn – h1 = (a2)2 - (a2)2

u2 2/2g (a1)2 (an)2

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 m2

a2: the cross sectional area of Venturimeter at piezometer position D = 0.020106 m2

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

determined

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

u22/2g : head loss at piezometer position D

Actual pressure = hn - h1

u22/2g


FLUIDS MECHANICS1 6

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.

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 per unit weight

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

a steady, incompressible, and one-dimensional flow.

Where,

P/ρg: Pressure Energy

V2/2g: 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

Pressure energy per unit weight

Potential energy per unit weight

Total energy per unit weight

a constant Kinetic energy per unit weight

+ + = =

Bernoulli’s equation: P1/ρg + V12/2g = P2/ρg + V2

2/2g

Continuity Equation: A1V1 =

A2V2


FLUIDS MECHANICS1 7

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)

Piezometer position

Diameter (mm)

A 26.00 B 23.20 C 18.40 D 16.00 E 16.80 F 18.47 G 20.16 H 21.84 J 23.53 K 25.24 L 26.00

Mass hanger

piezometers

A B C D E F G H J K L

26.00 mm 16.00 mm

Table I: Diameter of cross-section of venture metre at

different piezometer stations

Fig 1.2

Convergent-Divergent Venturi meter


FLUIDS MECHANICS1 8

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 h2 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.


FLUIDS MECHANICS1 9

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 an and ideal pressure for each section of the Venturimeter

4. Table 4: The values of Time, Volume Flow rate and u22/2g

5. Table 5: For Flow rate No 1, the values of hn, h1 and hn - h1 , for each section of the

Venturimeter.

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

Venturimeter.

7. Table 6: The values of h1, h2, √h1-h2 and Q.

8. Table 8: The values of Volumetric Flow rate, Q, h1, 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.

hn - h1

u22/2g

hn - h1

u22/2g


FLUIDS MECHANICS1 10

Tabulation of results

v Calculation of area of the different cross sections of the Venturimeter Area = ╥ x (diameter of the cross section of the venturimeter/2)2

Where ╥ = 22/7

The values of diameter and area of cross sections of venturimeter are tabulated below. Table 1 v 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 hn – h1 = (a2)2 - (a2)2

u2 2/2g (a1)2 (an)2

Piezometer position Diameter (m) Area (m2)x 10-4

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



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

v Calculating the ideal pressure at each section of the Venturimeter

Using the equation,

Manometer scale readings in millimetres Flow rate

No 1

2

3

4

5

6

7

8

9

10 Piezometer

Position A(h1) 219 224 225 230 235 237 241 243 246 249

B 218 220 220 225 226 228 232 234 237 238 C 210 205 200 195 188 185 180 176 173 169

D(h2) 192 180 168 148 132 120 105 94 81 67 E 196 195 174 154 172 134 121 112 103 93 F 198 204 189 179 188 168 159 154 150 142 G 208 204 199 193 188 187 184 179 176 173 H 205 205 202 200 199 196 196 195 194 192 J 210 204 207 201 205 205 206 204 205 204 K 214 212 211 202 210 214 203 213 213 214 L 212 219 203 214 204 216 217 218 218 219



Table 3: The values of an and ideal pressure for each section of the Venturimeter are tabulated

below.

Piezometer

Position

Cross sectional area of

Venturimeter, anx10-4

(m2)

Ideal Pressure at each

section of

Venturimeter

(Kg m-1 s-1)

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

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



A graph of ‘Ideal Pressure,’ is plotted against ‘Section of Venturimeter, an’



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,

Calculation of u2

2/2g:

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

7.5 kg/1000 kgm-3 = 0.0075 m3

Volume Flow rate = 0.0075 /Time taken to collect 0.0075m3 of water

Therefore,

u2 = (0.0075/Time taken to collect 0.0075m3 of water) / 2.0106 x 10-4

Thus,

u22/2g = [ (0.0075/Time taken to collect 0.0075m3 of water) / 2.0106 x 10-4]2 ÷ (2 x 9.810)

Unit: m

Actual pressure = hn - h1

u22/2g

Density of water = 1000kg/m3

Q=AV



Table 4: The values of Time, Volume Flowrate and u22/2g are tabulated below.

Flow rate No Time, t(s) Volume Flow rate,

Q*10-4 (m3/s) (u22/2g) 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

Table 5: For Flow rate No 1, the values of hn, h1 and hn - h1 , for each section of the Venturimeter

Note: For flow rate No 1, the value of u22/2g = 0.031928 m

Piezometer

Position h1 (m) hn (m)

Actual Pressure

hn – h1

U22/2g

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

hn - h1

u22/2g



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



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

Note: For flow rate No 4, the value of u22/2g = 0.094603 m

Piezometer

Position h1 (m) hn (m)

Actual Pressure

hn – h1

U22/2g

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

hn - h1

u22/2g



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



v 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 = gradient

Cd 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

Volume flow rate, Q*10-4 (m3/s)

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



A graph of √h1-h2 is plotted against Q



Using the graph, the gradient of the plotted line is calculated.

CALCULATION

From the equation,

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

(X1, Y1) - (1.5x10-4, 0.152)

M = (0.424 - 0.152) / (4.00 x 10 -4 - 1.5x10-4)

= 1088

1 √1-( a2 / a1)2 = gradient of plotted line

Cd a2 √ 2g

Then,

1 √1-( a2 / a1)2 = 1088 Cd a2 √ 2g

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)



v 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-‐( a2 / a1)2

a2 √ 2g (h1-‐h2)

Where,

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

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; h1; h2; and Coefficient of discharge, Cd are tabulated below.

Flow rate No

Volumetric Flow rate, Q x 10-4 (m3/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



A graph of Cd is plotted against Q



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 Cd 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 Cd against flow rate Q is a curve showing the variation of Cd with an increasing flow

rate.



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.



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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