iii

Table of Contents 1.0. INTRODUCTION ........................................................................................................... 1

1.1. AIMS AND OBJECTIVES ......................................................................................... 2

1.2. LITERATURE REVIEW ............................................................................................ 2

2.0. METHODOLOGY .......................................................................................................... 5

2.1. EXPERIMENTAL APPARATUS .............................................................................. 5

2.2. PROCEDURE ............................................................................................................. 6

3.0. DATA COLLECTION AND ANALYSIS ..................................................................... 7

4.0. DISCUSSION ................................................................................................................. 8

4.1. SOURCES OF ERROR ............................................................................................... 8

4.2. PRECAUTIONS .......................................................................................................... 9

4.3. IMPROVEMENTS ...................................................................................................... 9

5.0. RISK ASSESSMENT ................................................................................................... 10

6.0. CONCLUSION ............................................................................................................. 11

7.0. REFERENCES .............................................................................................................. 12

APPENDIX A: Derivation of formulas ................................................................................... 13

iv

LIST OF TABLES

Table 1: Tabulated recorded and calculated data ....................................................................... 7

LIST OF FIGURES

Figure 1: True orifice flow and features of orifice ..................................................................... 1

Figure 2: Observed jet during experiment .................................................................................. 2

Figure 3: Tank with small orifice in the side ............................................................................. 3

Figure 4: x and y distances on the jet ......................................................................................... 4

Figure 5: Head measurement through transparent tube.............................................................. 6

Figure 6: Diameter measurement of orifice using Vernier Calliper ........................................... 6

Figure 7: Dangling wires .......................................................................................................... 10

Figure 8:Slippery floor ............................................................................................................. 10

Figure 9:Tank with smallorifice in the side ............................................................................. 13

Figure 10: x and y distances on the jet ..................................................................................... 14

v

ABSTRACT

In this report it has been attempted to determine the coefficient of discharge, coefficient of

velocity and coefficient of contraction. This has been achieved by studying the flow through

an orifice found in the side of a tank and collecting the discharge. The results obtained have

been compared with known values for the coefficients which have been established from

previous experiments less affected by errors. The results have not been to the expectations as

they have exceeded these values. Flawed apparatus have been found to be the subject of most

of the errors which have been deduced to be systematic. All in all the results have been found

to be precise but inaccurate.

1

1.0. INTRODUCTION

An orifice is an opening which can be characterised by its shape or type of edge in the side or

base of a tank or reservoir through which a fluid can be freely discharged or discharged in a

submerged environment. True orifice flow occurs only when the upstream water level is well

above the orifice to reduce vortex flow through air entrainment. If this level drops below the

top of the opening, the discharge has to be considered as being through a weir (n.d.). A small

orifice is one which has a diameter or vertical dimension much smaller than the head causing

the flow so as there is negligible head between point to point along the orifice (Douglas et al).

The figure below shows the orifice characteristics and true orifice flow.

Figure 1: True orifice flow and features of orifice

(Douglas et al, 2005, pg 188-189)

In this experiment, a tank having a circular orifice has been considered. The orifice has a

sharp edge with the bevelled side facing outward. An orifice usually has a sharp edge so that

the fluid has minimum contact with the fluid and reduced frictional resistance at the sides of

the orifice. If sharp edges are not provided, the thickness of the orifice and its roughness

together affect the flow. A free jet has been considered and as such it occurs in air and is

acted upon by gravity. When a jet is surrounded with fluid, it is called a submerged jet. A

submerged jet has no gravity acting upon it when the surrounding fluid is the same.

2

1.1. AIMS AND OBJECTIVES

The aims of the experiment were to determine for a circular sharp edged orifice:

The coefficient of velocity,

The coefficient of contraction,

The coefficient of discharge

The objectives of the experiment were to:

Maintain a constant head for each experiment

Measure the x and y distances

Measure the discharge

1.2. LITERATURE REVIEW

When the fluid flows from the upstream level to the orifice there is an increase in velocity and

it will contract from all angles towards the orifice. Consequently the streamlines have a

perpendicular velocity directed towards the centre of the orifice causing the emerging jet to

contract resulting in it having a smaller diameter than the orifice it passed through. The

contraction continues till its maximum at the vena contracta where all the streamlines are

assumed to be horizontal (Spencer, 2013). Beyond the vena contracta, friction with the air

(fluid) slows down the jet and the cross sectional area increases per force. However this

divergence is negligible and the jet is considered as almost cylindrical with a constant

velocity. The surface tension maintains the jet, which has a stronger effect the smaller the

diameter of the jet. The figure below shows the observed jet during experiment.

Figure 2: Observed jet during experiment

3

The figure below shows a tank with a small orifice in the side. Calculations are based by

taking the datum at the centre of the orifice

Figure 3: Tank with small orifice in the side

(Douglas et al, 2005, pg. 188)

Applying the principle of conservation of energy in the form of the Bernoulli’s equation at A

and B, the velocity at the vena contracta can be obtained. Full derivation of the equations can

be found in appendix A.

……. Bernoulli’s equation

√ ……..velocity at vena contracta

This is the Torricelli’s theorem which states that the velocity of the issuing jet is proportional

to the square root of the head.

Due to energy losses, the coefficient of velocity is included in the equation and is found from

theory as:

√

4

Since the actual velocity, V, cannot be accurately measured, an alternate solution is used. The

figure below illustrates the method of calculating the coefficient of velocity used in the

experiment.

Figure 4: x and y distances on the jet

(Douglas et al, 2005, pg. 191)

From,

…….(1)

…….(2)

√

The coefficient of discharge is obtained from

√

Where A is area of orifice

The coefficient of contraction is obtained from the equation

5

Rearrangement gives

It should be noted that practically, the area at the vena contracta cannot be measured as it

cannot be accurately determined. Full derivation of the equations can be found in appendix

A.

Previous experiments have shown typical values for coefficient of velocity to be in the order

of 0.96. For the coefficient of discharge it is in the order of 0.62 and for the coefficient of

contraction it is about 0.64.

2.0. METHODOLOGY

2.1. EXPERIMENTAL APPARATUS

Tank with an orifice in the side

Channel canal

Collector tank

Stopwatch

Vernier calliper

Ruler

6

2.2. PROCEDURE

1) A constant head of H m was maintained above the centre line of the orifice by

adjusting the flow from the water supply using the tap as shown below.

Figure 5: Head measurement through transparent tube

2) After reaching steady conditions the discharge from the orifice was collected over a

convenient time measured using a stopwatch and the initial and final height, Z1 and Z2,

during this time of the collected discharge in the discharge tank was noted.

3) The vertical fall, y, of the centre-line of the jet over a horizontal distance x was

obtained by means of a hook gauge and scale.

4) The procedure was repeated for different heads H and the values tabulated.

5) The diameter of the orifice was measured by means of a Vernier calliper as shown

below.

Figure 6: Diameter measurement of orifice using Vernier Calliper

7

3.0. DATA COLLECTION AND ANALYSIS

Tank dimensions

Length = 0.91 m

Width = 0.66 m

Area = 0.6006 m2

Diameter of orifice = 12 mm

The equations below have been used to calculate the respective coefficients.

√

√

The table below shows the collected data and the calculated values for coefficient of velocity,

coefficient of discharge and coefficient of contraction for different heads.

Table 1: Tabulated recorded and calculated data

H/m t/s X/m Y/m Z1/m Z2/m Cv Cd Cc 0.395 38 0.20 0.027 0.060 0.080 0.968 0.994 1.027

0.25 0.040 0.994

0.500 59.51 0.20 0.020 0.200 0.225 1.000 0.705 0.714

0.25 0.032 0.988

0.620 62 0.20 0.016 0.160 0.190 1.000 0.730

0.25 0.024 1.024

0.7125 46 0.20 0.014 0.250 0.270 1.001 0.612 0.613

0.25 0.022 0.998

0.825 74 0.20 0.010 0.260 0.300 1.101 0.707

0.25 0.018 1.026

0.913 49 0.20 0.010 0.050 0.080 1.047 0.761

0.25 0.016 1.034

Note: Coefficients which have major errors have been neglected during calculation of the

average value.

8

4.0. DISCUSSION

Typical values of coefficient of velocity, coefficient of discharge and coefficient of

contraction include 0.96, 0.62 and 0.64 respectively. As it can be observed errors have been

introduced within the experiment most of them systematic as the values although precise are

not accurate. The experimental procedure has been observed to be flawed in the following

respect.

4.1. SOURCES OF ERROR

Adjusting the flow for a constant head resulted in the head being an estimation of the

true value due to the readings on the tank not being accurate enough.

The transparent tube through which water was entering for measurement of the head

was bent.

It was difficult to accurately measure the vertical fall y as the centreline of the jet was

not well defined.

Significant water losses were generated as the water tended to splash out of the

channel which was canalising the water to the collector tank.

The tank having the orifice may not be vertical and may not have a smooth surface

due to rust. It is known to increase the coefficient of discharge as much as 0.72

(Spencer, 2013).

The transparent tube on the collector tank which allowed measurement of the change

in height as the collected volume increased was bent.

The meniscus was difficult to locate due to lack of light and the tube being not

completely transparent.

Leakages were observed to be found at localised areas in the apparatus.

9

4.2. PRECAUTIONS

Within the limits of experimental procedure, the following precautions were taken:

The minimum head to obtain valid heads was not exceeded which stands at 0.20m.

Parallax error was avoided by carefully taking the readings

End errors was avoided by taking intermediate values

4.3. IMPROVEMENTS

The readings on the tank could be made more accurate.

A clean fully transparent tube and straight tube could be used.

The discharge could have been collected directly into the collector tank.

The tank from which flow is occurring should have clean vertical surfaces without

rust.

All leakages should be plugged.

The liquid could have been coloured.

10

5.0. RISK ASSESSMENT

A risk assessment is a systematic process of evaluating the potential dangers that can be

involved in a certain activity.

Health and safety issues were brought to our attention during a talk organised by the

department of Civil Engineering. Consequently, our team carried out a meticulous scan of the

Fluid Mechanics laboratory and a list of the hazards was produced as below.

Restrained mobility

Firstly, with the numerous apparatus in the laboratory, it was hard to move around

given the limited amount of space.

Slippery floor

While working, spilling of water was inevitable. However, the floor does not cater for

this as the surface is not non-skid.

Hanging wires

It was observed that electrical wires hung dangerously at head level and were not in

appropriate casings.

Figure X : Slippery floor

Figure 7: dangling wires

Figure 8: slippery floor

11

6.0. CONCLUSION

Experimental errors have significantly affected the values of the coefficients. The precision

of the results show that the errors are mostly systematic and hence procedure wise it is

deemed to be successful. However these are not reliable as evidenced by previous similar

experiments. The experimental values for the coefficients are therefore not a good

approximation of the true value.

12

7.0. REFERENCES

1. Anon, (n.d.). 1st ed. [ebook] pp.269-271. Available at:

http://content.alterra.wur.nl/Internet/webdocs/ilri-publicaties/publicaties/Pub20/pub20-

h8.0.pdf [Accessed 28 Apr. 2014]

2. Contraction of issuing jet [diagram] In: Douglas, J. F. ,Gasiorek, J. M. and Swaffield,

J. A. ed., 2005. Fluid Mechanics. England: Pearson Education Limited.

3. Determination of the coefficient of velocity [diagram] In: Douglas, J. F. ,Gasiorek, J.

M. and Swaffield, J. A. ed., 2005. Fluid Mechanics. England: Pearson Education

Limited.

4. Douglas, J. F. ,Gasiorek, J. M. and Swaffield, J. A. ed., 2005. Fluid Mechanics.

England: Pearson Education Limited.

5. Flow through small orifice [diagram] In: Douglas, J. F. ,Gasiorek, J. M. and Swaffield,

J. A. ed., 2005. Fluid Mechanics. England: Pearson Education Limited.

6. Spencer, P. (2013). INVESTIGATION OF DISCHARGE BEHAVIOUR FROM A

SHARP-EDGED CIRCULAR ORIFICE IN BOTH WEIR AND ORIFICE FLOW

REGIMES USING AN UNSTEADY EXPERIMENTAL PROCEDURE. 1st ed. [ebook]

pp.6-11. Available at:

http://ir.lib.uwo.ca/cgi/viewcontent.cgi?article=3020&context=etd [Accessed 28 Apr.

2014].

13

APPENDIX A: Derivation of formulas

Theoretical velocity at vena contracta

Figure 9: tank showing the bernoulli principle

(Douglas et al, 2005, pg. 188)

Using Bernoulli’s equation, the theoretical velocity is given by

Taking,

Since,

Replacing in the equation

√

14

Coefficient of velocity

However the actual velocity is less than the theoretical value which is given by,

√

To find CV, the actual velocity is calculated using the projectile motion analogy

Figure 10: x and y distances on jet

(Douglas et al, 2005, pg. 188)

…….(1)

…….(2)

15

Making v & t the subject of formula

……. (3) √

……. (4)

√

Eliminating t gives the actual velocity in terms of x and y:

√

Substituting in the equation for CV

√

√

√

√

16

Coefficient of discharge

Taking A as the area of the orifice, the theoretical discharge is given by

Where V is the theoretical velocity

√

√

√

Where A is area of orifice

Coefficient of contraction

The coefficient of contraction is given by

17

The actual discharge is given by

√

Comparing the equations for actual discharge

√

√