Bernoulli equation for venturi meter

EGME-306B-06

Spring 2015

Engineering Report

EXPERIMENT #1

Flow Through A Venturi Meter

Using Water As The Working Fluid

(Ref. Experimental Data Group 02- Taken on Feb 6, 2015)

By

02/20/2015

TABLE OF CONTENTS

List of Symbols………...…………………………………………………………………………………………2

Abstract…………………………………………………………………………………………................3

Procedure………………………………………………………………………………………………….4

Theory……………………………………………………………………………………………………..5

Results……………………………………………………………………………………………………10

Sample Calculations………………………………………………………………………………………………18

Error Analysis…………………………………………………………………………………………………..22

Discussion………………………………………………………………………………………………..26

References………………………………………………………………………………………………..28

LIST OF SYMBOLS

………………………………………………………………………………………..Mass flow rate ()

……………………………………………………………………………………………..flow rate ()

………………………………………………………………………………………...Mass density ()

……………………………………………………………………………………………...…gravity ()

……………………………………………………………………………………..…inlet pressure ()

………………………………………………………………………………………….inlet velocity ()

…………………………………………………………………………………………...inlet area ()

………………………………………………………………...inlet throat height from datum point ()

………………………………………………………………………………………outlet pressure ()

………………………………………………………………………………………...outlet velocity ()

………………………………………………………………………………………….outlet area ()

……………………………………………………………….outlet throat height from datum point ()

C…………………………………………………….………………discharge coefficient (dimensionless)

c………………………………………………………………………………………...speed of sound ()

Re……………………………………………………………………...Reynold’s number (dimensionless)

M…………………………………………………………………………...Mach number (dimensionless)

ABSTRACT

The motivation of this laboratory experiment is to analyze the flow of a fluid passing through a Venturi meter. The scheme is to compare the experimental volumetric flow rate to the theoretical volumetric flow rate. Deriving the Venturi velocity formula from the Bernoulli and continuity equations attains the theoretical volumetric flow rate.

It was found that the assumption of a frictionless surface through the Venturi meter was incorrect. When the water flows there is a head loss, which causes dissipation of energy throughout the system. Other factors such as human error, air bubbles in the fluid, and calculation round-offs led to discrepancies in the data.

RESULTS SUMMARY

Table 1: Pressure in piezometer tube and distance the fluid elevated to in Venturi meter

Venturi data and Piezometer Station

Piezo tube H2O height at station X , hX in.± .1in.

Distance from end, X in.

Venturi diameter D in. at X

Piezo. Station

Run 1

Run 2

Run 3

Run 4

Run 5

Run 6

hA-hD≈10 inH20

hA-hD≈8 inH20

hA-hD≈6 inH20

hA-hD≈4 inH20

hA-hD≈2 inH20

hA-hD≈.6 inH20

0

1

A-

9.3

8.8

8.0

7.0

6.1

5.5

1.125

1

A

9.3

8.8

8.0

7.0

6.1

5.5

1.625

1

A+

9.3

8.8

8.0

7.0

6.1

5.5

1.875

0.906

B

8.9

8.3

7.7

6.9

6.0

5.4

2.375

0.719

C

5.4

5.3

5.4

5.3

5.3

5.2

2.625

0.6445

D-

0.2

0.8

1.9

3.0

4.1

4.9

2.9375

0.6445

D

0.2

0.8

1.9

3.0

4.1

4.9

3.25

0.6445

D+

0.2

0.8

1.9

3.0

4.1

4.9

3.5

0.652

E

0.7

1.2

2.2

3.1

4.2

4.9

3.875

0.705

F

3.4

3.6

3.9

4.3

4.7

5.0

4.5

0.759

G

4.9

4.9

5.0

5.1

5.1

5.1

5

0.813

H

5.9

5.9

5.6

5.5

5.3

5.2

5.5

0.866

K

6.7

6.4

6.1

5.8

5.5

5.5

6

0.92

L

7.1

6.9

6.5

6.0

5.6

5.3

6.75

1

M-

7.8

7.4

6.9

6.3

5.7

5.3

6.875

1

M

7.8

7.4

6.9

6.3

5.7

5.3

7.875

1

M+

7.8

7.4

6.9

6.3

5.7

5.3

Graph 1: Piezometer tube versus distance of fluid in Venturi meter

Graph 2: Volume flow rate versus the square root of the height difference

Graph 3: Ideal Volume Flow Rate versus Discharge Coefficient

Graph 4: Free Stream and Throat velocity versus actual volume flow rate

Graph 5: Free stream and throat Mach number versus actual volume flow rate

Graph 6: Free stream and throat Mach number versus actual volume flow rate

DISCUSSION AND CONCLUSION

The Venturi meter is one of the most efficient systems with minimal error for use in experimentation surfaces. In comparison to the sharp-edged orifice of flow nozzle, the Venturi meter shows the most minimal head loss caused by friction and heat.

Analyzing the data acquired, the values attained for velocity and volumetric flow rate are in correlation with the Venturi meter experiment. There were slight deviations that occurred in conducting the experiment due to realistic conditions; thus, compared to the calculated values where ideal conditions were taken into account. The factors such as heat loss and friction affected the system and prevented the ideal-condition results. Another factor affecting the measurements is the readings taken from the manometer are the students’ readings, they may not have been as accurate as possible which have resulted in a percentage error.

Contained in the “Results” section the dimensionless measurements of Reynolds number and the discharge coefficient show the Reynolds number increases and so does the discharge coefficient. This concludes that the larger the cross-sectional area the more efficient the system becomes, this is true for a certain ratio.

Errors cannot be completely avoided therefore a margin of error is expected given the high possibility of factors affecting error. In our attempt to eliminate all air bubbles, it appeared as if they were all gone but our visual judgment only goes so far, causing small errors in data collected. Other errors arise in the contents of the water such as impurities; even though the water was distilled, the distillations might not have been 100% effective. Due to the minimal amount of theoretical background, an error in precision is inevitable. Calculation round-offs are another contributing factor in our error margin.

APPENDICES

APPENDIX A

EXPERIMENTAL DATA

APPENDIX B

THEORY

A Venturi meter is an instrument used to measure the flow rate through a pipe. The design includes divergent and convergent sections, which change the flow velocity of the fluid for experimental purposes. Below, in figure one, the setup for the experiment includes two tanks, one for the fluid storage, and the metal tank is for the collected fluid, which has passed through the Venturi meter and pipe.

Venturi meters are commonly preferred over other meters due to their minimal head loss corresponding to its streamline design. In the system, there is no loss of mass and the fluid is in steady flow throughout the experiment, therefore, the conservation of mass equation can be applied to the Venturi meter experiment:

Figure 2 – Venturi Meter

The mass flow at i and j are equal. Water is considered incompressible; therefore, the equation can be simplified:

From the equation it can be concluded that velocity decreases and increases at the diverging and converging sections, respectively, compensating for the continuity of mass.

To calculate pressure, piezometer tubes are integrated in the Venturi meter system. Using the Bernoulli’s equation, for steady, incompressible frictionless flow along a streamline gives a relationship between the constant measure of total energy in a system with static pressure, kinetic energy, and potential energy of a flow. Consider a fluid flowing through the pipe having reached equilibrium state, the fluid in the piezometer tube is a non-flow static condition, velocity equals zero. Under these conditions, the Bernoulli equation reduces to:

The expression gives the relationship between the change in pressure and change in elevation of a fluid within the tube. Utilizing the results achieved for the expressions for mass continuity and hydrostatic balance on a fluid column, and the linear momentum equation, the velocities for the Venturi meter at each station can be calculated. The linear momentum equation is:

All Venturi meters part of the experiment had an equal elevation; therefore, the equation can be simplified.

And since,

Then,

APPENDIX C

PROCEDURE

The equipment of the experiment consists of a flow bench, which allows water to flow through the Venturi meter. Under the flow bench a weighing tank is attached to one end of a lever where weight is added in order to calculate mass flow rate. Before beginning the experiment, the apparatus was calibrated by the lab technician. Air bubbles inside the system can cause discrepancies in the data; therefore, the flow control and bench supply valves were slightly opened by the technician to allow the water to flow and eliminate the air bubbles within the system.

In the next step we closed the apparatus flow control valve and opened the air purge valve on the manifold to allow the water to rise into the piezometer tube. The height of the water was recorded on our data sheets. Each team member was assigned a task prior to conducting the experiment; this allowed the experiment to flow smoothly. Once everyone was ready, the apparatus flow control valve was opened all the way.

The timer was started when the valve was opened, when the arm holding the weight rose, time was recorded and 4 lbs. were added each time until a total of 20 lbs. Knowing the amount of water it took to fill the tank and the time elapsed, we were able to determine the mass flow rate. At the end of the trial, the new piezometer tube pressures were carefully noted and the experiment was conducted again, six times. Different flow rates were collected at each trial.

APPENDIX D

CALCULATIONS

APPENDIX E

ERROR ANALYSIS

APPENDIX F

REFERENCES

1. Engineering 306B –Unified Laboratory Manual

2. Fox and McDonald’s Introduction to Fluid Mechanics (8th Edition)

3. http://en.wikipedia.org/wiki/Venturi_effect

4. http://www.hendersons.co.uk/wms/venturi_principle.html

Pressure vs. Distance

Run 1 0.0 1.125 1.625 1.875 2.375 2.625 2.9375 3.25 3.5 3.875 4.5 5.0 5.5 6.0 6.75 6.875 7.875 9.3 9.3 9.3 8.9 5.4 0.2 0.2 0.2 0.7 3.4 4.9 5.9 6.7 7.1 7.8 7.8 7.8 Run 2 0.0 1.125 1.625 1.875 2.375 2.625 2.9375 3.25 3.5 3.875 4.5 5.0 5.5 6.0 6.75 6.875 7.875 8.8 8.8 8.8 8.3 5.3 0.8 0.8 0.8 1.2 3.6 4.9 5.9 6.4 6.9 7.4 7.4 7.4 Run 3 0.0 1.125 1.625 1.875 2.375 2.625 2.9375 3.25 3.5 3.875 4.5 5.0 5.5 6.0 6.75 6.875 7.875 8.0 8.0 8.0 7.7 5.4 1.9 1.9 1.9 2.2 3.9 5.0 5.6 6.1 6.5 6.9 6.9 6.9 Run 4 0.0 1.125 1.625 1.875 2.375 2.625 2.9375 3.25 3.5 3.875 4.5 5.0 5.5 6.0 6.75 6.875 7.875 7.0 7.0 7.0 6.9 5.3 3.0 3.0 3.0 3.1 4.3 5.1 5.5 5.8 6.0 6.3 6.3 6.3 Run 5 0.0 1.125 1.625 1.875 2.375 2.625 2.9375 3.25 3.5 3.875 4.5 5.0 5.5 6.0 6.75 6.875 7.875 6.1 6.1 6.1 6.0 5.3 4.1 4.1 4.1 4.2 4.7 5.1 5.3 5.5 5.6 5.7 5.7 5.7 Run 6 0.0 1.125 1.625 1.875 2.375 2.625 2.9375 3.25 3.5 3.875 4.5 5.0 5.5 6.0 6.75 6.875 7.875 5.5 5.5 5.5 5.4 5.2 4.9 4.9 4.9 4.9 5.0 5.1 5.2 5.5 5.3 5.3 5.3 5.3
Distance (in.)

Pressure (psi)