Experiment to Verify Bernoulli's Principle

University of Mauritius FACULTY OF ENGINEERING CIVIL ENGINEERING DEPARTMENT B Eng (Hons) Civil Engineering-2008/2012 Engineering-2008/2012 Year 2- Semester 1 CIVE 2003Y Fluid Mechanics Assignment: Laboratory Report

Lecturer: Ms. Manta Nowbuth Submitted by: by: JUTTUN Ghanisha Devi (ID: 0812567) Date of submission: 29.12.09

Experiment to study flow a Venturi Meter Introduction The Bernoulli’s principle states that as the velocity of a fluid increases, its pressure decreases. Bernoulli’s principle can be demonstrated by the Bernoulli equation. However, Bernoulli’s principle can only be applied under certain conditions. Massey (2006 p.94) states that: “...the conditions to which Bernoulli’s equation applies are: the fluid must be frictionless (inviscid) and of constant density; the flow must be steady, and the relation holds in general for single streamlines” These are therefore assumptions that are made.

Objective The objective of the experiment is to verify the Bernoulli’s equation which relates the pressure energy, kinetic energy and potential energy per unit weight from point to point in the fluid. The equation is based upon the law of conservation of energy below. Douglas et al (2005 p.169)

Pressure energy + Kinetic energy + Potential energy = Total energy = a constant per unit weight per unit weight per unit weight per unit weight

In this experiment water is made to flow in a Perspex tank in which the centreline of the cross sections lies all on the same horizontal plane. 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 to be verified is as follows: Pressure energy + Kinetic energy = Total energy = A constant per unit weight per unit weight per unit weight

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P/w + V  /2g = A constant Where 1. w: ρg 2. P/w is the pressure head (i.e. – the height of the water in the Piezometer tube) 2 3. V  /2g is the velocity head The heads are expressed in mm of water since velocities of flow are small.

Methodology  To carry out the experiment, water was allowed to flow in the Perspex tank and

the outflow was adjusted so that the water levels in the piezometers are steady.  Piezometer readings for pressure head are taken 

The water is allowed to flow into the collecting vessel for a given time period, t seconds, and the height, h, of water collected in the measuring tank is recorded.

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The area of the tank, A is calculated.

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The velocity, V, of flow at any section is then obtained by dividing the following equation (A * h/t), by the area of the converging tube at that section.

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The experiment was repeated with a lower head h in the tank that is with a lower flow rate of fluid.

Calculation and Results 2

The calculation for V  /2g is done as follows: 2

V=576.39mm 2

g= acceleration due to gravity = 9810 mm  /s 2

V  /2g= 576.39  / (2x9810) = 17mm The same procedure is followed for all the other values of velocity for all the piezometers. 2

The calculation for P/w + V  /2g is done as follows: It is done by adding the piezometer reading to the corresponding velocity head in the table. The results are tabulated as follows: Section No. 2 Area (mm ) Peizometer Reading (mm) V (mm/s) 2 V  /2g (mm) 2 P/w + V   /2g (mm)

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1 300 418

2 275 414

576.39

3 250 409

4 225 401

5 200 395

6 175 282

7 150 365

618.98 680.87 756.53 851.10 972.68 1134.79 17 20 24 29 37 48 66 435 434 433 430 432 330 431

REPEATED EXPERIMENT

The experiment is repeated for a lower head in the tank 

Section No. 2 Area (mm ) Peizometer Reading (mm) V (mm/s) 2 V  /2g (mm) 2 P/w + V   /2g (mm)

1 300 268 494.16

2 3 4 5 6 7 275 250 225 200 175 150 265 261 256 249 240 226 539 592.98 658.87 741.23 847.13 988.31 12 15 18 22 28 37 50 280 280 279 278 277 277 276

Data Analysis 1. A graph of P/w is plotted against Section No.

From the plots it can be seen that there is an overall decrease in pressure head with decrease in cross sectional area of the piezometers. This is demonstrated clearly in the lower head h series. In the higher head series however, there is a sudden decrease in the pressure head at piezometer section no. 6.

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2. A graph of V  /2g is plotted against Section No.

It can be seen that the velocity head at each piezometer section is increasing with decrease in the cross sectional area of the piezometers according to the equation Q = AV while the flow velocity in the tank is being kept nearly constant. Q= discharge A= cross sectional area (of the piezometers) V= velocity of fluid at the piezometer

Upon comparison of the two plots of velocity head and pressure head above, it can be observed that when the velocity in a fluid increases the pressure decreases.

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3. A graph of (P/w + V  /2g) is plotted against Section No.

From the graph plotted for the lower head series it can be seen that the total energy per unit weight of the fluid remains more or less constant along the flow at the piezometer sections. However for the higher head series it can be seen that the total energy per unit head drops suddenly at section no. 6.

Discussion Upon comparison of the two plots of velocity head and pressure head above, it can be observed that when the velocity in a fluid increases the pressure decreases. Therefore it can be said that the Bernoulli’s equation and principle are both valid.

At the piezometer section no. 6 it can be observed that the pressure head has dropped abnormally low. This may be attributed to an experimental error. Under ideal conditions the total energy per unit weight of the fluid at each piezometer section should have been constant. However in practice they are not as shown in the experiment carried out. This may be because the flow rate causes formation of “eddies” which are fluid currents whose prevalence increase with increase in fluid flow velocity due to friction with the boundary causing loss of energy. Also there are assumptions that have been made to create idealistic conditions for Bernoulli’s principle to apply which could not be fully reproduced in this experiment. For example the formation of eddy currents indicate that the flow has not been completely steady as the eddies’ velocity fluctuates with time. In fact a high prevalence of “eddies” is associated with turbulent flow. However modifications could have been made on the apparatus to ensure a steady flow of  fluid. The coefficient of friction at the boundary could have been reduced by coating with a near-frictionless material for example carbon-coating. Also the diameter of the tube carrying the liquid from the perspex tank to the venturimeter can be made smaller to have a steady flow at higher velocities of fluid flow.

References th

Massey, B. (2006) Mechanics of Fluid. 8 ed. Taylor and Francis Press. New York, USA. Douglas, J. F., Gasiorek J.M, Swaffield J.A & Jack L.B. (2005) Fluid Mechanics. 5 Pearson Prentice Hall. Harlow, Essex, England.

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